On the cohomology of spaces

associated to

Davis-Januszkiewicz spaces

Rik Danko

MSc Thesis

under supervision of

Dr. D.R.A.W. Notbohm (Vrije Universiteit)

Universiteit van Amsterdam

On the cohomology of spaces associatedto Davis-Januszkiewicz spaces

MSc Thesis

Author: Rik Danko

Supervisor: Dr. D.R.A.W. Notbohm (Vrije Universiteit)

Second reader: Prof.Dr. E. M. Opdam (Universiteit van Amsterdam)

Member of examination board: Dr. J.V. Stokman (Universiteit vanAmsterdam)

Date: 16/12/2012

Abstract

In this thesis we construct an n-dimensional complex vector bundleξ ↓ DJ(K) over a Davis-Januszkiewicz -space DJ(K). We will de-termine cohomological properties of the homotopy �ber XK of theclassifying map DJ(K) → BU(n). The main theorem states thatthe cohomology of XK is zero in degrees greater then n2 + n andthat it is equal to Z in degree n2 + n, where n− 1 is the dimensionof the �nite abstract simplicial complex K. The main theorem alsotells us more about the relations between faces of K and elementsin H∗(XK).

rikdanko@hotmail.com

Korteweg-de Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdam

Contents

1 Introduction 1

2 Toric Spaces and Complexes 5

2.1 The Borel construction . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Davis-Januszkiewicz spaces and moment angle complexes . . . 6

3 Vector Bundles 8

3.1 Complex Vector Bundles . . . . . . . . . . . . . . . . . . . . . 8

3.2 The universal bundle EU(n)→ BU(n) . . . . . . . . . . . . . 10

3.3 A vector bundle over DJ(K) . . . . . . . . . . . . . . . . . . . 11

3.4 Splitting of a trivial part of λ . . . . . . . . . . . . . . . . . . 13

4 Fibrations and Spectral Sequences 16

4.1 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 Poincaré Series . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Cohomological properties of XK 22

5.1 The homotopy equivalence of U(m)/Tm and X∂∆[m] . . . . . . 22

5.2 The Stanley Reisner algebra and the elementary symmetricpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 K a polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.4 Polynomial relations between maximal faces . . . . . . . . . . 27

5.5 The highest non-zero cohomology group of XK . . . . . . . . . 30

ii

Chapter 1

Introduction

In March 1991 Davis and Januskiewicz published an article [5], in which theystudied certain toric spaces. They combined algebraic topology with combi-natorics and symplectic geometry to create a new �eld of mathematics whichis now known as Toric Topology. To each �nite abstract simplicial complex KDavis and Januszkiewicz assigned a family of homotopy equivalent spaces, forwhich a generic model shall be denoted by DJ(K). The cohomology ring ofthese Davis-Januskiewicz spaces is the Stanley-Reisner algebra R[K], whichis a quotient of a polynomial algebra in m generators, where m is the num-ber of vertices of K. A construction of this algebra will be given later in thischapter.

In section 6 of the mentioned paper, Davis and Januszkiewicz constructeda vector bundle λ over DJ(K) with Chern classes the elementary symmetricpolynomials in the generators of the Stanley Reisner algebra. The vectorbundle λ has dimension m, with m the number of vertices of K. This vectorbundle λ has some interesting properties. For example, if K is the dual of asimple polytope and we apply the Borel construction to the tangent bundle ofZK , a certain moment angle complex associated to K, we get a vector bundleτ stably isomorphic (i.e. isomorphic up to the sum of a trivial bundle) to thereali�cation of λ. The proofs of all these statements can be found in thementioned paper.

Several applications of Toric Topology can be found. The structure of thecohomology of ZK is closely related to the so called g-conjecture. By de-terming the cohomology of ZK for K the dual of a simple polytope K = KP ,Stanly managed to proof the g-conjecture for K the abstract dual of a sim-ple polytope. However, for the construction of the moment angle complexZK associated to K, one needs to choose a certain matrix A. This makesthe computations much harder. Later, Notbohm created an n-dimensional

1

vector bundle λ over DJ(K) in [12], where n is the dimension of K, sta-bly isomorphic to λ, i.e. λ ∼= ξ ⊕ Cm−n. Notbohm looked at the homotopy�ber XK of the classifying map of the vector bundle ξ. For the space XK

there is no choice of a matrix necessary, so in this sense the space XK comesmore natural. By studying the cohomology of XK , there is hope in provingthe g-conjecture for more general K. In this thesis we are interested in co-homological properties of XK . We give alternative, more elementary, proofsfor some of the Theorems proved in [12]. For more information about ToricTopology, we refer to [4] for a survey article.

To make our statements more exact, we need a little background informa-tion. Let [m] := {1, ...,m} ⊂ N be the set of the �rst m natural numbers.A �nite abstract simplicial complex K is a collection of subsets of [m] thatis closed under inclusions, that is, if β ∈ K and α ⊆ β, then α ∈ K. Notethat if K is not the empty set, then ∅ ∈ K. The elements of K are called thefaces of K, while the subsets of [m] that are not contained in K are calledthe missing faces of K. For i ∈ [m], we call {i} a vertex of K. The dimensionof a face α of K is de�ned by dim(α) = |α| − 1. The dimension of K isthe maximum of the dimensions of its faces. Unless otherwise stated, we willalways let K denote a �nite abstract simplicial complex with m vertices anddim(K) = n− 1.

In chapter 5 we will determine some cohomological properties of XK . Inone of the proofs we shall use induction on the dimension of a �nite abstractsimplicial complex K. We will use the so called link and star of a face α ofK:

De�nition 1.0.1. Let K be a �nite abstract simplicial complex and letα ∈ K

(i) stK(α) := {β ∈ K|α ∪ β ∈ K} is called the star of α in K.

(ii) linkK(α) := {β ∈ K|α ∪ β ∈ K,α ∩ β = ∅} is called the link of alphain K.

We also need the following elementary lemma:

Lemma 1.0.2. If K = KP for an n-dimensional simple polytope P and {i} avertex of K, then linkK({i}) = KFi, where Fi is the facet of P correspondingto {i}.

To each simple polytope P , we can assign a �nite abstract simplicial com-plex KP dual to P . Let P be a simple polytope with facets {F1, ..., Fm}. The

2

�nite abstract simplicial complex KP dual to P is de�ned by

KP = {σ = {i1, ..., ir} ⊆ [m] :r⋂j=1

Fij 6= ∅}

By R we denote a commutative ring with unit and we de�ne R[m] :=R[v1, ..., vm], the polynomial algebra in m generators, with the topologicalgrading deg(vi) = 2. For α ∈ K, we write vα :=

∏i∈α vi. One of our main

tools in the study of simplicial complexes is the Stanley-Reisner algebra,which is an algebra associated to a �nite abstract simplicial complex K thatcarries all the combinatorial information of K. Let IK be the ideal of R[m]generated by the set {vα : α /∈ K}. We call this ideal the Stanley-Reisnerideal.

De�nition 1.0.3. Let K be a fasc. To K we associate an algebra R[K],called the Stanley-Reisner algebra, de�ned by

R[K] := R[m]/IK

Now we have enough background information to state the main Theoremof this thesis.

