AN ELEMENTARY PROOF OF QUILLEN’S THEOREM FOR COMPLEXCOBORDISM

A thesis submitted byChristian Carrick,

advised byMichael Hopkins,

to the Department of Mathematicsat Harvard Universityon November 28, 2016,

in partial fulfillment of the requirementsfor the degree of Bachelor of Arts with Honors.

Contents

1. Introduction 22. Prerequisites 23. Geometric Cobordism 53.1. The Geometric Model of MU∗(X) 53.2. The Thom Isomorphism 83.3. Characteristic Classes in U∗ 93.4. Operations in U∗ 124. Localizing at the Fixedpoint Set 144.1. Fixedpoint Formula 144.2. Formal Group Laws and the Key Formula 195. Quillen’s Theorem 225.1. The Technical Lemma 225.2. The Main Theorem 275.3. Proof of Quillen’s Theorem 31Appendix A. Comments on Quillen’s Paper 33Appendix B. The Proof of Thom’s Theorem 33References 38

Acknowledgments

I would like to thank my thesis advisor, Michael Hopkins, for helping me come up with the idea ofthis paper and for teaching me so much over the last 2 and a half years. His course on algebraic topologychanged my life, and he has passed so much wisdom on to me every step of the way as I have gotten toknow homotopy theory. Mike is all-around one of the coolest guys I know, and it has been an honor to learnfrom him. I would also like to thank Eric Peterson for his course on cobordism theory and his generosity inhelping me with this paper. I want to thank the folks I’ve met in Currier house over the years for supportingme and making life interesting. Finally, I want to thank my parents and my brother for always giving meunquestioning love and support.

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1. Introduction

In ordinary cohomology of topological spaces, one has the Steenrod squares and powers. These arecohomology operations

Sqi : Hk(X; Z/2)→ Hk+i(X; Z/2)

Pip : Hk(X; Z/p)→ Hk+2i(p−1)(X; Z/p)

that can be characterized axiomatically. They are extremely useful and one only needs to know their axiomsin practice, but their construction is not very straightforward. The p-th power operation is constructed bynoticing that, for a space X, the p-fold cartesian product Xp is acted on cyclically by Z/p and fits into afibration

Xp → EZ/p×Z/p Xp → BZ/nThen if u ∈ H∗(X; Z/p), one constructs the k-fold cross-product u× · · · × u ∈ H∗(Xp, Z/p) in the obviousway on the chain level. This class is then extended to a class u ∈ H∗(EZ/p×Z/p Xp; Z/p) by a similarconstruction on the chain level, and one pulls u back along the map

EZ/p×Z/p ∆ : EZ/p×Z/p X → EZ/p×Z/p Xp

to give an element of H∗(BZ/p × X; Z/p). Applying the Kunneth formula and taking the classes inH∗(X; Z/p), one defines the classes Pi

p(u).Since this construction uses heavily the chain-level description of cohomology, it is not clear how to

extend such a construction to other generalized cohomology theories. If E is a highly commutative ringspectrum in the sense that it admits a factorization

E∧p → (E∧p)hZ/p → E

where subscript hZ/p denotes the homotopy quotient, and X is a space with u : Σ∞+X → E ∈ E∗(X), we

take the mapΣ∞+X ∆−→ (Σ∞

+X)∧p → E∧p

taking the homotopy quotient of this map with respect to Z/p gives a mapΣ∞+(X× BZ/p)→ (E∧p)hZ/p → E

in E∗(X× BZ/p). But this requires some relatively advanced concepts, like the language of highly commu-tative ring spectra. This approach would work and could prove all the results in this paper [8], but we takethe more elementary approach of Quillen. Namely, when E = MU and X is a smooth manifold, MU∗(X)has an elementary description as a particular set of smooth maps into X. If f : Z → X is such a map,one can simply take the map EZ/p×Z/p ( f×p) and pull it back along EZ/p×Z/p ∆, after approximatingthe infinite complex EZ/p by manifolds. Not much generality is lost here since every finite CW complex ishomotopy equivalent to a smooth manifold via a neighborhood of an embedding into Euclidean space.

These operations in MU∗ turn out to have useful properties like the Steenrod powers, and when onecompares them to the Landweber-Novikov operations - MU operations which are defined via MU Chernclasses - one finds they are closely related. In his 1971 paper Elementary Proofs of Some Results of CobordismTheory, Quillen discovers this relationship and uses it to give a remarkable description of MU∗(X) for X afinite CW complex [7]. He then uses it to re-prove his theorem that the map L→ π∗(MU) from the Lazardring classifying the formal group law on π∗(MU) is an isomorphism. His paper is notoriously difficult tofollow, and this paper is an exposition of Quillen’s results that fills in many of the details left out in hispaper, motivates some of his results, and attempts to give a sense of why his paper is important. I alsoinclude as an appendix a detailed proof that MU∗(X) has the claimed geometric description for manifolds.

2. Prerequisites

Since the spirit of Quillen’s paper is that his proof is elementary - in the sense that it does not make anyuse of the Adams’ Spectral Sequence or the structure of H∗(MU) as a module over the Steenrod Algebra,and only relies on one result from homotopy theory (5.2.1) - I cover some basic concepts in stable homotopytheory that I make use of in the paper. Most of this material is not crucial to the main arguments, but serveswell to put things in a broader context. I assume the reader has familiarity with smooth manifolds, singularcohomology of spaces, basic category theory, and vector bundles and principal bundles. The material covered

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in this section can be found in greater detail in [1] and [5].

Definition 2.1. A spectrum E is a sequence of pointed topological spaces Eii∈Z together with pointedstructure maps

ΣEi = S1 ∧ X → Ei+1

for all i. If X is a pointed space, we let Σ∞X - its suspension spectrum - be the spectrum with (Σ∞X)i =ΣiX = Si ∧ X.

One of the key motivations for defining spectra is Brown’s representability theorem. This says if hii∈Z

is a sequence of contravariant functors from the category of CW pairs to the category of abelian groupssatisfying certain axioms satisifed by ordinary cohomology like homotopy invariance and excision - whatone calls a generalized cohomology theory - then for each i there is a space Ei such that hi(X, x0) ∼= [X, Ei],where X is a CW complex with basepoint x0, and [−,−] denotes pointed homotopy classes. For ordinarycohomology, these Ei are the Eilenberg-Maclane spaces K(Z, i). Because every generalized cohomologytheory has suspension isomorphisms, the sequence Ei then forms a spectrum E. We would thus like todefine the category of spectra so that the spectrum E represents hi in the sense that a cohomology classin hi(X) corresponds to a morphism Σ∞X → E. That hi(X, x0) ∼= [X, Ei] tells us already that our notion ofmorphism should be defined modulo homotopy in a suitable sense. It is easy to define a function betweenspectra E and F - one takes a sequence of maps fi : Ei → Fi−k such that the diagram

ΣEi ΣFi−k

Ei Fi−k

Σ fi

fi

commutes for all i, and we say fi is a function of degree k. But it takes a bit more work to define amorphism that satisfies the above property. We take it as given that there is a notion of morphism betweenspectra such that

[Σ∞X, E]k ∼= colimi→∞[ΣiX, Ei−k]

where [−,−]k denotes morphisms of degree k, and when E comes from a generalized cohomology theoryhi, we have colimi→∞[ΣiX, Ei−k] ∼= [X, E−k] ∼= h−k(X, x0). If E is a spectrum, and we let (ΣkE)i = Ek+i,then one has isomorphisms

[E, F]∗ ∼= [ΣE, ΣF]∗ ∼= [Σ2E, Σ2F]∗And since forming ΣE is the same as forming the spectrum whose i-th space is S1 ∧ Ei, [E, F]k is an abeliangroup for all k by essentially the same reasoning that shows πk(X) is an abelian group for a space X andk > 1. The isomorphism h−k(X, x0) ∼= [Σ∞X, E]k given by Brown representability is then an isomorphism ofabelian groups, and if the hi are a multiplicative cohomology theory in the sense that h∗(X) forms a gradedring, we want to introduce a notion intrinsic to spectra that makes this isomorphism a ring isomorphism.We take the following result as given:

Proposition 2.2. There exists a bifunctor −∧− on the category of spectra that is associative, commuta-tive, and has the sphere spectrum S (i.e. Σ∞S0) as a unit, all up to coherent natural isomorphism. If X is apointed space then Σ∞X ∧ E is isomorphic to the spectrum whose i-th space is X ∧ Ei. A spectrum E withmaps µ : E∧ E→ E and η : S→ E such that µ is associative and unital with respect to η in a suitable senseis called a ring spectrum.

Definition 2.3. If E is a spectrum and X a pointed space, we define En(X) := [Σ∞X, E]−n ∼= colimi→∞[ΣiX, Ei+n]

and En(X) := [S, Σ∞X ∧ E]n ∼= colimi→∞πi(X ∧ Ei−n), the reduced E cohomology and homology of X re-spectively. When X is an unpointed space we define En(X) = En(X+) where X+ := X t ∗ and similarly forhomology. Homology and cohomology of a pair (X, A) are defined by taking reduced cohomology of X/A.One also has a splitting

E∗(X) ∼= E∗(X)⊕ E∗(∗)3

as in ordinary cohomology. E∗(∗) is often abbreviated E∗.

When E is a ring spectrum, E∗(X) forms a graded ring and the isomorphism from Brown representabilitywhen E is built from hi is then an isomorphism of graded rings. The ring structure on E∗(X) is given by

E∗(X)⊗ E∗(X) = [Σ∞X, E]⊗ [Σ∞X, E]→ [Σ∞X ∧ Σ∞X, E ∧ E]µ−→ [Σ∞(X ∧ X), E] ∆∗−→ [Σ∞X, E] = E∗(X)

where ∆ : X → X ∧ X is the diagonal and we use the fact that Σ∞X ∧ Σ∞X ∼= Σ∞(X ∧ X).

Example 2.4. Let EO(n) → BO(n) be the universal vector bundle of rank n and let MO(n) be itsThom space. Then since the Thom space construction is functorial, we have a map Thom(EO(n)⊕ 1) →Thom(EO(n+ 1)) = MO(n+ 1) where 1 denotes the trivial bundle of rank 1. However, Thom(EO(n)⊕ 1) =Thom(EO(n)) ∧ S1 = ΣMO(n), hence the sequence MO(n) forms a spectrum we call MO. Applying thesame construction to the universal bundles EU(n)→ BU(n) we obtain a spectrum MU, where in this casewe have maps Σ2MU(n) → MU(n + 1), so we let the even spaces be MU(n), and (MU)2n+1 = ΣMU(n).In fact if Gn is a sequence of topological groups with maps Gn → BO(kn), satisfying certain compatibilityconditions (e.g, BU(n) with kn = 2n, or BSO(n) with kn = n) we may form a spectrum MG in thesame way, and there is a canonical morphism MG → MO. We call these Thom spectra.

MO and MU are ring spectra because the direct sum maps BO(n)× BO(m)→ BO(n + m) under theThom construction give maps MO(n) ∧MO(m) → MO(n + m) and similarly for MU. These maps patchtogether in the limit to give a ring map µ.

Definition 2.5. A complex oriented cohomology theory is a ring spectrum E with a chosen class x ∈E2(CP∞) such that under the map

E2(CP∞)→ E2(CP1) = E2(S2) ∼= E0(∗)induced by inclusion CP1 → CP∞, x is sent to 1 in the ring E0(∗).

Arguing by universality building up from the universal complex line bundle, one can show that this isthe same as saying that there is a Thom class in E2n(V, V − X) for every complex vector bundle V → X ofrank n, and these Thom classes are multiplicative and natural under pullbacks. Then since every complexvector bundle is orientable in the sense of ordinary cohomology and thus has a Thom class, we see that theEilenberg-Maclane spectrum (HZ)n = K(Z, n) (i.e. the one that represents ordinary cohomology) is complexoriented. MU is a complex oriented cohomology theory because the zero section CP∞ = BU(1) → EU(1)induces a homotopy equivalence CP∞ → MU(1), which determines a class x ∈ [Σ∞CP∞, MU]−2, and itpulls back to the 1 ∈ MU0(∗) because the map CP1 → CP∞ → MU(1) induces a map S→ MU, and this isby definition the unit for the ring spectrum MU. MU is in fact the universal complex oriented cohomologytheory:

Proposition 2.7. Every complex oriented cohomology theory E receives a unique ring map from MU thatrespects the classes x.

Proof : We sketch a proof of this. Since the universal bundle over BU(n) is a complex vector bundle thereexists a Thom class u ∈ E2n(EU(n), EU(n) − BU(n)) ∼= E2n(MU(n)) and hence we have an element of[Σ−2nΣ∞ MU(n), E]0 ∼= E2n(MU(n)) for all n. Then since MU = colimnΣ−2nΣ∞ MU(n), we may ask ifthese maps define a map from the colimit MU. This can be deduced from the Milnor sequence [1], notingthat the lim1 terms vanish since the maps E∗(MU(n + 1)) → E∗(MU(n)) are surjections since the mapsE∗(BU(n + 1)) → E∗(BU(n)) are. We omit the proof that this map respects the ring structures, except tosay that it follows by building E∗ Chern classes in the method of 3.3 and applying the Cartan formula.

We finish off with an important property about complex oriented cohomology theories, which followsby a computation using the Atiyah-Hirzebruch spectral sequence.

Proposition 2.8. If E is a complex oriented cohomology theory, then E∗(CPn) ∼= E∗[x]/xn+1.

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3. Geometric Cobordism

3.1. The Geometric Model of MU∗(X).

In all that follows we assume all manifolds and vector bundles to be smooth. For a compact manifoldX, we define the unoriented bordism groups of X as the set

ΩOn (X) := f : M→ X : f is smooth, and dim M− dim X = n/ ∼

where ( f1 : M1 → X) ∼ ( f2 : M2 → X) if there is a smooth map g : W → X for a compact manifoldW, with ∂W = M1 t M2 and ∂g = f1 t f2. ΩO

n (X) becomes a group under disjoint union, and Thom’stheorem says that there is an isomorphism of groups ΩO

n (X) ∼= [S, MO ∧ Σ∞+X]n =: MOn(X), where MO

is the Thom spectrum of BO. Since MO is the universal Thom spectrum, this theorem may be seen as away to geometrically interpret a Thom spectrum MG as a homology theory by asking that the geometricinterpretation of MO factor through the map MG → MO in some sense.

For the proof of Thom’s theorem in the case X = ∗, one takes an embedding M → RN and formsthe normal bundle νN, with dim M = n. Then by the tubular neighborhood theorem [4] there is an openembedding νN → RN so that the following diagram commutes

νN

M RN

0−section

Since one point compactification is a contravariant functor with respect to open inclusions of locally compactspaces, we get a map SN ∼= (RN)+ → (νN)

+. Since νN is a vector bundle, it admits νN → EO(N − n), andapplying the Thom functor, we have a map (νN)

+ → MO(N − n), recalling that Thom(νN) ∼= ν+N since Mis compact. Putting these together, we have a map

SN → Thom(νN)→ MO(N − n)

which gives an element of [S, MO]n = MOn(∗) The choice of embedding of M and the cobordism relationcorresponds exactly to two maps obtained in the above way being stably homotopic, i.e. equivalent in[S, MO], where S is the sphere spectrum. Said another way, the element in [S, MO]n only depends on theclass of νN in [M, BO] and the cobordism class of M. For a different Thom spectrum MG, we may thereforeask that the classifying map ν : M→ BO factor through the map BG → BO that forgets the G structure ona vector bundle, and thus a factorization of the maps S→ MG and S→ MO through MG → MO.

This approach works to define geometric bordism (i.e. a geometric model for MG∗) for arbitrary MG.Defining geometric cobordism (i.e. a geometric model for MG∗) is a bit trickier, and one of the interestingparts of Quillen’s paper is that he constructs such a geometric model for MU∗. His model can be applied toother MG as well [2]. The model is as follows:

Definition 3.1.1. A complex oriented map is a pair ( f , ν) where f is a smooth proper map f : Z → X ofeven dimension (i.e. dim f := dim Z − dim X is even), and ν ∈ [Z, BU] is such that for some n, we havethat f factors as Z → X ×Cn → X for an embedding i : Z → X ×Cn, and νi has a complex structure sothat the class of

Zνi−→ BU((2n− dim f )/2)→ BU

in [Z, BU] is equal to ν. We say that a complex orientation on an even dimensional map f is a pair as above,and one on an odd dimensional map is a pair as above for the map ( f , 0) : Z → X×R.

