mod two homology and cohomology jean-claudehausmann · · 2016-05-30mod two homology and...

Mod Two Homology and Cohomology

Jean-Claude HAUSMANN

University of Geneva, Switzerland

[email protected]

May 30 2016

Contents

Chapter 1. Introduction 7

Chapter 2. Simplicial (co)homology 92.1. Simplicial complexes 92.2. Definitions of simplicial (co)homology 142.3. Kronecker pairs 162.4. First computations 212.4.1. Reduction to components 212.4.2. 0-dimensional (co)homology 212.4.3. Pseudomanifolds 222.4.4. Poincare series and polynomials 232.4.5. (Co)homology of a cone 232.4.6. The Euler characteristic 252.4.7. Surfaces 252.5. The homomorphism induced by a simplicial map 282.6. Exact sequences 332.7. Relative (co)homology 372.8. Mayer-Vietoris sequences 422.9. Appendix A: an acyclic carrier result 442.10. Appendix B: ordered simplicial (co)homology 452.11. Exercises for Chapter 2 49

Chapter 3. Singular and cellular (co)homologies 513.1. Singular (co)homology 513.1.1. Definitions 513.1.2. Relative singular (co)homology 573.1.3. The homotopy property 623.1.4. Excision 643.1.5. Well cofibrant pairs 663.1.6. Mayer-Vietoris sequences 733.2. Spheres, disks, degree 743.3. Classical applications of the mod 2 (co)homology 793.4. CW-complexes 813.5. Cellular (co)homology 863.6. Isomorphisms between simplicial and singular (co)homology 913.7. CW-approximations 943.8. Eilenberg-MacLane spaces 1003.9. Generalized cohomology theories 1043.10. Exercises for Chapter 3 105

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4 CONTENTS

Chapter 4. Products 1094.1. The cup product 1094.1.1. The cup product in simplicial cohomology 1094.1.2. The cup product in singular cohomology 1124.2. Examples 1144.2.1. Disjoint unions 1144.2.2. Bouquets 1144.2.3. Connected sum(s) of closed topological manifolds 1154.2.4. Cohomology algebras of surfaces 1164.3. Two-fold coverings 1184.3.1. H1, fundamental group and 2-fold coverings 1184.3.2. The characteristic class 1204.3.3. The transfer exact sequence of a 2-fold covering 1234.3.4. The cohomology ring of RPn 1244.4. Nilpotency, Lusternik-Schnirelmann categories and topological complexity1244.5. The cap product 1274.6. The cross product and the Kunneth theorem 1314.7. Some applications of the Kunneth theorem 1394.7.1. Poincare series and Euler characteristic of a product 1394.7.2. Slices 1394.7.3. The cohomology ring of a product of spheres 1404.7.4. Smash products and joins 1414.7.5. The theorem of Leray-Hirsch 1444.7.6. The Thom isomorphism 1514.7.7. Bundles over spheres 1584.7.8. The face space of a simplicial complex 1624.7.9. Continuous multiplications on K(Z2,m) 1644.8. Exercises for Chapter 4 166

Chapter 5. Poincare Duality 1695.1. Algebraic topology and manifolds 1695.2. Poincare Duality in polyhedral homology manifolds 1705.3. Other forms of Poincare Duality 1785.3.1. Relative manifolds 1785.3.2. Manifolds with boundary 1815.3.3. The intersection form 1835.3.4. Non-degeneracy of the cup product 1845.3.5. Alexander Duality 1855.4. Poincare duality and submanifolds 1865.4.1. The Poincare dual of a submanifold 1865.4.2. The Gysin Homomorphism 1895.4.3. Intersections of submanifolds 1915.4.4. The linking number 1945.5. Exercises for Chapter 5 198

Chapter 6. Projective spaces 2016.1. The cohomology ring of projective spaces - Hopf bundles 2016.2. Applications 2056.2.1. The Borsuk-Ulam theorem 205

CONTENTS 5

6.2.2. Non-singular and axial maps 2066.3. The Hopf invariant 2096.3.1. Definition 2096.3.2. The Hopf invariant and continuous multiplications 2106.3.3. Dimension restrictions 2126.3.4. The Hopf invariant and linking numbers 2136.4. Exercises for Chapter 6 215

Chapter 7. Equivariant cohomology 2177.1. Spaces with involution 2177.2. The general case 2297.3. Localization theorems and Smith theory 2377.4. Equivariant cross products and Kunneth theorems 2427.5. Equivariant bundles and Euler classes 2497.6. Equivariant Morse-Bott Theory 2597.7. Exercises for Chapter 7 268

Chapter 8. Steenrod squares 2718.1. Cohomology operations 2718.2. Properties of Steenrod squares 2758.3. Construction of Steenrod squares 2778.4. Adem relations 2838.5. The Steenrod algebra 2898.6. Applications 2948.7. Exercises for Chapter 8 295

Chapter 9. Stiefel-Whitney classes 2979.1. Trivializations and structures on vector bundles 2979.2. The class w1 – Orientability 3039.3. The class w2 – Spin structures 3079.4. Definition and properties of Stiefel-Whitney classes 3119.5. Real flag manifolds 3149.5.1. Definitions and Morse theory 3149.5.2. Cohomology rings 3189.5.3. Schubert cells and Stiefel-Whitney classes 3259.6. Splitting principles 3349.7. Complex flag manifolds 3379.8. The Wu formula 3439.8.1. Wu’s classes and formula 3439.8.2. Orientability and spin structures 3469.8.3. Applications to 3-manifolds 3489.8.4. The universal class for double points 3499.9. Thom’s theorems 3549.9.1. Representing homology classes by manifolds 3549.9.2. Cobordism and Stiefel-Whitney numbers 3579.10. Exercises for Chapter 9 359

Chapter 10. Miscellaneous applications and developments 36110.1. Actions with scattered or discrete fixed point sets 36110.2. Conjugation spaces 364

6 CONTENTS

10.3. Chain and polygon spaces 37010.3.1. Definitions and basic properties 37010.3.2. Equivariant cohomology 37410.3.3. Non-equivariant cohomology 38210.3.4. The inverse problem 38710.3.5. Spatial polygon spaces and conjugation spaces 39010.4. Equivariant characteristic classes 39210.5. The equivariant cohomology of certain homogeneous spaces 39610.6. The Kervaire invariant 40310.7. Exercises for Chapter 10 415

Chapter 11. Hints and answers for some exercises 41711.1. Exercises for Chapter 2 41711.2. Exercises for Chapter 3 41811.3. Exercises for Chapter 4 41911.4. Exercises for Chapter 5 42111.5. Exercises for Chapter 6 42211.6. Exercises for Chapter 7 42311.7. Exercises for Chapter 8 42611.8. Exercises for Chapter 9 42711.9. Exercises for Chapter 10 428

Chapter 12. Errata and Comments 431

Bibliography 441

Index 449

CHAPTER 1

Introduction

Mod2 homology first occurred in 1908 in a paper of Tietze [196] (see also[40, pp. 41–42]). Several results were first established using this mod 2 approach,like the linking number for submanifolds in Rn (see § 5.4.4), as well as Alexanderduality [7]. One argument in favor of the choice of the mod 2 homology was itssimplicity, as J.W. Alexander says in his introduction: “The theory of connectivity[homology] may be approached from two different angles depending on whether ornot the notion of sense [orientation] is developed and taken into consideration. Wehave adopted the second and somewhat simpler point of view in this discussion inorder to condense the necessary preliminaries as much as possible. A treatmentinvolving the idea of sense would be somewhat more complicated but would followalong much the same lines.”

Besides being simpler than its integral counterpart, mod 2 homology sometimesgives new theorems. The first historical main example is the generalization ofPoincare duality to all closed manifolds, whether orientable or not, a result obtainedby Veblen and Alexander in 1913 [200]. As a consequence, the Euler characteristicof a closed odd-dimensional manifold vanishes.

The discoveries of Stiefel-Whitney classes in 1936–38 and of Steenrod squaresin 1947–50 gave mod 2 cohomology the status of a major tool in algebraic topol-ogy, providing for instance the theory of spin structures and Thom’s work on thecobordism ring.

These notes are an introduction, at graduate student’s level, of mod 2 (co)homo-logy (there will be essentially no other). They include classical applications (Brouwerfixed point theorem, Poincare duality, Borsuk-Ulam theorem, Smith theory, etc)and less classical ones (face spaces, topological complexity, equivariant Morse the-ory, etc). The cohomology of flag manifolds is treated in details, including forGrassmannians the relationship between Stiefel-Whitney classes and Schubert cal-culus. Some original applications are given in Chapter 10.

Our approach is different than that of classical textbooks, in which mod 2(co)homology is just a particular case of (co)homology with arbitrary coefficients.Also, most authors start with a full account of homology before approaching coho-mology. In these notes, mod 2 (co)homology is treated as a subject by itself and westart with cohomology and homology together from the beginning. The advantagesof this approach are the following.

• The definition of a (co)chain is simple and intuitive: an (say, simplicial)m-cochain is a set of m-simplexes; an m-chain is a finite set of m-simplexes.The concept of cochain is simpler than that of chain (one less word in thedefinition...), more flexible and somehow more natural. We thus tend toconsider cohomology as the main concept and homology as a (useful) toolfor some arguments.

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8 1. INTRODUCTION

• Working with Z2 and its standard linear algebra is much simpler thanworking with Z. For instance, the Kronecker pairing has an intuitivegeometric interpretation occurring at the beginning which shows in anelementary way that cohomology is the dual of homology. Several com-putations, like the homology of surfaces, are quite easy and come earlyin the exposition. Also, the cohomology ring is commutative. The cupsquare a 7→ a a is a linear map and may be also non-trivial in odddegrees, leading to important invariants.• The absence of sign and orientation considerations is an enormous techni-cal simplification (even of importance in computer algorithms computinghomology). With much lighter computations and technicalities, the ideasof proofs are more apparent.

We hope that these notes will be, for students and teachers, a complement orcompanion to textbooks like those of A. Hatcher [82] or J. Munkres [155]. Fromour teaching experience, starting with mod 2 (co)homology and taking advantageof its above mentioned simplicity is a great help to grasp the ideas of the subject.The technical difficulties of signs and orientations for finer theories, like integral(co)homology, may then be introduced afterwards, as an adaptation of the moreintuitive mod 2 (co)homology.

Not in this book. The following tools are not used in these notes.

• Augmented (co)chain complexes. The reduced cohomology H∗(X) is de-fined as coker (H∗(pt)→ H∗(X)) for the unique map X → pt.• Simplicial approximation.• Spectral sequences (except in the proof of Proposition 7.2.17).

Also, we do not use advanced homotopy tools, like spectra, completions, etc.Because of this, some prominent problems using mod 2 cohomology are only brieflysurveyed, like the work by Adams on the Hopf-invariant-one problem (p. 295), theSullivan’s conjecture (pp. 202 and 295) and the Kervaire invariant (§ 10.6).

Prerequisites. The reader is assumed to have some familiarity with the followingsubjects:

• general point set topology (compactness, connectedness, etc).• elementary language of categories and functors.• simple techniques of exact sequences, like the five lemma.• elementary facts about fundamental groups, coverings and higher homo-topy groups (not much used).• elementary techniques of smooth manifolds.

Acknowledgments: A special thank is due to Volker Puppe who provided sev-eral valuable suggestions and simplifications. Michel Zisman, Pierre de la Harpe,Samuel Tinguely and Matthias Franz have carefully read several sections of thesenotes. The author is also grateful for useful comments to Jim Davis, Rebecca Goldin,Andre Haefliger, Tara Holm, Allen Knutson, Jerome Scherer, Dirk Schutz, AndrasSzenes, Vladimir Turaev, Paul Turner, Claude Weber and Saıd Zarati.

CHAPTER 2

Simplicial (co)homology

Simplicial homology was invented by H. Poincare in 1899 [162] and its mod 2version, presented in this chapter, was introduced in 1908 by H. Tietze [196]. It isthe simplest homology theory to understand and, for finite complexes, it may becomputed algorithmically. The mod 2 version permits rapid computations on easybut non-trivial examples, like spheres and surfaces (see § 2.4).

Simplicial (co)homology is defined for a simplicial complex, but is an invariantof the homotopy type of its geometric realization (this result will be obtained indifferent ways using singular homology: see § 3.6). The first section of this chapterintroduces classical techniques of (abstract) simplicial complexes. Since simplicialhomology was the only existing (co)homology theory until the 1930’s, simplicialcomplexes played a predominant role in algebraic topology during the first thirdof the XXth century (see the introduction of Chapter 5.1). Later developments of(co)homology theories, defined directly for topological spaces, made this combina-torial approach less crucial. However, simplicial complexes remain an efficient wayto construct topological spaces, also largely used in computer science.

2.1. Simplicial complexes

In this section we fix notations and recall some classical facts about (abstract)simplicial complexes. For more details, see [179, Chapter 3].

A simplicial complex K consists of

• a set V (K), the set of vertices of K.• a set S(K) of finite non-empty subsets of V (K) which is closed underinclusion: if σ ∈ S(K) and τ ⊂ σ, then τ ∈ S(K). We require thatv ∈ S(K) for all v ∈ V (K).

An element σ of S(K) is called a simplex of K (“simplexes” and “simplices” areadmitted as plural of “simplex”; we shall use “simplexes”, in analogy with “com-plexes”). If ♯(σ) = m+1, we say that σ is of dimension m or that σ is anm-simplex.The set of m-simplexes of K is denoted by Sm(K). The set S0(K) of 0-simplexes isin bijection with V (K), and we usually identify v ∈ V (K) with v ∈ S0(K). Wesay that K is of dimension ≤ n if Sm(K) = ∅ for m > n, and that K is of dimen-sion n (or n-dimensional) if it is of dimension ≤ n but not of dimension ≤ n− 1.A simplicial complex of dimension ≤ 1 is called a simplicial graph. A simplicialcomplex K is called finite if V (K) is a finite set.

If σ ∈ S(K) and τ ⊂ σ, we say that τ is a face of σ. As S(K) is closed underinclusion, it is determined by it subset Smax(K) of maximal simplexes (if K is finitedimensional). A subcomplex L of K is a simplicial complex such that V (L) ⊂ V (K)and S(L) ⊂ S(K). If S ⊂ S(K) we denote by S the subcomplex generated by S,

9

10 2. SIMPLICIAL (CO)HOMOLOGY

i.e. the smallest subcomplex of K such that S ⊂ S(S). The m-skeleton Km of Kis the subcomplex of K generated by the union of Sk(K) for k ≤ m.

Let σ ∈ S(K). We denote by σ the subcomplex of K formed by σ and all its

faces (σ in the above notation). The subcomplex σ of σ generated by the properfaces of σ is called the boundary of σ.

2.1.1. Geometric realization. The geometric realization |K| of a simplicial com-plex K is, as a set, defined by

|K| := µ : V (K)→ [0, 1]∣∣∑

v∈V (K) µ(v) = 1 and µ−1((0, 1]) ∈ S(K) .

We can thus see |K| as the set of probability measures on V (K) which are supportedby the simplexes (this language is just used for comments and only in this section).There is a distance on |K| defined by

d(µ, ν) =

√ ∑

v∈V (K)

[µ(v)− ν(v)]2

which defines the metric topology on |K|. The set |K| with the metric topology isdenoted by |K|d. For instance, if σ ∈ Sm(K), then |σ|d is isometric to the standardEuclidean simplex ∆m = (x0, . . . , xm) ∈ Rm+1 | xi ≥ 0 and

∑xi = 1.

However, a more used topology for |K| is the weak topology, for which A ⊂ |K|is closed if and only if A ∩ |σ|d is closed in |σ|d for all σ ∈ S(K). The notation|K| stands for the set |K| endowed with the weak topology. A map f from |K|to a topological space X is then continuous if and only if its restriction to |σ|d iscontinuous for each σ ∈ S(K). In particular, the identity |K| → |K|d is continuous,which implies that |K| is Hausdorff. The weak and the metric topology coincide ifand only if K is locally finite, that is each vertex is contained in a finite numberof simplexes. When K is not locally finite, |K| is not metrizable (see e.g. [179,Theorem 3.2.8]).

When a simplicial complex K is locally finite, has countably many vertices andis finite dimensional, it admits a Euclidean realization, i.e. an embedding of |K|into some Euclidean space RN which is piecewise affine. A map f : |K| → RN ispiecewise affine if, for each σ ∈ S(K), the restriction of f to |σ| is an affine map.Thus, for each simplex σ, the image of |σ| is an affine simplex of RN . If dimK ≤ n,such a realization exists in R2n+1 (see e.g. [179, Theorem 3.3.9]).

If σ ∈ S(K) then |σ| ⊂ |K|. We call |σ| the geometric simplex associated to σ.Its boundary is |σ|. The space |σ| − |σ| is the geometric open simplex associatedto σ. Observe that |K| is the disjoint union of its geometric open simplexes.

There is a natural injection i : V (K) → |K| sending v to the Dirac measurewith value 1 on v. We usually identify v with i(v), seeing a simplex v as a pointof |K| (a geometric vertex). In this way, a point µ ∈ |K| may be expressed as aconvex combination of (geometric) vertices:

(2.1.1) µ =∑

v∈V (K)

µ(v)v .

2.1.2. LetK and L be simplicial complexes. Their join is the simplicial complexK ∗ L defined by

(1) V (K ∗ L) = V (K) ∪V (L).(2) S(K ∗ L) = S(K) ∪ S(L) ∪ σ ∪ τ | σ ∈ S(K) and τ ∈ S(L).

2.1. SIMPLICIAL COMPLEXES 11

Observe that, if σ ∈ Sr(K) and τ ∈ Ss(L), then σ ∪ τ ∈ Sr+s+1(K ∗ L). Also,σ ∪ τ = σ ∗ τ and |K ∗ L| the topological join of |K| and |L| (see p. 143).

2.1.3. Stars, links, etc. Let K be a simplicial complex and σ ∈ S(K). Thestar St(σ) of σ is the subcomplex of K generated by all the simplexes containingσ. The link Lk(σ) of σ is the subcomplex of K formed by the simplexes τ ∈ S(K)such that τ ∩ σ = ∅ and τ ∪ σ ∈ S(K). Thus, Lk(σ) is a subcomplex of St(σ) and

St(σ) = σ ∗ Lk(σ) .More generally, if L is a subcomplex of K, the star St(L) of L is the subcomplexof K generated by all the simplexes containing a simplex of L. The link Lk(L) ofL is the subcomplex of K formed by the simplexes τ ∈ S(St(L))− S(L). One hasSt(L) = L ∗ Lk(L). The open star Ost(L) of L is the open neighbourhood of |L| in|K| defined by

Ost(L) = µ ∈ |K| | µ(v) > 0 if v ∈ V (L) .This is the interior of |St(L)| in |K|.

2.1.4. Simplicial maps. Let K and L be two simplicial complexes. A simplicialmap f : K → L is a map f : V (K)→ V (L) such that f(σ) ∈ S(L) if σ ∈ S(K), i.e.the image of a simplex of K is a simplex of L. Simplicial complexes and simplicialmaps form a category, the simplicial category, denoted by Simp.

A simplicial map f : K → L induces a continuous map |f | : |K| → |L| defined,for w ∈ V (L), by

|f |(µ)(w) =∑

v∈f−1(w)

µ(v) .

In other words, |f |(µ) is the pushforward of the probability measure µ on |L|. Thegeometric realization is thus a covariant functor from the simplicial category Simpto the topological category Top of topological spaces and continuous maps.

2.1.5. Components. Let K be a simplicial complex. We define an equivalencerelation on V (K) by saying that v ∼ v′ if there exists x0, . . . , xm ∈ V (K) withx0 = v, xm = v′ and xi, xi+1 ∈ S(K). A maximal subcomplex L of K such thatV (L) is an equivalence class is called a component of K. The set of componentsof K is denoted by π0(K). As the vertices of a simplex are all equivalent, K isthe disjoint union of its components and π0(K) is in bijection with V (K)/ ∼. Therelationship with π0(|K|), the set of (path)-components of the topological space|K|, is the following.

Lemma 2.1.6. The natural injection j : V (K) → |K| descends to a bijection

j : π0(K)≈→ π0(|K|).

Proof. The definition of the relation ∼ makes clear that j descends to amap j : π0(K) → π0(|K)|. Any point of |K| is joinable by a continuous pathto some vertex j(v). Hence, j is surjective. To check the injectivity of j, letv, v′ ∈ V (K) with j(v) = j(v′). There exists then a continuous path c : [0, 1]→ |K|with c(0) = j(v) and c(1) = j(v′). Consider the open cover Ost(w) | w ∈ V (K) of|K|. By compactness of [0, 1], there exists n ∈ N and vertices v0, . . . , vn−1 ∈ V (K)such that c([k/n, (k + 1)/n]) ⊂ Ost(vk) for all k = 0, . . . , n − 1. As c(0) = j(v)and c(1) = j(v′), one deduces that v0 = v and vn−1 = v′. For 0 < k ≤ n − 1,one has c(k/n) ∈ Ost(vk−1) ∩Ost(vk). This implies that vk−1, vk ∈ S(K) for allk = 1, . . . , n− 1, proving that v ∼ v′.


A simplicial complex is called connected if it is either empty or has one compo-nent. Note that |K| is locally path-connected for any simplicial complex K. Indeed,any point has a neighborhood of the form |St(v)| for some vertex v, and |St(v)| ispath-connected. Therefore, |K| is path-connected if and only if |K| is connected.Using Lemma 2.1.6, this proves the following lemma.

Lemma 2.1.7. Let K be a simplicial complex. Then K is connected if and onlyif |K| is a connected space.

Finally, we note the functoriality of π0. Let f : K → L be a simplicial map. Ifv ∼ v′ for v, v′ ∈ V (K), then f(v) ∼ f(v′), so f descends to a map π0f : π0(K)→π0(L). If f : K → L and g : L → M are two simplicial maps, then π0(gf) =π0gπ0f . Also, π0idK = idπ0(K). Thus, π0 is a covariant functor from the simplicialcategory Simp to the category Set of sets and maps.

2.1.8. Simplicial order. A simplicial order on a simplicial complex L is apartial order ≤ on V (L) such that each simplex is totally ordered. For example,a total order on V (L), as in examples where vertices are labeled by integers, isa simplicial order. A simplicial order always exists, as a consequence of the well-ordering theorem.

2.1.9. Triangulations. A triangulation of a topological space X is a homeo-morphism h : |K| → X , where K is a simplicial complex. A topological space istriangulable if it admits a triangulation. It will be useful to have a good process totriangulate some subspaces of Rn. A compact subspace A of Rn is a convex cell ifit is the set of solutions of families of affine equations and inequalities

fi(x) = 0, i = 1 . . . r and gj(x) ≥ 0, j = 1 . . . s .

A face B of A is a convex cell obtained by replacing some of the inequalities gj ≥ 0by the equations gj = 0. The dimension of B is the dimension of the smallest affinesubspace of Rn containing B. A vertex of A is a cell of dimension 0. By inductionon the dimension, one proves that a convex cell is the convex hull of its vertices(see e.g. [138, Theorem 5.2.2]).

A convex-cell complex P is a finite union of convex cells in Rn such that:

(i) if A is a cell of P , so are the faces of A;(ii) the intersection of two cells of P is a common face of each of them.

The dimension of P is the maximal dimension of a cell of P . The r-skeleton P r

is the subcomplex formed by the cells of dimension ≤ r. The 0-skeleton coincideswith the set V (P ) of vertices of P .

A partial order ≤ on V (P ) is an affine order for P if any subset R ∈ V (P )formed by affinely independent points is totally ordered. For instance, a total orderon V (P ) is an affine order. The following lemma is a variant of [104, Lemma 1.4].

Lemma 2.1.10. Let P be a convex-cell complex. An affine order ≤ for P de-

termines a triangulation h≤ : |L≤| ≈−→ P , where L≤ is a simplicial complex withV (L≤) = V (P ). The homeomorphism h≤ is piecewise affine and ≤ is a simplicialorder on L≤.

Proof. The order ≤ being chosen, we drop it from the notations. For eachsubcomplex Q of P , we shall construct a simplicial complex L(Q) and a piecewiseaffine homeomorphism hQ : |L(Q)| → Q such that,

2.1. SIMPLICIAL COMPLEXES 13

(i) V (L(Q)) = V (Q);(ii) if Q′ ⊂ Q, then L(Q′) ⊂ L(Q) and hQ′ is the restriction of hQ to |L(Q′)|.

The case Q = P will prove the lemma. The construction is by induction on thedimension of Q, setting L(Q) = Q and hQ = id if dimQ = 0.

Suppose that L(Q) and hQ have been constructed, satisfying (i) and (ii) above,for each subcomplex Q of P of dimension ≤ k − 1. Let A be a k-cell of K withminimal vertex a. Then A is the topological cone, with cone-vertex a, of theunion B of faces of A not containing a. The triangulation hB : |L(B)| → |B| beingconstructed by induction hypothesis, define L(A) to be the join L(B) ∗ a and hAto be the unique piecewise affine extension of hB. Observe that, if C is a face ofA, then hC is the restriction to L(C) of hA. Therefore, this process may be usedfor each k-cell of P to construct hQ : |L(Q)| → Q for each subcomplex Q of P withdimQ ≤ k.

2.1.11. Subdivisions. Let Z be a set and A be a family of subsets of Z. Asimplicial complex L such that

(a) V (L) ⊂ Z;(b) for each σ ∈ S(L) there exists A ∈ A such that σ ⊂ A;

is called a (Z,A)-simplicial complex, or a Z-simplicial complex supported by A.Let K be a simplicial complex. Let N be a (|K|,GS(K))-simplicial complex,

whereGS(K) = |σ| | σ ∈ S(K)

is the family of geometric simplexes of K. A continuous map j : |N | → |K| isassociated to N , defined by

j(µ) =∑

w∈V (N)

µ(w)w .

In other word, j is the piecewise affine map sending each vertex of N to to the cor-responding point of |K|. A subdivision of a simplicial complex K is a (|K|,GS(K))-simplicial complex N for which the associated map j : |N | → |K| is a homeomor-phism (in other words, j is a triangulation of |K|).

Let N be a (|K|,GS(K))-simplicial complex for a simplicial complex K. If Lis a subcomplex of K, then

NL = σ ∈ S(N)|σ ⊂ |L|is a (|L|,GS(L))-simplicial complex. Its associated map jL : |NL| → |L| is therestriction of j to |L|. The following lemma is useful to recognize a subdivision(compare [179, Ch. 3, Sec. 3, Th. 4]).

Lemma 2.1.12. Let N be a (|K|,GS(K))-simplicial complex. Then N is asubdivision of K if and only if, for each τ ∈ S(K), the simplicial complex Nτ isfinite and jτ : |Nτ | → |τ | is bijective.

Proof. If N is a subdivision of K, then jτ is bijective since j is a homeomor-phism. Also, |Nτ | = j−1(|τ |) is compact, so Nτ is finite.

Conversely, The fact that jτ is bijective for each τ ∈ S(K) implies that thecontinuous map j is bijective. If Nτ is finite, then jτ is a continuous bijectionbetween compact spaces, hence a homeomorphism. This implies that the map j−1,restricted to each geometric simplex, is continuous. Therefore, j−1 is continuoussince K is endowed with the weak topology.


Seeing V (K) as a subset of |K|, we get the following corollary.

Corollary 2.1.13. Let N be a subdivision of K. Then V (K) ⊂ V (N).

A useful systematic subdivision process is the barycentric subdivision. Letσ ∈ Sm(K) be an m-simplex of a simplicial complex K. The barycenter σ ∈ |K| ofσ is defined by

σ =1

m+ 1

∑

v∈σ

v .

The barycentric subdivision K ′ of K is the (|K|,GS(K))-simplicial complex where

• V (K ′) = σ ∈ |K| | σ ∈ S(K);• σ0, . . . , σm ∈ Sm(K ′) whenever σ0 ⊂ · · · ⊂ σm (σi 6= σj if i 6= j).

Using Lemma 2.1.12, the reader can check that K ′ is a subdivision of K. Observethat the partial order “≤” defined by

(2.1.2) σ ≤ τ ⇐⇒ σ ⊂ τis a simplicial order on K ′.

2.2. Definitions of simplicial (co)homology

Let K be a simplicial complex. In this section, we give the definitions of thehomology H∗(K) and cohomology H∗(K) of K under the various and peculiarforms available when the coefficients are in the field Z2 = 0, 1.

Definition 2.2.1 (subset definitions). .

(a) An m-cochain is a subset of Sm(K).(b) An m-chain is a finite subset of Sm(K).

The set of m-cochains of K is denoted by Cm(K) and that of m-chains byCm(K). By identifying σ ∈ Sm(K) with the singleton σ, we see Sm(K) as asubset of both Cm(K) and Cm(K). Each subset A of Sm(K) is determined by itscharacteristic function χA : Sm(K)→ Z2, defined by

χA(σ) =

1 if σ ∈ A0 otherwise.

This gives a bijection between subsets of Sm(K) and functions from Sm(K) to Z2.We see such a function as a colouring (0 = white and 1 = black). The following“colouring definition” is equivalent to the subset definition:

Definition 2.2.2 (colouring definitions). .

(a) An m-cochain is a function a : Sm(K)→ Z2.(b) An m-chain is a function α : Sm(K)→ Z2 with finite support.

The colouring definition is used in low-dimensional graphical examples to draw(co)chains in black (bold lines for 1-(co)chains).

Definitions 2.2.2 endow Cm(K) and Cm(K) with a structure of a Z2-vectorspace. The singletons provide a basis of Cm(K), in bijection with Sm(K). Thus,Definition 2.2.2.b is equivalent to

Definition 2.2.3. Cm(K) is the Z2-vector space with basis Sm(K):

Cm(K) =⊕

σ∈Sm(K)

Z2 σ .

2.2. DEFINITIONS OF SIMPLICIAL (CO)HOMOLOGY 15

We shall pass from one of Definitions 2.2.1, 2.2.2 or 2.2.3 to another withoutnotice; the context usually prevents ambiguity. We consider C∗(K) = ⊕m∈NCm(K)and C∗(K) = ⊕m∈NCm(K) as graded Z2-vector spaces. The convention C−1(K) =C−1(K) = 0 is useful.

We now define the Kronecker pairing on (co)chains

Cm(K)× Cm(K)〈 , 〉−→ Z2

by the equivalent formulae

(2.2.1)

〈a, α〉 = ♯(a ∩ α) (mod 2) using Definition 2.2.1.a and b

=∑

σ∈α a(σ) using Definitions 2.2.1.a and 2.2.2.b

=∑

σ∈Sm(K) a(σ)α(σ) using Definitions 2.2.2.a and b .

Lemma 2.2.4. The Kronecker pairing is bilinear and the map a 7→ 〈a, 〉 is anisomorphism between Cm(K) and Cm(K)♯ = hom(Cm(K),Z2).

Proof. The bilinearity is obvious from the third line of Equations (2.2.1). Let0 6= a ∈ Cm(K). This means that, as a subset of Sm(K), a is not empty. If σ ∈ a,then 〈a, σ〉 6= 0, which proves the injectivity of a 7→ 〈a, 〉. As for its surjectivity, leth ∈ hom(Cm(K),Z2). Using the inclusion Sm(K) → Cm(K) given by τ 7→ τ,define

a = τ ∈ Sm(K) | h(τ) = 1 .For each σ ∈ Sm(K) the equation h(σ) = 〈a, σ〉 holds true. As Sm(K) is a basis ofCm(K), this implies that h = 〈a, 〉.

We now define the boundary and coboundary operators. The boundary operator∂ : Cm(K)→ Cm−1(K) is the Z2-linear map defined by

(2.2.2) ∂(σ) = (m− 1)-faces of σ = Sm−1(σ) , σ ∈ Sm(K) .

Formula (2.2.2) is written in the language of Definition 2.2.1.b. Using Defini-tion 2.2.3, we get

(2.2.3) ∂(σ) =∑

τ∈Sm−1(σ)

τ .

The coboundary operator δ : Cm(K)→ Cm+1(K) is defined by the equation

(2.2.4) 〈δa, α〉 = 〈a, ∂α〉 .The last equation indeed defines δ by Lemma 2.2.4 and δ may be seen as theKronecker adjoint of ∂. In particular, if σ ∈ Sm(K) and τ ∈ Sm−1(K) then

(2.2.5) τ ∈ ∂(σ) ⇔ τ ⊂ σ ⇔ σ ∈ δ(τ) .The first equivalence determines the operator ∂ since Sm(K) is a basis for Cm(K).The second equivalence determines δ if Sm−1(K) is finite. Note that the definitionof δ may also be given as follows: if a ∈ Cm(K), then

δ(a) = τ ∈ Sm+1(K) | ♯ (a ∩ ∂(τ)) is odd .Let σ ∈ Sm(K). Each τ ∈ Sm−2(K) with τ ⊂ σ belongs to the boundary

of exactly two (m − 1)-simplexes of σ. Using Equation (2.2.3), this implies that∂∂ = 0. By Equation (2.2.4) and Lemma 2.2.4, we get δδ = 0. We define theZ2-vector spaces

• Zm(K) = ker(∂ : Cm(K)→ Cm−1(K)), the m-cycles of K.


• Bm(K) = image (∂ : Cm+1(K)→ Cm(K)), the m-boundaries of K.• Zm(K) = ker(δ : Cm(K)→ Cm+1(K)), the m-cocycles of K.• Bm(K) = image (δ : Cm−1(K)→ Cm(K)), the m-coboundaries of K.

For example, Figure 2.1 shows a triangulation K of the plane, with V (K) =Z × Z. The bold line is a cocycle a which is a coboundary: a = δB, with B =(m,n) | (m,n) ∈ V (K) and m ≤ 0, drawn in bold dots.

0

a

Figure 2.1.

Since ∂∂ = 0 and δδ = 0, one has Bm(K) ⊂ Zm(K) and Bm(K) ⊂ Zm(K).We form the quotient vector spaces

• Hm(K) = Zm(K)/Bm(K), the mth-homology vector space of K.• Hm(K) = Zm(K)/Bm(K), the mth-cohomology vector space of K.

As for the (co)chains, the notationsH∗(K) = ⊕m∈NHm(K) andH∗(K) = ⊕m∈NHm(K)stand for the (co)homology seen as graded Z2-vector spaces. By convention,H−1(K) =H−1(K) = 0. Also, the homology and the cohomology are in duality via the Kro-necker pairing:

Proposition 2.2.5 (Kronecker duality). The Kronecker pairing on (co)chainsinduces a bilinear map

Hm(K)×Hm(K)〈 , 〉−−→ Z2 .

Moreover, the correspondence a 7→ 〈a, 〉 is an isomorphism

Hm(K)k−−−→≈

hom(Hm(K),Z2) .

Proof. Instead of giving a direct proof, which the reader may do as an exercise,we will take advantage of the more general setting of Kronecker pairs, developed inthe next section. In this way, Proposition 2.2.5 follows from Proposition 2.3.5.

2.3. Kronecker pairs

All the vector spaces in this section are over an arbitrary fixed field F. Thedual of a vector space V is denoted by V ♯.

A chain complex is a pair (C∗, ∂), where

• C∗ is a graded vector space C∗ =⊕

m∈N Cm. We add the convention thatC−1 = 0.• ∂ : C∗ → C∗ is a linear map of degree −1, i.e. ∂(Cm) ⊂ Cm−1, satisfying∂∂ = 0. The operator ∂ is called the boundary of the chain complex.

2.3. KRONECKER PAIRS 17

A cochain complex is a pair (C∗, δ), where

• C∗ is a graded vector space C∗ =⊕

m∈NCm. We add the convention that

C−1 = 0.• δ : C∗ → C∗ is a linear map of degree +1, i.e. ∂(Cm) ⊂ Cm+1, satisfyingδδ = 0. The operator δ is called the coboundary of the cochain complex.

A Kronecker pair consists of three items:

(a) a chain complex (C∗, ∂).(b) a cochain complex (C∗, δ).(c) a bilinear map

Cm × Cm〈 , 〉−→ F

satisfying the equation

(2.3.1) 〈δa, α〉 = 〈a, ∂α〉 .for all a ∈ Cm and α ∈ Cm+1 and all m ∈ N. Moreover, we require thatthe map k : Cm → C♯m, given by k(a) = 〈a, 〉, is an isomorphism.

Example 2.3.1. Let K be a simplicial complex. Its simplicial (co)chain com-plexes (C∗(K), δ), (C∗(K), ∂), together with the pairing 〈 , 〉 of § 2.2 is a Kroneckerpair, with F = Z2, as seen in Lemma 2.2.4 and Equation (2.3.1).

Example 2.3.2. Let (C∗, ∂) be a chain complex. One can define a cochaincomplex (C∗, δ) by Cm = C♯m and δ = ∂♯ and then get a bilinear map (pairing) 〈,〉by the evaluation: 〈a, α〉 = a(α). These constitute a Kronecker pair. Actually, viathe map k, any Kronecker pair is isomorphic to this one. The reader may use thisfact to produce alternative proofs of the results of this section.

We first observe that, as the Kronecker pairing is non-degenerate, chains andcochains mutually determine each other:

Lemma 2.3.3. Let((C∗, δ), (C∗, ∂), 〈 , 〉

)be a Kronecker pair.

(a) Let a, a′ ∈ Cm. Suppose that 〈a, α〉 = 〈a′, α〉 for all α ∈ Cm. Then a = a′.(b) Let α, α′ ∈ Cm. Suppose that 〈a, α〉 = 〈a, α′〉 for all a ∈ Cm. Then

α = α′.(c) Let Sm be a basis for Cm and let f : Sm → F be a map. Then, there is a

unique a ∈ Cm such that 〈a, σ〉 = f(σ) for all σ ∈ Sm.

Proof. In Point (a), the hypotheses imply that k(a) = k(a′). As k is injective,this shows that a = a′.

In Point (b), suppose that α 6= α′. Let A ∈ (Cm)♯ such that A(α − α′) 6= 0.Then, 〈a, α〉 6= 〈a, α′〉 for a = k−1(A) ∈ Cm.

Finally, the condition a(σ) = f(σ) for all σ ∈ Sm defines a unique a ∈ C♯m anda = k−1(a).

As is § 2.2, we consider the Z2-vector spaces

• Zm = ker(∂ : Cm → Cm−1), the m-cycles (of C∗).• Bm = image (∂ : Cm+1 → Cm), the m-boundaries.• Zm = ker(δ : Cm → Cm+1), the m-cocycles.• Bm = image (δ : Cm−1 → Cm), the m-coboundaries.

Since ∂∂ = 0 and δδ = 0, one has Bm ⊂ Zm and Bm ⊂ Zm. We form thequotient vector spaces


• Hm = Zm/Bm, the mth-homology group (or vector space).• Hm = Zm/Bm, the mth-cohomology group (or vector space).

We consider the (co)homology as graded vector spaces: H∗ = ⊕m∈NHm and H∗ =⊕m∈NHm.

The cocycles and coboundaries may be detected by the pairing:

Lemma 2.3.4. Let a ∈ Cm. Then

(i) a ∈ Zm if and only if 〈a,Bm〉 = 0.(ii) a ∈ Bm if and only if 〈a, Zm〉 = 0.

Proof. Point (i) directly follows from Equation (2.3.1) and the fact that k isinjective. Also, if a ∈ Bm, Equation (2.3.1) implies that 〈a, Zm〉 = 0. It remainsto prove the converse (this is the only place in this lemma where we need vectorspaces over a field instead just module over a ring). We consider the exact sequence

(2.3.2) 0→ Zm → Cm → Bm−1∂−→ 0 .

Let a ∈ Cm such that 〈a, Zm〉 = 0. By (2.3.2), there exists a1 ∈ B♯m−1 such that〈a, 〉 = a1∂. As we are dealing with vector spaces, Bm−1 is a direct summand

of Cm−1. We can thus extend a1 to a2 ∈ C♯m−1. As k is surjective, there exists

a3 ∈ Cm−1 such that 〈a3, 〉 = a2. For all α ∈ Cm, one then has

〈δa3, α〉 = 〈a3, ∂α〉 = a2(∂α) = a1(∂α) = 〈a, α〉 .As k is injective this implies that a = δa3 ∈ Bm.

Let us restrict the pairing 〈 , 〉 to Zm × Zm. Formula (2.3.1) implies that

〈Zm, Bm〉 = 〈Bm, Zm〉 = 0 .

Hence, the pairing descends to a bilinear map Hm × Hm〈 , 〉−→ F, giving rise to a

linear map k : Hm → H♯m, called the Kronecker pairing on (co)homology. We see

H∗ and H∗ as (co)chain complexes by setting ∂ = 0 and δ = 0.

Proposition 2.3.5. (H∗, H∗, 〈 , 〉) is a Kronecker pair.

Proof. Equation (2.3.1) holds trivially since ∂ and δ both vanish. It remainsto show that k : Hm → H♯

m is bijective.Let a0 ∈ H♯

m. Pre-composing a0 with the projection Zm →→ Hm producesa1 ∈ Z♯m. As Zm is a direct summand in Cm, one can extend a1 to a2 ∈ C♯m.Since (C∗, C

∗, 〈 , 〉) is a Kronecker pair, there exists a ∈ Cm such that 〈a, 〉 = a2.The cochain a satisfies 〈a,Bm〉 = a2(Bm) = 0 which, by Lemma 2.3.4, implies thata ∈ Zm. The cohomology class [a] ∈ Hm of a then satisfies 〈[a], 〉 = a0. Thus, k issurjective.

For the injectivity of k, let b ∈ Hm with 〈b,Hm〉 = 0. Represent b by b ∈ Zm,which then satisfies 〈b, Zm〉 = 0. By Lemma 2.3.4, b ∈ Bm and thus b = 0.

Let (C∗, ∂) and (C∗, ∂) be two chain complexes. A map ϕ : C∗ → C∗ is amorphism of chain complexes or a chain map if it is linear map of degree 0 (i.e.ϕ(Cm) ⊂ Cm) such that ϕ∂ = ∂ϕ. This implies that ϕ(Zm) ⊂ Zm and ϕ(Bm) ⊂Bm. Hence, ϕ induces a linear map H∗ϕ : Hm → Hm for all m.

In the same way, let (C∗, δ) and (C∗, δ) be two cochain complexes. A linearmap φ : C∗ → C∗ of degree 0 is a morphism of cochain complexes or a cochain mapif φ δ = δφ. Hence, φ induces a linear map H∗φ : Hm → Hm for all m.

2.3. KRONECKER PAIRS 19

Let P = (C∗, ∂, C∗, δ, 〈 , 〉) and P = (C∗, ∂, C

∗, δ, 〈 , 〉−) be two Kronecker pairs.A morphism of Kronecker pairs, from P to P, consists of a pair (ϕ, φ) whereϕ : C∗ → C∗ is a morphism of chain complexes and φ : C∗ → C∗ is a morphism ofcochain complexes such that

(2.3.3) 〈a, ϕ(α)〉− = 〈φ(a), α〉 .Using the isomorphisms k and k, Equation (2.3.3) is equivalent to the commuta-tivity of the diagram

(2.3.4)

C∗

k≈

φ // C∗

k≈

C♯∗ϕ♯

// C♯∗

.

Lemma 2.3.6. Let P and P be Kronecker pairs as above. Let ϕ : C∗ → C∗be a morphism of chain complex. Define φ : C∗ → C∗ by Equation (2.3.3) (orDiagram (2.3.4)). Then the pair (ϕ, φ) is a morphism of Kronecker pairs.

Proof. Obviously, φ is a linear map of degree 0 and Equation (2.3.3) is sat-isfied. It remains to show that φ is a morphism of cochain-complexes. But, ifb ∈ Cm(K) and α ∈ Cm+1(K), one has

〈δφ(b), α〉 = 〈φ(b), ∂α〉 = 〈b, ϕ(∂α)〉− = 〈b, ∂ϕ(α)〉−= 〈δb, ϕ(α)〉− = 〈φ(δb), α〉 ,

which proves that δφ(b) = φ(δb).

A morphism (ϕ, φ) of Kronecker pairs determines a morphism of Kroneckerpairs (H∗ϕ,H

∗φ) from (H∗, H∗, 〈 , 〉) to (H∗, H

∗, 〈 , 〉−). This process is functorial:Lemma 2.3.7. Let (ϕ1, φ1) be a morphism of Kronecker pairs from P to P and

let (ϕ2, φ2) be a morphism of Kronecker pairs from P to P. Then

(H∗ϕ2H∗ϕ1, H∗φ1H

∗φ2) = (H∗(ϕ2ϕ1), H∗(φ2 φ1))

Proof. That H∗ϕ2H∗ϕ1 = H∗(ϕ2 ϕ1) is a tautology. For the cohomologyequality, we use that

〈H∗φ1 H∗φ2(a), α〉 = 〈H∗φ2(a), H∗ϕ1(α)〉 = 〈a,H∗ϕ2 H∗ϕ1(α)〉= 〈a,H∗(ϕ2 ϕ1)(α)〉 = 〈H∗(φ2 φ1))(a), α〉

holds for all a ∈ H∗ and all α ∈ H∗.

We finish this section with some technical results which will be used later.

Lemma 2.3.8. Let f : U → V and g : V →W be two linear maps between vectorspaces. Then, the sequence

(2.3.5) Uf−→ V

g−→W

is exact at V if and only if the sequence

(2.3.6) U ♯f♯

←− V ♯ g♯←−W ♯

is exact at V ♯.


Proof. As f ♯g♯ = (gf)♯, then f ♯g♯ = 0 if and only if gf = 0.On the other hand, suppose that ker g ⊂ image f . We shall prove that ker f ♯ ⊂

image g♯. Indeed, let a ∈ ker f ♯. Then, a(image f) = 0 and, using the inclusionker g ⊂ image f , we deduce that a(ker g) = 0. Therefore, a descends to a linearmap a : V/ ker g → F. The quotient space V/ ker g injects into W , so there existsb ∈ W ♯ such that a = bg = g♯(b), proving that a ∈ image g♯.

Finally, suppose that ker g 6⊂ image f . Then there exists a ∈ V ♯ such thata(image f) = 0, i.e., a ∈ ker f ♯, and a(ker g) 6= 0, i.e. a /∈ image g♯. This provesthat ker f ♯ 6⊂ image g♯.

Lemma 2.3.9. Let (ϕ, φ) be a morphism of Kronecker pairs from P = (C∗, ∂, C∗, δ, 〈 , 〉)

to P = (C∗, ∂, C∗, δ, 〈 , 〉−). Then the pairings 〈 , 〉 and 〈 , 〉− induce bilinear maps

cokerφ× kerϕ〈,〉−→ F and kerφ× cokerϕ

〈,〉−−−−→ F

such that the induced linear maps

cokerφk−→ (kerϕ)♯ and kerφ

k−→ (cokerϕ)♯

are isomorphisms.

Proof. Equation (2.3.3) implies that 〈φ(C∗), kerϕ〉 = 0 and 〈kerφ, ϕ(C∗)〉− =0, whence the induced pairings. Consider the exact sequence

0 // kerϕ // C∗ϕ // C∗ // cokerϕ // 0 .

By Lemma 2.3.8, passing to the dual preserves exactness. Using Diagram (2.3.4),one gets a commutative diagram

0 (kerϕ)♯oo C♯∗oo C♯∗ϕ♯

oo (cokerϕ)♯oo 0oo

0 cokerφoo

k

OO

H∗oo

k≈

OO

C∗φoo

k≈

OO

kerφoo

k

OO

0oo

.

By diagram-chasing, the two extreme up-arrows are bijective (one can also invokethe famous five-lemma: see e.g. [179, Ch.4, Sec.5, Lemma11]).

Corollary 2.3.10. Let (ϕ, φ) be a morphism of Kronecker pairs from(C∗, ∂, C

∗, δ, 〈 , 〉) to (C∗, ∂, C∗, δ, 〈 , 〉−). Then the pairings 〈 , 〉 and 〈 , 〉− on (co)homo-

logy induce bilinear maps

cokerH∗φ× kerH∗ϕ〈,〉−→ F and kerH∗φ× cokerH∗ϕ

〈,〉−−−−→ F

such that the induced linear maps

cokerH∗φk−→ (kerH∗ϕ)

♯ and kerH∗φk−→ (cokerH∗ϕ)

♯

are isomorphisms.

Proof. The morphism (φ, ϕ) induces a morphism of Kronecker pairs (H∗φ,H∗ϕ)from (H∗, H∗, 〈 , 〉) to (H∗, H∗, 〈 , 〉−). Corollary 2.3.10 follows then from Lemma 2.3.9applied to (H∗φ,H∗ϕ).

Corollary 2.3.10 implies the following

2.4. FIRST COMPUTATIONS 21

Corollary 2.3.11. Let (ϕ, φ) be a morphism of Kronecker pairs from (C∗, ∂, C∗, δ, 〈 , 〉)

to (C∗, ∂, C∗, δ, 〈 , 〉−). Then

(a) H∗φ is surjective if and only if H∗ϕ is injective.(b) H∗φ is injective if and only if H∗ϕ is surjective.(c) H∗φ is bijective if and only if H∗ϕ is bijective.

2.4. First computations

2.4.1. Reduction to components. Let K be a simplicial complex. We haveseen in 2.1.5 that K is the disjoint union of its components, whose set is denotedby π0(K). Therefore, Sm(K) =

∐L∈π0(K) Sm(L) which, by Definition 2.2.2 gives a

canonical isomorphism ⊕

L∈π0(K)

Cm(L)≈→ Cm(K) .

This direct sum decomposition commutes with the boundary operators, giving acanonical isomorphism

(2.4.1)⊕

L∈π0(K)

H∗(L)≈→ H∗(K) .

As for the cohomology, seeing an m-cochain as a map α : Sm(K) → Z2 (Defini-tion 2.2.2) the restrictions of α to Sm(L) for all L ∈ π0(K) gives an isomorphism

Cm(K)≈−→

∏

L∈π0(K)

Cm(L)

commuting with the coboundary operators. This gives an isomorphism

(2.4.2) H∗(K)≈−→

∏

L∈π0(K)

H∗(L) .

The isomorphisms of (2.4.1) and (2.4.2) permit us to reduce (co)homology com-putations to connected simplicial complexes. They are of course compatible withthe Kronecker duality (Proposition 2.2.5). A formulation of these isomorphismsusing simplicial maps is given in Proposition 2.5.3.

2.4.2. 0-dimensional (co)homology. Let K be a simplicial complex. Theunit cochain 1 ∈ C0(K) is defined by 1 = S0(K), using the subset definition. Inthe language of colouring, one has 1(v) = 1 for all v ∈ V (K) = S0(K), that is allvertices are black. If β = v, w ∈ S1(K), then

〈δ1, β〉 = 〈1, ∂β〉 = 1(v) + 1(w) = 0 ,

which proves that δ(1) = 0 by Lemma 2.2.4. Hence, 1 is a cocycle, whose cohomol-ogy class is again denoted by 1 ∈ H0(K).

Proposition 2.4.1. Let K be a non-empty connected simplicial complex. Then,

(i) H0(K) = Z2, generated by 1 which is the only non-vanishing 0-cocycle.(ii) H0(K) = Z2. Any 0-chain α is a cycle, which represents the non-zero

element of H0(K) if and only if ♯α is odd.


Proof. If K is non-empty the unit cochain does not vanish. As C−1(K) = 0,this implies that 1 6= 0 in H0(K).

Let a ∈ C0(K) with a 6= 0,1. Then there exists v, v′ ∈ V (K) with a(v) 6= a(v′).Since K is connected, there exists x0, . . . , xm ∈ V (K) with x0 = v, xm = v′ andxi, xi+1 ∈ S(K). Therefore, there exists 0 ≤ k < m with a(xk) 6= a(xk+1). Thisimplies that xk, xk+1 ∈ δa, proving that δa 6= 0. We have thus proved (i).

Now, H0(K) = Z2 since H0(K) ≈ H0(K)♯. Any α ∈ C0(K) is a cycle sinceC−1(K) = 0. It represents the non-zero homology class if and only if 〈1, α〉 = 1,that is if and only if ♯α is odd.

Corollary 2.4.2. Let K be a simplicial complex. Then H0(K) ≈ Zπ0(K)2 .

Here, Zπ0(K)2 denotes the set of maps from π0(K) to Z2. The isomorphism of

Corollary 2.4.2 is natural for simplicial maps (see Corollary 2.5.6).

Proof. By Proposition 2.4.1 and its proof, H0(K) = Z0(K) is the set of mapsfrom V (K) to Z2 which are constant on each component. Such a map is determinedby a map from π0(K) to Z2 and conversely.

2.4.3. Pseudomanifolds. An n-dimensional pseudomanifold is a simplicialcomplex M such that

(a) every simplex of M is contained in an n-simplex of M .(b) every (n− 1)-simplex of M is a face of exactly two n-simplexes of M .(c) for any σ, σ′ ∈ Sn(M), there exists a sequence σ = σ0, . . . , σm = σ′ of

n-simplexes such that σi and σi+1 have an (n − 1)-face in common fori ≤ 1 < m.

Examples 2.4.3. (1) Let m be an integer with m ≥ 3. The polygon Pm is the1-dimensional pseudomanifold for which V (Pm) = 0, 1, . . . ,m − 1 = Z/mZ andS1(Pm) = k, k + 1 | k ∈ V (Pm). It can be visualized in the complex plane asthe equilateral m-gon whose vertices are the mth roots of the unity.

(2) Consider the triangulation of S2 given by an icosahedron. Choose one pairof antipodal vertices and identify them in a single point. This gives a quotientsimplicial complex K which is a 2-dimensional pseudomanifold. Observe that |K|is not a topological manifold.

Pseudomanifolds were introduced in 1911 by L.E.J. Brouwer [22, p. 477], forhis work on the degree and on the invariance of the dimension. They are alsocalled n-circuits in the literature. Proposition 2.4.4 below and its proof, togetherwith Proposition 2.4.1, shows that n-dimensional pseudomanifolds satisfy Poincareduality in dimensions 0 and n.

Let M be a finite n-dimensional pseudomanifold. The n-chain [M ] = Sn(M) ∈Cn(M) is called the fundamental cycle of M (it is a cycle by Point (b) of theabove definition). Its homology class, also denoted by [M ] ∈ Hn(M) is called thefundamental class of M .

Proposition 2.4.4. Let M be a finite non-empty n-dimensional pseudomani-fold. Then,

(i) Hn(M) = Z2, generated by [M ] which is the only non-vanishing n-cycle.(ii) Hn(M) = Z2. Any n-cochain a is a cocycle, and [a] 6= 0 in Hn(M) if and

only if ♯a is odd.


Proof. We define a simplicial graph L with V (L) = Sn(M) by setting σ, σ′ ∈S1(L) if and only if σ and σ′ have an (n − 1)-face in common. The identificationSn(M) = V (L) produces isomorphisms

(2.4.3) Fn : Cn(M)≈−→ C0(L) and Fn : Cn(M)

≈−→ C0(L) .

(AsM is finite, so is L and C∗(L) is equal to C∗(L), using Definition 2.2.2.) On the

other hand, by Point (b) of the definition of a pseudomanifold, one gets a bijection

F : Sn−1(M)≈−→ S1(L). It gives rise to isomorphisms

(2.4.4) Fn−1 : Cn−1(M)≈−→ C1(L) and Fn−1 : Cn−1(M)

≈−→ C1(L) .

The isomorphisms of (2.4.3) and (2.4.4) satisfy

Fn−1∂ = δFn and ∂ Fn−1 = Fnδ .

Since Cn+1(M) = 0 by Point (a) of the definition of a pseudomanifold, the aboveisomorphisms give rise to isomorphisms

F∗ : Hn(M)≈−→ H0(L) and F ∗ : Hn(M)

≈−→ H0(L)

with F∗([M ]) = 1. By Point (c) of the definition of a pseudomanifold, the graph Lis connected. Therefore, Proposition 2.4.4 follows from Proposition 2.4.1.

The proof of Proposition 2.4.4 actually gives the following result.

Proposition 2.4.5. Let M be a finite non-empty simplicial complex satisfyingConditions (a) and (b) of the definition of an n-dimensional pseudomanifold. Then,M is a pseudomanifold if and only if Hn(M) = Z2.

2.4.4. Poincare series and polynomials. A graded Z2-vector space A∗ =⊕i∈NAi is of finite type if Ai is finite dimensional for all i ∈ N. In this case, the

Poincare series of A∗ is the formal power series defined by

Pt(A∗) =∑

i∈N

dimAiti ∈ N[[t]].

When dimA∗ < ∞, the series Pt(A∗) is a polynomial, also called the Poincarepolynomial of A∗.

A simplicial complexK is of finite (co)homology type ifH∗(K) (or, equivalently,H∗(K)) is of finite type. In this case, the Poincare series of K is that of H∗(K).The (co)homology of a simplicial complex of finite (co)homology type is, up toisomorphism, determined by its Poincare series, which is often the shortest way todescribe it. The number dimHm(K) is called the m-th Betti number of K. Thevector space C∗(K) is endowed with the basis S(K) for which the matrix of theboundary operator is given explicitly. Thus, the Betti numbers may be effectivelycomputed by standard algorithms of linear algebra.

2.4.5. (Co)homology of a cone. The simplest non-empty simplicial com-plex is a point whose (co)homology is obviously

(2.4.5) Hm(pt) ≈ Hm(pt) ≈0 if m > 0

Z2 if m = 0 .

In terms of Poincare polynomial: Pt(pt) = 1. Let L be a simplicial complex. Thecone on L is the simplicial complex CL defined by V (CL) = V (L) ∪ ∞ and

Sm(CL) = Sm(L) ∪ σ ∪ ∞ | σ ∈ Sm−1(L) .


Note that CL is the join CL ≈ L ∗ ∞.

Proposition 2.4.6. The cone CL on a simplicial complex L has its (co)homologyisomorphic to that of a point. In other words, Pt(CL) = 1.

Proof. By Kronecker duality, it is enough to prove the result on homology.The cone CL is obviously connected and non-empty (it contains ∞), so H0(CL) =Z2.

Define a linear map D : Cm(CL)→ Cm+1(CL) by setting, for σ ∈ Sm(CL):

D(σ) =

σ ∪ ∞ if ∞ /∈ σ0 if ∞ ∈ σ .

Hence, DD = 0. If ∞ /∈ σ, the formula

(2.4.6) ∂D(σ) = D(∂σ) + σ

holds true in Cm(CL) (and has a clear geometrical interpretation). Suppose that∞ ∈ σ and dimσ ≥ 1. Then σ = D(τ) with τ = σ − ∞. Using Formula (2.4.6)and that DD = 0, one has

D(∂σ) + σ = D(∂D(τ)) + σ = D(D(∂τ) + τ) +D(τ) = 0 .

Therefore, Formula (2.4.6) holds also true if ∞ ∈ σ, provided dimσ ≥ 1. Thisproves that

(2.4.7) ∂D(α) = D(∂α) + α for all α ∈ Cm(CL) with m ≥ 1 .

Now, if α ∈ Cm(CL) satisfies ∂α = 0, Formula (2.4.7) implies that α = ∂D(α),which proves that Hm(CL) = 0 if m ≥ 1.

As an application of Proposition 2.4.6, let A be a set. The full complex FAon A is the simplicial complex for which V (FA) = A and S(FA) is the family of allfinite non-empty subsets of A. If A is finite and non-empty, then FA is isomorphicto a simplex of dimension ♯A− 1. Denote by FA the subcomplex of FA generatedby the proper (i.e. 6= A) subsets of A. For instance, FA = FA if A is infinite.

Corollary 2.4.7. Let A be a non-empty set. Then

(i) FA has its (co)homology isomorphic to that of a point, i.e. Pt(FA) = 1.

(ii) If 3 ≤ ♯A ≤ ∞, then Pt(FA) = 1 + t♯A−1.

(iii) If ♯A = 2, then Pt(FA) = 2.

Proof. As A is not empty, FA is isomorphic to the cone over FA deprived ofone of its elements. Point (i) then follows from Proposition 2.4.6. Let n = ♯A− 1.The chain complex of FA looks like a sequence

0→ Cn(FA) ∂n−→ Cn−1(FA)∂n−1−−−→ · · · → C0(FA)→ 0 ,

which, by (i), is exact except at C0(FA). One has Cn(FA) = Z2, generated by the

A ∈ Sn(FA). Hence, ker ∂n−1 ≈ Z2. As the chain complex C∗(FA) is the same as

that of FA with Cn replaced by 0, this proves (ii). If ♯A = 2, then FA consists oftwo 0-simplexes and Point (iii) follows from (2.4.5) and (2.4.1).


2.4.6. The Euler characteristic. Let K be a finite simplicial complex. ItsEuler characteristic χ(K) is defined as

χ(K) =∑

m∈N

(−1)m ♯Sm(K) ∈ Z .

Proposition 2.4.8. Let K be a finite simplicial complex. Then

χ(K) =∑

m∈N

(−1)m dimHm(K) =∑

m

(−1)m dimHm(K) .

As in the definition of the Poincare polynomial, the number dimHm(K) isthe dimension of Hm(K) as a Z2-vector space. In other words, dimHm(K) is them-th Betti number of K. Proposition 2.4.8 holds true for the (co)homology withcoefficients in any field F, though the Betti numbers depend individually on F.

Proof. By Kronecker duality, only the first equality requires a proof. Letcm, zm, bm and hm be the dimensions of Cm(K), Zm(K), Bm(K) and Hm(K). El-ementary linear algebra gives the equalities

cm = zm + bm−1zm = bm + hm .

We deduce that

χ(K) =∑

m∈N

(−1)mcm =∑

m∈N

(−1)mhm +∑

m∈N

(−1)mbm +∑

m∈N

(−1)mbm−1 .

As b−1 = 0, the last two sums cancel each other, proving Proposition 2.4.8.

Corollary 2.4.9. Let K be a finite simplicial complex. Then

χ(K) = Pt(K)t=−1.

The following additive formula for the Euler characteristic is useful.

Lemma 2.4.10. Let K be a simplicial complex. Let K1 and K2 be two subcom-plexes of K such that K = K1 ∪K2. Then,

χ(K) = χ(K1) + χ(K2)− χ(K1 ∩K2) .

Proof. The formula follows directly from the equations Sm(K) = Sm(K1) ∪Sm(K2) and Sm(K1 ∩K2) = Sm(K1) ∩ Sm(K2).

2.4.7. Surfaces. A surface is a manifold of dimension 2. In this section, wegive examples of triangulations of surfaces and compute their (co)homology. Strictlyspeaking, the results would hold only for the given triangulations, but we allow usto formulate them in more general terms. For this, we somehow admit that

• a connected surface is a pseudomanifold of dimension 2. This will beestablished rigorously in Corollary 5.2.7 but the reader may find a proofas an exercise and this is easy to check for the particular triangulationsgiven below.• up to isomorphism, the (co)homology of a simplicial complex K dependsonly of the homotopy type of |K|. This will be proved in § 3.6. In partic-ular, the Euler characteristic of two triangulations of a surface coincide.

The 2-sphere. The 2-sphere S2 being homeomorphic to the boundary of a3-simplex, it follows from Corollary 2.4.7 that:

Pt(S2) = 1 + t2 .


0

1

23

4

5

1

2

3

45

a

Figure 2.2. A triangulation of RP 2

The projective plane. The projective plane RP 2 is the quotient of S2 by theantipodal map. The triangulation of S2 as a regular icosahedron being invariantunder the antipodal map, it gives a triangulation of RP 2 given in Figure 2.2. Notethat the border edges appear twice, showing as expected that RP 2 is the quotientof a 2-disk modulo the antipodal involution on its boundary.

Being a quotient of an icosahedron, the triangulation of Figure 2.2 has 6 ver-tices, 15 edges and 10 facets, thus χ(RP 2) = 1. Using that RP 2 is a connected2-dimensional pseudomanifold, we deduce that

(2.4.8) Pt(RP2) = 1 + t+ t2 .

To identify the generators of H1(RP 2) ≈ Z2 and H1(RP 2), we define

(2.4.9) a = α =1, 2, 2, 3, 3, 4, 4, 5, 5, 1

⊂ S1(RP 2) .

We see a ∈ C1(RP 2) and α ∈ C1(RP 2). The cochain a is drawn in bold onFigure 2.2, where it looks as the set of border edges, since each of its edges appearstwice on the figure. It is easy to check that δ(a) = 0 and ∂(α) = 0. As ♯α = 5 isodd, one has 〈a, α〉 = 1, showing that a is the generator of H1(RP 2) = Z2 and α isthe generator of H1(RP 2) = Z2.

The 2-torus. The 2-torus T 2 = S1 × S1 is the quotient of a square whoseopposite sides are identified. A triangulation of T 2 is described (in two copies) inFigure 2.3. This triangulation has 9 vertices, 27 edges and 18 facets, which impliesthat χ(T 2) = 0. Since T 2 is a connected 2-dimensional pseudomanifold, we deducethat

Pt(T2) = (1 + t)2 .

In Figure 2.3 are drawn two chains α, β ∈ C1(T2) given by

α =3, 8, 8, 9, 9, 3

and β =

5, 7, 7, 9, 9, 5

.

We also drew two cochains a, b ∈ C1(T 2) defined as

a =4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 4

and

b =2, 3, 3, 6, 6, 8, 8, 7, 7, 9, 9, 2

.

One checks that ∂α = ∂β = 0 and that δa = δb = 0. Therefore, they representclasses a, b ∈ H1(T 2) and α, β ∈ H1(T

2). The equalities

〈a, α〉 = 1 , 〈a, β〉 = 0 , 〈b, α〉 = 0 , 〈b, β〉 = 1


1 2 3 1

4

5

1 2 3 1

4

5

6

7

8

9

a

α

1 2 3 1

4

5

1 2 3 1

4

5

6

7

8

9

b

β

Figure 2.3. Two copies of a triangulation of the 2-torus T 2, show-ing generators of H1(T 2) and H1(T

2)

imply that a, b is a basis of H1(T 2) and α, β is a basis of H1(T2).

If we consider a and b as 1-chains (call them a and b), we also have ∂a = ∂b = 0.Note that

〈a, b〉 = 1 , 〈a, a〉 = 0 , 〈b, b〉 = 0 , 〈b, a〉 = 1

This proves that a = β and b = α in H1(T2).

The Klein bottle. A triangulation of the Klein bottle K is pictured in Fig-ure 2.4. As the 2-torus, the Klein bottle is the quotient of a square with opposite sideidentified, one of these identifications “reversing the orientation”. One checks thatχ(K) = 0. SinceK is a connected 2-dimensional pseudomanifold, the (co)homologyof K is abstractly isomorphic to that of T 2:

Pt(K) = (1 + t)2

(In Chapter 3, H∗(T 2) and H∗(K) will be distinguished by their cup product: seep. 117). In Figure 2.4 the dotted lines show two 1-chains α, β ∈ C1(K) given by

(2.4.10) α =3, 8, 8, 9, 9, 3

and β =

5, 7, 7, 9, 9, 5

.

The bold lines describe two 1-cochains a, b ∈ C1(K) defined as

(2.4.11) a =4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 5

and

(2.4.12) b =2, 3, 3, 6, 6, 8, 8, 7, 7, 9, 9, 2

.

One checks that ∂α = ∂β = 0 and that δa = δb = 0. Therefore, they representclasses a, b ∈ H1(K) and α, β ∈ H1(K). The equalities

〈a, α〉 = 1 , 〈a, β〉 = 1 , 〈b, α〉 = 0 , 〈b, β〉 = 1

imply that a, b is a basis of H1(K) and α, β is a basis of H1(K).

As in the case of T 2, we may regard a and b as 1-chains (call them a and b).

Here ∂b = 0 but ∂a = 4+ 5 6= 0.Other surfaces. Let K1 and K2 be two simplicial complexes such that |K1|

and |K2| are surfaces. A simplicial complex L with |L| homeomorphic to the con-nected sum |K1|♯|K2| may be obtained in the following way: choose 2-simplexesσ1 ∈ K1 and σ2 ∈ K2. Let Li = Ki − σi and let L be obtained by taking thedisjoint union of L1 and L2 and identifying σ1 with σ2. Thus, L = L1 ∪ L2 andL0 = L1 ∩ L2 is isomorphic to the boundary of a 2-simplex.


1 2 3 1

4

5

1 2 3 1

5

4

6

7

8

9

a

α

1 2 3 1

4

5

1 2 3 1

5

4

6

7

8

9

b

β

Figure 2.4. Two copies of a triangulation of the Klein bottle K,showing generators of H1(K) and H1(K)

By Lemma 2.4.10, one has

χ(L) = χ(L1) + χ(L2)− χ(L0)

= χ(K1)− 1 + χ(K2)− 1− 0

= χ(K1) + χ(K2)− 2 .(2.4.13)

The orientable surface Σg of genus gis defined as the connected sum of g copies ofthe torus T 2. By Formula (2.4.13), one has

(2.4.14) χ(Σg) = 2− 2g .

As Σg is a 2-dimensional connected pseudomanifold, one has

Pt(Σg) = 1 + 2gt+ t2 .

The nonorientable surface Σg of genus g is defined as the connected sum of gcopies of RP 2. For instance, Σ1 = RP 2 and Σ2 is the Klein bottle. Formula (2.4.13)implies

(2.4.15) χ(Σg) = 2− g .As Σg is a 2-dimensional connected pseudomanifold, one has

Pt(Σg) = 1 + gt+ t2 .

2.5. The homomorphism induced by a simplicial map

Let f : K → L be a simplicial map between the simplicial complexes K and L.Recall that f is given by a map f : V (K) → V (L) such that f(σ) ∈ S(L) ifσ ∈ S(K), i.e. the image of an m-simplex of K is an n-simplex of L with n ≤ m.We define C∗f : C∗(K) → C∗(L) as the degree 0 linear map such that, for allσ ∈ Sm(K), one has

(2.5.1) C∗f(σ) =

f(σ) if f(σ) ∈ Sm(L) (i.e. if f|σ is injective)

0 otherwise.

We also define C∗f : C∗(L)→ C∗(K) by setting, for a ∈ Cm(L),

(2.5.2) C∗f(a) =σ ∈ Sm(K) | f(σ) ∈ a

.

In the following lemma, we use the same notation for the (co)boundary operators∂ and δ and the Kronecker product 〈 , 〉, both for K of for L.

2.5. THE HOMOMORPHISM INDUCED BY A SIMPLICIAL MAP 29

Lemma 2.5.1. Let f : K → L be a simplicial map. Then

(a) C∗f ∂ = ∂C∗f .(b) δC∗f = C∗f δ.(c) 〈C∗f(b), α〉 = 〈b, C∗f(α)〉 for all b ∈ C∗(L) and all α ∈ C∗(K).

In other words, the couple (C∗f, C∗f) is a morphism of Kronecker pairs.

Proof. To prove (a), let σ ∈ Sm(K). If f restricted to σ is injective, it isstraightforward that C∗f ∂(σ) = ∂C∗f(σ). Otherwise, we have to show thatC∗f ∂(σ) = 0. Let us label the vertices v0, v1, . . . , vm of σ in such a way thatf(v0) = f(v1). Then, C∗f ∂(σ) is a sum of two terms: C∗f ∂(σ) = C∗f(τ0) +C∗f(τ1), where τ0 = v1, v2, . . . , vm and τ1 = v0, v2, . . . , vm. As C∗f(τ0) =C∗f(τ1), one has C∗f ∂(σ) = 0. Thus, Point (a) is established. Point (c) can beeasily deduced from Definitions (2.5.1) and (2.5.2), taking for α a simplex of K.Point (b) then follows from Points (a) and (c), using Lemma 2.3.6 and its proof.

By Lemma 2.5.1 and Proposition 2.3.5, the couple (C∗f, C∗f) determines linear

maps of degree zero

H∗f : H∗(K)→ H∗(L) and H∗f : H∗(L)→ H∗(K)

such that

(2.5.3) 〈H∗f(a), α〉 = 〈a,H∗f(α)〉 for all a ∈ H∗(L) and α ∈ H∗(K) .

Lemma 2.5.2 (Functoriality). Let f : K → L and g : L→M be simplicial maps.Then H∗(gf) = H∗g H∗f and H∗(gf) = H∗f H∗g. Also H∗idK = idH∗(K)

and H∗idK = idH∗(K)

In other words, H∗ and H∗ are functors from the simplicial category Simp tothe category GrV of graded vector spaces and degree 0 linear maps. The cohomol-ogy is contravariant and the homology is covariant.

Proof. For σ ∈ S(K), the formula C∗(gf)(σ) = C∗g C∗f(σ) follows di-rectly from Definition (2.5.1). Therefore C∗(gf) = C∗g C∗f and then H∗(gf) =H∗g H∗f . The corresponding formulae for cochains and cohomology follow fromPoint (c) of Lemma 2.5.1. The formulae for idK is obvious.

Simplicial maps and components. Let K be a simplicial complex. For eachcomponent L ∈ π0(K) of K, the inclusion iL : L → K is a simplicial map. Theresults of § 2.4.1 may be strengthened as follows.

Proposition 2.5.3. Let K be a simplicial complex. The family of simplicialmaps iL : L→ K for L ∈ π0(K) gives rise to isomorphisms

H∗(K)(H∗iL)

≈// ∏

L∈π0(K)H∗(L)

and

⊕L∈π0(K)H∗(L)

∑H∗iL

≈// H∗(K) .


The homomorphisms H0f and H0f . We use the same notation 1 ∈ H0(K)and 1 ∈ H0(L) for the classes given by the unit cochains.

Lemma 2.5.4. Let f : K → L be a simplicial map. Then H0f(1) = 1.

Proof. The formula C0f(1) = 1 in C0(K) follows directly from Definition (2.5.2).

Corollary 2.5.5. Let f : K → L be a simplicial map with K and L connected.Then

H0f : Z2 = H0(L)→ H0(K) = Z2

andH0f : Z2 = H0(K)→ H0(L) = Z2

are the identity isomorphism.

Proof. By Proposition 2.4.1, the generator of H0(L) (or H0(K)) is the unitcocycle 1. By Lemma 2.5.4, this proves the cohomology statement. The homologystatement also follows from Proposition 2.4.1, sinceH0(K) andH0(L) are generatedby a cycle consisting of a single vertex.

More generally, one has H0(L) ≈ Zπ0(L)2 and H0(K) ≈ Z

π0(K)2 by Corol-

lary 2.4.2. Using this and Lemma 2.5.4, one gets the following corollary.

Corollary 2.5.6. Let f : K → L be a simplicial map. Then H0f : Zπ0(L)2 →

Zπ0(K)2 is given by H0f(λ) = λπ0f .

The degree of a map. Let f : K → L be a simplicial map between two finiteconnected n-dimensional pseudomanifolds. Define the degree deg(f) ∈ Z2 by

(2.5.4) deg(f) =

0 if Hnf = 0

1 otherwise.

By Proposition 2.4.4, Hn(K) ≈ Hn(L) ≈ Z2. Thus, deg(f) = 1 if and only if Hnfis the (only possible) isomorphism between Hn(K) and Hn(L). By Kroneckerduality, the homomorphism Hnf may be used instead of Hnf in the definitionof deg(f). Our degree is sometimes called the mod 2 degree, since, for orientedpseudomanifolds, it is the mod 2 reduction of a degree defined in Z (see, e.g. [179,exercises of Ch. 4]).

Let f : K → L be a simplicial map between two finite n-dimensional pseudo-manifolds. For σ ∈ Sn(L), define(2.5.5) d(f, σ) = ♯τ ∈ Sn(K) | f(τ) = σ ∈ N.

As an example, let K = L = P4, the polygon of Example 2.4.3 with 4 edges. Letf be defined by f(0) = 0, f(1) = 1, f(2) = 2, f(3) = 1. Then, d(f, 0, 1) =d(f, 1, 2) = 2, d(f, 2, 3) = d(f, 3, 0) = 0 and deg(f) = 0. This exampleillustrates the following proposition.

Proposition 2.5.7. Let f : K → L be a simplicial map between two finiten-dimensional pseudomanifolds which are connected. For any σ ∈ Sn(L), one has

deg(f) = d(f, σ) mod 2 .

Proof. By Proposition 2.4.4, Hn(L) = Z2 is generated by the cocycle formedby the singleton σ and Cnf(σ) represents the non-zero element of Hn(K) if andonly if ♯Cnf(σ) = d(f, σ) is odd.

2.5. THE HOMOMORPHISM INDUCED BY A SIMPLICIAL MAP 31

The interest of Proposition 2.5.7 is 2-fold: first, it tells us that deg(f) may becomputed using any σ ∈ Sm(L) and, second, it asserts that d(f, σ) is independentof σ. Proposition 2.5.7 is the mod 2 context of the identity between the degreeintroduced by Brouwer in 1910, [22, p.419], and its homological interpretation dueto Hopf in 1930, [98, § 2]. For a history of the notion of the degree of a map, see[40, pp. 169–175].

Example 2.5.8. Let f : T 2 → K be the two-fold cover of the Klein bottle Kby the 2-torus T 2, given in Figure 2.5. In formulae: f(i) = i = f (i) for i = 1, . . . , 9.

f

1 2 3 1 2 3 1

2 3 1 2 3 1

4

5

6

7

8

9

5

4

6

7

8

9

4

5

1

a

α

1 2 3 1

4

5

1 2 3 1

5

4

6

7

8

9

a

α

Figure 2.5. Two-fold cover f : T 2 → K over the triangulation Kof the Klein bottle given in Figure 2.4.

The 1-dimensional (co)homology vector spaces of T 2 and K admit the bases:

(i) V = [a], [b] ⊂ H1(T 2), where a is drawn in Figure 2.5 and

b =2, 3, 3, 6, 6, 8, 8, 7, 7, 9, 9, 2

.

(ii) W = [α], [β] ⊂ H1(T2), where α is drawn in Figure 2.5 and

β =5, 7, 7, 9, 9, 4, 4, 7, 7, 9, 9, 5

.

(iii) V = [a], [b] ⊂ H1(K), where a and b are defined in Equations (2.4.11)and (2.4.12) (a drawn in Figure 2.5).

(iv) W = [α], [β] ⊂ H1(K), where α and β are defined in Equation (2.4.10)(α drawn in Figure 2.5).

The matrices for C∗f and C∗f in these bases are

C∗f =

(1 00 0

)and C∗f =

(1 00 0

).

Note that, under the isomorphism k : H1(−) ≈−→ H1(−)♯, the bases V and V are

dual of W and W ; therefore, the matrix of C∗f is the transposed of that of C∗f .Now, T 2 and K are 2-dimensional pseudomanifolds and d(f, σ) = 2 for each

σ ∈ S2(K). By Proposition 2.5.7, deg(f) = 0 and both H∗f : H2(K) → H2(T 2)and H∗f : H2(T

2)→ H2(K) vanish.

Contiguous maps. Two simplicial maps f, g : K → L are called contiguousif f(σ) ∪ g(σ) ∈ S(L) for all σ ∈ S(K). We denote by τ(σ) the subcomplex of Lgenerated by the simplex f(σ)∪g(σ) ∈ S(L). For example, the inclusion K → CKof a simplicial complex K into its cone and the constant map of K onto the conevertex of CK are contiguous.


Proposition 2.5.9. Let f, g : K → L be two simplicial maps which are con-tiguous. Then H∗f = H∗g and H∗f = H∗g.

Proof. By Kronecker Duality, using Diagram (2.3.4), it is enough to provethat H∗f = H∗g. By induction on m, we shall prove the following property:

Property H(m): there exists a linear map D : Cm(K)→ Cm+1(L) such that:

(i) ∂D(α) +D(∂α) = C∗f(α) + C∗g(α) for each α ∈ Cm(K).(ii) for each σ ∈ Sm(K), D(σ) ∈ Cm+1(τ(σ)) ⊂ Cm+1(L).

We first prove that Property H(m) for all m implies that H∗f = H∗g. Indeed,we would then have a linear map D : C∗(K)→ C∗+1(L) satisfying

(2.5.6) C∗f + C∗g = ∂D +D∂ .

Such a map D is called a chain homotopy from C∗f to C∗g. Let β ∈ Z∗(K). ByEquation (2.5.6), one has C∗f(β)+C∗g(β) = ∂D(β) which implies that H∗f([β])+H∗g([β]) in H∗(L).

We now prove that H(0) holds true. We define D : C0(K) → C1(L) as theunique linear map such that, for v ∈ V (K):

D(v) =f(v), g(v) = τ(v) if f(v) 6= g(v)

0 otherwise.

Formula (i) being true for any v ∈ S0(K), it is true for any α ∈ C0(K). Formula(ii) is obvious.

Suppose that H(m − 1) holds true for m ≥ 1. We want to prove that H(m)also holds true. Let σ ∈ Sm(K). Observe that D(∂σ) exists by H(m−1). Considerthe chain ζ ∈ Cm(L) defined by

ζ = C∗f(σ) + C∗g(σ) +D(∂σ)

Using H(m− 1), one has

∂ζ = ∂C∗f(σ) + ∂C∗g(σ) + ∂D(∂σ)

= C∗f(∂σ) + C∗g(∂σ) +D(∂∂σ) + C∗f(∂σ) + C∗g(∂σ)

= 0 .

On the other hand, ζ ∈ Cm(τ(σ)). As m ≥ 1, Hm(τ(σ)) = 0 by Corollary 2.4.7.There exists then η ∈ Cm+1(τ(σ)) such that ζ = ∂η. Choose such an η and setD(σ) = η. This defines D : Cm(K)→ Cm+1(L) which satisfies (i) and (ii), provingH(m).

Remark 2.5.10. The chain homotopy D in the proof of Proposition 2.5.9 is notexplicitly defined. This is because several of these exist and there is no canonicalway to choose one (see [155, p. 68]). The proof of Proposition 2.5.9 is an exampleof the technique of acyclic carriers which will be developed in § 2.9.

Remark 2.5.11. Let f, g : K → L be two simplicial maps which are contiguous.Then |f |, |g| : |K| → |L| are homotopic. Indeed, the formula F (µ, t) = (1−t)|f |(µ)+t|g|(µ) (t ∈ [0, 1]) makes sense and defines a homotopy from |f | to |g|.

2.6. EXACT SEQUENCES 33

2.6. Exact sequences

In this section, we develop techniques to obtain long (co)homology exact se-quences from short exact sequences of (co)chain complexes. The results are usedin several forthcoming sections. All vector spaces in this section are over a fixedarbitrary field F.

Let (C∗1 , δ1), (C∗2 , δ2) and (C∗, δ) be cochain complexes of vector spaces, giving

rise to cohomology graded vector spaces H∗1 , H∗2 and H∗. We consider morphisms

of cochain complexes J : C∗1 → C∗ and I : C∗ → C∗2 so that

(2.6.1) 0→ C∗1J−→ C∗

I−→ C∗2 → 0

is an exact sequence. We call (2.6.1) a short exact sequence of cochain complexes.Choose a GrV-morphism S : C∗2 → C∗ which is a section of I. The section Scannot be assumed in general to be a morphism of cochain complexes. The linearmap δS : Cm2 → Cm+1 satisfies

I δS(a) = δ2I S(a) = δ2(a) ,

thus δS(Zm2 ) ⊂ J(Cm+11 ). We can then define a linear map δ∗ : Zm2 → Cm+1

1 bythe equation

(2.6.2) J δ∗ = δS .

If a ∈ Zm2 , then J δ1(δ∗(a)) = δδ(S(a)) = 0. Therefore, δ∗(Zm2 ) ⊂ Zm+1

1 .

Moreover, if b ∈ Cm−12 and a = δ2(b), then

I δS(b) = δ2I S(b) = a ,

whence δS(b) = S(a) + J(c) for some c ∈ Cm1 . Therefore δ∗(a) = δ1(c), which

shows that δ∗(B∗2 ) ⊂ B∗1 . Hence, δ∗ induces a linear map

δ∗ : H∗2 → H∗+11

which is called the cohomology connecting homomorphism for the short exact se-quence (2.6.1).

Lemma 2.6.1. The connecting homomorphism δ∗ : H∗2 → H∗+11 does not depend

on the linear section S.

Proof. Let S′ : Cm2 → Cm be another section of I, giving rise to δ′∗ : Zm2 →Zm+11 , via the equation J δ′∗ = δS′. Let a ∈ Zm2 . Then

S′(a) = S(a) + J(u)

for some u ∈ Cm1 . Therefore, the equations

J δ′∗(a) = δ(S(a)) + δ(J(u)) = δ(S(a)) + J(δ1(u))

hold in Cm+1. This implies that δ′∗(a) = δ∗(a)+ δ1(u) in Zm+11 , and then δ′∗(a) =

δ∗(a) in Hm+11 .

Proposition 2.6.2. The long sequence

· · · → Hm1

H∗J−−−→ Hm H∗I−−−→ Hm2

δ∗−→ Hm+11

H∗J−−−→ · · ·is exact.

The exact sequence of Proposition 2.6.2 is called the cohomology exact sequence,associated to the short exact of cochain complexes (2.6.1).


Proof. The proof involves 6 steps.

1. H∗I H∗J = 0: As H∗I H∗J = H∗(I J), this comes from that I J = 0.

2. δ∗H∗I = 0: Let b ∈ Zm. Then I(b + S(I(b))) = 0. Hence, b+ S(I(b)) = J(c)for some c ∈ Cm1 . Therefore,

J δ∗I(b) = δ(S(I(b)) = δ(b+ J(c)) = δ(b) + J δ1(c) = J δ1(c) ,

which proves that δ∗I(b) = δ1(c), and then δ∗H∗I = 0 in H∗1 .

3. H∗J δ∗ = 0: Let a ∈ Zm2 . Then, J δ∗(a) = δ(S(a)) ⊂ Bm+1, soH∗J δ∗([a]) =0 in Hm+1(K).

4. kerH∗J ⊂ Image δ∗: Let a ∈ Zm+11 representing [a] ∈ kerH∗J . This means

that J(a) = δ(b) for some b ∈ Cm. Then, I(b) ∈ Zm2 and S(I(b)) = b + J(c) forsome c ∈ Cm1 . Therefore,

δS I(b) = δ(b) + δ(J(c)) = J(a) + J(δ1(c)) .

As J is injective, this implies that δ∗(I(c)) = a+δ1(c), proving that δ∗([I(c)]) = [a].

5. kerH∗I ⊂ ImageH∗J : Let a ∈ Zm representing [a] ∈ kerH∗I. This meansthat I(a) = δ2(b) for some b ∈ Cm−12 . Let c = δ(S(b)) ∈ Cm. One has I(a+ c) = 0,so a+ c = J(e) for some e ∈ Cm1 . As δ(a+ c) = 0 and J is injective, the cochain eis in Zm1 . As c ∈ Bm, H∗J([e]) = [a] in Hm.

6. ker δ∗ ⊂ ImageH∗I: Let a ∈ Zm2 representing [a] ∈ ker δ∗. This means that

δ∗(a) = δ1(b) for some b ∈ Cm1 . In other words,

δ(S(a)) = J(δ1(b)) = δ(J(b)) .

Hence, c = J(b) + S(a) ∈ Zm and H∗I([c]) = [a].

We now prove the naturality of the connecting homomorphism in cohomology.We are helped by the following intuitive interpretation of δ∗: first, we consider C∗1a cochain subcomplex of C∗ via the injection J . Second, a cocycle a ∈ Zm2 may berepresented by a cochain in a ∈ Cm such that δ(a) ∈ C∗1 . Then, δ∗([a]) = [δ(a)].More precisely:

Lemma 2.6.3. Let

0→ C∗1J−→ C∗

I−→ C∗2 → 0

be a short exact sequence of cochain complexes. Then

(a) I−1(Zm2 ) = b ∈ Cm | δ(b) ∈ J(Cm+11 ).

(b) Let a ∈ Zm2 representing [a] ∈ Hm2 . Let b ∈ Cm with I(b) = a. Then

δ∗([a]) = [J−1(δ(b))] in Hm+11 .

Proof. Point (a) follows from the fact that I is surjective and from the equalityδ2I = I δ. For Point (b), choose a section S : Cm2 → Cm of I. By Lemma 2.6.1,δ∗([a]) = [J−1(δ(S(a))]. The equality I(b) = a implies that b = S(a) + J(c) forsome c ∈ Cm1 . Therefore,

[J−1(δ(b))] = [J−1(δS(a))] + [δ1(c)] = δ∗([a]) .

2.6. EXACT SEQUENCES 35

Let us consider a commutative diagram

(2.6.3)

0 // C∗1

F1

J // C∗

F

I // C∗2

F2

// 0

0 // C∗1J // C∗ I // C∗2 // 0

of morphisms of cochain complexes, where the horizontal sequences are exact. Thisgives rise to two connecting homomorphisms δ∗ : H∗2 → H∗+1

1 and δ∗ : H∗2 → H∗+11 .

Lemma 2.6.4 (Naturality of the cohomology exact sequence). The diagram

· · · // Hm1

H∗F1

H∗J // Hm

H∗F

H∗ I // Hm2

H∗F2

δ∗ // Hm+11

H∗F1

H∗J // · · ·

. . . // Hm1

H∗J // Hm H∗I // Hm2

δ∗ // Hm+11

H∗J // · · ·

is commutative.

Proof. The commutativity of two of the square diagrams follows from thefunctoriality of the cohomology: H∗F H∗J = H∗J H∗F1 since F J = J F1 andH∗F2H

∗I = H∗I H∗F since F2 I = I F . It remains to prove that H∗F1 δ∗ =

H∗δ∗F2.Let a ∈ Zm2 representing [a] ∈ Hm

2 . Let b ∈ Cm with I(b) = a. Then,I F (b) = F2(a). Using Lemma 2.6.3, one has

δ∗H∗F2([a]) = [J−1δF (b)]

= [J−1F δ(b)]

= [F1 J−1 δ(b)]

= H∗F1 δ∗([a]) .

We are now interested in the case where the cochain complexes (C∗i , δi) and(C∗, δ) are parts of Kronecker pairs

P1 =((C∗1 , δ1), (C∗,1, ∂1), 〈 , 〉1

), P2 =

((C∗2 , δ2), (C∗,2, ∂2)〈 , 〉2

)

and

P =((C∗, δ), (C∗, ∂), 〈 , 〉

).

Let us consider two morphism of Kronecker pairs, (J, j) from P to P1 and (I, i)from P2 to P . We suppose that the two sequences

(2.6.4) 0→ C∗1J−→ C∗

I−→ C∗2 → 0

and

(2.6.5) 0→ C∗,2i−→ C∗

j−→ C∗,1 → 0

are exact sequences of (co)chain complexes. Note that, by Lemma 2.3.8, (2.6.4)is exact if and only if (2.6.5) is exact. Exact sequence (2.6.4) gives rise to thecohomology connecting homomorphism δ∗ : H∗2 → H∗+1

1 . We construct a homologyconnecting homomorphism in the same way. Choose a linear section s : C∗,1 → C∗


of j, not required to be a morphism of chain complexes. As in the cohomologysetting, one can defines ∂∗ : Zm+1,1 → Zm,2 by the equation

(2.6.6) i ∂∗ = ∂s .

We check that ∂∗(Bm+1,1) ⊂ Bm,2. Hence ∂∗ induces a linear map

∂∗ : H∗+1,1 → H∗,2

called the homology connecting homomorphism for the short exact sequence (2.6.5).

Lemma 2.6.5. The connecting homomorphism ∂∗ : H∗+1,1 → H∗,2 does notdepend on the linear section s.

Proof. The proof is analogous to that of Lemma 2.6.1 and is left as an exerciseto the reader.

Lemma 2.6.6. The connecting homomorphisms δ∗ : Hm2 → Hm+1

1 and ∂∗ : Hm+1,1 →Hm,1 satisfy the equation

〈δ∗(a), α〉1 = 〈a, ∂∗(α)〉2for all a ∈ Hm

2 , α ∈ Hm+1,1 and all m ∈ N. In other words, (δ∗, ∂∗) is a morphismof Kronecker pairs from (H∗1 , H∗,1, 〈 , 〉1) to (H∗2 , H∗,2, 〈 , 〉2).

Proof. Let a ∈ Zm2 represent a and α ∈ Zm+1,1 represent α. Choose linearsections S and s of I and j. Using Formulae (2.6.2) and (2.6.6), one has

〈δ∗(a), α〉1 = 〈δ∗(a), α〉1= 〈δ∗(a), j s(α)〉1= 〈J δ∗(a), s(α)〉= 〈S(a), ∂s(α)〉= 〈S(a), i ∂∗(α)〉= 〈I S(a), ∂∗(α)〉2= 〈a, ∂∗(α)〉2 = 〈a, ∂∗(α)〉2 .


· · · → Hm,2H∗i−−→ Hm

H∗j−−→ Hm,1∂∗−→ Hm−1,2

H∗i−−→ · · ·is exact.

The exact sequence of Proposition 2.6.7 is called the homology exact sequenceassociated to the short exact of chain complexes (2.6.5). It can be establisheddirectly, in an analogous way to that of Proposition 2.6.2. To make a change, weshall deduce Proposition 2.6.7 from Proposition 2.6.2 by Kronecker duality.

Proof. By our hypotheses couples (I, i) and (J, j) are morphisms of Kro-necker pairs, and so is (δ∗, ∂∗) by Lemma 2.6.6. Using Diagram (2.3.4), we get acommutative diagram

· · · (Hm,2)♯oo (Hm)♯

(H∗i)♯

oo (Hm,1)♯

(H∗j)♯

oo H♯m−1,2

∂♯∗oo · · ·oo

· · · Hm1

oo

k≈

OO

HmH∗Ioo

k≈

OO

Hm1

H∗Joo

k≈

OO

Hm−12

δ∗oo

k≈

OO

· · ·oo

.

2.7. RELATIVE (CO)HOMOLOGY 37

By Proposition 2.6.2, the bottom sequence of the above diagram is exact. Thus,the top sequence is exact. By Lemma 2.3.8, the sequence of Proposition 2.6.7 isexact.

Let us consider commutative diagrams

(2.6.7)

0 // C∗1

F1

J // C∗

F

I // C∗2

F2

// 0

0 // C∗1J // C∗

I // C∗2 // 0

and

(2.6.8)

0 C∗,1oo C∗joo C∗,2

ioo 0oo

0 C∗,1oo

f1

OO

C∗joo

f

OO

C∗,2ioo

f2

OO

0oo

such that the horizontal sequences are exact, Fi and F are morphisms of cochaincomplexes and fi and f are morphisms of cochain complexes.

Lemma 2.6.8 (Naturality of the homology exact sequence). Suppose that (Fi, fi)and (F, f) are morphisms of Kronecker pairs. Then, the diagram

· · · // Hm,2

H∗f2

H∗i // Hm

H∗f

H∗j // Hm,1

H∗f1

∂∗ // Hm−1,2

H∗f2

H∗i // · · ·

. . . // Hm,2H∗ i // Hm

H∗ j // Hm,1∂∗ // Hm−1,2

H∗ i // · · ·

is commutative.

Proof. By functoriality of the homology, the square diagrams not involving ∂∗commute. It remains to show that ∂∗H∗f1 = H∗f2∂∗. As H∗F1 δ

∗ = δ∗H∗F2

by Lemma 2.6.4, one has

〈a, ∂∗ H∗f1(α)〉2 = 〈H∗F1 δ∗(a), α〉1 = 〈δ∗ H∗F2(a), α〉1 = 〈a,H∗f1∂∗(α)〉2

for all a ∈ Hm−12 and α ∈ Hm,1. By Lemma 2.3.3, this implies that ∂∗H∗f1 =

H∗f2∂∗.

2.7. Relative (co)homology

A simplicial pair is a couple (K,L) where K is a simplicial complex and L is asubcomplex of K. The inclusion i : L → K is a simplicial map. Let a ∈ Cm(K).If, using Definition defiI.a of § 2.2, we consider a as a subset of Sm(K), thenC∗i(a) = a ∩ Sm(L). If we see a as a map a : Sm(K) → Z2, then C∗i(a) is therestriction of a to Sm(L). We see that C∗i : C∗(K)→ C∗(L) is surjective. Define

Cm(K,L) = ker(Cm(K)

C∗i−−→ Cm(L))

and C∗(K,L) = ⊕m∈NCm(K,L). This definition implies that

• Cm(K,L) is the set of subsets of Sm(K)− Sm(L);• if K is a finite simplicial complex, Cm(K,L) is the vector space with basisSm(K)− Sm(L).


As C∗i is a morphism of cochain complexes, the coboundary δ : C∗(K) →C∗(K) preserves C∗(K,L) and gives rise to a coboundary δ : C∗(K,L)→ C∗(K,L)so that (C∗(K,L), δ) is a cochain complex. The cocycles Z∗(K,L) and the cobound-aries B∗(K,L) are defined as usual, giving rise to the definition

Hm(K,L) = Zm(K,L)/Bm(K,L) .

The graded Z2-vector space H∗(K,L) = ⊕m∈NHm(K,L) is the simplicial relative

cohomology of the simplicial pair (K,L).When useful, the notations δK , δL and δK,L are used for the coboundaries of

the cochain complexes C∗(K), C∗(L) and C∗(K,L). We denote by j∗ the inclusionj∗ : C∗(K,L) → C∗(K), which is a morphism of cochain complexes, and use thesame notation j∗ for the induced linear map j∗ : H∗(K,L)→ H∗(K) on cohomol-ogy. We also use the notation i∗ for both C∗i and H∗i. We get thus a short exactsequence of cochain complexes

(2.7.1) 0→ C∗(K,L)j∗−→ C∗(K)

i∗−→ C∗(L)→ 0 .

If a ∈ Cm(L), any cochain a ∈ Cm(K) with i∗(a) = a is called a extension of a asa cochain in K. For instance, the 0-extension of a is defined by a = a ∈ Sm(L) ⊂Sm(K). Using § 2.6, Exact sequence (2.7.1) gives rise to a (simplicial cohomology)connecting homomorphism

δ∗ : H∗(L)→ H∗+1(K,L) .

It is induced by a linear map δ∗ : Zm(L) → Zm+1(K,L) characterized by the

equation j∗ δ∗ = δK S for some (or any) linear section S : Cm(L)→ Cm(K) of i∗,not required to be a morphism of cochain complex. For instance, one can take S(a)to be the 0-extension of a. Using that C∗(K,L) is a chain subcomplex of C∗(K),the following statement makes sense and constitutes a useful recipe for computingthe connecting homomorphism δ∗.

Lemma 2.7.1. Let a ∈ Zm(L) and let a ∈ Cm(K) be any extension of a as anm-cochain of K. Then, δK(a) is an (m+ 1)-cocycle of (K,L) representing δ∗(a).

Proof. Choose a linear section S : Cm(L)→ Cm(K) such that S(a) = a. The

equation j∗ δ∗ = δK S proves the lemma.

We can now use Proposition 2.6.2 and get the following result.


· · · → Hm(K,L)j∗−→ Hm(K)

i∗−→ Hm(L)δ∗−→ Hm+1(K,L)

j∗−→ · · ·is exact.

The exact sequence of Proposition 2.7.2 is called the simplicial cohomologyexact sequence, or just the simplicial cohomology sequence, of the simplicial pair(K,L).

We now turn our interest to homology. The inclusion L → K induces aninclusion i∗ : C∗(L) → C∗(K) of chain complexes. We define Cm(K,L) as thequotient vector space

Cm(K,L) = coker(i∗ : Cm(L) → Cm(K)

).

As i∗ is a morphism of chain complexes, C∗(K,L) = ⊕m∈NCm(K,L) inherits aboundary operator ∂ = ∂K,L : C∗(K,L)→ C∗−1(K,L). The projection j∗ : C∗(K)→


→ C∗(K,L) is a morphism of chain complexes and one gets a short exact sequenceof chain complexes

(2.7.2) 0→ C∗(L)i∗−→ C∗(K)

j∗−→ C∗(K,L)→ 0 .

The cycles and boundaries Z∗(K,L) and B∗(K,L) are defined as usual, giving riseto the definition

Hm(K,L) = Zm(K,L)/Bm(K,L) .

The graded Z2-vector space H∗(K,L) = ⊕m∈NHm(K,L) is the relative homologyof the simplicial pair (K,L). As before, the notations ∂K and ∂L may be used forthe boundary operators in C∗(K) and C∗(L) and i∗ and j∗ are also used for theinduced maps in homology.

Since the linear map i∗ : C∗(L) → C∗(K) is induced by the inclusion of basesS(L) → S(K), the quotient vector space C∗(K,L) may be considered as the vectorspace with basis S(K) − S(L). This point of view provides a tautological linearmap s : C∗(K,L) → C∗(K), which is a section of j∗ but not a morphism of chaincomplexes.

The Kronecker pairings for K and L are denoted by 〈 , 〉K and 〈 , 〉L, both atthe levels of (co)chains and of (co)homology. As 〈j∗(K,L), i∗(L)〉K = 0, we get abilinear map

Cm(K,L)× Cm(K,L)〈,〉K,L−−−−→ Z2 .

The formula

(2.7.3) 〈a, α〉K,L = 〈j∗(a), s(α)〉Kholds for all a ∈ Cm(K,L), α ∈ Cm(K,L) and all m ∈ N. Observe also that theformula

(2.7.4) 〈S(b), i∗(β)〉K = 〈b, β〉Lholds for all b ∈ Cm(L), β ∈ Cm(L) and all m ∈ N.

Lemma 2.7.3.(C∗(K,L), δK,L, C∗(K,L), ∂K,L, 〈 , 〉K,L

)is a Kronecker pair.

Proof. We first prove that 〈δK,L(a), α〉K,L = 〈a, ∂K,L(α)〉K,L for all a ∈Cm(K,L) and all α ∈ Cm+1(K,L) and all m ∈ N. Indeed, one has

〈δK,L(a), α〉K,L = 〈j∗δK,L(a), s(α)〉K(2.7.5)

= 〈δK j∗(a), s(α)〉K= 〈j∗(a), ∂K s(α)〉K

Observe that j∗∂K s(α) = ∂K,L(α) and therefore ∂K s(α) = s∂K,L(α) + i∗(c)for some c ∈ Cm(L). Hence, the chain of equalities in (2.7.5) may be continued

〈δK,L(a), α〉K,L = 〈j∗(a), ∂K s(α)〉K(2.7.6)

= 〈j∗(a), s∂K,L(α) + i∗(c)〉K= 〈j∗(a), s∂K,L(α)〉K + 〈j∗(a), i∗(c)〉K︸︷︷︸

0= 〈a, ∂K,L(α)〉K,L .It remains to prove that the linear map k : C∗(K,L) → C∗(K,L)

♯ given byk(a) = 〈a, 〉 is an isomorphism. As the inclusion i : L → K is a simplicial map, thecouple (C∗i, C∗i) is a morphism of Kronecker pairs by Lemma 2.5.1 and the resultfollows from Lemma 2.3.9.


Passing to homology then produces three Kronecker pairs with vanishing (co)boun-dary operators:

PL = (H∗(L), H∗(L), 〈, 〉L) , PK = (H∗(K), H∗(K), 〈, 〉K)

and

PK,L = (H∗(K,L), H∗(K,L), 〈, 〉K,L) .Using § 2.6, short exact sequence (2.7.2) gives rise to the (simplicial homology)

connecting homomorphism

∂∗ : H∗(K,L)→ H∗−1(L) .

It is induced by a linear map ∂ : Zm(K,L)→ Zm−1(L) characterized by the equa-tion

j∗ ∂∗ = ∂K s ,

using the section s of j∗ defined above (or any other one).

Lemma 2.7.4. The following couples are morphisms of Kronecker pairs:

(a) (i∗, i∗), from PL to PK .(b) (j∗, j∗), from PK to PK,L.(c) (δ∗, ∂∗), from PK,L to PL.

Proof. As the inclusion L → K is a simplicial map, Point (a) follows fromLemma 2.5.1. Point (c) is implied by Lemma 2.6.6. To prove Point (b), let a ∈Cm(K,L) and α ∈ Cm(K). Observe that s(j∗(α)) = α+ i∗(β) for some β ∈ Cm(L)and that 〈j∗(a), i∗(β)〉K = 0. Therefore:

〈a, j∗(α)〉K,L = 〈j∗(a), sj∗(α)〉K = 〈j∗(a), α〉K

Proposition 2.6.7 now gives the following result.


· · · → Hm(L)i∗−→ Hm(K)

j∗−→ Hm(K,L)∂∗−→ Hm−1(L)

i∗−→ · · ·is exact.

The exact sequence of Proposition 2.7.5 is called the (simplicial) homology exactsequence, or just the (simplicial) cohomology sequence, of the simplicial pair (K,L).

We now study the naturality of the (co)homology sequences. Let (K,L) and(K ′, L′) be simplicial pairs. A simplicial map f of simplicial pairs from (K,L)to (K ′, L′) is a simplicial map fK : K → K ′ such that the restriction of f toL is a simplicial map fL : L → L′. The morphism C∗fK : C∗(K ′) → C∗(K) thenrestricts to a morphism of cochain complexes C∗f : C∗(K ′, L′)→ C∗(K,L) and themorphism C∗fK : C∗(K) → C∗(K

′) descends to a morphism of chain complexesC∗f : C∗(K,L) → C∗(K

′, L′). The couples (C∗fK , C∗fK) and (C∗fL, C∗fL) aremorphisms of Kronecker pairs by Lemma 2.5.1. We claim that (C∗f, C∗f) is amorphism of Kronecker pair from (C∗(K,L), . . . ) to (C∗(K ′, L′), . . . ). Indeed, leta ∈ Cm(K ′, L′) and α ∈ Cm(K,L). One has

〈C∗f(a), α〉K,L = 〈j∗ C∗f(a), s(α)〉K= 〈C∗fK j′∗(a), s(α)〉K= 〈j′∗(a), C∗fK s(α)〉K′= 〈j′∗(a), C∗fK s(α)〉K′(2.7.7)


and

(2.7.8) 〈a, C∗f(α)〉K′,L′ = 〈j′∗(a), s′ C∗f(α)〉K′The equation j′∗ s

′C∗f(α) = j′∗ C∗fK s(α) = C∗f(α) implies that s′ C∗f(α) =

C∗fK s(α)+i′∗(β) for some β ∈ Cm(L′). As 〈j′∗(a), i′∗(β)〉K′ = 0, Equations (2.7.7)

and (2.7.8) imply that 〈C∗f(a), α〉K,L = 〈a, C∗f(α)〉K′,L′ .Lemma 2.6.4 and Lemma 2.6.8 then imply the following

Proposition 2.7.6. The cohomology and homology sequences are natural withrespect to simplicial maps of simplicial pairs. In other words, given a simplicialmap of simplicial pairs f : (K,L)→ (K ′, L′), the following diagrams

· · · // Hm(K ′, L′)

H∗f

j′∗ // Hm(K ′)

H∗fK

i′∗ // Hm(L′)

H∗fL

δ′∗ // Hm+1(K ′, L′)

H∗f

j′∗ // · · ·

. . . // Hm(K,L)j∗ // Hm(K)

i∗ // Hm(L)δ∗ // Hm+1(K,L)

j∗ // · · ·

and

· · · // Hm(L)

H∗fL

i∗ // Hm(K)

H∗fK

j∗ // Hm(K,L)

H∗f

∂∗ // Hm−1(L)

H∗fL

i∗ // · · ·

. . . // Hm(L′)i′∗ // Hm(K ′)

j′∗ // Hm(K ′, L′)∂′∗ // Hm−1(L

′)i′∗ // · · ·

are commutative.

We finish this section by the exact sequences for a triple. A simplicial triple isa triplet (K,L,M) where K is a simplicial complex, L is a subcomplex of K andM is a subcomplex of L. A simplicial map f of simplicial triples, from (K,L,M)to (K ′, L′,M ′) is a simplicial map fK : K → K ′ such that the restrictions of fK toL and M are simplicial maps fL : L→ L′ and fM : M →M ′.

A simplicial triple T = (K,L,M) gives rise to pair inclusions

(L,M)i−→ (K,M)

j−→ (K,L)

and to a commutative diagram

(2.7.9)

0 // C∗(K,L)

C∗j

j∗K,L // C∗(K)OOid=

i∗K,L // C∗(L)

i∗L,M

// 0

0 // C∗(K,M)j∗K,M // C∗(K)

i∗K,M // C∗(M) // 0

where the horizontal lines are exact sequences of cochain complexes. A diagram-chase shows that the morphism i∗K,L j

∗K,M , which sends C∗(K,M) to C∗(L), has

image C∗(L,M) and kernel the image of C∗j. This morphism coincides with C∗i.We thus get a short exact sequence of cochain complexes

(2.7.10) 0→ C∗(K,L)C∗j−−→ C∗(K,M)

C∗i−−→ C∗(L,M)→ 0 .

The same arguments with the chain complexes gives a short exact sequence

(2.7.11) 0→ C∗(L,M)C∗i−−→ C∗(K,M)

C∗j−−→ C∗(K,L)→ 0 .


As above in this section, short exact sequences (2.7.10) and (2.7.11) producesconnecting homomorphisms δT : H

∗(L,M) → H∗+1(K,L) and ∂T : H∗(K,L) →C∗−1(L,M). They satisfy 〈δT (a), α〉 = 〈a, ∂T (α)〉 as well as following proposition.

Proposition 2.7.7 ((Co)homology sequences of a simplicial triple). Let T =(K,L,M) be a simplicial triple. Then,

(a) the sequences

· · · → Hm(K,L)H∗j−−−→ Hm(K,M)

H∗i−−→ Hm(L,M)δT−→ Hm+1(K,L)

H∗j−−−→ · · ·and

· · · → Hm(L,M)H∗i−−→ Hm(K,M)

H∗j−−→ Hm(K,L)∂T−−→ Hm−1(L,M)

H∗i−−→ · · ·are exact.

(b) the exact sequences of Point (a) are natural for simplicial maps of simpli-cial triples.

Remark 2.7.8. AsH∗(∅) = 0, we get a canonicalGrV-isomorphismsH∗(K, ∅) ≈−→H∗(K), etc. Thus, the (co)homology sequences for the triple (K,L, ∅) give backthose of the pair (K,L)

(2.7.12) · · · → Hm(K,L)H∗j−−−→ Hm(K)

H∗i−−→ Hm(L)δ∗−→ Hm+1(K,L)

H∗j−−−→ · · ·and

(2.7.13) · · · → Hm(L)H∗i−−→ Hm(K)

H∗j−−→ Hm(K,L)∂∗−→ Hm−1(L)

H∗i−−→ · · ·where i : L → K and j : (K, ∅) → (K,L) denote the inclusions. This gives a moreprecise description of the morphisms j∗ and j∗ of Propositions 2.7.2 and 2.7.5.

2.7.9. Historical note. The relative homology was introduced by S. Lefschetzin 1927 in order to work out the Poincare duality for manifolds with boundary(see, e.g. [40, p. 58], [51, p. 47]). The use of exact sequences occurred in sev-eral parts of algebraic topology after 1941 (see, e.g. [40, p. 86], [51, p. 47]). The(co)homology exact sequences play an essential role in the axiomatic approach ofEilenberg-Steenrod, [51].

2.8. Mayer-Vietoris sequences

Let K be a simplicial complex with two subcomplexes K1 and K2. We supposethat K = K1 ∪K2 (i.e. S(K) = S(K1) ∪ S(K2)). We call (K,K1,K2) a simplicialtriad. Then, K0 = K1 ∩ K2 is a subcomplex of K1, K2 and K, with S(K0) =S(K1)∩S(K2). The Mayer-Vietoris sequences relate the (co)homology of X to thatof Xi, generalizing Lemma 2.4.10. The various inclusions are denoted as follows

(2.8.1)

K0

i2

i1 // K1

j1

K2j2 // K .

The notations i∗1, j∗1 , . . . , stand for both C∗i1, C

∗j1, etc, and H∗i1, H∗j1, etc.

The same holds for chains and homology: i1∗ for both C∗i1 and H∗i1, etc. Dia-gram (2.8.1) induces two diagrams

2.8. MAYER-VIETORIS SEQUENCES 43

C∗(K)

j∗2

j∗1 // // C∗(K1)

i∗1C∗(K2)

i∗2 // // C∗(K0)

and

C∗(K0)

i2∗

// i1∗ // C∗(K1)j1∗

C∗(K2) // j2∗ // C∗(K) .

The cohomology diagram is Cartesian (pullback) and the homology diagram isco-Cartesian (pushout). Therefore, the sequence

(2.8.2) 0→ C∗(K)(j∗1 ,j

∗2 )−−−−→ C∗(K1)⊕ C∗(K2)

i∗1+i∗2−−−→ C∗(K0)→ 0

is an exact sequence of cochain complexes and the sequence

(2.8.3) 0→ C∗(K0)(i1∗,i2∗)−−−−−→ C∗(K1)⊕ C∗(K2)

j1∗+j2∗−−−−−→ C∗(K)→ 0

is an exact sequence of chain complexes.Consider the Kronecker pairs (C∗(Ki), C∗(Ki), 〈 , 〉i) for i = 0, 1, 2, and the

Kronecker pair (C∗(K), C∗(K), 〈 , 〉). A bilinear map

〈 , 〉⊕ :[C∗(K1)⊕ C∗(K2)

]×

[C∗(K1)⊕ C∗(K2)

]→ Z2

is defined by

〈(a1, a2), (α1, α2)〉⊕ = 〈a1, α1〉1 + 〈a2, α2〉2 .We check that (C∗(K1) ⊕ C∗(K2), C∗(K1) ⊕ C∗(K2), 〈 , 〉⊕) is a Kronecker pairand that the couples ((j∗1 , j

∗2 ), j

∗1 + j∗2 ) and (i∗1 + i∗2, (i

∗1, i∗2)) are morphisms of Kro-

necker pairs. By § 2.6, there exist linear maps δMV : H∗(K0) → H∗+1(K) and∂MV : H∗(K) → H∗−1(K0) which, by Lemma 2.6.2 and 2.6.7, give the followingproposition.

Proposition 2.8.1 (Mayer-Vietoris sequences). The long sequences

· · · → Hm(K)(j∗1 ,j

∗2 )−−−−→ Hm(K1)⊕Hm(K2)

i∗1+i∗2−−−→ Hm(K0)

δMV−−−→ Hm+1(K)→ · · ·and

· · · → Hm(K0)(i1∗,i2∗)−−−−−→ Hm(K1)⊕Hm(K2)

j1∗+j2∗−−−−−→ Hm(K)∂MV−−−→ Hm−1(K0)→ · · ·

are exact.

The homomorphisms δMV and ∂MV are called the Mayer-Vietoris connectinghomomorphisms in (co)homology. By Lemma 2.6.6, they satisfy 〈δMV (a), α〉 =〈a, ∂MV (α)〉0 for all a ∈ Hm(K0), all α ∈ Hm+1(k) and all m ∈ N. To define theconnecting homomorphisms, one must choose a linear section S of i∗1 + i∗2 and s ofj1∗ + j2∗. One can choose S(a) = (S1(a), 0), where S1 : C

∗(K) → C∗(K1) is thetautological section of i∗1 given by the inclusion S(K0) → S(K1) (see § 2.7). Achoice of s is given, for σ ∈ S(K), by

s(σ) =

(σ, 0) if σ ∈ S(K1)

(0, 0) if σ /∈ S(K1) .

These choices produce linear maps δMV : Z∗(K0)→ Z∗+1(K) and ∂MV : Z∗(K)→Z∗−1(K0), representing δMV and ∂MV and defined by the equations

(j∗1 , j∗2 ) δMV = (δ1, δ2)S and (i1∗, i2∗) ∂MV = (∂1, ∂2)s .


(The apparent asymmetry of the choices has no effect by Lemma 2.6.1 and itshomology counterpart: exchanging 1 and 2 produces other sections, giving rise tothe same connecting homomorphisms.)

Finally, the Mayer-Vietoris sequences are natural for maps of simplicial triads.If T = (K,K1,K2) and T ′ = (K ′,K ′1,K

′2) are simplicial triads and if f : K → K ′

is a simplicial map such that f(Ki) ⊂ K ′i, then the Mayer Vietoris sequences ofT and T ′ are related by commutative diagrams, as in Proposition 2.7.6. This is adirect consequence of Lemma 2.6.4 and Lemma 2.6.8.

2.9. Appendix A: an acyclic carrier result

The powerful technique of acyclic carriers was introduced by Eilenberg andMacLane in 1953 [50], after earlier work by Lefschetz. Proposition 2.9.1 belowis a very particular example of this technique, adapted to our needs. For a fulldevelopment of acyclic carriers, see, e.g., [155, Ch. 1, § 13].

Let (C∗, ∂) and (C∗, ∂) be two chain complexes and let ϕ : C∗ → C∗ be amorphism of chain complexes. We suppose that Cm is equipped with a basis Smfor each m and denote by S the union of all Sm. An acyclic carrier A∗ for ϕ withrespect to the basis S is a correspondence which associates to each s ∈ S a subchaincomplex A∗(s) of C∗ such that

(a) ϕ(s) ∈ A∗(s).(b) H0(A∗(s)) = Z2 and Hm(A∗(s)) = 0 for m > 0.(c) let s ∈ Sm and t ∈ Sm−1 such that t occurs in the expression of ∂ s in the

basis Sm−1. Then A∗(t) is a subchain complex of A∗(s) and the inclusionA∗(t) ⊂ A∗(s) induces an isomorphism on H0.

(d) if s ∈ S0 ⊂ C0 = Z0, then H0ϕ(s) 6= 0 in H0(A∗(s)).

Proposition 2.9.1. Let ϕ and ϕ′ be two morphisms of chain complexes from(C∗, ∂) to (C∗, ∂). Suppose that ϕ and ϕ′ admit the same acyclic carrier A∗ withrespect to some basis S of C∗. Then H∗ϕ = H∗ϕ

′.

Proof. The proof is similar to that of Proposition 2.5.9. By induction on m,we shall prove the following property:

Property H(m): there exists a linear map D : Cm → Cm+1 such that:

(i) ∂D(α) +D(∂α) = ϕ(α) + ϕ′(α) for all α ∈ Cm.(ii) for each s ∈ Sm, D(s) ∈ Am+1(s).

Property H(m) for all m implies that H∗ϕ = H∗ϕ′. Indeed, we then have a

linear map D : C∗ → C∗+1 satisfying

(2.9.1) ϕ+ ϕ′ = ∂D +D∂ .

Let β ∈ Z∗. By Equation (2.9.1), one has ϕ(β) +ϕ(β) = ∂D(β) which implies thatH∗ϕ([β]) +H∗ϕ

′([β]) in H∗.Let us prove H(0). Let s ∈ S0. In H0(A∗(s)) = Z2, one has H0ϕ(s) 6= 0

and H0ϕ′(s) 6= 0. Therefore H∗ϕ(s) = H∗ϕ

′(s) in H0(A∗(s)). This implies thatϕ(s) + ϕ′(s) = ∂(ηs) for some ηs ∈ A1(s). We set D(s) = ηs. This procedure,for each s ∈ S0, provides a linear map D : C0 → C1, which, as ∂C0 = 0, satisfiesϕ(s) + ϕ′(s) = ∂D(α) +D(∂(α)).

We now prove that H(m− 1) implies H(m) for m ≥ 1. Let s ∈ Sm. The chainD(∂s) exists in Am(s) by H(m− 1). Let ζ ∈ Am(s) defined by

ζ = ϕ(s) + ϕ′(s) +D(∂s)

2.10. APPENDIX B: ORDERED SIMPLICIAL (CO)HOMOLOGY 45

Using H(m − 1), one checks that ∂ζ = 0. Since Hm(A∗(s)) = 0, there existsν ∈ Am+1(s) such that ζ = ∂ν. Choose such an element ν and set D(σ) = ν. Thisdefines D : Cm → Cm+1 which satisfies (i) and (ii), proving H(m).

2.10. Appendix B: ordered simplicial (co)homology

This technical section may be skipped in a first reading. It shows that sim-plicial (co)homology may be defined using larger sets of (co)chains, based on or-dered simplexes. This will be used for comparisons between simplicial and singular(co)homology (see § 3.6) and to define the cup and cap products in Chapter 4.

Let K be a simplicial complex. Define

Sm(K) = (v0, . . . , vm) ∈ V (K)m+1 | v0, . . . , vm ∈ S(K) .Observe that dimv0, . . . , vm ≤ m and may be strictly smaller if there are repeti-

tions amongst the vi’s. An element of Sm(K) is an ordered m-simplex of K.The definitions of ordered (co)chains and (co)homology are the same those for

the simplicial case (see § 2.2), replacing the simplexes by the ordered simplexes.We thus set


(a) An ordered m-cochain is a subset of Sm(K).

(b) An ordered m-chain is a finite subset of Sm(K).

The set of ordered m-cochains of K is denoted by Cm(K) and that of ordered

m-chains by Cm(K). As in § 2.2, Definitions I are equivalent to


(a) An ordered m-cochain is a function a : Sm(K)→ Z2.

(b) An ordered m-chain is a function α : Sm(K)→ Z2 with finite support.

Definitions 2.10.2 endow Cm(K) and Cm(K) with a structure of a Z2-vector

space. The singletons provide a basis of Cm(K), in bijection with Sm(K). Thus,Definition 2.10.2.b is equivalent to

Definition 2.10.3. Cm(K) is the Z2-vector space with basis Sm(K):

Cm(X) =⊕

σ∈Sm(X)

Z2 σ .

We consider the graded Z2-vector spaces C∗(K) = ⊕m∈NCm(K) and C∗(K) =

⊕m∈NCm(K). The Kronecker pairing on ordered (co)chains

Cm(K)× Cm(K)〈 , 〉−−→ Z2

is defined, using the various above definitions, by the equivalent formulae

(2.10.1)

〈a, α〉 = ♯(a ∩ α) (mod 2) using Definitions 2.10.1.a and b

=∑σ∈α a(σ) using Definitions 2.10.1.a and 2.10.2.b

=∑σ∈Sm(K) a(σ)α(σ) using Definitions 2.10.2.a and b .

As in Lemma 2.2.4, we check that the map k : Cm(K)→ Cm(K)♯, given by k(a) =〈a, 〉, is an isomorphism.


The boundary operator ∂ : Cm(K) → Cm−1(K) is the Z2-linear map defined,

for (v0, . . . , vm) ∈ Sm(K) by

(2.10.2) ∂(v0, . . . , vm) =

m∑

i=0

(v0, . . . , vi, . . . , vm) ,

where (v0, . . . , vi, . . . , vm) ∈ Sm−1 is the m-tuple obtained by removing vi. The

coboundary operator δ : Cm(K)→ Cm+1(K) is defined by the equation

(2.10.3) 〈δa, α〉 = 〈a, ∂α〉 .With these definition, (C∗(K), ∂, C∗(K), δ, 〈 , 〉) is a Kronecker pair. We de-

fine the vector spaces of ordered cycles Z∗(K), ordered boundaries B∗(K), ordered

cocycles Z∗(K), ordered coboundaries B∗(K), ordered homology H∗(K) and or-

dered cohomology H∗(K) as in § 2.3. By Proposition 2.3.5, the pairing on (co)chaindescends to a pairing

Hm(K)×Hm(K)〈 , 〉−−→ Z2

so that the map k : Hm → H♯m, given by k(a) = 〈a, 〉, is an isomorphism (ordered

Kronecker duality).

Example 2.10.4. Let K = pt be a point. Then, Sm(pt) contains one element

for each integer m, namely the (m + 1)-tuple (pt, . . . , pt). Then, Cm(pt) = Z2 forall m ∈ N and the chain complex looks like

· · · ≈−→ C2k+1(pt)0−→ C2k(pt)

≈−→ C2k−1(pt)0−→ · · · ≈−→ C1(pt)

0−→ C0(pt)→ 0 .

Therefore,

H∗(pt) ≈ H∗(pt) ≈

0 if ∗ > 0Z2 if ∗ = 0 .

One sees that, for a simplicial complex reduced to a point, the ordered (co)homologyand the simplicial (co)homology are isomorphic.

Example 2.10.5. The unit cochain 1 ∈ C0(K) is defined as 1 = S0(K). It is

a cocycle and defines a class 1 = H0(K). If K is non-empty and connected, then

H0(K) ≈ Z2 generated by 1. Then H0(K) ≈ Z2 by Kronecker duality; one has

Z0(K) = C0(K) and α ∈ Z0(K) represents the non-zero element of H0(K) if andonly if ♯α is odd. The proofs are the same as for Proposition 2.4.1.

Example 2.10.6. Let L be a simplicial complex and CL be the cone on L.Then

H∗(CL) ≈ H∗(CL) ≈

0 if ∗ > 0Z2 if ∗ = 0 .

The proof is the same as for Proposition 2.4.6, even simpler, since D : Cm(CL)→Cm+1(CL) is defined, for (v0, . . . , vm) ∈ Sm(CL) by the single line formulaD(v0, . . . , vm) = (∞, v0, . . . , vm).

Let f : L → K be a simplicial map. We define C∗f : C∗(L) → C∗(K) as thedegree 0 linear map such that

C∗f(v0, . . . , vm) = (f(v0), . . . , f(vm))

2.10. APPENDIX B: ORDERED SIMPLICIAL (CO)HOMOLOGY 47

for all (v0, . . . , vm) ∈ S(L). The degree 0 linear map C∗f : C∗(K) → C∗(L) isdefined by

〈C∗f(a), α〉 = 〈a, C∗f(α)〉 .By Lemma 2.3.6, (C∗f, C∗f) is a morphism of Kronecker pairs.

We now construct a functorial isomorphism between the ordered and non-ordered (co)homologies, its existence being suggested by the previous examples.

Define ψ∗ : C∗(K)→ C∗(K) by

ψ∗((v0, . . . , vm)) =

v0, . . . , vm if vi 6= vj for all i 6= j

0 otherwise.

We check that ψ is a morphism of chain complexes. We define ψ∗ : C∗(K) →C∗(K) by requiring that the equation 〈ψ∗(a), α〉 = 〈a, ψ∗(α)〉 holds for all a ∈C∗(K) and all α ∈ C∗(K). By Lemma 2.3.6, ψ∗ is a morphism of cochain com-

plexes and (ψ∗, ψ∗) is a morphism of Kronecker pairs between (C∗(K), C∗(K))

and (C∗(K), C∗(K)). It thus defines a morphism of Kronecker pairs (H∗ψ,H∗ψ)

between (H∗(K), H∗(K)) and (H∗(K), H∗(K)).To define a morphism of Kronecker pairs in the other direction, choose a sim-

plicial order ≤ on K (see 2.1.8). Define φ≤∗ : C∗(K)→ C∗(K) as the unique linearmap such that

φ≤∗(v0, . . . , vm) = (v0, . . . , vm) ,

where v0 ≤ v1 ≤ · · · ≤ vm. We check that φ≤∗ is a morphism of chain complexes

and define φ≤∗ : C∗(K) → C∗(K) by requiring that the equation 〈φ≤∗(a), α〉 =

〈a, φ≤∗(α)〉 holds for all a ∈ C∗(K) and all α ∈ C∗(K). By Lemma 2.3.6, (φ≤∗, φ≤∗)

is a morphism of Kronecker pairs between (C∗(K), C∗(K)) and (C∗(K), C∗(K)). Itthen defines a morphism of Kronecker pairs (H∗φ≤, H

∗φ≤) between (H∗(K), H∗(K))

and (H∗(K), H∗(K)).

Proposition 2.10.7. H∗ψH∗φ≤ = idH∗(K) and H∗φ≤ H∗ψ = idH∗(K).

Proof. As ψ∗φ≤∗ = idC∗(K), the first equality follows from Lemma 2.3.7.

For the second one, let (v0, . . . , vm) ∈ Sm(K). Let σ = v0, . . . , vm ∈ Sk(K) with

k ≤ m. Clearly, φ≤∗ψ∗(v0, . . . , vm) ∈ C∗(σ). By what was seen in Examples 2.10.5

and 2.10.6, the correspondence (v0, . . . , vm) 7→ C∗(v0, . . . , vm) is an acyclic carrier

A∗, with respect to the basis S∗(K), for both idC(K) and φ≤∗ψ∗. Therefore, the

equality H∗φ≤H∗ψ = idH∗(K) follows by Lemma 2.3.7 and Proposition 2.9.1.

Applying Kronecker duality to Proposition 2.10.7 gives the following

Corollary 2.10.8. H∗ψH∗φ≤ = idH∗(K) and H∗φ≤H

∗ψ = idH∗(K).

Corollary 2.10.9. H∗ψ and H∗ψ are isomorphisms.

Corollary 2.10.10. H∗φ≤ and H∗φ≤ are isomorphisms which do not dependon the simplicial order ≤.

Proof. This follows from Proposition 2.10.7 and Corollary 2.10.8, since H∗ψand H∗ψ do not depend on ≤.


We shall see in § 4.1 that H∗ψ and H∗φ≤ are isomorphisms of graded Z2-algebras. We now prove that they are also natural with respect to simplicial maps.Let f : L→ K be a simplicial map. Let C∗f : C∗(L)→ C∗(K) be the unique linearmap such that

C∗f((v0, . . . , vm)) = (f(v0), . . . , f(vm))

for each (v0, . . . , vm) ∈ Sm(K). Doing this for each m ∈ N produces a GrV-

morphism C∗f : C∗(L)→ C∗(K). The formula ∂ C∗f = C∗f ∂ is straightforward(much easier than that for non-ordered chains). Hence, we get a GrV-morphism

H∗f : H∗(L)→ H∗(K). A GrV-morphism C∗f : C∗(K)→ C∗(L) is defined by the

equation 〈C∗f(a), α〉 = 〈a, C∗f(α)〉 required to hold for all a ∈ Cm(L), α ∈ Cm(K)

and all m ∈ N. It is a cochain map and induces a GrV-morphism H∗f : H∗(K)→H∗(L), Kronecker dual to H∗f .

Proposition 2.10.11. Let f : L→ K be a simplicial map. Let ≤ be a simplicialorder on K and ≤′ be a simplicial order on L. Then the diagrams

H∗(L)

H∗ψ

H∗f // H∗(K)

H∗ψ

H∗(L)

H∗f //

H∗φ≤′

JJ

H∗(K)

H∗φ≤

JJand

H∗(K)

H∗φ≤

H∗f // H∗(L)

H∗φ≤′

H∗(K)

H∗f //

H∗ψ

OO

H∗(L)

H∗ψ

OO

are commutative.

Proof. By Kronecker duality, only the homology statement requires a proof.It is enough to prove that H∗f H∗ψ = H∗ψH∗f since the formula H∗f H∗φ≤′ =

H∗φ≤H∗f will follow by Corollary 2.10.8. Finally, the formula C∗f C∗φ≤′ =C∗φ≤C∗f is straightforward.

The above isomorphism results also work in relative ordered (co)homology. Let(K,L) be a simplicial pair. Denote by i : L → K the simplicial inclusion. We definethe Z2-vector space of relative ordered (co)chain by

Cm(K,L) = ker(Cm(K)

C∗i−−→ Cm(L))

andCm(K,L) = coker

(i∗ : Cm(L) → Cm(K)

).

These inherit (co)boundaries δ : C∗(K,L) → C∗(K,L) and ∂ = C∗(K,L) →C∗−1(K,L) which give rise to the definition of relative ordered (co)homology H∗(k, L)

and H∗(K,L). Connecting homomorphisms δ∗ : H∗(L)→ H∗+1(K,L) and

∂∗ : H∗(K,L) → H∗−1(L) are defined as in § 2.7, giving rise to long exact se-

quences. Our homomorphisms ψ∗ : C∗(K) → C∗(K) and φ≤∗ : C∗(K) → C∗(K)

satisfy ψ∗(C∗(L)) ⊂ C∗(L) and φ≤∗(C∗(L) ⊂ C∗(L), giving rise to homomorphisms

on relative (co)chains and relative (co)homology H∗ψ : H∗(K,L) → H − ∗(K,L),etc. Proposition 2.10.7 and Corollary 2.10.8 and their proofs hold in relative(co)homology. Hence, as for Corollary 2.10.9 and 2.10.10, we get

Corollary 2.10.12. H∗ψ : H∗(K,L) → H∗(K,L) and H∗ψ : H∗(K,L) →H∗(K,L) are isomorphisms.

Corollary 2.10.13. H∗φ≤ : H∗(K,L) → H∗(K,L) and H∗φ≤ : H∗(K,L) →

H∗(K,L) are isomorphisms which do not depend on the simplicial order ≤.

2.11. EXERCISES FOR CHAPTER 2 49

2.11. Exercises for Chapter 2

2.1. Let Fn be the full complex on the set 0, 1, . . . , n (see p. 24). What are the2-simplexes of the barycentric subdivision F ′2 of F2? How many n-simplexes doesF ′n contain?

2.2. Compute the Euler characteristic and the Poincare polynomial of the k-skeletonFkn of Fn.2.3. Let X be a metric space and let ε > 0. The Vietoris-Rips complex Xε of X isthe simplicial complex whose simplexes are the finite non-empty subset of X whosediameter is < ε (the diameter of A ⊂ X is the least upper bound of d(x, y) forx, y ∈ A). In particular, V (Xε) = X .

(a) Describe |Xε| for various ε when X is the set of vertices of a cube of edge 1

in R3. In particular, if√2 < ε ≤

√3, show that |Xε| is homeomorphic

to S3.(b) Let X be the space n-th roots of unity, with the distance d(x, y) being the

minimal length of an arc of the unit circle joining x to y. Suppose that4π/n < ε ≤ 6π/n.(i) If n = 6, show that |Xε| is homeomorphic to S2.(ii) If n ≥ 7 is odd, show that |Xε| is homeomorphic to a Mobius band.(iii) If n ≥ 7 is even, show that |Xε| is homeomorphic to S1 × [0, 1].

Note: the complex Xε was introduced by L Vietoris in 1927 [201]. After its re-introduction by E. Rips for studying hyperbolic groups, it has been popularizedunder the name of Rips complex. For some developments and applications, see[84, 129] and Wikipedia’s page “Vietoris-Rips complex”.

2.4. Let ℓ = (ℓ1, . . . , ℓn) ∈ Rn>0. A subset J of 1, . . . , n is called ℓ-short (or justshort) if

∑i∈J ℓi <

∑i/∈J ℓi. Show that short subsets are the simplexes of a sim-

plicial complex Sh(ℓ) with V (Sh(ℓ)) ⊂ J (used in § 10.3). Describe Sh(1, 1, 1, 1, 3),Sh(1, 1, 3, 3, 3) and Sh(1, 1, 1, 1, 1). Compute their Euler characteristics and theirPoincare polynomials.

2.5. Let K be the simplicial complex with V (K) = Z and S1(K) = r, r + 1 |r ∈ Z (|K| ≈ R). Then S1(K) is a 1-cocycle. Find all the cochains a ∈ C0(K)such that S1(K) = δ(a).

2.6. Find a simplicial pair (K,L) such that |K| is homeomorphic to S1 × I and|L| = Bd |K|. In the spirit of § 2.4.7, compute the simplicial cohomology of K andof (K,L) and find (co)cycles generating H∗(K), H∗(K,L), H

∗(K) and H∗(K,L).Write completely the (co)homology sequence of (K,L).

2.7. Same exercise as before with |K| the Mobius band and |L| = Bd |K|.2.8. Let f : K → L be a simplicial map between simplicial complexes. Supposethat L is connected and K is non-empty. Show that H0f is surjective.

2.9. Let m,n, q be positive integers. If m = nq, the quotient map Z → Z/nZdescends to a map Z/mZ → Z/nZ, giving rise to a simplicial map f : Pm → Pnbetween the simplicial polygons Pm and Pn (see Example 2.4.3). Compute H∗f .

2.10. Let M be an n-dimensional pseudomanifold. Let σ and σ′ be two distinctn-simplexes of M . Find a ∈ Cn−1(M) such that δ(a) = σ, σ′.


2.11. LetM be a finite non-empty n-dimensional pseudomanifold. Let γ ∈ Zn−1(M)which is a boundary. Prove that γ is the boundary of exactly two n chains.

2.12. Let f : M → N be a simplicial map between finite n-dimensional pseudoman-ifolds. Show that the following two conditions are equivalent.

(a) Hnf 6= 0.(b) There exists σ ∈ S(N) such that ♯f−1(σ) is odd.

2.13. Let ±1 be the 0-dimensional simplicial complex with vertices −1 and 1.Let K be a simplicial complex. The simplicial suspension ΣK is the join K ∗ ±1.

(a) Let P4 be the polygon complex with 4-edges (see Example 2.4.3). Showthat P4 ∗K is isomorphic to the double suspension Σ(ΣK). [Hint: showthat the join operation is associative: (K ∗ L) ∗M ≈ K ∗ (L ∗M).]

(b) Prove that the suspension of a pseudomanifold is a pseudomanifold.(c) Prove that the correspondence K 7→ ΣK gives a functor from Simp to

itself.

2.14. Let A be a finite set. Show that FA is a pseudomanifold.

2.15. LetM be an n-dimensional pseudomanifold which is infinite. What isHn(M)?

2.16. Let (K,K1,K2) be a simplicial triad. Suppose that K1 and K2 are connectedand that K1 ∩K2 is not empty. Show that K is connected.

2.17. Let (K,K1,K2) be a simplicial triad and let K0 = K1 ∩K2.

(a) Prove that the homomorphism H∗(K1,K0) → H∗(K,K2) induced by theinclusion is an isomorphism (simplicial excision).

(b) Write the commutative diagram involving the homology sequences of(K1,K0) and (K,K2). Using (a), construct out of this diagram the Mayer-Vietoris sequence for the triad (K,K1,K2).

2.18. Deduce the additivity formula for the Euler characteristic of Lemma 2.4.10from the Mayer-Vietoris sequence.

2.19. LetM1 andM2 be two finite n-dimensional pseudomanifolds. Let σi ∈ S(Mi)and let h : σ1 → σ2 be a bijection. The simplicial connected sum M = M1 ♯M2

(using h) is the simplicial complex defined by

V (M) = V (M1) ∪V (M2)/v ∼ h(v) for v ∈ σ1

andS(M) =

(S(M1)− σ1

)∪(S(M2)− σ2

).

Prove that M is a pseudomanifold. Compute H∗(M) in terms of H∗(M1) andH∗(M2).

CHAPTER 3

Singular and cellular (co)homologies

3.1. Singular (co)homology

Singular (co)homology provides a functor associating to a topological space Xa graded Z2-vector space, whose isomorphism class depends only on the homotopytype of X . Such functors, from Top to categories of algebraic objects, constitutethe main subject of algebraic topology.

Invented by S. Eilenberg in 1944 [49] after earlier attempts by Lefschetz, sin-gular homology is formally akin to simplicial homology. However, in order to makecomputations for non-trivial examples, we need to establish some properties, such ashomotopy and excision, which require some work. When K is a simplicial complex,the simplicial homology of K and the singular cohomology of |K| are isomorphicin several ways, some of them being functorial (see § 3.6). Singular (co)homologyis especially powerful and relevant for spaces having the homotopy type of a CW-complex, a notion introduced in § 3.4. For such spaces, singular (co)homology isisomorphic to other (co)homology theories (see § 3.7) and the cohomology functorHn is representable by the Eilenberg-MacLane space K(Z2, n) (see § 3.8).

3.1.1. Definitions. The standard Euclidean m-simplex ∆m is defined by

∆m = (x0, . . . , xm) ∈ Rm+1 | xi ≥ 0 and∑

xi = 1 ,

endowed with the induced topology from that of Rn+1. In particular, ∆m = ∅ ifm < 0. Let X be a topological space. A singular m-simplex of X is a continuousmap σ : ∆m → X . The set of singular m-simplexes of X is denoted by Sm(X).

The definitions of singular (co)chains and (co)homology are copied from thosefor the simplicial case (see § 2.2), replacing simplicial simplexes by singular ones.We thus set


(a) A singular m-cochain of X is a subset of Sm(X).(b) A singular m-chain of X is a finite subset of Sm(X).

The set of singular m-cochains of X is denoted by Cm(X) and that of singularm-chains by Cm(X). As in § 2.2, Definitions 3.1.1 are equivalent to


(a) A singular m-cochain is a function a : Sm(X)→ Z2.(b) A singular m-chain is a function α : Sm(X)→ Z2 with finite support.

Definitions 3.1.2 endow Cm(X) and Cm(X) with a structure of a Z2-vectorspace. The singletons provide a basis of Cm(X), in bijection with Sm(X). Thus,Definition 3.1.2.b is equivalent to

51

52 3. SINGULAR AND CELLULAR (CO)HOMOLOGIES

Definition 3.1.3. Cm(X) is the Z2-vector space with basis Sm(X):

Cm(X) =⊕

σ∈Sm(X)

Z2 σ .

We consider the graded vector spaces C∗(X) = ⊕m∈NCm(X) and C∗(X) =⊕m∈NCm(X). By convention, Cm(X) = Cm(X) = 0 if m < 0 (so the index mcould be taken in Z in the previous formulae).

The Kronecker pairing on singular (co)chains

Cm(X)× Cm(X)〈 , 〉−−→ Z2

is defined, using the various above definitions, by the equivalent formulae

(3.1.1)

〈a, α〉 = ♯(a ∩ α) (mod 2) using Definitions 3.1.1.a and b

=∑σ∈α a(σ) using Definitions 3.1.1.a and 3.1.2.b

=∑σ∈Sm(X) a(σ)α(σ) using Definitions 3.1.2.a and b .

As in Lemma 2.2.4, we check that the map k : Cm(X)→ Cm(X)♯, given by k(a) =〈a, 〉, is an isomorphism.

Let m, i ∈ N with 0 ≤ i ≤ m. Define the i-th face inclusion ǫi : ∆m−1 → ∆m

by

ǫi(x0, . . . , xm−1) = (x0, . . . , xi−1, 0, xi+1, . . . , xm−1) .

The boundary operator ∂ : Cm(X) → Cm−1(X) is the Z2-linear map defined, forσ ∈ Sm(X) by

(3.1.2) ∂(σ) =

m∑

i=0

σǫi .

Lemma 3.1.4. ∂∂ = 0.

Proof. By linearity, it suffices to prove that ∂∂(σ) = 0 for σ ∈ Sm(X). Onehas

(3.1.3) ∂∂(σ) = ∂ (m∑

i=0

σǫi) =∑

(i,j)∈A

σǫiǫj ,

where A = 0, . . . ,m × 0, . . . ,m − 1. The set B = (i, j) ∈ A | i ≤ j isin bijection with A − B, via the map (i, j) 7→ (j + 1, i). But if (i, j) ∈ B, thenǫiǫj = ǫj+1 ǫi, which implies that ∂∂ = 0.

The coboundary operator δ : Cm(X)→ Cm+1(X) is defined by the equation

(3.1.4) 〈δa, α〉 = 〈a, ∂α〉 .With these definition, ((C∗(X), ∂), (C∗(X), δ), 〈 , 〉) is a Kronecker pair. We de-

fine the vector spaces of singular cycles Z∗(X), singular boundaries B∗(X), singularcocycles Z∗(X), singular coboundaries B∗(X), singular homology H∗(X) and sin-gular cohomology H∗(X) as in § 2.3. By Proposition 2.3.5, the pairing on (co)chaindescends to a pairing

Hm(X)×Hm(X)〈 , 〉−−→ Z2

so that the map k : Hm → H♯m, given by k(a) = 〈a, 〉, is an isomorphism (Kronecker

duality in singular (co)homology). The Kronecker pairing extends to a bilinear map

3.1. SINGULAR (CO)HOMOLOGY 53

H∗(X) ×H∗(X)〈,〉−→ Z2 by setting 〈a, α〉 = 0 if a ∈ Hp(X) and α ∈ Hq(X) with

p 6= q.

Example 3.1.5. If X is the empty space, then Sm(X) = ∅ for all m and thusH∗(∅) = H∗(∅) = 0. Let X = pt be a point. Then, Sm(pt) contains one element foreach m ∈ N, namely the constant singular simplex ∆m → pt. Then, Cm(pt) = Z2

for all m ∈ N and the chain complex looks like

· · · ≈−→ C2k+1(pt)0−→ C2k(pt)

≈−→ C2k−1(pt)0−→ · · · ≈−→ C1(pt)

0−→ C0(pt)→ 0 .

Therefore,

(3.1.5) H∗(pt) ≈ H∗(pt) ≈

0 if ∗ > 0Z2 if ∗ = 0 .

Example 3.1.6. Let K be a simplicial complex. Choose a simplicial order “≤”for K. To an m-simplex σ = v0, . . . , vm ∈ Sm(K), with v0 ≤ · · · ≤ vm, weassociate the singular m-simplex R≤(σ) : ∆

m → |K| defined by

(3.1.6) R≤(σ)(t0, . . . , tm) =

m∑

i=0

tivi .

The linear combination in (3.1.6) makes sense since v0, . . . , vm is a simplex of K.This defines a map R≤ : Sm(K)→ Sm(|K|) which extends to a linear map

R≤,∗ : C∗(K)→ C∗(|K|) .This map will be used several times in this chapter. The formula ∂R≤,∗ = R≤,∗∂is obvious, so R≤,∗ is a chain map from (C∗(K), ∂) to (C∗(|K|), ∂). We shallprove, in Theorem 3.6.3, that R≤,∗ induces an isomorphism between the simplicial(co)homology) of K and the singular (co)homology) of |K|.

Example 3.1.7. As the affine simplex ∆0 is a point, one can identify a singular0-simplex of X with its image, a point of X . This gives a bijection S0(X) ≈ X anda bijection between subsets of X and singular 0-cochains. For B ⊂ X and x ∈ X ,one has 〈B, x〉 = χB(x), where χB stands for the characteristic function for B. The1-cochain δB is the connecting cochain for B: if β ∈ S1(X), then

(3.1.7) 〈δ(B), β〉 = 〈B, ∂β〉 = 〈B, β(1, 0)〉+ 〈B, β(0, 1)〉 .In other words 〈δ(B), β〉 = 1 if and only if the (non-oriented) path β connects apoint in B to a point in X −B. Observe that δ(B) = δ(X −B).

Following Example 3.1.7, the unit cochain 1 ∈ C0(X) is defined by 1 = S0(X) ≈X . By Equation (3.1.7) 〈δ1, β〉 = 0 for all β ∈ S1(X). This proves that δ(1) = 0by Lemma 2.2.4. Hence, 1 is a cocycle, whose cohomology class is again denotedby 1 ∈ H0(X).

Proposition 3.1.8. Let X be a non-empty path-connected space. Then,

(i) H0(X) = Z2, generated by 1 which is the only non-vanishing singular0-cocycle.

(ii) H0(X) = Z2. Any 0-chain α is a cycle, which represents the non-zeroelement of H0(X) if and only if ♯α is odd.


Proof. The proof is analogous to that of Proposition 2.4.1. If X is non-emptythe unit cochain does not vanish and, as C−1(X) = 0, 1 6= 0 in H0(X).

Let a ∈ C0(X) with a 6= 0,1. Then there exists x, y ∈ X = S(X) witha(x) 6= a(y). Since X is path-connected, there exists σ ∈ S1(X) with σ(1, 0) = xand σ(0, 1) = y. As in Equation (3.1.7), this proves that 〈δ(a), σ〉 6= 0 so a is not acocycle. This proves (i).

Now, H0(X) = Z2 since H0(X) ≈ H0(X)♯. Any α ∈ C0(X) is a cycle sinceC−1(X) = 0. It represents the non-zero homology class if and only if 〈1, α〉 = 1,that is if and only if ♯α is odd.

The reduced (singular) cohomology H∗(X) and homology H∗(X) of a topologicalspace X are the graded Z2-vector spaces defined by

(3.1.8)H∗(X) = coker

(H∗p : H∗(pt)→ H∗(X)

)

H∗(X) = ker(H∗p : H∗(X)→ H∗(pt)

)

where p : X → pt denotes the constant map to a point. In particular, H∗(pt) = 0 =

H∗(pt). One checks that the Kronecker pairing induces a bilinear map 〈 , 〉 : Hm(X)×Hm(X) → Z2 such that the correspondence a 7→ 〈a, 〉 gives an isomorphism

k : Hm(X)≈−→ Hm(X)♯.

The full strength of Definition (3.1.8) appears in other (co)homology theories,such as equivariant cohomology (see p. 223). For the singular cohomology, asH∗(pt) = Z21, one gets

Hm(X) =

H0(X)/Z21 if m = 0

Hm(X) if m 6= 0

and

Hm(X) =

ker

(H0(X)

〈1, 〉−−−→ Z2

)if m = 0

Hm(X) if m 6= 0 .

Thus, by Proposition 3.1.8, H0(X) = 0 = H0(X) if X is path-connected (see alsoCorollary 3.1.12).

Let f : Y → X be a continuous map between topological spaces. It induces amap Sf : S(Y )→ S(X) defined by Sf(σ) = f σ. The linear map C∗f : C∗(X)→C∗(Y ) is, using Definition 3.1.2, defined by C∗f(a) = aS(f). As for C∗f : C∗(Y )→C∗(X), it is the linear map extending Sf , using Definition 3.1.3. One checks thatthe couple (C∗f, C∗f) is a morphism of Kronecker pair. It thus defines linear mapsof degree zero H∗f : H∗(X) → H∗(Y ) and H∗f : H∗(Y ) → H∗(X). The functo-rial properties are easy to prove: H∗ and H∗ are functors from the category Topof topological spaces to the category GrV of graded vector spaces (see Proposi-tion 3.1.22 for a more general statement). Also, for any map f : Y → X , thediagram

(3.1.9)

Y

p

f // X

p~~⑥⑥⑥⑥⑥⑥⑥

pt


is obviously commutative. This implies that the reduced cohomology H∗ and ho-mology H∗ are also functors from Top to GrV. The notations H∗f , H∗f , H

∗fand H∗f are sometimes shortened in f∗ and f∗.

As in Lemma 2.5.4, we prove the following

Lemma 3.1.9. Let f : Y → X be a continuous map. Then H0f(1) = 1.

Lemma 3.1.9 implies the following result.

Lemma 3.1.10. Let (X,Y ) be a topological pair with X path-connected. Denoteby i : Y → X the inclusion. Then there are exact sequences

0→ H0(X)H∗i−−→ H0(Y )→ H0(Y )→ 0

and

0→ H0(Y )→ H0(Y )H∗i−−→ H0(X)→ 0 .

We now prove some general results useful to compute the (co)homology of aspace. Let X be a topological space which is a disjoint union:

X =⋃

j∈JXj .

By this we mean that the above equality holds as sets and that each Xj is open(and therefore closed) in X . Denote the inclusion by ij : Xj → X . The equality

Sm(X) =⋃j∈J ij(Sm(Xj)) implies the following proposition.

Proposition 3.1.11. The family of inclusions ij : Xj → X for j ∈ J givesrise to isomorphisms

H∗(X)(H∗ij)

≈// ∏

j∈J H∗(Xj)

and⊕

j∈J H∗(Xj)

∑H∗ij

≈// H∗(X) .

Corollary 3.1.12. Let X be a topological space which is locally path-connected.Then, the family of inclusions iY : Y → X for Y ∈ π0(X) gives rise to isomorphisms

H∗(X)(H∗iY )

≈// ∏

Y ∈π0(X)H∗(Y )

and⊕

Y ∈π0(K)H∗(Y )

∑H∗iY

≈// H∗(X) .

Proof. As X is locally path-connected, each Y ∈ π0(X) is open in X and X istopologically the disjoint union of its path-connected components. Corollary 3.1.12then follows from Proposition 3.1.11.

Corollary 3.1.13. Let X be a topological space which is locally path-connected.Then,

H0(X) = 0 ⇔ H0(X) = 0 ⇔ X is path-connected.

Also,H0(X) = H0(X)⊕ Z2 and H0(X) = H0(X)⊕ Z2

if X is not empty.


In the same spirit of reducing the computations of H∗(X) to those of smallersubspaces, another consequence of the definition of the singular (co)homology isthe following proposition.

Proposition 3.1.14. Let X be a topological space. Let K be the set of compactsubspaces of X, partially ordered by inclusion. Then, the natural homomorphisms

J∗ : lim−→

K∈K

H∗(K) −→ H∗(X)

and

J∗ : H∗(X) −→ lim←−

K∈K

H∗(K)

are isomorphisms.

Here, lim−→

denotes the direct limit (also called inductive limit or colimit) and lim←−

denotes the inverse limit (also called projective limit or just limit) in GrV.

Proof. Let A ∈ Hr(X), represented by α ∈ Zr(X). Then, α is a finite setof r-simplexes of X and K =

⋃σ∈α σ(∆

r) is a compact subspace of X . One cansee α ∈ Zr(K), so J∗ is onto. Now, let K be a compact subspace of X andA ∈ Hr(K) mapped to 0 under Hr(K)→ Hr(X). Represent A by α ∈ Zr(K) andlet β ∈ Cr+1(X) with α = ∂(β). As before, there exists a compact subset L of Xcontaining K with β ∈ Cr+1(L), so A is mapped to 0 under Hr(K)→ Hr(L). Thisproves that J∗ is injective. Finally, the bijectivity of J∗ is deduced from that of J∗by Kronecker duality.

Remark 3.1.15. In Proposition 3.1.14 the morphism J∗ is an isomorphism forthe homology with any coefficients. The morphism J∗ is always surjective but,in general not injective (except for coefficients in a field, like Z2). Its kernel isexpressible using the derived functor lim

←−

1 (see e.g. [82, Theorem 3F.8]). The same

considerations hold true for the following corollary.

Corollary 3.1.16. Let X be a topological space and let A be a family of sub-spaces of X, partially ordered by the inclusion. Suppose that each compact subspaceof X is contained in some A ∈ A. Then, the homomorphisms

j∗ : lim−→A∈A

H∗(A) −→ H∗(X)

and

j∗ : H∗(X) −→ lim←−A∈A

H∗(A)

are isomorphisms.

Proof. The hypothesis that each compact K ⊂ X is contained in some A ∈ Aimplies a factorization of the homomorphism J∗ of Proposition 3.1.14:

lim−→

K∈K

H∗(K)

β

%%

J∗

≈// H∗(X)

lim−→A∈A

H∗(A)

j∗

::


The same hypothesis implies that β is onto, whence j∗ is an isomorphism. Theassertion for j∗ comes from Kronecker duality.

3.1.2. Relative singular (co)homology. A (topological) pair is a couple(X,Y ) where X is a topological space and Y is a subspace of X . The inclusioni : Y → X is a continuous map. Let a ∈ Cm(X). If, using Definition 3.1.1.a,we consider a as a subset of Sm(X), then C∗i(a) = a ∩ Sm(Y ). If we see a as amap a : Sm(X) → Z2, then C∗i(a) is the restriction of a to Sm(Y ). We see thatC∗i : C∗(X)→ C∗(Y ) is surjective. Define

Cm(X,Y ) = ker(Cm(X)

C∗i−−→ Cm(Y ))

and C∗(X,Y ) = ⊕m∈NCm(X,Y ). As C∗i is a morphism of cochain complexes, thecoboundary δ : C∗(X)→ C∗(X) preserves C∗(X,Y ) and gives rise to a coboundaryδ : C∗(X,Y )→ C∗(X,Y ) so that (C∗(X,Y ), δ) is a cochain complex. The cocyclesZ∗(X,Y ) and the coboundaries B∗(X,Y ) are defined as usual, giving rise to thedefinition

Hm(X,Y ) = Zm(X,Y )/Bm(X,Y ) .

The graded Z2-vector space H∗(X,Y ) = ⊕m∈NHm(X,Y ) is the relative (singular)

cohomology of the pair (X,Y ). Observe that H∗(X, ∅) = H∗(X). We denote by j∗

the inclusion j∗ : C∗(X,Y ) → C∗(X), which is a morphism of cochain complexes,and use the same notation j∗ for the induced linear map j∗ : H∗(X,Y ) → H∗(X)on cohomology. We also use the notation i∗ for both C∗i and H∗i. We get thus ashort exact sequence of cochain complexes

(3.1.10) 0→ C∗(X,Y )j∗−→ C∗(X)

i∗−→ C∗(Y )→ 0 .

If a ∈ Cm(Y ), any cochain a ∈ Cm(X) with i∗(a) = a is called a extension ofa as a singular cochain in X . For instance, the 0-extension of a is defined bya = a ∈ Sm(Y ) ⊂ Sm(X).

With chains, the inclusion Y → X induces an inclusion i∗ : C∗(Y ) → C∗(X)of chain complexes. We define Cm(X,Y ) as the quotient vector space

Cm(X,Y ) = coker(i∗ : Cm(Y ) → Cm(X)

).

As i∗ is a morphism of chain complexes, C∗(X,Y ) = ⊕m∈NCm(X,Y ) inherits aboundary operator ∂ = ∂X,Y : C∗(X,Y )→ C∗−1(X,Y ). The projection j∗ : C∗(X)→→ C∗(X,Y ) is a morphism of chain complexes and one obtains a short exact se-quence of chain complexes

(3.1.11) 0→ C∗(Y )i∗−→ C∗(X)

j∗−→ C∗(X,Y )→ 0 .

The cycles and boundaries Z∗(X,Y ) and B∗(X,Y ) are defined as usual, giving riseto the definition

Hm(X,Y ) = Zm(X,Y )/Bm(X,Y ) .

The graded Z2-vector space H∗(X,Y ) = ⊕m∈NHm(X,Y ) is the relative (singular)homology of the pair (X,Y ). Observe that H∗(X, ∅) = H∗(X). The notations i∗and j∗ are also used for the induced maps in homology.

As in Sections 2.6 and 2.7 of Chapter 2, one gets a pairing 〈 , 〉 : Hm(X,Y ) ×Hm(X,Y )→ Z2 which makes (Hm(X,Y ), Hm(X,Y ), 〈 , 〉) a Kronecker pair. Also,the singular (co)homology connecting homomorphisms

δ∗ : H∗(Y )→ H∗+1(X,Y ) . and ∂∗ : H∗(X,Y )→ H∗−1(Y )


are defined and satisfy 〈δ∗(a), α〉 = 〈a, ∂∗(α)〉. The proof of the following lemma isthe same as that of Lemma 2.7.1.

Lemma 3.1.17. Let a ∈ Zm(Y ) and let a ∈ Cm(X) be any extension of a asa singular m-cochain of X. Then, δX(a) is a singular (m + 1)-cocycle of (X,Y )representing δ∗(a).

Remark 3.1.18. A class in A ∈ Hm(X,Y ) is represented by a relative singularcycle, i.e. a singular chain α ∈ Cm(X) such that ∂(α) is a singular chain (cycle)of Y . The homology class of ∂(α) in Hn−1(Y ) is ∂∗(A). This is the Kronecker dualstatement of Lemma 3.1.17.

As for the simplicial (co)homology (see § 2.7), the results of § 2.6 give the fol-lowing (singular) (co)homology exact sequences, or just the (co)homology sequence,of the pair (X,Y ).

Proposition 3.1.19 ((Co)homology exact sequences of a pair). Let (X,Y ) bea topological pair. Then, the sequences

· · · → Hm(X,Y )j∗−→ Hm(X)

i∗−→ Hm(Y )δ∗−→ Hm+1(X,Y )

j∗−→ · · ·

and

· · · → Hm(Y )i∗−→ Hm(X)

j∗−→ Hm(X,Y )∂∗−→ Hm−1(Y )

i∗−→ · · ·are exact.

These exact sequences are also available for reduced (co)homology. For this,the reduced (co)homology of a pair is defined as follows: when Y 6= ∅, then

H∗(X,Y ) = H∗(X,Y ) and H∗(X,Y ) = H∗(X,Y ); otherwise H∗(X, ∅) = H∗(X)

and H∗(X, ∅) = H∗(X).

Proposition 3.1.20 (Reduced (co)homology sequences of a pair). The exactsequences of Proposition 3.1.19 hold with reduced (co)homology.

Proof. An argument is only required around m = 0. For the homology exactsequence, consider the commutative diagram:

· · · // H1(X,Y )

=

∂∗ // H0(Y )

i∗ // H0(X)

j∗ // H0(X,Y )

=

// 0

· · · // H1(X,Y )∂∗ // H0(Y )

〈1, 〉

i∗ // H0(X)

〈1, 〉

j∗ // H0(X,Y ) // 0

Z2= // Z2

The commutativity of the bottom square is due to Lemma 3.1.9. As i∗∂∗ = 0,〈1, ∂(α)〉 = 0 for all α ∈ H1(X,Y ) and therefore ∂ : H1(X,Y ) → H1(Y ) exists.Since the sequence of the second line is exact, an easy diagram-chase shows thatthe sequence of the first line is exact as well.

The reduced cohomology sequence can be established in an analogous way ordeduced from the homology one by Kronecker duality, using Lemma 2.3.8.


Remark 3.1.21. Let (X,Y ) be a topological pair with Y path-connected andnon-empty. By Proposition 3.1.20 and its proof, we get the isomorphisms

(3.1.12) j∗ : H0(X)≈−→ H0(X,Y ) and j∗ : H0(X,Y )

≈−→ H0(X) .

Also, if Y = x, we get the isomorphisms

(3.1.13) j∗ : H∗(X)≈−→ H∗(X, x) and j∗ : H∗(X, x)

≈−→ H∗(X) .

A direct proof of (3.1.13), say for cohomology, is given by the diagram.

(3.1.14)

H∗(pt)

p∗

≈

$$

H∗(X, x)j∗ //

≈

%%

H∗(X)i∗ //

H∗(x)

H∗(X)

where the row and the column are exact and p : X → pt is the constant map onto apoint. We see that the choice of x ∈ X produces a supplementary vector subspaceto p∗(H∗(pt)) in H∗(X).

We now study the naturality of the relative (co)homology and of the exactsequences. Let (X,Y ) and (X ′, Y ′) be topological pairs. A map f of (topological)pairs from (X,Y ) to (X ′, Y ′) is a continuous map f : X → X ′ such f(Y ) ⊂ Y ′.With these maps, topological pairs constitute a categoryTop2. The correspondenceX 7→ (X, ∅) makes Top a full subcategory of Top2.

Let f : (X,Y ) → (X ′, Y ′) be a map of topological pairs. The morphismC∗f : C∗(X ′)→ C∗(X) then restricts to a morphism of cochain complexes C∗f : C∗(X ′, Y ′)→C∗(X,Y ) and the morphism C∗f : C∗(X) → C∗(X

′) descends to a morphismof chain complexes C∗f : C∗(X,Y ) → C∗(X

′, Y ′). As in § 2.7, we prove that(C∗f, C∗f) is a morphism of Kronecker pair. One then gets degree zero linearmaps H∗f : H∗(X ′, Y ′) → H∗(X,Y ) and H∗f : H

∗(X,Y ) → H∗(X′, Y ′) satisfy-

ing 〈H∗a, α〉 = 〈a,H∗α〉 for all a ∈ Hm(X ′, Y ′), α ∈ Hm(X,Y ) and all m ∈ N.Functorial properties are easy, so we get the following

Proposition 3.1.22. The relative singular cohomology H∗( , ) is a contravari-ant functor from the category Top2 to the category GrV of graded Z2-vector spaces.The relative singular homology H∗( , ) is a covariant functor between these cate-gories. The same holds true for the reduced singular (co)homology.

As for Proposition 2.7.6, we can prove the following

Proposition 3.1.23. The (co)homology sequences are natural with respect tomaps of topological pairs.

Here is a special form of the cohomology sequence of a pair.