Theorem 1.0.4. Let K be a �nite abstract simplicial complex dual to asimple polytope P , i.e. K = KP . Let the dimension of K be n − 1 and letK have m vertices. Furthermore, let L be a maximal face of K and let {ij}denote the vertices of L and σ a permutation in n variables. Then

(i)

H i(XK) =

{Z if i = n2 + n;0 if i > n2 + n,

(ii)v1i1· ... · vnin 6= 0.

is a generator for H∗(XK).

(iii)v1i1· ... · vnin = sign(σ)v1

iσ(1)· ... · vniσ(n) .

Corollary 1.0.5. For all maximal faces {i1, ..., in} and {j1, ..., jn}, we have

v1i1· ... · vnin = ±v1

j1· ... · vnjn

3

In chapter 2 we de�ne some toric spaces in which we are interested. Inparticular, we de�ne the Davis-Januszkiewicz space DJ(K). Also, we de�nethe Borel construction of a toric space X. These form the main tools of ourstudy.

In chapter 3 we will explain more about vector bundles. We construct theuniversal complex n-plane bundle γn ↓ BU(n) and the vector bundle λ′ overDJ(K), which was studied by Davis and Januszkiewicz in [5]. We will alsosplit of an (m− n)-dimensional part of λ′ to get the vector bundle λ studiedby Notbohm.

In chapter 4 we make a small detour to give more information aboutspectral sequences and �brations. Spectral sequences form our main tool togive us information about the cohomology of the total space, the base spaceand the �bers of a vector bundle. We �nish this chapter with some basic factsabout Poincaré series.

In chapter 5 we explicitely determine XK for K the boundary of a simplex∂∆[m]. We will also show that the cohomology of XK is the Stanly Reisneralgebra modulo the ideal generated by the �rst n symmetric polynomials.Finally, we will use our main theorem, using induction on the dimension ofK.

4

Chapter 2

Toric Spaces and Complexes

2.1 The Borel construction

In this thesis we will investigate the structure of toric spaces, that is, spacesthat carry a T n action. If X is a T n-space, we can look at the orbit spaceX/T n. The problem is that given an orbit spacen Y = X/T n, we can notreconstruct the original space X. This is a consequence of losing to muchinformation over the structure of X when we qutient out T n. Therefore, weusually consider a di�erent construction, called the Borel construction.

If X and Y are T n-spaces, then X×Y is again a T n-space via the diagonalaction. Let us de�ne the Borel construction XhTn of a T n-space X:

XhTn := (X × (S∞)n)/T n

Since S∞ is a contractible space, we see that X × (S∞)n is homotopy equiv-alent to X. On the other hand, since (S∞)n has a free T n-action, we can stillreconstruct X from the orbit space XhTn , i.e.the homotopy �ber (see section4.1) of the projection map XhTn → (S∞)n is equal to X:

X → XhTn → (S∞)n

Without going any deeper into it, we note that the projection map

(S∞)n → (S∞)n/T n = CP (∞)n

is the universal bundle ET n → BT n. Therefore, we shall write ET for (S∞)n

and BT for CP (∞)n. We also note that ET n is the up to homotopy uniquelydetermined contractible space with free T n-action.

5

2.2 Davis-Januszkiewicz spaces and moment

angle complexes

For a pair of spaces (X,A) and α ⊆ [m] we write (X,A)α = A[m]\α ×Xα ⊆X [m], where the last inclusion is coordinate wise. For a �nite abstract sim-plicial complex K, the partial product P (K;X,A) is given by

P (K;X,A) :=⋃α∈K

(X,A)α ⊆ X [m]

Note that we would get the same de�nition if we would only take the unionover the maximal faces of K. A Davis-Januszkiewicz space is an example ofa partial product:

De�nition 2.2.1. For a �nite abstract simplicial comples K we de�ne theDavis-Januszkiewicz space DJ(K) associated to K as

DJ(K) : = P (K;BT, ∗)

=⋃α∈K

BT α ⊆ BT [m],

where the last inclusion is coordinate wise.

It is well known that the integral cohomology of BT ∼= CP (∞) over Z isisomorphic to the polynomial algebra Z[x]. As a consequence of the Künnethformula, we have

H∗(BTm;R) ∼= R[m]

The following theorem is proved in section 6.5 of [3].

Theorem 2.2.2. Let K be a �nite abstract simplicial complex and letπ : R[m] → R[K] denote the canonical projection. Then there exists anisomorphism φ : R[K]

∼−→ H∗(DJ(K)), making the next diagram commute

R[m] R[K]

H∗(BTm) H∗(DJ(K))

π

∼=

q∗K

φ

Another example of a partial product that we will need is the momentangle complex ZK over a �nite abstract simplicial complex K:

6

De�nition 2.2.3. Let K be a �nite abstract simplicial complex. We de�nethe moment angle complex ZK over K by:

ZK : = P (K;D2, S1)

=⋃α∈K

(S1)[m]\α × (D2)α ⊆ (D2)[m]

Note that ZK carries a coordinate wise Tm action. Proposition 6.29 of [3]tells us that the Borel construnction of ZK is homotopy equivalent to DJ(K):

DJ(K) ' (ZK)hTn

7

Chapter 3

Vector Bundles

3.1 Complex Vector Bundles

De�nition 3.1.1. An n-dimensional complex vector bundle ξ = ξ(E,B, p)is a map p : E → B such that

(i) p−1(b) is a complex vectorspace for all b ∈ B.

(ii) ξ is locally trivial, which means there exists an open cover {Uα}A of B,such that for all α ∈ A there exists a homeomorphism hα : p−1(Uα)→Uα×Cn mapping p−1(b) via a vectorspace isomorphism onto {b}×Cn.

Any hα is called a local trivializations of the vector bundle. B is called thebase space, E the total space and Fb := p−1(b) are called the �bers of ξ. Ifdim(Fb) = n for all b ∈ B, we say that ξ is an n-dimensional vector bundle.

A bundle map between two vector bundles ξ and η is a continuous functionbetween the total spaces

f : E(ξ)→ E(η)

such that each �ber Fb is mapped to a �ber Ff(b) by a linear map. Note thatif f : E(ξ) → E(η) is a bundle map, then im(f |B(ξ)) ⊆ B(η) and we writef := f |B(ξ) for the corresponding map between base spaces. We say that thebundle map is a bundle isomorphism if f is a homeomorphism and we thenwrite ξ ∼= η. A bundle homotopy between two bundle maps f, g : ξ → η is acontinuous function:

H : E(ξ)× I → E(η)

8

such thatH|E(ξ)×c : E(ξ)→ E(η) is a bundle map for all c ∈ I andH|E(ξ)×0 =f and H|E(ξ)×1 = g.

Given a vector bundle ξ(E,B, p) and a map f : B′ → B, we de�ne

E ′ = {(b′, v) ∈ B′ × E|f(b′) = p(v)}.

By proposition 1.5 of [7],

p′ : E ′ → B′

(b′, v) 7→ b′

is a vector bundle that �ts into the commutative diagram

E ′ E

B′ B

f

p'

f

p

where f : (b′, v) 7→ v is a bundle map. Moreover, the proposition tells us thatthe vector bundle p′ : E ′ → B′ is unique up to isomorphism. This vectorbundle is called the pull back bundle of f and is denoted by f ∗(ξ).

We can also take the product of two vector bundles.