Example 3.1.2. If E→ Z is a complex vector bundle, then the zero section i : Z → E is a complex orientedmap (where E is playing the role of X in 3.1.1). Since i is an embedding, we may factor i as Z → E×C0 → Eand νi

∼= E. More generally, if i : Z → X is a closed embedding of manifolds such that νi has a complexstructure, then it is a proper map, and we may factor i through X×C0, and thus i is complex oriented.

Pullbacks:

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If f : Z → X has a complex orientation and g : Y → X is transverse to f , we have the maps

h : Y×X Z → Y

ν′ : Y×X Z → Z ν−→ BU

Since f factors through an embedding i into X×Cn, we can pullback this bundle over X to Y×Cn over Y,and Y×X Z factors through an embedding j into Y×Cn. Then νj is the pullback of νi along g : Y×X Z → Z,hence νj is the same class in [Y×X Z, BU] as ν′ since νi is in the same class as ν. Furthermore, there is nodependence on the choice of representative for the homotopy class ν ∈ [Z, BU], since all such representativeswill yield the same class for ν′. It is not hard to see that since all the spaces are Hausdorff, the pullback ofa proper map is proper. h is thus complex oriented.

Definition 3.1.3. Two complex oriented maps ( f0 : Z0 → X, ν0) and ( f1 : Z1 → X, ν1) are said to becobordant if there is a complex oriented map (g : W → X × [0, 1], τ) with εi : X → X × [0, 1] sendingx 7→ (x, i) transversal to g for i = 0, 1, and ε∗i (g, τ) = ( fi, νi).

Cobordism forms an equivalence relation because if f1 : Z1 → X, f2 : Z2 → X, and f3 : Z3 → X havecomplex orientations such that f1 is cobordant to f2, and f2 is cobordant to f3, there exist manifolds Wand W ′ with the conditions as above. In particular, Z2 is a submanifold of W and W ′, so we can form theconnected sum of W and W ′ along Z2, and the resulting manifold exhibits a cobordism between f1 and f3.Symmetry and reflexivity are clear. There is a bit more to be said here regarding orientations and gluing,but we refer the reader to [2] since these details are non-essential here. We thus define Uq(X), the complexcobordism groups of X to be the set of complex oriented maps into X of dimension −q modulo cobordism.The negative dimesion here is in place so that the gradings in U∗ and MU∗ coincide.

Group Structure:

If ( f , ν), ( f ′, ν′) ∈ Uq(X), we define their sum to be ( f t f ′, ν t ν′). This is a complex oriented mapbecause the factorizations of f and f ′ through embeddings i, i′ into trivial bundles n and n′ respectively givea factorization of f t f ′ through an embedding into their direct sum, and the corresponding normal bundleis (νi ⊕ n′) t (νi′ ⊕ n) → Z t Z′, whose class in [Z t Z′, BU] is equal to ν t ν′. The group law respectscobordism because a cobordism on either side of a map Z1 ä Z2 → X gives one on the sum of the maps byholding the other side constant. Regarding the empty map ∅→ X as a map of dimension −q gives an iden-tity element. We note that any ( f , ν) has an inverse given by the same map f , with the same factorizationvia an embedding i into n, but Cn is given the complex structure and νi is thus given the structure inducedon the quotient. This gives an inverse for ( f , ν) because the orientation on the manifold X × [0, 1] is theone with the orientation for X on X × 0 and the negative orientation (i.e. −X) on X × 1. When themanifold X × [0, 1] is oriented via a complex structure on the normal bundle of an embedding into Cn, thenegative orientation is induced on X× 1 by taking the negative of the n-th normal vector, as above. Thusthe horseshoe map X× [0, 1]→ X× [0, 1] has the corresponding complex orientation, and gives a cobordismbetween ( f , ν) + (−( f , ν)) and the empty map. For more details, see [2].

Ring Structure:

U∗ is a contravariant functor - at least to Sets for now - because for any g : Y → X and any( f , ν) ∈ U∗(X), we may find g homotopic to g that is transverse to f [4], and then define g∗(( f , ν)) to bethe pullback of ( f , ν) along g as above. This of course does not depend on the choice of g since all suchchoices will be homotopic and they will thus pull back ( f , ν) to the same cobordism class, and for the samereason we see that any two homotopic maps g, g′ have the same pullback.

For ( f , ν), ( f ′, ν′) ∈ U∗(X), we can define ( f × f ′, ν× ν′) ∈ U∗(X× X), where ν× ν′ is the homotopyclass

Z× Z′ ν×ν′−−→ BU × BU → BU

using the H-space structure on BU given by direct sums of vector bundles. This gives a map U∗(X) ×U∗(X) → U∗(X × X), and we postcompose this with the map ∆∗ : U∗(X × X) → U∗(X) to define U∗(X)

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as a ring, where ∆ : X → X × X is the diagonal map. The identity map of X with the obvious orientationserves as a multiplicative identity.

U∗(X) is moreover an algebra over U∗(∗) because we may use the same procedure to define a mapU∗(Y)×U∗(X)→ U∗(Y× X) for any Y, and when Y = ∗, we identify X ∼= ∗ × X to obtain an action.

Lemma 3.1.4. U∗ is a contravariant functor of U∗(∗)-algebras.

Proof : We have already shown U∗ to be functorial via pullbacks, it suffices to show that g∗ is a map ofU∗(∗)-algebras for g : Y → X smooth. g∗ is additive because g∗( f t f ′, ν t ν′) is just g∗ f t g∗ f ′ with thenormal bundles pulled back, as any map g with which we replace g that is transverse to f t f ′ must betransverse to both f and f ′. Hence when we pull back, we get the sum of the maps obtained from pullingback f and f ′ along g. g∗ is multiplicative since g∗( f1 · f2) = (∆Y g)∗( f1 × f2), and

g∗( f1) · g∗( f2) = ∆∗X(g× g)∗( f1 × f2) = (g× g ∆X)∗( f1 × f2) = (∆Y g)∗( f1 × f2)

g∗ is unital since the identity map pulls back to the identity. Finally, g∗ is U∗(∗)-linear since the choice of areplacement for g that is transverse to a map r× f : M×Z → ∗×X is the same as one that is transverse justto f , and when we pull back we just get r× g∗( f ). All of the normal bundle data carries through the abovearguments naturally because g - the map at the top of the pullback square involving g- pulls back the normalbundles to their vector bundle pullbacks, and such pullbacks are natural with respect to sums and products.

Pushforwards:

If ( f , ν) ∈ U∗(X) and (g : X → Y, τ) ∈ U∗(Y), we may define g∗( f ) to be the map g f . To see thatthis map has a complex orientation, recall that we can factor f as Z → X × Cn → X for an embedding iand g as X → Y×Cm → Y for an embedding j, and thus we may factor g f as

Z i−→ X×Cn j×1−−→ (Y×Cm)×Cn → Y

and the normal bundle of the embedding (j× 1) i is vi ⊕ f ∗νj and thus comes equipped with a complexstructure coming from those of νi and νj. We then have an element (g f , ν(j×1)i) ∈ U∗(Y) which dependson the choices of i and j only up to their class in [Z, BU] and [X, BU] since we take the class ν(j×1)i ∈ [Z, BU].It is easy to check that these pushforwards are group homomorphisms and are U∗(∗)-linear, but they arenot in general multiplicative. We record some basic facts about how the pushforwards and pullbacks interact.

Lemma 3.1.5. If ( f : Z → X, ν) ∈ U∗(X) and g : Y → X is transverse, and g′, f ′ are the maps in thefollowing diagram

Y×X Z Z

Y X

g′

f ′ f

g

and f ′ is given the pullback complex orientation, then g∗ f∗ = f ′∗g′∗.

Proof : Let (h : A → Z) ∈ U∗(Z), and let f and h factor through embeddings j and i, respectively. Theng′∗(h) is given by the pullback map (Y×X Z)×Z A → Y×X Z with the pullback complex orientation, andthus f ′∗g′∗(h) is the map

(Y×X Z)×Z A→ Y×X Zf ′−→ Y

with the pushforward complex orientation. The other way around, f∗(h) is the map A h−→ Zf−→ X, so

g∗ f∗(h) is the pullback map Y×X A→ Y. But Y×X A = (y, a) : g(y) = f (h(a)) and (Y×X Z)×Z A =(y, z, a) : g(y) = f (z) and z = h(a), and these are isomorphic and the maps are the same. Let g′′ bethe map Y×X A → A. The normal bundle given by going around counterclockwise is g′′∗νi ⊕ g′∗ f ∗νj, andgoing clockwise we have g′′∗(νi⊕ f ∗νj) hence the result follows from the fact that pullbacks of vector bundlescommute with direct sums.

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Lemma 3.1.6. Let f : Z → X be a complex oriented map, and let x ∈ U∗(X), then f∗ f ∗(x) = f∗(1) · x.

Proof : It is sometimes useful to refer to the class f also as f∗(1) to make clear we are referring to thecobordism class and not the map itself. Represent x by a map x : M → X, then f ∗(x) is the map from thefiber product of f ′ and x for f ′ homotopic to f with f ′ transverse to x. To say f ′ is transverse to x is thesame as saying the map f ′ × x : Z×M → X × X is transverse to the diagonal ∆ : X → X × X. Then it iseasy to check that the latter fiber product is just the pullback f∗ f ∗(x), thus by the definition of productsin U∗(X), we have the above equation. An argument similar to 3.1.5 shows that the normal bundles match.

With these definitions in place, we may now state the analogue of Thom’s theorem for MU∗. We givethe proof in Appendix B, since the details of it are not crucial to understand moving forward.

Theorem 3.1.7. Regarding MU∗ and U∗ as functors from the category of smooth manifolds to the categoryof graded rings, there is a natural isomorphism U∗ → MU∗. For any manifold X, U∗(X) has the structureof a graded algebra over U∗(∗), and similarly for MU∗(X). Under the isomorphism U∗(∗) → MU∗(∗),the ring isomorphism U∗(X) → MU∗(X) is an isomorphism of graded U∗(∗)-algebras. If A is a strongdeformation retract of an open neighborhood U in X, we may similarly identify

U∗(X, X− A) := Complex-oriented maps f : Z → X : f (Z) ⊂ A

with MU∗(X, X− A).

3.2. The Thom Isomorphism.

Since MU is a complex-oriented cohomology theory, if E → X is a complex vector bundle, we have aThom isomorphism MU∗(X)→ MU∗(E, E−X) which pulls back a class in MU∗(X) along E→ X and takesits product with the Thom class u ∈ MU∗(E, E− X). If i : X → E is the zero section of this bundle, it is acomplex oriented map as in 3.1.2, and we thus have a pushforward i∗ : U∗(X)→ U∗(E, E− X), which is ob-viously injective. It is also surjective because, for ( f , ν) ∈ U∗(E, E−X), since f lands in X, we can factor f as

Z X×Cn E×Cn Ei j

where i and j are embeddings. νji has a complex structure and splits as νi ⊕ i∗νj. νj is just the normalbundle of X → E (i.e. E), and hence we may give νi the complex structure of νji/i∗νj. This complex ori-ented map now pushes forward to recover ( f , ν). This argument goes through unchanged for an embeddingi : A → X with a complex structure on νi. It will be very useful to know that i∗ corresponds to the usualThom isomorphism in the following way.

Proposition 3.2.1. If π : E → X is a complex vector bundle with zero section i : X → E, the followingdiagram commutes

U∗(X) U∗(E, E− X)

MU∗(X) MU∗(E, E− X)

i∗

∼= ∼=

Thom-∼=

Proof : We first show that i∗( f ) = π∗( f ) · i∗(1) for f ∈ U∗(X), where 1 is the identity map of X withthe obvious complex orientation. f is transverse to π since π is a submersion, so we may take the fiberproduct and we see that π∗( f ) is the map f ∗E→ E, with the complex structure pulled back from π. Thenπ∗( f ) · i∗(1) is the map in the pullback square

P X× f ∗E

E E× E∆

8

after approximating ∆ by something transverse to the right-hand map. Assuming the maps are alreadytransverse it is easy to see that P→ E is just i f : Z → E with the desired complex orientation from pullingback the orientation on the right. But the maps are transverse because i and f ∗E → E are transverse, andthus their product is transverse to the diagonal submanifold.

Now since the Thom isomorphism in MU∗ has the form π∗(x) · u where u is the Thom class, andU∗ → MU∗ is natural and multiplicative, it suffices to show that i∗(1) is sent to the Thom class in MU∗.Looking at the universal case for line bundles, let Ln

1 → CPn be the tautological line bundle for n. Then inthis case, the Thom class in MU∗ is given by the inclusion map Σ∞(Ln

1 )+ → Σ∞ MU(1) → Σ2MU since

this is the map pulled back from the universal Thom class Σ∞ MU(1) → Σ2MU. Then this is sent to theclass in U∗ given by pulling back CPn → (Ln

1 )+ along the identity map of (Ln

1 )+. The maps are obviously

transverse, and we just get the zero section CPn → Ln1 with orientation as in 3.1.2, or i∗(1). Then since the

pullback of the zero section is the zero section, the classes i∗(1) are natural under pullbacks, so we have theresult for all line bundles.

To extend the result to all sums of line bundles, we just need to show the classes i∗(1) are multiplicativein the same sense that Thom classes are in MU∗, then by the splitting principle we may extend to all bundles.Thom classes in MU∗ are multiplicative in the sense that if L1 → X and L2 → X are bundles over X, theThom class of the pullback bundle L1 ⊕ L2 → X is the product of the pullbacks of the Thom classes of Lialong L1 ⊕ L2 → Li. The corresponding result holds in U∗ because if we let πi : L1 ⊕ L2 → Li, then πi is asubmersion, so we may form the fiber product and compute the pullback square as

Li X

L1 ⊕ L2 Li

ji ki

πi

letting ki : X → Li and ji : Li → L1 ⊕ L2 be the zero sections. Then we note that the maps j1 × j2 :L1× L2 → (L1⊕ L2)× (L1⊕ L2) and ∆ : L1⊕ L2 → (L1⊕ L2)× (L1⊕ L2) are transverse since j1 and j2 are.We thus compute the pullback as the fiber product (v1, v2, v3, v4) ∈ (L1 ⊕ L2)× (L1 ⊕ L2) : (v3, 0, 0, v4) =(v1, v2, v1, v2), which is just the zero section in L1 ⊕ L2.

Remark 3.2.2. In regular cohomology, one defines the Euler class of a complex vector bundle E → X asthe pullback of the Thom class along the zero section of a vector bundle. Since the classes i∗(1) correspondto Thom classes, we thus define i∗i∗(1) as the Euler class of the bundle.

Corollary 3.2.3. Euler classes are multiplicative and natural under pullbacks.

Proof : Thom classes in MU∗ are multiplicative and natural under pullbacks, then since i∗ is a ring homo-morphism, and zero sections always pull back to zero sections, Euler classes in MU∗ are multiplicative andnatural under pullbacks. But U∗ → MU∗ is natural and multiplicative, and it sends the classes i∗i∗(1) toEuler classes in MU∗.

3.3. Characteristic Classes in U∗.

It is easy to define Chern classes in U∗ (and therefore MU∗) using the Grothendieck construction,which we recall here. There is a clean way to organize these classes into polynomials - not the usual Chernpolynomials - so that on a line bundle, the polynomial is given by the sum of the powers of the Chern class.Via the Thom isomorphism, these polynomials thus define elements of MU∗(MU) := [MU, MU]∗ which arecalled the Landweber-Novikov operations. These operations are the key tools in Quillen’s proof.