Proposition 3.1.24. Let A and B be topological spaces. Then the cohomol-ogy sequence of the pair (A∪B,A) splits into short exact sequences and there is a


commutative diagram

(3.1.15)

0 // H∗(B) //

≈

H∗(A∪B) //

id

H∗(A) //

id

0

0 // H∗(A∪B,A) // H∗(A∪B) // H∗(A) // 0

.

Proof. If iA : A → A∪B and iB : B → A∪B denote the inclusions, Proposi-tion 3.1.11 provides a commutative diagram

H∗(A∪B)(i∗A,i

∗B)

≈//

i∗A %%

H∗(A)×H∗(B)

proj1ww♦♦♦♦♦♦♦♦♦♦

H∗(A)

.

This proves that i∗a is surjective, which splits the cohomology sequence of (A∪B,A),giving the bottom line of (3.1.15). Also, ker i∗A is the image of H∗(B) under themonomorphism j : H∗(B) → H∗A∪B) given by j(u) = (i∗A, i

∗B)−1(0, u), which we

placed in the top line of (3.1.15).

As in simplicial (co)homology, the exact sequences of a pair generalize to that ofa triple. A (topological) triple is a triplet (X,Y, Z) where X is a topological spacesand Y , Z are subspaces of X with Z ⊂ Y . A map f of triples, from (X,Y, Z) to(X ′, Y ′, Z ′) is a continuous map f : X → X ′ such that f(Y ) ⊂ Y ′ and f(Z) ⊂ Z ′.

A triple T = (X,Y, Z) gives rise to pair inclusions

(Y, Z)i−→ (X,Z)

j−→ (X,Y )

and to a commutative diagram

(3.1.16)

0 // C∗(X,Y )

C∗j

j∗X,Y // C∗(X)

id=

i∗X,Y // C∗(Y )

i∗Y,Z

// 0

0 // C∗(X,Z)j∗X,Z // C∗(X)

i∗X,Z // C∗(Z) // 0

where the horizontal lines are exact sequences of cochain complexes As in (2.7.9),we get a short exact sequence of cochain complexes

(3.1.17) 0→ C∗(X,Y )C∗j−−→ C∗(X,Z)

C∗i−−→ C∗(Y, Z)→ 0 .

The same arguments with the chain complexes gives a short exact sequence

(3.1.18) 0→ C∗(Y, Z)C∗i−−→ C∗(X,Z)

C∗j−−→ C∗(X,Y )→ 0 .

As in § 2.7 of Chapter 2, short exact sequences (3.1.17) and (3.1.18) producesconnecting homomorphisms δT : H

∗(Y, Z) → H∗+1(X,Y ) and ∂T : H∗(X,Y ) →C∗−1(Y, Z). They satisfy 〈δT (a), α〉 = 〈a, ∂T (α)〉 as well as following proposition.

Proposition 3.1.25 ((Co)homology sequences of a triple). Let T = (X,Y, Z)be a triple. Then,


(a) the sequences

· · · → Hm(X,Y )H∗j−−−→ Hm(X,Z)

H∗i−−→ Hm(Y, Z)δT−→ Hm+1(X,Y )

H∗j−−−→ · · ·and

· · · → Hm(Y, Z)H∗i−−→ Hm(X,Z)

H∗j−−→ Hm(X,Y )∂T−−→ Hm−1(Y, Z)

H∗i−−→ · · ·are exact.

(b) the exact sequences of Point (a) are natural for maps of triples.

Remark 3.1.26. AsH∗(∅) = 0, we get a canonicalGrV-isomorphismsH∗(X, ∅) ≈−→H∗(X), etc. Thus, the (co)homology sequences for the triple (X,Y, ∅) give backthose of the pair (X,Y )

(3.1.19) · · · → Hm(X,Y )H∗j−−−→ Hm(X)

H∗i−−→ Hm(Y )δ∗−→ Hm+1(X,Y )

H∗j−−−→ · · ·and

(3.1.20) · · · → Hm(Y )H∗i−−→ Hm(X)

H∗j−−→ Hm(X,Y )∂∗−→ Hm−1(Y )

H∗i−−→ · · ·where i : Y → X and j : (X, ∅)→ (X,Y ) denote the inclusions. This gives a moreprecise description of the morphisms j∗ and j∗ of Proposition 3.1.19.

We now draw a few consequences of Proposition 3.1.19. A topological pair(X,Y ) is of finite (co)homology type if its singular homology (or, equivalently, co-homology) is of finite type. In this case, the Poincare series of (X,Y ) is that ofH∗(X,Y ):

Pt(X,Y ) =∑

i∈N

dimHi(X,Y ) ti =∑

i∈N

dimHi(X,Y ) ti ∈ N[[t]].

Corollary 3.1.27. Let (X,Y, Z) be a topological triple. Suppose that two ofthe pairs (X,Y ), (Y, Z) and (X,Z) are of finite cohomology type. Then, the thirdpair is of finite cohomology type and there is Qt ∈ N[[t]] such that the equality

(3.1.21) Pt(X,Y ) +Pt(Y, Z) = Pt(X,Z) + (1 + t)Qt ,

holds in N[[t]].

Proof. This follows from the cohomology sequence of T = (X,Y, Z) and el-ementary linear algebra. If δk

T: Hk(Y, Z) → Hk+1(X,Y ) denotes the connecting

homomorphism, one checks that (3.1.21) holds true for

Qt =∑

tk codim δkT.

Corollary 3.1.27 implies straightforwardly the following result.

Corollary 3.1.28. Let (X,Y, Z) be a topological triple. Suppose that dimH∗(Y, Z) <∞ and that dimH∗(X,Y ) <∞. Then dimH∗(X,Z) <∞ and

dimH∗(X,Z) ≤ dimH∗(X,Y ) + dimH∗(Y, Z) .

Corollary 3.1.16 has the following generalization with relative (co)homology.

Proposition 3.1.29. Let (X,Y ) be a topological pair. Let A be family ofsubspaces of X, partially ordered by inclusion. Suppose that each compact subspaceof X is contained in some A ∈ A. Then, the natural homomorphisms

J∗ : lim−→A∈A

H∗(A,A ∩ Y )≈−→ H∗(X,Y )


and

J∗ : H∗(X,Y )≈−→ lim

←−A∈A

H∗(A,A ∩ Y )

are isomorphisms.

Proof. By Kronecker duality, only the bijectivity of J∗ must be proven. LetHr(Y ) = lim

−→A∈A

Hr(A∩ Y ), Hr(X) = lim−→A∈A

Hr(A), and Hr(X,Y ) = lim−→A∈A

Hr(A,A∩ Y ).

For each A ∈ A, one has the homology sequence of the pair (A,A∩Y ). By naturalityof these exact sequences under inclusions, one gets the diagram:

Hr(Y )

≈

// Hr(X)

≈

// Hr(X,Y )

J∗

∂ // Hr−1(Y )

≈

// Hr−1(X)

≈

Hr(Y ) // Hr(X) // Hr(X,Y )∂ // Hr−1(Y ) // Hr−1(X)

The top horizontal line is exact because the direct limit of exact sequences is exact.The bijectivity of the vertical arrows comes from Corollary 3.1.16. By the five-lemma, one deduces that J∗ is an isomorphism.

3.1.3. The homotopy property. Let f, g : (X,Y ) → (X ′, Y ′) be two mapsbetween topological pairs. Let I = [0, 1]. A homotopy between f and g is a map ofpairs F : (X × I, Y × I) → (X ′, Y ′) such that F (x, 0) = f(x) and F (x, 1) = g(x).If such a homotopy exists, we say that f and g are homotopic.

Proposition 3.1.30 (Homotopy property). Let f, g : (X,Y ) → (X ′, Y ′) betwo maps between topological pairs which are homotopic. Then H∗f = H∗g andH∗f = H∗g.

Proof. Note that H∗f = H∗g implies H∗f = H∗g by Kronecker duality, usingDiagram (2.3.4). We shall construct a Z2-linear map D : C∗(X)→ C∗+1(X

′) suchthat

(3.1.22) C∗f + C∗g = ∂D +D∂ ,

i.e. D is a chain homotopy from C∗f to C∗g. The map D will satisfy D(C∗(Y )) ⊂C∗+1(Y

′) and so will induce a linear map D : C∗(X,Y ) → C∗+1(X′, Y ′) satisfy-

ing (3.1.22). As in the proof of Proposition 2.5.9, this will prove that H∗f = H∗g.That H∗f = H∗g is then deduced by Kronecker duality, using Diagram (2.3.4).Let F : (X × I, Y × I)→ (X ′, Y ′) be a homotopy from f to g.

By linearity, it is enough to define D on singular simplexes. Let σ : ∆m → X bea singularm-simplex of X . Consider the convex-cell complex P = ∆m×I. One hasV (P ) = V (∆m)×0, 1. Using the natural total order on V (∆m), we can define anaffine order on P by deciding that the elements of V (∆m) × 1 are greater than

those of V (∆m)×0. Lemma 2.1.10 thus provides a triangulation h≤ : |L≤(P )| ≈−→P , with V (L≤(P )) = V (P ). Set L = L≤(P ) and h = h≤. The order ≤ becomes asimplicial order on L, giving rise to a chain map R≤,∗ : C∗(L) → C∗(P ) from thesimplicial chains of L to the singular chains of P (see Example 3.1.6). ConsiderSm+1(L) as an (m+ 1)-simplicial cochain of L and define D(σ) to be the image ofSm+1(L) under the composite map(3.1.23)

Sm+1(L)R≤,∗−−−→ Sm+1(|L|) C∗h−−→ Sm+1(P )

C∗(σ×id)−−−−−−→ Sm+1(X × I) C∗F−−−→ Sm+1(X′) .


Observe that, if τ ∈ Sm(L) such that h(|τ |) hits the interior of P , then τ is theface of exactly two (m+1)-simplexes of L. Therefore, ∂(Sm+1(L)) = Sm(L(BdP )).But

BdP = ∆m × 0 ∪∆m × 1 ∪ Bd∆m × I .As all the maps in (3.1.23) are chain maps, this permits us to prove that

(3.1.24) ∂D(σ) = C∗f(σ) + C∗g(σ) +D∂(σ)

As (3.1.24) holds true for all σ ∈ S(X), it implies (3.1.22).

Remark 3.1.31. In the proof of Proposition 3.1.30, the chain homotopy Dis not unique. Some authors (e.g. [179, 43, 155]) just give an existence proof,based on an easy case of the acyclic carrier’s technique (like in our proof of Propo-sition 2.5.9). We used above an explicit triangulation of ∆m × I. The same tri-angulation occurs in the proof of [82, p. 112], presented differently for the sake ofsign’s control. The idea of such triangulations of ∆m × I will be used again in theproof of the small simplex theorem 3.1.34.

A map of pairs f : (X,Y )→ (X ′, Y ′) is a homotopy equivalence if there exists amap of pairs g : (X ′, Y ′)→ (X,Y ) such that gf is homotopic to id(X,Y ) and f gis homotopic to id(X′,Y ′). The pairs (X,Y ) and (X ′, Y ′) are then called homotopyequivalent. Two spaces X and X ′ are homotopy equivalent if the pairs (X, ∅) and(X ′, ∅) are homotopy equivalent. Two homotopy equivalent spaces (or pairs) arealso said to have the same homotopy type.

By functoriality (Proposition 3.1.22), Proposition 3.1.30 implies that (co)homologyis an invariant of homotopy type:

Corollary 3.1.32 (Homotopy invariance of (co)homology). Let f : (X,Y )→(X ′, Y ′) be a homotopy equivalence. Then H∗f : H∗(X,Y ) → H∗(X

′, Y ′) andH∗f : H∗(X ′, Y ′)→ H∗(X,Y ) are isomorphisms.

A (non-empty) topological space X is contractible if there exists a homotopyfrom idX to a constant map. For instance, the cone CX over a space X

(3.1.25) CX =(X × I

)/(X × 1

),

with the quotient topology, is contractible. A homotopy from idCX to a constantmap is given by F ((x, τ), t) = [x, t+ (1− t)τ ].

Corollary 3.1.33. The (co)homology of a contractible space is isomorphic tothat of a point:

H∗(X) ≈ H∗(X) ≈0 if ∗ > 0

Z2 if ∗ = 0 .

Proof. Let x0 ∈ X such that there exists a homotopy from idX to the con-stant map onto x0 Then, the inclusion x0 → X is a homotopy equivalence andCorollary 3.1.33 follows from Corollary 3.1.32.

For a direct proof of Corollary 3.1.33, see Exercise 3.2.


3.1.4. Excision. Let X be a topological space. Let B be a family of subspacesof X . A map f : L → X is called B-small if f(L) is contained in an element ofB. Let SBm(X) be the set of singular m-simplexes of X which are B-small. Thevector spaces of (co)chains CmB (X) and CBm(X) are defined as in § 3.1, using B-small m-simplexes. We get, in the same way, a pairing 〈 , 〉 : CmB (X) × CBm(X) →Z2 identifying CmB (X) to CBm(X)♯. The boundary of a B-small simplex is a B-small chain, so (CB∗ (X), ∂) is a subcomplex of chains of (C∗(X), ∂), the inclu-sion being denoted by iB∗ : C

mB (X) → Cm(X). Define δ : CmB (X) → Cm+1

B (X) by〈δ(a), α〉 = 〈a, ∂(α)〉 and i∗BC

m(X) → CmB (X) by 〈i∗B(a), α〉 = 〈a, iB∗ (α)〉. Then,((C∗B(X), δ), (CB∗ (X), ∂), 〈 , 〉) is a Kronecker pair.

The (co)homologies obtained by these definitions are denoted by H∗B(X) andHB∗ (X). One uses the notations iB∗ : H

Bm(X)→ Hm(X) and i∗B : H

m(X)→ HmB (X)

for the induced linear maps. The following result is very useful.

Proposition 3.1.34 (Small simplexes theorem). Let X be a topological spacewith a family B of subspaces of X, whose interiors cover X. Then iB∗ : H

B∗ (X) →

H∗(X) and i∗B : H∗(X)→ H∗B(X) are isomorphisms.

The proof of Proposition 3.1.34 uses iterations of the subdivision operator,a chain map sd ∗ : C∗(X,Y ) → C∗(X,Y ) which replaces chains by chains with”smaller” simplexes. Intuitively, sd ∗ replaces a singular simplex σ : ∆m → X bythe sum of σ restricted to the barycentric subdivision of ∆m.

More precisely, consider the standard simplex ∆m as the geometric realizationof the full complex Fm over the set 0, 1, . . . ,m. The barycentric subdivision F ′mis endowed with its natural simplicial order ≤ of (2.1.2), p. 14. As explained inExample 3.1.6, we get a chain map

R∗ = R≤,∗ : C∗(F ′m)→ C∗(|F ′m|) = C∗(∆m) .

Let σ ∈ Sm(X). As a continuous map from ∆m to X , σ induces C∗σ : C∗(∆m)→

C∗(X). Definesd ∗(σ) = C∗σ(Sm(F ′m)) .

This formula determines a unique linear map sd ∗ : C∗(X)→ C∗(X) which is clearlya chain map. If Y is a subspace of X , then sd ∗(C∗(Y )) ⊂ C∗(X), so we get a chainmap sd ∗ : C∗(X,Y )→ C∗(X,Y ), giving rise to a GrV-morphism

sd ∗ : H∗(X,Y )→ H∗(X,Y ) .

By Kronecker duality, we get a cochain map sd ∗ : C∗(X,Y ) → C∗(X,Y ) and aGrV-morphism sd ∗ : H∗(X,Y ) → H∗(X,Y ) satisfying 〈sd ∗(a), α〉 = 〈a, sd ∗(α)〉for all a ∈ C∗(X,Y ) and α ∈ C∗(X,Y ).

Observe that sd sends CB∗ (X,Y ) into CB∗ (X,Y ) and thusHB∗ (X,Y ) intoHB∗ (X,Y )and H∗B(X,Y ) into H∗B(X,Y )

Lemma 3.1.35. The subdivision operators sd ∗ : H∗(X,Y )→ H∗(X,Y ) andsd ∗ : H∗(X,Y ) → H∗(X,Y ) are equal to the identity. The same holds true forsd ∗ : H

B∗ (X,Y )→ HB∗ (X,Y ) and sd ∗ : H∗B(X,Y )→ H∗B(X,Y ).

Proof. We shall construct a Z2-linear map D : C∗(X)→ C∗+1(X) such that

(3.1.26) id + sd ∗ = ∂D +D∂ .

In other words, D is a chain homotopy from id to sd ∗ (see p. 32). The map Dwill satisfy D(C∗(Y )) ⊂ C∗+1(Y ) and will so induce a linear map D : C∗(X,Y )→


C∗+1(X,Y ) satisfying (3.1.26). As in the proof of Proposition 2.5.9, this will provethat sd ∗ = id. That sd ∗ = id is then implied by Kronecker duality, using Dia-gram (2.3.4). Also, the map D will satisfy D(CB∗ (X)) ⊂ CB∗+1(X).

By linearity, it is enough to define D on singular simplexes. The proof issimilar to that of Proposition 3.1.30 (an idea of V. Puppe). Let σ : ∆m → Xbe a singular m-simplex of X . Consider the convex-cell complex P = ∆m × I,where the upper face ∆m × 1 is replaced by its barycentric subdivision |F ′m|.One has V (P ) = V (∆m) × 0 ∪V (F ′m) × 1. We use the natural total order onV (∆m)×0 and the natural simplicial order on V (F ′m)×1 (see (2.1.2)). Decidingin addition that the elements of V (F ′m)×1 are greater than those of V (∆m)×0provides an affine order ≤ on P . Lemma 2.1.10 thus constructs a triangulation

h≤ : |L≤(P )| ≈−→ P , with V (L≤(P )) = V (P ). Set L = L≤(P ) and h = h≤. Seeing≤ as a simplicial order on L gives rise to a chain map R≤,∗ : C∗(L)→ C∗(P ) fromthe simplicial chains of L to the singular chains of P (see Example 3.1.6). ConsiderSm+1(L) as an (m+ 1)-simplicial cochain of L and define D(σ) to be the image ofSm+1(L) under the composite map

(3.1.27) Sm+1(L)R≤,∗−−−→ Sm+1(|L|) C∗h−−→ Sm+1(P )

C∗(σ)−−−−→ Sm+1(X) ,

where σ : P → X is the map σ(x, t) = σ(x). Observe that the inclusion |F ′m| ⊂∆m×1 is already the piecewise affine triangulation of ∆m×1 determined by thesimplicial order on F ′m. Therefore, the construction of the proof of Lemma 2.1.10leaves ∆m × 1 unchanged. Formula (3.1.26) is then deduced as in the proofof Proposition 3.1.30. Finally, if σ is B-small, so is the map σ. Hence D(σ) ∈CBm+1(X), which proves the lemma for the B-small (co)homology.

Proof of Proposition 3.1.34. By Kronecker duality, using Corollary 2.3.11,only the homology statement must be proved. Let sd k = sd · · · sd (k times).We shall need the following statement.

Claim: let α ∈ C∗(X). Then, there exists k(α) ∈ N such that sd k(α) ∈ CB∗ (X)for all k ≥ k(α).

Let us show that the claim implies Proposition 3.1.34. Let α ∈ Hm(X,Y )represented by α ∈ Cm(X) with ∂(α) ∈ Cm−1(Y ). The claim implies that, for

k big enough, sd k(α) ∈ CBm(X), and thus ∂(α) ∈ Cm−1(Y ). This implies that

sd k(α) is in the image of iB∗ . By Lemma 3.1.35, sd k(α) = α, so α is in the imageof iB∗ , which proves that iB∗ is surjective. For the injectivity, let β ∈ HBm(X,Y ) withiB∗ (β) = 0. Represent β by β ∈ CBm(X) with ∂(β) ∈ Cm−1(Y ). The hypothesisiB∗ (β) = 0 says that β = ∂(γ) + ω with γ ∈ Cm+1(X) and ω ∈ Cm(Y ). The claim

tells us that, for k big enough, sd k(γ) ∈ CBm+1(X) (and, so, sd k(ω) ∈ CBm(Y )). This

implies that sd k(β) = 0 in HBm(X,Y ). But sd k(β) ∈ CBm(X) and Lemma 3.1.35

tells us that sd k coincides with the identity of HBm(X,Y ). Thus, β = 0 for allβ ∈ ker iB∗ .

It remains to prove the claim. Let ρ(m, k) be the maximal distance betweentwo points of a simplex of the k-th barycentric subdivision of ∆m. An elementaryargument of Euclidean geometry shows that

(3.1.28) ρ(m, k) ≤ ρ(m, 0)

(m

m+ 1

)k


(of course, ρ(m, 0) =√2). For details, see e.g., [155, Proof of Theorem 15.4] or

[82, p. 120]. By hypothesis, the family B = intB | B ∈ B is an open covering

of X . Consider the induced open covering σ−1B of ∆m. By (3.1.28), ρ(m, k) → 0

when k → ∞. Using a Lebesgue number for the open covering σ−1B, this provesthe claim.

The main application of the small simplexes theorem is the invariance underexcision (see also § 3.1.6).

Proposition 3.1.36 (Excision property). Let (X,Y ) be a topological pair. LetU be a subspace of X with U ⊂ intY . Then, the linear maps induced by inclusions

i∗ : H∗(X,Y )≈−→ H∗(X − U, Y − U) and i∗ : H∗(X − U, Y − U)

≈−→ H∗(X,Y )

are isomorphisms.

Proof. By Corollary 2.3.11, i∗ is an isomorphism if and only if i∗ is an iso-morphism. We shall prove that i∗ is an isomorphism.

Let B = Y,X − U. One has a commutative diagram

0 // C∗(Y )

id=

// CB∗ (X)

iB∗

// CB∗ (X)/C∗(Y )

IB∗

// 0

0 // C∗(Y ) // C∗(X) // C∗(X)/C∗(Y ) // 0

where all arrows are induced by inclusions and the horizontal lines are short ex-act sequences of chain complexes. As in § 2.6, this gives a commutative diagrambetween the corresponding long homology sequences

. . . // Hm(Y )

id=

// HBm(X)

iB∗

// Hm(CB∗ (X)/C∗(Y ))

IB∗

// Hm−1(Y )

id=

// . . .

. . . // Hm(Y ) // Hm(X) // Hm(X,Y ) // Hm−1(Y ) // . . .

As U ⊂ intY , the family B = Y,X − U satisfies the hypotheses of Proposi-tion 3.1.34 and iB∗ is an isomorphism. By the five-lemma, IB∗ is an isomorphism.Therefore, it suffices to show that H∗(X − U, Y − U) → H∗(C

B∗ (X)/C∗(Y )) is an

isomorphism. But it is easy to see that this is already the case at the chain level:

C∗(X − U, Y − U) = C∗(X − U)/C∗(Y − U)≈−→ CB∗ (X)/C∗(Y ) .

3.1.5. Well cofibrant pairs. Let (Z, Y ) be a topological pair and denote byi : Y → Z the inclusion. A (continuous) map r : Z → Y is called a retraction ifri = idY . It is a retraction by deformation if ir is homotopic to the identity ofZ. A retraction by deformation is thus a homotopy equivalence.

Note that Z retracts by deformation on Y if and only if there is a homotopyh : Z × I → Z which, for all (z, t) ∈ Z × I, satisfies h(z, 0) = z, h(z, 1) ∈ Y andh(y, t) = y when y ∈ Y . A topological pair (X,A) is called good if A is closed in Xand if there is a neighbourhood V of A which retracts by deformation onto A. Forinstance, (X, ∅) is a good pair (V = ∅ and h(x, t) = x).

Good pairs were introduced in [82] (with the additional condition that A isnon-empty). Earlier books rather rely on the notion of cofibration, developed in


the 1960’s essentially by D. Puppe and N. Steenrod (see [185] for references). Bothare useful in different circumstances, so we introduce below the mixed notion ofa well cofibrant pair, especially useful in equivariant homotopy theory (see e.g.Chapter 7). We begin with cofibrant pairs, starting with the following lemma.

Lemma 3.1.37. For a topological pair (X,A), the following conditions are equiv-alent.

(1) There is a retraction from X × I onto X × 0 ∪ A× I.(2) Let f : X → Z and FA : A × I → Z be continuous maps such that

FA(a, 0) = f(a). Then, FA extends to a continuous map F : X × I → Zsuch that F (x, 0) = f(x) for all x ∈ X.

Proof. We give below the easier proof available when A is closed in X (for aproof without this hypothesis: see [39, (1.19)]). Let r : X× I → X×0∪A× I bea retraction. Given f and FA as in (2), define the map g : X ×0∪A× I → Z byg(x, 0) = f(x) and g(a, t) = FA(a, t). If A is closed, then g is continuous and themap F = gr satisfies the required condition. Hence, (1) implies (2). Conversely, iff and FA are the inclusions of X and A× I into Z = X×0∪A× I, the extensionF given by (2) is a continuous retraction from X × I onto Z.

A pair (X,A) with A closed in X which satisfies (1) or (2) of Lemma 3.1.37 iscalled cofibrant. According to the literature, the inclusion A → X is a cofibration,or satisfies the absolute homotopy extension property (AHEP) (see e.g. [44, 73]).See e.g. [38, Chapter 5] for other characterizations and properties of cofibrantpairs.

As a motivation of our concept of well cofibrant pair, we first give an example.

Example 3.1.38. Mapping cylinder neighbourhoods. Let (X,A) be a topologicalpair. A neighbourhood V of A is called a mapping cylinder neighbourhood if there isa continuous map ϕ : V → A (where V is the frontier of V ) and a homeomorphismψ : Mϕ → V where

Mϕ = [(V ×I) ∪A]/(x, 0) ∼ ϕ(x) | x ∈ V

is the mapping cylinder of ϕ. The homeomorphism ψ is required to satisfy ψ(x, 1) =

x and ψ(x, 0) = ϕ(x) for all x ∈ V . Here are examples of mapping cylinderneighbourhoods

• if X is a smooth manifold and A is a smooth submanifold of codimension≥ 1, then a closed tubular neighbourhood of A [95, § 4.6] is a mappingcylinder neighbourhood.• if A is the boundary of a smooth manifold X , then a collar neighbourhoodof A [95, § 4.6] is a mapping cylinder neighbourhood.• a subcomplex of a CW-complex admits a mapping cylinder neighbour-hood. The proof of this will be given in Lemma 3.4.2.

Given a mapping cylinder neighbourhood as above, a continuous retractionF : X × I → X × 0 ∪ A× I is defined by

F (x, t) =

(ϕ(v), t(1 − 2τ)) if x = ψ(v, τ) with τ ≤ 1/2.

(ψ(v, 2τ − 1), 0) if x = ψ(v, τ) with τ ≥ 1/2.

(x, 0) if x ∈ X − intV .


Let u : X → I defined by

u(x) =

2τ if x = ψ(v, τ) with τ ≤ 1/2.

1 otherwise.

Let h : X × I → X defined by h = pX F , where pX : X × I → X is the projection.Then, u(h(x)) ≤ u(x) which implies that, for all T < 1, h restricts to a strongdeformation retraction from u−1([0, T ]) onto A = u−1(0). Hence, (X,A) is a goodand cofibrant pair.

A topological pair (X,A) is well cofibrant if there exists continuous mapsu : X → I and h : X × I → X such that

(1) A = u−1(0) (in particular, A is closed in X).(2) h(x, 0) = x for all x ∈ X .(3) h(a, t) = a for all (a, t) ∈ A× I.(4) h(x, 1) ∈ A for all x ∈ X such that u(x) < 1.(5) u(h(x, t)) ≤ u(x) for all (x, t) ∈ X × I.

We say that (u, h) is a presentation of (X,A) as a well cofibrant pair. Condi-tions (1)-(4) define a NDR-pair (neighbourhood deformation retract pair) in thesense of [185, 38, 140] (see also Remark 3.1.42 (b) below).

The pairs (X, ∅) and (X,X) are well cofibrant. One takes u(x) = 1 for (X, ∅),u(x) = 0 for (X,X) and h(x, t) = x for both pairs. Another basic example of wellcofibrant pairs is given by the following lemma.

Lemma 3.1.39. Suppose that A ⊂ X admits a mapping cylinder neighbourhoodin X. Then, (X,A) is well cofibrant.

Proof. The pair (u, h) in Example 3.1.38 is a presentation of (X,A) as a wellcofibrant pair.

Lemma 3.1.40. Let (X,A) and (Y,B) be two well cofibrant pairs. Then, the“product pair” (X × Y,A× Y ∪X ×B) is well cofibrant.

The following proof, coming from that of [185, Theorem 6.3], will be convenientfor the equivariant setting (see Lemma 7.2.12).

Proof. Let (u, h) and (v, j) be presentations of (X,A) and (Y,B) as wellcofibrant pairs. Define w : X×Y → I by w(x, y) = u(x)v(y). Define q : X×Y ×I →X × Y by

q(x, y, t) =

(x, y) if (x, y) ∈ A×B.(h(x, t), j(y, u(x)v(y) t)

)if v(y) ≥ u(x) and v(y) > 0.(

h(x, v(y)u(x) t), j(y, t))

if v(y) ≤ u(x) and u(x) > 0.

One checks that (w, q) is a presentation of (X×Y,A×Y ∪X×B) as a well cofibrantpair. Details for (1)-(4) are given in [185, p. 144] and (5) is obvious.

Lemma 3.1.41. Let (X,A) be a well cofibrant pair. Then, (X,A) is good andcofibrant.

Proof. Let (u, h) be a presentation of (X,A) as a well cofibrant pair. Asnoticed in Example 3.1.38, the condition u(h(x, t)) ≤ u(x) implies that, for allT < 1, h restricts to a strong deformation retraction from u−1([0, T ]) onto A. SinceA = u−1(0), it is closed. Hence, (X,A) is good. To see that (X,A) is cofibrant, let


(Y,B) = (I, 0) presented as well cofibrant pair by (v, j) where v(y) = y/2 andj(y, t) = (1 − t)y. Let (w, q) be the presentation of

(X × Y,A× Y ∪X ×B) = (X × I,X × 0 ∪ A× I)as a well cofibrant pair given in the proof of Lemma 3.1.40. As

w(x, y) = u(x)y/2 < 1 ,

the formula r(x, y) = q(x, y, 1) defines a retraction

(3.1.29) X × I r−→ X × 0 ∪ A× I .By Lemma 3.1.37, (X,A) is cofibrant.

Remarks 3.1.42. (a) The fact that the retraction r of (3.1.29) is a strongdeformation retraction should not be a surprise. If r = (r1, r2) : X× I → X×0∪A× I ⊂ X × I is any retraction, then the map R : X × I → X × I defined by

R(x, t, s) =(r1(x, (1− s)t), st+ (1− s)r2(x, t)

)

is a homotopy from idX×I to r [73, Lemma 16.28].

(b) The proof of Lemma 3.1.41 shows that a NDR-pair is cofibrant. The con-verse is also true (see [140, § 6.4]).

If (X,A) is a topological pair, we denote by X/A the quotient space where allpoints of A are identified in a single class. The projection π : (X,A)→ (X/A,A/A)is a map of pairs.

Lemma 3.1.43. Let (X,A) be a well cofibrant pair and let B ⊂ A. Then(X/B,A/B) is well cofibrant. In particular, the pair (X/A,A/A) is well cofibrant.

Proof. Let (u, h) be a presentation of (X,A) as a well cofibrant pair. By (1)and (3), u and h descend to continuous maps u : X/B → I and h : (X/B) × I →X/B, giving a presentation (u, h) of (X/B,A/B).

Lemma 3.1.44. Let (X,A) be a cofibrant pair such that A is contractible. Thenthe quotient map X → X/A is a homotopy equivalence.

Proof. If A is contractible, there is a continuous map FA : A × I → A ⊂ Xsuch that FA(a, 0) = a and F (A × 1) = a0. As (X,A) is cofibrant, there is acontinuous map F : X × I → X such that F (x, 0) = x and F (a, t) = FA(a, t) fora ∈ A. F|X×1 admits a factorization

X≈ //

q)) ))

X × 1

$$ $$

F|X×1 // X

X/A

g

==④④④④④④④

Using F , gq is homotopic to idX . On the other hand, as F (A× I) ⊂ A, the mapqF descends to a continuous map F : X/A× I → X/A which is a homotopy fromidX/A to qg.

Proposition 3.1.45. Let (X,A) be a well cofibrant pair. Then, the homomor-phisms

π∗ : H∗(X/A,A/A) −→ H∗(X,A) and π∗ : H∗(X,A) −→ H∗(X/A,A/A)

are isomorphisms.


Proof. By Corollary 2.3.11, π∗ is an isomorphism if and only if π∗ is anisomorphism. We shall prove that π∗ is an isomorphism (the proof is the same forboth). There is nothing to prove if A = ∅, so we assume that A is not empty.

By Lemma 3.1.41, (X,A) is cofibrant. Let r be a retraction from X × I toX × 0 ∪ A × I. Let CA =

(A × I

)/(A × 1

)be the cone over A and let

X = X ∪A CA. As A is closed, r extends (by the identity on CA × I) to a

continuous retraction from X × I onto X ×0∪CA× I. Hence, the pair (X, CA)is cofibrant. But

H∗(X, CA)≈

excision// H∗(X − [A× 1], CA− [A× 1]) ≈

homotopy// H∗(X,A) .

On the other hand, X/A = X/CA. Set X = X/CA and C = CA/CA ≈ pt. Thequotient map q : (X, CA)→ (X, C) provides a morphism of exact sequences

Hk−1(X) //

q∗≈

Hk−1(C) //

q∗≈

Hk(X, C) //

q∗

Hk(X) //

q∗≈

Hk(C)

q∗≈

Hk−1(X) // Hk−1(CA) // Hk(X, CA) // Hk(X) // Hk(CA)

.

As C and CA are contractible, q∗ : H∗(C) → H∗(CA) is an isomorphism and so

is q∗ : H∗(X) → H∗(X) by Lemma 3.1.44. By the five lemma, q∗ : Hk(X, C) →Hk(X, CA) is an isomorphism, which proves Proposition 3.1.45

Remark 3.1.46. The proof of Proposition 3.1.45 uses only that the pair (X,A)is cofibrant. Another proof exists using that (X,A) is a good pair (see [82, Propo-sition 2.22] or Proposition 7.2.15). It is interesting to note that these relativelyshort proofs both use almost all the axioms of a cohomology theory (see § 3.9):functoriality, homotopy, excision and functorial exactness.

Corollary 3.1.47. Let (X,A) be a well cofibrant pair with A non-empty.Then,

π∗ : H∗(X/A)≈−→ H∗(X,A) and π∗ : H∗(X,A)

≈−→ H∗(X/A)

are isomorphisms.

Proof. If A 6= ∅, then A/A is a point. Therefore, by (3.1.13), H∗(X/A)≈−→

H∗(X/A,A/A) and H∗(X/A,A/A)≈−→ H∗(X/A). The results then follows form

Proposition 3.1.45.

Corollary 3.1.48. Let (X,A) be a well cofibrant pair. Denote by i : A → Xthe inclusion and by j : X → X/A the quotient map. Then, there is a functorialexact sequence in reduced cohomology

· · · → Hk−1(X)H∗i−−→ Hk−1(A)

δ∗−→ Hk(X/A)H∗j−−−→ Hk(X)

H∗i−−→ Hk(A)→ · · ·The corresponding sequence exists in reduced homology.

Proof. The result is obvious if A is empty. Otherwise, this comes from the

exact sequence of Proposition 3.1.20 together with the isomorphism H∗(X/A)≈−→

H∗(X,A)≈−→ H∗(X,A) of Corollary 3.1.47.


One application of well cofibrant pairs is the suspension isomorphism. Let Xbe a topological space. The suspension ΣX of X is the quotient space

ΣX = CX/(X × 0

)

where CX is the cone overX (see (3.1.25)). The pairs (CX,X×0) and (ΣX,X× 12 )

are well cofibrant by Lemma 3.1.39, since the subspaces admits mapping cylinderneighbourhoods. The (cohomology) suspension homomorphism is the degree-1 lin-

ear map Σ∗ : Hm(X)→ Hm+1(ΣX) given by the composition(3.1.30)

Σ∗ : Hm(X)δ∗−→ Hm+1(CX,X × 0) ≈ Hm+1(ΣX, [X × 0])

j∗−→ Hm+1(ΣX) ,

where the middle isomorphism comes from Proposition 3.1.45.

Proposition 3.1.49. For any topological space X, the suspension homomor-phism

Σ∗ : Hm(X)→ Hm+1(ΣX)

is an isomorphism for all m.

Because of Proposition 3.1.49, the homomorphism Σ∗ is called the suspensionisomorphism (in cohomology). Observe that Formula (3.1.30) can also be used todefine Σ∗ : Hm(X) → Hm+1(ΣX). By Proposition 3.1.49, this unreduced suspen-sion homomorphism is an isomorphism if m ≥ 1.

Proof. If X is empty, so is ΣX and Proposition 3.1.49 is trivial. Supposethen that X 6= ∅. As [X× 0] is a point, its reduced cohomology vanish and j∗ is anisomorphism in Formula (3.1.30), by the reduced cohomology sequence of the pair(ΣX, [X × 0]). As CX is contractible, its reduced cohomology also vanish and δ∗

is an isomorphism in Formula (3.1.30), by the reduced cohomology sequence of the

pair (CX,X × 0). Finally, note that ΣX is path-connected, so Hm(ΣX) = 0 form ≤ 0, so Proposition 3.1.49 is also true for m ≤ 0.

Analogously, we define Σ∗ : Hm+1(ΣX)→ Hm(ΣX) by the composition

(3.1.31) Σ∗ : Hm+1(ΣX)j∗−→ Hm+1(ΣX, [X×0]) ≈ Hm+1(CX,X×0) ∂∗−→ Hm(X)

which satisfies 〈Σ∗(a), α〉 = 〈a,Σ∗(α)〉 for all a ∈ Hm(X) and all α ∈ Hm+1(X).By Proposition 3.1.49 (or directly), we deduce that Σ∗ is an isomorphism, calledthe suspension isomorphism (in homology).

Let (Xj , xj) (j ∈ J ) be a family of pointed spaces, i.e. xi ∈ Xi. Their bouquet(or wedge) X =

∨j∈J Xj is defined as the quotient space

X =∨

j∈J

Xj =⋃

j∈JXj

/⋃j∈Jxj .

By naming x ∈ X the equivalence class x =⋃j∈J xj, the couple (X, x) is a

pointed space. For each j ∈ J , one has a pointed inclusion ij : (Xj , xj) → (X, x).The bouquet plays the role of a sum in the category of pointed spaces and pointedmaps: if f j : (Xj , xj)→ (Y, y) are continuous pointed maps, then there is a uniquecontinuous pointed map f : (X, x)→ (Y, y) such that f ij = fj.

A well pointed space is a pointed space (X, x) such that (X, x) is a wellcofibrant pair. Observe that this definition is stronger than that in other textbooks.


Lemma 3.1.50. If (Xj , xj) (j ∈ J ) are well pointed spaces, then their wedge(X, x) is a well pointed space.

Proof. Let (uj , hj) be a presentation of (Xj , xj) as a well cofibrant pair. Then

(⋃j∈J u

j ,⋃j∈J h

j) is a presentation of (⋃j∈JX

j ,⋃j∈J xj) as a well cofibrant

pair. By Lemma 3.1.43, the quotient pair (X, x) is well cofibrant, so (X, x) is awell pointed space.

Proposition 3.1.51. Let (Xj , xj), with j ∈ J , be a family of well pointedspaces. Then, the family of inclusions ij : Xj → X =

∨j∈J Xj, for j ∈ J , gives

rise to isomorphisms on reduced (co)homology

H∗(X)(H∗ij)

≈// ∏

j∈J H∗(Xj)

and⊕

j∈J H∗(Xj)

∑H∗ij

≈// H∗(X) .

Proof. It is enough to establish that∑H∗ij is an isomorphism. The coho-

mology statement can be proved analogously or by Kronecker duality, using Dia-gram (2.3.4).

Write, as above, x =⋃j∈J xi ∈ X . The maps of pairs (Xj , xi) → (X, x)

give rise, for each m ∈ N, to a commutative diagram between exact sequences

⊕

j∈J Hm+1(xj)

≈

// ⊕j∈J Hm+1(Xj)

≈

// ⊕j∈J Hm+1(Xj , xj)

⊕H∗ij

∂∗ //

Hm+1( ˙⋃

j∈J xj) // Hm( ˙⋃

j∈JXj) // Hm+1( ˙⋃

j∈JXj , ˙⋃

j∈J xj)∂∗ //

∂∗ // ⊕j∈J Hm(xj)

≈

// ⊕j∈J Hm(Xj)

≈

∂∗ // Hm(⋃j∈J xj) // Hm(

⋃j∈JXj)

The isomorphisms for the vertical arrows are due to Proposition 3.1.11. By thefive-lemma,

⊕H∗ij is an isomorphism. As (Xj , xj) is well pointed, the pair

(⋃j∈JXj ,

⋃j∈J xj) is well cofibrant. By Proposition 3.1.45, the quotient map

q : (⋃j∈JXj,

⋃j∈J xj) → (X, x) induces an isomorphism on homology. One has

the commutative diagram

⊕j∈J H∗(Xj)

≈

∑H∗ij // H∗(X)

≈

⊕j∈J H∗(Xj , xj)

⊕H∗ij

≈// H∗(

⋃j∈JXj ,

⋃j∈J xj)

H∗q

≈// H∗(X, x) ,

where the vertical arrows are isomorphisms by Remark 3.1.21. Therefore,∑H∗ij

is an isomorphism.


The (co)homology of X =∨j∈J Xj may somehow be also controlled using the

projection πj : X → Xj defined by

(3.1.32) πj(z) =

z if z ∈ Xj

x otherwise,

where x =⋃j∈J xi ∈ X . As πj ij = idXj

, Proposition 3.1.51 implies the follow-ing

Proposition 3.1.52. Let (Xj , xj), with j ∈ J , be a family of well pointedspaces. Then, the composition

⊕j∈J H

∗(Xj)

∑H∗πj// H∗(X)

(H∗ij)

≈// ∏

j∈J H∗(Xj)

is the inclusion of the direct sum into the product. Also, the composition

⊕j∈J H∗(Xj)

∑H∗ij

≈// H∗(X)

(H∗πj)// ∏j∈J H∗(Xj)

is the inclusion of the direct sum into the product.In particular, if J is finite, then

⊕j∈J H

∗(Xj)

∑H∗πj

≈// H∗(X) and H∗(X)

(H∗πj)

≈// ∏

j∈J H∗(Xj)

are isomorphisms.

3.1.6. Mayer-Vietoris sequences. Let X be a topological space. Let B =X1, X2 be a collection of two subspaces of X . Write X0 = X1 ∩X2 and

X0

i2

i1 // X1

j1

X2j2 // X

for the inclusions. We call (X,X1, X2, X0) a Mayer-Vietoris data. The sequence ofcochain complexes

0→ C∗(X)(C∗j1,C

∗j2)−−−−−−−−→ C∗(X1)⊕ C∗(X2)C∗i1+C

∗i2−−−−−−−→ CB∗ (X0)→ 0

is then exact, as well as the sequence of chain complexes

0→ C∗(X0)(C∗i1,C∗i2)−−−−−−−→ C∗(X1)⊕ C∗(X2)

C∗j1+C∗j2−−−−−−−→ CB∗ (X)→ 0

By § 2.6, these short exact sequences give rise to connecting homomorphisms

δMV : H∗(X0)→ H∗+1B (X) and ∂MV : HB∗ (X)→ H∗−1(X0)

involved in long (co)homology exact sequences. If the interiors of X1 and X2

cover X , the theorem of small simplexes 3.1.34 implies that H∗B(X) ≈ H∗(X) andHB∗ (X) ≈ H∗(X). Therefore, we obtain the following proposition.

Proposition 3.1.53 (Mayer-Vietoris sequences I). Let (X,X1, X2, X0) be aMayer-Vietoris data. Suppose that X = intX1 ∪ intX2. Then, the long sequences

→ Hm(X)(H∗j1,H

∗j2)−−−−−−−−→ Hm(X1)⊕Hm(X2)H∗i1+H

∗i2−−−−−−−→ Hm(X0)δMV−−−→ Hm+1(X)→


and

→ Hm(X0)(H∗i1,H∗i2)−−−−−−−−→ Hm(X1)⊕Hm(X2)

H∗j1+H∗j2−−−−−−−→ Hm(X)∂MV−−−→ Hm−1(X0)→

are exact.

These Mayer-Vietoris sequences are natural for maps f : X → X ′ such thatf(Xi) ⊂ X ′i.

The hypotheses of Proposition 3.1.53 may not by directly satisfied in usualsituations. Here is a variant which is more useful in practice.

Proposition 3.1.54 (Mayer-Vietoris sequences II). Let (X,X1, X2, X0) be aMayer-Vietoris data, with Xi closed in X. Suppose that X = X1 ∪ X2 and that(Xi, X0) is a good pair for i = 1, 2. Then, the long sequences

→ Hm(X)(H∗j1,H

∗j2)−−−−−−−−→ Hm(X1)⊕Hm(X2)H∗i1+H

∗i2−−−−−−−→ Hm(X0)δMV−−−→ Hm+1(X)→

and

→ Hm(X0)(H∗i1,H∗i2)−−−−−−−−→ Hm(X1)⊕Hm(X2)

H∗j1+H∗j2−−−−−−−→ Hm(X)∂MV−−−→ Hm−1(X0)→

are exact.

Proof. Choose a neighbourhood Ui of X0 in Xi admitting a retraction bydeformation onto X0, called ρ

it : Ui → Ui, t ∈ I. Let X ′1 = X1 ∪ U2, X

′2 = X2 ∪ U1

and X ′0 = X ′1 ∩ X ′2 = X0 ∪ U1 ∪ U2. We claim that X = intX ′1 ∪ intX ′2. Indeed,as U2 is a neighbourhood of X0 in X2, there exists an open set V2 of X such thatX0 ⊂ V2 ∩X2 ⊂ U2. As X2 is closed, X1 −X2 = X −X2 is open in X . Therefore

X1 ⊂ (X1 −X2) ∪ (V2 ∩X2) = (X1 −X2) ∪ V2 ⊂ intX ′1 .

In the same way, X2 ⊂ intX ′2. Hence, X = intX ′1 ∪ intX ′2.As X = intX ′1 ∪ intX ′2, the Mayer-Vietoris sequences of Proposition 3.1.53

hold true with (X ′1, X′2, X

′0). But X ′1 retracts by deformation onto X1, using the

retraction ρ1t : X′1 → X ′1 given by

ρ1t =

ρ2t (x) if x ∈ U2

x if x ∈ X1 .

In the same way, X ′2 retracts by deformation onto X2 and X ′0 retracts by deforma-tion onto X0. This proves Proposition 3.1.54.

For Mayer-Vietoris sequences with other hypotheses, see Exercise 3.11, fromwhich Proposition 3.1.54 may also be deduced.

3.2. Spheres, disks, degree

So far, we have not encountered any space whose (co)homology is not zero inpositive dimensions. The unit sphere Sn in Rn+1 will be the first example. Theshortest way to describe the (co)homology of such simple spaces is by giving theirPoincare polynomials. The definitions are the same as for simplicial complexes.A topological pair (X,Y ) is of finite (co)homology type if its singular homology(or, equivalently, cohomology) is of finite type. In this case, the Poincare series of(X,Y ) (or of X if Y is empty) is that of H∗(X,Y ):

Pt(X,Y ) =∑

i∈N

dimHi(X,Y ) ti =∑

i∈N

dimHi(X,Y ) ti ∈ N[[t]].

3.2. SPHERES, DISKS, DEGREE 75

When the series is a polynomial, we speak of the Poincare polynomial of (X,Y ).

Proposition 3.2.1. The Poincare polynomial of the sphere Sn is

Pt(Sn) = 1 + tn .

Proof. The sphere S0 consists of two points, so the result for n = 0 followsfrom (3.1.5) and Corollary 3.1.12. We can then propagate the result by the suspen-

sion isomorphism Σ∗ : H∗(Sn)≈−→ H∗(Sn+1) (see Proposition 3.1.49), since Sn+1

is homeomorphic to ΣSn. The homology statement uses the homology suspension

isomorphism Σ∗ : H∗(Sn+1)

≈−→ H∗(Sn).

As a consequence of Proposition 3.2.1, the sphere Sn is not contractible, thoughit is path-connected if n > 0. Also Sn and Sp are not homotopy equivalent if n 6= p.A useful corollary of Proposition 3.2.1 is the following

Corollary 3.2.2. Pt(Dn, Sn−1) = tn.

Proof. This follows from Proposition 3.2.1 and the (co)homology exact se-quence of the pair (Dn, Sn−1).

It will be useful to have explicit cycles for the generators of Hn(Dn, Sn−1) and

Hn(Sn). Let ∆n = ∆n−int∆n be the topological boundary of the standard simplex

∆n. The identity map in : ∆n → ∆n is a relative cycle of (∆n, ∆n), representing a

class [in] ∈ Hn(∆n, ∆n). The boundary ∂(in) belongs to Zm−1(∆

n) and represents

∂∗([in]) in Hn−1(∆n).

Proposition 3.2.3. For all n ∈ N, the following two statements hold true:

An: [in] is the non-zero element of Hn(∆n, ∆n) = Z2.

Bn: [∂(in+1)] is the non-zero element of Hn(∆n+1) = Z2.

Proof. Statements An and Bn are proven together, by induction on n, asfollows:

(a) A0 and B0 are true.(b) An implies Bn.(c) Bn implies An+1.

As the affine simplex ∆0 is a point and ∆0 is empty, Statement A(0) follows from

the discussion in Example 3.1.5. To prove B0, observe that ∆1 consists of twopoints p and q. Identifying a singular 0-simplex with a point, one has ∂(i1) = p+ q,

which represents a non-vanishing element of H0(∆1). But 〈1, p + q〉 = 0, which

shows that [∂(i1)] 6= 0 in H0(∆1).

Let us prove (b). Consider the inclusion ǫ : ∆n → ∆n+1 given by ǫ(t0, . . . , tn) =

(t0, . . . , tn, 0). Let Λn = adh (∆n+1 − ǫ(∆n)). Consider the homomorphisms:

Hn(∆n+1)

j∗

≈// Hn(∆

n+1,Λn) Hn(∆n, ∆n) .

≈

H∗ǫoo

The arrow j∗ is bijective, as in (3.1.13), since Λm is contractible; the arrow H∗ǫ is

bijective by excision and homotopy. As in H∗(∆n+1,Λn) we neglect the singular

chains in Λn, one hasj∗([∂(in+1)]) = H∗ǫ ([in])

which proves (b).

To prove (c), we use that ∂∗ : Hn+1(∆n+1, ∆n+1)

≈−→ Hn(∆n) is an isomor-

phism, since ∆n+1 is contractible, and that ∂∗([in+1]) = [∂(in+1)].


For the sphere S1, Proposition 3.2.3 has the following corollary.

Corollary 3.2.4. Let σ : ∆1 → S1 given by σ(t, 1 − t) = e2iπt. Then σ ∈C1(S

1) is a singular 1-cycle of S1 and its homology class is the non-zero elementof H1(S

1) = Z2.

Proof. Since σ(1, 0) = σ(0, 1), the 1-cochain σ is a cycle. The map σ factorsas

∆1

p

##

σ // S1

∆1/∆1

s

≈

;;①①①①①①①

where s is a homeomorphism. Under the composite homomorphism

H1(∆1, ∆1)

H∗p

≈// H1(∆

1/∆1, [∆1])≈ // H1(∆

1)H∗s

≈// H1(S

1) ,

the class [i1] goes to [σ]. By Proposition 3.2.3, [i1] is a generator of H1(S1), which

proves Corollary 3.2.4.

Let f : Sn → Sn be a continuous map. The linear map Hnf : Hn(Sn) →

Hn(Sn) is a map between Z2 and itself. The degree deg(f) ∈ Z2 of f by

deg(f) =

0 if Hnf = 0

1 otherwise.

One can define the same degree using Hnf . For instance, the degree of a home-omorphism is 1 and the degree of a constant map is 0. Let f, g : Sn → Sn. ByProposition 3.1.30, deg(f) = deg(g) if f, g : Sn → Sn are homotopic. Also, usingthat Hn(gf) = HngHnf one gets

(3.2.1) deg(gf) = deg(g) · deg(f)These simple remarks have the following surprisingly strong consequences. (For arefinement of Proposition 3.2.5 below using the integral degree (see [155, Theo-rems 21.4 and 21.5].)

Proposition 3.2.5. Let f : Sn → Sn be a continuous map with deg f = 0.Then,

(a) f admits a fixed point.(b) there exists x ∈ Sn with f(x) = −x.Proof. Suppose that there is no fixed point. Then f is homotopic to the

antipodal map a(x) = −x: a homotopy is obtained by following the arc of greatcircle from f(x) to −x not containing x. Therefore deg f = deg a = 1 since a is ahomeomorphism. If f(x) 6= −x for all x, then deg f = 1 because f is homotopic tothe identity (following the arc of great circle from f(x) to x not containing −x).

We now give three recipes to compute the degree of a map from Sn to itself.A point u ∈ Sn is a topological regular value for f : Sn → Sn if there is a neigh-bourhood U of u such that U is “evenly covered” by f . By this, we mean thatf−1(U) is a disjoint union of Uj, indexed by a set J , such that, for each j ∈ J , therestriction of f to Uj is a homeomorphism from Uj to U . In particular, f−1(u) is adiscrete closed subset of Sn indexed by J , so J is finite since Sn is compact. For

3.2. SPHERES, DISKS, DEGREE 77

instance, a point u which is not in the range of f is a topological regular value of f(with J empty). For a topological regular value u of f , we define the local degreed(f, u) ∈ N of f at u by

d(f, u) = ♯f−1(u) .

Proposition 3.2.6. Let f : Sn → Sn be a continuous map. For any topologicalregular value u of f , one has

deg(f) = d(f, u) mod 2 .

Example 3.2.7. The map S1 → S1 given by z 7→ zk has degree the residueclass of k mod 2.

Proof of Proposition 3.2.6. When n = 0, each of the two points of S0

is a regular value of f and the equality of Proposition 3.2.6 is easy to check byexamination of the various cases. We then suppose that n > 0.

If u is a topological regular value, there is a neighbourhood B of u which isevenly covered by f and which is homeomorphic to a closed disk Dn. Its preimageB = f−1(B) is a finite disjoint union of n-disks Bj , indexed by j ∈ J . DefineJ = J ∪0 and set B0 = B.

For j ∈ J , define Vj = Sn −Bj and set V = Sn − B. Consider the quotient

spaces Snj = Sn/Vj (j ∈ J ), which are homeomorphic to Sn. Thus, Sn/V ≈∨j∈J S

nj is homeomorphic to a bouquet of ♯J copies of Sn. Denote the quotient

maps by j : Sn → Snj and : Sn → Sn/V ≈ ∨

j∈J Snj .

If uj ∈ Uj is the point such that f(uj) = u (u0 = u), then Sn − uj is aneighbourhood of Vj which retracts by deformation onto Vj . Therefore, (S

n, Vj) isa good pair and, as Sn−uj is homeomorphic to Rn, the space Vj is contractible.Also, Vj admits a mapping cylinder neighbourhood in Sn, so the pair (Sn, Vj) iswell cofibrant by Lemma 3.1.39. By the reduced homology sequence of the pairs(Sn, Vj) and (Snj , [Vj ]) and Proposition 3.1.45, we get three isomorphisms in thecommutative diagram

Hn(Sn)

≈

Hnj // Hn(Snj )

≈

Hn(Sn, Vj) ≈

// Hn(Snj , [Vj ]) ,

which shows that Hnj is an isomorphism.Let 0 6= α ∈ Hn(S

n), and, for j ∈ J , let 0 6= αj ∈ Hn(Snj ). The map f

descends to a continuous map f : Sn/V → Sn0 . Let us consider the commutativediagram:

(3.2.2)

Hn(Sn)

Hnf

Hn // ⊕j∈J Hn(S

nj )

Hn f

Hn(Sn)

Hn0

≈// Hn(S

n0 )

The restriction of f to Snj is a homeomorphism, soHnf(αj) = α0. Let πk :∨j∈J S

nj →

Snk be the projection onto the kth component (see Equation (3.1.32)). Then


j = πj . By Proposition 3.1.52, this implies that Hn(α) = (αj). Then,

Hnf Hn(α) = d(f, u)α0. On the other hand, Hn0Hnf(α) = deg(f)α0. AsDiagram (3.2.2) is commutative, this proves Proposition 3.2.6.

The second recipe is the following lemma.

Lemma 3.2.8. Let f : Sn → Sn be a continuous map, with n > 0. Let B1, . . . , Bkbe disjoint embedded closed n-disks of Sn with boundary Bi. Let V be the closureof Sn −⋃

Bi. Suppose that f sends V onto a single point v ∈ Sn and thus induces

continuous maps fi : Sn ≈ Bi/Bi → Sn. Then

deg f =

k∑

i=1

deg fi .

Proof. Let Sni = Bi/Bi, homeomorphic to Sn. The map f factors in thefollowing way:

Sn

p

&&

f // Sn

Sn/V ≈ ∨ki=1 S

n

∨fi

88♣♣♣♣♣♣♣♣♣♣

Obviously, H∗p([Sn]) =

∑ki=1[S

ni ]. Hence,

deg f [Sn] =k∑

i=1

H∗fi([Sni ]) =

( k∑

i=1

deg fi)[Sn] .

The third recipe concerns the self-maps of S1. Recall the elementary wayto prove that [S1, S1] ≈ Z. Let f : S1 → S1 be a (continuous) map. As t 7→exp(2iπt) is a local homeomorphism R → S1, there exists a map g : I → R suchthat f(exp(2iπt)) = exp(2iπg(t)). The integer

DEG (f) = g(1)− g(0) ∈ Z

depends only on the homotopy class of f . This defines a bijection

(3.2.3) DEG : [S1, S1]≈−→ Z .

For instance, for the map f(z) = zn, one can choose g(t) = nt. Thus, DEG (f) = nif and only if f is homotopic to z 7→ zn.

Proposition 3.2.9. For a map f : S1 → S1,

deg(f) = DEG(f) mod 2 .

Proof. If DEG (f) = n, then f is homotopic to z 7→ zn. This map satisfiesdeg(f) = n mod 2 by Proposition 3.2.6.

Remarks 3.2.10. (a) Our degree is the reduction mod 2 of the integral degreeobtained using integral homology (see e.g. [82, § 2.2]). Proposition 3.2.6 would alsohold for the integral degree, provided one takes into account the orientations in thedefinition of the local degree.

(b) A continuous map f : Sn → Sn may not have any topological regular value.For example, S. Ferry constructed a map f : S3 → S3 with (integral) degree 2 sothat the preimage of every point is connected [62].

3.3. CLASSICAL APPLICATIONS OF THE mod2 (CO)HOMOLOGY 79

(c) Suppose that f = |g|, where g : K → L is a simplicial map, with |K| and|L| homeomorphic to Sn. Let τ ∈ Sn(L) and u be a point in the interior of |τ |.Then u is a regular value and d(f, u) = d(g, τ) (see Equation (2.5.5)).

(d) Let f : Sn → Sn be a smooth map. Any smooth regular value is a topolog-ical regular value. Then, deg(f) coincides with the degree mod 2 of f as presentedin e.g. [152, Section 4].

3.3. Classical applications of the mod 2 (co)homology

At our stage of development of homology, textbooks usually present a couple ofclassical applications in topology. Several of them only require Z2-homology, otherneed the integral homology. We discuss this matter in this section.

Retractions and Brouwer’s fixed point Theorem. Let X be a topologicalspace with a subspace Y . A continuous retraction of X onto Y is a continuous mapr : X → Y extending the identity of Y . In other words, r i = idY , where i : Y → Xdenotes the inclusion. Therefore, H∗rH∗i = id and H∗iH∗r = id, which impliesthe following lemma.

Lemma 3.3.1. If there exists a continuous retraction from X onto Y , thenH∗i : H∗(Y ) → H∗(X) is injective and H∗i : H

∗(X) → H∗(Y ) is surjective. Thesame holds true for the reduced (co)homology.

As H∗(Dn) = 0 while Hn−1(S

n−1) = Z2, Lemma 3.3.1 has the following corol-lary.

Proposition 3.3.2. There is no continuous retraction of the n-disk Dn ontoits boundary Sn−1.

The most well known corollary of Proposition 3.3.2 is the fixed point theoremproved by Luitzen Egbertus Jan Brouwer around 1911 (see [40, Chapter 3]).

Corollary 3.3.3. A continuous map from the disk Dn to itself has at leastone fixed point.

Proof. Suppose that f(x) 6= x for all x ∈ Dn. Then. a retraction r : Dn →Sn−1 is constructed using the following picture, contradicting Proposition 3.3.2.

f(x)

r(x)

x

Brouwer’s theorem says that, for a map f : Dn → Dn, the equation f(x) = xadmits a solution under the only hypothesis that f is continuous. Given the possiblewildness of a continuous map, this is a very deep theorem. It is impressive thatsuch a result is due to the fact that Hn(D

n) = 0 and Hn(Sn−1) = Z2.


Invariance of dimension. An n-dimensional topological manifold is a topo-logical space such that each point has an open neighbourhood homeomorphic toRn. The following result is known as the topological invariance of the dimensionand goes back to the work of Brouwer in 1911 (see [40, Chapter II]).

Theorem 3.3.4. Suppose that a non-empty m-dimensional topological manifoldis homeomorphic to an n-dimensional topological manifold. Then m = n.

Proof. Let M be a non-empty m-dimensional topological manifold and N bean n-dimensional topological manifold. Let h : M → N be a homeomorphism. Letx ∈ M . Then h restricts to a homeomorphism from M − x onto N − h(x).Hence, H∗h : H∗(M,M − x)→ H∗(N,N − h(x)) is an isomorphism. But thiscontradicts the fact that(3.3.1)

Hk(M,M −x) =Z2 if k = 0,m

0 otherwiseand Hk(N,N−h(x)) =

Z2 if k = 0, n

0 otherwise.

Indeed, it enough to prove (3.3.1) in the case of M . As x has a neighbourhoodin M homeomorphic to Rm, it has a neighbourhood B homeomorphic to a closedm-ball. By the excision of M −B and homotopy, one has

H∗(M,M − x) ≈ H∗(B,B − x) ≈ H∗(B,BdB)

and (3.3.1) follows from Corollary 3.2.2.

Balls and spheres in spheres. The following results concerns the comple-ments of k-balls or k-spheres in Sn.

Proposition 3.3.5. Let h : Dk → Sn be an embedding. ThenH∗(S

n − h(Dk)) = 0.

Proof. We follow the classical proof (see e.g. [82, Proposition 2b.1]), whichgoes by induction on k. For k = 0, D0 is a point and Sn−h(D0) is then contractible.

For the induction step, suppose that Hi(Sn − h(Dk)) contains a non-zero element

α0. We use the homeomorphism Dk ≈ Dk−1 × I0 with I0 = [0, 1]. Then Sn −h(Dk) = A ∪B with A = Sn − (Dk−1 × [0, 1/2]) and B = Sn − (Dk−1 × [1/2, 1]).

Since, by induction hypothesis, H∗(A ∩ B) = H∗(Sn − h(Dk−1 × 1/2) = 0,

the Mayer-Vietoris sequence implies that for I1 = [0, 1/2] or I1 = [1/2, 1], thehomomorphism Hi(S

n−h(Dk× I0))→ Hi(Sn−h(Dk× I1))) sends α0 to 0 6= α1 ∈

Hi(Sn− (Dk−1× I1). Iterating this process produces a nested sequence Ij of closed

intervals converging to a point p ∈ I and a non-zero element αj ∈ lim→H∗(Xj)where Xj = Sn − (Dk−1 × Ij). Set X = Sn − (Dk−1 × p). As each compact

subspace of X is contained in some Xj , Corollary 3.1.16 implies that lim→H∗(Xj) is

isomorphic to H∗(X), contradicting the induction hypothesis.

Proposition 3.3.6. Let h : Sk → Sn be an embedding with k < n. ThenH∗(S

n − h(Sk)) ≈ H∗(Sn−k−1).Proof. The proof is by induction on k. The sphere S0 consisting of two points,

Sn − h(S0) is homotopy equivalent to Sn−1. For the induction step, write Sk asthe union of two hemispheres D±. The Mayer-Vietoris sequence for Sn − h(D±)together with Proposition 3.3.5 gives the isomorphism H∗(S

n−h(Sk)) ≈ H∗−1(Sn−h(Sk−1)).

3.4. CW-COMPLEXES 81

The case k = n− 1 in Proposition 3.3.6 gives the following corollary.

Corollary 3.3.7 (Generalized Jordan Theorem). Let h : Sn−1 → Sn be anembedding. Then Sn − h(Sn−1) has two path-connected components.

Remarks 3.3.8. (a) Topological arguments show that, in Corollary 3.3.7, h(Sn−1)is the common frontier of each of the components of its complement (see e.g. [155,Theorem 6.3]). For a discussion about the possible homotopy types of these com-ponents, see e.g. [155, § 36] or [82, § 2B].

(b) A well known consequence of the generalized Jordan theorem is the invari-ance of domain: if U is an open set in Rn, then its image h(U) under an embeddingh : U → Rn is an open set in Rn. This can be deduced from Corollary 3.3.7 by apurely topological argument (see, e.g. [155, Theorem 36.5] or [82, Theorem 2B.3]).

Unavailable applications. Some applications cannot be obtained using Z2-homology. The most well known are the following.

(a) The antipodal map in S2n is not homotopic to the identity and its con-sequence, the non-existence of non-zero vector fields on even-dimensionalspheres (see, e.g. [82, Theorem 2.28] or [155, Corollary 21.6]). This re-quires the (co)homology with coefficients in Z or in a field of characteristic6= 2.

(b) The determination of [S1, S1] and the fundamental theorem of algebra(using H1(S1;Z)) (see, e.g., [155, Exercise 2, § 21] or [82, Theorem 1.8]).

3.4. CW-complexes

CW-complexes were introduced and developed by J.H.C. Whitehead in theyears 1940-50 [40, p. 221]. The spaces having the homotopy type of a CW-complex(CW-space) are closed under several natural construction (see [147]). They are thespaces for which many functors of algebraic topology, like singular (co)homology,are reasonably efficient.

Let Y be a topological space and let (Z,A) be a topological pair. Let ϕ : A→ Ybe a continuous map. Consider the space

Z ∪ϕ Y = Z ∪ Y/z = ϕ(z) | z ∈ A ,

endowed with the quotient topology. The space Y is naturally embedded intoZ∪ϕY . We say that Z∪ϕY is obtained from Y by attachment (or adjunction) of Z,using the attaching map ϕ. When (Z,A) is homeomorphic to (Λ×Dn,Λ× Sn−1),where Λ is a set (considered as a discrete space), we say that Z is obtained from Yby attachment of n-cells, indexed by Λ. For λ ∈ Λ, the image of λ× intDn in Xis the open cell indexed by λ.

A CW-structure on the space X is a filtration

(3.4.1) ∅ = X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ X =⋃

n∈N

Xn ,

such that, for each n, the space Xn is homeomorphic to a space obtained fromXn−1 by attachment of n-cells, indexed by a set Λn = Λn(X). A space endowedwith a CW-structure is a CW-complex. We see Λn as the set of n-cells of X . Thespace Xn is called the n-skeleton of X . The topology of X is supposed to be theweak topology: a subspace A ⊂ X is open (or closed) if and only if A ∩Xn is open(or closed) for all k ∈ N.


IfX is a CW-complex, a subspace Y ⊂ X is a subcomplex ofX if Y n = Y ∩Xn isobtained from Y n−1 = Y ∩Xn−1 by attaching n-cells, indexed by Λn(Y ) ⊂ Λn(X)and using the same attaching maps. For instance, the skeleta ofX are subcomplexesof X . A topological pair (X,Y ) formed by a CW-complex X and a subcomplex Yis called a CW-pair.

Let X be a CW-complex. With the above definition, the following propertieshold true:

(1) X is a Hausdorff space.(2) for each n and each λ ∈ Λk, there exists a continuous map ϕλ : (D

n, Sn−1)→(Xn, Xn−1) ⊂ (X,Xn−1) such that its restriction to intDn is an embed-ding from intDn into X . Indeed, such a map, called a characteristic mapfor the n-cell λ, may be obtained by choosing a homeomorphism between(Xn, Xn−1) and (

[Λ×Dn

]∪ϕ Xn−1, Xn−1).

(3) a map f : X → Z to the topological space Z is continuous if and only ifits restriction to each skeleton is continuous. Also, f is continuous if andonly if f ϕλ is continuous for any characteristic map ϕλ and any cell λ.

(4) each subcomplex of X is a closed subset of X .(5) X0 is a discrete space.(6) A compact subset of a CW-complex meets only finitely many cells. In

consequence, a CW-complex is compact if and only if it is finite, i.e. itcontains a finite number of cells.

These properties are easy to prove (see, e.g. [82, pp. 519–523]).

Proposition 3.4.1. A CW-pair (X,A) is well cofibrant.

The literature contains many proofs that a CW-pair is good (see e.g. [82,Prop. A.5] or [64, Prop. 1.3.1]), or cofibrant (see e.g. [73, Prop. 14.13] or [38,Prop. 8.3.9]). The proof of Proposition 3.4.1 uses the following lemma.

Lemma 3.4.2. Let Z be a space obtained from a space Y by attaching a col-lection of n-cells. Then Y admits in Z a mapping cylinder neighbourhood (seeExample 3.1.38).

Proof. Let ϕ : Λ×Dn → Y be the attaching map. Let Cn = x ∈ Dn | |x| ≥1/2. Then, Z contains V = (Λ × Cn) ∪ϕ Y as a closed neighbourhood of Y . Thereader will check that V is homeomorphic to the mapping cylinder of ϕ.

Proof of Proposition 3.4.1. Let Xn = Xn∪A. By Lemmas 3.4.2 and 3.1.39,the pair (Xn, Xn−1) is well cofibrant for all n. Let (vn, gn) be a presentation of(Xn, Xn−1) as a well cofibrant pair. As X0 is the disjoint union of A with a discreteset, we may assume that v0(x) = 1 if x /∈ A. LetWn be the closure of (vn)−1([0, 1)).For n ≥ 1, by replacing vn by min2vn, 1 if necessary, we may assume that gn

restricts to a strong deformation retraction of Wn onto Xn−1.We now define a map u : X → I by constructing, inductively on n ∈ N, its

restriction to Xn, denoted by un. We set u0 = v0 and

un(x) =

min1, vn(x) + un−1(gn(x, 1)) if x ∈ Wn

1 if x ∈ Xn − intWn .

We check that un is continuous. If x ∈ Xn, then uk(x) = un(x) for k ≥ n, thereforeu is well defined and continuous. The space V = u−1(([0, 1)) =

⋃n V

n is a closedneighbourhood of A in X , where V n = V ∩ Xn ⊂Wn.


Define hn : V n × I → V n by

hn(x, t) =

x if t ≤ 1/2n+1

gn(x, 2n+1t− 1) if 1/2n+1 ≤ t ≤ 1/2n

gn(x, 1) if t ≥ 1/2n .

Define hnt : Vn → V n by hnt (x) = h(x, t). If x ∈ V n, define h(x, t) = h1t · · · hnt (x).

Note that hkt (x) = x for k > n so, if x ∈ V n ⊂ V m, then h1t · · · hnt (x) =h1t · · · hmt (x). Therefore, h : V × I → V is well defined and continuous.

The pair (u, h) satisfies all the conditions for being a presentation of (X,A)as a well cofibrant pair, except that h is only defined on V × I instead of X × I.To fix that, choose a continuous map α : I → I such that α([0, 1/2]) = 1 andα vanish on a neighbourhood of 1. Let h : X × I → X and u : X → I defined byh(x, t) = h

(x, α(u(x))t

)and u(x) = u(x). One checks that (u, h) is a presentation

of (X,A) as a well cofibrant pair.

Here are classical examples of CW-complexes.

Example 3.4.3. The sphere Sn has an obvious CW-structure with one 0-celland one n-cell (attached trivially).

Example 3.4.4. Observe that the sphere

Sn = x = (x0, x1, . . . , xn) ∈ Rn+1 | |x|2 = 1

is obtained from Sn−1 by adjunction of two (n+1)-cellsDn+1± , attached by the iden-

tity map of Sn−1. Indeed, the embeddings Dn± → Sn given by y = (y1, . . . , yn) 7→

(±√1− |y|2, y1, . . . , yn) extend the inclusion Sn−1 → Sn and provide a homeo-

morphism between Sn−1 ∪ Dn± and Sn. Starting from S0 = ±1, we thus get a

CW-structure on Sn with two cells in each dimension and whose k-skeleton is Sk.Taking the inductive limit S∞ of those Sn gives a CW-complex known as the in-finite dimensional sphere. This is a contractible space (see e.g. [82, example 1.B.3p. 88]).

Example 3.4.5. The CW-structure on Sn of Example 3.4.4 is invariant underthe antipodal map. It then descends to a CW-structure on the projective spaceRPn = Sn/x ∼ −x, having one cell in each dimension. Its k-th skeleton is RP k

and the (k + 1)-cell is attached to RP k by the projection map Sk → RP k. Thisis called the standard CW-structure on RPn. Taking the inductive limit RP∞ ofthese CW-complexes gives a CW-complex known as the infinite dimensional (real)projective space. Analogous CW-decompositions for complex and quaternionic pro-jective spaces are given in § 6.1.

Example 3.4.6. If X and Y are CW-complexes, a CW-structure on X×Y may

be defined, with (X × Y )n =⋃p+q=nX

p ×Xq and Λn(X × Y ) =⋃p+q=nΛp(X)×

Λq(Y ) (see [64, Theorem 2.2.2]). The weak topology may have more open setsthan the product topology so the identity i : (X × Y )CW → (X × Y )prod is onlya continuous bijection. If X or Y is finite, or if both are countable, then i isa homeomorphism (see [64, p. 60]). These consideration are not important for ussince the two topologies have the same compact sets. Therefore, they have the samesingular simplexes, whence i induces an isomorphism on singular (co)homology.


We now establish a few lemmas useful for the cellular (co)homology. Let X bea CW-complex. Fix an integer n and choose, for each λ ∈ Λn, a characteristic mapϕλ : (D

n, Sn−1)→ (Xn, Xn−1). These maps produce a global characteristic map

ϕn : (Λn ×Dn,Λn × Sn−1)→ (Xn, Xn−1) .

Lemma 3.4.7. Let X be a CW-complex and let n ∈ N. Let ϕn be a globalcharacteristic map for the n-cells. Then

(i) H∗ϕn : H∗(Λn ×Dn,Λn × Sn−1) ≈−→ Hk(X

n, Xn−1) is an isomorphism.

(ii) H∗ϕn : H∗(Xn, Xn−1)≈−→ Hk(Λn ×Dn,Λn × Sn−1) is an isomorphism.

Proof. By Kronecker duality, using Corollary 2.3.11, only statement (i) mustbe proved. The proof for n = 0 is easy and left to the reader, so we assume thatn ≥ 1.

By Lemma 3.4.2, (Xn, Xn−1) is a well cofibrant pair. As n ≥ 1, the spaceXn−1 is not empty (unless X = ∅, a trivial case). The continuous map

ϕ : Λn ×Dn/Λn × Sn−1 → Xn/Xn−1

induced by ϕn is a homeomorphism, both spaces being homeomorphic to a bouquetof copies of Sn indexed by Λn. In the commutative diagram

H∗(Λn ×Dn,Λn × Sn−1)

H∗ϕn

≈ // H∗(Λn ×Dn/Λn × Sn−1)

H∗ϕ≈

H∗(Xn, Xn−1)

≈ // H∗(Xn/Xn−1)

,

the horizontal maps are isomorphisms by Corollary 3.1.47. Therefore, H∗ϕn is an

isomorphism.

Corollary 3.4.8. Let X be a CW-complex and let n ∈ N. Then

(i)

Hk(Xn, Xn−1) ≈

⊕

Λn

Z2 if k = n.

0 if k 6= n.

(ii)

Hk(Xn, Xn−1) ≈

∏

Λn

Z2 if k = n.

0 if k 6= n.

Proof. Again, the easy case n = 0 is left to the reader. If n > 0, we use that,as noticed in the proof of Lemma 3.4.7, the map

Xn/Xn−1 →

∨

Λn

Sn

is a homeomorphism. Corollary 3.4.8 then follows from Proposition 3.1.51.

Lemma 3.4.9. Let X be a CW-complex and let n ∈ N. Then

(i) the homomorphism Hk(Xn)→ Hk(X) induced by the inclusion is an iso-

morphism for k < n and is surjective for k = n.


(ii) the homomorphism Hk(X) → Hk(Xn) induced by the inclusion is anisomorphism for k < n and is injective for k = n.

Proof. By Kronecker duality, using Corollary 2.3.11, only statement (i) mustbe proved. The homomorphisms induced by inclusions form a sequence

(3.4.2) Hk(Xk)→→ Hk(X

k+1)≈−→ Hk(X

k+2)≈−→ · · · → Hk(X) .

The bijectivity or surjectivity of Hk(Xr)→ Hk(X

r+1) is deduced from the homol-ogy sequence of the pair (Xr+1, Xr) and Corollary 3.4.8. By Proposition 3.1.29,H∗(X) is the direct limit of H∗(K), for all compact sets K of X . Using that eachcompact set of X is contained in some skeleton, one checks that H∗(X) is the directlimit of H∗(X

k). By (3.4.2), this proves (i).

Lemma 3.4.10. Let X be a CW-complex and let n ∈ N. Then Hk(Xn) =Hk(X

n) = 0 if k > n.

Proof. The proof is by induction on n. The lemma is true for n = 0 since X0

is a discrete set. The induction step uses the exact sequence of the pair (Xn, Xn−1)together with Corollary 3.4.8.

Let (X,Y ) be a CW-pair. LetM = (r, s) ∈ N× N | r ≥ s endowed with thelexicographic order. The pairs (Xr, Y s) ((r, s) ∈ M), together with the inclusion

(Xr, Y s) → (Xr′ , Y s′

) when (r, s) ≤ (r′, s′), forms a direct system. The inclusionsjr,s : (X

r, Y s) → (X,Y ) induce a GrV-morphism

J∗ : lim−→

(r,s)∈M

H∗(Xr, Y s) −→ H∗(X,Y )

and a GrA-morphism

J∗ : H∗(X,Y ) −→ lim←−

(r,s)∈M

H∗(Xr, Y s) .

To get a more general result, which will be useful, we can take the product with anarbitrary topological space Z.

Proposition 3.4.11. Let (X,Y ) be a CW-pair and M be as above. Let Z bea topological space. Then, the GrV-morphism

J∗ : lim−→

(r,s)∈M

H∗(Xr × Z, Y s × Z) ≈−→ H∗(X × Z, Y × Z)

and the GrA-morphism

J∗ : H∗(X × Z, Y × Z) ≈−→ lim←−

(r,s)∈M

H∗(Xr × Z, Y s × Z) .

are isomorphisms.

Proof. By Kronecker duality, only the homology statement needs a proof.Let K be a compact subspace of X × Z. By Property (6) of p. 82, K is containedin Xr×Z for some integer r. Hence, if Y is empty, Proposition 3.4.11 follows fromCorollary 3.1.16. When Y 6= ∅, we use the long exact sequences in homology andthe five lemma, as in the proof of Proposition 3.1.29.


3.5. Cellular (co)homology

Let X be a CW-complex. For m ∈ N, the m-cellular (co)chain vector spaces

Cm(X) and Cm(X) are defined as

Cm(X) = Hm(Xm, Xm−1) and Cm(X) = Hm(Xm, Xm−1) .

The cellular boundary operator ∂ : Cm(X)→ Cm−1(X) is defined by the compositehomomorphism

∂ : Hm(Xm, Xm−1)∂−→ Hm−1(X

m−1)→ Hm−1(Xm−1, Xm−2) .

The expression for ∂ ∂ contains the sequenceHm−1(Xm−1)→ Hm−1(X

m−1, Xm−2)∂−→

Hm−2(Xm−2) and then ∂ ∂ = 0

The cellular co-boundary operator δ : Cm(X) → Cm+1(X) is defined by thecomposite homomorphism

δ : Hm(Xm, Xm−1)→ Hm(Xm)δ−→ Hm+1(Xm+1, Xm) .

with again δ δ = 0. (Co)cycles Zm, Zm and (co)boundaries Bm, Bm are definedas usual, which leads to the definition

Hm(X) = Zm(X)/Bm(X) and Hm(X) = Zm(X)

/Bm(X) .

The graded Z2-vector space H∗(X) is the cellular homology of the CW-complex X

and the graded Z2-vector space H∗(X) is its cellular cohomology. The Kroneckerpairing

Hm(Xm, Xm−1)×Hm(Xm, Xm−1)〈 ,〉−−→ Z2

gives a pairing

Cm(X)× Cm(X)〈 ,〉−−→ Z2

which makes ((C∗(X), δ), (C∗(X), ∂), 〈 , 〉) a Kronecker pair.In the language of former sections, the cellular (co)chains admit the usual

equivalent definitions:


(a) A cellular m-cochain is a subset of is a subset of Λm.(b) A cellular m-chain is a finite subset of Λm.


(a) A cellular m-cochain is a function a : Λm → Z2.(b) A cellular m-chain is a function α : Λm → Z2 with finite support.

Definition 3.5.2.b is equivalent to

Definition 3.5.3. Cm(X) is the Z2-vector space with basis Λm:

Cm(X) =⊕

λ∈Λm(X)

Z2 λ .

The Kronecker pairing on (co)chains admits the usual equivalent formula

(3.5.1)〈a, α〉 = ♯(a ∩ α) (mod 2)

=∑

σ∈α a(σ) .

3.5. CELLULAR (CO)HOMOLOGY 87

We now give a formula for the cellular boundary operator ∂ : Cm(X)→ Cm−1(X).

By Definition 3.5.3, it is enough to define ∂(λ) for λ ∈ Λm. Choose an attaching

map ϕλ : Sm−1 → Xm−1 for the m-cell λ. When m = 1, the formula for ∂(λ) is

easy:

(3.5.2) ∂(λ) =

0 if ♯ϕλ(S

0) = 1

ϕλ(S0) otherwise (using the subset definition) .

Let us now suppose that m > 1. For µ ∈ Λm−1, define ϕλ,µ : Sm−1 → Sm−1 as the

composite map:

Sm−1ϕλ−−→ Xm−1 →→ Xm−1/Xm−2 ≈

∨

Λm−1

Sm−1πµ−−→ Sm−1 ,

where πµ is the projection onto the µ-th component. Using the colouring definition

of cellular chains, we must give, for each µ ∈ Λm−1, the value ∂(λ)(µ) ∈ Z2.

Lemma 3.5.4. For m > 1, the cellular boundary operator ∂ : Cm(X)→ Cm−1(X)is the unique linear map satisfying

(3.5.3) ∂(λ)(µ) = deg(ϕλ,µ) .

for each λ ∈ Λm.

Proof. The attaching map ϕλ : Sm−1 → Xm−1 extends to a characteristic

map ϕλ : Dm → Xm. Consider the commutative diagram:

Hm(Dm, Sm−1)

∂≈

H∗ϕλ// Hm(Xm, Xm−1)

∂

∂

))

Hm−1(Sm−1)

H∗ϕλ // Hm−1(Xm−1) // Hm−1(X

m−1/Xm−2)

H∗πµ

Hm−1(S

m−1)

Let α be the generator of Hm(Dm, Sm−1) = Z2 and let β be that of Hm−1(Sm−1).

Using Lemma 3.4.7 and its proof, one sees that

(a) λ ∈ Cm(X) corresponds to H∗ϕλ(α) ∈ Hm(Xm, Xm−1).(b) if γ ∈ Hm−1(X

m−1/Xm−2), then H∗πµ(γ) = γ(µ)·β (we use the colouringdefinition and see γ as a function from Λm−1 to Z2).

As ∂(α) = β, one has

∂(λ)(µ) · β = H∗πµ ∂H∗ϕλ(α)

= H∗πµH∗ϕλ(β)

= deg(ϕλ,µ) · β ,which proves the lemma.

Formulae (3.5.2) and (3.5.3) for the cellular boundary operator take a specialform when X is a regular CW-complex, i.e. when each cell λ admits a characteristicmap ϕλ which is an embedding onto a subcomplex of X . A cell of this subcomplexis called a face of λ.


Lemma 3.5.5. Let X be a regular CW-complex. Let λ ∈ Λm and µ ∈ Λm−1.Then

∂(λ)(µ) =

1 if µ is a face of λ

0 otherwise.

Proof. When m = 1, this follows from (3.5.2), where the case ♯ϕλ(S0) = 1

does not happen since X is regular. When m > 1, we use Lemma 3.5.4 andcompute the degree of ϕλ,µ using Proposition 3.2.6: since ϕλ is an embedding, anytopological regular value of ϕλ,µ has exactly one element in its preimage.

We now prove the main result of this section.

Theorem 3.5.6. Let X be a CW-complex. Then, the cellular and the singular(co)homology of X are isomorphic:

H∗(X) ≈ H∗(X) and H∗(X) ≈ H∗(X) .

Proof. We consider the commutative diagram:

(3.5.4)

Hm+1(Xm+1, Xm)

∂m+1

∂m+1

))

0 // Hm(Xm+1)

≈

Hm(Xm)uu

jm

uu

// //

77 77♥♥♥♥♥♥♥♥♥♥

0

Hm(X)

Hm(Xm, Xm−1)

∂m

∂m

((

Hm−1(Xm−1)

vvjm−1

vv

Hm(Xm−1, Xm−2)

The properties of arrows (surjective, injective, bijective) come from Lemma 3.4.9,3.4.10 and Corollary 3.4.8. From Diagram (3.5.4), we get

Hm(X)≈←− Hm(Xm+1) ≈ Hm(Xm)/Im ∂m+1 ≈

jm // ker ∂m/Im ∂m+1 = Hm(X)

As the isomorphism H∗(X) ≈ H∗(X) does not come from a morphism of chain complex,we cannot invoke Kronecker duality to deduce the isomorphism in cohomology. Instead,

3.5. CELLULAR (CO)HOMOLOGY 89

we consider the Kronecker dual of Diagram (3.5.4)

(3.5.5)

Hm+1(Xm+1, Xm)

Hm(Xm)

δm+1

iiHm(X)

≈

oooo

Hm(Xm, Xm−1)

Jm

55 55

δm+1

OO

Hm(Xm+1)

gg

ggPPPPPPPPPP

Hm−1(Xm−1)

δm

ii

0

OO

Hm(Xm−1, Xm−2)

Jm−166 66

δm

OO

which gives

Hm(X)≈−→ Hm(Xm+1)

≈−→ ker δm+1 ≈ J−1

m (ker δm+1)/Im δm = ker δm+1/Im δm = Hm(X) .

Here are some applications of the isomorphism between cellular and singular(co)homology.

Corollary 3.5.7. Let X be a CW-complex with no m-dimensional cell. ThenHm(X) = Hm(X) = 0.

Proof. If Λm(X) = ∅, then Cm(X) = Cm(X) = 0, which implies Hm(X) =

Hm(X) = 0, and then Hm(X) = Hm(X) = 0 by Theorem 3.5.6.

Corollary 3.5.8. Let X be a CW-complex with k cells of dimension m. ThendimHm(X) = dimHm(X) ≤ k.

Proof. One has dim Hm(X) ≤ dim Zm(X) ≤ dim Cm(X) = k. Therefore,dimHm(X) ≤ k by Theorem 3.5.6. The result on cohomology is proven in thesame way or deduced by Kronecker duality.

A CW-complex is finite if it has a finite number of cells.

Corollary 3.5.9. Let X be a compact CW-complex. Then

dimH∗(X) = dimH∗(X) <∞ .

Proof. By the weak topology, a compact CW-complex is finite (see Remark (6)

p. 82). Hence, C∗(X) is a finite dimensional vector space, and so is Z∗(X) and

H∗(X). Proposition 3.5.9 then follows from Theorem 3.5.6 and Kronecker dual-ity.

Let X be a finite CW-complex. Its Euler characteristic χ(X) is defined as

χ(X) =∑

m∈N

(−1)m ♯Λm(X) ∈ Z .


Proposition 3.5.10. Let X be a finite CW-complex. Then

χ(X) =∑

m∈N

(−1)m dimHm(X) =∑

m

(−1)m dimHm(X) .

Proof. If we use the cellular (co)homology, the proof of Proposition 3.5.10 isthe same as that of Proposition 2.4.8. The result then follows from Theorem 3.5.6.

A CW-complex X (or a CW-structure on X) is called perfect if the cellularboundary vanishes. For instance, if X does not have cells in consecutive dimensions,then it is perfect. Also, the standard CW-structure on RPn (n ≤ ∞) is perfect (see

e.g. Proposition 6.1.1). If X is a perfect CW-complex, C∗(X) = H∗(X) and theidentification between the singular and cellular homologies, out of Diagram (3.5.4),is particularly simple:

(3.5.6) Hm(X)≈←− Hm(Xm)

≈−→ Hm(Xm, Xm−1)≈−→ Hm(X)

≈←− Cm(X) .

The natural functoriality of cellular (co)homology is for cellular maps. If Xand Y are CW-complexes, a continuous map f : Y → X is cellular if f(Y m) ⊂ Xm

for all m ∈ N. We thus get GrV-morphisms C∗f and C∗f making the followingdiagrams commute

Cm(Y )

=

C∗f // Cm(X)

=

Hm(Y m, Y m−1)H∗f // Hm(Xm, Xm−1)

Cm(Y ) Cm(X)C∗foo

Hm(Y m, Y m−1)

=

OO

Hm(Xm, Xm−1)C∗foo

=

OO

They satisfy 〈C∗f(a), α〉 = 〈a, C∗f(α)〉 for all a ∈ C∗(X) and α ∈ C∗(X). It isuseful to have a formula for C∗f , using that

Cm(Y ) =⊕

λ∈Λm(Y )

Z2 λ and Cm(X) =⊕

µ∈Λm(X)

Z2 µ .

For λ ∈ Λm(Y ) and µ ∈ Λm(X), consider the map fλ,µ : Sm → Sm defined by the

composition

fλ,µ : Smjλ−→

∨

λ∈Λm(Y )

Sm ≈ Y m/Y m−1 f−→ Xm/Xm−1 ≈∨

µ∈Λm(X)

Smπµ−−→ Sm ,

where jλ is the inclusion of the λ-component and πµ the projection onto the µ-component.

Lemma 3.5.11. For m ≥ 1, C∗f : Cm(Y ) → Cm(X) is the unique linear mapsuch that

C∗f(λ) =∑

µ∈Λm(X)

deg(fλ,µ)µ .

3.6. ISOMORPHISMS BETWEEN SIMPLICIAL AND SINGULAR (CO)HOMOLOGY 91

Proof. The map f : (Y m, Y m−1)→ (Xm, Xm−1) induces a map f :∨λ∈Λm(Y ) S

m →∨µ∈Λm(X) S

m making the following diagram commute

Cm(Y )

C∗f

≈ // Hm(Y m, Y m−1)

H∗f

≈ // Hm(∨λ∈Λm(Y ) S

m)

H∗f

Cm(X)

≈ // Hm(Y m, Y m−1)≈ // Hm(

∨µ∈Λm(X) S

m)

As in the proof of Lemma 3.5.4, one checks that, under the top horizontal iso-morphisms, λ ∈ Cm(Y ) corresponds toH∗jλ([S

m−1]). Also, if γ ∈ Hm−1(Xm−1/Xm−2),

then H∗πµ(γ) = γ(µ) · [Sm−1] (seeing γ as a function from Λm(X) to Z2 by thecolouring definition). Hence,

C∗f(λ)(µ) = H∗πµ H∗f H∗jλ([Sm−1]) = H∗fλ,µ([S

m−1]) = deg(fλ,µ) [Sm−1] ,

which proves the lemma.

3.5.12. Homology-cell complexes. The results of this section and the previousone are also valid for complexes where cells are replaced by homology cells. A wellcofibrant pair (B, B) is a homology n-cell if H∗(B) = 0 and H∗(B) ≈ H∗(S

n−1).This GrV-isomorphism is “abstract”, i.e. not assumed to be given by any con-tinuous map. It follows that H∗(B, B) ≈ H∗(D

n, Sn−1). We also say that B is a

homology n-cell with boundary B. If λ ∈ Λ is indexing a family of homology n-cell(B(λ), B(λ)) and if ϕλ : B(λ) → Y is a family of continuous maps, we say thatthe quotient space

X = Y ∪ϕ(∪λ∈ΛB(λ)

)

is obtained from Y by attaching homology n-cells (they may be different for variousλ’s). We identify Λ with the set of homology n-cells.

A homology-cell complex is defined as in p. 81 with attachments of n-cells re-placed by attachments of a set Λn(X) of homology n-cells. The cellular (co)homology

H∗(X) and H∗(X) are defined accordingly and Theorem 3.5.6 holds true, with thesame proof. Homology-cell structures are used in the proof of Poincare duality (see§ 5.2).

3.6. Isomorphisms between simplicial and singular (co)homology

LetK be a simplicial complex. In this section, we prove three theorems showingthat the simplicial (co)homology of K and the singular (co)homology of |K| areisomorphic.

Theorem 3.6.1. Let K be a simplicial complex. Then

H∗(K) ≈ H∗(|K|) and H∗(K) ≈ H∗(|K|)

Proof. The geometric realization |K| of K is naturally endowed with a struc-ture of a regular CW-complex, with |K|m = |Km|, Λm(|K|) = Sm(K), with acanonical characteristic map for the m-cell σ ∈ Sm(K) given by the inclusion of |σ|


into |K|. Thus, Cm(|K|) = Cm(K) and, using Lemma 3.5.5, the diagram

Cm(|K|)

∂

= // Cm(K)

∂

Cm−1(|K|) = // Cm−1(K)

is commutative. Therefore, H∗(|K|) = H∗(K) and, by Theorem 3.5.6, the singularhomologyH∗(|K|) and the simplicial homologyH∗(K) are isomorphic. The equality

H∗(|K|) = H∗(K) is deduced from H∗(|K|) = H∗(K) by Kronecker duality and,using by Theorem 3.5.6 again, the singular cohomology H∗(|K|) and the simplicialcohomology H∗(K) are also isomorphic.

We now go to the second isomorphism theorem, which uses the ordered sim-plicial (co)homology of § 2.10. To an ordered m-simplex (v0, . . . , vm) ∈ Sm(K), weassociate the singular m-simplex R(v0, . . . , vm) : ∆m → |K| defined by

(3.6.1) R(v0, . . . , vm)(t0, . . . , tm) =

m∑

i=0

tivi .

The linear combination in (3.6.1) makes sense since v0, . . . , vm is a simplex of K.

This defines a map R : Sm(K)→ Sm(|K|) which extends to a linear map

R∗ : C∗(K)→ C∗(|K|) .

The formula ∂R = R ∂ is obvious, so R is a morphism of chain complexes

from (C∗(K), ∂) to (C∗(|K|), ∂). Define the linear map R∗ : C∗(|K|) → C∗(K)by 〈R∗(a), α〉 = 〈a,R∗(α)〉. By Lemma 2.3.6, (R∗, R∗) is a morphism of Kroneckerpair. We also denote by R∗ and R∗ the induced linear maps on (co)homology:

R∗ : H∗(K)→ H∗(|K|) and R∗ : H∗(|K|)→ H∗(K) .

If f : L→ K be a simplicial map, the formulae

R∗C∗f = C∗|f |R∗ and C∗f R∗ = R∗C∗|f |are easy to check. They induce the formulae

(3.6.2) R∗H∗f = H∗|f |R∗ and H∗f R∗ = H∗C∗|f |on (co)homology. In particular, if f is the inclusion of a subcomplex L of K, theabove considerations permit us to construct degree zero linear maps

R∗ : H∗(K,L)→ H∗(|K|, |L|) and R∗ : H∗(|K|, |L|)→ H∗(K,L)

so that (R∗, R∗) is a morphism of Kronecker pair. Finally, if f : (K,L)→ (K ′, L′)is a simplicial map of simplicial pairs, then Formulae (3.6.2) hold true in relative(co)homology.

Theorem 3.6.2. Let (K,L) be a simplicial pair. Then the linear maps

R∗ : H∗(K,L)≈−→ H∗(|K|, |L|) and R∗ : H∗(|K|, |L|) ≈−→ H∗(K,L)

are isomorphisms. They are functorial for simplicial maps of simplicial pairs

3.6. ISOMORPHISMS BETWEEN SIMPLICIAL AND SINGULAR (CO)HOMOLOGY 93

Proof. The functoriality has already been established. By Kronecker duality,it is enough to prove that R∗ is an isomorphism. The proof goes through a coupleof particular cases.

Case 1: (K,L) = (FA, FA), where FA is the full complex on the finite set of m+1elements A = v0, . . . , vm (see p. 24), which is isomorphic to an m-simplex. Then

(|FA|, |FA|) ≈ (Dm, Sm−1). By Corollary 2.4.7 and Corollary 3.2.2,

Hk(FA, FA) = Hk(FA, FA) = Hk(|FA|, |FA|) = 0

if k 6= m and

Hm(FA, FA) ≈ Hm(FA, FA) ≈ Hm(|FA|, |FA|) ≈ Z2 .

Thus, it is enough to prove that R∗ : Hm(FA, FA)→ Hm(|FA|, |FA|) is not triv-ial. The vector space Hm(FA, FA) is generated by the ordered simplex σ =

(v0, . . . , vm). Let r = R(σ) : (∆m, ∆m) → |FA|, |FA|). One has [r] = H∗r([im])

where im is the identity map of (∆m, ∆m). But [im] 6= 0 in Hm(∆m, ∆m) by Propo-

sition 3.2.3 and r is a homeomorphism of pairs. Thus R∗(σ) 6= 0 inHm(|FA|, |FA|).Case 2: (K,L) = (Km,Km−1) with m ≥ 1. The non-vanishing homology groupsare

Hm(Km,Km−1) ≈ Hm(Km,Km−1) ≈ Hm(|Km|, |Km−1|) ≈⊕

Sm(K)

Z2

For each σ ∈ Sm(K) choose an ordered simplex σ = (v0, . . . , vm) with v0, . . . , vm =σ. Then Hm(Km,Km−1) ≈ Hm(Km,Km−1) is the Z2-vector space with basisσ | σ ∈ Sm(K). Denote by σ the subcomplexes of K generated by σ and by ˙σthe subcomplex of the proper faces of σ. The map rσ : (|σ|, | ˙σ|)→ (|Km|, |Km−1|)is a characteristic map for the m-cells of |K| corresponding to σ. The union rm ofthe rσ is then a global characteristic map for the m-cells of |K|. Let us considerthe commutative diagram

⊕σ∈Sm(K) Hm(σ, ˙σ)

≈

R∗

≈// ⊕

σ∈Sm(K)Hm(|σ|, | ˙σ|)

≈ H∗rm

Hm(Km,Km−1)

R∗ // Hm(|Km|, |Km−1|)

The bijectivity of the left vertical arrow was seen above. That of the right verticalarrow is Lemma 3.4.7. The bijectivity of the top horizontal arrow comes fromCase 1. Hence, R∗ : Hm(Km,Km−1)→ Hm(|Km|, |Km−1|) is an isomorphism.

Case 3: (K,L) = (Km, ∅). This is proven by induction on m, the case m = 0being obvious. By the naturality of R∗, one has the commutative diagram of exactsequences:

H∗+1(Km,Km−1)

R∗ ≈

∂∗ // H∗(Km−1)

R∗ ≈

// H∗(Km)

R∗

// H∗(Km,Km−1)

R∗ ≈

∂∗ // H∗−1(Km−1)

R∗ ≈

H∗+1(Km,Km−1)

∂∗ // H∗(Km−1) // H∗(Km) // H∗(Km,Km−1)∂∗ // H∗−1(Km−1)

(one has to check that the diagrams with ∂∗ are commutative). The bijectivity ofthe vertical arrows come by induction hypothesis and by case 2. By the five-lemma,

R∗ : H∗(Km)→ H∗(|Km|) is an isomorphism.


General case. We first prove that R∗ : Hm(K) → Hm(|K|) is an isomorphism forall m. By the naturality of R∗, one has the commutative diagram:

Hm(Km+1)

≈

R∗

≈// Hm(|Km+1|)

≈

Hm(K)

R∗ // Hm(|K|) .

The bijectivity of the left vertical arrow is obvious. That of the right vertical arrowis Lemma 3.4.9. The bijectivity of the top horizontal arrow was established inCase 3. Therefore, the bottom horizontal arrow is bijective. Finally, the generalcase (K,L) is deduced from the absolute cases using, as in Case 3, the homologysequences of the pair (K,L) and the five-lemma.

For our third isomorphism theorem, choose a simplicial order ≤ on K. Definea map R≤ : S(K) → S(|K|) by R≤(σ) = R(σ) where, if σ = v0, . . . , vm, thenσ = (v0, . . . , vm) with v0 ≤ · · · ≤ vm. As above, we check that R≤ induces linearmaps R≤,∗ : H∗(K,L)→ H∗(|K|, |L|) and R∗≤ : H∗(|K|, |L|)→ H∗(K,L) of degreezero.

Theorem 3.6.3. Let (K,L) be a simplicial pair. For any simplicial order ≤on K, the linear maps

R≤,∗ : H∗(K,L)≈−→ H∗(|K|, |L|) and R∗≤ : H

∗(|K|, |L|) ≈−→ H∗(K,L)

are isomorphisms. Moreover, these isomorphisms do not depend on the simplicialorder ≤.

Proof. By Kronecker duality, only the homology statement requires a proof.By our definitions, one has the commutative diagram

H∗(K,L)R≤,∗ //

H∗φ≤

≈ &&

H∗(|K|, |L|)

H∗(K,L)

R∗

≈

77♣♣♣♣♣♣♣♣♣

The bijectivity of the arrows come from Corollary 2.10.13 and Theorem 3.6.3.Therefore, R≤,∗ is an isomorphism. As H∗φ≤ is independent of ≤ by Corol-lary 2.10.13, so is R≤,∗.

3.7. CW-approximations

It is sometimes useful to know that any space has the (co)homology of a CW-complex or of a simplicial complex (see e.g. p. 113, 136 and 272 in this book).We give below classical functorial results about that. Relationships with similarconstructions in the literature are discussed in Remark 3.7.5 at the end of thesection.

We shall need a standard notion of category theory: natural transformations.Let a and b be two (covariant) functors from a category C to a category C′. A

3.7. CW-APPROXIMATIONS 95

natural transformation associates to each object X in C a morphism φX : a(X)→b(X) in C′ such that the diagram

(3.7.1)

a(X)a(f) //

ΦX

a(Y )

ΦY

b(X)

b(f) // b(Y )

is commutative for every morphism f : X → Y in C.We first consider the category of CW-spaces and cellular maps. It is denoted

by CW and, as usual, by CW2 for pairs of CW-complexes. We denote by j be theinclusion morphism from CW2 to Top2.

Theorem 3.7.1. There is a covariant functor cw : (X,Y ) → (XCW, Y CW)from Top2 → CW2 and a natural transformation φ = φ(X,Y ) : (X

CW, Y CW) →(X,Y ) from jcw to the identity functor of Top2, such that H∗φ and H∗φ areisomorphisms.

The construction in the proof below is sometimes called in the literature thethick geometric realization of the singular complex of X .

Proof. We start with some preliminaries. If ∆m is the standard m-simplexand I ⊂ 0, 1, . . . ,m, we set

∆mI = (t0, . . . , tm) ∈ ∆m | ti = 0 if i /∈ I

which is a simplex of dimension ♯I − 1. This gives rise to an obvious inclusion mapǫI : ∆

♯I−1 → ∆m.Let X be a topological space. The space XCW is defined as the quotient space

(3.7.2) XCW =⋃

m≥0

(Sm(X)×∆m

)/∼

where ∼ is the equivalence relation (σ, ǫI(u)) ∼ (σǫI , u) for all σ ∈ Sm(X), I ⊂0, . . . ,m and u ∈ ∆♯I−1. Then XCW is a CW-complex whose k-skeleton is

(XCW)k =⋃

0≤m≤k

(Sm(X)×∆m

)/∼ .

In particular, (XCW)0 is just the space X endowed with the discrete topology. Thek-cells are indexed by Sk(X). The characteristic map for the k-cell correspondingto σ ∈ Sk(X) is the restriction to σ × ∆k of the quotient map from the disjointunion in (3.7.2) onto (XCW)k.

A continuous f : X1 → X2 determines a cellular map fCW : XCW1 → XCW

2

induced by fCW(σ, u) = (f σ, u). Note that, if Y is a subspace of X , then Y CW

is a subcomplex of XCW. We thus check that cw is a covariant functor (X,Y )→(XCW , Y CW ) from Top2 to CW2.

For σ ∈ Sm(X), one has a continuous map φσ : σ × ∆m → X defined byφσ(σ, u) = σ(u). The disjoint union of those φσ descends to a continuous mapφ : XCW → X , or φ : (XCW, Y CW)→ (X,Y ). One has

φfCW(σ, u) = φ(f σ, u) = f σ(u) = f φ(σ, u)

which amounts, using (3.7.1), to φ being a natural transformation from jcw tothe identity functor of Top2.


It remains to prove that H∗φ is a GrV-isomorphism (that H∗φ is a GrA-isomorphism will follow by Kronecker duality). We start with the absolute caseY = ∅. We shall construct a diagram

(3.7.3)

H∗(XCW)

H∗φ // H∗(X)

αyyrrrrrrrr

H∗(XCW)

β

ff

such that H∗φβα = id and α and β are isomorphisms. The bijection S(X)=−→

cells of XCW extends to a linear map α : C∗(X) → C∗(XCW) which satisfies

αδ = δα and thus induces the isomorphism α : H∗(X)≈−→ H∗(X

CW). For β, oneassociates to the k-cell of XCW indexed by σ the map

∆k ≈−→ σ ×∆k char.map−−−−−−→ XCW

which is an element of Sk(XCW). This extends to a linear map β : Ck(XCW) →

Ck(XCW). Again, we check that β δ = δβ. We thus get the linear map β : H∗(X

CW)→H∗(X

CW). The equation H∗φβα = id is straightforward.It remains to prove that β is an isomorphism. To simplify the notation,

write X = XCW. Note that β induces linear maps βk : H∗(Xk) → H∗(X

k) and

βk+1,k : H∗(Xk+1, Xk) → H∗(X

k+1, Xk). Obviously, H∗(X) = limk H∗(Xk). By

Corollary 3.1.16, one also has that H∗(X) = limkH∗(Xk). Therefore, it suf-

fices to show that βk is an isomorphism for all k. This is done by induction onk. It is obviously true for k = 0, since X0 is a discrete space. For the induc-tion step, suppose that βk : Hi(X

k) → Hi(Xk) is an isomorphism for all i ∈ N.

Then, βk+1 : Hi(Xk+1)→ Hi(X

k+1) is an isomorphism for all i, except perhaps fori = k, k + 1 where we must consider the commutative diagram(3.7.4)

0 // Hk+1(Xk+1)

βk+1

// Hk+1(Xk+1, Xk) //

βk+1,k

Hk(Xk) //

βk≈

Hk(Xk+1) //

βk+1

0

0 // Hk+1(Xk+1) // Hk+1(X

k+1, Xk) // Hk(Xk) // Hk(X

k+1) // 0

where the horizontal lines are the cellular and singular homology exact sequencesof the pair (Xk+1, Xk). By the five lemma, it thus suffices to prove that βk+1,k isan isomorphism. One has the commutative diagram

(3.7.5)

Hk+1(Xk+1, Xk)

βk+1,k

oo ≈⊕

σ∈Sk+1(X)

Hk+1(σ × (∆k+1,Bd∆k+1))

⊕βσ

Hk+1(Xk+1, Xk) oo ≈

⊕

σ∈Sk+1(X)

Hk+1(σ × (∆k+1,Bd∆k+1))

where βσ sends the (k+1)-cell σ×(∆k+1 (generator of Hk+1(σ×(∆k+1,Bd∆k+1)) =Z2) to the tautological singular simplex ∆k+1 → σ × ∆k+1. The latter is the


generator of Hk+1(σ × (∆k+1,Bd∆k+1)) = Z2 (see Proposition 3.2.3). Hence,βk+1,k is an isomorphism.

We have proven that H∗φ : H∗(XCW) → H∗(X) is a GrV-isomorphism for

all topological spaces X . Using the homology sequences and the five lemma, thisimplies thatH∗φ : H∗(X

CW, Y CW)→ H∗(X,Y ) is aGrV-isomorphism for all topo-logical pairs (X,Y ).

A slightly more sophisticated construction for the functor of Theorem 3.7.2gives the following result. Let RCW be the category of regular CW-complexesand cellular maps and let j be the inclusion morphism from RCW2 to Top2.

Theorem 3.7.2. There is a covariant functor rcw : (X,Y )→ (XRCW, Y RCW)from Top2 → RCW2 and a natural transformation φ = φ(X,Y ) : (X

RCW, Y RCW)→(X,Y ) from jrcw to the identity functor of Top2, such that H∗φ and H∗φ areisomorphisms.

The proof of Theorem 3.7.2 requires some preliminaries.Let FN be the full simplicial complex with vertex set the integers N. If X is a

topological space, the set of N-singular simplexes of X is defined by

NS(X) = (s, τ) | s ∈ S(FN) and τ : |s| → X is a continuous map .where s is the simplicial complex formed by s and all its faces (see p. 10). LetNSn(X) be the subset of NS(X) formed by those pairs (s, τ), where s is of dimensionn and let NCn(X) be the Z2-vector space with basis NSn(X). Using the facets of s,we define a boundary operator ∂ : NCn(X)→ NCn−1(X) making NC∗(X) a chaincomplex. The homology of this chain complex is the N-singular homology of X ,denoted by NH∗(X). The relative homology NH∗(X,Y ) is defined as in § 3.1.2.

The order on N provides a simplicial order on FN. Thus, if s ∈ Sn(FN),there is a canonical homeomorphism hs : |s| ≈−→ ∆n (see (3.1.6)). We define mapsµ : Sn(X)→ NSn(X) and ν : NSn(X)→ Sn(X) by:

• µ(σ) = (s0, σhs0(n)), where s0(n) = 0, 1, . . . , n and• ν(s, τ) = τ h−1s .

The linear extensions C∗µ : Cn(X) → NCn(X) and C∗ν : NCn(X) → Cn(X) com-mute with the boundary operators and are thus morphisms of chain complexes.The constructions extend to pairs and we get GrV-morphisms H∗µ : Hn(X,Y )→NHn(X,Y ) and H∗ν : NHn(X,Y )→ Hn(X,Y ).

Lemma 3.7.3. H∗µ : Hn(X,Y )→ NHn(X,Y ) andH∗ν : NHn(X,Y )→ Hn(X,Y )are isomorphisms, inverse to each other.

Proof. Clearly, ν µ = id on S(X), thus H∗ν H∗µ = id on H∗(X,Y ). Tosee that H∗µH∗ν = id on NH∗(X,Y ), we first restrict ourselves to the absolutecase Y = ∅. We shall prove that C∗µC∗ν and the identity of NC∗(X) admit acommon acyclic carrier A∗ with respect to the basis NS(X). The condition thatH∗µH∗ν = id on NH∗(X) then follows from Proposition 2.9.1.

For (s, τ) ∈ NSn(X) and k ∈ N, define Bk(s, τ) by

Bk(s, τ) = (t, τ |p|) | t ∈ Sk(FN) and p : t→ s is a simplicial map .Let Ak(s, τ) be the Z2-vector space with basis Bk(s, τ). Using the restriction of pto the facets of t, one defines a boundary operator ∂ : Ak(s, τ)→ Ak−1(s, τ) makingA∗(s, τ) a subchain complex of NC∗(X).


For (s, τ) ∈ NS(X), one has (sτ) ∈ A∗(s, τ) (p = ids and

µν(s, τ) = (s0, τ h−1s hs0(n)) ∈ A∗(s, τ) ,

since h−1s hs0(n) = |p| for p : s → s0 the unique simplicial isomorphism preserv-ing the order. The conditions for the correspondence (s, τ) 7→ A∗(s, τ) being anacyclic carrier (see § 2.9) are easy to check once we know that H0(A∗(s)) = Z2 andHm(A∗(s)) = 0 for m > 0 which we prove below.

Consider the simplicial complex K(s) with vertex set V (K(s)) = N×V (s) andwhose k-simplexes are the sets (n0, s0), . . . , (nk, sk) with n0 < · · · < nk. We checkthat the correspondence sending (t, τ |p|) to the graph of p : V (t)→ V (s) inducesan isomorphism of chain complex between A∗(s, τ) and the simplicial chain ofK(s).We have thus to prove that H∗(K(s)) ≈ H∗(pt). But K(s) is the union of Kn(s),whereKn(s) is the union of all simplexes of K(s) with vertices in 0, . . . , n×V (S).The inclusion V (Kn(s)) → V (Kn+1(s)) together with the map k 7→ (n + 1, k)

provides a bijection V (Kn(s)) ∪ V (s)≈−→ V (Kn+1(s)) and a simplicial isomorphism

Kn(s) ∗ s0 ≈−→ Kn+1(s) ,

where s0 is the 0-skeleton of s. Hence, the inclusion Kn(s) → Kn+1(s) factorsthrough the cones onKn(s) contained in the joinKn(s)∗s0. Therefore,H∗(K(s)) ≈lim→H∗(Kn(s)) ≈ H∗(pt).We have thus proved Lemma 3.7.3 in the case Y = ∅. Using the homology

sequences, this proves thatH∗µ : Hn(X,Y )→ NHn(X,Y ) andH∗ν : NHn(X,Y )→Hn(X,Y ) are both isomorphisms. But we have already noted that H∗ν H∗µ = idon H∗(X,Y ). Therefore, H∗µH∗ν = id on NH∗(X,Y ).

We are now ready for the proof of Theorem 3.7.2.

Proof of Theorem 3.7.2. If t ⊂ s are simplexes ofFN, we denote by it,s : t→s the simplicial map given by the inclusion. The space XRCW is defined as the quo-tient space

(3.7.6) XRCW =⋃

(s,τ)∈NS(X)

((s, τ)) × |s|

)/∼

where ∼ is the equivalence relation ((s, τ), |it,s|(u)) ∼ (t, τ |it,s|), u) for all (s, τ) ∈NS(X), all subsimplex t of s and all u ∈ |t|. As in the proof of Theorem 3.7.1,XRCW is a naturally a CW-complex. The characteristic map for the k-cell cor-responding to (s, τ) ∈ NSk(X) is the restriction to (s, τ)) × |s| of the quotientmap in (3.7.6). In particular, (XRCW)0 is the set N×X endowed with the discretetopology. Because of the role of N in the indexing of the cells, one checks thatXRCW is a regular CW-complex. For (s, τ) ∈ NSm(X), one has a continuous mapφ(s,τ) : (s, τ) × |s| → X defined by φ(s,τ)((s, τ), u) = τ(u). The disjoint union

of those evaluation maps descends to a continuous map φ : XRCW → X . if Y isa subspace of X , then Y RCW is a subcomplex of XRCW. The functoriality of thecorrespondence (X,Y ) 7→ (XRCW, Y RCW), as well as that φ is a natural transfor-mation from jrcw to the identity functor of Top2, are established as in the proofof Theorem 3.7.1.


We now prove that NH∗φ : NH∗(XRCW) → NH∗(X) is a GrV-isomorphism,following the pattern of the proof of Theorem 3.7.1. Similarly to (3.7.3), we con-struct the diagram

(3.7.7)

NH∗(XRCW)NH∗φ // NH∗(X)

Nαxxqqqqqqqqq

H∗(XRCW)

Nβ

gg

such that NH∗φNβNα = id and Nα and Nβ are isomorphisms. As in (3.7.3),

the bijection NS(X)=−→ cells of XRCW gives the isomorphism Nα. The GrV-

morphism β comes from associating to the k-cell of XRCW indexed by (s, τ) themap

|s| ≈−→ (s, τ) × |s| char.map−−−−−−→ XRCW

which is an element of NSk(XRCW). The equation NH∗φNβNα = id is straight-forward.

The proof that Nβ is an isomorphism is quite similar to to that (for β) inthe proof of Theorem 3.7.1. Indeed, using Lemma 3.7.3, NH∗( ) is a homologytheory and thus diagrams as in (3.7.4) and (3.7.5) do exist; the isomorphismH∗µ : NH∗(∆k−1,Bd∆k+1)fl≈H∗(∆k−1,Bd∆k+1) is also explicit enough and per-mits to proceed as in the proof of Theorem 3.7.1. Details are left to the reader.

The GrV-isomorphism H∗µ of Lemma 3.7.3 is natural; one has thus a com-mutative diagram

NH∗(XRCW )

µ≈

NH∗φ

≈// NH∗(X)

µ≈

H∗(XRCW )

H∗φ // H∗(X)

which shows that H∗φ is an isomorphism. The relative case is obtained as at theend of the proof of Theorem 3.7.1 and that H∗φ is a GrA-isomorphism comes fromKronecker duality.

Theorem 3.7.4. There is a covariant functor symp: (X,Y )→ (KX ,KY ) fromTop2 → Simp2 and a natural transformation φ = φ(X,Y ) : (|KX |, |KY |) → (X,Y )from | symp| to the identity functor of Top2 such that H∗φ and H∗φ are isomor-phisms.

Proof. Let X be a topological space. By the proof of Theorem 3.7.2, the regu-lar CW-complexXRCW comes equipped with characteristic embeddings ϕ(s,τ) : (s, τ))×|s| → XRCW ((s, τ)) ∈ NS(X)), satisfying the following condition: if t is a face of swith simplicial inclusion it,s : t→ s, then ϕ(t,τ |it,s|) = ϕ(s,τ) |it,s|. The only miss-

ing thing to make XRCW a simplicial complex is that several simplexes may havethe same boundary. But this can be avoided by taking the barycentric subdivisionof each cell, with the characteristic embedding ϕ′(s,τ) : (s, τ)) × |(s)′| → XRCW .

We thus get a functorial triangulation of XRCW .

Remarks 3.7.5. The following facts about the above constructions should benoted.


(a) The proof of Theorem 3.7.1 goes back to Giever [67]; for a more recenttreatment, see [73, p. 146]. Theorem 3.7.2 may be obtained from Theo-rem 3.7.1 by subdivision techniques in semi-simplicial complexes (see [73,Theorem 16.41]). Our proof of Theorem 3.7.2 is different.

(b) The construction XCW in the proof of Theorem 3.7.1 is sometimes calledin the literature the thick geometric realization of the singular complex

of X . A quotient XCW of XCW was introduced by J. Milnor [146], in

which the degenerate simplexes are collapsed. Thus, XCW has one k-cellfor each non-degenerate singular k-simplex of X . Under mild conditions,the Milnor functor behaves well with products (see [146, § 2]).

(c) The maps φ of Theorem 3.7.1, 3.7.2 and 3.7.4 are actually weak homotopyequivalences (see e.g. [73, Corollary 16.43]). Such maps are called CW-approximations [82] or resolutions [73]. In particular, if X is itself aCW-complex, these maps are homotopy equivalences by the Whiteheadtheorem [82, Theorem 4.5] (but they are not homeomorphisms). Somehowsimpler (but not functorial) proofs that any spaces has the weak homotopytype of a CW-complex may be found in e.g. [82, Proposition 4.13] or [73,Proposition 16.4].

(d) By its construction in the proof of Theorem 3.7.2, XRCW is a regular ∆-set is the sense of [82, p. 533–34]. In this appendix of [82], the readermay find enlightening considerations related to our constructions in thissection.

3.8. Eilenberg-MacLane spaces

The Eilenberg-MacLane spaces are used to make the cohomology H∗(−) arepresentable functor. With Z2 as coefficients, they admit an ad hoc presentationgiven below, which only uses the material developed in this book. The equivalencewith the usual definition using the homotopy groups is proven at the end of thesection.

A CW-complex K is an Eilenberg-MacLane space in degree m if

(i) Hm(K) = Z2; we denote by ι the generator of Hm(K).(ii) for any CW complex X , the correspondence f 7→ H∗f(ι) gives a bijection

φ : [X,K] ≈−→ Hm(X) ,

where [X,K] denotes the set of homotopy classes of continuous maps fromX to K.

If f : X → K is a map, the class H∗f(ι) is said to be represented by f . Property(ii) says that the functorH∗(−) would be representable by K in the sense of categorytheory [134].

The notation K(Z2,m) is usual for a CW-complex which is an Eilenberg-MacLane space in degree m. We shall also use the notation Km. The unambiguityof these notations is guaranteed by the following existence and uniqueness result.

Proposition 3.8.1. (a) For any integer m, there exists an Eilenberg-MacLane space in degree m.

(b) Let Km and K′m be two Eilenberg-MacLane spaces in degree m. Then,there exists a homotopy equivalence g : K′m → Km whose homotopy classis unique.

3.8. EILENBERG-MACLANE SPACES 101

Example 3.8.2. By Corollary 3.1.12, we see that the point is an Eilenberg-MacLane space in degree 0.

Proof. We start with the uniqueness statement (b). Let K and K′ be twoEilenberg-MacLane spaces in degree m. Then, there is a bijection

Z2 = Hm(K′) ≈ [K′,K]under which the constant maps corresponds to 0. Let g : K′ → K be a continuousmap representing the non-vanishing class (unique up to homotopy). In the sameway, let h : K → K′ represent the non-vanishing class of Z2 = Hm(K) ≈ [K,K′].Then, gh represents the non-vanishing class of Z2 = Hm(K) ≈ [K,K] and hgrepresents the non-vanishing class of Z2 = Hm(K′) ≈ [K′,K′]. As idK and idK′ dothe same, we deduce that gh is homotopic to idK and hg is homotopic to idK′ .Therefore, h and g are homotopy equivalences.

We now construct an Eilenberg-MacLane space K in degree m ≥ 1 (K0 = pt,as noticed in Example 3.8.2). Its m-skeleton Km is the sphere Sm, with one 0-cellv and one m-cell called ε. Then, for each map ϕ : Sm → Km of degree 0, an(m+1)-cell is attached to to Km via ϕ, thus getting Km+1. Finally, for k ≥ m+2,Kk is constructed by induction by attaching to Kk−1 a k-cell for each continuousmap f : Sk−1 → Kk−1.

As the (m+ 1)-cells of K are attached to Km by maps of degree 0, the cellular

boundary ∂ : Cn+1(K)→ Cn(K) vanishes by Lemma 3.5.4. Therefore,Hm(K) = Z2

by Theorem 3.5.6 and Hm(K) = Z2 by Kronecker duality. The singleton ε, seenas a cellular m-cycle of K, is called ι ∈ Zm(K). Seen as an m-cocycle, we denote it

by ι ∈ Zm(K) (it represents ι ∈ Hm(K)).Let us prove the surjectivity of φ : [X,K] → Hm(X). Let a ∈ Hm(X), repre-

sented by a cellular cocycle a ∈ Cm(X) ⊂ Cm(Xm). We shall construct a mapf : X → K such that H∗f(ι) = a. Let j : Dm → K be a characteristic map for theunique m-cell of K. The map f sends Xm−1 to the point v = K0. Its restrictionto an m-cell e of X is equal to j if e ∈ a and the constant map onto v otherwise.This gives a map f : Xm → Km which, by construction and Lemma 3.5.11, satisfies

(3.8.1) C∗f(α) = 〈a, α〉 ιfor all α ∈ Cm(X). Hence

〈C∗f(ι), α〉 = 〈ι, C∗f(α)〉 = 〈ι, 〈a, α〉ι〉 = 〈a, α〉for all α ∈ Cm(X). By Lemma 2.3.3, we deduce that C∗f(ι) = a.

To extend f to Xm+1, let λ ∈ Λm+1(X) with attaching map ϕλ : Sm → Xm.

As a is a cocycle, one has 〈a, ∂λ〉 = 〈δ(a), λ〉 = 0. Using Lemmas 3.5.4 and 3.5.11together with Equation (3.8.1), we get that fλ = f ϕλ : S

m → Km ≈ Sm is a mapof degree 0. By construction of K, an (m + 1)-cell e is attached to Km via fλ,so f may be extended to λ, using a characteristic map for e extending fλ. Thisproduces a cellular map fm+1 : Xm+1 → Km+1. Finally, suppose, by inductionon k ≥ m + 1, that fm+1 extends to fk : Xk → Kk. Let λ ∈ Λk+1(X) withattaching map ϕλ : S

k → Xk. Set gλ = fk ϕλ. By construction of K, there existsegλ ∈ Λk+1(K) with attaching map gλ. Thus f

k may be extended to the cell λ, usinga characteristic map for eλ extending gλ. Doing this for each λ ∈ Λk+1(X) producesthe desired extension fk+1 : Xk+1 → Kk+1. the surjectivity of φ : [X,K]→ Hm(X)is thus established.


For the injectivity of φ, let f0, f1 : X → K such that H∗f0(ι) = H∗f1(ι).Since any map between CW-complexes is homotopic to a cellular map (see e.g. [64,Theorem 2.4.11]), we may assume that f0 and f1 are cellular. We must constructa homotopy F : X × I → K between f0 and f1, which will be done cell by cell. Asf0(X

m−1) = f1(Xm−1) = v, The maps f0 and f1 descend to cellular maps from

X/Xm−1 to K. Hence, we can assume that Xm−1 = X0 is a single point w, withf0(w) = f1(w) = v. The homotopy F is defined to be constant on w: F (w, t) = v.

As Xm−1 is a point, the homology class H∗f0(ι) = H∗f1(ι) is represented by a

single cellular cocycle a ∈ Cm(X) (Bm(X) = 0). Let λ ∈ Λm(X) with characteristicmap ϕλ : S

m → X . By Lemma 3.5.11, one has, for j = 0, 1:

(3.8.2) 〈a, λ〉 = 〈C∗fj(ι), λ〉 = 〈ι, C∗fj(λ)〉 = 〈ι, deg(fj ϕλ)ι〉 = deg(fj ϕλ) .

Let Σm be the boundary of Dm × I, homeomorphic to Sm. A map Fλ : Σm → Km

is defined by

(3.8.3) Fλ(x, t) =

f0(ϕλ(x)) if t = 0

f1(ϕλ(x)) if t = 1

v if x ∈ Sm−1 .Using (3.8.2) together with Lemma 3.2.8 (with B1 = Dm×0 and B2 = Dm×1),we deduce that degFλ = 0. Then, there is an (m+ 1)-cell of K is attached to Kmwith Fλ. This implies that F extends to Fλ : D

m× I → Km+1 which is a homotopyfrom f0 to f1 over Xm union the cell λ. Doing this for each λ ∈ Λm(X) producesa homotopy Fm : Xm × I → Km+1 between f0 and f1. We can thus assume, byinduction on k ≥ m, that a homotopy F k : Xk × I → Kk+1 between f0 and f1has been constructed. We must extend it to F k+1 : Xk+1 × I → Kk+2, which canbe done individually over each cell λ ∈ Λk+1(X). We define Fλ : Σ

k+1 → Kk+1 asin (3.8.3). As k + 1 > m, a (k + 2)-cell of K is attached to Kk+1 with Fλ, whichpermits us, as above, to extend the homotopy F k over the cell λ.

The proof of Proposition 3.8.1 is now complete.

The above construction of an Eilenberg-MacLane space uses a lot of cells sowe may expect that the (co)homology of Kn is complicated. It was computed byJ-P. Serre [175, § 2], whose result will be given in Theorem 8.5.5. In degree 1however, we have the following simple example of an Eilenberg-MacLane space.

Proposition 3.8.3. The projective space RP∞ is an Eilenberg-MacLane spacein degree 1 (RP∞ ≈ K(Z2, 1)).

Proof. We use the standard CW-structure on K = RP∞ of Example 3.4.5,with one cell in each dimension and so that Kk = RP k. Let pk : S1 → S1 given bypk(z) = zk. The following properties hold true:

(i) the 2-cell of K is attached to K1 ≈ S1 by the map p2 which, by Proposi-tion 3.2.6 is of degree 0.

(ii) each map g : S1 → K1 of degree 0 is null-homotopic (i.e. homotopic toa constant map) in K2. Indeed, it is classical that any map from S1 toS1 ≈ K1 is homotopic to pk for some integer k (see e.g. [136, Theorem 5.1]or [82, Theorem 1.7]). By Proposition 3.2.6, deg pk = 0 if and onlyif k = 2r. Point (i) implies that g = p2 is null homotopic and so isp2r = p2pr.

3.8. EILENBERG-MACLANE SPACES 103

(iii) for k ≥ 2, each map g : Sk → Kk is null-homotopic into Kk+1. Indeed,the lifting property of covering spaces tells us that g admits a lifting

Sk

p

Sk

g<<②

②②

②g // RP k

and the (k + 1)-cell of K is attached via the covering map p.

By Point (i), H1(K) = Z2. Points (ii) and (iii) imply that the argument of theproof of Proposition 3.8.1 may be used to prove that φ : [X,RP∞] → H1(X) is abijection. Hence, K = RP∞ is an Eilenberg-MacLane space in degree 1.

Corollary 3.8.4. Let f : RPn → RP k be a continuous map, with n < k ≤ ∞.Then f is either homotopic to a constant map or to the inclusion RPn → RP k.

Proof. The lemma is true for k = ∞ by Proposition 3.8.3. Therefore, thereis a homotopy from the composition of f with the inclusion RP k → RP∞ toeither a constant map or the inclusion. Making this homotopy cellular (see [207,Corollary 4.7, p. 78]) produce a homotopy whose range is in RPn+1.

We finish this section with the relationship between our definition of Eilenberg-MacLane spaces and the usual one involving the homotopy groups. Recall that thei-th homotopy group πi(X, x) of a pointed space (X, x) is defined by πi(X, x) =[Si, X ]•, for some fixed base point in Sn. Below, the base points are omitted fromthe notation.

Proposition 3.8.5. A CW-complex X is an Eilenberg-MacLane space Km ifand only if πi(X) = 0 if i 6= m and πm(X) = Z2.

Proof. We first prove that Km satisfies the conditions. By Propositions 3.8.1and 3.8.3, the space K1 is homotopy equivalent to RP 1. The statement then followsusing the 2-fold covering S∞ → RP∞ and the fact that S∞ is contractible (see [82,example 1.B.3, p. 88]). By Proposition 3.8.1 and its proof, the space Km admits aCW-structure whose (m− 1)-skeleton is a point. Thus, when m > 1, Km is simplyconnected and [Si,Km]• ≈ [Si,Km] (see [82, Proposition 4A.2]). The cohomologyof Si, computed in Proposition 3.8.1, implies that the set [Si,Km] ≈ Hm(Si)contains one element if i 6= m and two elements if i = m.

Conversely, if X is a CW-complex satisfying πi(X) = 0 if i 6= m and πm(X) =Z2, we must prove that X is homotopy equivalent to Km. This requires techniquesnot developed in this book. When m = 1, there exists a map f : X → RP∞ ≈ K1

inducing an isomorphism on the fundamental group (see (4.3.1)). The map f theninduces an isomorphism on all the homotopy groups, what is called a weak homotopyequivalence. By the Whitehead theorem [82, Theorem 4.5], a weak homotopy equiv-alence between connected CW-complexes is a homotopy equivalence. When m > 1,let α : Sm → X representing the non-zero element of πm(X). By the Hurewicztheorem [82, Theorem 4.32], the integral homology Hm(X ;Z) = Z2 and, from theuniversal coefficient theorem [82, Theorem 3B.5], it follows that Hm(X) = Z2 andH∗α : Hm(Sm)→ Hm(X) is an isomorphism. By Kronecker duality, Hm(X) = Z2

and H∗α : Hm(X) → Hm(Sm) is an isomorphism. Let g : X → Km representingthe non-zero element of Hm(X). As H∗α : Hm(X)→ Hm(Sm) is an isomorphism,


the map g induces an isomorphism from πm(X) to πm(Km) (we have proved abovethat πm(Km) = Z2). Hence, g is a weak homotopy equivalence and therefore ahomotopy equivalence by the Whitehead theorem.

3.9. Generalized cohomology theories

The axiomatic viewpoint for (co)homology was initiated by Eilenberg andSteenrod in the late 1940’s [52, 51] and had a great impact on the general un-derstanding of the theory. We give below a version in the spirit of [82, Sections 2.3and Chapter 3]. Our application will be the Kunneth theorem 4.6.7.

A cohomology theory is a contravariant functor h∗ from the category Top2 oftopological pairs to the category GrV of graded Z2-vector spaces, together witha natural connecting homomorphism δ∗ : h∗(A)→ h∗+1(X,A) (the notation h∗(A)stands for h∗(A, ∅)). In addition, the following axioms must be satisfied.

(1) Homotopy axiom: if f, g : (X,A) → (X ′, A′) are homotopic, then h∗f =h∗g.

(2) Exactness axiom: for each topological pair (X,A) there is a long exactsequence

· · · δ∗−−→ hm(X,A)→ hm(X)→ hm(A)δ∗−→ hm+1(X,A)→ · · ·

where the unlabeled arrows are induced by inclusions. This exact sequenceis functorial, i.e. if f : (X ′, A′) → (X,A) is a map of pair, there is acommutative diagram

· · · // h∗(X)

h∗f

// h∗(A)

h∗f

δ∗ // h∗+1(X,A)

h∗f

// h∗+1(X)

h∗f

// · · ·

· · · // h∗(X ′) // h∗(A′)δ∗ // h∗+1(X ′, A′) // h∗+1(X ′) // · · ·

(3) Excision axiom: let (X,A) be a topological pair, with U be a subspace ofX satisfying U ⊂ intA. Then, the GrV-morphism induced by inclusionsi∗ : h∗(X,A)→ h∗(X − U,A− U) is an isomorphism.

(4) Disjoint union axiom: for a disjoint union (X,A) =⋃j∈J (Xj , Aj) the

homomorphism

h∗(X,A)→∏

j∈J

h∗(Xj , Aj)

induced by the family of inclusions (Xj , Aj) → (X,A) is an isomorphism.

Examples 3.9.1. The singular cohomology H∗ is a generalized cohomologytheory. Axioms (1)–(3) are fulfilled, as seen in § 3.1.2–3.1.4. The disjoint unionaxiom corresponds to Proposition 3.1.11 for a pair (X, ∅); it may be extended toarbitrary topological pairs, using the exactness axiom and the five lemma.

K-theory and cobordism are other examples of generalized cohomology theo-ries.

Let h∗ and k∗ be two cohomology theories A natural transformation µ from h∗

to k∗ is a natural transformation of functors commuting with the connecting homo-morphisms. In particular, for each topological pair (X,A), one has a commutative


diagram of exact sequences:

(3.9.1)

· · · // h∗(X)

µ

// h∗(A)

µ

δ∗ // h∗+1(X,A)

µ

// h∗+1(X)

µ

// · · ·

· · · // k∗(X) // k∗(A)δ∗ // k∗+1(X,A) // k∗+1(X) // · · ·

The aim of this section is to prove the following theorem.

Proposition 3.9.2. Let h∗ and k∗ be two cohomology theories and let µ be

a natural transformation from h∗ to k∗. Suppose that µ : h∗(pt)≈−→ k∗(pt) is an

isomorphism. Then µ : h∗(X,A)≈−→ k∗(X,A) is an isomorphism for all CW-pairs

(X,A) where X is finite dimensional.

The hypothesis thatX is finite dimensional is not necessary in Proposition 3.9.2(see [82, Proposition 3.19]), but it simplifies the proof considerably. Proposi-tion 3.9.2 is enough for the applications in this book (see § 4.6).

Proof. We essentially recopy the proof of [82, Proposition 3.19]. By Di-agram (3.9.1) and the five-lemma, it suffices to show that µ is an isomorphismwhen A = ∅. The proof goes by induction on the dimension of X . When X is0-dimensional, the result holds by hypothesis and by the axiom for disjoint unions.Diagram (3.9.1) for (X,A) = (Xm, Xm−1) and the five-lemma reduce the induc-tion step to showing that µ is an isomorphism for the pair (Xm, Xm−1). Letϕm : Λm × (Dm, Sm−1)→ (X ,Xm−1) be a global characteristic maps for all the mcells of X. Like in the proof of Lemma 3.4.7, the axioms (essentially excision) implythat h∗ϕm and k∗ϕm are isomorphisms so, by naturality, it suffices to show thatµ is an isomorphism for Λm × (Dm, Sm−1). The axiom for disjoint unions gives afurther reduction to the case of the pair (Dm, Sm−1). Finally, this case follows byapplying the five-lemma to Diagram (3.9.1), since Dm is contractible and hence iscovered by the 0-dimensional case, and Sn−1 is (n− 1)-dimensional.


3.1. Give the list of the (maximal) simplexes of the triangulation of ∆m × I usedin the proof of Proposition 3.1.30. Draw them for m = 1, 2. Same question for thetriangulation used in the proof of Lemma 3.1.35.

3.2. Let X be a topological space.

(a) Show that X is contractible if and only if there is a correspondence f 7→f associating to a continuous map f : A → X a continuous extension

f : CA→ X , where CA is the cone over A. This correspondence is natural

in the following sense: if g : B → A is a continuous map, then f g = f Cg.(b) Show that if X is contractible then, for any CW-pair (A,B), any contin-

uous map f : B → X admits a continuous extension g : A→ X .(c) Using (a), find a direct proof of that the (co)homology of a contractible

space is isomorphic to that of a point (Corollary 3.1.33). [Hint: use that∆n+1 ≈ C∆n.]

3.3. Let X be a 2-sphere or a 2-torus. Let A be a non-empty subset of X containingn points. Compute H∗(X −A) and H∗(X,A).


3.4. Find topological pairs (X,Y ) and (X ′, Y ′) such that H∗(X,Y ) 6≈ H∗(X′, Y ′)

while X is homeomorphic to X ′ and Y is homeomorphic to Y ′.

3.5. Let X be a topological space. Let A be a subspace of X which is open andclosed. Show that (X,A) is well cofibrant.

3.6. Show that there is no continuous retraction from the Mobius band onto itsboundary.

3.7. Show that the Klein bottle K is made out of two copies of the Mobius bandglued along their common boundaries. Compute H∗(K), using the Mayer-Vietorisexact sequence for this decomposition.

3.8. Let f : Sn → Sn be a continuous map such that no antipodal pair of pointsgoes to an antipodal pair of points. Show that the degree of f is 0.

3.9. Let (X,Y, Z) be a topological triple. Draw a commutative diagram linkingthe cohomology sequences of the pairs (X,Y ), (X,Z), (Y, Z) and that of the triple(X,Y, Z).

3.10. Let (X,X1, X2, X0) be a Mayer-Vietoris data with X = X1 ∪ X2. Supposethat X is a CW-complex and that Xi are subcomplexes. Find a short proof of theexistence of the Mayer-Vietoris for the cellular (co)homology. [Hint: analogous tothe simplicial case.]

3.11. Let (X,X1, X2, X0) be a Mayer-Vietoris data with X = X1 ∪ X2. Sup-pose that the homomorphism H∗(X1, X0) → H∗(X,X2) induced by the inclu-sion is an isomorphism. Deduce the Mayer-Vietoris (co)homology sequences for(X,X1, X2, X0).

3.12. Using a tubular neighbourhood and the Mayer-Vietoris sequence, computethe homology of the complement of a (smooth) knot in S3.

3.13. Let X be a countable CW-complex. Show that H∗(X) is countable. Is ittrue for H∗(X)?

3.14. For n ∈ N≥1, consider the circle Cn := z ∈ C | |z − 1/n| = 1/n. TheHawaiian earring is the subspace B of C consisting of the union of Cn for n ≥ 1.

(a) Show that H1(B) surjects onto∏∞n=1 Z2.

(b) Show that [B,RP∞] is countable.(c) Deduce from (a) and (b) that B does not have the homotopy type of a

CW-complex.

3.15. Let Rq, q ∈ N≥1 be a sequence of Z2-vector spaces. Find a path-connectedspace X such that Hq(X) ≈ Rq.3.16. Let X be a 2-dimensional CW-complex with a single 0-cell, m 1-cells and n2-cells. Show that m = n if and only if b1(X) = b2(X).

3.17. Find perfect CW-decompositions for the 2-torus and the Klein bottle.

3.18. Let P = 〈A|R〉 be a presentation of a group G with a set A of generatorsand a set R of relators. The presentation complex XP is the 2-dimensional complexobtained from a bouquet CA of circles indexed by A by attaching, for each relatorr ∈ R, a 2-cell according to r ∈ π1(CA). Hence, π1(XP ) ≈ G. Compute H∗(XP )in the following cases.


• P1 = 〈a, b, c | abc−1b−1, bca−1c−1〉 and P2 = 〈x, y |xyxy−1x−1y−1〉 (twopresentations of the trefoil knot group).• P3 = 〈a, b, c | a5, b3, (ab)2〉 (a presentation of the alternating group A5).

3.19. Let K = K(Z2, n) be an Eilenberg-MacLane space. Let f : K → K be acontinuous map. Show that f is either homotopic to the identity or to a constantmap.

3.20. Let Kn = K(Z2, n) be an Eilenberg-MacLane space. Let X be a non-

contractible CW-complex. Suppose that there are continuous maps Xj−→ Kn r−→ X

such that rj is homotopic to the identity. Show that X and Kn have the samehomotopy type.

CHAPTER 4

Products

So far, the reader may not have been impressed by the essential differences be-tween homology and cohomology: the latter is dual to the former via the Kroneckerpairing, so they are even isomorphic for spaces of finite homology type. However,cohomology is a definitely more powerful invariant than homology, thanks to itscup product, making H∗(−) a graded Z2-algebra. Thus, the homotopy types of twospaces with isomorphic homology may sometimes be distinguished by the algebra-structure of their cohomology. Simple examples are provided by RP 2 versus S1∨S2,or by the 2-torus versus the Klein bottle.

In this chapter, we present the cup product for simplicial and singular coho-mology, out of which the cap and cross products are derived, with already manyapplications (more will come in other chapters).

Cohomology and its cup product occurred in 1935 (40 years after homology) inthe independent works of Kolmogoroff and Alexander, soon revisited and improvedby Cech and by Whitney [29, 209]. These people were all present at the interna-tional topology conference held in Moscow, September 1935. Vivid recollections ofthis memorable meeting were later written by Hopf and by Whitney [102, 211].For surveys of the interesting history of cohomology and products, see [40, ChapterIV] and [137].

4.1. The cup product

4.1.1. The cup product in simplicial cohomology. Let K be a simplicialcomplex. Choose a simplicial order ≤ on K. Let a ∈ Cp(K) and b ∈ Cq(K). UsingPoint (c) of Lemma 2.3.3, we define a cochain a ≤ b ∈ Cp+q(K) by the formula

〈a ≤ b, σ〉 = 〈a, v0, . . . , vp〉〈b, vp, . . . , vp+q〉 ,required to be valid for all σ = v0, . . . , vp+q ∈ Sp+q(K), with v0 < v1 < · · · <vp+q. This defines a map

Cp(K)× Cq(K)≤−−→ Cp+q(K) .

We can see ≤ as a composition law on C∗(K):

C∗(K)× C∗(K)≤−−→ C∗(K) .

Lemma 4.1.1. (C∗(K),+,≤) is a (non-commutative) graded Z2-algebra.

Proof. The associativity and distributivity properties are obvious. The neu-tral element for ≤ is the unit cochain 1 ∈ C0(K).

Lemma 4.1.2. δ(a ≤ b) = δa ≤ b+ a ≤ δb.

109

110 4. PRODUCTS

Proof. Set a ∈ Cp(K), b ∈ Cq(K) and σ = v0, . . . , vp+q+1 ∈ Sp+q+1(K)with v0 < v1 < · · · < vp+q+1. One has

〈δa ≤ b, σ〉 = 〈δa, v0, . . . , vp+1〉〈b, vp+1, . . . , vp+q+1〉= 〈a, ∂v0, . . . , vp+1〉〈b, vp+1, . . . , vp+q+1〉

=

p+1∑

i=0

〈a, v0, . . . , vi, . . . , vp+1〉〈b, vp+1, . . . , vp+q+1〉 .(4.1.1)

In the same way,

(4.1.2) 〈a ≤ δb, σ〉 =p+q+1∑

i=p

〈a, v0, . . . , vp〉〈b, vp, . . . , vi, . . . , vp+q+1〉 .

The last term in the sum of (4.1.1) is equal to the first term in the sum of (4.1.2).Hence, these terms cancel when adding up the two sums and the remaining termsare those of 〈a ≤ b, ∂(σ)〉 = 〈δ(a ≤ b), σ〉.

Lemma 4.1.2 implies that Z∗(K) ≤ Z∗(K) ⊂ Z∗(K), B∗(K) ≤ Z

∗(K) ⊂B∗(K) and Z∗(K) ≤ B

∗(K) ⊂ B∗(K). Therefore, ≤ induces a map Hp(K) ×Hq(K)

−→ Hp+q(K), seen as a composition law on H∗(K):

H∗(K)×H∗(K)−→ H∗(K)

called the cup product on simplicial cohomology. The notation and the namecup product (the latter due to the former) were first used by Whitney [209]. Itfollows from Lemma 4.1.1 that (H∗(K),+,) is a graded Z2-algebra. Droppingthe index “≤” is justified by the following proposition.

Proposition 4.1.3. The cup product on H∗(K) does not depend on the sim-plicial order “≤”.

Proof. The procedure to define the cup product may be done with the orderedcochains. For a ∈ Cp(K) and b ∈ Cq(K), we define a b ∈ Cp+q(K) by theformula

〈a b, σ〉 = 〈a, (v0, . . . , vp)〉〈b, (vp, . . . , vp+q)〉 ,required to be valid for all (v0, . . . , vp+q) ∈ Sp+q(K). This defines a graded Z2-

algebra structure on C∗(K). The formula δ(a b) = δa b+ a δb is proven as

for Lemma 4.1.2, whence a graded algebra structure on H∗(K). These definitionsimply that the isomorphism

H∗φ≤ : (H∗(K),+,)≈−→ (H∗(K),+,≤)

of § 2.10 is an isomorphism of graded algebras. As, by Corollary 2.10.10, H∗φ≤ isindependent of the simplicial order “≤”, so is the cup product on H∗(K).

Corollary 4.1.4 (Commutativity of the cup product). The cup product insimplicial cohomology is commutative, i.e. a b = b a for all a, b ∈ H∗(K).

Proof. Let a, b ∈ Z∗(K) representing a, b. Let “≤” be a simplicial order onK. In Z∗(K), one has

a ≤ b = b ≥ a ,

where “≥” is the opposite order of “≤”. By Proposition 4.1.3, this proves Corol-lary 4.1.4.

4.1. THE CUP PRODUCT 111

The commutativity of the cup product is an important feature of the mod 2cohomology. In other coefficients, this holds true only up to signs.

Let GrA be the category whose objects are commutative graded Z2-algebrasand whose morphisms are algebra maps. Corollary 4.1.4 says that H∗(K) is anobject of GrA. There is an obvious forgetful functor from GrA to GrV.

Proposition 4.1.5 (Functoriality of the cup product). Let f : L → K be asimplicial map. Then H∗f : H∗(K) → H∗(L) is multiplicative: H∗f(a b) =H∗f(a) H∗f(b) for all a, b ∈ H∗(K).

Proof. The proof of Proposition 4.1.3 shows that H∗(K) is an object of GrA.

Using Corollary 2.10.8, it also shows that the isomorphism H∗ψ : H∗(K)→ H∗(K)

is a GrA-isomorphism. Let a ∈ Cp(K) and b ∈ Cq(K). Then, for all σ =

(v0, . . . , vp+q) ∈ Sp+q(K), one has

〈C∗f(a b), σ〉 = 〈a b, (f(v0), . . . , f(vp+q))〉= 〈a, (f(v0), . . . , f(vp))〉〈b, (f(vp), . . . , f(vp+q))〉= 〈C∗f(a), (v0, . . . , vp)〉〈C∗f(b), (vp, . . . , vp+q)〉= 〈C∗f(a) C∗f(b), σ〉 .

By Lemma 2.3.3, this implies that C∗f(a b) = C∗f(a) C∗f(b). We de-

duce that H∗f : H∗(K)→ H∗(L) is multiplicative. Using Proposition 2.10.11, thisimplies that H∗f is multiplicative.

Corollary 4.1.6. The simplicial cohomology is a contravariant functor fromSimp to GrA.

The cup product may also be defined in relative simplicial cohomology. Let L1

and L2 be two subcomplexes of K. For any simplicial order “≤” on K, one has

C∗(K,L1)≤ C∗(K,L2) ⊂ C∗(K,L1 ∪ L2) .

Hence, we get a map

H∗(K,L1)×H∗(K,L2)−→ H∗(K,L1 ∪ L2)

which is bilinear and commutative. In particular, we get relative cup products

H∗(K,L)×H∗(K)−→ H∗(K,L) and H∗(K)×H∗(K,L) −→ H∗(K,L)

which are related as described by the following two lemmas.

Lemma 4.1.7. Let (K,L) be a simplicial pair. Denote by j : (K, ∅) → (K,L)the inclusion. Let a ∈ Hp(K,L) and b ∈ Hq(K,L). Then, the equality

H∗j(a) b = a b = a H∗j(b)

holds in Hp+q(K,L).

Proof. Denote also by a ∈ Zp(K,L) and b ∈ Zq(K,L) cocycles representingthe cohomology classes a and b. Choose a simplicial order on K and let σ =v0, . . . , vp+q ∈ Sp+q(K) − Sp+q(L) with v0 < · · · < vp+q. Let σ1 = v0, . . . , vpand σ2 = vp, . . . , vp+q. One has

(4.1.3) 〈C∗j(a) b, σ〉 = 〈C∗j(a), σ1〉〈b, σ2〉 = 〈a, C∗j(σ1)〉〈b, σ2〉and

(4.1.4) 〈a b, σ〉 = 〈a, σ1〉〈b, σ2〉 .

112 4. PRODUCTS

If σ1 ∈ Sp(L), then C∗j(σ1) = 0 and the right hand sides of (4.1.3) and (4.1.4)both vanish. If σ1 /∈ Sp(L), then C∗j(σ1) = σ1 and the right hand sides of (4.1.3)and (4.1.4) are equal. As Cp+q(K,L) is the vector space with basis Sp+q(K) −Sp+q(L), this proves that H∗j(a) b = a b. The other equation is provensimilarly.

The proof of the following lemma, quite similar to that of Lemma 4.1.7, is leftto the reader (Exercise 4.1).

Lemma 4.1.8. Let (K,L) be a simplicial pair. Denote by j : (K, ∅) → (K,L)the inclusion. Let a ∈ Hp(K) and b ∈ Hq(K,L). Then, the equality

H∗j(a b) = a H∗j(b)

holds in Hp+q(K).

There is also a relationship between the relative cup product and the connectinghomomorphism δ∗ of a simplicial pair.

Lemma 4.1.9. Let (K,L) be a simplicial pair. Denote by i : L→ K the inclu-sion. Let a ∈ Hp(K) and b ∈ Hq(L). Then, the equality

δ∗(b H∗i(a)) = δ∗b a

holds true in Hp+q+1(K,L).

Proof. Denote also by a ∈ Zp(L) and b ∈ Zq(L) the cocycles represent-ing the cohomology classes a and b. Let b ∈ Cq(K) be an extension of thecochain b. The cochain b a ∈ Cp+q(K) is then an extension of b C∗i(a).By Lemma 2.7.1, δK(b a) ∈ Zp+q+1(K,L) represents δ∗(b H∗i(a)), whereδK : C∗(K) → C∗+1(K) is the coboundary homomorphism for K. As a is a cocy-cle, one has δK(b a) = δK(b) a. By Lemma 2.7.1 again, δK(b) a representsthe cohomology class δ∗b a. This proves the lemma.

4.1.2. The cup product in singular cohomology. Let X be a topologicalspace and let σ : ∆m → X be an element of Sm(X). For 0 ≤ p, q ≤ m, we definepσ ∈ Sp(X) and σq ∈ Sq(X) by

pσ(t0, . . . , tp) = σ(t0, . . . , tp, 0 . . . , 0) and σq(t0, . . . , tq) = σ(0, . . . , 0, t0, . . . , tq) .

The singular simplexes pσ and σq are called the front and back faces of σ. Leta ∈ Cp(X) and b ∈ Cq(X). Using Point (c) of Lemma 2.3.3, we define a cochaina b ∈ Cp+q(X) by the formula

(4.1.5) 〈a b, σ〉 = 〈a, pσ〉〈b, σq〉 ,required to be valid for all σ ∈ Sp+q(X). This defines a bilinear map

(4.1.6) Cp(X)× Cq(X)−→ Cp+q(X) .

The formula of Lemma 4.1.2 holds true, with the same proof. Hence, we get a cupproduct in singular cohomology: Hp(X) × Hq(X)

−→ Hp+q(X), giving rise to acomposition law

H∗(X)×H∗(X)−→ H∗(X) .

Proposition 4.1.10. (H∗(X),+,) is a commutative graded Z2-algebras.

4.1. THE CUP PRODUCT 113

Proof. The associativity and distributivities are easily deduced from the def-initions, like for the cup product in simplicial cohomology. If X is empty, thenH∗(X) = 0 and there is nothing to prove. Otherwise, the neutral element for is the class of the unit cochain 1 ∈ H0(X). Proving the commutativity directly israther difficult. We shall use that the singular cohomology of X is that of a sim-plicial complex (see Theorem 3.7.4), together with Proposition 4.1.11 below, whoseproof is straightforward.

Proposition 4.1.11. Let K be a simplicial complex. For any simplicial order≤ on K, the isomorphism

R∗≤ : H∗(|K|) ≈−→ H∗(K)

of Theorem 3.6.3 is an isomorphism of graded algebras.

Proposition 4.1.12. The singular cohomology is a contravariant functor fromTop to GrA.

Proof. By Proposition 4.1.10, we already know that H∗(X) is an object ofGrA. We also know, by Proposition 3.1.22, that H∗() is a contravariant functorfrom Top to GrV. It remains to prove the multiplicativity of H∗f : H∗(X) →H∗(Y ) for a continuous map f : Y → X . If σ ∈ Sp+q(X), then f pσ = p(f σ) andf σq = (f σ)q. Thus, the proof that C∗f(a b) = C∗f(a) C∗f(b) is the sameas for Proposition 4.1.5.

To get relative cup products as in simplicial cohomology, some hypothesis re-lated to the techniques of small simplexes (§ 3.1.4) is required. Let Y1 and Y2 besubspaces of a topological space X . Let Y = Y1 ∪ Y2 and B = Y1, Y2. We saythat (Y1, Y2) is an excisive couple if H∗(Y )→ H∗B(Y ) is an isomorphism.

Lemma 4.1.13. A couple (Y1, Y2) of subspaces of X is excisive if and only if theinclusion (Y1, Y1 ∩ Y2) → (Y1 ∪ Y2, Y2) induces an isomorphism in (co)homology.

Proof. Let Y = Y1 ∪ Y2 and B = Y1, Y2 as above. There is a morphism

0 // C∗(Y, Y2) //

C∗(Y ) //

C∗(Y2) //

0

0 // C∗(Y1, Y1 ∩ Y2) // C∗B(Y ) // C∗(Y2) // 0

of short exact sequences of singular cochain complex. It induces a morphism of theassociated long exact sequences on cohomology which, by the five-lemma, impliesthe result.

Lemma 4.1.14. Let (Y1, Y2) be an excisive couple of a topological space X.Then, (4.1.6) defines a relative cup product

(4.1.7) H∗(X,Y1)×H∗(X,Y2) −→ H∗(X,Y1 ∪ Y2)which is bilinear. The analogues of Lemmas 4.1.7–4.1.9 hold true.

Proof. Let Y = Y1 ∪ Y2 and B = Y1, Y2. Equation (4.1.5) gives a bilinearmap

C∗(X,Y1)× C∗(X,Y2) −→ C∗(X,Y B)

114 4. PRODUCTS

where C∗(X,Y B) = ker(C∗(X)→ C∗B(Y )). There is a commutative diagram

Hk(X)

≈

// Hk(Y )

// Hk+1(X,Y )

// Hk+1(X)

≈

// Hk+1(Y )

Hk(X) // HkB(Y ) // Hk+1(X,Y B) // Hk+1(X) // Hk+1

B (Y )

where the lines are exact. By the five-lemma, ifH∗(Y )→ H∗B(Y ) is an isomorphism,so is H∗(X,Y ) → H∗(X,Y B), which gives (4.1.7). The properties of the relativecup product listed at the end of Lemma 4.1.14 are proved as in the simplicialcase.

Remark 4.1.15. The couple (Y1, Y2) is excisive in X if and only if it is excisivein Y1 ∪ Y2. Thus, by Proposition 3.1.34, (Y1, Y2) is excisive when Y1 and Y2 areboth open. Also, (Y1, Y2) is excisive when one of the subspaces Yi is containedin the other, for instance if one is empty or if Y1 = Y2. In some situations, thehypothesis can be fulfilled by enlarging Yi to Y

′i without changing the homotopy

type, and then (4.1.7) makes sense. As in Proposition 3.1.54 and its proof, this isthe case if X is a CW-complex and Yi are subcomplexes. Note that, if (Y1, Y2) isexcisive, then the Mayer-Vietoris sequence for (Y1 ∪ Y2, Y1, Y2, Y1 ∩ Y2) holds true,by Lemma 4.1.13 and Exercise 3.11.

4.2. Examples

4.2.1. Disjoint unions. Let X be a topological space which is a disjointunion:

X =⋃

j∈JXj .

By Proposition 3.1.11, the family of inclusions ij : Xj → X induce an isomorphismin GrV

H∗(X)(H∗ij)

≈// ∏

j∈J H∗(Xj) .

By Proposition 4.1.12, H∗ij is a homomorphism of algebras for each j ∈ J . Hence,the above map (H∗ij) is an isomorphism of graded algebras.

4.2.2. Bouquets. Let (Xj , xj), with j ∈ J , be a family of well pointed spaceswhich are path-connected. By Proposition 3.1.51, the family of inclusions ij : Xj →X =

∨j∈J Xj, for j ∈ J , gives rise to isomorphisms on reduced cohomology

H∗(X)(H∗ij)

≈// ∏

j∈J H∗(Xj) .

The reduced and unreduced cohomologies share the same positive parts: H>0( ) =

H>0( ). As each space Xj is path-connected, so is the their bouquet X . Thus

H>0(X) = H∗(X) and we get a GrV-morphism

H>0(X)(H∗ij)

≈// ∏

j∈J H>0(Xj) .

4.2. EXAMPLES 115

Being induced by continuous maps, (H∗ij) is multiplicative. AsX is path-connected,this produces the GrA-isomorphism

(4.2.1) H∗(X)≈−→ Z2 1⊕

∏

j∈J

H>0(Xj) .

When J is finite, one can also use the projections πj : X → Xj defined in (3.1.32).By Proposition 3.1.52, they produce a GrV-isomorphism

⊕j∈J H

>0(Xj)

∑H∗πj

≈// H>0(X) (J finite) .

Being induced by continuous maps,∑H∗πj is multiplicative. As X is path-

connected, this produces the GrA-isomorphism

(4.2.2) Z2 1⊕⊕

j∈J

H>0(Xj)≈−→ H∗(X) (J finite) .

4.2.3. Connected sum(s) of closed topological manifolds. A closed n-dimensional topological manifold is a compact space such that each point has anopen neighbourhood homeomorphic to Rn.

LetM1 andM2 be two closed n-dimensional topological manifolds. We supposethat M1 and M2 are connected. Let Bj ⊂ Mj be two embedded compact n-ballswith boundary Sj. We suppose that each ball Bj is nicely embedded in a bigger ball;this implies that (Mj , Bj) and (Mj, Sj) are good pairs. Given a homeomorphism

h : B1≈−→ B2, form the closed topological manifold

M =M1♯hM2 = (M1 − intB1) ∪h (M2 − intB2) .

The manifold M is called a connected sum of M1 and M2. Though connectedtopological manifolds are homogeneous for nicely embedded balls (see e.g. [95,Theorem 6.7]), the homeomorphism type of M may depend on h: for example, if his obtained from h by precomposition with a homeomorphism of B1 which reversesthe orientation, then M1♯hM2 does not have, in general, the same homotopy typeas M1♯hM2 (for instance, if M1 = M2 = CP 2, the two cases are distinguishedby their integral intersection form). In most applications in the literature, theconnected sum is defined for oriented manifolds and one requires that h reverses theorientation; this makes the oriented homeomorphism type of M1♯M2 well defined.However, by Proposition 4.2.1 below, the mod2-cohomology algebra of M1♯hM2

does not depend on h, up to algebra isomorphism.If each Mj admit a triangulation |Kj| ≈ Mj, then Kj is a connected n-

dimensional pseudomanifold (see Corollary 5.2.7). The connected sum may be donein the world of pseudomanifolds, using n-simplexes for the balls Bj . By Proposi-tion 2.4.4, Hn(Mj) = Z2, generated by the fundamental class [Mj ]. The statementHn(Mj) = Z2 also holds for closed connected topological manifolds (see e.g. [82,Theorem 3.26]). We denote by [Mj ]

♯ the generator of Hn(Mj) = Z2.

Proposition 4.2.1. Under the above hypotheses, the cohomology ring H∗(M1♯hM2)is isomorphic to the quotient of Z2 1⊕H>0(M1)⊕H>0(M2) by the ideal generatedby [M1]

♯ + [M2]♯:

H∗(M1♯hM2) ≈ Z2 1⊕H>0(M1)⊕H>0(M2)/([M1]

♯ + [M2]♯) .

116 4. PRODUCTS

In particular, under this isomorphism, the classes [M1]♯ and [M2]

♯ both correspondto the fundamental class [M ]♯ of M .

Proof. Form the space M = M1 ∪h M2 and let B ⊂ M be the commonimage of B1 and B2 in M , with boundary S. As (Mj , Bj) are good pairs, Proposi-

tion 3.1.54 provides a Mayer-Vietoris sequence for (M,M1,M2, B). As B has thecohomology of a point, one gets a multiplicative GrV-isomorphism:

α : H>0(M)≈−→ H>0(M1)⊕H>0(M2) .

As Mj is connected, so is M and α extend to a GrA-isomorphism

α : H∗(M)≈−→ Z2 1⊕H>0(M1)⊕H>0(M2) .

Let M = M1♯hM2 ⊂ M . The pair (M,M) is obviously a good pair, whence, by

excision and homotopy, the isomorphism H∗(M,M) ≈ H∗(B,S). The non-zero

part of H∗(B,S) is Hn(B,S) = Z2. Therefore, the homomorphism β∗ : Hk(M)→Hk(M) induced by the inclusion is an isomorphism, except possibly for k = n− 1

or n. In these degrees, the cohomology sequence of (M,M) looks like

Hn−1(M)β∗ // Hn−1(M)

δ∗ // Hn(M,M) //

≈

Hn(M)

≈

β∗ // Hn(M)

≈

// 0

Z2 Z2 ⊕ Z2 Z2

Therefore, β∗ : Hk(M)→ Hk(M) is an isomorphism for k ≤ n−1 and theGrA ho-

momorphism β∗ : H∗(M)→ H∗(M) is onto. The kernel of β∗ : Hn(M)→ Hn(M)is of dimension 1 and, by symmetry (M1∪hM2 =M2∪h−1M1), it must be generatedby [M1]

♯ + [M2]♯.