De�nition 3.1.2. The carthesian product of two vector bundles ξ1 and ξ2 isthe product of π1 and π2:

π1 × π2 : E1 × E2 → B1 ×B2

(π1×π2)−1(b1, b2) = Fb1(ξ1)×Fb2(ξ2) has an obvious complex vector spacestructure and clearly ξ1 × ξ2 is locally trivial.

For n ≥ 0 and k ≥ 0 de�ne the complex Stiefel manifold as

Vn(Cn+k) = {(z1, ..., zn) ∈ Cn+k × ...× Cn+k| < zi, zj >= δij}= U(n+ k)/U(k)

Now let Gn(Cn+k) be the set of all n-dimensional planes of Cn+k throughthe origin. We can topologize Gn(Cn+k) by giving it the quotient topologyof the surjective map f : Vn(Cn+k) → Gn(Cn+k) mapping each n-frame tothe subspace it spanns. With this topology Gn(Cn+k) is called the complexGrassmann manifold, which is a compact topological manifold of dimensionn(n+ k), by Lemma 5.1 of [9]. We de�ne the in�nite Grassmann manifold as

9

Gn(C∞) = ∪∞k=0Gn(Cn+k) with the direct limit topology, where the inclusionsGn(Cn+k) ⊆ Gn(Cn+k+1) come from the inclusions Cn+k ⊆ Cn+k+1.

We now de�ne En(Cn+k) = {(v, x) ∈ Gn(Cn+k) × Cn+k|x ∈ v}, i.e.En(Cn+k) is the space consisting of n-planes v in Cn+k together with avector x in that plane. Again, we take the in�nite dimensional versionEn(C∞) = ∪∞k=0En(Cn+k) with the direct limit topology.

Lemma 1.15 of [7] tells us that the map

γn(Cn+k) : En(Cn+k)→ Gn(Cn+k)

(l, v) 7→ l

is a vector bundle for k �nite and in�nite. We write γn for γn(C∞), E(γn)for En(C∞) and Gn for Gn(C∞).

In �14 of [9] the following theorem is proved:

Theorem 3.1.3. Let B be a paracompact topological space and let ξ : E → Bbe an n-dimensional complex vector bundle. Then there exists a bundle mapf : ξ → γn, such that ξ is isomporphic to the pull back of f . Moreover, ifg : ξ → γn is another bundle map, then g∗(γn) ∼= f ∗(γn) if and only if g ' f .

So any complex n-dimensional vector bundle ξ determines a unique ho-motopy class of maps fξ : B → Gn. A map from this class is called theclassifying map of the vector bundle ξ. The bundle γn ↓ Gn is called theuniversal complex n-plane bundle. The base space Gn is called the classifyingspace for complex n-plane bundles.

3.2 The universal bundle EU(n)→ BU(n)

The total space of universal bundle EU(n)→ BU(n) is

EU(n) = ∪∞k=0U(n+ k)/U(k)

= ∪∞k=0{(z1, ..., zn)|zi ∈ Cn+k, < zi, zj >= δij}.

This space has a U(n)-action via the inclusion U(n) ↪→ U(n+ k), and there-fore we get

E(γn) = EU(n)×U(n) Cn

The orbit space BU(n) = EU(n)/U(n) is equal to

BU(n) = EU(n)/U(n)

= ∪∞k=0U(n+ k)/(U(n)× U(k)

= Gn

10

From now on we will denote the universal n-plane bundle by γn : E(γn) ↓BU(n) : (z1, ..., zn, λ) 7→

∑i λivi.

We have BU(1) = CP∞ = S∞/T = BT . This leads us to the followingexample

Example 3.2.1. The m-fold carthesian product of the universal line bun-dle γ1 ↓ BU(1) is the pull back of the maximal torus inclusion BU(1)m =BTm → BU(m).

This is easily seen to be true, as the maximal torus inclusion BTm →BU(m) maps m orthogonal lines to the plane they span.

3.3 A vector bundle over DJ(K)

In this section we will show that the vector bundle λ ↓ DJ(K) is isomorophicto the pullback of the universal bundle γ ↓ BU(m) along the composition ofDJ(K) → BTm with the maximal torus inclusion BT → BU(m). For thiswe �rst need the notion of a T n-equivariant vector bundles.

De�nition 3.3.1. A T n-equivariant vector bundle is a vector bundle p :E → B with an action of T n on E and on B, such that

(i) p(tv) = tp(v) for all v ∈ E and t ∈ T n.

(ii) The �ber map t : Fb → Ftb is linear for all t ∈ T n and b ∈ B.

De�nition 3.3.2. A G-equivariant bundle map is a bundle map f : E(ξ)→E(η) between G-equivariant vector bundles, such that for all g ∈ G andv ∈ E we have

f(gv) = gf(v)

Example 3.3.3. Let us equip Cm with a Tm-action via coordinate wisemultiplication. This induces a Tm-equivariant complex vector bundle λ′ overZK , via the diagonal action of Tm:

∆ : Tm × Cm × ZK → Cm × ZK ,(t, c, z) 7→ (t(c), t(z))

By [13], the Borel construction λ := λ′hTm is anm-dimensional complex vectorbundle over (ZK)hTm ∼= DJ(K).

Proposition 3.3.4. The vector bundle λ ↓ DJ(K) is isomorphic to thepullback of the universal bundle γm ↓ BU(m) along the composition ofDJ(K)→ BTm with the maximal torus inclusion BTm → BU(m).

11

Proof. Let λ0 ↓ BT denote the vector bundle λ ↓ DJ(K) for K a point.The inclusion ZK ⊆ (D2)m induces the following pullback diagram of vectorbundles

(Cm × ZK)hTm (Cm × (D2)m)hTm ((C×D2)hT )m

(ZK)hTm ((D2)m)hTm ((D2)hT )m

λ λm0 (3.3.1)

where the right two horizontal maps are the natural identi�cations. Theprojection (C×D2)hT → (∗× ∗)hT can be written as the composition of twoprojections, one for each coördinate. Since the order of the projections doesnot matter, we have the following commuting diagram

(C×D2)hT (C× ∗)hT

(∗ ×D2)hT (∗ × ∗)hT

λ0 π

'

where the bottom map is a T -invariant homotopy equivalence. Since

ChT = (C× S∞)/T = E(γ1),

it follows that

π : E(γ1)→ BT = S∞/T = BU(1)

(v, x) 7→ v

i.e. π is the universal plane bundle γ1. So we see that λ∗ is the pullback ofγ1 ↓ BU(1):

(C×D2)hT E(γ1)

BT BU(1)

λ∗ γ1

'

We now have λm0 ↓ BTm = (γ1)m ↓ BTm, which by example 3.2.1 is thepullback of the universal m-plane bundle γm ↓ BU(m), along the maximaltorus inclusion BTm → BU(m).

Thus, by diagram 3.3.1, we see that λ ↓ DJ(K) is the pullback of theuniversal m-plane bundle, along the composition of DJ(K) → BTm withthe maximal torus inclusion BTm → BU(m).

12

3.4 Splitting of a trivial part of λ

Now that we established the vector bundle λ, we will split of a trivial part.Proposition 3.4.5 shows us how the classifying map of this new vector bundlerelates to the classifying map of λ.