Proposition 3.3.1. Let E → X be a complex vector bundle of rank n with zero section i : X → E. Thereexist unique classes ci ∈ U2i(X) for i = 0, 1, . . . , n such that the following hold:

(1) c0(E) = 1

9

(2) For f : Y → X, f ∗(ci(E)) = ci( f ∗E)

(3) ck(E1 ⊕ E2) = ∑i+j=k

ci(E1) · cj(E2).

(4) If LN1 → CPN is the tautological line bundle, c1(L

N1 ) = i∗i∗(1).

Proof : Taking the projective bundle of E, one has a fibration

CPn−1 → P(E)→ X

Then since MU∗(CPn−1) is a free MU∗(∗) module by 2.8, the Leray-Hirsch theorem says that there is aclass y ∈ MU2(P(E)) that restricts to the element xn−1 ∈ MU2(CPn−1) = MU

2(CPn−1) that is pulled

back from the complex orientation x ∈ MU2(CP∞). Here we are using the fact that MU2(CPn−1) =

MU2(CPn−1)⊕ MU2(∗), and MU2(∗) ∼= U2(∗) = 0 since there are no manifolds of negative dimension.

One also has that the set 1, y, . . . , yn−1 forms a basis for MU∗(P(E)) as a module over MU∗(B). Therethus exist unique classes ci(E) ∈ MU2i(B) such that

yn − c1(E)yn−1 + · · ·+ (−1)ncn(E) = 0

and these are the Chern classes of E, defining c0(E) = 1. Properties (2) and (3) are proved in the usual wayas in cohomology. To show (4), we note that when we take the line bundle L∞

1 → CP∞, we get the fibration

∗ → P(L∞1 )→ CP∞

and so the latter map is a homotopy equivalence, and the Leray-Hirsch theorem just tells us that the class yas above is just the complex orientation x pulled back along the homotopy equivalence. The above equationthen becomes x− c1(L

∞1 ) · 1 = 0. Pulling back along the isomorphism U∗ → MU∗, we get classes in U∗ for

smooth E → X. Now the restriction of the complex orientation x ∈ MU2(CP∞) to MU2(CPN) pulls backto i∗i∗(1) ∈ U∗(CPN) by the arguments of 3.2 as Thom classes in MU are defined so that x is the pullbackof the Thom class u along the zero section i.

A Basis of MU∗BU

Just as the Steenrod operations are given by elements of [HZ/2, HZ/2] = HZ/2∗HZ/2, one is ledto look at elements of MU∗MU for cobordism operations. But by the Thom isomorphism, it suffices to lookat elements of MU∗BU. In ordinary cohomology, one has H∗(BU(1)) ∼= Z[x] with x ∈ H2(BU(1)), and

H∗(BU(n)) ⊂ H∗(BU(1))⊗n ∼= Z[x1, . . . , xn]

is the subalgebra of symmetric functions. Moreover, the Chern classes of the tautological bundle over BU(n)correspond to the elementary symmetric polynomials in the xi’s. Thus in the limit we have

H∗(BU) ∼= lim←

Z[σ1, . . . , σn]

where σi(x1, . . . , xn) is the i-th elementary symmetric polynomial. This gives a Z-basis for H∗(BU) consistingof the monomials in the elementary symmetric functions in the xi, or equivalently the monomials in the Chernclasses. Since BU(1) ∼= CP∞, and MU has the complex orientation x ∈ MU2(BU(1)), the same results holdby using the Chern classes we have built above and 2.8. We thus get an MU∗ basis for MU∗BU. In fact,for this reason we refer primarily to H∗(BU) in this section for simplicity, but all of the same methods workfor MU∗.

This is perhaps the first basis - and the simplest - that one would try, but it is not the one that is usedmost commonly because there is a different basis that behaves much more naturally with respect to directsums of vector bundles. In particular, H∗BU is a ring under the cup product furnished by the diagonalmap ∆ : BU → BU × BU. H∗MU, however does not have a cup product because MU is not a space andthus does not have a diagonal map. This reflects the fact that the Thom isomorphism is an isomorphism ofZ-modules, not Z-algebras. But the basis we have chosen above is built from the cup product structure onspaces - we took polynomials in the classes xi, so this is not a particularly natural basis to choose.

10

H∗MU does have the structure of a Z-coalgebra, however, as we have a map[MU, HZ]→ [MU ∧MU, HZ] ∼= [MU, HZ]⊗HZ∗ [MU, HZ]

induced by the ring map µ : MU ∧MU → MU, which comes from direct sums on vector bundles. We thuslook for a basis that fits more naturally into this structure. Starting again with the basis 1, x, x2, . . . ofH∗(BU(1)), we take the dual basis b0, b1, . . .. Via the injection BU(1) → BU, we send the bi’s alongthe map H∗(BU(1)) → H∗(BU). This map is an injection because U(1) is a retract of U(n) for all n viacontinuous group maps, namely we have the sequence

U(1) → U(n) det−→ U(1)

so we know the bi’s are linearly independent in H∗(BU). Now since BU is an H-space via direct sum ofvector bundles, the bi’s can be multiplied in the ring H∗(BU). So we take the set of all monomials in the bi:each one is given by bα = bα1

1 · · · bαnn for some n and some sequence of nonnegative integers α = (α1, α2, . . .)

with all but finitely many αi = 0. Using the fact that H∗(BU(1)) → H∗(BU) is a monomorphism, the factthat the bi’s form a basis of H∗(BU(1)), and iteratively running the Serre spectral sequence on the fibrationS2n−1 → BU(n− 1)→ BU(n), one may prove that the bα’s form a basis of H∗(BU). Our basis of H∗(BU)will be the one dual to this: we define cα to be the element dual to ∏ bαi

i . We already see that dircet sumsare built into this basis since we defined the monomials ∏ bαi

i using multiplication in H∗(BU) from directsums.

Bases of the Ring of Symmetric Functions

We explain the relationship between the two bases of H∗(BU) discussed. The ring of symmetricfunctions in n variables Λn is very special in that it is a polynomial ring, it is self-dual, and there are manystandard bases of it. The most commonly used bases are the elementary symmetric polynomials, the completehomogeneous symmetric polynomials, the monomial symmetric functions, and the Schur polynomials. Sincethe standard Chern classes in H∗(BU) correspond to the elementary symmetric polynomials in this ring,we may ask if the basis we have cooked up corresponds to one of these other standard bases. If we canprove such a thing, then we may use facts from algebra to go between our two bases. We briefly explainhow one might prove this, and the full result can be pieced together from [1] and [6]. In the case of theelementary symmetric polynomials, we have the Chern classes c1, . . . , cn ∈ H∗(BU(n)), and the direct summap BU(1)×n → BU(n), and thus a sequence of maps

Z[c1, . . . , cn]→ H∗(BU(n)) ⊕∗−→ H∗(BU(1)×n) ∼= H∗(BU(1))⊗n ∼= Z[x]⊗n ∼= Z[x1, . . . , xn]

where x is the generator as above. One finds that ci is sent to the elementary symmetric polynomial σi inthis sequence, and the first map in the sequence is an isomorphism. Taking duals, we have

H∗(BU(1)×n)⊕∗−→ H∗(BU(n))→ (Z[c1, . . . , cn])

∗

and replacing xji with its dual (bj)i (by which we mean the element bj ∈ H∗(BU(1)) on the i-th factor), we

have that the "polynomial" dual to σi in the (bj)i’s is sent to the element dual to the i-th Chern class ci.The quotes here are meant to reflect the fact that we are not taking powers of the bi’s, the "polynomial"∑ aI t

α11 · · · t

αnn in indeterminates t1, . . . , tn refers to the sum ∑ aI(bα1)1 · · · (bαn)n. Applying ⊕∗ to this then

sends (bi)j to bi ∈ H∗(BU(n)), and so may write the elements dual to the monomials in the ci’s as apolynomial in the bi’s, under direct sum. Taking duals and then taking limits, we can write the basis ofH∗(BU) consisting of monomials in the elementary symmetric polynomials in terms of our new basis. Wefind that for each integer sequence α, there is a unique polynomial Pα such that cα, as defined above, can bewritten as Pα(c1, c2, . . .). Moreover, if we set n = ∑ αi, then the Pα are the Schur polynomials defined by theproperty that if σi(x1, . . . , xn) is the i-th elementary symmetric polynomial, then

Pα(σ1, . . . , σn) = ∑ xm11 · · · x

mnn

where the sum ranges over n-tuples (m1, . . . , mn) such that α1 of the mi’s are 1, α2 of the mi’s are 2, and soon.

Perhaps a clearer way to describe the Schur polynomials is to say that for each nonnegative integersequence α = (α1, α2, . . .) with all but finitely many αi = 0 we may find k s1uch that αi = 0 for all i > k,

11

then we can associate a nonincreasing integer sequence β = (β1, β2, . . .) which begins with αk many k’s, thenαk−1 many (k− 1)’s, and so on. This nonincreasing integer sequence has the property that βi = 0 for alli > n, again setting n = ∑ αi. Then to this nonincreasing integer sequence with βi = 0 for all i > n, we mayassociate a monomial symmetric function given by the formula(

∏i

1αi!

)∑

i1,...,in

xβ1i1· · · xβn

in = cα

As an example, if we begin with the sequence α = (1, 2, 0, 0, . . .), we get the nonincreasing sequence β =(2, 2, 1, 0, . . .) and the monomial symmetric function

12 ∑

i,j,kx2

i x2j xk = x2

1x22x3 + x2

1x2x23 + x1x2

2x23 + · · ·

We can check for instance that when we take α = (i, 0, . . .), we recover the i-th elementary symmetricpolynomial in the xi’s, and therefore the i-th Chern class. We also see that the classes cα live in degree2 ·∑j jαj.

3.4. Operations in U∗.

Building the classes cα’s in MU∗(BU) in exactly the same way, they give characteristic classes forcomplex vector bundles by pulling back from universal bundles, and thus they give characteristic classes inU∗. By analogy with the other basis of H∗(BU), we define a "total" Chern class so that we can state a sortof Cartan formula satisfied by the cα. This formula will be much easier to prove, however, because we havebuilt direct sums into this basis. From there we will define operations in U∗ and see that the fruit of ourlabor in defining this basis is that our operations will be ring homomorphisms.

Definition 3.4.1. The total Chern class of a complex vector bundle E→ X is the sum

ct(E) := ∑α

cα(E)tα ∈ U∗(X)[t1, t2, . . .]

where the cα(E) are defined by pulling back the classes cα defined above, and tα := tα11 · · · t

αnn where the ti’s

are indeterminates of degree −2i, and t0 = 1.

Proposition 3.4.2. The total Chern ct(E) class of a complex vector bundle E → X with zero sectioni : X → E satisfies the following properties

(1) If E = L is a complex line bundle, ct(L) = ∑j≥0

tje(L)j where e(L) = i∗i∗(1) is the Euler class of L.

(2) For vector bundles E1, E2, ct(E1 ⊕ E2) = ct(E1) · ct(E2)

Proof : For item 1, since the complex orientation x ∈ MU2(CP∞) pulls back to e(LN1 ) under the map

CPN → BU(1) and U∗ → MU∗ by 3.3.1, it suffices to prove

ct(L∞1 ) = ∑

j≥0tjxj ∈ MU∗(BU(1))[t1, t2, . . .]

defining total Chern classes in MU∗ in exactly the same way. We again prove these in H∗ for simplicity, notingthe arguments may be carried out identically in MU∗. Since the isomorphisms H∗(BU(n)) ∼= Λn, whereΛn is the ring of symmetric functions, are compatible with the restrictions H∗(BU(n)) → H∗(BU(k)) fork < n, we get that the i-th elementary symmetric polynomial in n variables pulls back to the i-th elementarysymmetric polynomial in k variables. In particular, since the i-th elementary symmetric polynomial in kvariables vanishes for i > k, the map r : H∗(BU(n)) → H∗(BU(1)) sends σi to 0 for i > 1 and σ1 to thegenerator x. It is a general fact about Schur polynomials that for α = (0, 0, . . . , 1, . . .) where 1 is in the i-thspot,

Pα(σ1, σ2, . . .) = ∑j

σij

12

Moreover, for any α not of this form, for each monomial in the sum Pα(σ1, σ2, . . .), there exists an i > 1such that σi is a factor of this monomial. Therefore for α = (0, 0, . . . , 1, . . .), r(cα) = xi, for α = (0, 0, . . .),r(cα) = 1, and for any other α, r(cα) = 0, thus proving 1.

To show item 2, one may use the Cartan formula for integer Chern classes and prove combinatoriallyusing the formulas defining Schur polynomials that we have the desired formula for ct. To help motivate theuse of this basis, however, we prove this directly from the definitions using the fact that H∗(BU) is a ringunder direct sum ⊕. Writing an element z ∈ H∗(BU) in terms of the monomials bα in the bi’s is anotherway of writing down the identity map of H∗(BU), and the fact that the identity map is a ring map meansthe following diagram commutes

H∗(BU)⊗ H∗(BU) H∗(BU)⊗ H∗(BU)

H∗(BU) H∗(BU)

1⊗1

⊕∗ ⊕∗1

This seems trivial, but since the basis elements are monomials in the bi’s, this says that if we express twoelements of H∗(BU) in terms of this basis, multiplying them corresponds to multiplying the correspondingmonomials, and this multiplication comes from direct sums of vector bundles. In particular, under theisomorphism Hom(H∗(BU), H∗(BU)) ∼= Hom(Z, H∗(BU)⊗H∗(BU)) (this isomorphism holds for projectivemodules M over a ring and it comes via the isomorphism M⊗M∗ → End(M)), this diagram correspondsto the diagram

Z H∗(BU)⊗ H∗(BU)

H∗(BU)⊗ (H∗(BU))⊗2 H∗(BU)⊗ H∗(BU)

(H∗(BU)⊗ H∗(BU))⊗2 (H∗(BU))⊗2 ⊗ (H∗(BU))⊗2

(1)⊗(⊕∗)1⊗µ

flip

(⊕∗)⊗(1)

The map 1⊗ µ does not come from the isomorphism described, we just include it for reference. The map inthe top row sends 1 7→ ∑α bα ⊗ cα since the bα and cα are dual, and interpreting ∑α bα ⊗ cα as ct = ∑α tαcα,the clockwise map followed by 1⊗ µ expresses ct(E) as ct(E1 ⊕ E2) after collecting terms via 1⊗ µ. Theother way around followed by 1⊗ µ represents ct(E1)ct(E2). We are relying heavily on the facts that

⊕∗(bi ⊗ bj) = bibj

⊕∗(cα) = ∑β+γ=α

cβ ⊗ cγ

where by addition of integer sequences in the above sum we mean entry-wise addition. These come for freeto us since we have defined these elements directly using direct sums, and the cα are dual to the bα.

Operations via the Chern Classes

We have defined classes cα ∈ MU∗(MU) and made sense of them as characteristic classes in U∗(X)by pulling back from the universal bundles. But we want to interpret these operations MU → MU geo-metrically as operations U∗(X)→ U∗(X). The natural thing to do would be to associate a complex vectorbundle bundle over X to ( f , ν) ∈ U∗(X) that depends on both f and ν, and then take its Chern classes toget another element of U∗(X). We of course have such a bundle since we can factor f through an embeddingi : Z → X×Cn and take the complex vector bundle νi.

Definition 3.4.3. The Landweber-Novikov operation on X is the map

st : U∗(X)→ U∗(X)[t1, t2, . . .]

( f , ν) 7→∑ tαsα( f , ν) := ∑ tα f∗(cα(νi))

13

which we can think of as f∗(ct(νi)) if we adopt the convention that f∗ moves past the indeterminates. Itdoes not depend on the choice of νi because total Chern classes are invariant under vector bundle isomor-phism, and stably so because they are multiplicative with respect to direct sums and trivial bundles havevanishing Chern classes. We will relate these to our other key U∗ operations, which are much easier to define.

Remark 3.4.4. The Landweber-Novikov operations are additive in the sense that sα : U∗(X) → U∗(X)are group homomorphisms because the cα’s are additive which can be seen directly from the definition ofaddition in U∗(X). They are multiplicative in the sense that st is a ring homomorphism, which is the contentof (2) in 3.4.2. This fact is the best justification for using the basis of MU∗(BU) we chose.