Remark 4.2.2. If we work simplicially with pseudomanifolds, the fact thatker(β∗ : Hn(M) → Hn(M)) contains [M1]

♯ + [M2]♯ may be seen directly. Indeed

the n-cocycle consisting of the n-simplex Bj represents [Mj ]♯ by Proposition 2.4.4.

Hence, the n-cocycle B represents [M1]♯ + [M2]

♯ in Hn(M) and is in kerβ∗.

4.2.4. Cohomology algebras of surfaces. We start with the triangulationM of RP 2 drawn in Figure 2.2 p. 26. We use the simplicial order given by thenumeration 0, . . . , 5 of the vertices. The computation of H∗(M) is given in (2.4.8)and the generator of H1(M) = Z2 is given by the cocycle a given in (2.4.9):

a = α =1, 2, 2, 3, 3, 4, 4, 5, 5, 1

⊂ S1(RP 2) .

We see that, in C2(M),

a a =1, 2, 3, 2, 3, 4, 3, 4, 5

.

As a a contains an odd number of 2-simplexes, Proposition 2.4.4 implies thata a is the generator [M ]♯ of H2(M). Therefore, one gets a GrA-isomorphism

H∗(RP 2) ≈ Z2[a]/(a3)

from H∗(RP 2) to the quotient of the polynomial ring Z2[a] by the ideal generatedby a3. Using (4.2.1), this shows that RP 2 and S1 ∨ S2 do not have the samehomotopy type though they have the same Betti numbers.

4.2. EXAMPLES 117

Our next example is the torus T 2. We use the triangulation given in Figure 2.3on p. 27 which shows two 1-cocycles a, b ∈ C1(T 2) whose cohomology classes, againdenoted by a and b, form a basis of H1(T 2) ≈ Z2 ⊕ Z2. One checks that thefollowing equations hold in C2(T 2):

a a =4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9

b b =2, 3, 6, 3, 6, 8

a b =6, 7, 8

b a =7, 8, 9

.

In H2(T 2) = Z2, generated by [T 2]♯, Proposition 2.4.4 implies that

a a = b b = 0 and a b = b a = [T 2]♯ .

Observe that a b 6= b a in C2(T 2), the equality only holding true in cohomol-ogy. We get a GrA-isomorphism

H∗(T 2) ≈ Z2[a, b]/(a2, b2) .

Our third example is the Klein bottle K, using the triangulation given in Fig-ure 2.4 on p. 28: analogously to the case of the torus, Figure 2.4 shows two 1-cocyclesa, b ∈ C1(K) whose cohomology classes, again denoted by a and b, form a basis ofH1(K) ≈ Z2 ⊕ Z2. The following equations hold in C2(K):

a a =4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 4, 5, 9

b b =2, 3, 6, 3, 6, 8

a b =6, 7, 8

b a =7, 8, 9

.

In H2(K) = Z2, generated by [K]♯, Proposition 2.4.4 implies

a a = [K]♯ , b b = 0 and a b = b a = [K]♯ .

Though H∗(T 2) and H∗(K) are GrV-isomorphic, we see that they are notGrA-isomorphic. Indeed, for a space X , consider the cup-square map

H∗(X)2

−−→ H∗(X)

given by2 (x) = x x. Note that this map is linear, since the ground field is Z2.Our above computations show that 2= 0 for X = T 2 but not for X = K. It doesnot vanish either for X = RP 2, as seen above. Now, it is classical that a connectedclosed surfaceX is a connected sum of tori if X is orientable and a connected sum ofprojective spaces otherwise. Hence, Proposition 4.2.1 implies that the orientabilityof a connected surface may be seen on its cohomology algebra:

Proposition 4.2.3. LetM be a closed connected surface. ThenM is orientableif and only if its cup-square map H1(M)→ H2(M) vanishes.

Remark 4.2.4. As a consequence of Wu’s formula, we shall see in Corol-lary 9.8.5 that Proposition 4.2.3 generalizes in the following way: a closed connectedn-dimensional manifoldM is orientable if and only if the linear map Sq1 : Hn−1(M)→Hn(M) vanishes.

118 4. PRODUCTS

Finally, we see that closed surfaces are distinguished by their cohomology al-gebra.

Proposition 4.2.5. Two closed surfaces are diffeomorphic if and only if theircohomology algebra are GrA-isomorphic.

Proof. By Proposition 4.2.3, the cohomology algebra determines whether aclosed surfaceM is orientable or not. IfM is orientable, thenM is a connected sumofm tori and, by Proposition 4.2.1H1(M) ≈ Z2m

2 . IfM is not orientable, thenM isa connected sum ofm projective planes and, by Proposition 4.2.1H1(M) ≈ Zm2 .

4.3. Two-fold coverings

4.3.1. H1, fundamental group and 2-fold coverings. Let (Y, y) and (Y ′, y′)be two pointed spaces. Let [Y, Y ′]• be the set of homotopy classes of pointed mapsfrom Y to Y ′ (the homotopies also preserving the base point). Let F : [Y, Y ′]• →[Y, Y ′] be the obvious forgetful map.

Let (X, x) be a pointed topological space. We first define a map e : H1(X)→map(π1(X, x),Z2). Let a ∈ H1(X). If c : S1 → X is a pointed map representing[c] ∈ [S1, X ]• = π1(X, x), we set e(a)([c]) = H∗c(a) ∈ H1(S1) = Z2. As H∗c =H∗c′ if c is homotopic to c′, the map is well defined. Observe that map(π1(X, x),Z2)is naturally a Z2-vector space, containing hom(π1(X, x),Z2) as a linear subspace.

Lemma 4.3.1. Let X be a connected CW-complex, pointed by x ∈ X0. Thenthe map e is an isomorphism

e : H1(X)≈−→ hom(π1(X, x),Z2) .

Proof. We first prove that the image of e lies in hom(π1(X, x),Z2). Themultiplication in π1(X, x) = [S1, X ]• may be expressed using the comultiplicationµ : S1 →→ S1/S0 ≈ S1 ∨ S1. Then [c][c′] = [(c ∨ c′)µ]. Using that H1(S1 ∨ S1) ≈H1(S1)×H1(S1) (see Proposition 3.1.51), one has

e(a)([c][c′]) = H∗µ(e(a)([c]), e(a)([c′])

)= e(a)([c]) + e(a)([c′])

for all a ∈ H1(X). This proves that e([c][c′]) = e([c]) + e([c′]). The equalitye(a+ b)([c]) = e(a)([c]) + e(b)([c]) is obvious, so e is a homomorphism.

Let us consider RP∞ with its standard CW-structure of Example 3.4.5, withone cell in each dimension, pointed by its 0-cell a. Van Kampen’s Theorem impliesthat π1(RP∞, a) = Z2. The fundamental group functor gives rise to a map

(4.3.1) [X,RP∞]•≈−→ hom(π1(X, x),Z2)

which is a bijection. Indeed, the bijectivity is established in the same way as, inProposition 3.8.3, the fact that φ : [X,RP∞]→ H1(X) is a bijection. The forgetfulmap F : [X,RP∞]• → [X,RP∞] and the homomorphism e fit in the commutativediagram

(4.3.2)

[X,RP∞]•F //

≈

[X,RP∞]

≈

hom(π1(X, x),Z2) H1(X)eoo

The map F is surjective: if f : X → RP∞, any path γ from f(x) to a extendsto homotopy from f to a pointed map. This follows from the fact that (X, a)

4.3. TWO-FOLD COVERINGS 119

is cofibrant (see Proposition 3.4.1). Hence, the commutativity of Diagram (4.3.2)implies that e (and F ) are bijective.

We now turn our attention to 2-fold coverings. The reader is assumed to havesome familiarity with the theory of covering spaces, as presented in many textbooks(see e.g. [179, Chapter 2] or [82, section 1.3]).

Let X be a connected CW-complex, pointed by x ∈ X0. Two covering pro-jections pi : Xi → X are equivalent if there exists a homeomorphism h : X1 → X2

such that p2h = p1. Denote by Cov2(X) the set of equivalence classes of 2-foldcoverings of X .

Let p : X → X be a 2-fold covering. Choose x ∈ p−1(x). Then p∗(π1(X, x))is a subgroup of index ≤ 2 of π1(X, x). Let Grp2(π1(X, x)) be the set of such

subgroups. A subgroup of index ≤ 2 being normal, the subgroup p∗(π1(X, x)) doesnot depend on the choice of x ∈ p−1(x). We thus get a map

Cov2(X)≈−→ Grp2(π1(X, x))

which is a bijection (see, e.g, [82, Theorem 1.38]). For example, the trivial 2-foldcovering ±1×X → X corresponds to the whole group π1(X, x) which is of index1 ≤ 2. An element H ∈ Grp2(π1(X, x)) is the kernel of a unique homomorphismπ1(X, x)→→ π1(X, x)/H → Z2. This gives a bijection

Grp2(π1(X, x))≈−→ hom(π1(X, x),Z2) .

If f : X → RP∞ is a continuous map, one can form the pullback diagram

(4.3.3)

Xf //

p

S∞

p∞

Xf // RP∞ .

In detail, X = (u, z) ∈ X×S∞ | f(u) = p∞(z), with p(u, z) = u and f(u, z) = z.We say that the covering projection p is induced from p∞ by the map f and writep = f∗p∞. Observe that p correspond to the subgroup kerπ1f . Thus, homotopicmaps induce equivalent coverings and we get a map ind: [X,RP∞] → Cov2(X).These various maps, together with those of (4.3.2) sit in the commutative diagram

(4.3.4)

[X,RP∞]ind //

φ

≈ &&

Cov2(X)

≈

H1(X)

e

≈ ''PPPP

PPPP

PPGrp2(π1(X, x))

≈

[X,RP∞]•

≈F

OO

≈ // hom(π1(X, x),Z2)

The commutativity of Diagram (4.3.4) implies the following proposition.

Proposition 4.3.2. ind: [X,RP∞]→ Cov2(X) is a bijection.

Let p : X → X be a 2-fold covering. A continuous map f : X → RP∞ suchthat p is equivalent to f∗p∞ is called a characteristic map for the covering p.Proposition 4.3.2 implies the following corollary.

120 4. PRODUCTS

Corollary 4.3.3. Let X be a connected CW-complex. Then, any 2-fold cov-ering admits a characteristic map. Two such characteristic maps are homotopic.

Let p : X → X be a 2-fold covering. The correspondence which, over eachx ∈ X , exchanges the two points of p−1(x) defines a homeomorphism τ : X → X,which is an involution (i.e. τ τ = id) without fixed point. Also, τ is a decktransformation, i.e. pτ = p. We call τ the deck involution of p. For the coveringp∞ : S∞ → RP∞, one has τ(z) = −z.

Lemma 4.3.4. A continuous map f : X → RP∞ is a characteristic map for the2-fold covering p : X → X if and only if there exists a commutative diagram

Xf //

p

S∞

p∞

Xf // RP∞ ,

where f is a continuous map such that f τ(v) = −f(v).

Proof. Let X → X be the covering induced by f (see (4.3.3)). If f is a

characteristic map for p, there is a homeomorphism g : X≈−→ X such that pg = p.

Therefore, g satisfies gτ = τ g. As f τ(v) = −f(v), the map f = f g satisfies

the requirements of Lemma 4.3.4. Conversely, given f , let g : X → X given byg(v) = (p(v), f (v)). The map g satisfies pg = p and gτ = τ g. Hence, g issurjective and is a covering projection. Since both p and p are 2-folds coverings, gis a homeomorphism and p is equivalent to p.

Example 4.3.5. The inclusion i : RPn → RP∞ is covered by the τ -equivariantmap i : Sn → S∞. By Lemma 4.3.4, the map i is characteristic map for thecovering Sn → RPn. In particular, the identity of RP∞ is a characteristic map forthe covering S∞ → RP∞.

4.3.2. The characteristic class. Diagram (4.3.4) together with Proposi-tion 4.3.2 provides a bijection

(4.3.5) w : Cov2(X)≈−→ H1(X) .

This associates to a 2-fold covering p : X → X its characteristic class w(p) ∈H1(X). For instance, the characteristic class w(p∞) for the covering p∞ : S∞ →RP∞ is the non-zero element ι ∈ H1(RP∞) = Z2. Indeed, as H

1(RP∞) = Z2, theset Cov2(RP∞) has two elements and the trivial covering corresponds to 0. As S∞

is connected, p∞ is not the trivial covering, hence w(p∞) = ι. The following lemmais obvious.

Lemma 4.3.6. Let p : X → X be a 2-fold covering over a CW-complex. Then

(1) if f : X → RP∞ is a characteristic map for the covering p, thenw(p) = H∗f(w(p∞)) = H∗f(ι).

(2) if g : Y → X is a continuous map, then w(g∗p) = H∗g(w(p)).(3) p is the trivial covering if and only if w(p) = 0.


Let us give geometric descriptions of the characteristic class w(p). Choose a

set-theoretic section b : X → X of p and let B = b(X) ⊂ X . We consider B

as a singular 0-cochain of X. Using the coboundary δ : C0(X) → C1(X), we get

δB ∈ C1(X) which is the connecting 1-cochain for B: for σ ∈ S1(X), 〈δ(B), σ〉 = 1if and only if the (non-oriented) path σ connects a point in B to a point in X −B(see Example 3.1.7). Observe that δ(B) = δ(X − B) = δ(τ(B)), where τ is thedeck involution of p. Hence

〈δ(B), τ σ〉 = 〈δ(B), C∗τ(σ)〉 = 〈C∗τ δ(B), σ〉= 〈δC∗τ(B), σ〉 = 〈δ(τ(B)), σ〉= 〈δ(B), σ〉 .

Thus, 〈δ(B), σ〉 depends only on pσ ∈ S1(X). This permits us to define a singular1-cochain wb(p) ∈ C1(X) by the formula

〈wb(p), σ〉 = 〈δ(B), σ〉

where σ ∈ S1(X) is any lifting of σ ∈ S1(X).

Proposition 4.3.7. wb(p) is a 1-cocycle representing w(p) ∈ H1(X).

Proof. Let wb = wb(p). Let σ2 ∈ S2(X). If σ2 ∈ S2(X) is a lifting of σ2, then

the 1-simplexes in ∂(σ2) are liftings of those in ∂(σ2). Therefore

(4.3.6) 〈δ(wb), σ2〉 = 〈wb, ∂(σ2)〉 = 〈δ(B), ∂(σ2)〉 = 0 ,

which proves that wb is a cocycle.We next prove that the cohomology class wb ∈ H1(X) of wb does not depend

on the set-theoretic section b. Let b′ : X → X ′ another such section, giving B′ =b′(X) ∈ C0(X). Define r ∈ C0(X) by

〈r, x〉 = 〈B, x〉+ 〈B′, x〉 ,

where x is a chosen element in p−1(x). If ˜x is another choice, one has 〈B, ˜x〉 =〈B, x〉+ 1 and 〈B′, ˜x〉 = 〈B′, x〉+ 1 in Z2, so r is well defined. Let σ ∈ S1(X) with

end points u and v. Let σ ∈ S1(X) be a lifting of σ with end points u and v. Then

〈wb + wb′ , σ〉 = 〈δ(B) + δ(B′), σ〉= 〈B, u〉+ 〈B′, u〉+ 〈B, v〉+ 〈B′, v〉= 〈δ(r), σ〉 .

This proves that wb′ = wb + δ(r) and thus [wb′ ] = [wb]. Denote by w(p) ∈ H1(X)the cohomology class [wb].

We can now prove that w(p) = w(p′) if p′ : X ′ → X is a 2-fold covering equiv-

alent to p. Indeed, if h : X≈−→ X ′ is a homeomorphism such that p′ h = p, then,

wb(p) = wh b(p′), which implies that w(p) = w(p′).

Choosing a characteristic map f : X → RP∞ for p, we now have w(p) = w(p),

where p : X → X is the induced covering of Diagram (4.3.3). Choose a set-theoreticsection b0 : RP∞ → S∞ of p∞ and set B0 = p∞(RP∞) and w0 = [wb0(p∞)] ∈H1(RP∞). This gives rise to a set-theoretic section b of p by the formula b(x) =

(x, b0f(x)). It satisfies B = b(X) = f−1(B0), where f : X → S∞ is the map

122 4. PRODUCTS

covering f , as in (4.3.3). Let σ ∈ S1(X) with a lifting σ ∈ S1(X). Then, f σ is alifting of f σ in S1(S∞) and we have

〈C ∗ f(wb0), σ〉 = 〈wb0 , f σ〉 = 〈δ0(B0), f σ〉= 〈δ0(B0), C∗f(σ)〉 = 〈C∗f δ0(B0), σ〉= 〈δC∗f(B0), σ〉 = 〈δ(f−1(B0)), σ〉 = 〈δ(B), σ〉= 〈wb(p), σ〉 .

Hence,

(4.3.7) w(p) = C∗f(w0) .

Together with Lemma 4.3.6, Equation (4.3.7) reduces the proof of Proposition 4.3.7,to showing that w0 = w(p∞). As 0 6= w(p∞) ∈ H1(RP∞) = Z2, it is enough toprove that w0 6= 0. By Equation (4.3.7) again, it is enough to find a covering

q : Y → Y over some CW-complex Y for which w(q) 6= 0.We take for q the double covering q : S1 → S1. Let σ : ∆1 → S1 given by

σ(t, 1 − t) = e2iπt. As σ(0, 1) 6= σ(1, 0), one has 〈w(q), σ〉 6= 0. But σ is a 1-cocycle representing the generator of H1(S1) = Z2 (see Corollary 3.2.4). Hence,w(q) 6= 0.

Remark 4.3.8. Let p : X → X be a two-fold covering over a CW-complex X .To describe the characteristic class w(p) in the cellular cohomology of X , choose a

section b : X0 → X and see B = b(X0) ⊂ X0 as a cellular 0-cochain of X (for thecellular decomposition induced from that of X). Let ϕ : Λ1(X)×I → X be a global

characteristic maps for the 1-cells of X . Let ϕ : Λ1(X) × I → X be the lifting of

ϕ for which ϕ(λ, 0) ∈ B. Consider the cellular 1-cochain wB ∈ C1(X) defined, forλ ∈ Λ1(X), by

〈wB , λ〉 =1 if ϕ(λ, 1) /∈ B0 otherwise.

Note that 〈wB , λ〉 = 〈δ(B), λ〉 where λ is any one-cell of X above λ, which, as

in (4.3.6), proves that wB is a cellular cocycle. We claim that [wB ] ∈ H1(X)

corresponds to w(p) ∈ H1(X), under the identification of H1(X) and H1(X) asthe same subgroup of H1(X1) (see (3.5.5)). We can thus suppose that X = X1.We can also suppose that X is connected. If T is a maximal tree of X1, thenthe quotient map X1 → X1/T is a homotopy equivalence by Proposition 3.4.1and Lemma 3.1.44. The covering p is then induced from one over X1/T , so wecan assume that X1 is a bouquet of circles indexed by Λ1. For each one cell λ, acharacteristic map ϕλ : D

1 → X1 gives a singular 1-simplex of X1 (identifying D1

with ∆1). If ϕλ is a lifting of ϕλ, one has

(4.3.8) 〈wB , λ〉 = 〈wb, ϕλ〉 =1 if ϕλ is a loop

0 otherwise,

where b : X → X is a set theoretic section of p extending b. As [ϕλ] | λ ∈ Λ1 isa basis for H1(X1), Equation (4.3.8) implies that [wB ] ∈ H1(X1) corresponds tow(p) ∈ H1(X).


4.3.3. The transfer exact sequence of a 2-fold covering. Let p : X → Xbe a 2-fold covering projection with deck involution τ . To each singular simplexσ : ∆m → X , one can associate the set of the two liftings of σ into X . This defines amap from Sm(X) to Cm(X), extending to a linear map tr∗ : Cm(X)→ Cm(X). Themap tr∗ is clearly a chain map. By § 2.3, this gives risen to two GrV-morphisms

tr∗ : Hm(X)→ Hm(X) and tr∗ : Hm(X)→ Hm(X)

satisfying 〈tr∗(a), α〉 = 〈a, tr∗(α)〉. The linear maps tr∗ and tr∗ are called thetransfer homomorphisms for the covering p.

The transfer homomorphism in cohomology and the characteristic class w(p) ∈H1(X) are related by the following exact sequence.

Proposition 4.3.9 (Transfer exact sequence). The sequence

· · · → Hm(X)H∗p−−−→ Hm(X)

tr∗−−→ Hm(X)w(p)−−−−−−→ Hm+1(X)

H∗p−−−→ · · ·is exact. It is functorial with respect to induced coverings.

Proof. The sequence

0→ C∗(X)tr∗−−→ C∗(X)

C∗p−−→ C∗(X)→ 0 .

is clearly an exact sequence of chain complexes and it is functorial with respect toinduced coverings. By Kronecker duality, it gives a short exact sequence of cochaincomplexes

(4.3.9) 0→ C∗(X)C∗p−−→ C∗(X)

tr∗−−→ C∗(X)→ 0 .

By Proposition 2.6.2, this gives rise to a connecting homomorphism d∗ : H∗(X)→H∗+1(X) and a functorial long exact sequence

· · · → Hm(X)H∗p−−−→ Hm(X)

tr∗−−→ Hm(X)d∗−→ Hm+1(X)

H∗p−−−→ · · · .It just remains to identify d∗ with w(p) −.

To construct the connecting homomorphism d∗ we need a GrV-section of tr∗ inSequence (4.3.9). Choose a set-theoretic section b : X → X of p. If σ : ∆m → X is

a singular 1-simplex of X , define b×(σ) : ∆m → X to be the unique lifting of σ with

b×(σ)(1, 0, . . . , 0) ∈ b(X). This defines a map b× : S(X) → S(X). If a ∈ Cm(X),

we consider a as a subset of Sm(X) and so its direct image b×(a) ⊂ Sm(X) is an m-

cochain of X . This determines a GrV-morphism b× : C∗(X)→ C∗(X) which is asection of tr∗. By Equation (2.6.2), the connecting homomorphism d∗ is determinedby the equation

(4.3.10) 〈C∗pd∗(a), β〉 = 〈δb×(a), β〉 ,for all a ∈ Cm(X) and β ∈ Sm(X), where δ : C∗(X)→ C∗+1(X) is the coboundary.The equality 〈C∗pd∗(a), β〉 = 〈d∗(a), C∗p(β)〉 together with (4.3.10) shows that

〈δb×(a), β〉 depends only on C∗p(β) = pβ = β. Therefore, by taking τ β insteadof β if necessary, we may assume that β /∈ b×(Sm(X)). Then, the faces βǫi of βare not in b×(Sm(X)), except possibly for i = 0 and

(4.3.11) 〈δb×(a), β〉 = 〈b×(a), ∂(β)〉 = 〈b×(a), βǫ0〉 .The number 〈b×(a), βǫ0〉 equals 1 if and only if β(0, 1, 0, . . . , 0) ∈ b(X) and βǫ0 ∈a. In other words, if and only if the front and back faces of β satisfy 1β ∈ wb(p)

124 4. PRODUCTS

(see previous subsection) and βm−1 ∈ a. Hence, Equations (4.3.10) and (4.3.11)imply that

〈d∗(a), β〉 = 〈b×(a), βǫ0〉 = 〈wb(p) a, β〉for all a ∈ Cm(X) and β ∈ Sm(X). This proves that d∗(−) = wb(p) − in C∗(X).By Proposition 4.3.7, this implies that d∗(−) = w(p) − in H∗(X).

An important application of the transfer exact sequence is the determinationof the cohomology ring of RPn.

4.3.4. The cohomology ring of RPn. Let Z2[a] be the polynomial ringover a formal variable a in degree 1. This is an object of GrA, as well as itstruncation Z2[a]/(a

n+1), the quotient of Z2[a] by the ideal generated by an+1. ByProposition 3.8.3, H1(RP∞) = Z2, generated by the class ι. Therefore, there is aGrA-morphism Z2[a]→ H∗(RP∞) sending ak to ιk, where the latter denotes thecup product of k copies of ι. The composition Z2[a] → H∗(RP∞) → H∗(RPn)factors by a GrA-morphism Z2[a]/(a

n+1)→ H∗(RPn).

Proposition 4.3.10. The above GrA-morphisms

Z2[a]→ H∗(RP∞) and Z2[a]/(an+1)→ H∗(RPn)

are GrA-isomorphisms. In particular, the GrA-morphism H∗(RP∞)→ H(RPn),induced by the inclusion, is surjective.

Proof. As S∞ is contractible [82, example 1.B.3 p. 88], the transfer ex-act sequence of the covering p∞ : S∞ → RP∞ shows that the cup product with

w(p∞) ∈ H1(RP∞) gives an isomorphism H∗(RP∞)≈−→ H∗+1(RP∞). In particu-

lar, w(p∞) is the generator of H1(RP∞). This proves the statement for RP∞.For RPn we use the covering p : Sn → RPn. The transfer exact sequence

proves at once that the cup product with w(p) ∈ H1(RPn) gives an isomorphism

Hm(RPn)≈−→ Hm+1(RPn) for 0 ≤ m < n− 1. As RPn has a CW-structure with

one k-cell for 0 ≤ k ≤ n, the end of the transfer exact sequence of the coveringp : Sn → RPn involves the Z2-vector spaces

0→ Hn−1(RPn)︸︷︷︸dim=1

w(p)−−−−−−→ Hn(RPn)︸︷︷︸dim≤1

H∗p−−−→ Hn(Sn)︸︷︷︸dim=1

tr∗−−→ Hn(RPn)︸︷︷︸dim≤1

→ 0

Thus, the cup product with w(p) is also an isomorphismHn−1(RPn)≈−→ Hn(RPn).

This proves the proposition for RPn.

4.4. Nilpotency, Lusternik-Schnirelmann categories and topologicalcomplexity

Let X be a topological space. A subspace U of X is categorical if the in-clusion U → X is homotopic to a constant map. The Lusternik-Schnirelmanncategory cat (X) is the minimal cardinality of an open covering of X with categori-cal subspaces. Some authors (see, e.g. [35]) adopt a different normalization for theLusternik-Schnirelmann category, equal to one less than the definition above.

For a survey paper about the Lusternik-Schnirelmann category, see [108].Amongst its properties, cat (X) is an invariant of the homotopy type of X . Forexample, cat (X) = 1 if and only if X is contractible and cat (Sn) = 2. Moregenerally, one has the following result (see [108, Proposition 1.2] for a more generalstatement and a different proof).

4.4. NILPOTENCY, LUSTERNIK-SCHNIRELMANN CATEGORIES, ETC 125

Proposition 4.4.1. Let X be a connected CW-complex of dimension n. Thencat (X) ≤ n+ 1.

Proof. By induction on the dimension of X , the statement being obvious ifdimX = 0. By Proposition 3.4.1, we can write X as the union of two open setsX = C ∪Z, where C is the disjoint union of the open n-cells of X and Z retractingby deformation on X(n−1). By induction hypothesis, Z admits an open coveringwith ≤ n categorical subspaces. If n ≥ 1, C is a categorical open set of X , whichproves that cat (X) ≤ n+ 1.

Let X be a topological space and B be a vector subspace of H∗(X). Thenilpotency class nilB of B is the minimal integer m such that

B · · · B︸︷︷︸m

= 0 .

If no such integer exists, we set nilB =∞.

Proposition 4.4.2. Let X be a topological space. Then nilH>0(X) ≤ cat (X).

Proof. Let U1, . . . , Um ⊂ X be open subspaces of X which are categorical.By the homotopy property, the homomorphism H>0(X) → H>0(Ui) induced bythe inclusion vanishes. Hence, the exact sequence of the pair (X,Ui) implies thatthe restriction homomorphism H>0(X,Ui) → H>0(X) is surjective. Then, in thediagram

m∏

i=1

H>0(X,Ui)

// //m∏

i=1

H>0(X)

H>0(X,U1 ∪ · · · ∪ Um) // H∗(X)

,

which is commutative by the functoriality of the cup product, the upper horizontalarrow is surjective. But, if X = U1 ∪· · · ∪ Um, the lower left vector space vanishes.This proves that nilH>0(X) ≤ m.

Corollary 4.4.3. The Lusternik-Schnirelmann category cat (RPn) of RPn isequal to n+ 1.

Proof. As seen in Example 3.4.5, the projective space RPn is a CW-complexof dimension n. Therefore, cat (RPn) ≤ n+ 1 by Proposition 4.4.1. On the otherhand, by Proposition 4.4.2, cat (RPn) ≥ nilH>0(RPn) and nilH>0(RPn) = n+1by Proposition 4.3.10.

Another classical consequence of Proposition 4.4.2 is the vanishing of the cupproducts in a suspension.

Corollary 4.4.4. Let Y be a topological space. Then, all cup products inH>0(ΣY ) vanish.

Proof. As ΣY is the union of two cones, cat (ΣY ) ≤ 2, which proves thecorollary.

126 4. PRODUCTS

The Lusternik-Schnirelmann category admits several generalizations, for in-stance the category of a map (see, e.g. [108, § 7]). Here, we introduce the categorycat (X,A) of a topological pair (X,A). A subspace U of a topological space X isA-categorical if the inclusion U → X is homotopic to a map with value in A. Thencat (X,A) is defined to be the minimal cardinality of an open covering of X withA-categorical subspaces. For instance, X is path-connected, then

(4.4.1) cat (X,A) ≤ cat (X) = cat (X, pt)

Lemma 4.4.5. cat (X,A) is an invariant of the homotopy type of the pair(X,A).

Proof. Let f : (X,A)→ (X ′, A′) be a homotopy equivalence of pair. It sufficesto prove that, if U ′ is a subset of X ′ which is A′-categorical, then U = f−1(U ′)is A-categorical in X . This will imply that cat (X,A) ≤ cat (X ′, A′). Homotopyequivalence being an equivalence relation, we also get cat (X ′, A′) ≤ cat (X,A).

Let g : (X ′, A′) → (X,A) be a homotopy inverse of f . Let βt : U′ → X ′ be a

homotopy with β0(v) = v and β1(U′) ⊂ A′. Then, the map αt : U → X defined by

αt(u) = gβtf(u) is a homotopy satisfying α0(u) = gf(u) and α1(U) ⊂ A. Asgf is homotopic to id(X,A), this proves that U is A-categorical.

Proposition 4.4.2 generalizes in the following statement.

Proposition 4.4.6. Let (X,A) be a topological pair with X path-connected.Then

nilB ≤ cat (X,A) ,

where B = kerH∗(X)→ H∗(A) .Proof. As X is path-connected, B ⊂ H>0(X). Let U1, . . . , Um ⊂ X be open

subspaces of X which are A-categorical. Then, the homomorphism H∗(X) →H∗(Ui) factors:

H∗(X)

$$

// H∗(Ui)

H∗(A)

99.

Therefore, if a ∈ B, then a is in the image of H>0(X,Ui). The proof of Proposi-tion 4.4.6 is then the same as that of Proposition 4.4.2.

This category for pairs is related to the topological complexity, a notion ofmathematical robotics introduced by M. Farber [54, 55]. Let Y be a topologicalspace and PY be the space of continuous paths γ : I → Y , endowed with thecompact-open topology. Let π : PY → Y × Y be the origin-end map: π(γ) =(γ(0), γ(1)). A motion planning algorithm is a section of π. It is not possible to finda continuous motion planning algorithm unless Y is contractible [54, Theorem 1].The topological complexity TC (Y ) is the minimal cardinality of an open covering Uof Y × Y such that π : PY → Y × Y admits a continuous section over each U ∈ U .Let ∆Y be the diagonal subset of Y × Y . The following proposition is the contentsof [55, Corollary 18.2].

Proposition 4.4.7. TC (Y ) = cat (Y × Y,∆Y ).

In consequence, TC (Y ) is an invariant of the homotopy type of Y .

4.5. THE CAP PRODUCT 127

Proof. Let U ⊂ Y × Y . Suppose that a continuous section s : U → PYof π exists. Then σ(y, y′, t) = (s(y, y′)(t), y′) satisfies σ(y, y′, 0) = (y, y′) andσ(y, y′, 1) = (y′, y′) ∈ ∆Y , showing that U is ∆Y -categorical. Conversely, ifc(t) = (c1(t), c2(t)) ∈ Y × Y is a path from (y, y′) to (u, u) ∈ ∆Y , then the pathc1 c−12 joins y to y′. This process being continuous in (y, y′), it provides a sections

of π over ∆Y -categorical subsets of Y × Y .

Proposition 4.4.7 together with (4.4.1) implies that TC (Y ) ≤ cat (Y × Y ).The inequality cat (Y ) ≤ TC (Y ) also holds true [55, Lemma 9.2], but is not aconsequence of Proposition 4.4.7.

If Y is path-connected, Propositions 4.4.7 and 4.4.6 give the inequality

(4.4.2) TC (Y ) ≥ nil kerH∗j ,

where j : ∆Y → Y × Y denotes the inclusion. We shall see with the Kunneththeorem that there is a commutative diagram (see Remark 4.6.1):

(4.4.3)

H∗(Y × Y )

×≈

H∗j // H∗(∆Y )

≈

H∗(Y )⊗H∗(Y ) // H∗(Y )

.

(to use the Kunneth theorem, we need that Y is of finite cohomology type). Underthe cross product, the image of kerH∗j in the ring H∗(Y )⊗H∗(Y ) is the ideal ofthe divisors of zero for the cup product. The inequality (4.4.2) thus corresponds to[54, Theorem 7].

For results concerning the topological complexity of projective spaces, see theend of § 6.2.2.

4.5. The cap product

Let K be a simplicial complex. Choose a simplicial order ≤ on K. We definethe cap product

Cp(K)× Cn(K)≤−−→ Cn−p(K)

to be the unique bilinear map such that

(4.5.1) a ≤ v0, . . . , vn = 〈a, v0, . . . , vp〉 vp, . . . , vnfor all a ∈ Cp(K) and all v0, . . . , vn ∈ Sn(K), with v0 < v1 < · · · < vn (thismakes sense if n ≥ p; otherwise, the cap product just vanishes). If a ∈ Cp(K), b ∈Cn−p(K) and γ ∈ Cn(K) the following formula follows directly from the definitions

(4.5.2) 〈a ≤ b, γ〉 = 〈b, a ≤ γ〉 .

Lemma 4.5.1. If a ∈ Cp(K) and γ ∈ Cn(K), then

∂(a ≤ γ) = δ(a)≤ γ + a ≤ ∂(γ) .

Proof. Let q = n− p and b ∈ Cq(K). Denote ≤ and ≤ by just and .Using (4.5.2), one has

(4.5.3) 〈δ(a b), γ〉 = 〈(a b, ∂(γ)〉 = 〈b, a ∂(γ)〉 .

128 4. PRODUCTS

In the other hand

〈δ(a b), γ〉 = 〈δ(a) b, γ〉+ 〈a δ(b), γ〉(4.5.4)

= 〈b, δ(a) γ〉+ 〈δ(b), a γ〉= 〈b, δ(a) γ〉+ 〈b, ∂(a γ)〉 .

Equations (4.5.3) and (4.5.4) imply that

〈b, ∂(a γ)〉 = 〈b, δ(a) γ + a ∂(γ)〉 .for all b ∈ Cq(K). By Lemma 2.3.3, this implies Lemma 4.5.1.

Lemma 4.5.1 implies that Z∗(K) ≤ Z∗(K) ⊂ Z∗(K), B∗(K) ≤ Z∗(K) ⊂B∗(K) and Z∗(K) ≤ B∗(K) ⊂ B∗(K). Therefore, ≤ induces a map Hp(K) ×Hn(K)

−→ Hn−p(K), or

H∗(K)×H∗(K)−→ H∗(K)

called the cap product (on simplicial cohomology). As in the case of the cup productwe drop the index “≤” from the notation because of the following proposition.

Proposition 4.5.2. The cap product on H∗(K)×H∗(K)−→ H∗(K) does not

depend on the simplicial order “≤”.Proof. Let ≤ and ≤′ be two simplicial orders on K. Let a ∈ Hp(K) and

γ ∈ Hn(K). For any b ∈ Hn−p(K), Formula (4.5.2) and Proposition 4.1.3 implythat

〈b, a ≤ γ〉 = 〈a ≤ b, γ〉= 〈a ≤′ b, γ〉= 〈b, a ≤′ γ〉 .

By Part (b) of Lemma 2.3.3, this implies that a ≤ γ = a ≤′ γ in Hn−p(K).

Proposition 4.5.3. The cap product H∗(K) × H∗(K)−→ H∗(K) endows

H∗(K) with a structure of H∗(K)-module.

Proof. By definition, is bilinear and the equality 1 γ = γ is obvious. Itremains to prove that

(4.5.5) (a b) γ = a (b γ)

for all a ∈ Hp(K), b ∈ Hq(K) and γ ∈ Hn(K). As the cup product is associativeand commutative (Corollary 4.1.4), one has, for any c ∈ Hn−p−q(K),

〈c, (a b) γ〉 = 〈c (a b), γ〉= 〈(c a) b, γ〉= 〈c a, b γ〉= 〈c, a (b γ)〉 .

By Lemma 2.3.3, this proves Equation (4.5.5).

Proposition 4.5.4 (Functoriality of the cap product). Let f : L → K be asimplicial map. Then, the formula

a H∗f(γ) = H∗f(H∗f(a) γ)

holds in H∗(K) for all a ∈ H∗(K) and all γ ∈ H∗(L).

4.5. THE CAP PRODUCT 129

Proof. Suppose that a ∈ Hp(K) and γ ∈ Hn(L). Using the functoriality ofthe cup product established in Proposition 4.1.5, one has, for any b ∈ Hn−p(K),

〈b, a H∗f(γ)〉 = 〈a b,H∗f(γ)〉= 〈H∗f(a b), γ〉= 〈H∗f(a) H∗f(b), γ〉= 〈H∗f(b), H∗f(a) γ〉= 〈b,H∗f(H∗f(a) γ)〉 .

By Part (b) of Lemma 2.3.3, this proves Proposition 4.5.4.

There are several version of the cap product in relative simplicial (co)homology.Let (K,L) be a simplicial pair. Choose a simplicial order ≤ on K. We note firstthat C∗(K)≤ C∗(L) ⊂ C∗(L), whence a cap product

(4.5.6) Hp(K)×Hn(K,L)−→ Hn−p(K,L) .

One may also compose

Hp(K,L)×Hn(K)j∗×id−−−−→ Hp(K)×Hn(K)

−→ Hn−p(K)

to obtain a cap product

(4.5.7) Hp(K,L)×Hn(K)−→ Hn−p(K) .

The latter cap product may be post-composed with Hn−p(K) → Hn−p(K,L) andget a cap product

(4.5.8) Hp(K,L)×Hn(K)−→ Hn−p(K,L) .

As the restriction of Cp(K)× Cn(K)−→ Cn−p(K) to Cp(K,L)× Cn(L) vanishes,

we obtain a cap product

(4.5.9) Hp(K,L)×Hn(K,L)−→ Hn−p(K) .

As for Formula (4.5.5), the equation

(4.5.10) (a b) γ = a (b γ)

holds true in Hn−p−q(K) for all a ∈ Hp(K,L), b ∈ Hq(K,L) and γ ∈ Hn(K,L).The cap products (4.5.7) and (4.5.9) are used in (4.5.10).

More generally, suppose that L is the union of two subcomplexes L = L1 ∪L2.Then, the restriction of Cp(K) × Cn(K)

−→ Cn−p(K) to Cp(K,L1) × Cn(L) hasimage contained in Cn−p(L2). This gives a cap product

(4.5.11) Hp(K,L1)×Hn(K,L)−→ Hn−p(K,L2) .

The functoriality holds for a simplicial map f : (K ′, L′)→ (K,L) satisfying f(L′i) ⊂Li for i = 1, 2: the formula

(4.5.12) a H∗f(γ) = H∗f(H∗f(a) γ)

holds in H∗(K,L2) for all a ∈ H∗(K,L1) and all γ ∈ H∗(K ′, L′). The proof is thesame as for Proposition 4.5.4.

The next two lemmas express the compatibility between these relative capproducts, the absolute one and the connecting homomorphisms for a simplicialpair (K,L).

130 4. PRODUCTS

Lemma 4.5.5. Let (K,L) be a simplicial pair. Denote by i : L → K andj : (K, ∅) → (K,L) the inclusions. Let x ∈ Hn(K,L). Then, for all integer p,the diagram

Hp(K,L)

x

H∗j // Hp(K)

x

H∗i // Hp(L)

∂∗x

δ∗ // Hp+1(K,L)

x

Hn−p(K)H∗j // Hn−p(K,L)

∂∗ // Hn−p−1(L)H∗i // Hn−p−1(K)

is commutative.

Proof. For the left hand square diagram, let a ∈ Hp(K,L) and b ∈ Hn−p(K,L).One has

〈b,H∗j(a) x〉 = 〈b H∗j(a), x〉and

〈b,H∗j(a x)〉 = 〈H∗j(b), a x〉 = 〈H∗j(b) a, x〉 .Hence, the left hand square diagram commutes if and only if b H∗j(a) =H∗j(b) a, which was established in Lemma 4.1.7.

For the middle square diagram, let a ∈ Hp(K) and b ∈ Hn−p−1(L). One has

〈b,H∗i(a) ∂∗x〉 = 〈b H∗i(a), ∂∗x〉 = 〈δ∗(b H∗i(a)), x〉 .On the other hand:

〈b, ∂∗(a x)〉 = 〈δ∗(b), a x〉 = 〈δ∗(b) a, x〉 .The commutativity of the middle square diagram is thus equivalent to the formulaδ∗(b H∗i(a)) = δ∗(b) a holding true in Hn−p(K,L) for all a ∈ Hp(K) andb ∈ Hn−p−1(L). This formula was proven in Lemma 4.1.9. In the same way, wesee that the commutativity of the right hand square diagram is a consequence ofLemma 4.1.9 (intertwining the role of a and b).

Lemma 4.5.6. Let (K,L) be a simplicial pair. Denote by j : (K, ∅) → (K,L)the pair inclusion. Then, the equation

H∗j(a α) = a H∗j(α) .

holds true in Hn−p(K,L) for all a ∈ Hp(K) and all α ∈ Hn(K).

Proof. It is then enough to prove that 〈b,H∗j(a α)〉 = 〈b, a H∗j(α)〉 forall b ∈ Hn−p(K,L). But,

〈b,H∗j(a α)〉 = 〈H∗j(b), a α〉= 〈H∗j(b) a,α〉= 〈H∗j(b a), α〉 by Lemma 4.1.8

= 〈b a,H∗j(α)〉= 〈b, a H∗j(α)〉 .

The cap product is also defined in the singular (co)homology of a space X . Onthe (co)chain level, it is the unique bilinear map

Cp(X)× Cn(X)−→ Cn−p(X)

4.6. THE CROSS PRODUCT AND THE KUNNETH THEOREM 131

such that

a σ = 〈a, pσ〉σq

for all a ∈ Cp(X) and all σ ∈ Sn(X), where the back and front faces pσ and σq

are defined as in p 112. If a ∈ Cp(X), b ∈ Cn−p(X) and γ ∈ Cn(X) the followingformula follows directly from the definition

(4.5.13) 〈a b, γ〉 = 〈b, a γ〉 .Therefore, as for the simplicial cap product, properties follow from those of thecup product. The formula ∂(a γ) = δ(a) γ + a ∂(γ) is proved as for

Lemma 4.5.1 and we get an induced bilinear map Hp(X)×Hn(X)−→ Hn−p(X),

or

H∗(X)×H∗(X)−→ H∗(X)

called the cap product in singular (co)homology. This cap product endows H∗(X)with a structure of H∗(X)-module, as in Proposition 4.5.3 and is functorial forcontinuous maps f : Y → X , as for Proposition 4.5.4.

For a topological pair (X,Y ), the three relative versions of the cap products:

(4.5.14) Hp(X)×Hn(X,Y )−→ Hn−p(X,Y ) ,

(4.5.15) Hp(X,Y )×Hn(X)−→ Hn−p(X,Y )

and

(4.5.16) Hp(X,Y )×Hn(X,Y )−→ Hn−p(X)

hold true, as for (4.5.6)–(4.5.9). When Y = Y1∪Y2, a relative cap product analogousto (4.5.11)

(4.5.17) Hp(X,Y1)×Hn(X,Y )−→ Hn−p(X,Y2) .

is available under some conditions, for instance if (Y, Yi) is a good pair for i = 1, 2,so one can use the small simplexes technique, as for the Mayer-Vietoris sequencein Proposition 3.1.54. The functoriality formula (4.5.12) as well as the analoguesof Lemmas 4.5.5 and 4.5.6 hold true.

Finally, the simplicial and singular cap products are intertwined by the isomor-phisms

R≤,∗ : H∗(K)≈−→ H∗(|K|) and R∗≤ : H

∗(|K|) ≈−→ H∗(K)

of Theorem 3.6.3. For any simplicial order ≤, the equation

(4.5.18) a R≤,∗(γ) = R≤,∗(R∗≤(a) γ)

holds in H∗(|K|) for all a ∈ H∗(|K|) and all γ ∈ H∗(K): The proof of (4.5.18) isstraightforward for γ a simplex of K.

4.6. The cross product and the Kunneth theorem

Let X and Y be topological spaces. Results computing H∗(X × Y ) in terms ofH∗(X) and H∗(Y ) are known as Kunneth theorems (or Kunneth formulas). Thisgeneric name comes from the thesis of Hermann Kunneth in 1923 (see [40, pp. 55–56]). To give an example, when X and Y are discrete spaces, the cohomology ringsare concentrated in dimension 0 and

(4.6.1) H0(X) = ZX2 , H0(Y ) = ZY2 , H0(X × Y ) = Z2X×Y .

132 4. PRODUCTS

The cross product of maps

ZX2 × ZY2×−→ Z2

X×Y

defined by (f × g)(x, y) = f(x)g(y) is bilinear. The associated linear map

(4.6.2) ZX2 ⊗ ZY2×−→ Z2

X×Y

is also called the cross product. The map (4.6.2) is clearly injective. It is notsurjective if both X and Y are infinite; for instance, if X = Y is infinite, it is easyto see that the characteristic function of the diagonal in X×X is not in the image of×. On the other hand, suppose that X or Y is finite (say Y ). Let F : X×Y → Z2.For y ∈ Y , define Fy : X → Z2 by Fy(x) = F (x, y) and let χy be the characteristicfunction of y. Then

F =∑

y∈Y

Fy × χy .

Thus, if Y is finite, the cross product of (4.6.2) is an isomorphism.Such finiteness conditions will occur in the statements of this section, under

the form that Y should be of finite cohomology type (see definition p. 74).Observe that, under the identification of (4.6.1), the cross product H0(X) ×

H0(Y )→ H0(X × Y ) satisfies the formula

f × g = π∗Xf π∗Y g ,

where πX and πY are the projections of X × Y onto X and Y .More generally, let X and Y be two topological spaces. Using the usual tensor

product ⊗ of vector spaces over Z2, we define the tensor product of the Z2-algebras(H∗(X),+,) and (H∗(Y ),+,) as the Z2-algebra (H

∗(X)⊗H∗(Y ),+, •) definedby

[H∗(X)⊗H∗(Y )]m =⊕

i+j=m

Hi(X)⊗Hj(Y ) ,

with the product

(4.6.3) (a1 ⊗ b1) • (a2 ⊗ b2) = (a1 a2)⊗ (b1 b2) .

The projections πX : X × Y → X et πY : X × Y → Y give GrA-morphismsπ∗X : H∗(X)→ H∗(X×Y ) et π∗Y : H∗(Y )→ H∗(X×Y ). This permits us to definea bilinear map

H∗(X)×H∗(Y )×−→ H∗(X × Y )

by

(4.6.4) a× b = ×(a, b) = π∗X(a)π∗Y (b)

called the cross product. By the universal property of the tensor product (analogousto that for vector spaces), this gives a GrV-morphism

H∗(X)⊗H∗(Y )×−→ H∗(X × Y ) ,

also called the cross product.

Remark 4.6.1. Let ∆ : X → X ×X be the diagonal map ∆(x) = (x, x). Thecomposition

(4.6.5) H∗(X)×H∗(X)×−→ H∗(X ×X)

∆∗−−→ H∗(X)

is equal to the cup product (see also Diagram (4.4.3)) This relation, due to Lef-schetz (see [183, pp. 38–41] for historical considerations), was quite influential: in


some books (e.g. [179, 136]), the cross product is introduced first using homo-logical algebra (the Eilenberg-Zilber theorem) and the cup product is defined viaFormula (4.6.5). Our opposite approach follows the viewpoint of [74, 82].

Under some hypotheses, the cross product may be defined in relative coho-mology. Let (X,A) and (Y,B) be topological pairs. The projections πX and πYgive homomorphisms π∗X : H∗(X,A) → H∗(X × Y,A × Y ) and π∗Y : H∗(Y,B) →H∗(X × Y,B ×X). Suppose that A or B is empty, or one of the pairs (X,A) or(Y,B) is a good pair. Then formula (4.6.4) defines a relative cross product

(4.6.6) H∗(X,A)⊗H∗(Y,B)×−→ H∗(X × Y,A× Y ∪X ×B) .

Indeed, we must just check that the relative cup product

H∗(X × Y,A× Y )⊗H∗(X × Y,B ×X)−→ H∗(X × Y,A× Y ∪X ×B) .

is defined. By Lemma 4.1.14, it is enough to show that (A×Y,X×B) is excisive inX×Y . This is obvious if A or B is empty. Otherwise, suppose that one of the pair,say (Y,B), is a good pair. Let V be a neighbourhood of B in Y which retracts by

deformation onto B. Let Z = A× Y ∪X ×B. Then, A× (Y − V ) ⊂ intZ(A× Y ).By excision of A× (Y − V ) and homotopy, we get isomorphisms

H∗(Z,A× Y )≈−→ H∗(A× V ∪X ×B,A× V )

≈−→ H∗(X ×B,A×B) .

By Lemma 4.1.13, this implies that (A× Y,X ×B) is excisive.We first establish the functoriality of the cross product. In Lemmas 4.6.2

and 4.6.3 below, we assume that the conditions for the relative cross product to bedefined are satisfied.

Lemma 4.6.2. Let f : (X ′, A′) → (X,A) and g : (Y ′, B′) → (Y,B) be maps ofpairs. Then, for all a ∈ H∗(X,A) and b ∈ H∗(Y,B) the following formula holds:

H∗(f × g)(a× b) = H∗f(a)×H∗g(b) .Proof. As πX (f × g) = f πX′ and πY (f × g) = gπY ′ , one has

H∗(f × g)(a× b) = H∗(f × g)(H∗πX(a) H∗πY (b)

)

= H∗(f × g)(H∗πX(a)) H∗(f × g)(H∗πY (b)))= H∗πX′ H

∗f(a) H∗πY ′ H∗g(b)

= H∗f(a)×H∗g(b) .

Formula (4.6.3) provides a product “•” on H∗(X,A)⊗H∗(Y,B).

Lemma 4.6.3. The cross product H∗(X,A) ⊗ H∗(Y,B)×−→ H∗(X × Y,A ×

Y ∪X ×B) is multiplicative. In particular, the cross product H∗(X)⊗H∗(Y )×−→

H∗(X × Y ) is a GrA-morphism.

Proof.

×((a1 ⊗ b1) • (a2 ⊗ b2)

)= (a1 a2)× (b1 b2)

= π∗X(a1 a2) π∗Y (b1 b2)

= π∗X(a1) π∗X(a2) π∗Y (b1) π∗Y (b2)

= π∗X(a1) π∗Y (b1) π∗X(a2) π∗Y (b2)

= (a1 × b1) (a2 × b2)= ×(a1 ⊗ b1) ×(a2 ⊗ b2)

134 4. PRODUCTS

Remark 4.6.4. In the proof of Lemma 4.6.3, we have established that

(a1 a2)× (b1 b2) = (a1 × b1) (a2 × b2)for all ai ∈ H∗(X,A) and bj ∈ H∗(Y,B).

Observe that the Kronecker pairing

[H∗(X)⊗H∗(Y )]× [H∗(X)⊗H∗(Y )]〈 , 〉−−→ Z2

given by

(4.6.7) 〈a⊗ b, α⊗ β〉 = 〈a, α〉〈b, β〉is a bilinear map (By convention, 〈a, α〉 = 0 if a ∈ Hp(−) and α ∈ Hq(−) withp 6= q.)

Lemma 4.6.5. Let X and Y be topological spaces with Y of finite cohomologytype. Then, for all n ∈ N, the linear map

⊕

p+q=n

Hp(X)⊗Hq(Y )k−→ [

⊕

p+q=n

Hp(X)⊗Hq(Y )]♯

given by k(a⊗ b) = 〈a⊗ b,−〉 is an isomorphism.

Proof. It suffices to prove that k : Hp(X)⊗Hq(Y )→ [Hp(X)⊗Hq(Y )]♯ is anisomorphism for all integers p, q. As Hr(−) ≈ Hr(−)♯ via the Kronecker pairing,this amounts to proving that, for vector spaces V and W , the homomorphism

k : V ♯ ⊗W ♯ → [V ⊗W ]♯ ,

given by k(r⊗s)(v⊗w) = r(v)s(w), is an isomorphism whenW is finite dimensional.This classical fact (true over any base field) is easily proven by induction on dimW(see, e.g., [43, Chapter VI, Proposition 10.18] for a proof in a more general setting).

The following lemma permits us to define a Kronecker dual × to the crossproduct, called the homology cross product.

Lemma 4.6.6. Let X and Y be topological spaces, with Y of finite cohomologytype. Then, there exists a unique GrV-homomorphism

× : H∗(X × Y )→ H∗(X)⊗H∗(Y )

such that the equation

(4.6.8) 〈a× b, γ〉 = 〈a⊗ b,×(γ)〉holds true for all a ∈ H∗(X), b ∈ H∗(Y ) and γ ∈ H∗(X × Y ).

Proof. Let γ ∈ H∗(X × Y ). The uniqueness of ×(γ) is guaranteed byLemma 4.6.5, using Lemma 2.3.3. For its existence, let M be a basis for H∗(Y )and letM∗ = m∗ ∈ H∗(Y ) | m ∈M be the dual basis for the Kronecker pairing.We define

(4.6.9) ×(γ) =∑

m∈M

H∗πX

(H∗πY (m

∗) γ)⊗m.


We must check that (4.6.8) holds true for all a ∈ H∗(X), b ∈ H∗(Y ). It is enoughto check it for b = n∗ with n ∈ M. As 〈n∗,m〉 = 1 if m = n and 0 otherwise, onehas

〈a⊗ n∗, ×(γ)〉 = 〈a⊗ n∗,∑m∈MH∗πX

(H∗πY (m

∗) γ)⊗m〉

=∑

m∈M〈a,H∗πX

(H∗πY (m

∗) γ)〉〈n∗,m〉

= 〈H∗πX(a), H∗πY (n

∗) γ〉= 〈H∗πX(a) H∗πY (n

∗), γ〉= 〈a× n∗, γ〉 .

(Remark: the uniqueness of ×(γ) shows that the right member of (4.6.9) does notdepend on the choice of the basisM.)

Theorem 4.6.7 (Kunneth Theorem). Let X and Y be topological spaces. Sup-pose that Y is of finite cohomology type. Then, the cross product

× : (H∗(X)⊗H∗(Y ),+, •)≈−→ (H∗(X × Y ),+,)

is a GrA-isomorphism and the homology cross product

× : H∗(X × Y )→ H∗(X)⊗H∗(Y )

is a GrV-isomorphism.

The finiteness condition on one of the space (here Y ) is necessary in the coho-mology statement, as seen in the beginning of the section. It is used in the proofthrough the following lemma.

Lemma 4.6.8. Let V be a family of vector spaces over a field F. Let W be afinite dimensional F-vector space. Then the linear map

Φ:( ∏

V ∈V

V)⊗W −→

∏

V ∈V

(V ⊗W

)

given byΦ((v) ⊗ w) = (v ⊗ w)

is an isomorphism.

Proof. The proof is by induction on n = dimW . The case n = 1 followsfrom the canonical isomorphism T ⊗ F ≈ T for any vector space T . The inductionstep uses that, in the category of F-vector spaces, tensor and Cartesian productscommute with direct sums.

Proof of the Kunneth theorem. By Lemma 4.6.3, we know that the crossproduct is aGrA-morphism. It is then enough to prove that it is aGrV-isomorphism.Assuming that Y is of finite cohomology type, the proof goes as follows.

(1) We prove that the cross product is a GrV-isomorphism when X is a finitedimensional CW-complex.

(2) By Kronecker duality, Point (1) implies that the homology cross productis a GrV-isomorphism when X is a finite dimensional CW-complex. Anycompact subspace of X × Y is contained in Xn × Y for some n ∈ N.Therefore,H∗(X×Y ) is the direct limit ofH∗(X

n×Y ) by Corollary 3.1.16.Also, H∗(X) is the direct limit of H∗(X

n). The homology cross productbeing natural by Lemma 4.6.2 and Kronecker duality, we deduce that × is

136 4. PRODUCTS

a GrV-isomorphism when X is any CW-complex. By Kronecker duality,the cross product is a GrV-isomorphism for any CW-complex X .

(3) If X is any space, there is a map fX : X → X , where X is a CW-complexand f is a weak homotopy equivalence, i.e. the induced map on the

homotopy groups π∗f : π∗(X, u)≈−→ π∗(X, f(u)) is an isomorphism for all

u ∈ X (see [82, p.352] or Remark 3.7.5). As π∗(A×B, (a, b)) ≈−→ π∗(A, a)×π∗(B, b), the map fX × id : X × Y → X × Y is also a weak homotopyequivalence. But, weak homotopy equivalences induce isomorphisms onsingular (co)homology (see [82, Prop. 4.21]). The diagram

H∗(X)⊗H∗(Y )

H∗fX⊗ id ≈

× // H∗(X × Y )

H∗(fX×id)≈

H∗(X)⊗H∗(Y )×

≈// H∗(X × Y )

is commutative by Lemma 4.6.2. This proves that the cross product insingular cohomology is a GrA-isomorphism for any space X . The corre-sponding diagram for the homology cross product, or Kronecker duality,proves that the homology cross product is a GrV-isomorphism.

It thus remains to prove Point (1). We follow the idea of [82, p. 218]. Let usfix the topological space Y . To a topological pair (X,A), we associate two gradedZ2-vector spaces:

h∗(X,A) = H∗(X,A)⊗H∗(Y ) and k∗(X,A) = H∗(X × Y,A× Y ) .

Using Proposition 3.9.2, Point (1) follows from the following lemma.

Lemma 4.6.9. Let Y be a topological space of finite cohomology type. Then

(a) k∗ and h∗ are two generalized cohomology theories in the sense of § 3.9,with h∗(pt) ≈ k∗(pt) ≈ H∗(Y ).

(b) The cross product provides a natural transformation from h∗ to k∗, re-

stricting to an isomorphism h∗(pt)≈−→ k∗(pt).

Proof. If f : (X,A)→ (X ′, A′) is a continuous map of pairs, we define h∗f =H∗f⊗idH∗(Y ) and k

∗f = H∗(f×idY ). This makes h∗ and k∗ functors fromTop2 to

GrV. The connecting homomorphism δ∗h : h∗(A) → h∗+1(X,A) and δ∗k : k

∗(A) →k∗+1(X,A) are defined by

δ∗h = δ∗ ⊗ idH∗(Y ) and δ∗k = δ∗ : H∗(A× Y )→ H∗+1(X × Y,A× Y ) ,

using the homomorphism δ∗ of singular cohomology; δ∗h and δ∗k are then functorialfor continuous maps.

We now check that Axioms (1)–(3) of p. 104 hold both for h∗ and k∗. Thehomotopy and excision axioms are clear. For a topological pair (X,A), the longexact sequence for k∗ is that in singular cohomology for the pair (X × Y,A × Y ).The exact sequence for h∗ is obtained by tensoring with H∗(Y ) the exact sequenceof (X,A) for H∗. We use that a direct sum of exact sequences is exact and that,over a field, tensoring with a vector space preserves exactness. The disjoint unionaxiom holds trivially for k∗. For h∗, we use that Hm(Y ) if of finite dimension forall m and Lemma 4.6.8. Thus, both h∗ and k∗ are generalized cohomology theories.


We now check Point (b). Let f : (X ′, A′) → (X,A) be a continuous map ofpairs. We must prove that the following diagram

(4.6.10)

h∗(X,A)

h∗f

× // k∗(X,A)

k∗f

h∗(X ′, A′)× // k∗(X ′, A′)

is commutative. This amounts to showing that

(4.6.11) H∗f(a)× y = H∗(f × idY )(a× y)for all a ∈ H∗(X,A) and y ∈ H∗(Y ). This follows from Lemma 4.6.2.

For the second part of Point (b), we must show the commutativity of thediagram

(4.6.12)

h∗(A)δ∗h //

×

h∗+1(X,A)

×

k∗(A)δ∗k // k∗+1(X,A) .

This is equivalent to the commutativity of the diagram

(4.6.13)

Hp(A)×Hq(Y )δ∗×id //

×

Hp+1(X,A)×Hq(Y )

×

Hp+q(A× Y )δ∗× // Hp+q+1(X × Y,A× Y ) ,

for all p, q ∈ N. Here, we have introduced more precise notations, distinguishing theconnecting homomorphisms in singular cohomology δ∗ : H∗(A)→ H∗+1(X,A) andδ∗× : H

∗(A × Y ) → H∗+1(X × Y,A × Y ). We shall also distinguish the homomor-phisms π∗X : H∗(X) → H∗(X × Y ) and π∗X : H∗(A) → H∗(A × Y ) induced by theprojections onto A and X , as well as the homomorphisms π∗Y : H∗(Y )→ H∗(X×Y )and π∗Y : H∗(Y ) → H∗(A × Y ) induced by the projections onto Y . Analogousnotations are used for cochains. The commutativity of Diagram (4.6.13) is thusequivalent to the formula(4.6.14)

π∗X δ∗(a) π∗Y (y) = δ∗×

(π∗X(a) π∗Y (y)

)for all a ∈ Hp(A) , y ∈ Hq(Y ) .

Let a ∈ Zp(A) and y ∈ Zq(Y ) represent a and y. Let a ∈ Cp(X) be an extension of aas a p-cochain of X . By the recipe of Lemma 3.1.17, δ(a) is a cocycle of Cp+1(X,A)representing δ∗(a). Thus, the left hand member of (4.6.14) is represented by thecocycle

(4.6.15) π∗X δ(a) π∗Y (y) .

To compute the right hand member, we need an extension of π∗X(a) π∗Y (y) asa cochain of X × Y . But, as cochains of X × Y , π∗X(a) is an extension of π∗X(a)and the cocycle πY (y) is an extension of π∗Y (y). Therefore, π∗X(a) πY (y) is anextension of π∗X(a) π∗Y (y). By Lemma 3.1.17, the right hand member of (4.6.14)is then represented by the cocycle δ×

(π∗X(a) πY (y)

). As δ×(πY (y)) = 0, one has

(4.6.16) δ×(π∗X(a) πY (y)

)= δ×π

∗X(a) πY (y) = π∗X δ(a) π∗Y (y)

138 4. PRODUCTS

Comparing (4.6.15) and (4.6.16) proves Formula (4.6.14) and then the commuta-tivity of Diagram (4.6.12).

Under some hypotheses, there are relative versions of the Kunneth theorem,generalizing Theorem 4.6.7.

Theorem 4.6.10 (Relative Kunneth theorem). Let (X,A) be a topological pair.Let (Y,B) be a good pair such that Y and B are of finite cohomology type. Then,the cross product

(4.6.17) × : H∗(X,A)⊗H∗(Y,B)≈−→ H∗(X × Y,A× Y ∪X ×B)

is a GrA-isomorphism.

The classical proof of the Kunneth theorem (see e.g. [179]) gives the moregeneral statement that (4.6.17) is an isomorphism if (X ×B,A × Y ) is excisive inX × Y and (Y,B) is of finite cohomology type. If (X,A) and (Y,B) are CW-pairs,the condition that (Y,B) is of finite cohomology type is also sufficient (see [82,Theorem 3.21]; see also Corollary 4.7.25 below).

Proof. As (Y,B) is a good pair, the relative cross product (4.6.6) is defined.Let Z = A× Y ∪X ×B and let p ∈ N. Let us consider the commutative diagram.

Hp(X,A)⊗Hq−1(Y )×

≈//

Hp+q−1(X×Y,A×Y )

Hp(X,A)⊗Hq−1(B)

×

≈//

Hp+q−1(X×B,A×B) oo J∗

≈Hp+q−1(Z,A×Y )

Hp(X,A)⊗Hq(Y,B)

× //

Hp+q(X×Y,Z)

Hp(X,A)⊗Hq(Y )

×

≈//

Hp+q(X×Y,A×Y )

Hp(X,A)⊗Hq(B)

×

≈// Hp+q(X×B,A×B) oo J∗

≈Hp+q(Z,A×Y )

The left column is the cohomology sequence for (Y,B) tensored by Hp(X,A). It isstill exact since we work in the category of Z2-vector spaces. The right column isthe cohomology sequence for the triple (X × Y, Z,A× Y ). The homomorphism J∗,induced by inclusion, is an isomorphism: if V is a neighbourhood of B in Y whichretracts by deformation onto B, J∗ is the composition

H∗(A×Y ∪X×B,A×Y ) ≈−→ H∗(A×V ∪X×B,A×V ) ≈−→ H∗(X×B,A×B) .

The left arrow is an isomorphism by excision of A × (Y − V ) and the right oneby the homotopy property. As Y and B are of finite cohomology type, the crossproducts involving the absolute cohomology H∗(Y ) or H∗(B) are isomorphisms, asestablished during the proof of Theorem 4.6.7. By the five-lemma, this proves thatthe middle cross product is an isomorphism.

4.7. SOME APPLICATIONS OF THE KUNNETH THEOREM 139

4.7. Some applications of the Kunneth theorem

4.7.1. Poincare series and Euler characteristic of a product. One ap-plication of the Kunneth theorem is the multiplicativity of Poincare series and Eulercharacteristic.

Proposition 4.7.1. Let X and Y be spaces of finite cohomology type. Then,X × Y is of finite cohomology type and

(4.7.1) Pt(X × Y ) = Pt(X) ·Pt(Y ) .

If X and Y are finite complexes, then

(4.7.2) χ(X × Y ) = χ(X) · χ(Y ) .

Proof. Let ai = dimHi(X), bi = dimHi(Y ). The Kunneth theorem impliesthat dimHn(X ×Y ) =

∑i+j=n aibj which proves (4.7.1). Equation (4.7.2) follows,

since χ is the evaluation of Pt at t = −1. Note that (4.7.2) also follows moreelementarily from the cellular decomposition of X × Y (see Example 3.4.6).

4.7.2. Slices. Let y0 ∈ Y . The slice inclusion sX : X → X × Y at y0 is thecontinuous map defined by sX(x) = (x, y0). The slice inclusion sY : Y → X × Y atx0 ∈ X is defined accordingly.

Using the bijection Y ≈ S0(Y ), we see y0 ∈ Y as a 0-homology class [y0] ∈H0(Y ). Hence, for b ∈ H0(Y ) the number 〈b, y0〉 ∈ Z2 is defined.

Lemma 4.7.2. Let sX : X → X × Y be the slice inclusion at y0 ∈ Y . Leta ∈ Hm(X) and b ∈ Hn(Y ). Then,

H∗sX(a× b) =〈b, y0〉 a if n = 0

0 otherwise.

Proof. One has πX sX = idX , while πY sX is the constant map c onto y0.Thus, H∗c (b) = 0 if n 6= 0. When n = 0, H∗c (b) = 〈b, y0〉1. Thus,

H∗sX(a× b) = H∗sX(π∗X(a) π∗

Y(b)) = a H∗c (b) = 〈b, y0〉 a .

Here are two corollaries of Lemma 4.7.2 which enable us to detect cohomologyclasses via the slice homomorphisms.

Corollary 4.7.3. Let X and Y be path-connected topological spaces such thatHk(X) = 0 for k < n. Then, the equation

a = 1×H∗sY (a) +H∗sX(a)× 1

is satisfied for all a ∈ Hn(X × Y ).

Proof. By the hypotheses and the Kunneth theorem, the cross product pro-vides an isomorphism

× : H0(X)⊗Hn(Y )⊕Hn(X)⊗H0(Y )≈−→ Hn(X × Y ) .

and H0(X) ≈ Z2 ≈ H0(Y ). This implies that a = 1×v+u×1 for some unique u ∈Hn(X) and v ∈ Hn(Y ). By Lemma 4.7.2, one has H∗sY (a) = v and H∗sX(a) = u,which proves the corollary.

The case n = 1 in Corollary 4.7.3 gives the following statement.

140 4. PRODUCTS

Corollary 4.7.4. Let X and Y be path-connected spaces. Then, the equation

a = 1×H∗sY (a) +H∗sX(a)× 1

is satisfied for all a ∈ H1(X × Y ).

4.7.3. The cohomology ring of a product of spheres. We first note theassociativity of the cross product.

Lemma 4.7.5. Let X, Y and Z be three topological spaces. In H∗(X ×Y ×Z),the cross product is associative: (x× y)× z = x× (y× z) for all x ∈ H∗(X), y ∈ Yand z ∈ Z.

Proof. We have to consider the various projections π12 : X×Y ×Z → X×Y ,π23 : X × Y × Z → Y × Z, π1 : X × Y × Z → X , etc. Also, π12

1 : X × Y → X , etc.

They satisfy πijj πij = πj . Using the associativity and the functoriality of the cupproduct, we get

(x× y)× z = π∗12(π121∗(x) π12

2∗(y)) π∗3(z) = π∗1(x) π∗2(y) π∗3(z) .

In the same way, x× (y × z) = π∗1(x) π∗2(y) π∗3(z).

The cohomology of the sphere Sd being concentrated in dimension 0 and d, onehas a GrA-isomorphism

(4.7.3) Z2[x]/(x2)≈−→ H∗(Sd) (x of degree d) ,

sending x to the generator [Sd]♯ ∈ Hd(Sd). Here, Z2[x]/(x2) denotes the quotient

of the polynomial ring Z2[x], where x is a formal variable (here of degree d), bythe ideal generated by x2. The following proposition then follows directly from theKunneth theorem.

Proposition 4.7.6. Let X be a topological space. The GrA-homomorphism

H∗(X)[x]/(x2) −→ H∗(X × Sd) (x of degree d) ,

induced by a 7→ a× 1, for a ∈ H∗(X), and x 7→ 1× [Sd]♯, is a GrA-isomorphism.

Using Proposition 4.7.6 together with Lemma 4.7.5, we get the following propo-sition.

Proposition 4.7.7. For i = 1 . . . ,m, let xi be a formal variable of degree di.Then, the GrA-homomorphism

Z2[x1, . . . , xm]/(x21, . . . , x

2m) −→ H∗(Sd1 × · · · × Sdm)

induced by

xi 7→ 1× · · · × 1× [Sdi ]♯ × 1× · · · × 1

is a GrA-isomorphism.


4.7.4. Smash products and joins. Let (X, x) and (Y, y) be two pointedspaces. The base points provide an inclusion

X ∨ Y ≈ X × y ∪ x × Y → X × Y .The smash product X ∧ Y of X and Y is the quotient space

X ∧ Y = X × Y/X ∨ Y .

It is pointed by x ∧ y, the image of X ∨ Y in X ∧ Y .Recall that (X, x) is well pointed if the pair (X, x) is well cofibrant. The

following lemma is the first place where the full strength of this definition is used.

Lemma 4.7.8. If (X, x) and (Y, y) are well pointed, so is (X ∧ Y, x ∧ y).Proof. By Lemma 3.1.40, the pair (X × Y,X ∨ Y ) is well cofibrant and, by

Lemma 3.1.43, so is (X ∧ Y, x ∧ y).

By Proposition 3.1.45 and (3.1.13), one has the isomorphisms

(4.7.4) H∗(X × Y,X ∨ Y ) ≈ H∗(X ∧ Y, x ∧ y) ≈ H∗(X ∧ Y ) .

Proposition 4.7.9. Let (X, x) and (Y, y) be well pointed spaces. Then, thehomomorphisms induced by the inclusion i : X ∨ Y → X × Y and the projectionp : X × Y → X ∧ Y give rise to the short exact sequence

0→ H∗(X ∧ Y )H∗p−−−→ H∗(X × Y )

H∗i−−→ H∗(X ∨ Y )→ 0 .

Proof. Using the isomorphism (4.7.4) and the exact sequence of Corollary 3.1.48,

it is enough to prove that H∗i is onto. Consider the commutative diagram

H∗(X)id //

H∗π∗X

&&

H∗(X)

H∗(X × Y )H∗i // H∗(X ∨ Y )

H∗j∗Y &&

H∗jX88rrrrrrrr

H∗(Y )id //

H∗π∗Y

88rrrrrrrrr

H∗(Y )

where πX , πY are the projections and jX , jY the inclusions. We note that πY jXand πX jY are constant maps. By Proposition 3.1.51, the homomorphism H∗(X ∨Y )

(H∗jX ,H∗j∗Y )−−−−−−−−−→ H∗(X)× H∗(Y ) is an isomorphism. Hence H∗i is onto.

Remark 4.7.10. Using the relationship between the exact sequence of the pair(X × Y,X ∨ Y ) and that of Corollary 3.1.48, Proposition 4.7.9 implies that thehomomorphism H∗i : H∗(X × Y ) → H∗(X ∨ Y ) is surjective, whence the shortexact sequence

(4.7.5) 0→ H∗(X × Y,X ∨ Y ) −→ H∗(X × Y )H∗i−−→ H∗(X ∨ Y )→ 0 .

As (X, x) and (Y, y) are good pairs, the relative cross product

H∗(X, x)⊗H∗(Y, y) ×−→ H∗(X × Y,X ∨ Y )

142 4. PRODUCTS

is defined by (4.6.6). Using the isomorphisms of (3.1.13), one constructs the com-mutative diagram

(4.7.6)

H∗(X, x)⊗H∗(Y, y)

×

≈ // H∗(X)⊗ H∗(Y )

×

H∗(X × Y,X ∨ Y )≈ // H∗(X ∧ Y )

which defines the reduced cross product ×. The relative Kunneth theorem 4.6.10gives the following reduced Kunneth theorem.

Proposition 4.7.11. Let (X, x) and (Y, y) be well pointed spaces, with Y offinite cohomology type. Then, the reduced cross product

× : H∗(X)⊗ H∗(Y )≈−→ H∗(X ∧ Y )

is a multiplicative GrV-isomorphism.

For a pointed space (Z, z), Diagram (3.1.14) provides an injective homomor-

phism H∗(Z)→ H∗(Z). Using this together with Proposition 4.7.9 (or Remark 4.7.10),the Kunneth theorem and its reduced form are summed up by the diagram

(4.7.7)

H∗(X)⊗ H∗(Y )

×≈

// // H∗(X)⊗H∗(Y )

×≈

H∗(X ∧ Y ) // // H∗(X × Y )

.

Example 4.7.12. Proposition 4.7.11 says that Hk(Sp ∧ Sq) = 0 for k 6= p+ qand Hp+q(Sp ∧ Sq) = Z2. Actually, Sp ∧ Sq is homeomorphic to Sp+q by thefollowing homeomorphisms. Let Dr be the compact unit disk of dimension r withboundary ∂Dr = Sr−1. Then Dr/∂Dr is homeomorphic to Sr and

Sp ∧ Sq ≈−→(Dp/∂Dp ×Dq/∂Dq

)/[∂Dp]×Dq ∪Dp × [∂Dq]

≈−→ Dp ×Dq

/∂Dp ×Dq ∪Dp × ∂Dq

≈−→ Dp ×Dq/∂(Dp ×Dq) ≈ Sp+q .

Let (X, x) be a well pointed space. The smash product X ∧ S1 is called thereduced suspension of X , which has the same homotopy type than the suspensionΣX . Indeed, let ∂I = 0, 1. The map

F : ΣX = (X × I)/(X × ∂I)→ X ∧ S1

given by F (x, t) = [(x, e2iπt)] descends to a homeomorphism

F : ΣX = ΣX/(x × I) ≈−→ X ∧ S1 .

This homeomorphism preserves the base points, if we choose those to be [x] ∈ ΣXand 1 ∈ S1. The pair (I, ∂I) is well cofibrant by Lemma 3.1.39. By Lemma 3.1.40,so is the product pair (X × I,X × ∂I ∪ x × I). By Lemma 3.1.43, the pair(ΣX, x×I) is well cofibrant. As x×I is contractible, the projection F : ΣX →→ ΣX ≈ X ∧ S1 is a homotopy equivalence by Lemma 3.1.44.


Let b be the generator of H1(S1) = Z2. By Propositions 4.7.11 and 3.1.49 andthe above, the three arrows in the following proposition are isomorphisms.

Lemma 4.7.13. The diagram

Hn(X)

−×b

≈ww♦♦♦♦♦♦♦♦♦ Σ∗

≈ &&

Hn+1(X ∧ S1)H∗F

≈// Hn+1(ΣX)

is commutative.

Proof. As all these isomorphisms are functorial, it is enough to prove thelemma for X = Kn. This is possible since the cellular decomposition of Kn given inProposition 3.8.1 has 0-skeleton K0

n = x0, so (Kn, x0) is a well pointed space byProposition 3.4.1. In this particular case, the statement is obvious since the threegroups are isomorphic to Z2.

The smash product gives a geometric interpretation of the cup product. Leta ∈ Hm(X) and b ∈ Hm(X), given by maps fa : X → Km and fm : X → Kn toEilenberg-MacLane spaces. By Proposition 4.7.11, Hm+n(Km ∧ Kn) = Z2, withgenerator corresponding to g : Km ∧ Kn → Km+n.

Proposition 4.7.14. The composite map

X(fa,fb)−−−−→ Km ×Kn → Km ∧ Kn g−→ Km+n

represents the class a b ∈ Hm+n(X).

Proof. By Proposition 4.7.11, the generator of Hm+n(Km ∧ Kn) = Z2 isthe reduced cross product ım×ın. By Diagram (4.7.7), it is send to ım × ın inHm+n(Km × Kn). Now, the composite map of Proposition 4.7.14 coincides withthe composition

X∆−→ X ×X fa×fb−−−−→ Km ×Kn → Km ∧ Kn g−→ Km+n .

Proposition 4.7.14 then follows from Remark 4.6.1.

If we consider the composite map f : X(fa,fb)−−−−→ Km × Kn → Km ∧ Kn, Propo-

sition 4.7.14 gives the following corollary.

Corollary 4.7.15. The diagram

Hm(Km)⊗Hn(Kn)

f∗a⊗f∗b

×

≈// Hm+n(Km ∧ Kn)

f∗

Hm(X)⊗Hn(X)

// Hm+n(X)

is commutative.

Let X and Y be two topological spaces. Their join X ∗ Y is the quotientof X × Y × I by the equivalence relation (x, y, 0) ∼ (x, y′, 0) for y, y′ ∈ Y and(x, y, 1) ∼ (x′, y, 1) for x, x′ ∈ X . This topological join is related to the simplicialjoin in the following way: if K and L are locally finite simplicial complexes, then|K ∗ L| is homeomorphic to |K| ∗ |L| (see [155, Lemma 62.2]).

144 4. PRODUCTS

The two open subspacesX×Y ×[0, 1) andX×Y ×(0, 1] ofX×Y ×I define opensubspaces UX and UY of X ∗Y . The space UX retracts by deformation onto X andUY retracts by deformation onto Y . Moreover, UX ∩ UY retracts by deformationonto X × Y × 12. The following diagram is homotopy commutative,

UX ∩ UY incl // UX

X × Y πX //

≃

OO

X

≃

OO

as well as the corresponding diagram for Y . Consider the homomorphism

Hk(X)⊕Hk(Y )π∗X+π∗Y−−−−−→ Hk(X × Y ) .

If k > 0, then (π∗X+π∗

Y)(a, b) = a×1+1×b and, by the Kunneth theorem, π∗

X+π∗

Y

is injective. As X ∗ Y is path-connected, the Mayer-Vietoris sequence for the data(X ∗Y, UX , UY , UX ∩UY ) splits and gives, for all integers k ≥ 0, the exact sequence

(4.7.8) 0→ Hk(X)⊕ Hk(Y )π∗X+π∗Y−−−−−→ Hk(X × Y ) −→ Hk+1(X ∗ Y )→ 0 .

Example 4.7.16. The join Sp ∗ Sq is homeomorphic to Sp+q+1. Considering

Sp+q+1 ⊂ Rp+1 ×Rq+1, a homeomorphism Sp+q+1 ≈−→ Sp ∗ Sq is given by (x, y) 7→[(x, y, |x|]. The reader can check (4.7.8) on this example, including the case p =q = 0.

Observe that UX and UY are contractible in X ∗ Y . Hence, the Lusternik-Schnirelmann category of X ∗Y is equal to 2. By Proposition 4.4.2, the cup productin H>0(X ∗ Y ) vanishes.

When the Kunneth theorem is valid, one sees that the cohomology ring ofX ∗ Y is isomorphic to that of Σ(X ∧ Y ). Actually, under some hypotheses, thesetwo spaces have the same homotopy type (see [82, Ex. 24, p. 20]).

4.7.5. The theorem of Leray-Hirsch. An important generalization of aproduct space is a locally trivial fiber bundle. A map p : E → B is a locally trivialfiber bundle with fiber F (in short: a bundle) if there exists an open covering Uof B and, for each U ∈ U , a homeomorphism ψU : U × F ≈−→ p−1(U) such thatpψ(x, v) = x for all (x, v) ∈ U × F . The space E is the total space and B is thebase space of the bundle. If A is a subspace of B, we set EA = p−1(A), getting abundle p : EA → A. If b ∈ B, we set Eb = Eb and denote by ib : Eb → E theinclusion. A fiber inclusion is an embedding i : F → E which is a homeomorphismonto some fiber Eb. As elsewhere in the literature, we shall often speak about a

(locally trivial) bundle Fi−→ E

p−→ B, meaning a locally trivial bundle p : E → Bwith fiber F together with a chosen fiber inclusion i.

If p : E → B is a bundle, then the homomorphism p∗ = H∗p : H∗(B)→ H∗(E)provides a structure of graded H∗(B)-module on H∗(E).

A cohomology extension of the fiber is a GrV-morphism θ : H∗(F ) → H∗(E)such that, for each b ∈ B, the composite map

H∗(F )θ−→ H∗(E)

H∗ib−−−→ H∗(Eb)


is a GrV-isomorphism. We do not require that θ is multiplicative. In the presenta-

tion of a bundle by a sequence Fi−→ E

p−→ B with B path-connected, a cohomologyextension θ of the fiber exists if and only if H∗i is surjective.[see Comment 12.0.2.]

A cohomology extension θ of the fiber provides a morphism of graded H∗(B)-modules

H∗(B)⊗H∗(F ) θ−→ H∗(E)

given by θ(a⊗ b) = p∗(a) θ(b).Suppose that F is of finite cohomology type. As in Lemma 4.6.6, there is a

unique GrV-homomorphism

θ : H∗(E)→ H∗(B)⊗H∗(F ) .such that the formula

(4.7.9) 〈θ(b⊗ u), γ〉 = 〈b ⊗ u, θ(γ)〉holds true for all b ∈ H∗(B), u ∈ H∗(F ) and γ ∈ H∗(E). As in the proof ofLemma 4.6.6, we show that

θ(γ) =∑

m∈M

H∗p(θ(m∗) γ

)⊗m

where M be a basis for H∗(F ) and M∗ = m∗ ∈ H∗(F ) | m ∈ M is the dualbasis for the Kronecker pairing.

Theorem 4.7.17 (Leray-Hirsch). Let Ep−→ B be a locally trivial fiber bundle

with fiber F . Suppose that F is of finite cohomology type. Let θ : H∗(F )→ H∗(E)

be a cohomology extension of the fiber. Then, θ is an isomorphism of graded H∗(B)-modules and θ is a GrV-isomorphism.

Proof. By Kronecker duality, only the cohomology statement must be proven.Let A ⊂ B and let h∗(A) = H∗(A) ⊗H∗(F ). The composition

θA : H∗(F )θ−→ H∗(E) −→ H∗(EA)

is a cohomology extension of the fiber for the bundle p : EA → A, giving rise to

θA : h∗(A) → H∗(EA). We want to prove that θB is an isomorphism. Consideringthe commutative diagram

h∗(B)θB //

≈

H∗(E)

≈

(∏A∈π0(B)H

∗(A))⊗H∗(F )

Φ ≈

∏A∈π0(B) h

∗(A)

∏θA // ∏

A∈π0(B)H∗(EA)

where Φ is the linear map of Lemma 4.6.8, which is an isomorphism since Hk(F )is finite dimensional for all k, permits us to reduce to the case where the base ispath-connected.

We now suppose that B is path-connected and that the bundle E → B istrivial, i.e. there exists a homeomorphism ϕ : B ×F → E such that pϕ = πB, the

146 4. PRODUCTS

projection to the factor B. Since F is of finite cohomology type, one may use H∗ϕtogether with the Kunneth formula to identify H∗(E) with H∗(B)⊗H∗(F ).

Fix an integer n and consider the vector subspace Pnk ofH∗(B)⊗H∗(F ) definedby

Pnk =⊕

p,q

Hp(B)⊗Hq(F ) | p+ q = n and q ≤ k.

These subspaces provide a filtration

(4.7.10) 0 = Pn−1 ⊂ Pn0 ⊂ · · · ⊂ Pnn = Hn(E) .

We denote by ψ : H∗(F )→ H∗(F ) the GrV-morphism defined by ψ = H∗iθ.The fiber inclusion i is a slice over one point. Let u ∈ Hq(F ). As B is path-connected, Lemma 4.7.2 implies that

(4.7.11) θ(u) = 1⊗ ψ(u) mod P qq−1 .

Hence, if a ∈ Hp(B) with p+ q = n, one has

(4.7.12)θ(a⊗ u) = (a⊗ 1) θ(u)

= (a⊗ 1) (1⊗ ψ(u)) mod Pnn−1= a⊗ ψ(u) mod Pnn−1

In particular, θ preserves the filtration (4.7.10). It thus induces homomorphismsθ : Pnk /P

nk−1 → Pnk /P

nk−1. Moreover, one has a natural identification Pnk /P

nk−1 ≈

Hn−k(B) ⊗Hk(F ) under which θ(a⊗ u) = a⊗ ψ(u). This enables us to prove by

induction on k that θ : Pnk → Pnk is an isomorphism, using the five-lemma in thediagram

0 // Pnk−1 //

θ≈

Pnk//

θ

Hn−k(B)⊗Hk(F ) //

id⊗ψ≈

0

0 // Pnk−1 // Pnk // Hn−k(B)⊗Hk(F ) // 0

Indeed, the left vertical arrow is an isomorphism by induction hypothesis. SincePn−1 = 0, the induction starts with k = 0, using (4.7.12). Therefore, the Leray-Hirsch theorem is true for a trivial bundle.

Let Bi, i = 1, 2, be two open sets of B with B = B1∪B2. Let B0 = B1∩B2 andEi = p−1(Bi). The Mayer-Vietoris cohomology sequence for (B,B1, B2, B0) maybe tensored with Hk(F ) and remains exact, since we are dealing with Z2-vectorspaces. The sum of these sequences provides the exact sequence of the top line ofthe commutative diagram

hk−1(B1)⊕ hk−1(B2)

θ1⊕θ2

// hk−1(B0)

θ0

// hk(B)

θ

//

Hk−1(E1)⊕Hk−1(E2) // Hk−1(E0) // Hk(E) //

// hk(B1)⊕ hk(B2) //

θ1⊕θ2

hk(B0)

θ0

// Hk(E1)⊕Hk(E2) // hk(E0)


The bottom line is the Mayer-Vietoris sequence for the data (B,B1, B2, B0). By

the five-lemma, this shows that, if θi are isomorphisms for i = 0, 1, 2, then θ is anisomorphism.

What has been done so far implies that the θ is an isomorphism for a bundle of

finite type, i.e. admitting a finite covering U such that EUp−→ U is trivial for U ∈ U .

By Kronecker duality, θ is an isomorphism in this case. As in Point (2) of theproof of the Kunneth theorem (p. 135), θ is the direct limit of θA for A ⊂ B suchthat EA → A is of finite type. Therefore, θ is an isomorphism and, by Kronecker

duality, θ is an isomorphism for any bundle.

The Leray-Hirsch theorem also has the following version, in which the finitetype hypothesis is on the base rather than on the fiber. The proof, involving theSerre spectral sequence, may be found in [141, Theorem 10].

Theorem 4.7.18 (Leray-Hirsch II). Let Ep−→ B be a locally trivial fiber bun-

dle with fiber F . Suppose that B is path-connected and of finite cohomology type.

Let θ : H∗(F ) → H∗(E) be a cohomology extension of the fiber. Then, θ is anisomorphism of graded H∗(B)-modules and θ is a GrV-isomorphism.

Here are a few corollaries of the above Leray-Hirsch theorems.

Corollary 4.7.19. Let Fi−→ E

p−→ B be a locally trivial fiber bundle whosebase B is path-connected and whose fiber F (or base B) is of finite cohomologytype. Suppose that H∗i : H∗(E)→ H∗(F ) is surjective. Then

(1) H∗p : H∗(B)→ H∗(E) is injective.(2) kerH∗i is the ideal generated by the elements of positive degree in the

image of H∗p.

Proof. Let θ : Hk(F ) → Hk(E) be a cohomology extension of the fiber such

that H∗iθ is the identity of H(F ). As H∗p(b) = θ(b⊗1), the homomorphism H∗pis injective.

To prove (2), we note that any element in A ∈ H∗(E) may be written uniquely

as a finite sum A =∑

k∈K θ(bk ⊗ ak). Let Ko = k ∈ K | bk = 1. As pi isa constant map, one has H∗(pi)(1) = 1 and H∗(pi)(bk) = 0 if bk has positivedegree. Therefore,

H∗i(A) =∑

k∈KH∗i(H∗p(bbk) θ(ak))

=∑

k∈K

(H∗iH∗p(bk) H∗i(θ(ak))

)

=∑

k∈K

(H∗(pi)(bk) ak

)

=∑

k∈K0ak .

Hence, H∗i(A) = 0 if and only if Ko = ∅, which proves (2).

As for Proposition 4.7.1, the Leray-Hirsch theorem implies the following result.

Corollary 4.7.20. Let Fi−→ E

p−→ B be a locally trivial fiber bundle whosebase B is path-connected. Suppose that H∗i : H∗(E) → H∗(F ) is surjective. Sup-pose that F and B are of finite cohomology type. Then, E is of finite cohomologytype and the Poincare series of F , E and B satisfy

(4.7.13) Pt(E) = Pt(F )Pt(B) .

148 4. PRODUCTS

Actually, Equation (4.7.13) is equivalent to H∗i being surjective (see [15,Proposition 2.1]). Here is another kind of corollary of the Leray-Hirsch theorem.

Corollary 4.7.21. Let p : E → B be a locally trivial fiber bundle with fiber F .Suppose that H∗(F ) = 0. Then, H∗p : H∗(B) → H∗(E) is a GrA-isomorphism.

Remarks 4.7.22. (1) By Corollary 4.7.19, the existence of a cohomology ex-tension of the fiber implies that p∗ : H∗(B) → H∗(E) is injective. The converse isnot true, even if the map p has a section (see e.g. [72]).

(2) In the Leray-Hirsch theorem the isomorphism θ is not a morphism of alge-bras, unless θ is multiplicative. It is possible that there exists cohomology extensionsof the fiber but that none of them is multiplicative (see Examples 4.7.45 or 7.1.16).

(3) The proof of the Leray-Hirsch theorem shows the following partial result.Let θ : Hk(F ) → Hk(E) be a linear map defined for all k ≤ n. Suppose that,for each b ∈ B, the composition H∗ibθ : H

k(F ) → Hk(Eb) is an isomorphismfor k ≤ n. Then, with the notation of the proof of Theorem 4.7.17, the linear map

θ : hk(B)→ Hk(E) is an isomorphism for k ≤ n. For instance, we get the followingproposition.

Proposition 4.7.23. Let p : E → B be a locally trivial fiber bundle with fiber F .Suppose that Hk(F ) = 0 for all k ≤ m. Then, H∗p : Hk(B) → Hk(E) is anisomorphism for k ≤ m.

The Leray-Hirsch theorem admits a version for bundle pairs. A bundle pairwith fiber (F, F ′) is a topological pair (E,E′) and a map p : (E,E′)→ (B,B) suchthat there exists an open covering U of B and, for each U ∈ U , a homeomorphism

ψU : U × (F, F ′)≈−→ (p−1(U), p−1(U) ∩ E′) such that pψ(x, v) = x for all (x, v) ∈

U × F . In consequence, p : E → B is a bundle with fiber F and the restriction ofp to E′ is a bundle with fiber F ′. A cohomology extension of the fiber is a GrV-morphism θrel : H

∗(F, F ′)→ H∗(E,E′) such that, for each b ∈ B, the composite

H∗(F, F ′)θrel−−→ H∗(E,E′)

H∗ib−−−→ H∗(Eb, E′b)

is a GrV-isomorphism. A cohomology extension θ of the fiber provides a morphismof graded H∗(B)-modules

H∗(B)⊗H∗(F, F ′) θrel−−→ H∗(E,E′)

given by θrel(a⊗ b) = p∗(a) θrel(b).Suppose that F is of finite cohomology type. As in Lemma 4.6.6, there is a

unique GrV-homomorphism

θrel : H∗(E,E′)→ H∗(B)⊗H∗(F, F ′) .

such that the formula

(4.7.14) 〈θrel(b⊗ u), γ〉 = 〈b ⊗ u, θrel(γ)〉holds true for all b ∈ H∗(B), u ∈ H∗(F, F ′) and γ ∈ H∗(E,E′). The formula

(4.7.15) θrel(γ) =∑

m∈M

H∗p(θrel(m

∗) γ)⊗m.

is satisfied for all γ ∈ H∗(E,E′), where M is a basis for H∗(F, F

′) and M∗ =m∗ ∈ H∗(F, F ′) | m ∈M is the dual basis for the Kronecker pairing.


Theorem 4.7.24 (Leray-Hirsch relative). Let p : (E,E′)→ (B,B) be a bundlepair with fiber (F, F ′). Suppose that (F, F ′) is a well cofibrant pair and is of finitecohomology type. Let θrel : H

∗(F, F ′) → H∗(E,E′) be a cohomology extension of

the fiber. Suppose that (E,E′) is a well cofibrant pair. Then, θrel is an isomorphismof graded H∗(B)-modules and θrel is a GrV-isomorphism.

The hypothesis that (E,E′) is well cofibrant may be removed but this wouldnecessitate some preliminary work. Besides, this hypothesis is easily fulfilled in ourapplications.

Proof. By Kronecker duality, only the cohomology statement must be proven.We first reduce to the case where F ′ is a point. Let E = E/∼ where ∼ is theequivalence relation

x ∼ y ⇐⇒ p(x) = p(y) and x, y ∈ E′

and let E′ = E′/∼. Then the map p descends to a map p : (E, E′)→ (B,B) whichis a bundle pair with fiber (F/F ′, y0), where y0 is the point given by F ′ in F/F ′. In

particular, p : E′ → B is a homeomorphism. Consider the commutative diagram:

(4.7.16)

H∗(B)⊗H∗(F/F ′, y0)

θrel

≈ // H∗(B)⊗H∗(F, F ′)

θrel

H∗(E, E′)≈ // H∗(E,E′)

We shall show that the horizontal homomorphisms, induced by the quotient maps,are isomorphisms. Therefore, the right vertical arrow is bijective if and only if theleft one is.

The top horizontal homomorphism of Diagram (4.7.16) is an isomorphism byProposition 3.1.45 since (F, F ′) is a well cofibrant pair. To see that the bottomhorizontal map is also bijective, consider the commutative diagram

H∗(E/E′, [E′])≈ //

≈

H∗(E/E′, [E′])

≈

H∗(E, E′) // H∗(E,E′)

The top horizontal map is an isomorphism because the quotient spaces E/E′ and

E/E′ are equal. As (E,E′) is well cofibrant, the right hand vertical map is bijective

by Proposition 3.1.45. Also, Lemma 3.1.43 implies that (E, E′) is well cofibrantand thus, the left hand vertical map is an isomorphism by Proposition 3.1.45. Now,the diagram

H∗(F/F ′, [F ′])

≈

θrel // H∗(E, E′)

≈

// H∗(Eb, E′b)

≈

H∗(F, F ′)θrel //

≈

33H∗(E,E′) // H∗(Eb, E′b)

shows that the bundle pair p inherits a cohomology extension of the fiber θrel.

150 4. PRODUCTS

We are then reduced to the case F ′ = y0 being a single point. Consider thecommutative diagram:

H∗(B)⊗H∗(F, y0)

θrel

// // H∗(B)⊗H∗(F )

θ≈

// // H∗(B)⊗H∗(y0)

θ≈

H∗(E,E′) // // H∗(E) // // H∗(E′)

The top line is the exact sequence of the pair (F, y0) tensored by H∗(B). It is exactsince we are dealing with Z2-vector spaces and its splits since y0 is a retract ofF ′. The bottom exact sequence of the pair (E,E′) also splits since p : E → B ≈ E′is a retraction of E onto E′. We shall check below the existence of a cohomology

extension of the fiber θ : H∗(F )→ H∗(E), whence the middle vertical map θ. The

two maps θ are bijective by the absolute Leray-Hirsch theorem 4.7.17. By thefive-lemma, θrel is then also an isomorphism.

The existence of a cohomology extension of the fiber θ : H∗(F )→ H∗(E) comesfrom θrel : H

∗(F, y0) → H∗(E,E′) when ∗ > 0, since Hk(F, y0) ≈ Hk(F ) andHk(Eb, E

′b) ≈ Hk(Eb) for k > 0. When k = 0, we consider the diagram

H0(F, y0)

//θrel

//

≈,,

H0(E,E′)

H∗j

// // H0(Eb, E′b)

H0(F )

θ // H0(E)

// // H0(Eb)

H0(y0)

θ′ //

≈

22

r∗

XX

H0(E′) //

p∗

XX

H0(E′b)

,

where j : (E, ∅) → (E,E′) denotes the inclusion. The retraction r : F → y0produces a section r∗ : H0(y0)→ H0(F ) of the homomorphism induced by the in-clusion; this section provides an isomorphism H0(F ) ≈ H0(F, y0) ⊕ H0(y0). Asp : E′ → B is a homeomorphism, one gets a section p∗ : H0(E′) → H0(E) ofthe homomorphism induced by the inclusion. The homomorphism θ : H0(F ) ≈H0(F, y0)⊕H0(y0)→ H0(E) given by θ(a, b) = H∗j θrel(a) + p∗θ′(b) completesthe definition of the cohomology extension of the fiber θ in degree 0.

Corollary 4.7.25 (Relative Kunneth theorem II). Let X be a topological spaceand (Y,C) be a well cofibrant pair which is of finite cohomology type. Then, thecross product

× : H∗(X)⊗H∗(Y,C) ≈−→ H∗(X × Y,X × C)is a GrV-isomorphism.

Proof. We see the projection π1 : (X × Y,X × C) → (X,X) as a trivialbundle pair with fiber (Y,C). Then θ = H∗π2 : H

∗(Y,C) → H∗(X × Y,X × C)is a cohomology extension of the fiber. As (Y,C) is a well cofibrant pair, so is(X × Y,X × C). The cohomological result then follows from the relative Leray-Hirsch theorem 4.7.24.


4.7.6. The Thom isomorphism. We start with some preliminary results.

Lemma 4.7.26. Let p : (E, E)→ (B,B) be a bundle pair whose fiber (F , F ) is a

well cofibrant pair. Suppose that Hk(F , F ) = 0 for k < r and that Hr(F , F ) = Z2.

Then, Hk(E, E) = 0 for k < r and there is a unique isomorphism Φ∗ : Hr(E, E)≈−→

H0(B) such that, for each b ∈ B, the diagram

(4.7.17)

Hr(Eb, Eb)

=

// // Hr(E, E)

Φ∗≈

H0(b) // // H0(B)

is commutative, where the horizontal homomorphisms are induced by the inclusions.

In Diagram (4.7.17), the left vertical isomorphism is abstract but well defined,

since both Hr(Eb, Eb) ≈ Hr(F , F ) and H0(b) are equal to Z2.

Proof. We first prove the uniqueness of Φ∗, if it exists. Indeed, for each path-

connected component A of B, Diagram (4.7.17) implies that Φ∗(Hr(EA, EA)) =

H0(A) and, as H0(A) = Z2, the isomorphism Φ∗ is unique.If the bundle pair is trivial, the lemma follows from the relative Kunneth the-

orem 4.7.25. Suppose that B = B1 ∪B2, where B1 and B2 are two open sets withB1 ∩ B2 = B0. Suppose that the conclusion of the lemma is satisfied for (Ei, Ei)

for i = 0, 1, 2. Then, the Mayer-Vietoris sequence for the data (Ei, Ei) implies that

Hk(E, E) = 0 for k < r and gives the diagram

Hr(E0, E0) //

Φ∗≈

Hr(E1, E1)⊕Hr(E2, E2) //

Φ∗≈

Hr(E, E)

Φ∗≈

// 0

H0(B0) // H0(B1)⊕H0(B2) // H0(B) // 0

.

Diagram (4.7.17) for each b ∈ B0 implies that the left square is commutative.Therefore, the middle vertical isomorphism descends to a unique homomorphismΦ∗ : Hr(E, E) → H0(B) making the right square commutative, which an isomor-phism by the five-lemma. It remains to prove the commutativity of Diagram (4.7.17)for Φ∗. Let b ∈ B. Without loss of generality, we may suppose that b ∈ B1. Con-sider the diagram

Hr(Eb, Eb) //

=

Hr(E1, E1) //

Φ∗≈

Hr(E, E)

Φ∗≈

H0(b) // H0(B1) // H0(B)

.

As both square commute, this gives the commutativity of Diagram (4.7.17) for Φ∗.

We have so far proven the lemma when the bundle pair (E, E) → (B,B)is of finite type. Let A be the set of subspaces A of B such that the bundle pair(EA, EA)→ (A,A) is of finite type. Each compact of B is contained in some A ∈ Aand each compact of E is contained in EA for some A ∈ A. By Proposition 3.1.29,

152 4. PRODUCTS

this provides isomorphisms

(4.7.18) lim−→A∈A

Hr(EA, EA) ≈ Hr(E, E) and lim−→A∈A

H0(A) ≈ H0(B) .

Now, if A,A′ ∈ A with A ⊂ A′, Diagram (4.7.17) for each b ∈ A implies that thediagram

(4.7.19)

Hr(EA, EA)

Φ∗≈

// Hr(EA′ , EA′)

Φ∗≈

H0(A) // H0(A′)

is commutative. We therefore get isomorphisms

lim−→A∈A

Hr(EA, EA)≈−→ lim

−→A∈A

H0(A)

which, together with the isomorphisms of (4.7.18), produce the required isomor-

phism Φ∗ : Hr(E, E)≈−→ H0(B).

By Kronecker duality, Lemma 4.7.26 gives the following lemma.

Lemma 4.7.27. Let p : (E, E)→ (B,B) be a bundle pair whose fiber (F , F ) is a

well cofibrant pair. Suppose that Hk(F , F ) = 0 for k < r and that Hr(F , F ) = Z2.

Then, Hk(E, E) = 0 for k < r and there is a unique isomorphism Φ∗ : H0(B)≈−→

Hr(E, E) such that, for each b ∈ B, the diagram

(4.7.20)

H0(B)

Φ∗≈

// // H0(b)

=

Hr(E, E) // // Hr(Eb, Eb)

is commutative, where the horizontal homomorphisms are induced by the inclusions.

Let p : (E, E)→ (B,B) a bundle pair satisfying the hypotheses of Lemma 4.7.27.

The class U = Φ∗(1) ∈ Hr(E, E) is called the Thom class of the bundle pair p. If B

is path-connected, the Thom class is just the non-zero element of Hr(E, E) = Z2,whence the following characterization of the Thom class.

Lemma 4.7.28. The Thom class of p is the unique class in Hr(E, E) which

restricts to the generator of Hr(Eb, Eb) for all b ∈ B.

Let Σ be a topological space having the same homology (mod 2) as the sphereSk−1. For example, Σ = Sk−1 or a lens space with odd fundamental group. Letp : E → B be a bundle with fiber Σ. Let E be the mapping cylinder of p:

E = (E × I) ∪B/(x, 1) ∼ p(x) .

Let CΣ be the cone over Σ. Then p extends to a bundle pair p : (E, E) → (B,B)with fiber (CΣ,Σ), called themapping cylinder bundle pair of p. As (CΣ,Σ) is a wellcofibrant pair (by Lemma 3.1.39) and Hk(CΣ,Σ) = 0 for k 6= r and Hr(CΣ,Σ) =

Z2, the Thom class U ∈ Hr(E, E) is defined.


Theorem 4.7.29 (The Thom isomorphism theorem). Let p : E → B be a bundle

with fiber Σ, where Σ has the homology of the sphere Sr−1. Let p : (E, E)→ (B,B)

be its mapping cylinder bundle pair. Let U ∈ Hr(E, E) be the Thom class. Then,the homomorphisms

Φ∗ : Hk(B)→ Hk+r(E, E) and Φ∗ : Hk(E, E)→ Hk−r(B)

given by

Φ∗(a) = H∗p(a) U and Φ∗(γ) = H∗p(U γ)

are isomorphism for all k ∈ Z.

Observe that Lemma 4.7.27 gives the result for k ≤ 0.

Proof. As Hj(CΣ,Σ) = 0 for j 6= r, the homomorphism θrel : H∗(CΣ,Σ) →

H∗(E, E) sending the generator of Hr(CΣ,Σ) = Z2 onto the Thom class U is a

cohomology extension of the fiber. Also, (E, E) and the fiber (CΣ,Σ) are wellcofibrant by Lemma 3.1.39. The relative Leray-Hirsch theorem 4.7.24 then pro-

vides a GrV-isomorphism θrel : H∗(B) ⊗H∗(CΣ,Σ) ≈−→ H∗(E, E). Let Φ∗ be the

composite isomorphism

Φ∗ : Hk(B) ≈ Hk(B)⊗Hr(CΣ,Σ)θrel−−→ Hk+r(E, E)

satisfy, by definition of θrel, the formula Φ∗(a) = H∗p(a) U . This proves thecohomology statement.

For the isomorphism Φ∗, let 0 6= m ∈ Hr(CΣ,Σ). Then m and U areKronecker dual bases for (co)homology of (CΣ,Σ) in degree r. Theorem 4.7.24 andFormula (4.7.15) implies that the composite isomorphism

Φ∗ : Hk+r(E, E)θrel−−→ Hk(B)⊗Hr(CΣ,Σ) ≈ Hk(B)

satisfies Φ∗(γ) = H∗p(U γ).

Let q : E → B be a bundle with fiber F and let f : A → B be a continuousmap. The induced bundle f∗q : f∗E → A is defined by

f∗E = (a, y) ∈ A× E | f(a) = q(y) , f∗q(a, y) = a ,

where f∗E is topologized as a subspace of A × E. Then f∗q is a bundle over Awith fiber F . The projection onto E gives a map f : f∗E → E and a commutativediagram

f∗E

f∗q

f // E

q

A

f // B

.

Let p : E → B be a bundle with fiber Σ, where Σ has the homology of thesphere Sr−1 and let p : (E, E) → (B,B) be its mapping cylinder bundle pair. Let

f : A → B be a map. Then (f∗E, f∗E) → (A;A) is the mapping cylinder bundlepair of the induced bundle f∗E. The following lemma states the functoriality ofthe Thom class.

Lemma 4.7.30. If U ∈ Hr(E, E) is the Thom class of p, then H∗f(U) ∈Hr(f∗E, f∗E) is the Thom class of f∗p.

154 4. PRODUCTS

Proof. For a ∈ A, let consider the commutative diagram

Hr(E, E)

H∗ f // Hr(f∗E, f∗E)

Hr(Ef(a), Ef(a))H∗ f

≈// Hr((f∗E)a, (f

∗E)a)

.

Both cohomology groups downstairs are equal to Z2. The left vertical arrow sendsthe Thom class U to the non-zero element. Therefore, H∗f(U) goes, by the rightvertical arrow, to the non-zero element. As this is true for all a ∈ A, we deducefrom Lemma 4.7.28 that H∗f(U) is the Thom class of f∗p.

Let p : E → B be a bundle with fiber Σ, where Σ has the homology of thesphere Sr−1. Let p : (E, E) → (B,B) be its mapping cylinder bundle pair. As

CΣ is contractible, Proposition 4.7.23 implies that H∗p : H∗(B) → H∗(E) is aGrA-isomorphism. Therefore, there is a unique class e ∈ Hr(B) such that

(4.7.21) H∗j(U) = H∗p(e) ,

where U ∈ Hr(E, E) is the Thom class and j : (E, ∅)→ (E, E) is the pair inclusion.The class e = e(p) is called the Euler class of the bundle p. If Φ∗ : Hr(B) →H2r(E, E) is the Thom isomorphism, one has the formula

(4.7.22) Φ∗(e) = U U .

Indeed:

Φ∗(e) = H∗p(e) U = H∗j(U) U = U U ,

the last equality coming from Lemma 4.1.7. The Euler class is functorial by thefollowing lemma.

Lemma 4.7.31. Let p : E → B be a bundle with fiber Σ, where Σ has thehomology of the sphere Sr−1. Let f : A → B be a map. If e ∈ Hr(B) is the Eulerclass of p, then H∗f(e) ∈ Hr(A) is the Euler class of f∗p.

Proof. This follows from the definition of the Euler class, Lemma 4.7.30 andthe commutativity of the diagram.

Hr(E, E)

H∗ f

// H(E)

H∗ f

oo H∗p

≈Hr(B)

H∗f

Hr(f∗E, f∗E) // Hr(f∗E) ooH∗f∗p

≈Hr(A)

.

The Euler class occurs in the Gysin exact sequence.

Proposition 4.7.32 (Gysin exact sequence). Let p : E → B be a bundle withfiber Σ, where Σ has the homology of the sphere Sr−1. Let e ∈ Hr(B) be its Eulerclass. Then, there is a long exact sequence

· · · → Hk−1(B)H∗p−−−→ Hk−1(E)→ Hk−r(B)

−e−−−→ Hk(B)H∗p−−−→ Hk(E)→ · · ·

which is functorial with respect to induced bundles. [See Comment 12.0.1].


Proof. Let p : (E, E) → (B,B) be the mapping cylinder pair of p. One uses

the cohomology sequence of the pair (E, E) and the commutative diagram

· · · // Hk−1(E) // Hk(E, E)H∗j // Hk(E) // · · ·

Hk−r(B)

Φ∗ ≈

OO

−e // Hk(B)

≈ H∗p

OO

where j : (E, E) → (E, ∅) denotes the inclusion and Φ∗ is the Thom isomorphism.The diagram is commutative since, for a ∈ Hk−r(B),

H∗j Φ∗(a) = H∗j(H∗p(a) U

)

= H∗p(a) H∗j(U)= H∗p(a) H∗p(e)= H∗p(a e) .

(The second equality is the singular analogue of Lemma 4.1.8). The functorialityof the Gysin exact sequence comes from Lemma 4.7.30 and 4.7.31.

Corollary 4.7.33. Let p : E → B be a bundle with fiber Σ, where Σ has thehomology of the sphere Sr−1. If p admits a continuous section, then the Euler classof p vanishes.

Proof. In the following segment of the Gysin sequence:

H0(B)−e−−−→ Hr(B)

H∗p−−−→ Hr(E) ,

the class 1 ∈ H0(B) is sent to the Euler class e. If p admits a section, then H∗p isinjective, which implies that e = 0.

Remark 4.7.34. The vanishing of the Euler class of p : E → B does not implythat p admits a section. As an example, let p : SO(3) → S2 the map sending amatrix to its first column vector. Then p is an S1-bundle, equivalent to the unittangent bundle of S2. The Gysin sequence gives the exact sequence

H0(S2)−e−−−→ H2(S2)

H∗p−−−→ H2(SO(3))→ 0

As SO(3) is homeomorphic to RP 3, H2(SO(3)) = Z2 by Proposition 4.3.10 and allthe cohomology groups in the above sequence are equal to Z2. Hence, e = 0. Butit is classical that S2 admits no nowhere zero vector field [82, Theorem 2.28].

Proposition 4.7.35. Let p : E → B be a bundle with fiber Σ, where Σ has thehomology of the sphere Sr−1. Let e ∈ Hr(B) be its Euler class. Then, the followingassertions are equivalent.

(1) e = 0.(2) The restriction homomorphism H∗(E)→ H∗(Σ) is surjective.(3) Hr−1(E) ≈ Hr−1(B) ⊕ Z2.

156 4. PRODUCTS

Proof. Let p : (E, E)→ (B,B) be the mapping cylinder pair of p. IdentifyingΣ as the fiber over some point of B, we get a commutative diagram

0 // Hr−1(E) //OOH∗p≈

Hr−1(E) //

Hr(E, E) //

≈

Hr(E)

Hr−1(B) Hr−1(Σ)≈ // Hr(CΣ,Σ)

where the top line is the cohomology sequence of (E, E). But Hr(E, E) ≈ Z2 gen-

erated by the Thom class which, under the homomorphism Hr(E, E)→ Hr(E) ≈Hr(B), goes to the Euler class. This proves the proposition.

Let us consider the particular case of the Gysin sequence for an S0-bundle.Such a bundle is simply a 2-fold covering ξ = (p : X → X). The Gysin sequencemay thus be compared to the transfer exact sequence of Proposition 4.3.9.

Proposition 4.7.36. Let ξ = (p : X → X) be a 2-fold covering (an S0-bundle).Then, the Gysin and the transfer exact sequences of ξ coincide, i.e. the diagram

· · · // Hk(X)OOid

H∗p // Hk(X)OOid

tr∗ // Hk(X)w(ξ)−//

OOid

Hk+1(X)OOid

// · · ·

· · · // Hk(X)H∗p // Hk(X) // Hk(X)

e(p)−// Hk+1(X) // · · ·

is commutative. In particular, the Euler class e(ξ) ∈ H1(X) and the characteristicclass w(p) ∈ H1(X) are equal.

Proof. By Corollary 4.3.3, ξ is induced from ξ∞ = (p∞ : S∞ → RP∞) by acharacteristic map f : X → RP∞. Both the Gysin and transfer exact sequencesbeing functorial with respect to induced bundles, it suffices to prove the propositionfor ξ∞. This is trivial since the vector spaces occurring in the diagram are eitherequal to 0 or Z2.

The Thom isomorphism is classically used for vector bundles. Recall that a(real) vector bundle ξ of rank r is a map p : E → B together with a R-vector spacestructure on Eb = p−1(b) for each b ∈ B, satisfying the following local trivialitycondition: there is an open covering U of B and for each U ∈ U , a homeomorphism

ψU : U × Rr≈−→ p−1(U) such that, for all (b, v) ∈ U × Rr, pψ(b, v) = b and

ψU : b × Rr → Eb is a R-linear isomorphism. In consequence, p is a bundlewith base B = B(ξ), total space E = E(ξ) and fiber Rr. The map σ0 : B → Esending b ∈ B to the zero element of Eb is called the zero section of ξ (it satisfiespσ0(b) = b).

An Euclidean vector bundle is a vector bundle ξ together with a continuousmap v 7→ |v| ∈ R≥0 defined on E(ξ) whose restriction to each fiber is quadraticand positive definite. Such a map is called an Euclidean structure (or Riemannianmetric) on ξ. It is of course the same as defining a positive definite inner product oneach fiber which varies continuously. Vector bundles with paracompact basis admita Euclidean structure, [105, Chapter 3, Theorems 9.5 and 5.5]. If ξ = (p : E → B)is a Euclidean vector bundle, the restriction of p to

S(E) = v ∈ E | |v| = 1 and D(E) = v ∈ E | |v| ≤ 1


gives the associated unit sphere and disk bundles. These bundles do not dependon the choice of the Euclidean structure on ξ. Indeed, using the map (v, t) 7→ tvfrom S(E) × I → D(V ) together with the zero section, the reader will easily con-

struct a homeomorphism (S(E), S(E))≈−→ (D(E), S(E)) over the identity of B,

where (S(E), S(E)) → (B,B) is the mapping cylinder bundle pair of S(E) → B.Thus, the Thom class U ∈ Hr(D(E), S(E)) exists by Lemma 4.7.27 and, by Theo-

rem 4.7.29, gives rise to the Thom isomorphisms Φ∗ : Hk(B)≈−→ Hk+r(D(E), S(E))

and Φ∗ : Hk+r(D(E), S(E))≈−→ Hk(B). Let E0 = E − σ0(B) and D(E)0 =

D(E) ∩ E0. By excision and homotopy, one has

H∗(E,E0)≈−→ H∗(D(E), D(E)0)

≈−→ H∗(D(E), S(E)) .

Hence the Thom class may be seen as an element U(ξ) ∈ Hr(E,E0) and one hasthe following theorem.

Theorem 4.7.37 (The Thom isomorphism theorem for vector bundles). Letξ = (p : E → B) be a vector bundle of rank r with B paracompact. Let U(ξ) ∈Hr(E,E0) be the Thom class. Then, the homomorphisms

Φ∗ : Hk(B)→ Hk+r(E,E0) and Φ∗ : Hk(E,E0)→ Hk−r(B)

given by

Φ∗(a) = H∗p(a) U(ξ) and Φ∗(γ) = H∗p(U(ξ) γ)

are isomorphism for all k ∈ Z.

Let ξ = (p : E → B) be a vector bundle of rank r. The map E × I → E givenby (v, t) 7→ tv is a retraction by deformation of E onto the zero section of ξ. Hence,H∗p : H∗(B) → H∗(E) is a GrA-isomorphism. Therefore, there is a unique classe(ξ) ∈ Hr(B) such that H∗j(U(ξ)) = H∗p(e(ξ)), where j : (E, ∅) → (E,E0). Theclass e(ξ) is called the Euler class of ξ (it coincides with the Euler e(S(E)) definedabove). Lemma 4.7.31 and Corollary 4.7.33 imply the following two lemmas.

Lemma 4.7.38. Let ξ = (p : E → B) be a vector bundle of rank r, with Bparacompact. Let f : A → B be a continuous map. Then the equality e(f∗ξ) =H∗f(e(ξ)) holds in Hr(A).

Lemma 4.7.39. Let ξ = (p : E → B) be a vector bundle of rank r, with Bparacompact. If ξ admits a nowhere zero section, then e(ξ) = 0.

Let ξi = (pi : Ei → Bi) (i = 1, 2) be two vector bundles of rank ri, with Biparacompact. The product bundle ξ1× ξ2 is the vector bundle of rank r1 + r2 givenby p1 × p2 : E1 × E2 → B1 × B2. If B1 = B2 = B, the Whitney sum ξ1 ⊕ ξ2 isthe vector bundle of rank r1 + r2 over B given by ξ1 ⊕ ξ2 = ∆∗(ξ1 × ξ2) where∆: B → B×B is the inclusion of the diagonal, ∆(x) = (x, x). The behavior of theEuler class under these constructions is as follows.

Proposition 4.7.40. .

(1) e(ξ1 × ξ2) = e(ξ1)× e(ξ2).(2) e(ξ1 ⊕ ξ2) = e(ξ1) e(ξ2).

Proof. Using Euclidean structures on ξi the Thom class U(ξi) may be seen asan element of Hri(D(Ei), S(Ei)). Let E = E1 ×E2, B = B1×B2 and r = r1 + r2.

158 4. PRODUCTS

Let ji : (D(Ei), ∅) → (D(Ei), S(Ei)) and j : (D(E), ∅) → (D(E), S(E)) denotethe inclusions. There are homeomorphisms of pairs making the following diagramcommutative

(D(E), ∅)≈

j // (D(E), S(E))

≈

(D(E1), ∅)× (D(E2), ∅)j1×j2 // (D(E1), S(E1))× (D(E2), S(E2))

.

In the same way, if b = (b1, b2) ∈ B, there is a homeomorphism of pairs

(4.7.23)(D(E)b, S(E)b

)≈

(D(E1)b1 , S(E1)b1

)×(D(E2)b2 , S(E2)b2

).

By the relative Kunneth theorem 4.6.10, the generator of Hr(D(E)b, S(E)b) =Z2 is the cross product of the generators ofH

ri(D(Ei)bi , S(Ei)bi). Using Lemma 4.7.28,we deduce that

(4.7.24) U(ξ1 × ξ2) = U(ξ1)× U(ξ2) .

Using Lemma 4.6.2, one has

H∗(p1 × p2)(e(ξ)) = H∗j(U(ξ))= H∗j(U(ξ1)× U(ξ2))= H∗j1(U(ξ1))×H∗j2(U(ξ2))= H∗p1(e(ξ1))×H∗p2(e(ξ2))= H∗(p1 × p2)(e(ξ1)× e(ξ2)) .

As H∗(p1 × p2) is an isomorphism, this proves (1). Point (2) is deduced from (1)using the definition of ξ1 ⊕ ξ2 and Remark 4.6.1:

e(ξ1 ⊕ ξ2) = H∗∆(e(ξ1 × ξ2)) = H∗∆(e(ξ1)× e(ξ2)) = e(ξ1) e(ξ2) .

The Thom class of a product bundle was computed in (4.7.24). For the Whitneysum, we use the projections πi : E(ξ1 ⊕ ξ2)→ E(ξi).

Proposition 4.7.41. Let ξ1 and ξ2 be two vector bundles over a paracompactbasis. Let U(ξi) ∈ Hri(D(Ei), S(Ei)) be the Thom classes (for a Euclidean struc-ture). Then

U(ξ1 ⊕ ξ2) = H∗π1(U(ξ1)) H∗π2(U(ξ2)) .

Proof. Restricted to the fiber over b ∈ B, the right hand side of the formulagives the cross product of the generators of Hri(D(Ei)b, S(Ei)b). The latter is thegenerator of Hr(D(E)b, S(E)b). The proposition thus follows from Lemma 4.7.28.

4.7.7. Bundles over spheres. In this section, we study bundles ξ = (p : E →Sm) over the sphere Sm with fiber F . If A ⊂ Sm we set EA = p−1(A). Consider thecellular decomposition of Sm with one 0-cell b and one m-cell with characteristicmap ϕ : Dm → Sm sending Sm−1 onto b. We denote by φ : Sm−1 → b thisconstant map. We identify F with Eb, getting thus an inclusion i : F → E. AsDm is contractible, any bundle over Dm is trivial [181, Corollary 11.6]. Therefore,there exists a trivialization ϕ∗E ≈ Dm × F of the induced bundle ϕ∗ξ. The map(ϕ, φ) are covered by a bundle maps ϕ : Dm×F → E and φ : Sm−1×F → F . The


latter satisfies, for each x ∈ Sm−1, that φ : x × F ≈−→ F is a homeomorphism.Observe that

(4.7.25) E = (Dm × F ) ∪ϕ F .Let x0 ∈ Sm−1 be the base point corresponding to 1 ∈ S0 ⊂ Sm−1. By changingthe trivialization of ϕ∗ξ if necessary, we shall assume that φ : x0 × F → F is theprojection onto F . The map ϕ is called the bundle characteristic map and the mapφ is called the bundle gluing map of the bundle ξ.

Lemma 4.7.42. The bundle characteristic map ϕ : Dm × F → E induces anisomorphism

H∗ϕ : H∗(E,F )≈−→ H∗(Dm × F, Sm−1 × F ) .

Proof. Consider the decomposition Dm = B ∪ C, where B is the disk withcenter 0 and radius 1/2 and C the closure of Dm − B; let S = B ∩ C. As ϕ(C)is a disk around b, the bundle ξ is trivial above ϕ(C): Eϕ(C) ≈ ϕ(C) × F . Asϕ : B×F → Eϕ(B) is a homeomorphism, the lemma follows from the commutativediagram

H∗(E,F )H∗ϕ // H∗(Dm × F, Sm−1 × F )

H∗(E,Eϕ(C))

≈ excision

≈ excision

OO

H∗(Dm × F,C × F )

≈ excision

OO

≈ excision

H∗(Eϕ(B), Eϕ(S))H∗ϕ

≈// H∗(B × F, S × F )

Proposition 4.7.43. Let p : E → Sm be a bundle with fiber F . There is a longexact sequence

· · · → Hk−1(E)H∗i−−→ Hk−1(F )

Θ−→ Hk−m(F )J−→ Hk(E)

H∗i−−→ Hk(F )→ · · · .The exact sequence of Proposition 4.7.43 is called the Wang exact sequence.

Proof. We start with the exact sequence of the pair (E,F )

(4.7.26) · · · → Hk−1(E)H∗i−−→ Hk−1(F )

δ∗−→ Hk(E,F )H∗j−−−→ Hk(E)→ · · ·

where j : (E, ∅) → (E,F ) denotes the pair inclusion. The following commutativediagram defines the homomorphism Θ and J .

(4.7.27)

Hk−1(F )

Θ

''

δ∗ // Hk(E,F )H∗j //

H∗ϕ≈

Hk(E)

Hk(Dm × F, Sm−1 × F )

Hk−m(F )

e×−≈

OO J

DD

160 4. PRODUCTS

Here, H∗ϕ is an isomorphism by Lemma 4.7.42, e ∈ Hm(Dm, Sm−1) = Z2 is thegenerator and the map e × − is an isomorphism by the relative Kunneth theo-rem 4.7.25.

We now give some formulae satisfied by the homomorphism Θ: Hk−1(F ) →Hk−m(F ). We start with the case m = 1 which deserves a special treatment. Thebundle gluing map φ : S0 × F → F satisfies φ(1, x) = x and φ(−1, x) = h(x) forsome homeomorphism h : F → F . The decomposition of (4.7.25) amounts to sayingthat E is the mapping torus Mh of h:

E =Mh =([−1, 1]× F

)/(1, x) ∼ (−1, h(x)) .

The bundle projection p : Mh → S1 is given by p(t, x) = exp(2iπt). The corre-spondence x → [(x, 0)] gives an inclusion j : F → Mh. Let e ∈ H1(S1) = Z2 bethe generator. Proposition 4.7.43 may be rephrased and made more explicit in thefollowing way.

Proposition 4.7.44 (Mapping torus exact sequence). Let h : F → F be ahomeomorphism. Then, there is a long exact sequence

· · ·Hk−1(Mh)H∗i−−→ Hk−1(F )

Θ−→ Hk−1(F )J−→ Hk(Mh)

H∗i−−→ Hk(F )→ · · · ,with Θ = id +H∗h.

Proof. We use the exact sequence (4.7.26) with E =Mh and Diagram (4.7.27).It remains to identify Θ with id +H∗h. Let i± : ±1 × F → S0 × F denote theinclusions. Let α : Hk−1(F ) → Hk−1(S0 × F ) be the homomorphism such thatH∗i+α(a) = a and H∗i−α(a) = H∗h(a). Consider the diagram.

Hk−1(F )

δ∗

α // Hk−1(S0 × F )

δ∗

i∗

≈// Hk−1(1 × F )⊕Hk−1(−1 × F )

+

Hk(Mh, F )H∗ϕ

≈// Hk(D1 × F, S0 × F ) Hk−1(F )

e×−

≈oo

where i∗ = (H∗i+, H∗i−). Let Ψ± : H

k−1(F ) → Hk−1(F ) be the composite ho-momorphisms through the upper right or lower left corners. Then ψ+ = id +H∗hand ψ− = Θ. The left square of the diagram being commutative by construction ofMh, it then suffices to prove the commutativity of the right square, that is δ∗ = ψ,where ψ(a) = e ×

(H∗i+(a) +H∗i−(a)

). The homomorphisms δ∗ and ψ are both

functorial. As, by § 3.8, a class a ∈ Hk−1(F ) is represented by a map F → Kk−1,it suffices to prove that δ∗ = ψ for F = Kk−1. Observe that δ∗ and ψ are bothsurjective and have the same kernel, the image of Hk−1(D1×F )→ Hk−1(S0×F ).As Hk−1(Kk−1) = Z2, this proves that δ

∗ = ψ when F = Kk−1.

Example 4.7.45. Let h : S1 → S1 be the complex conjugation. Then, Mh is

homeomorphic to the Klein bottle K and we get a bundle S1 → Kp−→ S1. The

homomorphism Θ of Proposition 4.7.44 satisfies Θ = id+H∗h = 0. By the mappingtorus exact sequence, we deduce that H∗(K)→ H∗(S1) is surjective (this can alsobe obtained using a triangulation like on p. 28 and computations like on p. 117).A cohomology extension of the fiber σ : H∗(S1)→ H∗(K) produces, by the Leray-

Hirsch theorem 4.7.17, a GrV-isomorphism σ : H∗(S1) ⊗H∗(S1)≈−→ H∗(K). But

σ is not a morphism of algebra. Indeed, the square map x 7→ x•x vanishes in


H∗(S1)⊗H∗(S1) while the cup-square map x 7→ x x does not vanish in H∗(K)(see, Proposition 4.2.3).

When m > 1, some information about the homomorphism Θ: Hk−1(F ) →Hk−m(F ) may be obtained via the composition

Hk−1(F )Θ // Hk−m(F ) // e×− // Hk−1(Sm−1 × F ) ,

where e ∈ Hm−1(Sm−1) = Z2 is the generator. The map e× − is injective by theKunneth theorem.

Proposition 4.7.46. Suppose that m > 1. Then

e×Θ(a) = H∗φ(a)−H∗p2(a) ,where φ, p2 : S

m−1 × F → F are the bundle gluing map and the projection onto F .

Proof. As F is a retract of Sm−1 × F , the cohomology sequence of the pair(Sm−1 × F, F ) splits into short exact sequences and, by the Kunneth theorem andLemma 4.7.2, there is a commutative diagram(4.7.28)

0 // Hk−m(F )

≈

e×−

**

// Hk−m(F )⊕Hk−1(F )

α≈

+ // Hk−1(F )

id≈

// 0

0 // Hk−1(Sm−1 × F, F ) // Hk−1(Sm−1 × F ) H∗i // Hk−1(F ) // 0

where i : F → Sm−1 × F is the slice inclusion at the base point x0 ∈ Sm−1 andα(a, b) = e× a+ 1× b. Recall that we assume the restriction of φ to x0 × F tocoincide with the projection p2. Therefore, the composition

Hk−1(F )H∗φ−H∗p2−−−−−−−→ Hk−1(Sm−1 × F )→ Hk−1(F )

vanishes. Using Diagram (4.7.28), we get a factorization

Hk−1(F )H∗φ−H∗p2 //

Θ′

&&

Hk−1(Sm−1 × F )

Hk−m(F )

e×−

66♠♠♠♠♠♠♠♠♠♠

which we introduce in the diagram

(4.7.29)

Hk−1(F )

Θ′

&&

δ∗ //

H∗φ+H∗p2

Hk(E,F )

H∗ϕ≈

Hk−1(Sm−1 × F ) δ∗ // Hk(Dm × F, Sm−1 × F )

Hk−m(F )= //

e×−

OO

Hk−m(F )

e×−≈

OO.

We claim that the two squares of Diagram (4.7.29) are commutative. By Dia-gram (4.7.27), this will imply that Θ′ = Θ and will prove the lemma.

162 4. PRODUCTS

As φ is the restriction of ϕ, the naturality of the connecting homomorphism δ∗

implies that δ∗ H∗φ = H∗ϕδ∗. Since p2 extends to Dn × F , the homomorphismH∗p2 : H

k−1(F ) → Hk−1(Sm−1 × F ) factors through Hk−1(Dm × F ) and thusδ∗H∗p2 = 0. Hence, the top square is commutative. For the bottom one, leta ∈ Hk−1(F ). By § 3.8, a = H∗f(ι) for some map f from F into the Eilenberg-MacLane space Kk−1. The bottom square being functorial for the map f , it sufficesto prove its commutativity for F = Kk−1. As the source and range vector spaceare both then isomorphic to Z2, the commutativity holds trivially.

As an exercise, the reader may adapt the proof of Proposition 4.7.46 to the casem = 0, thus getting an alternative proof of Proposition 4.7.44. The main point isto replace e (which has no meaning in H0(S0) by the class of −1.

The family of homomorphisms Θ: Hk−1(F ) → Hk−m(F ) forms an endomor-phism of H∗(F ) of degree m− 1 (it sends Hq(F ) to Hq−m+1(F )).

Proposition 4.7.47. As an endomorphism of H∗(F ), Θ satisfies

Θ(a b) = Θ(a) b + a Θ(b) .

Proof. Proposition 4.7.46 may be rephrased as

H∗φ(a) = H∗p2(a) + e ×Θ(a) = 1× a+ e×Θ(a) .

Therefore, if a ∈ Hp(F ) and b ∈ Hq(F ),

H∗φ(a b) = 1× (a b) + e×Θ(a b)

and, using Lemma 4.6.3,

H∗φ(a) H∗φ(b) =[1× a+ e×Θ(a)

]

[1× b+ e×Θ(b)

]

= 1× (a b) + e× (Θ(a) b) + e× (a Θ(b))= 1× (a b) + e×

[Θ(a) b+ a Θ(b)

].

As H∗φ(ab) = H∗φ(a)H∗φ(b) and e×− is injective, this proves the proposi-tion.

Remark 4.7.48. The material of this section was inspired by [207, § 1, Chap-ter VII]. As in this reference, the following facts can also be proved:

(1) The Wang exact sequence holds for Serre fibrations. It also has a gener-alization to bundles over a suspension.

(2) A Wang exact sequence for homology exists.

Further properties of the Wang sequences are given in [207, § 2, Chapter VII].

4.7.8. The face space of a simplicial complex. Let K be simplicial com-plex. Fix an integer d > 0. For each v ∈ V (K), consider a copy Sdv of the sphereSd. It is pointed by e1 = (1, 0, . . . , 0) ∈ Sdv . For σ ∈ S(K), consider the space

Fd(σ) = (zv) | zv = e1 if v /∈ σ ⊂∏

v∈V (K)

Sdv ,

which is homeomorphic to∏v∈σ S

dv . The face space ofK is the subset of

∏v∈V (K) S

dv

defined by

Fd(K) =⋃

σ∈S(K)

Fd(σ) ⊂∏

v∈V (K)

Sdv .


Remark 4.7.49. Let K be a flag simplicial complex (i.e. if K contains a graphL isomorphic to the 1-skeleton of an r-simplex, then L is contained in an r-simplexof K). Then the complex F1(K) is the Salvetti complex of the right-angled Coxetergroup determined by the 1-skeleton of K (see [30]).

The interest of the face space appears in the following proposition, based on aalgebraic theorem of J. Gubeladze.

Proposition 4.7.50. Let K and K ′ be two finite simplicial complexes. Let dbe a positive integer. Then, K is isomorphic to K ′ if and only if H∗(Fd(K)) andH∗(Fd(K

′)) are GrA-isomorphic.

To explain the proof of Proposition 4.7.50, we compute the cohomology algebraof Fd(K) for a finite simplicial complex K. Let us number the vertices of K:V (K) = 1, . . . ,m. Consider the polynomial ring Z2[x1, . . . , xm] with formalvariables x1, . . . , xm which are of degree d. If J ⊂ 1, . . . ,m, we denote by xJ ∈Z2[x1, . . . , xm] the monomial

∏j∈J xj . Let I(K) be the ideal of Z2[x1, . . . , xm]

generated by the squares x2i of the variables and the monomials xJ for J /∈ S(K)(non-face monomials). The quotient algebra

Λd(K) = Z2[x1, . . . , xm]/I(K)

is called the face exterior algebra (because u2 = 0 for all u ∈ Λd(K); however,because the ground field is Z2, Λd(K) is commutative).

Lemma 4.7.51. The ring H∗(Fd(K)) is isomorphic to Λd(K).

Proof. (Compare [59, Proposition 4.3].) Let ∆K = F(V (K)) be the fullcomplex over the set V (K) = 1, . . . ,m. The simplicial inclusionK ⊂ ∆K inducesan inclusion

j : Fd(K) → Fd(∆K) =∏

v∈V (K)

Sdv .

For σ ⊂ 1, . . . ,m, the fundamental class [Fd(σ)] ∈ H(dimσ+1)d(Fd(σ)) deter-mines a class [σ] ∈ H(dimσ+1)d(Fd(∆K)) (by convention, [∅] is the generator ofH0(Fd(∆K))). If σ ∈ S(K), [σ] is the image underH∗j of a class inH(dimσ+1)d(Fd(K)),also called [σ]. Let

A = [σ] ∈ H∗(Fd(K)) | σ ∈ S(K) ∪ ∅ ⊂ H∗(Fd(K))

andB = [σ] ∈ H∗(Fd(∆K)) | σ ⊂ 1, . . . ,m ⊂ H∗(Fd(∆K)) .

By the Kunneth theorem and Corollary 3.1.16, H∗(Fd(K)) is generated by A andB is a basis of H∗(Fd(∆K)). It follows that A is a basis of H∗(Fd(K)) and thatH∗j is injective. By Kronecker duality, H∗j is surjective and the Kronecker-dualbasis B♯ of B is sent onto Kronecker-dual basis A♯ of A by

(4.7.30) H∗j([σ]♯) =

[σ]♯ if σ ∈ S(K)

0 otherwise.

By the Kunneth theorem again,

(4.7.31) H∗(Fd(∆K)) ≈ Z2[x1, . . . , xm]/(x21, . . . , x

2m)

and, if σ ⊂ 1, . . . ,m, then [σ]∗ = xσ. By (4.7.30), kerH∗j is the Z2-vectorspace in H∗(Fd(∆K)) with basis xσ | σ /∈ S(K). Using (4.7.31), we check that,

164 4. PRODUCTS

under the epimorphism Z2[x1, . . . , xm] →→ H∗(Fd(∆K)), kerH∗j is the image ofI(K).

The proof of Lemma 4.7.51 provides the following corollary.

Corollary 4.7.52. The Poincare polynomial of the algebra Λd(K) is

Pt(Λd(K)) = 1+∑

σ∈S(K)

t(dimσ+1)d .

The proof of Proposition 4.7.50 follows from Lemma 4.7.51 and the followingtheorem of J. Gubeladze. For a proof, see [76, Theorem 3.1].

Theorem 4.7.53 (J. Gubeladze). Let K and K ′ be two finite simplicial com-plexes. Suppose that Λd(K) = Z2[x1, . . . , xm]/I(K) and Λd(K

′) = Z2[y1, . . . , ym′ ]/I(K ′)are isomorphic as graded algebras. Then m = m′ and there is a bijection

φ : x1, , . . . , xm ≈−→ y1, . . . , ym′such that φ(I(K)) = I(K ′).

4.7.9. Continuous multiplications on K(Z2,m). A continuous multiplica-tion µ : X ×X → X on a space X is homotopy commutative if the maps (x, y) 7→µ(x, y) and (x, y) 7→ µ(y, x) are homotopic. An element u ∈ X is a homotopy unitfor µ if the maps x 7→ µ(u, x) and x 7→ µ(x, u) are homotopic to the identity of X .Note that, if u0 ∈ X is a homotopy unit for µ and if X is path-connected, then anyu ∈ X is also a homotopy unit.

Let K ≈ K(Z2,m) be an Eilenberg-MacLane space in degree m, with its class0 6= ι ∈ Hm(K). Recall from § 3.8, the map φ : [X,K] −→ Hm(X) given by φ(f) =H∗f(ι) is a bijection. In particular, if K and K′ are two Eilenberg-MacLane spacesin degree m, there is a homotopy equivalence g : K′ → K whose homotopy class isunique.

Proposition 4.7.54. Let K be an Eilenberg-MacLane space in degree m.

(1) There exists a continuous multiplication on K admitting a homotopy unitand which is homotopy commutative.

(2) Any two continuous multiplications on K admitting a homotopy unit arehomotopic.

(3) Let (K, µ) and K′, µ′) be two Eilenberg-MacLane spaces in degree m withcontinuous multiplications admitting homotopy units, Let g : K′m → Km a(unique up to homotopy) homotopy equivalence. Then, the diagram

K′ ×K′ µ′ //

g×g

K′

g

K ×K µ // K

commutes up to homotopy.

Proof. Consider the class

(4.7.32) p = ι× 1+ 1× ι ∈ Hm(K ×K) .Since [K × K,K] is in bijection with Hm(K × K), one has p = H∗µ(ι) for somecontinuous map µ : K × K → K, which we see as a continuous multiplication. The


involution τ exchanging the coordinates on K × K satisfies H∗τ(p) = p and thenH∗(µτ) = H∗µ. Hence, µτ is homotopic to µ, which says that µ is homotopycommutative.

Choose u ∈ K and let i1, i2 : K → K × K be the slice inclusions i1(x) = (x, u)and i2(x) = (u, x). By Lemma 4.7.2, i∗j H

∗µ(ι) = ι for j = 1, 2. Hence, µij ishomotopic to the identity, which proves that u is a homotopy unit. Point (1) isthus established.

For Point (2), let µ is continuous multiplication on K admitting a homotopyunit u. Let i1, i2 : K → K×K be the slice inclusions i1(x) = (x, u) and i2(x) = (u, x).As u is a homotopy unit, hij is homotopic to the identity for j = 1, 2, and thusH∗ij H

∗µ(a) = a for all a ∈ H∗(X). By Lemma 4.7.2, this implies that

(4.7.33) H∗µ(a) = a× 1+ 1× a+∑

y × y′ ,where the degrees of y and y′ are both positive. By the Kunneth theorem, the crossproduct gives an isomorphism isomorphism Hm(K)⊗H0(K)⊕H0(K)⊗Hm(K) ≈Hm(K). Therefore, H∗µ(ι) = p, which says that the homotopy class of µ is welldetermined.

For Point (3), let h : K → K ′ be a homotopy inverse for g. Then, the formulaµ′′(x, y) = hµ(g(x), g(y)) is a continuous multiplication of K′ with a homotopyunit. By (2), µ′′ is homotopic to µ′, which proves (3).

Examples 4.7.55. The following classical multiplications occur in Eilenberg-MacLane spaces Km ≈ K(Z2,m) (or more generally on K(G,m) for an abeliangroup G).

• The loop space ΩKm+1 is an Eilenberg-MacLane space in degree m [82,pp. 407 and ff.]. One can use the loop multiplication.• Using semi-simplicial techniques, J. Milnor has shown that there exists anEilenberg-MacLane space Km which is an abelian topological group [146,§ 3].

The following property of the multiplication µ of Proposition 4.7.54 will beuseful in § 8.3.

Lemma 4.7.56. Let K be an Eilenberg-MacLane space in degree m. Let a ∈Hk(K) for m ≤ k < 2m. Then

(4.7.34) H∗µ(a) = a× 1+ 1× a .Proof. This comes from (4.7.33) since

(4.7.35) Hk(K)⊗H0(K) ⊕ H0(K) ⊗Hk(K) ≈−→ Hk(K ×K)for m ≤ k < 2m by the Kunneth theorem.

Remark 4.7.57. Together with the cup product, the mapH∗µmakesH∗(K(Z2,m))a Hopf algebra (see [82, Section 3.C]). In this setup, an element a ∈ Hk(K(Z2,m))satisfying (4.7.34) is called primitive.

Let X be a CW-complex. The multiplication µ on K = K(Z2;m) induces acomposition law

[X,K]× [X,K] ⋆−→ [X,K]given by f ⋆ g (x) = µ(f(x), g(x)). It admits the following interpretation.

166 4. PRODUCTS

Proposition 4.7.58. Let X be a CW-complex. Then, the bijection φ : Hm(X)≈−→

[X,K] satisfiesφ(a) ⋆ φ(b) = φ(a+ b) .

for all a, b,∈ Hm(X).

Proof. Let f, g : X → K represent φ(a) and φ(b). Then φ(a) ⋆ φ(b) is repre-sented by the composition

X(f,g)−−−→ K×K µ−→ K .

The two projections π1, π2 : K × K → K satisfy π1 (f, g) = f and π2(f, g) = g.Using that H∗µ(ι) = ι× 1+ 1× ι (see the proof of Proposition 4.7.54), one has

φ(a) ⋆ φ(b) = H∗(f, g)H∗µ(ι)

= H∗(f, g)(ι× 1+ 1× ι)= H∗(f, g)(H∗π1(ι) +H∗π2(ι))

= H∗f(ι) +H∗g(ι)

= φ(a) + φ(b) .


4.1. Write the proof of Lemma 4.1.8.

4.2. As H∗(S1 ∨ S1) has 4 elements, the bouquet of two circle has 4 inequivalent2-fold coverings by the bijection (4.3.5). For each of them, describe the total spaceand the transfer exact sequence.

4.3. Same exercise as the previous one, replacing S1 ∨ S1 by the Klein bottle.Compare with the discussion on p. 31.

4.4. Write the transfer exact sequence for a trivial 2-fold covering.

4.5. Let p : X → X be finite covering with an odd number of sheets. Prove thatH∗p is injective.

4.6. Let M and N be closed surfaces, with M orientable and N non-orientable.Prove that there is no continuous map f : M → N which is of degree one.

4.7. Show that there are no continuous map of degree one between the torus T andthe Klein bottle K, in either direction. Same things for S1 × S2 and RP 3.

4.8. Let M be a closed topological manifold of dimension n. Let h : Dn → M bean embedding of the closed disk Dn into M . Form the manifold M as the quotientof M − inth(Dn) by the identification h(x) ∼ h(−x) for x ∈ BdDn. Compute the

ring H∗(M). [Hint: express M as a connected sum.]

4.9. Show that the cohomology algebras of (S1×S1) ♯RP 2 and of RP 2 ♯RP 2 ♯RP 2

are GrA-isomorphic. (It is classical that these two spaces are homeomorphic: see[136, Lemma 7.1]).

4.10. Using the triangulation of the Klein bottle given in Figure 2.4, compute allthe simplicial cap products. Check the formula 〈a b, γ〉 = 〈a, b γ〉.4.11. Show that the smash product and the join of two homology spheres is ahomology sphere.


4.12. Compute the cohomology ring of (a) X = RP∞ × · · · × RP∞ (n times); (b)Y = CP 2 ∧ CP 3; (c) Z = CP 2 ∗ CP 3.

4.13. Write the Mayer-Vietoris cohomology sequence for the decomposition

S1 × Sn = [(S1 − 1))× Sn]× [(S1 − −1))× Sn] .and describe its various homomorphisms. If a ∈ H1(S1) and b ∈ Hn(Sn) are thegenerators, describe how the elements a× 1, 1× b and a× b behave with respect tothe homomorphisms of the Mayer-Vietoris sequence.

4.14. Show that the product of two perfect CW-complexes is a perfect CW-complex.

4.15. What is the Lusternik-Schnirelmann category of RP 2 × RP 3?

4.16. What is the Lusternik-Schnirelmann category of the n-dimensional torusT n = S1 × · · · × S1 (n times)?

4.17. Prove the relevant functoriality property for the homology cross product.

4.18. For m a positive integer, let B(m) be a bouquet of m circles. Let X =∏ri=1 B(ai) and Y =

∏sj=1 B(bj). Suppose that X and Y have the same Poincare

polynomial. Prove that r = s and that bi = aα(i) for some permutation α.

4.19. Cap product in the (co)homology of X × Y . Let X and Y be topologicalspaces, with Y being of finite cohomology type. Let a ∈ H∗(X), b ∈ H∗(Y ),α ∈ H∗(X) and β ∈ H∗(Y ). Prove that the formula

(4.8.1) ×((a× b) ×−1(α⊗ β)

)= (a α)⊗ (b β)

holds in H∗(X)⊗H∗(Y ), using the (co)homology cross products × and × of § 4.6.4.20. Slices in homology. Let X and Y be topological spaces, with Y being offinite cohomology type. Let y0 ∈ Y and let sX : X → X × Y be the slice inclusionof X at y0. Let α ∈ H∗(X). Prove that

×(H∗sX(α)

)= α⊗ y0 ,

where y0 is seen as a 0-homology class of Y , using the bijection Y ≈ S0(Y ).

4.21. Let F → Ep−→ B be a locally trivial bundle containing a subbundle F0 →

E0p0−→ B. Prove that the cohomology sequence of (E,E0) is a sequence of H∗(B)-

modules.

4.22. Let K be a finite simplicial complex and let Fd(K) its face complex for aninteger d > 0. What is the relationship between the Euler characteristic of Fd(K)and that of K?

4.23. Let p : E → Sn be a bundle with fiber Σ, where Σ has the homology of thesphere Sn−1. Let e ∈ Hn(Sn) be its Euler class. Prove that

(a) e 6= 0 if and only if H∗(E) ≈ H∗(S2n−1).(b) e = 0 if and only if there is a GrV-isomorphism φ : H∗(E) ≈ H∗(Sn ×

Sn−1).(c) if n > 2, prove that theGrV-isomorphism φ in (b) is aGrA-isomorphism.

4.24. Let i : Q→M be the inclusion of a smooth submanifold of codimension r ina smooth manifoldM . Let ν be the normal bundle to Q. Suppose that Hr(M) = 0.Prove that the Euler class e(ν) ∈ Hr(Q) vanishes.

CHAPTER 5

Poincare Duality

5.1. Algebraic topology and manifolds

Manifolds studied by algebraic topology tools occur in several categories: smooth,piecewise linear, topological, homology manifolds, etc. Below are a few words aboutthis matter.

Henri Poincare’s paper Analysis situs [161], published 1895, is considered asthe historical start of algebraic topology (for the “prehistory” of the field, see [163]).The aim of Poincare was to use tools of algebraic topology in order to distinguishsmooth manifolds up to diffeomorphism (which he called “homeomorphism”). So,differential and algebraic topology were born together. The importance of studyingsmooth manifolds up to diffeomorphism was reaffirmed throughout the twentiethcentury by many great mathematicians (Thom, Smale, Novikov, Atiyah, etc). It isbased on the deep role played by global properties of smooth manifolds in analysis,differential geometry, dynamical systems and physics.

After the failure of defining homology using submanifolds (see [40, § I.3]),Poincare initiated a new approach [162], in which smooth manifolds are equippedwith a triangulation. This permitted him to define what will later become simplicialhomology. The existence and essential uniqueness of smooth triangulations wereof course a problem, solved only in 1940 by J.H.C. Whitehead [208, Theorems 7and 8]. Also, besides some developments in the twenties (Veblen, Morse), thereal foundations of differential topology arose only after 1935 with the works ofH. Whitney. As a result, homology was seen for three decades as combinatorial innature and smooth manifolds were not considered as the right objects of study. Inthe prominent book written in 1934 by H. Seifert and W. Threlfall [174], smoothmanifolds are not even mentioned, but replaced by a simplicial counterpart, i.e.combinatorial or piecewise linear (PL) manifolds (see definition in § 5.2 below).Techniques analogous to those for smooth manifolds were later developed in thePL-framework (see [104]). Polyhedral homology manifolds were later introduced(see § 5.2), whose importance may grow with the development of computationalhomology. For even more general objects, like ANR homology manifolds, see e.g.[206].

Topological manifolds have also long attracted the attention of topologists,mostly to know whether they carry smooth or piecewise linear structures (see,e.g. [8, p. 235], [132, p. 183]). Their status however remained mysterious untilthe 1960s. R. Kirby and L.C. Siebenmann produced examples in all dimension ≥ 5of topological manifolds without PL-structures and developed many techniques todeal with these topological manifolds [116]. The field of topological versus smoothmanifolds developed very much in dimension four, after 1980, with the work ofM. Freedman and S. Donaldson.

169

170 5. POINCARE DUALITY

Poincare duality is one of the most remarkable properties of closed manifolds.In its strong form, it gives, for a compact n-manifoldM , that Hk(M) andHn−k(M)are isomorphic under the cap product with the fundamental class [M ]. This resultcan be obtained in two contexts:

• by working with homology manifolds, using simplicial topology and dualcells. Taking its origin in the early work of Poincare, this was achievedaround 1930 in the work by L. Pontryagin, L. Vietoris and S. Lefschetz(see [40, § II.4.C]). In the next sections, we follow this approach, akinto the presentation of [155, Chapter 8]. This proves Poincare duality fortriangulable topological manifold, whence for smooth manifolds. Observethat smooth manifolds techniques (Morse theory or handle presentations)give an isomorphism from Hk(M) to Hn−k(M) but not the identificationof this isomorphism with a cap product (see e.g. [120, §VII.6]).• by working with topological manifolds, using Cech cohomology techniques(see, e.g. [179, § 6.2] or [82, § 3.3]). This is not done in this book.

5.2. Poincare Duality in polyhedral homology manifolds

A polyhedral homology n-manifold is a simplicial complex such that, for eachσ ∈ Sk(M), the link Lk(σ) of σ in M is a simplicial complex of dimension n−k− 1which has the homology of the sphere Sn−k−1. (Recall that our homology is mod 2by default; thus, in a broader context, these objects may more accurately be calledpolyhedral Z2-homology manifolds).

Remark 5.2.1. (1) Let X be topological space satisfying the followinglocal property: for any x ∈ X ,

Hj(X,X − x) =Z2 j = n

0 j 6= n .

Such a space is called a homology n-manifold. For instance, an n-dimensionaltopological manifold is a homology n-manifold by (3.3.1). The followingresult is proven in e.g. [155, Theorem 63.2]: if K is a simplicial complexsuch that |K| is a homology n-manifold, then K is a polyhedral homologyn-manifold.

(2) Special kind of polyhedral homology n-manifolds are PL-manifolds. Asimplicial complexM is a PL-manifold, or a combinatorial manifold if, foreach σ ∈ Sk(M), the link Lk(σ) of σ in M has a subdivision isomorphicto a subdivision of the boundary of the (n − k)-simplex. PL-manifoldswere the combinatorial objects replacing smooth manifolds for algebraictopologists around 1930.

(3) A smooth manifold M admits a so-called C1-triangulation, making M aPL-manifold. Two C1-triangulations have isomorphic subdivisions. Thiswas proven by J.H.C. Whitehead in [208, Theorems 7 and 8].

(4) By a result of R. Edwards (see [128]), any PL-manifold of dimension ≥ 5admits non-PL triangulations (which are then polyhedral homology man-ifolds by (1) above). It is an open problem whether a closed topologicalmanifold of dimension ≥ 5 admits a (possibly non-PL) triangulation. Thisis wrong in dimension 4 (see [168, § 5]).

5.2. POINCARE DUALITY IN POLYHEDRAL HOMOLOGY MANIFOLDS 171

(5) There are polyhedral homological manifolds M such that |M | is not atopological manifold. For instance, the suspension of a homology n-manifold N which has the mod 2 homology of Sn (a homology sphere) is an(n+1)-dimensional homology manifold. But there are many PL-homologyspheres (even for integral homology) with non-trivial fundamental group[115]. More examples are given, for instance, by lens spaces with oddfundamental groups.

Here are two first consequences of the definition of a polyhedral homology n-manifold.

Lemma 5.2.2. Let M be a polyhedral homology n-manifold. Then

(1) any simplex of M is contained in some n-simplex of M .(2) any (n− 1)-simplex of M is a face of exactly two n-simplexes of M .

Proof. If v is a vertex of M , then Lk(v) is n − 1 dimensional, so M is n-dimensional. Let σ ∈ Sk(M). If Lk(σ) = ∅, σ must be an n-simplex by the above.If Lk(σ) is not empty, it must contain an (n−k−1)-simplex τ . Then, σ is containedin the join σ ∗ τ which is an n-simplex. This proves (1).

If σ ∈ Sn−1(M) then Lk(σ) is a 0-dimensional complex having the homologyof S0. Hence, Lk(σ) consists of two points, which proves (2).

Let M be a finite polyhedral homology n-manifold. It follows from Point (2)of Lemma 5.2.2 that the n-chain Sn(M) is a cycle and represents a homology class[M ] ∈ Hn(M), called the fundamental class of M .

Theorem 5.2.3 (Poincare Duality). Let M be a finite polyhedral homology n-manifold. Then, for any integer k, the linear map

−[M ] : Hk(M) −→ Hn−k(M)

is an isomorphism.

The proof of this Poincare duality theorem will start after Proposition 5.2.8below. We first give some corollaries of Theorem 5.2.3. By Kronecker duality, weget

Corollary 5.2.4 (Poincare Duality, weak form). Let M be a finite polyhedralhomology n-manifold. Then, for any integer k,

dimHk(M) = dimHn−k(M) .

Thus, in the computation of the Euler characteristic of M , the Betti numbersessentially come in pairs, which gives the following corollary.

Corollary 5.2.5. Let M be a finite polyhedral homology n-manifold. Then,the Euler characteristic χ(M) satisfies the following:

(1) if n is odd, then χ(M) = 0.(2) if n = 2m, then χ(M) ≡ dimHm(M) (mod 2).

Expressed in terms of Poincare polynomial, Corollary 5.2.4 has the followingform.

Corollary 5.2.6. Let M be a finite polyhedral homology n-manifold. Then,

Pt(M) = tnP1/t(M) .


Another easy consequence of Poincare duality is the following.

Corollary 5.2.7. A finite polyhedral homology n-manifold which is connectedis an n-dimensional pseudomanifold.

Proof. Let M be a finite polyhedral homology n-manifold. We may supposethat M is non-empty, otherwise there is nothing to prove. By Lemma 5.2.2, Msatisfies Conditions (a) and (b) of the definition of an n-dimensional pseudoman-ifold. If M is connected, then H0(M) = Z2. By Poincare duality, this impliesthat Hn(M) = Z2. Using Proposition 2.4.5, we deduce that M is an n-dimensionalpseudomanifold.

By Corollary 5.2.7, a continuous map between connected finite polyhedral man-ifolds of the same dimension has a degree (see (2.5.4)).

Proposition 5.2.8. Let f : M ′ → M be a continuous map of degree one be-tween connected finite n-dimensional polyhedral manifolds. Then H∗f : H∗(M

′)→H∗(M) is surjective and H∗f : H∗(M)→ H∗(M ′) is injective

Proof. By Kronecker duality, only the homology statement needs a proof.The hypotheses imply that H∗f([M ′]) = [M ]. By Proposition 4.5.4, this impliesthat the diagram

Hk(M ′) oo H∗f

−[M ′]

Hk(M)

−[M ]

Hn−k(M′)

H∗f // Hn−k(M)

is commutative for all integer k ≥ 0. This provides a section for H∗f .

The remainder of this section is devoted to the proof of Theorem 5.2.3. We shallintroduce several simplicial or homology-cell complexes all having the homologyof M . Let M ′ be the barycentric subdivision of M , with the notations introducedin p. 14. The simplicial complex M ′ is endowed with its natural simplicial order ≤defined in (2.1.2). For σ ∈ S(M), define D(σ) ⊂ S(M ′) by

D(σ) = t ∈ S(M ′) | σ = min t .The simplicial subcomplex D(σ) of M ′ generated by D(σ) is called the dual cell

of σ. Observe that dimD(σ) = n − dim(σ). The simplicial subcomplex D(σ) =Lk(σ,D(σ)) is called the boundary of D(σ). Its dimension is one less than that of

D(σ). We are interested in the topological spaces E(σ) = |D(σ)| and E(σ) = |D(σ)|.Lemma 5.2.9. Let σ ∈ Sk(M), where M is a polyhedral homology n-manifold.

Then,

(a) the space E(σ) is a homology (n− k)-cell with boundary E(σ);(b) D(σ) is an (n− k − 1)-dimensional pseudomanifold.

Proof. The space E(σ) is compact. Observe that D(σ) is the cone over D(σ),with cone vertex σ. Hence, E(σ) is the topological cone over E(σ). Therefore,

(E(σ), E(σ)) is a good pair and H∗(E(σ)) = 0. It then suffices to prove that

H∗(E(σ)) ≈ H∗(Sn−k−1). We shall see below that D(σ) and Lk(σ,M)′ are iso-

morphic simplicial complexes. As |Lk(σ,M)′| = |Lk(σ,M)| and M is a polyhedral


b

b

b

b

b

b

b

1

2

3

4

5

6

7

16

15615715

D(15)

D(7)

s(156)

Figure 5.1. Dual cells and the map s : S(M) → S(M ′) of(5.2.10). In the simplex notation, brackets and commas have beenomitted: 156 = 1, 5, 6, etc.

homology n-manifold, this implies that

H∗(E(σ)) ≈ H∗(|Lk(σ,M)′|) ≈ H∗(Lk(σ,M)) ≈ H∗(Sn−k−1) .

The simplicial isomorphisms p : D(σ) → Lk(σ,M)′ and q : Lk(σ,M)′ → D(σ) aredefined as follows.

• Let τ ∈ V (D(σ)). This implies that σ ⊂ τ and τ ∈ S(M). Hence,κ = τ − σ ∈ S(Lk(σ,M)). We set p(τ ) = κ.• Let ω ∈ V (Lk(σ,M)′). Then, ω ∈ S(Lk(σ,M)), whence ω ∪ σ ∈ S(M).

We set q(ω) = ω ∪ σ.We check that p and q are simplicial maps and are inverse of each other. Thisproves Point (a).


To prove Point (b), let σ ∈ Sk(M). We leave as an exercise to the reader thata simplicial complex K is an m-dimensional pseudomanifold if and only if K ′ is so.Therefore, by Point (a) above and its proof, it is enough to prove that L = Lk(σ,M)is (n− k − 1)-dimensional pseudomanifold.

Let τ ∈ S(L). By Lemma 5.2.2, the simplex σ ∗ τ is contained in some n-simplex of M , which is of the form σ ∗ τ ∗ κ. Therefore, τ ⊂ τ ∗ κ ∈ Sn−k−1(L).Now, if τ ∈ Sn−k−2(L), then σ ∗ τ is a common face of exactly two n-simplexesof M (by Lemma 5.2.2). Hence, τ is a common face of exactly two (n − k − 1)-simplexes of L. We have proven that L satisfies Conditions (a) and (b) of thedefinition of an (n − k − 1)-dimensional pseudomanifold. By Proposition 2.4.5, Lis an (n− k − 1)-dimensional pseudomanifold.

Lemma 5.2.9 permits us to see |M | as a homology-cell complex (see p. 91). Ther-skeleton |M |r is defined by

(5.2.1) |M |r =⋃

σ∈Ss(M)s≥n−r

E(σ) .

Indeed, the space |M ′| is the disjoint union of its geometric open simplexes

|M ′| =⋃

t∈S(M ′)

(|t| − |t|

)

and each |t|−|t| is contained in a single open dual cell E(σ)−E(σ), the one associatedto σ for which σ = min t. This shows that |M |n = |M |. If σ ∈ Sn−r(M), then

E(σ) = E(σ) ∩ |M |r−1; if σ′ ∈ Sn−r(M) is distinct from σ, the open dual cells of σand σ′ are disjoint. This shows that |M |r+1 is obtained by from |M |r by adjunctionof the family of r-homology cells:

|M |r+1 = |M |r ∪ϕ(∪σ∈Sn−r(M) E(σ)

),

where ϕ is the attaching map

ϕ : ∪σ∈Sn−r(M) E(σ)→→ ∪σ∈Sn−r(M) E(σ) ⊂ |M |r .We denote byX the space |M | endowed with this (regular) homology-cell struc-

ture. As noted in p. 91, the cellular homology H∗(X) (defined with the homologycells) is isomorphic to the singular homology H∗(|M |) of |M |. If σ ∈ Sk(M), then

E(σ) is the union of those E(τ) for which τ ∈ Sk+1(M) has σ as a face. UsingFormula (2.2.5), this amounts to

(5.2.2) E(σ) =⋃

τ∈δ(σ)

E(τ) .