Let A ∈ C(m−n)×m be an (m − n) × m-matrix. For any face α ∈ K, wede�ne Am−α ∈ C(m−n)×(m−|α|) to be the submatrix consisting of the columnsof A indexed by the elements not in α. We call A K-admissible, if Am−α isan epimorphism for all faces α ∈ K.

Example 3.4.1. For A we could take the Vandermonde matrix given byAij = ai−1

j , for some distinct integers ai > 0: a01 . . . a0

m...

...

a(m−n)−11 . . . a

(m−n)−1m

A Vandermonde matrix A ∈ Cm×n with m ≤ n, has rank m if and only ifall ai are distinct. Since m − n < m, we have that A ∈ C(m−n)×m has rank(m−n), if and only if all the ai are distinct. For any face α ∈ K, the matrixAm−α ∈ C(m−n)×(m−|α|) is again a Vandermonde matrix, and since |α| ≤ n,we have m − n ≤ m − |α|, so we see that Aα is indeed an epimorphism forall faces α ∈ K.

Let A be the Vandermonde matrix with ai = i. De�ne the bundle map

fK : Cm × ZK → Cm−n × ZK(x, z) 7→ (A(xz), z),

where xz denotes the coordinate wise product of x and z, with z the coör-dinate wise complex conjugate of z ∈ ZK ⊂ Cm. Now let E(m − n) :=Cm−n × ZK be the total space of the (m − n)-dimensional Tm-bundleCm−n × ZK ↓ ZK , where Tm acts trivial on the �rst coördinate and nat-ural on the second coördinate, so t(x, z) = (x, tz). The following propositionis also part of Proposition 3.1 of [12].

Proposition 3.4.2.

(i) fK : E(λ′)→ E(m− n) is a Tm-equivariant bundle epimorphism.

(ii) There exists a Tm-equivariant n-dimensional complex vector bundle ξ′ ↓ZK such that λ′ ∼= Cm−n⊕ξ′ as Tm-equivariant vector bundles over ZK.

13

(iii) λ ∼= ξ′hTm ⊕ Cm−n as vector bundles over DJ(K)

Proof. We have

fK(t(x, z)) = fK(tx, tz)

= (A(txtz), tz)

= (A(xz), tz)

= t(A(xz), z)

= tfK(x, z),

so fA is Tm-equivariant.

Now let (y, z) ∈ Cm−n × ZK be arbitrary. There exists a face α ∈ K,such that z ∈ S[m]−α × (D2)α and therefore zi 6= 0 for i /∈ α. Let z′ =(zi1 , ..., zim−|α|) ∈ Cm−|α| denote the element indexed by the vertices ij /∈ αand let x ∈ Cm be an element with xi = 0 for i ∈ α. Then

A(xz) = Aα((xi1 zi1 , ..., xim−n zim−|α|))

Since Aα is an epimorphism, there exists an element w ∈ Cm−|α|, such thatAα(w) = y. Furtheremore, because zij 6= 0 for all j, we see that there exists

x′ ∈ Cm−|α|, such that x′z′ = w. This shows us that

y = Aα((xi1 zi1 , ..., xim−n zim−|α|))

has a solution. We conclude that fK is an epimorphism.

Let ξ′ := ker(fK). This is again a Tm-equivariant vector bundle over ZKand the inclusion E(ξ′)→ E(λ′) is a Tm-equivariant bundle map, so we havethe next short exact sequence of Tm-equivariant complex vector bundles

0→ ξ′ ↪→ E(λ′)fA−→ E(m− n)→ 0.

For a compact Lie group G, every short exact sequence of G-equivariantvector bundles over compact spaces splits by [13]. It follows that we haveλ′ ∼= ξ′ ⊕ Cm−n.

As a direct consequence of part (ii) we see that λ ∼= ξ′hTm⊕Cm−n as vectorbundles over DJ(K).

We de�ne ξ := ξ′hTm , an n-dimensional complex vector bundle overDJ(K). Let fK : DJ(K)→ BU(n) denote the classifying map of the vectorbundle ξ ↓ DJ(K).

14

De�nition 3.4.3. Let XK be the homotopy �ber of the map fK : DJ(K)→BU(n):

XK → DJ(K)→ BU(n)

Example 3.4.4. If |K| = ∂∆[m], we will later see that XK ' U(m)/Tm.

Proposition 3.4.5. The commuting diagram

DJ(K) BU(n)

BTm BU(m)

fK

is the topological realization of

Z[K] Z[c1, ..., cn]

Z[m] Z[c1, ..., cm]

f∗K

ψ

q p ,

where f ∗K maps the i-th Chern class ci to the i-th symmetric polynomial σi(n)

Proof. The cohomology of BU(n) is the polynomial algebra generated bythe Chern classes, H∗(BU(n);Z) = Z[c1, ..., cn] (see [9] �14). In �7 of [9] itis shown that the map ψ : Z[c1, ..., cm] → Z[m], induced by the maximaltorus inclusion BTm ↪→ BU(m), maps the i-th Chern class ci to the i-thsymmetric polynomial σi(m). The map p, induced by the coördinatewiseinclusion U(n) ↪→ U(m) maps ci to ci if i ≤ n and maps ci to 0 if i > n.From the commutativity of the diagram it now follows that f ∗K maps the i-thChern class ci to the i-th symmetric polynomial σi(n)

15

Chapter 4

Fibrations and Spectral Sequences

4.1 Fibrations

A map f : E → B is said to have the homotopy lifting property for a spaceX, if for eacht homotopy h : X × I → B and each map a : X → E, suchthat the following diagram commutes

X E

X × I B

a

i f

h

where i(x) = (x, 0), there exists a lift of h to H : X × I → E. i.e., thefollowing diagram commutes

X E

X × I B

a

i f

h

H

De�nition 4.1.1. A map f : E → B is a �bration if it has the homotopylifting property for all spaces X.

If we start of with an arbitrary map f : X → Y , by changing the spaceX up to homotopy, it is possible to make f into a �bration. More details onthis construction can be found in chapter 5.7 of [6]. So up to homotopy, every

16

map is a �bration. The up to homotopy determined �ber of this �bration iscalled the homotopy �ber of f . The following Proposition is Corollary 6.30of [3]:

Proposition 4.1.2. The homotopy �ber of

DJ(K)→ BTm

is ZK.

Corollary 4.1.3. If K = ∂∆[m], then the homotopy �ber of DJ(K)→ BTm

is S2m−1

Proof.

S2m−1 = ∂(D2m)

' ∂(D2)m

=⋃

α⊂[m],α 6=[m]

(S1)[m]−{α} × (D2)α

=⋃α∈K

(S1)[m]−{α} × (D2)α

= ZK

Suppose that we are given two �brations, both with the same total spaceE, with a �ber map between them, i.e. we have a commuting diagram

F0 E B0

F1 E B1

=

Lemma 4.1.4. Let Fi → E → Bi be a �bration for i = 0 and i = 1. Supposewe have the following commuting diagram

X

F0 E B0

F1 E B1

=

17

where X → B0 → B1 is a �bration. Then we have a �bration

F0 → F1 → X

The following lemma is proved in 7.6.1 of [14].