Definition 3.4.5. If Q → B is a smooth principal Z/p-bundle, then the power operation P associated toQ is the map

P : U−2q(X)→ U−2qp(Q×Z/p Xp)(Q×Z/p∆)∗−−−−−−→ U−2qp(B× X)

f 7→ Q×Z/p f×p 7−→ (Q×Z/p ∆)∗(Q×Z/p f×p)

where Z/p acts cyclically on Xp. We defined these as suggested in the introduction, except we have notused the bundle EZ/p → BZ/p since these are not manifolds, but we will model this bundle with varioussmooth Q → B. We are giving X the trivial G-action, and identifying Q×Z/p X ∼= B× X, as is true forany trivial G-space X. q can be any integer, but we take U−2q in the above definition since it will makecalculations clearer in our main theorem.

Remark 3.4.6. We are only considering maps of even dimension here because in order for Q×Z/p f×p tobe a complex oriented map, we need the factorization of f as Z → X ×Cn → X to be equivariant, namelyX×Cn needs to be a G-bundle in order for Q×Z/p (X×Cn) to make sense, and the complex structure on νineeds to be equivariant in order for νQ×Z/pi to have a complex structure. These conditions are not necessarilysatisfied when f has odd dimension because if f is complex oriented, what we really mean is that the mapZ → X×R is complex oriented. Taking p = 2, for instance, this tells us the map Z× Z → X×R× X×R

is complex oriented, but in order to say the map of odd dimension f × f is complex oriented, we need anorientation on Z × Z → X × X ×R. Our only hope is to project R2 → R and give X × X ×R the swapaction of Z/2 on the first two factors, so that the bundle E× E → (X ×R)2 may be regarded as a bundleover X × X ×R. However, this projection map is not equivariant, since the action of Z/2 on (X ×R)2 isthe swap map.

Lemma 3.4.7. If b ∈ B and i : b → B, then i∗(Px) = xp.

Proof : Let d be an approximation to ∆ transverse to x×p. The map (Q ×Z/p d) i is then an approxi-mation to (Q×Z/p ∆) i transverse to Q×Z/p x×p. Thus i∗(Px) is given by forming the fiber product of(Q×Z/p d) i with Q×Z/p x×p, which gives d∗(x×p) = ∆∗(x×p) = xp.

4. Localizing at the Fixedpoint Set

The point of this section is to relate the two U∗ operations we have just defined. The basic idea willbe to modify the formula of 3.1.5 when g and f are not transverse and then specialize to the case of thediagram with g = Q×Z/p ∆ and f = Q×Z/p x×p defining the operations P.

4.1. Fixedpoint Formula.

We now have our power operations P : U−2q(X) → U−2qp(B × X) for Q → B a principal Z/p-bundle. These operations factor through U−2qp(Q×Z/p Xp), by sending an even dimensional f : Z → Xto Q ×Z/p f×p and then pulling back along the diagonal. Zp and Xp carry an action of Z/p by cyclicpermutation of the factors and f×p is equivariant with respect to this action. Moreover, the fixed points ofthe action are given by the diagonals, hence pulling back along the diagonal can be thought of as restricting

14

to the fixed point set. The power operations thus carry certain equivariance data which we exploit to deriveour key formula relating the power operations and the Landweber-Novikov operations.

Since f×p is Z/p-equivariant, and the diagonal maps are the inclusions of the fixed points, we havethe diagram

B× Z Q×Z/p Zp

B× X Q×Z/p Xp

Q×Z/p∆Z

Q×Z/p f Q×Z/p f×p

Q×Z/p∆X

If we knew the maps Q ×Z/p ∆X and Q ×Z/p f×p were transverse, by 3.1.5, we would have f∗∆∗Z(a) =

∆∗X f p∗ (a) where a ∈ U∗(Q×Z/p Zp), suppressing the Q×Z/p − notation. Letting a = 1 ∈ U∗(Q×Z/p Zp),

we would have the formulaP( f ) = ∆∗X f p

∗ (1) = f∗∆∗Z(1)

We of course don’t have that the maps are transverse in general, so we look for a way to measure how farthe maps are from being transverse.

Definition 4.1.1. For a diagram

W Z

Y X

j′

i′ ij

we define the excess bundle F to be the cokernel of the map TZ|W ⊕ TY|W → TX|W . Thus i and j aretransverse iff F = 0.

Remark 4.1.2. We want to derive a formula involving F that specializes to 3.1.5 when F = 0 so as tounpack P( f ) in the above manner. We ignore the equivariance data for a moment and look at the simplecase when all the maps are embeddings of closed submanifolds, and W = Z ∩Y.

Lemma 4.1.3. In the diagram 4.1.1, let all maps be closed embeddings with W = Z ∩ Y, and let TwW =TwY ∩ TwZ for all w ∈W - such an intersection of submanifolds is called a clean intersection. Then there isa short exact sequence of vector bundles over W

0→ νi′ → j′∗νi → F → 0

Proof : For subspaces U, U′ ⊂ V of a vector space, one has an isomorphismV/U

U′/U ∩U′→ V

U ⊕U′

Letting V = TwX, U = TwZ, and U′ = TwY, the result follows by the fact that W is a clean intersection.

Proposition 4.1.4. In the conditions of 4.1.3, let νi and νi′ have complex structures so that the mapνi′ → j′∗νi in 4.1.3 is a morphism of complex vector bundles, and give F the complex structure that iscompatible with the short exact sequence. Then, if z ∈ U∗(Z),

j∗i∗(z) = i′∗(e(F) · j′∗(z))

in U∗(Y, Y−W).

Proof : We first prove the claim for the following diagram:

W νj′

νi′ νij′∼= νi′ ⊕ νj′ ⊕ F

l′

k′ k

l

15

where all the maps are zero sections. To see why the isomorphism in the bottom right corner holds, notethat from the sequence

νi′ → j′∗νi → νij′

we have a short exact sequence

0→ j′∗νiνi′→

νij′

νi′→

νij

j′∗νi∼= νj′ → 0

Moreover, the embedding k : νj′ → νi′ ⊕ νj′ ⊕ F has normal bundle νi′ ⊕ F, which has a given complexstructure matching the one on j′∗νi via the sequence 4.1.3, by assumption. νi′ also has by assumption acomplex structure, and so νk′ and νk have complex structures, and thus k and k′ are complex oriented,by 2.2, so we may speak of their pushforwards. Define h1 : νi′ → νi′ ⊕ νj′ , h2 : νj′ → νi′ ⊕ νj′ , andh3 : νi′ ⊕ νj′ → νi′ ⊕ νj′ ⊕ F to be the zero sections, and π : νi′ ⊕ νj′ ⊕ F → νi′ ⊕ νj′ to be projection. Using3.2.1, 3.2.3, and functoriality, we have the following computation

l∗k∗(z) = h∗1h∗3h3∗h2∗(z)

= h∗1h∗3(π∗(h2∗(z)) · h3∗(1))

= h∗1((π h2)∗(h2∗(z)) · h∗3h3∗(1))

= h∗1(h∗3h3∗(1) · h2∗(z))

= h∗1(e(νi′ ⊕ νj′ ⊕ F → νi′ ⊕ νj′) · h2∗(z))

= h∗1(e(νi′ ⊕ νj′ ⊕ F → νi′ ⊕ νj′)) · h∗1h2∗(z)

But then by the following diagrams

νi′ ⊕ F νi′ ⊕ νj′ ⊕ F

νi′ νi′ ⊕ νj′

W

h1

p1

p2

νi′ ⊕ νj′ ⊕ F F

νi′ ⊕ νj′ Wp2

we haveh∗1(e(νi′ ⊕ νj′ ⊕ F → νi′ ⊕ νj′)) = h∗1(p∗2(e(F))) = p∗1(e(F))

Then since h1 and h2 are transverse, we apply 3.1.5, and we havel∗k∗(z) = p∗1(e(F)) · h∗1h2∗(z)

= p∗1(e(F)) · k′∗l′∗(z)= k′∗(e(F) · l′∗(z))

Now we need to reduce the general case to the above diagram, which we do by way of replacing X witha tubular neighborhood of W. By pulling back the tubular neighborhood diagram for i j′ : W → X to Yand Z, we can arrange the following commutative diagram

W νj′

W Z

νi′ νij′

Y X

idW

f1j′

i′

f2 f3j

i

Then since the equation of interest lands in U∗(Y, Y−W) and the image of νi′ is an open set containing W,we may use excision to show that the induced map f ∗2 : U∗(Y, Y −W) → U∗(νi′) is an monomorphism. It

16

thus suffices to show that j∗i∗(z) and i′∗(e(F) · j′∗(z)) are equal after applying this map. We remark that theleft facing square is a transverse pullback square since f2 is an embedding and νi′ has the same dimensionas a manifold as X, and f2(νi′) ∩W = W, the map W → f2(νi) being the image of the zero section. 3.1.5then gives f ∗2 i′∗ = k′∗, hence we have

f ∗2 (i′∗(e(F) · j′∗(z))) = f ∗2 (i

′∗(e(F) · l′∗ f ∗1 (z)))

= k′∗(e(F) · l′∗ f ∗1 (z))

= l∗k∗( f ∗1 (z)

= l∗( f ∗3 i∗(z))

= f ∗2 (j∗i∗(z))

where in line 4 we used the fact that the right facing square is a transverse pullback square, which followsfrom an argument similar to the above one since we obtained f1 by pulling back the tubular neighborhooddiagram involving f3 to Z.

We now look to bring equivariance back into the picture. We consider the fixedpoint diagram of anequivariant closed embedding of G-manifolds i : Z → X for a compact Lie group G

ZG Z

XG X

rZ

iG irX

where νi is a G-bundle with an equivariant complex structure. Our case of interest of a map f×p : Zp → Xp

is a special case of this when f is an embedding and has even dimension.

Lemma 4.1.5. A fixedpoint diagram as above is a clean intersection.

Proof : Since tangent spaces are determined locally, we may choose Euclidean neighborhoods, whereby theaction of G becomes a representation, and looking at the tangent bundles over these neighborhoods, theproblem is reduced to the fact that the intersection of the fixedpoints of a representation with a subrepre-sentation is the fixedpoints of the subrepresentation.

In order to apply 4.1.4, we look for a description of the excess bundle in this case. We have that νi is aG-bundle, and thus r∗Z(νi) is a G-bundle over a trivial G-space. As such, the action of G on r∗Z(νi) is just arepresentation on each fiber. Then since νiG is the sub-bundle fixed by G, we have r∗Z(νi) ∼= νiG ⊕ µi, whereµi is the subbundle of νi on which G acts nontrivially, or the direct sum of all the nontrivial irreduciblerepresentations coming from the action of G. Since the complex structure of νi is equivariant, νiG inherits acomplex structure, and by 4.1.3, µi is the excess bundle, and the conditions of 4.1.4 are satisfied. We thushave the formula

r∗Xi∗(z) = iG∗ (e(µi) · r∗Z(z))

Remark 4.1.6. If instead of U∗(−), we used U∗(Q×G −) for a principal G-bundle Q → B on a diagramas in 4.1.5, the proof of 4.1.4 would go through unchanged since we have made all the necessary equivarianceassumptions. In this setting, we suppress the Q×G − notation for pushforwards, so that we write f∗ for(Q×G f )∗ and e(E) for (Q×G i)∗(Q×G i)∗(1) where i the zero section of a bundle E→ X, when it is clearfrom context what is meant.

We proceed in this way, and we remove the assumption that i need be an embedding with that of anequivariant complex oriented map. Namely, let f : Z → X be an complex oriented G-map of even dimensionsuch that it factors equivariantly through i : Z → X×Cn where X×Cn is a complex G-bundle, and νi hasan equivariant complex structure. Then define µi as above, and τi as above on the bundle r∗X(X×Cn).

Proposition 4.1.7. In a diagram as in 4.1.5 with f an equivariant complex oriented map f replacing theembedding i, we have

e(τi) · r∗X f∗(z) = f G∗ (e(µi) · r∗Zz)

17

in U∗(B× ZG).

Proof : Let j : X → n be the zero section. First working in U∗(−), applying 4.1.4 to the diagrams

XG X

nG n

rX

jG jrn

ZG Z

nG n

rZ

iG irn

where we know i is a closed embedding because f is proper, we have the equations

r∗n j∗(x) = jG∗ (e(τi) · r∗X(x))

r∗ni∗(z) = iG∗ (e(µi) · r∗Z(z))

Then letting x = f∗(z), since j∗ f∗ = i∗, jG∗ f G∗ = iG

∗ , we have

jG∗ f G∗ (e(µi) · r∗Z(z)) = iG

∗ (e(µi) · r∗Z(z))= r∗n j∗ f∗(z)

= jG∗ (e(τi) · r∗X f∗(z))

If we choose a Riemannian metric on n so that i(Z) ⊂ Dn where Dn is the disk bundle with respectto the metric, then we have that i∗ : U∗(Z) → U∗(n, n − Dn). Then we have j∗ f∗ = i∗ as mapsU∗(Z) → U∗(n, n − Dn), and when regarded this way, j∗ is an isomorphism as in 3.1. The same canbe said of jG and iG, and thus we have jG

∗ is an isomorphism, and so we may cancel it from both sides ofthe above. As before, we may carry out the arguments identically in U∗(Q×G −) since we have assumedeverything to be equivariant.

Proposition 4.1.8. Fix a principal Z/p-bundle Q → B. Plugging in the data of the power operationsto 4.1.6, let f : Z → X be a complex oriented map of dimension −2q that factors through an embeddingi : Z → X×Cn. Let ρ be the reduced regular representation of Z/p, i.e. the quotient of the representationof Z/p on Cp given by cyclic permutation of the factors by the subrepresentation given by the diagonal in Cp.Let ρ also denote the corresponding trivial Z/p-bundle over a trivial Z/p-space: i.e. for a representation ρof Z/p on a complex vector space V, we can form the vector bundle X ×V → X that carries the action ofZ/p coming from ρ. Then

e(ρ)nP( f ) = f∗(e(ρ⊗ νi))

in U2n(p−1)−2qp(B× X).

Proof : First working in U∗, we apply 4.1.7 to the diagram

Z Zp

X Xp

∆Z

f f p

∆X

and we havee(τip) · ∆∗X( f p) = e(τip) · ∆∗X f p

∗ (1) = f∗(e(µip) · ∆∗Z(1)) = f∗(e(µip))

Let j : X → X×Cn be the zero section. The normal bundle of jp is just Xp × (Cn)p, hence its restriction toX is just X× (Cn)p, with the action of Z/p cyclically permuting the factors of Cn. τi is the quotient of thisbundle by the subbundle fixed by Z/p, namely the diagonal in (n)p. Thus we have τi

∼= ρ⊗ n, and thuse(τi) = e(ρ)m. Similarly, the normal bundle of ip is just ν

×pi , and its restriction to Z is ν

⊕pi with the cyclic

action of Z/p, so we have µi∼= ρ⊗ νi. Applying the same reasoning to U∗(Q×Z/p −), we have the stated

formula.

Remark 4.1.9. The right side of this equation already looks very close to the definition of the Landweber-Novikov operations. If we took νi to be a line bundle, we could break ρ⊗ νi into a sum of line bundles, usethe result 3.4.2 relating the total Chern class of a line bundle to a power series on its Euler class, then use

18

the splitting principle to derive a formula relating the right hand side to the total Chern class of νi, and thusto the Landweber-Novikov operation. There is, however, a better way to say all of this using the languageof formal group laws. As we will show there is a formal group law - a certain bivariate power series - overthe ring U∗(∗) that comes from Euler classes of tensor products of line bundles, and this is the situationwe have in 4.1.8 when νi is a line bundle. This also has the advantage of giving us reason to believe thata formula like 4.1.8 may be used to say something about the structure of U∗(∗), if we are already familiarwith the fact that it is generated by the coefficients of the formal group law.

4.2. Formal Group Laws and the Key Formula.

We review briefly the notion of a formal group law and the one naturally occurring over the ring U∗(∗),and use it to restate 4.1.8 in terms of the Landweber-Novikov operations.