On the other hand, since D(σ) is a (n − k − 1)-dimensional pseudomanifold by

Lemma 5.2.9, Proposition 2.4.4 tells us that Hn−k−1(D(σ)) = Z2 is generated by

[D(σ)] = Sn−k−1(D(σ)) and the generator of Hn−k−1(D(σ)) = Z2 is represented by

any cochain formed by a single (n−k−1)-simplex. Hence, Hn−k(D(σ), D(σ)) = Z2

is generated by [D(σ)] = Sn−k(D(σ)) and the generator Hn−k(D(σ), D(σ)) = Z2 isrepresented by any cochain formed by a single (n− k)-simplex of D(σ). The proof

of Lemma 3.5.5 thus works and, using (5.2.2), ∂ : Cn−k(X)→ Cn−k−1(X) satisfies

(5.2.3) ∂(D(σ)) =∑

τ∈δ(σ)

[D(τ)] .


As M is a finite simplicial complex, Ck(M) is isomorphic to the vector space gen-erated by Sk(M). Therefore, the correspondence σ 7→ E(σ) gives a linear map

Φ1 : Ck(M)→ Cn−k(X) which, by (5.2.3), satisfies

(5.2.4) ∂Φ1 = Φ1δ .

As Φ1 is bijective, the induced map

(5.2.5) Φ1 : Hk(M)

≈−→ Hn−k(X)

is an isomorphism. Observe that this proves the weak form of Poincare duality ofCorollary 5.2.4.

To prove Theorem 5.2.3, we now need to identify Φ1 with a cap product. Thecorrespondence E(σ) 7→ [D(σ)] provides a linear map Φ2 : Cn−k(X) → Cn−k(M

′).

By (5.2.2), Φ2 is a chain map, thus inducing a linear map Φ2 : Hn−k(X)→ Hn−k(M′).

Lemma 5.2.10. Φ2 : Hn−k(X)→ Hn−k(M′) is an isomorphism.

Proof. The r-skeleton Xr of the homology-cell decomposition of X was givenin (5.2.1). Note that Xr = |Kr| where Kr is the subcomplex of M ′ given by

(5.2.6) Kr =⋃

σ∈Ss(M)s≥n−r

D(σ) .

Thus, Kr is a simplicial complex of dimension r. We can use the simplicialpairs (Kr,Kr−1) to compute the simplicial homology of M ′. Define Cr(M

′) =

Hr(Kr,Kr−1) with the boundary ∂ : Cr((M′)→ Cr−1((M

′) given by the composi-tion

Hr(Kr,Kr−1)→ Hr−1(Kr−1)→ Hr−1(Kr−1,Kr−2) .

One has ∂ ∂ = 0. Set H∗(M′) = ker ∂/Image∂. The correspondence E(σ) 7→ [D(σ)]

gives an isomorphism Φ′2 : Hr(X)≈−→ Hr(M

′). Note that

Cr(M′) = Hr(Kr,Kr−1)

= ker

(Cr(Kr)/Cr(Kr−1)︸︷︷︸

Cr(Kr)

∂−→ Cr−1(Kr)/Cr−1(Kr−1)

),

whence

Cr(M′) =

α ∈ Cr(Kr) | ∂α ∈ Cr−1(Kr−1)

⊂ Cr(Kr) ⊂ Cr(M ′) .


The inclusion Φ′′2 : C∗(M′) → C∗(M

′) is clearly a morphism of chain complexes.

It induces a homomorphism Φ′′2 : H∗(M′) → H∗(M

′). As in the proof of Theo-rem 3.5.6, we have the commutative diagram

(5.2.7)

Hr+1(Kr+1,Kr)

∂r+1

∂r+1

((

0 // Hr(Kr+1)

≈

Hr(Kr)vv

j

vv♠♠♠♠♠♠♠♠

♠♠♠

// //

µ77 77♦♦♦♦♦♦♦♦♦

0

Hr(M′)

Hr(Kr,Kr−1)

∂r

∂r

((

Hr−1(Kr−1)vv

j

vv♠♠♠♠♠♠♠

♠♠♠♠

Hr(Kr−1,Kr−2)

which permits us to compute H∗(M′). As Φ′′

2 is just the inclusion, the diagram

Hr(Kr)/Im∂r+1

j ≈

µ

≈// ker∂r/Im∂r+1

= // ker ∂r/Im∂r+1

=

Hr(M

′) Hr(M′)

Φ′′2oo

is commutative, which proves that Φ′′2 is an isomorphism. Finally, the commutative dia-

gram

Hr(X)

Φ′2

≈ $$

Φ2 // Hr(M′)

Hr(M′)

Φ′′2

≈

99tttttttt

shows that Φ2 is an isomorphism.

We now need a good identification of the simplicial (co)homology of M withthat of M ′. Choose a simplicial order on M . One has a simplicial map g :M ′ →M given, for σ ∈ Sm(M), by

(5.2.8) g(σ) = max σ .

In the other direction, one has a chain map sd : Cm(M) → Cm(M ′) given, forσ ∈ Sm(M), by

(5.2.9) sd(σ) = Sm(σ′) .

(This chain map is in fact defined for any subdivision and is called the subdivisionoperator). Observe that, for any σ ∈ Sm(M), there exists a unique τ ∈ Sm(M ′)such that C∗g(τ) = σ. Indeed, if σ = v0, v1, . . . , vm with v0 v1 · · · vm,then τ = σ0, σ1, . . . , σm, where σi is the barycenter of v0, v1, . . . , vi. The otherm-simplexes of σ′ are mapped to proper faces of σ. This defines a map

(5.2.10) s : S(M)→ S(M ′)


by s(σ) = τ . For σ ∈ S(M), one has

(5.2.11) C∗gC∗sd(σ) = C∗gs(σ) = σ ,

which proves that H∗gH∗sd = idH∗(M).On the other hand, if t = σ0, σ1, . . . , σm ∈ Sm(M ′), with σ0 ⊂ · · · ⊂ σm,

then sdg(t) ∈ C∗(σ′m). As, also t ∈ C∗(σ′m), the correspondence t 7→ C∗(σ′m) is an

acyclic carrier for both sdg and idC∗(M ′). By Proposition 2.9.1, this implies thatH∗sdH∗g = idH∗(M ′). Therefore, g and sd induce isomorphisms in (co)homologywhich are inverse of each other. In particular, H∗g and H∗g do not depend on theorder since this is the case for sd.

It is straightforward that sd([M ]) = [M ′]. As g :M ′ →M is a simplicial map,Proposition 4.5.4 gives the formula

H∗g(H∗g(a) [M ′]

)= a [M ] ,

which is equivalent to the commutativity of the diagram.

(5.2.12)

Hk(M)

≈H∗g

[M ]// Hn−k(M)

Hk(M ′)[M ′]// Hn−k(M

′)

≈ H∗ g

OO

The identification of the isomorphism Φ1 with the cap product with the fundamentalclass then follows from the following lemma.

Lemma 5.2.11. The diagram

Hk(M)Φ1

≈//

≈ H∗g≈

Hn−k(X)

Φ2≈

Hk(M ′)−[M ′]// Hn−k(M

′)

is commutative.

Proof. Let σ ∈ Sk(M). The properties of the map s : S(M)→ S(M ′) definedin (5.2.10) imply that C∗g(σ) = s(σ) and max s(σ) = σ (for the natural simplicialorder ≤ on M ′ defined in (2.1.2)). The isomorphism Ψ∗ = Φ2Φ1 comes from themorphism of cochain-chains Ψ : C∗(M)→ Cn−∗(M

′) such that

Ψ(σ) = [D(σ)] = t ∈ Sn−k(M ′) | min t = σ= t ∈ Sn−k(M ′) | s(σ) ∪ t ∈ Sn(M ′) .

On the other hand, if τ = σ0, . . . σn ∈ Sn(M ′) with σ0 ⊂ σ1 ⊂ · · · ⊂ σn,Formula (4.5.1) gives

C∗g(σ) τ = s(σ)≤ τ = 〈s(σ), σ0, . . . , σk〉 σk, . . . , σn .But

〈s(σ), σ0, . . . , σk〉 =

1 if s(σ) = σ0, . . . , σk0 otherwise

=

1 if s(σ) ∪ σk, . . . , σn ∈ Sn(M ′)0 otherwise.


Therefore

C∗g(σ) [M ′] = s(σ) [M ′] = Ψ(σ) .

The proof of Poincare Duality Theorem 5.2.3 is now complete.

5.3. Other forms of Poincare Duality

5.3.1. Relative manifolds. A topological pair (X,Y ) such that

Hj(X,X − x) =Z2 j = n

0 j 6= n .

for any x ∈ X − Y is called a relative homology n-manifold. The condition is forinstance fulfilled if X − Y is n-dimensional topological manifold, by (3.3.1).

A simplicial pair (M,A) is a relative polyhedral homology n-manifold if, for eachσ ∈ Sk(M)−Sk(A), the link Lk(σ) of σ in M is a simplicial complex of dimensionn− k − 1 which has the homology of the sphere Sn−k−1. For instance, (M, ∅) is arelative polyhedral homology n-manifold if and only if M is a polyhedral homologyn-manifold.

The following result is proven in e.g. [155, Theorem 63.2].

Proposition 5.3.1. If (K,L) is a simplicial pair such that (|K|, |L|) is a rel-ative homology n-manifold, then K is a relative polyhedral homology n-manifold.

A topological pair (X,Y ) is triangulable if there exists a simplicial pair (K,L)and a homeomorphism of pair h : (|K|, |L|)→ (X,Y ). Such a homeomorphism h iscalled a triangulation of (X,Y ).

Theorem 5.3.2 (Lefschetz duality). Let (X,Y ) be a compact relative homologyn-manifold which is triangulable. Then, for any integer k, there is an isomorphism

Φ: Hk(X,Y ) ≈ Hn−k(X − Y ) .

For a more general result, see [43, Proposition 7.2 in Chapter VII].

Proof. Let (M,A) be a simplicial pair such that (|M |, |A|) is homeomorphic to(X,Y ). By Proposition 5.3.1, (M,A) is a relative polyhedral homology n-manifold.We shall construct an isomorphism

(5.3.1) Φ0 : Hk(M,A)

≈−→ Hn−k(|M | − |A|) ,where Hk(M,A) is the simplicial cohomology. The proof is close to that of The-orem 5.2.3, so we just sketch the argument. For more details (see [155, Theo-rem 70.2]).

Let M∗ be the subcomplex of the first barycentric subdivision of M consistingof all simplexes of M ′ that are disjoint from A. As in the proof of Theorem 5.2.3,consider the dual cell D(σ) for each σ ∈ S(M)−S(A) and its geometric realizationE(σ) = |D(σ)|. Lemma 5.2.9 holds for these dual cells and, as in the proof ofTheorem 5.2.3, they provide a structure of a homology-cell complex on |M∗|. CallX∗ the space |M∗| endowed with this homology-cell decomposition. As M is afinite complex, then Ck(M,A) is the vector space with basis Sk(M) − Sk(A) (seep. 37). As in (5.2.5), the correspondence σ 7→ E(σ) produces an isomorphism

(5.3.2) Φ1 : Hk(M,A)

≈−→ Hn−k(X∗) .

5.3. OTHER FORMS OF POINCARE DUALITY 179

To get the isomorphism Φ0 from Φ1, we use that |M∗| ≈ X∗ is a deformationretract of |M | − |A| (see [155, Lemma 70.1]).

Corollary 5.3.3. Let (X,Y ) be a connected compact relative homology n-manifold which is triangulable. If Y 6= ∅, then Hn(X − Y ) ≈ Hn(X − Y ) = 0.

Proof. By Kronecker duality, it is enough to prove that Hn(X − Y ) = 0. ByTheorem 5.3.2, Hn(X−Y ) ≈ H0(X,Y ) and, as X is path-connected, H0(X,Y ) = 0if Y 6= ∅. (Corollary 5.3.3 may also be obtained using cohomology with compactsupports: see [82, Theorem 3.35]).

The following consequence of Corollary 5.3.3 is often referred to as the Z2-orientability of finite polyhedral homology n-manifolds (see e.g. [82, pp. 235–236]).

Corollary 5.3.4. Let M be a finite polyhedral homology n-manifold and letx ∈M . We denote by j : (M, ∅)→ (M,M − x) the pair inclusion. Then

H∗j : Hm(M)→ Hm(M,M − x)sends [M ] onto the generator of Hn(M, ,M − x) ≈ Z2. In particular, if M isconnected, H∗j is an isomorphism.

Proof. The fundamental class of M being the sum of those of its connectedcomponents, it is enough to consider the case whereM is connected. Corollary 5.3.4then follows from the exact sequences

Hm(M − x)→ Hm(M)H∗j−−→ Hm(M,M − x) ,

using that Hm(M − x)) = 0 by Corollary 5.3.3.

Let (X,Y ) be a compact triangulable relative homology n-manifold. Choosea simplicial pair (M,A) such that (|M |, |A|) is homeomorphic to (X,Y ). Then,(M,A) is a finite relative polyhedral homology n-manifold by Proposition 5.3.1.Lemma 5.2.2 holds true for the simplexes of M which are not in A. As a conse-quence, the n-chain Sn(M)−Sn(A) is a cycle relative to A and represent a homologyclass [M ] ∈ Hn(M,A) called the fundamental class of (M,A). Under the isomor-phism between simplicial and singular homology of Theorem 3.6.3, the class [M ]corresponds to a singular class [X ] ∈ Hn(X,Y ) called fundamental class of (X,Y ).Let i : X − Y → X denote the inclusion. The isomorphism Φ of Theorem 5.3.2 isrelated to the cap product with [X ] in the following way.

Proposition 5.3.5. Let (X,Y ) be a compact relative homology n-manifoldwhich is triangulable. Then the diagram

Hk(X,Y )

[X] ''

Φ

≈// Hn−k(X − Y )

H∗i

Hn−k(X)

.

is commutative.

Proof. As in the proof of Theorem 5.3.2, we choose a finite relative polyhedralhomology n-manifold (M,A) such that such that (|M |, |A|) is homeomorphic to(X,Y ) and we use the same definitions and notations, such that X∗ ≈ M∗. The

isomorphism Φ2 : Hn−k(X∗)→ Hn−k(M

∗) may be established as in Lemma 5.2.10.


The subdivision operator sd : Cm(M) → Cm(M ′) of (5.2.9) is defined, as well asthe simplicial map g : M ′ →M of (5.2.8), choosing for the latter a simplicial orderon M . They induce reciprocal isomorphisms on (co)homology. Ons has sd([M ]) =[M ′], where [M ′] ∈ Hn(M

′, A′) is the class of the relative cycle Sn(M ′)− Sm(A′).The commutative diagram (5.2.12) becomes

(5.3.3)

Hk(M,A)

≈H∗g

[M ] // Hn−k(M)

Hk(M ′, A′)[M ′]// Hn−k(M

′)

≈ H∗ g

OO

If i : M∗ →M ′ denotes the simplicial map given by the inclusion, the commutativityof the diagram

Hk(M,A)Φ1

≈//

≈ H∗g≈

Hn−k(X∗)

Φ2

≈// Hn−k(M

∗)

H∗i≈

Hk(M ′, A′)−[M ′] // Hn−k(M

′)

is proven as in Lemma 5.2.11. Finally, as mentioned in the proof of Theorem 5.3.2,|M∗| ≈ X∗ is a deformation retract of |M | − |A|, hence we obtain a commutativediagram involving simplicial and singular homology:

(5.3.4)

H∗(M∗)

≈ //

H∗i

H∗(|M | − |A|)H∗j

H∗(M′)

≈ // H∗(|M |)

.

In the definition of a relative homology n-manifold (X,Y ), it is not requiredthat X itself is a homology n-manifold. If this is the case (and if X is compact andtriangulable), the fundamental class [X ] ∈ Hn(X) is defined. To distinguish, call[X ]rel ∈ Hn(X,Y ) the class of Proposition 5.3.5. If (M,A) is a simplicial pair with(|M |, |A|) homeomorphic to (X,Y ), then H∗j([M ]) = [M ]rel, where j : (M, ∅) →(M,A) (or j : (X, ∅) → (X,Y )) denote the pair inclusion. Therefore H∗j([X ]) =[X ]rel.

Proposition 5.3.6. Let X be a compact homology n-manifold and let Y be aclosed subset of X. Assume that the pair (X,Y ) is triangulable. Then (X,Y ) is arelative homology n-manifold and the diagram

Hk(X,Y )

[X]rel

''

Φ

≈//

H∗j

Hn−k(X − Y )

H∗i

Hk(X)[X] // Hn−k(X)

.

is commutative. Here, j : (X, ∅) → (X,Y ) is the inclusion and Φ is the Lefschetzduality isomorphism of Theorem 5.3.2.


Proof. Only the commutativity of the diagram requires a proof. The com-mutativity of the upper triangle is established in Proposition 5.3.5. For the lowertriangle, let a ∈ Hk(X,Y ) and u ∈ Hn−k(X). One has

(5.3.5)

〈u, a [X ]rel〉 = 〈u a, [X ]rel〉= 〈u a,H∗j([X ])〉 as [X]rel = H∗j([X])

= 〈H∗j(u a), [X ]〉= 〈u H∗j(a), [X ]〉 by Lemma 4.1.8

= 〈u,H∗j(a) [X ]〉 ,which is, in formula, the commutativity of the lower triangle.

5.3.2. Manifolds with boundary. LetX be a compact topological n-manifoldwith boundary Y = BdX . Then (X,Y ) is a compact relative homology n-manifold.As seen in the previous subsection, if the pair (X,Y ) is triangulable, the funda-mental class [X ] ∈ Hn(X,Y ) is defined.

Theorem 5.3.7. Let X be a compact topological n-manifold with boundaryY = BdX. Suppose that the pair (X,Y ) is triangulable. Then, for any integer k,the linear maps

−[X ] : Hk(X) −→ Hn−k(X,Y )

and

−[X ] : Hk(X,Y ) −→ Hn−k(X)

given by the cap product with [X ] ∈ Hn(X,Y ) are isomorphisms.

Theorem 5.3.7 is also true without the hypothesis of the triangulability of(X,Y ), [82, Theorem 3.43].

Proof. We first establish the isomorphism.

−[X ] : Hk(X,Y ) −→ Hn−k(X) .

As X is a topological manifold, its boundary admits a collar neighbourhood, i.e.there exists a embedding h : Y × [0, 1) → X , extending the identity on Y (see,e.g. [82, Proposition 3.42]). Then, X − h(Y × [0, 1/2]) is a deformation retractof both X and X − Y . It follows that the inclusion X − Y → X is a homotopyequivalence. Hence, the result follows from Proposition 5.3.5.

The other isomorphism comes from the five lemma applied to the diagram

// Hk(X,Y )

[X]≈

// Hk(X)

[X]

// Hk(Y )

[Y ]≈

δ∗ // Hk+1(X,Y )

[X]≈

// Hn−k(X) // Hn−k(X,Y )∂∗ // Hn−k−1(Y ) // Hn−k−1(X)

The commutativity of the above diagram comes from Lemma 4.5.5, since

(5.3.6) ∂∗([X ]) = [Y ] .

Indeed, if (M,N) be a finite simplicial pair triangulating (X,Y ), the fundamentalclass [M ] is represented by the chain Sn(M) ∈ C∗(M) and ∂∗([M ]) is representedby ∂(Sn(M)) = Sn−1(N).


Corollary 5.3.8. Let X be a compact triangulable topological n-manifold withboundary BdX = Y . Suppose that Y = Y1∪Y2 is the union of two compact (n−1)-manifolds with common boundary Y1 ∩ Y2 = BdY1 = BdY2. Then, for any integerk, the linear map

−[X ] : Hk(X,Y1) −→ Hn−k(X,Y2)

given by the cap product with [X ] ∈ Hn(X,Y ) is an isomorphism.

Again, Corollary 5.3.8 is true without the hypothesis of triangulability (see [82,Theorem 3.43]).

Proof. Corollary 5.3.8 reduces to Theorem 5.3.7 by applying the five lemmato the diagram

// Hk(X,Y )

[X]≈

// Hk(X,Y1)

[X]

// Hk(Y, Y1)

µ≈

// Hk+1(X,Y )

[X]≈

//

// Hn−k(X) // Hn−k(X,Y2) // Hn−k−1(Y2) // Hn−k−1(X) //

The top line is the cohomology sequence for the triple (X,Y, Y1) and the bottomline is the homology sequence for the pair (X,Y2). The isomorphism µ is thecomposition

µ : Hk(Y, Y1)≈−→ Hk(Y2,BdY2)

[Y2]−−−−→ Hn−k−1(Y2) .

The commutativity of the above diagram is obtained as for those in the proofsof § 5.3.1.

Here are some applications of the Poincare duality for compact manifolds withboundary.

Proposition 5.3.9. Let X be a compact triangulable manifold of dimension2n+ 1, with boundary Y . Let B = Image

(Hn(X)→ Hn(Y )

). Then

(1) Let u ∈ Hn(Y ). Then

u ∈ B ⇐⇒ 〈u B, [Y ]〉 = 0 .

In particular, 〈B B, [Y ]〉 = 0.(2) dimHn(Y ) = 2 dimB.

For example, RP 2n is not the boundary of a compact manifold.

Proof. We follow the idea of [133, Lemma 4.7 and Corollary 4.8]. Let i : Y →X denote the inclusion and let a, b ∈ Hn(X). Then,

〈H∗i(a) H∗i(b), [Y ]〉 = 〈H∗i(a b), [Y ]〉 = 〈a b,H∗i([Y ])〉 = 0 ,

since H∗i([Y ]) = 0 by (5.3.6). This proves the implication ⇒ of (1). Conversely,suppose that 〈u B, [Y ]〉 = 0 for u ∈ Hn(Y ). Since B = ker(δ : Hn(Y ) →Hn+1(X,Y ), it suffices to prove that δ(u) = 0. Let v ∈ Hn(X). One has

0 = 〈u H∗i(v), [Y ]〉= 〈u H∗i(v), ∂[X ]〉 by (5.3.6)

= 〈δ(u H∗i(v)), [X ]〉= 〈δ(u) v, [X ]〉 by Lemma 4.1.9.


This equality, holding for any v ∈ Hn(X), implies, by Theorem 5.3.12 below, thatδ(u) = 0.

To prove (2), let us consider the linear map Φ : Hn(Y ) → Hn(Y )♯ given byΦ(a)(b) = 〈a b, [Y ]〉). Let ΦB be the restriction of Φ to B. The map Φ is anisomorphism by Theorem 5.3.13 below. By (1), ΦB(B) = A♯, where A = Hn(Y )/B.

Thus, there is a quotient map Φ fitting in the commutative diagram

(5.3.7)

0 // B //

ΦB≈

Hn(Y ) //

Φ≈

A //

Φ≈

0

0 // A♯ // Hn(Y )♯ // B♯ // 0

(whose rows are exact) and Φ is also an isomorphism. Therefore,

dimHn(Y ) = dimB + dimA = 2dimB .

(Remark: the proof does not use the map Φ, only that ΦB is an isomorphism; butDiagram (5.3.7) will be useful later.)

Proposition 5.3.9 and Corollary 5.2.5 have the following consequence on theEuler characteristic of bounding manifolds.

Corollary 5.3.10. Let Y be a closed triangulable n-manifold. If Y is theboundary of a compact triangulable manifold, then χ(Y ) is even.

5.3.3. The intersection form. Let X be a compact topological n-manifoldwith boundary Y = BdX . We assume that the pair (X,Y ) is triangulable.From Theorem 5.3.7, the cap product with [X ] ∈ Hn(X,Y ) induces isomorphisms

Hq(X)≈−→ Hn−q(X,Y ) and Hq(X,Y )

≈−→ Hn−q(X). We denote by PD the inverseof these isomorphisms. Thus, if α ∈ Hq(X) and β ∈ Hq(X,Y ), their Poincare dualPD(α) ∈ Hn−q(X,Y ) and PD(β) ∈ Hn−q(X) are the classes determined by theequations

PD(α) [X ] = α and PD(β) [X ] = β .

(The first equation uses the cap product of (4.5.16) and the second that of (4.5.14)).This permits us to define two intersection forms on the homology of X .

(1) If α ∈ Hq(X) and β ∈ Hn−q(X), we set

α ·a β = 〈PD(α) PD(β), [X ]〉 .

This defines the (absolute) intersection form H∗(X)⊕Hn−∗(X)·a−→ Z2.

(2) Similarly, if α ∈ Hq(X) and β ∈ Hn−q(X,Y ), the same formula defines

the (relative) intersection form H∗(X)⊕Hn−∗(X,Y )·r−→ Z2.

The name “intersection form” will be justified by Corollary 5.4.13 below.Let j : (X, ∅) → (X,Y ) denote the pair inclusion. For α ∈ Hq(X) and β ∈

Hn−q(X), the absolute and relative intersection forms are related by the formula

α ·a β = H∗j(α) ·r β = H∗j(β) ·r α .


Indeed:

H∗j(α) ·r β = 〈PD(H∗j(α)) PD(β), [X ]〉= 〈(H∗j(PD(α)) PD(β), [X ]〉 by Lemma 4.5.5

= 〈PD(α) PD(β), [X ]〉 by Lemma 4.1.7

= α ·a βand the other equality is proven the same way.

The absolute and relative intersection forms coincide when Y is empty. Evenwhen Y 6= ∅, we shall usually not distinguish between the two forms and just writeα · β when the context makes it clear. In both cases, since 〈a b, γ〉 = 〈a, b γ〉(see (4.5.2)), one has

(5.3.8) α · β = 〈PD(α), β〉 = 〈PD(β), α〉 .By Theorem 5.3.13 below, the relative intersection form is non-degenerate, i.e.

induces an isomorphism Hq(X)≈−→ Hn−q(X,Y )♯ for all q. If Y 6= ∅, the absolute

intersection form may be degenerate (example: X = S1 × D2). In fact, if X isconnected, it is always degenerate for q = 0, since Hn(X) = 0. However, one hasthe following proposition.

Proposition 5.3.11. Suppose that X is connected and that Y is not empty.Then, the following conditions are equivalent.

(a) The absolute intersection form induces an isomorphism Hq(X)≈−→ Hn−q(X)♯

for 1 ≤ q ≤ n− 1.(b) Y = BdX is a Z2-homology sphere.

Proof. Let j : (X, ∅) → (X,Y ) denote the pair inclusion. By (5.3.8), thecomposite homomorphism

Hq(X)Hqj // Hq(X,Y )

PD

≈// Hn−q(X)

k

≈// Hn−q(X)♯

is just the absolute intersection form of X . Thus, (a) is equivalent to Hqj being anisomorphism for 1 ≤ q ≤ n− 1. By the exact homology sequence of (X,Y ) this isequivalent to (b) if X is connected.

5.3.4. Non-degeneracy of the cup product.

Theorem 5.3.12. Let M be a finite polyhedral homology n-manifold. Then, forany integer k, the bilinear map

Hk(M)×Hn−k(M)−→ Hn(M)

〈−,[M ]〉−→ Z2

induces an isomorphism Hk(M)≈−→ Hn−k(M)♯.

Proof. By Corollary 5.2.4, it suffices to prove that the linear map Φ : Hk(M)→Hn−k(M)♯ given by

aΦ7→

(b 7→ 〈a b, [M ]〉

)

is injective. Suppose that a ∈ kerΦ. Then

0 = 〈a b, [M ]〉 = 〈b, a [M ]〉for all b ∈ Hn−k(M). By Point (a) of Lemma 2.3.3, we deduce that a [M ] = 0,which implies that a = 0 by Theorem 5.2.3.


The same proof, using Corollary 5.3.8, gives the following result.

Theorem 5.3.13. Let X be a compact triangulable topological n-manifold withboundary BdX = Y . Suppose that Y = Y1 ∪ Y2 is the union of two compact(n − 1)-manifolds Yi with common boundary Y1 ∩ Y2 = BdY1 = BdY2. Then, forany integer k, the bilinear map

Hk(X,Y1)×Hn−k(X,Y2)−→ Hn(X,Y )

〈−,[X]〉−→ Z2

induces an isomorphism Hk(X,Y1)≈−→ Hn−k(X,Y2)

♯.

5.3.5. Alexander Duality. The first version of Alexander Duality was provenin a paper [7] of James Waddell Alexander II (1888–1971). This article pioneeredseveral new methods and was very influential at the time (see [40, p. 56]). In hispaper, Alexander used the mod2 homology. Classical Alexander duality relates thecohomology of a closed subset A or Sn to the homology of Sn−A. We give below aversion where Sn is replaced by a homology sphere (for instance a lens space withodd fundamental group).

Theorem 5.3.14 (Alexander Duality). Let (X,A) be a compact triangulablepair with ∅ 6= A 6= X. Suppose that X is a relative homology n-manifold andhas its homology isomorphic to that of Sn. Then, for all integer k, there is anisomorphism

Hk(A) ≈ Hn−k−1(X −A) .Particular cases of Alexander duality were encountered in Proposition 3.3.6

and Corollary 3.3.7. For a version of Theorem 5.3.14 without the assumption oftriangulability (see [82, Theorem 3.44]).

Proof. The case n = 0 being trivial, we assume n > 0. The pair (X,A) satis-fies the hypotheses of Lefschetz duality Theorem 5.3.2. This gives an isomorphism

Φ: Hk+1(X,A) ≈ Hn−k−1(X −A) .Suppose that k 6= n, n−1. Since H∗(X) ≈ H∗(Sn), the connecting homomorphism

δ∗ : Hk(A)→ Hk+1(X,A) is an isomorphism and Hn−k−1(X −A) ≈ Hn−k−1(X −A). This proves the result in this case.

Let (M,L) be a simplicial pair such that (|M |, |L|) is homeomorphic to (X,A).As M is a relative polyhedral homology n-manifold by Proposition 5.3.1. As Lis a proper subcomplex of M , one has Hn(L) = 0, since Sn(M) is the only non-

vanishing n-cycle of M . Hence, Hn(A) ≈ Hn(A) = 0 by Kronecker duality. As,

Hn−k−1(X −A) = H−1(X −A) = 0, the theorem is true for k = n.When k = n− 1, consider the diagram

Hn−1(X)H∗i // Hn−1(A)

Φ

// Hn(X,A)

Φ≈

// Hn(X)

[X]≈

// 0

0 // H0(X −A) // H0(X −A)H∗j // H0(X) // 0

where i : A→ X and j : (X−A)→ X denote the inclusions. The bottom line is theexact sequence of Lemma 3.1.10 and the commutativity of the right hand square is

the contents of Proposition 5.3.6. Then the homomorphism φ : Hn−1(A)→ H0(X−A) exists, making the diagram commutative. If n > 1, Hn−1(A) = Hn−1(A) and,


as Hn−1(X) = 0, the map φ is an isomorphism by the five lemma. Finally, when

n = 1, then cokerH∗i = H0(A) by Lemma 3.1.10 and φ induces an isomorphism

from Hn−1(A) to H0(X −A).

5.4. Poincare duality and submanifolds

In this section, we assume some familiarity of the reader with standard tech-niques of smooth manifolds, as exposed in e.g. [95].

5.4.1. The Poincare dual of a submanifold. LetM be a smooth compactn-manifold and let Q ⊂M be a closed smooth submanifold of codimension r. Recallthat smooth manifolds admit PL-triangulations [208], so the fundamental classes[Q] ∈ Hn−r(Q) and [M ] ∈ Hn(M,BdM) do exist. We are interested in the Poincaredual PD(H∗i([Q])) ∈ Hr(M,BdM) (see § 5.3.3) of the class H∗i([Q]) ∈ Hn−r(M),where i : Q→M denotes the inclusion. We write PD(Q) for PD(H∗i([Q])) and callit the Poincare dual of Q. It is thus characterized by the equation

PD(Q) [M ] = H∗i([Q]) ,

Two simple examples are given in Figure 5.2.

1 2 3 1

4

7

1 2 3 1

4

7

5

8

6

9PD(Q)

Q

W

1 2 3 1

4

7

1 2 3 1

7

4

5

8

6

9

Q

W

PD(Q)

Figure 5.2. The Poincare dual PD(Q) of a circle Q in the torus(left) or the Klein bottle (right). This illustrates the localizationprinciple of Remark 5.4.3: PD(Q) is supported in a tubular neigh-bourhood W of Q.

Example 5.4.1. For a more elaborate example, let Q be a smooth closed con-nected manifold, seen as the diagonal submanifold of M = Q × Q. Let A =a1, a2, . . . ⊂ H∗(Q) be an additive basis of H∗(Q). By Theorem 5.3.12, there isa basis B = b1, b2, . . . of H∗(Q) which is dual to A for the Poincare duality, i.e.〈ai bj, [Q]〉 = δij . We claim that

(5.4.1) PD(Q) =∑

i

ai × bi .

Indeed, by the Kunneth theorem, the elements ai × bj form a basis of H∗(M), sothere are unique coefficients γij ∈ Z2 such that

PD(Q) =∑

i,j

γij ai × bj .

5.4. POINCARE DUALITY AND SUBMANIFOLDS 187

Let ∆: Q→M be the diagonal inclusion, ap ∈ A and bq ∈ B. As H∗∆(bp × aq) =bp aq (see Remark 4.6.1), one has

(5.4.2) 〈bp × aq, H∗∆([Q])〉 = 〈H∗∆(bp × aq), [Q]〉 = 〈bp aq, [Q]〉 = δpq .

Without loss of generality, we may suppose that Q is connected. Let [Q]♯ be thenon-zero element of HdimQ(Q). One has 〈[Q]♯ × [Q]♯, [M ]〉 = 1 and(5.4.3)〈bp × aq, H∗∆([Q])〉 = 〈(bp × aq) PD(Q), [M ]〉

=∑

i,j γij 〈(bp × aq) (ai × bi), [M ]〉=

∑i,j γij 〈(bp ai)× (aq bi), [M ]〉 by Remark 4.6.4

=∑

i,j γij 〈δpi[Q]♯ × δqj [Q]♯, [M ]〉= γpq .

Thus, Equation (5.4.1) follows from (5.4.2) and (5.4.3).

The next two lemmas are recipes to compute PD(Q). Let us denote by ν =ν(M,Q) the normal bundle of Q in M . A Riemannian metric provides a smoothbundle pair (D(ν), S(ν)) with fiber (Dr, Sr−1) and there is a diffeomorphism fromD(ν) to a closed tubular neighbourhood W of Q in M . By excision,

H∗(M,M −Q)≈−→ H∗(W,BdW ) ≈ H∗(D(ν), S(ν)) .

Hence, the Thom class U(ν) ∈ Hr(D(ν), S(ν)) determines an element U(M,Q) ∈Hr(M,M −Q). Let j : (M,BdM)→ (M,M −Q) denote the pair inclusion.

Lemma 5.4.2. PD(Q) = H∗j(U(M,Q)).

Proof. We first reduce to the case where Q is connected. Indeed, as Q is thefinite union of components Qi, with tubular neighbourhood Wi, then

Hr(M,M −Q)≈−→ Hr(W,BdW ) ≈

⊕Hr(Wi,BdWi) ≈

⊕Hr(M,M −Qi)

and U(M,Q) =∑U(M,Qi). On the other hand, PD(Q) =

∑PD(Qi). Thus, we

shall assume that Q is connected.Let us consider the case where M = D(ν) and Q is the image of the zero

section. As Hn−r(Q) = Z2 = Hn(D(ν), S(ν)), the Thom isomorphism of Theo-rem 4.7.29 says that U(ν) [D(ν)] = [Q]. This proves the lemma for any tubularneighbourhood of Q, for instance W or a smaller tube W ′ contained in the interiorof W .

Let us choose a triangulation ofM for whichW andW ′ are subcomplexes. Theclass H∗j(U(M,Q)) ∈ Hr(M) may then be represented by a simplicial cocycle q ⊂Sr(W ′). The n-simplexes of M involved in the computation of H∗j(U(M,Q)) [M ] are then all simplexes of W . Therefore

H∗j(U(M,Q)) [M ] = H∗i(U(W,Q)

) [W ] = H∗i([Q]) .

Remark 5.4.3. We see in the proof of Lemma 5.4.2 that the Poincare dualPD(Q) of a submanifold Q ⊂ M is supported in an arbitrary small tubular neigh-bourhood of Q. This localization principle is illustrated in Figure 5.2 for Q a circlein the torus or the Klein bottle. For the analogous localization principle in de Rhamcohomology, see [19, Proposition 6.25].

Lemma 5.4.4. The image of PD(Q) under the homomorphism Hr(M,BdM)→Hr(M)→ Hr(Q) is equal to the Euler class of the normal bundle ν = ν(M,Q).


Proof. We use the notations of the proof of Lemma 5.4.2. Let k : (M, ∅) →(M,M − Q) denote the pair inclusion and let σ0 : Q → D(ν) be the zero section.The various inclusions give rise to the commutative diagram

Hr(M,M −Q)

≈

H∗j //

H∗k

22Hr(M,BdM)) // Hr(M)

H∗i

%%

Hr(W,BdW )

≈

Hr(W )

≈

Hr(Q)

Hr(D(ν), S(ν))H∗jν // Hr(D(ν))

H∗σ0

99rrrrrrrr

The Euler class e(ν) ∈ Hr(Q) is characterized by the equation H∗jν(U(ν)) =H∗p(e(ν)), where p : D(ν)→ Q is the bundle projection (see p. 154). Since pσ0 =idQ, the previous diagram and Lemma 5.4.2 yield

H∗i(PD(Q)) = H∗iH∗k(U(M,Q)) = H∗σ0 H∗jν(U(ν)) = H∗σ0 H∗p(e(ν)) = e(ν) .

We now compute ker(H∗i : H∗(M)→ H∗(Q)

).

Proposition 5.4.5. Let M be a smooth compact n-manifold and let i : Q →Mbe the inclusion of a closed smooth submanifold Q of codimension r. Then

kerH∗i ⊂ Ann (PD(Q)) = x ∈ H∗(M) | x PD(Q) = 0(for a class b, the ideal Ann (b) is called the annihilator of b). The above inclusionis an equality if and only if H∗i is surjective.

Proof. Let a ∈ H∗(M) and let q = PD(Q). Then

H∗i(H∗i(a) q)

)= a H∗i([Q]) by (4.5.5)

= a (q [M ]) by definition of q

= (a q) [M ] by Proposition 4.5.4.

Therefore, if H∗i(a) = 0, then a ∈ Ann (q) and the converse is true if and only ifH∗i is injective. By Kronecker duality (see Corollary 2.3.11), the latter is equivalentto H∗i being surjective.

Example 5.4.6. Let i be the standard inclusion of Q = RP k in M = RP k+r.By Proposition 4.3.10, one has a commutative diagram

Z2[a]/(ak+r+1)

≈

// // Z2[a]/(ak+1)

≈

H∗(RP k+r)H∗i // // H∗(RP k)

where a is of degree 1. Then, via the above vertical isomorphisms,

kerH∗i = (ak+1) = Ann (ar) = Ann (PD(RP k)) .

The following proposition states the functoriality of the Poincare dual for trans-verse maps.


Proposition 5.4.7. Let f : M → N be a smooth map between smooth closedmanifolds. Suppose that f is transverse to a closed submanifold Q of N . Then

PD(f−1(Q)) = H∗f(PD(Q)) .

Proof. Let P = f−1(Q). We consider the commutative diagram

H∗(N,N −Q)H∗f //

H∗i

H∗(M,M − P )

H∗j

H∗(N)H∗f // H∗(M)

where the vertical arrow are induced by the inclusions i : (N, ∅)→ (N,N −Q) andj : (M, ∅)→ (M,M − P ). Then,

PD(P ) = H∗j(U(M,P )) by Lemma 5.4.2

= H∗j H∗f(U(N,Q)) by transversality and Lemma 4.7.30

= H∗f H∗i(U(N,Q))

= H∗f(PD(Q)) by Lemma 5.4.2.

5.4.2. The Gysin Homomorphism. Let (M,Q) be a pair of smooth com-pact manifolds. Let i : Q → M denote the inclusion. Set q = dimQ and m =dimM = q + r. The Gysin homomorphism Gys: Hp(Q) → Hp+r(M,BdM) isdefined for all p ∈ N by the composite homomorphism

Hp(Q)[Q]

≈// Hq−p(Q)

H∗i // Hq−p(M) oo [M ]

≈Hp+r(M,BdM) .

The notation i! and the terminology umkehr homomorphism are also used in theliterature.

For example, Gys(1) = PD(Q), the Poincare dual of Q. More generally:

Lemma 5.4.8. For a ∈ Hp(M), one has Gys(H∗i(a)

)= a PD(Q).

Proof.

(a PD(Q)) [M ] = a (PD(Q)) [M ])

= a H∗i([Q])

= H∗i(H∗i(a) [Q]

)by Proposition 4.5.4

= Gys(H∗i(a)

) [M ]

As − [M ] is an isomorphism, this proves the lemma.

Example 5.4.9. Let M be the total space of an r-disk bundle π : M → Q. Wesee Q as a submanifold of M via the 0-section i : Q→ M . Let U ∈ Hr(M,BdM)be the Thom class. Since U [M ] = H∗i([Q]), we see that U = PD(Q). Forb ∈ Hp(Q), one has

Gys(b) [M ] = Gys(H∗iH∗π(b)

) [M ] since π i = idQ

=(H∗π(b) PD(Q)

) [M ] by Lemma 5.4.8

=(H∗π(b) U

) [M ] since U = PD(Q)

= Thom(b) [M ] .


Since − [M ] is an isomorphism, we see in this example that the Gysin homo-morphism is identified with the Thom isomorphism.

Proposition 5.4.10. Let (M,Q) be a pair of smooth closed manifolds, with Qof codimension r. Let W be a closed tubular neighbourhood of Q in M . There is acommutative diagram

// Hp−1(M −Q) //

Hp−r(Q)Gys //

=

Hp(M) //

Hp(M −Q) //

// Hp−1(BdW ) // Hp−r(Q) // Hp(Q) // Hp(BdW ) //

where the vertical arrows are induced by inclusions. The horizontal lines are exactsequences and the bottom one is the Gysin sequence of the sphere bundle BdW → Q.

Proof. We start with the commutative diagram

// Hp−1(M −Q) //

Hp(M,M −Q) //

Hp(M) //

Hp(M −Q) //

// Hp−1(BdW ) // Hp(W,BdW ) // Hp(W ) // Hp(BdW ) //

using the cohomology sequences of the pairs (M,M − Q) and (W,BdW ). To getthe diagram of the proposition, we use the identification

Hp−r(Q)Thom

≈// Hp(W,BdW ) oo

≈Hp(M,M −Q)

and H∗(W ) ≈ H∗(Q). Thus the bottom line is the Gysin sequence of BdW → Q.It remains to identify the homomorphism Hp−r(Q) → Hp(M) with the Gysinhomomorphism. This amounts to the commutativity of the diagram

Hp−r(Q)Thom

≈//

−[Q]≈

Hp(W,BdW ) oo≈

−[W ]≈

Hp(M,M − intW ) // Hp(M)

−[M ]≈

Hq−p+r(Q) // Hq−p+r(W ) // Hq−p+r(M)

The commutativity of the left square was observed in Example 5.4.9. That of theright square may be checked using simplicial (co)homology for a triangulation ofM extending one of W .

Proposition 5.4.11. Let f : M ′ → M be a smooth map between closed man-ifolds. Let Q be a closed submanifold of codimension r in M . Suppose that f istransverse to Q. Then, for all p ∈ N, the diagram

Hp(Q)Gys //

H∗f

Hp+r(M)

H∗f

Hp(f−1(Q))Gys // Hp+r(M ′)

is commutative.


Proof. Let Q′ = f−1(Q). By transversality, f : Q′ → Q is covered by a

morphism of vector bundle f : ν(M ′, Q′)→ ν(M,Q). Put a Riemannian metric on

ν(M,Q) and pull it back on ν(M ′, Q′), so that f is an isometry on each fiber. Bystandard technique of Riemannian geometry, one can find a tubular neighbourhoodW of Q and a tubular neighbourhoodW ′ of Q′ and modify f by a homotopy relativeto Q′ so that f(W ′) ⊂ W , f(BdW ′) ⊂ BdW , f(M ′ − intW ′) ⊂ M − intW and

f : W ′ → W coincides with f via the exponential maps of W and W ′. We thus geta diagram.

Hp−r(Q)Thom

≈//

H∗f

Hp(W,BdW ) oo≈

H∗f

Hp(M,M − intW ) //

H∗f

Hp(M)

H∗f

Hp−r(Q′)Thom

≈// Hp(W ′,BdW ′) oo

≈Hp(M ′,M ′ − intW ′) // Hp(M ′)

.

The left square is commutative by construction and the functoriality of the Thomisomorphism (coming from Lemma 4.7.30). The other squares are obviously com-mutative. But, as seen in the proof of Proposition 5.4.10, the compositions from theleft end to the right end of the horizontal lines are the Gysin homomorphisms.

5.4.3. Intersections of submanifolds. Consider two closed submanifoldsQi (i = 1, 2) of the compact smooth n-manifold M , Qi being of codimension ri.We suppose that Q1 and Q2 intersect transversally. Then, Q = Q1 ∩Q2 is a closedsubmanifold of codimension r = r1 + r2.

Proposition 5.4.12. Under the above hypotheses

PD(Q) = PD(Q1) PD(Q2) .

Proof. As (M−Q1)∪(M−Q2) =M−Q, the cup product provides a bilinearmap

Hr1(M,M −Q1)×Hr2(M,M −Q2)−→ Hr(M,M −Q) .

In virtue of Lemma 5.4.2, it suffices to prove that

(5.4.4) U(M,Q1) U(M,Q2) = U(M,Q) .

We may suppose that Q 6= ∅ for, otherwise, the proposition is trivially true, sinceq = 0 and r > n.

If A is a submanifold of B, we denote by ν(B,A) the normal bundle of A in B.Choose an embedding µ : D(ν(B,A)) → B parameterizing a tubular neighbourhoodW (B,A). If V ⊂ B the notation W (A,B)V means µ(D(ν(B,A)V )). As Q1 andQ2 intersect transversally, one has

ν(M,Q) = ν(Q1, Q)|Q ⊕ ν(Q2, Q)|Q .

Let b ∈ Q. One may choose convenient tubular neighbourhood parameterizationsso that W (M,Q)b ∩ Qj = W (Qj , Q)b. Let W1 = W (Q1, Q)b ≈ Dr2 , W2 =W (Q2, Q)b ≈ Dr1 and W =W (M,Q)b ≈W1 ×W2 ≈ Dr. Let πi : W →Wi bethe projection. By Lemma 5.4.2:

• the class U(M,Q1) ∈ Hr1(M,M − Q1) restricts to the non-zero elementa1 ∈ Hr1(W,W1 × BdW2) = Z2;• the class U(M,Q2) ∈ Hr2(M,M − Q2) restricts to the non-zero elementa2 ∈ Hr2(W,BdW1 ×W2) = Z2.


Hence, U(M,Q1) U(M,Q2) restricts to a1 a2 ∈ Hr(W,BdW ) = Z2. Wehave to prove that a1 a2 6= 0. Let 0 6= ai ∈ Hri(Wj ,BdWj) (i 6= j). Then

a1 a2 = H∗π1(a1) H∗π2(a2) = a1 × a2 .

By the relative Kunneth theorem 4.6.10, a1 × a2 6= 0 in Hr(W,BdW ). Hence,U(M,Q1) U(M,Q2) restricts to the non-zero element ofHr(W (M,Q)b,BdW (M,Q)b) for all b ∈ Q. By Lemma 5.4.2, this proves(5.4.4).

An interesting case is when dimQ1 +dimQ2 = dimM . If Q1 and Q2 intersecttransversally, then Q1 ∩Q2 is a finite collection of points. Let ji : Qi →M denotethe inclusion. The following result says that the parity of this number of pointsdepends only on [Qi]M = H∗ji([Qi]) and justifies the terminology of intersectionform.

Corollary 5.4.13. Let Qi (i = 1, 2) be two closed submanifolds of the compactsmooth n-manifold M , with dimQ1 + dimQ2 = n. Let qi = PD(Qi). Suppose thatQ1 and Q2 intersect transversally. Then

♯ (Q1 ∩Q2) ≡ 〈q1 q2, [M ]〉 mod 2 .

In other words,

♯ (Q1 ∩Q2) ≡ H∗j1([Q1)] ·H∗j2([Q2)] mod 2 ,

where “·” denotes the (absolute) intersection form (see § 5.3.3).

Proof. One has

〈q1 q2, [M ]〉 = 〈1, (q1 q2) [M ]〉 = 〈1, [Q1 ∩Q2]〉 ≡ ♯ (Q1 ∩Q2) mod 2 .

Lemma 5.4.14. Let ξ = (p : E → N) be a smooth vector bundle over a closedsmooth manifold N . Let σ, σ′ : N → E be two smooth sections of ξ which aretransverse. Let Q be the submanifold of N defined by Q = σ−1

(σ(N) ∩ σ′(N)

).

Then, the Poincare dual of Q in N is the Euler class e(ξ) of ξ.

Proof. We us the following notation: if λ : Y → X is a continuous map andY a closed manifold, we write [Y ]X = H∗λ([Y ]) ∈ HdimY (X); the map λ is usuallyimplicit, being an inclusion or an embedding obvious from the context.

Endow ξ with a Euclidean structure and consider the pair (D,S) = (D(ξ), S(ξ))of the associated unit disk and sphere bundle. Using a homotopy in each fiber, wecan assume that σ(N) and σ′(N) are contained in the interior of D. All the sectionsof a bundle are homotopic. By Lemma 5.4.2 and its proof, the Thom class U of ξ isthe Poincare dual in D of [N ]D = H∗σ([N ]) = H∗σ

′([N ]). By Proposition 5.4.12,U U is the Poincare dual in D of [Q]D. Let j : (D, ∅)→ (D,S) denote the pair


inclusion. As pσ = idN , one has

[Q]N = H∗p([Q]D)

= H∗p((U U) [D]

)cap product of (4.5.9)

= H∗p(U (U [D])

)by Formula (4.5.10)

= H∗p(U [N ]D

)

= H∗p(H∗j(U) [N ]D

)by definition of the cap product (4.5.7)

= H∗p(H∗p(e(ξ)) [N ]D

)by definition of the Euler class

= e(ξ) H∗p([N ]D)

= e(ξ) [N ] .

which proves the lemma.

When, in Lemma 5.4.14, the rank of ξ is equal to the dimension of the mani-fold N , then σ(N) ∩ σ′(N) is a finite collection of point and one gets the followingcorollary.

Corollary 5.4.15. Let ξ = (p : E → N) be a smooth vector bundle of rank nover a closed smooth n-manifold N . Let σ, σ′ : N → E be two smooth sections of ξwhich are transversal. Then

♯ (σ(N) ∩ σ′(N)) ≡ 〈e(ξ), [N ]〉 mod 2 .

The following corollary is a justification for the name Euler class.

Corollary 5.4.16. Let N be a smooth closed manifold. Then the followingcongruences mod 2 hold:

〈e(TN), [N ]〉 ≡ χ(N) ≡ dimH∗(N) mod 2 .

Proof. As χ(N) =∑

i(−1)i dimHi(N), the second congruence is obvious.Let σ0 : N → D(TN) be the zero section and let σ : N → D(TN) be another smoothsection (i.e. a vector field on N) which is transverse to σ0. By Corollary 5.4.15, thenumber of zeros of σ is congruent mod 2 to 〈e(TN), [N ]〉. It then suffices to findsome vector field transverse to σ0 for which we know that its number of zeros iscongruent mod 2 to χ(N). Observe that, for a finite CW-complex X , the followingcongruence mod 2 holds

χ(X) ≡ dimH∗(X) ≡ ♯Λ(X) mod 2 ,

where Λ(X) is set of cells of X . For the required vector field, one can take thegradient vector field σ = gradf of a Morse function f : N → R. Then σ−10

(σ(N) ∩

σ0(N))= Crit f , the set of critical points of f . The transversality of σ with σ0

is equivalent to f being a Morse function (see [95, Chapter 6]). By Morse theory,N has then the homotopy type of a CW-complex X with ♯Λ(X) = ♯Crit f , [95,Chapter 6, Theorem 4.1]. One can also use the classical vector field associated toa C1-triangulation of N , with one zero at the barycenter of each simplex (see, e.g.[180, pp. 611–612]).

We give below a second proof of Corollary 5.4.16, using the following lemma.

Lemma 5.4.17. Let ∆N be the diagonal submanifold ofM = N×N , with normalbundle ν(M,∆N ). Then, there is a canonical isomorphism of vector bundles

ν(M,∆N ) ≈ TN .


Proof. Let p1, p2 : N × N → N be the projections onto the first and secondfactor. For x ∈ N , consider the commutative diagram in the category of real vectorspaces

0 // T(x,x)(∆N ) //

T(x,x)(N ×N) //

φ

ν(x,x)(M,∆N )

φ

// 0

0 // ∆(TxN) // TxN × TxN − // TxN // 0

where the rows are exact and φ(v) = (Tp1(v), T p2(v)). The map φ is an isomor-phism and sends T(x,x)(∆N ) onto ∆(TxN). Hence, φ descends to the isomorphism

φ : ν(M,∆N )≈−→ TxN which, of course, depends continuously on x.

Second proof of Corollary 5.4.16. We consider N as the diagonal sub-manifold of M = N × N , with the diagonal inclusion ∆: N → M . The normalbundle ν(M,N) is isomorphic to the tangent bundle of M by Lemma 5.4.17.

By (5.4.1), the Poincare dual of N is equal to∑i ai×bi, where A = a1, a2, . . .

and B = b1, b2, . . . are bases of H∗(N) dual one to the other for the Poincareduality.

〈e(TM), [N ]〉 = 〈e(ν(M,N)), [N ]〉= 〈H∗∆(

∑i ai × bi), [N ]〉 by Lemma 5.4.4

= 〈∑i ai bi, [N ]〉 ≡ dimH∗(N) mod 2 .

Example 5.4.18. As χ(Sn) ≡ 0 mod 2, the Euler class of TSn vanishes byCorollary 5.4.16. Let T 1Sn be the associated sphere bundle. By Proposition 4.7.35and the Leray-Hirsch theorem, we get an isomorphism of H∗(Sn)-module

H∗(T 1Sn) ≈ H∗(Sn)⊗H∗(Sn−1) .If n ≥ 3, Poincare duality implies that this isomorphism is a ring-isomorphism.This is not true if n = 2 (see Remark 4.7.34).

Thus, for n ≥ 3, T 1Sn has the same cohomology ring as Sn × Sn−1. However,by [109, Theorem 1.12], these two spaces have the same homotopy type if and onlyif there exists a map f : S2n+1 → Sn+1 with Hopf invariant one (see § 6.3). ByTheorem 8.6.6, such an f exists if and only if n = 1, 3, 7.

5.4.4. The linking number. Let Q and Q′ be two disjoint closed subman-ifold of a closed manifold Σ (say, in the smooth category), with q = dimQ,q′ = dimQ′ and s = dimΣ. We assume that

(1) q + q′ = s− 1.(2) Σ is a Z2-homology sphere, i.e. H∗(Σ) ≈ H∗(Ss).(3) 〈1Q, [Q]〉 = 〈1Q′ , [Q′]〉 = 0. This condition is always satisfied when q and

q′ are not zero. If, say, q = 0, it means that Q has an even number ofpoints, so that [Q] ∈ H0(Q) = ker〈1Q, 〉.

Thanks to (2), Alexander duality (see Theorem 5.3.14) provides an isomorphism

A : Hq(Q)≈−→ Hs−q−1(Σ−Q) .

Note that s− q− 1 = q′. By Condition (3), [Q] ∈ Hq(Q) and [Q′] ∈ Hq′ (Q′), so we

can define the linking number (sometimes called the linking coefficient) l(Q,Q′) of


Q and Q′ in Σ by

(5.4.5) l(Q,Q′) = 〈A[Q], H∗i([Q′])〉 ∈ Z2 ,

where i : Q′ → Σ denotes the inclusion. Despite the asymmetry of the definition,the equality l(Q,Q′) = l(Q′, Q) will be proven in Proposition 5.4.25 below.

The linking number l(Q,Q′) was introduced in 1911 by Lebesgue [131, pp. 173–175], with a definition in the spirit of Proposition 5.4.22 below. Lebesgue calledQ and Q′ “enlacees” if l(Q,Q′) = 1. One year later, Brouwer [22, pp. 511–520]refined the idea when Σ, Q and Q′ are oriented, defining an integral linking numberwhose reduction mod 2 is l(Q,Q′) (for the philosophy of Brouwer’s definition, seeExercise 5.16). More history and references about the linking numbers may befound in [40, pp. 176–179 and 185].

As Q is a submanifold of Σ, the isomorphism A may be described in the fol-lowing way, which will be useful for computations. Let V be a closed tubularneighbourhood of Q in Σ−Q′ and let X = Σ− intV . The cohomology sequence of(Σ, X)

Hs−q−1(Σ) // Hs−q−1(X)δ∗ // Hs−q(Σ, X) // Hs−q(Σ)

shows that the connecting homomorphism δ∗ descends to an injection δ∗ of

coker(Hs−q−1(Σ)→ Hs−q−1(X)

)≈ Hs−q−1(X)

into Hs−q(Σ, X). Let j : (V,BdV )→ (Σ, X) denote the pair inclusion. IdentifyingH∗(V ) with H∗(Q), one has the diagram

(5.4.6)

Hs−q−1(X)

δ∗

oo A

≈Hq(Q)

Hs−q(Σ, X)

H∗j

≈//

Hs−q(V,BdV )[V ]

≈// Hq(Q)

Hs−q(Σ)

[Σ]

≈// Hq(Σ)

whose columns are exact (the right hand one by Lemma 3.1.10) and whose bottompart is commutative by Proposition 5.3.6. Then A is the unique isomorphismmaking the top rectangle commutative (compare the proof of Theorem 5.3.14).

Remark 5.4.19. Diagram (5.4.6) uses the singular (co)homology. But, via atriangulation of Σ, the various spaces may be the geometric realizations of simplicialcomplexes (by abuse of notations we use the same letters, i.e. Σ = |Σ|, etc).Then, Diagram (5.4.6) makes sense for simplicial (co)homology; the isomorphismHs−q(Σ, X) ≈ Hs−q(V,BdV ) is just the simplicial excision (see Exercise 2.17).

Remark 5.4.20. Suppose that Q is connected or consists of two points. ThenHq′(Σ−Q) ≈ Hq(Q) ≈ Z2. Therefore, l(Q,Q

′) = 1 if and only if H∗i([Q′]) 6= 0 in

Hq′(Σ−Q). In this case, l(Q,Q′) determines H∗i([Q′]).

The following lemma shows that l(Q,Q′) is not always zero. We say that Q′ isa meridian sphere for Q if Q′ is the boundary of a (s− q)-disk ∆ in Σ intersectingQ transversally in one point.


Lemma 5.4.21. Let Q, Q′ and Σ satisfying (1)–(3) above. Suppose that Q′ isa meridian sphere for Q. Then l(Q,Q′) = 1.

Proof. Let ν(Q,Σ) be the normal bundle of Q in Σ. A Riemannian met-ric provides a smooth bundle pair (D(ν), S(ν)) with fiber (Dm−q, Sm−q−1) and

a diffeomorphism φ : (D(ν), S(ν))≈−→ (V,BdV ), where V is a tubular neighbour-

hood of Q in Σ. By choosing the Riemannian metric conveniently, we may assumethat ∆ ∩ V is the image by φ of a fiber Dm−q of D(ν). One has H∗i([Q

′]) =

H∗j H∗φ([Sm−q−1]), where j : (V,BdV ) → (Σ,Σ −Q) denotes the pair inclusion,

so l(Q,Q′) = l(Q,φ(Sm−q−1)) by (5.4.5).

In Diagram (5.4.6), one has H∗φH∗j δ∗A([Q]) = U , the Thom class of ν,as can be checked on each connected component of Q. Therefore,

l(Q,S) = 〈A([Q]), H∗i([Q′])〉

= 〈A([Q]), H∗j H∗φ([Sm−q−1])〉

= 〈A([Q]), ∂∗H∗j H∗φ([Dm−q ])〉

= 〈H∗φH∗j δ∗A([Q]), [Dm−q]〉= 〈U, [Dm−q]〉

and 〈U, [Dm−q]〉 = 1 by Lemma 4.7.28.

The following proposition gives a common way to compute a linking number,related to the original definition of Lebesgue [131, pp. 173–175]. Let Q, Q′ andΣ satisfying (1)–(3) above. Suppose that there exists a compact manifold W withBdW = Q′ so that the inclusion of Q′ into Σ extends to a map j : W → Σ which istransverse to Q (j needs not to be an embedding). Then j−1(Q) is a finite numberof points in W .

Proposition 5.4.22. l(Q,Q′) = ♯ (j−1(Q)) mod 2 .

Proof. Let k = ♯ (j−1(Q)) ∈ N. Let W0 be the manifold W minus an opentubular neighbourhood of j−1(Q). By (5.4.5), l(Q,Q′) depends only on the homol-ogy classH∗i([Q

′]) ∈ Hq′(Σ−Q) which, thanks to the map j (see Exercise 5.6), is thesame as that of k meridian spheres. The result then follows from Lemma 5.4.21.

We now introduce some material for Lemma 5.4.23 below, which will enable usto compute linking numbers using convenient singular cochains. Let Q, Q′ and Σsatisfying (1)–(3) above. Let W be a closed tubular neighbourhood of Q in Σ−Q′and let V be a closed tubular neighbourhood of Q in intW . We also considerthe symmetric data Q′ ⊂ V ′ ⊂ W ′ ⊂ Σ − Q, assuming that W ∩ W ′ = ∅. LetB = W,W ′,Σ− (V ∪ V ′); note that the small simplex theorem 3.1.34 holds truefor B.

Let c ∈ Zs−q(W,W − intV ) be a singular cocycle representing the Poincaredual class of Q in Hs−q(W,W − intV ) ≈ Hs−q(V,BdV ). We can see c as a cocycle

of W and take its zero extension c ∈ Cs−qB (Σ), i.e.

〈c, σ〉 =〈c, σ〉 if σ ∈ Ss−q(W )

0 otherwise.

Since c vanishes on Ss−q(W,W − intV ), the cochain c is a B-small cocycle, i.e.

c ∈ Zs−qB (Σ). We claim that we can choose a ∈ Cs−q−1B (Σ) such that δa = c.


Indeed, by (2) and the small simplex theorem 3.1.34, 0 = Hs−q(Σ) ≈ Hs−qB (Σ) when

q > 0. When q = 0, c represents the Poincare dual class of Q in HsB(Σ) ≈ Hs(Σ).

But, by (3), [Q] represents 0 in H0(Σ), so c represents 0 in HsB(Σ). A cochain

a′ ∈ Cs−q′−1

B (Σ) with δ(a′) = c′ may be also chosen, using the symmetric dataQ′ ⊂ V ′ ⊂W ′ ⊂ Σ−Q.

Finally, let µ′ ∈ Zq′(Σ− intV ) represent H∗i([Q′]) and let ν ∈ ZBs (Σ) represent

[Σ] in HBs (Σ) ≈ Hs(Σ).

Lemma 5.4.23. The following equalities hold true.

(a) l(Q,Q′) = 〈a, µ′〉.(b) l(Q,Q′) = 〈a c′, ν〉.Proof. We first establish some preliminary steps.

Step 1: If a1, a2 ∈ Cs−q−1(Σ) satisfy δ(a1) = δ(a2), then 〈a1, µ′〉 = 〈a2, µ′〉. In-deed, one has then δ(a1 + a2) = 0. If s− q− 1 = q′ > 0, Condition (2) implies thatthere exists b ∈ Cs−q−2(Σ) such that δb = a1 + a2. Hence,

〈a1, µ′〉+ 〈a2, µ′〉 = 〈a1 + a2, µ′〉 = 〈δb, µ′〉 = 〈b, ∂µ′〉 = 0 .

If q′ = 0, then δ(a1 + a2) = 0 implies that a1 + a2 = 1 by Proposition 3.1.8, thus〈a1 + a2, µ

′〉 = 0 by Condition (3).

Step 2: Let ci ∈ Zs−q(W,W − intV ) (i = 1, 2) be singular cocycles both representing

the Poincare dual class of Q in Hs−q(W,W − intV ). Let ai ∈ Cs−q−1B (Σ) such thatδai = ci as above. Then 〈a1, µ′〉 = 〈a2, µ′〉. Indeed, there exists b ∈ Cs−q−1(W,W−intV ) such that δ(b) = c1 + c2. Its zero extension b ∈ Cs−q−1B (Σ) then satisfiesδ(b) = c1 + c2 and then δ(a1 + b) = c2. Thus

〈a2, µ′〉 = 〈a1 + b, µ′〉 by Step 1

= 〈a1, µ′〉 since 〈b, µ′〉 = 0 .

We can now start the proof of Lemma 5.4.23. Given Steps 1 and 2, it is enoughto prove (a) for a particular choice of c and a. We use Diagram (5.4.6) and see A

as an isomorphism from Hq(Q) onto Hs−q−1(Σ− intV ). Let a ∈ Zs−q−1(Σ− intV )

represent A([Q]). Let a ∈ Cs−q−1B (Σ) be its zero extension and let c = δ(a) ∈Zs−qB (Σ). By Lemma 3.1.17, c represents δ∗(A([Q]). Also, c is the zero extension ofthe cocycle c ∈ Zs−q(W,W − intV ) which, by definition of A and Diagram (5.4.6),represents the Poincare dual class of Q in Hs−q(W,W − intV ). Therefore, since arepresents A([Q]) and µ′ ∈ Zq′(Σ− intV ) represents H∗i([Q

′]), one has l(Q,Q′) =〈a, µ′〉 = 〈a, µ′〉.

To prove (b), consider the pair inclusions j1 : (Σ, ∅) → (Σ,Σ − intV ′) andj2 : (W

′,W ′ − intV ′) → (Σ,Σ − intV ′). Since ν ∈ ZBs (Σ), there exists a (unique)ν′ ∈ Zs(W

′,W ′ − intV ′) such that C∗j2(ν′) = C∗j1(ν). As H∗j1 and H∗j2 are

isomorphisms, ν′ represents the generator of Hs(W′,W ′− intV ′) = Z2. Therefore,

c′ ν = c′ ν′ represents H∗i([Q′]) and, by (a),

l(Q,Q′) = 〈a, c′ ν〉 = 〈a c′, ν〉 .

Remark 5.4.24. The proof of Lemma 5.4.23 in the simplicial category (seeRemark 5.4.19) is somewhat simpler. It uses only the tubular neighbourhoodsVi and not the Wi’s, and, of course, does not require the use of small simplextechniques. Also, ν may be taken explicitly as Ss(Σ). Writing the details is left tothe reader as an exercise.


Lemma 5.4.23 will be used for the Hopf invariant (see § 6.3.4). For the moment,its main consequence is the following proposition.

Proposition 5.4.25. Let Q, Q′ and Σ satisfying (1)–(3) above. Then

l(Q,Q′) = l(Q′, Q) .

Proof. By Point (b) of Lemma 5.4.23, one has l(Q,Q′) = 〈a c′, ν〉 andl(Q′, Q) = 〈a′ c, ν〉. Thenl(Q,Q′) + l(Q′, Q) = 〈a c′ + a′ c, ν〉 = 〈δ(a a′), ν〉 = 〈a a′, ∂(ν)〉 = 0 .


5.1. Prove that the product of two homology manifolds is a homology manifold.

5.2. Let M be a compact manifold with boundary such that H∗(M) = 0. Showthat the boundary of M is a homology sphere.

5.3. Check the Poincare duality (Theorem 5.3.7) for the manifolds S1 × I and theMobius band.

5.4. Show that there is no continuous retraction of a non-empty compact manifoldonto its boundary.

5.5. Let M be a closed triangulable manifold of dimension n. Prove that thehomomorphismHk(M−pt)→ Hk(M) induced by the inclusion is an isomorphismfor k < n.

5.6. LetM be a compact triangulable topological n-manifold with boundary BdM =N . Suppose that is N = N1∪N2 the union of two closed (n − 1)-manifolds. Letf : M → X be a continuous map. Show that H∗f([N1]) = H∗f([N2]) in Hn−1(X).What happens if N2 = ∅?5.7. Let f : M → N be a map between closed n-dimensional manifolds of the samedimension. Show that the degree of f may be computed locally, using a topologicalregular value, like in Proposition 3.2.6.

5.8. Let Σm be the orientable surface of genus m and let Σn be the nonorientablesurface of genus n. For which m and n does there exist a continuous map of degreeone from Σm to Σn or from Σn to Σm?

5.9. LetM be a closed manifold of dimension m which is the products of two closedmanifolds of positive dimensions. Does there exist a degree one map f : M → RPm?

5.10. Let Q1 and Q2 be closed submanifolds of a closed manifoldM (in the smoothcategory). Suppose that Q1 and Q2 intersect transversally in an odd number ofpoints. Show that [Q1] and [Q2] represent non-zero classes in H∗(M).

5.11. Let i : Q→M be the inclusion of a smooth closed submanifold of dimensionq in a smooth closed manifold M . Suppose that H∗i([Q]) = 0 in Hq(M). Provethat the Euler class of the normal bundle to Q vanishes.

5.12. For A ⊂ 0, 1, . . . , n, let PA = [x0 : · · · : xn] ∈ RPn | xi = 0 when i /∈ A.What is the diffeomorphism type of PA? Show that, if A ∪ B = 0, 1, . . . , n,then PA and PB intersect transversally (what is the intersection?). How doesProposition 5.4.12 apply in this example?


5.13. Poincare dual classes in a product. Let M1 and M2 be smooth compactmanifolds. Let Qi be a closed submanifold of Mi (i = 1, 2). Then Q1 × Q2 is aclosed submanifold of M1 ×M2. Prove that

(5.5.1) PD(Q1 ×Q2) = PD(Q1)× PD(Q2)

in H∗(M1 ×M2).

5.14. Poincare dual classes in a product II. Let M and M ′ be smooth closedmanifolds and let x ∈M and x′ ∈M ′. What are PD(x×M ′) and PD(M ×x′)in H∗(M ×M ′)? Check that PD(x ×M ′) PD(M × x′) = PD((x, x′)).5.15. Let Q and Q′ be disjoint submanifolds of S2, where Q consists of two circlesand Q′ of four points. Using Proposition 5.4.22, compute the linking numbersl(Q,Q′) and l(Q′, Q) for the various possibilities.

5.16. Brouwer’s definition of the linking number. Let Q and Q′ be two disjointclosed submanifolds of Sn, of dimension respectively q and q′ satisfying p+q = n−1.If Q (or Q′) is of dimension 0, it should consist of an even number of points. SeeQ and Q′ as submanifolds of Rn via a stereographic projection of Sn − pt ontoRn. Consider the Gauss map λ : P × Q → Sn−1 given by λ(x, y) = x−y

||x−y|| . Show

that the degree of λ is equal to the linking number l(Q,Q′) (see § 5.4.4). [Hint: useProposition 5.4.22.]

5.17. Write the proof of Lemma 5.4.23 in the simplicial category (see Remarks 5.4.19and 5.4.24).

5.18. Let Σ be the unit sphere in Rq+1 × Rq′+1. Let Q = Sq × 0 ⊂ Σ and

Q′ = 0 × Sq′ ⊂ Σ. Compute the linking number l(Q,Q′) in Σ.

CHAPTER 6

Projective spaces

Coming from algebraic geometry, projective spaces and their Hopf bundles playan important role in homotopy theory, as already seen in Sections 3.8 and 4.3. Theprecise knowledge of their cohomology algebra has interesting applications, like theBorsuk-Ulam theorem, continuous multiplications in Rm and the Hopf invariant,which are presented in § 6.2.

6.1. The cohomology ring of projective spaces - Hopf bundles

The cohomology ring of RPn for n ≤ ∞ was established in Proposition 4.3.10,using the transfer (or Gysin) exact sequence for the double cover (S0-bundle) Sn →RPn. It gives a GrA-isomorphism Z2[a]/(a

n+1) → H∗(RPn). We give below acompletely different proof of this fact, which is based on Poincare duality (as RPn

is a smooth closed manifold, it can be triangulated as a polyhedral homology n-manifold: see p 170). We shall also discuss the cases of complex and quaternionicprojective spaces CPn and HPn, and of the octonionic projective plane OP 2.

Proposition 6.1.1. The cohomology algebra of RPn (n ≤ ∞) is given by

H∗(RPn) ≈ Z2[a]/(an+1) , H∗(RP∞) ≈ Z2[a] ,

with a ∈ H1(RPn).

Proof. We prove the first statement by induction on n. It is true for n = 1since RP 1 = S1 (and also proven for n = 2 on p. 116). Suppose, by induction, thatit is true for RPn−1.

In Example 3.4.5 is given the standard CW-structure of RPn, with one k-cell for each k = 0, 1 . . . , n. It follows that Hk(RPn,RPn−1) = 0 for k ≤ n − 1and Hn(RPn,RPn−1) = Z2. By Poincare duality, Hn(RPn) = Z2, so the exactsequence for the pair (RPn,RPn−1) gives

0→ Hn−1(RPn)→ Hn−1(RPn−1)︸︷︷︸Z2

→ Hn(RPn,RPn−1)︸︷︷︸Z2

→ Hn(RPn)︸︷︷︸Z2

→ 0 .

All that implies that the inclusion induces an isomorphismHk(RPn)→ Hk(RPn−1)for k ≤ n − 1. By the induction hypothesis and functoriality of the cup product,Hk(RPn) = Z2 for k ≤ n− 1, generated by ak.

By Poincare duality, Hn(RPn) = Z2 with generator [RPn] and, by Theo-rem 5.3.12, the bilinear map

H1(RPn)×Hn−1(RPn)→ Hn(RPn)[RPn]−−−−−→ Z2

is non-degenerate. Therefore, a an−1 6= 0, which proves that H∗(RPn) ≈Z2[a]/(a

n+1).

201

202 6. PROJECTIVE SPACES

Finally, by the standard CW-structure of RP∞, one has, for all integer n, thatHk(RP∞,RPn) = 0 for k < n. The first statement then implies the second. Notethat we have also proven that the standard CW-structure of RPn (n ≤ ∞) isperfect.

Corollary 6.1.2. The Poincare series of RPn (n ≤ ∞) are

Pt(RPn) = 1 + t+ · · ·+ tn =

1− tn+1

1− t and Pt(RP∞) =

1

1− t .

Remark 6.1.3. Proposition 6.1.1 and its proof show that theGrA-homomorphismHj(RPn+k) → Hj(RPn) induced by the inclusion RPn → RPn+k is surjective(k ≤ ∞). In particular, it is an isomorphism for j ≤ n.

Remark 6.1.4. The polynomial structure on H∗(RP∞) implies the followingfact: if f : RP∞ → X is a continuous map withX a finite dimensional CW-complex,then H∗f = 0. Actually, f is homotopic to a constant map. This result is a weakversion of the original Sullivan conjecture [188, p. 180], which led to importantresearches in homotopy theory (see, e.g. [171]) and was finally proven, in a moregeneral form, by H. Miller [143].

We now pass to the complex projective space CPn, the space of complex lines inCn+1. Such a line is represented by a non-zero vector z = (z0, . . . , zn) ∈ Cn+1−0,and two such vectors z and z′ are in the same line if and only if z′ = λz withλ ∈ C∗ = C− 0. If |z| = |z′| = 1, then λ ∈ S1. Thus

CPn = (Cn+1 − 0)/C∗ = S2n+1

/S1 .

The image of (z0, . . . , zn) in CPn is denoted by [z0 :z1 : . . . :zn]. As S1 acts smoothly

on S2n+1, the quotient CPn is a closed smooth manifold and the quotient map

p : S2n+1 → CPn

is a principal S1-bundle [82, Example 4.44], called the Hopf bundle. In this simpleexample, this can be proved directly. Consider the open set Vk ⊂ Cn+1−0 givenby Vk = (z0, . . . , zn) ∈ Cn+1 | zk 6= 0. Its image in CPn is an open set Uk,

domain of the chart ϕk : Cn≈−→ Uk given by

(6.1.1) ϕk(z0, . . . , zn−1) = [z0 :z1 : . . . :zk−1 : 1 : zk : . . . : zn−1] .

On the other hand, a trivialization ϕk : Uk × S1 ≈−→ p−1(Uk) is given by(6.1.2)

ϕk(ϕk(z0, . . . , zn−1), g) =1√

1 +∑n−1i=0 |zi|2

(z0, z1, . . . , zk−1, g, zk, . . . , zn−1) .

It is also classical that CPn is obtained from CPn−1 by attaching one cell ofdimension 2n,

CPn = CPn−1 ∪p D2n ,

with the attaching map p : S2n−1 → CPn−1 being the quotient map (see e.g. [82,Example 0.6] or [155, Theorem 40.2]). This gives a standard CW-structure onCPn with one cell in each even dimension ≤ 2n. For the direct limit CP∞, weget CW-structure with one cell in each even dimension. For these CW-structure,

6.1. THE COHOMOLOGY RING OF PROJECTIVE SPACES - HOPF BUNDLES 203

the vector space of cellular chains vanish in odd degree, so the cellular boundary isidentically zero. Therefore,

Pt(CPn) = 1 + t2 + · · ·+ t2n =

1− t2(n+1)

1− t2 and Pt(CP∞) =

1

1− t2 .

As CPn is a smooth manifold, the same proof as for Proposition 6.1.1, usingPoincare duality, gives Proposition 6.1.5 below. One can also adapt the proof ofProposition 4.3.10, using the Gysin exact sequence of the Hopf bundle (see Exer-cise 6.2):

· · ·Hk−1(S2n+1)→ Hk−2(CPn)−e(ξ)−−−−−→ Hk(CPn)

H∗p−−−→ Hk−1(S2n+1)→ · · · .Proposition 6.1.5. The cohomology algebra of CPn (n ≤ ∞) is given by

H∗(CPn) ≈ Z2[a]/(an+1) , H∗(CP∞) ≈ Z2[a] ,

with a ∈ H2(CPn). The class a is the Euler class of the Hopf bundle S2n+1 → CPn.

If we replace the field of complex numbers by that of quaternions H, we getquaternionic projective space HPn:

HPn = (Hn+1 − 0)/H∗ = S4n+3

/S3

(it is usual to take the right H-vector space structure on Hn+1). The space HPn

is obtained from HPn−1 by attaching one cell of dimension 4n, with the attachingmap p : S4n−1 → HPn−1 being the quotient map. The map p is an S3-bundle calledthe Hopf bundle.This gives a standard CW-structure on HPn with one cell in eachdimension 4k ≤ 4n. For the direct limit HP∞, we get CW-structure with one cellin each dimension 4k and

Pt(HPn) = 1 + t4 + · · ·+ t4n =

1− t4(n+1)

1− t4 and Pt(HP∞) =

1

1− t4 .

Proposition 6.1.6 below is proven as Proposition 6.1.5, either using Poincareduality (HPn is a smooth 4n-manifold), or the Gysin exact sequence of the Hopfbundle (see Exercise 6.2).

Proposition 6.1.6. The cohomology algebra of HPn (n ≤ ∞) is given by

H∗(HPn) ≈ Z2[a]/(an+1) , H∗(HP∞) ≈ Z2[a] ,

with a ∈ H4(HPn). The class a is the Euler class of the Hopf bundle S4n+3 → HPn.

Let K = R, C or H and let d = d(K) = dimR K. The space KP 1 has a CW-structure with one 0-cell and one d-cell and is then homeomorphic to Sd. Thequotient maps S2d−1 →→ KP 1 thus give maps

h1,1 : S1 →→ S1 , h3,2 : S

3 →→ S2 and h7,4 : S7 →→ S4

called the Hopf maps. Note that h1,1 is just a 2-fold covering. Using the homeo-

morphism Sd ≈ K = K∪∞ given by a stereographic projection, these Hopf mapsadmit the formula

(6.1.3) hi,j(v, w) =

vw−1 if w 6= 0

∞ otherwise.


This formula also makes sense for K = O, the octonions, whose multiplicationadmits inverses for non-zero elements. This gives one more Hopf map h15,8 : S

15 →S8. One can prove that h15,8 is an S7-bundle (see [82, Example 4.47]), also calledthe Hopf bundle. Attaching a 16-cell to S8 using h15,8 produces the octonionicprojective plane OP 2 (because of non-associativity of the octonionic multiplication,there are no higher dimensional octonionic projective spaces).

Proposition 6.1.7. The cohomology algebra of OP 2 is given by

H∗(OP 2) ≈ Z2[a]/(a3)

with a ∈ H8(OP 2) = Z2. In particular, Pt(OP 2) = 1 + t8 + t16.

Proof. By its cellular decomposition, Hk(OP 2) = Z2 for k = 0, 8, 16 andzero otherwise. Let a ∈ H8(OP 2) and b ∈ H8(S8) be the non-zero elements. The

mapping cylinder E of h15,8 is the disk bundle associated to the Hopf bundle and

OP 2 has the homotopy type of E ∪D16, with E ∩D16 = S15. The Thom class ofthe Hopf bundle h15,8

U ∈ H8(E, S15) ≈ H8(OP 2, intD16) ≈ H8(OP 2) ,

is not zero, so corresponds to a ∈ H8(OP 2). The diagram

H8(E)−U

≈// H16(E, S15)

H8(OP 2) //

−a

33

≈

OO

H16(OP 2, intD16)

≈

OO

≈ // H16(OP 2)

is then commutative by the analogue in singular cohomology of Lemma 4.1.8. Thisproves that − a : H8(OP 2)→ H16(OP 2) is bijective.

Remark 6.1.8. Proposition 6.1.7 may also be proved using Poincare duality,since OP 2 has the homotopy type of a closed smooth 16-manifold, in fact a homo-geneous space of the exceptional Lie group F4 (see [207, Theorem 7.21, p. 707]).

The computations of the cohomology algebra H∗(KP 2) have the following con-sequence.

Corollary 6.1.9. The Hopf maps

h1,1 : S1 → S1 , h3,2 : S

3 → S2 , h7,4 : S7 → S4 and h15,8 : S

15 → S8

are not homotopic to constant maps.

Using the Steenrod squares, we shall prove in Chapter 8 that no suspension ofthese Hopf maps is homotopic to a constant map (see Proposition 8.6.1).

Proof. One has S1 ∪h1,1 D2 ≈ RP 2. If h1,1 were null-homotopic, RP 2 would

have the homotopy type of S1∨S2 (see [82, Proposition 0.18]). But, in H∗(S1∨S2),the cup-square map vanishes by (4.2.1), which is not the case in H∗(RP 2). Thesame proof works for the other Hopf maps.

6.2. APPLICATIONS 205

We now consider the vector bundle γK over KPn associated to the Hopf bundle,where K = R,C,H and n ≤ ∞. For n ∈ N, the total space of γK is

E(γK) = (a, v) ∈ KPn ×Kn+1 | v ∈ a ,with bundle projection (a, v) 7→ a. For instance, for n = 1, E(γR) is the Mobiusband. The correspondence (a, v) → V defines a map E(γK) → Kn+1 whose re-striction to each fiber is linear and injective. This endows γK with a Euclideanstructure, whose unit sphere bundle is the Hopf bundle. The vector bundle γK iscalled the Hopf vector bundle or the tautological bundle over KPn. Its (real) rankis d = d(K) = dimR K. By passing to the direct limit when n → ∞, we get atautological bundle γK over KP∞. Propositions 6.1.5, 6.1.6 (and 4.7.36 for K = R)gives the following result.

Proposition 6.1.10. For n ≤ ∞, the Euler class e(γK) is the non-zero elementad ∈ Hd(KPn) = Z2.

The inclusions R ⊂ C ⊂ H induce inclusions

RPnj1−→ CPn

j2−→ HPn , n ≤ ∞ .

The above proposition permits us to determine the GrA-homomorphism inducedin cohomology by these inclusions.

Proposition 6.1.11. H∗jd (a2d) = a2d.

Proof. Observe that γC is a complex vector bundle, so the multiplication byi ∈ C is defined on each fiber. We notice that

(6.1.4) i∗1γC = γR ⊕ i γR ≈ γR ⊕ γR .

ThenH∗j1 (a2) = H∗j1 (e(γC)) by Proposition 6.1.11

= e(j∗1γC) by Lemma 4.7.31

= e(γR ⊕ γR) by (6.1.4)

= e(γR) e(γR) by Proposition 4.7.40

= a21 by Proposition 6.1.11.

The proof that H∗j2 (a4) = a22 is the same, using the multiplication by j ∈ H onthe fiber of γH which is a quaternionic vector bundle.

The Hopf bundles are sphere bundles over Sp such that the total space is also asphere. We shall see in Proposition 6.3.5 that p = 1, 2, 4, 8 are the only dimensionswhere such examples may occur.

6.2. Applications

6.2.1. The Borsuk-Ulam theorem. A (continuous) map f : Rm → Rn orf : Sm → Sn such that f(−x) = −f(x) is called an odd map.

Theorem 6.2.1. Let f : Sm → Sn be an odd map. Then:

(1) n ≥ m.(2) if m = n, then deg f = 1.


Proof. If f is odd, it descends to a map f : RPm → RPn with a commutativediagram

Sm

pm

f // Sn

pn

RPmf // RPn

.

The two-fold covering pn is induced from p∞ : S∞ → RP∞ by the inclusion RPn →RP∞. By Lemma 4.3.6 and Proposition 4.3.10, the characteristic classes w(pm) ∈H1(RPm) and w(pn) ∈ H1(RPn) are the generators of these cohomology groupsandH∗f(w(pn)) = w(pm). By Proposition 4.3.10 again, one has that 0 = H∗f(w(pn)

n+1) =w(pm)n+1 which implies that n ≥ m.

If m = n, observe that H∗pn : Hn(RPn)→ Hn(Sn) is the zero homomorphism

since pn is of local degree 2. The transfer exact sequence of (4.3.9), which isfunctorial, gives the commutative diagram

Hn(Sn)

H∗f

tr∗

≈// Hn(RPn)

H∗ f≈

Hn(Sn)tr∗

≈// Hn(RPn)

,

proving that deg f = 1.

As a corollary, we get the theorem of Borsuk-Ulam.

Corollary 6.2.2 (Borsuk-Ulam theorem). Let g : Sn → Rn be a continuousmap. Then, there exists z ∈ Sn such that g(z) = g(−z).

Proof. Otherwise, the map f : Sn → Sn−1 defined by

f(z) =g(z)− g(−z)|g(z)− g(−z)|

is continuous and odd, which contradicts Theorem 6.2.1.

A famous consequence is the ham sandwich theorem. For an early history ofthis theorem, see [14].

Corollary 6.2.3. Let A1, . . . , An be n bounded Lebesgue measurable subsetsof Rn. Then, there exists a hyperplane which bisects each Ai.

Proof. Identify Rn by an isometry with an affine n-subspace W of Rn+1 notpassing through the origin, and thus see A1, . . . , An ⊂ W . For each unit vectorv ∈ Rn+1, consider the half-space Q(v) = x ∈ Rn+1 | 〈v, x〉 > 0. Let gi : Sn → Rdefined by gi(v) = measure(Ai ∩ Q(v)). The maps gi are the coordinates of acontinuous map g : Sn → Rn. By Corollary 6.2.2, there is z ∈ Sn such thatg(z) = g(−z), which means that gi(z) = 1

2 measure(Ai). Then, P (z) ∩W is thedesired bisecting hyperplane, where P (z) is the orthogonal complement of z.

6.2.2. Non-singular and axial maps. A continuous map µ : Rm×Rm → Rk

is called non-singular if

(1) µ(αx, βy) = αβ µ(x, y) for all x, y ∈ Rm and all α, β ∈ R, and(2) µ(x, y) = 0 implies that x = 0 or y = 0.

6.2. APPLICATIONS 207

Non-singular maps generalize bilinear maps without zero divisors. They wereintroduced in [61], from where the results of this section are extracted. For refer-ences about the earlier literature, see also [61].

Non-singular maps are related to axial maps. A continuous map g : RPm ×RPm → RP ℓ, with ℓ ≥ m, is called axial if the restriction of g to each slice is nothomotopic to a constant map. By Corollary 3.8.4, this is equivalent to asking thatthese restrictions g1x : x × RPm → RP ℓ or g2x : RP

m × x → RP ℓ be homotopicto the inclusion RPm → RP ℓ. Using Corollary 3.8.4, this is equivalent to askingthat H∗gix(aℓ) = am, where aj is the generator of H1(RP j). By Corollary 4.7.4,we deduce that a continuous map g : RPm × RPm → RP ℓ is axial if and only if

(6.2.1) H∗g(aℓ) = 1× am + am × 1 .

The name of axial map appeared in [6] where references about the earlierliterature on the subject may be found. It started with the work of Stiefel andHopf [101].

Let µ : Rm × Rm → Rk be a non-singular map. By Point (2) of the definition,we get a continuous map µ : Sm−1 × Sm−1 → Sk−1 defined by

(6.2.2) µ(x, y) =µ(x, y)

|µ(x, y)| .

Point (1) above implies that µ descends to a map

(6.2.3) µ : RPm−1 × RPm−1 → RP k−1 .

For x ∈ RPm−1, the restriction of µx to the slice x × RPm−1 is covered bytwo-fold covering maps:

x × Sm−1

pm−1

// RPm−1 × Sm−1

// Sk−1

pk−1

x × RPm−1 //

µx

11RPm−1 × RPm−1µ // RP k−1

.

By Lemma 4.3.6 and Proposition 4.3.10, the characteristic classes w(pm−1) ∈H1(RPm−1) and w(pk−1) ∈ H1(RP k−1) are the generators of these cohomologygroups and H∗µx(w(pk−1)) = w(pm−1). Hence, µx is not homotopic to a constantmap. The same reasoning holds for the slices RPm−1 × x. Therefore, µ is axial.Conversely, if g : RPm−1×RPm−1 → RP k−1 is an axial map, it induces on universalcovers a map g : Sm−1 × Sm−1 → Sk−1 satisfying g(−x, y) = g(x,−y) = −g(x, y).The map µ : Rm × Rm → Rk defined by

µ(x, y) = |x| · |y| · g( x|x| ,y

|y| )

is a non-singular map. This proves the following lemma.

Lemma 6.2.4. The correspondence µ 7→ µ provides a bijection between non-singular maps Rm × Rm → Rk (up to multiplication by non-zero constants) andaxial maps RPm−1 × RPm−1 → RP k−1.

Let µ : Rm × Rm → Rk be a non-singular map. The restriction of µ to eachslice is odd. Hence, if a non-singular map µ : Rm×Rm → Rk exists, it follows fromTheorem 6.2.1 that k ≥ m. When m = k, the following proposition is attributedto Stiefel. For other proofs (see [153, Theorem 4.7] or Remark 8.6.7).


Proposition 6.2.5. Let µ : Rm × Rm → Rm be a non-singular map. Thenm = 2r.

In fact, by a famous result of J.F. Adams (see Remark 8.6.7), non-singularmaps Rm × Rm → Rm exist only if m = 1, 2, 4, 8.

Proof. We consider the associated axial map µ : RPm−1×RPm−1 → RPm−1

and denote by a the generator of Hm−1(RPm−1). The Kunneth theorem impliesthat the correspondence x 7→ 1× a and y 7→ a× 1 provides a GrA-isomorphism

Z2[x, y]/(xm, ym)

≈−→ H∗(RPm−1 × RPm−1) .

By (6.2.1), H∗µ(a) = x + y. Therefore, (x + y)m = 0. As xm and ym also vanish,one has

(x+ y)m =

m∑

i=0

(mi

)xiym−i =

m−1∑

i=1

(mi

)xiym−i .

This implies that(mi

)≡ 0 mod 2 for all i = 1, . . . ,m − 1 which, by Lemma 6.2.6

below, happens only if m = 2r.

For n ∈ N, denote its dyadic expansion in the form n =∑j∈J(n) 2

j where

J(n) ⊂ N.

Lemma 6.2.6 (Binomial coefficients mod 2). Let m, r ∈ N. Then(mr

)≡ 1 mod 2 ⇐⇒ J(r) ⊂ J(m) .

In other words,(mr

)≡ 1 mod 2 if and only if the dyadic expansion of r is a

sub-sum of that of m.

Proof. In Z2[x], the equation (1 + x)2 = 1 + x2 holds, whence (1 + x)2j

=

1 + x2j

. Thereforem∑

r=0

(mr

)xr = (1 + x)m =

∏

j∈J(m)

(1 + x)2j

=∏

j∈J(m)

(1 + x2j

) .

The identification of the coefficient of xr gives the lemma.

The technique of the proof of Proposition 6.2.5 also gives a result of H. Hopf,[101, Satz I.e].

Proposition 6.2.7. Let µ : Rm×Rm → Rk be a non-singular map. If m > 2r,then k ≥ 2r+1.

Proof. We already know that k > m. As in the proof of Proposition 6.2.5, weconsider the associated axial map µ : RPm−1×RPm−1 → RP k−1 and use the samenotations. We get the equation (x+y)k = 0 in Z2[x, y]

/(xm, ym), which, asm > 2r,

implies that(ki

)= 0 for all 1 ≤ i ≤ 2r. By Lemma 6.2.6, the dyadic expansion

k =∑

j kj2j must satisfy kj = 0 for j ≤ r, which is equivalent to k ≥ 2r+1.

Remark 6.2.8. There always exists a non-singular map µ : Rm×Rm → R2m−1

(see [61, § 5]).

We finish this section by mentioning two results relating non-singular or axialmaps to the immersion problem and the topological complexity of projective spaces.The following proposition was proven in [6].

6.3. THE HOPF INVARIANT 209

Proposition 6.2.9. There exists an axial map g : RPn×RPn → RP k (k > n)if and only if there exists an immersion of RPn in Rk.

We shall not talk about the large literature and the many results on the problemof immersing or embedding RPm in Rq (see however Proposition 9.5.23). Tablesand references are available in [36]. The existence of non-singular maps is alsorelated to the topological complexity of projective spaces. The following is provenin [61, Theorem 6.1].

Theorem 6.2.10. The topological complexity TC (RPn) is equal to the smallestinteger k such that there is a non-singular map µ : Rn+1 × Rn+1 → Rk.

Symmetric non-singular maps (i.e. µ(x, y) = µ(y, x)) are, in some range, re-lated to embeddings of RPn in Euclidean spaces or to the symmetric topologicalcomplexity. For results and references, see [68].

6.3. The Hopf invariant

6.3.1. Definition. Let f : S2m−1 → Sm be a continuous map. The spaceCf = D2m ∪f Sm is a CW-complex with one cell in dimension 0, m and 2m.Consider the cup-square map 2

m : Hm(Cf )→ H2m(Cf ), given by 2m (x) = x

x. The Hopf invariant Hopf (f) ∈ Z2 is defined by

Hopf (f) =

1 if 2

m is surjective for Cf .

0 otherwise.

The space Cf depends only on the homotopy class of f (see, e.g. [82, Proposi-tion 0.18]), then so does the Hopf invariant. A constant map has Hopf invariant 0.The computation of the cohomology ring of the various projective planes in § 6.1shows that the 2-fold cover S1 → S1 as well as the other Hopf maps h3,2 : S

3 → S2,h7,4 : S

7 → S4 and h15,8 : S15 → S8 have Hopf invariant 1.

Our Hopf invariant is just the mod 2 reduction of the classical integral Hopfinvariant defined in e.g. [82, § 4.B]. The form of our definition is motivated byextending the statements to the case m = 1, usually not considered by authors.

Note that Hopf defined his invariant in 1931–35 [99, 100], before the inventionof the cup product. He used linking numbers (see § 6.3.4 below).

For m = 1, recall the bijection DEG : [S1, S1]≈−→ Z given in (3.2.3).

Proposition 6.3.1. Let f : S1 → S1. Then

Hopf (f) =

0 if DEG(f) ≡ 0 mod 4

1 otherwise.

Proof. Let C = Cf . If DEG (f) is odd, then deg(f) = 1 by Proposition 3.2.9.The computation of the cellular cohomology of C using Lemma 3.5.4 shows thatH∗(C) = 0, so2

1 is surjective and Hopf (f) = 1. If DEG (f) = 2k, then, H1(C) ≈Z2 ≈ H2(C). Consider the 2-fold covering p : C → C whose characteristic class isthe non-zero element a ∈ H1(C). Its transfer exact sequence looks like

0→ H0(C)︸︷︷︸Z2

→ H1(C)︸︷︷︸Z2

H∗p−−−→ H1(C)tr∗−−→ H1(C)︸︷︷︸

Z2

−a−−−→ H2(C)︸︷︷︸Z2


By Van Kampen’s theorem, π1(C) is cyclic of order 2k. Thus, π1(C) is cyclic of

order k and H1(C) = Z2 if k is even while it vanishes if k is odd. The propositionthus follows from the above transfer exact sequence.

6.3.2. The Hopf invariant and continuous multiplications. A classicalconstruction associates a map fκ : S

2m−1 → Sm to a “continuous multiplication”κ : Sm−1 × Sm−1 → Sm−1. Let D = Dm and S = Sm−1 = ∂D. Divide Sm intothe upper and lower hemisphere: Sm = B+ ∪ B− ⊂ Rm × R, with B+ ∩ B− =Sm ∩ Rm × 0 = S. Using the decomposition

∂(D ×D) = ∂D ×D ∪ D × ∂D = S ×D ∪ D × S ,the map fκ : ∂(D ×D)→ Sm is defined, for x, y ∈ S et t ∈ [0, 1], by

fκ(tx, y) = (t κ(x, y),√

1− t2) and fκ(x, ty) = (t κ(x, y),−√1− t2) .

For u, v,∈ Sm−1, we consider the hypothesis H(u, v) on κ:H(u, v): κ(u,−x) = −κ(u, x) and κ(−x, v) = −κ(x, v) for all x ∈ Sm−1.

For example, H(u, v) holds for all u, v ∈ Sm−1 if κ = µ, the map associatedusing (6.2.2) to a non-singular map µ : Rm×Rm → Rm. Also, H(e, e) is satisfied ife is a neutral element for κ.

Example 6.3.2. Let κ : S0×S0 → S0 be the usual sign rule (S0 = S = ±1).Then D = [−1, 1] and the map fκ : ∂(D ×D)→ S1 is pictured in Figure 6.1.

(−1,−1) (1,−1)

(1, 1)(−1, 1) A

C

B D

fκ1−1

fκ(A) = fκ(C)

fκ(B) = fκ(D)

Figure 6.1. The map fκ for the usual sign rule.One sees that fκ has degree 2. By Proposition 6.3.1, this implies that Hopf (fκ) =

1. Actually, the map fκ is topologically conjugate to the projection S1 →→ S1/x ∼−x, so Cfκ is homeomorphic to RP 2.

The same exercise may be done for the other possible multiplications on S0.By changing the sign of κ if necessary, we may assume that κ(1, 1) = 1. There arethen 8 cases.

κ(1, 1) κ(−1, 1) κ(−1,−1) κ(1,−1) DEG (fκ) Hopf (fκ) satisfies

1 1 1 1 1 0 0

2 1 1 1 -1 1 1 H(1,−1)

3 1 1 -1 1 -1 1 H(−1,−1)

4 1 1 -1 -1 0 0

5 1 -1 1 1 1 1 H(−1, 1)

6 1 -1 1 -1 2 1 H(u, v) ∀ u, v

7 1 -1 -1 1 0 0

8 1 -1 -1 -1 1 1 H(1, 1)

One sees that Hopf (fκ) = 1 if and only if H(u, v) is satisfied for some u, v ∈ S0.This is partially generalized in the following result.


Proposition 6.3.3. Let κ : Sm−1×Sm−1 → Sm−1 be a continuous multiplica-tion. Suppose that H(u, v) is satisfied for some u, v ∈ Sm−1. Then Hopf (fκ) = 1.

Proof. The casem = 1 was done in Example 6.3.2. We may thus assume thatm > 1. The following proof is inspired by that of [81, Lemma 2.18]. Let f = fκ.Consider the commutative diagram

Hm(Cf )⊗Hm(Cf ) // H2m(Cf )

Hm(Cf , B+)⊗Hm(Cf , B−)

≈

OO

φ∗⊗φ∗

// H2m(Cf , Sm)

≈

OO

≈ Φ∗

Hm(D ×D,S ×D)⊗Hm(D ×D,D × S) // H2m(D ×D, ∂(D ×D))

Hm(D,S)⊗Hm(D,S)

×

≈

33π∗1⊗π

∗2 ≈

OO

where φ : D × D → Cf is the characteristic map for the 2m-cell of Cf and φ∗ =H∗φ. The cross-product map at the bottom of the diagram is an isomorphismby the relative Kunneth theorem 4.6.10. Hence, Hopf (fκ) = 1 if and only if thehomomorphism φ∗ ⊗ φ∗ in the left column is an isomorphism. By symmetry, it isenough to prove that φ∗ : Hm(Cf , B+)→ Hm(D×D,S×D) is not zero. Considerthe commutative diagram

Hm(Cf , B+)

φ∗

≈ // Hm(Sm, B+)

f∗κ

≈ // Hm(B−, S)

f∗κ

Hm(D ×D,S ×D)≈ // Hm(∂(D ×D), S ×D)

≈ // Hm(D × S, S × S) .

The left horizontal maps are isomorphism since m > 1 and the right ones byexcision. It then suffices to prove that f∗κ : H

m(B−, S)→ Hm(D× S, S × S) is notzero.

As the restriction of fκ to S ×S is equal to κ, one has a commutative diagram

Hm−1(S)

κ∗

≈ // Hm(B−, S)

f∗κ

Hm−1(D × S) // Hm−1(S × S) δ∗ // Hm(D × S, S × S)

Hm−1(S)

π∗2≈

OOπ∗2

66♠♠♠♠♠♠♠♠♠♠

where the second line is the cohomology sequence of the pair (D × S, S × S). Letu, v ∈ Sm−1 such that H(u, v) is satisfied. Let s1, s2 : S → S × S be the sliceinclusions given by s1(x) = (x, v) and s2(x) = (u, x). The composition κsi isthus an odd map. Therefore, Theorem 6.2.1 implies that H∗(κsi)(a) = a. By


Corollary 4.7.3, one deduces that

(6.3.1) H∗κ(a) = 1× a+ a× 1 .

On the other hand, ker δ∗ = Imageπ∗2 = 0,1× a. Therefore, f∗κ does not vanish.

In the proof of Proposition 6.3.3, the hypothesis on H(u, v) is only used toobtain Equation (6.3.1). Therefore, one has the following proposition.

Proposition 6.3.4. Let κ : Sm−1 × Sm−1 → Sm−1 be a continuous multi-plication. Suppose that H∗κ(a) = 1 × a + a × 1 for a ∈ Hm−1(Sm−1). ThenHopf (fκ) = 1.

6.3.3. Dimension restrictions. We shall prove in Corollary 8.6.4 that, ifthere exists a map f : S2m−1 → Sm with Hopf invariant 1, then m = 2r. Actually,m = 1, 2, 4, 8 by a famous theorem of Adams (see Theorem 8.6.6). This theoremimplies the following result.

Proposition 6.3.5. Let Sqi−→ E

π−→ Sp be a locally trivial bundle. Supposethat H∗(E) ≈ H∗(Sp+q). Then q = p− 1 and p = 1, 2, 4 or 8.

Proof. If p = 1 and q > 0, then H∗(E) is GrV-isomorphic to H∗(S1) ⊗H∗(Sq) by the argument of Example 4.7.45. Thus, we must have q = 0 and π is anon-trivial double cover of S1.

Let us suppose that p ≥ 2. If H∗(E) ≈ H∗(Sp+q), then H∗i is not surjective;otherwise, H∗(E) is GrV-isomorphic to H∗(Sp) ⊗ H∗(Sq) by the Leray-Hirschtheorem. The Wang exact sequences (see Proposition 4.7.43)

· · · → Hq(E)H∗i−−→ Hq(Sq)

Θ−→ Hq+1−p(Sq)→ · · · .then implies that q + 1− p = 0 (since p > 1). Therefore, q = p− 1.

The bundle gluing map φ : Sq × Sq → Sq (see p 159) may thus be seen asa continuous multiplication, to which a map fφ : S

2p−1 → Sp may be associated

using (6.1.2). We shall prove that Hopf (fφ) = 1. By Theorem 8.6.6, this impliesthat p = 1, 2, 4 or 8.

Let a ∈ Hq(Sq) be the generator. The restriction of φ to a slice x×Sq beinga homeomorphism, one has, using Lemma 4.7.2, that

H∗φ(a) = 1× a+ λ(1× a)for some λ ∈ Z2. As Θ 6= 0 and p > 1, one gets from Proposition 4.7.46 that

0 6= e×Θ(a) = H∗φ(a) + 1× a ,where e ∈ Hp(Sp) is the generator. Therefore, H∗φ(a) = a × 1 + 1 × a. ByProposition 6.3.4, this implies that Hopf (fφ) = 1.

Examples 6.3.6. Consider the bundle S1 → E → S2 with gluing map φ(u, z) =ukz. The total space E, obtained by gluing two copies of D2 × S1 using the mapφ, is then a lens space with fundamental group of order k. Thus, if k is odd,E satisfies the hypotheses of Proposition 6.3.5. Other famous examples are thebundles S3 → E → S4 which were used by J. Milnor to produce his exotic 7-spheres [145, § 3]. Indeed, with a well chosen gluing map, the total space E is asmooth 7-manifold homeomorphic but not diffeomorphic to S7.


6.3.4. The Hopf invariant and linking numbers. Let f : S2m−1 → Sm

be a smooth map. Let y, y′ ∈ Sm be two distinct regular values of f . ThenQ = f−1(y) and Q′ = f−1(y′) are two disjoint closed submanifolds of S2m−1,both of dimension m−1. Therefore, their linking number l(Q,Q′) ∈ Z2 (see § 5.4.4)is defined, at least if m > 1 (see also Remark 6.3.8 below).

Proposition 6.3.7. If m > 1, then Hopf (f) = l(Q,Q′).

Actually, l(Q,Q′) is the original definition by H. Hopf of his invariant [99, 100].Proposition 6.3.7 goes back to the work of N. Steenrod [182], after which thedefinition of Hopf (f) with the cup product in Cf was gradually adopted.

Proof. Let Σ = S2m−1. We consider the mapping cylinder Mf of f

Mf = [(S2m−1×I) ∪Sm]/(x, 0) ∼ f(x) | x ∈ S2m−1 .

The correspondence (x, t) 7→ f(x) descends to a retraction by deformation ρ : Mf →Sm. We identify Σ with the subspace Σ× 1 of Mf . The mapping cone Mf of f ,defined as

Mf =Mf ∪Σ CΣ ≈Mf ∪Σ D2m ,

whereCΣ ≈ D2m is the cone over Σ, is homotopy equivalent to the CW-complex Cf .We first introduce some material in order to compute l(Q,Q′) using Lemma 5.4.23.

Let W0 be a closed tubular neighbourhood of y in Sm−y′ and let V0 be a closedtubular neighbourhood of y in intW0 (W0 and V0 are just m-disks). Let y′ ∈ V ′0 ⊂W ′0 be a symmetric data for y′ withW0∩W ′0 = ∅. Let B0 = W0,W

′0, S

m−(V0∪V ′0 ).As y and y′ are regular values of f , we may assume, providedW0 and W ′0 are smallenough, that W = f−1(W0) and V = f−1(V0) are nested tubular neighbourhoodsof Q and thatW ′ = f−1(W ′0) and V

′ = f−1(V ′0 ) are nested tubular neighbourhoodsof Q′. Let B = f−1(B0) = W,W ′,Σ− (V ∪ V ′).

Let us briefly repeat the preliminary constructions for Lemma 5.4.23 (see p. 196for more details), in our context, with the dimensions q = q′ = m − 1 and s =2m− 1. Let c0 ∈ Zm(W0,W0 − intV0) represent the Poincare dual class of y inHm(W0,W0 − intV0) ≈ Hm(V0,BdV0) and let c′0 ∈ Zm(W ′0,W ′0 − intV ′0) representthe Poincare dual class of y′ in Hm(W ′0,W

′0 − intV ′0) ≈ Hm(V ′0 ,BdV

′0). Let

c0, c′0 ∈ ZmB0

(Sm) be their zero extensions. Then c = C∗f(c0) ∈ Zm(W,W − intV )represents the Poincare dual class of Q in Hm(W,W − intV ) ≈ Hm(V,BdV ) andc′ = C∗f(c′0) ∈ Zm(W ′,W ′ − intV ′) represents the Poincare dual class of Q′ inHm(W ′,W ′ − intV ) ≈ Hm(V ′,BdV ′). Also, c = C∗f(c0) ∈ ZmB (Σ) and c′ =

C∗f(c′0) ∈ ZmB (Σ) are the zero extensions of c and c′. Choose a ∈ Cm−1B (Σ)such that δa = c. Let ν ∈ ZB2m−1(Σ) represent [Σ]B in HB2m−1(Σ) ≈ H2m−1(Σ).According to Lemma 5.4.23, one has l(Q,Q′) = 〈a c′, ν〉. We note that a c′ ∈Z2m−1B (Σ). Indeed, for any σ ∈ S2m−1(Σ), one has

(6.3.2) 〈δ(a c′), σ〉 = 〈c c′, σ〉 = 0

since the support of c is in W and that of c′ is in W ′. Therefore, δ(a c′) = 0and a c′ represents a cohomology classes |a c′| ∈ H2m−1

B (Σ) (in this proof, weuse the notation | | for the cohomology class of a cocycle). The equality l(Q,Q′) =〈a c′, ν〉 is equivalent to

(6.3.3) |a c′| = l(Q,Q′) [Σ]♯B ,


an equality holding in H2m−1B (Σ), where [Σ]♯B is the generator of H2m−1

B (Σ) ≈H2m−1(Σ) = Z2.

Let B1 = ρ−1(B). The inclusion i : Σ → Mf induces a morphism of cochaincomplexes C∗i : C∗B1

(Mf ) → C∗B(Σ) whose kernel is denoted by C∗B1(Mf ,Σ). Note

that C∗B1(Mf , ∅) = C∗B1

(Mf ) and so the inclusion C∗B1(Mf ,Σ) → C∗B1

(Mf) coincideswith C∗j, the morphism induced by the pair inclusion j : (Mf , ∅) → (Mf ,Σ) (seeRemark 3.1.26). One has the commutative diagram

(6.3.4)

0 // C∗B1(Mf ,Σ)

C∗j // C∗B1(Mf )

C∗i //OO

C∗ρ

C∗B(Σ)// 0

C∗B0(Sm)

C∗f

99ssssssss

where the top row is an exact sequence of cochain complexes. This sequence givesrise to a connecting homomorphism δ∗ sitting in the exact sequence

H2m−1B1

(Mf )H∗i // H2m−1

B (Σ)δ∗ // H2m

B1(Mf ,Σ)

H∗j // H2mB1

(Mf ) .

As m > 1, one has H2m−1B1

(Mf ) ≈ H2m−1(Mf ) = 0 and H2mB1

(Mf ) ≈ H2m(Mf ) =

0. Therefore δ∗ : H2m−1B (Σ) → H2m

B1(Mf ,Σ) is an isomorphism. Let b = δ∗([Σ]♯B)

be the generator of H2mB1

(Mf ,Σ). By (6.3.3), the linking number l(Q,Q′) is thendetermined by the equation

(6.3.5) δ∗(|a c′|) = l(Q,Q′) b .

To compute δ∗(|a c′|), write δM for the coboundary operator in C∗B1(Mf ).

Let c1 = C∗ρ(c0) and c′1 = C∗ρ(c′0), both in ZmB1(Mf ). Let a1 ∈ Cm−1(Mf) such

that C∗i(a1) = a. By the commutativity of diagram (6.3.4), one has C∗i(a1 c′1) = a c′. Then δM (a1 c′1) is a cocycle in kerC∗i, so there is a uniqueu ∈ Z2m

B1(Mf ,Σ) such that C∗j(u) = δM (a1 c′1). As in Lemma 2.7.1, one has

(6.3.6) |u| = δ∗(|a c′|) .The cohomology class |u| may be described in another way. As for (6.3.2), one hasc1 c′1 = 0 for support reasons. Therefore,

(6.3.7) (δM (a1) + c1) c′1 = δM (a1) c′1 = δM (a1 c′1) .

Now, C∗i(δM (a1)+ c1) = 0, thus there is a unique w ∈ ZmB1(Mf ,Σ) with C

∗j(w) =δM (a1)+ c1. The first cup product of (6.3.7) may be understood as relative cochaincup product giving rise to a relative cohomology cup product

H∗B1(Mf ,Σ)×H∗B1

(Mf )−→ H∗B1

(Mf ,Σ)

analogous to that of Lemma 4.1.14 (in the case Y2 = ∅). Equation (6.3.6) isequivalent to

(6.3.8) |w| |c′1| = δ∗(|a c′|)and, using (6.3.6), we get the equality

(6.3.9) |w| |c′1| = l(Q,Q′) b

holding in H2mB1

(Mf ,Σ).


Now, under the isomorphism Hm(Sm)≈−→ Hm

B0(Sm) (due to the small sim-

plex theorem 3.1.34), the cohomology class |c0| corresponds to the Poincare dualPD(y) ∈ Hm(Sm), which is [Sm]♯. Analogously, |c′0| also corresponds to [Sm]♯.

Hence, under the isomorphism Hm(Mf )≈−→ Hm

B1(Mf ), |c1|, |c′1| and H∗j(|w|) all

correspond to H∗ρ([Sm]♯), that is the generator e of Hm(Mf ) ≈ Z2. The diagram

(6.3.10)

HmB1(Mf )×Hm

B1(Mf ,Σ)

//OO≈

H2mB1

(Mf ,Σ)OO≈

Hm(Mf )×Hm(Mf ,Σ) //

OO≈

H2m(Mf ,Σ)OO≈

Hm(Mf )×Hm(Mf , CΣ) //

≈

H2m(Mf , CΣ)

≈

Hm(Mf )×Hm(Mf ) // H2m(Mf )

is commutative, where the vertical arrows are the obvious ones or induced by theinclusions (the commutativity of the bottom square is the content of the singular

analogue of Lemma 4.1.8). Let k : Mf → Mf denote the inclusion. ThenH∗k (e) = e,

the generator of Hm(Mf ) ≈ Z2. Equation (6.3.9), obtained using the top lineof (6.3.10), becomes, using the bottom line

(6.3.11) e e = l(Q,Q′) b

where b is the generator of H2m(Mf ) ≈ Z2. But, as m > 1, the equality e e =

Hopf (f) b holds true, by definition of the Hopf invariant. Therefore, Hopf (f) =l(Q,Q′).

Remark 6.3.8. For a map f : S1 → S1 with even degree, the equality Hopf (f) =l(Q,Q′) holds true, using Proposition 6.3.1 (see Exercise 6.11). When deg f is odd,the linking number l(Q,Q′) is not defined. Indeed, both Q and Q′ have an oddnumber of points and condition (3) of p. 194 is not satisfied.


6.1. What is the Lusternik-Schnirelmann category of KPn for K = C or H?

6.2. Compute the cohomology ring of CPn and HPn, using the Gysin exact se-quence for the Hopf bundles. [Hint: like in § 4.3.4.]6.3. For K = C or H, prove that H∗(KP∞)→ H∗(RP∞) is injective.

6.4. Let f : X → Y be a map. The double mapping cylinder CCf of f is the unionof two copies of the mapping cylinder Cf of f glued along X . Compute H∗(CCj)where j is the inclusion of CP∞ → HP∞.

6.5. Prove that X = S4× S4 and Y = HP 2 ♯HP 2 have the same Poincare polyno-mial. Are H∗(X) and H∗(Y ) GrA-isomorphic?

6.6. For any positive integer n, construct a vector bundle ξ of rank n over a closedn-dimensional manifold such that e(ξ) 6= 0.


6.7. Let X be a CW-complex of dimension n = 1, 2, 4, 8 and let a ∈ Hn(X). Provethat there exists a vector bundle ξ over X with e(ξ) = a.

6.8. Prove that there is no continuous injective map f : Rn → Rk if n > k. [Hint:use the Borsuk-Ulam theorem.]

6.9. Check the table of p. 210.

6.10. Show that the Hopf vector bundle over KP 1 ≈ Sd (d = dimR K) cannot bethe normal bundle of an embedding of Sd into a manifold M of dimension 2d withHd(M) = 0.

6.11. Let f : S1 → S1 be a smooth map with even degree. Show that the Hopfinvariant of f is equal to the linking number of the inverse image of two regularvalues of f , as in Proposition 6.3.7. [Hint: use Proposition 6.3.1.]

6.12. Using the linking numbers and Proposition 6.3.7, show that the various Hopfmaps have Hopf invariant 1. [Hint: use Formula (6.1.3) and Exercise 5.18.]

6.13. Let g : S2m+1 → Sm be a continuous map, as well as f : Sm → Sm andh : S2m+1 → S2m+1. Prove that Hopf (f gh) = deg(h) deg(f)2 Hopf (g). (Re-mark: of course, deg(f)2 ≡ deg(f) mod 2 but the formula is the one which is validfor the cohomology with any coefficients.)

6.14. Let f : S2m−1 → Sm be a smooth map and let y ∈ Sm be a regular valuefor f . The closed (m − 1)-manifold Q = f−1(y) bounds a compact manifoldW (see Exercise 9.14). As S2m−1 − pt is contractible, the inclusion of Q intoS2m−1 extends to a smooth map j : W → S2m−1 (see Exercise 3.2). We thus geta homomorphism H∗(f j) : Hm(W,Q) → Hm(Sm, y). As both these homologygroups are isomorphic to Z2, this defines a degree for f j, as in (2.5.4). Prove thatHopf (f) = deg(f j).

CHAPTER 7

Equivariant cohomology

In ordinary life, the symmetries of an object (like a ball or a cube) help us toapprehend it. The same should happen in topology when studying spaces with sym-metries, i.e. endowed with actions of topological groups. Equivariant cohomologyis one tool for such a purpose.

Our aim here is mostly to develop enough material needed in the forthcomingchapters. For instance, the definition and most properties of the Steenrod squaresuse equivariant cohomology for spaces with involution. This case is treated in detailin § 7.1, at an elementary level and with ad hoc techniques. A second section dealswith Γ-spaces for any topological group Γ (the proof of the Adem relations requiresΓ-equivariant cohomology with Γ the symmetric group Sym4). Equivariant crossproducts, treated in § 7.4, will also be used. Only § 7.3 is written uniquely forits own interest, devoted to some simple form of localization theorems and Smiththeory. A final section presents the equivariant Morse-Bott theory, used in § 9.5 tocompute the cohomology of flag manifolds (see also § 10.3.5). For further readingon equivariant cohomology, see e.g. [37, 9, 103].

7.1. Spaces with involution

An involution on a topological space X is a continuous map τ : X → X suchthat τ τ = id. The letter τ is usually used for all encountered involutions. Wealso use the symbol τ for the non-trivial element of the cyclic group G = id, τ oforder 2; an involution on X is thus equivalent to a continuous action of G on Xand a space with involution is equivalent to a G-space, i.e. a space together withan action of G. We often pass from one language to the other. If X is a G-space,its fixed point subspace XG is defined by

XG = x ∈ X | τ(x) = x .As G has only two elements, the complement of XG is the subspace where theaction is free.

A continuous map f : X → Y between G-spaces is a G-equivariant map, orjust a G-map if it commutes with the involutions: f τ = τ f . Two G-mapsf0, f1 : X → Y are G-homotopic if there is a homotopy F : X × I → Y connectingthem which is a G-map. Here, the involution on X × I is τ(x, t) = (τ(x), t). Thispermits us to define the notion of G-homotopy equivalence and of G-homotopy type.For instance, a G-space is G-contractible if it has the G-homotopy type of a point.

Let X be a G-space. A CW-structure on X with set of n-cells Λn is a G-CW-structure if the following condition is satisfied: for each integer n, there is aG-action on Λn and a G-equivariant global characteristic map ϕn : Λn ×Dn → X,where the G-action on Λn × Dn is given by τ(λ, x) = (τ(λ), x). In particular, ifλ ∈ Λn satisfies τ(λ) = λ, then τ restricted to λ×Dn is the identity. These cells

217

218 7. EQUIVARIANT COHOMOLOGY

are called the isotropic cells; they form a G-CW-structure for XG. The other cells,the free cells, come in pairs (λ, τ(λ)). A G-space endowed with a G-CW-structureis a G-CW-complex, or just a G-complex. Observe that, if X is a G-complex,then the quotient space X/G inherits a CW -structure (with set of n-cells equalto Λn/G) for which the quotient map is cellular. A smooth G-manifold admits aG-CW-structure, in fact a G-triangulation (see [106]).

Example 7.1.1. LetX = Sn (n ≤ ∞) be the standard sphere endowed with theCW-structure where them-skeleton is Sm and having twom-cells in each dimensionm ≤ n (see Example 3.4.4). This is a G-CW-structure for the free involution givenby the antipodal map z 7→ −z. The quotient space X/G is RPn with its standardCW-structure.

Let X be a space with an involution τ . The Borel construction XG, also knownas the homotopy quotient, is the quotient space

(7.1.1) XG = S∞ ×G X = (S∞ ×X)/∼

where∼ is the equivalence relation (z, τ(x)) ∼ (−z, x). IfX and Y areG-spaces andif f : Y → X is a continuous G-equivariant map, the map id×f : S∞×Y → S∞×Xdescends to a map fG : YG → XG. This makes the Borel construction a covariantfunctor from the category TopG to Top, where TopG is the category of G-spacesand G-equivariant maps. Using the obvious homeomorphism between (X × I)Gand XG× I, a G-homotopy between two G-maps f0 and f1 : X → Y descends to ahomotopy between f0

G and f1G. Hence, XG and YG have the same homotopy type

if X and Y have the same G-homotopy type.Let p : S∞ → RP∞ be the quotient map (this is a 2-fold covering projection).

A map p : XG → RP∞ is then given by p([z, x]) = p(z). Observe that p coincideswith the map fG : XG → ptG = RP∞ induced by the constant map X → pt.

Example 7.1.2. Suppose that the involution τ is trivial, i.e. τ(x) = x for allx ∈ X . The projection S∞ ×X → X then descends to XG → X . Together with

the map p, this gives a homeomorphism XG≈−→ RP∞ ×X .

Lemma 7.1.3. (1) The map p : XG → RP∞ is a locally trivial fiber bundlewith fiber homeomorphic to X.

(2) If f : Y → X is a G-equivariant map, then the diagram

YGp

##

fG // XG

p

RP∞

is commutative.(3) If τ has a fixed point, then p admits a section. More precisely, the choice

of a point v ∈ XG provides a section sv : RP∞ → XG of p.(4) The quotient map S∞ × X → XG is a 2-fold covering admitting p as a

characteristic map.

Proof. We use that p : S∞ → RP∞ is a principal G-bundle, i.e. a 2-foldcovering. Denote by z = (z0, z1, . . . ) the elements of S∞. The set Vi = z ∈ S∞ |zi 6= 0 is an open subspace of S∞. As p is an open map, Ui = p(Vi) is an openset of RP∞. A trivialization ψi : Vi → Ui×±1 is given by ψi(z) = (p(z), zi/|zi|).

7.1. SPACES WITH INVOLUTION 219

Using the group isomorphism ±1 ≈−→ G, this gives a trivialization ψi : Vi →Ui × G. Now, ψi × id : Vi × X

≈−→ Ui × G × X descends to a homeomorphism

p−1(Ui)≈−→ Ui × (G ×G X). Here, G ×G X denotes the quotient of G × X by

the equivalence relation (g, τ(x)) ∼ (gτ, x). But the map x 7→ (id, x) provides ahomeomorphism from X onto G×GX . This proves Point (1). This also shows that,over p−1(Ui), the map S∞ × X → XG looks like the projection G × p−1(Ui) →p−1(Ui). Therefore, S

∞×X → XG is a 2-fold covering, with the product involutionτ×(z, x) = (−z, τ(x)) as deck transformation. The diagram

S∞ ×X

projS∞ // S∞

XG

p // RP∞

is commutative and projS∞(τ×(y)) = −projS∞(y). By Lemma 4.3.4, this impliesthat p is a characteristic map for the covering S∞ × X → XG. Point (4) is thusestablished.

Point (2) is obvious from the definitions. For Point (3), let v ∈ XG. ByPoint (2), the inclusion i : v → X gives rise to a commutative diagram

vG≈

$$

iG // XG

p

RP∞

.

Hence, iG provides a section sv : RP∞ → XG of p, depending on the choice of thefixed point v.

The projection q : S∞ ×X → X seen in Example 7.1.2 descends to q : XG →X/G.

Lemma 7.1.4. Let X be a free G-space such that X is Hausdorff. Then, theGrA-morphism H∗q : H∗(X/G)→ H∗(XG) is an isomorphism. Moreover, if X isa free G-complex, then the map q : XG → X/G is a homotopy equivalence and themap p : XG → RP∞ is homotopic to the composition of q with a characteristic mapfor the covering X → X/G.

Proof. If X is Hausdorff, such a projection X → X/G is a 2-fold coveringand X/G is Hausdorff. Over a trivializing open set U ⊂ X/G, this covering isequivalent to G× U → U . Then (G× U)G ≈ S∞ × U since any class has a unique

representative of the form (z, id, u) ∈ S∞ ×G× U . Hence, XGq−→ X/G is a locally

trivial bundle with fiber S∞. As H∗(S∞) = 0, the map q∗ : H∗(X/G)→ H∗(XG)is a GrA-isomorphism by Corollary 4.7.21. Actually. as S∞ is contractible [82,example 1.B.3 p. 88], the homotopy exact sequence of the bundle [82, Theorem 4.41and Proposition 4.48] implies that q is a weak homotopy equivalence. If X is a G-complex, then X/G is a CW -complex. Also, S∞ × X is a free G-complex andthus XG is a CW-complex. Therefore, a weak homotopy equivalence is a homotopyequivalence by the Whitehead theorem [82, Theorem 4.5]. Also, again since S∞

is contractible and X/G is a CW-complex, a direct proof that q is a homotopyequivalence is available using [42, Theorem 6.3].


Let f : X/G→ RP∞ be a characteristic map for the covering X → X/G. Thediagram

S∞ ×Xcovering

// X

covering

f // S∞

XG

q // X/Gf // RP∞

is commutative and the upper horizontal arrows commute with the deck involutions.By Lemma 4.3.4, this implies that f q is a characteristic map for the covering S∞×X → XG. By Lemma 7.1.3, so is p. By Corollary 4.3.3, two characteristic maps ofa covering are homotopic. Therefore, the maps p and f q are homotopic.

Corollary 7.1.5. Let X be a finite dimensional G-complex. Then, the fol-lowing conditions are equivalent.

(1) X has a fixed point.(2) The morphism H∗p : H∗(RP∞)→ H∗(XG) is injective.

Proof. If X has a fixed point, then p admits a section by Lemma 7.1.3,so H∗p is injective. If X has no fixed point, then X is a free G-complex and,by Lemma 7.1.4, H∗(XG) ≈ H∗(X/G). Also, X/G is a finite dimensional CW-complex, so H∗p is not injective.

Let X be a space with an involution τ and let Y ⊂ X be an invariant sub-space. Then YG ⊂ XG. The (relative) G-equivariant cohomology H∗G(X,Y ) is thecohomology algebra

H∗G(X,Y ) = H∗(XG, YG) .

We shall mostly concentrate on the absolute case H∗G(X) = H∗(XG) = H∗G(X, ∅).The map p : XG → RP∞ induces aGrA-homomorphism p∗ : H∗(RP∞)→ HG(X).By Proposition 6.1.1, H∗(RP∞) is GrA-isomorphic to the polynomial ring Z2[u],where u is a formal variable in degree 1. Hence, the GrA-homomorphism p∗ giveson H∗G(X) a structure of Z2[u]-algebra. In particular, H∗G(pt) = Z2[u].

As an important example, let us consider the case of a G-space Y with Y = Y G,i.e. the involution τ is trivial. As seen in Example 7.1.2, we get an identificationYG = RP∞ × Y . By the Kunneth theorem,

(7.1.2) H∗G(Y ) ≈ Z2[u]⊗H∗(Y ) ≈ H∗(Y )[u] .

The GrA-homomorphismH∗(Y )→ H∗G(Y ) induced by the projection RP∞×Y →Y corresponds to the inclusion of the “ring of constants” H∗(Y ) into H∗(Y )[u].

The functoriality of the Borel construction and of the cohomology algebra, to-gether with Point (2) of Lemma 7.1.3, says that, if f : Y → X is a G-equivariantmap between G-spaces, then H∗fG : H∗G(X) → H∗G(Y ) is a GrA-homomorphismcommuting with the multiplication by u. We are then driven to consider the cat-egory GrA[u] whose objects are graded Z2[u]-algebras and whose morphisms areGrA-homomorphisms commuting with the multiplication by u. Hence, the cor-respondence X 7→ H∗G(X) is a contravariant functor from TopG to GrA[u]. Iff : Y → Y is a G-equivariant map between trivial G-spaces (i.e., any continuous


map), the following diagram is commutative:

(7.1.3)

H∗(Y )[u]

H∗f [u]

≈ // H∗G(Y )

H∗Gf

H∗(Y )[u]≈ // H∗G(Y ) .

Choosing a point z ∈ S∞ provides, for each G-space X , a map iz : X → XG

defined by iz(x) = [z, x]. As S∞ is path-connected, the homotopy class of iz doesnot depend on z. Therefore, we get a well defined GrA-homomorphism

(7.1.4) ρ : H∗G(X)→ H∗(X)

given by ρ = H∗iz for some (and so any) z ∈ S∞. We can call ρ the forgetfulhomomorphism (it forgets the G-action). Observe that ρ is functorial. Indeed, iff : X → X is a G-equivariant map, the diagram

X

f

iz // XG

fG

Xiz // XG

is commutative, and so is the diagram

(7.1.5)

H∗(X)

H∗f

H∗G(X)ρXoo

H∗Gf

H∗(X) H∗G(X)ρXoo .

If Y is a G-space with Y = Y G, we get, using the GrA[u]-isomorphismof (7.1.2), the commutative diagram

(7.1.6)

H∗(Y )[u]

ev0 %%

≈ // H∗G(Y )

ρzztttttttt

H∗(Y )

where ev0 is the evaluation of a polynomial at u = 0, i.e. the unique algebrahomomorphism extending the identity on H∗(Y ) and sending u to 0.

We now explain how some information on H∗G(X) may be obtained from trans-fer exact sequences. Observe that ρ = H∗ρ where ρ : X → XG is the composition

ρ : Xslice−−−→ X × S∞ → XG .

As S∞ is contractible [82, example 1.B.3 p. 88], the slice inclusion is a homotopyequivalence. By Lemma 7.1.3, the map X × S∞ → XG is a 2-fold covering. Itscharacteristic map is p : XG → RP∞ and, by Lemma 4.3.6, its characteristic classcoincides with u ∈ H1

G(X). Therefore, the transfer exact sequence of the covering(see Proposition 4.3.9) gives the exact sequence(7.1.7)

· · · → Hm−1G (X)

−·u−−→ HmG (X)

ρ−→ Hm(X)tr∗−−→ Hm

G (X)−·u−−→ Hm+1

G (X)→ · · · .


Denote by (u) the ideal of H∗G(X) generated by u and by

Ann (u) = x ∈ H∗G(X) | ux = 0the annihilator of u. The information carried by Sequence (7.1.7) may be concen-trated in the following short exact sequence of graded Z2[u]-modules:

(7.1.8) 0→ H∗G(X)/(u)ρ−→ H∗(X)

tr∗−−→ Ann (u)→ 0 .

A G-space X is called equivariantly formal if ρ : H∗G(X)→ H∗(X) is surjective.For instance, X is equivariantly formal if the G-action is trivial. See 7.2.9 for adiscussion of this definition in a more general setting.

Proposition 7.1.6. For a G-space X, the following conditions are equivalent.

(1) X is equivariantly formal.(2) H∗G(X) is a free Z2[u]-module.(3) Ann (u) = 0.

Proof. That (2) ⇒ (3) is obvious and (3) ⇔ (1) follows from (7.1.8). For (1)⇒ (2), choose a GrV-section θ : H∗(X)→ H∗G(X) of ρ (as ρ is surjective). Then θ

is a cohomology extension of the fiber for the fiber bundle X → XGp−→ RP∞. As

RP∞ is path-connected and of finite cohomology type, the Leray-Hirsch theoremII (Theorem 4.7.18) implies that H∗G(X) is a free Z2[u]-module generated by θ(B),where B is a Z2-basis of H

∗(X).

Remark 7.1.7. As noted before, H∗(X) ⊗ Z2[u] is isomorphic, as a Z2[u]-algebra, to H∗(X)[u]. If X is equivariantly formal, the Leray-Hirsch theorem II(Theorem 4.7.18) thus provides an isomorphism of Z2[u]-modules between H∗G(X)andH∗(X)[u]. This isomorphism depends on the choice of aGrV-section θ : H∗(X)→H∗G(X) of ρ : H∗G(X)→ H∗(X) and is not, in general, an isomorphism of algebras.However, as in the case of a trivial G-action, Diagram (7.1.6) is commutative.

Corollary 7.1.8. Let X be a G-space. Suppose that r : H∗G(X) → H∗G(XG)

is injective. Then X is equivariantly formal.

Proof. Since H∗G(XG) is a free Z2[u]-module, the hypothesis implies that

Ann (u) = 0.

The converse of Corollary 7.1.8 is true in many cases (see Proposition 7.3.9), butnot in general. For example, S∞ is equivariantly formal since H∗(S∞) ≈ H∗(pt).But H∗G(S

∞) = H∗(RP∞) by Lemma 7.1.4 and H∗G((S∞)G) = H∗G(∅) = 0.

For X a G-space, let r : H∗G(X) → H∗G(XG) and r : H∗(X) → H∗(XG) be

the GrA[u]-homomorphism induced by the inclusion XG → X . One can composetr∗ : H∗(X)→ H∗G(X) with r.

Proposition 7.1.9. rtr∗ = 0.

Proof. As G acts trivially on XG, one has H∗G(XG) ≈ H∗(RP∞)⊗H∗(XG).

The commutative diagram

H∗(S∞ ×X)

tr∗

r // H∗(S∞ ×XG)

tr∗

H∗(S∞)⊗H∗(XG)

tr∗⊗id

≈oo

H∗(S∞ ×G X)r // H∗(RP∞ ×XG) H∗(RP∞)⊗H∗(XG)

≈oo


proves the proposition since tr∗ : H∗(S∞) → H∗(RP∞) is the zero map (even indegree 0, by the transfer exact sequence).

To say more about the image of ρ : H∗G(X)→ H∗(X), define

H∗(X)G = a ∈ H∗(X) | H∗τ(a) = a ⊂ H∗(X) .

As H∗τ is a GrA-morphism, H∗(X)G is a Z2-graded subalgebra of H∗(X).

Lemma 7.1.10. ρ(H∗G(X)) ⊂ H∗(X)G.

Proof. Let z ∈ S∞ and b ∈ H∗G(X). Then,

H∗τ ρ(b) = H∗τ H∗iz(b) = H∗(iz τ)(b) = H∗i−z(b) = ρ(b) .

The reverse inclusion in Lemma 7.1.10 may be wrong, as shown by the followingexample.

Example 7.1.11. Let τ be the antipodal map on the sphere X = Sn, makingX a free G-complex, as seen in Example 7.1.1. The equality Hn(X) = Hn(X)G

holds true since Hn(X) = Z2. For any z ∈ S∞, the composition Xiz−→ XG

q−→ X/Gcoincides with the quotient map X →→ X/G = RPn. By Lemma 7.1.4, the mapq is a homotopy equivalence. But, since Sn →→ RPn is of local degree 2, thehomomorphism Hn(RPn) → Hn(Sn) vanishes. Thus ρ = 0. In this example, thenon-existence of fixed points is important (see Proposition 7.1.12 below).

LetX be aG-space. The reduced equivariant cohomology H∗G(X) is theGrA[u]-algebra defined by

(7.1.9) H∗G(X) = coker(H∗pG : H∗G(pt)→ H∗G(X)

)

where pG : X → pt denotes the constant map to a point (which is G-equivariant).

Warning: H∗G(X) 6= H∗(XG). Here are some examples.

(1) H∗G(pt) = 0.(2) Let X = Sn with the antipodal involution. By Lemma 7.1.4, XG has

the homotopy type of RPn and H∗p : H∗(RP∞)→ H∗G(X) is surjective.

Therefore, H∗G(X) = 0.(3) If Y is a space with trivialG-action, one has a naturalGrA[u]-isomorphism

(7.1.10) H∗G(Y ) = H∗(Y )[u]/Z2[u] ≈ H∗(Y )[u]

(4) If X is equivariantly formal, we get, as in Remark 7.1.7, an isomorphism of

Z2[u]-modules between H∗G(X) and H∗(X)[u]. This isomorphism dependson the choice of a section of ρ : H∗G(X) → H∗(X) and is not, in general,an isomorphism of algebras.

Any G-equivariant map f : Y → X satisfies pf = p, so H∗G is a contravariantfunctor from TopG to GrA[u]. One checks that the homomorphisms ρ : H∗G(X)→H∗(X)G and tr∗ : H∗(X) → H∗G(X) descend to ρ : H∗G(X) → H∗(X)G and to

tr∗: H∗(X)→ H∗G(X).The equivariant reduced cohomology will be further developed in a more general

setting (see 7.2.10 in the next section). Here, we shall only prove the followingproposition, which plays an important role in the construction of the Steenrodsquares in Chapter 8. Note that H∗(X)G contains the classes a + τ∗(a) for alla ∈ H∗(X).


Proposition 7.1.12. Let X be a G-space with XG 6= ∅. Suppose that Hj(X) =

0 for 0 ≤ j < r. Then, ρ : HrG(X) → Hr(X)G is an isomorphism. Moreover,

ρ−1(a+ τ∗(a)) = tr∗(a) for all a ∈ Hr(X).

Proof. The last assertion follows from the main one since ρ tr∗(a) = a+τ∗(a)

by definition of the transfer. We shall prove that the sequence

(7.1.11) 0→ HrG(pt)

H∗p−−−→ HrG(X)

ρ−→ Hr(X)G → 0

is exact. This will prove the main assertion.For 0 ≤ k ≤ ∞, let

Zk = Sk ×G X .

Thus, XG = Z∞ and, as in Lemma 7.1.3, there is a natural locally trivial bundlep : Zk → RP k with fiber X .

Choosing a point z ∈ S1 provides a map iz : X → Z1 ⊂ Zk, defined by iz(x) =[z, x], which induces a GrA-homomorphism ρ : H∗(Zk)→ H∗(X) (independent ofz) given by ρ = H∗iz. As in Lemma 7.1.10, one proves that ρ(H∗(Zk)) ⊂ H∗(X)G.We shall prove, by induction on k, that the sequence

(7.1.12) 0→ Hr(RP k)H∗p−−−→ Hr(Zk)

ρ−→ Hr(X)G → 0

is exact for each k ≥ 1. Since any compact subset of XG = Y∞ is contained in Zkfor some k, the exactness of (7.1.11) will follow, using Corollary 3.1.16.

Observe that, in Sequence (7.1.12), the homomorphism H∗p is injective sincethe choice of a G-fixed point in X provides a section of p. It is also clear thatρH∗p = 0.

We start with k = 1. The space Z1 is the mapping torus of τ . The mappingtorus exact sequence of Proposition 4.7.44 is of the form

· · · → Hr−1(X)Θ−→ Hr−1(X)

J−→ Hr(Z1)ρ−→ Hr(X)

Θ−→ Hr(X)→ · · · ,

where Θ = id + H∗τ . Hence, ker(Hr(X)Θ−→ Hr(X)) = Hr(X)G. If r ≥ 2, then

H1(X) = 0, which proves that ρ : Hr(Z1)→ Hr(X)G is an isomorphism and thusSequence (7.1.12) is exact (for k = 1). If r = 1, then Θ: H0(X) → H0(X) is the

null-homomorphism, since X is path-connected. Then, ker(H1(Z1)ρ−→ H1(X)G) ≈

Z2 which implies that Sequence (7.1.12) is also exact when k = r = 1.Take, as induction hypothesis, that Sequence (7.1.12) is exact for k = ℓ−1 ≥ 1.

We have to prove that it is exact for k = ℓ. The space Zℓ is obtained from Zℓ−1 bygluing Dℓ ×X using the projection Sℓ−1 ×X →→ Zℓ−1. Let e be the generator ofHℓ(Dℓ, Sℓ−1) = Z2. Using excision and the relative Kunneth theorem, we get thecommutative diagram

H∗(RP ℓ,RP ℓ−1)

≈// H∗(Dℓ, Sℓ−1)

H∗−ℓ(pt)≈

e×−oo

H∗(Zℓ, Zℓ−1) ≈

// H∗(Dℓ ×X,Sℓ−1 ×X) H∗−ℓ(X)≈

e×−oo


This diagram, together with the cohomology sequences for the pairs (RP ℓ,RP ℓ−1)and (Zℓ, Zℓ−1) gives the commutative diagram:

(7.1.13)

Hr−ℓ(pt)

// Hr(RP ℓ)H∗p

// Hr(RP ℓ−1)H∗p

// Hr+1−ℓ(pt)

Hr−ℓ(X) // Hr(Zℓ)

ρ

// Hr(Zℓ−1)

ρ

// Hr+1−ℓ(X)

Hr(X)G= // Hr(X)G

.

where the two long lines are exact. The induction step follows by comparing thetwo middle columns. The argument divides into four cases.Case 1: ℓ < r. As ℓ ≥ 2, one has 0 < r − ℓ ≥ r − 2. By hypothesis, Hj(X) = 0 for1 ≤ j < r. Therefore the left and right columns vanish and the two middle columnsare isomorphic.Case 2: ℓ = r. Since ℓ ≥ 2, one hasH1(X) = 0 and the right column vanishes. Also,

Hr(RP ℓ−1) = 0 and the left vertical arrow is an isomorphism (since H0(X) = 0).The induction step follows. Observe that H0(X)→ Hr(Zℓ) is injective.Case 3: ℓ = r+1. The left column vanishes. By step 2, Diagram (7.1.13) continueson the right by injections

H0(pt)

≈

// // Hr+1(RP ℓ)H∗p

H0(X) // // Hr+1(Zℓ)

.

Hence, the two middle columns are isomorphic.Case 4: ℓ > r + 1. The left and right columns vanish for dimensional reasons, sothe two middle columns are isomorphic.

Remark 7.1.13. The Serre spectral sequence for the bundle X → XG → RP∞

provides a shorter proof of the exactness of sequences (7.1.11) and (7.1.12). Thiswill be used to prove the more general Proposition 7.2.17 in the next section.

Example 7.1.14. Linear involution on spheres. Let Sn be the standard sphereequipped with an involution τ ∈ O(n+ 1). In Rn+1, the equality

x =x+ τ(x)

2+x− τ(x)

2

gives the decomposition Rn+1 = V+ ⊕ V− with V± being the eigenspace for theeigenvalue ±1. As τ is an isometry, the vector spaces V+ and V− are orthogonal.Therefore, two elements τ, τ ′ ∈ O(n + 1) of order 2 are conjugate in O(n + 1) if

and only if dim(Sn)τ = dim(Sn)τ′

. We write Snp , (−1 ≤ p ≤ n) for the sphereSn equipped with an involution τ ∈ O(n + 1) such that dim(Sn)τ = p. Hence,(Snp )

τ ≈ Sp. The equivariant CW-structure on Sn (see Example 3.4.4) provides aG-CW-structure on Snp for all p ≤ n.


The involution on Sn−1 is just the antipodal map and, by Lemma 7.1.4, (Sn−1)G ≈RPn. For p ≥ 0, the inclusion Sp = (Snp )

G → Snp gives rise to GrA[u]-morphisms

r : H∗G(Snp )→ H∗G((S

np )G) = H∗G(S

p) ≈ H∗(Sp)[u]and

r : H∗G(Snp )→ H∗G((S

np )G) = H∗G(S

p) ≈ H∗(Sp)[u] .If n ≥ 1, then Hj(Snp ) = 0 for 0 ≤ j < n and Proposition 7.1.12 asserts that

ρ : HnG(S

np ) → Hn(Snp )

G = Hn(Snp ) is an isomorphism (this is also true if n = p =

0). Let a ∈ Hn(Sn) and b ∈ Hp(Sp) be the generators.

Proposition 7.1.15. When p ≥ 0 the GrA[u]-morphisms r and r are injective.Moreover, r ρ−1(a) = b un−p.

Proof. The proposition is trivial if n = p, so we can suppose that n > p ≥ 0.Using the commutative diagram

0 // H∗G(pt)

=

// H∗G(Snp )

r

// H∗G(Snp )

r

// 0

0 // H∗G(pt) // H∗G(Sp) // H∗G(S

p) // 0

,

the five-lemma technique show that r is injective if and only if r is injective. Thuswe shall prove that r is injective.

We first prove that r is injective when p = 0. One can see Sn0 as the suspensionof Sn−1, with (Sn0 )

G = ω+, ω− ≈ S0. Then, X = Sn0 is the union of the G-equivariantly contractible open sets X+ = Sn0 − ω− and X− = Sn0 − ω+, withintersection X0 having the G-equivariant homotopy type of Sn−1. Hence, X±G hasthe homotopy type of ω±G and

H∗G(S0) ≈ Hk

G(ω−)⊕HkG(ω+) ≈ Hk

G(X−)⊕Hk

G(X+) .

By Lemma 7.1.4, X0G has the homotopy type of RPn−1. By Proposition 3.1.53, the

Mayer-Vietoris data (XG, X+G , X

−G , X

0G) gives rise to the long exact sequence

(7.1.14)

Hk−1(RPn)δMV // Hk

G(Sn0 )

r // H∗G(S0)

J // Hk(RPn)δMV // · · ·

with J = H∗Gj+ + H∗Gj

−, where j± : X0 → X± denotes the inclusion. Themap j± is G-homotopy equivalent to the constant map Sn−1 7→ ω±. As noted

before Example 7.1.2, the induced map H∗G(ω±)→ H∗G(X0) is the GrA-morphism

H∗p : H∗(RP∞) → H∗(RPn−1). By Lemma 7.1.4, p is the characteristic map forthe covering Sn−1−1 → (Sn−1−1 )G = RPn−1. As noticed in Example 4.3.5, this map

is just the inclusion RPn−1 → RP∞ and, then, H∗Gj± is surjective by Proposi-

tion 4.3.10. Hence, J is surjective and the exact sequence (7.1.14) splits. Thisproves that r : H∗G(S

n0 ) → H∗G(S

0) is injective. For a more precise analysis ofH∗G(S

n0 ), see Examples 7.1.16 or 7.6.9.

Suppose, by induction on p ≥ 1, that r : HiG(S

mp−1) → Hi

G(Sp−1) is injective

for all i ∈ N and all m ≥ p − 1. We have to prove that r : HkG(S

np ) → Hk

G(Sp)

is injective for all k ∈ N and all n ≥ p. As (Snp )G and (Sp)G are path-connected,the required assertion is true for k = 0 by Lemma 3.1.9. Thus, we may supposethat k ≥ 1. As n ≥ p ≥ 1, the G-sphere Sn−1p−1 exists and Snp is the suspension


of Sn−1p−1 , with (Snp )G being the suspension of (Sn−1p−1 )

G. Let Y = Y G = ω−, ω+be the suspension points. As above, we decompose the X = Snp = X− ∪X+ with

X− ∩X+ = X − Y . The maps r sit in a commutative diagram

(7.1.15)

Hk−1G (Y )

≈ r

// Hk−1G (Sn−1p−1 )

δMV //r

HkG(S

np )

r

// HkG(Y )

≈ r

Hk−1G (Y G) // Hk−1

G (Sp−1)δMV // Hk

G(Sp) // Hk

G(YG)

where the horizontal line are the Mayer-Vietoris sequences for the data(X,X+, X−, X − Y ) and (XG, (X+)G, (X−)G, (X − Y )G). Hence, r : Hk

G(Snp ) →

HkG(S

p) is injective by the proof of the five lemma (see [82, p. 129]).

The last assertion is now obvious, since r ρ−1 is injective and, as H∗G(Sp) ≈

H∗(Sp)[u], one has HnG(S

p) = Z2un−p.

Example 7.1.16. We use the notations of the proof of Proposition 7.1.15 in thecase p = 0, with (Sn0 )

G = S0 = ω±. The isomorphism σ− : Hn(Sn0 ) → Hn

G(Sn0 )

defined by the commutative diagram

Hn(Sn0 )ρ−1

≈// Hn

G(Sn0 ) // r // Hn

G(S0)

HnG(S

n0 , ω−)

≈

OO

// r //

HnG(S

0, ω−)

≈

OO

Hn(Sn0 )σ− //

≈

OO

HnG(S

n0 ) // r // Hn

G(S0) oo ≈ // Hn

G(ω−)⊕HnG(ω+)

is an extension of the fiber for the bundle Sn0 → (Sn0 )G → RP∞. Another one,σ+, is obtained using ω+ (there are two of them by the exact sequence (7.1.11) andσ+(a) = σ−(a) + un). Then, rσ−(a) = (un, 0) and rσ+(a) = (0, un). Hence,neither σ− nor σ+ is multiplicative. We see the relation rσ±(a)

2 = un rσ±(a).Hence, as r is a monomorphism of Z2[u]-module, the relation σ±(a)

2 = un σ±(a)holds in H∗G(S

n0 ). By the Leray-Hirsch Theorem 4.7.17, H∗G(S

n0 ) is a free Z2[u]-

module generated by A = σ+(a) (or, by B = σ−(a)). By dimension counting, wecheck that H∗G(S

n0 ) admits, as a Z2[u]-algebra, the presentation

(7.1.16) H∗G(Sn0 ) ≈ Z2[u][A]

/(A2 + unA) .

As σ−(a)σ+(a) = 0, a more symmetric presentation is obtained using the twogenerators A and B:

H∗G(Sn0 ) ≈ Z2[u][A,B]

/I

where I is the ideal generated by

A+B + un , and A2 + unA .

Note that AB and B2 + unB are in I. Indeed, mod I, one has

AB = A(A+ un) = A2 + unA = 0

and

B2 = (A+ un)2 = A2 + u2n = unA+ u2n = un(A+ un) = unB .


Corollary 7.1.17. If p ≥ 0, then Snp is equivariantly formal. There is a

section σ : Hn(Snp ) → HnG(S

np ) of ρ such that rσ : Hn(Snp ) → Hn

G((Snp )G) →

H∗(Sp)[u] satisfies

(7.1.17) rσ(a) = b un−p .

Proof. By Proposition 7.1.15 ρ : HnG(S

np ) → Hn(Snp ) is an isomorphism, so

the commutative diagram

HnG(S

np )

ρ //

Hn(Snp )

≈

HnG(S

np )

ρ

≈// Hn(Snp )

shows that ρ is surjective and thus Snp is equivariantly formal. Choose a section σ

of ρ. By Proposition 7.1.15, Equation (7.1.17) holds true modulo ker(HnG(S

np ) →

HnG(S

np ))= Z2u

n. By changing σ(a) by σ(a) + un if necessary, (7.1.17) will holdtrue strictly.

As another example, we consider CPn as a G-space with the involution τ beingthe complex conjugation. Thus, (CPn)G = RPn. Let 0 6= a ∈ H2(CPn) and0 6= b ∈ H1(RPn).

Proposition 7.1.18. For n ≤ ∞, CPn is equivariantly formal. Moreover,there is a section σ : H∗(CPn) → H∗G(CP

n) which is multiplicative and satisfiesrσ(a) = bu+ b2.

Proof. As Hi(CPn) = 0 for i ≤ 1, Proposition 7.1.12 implies that ρ : H2G(CP

n)→H2(CPn) is an isomorphism. As in the proof of Corollary 7.1.17, this implies thatρ : H2

G(CPn) → H2(CPn) is surjective. As H∗(CPn) is generated by a as an al-

gebra and ρ is a GrA-morphism, we deduce that ρ : H∗G(CPn) → H∗(CPn) is

surjective. Thus CPn is equivariantly formal.Choose a section σ2 : H

2(CPn)→ H2G(CP

n) of ρ. AsH∗G(RPn) ≈ H∗(RPn)[u],

there exists λ, µ and ν in Z2 such that

rσ2 = λu2 + µbu+ νb2 .

By changing σ2(a) by σ′2(a) = σ(a) + λu2, we may assume that λ = 0. We mustprove that µ = ν = 1. The inclusions i : RPn → CPn and j : CP 1 → CPn providecommutative diagrams

H2G(CP

n)r //

ρ

H2G(RP

n)

ρG

H2(CPn)

H∗i // H2(RPn)

H2G(CP

n)ρ //

H∗Gj

H2(CPn)

H∗j≈

σmm

H2G(CP

1)ρ1 // H2(CP 1)σ1

mm

with r = H∗Gi. By Proposition 6.1.11, H∗i(a) = b2, so ν = 1. Note that σ1 =H∗Gj σ(H

∗j)−1 is a section of ρ1. As CP 1 with the complex conjugation is G-diffeomorphic to S2

1 (via the stereographic projection of S2 onto C ∪ ∞ ≈ CP 1),Corollary 7.1.17 implies that r1 σ1(a) = au, which proves that µ = 1.

7.2. THE GENERAL CASE 229

For n ≤ ∞, we now define σ[n] : H∗(CPn)→ H∗G(CPn) by σ[n](ak) = σ2(a)

k.This is a section of ρ and σ[∞] is clearly multiplicative. As σ[n] is the composi-tion of σ[∞] with the morphism H∗G(RP

∞)→ H∗G(RPn) induced by the inclusion,

the section σ = σn of ρ is multiplicative and satisfies the requirements of Proposi-tion 7.1.18.

Corollary 7.1.19. For n ≤ ∞, let σ : H∗(CPn) → H∗G(CPn) be the section

of Proposition 7.1.18. Then, the correspondence a 7→ σ(a) provides a GrA[u]-isomorphism

Z2[u, a]/(an+1)

≈−→ H∗G(CPn) .

Corollary 7.1.20. For n ≤ ∞, the restriction morphism r : HG(CPn) →HG(RPn) is injective.

Proof. Let x ∈ Hm(CPn) with x ∈ ker r. Write x under the form x =σ(ak)ur + ℓtr (k + r = m), where σ is given by Proposition 7.1.19 and ℓtr denotessome polynomial in the variable u of degree less than r. Then, the equation

(7.1.18) 0 = r(x) = bkur+k + ℓtr+k

holds in H∗G(RPn) = H∗(RPn)[u]. This first proves that k > 0. Choose x so that

k is minimal. Then, (7.1.18) again implies that bk = 0. Hence, n <∞ and k > n.As σ is multiplicative, one has σ(ak) = 0 and x = ℓtr, contradicting the minimalityof k.

The proof of Corollary 7.1.20 generalizes for conjugation spaces (see Lemma 10.2.8).For n < ∞, Corollary 7.1.20 is a consequence of the equivariant formality of CPn

(see Proposition 7.3.9).

Remark 7.1.21. As an exercise, the reader may develop the analogue of Propo-sition 7.1.18 and Corollaries 7.1.19 and 7.1.20 for the G-space X = HPn, where Gacts via the involutions on H defined by τ(x+ iy+ jz+ kt) = x+ iy− jz− kt (thusXG ≈ CPn), or τ(x + iy + jz + kt) = x − iy − jz − kt (thus XG ≈ RPn). Thesame work may be done with X = OP 2 with various R-linear involutions on O sothat XG ≈ HP 2, CP 2 or RP 2.

7.2. The general case

Let Γ be a topological group. Let p : EΓ → BΓ be the universal principalΓ-bundle constructed by Milnor [144]. The space EΓ is contractible, being thejoin of infinitely many copies of Γ with a convenient topology. An element of EΓis represented by a sequence (tiγi) (i ∈ N) with (ti) ∈ ∆∞ and γi ∈ Γ; two suchsequences (tiγi) and (t′iγ

′i) represent the same class in EΓ if ti = t′i and γi = γ′i

whenever ti 6= 0. There is a free right action of Γ on EΓ given by (tiγi) g =(tiγi g). One defines BΓ = EΓ/Γ. The quotient map p : EΓ → BΓ enjoys localtriviality, in other words is a principal Γ-bundle. These constructions are functorial:a continuous homomorphism α : Γ′ → Γ induces a continuous map Eα : EΓ′ → EΓ,defined by Eα(tiγi) = (tiα(γi)), which descends to a continuous map Bα : BΓ′ →BΓ.

Example 7.2.1. Consider the case Γ = G = I, τ. Then, EG → BG ishomotopy equivalent to S∞ → RP∞. This is because the join of a space Y withG ≈ S0 is homeomorphic to the suspension of Y . In the same way, ES1 → BS1 ishomotopy equivalent to S∞ → CP∞.


Lemma 7.2.2. Let Γ be a finite group of odd order. Then H∗(BΓ) ≈ H∗(pt).Proof. By Kronecker duality, it is equivalent to prove that H∗(BΓ) ≈ H∗(pt).

When Γ is a discrete group, the principal Γ-bundle p : EΓ → BΓ is the universalcovering of BΓ. One has the transfer chain map tr∗ : Cm(BΓ) → Cm(EΓ) as in§ 4.3.3, sending a singular simplex σ : ∆m → BΓ to the set of its liftings in EΓ. Ifthe number of sheets is odd, the composition

H∗(BΓ)tr−→ H∗(EΓ)

H∗p−−−→ H∗(BΓ)

is the identity. Since EΓ is contractible, this proves the lemma.

When Γ is a discrete group, the cohomology of BΓ is isomorphic to the co-homology H∗(Γ;Z2) of the group Γ in the sense of [26, 3]. The isomorphismH∗(BΓ) ≈ H∗(Γ;Z2) is proven in e.g. [3, § II.2]. The following proposition,proven in [26, Proposition III.8.3] will be useful.

Proposition 7.2.3. Let Γ be a discrete group. Let α be an inner automorphismof Γ, i.e. α(g) = g0gg

−10 for some g0 ∈ Γ. Then H∗Bα(a) = a for all a ∈ H∗(BΓ).

Let X be a left Γ-space. The Borel construction XΓ or homotopy quotient, isthe quotient space

XΓ = EΓ×Γ X = (EΓ×X)/∼

where ∼ is the equivalence relation (z, γx) ∼ (zγ, x) for all x ∈ X , z ∈ EΓ andγ ∈ Γ. A map p : XΓ → BΓ is then given by p(z, x) = p(z). If Γ = G = I, τ, thisBorel construction coincides with that defined in (7.1.1).

As in § 7.1, one proves the following statements.

(1) The Borel construction is a covariant functor from the category TopΓ toTop, where TopΓ is the category of Γ-spaces and Γ-equivariant maps.The map p : XΓ → BΓ coincides with the map XΓ → ptΓ = BΓ inducedby the constant map X → pt.

(2) XΓ and YΓ have the same homotopy type if X and Y have the sameΓ-homotopy type.

(3) The map p : XΓ → BΓ is a locally trivial fiber bundle with fiber homeo-morphic to X .

(4) If f : Y → X is a G-equivariant map, then the diagram

YΓp

!!

fΓ // XΓ

p

||③③③③③③③

BΓ

is commutative.(5) If the Γ action on X has a fixed point, then p admits a section. More

precisely, the choice of a point v ∈ XΓ provides a section sv : BΓ → XΓ

of p. (see the proof of Point (3) of Lemma 7.1.3).(6) If Γ acts trivially on X , then XΓ has the homotopy type of BΓ×X (see

Example 7.1.2).(7) The projection EΓ × X → XΓ is a Γ-principal bundle induced from the

universal bundle by p.


Example 7.2.4. Let i : Γ0 → Γ denote the inclusion of a closed subgroup Γ0

of Γ. We consider the homogeneous Γ-space X = Γ/Γ0. Then, the map h : EΓ ×X → EΓ/Γ0 given by h(z, [γ]) = [zγ] descends to a homeomorphism h : EΓ×ΓX

≈−→EΓ/Γ0. Consider the commutative diagram

Γ0//

=

EΓ0//

Ei

EΓ0/Γ0oo = //

Ei

BΓ0

Bi

gzz

Γ0// EΓ //

OO

=

EΓ/Γ0

EΓ // EΓ/Γ oo = // BΓ

The two upper lines are Γ0-principal bundles. As both EΓ0 and EΓ have vanishinghomotopy groups, the map Ei is a weak homotopy equivalence, and so is g. HenceXΓ has the weak homotopy type of BΓ0. In addition, the map Bi : BΓ0 → BΓ isweakly homotopy equivalent to the locally trivial bundle XΓ → BΓ with fiber X .More generally, let Y be a Γ-space and consider Γ/Γ0×Y endowed with the diagonalΓ-action; then H∗Γ(Γ/Γ0 × Y ) ≈ H∗Γ0

(Y ) (see the proof of Theorem 7.4.3).

For a general Γ-space Y , the quotient map q : Y → Γ\Y descends to a surjectivemap q : YΓ → Γ\Y such that q−1([y]) has the weak homotopy type of BΓy for ally ∈ Y , where Γy is the stabilizer of y.

Let (X,Y ) be a Γ-pair, i.e. a Γ-space X with a Γ-invariant subspace Y . TheΓ-equivariant cohomology H∗Γ(X,Y ) is the cohomology algebra

H∗Γ(X,Y ) = H∗(XΓ, YΓ) and H∗Γ(X) = H∗(XΓ) = HΓ(X, ∅) .In particular,H∗Γ(pt) = H∗(BΓ). The map p : XΓ → BΓ induces aGrA-homomorphismH∗p : H∗(BΓ)→ H∗Γ(X), endowing the latter with the structure of H∗Γ(pt)-algebra.

7.2.5. Changing spaces and groups. Let α : Γ′ → Γ be a continuous homomor-phism. Let X be a Γ-space and X ′ be a Γ′-space. A continuous map f : X ′ → Xsatisfying

f(γx) = α(γ)f(x)

for all x ∈ X ′ and γ ∈ Γ′ is called equivariant with respect to α. The continuous

map EΓ′×X ′ Eα×f−−−−→ EΓ×X then descends to a continuous map fΓ′,Γ : XΓ′ → XΓ

(depending on α). There is a commutative diagram

(7.2.1)

XΓ′fΓ′,Γ //

XΓ

BΓ′

Bα // BΓ

.

The map fΓ′,Γ induces a GrA-homomorphism

(7.2.2) f∗Γ′,Γ : H∗Γ(X)→ H∗Γ′(X

′) .

By commutativity of the Diagram (7.2.1), one has

f∗Γ′,Γ(av) = f∗Γ′,Γ(a)(Bα)∗(v) for all a ∈ H∗Γ(X) and v ∈ H∗Γ(pt) .