Lemma 4.1.5. Suppose X, Y and Z are spaces with the homotopy type ofconnected CW - complexes and we have the following homotopy commutativediagram

W X

Y Z

f ′

g′ g

f

Then the following are equivalent

(i) the map induced on the homotopy �bers Fg′ → Fg is a homotopy equiv-alence

(ii) the map induced on the homotopy �bers Ff ′ → Ff is a homotopy equiv-alence

In chapter 5 we will need the following lemma, which is proved on [8,p.212], a couple of times.

Lemma 4.1.6. Let G be a topological group and H ⊂ G a topological sub-group. Then we have a �bration

G/H → BHBi−→ BG

where the second map is the map induced by the inclusion i : H ↪→ G.

4.2 Spectral Sequences

One of our main tools in the study of vector bundles and �brations, are spec-tral sequences. Spectral sequences allow us to get cohomological informationof the �ber of a �bration from cohomological information of the total spaceand the base space.

De�nition 4.2.1. Let p,q,r be integers. A di�erential bigraded module overa ring R, is a collection R-modules Ep,q, together with an R-linear mappingd : E∗,∗ → E∗,∗ of bidegree (r, 1− r) or (−r, r − 1) with d ◦ d = 0. The mapd is called the di�erential.

18

De�nition 4.2.2. Let r > 0 be an integer. A spectral sequence of homo-logical, respectively cohomological type is a collection of di�erential bigradedmodules {Ep,q

r , dr}, where the bidegree of d is (−r, r−1), respectively (r, r−1)and Ep,q

r+1∼= Hp,q(E∗,∗r , dr) for all p, q, r.

If there exist an integer n such that Ep,qr = Ep,q

r+1 for all r ≥ n, then wecall Ep,q

n the limit term of the spectral sequence and we write Ep,q∞ := Ep,q

n

A sequence of submodules F nH of an R-module H is called an increasing�ltration if

... ⊆ F nH ⊆ F n+1H ⊆ ... ⊇ H,

holds and a decreasing �ltration if

... ⊆ F nH ⊆ F n−1H ⊆ ... ⊆ H,

holds. If H∗ is a graded R-module with a �ltration, then we de�ne

Ep,q0 (H∗, F ) =

{F pHp+q/F p−1Hp+q if F is increasing;F pHp+q/F p+1Hp+q if F is decreasing.

De�nition 4.2.3. Let H∗ be a graded R-module. A spectral sequence{E∗,∗r , dr} converges to H∗ if there exists a �ltration F on H∗, such that

Ep,q∞∼= Ep,q

0 (H∗, F ).

Since the cohomology H∗(XK) also has a multiplicative structure, we areinterested in spectral sequences of algebras.

De�nition 4.2.4. A spectral sequence of algebras is a spectral sequence{E∗,∗r , dr} together with an algebra structure ψr : E∗,∗r ⊗R E∗,∗r → E∗,∗r for allr, such that ψr+1 is given by the composition

E∗,∗r+1 ⊗R E∗,∗r+1

∼=−→ H(E∗,∗r )⊗R H(E∗,∗r )

p−→ H(E∗,∗r ⊗R E∗,∗r )H(ψr)−−−→ H(E∗,∗r )

∼=−→ E∗,∗r+1,

where the homomorphism p maps [u]⊗ [v] to [u⊗ v].

A spectral sequence of algebras is said to converge to a graded algebra H∗

as an algebra, if there exists a �ltration F on H∗, such that E∗,∗∞∼= E∗,∗0 (H,F )

as algebras. If Ep,qr = 0 when either p < 0 or q < 0, we say that the spectral

sequence is a �rst quadrant spectral sequence.

The following theorem is a direct consequence of Theorem 7.1 of [8]:

19

Theorem 4.2.5. Let F → E → B be a �bration, with B simply connected.If H∗(E) is a free �nitely generated H∗(B)-module, then

H∗(F ) = H∗(E)⊗H∗(B) Z

The following theorem is proved in theorems 5.2 and 5.6 of [8].

Theorem 4.2.6. Let F → E → B be a �bration, with B path-connectedand F connected. Suppose that Hp(B) and Hq(F ) are free �nitely generatedZ-modules and that B is simply connected. Then there exists a �rst quadrantspectral sequence of algebras {E∗,∗r , dr} converging to H∗(E) as an algebrawith

E∗,∗2∼= H∗(B)⊗R H∗(F )

as a bigraded algebra.

Proposition 4.2.7. Let F → E → B be a �bration such that H i(F ) = 0 =H i(B) for i odd. Then E∗,∗2 = E∗,∗∞

Proof. Let r ≥ 2. Suppose Ep,qr 6= 0, so p and q must be even. Consider the

mapsdp−r,q+r−1r : Ep−r,q+r−1

r → Ep,qr dp,qr : Ep,q

r → Ep+r,q−r+1r

Clearly either p− r or q+ r−1 must be odd and also either p+ r or q− r+ 1must be odd, and therefore Ep−r,q+r−1

r = Ep+r,q−r+1r = 0. We conclude that

dr = 0 for all r, so

Ep,qr+1 = ker(dp,qr )/im(dp−r,q+r−1

r )

= Ep,qr

The following is proved in [8] Example 5.D.

Proposition 4.2.8. Let F → E → B be a �bration with B path-connectedand H i(B) = 0 for 0 < i < p and H i(F ) = 0 for 0 < i < q. Then there existsan exact sequence

0→ H1(B)→H1(E)→ H1(F )→ H2(B)→ ...

...→ Hp+q−1(B)→ Hp+q−1(E)→ Hp+q−1(F ),

where the maps H i(B)→ H i(E)→ H i(F ) are induced by F → E → B.

20

4.3 Poincaré Series

For a topological space X, the Poincaré series P (X) associated to X is theformal in�nite power series

P (X) :=∑i

dim(H i(X))ti

The cohomology of BU(n) is a polynomial algebra in n generators of degree2. For n = 1, we have

P (BU(1)) =∞∑i=0

(t2)i

=1

1− t2

and this generalizes to

P (BU(n)) =n∏i=1

1

(1− t2)i

We will also need to know the Poincaré series of DJ(K). In [3, p.37] it isshown that

P (DJ(K)) =h0 + h1t

2 + ...+ hnt2n

(1− t2)n,

where, without going into details, (h0, ..., hn) is the h-vector of K. The onlything we need to know is that hn = 1, which follows from the so called DehnSommer�eld equations, which are proved in Theorem 1.30 of [3].

The following proposition is a direct consequence of Theorem 4.2.6 andProposition 4.2.7:

Proposition 4.3.1. Let F → E → B be a �bration, with B path-connectedand F connected. Suppose that Hp(B) and Hq(F ) are free �nitely generatedZ-modules and that B is simply connected and H i(F ) = H i(b) = 0 for i odd.Then

P (F ) =P (E)

P (B)

21

Chapter 5

Cohomological properties of XK

5.1 The homotopy equivalence of U(m)/Tm and

X∂∆[m]

In example 3.4.4 we claimed that X∂∆[m] is homotopically equivalent toU(m)/Tm. In this section we will construct a homotopy equivalence betweenthese two spaces.

