Stiefel-Whitney Classes and Thom’s Theorem

SJÄLVSTÄNDIGA ARBETEN I MATEMATIK

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET

Stiefel-Whitney Classes and Thom's Theorem

av

Marco Bernagozzi

2017 - No 14

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM

Stiefel-Whitney Classes and Thom's Theorem

Marco Bernagozzi

Självständigt arbete i matematik 30 högskolepoäng, avancerad nivå Handledare: Alexander Berglund

2017

Stiefel-Whitney Classes and Thom’s Theorem

Marco Bernagozzi

Abstract

This Master thesis contains an introduction to the Stiefel-Whitney classes and describes how they can be used to determine if two manifolds belong (or do not belong) to the same cobordism group. Two manifolds are cobordant if they can be expressed as the boundary of a third one, and the relation between Stiefel-Whitney classes and cobordism is given by Thom’s theorem. Classification of manifolds is an important problem, and in general hard for high dimensional manifolds.

Diffeomorphisms and homotopy equivalence can be very difficult to compute, and cobordism can be a valid alternative.

iii

Contents

Introduction vii

1 Preliminary Notions 1

1.1 Vector bundles . . . 1

1.2 Orientability in Z² and the Poincar´e Duality theorem . . . 5

2 Stiefel-Whitney Classes 9 2.1 Axioms . . . 9

2.2 Stiefel-Whitney numbers . . . 11

2.3 Grassmann manifolds . . . 13

2.4 Uniqueness of Stiefel-Whitney classes . . . 16

2.5 Existence of Stiefel-Whitney classes . . . 19

3 Cobordism 23 3.1 Cobordism groups of smooth closed manifolds . . . 23

3.2 h-cobordism . . . 24

3.3 Surgery . . . 24

4 Thom’s Theorem 27 4.1 Outline of the proof . . . 30

4.2 Nn→ πn(T O) is an isomorphism . . . 30

4.3 πn(T O) → Hn(T O) is a monomorphism . . . 34

4.4 Hn(T O)→ Hn(BO) is an isomorphism . . . 35

4.5 The diagram commutes . . . 36

Appendix 39

Bibliography 43

v

Introduction

The aim of this thesis is to give a straightforward introduction to Stiefel- Whitney classes, and apply them to the cobordism group later.

The reader should have a basic notion of topology and homological algebra:

homology, cohomology groups, relative homology and cohomology, and homol- ogy/cohomology with coefficient. Basic knowledge of differential geometry will be helpful, but is not required.

Cobordism first appears in 1895, in a paper where Poincar´e defined homotopy groups.

Bordism was then introduced explicitly by Pontryagin. Thom then used Poincar´e work to define the structure of the graded unoriented cobordism ring, and he later defined the oriented cobordism. In 1936-1940 Stiefel and Whitney described the Stiefel-Whitney classes, which are an important invariant of the manifolds that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. A few years later, in 1952, Thom showed the relation that exists between the fundamental class and the Steenrod operations. In 1954, he described the cobordism group as the homotopy group of the Thom space. This result is a theorem called Thom’s theorem. This is where this thesis ends, since Thom’s theorem gives us the last piece to be able to prove that two manifolds are cobordant if and only if they have the same Stiefel-Whitney numbers.

The paper is divided into two main parts: the first one, that goes from Chapter 1 to Chapter 7, is about the Stiefel-Whitney classes. It defines them and proves uniqueness and existence. The second part shows the relation between the Stiefel- Whitney classes and the cobordism group.

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The first part starts with an introduction to vector bundles, and basic opera- tions between them. It then proceeds with a brief but thorough discussion about the Poincar´e Duality, that gives a definition of orientability in terms of homology, and shows the reason why the coefficients in Z2 are used in the Stiefel-Whitney classes, since the chapter ends proving that every manifold is orientable in Z2.

In Chapter 3, Stiefel-Whitney classes are defined by a set of four Axioms, and the total Stiefel-Whitney class is introduced.

Chapter 4 introduces Stiefel-Whitney numbers. These numbers have a special role, since they are something we can compute, and they are strongly related to the cobordism group. In fact, the chapter ends with the main theorem of the paper:

two smooth, closed n−manifolds belong to the same cobordism class if and only if all of their Stiefel-Whitney numbers are equal.

Given this important result, the paper continues proving the existence and uniqueness of the Stiefel-Whitney classes, after a brief introduction on the Grass- manian manifolds, which are widely used in those proofs.

The first part ends here, and the second part describes briefly the cobordism group and gives insight on the proof of Thom’s theorem: if all of the Stiefel- Whitney numbers of a manifold M are zero, then M can be realised as the bound- ary of some smooth compact manifold.

Chapter 1

Preliminary Notions

1.1 Vector bundles

This chapter contains basic notions of differential geometry, that will be used throughout the thesis. In particular, the focus will be on vector bundles and op- erations over vector bundles.

Definition 1.1.1. A vector bundle ξ consists of:

• a pair of topological spaces E (called the total space) and B (called the base space);

• a projection map π : E → B;

• the structure of a real vector space on the fiber π⁻¹(b) (∀b ∈ B)

with the condition of local triviality, that is: ∀b ∈ B there exist an open neigh- bourhood U ⊂ B, an integer n ≥ 0 and a homeomorphism

h : U × Rⁿ→ π⁻¹(U ) such that:

1. U 3 u 7→ π(h(u, x)) is the identity map on U (∀x ∈ Rⁿ)) (in other words, u7→ h(u, x) is a section of U);

2. Rⁿ3 x 7→ h(u, x) is an isomorphism of vector spaces between Rⁿ and π⁻¹(u) (∀u ∈ U).

1

A pair (U, h) is called a local coordinate system for ξ about b.