We say that f∗Γ′,Γ preserves the module structures via α. More simply, the Γ-space

X becomes a Γ′-space via α, thus H∗Γ(X) becomes a H∗(BΓ′)-algebra and f∗Γ′,Γ is

a morphism of H∗(BΓ′)-algebras. An important case is given by f = id: X → X ,Setting id∗Γ,Γ′ = α∗, we get a map α∗ : H∗Γ(X) → H∗Γ′(X) which is a morphism ofH∗(BΓ′)-algebras.

7.2.6. Free actions. Let Γ0 be a closed normal subgroup of Γ and let X be aΓ-space. For x ∈ X , γ ∈ Γ and γ0, γ

′0 ∈ Γ0, the equation

(γγ0) (γ′0x) = (γγ0γ

′0γ−1) γx

shows that the Γ-action on X descends to a (Γ/Γ0)-action on Γ0\X . By the func-toriality of equivariant cohomology (see (7.2.5)), we get a map

(7.2.3) H∗Γ/Γ0(Γ0\X)→ H∗Γ(X)

which is a homomorphism of H∗(B(Γ/Γ0))-algebras. The following lemma general-izes Lemma 7.1.4. To avoid point-set topology complications, we restrict ourselvesto the smooth action of a Lie group.

Lemma 7.2.7. Let Γ0 be a compact normal subgroup of a Lie group Γ. LetX be a smooth Γ-manifold on which Γ0 acts freely. Then, the map (7.2.3) is anisomorphism of H∗(B(Γ/Γ0))-algebras.

Proof. Let Y = E(Γ/Γ0)×Γ/Γ0(Γ0\X). Consider the commutative diagram

EΓ×X q //

p

EΓ×Γ X

p

E(Γ/Γ0)× (Γ0\X)

q // Y

.

Let a ∈ Y represented by ((tiγi), x) in EΓ×X . Then,

(qp)−1(a) = ((tiγiδi), δx

)| δi, δ ∈ Γ0 .

Therefore,

(7.2.4) p−1(a) = q((qp)−1(a)) ≈ ((tiγiδi), x

)| δi ∈ Γ0 ≈−→ EΓ0 ,

the last homeomorphism being given by((tiγiδi), x

)7→ (tiδi).

As Γ0 acts smoothly and freely on X , the quotient map X → Γ0\X is alocally trivial bundle (this follows from the slice theorem: see [12, Theorem 2.2.1]).Hence, p is homotopy equivalent to a locally trivial bundle, which is numerable([179, p. 94]) since Γ0\X is paracompact. The map q is also a numerable locallytrivial bundle (see (7) on p. 230). Therefore, qp is a fibration (i.e. satisfies thehomotopy covering property for any space: see [179, Theorem 12, p. 95]), and sodoes p. As Γ0\X is a manifold and E(Γ/Γ0) is contractible, the space Y admits anumerable covering Vλλ∈Λ such that each inclusion Vλ → Y is null-homotopic.As each fiber of p is contractible by (7.2.4), [42, Theorem 6.3] implies that p is ahomotopy equivalence, which proves the lemma.

7.2.8. The forgetful homomorphism. Choosing a point ζ ∈ EΓ provides a mapiζ : X → XΓ defined by iζ(x) = [ζ, x]. As EΓ is path-connected, the homotopy class


of iζ does not depend on ζ. For instance, we can take ζ = ζ0 = (1e, 0, 0, . . . ) wheree ∈ Γ is the unit element. Therefore, we get a well defined GrA-homomorphism

(7.2.5) ρ : H∗Γ(X)→ H∗(X)

given by ρ = H∗iζ for some ζ ∈ EΓ. As in (7.1.5), one proves that ρ is functorial.In fact, using 7.2.5, the homomorphism ρ coincides with the homomorphism ide,Γinduced by inclusion of the trivial group e into Γ:

(7.2.6) ρ = ide,Γ : H∗Γ(X)→ H∗e(X) = H∗(X) .

Indeed, iζ0 factors through X → Xe ≈ X . Hence, ρ may be seen as a forgetfulhomomorphism (one forgets the Γ-action).

A consequence of (7.2.5) is that ρ is functorial for the changing of groups: ifα : Γ′ → Γ is a continuous homomorphism and X a Γ-space, the diagram

H∗Γ(X)α∗ //

ρ %%

H∗Γ′(X)

ρyyssssssss

H∗(X)

is commutative.

7.2.9. Equivariant formality. A Γ-space X is called equivariantly formal ifρ : HΓ(X) → H∗(X) is surjective. For instance, X is equivariantly formal if theΓ-action is trivial. For relationships with other kind of “formal” spaces, see [173].If X is equivariantly formal, one can choose, as in the proof of Proposition 7.1.6,a GrV-section θ : H∗(X) → H∗Γ(X) of ρ. Then θ is a cohomology extension of

the fiber for the fiber bundle XΓp−→ BΓ. If X is of finite cohomology type, the

Leray-Hirsch theorem 4.7.17 then gives a map (depending on θ)

(7.2.7) H∗Γ(pt)⊗H∗(X)≈−→ H∗Γ(X)

which is an isomorphism of H∗Γ(pt)-modules.

7.2.10. Reduced cohomology. Let X be a Γ-space. The reduced equivariantcohomology H∗Γ(X) is the H∗Γ(pt)-algebra defined by

(7.2.8) H∗Γ(X) = coker(H∗pΓ : H

∗Γ(pt)→ H∗Γ(X)

)

where p : X → pt denotes the constant map to a point (which is Γ-equivariant).

Warning: H∗Γ(X) 6= H∗(XΓ). Examples:

(1) H∗Γ(pt) = 0.(2) If Y is a space with trivial Γ-action, there is an isomorphism of H∗Γ(pt)-

algebras

(7.2.9) H∗Γ(Y ) =(H∗(Y )⊗H∗Γ(pt)

)/(1⊗H∗Γ(pt)) ≈ H∗(Y )⊗H∗Γ(pt) .

(3) IfX is equivariantly formal and is of finite cohomology type, one uses (7.2.7)

to provides an isomorphism ofH∗Γ(pt)-modules between H∗Γ(X) and H∗(X)⊗H∗Γ(pt). This isomorphism depends on the choice of a section of ρ : H∗(X)→H∗Γ(X) and is not, in general, an isomorphism of algebras.


Any Γ-equivariant map f : Y → X satisfies pf = p, so H∗Γ is a contravariantfunctor from TopΓ to the category of H∗Γ(pt)-algebra.

Let v ∈ XΓ. As for (3.1.14), one has the following diagram

(7.2.10)

H∗Γ(pt)

p∗Γ

≈

$$

H∗(XΓ, vΓ)j∗ //

≈

&&

H∗Γ(X)

i∗Γ //

H∗Γ(v)

H∗Γ(X)

where the row and the column are exact. This proves that

(7.2.11) H∗(XΓ, vΓ) ≈−→ H∗Γ(X) .

Observe that, in (7.2.10) , i∗Γ coincides with the section sv of p∗Γ. We see thatthe choice of v ∈ XΓ produces a supplementary vector subspace to p∗

Γ(H∗Γ(pt)) in

H∗Γ(X).

A pair (X,A) of Γ-spaces is called equivariantly well cofibrant if it admits apresentation (u, h) as a well cofibrant pair which is Γ-equivariant, i.e. u(γx) = u(x)and h(γx, t) = γ h(x, t) for all γ ∈ Γ, x ∈ X and t ∈ I.

Lemma 7.2.11. Let (X,A) be a pair of Γ-spaces which is equivariantly wellcofibrant. Then (XΓ, AΓ) is well cofibrant.

Proof. Let (u, h) be a presentation of (X,A) as an equivariantly well cofibrant

pair. Define u : EΓ × X → I and h : EΓ × X × I → EΓ × X by u(z, x) = u(x)

and h(z, x, t) = (z, h(x, t)). We check that these maps descend to uΓ : XΓ → I andhΓ : XΓ× I → XΓ and that (uΓ, hΓ) is a presentation of (XΓ, AΓ) as a well cofibrantpair.

Lemma 7.2.12. Let (X,A) be a Γ1-equivariantly well cofibrant pair of Γ1-spaces.Let (Y,B) be a Γ2-equivariantly well cofibrant pair of Γ2-spaces. Then, (X×Y,A×Y ∪ X × B) is a Γ12-equivariantly well cofibrant pair of Γ12-spaces, where Γ12 =Γ1 × Γ2.

Proof. One checks that the proof of Lemma 3.1.40 works equivariantly.

If (X,A) is a pair of Γ-spaces, the quotient space X/A inherits a Γ-action, with[A] ∈ (X/A)Γ, where [A] denotes the set A as a class in X/A. The proof of thefollowing lemma is the same as that of Lemma 3.1.43.

Lemma 7.2.13. If (X,A) is an equivariantly well cofibrant pair of Γ-spaces, sois the pair (X/A,A/A).

Example 7.2.14. Let (X, x) be a pointed space. The group G = I, τ acts onX ×X by exchanging the coordinates and this action descends to X ∧X . If (X, x)is well pointed, the proof of Lemma 3.1.40 shows that the pair (X ×X,X ∨X) isG-equivariantly well cofibrant. By Lemma 7.2.13, (X ∧X, x∧x) is G-equivariantlywell pointed.


The quotient map π : (X,A) → (X/A,A/A) is a Γ-equivariant map of pairswhich induces πΓ : (XΓ, AΓ)→ ((X/A)Γ, (A/A)Γ).

Proposition 7.2.15. Let (X,A) be a pair of Γ-spaces which is equivariantlywell cofibrant. Then,

π∗Γ: H∗((X/A)Γ, (A/A)Γ)

≈−→ H∗(XΓ, AΓ)

is an isomorphism.

Proof. Let (K,L) = (X/A,A/A). Let (u, h) be a presentation of (X,A) asan equivariant well cofibrant pair and let (u, h) be the induced presentation of(K,L). Let V = u−1([0, 1/2]) and W = u−1([0, 1/2]) = π(V ). As noticed in theproof of Lemma 3.1.41, the condition u(h(x)) ≤ u(x) implies that h and h restrictto Γ-equivariant deformation retractions from V to A and from W to L. Thetautological homeomorphism from EΓ× (V × I) onto (EΓ× V )× I descends to a

homeomorphism (V × I)Γ ≈−→ VΓ × I. Using this, h and h descend to deformationretractions hΓ : VΓ×I → VΓ, and hΓ : WΓ×I →WΓ onto A and L, making (XΓ, AΓ)and (KΓ, LΓ) good pairs.

The inclusion (K,L)→ (K,W ) gives rise to a morphism of exact sequences

Hk−1(KΓ) //

=

Hk−1(LΓ) //

≈

Hk(KΓ, LΓ) //

Hk(KΓ) //

=

Hk(LΓ)

≈

Hk−1(KΓ) // Hk−1(WΓ) // Hk(KΓ,WΓ) // Hk(KΓ) // Hk(WΓ)

which, by the five lemma, implies that Hk(KΓ, LΓ)≈−→ Hk(KΓ,WΓ) is an iso-

morphism. The same proof gives the isomorphism Hk(XΓ, AΓ)≈−→ Hk(XΓ, VΓ).

Proposition 7.2.15 then comes from the commutativity of the following diagram(where the vertical arrows are induced by inclusions)

H∗(KΓ, LΓ)OO

≈

π∗Γ // H∗(XΓ, AΓ)OO

≈

H∗(KΓ,WΓ)π∗Γ // H∗(XΓ, VΓ)

H∗(KΓ − LΓ,WΓ − LΓ)π∗Γ

≈//

≈ excision

H∗(XΓ −AΓ, VΓ −AΓ) .

≈ excision

The bottom horizontal arrow is indeed an isomorphism since π : (X −A, V −A)→(K − L,W − L) is a Γ-equivariant homeomorphism.

Corollary 7.2.16. Let (X,A) be a pair of Γ-spaces which is equivariantly wellcofibrant. If A is non-empty, there is a functorial isomorphism of H∗Γ(pt)-algebras

H∗(XΓ/AΓ)≈−→ H∗Γ(X/A) .

The hypothesis A 6= ∅ is necessary since H∗(XΓ) is not isomorphic to H∗Γ(X).


Proof. This follows from the diagram

H∗((X/A)Γ, (A/A)Γ)

π∗Γ≈

≈ // H∗Γ(X/A)

H∗(XΓ, AΓ)≈ // H∗(XΓ/AΓ)

.

The bijectivities come

• from (7.2.11) for the top horizontal arrow, since, as A 6= ∅, A/A is a point.• from Lemma 7.2.11 and Proposition 3.1.45 for the bottom horizontal ar-row.• from Proposition 7.2.15 for the vertical arrow.

As in 7.2.5, the reduced equivariant cohomology is functorial for changinggroups. In particular, as in 7.2.8, the inclusion of the trivial group e into Γprovides the forgetful homomorphism

ρ : H∗Γ(X)→ H∗e(X) ≈ H∗(X)

which is functorial.

As in Lemma 7.1.10, one proves that ρ(H∗Γ(X)) ⊂ H∗(X)Γ. The followingstatement generalizes Proposition 7.1.12.

Proposition 7.2.17. Let X be a Γ-space with XΓ 6= ∅. Suppose that Hj(X) =

0 for 0 ≤ j < r. Then, ρ : HrΓ(X)→ Hr(X)Γ is an isomorphism.

Proof. (Using a spectral sequence). In the E2-term of the Serre spectralsequence of the bundle X → XΓ → BΓ, the lines from 1 to n− 1 vanish:

H0(BΓ) H1(BΓ) H2(BΓ)· · ·0

...

0

Hr(X)Γ

0

Therefore, it gives rise to the edge exact sequence:

(7.2.12) 0→ Hr(BΓ)H∗p−−−→ Hr

Γ(X)ρ−→ Hr(X)Γ −→ Hr+1(BΓ)

H∗p−−−→ Hr+1Γ (X) .

The choice of a fixed point v ∈ XΓ provides a section sv : BΓ → XΓ, as seen in

Lemma 7.1.3. Therefore, H∗p is injective and HrΓ(X)

ρ−→ Hr(X)Γ is surjective.Proposition 7.2.17 follows from this, as in the proof of Proposition 7.1.12.

Remark 7.2.18. When Γ is discrete, a proof of Proposition 7.2.17 without usinga spectral sequence is possible, following the pattern of that of Proposition 7.1.12.The role of RP k is played by BkΓ, the quotient by Γ of the join EkΓ = Γ ∗ · · · ∗ Γ(k + 1 times) (see [144, § 3]). The space Yk is defined to be EkΓ×Γ X . We leavethis proof as an exercise to the reader.

7.3. LOCALIZATION THEOREMS AND SMITH THEORY 237

7.3. Localization theorems and Smith theory

As in § 7.1, we consider in this section the group G = I, τ of order 2, soBG ≈ RP∞ and H∗G(pt) = Z2[u] with u of degree one. Strong results come outif we invert u, namely if we tensor the Z2[u]-modules with the ring of Laurentpolynomials Z2[u, u

−1]. For a pair (X,Y ) of G-spaces, we thus define

h∗G(X,Y ) = Z2[u, u−1]⊗Z2[u] H

∗G(X,Y ) ,

with the notation h∗G(X) = h∗G(X, ∅). Note that Z2[u, u−1] is Z-graded, with

Z2[u, u−1]k = Z2u

k and we use the graded tensor product. Hence, h∗G(X,Y ) is aZ-graded Z2[u, u

−1]-algebra, with

(7.3.1) hkG(X,Y ) =⊕

i+j=k

Z2ui ⊗Hj

G(X,Y ) ≈⊕

ℓ∈Z

Hk−ℓG (X,Y ) .

[Actually, hkG(X,Y ) is a quotient of the right members of (7.3.1) (see Erratum 12.0.6and Example 7.3.2 below).]

The theorem below is an example of the so called localization theorems. Formore general statements, see e.g. [37, Chapter 3] or [9, Chapter 3].

Theorem 7.3.1. Let X be a finite dimensional G-complex. Then, the inclusionXG ⊂ X induces an isomorphism

h∗G(X)≈−→ h∗G(X

G)

of Z-graded Z2[u, u−1]-algebras.

Before proving Theorem 7.3.1, we discuss a few examples.

Example 7.3.2. Suppose, in Theorem 7.3.1, that X is a free G-complex. ByLemma 7.1.4, H∗G(X) ≈ H∗(X/G). As X/G is a finite dimensional CW-complex,there exists an integerm such that um ·H∗G(X) = 0. As u is invertible in Z2[u, u

−1],this proves that h∗G(X) = 0, as predicted by Theorem 7.3.1, since XG = ∅. We seehere that the finite dimensional hypothesis is necessary in Theorem 7.3.1. Indeed,the free G-complex S∞ = EG satisfies H∗G(EG) = H∗(BG) = Z2[u], so h

∗G(EG) =

Z2[u, u−1].

Example 7.3.3. Consider the G-space Snp of Example 7.1.14, i.e. the sphere

Sn endowed with a linear G-action with (Snp )G ≈ Sp. We assume that 0 ≤ p ≤

n. Using Corollary 7.1.17, there are elements a ∈ HnG(S

np ) and b ∈ Hp

G((Snp )G)

generating respectively H∗G(Snp ) and H

∗G((S

np )G) as free Z2[u]-modules and r(a) =

bun−p. Then, as predicted by Theorem 7.3.1, r : h∗G(Snp ) → h∗G((S

np )G) admits an

inverse, sending b to aup−n.

Example 7.3.4. Consider the G-space CPn (n ≥ 1), where G acts via the com-plex conjugation, with (CPn)G = RPn. By Proposition 7.1.18 and Corollary 7.1.19,there is a commutative diagram

Z2[u, u−1, a]

/(an+1)

r //

≈

Z2[u, u−1, b]

/(bn+1)

≈

h∗G(CPn)

r // h∗G(RPn)


with a of degree 2, b of degree 1 and r(a) = bu+ b2. If n <∞, the correspondence

(7.3.2) b 7→ au−1 + a2u−3 + a4u−7 + · · · =∑

i≥0

a2i

u2i+1−1

extends to aGrA[u]-isomorphism r−1 : Z2[u, u−1, b]

/(bn+1)→ Z2[u, u

−1, a]/(an+1)

which is the inverse of r.

Example 7.3.5. If n =∞, the right hand member of (7.3.2) is not a polynomialand no inverse of r may be defined this way. In fact, r (and then r) is not anisomorphism. Indeed, the composition of r with the epimorphism Z2[u, u

−1, b]→ Z2

sending both b and u to 1 is the zero map. Of course, CP∞ violates the finitedimensional hypothesis in Theorem 7.3.1.

Proof of Theorem 7.3.1. The proof is by induction on the dimension of X ,which starts trivially with X = ∅ (dimension −1). The induction step reduces toproving that, if the theorem is true for X , it is then true for Z = X ∪ C where Cis a family of G-cells. We consider the commutative diagram

hk−1G (X) //

rX≈

hkG(Z,X) //

rZ,X

hkG(Z) //

rZ

hkG(X) //

rX≈

hk+1G (Z,X)

rZ,X

hk−1G (XG) // hk

G(ZG, XG) // hk

G(ZG) // hk

G(XG) // hk+1

G (ZG, XG)

The two lines are exact sequences, obtained by tensoring with Z2[u, u−1] the exact

sequence of (Z,X) for H∗G (as in the proof of Lemma 4.6.9, we use that a directsum of exact sequences is exact and that, over a field, tensoring with a vector spacepreserves exactness [this justification is incorrect (see Erratum 12.0.6)]). If rX is anisomorphism by induction hypothesis, it is enough, using the five lemma, to provethat rZ,X is an isomorphism. Note that C is a disjoint union of free G-cells Cf andof isotropic G-cells Ci. By excision, one has the commutative diagram

h∗G(Z,X)≈ //

rZ,X

h∗G(C,BdC)≈ // h∗G(Cf ,BdCf )× h∗G(Ci,BdCi)

r

h∗G(Z

G, XG)≈ // h∗G(Ci,BdCi)

where r(a, b) = b. It is then enough to prove that h∗G(Cf ,BdCf ) = 0. But thisfollows from the exact sequence of (Cf ,BdCf ) for h

∗G and, since Cf and BdCf are

free G-spaces (see Example 7.3.2).

We are now leading toward the Smith inequalities. Let us extend our groundring Z2[u, u

−1] to the fraction field Z2(u) of Z2[u] (this is just a field of character-istic 2, the grading is lost). For a space X , the total Betti number b(X) of X isdefined by

b(X) =

∞∑

i=0

dimH∗(X) ∈ N ∪ ∞ .

Lemma 7.3.6. Let X be a finite dimensional G-complex with b(X) <∞. Then,as a vector space over Z2(u),

dimZ2(u)⊗Z2[u] H∗G(X) ≤ b(X)


with equality if and only if X is equivariantly formal.

Proof. From the transfer exact sequence (7.1.7), we extract the exact se-quence

Hk−1G (X)

u−−→ HkG(X)

ρ−→ Hk(X) −→ HkG(X)

u−−→ Hk+1G (X) .

We deduce that HkG(X) is generated by u ·Hk−1

G (X) and a number of elements ≤dimHk(X), which proves the first assertion. Moreover, dimZ2(u)⊗Z2[u] H

∗G(X) =

b(X) if and only if ρ : H∗G(X) → H∗(X) is surjective, that is X is equivariantlyformal.

Proposition 7.3.7. Let X be a finite dimensional G-complex with b(X) <∞.Then

(7.3.3) b(XG) ≤ b(X)


Proof.

b(X) ≥ dimZ2(u)⊗Z2[u] H∗G(X) by Lemma 7.3.6

= dimZ2(u)⊗Z2[u] H∗G(X

G) by Theorem 7.3.1

= b(XG) ,

the last equality coming from Lemma 7.3.6, since XG is equivariantly formal. FromLemma 7.3.6 again, the above inequality is an equality if and only if X is equivar-iantly formal.

Formula (7.3.3) is an example of Smith inequalities, a development of the workof P. Smith started in 1938 [178]. The following corollary is a classical result in thetheory.

Corollary 7.3.8. Let X be a finite dimensional G-complex. Then,

(1) If H∗(X) ≈ H∗(pt), then H∗(XG) ≈ H∗(pt).(2) If X has the cohomology of a sphere, so does XG.

Proof. If X has the cohomology of a point, it is equivariantly formal and, byProposition 7.3.7, b(XG) = 1 which proves (1). For Point (2), Proposition 7.3.7implies that b(XG) ≤ 2. Statement (2) is true if b(XG) = 2 or if b(XG) = 0 (since∅ = S−1). It remains to prove that b(XG) = 1 is impossible if H∗(X) ≈ H∗(Sn).

If b(XG) = 1, then X is not equivariantly formal. Using exact sequence (7.1.7),this implies that H∗G(X) ≈ H∗(RPn). As in Example 7.3.2, we deduce thath∗G(X) = 0, contradicting Theorem 7.3.1 (h∗G(X

G) = Z2[u, u−1] if b(XG) = 1).

Here is another consequence of Theorem 7.3.1.

Proposition 7.3.9. Let X be a finite dimensional G-complex. Then, the fol-lowing statements are equivalent.

(1) X is equivariantly formal.(2) r : H∗G(X)→ H∗G(X

G) is injective.


Proof. From Corollary 7.1.8, we already know that (2) implies (1). For theconverse, suppose that X is equivariantly formal. Then H∗G(X) is a free Z2[u]-module by Proposition 7.1.6 and thus j : H∗G(X) → Z2[u, u

−1] ⊗Z2[u] H∗G(X) =

h∗G(X) is injective. Therefore, in the commutative diagram

H∗G(X)r //

j

H∗G(XG)

jG

h∗G(X)

r // h∗G(XG)

the left vertical arrow is injective. When X is a finite dimensional G-complex, thebottom arrow is an isomorphism by Theorem 7.3.1. Hence j is injective.

We shall now prove a localization theorem analogous to Theorem 7.3.1 for S1-spaces. Since we are working with Z2-cohomology, an important role is played bythe subgroup ±1 = S0 of S1. We also need the notion of a Γ-CW-complex for Γa topological group. If Γ0 is a closed subgroup of Γ, the Γ-space Γ/Γ0×Dn is calleda Γ-cell of dimension n (of type Γ0), with boundary Γ/Γ0×Sn−1 (the group Γ actson the left on Γ/Γ0 and trivially on Dn). One can attach a Γ-cell to a Γ-space Yvia a G-equivariant map ϕ : Γ/Γ0×Sn−1 → Y . A Γ-CW-structure on a Γ-space Xis a filtration

(7.3.4) ∅ = X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ X =⋃

n∈N

Xn

by Γ-subspaces, such that, for each n, the space Xn (the n-skeleton) is Γ-homeo-morphic to a Γ-space obtained from Xn−1 by attachment of a family Γ-cells ofdimension n (of various type). A Γ-space endowed with a Γ-CW-structure is aΓ-CW-complex (or just a Γ-complex).The topology of X is the weak topology withrespect to the filtration (7.3.4).

If X is a Γ-complex, then X/Γ admits a CW-structure so that the projectionX → X/Γ is cellular. For Γ = G of order 2, the above definition is easily madeequivalent to that of p. 218 (compare also [37, pp. 101–102]). If Γ is a compact Liegroup acting smoothly on a smooth manifold X , then X admits a Γ-CW-structure(see [107]).

The Milnor classifying space BS1 for principal S1-bundles is homotopy equiv-alent to CP∞. Then, by Proposition 6.1.5, H∗S1(pt) = Z2[v] with v of degree 2. Fora pair (X,Y ) of S1-spaces, we thus define

h∗S1(X,Y ) = Z2[v, v−1]⊗Z2[v] H

∗S1(X,Y ) ,

with the notation h∗S1(X) = h∗S1(X, ∅). As in (7.3.1), h∗S1(X,Y ) is a Z-gradedZ2[v, v

−1]-algebra.

Theorem 7.3.10. Let X be a finite dimensional S1-complex such that XS1

=

XS0

. Then, the inclusion XS1 ⊂ X induces an isomorphism

h∗S1(X)≈−→ h∗S1(XS1

)

of Z-graded Z2[v, v−1]-algebras.

The hypothesis XS1

= XS0

is necessary in the above localization theoremTheorem 7.3.10. For example, let X = S1 with S1-action g · z = g2 z. Then

XS1 ≈ BS0 ≈ RP∞ by Example 7.2.4, so h∗S1(X) ≈ Z2[v, v−1, u] while XS1

= ∅.


Proof. The proof follows the plan of that of Theorem 7.3.1, by induction onskeleta of X , starting trivially with the (−1)-skeleton which is the empty set. Theinduction step reduces to proving that, if the theorem is true for X , it is thentrue for Z = X ∪ C where C is a family of S1-cells. As for Theorem 7.3.1, thiseventually reduces to proving that h∗G(C,BdC) = 0 when C is not an isotropy cell.

As XS1

= XS0

, the isotropy group Γ of C is then a finite group of odd order. Thepair (C,BdC) is of the form (S1/Γ×Dn, S1/Γ×Sn−1) and, as seen in Example 7.2.4,CS1 ≈ BΓ and (BdC)S1 ≈ BΓ × Sn−1. By Lemma 7.2.2, H∗S1(C) = H∗(pt) andH∗S1(BdC) = H∗(Sn−1). In particular, the multiplication by u is the zero map andthus h∗S1(C) = h∗S1(BdC) = 0. From the exact sequence of (C,BdC) for h∗G, itfollows that h∗G(C,BdC) = 0.

The Smith theory for S1-complexes with XS1

= XS0

is very similar to that ofS0-spaces. Let Z2(v) be the fraction field of Z2[v].

Lemma 7.3.11. Let X be a finite dimensional S1-complex with b(X) <∞ and

XS1

= XS0

. Then, as a vector space over Z2(v),

dimZ2(v)⊗Z2[v] H∗S1(X) ≤ b(X)


Proof. The proof is the same as that of Lemma 7.3.6. The transfer exactsequence is replaced by the Gysin exact sequence of the S1-bundle X ×S∞ → XS1

which, as indicated in (7) p. 230, is induced from the universal bundle by p : XS1 →BS1 ≈ CP∞. Therefore, this Gysin sequence looks like

Hk−1S1 (X)

v−−→ Hk+1S1 (X)

ρ−→ Hk+1(X) −→ HkS1(X)

v−−→ Hk+2S1 (X)

and permits us the same arguments as for Lemma 7.3.6.

The proofs of 7.3.12–7.3.14 below are then the same as those of 7.3.7–7.3.9,replacing Theorem 7.3.1 by Theorem 7.3.10.

Proposition 7.3.12. Let X be a finite dimensional S1-complex with b(X) <∞and XS1

= XS0

. Then

(7.3.5) b(XS1

) ≤ b(X)


Corollary 7.3.13. Let X be a finite dimensional S1-complex with XS1

=

XS0

. Then,

(1) If H∗(X) ≈ H∗(pt), then H∗(XS1

) ≈ H∗(pt).(2) If X has the cohomology of a sphere, so does XS1

.

Proposition 7.3.14. Let X be finite dimensional S1-complex with XS1

= XS0

.Then, the following statements are equivalent.

(1) X is equivariantly formal.

(2) r : H∗S1(X)→ H∗S1(XS1

) is injective.


7.4. Equivariant cross products and Kunneth theorems

Let Γ1 and Γ2 be two topological groups; we set Γ12 = Γ1 × Γ2. Let X be aΓ1-space and Y be a Γ2-space. Then X × Y is a Γ12-space by the product action(γ1, γ2) · (x, y) = (γ1x, γ2y). The projections P1 : X × Y → X and P2 : X × Y → Yare equivariant with respect to the projection homomorphisms Γ12 → Γi. Passingto the Borel construction gives a map

(7.4.1) (X × Y )Γ12

P−→ XΓ1 × YΓ2

The map P is a homotopy equivalence, being induced by the homotopy equivalence

(7.4.2) P : EΓ12 × (X × Y )→ (EΓ1 ×X)× (EΓ2 × Y )

[see Comment 12.0.8] given by

P((ti(ai, bi), (x, y)

)=

((tiai, x), (tibi, y)

),

where (ti) ∈ ∆∞, (ai, bi) ∈ Γ12 and (x, y) ∈ X × Y . The case X = Y = pt providesa homotopy equivalence P0 : B(Γ12)→ BΓ1 ×BΓ2 and a commutative diagram

(7.4.3)

(X × Y )Γ12

P // XΓ1 × YΓ2

B(Γ12)

P0 // BΓ1 ×BΓ2

.

The cross product H∗(BΓ1) ⊗H∗(BΓ2)×−→ H∗(BΓ1 × BΓ2) post-composed with

H∗P0 gives a ring homomorphism

h : H∗Γ1(pt)⊗H∗Γ2

(pt) −→ H∗Γ12(pt) .

Note that, if BΓ1 or BΓ2 is of finite cohomology type, the Kunneth theorem impliesthat h is an isomorphism. The homotopy equivalence (7.4.1) together with (7.4.3)and the Kunneth theorem gives the following lemma.

Lemma 7.4.1. The composite map

×Γ12 : H∗Γ1(X)⊗H∗Γ2

(Y )× // H∗(XΓ1 × YΓ2)

H∗P

≈// H∗Γ12

(X × Y ) .

is an homomorphism of algebras. The (H∗Γ1(pt) ⊗ H∗Γ2

(pt))-module structure onH∗Γ1

(X)⊗H∗Γ2(Y ) and the H∗Γ12

(pt)-module structure on H∗Γ12(X×Y ) are preserved

via h. If YΓ2 is of finite cohomology type, then ×Γ12is an isomorphism.

Example 7.4.2. Let Γ1 = Γ2 = G = ±1. We let Γ1 act on the linear sphereX = Sm0 with XΓ1 = ω1

±, and let Γ2 act on Y = Sn0 with Y Γ2 = ω2± (see

Example 7.1.16). Set H∗Γ1(pt) = Z2[u1] and H

∗Γ2(pt) = Z2[u2] (ui of degree 1). As

seen in Example 7.1.16, H∗Γ1(X) and H∗Γ2

(Y ) admit the following presentations

H∗Γ1(X) ≈ Z2[u1, A1, B1]

/(A1 +B1 + um1 , A

21 + um1 A1)

andH∗Γ2

(Y ) ≈ Z2[u2, A2, B2]/(A2 +B2 + un2 , A

22 + un2A2) ,

where A1, B1 are of degree m and A2, B2 are of degree n. To shorten the formulae,we also denote by A1 the element A1 ×Γ12

1 ∈ H∗Γ12(X × Y ), by A2 the element

1×Γ12A2 ∈ H∗Γ12

(X × Y ), etc. By Lemma 7.4.1. we thus get the presentation

(7.4.4) H∗Γ12(X × Y ) ≈ Z2[u1, u2, A1, B1, A2, B2]

/I ,

7.4. EQUIVARIANT CROSS PRODUCTS AND KUNNETH THEOREMS 243


A1 +B1 + um1 , A21 + um1 A1 , A2 +B2 + un2 and A2

2 + un2A2 .

One can of course eliminate the Bi’s and get the shorter presentation

H∗Γ12(X × Y ) ≈ Z2[u1, u2, A1, A2]

/(A2

1 + um1 A1, A22 + un2A2) .

The commutative diagram

(7.4.5)

H∗Γ1(X)⊗H∗Γ2

(Y )×Γ12

≈//

rX⊗rY

H∗Γ12(X × Y )

r

H∗Γ1(XΓ1)⊗H∗Γ2

(Y Γ2)×Γ12

≈// H∗Γ12

((X × Y )Γ12 )

permits us to compute the image under r of the various classes of H∗Γ12(X × Y ).

Set

H∗Γ1(XΓ1) = Z2[u1]ω

1− ⊕ Z2[u1]ω

1+ and H∗Γ2

(Y Γ2) = Z2[u2]ω2− ⊕ Z2[u2]ω

2+ .

Denote the four points of (X × Y )Γ12 = XΓ1 × Y Γ2 by ω−− = (ω1−, ω

2−), ω−+ =

(ω1−, ω

2+), etc. With the notation R = Z2[u1, u2], one has

(7.4.6) H∗Γ12((X × Y )Γ12) ≈ Rω−− ⊕Rω+− ⊕Rω−+ ⊕Rω++

One has

r(A1) = rX(A1)×Γ12rY (1) = um1 ω

1+ ×Γ12

(1ω2− + 1ω2

+) = um1 ω+− + um1 ω++ .

Hence, the coordinates of r(A1) using (7.4.6) are (0, um1 , 0, um1 ). Similar computa-

tions provide the following table.

x coord. of r(x) in (7.4.6)

1 1 1 1 1

ui ui ui ui uiA1 0 um1 0 um1A2 0 0 un2 un2A1A2 0 0 0 um1 u

n2

B1 um1 0 um1 0

B2 un2 un2 0 0

B1B2 um1 un2 0 0 0

We now concentrate our interest on the case where Γ1 = Γ2 = Γ and seeX × Y as a Γ-space using the diagonal homomorphism ∆: Γ → Γ × Γ. We get ahomomorphism ∆∗ : H∗Γ×Γ(X × Y )→ H∗Γ(X × Y ). The composite map(7.4.7)

H∗Γ(X)⊗H∗Γ(Y )

×Γ

22×Γ×Γ // H∗(XΓ × YΓ) H∗P

≈// H∗Γ×Γ(X × Y )

∆∗ // H∗Γ(X × Y )


is called the equivariant cross product. For X = Y = pt, one has the commutativediagram

(7.4.8)

H∗Γ(pt)⊗H∗Γ(pt)×Γ //

≈

H∗Γ(pt)

≈

H∗(BΓ)⊗H∗(BΓ) // H∗(BΓ)

.

Indeed, one has

(7.4.9)BΓ

B∆ //

∆BΓ

55B(Γ× Γ)

P

≈// BΓ×BΓ

and H∗∆BΓ(a× b) = a b by (4.6.5).

The equivariant cross product ×Γ will be useful in § 8.3 but one may wish toget some Kunneth theorem. As this is not even the case for X = Y = pt, someadaptation is needed. Lemma 7.4.1 together with diagram (7.4.8) implies that

(7.4.10) (w · a)×Γ b = a×Γ (w · b) = w · (a×Γ b) .

for all a ∈ H∗Γ(X), b ∈ H∗Γ(Y ) and w ∈ H∗Γ(pt). Therefore, ×Γ descend to thestrong equivariant cross product

×Γ : H∗Γ(X)⊗H∗Γ(pt) H∗Γ(Y )→ H∗Γ(X × Y ) .

The tensor product H∗Γ(X)⊗H∗Γ(pt)H∗Γ(Y ) still carries an H∗Γ(pt)-action, defined by

w · (a ⊗ b) = (w · a) ⊗ b = a ⊗ (w · b). Lemma 7.4.1 together with (7.4.10) impliesthat ×Γ is a morphism of H∗Γ(pt)-algebras.

Theorem 7.4.3 (Equivariant Kunneth theorem). Let Γ be a topological groupsuch that BΓ0 is of finite cohomology type for any closed subgroup Γ0 of Γ. LetX and Y be Γ-spaces, where X is a finite dimensional Γ-CW-complex. Supposethat Y is of finite cohomology type and is equivariantly formal. Then, the strongequivariant ×Γ cross product is an isomorphism of H∗Γ(pt)-algebras.

Proof. As ×Γ is a morphism of H∗Γ(pt)-algebra, it suffices to prove that itis a GrV-isomorphism. We follow the idea of the proof of the ordinary Kunneththeorem 4.6.7, fixing the Γ-space Y and comparing the “equivariant cohomologytheories”

h∗(X,A) = H∗Γ(X,A)⊗H∗Γ(pt) H∗Γ(Y ) and k∗(X,A) = H∗Γ(X × Y,A× Y )

defined for a Γ-pair (X,A). The definition of the strong equivariant cross productextends to pairs and we get a morphism of H∗Γ(pt)-algebras

×Γ : h∗(X,A)→ k∗(X,A)

One gets a commutative diagram(7.4.11)

h∗(X)

×Γ

// h∗(A)

×Γ

δ∗ // h∗+1(X,A)

×Γ

// h∗+1(X)

×Γ

// h∗+1(A)

×Γ

h∗(X) // k∗(A)

δ∗ // k∗+1(X,A) // k∗+1(X) // k∗+1(A)


where the lines are exact [see Remark 12.0.7]. That the square diagram with theδ∗’s commutes comes from the definition of ×Γ, using the commutativity of Dia-gram (4.6.13).

The theorem is proven by induction on the dimension ofX . IfX is 0-dimensional,it is a disjoint union of homogeneous Γ-spaces. As the disjoint union axiom holds forour theories, the induction starts by proving the theorem for X = Γ/Γ0, where Γ0

is a closed subgroup of Γ. As Y is Γ-equivariantly formal, it is also Γ0-equivariantlyformal and one has

h∗(X) = H∗Γ(Γ/Γ0)⊗H∗Γ(pt) H∗Γ(Y )

≈ H∗(BΓ0)⊗H∗(BΓ)

(H∗(BΓ)⊗H∗(Y )

)

≈ H∗(BΓ0)⊗H∗(Y )

≈ H∗Γ0(Y ) .

On the other hand, consider the map α : EΓ × (Γ× Y )→ EΓ × Y given by

α(z, (γ, y)) = (zγ, γ−1y) .

It satisfies α(z, (δγ, δy)) = α(zδ, (γ, y)) and α(z, (γγ0, y)) = (zγγ0, γ−10 γ−1y); it

thus descends to a map

α : EΓ ×Γ (Γ/Γ0 × Y )≈−→ EΓ×Γ0 Y

which is a homeomorphism: its inverse is induced by the map β(z, y) = (z, ([e], y)),where e ∈ Γ is the unit element. Hence, k∗(X) is also isomorphic, as an H∗Γ(pt)-algebra, to H∗Γ0

(Y ). It remains to show that ×Γ is a GrV-isomorphism. As Yand BΓ0 are both of finite cohomology type, the graded vector space HΓ0(Y ) ≈H∗(BΓ0) ⊗ H∗(Y ) is finite dimensional in each degree. Therefore, it suffices toprove that ×Γ is surjective.

If Z is a Γ-space, we denote by i : Z → ZΓ the inclusion i(z) = [(1e, 0, ...), z](it induces the forgetful homomorphism H∗i = ρ : H∗Γ(Z) → H∗(Z)). One has acommutative diagram

Γ/Γ0 × Y oo s

i

Y

i

EΓ ×Γ (Γ/Γ0 × Y ) oo β

EΓ×Γ0 Y

where s is the slice inclusion s(y) = ([e], y]). We thus get a commutative diagram

H∗(Γ/Γ0)⊗H∗(Y )× // H∗(X × Y )

H∗s // H∗(Y )

H∗Γ(Γ/Γ0)⊗Γ H∗(Y )Γ

ρ⊗ρ

OO

×Γ // H∗Γ(X × Y )

ρ

OO

H∗β

≈// H∗Γ0

(Y )

ρ

OO

Let B be a GrV-basis of H∗(Y ). Let σ : Hk(Y ) → HkΓ(Y ) be a section of ρ.

For b ∈ B, one has

H∗s(ρ(1)× ρ(σ(b)) = H∗s(1×Γ b) = b ,


the last equality coming from Lemma 4.7.2. Therefore, H∗s× (ρ⊗ρ) is surjectiveand the formula

σ(a) = H∗β(1×Γ σ(a))

defines a section σ : H∗(Y )→ H∗Γ0(Y ) of ρ. The Leray-Hirsch theorem then implies

that H∗Γ0(Y ) is generated, as a H∗Γ(pt)-module, by σ(B). Hence, ×Γ is surjective.

The induction step reduces to proving that, if the theorem is true for A, it isthen true for X = A ∪ C where C is a family of Γ-cells. By the five lemma inDiagram (7.4.11), it suffices to prove that ×Γ : h

∗(X,A) → k∗(X,A) is an isomor-phism. By excision and the disjoint union axiom one can restrict ourselves to thecase of a pair (X0, A0) = Γ/Γ0 × (Dn, Sn−1) (a Γ-cell). By the five lemma in Dia-gram (7.4.11) for the pair (X0, A0), it suffices to prove the theorem for X0 and forA0. The former is covered by the 0-dimensional case (sinceX0 is Γ-homotopy equiv-alent to Γ/Γ0) and the latter is (n− 1)-dimensional, thus covered by the inductionhypothesis.

Remark 7.4.4. If Γ = e, Theorem 7.4.3 reduces to the ordinary Kunneththeorem 4.6.7. Therefore, the hypothesis that Y is of finite cohomology type isessential. Theorem 7.4.3 may also be wrong if Y is not equivariantly formal. Forexample, set Γ = ±1, X = S1 and Y = S2, with the antipodal involution. Theseare free Γ-spaces and, by Lemma 7.1.4,

H∗Γ(X) ≈ H∗(S1/± 1) ≈ Z2[u]/(u2) and H∗Γ(Y ) ≈ H∗(S2/± 1) ≈ Z2[u]/(u

3) .

Moreover,H∗Γ(pt) = H∗(RP∞) ≈ Z2[u] and, using Lemma 7.1.4 again together withProposition 4.3.10, the Z[u]-morphismsH∗Γ(pt)→ H∗Γ(X) and H∗Γ(pt)→ H∗Γ(Y ) aresurjective. Therefore,

H∗Γ(X)⊗H∗Γ(pt) H∗Γ(Y ) ≈ Z2[u]/(u

2)×Z2[u] Z2[u]/(u3) ≈ Z2[u]/(u

2) .

In particular, H∗Γ(X)⊗H∗Γ(pt) H∗Γ(Y ) vanishes in degree 3, while H3Γ(X × Y ) =

Z2. Indeed H∗Γ(X ×Y ) ≈ H∗((X × Y )/± 1) and (X ×Y )/± 1 is a closed manifold

of dimension 3.The hypothesis that BΓ0 is of finite cohomology type is fulfilled if Γ0 is a

compact Lie group. Note that it is only used in the proof for the stabilizers ofpoints of X . For other kinds of equivariant Kunneth theorems, see [176].

Example 7.4.5. Consider the diagonal action of the group G = ±1 on theproduct of linear spheres Sm0 × Sn0 . Set H∗G(pt) = Z2[u], with u of degree 1. ByExample 7.4.2 and Theorem 7.4.3, one has

H∗G(Sm0 × Sn0 ) ≈ Z2[u,A1, B1, A2, B2]

/I ,


A1 +B1 + um , A21 + umA1 , A2 +B2 + un and A2

2 + unA2 .

Using the notations of Example 7.4.2 for the fixed points, one has

(7.4.12) H∗G((Sm0 × Sn0 )G) ≈ Z2[u]ω−− ⊕ Z2[u]ω+− ⊕ Z2[u]ω−+ ⊕ Z2[u]ω++


and one has the following table for r : H∗G(Sm0 × Sn0 )→ H∗G((S

m0 × Sn0 )G).

x coord. of r(x) in (7.4.12)

1 1 1 1 1

u u u u u

A1 0 um 0 um

A2 0 0 un un

A1A2 0 0 0 um+n

B1 um 0 um 0

B2 un un 0 0

B1B2 um+n 0 0 0

For a generalization of this example, see Proposition 10.3.5.

We now define the equivariant reduced cross product, related to the equivariantcohomology of a smash product. Let X be a Γ1-space and Y be a Γ2-space, pointedby x ∈ XΓ1 and y ∈ Y Γ2 . Then, X ∨ Y is a Γ12-invariant subspace of X × Y .Consider the space

XΓ1∨YΓ2 = (XΓ1 × yΓ2) ∪ (xΓ1 × YΓ2) ⊂ XΓ1 × YΓ2 .

If the pairs (X, x) and (Y, y) are equivariant well cofibrant pairs, we saythat (X, x) and (Y, y) are equivariantly well pointed.

Lemma 7.4.6. Let (X, x) be an equivariantly well pointed Γ1-space and (Y, y) bean equivariantly well pointed Γ2-space. Then, the map P : (X×Y )Γ12 → XΓ1×YΓ2

of (7.4.1) sends (X ∨ Y )Γ12 onto XΓ1∨YΓ2 and induces an isomorphism

H∗P : H∗(XΓ1 ∨YΓ2)≈−→ H∗Γ12

(X ∨ Y )

Proof. That P ((X ∨ Y )Γ12) = XΓ1∨YΓ2 follows directly from the definitionof P , using (7.4.2). This gives a commutative diagram(7.4.13)

xΓ1 × yΓ2//

xΓ1 × YΓ2

(x × y)Γ12

P

h.e.

hh//

(x × Y )Γ12

P

h.e.

66♥♥♥♥♥♥♥♥♥♥

(X × y)Γ12

P

h.e.vv♠♠♠♠♠♠♠

♠♠♠♠

// (X ∨ Y )Γ12

P

((PPPP

PPPP

PP

XΓ1 × yΓ2// XΓ1 ∨YΓ2

where the unlabeled arrows are inclusions and h.e. means “homotopy equivalence”.Our hypotheses and Lemma 7.2.11 imply that pairs like ((X × y)Γ12 , (x ×y)Γ12), etc, are good. Hence, the hypotheses of Proposition 3.1.54 to get Mayer-Vietoris sequences are fulfilled. We thus get a morphism from the Mayer-Vietorissequence for the outer square of (7.4.13) to that of the inner square, and the propo-sition follows from the five-lemma.


Remark 7.4.7. The map P : (X ∨ Y )Γ12 → XΓ1 ∨YΓ2 of Lemma 7.4.6 is ac-tually a weak homotopy equivalence, since the squares in (7.4.13) are homotopyco-Cartesian diagrams (see [38, Prop. 5.3.3]).

As X ∨ Y is a Γ12-invariant subspace of X × Y , the wedge product X ∧ Yinherits a Γ12-action.

Lemma 7.4.8. Let (X, x) be an equivariantly well pointed Γ1-space and (Y, y)be an equivariantly well pointed Γ2-space. Then, there is a natural isomorphism

H∗(XΓ1 × YΓ2 , XΓ1∨YΓ2)≈−→ H∗Γ12

(X ∧ Y )) .

Proof. By Lemma 7.4.6, the map P produces a morphism from the cohomol-ogy sequence of the pair (XΓ1×YΓ2 , XΓ1∨ YΓ2) to that of the pair

((X×Y )Γ12 , (X∨

Y )Γ12

). By Lemma 7.4.6 again and the fact that the map P of (7.4.1) is a homotopy

equivalence, the five lemma implies that

H∗P : H∗(XΓ1 × YΓ2 , XΓ1∨YΓ2)→ H∗((X × Y )Γ12 , (X ∨ Y )Γ12

)

is an isomorphism. By Lemma 7.2.12, the pair (X × Y,X ∨ Y ) is Γ12-equivariantlywell cofibrant. As X ∧ Y is not empty, Corollary 7.2.16 provides a natural isomor-phism between H∗((X × Y )Γ12 , (X ∨ Y )Γ12

)and H∗Γ12

(X ∧ Y )).

Using the isomorphism of Lemma 7.4.8 as well as those of (7.2.11), one con-structs the commutative diagram

(7.4.14)

H∗(XΓ1 , xΓ1)⊗H∗(YΓ2 , yΓ2)

×

≈ // H∗Γ1(X)⊗ H∗Γ2

(Y )

×Γ12

H∗(XΓ1 × YΓ2 , XΓ1 ∨YΓ2)≈ // H∗Γ12

(X ∧ Y )

which defines the equivariant reduced cross product ×Γ12 . The relative cross product(left vertical arrow) is indeed defined as in (4.6.6), since, as (Y, y) is equivariantlywell pointed, the couple (YΓ2 , yΓ2) is a good pair by Lemma 7.2.11.

In the case where Γ1 = Γ2 = Γ, one can see X ∧ Y as a Γ-space via thediagonal homomorphism ∆: Γ→ Γ×Γ. Composing ×Γ12 with ∆∗ : H∗Γ×Γ(X∧Y )→H∗Γ(X ∧ Y ), we get the equivariant reduced cross product

(7.4.15) H∗Γ(X)⊗ H∗Γ(Y )×Γ−−→ H∗Γ(X ∧ Y ) .

Lemma 7.4.9. Let (X, x) and (Y, y) be equivariantly well pointed Γ-spaces.Then,

(1) there is an equivariant reduced cross product

H∗Γ(X)⊗ H∗Γ(Y )×Γ−−→ H∗Γ(X ∧ Y )

which is a bilinear map.(2) the diagram

(7.4.16)

H∗Γ(X)⊗ H∗Γ(Y )

ρ⊗ρ

×Γ // H∗Γ(X ∧ Y )

ρ

H∗(X)⊗ H∗(Y )× // H∗(X ∧ Y )

7.5. EQUIVARIANT BUNDLES AND EULER CLASSES 249

is commutative, where ρ is the forgetful homomorphism.(3) the hypotheses on X, Y are inherited by XΓ, Y Γ and there is a commu-

tative diagram(7.4.17)


r⊗r

×Γ // H∗Γ(X ∧ Y )

r

H∗Γ(XΓ)⊗ H∗Γ(Y Γ)

OO≈

×Γ // H∗Γ(XΓ ∧ Y Γ)OO≈

[H∗(XΓ)⊗H∗(BΓ)]⊗ [H∗(Y Γ)⊗H∗(BΓ)]×⊗ // H∗(XΓ ∧ Y Γ)⊗H∗(BΓ)

Proof. The equivariant reduced cross product of (1) is obtained by post-

composing ×Γ12 of (7.4.14) (with Γ1 = Γ2 = Γ) with ∆∗ : H∗Γ×Γ(X ∧Y )→ H∗Γ(X ∧Y ).

Let α : Γ′ → Γ is a continuous homomorphism. Then (X, x) and (Y, y) areΓ′-equivariantly well cofibrant. Our constructions are natural enough so that thereis a commutative diagram

(7.4.18)


α∗⊗α∗

×Γ // H∗Γ(X ∧ Y )

α∗

H∗Γ′(X)⊗ H∗Γ′(Y )×Γ′ // H∗Γ′(X ∧ Y )

.

For Γ′ = I, the homomorphism α∗ coincides with the forgetful homomorphism ρ(see 7.2.8), which proves (2).

To prove (3), we note that the upper square of (7.4.16) commutes by obviousnaturality of the equivariant reduced cross product with respect to equivariantmaps. The commutativity of the lower square is obtained using the considerationsof (7.4.8) and (7.4.9).

Example 7.4.10. Let (Z, z) be a well pointed space, considered with the trivial

action of G = I, τ. Then, H∗G(Z) ≈ H∗(Z)[u] and the bottom square in (7.4.17)becomes

H∗G(Z)⊗ H∗G(Z)×G //

OO≈

H∗G(Z ∧ Z)OO≈

H∗(Z)[u]⊗ H∗(Z)[u]×[u]// H∗(Z ∧ Z)[u]

,

where, for a, b ∈ H∗(Z), ×[u] is defined by

aum ×[u] bun = (a×b)um+n .

7.5. Equivariant bundles and Euler classes

Although we are mostly interested in equivariant vector bundles, passing throughequivariant principal bundles is easier and more powerful. Let A be a topologicalgroup. A principal A-bundle ζ over over a space Y consists of a continuous map


p : P → Y , a continuous right action of A on P such that p(uα) = p(u) for allu ∈ P and all α ∈ A; in addition, the following local triviality should hold: for eachx ∈ X there is a neighbourhood U of x and a homeomorphism ψ : U ×A→ p−1(U)such that pψ(x, α) = x and ψ(x, αβ) = ψ(x, α)β. In consequence, A acts freelyon P and transitively on each fiber. Also, p is a surjective open map, descending to

a homeomorphism P/A≈−→ X (use [44, § I Chapter VI]). Two principal A-bundles

ζ = (Pp−→ X) and ζ = (P

p−→ X) are isomorphic if there exists an A-equivarianthomeomorphism h : P → P such that ph = p.

Let Γ be a topological group and let X be a (left) Γ-space. An A-principal

bundle ζ : Pp−→ X is called a Γ-equivariant principal A-bundle if it is given a left

action Γ × P → P commuting with the free right action of A and such that theprojection p is Γ-equivariant (a more general setting is considered in e.g. [126, 37,

139]). Two Γ-equivariant principal A-bundles ζ = (Pp−→ X) and ζ′ = (P

p−→ X)are isomorphic if there exists a (Γ,A)-equivariant homeomorphism h : P → P suchthat ph = p.

Example 7.5.1. Let p : P → pt be a Γ-equivariant principal A-bundle over apoint. The A-action on P is free and transitive. Hence, choosing a point s ∈ Pprovides a continuous map µ : Γ → A by the equation γs = sµ(γ). For γ, γ′ ∈ Γ,one has

sµ(γγ′) = (γγ′)s = γ(γ′s) = (γs)µ(γ′) = (sµ(γ))µ(γ′) = s(µ(γ)µ(γ′)) ,

which proves that µ is a homomorphism. Another point s ∈ P is of the form s = sαfor some α ∈ A. The map µ obtained from s is related to µ by

sαµ(γ) = sµ(γ) = γs = γsα = sµ(γ)α

and hence µ(γ) = α−1µ(γ)α. If p : P → pt is another Γ-equivariant principal A-bundle and if h : P → P is a (Γ, A)-equivariant homeomorphism, then γh(s) =h(s)µ(γ). This provides a map from the isomorphism classes of Γ-equivariant prin-cipal A-bundles over a point to the set hom(Γ, A)/A of the conjugation classes ofcontinuous homomorphisms from Γ to A. This map is a bijection. A homomor-phism µ : Γ→ A is realized by the bundle A→ pt with the Γ-action γ · α = µ(γ)α(hence, if e is the unit element of A, one has indeed γ · e = eµ(γ)). This proves thesurjectivity. The proof of the injectivity is left to the reader.

Let ζ : Pp−→ X be a Γ-equivariant principal A-bundle. Being Γ-equivariant, the

map p induces a map pΓ : PΓ → XΓ. Let i : X → XΓ be an inclusion as in (7.2.8).

Lemma 7.5.2. The map pΓ : PΓ → XΓ is a principal A-bundle, denoted by ζΓ.Moreover, ζ is isomorphic to the induced principal A-bundle i∗ζΓ.

Example 7.5.3. Let ξ be a Γ-equivariant principal A-bundle over a point,corresponding to [µ] ∈ hom(Γ, A)/A (see Example 7.5.1). It is then isomorphic toA→ pt with the Γ-action γ ·α = µ(γ)α. Then ξΓ is the principal A-bundle over BΓinduced by the map Bµ : BΓ→ BA. Indeed, the map f : EΓ×ΓA→ EA given byf([(tiγi), α]) = (tiµ(γi)α) is A-equivariant and covers the map Bµ.

Before proving Lemma 7.5.2, let us recall the standard local cross-sections forthe Milnor construction of the universal Γ-bundle p : EΓ → BΓ. For i ∈ N, let(EΓ)i = (tjγj) ∈ EΓ | ti 6= 0 and let (BΓ)i = p((EΓ)i). There is a cross-section si of p over (BΓ)i sending b ∈ (BΓ)i to the unique element in (tjΓj) ∈


p−1(b) with γi = 1. If Z is a Γ-space, the map ψi : (BΓ)i × Z≈−→ (EΓ)i ×Γ Z

given by ψi(b, u) = [si(b), u] is a homeomorphism: its inverse is induced by thecorrespondence [(tjγj), u] 7→ (p(tjγj), γiz).

Proof of Lemma 7.5.2. The right A-action on PΓ = EΓ×Γ P is defined by[z, u]α = [z, uα]. For i ∈ N, one has the commutative diagram

(BΓ)i × Pψi

≈//

id×p

(EΓ)i ×Γ P

pΓ

(BΓ)i ×Xψi

≈// (EΓ)i ×Γ X

.

The upper homeomorphism is A-equivariant and id× p is a principal A-bundle. As(EΓ)i×ΓXi∈N is an open covering of XΓ, the map pΓ admits local trivializationsof a principal A-bundle.

We have proven that pΓ is a principal A-bundle. If z ∈ EΓ, let iz : X → XΓ

be the inclusion defined in (7.2.8). Then, iz : P → PΓ is an A-equivariant mapcovering iz, inducing an isomorphism of principal A-bundles ζ ≈ i∗zζΓ.

Remark 7.5.4. When A is abelian, the last assertion of Lemma 7.5.2 may bestrengthened: a principal A-bundle over a Γ-space X admits a structure of a Γ-equivariant bundle if and only if it is induced from a principal A-bundle over XΓ

(see [127]).

The construction ζ 7→ ζΓ enjoys some functorial properties. Let µ : Γ′ → Γ bea continuous homomorphism between topological groups. Let X ′ be a Γ′-space, Xa Γ-space and let f : X ′ → X be a continuous map which is Γ′-equivariant withrespect to µ. Recall from (7.2.1) that f induces a map fΓ′,Γ : XΓ′ → XΓ. If ζ isa Γ-equivariant principal A-bundle over X , then f∗ζ is a Γ′-equivariant principalA-bundle over X ′.

Lemma 7.5.5. (f∗ζ)Γ′ ≈ f∗Γ′,ΓζΓ.

Proof. The map f is covered by a Γ′-equivariant map of principal A-bundlef : P (f∗ζ) → P , where P (f∗ζ) denotes the total space of f∗ζ. By functoriality ofthe Borel construction (see 7.2.5), there is a commutative diagram

P (f∗ζ)Γ′

fΓ′,Γ // PΓ

oo = // P (ζΓ)

X ′Γ′

fΓ′,Γ // XΓoo = // XΓ

Thanks to the description of the A-actions (see the proof of Lemma 7.5.2), the

map fΓ′,Γ A-equivariant. Hence, fΓ′,Γ factor through an isomorphism (f∗ζ)Γ′ ≈f∗Γ′,ΓζΓ.

For another functoriality of ζΓ let ϕ : A → A′ be a continuous homomorphismbetween topological groups. It makes A′ a left A-space (α ·α′ = ϕ(α)α′). If ζ : P →X is a Γ-equivariant A-principal bundle, P ×A A′ is, in an obvious way, the total


space of a Γ-equivariant A′-principal bundle ϕ∗ζ. The tautological homeomorphismEΓ×Γ (P ×AA′) ≈ (EΓ×Γ P )×AA′ gives an isomorphism of A′-principal bundles

(7.5.1) (ϕ∗ζ)Γ ≈ ϕ∗ζΓ .

By a Γ-equivariant K-vector bundle ξ over X (K = R or C), we mean a Γ-equivariant map p : E = E(ξ)→ X which is a K-vector bundle, such that, for eachγ ∈ Γ and each x ∈ X , the map y 7→ γy is a K-linear map from p−1(x) to p−1(γx).The tangent bundle to a smooth Γ-manifold is an example for K = R.

It is convenient here to see a K-vector bundle ξ of rank r as associated to aprincipal GL(Kr)-bundle, the bundle Fra(ξ) of frames of ξ. Its total space is

Fra(ξ) = ν : Kr → E(ξ) | ν is a K-linear isomorphism onto some fiber of ξ .with the map pFra : Fra(ξ) → X given by pFra(ν) = pν(0). The right GL(Kr)-action on Fra(ξ) is by precomposition (we use the same notation for the bundleFra(ξ) and for its total space). The evaluation map sending [ν, t] ∈ Fra(ξ)×GL(Kr)

Kr to ν(t) ∈ E(ξ) defines an isomorphism of K-vector bundles

(7.5.2) Fra(ξ)×GL(Kr) Kr ≈−→ E(ξ) .

For more details and developments, see 9.1.9. If ξ is a Γ-equivariant K-vectorbundle, then Γ acts on Fra(ξ) by (γν)(t) = γ ·ν(t). Hence, Fra(ξ) is a Γ-equivariantprincipal GL(Kr)-bundle and (7.5.2) is an isomorphism of Γ-equivariant K-vectorbundles. The tautological homeomorphism

EΓ×Γ (Fra(ξ)×GL(Kr) Kr) ≈ (EΓ×Γ Fra(ξ))×GL(Kr) K

r

implies that

(7.5.3) E(ξ)Γ ≈ Fra(ξ)Γ ×GL(Kr) Kr .

Using Lemma 7.5.2, this proves the following lemma.

Lemma 7.5.6. Let ξ = (p : E(ξ) → X) be a Γ-equivariant K-vector bundle ofrank r. Then, the map pΓ : E(ξ)Γ → XΓ is a K-vector bundle of rank r, denoted byξΓ. Moreover, ξ is isomorphic to the induced vector bundle i∗ξΓ.

Let µ : Γ′ → Γ and f : X ′ → X as for Lemma 7.5.5. If ξ is a Γ-equivariant K-vector bundle of rank r over X , then f∗ξ is a Γ′-equivariant K-vector bundle overX ′ of the same rank. One has an isomorphism of Γ′-equivariant GL(Kr)-principalbundles f∗Γ′,ΓFra(ξ) ≈ Fra(f∗ξ). Therefore, Lemma 7.5.5 gives an isomorphism ofK-vector bundles

(7.5.4) (f∗ξ)Γ′ ≈ f∗Γ′,ΓξΓ .The correspondence ξ 7→ ξΓ commutes with some operations on vector bundles,

like the Whitney sum or the tensor product. We first define these operations in thecategory of Γ-equivariant K-vector bundles. Let ξ (respectively ξ′) be two two suchbundles over a Γ-spaceX , of ranks r (respectively r′). Set F = Fra(ξ), F ′ = Fra(ξ′),

G = GL(Kr) and G′ = GL(Kr′

). The diagonal inclusion ∆X : X → X ×X is Γ-equivariant with respect to the diagonal homomorphism ∆Γ : Γ → Γ × Γ. Hence,∆∗X(F × F ′) is a Γ-equivariant principal (G × G′)-bundle. The linear (G × G′)-action on Kr ⊕ Kr

′

given by (R,R′) · (v, v′) = (Rv,R′v′) defines a continuous

homomorphism ϕ⊕ : G×G′ → GL(Kr⊕Kr′). This permits us to define the Whitneysum

ξ ⊕ ξ′ = ϕ⊕∗ ∆∗X(F × F ′)×GL(Kr⊕Kr′ ) (K

r ⊕Kr′

)


as a Γ-equivariant K-vector space. The Γ-equivariant tensor product is definedaccordingly

(7.5.5) ξ ⊗ ξ′ = ϕ⊗∗ ∆∗X(F × F ′)×GL(Kr⊗Kr′ ) (K

r ⊗Kr′

) ,

using the homomorphism ϕ⊗ : G×G′ → GL(Kr⊗Kr′) induced by the unique linear

action of G×G on Kr ⊗Kr′

satisfying (R,R′) · (v ⊗ v′) = (Rv ⊗R′v′).Lemma 7.5.7. (ξ ⊕ ξ′)Γ ≈ ξΓ ⊕ ξ′Γ and (ξ ⊗ ξ′)Γ ≈ ξΓ ⊗ ξ′Γ.Proof. One has

(ξ ⊕ ξ′)Γ =[ϕ⊕∗ ∆

∗X(F × F ′)×GL(Kr⊕Kr′ ) (K

r ⊕Kr′

)]Γ

≈[ϕ⊕∗ ∆

∗X(F × F ′)

]Γ×GL(Kr⊕Kr′ ) (K

r ⊕Kr′

) by (7.5.3)

≈ ϕ⊕∗[∆∗X(F × F ′)

]Γ×GL(Kr⊕Kr′ ) (K

r ⊕Kr′

) by (7.5.1)

whileξΓ ⊕ ξ′Γ = ϕ⊕∗ ∆

∗XΓ

(FΓ × F ′Γ)×GL(Kr⊕Kr′ ) (Kr ⊕Kr

′

) .

Therefore, it is enough to construct an isomorphism of principal (G×G′)-bundles(7.5.6) ∆∗XΓ

(FΓ × F ′Γ) ≈[∆∗X(F × F ′)

]Γ.

This will prove the lemma for Whitney sum, and also for the tensor product, usingϕ⊗ instead of ϕ⊕.

As ∆X : X → X ×X is Γ-equivariant with respect to the diagonal homomor-phism ∆Γ : Γ→ Γ× Γ, it induces a map Φ = (∆X)Γ,Γ×Γ : XΓ → (X ×X)Γ×Γ. ByLemma 7.5.5, one has

(7.5.7)[∆∗X(F × F ′)

]Γ≈ Φ∗(F × F ′)Γ .

For Γ-spaces Z and Z ′, a natural homotopy equivalence P : (Z × Z ′)Γ ≃−→ ZΓ × Z ′Γwas constructed in (7.4.1). For Z = F and Z ′ = F ′, we thus get a homotopy

equivalence P : (F × F ′)Γ ≃−→ FΓ × F ′Γ which is (G×G′)-equivariant. The diagram

Φ∗(F × F ′)Γ

Φ // (F × F ′)Γ

P

≃// FΓ × F ′Γ

XΓ

Φ // (X ×X)Γ×ΓP

≃// XΓ ×XΓ

is commutative, thus each square is a morphism of principal (G × G′)-bundles.By the definition of P in § 7.4, one has P Φ = ∆XΓ . Therefore, Φ∗(F × F ′)Γ ≈∆∗XΓ

(FΓ×F ′Γ). This together with (7.5.7) gives the required isomorphism of (7.5.6).

Let ξ be a Γ-vector bundle of rank r over X . The equivariant Euler class eΓ(ξ)is the Euler class of ξΓ:

eΓ(ξ) = e(ξΓ) ∈ HrΓ(X) .

Example 7.5.8. Let χ : Γ→ GL(V ) be a representation of Γ on a vector spaceV of dimension r. This makes V a Γ-space. One can also see V as a vector bundlewith basis a point. This gives a Γ-vector bundle χ of rank r over a point and thena vector bundle χΓ = (VΓ → BΓ) of rank r over BΓ. Its equivariant Euler classeΓ(χ) is an element of Hr

Γ(pt) = Hr(BΓ). Its vanishing is related to the existenceof a non-zero fixed vector in V , as seen in the following lemma.


Lemma 7.5.9. If V Γ 6= 0, then eΓ(χ) = 0.

Proof. A non-zero fixed vector 0 6= v ∈ V Γ determines a nowhere-zero sectionof p : VΓ → BΓ (see (5) p. 230). This implies that eΓ(χ) = 0 by Lemma 4.7.39.

We now give a few recipes to compute an equivariant Euler class. A Γ-equivariant vector bundle p : E → X is called rigid if the Γ-action on X is trivial.

Lemma 7.5.10. Let ξ = (p : E → X) be a rigid Γ-vector bundle of rank r. Letχ : Γ → GL(Ex) be the representation of Γ on the fiber Ex over x ∈ X. Suppose

that Hk(X) = 0 for k < r. Then, the equation

eΓ(ξ) = 1× e(ξ) + eΓ(χ)× 1

holds in HrΓ(X) = Hr(BΓ×X).

Proof. The inclusion j : x → X satisfies j∗ξ = χ. It induces jΓ : BΓ =xΓ → XΓ satisfying j∗ΓξΓ = χΓ. Hence, H

∗jΓ(eΓ(ξ)) = eΓ(χ). By construction ofξΓ, one has i

∗ξΓ = ξ, where i : X → XΓ denotes the inclusion. Hence, H∗i(eΓ(ξ)) =e(ξ). Using the homeomorphism XΓ ≈ BΓ × X , the maps jΓ and i are sliceinclusions. The lemma then follows from Corollary 4.7.3.

Let χ : Γ → O(1) be a continuous homomorphism, permitting Γ to act on R.This gives a Γ-line bundle χ over a point (see Example 7.5.8). Its equivariantEuler class lives in H1

Γ(pt) = H1(BΓ). As χ is continuous, it factors through thehomomorphism π0χ : π0(Γ) → O(1) ≈ Z2. As EΓ is contractible, the homotopyexact sequence of Γ → EΓ → BΓ identifies π0(Γ) with π1(BΓ). One thus gets(using Lemma 4.3.1) the isomorphism(7.5.8)

homcont(Γ, O(1)) ≈ hom(π0(Γ),Z2) ≈ hom(π1(BΓ),Z2) ≈ H1(BΓ) = H1Γ(pt) .

Lemma 7.5.11. Under the isomorphism of (7.5.8), one has eΓ(χ) = χ.

Proof. Note that EΓ×Γ O(1)→ BΓ is a 2-fold covering and that

E(χ) = EΓ×Γ R = (EΓ×Γ O(1))×O(1) R .

Then, EΓ ×Γ O(1) → BΓ is the sphere bundle S(χΓ) for the Euclidean structureon χΓ given by the standard Euclidean structure on R. By Proposition 4.7.36,the Euler class eΓ(χ) coincides with the characteristic class w(S(χΓ)) of the 2-foldcovering S(χΓ)→ BΓ. But EΓ×Γ O(1) ≈ B kerχ and thus

π1(S(χΓ)) = π1(B kerχ) = π0(kerχ) = kerπ0χ = kerπ1Bχ .

Therefore, S(χΓ)→ BΓ is the 2-fold covering with fundamental group correspond-ing to kerπ1Bχ ⊂ π1(BΓ). Diagram (4.3.4) then implies that w(S(χΓ)) = χ.

A discrete group Γ is a 2-torus if it is finitely generated and if every elementhas order 2. It follows that Γ is isomorphic to ±1m, the integer m being calledthe rank of Γ. As seen in § 7.4,(7.5.9) BΓ ≃ B(±1m) ≃ (B±1)m ≃ (RP∞)m .

Hence, H∗Γ(pt) is isomorphic to a polynomial algebra

(7.5.10) H∗Γ(pt) ≈ H∗((RP∞)m) ≈ H∗(RP∞)⊗· · ·⊗H∗(RP∞) ≈ Z2[u1, . . . , um] ,

where degree(ui) = 1. Under the identifications of (7.5.8), ui ∈ H1Γ(pt) corre-sponds to the homomorphism ±1n → ±1 which is the projection onto the ithfactor.


Example 7.5.12. Let Γ be the 2-torus formed by the diagonal matrices ofO(n). Then H∗(BΓ) ≈ Z2[u1, . . . , un], where ui ∈ H1(BΓ) corresponds to thehomomorphism πi : Γ → ±1 given by the i-th diagonal entry. The inclusionχ : Γ→ O(n) provides a Γ-equivariant vector bundle χ of rank n over a point. Notethat χ is a direct sum of 1-dimensional representations πi. Using Lemmas 7.5.7and 7.5.11, we get that

w(χ) =

n∏

i=1

(1 + ui) .

Lemma 7.5.13. Let χ : Γ→ GL(V ) be a representation of a 2-torus Γ on a finitedimensional vector space V . Then the following two conditions are equivalent.

(1) V Γ = 0.(2) eΓ(χ) 6= 0.

Proof. That (2) implies (1) is given by Lemma 7.5.9. To prove the converse,we use the fact that χ is diagonalizable, with eigenvalues ±1: indeed, this is truefor a linear involution (see Example 7.1.14) and, if a, b ∈ GL(V ) commute, then bpreserves the eigenspaces of a. Thus, V = V1⊕ · · · ⊕Vr and Γ acts on Vj through ahomomorphism χj : Γ→ ±1 = O(1). Hence, χ is the Whitney sum χ1⊕· · ·⊕ χr.Therefore,

eΓ(χ) = e(χΓ)

= e((χ1)Γ ⊕ · · · ⊕ (χr)Γ) by Lemma 7.5.7

= e((χ1)Γ) · · · e((χr)Γ) by Proposition 4.7.40

= χ1 · · · χr by Lemma 7.5.11.

The condition V Γ = 0 implies that none of the χj vanishes. As H∗(BΓ) is apolynomial algebra, this implies that eΓ(χ) 6= 0.

Proposition 7.5.14. Let ξ = (Ep−→ X) be a rigid Γ-vector bundle of rank r,

where Γ is a 2-torus. Suppose that EΓ consists only of the image of the zero section.Then, the cup-product with the equivariant Euler class

H∗Γ(X)−eΓ(ξ)−−−−−−→ H∗+rΓ (X)

is injective.

Proof. Without loss of generality, we may suppose that X is path-connectedand non-empty. Let x ∈ X . Consider the slice inclusion s : BΓ → BΓ × X withimage BΓ × x. Then, H∗

Γs(eΓ(ξ)) = eΓ(ξx), where ξx = (Ex → x) is the

restriction of ξ over the point x. As X is path-connected, Lemma 4.7.2 implies thatthe component of eΓ(ξ) ∈ Hr(BΓ⊗X) in Hr(BΓ)×H0(X) is equal to eΓ(ξx)× 1(as BΓ is of finite cohomology type, we identify Hr(BΓ×X) with H∗(BΓ)⊗H∗(X)by the Kunneth theorem).

Now let 0 6= a ∈ Hk(BΓ ×X). We isolate its minimal component amin by theformula

a = amin +A

with

0 6= amin ∈ Hk−p(BΓ)⊗Hp(X) and A ∈⊕

q>p

Hk−q(BΓ)⊗Hq(X) .


Then

a eΓ(ξ) = amin (eΓ(ξx)⊗ 1) +A′

with

amin (eΓ(ξx)⊗ 1) ∈ Hk−p+r(BΓ)⊗Hp(X) and A′ ∈⊕

q>p

Hk−q+r(BΓ)⊗Hq(X) .

Therefore it suffices to prove that amin (eΓ(ξx) ⊗ 1) 6= 0. The condition onEΓ implies that EΓ

x = 0 and thus, by Lemma 7.5.13, eΓ(ξx) 6= 0. As H∗(BΓ)is a polynomial algebra, this implies that a eΓ(ξ) 6= 0. Let B be a basis ofHk−p(BΓ) and C be a basis of Hp(X). Then, b ⊗ c | (b, c) ∈ B × C is a basis ofHk−p(BΓ)⊗Hp(X). As H∗(BΓ) is a polynomial algebra, the family b eΓ(ξx) |b ∈ B is free in Hk−p+r(BΓ). Hence, if

0 6= amin =∑

(b,c)∈B×C

λbc (b ⊗ c) (λbc ∈ Z2) ,

then

amin (eΓ(ξx)⊗ 1) =∑

(b,c)∈B×C

λbc((b eΓ(ξx))⊗ c

)6= 0 .

Statements 7.5.11, 7.5.13 and 7.5.14 have analogues, replacing O(1) by SO(2)and 2-tori by tori. Let Γ be a topological group and let χ : Γ → SO(2) be acontinuous homomorphism, making Γ act on R2. This gives a Γ-vector bundle χof rank 2 over a point (see Example 7.5.8). Its equivariant Euler class lives inH2

Γ(pt) = H2(BΓ). As SO(2) ≈ S1, one has BSO(2) ≈ CP∞ (see Example 7.2.1).Define

(7.5.11) κ(χ) = H∗Bχ(ı) ∈ H2Γ(pt) = H2(BΓ) ,

where ı is the non-zero element of H2(BSO(2)) = Z2.

Lemma 7.5.15. eΓ(χ) = κ(χ).

Proof. The homomorphism χ : Γ→ SO(2) makes SO(2) a Γ-space. The mapEχ : EΓ → ESO(2) descends to a continuous maps EΓ×Γ SO(2) → ESO(2) andthere is a commutative diagram

(7.5.12)

EΓ×Γ SO(2) //

ESO(2) oo ≈ //

S∞

BΓ

Bχ // // BSO(2) oo ≈ // CP∞

Note that EΓ×Γ SO(2)→ BΓ is an SO(2)-principal bundle and that

E(χ) = EΓ×Γ R2 = (EΓ×Γ SO(2))×SO(2) R2 .

As SO(2) ≈ S1, EΓ×ΓSO(2) is the sphere bundle S(χΓ) for the Euclidean structureon χΓ given by the standard Euclidean structure on R2. Diagram (7.5.12) impliesthat S(χΓ) is induced by Bχ from the Hopf bundle S∞ → CP∞, whose Euler classis ı ∈ H2(CP∞) (see Proposition 6.1.5). Hence,

eΓ(χ) = e(S(χΓ)) = H∗Bχ(ı) = κ(χ) .


A torus Γ is a Lie group isomorphic to (S1)m, the integer m being called therank of Γ. For instance, SO(2) is a torus of rank 1. As seen in § 7.4,(7.5.13) BΓ ≃ B((S1)m) ≃ (BS1)m ≃ (CP∞)m .

Hence, H∗Γ(pt) is isomorphic to a polynomial algebra

(7.5.14) H∗Γ(pt) ≈ H∗((CP∞)m) ≈ H∗(CP∞)⊗· · ·⊗H∗(CP∞) ≈ Z2[v1, . . . , vm] ,

where degree(vi) = 2. One has vi = κ(χi) where χi : (S1)n → S1 ≈ SO(2) is

the projection onto the ith factor. If Γ is a torus, its associated 2-torus Γ2 is thesubgroup of elements of order 2 in Γ.

Lemma 7.5.16. Let Γ be a torus and Γ2 be its associated 2-torus. Let χ : Γ →GL(V ) be a representation of Γ on a finite dimensional vector space V . Then thefollowing two conditions are equivalent.

(1) V Γ2 = 0.(2) eΓ(χ) 6= 0.

Moreover, if (1) or (2) holds true, then dimV is even.

Proof. As Γ is a torus, χ(Γ) is contained in a maximal torus T of GL(V ).Those are all conjugate (see [21, § IV.1]). If dim V = 2s+1, there is an isomorphismV ≈ R2⊕· · ·⊕R2⊕R intertwining T with SO(2)×· · ·SO(2)×1 (see [21, Chapter IV,Theorem 3.4]). This contradicts the condition V Γ 6= 0. We can then suppose thatdimV = 2s, in which case there is an isomorphism V ≈ R2 ⊕ · · · ⊕ R2 conjugatingT with SO(2)× · · ·SO(2). The homomorphism χ takes the form χ = (χ1, . . . , χs)where χj : Γ → SO(2). Hence χ = χ1 ⊕ · · · ⊕ χs and, using Lemma 7.5.7 andProposition 4.7.40,

eΓ(χ) = eΓ(χ1) · · · eΓ(χs) = κ(χ1) · · · κ(χs) ,

the last equality coming from Lemma 7.5.15. SinceH∗(BΓ) is a polynomial algebra,the condition eΓ(χ) 6= 0 is equivalent to κ(χj) 6= 0 for all j. The condition V Γ =

0 = V Γ2 is equivalent to V Γj = 0 = V Γ2

j for all j, where Vj is the 2-dimensional

vector space corresponding to the jth summand R2 in the decomposition of V .We are thus reduced to the case dimV = 2 and χ : Γ→ SO(2). We start with

preliminaries. Choose isomorphisms Γ ≈ (S1)m, SO(2) ≈ S1 and S1 ≈ R/Z. Weget a commutative diagram

(7.5.15)

Zm // //

π1χ

Rm // //

χ

(S1)m

χ

Z // // R // // S1

where the vertical arrows are homomorphisms. Therefore, χ(x1, . . . , xm) =∑

i bixiwith bi ∈ Z and

(7.5.16) χ(γ1, . . . , γm) = γb11 · · · γbmm .

We deduce that

(7.5.17) V Γ = 0 ⇐⇒ π1χ non-trivial ⇐⇒ χ surjective⇐⇒ bj 6= 0 ∀ j .If χ is surjective, one gets a fiber bundle kerχ→ Γ→ S1 and, using its homotopyexact sequence and (7.5.16), we get

(7.5.18) V Γ = 0 =⇒ cokerπ1χ ≈ π0(kerχ) ≈ Z/gcd(b1, . . . , bm)Z .


Since 0 ∈ V Γ ⊂ V Γ2 , the condition V Γ2 = 0 implies that V Γ = 0. Hence,using (7.5.15)–(7.5.18),

(7.5.19)

V Γ2 = 0 ⇐⇒ Γ2 ⊂ kerχ

⇐⇒ 12Z ⊂ kerπ1χ

⇐⇒ 2 | gcd(b1, . . . , bm)⇐⇒ hom(π0(kerχ);Z2) = 0

.

As in the proof of Lemma 7.5.15, we consider S = EΓ ×Γ S1, which is the total

space for the sphere bundle S(χ). One has a commutative diagram

Γ

χ // EΓ

// BΓ

=

S1 // S // BΓ

whose rows are fiber bundles. Passing to the homotopy exact sequences, we get acommutative diagram

π2(BΓ)

=

≈ // π1(Γ)

π1χ

// 0

π2(BΓ)

≈ // π1(S1) // π1(S) // 0

whose rows are exact sequences. Hence,

(7.5.20) π1(S) ≈ cokerπ1χ

Now, the Gysin sequence for S → BΓ gives

(7.5.21) H1(BΓ)︸︷︷︸0

→ H1(S)→ H0(BΓ)︸︷︷︸Z2

eΓ(χ)−−−−−→ H2(BΓ) .

By Lemma 7.5.9, one knows that eΓ(χ) 6= 0 implies V Γ = 0. Therefore, us-ing (7.5.18)–(7.5.21),

(7.5.22)

eΓ(χ) 6= 0 ⇐⇒ H1(S) = 0

⇐⇒ hom(π1(S);Z2) = 0

⇐⇒ hom(cokerπ1χ;Z2) = 0

⇐⇒ hom(π0(kerχ);Z2) = 0

⇐⇒ V Γ2 = 0

.

Proposition 7.5.17. Let Γ be a torus and Γ2 be its associated 2-torus. Letξ = (p : E → X) be a rigid Γ-vector bundle of rank r. Suppose that EΓ2 consistsonly of the image of the zero section. Then r is even and the cup-product with theequivariant Euler class

H∗Γ(X)−eΓ(ξ)−−−−−−→ H∗+rΓ (X)

is injective.

Proof. The proof is the same as that of Proposition 7.5.14, using Lemma 7.5.16instead of Lemma 7.5.13.

7.6. EQUIVARIANT MORSE-BOTT THEORY 259

7.6. Equivariant Morse-Bott Theory

Let f : M → R be a smooth function defined on a smooth manifoldM . A pointx ∈ M is critical for f if df(x) = 0. Let Crit f ⊂ M be the subspace of criticalpoints for f . Then, f(Crit f) ⊂ R is the set of critical values of f . We say that fis Morse-Bott if the following two conditions hold:

• Crit f is a disjoint union of submanifolds. A connected component ofCrit f is called a critical manifold of f .• the kernel of the Hessian Hx at a critical point x equals the tangent spaceto the critical manifold N containing x.

This definition coincides with that of a Morse function when Crit f is a discrete set.See e.g. [151, 149, 95, 18, 13] for presentations of Morse and Morse-Bott theory.The index of x ∈ Crit f is the number of negative eigenvalues of Hx. This numberis constant over a critical manifold and thus defines a function ind: π0(Crit f)→ N.Also, the normal bundle νN to a critical manifold N decomposes into a Whitneysum νN ≈ ν−N⊕ν+N of the negative and positive normal bundles, i.e. the bundlesspanned at each x ∈ N respectively by the negative and positive eigenspaces of Hx.Note that rank ν−(N) = indN .

A (continuous) map g : X → Y is called proper if the pre-image of any compactset is compact. For instance, if X is compact, then any map g is proper. Letf : M → R be a proper Morse-Bott function and let a < b be two regular values.Define Ma,b = f−1([a, b]), a compact manifold whose boundary is the union ofMa = f−1(a) and Mb = f−1(b). Denote by fa,b the restriction of f to Ma,b. TheMorse-Bott polynomial Mt(Ma,b) is defined by

Mt(fa,b) =∑

N∈π0(Crit fa,b)

t indNPt(N)

(the sum is finite since, as f is proper, Crit fa,b is compact).

Proposition 7.6.1 (Morse-Bott inequalities). There is a polynomial Rt, withpositive coefficients, such that

(7.6.1) Mt(fa,b) = Pt(Ma,b,Ma) + (1 + t)Rt .

Equation (7.6.1) implies that the coefficients of Mt(fa,b) are greater or equalto those of Pt(Ma,b,Ma) (whence the name of Morse-Bott inequalities). For theequivalence of (7.6.1) with other classical and more subtle forms of the Morse-Bottinequalities, see [13, § 3.4].

Proof. The map fa,b has a finite number of critical values, all in the interiorof [a, b]. Let a = a0 < a1 < · · · < ar = b be regular values such that [ai, ai + 1]contains a single critical value. We shall prove by induction on i that (7.6.1) holdstrue for fa,ai . The induction starts trivially for i = 0, with the three terms of (7.6.1)being zero.

As [ai, ai+1] contains a single critical level, there is a homotopy equivalence

(7.6.2) Mai,ai+1 ≃Mai ∪SiDi

where (Di, Si) is the disjoint union over N ∈ π0(Crit fai,ai+1) of the pairs formedby the disk and sphere bundles of ν−(N) (see [18, pp. 339–344]). By excision and


the Thom isomorphism,

(7.6.3) H∗(Mai,ai+1 ,Ma) ≈ H∗(Di, Si) ≈∏

N∈π0(Crit fai,ai+1)

H∗−indN (N) .

Therefore,

(7.6.4) Mt(fai,ai+1) = Pt(Mai,ai+1,Mai) .

On the other hand, Corollary 3.1.27 applied to the triple (Ma,ai+1 ,Ma,ai ,Ma) givesthe equality

(7.6.5) Pt(Ma,ai+1 ,Ma) + (1 + t)Qt = Pt(Ma,ai+1 ,Ma,ai) +Pt(Ma,ai ,Ma) ,

for some Qt ∈ N[t]. By excision and (7.6.4), one gets

(7.6.6) Pt(Ma,ai+1 ,Ma,ai) = Pt(Mai,ai+1,Mai) = Mt(fai,ai+1) .

Thus, (7.6.5) and (7.6.6) provide the induction step.

A proper Morse-Bott function f : M → R is called perfect if for any two regularvalues a < b, Equation (7.6.1) reduces to

(7.6.7) Mt(fa,b) = Pt(Ma,b,Ma) .

The easiest occurrence of perfectness is the following lacunary principle.

Lemma 7.6.2. Suppose that no consecutive powers of t occur in Mt(f). Then,f is perfect.

Proof. Suppose that Rt 6= 0 in (7.6.1). Then, two successive powers of t occurin (1 + t)Rr. The same happens then in Mt(fa,b), and then in Mt(f).

Other simple criteria for perfectness are given by the following three results.For a regular value x of f : M → R, set Wx = f−1(−∞, x].

Lemma 7.6.3. Let f : M → R be a proper Morse-Bott function. Then, thefollowing two conditions are equivalent.

(1) f is perfect.(2) For any regular values a < b < c of f , the cohomology sequence of the

triple (Wc,Wb,Wa) reduces to a global short exact sequence

0→ H∗(Wc,Wb)→ H∗(Wc,Wa)→ H∗(Wb,Wa)→ 0 .

Proof. Suppose that f is perfect. Then, by excision,

Pt(Wc,Wa) = Pt(Ma,c,Ma) = Mt(fa,c)

and analogously for Pt(Wb,Wa) and Pt(Wc,Wb). As

Mt(fc,a) = Mt(fc,b) +Mt(fb,a) ,

one has

Pt(Wc,Wa) = Pt(Wc,Wb) +Pt(Wb,Wa) .

By Corollary 3.1.27 and its proof, this implies that H∗(Wb,Wa)→ H∗(Wc,Wb) issurjective, whence (2).

Conversely, suppose that (2) holds true. For two regular values a < c, we provethat Pt(Ma,c,Ma) = Pt(Wc,Wa) = Mt(fa,c), by induction on the number na,c ofcritical values in the segment [a, c]. This is trivial for na,c = 0, since Ma,c is thendiffeomorphic to Ma × [a, c] (see [95, Chapter 6, Theorem 2.2]). When na,c = 1,


one uses (7.6.4). For the induction step, when na,c ≥ 2, choose a regular valueb ∈ (a, c) such that na,b = na,c − 1. Then,

Pt(Wc,Wa) = Pt(Wc,Wb) +Pt(Wb,Wa) by (2)

= Mt(fc,b) +Mt(fb,a) by induction hypothesis

= Mt(fa,c) ,

which proves the induction step.

Lemma 7.6.4. Let f : M → R be a proper Morse-Bott function. Then for anytwo regular values a < b, one has

(7.6.8) dimH∗(Ma,b,Ma) ≤ dimH∗(Crit fa,b) .

Moreover, f is perfect if and only if (7.6.8) is an equality for all such a and b.

Proof. The evaluation of (7.6.1) at t = 1 implies (7.6.8) and the equality isequivalent to Rt = 0.

In the case where M is a closed manifold, one has the following result.

Proposition 7.6.5. Let f : M → R be a Morse-Bott function, where M is aclosed manifold. Then f is perfect if and only if

(7.6.9) dimH∗(M) = dimH∗(Crit f) .

Proof. Equation (7.6.8) implies (7.6.9) when f(M) ⊂ (a, b). Conversely, leta < b be two regular values of f . Let a′ < a and b′ > b such that f(M) ⊂ (a′, b′).Using Corollary 3.1.28 and excision, we get

(7.6.10)dimH∗(M) = dimH∗(Ma′,b′ ,Ma′)

≤ dimH∗(Ma′,b,M′a) + dimH∗(Ma′,b′ ,Ma′,b)

= dimH∗(Ma′,b,M′a) + dimH∗(Mb,b′ ,Mb) .

Doing the same for dimH∗(Ma′,b,M′a) and using (7.6.1) gives

(7.6.11)dimH∗(M) ≤ dimH∗(Ma′,a,M

′a) + dimH∗(Ma,b,Ma) + dimH∗(Mb,b′ ,Mb)

≤ dimH∗(Crit fa′,a) + dimH∗(Crit fa,b) + dimH∗(Crit fb,b′)= dimH∗(Crit f) .

Now, if (7.6.9) holds true, then all the inequalities occurring in (7.6.10) and (7.6.10)are equalities, including dimH∗(Ma,b,Ma) = dimH∗(Crit fa,b).

Theorem 7.6.6. Let M be a smooth Γ-manifold, where Γ is a 2-torus. Letf : M → R be a proper Γ-invariant Morse-Bott function which is bounded below.Suppose that Crit f =MΓ. Then

(1) f is perfect.(2) M is Γ-equivariantly formal.(3) the restriction morphism H∗Γ(M)→ H∗Γ(M

Γ) is injective.

Remark 7.6.7. When Γ = ±1, Theorem 7.6.6 follows from Smith theory.Indeed, for any regular values a < b of f ,

dimCrit fa,b = dimH∗(MΓa,b) by hypothesis

≤ dimH∗(Ma,b) by Proposition 7.3.7

≤ dimCrit fa,b by Lemma 7.6.4.


Therefore, the above inequalities are equalities and f is perfect by Lemma 7.6.4and equivariantly formal by Proposition 7.3.7. Point (3) then follows from Propo-sition 7.3.9 [see Comment 12.0.10].

Remark 7.6.8. Under the hypotheses of Theorem 7.6.6, when Γ = ±1 andf is a Morse function, M. Farber and D. Schutz have proven that each integralhomology group Hi(M ;Z) is free abelian with rank equal to the number of crit-ical points of index i [60, Theorem 4]. By the universal coefficient theorem [82,Theorem 3.2], such a function is perfect.

Proof of Theorem 7.6.6. For a regular value x of f , setWx = f−1(−∞, x].We first prove that

(7.6.12) H∗Γ(Wx)→ H∗Γ(WΓx ) is injective for all regular value x.

This is proven by induction on the number nx of critical values in the interval(−∞, x], following the argument of [198, proof of Proposition 2.1]. If nx = 0, thenWx = ∅ and H∗Γ(Wx) = 0, which starts the induction as f is bounded below.

Suppose that n ≥ 1 and that (7.6.12) holds true when nx < n. Let y be aregular value of f with ny = n. Choose z < y such that nz = n− 1 (this is possiblesince the set of critical values of a proper Morse-Bott function is discrete). Asin (7.6.2), one has a homotopy equivalence

Mz,y ≃My ∪S D ,

where (D,S) is the disjoint union over N ∈ π0(Crit fz,x) of the pairs formed by thedisk and sphere bundles of ν−(N). Using (7.6.3) and the proof of Proposition 4.7.32,we get the commutative diagram(7.6.13)

H∗Γ(Wy ,Wz) oo ≈excision

H∗Γ(Mz,y,Mz) oo ≈excision

H∗Γ(D,S)oo ≈Thom

∏N H

∗−indNΓ (N)

−(eΓ(ν

−(N)))

H∗Γ(Wy) // H∗Γ(Mz,y) // H∗Γ(MΓz,y)

≈ // ∏N H

∗Γ(N)

where N runs over π0(Crit fz,y) = π0(MΓz,y). As MΓ = Crit f , the linear Γ-action

of ν−(N) has fixed point set consisting only of the image of the zero section. ByLemma 7.5.14, the right vertical arrow of (7.6.13) is injective. Thus, we deducefrom (7.6.13) thatH∗Γ(Wy ,Wz)→ H∗Γ(Wy) is injective. This splits the Γ-equivariantcohomology sequence of (Wy,Wz) into short exact sequences. The same cuttingoccurs for the pair (WΓ

y ,WΓz ) using Proposition 3.1.24, and one has a commutative

diagram

(7.6.14)

0 // H∗Γ(Wy ,Wz) //

rz,y

H∗Γ(Wy) //

ry

H∗Γ(Wz) //

rz

0

0 // H∗Γ(MΓz,y) // H∗Γ(W

Γy ) // H∗Γ(W

Γz ) // 0

where the vertical arrows are induced by the inclusions. The left vertical arrow isinjective by (7.6.13). Since nz = n− 1, the right one is injective by induction hy-pothesis. By diagram-chasing, we deduce that the middle vertical arrow is injective,which proves (7.6.12).


Warning: As WΓy is the disjoint union of MΓ

z,y and WΓz , one has H∗Γ(W

Γy ) ≈

H∗Γ(MΓz,y) ⊕ H∗Γ(W

Γz ). Consider the image Im ry of ry under under this decom-

position. The above arguments imply that Im rz,y × 0 ⊂ Im ry. But, in general0× Im rz 6⊂ Im ry (see Example 7.6.9 below).

We deduce Point (3) from (7.6.12). Indeed as M =⋃xWx Corollary 3.1.16

provides a commutative diagram

(7.6.15)

H∗Γ(M)

≈ // lim←−x

H∗Γ(Wx)

H∗Γ(MΓ)

≈ // lim←−x

H∗Γ(WΓx )

.

As the right vertical arrow is injective by (7.6.12), so is the left one.For Point (2), we first prove that ρx : H

∗Γ(Wx) → H∗(Wx) is surjective for

all regular value x. This is also done by induction on nx, starting trivially whennx = 0. For the induction step, consider as above two regular values z < y such thatny = nz+1. The cohomology sequences of the pair (Wy,Wz) give the commutativediagram(7.6.16)

0 // HkΓ(Wy ,Wz) //

ρy,z

HkΓ(Wy) //

ρy

HkΓ(Wz) //

ρz

0

Hk−1(Wz) // Hk(Wy ,Wz) // Hk(Wy) // Hk(Wz) // Hk+1(Wy ,Wz)

,

the top sequence being cut as seen above. Similarly to (7.6.13), we get a commu-tative diagram(7.6.17)

H∗Γ(Wy,Wz) oo ≈excision

ρy,z

H∗Γ(Mz,y,Mz) oo ≈excision

H∗Γ(D,S)oo ≈Thom

∏N H

∗−indNΓ (N)

ρCrit

H∗(Wy,Wz) oo ≈excision

H∗(Mz,y,Mz) oo ≈excision

H∗(D,S) oo ≈Thom

∏N H

∗−indNΓ (N)

.

Since Crit f ⊂ MΓ, the map ρCrit is surjective and so is ρy,z. If ρz is surjectiveby induction hypothesis, a diagram-chase proves that ρy is surjective. Now, recallthat, for a Γ-spaceX , ρ : : H∗Γ(X)→ H∗(X) is equal to H∗i for some fiber inclusioni : X → XΓ. One can thus consider the Kronecker dual ρ∗ : H∗(X)→ H∗(XΓ). Onehas a commutative diagram

(7.6.18)

lim−→x

H∗(Wx)

lim−→

ρx,∗

≈ // H∗(M)

ρ∗

lim−→x

H∗((Wx)Γ)≈ // H∗(MΓ)

.


As ρx is surjective, ρx,∗ is injective and thus lim−→

ρx,∗ is injective. By Diagram (7.6.18)

and Kronecker duality, ρ : H∗Γ(M) → H∗(M) is surjective and M is equivariantlyformal.

Let us finally prove Point (1). Consider two regular values z < y such thatny = nz + 1. As in (7.6.16) the vertical maps are surjective, the cohomologysequence of (Wy,Wz) reduces to a global short exact sequence. By the proof that (2)implies (1) in Lemma 7.6.3, this implies that f is perfect.

Example 7.6.9. Consider the action of Γ = ±1 on M = Sn ⊂ R×Rn givenby γ · (t, x) = (t, γ x), with fixed points p± = (±1, 0). Note that M is a sphere withlinear involution Sn0 in the sense of Example 7.1.14. The formula f(t, x) = t definesa Morse function M satisfying the hypotheses of Theorem 7.6.6. Taking z = 0 andy = 2 as regular values of f , one has Wy =M and Diagram (7.6.14) becomes

0 // H∗Γ(M,W0)j //

r+

H∗Γ(M) //

r

H∗Γ(W0) //

r−≈

0

0 // H∗Γ(p+) // H∗Γ(p+)⊕H∗Γ(p−) // H∗Γ(p−) // 0

.

Set H∗(BΓ) = Z2[u]. By Lemma 7.5.13, eΓ(ν−(p+)) = un. Together with Dia-

gram (7.6.13), this shows that rj(U) = (un, 0), where

U ∈ HnΓ (M,W0) ≈ Hn

Γ (M0,2,M0) ≈ H0Γ(p+)

is the Thom class of ν−(p+). Let B = j(U). Using the diagram

H0Γ(p+)

Thom

≈//

≈

HnΓ (M,W0)

j //

HnΓ (M)

ρ

H0(p+)

Thom

≈// Hn(M,W0)

≈ // Hn(M)

one sees that ρ(B) is the generator of Hn(Sn) = Z2. Hence, ρ is surjective, asexpected by Theorem 7.6.6. By the Leray-Hirsch theorem, H∗Γ(M) is the free Z[u]-module generated by B. Now, r(B) = (un, 0) and r(u) = (u, u). As r is injective byTheorem 7.6.6, the relation B2 = unB holds true in H∗Γ(M). Given the dimensionof Hk

Γ(M), this establishes the GrA[u]-isomorphism

Z2[B, u]/(B2 + unB) ≈ H∗Γ(M) .

Taking the image by ρ adds the relation u = 0 and we recover that H∗(Sn) ≈Z2[B]

/(B2). Note that (0, u) is not in the image of r, confirming the warning in the

proof of Theorem 7.6.6. But (0, un) = r(B+un) is in the image of r, correspondingto the generator A = B + un of H∗Γ(M) (see Example 7.1.16). Had we considered−f instead of f , the above discussion would have selected the generator A first.The relation A2 = unA also holds true and recall from Example 7.1.16 that H∗Γ(M)admits the presentation

Z2[u,A,B]/(A2 + unA,A+B + un) ≈ H∗Γ(M) .


Example 7.6.10. Let Γ be a 2-torus and let χ : Γ→ ±1 be a homomorphism,identified with χ ∈ H1(BΓ) under the bijection (7.5.8). Consider the Γ-action onM = S1 ⊂ R2 given by γ · (t, x) = (t, χ(γ)x), with fixed points p± = (±1, 0). Wecall M a χ-circle. As in Example 7.6.9, one sees that the image of

r : H∗Γ(M)→ H∗Γ(p±) ≈ H∗Γ(p−)⊕H∗Γ(p+) ≈ H∗(BΓ)⊕H∗(BΓ)

is the H∗(BΓ)-module generated by B = (χ, 0) and that H∗Γ(M) admits the pre-sentation

H∗Γ(M) ≈ H∗(BΓ)[B]/(B2 + χB) .

Moreover, the image of r is the set of classes (a, b) such that b − a is a multipleof χ.

Theorem 7.6.6 admits the following analogue for torus actions.

Theorem 7.6.11. Let Γ be a torus and Γ2 be its associated 2-torus. Let M bea smooth Γ-manifold. Let f : M → R be a proper Γ-invariant Morse-Bott functionwhich is bounded below. Suppose that Crit f =MΓ =MΓ2 . Then

(1) f has only critical manifolds of even index. In particular, if f is a Morsefunction, then M is of even dimension.

(2) f is perfect.(3) M is Γ-equivariantly formal.(4) the restriction morphism H∗Γ(M)→ H∗Γ(M

Γ) is injective.

Proof. The proof is the same as that of Theorem 7.6.6. The hypothesisCrit f = MΓ implies that the negative normal bundles are Γ-vector bundles andthe hypothesis Crit f =MΓ2 permits us to use Proposition 7.5.17 instead of Propo-sition 7.5.14.

In Theorem 7.6.11, note that the perfectness of f is implied by (1). WhenΓ = S1, Points (3) and (4) follows from Smith theory, in the same way as inRemark 7.6.7.

Example 7.6.12. Let Γ be a torus with associated 2-torus Γ2. Let χ : Γ→ S1

be a continuous homomorphism. Consider the Γ-action on M = S2 ⊂ R× C givenby γ · (t, x) = (t, χ(γ)x), with fixed points p± = (±1, 0). We call M a χ-sphere.Let us assume that the restriction of χ to the associated 2-torus Γ2 of Γ is nottrivial. This implies that MΓ = MΓ2 , so we can apply Theorem 7.6.11 and, as inExample 7.6.9, one sees that the image of

r : H∗Γ(M)→ H∗Γ(p±) ≈ H∗Γ(p−)⊕H∗Γ(p+) ≈ H∗(BΓ)⊕H∗(BΓ)

is theH∗(BΓ)-module generated by B = (κ(χ), 0), where κ(χ) ∈ H2(BT ) is definedin (7.5.11). Also, H∗Γ(M) admits the presentation

H∗Γ(M) ≈ H∗(BΓ)[B]/(B2 + κ(χ)B) .

Moreover, the image of r is the set of classes (a, b) such that b − a is a multipleof κ(χ).

A consequence of Theorems 7.6.6 an 7.6.11 are the surjectivity theorems a laKirwan (see Remark 7.6.16 below). For f : M → R a continuous map, we set


M− = f−1((−∞, 0]), M+ = f−1([0 − ∞)) and M0 = M− ∩M+ = f−1(0). Theinclusions form a commutative diagram.

(7.6.19)

M0

i+ //

i−

i

""

M+

j+

M−j− // M

.

Proposition 7.6.13. Let M be a closed smooth Γ-manifold, where Γ is a 2-torus. Let f : M → R be a Γ-invariant Morse-Bott function satisfying Crit f =MΓ.Suppose that 0 is a regular value of f . Then

H∗Γi : H∗Γ(M)→ H∗Γ(M0)

is surjective and its kernel is the ideal kerH∗Γj−+kerH∗Γj+, generated by kerH∗Γj−and kerH∗Γj+.

For applications of this proposition, see § 10.3.2.

Proof. We use the abbreviations i∗± = H∗Γi±, j∗± = H∗Γj±, etc. As M is com-

pact, f is proper and bounded. By Theorem 7.6.6, the restriction homomorphismH∗Γ(M)→ H∗Γ(M

Γ) = H∗Γ(Crit f) is injective. The commutative diagram

H∗Γ(M)(j∗−,j

∗+)//

H∗Γ(M−)⊕H∗Γ(M+)

H∗Γ(M

Γ)≈ // H∗Γ(M

Γ−)⊕H∗Γ(MΓ

+)

shows that the Mayer-Vietoris sequence in equivariant cohomology for Diagram (7.6.19)splits into a global short exact sequence

(7.6.20) 0→ H∗Γ(M)(j∗−,j

∗+)−−−−−→ H∗Γ(M−)⊕H∗Γ(M+)

i∗−+i∗+−−−−→ H∗Γ(M0)→ 0 .

Suppose that x1 < x2 < x3 < · · · are regular values of f such that fxi,xi+1 hasonly one critical level (we use the notations of Theorem 7.6.6 and its proof). Then,by (7.6.14), H∗Γ(Wi+1) → H∗Γ(Wi) is surjective. As W0 = M− and M is compact,this argument shows that j∗− is surjective. By symmetry, replacing f by −f (usingagain that M is compact), one also has that j∗+ is surjective.

Let a ∈ H∗Γ(M0). Using (7.6.20), choose a± ∈ H∗Γ(M±) such that a =i∗−(a−) + i∗−(a+). As j∗± is surjective, there exist b± ∈ H∗Γ(M) with i∗±(b±) = a±.Then

i∗(b− + b+) = i∗(b−) + i∗(b+) = i∗−j∗−(b−) + i∗+j

∗+(b+) = i∗−(a−) + i∗+(a+) = a ,

which proves that i∗ = H∗Γi is surjective. As i∗ = i∗±j±, we have also proven that

i∗± is surjective. Therefore, one has a commutative diagram

(7.6.21)

H∗Γ(M,M+) // //

≈

H∗Γ(M)j∗+ // //

j∗−i∗

&& &&

H∗Γ(M+)

i∗+H∗Γ(M−,M0) // // H∗Γ(M−)

i∗− // // H∗Γ(M0)


where the horizontal and vertical sequences are exact and the left hand verticalarrow is an isomorphism by excision. Hence, ker i∗ = (j∗−)

−1(ker i∗−), which yieldsan exact sequence

0→ ker j∗− → ker i∗ → ker i∗− → 0 .

But Diagram (7.6.21) provides a section of ker i∗ → ker i∗−, whose image is equalto ker j∗+. This proves the assertion on ker i∗ (which is actually GrV-isomorphic toker j∗− ⊕ ker j∗+).

Example 7.6.14. Consider the action of Γ = ±1 on M = RPn given by

γ · [x0, . . . , xn] = [x0, . . . , xn−1, γxn] .

The Morse-Bott function defined on M by f([x0, . . . , xn]) = 1 − 2x2n satisfies thehypotheses of Theorem 7.6.6 and Proposition 7.6.13. SetMc = f−1(c). The criticalsubmanifolds are M±1 and one has M− ≃M−1 = pt and M+ ≃M1 = RPn−1. Letu ∈ H1(BΓ) = Z2 be the generator.

The bundle projection p : MΓ → BΓ and its restriction pc to (Mc)Γ give ele-ments H∗pc(u) = uc ∈ H∗Γ(Mc) and H

∗p(u) = v ∈ H∗Γ(M). One has

H∗Γ(M−) ≈ Z2[u−1] and H∗Γ(M+) ≈ H∗(BΓ×M1) ≈ Z2[b, u1]/(bn) ,

with the degree of b equal to 1. Consider the commutative diagram

H0Γ(M1)

Thom

≈//

α

33H1Γ(M,M−) // H1

Γ(M)

r1

ρ // H1(M)

≈

H1Γ(M1)

ρ1 // H1(M1)

.

By Diagram (7.6.13), r1 α(1) = eΓ(ν−), the equivariant Euler class of the (nega-

tive) normal bundle ν−(M1). By Lemma 7.5.10,

eΓ(ν−) = 1× e(ν−(M1)) + e(χ)× 1 ∈ H1(BΓ×M1) ,

where χ is the representation of Γ on a fiber of ν−(M1). Note that ν−(M1) = ν(M1)

is the canonical line bundle over RPn−1 (since M = RPn is obtained by attachingan n-cell to M1 = RPn−1 by the Hopf map (two-fold covering) Sn−1 → RPn−1).Then, by Proposition 4.7.36 and its proof, e(ν−(M1)) = b. As χ has no non-zerofixed vector, eΓ(χ) = u by Lemma 7.5.13. Therefore,

eΓ(ν−) = 1× b+ u× 1 = b+ u1 ,

the last formula making sense in the presentation H∗Γ(M1) ≈ Z2[b, u1]/(bn). As

ρ1(b+u1) = b, this proves that ρ is surjective, as already known by Theorem 7.6.6.Let a = α(1) + v. One also has ρ1r1(a) = b so, by the Leray-Hirsch theorem,H∗Γ(M) is the free Z2[v]-module generated by a, a2, . . . , an−1 and the Poincare seriesof MΓ is

(7.6.22) Pt(MΓ) = Pt(M) ·Pt(BΓ) =1− tn+1

(1− t)2 .

By Diagram (7.6.14), r−1α(1) = 0 in H1Γ(M−) ≈ H1

Γ(M−1). Therefore, thehomomorphism r : H∗Γ(M) → H∗Γ(M

Γ) ≈ H∗Γ(M−1) ⊕ H∗Γ(M1) satisfies r(a) =(u−1, b) and r(v) = (u−1, u1). Finally, we claim that there is a GrA-isomorphism

(7.6.23) Z2[v, a]/(an+1 + va)

≈−→ H∗Γ(M) .


Indeed, one already knows that H∗Γ(M) is GrA-generated by v and a, and, usingthe injective homomorphism r, one checks that the relation an+1 = va holds true.This gives the GrA-morphism of (7.6.23) which is surjective, and hence bijectivesince both sides of (7.6.23) have the same Poincare series, computed in (7.6.22).

Replacing Theorem 7.6.6 by Theorem 7.6.11 in the proof of Proposition 7.6.13gives the following result.

Proposition 7.6.15. Let M be a closed smooth Γ-manifold, where Γ is a toruswith associated 2-torus Γ2. Let f : M → R be a Γ-invariant Morse-Bott functionsatisfying Crit f =MΓ =MΓ2 . Suppose that 0 is a regular value of f . Then

H∗Γi : H∗Γ(M)→ H∗Γ(M0)

is surjective and its kernel is the ideal kerH∗Γj−+kerH∗Γj+, generated by kerH∗Γj−and kerH∗Γj+.

As an example, one can take the complex analogue of Example 7.6.14, i.e.Γ = S1 acting on M = CPn given by γ · [x0, . . . , xn] = [x0, . . . , xn−1, γxn] and theMorse-Bott function f([x0, . . . , xn]) = 1− 2x2n. All the formulae of Example 7.6.14hold true, with the degrees of all the classes multiplied by 2.

Remark 7.6.16. For Γ = S1, the hypotheses of Proposition 7.6.15 are realizedwhen f is the moment map of a Hamiltonian circle action (see [12]). In this case, itfollows from F. Kirwan’s thesis [117, § 5] that H∗Γ(M ;Q)→ H∗Γ(M0;Q) is surjective(see e.g. [198, Theorem 2]). This justifies the terminology of surjectivity theorem ala Kirwan used above to introduce Proposition 7.6.13 and 7.6.15. For the assertionon kerHΓi in these propositions, compare [199, Theorem 2]; our proofs followedthe hint of [199, Remark 3.5].


Notations. As in § 7.1, G denotes the group with 2 elements G = id, τ. AG-space is thus a space endowed with an involution τ . The notation Snp stands for

the G-linear sphere Sn with (Snp )G = Sp, as in Example 7.1.14.

7.1. Let G acting on X = S1 with τ(z) = z. Prove that XG is homeomorphic tothe double mapping cylinder CCq, where q : S

∞ → RP∞ is the covering projection.Prove that XG has the homotopy type of RP∞ ∨RP∞ (use [82, Proposition 0.17]and that S∞ is contractible).

7.2. If p ≥ 1, prove that H∗G(Snp ) admits, as a Z2[u]-algebra, the presentation

H∗G(Snp ) ≈ Z2[u][A]

/(A2), where A is of degree n.

7.3. Let X be an equivariantly formal G-space which is of finite cohomology type.Find a formula giving Pt(XG) and Pt(u ·H∗(XG)) in terms of Pt(X).

7.4. Write the details for Remark 7.1.21.

7.5. What is H∗G(Snp )?

7.6. Let X = Sd ∨ Sd with d ≥ 1, endowed with the G action intertwining the twospheres. Prove that H∗G(X) is, as a GrA[u]-algebra, isomorphic to Z2[u, a]

/(ua),

with a of degree d. Prove that a2 = 0.


7.7. Let ∆ be the subgroup of SU(2) formed by the diagonal matrices. Prove thatthe map B∆→ BSU(2) induced on the Milnor classifying spaces by the inclusionis, up to homotopy type, equivalent to the inclusion CP∞ → HP∞.

7.8. Let Γ be a topological group acting on a space X . Let Y be a space of finitecohomology type, considered as a Γ-space with trivial Γ-action (Y = Y Γ). Provethat H∗Γ(X × Y ) ≈ H∗Γ(X)⊗H∗(Y ) (tensor product over Z2).

7.9. Let Γ be a compact Lie group. Let X and Y be two Γ-space which are equi-variantly formal. We suppose that X is a finite dimensional Γ-complex and thatY is of finite cohomology type. Prove that X × Y (with the diagonal Γ-action) isequivariantly formal. [see Comment 12.0.11.]

7.10. Let Y be a G-space with Y G = Y . We suppose that Y is of finite cohomologytype. LetX = S1

0×Y , with the diagonalG-action. Give a presentation ofH∗G(X) asa Z2[u]-algebra. Describe, for H∗G, the Mayer-Vietoris sequence analogous to thatof Exercise 4.13. Describe the injective restriction homomorphism r : H∗G(X) →H∗G(X

G).

7.11. Let X = S10 ×S1

0 , with the diagonal G-action. Give a presentation of H∗G(X)as a Z2[u]-algebra. Prove that the map f : X → R given by f(eiα, eiβ) = cosα +2 cosβ is an equivariant Morse function satisfying the hypotheses of Theorem 7.6.6and, with the help of this theorem, describe the injective restriction homomorphismH∗G(X)→ H∗G(X

G).

7.12. Find a connected equivariantly formal G-spaceX such that XG is the disjointunion of a point and of a sphere.

7.13. For 0 ≤ p ≤ n, let Pnp denote the projective space RPn endowed with theinvolution

τ(x0 : x1 : · · · : xn) = (−x0 : · · · : −xp : xp+1 : · · · : xn) .(a) Prove that Pnp is G-equivariantly formal.

(b) Describe the restriction homomorphism r : HG(Pnp ) → HG((P

np )

G) (it isinjective by (a) and Proposition 7.3.9).

(c) Prove that H∗G(Pn0 ) admits, as Z2[u]-algebra, the presentations

H∗G(Pn0 ) ≈ Z2[u][A]

/(An+1 + uAn) or Z2[u][B]

/((B + u)n+1 + u(B + u)n)

(d) Prove the GrA[u]-isomorphisms

(d.1) Z2[u][A]/(An+1 + u2An−1)

≈−→ H∗G(Pn1 ) and

(d.2) Z2[u][A]/(A12 + uA11 + u4A8 + u5A7)

≈−→ H∗G(P114 ).

7.14. Prove that the algebras

R = Z2[u,A]/(A3 + uA2) and S = Z2[u,B]

/(B3 + u2B)

are GrA[u]-isomorphic (A and B of the same degree). Find a G-space X such thatH∗G(X) is GrA[u]-isomorphic to R, with A of degree 4.

7.15. Let Pnp be the G-space of Exercise 7.13. One checks that the functionf : Pnp → R given by

f(x0 : x1 : · · · : xn) = x20 + · · ·+ x2p


is a Morse-Bott function. Prove that it satisfies the hypotheses of Theorem 7.6.6.With the help of the proof of this theorem (as in Example 7.6.9), describe therestriction homomorphism r : HG(P

np )→ HG((P

np )

G). Compare Exercise 7.13 (b).

7.16. What would be the analogue of Exercises 7.13 and 7.15 for S1-actions?

7.17. Let Γ be a topological group. Prove that any functor J from Top to Top ex-tends to a functor JΓ from TopΓ to TopΓ. What is here the meaning of “extends”?

7.18. We apply Exercise 7.17 to the suspension functor X 7→ ΣX . Let X be aΓ-space.

(a) Prove that there exists a suspension homomorphism

Σ∗Γ : H∗Γ(X)→ H∗+1

Γ (ΣΓX)

which is a morphism of H∗Γ(pt)-module and which is injective. Discuss itsfunctoriality.

(b) Find an example where Σ∗Γ is not surjective.(c) Suppose that X is Γ-equivariantly formal. Prove that ΣΓX is Γ-equivar-

iantly formal and that Σ∗Γ is an isomorphism.

7.19. Let X be a G-space. The G-action on X may be extended to a G-action onΣX , permuting the suspension points, giving rise to a G-space ΣX (note that ΣX ≈X ∗ G). Suppose that X is a connected finite dimensional G-complex satisfyingb(X) = b(XG) < ∞. Let i : X → ΣX denote the inclusion. Prove that thesequence

0 // H∗G(ΣX)H∗Gi // H∗G(X)

ρ // H∗(X) // 0

is exact.

7.20. Let Γ = SU(2) acting on X = SU(2) by conjugation.

(a) Show that XΓ has the homotopy type of the double mapping cylinder CCjwhere j is the inclusion of CP∞ → HP∞ (see Exercise 6.4). [Hint: usethat X/Γ is homeomorphic to a segment]

(b) Deduce from Exercise 6.4 that X is Γ-equivariantly formal.(c) Prove that there is a (unique) isomorphism of H∗Γ(pt)-algebras

H∗Γ(X) ≈ H∗Γ(pt)[b]/(b2) ≈ Z2[a, b]/(b2) ,

where a is of degree 4 and b of degree 3.

7.21. Let (X,X1, X2, X0) be a Mayer-Vietoris data. Suppose that X is a Γ-space(Γ a topological group) and that Xi are closed Γ-invariant subspaces of X . Sup-pose that X = X1 ∪X2 and that (Xi, X0) is a Γ-equivariantly well cofibrant pairfor i = 1, 2. Prove that there is a Mayer-Vietoris sequence for the Γ-equivariantcohomology.

7.22. Let X and Y be Γ-spaces, equivariantly well pointed by x ∈ XΓ and y ∈ Y Γ.Thus, X∨Y (using these base points) is a Γ-space. Prove that there is a H∗Γ-algebra

isomorphism H∗Γ(X ∨ Y ) ≈ H∗Γ(X)⊕ H∗Γ(Y ).

CHAPTER 8

Steenrod squares

In Chapter 4, the power of cohomology was much increased by the introductionof the cup product, making H∗(X) a graded algebra. Another rich structure onH∗(X) comes from cohomology operations, i.e. the natural self-transformations ofthe mod 2 cohomology functor (see § 8.1). The basic examples of such operations,

the Steenrod squares Sqi : H∗(X)→ H∗+i(X), were discovered by Norman Steenrodand Henri Cartan in the late 1940s (see, e.g. [40, pp. 510–523] for historical details).The GrA-morphism induced by any continuous map must then commute with allthe Steenrod squares, which imposes strong restrictions. For instance, the spacesY = S2 ∨ S3 and Y ′ = ΣRP 2 do not have the same homotopy type, althoughtheir cohomology are GrA-isomorphic. Indeed, Sq1 vanishes on H∗(Y ) but not onH∗(Y ′). In the same way, we show that all suspensions of Hopf maps are essential,i.e. not homotopic to a constant map (see § 8.6).

After an introductory section on cohomology operations, we state in § 8.2 thebasic properties and make some computations of Steenrod squares. Their construc-tions and the proof of Adem relations are given in Sections 8.3 and 8.4. Based onequivariant cohomology, these two technical sections may be skipped on first read-ing, since the applications of Steenrod squares are consequences of the propertiespresented in § 8.2.

The last two sections of this chapter treat applications of Steenrod squares.Prominent among them are Adams’ theorem on “the Hopf invariant one prob-lem” and Serre’s computation of the cohomology algebra of Eilenberg-MacLanespaces K(Z2, n). The latter implies that mod 2 cohomology operations are, in somesense, generated by sums, cup products and iterations of Steenrod squares (seeRemark 8.5.7). More applications will appear in Chapter 9, for instance Thom’sdefinition of Stiefel-Whitney classes and Wu’s formula.

8.1. Cohomology operations

This section contains some generalities on mod 2 cohomology operations, inorder to present Steenrod squares. We take a global approach which may shed anew light with respect to existing texts on the subject.

A cohomology operation is a map

Q = Q(X,Y ) : H∗(X,Y )→ H∗(X,Y )

defined for any topological pair (X,Y ), satisfying the following two conditions:

(1) Q is functorial, i.e. if g : (X ′, Y ′)→ (X,Y ) is a continuous map of pairs,then

(8.1.1) H∗gQ = QH∗g .

271

272 8. STEENROD SQUARES

(2) Q(X,Y ) =∑Q[i](X,Y ) where Q[i](X,Y ) is the restriction of Q(X,Y )

to Hi(X,Y ).

We may restrict the definition to some classes of pairs, like CW-pairs, etc. Forinstance, restricting to pairs (X, ∅) gives operations on absolute cohomology, since

H∗(X, ∅) ≈−→ H∗(X). Point (2) is a partial linearity (Q is not supposed to be linear)and permits us to define Q via its restrictions Q[i].

Examples of cohomology operations are given by Q = 0 or Q = id . A lesstrivial example is the cohomology operation Q such that by Q[n](a) = an for alln ∈ N, where an = a · · · a (n times). Cohomology operations may be added,multiplied by cup products and composed, giving rise to more examples.

Here are a few remarks about cohomology operations. They are used through-out this section, without always an explicit mention.

8.1.1. By Theorem 3.7.1, a topological pair has, in a functorial way, the samecohomology as a CW-pair. Hence, when studying cohomology operations, we do notlose generality by restricting to CW-pairs. For instance, a cohomology operationdefined for CW-pairs extends in a unique way to a cohomology operation definedfor all topological pairs.

8.1.2. Let (X,Y ) be a CW-pair with Y non-empty. The quotient map (X,Y )→(X/Y, [Y ]) induces an isomorphismH∗(X/Y, [Y ])

≈−→ H∗(X,Y ) (Proposition 3.1.45).Most questions on cohomology operations may thus be settled by considering theCW-pairs of type (X, ∅) and (X, pt). In particular, a cohomology operation definedfor these pairs extends to a unique cohomology operation. Moreover, a cohomologyoperation Q defined on absolute cohomology for CW-complexes extends to a uniquecohomology operation for CW-pairs, using the commutative diagram

(8.1.2)

0 // H∗(X, pt) //

Q

H∗(X) //

Q

H∗(pt) //

Q

0

0 // H∗(X, pt) // H∗(X) // H∗(pt) // 0

,

where (X, pt) is a CW-pair.

8.1.3. Let (X,Y ) be a CW-pair. By Corollary 3.1.12, the family of inclusionsiA : A→ X for A ∈ π0(X) gives rise to the commutative diagram

H∗(X, ∅)

Q

(H∗iA)

≈// ∏

A∈π0(X)H∗(A, ∅)

∏Q

H∗(X, ∅)(H

∗iA)

≈// ∏

A∈π0(X)H∗(A, ∅)

or, if pt ∈ A0 ∈ π0(X),

H∗(X, pt)

Q

(H∗iA)

≈// H∗(A0, pt)×

∏A∈π0(X)−A0

H∗(A, ∅)∏Q

H∗(X, pt)

(H∗iA)

≈// H∗(A0, pt)×

∏A∈π0(X)−A0

H∗(A, ∅)

8.1. COHOMOLOGY OPERATIONS 273

In other words, a cohomology operation preserves the connected components. To-gether with 8.1.2, this permits us often to restrict, without loss of generality, acohomology operation to pairs (X,Y ) where X is path-connected.

Lemma 8.1.4. If Q is a cohomology operation, then Q(0) = 0.

Proof. The class 0 ∈ H∗(X,Y ) is in the image of H∗(X,X) → H∗(X,Y ).As, H∗(X,X) = 0, the lemma follows from functoriality.

An important property of cohomology operations is that it does not decreasedimensions.

Lemma 8.1.5. Let Q be a cohomology operation. Then, there is a functionN : N→ N, satisfying N(0) = 0 and N(m) ≥ m, such that

(8.1.3) Q(Hm(X,Y )) ⊂N(m)⊕

k=m

Hk(X,Y )

for all topological pairs (X,Y ).

Proof. By 8.1.1–8.1.3 above, it is enough to prove the lemma for CW-pairs(X,Y ) with X connected and Y = pt or ∅. As H0(X, pt) = 0, Q(H0(X, pt)) = 0by Lemma 8.1.4. The constant map X → pt induces an isomorphism H0(pt, ∅)→H0(X, ∅). As H>0(pt, ∅) = 0, the functoriality implies (8.1.3) for m = 0, withN(0) = 0.

If m > 0, then Hm(X, pt)≈−→ Hm(X, ∅) ≈−→ Hm(X), so it is enough to

prove (8.1.3) in the absolute case. By functoriality, the following diagram is com-mutative.

Hm(X,Xm−1)

Q

// // Hm(X)

Q

H∗(X,Xm−1) // H∗(X)

As Hk(X,Xm−1) = 0 for k < m, this proves that the direct sum in (8.1.3) startsat k = m. Also, any class a ∈ Hm(X) is of the form a = H∗f(ι) for some mapf : X → Km. Thus, N(m) is the maximal degree of Q(ι) ∈ H∗(Km).

A cohomology operation Q restricts to a cohomology operation Q′ on absolutecohomology by Q′X = Q(X,∅). Not every absolute cohomology operation Q′ is sucha restriction because it may not satisfy Q′(0) = 0, contradicting Lemma 8.1.4. Asan example, we can, for X path-connected and non-empty, define Q′ by Q′(a) = 0for a ∈ Hm(X) (m 6= 1) and Q′(H1(X)) = 1 (the functoriality of Q′ coming fromLemma 2.5.4). In fact, we have the following lemma.

Lemma 8.1.6. Let Q′ be a cohomology operation defined on absolute cohomologyfor connected CW-complexes. Then, there exists a cohomology operation Q suchthat Q′X = Q(X,∅) if and only if Q′(0) = 0. The cohomology operation Q is unique.

Proof. The condition is necessary by Lemmas 8.1.4. For the converse, using8.1.1–8.1.3 above, it suffices to define Q(X,pt) for an path-connected non-empty

CW-complex X . On H0(X, pt) = 0, Q is defined by Q(0) = 0 (this is compulsoryby Lemma 8.1.4). The functoriality for the inclusion (X, ∅) → (X, pt) on H0 isguaranteed by the condition Q′(0) = 0. Let P : H∗(X)→ H0(X) be the projection


onto the component of degree 0. Let j : pt → X be an inclusion of a point in X .As Q′(0) = 0, the commutative diagram

H>0(X)Q′ //

H∗j

H∗(X)P //

H∗j

H0(X)

H∗j≈

H>0(pt)Q′=0 // H∗(pt) P // H0(pt)

shows that Q′(H>0(X)) ⊂ H>0(X). Hence, the commutative diagram

H>0(X, pt)

Q

≈ // H>0(X)

Q′

H>0(X, pt)

≈ // H>0(X)

defines Q and shows its uniqueness.

The notion of cohomology operation makes sense for the reduced cohomology,with the same definition.

Lemma 8.1.7. A cohomology operation Q descends to a unique cohomologyoperation on reduced cohomology, also called Q.

Proof. Let p : X → pt be the unique map from X to a point. Consider thediagram

H∗(pt)

Q

H∗p // H∗(X)

Q

// // H∗(X)

Q

H∗(pt)H∗p // H∗(X) // // H∗(X)

where the line are exact. As Q is a cohomology operation, the left square is commu-tative, so there is a unique Q : H∗(X)→ H∗(X) so that the right square commutesand this construction is functorial. (Recall that, if X is path-connected and Y is

non-empty, then H∗(X,Y ) = H∗(X,Y )).

We now study the multiplicativity of a cohomology operation Q. Note that,by Lemma 4.1.14, the relative cup product H∗(X,Y ) ⊗H∗(X,Y ) → H∗(X,Y ) isdefined for all topological pairs (X,Y ). Consider the following four statements.

(a) Q(a b) = Q(a) Q(b) for all a, b ∈ H∗(X) and all spaces X .(b) Q(a b) = Q(a) Q(b) for all a, b ∈ H∗(X,Y ) and all topological pairs

(X,Y ).(c) Q(a × b) = Q(a) × Q(b) for all a ∈ H∗(X1), b ∈ H∗(X2) and all spaces

X1 and X2.(d) Q(a × b) = Q(a) ×Q(b) for all a ∈ H∗(X1), b ∈ H∗(X2) and all pointed

spaces X1 and X2.

Proposition 8.1.8. For a cohomology operation Q, Conditions (a), (b) and(c) are equivalent and (a) implies (d). If Q(1) = 1, then (d) implies (a).

8.2. PROPERTIES OF STEENROD SQUARES 275

Proof. Without loss of generality, we may suppose that the spaces X and Xi

are connected CW-complexes. Statement (b) is stronger than (a) since H∗(X) =H∗(X, ∅). To prove that (a) implies (b), it suffices to consider the case Y = pt,which is obvious.

Using the functoriality of Q, (a)⇒(c) follows from the definition of the crossproduct and (c)⇒(a) from the formula a b = ∆∗(a × b) (see Remark 4.6.1).That (c)⇒(d) is obvious, so (a)⇒(d). Now, (d) implies (c) for classes of positivedegree. As (c)⇒(a), property (d) implies that Q(a b) = Q(a) Q(b) exceptpossibly for a or b equal to 1. If, say a = 1, then

Q(1 b) = Q(b) = 1 Q(b) = Q(1) Q(b) ,

since Q(1) = 1. Thus, (d)⇒(a) if Q(1)) = 1.

Corollary 8.1.9. Let Q be a cohomology operation with Q(1) = 1. Then, (a)is equivalent to

(d’) the diagram

Hm(Km)⊗ Hn(Kn)

Q⊗Q

× // Hm+n(Km ∧ Kn)

Q

H∗(Km)⊗ H∗(Kn)× // H∗(Km ∧ Kn)

is commutative for all positive integers m and n.

Proof. It is clear that (d)⇒(d’). The corollary will then follow from Propo-sition 8.1.8 if we prove that (d’)⇒(d).

It suffices to prove (d) for a ∈ Hm(X1) and b ∈ Hn(X2) where X1 and X2

are connected CW-complexes, so m,n > 0. Then a = f∗a (ιm) and a = fb ∗ (ιn)for maps fa : X1 → Km and fb : X2 → Kn. Condition (d’) says that Q(ιm × ιn) =Q(ιm) × Q(ιn) and Q(a × b) = Q(a) × Q(b) from this special case, using thefunctoriality of× and of Q.

8.2. Properties of Steenrod squares

One of the most remarkable features of mod 2 cohomology is the existenceof cohomology operations, introduced by N. Steenrod and H. Cartan in the late1940s (see, e.g. [40, pp. 510–523]), called the Steenrod squares Sqi : H∗(X,Y ) →H∗(X,Y ) (i ∈ N). For a ∈ Hm(X,Y ), one has Sqi(a) = 0 for i > m (see (2.a) in

Theorem 8.2.1). Hence, the sum∑i∈N Sqi(a) has only a finite number of non-zero

terms and thus defines the total Steenrod square

(8.2.1) Sq: H∗(X,Y )→ H∗(X,Y ) , Sq(a) =∑

i∈N

Sqi(a) .

Here is the main theorem of this chapter.

Theorem 8.2.1. There exists cohomology operations Sq and Sqi as in (8.2.1),which enjoys the following properties:

(1) Sq is Z2-linear.

(2) if a ∈ Hn(X,Y ) then Sqi(a) ∈ Hn+i(X,Y ) and

(a) Sqi(a) = 0 for i < 0 and i > n.(b) Sq0(a) = a.


(c) Sqn(a) = a a.

(3) Sq(a b) = Sq(a) Sq(b). This is equivalent to the formula

Sqk(a b) =∑

i+j=k

Sqi(a) Sqj(b) (Cartan’s formula).

(4) Σ∗Sq = SqΣ∗, where Σ∗ : H∗(X)≈−→ H∗+1(ΣX) is the suspension iso-

morphism of Proposition 3.1.49.

(5) The Adem relations:

SqiSqj =

[i/2]∑

k=0

(j−k−1i−2k

)Sqi+j−kSqk (0 < i < 2j) .

The Steenrod squares are characterized amongst cohomology operations bysome of these properties (see Proposition 8.5.12). Also, the Adem relations generateall the polynomial relations amongst the compositions of Sqi’s which hold true forany space (see Corollary 8.5.11).

Example 8.2.2. Theorem 8.2.1 permits us to compute Sqi easily for the pro-jective spaces RPn, CPn and HPn. Indeed, one has the following results.

(a) Let a ∈ H1(X). Then (2) implies that Sq(a) = a + a2 (we write an forthe cup product of n copies of a). Then, (3) implies that

Sq(an) = (a+ a2)n = an (1+ a)n = an

n∑

i=1

(ni

)ai .

Therefore

(8.2.2) Sqi(an) =(ni

)an+i .

(b) If a ∈ H2(X) satisfies Sq1(a) = 0, then Sq(a) = a+ a2 and, as in (a), onehas

(8.2.3) Sq2i(an) =(ni

)an+i and Sq2i+1(an) = 0 .

(c) If a ∈ H4(X) satisfies Sq(a) = a+ a2, then

(8.2.4) Sq4i+k(an) =

(ni

)an+i if k ≡ 0 mod 4.

0 otherwise.

Besides the trivial case of OP 2, there are no more such examples. Indeed, byCorollary 8.6.3 and Theorem 8.6.6 below, if a ∈ Hm(X) satisfies Sq(a) = a + a2

with a2 6= 0, then m = 1, 2, 4 or 8.

We finish with two more properties of Steenrod squares. As Sq(a b) =Sq(a) Sq(b), Proposition 8.1.8 implies the following result.

Proposition 8.2.3. Let X1 and X2 be topological spaces (pointed for (2)).Then,

(1) Sq(a× b) = Sq(a)× Sq(b) for all a ∈ H∗(X1), b ∈ H∗(X2).

(2) Sq(a×b) = Sq(a)×Sq(b) for all a ∈ H∗(X1), b ∈ H∗(X2).

8.3. CONSTRUCTION OF STEENROD SQUARES 277

Proposition 8.2.4. Let (X,Y ) be a topological pair. Then

Sqδ∗ = δ∗Sq ,

where δ∗ is the connecting homomorphism δ∗ : H∗(Y )→ H∗+1(X,Y ).

Proof. By 8.1.1, we may suppose that (X,Y ) is a CW-pairs. Let Z = X ∪[−2, 1]× Y and Z ′ = X ∪ [−2,−1]× Y ∪ 1× Y . The projection p : [−2, 1]× Y →−2 × Y extends to a homotopy equivalence of pairs p : (Z, 1 × Y ) → (X,Y ).The commutative diagram

H∗(1 × Y )

δ∗

oooo p∗

H∗(S0 × Y )

δ∗

H∗+1(Z, 1 × Y ) oo H∗+1(Z,Z ′) oo ≈

excision// H∗+1(D1, S0 × Y )

shows that it is enough to prove that Sqδ∗ = δ∗Sq for a CW-pair of the type(D1 × Y, S0 × Y ). In the proof of Proposition 4.7.44, we showed that

δ∗(a) = e×(H∗i+(a) +H∗i−(a)

),

where i± : ±1×F → S0×F denote the inclusions and 0 6= e ∈ H1(D1, S0) = Z2.One has Sq(e) = Sq0(e) = e. Using the linearity of Sq and Proposition 8.2.3, weget

Sqδ∗(a) = Sq(e×

(H∗i+(a) +H∗i−(a)

))

= Sq(e)× Sq(H∗i+(a) +H∗i−(a))

= e×(SqH∗i+(a) + SqH∗i−(a)

)

= e×(H∗i+Sq(a) +H∗i−Sq(a)

)

= δ∗Sq(a) .

Sections 8.3 and 8.4 contain the proof of Theorem 8.2.1. They are based on theideas of Steenrod (see [184, VII.1 and VIII.1]) using the equivariant cohomology.Other treatments of similar ideas are developed in [3, VI.7] and [82, Section 4.L].

8.3. Construction of Steenrod squares

The involution τ on X ×X given by τ(x, y) = (y, x) makes X ×X a G-spacefor G = id, τ. We consider the cross-square map β : Hn(X) → H2n(X × X)defined by β(a) = a × a. Its image is obviously contained in H2n(X × X)G. ByLemma 7.1.10, the image of ρ : H∗G(X × X) → H∗(X × X) is also contained inH∗(X ×X)G. The same considerations are valid for the reduced cross-square map

β : Hn(X)→ H2n(X ∧X)G, defined for a space X which is well pointed by x ∈ X .

The maps β and β are not linear but they are functorial: if f : X ′ → X is acontinuous maps, then H∗(f × f)β = βH∗f and H∗(f ∧ f) β = βH∗f . UsingDiagram (4.7.7), the use of base points x ∈ X and (x, x) ∈ X ∧ X provides a


commutative diagram

(8.3.1)

H∗(X ∧X) // // H∗(X ×X)

H∗(X) // //

β

OO

H∗(X)

β

OO

Lemma 8.3.1. Let X be a connected CW-complex, pointed by x ∈ X0. Then,the cross-square maps β and β admit liftings βG and βG so that the diagram

H∗(X ∧X) // // H∗(X ×X)

H∗G(X ∧X)

ρ

gg

// H∗G(X ×X)

ρ

gg

H∗(X) // //

β

OO

βG

77

H∗(X)

β

OO

βG

77

is commutative. These liftings are functorial and satisfy βG(0) = 0 and βG(0) = 0.Such liftings are unique.

Proof. The maps β, ρ, β and ρ preserving the connected components, one maysuppose that X is connected. Lemma 8.3.1 is obvious when n = 0, giving βG(0) = 0

and βG(1) = 1. We can then assume n > 0. In this case, Hn(X) → Hn(X) is an

isomorphism, so it is enough to define βG.We first define βG when X = Kn = K(Z2, n) with its CW-structure given in

the proof of Proposition 3.8.1, whose 0-skeleton consists in a single point x. ByProposition 4.7.11, the G-space Kn ∧ Kn satisfies Hk(Kn ∧ Kn) = 0 for k < 2n.

Hence, Proposition 7.1.12 implies that ρ : H2nG (Kn ×Kn)→ H2n(Kn × Kn)G is an

isomorphism. We define βG = ρ−1 β.Now, a cohomology class a ∈ Hn(X) is of the form a = H∗fa(ι) for a map

fa : X → Kn, well defined up to homotopy. We define

(8.3.2) βG(a) = H∗G(fa × fa) βG(ι) .

This definition makes βG functorial. Indeed, let g : Y → X be a continuous mapand a = H∗fa(ι) ∈ Hn(X). If b = H∗g(a), then b = H∗fb(ι) with fb = fag. By

definition of βG, one has

H∗G(g × g) βG(a) = H∗G(g × g)H∗G(fa × fa)βG(ι) = H∗G(fb × fb)βG(ι) = βG(b) .

The functoriality of βG follows from that of βG. The uniqueness of βG (then, thatof βG) is obvious, since there was no choice for X = Kn and Definition (8.3.2) iscompulsory by the required functoriality.

Remark 8.3.2. When Hi(X) = 0 for i < n, then Hj(X ∧X) = 0 for j < 2nby Proposition 4.7.11. The homomorphism ρ : H2n(X ∧ X) → H2n(X ∧ X)G is

an isomorphism which is natural. Then, by functoriality, the formula βG = ρ−1 βholds true.


The map βG : Hn(X) → H2n

G (X ×X) extends to βG : H∗(X) → H∗G(X × X)

by

(8.3.3) βG(∑

j∈N

aj) =∑

j∈N

βG(aj) , where aj ∈ Hj(X) .

The inclusion of (X×X)G intoX×X induces aGrA[u]-morphism r : H∗G(X×X)→H∗G((X × X)G). Observe that (X × X)G is the diagonal subspace (x, x) ofX × X , hence homeomorphic to X . We thus write r : H∗G(X × X) → H∗G(X),considering X as a G-space with trivial G-action. Using (7.1.2), we thus get aGrA[u]-isomorphism H∗G(X) ≈ H∗(X)[u]. We also consider the (non-graded) ringhomomorphism ev1 : H

∗(X)[u]→ H∗(X) which extends the identity on H∗(X) bysending u to 1 (evaluation of a polynomial at 1).

Let X be a CW -complex. By definition, the Steenrod square Sq : H∗(X) →H∗(X) is the composition

(8.3.4) Sq = ev1rβG

making the following diagram commutative

(8.3.5)

H∗G(X ×X)r // H∗G(X) oo ≈ // H∗(X)[u]

ev1

H∗(X)

βG

OO

Sq // H∗(X)

.

By (8.3.3) and Lemma 8.3.1, the map Sq is a cohomology operation, de-fined so far on absolute cohomology for connected CW-complexes. As βG(0) = 0,Lemma 8.1.6 implies that this partial definition of Sq extends to a unique cohomol-ogy operation Sq: H∗(X,Y )→ H∗(X,Y ) defined for all topological pairs (X,Y ).

For a ∈ Hn(X,Y ) let Sqi(a) be the component of Sq(a) in Hn+i(X,Y ). Again,

defining Sqi : H∗(X,Y )→ H∗(X,Y ) by

(8.3.6) Sqi(∑

j∈N

aj) =∑

j∈N

Sqi(aj) , where aj ∈ Hj(X)

provides a family of cohomology operation Sqi. A priori, i ∈ Z but, by Lemma 8.1.5,Sqi = 0 if i < 0. It follows from these definitions that Sq =

∑i∈N Sqi.

By Lemma 8.1.7, the Steenrod squares Sq and Sqi are also cohomology opera-tions on reduced cohomology. In this case, the following diagram is commutative.

(8.3.7)

H∗G(X ∧X)r // H∗G(X) oo ≈ // H∗(X)[u]

ev1

H∗(X)

βG

OO

Sq // H∗(X)


In the important case where Hj(X) = 0 for j < n (e.g. X = Kn), one can useRemark 8.3.2 to get the following commutative diagram

(8.3.8)

H2n(X ∧X)Gρ−1

≈// H2n

G (X ∧X)r // H∗G(X) oo ≈ // H∗(X)[u]

ev1

Hn(X)

β

OO

Sq //βG

66♥♥♥♥♥♥♥♥♥♥♥H∗(X)

We now prove Properties (1)-(4) of Theorem 8.2.1, for the absolute cohomologyH∗(X) of a connected CW-complex X .

Proof of (2). That Sqi sends Hn(X) to Hn+i(X) is by definition and we

already noticed that Sqi = 0 for i < 0. If a ∈ Hn(X), then rβG(a) ∈ H2nG (X),

which implies that Sqi = 0 for i > n. Note that, from (8.3.4) and the definition of

the Sqi, one has, for a ∈ Hn(X):

(8.3.9) rβG(a) =

n∑

i=0

Sqi(a)un−i .

Let us prove that Sqn(a) = a a. From the above equation, we deduce thatSqn(a) = ev0rβG(a). Using Diagrams (7.1.5) and (7.1.6), we get the diagram

H∗(X)βG //

OO=

H∗G(X ×X)

ρ

r // H∗G(X) oo ≈ // H∗(X)[u]

ev0

H∗(X)

β // H∗(X ×X)∆∗ // H∗(X)

where ∆∗ is induced by the diagonal map ∆: X → X ×X . By (4.6.5) p. 132, onehas

Sqn(a) = ev0rβG(a) = ∆∗β(a) = ∆∗(a× a) = a a .

It remains to prove that Sq0(a) = a. By naturality and since Hn(Kn) = Z2, itsuffices to find, for each integer n, some space X with a class a ∈ Hn(X) such thatSq0(a) 6= 0. The space X will be the sphere Sn. Indeed, there is a homeomorphism

h : Sn ∧ Sn ≈−→ S2n (see Example 4.7.12). We leave as an exercise to the reader toconstruct such a homeomorphism h which conjugates the G-action on Sn ∧ Sn toa linear involution on S2n. Therefore, Sn ∧ Sn is G-homeomorphic to the sphereS2nn of Example 7.1.14. We now use Diagram (8.3.8). By the reduced Kunneth

theorem, β(a) = a, the generator of H2n(Sn ∧ Sn) = Z2. By Proposition 7.1.15,r ρ−1(a) = a un, whence Sq0(a) = a.

Linearity of Sq. We have to prove that for each n ∈ N, the map Sq: Hn(X)→H∗(X) is linear. By Point (2) already proven, the restriction of Sq toH0 is Sq0 = id,so we may assume that n ≥ 1. By functoriality of Sq, the diagram

Hn(X/Xn−1)

Sq

// // Hn(X)

Sq

H∗(X/Xn−1) // H∗(X)


is commutative, where the horizontal maps are induced by the projection X →X/Xn−1. Therefore, it is enough to prove the linearity of Sq: Hn(X/Xn−1) →H∗(X/Xn−1). We may thus assume that Hk(X) = 0 for k < n and use Dia-

gram (8.3.8) to define Sq. It is then enough to show that r ρ−1 β : Hn(X) →H2nG (X) is linear. One has

β(a+ b) = β(a) + β(b) + a × b+ b × a= β(a) + β(b) + a × b+ τ∗(a × b) .

Using Proposition 7.1.12, we get

ρ−1 β(a+ b) = β(a) + β(b) + tr∗(a × b) .As rtr∗ = 0 by Proposition 7.1.9, this prove that r ρ−1 β is linear.

From the already proven properties of Sq, we now deduce a structure resultabout H∗G(X×X) (Proposition 8.3.3 below) which will be used to prove the multi-plicativity of Sq. Let N : H∗(X×X)→ H∗(X×X) be the GrV-morphism definedby N(x× y) = x× y + y × x and let N be the image of N . Note that

kerN = H∗(X ×X)G = D ⊕Nwhere D is the subgroup generated by x × x | x ∈ H∗(X). By definition of thetransfer map tr∗ : H∗(X ×X)→ H∗G(X ×X) (see (7.1.7)), one has

ρtr(x× y) = x× y + τ∗(x× y) = N(x× y) .The correspondence (x × x,N(y × z)) 7→ βG(x) + tr∗(y × z) produces a sectionσ : D ⊕N → H∗G(X ×X) of ρ : H∗G(X ×X)→ H∗(X ×X)G. We identify D ⊕Nwith σ(D ⊕ N ) and thus see D ⊕ N as a subgroup of H∗G(X ×X). Let D be theZ[u]-module generated in H∗G(X×X) by D. The following result is due to Steenrod(unpublished, but compare [78, § 2]).

Proposition 8.3.3. With the above identifications, the following propertieshold true.

(a) The restriction of ρ to D ⊕ N coincides with the identity D ⊕ N id−→H∗(X ×X)G.

(b) N = Ann (u).

(c) D is a free Z2[u]-module with basis D−0. In particular, D is isomorphicto Z2[u]⊗D.

(d) As a Z2[u]-module, H∗G(X ×X) = D ⊕ N .

Proof. Point (a) is obvious from the identification via σ. Hence, H∗(X×X)G

is the image of ρ and one has the commutative diagram

0 // H∗G(X ×X)/(u) //

ρ

H∗(X ×X)tr∗ //

OO=

Ann (u) //

ρ

0

0 // H∗(X ×X)G // H∗(X ×X) // N // 0

where the lines are exact (the upper line is the transfer exact sequence (7.1.8)).By the techniques of the five lemma, we deduce that ρ : Ann (u) → N is an iso-morphism. As σ(N) is contained in the image Ann (u) of the transfer map, this


proves (b). Also, we get the isomorphism

(8.3.10) ρ : H∗G(X ×X)/(u)≈−→ H∗(X ×X)G .

Let B be Z2-basis H∗(X) formed by homogeneous classes. To prove (c), one

has to show that

(8.3.11)∑

a∈B0

βG(a)uk(a) 6= 0 .

for any non-empty finite subset B0 of B and any function k : B0 → N. Let B0min be

the subset formed by the elements in B0 which are of minimal degree. Then

ev1r( ∑

a∈B0

βG(a)uk(a)

)=

∑

a∈B0

Sq(a) =∑

a∈B0min

a+ terms of higher degrees 6= 0 .

To prove (d), let A = H∗G(X×X) and B = D ⊕N . If B 6= A, let a ∈ A−B beof minimal degree. By the above and Sequence (7.1.8), one has B/uB = A/uA ≈H∗(X ×X)G. Hence, there exists b ∈ B and c ∈ A such that a = b + uc. By theminimality hypotheses on a, one has c ∈ B and thus a ∈ B (contradiction).

Proof that Sq(a b) = Sq(a) Sq(b) (multiplicativity). One has

β(a b) = (a b)× (a b) = (a a)× (b b) = β(a) β(b) .

Hence,

βG(a b) = βG(a) βG(b) + x

with x ∈ ker ρ = (u) (the last equality was established in (8.3.10)). We maysuppose that a ∈ Hm(X) and b ∈ Hn(X). Let V be Z2-basisH

<m+n(X) formed byhomogeneous classes. By Proposition 8.3.3, x =

∑v∈V0 βG(a)u

k(v) for some finite

subset V0 of V , with k(v) > 0. Let V0min be the subset formed by the elements in

V0 which are of minimal degree. Then,

Sq(a b) = ev1 rβG(a b) =∑

v∈V0min

a+ terms of higher degrees .

Since Sqi(a b) = 0 for i < 0, one has V0min = ∅ and therefore V0 = ∅. This

implies that βG(a b) = βG(a) βG(b) and thus Sq(a b) = Sq(a) Sq(b).

Proof that Σ∗Sq = SqΣ∗. By Lemma 4.7.13, using the reduced suspensionS1 ∧ X , the relation Σ∗Sq = SqΣ∗ is equivalent to the following equation inH∗(S1 ∧X):

(8.3.12) b × Sq(c) = Sq(b × c) ∀c ∈ Hn(X) ,

where b is the generator of H1(S1) = Z2. As noticed in Proposition 8.2.3, theformula Sq(b c) = Sq(b) Sq(c) already proven implies that Sq(b × c) =Sq(b) ×Sq(c). Also, by (2) already established, Sq(b) = b. Hence, Equation (8.3.12)holds true.

The proof of Points (1)–(4) in Theorem 8.2.1 is now complete for pairs (X, ∅)with X a connected CW-complex. We check easily that the extension of Sq andSqi to topological pairs given by Lemma 8.1.6 satisfies the same properties.

8.4. ADEM RELATIONS 283

8.4. Adem relations

The Adem relations are relations amongst the compositions SqiSqj . They wereconjectured by Wu Wen-tsun around 1950 and first proved in 1952 by J. Ademin his thesis at Princeton University (summary in [4] and full proofs in [5]). Wepresent below a proof based on the idea of Steenrod [184, Chapter VIII], usingthe equivariant cohomology for the symmetric group Sym4. The proof in [82,§ 4.L] is another adaptation of the same idea. For different proofs, see [33] andRemark 8.5.10 below.

Let X be a topological space. Consider the map

Sq : H∗(X)βG−−→ H∗G(X ×X)

r−→ H∗G((X ×X)G) ≈ H∗(BG×X)

(recall that BG ≈ RP∞: see Example 7.2.1). The map Sq would sit diagonally

in (8.3.5), the diagram used to define Steenrod squares. By § 8.3, the map Sq isfunctorial, Z2-linear and multiplicative.

Consider now the iterated map

Sq Sq : H∗(X)→ H∗(BG×BG×X) ≈ H∗(X)[u, v] ,

using the Kunneth theorem and that H∗(BG × BG) ≈ Z2[u, v] with u and v indegree 1.

Proposition 8.4.1. For any a ∈ H∗(X), the polynomial Sq Sq(a) is symmet-ric in the variables u and v.

Before proving Proposition 8.4.1, we do some preliminaries. Let Y be a K-space for a topological group K. The equivariant cross product ×K of § 7.4 givesrise to an equivariant cross square map βK : Hn

K(Y ) → H2nK (Y × Y )G, defined by

βK(a) = a×K a, where G = I, τ acting on Y × Y by exchanging the factors. A

map βK : HnK(Y )→ H2n

K (Y ∧Y )G is similarly defined, using the reduced equivariantcross product×K. The following is a generalization of Lemma 8.3.1.

Lemma 8.4.2. Let Y be a K-space, equivariantly well pointed by y ∈ Y K . Then,

(1) The cross-square maps βK and βK admit liftings βKG

and βKG

so that thediagram

H2nK (Y ∧ Y )G // // H2n

K (Y × Y )G

H2nG×K(Y ∧ Y )

ρhh

// H2nG×K(Y × Y )

ρhh

HnK(Y ) // //

βK

OO

βKG

66

HnK(Y )

βK

OO

βKG

66

is commutative, where ρ and ρ are induced by the homomorphism K →G×K. When K is the trivial group, these liftings coincide with those ofLemma 8.3.1.

(2) The lifting βKG

is functorial in Y and G, i.e. if Y ′ is a K ′-space, f : Y →Y ′ is equivariant with respect to a continuous homomorphism ϕ : K → K ′,


then the diagram

H2nG×K′(Y

′ × Y ′) f∗×f∗// H2nG×K(Y × Y )

HnK′(Y

′)f∗ //

βK′

G

OO

HnK(Y )

βKG

OO

is commutative. The analogous property holds for βKG.

(3) Suppose that the K-action on Y is trivial. Then, the following diagram iscommutative

H2nG×K(Y × Y )

r // H2nG×K(Y )

≈ // H2n(B(G ×K)× Y )

HnK(Y ) oo ≈ //

βKG

OO

Hn(BK × Y )Sq // H2n(BG×BK × Y )

P ≈

OO

Proof. The lifting βKG

is defined using the lifting βG of Lemma 8.3.1 and thecommutative diagram(8.4.1)

H2nG×K(Y × Y )

r //

ρ

rr

H2nG×K(Y )

H2nK (Y × Y )G oo

=// H2n((Y × Y )K)G oo ρ

H2nG ((Y × Y )K)

r //

≈

OO

H2nG (YK)

≈

OO

H2n((Y × Y )K×K)G oo ρ

∆∗K

OO

H2nG ((Y × Y )K×K)

r //

∆∗K

OO

H2nG ((∆Y )∆K)

≈

OO

H2n(YK × YK)G oo ρ

P

OO

H2nG (YK × YK)

r //

P

OO

H2nG (∆(Y )K))

≈

OO

HnK(Y ) oo = //

βK

OO

Hn(YK)

β

OOβG

55 Sq

22

(we do not need the last column for the definition of βKG

but it will be usedlater). The commutativity of the right rectangle is the definition of βK , using theformula (7.4.7) for the equivariant cross product. The top vertical isomorphismscome from the following fact: if Z is a (G ×K)-space, the homotopy equivalenceE(G × K) × Z → EG × EK × Z given by

((ti(gi, ki), z

)7→

((tigi, z), (tiki, z)

)

descends to a homotopy equivalence E(G × K)G×K × Z → EG ×G (EK ×K Z).The same homotopy equivalence is used to get the map

(8.4.2) ZK = EK ×K Z → EG×G (EK ×K Z) ≈ E(G×K)×G×K Z = ZG×K

giving rise (for Z = Y × Y ) to the homomorphism ρ. For βKG, we use the commu-

tative diagram


(8.4.3)

H2nG∧K(Y ∧ Y )

ρ

qq

H2nK (Y ∧ Y )G oo ≈

(1)// H2n((Y × Y )K , (Y ∨ Y )K)G oo ρ

H2nG ((Y × Y )K , (Y ∨ Y )K)

≈(2)

OO

H2n((Y × Y )K×K , (Y ∨ Y )K×K)G oo ρ

∆∗K

OO

H2nG ((Y × Y )K×K , (Y ∨ Y )K×K)

∆∗K

OO

H2n(YK × YK , YK∨YK)G oo ρ

P

OO

H2nG (YK × YK , YK∨YK)

OO

HnK(Y ) oo ≈ //

βK

OO

Hn(YK , yK)

β

OO

βG

33

As Y is equivariantly well pointed by y, the pair (YK , yK) is well cofibrant, sothe cross-square map β is defined. Also, the pair (Y × Y, Y ∨Y ) is K-equivariantlywell cofibrant (the proof is the same as for Lemma 7.2.12). Hence, Identification(1) then comes from Corollary 7.2.16. The same argument gives Identification (2),

using (8.4.2) for Z = Y × Y and Z = Y ∨ Y . The lifting βG is defined using thefollowing diagram.

H∗K(YK ∧ YK)G // H2n

K (YK × YK , YK∨YK)G

H2nG (YK ∧ YK)

ρhh

// H2nG×K(YK × YK , YK∨YK)

ρjj❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱

Hn(YK)≈ //

β

OO

βG

66♠♠♠♠♠♠♠♠♠♠♠H∗(YK , yK)

β

OO

βG

44

With these definitions, Diagram (8.4.3) is mapped into Diagram (8.4.1), giving rise

to the diagram of Point (1) of our lemma. That βKG

and βKG

coincide with βG andβG when K is the trivial group follows from the definitions, as well as Point (2).

Point (3) comes from (8.4.1), where the occurrence of Sq is noticed.

Proof of Proposition 8.4.1. By naturality, it is enough to prove the propo-sition for X = Kn and a the generator of Hn(Kn) = Z2 (n ≥ 1). We consider X aspointed by x ∈ X0, using the CW-structure given in the proof of Proposition 3.8.1,whose 0-skeleton consists in the single point x.

Let the symmetric group Σ = Sym4 act on X∧4 = X ∧ X ∧ X ∧ X andX4 = X×X×X×X by permutation of the factors. This action may be restrictedto the subgroup of Γ of Σ generated by s = (1, 2)(3, 4) and t = (1, 3)(2, 4). Asfor (8.3.1), the use the base points x ∈ X and (x, x, x, x) ∈ X ∧ X provide a


commutative diagram

(8.4.4)

H∗(X∧4) // // H∗(X4)

H∗(X) // //

β β

OO

H∗(X)

ββ

OO.

By Proposition 4.7.11, Hk(X∧4) = 0 for k < 4n and H4n(X∧4)Σ = H4n(X∧4) =Z2. As in Lemma 8.3.1, using Proposition 7.2.17, we get liftings

(β β)Σ : Hn(X)→ H4nΣ (X∧4) and (ββ)Σ : Hn(X)→ H4n

Σ (X4)

of β β and ββ. Consider the composite map

Φ: Hn(X)(ββ)Σ−−−−−→ H4n

Σ (X4)r−→ H∗Σ((X

4)Σ) ≈ H∗(X)⊗H∗(BΣ) .

Let Γ be the subgroup of Γ generated by s = (1, 2)(3, 4) and t = (1, 3)(2, 4). As sand t commute, Γ is isomorphic to G×G. Let i : Γ→ Σ denote the inclusion. Weshall prove that the diagram

(8.4.5)

H∗(X)⊗H∗(BΓ)

id⊗H∗Bi

H∗(X)⊗H∗(BΓ0)OO≈

Hn(X)

Φ

99

Sq Sq // H∗(X)[u, v]

.

is commutative. For the moment, let us show that this property implies Propo-sition 8.4.1. It is enough to prove that the image of H∗(BΣ) → Z2[u, v] consistsof symmetric polynomials. Under the isomorphism Z2[u, v] ≈ H∗(BΓ) the auto-morphism exchanging u and v corresponds to that induced by exchanging s andt. But in Σ, exchanging s and t is achieved by the conjugation by the transposi-tion (2, 4). Such an inner automorphism of Σ induces the identity on H∗(BΣ) byProposition 7.2.3.

It remains to prove that Diagram (8.4.5) is commutative. Let G1 and G2

be the subgroups of Γ generated by s and t respectively, so Γ ≈ G1 × G2. The


commutativity of Diagram (8.4.5) comes from that of the diagram(8.4.6)

H4nΓ (X4)

≈

''

r // H4nΓ (X)

≈

H4nG2×G1

(X4)r //

r×r

((

II

H4nG2×G1

(X)OO=

H4nG2×G1

(X2)r //

III

H4nG2×G1

(X)

Hn(X)βG1 //

(ββ)Γ

OO

Sq

++❲❲❲❲❲❲❲❲

❲❲❲❲❲❲❲❲

❲❲❲❲❲❲❲❲

I

H2nG1

(X2)r //

βG1G2

OO

H2nG1

(X)OO≈

βG1G2

OO

H2n(BG1 ×X)Sq // H4n(BG2 ×BG1 ×X)

≈

OO

together with the obvious commutativity of the diagram

H4nΣ (X4)

j∗

&&

r // H4nΣ (X)

j∗

≈ // H4n(BΣ×X)

(Bj×id)∗

Hn(X)

β2Γ //

β2Σ

OO

H4nΓ (X4)

r // H4nΓ (X)

≈ // H4n(BΓ×X)

.

The commutativity of Diagram (8.4.6) comes from that of its subdiagrams I, IIand III (the commutativity of the other diagrams is obvious). Diagrams II and IIIcommute because of Points (2) and (3) of Lemma 8.4.2. To verify the commutativityof Diagram (I), we put it as the inner square of the following diagram.

H4nΓ (X∧4)

≈ //

&&

(A)

(D)

H4nG2×G1

(X∧4)

ww♥♥♥♥♥♥♥♥♥♥

H4nΓ (X4)

≈ // H4nG2×G1

(X4)

Hn(X)βG1 //

(ββ)Γ

OO

H2nG1

(X2)

βG1G2

OO

Hn(X)βG1 //

(β β)Γ

OO

≈88qqqqqqqqq

H2nG1

(X∧2)

βG1G2

OO

ggPPPPPPPPPP

(C)

(B)

Diagrams (A), (B) and (C) commute because of Lemmas 8.3.1 and 8.4.2, and Dia-

gram (A) is obviously commutative. As Hn(X)→ Hn(X) is an isomorphism, theinner square will commute if the outer does. But the outer square commutes bydefault, since, using Proposition 7.2.17, all the groups are equal to Z2 and the mapsare non-trivial (for βG1 and βG1

G2, this is checked using the restrictions to the trivial

group and Point(2) of Lemma 8.4.2).