Let K be an (n−1)-dimensional �nite abstract simplicial complex with mvertices. Recall from section 3.3 that we had the m-dimensional complex vec-tor bundle λ ↓ DJ(K), which was isomorphic to the pull back of the universalbundle γm ↓ BU(m) along the composition of the inclusion DJ(K) ↪→ BTm

and the maximal torus inclusion BTm → BU(m). Of course, the vector bun-dle γn ⊕ Rm−n ↓ BU(n) can be obtained from the pullback of γm underthe coördinate wise inclusion BU(n) → BU(m). Also, the vector bundle ξover DJ(K) can be obtained as the pullback of the universal vector bundleγn ↓ BU(n). Thus we see that the vector bundle λ ↓ DJ(K) is the pull backof the universal bundle over BU(m) along the the map

DJ(K)→ BU(n)→ BU(m)

as well as the map

DJ(K)→ BT (m)→ BU(m).

Using Theorem 3.1.3, we see that the following diagram commutes up to

22

homotopy

DJ(K) BU(n)

BT (m) BU(m)

(5.1.1)

Theorem 5.1.1. The homotopy �ber of DJ(∂∆[m]) → BU(m − 1) is ho-motopy equivalent to U(m)/Tm:

X∂∆[m] ' U(m)/Tm

Proof. If K = ∂∆[m], then n = m− 1 and the homotopy �ber of

BU(m− 1)→ BU(m)

is U(m)/U(m − 1) ' S2m−1. Together with Corollary 4.1.3, diagram 5.1.1gives us the following diagram of �brations:

S2m−1 S2m−1

DJ(∂∆[m]) BU(m− 1)

BTm BU(m)

By proposition 4.2.8 we �nd two short exact sequences, which �t in thefollowing commuting diagram

0 H2m−1(S2m−1) H2m(BU(m)) H2m(BU(m− 1)) 0

0 H2m−1(S2m−1) H2m(BTm) H2m(DJ(∂∆[m])) 0

h

f p

g

We have that p maps the i-th Chern class ci to the i-th Chern class ofBU(m − 1) for i < m and maps cm to zero. Since the �rst row is exact, wesee that f maps a generator t for H2m−1(S2m−1) to cm, the m-th Chern class.Since g maps cm to σm, the m-th symmetric polynomial, we see that h mapst to a generator of H2m−1(S2m−1), so this map is an isomorphism. Whitehead

23

Theorem now tells us that S2m−1 → S2m−1 is a homotopy equivalence. Wenow have the next commuting diagram �brations:

S2m−1 S2m−1

X∂∆[m] DJ(∂∆[m]) BU(m− 1)

U(m)/Tm BTm BU(m)

'

and by Lemma 4.1.5 we see that X∂∆[m] → U(m)/Tm is a homotopy equiva-lence.

5.2 The Stanley Reisner algebra and the ele-

mentary symmetric polynomials

In this section we proof that the cohomology of XK is isomorphic to theStanley Reisner algebra Z[K] modulo the ideal generated by the �rst n ele-mentary symmetric polynomials. For this we need to know that H∗(DJ(K))is a free �nitely generated H∗(BU(m))-module. We start with the notion ofa regular sequence.

Let A be a Z≥0-graded algebra. A sequence a1, ..., ar ∈ A+ is called aregular sequence if ai ∈ A/ < a1, ..., ai−1 > is not a zero divisor. Theorem 4.1of [10] states that

Proposition 5.2.1. The following are equivalent

(i) a1, ..., ar ∈ A+ is a regular sequence.

(ii) A is a free �nitely generated module over the polynomial algebrak[a1, ..., ar]

Since σ1, ..., σm form a regular sequence in Z[m], we conclude thatH∗(DJ(K)) is a free �nitely generated H∗(BU(m))-module.

We can now proof the main theorem of this section.

24

Theorem 5.2.2. Let K = KP be a �nite abstract simplicial complex dualto a simple polytope P . The cohomology of XK is isomorphic to the StanleyReisner algebra Z[K] modulo the �rst n elementary symmetric polynomials

H∗(XK) ∼= H∗(DJ(K))⊗BU(n) Z∼= Z[K]/ < σ1, ..., σn > .

(5.2.1)

Proof. Recall from section 5.1 that we have the following commuting diagram

DJ(K) BU(n)

BT (m) BU(m)

so the map DJ(K)→ BU(m) factors through BU(n). Since H∗(DJ(K)) isa �nitely generated BU(m)-module, H∗(DJ(K)) is also a �nitely generatedH∗(BU(n))-module.

Since σ1, ..., σn is a regular sequence, Proposition 5.2.1 tells usthat H∗(DJ(K)) is a free �nitely generated H∗(BU(n))-module. SinceH∗(BU(n)) is a polynomial algebra in n generators, Theorem 4.2.5 now tellsus that

H∗(XK) ∼= H∗(DJ(K))⊗H∗(BU(n)) Z ∼= Z[K]/ < σ1, ..., σn >

Corollary 5.2.3. The cohomology of XK is torsion free and sits in evendegrees.

Proof. Since both H∗(DJ(K)) and H∗(BU(n)) are zero in odd degrees,equation 5.2.1 shows us that H∗(XK) sits in even degrees as well. SinceH∗(DJ(K)) is torsion free and a free H∗(BU(n))-module, equation 5.2.1also shows us that H∗(XK) is torsion free.

Example 5.2.4. For K = ∂∆[m], we have m− 1 = n and

H∗(XK) ∼= Z[K]/ < σ1, ..., σn >∼= (Z[m]/ < σm >)/ < σ1, ..., σm−1 >∼= Z[m]/ < σ1, ..., σm >

25

5.3 K a polygon

Later on, we shall determine cohomological properties of XK by using induc-tion on the dimension of the �nite abstract simplicial complex K. Therefore,it is important to have an explicit calculation of the cohomology of XK for Kone dimensional, i.e. a polygon. In this section, let K be an m-gon. We recallfrom Theorem 5.2.2 that for a one dimensional complex the cohomology isgiven by

H∗(XK) ∼= Z[K]/ < σ1, σ2 > .

If we de�neN := {vivj|1 ≤ i < j − 1 ≤ m}\v1vm

Then the cohomology of XK is given by

H∗(XK) ∼= Z[v1, ..., vm]/ < vivj ∈ N, σ1, σ2 >

In degree two we have one relation:

v1 + ...+ vm = 0 (5.3.1)

So we see that choosing an integer 1 ≤ k ≤ m gives us m − 1 generators{vi|i 6= k} in degree two.

In degree four we have the relations vivj = 0, for v1vj ∈ N , as well as foreach j not equal to k − 1, k or k + 1 we have the relation

0 = vj∑

vi

= vj−1vj + v2j + vjvj+1,

(5.3.2)

where we identify v0 := vm and vm+1 := v1. Note that for j equal to k− 1 ork+ 1 this relation becomes the zero relation, so this gives us m− 3 relations.Using relations 5.3.2, we can eliminate all squares, exept for v2

k−1 and v2k+1.

The �nal relation we have in degree four is

0 = σ2

=∑

i 6=k−1,k

vivi+1 −(v2k−1 + v2

k+1 + vk−2vk−1 + vk+1vk+2

)2

= −∑

i 6=k−2,...,k+1

vivi+1 − v2k−1 − v2

k+1

(5.3.3)

Which allows us to eliminate either v2k−1 or v2

k+1. If we choose to eliminatev2k−1, a set of generators for degree four is given by

{vivi+1|i 6= k − 1, k} ∪ {v2k+1}.