Definition 4.2.1. A formal group law over a ring R is a power series F(x, y) ∈ R[[x, y]] satisfying axiomsmeant to mirror those of an abelian group, namely:

(1) F(0, 0) = 0

(2) F(x, 0) = x and F(0, y) = y

(3) F(x, F(y, z)) = F(F(x, y), z)

(4) F(x, y) = F(y, x)

One can show that there always exists a power series ι(x) ∈ R[[x]] called the inverse, satisfying F(x, ι(x)) = 0,hence one may think of F as addition in an abelian group. We can interpret the action of Z on an abeliangroup in this context and define [1]F(x) = x and [n]F(x) = F([n− 1]F(x), x) and extend to negative integersvia ι. One checks for instance that the leading term of [n]F(x) is nx. We also define 〈n〉F(x) = (1/x)[n]F(x)and omit the subscript F when understood.

Proposition 4.2.2. There is a unique formal group law F(x, y) = ∑i,j≥0

cijxiyj over U∗(∗) with cij ∈

U2−2i−2j(∗) such thate(L1 ⊗ L2) = F(e(L1), e(L2))

for line bundles L1, L2 over the same base. This is usually stated in terms of the bundles’ first Chern classes,but of course these coincide with Euler classes, and the classes of interest in this paper are Euler classes. Welet C denote the subring of U∗(∗) generated by the cij’s.

Proof : We know by 2.8 and 3.3.1 that U∗(CPn) ∼= U∗[x]/xn+1 and x = e(Ln1 ), hence by the Kunneth

formula, we have U∗(CPn ×CPn) ∼= U∗[x1, x2]/(xn+11 , xn+1

2 ) with xi = pr∗i e(Ln1 ) where pi : CPn ×CPn →

CPn for i = 1, 2 are the projection maps. It follows that

e(pr∗1Ln1 ⊗ pr∗2Ln

1 ) = ∑i,j≤n

cnijx

i1xj

2

for some coefficients cnij ∈ U∗(∗). The coefficients do not change as n→ ∞ and hence one has a well-defined

power seriesF(x, y) = ∑

i,jcijxiyj ∈ U∗[[x, y]]

Then since every line bundle is pulled back from the tautological bundle Ln1 → CPn for some n, every tensor

product of line bundles can be pulled back frompr∗1Ln

1 ⊗ pr∗2Ln1 → CPn ×CPn

for some n. Then since Euler classes are natural under pullbacks, the same expression F relates the Eulerclasses for an arbitrary tensor product of line bundles. By homogeneity the coefficients cij are forced to have

19

degree 2− 2i− 2j, and since tensor products of vector bundles are associative, commutative, and unital withrespect to the trivial line bundle, and the trivial line bundle has a trivial Euler class, the axioms of 4.2.1 hold.

Proposition 4.2.3. The functor sending a commutative ring R to the set of formal group laws over R iscorepresentable. That is, there exists a ring L, called the Lazard ring, equipped with a formal group lawFUniv with coefficients aij over L such that for any ring R, there is a bijection

HomCRing(L, R)→ Formal Group Laws over R

that is natural in R, and is given by taking a map f : L→ R and sending it to the formal group law over Rwhose coefficients are f (aij). [3]

The statement of Quillen’s theorem is that the map L→ U∗(∗) corresponding to the formal group lawof 6.2 is a ring isomorphism. Before we use the formal group law to unpack 6.2, we prove some results aboutthe representation ρ, in order to break it up as a sum of line bundles.

Lemma 4.2.4. Let σ be the representation of Z/p on the subspace of Cp given by

W := (z1, . . . , zp) : ∑ zi = 0

where Z/p acts cyclically on the coordinates. Then σ and ρ are isomorphic representations.

Proof : Define a linear map T : Cp →W by

(v1, . . . , vp) 7→ (v1 −1p ∑ vi, . . . , vp −

1p ∑ vi)

T is equivariant since the sums (1/p)∑ vi are symmetric, and the kernel of T is the diagonal, so we get amonomorphism of representations ρ→ σ of dimension p− 1, and thus an isomorphism of representations.

Lemma 4.2.5. Let σ be as above and let η be the representation of Z/p on C given by multiplication bye(2πi)/p. Then there is an isomorphism of representations

p−1⊕k=1

η⊗k → σ

Proof : Let ζ := e(2πi)/p. We first remark that the representation η⊗p is isomorphic to the one on C given bymultiplication by ζk. We define a linear map T : Cp−1 →W by ek 7→ (1, ζk, . . . , ζ(p−1)k), where ek is the k-thstandard basis vector. T lands in W because since p is prime and 1 ≤ k ≤ p− 1, 1, ζk, . . . , ζk(p−1) is theset of all p-th roots of unity, whose sum vanishes. Now, T is injective because each T(ek) is an eigenvectorfor the action of 1 ∈ Z/p on W with eigenvalue ζk, and eigenvectors with distinct eigenvalues are linearlyindependent. Since the dimensions match, it suffices to show T is equivariant, and for that it suffices tocheck basis vectors. If l ∈ Z/p, we have

T(l · ek) = T(ζ lkek) = ζ lkT(ek) = (ζkl , ζk(l+1), . . . , ζk(l+p−1))

which is what we would get if we cyclically permuted T(ek) l times.

Plugging into 4.1.8

20

Let v = e(η) and w = e(ρ). With this information, we take a line bundle L, and since an isomorphismof representations gives an isomorphism of their corresponding vector bundles as in 4.1.8, we have

e(ρ⊗ L) =p−1

∏k=1

e(η⊗k ⊗ L)

=p−1

∏k=1

F(e(η⊗i), e(L))

=p−1

∏k=1

F([k]F(v), e(L))

=p−1

∏k=1

([k]F(v) + ∑

j≥1bj(c)e(L)j

)

=p−1

∏k=1

([k]F(v)

)+ ∑

j≥1aj(v)e(L)j

= w + ∑j≥1

aj(v)e(L)j

for aj(x), bj(x) ∈ C[[x]], where C is the ring generated by the coefficients of the formal group law F as in4.2.2, and we have

w =p−1

∏k=1

[k]F(v) = (p− 1)!vp−1 + ∑j≥p

djvj

for dj ∈ C. Building up from line bundles, and working equivariantly with U∗(Q×Z/p −), we obtain thefollowing:

Theorem 4.2.6. If P is the q-th Power operation associated to a principal Z/p-bundle Q→ B, and v andw are the Euler classes as above, there exists n sufficiently large with respect to q and the dimension of Xsuch that for x ∈ U−2q(X), we have

wn+qPx = ∑l(α)≤n

wn−l(α)(

∏j≥1

aj(v)αj

)sα(x)

in U∗(B× X), where l(α) = ∑ αi and aj(v) ∈ C[[v]].

Proof : We will derive this result from the above discussion and 4.1.8 and note that it suffices to prove thisresult for νi a sum of line bundles by the splitting principle. The left side of the above equation is the sameas the left side of 4.1.8, replacing n with n + q to make calculations easier in our main theorem, but 4.1.8came from a factorization through X×Cn. The statement about n being sufficiently large is just to say thatwe can bound n in this factorization for all x because if the map factors through a large trivial bundle overX, we can push it into a smaller one n + q for some n without loss of generality since the dimension of Xand q are finite. Then we may take all x to factor through n + q by adding the necessary number of trivialbundles to any given factorization.

21

Now let νi be a sum of n line bundles Li (we know νi has dimension n because f has dimension 2q andi is an embedding into n + q). With the convention that a0(v) = 1, we compute

e(ρ⊗ νi) =n

∏i=1

e(ρ⊗ Li)

=n

∏i=1

(w + ∑j≥1

aj(v)e(Li)j)

=n

∑k=0

wn−k(

∑j≥1

aj(v)e(Li)j)

=n

∑k=0

wn−k(

∑i1,...,ik≥1j1,...,jk≥1

( k

∏l=1

ajl (v))

e(Li1)j1 · · · e(Lik )

jk)

=n

∑k=0

wn−k(

∑j1,...,jn≥0

ji=0 for exactlyn−k values of i

( k

∏l=1

ajl (v))

e(L1)j1 · · · e(Ln)

jn)

Now working backward, we know that for an integer sequence α with length k, if we let γ(i) be the sequencewith 0’s in every entry except a 1 in the i-th entry and letting γ(0) be the zero sequence, we have

cα(L1 ⊕ · · · ⊕ Ln) = ∑β(1)+···+β(n)=α

cβ(1)(L1) · · · cβ(n)(Ln)

for sequences β(i) by 3.4.2. Then since sequences not of the form γ(i) give vanishing Chern classes on linebundles and cγ(i)(L) = e(L)i for a line bundle L, we have

cα(L1 ⊕ · · · ⊕ Ln) = ∑γ(j1)+···+γ(jn)=α

e(L1)j1 · · · e(Ln)

jn

= ∑γ(j1)+···+γ(jn)=α

ji=0 for exactlyn−k values of i

e(L1)j1 · · · e(Ln)

jn

= ∑j1,...,jn≥0

ji=0 for exactlyn−k values of i,|jk : jk=ji|=αji

e(L1)j1 · · · e(Ln)

jn

The second line follows because l(α) = l(γ(j1)) + · · ·+ l(γ(jn)) and l(γji ) = 1 or 0, and the third becauseto say γ(j1) + · · ·+ γ(jn) = α is by definition of the γ(ji)’s to say that |jk : jk = ji| = αji . We check thatthe terms appearing in the sum where we left off having the property that |jk : jk = ji| = αji all havecoefficient ∏

j≥1aj(v)

αj .

5. Quillen’s Theorem

5.1. The Technical Lemma.

The purpose of this section is to show the existence of certain exact sequences that will be of use inapplying 4.2.6 to prove our main theorem. It is difficult to motivate this section without having seen the proofof our main theorem that will follow, other than to say that 4.2.6 tells us that there is a formula involvingany x ∈ U∗(X) and the coefficients of the formal group law, i.e. the ring C, via the power operations. Whenwe probe this formula for more information by, for instance, looking at the equation p-locally, we are led tosome amazing consequences if we know there is an exact sequence involving multiplication by the class v as

22

in 6.5. In fact, we could prove without much difficulty that there is an exact sequence

MUq(X)·〈p〉(v)−−−−→ MUq(BZ/p× X)

·v−→ MUq+2(BZ/p× X)

where 〈p〉(v) is as in 6.1. But we wish to make sense of this in U∗, so we prove the existence of a relationinvolving manifolds that approximate BZ/p, namely S2n+1/Z/p so that this relation approximates theabove exact sequence as n → ∞. One can think of these as approximating BZ/p because Z/p acts freelyon S2n+1 ⊂ Cn, the connectivity of which strictly increases as n→ ∞, hence these approach EZ/p.

Proposition 5.1.1. Let B be compact, f : Q → B a principal Z/p-bundle, and L = Q×Z/k C → B theline bundle associated to the representation η as in 4.2.5. Then f = 〈p〉(e(L)) ∈ U0(B).

Proof : Let i : B → L be the zero section and j : Q → L be the embedding sending q 7→ (q, 1). f is properand factors through L, and its normal bundle has a complex structure since that of i does. Similar remarksapply to j, so these maps are complex oriented, and we may speak of their pushforwards. Letting g : L→ Bbe the projection map, we form the line bundle g∗L = L×B L→ L. Letting s be the diagonal section, notethat s is nonvanishing away from the zero section of L, hence it trivializes g∗L away from the zero section.We may therefore extend g∗L to a bundle M over L+ (i.e. L t ∞) since g∗L is trivial near infinity. s thenextends to a section of M and is homotopic to the zero section since all sections are, and it is transverse tothe zero section since the diagonal and the component on the left together span each tangent space. Thuswe may compute the e(M) by forming the fiber product of s with the zero section, which is easily seen tobe i∗(1).

We may similarly trivialize the bundle g∗(L⊗p) away from the zero section via s⊗p and form the bundleM⊗p over L+. We now define a section t of M⊗p by t(q, z) = ((q, z), (q, zp)). This is transverse to the zerosection since the map on the right (q, z) 7→ (q, zp − 1) is a submersion fiberwise, and the fiber product ofthis with the zero section is (q, z) : zp = 1 = j(Q) since pairs in L are subject to the equivalence relation(q, e2πi/pz) ∼ (qσ, z) where σ is the generator of Z/p. Thus

j∗(1) = e(M⊗p)

= [p]F(i∗(1))

= i∗1 · 〈p〉(i∗(1))= i∗(〈p〉(i∗i∗(1)))= i∗(〈p〉(e(L)))

the last line following from 3.1.6. Now, we note that i∗ : Uq(B) → Uq+2(L+, ∞) is an isomorphism, by3.2.1 and the fact that the usual Thom isomorphism in MU can be seen as MUq(B)→ MU

q+2(Thom(L)),

and since B is compact L+ ∼= Thom(L). If p : L → B is the projection, then p∗ is an inverse to i∗, henceapplying i−1

∗ to both sides of the above we have f∗(1) = (p j)∗(1) = 〈p〉(e(L)) since f = p j. Note thatwe could not have just used the fact that i∗ : Uq(B)→ Uq+2(L, L− B) is an isomorphism - it was necessaryto extend the bundles over a point at infinity since only then would all of the relevant classes live in U∗(L+),allowing us to apply the inverse of i∗ to j∗(1). In particular, if we ran the argument with the isomorphisminto Uq+2(L, L− B), we could not have applied i−1

∗ to j∗(1) since the image of j is not contained in the zerosection. Note that L+ is not necessarily a smooth manifold, so U∗(L+) doesn’t make sense, but we can passto MU∗(L+), and then since U∗ → MU∗ defines pushforwards for complex oriented maps in MU∗, sendsEuler classes to Euler classes, is natural, etc. the result holds for U∗.

Remark 5.1.2. In the form of 5.1.1, this result doesn’t seem particularly useful with respect to studyingthe formula in 4.2.6, because the Euler class in 5.1.1 is of the bundle Q ×Z/p C → B associated to therepresentation η, whereas the Euler class v in 4.2.6 is of the bundle Q×Z/p (X×C)→ Q×Z/p X ∼= B× X.However, these are essentially the same thing because since X is a trivial Z/p space, the two bundles comingfrom η

Q×Z/p (X×C)→ B× X

(Q×Z/p C)× X → B× X

23

are the same, and hence their Euler classes are the same. But if pr1 : B× X → B is projection, then thelatter bundle is just pr∗1(L), where L is the bundle of 5.1.1. Hence v = pr∗1(e(L)), where v is as in 4.2.6. Onefinds, for instance that the inclusion of a point i : b → B gives a ring homomorphism

U∗(B× X)i∗−→ U∗(b × X) ∼= U∗(X)

that sets v = 0 because i∗pr∗1(e(L)) = (pr1 i)∗(e(L)) ∈ U2(X), pr1 i is null, and U2(∗) = 0.