From Proposition 8.4.1, we deduce relations between SqiSqj called the Ademrelations.

Theorem 8.4.3 (Adem relations). Let X be a topological space. For eachi, j ∈ N with i < 2j, the relation

SqiSqj =

[i/2]∑

k=0

(j − k − 1

i− 2k

)Sqi+j−kSqk (0 < i < 2j)

holds amongst the non-graded endomorphisms of H∗(X).

Examples 8.4.4. (1) When i = 1, the right hand member in the Adem relation

reduces to(j−11

)Sqj+1. Hence

Sq1Sqj =

Sqj+1 if j is even

0 if j is odd.

For instance, Sq1Sq1 = 0 and Sq1Sq2 = Sq3.

(2) In the limit case i = 2j − 1 with j > 0, the binomial coefficient(j−k−1

2j−1−2k

)

vanishes if k ≤ j − 1, since 2j − 1− 2k > j − k − 1. Then

Sq2j−1Sqj = 0 if j > 0 .

Proof of the Adem relations. Let a ∈ Hn(X). The homomorphism Sq: Hn(X)→H∗(RP∞ ×X)

≈−→ H∗(X)[u] satisfies

Sq(a) =

n∑

µ=0

Sqµ(a)un−µ =∑

µ∈Z

Sqµ(a)un−µ .

The range extension µ ∈ Z is possible since the other summands vanish. We shalldo that repeatedly in the computations below and, without other specification, thesummations will be over the integers (with only a finitely many non-zero terms).This permits us to exchange the summation symbols.

Observe that Sq(u) = u2 + uv = u(u + v) thus, by multiplicativity of Sq, one

has Sq(uk) = Sq(u)k = uk(u+ v)k. Therefore

Sq Sq(a) =∑µ Sq(u

n−µ)Sq(Sqµ(a))

=∑µ u

n−µ(u+ v)n−µ∑λ Sq

λSqµ(a)vn+µ−λ

=∑µ,λ u

n−µvn+µ−λ(u+ v)n−µSqλSqµ(a)

=∑µ,λ,ν

(n−µν

)un−µ+νv2n−λ−νSqλSqµ(a)

Setting λ+ ν = i yields

Sq Sq(a) =∑

ν,µ,i

(n−µν

)un−µ+νv2n−iSqi−νSqµ(a) .

Setting n− µ+ ν = 2n− q yields

(8.4.7) Sq Sq(a) =∑ν,q,i

(2n−q−ν

ν

)u2n−qv2n−iSqi−νSqq+ν−n(a) .

By Proposition 8.4.1, for each i and q, the coefficient of u2n−qv2n−i in (8.4.7) mustbe equal to that of u2n−iv2n−q. This leads to the equation

∑ν

(2n−q−ν

ν

)Sqi−νSqq+ν−n(a) =

∑ν

(2n−i−ν

ν

)Sqq−νSqi+ν−n(a) .

8.5. THE STEENROD ALGEBRA 289

In the left hand member, set j = q−n. In the right hand member, set k = i+ν−nand use the relation

(xy

)=

(x

x−y

). In both sides, restrict the range of summation so

that the summands are not zero for obvious reasons. This gives

(8.4.8)i∑

ν=0

(n−j−ν

ν

)Sqi−νSqj+ν(a) =

[i/2]∑

k=0

(n−ki−2k

)Sqj+i−kSqk(a) .

Now, i and j being fixed, suppose that n = 2r − 1 + j for r large. Then, byLemma 6.2.6, (

n−j−νν

)=

(2r−1−ν

ν

)= 0 if ν 6= 0

for, the dyadic expansion of 2r− 1− ν has a zero where that of ν has a one. Hence,the left hand member of (8.4.8) reduces to the single term SqiSqj(a). Also,

(n−ki−2k

)=

(2r+j−k−1

i−2k

)=

(j−k−1i−2k

)if i < 2j

since the length of the dyadic expansions of i − 2k is not more than that of j − 1and adding 2r to j − k − 1 only puts a single 1 far to the left.

Thus, (8.4.8) proves the Adem relations classes of degree 2r−1+ j with r large(2r > maxi, j). As Σ∗Sq = SqΣ∗, Equation (8.4.8) holds for a if and onlyif it holds for Σ∗(a). But the suspension isomorphism may be iterated on a classa ∈ Hn(X) till its degree becomes of the form 2r − 1 + j with r large. This provesthe Adem relations.

8.5. The Steenrod algebra

The Steenrod algebra A is the graded Z2-algebra generated by indeterminatesSqi (in degree i) and subject to the Adem relations and to Sq0 = 1. The propertiesof Steenrod squares imply that the cohomologyH∗(X) of a spaceX is anA-module.The algebraic study of A-modules is a rich subject (see, e.g. [212] for a survey).

Lemma 8.5.1. As an algebra, A is generated by Sqn | n = 2r.

Proof. Let m = 2r + s with s < 2r (r ≥ 1). As(2r−1s

)≡ 1 mod 2, the Adem

relation

SqsSq2r

= Sqm +

[s/2]∑

k=1

(2r−k−1s−2k

)Sqm−kSqk (0 < i < 2j)

expresses Sqm as a sum of SqiSqj . If s > 0 then i, j < m, which permits us toprove the lemma by induction on m.

Here are a few examples of decompositions of Sqi according to Lemma 8.5.1and its proof:

(8.5.1)

Sq3 = Sq1Sq2

Sq5 = Sq1Sq4

Sq6 = Sq2Sq4 + Sq5Sq1 = Sq2Sq4 + Sq1Sq4Sq1

Sq7 = Sq1Sq6 = Sq1Sq2Sq4

For 0 6= a ∈ H1(RP∞), the formula Sq(a2n

) = a2n

+ a2n+1

shows that Sq2n

is

not a sum of SqiSqj with i, j < 2n. On the other hand, the Adem relations implythat

Sq2Sq2 = Sq3Sq1 = Sq1Sq2Sq1 .


Therefore, the Sq2r

do not generate A freely. In order to achieve that, we shall takeanother system of generators.

For a sequence I = (i1, . . . , ik) of positive integers, we set SqI = Sqi1 · · · Sqik ∈

A. The degree of I is i1+ · · ·+ ik. The sequence I is called admissible if ij ≥ 2ij+1.

Let Admn be the (finite) set of admissible sequences of degree n. A monomial SqI

is called admissible if I is admissible.

Proposition 8.5.2. A is the polynomial algebra over the admissible monomi-als. [Uncorrect! see Erratum 12.0.12]

The family of admissible monomials is sometimes called the Cartan-Serre basisof A.

Before proving Proposition 8.5.2, we develop some preliminaries. Fix an inte-ger n and consider

(8.5.2) wn = x1 · · ·xn ∈ H∗((RP∞)n) ≈ Z2[x1, . . . , xn] .

As Sq(xi) = xi + x2i = xi(1 + xi), one has

Sq(wn) =

n∏

i=1

Sq(xi) = wn

n∏

i=1

(1 + xi) .

Hence,

(8.5.3) Sqk(wn) = wnσk

where σk is the k-th elementary symmetric polynomial:

(8.5.4)

σ0 = 1σ1 = x1 + · · ·+ xnσ2 = x1x2 + · · ·+ xn−1xnσk =

∑i1<···<ik

xi1 · · ·xik .For an integer p≥ 0, we use the joker notation Lp for any polynomial in σ0, . . . , σp.

For instance, the equations Lp = Lp+q [see Comment 12.0.13] and Sqi(Lp) = Lp+ihold true but, as Sqp(σp) = σ2

p and Sqi(σp) = 0 for i > p, one has Sqi(Lp) = L2p−1

for all i [see also Comment 12.0.14].

Lemma 8.5.3. [See Comment 12.0.16.] Let I = (i1, . . . , ir) be an admissiblesequence. Then, for w = wn with n ≥ i1, one has

SqI(w) = w(σi1Li1−1 + Li1−1) = wLi1 .

Proof. Only the first equation has to be proven. We proceed by inductionon r. For r=1, the lemma follows from (8.5.3). Suppose that r ≥ 2.

Let I = (i2, . . . , ir). By induction hypothesis, Equation (8.5.3) and the Cartanformula, one has

SqI(w) = Sqi1SqI(w)

= Sqi1(wLi2 )

= Sqi1(w)Li2 +∑

j≥1 Sqi1−j(w)Sqj(Li2)

= wσi1Li2 + w∑j≥1 σi1−jL2i2−1

= w(σi1Li1−1 + Li1−1)

since i1 ≥ 2i2.


Proof of Proposition 8.5.2. [This proof does not work: see Erratum 12.0.12and Comment 12.0.16 for a new proof.] The Adem relations imply that any mono-

mial SqI is a sum of admissible ones. It remains to see that the admissible mono-mials are linearly independent.

Suppose that∑

I∈I λISqI = 0 for I ⊂ Admn and λI ∈ Z2. We must prove that

λI = 0 for all I. We proceed by induction on the cardinality of I. Lemma 8.5.3proves the assertion for I = (i1, . . . , ik) (and/or there is nothing to prove ifI = ∅).

Let maxj I = max0, ij | (i1, . . . , ik) ∈ I and let w = wn as in (8.5.2) withn ≥ max1 I. Set I = I0∪ I1 where I0 = (i1, . . . , ik) ∈ I | i1 = max1 I.By Lemma 8.5.3, one has

0 =∑

I∈I

λISqI(w) =

∑

I∈I0

λISqI(w)+

∑

I∈I1

λISqI(w) = w(σmax1 ILmax1 I−1+Lmax1 I−1) .

Therefore,

(8.5.5)∑

I∈I0

λISqI(w) = 0 and

∑

I∈I1

λISqI(w) = 0 .

(This uses a very easy case of the uniqueness of the expression of a symmetricpolynomial as a polynomial in the σi’s) [this deduction is not correct: see Erra-tum 12.0.12].

If the decomposition I = I0 ∪ I1 is non-trivial (i.e. I 6= I0), (8.5.5) permitsus to apply the induction hypothesis. Otherwise, we decompose I0 with respectto max2(I0): I0 = I00∪I01, where I0 = (i1, . . . , ik) ∈ I | i2 = max2 I0 and, asin (8.5.5), obtain that

(8.5.6)∑

I∈I00

λISqI(w) = 0 and

∑

I∈I01

λISqI(w) = 0 .

If ♯ I > 1, iterating this process will once produce a non-trivial decomposition,enabling us to use the induction hypothesis.

The proof of Proposition 8.5.2 shows that the map A 7→ A(w) sends SqI | I ∈Admn into a free family of H∗((RP∞)n). This proves the following result.

Proposition 8.5.4. Let 0 6= a ∈ H1(RP∞). The evaluation map A →H∗((RP∞)n) given by A 7→ A(a× · · · × a) (n times) is injective in degree ≤ n.

We now turn our interest to the cohomology ring H∗(Km) of the Eilenberg-

MacLane complex Km. It contains the classes SqI(ι) (0 6= ι ∈ Hm(Km) = Z2)and the admissible monomials play an important role. Define the excess e(I) of anadmissible sequence I = i1, . . . , ik by

e(I) = (i1 − 2i2) + (i2 − 2i3) + · · ·+ (ik−1 − 2ik) + ik = i1 − i2 − · · · − ik .

The excess of an admissible monomial SqI is the excess of I. Here is a famoustheorem of J-P. Serre [175, § 2].

Theorem 8.5.5. H∗(Km) is the polynomial algebra over SqI(ι) for I admissibleof excess < m.


The proof of this theorem uses spectral sequences and will not be given here.The condition e(I) < m is natural: SqI(ι) = 0 if e(I) > m since i1 = e(I) + i2 +· · ·+ ik > n+ i2 + · · ·+ ik. If e(I) = m, then

SqI(ι) = (Sqi2 · · · Sqik)2(ι) = · · · = (Sqir · · ·Sqik)2r−1

(ι)

where e(ir, . . . , ik) < m.

Example 8.5.6. (1) Only the empty sequence has excess 0. Then H∗(K1)is the polynomial algebra generated by ι ∈ H1(K1). This is not a surprise sinceK1 ≈ RP∞ by Proposition 3.8.3.

In order to formulate the other examples, observe that if I = (i1, . . . , ik) isadmissible, so is I+ = (2i1, i1, . . . , ik) and e(I+) = e(I). We denote by F(I) thefamily of admissible sequences obtained from I by iterating this construction.

(2) The family of admissible monomials with excess 1 is F(1). Thus H∗(K2) isa polynomial algebra with one generator in each degree 2i + 1, i ∈ N. Its Poincareseries is

Pt(K2) =∏

i∈N

1

1− t2i+1.

For the Poincare series of Km, see Lemma 8.5.13 below.(3) The set of admissible monomials with excess 2 is the union of the families

F(2r + 1, 2r−1, . . . , 2, 1) for r ≥ 0.(4) The Poincare series of Km is computed in [175, § 17].

Remark 8.5.7. The coefficient exact sequence 0 → Z2 → Z4 → Z2 → 0 givesrise to a Bockstein homomorphism β : H∗(X)→ H∗+1(X) (see [82, § 3.E]). As β isfunctorial and not trivial, one has β(ι) 6= 0 in Hn+1(Kn). But, by Theorem 8.5.5,the only non-trivial element in Hn+1(Kn) is Sq1(ι). By naturality of β and Sq1,this proves that β = Sq1. This argument illustrates the following corollary ofTheorem 8.5.5, saying that the actions of the Steenrod algebra on Hn(−) for alln ∈ N generate all the mod 2 cohomology operations.

Corollary 8.5.8. Let Q be a cohomology operation, and let Q[n] its restrictionto Hn(−). Then, there exists An ∈ A such that Q[n](x) = Anx for all x ∈ Hn(X)and all spaces X.

Proof. By functoriality, it suffices to prove the statement for X = K =K(Z2, n) and x = ι, the generator ofHn(K). ButQ[n](ι) ∈ H≥n(K) by Lemma 8.1.5

and H≥i(K) = A · ι by Theorem 8.5.5.

We now list other corollaries of Theorem 8.5.5. The following one comes fromProposition 8.5.4.

Corollary 8.5.9. Let 0 6= a ∈ H1(RP∞) and let y = a×· · ·×a ∈ Hn((RP∞)n).Let f : (RP∞)n → Kn be such that H∗f(ι) = y. Then, H∗f : Hi(Kn)→ Hi((RP∞)n)is injective for i ≤ 2n.

Remark 8.5.10. The proofs of both Theorem 8.5.5 and Proposition 8.5.4, andthen that of Corollary 8.5.9, do not use the Adem relations. Thus, one can useCorollary 8.5.9 to give an alternative proof of the Adem relations, as in [175, § 33],[154, pp. 29–31] or [27].

Corollary 8.5.11. The Adem relations are the only relations amongst theSqI ’s which hold true for all spaces.


Proof. A relation amongst the SqI ’s would be of the form P (SqI1 , . . . , SqIr ) =0, where P is a Z2-polynomial in r variables. The Adem relations imply that anymonomial SqI is a sum of admissible ones. Therefore, there is a relation of the formP (SqJ1 , . . . , SqJs) = 0, where P is a Z2-polynomial in s variables and J1, . . . , Jsare admissible sequences. Let m be the maximal excess of J1, . . . , Js. For ι thegenerator of Hm(K(Z2,m)), the equation P (SqJ1(ι), . . . , SqJs(ι)) = 0 implies, by

Theorem 8.5.5, that P = 0. Hence, the original relation P (SqI1 , . . . , SqIr ) = 0 was aconsequence of the Adem relations. [For a simpler argument, see Comment 12.0.18.]

Another consequence of Theorem 8.5.5 is that Steenrod squares are character-ized by some of their properties listed in Theorem 8.2.1.

Proposition 8.5.12. Suppose that for each CW -complex X, there exists a mapP : H∗(X)→ H∗(X) satisfying the following properties.

(a) If g : Y → X is a continuous map, then H∗gP = P H∗g.(b) P (Hn(X)) ⊂ H≤2n(X).(c) If a ∈ H1(RP∞) then P (a) = a+ a a.(d) P (x y) = P (x) P (y) for all x, y ∈ H(X).

Then P = Sq.

Proof. Using (a) and (d) together with the definition of the cross product,we get

(d’) P (x× y) = P (x)× P (y) for all x ∈ H(X) and y ∈ H(Y ).

Let w = x1 · · ·xn ∈ H∗((RP∞)n) ≈ Z2[x1, . . . , xn]. Using (c) and (d’) we prove,as for (8.5.3), that P (w) = w σk. But, by Formula (8.5.3) again, this shows thatP (w) = Sq(w). Using (a), (b) and Corollary 8.5.9, we deduce that P = Sq onHn(Kn). By (a), this proves that P = Sq in general.

As a last application of Theorem 8.5.5, we compute the Poincare series of Km,following [175, § 17]. By Theorem 8.5.5, one has

Pt(Km) =∞∏

r=0

(1

1− tm+r

)a(r),

wherea(r) = ♯ I | I admissible, e(I) < m and deg(I) = r .

To compute a(r), we note that an admissible sequence I = (i1, . . . , ik) is determinedby its excess components α1 = i1 − 2i2, . . . , αk−1 = ik−1 − 2ik, αk = ik. Therefore,

(8.5.7) a(r) = ♯ (α1, . . . , αk) |k∑

i=1

αi < m and

k∑

i=1

αi(2i − 1) = r .

Set α0 = m− 1−∑ki=1 αi. Then

(8.5.8) m+ r = 1 +k∑

i=0

αi2i = 1 + 20 + · · ·+ 20︸︷︷︸

α0

+ · · ·+ 2k + · · ·+ 2k︸︷︷︸αk

.

Using that∑ki=0 αi = m− 1 and writing the exponents of 2 in (8.5.8) in decreasing

order h1 ≥ · · · ≥ hm−1, we get

a(r) = ♯ (h1, . . . , hm−1) ∈ Nm−1 | h1 ≥ · · · ≥ hm−1 and 2h1 + · · ·+ 2hm−1 + 1 = m+ r .


This proves the following result of [175, § 17].

Lemma 8.5.13. The Poincare series of Km is

Pt(Km) =∏

h1≥···≥hm−1≥0

1

1− t2h1+···+2hm−1+1.

Let r < m. If I is an admissible sequence with deg(I) = r, the conditione(I) < m is automatic since e(I) ≤ deg(I). Using (8.5.7), we see that a(r) is equalto the number of partitions of r into integers of the form 2i − 1. Also, Hm+r(Km)

only contains classes of the form SqI(ı) (products like SqI(ı)SqJ(ı) have higherdegree). This proves the following result of [191, p. 37].

Lemma 8.5.14. If r < m, then dimHm+r(Km) is equal to the number of par-titions of r into integers of the form 2i − 1.

8.6. Applications

Suspensions of the Hopf maps. Recall that the non-triviality of the cup-square map α(a) = a a is H∗(KP 2) for K = R,C,H or O implies that the Hopfmaps

h1,1 : S1 → S1 , h3,2 : S

3 → S2 , h7,4 : S7 → S4 and h15,8 : S

15 → S8

are not homotopic to constant maps (see Corollary 6.1.9). This argument cannotbe applied to the suspensions of the Hopf maps Σkhp,q : S

p+k → Sq+k since the cupproduct in H>0(ΣkKP 2) vanishes by dimensional reasons (also by Corollary 4.4.4).But, for instance in RP 2, α(a) = Sq1(a). As Σ∗Sq = SqΣ∗, one deduces thatSq1 is not trivial on ΣkRP 2 and therefore Σkh1,1 is not homotopic to a constantmap for all k ∈ N (though Hk+1Σkh1,1 vanishes on Hk+1(ΣkRP 2)). The sameargument applies for the other Hopf maps, so we get the following proposition.

Proposition 8.6.1. For all k ≥ 0, the k-th suspensions of the Hopf maps

Σkh1,1 : Sk+1 → Sk+1 , Σkh3,2 : S

k+3 → Sk+2 ,

Σkh7,4 : Sk+7 → Sk+4 and Σkh15,8 : S

k+15 → Sk+8

are not homotopic to constant maps.

Actually, for k ≥ 1, Σkh3,2 represents the generator of πk+3(Sk+2) ≈ Z2 (see

[197, Proposition 5.1]).Restrictions on cup-squares. The action of the Steenrod algebra on the co-

homology imposes strong restrictions for the existence of classes with non-vanishingcup-square. Let A<n be the subalgebra of A generated by Sqi | i < n.

Proposition 8.6.2. Let X be a topological space and let a ∈ Hm(X). If m isnot a power of 2, then a a ∈ A<m(a).

Proof. By Lemma 8.5.1, A is generated by Sqk | k = 2i, so Sqm ∈ A<m ifm is not a power of 2. As a a = Sqm(a), this proves the proposition.

Corollary 8.6.3. Let X be a topological space. Let a ∈ Hm(X) such that

a a 6= 0. Then, there exists k ≤ m with k = 2i such that Sqk(a) 6= 0.


As a consequence of Proposition 8.6.2, if a ∈ Hm(X) satisfies a a 6= 0 withm not a power of 2, there must be a non-zero group Hm+k(X) with 0 < k < m.This is not the case for the space X = Cf = D2m ∪f Sm used in § 6.3 to define theHopf invariant of f : S2m−1 → Sm. Therefore, Proposition 8.6.2 has the followingcorollary, a result due to Adem [4].

Corollary 8.6.4. Suppose that f : S2m−1 → Sm satisfies Hopf (f) = 1. Thenm = 2r.

When m is a power of 2, Proposition 8.6.2 is wrong, as seen with the projectivespaces KP 2. Using secondary cohomology operations, J.F. Adams proved deeperresults [2, Theorem 4.6.1 and §1.2] implying the following theorem.

Theorem 8.6.5. Let X be a space such that Hm+k(X) = 0 for 1 ≤ k ≤ 2r. If

r ≥ 3, then Sq2r+1

(Hm(X)) = 0.

Combining this theorem with Corollary 8.6.4, Adams got his famous result [2,Theorem 1.1.1]:

Theorem 8.6.6. Continuous maps f : S2m−1 → Sm with Hopf invariant 1 existonly for m = 1, 2, 4, 8.

Remark 8.6.7. Recall that a non-singular map µ : Rm×Rm → Rm (see § 6.2.2)determines, using (6.2.2), a continuous multiplication µ : Sm−1 × Sm−1 → Sm−1,giving rise to a map fµ : S

2m−1 → Sm with Hopf (fµ) = 1 (Proposition 6.3.3). Thus,by Theorem 8.6.6, non-singular maps Rm ×Rm → Rm exist only for m = 1, 2, 4, 8.Note that Corollary 8.6.4 gives another proof of Proposition 6.2.5.

Somehow related to Theorem 8.6.6 are the results of E. Thomas on the cohomol-ogy of H-spaces (see e.g. [194]). They also heavily use Steenrod square techniques.

A variant of the Sullivan conjecture. If f : R → S is a continuous map,then H∗f is a morphism of algebras over the Steenrod algebra A. The followingresult, conjectured by H. Miller, was proven by J. Lannes [124, Theorem 0.4].

Theorem 8.6.8. Let Y be a simply connected space of finite cohomology type.Let B = (RP∞)n. Then the map

[B, Y ]→ homA(H∗(Y ), H∗(B))

is a bijection.

In particular, if Y is a finite dimensional CW -complex, then [B, Y ] is a single-ton. This is a weak version of the Sullivan conjecture (see Remark 6.1.4).


8.1. Let T = S1 × S1 and let K be the Klein bottle. Is ΣT homotopy equivalentto ΣK?

8.2. In H∗(RP∞ × RP∞) ≈ Z2[a, b], compute Sq6(a5b7).

8.3. Let a ∈ Hm(X) be such that Sq(a) 6= a. Prove that the smallest integer i suchthat Sqi(a) 6= 0 is a power of 2.

8.4. Verify the relations in the Steenrod algebra given in (8.5.1).

8.5. Check that Sq2Sq2Sq2Sq2 = 0 and that Sq3Sq3Sq3 = 0


8.6. Find a minimal set of generators of H∗(RP∞) as a module over the Steenrodalgebra.

8.7. Let X be a space such that there is a GrA-epimorphism Z2[a] →→ H∗(X),with a of degree r. Prove that r = 2k with k ≤ 3.

8.8. Let Sr−1 → Sn−1p−→M be a locally trivial fiber bundle, where M is a closed

manifold of positive dimension. Prove that n = kr and r = 1, 2, 4, 8.

8.9. Let M be a closed connected manifold with total Betti number 3. What canbe said about its dimension?

8.10. Let M be a smooth closed G-manifold (G of order 2). Let f : M → Rbe a G-invariant Morse-Bott function satisfying the hypotheses of Theorem 7.6.6.Suppose that f has only two critical values ±1, with M−1 = pt and dimM1 > 0.Suppose that n = dimM is odd. Prove that the pair (M,M1) is homeomorphic to(RPn,RPn−1). [Hint: this uses Exercise 8.8.]

8.11. Let Sq∗ be the cohomology operation defined as follows: for a ∈ Hn(X), we

set Sqi(a) = Sqn−i(a) (these Sqi’s were the first operations defined by Steenrod:see [40, Chapter 6]).

(a) Prove that Image(Sq0) ⊂ ker Sq1.(b) Prove that Image(Sq0) = ker Sq1 for X = RP∞.(c) Is (b) true for X = CP∞?

CHAPTER 9

Stiefel-Whitney classes

In § 4.7.6, we encountered the Euler class e(ξ) ∈ Hn(B) of a vector bundle ξof rank n over B, which sometimes distinguishes between non-isomorphic vectorbundles. This is an example of a characteristic class, meaning that it is naturalwith respect to induced bundles: e(f∗ξ) = H∗f(e(ξ)) for any continuous mapf : B′ → B. Characteristic classes play a major role in the study of vector bundles.

In § 9.4 below, we shall see that the Euler class belongs to a family wi(ξ) ∈Hi(B) of characteristic classes called the Stiefel-Whitney classes. Historically, thesewere the first characteristic classes discovered, in the simultaneous work of EduardStiefel and Hassler Whitney starting around 1935 (see [40, pp. 421–426]). Us-ing Morse theory, we shall see in § 9.5 that the cohomology of the Grassmanni-ans is GrA-generated by the Stiefel-Whitney classes of their tautological bundles.This implies that any characteristic class in mod 2 cohomology is a polynomial inthe Stiefel-Whitney classes (see Corollary 9.5.10). The Chern classes of a com-plex vector bundle, when reduced mod 2, are also Stiefel-Whitney classes (see Re-mark 9.7.6).

Geometric interpretations of w1 and w2 are given in Sections 9.2 and 9.3, re-lated to orientations and spin structures. Stiefel-Whitney classes have importantapplications in the topology of manifolds, for instance via the Wu formula (see§ 9.8) or in the work by R. Thom on cobordism, surveyed in § 9.9.2.

9.1. Trivializations and structures on vector bundles

We recall below some classical facts about vector bundles (defined on p. 156).Two vector bundles ξ = (p : E → X) and ξ′ = (p′ : E′ → X) over the samespace X are isomorphic if there exists a homeomorphism h : E → E′ such thatp′ h = p and such that the restriction of h to each fiber is linear. Two suchisomorphisms h0, h1 : E → E′ are isotopic if there exists a family of isomorphismsht : E → E′ depending continuously on t ∈ [0, 1] and joining h0 to h1. Unlessotherwise mentioned, the total space of a vector bundle ξ is denoted by E(ξ) andthe bundle projection by p.

Let ξ be a vector bundle of rank r over Y and let f : X → Y be a continuousmap. The induced vector bundle f∗ξ is the vector bundle of rank r over X definedby

E(f∗ξ) = (x, z) ∈ X × E(ξ) | f(x) = p(z) ,

with the projection onto X as the bundle projection. The projection to E(ξ) gives

a map f : E(f∗ξ)→ E(ξ) which is a linear isomorphism on each fiber and such that

297

298 9. STIEFEL-WHITNEY CLASSES

the diagram

(9.1.1)

E(f∗ξ)f //

prX

E(ξ)

p

X

f // Y

is commutative. Note that if f and g coincide over A ⊂ X , the induced bundlesf∗ξ and g∗ξ restrict to the same bundle over A. Two such maps are said homotopicrelative to A if there is a homotopy F : X × I → X between f and g such thatF (a, t) = f(a) = g(a) for all a ∈ A. The next proposition is proven in [181,Theorem 11.3].

Proposition 9.1.1. Let ξ be a vector bundle over a space B. Let f, g : X → Bbe two maps, where X is a paracompact space. Suppose that f and g coincide overA ⊂ X. If f and g are homotopic relative to A, then there exists an isomorphismbetween f∗ξ and g∗ξ which is the identity over A.

A trivialization of a vector bundle ξ over X is an isomorphism of ξ with theproduct bundle ηr = (prX : X×Rr → X). A vector bundle admitting a trivialization

is called trivial. Denote by T (ξ) the space of trivializations of ξ (endowed with the

compact-open topology). Then, T (ξ) = π0(T (ξ)) is the set of isotopy classes of

trivializations of ξ is π0(T (ξ)).By post-composition, the topological group Aut (ηr) of the automorphisms of

the product bundle ηr acts continuously on the left on T (ξ). As usual we denote byGL(r,R) the general linear group, i.e. the topological group of automorphisms ofRr. Alternatively, GL(r,R) is the group of invertible (r×r)-matrices with real coef-

ficients, topologized as an open set of Rr2

. Note that Aut (ηr) ≈ Map(X,GL(r,R)),so the group π0(Map(X,GL(r,R))) = [X,GL(r,R)] acts on T (ξ).

Lemma 9.1.2. Let ξ be a trivial vector bundle over a topological space X. Thenthe action

(9.1.2) [X,GL(r,R)]× T (ξ)→ T (ξ)is simply transitive. In particular, given a trivialization T0 of ξ, the correspondence

λ 7→ λ · T0 induces a bijection [X,GL(r,R)]≈−→ T (ξ).

Proof. Let h, h′ ∈ T (ξ). Then h′ = (h′ h−1)h and h′ h−1 ∈ Aut (ηr). Thisshows that Aut (ηr) acts transitively on T (ξ), whence the action (9.1.2) is transitive.

On the other hand, if h1 = λ · h0 ∈ T (ξ) is isotopic to h0 via a homotopyht ∈ T (ξ), the formula λt = hth

−10 defines a continuous path λt ∈ Aut (ηr) joining

λ0 = id to λ1 = λ. This shows that [X,GL(r,R)] acts simply on T (ξ).

Remark 9.1.3. The bijection of Lemma 9.1.2 depends on the choice of [h0] ∈T (ξ). In general, there is no natural choice, except for the product bundle (like inthe next example).

Example 9.1.4. We see the projection S1 × C → S1 as the product vectorbundle ξ of rank 2 over the circle, identifying C with R2. By Lemma 9.1.2

T (ξ) ≈ [S1, GL(2,R)] ≈ [S1, O(2)] ≈ ±1 × Z

9.1. TRIVIALIZATIONS AND STRUCTURES ON VECTOR BUNDLES 299

(by Gram-Schmidt orthonormalization, GL(2,R) has the homotopy type of O(2)which is homeomorphic to ±1×S1). The fiber linear map corresponding to (1, k)is given by (x, z) 7→ xkz and that corresponding to (−1, k) is given by (x, z) 7→ xkz.

Example 9.1.5. Let ξ be a vector bundle over Y and let f : X → Y be acontinuous map. If ξ is trivial, so is f∗ξ. More precisely, let h : E(ξ)→ Y ×Rr be a

trivialization of ξ. Write h under the form h(z) = (p(z), h(z)) where h : E(ξ)→ Rd

is a map whose restriction to each fiber is a linear isomorphism. Then the map

f∗h : E(f∗ξ) → X × Rr given by f∗h(x, z) = (x, h(z)) is a trivialization of f∗ξ.This process descends to a map

f∗ : T (ξ)→ T (f∗ξ)which is equivariant for the actions defined in (9.1.2), via the homomorphismf∗ : [Y,GL(r,R)] → [X,GL(r,R)] induced by λ 7→ λf . Indeed, for λ : Y →GL(r,R) and T ∈ T (ξ), the following formula holds true in T (f∗ξ)(9.1.3) f∗(λT ) = (f λ) f∗T .

Proposition 9.1.6. Let ξ be a vector bundle over a space X. If X is paracom-pact and contractible, then ξ is trivial. Moreover, any trivialization of the restric-tion of ξ over a point x0 ∈ X extends to a trivialization of ξ which is unique up toisotopy.

Proof. Let ξ = (p : E → X) and let ξ0 = (p : E0 → x0) be the restrictionof ξ over the point x0. The vector bundle ξ is induced by the identity of X :ξ = id∗Xξ. As X is contractible, idX is homotopic to a constant map c onto x0and, by Proposition 9.1.1, ξ ≈ c∗ξ0. The contractability of X also implies thatc∗ : [x0, GL(r,R)] → [X,GL(r,R)] is an isomorphism. Proposition 9.1.6 thenfollows from Lemma 9.1.2 and the considerations of Example 9.1.5.

We now consider a vector bundle ξ over a space X = Z ∪ϕ Y obtained byattaching a pair (Z,A) to a space Y using an attaching map ϕ : A→ Y (see p. 81).Let jY : Y → X and jZ : Z → X be the natural maps. Let ξY = j∗Y ξ, ξZ = j∗Zξand denote by ξA the restriction of ξZ over A.

Lemma 9.1.7. Let ξ be a vector bundle over X = Z∪ϕY as above. Suppose that

there exists trivializations hY ∈ T (ξY ) and hZ ∈ T (ξZ) such that the restrictionof hZ to ξA is isotopic to ϕ∗hY . Suppose that Z is paracompact and that the pair(Z,A) is cofibrant. Then, the bundle ξ is trivial.

Proof. Let hA ∈ T (ξA) be the restriction of hZ to ξA. An isotopy from hAto ϕ∗hY produces a trivialization h ∈ T (ξA× I) (where ξA× I is the vector bundlep× id : E(ξA)× I → A× I) restricting to hA on ξA×0 and to ϕ∗hY on ξA×1.The pair (Z,A) being cofibrant, there exists a retraction r : Z×I → Z×0∪A×I ofZ×I onto Z×0∪A×I (see Lemma 3.1.37). Let ξ1 be the restriction of r∗ξZ aboveZ×1 ≈ Z. As Z is paracompact and r is actually a strong deformation retraction(see Remark 3.1.42), ξ1 is isomorphic to ξZ relative to A by Proposition 9.1.1. Then,r∗h produces at level 1 a trivialization hZ of ξZ which is equal to ϕ∗hY on E(ξA).This condition implies that the formula

H(u) =

hZ(z, u) if p(u) = iZ(z)

hY (y, u) if p(u) = iY (y)


defines a trivialization H : E(ξ)→ X × Rr of ξ.

The above lemmas have analogues for principal bundles. Let A be a topologicalgroup. Recall that a A-principal bundle overX consists of a continuous map p : P →X , a continuous right action of A on P such that p(zα) = p(z) for all z ∈ P and allα ∈ A. In addition, the following local triviality should hold: for each x ∈ X thereis a neighbourhood U of x and a homeomorphism h : U × A → p−1(U) such thatph(x, a) = x and h(x, aα) = h(x, a)α. In consequence, A acts simply transitivelyon each fiber and p is a surjective open map, thus descending to a homeomorphism

P/A≈−→ X (use [44, § I of Chapter VI]). An isomorphism of A-principal bundles

from p : P → X to p′ : P ′ → X is a A-equivariant homeomorphism h : P → P ′

such that p′ h = p. Two such isomorphisms h0, h2 : P → P ′ are isotopic if thereexists a family of isomorphisms ht : P → P ′ depending continuously on t ∈ [0, 1]and joining h0 to h1. The notion of induced A-principal bundle works as for vectorbundles and Proposition 9.1.1 also follows from [181, Theorem 11.3].

A trivialization of an A-principal bundle is an isomorphism with the productbundle X × A → X . An A-principal bundle is trivial if and only if it admits asection (see, e.g. [181, I.8]). This gives a bijection between sections and trivializa-tions and between homotopy classes of sections and isotopy classes of trivialization.If σ1, σ2 : X → P are two sections of p, then σ2(x) = σ1(x)α(x) for a uniqueα : Map(X,A) whence the analogue of Lemma 9.1.2: the action of Map(X,A) onthe trivializations of P is simply transitive. Also, Lemma 9.1.6 holds true for prin-cipal bundles.

Let p : P → X be an A-principal bundle, p′ : P ′ → X be an A′-principal bundleand let ϕ : A→ A′ be a continuous homomorphism. A continuous map g : P → P ′

is called a ϕ-map if p′ g = p and g(zα) = g(z)ϕ(α) for all z ∈ P and α ∈ A.Let ρ : A → GL(r,R) be a representation (i.e. a continuous homomorphism)

of the topological group A. Let P → X be an A-principal bundle. Then theRr-associated bundle with total space

P ×(A,ρ) Rr = P × Rr

/(zα, u) ∼ (z, ρ(α)u)

is a vector bundle of rank r over X . Most of the time, the representation ρ isimplicit and we just write P ×A Rr for P ×(A,ρ) R

r. A (A, ρ)-structure (or just A-structure) for a vector bundle ξ of rank r over X is an A-principal bundle P → Xtogether with a vector bundle isomorphism f : P ×ARr → E(ξ). Two A-structures(P, f) and (P ′, f ′) are

• strongly equivalent if there exists an isomorphism of A-principal bundlesh : P → P ′ such that f ′ (h× idRr) = f .• weakly equivalent if there exists an isomorphism of A-principal bundlesh : P → P ′ such that f ′ (h× idRr) is isotopic to f

Here are a few examples.

9.1.8. A = 1, the trivial group. An 1-structure on a vector bundle ξ isjust a trivialization of ξ. Strong equivalence coincides here with equality. Weakequivalence classes of 1-structures correspond to isotopy classes of trivializations.

9.1.9. Each vector bundle ξ of rank r admits a GL(r,R)-structure which isunique up to strong equivalence. Indeed, consider the space of frames of ξ:

Fra(ξ) = ν : Rr → E(ξ) | ν is a linear isomorphism onto some fiber of ξ .

9.1. TRIVIALIZATIONS AND STRUCTURES ON VECTOR BUNDLES 301

with the map pFra : Fra(ξ) → X given by pFra(ν) = pν(0). (The image by ν ∈Fra(ξ) of the standard basis of Rr is a frame (basis) of ν(Rr), whence the name spaceof frames). By precomposition, the topological group GL(r,R) acts continuouslyand freely on the right upon Fra(ξ). The map pFra descends to a continuous bijectionFra(ξ)/GL(r,R) → X . Using local trivializations, one checks that this map is ahomeomorphism. Also, a local trivialization of ξ gives rise to a local section of pFra.Hence, pFra is a GL(r,R)-principal bundle, called the framed bundle of ξ.

Consider the evaluation map fFra : Fra(ξ) ×GL(r,R) Rr → E(ξ) sending [ν, t] toν(t). This map is a continuous bijection which is linear on each fiber and, usinglocal trivializations of ξ again, one checks that it is a homeomorphism. Hence,fFra is a GL(r,R)-structure on ξ. We claim that fFra is a universal structure inthe following sense. Each A-structure (for a representation ρ : A → GL(r,R))

f : P ×Rr → E(ξ) determines a ρ-map f : P → Fra(ξ). The map f sends z ∈ P tothe map f(z,−) ∈ Fra(ξ). We check that two A-structure (P, f) and (P ′, f ′) are

(a) strongly equivalent if there exists an isomorphism of A-principal bundles

h : P → P ′ such that f ′ h = f .(b) weakly equivalent if there exists an isomorphism of A-principal bundles

h : P → P ′ such that f ′ h is homotopic to f .

The case A = GL(r,R) give the uniqueness claimed above: any GL(r,R)-structureon ξ (for ρ = id) is strongly equivalent to fFra.

9.1.10. GL+(r,R)-structures and orientations (see also § 9.2 below). Recallthat an orientation of a finite dimensional real vector space is an equivalence class of(ordered) bases, where two bases are equivalent if their change-of-basis matrix is inGL+(r,R), i.e. has positive determinant. An orientation of a vector bundle ξ is anorientation of each fiber which varies continuously, i.e. there are local trivializations

p−1(U)≈−→ U ×Rr whose restriction to each fiber is orientation-preserving (for the

standard orientation of Rr). A vector bundle admitting an orientation is calledorientable and the choice of an orientation makes it oriented.

If P is a GL+(r,R)-principal bundle, then P ×GL+(r,R)Rr is oriented, using the

standard orientation of Rr. Hence, a GL+(r,R)-structure (P, f) on a vector bundleξ makes it oriented. Conversely, an oriented vector bundle ξ admits a GL+(r,R)-structure: one just restricts the canonical (Fra(ξ), fFra) of 9.1.9 to Fra+(ξ), where

(9.1.4) Fra+(ξ) = ν ∈ Fra(ξ) | ν preserves the orientation .(for the standard orientation of Rr). As in 9.1.9, we check that (Fra+(ξ), fFra) isuniversal for the A-structures on ξ with a representation ρ : A→ GL+(r,R).

9.1.11. O(r)-structures. Consider the orthogonal group O(r) with its standard

representation O(r) → GL(r,R). Let f : P ×O(r) Rr ≈−→ E(ξ) be an O(r)-structure

on ξ. Then, the standard inner product on Rr gives, via f , a Euclidean structureon ξ (see p. 156) for which f is an isometry on each fiber. On the other hand, aEuclidean bundle ξ of rank r admits an O(r)-structure: one restricts fFra to thesubbundle of Fra(ξ) formed by orthonormal frames:

Fra⊥(ξ) = ν : Rr → E(ξ) | ν is a linear isometry onto some fiber of ξ .As in (9.1.9), one shows that such an O(r)-structure compatible with a given Eu-clidean structure on ξ is unique up to strong equivalence. This process provides


a bijection between Euclidean structures on ξ and strong equivalences of O(r)-structures.

On the other hand, a vector bundle over a paracompact space admits Euclideanstructures which form a convex space. Let (ξ, et) be the vector bundle ξ endowedwith a Euclidean structure et depending continuously on t ∈ I. Then

Frae(ξ) = νt : Rr → E(ξ) | t ∈ I, νt is a linear isometry onto some fiber of (ξ, et)

is the total space of an O(r)-principal bundle over X × I. Note that Frae(ξ) isthe union indexed by I of Fra⊥(ξ, et), the bundle of orthonormal frames for theEuclidean structure et. If X is paracompact, Frae(ξ)→ X × I is isomorphic to theprincipal bundle Fra⊥(ξ, e0) × I → X × I [105, Chapter 4, Theorem 9.8]. Hence,there is an isomorphism h : Fra⊥(ξ, e0)→ Fra⊥(ξ, e1) such that i1h is homotopic toi0 (where it : Fra⊥(ξ, et)→ Fra(ξ) denotes the inclusion). This proves the followingstatement: a vector bundle over a paracompact space admits an O(r)-structurewhich is unique up to weak equivalence.

9.1.12. The case of orthogonal representations. Let ρ : A → O(r) be an or-thogonal representation of a topological group A. Let f : P ×A Rr → E(ξ) bean A-structure on the vector bundle ξ (for the representation ρ). As ρ is orthog-onal, P ×A Rr inherits a natural Euclidean structure which is transported on ξvia f (see 9.1.11). As in 9.1.9, there is an O(r)-structure (PO, fO) on ξ such that

f = fO f for a ρ-map f : P → PO. If h : P → P ′ induces a strong equivalence

between the A-structures (P, f) and (P ′, f ′), then PO = P ′O and f ′ h = f . If honly induces a weak equivalence, it descends to an isomorphism hO : PO → P ′Omaking the following diagram commutative

Ph //

f

P ′

f ′

PO

hO // P ′O

.

We can pull back f ′ over PO via hO, getting another representative of the weakequivalence class of (P ′, f ′). This permits us to assume that PO = P ′O and hO = id;this also means that the Euclidean structures induced by f and f ′ coincide. Now,using the Gram-Schmidt orthonormalization process in each fiber of ξ, the isotopybetween f ′ (h× idRr ) and f may be deformed into an isotopy of isometries. This

implies that f and f ′ h are homotopic.These considerations drive us to the following point of view for an A-structure

on ξ, in the case of an orthogonal representation. We first fix a Euclidean structureon ξ and consider only A-structures (P, f) on ξ for which f is an isometry. In other

words, an A-structure may be seen as a ρ-map f : P → Fra⊥(ξ) and strong or weakequivalences are described as in (a) and (b) of 9.1.9.

Note that, if ξ is an oriented Euclidean bundle, one can consider the orientedorthonormal frames

(9.1.5) Fra+⊥ (ξ) = Fra⊥(ξ) ∩ Fra+(ξ)

9.2. THE CLASS w1 – ORIENTABILITY 303

which is the total space of an SO(r)-principal bundle over X . Then (Fra+⊥ (ξ), fFra)

is an SO(r)-structure which is universal for the A-structures associated to a rep-resentation A → SO(r). The special case of the representation Spin(r) → SO(r)(spin structures) is treated in § 9.3 (compare [130, § II.1]).

The best known example of rep+⊥ (ξ) is for ξ = TSn, the tangent bundle to the

standard unit sphere Sn. One sees TSn as the space of couples (v, w) ∈ Rn+1×Rn+1

such that |v| = 1 and 〈v, w〉 = 0 (up to translation, w is tangent to v ∈ Sn). Thenthe map q : SO(n + 1) → Sn defined by q(A) = Ae1 (first column vector) is the

oriented frame bundle for TSn. The bundle isomorphism SO(n+1)≈−→ Fra+

⊥ (TSn)

sends A to the map νA : Rn → TSn defined by

νA(t1, . . . , tn) =(Ae1,

n∑

i=1

tiAei+1

).

9.1.13. Complex vector bundles. By replacing R by C in the definition of avector bundle, we get the notion of a complex vector bundle. The notion of a A-structure is defined accordingly, using a complex representation ρ : A → GL(r,C).As in 9.1.9, a complex vector bundle ξ of rank r admits a GL(r,C)-structure whichis unique up to strong equivalence. Also, as in 9.1.11, using a Hermitian structureon ξ (those form a contractible space), ξ admits an U(r)structure which is uniqueup to weak equivalence.

9.1.14. Classifying spaces and structures. Let EGL(r,R)→ BGL(r,R) be theuniversal bundle for the principal GL(r,R)-bundles and let ζ its associated vectorbundle. Recall that, for X paracompact, the correspondence [c] → c∗ζ provides abijection from [X,BGL(r,R)] and the set of isomorphism classes of vector bundlesover X of rank r (using (9.1.9)). We shall use the following consequence of thisresult.

Lemma 9.1.15. Let (X,A) be a cofibrant pair with X paracompact. Let ξ be avector bundle over X whose restriction over A is trivial. Then, there exists a vectorbundle ξ over X/A such that ξ ≈ π∗ξ, where π : X → X/A is the quotient map.

Proof. Let c : X → BGL(r,R) be a classifying map for ξ (r = rank of ξ),i.e. ξ ≈ c∗(ζ). In the definition of a cofibrant pair, A is supposed to be closed,so A is paracompact. The restriction of c to A is then null-homotopic. As (X,A)is cofibrant, there is a homotopy from c to c such that c|A is a constant map.Therefore, c descends to c : X/A→ BGL(r,R). Hence, ξ ≈ c∗ζ ≈ π∗(c∗ζ).

Structures on vector bundles and the classifying spaces are related as follows.Let ξ be a vector bundle of rank r over a paracompact space X . Fix a characteristicmap c : X → BGL(r,R) for Fra(ξ). A representation ρ : A→ GL(r,R) of A inducesa continuous map Bρ : BA→ BGL(r,R) which may be made a Serre fibration. Alifting of c is a continuous map c : X → BA such that Bρ c = c and two such liftingsc0 and c1 are homotopic if there exists a homotopy h : X × I → BA between c0and c1 such that Bρh(x, t) = c(x) for all (x, t) ∈ X × I. Then, the set of weakequivalence of (A, ρ)-structures on ξ is in bijection with the homotopy classes ofliftings of c. For details, see [23, § 4].

9.2. The class w1 – Orientability

Let O(V ) be the set of the two orientations of a finite dimensional vector spaceV . Let ξ be a vector bundle of rank r over a topological space X . Using the


canonical GL(r,R)-structure (Fra(ξ), fFra) of 9.1.9, we define

O(ξ) = Fra(ξ)×GL(r,R) O(Rr) .The projection O(ξ)→ X is a locally trivial bundle whose fiber over x ∈ X is, viafFra, in bijection with O(p−1(x)). In consequence, O(ξ)→ X is a two-fold covering.An orientation of ξ (see 9.1.10) is clearly a continuous section of O(ξ)→ X .

The characteristic class w(O(ξ)→ X) ∈ H1(X) (see § 4.3.2), is called the firstStiefel-Whitney class of ξ and is denoted by w1(ξ).

Proposition 9.2.1. A vector bundle ξ over a CW-complex X is orientable ifand only if w1(ξ) = 0 in H1(X). The set of orientations of an orientable bundle isin bijection with H0(X).

Proof. The 2-fold covering O(ξ) is trivial if and only if it admits a continuoussection, that is to say if and only if ξ is orientable. On the other hand, if X is aCW-complex, the 2-fold covering O(ξ)→ X is trivial if and only if its characteristicclass vanishes (see Lemma 4.3.6). There are two orientation for the restriction of ξover each connected component, whence the last assertion.

Remark 9.2.2. If η is a trivial vector bundle, then w1(ξ⊕ η) = w1(ξ). Indeed,there exists a natural map O(ξ) → O(ξ ⊕ η) which is an isomorphism of 2-foldcoverings.

Proposition 9.2.3. Vector bundles over a 1-dimensional CW-complex are clas-sified by their rank and their first Stiefel-Whitney class.

Proof. Let ξ and ξ′ be two vector bundles over a CW-complex X . If ξ and ξ′

are isomorphic, they have the same rank and the 2-fold coverings O(ξ) and O(ξ′)are isomorphic, which implies that w1(ξ) = w1(ξ

′).To prove the converse, note that, when X is 1-dimensional, any vector bundle

ζ over X is the Whitney sum of a line bundle λ with the trivial vector bundle [105,Chapter 8, Theorem 1.2]. By Remark 9.2.2, w1(ζ) = w1(λ). We are thus reducedto ξ and ξ′ being both of rank 1, in which case we use Proposition 9.2.4 below.

Let L(X) be the set of isomorphism classes of real lines bundles over a spaceX . The tensor product (see (7.5.5)) of two line bundles is again a line bundle. Thisprovides an operation ⊗ on L(X).

Proposition 9.2.4. Let X be a CW-complex. Then the first Stiefel-Whitneyclass provides an isomorphism

w1 : (L(X),⊗) ≈−→ (H1(X),+) .

In particular, (L(X),⊗) is an elementary abelian 2-group.

Proof. Let ξ = (p : E → B) be a line bundle over X . Endow ξ with aEuclidean structure. The unit sphere bundle S(E) → X is then a 2-fold coveringwhich is clearly isomorphic to O(ξ). By (4.3.5), w(O(ξ)) determines O(ξ) and then

S(E). But S(E) determines ξ by the isomorphism S(E) ×O(1) R≈−→ E (an O(1)-

structure on ξ). Hence, w1 is injective. For the surjectivity, let a ∈ H1(X). By

(4.3.5), there is a 2-fold covering X → X with characteristic class a. We see it

as an O(1)-principal bundle. Then, X ×O(1) R → X is a Euclidean line bundle ξ

whose sphere bundle is X. By the above, w1(ξ) = a.

9.2. THE CLASS w1 – ORIENTABILITY 305

It remains to show that w1(ξ⊗ξ′) = w1(ξ)+w1(ξ′) for two line bundles ξ and ξ′

over X . We start with some preliminaries. The linear group GL(R) is isomorphicthe multiplicative group R× = R − 0. Set K = R× × R×. Using the R-vector

space isomorphism R⊗ R≈−→ R such that x⊗ y 7→ xy, the diagram

GL(R)×GL(R)≈

ϕ⊗ // GL(R⊗ R)

≈

Kϕ // R×

is commutative, where ϕ⊗ is the natural homomorphism (see p. 253) and ϕ is justthe standard product, which is a continuous homomorphism.

Let F = Fra(ξ), F ′ = Fra(ξ′) and F⊗ = Fra(ξ ⊗ ξ′) seen, using the above, asprincipal R×-bundles. Let ∆: X → X×X be the diagonal inclusion. The definitionof F⊗ in (7.5.5) becomes

F⊗ = ϕ⊗∗ ∆∗(F × F ′) ≈ ∆∗(ϕ⊗∗ (F × F ′)) .

Let β, β′ : X → BR× be characteristic maps for F and F ′. One has morphismsof R×-principal bundles, i.e. commutative diagrams

(9.2.1)

E(F⊗)

// E(F × F ′)×K R×

// (ER× × ER×)×K R×

X

∆ //(β,β′) 33

X ×X β×β′ // BR× ×BR×

where the maps on the top line are R×-equivariant.Recall from (7.4.3) that the two projections of K onto R× induce a homotopy

equivalence P : BK≃−→ BR××BR×. Let P ′ be a homotopy inverse of P . One has

more morphisms of R×-principal bundles:

(9.2.2)

(ER× × ER×)×K R× //

EK ×K BR×κ //

ER×

BR× ×BR×

P ′

≃// BK

Bϕ // BR×

where κ is defined by κ([(ti(gi, g′i), λ]) = (ti, gig

′iλ). Diagrams (9.2.1) and (9.2.2)

imply that BϕP ′ (β, β′) is a characteristic map for the R×-principal bundle F⊗.As the inclusion ±1 → R× is a homomorphism and a homotopy equivalence,

one has BR× ≃ B±1 ≃ RP∞ ≃ K(Z2, 1). The map BϕP ′ thus defines con-tinuous multiplication on K(Z2, 1). One checks that it is homotopy commutativeand admits a homotopy unit. Let u be the generator of H1(BR×) = Z2. ByProposition 4.7.54 and Lemma 4.7.56, one has

(9.2.3) H∗(BϕP ′)(u) = u× 1+ 1× u .

Hence,

w1(ξ ⊗ ξ′) = H∗(β, β′)(u × 1+ 1× u) = H∗β(u) +H∗β′(u) = w1(ξ) + w1(ξ′) .


The above shows that the correspondence [X → X ] 7→ [X×O(1)R→ X ] inducesa bijection between the set Cov2(X) of equivalence classes of 2-fold coverings of Xand L(X), with a commutative diagram

Cov2(X)≈ //

≈

w %%

L(X)

≈

w1zz

H1(X)

.

By Proposition 4.7.36, we get the following corollary.

Corollary 9.2.5. Let λ be a line bundle over a CW-complex X. Then w1(λ) ∈H1(X) coincides with the Euler class e(λ) of λ.

Example 9.2.6. The tautological line bundle over RPn. The 2-fold coveringζ = (Sn → RPn) (1 ≤ n ≤ ∞) is an O(1)-principal bundle. Its associated linebundle λ = (Sn×O(1)R→ RPn) satisfies w1(λ) = w(ζ) 6= 0 in H1(RPn) = Z2 (see

the proof of Proposition 4.3.10). Seeing RPn as the space of vector lines in Rn+1,λ identifies itself with the tautological line bundle over RPn, i.e.

E(λ) = (a, v) ∈ RPn × Rn+1 | v ∈ a , p(a, v) = a .

The identification Sn ×O(1) R≈−→ E(λ) is given by [z, t] 7→ (R z, tz).

Proposition 9.2.7. Let ξ be a vector bundle over a CW-complex X. Then,the following conditions are equivalent.

(1) The restriction of ξ over the 1-skeleton of X is trivial.(2) w1(ξ) = 0.

Proof. Let i : X1 → X denote the inclusion and let ξ1 ≈ i∗ξ be the restric-tion of ξ over X1. By Lemma 4.3.6, w1(ξ1) = i∗(w1(ξ)) and w1(ξ1) = 0 if ξ1 istrivial. As i∗ : H1(X)→ H1(X1) is injective, this proves the implication (1)⇒ (2).Conversely, if w1(ξ) = 0, then w1(ξ1) = 0 and ξ1 is trivial by Proposition 9.2.3.

We finish this section by describing a singular cocycle representing w1(ξ) interms of transporting orientations. Let c : I → X be a path and let α ∈ O(Ec(0))(where Ex is the fiber of ξ over x ∈ X). We see α as an element of the fiber ofFra(ξ) over c(0). The unique lifting c : I → Fra(ξ) of c with c(0) = α provides anorientation c∗α = c(1) ∈ O(Ec(1)). We say that c∗α is obtained from α by transportalong c.

Choose a set-theoretic section α : X → Fra(ξ), i.e. an assignation of an orien-tation of Ex for each x ∈ X (which need not vary continuously). We see a singularsimplex σ ∈ S1(X) as a path σ : I → X via the identification I ≈ ∆1 sending t to(t, 1− t). A cochain w1(ξ, α) is defined by

〈w1(ξ, α), σ〉 =1 if σ∗α(σ(0)) 6= α(σ(1))

0 otherwise.

The cochains w1(ξ, α) and w(Fra(ξ), α) of § 4.3.2 clearly coincide. Hence, by Propo-sition 4.3.7, w1(ξ, α) is a cocycle representing w1(ξ).

Let x0 ∈ X and α0 ∈ O(Ex0). We say that a loop c : I → X at x0 preservesthe orientation if c∗α0 = α0 (this condition does not depend on the choice of α0).Note that, if d is another loop at x0, then (dc)∗α0 = d∗(c∗α0). Also, c∗α0 depends

9.3. THE CLASS w2 – SPIN STRUCTURES 307

only on the homotopy class of the loop c, by the homotopy lifting property of thecovering Fra(ξ)→ X . Hence the correspondence

c 7→0 if c preserves the orientation

1 otherwise.

provides a homomorphism w : π1(X, x0)→ Z2 which corresponds to w1(ξ) ∈ H1(X)

via the isomorphism H1(X)≈−→ hom(π1(X, x0),Z2) of Lemma 4.3.1.

9.3. The class w2 – Spin structures

In this section, we define a cellular second Stiefel-Whitney class w2(ξ) ∈ H2(X),when ξ is an orientable vector bundle over a regular CW-complex X (recall that

H∗ denotes the cellular cohomology introduced in § 3.5). A more general w2(ξ) ∈H2(X) (X any space) is considered in § 9.4, but the results below are used toestablish the relationship between w2(ξ) and the existence of spin structures on ξ.

Let ξ be an orientable vector bundle of rank r ≥ 2 over a CW-complex X .Denote by ξi the restriction of ξ over the i-skeleton X i of X . By Propositions 9.2.1and 9.2.7, ξ1 is trivial. Fix an orientation of ξ. Choose a trivialization T1 of ξ1which is compatible with this orientation. The restriction T0 of T1 over X0 is thusuniquely determined up to isotopy.

Let e be a 2-cell of X with characteristic map ϕ : (D2, S1) → (X,X1). AsD2 is contractible, there is a unique (up to isotopy) trivialization Te of ϕ∗ξ overD2 which is compatible with the orientation (see Proposition 9.1.6). Thus, ϕ∗T1and Te provides two trivialization of ϕ∗ξ restricted to the boundary S1 of D2.By Lemma 9.1.2, these two trivializations differ by the action of a map from S1

to GL(r,R), whose range is contained in GL+(r,R) since ϕ∗T1 and Te are bothcompatible with the chosen orientation. Let

(9.3.1) w2(e) ∈ [S1, GL+(r,R)] ≈ π1(GL+(r,R)) ≈ π1(SO(r))be the homotopy class of this map. Here, the isomorphism [S1, GL+(r,R)] ≈π1(GL

+(r,R)) holds true since GL+(r,R) is a topological group and GL+(r,R) hasthe homotopy type of SO(r) by the Gram-Schmidt orthonormalization process. Ifr ≥ 3, then π1(SO(r)) = Z2. If r = 2 then π1(SO(r)) ≈ Z and, by convention, wetake w2(e) mod 2. The correspondence e 7→ w2(e) thus defines a cellular 2-cochain

w2 = w2(ξ, T1) ∈ C2(X).

Lemma 9.3.1. Suppose that X has no 3-cells or that X3 is a regular complex.Then, the cochain w2(ξ, T1) is a cellular cocycle. Its cohomology class w2(ξ) ∈H2(X) depends only of the isomorphism class of ξ.

The class w2(ξ) ∈ H2(X) is called the second Stiefel-Whitney class of the vectorbundle ξ. If the rank of ξ is ≤ 1, we set by convention that w2(ξ) = 0.

Proof. We may assume that X is connected. Observe first that the cochainw2(ξ, T1) does not depend on the orientation of ξ, since the other choice would justchange all the orientations under consideration. Let us see how w2(ξ, T1) depends

on the isotopy class of the trivialization [T1] ∈ T (ξ1). Let T ′1 ∈ T (ξ1) be anothertrivialization compatible with the orientation. As seen above, this compatibilityimplies that the restriction T ′0 of T ′1 to ξ0 coincides with T0 up to isotopy. Such anisotopy may be realized in a neighbourhood of X0 in X1, so one may assume that


T0 = T ′0. By Lemma 9.1.2, there is a unique map a : X1 → GL+(r,R) such thatT ′1 = a ·T1. As T0 = T ′0, the restriction of a to X0 is constant to the identity of Rr.Hence, each 1-cell ε with characteristic map ϕε : D

1 → X1 gives rise to a homotopyclass

a(ε) = [aϕε] ∈ [(D1, S0), (GL+(r,R), id)] ≈ π1(GL+(r,R), id) ≈ π1(SO(r), id) = Z2

which does not depend on the choice of ϕε (since π1(SO(r), id) = Z2; again, ifr = 2, one takes by convention the value mod 2 of a ∈ π1(SO(2)) ≈ Z). The corre-

spondence ε 7→ a(ε) determines a cellular 1-cochain a ∈ C1(X). Let e be a 2-cellof X with characteristic map ϕ : (D2, S1)→ (X,X1). Using the cellular boundaryformula (3.5.4) and writing the abelian group π1(GL

+(r,R)) = Z2 additively, weget that [aϕ] = δ(a). In the same way, using (9.1.3), one gets the formula

(9.3.2) w2(ξ, T′1) = w2(ξ, a T1) = w2(ξ, T1) + δ(a) .

We now prove that w2 is a cellular cocycle. If X has no 3-cells, there is nothingto prove. Let ǫ be a 3-cell of X which, as X3 is now supposed to be regular, canbe identified with a subcomplex (also called ǫ) of X . As ǫ is contractible, there isa unique (up to isotopy) trivialization T ǫ of ξ|ǫ, compatible with the orientation.

As above, one may assume that T ǫ coincides with T0 over ǫ0. A trivializationT ǫ1 ∈ T (ξ1) may be thus defined by

T ǫ1 (z) =

T ǫ(z) if p(z) ∈ ǫT1(z) otherwise.

By (9.3.2) and the construction of T ε1 , one has

(9.3.3) δ(w2(ξ, T1))(ǫ) = δ(w2(ξ, Tǫ1 ))(ǫ) = 0 .

Equation (9.3.3) shows that δ(w2(ξ, T1)))(ǫ) = 0 for all 3-cell ǫ, proving that

δ(w2(ξ, T1)) = 0. Also, formula (9.3.2) show that [w2(ξ, T1)] ∈ H2(X) does notdepend on the choice of T1 and thus depends only on ξ. Finally, if ξ′ is isomorphic

to ξ via a homeomorphism h : E(ξ′)≈−→ E(ξ) over the identity of X , the trivializa-

tion T ′1 = T1h satisfies w2(ξ′, T ′1) = w2(ξ, T1). Therefore, w2(ξ

′) = w2(ξ).

Lemma 9.3.2. Let ξ be an orientable vector bundle of rank r ≥ 2 over a CW-complex X satisfying the hypotheses of Lemma 9.3.1. If η is a trivial vector bundleover X, then w2(ξ ⊕ η) = w2(ξ).

Proof. Note that ξ ⊕ η is orientable since ξ is so, thus w2(ξ ⊕ η) is defined.

Let us represent w2(ξ) by a cocycle w2(ξ, T1) as above. If Tη : X × Rs≈−→ E(η)

is a fixed trivialization of η, one checks that w(ξ, T1 ⊕ Tη) = π1j w(ξ, T1), wherej : SO(r) → SO(r + s) is the inclusion. But π1j is an isomorphism if r ≥ 3 or thereduction mod 2 if r = 2, which proves the lemma in these cases.

If r = 1, then w2(ξ) = 0 by convention. Also, as ξ is orientable, it is trivial byProposition 9.2.4. Thus, ξ ⊕ η is trivial and w2(ξ ⊕ η) = 0 by the proof that (2)implies (3) in the next proposition (this part of the proof is valid for any r).

Example 9.3.3. Let X = S2, with its cellular decomposition with one 0-celland one 2-cell: X = D2 ∪S1 pt. Let α : S1 → GL+(r,R), representing [α] ∈π1(GL

+(r,R)). Then

E(ξ[α]) = D2 × Rr ∪α pt× Rr

9.3. THE CLASS w2 – SPIN STRUCTURES 309

is the total space of a vector bundle ξ[α] of rank r over X . This process gives a

bijection between π1(GL+(r,R)) and the isomorphism classes of vector bundles of

rank r over S2 (compare [181, § 18]). The two cochain associating to the 2-cell ofX the element [α] ∈ π1(GL+(r,R)) = Z2 if r ≥ 3 (or its reduction mod 2 if r = 2)

represents w2(ξ[α]) ∈ H2(X) = Z2. Summing up, there are two vector bundles (up

to isomorphism) of rank ≥ 3 over S2: the trivial bundle η, satisfying w2(η) = 0,and the non-trivial bundle ξ, characterized by the property that w2(ξ) 6= 0. Thisis an example of Proposition 9.3.4 below.

Proposition 9.3.4. Let ξ be a vector bundle of rank r ≥ 3 over a CW-complexX. Suppose that X has no 3-cell or that X3 is a regular complex. Let ξi be therestriction of ξ over the X i. Then, the following conditions are equivalent.

(1) ξ3 is trivial.(2) ξ2 is trivial.(3) w1(ξ) = 0 and w2(ξ) = 0.

In the next section, Proposition 9.3.4 will be generalized, using the singularsecond Stiefel-Whitney class (see Proposition 9.4.6).

Proof. By Proposition 9.2.7, w1(ξ) = 0 if ξ2 is trivial. Also, ξ is orientableby Proposition 9.2.1, so w2(ξ) is defined. We can represent w2(ξ) by a cocycle

w2(ξ, T1) where T1 is the restriction of Γ ∈ T (ξ2). For each 2-cell e, we thus haveT e = Γ|e, thus w2(ξ, T1) = 0, proving that w2(ξ) = 0. Thus, (2) implies (3).

Conversely, suppose that w1(ξ) = 0 and w2(ξ) = 0. Choose a trivialization T1of ξ1 and let ϕ : Λ2 × (D2, S1)→ (X,X1) be a global characteristic map for the 2-

cells of X . Then, w2(ξ, T1) = δ(a) for a ∈ C1(X). As in the proof of Lemma 9.3.1,the cochain a may be used to modify (relative to ξ0) T1 into a trivialization T ′1such that w2(ξ, T

′1) = 0 (this uses that r ≥ 3). This means that, over Λ2 × S1,

the trivialization ϕ∗T ′1 coincides up to isotopy with the unique (up to isotopy)trivialization over Λ2 ×D2 compatible with the orientation. By Lemma 9.1.7, thisimplies that ξ2 is trivial.

We have so far proven that (2)⇔ (3). We now prove the equivalence (1)⇔ (2),which is true for any CW-complex. The implication (1)⇒ (2) is trivial. Conversely,let Γ ∈ T (ξ2). Let ϕ : Λ3 × (D3, S2)→ (X,X2) be a global characteristic map forthe 3-cells of X . As above, one compares the trivialization ϕ∗Γ over Λ3 × S2

with the unique (up to isotopy) trivialization over Λ3 × D3 which is compatiblewith the orientation. Their isotopy classes differ by the action of an element of[Λ3 × S2, GL+(r,R)]. But π2(SO(r)) = 0 (see, e.g. [181, 22.10]), thus

[S2, GL+(r,R)] ≈ [S2, SO(r)] ≈ π2(SO(r)) = 0 .

As above, using Lemma 9.1.7, this implies that ξ3 is trivial.

We now see the second Stiefel-Whitney class w2(ξ) as the obstruction to theexistence of the existence of a spin structure on ξ. This refers to the standardorthogonal representation ρ0 : Spin(r) → SO(r) which is a 2-fold covering. Asin 9.1.12, it is natural to fix a Euclidean structure and an orientation on ξ, in which

case, a spin structure is seen a ρ0-map f : P → Fra+⊥ (ξ), where P is a Spin(r)-

principal bundle. Note that f is a 2-fold covering whose restriction to each fiberis modeled by ρ0, and any such covering would be a spin structure. Alternativeequivalent definitions of spin structures are to be found in [150] or [130, § 2.1].


Strong and weak equivalences for spin structures may be expressed as in (a)and (b) in 9.1.9 but then, by the homotopy lifting property for coverings, thesetwo notions of equivalence coincide. Since π1(SO(r)) is cyclic, ρ0 is, up to equiv-alence, the only 2-fold covering of SO(r) which is non-trivial (see § 4.3.1). Theseconsiderations prove the following lemma.

Lemma 9.3.5. Let ξ be an oriented Euclidean bundle ξ of rank r over a CW-complex X. Then, there is a bijection between

• the strong (or weak) equivalence classes of spin structures on ξ.• the isomorphism classes of 2-fold coverings of Fra+

⊥ (ξ) whose restrictionto the fibers of pFra is non-trivial.

In general, the above sets are empty. For the existence of a spin structure, onehas the following proposition (see Proposition 9.4.6 for a more general framework).

Proposition 9.3.6. Let ξ be a vector bundle of rank r ≥ 2 over a regularCW-complex X. Then, the following conditions are equivalent.

(a) ξ admits a spin structure.(b) w1(ξ) = 0 and w2(ξ) = 0.

Moreover, if (a) or (b) holds true, then the set of strong (or weak) equivalenceclasses of spin structures on ξ is in bijection with H1(X).

Proof. Let (P, f) be a spin structure on ξ and let P2 → X2 be the restrictionof P onto the 2-skeleton of X . Suppose first that r ≥ 3. As π1(Spin(r)) = 0, theprincipal bundle P2 → X2 admits a section by [181, Corollary 34.4] and is thereforetrivial. This implies that ξ2 is trivial and (b) holds by Proposition 9.3.4. Whenr = 2, we apply the above argument to ξ ⊕ η where η is a trivial bundle and useRemark 9.2.2 and Lemma 9.3.2.

Conversely, suppose that w1(ξ) and w2(ξ) vanish. Fix an orientation and aEuclidean structure on ξ (thus inducing an orientation and a Euclidean structureon ξk). Suppose first that r ≥ 3. By Proposition 9.3.4, ξ2 is trivial and thus admits

a spin structure f2 : P2 → Fra+⊥ (ξ2). Consider the commutative diagram

(9.3.4)

π2(X2) //

π1(SO(r))i∗ //

≈

π1(Fra+⊥ (ξ2))

//

π1(X2) //

≈

π0(SO(r))

≈

π2(X) // π1(SO(r))

i∗ // π1(Fra+⊥ (ξ))

// π1(X) // π0(SO(r))

where the rows are the homotopy exact sequences of the bundles Fra+⊥ (ζ) for ζ = ξ2

or ξ. By Lemma 9.3.5, the set of strong equivalence classes of spin structures on ζis in bijection with the set

E(ζ) = κ ∈ hom(π1(Fra+⊥ (ζ)),Z2) | κi∗ 6= 0 .

Diagram (9.3.4) implies that π1(Fra+⊥ (ξ2))→ π1(Fra

+⊥ (ξ)) is an isomorphism (five-

lemma) and that E(ξ) = E(ξ2). Hence, the spin structure on ξ2 extends to ξ.In the case r = 2, we apply the above argument to ξ ⊕ η where η is a trivial

bundle of rank 1. Note that w1(ξ ⊕ η) and w2(ξ ⊕ η) vanish by Remark 9.2.2 andLemma 9.3.2. Hence, ξ ⊕ η admits a spin structure. Taking the sum with a fixed

9.4. DEFINITION AND PROPERTIES OF STIEFEL-WHITNEY CLASSES 311

frame of η gives a (SO(2) → SO(3))-map Fra+⊥ (ξ) → Fra+

⊥ (ξ ⊕ η) and thus acommutative diagram(9.3.5)

π2(X) //

≈

π1(SO(2))i∗ //

π1(Fra+⊥ (ξ))

//

π1(X) //

≈

π0(SO(2))

≈

π2(X) // π1(SO(3))

i∗ // π1(Fra+⊥ (ξ ⊕ η)) // π1(X) // π0(SO(3))

As E(ξ ⊕ η) 6= ∅, Diagram (9.3.5) shows that E(ξ) 6= ∅ and hence ξ admits aSpin(2)-structure.

Finally, note that, in (9.3.4), the last horizontal map vanishes (π0(SO(r)) →π0(Fra

+⊥ (ξ)) is injective). Also, hom(π1(Fra

+⊥ (ζ)),Z2) is an Abelian group. Hence,

if E(ξ) 6= ∅, it is in bijection with

κ ∈ hom(π1(Fra+⊥ (ζ)),Z2) | κi∗ = 0 ≈ hom(π1(X),Z2) ≈ H1(X) ,

the last isomorphism being established in Lemma 4.3.1.

9.4. Definition and properties of Stiefel-Whitney classes

Let ξ = (p : E → X) be a vector bundle of rank r over a paracompact spaceX . In Proposition 4.7.37, we established the Thom isomorphism

Φ∗ : Hk(X)≈−→ Hk+r(E,E0) ,

where E0 ⊂ E is the complement of the zero section. Using the Steenrod squareSq: H∗(E,E0) → H∗(E,E0), the (total) Stiefel-Whitney class w(ξ) ∈ H∗(X) of ξis defined by

(9.4.1) w(ξ) = Φ−1SqΦ(1) ,

where 1 ∈ H0(X) is the unit class. The component of w(ξ) in Hi(X) is denotedby wi(ξ) and is called the i-th Stiefel-Whitney class of ξ.

Equation (9.4.1) is one of the multiple definitions of the Stiefel-Whitney classesand it is due to Thom [190, § II and III]. For a history of Stiefel-Whitney classes,see [153, p. 38] and [40, Chapter IV, § 1]. The main properties of these classes aregiven in the following proposition.

Theorem 9.4.1. Let ξ = (p : E → X) be a vector bundle of rank r over aparacompact space X.

(1) w(ξ) = 1+ w1(ξ) · · ·+ wr(ξ). In particular, wi(ξ) = 0 if i > r.(2) If f : Y → X is a continuous map, then w(f∗ξ) = H∗f(w(ξ)). In partic-

ular, if ξ is isomorphic to ξ′, then w(ξ) = w(ξ′).(3) If ξ is trivial, then w(ξ) = 1.(4) Let ξ′ be a vector bundle over a paracompact space X ′. Then,

(9.4.2) w(ξ × ξ′) = w(ξ) × w(ξ′) ∈ H∗(X ×X ′) .If X ′ = X, then

(9.4.3) w(ξ ⊕ ξ′) = w(ξ) w(ξ′) ∈ H∗(X) .

(5) If η is a trivial vector bundle over X, then w(ξ) = w(ξ ⊕ η).(6) wr(ξ) is the Euler class of ξ.


Property (2) says that the total Stiefel-Whitney class and its homogeneouscomponents are characteristic classes. Historically, they were the first characteristicclasses defined, in the simultaneous work of Eduard Stiefel and Hassler Whitneystarting starting around 1935 (see [40, pp. 421–426]).

Proof. Recall that Φ(1) is the Thom class U(ξ) ∈ Hr(E,E0); thus, (9.4.1) isequivalent to

(9.4.4) w(ξ) = Φ−1Sq(U(ξ)) .

Now, Theorem 9.4.1 comes from the properties of Sq established in Theorem 8.2.1.Since Sq0 = id, one has w0(ξ) = φ−1Sq0(U(ξ)) = 1. As Sqi(U(ξ)) = 0 for i > r,this proves (1). The naturality (2) comes from the naturality of all the ingredientsof (9.4.4): the Thom class is natural (Lemma 4.7.30), and so is Φ, and Sq is alsonatural, being a cohomology operation. Now, (3) is a consequence of (2) since atrivial bundle is induced by a map to a point.

To prove (4), one has

(9.4.5)w(ξ × ξ′) = φ−1Sq(U(ξ × ξ′))

= φ−1Sq(U(ξ)× U(ξ′)

)using (4.7.24)

= φ−1(Sq(U(ξ))× Sq(U(ξ′))

)by (3) of Theorem 8.2.1.

On the other hand, if a ∈ H∗(X) and a′ ∈ H∗(X ′), one has

(9.4.6)

φ(a× a′) = H∗(p× p′)(a× a′) U(ξ × ξ′)= [H∗p(a)×H∗p′(a′)] [U(ξ)× U(ξ′)]= [H∗p(a) U(ξ)]× [H∗p(a′) U(ξ′)]= Φ(a)× Φ(a′) .

Thus, (9.4.5) together with (9.4.6) proves (9.4.2). IfX ′ = X , then ξ⊕ξ′ = ∆∗(ξ×ξ′)where ∆: X → X ×X is the diagonal inclusion. Therefore, (9.4.3) comes from (2)already proven, (9.4.2) and Remark 4.6.1:

w(ξ ⊕ ξ′) = H∗∆(w(ξ × ξ′)) = H∗∆(w(ξ) × (ξ′)) = w(ξ) w(ξ′) .

Property (5) is a consequence of (3) and (4). Finally, (6) follows from

wr(ξ) = Φ−1Sqr(U(ξ)) = Φ−1(U(ξ) U(ξ)) = e(ξ) ,

the last equality coming from (4.7.22).

Remark 9.4.2. Versions of Properties (1), (2), (9.4.3) and (6) uniquely char-acterize the total Stiefel-Whitney class. See Proposition 9.6.4, [153, Theorem 7.3]or [105, Chapter 16,§ 5]. This is the philosophy of the axiomatic presentation ofStiefel-Whitney classes (see [153]), inspired by that of the Chern classes introducedby Hirzebruch [96, p. 58].

Remark 9.4.3. As the Steenrod squares are used for Definition (9.4.1), theAdem relations provide constraints amongst Stiefel-Whitney classes. For instance,the relation Sq2i+1 = Sq1Sq2i (see Example 8.4.4) implies that w2i+1(ξ) = 0 ifw2i(ξ) = 0. Also, if w2k(ξ) = 0 for k = 1 . . . , r, then, by Lemma 8.5.1, wj(ξ) = 0for 0 < j < 2r+1.

We now discuss the relationship with the classes w1 and w2 defined in § 9.2and 9.3.

9.4. DEFINITION AND PROPERTIES OF STIEFEL-WHITNEY CLASSES 313

Proposition 9.4.4. Let ξ be a vector bundle over a CW-complex X. Then, thefirst Stiefel-Whitney class w1(ξ) ∈ H1(X) defined above coincides with that definedin § 9.2. In particular, w1(ξ) = 0 if and only if ξ is orientable.

Proof. Both definitions enjoy naturality for induced bundles. We can thenrestrict ourselves to X being 1-dimensional, since H1(X) → H1(X1) is injective.In this case, ξ ≈ λ ⊕ η where λ is a line bundle and η a trivial vector bundle (seee.g. [105, Chapter 8, Theorem 1.2]). By Remark 9.2.2 and (5) of Theorem 9.4.1,we are reduced to the case of a line bundle. Then, both definitions coincide withthe Euler class by Corollary 9.2.5 and Point (6) of Theorem 9.4.1.

A similar result holds for the second Stiefel-Whitney class.

Proposition 9.4.5. Let ξ be an orientable vector bundle over a CW-complexX. Suppose that X has no 3-cells or that X3 is a regular complex. Then, the secondStiefel-Whitney class w2(ξ) ∈ H2(X) defined above coincides with the cellular one

w2(ξ) ∈ H2(X) defined in § 9.3.Proof. Recall that the condition on X (and the orientability of ξ) was nec-

essary for us to define w2(ξ). The coincidence between w2(ξ) ∈ H2(X) and

w2(ξ) ∈ H2(X) holds under the identification of H2(X) and H2(X) as the samesubgroup of H2(X2) (see (3.5.5)). The class w2 is natural by Point (2) of Theo-rem 9.4.1 and, by construction, w2 is natural for the restriction to a subcomplex.We can thus suppose that X = X2 and that X is connected.

As ξ is orientable, its restriction over X1 is trivial. By Lemma 9.1.15, ξ ≈ p∗ξ,where p : X → X = X/X1. Again, w2(ξ) = H∗p(w2(ξ)) and, by construction of

w2, w2(ξ) = H∗p(w2(ξ)). We can thus suppose that X is a bouquet of 2-spheres,or even that X = S2 with its minimal cell decomposition.

If η is a trivial bundle, both equations w2(ξ⊕η) = w2(ξ) and w2(ξ⊕η) = w2(ξ)hold true, by Point (5) of Theorem 9.4.1 and Lemma 9.3.2. We can thus supposethat ξ has rank ≥ 3. As seen in Remark 9.3.3, there is only one non-trivial suchbundle over S2, characterized by w2(ξ) 6= 0. Let γC be the tautological bundle overCP 1 ≈ S2. By Proposition 6.1.10, one has

0 6= e(γC) = w2(γC) = w2(γC ⊕ η) .which finishes the proof of our proposition. Incidentaly, we have proven that γC isstably non-trivial.

Proposition 9.4.5 permits us to generalize the framework of Propositions 9.3.4and 9.3.6.

Proposition 9.4.6. Let ξ be a vector bundle of rank r ≥ 3 over a CW-complexX. Then, the following conditions are equivalent.

(1) w1(ξ) = 0 and w2(ξ) = 0.(2) the restriction ξ3 of ξ over X3 is trivial.

Proof. By Theorem 9.4.1, (1) implies w1(ξ2) = 0 and w2(ξ2) = 0. As X2 hasno 3-cells, w2(ξ2) is defined and, by Proposition 9.4.5, w2(ξ2) = 0. By Proposi-tion 9.3.4, ξ2 is trivial which, as seen in the proof of Proposition 9.3.4, implies thatξ3 is trivial. Thus, (1) implies (2). To prove that (2) implies (1), let j : X3 → Xdenote the inclusion. Then j∗(wi(ξ)) = wi(ξ3)) = 0 for and j∗ : Hk(X)→ Hk(X3)is injective for k ≤ 3. (We have also proven that (1) implies w3(ξ) = 0, but this isalready known by Remark 9.4.3).


Proposition 9.4.7. Let ξ be a vector bundle of rank r ≥ 2 over a CW-complexX. Then, the following conditions are equivalent.

(1) w1(ξ) = 0 and w2(ξ) = 0.(2) ξ admits a spin structure.

Moreover, if (2) holds true, then the set of strong (or weak) equivalence classes ofspin structures on ξ is in bijection with H1(X).

Proof. Suppose first that r ≥ 3. If ξ admits a spin structure, then ξ2 istrivial (see the proof of Proposition 9.3.6), which implies (1) by Proposition 9.4.6.Conversely, if (1) holds true, then ξ2 is trivial by Proposition 9.4.6 and thus ξ2admits a spin-structure. That this structure extends to ξ is established as in theproof of Proposition 9.3.6. For the case r = 2 as well as for the last assertion of theproposition, the proofs are the same as those for for Proposition 9.3.6.

9.5. Real flag manifolds

Most of the results of this section come from [15], but we do not use spectral se-quences. The Leray-Hirsch theorem 4.7.17 for locally trivial bundles, together withsome perfect Morse theory, is sufficient for our needs. We shall deal with homoge-neous spaces of the form Γ/Γ0, where Γ is a Lie group and Γ0 a compact subgroup(therefore, a Lie subgroup). Then Γ/Γ0 inherits a smooth manifold structure [37,Chapter 1, Proposition 5.3]. More generally, [20, Chapter II, Theorem 5.8] impliesthe following lemma.

Lemma 9.5.1. Let Γ be a Lie group and H ⊂ G be compact subgroups of Γ.Then, the quotient map Γ/H → Γ/G is a smooth locally trivial fiber bundle withfiber G/H. If H = 1, then the quotient map Γ → Γ/G is a smooth G-principalbundle.

9.5.1. Definitions and Morse theory. Let n1, . . . nr be positive integersand let n = n1 + n2 + · · ·nr. By the flag manifold Fl(n1, . . . , nr), we mean anysmooth manifold diffeomorphic to the homogeneous space

(9.5.1) Fl(n1, . . . , nr) ≈ O(n)/O(n1)×O(n2)× · · · ×O(nr) .

Here are some examples.

(1) Nested subspaces. Fl(n1, . . . , nr) is the set of nested vector subspaces V1 ⊂· · · ⊂ Vr ⊂ Rn with dimVi =

∑ij=1 nj .

(2) Mutually orthogonal subspaces. Fl(n1, . . . , nr) is the set of r-tuples (W1, . . . ,Wr)of vector subspacesRn which are mutually orthogonal and satisfy dimWi =ni. The correspondence from this definition to Definition (1) associates to(W1, . . . ,Wr) the nested family Vi where Vi is the vector space generatedby W1 ∪ · · · ∪Wi.

(3) Isospectral symmetric matrices. Let λ1 > · · · > λr be real numbers. Con-sider the manifold SM(n) of all symmetric real (n×n)-matrices, on whichO(n) acts by conjugation. Then Fl(n1, . . . , nr) occurs as the orbit of thediagonal matrix having entries λi with multiplicity ni.

(9.5.2) Fl(n1, . . . , nr) =R dia

(λ1, . . . , λ1︸︷︷︸

n1

, · · · , λr, . . . , λr︸︷︷︸nr

)R−1 | R ∈ O(n)

.

In other words, Fl(n1, . . . , nr) is here the space of symmetric real (n×n)-matrices with characteristic polynomial equal to

∏ri=1(x− λi)ni . Indeed,

9.5. REAL FLAG MANIFOLDS 315

elementary linear algebra teaches us that two matrices in SM(n) are in thesame O(n)-orbit if and only if they have the same characteristic polyno-mial. The correspondence from this definition to Definition (2) associates,to a matrix M , its eigenspaces for the various eigenvalues.

Concrete definition (3) will be our working definition for Fl(n1, . . . , nr) throughoutthis section. Special classes of flag manifolds are given by the Grassmannians

Gr(k;Rn) = Fl(k, n− k) ≈ O(n)/O(k) ×O(n− k)

of k-planes in Rn. This is a closed manifold of dimension

dimGr(k;Rn) = dimO(n)− dimO(k) − dimO(n− k) = k(n− k) .For example, Gr(1;Rn) ≈ RPn−1, of dimension n− 1. Using Definition (3) above,our “concrete Grassmannian” will be

(9.5.3) Gr(k;Rn) =R dia

(1, . . . , 1︸︷︷︸

k

, 0, . . . , 0︸︷︷︸n−k

)R−1 | R ∈ O(n)

.

In other words, Gr(k;Rn) is the space of orthogonal projectors on Rn of rank k.Another interesting flag manifold is the complete flag manifold

Fl(1, . . . , 1) ≈ O(n)/O(1) × · · · ×O(1)

with dimFl(1, . . . , 1) = dimO(n) = n(n−1)2 .

We now define real functions on the flag manifolds by restriction of the weightedtrace on f : SM(n)→ R defined by

f(M) =

n∑

j=1

j Mjj

where Mij denotes the (i, j)-entry of M .

Proposition 9.5.2. Let Fl(n1, . . . , nr) ⊂ SM(n) be the flag manifold as pre-sented in (9.5.2). Then, the restriction f : Fl(n1, . . . , nr) → R of the weightedtrace is a perfect Morse function whose critical points are the diagonal matrices inFl(n1, . . . , nr). The index of the critical point dia(x1, . . . , xn) is the number of pairs(i, j) with i < j and xi < xj.

For a general discussion about such Morse functions on flag manifolds, see [13,Chapter 8].

Example 9.5.3. For Gr(2;R5) = Fl(2, 3), we get the following (52) = 10 criticalpoints, with their index and value by f .

critical point index value

dia(1, 1, 0, 0, 0) 0 3

dia(1, 0, 1, 0, 0) 1 4

dia(1, 0, 0, 1, 0), dia(0, 1, 1, 0, 0) 2 5

dia(1, 0, 0, 0, 1), dia(0, 1, 0, 1, 0) 3 6

dia(0, 0, 1, 1, 0), dia(0, 1, 0, 0, 1) 4 7

dia(0, 0, 1, 0, 1) 5 8

dia(0, 0, 0, 1, 1) 6 9


Remark 9.5.4. The function f : Gr(k;Rn)→ R given by

f(M) = −k(k + 1)

2+ f(M)

is a Morse function which is self-indexed, i.e. f(M) = j if M is a critical point ofindex j.

Proof of Proposition 9.5.2. We introduce precise notations which will beused later. For 1 ≤ i < j ≤ n, let rij :M2(C)→Mn(C) defined by requiring thatthe entries of rij(N) are those of the identity matrix In, except for

rij(N)ii = N11 , rij(N)ij = N12 , r

ij(N)ji = N21 , rij(N)jj = N22 .

The restriction of rij to SO(2) gives an injective homomorphism rij : SO(2) →SO(n) whose image is formed by the matrices

Rijt = rij(

cos t − sin tsin t cos t

)(t ∈ R) .

The action of Rijt on Fl(n1, . . . , nr) ⊂ SM(n) by conjugation produces a flow and

thus a vector field V ij on Fl(n1, . . . , nr), whose value V ijM at M ∈ Fl(n1, . . . , nr) is

V ijM = ddt(R

ijt MRij−t)|t=0 (we identify TMFl(n1, . . . , nr) as a subspace of SM(n)).

A direct computation gives that

(9.5.4)

(Rijt MRij−t)ii = Mii cos2 t−Mij sin 2t+Mjj sin

2 t

(Rijt MRij−t)jj = Mii sin2 t+Mij sin 2t+Mjj cos

2 t

(Rijt MRij−t)ij = Mij cos 2t+ (Mii −Mjj) sin t cos t .

Moreover,

(9.5.5)

(Rijt MRij−t)ik = Mik cos t−Mjk sin t if i 6= k 6= j

(Rijt MRij−t)kj = Mki sin t+Mkj cos t if i 6= k 6= j

(Rijt MRij−t)kl = Mkl if i 6= k and j 6= l .

Let gij(t) = f(Rijt MRij−t). The first derivative gij(t) satisfies

gij(t) = (j − i)(Mii −Mjj) sin 2t+ 2(j − i)Mij cos 2t .

Hence,

(9.5.6) V ijM f = gij(0) = 2(j − i)Mij ,

which proves that only the diagonal matrices in Fl(n1, . . . , nr) may be critical pointsof the weighted trace.

Suppose that ∆ ∈ Fl(n1, . . . , nr) is a diagonal matrix. Let

J∆ = (i, j) | 1 ≤ i < j ≤ n and ∆ii 6= ∆jj .

and let V∆ = V ij∆ | (i, j) ∈ J∆ ⊂ T∆Fl(n1, . . . , nr). By (9.5.4) and (9.5.5),ddt(R

ijt ∆R

ij−t)(0) has only non-zero terms away from the diagonal, namely d

dt (Rijt ∆R

ij−t)ij(0) =

∆ii −∆jj . Hence, vectors of V∆ are linearly independent. But

♯J∆ =n(n− 1)

2−

r∑

k=1

nk(nk − 1)

2= dimO(n)−

r∑

k=1

dimO(nk) = dimFl(n1, . . . , nr) .


Therefore, V∆ is a basis of T∆Fl(n1, . . . , nr). Using (9.5.6), this proves that thediagonal matrices in Fl(n1, . . . , nr) are exactly the critical points of the weightedtrace. The matrix of the Hessian form Hf on T∆Fl(n1, . . . , nr) is

Hf(V kl∆ , V ij∆ ) = V kl∆ (V ijf)

= V kl∆

(M 7→ 2(j − i)Mij

)by (9.5.6)

= 2(j − i)[ddt (R

klt ∆Rkl−t)|t=0

]ij.

Using (9.5.4) and (9.5.5), we see that the matrix of Hf in the basis V∆ is diagonal,with diagonal term

Hf(V ij∆ , V ij∆ ) = 2(j − i)(∆ii −∆jj) .

As (i, j) ∈ J∆, this proves that f is a Morse function as well as the assertion onthe Morse index of ∆.

It remains to prove that f is perfect. Let Γ be the subgroup of O(n) formed bythe diagonal matrices (with coefficients ±1). The O(n)-action on SM(n) by conju-gation may be restricted to Γ and f is Γ-invariant. Moreover, the diagonal matricesin Fl(n1, . . . , nr) are exactly the fixed points of the Γ-action. The perfectness of fthen follows from Theorem 7.6.6.

Here is a first consequence of Proposition 9.5.2.

Corollary 9.5.5.

dimH∗(Fl(n1, . . . , nr)) =n!

n1! · · ·nr!.

In particular,

dimH∗(Fl(k, n− k)) = dimGr(k;Rn) = (nk ) and dimH∗(Fl(1, · · · , 1)) = n! .

Proof. By Proposition 9.5.2, the weighted trace f : Fl(n1, . . . , nr) → R is aperfect Morse function. Hence, by Proposition 7.6.5, dimH∗(Fl(n1, . . . , nr)) =♯Crit f . But Crit f consists of the diagonal matrices in Fl(n1, . . . , nr), which are allconjugate to

dia(λ1, . . . , λ1︸︷︷︸

n1

, · · · , λr, . . . , λr︸︷︷︸nr

)

by a permutation matrix. Hence, Crit f is an orbit of the symmetric group Symn,with isotropy group Symn1

× · · · × Symnr , whence the formulae.

Remark 9.5.6. The critical points of f in Proposition 9.5.2 are related to theSchubert cells (see § 9.5.3).

Consider the inclusion SM(n) ⊂ SM(n + 1) with image the matrices withvanishing last row and column. Seeing Gr(k;Rn) ⊂ SM(n) as in (9.5.3), this givesan inclusion Gr(k;Rn) ⊂ Gr(k;Rn+1).

Lemma 9.5.7. The homomorphism Hj(Gr(k;Rn+1))→ Hj(Gr(k;Rn)) inducedby the inclusion is surjective for all j and is an isomorphism for j ≤ n− k.

Proof. Let us use the Morse function f : Gr(k;Rn+1) → R of Remark 9.5.4and let f ′ be its restriction to Gr(k;Rn). Then, f ′ and f are self-indexed andCritf ′ ⊂ Critf ⊂ N. For m ∈ N, let Wm = f−1((∞,m + 1/2]) and W ′m =(f ′)−1((∞,m + 1/2]). For the first assertion, we prove, by induction on m that


H∗(Wm)→ H∗(W ′m) is surjective for all m ∈ N. The induction starts with m = 0,since W0 ≃W ′0 ≃ pt. The induction step involves the cohomology sequences

(9.5.7)

0 // H∗(Wm,Wm−1) //

i∗m,m−1

H∗(Wm) //

i∗m

H∗(Wm−1) //

i∗m−1

0

0 // H∗(W ′m,W′m−1)

// H∗(W ′m) // H∗(W ′m−1) // 0

obtained by Lemma 7.6.3, since f and f ′ are perfect by Proposition 9.5.2. FromProposition 9.5.2 again and its proof, the critical points of f ′ have the negativenormal directions in Wm−1 or in W ′m−1. Hence, using excision, the Morse lemmaand Thom isomorphisms, we get the commutative diagram

H∗(Wm,Wm−1)≈ //

i∗m,m−1

∏C∈Critf∩f−1(m)H

∗−m(C)

proj

H∗(W ′m,W′m−1)

≈ // ∏C∈Critf′∩f−1(m)H

∗−m(C)

which proves that i∗m,m−1 is onto. If i∗m−1 is surjective by induction hypothesis, weget that i∗m is surjective by diagram-chasing.

Note that the point D ∈ Critf − Critf ′ of lowest index is

D = dia(1, . . . , 1, 0, . . . , 0, 1) ∈ SM(n+ 1)

satisfies f(D) = index(D) = n − k + 1 (the number of zeros in D). Hence,Critf ∩ Wn−k = Critf ′ ∩ W ′n−k. The same induction argument as above shows

that Hj(Gr(k;Rn+1))→ Hj(Gr(k;Rn)) is an isomorphism for j ≤ n− k.

9.5.2. Cohomology rings. The cohomology ring of a flag manifold V will begenerated by Stiefel-Whitney classes of some tautological bundles over V . Considera flag manifold Fl(n1, . . . , nr), with n = n1+ · · ·+nr. Consider the following closedsubgroups of O(n)

Bi = O(n1)× · · · × 1 × · · · ×O(nr) ⊂ O(n1)× · · · ×O(nr) ⊂ O(n) ,where 1 sits at the i-th place. Then

Pi = O(n)/Bi →→ O(n)/O(n1)× · · · ×O(nr) = Fl(n1, . . . , nr)

is an O(ni)-principal bundle over Fl(n1, . . . , nr). Indeed, if K is a compact sub-group of a Lie group G, then G → G/K is a principal K-bundle (see, e.g. [12,Theorem 2.1.1, Chapter I]). Let ξi be the vector bundle of rank ni associated toPi, i.e. E(ξi) = Pi ×O(ni) R

ni . The vector bundle ξi is called the i-th-tautologicalvector bundle over Fl(n1, . . . , nr). Being associated to an O(ni)-principal bundle, ξiis endowed with a Euclidean structure and its space of orthogonal frames Fra⊥(ξi)is equal to Pi.

In the mutually orthogonal subspaces description (presentation (2), p. 314) ofFl(n1, . . . , nr), we see that

E(ξi) = (W1 . . . ,Wr, v) ∈ Fl(n1, . . . , nr)× Rn | v ∈ Wi .Note that ξ1 ⊕ · · · ⊕ ξr is trivial. Indeed,

E(ξ1⊕ · · · ⊕ ξr) = ((W1, . . . ,Wr), (v1, . . . , vr)) ∈ Fl(n1, . . . , nr)× (Rn)r | vi ∈ Wi


and the correspondence

(9.5.8) ((W1, . . . ,Wr), (v1, . . . , vr)) 7→ v1 + · · ·+ vr

restricts to a linear isomorphism on each fiber. Such a map thus provides a trivial-ization of ξ1 ⊕ · · · ⊕ ξr.

If one sees Fl(n1, . . . , nr) as the space of matrices M ∈ SM(n) with character-istic polynomial equal to

∏ri=1(x− λi)ni (presentation (3), p. 314), then

(9.5.9) E(ξi) = (M, v) ∈ Fl(n1, . . . , nr)× Rn |Mv = λiv .The vector bundle ξ1 over Fl(k, n − k) = Gr(k;Rn) is called the tautological

vector bundle over the Grassmannian Gr(k;Rn); it is of rank k and is denoted by ζ,ζk or ζk,n. The space of Fra⊥(ζk) is the Stiefel manifold Stief(k,Rn) of orthonormalk-frames in Rn.

The inclusion Rn ≈ Rn × 0 → Rn+1 induces an inclusion Gr(k;Rn) →Gr(k;Rn+1) and we may consider the direct limit

Gr(k;R∞) = lim−→n

Gr(k;Rn)

which is a CW-space. The tautological vector bundle ζk is also defined overGr(k;R∞) and induces that over Gr(k;Rn) by the inclusion Gr(k;Rn) → Gr(k;R∞).It is classical that πi(Stief(k,Rn)) = 0 for i < n − k (see [181, Theorem 25.6]),thus Stief(k,R∞) = Fra(ζk) is contractible. Hence, the O(k)-principal bundleStief(k,R∞) → Gr(k;R∞) is a universal O(k)-principal bundle (see [181, § 19.4])and thus homotopy equivalent to the Milnor universal bundle EO(k) → BO(k).In particular, Gr(k;R∞) has the homotopy type of BO(k). As a consequence, anyvector bundle of rank k over a paracompact space X is induced from ζk by a mapX → Gr(k;R∞) (for a direct proof of that, see [153, Theorem 5.6]).

Theorem 9.5.8. The cohomology ring of BO(k) is GrA-isomorphic to thepolynomial ring

H∗(BO(k)) ≈ H∗(Gr(k;R∞)) ≈ Z2[w1, . . . , wk]

generated by the Stiefel-Whitney classes wi = wi(ζk) of the tautological bundle ζk.

Proof. Slightly more formally, we consider the polynomial ring Z2[w1, . . . , wk]with formal variables wi of degree i. The correspondence wi 7→ wi(ζk) provides aGrA-morphism ψ : Z2[w1, . . . , wk]→ H∗(BO(k)) which we shall show is bijective.

For the injectivity, we consider the tautological line bundle γ over RP∞ andits n-times product γn over (RP∞)n. As seen above, ζk is universal so γn isinduced by a map f : (RP∞)n → BO(n). Recall from Proposition 4.3.10 thatH∗(RP∞) = Z2[a] with a of degree 1 and, by Theorem 9.4.1, w(γ) = 1 + a. By

the Kunneth theorem, there is a GrA-isomorphism Z2[a1, . . . , an]≈−→ H∗((RP∞)n)

and, by Theorem 9.4.1, w(γn) = (1 + a1) · · · (1 + an). As H∗f(w(ζk)) =w(γn), there is a commutative diagram

Z2[w1, . . . , wk]

φ

ψ // H∗(BO(k))

H∗f

Z2[a1, . . . , an]≈ // H∗((RP∞)n))


with

φ(w(ζk)) = (1 + a1) · · · (1 + an) = 1 + σ1 + · · ·+ σn ,

where σi = σi(a1, . . . , an) is the i-th elementary symmetric polynomial in the vari-ables aj (see (8.5.4)). Thus, φ(wi) = σi. Now, if 0 6= A ∈ Z2[w1, . . . , wk] satisfiesψ(A) = 0, then φ(A) = 0 would be a non-trivial polynomial relation between theσi’s. But the elementary symmetric polynomials are algebraically independent (seee.g. [122]). Thus, ψ is injective.

For d ∈ N, let

Bd = (d1, . . . , dk) ∈ Nk |k∑

j=1

j dj = d .

The correspondence (d1, . . . , dk) 7→ wd11 · · ·wdkk is a bijection from Bd onto a basis

of the vector subspace Z2[w1, . . . , wk][d] formed by the elements in Z2[w1, . . . , wk]

which are of degree d. On the other hand, consider Gr(k;Rn) ⊂ SM(n) as in (9.5.3),with n large. Let Critdf ⊂ Gr(k;Rn) be the set of critical points of index d for theweighted trace. Then the correspondence

(d1, . . . , dk) 7→ dia(0, . . . , 0︸︷︷︸dk

, 1, 0, . . . , 0︸︷︷︸dk−1

, 1, . . . , 0, . . . , 0︸︷︷︸d1

, 1, 0, . . . , 0)

provides a bijection Bd ≈−→ Critdf . As f is a perfect Morse function by Proposi-tion 9.5.2, one has

♯Bd = ♯Critdf = dimHd(Gr(k;Rn)) = dimHd(BO(k)) ,

the last equality coming from Lemma 9.5.7 when n is large enough. Therefore,

dimZ2[w1, . . . , wk][d] = dimHd(BO(k)) .

As ψ is injective, it is then bijective.

Define

(9.5.10) Qr(t) =1

1− tr = 1 + tr + t2r + · · · ∈ Z[[t]] ,

which is the Poincare series of Z2[x] if x is of degree r. Here is a direct consequenceof Theorem 9.5.8.

Corollary 9.5.9. The Poincare series of BO(k) is

Pt(BO(k)) = Q1(t) · · ·Qk(t) .

As any vector bundle of rank k over a paracompact space is induced from theuniversal bundle ζk [153, Theorem 5.6], Theorem 9.5.8 has the following corollary.

Corollary 9.5.10. Any characteristic class in mod 2 cohomology for vectorbundles of finite rank over paracompact spaces is a polynomial in the Stiefel-Whitneyclasses wi.

Also, Theorem 9.5.8 together with Lemma 9.5.7 gives the following corollary.

Corollary 9.5.11. The cohomology ring H∗(Gr(k;Rn)) is generated, as aring, by the Stiefel-Whitney classes w1(ζk), . . . , wk(ζk) of the tautological bundleζk.


Theorem 9.5.8 permits us to compute the cohomology of BSO(k). The latter

also has a tautological bundle ζk = (ESO(k) ×SO(k) Rk → BSO(k)) which is

orientable.

Corollary 9.5.12. The cohomology ring of BSO(k) is GrA-isomorphic tothe polynomial ring

H∗(BSO(k)) ≈ Z2[w2, . . . , wk]

generated by the Stiefel-Whitney classes wi = wi(ζk) of the tautological bundle ζk.

Proof. Let i : SO(k) → O(k) denote the inclusion. By Example 7.2.4, themap Bi : BSO(k) → BO(k) is homotopy equivalent to a two-fold covering, whichis non-trivial since BSO(k) is connected. By Lemma 4.3.6, its characteristic classw(Bi) ∈ H1(BO(k)) is not trivial. By Theorem 9.5.8, the only non-zero elementin H1(BO(k)) is w1(ξk), so w(Bi) = w1(ξk).

By Theorem 9.5.8 and the transfer exact sequence (Proposition 4.3.9), the ringhomomorphism H∗Bi : H∗(BO(k)) → H∗(BSO(k)) is surjective with kernel the

ideal generated by w1(ζk). As Bi is covered by a bundle map from ζk to ζk, one

has H∗Bi(wi(ζk) = wi(ζk). The corollary follows.

Remark 9.5.13. In contrast with the simplicity of H∗(BSO(k)), the coho-mology ring H∗(BSpin(k)) is complicated and its computation requires spectralsequences (see [167]). The stable case BSpin = limk BSpin(k) is slightly simpler(see [193]).

We are now in position to give a GrA-presentation of H∗(Fl(n1, . . . , nr)). Let

(9.5.11) w(ξj) = 1+ w1(ξj) + · · ·+ wnj(ξj) ∈ H∗(Fl(n1, . . . , nr))

be the Stiefel-Whitney class of the tautological vector bundle ξj . As seen in (9.5.8),ξ1 ⊕ · · · ⊕ ξr is trivial. By Theorem 9.4.1, the equation

(9.5.12) w(ξ1) · · · w(ξr) = 1

holds true. Hence, the homogeneous components of w(ξ1) · · · w(ξr) in positivedegrees vanish, giving rise to n equations.

Theorem 9.5.14. The cohomology algebra H∗(Fl(n1, . . . , nr)) is GrA-isomorphicto the quotient of the polynomial ring

Z2[wi(ξj)] , 1 ≤ i ≤ rj , j = 1, . . . , r

by the ideal generated by the homogeneous components of w(ξ1) · · ·w(ξr) in positivedegrees.

Proof. We first prove that H∗(Fl(n1, . . . , nr)) is, as a ring, generated by theStiefel-Whitney classes wi(ξj) (1 ≤ i ≤ rj , j = 1, . . . , r). This is done by inductionon r (note that r ≥ 2 in order for the definition of Fl(n1, . . . , nr) to make sense).For r = 2, as Fl(n1, n2) = Gr(n1;Rn1+n2), the result comes from Corollary 9.5.11.For the induction step, let us define a map π : Fl(n1, . . . , nr) → Fl(n − nr, nr) byπ(W1 . . . ,Wr) = (W1⊕· · ·⊕Wr−1,Wr) (using the mutually orthogonal definition (2)of the flag manifolds). By Lemma 9.5.1, this gives a locally trivial bundle

Fl(n1, . . . , nr−1)ι−→ Fl(n1, . . . , nr)

π−→ Fl(n− nr, nr) .By induction hypothesis,H∗(Fl(n1, . . . , nr−1)) is generated, as a ring, by the Stiefel-Whitney classes of its tautological bundles, say wi(ξj). Note that these bundles are


induced by the tautological bundles (called ξj) over Fl(n1, . . . , nr): ξj = ι∗ξj . Then,H∗ι is surjective and wi(ξj) 7→ wi(ξj)) is a cohomology extension of the fiber (seep. 144). On the other hand,

Fl(n− nr, nr) ≈ Gr(n− nr;Rn) ≈ Gr(nr;Rn) ,

the last isomorphism sending an (n − nr)-dimensional subspace of Rn to its or-thogonal complement. By Corollary 9.5.11, H∗(Gr(nr;Rn)) is GrA-generatedby w1(ζnr

), . . . , wnr(ζnr

) and H∗π(wi(ζnr)) = wi(ξr). By the Leray-Hirsch the-

orem 4.7.17, H∗(Fl(n1, . . . , nr)) is then GrA-generated by wi(ξj) (1 ≤ i ≤ nj andj = 1 . . . r).

Let Γ = O(n1)× · · · ×O(nr) ⊂ O(n) and consider the commutative diagram

(9.5.13)

O(n) //

EO(n) //

BO(n)

=

O(n)/Γ // EO(n)/Γ // BO(n)

where the top line is the O(n)-universal bundle. Hence, the bottom line is a locallytrivial bundle with fiber equal to O(n)/Γ = Fl(n1, . . . , nr); as EO(n) is contractible,there are homotopy equivalences

EO(n)/Γ ≃ BΓ ≃ BO(n1)× · · · ×BO(nr) ,the last homotopy equivalence coming from (7.4.3). Hence, Diagram (9.5.13) maybe rewritten in the following way

(9.5.14)

Γ= //

Γ

O(n) //

EO(n) //

BO(n)

=

Fl(n1, . . . , nr)β // BΓ

Bα // BO(n)

where α denotes the inclusion of Γ in O(n). The left column is a Γ-principalbundle which is a Γ-structure on ξ = ξ1 ⊕ · · · ⊕ ξr. The central column is theΓ-principal bundle associated to the vector bundle ζ = ζn1 × · · · × ζnr

over BΓ ≃BO(n1)×· · ·×BO(nr). Thus, the map β is a classifying map for the Γ-structure onξ: it lifts the map Bαβ, which is classifying for ξ as a vector bundle (that Bαβis null-homotopic is consistent with the triviality of ξ, seen in (9.5.8)). Hence

β∗ζnj≈ ξj , β∗ζ ≈ ξ

and thusH∗β(wi(ζnj

)) = wi(ξj) , H∗β(wi(ζ)) = wi(ξ) .

As H∗(Fl(n1, . . . , nr)) is GrA-generated by the classes wi(ξj), H∗β is surjective

and one may apply the Leray-Hirsch theorem 4.7.17 and its corollaries. By Theo-rem 9.5.8 and the Kunneth theorem, there is a GrA-isomorphism

Z2[wi(ζnj)]≈−→ H∗(BΓ) , (1 ≤ i ≤ rj , j = 1, . . . , r) .

On the other hand, H∗(BO(n)) ≈ Z2[wi(ζn)] and H∗Bα(wi(ζn)) = wi(ζ). ByCorollary 4.7.19, H∗(Fl(n1, . . . , nr)) is then GrA-isomorphic to the quotient of


H∗(BΓ) by the ideal generated by wi(ζ) (i > 0). Hence, one has the commutativediagram

Z2[wi(ζj)]/(

wi(ζ), i > 0)

≈ **

≈ // Z2[wi(ξj)]/(

wi(ξ), i > 0)

tt

H∗(Fl(n1, . . . , nr))

which proves Theorem 9.5.14.

In the following corollary, we use the notations Qi of (9.5.10).

Corollary 9.5.15. The Poincare polynomial of Fl(n1, . . . , nr) is given by theformula

Pt(Fl(n1, . . . , nr)) =

∏rj=1[Q1(t) · · ·Qnj

(t)]

Q1(t) · · ·Qn(t).

In particular,

Pt(Gr(k;Rn)) = Pt(Fl(k, n− k)) =Q1(t) · · ·Qk(t)

Qn−k+1(t) · · ·Qn(t)and

Pt(Fl(1, . . . , 1)) =Q1(t)

n

Q1(t) · · ·Qn(t)=

(1− t)(1 − t2) · · · (1− tn)(1 − t)n .

Remark 9.5.16. The above formulae, evaluated at t = 1 using L’Hospital’srule, give dimH∗(Fl(n1, . . . , nr)), etc, giving again the formulae of Corollary 9.5.5.

Proof of Corollary 9.5.15. We have seen in the proof of Theorem 9.5.14that

Fl(n1, . . . , nr)β−→ BΓ

Bα−−→ BO(n)

is a locally trivial bundle satisfying the hypotheses of the Leray-Hirsch theorem.We know by Corollary 9.5.9 that Pt(BO(k)) = Q1(t) · · ·Qk(t). By the Kunnethformula, we get that

Pt(BΓ) = Pt(BO(n1)× · · · ×BO(nr)) =r∏

j=1

[Q1(t) · · ·Qnj(t)] .

The first formula then comes from Corollary 4.7.20. The other formulae are conse-quences of the first one.

We now give some illustrations of Theorem 9.5.14.

Example 9.5.17. Consider the case of the complete flag manifold Fl(1, . . . , 1).Theorem 9.5.14 says that H∗(Fl(1, . . . , 1)) is generated by xi = w1(ξi) for i =1 . . . , n. In this generating system, wi(ξ1 ⊕ · · · ⊕ ξn) = σi, the ith elementarysymmetric polynomial in the variables xi. Hence, by Theorem 9.5.14,

H∗(Fl(1, . . . , 1)) ≈ Z2[x1, . . . , xn]/(σ1, . . . , σn

).

Example 9.5.18. Consider the case of the Grassmannian

Gr(k;Rn) = Fl(k, n− k) .Set ζ = ξ1, with Stiefel-Whitney class w(ζ) = w = 1+w1 +w2 + · · · , and ζ⊥ = ξ2,with w(ζ⊥) = w = 1 + w1 + w2 + · · · . Note that the fiber of the vector bundle


ζ⊥ over P ∈ Gr(k;Rn) is the set of vectors in Rn which are orthogonal to P .Equation (9.5.12) becomes

(9.5.15) w w = 1 ,

which is equivalent to the following system of equations:

(9.5.16) wi =

k∑

r=1

wrwi−r (i = 1, . . . , n− k) and wi = 0 if i > n− k .

This system has the following unique solution.

Lemma 9.5.19. With the convention wi = wi = 0 for i < 0, the equation

wr =

∣∣∣∣∣∣∣∣∣∣∣

w1 1w2

...wr−1 1wr wr−1 · · · w2 w1

∣∣∣∣∣∣∣∣∣∣∣

= det(wi+1−j

)1≤i,j≤r

.

0

holds true in Hr(Gr(k;Rn)). The symmetric formula wr = det(wi+1−j

)1≤i,j≤r

holds true as well. These equalities are both equivalent to Equation (9.5.15).

Proof. The first equation is proved by induction on r, starting, for r = 1,with w1 = w1 (this also gives the uniqueness of the solution). The induction stepis achieved by expanding the determinant with respect to the first column: the(s, 1)-th minor is equal to wr−s by induction hypothesis and the result followsfrom (9.5.16). The symmetric equation follows from the symmetry in wi and wiof (9.5.16) (coming from the symmetry of (9.5.15)).

Below are two special case of Example 9.5.18.

Example 9.5.20. Consider the case of Gr(1;Rn) = Fl(1, n− 1) ≈ RPn−1. Therelation w w = 1 gives rise to the system of equations

w1 + w1 = 0wi + wi−1w1 = 0 (i = 2, . . . , n− 1)wn−1w1 = 0

from which we deduce the usual presentation H∗(RPn−1) ≈ Z2[w1]/(wn1 ).

Example 9.5.21. In the case Gr(2;R4) = Fl(2, 2), the relation w w = 1gives rise to four equations

(9.5.17)

w1 = w1

w2 = w21 + w2

w31 = 0

w2w21 + w2

2 = 0 .

and Theorem 9.5.14 says that H∗(Gr(2;R4)) is generated by w1, w2, w1 and w2,subject to Relations (9.5.17). The first two equations imply that H∗(Gr(2;R4))is generated by w1 and w2, as known since Corollary 9.5.11. We check that an


additive basis of H∗(Gr(2;R4)) is given by 1, w1, w2, w21 , w2w1 and w2w

21 = w2

2 .The Poincare polynomial of Gr(2;R4) is given by Corollary 9.5.15:

Pt(Gr(2;R4)) =Q1(t)Q2(t)

Q3(t)Q4(t)=

(1− t3)(1− t4)(1− t)(1− t2) = 1 + t+ 2t2 + t3 + t4 .

For any bundle of ξ rank k over a space X the dual (or normal) Stiefel-Whitneyclass wr(ξ) are defined by the equation of Lemma 9.5.19. Set w(ξ) = 1+ w1(ξ) +· · · for the total dual Stiefel-Whitney class. Equations (9.5.15) and (9.5.16) aresatisfied. If there exists a vector bundle η over X such that ξ ⊕ η is trivial, thenw(η) = w(ξ). Thus, if η is of rank r, then wi(ξ) = 0 for i > r. The samecondition is necessary for ξ being induced from the tautological bundle ζ by a mapf : X → Gr(k;Rk+r).

For example, let M be a smooth manifold of dimension k which admits animmersion β : M → Rk+r. Let x ∈ M . By identifying Tβ(x)R

k+r with Rk+r, the

k-vector space Txβ(TM) becomes a element of Gr(k;Rk+r). This produces a map

β : M → Gr(k;Rk+r) and TM = β∗ζ. We thus get the following result.

Proposition 9.5.22. If a smooth manifold M of dimension k admits an im-mersion into Rk+r, then wi(TM) = 0 for i > r.

For improvements of Proposition 9.5.22 concerning also smooth embeddings,see Proposition 9.8.23 and Corollary 9.8.24. Usually, Proposition 9.5.22 does notgive the smallest integer r for which M immerses into Rk+r. This is however thecase in the following example, taken from [153, Theorem 4.8].

Proposition 9.5.23. For k = 2j (j ≥ 1), the projective space RP k immersesinto RN if and only if N ≥ 2k − 1.

Proof. That a manifold of dimension k ≥ 2 immerses into R2k−1 is a classicaltheorem of H. Whitney [210]. Conversely, we shall see in Proposition 9.8.10 that

w(TRP 2j ) = (1 + a)2j+1 = 1 + a+ a2

j

,

where 0 6= a ∈ H1(RP 2j ) = Z2. Hence,

w(TRP 2j) = 1 + a+ a2 + · · ·+ a2j−1 ,

which, using Proposition 9.5.22 implies that RP 2j does not immerse into R2j+1−2.

9.5.3. Schubert cells and Stiefel-Whitney classes. Let f : M → R be aMorse function on a manifold M . It is classical that M has the homotopy typeof a CW-complex whose r-cells are in bijection with the critical points of indexr of f (see, e.g. [13, Theorem 3.28]). For the weighted trace f (or f) definedon M = Gr(k;Rn) in Proposition 9.5.2 (or Remark 9.5.4), a very explicit suchCW-structure is given, using the Schubert cells (there are generalizations for flagmanifolds). Inspired by works of H. Schubert on enumerative geometry in theXIXth century (see e.g. [169]), Schubert cells were introduced in 1934 (for complexGrassmannians) by Ch. Ehresmann [47] ([48] for the real Grassmannians). See[40, pp. 224–25] for a history. We restrict ourselves here to a very elementary pointof view, Schubert calculus being a huge subject in algebraic geometry.

Recall that Crit f are diagonal matrices in SM(n). We write dia(λ1, . . . , λn) =dia(λ), where λ = λ1 · · ·λn is a binary word of length n. Let [nk ] be the set of


such words with∑λi = k (they are (nk ) in number). The correspondence λ 7→ λ0

identifies [nk ] with a subset of [n+1k ], permitting us to define [∞k ] as the direct limit

of [nk ].Let F = (F1 ⊂ · · · ⊂ Fn) be a complete flag in Rn (adding the convention that

F0 = 0). For λ ∈ [nk ], the Schubert cell CFλ with respect to F is defined by

CFλ = P ∈ Gr(k;Rn) | dim(P ∩ Fi) =i∑

j=1

λj ⊂ Gr(k;Rn) .

(This convention is close to that of [119], except for the binary words being writtenin the reverse order, so it works for n =∞, in the spirit of [153, § 6]). The followingfacts may be proven.

(1) The Schubert cells CFλ | λ ∈ [nk ] are the open cells of a CW-structureXF on Gr(k;Rn) (see [153, § 6]). The dimension of CFλ is

d(λ) = index (dia(λ)) = f(dia(λ)) = −k(k + 1)

2+∑

i≥1

λi .

By Proposition 9.5.2, the cellular chains have then the same Poincarepolynomial as the homology. Therefore, XF is a perfect CW-structure.

(2) The closure CFλ , called the Schubert variety, satisfies

CFλ = P ∈ Gr(k;Rn) | dim(P ∩ Fi) ≥i∑

j=1

λj ⊂ Gr(k;Rn)

and is a subcomplex of XF (see e.g. [47, § 10]). As XF is perfect, so is CFλand thus CFλ defines a homology class

[λ] = [CFλ ] ∈ Hd(λ)(Gr(k;Rn)) (n ≤ ∞)

which does not depend on F since, by Proposition 9.5.2, the complete flagmanifold is path-connected. It corresponds, under the isomorphism (3.5.6)between cellular and singular homology, to the cellular homology class forXF indexed by λ. It follows that the S = [λ] ∈ H∗(Gr(k;Rn)) | λ ∈ [nk ]is a basis of H∗(Gr(k;Rn)) (n ≤ ∞).

(3) Let P ∈ Gr(k;Rn). Using a basis of Rn compatible with the flag F , letMP be the matrix of a linear epimorphism Rn → Rn−k with kernel P .The condition P ∈ CFλ is equivalent to the vanishing of various minorsof MP . Therefore, P ∈ CFλ is a compact real algebraic variety. This isanother proof of the existence of the class [λ], since such a variety carriesa fundamental class (see [192, p. 67] or [16, Theorem 3.7 and § 3.8]).

(4) Suppose that F is the standard flag (Fi = Ri×0). Then f(CFλ ) = [0, d(λ)]and CFλ ∩ f−1(d(λ)) = dia(λ). Recall from Proposition 9.5.2 that d(λ) isequal to the number of pairs (i, j) with 1 ≤ i < j ≤ n such that λi < λj .

For such a pair (i, j), let Rijt be the one-parameter subgroup of SO(n)

considered in the proof of Proposition 9.5.2. Then, the curveRijt dia(λ)Rij−t

is contained in CFλ for t ∈ R and stays in CFλ when |t| < π/2. By the proofof Proposition 9.5.2, these curves generate the negative normal bundle forf at dia(λ).


Example 9.5.24. Consider the case of Gr(1;Rn+1) ≈ RPn. For F the standardflag in Rn+1, the Schubert cells give the standard CW-structure on RPn, the cellCFλ for λ = 0r10n−r being of dimension r. The Schubert variety CFλ is equal toRP r (a rare case where it is a smooth manifold).

Note that H∗α([λ]) = [λ0] where α : Gr(k;Rn) → Gr(k;Rn+1) is induced bythe inclusion Rn ≈ Rn ⊕ 0 → Rn ⊕R. We often identify [λ] with [λ0]. In this way,for instance, [100101] determines a class in H5(Gr(k;Rn)) for n ≥ 6.

Let S♯ = [λ]♯ | λ ∈ [nk ] (n ≤ ∞) be the additive basis of H∗(Gr(k;Rn)) whichis dual for the Kronecker pairing to the basis S (see (2) above): the class [λ]♯ isdefined by

〈[λ]♯, [µ]〉 = δλµ,

where δλµ is the Kronecker symbol. The basis S♯ was studied in [31, 32]. Becauseof intersection theory, a more widely used additive basis for H∗(Gr(k;Rn)) (definedonly or n <∞) is SPD, formed by the Poincare duals [λ]PD for all λ ∈ [nk ]. Thoughsome intersection theory in used in the proof of Proposition 9.5.29 below, we shallnot use SPD. We just note the following result.

Lemma 9.5.25. For any k ≤ n <∞, the two sets S♯ and SPD in H∗(Gr(k;Rn))are equal.

Proof. Let F be the standard flag (Fi = Ri × 0) and let F− be the anti-standard one (F−i = 0 × Ri). For λ ∈ [nk ], define λ− ∈ [nk ] by λ−i = λn+1−i.

The cycles CFλ and CF−

λ− are of complementary dimensions and, by (4) above,they intersect transversally in a single point (the k-plane generated by λiei fori = 1, 2, . . . , n). In the same way, if µ ∈ [nk ] satisfies d(λ) = d(µ) but µ 6= λ, then

CFλ ∩ CF−

µ− = ∅. Analogously to Proposition 5.4.12, one has

(9.5.18) [µ−]PD [λ]PD = [CFλ ∩ CF−

λ− ]

(see Remark 9.5.26 below). This implies that

〈[µ−]PD [λ]PD, [Gr(k;Rn)]〉 = δµλ

But, using (4.5.13),

〈[µ−]PD [λ]PD, [Gr(k;Rn)]〉 = 〈[µ−]PD, [λ]PD [Gr(k;Rn)]〉= 〈[µ−]PD, [λ]〉 .

This proves that [λ]♯ = [λ−]PD (or [λ]PD = [λ−]♯).

Remark 9.5.26. In the above proof, (9.5.18) is not a consequence of Proposi-

tion 5.4.12, which would require that CFλ and CF−

λ− are submanifolds of Gr(k;Rn). Inthis simple situation, one could use the Morse function f to isolate the intersectionpoint around the critical level d(λ) and deal with an intersection of submanifolds(with boundaries). For more general situations (see the proof of Proposition 9.5.29),one must rely on the intersection theory for real algebraic varieties (see, e.g. [16,(1.12) and § 5]).

In addition to the above ambient inclusion α : Gr(k;Rn) → Gr(k;Rn+1) wealso consider the fattening inclusion β : Gr(k;Rn)→ Gr(k + 1;Rn+1) sending P toR⊕P ⊂ R⊕Rn. Then H∗j([λ]) = [1λ] for all λ ∈ [kn]. This drives us to decompose


a word λ ∈ [nk ] into its prefix, stem and suffix, delimited by the first 0 and the last1 of λ:

λ = 111111111︸︷︷︸prefix

00101101︸︷︷︸stem

0000︸︷︷︸suffix

.

Given n and k, a word λ ∈ [nk ] (and then a class [λ] ∈ H∗(Gr(k;Rn)) or [λ]♯ ∈H∗(Gr(k;Rn))) is determined by its stem. For example, 0101 is, for k = 4 andn = 8, the stem of the unique class [11010100]♯ ∈ H3(Gr(4;R8)). The stem of 1 ∈H0(Gr(k;Rn)) is just 0. Here is a first use of the prefix-stem-suffix decomposition.

Proposition 9.5.27. Let n, k and i be integers with 0 ≤ i ≤ k. Then, fork+1 ≤ n ≤ ∞, the Stiefel-Whitney class wi = wi(ζk) is the class in Hi(Gr(k;Rn))with stem 01i. For example, w3(ζ4) = [1011100]♯ ∈ H3(Gr(4;R7)).

Proof. The proposition is true if i = 0, since w0(ζk) = 1. Let us as-sume that i ≥ 1. We first prove that wi(ζk) = [01i]♯ in Hi(Gr(i;R∞)). RecallH∗(Gr(i;R∞)) ≈ Z2[w1, . . . , wi] (where wj = wj(ζi)) so

K∗ =(kerH∗β : H∗(Gr(i;R∞))→ H∗(Gr(i− 1;R∞))

)

is the ideal generated by wi. Hence, Ki is one-dimensional generated by wi. AsH∗β([λ]) = [1λ], one has

(9.5.19) H∗β([µ]♯) =

[λ]♯ if µ = 1λ

0 otherwise.

Hence 0 6= [01i]♯ ∈ Ki. Therefore, wi(ζk) = [01i]♯. Proposition 9.5.27 followsfrom the above particular case since H∗α(wi(ζk)) = wi(ζk) and H∗β(wi(ζk)) =wi(ζk−1).

Let λ, µ ∈ [nk ]. As S is a basis for H∗(Gr(k;Rn)) and S♯ is the Kronecker dualbasis for H∗(Gr(k;Rn)), we can write

[λ]♯ [µ]♯ =∑

ν∈[nk]

Γνλµ [ν]♯

where

Γνλµ = 〈[λ]♯ [µ]♯, [ν]〉 ∈ Z2 .

Computing the ”structure constants” Γνλµ is a version of the Schubert calculus

(mod 2). The usual Schubert calculus deals with the structure constants Cνλµ for

the basis SPD, defined by

[λ]PD [µ]PD =∑

ν∈[nk]

Cνλµ [ν]PD .

By Lemma 9.5.25 and its proof, one has Γνλµ = Cν−

λ−µ− . Again, Schubert calculus

was initiated by Ch. Ehresmann in [47, 48] and further developed in e.g. [31, 32,75, 66]. For a more recent as well as an equivariant version, see [119]. Note thatΓνλµ = 0 unless d(λ) + d(µ) = d(ν).

A binary word λ ∈ [nk ] is determined by its Schubert symbol, i.e. the k-tuple ofintegers indicating the positions of the 1’s in λ. For instance, the Schubert symbolof 0100101 is (2, 5, 7). We use the Schubert symbol of λ for all the cohomology


classes [λ0j ]♯ (λ and λ0 having the same symbol). For the reverse correspondence,we decorate the Schubert symbol by a flat sign . Example:

[01001010r]♯ = (2, 5, 7) in H7(Gr(3;R7+r))

(2, 5, 7) = [01001010r] in H7(Gr(3;R7+r))

Our notation for Schubert symbols is that of [153], close to the original one of[47]. Other conventions are used in e.g. [32, 75].

Remark 9.5.28. Fix the integers k ≤ n and let a = (a1, . . . , ak) be a k-tuple of integers. In order for a to be a Schubert symbol determining a class inH∗(Gr(k;Rn)), it should satisfy

(9.5.20) 1 ≤ a1 < a2 < · · · < an ≤ n .

When this is not the case, we decide by convention, that a represents the class 0.

A Schubert cell CFλ will be also labeled by the Schubert symbol of λ: CFλ = CFaif a = [λ]♯. For the Poincare duality (see Lemma 9.5.25), we set

a− = [λ−]♯ = [λ]PD .

If a = (a1, . . . , ak) then a− = (n+ 1− ak, . . . , n+ 1− a1). The definition of Γνλµ is

also transposed for Schubert symbols:

a b =∑

c

Γcab c

where the sum runs over all Schubert symbols c and

Γcab = 〈a b, c〉 ∈ Z2 .

The following proposition and its proof is a variant, in our language, of ReductionFormula I of [75, p. 202].

Proposition 9.5.29 (Reduction formula). Let k ≥ 2 be an integer. Let r, sand t be positive integers ≤ k satisfying t = r + s − 1. Let a, b and c be Schubertk-symbols. Then

Γcab =

0 if ct < ar + bs − 1

Γcab

if ct = ar + bs − 1,

where a, b and c are the Schubert (k − 1)-symbols

a = (a1, . . . , ar−1, ar+1 − 1, . . . , ak − 1)

b = (b1, . . . , bs−1, bs+1 − 1, . . . , bk − 1)

c = (c1, . . . , ct−1, ct+1 − 1, . . . , ck − 1)

Example 9.5.30. Let us use the formula for s = 1 and suppose that bs = 1.Thus b = [µ] = [1µ] with µ ∈ [n−1k−1 ]. The condition t = r + s− 1 reduces to t = r

and ct = ar + bs − 1 becomes cr = ar. Writing it in terms of a = [λ] and c = [ν]


this means that if λr = νr = 1 for some r, one can remove λr from λ and νr fromν and replace µ by µ. For instance,

Γ0110110101,11010 = Γ0110

1010,1010 s = 1 and r = t = 5

= Γ010100,010 s = 1 and r = t = 3

= 1 since [100]♯ = 1 in H0(Gr(1;R3)).

Proof of Proposition 9.5.29. Let F , F ′ and F ′′ be three complete flags inRn. If chosen generically, then CFx , C

F ′

y and CF′′

z are pairwise transverse for any

Schubert symbols x, y and z. Therefore, if d(a) + d(b) = d(c), Ca− ∩ Cb− ∩ Cc is0-dimensional and

(9.5.21)

Γcab = 〈a b, c〉= 〈a b, (c)PD [Gr(k;Rn)]〉= 〈a b (c)PD, [Gr(k;Rn)]〉= ♯(Ca− ∩ Cb− ∩ Cc) mod 2 ,

the last equality coming from the intersection theory analogous to Proposition 5.4.12but for algebraic cycles (see Remark 9.5.26).

Let P ∈ Gr(k;Rn). If P ∈ Ca− ∩ Cb− ∩ Cc thendim(P ∩ Fn+1−ar ) ≥ k + 1− rdim(P ∩ F ′n+1−bs

) ≥ k + 1− sdim(P ∩ F ′′ct) ≥ t .

Therefore, the condition t = r + s− 1 implies that

dim(P ∩ Fn+1−ar ∩ F ′n+1−bs ∩ F ′′ct) ≥ 1 .

On the other hand, as F , F ′ and F ′′ are transverse flags,

dim(Fn+1−ar ∩ F ′n+1−bs ∩ F ′′ct) = t− r − s+ 2 .

Thus, Γcab = 0 if ct < ar + bs − 1. If ct = ar + bs− 1, then Fn+1−ar ∩F ′n+1−bs∩F ′′ct

is a line L, which must be contained in any P ∈ Ca− ∩ Cb− ∩ Cc. Let L⊥ be theorthogonal complement of L and let π : Rn → L⊥ be the orthogonal projection.For 1 ≤ i ≤ n− 1, define

Fi =

π(Fi) if i ≤ n− arπ(Fi+1) if i ≥ n+ 1− ar .

As L ⊂ Fn+1−ar but L 6⊂ Fn−ar , the sequence of vector spaces Fi constitutes acomplete flag F in for L⊥. Define F ′ accordingly and F ′′ by

F ′′i =

π(F ′′i ) if i ≤ ctπ(F ′′i+1) if i ≥ ct + 1 .

Then, F , F ′ and F ′′ are transverse flags and, by linear algebra, one checks that

P = π(P )⊕ L ∈ Ca− ∩ Cb− ∩ Cc ⇐⇒ π(P ) ∈ C(a)− ∩ C(b)− ∩ Cc .Hence,

♯(CFa− ∩ CF′

b− ∩ CF′′

c ) = ♯(CF(a)− ∩ CF′

(b)− ∩ CF′′

c )

which, using (9.5.21), proves that Γcab = Γcab.


Corollary 9.5.31. Let Let a, b and c be Schubert k-symbols. Then Γcab = 0unless ci ≥ maxai + b1 − 1, bi + a1 − 1 for all 1 ≤ i ≤ k. In particular, Γcab = 0unless ci ≥ maxai, bi for all 1 ≤ i ≤ k.

Proof. If cr < ar + b1 − 1 for some integer r, then Γcab = 0 by the reductionformula for s = 1. As Γcab = Γcba, this proves the corollary.

We now compute, for a Schubert symbol a, the expression of wi a in thebasis S♯. For J ⊂ 1, 2, . . . , k, we define a map a 7→ aJ from Nk to itself by

aJi =

ai + 1 if i ∈ Jai if i /∈ J .

Proposition 9.5.32. Let a be a Schubert k-symbol. The equation

(9.5.22) wi a =∑

J⊂1,2,...,k♯J=i

aJ

holds in H∗(Gr(k;Rn)) (with the convention of Remark 9.5.28).

Since in the right side of (9.5.22), we use the convention of Remark 9.5.28,Proposition 9.5.32 holds true for any n and any i (wi = 0 if i ≥ k).

Example 9.5.33.

w2 (1, 3, 4, 6) =

(2, 3, 5, 6) + (2, 3, 4, 7) + (1, 3, 5, 7) in H6(Gr(4;Rn)) for n ≥ 7.

(2, 3, 5, 6) in H6(Gr(4;R6)).

Proof. It suffices to prove the proposition for n =∞. We identify wi with itSchubert symbol which, by Proposition 9.5.27, is

wi = (1, 2, . . . , k − i, k − i+ 1, . . . , k + 1) .

Then, the notation Γca,wiis meaningful. Let c be a Schubert k-symbol such that

Γca,wi6= 0. Then d(c) = d(a) + i and, as cj ≥ aj by Lemma 9.5.31, there is

K ⊂ 1, 2, . . . , k with ♯K = k − i such that cj = aj for j ∈ K. By iterating thereduction formula for s = 1 with the indices in K, we get that Γca,wi

= Γca,wi, where

a and c are Schubert i-symbols and wi = (2, 3, . . . , i + 1). By Lemma 9.5.31, wehave cj ≥ aj+1 for all 1 ≤ j ≤ i. This implies that c = aJ for J = 1, 2, . . . , k−K.

Conversely, let J ⊂ 1, 2, . . . , k with |J | = i. We have to prove that Γ(J) =

ΓaJ

a,wi= 1 if aJ is a Schubert symbol for Gr(k;R∞). By repeating the reduction

formula for s = 1 with all the indices not in J , we get that

Γ(J) = ΓaJ

a,wi

where a is a Schubert i-symbol, J = 1, 2, . . . , i and wi = (2, 3, . . . , i + 1). Byiterating again the reduction formula for s = 1 with the indices i, i− 1, etc, till 2,we get that

Γ(J) = Γ(a1+1)(a1),(2)

.

This coefficient is equal to 1, as (u) w1 = (u+1) in H∗(Gr(1;R∞)) ≈ H∗(RP∞).

For λ ∈ [nk ], let λ⊥ ∈ [nn−k] be obtained from λ by exchanging 0’s and 1’s and

reversing the order: 100101⊥ = 010110; in formula:

(9.5.23) λ⊥j = 0 ⇐⇒ λn+1−j = 1 .


Note that d(λ⊥) = d(λ). This formal operation is related to the homeomorphismh : Gr(k;Rn)→ Gr(n−k;Rn) sending k-plane P to its orthogonal complement P⊥.

Lemma 9.5.34. H∗h([λ]♯) = [λ⊥]♯.

Proof. Let F = (F1 ⊂ · · · ⊂ Fn) be a complete flag in Rn and let F− be thedual flag, defined by F−i = F⊥n−i (we add the convention that F0 = 0 = F−0 ). To

establish Lemma 9.5.34, we shall prove that h(C(F )λ ) = C

(F−)

λ⊥.

Let P ∈ Gr(k;Rn). Write P = (P ∩ Fi)⊕Qi. ThenP⊥ ∩ F−n−i = v ∈ F−n−i | 〈v,Qi〉 = 0 .

Hence

(9.5.24) codimF−n−i(P⊥ ∩ F−n−i) = dimQi = codimP (P ∩ Fi) .

Suppose that P ∈ C(F )λ for λ ∈ [nk ]. Then, P⊥ ∈ C(F−)

µ µ ∈ [nn−k]. We must prove

that µ = λ⊥, that is to say (λi = 0 ⇐⇒ µn+1−i = 1). But, using (9.5.24)

λi = 0 ⇐⇒ dim(P ∩ Fi) = dim(P ∩ Fi−1)⇐⇒ codimF−n−i

(P⊥ ∩ F−n−i) = codimF−n+1−i(P⊥ ∩ F−n+1−i)

⇐⇒ dim(P⊥ ∩ F−n+1−i) = dim(P⊥ ∩ F−n−i) + 1

⇐⇒ µn+1−i = 1 .

Let w = w(ζ⊥k ) = 1 + w1 + · · · wn−k be the total Stiefel-Whitney class of thetautological (n− k)-vector bundle over Gr(k;Rn) (see Example 9.5.18).

Proposition 9.5.35. Suppose that n ≥ i + k. Then wi ∈ Hi(Gr(k;Rn) is theclass of stem 0i1. Its Schubert symbol is (1, 2, . . . , k − 1, k + i).

Example: w5 = [111000001]♯ = (1, 2, 3, 9) in H5(Gr(4;Rn) for n ≥ 9.

Proof. The homeomorphism h : Gr(k;Rn) → Gr(n − k;Rn) is covered bythe tautological bundle map ζ⊥k → ζn−k. Hence, h∗ζn−k = ζ⊥k and thus wi =H∗h(wi(ζn−k)). Therefore, wi = (wi)

⊥ by Lemma 9.5.34. As stem (λ⊥) =stem (λ)⊥, Proposition 9.5.35 follows from Proposition 9.5.27.

We now give the expression of wi a for a Schubert k-symbol a. As w1 = w1,we can use Formula (9.5.22) for i = 1:

(9.5.25) w1 a = w1 a =∑

b

b

where the sum runs over all the Schubert k-symbols b such that

aj ≤ bj ≤ aj + 1 andk∑

j=1

(bj − aj) = 1 .

Example:

w21 = w1 (1, 2, . . . , k − 1, k + 1)

= (1, 2, . . . , k − 2, k, k + 1) + (1, 2, . . . , k − 1, k + 2)

= w2 + w2 .

Formula (9.5.25) admits the following generalization, called the Pieri formula,which is a sort of a dual of Proposition 9.5.32.


Proposition 9.5.36 (Pieri’s formula). Let a be a Schubert k-symbol. Theequation

(9.5.26) wi a =∑

b

b

holds in H∗(Gr(k;Rn)), where the sum runs over all the Schubert k-symbols b suchthat

(9.5.27) aj ≤ bj < aj+1 andk∑

j=1

(bj − aj) = i

(with the convention of Remark 9.5.28).

Proof. For Schubert k-symbols a, b, the proposition says that

(9.5.28) 〈wi a, b〉 = 1⇐⇒ (a, b) satisfies (9.5.27) .

(Note that the implication ⇒ follows from Lemma 9.5.31 and from wi being ofdegree i). Rewriting (9.5.28) with λ, µ ∈ [nk ] gives that 〈wi [λ]♯, [µ]〉 = 1 if andonly if λ and µ satisfy the following pair of conditions

(i) λ = A110r1A210

r2 · · ·As10rsAs+1, with∑s

j=1 rj = i, and

(ii) µ = A10r11A20

r21 · · ·Ars0rs1As+1.

(Intuitively: a certain quantity of 1’s are shifted by one position to the right oftotal amount shifting being i). The pair of conditions (i) and (ii) is equivalent tothe following ones

(i)⊥ λ⊥ = A⊥s+11rs0A⊥s 1

rs−10 · · ·A⊥2 1r10A⊥1 with∑sj=1 rj = i, and

(ii)⊥ µ⊥ = A⊥s+101rs A⊥s 01

rs−1 · · ·A⊥2 01r1.Recall from Lemma 9.5.34 and the proof of Proposition 9.5.35 that the homeo-morphism h : Gr(k;Rn)→ Gr(n−k;Rn) satisfiesH∗([ν]) = [ν⊥] and wi = H∗(wi(ζn−k)).Therefore

(9.5.29) 〈wi [λ]♯, [µ]〉 = 1⇐⇒ 〈wi(ζn−k) [λ⊥]♯, [µ⊥]〉 = 1 .

By Proposition 9.5.32, the right hand equality in (9.5.29) is equivalent to the pairof conditions (i)⊥ and (ii)⊥, which proves Proposition 9.5.36.

We finish this subsection by mentioning Giambelli’s formula, which express acohomology class given by a Schubert symbol as a polynomial in the wi’s. TheGiambelli and the generalized Pieri formulae together provides a procedure forcomputing the structure constants Γcab.

Proposition 9.5.37 (Giambelli’s formula).

(a1, . . . , ak) = det(wai−j

)1≤i,j≤k

.

with the convention that wu = 0 if u < 0.

For wr = (1, 2, . . . , k − r, k − r + 1, . . . , k + 1), Proposition 9.5.37 reproves thesecond formula of Lemma 9.5.19.

Proof. By induction on k, starting trivially if k = 1. The lengthy inductionstep, using the Pieri formula, may be translated in our language from [75, pp. 204–205] (see also [32, p. 366]).


9.6. Splitting principles

Let α : Γ→ O(n) denotes the inclusion of the diagonal subgroup of O(n)

Γ ≈ O(1)× · · · ×O(1) ⊂ O(n) .This induces an inclusion Bα : BΓ → BO(n) between the classifying spaces. Thesymmetric group Symn acts on O(1) × · · · × O(1) by permuting the factors, andthen on BΓ. As in § 9.5, ζn denotes the tautological vector bundle on BO(n) ≃Gr(n;R∞). It is the vector bundle associated to the universalO(n)-bundle EO(n)→BO(n).

Theorem 9.6.1. The GrA-morphism H∗Bα : H∗(BO(n)) → H∗(BΓ) is in-jective and its image is H∗(BΓ)Symn . The induced vector bundle Bα∗ζn splits intoa Whitney sum of line bundles.

Proof. We have seen in (9.5.13) that the homotopy equivalence

BΓ ≃ EO(n)/O(1)× · · · ×O(1)

makes Bα homotopy equivalent to the locally trivial bundle

(9.6.1) Fl(1, . . . , 1)β−→ BΓ

Bα−−→ BO(n) .

We have also established in the proof of Theorem 9.5.14 that H∗β is surjective.Hence, by Corollary 4.7.19, H∗Bα is injective. Also, using (7.4.3) and that O(1) ≈±1. one has a homotopy equivalence

BΓψ

≃// BO(1)× · · · ×BO(1) ≃ RP∞ × · · · × RP∞

and thus a GrA-isomorphism

ψ∗ : Z2[x1, · · · , xn] ≈−→ H∗(BΓ)

where xi has degree 1. By Theorem 9.5.8,

H∗(BO(n)) ≈ Z2[w1(ζn), . . . , wn(ζn)] .

Note that Bα is covered by a morphism of principal bundles

EO(1)× · · · × EO(1) Eα //

EO(n)

BO(1)× · · · ×BO(1) Bα // BO(n) .

One has a similar diagram for the associated vector bundles γ = (EO(1)×O(1)R→BO(1)) (corresponding to the tautological line bundle over RP∞) and ζn. Thisimplies that Bα∗ζn ≈ γ × · · · × γ. As w(γ × · · · × γ) =

∏ni=1(1 + xi), one has

H∗Bα(wi(ζn)) = wi(γ × · · · × γ) = σi ,

where σi is i-th elementary symmetric polynomial in the variables xj . The secondassertion of Theorem 9.6.1 follows, since the elementary symmetric polynomialsGrA-generate Z2[x1, · · · , xn]Symn ≈ H∗(BΓ)Symn .

Finally, the homotopy equivalence ψ is of the form ψ = (ψ1, . . . , ψn), withψi : BΓ→ BO(1). In other words, ψ coincides with the composition

BΓ∆−→ BΓ× · · · ×BΓ

ψ1×···×ψn−−−−−−−→ BO(1) × · · · ×BO(1) ,

9.6. SPLITTING PRINCIPLES 335

where ∆ is the diagonal map. Hence,

Bα∗ζn ≈ ψ∗(γ × · · · × γ) = ∆∗(ψ∗1γ × · · · × ψ∗nγ) = ψ∗1γ ⊕ · · · ⊕ ψ∗nγ ,which shows that Bα∗ζn is isomorphic to a Whitney sum of line bundles.

Theorem 9.6.1 may be generalized as follows. Consider the inclusion homomor-phism

αn1,...,nr: O(n1)× · · · ×O(nr)→ O(n)

sending (A1, . . . , Ar) to the diagonal-blockmatrix with blocksA1, . . . , Ar. Using thehomotopy equivalence BO(n1)×· · ·×BO(nr) ≃ B(O(n1)×· · ·×O(nr)) (see (7.4.3)),the homomorphism αn1,...,nr

induces a continuous map

Bαn1,...,nr: BO(n1)× · · · ×BO(nr)→ BO(n) .

Theorem 9.6.2. The map Bαn1,...,nrsatisfies the following properties.

(1) The GrA-morphism

H∗Bαn1,...,nr: H∗(BO(n))→ H∗(BO(n1)× · · · ×BO(nr))

is injective.(2) H∗Bαn1,...,nr

(wi) = wi(ζn1 × · · · × ζnr) for each i ≥ 0. In particular, the

image of H∗Bαn1,...,nris generated by wi(ζn1 × · · · × ζnr

) (i ≥ 0).(3) The induced vector bundle Bα∗n1,...,nr

ζn splits into a Whitney sum of vectorbundles of ranks n1, . . . , nr.

Proof. Using the inclusion factorization

O(1)n≈ // O(1)n1 × · · · ×O(1)nr // O(n1)× · · · ×O(nr) α // O(n) ,

where α = αn1,...,nr, the injectivity of H∗Bα comes from that of H∗Bα1,...,1, es-

tablished in Theorem 9.6.1. As Bα is covered by a morphism of principal bundles

EO(n1)× · · · × EO(nr) α //

EO(n)

BO(n1)× · · · ×BO(nr) Bα // BO(n)

,

one deduces (2) and (3) as in the proof of Theorem 9.6.1.

Proposition 9.6.3. Let ξ be a vector bundle over a paracompact space X.Then, there is a map f : Xξ → X such that

(1) H∗f is injective.(2) f∗ξ splits into a Whitney sum of line bundles.

Proposition 9.6.3 is called the splitting principle. For ξ = ζn over BO(n),Theorem 9.6.1 says that on can take BO(n)ζn and f = Bα.

Proof. As X is paracompact, ξ admits a Euclidean structure and there isa classifying map ϕ : X → BO(n) for ξ, i.e. ξ ≈ ϕ∗ζn. Consider the pull backdiagram

Xξf //

ϕ

X

ϕ

BΓ

Bα // BO(n)

,


where Bα is defined as in (9.6.1). As Bα is a locally trivial bundle with fiberFl(n1, . . . , nr), so is f (this is the Fl(n1, . . . , nr)-bundle associated to Fra⊥ξ). Wesaw in the proof of Theorem 9.6.1 that H∗(BΓ)→ H∗(Fl(n1, . . . , nr)) is surjective.Then, so is H∗(Xξ) → H∗(Fl(n1, . . . , nr)). Hence, by Corollary 4.7.19, H∗f isinjective. Now,

f∗ξ = f∗ϕ∗ζn = ϕ∗Bα∗ζn .

As, by Theorem 9.6.1, Bα∗ζn is a Whitney sum of line bundles, so does f∗ξ.

One consequence of the splitting principle is the uniqueness of Stiefel-Whitneyclasses (compare [153, Theorem 7.3] or [105, Chapter 16,§ 5]).

Proposition 9.6.4. Suppose that w is a correspondence associating, to eachvector bundle ξ over a paracompact space X, a class w(ξ) ∈ H∗(X), such that

(1) if f : Y → X be a continuous map, then w(f∗ξ) = H∗f(w(ξ)).(2) w(ξ ⊕ ξ′) = w(ξ) w(ξ′).(3) if γ is the tautological line bundle over RP∞, then w(γ) = 1 + a, where

0 6= a ∈ H1(RP∞).

Then w = w, the total Stiefel-Whitney class.

Proof. Condition (2) implies that

(2.bis) w(ξ1 ⊕ · · · ⊕ ξn) = w(ξ1) · · · w(ξn).

As in the proof of Theorem 9.6.1, Conditions (1), (2.bis) and (3) imply that themap Bα : BΓ→ BO(n) satisfies

H∗Bα(w(ζn)) = (1 + xi)n ∈ H∗(BΓ) ≈ Z2[x1, · · · , xn] .

Thus, still by the proof of Theorem 9.6.1, H∗Bα(w(ζn)) = H∗Bα(w(ζn)). AsH∗Bα is injective, this implies that w(ζn) = w(ζn). The bundle ζn being universal(see 9.1.14), Condition (1) implies that w(ξ) = w(ξ) for any vector bundle ξ over aparacompact space X .

Another consequence of the splitting principle is the action of the Steenrodalgebra on Stiefel-Whitney classes. The following proposition was proved by WuWen-tsun [214].

Proposition 9.6.5. Let ξ be a vector bundle over a paracompact space X.Then

(9.6.2) Sqiwj(ξ) =∑

0≤k≤i

(j−i+k−1

k

)wi−k(ξ)wj+k(ξ) .

Example 9.6.6. Setting wi = wi(ξ), we get

Sq1wj = w1wj + (j − 1)wj+1

Sq2wj = w2wj + (j − 2)w1wj+1 +(j−12

)wj+2

Sq3wj = w3wj + (j − 3)w2wj+1 +(j−22

)w1wj+2 +

(j−33

)wj+3 .

Proof of Proposition 9.6.5. By naturality ofw and Sq, it suffices to prove (9.6.2)for ξ = ζn, the tautological vector bundle on BO(n). By Theorem 9.6.1 and itsproof,

H∗Bα(wj(ζn)) = σj ∈ Z2[x1, · · · , xn]

9.7. COMPLEX FLAG MANIFOLDS 337

where xj corresponds to the non-trivial element in H1(RP∞), and σj is the j-th el-ementary symmetric polynomial in the variables xr. As H

∗Bα is injective by Theo-rem 9.6.1, Formula (9.6.2) reduces to the computation of Sqiσj in Z2[x1, · · · , xn]Symn ,using that Sq(xr) = xr + x2r . This technical computation may be found in full de-tails in [15, Theorem 7.1]. The reader may, as an exercise, prove the special casesof Example 9.6.6.

The splitting principle also gives the following result about the Stiefel-Whitneyclasses of a tensor product (for a more general formula, see [153, p. 87]).

Lemma 9.6.7. Let η and ξ be vector bundles over a paracompact space X.Suppose that ξ is of rank r and that η is a line bundle. Then

(9.6.3) w(η ⊗ ξ) =r∑

k=0

(1 + w1(η))kwr−k(ξ) .

Proof. Set u = w1(η). Suppose first that ξ splits into a Whitney sum of r linebundles ξj , of Stiefel-Whitney class 1+ vj . Then, letting σk = (v1, . . . , vr) denotethe kth elementary symmetric polynomial, one has

w(η ⊗ ξ) = w(⊕rj=1(η ⊗ ξj))=

∏rj=1 w(η ⊗ ξj) by (9.4.3)

=∏rj=1(1+ u+ vj) by Proposition 9.2.4

=∑r

k=0(1+ u)kσr−k(v1, . . . , vr).

Since σr−k(v1, . . . , vr) = wr−k(ξ), we have shown (9.6.3) when ξ splits into a Whit-ney sum of r line bundles. If this is not the case, Formula (9.6.3) still holds trueby the splitting principle of Proposition 9.6.3.

9.7. Complex flag manifolds

The plan of this section follows that of § 9.5. We shall indicate the slightchanges to get from the real flag manifolds to the complex ones, without repeatingall the proofs.

Let n1, . . . nr be positive integers and let n = n1 + n2 + · · ·nr. By the complexflag manifold FlC(n1, . . . , nr), we mean any smooth manifold diffeomorphic to thehomogeneous space

(9.7.1) FlC(n1, . . . , nr) ≈ U(n)/U(n1)× U(n2)× · · · × U(nr) .

The most usual concrete occurrence of complex flag manifolds are as below.

(1) Nested subspaces. FlC(n1, . . . , nr) is the set of nested complex vector sub-

spaces V1 ⊂ · · · ⊂ Vr ⊂ Cn with dimC Vi =∑ij=1 nj .

(2) Mutually orthogonal subspaces. FlC(n1, . . . , nr) is the set of r-tuples (W1, . . . ,Wr)of complex vector subspaces Cn which are mutually orthogonal (for thestandard Hermitian product on Cn) and satisfy dimWi = ni. The corre-spondence from this definition to Definition (1) associates to (W1, . . . ,Wr)the nested family Vi where Vi is the complex vector space generated byW1 ∪ · · · ∪Wi.

(3) Isospectral Hermitian matrices. Let λ1 > · · · > λr be real numbers. Con-sider the manifold HM(n) of all Hermitian (n × n)-matrices, on which


U(n) acts by conjugation. Then FlC(n1, . . . , nr) occurs as the orbit of thediagonal matrix having entries λi with multiplicity ni.

(9.7.2) FlC(n1, . . . , nr) =R dia

(λ1, . . . , λ1︸︷︷︸

n1

, · · · , λr, . . . , λr︸︷︷︸nr

)R−1 | R ∈ U(n)

.

In other words, FlC(n1, . . . , nr) is here the space of Hermitian (n × n)-matrices with characteristic polynomial equal to

∏ri=1(x− λi)ni . Indeed,

two matrices in HM(n) are in the same U(n)-orbit if and only if theyhave the same characteristic polynomial. The correspondence from thisdefinition to Definition (2) associates, to a matrix M , its eigenspaces forthe various eigenvalues.

Concrete definition (3) is our working definition for FlC(n1, . . . , nr) throughout thissection. Special classes of flag manifolds are given by the complex Grassmannians

Gr(k;Cn) = FlC(k, n− k) ≈ U(n)/U(k)× U(n− k)

of complex k-planes in Cn. This is a closed manifold of dimension

dimGr(k;Cn) = dimU(n)− dimU(k)− dimU(n− k) = 2k(n− k) .For example, Gr(1;Cn) ≈ CPn−1, of dimension 2(n− 1).

Using Definition (3) above, our “concrete Grassmannian” will be

(9.7.3) Gr(k;Cn) =R dia

(1, . . . , 1︸︷︷︸

k

, 0, . . . , 0︸︷︷︸n−k

)R−1 | R ∈ U(n)

.

As, in the real case, we define the complete complex flag manifold

FlC(1, . . . , 1) ≈ U(n)/U(1)× · · · × U(1)

with dimFlC(1, . . . , 1) = dimU(n)− n = n2 − n = n(n− 1).As in § 9.5, we define real functions on the flag manifolds by restriction of the

weighted trace on f : HM(n)→ R defined by

f(M) =

n∑

j=1

j Mjj

where Mij denotes the (i, j)-entry of M .

Proposition 9.7.1. Let FlC(n1, . . . , nr) ⊂ HM(n) be the complex flag man-ifold as presented in (9.5.2). Then, the restriction f : FlC(n1, . . . , nr) → R ofthe weighted trace is a perfect Morse function whose critical points are the diag-onal matrices in FlC(n1, . . . , nr). The index of the critical point dia(x1, . . . , xn)is twice the number of pairs (i, j) with i < j and xi < xj. In consequence,dimFlC(n1, . . . , nr) = 2 dimFl(n1, . . . , nr) and

(9.7.4) Pt(FlC(n1, . . . , nr)) = Pt2(Fl(n1, . . . , nr)) .

Recall that dimFl(n1, . . . , nr) was computed in Corollary 9.5.5 and that thePoincare polynomial Pt(Fl(n1, . . . , nr)) was described in Corollary 9.5.15. Equal-ity (9.7.4) implies the following corollary.

Corollary 9.7.2. The cohomology groups of FlC(n1, . . . , nr) vanish in odddegrees.


Remark 9.7.3. The manifold FlC(n1, . . . , nr) ⊂ HM(n) admits a U(n)-invariantsymplectic form, induced from the non-degenerate symmetric form (X,Y ) 7→ trace (XY )on HM(n) (see [12, Chapter II, Example 1.4]). The weighted trace is the momentmap for the Hamiltonian circle action given by the conjugation by dia(eit, e2it, . . . , enit).The involution τ given on FlC(n1, . . . , nr) by the complex conjugation is anti-symplectic and anti-commutes with the circle action. Its fixed point set is Fl(n1, . . . , nr).Note that f is τ -invariant and the critical point of f and f |Fl(n1, . . . , nr) are thesame. This, together with (9.7.4), is a particular case of a theorem of Duistermaat[45] (see also Remark 9.7.9).

Proof of Proposition 9.7.1. We use the injective homomorphism rij : SU(2)→U(n), introduced in the proof of Proposition 9.7.1, whose image contains the ma-trices

Rijt = rij(

cos t − sin tsin t cos t

)and Rijt = rij

(cos t

√−1 sin t√

−1 sin t cos t

)(t ∈ R) .

Suppose that ∆ ∈ FlC(n1, . . . , nr) is a diagonal matrix. Then, a basis ofT∆FlC(n1, . . . , nr) is represented by the curves

∆ij(t) = Rijt ∆Rij−t and ∆ij(t) = Rijt ∆R

ij−t .

As in the proof of Proposition 9.5.2, this shows that the critical points of f areexactly the diagonal matrices and computes the indices.

As the critical points are all of even index, the function f is a perfect Morsefunction by Lemma 7.6.2. One can also proceed as in the proof of Proposition 9.5.2,using that f is invariant for the action of the diagonal subgroup T of U(n), whichis the torus U(1)× · · · × U(1), and use Theorem 7.6.11.

As in § 9.5, consider the inclusionHM(n) ⊂ HM(n+1) with image the matriceswith vanishing last row and column. Seeing Gr(k;Cn) ⊂ HM(n) as in (9.5.3), thisgives an inclusion Gr(k;Cn) ⊂ Gr(k;Cn+1). The proof of the following lemma isthe same as that of Lemma 9.5.7.

Lemma 9.7.4. The homomorphism Hj(Gr(k;Cn+1))→ Hj(Gr(k;Cn)) inducedby the inclusion is surjective for all j and is an isomorphism for j ≤ 2(n− k).

Tautological bundles. Consider a complex flag manifold FlC(n1, . . . , nr), with n =n1 + · · ·+ nr and the following closed subgroups of U(n)

Bi = U(n1)× · · · × 1 × · · · × U(nr) ⊂ U(n1)× · · · × U(nr) ⊂ U(n) .

Then

Pi = U(n)/Bi →→ U(n)/U(n1)× · · · × U(nr) = FlC(n1, . . . , nr)

is an U(ni)-principal bundle (see p. 318) over FlC(n1, . . . , nr). Its associated com-plex vector bundle of rank ni, i.e. E(ξi) = Pi ×U(ni) C

ni , is called i-th tautologicalvector bundle over FlC(n1, . . . , nr). Being associated to an U(ni)-principal bundle,ξi is endowed with an Hermitian structure and its space of orthonormal framesFra⊥(ξi) is equal to Pi. In the mutually orthogonal subspaces presentation (2) ofFlC(n1, . . . , nr), we see that

E(ξi) = (W1 . . . ,Wr, v) ∈ FlC(n1, . . . , nr)× Cn | v ∈ Wi .Note that ξ1 ⊕ · · · ⊕ ξr is trivial (see § 9.5, p. 318).


The complex vector bundle ξ1 over FlC(k, n−k) = Gr(k;Cn) is called the tauto-logical vector bundle over the complex Grassmannian Gr(k;Cn); it is of (complex)rank k and is denoted by ζ or ζk. The space of Fra⊥(ζk) is the complex Stiefelmanifold Stief(k,Cn) of orthonormal k-frames in Cn.

The inclusion Cn ≈ Cn × 0 → Cn+1 induces an inclusion Gr(k;Cn) →Gr(k;Cn+1) and we may consider the direct limit

Gr(k;C∞) = lim−→n

Gr(k;Cn)

which is a CW-space. The tautological vector bundle ζk is also defined overGr(k;C∞) and induces that over Gr(k;Cn) by the inclusion Gr(k;Cn) → Gr(k;C∞).It is classical that πi(Stief(k,C

n)) = 0 for i < 2(n − k) + 1 (see [181, 25.7]),thus Stief(k,C∞) = Fra(ζk) is contractible. Hence, the U(k)-principal bundleStief(k,C∞) → Gr(k;C∞) is a universal U(k)-principal bundle (see [181, § 19.4])and thus homotopy equivalent to the Milnor universal bundle EU(k) → BU(k).In particular, Gr(k;C∞) has the homotopy type of BU(k). As a consequence, anycomplex vector bundle of rank k over a paracompact space X is induced from ζkby a map X → Gr(k;C∞).