26

Now let us look at monomials vivjvk of degree 6. Clearly if i, j and k areall di�erent, then vivjvk = 0, so it su�ces to look at elements of the formv2i vj, where j = i− 1 or j = i+ 1 and the element v3

k+1. First of all, for all iwe have

0 = −viσ2

= vi−1v2i + v2

i vi+1

(5.3.4)

Now suppose that i is not equal to k − 1 or k, then equation 5.3.2 tells usthat

−v2i vi+1 = (vi−1vi + vivi+1)vi+1

= viv2i+1

(5.3.5)

From equations 5.3.4 and 5.3.5 we see that there are only two possible gen-erators left, being vk−1v

2k and v

3k+1. Using 5.3.3, we see that

−v3k+1 = vk+1 ·

( ∑i 6=k−2,...,k+1

vivi+1 + v2k−1

)= 0,

(5.3.6)

Since there are no further relations, we conclude that there is just one gen-erator in degree six,

vkv2k+1.

There are no generators in degree eight. A set of generators for each degreeis given in the following table

i set of generators of H i(XK) dim(H2n(XK))0 1 12 {vi|i 6= k} m-14 {vivi+1|i 6= k − 1, k} ∪ {v2

k+1} m-16 vkv

2k+1 1

where we still identify v0 = vm and vm+1 = v1 and 1 ≤ k ≤ m is arbitrary.The algebra structure of H∗(XK) is determined by the relations 5.3.2. 5.3.3,5.3.4, 5.3.5 and 5.3.6.

5.4 Polynomial relations between maximal

faces

In this section, let K = KP be the abstract dual of a simple polytope P .

27

Lemma 5.4.1. Let 1 ≤ k ≤ n be an integer and q = vr1i1 ·...·vrnin

be a monomialin Z[K]/ < σ1, ..., σn > with ri ≤ ri+1 for all i. Suppose that rj = j for allj < k. If rk > k, then q = 0.

Proof. We will use induction on k. Suppose k = 1, then

vr1i1 · ... · vrnin

= (vi1 · ... · vin)2(vr1−2i1· ... · vin−2

in)

On the other hand we have

0 = σn · vi1 · ... · vin= (vi1 · ... · vin)2,

where the second equallity follows from the fact any product of n+1 di�erentgenerators is zero. We conclude that q = 0.

Now let m ≤ n be an integer and suppose that the lemma is true fork < m and suppose that rm > m. We have

vr1i1 · ... · vrnin

= (v1i1· v2

i2· ... · vm−1

im−1 · vm+1im· ... · vm+1

n ) · f,

where f = vim−(m+1)im

· ... · vin−(m+1)in

But this time we have

0 = σn−N+1(v1i1· v2

i2· ... · vm−1

im−1· vmim · ... · v

mn )

= (v1i1· v2

i2· ... · vm−1

im−1 · vm+1im· ... · vm+1

n ),

where the second equallity follows from the induction hypothesis.

Proposition 5.4.2. Let {i1, ..., in} be a maximal face of K and σ be a per-mutation of {1, ..., n}. Then

v1i1· ... · vnin = sign(σ)v1

iσ(1)· ... · vniσ(n)

Proof. Let τ = (j, j + 1) be a transposition. Then

0 = σn−iv1i1· ... · vjij · v

(j+1)−1ij+1

· ... · vn−1in

= v1i1· ... · vnin + v1

i1· ... · vj−1

ij−1· vj+1

ij· vjij+1

· vj+2ij+2· ... · vnin

where the last equallity follows with induction from Lemma 5.4.1. This meansthat we have

v1i1· ... · vnin = sign(τ)v1

iτ(1)· ... · vniτ(n)

Since the transpositions (i, i+ 1) generate Sn, we are now done.

28

Lemma 5.4.3. Let {i1, ..., in} and {i1, , ..., ik−1, jk, ik+1, ..., in} be two maxi-mal faces of K having only 1 vertex distinct. Then

v1i1· ... · vnin = −v1

i1· ... · vk−1

ik−1· vkjk · v

k+1ik+1· ... · vnin

Proof. By Proposition 5.4.2, it su�ces to proof the statement for k = 1. Theface {i2, ..., in} has codimension 1. This means that the corresponding face αof P has dimension 1, so we see that α has two vertices. The maximal facesof K corresponding to these vertices are exactly the only two maximal facesof K that have {i2, ..., in} as a facet. It now follows that

0 = σn · v1i2· ... · vn−1

in

= v1i1· ... · vnin + v1

j1· v2

i2... · vnin ,

so vi1 · ... · vnin = −v1j1· v2

i2... · vnin

Proposition 5.4.4. Let {i1, ..., in} and {j1, ..., jn} be two (not necessarilydistinct) maximal faces. Then

v1i1· ... · vnin = ±v1

j1· ... · vnjn

Proof. We have K = KP with P a simple polytope. Let us consider thegraph P (1), i.e. the 1-skeleton of P . Two vertices n and m in P (1) that arelinked by an edge v correspond to two maximal faces Fn and Fm of K with|Fn ∩ Fm| = n− 2. So if Fn = {i1, ..., in} and Fm = {j1, i2, ..., in}, then

v1i1· ... · vnin = −v1

j1· v2

i2... · vnin

by lemma 5.4.3.

Now let Fi = {i1, ..., in} and Fj = {j1, ..., jn} be two arbitrary maximalfaces of K. If Fi = Fj the Proposition follows form Proposition 5.4.2. Leti and j be the vertices of P corresponding to the maximal faces Fi andFj respectively. In P (1) we can �nd a �nite walk from i to j. This walkcorresponds to a �nite chain (Fi = G0, G1, ..., Gk = Fj) of maximal faces ofK, with |Gi ∩Gi+1| = n− 2. It follows by induction on the cardinality of theset {G0, ..., Gk} together with Proposition 5.4.2 that

v1i1· ... · vnin = ±v1

j1· ... · vnjn

29

5.5 The highest non-zero cohomology group of

XK

In this section we will proof Theorem 1.0.4. Let K = KP , with P a simplepolytope and {i} a vertex of K. We know from Proposition 1.0.2 that thelink of a vertex of K is the abstract dual of a simple polytope as well andthat its dimension is n− 1. We already have an explicit construction of thecohomology of XK for 1-dimensional K from section 5.3, so if we can deter-mine cohomological properties of XK from information of the cohomology ofXlinkK({i}), we can use induction on the dimension of K. We will do this intwo steps. From the cohomology of XlinkK({i}) to the cohomology of XstK({i})and from there we go to the cohomology of XK .

Let us start out by considering the �bration

XlinkK({i}) → DJ(linkK({i}))→ BU(n− 1).