The Gysin Sequence

If p : E→ B is a complex vector bundle of rank n, and π : SE→ B is the unit sphere bundle associatedto E, one in general has the Gysin sequence

· · · → Hq−2n(B)·e(E)−−→ Hq(B) π∗−→ Hq(SE)

φ−→ Hq−2n+1(B)→ · · ·where φ is the connecting map Hq(SE) ∼= Hq(E− 0)→ Hq+1(E, E− 0) followed by the inverse of the Thomisomorphism. The same sequence holds for any cohomology theory with Thom isomorphisms, i.e. complexoriented cohomology theories, so it holds for MU∗. In U∗, however, we have a more geometric interpretationof the Thom isomorphism as the pushforward i∗, and its inverse is p∗|0, and in fact it is not hard to showthat φ = p∗|E−0 = π∗. This pushforward makes sense because π is proper as its fibers are spheres whichare compact, and it factors through the embedding SE → E. In U∗ the sequence thus becomes

· · · → Uq−2n(X)·e(E)−−→ Uq(B) π∗−→ Uq(SE) π∗−→ Uq−2n+1(B)→ · · ·

We apply this sequence to the following situation. Let Z/p act on Cn by having 1 ∈ Z/p act asmultiplication by ζ := exp( 2πi

p ). Then if X is a trivial Z/p space, we have the bundles En := Q×Z/p (X×Cn) → B× X and (Q×Z/p Cn)× X → B× X, which are the same by 5.1.2. There is an induced action ofZ/p on S2n−1 ⊂ Cn since we may think of S2n−1 as z ∈ Cn : ∑ zizi = 1, making it clear that S2n−1 isinvariant under Z/p as ζζ = 1. Letting Q→ B be the bundle S1 → S1/Z/p, we thus conclude that

S(En) = (S2n−1 ×Z/p S1)× X

Now let pn : (S2n−1 ×Z/p S1)× X → S1/Z/p× X be the projection of this spherical bundle. Then we havea Gysin sequence

· · ·Uq−2n(S1/Z/p× X)·e(En)−−−→ Uq(S1/Z/p× X)

p∗n−→ Uq((S2n−1 ×Z/p S1)× X)pn∗−−→ Uq−2n+1(S1/Z/p) · · ·

However, it is clear that the representation of Z/p on Cn described above is just η⊕n, and their correspondingbundles and unit sphere bundles are thus the same. In particular, e(En) = vn

1 , where vi is the Euler classin 4.2.6 with respect to the principal Z/p-bundle S2i−1 → S2i−1/Z/p. Now we want to see what happenswhen we stabilize these Gysin sequences against n, and we claim there is the following commutative diagram

Uq−2n−2(S1/Z/p× X) Uq(S1/Z/p× X) Uq((S2n+1 ×Z/p S1)× X) Uq−2n−1(S1/Z/p× X)

Uq−2n(S1/Z/p× X) Uq(S1/Z/p× X) Uq((S2n−1 ×Z/p S1)× X) Uq−2n+1(S1/Z/p× X)

·vn+11

·v1 id

p∗n+1

j∗n

pn+1∗

·v1

·vn1 p∗n pn∗

where jn : (S2n−1 ×Z/p S1)× X → (S2n+1 ×Z/p S1)× X is induced by the inclusion ι : Cn → Cn+1. Theleft square obviously commutes, and the middle commutes as pn = pn+1 jn. The right commutes by thefollowing lemma.

Lemma 5.1.3. Let E, F be complex vector bundles over X, with f : S(E ⊕ F) → X, g : SE → X, andj : SE → S(E⊕ F). Then, for z ∈ U∗(S(E⊕ F)),

g∗ j∗(z) = e(F) · f∗(z)

Proof : We remark that f and g are proper as their fibers are spheres, and hence they are all complex orientedby 3.1.2 since they factor through obvious bundles, and we may take these bundles to be trivial by possiblyembedding them in large trivial bundles. j is proper as a closed embedding, and it has a complex structure

24

on its normal bundle coming from the fact that the embedding E → E⊕ F does. Hence we may speak ofthe pushforwards of each of these maps. f factors as

S(E⊕ F)→ E⊕ F → X

so that g = f j. Let s : X → F be the zero section, then the projection p : S(E⊕ F)→ F is transverse to fbecause locally p is given by

X× S2n+2m−1 → X×Cn ×Cm → X×Cm

If n = 0, this map never intersects the zero section so there is nothing to check. When n > 0, themap S2n+2m−1 → Cn × Cm → Cm has 0 as a regular value, since a point sent to zero looks like v =(z1, . . . , zn, 0, . . . , 0) ∈ Cn+m, i.e. a point in S2n−1 ⊂ S2n+2m−1, and after identifying the tangent spaces withthe vector spaces themselves, TvS2n+2m−1 is identified with the subspace of Cn ×Cm orthogonal to v, whichcontains the factor of Cm on the right. We compute the fiber product p∗(s) = j and therefore

j∗1 = p∗s∗1 = f ∗s∗s∗1 = f ∗(e(F))

using the fact that the map s f : S(E⊕ F)→ X → F is homotopic to p since F deformation retracts onto thezero section. We thus make the following calculation

g∗ j∗(z) = f∗ j∗ j∗(z)

= f∗(j∗(1) · z)= f∗( f ∗(e(F)) · z)= e(F) · f∗(z)

using 3.1.6 in lines 2 and 4, and using the above equation in line 3. Letting g = pn, j = jn, and f = pn+1,since En = En−1 ⊕ E1 in the above notation, this result proves that the above square commutes.

The Key Lemma

With these in place, we may now prove the main result of this section, as referenced in the introduction.We remark that there is an action of U∗(X) on U∗(Y × X) given by essentially the same action of U∗(∗)on U∗(Y), when Y has a basepoint. Namely, in our case of interest the inclusion of the p-th roots of unityZ/p → S1 → S2n+1 induces a map ∗ = Z/p/Z/p → S2n+1/Z/p, i.e. we have a choice of basepoint inS2n+1/Z/p, and hence any complex oriented map f : Z → X may be considered as

∗ × Z → S2n+1/Z/p× X

which thus may be multiplied by anything in U∗(S2n+1/Z/p×X). By the same reasoning as in the action ofU∗(∗), one sees that pushforwards of the form ( f × idX)∗ for f ∈ U∗(S2n+1/Z/p) and pullbacks (g× idX)

∗

are U∗(X)-linear.Let jn be the map as above and define j′n : S2n−1/Z/p× X → S2n+1/Z/p× X also induced from the

map ι : Cn → Cn+1.

Proposition 5.1.4. If x ∈ Uq(S2n+1/Z/p× X) such that x · vn+1 = 0, then there exists y ∈ Uq(X) suchthat y · 〈p〉(vn) = j′∗n (x). Since the S2n+1/Z/p approximate BZ/p, this may be thought of as approximat-ing the exact sequence in the beginning of this section.

Proof : Let πn+1 : S2n+1 ×Z/p S1 × X → S2n+1/Z/p × X be the unit sphere bundle of the line bundleinduced from η, i.e. this is the sphere bundle of the bundle Fn+1 such that e(Fn+1) = vn+1. Applying theGysin sequence to this sphere bundle, we have an exact sequence

Uq+1((S2n+1 ×Z/p S1)× X)πn+1∗−−−→ Uq(S2n+1/Z/p× X)

·vn+1−−−→ Uq+2(S2n+1/Z/p× X)25

Hence if vn+1 · x = 0, there exists z ∈ Uq+1((S2n+1 ×Z/p S1)× X) such that x = πn+1∗(z). Since jn and j′nare both induced from ι, the following square commutes

(S2n−1×Z/p)× X (S2n+1×Z/p)× X

S2n−1/Z/p× X S2n+1/Z/p× X

jn

πn πn+1

j′n

But πn+1 is a submersion and the square is a pullback, hence we have

j′∗n (x) = j∗nπn+1∗(z) = πn∗ j∗n(z)

by 3.1.5.Recall that by 5.1.2, v1 ∈ U2(S1/Z/p × X) is equal to pr∗1(e(L)), where pr1 : S1/Z/p × X →

S1/Z/p is projection, and L is the bundle S1 ×Z/p C → S1/Z/p induced from η, as in 5.1.1. Thereforee(L) ∈ U2(S1/Z/p) ∼= U2(S1) = 0 since there are no manifolds of negative dimension, and hence v1 = 0.

Since v1 = 0, in the diagram preceding 5.1.3, we have pn∗ j∗n(z) = v1 · pn+1∗(z) = 0, so j∗nz ∈ ker pn∗,

and thus there exists z′ ∈ Uq+1(S1/Z/p× X) such that p∗n(z′) = j∗n(z). Hence

j′n∗(x) = πn∗p∗n(z

′)

Now by identifying S1/Z/p ∼= S1, we may regard φ : S1/Z/p×X → X as the unit sphere bundle of C×X,whose Euler class is trivial. We thus have that the corresponding Gysin sequence splits into short exactsequences

0→ Uq+1(X)φ∗−→ Uq+1(S1/Z/p× X)

φ∗−→ Uq(X)→ 0

These sequences split on the right via the map y 7→ y · i∗(1) where i : X ∼= ∗ × X → S1/Z/p× X is givenby the inclusion of the basepoint in S1/Z/p, and y· is shorthand for the action of U∗(X) described above.Computing φ∗(y′) = y′ · 1 where 1 is the cobordism class of the identity in U∗(S1/Z/p× X), we thus maywrite z′ = y′ · 1 + y · i∗(1). Hence

j′n∗(x) = πn∗p∗n(y

′ · 1) + πn∗p∗n(y · i∗(1))

We have by U∗(X)-linearity that

πn∗p∗n(y′ · 1) = πn∗(y · p∗n(1))

= πn∗(y · 1)= y · πn∗(1)

= y · πn∗(π∗n(1))

= 0

using the Gysin sequence involving πn. One easily checks that p∗ni∗(1) is the cobordism class of S2n−1×X ∼=(S2n−1×Z/p Z/p)×X → (S2n−1×Z/p S1)×X, and hence πn∗p∗ni∗(1) is the cobordism class of S2n−1×X →S2n−1/Z/p× X, which is 〈p〉(e(L)) where L is the bundle (S2n−1 ×Z/p C)× X → S2n−1/Z/p× X, i.e. Fn,noting that the proof of 5.1.1 goes through unchanged taking cartesian products with X everywhere. Hencewe have

j′n∗(x) = πn∗p∗n(y

′ · 1) + πn∗p∗n(y · i∗(1))= 0 + y · 〈p〉(vn)

recalling that vn is by definition the Euler class of Fn.

26

5.2. The Main Theorem.

We now have all we need to prove the main result of Quillen’s paper, from which we will derive Quillen’stheorem as a corollary in the following section. The main maneuver will be to look at the equation 4.2.6p-locally for each prime p and to use the key lemma 5.1.4. We will need the following result from homotopytheory, which follows from the Serre finiteness theorem, and the fact that MU has only finitely many cellsin each degree. We also record an easy lemma to be used in our proof.

Proposition 5.2.1. If X is a finite CW complex with basepoint x0, MUq(X) is a finitely generated abeliangroup for all q.

Definition 5.2.2. Let X be a path-connected space with basepoint x0, and let MU∗(X) ⊂ MU∗(X) be

the ideal consisting of elements that vanish when restricted to the basepoint x0. This agrees with the usualdefinition of MU

∗(X) up to isomorphism since one always has the splitting

MU∗(X) ∼= MU∗(X)⊕MU∗(x0)

but we are using this isomorphism to identify MU∗(X) and MU∗(x0) as subsets of MU∗(X) so that the

splitting is an equality. Via the isomorphism U∗ → MU∗, we identify MU∗(X) with U∗(X) := U∗(X, x0)

when X is a manifold.

Lemma 5.2.3. If X is a space with basepoint x0, one has MU∗(∗)-linear suspension isomorphisms

(1) MU2q−1(X) ∼= MU2q(S1 ∧ X+)

(2) MU2q(X) ∼= MU2q+2

(S2 ∧ X+)

(3) MU2q−1

(X) ∼= MU2q(S1 ∧ X)

Proof : (3) is the usual suspension isomorphism carried by any cohomology theory, which is almost tautolog-ical by 2.3. Recall that MU(−) is the functor sending a pointed space X to [Σ∞X, MU] and MU(−) sends aspace X to X+, i.e. X with a disjoint basepoint, followed by MU(−). Hence (1) and (2) follow immediatelyfrom (3) since MU∗(X) = MU

∗(X+), and it is easy to see that (3) is MU∗(∗)-linear, hence so are (1) and

(2).

Lemma 5.2.4. If X is a pointed space with basepoint x0 that is homotopy equivalent to a finite CW com-plex, the ideal MU

0(X) ⊂ MU0(X) is nilpotent.

Proof : Identifying MU0(X) ∼= MU∗(X, x0), we recall that the relative cup product is the map

MU0(X)⊗ MU

0(X) ∼= MU0(X, x0)⊗MU0(X, x0)→ MU∗(X× X, X ∨ X) ∼= MU

0(X ∧ X)

∆∗2−→ MU0(X)

where ∆2 is the map X → X × X → X ∧ X. Then ∆n is nullhomotopic for n sufficiently large since we cantake X to be a CW complex, and take the map to be null by cellular approximation since forming X ∧ Xraises the degrees of the nonzero cells in X.

Theorem 5.2.5. If X is a connected space homotopy equivalent to a finite CW complex, then

MU∗(X) = C ·⊕q≥0

MUq(X)

MU∗(X) = C ·

⊕q>0

MUq(X)

where C is the ring as in 4.2.2 that is generated by the coefficients of the formal group law.

27

Proof :

(i) Reducing to the Even Case: Suppose we have proven that

MUev(X) = C ·

⊕q>0

MU2q(X)

where MUev(X) is the subring of MU

∗(X) consisting of elements of even degree. Then since C is concentrated

in even degrees, we have

C ·⊕q>0

MUq(X) =

(C ·⊕q>0

MU2q(X)

)⊕(

C ·⊕q>0

MU2q−1(X)

)= MU

ev(X)⊕

(C ·⊕q>0

MU2q−1(X)

)By the splitting in 5.2.2, MU2q−1(X) = MU

2q−1(X) ⊕ MU2q−1(x0), and since q > 0, MU2q−1(x0)

vanishes since MU2q−1(x0) ∼= U2q−1(∗) = 0 since q > 0. Hence we have

C ·⊕q>0

MU2q−1(X) = C ·⊕q>0

MU2q−1

(X)

Applying the isomorphism (3) of 5.2.3 to MU∗(X), we have that the images of C · ⊕

q>0MU

2q−1(X) and

MUodd

(X) areC ·⊕q>0

MU2q(S1 ∧ X) and MU

ev(S1 ∧ X)

respectively, noting thatC ·⊕q>0

MU2q−1

(X) ⊂ MU∗(X)

because the restriction map MU∗(X)→ MU∗(x0) is MU∗(∗)-linear. By the above assumption, these areequal, again using the fact that MU

2q(S1 ∧ X) = MU2q(S1 ∧ X) when q > 0. Thus the two preimages are

equal, and we have

C ·⊕q>0

MUq(X) = MUev(X)⊕

(C ·⊕q>0

MU2q−1(X)

)= MU

ev(X)⊕ MU

odd(X) = MU

∗(X)

hence we recover the second claim of the theorem. To recover the first claim, we note that

MUev(X) = MUev(X)⊕MUev(x0) =

(C ·⊕q>0

MU2q(X)

)⊕MUev(x0)

If we apply the isomorphism (2) of 5.2.3 to C ·MU0(x0) and MUev(x0), we get

C · MU2(S2) and MU

ev(S2)

respectively, and by our key assumption, the latter is

C ·⊕q>0

MU2q(S2) = C · MU2(S2)

using the splitting 5.2.2 and the fact that πk(MU) vanishes for k < 0, as we showed since there are nomanifolds of negative dimension. Putting these facts together we have

MUev(X) =

(C ·⊕q>0

MU2q(X)

)⊕(

C ·MU0(x0))

= C ·((⊕

q>0MU2q(X)

)⊕MU0(x0)

)= C ·

⊕q≥0

MU2q(X)

28

the last equality following from the fact that

MU0(X) = MU0(X)⊕MU0(x0) =

[C ·⊕q>0

MU2q(X)

]0⊕MU0(x0)

where [−]0 denotes all terms of degree 0. Using an identical argument with the isomorphism (1) in 5.2.3, werecover the first statement of the theorem.

(ii) Setting up Induction on the Key Formula: We now take a smooth manifold homotopy equivalentto X, which we also call X, and thereby reduce our statements to U∗. By (i), letting

R := C ·⊕q>0

U2q(X)

it suffices to show Uev(X) = R, and it suffices to show this p-locally for every prime p by a localization-globalization argument since Uev(X) and R are sub Z-modules of U∗(X), and the prime ideals (p) ⊂ Z areall the maximal ideals in Z. Noting that both of these are concentrated in even degrees only, we assumeas an inductive hypothesis that wi v. For the base case, if q = 0 and j < q, then −2j is positive, henceU−2j(X) = U−2j(X), and R−2j ⊂ U−2j(X) since Ui(∗) ·U j(X) ⊂ Ui+j(X) and U−2j(X) ⊂ R−2j since 1 ∈ Cand U−2j(X) appears in the sum defining R as −j > 0.