To emphasize the analogy with § 9.5, we introduce the total Chern classesc(ξ) ∈ H2∗(X) of a complex vector bundle ξ of rank k over a space X by

c(ξ) =

k∑

j=1

w2j(ξR) ,

where ξR is the vector bundle ξ seen as a real vector bundle of (real) rank 2k. Thecomponent of c(ξ) in H2j(X) is

(9.7.5) cj(ξ) = w2j(ξR) ∈ H2j(X)

is called the i-th Chern class of ξ.

Theorem 9.7.5. The cohomology ring of BU(k) is GrA-isomorphic to thepolynomial ring

H∗(BU(k)) = H∗(Gr(k;C∞)) ≈ Z2[c1, . . . , ck]

generated by the Chern classes ci = ci(ζk) of the tautological bundle ζk.

Remark 9.7.6. Our Chern classes cj(ξ) are the reduction mod 2 of the integralChern classes (see [153, § 14] or [105, Chapter 16]). That the restriction mod 2of cj(ξ) coincides with w2j(ξR) (whence our definition (9.7.5)) is proven in [181,Theorem 41.8]. Note that, by Theorem 9.7.5, w2j+1(ξR) = 0.

Proof of Theorem 9.7.5. It is the same as that of Theorem 9.5.8, usingProposition 9.7.1. To see that the Chern classes are algebraically independent, weuse the tautological complex line bundle γ over CP∞ and its n-times product γn

over (CP∞)n.

Theorem 9.7.5 together with Lemma 9.7.4 gives the following corollary.

Corollary 9.7.7. The cohomology ring H∗(Gr(k;Cn)) is generated, as a ring,by the Chern classes c1(ζk), . . . , ck(ζk) of the tautological bundle ζk.


Let c(ξj) = 1+ c1(ξj) + · · ·+ cnj(ξj) ∈ H∗(FlC(n1, . . . , nr)) be the Chern class

of the tautological vector bundle ξj . The following theorem is proven in the sameway as Theorem 9.5.14. Actually, replacing Z2[ci(ξj)] by Z[ci(ξj)], the statement istrue for the integral cohomology (as we wrote a minus sign in the last expression).

Theorem 9.7.8. The cohomology algebra H∗(FlC(n1, . . . , nr)) is GrA-isomorphicto the quotient of the polynomial ring

Z2[ci(ξj)] , 1 ≤ i ≤ rj , j = 1, . . . , r

by the ideal generated by the homogeneous components of 1− c(ξ1) · · · c(ξr).

Remark 9.7.9. By Theorems 9.7.8 and 9.5.14, the correspondence ci(ξj) 7→wi(ξj) provides an abstract ring isomorphism

H2∗(FlC(n1, . . . , nr))≈−→ H∗(Fl(n1, . . . , nr)) .

Actually, FlC(n1, . . . , nr) with the complex conjugation is a conjugation space (see§ 10.2). Given Remark 9.7.3, this is established in [87, Theorem 8.3].

We have seen in Proposition 9.2.4 that the first Stiefel-Whitney class classifiesthe real lines bundles. The full analogue for complex line bundles requires coho-mology with Z-coefficients: the first integral Chern class provides an isomorphism

(LC(X),⊗) ≈−→ H2(X ;Z), where LC(X) be the set of isomorphism classes of com-plex lines bundles over a CW-complex X (see [96, pp. 62–63]). But, staying withinthe mod 2 cohomology, one can prove the following result.

Proposition 9.7.10. Let ξ and ξ′ be two complex line bundles over a CW-complex X. Then c1(ξ ⊗ ξ′) = c1(ξ) + c1(ξ

′).

Proof. The argument follows the end of the proof of Proposition 9.2.4. Onehas to replace R× by C× and K by KC = C× × C×. The only thing to prove isthat the composite map

BC× ×BC×P ′

≃// BKC

Bϕ // BC×

corresponding to that of Diagram (9.2.2) satisfies

(9.7.6) H∗(BϕP ′)(v) = v × 1+ 1× v ,where v is the generator of H2(BC×) = H2(BU(1)) = H2(CP∞). The complexconjugation of C× induces an involution τ on BC× corresponding to the conjugationon CP∞, with fixed point RP∞ = BR×. The map BϕP ′ is τ -equivariant. Hence,Equation (9.7.6) follows from (9.2.3), using that the inclusion j : RP∞ → CP∞

satisfies H∗j(v) = u2 (see Proposition 6.1.11).

Finally, the splitting principle results of § 9.6 have their correspondents forcomplex bundles. One uses the inclusion of the diagonal subgroup of U(n)

Γ ≈ U(1)× · · · × U(1) ⊂ U(n) .

The following result is proven in the same way as for Theorem 9.6.1.

Theorem 9.7.11. The GrA-morphism H∗Bα : H∗(BU(n)) → H∗(BΓ) is in-jective and its image is H∗(BΓ)Symn . The complex vector bundle Bα∗ζn inducedfrom the universal bundle ζn splits into a Whitney sum of complex line bundles.


As for Theorem 9.6.2, Theorem 9.7.11 generalizes in the following way for theinclusion

αn1,...,nr: U(n1)× · · · × U(nr)→ U(n) .

Theorem 9.7.12. The map Bαn1,...,nrsatisfies the following properties.

(1) The GrA-morphism

H∗Bαn1,...,nr: H∗(BU(n))→ H∗(BU(n1)× · · · ×BU(nr))

is injective.(2) H∗Bαn1,...,nr

(ci) = ci(ζn1 × · · · × ζnr) for each i ≥ 0. In particular, the

image of H∗Bαn1,...,nris generated by ci(ζn1 × · · · × ζnr

) (i ≥ 0).(3) The induced complex vector bundle Bα∗n1,...,nr

ζn splits into a Whitney sumof complex vector bundles of ranks n1, . . . , nr.

As in § 9.6, we deduce from Theorem 9.7.11 the following proposition (splittingprinciple for complex bundles).

Proposition 9.7.13. Let ξ be a complex vector bundle over a paracompactspace X. Then, there is a map f : Xξ → X such that

(1) H∗f is injective.(2) f∗ξ splits into a Whitney sum of complex line bundles.

As in Proposition 9.6.4, we get an axiomatic characterization of Chern classes.

Proposition 9.7.14. Suppose that c is a correspondence associating, to eachcomplex vector bundle ξ over a paracompact space X, a class c(ξ) ∈ H2∗(X), suchthat

(1) if f : Y → X is a continuous map, then c(f∗ξ) = H∗f(c(ξ)).(2) c(ξ ⊕ ξ′) = c(ξ) c(ξ′).(3) if γ is the tautological complex line bundle over CP∞, then c(γ) = 1+ a,

where 0 6= a ∈ H2(CP∞).

Then c = c, the total Chern class.

Thanks to our definition of Chern classes via Stiefel-Whitney classes, the fol-lowing proposition is a direct consequence of Proposition 9.6.5.

Proposition 9.7.15. Let ξ be a complex vector bundle over a paracompactspace X. Then

(9.7.7) Sq2icj(ξ) =∑

0≤k≤i

(j−i+k−1

k

)ci−k(ξ) cj+k(ξ) .

Remark 9.7.16. As in § 9.5.3, the Schubert calculus may be developed forcomplex Grassmannians. The degrees of (co)homology classes are doubled. TheStiefel-Whitney classes wi are replaced by the Chern classes ci. The Stiefel-Whitneyclasses wi corresponds, in the literature, to the Segre classes.

We finish this section with the complex analogue of Lemma 9.6.7.

Lemma 9.7.17. Let η and ξ be complex vector bundles over a paracompact spaceX. Suppose that ξ is of rank r and that η is a line bundle. Then

(9.7.8) c(η ⊗ ξ) =r∑

k=0

(1 + c1(η))kcr−k(ξ) .

9.8. THE WU FORMULA 343

Proof. The proof is the same as that of Lemma 9.6.7. The use of Proposi-tion 9.2.4 has to be replaced by that of Proposition 9.7.10.

9.8. The Wu formula

9.8.1. Wu’s classes and formula. Let Q be a closed manifold of dimensionn. The map

Hn−k(Q)Sqk

−−→ Hn(Q)〈−,[Q]〉−−−−−→ Z2

is a linear form on Hn−k(Q). By Poincare duality (see Theorem 5.3.12), there is

a unique class vk(Q) ∈ Hk(Q) such that 〈Sqk(a), [Q]〉 = 〈vk(Q) a, [Q]〉 for alla ∈ Hn−k(Q). In other words,

(9.8.1) Sqk(a) = vk(Q) a

for all a ∈ Hn−k(Q). The left hand side of (9.8.1) vanishing if k > n− k, one hasvk(Q) = 0 if k > n/2. The class vi(Q) is the i-th Wu class of Q (for Wu classes ina more general setting, see [123, § 3]). Note that v0(Q) = 1. The total Wu classv(Q) is defined by

v(Q) = 1+ v1(Q) + · · ·+ v[n/2](Q) ∈ H∗(Q) .

As an example the next lemma shows the role of vk(Q) when n = 2k. Let Vbe a Z2-vector space and let B : V × V → V be a bilinear form. A symplectic basisof V for B is a basis a1, . . . , ak, b1 . . . , bk of V such that B(ai, aj) = B(bi, bj) = 0and B(ai, bj) = B(bj , aj) = δij . By convention, the empty basis for V = 0 is alsosymplectic.

Lemma 9.8.1. Let Q be a closed smooth manifold of dimension 2k such that itsWu class vk(Q) vanishes. Then the bilinear form B : Hk(Q)×Hk(Q)→ Z2 givenby B(x, y) 7→ 〈x y, [Q]〉 admits a symplectic basis.

Note that the lemma implies that Hk(Q) has even dimension and thus, byPoincare duality, χ(Q) is even. This can be also deduced from Corollary 5.4.16 and

Theorem 9.4.1 since, by the Wu formula (see below), w2k(TQ) = Sqk(vk(Q)) = 0.

Proof. By definition of the Wu class, vk(Q) = 0 is equivalent to B beingalternate, i.e. B(x, x) = 0 for all x ∈ Hk(Q). By Theorem 5.3.12, B is non-degenerate. We are thus reduced to proving the following classical claim: on aZ2-vector space V of finite dimension, a non-degenerate bilinear form B which isalternate admits a symplectic basis. As B is non-degenerate, there exists a1, b1 ∈ Vsuch that B(a1, b1) = 1. Hence B(b1, a1) = 1 since alternate implies symmetric incharacteristic 2. One has an exact sequence

0→ A→ Vφ−→ Z2 ⊕ Z2 → 0 ,

where φ is the linear map φ(v) = (B(a1, v), B(v, b1)) and A = kerφ. As B is non-degenerate, so is its restriction to A × A. This permits us to prove the claim byinduction on the dimV .

Wu’s formula below relates the Wu class of Q to the Stiefel-Whitney classw(TQ) of the tangent bundle TQ of Q (often called the Stiefel-Whitney class of Q).

Theorem 9.8.2 (Wu’s formula). For any smooth closed manifold Q, one has

w(TQ) = Sq(v(Q)) .


Wu’s formula was proved byWu Wen-tsun in 1950 [213] by direct computationsin H∗(Q×Q). We follow below the proof of Milnor-Stasheff [153, Theorem 11.14](for a proof using equivariant cohomology, see Remark 9.8.21). The computationsare lightened by the use of the slant product

(9.8.2) H∗(X × Y )⊗H∗(Y )/−→ H∗(X)

(actually: Hk(X×Y )⊗Hm(Y )/−→ Hk−m(X)) which is defined as follows. Consider

the map H∗(X)⊗H∗(Y )⊗H∗(Y )→ H∗(X) defined by the correspondence a⊗ b⊗β 7→ 〈b, β〉a, using the Kronecker pairing 〈 , 〉. As H∗(X×Y ) ≈ H∗(X)⊗H∗(Y ) bythe Kunneth theorem (we assume that Y is of finite cohomology type), this givesthe linear map (9.8.2). The slant product is characterized by the equation

(a× b)/β = 〈b, β〉 afor all a ∈ H∗(X), b ∈ H∗(Y ) and β ∈ H∗(Y ). It is also a morphism of H∗(X)-modules, i.e.

(9.8.3) [(u× 1) c]/β = u (c/β)

for all u ∈ H∗(X), c ∈ H∗(X × Y ) and β ∈ H∗(Y ).

Proof of Wu’s formula. Consider Q as the diagonal submanifold of M =Q × Q, with normal bundle ν = ν(Q,M). By Lemma 5.4.17, TQ is isomorphicto ν.

A Riemannian metric provides a smooth bundle pair (D(ν), S(ν)) with fiber(Dr, Sr−1) and there is a diffeomorphism from D(ν) to a closed tubular neighbour-hood W of Q in M . One has the diagram

Q∆ //

i

M==j

⑤⑤⑤⑤⑤⑤⑤

Wπ

YY

where π is the bundle projection and the other maps are inclusions. By excision,

H∗(M,M −Q) oo j∗

≈H∗(W,BdW ) ≈ H∗(D(ν), S(ν)) .

Hence, the Thom class of ν may be seen as an element U ∈ Hq(W,BdW ) satisfyingU = j∗(U ′) for a unique U ′ ∈ Hq(M,M − Q). Let U ′′ ∈ Hq(M) be the image ofU ′ under the restriction homomorphisms Hq(M,M −Q)→ Hq(M).

By definition of the Stiefel-Whitney class w = w(TQ) = w(ν), one has

π∗(w) U = SqU .

One has ∆∗(1×w) = 1 w = w, whence j∗(1×w) = π∗(w). Hence, the equation(1× w) U ′ = SqU ′ holds true in H∗(M,M −Q) which, in H∗(M), implies

(9.8.4) (1× w) U ′′ = SqU ′′ .

Without loss of generality, we may assume thatQ is connected. LetA = a1, a2, . . . and B = b1, b2, . . . be additive bases of H∗(Q) which are Poincare dual, i.e.〈ai bj , [Q]〉 = δij . We suppose that a0 = 1. By Lemma 5.4.2 and (5.4.1), onehas

U ′′ =∑

i

ai × bi


and thereforeU ′′/[Q] = (1× b0)/[Q] = 1 .

Applying this together with Equations (9.8.4) and (9.8.3) gives

(9.8.5) SqU ′′/[Q] = [(1× w) U ′′]/[Q] = w (U ′′/[Q]) = w .

We now express the Wu class v = v(M) in the A-basis: v =∑

i λiai. Then,〈v bj , [Q]〉 = λj , which implies that

(9.8.6) v =∑

i

〈v bi, [Q]〉 ai =∑

i

〈Sq bi, [Q]〉ai, .

Hence, using (9.8.5), we get

Sq v =∑i〈Sq bi, [Q]〉Sq ai

=∑i

(Sq ai × Sq bi

)/[Q]

= SqU ′′/[Q]

= w .

The remainder of this subsection is devoted to general corollaries of Wu’s for-mula. The first one says that the Stiefel-Whitney class w(TQ) depends only on thehomology type of Q.

Corollary 9.8.3. Let f : Q′ → Q be a continuous map between smooth closedmanifolds Q and Q′ of the same dimension. Suppose that H∗f : H∗(Q)→ H∗(Q′)is surjective. Then H∗f(w(TQ)) = w(TQ′).

Proof. By Kronecker duality, H∗f is injective and thus π0f is injective. Theconnected components of Q out of the image of f play no role, so one may assumethat π0f is a bijection. This implies that H∗f([Q

′]) = [Q].Let v′ = H∗f(v(Q)). For b ∈ H∗(Q), one has

〈v(Q) b, [Q]〉 = 〈v(Q) b,H∗f [Q′]〉

= 〈H∗f(v(Q) b), [Q′]〉= 〈v′ H∗f(b), [Q′]〉 .

On the other hand

〈v(Q) b, [Q]〉 = 〈Sq b,H∗f [Q′]〉= 〈Sq(H∗f(b)), [Q′]〉= 〈v(Q′) H∗f(b), [Q′]〉 .

Therefore, the equality

〈v′ H∗f(b), [Q′]〉 = 〈v(Q′) H∗f(b), [Q′]〉is valid for all b ∈ H∗(Q). As H∗f is surjective, Theorem 5.3.12, this implies thatv′ = V (Q′), so H∗f(v(Q)) = v(Q′). By Wu’s formula,

w(TQ′) = Sq v(Q′) = SqH∗f(v(Q)) = H∗f(Sq v(Q)) = H∗f(w(TQ)) .

Corollary 9.8.4. Let f : Q → Q be a continuous map of degree one. ThenH∗f(w(TQ)) = w(TQ).

Proof. By Proposition 5.2.8, H∗f is surjective and then H∗f is injective byKronecker duality. As H∗(Q) is a finite dimensional vector space, this implies thatH∗f (and H∗f) are bijective. The results then follows from Corollary 9.8.3.


9.8.2. Orientability and spin structures. A smooth manifold is orientableif its tangent bundle is orientable. The following corollary generalizes Proposi-tion 4.2.3.

Corollary 9.8.5. Let Q be a smooth closed n-dimensional manifold. Then Qis orientable if and only if Sq1 : Hn−1(Q)→ Hn(Q) vanishes.

Proof. By Proposition 9.4.4, Q is orientable if and only if w1(TQ) = 0 which,by Wu’s formula, is equivalent to v1(Q) = 0. By the definition of v1(Q), its vanish-ing is equivalent to Sq1 : Hn−1(Q)→ Hn(Q) being zero.

The same argument, using Proposition 9.4.7, implies the following result.

Corollary 9.8.6. Let Q be a smooth closed n-dimensional manifold which isorientable. Then TQ admits a spin structure if and only if Sq2 : Hn−2(Q)→ Hn(Q)vanishes.

Example 9.8.7. A closed manifold M such that H∗(M) is GrA-isomorphic toH∗(RPn) is orientable if and only if n is odd. Indeed, let 0 6= a ∈ H1(M) = Z2.By the Cartan formula, Sq1(an−1) = 0 if and only if n is odd. As Hn−1(M) isgenerated by an−1, the assertion follows from Corollary 9.8.5. A similar argument,using Corollary 9.8.6, proves that a closed manifold M such that H∗(M) is GrA-isomorphic to H∗(CPn) admits a spin structure if and only if n is odd.

In the particular case n = 4, Corollary 9.8.6 gives the following result.

Corollary 9.8.8. Let Q be a smooth connected closed 4-dimensional manifoldwhich is orientable. Then,

(1) a a = w2(TQ) a for all a ∈ H2(Q).(2) TQ admits a spin structure if and only if the cup-square map H2(Q) →

H4(Q) vanishes.(3) w4(TQ) = w2(TQ) w2(TQ).

Point (2) is the analogue of Proposition 4.2.3 for surfaces. In particular, TCP 2

does not admit a spin structure.

Proof. If w1(TQ) = 0, then w2(TQ) = v2(Q) by Wu’s formula. Hence,a a = Sq2(a) = v2(Q) a = w2(TQ) a for all a ∈ H2(Q), which proves (a).Point (b) thus follows from Corollary 9.8.6. For Point (c), we use Wu’s formulaagain:

w4(TQ) = Sq2(v2(Q)) = Sq2(w2(TQ)) = w2(TQ) w2(TQ) .

Corollary 9.8.9. Let Q be a smooth closed manifold of dimension n ≤ 7. IfTQ admits a spin structure, then w(TQ) = 1.

Proof. The proposition is obvious for n ≤ 2. Otherwise, by Proposition 9.4.6,the existence of a spin structure implies that the restriction of TQ over the 3-skeleton of a triangulation of Q is trivial. Hence w3(TM) also vanishes which, byWu’s formula, implies that vi(M) vanishes for i ≤ 3. As n ≤ 7, this implies thatv(Q) = 1 and thus w(TQ) = Sq v(Q) = 1.

An interesting example is given by the projective spaces.


Proposition 9.8.10. Let 0 6= a ∈ H1(RPn). The Stiefel-Whitney class of thetangent space of RPn is

w(TRPn) = (1+ a)n+1

and the Wu class of RPn is

v(RPn) =

[n/2]∑

i=0

(n−ii

)ai .

Here are a few examples.

n v(RPn) w(TRPn)

2 1+ a 1+ a+ a2

3 1 14 1+ a+ a2 1+ a+ a5

5 1+ a2 1+ a2 + a4

6 1+ a+ a3 1+ a+ a2 + a3 + a4 + a5 + a6

7 1 1

Remark 9.8.11. The formulae of Proposition 9.8.10 imply the following.

(1) RPn is orientable if and only if n is odd (this is not a surprise!). Moregenerally, w2i+1(TRP 2k+1) = 0.

(2) TRPn admits a spin structure if and only if n ≡ 3 mod 4. In this case,there are two spin structures, sinceH1(RPn) = Z2. For a discussion aboutthese two structures for RP 3 ≈ SO(3), see [130, Example 2.5, p. 87].

(3) w(TRPn) = 1 if and only if n = 2k − 1. But TRPn is trivial if and onlyif n = 1, 3, 7 by Adams’ Theorem [2, p. 21].

Proof of Proposition 9.8.10. The two formulae will be proved separately.Checking Wu’s formula is left as an exercise. By (8.2.2),

vi(RPn) an−i = Sqian−i =

(n−ii

)an =

(n−ii

)ai an−i .

which proves that vi(RPn) =(n−ii

)ai. This proves the formula for the Wu class.

As for the Stiefel-Whitney class, the idea is the following. Recall that RPn =Gr(1;Rn+1) = Fl(1, n). Write γ = ξ1 and γ⊥ = ξ2 for the tautological bundles.Then,

(9.8.7) TRPn ≈ hom(γ, γ⊥)

(see [153, Lemma 4.4] for a proof). But γ ⊕ γ⊥ is the trivial bundle ηn+1 of rankn+ 1. Adding to both side of (9.8.7) the bundle hom(γ, γ) ≈ η1, we get

TRPn ⊕ η1 ≈ hom(γ, ηn+1) .

The latter is the Whitney sum of (n + 1) copies of γ∗ = hom(γ, η1). But γ∗ ≈ γ,using a Euclidean metric on γ. For details (see [153, Theorem 4.5]). Hence, theformula for w(TRPn) follows from (9.4.3).

Remark 9.8.12. The argument of the proof of Proposition 9.8.10 essentiallyworks for computing the Chern class of the tangent bundle to TCPn (which is acomplex vector bundle). The slight difference is that γ∗ = hom(γ, η1) is not, ascomplex vector bundle, isomorphic to γ but to the conjugate bundle γ (the complexstructure on each fiber is the conjugate of that of γ (see [153, pp. 169–170]). But this


does not alter our Chern classes which are defined mod 2: ci(TCPn) = w2i(TCPn).Thus

c(TCPn) = (1+ a)n+1 and v(CPn) =

[n/2]∑

i=0

(n−ii

)ai

where 0 6= a ∈ H2(CPn). The first formula holds as well for the integral Chern classwith a suitable choice of a generator of H2(CPn;Z) (see [153, Theorem 14.10]).

9.8.3. Applications to 3-manifolds. Wu’s formula has two important con-sequences for closed 3-dimensional manifolds. The first one is the following.

Proposition 9.8.13. Let Q be a smooth closed 3-dimensional manifold which isorientable. Then TQ is a trivial vector bundle (in other words: Q is parallelizable).

Proof. For any smooth closed manifold, one has w1(TQ) = v1(Q) by Wu’sformula. Thus, v1(Q) = 0 if Q is orientable. In dimension 3, this implies thatv(Q) = 1 and, by Wu’s formula again, w(TQ) = 1. The result then follows fromProposition 9.4.6.

The second application is Postnikov’s characterization of the cohomology ringof a closed connected 3-dimensional manifold [164]. Let M be such a manifold.Consider the symmetric trilinear form πM : H1(M)×H1(M)×H1(M)→ Z2 definedby

πM(a, b, c) = 〈a b c, [M ]〉 .The first observation is that πM determines the ring structure of H∗(M).

Lemma 9.8.14. LetM and M be two closed connected 3-dimensional manifolds.Suppose that there exists an isomorphism h1 : H1(M)→ H1(M) such that

πM(h1(a), h1(b), h1(c)) = πM (a, b, c) .

Then, h1 extends to a GrA-isomorphism h∗ : H∗(M)→ H∗(M).

Proof. LetA = a1, . . . , am be a Z2-basis ofH1(M). The set A = a1, . . . , am

where ai = h1(ai) is then a Z2-basis of H1(M). Let B = b1, . . . , bm andB = b1, . . . , bm be the bases of H2(M) and H2(M) which are Poincare dualto A and A, i.e. the equations

(9.8.8) 〈ai bj , [M ]〉 = δij and 〈ai bj , [M ]〉 = δij

are satisfied for all i, j. Let h2 : H2(M) → H2(M) be the isomorphism such thath2(bi) = bi and let h3 be the unique isomorphism from H3(M) to H3(M). Thisproduces a GrV-isomorphism h∗ : H∗(M)→ H∗(M). To prove that h∗ is a GrA-morphism, write ai aj =

∑ml=1 λ

lijbl and, using (9.8.8), note that

πM(ai, aj , ak) = 〈ai aj ak, [M ]〉 =m∑

l=1

λlij〈bl ak, [M ]〉 = λkij .

Therefore,

〈h2(ai aj) ak, [M ]〉 =∑m

l=1 πM (ai, aj , al)〈h2(bl) ak, [M ]〉=

∑ml=1 πM (ai, aj , al)〈bl ak, [M ]〉

= πM (ai, aj , ak)


and

〈h1(ai) h1(aj) ak, [M ]〉 = 〈h1(ai) h1(aj) h1(ak), [M ]〉= πM (h1(ai), h

1(aj), h1(ak)) .

Since πM (h1(ai), h1(aj), h

1(ak)) = πM(ai, aj, ak), this proves that h2(ai aj) =h1(ai) h1(aj). On the other hand, h3 formally satisfies 〈h3(u), [M ]〉 = 〈u, [M ]〉.Hence, the equality h1(ai) h2(bj) = h3(ai bj) follows from (9.8.8). We havethus established that h∗ is a GrA-morphism.

The trilinear form πM is linked to the Wu class v(M) ∈ H1(M).

Lemma 9.8.15. Let M be a closed connected 3-dimensional manifold. Then,the Wu class v = v1(M) satisfies

(9.8.9) πM(v, b, c) = πM(b, b, c) + πM(b, c, c)

for all b, c ∈ H1(M).

Proof. This comes from that

v1(M) (b c) = Sq1(b c) = Sq1(b) c + b Sq1(c) = b b c+b c c .

The following “realizability result” is due to Postnikov [164].

Proposition 9.8.16. Let (V, π) a symmetric trilinear form, with V a finitedimensional Z2-vector space. Let v ∈ V satisfying (9.8.9). Then, there exists aclosed connected 3-manifold M with an isometry (H1(M), πM) ≈ (V, π), sendingv1(M) onto v.

Proof (indications). The full proof may may be found in [164]. When Mis orientable, the form πM is alternating since the left hand side of (9.8.9) vanishes

(v = w1(TM) = 0). Hence, πM ∈∧3H1(M). An alternating form π ∈ ∧3 V

may be lifted to π ∈ ∧3V , where V is a free abelian group with V ⊗ Z2 ≈ V .

In [187], D. Sullivan constructed a closed connected orientable 3-manifold M with

(H1(M ;Z), πM ) ≈ (V , π), which thus proves Proposition 9.8.16 in the orientablecase.

9.8.4. The universal class for double points. The material of this sectionis essentially a rewriting in our language of results of A. Haefliger [78]. Let Mbe a smooth closed manifold of dimension m. Let G = 1, τ act on M ×M byτ(x, y) = (y, x), with fixed point set (M×M)G = ∆M , the diagonal submanifold of

M×M . The diagonal inclusion δ : M →M×M induces a diffeomorphism δ : M≈−→

∆M . For N > 1, SN ×G (M ×M) is a closed manifold containing RPN ×∆M asa closed submanifold of codimension m. Let PDG,N (M) = PD(RPN × ∆M ) ∈Hm(SN ×G (M ×M)), the Poincare dual of RPN ×∆M (see § 5.4.1). If N is bigenough,

HmG (M ×M) ≈ Hm(S∞ ×G (M ×M))→ Hm(SN ×G (M ×M))

is an isomorphism. Therefore, there is a unique class PDG(M) ∈ HmG (M × M)

whose image in Hm(SN ×G (M ×M)) is equal to PDG,N (∆M ).The class PDG(M) is called the universal class of double points for continuous

maps into M , a terminology justified by Lemma 9.8.17 below. For a space X ,denote by j : X0 → (X×X) the inclusion of X0 = (X×X)−∆X into (X×X). As


G acts freely on X0, the quotient space X∗ = X0/G, called the reduced symmetricsquare of X , has the homotopy type of X0

G by Lemma 7.1.4.

The diffeomorphism δ : M≈−→ ∆M is covered by a bundle isomorphism δ : TM

≈−→ν(M ×M,∆M ) (see Lemma 5.4.17), which intertwines τ with the antipodal invo-

lution on TM . Via δ, the sphere bundle T 1M becomes G-diffeomorphic with theboundary BdW of a G-invariant tubular neighbourhood of ∆M in M ×M . Thus,

(9.8.10) W ∗ ≈ (W −∆M )G ≈ (BdW )G ≈ (T 1M)G ≈ (T 1M)/G .

As j is G-equivariant, it induces H∗Gj : H∗G(X×X)→ H∗G(X

0). Let f : Q→Mbe a continuous map. Consider the homomorphism

Ψ: HmG (M ×M)

H∗G(f×f)−−−−−−→ HmG (Q×Q)

H∗Gj−−−→ HmG (Q0)

≈−→ Hm(Q∗) .

We denote by Ψ : HmG (M×M)→ Hm(T 1Q/G) the post-composition of Ψ with the

homomorphism Hm(Q∗)→ Hm(W ∗) ≈ Hm(T 1N/G). Define

OfG = Ψ(PDG(M)) ∈ HmG (Q∗) and OfG = Ψ(PDG(M)) ∈ Hm

G ((T 1Q)/G) .

Lemma 9.8.17. Let f : Q→M be a continuous map between closed manifolds.Then

(1) if f is homotopic to an embedding, then OfG = 0.

(2) if f is homotopic to an immersion, then OfG = 0.

Proof. The classes OfG and OfG depend only on the homotopy class of f . If fis injective, then (f × f)(Q0) ⊂M0 and thus (f × f)G(Q0

G) ⊂M0G = (M ×M)G −

(∆M )G. By Lemma 5.4.2, PDG(M) has image zero in HmG (M0), which proves (1)

(this does not use that Q is a manifold). Suppose that f is an immersion, so f islocally injective. As Q is compact, there is a G-invariant tubular neighbourhoodWof ∆Q in Q×Q such that (f × f)(W 0) ⊂M0. We deduce (2) as above for (1).

In order to get applications of Lemma 9.8.17, we now express PDG,N(M) withinthe description of Hm

G (M ×M) given by Proposition 8.3.3, which can be rephrasedas follows. There is a GrA[u]-isomorphism from H∗G(M ×M) to (Z2[u]⊗D) ⊕ N ,where D is the Z2-vector space generated by x × x | x ∈ Hk(M), k ≥ 0 so thatρ : H∗G(M ×M) → H∗(M ×M)G sends the elements of D ⊕ N (elements of u-degree 0) isomorphically to H∗(M ×M)G. The Z2-vector space N is generated byx× y + y × x | x, y ∈ H∗(M) and coincides with the ideal Ann (u).

Proposition 9.8.18. Using the isomorphism HmG (M×M) ≈ (Z2[u]⊗D) ⊕ N ,

we have

(1) PDG(M) ≡∑[m/2]k=0 um−2k(vk(M)× vk(M)) mod N , where vk(M) is the

k-th Wu class of M .

(2) ρ(PDG(M)) = PD(∆M ), the Poincare dual of ∆M in M ×M .

Example 9.8.19. Let M = RP 2. One has H∗(M) = Z2[a]/(a3) and v(M) =

1 + a. Then, PDG(M) ≡ u2 + a × a mod N and, according to Equation (5.4.1),ρ(PDG(M)) = PD(∆M ) = 1× a2 + a× a+ a2 × 1. Therefore,

PDG(RP2) = u2 + a× a+N(1× a2) .


Proof. It is enough to prove (1) for PDG,N (M) with N big enough. Leti : Q→ P be the inclusion of a closed manifold Q into a compact manifold P . Forx ∈ Hj(P ), one has

(9.8.11) 〈x PD(Q), [P ]〉 = 〈x,PD(Q) [P ]〉 = 〈x,H∗i([Q])〉 = 〈H∗i(x), [Q]〉 .We shall apply (9.8.11) to Q = RPN ×M and P = SN ×G (M ×M) and x =uN−m+2i(a× a) ∈ HN+m(P ), where a ∈ Hm−i(M). One has(9.8.12)〈H∗i(x), [Q]〉 = 〈uN−m+2iH∗i(a× a), [Q]〉

= 〈uN−m+2i∑m−i

j=0 um−i−j Sqj(a), [Q]〉 by definition of Sq(a)

= 〈uNSqi(a), [Q]〉 only non-zero term

= 〈Sqi(a), [M ]〉 .For y, z ∈ Hj(M), we have (y × y) + (z × z) ≡ (y + z) × (y + z) mod N and

uN = 0. Therefore, for k > 0, uk PDG,N(Q) admits an expression of the form

uk PDG,N(Q) = uk∑[m/2]

j=0 um−2j(yj × yj) with yj ∈ Hj(M). Hence,

(9.8.13)

〈x PDG,N(Q), [P ]〉 = 〈uN−m+2i(a× a) ∑[m/2]j=0 um−2j(yj × yj), [P ]〉

= 〈uN (a× a) (yi × yi), [P ]〉= 〈uN(a yi)× (a yi), [P ]〉= 〈a yi, [M ]〉

Using (9.8.11), Formulae (9.8.12) and (9.8.13) imply that yi = vi(M) for all i =0 . . . , [m/2]. This proves (1).

For Point (2) we must prove that ρN(PDG,N) = PD(M) where ρN is inducedby the fiber inclusion M ×M → SN × (M ×M)→ P . But this map is transverseto RPN ×∆M . Point (2) thus comes from Proposition 5.4.7.

Proposition 9.8.18 enables us to compute the image of PDG(M) under thehomomorphism

H∗G(M ×M)r // H∗G((M ×M)G)

≈ // H∗(M)[u]ev1 // H∗(M)

Corollary 9.8.20. ev1 r(PDG(M)) = w(TM), the total Stiefel-Whitneyclass of the tangent bundle TM .

Proof.

ev1 r(PDG(M)) =∑[m/2]k=0 ev1r

(vk(M)× vk(M)

)by Proposition 9.8.18

=∑[m/2]k=0 Sq(vk(M)) by (8.3.5)

= Sq(v(M))

= w(TM) by the Wu formula.

Remark 9.8.21. The formula of Corollary 9.8.20 may be proven directly inthe following way. By Lemma 5.4.4 and the considerations before (9.8.10), one hasr(PDG(M)) = eG(TM), where G acts on TM by the antipodal action on eachfiber. This equivariant Euler class satisfies ev1(eG(TM)) = w(TM) (see (10.4.7)


in § 10.4), which proves Corollary 9.8.20. Moreover, using the proof of Corol-lary 9.8.20, with the last line removed, gives a new proof of the Wu formula.

Lemma 9.8.22. Let Q be a closed manifold of dimension p. There is a commu-tative diagram

0 // H∗−pG (∆Q)GysG //

H∗G δ≈

H∗G(Q ×Q) //

r

H∗(Q∗)

// 0

0 // H∗−pG (Q)Φ // H∗G(Q) // H∗((T 1Q)/G) // 0

where the rows are exact sequences. Here, Φ(a) = a eG(TQ), where G acts onTQ via the antipodal involution.

Proof. The diagram comes from Proposition 5.4.10 applied, for N big, to thepair (SN ×G (Q ×Q),RPN ×∆Q). The long diagram of Proposition 5.4.10 splitsinto the above diagram because Φ is injective (see Proposition 7.5.14). The bottomline is the Gysin sequence for the sphere bundle (T 1Q)G → QG.

The above results permit us to express an obstruction to embedding in termsof the dual Stiefel-Whitney classes. Let f : Q → M be a smooth map. Definewf = w(TQ) H∗f(w(TM)) ∈ H∗(Q) where w(TQ) is the dual Stiefel-Whitneyclasses of TQ (see p. 325).

Proposition 9.8.23. Let f : Qq →Mm be a smooth map between closed man-ifolds.

(1) If f is homotopic to a smooth immersion, then wfk = 0 for k > m− q.(2) If f is homotopic to a smooth embedding, then

(wfm−q × 1) PD(∆Q) = H∗(f × f)(PD(∆M ))

in Hm(Q×Q).

Proof. We shall argue with the help of the diagram

(9.8.14)

Hm−pG (∆Q)

GysG

H∗G δ

≈// Hm−p

G (Q)ev1 //

−eG(TQ)

H∗(Q)

−w(TQ)

HmG (Q×Q)

r // HmG (Q)

ev1 // H∗(Q)

The left square comes from Lemma 9.8.22 and is thus commutative. So is the rightsquare by Remark 9.8.21.

If f is homotopic to a smooth immersion, then OfG = 0 by Lemma 9.8.17. By

Lemma 9.8.22, this implies that there exists b ∈ Hm−qG (Q) such that the equations

ev1rH∗G(f × f)(PDG(M)) = ev1(b eG(TQ)) = ev1(b) w(TQ)

hold true in H∗(Q). But

ev1 rH∗G(f × f)(PDG(M)) = H∗f ev1r(PDG(M))

= H∗f(w(TM)) by Corollary 9.8.20.

Hence

(9.8.15) ev1(b) w(TQ) = H∗f(w(TM)) .


Since w(TQ) w(TQ) = 1, multiplying both sides of (9.8.15) by w(TQ) gives

(9.8.16) ev1(b) = H∗f(w(TM)) w(TQ) = wf .

As b is of degree m− q, Equation (9.8.16) implies that wfk = 0 for k > m− q. Wehave thus proven (1). Also, (9.8.16) is equivalent to

(9.8.17) b =

m−q∑

i=0

wfi um−q−i .

To prove Point (2), we use the diagram

(9.8.18)

Hm−pG (Q) oo H

∗G δ

≈

ρ

Hm−pG (∆Q)

GysG //

ρ

HmG (Q×Q)

ρ

Hm−p(Q) oo H

∗δ

≈Hm−p(∆Q)

Gys // Hm(Q×Q)

The left square is obviously commutative and the right square is so by Propo-sition 5.4.11, since the fiber inclusion Q × Q → SN ×G (Q × Q) is transverse

to RPN × ∆Q. If f is homotopic to a smooth embedding, then OfG = 0 byLemma 9.8.17. Also, (1) holds and, by the above and Diagram (9.8.14), one has

(9.8.19) ρH∗G(f × f)(PDG(M)) = ρGysG (H∗Gδ)−1(b)

By Point (2) of Proposition 9.8.18, one has(9.8.20)

ρH∗G(f × f)(PDG(M)) = H∗(f × f)ρ(PDG(M)) = H∗(f × f)(PD(∆M )) .

Let ı : ∆Q → Q ×Q be the inclusion and pr2 : Q × Q → Q be the projection ontothe first factor. Then pr1ı δ = idQ. Hence, any a ∈ H∗(Q) satisfies

(9.8.21) a = H∗δH∗ıH∗pr1(a) = H∗δH∗ı(a× 1) .

Hence,(9.8.22)ρGysG(H

∗Gδ)−1(b) = Gys(H∗δ)−1ρ (b) by commutativity of (9.8.18)

= Gys(H∗δ)−1(wfm−q) by (9.8.17)

= GysH∗ı(wfm−q × 1) by (9.8.21)

= (wfm−q × 1) PD(∆Q) by Lemma 5.4.8

Combining (9.8.19), (9.8.20) and (9.8.22) provides the proof of Point (2).

Corollary 9.8.24. Let Q be a closed manifold of dimension q.

(1) If Q may be immersed in Rm, then wk(TQ) = 0 for k > m− q.(2) If Q may be embedded in Rm, then wm−q(TQ) = 0.

Point (1) was already proven in Proposition 9.5.22.

Proof. Let f0 : Q → Rm be a smooth map. Composing with the inclusionRm → Sm = Rm ∪ ∞ gives a smooth map f : Q→ Sm, homotopic to a constantmap. Then wf = w(TQ) and H∗(f × f) = 0. Corollary 9.8.24 thus follows fromProposition 9.8.23.


Examples 9.8.25. 1. LetQ be a closed non-orientable surface. Then, w1(TQ) =w1(TQ) 6= 0. Therefore, Q cannot be embedded in R3. Note that M can be em-bedded in R4 by Whitney’s theorem and immersed in R3 using a connected sum ofBoy’s surface, the latter being an immersion of RP 3.

2. Let Q be a closed 4-dimensional orientable manifold which is not spin (forinstance CP 2). Then, w2(TQ) = w2(TQ) 6= 0. Therefore, Q cannot be embed-ded in R6. Note that CP 2 embeds in R7. Indeed, CP 2 is diffeomorphic to thespace FlC(1, 2) of Hermitian (3 × 3)-matrices with characteristic polynomial equalto x2(x − 1) (see (3) on p. 337). The vector space of Hermitian (3 × 3)-matriceswith trace 1 is isomorphic to R8 and each radius intersects FlC(1, 2) at most once.This gives an embedding of CP 2 in S7 = R7 ∪ ∞ and thus in R7.

3. The quaternionic projective plane Q = HP 2 has Wu class v4(Q) 6= 0. Hence,w4(TQ) = w4(TQ) 6= 0. Therefore, HP 2 cannot be embedded in R12. In thesame way, the octonionic projective plane OP 2 of dimension 16 cannot be embed-ded in R24. Improving the technique explained in the previous example producesembeddings HP 2 → R13 and OP 2 → R25 (see [135, § 3]).

9.9. Thom’s theorems

This section is a survey of some results of Thom’s important paper [191], which,amongst other things, was the foundation of cobordism theory. Some proofs arealmost complete and others are just sketched.

9.9.1. Representing homology classes by manifolds.

Theorem 9.9.1. Let X be a topological space and α ∈ Hk(X). Then, thereexists a closed smooth manifoldM of dimension k and a continuous map f : M → Xsuch that H∗f([M ]) = α.

This theorem is due to Thom [191, Theorem III.2]. The result is wrong forintegral cohomology (see [191, Theorem III.9]). This section is devoted to the proofof Theorem 9.9.1. We start with some preliminaries.

Let ξ be a vector bundle of rank r over a paracompact space Y . Let (D(ξ), S(ξ))be the pair of the disk and sphere bundles associated to ξ via a Euclidean structure.The Thom space T (ξ) of ξ is defined by

T (ξ) = D(ξ)/S(ξ) .

The homeomorphism class of T (ξ) does not depend on the choice of the Euclideanstructure (see p. 157). Also, S(ξ) has a collar neighborhood in D(ξ). Hence, byLemma 3.1.39, the pair (D(ξ), S(ξ)) is well cofibrant. Using Proposition 3.1.45together with the Thom isomorphism theorem provides the following isomorphisms

Hk(B)≈−→ Hk+r(D(ξ), S(ξ))

≈−→ Hk+r(T (ξ)) .

We now specialize to ξ = ζr,N , the tautological vector bundle over the Grass-mannian Gr(r;RN ) (N ≤ ∞). (The Thom space T (ζr,∞) is also called MO(r)in the literature; it is the r-th space of the Thom spectrum MO). We get someinformation on the cohomology ring H∗(T (ζr,N )) using the Gysin exact sequenceof the sphere bundle q : S(ζr,N )→ Gr(r;RN ), whose Euler class is wr(9.9.1)

Hi−r(Gr(r;RN ))wr−−−→ Hi(Gr(r;RN ))

H∗q−−−→ Hi(S(ζr,N ))→ Hi−r+1(Gr(r;RN ))

wr−−−→ · · ·

9.9. THOM’S THEOREMS 355

together with the exact sequence of Corollary 3.1.48 for the pair (D(ζr,N ), S(ζr,N ))(using that D(ζr,N ∼= Gr(r;RN ))

(9.9.2) Hi(T (ζr,N ))→ Hi(Gr(r;RN ))H∗q−−−→ Hi(S(ζr,N ))→ Hi+1(T (ζr,N )) .

For N = ∞, − wr is injective by Theorem 9.5.8. Together with Lemma 9.5.7,sequences (9.9.1) and (9.9.2) gives the following lemma.

Lemma 9.9.2. (1) The GrA-morphism

H∗p : H∗(T (ζr,∞))→ H∗(Gr(r;R∞))

is injective and its image in positive degrees is the ideal generated by wr.In particular, H∗p(U) = wr.

(2) The GrA-morphism Hi(T (ζr,∞))→ Hi(T (ζr,N )) generated by the inclu-sion is bijective for N ≥ N − r − 1.

(Note that the equation H∗p(U) = wr is consistent with (4.7.21)). Considerthe diagram

(9.9.3)

Gr(r;R∞) oo g

fwr

&&

p

(RP∞)r

fa×···×a

T (ζr,∞)

fU // Kr

in which the following notations are used. If X is a CW-complex and y ∈ Hr(X),then fy : X → Kr = K(Z2, r) denotes a map representing y. Then wr ∈ Hr(Gr(r;R∞))is the r-th Stiefel-Whitney class, U ∈ Hr(Tζr,∞) is the Thom class and 0 6= a ∈H1(RP∞). The map g classifies the r-th product of the tautological line bundleover RP∞ and p : Gr(r;R∞) ∼= D(ζr,∞) → T (ζr,∞) is the quotient map. Dia-gram (9.9.3) is homotopy commutative. The commutativity of the lower trianglecomes from the already mentioned equation H∗p(U) = e(ζr,∞) = wr. For the uppertriangle (see e.g. the proof of Theorem 9.5.8).

Applying the cohomology functor to Diagram (9.9.3) provides the commutativediagram

(9.9.4)

H∗(Gr(r;R∞)) //H∗g //

hhH∗fwr

OO

H∗pOO

H∗((RP∞)r)OOH∗fa×···×a

H∗(T (ζr,∞)) oo H∗fU

H∗(Kr)

.

By Corollary 8.5.9, H∗fa×···×a : Hi(Kr)→ Hi((RP∞)r) is injective for i ≤ 2r.

Therefore, H∗fU : Hi(Kr)→ Hi(T (ζr,∞)) is also injective for i ≤ 2r. As indicatedin Diagram (9.9.4), H∗p and H∗g are injective. The injectivity of H∗p was provenin Lemma 9.9.2 and that of H∗g was established in the proof of Theorem 9.5.8 orin Theorem 9.6.1 and its proof.

The map H∗fU is of course not surjective. By Lemma 9.9.2, dimHr+j(Tζr,∞)is the number of partitions of j while, for i < r, dimHr+j(Kr) is equal, byLemma 8.5.14, to the number of partitions of j into integers of the form 2i− 1. LetD(j) be the number of partitions of j into integers with none of them of the form


2i − 1. For each ω ∈ D(j) with j ≤ r, Thom constructs a class Xω ∈ Hr+j(Tζr,∞)represented by a map fω : Tζr,∞ → Kj . Together with fU , this gives a map

(9.9.5) F : Tζr,∞ → Y = Kr ×r∏

j=1

K♯D(j)j = Kr ×Kr+2 × · · · .

Thom proves that H∗F is an isomorphism up to degree 2r. Some analogous resultis proved for the cohomology with coefficients in a field of characteristic 6= 2 andboth Tζr,∞ and Y are simply connected. This enables Thom to prove the followingresult (see [191, pp.35 42]).

Lemma 9.9.3. If N ∈ N ∪ ∞ is big enough, there exists a map ψ from the2r-skeleton of Y to T (ζr,N ) such that the restrictions of ψF and F ψ to the(2r − 1)-skeleta of Y and Tζr,N respectively are homotopic to the identity.

As a corollary, we get the following result (see [191, Corollary II.12]).

Corollary 9.9.4. If N ∈ N ∪ ∞ is big enough, there exists a map ψ fromthe 2r-skeleton of Kr to Tζr,N such that H∗ψ(U) = ı, the fundamental class of Kr.

We are now ready to prove Theorem 9.9.1.

Proof of Theorem 9.9.1. By Theorem 3.7.4, there is a simplicial complexKX and a map φ : |KX | → X such that H∗φ is an isomorphism. By § 3.6, Thehomology of X is isomorphic to the simplicial homology of KX . By the definitionof the simplicial homology, there is a finite simplicial subcomplex K of KX , ofdimension k, such that α is in the image of Hk(|K|) → Hk(X). Now, there isa PL-embedding ψ : |K| → R2k+1 (see e.g. [179, Theorem 3.3.9]) and the theoryof smooth regular neighborhoods [94] produces a smooth compact codimension 0submanifold W of R2k+1 which is a regular neighborhood of ψ(|K|) for some C1-triangulation of R2k+1. In particular, W retracts by deformation on ψ(K). Bygeneral position, ψ is isotopic to an embedding ψ′ such that ψ′(|K|) avoids someregular neighborhood W ′ of ψ(|K|) contained in the interior of W . The closure ofW ′−W is homeomorphic toM×[0, 1], whereM = BdW (see [104, Corollary 2.16.2,p. 74]). We can thus construct a map ψ′′ : |K| →M such that the composite map

|K| ψ′′

−−→M →W → |K|is isotopic to id|K|. Hence α is in the image of H∗(M) → Hk(X). Therefore, it isenough to prove Theorem 9.9.1 when X is a closed manifold of dimension 2k.

Let a ∈ Hk(X) be the cohomology class which is Poincare dual to α. As asmooth manifold, X admits a C1-triangulation by a simplicial complex of dimension2k. There exists thus a continuous map fa : X → Kk representing the class a and,by cellular approximation, one may suppose that fa(X) is contained in the 2k-

skeleton of K. Let f = ψfa : X → T = T (ζk,N ), where ψ is a map as provided

by Corollary 9.9.4 for N large enough. Then fU f is homotopic to fa. Note thatT is a smooth manifold except at the point [S(ζk,N )]. Using standard techniques

of differential topology (see [95, § 2.2 and 3.2]), f is homotopic to f1 such that f1is a smooth map around f−11 (Gr(r;RN ))) which is transverse to Gr(r;RN ). ThenM = f−11 (Gr(r;RN ))) is a closed submanifold of codimension k in X with normalbundle ν = f∗1 ζk,N .

9.9. THOM’S THEOREMS 357

Let a′ ∈ Hk(X) be the Poincare dual of the homology class generated by[M ]. As in § 5.4.1, we consider the Thom class U(X,M) of ν as an element ofHk(X,X −M) and, if j : (X, ∅)→ (X,X −M) denotes the pair inclusion, one has

a′ = H∗j(U(X,M)) by Lemma 5.4.2

= H∗f1(U)

= H∗(ψfa)(U)

= H∗faH∗ψ(U)

= H∗fa(ı)

= a .

As Poincare duality is an isomorphism, this proves that [M ] = α.

Observe that, in the proof of Theorem 9.9.1, we have established the followingresult, due to Thom [191, Theorem II.1, p. 29].

Proposition 9.9.5. Let α ∈ Hk(X), where X is a closed smooth manifold ofdimension k + q > 2k. Let a = PD(α) ∈ Hq(X) be the Poincare dual of α. Then,the following statements are equivalent.

(1) There exists a closed submanifold M in X such that [M ] represents α.(2) There exists a continuous map F : X → T (ζq,∞) such that H∗F (U) = a.

9.9.2. Cobordism and Stiefel-Whitney numbers. Let M be a (smooth,possibly disconnected) manifold of dimension n. For a polynomial P ∈ Z2[X1, . . . , Xn],we set PM = P (w1(TM), . . . , wn(TM)) ∈ H∗(M). If M is closed, the numbermod 2

〈PM , [M ]〉 ∈ Z2

is called the Stiefel-Whitney number of M associated to P . We use the conventionthat 〈a, α〉 = 0 if a ∈ Hr(X) and α ∈ Hs(X) with r 6= s.

Two closed manifolds of the same dimension are called cobordant if their dis-joint union is the boundary of a compact manifold. One fundamental result ofThom [191, Theorema IV.3 and IV.10] is the following theorem, generalizing Corol-lary 5.3.10.

Theorem 9.9.6. Two closed manifolds of the same dimension are cobordant ifand only if their Stiefel-Whitney numbers coincide.

Example 9.9.7. Let M be a closed 3-dimensional manifold. Its Wu class isv(M) = 1 + v1(M) = 1 + w1(TM). By Wu’s formula, w(TM) = Sq(v(M)) =1 + w1(TM) + w1(TM)2, so w2(TM) = w1(TM)2. The only possible non-zeroStiefel-Whitney number is then 〈w1(TM)3, [M ]〉. But

w1(TM)3 = w1(TM)2 v1(M) since w1(TM) = v1(M)

= Sq1(w1(TM)2) by definition of v1, since dimM = 3

= 0 by the Cartan formula.

Therefore,M is the boundary of a compact manifold. Note that, ifM is orientable,the vanishing of its Stiefel-Whitney numbers follows from Proposition 9.8.13.

Example 9.9.8. The complex projective space CP 2 and the manifold RP 2 ×RP 2 have the same Stiefel-Whitney numbers by Proposition 9.8.10 and Remark 9.8.11.Therefore, they are cobordant.


Example 9.9.9. Let M be a closed orientable 4-dimensional manifold. Then,w1(TM) = 0 and w4(TM) = w2(TM)2 (see Corollary 9.8.8). Its only possiblenon-vanishing Stiefel-Whitney number is thus

〈w4(TM), [M ]〉 = 〈e(TM), [M ]〉 = χ(M) mod 2

(using Corollary 5.4.16). Therefore, M a boundary of a (possibly non-orientable)compact 5-manifold if and only if its Euler characteristic is even.

Proof of Theorem 9.9.6. Let M1 and M2 be two closed manifolds of thesame dimension and let M =M1∪M2. For any P ∈ Z2[X1, . . . , Xn], one has

〈PM , [M ]〉 = 〈PM1 , [M1]〉+ 〈PM2 , [M2]〉 .

Hence, Theorem 9.9.6 is equivalent to the following statement: a closed manifoldM bounds if and only if its Stiefel-Whitney numbers vanishes.

Suppose that M = BdW for some compact manifold W . Then TM ⊕ η ≈TW|M where η is the trivial bundle of rank 1 over M . If j : M → W denotes theinclusion, one has

〈PM , [M ]〉 = 〈H∗j(PW ), [M ]〉 = 〈PW , H∗j([M ])〉 = 0 ,

sinceH∗j([M ]) = 0 (see Equation (5.3.6) and the end of the proof of Theorem 5.3.7).For the converse, we shall prove that if a closed manifold M of dimension n

does not bound, then at least one of its Stiefel-Whitney numbers is not zero. Letus embed M into Rn+r for r large, with normal bundle ν. Let f : M → Gr(r;R∞)be a map such that ν ≈ f∗ζr,∞. The map f induces a map Tf : Tν → Tζr,∞.A closed tubular neighbourhood N of M is diffeomorphic to D(ν). We considerRn+r ⊂ Sn+r. The projection N ≈ D(ν) → T (ν) extends to a continuous mapπ : Sn+r → Tν by sending the complement of N onto the point [S(ν)]. This gives

a map f = Tf π : Sn+r → Tζr,∞ (called the Pontryagin-Thom construction) Byan argument based on transversality, one can prove that, for r large enough, M

bounds if and only if f is homotopic to a constant map [191, Lemma IV.7 and itsproof].

Let us compose f with the map F : Tζr,∞ → Y of (9.9.5). By Lemma 9.9.3,

f is not homotopic to a constant map if and only if F f is not homotopic to

a constant map. As Y is a product of Eilenberg-MacLane spaces, F f is not

homotopic to a constant map if and only if H∗(F f) 6= 0. The latter impliesthat H∗Tf : Hn+r(Tζr,∞)→ Hn+r(Tν) does not vanish. Using the Thom isomor-phisms, this implies that H∗f : Hn(Gr(r;R∞)) → Hn(M) does not vanish. Thisimplies that there is a polynomial P in the Stiefel-Whitney classes of ν such that〈P , [M ]〉 6= 0. These classes wi are the normal Stiefel-Whitney classes ofM and, byLemma 9.5.19, the Stiefel-Whitney classes wj = wj(TM) have polynomial expres-sions in the wi. Therefore, there is a polynomial P in wj such that 〈P, [M ]〉 6= 0,producing a non-zero Stiefel-Whitney number for M .

Corollary 9.9.10. Let M and M ′ be two closed smooth manifolds of thesame dimension. Suppose that there exists a map f : M ′ →M such that H∗f is anisomorphism. Then, M and M ′ are cobordant.

As a consequence of Corollary 9.9.10, a Z2-homology sphere bounds.


Proof. As H∗f is an isomorphism, π0f is a bijection and then H∗f([M′]) =

[M ]. Let P ∈ Z2[X1, . . . , Xn]. By Corollary 9.8.3, H∗f(w(TM)) = w(TM ′) andthen H∗f(PM ′) = PM . Therefore,

〈PM ′ , [M ′]〉 = 〈PM ′ , H∗f([M ])〉= 〈H∗f(PM ′), [M ]〉= 〈PM , [M ]〉 .

Hence,M andM ′ have the same Stiefel-Whitney numbers. By Theorem 9.9.6, theyare cobordant.

For closed manifolds of dimension n, “being cobordant” is an equivalence re-lation. The set of equivalence classes (cobordism classes) is denoted by Nn. Thedisjoint union endows Nn with an abelian group structure, actually a Z2-vectorspace structure since 2M =M ∪M is diffeomorphic to the boundary of M × [0, 1].The Cartesian product of manifolds makes N∗ =

⊕nNn a Z2-algebra, called the

cobordism ring. A development of the results of this section and the previous one en-abled Thom to compute the cobordism ring N∗ [191, § IV]; the results are summedup in the following theorem.

Theorem 9.9.11. (1) Nn is isomorphic to lim→k

πn+k(T (ζk,∞)).

(2) dimNn is the number of partitions of n into integers with none of themof the form 2i − 1.

(3) N∗ is GrA-isomorphic to a polynomial algebra Z2[X2, X4, X5, X6, X8, X9, . . . ]with one generator Xk for each integer k not of the form 2i − 1.

A representative for X2k is given by the cobordism class of RP 2k [191, p. 81].Odd dimensional generator of dimension 6= 2i − 1 were first constructed by Dold[41]. For details and proofs (see [191] or [186, Chapter VI]).

For example, N3 = 0, confirming Example 9.9.7. Another simple consequenceof Theorem 9.9.11 is the following corollary.

Corollary 9.9.12. Let M and N be closed manifolds which are not bound-aries. Then M ×N is not a boundary.


9.1. Let ξ be a vector bundle. Prove that ξ ⊕ ξ is orientable. If ξ is orientable,prove that ξ ⊕ ξ admits a spin structure.

9.2. Let p : X → X be a smooth covering of a smooth manifold X , with an oddnumber of sheets. Show that X is orientable if and only if X is orientable. Showthat X admits a spin structure if and only if X does.

9.3. Let p : M →M be a 2-fold covering of a smooth connected manifold M , withcharacteristic class ω ∈ H1(M). Suppose that M is orientable and that M is notorientable. Prove that ω = w1(TM).

9.4. Let M1 and M2 be two closed connected manifold of the same dimensionand let M be one of their their connected sums (see p. 115). What is the totalStiefel-Whitney class w(T (M))?

9.5. Prove that TRP 4 and TCP 2 are indecomposable as a Whitney sum of vectorbundles of smaller ranks.


9.6. Let ξ be a vector bundle over a space X , with w(ξ) 6= 1. Show that thesmallest integer i > 0 such that wi(ξ) 6= 0 is a power of 2.

9.7. List the critical points with their index for the weighted trace on Gr(2;R4)and Gr(2;R6). Using also Example 9.5.3, verify the statement of Lemma 9.5.7 andthe second formula of Corollary 9.5.15.

9.8. Same exercise as the previous one for Fl(1, 1, 1).

9.9. Let j : BO(n)→ BU(n) denote the inclusion. Prove that H∗j(ci) = w2i .

9.10. Write the details for Remark 9.5.16.

9.11. Like in Example 9.5.21, find an additive basis of H∗(Gr(2;R5)) in terms ofproducts of Stiefel-Whitney classes. Express each of these elements in terms ofSchubert symbols.

9.12. Let M be an orientable smooth closed manifold of dimension 6 or 10. Provethat χ(M) is even.

9.13. Let f : P → Q be a continuous map between n-dimensional connected closedsmooth manifolds. Suppose that one of the following conditions is satisfied:

(a) P is orientable while Q is non-orientable(b) P is spin while Q is non-spin (w2(TQ) 6= 0).

Then Hnf : Hn(Q)→ Hn(P ) is trivial

9.14. Let f : Sn+k → Sk be a smooth map. Let x ∈ Sk be a regular value. Showthat the closed manifold f−1(x) is the boundary of a (possibly non-orientable)compact manifold.

9.15. Prove that RP 2 × RP 2 and RP 4 are not cobordant.

9.16. Let M be a closed n-dimensional manifold whose cohomology ring is isomor-phic to that of RPn. Prove that M and RPn are cobordant.

CHAPTER 10

Miscellaneous applications and developments

This chapter, contains various applications and developments of the techniquesof mod2 (co)homology. Most of them are somewhat original.

10.1. Actions with scattered or discrete fixed point sets

Let X be a finite dimensional G-complex (G = id, τ) with b(X) < ∞. BySmith theory (Proposition 7.3.7), we know that b(XG) ≤ b(X), which implies that♯(π0(X

G)) ≤ b(X). Inspired by the work of V. Puppe [165], we study in this sectionthe extremal case ♯(π0(X

G)) = b(X) (scattered fixed point set). Analogous resultsfor S1-actions are presented at the end of this section.

Proposition 10.1.1. Let X be a finite dimensional G-complex with b(X) <∞.

Suppose that ♯(π0(XG)) = b(X). Let a ∈ Hk

G(X). Then Sqi(a) = (ki )uia.

Proof. By Proposition 7.3.7, H>0(XG) = 0 and X is equivariantly formal.Therefore, XG has the cohomology of b(X) points and (XG)G ≈ BG × XG ishomotopy equivalent to a disjoint union of b(X) copies of RP∞. By (8.2.2), anyclass b ∈ Hk

G(XG) satisfies Sqi(b) = (ki )u

ib. As the restriction homomorphismH∗G(X)→ H∗G(X

G) is injective by Proposition 7.3.9, this proves the assertion.

As seen in the above proof, the G-space X of Proposition 10.1.1 is equivariantlyformal. Thus, ρ : H∗G(X)→ H∗(X) is surjective. As kerρ = u ·H∗G(X) by (7.1.7),Proposition 10.1.1 has the following corollary (compare [165, Corollary 1]).

Corollary 10.1.2. Let X be a finite dimensional G-complex with b(X) <∞.Suppose that ♯(π0(X

G)) = b(X). Then, any a ∈ H∗(X) satisfies Sq(a) = a (i.e.

Sqi = 0 for i > 0). In particular, a a = 0 if a ∈ H>0(X).

Let us restrict the above results to the case where X is a smooth closed G-manifold. Then, XG is a union of closed manifolds (see, e.g. [12, Corollary 2.2.2]).We have seen in the proof of Proposition 10.1.1 that each component of XG hasthe cohomology of a point. Hence XG is a discrete set of b(X) points (the smoothinvolution τ is called an m-involution in [165]). Examples include linear spheresSn0 ; if X1 and X2 are such G-manifolds, so is X1×X2 with the diagonal involution;if dimX1 = dimX2, the G-equivariant connected sum X1♯X2 around fixed pointscarries an m-involution. Thus, an orientable surface carries an m-involution. Also, ifX admits a G-invariant Morse function, then τ is an m-involution by Theorem 7.6.6.Corollary 10.1.2 has the following consequence for the Stiefel-Whitney class w(TX)of a manifold X admitting an m-involution.

Corollary 10.1.3. Let X be a smooth closed G-manifold such that ♯(π0(XG)) =

b(X). Then w(TX) = 1.

361

362 10. MISCELLANEOUS APPLICATIONS AND DEVELOPMENTS

In consequence, a closed manifold X carrying an m-involution is orientable andadmits a spin structure. Also, X is the boundary of a (possibly non-orientable)compact manifold by Thom’s Theorem 9.9.6.

Proof. By Corollary 10.1.2, Sqi = 0 for i > 0. Hence, the Wu class v(X) isequal to 1. Therefore, using Wu’s formula 9.8.2, w(TX) = Sq(v(X)) = 1.

We now generalize to smooth closed G-manifolds with XG discrete (withoutasking that ♯XG = b(X)). This will lead us toward the link between closed G-manifolds with discrete fixed point set and coding theory; such a link was initiatedin [165] and further developed in [121]. We start with the following lemma.

Lemma 10.1.4. Let X2k+1 be a smooth closed G-manifold such that XG isdiscrete. Then, ♯XG is even.

Proof. Let r = ♯XG. Let W = X−intD whereD is a closedG-invariant tubu-lar neighborhood of XG. Then W is a compact free G-manifold with boundary V .The orbit space W = W/G is a compact manifold whose boundary V = V /Gis a disjoint union of r copies of RP 2k. By Proposition 5.3.9, the image B ofHk(W )→ Hk(V ) satisfies

2 dimB = dimHk(BdW ) = r

which shows that r is even.

Remark 10.1.5. If X is a finite dimensional G-CW-complex with b(X) < ∞,it is known that b(X) ≡ b(XG) mod 2 [9, Corollary 1.3.8] [see Comment 12.0.5].If X is an odd dimensional closed manifold, then b(X) is even by Poincare duality.This provides another proof of Lemma 10.1.4.

We use the notation of the proof of Lemma 10.1.4. Let 〈〈−,−〉〉 be the bilinearform on Hk(V ) given by 〈〈a, b〉〉 = 〈a b, [V ]〉. By Proposition 5.3.9 and its proof,one has 〈〈B,B〉〉 = 0 and r = 2dimB. Labeling the r points of XG produces anisomorphism Hk(V ) ≈ Zr2 intertwining 〈〈−,−〉〉 with the standard bilinear form onZr2. Hence, in terms of coding theory (see, e.g. [46]), B is a binary self-dual linearcode on Zr2. Choosing another labeling for the points of XG changes B by anisometry of Zr2 obtained by coordinate permutations. The class of B modulo theseisometries thus provides an invariant of the G-action.

The self-dual code B has other descriptions. For instance, the diagram ofinclusions

Vi //

j

D

W // X

gives rise to the commutative diagram

(10.1.1)

HkG(X) //

=

HkG(W )⊕Hk

G(D) //

≈

HkG(V )

≈

HkG(X) // Hk(W )⊕Hk

G(XG) // Hk(V )

10.1. ACTIONS WITH SCATTERED OR DISCRETE FIXED POINT SETS 363

whose rows are the Mayer-Vietoris exact sequences. Lemma 7.1.4 guarantees thatthe vertical maps are isomorphisms and, together with Corollary 3.8.4, implies thatthe map V ≃ VG → (XG)G is, on each component, homotopy equivalent to theinclusion RP 2k → RP∞. Therefore, the homomorphism Hk

G(XG) → Hk(V ) is an

isomorphism. Hence, diagram-chasing in (10.1.1) shows that B = image(Hk(W )→Hk(V )) coincides with

image(HkG(X)→ Hk

G(XG))

(pushed into Hk(V )). For other descriptions of B, see [165, § 2]. The followingtheorem is proved in [121, Theorem 3].

Theorem 10.1.6. Every binary self-dual linear code may be obtained from aclosed smooth 3-dimensional G-manifold X with scattered fixed point set.

As in § 7.3, the above results have analogues for S1-actions. The proofs ofProposition 10.1.7 and Corollary 10.1.8 below are the same as for Proposition 10.1.1and Corollary 10.1.2, replacing Propositions 7.3.7 and 7.3.9 by Propositions 7.3.12and 7.3.14, etc. Recall that H∗S1(pt) ≈ Z2[v] with v of degree 2.

Proposition 10.1.7. Let X be a finite dimensional S1-complex with b(X) <∞and XS1

= XS0

. Suppose that ♯(π0(XS1

)) = b(X). Then HoddS1 (X) = 0 and, if

a ∈ H2kS1(X), then Sq2i(a) = (ki ) v

ia.

Corollary 10.1.8. Let X be a finite dimensional S1-complex with b(X) <∞and XS1

= XS0

. Suppose that ♯(π0(XS1

)) = b(X). Then, Hodd(X) = 0 and anya ∈ H∗(X) satisfies Sq(a) = a

Analogously to Corollary 10.1.3, one has the following result, with the sameproof.

Corollary 10.1.9. Let X be a smooth closed S1-manifold such that XS1

=

XS0

and ♯(π0(XS1

)) = b(X). Then, Hodd(X) = 0 and w(TX) = 1.

In particular, the manifold X of Corollary 10.1.9 is even-dimensional. Notethat this is necessary for an S1-action admitting an isolated fixed point (the action,restricted to an invariant sphere around the fixed point has discrete stabilizers, sothe sphere is odd-dimensional). The analogue of Lemma 10.1.4 is Lemma 10.1.10below. To simplify, we restrict ourselves to semi-free actions (A Γ-action is calledsemi-free if the stabilizer of any point is either id or Γ). Incidentally, the hypoth-esis XS1

= XS0

is not required.

Lemma 10.1.10. Let X be a smooth closed S1-manifold such that XS1

is dis-

crete. Suppose that the action is semi-free. Then, ♯XS1

is even.

Proof. As seen above, X is even-dimensional, say dimX = 2k + 2. Let r =

♯XS1

. Let W = X − intD where D is a closed S1-invariant tubular neighborhood

of XS1

. Then W is a compact free S1-manifold with boundary V . The orbitspace W = W/G is then a compact manifold of dimension 2k+ 1 whose boundary

V = V /G is a disjoint union of r copies of CP k. By Proposition 5.3.9, the imageB of Hk(W )→ Hk(V ) satisfies

2 dimB = dimHk(BdW ) = r

which shows that r is even.


As for the case of an involution, Lemma 10.1.10 permits us to associate, to a

closed smooth semi-free S1-manifold with XS1

discrete, the self-dual linear code

B ⊂ Hk(V ) ≈ Zr2. One can also see B as the image of HkS1(X) in Hk

S1(XS1

).

10.2. Conjugation spaces

Introduced in [87], conjugation spaces are equivariantly formal G-spaces (G =id, τ) quite different from those with scattered fixed point sets studied in § 10.1.Here, the cohomology ring of the fixed point set most resembles that of the to-tal space. This similarity should be given by a “cohomology frame”, a notionwhich we now define. We use the notations of § 7.1 for a G-space X , for exam-ple the forgetful homomorphism ρ : H∗G(X) → H∗(X) and the GrA[u]-morphismr : H∗G(X)→ H∗G(X

G) ≈ H∗(XG)[u] induced by the inclusion XG ⊂ X .Let (X,Y ) be a G-pair. A cohomology frame or an H∗-frame for (X,Y ) is a

pair (κ, σ), where

(a) κ : H2∗(X,Y ) → H∗(XG, Y G) is an additive isomorphism dividing thedegrees in half; and

(b) σ : H2∗(X,Y ) → H2∗G (X,Y ) is an additive section of the natural homo-

morphism ρ : H∗G(X,Y )→ H∗(X,Y ),

satisfying, in H∗(XG) ≈ H∗(XG)[u], the conjugation equation

(10.2.1) rσ(a) = κ(a)um + ℓtm

for all a ∈ H2m(X,Y ) and all m ∈ N; in (10.2.1), ℓtm denotes any elementin H∗(X,Y )[u] which is of degree less than n in the variable u. An involutionadmitting an H∗-frame is called a conjugation. An even cohomology pair (i.e.Hodd(X,Y ) = 0) together with a conjugation is called a conjugation pair. A G-space X is a conjugation space if the pair (X, ∅) is a conjugation pair. The existenceof the section σ implies that a conjugation space is equivariantly formal. Note thatthere are examples of G-spaces which admit pairs (κ, σ) satisfying (a) and (b) abovebut none of them satisfying the conjugation equation (see [63, Example 1]). Forsimplicity, we shall mostly restrict this survey to conjugation spaces; the corre-sponding statements for conjugation pairs may be found in [87].

Any space X such that H∗(X) = H∗(pt) = H∗(XG) is a conjugation space.For instance, a finite dimensional G-CW-complex X satisfying H∗(X) = H∗(pt)and XG 6= ∅ is a conjugation space by Corollary 7.3.8. Another easy example isthe G-sphere S2m

m of Example 7.1.14. Indeed, one has the following lemma.

Lemma 10.2.1. Let X be a finite dimensional G-CW-complex. Suppose thatH∗(X) ≈ H∗(S2n) and Hn(XG) 6= 0. Then X is a conjugation space.

Proof. By Corollary 7.3.8, H∗(XG) ≈ H∗(Sn). By Proposition 7.3.7, X isequivariantly formal and, by Proposition 7.3.9, r : H∗G(X) → H∗G(X

G) is injec-tive. The proof of the existence of an H∗-frame then proceeds as in the proof ofCorollary 7.1.17.

Another important example is the complex projective space.

Example 10.2.2. Let a ∈ H2(CPm) and b ∈ H1(RPm) (m ≤ ∞). Then,the section σ : H∗(CPm) → H∗G(CP

m) of Proposition 7.1.18, together with theisomorphism κ : H2∗(CPm)→ H∗(RPm) sending a to b makes an H∗-frame for the

10.2. CONJUGATION SPACES 365

complex conjugation on CPm. By Corollary 7.1.18, the conjugation equation takesthe form

(10.2.2) rσ(ak) = (κ(a)u + b2)k = κ(ak)uk + ℓtk .

The same treatment may be done for HPm or OP 2 (see Remark 7.1.21).

These examples are actually spherical conjugation complexes, i.e. G-spacesobtained from the empty set by countably many successive adjunctions of collectionsof conjugation cells. A conjugation cell (of dimension 2k) is a G-space which is G-

homeomorphic to the cone over S2k−1k−1 , i.e. to the closed disk of radius 1 in R2k,

equipped with a linear involution with exactly k eigenvalues equal to −1. At eachstep, the collection of conjugation cells consists of cells of the same dimension but,as in [74], the adjective “spherical” is a warning that these dimensions do not needto be increasing. For less standard examples of spherical conjugation complexes,see [87, 5.3.3, p. 944].

It is proven in [87, Proposition 5.2] that a spherical conjugation complex is aconjugation space. For example, complex flag manifolds (with τ being the complexconjugation) are conjugation spaces because the Schubert cells (see § 9.5.3) areconjugation cells. This example generalizes to co-adjoint orbits of compact Liegroups for the Chevalley involution (see [87, § 8.3]) and more examples comingfrom Hamiltonian geometry (see [87, § 8.2 and 8.4] and [86]).

Other examples may be obtained from the previous ones since the class ofconjugation spaces is closed under many constructions, such as

• direct products, with the diagonal G-action, when one of the factor is offinite cohomology type (see [87, Proposition 4.5]).• inductive limits (see [87, Proposition 4.6]).• if (X,Y, Z) is a G-triple so that (X,Y ) and (Y, Z) are conjugation pairs,then (X,Z) is a conjugation pair. A direct proof using H∗-frames is givenin [87, Proposition 4.1]; a shorter proof using the conjugation criterion of[158, Theorem 2.3] is provided in [157, Proposition 2.2.1].• if F → E → B is G-equivariant bundle (with a compact Lie group asstructure group) such that F is a conjugation space and B is a sphericalconjugation complex, then E is a conjugation space (see [87, Proposi-tion 5.3]).

We now show the naturality of H∗-frames under G-equivariant maps, as provenin [87, Proposition 3.11]. Let X and Y be two conjugation spaces, with H∗-frames(κX, σX) and (κY , σY ). Let f : Y → X be a G-equivariant map. We denote byfG : Y G → XG the restriction of f to Y G.

Proposition 10.2.3 (Naturality of H∗-frames). The equations H∗Gf σX =σY H

∗f and H∗fG κX = κY H∗f hold true.

Proof. Let a ∈ H2k(X). As σX and σY are sections of ρX : H∗G(X)→ H∗(X)and ρY : H∗G(Y )→ H∗(Y ) respectively, one has

(10.2.3) ρY H∗Gf σX(a) = H∗f ρX σX(a) = H∗f(a) = ρY σY H

∗f(a) .

This implies that

H∗Gf σX(a) ≡ σY H∗f(a) mod ker

(ρ2k

Y: H2k

G (Y )→ H2k(Y )).


The section σ produces an isomorphism H∗G(X) ≈ H∗(X)[u] and ker ρY is the idealgenerated by u (see Remark 7.1.7). As Hodd(Y ) = 0, we deduce that there existsdi ∈ Hi(Y ) such that

(10.2.4) H∗Gf σX(a) = σY H∗f(a) + σY (d2k−2)u

2 + · · ·+ σY (d0)u2k .

Let us apply rY to both sides of (10.2.4). For the left hand side, we get(10.2.5)rY H

∗Gf σX(a) = H∗Gf

GrX σX(a)

= H∗GfG(κX(a)u

k + ℓtk) by the conjugation equation

= H∗GfG(κX(a))u

k + ℓtk H∗GfG being a GrA[u]-morphism.

But, using the right hand side of (10.2.4), we get

(10.2.6) rY H∗Gf σX(a) = κY (d0)u

2k + ℓt2k.

Comparing (10.2.5) with (10.2.6) and using that κY is injective implies that d0 = 0.Then, (10.2.6) may be replaced by

(10.2.7) rY H∗Gf σX(a) = κY (d2)u

2k−2 + ℓt2k−2.

Again, the comparison with (10.2.5) implies that d2 = 0. This process may becontinued, eventually giving that H∗Gf σX(a) = σY H

∗f(a). Applying rY to theright hand member of this equation gives

(10.2.8) rY σY H∗f(a) = κY H

∗f(a)uk + ℓtk

by the conjugation equation. Comparing the leading terms of (10.2.8) and (10.2.5)gives that H∗fGκX(a) = κY H

∗f(a).

Applying Proposition 10.2.3 to X = Y and f = id, we get the following corol-lary.

Corollary 10.2.4 (Uniqueness of H∗-frames). Let (κ, σ) and (κ′, σ′) be twoH∗-frames for the conjugation space X. Then (κ, σ) = (κ′, σ′).

We can thus speak about the H∗-frame of a conjugation space.

Proposition 10.2.5. Let (κ, σ) be the H∗-frame of a conjugation space X.Then κ and σ are multiplicative.

Proof. Let a ∈ H2m(X) and b ∈ H2n(X) One has a b = ρσ(a b) andρ(σ(a) σ(b)) = ρ(σ(a)) ρ(σ(b)) = a b. Hence, σ(a) σ(b) is congruent toσ(a b) modulo kerρ. The same proof as for Proposition 10.2.3 then proves theproposition (details may be found in [87, Theorem 3.3]).

Much more difficult to prove, the following result was established in [63, The-orem 1.3].

Proposition 10.2.6. Let (κ, σ) be the H∗-frame of a conjugation space X.Then Sqiκ = κSq2i for all integers i.


Remark 10.2.7. It is not true in general that σSq = Sqσ. For example,consider the conjugation space CPm for 1 ≤ m ≤ ∞, with the notations of Exam-ple 10.2.2. Of course, Sq1(a) = 0 and then σSq1(a) = 0. On the other hand,

rSq1σ(a) = Sq1rσ(a)

= Sq1((bu+ b2)

)by the conjugation equation (10.2.2)

= Sq1(b)u+ bSq1(u) by the Cartan formula

= b2u+ bu2

= r(σ(a)u) .

Since r is injective (see Lemma 10.2.8 below), this proves that

Sq1(σ(a)) = σ(a)u .

The following lemma is recopied with its short proof from [87, Lemma 3.8].

Lemma 10.2.8. Let X be a conjugation space. Then the restriction homomor-phism r : H∗G(X)→ H∗G(X

G) is injective.

Proof. Suppose that r is not injective. Let 0 6= x = σ(y)uk+ℓtk ∈ H2n+kG (X)

be an element in ker r. The conjugation equation guarantees that k 6= 0. Wemay assume that k is minimal. By the conjugation equation again, we have 0 =r(x) = κ(y)un+k + ℓtn+k. Since κ is an isomorphism, we get y = 0, which is acontradiction.

TheH∗-frame of a conjugation space behaves well with respect to the character-istic classes of G-conjugate-equivariant bundles. A G-conjugate-equivariant bundle

over a G-space X (with an involution τ) is a complex vector bundle η = (Ep−→ X),

together with an involution τ on E such that p τ = τ p and τ is conjugate-linearon each fiber: τ (λx) = λτ (x) for all λ ∈ C and x ∈ E. This was called a “real

bundle” by Atiyah [11]. Note that ηG = (EGp−→ XG) is a real vector bundle and

rankR ηG = rankC η. The following result is proven in [87, Proposition 6.8].

Proposition 10.2.9. Let η be a G-conjugate-equivariant bundle over a spher-ical conjugation complex X. Then κ(c(η)) = w(ητ ).

A theory of (integral) equivariant Chern classes for G-conjugate-equivariantbundles over a conjugation space is developed in [160].

Another relationship between conjugation spaces and the Steenrod squares wasdiscovered by M. Franz and V. Puppe in [63]. It is illustrated by the case of CPm,with the notations of Example 10.2.2, where the conjugation equation (10.2.2) forCPm may be written as follows

(10.2.9) rσ(ak) = (κ(a)u+ κ(a)2)k =

k∑

j=0

(kj )κ(a)k+juk−j =

k∑

j=0

Sqj(κ(ak))uk−j .

It was proven in [63, Theorem 1.1] that (10.2.9) holds true in general, leading tothe following universal conjugation equation.


Theorem 10.2.10. Let X be a conjugation space, with H∗-frame (κ, σ). Then,for x ∈ H2k(X), one has

(10.2.10) rσ(x) =

k∑

j=0

Sqj(κ(x))uk−j .

Note the resemblance between the right member of (10.2.10) and that of (8.3.9).For other occurrences of such an expression, see [125, § 2.4].

Let r : H∗(X)→ H∗(XG) be the restriction homomorphism in non-equivariantcohomology. The following corollary was observed in [63, Corollary 1.2].

Corollary 10.2.11. For x ∈ H∗(X), one has r(x) = κ(x) κ(x).

Proof. Suppose that x ∈ H2k(X). Denote by ρG : H∗G(XG) → H∗(XG) the

forgetful homomorphism for XG. Then

r(x) = rρσ(x) since ρσ = id

= ρGrσ(X) using (7.1.5)

= ρG(∑k

j=0 Sqj(κ(x))uk−j

)by Theorem 10.2.10

= Sqk(κ(x)) since ρG = evu=0, see (7.1.6)

= κ(x) κ(x) .

Another consequence of Theorem 10.2.10 is the commutativity of the diagram

(10.2.11)

H∗(X)σ //

κ

H∗G(X)r // H∗G(X

G) oo ≈ // H∗(XG)[u]

ev1

H∗(X)

Sq // H∗(XG)

We now turn our attention to conjugation manifolds, i.e. closed manifolds Xwith a smooth conjugation τ . Then, XG is a closed manifold (see, e.g. [12, Corol-

lary 2.2.2]) whose dimension, because of the isomorphism κ : H∗(X)≈−→ H∗(XG),

is half of the dimension of X . By looking at the derivative of τ around a fixed point,one checks that τ preserves the orientation if and only if dimX ≡ 0mod4. Forvarious properties of conjugation manifolds, see [80, § 2.7], from which we extractthe following results (see also [160, Appendix A]).

Proposition 10.2.12. Let X be a smooth conjugation manifold of dimension2n, with H∗-frame (κ, σ). Then κ preserves the Wu and Stiefel-Whitney classes:

κ(v(X)) = v(XG) and κ(w(TX)) = w(TXG) .

Proof. The Wu class v2i(X) is characterized by the equation

(10.2.12) v2i(X) a = Sq2i(a) for all a ∈ H2n−2i(X) .

Applying the ring isomorphism κ to (10.2.12) and using Proposition 10.2.6 gives

(10.2.13) κ(v2i(X)) κ(a) = Sqi(κ(a)) for all a ∈ H2n−2i(X) .

As κ is bijective, (10.2.13) implies that

κ(v2i(X)) b = Sqi(b) for all b ∈ Hn−i(XG) ,


which implies that κ(v2i(X)) = vi(XG), and, as Hodd(X) = 0, that κ(v(X)) =

v(XG). Using this and the Wu formula, one gets

κ(w(TX)) = κSq(v(X)) by the Wu formula

= Sqκ(v(X)) Proposition 10.2.6

= Sq(v(XG))) as already seen

= w(TXG) by the Wu formula.

In particular, X admits a spin structure if and only if XG is orientable. Also,the Stiefel-Whitney numbers of X all vanish if and only if those of XG do so. ByThom’s theorem 9.9.6, this gives the following

Corollary 10.2.13. Let X be a conjugation manifold. Then X bounds acompact manifold if and only if XG does so.

Two natural problems occur for conjugation manifolds.

(i) Given a closed connected smooth manifold Mn, does there exist a conju-gation 2n-manifold X with XG diffeomorphic to M ?

(ii) Classify, up to G-diffeomorphism, conjugation manifolds with a given fixedpoint set.

The circle is the fixed point set of a unique conjugation 2-manifold, namelyS21 ; the uniqueness may be proved using the Schoenflies theorem (compare [34,

Theorem 4.1]).For n = 2, recall that RP 2 is the fixed point set of the conjugation manifold

CP 2 and S1×S1 is that of S2×S2. The equivariant connected sum (around a fixedpoint) of conjugation manifolds being again a conjugation manifold (see [87, Propo-sition 4.7]), any closed surface is the fixed point set of some conjugation 4-manifold(of course, S2 = (S4

2)G). Answering Question (ii) is the main object of [80], using

the following idea. For a smooth G-action on a manifold X with XG being ofcodimension 2, the quotient space X/G inherits a canonical smooth structure. If Xis a conjugation 4-manifold, then H∗(X/G) ≈ H∗(S4). Conversely, let (Y,Σ) be amanifold pair such that Y is a 4-dimensional Z2-homology sphere containing Σ asa codimension 2 closed submanifold. By Alexander duality (Theorem 5.3.14), onehas H1(Y −Σ) = Z2. Thus, Y −Σ admits a unique non-trivial 2-fold covering (see§ 4.3); the latter extends to a unique branched covering X → Y , with branchedlocus Σ, and X turns out to be a conjugation 4-manifold with XG =M . The finalstatement is thus the following ([80, Theorem A]).

Theorem 10.2.14. The correspondence X 7→ (X/G,XG) defines a bijectionbetween

(a) the orientation-preserving G-diffeomorphism classes of oriented connectedconjugation 4-manifolds, and

(b) the orientation-preserving diffeomorphism classes of smooth manifold pairs(Y,Σ), where Y is an oriented 4-dimensional homology sphere and Σ is aclosed connected surface embedded in M .

The conjugation sphere S42 corresponds to the trivial knot S2 ⊂ S4. Under

the bijection of Theorem A, any knot S2 → S4 corresponds to a conjugation 4-manifold X with XG ≈ S2. In general, X is not simply connected. On the otherhand, Gordon [69], [70] and Sumners [189] found infinitely many topologically


distinct knots in S4 which are the fixed point set of smooth involutions. Theseexamples produce infinitely many topologically inequivalent smooth conjugationson S4 (see [80, Proposition 5.12]).

If X is a simply-connected conjugation 4-manifold, it is known that X/G is atleast homeomorphic to S4 (see [80, Proposition 5.3]). In addition, X is homeo-morphic (not necessarily equivariantly) to a connected sum of copies of S2 × S2,

CP 2, and CP2(see [80, Proposition 2.17]). These are severe restrictions on a

simply-connected closed smooth 4-manifold to carry a smooth conjugation.M. Olbermann, in his thesis [157], was the first to address Question (i); he

proved the following result (see [158, Theorem 1.2]).

Theorem 10.2.15. Any closed smooth orientable 3-manifold is diffeomorphicto the fixed point-set of a conjugation 6-manifold.

The case of non-orientable 3-manifolds is not known. Any 3-dimensional Z2-homology sphere is the fixed point of infinitely many inequivalent conjugations onS6 [159]; this gives a partial answer to Question (ii) in this case.

Remark 10.2.16. As observed by W. Pitsch and J. Scherer, the answer toQuestion (i) is not always positive. For example, the octonionic projective planeOP 2, which is a smooth closed 16-manifold (see Remark 6.1.8), is not the fixed pointset of any conjugation space. Indeed, H∗(OP 2) ≈ Z2[x]/(x

3) by Proposition 6.1.7,with degree (x) = 8, but, by Theorem 8.6.5, Z2[x]/(x

3) is not the cohomology ringof a topological space if degree (x) > 8.

10.3. Chain and polygon spaces

Chain and polygon spaces are examples of configuration spaces, a main conceptof classical mechanics. In recent decades, starting in [202, 83] (inspired by talksof W. Thurston on linkages [195]), new interests arose for polygon spaces, in con-nections with Hamiltonian geometry (see e.g. [118, 111, 88, 91]), mathematicalrobotics [142, 56] and statistical shape theory [90]. This section contains originalresults on the equivariant cohomology of chain spaces, giving new proofs for knownstatements about their ordinary cohomology.

We use the notations of [90, 85, 59], inspired by those of statistical shapetheory [112]. In order to make some formulae more readable, we may write |J | forthe cardinality ♯J of a finite set J .

10.3.1. Definitions and basic properties. Let ℓ = (ℓ1, . . . , ℓn) ∈ Rn>0 andlet d be an integer. We define the subspace Cnd (ℓ) of (Sd−1)n−1 by

Cnd (ℓ) =z = (z1, . . . , zn−1) ∈ (Sd−1)n−1 |

n−1∑

i=1

ℓizi = ℓn e1,

where e1 = (1, 0, . . . , 0) is the first vector of the standard basis e1, . . . , ed of Rd. Anelement of Cnd (ℓ), called a chain, may be visualized as a configuration of (n − 1)successive segments in Rd, of length ℓ1, . . . , ℓn−1, joining the origin to ℓne1. Thevector ℓ is called the length vector. The chain space Cnd (ℓ) is contained in the bigchain space BCnd (ℓ) defined as follows:

BCnd (ℓ) =z = (z1, . . . , zn−1) ∈ (Sd−1)n−1 | 〈

n−1∑

i=1

ℓizi, e1〉 = ℓn,

10.3. CHAIN AND POLYGON SPACES 371

(successions of (n− 1) segments in Rd, of length ℓ1, . . . , ℓn−1, joining the origin tothe affine hyperplane with first coordinate ℓn).

The group O(d − 1), viewed as the subgroup of O(d) stabilizing the firstaxis, acts naturally (on the left) upon the pair (BCnd (ℓ), Cnd (ℓ)). The quotientCnd (ℓ)

/SO(d − 1) is the polygon space Nn

d (ℓ), also defined as

Nnd (ℓ) = Nn

d (ℓ)/SO(d) ,

where

Nnd (ℓ) =

z ∈ (Sd−1)n

∣∣∣∣∣n∑

i=1

ℓizi = 0

is the free polygon space (called “space of polygons” in [57, 65]).

The map from SO(d) × Cnd (ℓ) to Nnd (ℓ) given by

(A, (z1, . . . , zn−1)) 7→ (Az1, . . . , Azn−1,−Ae1)descends to an SO(d)-homeomorphism

(10.3.1) SO(d) ×SO(d−1) Cnd (ℓ)≈−→ Nn

d (ℓ) .

Recall that the map SO(d)→ Sd−1 given by A 7→ −Ae1 is the orthonormal orientedframe bundle for the tangent bundle to Sd−1 (see p. 303). Thus, by (10.3.1), weget a locally trivial bundle

(10.3.2) Cnd (ℓ)→ Nnd (ℓ)→ Sd−1 .

When d = 2 the space of chains Cn2 (ℓ) coincides with the polygon space Nn2 (ℓ).

The axial involution τ on Rd = R × Rd−1 given by τ(t, y) = (t,−y) inducesan involution, still called τ , on the pair (BCnd (ℓ), Cnd (ℓ)) and on (Sd−1)n−1. As τcommutes with the O(d − 1)-action on Cnd (ℓ), it descends to a G-action on Nn

d (ℓ),where G = id, τ). A bar above a G-space denotes its orbit space:

BCnd (ℓ) = BCnd (ℓ)/G , Cnd (ℓ) = Cnd (ℓ)/G , Nnd (ℓ) = Nn

d (ℓ)/G .

We shall compute the G-equivariant cohomology of BCnd (ℓ) and Cnd (ℓ), as alge-bras over H∗G(pt) = Z2[u] (u of degree 1). This uses some G-invariant Morse theoryon M = (Sd−1)n−1. We start with the robot arm map Fℓ : M → Rd defined by

(10.3.3) Fℓ(z) =

n−1∑

i=1

ℓizi , z = (z1, . . . , zn−1) .

Consider Rd as the product R × Rd−1, which defines the projections p1 : Rd → Rand pd−1 : Rd → Rd−1. Define fℓ : M → R by

fℓ(z) = −p1(Fℓ(z)) = −n−1∑

i=1

ℓi〈zi, e1〉 .

Note that Fℓ is O(d)-equivariant and fℓ is O(d− 1)-invariant. For n = 2, it is clearthat fℓ is a Morse function on Sd−1, with two critical points, namely e1 of index 0and −e1 of index d− 1. The following lemma follows easily.

Lemma 10.3.1. The function fℓ : (Sd−1)n−1 → R defined by

fℓ(z1, . . . , zn−1) = −n−1∑

i=1

ℓi〈zi, e1〉


is a G-invariant Morse function, with one critical point PJ for each J ⊂ 1, . . . , n−1, where PJ = (z1, . . . , zn−1) with zi equal to −e1 if i ∈ J and e1 otherwise (acollinear chain). The index of PJ is (d− 1) |J |.

A length vector ℓ ∈ Rn>0 is generic if Cn1 (ℓ) = ∅, that is to say there are nocollinear chains or polygons. In this section, we shall only deal with generic lengthvectors.

Corollary 10.3.2. If ℓ is a generic length vector, then BCnd (ℓ), Cnd (ℓ) and

Nnd (ℓ) are smooth closed orientable manifolds of dimension

dimBCnd (ℓ) = dim Nnd (ℓ) = (n− 1)(d− 1)− 1 and dim Cnd (ℓ) = (n− 2)(d− 1)− 1 .

Proof. If ℓ is generic, then−ℓn is a regular value of fℓ. Indeed, if pd−1(Fℓ(z)) 6=0, this follows from the O(d)-equivariance of Fℓ. If pd−1(Fℓ(z)) = 0, then, as ℓis generic, z is not a critical point of Fℓ (these are the collinear configurationszi = ±zj: see [83, Theorem 3.1]). Since BCnd (ℓ) = f−1ℓ (−ℓn), this proves the asser-tion on BCnd (ℓ) (which is orientable, having trivial normal bundle in the orientablemanifold (Sd−1)n−1).

Define P : BCnd (ℓ) → Rd−1 by P (z) = pd−1(Fℓ(z)). As seen above, as ℓ isgeneric, P−1(0) contains no critical points of Fℓ. Therefore, P is transverse to 0and thus Cnd (ℓ) = P−1(0) is a closed submanifold of codimension d − 1 of BCnd (ℓ),with trivial normal bundle [see a simpler argument in Comment 12.0.23].

When ℓ is generic, the O(d − 1)-action on Cnd (ℓ) is smooth. Hence, the bun-

dle (10.3.2) is a smooth bundle and the assertion on Nnd (ℓ) follows from (10.3.1).

For another proof that Nnd (ℓ) is a manifold, see [57, Proposition 3.1].

We now see how chain and polygon spaces are determined by some combina-torics of their length vector ℓ = (ℓ1, . . . , ℓn). A subset J of 1, . . . , n is calledℓ-short (or just short) if

∑i∈J ℓi <

∑i/∈J ℓi. The complement of a short sub-

set is called long [NO! see Erratum 12.0.22]. If ℓ is generic, subsets are eithershort or long. Short subsets form, with the inclusion, a poset Sh(ℓ). DefineShn(ℓ) = J ⊂ 1, . . . , n− 1 | J ∪ n ∈ Sh(ℓ) .

For J ⊂ 1, . . . , n, let HJ be the hyperplane (wall) of Rn defined by

HJ :=(ℓ1, . . . , ℓn) ∈ Rn

∣∣∣∑

i∈J

ℓi =∑

i/∈J

ℓi

.

[See Comment 12.0.24.] The union H(Rn) of all these walls determines a set of openchambers in (R>0)

n whose union is the set of generic length vectors (a chamber isa connected component of (R>0)

n −H(Rn)). We denote by Ch(ℓ) the chamber ofa generic length vector ℓ. Note that Ch(ℓ) = Ch(ℓ′) if and only if Sh(ℓ) = Sh(ℓ′).

Let Symn be the group of bijections of 1, . . . , n; we see Symn−1 as thesubgroup of Symn formed by those bijections fixing n. If X is a set, the groupSymn acts on the Cartesian product Xn by

(x1, . . . , xn)σ = (xσ(1), . . . , xσ(n)) .

The notation emphasizes that this action is on the right: an element x ∈ Xn

is formally a map x : 1, . . . , n → X (xi = x(i)) and σ ∈ Symn acts by pre-composition, i.e. xσ = xσ. We shall use this action on various n-tuples, inparticular on length vectors.


Lemma 10.3.3. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ

′n) be two generic length

vectors. Then, the following conditions are equivalent

(1) Shn(ℓ) and Shn(ℓ′) are poset isomorphic.

(2) Sh(ℓ) and Sh(ℓ′) are poset isomorphic via a bijection σ ∈ Symn−1.(3) Ch(ℓ′) = Ch(ℓσ) for some σ ∈ Symn−1.

Moreover, if one of the above conditions is satisfied, there are O(d−1)-diffeomorphismsof manifolds pairs

h : (BCnd (ℓ), Cnd (ℓ))≈−→ (BCnd (ℓ′), Cnd (ℓ′))

andh : (BCnd (ℓ), Cnd (ℓ))

≈−→ (BCnd (ℓ′), Cnd (ℓ′)) .Proof. Implications (1)⇐ (2)⇔ (3) are obvious. Let us prove that (1)⇒ (2).

Let σ ∈ Symn−1 be the permutation giving the poset isomorphism Shn(ℓ) ≈Shn(ℓ

′). Replacing ℓ by ℓσ, we may assume that Shn(ℓ) = Shn(ℓ′). We now observe

that Shn(ℓ) determines Sh(ℓ). Indeed, let J ⊂ 1, . . . , n. Then, either n ∈ J orn ∈ J = 1, . . . , n − J , and thus Shn(ℓ) tells us whether J ∈ Sh(ℓ) (or J ∈ Sh(ℓ)).

It remains to prove that (3) implies the existence of the O(d−1)-diffeomorphismsh and h. As the G action commutes with the O(d − 1)-action (G is naturallyin the center of O(d − 1)), it suffices to construct h, which will induce h. Ifσ ∈ Symn−1, then the correspondence z → zσ defines an O(d− 1)-diffeomorphism

(BCnd (ℓ), Cnd (ℓ))≈−→ (BCnd (ℓσ), Cnd (ℓσ)). Replacing ℓ by ℓσ, we may thus assume that

Ch(ℓ) = Ch(ℓ′) and σ = id.Consider the smooth map L : (Rd−0)n−1 → (R>0)

n given, for x = (x1, . . . , xn−1)by

L(x) =(|x1|, . . . , |xn−1|, 〈F (x), e1〉

),

where F (x) =∑n−1

i=1 xi. Observe that the map (Sd−1)n−1 → (Rd − 0)n−1 send-

ing (z1, . . . , zn−1) to (ℓ1z1, . . . , ℓn−1zn−1) induces a diffeomorphism γℓ : BCnd (ℓ)≈−→

L−1(ℓ) such that F γ = F , the robot arm map of (10.3.3). If ℓ is generic, thenℓ is a regular value of L. Indeed, let x = (x1, . . . , xn−1) ∈ L−1(ℓ). For each i =1, . . . , n− 1, one can construct a path xi(t) = (xi1(t), . . . , x

in−1(t)) ∈ (Rd −0)n−1

with x(0) = x such that L(xi(t)) = (ℓi1(t), . . . , ℓin(t)) satisfies ℓij(t) = ℓj for j 6= i

and ℓii(t) = ℓi + αt with α 6= 0. For i = n, this follows from the proof of Corol-lary 10.3.2. Suppose that i ≤ n − 1. If x is not a lined configuration, then xi and∑j 6=i xj are linearly independent and generate a 2-dimensional plane Π, containing

F (x). There are rotations ρit and ρt of Π, depending smoothly on t, such that

ρit((1 + t)xi) + ρt(∑

j 6=i xj) = F (x). Hence, X i(t) may be defined as

xij(t) =

ρit((1 + t)xi) if j = i

ρt(xj) if j 6= i .

Finally, if x is a lined configuration, then F (x) and e1 are linearly independent(since ℓ is generic). They thus generate a 2-dimensional plane Ω. Let xi(t) definedby xii(t) = (1 + t)xi and xij(t) = xj when j 6= i. If t is small enough, there is

a unique rotation rt of Ω such that 〈rt(F (xi(t)), e1〉 = ℓn. We can thus definexij(t) = ρt(x

ij(t)).

As Ch(ℓ) is convex, it contains the segment [ℓ, ℓ′] consisting of only genericlength vectors. What has been done above shows that the map L is transverse to


[ℓ, ℓ′]. Therefore, X = L−1([ℓ, ℓ′]) is O(d)-cobordism between BCnd (ℓ) and BCnd (ℓ′).Let pd−1 : Rd → Rd−1 be the projection onto the last d− 1 coordinates. As in the

proof of Corollary 10.3.2, the map P : X → Rd−1 defined by P (x) = pd−1(F (x))is transverse to 0. Thus, Y = P−1(0) is a submanifold of X of codimensionn − 1 and the pair (X,Y ) is a cobordism of pairs between (BCnd (ℓ), Cnd (ℓ)) and(BCnd (ℓ′), Cnd (ℓ′)). The map L : X → [ℓ, ℓ′] has no critical point. The standardRiemannian metric on (Rd)n induces an O(d − 1)-invariant Riemannian metric on(X,Y ). Following the gradient lines of π for this metric provides the requiredO(d− 1)-equivariant diffeomorphism h.

For n ≤ 9, a list of all chambers (modulo the action of Symn) was obtainedin [90]. Their numbers are as follows (for n = 10, it was computed independentlyby Minfeng Wang and Dirk Schutz: see the Web complement to [90]). [See Com-ment 12.0.24.]

n 3 4 5 6 7 8 9 10

Nb of chambers 2 3 7 21 135 2’470 175’428 52’980’624

Geometric descriptions of several chain and polygon spaces for generic ℓ areprovided in [85], as well as some general constructions. Among them, the operationof “adding a tiny edge”, which we now describe. Let ℓ = (ℓ2, . . . , ℓn) be a genericlength vector. If ε > 0 is small enough, the n-tuple ℓ+ := (δ, ℓ2, . . . , ℓn) is a genericlength vector for 0 < δ ≤ ε.

Lemma 10.3.4. There are O(d − 1)-equivariant diffeomorphisms

BΦ: BCnd (ℓ+)≈−→ Sd−1 × BCn−1d (ℓ) and Φ: Cnd (ℓ+)

≈−→ Sd−1 × Cn−1d (ℓ) ,

where Sd−1×BCmd−1(ℓ) and Sd−1×Cmd−1(ℓ) are equipped with the diagonal O(d−1)-action.

Proof. The diffeomorphism Φ is constructed in [85, Proposition 2.1]. Theconstruction can be easily adapted to give BΦ.

10.3.2. Equivariant cohomology. Let M = (Sd−1)n−1. The G-invariantMorse function f = fℓ : M → R of Lemma 10.3.1 satisfies the hypotheses of Propo-sition 7.6.13, i.e. MG = Crit f . Therefore, M is G-equivariantly formal and therestriction morphism

(10.3.4) r : H∗G(M)→ H∗G(MG) ≈

⊕

J

H∗G(PJ ) ≈⊕

J

Z2[uJ ]

is injective (this also follows from Lemma 7.3.6 and Proposition 7.3.9). The vari-ables uJ are of degree one and the Z2[u]-module structure on H∗G(M

G) is given bythe inclusion u 7→∑

J uJ .In the remainder of this section, whenever xi (i ∈ N) are formal variables in a

polynomial ring and J ⊂ N, we set xJ =∏j∈J xj . In particular, x∅ = 1.

Proposition 10.3.5. For n ≥ 2, there is a GrA[u]-isomorphism

(10.3.5) Z2[u,A1 . . . , An−1, B1, . . . , Bn−1]/I ≈−→ H∗G((S

d−1)n−1)

where the variables Ai and Bi are of degree d − 1 and I is the ideal generated bythe families of relators


(a) Ai +Bi + ud−1 i = 1, . . . , n− 1(b) A2

i +Aiud−1 i = 1, . . . , n− 1

Moreover, using (10.3.4), one has for J ⊂ 1, 2, . . . , n− 1:

(10.3.6)

r(AJ ) =∑

J⊂K

u|J|(d−1)K

r(BJ ) =∑

J∩K=∅

u|J|(d−1)K

Proposition 10.3.5 generalizes Examples 7.6.9 and 7.4.5, with the slightly dif-ferent notations of (10.3.4) for the equivariant cohomology of the fixed point set.

Proof. The proof proceeds by induction on n. It starts with n = 2, usingExample 7.6.9. For the induction step, set M = (Sd−1)n−1 = M × M0, where

M = (Sd−1)n−2 and M0 = Sd−1. The induction hypothesis implies that

H∗G(M) ≈ Z2[u, A1 . . . , An−2, B1, . . . , Bn−2]/I

where I is the ideal generated by the families Ai + Bi + ud−1 and A2i + Aiu

d−1

(i = 1, . . . , n− 2). The Z2[u]-module structure is obtained by identifying u with u.Also,

H∗G(M0) ≈ Z2[u0, A,B]/(A+B + ud−10 , A2 = ud−10 A)

and the Z2[u]-module structure is obtained by identifying u with u0. By Theo-rem 7.4.3, the strong equivariant cross product provides an isomorphism

×G : H∗G(M)⊗Z2[u] H

∗G(M0)

≈−→ H∗G(M) .

Setting Ai = Ai ×G 1, Bi = Bi ×G 1 (i = 1, . . . , n − 2), An−1 = 1 ×G A andBn−1 = 1×G B gives the induction step for the isomorphism (10.3.5).

We now prove the induction step for (10.3.6). The fixed points of M are

denoted by PJ , indexed by J ⊂ 1, . . . , n − 2. We denote the fixed point ofM0 = Sd−1 ⊂ R×Rd−1 by ωmin = (1, 0) and ωmax = (−1, 0) (corresponding to theextrema of the Morse function (t, x) 7→ −t). Set H∗G(M

G0 ) ≈ Z2[umin] ⊕ Z2[umax].

For J ⊂ 1, . . . , n− 2, then PJ = PJ × ωmin and PJ∪n−1 = PJ × ωmax. Hence,for i = 1, . . . , n− 2, one has, using the obvious notations, that

r(Ai) = r(Ai×G 1)

= r(Ai)×G r0(1) by naturality of ×G

=[ ∑

J⊂1,...,n−2i∈J

u|J|(d−1)J

]×G [1min + 1max] by induction hypothesis

=∑

Ji∈J

u|J|(d−1)J +

∑

J∪n−1i∈J

u|J|(d−1)J (J ⊂ 1, . . . , n− 2)

=∑

J⊂1,...,n−1i∈J

u|J|(d−1)J .


As for i = n− 1, one has

r(An−1) = r(1×G A)

= r(1)×G r0(A)

=[ ∑

J⊂1,...,n−2

1J]×G ud−1max

=∑

J⊂1,...,n−2

ud−1J∪n−1

=∑

J⊂1,...,n−1n−1∈J

u|J|(d−1)J .

This proves (10.3.6) for r(Ai), i = 1, . . . , n− 1. The formula for r(Bi) are deducedusing relators (a). The formulae for r(AJ ) and r(BJ ) follow since r is multiplicative.

We are now ready to compute the G-equivariant cohomology of BCnd (ℓ).

Theorem 10.3.6. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. There is aGrA[u]-isomorphism

Z2[u,A1 . . . , An−1, B1, . . . , Bn−1]/Iℓ ≈−→ H∗G(BCnd (ℓ)) ≈ H∗(BC

n

d (ℓ))

where the variables Ai and Bi are of degree d − 1 and Iℓ is the ideal generated bythe families of relators

(a) Ai +Bi + ud−1 i = 1, . . . , n− 1(b) A2

i +Aiud−1 i = 1, . . . , n− 1

(c) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long(d) BJ J ⊂ 1, . . . , n− 1 and J is long.

Proof. LetM = (Sd−1)n−1,M− = f−1((−∞,−ℓn]) andM+ = f−1([−ℓn,∞]),with the inclusions j± : M± → M (f = fℓ, the Morse function of Lemma 10.3.1).One has M− ∩ M+ = B = BCnd (ℓ) = f−1(−ℓn). The G-invariant Morse func-tion f : M → R satisfies the hypotheses of Proposition 7.6.13. The latter impliesthat the morphism H∗G(M) → H∗G(B) induced by the inclusion is surjective withkernel equal to kerH∗Gj− + kerH∗Gj+. By Proposition 10.3.5, H∗G(M) is GrA[u]-isomorphic to Z2[u,A1 . . . , An−1, B1, . . . , Bn−1]

/I where I is the ideal generated

by families (a) and (b). We shall prove that kerH∗Gj− is the ideal generated byrelators (c) and that kerH∗Gj+ is the ideal generated by relators (d).

The critical point PJ satisfies

f(PJ) =∑

i∈J

ℓi −∑

i/∈J

ℓi .

Therefore,

(10.3.7) PJ ∈M− ⇐⇒ f(PJ) < −ℓn ⇐⇒ J ∪ n is short .


Therefore, one has a commutative diagram

(10.3.8)

H∗G(M)

H∗Gj−

// r // H∗G(MG)

H∗GjG−

≈ //⊕

J⊂1,...,n−1

Z2[uJ ]

pr−

H∗G(M−)// r− // H∗G(M

G− )

≈ //⊕

J⊂1,...,n−1J∪n short

Z2[uJ ]

That r and r− are injective follows from Theorem 7.6.6. Hence, for x ∈ H∗G(M),H∗j−(AJ ) = 0 if and only if pr−r(x) = 0. Since M is equivariantly formal (byTheorem 7.6.6 again), Theorem 10.3.6 implies that

H∗(M) ≈ H∗G(M)/(u) ≈ Z2[A1, . . . , An−1]/(A2i ) .

By the Leray-Hirsch theorem, H∗G(M)/(u) is then isomorphic to the free Z2[u]-module with basis Aj | J ⊂ 1, . . . , n− 1 (or Bj | J ⊂ 1, . . . , n− 1). Thus,x ∈ H∗G(M) may be uniquely written as x =

∑J⊂1,...,n−1 λJAJ , with λJ ∈ Z2[u].

Let J0 ⊂ 1, . . . , n− 1 minimal (for the inclusion) such that λJ0 6= 0. By (10.3.7),one has

r(x) = λJ0uJ0 mod⊕

J⊂1,...,n−1J 6=J0

Z2[uJ ] .

Hence, if x ∈ kerH∗Gj−, we deduce using Diagram (10.3.8) that J0 ∪ n is long.Therefore, λJ0AJ0 ∈ kerH∗Gj− and x + λJ0AJ0 ∈ kerH∗Gj−. Repeating the aboveargument with x+ λJ0 and so on proves that

x =∑

J⊂1,...,n−1

λJAJ ∈ kerH∗Gj− ⇐⇒ λJ = 0 whenever J ∪ n is short .

This proves that kerH∗Gj− is the Z2[u]-module generated by relators (c) (sinceAJAK = AJ∪K , this is an ideal).

In the same way, we prove that kerH∗Gj+ is the Z2[u]-module generated byrelators (d). Details are left to the reader.

Corollary 10.3.7. For a generic length vector ℓ = (ℓ1, . . . , ℓn), there is aGrA[u]-isomorphism

Z2[u,A1 . . . , An−1]/Iℓ ≈−→ H∗G(BCnd (ℓ))

where the variables Ai are of degree d−1 and Iℓ is the ideal generated by the familiesof relators

(1) A2i +Aiu

d−1 i = 1, . . . , n− 1(2) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long

(3) ud−1∑

K⊂J

AK u(|J−K|−1)(d−1) J ⊂ 1, . . . , n− 1 and J is long

Note that, by (2), only the sets K ⊂ J with K ∪ n being short occur in thesum of relators (3).


Proof. This presentation of H∗G(BCnd (ℓ)) is algebraically deduced from that ofTheorem 10.3.6. The generators Bi are eliminated using relators (a). Realtors (b)and (c) become respectively (1) and (2). Relators (d) become relators (3). Indeed,

BJ =∏i∈J Bi

=∏i∈J(Ai + ud−1) using (a)

=∑

K⊂J AK u|J−K|(d−1) plain expansion

= ud−1∑

K⊂J

AK u(|J−K|−1)(d−1) as AJ = 0 since J is long.

Example 10.3.8. Elementary geometry easily shows that BCnd (ℓ) = ∅ if andonly if n is long (see also Example 10.3.23 below). In Corollary 10.3.7, we see that

if n is long, then relator (2) for J = ∅ implies that 1 ∈ Iℓ and thus H∗G(BCnd (ℓ)) =0. Compare Example 10.3.21.

Example 10.3.9. Suppose that ℓ is generic and that ℓn = −α+∑n−1i=1 ℓi, with

α > 0 small enough such that J ∪ n is short only for J = ∅. Hence, relators (2)imply that Ai = 0 for i = 1, . . . , n− 1. The only subset J of 1, . . . , n− 1 which islong is 1, . . . , n− 1 itself. Thus, the family of relators (3) contains one element,in which the only non-zero term in the sum occurs for K = ∅. This relator thushas the form u(n−1)(d−1) and we get

(10.3.9) H∗G(BCnd (ℓ)) ≈ Z2[u]/(u(n−1)(d−1)) .

Notice that, with our hypothesis, −ℓn is a regular value of f which is just above aminimum. Thus, BCnd (ℓ) = f−1(−ℓ) is G-diffeomorphic to the sphere S(n−1)(d−1)−1

endowed with the antipodal involution (the isotropy representation of G on thetangent space to M at the minimum P∅ of f). As the G-action on BCnd (ℓ) is free,one has

H∗G(BCnd (ℓ)) ≈ H∗(BCn

d (ℓ)/G) ≈ H∗(RP (n−1)(d−1)−1)

which is compatible with (10.3.9).

Proposition 10.3.9 will help us to compute H∗G(Cnd (ℓ)), after introducing somepreliminary material. We use the robot arm map Fℓ : M = (Sd−1)n−1 → Rd =R × Rd−1 defined in (10.3.3). Let N = F−1ℓ (R × 0). If ℓ′ = (ℓ1, . . . , ℓn−1) isitself generic, then N is a closed submanifold of codimension d − 1 in M . Indeed,except at F−1ℓ (0), the robot arm map is clearly transverse to R× 0 (use that F isSO(d)-equivariant). If ℓ′ is generic, then 0 is a regular value of Fℓ (see the proof ofCorollary 10.3.2). Hence, Fℓ is everywhere transverse to R× 0.

A slight change of e.g. ℓ1 (which does not change the G-diffeomorphism type ofthe pair (BCnd (ℓ), Cnd (ℓ)) by Lemma 10.3.3) will make ℓ′ is generic. Hence, withoutloss of generality, one may assume that N is a closed G-invariant submanifold ofM . One has Cnd (ℓ) = N− ∩ N+ where N± = N ∩M± (notation of the proof ofProposition 10.3.7).

There is a G-equivariant map φt : M− →M− such that φ0 = id and φ1(M−) =N−. Indeed, for z ∈ M−, denote by Πz the 2-plane in Rd generated by e1 andFℓ(z). Define ρt(z) ∈ SO(d) to be the rotation of angle cos−1(t|Fℓ(z)|/|fℓ(z)|)on Πz, the identity on Π⊥z and such that 〈ρtFℓ(z), e1〉 ≥ fℓ(z). The retractionby deformation φt is defined by φt(z1, . . . , zn−1) = (ρt(z1), . . . , ρt(zn−1)). The


existence of φt implies that

(10.3.10) H∗G(M−) ≈ H∗G(N−) .

The restriction of fℓ,N : N → R of fℓ is also a G-invariant Morse function onM , with Crit fℓ,N = Crit fℓ (see [83, §3]; the index of a critical point P is differentfor fℓ,N and f when fℓ(P ) < 0).

Proposition 10.3.10. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector and leti : Cnd (ℓ) → BCnd (ℓ) be the inclusion. Then, H∗Gi : H

∗G(BCnd (ℓ)) → H∗G(Cnd (ℓ)) is

surjective, with kernel equal to Ann (ud−1), the annihilator of ud−1.

Proof. Let (B, C) = (BCnd (ℓ), Cnd (ℓ)). Consider the commutative diagram

H∗G(M−)// //

H∗Gj≈

H∗G(B)H∗Gi

H∗G(N−)// // H∗G(C)

where all the arrows are induced by the inclusions. The horizontal maps are indi-cated to be surjective: this follows from Proposition 7.6.13 since Crit f = MG =NG. That H∗Gj is an isomorphism was noticed in (10.3.10). Hence, H∗Gi is surjec-tive.

Let B = B/G and C = C/G. As the G-action on (B, C) is free, the vertical mapsin the diagram

CGiG //

BG

C i // B

are homotopy equivalences (see Lemma 7.1.4). As B and C are smooth closedmanifolds, Proposition 5.4.5 implies that kerH∗Gi is the annihilator of the Poincare

dual PD(C) ∈ Hd−1(B) ≈ Hd−1G (B). It thus remains to show that PD(C) = ud−1.

Let pd−1 : Rd = R × Rd−1 → Rd−1 be the projection onto the second factor.The map ϕ : B → (Rd−1)n−1 − 0 defined by

ϕ(z1, . . . , zn−1) = (pd−1(z1), . . . , pd−1(zn−1))

is smooth, G-equivariant (for the involution x 7→ −x on (Rd−1)n−1) and satis-fies C = ϕ−1((Rd−1)n−2 − 0). It thus descends to a smooth map ϕ : B →RP (n−1)(d−1)−1, such that C = ϕ−1(RP (n−2)(d−1)−1). As in the proof of Corol-lary 10.3.2, one shows that ϕ is transverse to RP (n−2)(d−1)−1. By Proposition 5.4.7,one has

PD(C) = H∗ϕ(PD(RP (n−2)(d−1)−1)) = H∗ϕ(ud−1) = ud−1 .

The two occurrences of the letter u in the above formulae is a slight abuse oflanguage, permitted by the considerations of Lemma 7.1.4: the G-actions under


consideration are all free and in the commutative diagram

H∗(RP (n−1)(d−1)−1)≈ //

H∗ϕ

H∗G((Rd−1)n−1 − 0)

H∗Gϕ

H∗(B) ≈ // H∗G(B)

the generator of H1(RP (n−1)(d−1)−1) is sent to u.

We are now ready to compute H∗G(Cnd (ℓ)).

Theorem 10.3.11. For a generic length vector ℓ = (ℓ1 . . . , ℓn), there is aGrA[u]-isomorphism

Z2[u,A1 . . . , An−1]/Iℓ ≈−→ H∗G(Cnd (ℓ)) ≈ H∗(Cnd (ℓ))

where the variables Ai are of degree d−1 and Iℓ is the ideal generated by the familiesof relators

(1) A2i +Aiu

d−1 i = 1, . . . , n− 1(2) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long

(3′)∑

K⊂J

AK u(|J−K|−1)(d−1) J ⊂ 1, . . . , n− 1 and J is long

Proof. We use the notations of the proof of Theorem 10.3.6, with M =(Sd−1)n−1, etc. Recall from Proposition 10.3.5 that

H∗G(M) ≈ Z2[u,A1 . . . , An−1]/I1

where I1 is the ideal generated by relators (1). Denote by I2, I3′ and I3 theideals of H∗(M) generated by, respectively, relators (2), (3′) and relators (3) ofCorollary 10.3.7. It was shown in Theorem 10.3.6 and Corollary 10.3.7 that

(10.3.11) J = ker(H∗G(M)→ H∗G(BCnd (ℓ)) = I2 + I3 .In view of Proposition 10.3.10, we have to prove that the “quotient ideal”

J = x ∈ H∗G(M) | ud−1x ∈ J is equal to I2 + I3′ .

That I2+I3′ ⊂ J is obvious. For the reverse inclusion, let x ∈ J . By (10.3.11),one has ud−1x = y2 + y3 for some y2 ∈ I2 and y3 ∈ I3. As I3 = ud−1I3′ , we canwrite y3 = ud−1y3′ with y3′ ∈ I3′ . Let z = y + y3′ . Then ud−1z ∈ I2. We shallprove that z ∈ I2.

As noticed in the proof of Theorem 10.3.6, H∗G(M) is the free Z[u]-modulegenerated by AJ (J ⊂ 1, . . . , n− 1). Thus, z admits a unique expression

z =∑

J⊂1,...,n−1

λJAJ ,

with λJ ∈ Z2[u]. Hence,

(10.3.12) ud−1z =∑

J⊂1,...,n−1

(ud−1λJ )AJ .


But, as ud−1z ∈ I2, one has

(10.3.13) ud−1z =∑

J⊂1,...,n−1J∪nlong

µJAJ .

As, H∗G(M) is the free Z[u]-module generated by the classes AJ , one deducesfrom (10.3.12) and (10.3.13) that λJ = 0 if J ∪ n is short. Thus, z ∈ I2.

The equality J = I2 + I3′ may be also obtained using a partial Groebnercalculus with respect to the variable u, as presented in [89, §6].

Remark 10.3.12. In the case d = 2, where Cn2 (ℓ) = Nn2 (ℓ), the presentation of

H∗(Cn2 (ℓ)) of Theorem 10.3.11 was obtained in [89, Corollary 9.2], using techniquesof toric manifolds.

Example 10.3.13. It is easy to see that Cnd (ℓ) = ∅ if and only k is long forsome k ∈ 1, . . . , n. If k = n then relator (2) for J = ∅ implies that 1 ∈ Iℓ andthus H∗G(Cnd (ℓ)) = 0. If k < n, it is relator (3′) for J = k which implies that1 ∈ Iℓ.

Example 10.3.14. Let ℓ = (1, 1, 1, ε), with ε < 1. The presentation ofH∗G(BC4d(ℓ))given by Corollary 10.3.7 takes the form

H∗G(BCnd (ℓ)) ≈ Z2[u,A1, A2, A3]/Iℓ

with Iℓ being the ideal generated by A2i +Aiu

d−1 (i = 1, 2, 3), AJ for |J | = 2, andrelators (3) for J = 1, 2, 1, 3 and 2, 3, which are

ud−1(ud−1 +A1 +A2)

ud−1(ud−1 +A2 +A3)

ud−1(ud−1 +A1 +A3) .

The sum of these relators equals u2(d−1) which thus belongs to Iℓ. Relator (3) forJ = 1, 2, 3 does not bring new generators for Iℓ.

The presentation of H∗G(C4d(ℓ)) given by Theorem 10.3.11 is similar, with rela-tors (3) replaced by relators (3′):

ud−1 +A1 +A2

ud−1 +A2 +A3

ud−1 +A1 +A3 .

The sum of these relators being equal to ud−1, we get that the three classes Ai ∈Hd−1G (BC4d(ℓ)) are mapped to the same class A ∈ Hd−1

G (C4d(ℓ)). Therefore,(10.3.14) H∗G(C4d(ℓ)) ≈ Z2[u,A]

/(ud−1, A2) .

Note that C4d(ℓ) is G-diffeomorphic to the unit tangent space T 1Sd−1 (by orthonor-malizing (z1, z2)). Thus, (10.3.14) is a presentation ofH∗(C4d(ℓ)) ≈ H∗((T 1Sd−1)/G).

In the presentations of H∗G(BCnd (ℓ)) and H∗G(Cnd (ℓ)) given in Corollary 10.3.7and Theorem 10.3.11, the integer d is only used to fix the degree of the variablesAi. Here is an application of that.


Lemma 10.3.15. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. For d ≥ 2,there is an isomorphism of graded rings

ΨBCd : H∗G(BCn2 (ℓ))≈−→ H

∗(d−1)G (BCnd (ℓ)) and ΨCd : H

∗G(Cn2 (ℓ))

≈−→ H∗(d−1)G (Cnd (ℓ))

which multiplies the degrees by d− 1.

Proof. For an integer a ≥ 2, set Ma = (Sa−1)n−1 and BCa = BCna (ℓ). AsMa is equivariantly formal, Corollary 4.7.20 applied to the bundle Ma → (Ma)G →RP∞ implies that

Pt(H∗G(Ma)) = Pt(Ma) ·Pt(RP

∞) =(1 + ta−1)n−1

1− t .

Hence

Pt(H∗(d−1)G (Md)) =

(1 + td−1)n−1

1− td−1 = Ptd−1(H∗G(M2))

which implies that

(10.3.15) dimHpG(M2) = dimH

p(d−1)G (Md)

for all p ∈ N.By eliminating the variables Bi in the presentation of H∗G(Ma) given in Propo-

sition 10.3.5, we get the presentation

H∗G(Ma) ≈ Z2[ua, Aa1 . . . , A

an−1]

/((Aai )

2 = ua−1a Aai),

where Aai is of degree a− 1 and ua is of degree 1. Therefore, the correspondences

u2 7→ ud−1d and A2i 7→ Aai define a homomorphism of graded rings Ψd : H

∗G(M2)→

H∗(d−1)G (Md), multiplying the degrees by d−1, which is clearly surjective. By (10.3.15),

Ψd is an isomorphism.By Corollary 10.3.7, H∗G(BCa) is the quotient of H∗G(Ma) by Ia2 + Ia3 , where

Iaj is the ideal generated by relators (j) of Corollary 10.3.7. As Ψd(I2j ) = Idj ∩H∗(d−1)G (Md), the isomorphism Ψd descends to the required isomorphism ΨBCd . In

the same way, we construct ΨCd using Theorem 10.3.11.

10.3.3. Non-equivariant cohomology. TheG-cohomology computations ofCorollary 10.3.7 and Theorem 10.3.11 give some information on the non-equivariantcohomology of BCnd (ℓ) and Cnd (ℓ). We start with the big chain space.

Theorem 10.3.16. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. The Poincarepolynomial of BCnd (ℓ) is

(10.3.16) Pt(BCnd (ℓ)) =∑

J⊂1,...,n−1J∪nshort

t|J|(d−1) +∑

K⊂1,...,n−1K long

t|K|(d−1)−1 .

The proof of Theorem 10.3.16 makes use of the simplicial complex Sh×n (ℓ) whosesimplexes are the non-empty subsets of the poset Shn(ℓ).

Proof. Let BCd = BCnd (ℓ). By (7.1.8), one has a short exact sequence

(10.3.17) 0→ H∗G(BCd)/(u)ρ−→ H∗(BCd) tr∗−−→ Ann (u)→ 0 ,

whence

(10.3.18) Pt(BCd) = Pt(H∗G(BCd)/(u)) +Pt(Ann (u)) .


The presentation of H∗G(BCd) given in Corollary 10.3.7 implies that

(10.3.19) H∗G(BCd)/(u) ≈ Z2[A1, . . . , An−1]/J

where J is the ideal generated by the squares A2i of the variables and the monomials

AJ when J ∪ n is long. Therefore,(10.3.20) H∗G(BCd)/(u) ≈ Λd−1(Sh

×n (ℓ)) ,

the face exterior algebra of the simplicial complex Sh×n (ℓ) (see § 4.7.8). Then, byCorollary 4.7.52,(10.3.21)

Pt(H∗G(BCd)/(u)) = Pt(Λd−1(Sh

×n (ℓ))) =

∑

σ∈S(Sh×n (ℓ))

t(dimσ+1)(d−1) =∑

J∈Shn(ℓ)

t|J|(d−1) .

Let us assume that d ≥ 3. The graded algebra H∗G(BCd)/(u) is concentratedin degrees ∗(d − 1). We claim that Ann (u) is concentrated in degrees ∗(d − 1) −1. Indeed, let us write H∗G(BCd) as the quotient H∗(M)/J as in the proof ofTheorem 10.3.11. A class 0 6= z ∈ Hp

G(BCd) is the image of z ∈ H∗G(M). As M isequivariantly formal, one has uz 6= 0. Hence, if z ∈ Ann (u), one has 0 6= uz ∈ J .As the ideal J is concentrated in degrees ∗(d− 1), we deduce that p = q(d− 1)− 1.Together with (10.3.17), this implies that

(10.3.22) H∗(d−1)(BCd) ≈ H∗G(BC)/(u) , H∗(d−1)−1(BCd) ≈ Ann (u)

and H∗(BCd) vanishes in other degrees. Since dimBCd = (n−1)(d−1)−1, Poincareduality gives the formula

(10.3.23) Pt(BCd) =∑

J∈Shn(ℓ)

t|J|(d−1) +∑

J∈Shn(ℓ)

t(n−1−|J|)(d−1)−1 .

Using (10.3.17), we thus get, for d ≥ 3, that(10.3.24)

Pt(Ann (u)) =∑

J∈Shn(ℓ)

t((n−1−|J|)(d−1)−1)(d−1) =∑


t|K|(d−1)−1 ,

where the last equality is obtained by re-indexing the sum with K = 1, . . . , n −1 − J . It remains to prove that (10.3.24) is also valid when d = 2.

Let us fix some integer d ≥ 3. By Lemma 10.3.15 and its proof, there is an

isomorphism of graded rings ΨBCd : H∗(BC2) ≈−→ H∗(d−1)(BCd) such that

(10.3.25) ΨBCd(Ann (u;H∗(BC2)

)= Ann (ud−1;H∗(d−1)(BCd)) ,

where the second argument in Ann( ) specifies the ring in which the first argument isconsidered. As the relators of the presentation of H∗(BCd) given in Corollary 10.3.7are in degree ∗(d− 1), the correspondence x 7→ ud−2x provides, for every p ≥ 0, anisomorphism of Z2-vector spaces

Φd : Hp(d−1)(BCd) ≈−→ H(p+1)(d−1)−1(BCd) .

We thus get an isomorphism of Z2-vector spaces

Φd : H∗(d−1)(BCd) ≈−→ H∗(d−1)−1(BCd)

multiplying the degrees by d− 2 and satisfying

(10.3.26) Ψd(Ann (ud−1;H∗(d−1)(BCd))

)= Ann (u;H∗(BCd)) .


From (10.3.25) and (10.3.26), we get

(10.3.27) td−2 Ptd−1

(Ann (u;H∗(BC2))

)= Pt(Ann (u;H

∗(BCd))) .The right hand of (10.3.27) being given by (10.3.24), we check that

Pt(Ann (u;H∗(BC2))) =

∑


t|K|−1

is the unique solution of Equation (10.3.27).We have thus proven that (10.3.24) is valid for all d ≥ 2. Together with (10.3.21)

and (10.3.18), this establishes the proposition.

Below is the counterpart of Theorem 10.3.16 for chain spaces. It requires thelength vector ℓ be dominated, i.e. satisfying ℓn ≥ ℓi for i ≤ n.

Theorem 10.3.17. Let ℓ be a generic length vector which is dominated. Then,the Poincare polynomial of Cnd (ℓ) is

(10.3.28) Pt(Cnd (ℓ)) =∑

J⊂1,...,n−1J∪nshort

t|J|(d−1) +∑


t(|K|−1)(d−1)−1.

Theorem 10.3.17 above reproves the computations of the Betti numbers of Cnd (ℓ)obtained by other methods in [60, Theorem 1] and [58, Theorem 2.1].

Proof. The proof is the same as that of Theorem 10.3.16, using the isomor-phism ΨCd of Lemma 10.3.15, instead of ΨBCd . The hypothesis that ℓ is dominatedis used to obtain the analogue of Equation (10.3.20), namely

(10.3.29) H∗G(Cnd (ℓ))/(u) ≈ Λd−1(Sh

×n (ℓ)) .

Indeed, let J ⊂ 1, . . . , n− 1 be a long subset and k ∈ J . As ℓ is dominated, theset (J − k) ∪ n is long. Therefore, the constant terms in relators (3′) of Theo-rem 10.3.11 vanish and these relators are all multiples of ud−1. Equation (10.3.29)thus follows from Theorem 10.3.11.

Example 10.3.18. The length vector ℓ = (1, 1, . . . , 1) is dominated and isgeneric if n = 2r+1. A subset J of 1, . . . , n is short if and only if |J | ≤ r. Hence,for d = 2, Equation (10.3.28) gives

Pt(C2r+12 (1, . . . , 1)) = Pt(N 2r+1

2 (1, . . . , 1)) =∑

k≤r−1

(n−1k

)tk +

∑

k≥r−1

(n−1k+2

)tk .

This formula was first proven in [110, Theorem C].

Remark 10.3.19. The hypothesis that ℓ is dominated is necessary (for anyd) in Theorem 10.3.17, as shown by the example C = C4d(ℓ) for ℓ = (1, 1, 1, ε)(see Example 10.3.14). As C is diffeomorphic to the unit tangent space T 1Sd−1,one has Pt(C) = 1 + td−2 + td−1 + t2(d−1)−1, as seen in Example 5.4.18, whileTheorem 10.3.17 would give 1 + 3td−2 + 3td−1 + t2(d−1)−1. What goes wrong isFormula (10.3.29). Using the presentation of H∗G(C) given in Theorem 10.3.11,

one gets that H∗(d−1)(C) is the quotient of Λd−1(Sh×n (ℓ)) by the constant terms of

relators (3′) in Theorem 10.3.11, namely∑j∈J AJ−j for all J ⊂ 1, . . . , n − 1

which are long.


Remark 10.3.20. Let C = Cnd (ℓ) with ℓ generic and dominated. As observed

in [65, Proposition A.2.4], H∗(C) is determined by H∗(d−1)(C) when d > 3, usingPoincare duality. Indeed, by Theorem 10.3.11 and Equation (10.3.29), Z = ZJ =ρ(AJ) | J ∈ Shn(ℓ) is a Z2-basis of H∗(d−1)(C) (Z∅ = 1). The bilinear mapH∗(d−1)(C)×H∗(d−1)−1(C)→ Z2 given by

(x, y) 7→ 〈x y, [C]〉

is non-degenerate (see Theorem 5.3.12) and thus identifiesH∗(d−1)−1(C) withH∗(d−1)(C)♯.Let Y = YJ | J ∈ Shn(ℓ) be the Z2-basis ofH

∗(d−1)−1(C) which is dual to Z underthis identification. In particular, Y∅ = [C], the generator of H(n−2)(d−1)−1(C) = Z2

(we say that Y is the Poincare dual basis to the basis Z). One has then the multi-plication table:

(10.3.30) ZJ ZK =

ZJ∪K if J ∩K = ∅ and J ∪K ∈ Shn(ℓ)

0 otherwise,

(10.3.31) ZJ YK =

YJ−K if K ⊂ J0 otherwise

and

(10.3.32) YJ YK = 0 .

Indeed, (10.3.30) comes from the corresponding relation amongst the classes AJ .Formula (10.3.32) is true for dimensional reasons, since d > 3. For (10.3.31), notethat, ZJ YK ∈ H(n−2−(|K|−|J|)(d−1)−1(C) and hence may be uniquely written asa linear combination

ZJ YK =∑

L∈L

λLYL ,

where L is the set L ∈ Shn(ℓ) with |L| = |K| − |J |. If I ∈ L, one has on one hand

〈ZI ∑

L∈L

λLYL, [C]〉 = λI

and on the other hand

〈ZI (ZJ YK), [C]〉 = 〈ZI∪J YK , [C]〉 =1 if J ∩K = ∅ and I ∪ J = k

0 otherwise.

This shows that λL = 1 if and only if L = K − J .

We finish this subsection with some illustrations and applications of Theo-rems 10.3.16 and 10.3.17. The lopsidedness lops (ℓ) of a length vector ℓ = (ℓ1, . . . , ℓn)is defined by

lops (ℓ) = infk | ∃ J ⊂ 1, . . . , n− 1 with J long and |J | = k ,with the convention that inf ∅ = 0. The terminology is inspired by that of [91]. Iflops (ℓ) > 0, one has

(10.3.33) dimSh×n (ℓ) = n− 2− lops (ℓ) .


Example 10.3.21. For a generic length vector ℓ = (ℓ1, . . . , ℓn), the conditionlops (ℓ) = 0 is equivalent to n being long. By Theorem 10.3.16 this is equivalentto BCnd (ℓ) = ∅: otherwise J = ∅ produces a non-zero summand in the first sumof (10.3.16) (Compare Example 10.3.8). The chamber of ℓ is unique, representedby e.g. ℓ0 = (ε, . . . , ε, 1) with ε < 1/(n− 1).

Example 10.3.22. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector with lops (ℓ) =

1. From the second sum of (10.3.16), we see that this is equivalent to Hd−2(BCnd (ℓ)) 6=0 (the reduced cohomology is relevant for the cases d = 2, where it says that BCn2 (ℓ)is not connected). We check that Shn(ℓ) is poset isomorphic to Shn(ℓ

0) whereℓ0 = (ε, . . . , ε, 2, 1), with ε < 1/(n− 2). By Lemma 10.3.3, the chamber of ℓ is welldetermined modulo the action of Symn−1. The O(d − 1)-diffeomorphism type of

BCnd (ℓ0) may be easily described. It is clear that BC2d(2, 1) ≈ Sd−2. Therefore, byLemma 10.3.4, BCnd (ℓ0) ≈ (Sd−1)n−2 × Sd−2. When d = 2, this is the only casewhere BCn2 (ℓ) is not connected, as shown by Theorem 10.3.16. Note that Cnd (ℓ0) isempty by Theorem 10.3.17, which is consistent with Proposition 10.3.10.

Example 10.3.23. Let ℓ = (ℓ1, . . . , ℓn) be a dominated generic length vectorwith lops (ℓ) = 2. We check that Shn(ℓ) is poset isomorphic to Shn(ℓ

0) where ℓ0 =(ε, . . . , ε, 1, 1, 1), with ε < 1/(n−3) (compare [85, Remark 2.4]). By Lemma 10.3.3,the chamber of ℓ is well determined modulo the action of Symn−1. As in the previ-ous example, we can describe the O(d−1)-diffeomorphism type of BCnd (ℓ0). Supposefirst that n = 3. The Morse function f : (S(d−1)2 → [−3, 3] of Lemma 10.3.1 hasno critical point between its minimum and the level set f−1(−1) = BC3d(1, 1, 1). Bythe Morse Lemma, BC3d(1, 1, 1) is diffeomorphic to S2(d−1)−1. Using Lemma 10.3.4,

we deduce that BCnd (ℓ0) ≈ (Sd−1)n−3 × S2(d−1)−1. In the same way, one provesthat Cnd (ℓ0) ≈ (Sd−1)n−3× Sd−2. Using Formula (10.3.28), we see that lops (ℓ) = 2if and only if Cn2 (ℓ) is not connected.

The following lemma uses the nilpotency class nil introduced in § 4.4.Lemma 10.3.24. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. Then

(a) If lops (ℓ) > 1, then lops (ℓ) = n− nilH>0(BCnd (ℓ)) + 1.

(b) Suppose that ℓ is dominated. If d > 2 or lops (ℓ) > 2, thenlops (ℓ) = n− nilH>0(Cnd (ℓ)) + 1.

Proof. Let BC = BCnd (ℓ). Suppose that lops (ℓ) = k ≥ 2. By (10.3.17)

and (10.3.20), the algebra H∗(BC) contains a copy of H∗G(BC)/(u) ≈ Λd−1(Sh×n (ℓ)).

By (10.3.33), dimSh×n (ℓ) = n − 2 − k. Therefore, there exists x1, . . . , xn−k−1 ∈Hd−1(BC) whose cup product v does not vanish in H(n−k−1)(d−1)(BC). By Poincareduality Theorem 5.3.12, there is w ∈ Hk(d−1)−1(BC) such that v w 6= 0 inH(n−1)(d−1)−1(BC). As k ≥ 2 the number k(d − 1) − 1 is strictly positive sinced ≥ 2. Thus, v w is a non-vanishing cup product of length n−k. Such a length isthe maximal possible, as seen using Sequence (10.3.17). Hence, nilH>0(BCnd (ℓ)) =n− k + 1. This proves (a).

The proof of (b) is similar, using Theorem 10.3.17 and its proof instead ofTheorem 10.3.16. As dim C = (n− 2)(d− 1)− 1, the class v is of degree (k− 1)(d−1)− 1. The latter is strictly positive if d > 2 or k > 2.

Corollary 10.3.25. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ′n) be two generic

length vectors. Suppose that, for some d ≥ 2, there exists a GrA-isomorphism


H∗(BCnd (ℓ)) ≈ H∗(BCnd (ℓ′)). Then

lops (ℓ) = lops (ℓ′) .

If ℓ and ℓ′ are both dominated, then the above equality holds true if there exists aGrA-isomorphism H∗(Cnd (ℓ)) ≈ H∗(Cnd (ℓ′)).

Proof. For the big chain space BCnd ( ), this is follows from Lemma 10.3.24,except when d = 2 and lops (ℓ) ≤ 1, cases which are covered by Examples 10.3.21and 10.3.22. The argument for Cnd ( ) is quite similar. The case lops (ℓ) = 2 iscovered by Example 10.3.23. The case lops (ℓ) = 1 is not possible if ℓ is dominated,so lops (ℓ) = 0 is equivalent to Cnd (ℓ) = ∅.

10.3.4. The inverse problem. By Lemma 10.3.3, the diffeomorphism typeof BCnd (ℓ) or Cnd (ℓ) is determined by the chamber Ch(ℓ) (up to the action ofSymn−1). The inverse problem consists of recovering Ch(ℓ) by algebraic topol-ogy invariants of BCnd (ℓ) (or Cnd (ℓ)). We start by the big chain space.

Proposition 10.3.26. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ

′n) be two generic

length vectors. Then, the following conditions are equivalent.

(1) Ch(ℓ′) = Ch(ℓσ) for some σ ∈ Symn− 1.

(2) BCnd (ℓ) and BCnd (ℓ′) are O(d − 1)-diffeomorphic.

(3) H∗G(BCnd (ℓ)) and H∗G(BCnd (ℓ′)) are GrA[u]-isomorphic.

Moreover, if d > 2 or n > 3, any condition (1)–(3) above is equivalent to

(4) H∗(BCnd (ℓ)) and H∗(BCn

d (ℓ′)) are GrA-isomorphic.

Finally, if d > 2 or if lops (ℓ) 6= 2, then any condition (1)–(3) above is equiva-lent to

(5) H∗(BCnd (ℓ)) and H∗(BCnd (ℓ′)) are GrA-isomorphic.

That (5) implies (1) is not known in general if d = 2. That (4) implies (3) is

wrong if n = 3 and d = 2. Indeed, BC32(1, 3, 1) and BC3

2(1, 1, 1) are connected closed1-dimensional manifolds, thus both diffeomorphic to S1 but, by Corollary 10.3.7,one has

H∗G(BC32(1, 3, 1)) ≈ Z2[u,A1]/(A2

1, u) while H∗G(BC32(1, 1, 1)) ≈ Z2[u]/(u2) .

Implications like (4) ⇒ (2) or (5) ⇒ (2) are in the spirit of Proposition 4.2.5:characterizing a closed manifold (within some class) by algebraic topology tools.This was the historical goal of algebraic topology (see p. 169).

Proof. As ℓ and ℓ′ are generic, one has H∗(BCnd (ℓ)) ≈ H∗G(BCnd (ℓ)) and thesame for ℓ′. The following implications are then obvious, except (a) which wasestablished in Lemma 10.3.3.

(1)(a) +3 (2) +3

$

(3) +3 (4)

(5)

We shall now prove that (3) ⇒ (1), (4) ⇒ (3) and finally (5) ⇒ (1).


(3) ⇒ (1). A GrA[u]-isomorphism H∗G(BCnd (ℓ))≈−→ H∗G(BCnd (ℓ′)) descends to a

GrA-isomorphism : H∗G(BCnd (ℓ))/(u)≈−→ H∗G(BCnd (ℓ′))/(u). By (10.3.20), this im-

plies that Λd−1(Sh×n (ℓ)) and Λd−1(Sh

×n (ℓ′)) areGrA-isomorphic. Using Lemma 4.7.51

and Proposition 4.7.50, we deduce that the simplicial complexes Sh×n (ℓ) and Sh×n (ℓ′)

are isomorphic. It follows that Shn(ℓ) and Shn(ℓ′) are poset isomorphic. By

Lemma 10.3.3, this implies (1).

(4) ⇒ (3). Let β : H∗(BCnd (ℓ))≈−→ H∗(BCnd (ℓ′)) is a GrA-isomorphism and let

β(u) = v. We must prove that v = u. This is obvious for d > 2 since H1(BCnd (ℓ′)) =Z2. If d = 2, Corollary 10.3.7 implies that

(10.3.34) xu = xv = x2

for all x ∈ H1(BCn2 (ℓ′)). By Corollary 10.3.7 again, H∗(BCn2 (ℓ′)) is generated in

degree 1, so (10.3.34) implies that (u + v)x = 0 for all x ∈ H∗(BCn2 (ℓ′)). By

Corollary 10.3.2, BCn2 (ℓ′) is a closed manifold of dimension > 1 (since n ≥ 4). Weconclude that v = u by Poincare duality, using Theorem 5.3.12.

(5) ⇒ (1). Suppose first that d > 2. By (10.3.22) and (10.3.20), Condition (5)implies that Λd−1(Sh

×n (ℓ)) and Λd−1(Sh

×n (ℓ′)) are GrA-isomorphic. The argument

is then the same as that for (3) ⇒ (1).We now assume that d = 2. By Corollary 10.3.25, Condition (5) implies that

lops (ℓ) = lops (ℓ′). Let L = lops (ℓ) = lops (ℓ′). The cases L = 0, 1 were treatedin Examples 10.3.21 and 10.3.22. Let us assume that L > 2. By Theorem 10.3.16and its proof, the subalgebra of H∗(BCn2 (ℓ)) (respectively: H∗(BCn2 (ℓ′))) gener-ated by the elements of degree one is isomorphic to Λ1(Sh

×n (ℓ)) (respectively:

Λ1(Sh×n (ℓ′))). By Condition (5), this implies that Λ1(Sh

×n (ℓ)) and Λ1(Sh

×n (ℓ′))

are GrA-isomorphic and the proof that Ch(ℓ′) = Ch(ℓσ) proceeds as that for (3)⇒ (1).

Here is the analogue of Proposition 10.3.26 for the chain spaces.

Proposition 10.3.27. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ

′n) be two generic

length vectors. Suppose that ℓ and ℓ′ are dominated. Then, the following conditionsare equivalent.

(1) Ch(ℓ′) = Ch(ℓσ) for some σ ∈ Symn− 1.

(2) Cnd (ℓ) and Cnd (ℓ′) are O(d − 1)-diffeomorphic.

(3) H∗G(Cnd (ℓ)) and H∗G(Cnd (ℓ′)) are GrA[u]-isomorphic.

Moreover, if d > 2 or n > 4, then any condition (1)–(3) above is equivalent to

(4) H∗(Cnd (ℓ)) and H∗(Cnd (ℓ′)) are GrA-isomorphic.

Finally, if d > 2 or if lops (ℓ) 6= 3, then any condition (1)–(3) above is equiva-lent to

(5) H∗(Cnd (ℓ)) and H∗(Cnd (ℓ′)) are GrA-isomorphic.

Proof. The proof is the same as that of Proposition 10.3.26, except for thefollowing small differences. For (3) ⇒ (1), instead of (10.3.20), one uses Equa-tion (10.3.29), using that ℓ and ℓ′ are dominated. For (4) ⇒ (3), the hypothesisthat n > 4 guarantees that dim Cn2 ( ) > 1. For (5)⇒ (1), one uses Theorem 10.3.17instead of Theorem 10.3.16.


Remark 10.3.28. In Proposition 10.3.27, implication (4) ⇒ (1) is wrong ford = 2 and n = 4: C42(1, 1, 1, 2) and C42(1, 2, 2, 2) are connected closed 1-dimensionalmanifolds, thus both diffeomorphic to S1. Implication (5) ⇒ (1) is not known ingeneral if d = 2. It is however true if one uses the integral cohomology: this difficultresult, conjectured by K. Walker in 1985 [202] was proved by D. Schutz in 2010[170], after being established when lops (ℓ) 6= 3 in [58, Theorem 4] (length vectorswith lopsidedness > 3 are called normal in [58, 170]).

The hypothesis that ℓ is dominated in Proposition 10.3.27 is essential, as seenby Proposition 10.3.29 and Lemma 10.3.30 below.

Proposition 10.3.29. Let ℓ be a generic length vector and let σ ∈ Symn. Ifd 6= 3, then H∗(Cnd (ℓ)) and H∗(Cnd (ℓσ)) are GrA–isomorphic.

Proposition 10.3.29 was first proved by V. Fromm in his thesis [65, Cor. 1.2.5].It is wrong if d = 3: for ε small, C43(ε, 1, 1, 1) is diffeomorphic to S2 × S1 (seeExample 10.3.23) while C43(1, 1, 1, ε) is diffeomorphic to T 1S2 ≈ RP 3 (see Exam-ple 10.3.14).

We give below a proof of Proposition 10.3.29 based on an idea of D. Schutz,using the following lemma.

Lemma 10.3.30. If d = 2, 4, 8, then Cnd (ℓ) is diffeomorphic to Cnd (ℓσ) for anyσ ∈ Symn.

The hypothesis d = 2, 4, 8 is essential in the above lemma. Indeed, for εsmall, C4d(ε, 1, 1, 1) is diffeomorphic to Sd−1 × Sd−2 (see Example 10.3.23) whileC4d(1, 1, 1, ε) is diffeomorphic to T 1Sd−1 (see Example 10.3.14). As d ≥ 2, these twospaces have the same homotopy type only when d = 2, 4, 8 (see Example 5.4.18).

Proof. Identifying Rd with C, H or O, we get a smooth multiplication onSd−1 with e1 as unit element. Consider the smooth map π : Nn

d (ℓ) → Cnd (ℓ) givenby

π(z1 . . . , zn) = −z−1n (z1, . . . , zn−1) .

The embedding j : Cnd (ℓ)→ Nnd (ℓ) given by

j(z1, . . . , zn−1) = (z1, . . . , zn−1,−e1)is a section of π. Consider the composite map

Cnd (ℓ)j // Nn

d (ℓ)hσ

≈// Nn

d (ℓσ)

π // Cnd (ℓσ)

where hσ(z) = zσ. Then, πhσ j is a diffeomorphism: a direct computation showsthat its inverse is π(hσ)−1j.

Proof of Proposition 10.3.29. The case d = 2 is covered by Lemma 10.3.30,so we assume that d ≥ 4. As observe in Remark 10.3.19, one has aGrA-isomorphism

(10.3.35) H∗(d−1)(Cnd (ℓ) ≈ Ωd(ℓ)

where Ωd(ℓ) is the quotient of Z2[A1, . . . , An−1] (Ai of degree d − 1) by the idealgenerated by A2

i , AJ when J ∪n is ℓ-long and∑

j∈J AJ−j when J is ℓ-long. By

Lemma 10.3.30, there exists a ring isomorphism q4 : Ω4(ℓ)≈−→ Ω4(ℓ

σ). But, in thedefinition of Ωd( ), the integer d is only used to fix the degree of the variables Ai,and thus the grading of Ωd( ) (as non-graded rings, the rings Ωd(ℓ) are isomorphic


for all d). Therefore, the isomorphism q4 defines a GrA-isomorphism qd : Ωd(ℓ)≈−→

Ωd(ℓσ) for all d ≥ 2. Together with (10.3.35), this gives a GrA-isomorphism

qd : H∗(d−1)(Cnd (ℓ))

≈−→ H∗(d−1)(Cnd (ℓσ)) when d ≥ 3.Without loss of generality, we may assume that ℓ is dominated. Remark 10.3.20

thus provides an additive basis Z ∪ Y of H∗(Cnd (ℓ)). The set Z ′ = ZJ = qd(ZJ) |J ∈ Sh(ℓ) is then a Z2-basis of H∗(d−1)(Cnd (ℓσ). Let Y ′ = Y ′J | J ∈ Sh(ℓ) be

the basis of H∗(d−1)−1(Cnd (ℓσ) which is Poincare dual to Z ′, as in Remark 10.3.20.Relations (10.3.30)–(10.3.32) hold true both for Z ∪Y in H∗(Cnd (ℓ)) and for Z ′∪Y ′in H∗(Cnd (ℓσ)). Therefore, qd extends to a GrA-isomorphism qd : H

∗(Cnd (ℓ))≈−→

H∗(Cnd (ℓσ)) when d > 3.

10.3.5. Spatial polygon spaces and conjugation spaces. The integralcohomology ring of the spatial polygon space Nn

3 (ℓ) has been computed in [89,Theorem 6.4]. The result is as follows.

Theorem 10.3.31. Let ℓ = (ℓ1 . . . , ℓn) be a generic length vector. Then, thereis a graded ring isomorphism

Z[v,A1 . . . , An−1]/Iℓ ≈−→ H∗(Nn

3 (ℓ);Z)

where the variables v and Ai are of degree 2 and Iℓ is the ideal generated by thefamilies of relators

(1) A2i +Aiv i = 1, . . . , n− 1

(2) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long

(3′)∑

K⊂J

AK v|J−K|−1 J ⊂ 1, . . . , n− 1 and J is long

Theorem 10.3.31 says in particular that Hodd(Nn3 (ℓ);Z) = 0. The Bockstein

exact sequence for 0 → Z → Z → Z2 → 0 (see [179, Ch. 5, Sec. 2, Theorem 11])thus implies that H∗(Nn

3 (ℓ)) ≈ H∗(Nn3 (ℓ);Z) ⊗ Z2. Using Theorem 10.3.11, the

correspondence Ai 7→ Ai and v 7→ u thus provides a graded ring isomorphism

H2∗(Nn3 (ℓ))

≈−→ H∗(Cn2 (ℓ)) ≈ H∗(Nn2 (ℓ))

which divides the degrees by half. This suggests that Nn3 (ℓ) is a conjugation space,

which we shall prove below. The involution τ on R3 given by the reflection throughthe horizontal plane induces an involution, still called τ , on Nn

3 (ℓ), with fixed pointset equal to Nn

2 (ℓ).

Proposition 10.3.32. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. Then,the space Nn

3 (ℓ) endowed with the involution τ is a conjugation manifold.

The proofs of this proposition use some Hamiltonian geometry, directly or indi-rectly, so their understanding requires some knowledge in the subject, as presentedin e.g. [12, Chapters II and III]. Recall that, if ℓ is generic, Nn

3 (ℓ) is a symplecticmanifold [118, 111, 88, 89]. The proof of Proposition 10.3.32 uses the followinglemma.

Lemma 10.3.33. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. Then,

(a) Nn3 (ℓ) is simply connected.

(b) τ induces the multiplication by (−1) on H2(Nn3 (ℓ);Z).

(c) τ is anti-symplectic.


Proof. The proof of (a) and (b) proceeds by induction on n. When n =3, there are two chambers up to permutation, represented by ℓ0 = (1, 1, 2) andℓ1 = (1, 1, 1). As N 3

3 (ℓ0) = ∅ and N 33 (ℓ1) = N 3

2 (ℓ1) = pt, (a) and (b) are true.For the induction step, we consider the diagonal-length function δ : Nn

3 (ℓ) → Rgiven by δ(z) = |ℓnzn − ℓn−1zn−1|. By Lemma 10.3.3, we can slightly changeℓn−1 without modifying the O(2)-diffeomorphism type of Cn3 (ℓ) and thus the τ -equivariant diffeomorphism type of Nn

3 (ℓ). Therefore, we can assume that ℓn−1 6=ℓn, in which case δ is a smooth map. Using [83, Theorem 3.2], we deduce that δ isMorse-Bott function. The critical points are of even index and are isolated, exceptpossibly for the two extrema. The preimage of the maximum is either a point orNn−1

3 (ℓ1, . . . , ℓn−2, ℓn−1 + ℓn) and the preimage of the minimum is either a point

or Nn−13 (ℓ1, . . . , ℓn−2, |ℓn−1 − ℓn|). This proves (a) by induction.The restriction δ′ of δ to Nn

2 (ℓ) ⊂ Nn−13 (ℓ) is also Morse-Bott with Crit δ′ =

Crit δ but the index of each critical point is divided in half. Thus, passing a criticalpoint of δ index 2 corresponds to add a conjugation 2-cell. This proves (b) byinduction on n.

To prove (c), we use that Nn−13 (ℓ) is the SO(3)-symplectic reduction at 0 of∏n

i=0 S2ℓiwhere S2

ℓiis the standard 2 sphere equipped with the SO(3)-homogeneous

symplectic form with symplectic volume equal to 2ℓi (see [89]). The involutionτ is clearly anti-symplectic on S2

ℓiand this property descends to the symplectic

reduction.

Proof of Proposition 10.3.32. We give below three proofs. The first oneis the one indicated in [87, Example 8.7].

1st proof. By induction on n, using the function δ : Nn3 (ℓ) → R of the proof

of Lemma 10.3.33. For n = 3, N 33 (ℓ) is either empty or a point, which starts the

induction. The induction step uses that δ is the moment map of an S1-Hamiltonianaction on Nn

3 (ℓ) [118] and called the bending flow [111]. It may be visualizedas a rotation of zn−1 and zn at constant speed around of axis ℓnzn + ℓn−1zn−1,leaving the other other zi’s fixed. As a moment map for a circle action, it satisfies

Crit δ = (Nn3 (ℓ))

S1

. We have seen that the critical points of δ are isolated or

polygon spaces with fewer edges. Hence, by induction hypothesis, (Nn3 (ℓ))

S1

isa conjugation space. The involution τ is anti-symplectic by Lemma 10.3.33 andsatisfies τ (γz) = γ−1τ (z) for all z ∈ N 3

3 (ℓ) and γ ∈ S1. That Nn3 (ℓ) is a conjugation

manifold thus follows from [87, Theorem 8.3].

2nd proof. We use that Nn3 (ℓ) is a symplectic reduction for the Hamiltonian action

of the maximal torus of U(n) on Gr(2;Cn) (see [88, Theorem 4.4]). The complexconjugation on Gr(2;Cn) is anti-symplectic, with fixed point set equal to Gr(2;Rn),descends to the involution τ on Nn

3 (ℓ). The manifold Gr(2;Cn) with the complexconjugation is a conjugation space (see p. 365 or Remark 9.7.9). That Nn

3 (ℓ) is aconjugation space thus follows from [87, Theorem 8.12].

3rd proof. By Theorem 10.3.31, Hodd(Nn3 (ℓ)) = 0 and H∗(Nn

3 (ℓ)) is generated byH2(Nn

3 (ℓ)). By Lemma 10.3.33, Nn3 (ℓ) is simply connected and H2τ is multiplica-

tion by (−1). Therefore, Nn3 (ℓ) is a conjugation manifold by results of V. Puppe

(see [166, Theorem 5 and Remark 2]).

Remark 10.3.34. The quotient BNn3 (ℓ) = BCn3 (ℓ)/SO(2) is called in [89]

the abelian polygon space (being an S1-symplectic reduction while Nn3 (ℓ) is an


SO(3)-symplectic reduction). In [89, Theorem 6.4], the integral cohomology ringof BNn

3 (ℓ) is computed: the statement is as that of Theorem 10.3.31 but with therelators of Corollary 10.3.7. The involution τ is defined on BNn

3 (ℓ) with fixed pointset BCn2 (ℓ). The 3rd proof of Proposition 10.3.32 may be easily adapted to showthat τ is a conjugation on BNn

3 (ℓ).

10.4. Equivariant characteristic classes

Let Γ be a topological group and let X be a Γ-space. Let ξ = (p : E → X) be aΓ-equivariant vector bundle overX . Recall from Lemma 7.5.6 that ξ is then inducedby i : X → XΓ from the vector bundle ξΓ over XΓ. The Stiefel-Whitney classes ofξΓ are called the equivariant Stiefel-Whitney classes of ξ. Hence, w(ξ) = ρ(w(ξΓ))where ρ : H∗Γ(X)→ H∗(X) is the forgetful homomorphism.

An important example is given by the tautological bundle ξj (j = 1, . . . , r) overthe flag manifold Fl(n1, . . . , nr) (see § 9.5.2), which is an O(n)-equivariant vectorbundle of rank nj .

Proposition 10.4.1. Let R be a closed subgroup of O(n). Then, as an H∗(BR)-algebra, H∗R(Fl(n1, . . . , nr)) is generated by the equivariant Stiefel-Whitney classeswi((ξj)R) (j = 1 . . . , r, i = 1, . . . , nj) of the tautological bundles. In particular,Fl(n1, . . . , nr) is R-equivariantly formal.

Proof. We have noticed above that wi(ξj) = ρ(wi((ξj)R)). By Theorem 9.5.14,H∗(Fl(n1, . . . , nr)) is additively generated by the monomials in the wi(ξj)’s. Sucha monomial is the image by ρ of the corresponding monomial in the wi((ξj)R)’s andFl(n1, . . . , nr) is thus R-equivariantly formal. By the Leray-Hirsch theorem 4.7.17,the monomials in the wi((ξj)R)’s generate H∗R(Fl(n1, . . . , nr)) as an H∗(BR)-module, whence the proposition.

We specialize to the Grassmannian Gr(k;Rn) = Fl(k, n− k) with R = T2, themaximal 2-torus of diagonal matrices inO(n). By (7.5.10), H∗(BT2) ≈ Z2[u1, . . . , un],with deg(ui) = 1. The tautological bundles are ζ = ξ1 and ζ⊥ = ξ2 (see Exam-ple 9.5.18). As Gr(k;Rn) is T2-equivariantly formal, the restriction to the fixedpoints r : H∗T2

(Gr(k;Rn)) → H∗T2(Gr(k;Rn)T2) is injective by Theorem 7.6.6. The

fixed point set Gr(k;Rn)T2 is discrete and in bijection with [nk ], the set of binarywords λ = λ1 · · ·λn such that

∑λi = k. We identify Gr(k;Rn)T2 with [nk ] via this

bijection, which associates to λ the coordinate k-plane

Πλ = (t1, . . . , tn) ∈ Rn | ti = 0 if λi = 0 .Note that Πλ is the “center” of the Schubert cell CFλ for the standard complete flagF in Rn (see § 9.5.3).

Proposition 10.4.2. For λ = λ1 · · ·λn ∈ [nk ], the restriction homomorphism

rλ : H∗T2(Gr(k;Rn))→ H∗T2

(λ) ≈ Z2[u1, . . . , un]

satisfies

rλ(w(ζT2)) =

∏

λi=1

(1 + ui) and rλ(w((ζ⊥)T2

) =∏

λi=0

(1 + ui) .