If we take the product of DJ(linkK({i})) and BU(n − 1) with BT , we getanother �bration

XlinkK({i}) → BT ×DJ(linkK({i}))→ BT ×BU(n− 1) = B(T ×U(n− 1))

We have

BT ×DJ(linkK({i})) = BT ×⋃

α∈linkK({i})

BT α

=⋃

α∈linkK({i})

B(T 1 × Tα)

=⋃

α∈stK({i})

BTα

= DJ(stK{i})

and therefore, we get the following commuting diagram of �brations

XlinkK({i}) BT ×DJ(linkK({i})) B(T × U(n− 1))

XstK({i}) DJ(stK({i})) BU(n)

=

From lemma 4.1.6 it follows that the righthand-side arrow in the abovediagram has �ber equal to

(U(n)/(T × U(n− 1)) ∼= S2n−1/T∼= CP (n− 1)

30

where the �rst isomorphism is shown in [2, p.464]. Lemma 4.1.4 tells us thatwe have the following commuting diagram of �brations:

S2(n−1)−1/T ∼= CP (n− 1)

XlinkK({i}) BT ×DJ(linkK({i})) B(T × U(n− 1))

XstK({i}) DJ(stK({i})) BU(n)

CP (n− 1)

=

Proposition 5.5.1. Suppose that for all vertices in ∈ K the highest non-zerocohomology group of XlinkK({in}) is generated by vi1 ·...·vn−1

in−1, with {i1, ..., in−1}

any maximal face of linkK({in}). Then for any vertex jn of K the highestnon-zero cohomology group of XstK({jn}) is generated by

v1j1· ... · vn−1

jn−1· vn−1

jn,

with {ji, ..., jn} any maximal face of stK({jn}). Furthermore, the cohomologyof XstK({jn}) sits in even degrees.

Proof. Let {j1, ..., jn} be an arbitrary maximal face of stk({jn}). Note that jncan be any vertex of K, since K is the dual of a simple polytope P . Considerthe �bration

XlinkK({jn}) → XstK({jn}) → CP (n− 1)

The cohomology of CP (n− 1) sits in even degrees and by Proposition 5.2.3together with Proposition 1.0.2, the cohomology of XlinkK({jn}) sits in evendegrees as well. Proposition 4.2.7 now tells us that E∗,∗2 = E∗,∗∞ and thereforeby Theorem 4.2.6 we have

H∗(XstK({jn}))∼= H∗(CP (n− 1))⊗H∗(XlinkK({jn})), (5.5.1)

as abelian groups. This shows us that the cohomology of XstK({jn}) sits ineven degrees. Proposition 4.2.8 implies that

0→ H2(CP (n− 1))π∗−→ H2(XK)

i∗−→ H2(XlinkK({jn}))→ 0

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is an exact sequence. Since the middle term is torsion free, we see that π∗

maps a generator t of CP (n − 1), to a generator vjk of H∗(XstK({jn})). Wealso have the folliwing commuting diagram of �brations:

XstK({i}) DJ(stK({i})) BU(n)

CP (n− 1) BT ×BU(n− 1) BU(n)

From here we see that π∗ maps t to vin .

We now use that the maximal non zero cohomology group ofH∗(XlinkK({jn})) is the group generated by v1

j1· ... · vn−1

jn−1, with {j1, ..., jn−1}

any of the maximal faces of linkK({jn}). We conclude that the highest nonzero cohomology group of XK is the group generated by v1

j1· ... · vnjn

Now that we can construct the highest nonzero cohomology group ofXstK({i}), we will use this to determine the higest non zero cohomology groupof XK . Consider the map

vi : Z[K]→ Z[K]

of multiplying by vi. Let w = vi1 · ... · vir 6= 0 be a monomial in Z[K]. Wehave v ∈ ker(vi) if and only if {i1, ..., ir} /∈ KstK({i}), so we see that the mapvi factorizes through Z[stK({i})]:

Z[K] Z[K]

Z[stK({i})]

vi

q f

(5.5.2)

with q the quotient map and f the induced map.

Since all the maps are H∗(BU(n))-linear, we can take the tensor productover BU(n) with Z to get the diagram

Z[K]⊗BU(n) Z Z[K]⊗BU(n) Z

Z[stK({i})]⊗BU(n) Z

vi ⊗ id

q ⊗ id f ⊗ id(5.5.3)

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Since H∗(DJ(K)) and H∗(DJ(stK({i}))) are free �nitely generatedH∗(BU(n))-modules, Theorem 4.2.5 tells us that diagram 5.5.3 is the sameas

H∗(XK) H∗(XK)

H∗(XstK{i})

(5.5.4)

Proposition 5.5.2. Suppose that for all vertices i of K we haveHr(XlinkK({i})) = 0 for r > (n− 1)2 + n− 1 = n2 − n. Then

H i(XK) = 0, for i > n2 + n

Proof. Let q ∈ H∗(XK) be a monomial with deg(q) > 0 and q 6= 0. Sincedim(K) = n− 1, we know that q is of the form vr1j1 · ... · v

rnjn, with {j1, ..., jn}

a maximal face of K. Without loss of generality, we may and do assumethat rn > 0 and ri ≥ 0 for all i. Clearly we have q ∈ im(vjn ⊗ id), so letq = vjn⊗id(q′). Since y := π⊗id(q′) is an element of H∗(XstK{jn})

+, we knowfrom Proposition 5.5.1 that the degree of y is less then or equal to n2 +n−2.If we now consider the polynomial y as a polynomial y in H∗(XK), then wesee that

q = vjn ⊗ id(q′)

= vjn ⊗ id(y)

This shows us that deg(q) ≤ n2+n−2+2 = n2+n and thus that H i(XK) = 0for i > n2 + 2.

Proposition 5.5.3. Suppose that for all vertices in of K the highest non-zerocohomology group of XlinkK({in}) is generated by vi1 ·...·vn−1

in−1, with {i1, ..., in−1}

any maximal face of linkK({i}). Then Hn2+n(XK) 6= 0

Proof. We will use Poincaré series to proof this. Recall the Poincaré series ofDJ(K) and BU(n) from section 4.3. By Proposition 4.3.1 we have

P (XK) = P (DJ(K))/P (BU(n))

=n∏i=1

(1− (t2)i)(h0 + h1(t2)1 + ...+ hn(t2)n)

(1− t2)n

This polynomial has degree∑n

i=1 2i = n2 + n, and we have top coë�cienthn = 1.

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Proof of Theorem 1.0.4We know the theorem to be true for n = 2. Supposethe theorem is true for i ≤ n and let K be a �nite abstract simplicial complexof dimension n−1. Let {k0, ..., kn} be an arbitrary maximal face of K. By theinduction hypothesis, for all vertices in ofK the highest non-zero cohomologygroup of XlinkK({in}) is generated by vi1 · ... · vn−1

in−1, with {i1, ..., in−1} any

maximal face of linkK({in}). By Proposition 5.5.2, this proofs H i(XK) = 0for all (i > n2 + n).

Let q ∈ Hn2+n(XK) be a non zero monomial. Just as in the proof ofProposition 5.5.2, we know that there exists a vertex in, and an element yin H∗(XstK({in})), such that q = vin ⊗ id(y). Since y has degree n2 + n − 2

and Hn2+n−2(XstK({in})) is generated by v1i1· ... · vn−1

in−1· vn−1

in, we see that

y = λv1i1· ... · vn−1

in−1· vn−1

in. If we lift y to y = v1

i1· ... · vn−1

in−1· vn−1

inin H∗(XK),

we see that

q = vjn ⊗ id(y)

= vjn ⊗ id(λv1i1· ... · vn−1

in−1· vn−1

in)

We conclude that q = ±v1i1· ... · vnin . By Proposition 5.4.4 we see that

q = ±v1k1· ... · vnkn

This, together with Proposition 5.5.3 proof part (i) and part (ii). Part (iii)was already proved in Proposition 5.4.2.

34

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