Applying 4.2.6 to the principal Z/p-bundle S2m+1 → S2m+1/Z/p, there exists n >> 0 such that ifx ∈ U−2q(X) ⊂ U−2q(X)

wn+qPx = ∑l(α)≤n

wn−l(α)(

∏j≥1

aj(v)αj

)sα(x)

with aj(v) ∈ C[[v]], where in this case v = vm+1 as in 5.1.4. We also have that

w = (p− 1)!vp−1 + ∑j≥p

djvj =⇒ vp−1 = vp−1((p− 1)! + ∑j≥p

djvj−p+1) =: vp−1θ(v)

where dj ∈ C and hence θ(v) ∈ C[[v]] and after localizing at p, θ(v) becomes multiplicatively invertiblebecause its leading term (p− 1)! is a unit after localizing at p. When α = (0, 0, . . .), sα(x) = x, so that

wn+qPx = wnx + ∑0<l(α)≤n

wn−l(α)(

∏j≥1

aj(v)αj

)sα(x)

Remarking that for α 6= 0, sα raises degree by an even number as in this case the Chern class cα has positiveeven degree, localizing the above equation at p, we have sα(x) ∈ R−2j

(p) for j < q. Hence the above summationon the right lies in R(p)[[v]]. Furthermore since θ(v) is invertible, we have

wn(wqPx− x) = ∑0<l(α)≤n

wn−l(α)(

∏j≥1

aj(v)αj

)sα(x) =⇒ (vp−1)n(wqPx− x) = ψ(v)

where ψ(v) ∈ R(p)[[v]]. Now we set m := n(p− 1) > 0, and we have

vm(wqPx− x) = ψ(v) ∈ U∗(S2m+1/Z/p× X)(p)

Hence the set of integers m > 0 such that an equation as above - with the terms on the left and a powerseries in v with coefficients in R(p) on the right, all living in U∗(S2m+1/Z/p × X)(p) is nonempty, so wechoose m to be minimal. There is no cause for concern here in defining m as above because n depends onlyon the dimension of X, and not on the dimension of the sphere defining the principal bundle in use.

(iii) m = 1: If i is as in 5.1.2, then i∗ is a ring homomorphism that sets v = 0, hence the above equationshows ψ(0) = 0, so we may define the power series ψ1(t) = (1/t)ψ(t), and thus we have

v(vm−1(wqPx− x)− ψ1(v)) = 029

Then since |v| = 2, |w| = 2(p− 1), and |Px| = −2qp, |vm−1(wqPx− x)− ψ1(v)| = 2(m− 1)− 2q, hence5.1.4 tells us there exists y ∈ U2(m−1)−2q(X)(p) with y · 〈p〉(vm) = j′m

∗(vm−1(wqPx− x)− ψ1(v)). But sincej′m∗(vm+1) = vm and by v we have meant vm+1, we have that since j′m

∗ is a ring homomorphism

vm−1(wqPx− x) = ψ1(v) + y · 〈p〉(v)

in U∗(S2m−1/Z/p×X)(p). Now restricting this equation from X to its basepoint x0 is a ring homomorphismthat sets x = 0 since x ∈ U−2q(X), the reduced cohomology of a point always vanishing. Similarly ψ1(v) isset to zero because its coefficients lie in R(p), which vanishes when restricted to a point since U2k(x0) = 0for k > 0. Hence if y′ is the restriction of y to U∗(x0)(p), we have y′〈p〉(v) = 0. Thus

vm−1(wqPx− x) = ψ1(v) + (y− y′)〈v〉

and so we may assume y ∈ U∗(X) since y− y′ is. Now if m > 1, then y ∈ R(p) because |y| = 2(m− 1− q)and m− 1− q > −q, so y is in the inductive range. Then the above equation would be of the type we areconsidering, and m > 1 =⇒ m− 1 ≥ 1, which would then contradict the fact that m is minimal. We thusconclude m = 1.

(iv) Finishing the Inductive Step: Since m = 1, we have wqPx− x = ψ1(v) + y · 〈v〉, hence applying i∗

again we have i∗(y · 〈v〉) = y · i∗(〈v〉) = p · y since the leading term of 〈v〉 is p, and we have i∗Px = xp by3.4.7. Therefore

ψ1(0) + py =

−x if q > 0xp − x if q = 0

In the q > 0 case, we thus have

U−2q(X)(p) ⊂ R−2q(p) + pU−2q(X)(p) ⊂ R−2q

(p) + p2U−2q(X)(p) ⊂ · · ·

then since U−2q(X) is a finitely generated abelian group by 5.2.1, pn eventually kills U−2q(X)(p), and we haveU−2q(X)(p) ⊂ R−2q

(p) , thus completing induction for q > 0, since the inclusion in the other direction is obvious.In the q = 0 case, using the same argument, we can reduce to showing that U0(X)(p) ⊂ R0

(p) + pU0(X)(p).The function x 7→ xp − x on U0(X)(p) lands in R0

(p) + pU0(X)(p), and it descends to a function on thequotient

U0(X)(p)

pU0(X)(p)→

R0(p) + pU0(X)(p)

pU0(X)(p)

Then since U0(X) is a nilpotent ideal by 5.2.4, x 7→ xp is a nilpotent endomorphism of U0(X)(p)/pU0(X)(p),hence there exists N >> 0 such that xpN

= 0 for all x. Thus applying x 7→ xp − x N times, we havexpN − xpN−1

+ · · · , the sum lying in S := (R0(p) + pU0(X)(p))/pU0(X)(p), but since xpN

= 0 and the terms

after −xpN−1 all lie in S as well since they are in the image of x 7→ xp − x, we thus have xpN−1 ∈ S.Arguing similarly in the next term in the sum, we see that xpN−2 ∈ S and so on, so that x ∈ S. Thusx ∈ R0

(p) + pU0(X)(p) up to a term in pU0(X)(p), so x ∈ R0(p) + pU0(X)(p). This completes the induction

and the proof of the theorem.

Corollary 5.2.6. The map L → MU∗(∗) = π∗(MU) corresponding to the formal group law of 4.2.2 is asurjection.

Proof : The image of the above map is the ring C, and 5.2.5 says MU∗(∗) = C · MU0(∗) since MUq(∗)vanishes for q > 0. C ⊂ C ·MU0(∗) since 1 ∈ MU0(∗), and U0(∗) ∼= Z since the manifolds of dimension0 are the finite discrete sets. Therefore MU0(∗) is generated by 1, so we have C ·MU0(∗) ⊂ C, and thusMU∗(∗) = C. Since C is concentrated in even degrees, we also see that MUev(∗) = C and MUodd(∗) = 0.

30

We state two more corollaries of 5.2.5 that use duality in MU to bound the degrees of generators ofMU∗(X) for a finite CW complex X. These are not of much interest to this exposition, and the proofs in [7]are clear, so we omit them here.

Corollary 5.2.7. Let X be a finite CW complex that can be embedded into an almost complex manifoldM of dimension n. Then MU∗(X) is generated as a MU∗(∗)-module by elements of degree ≤ n, and byelements of degree < n if none of the components of M are compact.

Corollary 5.2.8. If X is a finite CW complex of dimension r, then MU∗(X) is generated as an MU∗(∗)-module by elements of degree ≤ 2r.

5.3. Proof of Quillen’s Theorem.

With 5.2.5, we are already very close to proving Quillen’s theorem: we know the map f : L→ π∗(MU)from 4.2.3 is a surjection. To prove it is an injection, we will need to know that the Lazard ring is torsionfree. In fact we have the following purely algebraic result about the Lazard ring.

Proposition 5.3.1. (Lazard’s Theorem) The Lazard ring L is a polynomial ring over Z with a generatorin degree q for each q > 0. [3]

The basic idea of our proof that f is an injection will be to use our Landweber-Novikov operations tobuild a ring map π∗(MU) → R for an easier ring R and show that the composition L → π∗(MU) → R isan injection by showing it is an injection after tensoring with Q, and then using the fact that L is torsionfree.

As we noted in the construction of our total Chern classes cα, since H∗(CP∞) ∼= Z[x] and MU∗(CP∞) ∼=Z[x], the construction of the cα can be carried out identically in H∗. In fact since by construction the ringmap φ : MU → HZ of 2.7 sends the class x ∈ MU

2(CP∞) to x ∈ HZ

2(CP∞), φ carries the classes cU

α tocH

α . We thus have a ring map φ : U∗(X) → H∗(X) for a manifold X that preserves the cα′s. We define the

following map:β : U∗(X)

st−→ U∗(X)[t1, t2, . . .]φ−→ H∗(X)[t1, t2, . . .]

where st is the Landweber-Novikov operation. Then β is a ring homomorphism since st and φ are.

Lemma 5.3.2. If L is a line complex line bundle, then

β(eU(L)) = ∑j≥0

tj(eH(L))j+1, t0 = 1

Proof : Let i be the zero section of L, and let i be homotopic and transverse to i. Then, as in 3.1.2, we mayfactor i through itself, and then st(i∗(1)) = i∗(ct(νi)) and νi = L, and similarly we may factor eU(L) = i∗i∗(1)through i∗L and the normal bundle is just i∗L. We have

st(eU(L)) = eU(L)∗(ct(i∗L))

= eU(L)∗

(∑j≥0

tji∗((eU(L))j)

)= ∑

j≥0tji∗i∗(i∗((eU(L))j))

= ∑j≥0

tji∗(i∗(1)) · (eU(L))j

= ∑j≥0

tj(eU(L))j+1

and t0 = 1. In line 2 we used homotopy invariance, the fact that i∗ is a ring homomorphism, and the formulafor the total Chern class of a line bundle given in 3.4.2. In line 3 we used that the st are defined so that

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eU(L)∗ = i∗i∗ moves past the indeterminates, and line 4 is direct application of 3.1.6. Finally since φ is aring homomorphism and it sends cU

α 7→ cHα , it therefore sends eU 7→ eH, the result follows.

For line bundles L1, L2, setting θ(x) to be the power series ∑j≥0

tjxj+1, we plug in L1 ⊗ L2 into 5.3.2 and

we have(βF)(θ(eH(L1)), θ(eH(L2))) = ∑

i,jβ(cij)θ(eH(L1))

iθ(eH(L2))j

= β(F(eU(L1), eU(L2)))

= β(eU(L1 ⊗ L2))

= θ(eH(L1 ⊗ L2))

= θ(eH(L1) + eH(L2))

using the fact that Euler classes of tensor products add in ordinary cohomology. Thus the formal group lawβF satisfies (βF)(θ(x), θ(y)) = θ(x + y). Then since θ has constant term zero and degree 1 term 1, it has acompositional inverse θ−1, and we find

(βF)(x, y) = (βF)(θ(θ−1(x)), θ(θ−1(y))) = θ(θ−1(x) + θ−1(y))

and for a formal group law G and a compositionally invertible power series f , we define f ∗G, the conjugationof G by f to be f−1G( f (x), f (y)), so that βF = θ−1∗Ga, where Ga(x, y) = x + y is the additive formal grouplaw. We therefore have a ring homomorphism

Lf−→ U∗(∗) β−→ H∗(∗)[t1, t2, . . .] ∼= Z[t1, t2, . . .]

FUniv 7→ F 7−→ θ−1∗Ga

where when we say a formal group law is sent to another, we mean that the coefficients of one are sent tothose of the other.

Theorem 5.3.3. (Quillen’s Theorem) The above composition is an injection, and therefore f is an isomor-phism. It follows also by Lazard’s theorem that π∗(MU) is a polynomial ring over Z with one generator ofdegree −2q for each q > 0, since f (Lq) ⊂ U−2q(∗) as |cij| = 2− 2i− 2j.

Proof : Since L is torsion free by 5.3.1, it suffices to show that Q⊗ (β f ) is an injection. Consider the naturaltransformation induced by β f :

HomCRing(Z[t1, t2, . . .],−) (β f )∗−−−→ HomCRing(L,−)Plugging in some ring R, we have that ring maps u : Z[t1, t2, . . .] → R are in bijection with power seriesθu(x) = ∑ u(tj)xj+1, and ring maps L → R are in bijection with formal group laws over R. Under theseidentifications and (β f )∗, θu is sent to the formal group law over R given by θ−1

u∗Ga. It is a general fact

that for each formal group law G over a Q-algebra R, there is a unique power series logG(x) over R - calledthe logarithm of G by analogy with the power series log and exp - such that G = log∗G Ga [3]. Hence thismap is an isomorphism for Q-algebras R. Tensoring up with Q gives the adjunction

HomQ−Alg(Q[t1, t2, . . .], R) ∼= HomCRing(Z[t1, t2, . . .], R)

when R is a Q-algebra, and we have a similar statement for L. We thus have an isomorphism of functorsHomQ−Alg(Q[t1, t2, . . .],−)→ HomQ−Alg(Q⊗ L,−)

and it is induced by Q⊗ (β f ), hence by the Yoneda lemma, Q⊗ (β f ) is an isomorphism.

Remark 5.3.4. This result is powerful because if E is a ring spectrum and there is a ring map φ : MU → E(i.e. E is a complex-oriented cohomology theory), then there is a formal group law over the ring π∗(E)coming from tensor products of line bundles, just as in π∗(MU), and the map φ∗ : π∗(MU) → π∗(E)carries the formal group law over π∗(MU) to the one over π∗(E). Then 5.3.3 says that this map is just the

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map from the Lazard ring corresponding to the formal group law on π∗(E).

Remark 5.3.5. Quillen proves a similar result at the end of his paper for unoriented cobordism - i.e.π∗(MO). In fact, MO∗(X) can be identified with the ring of cobordism classes of smooth proper mapsf : Z → X - there is no need to add any conditions to this since f admits a stable normal bundle, and itdoesn’t need to have any additional structure for Thom’s theorem to go through. Then the arguments ofthis paper can be carried out in identical fashion, except that they become much simpler because one onlyneeds the power operations for p = 2 since everything is in characteristic 2. The proof of the main theoremthen goes through unchanged, except that the py term vanishes since the tensor square of a real line bundleis trivial, so that [2]F(x) = 0, where F is the analogous formal group law coming from tensor products of linebundles. Quillen then uses the fact that any formal group law in characteristic 2 with a vanishing 2 serieshas a canonical logarithm to prove the following result:

Theorem 5.3.6. If Λ is the ring corepresenting formal group laws over commutative rings of characterstic2 with a vanishing 2 series, the corresponding map Λ→ π∗(MO) is a ring isomorphism, and moreover thereis an isomorphism of π∗(MO) algebras

π∗(MO)⊗Z/2 H∗(X; Z/2)→ MO∗(X)

for X a finite CW complex.

Appendix A. Comments on Quillen’s Paper

Other than including details left out in Quillen’s paper, there are few places where this paper differsfrom Quillen’s, so that the reader who would like to understand Quillen’s paper may use this as a guide.There are, however, a few differences, which I think have made the arguments easier to understand. Thekey differences are as follows:

(1) My definition of complex oriented map is much simpler in that it considers only factorizations throughtrivial bundles and all the isotopy information is wrapped up in saying that we choose a single classin [Z, BU] for the normal bundle. These definitions are of course equivalent.

(2) I do not define a virtual bundle ν f as he does for the purposes of defining the Landweber-Novikovoperations, because it is not necessary since we can always take a complex oriented map f to factorthrough a trivial bundle, as in my definition of a complex orientation.

(3) I distinguish between the theories U∗ on manifolds and MU∗ on spaces. For Quillen these are thesame and it is understood that when applied to a manifold, one can use the geometric model.

(4) Quillen uses h∗-theories, which could be understood as a geometric version of complex orientedcohomology theories. However, all the ones used in the paper are U∗(−), U∗(Q ×Z/p −) andU∗(B×−), and I think it is easier to understand the arguments working with these directly, so Idon’t use this terminology.

Quillen’s paper is very important for a number of reasons - it is much more than a novel proof ofQuillen’s theorem. His geometric construction of complex cobordism in this paper has since been general-ized to other cobordism theories and used in other contexts. The main theorem 5.2.5 was a new result thatwas totally unexpected at the time. The paper was ahead of its time with respect to power operations aswell. The formula 4.2.6 can be shown to be of a standard form in the theory of power operations, followingthe method of defining these as in the introduction and looking at the Z/p Tate construction applied to E.The fact that Quillen derived this by hand is astounding, and his formula is also a significant improvementon this general formula in the case of E = MU, since he finds for instance that m = 1 in 5.2.5 (ii). We referthe reader to [8] for more details.