We may note that the classes rλ(w(ζT2)) and rλ(w(ζ

⊥T2)) satisfies the GKM-

conditions (see p. 402).

10.4. EQUIVARIANT CHARACTERISTIC CLASSES 393

Proof. Let E(ζ)λ be the fiber of ζ over πλ, seen as a T2-equivariant vectorbundle over λ. One has a T2-equivariant isomorphism

E(ζ)λ ≈⊕

λi=1

L(ui)

where L(ui) is the T2-equivariant line bundle over λ, on which T2 acts via the homo-morphism dia(δ1, . . . , δn) 7→ δi. This homomorphism is associated to ui under thebijection of (7.5.8). By Lemma 7.5.11, w(L(ui)T2) = 1+ui and, using Lemma 7.5.7and (9.4.3), we get

rλ(w(ζT2)) = w((E(ζ)λ)T2) =

∏

λi=1

(1 + ui) .

The proof of the assertion for ζ⊥ is similar, since

E(ζ⊥)λ = (t1, . . . , tn) ∈ Rn | ti = 0 if λi = 1 .

Remark 10.4.3. Proposition 10.4.2 implies that the relation

w(ζT2)w((ζ⊥)T2

) =

n∏

i=1

(1 + ui)

holds true in H∗T2(Gr(k;Rn). In fact, this provides a presentation of H∗T2

(Gr(k;Rn)(see Corollary 10.5.7 and Remark 10.5.8).

Example 10.4.4. Consider Gr(1;R3) ≈ RP 2. The T 2-fixed points are in bi-jection with 100, 010, 001. As a Z2[u1, u2, u3]-algebra, H

∗T2(RP 2) is generated by

w1 = w1(ζT2) and wi = wi(ζ

⊥T2) (i = 1, 2). The table of rλ(−) for these classes is

w1 w1 w2

100 u1 u2 + u3 u2u3010 u2 u1 + u3 u1u3001 u3 u1 + u2 u1u2

One checks that the relations

(10.4.1) w1 + w1 = σ1 , w2 + w1w1 = σ2 , w1w2 = σ3 .

are satisfied. For a generalization to Gr(1;Rn), see Example 10.5.9.

Example 10.4.5. The Z2[u1, . . . , u4]-algebra H∗T2(Gr(2;R4)) is generated by

wi = wi(ζT2) and wi = wi(ζ

⊥T2) (i = 1, 2). The table of rλ(−) for these classes is

w1 w1 w2 w2

1100 u1 + u2 u3 + u4 u1u2 u3u41010 u1 + u3 u2 + u4 u1u3 u2u41001 u1 + u4 u2 + u3 u1u4 u2u30110 u2 + u3 u1 + u4 u2u3 u1u40101 u2 + u4 u1 + u3 u2u4 u1u30011 u3 + u4 u1 + u2 u3u4 u1u2

The following relations are thus satisfied (compare Example 10.5.10).

(10.4.2)

w1 + w1 = σ1w2 + w1w1 + w2 = σ2w2w1 + w1w2 = σ3

w2w2 = σ4 .


The above results have their analogues in the complex case. The same proofas for Proposition 10.4.1 gives the following proposition.

Proposition 10.4.6. Let R be a closed subgroup of U(n). Then, as an H∗(BR)-algebra, H∗R(FlC(n1, . . . , nr)) is generated by the equivariant Chern classes ci((ξj)R)(j = 1 . . . , r, i = 1, . . . , nj) of the tautological bundles. In particular, FlC(n1, . . . , nr)is R-equivariantly formal.

As in the real case, we specialize to the Grassmannian Gr(k;Cn) = FlC(k, n−k)and their tautological bundles ζ = ξ1 and ζ⊥ = ξ2, with R = T , the maximaltorus of diagonal matrices in U(n). As seen in (7.5.14), H∗(BT ) ≈ Z2[v1, . . . , vn],with deg(vi) = 2. The associated 2-torus is the maximal 2-torus T2 of diagonalmatrices in O(n), and the action of T on Gr(k;Cn) shows that Gr(k;Cn)T =Gr(k;Cn)T2 . As Gr(k;Cn) is T -equivariantly formal, the restriction to the fixedpoints r : H∗T (Gr(k;Cn)) → H∗T (Gr(k;Cn)T ) is injective by Theorem 7.6.11. Thesame proof of Proposition 10.4.2 gives the analogous formulae

rλ(c(ζT )) =∏

λi=1

(1 + vi) and rλ(c(ζ⊥)T ) =

∏

λi=0

(1 + vi) .

We now study the equivariant characteristic classes of a rigid Γ-bundle (con-versations with T. Holm were useful for this part). Recall that a Γ-equivariant

vector bundle ξ = (Ep−→ X) is called rigid if the Γ-action on X is trivial (see

p. 254). Since then XΓ ≈ BΓ × X , the equivariant Stiefel-Whitney class w(ξΓ)belongs to H∗(XΓ) ≈ H∗(BΓ)⊗H∗(X). If η is a Γ-equivariant vector bundle overa space Y which is Γ-equivariantly formal, and if Γ is a 2-torus for instance, thenr(w(ηΓ)) = w(η|Y Γ), r : H∗Γ(Y )→ H∗(Y Γ) is injective (see Theorem 7.6.6) and η|Y Γ

is rigid. Hence, in such cases, rigid equivariant vector bundles play an importantrole.

Let ξ = (Ep−→ X) be a rigid Γ-equivariant vector bundle, where Γ is a 2-torus.

Let χ : Γ → O(1) ≈ ±1 be a homomorphism. We call ξ a weight Γ-bundle withrespect to χ (or just a χ-weight Γ-bundle) if γ · v = χ(γ)v for all γ ∈ Γ and v ∈ E.

Lemma 10.4.7. If Γ is 2-torus, then any rigid Γ-equivariant vector bundle overa locally contractible space decomposes into a Whitney sum ξ =

⊕χ∈hom(Γ,O(1)) ξ

χ

of weight subbundles.

Proof. Let ξ = (Ep−→ X) and let x ∈ X . Let ϕ : p−1(U)

≈−→ U × Rr be atrivialization of ξ over an open set U ⊂ X . Such a trivialization is of the formϕ(v) = (p(v), ϕ2(v)), where ϕ2 : p

−1(U)→ Rr is a continuous map which is a linearisomorphism on each fiber, and there is a bijection from the set of trivializationsof ξ over U and such maps. A continuous map y 7→ Aϕy from U to O(r) is thus

defined by the equation ϕ2(γv) = Aϕp(v)(γ)ϕ2(v), required to be valid for all γ ∈ Γ

and v ∈ p−1(U). If U retracts by deformation onto b ∈ U , a classical folklore factabout representation theory says that there is a continuous map y 7→ gy from U toO(r), with gb = id, such that Aϕy (γ) = gyA

ϕb (γ)g

−1y (see e.g. [79, Lemma 1.2]). We

can thus construct a new trivialization ϕ(v) = (p(v), ϕ2(v)) of ξ over U by settingϕ2(v) = g−1p(v)ϕ2(v). One checks that

(10.4.3) ϕ2(γv) = Aϕb (γ)v ,

10.4. EQUIVARIANT CHARACTERISTIC CLASSES 395

in other words, the map y 7→ Aϕy is constant over U . As X is locally contractible,there are such trivializations (Ux, ϕx) as above around each x ∈ X . Define

Eχ = v ∈ E | γ · v = χ(γ)v, ∀ γ ∈ Γ .As Γ as 2-torus, the vector space Ex decomposes into a direct sum of weight sub-spaces

(10.4.4) Ex =⊕

χ∈hom(Γ,O(1))

Eχx

(see the proof of Lemma 7.5.13). Clearly, Eχ ∩ p−1(x) = Eχx and, using (10.4.3),Eχ is the total space of a χ-weight subbundle of ξ. By (10.4.4), this proves theproposition.

By Lemma 7.5.7, ξΓ is the Whitney sum of the bundles ξχΓ . Hence, to computethe Stiefel-Whitney classes of rigid Γ-equivariant vector bundles, it suffices to knowthose of χ-weight bundles. The following proposition provides the answer. Weidentify hom(Γ, O(1)) with H1(BΓ) via the bijection of (7.5.8).

Proposition 10.4.8. Let Γ be a 2-torus and let χ ∈ H1(BΓ). Let ξ be aχ-weight Γ-bundle of rank r over X. Then, in H∗Γ(X) ≈ H∗(BΓ) ⊗ H∗(X), onehas

(10.4.5) w(ξΓ) =r∑

k=0

[(1 + χ)k × wr−k(ξ)

].

Proof. Let χ = (R → pt) be the χ-weight line bundle over a point and letχX = q∗χ, where q : X → pt is the constant map. Hence, χX is a χ-weight bundleoverX whose underlying bundle is a trivial line bundle. Note that any bundle η overX may be endowed with a structure of a χ-weight bundle using the isomorphismη ≈ η ⊗ χX . In fact, using the definition of the tensor product bundle of (7.5.5),

we see that any χ-weight bundle ξ is obtained this way: ξ ≈Γ ξ ⊗ χX , where ξ isthe bundle ξ endowed with the trivial Γ-action. By Lemma 7.5.7, one has

(10.4.6) ξΓ ≈ ξΓ ⊗ (χX)Γ .

Consider the two projections πBΓ : XΓ → BΓ and πX : XΓ → X (using that XΓ ≈BΓ × X). As Γ-acts trivially on E(ξ), one checks easily that ξΓ = π∗

Xξ and thus

w(ξΓ) = 1×w(ξ). By (7.5.4), (χX)Γ = π∗BΓχΓ and, by Lemma 7.5.11, w(χΓ) = 1+χ.

Hence, w((χX)Γ) = (1+ χ)× 1. As (χX)Γ is a line bundle, the proposition followsfrom (10.4.6) together with Lemma 9.6.7.

Example 10.4.9. Let ξ be a vector bundle of rank r over X . We let G = 1, τact on ξ by τ(v) = −v. Hence, ξ is an u-weight for u the generator of H1(BG).Using the identification H∗G(X) ≈ H∗(X)[u], Formula (10.4.5) becomes

w(ξG) =

r∑

k=0

[wr−k(ξ)(1 + u)k

].

Thus,wj(ξG) = wj(ξ) + wj−1(ξ)u + · · ·+ w1(ξ)u

j−1 + uj .

In particular, the evaluation ev1 at u = 1 of the equivariant Euler class eG(ξ) =e(ξG) satisfies

(10.4.7) ev1(eG(ξ)) = ev1(e(ξG)) = w(ξ) .


Proposition 10.4.8 has its analogue for T -equivariant complex vector bundles,where T is a torus. Let χ ∈ hom(T, U(1)), giving rise to κ(χ) ∈ H2(BT ) (see(7.5.11)). Then, if ξ be a χ-weight complex vector T -bundle of rank r over X , itsequivariant Chern class c(ξT ) satisfies

(10.4.8) c(ξT ) =r∑

k=0

[(1 + κ(χ))k × cr−k(ξ)

].

in H∗T (X) ≈ H∗(BT ) ⊗ H∗(X). The proof is the same as for Proposition 10.4.8,using at the end Lemma 9.7.17 instead of Lemma 9.6.7.

10.5. The equivariant cohomology of certain homogeneous spaces

The title of the section paraphrases that of A. Borel’s famous paper [15]. Themain goal is to prove and exemplify Theorem 10.5.1 below, due to A. Knutson(unpublished). In addition, at the end of the section, we study the so-called GKM-conditions for the flag manifolds.

Theorem 10.5.1 (A. Knutson). (1) Let Γ1,Γ2 be two closed subgroups ofa compact Lie group Γ. Suppose that Γ/Γ1 or Γ/Γ2 is Γ-equivariantlyformal. Then, there is a GrA-isomorphism

(10.5.1) ΞΓ1,Γ,Γ2 : H∗Γ1(Γ/Γ2)

≈−→ H∗(BΓ1)⊗H∗(BΓ) H∗(BΓ2) .

(2) Let (Γ,Γ1,Γ2) and (Γ′,Γ′1,Γ′2) be two data as in (1). Let Φ: Γ→ Γ′ be a

continuous homomorphism such that Φ(Γi) ⊂ Γ′i. Denote by Φi : Γi → Γ′ithe restriction of Φ. Set Ξ = ΞΓ1,Γ,Γ2 and Ξ′ = ΞΓ′1,Γ

′,Γ′2. Then the

diagram

(10.5.2)

H∗Γ′1(Γ′/Γ′2)

Ξ′

≈//

Φ∗

H∗(BΓ′1)⊗H∗(BΓ′) H∗(BΓ′2)

Φ∗1⊗Φ∗2

H∗Γ1

(Γ/Γ2)Ξ

≈// H∗(BΓ1)⊗H∗(BΓ) H

∗(BΓ2)

is commutative (the vertical arrows are induced by Φ and Φi, using thefunctorialities of the Borel construction).

Remark 10.5.2. Point (2) says in particular that ΞΓ1,Γ,Γ2 is an isomorphismof H∗(BΓ1)-modules with respect to the isomorphism ΞΓ1,Γ,Γ. With the obviousvertical identification in the commutative diagram

H∗Γ1(Γ/Γ)

ΞΓ1,Γ,Γ

≈// H∗(BΓ1)⊗H∗(BΓ) H

∗(BΓ)

≈

H∗(BΓ1)

≈

OO

ΞΓ1

≈// H∗(BΓ1)

,

the isomorphism ΞΓ1,Γ,Γ is identified with a GrA-automorphism ΞΓ1 of H∗(BΓ1).We do not know in general whether ΞΓ1 coincides with the identity. This is howevertrue in the following cases:

(1) Γ1 = ±1, since H∗(B±1) ≈ Z2[u].(2) Γ1 a 2-torus. One uses (1), the naturality of ΞΓ1 and that H1(BΓ1) ≈

hom(Γ1, ±1 (see (7.5.8)).

10.5. THE EQUIVARIANT COHOMOLOGY OF CERTAIN HOMOGENEOUS SPACES 397

(3) Γ1 = O(n). One uses (2), the naturality of ΞΓ1 and that H∗(BO(n)) →H∗(BT2) is injective, where T2 is a maximal 2-torus of O(n) (see Theo-rem 9.6.1).

(4) Γ1 a torus or U(n). The argument is analogous to (1)–(3) above.

Proof of Theorem 10.5.1. Let Γ1 × Γ2 acts on Γ by (γ1, γ2) · γ = γ1γγ−12 .

The kernel of the projection Γ1 × Γ2 → Γ1 acts freely on Γ. By Lemma 7.2.7, onehas a GrA-isomorphism

(10.5.3) H∗Γ1(Γ/Γ2)

≈−→ H∗Γ1×Γ2(Γ) .

Let Γ1 × Γ × Γ2 acts on Γ × Γ by (γ1, β, γ2) · (γ′, γ′′) = (γ1γ′β−1, βγ′′γ−12 ). Then

Γ = 1×Γ×1 acts freely on Γ×Γ and the multiplication µ : Γ×Γ→ Γ coincides withthe quotient map Γ×Γ→ Γ×Γ)/Γ. Hence, as above, one has a GrA-isomorphism

(10.5.4) µ∗ : H∗Γ1×Γ2(Γ)

≈−→ H∗Γ1×Γ×Γ2(Γ× Γ) .

The kernel of the projection Γ1×Γ×Γ2 → Γ acts freely on Γ×Γ. By Lemma 7.2.7again, one has a GrA-isomorphism

(10.5.5) H∗Γ(Γ1\Γ× Γ/Γ2)≈−→ H∗Γ1×Γ×Γ2

(Γ× Γ) .

Now, Γ1\Γ and Γ/Γ2 being closed smooth manifolds, they are equivalent to finiteΓ-CW-complexes (see [107]). If Γ1\Γ or Γ/Γ2 is Γ-equivariantly formal, then theequivariant Kunneth theorem 7.4.3 holds true, telling us that the strong equivariantcross product gives a GrA-isomorphism

(10.5.6) H∗Γ(Γ1\Γ)⊗H∗Γ(pt) H∗Γ(Γ/Γ2)

≈−→ H∗Γ(Γ1\Γ× Γ/Γ2) .

Using Example 7.2.4, we get a final GrA-isomorphism

(10.5.7) H∗Γ(Γ1\Γ)⊗H∗Γ(pt) H∗Γ(Γ/Γ2) ≈ H∗(BΓ1)⊗H∗(BΓ) H

∗(BΓ2) .

Combining the isomorphisms (10.5.3)–(10.5.7) provides theGrA-isomorphism ΞΓ1,Γ,Γ2 .Point (2) comes from the functoriality of the above constructions and Point (3) isjust an observation about Diagram (10.5.2).

Remark 10.5.3. As the right member of (10.5.1) is symmetric in Γ1 and Γ2,one has a GrA-isomorphism

H∗Γ1(Γ/Γ2) ≈ H∗Γ2

(Γ/Γ1) .

This can be more easily deduced from (10.5.3) and thus, does not require Γ/Γ1 orΓ/Γ2 being Γ-equivariantly formal.

The first two applications of Theorem 10.5.1 concern the extreme cases Γ1 = Γand Γ1 = 1. The space Γ/Γ = pt is Γ-equivariantly formal. Therefore, for anyclosed subgroup Γ2 of Γ, one has the GrA-isomorphism

(10.5.8) H∗Γ(Γ/Γ2) ≈ H∗(BΓ)⊗H∗(BΓ) H∗(BΓ2) ≈ H∗(BΓ2) .

Note that one already has an identification H∗Γ(Γ/Γ2) ≈ H∗(BΓ2) established inExample 7.2.4. We do not know whether these two identifications are the same (itcan be proved for special cases, as in Remark 10.5.2). In the case Γ1 = 1, we mustassume that Γ/Γ2 is Γ-equivariantly formal. We then get a GrA-isomorphism(10.5.9)H∗(Γ/Γ2) ≈ Z2 ⊗H∗(BΓ) H

∗(BΓ2) ≈ H∗(BΓ2)/image

(H∗(BΓ)→ H∗(BΓ2)

).


We now concentrate on flag manifolds. The real flag manifold is O(n)-equivar-iantly formal by Proposition 10.4.1. The complex flag manifold is U(n)-equivar-iantly formal by Proposition 10.4.6. Below is a choice of examples.

Example 10.5.4. Let β1 : Γ1 → O(n) denote the inclusion of the closed sub-group Γ1 in O(n). Theorem 10.5.1 together with Proposition 10.4.1 implies that

H∗Γ1(Fl(n1, . . . , nr)) ≈ H∗(BΓ1)⊗H∗(BO(n)) H

∗(BO(n1)× · · · ×BO(nr)) .Recall from Theorem 9.5.8 that H∗(BO(n)) ≈ Z2[w1, . . . , wn], where wi = wi(ζn),the Stiefel-Whitney classes of the tautological bundle ζn. Also, H∗(BO(nj)) ≈Z2[w1(ζnj

), . . . , wnj(ζnj

)]. Using the Kunneth formula, one has

H∗(BO(n1)× · · · ×BO(nr)) ≈ Z2[wi(ζnj)] (j = 1, . . . , r , i = 1, . . . , nj) .

By Theorem 9.6.2, the homomorphism H∗(BO(n))→ H∗(BO(n1)×· · ·×BO(nr))sends wi to wi(ζn1 × · · · × ζnr

). Hence(10.5.10)

H∗Γ1(Fl(n1, . . . , nr)) ≈ H∗(BΓ1)[wi(ζnj

) | j = 1, . . . , r , i = 1, . . . , nj]/I ,


w∗(ζn1 × · · · × ζnr) +H∗β1(w∗) (w∗ = 1 + w1 + w2 + · · · ) .

In the particular case of the complete flag manifold Fl(1, . . . , 1), Isomorphism (10.5.10)takes the form(10.5.11)

H∗Γ1(Fl(1, . . . , 1)) ≈ H∗(BΓ1)[x1, . . . , xn]

/(σi(x1, . . . , xn) = H∗β1(wi)) ,

where σi is the i-th elementary symmetric polynomial in the variables xj (seeExample 9.5.17).

Example 10.5.5. Let Γ1 = T2 ≈ O(1) × · · · × O(1) be the maximal 2-torusof the diagonal matrices in O(n). By Theorem 9.6.1 and its proof, H∗(BT2) ≈Z2[u1, . . . , un], with deg(ui) = 1, and H∗β1(wi) = σi(u1, . . . , un), where σi denotesthe i-th elementary symmetric polynomial. Also, O(n)/T2 ≈ Fl(1, . . . , 1) is O(n)-equivariantly formal by Proposition 10.4.1. Therefore, for any closed subgroup Γ2

in O(n), Theorem 10.5.1 provides an isomorphism

(10.5.12) Ξ = ΞT2,O(n),Γ2: H∗T2

(O(n)/Γ2)≈−→ Z2[u1, . . . , un]⊗H∗(BO(n))H

∗(BΓ2) .

For instance, (10.5.10) implies that(10.5.13)

H∗T2(Fl(n1, . . . , nr)) ≈ Z2[u1, . . . , un][wi(ζnj

) | j = 1, . . . , r , i = 1, . . . , nj]/I ,

where I is the ideal generated by wi(ζn1 ⊕· · ·⊕ ζnr)−σi(u1, . . . , un) (i = 1, . . . , n).

In the particular case of the full flag manifold Fl(1, . . . , 1), we thus get that

(10.5.14) H∗T2(Fl(1, . . . , 1)) ≈ Z2[u1, . . . , un, x1, . . . , xn]

/I ,


σi(u1, . . . , un, ) = σi(x1, . . . , xn, ) (i = 1, . . . , n) .

Another example is given by the Stiefel manifold Stief(k,Rn) = O(n)/O(n−k)of orthonormal k-frames in Rn. Here, H∗(BO(n)) → H∗(BO(n − k)) is just the


obvious epimorphism Z2[w1 . . . , wn] → Z2[w1 . . . , wn−k]. Hence, (10.5.12) impliesthat

(10.5.15) H∗T2(Stief(k,Rn)) ≈ Z2[u1, . . . , un]

/(σn−k+1, . . . , σn) .

Note that Stief(k,Rn)) is not O(n)-equivariantly formal. To the contrary,

ρ : H∗T2(Stief(k,Rn))→ H∗(Stief(k,Rn))

is the zero homomorphism in positive degrees.

It is reasonable to conjecture that the isomorphism ΞΓ1,O(n),O(n1)×···×O(nr)

of Theorem 10.5.1 identifies the equivariant Stiefel-Whitney class wi((ξj)Γ1) (see§ 10.4) with 1⊗wi(ζj). The following proposition proves it for the maximal 2-torusT2:

Proposition 10.5.6. Under the isomorphism Ξ = ΞT2,O(n),O(n1)×···×O(nr), onehas

Ξ(wi((ξj)T2)) = 1⊗ wi(ζj) .Proof. We start with the Grassmannian Gr(k;Rn) = Fl(k, n − k). For the

inclusion Φ: (T2, O(n), 1×O(n−k))→ (T2, O(n), O(k)×O(n−k)), Diagram (10.5.2)has the form

H∗T2(Gr(k;Rn))

Ξ

≈//

Φ∗

H∗(BT2)⊗H∗(BO(n)) H∗(BO(k) ×BO(n− k))

id⊗Φ∗2

H∗T2(Stief(k,Rn))

Ξ

≈// H∗(BT2)⊗H∗(BO(n)) H

∗(BO(n − k))

(we do not write the indices to the isomorphisms Ξ). The O(n−k)-principal bundleStief(k,Rn) → Gr(k;Rn) is ξ1, so Stief(k,Rn)T2 → Gr(k;Rn)T2 is (ξ1)T2 . Hence,w1((ξ1)T2) ∈ kerΦ∗. Using (10.5.15), one sees that ker(id ⊗ Φ∗2) is the Z2-vectorspace generated by 1⊗w1(ζ1). Therefore, Ξ(w1((ξ1)T2)) = 1⊗w1(ζ1). A symmetricargument shows that Ξ(w1((ξ2)T2)) = 1⊗ w1(ζ2).

This starts an induction argument on k to prove the proposition for Gr(k;Rk+1) =Fl(k, 1). The induction step uses Diagram (10.5.2) for the inclusion

Φ: (T2, O(k + 1), O(k)×O(1))→ (T2, O(k + 2), O(k + 1)×O(1))which looks like

H∗T2(Gr(k + 1;Rk+2))

Ξ

≈//

Φ∗

H∗(BT2)⊗H∗(BO(k+2)) H∗(BO(k + 1)×BO(1))

id⊗Φ∗2

H∗T2(Gr(k;Rk+1))

Ξ

≈// H∗(BT2)⊗H∗(BO(k+1)) H

∗(BO(k) ×BO(1))

.

By induction hypothesis, Ξ(wi((ξ1)T2)) = 1⊗wi(ζ1) for i ≤ k and Ξ(w1((ξ2)T2)) =1 ⊗ w1(ζ2). By Proposition 10.4.1, wk+1((ξ1)T2) is in kerΦ∗. On the other hand,ker(id ⊗ Φ∗2) is, in degree k + 1, the Z2-vector space generated by 1 ⊗ wk+1(ζ1).This proves that Ξ(wk+1((ξ1)T2)) = 1⊗ wk+1(ζ1).

We now prove the proposition for Gr(k;Rn) = Fl(k, n − k), by induction onn. The previous argument starts the induction for n = k + 1. The induction stepproceeds as above, using Diagram (10.5.2) for the inclusion

Φ: (T2, O(n), O(k) ×O(n− k))→ (T2, O(n+ 1), O(k)×O(n+ 1− k))


and checking kerΦ∗ and ker(id ⊗ Φ∗2) in degree n− k + 1.Finally, consider the map πj : Fl(n1, . . . , nr)→ Gr(nj ;Rn) defined, in the mu-

tually orthogonal subspaces presentation of Fl(n1, . . . , nr) given in (2) p. 314, byπj(W1, . . . ,Wr) = Wj . Take the simplest permutation matrix σ ∈ O(n) so thatσ(0× Rnj × 0) = Rnj × 0. The conjugation with σ gives an homomorphism

Φ: (T2, O(n), O(n1)× · · · ×O(nr))→ (T2, O(n), O(n1), O(nj)×O(n− nj))such that Φ2(1 × O(nj) × 1) = O(nj) × 1 and, for k 6= j, Φ2(1 × O(nk) × 1) ⊂1×O(n− nj). Diagram (10.5.2) for Φ has the form

H∗T2(Gr(nj ;Rn))

Ξ

≈//

π∗j

H∗(BT2)⊗H∗(BO(nj)) H∗(BO(nj)×BO(n − nj))

id⊗Φ∗2

H∗T2(Fl(n1, . . . , nr))

Ξ

≈// H∗(BT2)⊗H∗(BO(n)) H

∗(BO(n1)× · · · ×BO(nr))

Therefore,

Ξ(wi((ξj)T2)) = Ξπ∗j (wi((ξ1)T2)) since ξj = π∗j ξ1

= (id ⊗ Φ∗2)Ξ(wi((ξ1)T2))

= (id ⊗ Φ∗2)(1⊗ wi(ζ1)) (case Gr(nj ;Rn) done above)

= 1⊗ wi(ζj) since ξj = π∗j ξ1

Using (10.5.13), Proposition 10.5.6 has the following corollary.

Corollary 10.5.7. One has an isomorphism of Z2[u1, . . . , un]-algebra

H∗T2(Fl(n1, . . . , nr)) ≈ Z2[u1, . . . , un][wi((ξj)T2) | j = 1, . . . , r , i = 1, . . . , nj ]

/I ,


(10.5.16) wi((ξ1)T2 ⊕ · · · ⊕ (ξr)T2)− σi(u1, . . . , un) (i = 1, . . . , n) .

Remark 10.5.8. The vanishing of the generators of I is equivalent to therelation

w((ξ1)T2) · · ·w((ξr)T2) =

n∏

i=1

(1 + ui)

holding in H∗T2(Fl(n1, . . . , nr)). This relation may be obtained in the following way.

The trivializing map of (9.5.8) provides a morphism of T2-equivariant bundles

E(ξ1 ⊕ · · · ⊕ ξr) //

Rn

Fl(n1, . . . , nr)

f // pt

where T2 acts on Rn via the standard action of O(n). Using Example 7.5.12, onehas

w((ξ1)T2 ⊕ · · · ⊕ (ξr)T2) = f∗(w((Rn)T2) =n∏

i=1

(1 + ui) .


Example 10.5.9. Let Γ = O(n) and Γ2 = O(1)×O(n− 1), so Γ/Γ2 ≈ RPn−1.Then H∗(BO(1)) ≈ Z2[w1] and H∗(BO(n − 1)) ≈ Z2[w1, . . . , wn−1], where wiand wi are the Stiefel-Whitney classes of the tautological bundles ζ1 and ζn−1. IfΓ1 = T2, we get, as in (10.5.14) that H∗T2

(RPn−1) is the quotient of

Z2[u1, . . . , un][w1, w1, . . . , wn−1]

by the relations

(10.5.17) w1 + w1 = σ1 , wk + w1wk−1 = σk (k = 2, . . . , n− 1) , w1wn−1 = σn .

These relations are the same as those of (10.4.1), no wonder given Corollary 10.5.7.As RPn−1 is T2-equivariantly formal, H∗(RPn−1) is the quotient of H∗T2

(RPn−1)

by the relations ui = 0. Relations (10.5.17) becomes wk = wk1 and wn1 = 0. Thus,H∗(RPn−1) ≈ Z2[w1]/(w

n1 ) as expected.

Example 10.5.10. Let Γ = O(4) and Γ2 = O(2)×O(2), so Γ/Γ2 ≈ Gr(2;R4).As in Example 10.5.9, we see that H∗T2

(Gr(2;R4)) is isomorphic to the quotient ofZ2[u1, . . . , u4][w1, w2, w1, w2] by the relations

(10.5.18)

w1 + w1 = σ1w2 + w1w1 + w2 = σ2w2w1 + w1w2 = σ3

w2w2 = σ4 .

These relations are the same as those of (10.4.2) which is consistent with Corol-lary 10.5.7. As in Example 10.5.9, we get a presentation of H∗(Gr(2;R4)) by settingui = 0. This presentation is equivalent to that of Example 9.5.21.

Example 10.5.11. In the case Γ1 = 1 (the trivial subgroup), one hasH∗1(Fl(n1, . . . , nr)) ≈ H∗(Fl(n1, . . . , nr)) and the above coincides with some the-

orems of § 9.5.2 (e.g. Theorem 9.5.14 and Example 9.5.17).

Example 10.5.12. The analogues of the above examples works for the complexflag manifolds FlC(n1, . . . , nr), where O(n) is replaced by U(n), T2 is replaced by themaximal torus T ≈ (S1)n of the diagonal matrices in U(n) and the Stiefel-Whitneyclasses are replaced by the Chern classes. The variables xi and yi have degree 2instead of 1. The analogue of Proposition 10.5.6 says that Ξ(ci((ξj)T2)) = 1⊗ci(ζj).One can show that these results are valid for the cohomology with coefficients inany field.

Example 10.5.13. Consider the case where Γ = S3 ⊂ C2 and Γ1 = Γ2 =S1 ⊂ R2. Thus, Γ/Γ2 ≈ S2. By (7) p. 230, the inclusion i : S2 → (S2)Γ ishomotopy equivalent to a principal bundle with structure group S3. By Propo-sition 4.7.23, H2i is an isomorphism and therefore S2 is S3-equivariantly formal.Now, BS1 ≈ CP∞ and BS3 ≈ HP∞. Hence, H∗(BΓ1) ≈ Z2[x], H

∗(BΓ2) ≈ Z2[y]and H∗(BΓ) ≈ Z2[p], where x and y are of degree 2 and p of degree 4. The inclusionαi : Γi → Γ satisfies H∗Bα1(p) = x2 and H∗Bα2(p) = y2 (see Proposition 6.1.11).Therefore, using Theorem 10.5.1, one gets

H∗S1(S2) ≈ Z2[x, y]/(x2 + y2) .

We finish this section by studying the GKM-conditions for the flag manifolds.Viewing Fl(n1, . . . , nr) ⊂ SM(n) using (9.5.2), the fixed point set Fl(n1, . . . , nr)

T2

is clearly formed by the diagonal matrices. If ∆ = dia(x1 . . . , xn) is a diagonal


matrix and σ ∈ Symn, we set ∆σ = dia(xσ(1) . . . , xσ(n)). We say that a class

a ∈ H∗T2(Fl(n1, . . . , nr)

T2) satisfies the GKM-conditions if, for all transpositionτ = (i, j), the class a∆ − a∆τ is a multiple ui − uj. This is an ad hoc formulationfor Fl(n1, . . . , nr) (in the spirit of [119]) of the conditions introduced in [71] byM. Goresky, R. Kottwitz and R. MacPherson (whence the initials GKM). Theimportance of the GKM-conditions is illustrated in the following result.

Proposition 10.5.14. The image of

r : H∗T2(Fl(n1, . . . , nr))→ H∗T2

(Fl(n1, . . . , nr)T2)

is the set of classes satisfying the GKM-conditions.

Proof. We first prove that the classes in the image of r satisfies the GKM-conditions. Let τ = (i, j) with i < j. The GKM-condition for τ is trivial if∆τ = ∆ (i.e, when xi = xj). We may thus assume that xi 6= xj . In the proofof Proposition 9.7.1, we have introduced an embedding rij : SO(2)→ SO(n). Theorbit of ∆ under the action of rij(SO(2)) on Fl(n1, . . . , nr) by conjugation is a circleCij joining ∆ to ∆τ . This circle is T2-invariant and is actually an (ui − uj)-circlein the sense of Example 7.6.10, with fixed points ∆,∆τ. One has a commutativediagram

H∗T2(Fl(n1, . . . , nr))

r //

H∗T2(Fl(n1, . . . , nr)

T2)

≈ //⊕

Fl(n1,...,nr)T2

Z2[u1, . . . , un]

H∗T2(Cij)

r // H∗T2(CT2

ij )≈ //

⊕

∆,∆τ

Z2[u1, . . . , un]

where all the vertical arrows are induced by inclusions. The right vertical arrowis just the projection. Therefore, the GKM-condition for τ comes from Exam-ple 7.6.10.

For the converse, we use the weighted trace f(M) =∑n

j=1 j Mjj which is,

by Proposition 9.5.2, a Morse function with Critf = Fl(n1, . . . , nr)T2 . Let Wx =

f−1(−∞, x] and let Tx be the set of those transpositions τ such that ∆ and ∆τ

are in Wx. We claim that a class a ∈ H∗T2(WT2

x ) is in the image of r : H∗T2(Wx)→

H∗T2(WT2

x ) if and only if it satisfies the GKM-conditions for Tx. The “only if” partis proven as above since, if (i, j) ∈ Tx, then Cij ⊂ Wx. The proof of the “if” partproceeds by induction on the number nx of critical values of f in Wx, startingtrivially if nx = 0 or 1. For the induction step, choose z < y such that nz = ny− 1.Let Mz,y =Wy −Wz . As in (7.6.14), one has the commutative diagram

(10.5.19)

0 // H∗T2(Wy,Wz)

α //

rz,y

H∗T2(Wy)

β //

ry

H∗T2(Wz) //

rz

0

0 // H∗T2(Mz,y)

α // H∗T2(WT2

y )β // H∗T2

(WT2z ) // 0

where all arrows are induced by the inclusions and where the horizontal lines areexact.

If a ∈ H∗T2(WT2

y ) satisfies the GKM-conditions for Ty, so does a = β(a) for

Tz. By induction hypothesis, a = rz(b) for some b ∈ H∗T2(Wz). As β is surjective,

10.6. THE KERVAIRE INVARIANT 403

there exists b ∈ H∗T2(Wy) such that a − ry(b) = α(c) for some c ∈ H∗T2

(WT2z,y). By

the “only if” part the class ry(b) satisfies the GKM-conditions for Ty, and then sodoes a − ry(b). let D ∈ MT2

z,y. Let TD be the set of transpositions in Ty such thatDτ 6= D. For each (i, j) ∈ TD, the class (a− ry(b))D is a multiple of ui − uj (since(a − ry(b))∆ = 0 when ∆ 6= D). Since α is injective, the class cD is a multiple ofui− uj for each (i, j) ∈ TD. By the proof of Proposition 9.5.2, the negative normalbundle ν−(D) for f at D is the Whitney sum

ν−(D) =⊕

(i,j)∈TD

TDCij .

As Z2[u1, . . . , un] is a unique factorization domain, the class cD is a multiple of∏

(i,j)∈TD

(ui − uj) =∏

(i,j)∈TD

e(TDCij) = e(ν−(D)) .

This can be done for any D ∈MT2z,y. Using Diagram (7.6.13), we deduce that there

exists c ∈ H∗T2(Wy ,Wz) such that rz,y(c) = c. Hence, a = ry(α(c) + b).

Analogous GKM-relations hold true for the T -equivariant cohomology of thecomplex flag manifold FlC(n1, . . . , nr). Here, T is the maximal torus of diagonalmatrix in U(n). It is naturally isomorphic to U(1)n and thus H∗(BT ) is isomorphicto Z2[v1, . . . , vn] where vi ∈ H2(BT ) is the class associated, under the map κof (7.5.11), to the projection of U(1)n onto its i-th factor. As in the real case, wesee FlC(n1, . . . , nr) ⊂ HM(n) (using (9.7.2)), then FlC(n1, . . . , nr)

T is formed bythe diagonal matrices. We say that a class a ∈ H∗T (FlC(n1, . . . , nr)

T ) satisfies theGKM-conditions if, for all transposition τ = (i, j), the class a∆ − a∆τ is a multiplevi − vj .

Proposition 10.5.15. The image of the injective homomorphism of Z2[v1, . . . , vn]-algebras

r : H∗T (FlC(n1, . . . , nr))→ H∗T (FlC(n1, . . . , nr)T )

is the set of classes satisfying the GKM-conditions.

Proof. We use the injective homomorphism rij : SU(2) → U(n), introducedin the proof of Proposition 9.7.1. The orbit of ∆ under the action of rij(SU(2)) onFlC(n1, . . . , nr) by conjugation is a 2-sphere Sij (diffeomorphic to SU(2)/U(1) ≈S2), whose intersection with FlC(n1, . . . , nr)

T is ∆,∆τ. This sphere is T -invariantand is actually a χ-sphere in the sense of Example 7.6.12, with κ(χ) = vi−vj . Notethat FlC(n1, . . . , nr)

T = FlC(n1, . . . , nr)T2 . The proof of Proposition 10.5.15 then

goes as that of Proposition 10.5.14, replacing Cij by Sij and the material of Proposi-tion 9.5.2 and Example 7.6.10 by that of Proposition 9.7.1 and Example 7.6.12.

10.6. The Kervaire invariant

In 1960, Michel Kervaire introduced an invariant for framed manifolds which en-abled him to construct the first topological manifold admitting no smooth structure(see Theorem 10.6.12 below). The computation of the Kervaire invariant then ledto one of the most important problems in homotopy theory (see Theorem 10.6.11).In this section, we give a survey of the geometric side of the Kervaire invariant,using the surgery point of view of Wall (though well known by specialists, such apresentation is new in the literature). The stable homotopy aspect of the invari-ant is only briefly mentioned, being beyond the scope of this book. The notes of


Cl. Weber [205] were helpful for preparing this section. We start with the invariantintroduced by C. Arf [10] in order to classify quadratic forms in characteristic 2.

Let V be a finite dimensional Z2-vector space. A quadratic form on V is a mapq : V → Z2 such that the expression

(10.6.1) B(x, y) = q(x) + q(y) + q(x+ y)

defines a bilinear form B on V , the bilinear form associated to q. It is obviouslysymmetric and alternate, i.e. B(x, x) = 0.

For i ∈ Z2, let αi(q) = ♯ q−1(i). The majority (or democratic) invariant of q isthe element maj(q) ∈ Z2 defined by

maj(q) =

1 if α1(q) > α0(q)

0 otherwise.

In fact, α0(q) 6= α1(q) when B is non-degenerate (see Proposition 10.6.1 below).Suppose that the associated bilinear form B is non-degenerate. Since it is

alternate, there exists a symplectic basis a1, . . . , ak, b1 . . . , bk of V for B (see theproof of Lemma 9.8.1). The Arf invariant Arf(q) ∈ Z2 of q is defined by

Arf(q) =

k∑

i=1

q(ai)q(bi) .

That Arf(q) is independent of the choice of the symplectic basis follows fromPoint (2) of the following proposition. Two quadratic forms q and q′ on V areequivalent if there is an automorphism h of V such that q′ = qh.

Proposition 10.6.1. Let V be a finite dimensional Z2-vector space. Let q andq′ be two quadratic forms on V with non-degenerate associated bilinear forms. Then

(1) α0(q) 6= α1(q).(2) Arf(q) = maj(q).(3) q and q′ are equivalent if and only if Arf(q) = Arf(q′).

Proof. (Compare [24, § III.1].) Suppose first that dimV = 2. Let A = a, bbe a symplectic basis forB on V with which we compute Arf(q) = q(a)q(b). Supposethat Arf(q) = 0. There are two cases:

(i) q(a) + q(b) = 0, hence maj(q) = 0.(ii) q(a) + q(b) = 1. By (10.6.1), one has q(a + b) = 0 and maj(q) = 0. By

symmetry, one may assume that q(a) = 0 and q(b) = 1. Then, the changeof basis a′ = a and b′ = a+ b makes this form equivalent to that of (i).

If Arf(q) = 1, then q(a) = q(b) = 1, thus q(a+ b) = 1 and maj(q) = 1. Points (1),(2) and (3) are thus proven when dimV = 2. We denote by qi the quadratic formfor which Arf(qi) = i.

We now prove (1) and (2) by induction on dim V . Let B = a1, . . . , ak, b1 . . . , bkbe a basis of V which is symplectic for B. One has V = V ⊕ V where V is generatedby aj, bj | j ≤ k − 1 and V is generated by ak, bk. The restriction of q to V

(respectively: V ) is denoted by q (respectively: q). We have

Arf(q) = Arf(q) + Arf(q) using the basis B= maj(q) + maj(q) by induction hypothesis.


It thus suffices to prove that maj(q) +maj(q) = maj(q). Suppose that q = q1. Onehas α1(q1) = 3 and α0(q1) = 1. Therefore,

α1(q) = 3α0(q) + α1(q)α0(q) = 3α1(q) + α0(q) .

By induction hypothesis, α0(q) 6= α1(q). Therefore, maj(q) 6= maj(q), which impliesthat

maj(q) = maj(q) + 1 = maj(q) + maj(q1) .

If q = q0, a similar argument shows that maj(q) = maj(q) = maj(q) + maj(q0).Point (1) is thus established.

It remains to prove Point (3). If q and q′ are equivalent, it is obvious thatmaj(q) = maj(q′). Conversely, we easily deduce from above that q is an orthogonalsum of q0’s and q1’s and that Arf(q) is the numbers of q1’s mod 2. That q and q′

are equivalent when Arf(q) = Arf(q′) then comes from the equivalence

q1 ⊞ q1 ≃ q0 ⊞ q0

which is achieved by the automorphism

(10.6.2) h(a1) = a1 + a2 , h(a2) = a2 , h(b1) = b1 , h(b2) = b1 + b2 .

We now make some preparations for the Kervaire invariant. Let ξ and ξ′ betwo vector bundles over the same space X and let ηr = (prX : X×Rr → X) denotethe product vector bundle of rank r over X . A stable isomorphism from ξ to ξ′ is afamily of isomorphisms hr,r′ : ξ ⊕ ηr → ξ′ ⊕ ηr′ for each r, r′ sufficiently large, suchthat if s ≥ r and s′ ≥ r′, the diagram

ξ ⊕ ηrhr,r′

≈//

ξ′ ⊕ ηr′

ξ ⊕ ηshs,s′

≈// ξ′ ⊕ ηs′

is commutative, where the vertical arrows are the inclusion morphisms. A stabletrivialization of a vector bundle ξ overX is a stable isomorphism of ξ with a productbundle. A vector bundle admitting a stable trivialization is called stably trivial.

A framed manifold is a smooth manifold M together with a smooth stabletrivialization of its tangent bundle TM , called a stable framing of M . Two framedmanifolds M1 and M2 of the same dimension m and with BdM1 = BdM2 areframed cobordant if there is exists a framed manifold Wm+1 such that

BdW =M1 ∪M2 and M1 ∩M2 = BdM1 = BdM2

and whose stable framing extends those ofM1 andM2. The set of framed cobordismclasses of framed closedmanifolds of dimensionm is denoted by Ωfrm . It is an abeliangroup for the disjoint union.

Let Mm be a closed manifold. Let νkM be the normal bundle of an embeddingofM into Rn+k. If k > m, such an embedding is unique up to isotopy, so the stableisomorphism class of νkM is well defined. There is a canonical stable trivialization hM

of TM⊕νkM since the latter is the restriction of TRn toM . If h : TM⊕ηr ≈−→ ηm+r

is represents a stable isomorphism, the stable isomorphism represented by

ηm+k+r oo hM

≈TM ⊕ νkM ⊕ ηr ≈

// TM ⊕ ηr ⊕ νkMh

≈// ηm+r ⊕ νkM


represents a stable trivialization of νkM . For k large enough, such a stable trivial-ization gives a vector bundle morphism

E(νkM ) //

Rk

M // pt

.

Applying the Pontryagin-Thom construction (see p. 358) to this morphism gives anelement of πm+k(S

k). This produces an isomorphism Ωfrm ≈ πSm from the framedcobordism onto the m-stem

(10.6.3) πSm = lim−→k

πm+k(Sk)

(see [152, § 7]).Example 10.6.2. Let (Sn, F ) be the standard sphere equipped with a stable

framing F . The comparison between F and the standard stable framing F0 (ex-tending to Dn+1) takes the form F = λF · F0, where λF : Sn → SO is a smoothmap, whose class [λF ] ∈ πn(SO) is unique (compare Lemma 9.1.2). The correspon-dence λF 7→ [Sn, F ] ∈ Ωnfr ≈ πSn gives a map Jn : πn(SO) → πSn which coincides

with the J-homomorphism of Whitehead [207].

We now describe framed surgery, following the point of view of Wall [204] (foranother approach, see p. 412). Let β : Sj ×Dm−j →Mm (0 ≤ j ≤ m) be a smoothembedding. We consider the (m+ 1)-dimensional manifold

Wβ =M × [0, 1] ∪β Dj+1 ×Dm−j

where β is seen having image in M × 1. The corners of Wβ may be smoothedin a canonical way (see [17, Appendix, Theorem 6.2]) and thus W is a smoothcobordism between M and Mβ where

Mβ =M − int(imβ) ∪β|Sj×Sm−j−1 Dj+1 × Sm−j−1 .The manifold Mβ is said being obtained from M by a surgery using β. If M isendowed with a stable framing F which extends to a stable framing of Wβ , we saythat the surgery on β is a stably framed surgery (for the framing F ).

LetMm be a manifold and let α : Sj →M be a continuous map, with j < m−1.We wish to perform a surgery onM using a smooth embedding β : Sj×Dm−j →Mso that the restriction of β to Sj×0 is homotopic to α. Let Imm(Sj×Dm−j ,M)be the set of regular homotopy classes of smooth immersions from Sj ×Dm−j intoM . The restriction to Sj × 0 provides a map

ρ : Imm(Sj ×Dm−j ,M)→ [Sj ,M ] .

Proposition 10.6.3. Let Mm be a manifold and let j < m− 1. Then

(a) a stable framing F of M provides a map

φF : [Sj ,M ]→ Imm(Sj ×Dm−j ,M)

which is a section of ρ, i.e. ρφF (a) = a for all a ∈ [Sj ,M ].(b) Suppose that φF (a) contains an embedding β. Then, the surgery on β is

a stably framed surgery for the framing F .


Proof. Let α : Sj → M be a continuous map, which we extend to α1 : Sj ×

Dm−j → M by α1(x, z) = α(x). The framing F of M gives rise to a stabletrivialization of α∗1TM . On the other hand, T (Sj ×Dm−j) has a canonical stabletrivialization. Comparing these two trivializations gives a stable isomorphism αS1from T (Sj×Dm−j) to α∗1TM . Since j < m−1, πj(GL(m;R))→ πj(GL(m+N ;R))is an isomorphism and thus αS1 is induced by a unique isomorphism from T (Sj ×Dm−j) to α∗1TM , giving rise to an injective bundle map α′1 : T (S

j×Dm−j)→ TM .Assertion (a) then follows from the classification of immersions [93, § 5], saying thatImm(Sj ×Dm−j ,M) is (by the tangent map) in bijection with the set of injectivebundle maps from T (Sj ×Dm−j) into TM (this also uses that j < m− 1).

If β is as in (b), the equation Tβ = α′1 is satisfied. This is exactly whatis needed to extend the stable framing F over Wβ . For more details, see [204,Theorem 1.1].

Proposition 10.6.4 (Surgery below the middle dimension). A stably framedcompact m-dimensional manifold Mm is stably framed cobordant to a manifold M ′

which is ([m/2]− 1)-connected (i.e. πi(M′) = 0 for i ≤ m/2− 1).

Proof. Let F be the stable framing of M and let a ∈ [Sk,M ]. If k < m/2,general position implies that φF (a) contains an embedding α : Sk × Dm−k → Musing which, by Proposition 10.6.3, a stably framed surgery may be performed onM . Up to homotopy equivalence,Wα is obtained fromM by attaching a (k+1)-celland from Mα by attaching a (m − k)-cell. Hence, the inclusions i : M → Wα andj : Mα → Wα satisfy

• π∗i : πp(M) → πp(Wα) is an isomorphism for p ≤ k − 1. For p = k, it issurjective and kills [α].• π∗j : πp(Mα)→ πp(Wα) is an isomorphism for p ≤ m− k − 1.

Therefore, if k ≤ m/2 − 1, then πp(Mα) is isomorphic to πp(M) for p ≤ k − 1and, if [α] 6= 0, πk(Mα) is isomorphic to a strict quotient of πk(M). In particu-lar, if M is (k − 1)-connected, so is Mα and, in addition, the class [α] has been“killed”. Therefore, by a finite sequence of framed surgeries, one may obtain astably framed manifold M ′ which is [m/2]− 1-connected. For more details, see e.g.[204, Theorem 1.2].

We now treat the middle dimensional surgery for an (k − 1)-connected stablyframed closed manifold Mm with m = 2k (k odd). Consider the Hurewicz homo-morphisms h : πk(M) → Hk(M ;Z) and h2 : πk(M) → Hk(M) (sending [γ : Sk →M ] to H∗γ([S

k])); since k ≥ 3, we do not worry about base points). By theHurewicz theorem [82, Theorem 4.32], h is an isomorphism and, the universal coef-ficient theorem [82, Theorem 3B.5], Hk(M) ≈ Hk(M ;Z)⊗Z2. Hence, h2 descendsto an isomorphism

(10.6.4) πk(M)/2 πk(M)

≈−→ Hk(M) .

Let β : V v → Nn be an immersion of smooth manifolds. The self-intersectionSI(β) of β is defined by

SI(β) = x ∈ β(V ) | ♯β−1(x) > 1 .When β is in general position and 3v < 2n, then ♯β−1(x) ≤ 2 and SI(β) is a(2v − n)-dimensional submanifold of N (see [77, Theorem 2.5])


Let a ∈ πk(M). Choose an immersion α : Sk × Dk → M in general positionrepresenting φF (a) (where F is the stable framing ofM). Let α0 be the restrictionsof α to Sk × 0. The self-intersection SI(α0) is thus a finite number of points.Define

q(α) = ♯SI(α0) mod 2 .

Proposition 10.6.5. Let m = 2k ≥ 6 with k odd. Let Mm be (k−1)-connectedmanifold endowed with a stable framing F . Then

(a) the above correspondence a 7→ q(α) induces, via (10.6.4), a well definedmap

q = qM : Hk(M)→ Z2 .

(b) for all a, b ∈ Hk(M) one has

q(a+ b) = q(a) + q(b) + a · bwhere a · b denotes the (absolute) intersection form (see § 5.3.3).

(c) q(a) = 0 if and only if φF (a) contains an embedding.

Proof. Let α and α be two immersions representing φF (a) which are in generalposition. These are thus joined by an immersion A : Sk×Dk×I →M×I which wealso assume to be in general position. Then, SI(A0) is a compact 1-manifold withboundary SI(α0) ∪SI(α0). As an arc has two ends, one has q(α) = q(α). Thus,q(α) depends only on a ∈ πk(M), so we can write q(a). In order to prove Point (a),it remains to establish that q(2a) = 0, which will be done together with the proofof (b).

Let a, b ∈ πk(M). Let α ∈ φF (a) and β ∈ φF (b) be two immersions in generalposition. Then, φf (a + b) may be represented by an immersion γ obtained byconnected sum of α and β along a tube Sk−1×Dk × I, disjoint from the images ofα and β except at its ends. Let B0(α, β) ∈ Z2 defined by

B0(α, β) = ♯ [(α0(Sk) ∩ β0(Sk)] mod 2 .

Obviously, one has

(10.6.5) q(a+ b) = q(a) + q(b) +B0(α, β) .

We claim that

(10.6.6) B0(α, β) = a · b .Indeed, if α0 and β0 are embeddings, this is just Corollary 5.4.13. We claim that aand b may be represented by embeddings α0 and β0 such that α0(S

k) ∩ β0(Sk) =α0(S

k)∩ β0(Sk). Suppose first that ♯SI(α0) is even. The points of SI(α0) may bebe pairwise eliminated by the Whitney procedure (sinceM is simply connected: see[210, Theorem 4 and its proof]). This produces a regular homotopy αt0 : S

k → Mwith α0

0 = α0, so that α0 = α10 is an embedding. The control on the Whitney

process guarantees that the intersection of αt0(Sk) with β(Sk) is constant in t. If

♯SI(α0) is odd, we use that there exists an immersion µ : Sk → Rm with ♯SI(µ) = 1(see [210, § I.2]). We can compose µ with a chart Rm → M whose range is awayfrom α0(S

k) ∪ β0(Sk), obtaining an immersion µ′ : Sk → M which, as a map, isnull-homotopic. Let α : Sk → M be the immersion obtained by connected sum ofα0 and µ′. Then, α0 represents a and ♯SI(α0) is even, so we can proceed as in theprevious case. The whole process may be independently applied to β0. This provesthe claim and then Equation (10.6.6).


Equations (10.6.5) and (10.6.6) imply that q(2a) = 0, because

a · a = 〈PD(a) PD(a), [M ]〉 = 0 .

Indeed, each x ∈ Hk(M) satisfies x x = Sqk(x) = x vk(M), where vk(M)is the Wu class. But, as TM is stably trivial, its Stiefel-Whitney class satisfiesw(TM) = 1, which implies that v(M) = 1 by the Wu formula. This proves (a)and, thus, Equations (10.6.5) and (10.6.6) imply (b).

For Point (c), suppose that q(a) = 0 and let α ∈ φF (a) be an immersion ingeneral position. The self-intersection SI(α0) then consists of an even numberof double points. These points can then be pairwise eliminated by the Whitneyprocedure as explained above.

By Point (b) of Proposition 10.6.5, qM : Hk(M) → Z2 is a quadratic formassociated to the absolute intersection form ofM . If BdM is either empty or a Z2-homology sphere, the intersection form is non-degenerate (see Proposition 5.3.11).Its Arf invariant Arf(qM ) is thus defined and is called the Kervaire invariant c(M)ofM . The case where BdM is a Z2-homology sphere is too general for our purposeso we introduce the following definition: a compact manifold M is almost closed ifBdM is either empty or a homotopy sphere.

Proposition 10.6.6. Let m = 2k ≥ 6 (k odd). Let M0 and M1 be two (k− 1)-connected stably framed almost closed manifolds of dimension m. If M0 and M1

are stably framed cobordant, then c(M0) = c(M1).

Proof. Let Wm+10 be a stably framed cobordism between M0 and M1. If

BdM0 (and thus BdM1) is empty, we remove a tube Dm × I out of W0, getting astably framed manifoldWm+1 whose boundary is the connected sumM =M0♯M1.If BdM0 (and thus BdM1) is a homotopy sphere, we set W =W0 and M = BdW .The manifold M is a (k− 1)-connected stably framed closed manifold and, clearly,c(M) = c(M0) + c(M1). It thus suffices to prove that c(M) = 0. By surgerybelow the middle dimension (see Proposition 10.6.4), we may assume that W is(k − 1)-connected. To prove that c(M) = 0, it is enough to show that q(B) =0 where B = ker(Hk(M) → Hk(W )). Indeed, Proposition 5.3.9 and Kroneckerduality imply that 2 dimB = dimHk(M). The vanishing of q(B) thus implies,using Proposition 10.6.1, that c(M) = Arf(q) = maj(q) = 0.

Let h∗ and h∗ denote the integral (co)homology. As M is (k − 1)-connectedand almost closed, the Hurewicz theorem, the integral Poincare duality and theuniversal coefficient theorem imply that

(10.6.7) πk(M) ≈ hk(M) ≈ hk(M,BdM) ≈ hk(M) ≈ hom(hk(M);Z) .

Therefore, all the groups in (10.6.7) are free abelian and the isomorphism hk(M) ≈hk(M) sends BZ = ker(hk(M)→ hk(W )) onto BZ = Image

(hk(W )→ hk(M)

). As

hk(M) is free abelian, the same proof as for Proposition 5.3.9 shows that BZ is adirect summand of hk(M) and rankhk(M) = 2 rankBZ (all groups in the analogueof Diagram (5.3.7) are free abelian). Hence, BZ is a direct summand of hk(M).The homomorphism BZ/2BZ → B is then injective, where B = Image

(Hk(W ) →

Hk(M)). By Proposition 5.3.9, dimHk(M) = 2 dimB, so B and BZ/2BZ have

the same dimension. This shows that the homomorphism BZ → B is surjective.


Consider the commutative diagram

πk+1(W ) //

πk+1(W,M) //

πk(M) //

≈

πk(W )

≈

hk+1(W ) // hk+1(W,M) // hk(M) // hk(W )

whose rows are exact. The bijectivities and surjectivity seen on vertical arrowscome from the Hurewicz-Whitehead theorem [179, Ch. 7, Section 5, Theorem 9],since M and W are (k− 1)-connected. By a five-lemma arguments, we deduce thatπk+1(W,M)→ BZ → B is surjective.

Let b ∈ B. By the above, there is a map β : Sk → M , representing b, whichextends to γ : Dk+1 →W . Using the stable framing ofW , the pair if maps (γ, β) de-

termines, in its homotopy class, a pair of immersions (γ, β) : (Dk+1, Sk)→ (W,M)which we may assume to be in general position. The self-intersection SI(γ0) is

a compact 1-dimensional manifold whose boundary is SI(β0). As an arc has two

ends, one has q(b) = q(β0) = 0.

Example 10.6.7. The sphere Sk has a standard stable framing. LetM = Sk×Sk (k odd), with the product stable framing. The manifold M is (k− 1)-connectedand, by the Kunneth formula, Hk(M) ≈ Hk(Sk) ⊗ H0(Sk) ⊕ H0(Sk) ⊗ Hk(Sk),with generators a = [Sk]⊗1 and b = 1⊗ [Sk]. As M is the boundary of Sk×Dk+1

and of Dk+1 × Sk, one has c(M) = 0 and q(a) = q(b) = 0 by Proposition 10.6.6and its proof. As a · b = 1, one has q(a+ b) = 1. Note that a+ b is represented bythe diagonal manifold of Sk × Sk.

Proposition 10.6.5 permits us to define the Kervaire invariant for any stablyframed almost closed manifold M2k (k odd), as c(M) = c(M ′) where M ′ is a(k − 1)-connected manifold stably framed cobordant to M . For closed manifolds,this gives a map

c : Ωfr2k → Z2 (k odd) .

which is a homomorphism. Indeed, the sum in Ωfr2k may be represented by theconnected sum. IfM1 andM2 are (k−1) connected stably framed closed manifolds,then M =M1♯M2 is (k − 1)-connected and c(M) = c(M1) + c(M2).

Proposition 10.6.8. Let m = 2k ≥ 6 (k odd). Let Mm be a stably framedalmost closed manifold. Then, c(M) = 0 if and only if M is stably framed cobordantto a contractible manifold (if BdM is not empty) or to a homotopy sphere (if BdMis empty).

Proof. The “if” part is obvious since a contractible manifold of a homotopysphere is (k − 1)-connected and its middle dimensional homology vanish.

The proof of the converse uses the integral (co)homology, denoted by h∗ and h∗.

By surgery below the middle dimension, we may suppose that M is (k − 1)-connected. As seen in (10.6.7), hk(M) is free abelian. Consider the integral in-tersection form on hk(M) given by a · b = 〈PD(a) PD(b), [M ]Z〉 (for some choiceof a generator [M ]Z ∈ Hm(M,BdM)). This form is unimodular since M is almostclosed (same proof as for Proposition 5.3.11, or see [97, p. 58]). As k is odd, theintegral intersection form is alternate. Hence, hk(M) admits a skew-symplectic

basis, i.e. a basis a1, b1, . . . , ap, bp such that ai · aj = bi · bj = 0 and ai · bj = ±δij


(same proof as that of Lemma 9.8.1, or see [156, IV.1]). Under the isomorphism

hk(M)/2hk(M) ≈ Hk(M) the basis ai, bi gives a symplectic basis ai, bi for theZ2-intersection form.

By changing the basis ai, bi, we may assume that q(a1) = 0. Indeed, this can

be achieved if q(a1)q(b1) = 0 (by exchanging a1 with b1 if necessary). Otherwise,as Arf(q) = c(M) = 0, there exists j 6= 1 such that q(aj)q(bj) = 1 (say j = 2). Thebasis change of (10.6.2) then does the job.

We are thus in position to perform a stably framed surgery on an embeddingα : Sk ×Dk →M representing a1, giving a stably framed manifold M ′. Let M0 =M − int(α(Sk ×Dk)), contained in both M and M ′.

By excision, hj(M,M0) ≈ hj(Sk × Dk, Sk × Sk−1) vanishes except for j = k

where it is infinite cyclic. AsM is (k−1)-connected, the integral homology sequenceof the pair (M,M0) yields the exact sequence

(10.6.8) 0→ hk(M0)→ hk(M)h∗j−−→ hk(M,M0)→ hk−1(M0)→ 0

where j : (M, ∅) → (M.M0) denotes the pair inclusion. Since M is almost closed,Poincare and Kronecker dualities provide the commutative diagram

hk(M)h∗j //

OO[M ]≈

hk(M,M0) oo ≈hk(S

k ×Dk, Sk × Sk−1)OO[Sk×Dk]≈

hk(M,BdM)≈ // hk(M)

h∗α //

k≈

hk(Sk ×Dk)

k≈

hom(hk(M);Z)(h∗α)

♯

// hom(hk(Sk ×Dk);Z)

As hk(M) is free abelian, h∗α : hk(Sk × Dk) → hk(M) is injective, so h∗α is

surjective. Hence, h∗j is surjective and, by (10.6.8), hk−1(M0) = 0. Note thathi(M0) ≈ hi(M) = 0 for i < k and, since k ≥ 2, van Kampen’s theorem applied to

M ≈M0 ∪ (Sk ×Dk) and M0 ∩ (Sk ×Dk) ≈ Sk × Sk−1

implies that M0 is simply connected. Therefore, M0 is (k − 1)-connected. Also,from (10.6.8), we deduce that hk(M0) is free with rankhk(M0) = rankhk(M)− 1.

From similar considerations with the pair (M ′,M0), we deduce that M ′ is(k−1)-connected and that hk(M

′) is free with rankhk(M′) = rankhk(M)−2. The

above process may be repeated with M ′ showing that, thanks to a finite number ofstably framed surgeries, M is stably framed cobordant to M which is k-connected.By Poincare duality, h∗(M) = 0 if ∗ ≤ m− 1. If BdM is not empty, then h∗(M) ≈h∗(pt) and, as M is simply connected, M is contractible by the Hurewicz-Whiteheadtheorem [82, Proposition 4.74]. If M is closed, then h∗(M) ≈ h∗(S

m) and by theHurewicz theorem, πm(M) ≈ hm(M) ≈ Z. If γ : Sm → M represents a generatorof πm(M), then γ is a homotopy equivalence by the Hurewicz-Whitehead theorem.Hence M is a homotopy sphere.

Corollary 10.6.9. Let m = 2k ≥ 6 with k odd. Let Mm be a compact stablyframed manifold whose boundary is a homotopy sphere. Suppose that c(M) = 0.Then Σ = BdM is diffeomorphic to the standard sphere Sm−1.


Proof. By Proposition 10.6.8, the homotopy sphere Σ is also equal to Bd Mwhere M is contractible. Since Σ is simply connected, it is a consequence of theh-cobordism that M is diffeomorphic to Dm (see [151, Proposition A, p. 108]).Hence, Σ is diffeomorphic to Sm−1.

We have so far presented the stably framed surgery from the point of view ofWall [204], using the theory of immersions. An earlier approach was introducedby Milnor in [148] and developed in [113, 24] (for a presentation of the Kervaireinvariant in this framework, see [133, 120]). With this method, in order to performa stably framed surgery on a class a ∈ πj(Mm), we first represent a by an embeddingα : Sj → M (this is possible when m > 2j and when m = 2j if M is simplyconnected [77]). The stable framing of M gives a trivialization of ηN ⊕ TM whereηN is the trivial bundle of rank N (N large). We thus get a trivialization F of ηN ⊕α∗TM ≈ ηN⊕TSj⊕να. The vector bundle η1⊕TSj is the restriction of the tangentbundle to Rj+1 and therefore has a canonical field of orthonormal frames. Thus,ηN ⊕TSj = η1⊕ ηN−1⊕TSj has a canonical field of orthonormal frames. Togetherwith the above trivialization F , this gives an element [α] of the homotopy groupπj(Stief(j + N,Rm+N)) of the Stiefel manifold Stief(j+N,Rm+N ) which is shown tobe the obstruction to performing a stably framed surgery on the class a. Homotopygroups of Stiefel manifolds are known in a certain range (see [181, Theorem 25.6]).For 2j < m, πj(Stief(j + N,Rm+N)) = 0, whence the surgery below the middledimension. For m = 2k with k odd, one has πk(Stief(k + N,R2k+N)) = Z2. Thisgives the quadratic form q : Hk(M) → Z2, by q(a) = [α]. Consider the principalbundle

P =(SO(k)→ SO(2k +N)→ Stief(k +N,R2k+N )

)

and its homotopy exact sequence

πk(SO(2k +N))j∗−→ πk(Stief(k + N,R2k+N))

∂−→ πk−1(SO(k))

i∗−→ πk−1(SO(2k +N)) .

Changing the stable framing ofM adds to q(a) an element in the image of j∗. TheSO(k)-principal bundle α∗P is the bundle of orthonormal frames in να. Hence,∂(q(a)) ∈ πk−1(SO(k)) ≈ πk(BSO(k)) classifies να. If k is odd and k 6= 1, 3, 7,then ker i∗ ≈ Z2 (see [24, Corollary IV.1.11]) and thus ∂ is injective. This provesthe following result.

Lemma 10.6.10. Let M be a (k − 1)-connected 2k-dimensional stably framedmanifold, with k odd and k 6= 1, 3, 7. Then, a class a ∈ Hk(M) satisfies q(a) = 0 ifand only if it is representable by a k-sphere in M with trivial normal bundle.

The Kervaire invariant has several other definitions (see, e.g. [23, 123] and alsothe proof of Theorem 10.6.12 below). These more homotopy-theoretic descriptionswere much used for computing the image of the Kervaire invariant c : Ωmfr ≈ πSm →Z2, an outstanding problems in stable homotopy theory. A main advance was dueto Browder [23]. An almost complete solution (except for m = 126) was providedby Hill, Hopkins and Ravenel in 2009, who proved the following theorem (see [92]).

Theorem 10.6.11. Let m = 2k with k odd. Then, the Kervaire invariantc : Ωmfr → Z2 is surjective if m = 2, 6, 14, 30, 62 and possibly 126. In all otherdimensions, the Kervaire invariant vanishes.

The vanishing of the Kervaire invariant in dimension m implies the existenceof a closed PL-manifold Km which does not have the homotopy type of a closed


smooth manifold. Below is a description of the construction of Km, originally dueto M. Kervaire [114].

Let pi : Di → Sk (i = 1, 2) be two copies of the unit disk bundle associated tothe tangent disk bundle to Sk (k ≥ 3). Let A be a closed k-disk in Sk. Choose

trivializations µi : p−1i (A)

≈−→ A × Dk ≈ Dk × Dk over A (they are unique up toisotopy: see Lemma 9.1.2). Let K0 be the quotient space of D1 ∪ D2 under theidentification µ1(x, y) = µ2(y, x). After rounding the corners (see [17, Appendix,Theorem 6.2]), K0 is a smooth compact manifold of dimension m = 2k, withboundary Σ0 = Σm−10 . This is an example of the so called plumbing technique (see[97, § 8], [120]).

The boundary manifold Σ0 is a homotopy sphere. Indeed, K0 is (k − 1)-connected and hk(K0) is free abelian. The integral intersection form clearly induces

an isomorphism hk(K0)≈−→ hom(hk(K0),Z). By the analogue for the integral

homology of Proposition 5.3.11, we deduce that h∗(Σ0) ≈ h∗(Sm−1) (compare [97,p. 58]). Moreover, K0 is a thickening of Sk ∨Sk. By general position (since k ≥ 3),one has π1(Σ0) ≈ π1(K0) = 1. Hence, Σ0 is a homotopy sphere (see the proof ofProposition 10.6.8).

We claim that c(K0) = 1. Indeed, Hk(K0) has a symplectic basis a, b rep-resented by the two copies of Sk. Recall that TSk is isomorphic to the normalbundle of the diagonal sphere in Sk × Sk (see Lemma 5.4.17). It then follows fromExample 10.6.7 (or from Lemma 10.6.10 if k 6= 1, 3, 7) that qK0(a) = qK0(b) = 1.Therefore c(K0) = 1.

The smooth structure on K0 determines a unique PL-structure (see (3) onp. 170). The homotopy sphere Σ0 is PL-isomorphic to the standard sphere bySmale’s theorem [177]. Let K be the PL-manifold obtained by gluing to K0 thecone over Σ0. The homotopy sphere Σ0 = Σm−10 is called the Kervaire sphere whilethe PL-manifold K = Km is called the Kervaire manifold.

Theorem 10.6.12. Let m = 2k ≥ 10 with k odd. Then the following assertionsare equivalent.

(a) The Kervaire invariant c : Ωmfr → Z2 vanishes.

(b) The Kervaire sphere Σm−10 is not diffeomorphic to the standard sphere.(c) The Kervaire manifold Km does not have the homotopy type of a smooth

closed manifold.

Thus, according to Theorem 10.6.11, (b) and (c) are true for 10 ≤ m 6= 14, 30, 62and possibly 126. Theorem 10.6.12 goes back to Kervaire [114] who proved it in1960 for m = 10, constructing the first example in history of a topological closedmanifold not admitting any smooth structure (even up to homotopy type). Togetherwith the discovery by J. Milnor in 1956 of several smooth structures on the 7-sphere,Kervaire’s result, was quite influential in the history of differential topology (see[53]).

Proof. (c) ⇒ (b). Suppose that (b) is not true, so there is a diffeomorphismϕ : Sm−1 → Σ0. Then K0 ∪ϕ Dm is a smooth closed manifold which is homeomor-phic to K. This contradicts (c).

(b) ⇒ (a). If (a) is not true, there is a closed smooth stably framed manifoldNm with c(N) = 1. Let N0 be N with an open m-disk removed. Thus, N0

is a smooth almost closed stably framed manifold with c(N0) = 1. Let P be


the boundary connected sum of N0 and K0. The boundary of P is Σ0 ♯ Sm−1,

diffeomorphic to Σ0. But P is a smooth almost closed stably framed manifold withc(P ) = c(N0) + c(K0) = 0. By Corollary 10.6.9, Σ0 is diffeomorphic to Sm−1.

(a) ⇒ (c). This is more complicated and we just sketch the idea of the proof.Suppose that (c) is not true, so there is a smooth closed manifoldM and a homotopyequivalence f : Km → M . Hence, M is of dimension m and is (k − 1)-connected.Let x be the cone point of K and y = f(x). Let D be an open m-disk in M aroundy and let M0 = M −D. Using boundary collars in K0 and M0, one can constructcontinuous maps of pairs fK : (K0,Σ0) → (K,x) and fM : (M0, S

m−1) → (M, y).These maps induce isomorphism

(10.6.9)

Hk(K)H∗f

≈//

H∗fK≈

Hk(M)

H∗fM≈

Hk(K0) oo Ψ

≈Hk(M0)

where Ψ is defined to make the diagram commutative. For the sake of this proof, anm-dimensional relative smooth manifold (X,Y ) is called acceptable if X is (k− 1)-connected and Hi(X,Y ;G) = 0 for k < i < n for all coefficient groups G. Theabove pairs (K,x), (M, y), (K0,Σ0) and (M0, S

m−1) are acceptable. If (X,Y )is an acceptable pair, Kervaire and Milnor in [113, pp. 531–534] defined a mapψX,Y : Hk(X) → Z2 (this is the map ψ0 : Hk(X ;Z) → Z2 of [113, p. 534] whichdescends to Hk(X)). The following two properties hold true.

(i) Let g : (X,Y )→ (X ′, Y ′) be a continuous map between acceptable pairs.If g induces an isomorphism H∗(X ′, Y ′;Z)→ H∗(X,Y ;Z), then

ψX,Y H∗g = ψX′,Y ′ .

(ii) Let (X,Y ) be an acceptable pair with X stably parallelizable. ThenψX,Y (a) = 0 if and only if a is representable by an embedded k-spherewith trivial normal bundle [113, Lemma 8.3 and p. 534].

Another fact is that, as M is homotopy equivalent to K, it is stably paral-lelizable (see [25, Theorem A2.1]). Therefore, the quadratic form qM and qM0 aredefined. Since (a) is true and m ≥ 10, one has m 6= 14 by Theorem 10.6.11 andthus Lemma 10.6.10 applies. Therefore, for a ∈ Hk(M0) ≈ Hk(M), one has

qM (a) = qM0(a)

= ψM,M0(a) by (ii) and Lemma 10.6.10

= ψK,K0 Ψ(a) by (10.6.9) and (i)

= qK0 Ψ(a) by (ii) and Lemma 10.6.10.

Hence c(M) = maj(qM ) = maj(qK0) = 1, which contradicts Assertion (a).

Besides giving the Kervaire invariant for a stably framed manifold, the Arfinvariant of a quadratic form determines the surgery obstruction group L2(π) ≈Z2 for π of order ≤ 2 [24, 203]. It also has applications in classical and highdimensional knot theory (see [205] for a survey on these works).



10.1. Which closed surfaces admit a involution with scattered fixed point set?

10.2. Let X be a conjugation space. Prove that κ : H0(X) → H0(XG) coincideswith the homomorphism induced by the inclusion XG → X . Deduce that a G-space Y is a conjugation space if and only if each path-connected component of Yis a conjugation space.

10.3. Prove that the definition of a conjugation space may be expressed with thereduced cohomology.

10.4. Let X and Y be two conjugation spaces which are G-equivariantly wellpointed. Prove that X ∨ Y is a conjugation space.

10.5. Let Xi (i = 1, 2) be two conjugation spaces which are G-equivariantly wellpointed. Suppose that X2 has finite cohomology type. Prove that X1 ∧ X2 is aconjugation space.

10.6. Let X be a conjugation space which is equivariantly well pointed. Constructa conjugation space Z such that ZG is homeomorphic to XG ∧ S1.

10.7. Prove that there is no conjugation space X such that XG is homotopy equiv-alent to ΣkOP 2. [Hint: see Remark 10.2.16.]

10.8. Consider the generic length vector ℓ = (1, . . . , 1, n− 2) ∈ Rn>0. Let C = Cnd (ℓ)and C = Cnd (ℓ). Compute Pt(C) using Proposition 10.3.17. Compute H∗(C) usingthe transfer exact sequence or Theorem 10.3.11.

10.9. Let ℓ = (1, 1, 2, 2, 3). Compute Pt(C5d(ℓ)). What are C52(ℓ) and C52(ℓ)?10.10. For which generic length vector ℓ is the chain space Cn2 (ℓ) an orientablemanifold? [Hint: this depends on n and on the lopsidedness lops (ℓ).]

10.11. For which generic length vector ℓ does the spatial polygon space Nn3 (ℓ)

admit a spin structure?

10.12. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector with lops (ℓ) ≥ 2. Prove thatulops (ℓ)−2 6= 0 in H∗(Cn2 (ℓ)) ≈ H∗G(Cn2 (ℓ)).10.13. Do Example 10.4.5 for Gr(2;R5).

10.14. Use Theorem 10.5.1 to compute H∗Γ1(Γ/Γ2) for Γ1 = Γ2 = ±1, the center

of Γ = SU(2). Prove that H∗Γ1(Γ/Γ2) ≈ H∗(RP 3)[u]. Is that surprising?

CHAPTER 11

Hints and answers for some exercises


2.1. F ′n contains (n+ 1)! n-simplexes (by induction on n).

2.2. Pt(Fkn) = 1 +(nk+1

)tk. By induction on k, starting with Pt(F0

n) = 1 + n andwith

bk+1(Fk+1n ) = dimCk+1(Fn)− bk(Fkn) =

(n+1k+2

)−(nk+1

)=

(nk+2

).

as induction step. For the Euler characteristic, evaluate Pt(Fkn) at t = −1.2.3. (a) X is isometric to A∪B where A = (0, 0, 0), (0, 1, 0), (1, 1, 1), (1, 0, 1)and B = (1, 0, 0), (1, 1, 0), (0, 1, 1), (0, 0, 1). The complex Aε and Bε are bothquadrilaterals, and Xε = Aε ∗Bε. Thus Xε is isomorphic to the double suspensionof Bε, so |Xε| ≈ S3.

2.4. ℓ = (1, 1, 1, 1, 3): Sh(ℓ) = 5∪F1, 2, 3, 4. χ(Sh(ℓ)) = 3. Pt(Sh(ℓ)) = 2+ t2.ℓ = (1, 1, 3, 3, 3): the maximal simplexes of Sh(ℓ) are 1, 2, 3, 1, 2, 4 and

1, 2, 5. χ(Sh(ℓ)) = 1. Pt(Sh(ℓ)) = 1.ℓ = (1, 1, 1, 1, 1): Sh(ℓ) is the 1-skeleton of F1, 2, 3, 4, 5. χ(Sh(ℓ)) = −5.

Pt(Sh(ℓ)) = 1 + 6t.

2.5. a = 2z | z ∈ Z and a = 2z + 1 | z ∈ Z.2.10. By definition of a pseudomanifold, there exists a sequence σ = σ0, . . . , σm =σ′ of n-simplexes such that, for i ≤ 1 < m, σi and σi+1 have an (n − 1)-face incommon, called µi. Then, a = µ0, . . . , µm−1 satisfies δ(a) = σ, σ′.2.11. If γ = ∂(α), then γ = ∂(α+ [M ]). If also γ = ∂(α′), then α+ α′ ∈ Zm(M),therefore α+ α′ = 0 or [M ] by Proposition 2.4.4.

2.12. This is the content of Proposition 2.5.7.

2.15. Hn(M) = 0. Indeed, if 0 6= a ∈ Cn(M), there are σ ∈ a and σ′ ∈ S(M)− a(since M is infinite). Using Point (c) of the definition of a pseudomanifold, one cansuppose that σ ∩ σ′ = µ ∈ Sn−1(M). Then µ ∈ ∂a. Thus, Zn(M) = 0.

2.17. One has S(K)−S(K2) = S(K1)−S(K0), whence H∗(K1,K0) ≈ H∗(K,K2).The diagram for (b) is

Hr(K0) //

Hr(K1) //

Hr(K1,K0)∂01∗ //

i∗≈

Hr−1(K0)

Hr(K2) // Hr(K)

j∗ // Hr(K,K2)∂∗ // Hr−1(K2)

.

417

418 11. HINTS AND ANSWERS FOR SOME EXERCISES

One checks that ∂MV = ∂01∗ i−1∗ j∗ works as a Mayer-Vietoris connecting homo-

morphism.

2.19. By Mayer-Vietoris, Pt(M) = Pt(M1) +Pt(M2)− tn − 1.


3.1. In the triangulation of ∆m × I used in the proof of Proposition 3.1.30,the number of (m + 1)-simplexes is equal to m + 1. In that used in the proof of

Lemma 3.1.35, this number is∑m+1k=1 k!.

3.2. Point (b) is proven by induction on the skeleta of (A,B), using (a) and thatDn = CSn−1. For (c), let X be a contractible space. Thus, there exists x0 ∈ Xand a continuous map F : X × I → X with F (x, 0) = x and F (x, 1) = x0. Notethat the standard simplex ∆n+1 is homeomorphic to the cone on ∆n. Therefore, apoint of ∆n+1 has coordinates [z, t], where (z, t) ∈ ∆n×I, with [z, 1] = [z′, 1] for allz, z′ ∈ ∆n. A linear map D : Cm(X)→ Cm+1(X) is then defined, for σ ∈ Sm(X),by D(σ)([z, t]) = F (σ(z), t). It satisfies ∂D(α) = D(∂α)+α for all α ∈ Cm(X) andall m ≥ 1. This implies that Zm(X) = Bm(X) for m ≥ 1.

3.3. Using the homology sequence of (X,X −A): Pt(S2 −A) = 1 + (n− 1)t and

Pt(T2 −A) = 1 + (n+ 1)t.

3.4. (RP 2,RP 1) and (RP 2, S) where S is the boundary of a 2-disk in RP 2.

3.5. Define h(x, t) = x, u(A) = 0 and u(X −A) = 1. Then (u, h) is a presentationof (X,A) as a well cofibrant pair.

3.6. Let M be the Mobius band. Then H1(BdM) → H1(M) is the zero homo-morphism, contradicting Lemma 3.3.1 if there were a continuous retraction of Monto BdM).

3.8. There is a unique smaller arc of great circle joining f(x) to f(−x), whence ahomotopy to a map f satisfying f(x) = f(−x). This means that f factors throughSn → RPn which is of degree 0.

3.9. It is a braid diagram looking locally like

Hk(X) //

&&

Hk(Z) //δ∗

((

Hk+1(Y,Z)

''PPPP

PPP

Hk(Y )

77♦♦♦♦♦♦♦

δ∗

''

Hk+1(X,Z)

66♠♠♠♠♠♠♠♠

((

Hk+1(Y )

Hk(Y,Z)δ∗ //

88qqqqqqqHk+1(X,Y ) //

66♠♠♠♠♠♠♠♠Hk+1(X)

77♥♥♥♥♥♥♥

3.11. Like for Exercise 2.17.

3.13. If X is countable, so is its cellular chain complex and then H∗(X) iscountable. But if, for example, X has infinitely many connected components, thenH0(X) is not countable by Corollary 3.1.12.

3.14. There is a retraction rn : B → Cn sending B − Cn to 0. This gives ahomomorphism r∗ =

∏H∗rn : H1(B) → ∏∞

n=1H1(Cn) =∏∞n=1 Z2. Dividing [0, 1]

using the intervals In = [1/(n + 1), 1/n], we can construct various paths in B so


that the generator of H1(In,Bd In) is sent to that of H1(Cn) or to 0. Any elementof

∏∞n=1H1(Cn) may thus be realized, proving that r∗ is surjective.For (b), let Rk be the set of homotopy classes of maps from B to RP∞ sending

Cn onto a point for n > k. This set is finite. As RP∞ is locally contractible, themap

⋃∞n=1Rn → [B,RP∞] is surjective, which proves (b).

If B has the homotopy type of a CW-complex, then H1(B) ≈ [B,RP∞] byProposition 3.8.3. As the latter is countable by (b), this contradicts (a) by Kro-necker duality.

3.15. Let Bq be a Z2 basis of Rq and let Xq be a bouquet of q-spheres indexed byBq. The bouquet X =

∨q≥1Xq satisfies Hq(X) ≈ Rq.

3.16. Since 1−m+ n = χ(X) = 1− b1(X) + b2(X).

3.18. Note that H1(Xp) ≈(π1(XP )/[π1(XP ), π1(XP )]

)⊗ Z2. Thus, b1(XP1) =

b1(XP2) = 1 and b1(XP3) = 0. As χ(XP1) = χ(XP2) = 0 and χ(XP3) = 1, we getPt(XP1) = Pt(XP2) = 1 + t and Pt(XP3) = 1.

3.20. Since X is not contractible, rj is not homotopic to a constant map, andnor is j. One deduces that Hn(X) ≈ Z2 and then j r is homotopic to the identityby Exercise 3.19.


4.5. As in § 4.3.3, one defines a transfer homomorphism tr∗H∗(X) → H∗(X) by

sending each singular simplex σ : ∆m → X to the set of its liftings into X . As thenumber of these liftings is odd, one has H∗ptr∗ = id. Hence H∗p is surjective andH∗p is injective by Corollary 2.3.11.

4.6. Use Proposition 4.2.3.

4.7. From T to K, there is no map of degree one by Exercise 4.6. The same kindof argument shows that there is no map of degree one from S1 × S2 to RP 3 (themap a 7→ a3 is non-trivial in H∗(RP 3) while it is trivial in H∗(S1 × S2)). For theother directions, let f : K → T be a map. By Proposition 4.2.3, H1f cannot beinjective which, given the ring structure of H∗(T ), implies that f is not of degreeone. For a map g : S1 × S2 → RP 3, we see above that H1f = 0. Thus, thecomposition of g with the inclusion j : RP 3 → RP∞ is homotopic to a constantmap by Proposition 3.8.3. But H3j is an isomorphism by Proposition 4.3.10, so gis of degree 0.

4.8. M is a connected sum of M and RPn. By Propositions 4.2.1 and 4.3.10, onehas H∗(M) ≈ H∗(M)[a]/(am + [M ]♯).

4.9. By § 4.2.4 and Proposition 4.2.1, one has

H∗((S1 × S1) ♯RP 2) ≈ Z2[a, b, c]/(a2, b2, c3, ab+ c2, ac, bc) ,

with a, b and c of degree 1, and

H∗(RP 2 ♯RP 2 ♯RP 2) ≈ Z2[x, y, z]/(x3, y3, z3, x2 + y2, y2 + z2, xy, xz, yz)

with x, y and z of degree 1. An isomorphism is given by a 7→ x+ y, b 7→ x+ z andc 7→ x+ y + z.

4.11. Use Proposition 4.7.11 and (4.7.8)


4.12. (a) H∗(X) ≈ Z2[x1, . . . , xn], with xi of degree 1 (Kunneth theorem). For(b) and (c), set H∗(CP 2) ≈ Z2[a]/(a

3) and H∗(CP 3) ≈ Z2[b]/(b4), with a and b of

degree 2. Then

(b) As a Z2-vector space, H∗(Y ) admits the basis ab, a2b, ab2, ab3, a2b2, a2b3

(see Proposition 4.7.11; we write xy for the reduced cross product of xand y). The only non-trivial cup products are (ab)2 = a2b2 and (ab)(ab2)) =a2b3.

(c) H∗(Z) ≈ H∗(ΣY ) (see p. 144). All cup products vanish.

4.15. The space X = RPm × RPn is a connected CW-complex of dimensionm + 1, thus cat (X) ≤ m + n + 1 by Proposition 4.4.1. By Proposition 4.3.10and the Kunneth theorem, H∗(X) ≈ Z2[a, b]/(a

m+1, bn+1). Thus, ambn 6= 0, sonilH>0(X) ≥ m+n+1. Proposition 4.4.2 thus implies that cat (X) = m+n+1.

4.16. cat (T n) = n+ 1 (same arguments as for Exercise 4.15).

4.17. For continuous maps f : X → X ′ and g : Y → Y ′, the diagram

H∗(X × Y )×

//

H∗(f×g)

H∗(X)⊗H∗(Y )

H∗f⊗H∗g

H∗(X′ × Y ′)

×// H∗(X ′)⊗H∗(Y ′)

is commutative. This is proven using Formula (4.6.8).

4.18. Using the Kunneth formula, the condition Pt(X) = Pt(Y ) amounts tothe polynomial equality

∏ri=1(1 + ai t) =

∏sj=1(1 + bj t). Therefore r = s and

σk(a1, . . . , ar) = σk(b1, . . . , br) for k = 1 . . . , r, where σk is the k-th elementarysymmetric function. This implies the polynomial equality

r∏

i=1

(x− ai) =r∏

i=1

(x− bi) ,

which implies that bi = aα(i).

4.19. For u ∈ H∗(X) and v ∈ H∗(Y ), one has

〈u⊗ v, (a α)⊗ (b β)〉 = 〈u, a α〉〈v, b β〉 by (4.6.7)

= 〈u a,α〉〈v b, β〉

= 〈(u a)⊗ (v b), α⊗ β〉 by (4.6.7)

= 〈(u a)× (v b),×−1(α⊗ β)〉 by (4.6.8)

= 〈(u× v) (a× b),×−1(α⊗ β)〉 by Remark 4.6.4

= 〈u× v, (a× b) ×−1(α⊗ β)〉

= 〈u⊗ v,×(

(a× b) ×−1(α⊗ β))

〉 by (4.6.8).

As the elements u⊗ v generate H∗(X)⊗H∗(Y ), this proves the formula.

4.20. Using Lemma 4.7.2 and Equation (4.6.8).

4.21. let i : E0 → E and j : (E, ∅) → (E,E0) denote the inclusions. Themaps H∗i and H∗j are morphisms of H∗(B)-algebras. To see that δ∗ : H∗(E0)→


H∗+1(E,E0) is a morphism of H∗(B)-modules, let b ∈ Hr(B) and v ∈ Hs(E,E0).Then,

δ∗(H∗p0(b) v) = δ∗(H∗iH∗p(b) v) = H∗p(b) δ∗(v)

holds true inHr+s(E,E0), thanks to the singular cohomology analogue of Lemma 4.1.9.

4.22. If d is odd, then χ(Fd(K)) = 1 − χ(K). If d is even, χ(Fd(K)) = 1 +b(K), where (K) is the total Betti number of K. This may be deduced fromCorollary 4.7.52.

4.23. The Gysin sequence implies (a) and (b). Actually, e = 0 implies that therestriction homomorphism H∗(E) → H(Σ) is surjective (see Proposition 4.7.35),so the isomorphism φ of (b) is an isomorphism of H∗(Sm)-modules by the Leray-Hirsch Theorem. If n > 2, φ is a GrA-isomorphism for reasons of degree. Thehypothesis n > 2 in (c) is necessary (see Remark 4.7.34).

4.24. Via a Riemannian metric, we identify D(ν) with a closed tubular neighbour-hood of Q in M . For x ∈ Q, let (Dx, Sx) ≈ (Dr, Sr−1) be the fiber of (D(ν), S(ν))over x. The pair inclusions (Dx, Sx) → (D(ν), S(ν)) → (M,M − Q) gives rise tothe commutative diagram

Hr−1(M −Q) //

δ∗

Hr−1(S(ν)) //

δ∗

Hr−1(Sx)

δ∗≈

Hr(M,M −Q)≈

// Hr(D(ν), S(ν)) // // Hr(Dx, Sx)

.

The left vertical arrow is surjective since Hr(M) = 0. Thus, Hr−1(S(ν)) →Hr−1(Sx) is onto, which implies that e(ν) = 0 by Proposition 4.7.35. CompareExercise 5.11.


5.1. Let X and Y be two homology manifolds and let (x, y) ∈ X × Y . As(X×(Y−y))∪

((X−x)×Y

)= X×Y−(x, y), one can use a relative Kunneth theorem

like Theorem 4.6.10 (see the comments after this theorem for the hypotheses thatwe use, i.e. that (X × (Y − y), (X − x) × Y

)be excisive in X × Y , which is true

since they are both open).

5.4. It is enough to prove it for a connected manifold X with Y = BdX 6= ∅. Ifn = dimX , Theorem 5.3.7 implies that Hn(X) ≈ H0(X,Y ) = 0 and Hn(X,Y ) ≈H0(X) = Z2. Therefore, Hn−1(Y ) → Hn−1(X) is not injective by the homologysequence of (X,Y ), which would be the case if there were a retraction from Xonto Y (see Lemma 3.3.1).

5.5. Only the connected component of the point plays a role, so we can supposethat M is connected. Instead of removing a point, we remove a closed n-disk Daround it. Thus M = M0 ∪ D, where M0 = M − intD. As Hn(M) ≈ Z2 ≈Hn(M,M0), the result follows from the homology sequence of (M,M0).

5.6. Since [N ] = [N1 + [N2] = ∂[M ] in Hn−1(M). If N2 = ∅, then H∗f([N1]) = 0.

5.8. There exists no degree one map from Σm to Σn by Propositions 5.2.8 and 4.2.3.The same arguments show that if there is a degree one map from Σn to Σm, then


m ≤ [(n − 1)/]. The converse is true, using that RP 2 ♯RP 2 ♯RP 2 and (S1 ×S1) ♯RP 2 are homeomorphic (see [136, Lemma 7.1]).

5.9. No, becauseH∗f would be injective by Proposition 5.2.8, and, by the Kunneththeorem, there is no element a ∈ H1(M) with am 6= 0. THIS ANSWER IS WRONG

(see Erratum 12.0.3).

5.10. Follows from Corollary 5.4.13.

5.11. Let ν be the normal bundle to Q. That H∗i([Q]) = 0 implies that thePoincare dual PD(Q) of Q vanishes, whence e(ν) = 0 by Lemma 5.4.4. CompareExercise 4.24.

5.13. Let A = PD(Q1) × PD(Q2). For dimensional reasons, [M1 × M2] =×−1([M1]⊗ [M2]) and [Q1 ×Q2] = ×−1([Q1]⊗ [Q2]). Therefore,

×(A [M1 ×M2]

)= ×

(PD(Q1)× PD(Q2) ×−1([M1]⊗ [M2])

)

= (PD(Q1) [M1])⊗ (PD(Q2) [M2]) by (4.8.1)

= H∗i1([Q1])⊗H∗i2([Q2])

= (H∗i1 ⊗H∗i2)([Q1]⊗ [Q2])

= ×(H∗i([Q1 ×Q2])

)by Exercise 4.17

where i : Q1 ×Q2 →M1 ×M2 denotes the inclusion. As × is an isomorphism, thisproves (5.5.1)

5.14. By Exercise 5.13, PD(x ×M ′) = PD(x)× PD(M ′) = [Mx]♯ × 1, where

Mx is the connected component of x in M .

5.18. Let a ∈ Q. The map associating to (t, x) ∈ I ×Q′ the point (√1− t2 a, t2x)

parameterizes a smooth (q′ + 1)-disk D′a in Σ, with boundary Q′, which intersectsQ transversally at a. By Proposition 5.4.22, l(Q,Q′) = 1.


6.1. KPn may be covered by n + 1 chart domains which are contractible (see(6.1.1) for K = C; the same formalism works for K = H). On the other hand,nilH>0(KPn) = n + 1, thus cat (KPn) = n + 1 by Proposition 4.4.2, like forK = R: see Corollary 4.4.3).

6.3. Follows from Proposition 6.1.11.

6.4. The two parts of CCf give rise to the Mayer-Vietoris sequence

// Hk(CCf ) // Hk(Y )⊕Hk(Y )ϕ // Hk(X) // Hk+1(CCf ) //

where ϕ = H∗f +H∗f . For f = j, this sequence together Proposition 6.1.11 givesthe Poincare series of CCj :

Pt(CCj) =

∞∑

k=0

t4k +

∞∑

k=1

t4k+3 =1 + t3

1− t4 .

Actually, one has a GrA-isomorphism H∗(CCj) ≈ Z2[a, b]/(b2) where a is of de-

gree 4 and b of degree 3 (see Exercise 7.20).


6.5. By the Kunneth theorem and Proposition 4.2.1, H∗(X) ≈ Z2[a, b]/(a2, b2)

and H∗(X) ≈ Z2[c, d]/(c3, d3, cd, c2 + d2) (generators in degree 4). Thus, Pt(X) =

Pt(Y ) = 1 + 2t4 + t8. But the cup-square map vanishes in H∗(X) while not inH∗(Y ). (If the degree of the generators are set to 1, we get the cohomology of thetorus for X and of the Klein bottle for Y ).

6.6. Use the Hopf vector bundle γK over KP 1. For n = 2k, one can take ξ =γC × · · · × γC (k times) over (CP 1)k. For n = 2k + 1, take ξ = γC × · · · × γC × γR

over (CP 1)k × RP 1.

6.7. There exists a map f : X → Sn such that a = H∗f(ι) (see the proof ofProposition 3.8.1). One can then take ξ = f∗γK where γK is the vector Hopf bundleover KP 1 for K = R,C,H and O.

6.10. Follows from Exercise 4.24.

6.13. The maps h and f extend to maps on mapping cones Cghh−→ Cg and

Cgf−→ Cf g. The formula follows from the functionality of the cup product.

6.14. Let y′ ∈ Sm be another regular value for f . We may suppose that j istransverse to Q′ = f−1(y′). Then,

Hopf (f) = l(Q,Q′) by Proposition 6.3.7

= ♯ j−1(Q′) mod 2 by Proposition 5.4.22

= ♯ (f j)−1(y′) mod 2= deg(f j) as in Proposition 3.2.6.


7.2. Follows from Corollary 7.1.17. Note that this is wrong for p = 0 (seeExample 7.1.16).

7.3. AsX is equivariantly formal, there is aGrV[u]-isomorphism betweenH∗(XG)and Z2[u][B], where B is a GrV basis of H∗(X). Therefore,

Pt(XG) =Pt(X)

1− t and Pt(u ·H∗(XG)) =tPt(X)

1− t .

7.6. By Proposition 7.1.12, ρ : HdG(X)→ Hd(X)G is an isomorphism and, as d ≥ 1,

Hd(X)G is the diagonal subgroup of Hd(X) ≈ Z2⊕Z2. By Sequence (7.1.8), thereexists 0 6= a ∈ Hd

G(X) and Ann (u) = Z2a. Sequence (7.1.8) also implies thatH∗G(X) ≈ Z2[u, a]

/(ua). If a2 6= 0, then a2 = u2d which would contradict the

equation ua = 0; indeed ua2 = u2d+1 6= 0 by Corollary 7.1.5.

7.7. The pair (SU(2),∆) is isomorphic to (S3, S1), the groups of units quaternionsand complex numbers. For Γ a topological group, let EnΓ be the join of n copiesof Γ, on which Γ acts diagonally. Then, for n ≥ 1, En(S

3)/S3 ≈ S4(n−1)+3/S3 ≈HPn−1 and En(S

1)/S1 ≈ S4(n−1)+1/S1 ≈ CPn−1. Thus, there are homotopyequivalences BS3 ≃ E∞S3/S3 ≃ HP∞ and BS1 ≃ E∞S1/S1 ≃ CP∞.

7.8. One check that (X × Y )Γ ≈ XΓ × Y . The result then follows from theKunneth theorem 4.6.7. One can also use the equivariant Kunneth theorem 7.4.3with an additional hypothesis on Γ.


7.9. The hypotheses of the Kunneth theorems 4.6.7 and 7.4.3 are satisfied (see theend of Remark 7.4.4). Therefore, there is a commutative diagram

H∗Γ(X)⊗H∗Γ(pt) H∗Γ(Y )×Γ

≈//

ρX⊗ρY

H∗Γ(X × Y )

ρX×Y

H∗(X)⊗H∗(Y )

×

≈// H∗(X × Y )

.

As X and Y are equivariantly formal, ρX ⊗ ρY is surjective and thus, ρX×Y issurjective.

7.10. By Exercise 7.9 and (7.1.16), one has a GrA[u]-isomorphism

H∗G(X) ≈ H∗G(S10 )⊗H∗(Y ) ≈ Z2[u, a]

/(a2 + ua)⊗H∗(Y ) ≈ H∗(Y )[a]

/(a2 + ua)

with a of degree 1. The Mayer-Vietoris sequence in question is equivalent to

0 // H∗G(X)r // H∗G(X

G)J // H∗(Y ) // 0

with theGrA[u]-isomorphismH∗G(XG) ≈ H∗(Y )[u1]⊕H∗(Y )[u2]. One has J(b, c) =

b + c for b, c ∈ H∗(Y ) (proving that J is onto) and J(u1, 0) = J(0, u2) = 0. Onehas r(u) = (u1, u2), r(b) = (b, b) for b ∈ H∗(Y ) and r(a) = (0, u2).

7.11. By the equivariant Kunneth theorem 7.4.3, H∗G(X) ≈ Z2[u, a, b]/(a2 +

ua, b2 + ub) with a and b of degree 1. The critical points of f , labeled by therevalue, are x−3 = (−1,−1) (index 0), x−1 = (1,−1) and x1 = (−1, 1) (index 1) andx3 = (1, 1) (index 2). Therefore

H∗G(XG) ≈ Z2[u−3]⊕ Z2[u−1]⊕ Z2[u1]⊕ Z2[u3] .

One has r(u) = (u−3, u−1, u1, u3), r(a) = (0, u−1, 0, u3) and r(b) = (0, 0, u1, u3),whence r(ab) = (0, 0, 0, u23).

7.12. X = KP 2 (K = R,C,H or O) with the involution τ(x0 : x1 : x2) = (−x0 :x1 : x2). Related to this exercise is Exercise 8.9.

7.13. One has (Pnp )G = P pp ∪Pn−p−1n−p−1 . The total Betti numbers thus satisfy

b(Pnp ) = b((Pnp )G). This proves (a) by Proposition 7.3.7.

For (b), set H∗(P pp ) = Z2[u, b1]/(bp+11 ), H∗(Pn−p−1n−p−1 ) = Z2[u, b2]/(b

n−p2 ) and

H∗(Pnp ) = Z2[a]/(an+1). Thus, as a Z2-module, H∗G(P

np ) is generated by the powers

of an element A ∈ H1g (P

np ) such that ρ(A) = a. There are two such elements, A and

B, with the relation B = A+ u. One has r(A) = (b1 + u, b2) or r(A) = (b1, b2 + u).(use that r : H∗(Pnp ) → H∗((Pnp )

G) satisfies r(a) = (b1, b2); but (b1, b2)n = 0 in

HnG((P

np )

G) and (b1 + u, b2 + u)n ∈ (u), contradicting ρ(An) = an 6= 0). Byexchanging A and B if necessary, one may suppose that r(A) = (b1 + u, b2).

For (c), one uses that r(A) = (u, b2) (see (b)). Hence, r(Ak) = (uk, 0) fork ≥ n − 1. Therefore, the relation r(An+1) = r(uAn) holds true in H∗G((P

np )

G).

As r is injective, the relation An+1 = uAn holds true in H∗G(Pnp ). Therefore,

there is a surjective GrA[u]-morphism h : Z2[u][A]/(An+1+uAn)→ H∗G(P

np ). But

Z2[u][A]/(An+1 + uAn) and H∗G(P

np ) (which is a free Z2-module over A, . . . , An)

have the same Poincare series. Hence, h is an isomorphism. The second presentationfollows by substituting B + u to A. Point (d) is proved in the same way.


7.14. The isomorphism is given by A 7→ B + u, with inverse given by B 7→ A+ u.Note that R and S are the two presentations for H∗G(P

20 ) given in Exercise 7.14 (c).

Take X = HP 2 with τ(x0 : x1 : x2) = (−x0 : x1 : x2).

7.16. If we consider S1 acting on CPn by

g · (x0 : · · · : xp : xp+1 : · · · : xn) = (gx0 : · · · : gxp : xp+1 : · · · : xn) ,the statements are all the same, the degree of each cohomology class being doubled.

7.18. Write ΣX for ΣΓX . We seeX as the union of two cones C−X and C+X gluedover their common base X . The homomorphism Σ∗Γ comes from the composition

H∗Γ(X)δ∗ // H∗+1

Γ (C−X,X) oo ≈ H∗+1Γ (ΣX,C+X) // H∗+1

Γ (ΣX)

The homomorphism δ∗ comes from the cohomology sequence of the pair ((C−X)Γ, XΓ).As CX is Γ-equivariantly contractible, the homomorphism δ∗ descends to an in-jection δ∗ : H∗Γ(X) → H∗+1

Γ (C−X,X). The middle arrow is induced by the pairinclusion (C−X,X)→ (ΣX,C+X) and is an isomorphism by excision on the Borelconstructions. By (7.2.10), the last arrow becomes an isomorphism when composed

by H∗+1Γ (ΣX) → H∗+1

Γ (ΣX). Whence the homomorphism Σ∗Γ which is injective.That Σ∗Γ is a morphism of H∗Γ(pt)-module comes from Exercise 4.21 applied to thebundle pair ((C−X)Γ, XΓ) over BΓ. The morphism Σ∗Γ is functorial in both Xand in Γ. In particular, the trivial homomorphism Γ→ 1 gives the commutativediagram

(11.6.1)

H∗Γ(X)Σ∗Γ //

ρ

H∗+1Γ (ΣΓX)

ρΣ

H∗(X)Σ∗

≈// H∗+1(ΣX)

.

When Γ = G = X , one has H∗G(G) = 0 by Lemma 7.1.4 while H1G(ΣGG) = Z2,

since ΣGG is G-homeomorphic to the sphere S10 of Example 7.1.14.

For (c), let σ : H∗(X)→ H∗Γ(X) be a section of ρ. Using the commutativity ofDiagram (11.6.1), one defines a section σΣ of ρΣ by σΣ = Σ∗Γσ(Σ

∗)−1. Therefore,

ΣX is Γ-equivariantly formal and, as H∗Γ(pt)-module, H∗+1Γ (ΣX) is generated by

Image(σΣ). As Image(σΣ) ⊂ Image(Σ∗Γ), the morphism Σ∗Γ is an isomorphism.

7.19. The G-space ΣX is a finite dimensional G-complex satisfying b(ΣX) =b((ΣX)G) < ∞ (since (ΣX)G = XG). Therefore, ΣX is equivariantly formal by

Proposition 7.3.7 and thus r : H∗G(ΣX) → H∗G((ΣX)G) is injective by Proposi-

tion 7.3.9. As r factor through H∗Gi, we know that H∗Gi is injective. By Propo-sition 7.3.7 again, X is itself equivariantly formal and thus ρ is surjective. AsH∗i = 0, the image of H∗Gi is contained in ker ρ. By (7.1.8), the latter is equal to

the ideal (u) generated by u. By comparing the Poincare polynomials of H∗G(ΣX)with that of (u) (see Exercise 7.3) we deduce that the image of H∗Gi is actuallyequal to (u).

7.20. Let ∆ be the subgroup of SU(2) formed by the diagonal matrices andlet ∆+ = dia(eiα, e−iα) | α ∈ [0, π]. Each conjugation class in Γ has a uniquerepresentative in ∆+, so X/Γ is homeomorphic to [0, π]. Hence, each class in


EΓ ×Γ X has representatives of the form (z, b) with b ∈ ∆+. We deduce that XΓ

is homeomorphic to (EΓ/∆ × [0, π]) ∪ (EΓ/Γ × 0, π), that is XΓ ≈ CCj0 wherej0 is the quotient map EΓ/∆ → EΓ/Γ. The latter is homotopy equivalent to themap j1 : B∆ → BΓ induced by the inclusion ∆ → Γ (see Example 7.2.4). ByExercise 7.7, j1 is equivalent to the inclusion j : CP∞ → HP∞. This proves (a).From the correction of Exercise 6.4 (see p. 422), we have

Pt(H∗Γ(X)) =

(1 + t3)

1− t4 = Pt(H∗Γ(pt)) ·Pt(X) .

This equality implies that X is Γ-equivariantly formal (see the comment followingCorollary 4.7.20). The unique section H∗(X) → H∗Γ(X) of ρ is multiplicativefor reasons of degree. Hence, the Leray-Hirsch theorem gives the H∗Γ(pt)-algebraisomorphisms of (c).

7.21. (XΓ, (X1)Γ, (X2)Γ, (X0)Γ) is a Mayer-Vietoris data and XΓ = (X1)Γ∪(X2)Γ.By Lemma 7.2.11, ((Xi)Γ, (X0)Γ) is a well cofibrant for i = 1, 2. The Mayer-Vietorisis then given by Lemma 3.1.41 and Proposition 3.1.54.

7.22. Let z ∈ X ∨ Y be the image of x and y. Using the Mayer-Vietoris sequenceof Exercise 7.21, one has a commutative diagram

0 // H∗Γ(z)∆ //

H∗Γ(z)⊕H∗Γ(z)+ //

H∗Γ(z)//

≈

0

0 // H∗Γ(X ∨ Y ) // H∗Γ(X)⊕H∗Γ(Y ) // H∗Γ(z) // 0

where the vertical arrows are induced by the constant maps. Since the right ver-tical map is an isomorphism, we check that the quotient of the second row bythe first one is an exact sequence (a particular case of the snake lemma: see [28,

Lemma 3.3]). Thus, the H∗Γ-algebra homomorphism (H∗ΓiX , H∗ΓiY ) : H

∗Γ(X ∨ Y )→

H∗Γ(X)⊕ H∗Γ(Y ) induced by the inclusions is an isomorphism.


8.1. No, since Sq1 vanishes in H∗(ΣT ) while not in H∗(ΣK).

8.2. Sq6(a5b7) = a10b8 + a9b9 + a6b12 + a5b13

(use that Sq(a5b7) = Sq(a)5Sq(b)7 = (a+ a2)5(b + b2)7).

8.3. Comes form Lemma 8.5.1.

8.5. Follows from the Adem relations. Remark: by [1, § 5], any finite set in Agenerates a finite subalgebra; hence, all elements of A are nilpotent. For estimatesof nilpotency heights, see [212, § 6.2].8.6. If a ∈ H1(RP∞) is the generator then, as A-module, H∗(RP∞) admitsar | r = 2n−1 as a minimal set of generators. This follows from (8.2.2). Note: ifb(X) is infinite, then H∗(X) is not finitely generated as an A-module [172].

8.7. Follows from Proposition 8.6.2 and Theorem 8.6.5.

8.8. We may assume that r > 1, otherwise there is nothing to prove. If U is anopen set of M over which p is trivial, then p−1(U) is homeomorphic to U × Sr−1,


which implies that dimM = n − r (using Theorem 3.3.4). For r > 1, the Gysinexact sequence implies that H∗(M) is a GrA-quotient of Z2[e], where e ∈ Hr(M)is the Euler class of p. Hence, n = kr and, using Exercise 8.7, r = 1, 2, 4, 8.

8.9. One has n = dimM = 2r by Corollary 5.2.5. By Poincare duality, Pt(M) =1 + tr + t2r with non-trivial cup-square. By Exercise 8.7, n = 2s with s = 1, 2, 3, 4.

8.10. The map f must have at least two critical points on each connected com-ponent of M . Since M−1 = pt, M is connected. We use the notation of § 7.6:Mx,y = f−1([x, y]), and Mx = f−1(x). There are G-equivariant diffeomorphismsφ− : (D(ν−1), S(ν−1))→ (M−1,0,M0) and φ+ : (D(ν1), S(ν1))→ (M0,1,M0), whereνx is the normal bundle to Mx in M . The pair (D(ν−1), S(ν−1)) is G-equivariantlydiffeomorphic to (Dn, Sn−1) endowed with the antipodal involution. We thus have

a smooth locally trivial bundle Sr−1p−→ Sn−1 → M1, where r is the codimen-

sion of M1 in M . Since n is odd, Exercise 8.8 implies that r = 1, so p is a2-fold covering. But p is G-invariant, so it can be identified with the quotientmap Sn−1 → M1. Therefore, M1 is diffeomorphic to RPn−1 and ν1 is the Hopfbundle η. Thus, M is diffeomorphic to Dn ∪φ−1

+ φ−D(η) which is homeomorphic

to RPn. Note that M may not be diffeomorphic to RPn if the diffeomorphismφ−1+ φ− : S

n−1 → S(η) = Sn−1 does not extend to a diffeomorphism of Dn.


9.1. Use that w1(ξ ⊕ ξ) = w1(ξ) + w1(ξ) = 0 and, if w1(ξ) = 0, that w2(ξ ⊕ ξ) =w2(ξ) + w2(ξ) = 0.

9.2. As p is a local diffeomorphism, one has T X = p∗TX . Hence wi(T X) =H∗p(wi(TX)). But H∗p is injective by Exercise 4.5. The result follows from Propo-sitions 9.4.4 and 9.4.7.

9.3. Since p is a local diffeomorphism, one has TM ≈ p∗TM . Hence,H∗p(w1(TM)) =

w1(TM) = 0. Thus, 0 6= w1(TM) ∈ ker(H∗p : H1(M) → H1(M)) which, by thetransfer exact sequence, is generated by w.

9.4. Let m = dimM . Proposition 4.2.1 and its proof provide an epimorphismα : H>0(M1)⊕H>0(M2)→ H>0(M). For 0 < i < m, α is induced by the inclusionsof Mi minus a disk into M . This implies that, for 0 < i < dimMi, wi(M) is theimage of (wi(M1), wi(M2). The same holds true for i = m, using that χ(M) ≡χ(M1) + χ(M2) mod 2 together with Corollary 5.4.16.

9.5. The Stiefel-Whitney classes w(TRP 4) (see Proposition 9.8.10) and w(TCP 2)(see Remark 9.8.12) are incompatible with such a decomposition, given Formula (9.4.3).

9.6. Given Formula (9.4.1), this follows from Exercise 8.3.

9.9. Using the tautological bundles, as for Proposition 6.1.11. Using that BU(n)is a conjugation space (see p. 365), this also follows from Corollary 10.2.11.

9.11. H∗(Gr(2;R5)) has 10 basis elements: 1 = (1, 2), w1 = (1, 3), w21 = (2, 3) +

(1, 4), w2 = (2, 3), w31 = (1, 5), w1w2 = (2, 4), w2

2 = (3, 4), w21w2 = (3, 4) + (2, 5)

(w41 = w2

2+w21w2), w

51 = w1w

22 = (3, 5) (w3

1w2 = 0), w22w

21 = (4, 5). The expressions

in the Schubert symbols were found using Propositions 9.5.27 and 9.5.32.


9.12. Since M is orientable, Sq1 : H5(M) → H6(M) vanishes (see Corol-lary 9.8.5). By Wu’s formula and the Adem relations, w6(TM) = Sq3(v3(M)) =Sq1Sq2(v3(M)) = 0. The result follows from Corollary 5.4.16. The same argumentworks when dimM = 10, using the Adem relation Sq5 = Sq1Sq4.

9.13. By Corollary 9.8.5, Sq1 : Hn−1(Q)→ Hn(Q) is surjective while Sq1 : Hn−1(P )→Hn(P ) vanishes. As Sq1Hn−1f = Hnf Sq1, this proves the statement under Hy-pothesis (a). The argument for Hypothesis (b) is similar, using Corollary 9.8.6.

9.14. The normal bundle νM of M = f−1(x) in Sn+k is trivial, since νM =f∗TxS

k. Thus, M admits an embedding into Rn+k+1 with trivial normal bundle.Therefore, w(TM) = 1 and thus w(TM) = 1. The results follows from Thom’stheorem 9.9.6.

9.15. By Thom’s theorem 9.9.6, since they do not have the same Stiefel-Whitneynumbers.

9.16. The manifold M is of dimension n. Let a ∈ H1(M) be the non-zero class.By Proposition 9.13, there exists a map f : M → RP∞ such that a = H∗f(ι).By cellular approximation (or by the proof of Proposition 9.13), the image of fis contained in RPn and thus H∗f : H∗(RPn) → H∗(M) is an isomorphism. Theresult then follows from Corollary 9.9.10.


10.1. As usually drawn in R3, the orientable surface Σg of genus g admits an axialsymmetry, with a scattered fixed point set, i.e. 2g + 2 = b(Σg) fixed points (thiscorresponds to a connected sum of g copies of S1×S1 with the involution τ(z1, z2) =(z1, z2)). A non-orientable surface does not admit any involution with scatteredfixed point set by Proposition 4.2.3 and Corollary 10.1.2 (or by Corollary 10.1.3when the involution is smooth).

10.2. The first assertion comes from Diagram (10.2.11) (for a more elementaryargument, see [87, Remark 3.1]). It implies that the H∗-frame respects the path-connected component of X , proving the “only if” part of last assertion. The “if”part is obvious.

10.3. Let (κ, σ) be the H∗-frame of a conjugation space X . The isomorphism κsatisfies κ(1) = 1 by Exercise 10.2 (or by Proposition 10.2.5). Therefore, κ descends

to a GrV-isomorphism κ : H2∗(X) → H∗(XG). Also, σ(1) = 1, so it descends to

σ : H∗(X) → H∗G(X) which is a section of ρ : H∗G(X) → H∗(X). The conjugation

equation holds true for (κ, σ) (call (κ, σ) a H∗-frame for X). Conversely, such a

H∗-frame determines an H∗-frame (it helps to assume X path-connected, whichwe can do by Exercise 10.2).

10.4. Use Exercises 7.22 and 10.3.

10.5. By Exercise 10.3, one can use the reduced cohomology. Let (κi, σi) be the

H∗-frame forXi. Obviously, (X1∧X2)G = XG

1 ∧XG2 andXG

2 is of finite cohomologytype (using κ2). Using Proposition 4.7.11 and Lemma 7.4.9, one has commutative


diagrams

H2∗(X1)⊗ H2∗(X2)×

≈//

κ1⊗κ2≈

H2∗(X1 ∧X2)

κ≈

H∗(XG1 )⊗ H∗(XG

2 )×

≈// H∗(XG

1 ∧XG2 )

and

H2∗(X1)⊗ H2∗(X2)σ1⊗σ2//

×≈

H2∗G (X1)⊗ H2∗

G (X2)

×G

H2∗(X1 ∧X2)σ // H2∗

G (X1 ∧X2)

which define (κ, σ). Checking the conjugation equation is straightforward.

10.6. Z = X ∧ S21 , where S

21 is the 2-sphere with the linear involution of Exam-

ple 7.1.14. This uses Exercise 10.5 .

10.7. In H∗(X), Sq16 should not vanish by Proposition 10.2.6. This contradictsTheorem 8.6.5.

10.8. The two sums in Proposition 10.3.17 have only one term and Pt(C) =1 + t(n−2)(d−1)−1. Hence, H∗(C) ≈ H∗(S(n−2)(d−1)−1). By the transfer exact se-quence, H∗(C) is GrA-isomorphic to H∗(RP (n−2)(d−1)−1). Theorem 10.3.11 givesthe GrA[u]-presentation H∗G(C) ≈ Z2[u]/(u

(n−2)(d−1)). Actually, arguments like in

Example 10.3.9 enable us to prove that C ≈ S(n−2)(d−1)−1 and C ≈ RP (n−2)(d−1)−1

(see e.g. [85, Example 2.6]).

10.9. By Proposition 10.3.17, Pt(C5d(ℓ)) = 1+2td−1+2t2(d−1)−1+t3(d−1)−1. Whend = 2, C52(ℓ) is an orientable surface by Corollary 10.3.2. As Pt(C52 (ℓ) = 1+4t+ t2,C52(ℓ) is an orientable surface of genus 2 and thus C52(ℓ) is a non-orientable surfaceof genus 3.

10.10. Let C = Cn2 (ℓ) and C = Cn2 (ℓ). Since ℓ is generic, H∗(C) ≈ H∗G(C) and Cis a closed manifold of dimension n − 3 (see Corollary 10.3.2). By Lemma 10.3.3,we may suppose that ℓ is dominated. If lops (ℓ) ≤ 1, C is empty (see Exam-ples 10.3.21 and 10.3.22). If lops (ℓ) = 2, then Cnd (ℓ) is orientable, being diffeomor-phic to (S1)n−3 (see Example 10.3.23). Suppose that lops (ℓ) ≥ 3. By (10.3.28), Cis connected and, since p : C → C is of degree 0, one has Hn−3(C) = u · Hn−4(C)by the transfer exact sequence. As H∗G(C) is GrA-generated in degree 1 (see The-orem 10.3.11), there exist x1, . . . , xn−4 ∈ H1(C) such that x1 · · ·xn−4u 6= 0 inHn−3G (C) ≈ Z2. Then Sq1(x1 · · ·xn−4) = (n − 4)x1 · · ·xn−4u by Theorem 10.3.11.

Note that the monomials of the form x1, . . . , xn−4 generate Hn−4(C). Using Corol-lary 9.8.5, we see that, for ℓ generic and dominated, Cn2 (ℓ) is non-orientable if andonly if lops (ℓ) ≥ 3 and n is odd.

10.11. By Proposition 10.3.32, Nn3 (ℓ) is a conjugation manifold with Nn

3 (ℓ)G =

Cn2 (ℓ). Using Proposition 10.2.12, the answer is analogous to that of Exercise 10.10:for ℓ generic and dominated, Nn

3 (ℓ) has no spin structure if and only if lops (ℓ) ≥ 3and n is odd.


10.12. Let C = Cn2 (ℓ). By (10.3.33), the simplicial complex Sh×n (ℓ) (definitionp. 382) satisfies dim Sh×n (ℓ) = n − 2 − lops (ℓ). Therefore, in the face exterioralgebra Λ1(Sh

×n (ℓ)) (see § 4.7.8), there exist at most n−2− lops (ℓ) elements whose

product is not 0. By (10.3.20), H∗G(C)/(u) ≈ Λ. As Hn−3(C) 6= 0, this implies that

ulops (ℓ)−2 6= 0.

10.14. Like in Example 10.5.13, Theorem 10.5.1 gives the presentationH∗Γ1(Γ/Γ2) ≈

Z2[u, a]/(u4 + a4), where a and u are of degree one. Setting b = a + u, we get

H∗Γ1(Γ/Γ2) ≈ Z2[u, b]/(b

4) which is isomorphic to H∗(RP 3)[u]. This is not surpris-

ing since SU(2)/±1 ≈ SO(3) ≈ RP 3 and ±1 acts trivially on SU(2)/±1.

CHAPTER 12

Errata and Comments

This chapter contains errata and comments for both the Springer edition (SE)and the present pre-published version (PPV).

Contents

Errata and Comments p. 431Typos p. 439Acknowledgments p. 440

Comment 12.0.1. Euler characteristic of homology sphere bundles. Let p : E →B be a bundle with fiber Σ, where Σ has the homology of the sphere Sr−1. Supposethat the total Betti number b(B) is finite. Then b(E) is finite and χ(E) is even. Ifr is even, then χ(E) = 0.

This follows from the Gysin exact sequence. Indeed, if P ∗, Q∗ and R∗ are finitedimensional positively graded vector spaces and if there is a long exact sequence(see 4.7.32).

· · · → Rs−1 → P s → Qs → Rs → P s+1 → · · ·then χ(Q) = χ(P )+χ(R); this may be proven by induction on the smallest integermsuch that P q = Qq = Rq = 0 for q > m. Applied to the Gysin sequence 4.7.32, thisgives χ(H∗(E)) = χ(H∗(B))+χ(H∗+1−r(B)) which proves the assertion. Remark:the finiteness assumption on b(B) is necessary, as seen by χ(S∞) = 1.

Comment 12.0.2. SE p. 172 line -5, PPV p. 145, line 3. The assertion “In

the presentation of a bundle ξ by a sequence Fi−→ E

p−→ B with B path-connected,a cohomology extension θ of the fiber exists if and only if H∗i is surjective” followsfrom the following fact: if a, b ∈ B, then any fiber inclusion F → Ea is homotopicto some fiber inclusion F → Eb. This is obvious if a, b ∈ U where U belongs toa cover U of B by open sets over which ξ is trivial. Otherwise, we choose a pathc : I → B joining a to b, which, using a Lebesgue number for c−1(U), may bedecomposed into subpaths within elements of U .

Erratum 12.0.3. Answer to Exercise 5.9: PPV p. 422 (statement of the prob-lem p. 198); SE p. 506 (statement of the problem p. 236). This answer “no” iswrong. For instance, there exists a degree one map from M = RP 1 × RP 2k toRP 2k+1. Indeed, let a ∈ H1(RP 1) and b ∈ H1(RP 2k) be the generators. Then,1 × b + a × 1 ∈ H1(M) admits a characteristic map f : M → RP∞. As f is ho-motopic to a cellular map (see e.g. [64, Theorem 2.4.11]), one may assume thatthe the range of f lies in RP 2k+1. Since a2 and b2k+1 vanish, one has, using theformula of Remark 4.6.4, that

(12.0.1) (1× b+ a× 1)2k+1 =(2k+1

1

)a× b2k = a× b2k = [M ]♯ ,

431

432 12. ERRATA AND COMMENTS

which shows that f is of degree one.In the same way, one can prove the following result.

Proposition 12.0.4. Let X and Y be connected closed manifolds, of dimensionm (respectively n). Then, there exists a degree one map f : X × Y → RPm+n ifand only if the following three conditions hold true.

(1) There exists a ∈ H1(X) with am 6= 0.(2) There exists b ∈ H1(Y ) with bn 6= 0.(3)

(m+nm

)≡ 1 mod 2.

Comment 12.0.5. Euler characteristic of G-complexes. Let X be a finiteG-complex, with G = id, τ. As indicated in Remark 10.1.5, one has

χ(X) ≡ χ(XG) mod 2 .

This is actually obvious, since the free cells come in pairs. In particular, XG 6= ∅when χ(X) is odd.

Erratum 12.0.6. Formula (7.3.1) (SE, p. 283) should be

hkG(X,Y ) =⊕

i+j=k

Z2ui ⊗Hj

G(X,Y )/∼

where ∼ is the equivalence relation generated by ui+1⊗a ∼ ui⊗ua. Example 7.3.2shows that this equivalence relation may be quite coarse.

Formula (7.3.1) was implicitly used in the proof of Theorem 7.3.1 to establishedthe exactness of the sequence

(12.0.2) · · · → hk−1G (X)→ hkG(Z,X)→ hkG(Z)→ hkG(X)→ hk+1G (Z,X)→ · · · ,

so a different argument is needed to justify this exactness. We use that the Laurentpolynomial ring Z2[u, u

−1] is isomorphic to the localized ring U−1Z2[u], where Uis the multiplicative part U = ur | r ≥ 0. Recall that an element of U−1Z2[u] isthen represented by a fraction P/ur with P ∈ Z2[u], with P/ur being equivalentto Q/us if and only if there exists t ≥ 0 such that us+tP = ur+tQ [Ei, § 2.1]. Onechecks that the correspondence

uk 7→uk/1 k ≥ 0

1/uk k ≤ 0

extends to a ring homomorphism α : Z2[u, u−1]→ U−1Z2[u] and that P/uk → u−kP

defines a set-map inverse to α.To get Exact sequence (12.0.2), we start with the doubly long exact sequence

on Z2[u]-modules

→ H∗G(X)δ // H∗G(Z,X) // H∗G(Z) // H∗G(X)

δ // H∗G(Z,X)→

(infinite in both directions), where δ raises the degree by 1. As Z2[u, u−1] ≈

U−1Z2[u] is a flat Z2[u]-module [Ei, Proposition 2.5], tensoring the above sequencewith Z2[u, u

−1] gives the exact sequence

→ h∗G(X)δ // h∗G(Z,X) // h∗G(Z) // h∗G(X)

δ // h∗G(Z,X)→ .

12. ERRATA AND COMMENTS 433

As the maps are graded, Sequence (12.0.2) is obtained by restricting the last se-quence somewhere to hkG(Z,X).

Reference:

[Ei] D. Eisenbud. Commutative Algebra: With a View Toward Algebraic Geometry.Springer 1995.

Remark 12.0.7. A similar argument would give the exact sequence for h∗(X) =H∗Γ(X)⊗H∗Γ(pt)H∗Γ(Y ), used in the proof of the equivariant Kunneth theorem The-

orem 7.4.3 (SE, p. 292). Indeed, Y is assumed here to be of finite cohomology typeand equivariantly formal. Therefore, H∗Γ(Y ) is a finitely generated free H∗Γ(pt)-module, whence flat over H∗Γ(pt). Note that the decomposition

hk(X) ≈⊕

i+j=k

HiΓ(X)⊗Hj

Γ(Y )

is here valid, so the argument given in the proof of Theorem 7.4.3 is correct.

Comment 12.0.8. Formula (7.4.2) SE p. 289, PPV p. 242. Note that the map

P is Γ12-equivariant, so it descends to P . That P is a homotopy equivalence followsfrom the commutativity of the diagram

EΓ12 × (X × Y )P //

projX×Y ''PPPP

PPPP

PP(EΓ1 ×X)× (EΓ2 × Y )

projX×projYuu

X × Y

where the projections are homotopy equivalence, since their fibers are contractible.

Comment 12.0.9. The formulation of Lemma 7.6.4 (SE, p. 312) may be im-proved (see PPV, p. 261).

Comment 12.0.10. Remark 7.6.7, SE p. 313. The hypothesis of Proposi-tion 7.3.7 (finiteness of the total Betti number) is fulfilled forMa,b since f is proper.Note that we do not need in this remark that f is ±1-invariant. The same appliesto the context of Theorem 7.6.11 (see PPV p. 265, SE p. 317).

Comment 12.0.11. SE p. 321 Exercise 7.9, VVP p. 269. Amongst possiblegeneralizations of this statement, see [M. Franz: Symmetric products of equivari-antly formal spaces, arXiv:1604.08273].

Erratum 12.0.12. SE p. 347, Proposition 8.5.2. The Steenrod algebra Acannot be a polynomial algebra over the admissible monomials. Already, the def-inition SqI = Sqi1 · · · Sqik , for I = (i1, . . . , ik) admissible, is a polynomial relationbetween admissible monomials. Many other relations occurs (see, e.g. (8.5.1)). Thestatement of Proposition 8.5.2 should be

The family of admissible monomials form a Z2-basis for AThis is anyway what the proof of Proposition 8.5.2 (SE, p. 348, PPV p. 291)intended to establish. But this proof itself was incomplete: the deduction ofEquations (8.5.5) is not justified. For a new proof of Proposition 8.5.2, see Com-ment 12.0.16.


Comment 12.0.13. SE p. 348 line 2, PPV p. 290. The “equality” Lp =Lp+q simply means that one can replace Lp by Lp+q, of course not the opposite.A perhaps cleaner way to present these matters would be to define Lp as the

subalgebra generated by σ1, . . . , σp. One would thus have Lp ⊂ Lp+q, Sqi(Lp) ⊂

Lp+i, etc.

Comment 12.0.14. A computation of Sqi(σp). We recall that σ0 = 1 andσk = 0 if k > n.

Proposition 12.0.15. (a) For all integers i, p and j with 0 ≤ j ≤ i, there

exists µjip ∈ Z2 such that the formula

Sqi(σp) =

i∑

j=0

µjipσi−jσp+j

holds true in Hp+i((RP 1)n) for all n ∈ N.

(b) For i ≤ p, the coefficients µjip satisfy µ0ip = 1 and the recursive formula

µjip =

j−1∑

k=0

(p− i+ 2j

j − k

)µkip .

(c) For j ≤ i ≤ p and 2k−1 ≤ j < 2k, the coefficients µjip are periodic in p of

period exactly 2k.

Of course, µjip = 0 if i > p. Proposition 12.0.14 gives, for example, the formula

(12.0.3) Sq1(σp) = σ1σp + (p+ 1)σp+1 .

Note that, if n = p, then Sq1(σp) = σ1σp since then σp+1 = 0. The formulaSqp(σp) = σ2

p, valid for all n, implies that µjpp = 0 when j ≥ 1. This may also beobtained recursively by Part (b) of Proposition 12.0.14 since, for all j, the coefficient

of µ0pp is (2jj ) = 0 (using Lemma 6.2.6). Here are a few other computations, using

Part (b) of Proposition 12.0.14.

Coefficients µj2,p for Sq2(σp)

p 2 3 4 5 · · ·µ02,p 1 1 1 1 · · · period 1

µ12,p 0 1 0 1 · · · period 2

µ22,p 0 1 1 0 · · · period 4


p 3 4 5 6 · · ·µ03,p 1 1 1 1 · · · period 1

µ13,p 0 1 0 1 · · · period 2

µ23,p 0 1 1 0 · · · period 4

µ33,p 0 1 0 0 · · · period 4


p 4 5 6 7 8 9 10 11 · · ·µ04,p 1 1 1 1 1 1 1 1 · · · period 1

µ14,p 0 1 0 1 0 1 0 1 · · · period 2

µ24,p 0 1 1 0 0 1 1 0 · · · period 4

µ34,p 0 1 0 0 0 1 0 0 · · · period 4

µ44,p 0 1 1 1 1 0 0 0 · · · period 8


Proof of Proposition 12.0.14. It is enough to prove the proposition for nlarge: it will then hold true for m < n by naturality for the inclusion (RP 1)m ⊂(RP 1)n. We thus fix n large (≥ p+ i) and write [n] = 1, 2, . . . , n. More generally,if p, q are integers with p ≤ q, we set [p, q] = p, p + 1, . . . , q (thus, [n] = [1, n]).For J ⊂ [n], we set xJ =

∏j∈J xj ∈ Z2[x1, . . . , xn].

We start with some combinatorial preliminaries. For integers u, v with u ≤ v,we consider the set

Mu,v = (U, V ) | U, V ⊂ [n], |U | = u and |V | = v .There is a map m : Mu,v → Z2[x1, . . . , xn] given by m(U, V ) = xUxV . For K ⊂Mu,v, we write Σ(K) for

∑(U,V )∈Km(U, V ). As u ≤ v, Mu,v decomposes as

Mu,v =⋃u

w=0Mwu,v ,

where

Mwu,v = (U, V ) ∈Mu,v | |U − V | = w .

There is a map βwu,v : Mwu,v →M0

u−w,v+w given by

βwu,v(U, V ) = (U ∩ V, U ∪ V ) = (U ∩ V, V ∪(U − V ))

which commutes with the map m. It also commutes with the canonical actions ofthe symmetric group; thus, the number νwu,v = |(βwu,v)−1(Z)| does not depend on

Z ∈M0u−w,v+w. Choosing Z = ([u − w], [v + w]), we see that

(βwu,v)

−1([u−w], [v+w]) =

([u−w]∪W, [v+w]−W ) |W ⊂ [u−w+1, v+w] with |W | = w

.

Therefore

(12.0.4) νwu,v =

(v − u+ 2w

w

).

We are ready to prove Proposition 12.0.14. By the Cartan Formula, one hasSqi(σp) = Σ(M0

i,p), while σiσp = Σ(Mi,p). Therefore,

P1 := Sqi(σp) + σiσp = Σ(M≥1i,p ) .

We can get rid of the terms indexed by M1i,p using the map β1

i,p : M1i,p →M0

i−1,p+1,

for which each preimage contains ν1i,p elements. Define µ0ip = 1 and

µ1ip = µ0

ip ν1i,p = µ0

ip

(p− i+ 2

1

)∈ Z2 .

NowP2 := P1 + µ1

ip σi−1σp+1

= µ0ipΣ(M

≥2i,p ) + µ1

ip Σ(M≥1i−1,p+1) since µ0

ip = 1.

=∑1

k=0 µkipΣ(M

≥2−ki−k,p+k)

Again, we can get rid of the summands indexed by M2i,p and those indexed by

M1i−1,p+1 using the maps β2

i,p and β1i−1,p+1, both with range M0

i−2,p+2. We get

P3 := P2 + µ2ip σi−2σp+2 =

2∑

k=0

µkipΣ(M≥3−ki−k,p+k) ,


provided

µ2ip = µ0

ipν2i,p + µ1

ipν1i−1,p+1 =

1∑

k=0

µkipν2−ki−k,p+k .

This process may be pursued till we get the polynomial

(12.0.5) Pi =

i−1∑

k=0

µkipΣ(Mi−ki−k,p+k) = µiipσp+i

with

µiip =

i−1∑

k=0

µkipνi−ki−k,p+k .

Indeed, M≥i−ki−k,p+k =M i−ki−k,p+k; the last equality of (12.0.5) uses the maps

βi−ki−k,p+k : Mi−ki−k,p+k →M0

0,p+i and that Σ(M00,p+i) = σp+i.

Using (12.0.4), this proves Parts (a) and (b) of Proposition 12.0.14. Part (c)follows from Part (b) since, by Lemma 6.2.6, the binomial coefficient (rs) mod 2 isperiodic in r; the period divides 2k if s < 2k and, if 2k ≤ s < 2k, the period isexactly 2k (the period of the last k bits in the dyadic expansion of r).

Comment 12.0.16. New proof of Proposition 8.5.2 We first replace Lemma 8.5.3by Lemma 12.0.17 below. Define

N = (n1, n2, . . . ) ∈ N∞ | ni ≥ ni+1 and nk = 0 for some k .Any finite sequence (n1, n2, . . . , nr) ∈ Nr is considered as an element ofN by addingzeros to its right. The definition of admissibility (xi ≥ 2xi+1) makes sense for theelements of N . The set N is totally ordered by the lexicographic order ≤.

For any J = (j1, j2, . . . ) ∈ N , the expressions SqJ = Sqj1Sqj2 · · · and σJ =σj1σj2 · · · make sense since Sq0 = id and σ0 = 1. Any polynomial L in the σi’s maybe written in the form L =

∑J∈J σJ for a unique J ⊂ N . We write σI ≤ σJ if

I ≤ J . We write P ≤ σJ if σI ≤ σJ for all monomials σI occurring in the symmetricpolynomial P , and maxP = σJ if in addition σJ occurs in P . An easy argument(or Proposition 12.0.15) shows that

(12.0.6) Sqi(σp) ≤ σ2p ,the inequality being strict if p > 0.

Lemma 12.0.17. Let I = (i1, ..., ir) be an admissible sequence. Then, for w =wn with n ≥ i1, one has

SqI(w) = wP ,

where P is a symmetric polynomial satisfying maxP = σI .

For example, for I = (7, 2, 1) and n = 7, Proposition 12.0.15 and (12.0.3) implythat

Sq(7,2,1)(w) = w(

σ(7,2,1) + σ(7,1,1,1) + σ(6,3,1) + σ(6,2,1,1) + σ(6,1,1,1,1) + L)

with L < σ6.

Proof. The proof is by induction on r. For r = 0, one has P = σ∅ = 1, sothe claim holds. For r = 1, the statement follows from (8.5.3). Suppose that r ≥ 2


and set I2 = (i2, i3, . . . ). Using (8.5.3), the induction hypothesis and the Cartanformula, one has

SqI(w) = Sqi1SqI2(w)

= Sqi1(wP2) with maxP2 = σI2

= Sqi1(w)P2 +∑i1j=1 Sq

i1−j(w)Sqj(P2)

= w(σi1P2 +

∑i1j=1 σi1−jSq

j(P2)).

Since j ≥ 1, one has σi1−j < σI . Also, Sqj(P2) < σ2i2 ≤ σi1 ≤ σI , us-ing (12.0.6) and that I is admissible. The claim follows since max(σi1P2) = σi.

New proof of Proposition 8.5.2. The Adem relations imply that any mono-mial SqI is a sum of admissible ones. It remains to see that the admissible mono-mials are linearly independent. Suppose that there is a finite subset I of admissibleelements of N such that

(12.0.7)∑

I∈I

SqI = 0

Suppose that I is non-empty. Let J be the maximum of I. Applying (12.0.7) to wfor n large enough and using Lemma 12.0.17, one gets

0 =∑

I∈I

SqI(w) = wP (I) ,

where maxP (I) = σJ , which is impossible, since the above equation implies thatP (I) = 0.[Compare L. Smith, An algebraic introduction to the Steenrod algebra, Geometryand Topology Monographs 11 (2007) 327–348, Theorem 3.1.]

Comment 12.0.18. Proof of Corollary 8.5.11 (SE p. 350). This corollary isactually a direct consequence of Proposition 8.5.4: if there were another relation,say of degree n, then the map given in Proposition 8.5.4 would not be injective forthis n.

Erratum 12.0.19. SE p. 351, PPV p. 293. The central display equation forPt(Km) should be

Pt(Km) =

∞∏

r=0

(1

1− tm+r

)a(r).

(Corrected in PPV p. 293.)

Erratum 12.0.20. SE p. 351, PPV p. 293. The last equation on the pagecontains several typos. It should be

a(r) = ♯ (h1, . . . , hm−1) ∈ Nm−1 | h1 ≥ · · · ≥ hm−1 and 2h1 + · · ·+ 2hm−1 + 1 = m+ r .

(Corrected in PPV p. 293.)

Erratum 12.0.21. SE p. 441, paragraph after Diagram (10.2.11). Theo-rem 10.2.10 shoulod not be quoted here. See new text on PPV p. 368.

Erratum 12.0.22. The definition of a long subsets (SE, p. 446, PPV p. 372)should be read as follows. A subset J of 1, . . . , n is called ℓ-short (or just short)if∑i∈J ℓi <

∑i/∈J ℓi and long if the reverse inequality holds. If ℓ is generic, subsets

are either short or long.


Comment 12.0.23. SE p. 446, proof of Corollary 10.3.2. The first two para-graphs of the proof may be simplified. For the first one, since ℓ is generic, theassertion on BCnd (ℓ) follows directly from Lemma 10.3.1.

For the 2nd paragraph, by O(d)-equivariance of Fℓ and since ℓ is generic, thepoint (−ℓn, 0) is a regular value of Fℓ. Since A = R × 0 and B = 0 × Rd−1 aretransverse submanifolds of Rd, we deduce, using the implicit function theorem, thatCnd (ℓ) = F−1ℓ ((−ℓn, 0)) is a closed submanifold of codimension d − 1 of F−1ℓ (B) =BCnd (ℓ), with trivial normal bundle. [See also typo about p. 446.]

Comment 12.0.24. Number of chambers relevant for the classification of poly-gon or chain spaces with n edges (SE p. 448, PPV, p. 374). These numbers co-incide with the numbers of equivalence classes of self dual of threshold functionsof n variables or of majority (i.e., decisive and weighted) games with n players.See [J-Cl. Hausmann: Counting polygon spaces, Boolean functions and majoritygames, arXiv:1501.07553]. In this note, we propose to call the hyperplane HJ (seeSE, p. 446, PPV, p. 372) the J-tie hyperplane and H the tie-arrangement.

http://arxiv.org/abs/1501.07553


Typos

The following typos are already corrected in PPV. More small corrections(spelling, English usage, obvious typos) had been performed in PPV, without noticebelow when the comprehension is not really impaired.

• SE p. 5, 2nd paragraph. “12th century” should be “20th century”.

• SE p. 77, Proof of Proposition 3.1.34, line 4. “Claim let...” should be“Claim: let...”.

• SE p. 80, line 3 below the second display. “retraction form” should be“retraction from”.

• SE p. 101, second display morphism in the middle of the page and Propo-sition 3.4.11. The inverse limit should be taken over (r, s) ∈ M and not overK ∈ K.• SE p. 111, line -7. “Four” should be “For”.

• SE p. 117, line 2 below Diagram (3.7.7). “grv-morphism” should be “GrV-morphism”.

• SE p. 118, line 4 of Remark 3.7.5 (b). XCW should be XCW. Also, on line5 of Remark 3.7.5 (c), “a spaces” should be “a space” or, better, “any space”.

• SE pp. 121–22, proof of Proposition 3.8.3.

- line 2 of (ii): “S1 ≈ S1” should be “S1 ≈ K1”.- line -2: “RP 1” should be “RP∞” in the definition of φ.

• SE p. 122, proof of Proposition 3.8.5, line 11. “ro∞” should be “RP∞”.

• SE p. 126, Exercise 3.18. “a set B” should be “a set R”.

• SE p. 230, Point (6). The sentence in brackets actually applies to Point (5).A reference for Point (6) is Example 7.1.2.

• SE p. 240, second display on page. The numerator should be 1− t4(n+1).

• SE p. 244, line 2. “total space” should be “the total space”.

• SE p. 246, proof of Corollary 6.2.3. The last sentence should be: “Then,P (z) ∩W is the desired bisecting hyperplane, where P (z) is the orthogonal com-plement of z”.

• SE p. 260, line 4 after Equation (7.1.1). “TopG” and “where” should beexchanged.

• SE p. 264, first diagram on page. The bottom arrow should also be labeled iz.

• SE p. 284, line 3 of the proof of Theorem 7.3.1. Remove the adjective“finite”.

• SE p. 304, line -1 of the proof of Lemma 7.5.11. “fundamental group to...”should be “fundamental group corresponding to...”.

• SE p. 311, Lemma 7.6.2, “powers of f” should be “powers of t”.

• SE p. 311, proof of Lemma 7.6.3, line below the last display. “H∗(Wb,Wa)→H∗(Wc,Wb)” should be “H∗(Wc,Wa)→ H∗(Wb,Wa)”.

• SE p. 313, line -6, The sentence between brackets should be: “(this ispossible since the set of critical values of a proper Morse-Bott function is discrete)”.

• SE p. 327, Lemma 8.1.4. “be” should be “is”.


• SE p. 332, Proposition 8.2.3. “pointed for (b)” should be “pointed for (2)”.

• SE p. 340, Lemma 8.4.2. On the last line of (2), “βG

K” should be “βKG”. The

label of the left vertical arrow in (3) should be βKG• SE p. 348, proof of Proposition 8.5.2, lines 6–10. There are a couple of

typos there, e.g. “maxI” should be “max1 I”, etc (see p. 291). But, as this proofis not complete anyway, see Comment 12.0.16.

• SE p. 349, paragraph after Theorem 8.5.5. Several typos are there, e.g. “n”should be “m” and (ι) is missing twice in the displayed equation (see p. 292 inPPV).

• SE p. 349, line -2. “one generator degree...” should be “one generator ineach degree...”.

• SE p. 350, Part (3) of Example 8.5.6. “F(2r + 1, 2r, . . . , 2, 1)” should be“F(2r + 1, 2r−1, . . . , 2, 1)”.

• SE p. 350, Corollary 8.5.9. “fy” and “fw” should both be “f”.

• SE p. 376, 2 lines below (9.5.2). The summation in the characteristicpolynomial should be replaced by a product. This also happens on pp. 381 and 405.

• SE p. 410, first line. “As for Theorems 9.6.2 and 9.7.11...” should be “Asfor Theorem 9.6.2, Theorem 9.7.11...”

• SE p. 413, 2 lines below Equation (9.8.4). The sentence should be “LetA = a1, a2, . . . and B = b1, b2, . . . be additive bases of H∗(Q) which are...”

• SE p. 425, Example 9.8.25 1. The sentence “...using Boy’s surface” maybe improved in “...using a connected sum of Boy’s surface, the latter being animmersion of RP 3.”

• SE p. 434, Remark 10.1.5. XG should be XG. See also Comment 12.0.5.

• SE p. 437, Equation (10.2.2). “ltk” should be “ℓtk”.

• SE p. 443, Theorem 10.2.14, Part (b). “G-diffeomorphism” should be“diffeomorphism”.

• SE p. 444, line -8. “defines” should be “defined”. Also, in “(inspired bytalks of W. Thurston on linkages [195])” (first paragraph of § 10.3), the closingbracket is missing.

• SE p. 446, proof of Corollary 10.3.2, line 1. This is −ℓn which is a a regularvalue of fℓ (see also Comment 12.0.23).

• SE p. 447, proof of Lemma 10.3.3, line 4-5. J denotes the complement of Jin the set 1, . . . , n.• SE p. 466, Remark 10.3.28, line 2. “BC42(1, 1, 1, 2) and BC42(1, 2, 2, 2)” should

be “C42(1, 1, 1, 2) and C42(1, 2, 2, 2)”.• SE p. 470, proof of Proposition 10.4.1, line 1. “Theorem 9.5.15” should be

“Theorem 9.5.14”.

• SE p. 495, line -7. “νk” should be “να”.

Acknowledgments: I am highly grateful to Matthias Franz and Peter Landweberfor their careful reading, providing many corrections and valuable suggestions.

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[2] J. F. Adams. On the non-existence of elements of Hopf invariant one. Ann. of Math. (2),72:20–104, 1960.

[3] A. Adem and R. J. Milgram. Cohomology of finite groups, volume 309 of Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].Springer-Verlag, Berlin, 1994.

[4] J. Adem. The iteration of the Steenrod squares in algebraic topology. Proc. Nat. Acad. Sci.U. S. A., 38:720–726, 1952.

[5] J. Adem. The relations on Steenrod powers of cohomology classes. Algebraic geometry andtopology. In A symposium in honor of S. Lefschetz, pages 191–238. Princeton UniversityPress, Princeton, N. J., 1957.

[6] J. Adem, S. Gitler, and I. M. James. On axial maps of a certain type. Bol. Soc. Mat.Mexicana (2), 17:59–62, 1972.

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Index

actionwith scattered fixed point set, 361

code associated to an, 362semi-free, 363

acyclic carrier, 44

Adem relations, 288admissible sequence, 290

affine order, 12Alexander duality, 185

almost closed manifold, 409

annihilator, 188, 222Arf invariant, 404

axial map, 207axioms of a cohomology theory, 104

barycenter, 14

barycentric subdivision, 14Betti number, 23

total, 238

binomial coefficients mod 2, 208Bockstein homomorphism, 292

Borel construction, 218, 230Borsuk-Ulam theorem, 206

boundary of a simplex, 10

boundary operatorcellular, 86

in a chain complex, 16ordered, 46

simplicial, 15

singular, 52bouquet, 71

cohomology ring of, 114bundle, 144

induced, 153, 297of finite type, 147

pair, 148

principal, 229, 249, 300trivialization, 300

universal, 229, 250vector, 156–158, 297–300

bundle characteristic map, 159

bundle gluing map, 159

C1-triangulation, 170

cap productfunctoriality, 128, 129in relative simplicial cohomology, 129simplicial, 127singular, 131

categoryCW, 95GrA, 111GrA[u], 220GrV, 29RCW, 97Simp, 11Top, 11Top2, 59TopG, 218TopΓ, 230Lusternik-Schnirelmann, 124

cellattaching a, 81characteristic map of a, 82convex, 12homology, 91open, 81

cellular(co)chain, 86chain, 86cochain, 86cohomology, 86homology, 86map, 90

chaincellular, 86complex, 16

morphism of, 18homotopy, 32, 62map, 18ordered, 45simplicial, 14singular, 51

chain space, 370

big, 370chamber of a length vector, 372characteristic class

of a 2-fold covering, 120

449

450 INDEX

of a vector bundle, 297, 312

characteristic mapglobal, 84

of a 2-fold covering, 119

characteristic map of a cell, 82Chern classes

mod 2, 340uniqueness, 342

action of the Steenrod algebra on, 342

integral, 340circuit, 22

cobordant manifolds, 357cobordism, 359

coboundary operator

cellular, 86in a chain complex, 17

ordered, 46simplicial, 15

cochain

cellular, 86complex, 17

morphism of, 18map, 18

ordered, 45

simplicial, 14singular, 51

unit, 21, 53cocycles

simplicial, 16

singular, 52code, 362

cofibrant pair, 67cohomology

cellular, 86

connecting homomorphism, 33simplicial, 38

singular, 57extension of the fiber, 144

of a cochain complex, 18

operation, 271ordered, 46

positive parts, 114reduced

singular, 54

relativesimplicial, 38

singular, 57

sequencesimplicial, 38


theory, 104

natural transformation, 104colouring definitions

of cellular (co)chains, 86combinatorial manifold, 170

complex

chain, 16

cochain, 17

convex-cell, 12full, 24

simplicial, 9

componentof a simplicial complex, 11

coneon a simplicial complex, 23

on a topological space, 63

conjugationcell, 365

complex, 365equation, 364

universal, 367

manifold, 368space, 364

conjugation space, 341connected sum, 115

cohomology ring of, 115

connecting1-cochain, 53, 121

homomorphismcohomology, 33

homology, 36

connecting homomorphismsimplicial, 40

contiguous simplicial maps, 31contractible space, 63

convex cell, 12

convex-cell complex, 122-fold covering

characteristic class of, 120deck involution, 120

transfer exact sequence, 123

transfer homomorphism, 123cross product

associativity, 140equivariant, 244

strong, 244

functoriality, 133in cohomology, 132

in homology, 134in relative cohomology, 133

of maps, 132

reduced, 142equivariant, 248

cross-square map, 277

equivariant, 283cup product

commutativity, 110commutativity of, 112

functoriality, 111, 113

in relative simplicial cohomology, 111in relative singular cohomology, 113, 131


cup-square map, 117, 209, 294

CW

INDEX 451

complex, 81Cartesian product, 83equivariant, 218, 240finite, 89homology-cell complex, 91regular, 87weak topology, 81

pair, 82space, 81structure, 81

on Sn, 83on projective spaces, 83perfect, 90

cyclessimplicial, 15singular, 52

deck involutionof a 2-fold covering, 120

degreelocal, 77

degree of a mapbetween manifolds, 172between pseudomanifold, 30between spheres, 76

democratic invariant, 404dimension

of a simplex, 9of a simplicial complex, 9topological invariance, 80

disjoint union axiom, 104duality

Alexander, 185Kronecker, 16–21

ordered, 46singular, 52

Lefschetz, 178Poincare, 171

Eilenberg-MacLane space, 100cohomology of, 291

equivariantcross product, 244CW-complex, 218, 240Stiefel-Whitney classes, 392vector bundle, 252

equivariant cohomologyfor a pair with involution, 220in general, 231

equivariantly formal, 222, 233Euclidean bundle, 156Euler

characteristicof a finite CW-complex, 89of a finite simplicial complex, 25

of a manifold, 171, 183, 193class, 154, 157

equivariant, 253exact sequence

cohomology, 33homology, 36of cochain complexes, 33simplicial (co)homology, 38, 40

of a triple, 42singular (co)homology, 58

of a triple, 60exactness axiom, 104excess, 291excision

axiom, 104property, 66simplicial, 50

excisive couple, 113extension of a cochain


faceexterior algebra, 163front, back, 112in a regular CW-complex, 87inclusion, 52of a simplex, 9space, 162

fiber inclusion, 144finite (co)homology type, 23, 61, 74finite type (graded vector space), 23flag complex, 163flag manifold, 314

complete, 315, 338complex, 337

tautological bundle over..., 339tautological bundle over..., 318

forgetful homomorphism (in equiv.cohomology), 221, 233

framedcobordism, 405manifold, 405

framed bundle (of a K-vector bundle), 252framed bundle (of a vector bundle), 301framing (stable), 405full complex, 24fundamental class

of a polyhedral homology manifold, 171of a pseudomanifold, 22of a relative homology manifold, 179

G-contractible, 217generic length vector, 372genus

of a nonorientable surface, 28geometric

open simplex, 10simplex, 10

geometric realizationof a simplicial complex, 10

metric topology, 10weak topology, 10

452 INDEX

G-homotopy, 217

Giambelli formula, 333

GKM-conditions, 402complex case, 403

global characteristic map, 84

G-map, 217

good pair, 66GrA, 111

graded algebras

category of, 111

GrA[u], 220graph

simplicial, 9

Grassmannian, 315

complex, 338infinite, 319, 340

tautological bundle over, 319, 340

GrV, 29

Gubeladze’s theorem, 164Gysin

exact sequence, 154, 156

homomorphism, 189

ham sandwich theorem, 206

Hawaiian earring, 106homogeneous space, 314

homology

cell, 91

complex, 91cellular, 86

connecting homomorphism, 36

of a chain complex, 18

ordered, 46reduced

singular, 54

relative


sequence

simplicial, 40


sphere, 185

homology manifold, 170

relative, 178

homotopic maps, 62homotopy, 62

axiom, 104

chain, 32, 62

equivalence, 63equivalent pairs or spaces, 63

property (in singular (co)homology), 62

quotient, 218, 230

relative, 298type, 63

Hopf

bundle, 202–205

vector, 205

invariant, 209, 295

map, 203

intersection form, 183

invariance of dimension, 80inverse problem, 387

join

simplicial, 10topological, 143

Jordan Theorem, 81

Kervaire

invariant, 409, 410manifold, 413

sphere, 413Kirwan surjectivity theorems, 265–268

Klein bottlecohomology algebra of, 117

triangulation of, 27Kronecker

duality, 16–21ordered, 46

singular, 52pair, 17

morphism of, 19

pairingextended, 53

on (co)chains, 15on (co)homology, 18

on cellular (co)chains, 86on ordered (co)chains, 45

on singular (co)chains, 52Kunneth theorem, 135

equivariant, 244reduced, 142

relative, 138, 150

Lefschetz duality, 178

length vector, 370chamber of a, 372

dominated, 384generic, 372

lopsidedness of a, 385normal, 389

Leray-Hirsch Theorem, 145, 1492nd version, 147

link, 11linking number, 194localization

principle, 187theorem, 237, 240

long subset, 372lopsidedness, 385

Lusternik-Schnirelmann category, 124

m-involution, 361majority invariant, 404

manifold

INDEX 453

combinatorial, 170PL, 170polyhedral homology, 170relative polyhedral homology, 178topological, 80, 115

mapcellular, 90of simplicial triple, 41of topological pairs, 59piecewise affine, 10

mapping cylinderbundle pair, 152double, 215neighbourhood, 67

mapping torus, 160exact sequence, 160

maximal simplex, 9Mayer-Vietoris

connecting homomorphisms


data, 73sequence


meridian sphere, 195Morse function, 259Morse-Bott

function, 259critical manifold, 259perfect, 260self-indexed, 316

inequalities, 259lacunary principle, 260

polynomial, 259

natural transformation, 95NDR-pair, 68negative normal bundle, 259nilpotency class, 125non-singular map, 206, 295null-homotopic, 102

octonionic projective planecohomology algebra of OP 2, 204

odd map, 205open simplex, 10order

affine, 12simplicial, 12

orderedchain, 45cochain, 45cohomology, 46homology, 46

simplex, 45orientation

of a vector bundle, 301of a vector space, 301

transport along a path, 306

pair

CW, 82

good, 66

Kronecker, 17simplicial, 37

topological, 57

map of, 59

perfectCW-structure, 90

Morse-Bott function, 260

piecewise affine map, 10

Pieri formula, 333PL-manifold, 170

Poincare dual

functoriality, 188

of a homology class, 183

of a submanifold, 186Poincare duality, 171

Poincare series/polynomial, 23, 61, 74

pointed space, 71

well, 71, 141polygon space, 371

abelian, 391

free, 371

spatial, 390polyhedral homology

manifold, 170

relative, 178

Pontryagin-Thom construction, 358prefix, 328

product

cap, see also cap product

in relative simplicial cohomology, 129

cross, 132cup, see also cup product

in relative singular cohomology, 131

of CW-complexes, 83

slant, 344tensor, 132

projective space

RP∞ as Eilenberg-space, 102

cohomology algebra of CPn, 203cohomology algebra of HPn, 203

cohomology algebra of RPn, 124, 201

complex, 202

octonionic (projective plane), 204quaternionic, 203

real, 83

simplicial cohomology algebra of RP 2,116

standard CW-structure on RPn, 83

tautological line bundle, 306triangulation of RP 2, 26

proper map, 259

pseudomanifold, 22

fundamental class of, 22

454 INDEX

fundamental cycle of, 22

quadratic form, 404

reducedequivariant cohomology, 223, 233

singular (co)homology, 54suspension, 142

reduced symmetric square, 350regular

CW-complex, 87

value (topological), 76relative

cohomologyordered, 48


homologysimplicial, 39singular, 57

singular cycle, 58representable functor, 100retraction, 66, 79

by deformation, 66

Riemannian metric, 156robot arm map, 371

scattered fixed point set, 361Schubert

cell, 326variety, 326

Schubert calculus, 328–333complex, 342Giambelli formula, 333Pieri formula, 333

Segre class, 342

self-intersection, 407semi-free action, 363sequence, see also exact sequenceshort subset, 49, 372

Shn(ℓ), 372Sh×

n (ℓ), 382Simp, 11simplex, 9

boundary of a, 10

geometric, 10geometric open, 10maximal, 9ordered, 45

singular, 51small, 64standard, 51

simplicial(co)homology

exact sequence, 38, 40, 42category, 11chain, 14cochain, 14

cocycles, 16

cohomology, 16

cohomology connecting homomorphism,38

cycles, 15

graph, 9

homology, 16homology connecting homomorphism, 40

map, 11of simplicial pairs, 40

order, 12

for the barycentric subdivision, 14pair, 37

pairs

map of, 40relative

cohomology, 38homology, 39

suspension, 50

triad, 42triple, 41

map of, 41simplicial complex, 9

component of, 11

connected, 12Euclidean realization, 10

finite, 9

geometric realization, 10locally finite, 10

simplicial excision, 50singular

chain, 51

cochain, 51cocycles, 52

cohomology, 52cycles, 52

homology, 52

reduced (co)homology, 54relative

cohomology, 57homology, 57

simplex, 51

small, 64skeleton

of a CW-complex, 81

of a simplicial complex, 10slant product, 344

slice inclusion, 139small

map, 64

singular simplex, 64smash product, 141

Smith inequality, 239spin structure, 309

splitting principle, 335, 341

for complex bundles, 342generalized, 335, 342

star, 11

open, 11

INDEX 455

Steenrod

algebra, 289

squares, 279characterization, 293

stem, 328

Stiefel manifold, 319, 340, 398

Stiefel-Whitney classes, 311action of the Steenrod algebra on, 336

dual, 325

equivariant, 392

first, 304

of projective spaces, 347–348second, 307

uniqueness, 336

Stiefel-Whitney numbers, 357

subcomplexof a CW-complex, 82

of a simplicial complex, 9

subdivision, 13

barycentric, 14operator, 176

subset definitions

of (co)chains, 14

of cellular (co)chains, 86of ordered (co)chains, 45

of singular (co)chains, 51

suffix, 328

Sullivan conjecture, 202, 295surface, 25

cohomology algebra of, 116, 117

genus, 28

nonorientable, 28

orientable, 28surgery, 406

suspension, 71

isomorphism, 71

reduced, 142simplicial, 50

symplectic basis, 343

tautological vector bundle, 205, 306, 318,319

complex, 339, 340

tensor product, 132

Thomclass, 152

isomorphism theorem, 153, 157

space, 354

Thom-Pontryagin constructionsee Pontryagin-Thom construction, 358

Top, 11

Top2, 59

TopG, 218

TopΓ, 230topological

complexity, 126

pair, 57

regular value, 76

2-torus (2-elementary abelian group), 254torus (Lie group), 257

associated 2-torus, 257torus (manifold)

cohomology algebra of T 2, 117triangulation of T 2, 26

transfer, 123exact sequence, 123, 156

transport of an orientation, 306triangulable

pair, 178space, 12

triangulation, 12, 178triple

topological, 60map of, 60

umkehr homomorphism, 189unit cochain, 21, 46, 53

vector bundle, 156–158, 297–300associated framed bundle, 252, 301complex, 303equivariant, 252

rigid, 254, 394Euclidean, 156, 301induced, 157, 297isomorphism of, 297orientable, 301product, 298stable isomorphism of, 405structures on, 300tensor product, 253

for line bundles, 304, 341Thom space, 354trivial, 298trivialization, 298

stable, 405Whitney sum, 157

equivariant, 252vertex,vertices, 9

Wang exact sequence, 159weak homotopy equivalence, 103weak topology (for CW-complexes), 81wedge, see also bouquetweight

bundle, 394weighted trace, 315, 338well cofibrant pair, 68

equivariantly, 234

presentation of, 68well pointed space, 71, 141

equivariantly, 247Wu

class, 343formula, 343

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