Appendix B. The Proof of Thom’s Theorem

We now restate and give the proof of Theorem 3.1.7. It closely follows the proof of Thom’s theorem,but requires a bit more detail to handle the complex structures.

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Theorem 3.1.7. Regarding MU∗ and U∗ as functors from the category of smooth manifolds to the categoryof graded rings, there is a natural isomorphism U∗ → MU∗. For any manifold X, U∗(X) has the structureof a graded algebra over U∗(∗), and similarly for MU∗(X). Under the isomorphism U∗(∗) → MU∗(∗),the ring isomorphism U∗(X) → MU∗(X) is an isomorphism of graded U∗(∗)-algebras. If A is a strongdeformation retract of an open neighborhood U in X, we may similarly identify

U∗(X, X− A) := Complex-oriented maps f : Z → X : f (Z) ⊂ A

with MU∗(X, X− A).

Proof : We first prove the result for X a compact manifold, and explain briefly how this generalizes to anarbitrary manifold. When X is compact, since manifolds are Hausdorff, a complex oriented map f : Z → Xbeing proper just amounts to Z and X being compact and f smooth.

(i) The Map: Let X be a compact smooth manifold and ( f : Z → X, ν) ∈ U2q(X). Then f factorsthrough an embedding i : Z → X×Cn and νi has a complex structure so that the class of νi in [Z, BU] is ν.By the tubular neighborhood theorem, we have a diagram of embeddings

νi

Z X×Cni

Using Thom’s method as in the case of a point, noticing that (X × Cn)+ ∼= X+ ∧ S2n, we obtain a mapS2n ∧ X+ → ν+i

∼= Thom(νi) → Thom(EU(n + q)) = MU(n + q), and taking its class in [Σ∞+X, MU]−2q

gives us a map Φq : U2q(X) → MU2q(X). We need to check that this map does not depend on the choiceof νi ∈ ν nor the choice of representative of the cobordism class of ( f , ν).

(ii) Well-Defined: If we chose a different factorization of f through an embedding, say j : Z → X×Cm,to say that νj ∈ [νi] ∈ [Z, BU] is to say that the normal bundles νi and νj are stably isomorphic. Therefore,there exist k1, k2 with νi ⊕ k1

∼= νj ⊕ k2, and we set N := n + k1 = m + k2. We note that the normal bundleof the embedding

l1 : Z i−→ X×Cn idX×0−−−→ (X×Cn)×Ck1

is νi ⊕ k1 and similarly for j and k2, we may define l2. If we apply Φq to ( f , ν) where we represent ( f , ν) withthe factorization l1, the tubular neighborhood map νi ⊕ k1 → X ×Cn ×Ck1 is just the one from before onthe left and the identity on the right, and the resulting map S2k1 ∧ S2n ∧ X+ → S2k1 ∧Thom(νi) is just the2k1-fold suspension of the one from before, hence they produce the same class in [Σ∞

+X, MU]−2q. We canmake the same remarks about j and l1, and it thus suffices to show that the factorizations of f through l1and l2, respectively, determine the same class in [Σ∞

+X, MU]. But since νl1 and νl2 are isomorphic as vectorbundles, the corresponding maps Z → BU(N) are homotopic, hence so are the maps Thom(νli )→ MU(N).It would thus suffice to know that the two tubular neighborhood maps νi⊕ k ∼= νj⊕ k→ X×CN are isotopic,but we can arrange this by taking N as large as necessary.

Now let ( f0, ν0), ( f1, ν1) ∈ U2q(X) be cobordant, then we have a complex oriented map h : W →X × [0, 1] with the maps εi : X → X × [0, 1] both transverse to h, so that the respective pullbacks yield( f0, ν0) and ( f1, ν1). Fix an embedding W → X × [0, 1]× CN, and let ν be its normal bundle. Then νi isthe pullback of ν along Zi → W coming from pulling back h along εi, hence we can pull back the tubularneighborhood diagram for ν along εi since the above isotopy argument shows that there is no dependence

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on the choice of tubular neighborhoodν

W X× [0, 1]×CN

νi

Zi X×CN

(idX ,i,idCN )

Applying compactification to the back right square gives us a commutative diagram

S2n ∧ I+ ∧ X+ Thom(ν) MU(N + q)

S2n ∧ ∗ t i ∧ X+ Thom(νi) MU(N + q)

=

The left square commutes because the back right square in the above diagram commutes, and the rightsquare commutes because νi is the pullback of ν. The commutativity of this diagram tells us that we have amap S2n ∧ I+ ∧ X+ → MU(N + q) that agrees on the endpoint i with the map we obtain from the tubularneighborhood for νi. This data amounts to a pointed homotopy between the two maps obtained from Φq.

(iii) Natural: Let ( f , ν) ∈ U2q(X) and g : Y → X with g transverse to f , we may form the pullbackg∗( f , ν) ∈ U2q(Y). Represent ν by the normal bundle of a factorization of f as Z → X × Cn → X,then pulling everything back along g gives us the pullback map of f along g and ν′ as the pullback of ν,representing the class of g∗( f , ν). We would like to say that the following diagram commutes

Y+ ∧ S2n Thom(ν′) MU(n + q)

X+ ∧ S2n Thom(ν) MU(n + q)

=

The right square commutes since ν′ is the pullback of ν, it thus suffices to know the left square commutes,and for that it suffices to know the square

ν′ ν

Y×Cn X×Cng×1

after replacing the vertical arrows with ones that are isotopic. But we showed that the choice of tubularneighborhood only depends on the isotopy class after taking n sufficiently large, so as above we just pullback any tubular neighborhood for ν, and the above square commutes.

(iv) Additive: Recall that the sum of ( f , ν), ( f ′, ν′) ∈ U2q(X) is defined as ( f t f ′, ν t ν′). We factorf and f ′ through embeddings i : Z → X × Cn and i′ : Z′ → X × Cn′ , and take their sum Z t Z′ →X×Cn+n′ , the normal bundle of which is (νi ⊕ n′) t (νi′ ⊕ n). The one-point compactification of this spaceis (S2n′ ∧ ν+i ) ∨ (S2n ∧ ν+i′ ), hence the map given by Φq is

S2n+2n′ ∧ X+ → (S2n′ ∧ ν+i ) ∨ (S2n ∧ ν+i′ )→ MU(n + n′ + q)

but this is the definition of Φq( f , ν) + Φq( f ′, ν′) using the H-cogroup structure on spheres to define additionin [Σ∞

+X, MU]−2q.

(v) Multiplicative and U∗(∗)-linear : Let ( f : Z → X, ν) ∈ U2q(X) and (g : Y → X, τ) ∈ U2p(X),their product is given by ∆∗( f × g, ν× τ). Since Φq is natural, it suffices to show that µMU (Φq( f , ν) ∧Φq(g, τ)) = Φq( f × g, ν× τ), where µMU : MU ×MU → MU is the ring map. This is almost tautological

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as the class ν× τ is given by Z × Y ν×τ−−→ BU × BU → BU, where the last map is induced on the colimitBU = colimn→∞BU(n) by the direct sum maps BU(m)× BU(n) → BU(n + m), and the ring map MU ×MU → MU is induced by the maps MU(n) ∧MU(n)→ MU(n + m) obtained from Thomifying the directsum maps. The multiplicative unit in [Σ∞

+X, MU] is the Thomification of the map classifying the trivialbundle over X, so it is clear that Φq is unital.

U∗(∗)-linearity is similarly tautological: the product of ( f , ν) ∈ U2p(∗) on (g, τ) ∈ U2q(X) is( f × g, ν × τ), which Φq sends to S2(n+m) ∧ (∗ × X)+ ∼= S2(n+m) ∧ S0 ∧ X+ → Thom(ν) ∧ Thom(τ) →MU(n+ p)∧MU(m+ q)→ MU(m+n+ p+ q) where we have factored f through an embedding into ∗×Cn

and g through an embedding into X ×Cm. But the product of the classes of the maps S2n ∧ (∗)+ = S2n ∧S0 → Thom(ν)→ MU(n + p) ∈ [S, MU]−2p and S2m ∧X+ → Thom(τ)→ MU(m + q) ∈ [Σ∞

+X, MU]−2q isgiven by smashing the maps together and identifying S∧Σ∞

+X ∼= Σ∞+X, then using the ring structure on MU.

(iv) The Inverse: Let f ∈ MU2q(X) = colimn→∞[S2n ∧ X+, MU(r + q)] and represent f by a mapf : S2n ∧ X+ → MU(n + q). Since the space Snr ∧ X+ is compact, it lands in some (LN

n+q)+ where

LNn+q → Grn+q(CN) is the tautological bundle. (LN

n+q)+ is a smooth manifold away from the point at

infinity and the zero section Grn+q(CN) → LNn+q is proper. In particular, we can replace f with a map

homotopic to f that is transverse to the zero section Grn+q(CN) → LNn+q → (LN

n+q)+, which stays away

from the point at infinity [4]. The pullback square

Z Grn+q(CN)

S2n ∧ X+ (LNn+q)

+f

thus defines a smooth proper map f : Z → X. Since the zero section on the right stays away from thepoint at infinity, so does the map from Z, hence we may regard it as a map Z → X ×Cn, and composingwith projection to X, we obtain a complex oriented map. We check that it is well-defined with respect tosuspension and homotopy.

If we suspend f , we have the following diagram

Grn+q(CN) Grn+q+1(CN)

S2 ∧ (LNn+q)

+ (LNn+q+1)

+

BU(n + q) BU(n + q + 1)

S2 ∧ S2n ∧ X+ S2 ∧MU(n + q) MU(n + q + 1)

using the definition of the structure maps in the spectrum MU, identifying Thom(LNn+q⊕ 1) ∼= S2 ∧ (LN

n+q)+.

All of the squares in the commutative cube are pullbacks, hence pulling back from Grn+q+1(CN) →

(LNn+q+1)

+ will be the same as pulling back from Grn+q(CN) → S2 ∧ (LNn+q+1)

+. Since the map S2 ∧ S2n ∧X+ → S2 ∧ (LN

n+q)+ is the identity on the left factor, we can choose the same map as before and its suspen-

sion will give us a transverse map. But then the pullback is easily seen to be Z → S2n ∧X+ → S2 ∧ S2n ∧X+,hence we have the same map, with the complex orientation given by the direct sum of the normal bundlewith a trivial bundle, which gives the same complex orientation. By similar reasoning, we see that our mapdoes not depend on the choice of N.

If fi : S2n ∧ X+ → MU(n + q) for i = 0, 1 are homotopic, we have a map H : S2n ∧ X+ ∧ I+ →MU(n + q) which lands in some LN

n+q such that on the i-th endpoint it is equal to fi. We can replace each fi

by a map that is transverse to the zero section Grn+k(CN)→ LN

n+q, and then choose a map homotopic to H36

that is transverse to the zero section and matches the transverse replacements of fi on the endpoints. Thisfollows from the general fact that if M is a compact manifold and f : M → N is a map with ∂ f transverseto some map Z → N, then one can replace f up to homotopy by a map that is transverse to Z → N thatagrees with f on the boundary [4]. We thus form the pullback square

W Grn+k(CN)

S2n ∧ X+ ∧ I+ LNn+q

H

and obtain a map W → X × Cn × I → X × I, and the fact that H agrees with the fi on the endpointsmeans that when we pull back W → X × I along εi we get the complex oriented map obtained from fi.This shows that the map is well-defined with respect to homotopy and does not depend on the choice oftransverse replacement of f , since all such maps are homotopic.

The map we have described is an inverse to Φq because if we take f : S2n ∧ X+ → MU(n + q) andrestrict to S2n ∧X+ → LN

n+q for some N, pulling back Grn+q(CN) → LNn+q gives a map Z → X×CN → X,

and we take the tubular neighborhood diagram for the normal bundle of Z → X×CN to be the one pulledback from the one for the zero section, which we can take to be

LNn+q

Grn+k(CN) LN

n+q

=

since taking the normal bundle of a zero section always recovers the vector bundle. We thus have a diagramν

X×Cn LNn+q

where the vertical arrow is the pulled-back tubular neighborhood, the diagonal arrow is the upstairs map inthe pullback square classifying ν, and the horizontal arrow is the restriction of f to the complements of thepoints at infinity. But then since Φq sends this data to the compactification of the vertical arrow, followedby the Thomification of the diagonal arrow, the commutativity of the diagram tells us we recover f .

Going the other way, Φq( f : Z → X, ν) is the map S2n ∧ X+ → ν+ → LNn+q, but in the diagram

Z Z Grn+q(CN)

S2n ∧ X+ ν+ LNn+q

the bottom maps are already transverse to the corresponding vertical maps, and the right square is a pullbackbecause the zero section pulls back to the zero section, and the left is a pullback because it is obtained fromthe tubular neighborhood commutative diagram. Hence the complex oriented map obtained from Φq( f , ν)is just the left vertical arrow, which is ( f , ν).

(v) Odd-dimensional Maps: All of our remarks about complex oriented maps of even dimension carryover to maps of odd dimension, since by definition, a complex orientation on a map Z → X of odd dimensionis a complex orientation on the map of even dimension Z → X×R.

(vi) Non-compact Manifolds: The above proof can be modified to a noncompact manifold X by notingthat we can write noncompact X = colimαXα as a colimit of compact subspaces Xα. Then since f is assumedproper, we set Zα := f−1(Xα) ⊂ Z, which is compact and hence the Thom space of να := νi|Zα is ν+α . Byrestricting the tubular neighborhood corresponding to ν to να, we get a map S2n ∧ Xα+ → Thom(να) foreach α. Then since Thom(ν) = colimαThom(να), we get a map between the colimits. The only other place

37

where compactness is used is to claim that any map φ : S2n ∧ X+ → MU(n + q) lands in some (LNn+q)

+.This claim is modified similarly - write X as the colimit of the Xα’s, this time choosing the Xα’s to be anascending union of compact submanifolds, then we choose a map homotopic to φ whose restriction φ|S2n∧Xα+

is transverse to the zero section of the LNn+q that it lands in, for each α. Then taking the union of the

preimages gives a manifold with a closed embedding into X×Cn.

(vii) The Relative Case: When A → X is an open inclusion, the functoriality of one-point compact-ification comes from the fact that one has a homeomorphism X+/(X+ − A) → A+. Thus, if Z → X is acomplex oriented map whose image is contained in such an A, the tubular neighborhood embedding factors asν → A×Cn → X×Cn, and when we apply one-point compactification, we get a map X+/(X+ − A)→ ν+

and following that map with the usual one into MU, we obtain an element of MU∗(X, X − A). Runningthrough the above argument with slight modifications shows that we may identify MU∗(X, X− A) with theset of cobordism classes of complex oriented maps into X whose images are contained in A, which we willdenote U∗(X, X− A).

If A is a strong deformation retract of an open neighborhood U in X, then we may replace any com-plex oriented map into X whose image is contained in A, with the one whose image is contained in U,and the homotopy invariance of U∗ tells us these will be isomorphic. We conclude that we may identifyMU∗(X, X− A) and U∗(X, X− A) when A has this property.

References[1] J. Frank Adams, Stable homotopy and generalised homology, University of Chicago Press, 1974.[2] A Bojanowski and S Jackowski, Geometric bordism and cobordism.[3] A Frohlich, Formal groups, Springer-Verlag, 1968.[4] Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, 1974.[5] Jacob Lurie, Chromatic homotopy theory.[6] John Willard Milnor and James Stasheff, Characteristic classes, Princeton University Press, 1974.[7] Daniel Quillen, Elementary proofs of some results of cobordism theory using steenrod operations, Advances in Mathematics

(1971), no. 7, 29–56.[8] Yuli B. Rudyak, On thom spectra, orientability, and cobordism, Springer, 1998.[9] Norman Steenrod, Cohomology operations, Princeton University Press, 1962.

Currier House, Harvard College, Cambridge, MA 02138E-mail address: christiancarrick@college.harvard.edu

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