If it is possible to choose U as the entire base space, we say that ξ is a trivial bundle.

There always is a section for every open subset U ⊂ B, namely the map sending every u∈ U to the zero vector of π⁻¹(u), and it is called the zero section.

An n-plane bundle is an n-dimensional vector space bundle.

Definition 1.1.2. We call Fb(ξ) = π⁻¹(b) the fiber of ξ over b∈ B.

Definition 1.1.3. The tangent bundle T M of a smooth manifold M is defined as:

T M = G

x∈M

TxM

where TxM is the tangent space of M at x.

We recall that the tangent spaces are defined via derivations. A linear map v : C^∞(M )→ R is a derivation at x ∈ M if

v(f g) = f (x)v(g) + g(x)v(f ) ∀f, g ∈ C^∞(M ).

and the tangent space is defined as

TxM ={(x, v) : v is a derivation of M at x}.

We have the projection map:

π : T M → M that is defined by the mapping

π(x, v) = x.

We now define the topology on the tangent bundle. First step, is to define what is an atlas on a manifold.

Definition 1.1.4. A chart (U, φ) is a homeomorphism φ from an open set U ⊂ M, onto an open set ofRⁿ.

An atlas A on a manifold M is a collection of charts such that S

Uα = M .

1.1 Vector bundles 3 Clearly, a manifold comes with an atlas (Uα, φα) where φα are diffeomorphisms:

φα : Uα→ Rⁿ We define:

φ_α : π⁻¹(U_α)→ R²ⁿ by

φ_α(x, vⁱ∂i) = (φα(x), v¹, . . . , vⁿ) And finally, we define a set V to be open in T M if and only if:

φ_α(V ∩ π⁻¹(Uα)) is open in R²ⁿ.

Definition 1.1.5. An induced bundle is defined as follows:

Let ξ be a vector bundle with topological space E and base space B.

Let B₁ be a topological space. Given a map f : B₁ → B it is defined the induced bundle

f^∗ξ

• The base space is B¹.

The total space E1 ⊂ B¹× E is defined as E¹ ={(b, e)s.t. f(b) = π(e)}

• π¹ : E1 → B¹ is defined as: π1(b, e) = b

Definition 1.1.6. The Cartesian product between ξ₁, ξ₂ is defined as to be (in- dexes are according to which vector bundle they are related to)

ξ₁× ξ2

with projection map

π1× π2 : E1× E2 → B1× B2

and fiber

(π1× π²)⁻¹(b1, b2) = Fb1(ξ1)× F^b2(ξ2) Definition 1.1.7. A bundle map from η to ξ is a map:

f : E(η) → E(ξ)

that maps Fb(η) isomorphically into Fb⁰(ξ), for a b⁰ ∈ B(ξ).

Let f : E(η)→ E(ξ) be a bundle map. Then we define f as to be the corresponding map between base spaces, i.e.

f : B(η)→ B(ξ)

An Euclidean vector bundle is a pair (ξ, µ), where ξ is a real vector bundle and µ : E(ξ)→ R is a continuous function, such that the restriction of µ to a fiber of ξ is a positive definite, quadratic function.

Definition 1.1.8. Let ξ₁.ξ₂ be two bundles over the same base space, and d : B → B × B

denote the diagonal embedding.

The Whitney sum of ξ1 and ξ2 is:

ξ1⊕ ξ2 := d^∗(ξ1× ξ2)

1.2 Orientability in Z² and the Poincar´e Duality theorem 5

1.2 Orientability in Z

and the Poincar´ e Duality theorem

In this chapter, R will be a commutative ring.

This chapter will describe the orientability of manifolds, and its relation to homol- ogy groups. It also introduces the fundamental classes, that will be encountered again in the next chapters.

Let X be a CW−complex. We can define the cellular (co)chains with coeffi- cients, that are given by

Cⁿ(X; R) = Hⁿ(Xⁿ, Xⁿ⁻¹; R)

From here, we can tensor product a chain and a cochain with coefficients:

Cⁿ(X; R)⊗ Cn(X; R)→ R ⊗ R And, passing to the (co)homology:

Hⁿ(X; R)⊗ Hn(X; R)→ R ⊗ R UsingZ², we have:

Hⁿ(X;Z²)⊗ Hn(X;Z²)→ Z2⊗Z2 Z² ∼=Z²

Let M be a manifold, and for x∈ M let U^x∼=Rⁿ be a coordinate chart. Then, by excision we have

Hi(M, M\ {x}) ∼= ˜Hi−1(Sⁿ⁻¹) Therefore, we have:

H_i(M, M\ {x}; R) =

0 i < n R i = n

Definition 1.2.1. We call R−fundamental class of M at a subspace X the element z ∈ Hn(M, M \ X; R)

such that, ∀x ∈ X, the image of z under the following map (induced by the inclusion) is a generator.

Hn(M, M\ X) → Hn(M, M\ x) When X = M , we call the fundamental class of M z ∈ Hⁿ(M )

Definition 1.2.2. An R−orientation is an open cover {Uⁱ} and R−fundamental classes zi of M at Ui such that , if Ui ∩ Uj 6= 0 then zi and zj map to the same element in Hn(M, M \ (Ui∩ Uj))

Definition 1.2.3. M is R−orientable if it admits an R−orientation.

Theorem 1.2.1. Let M be a compact n−manifold. An R−orientation of M has a bijective correspondence with an R−fundamental class.

Suppose we have an R−fundamental class.

Proof. The R−orientation is simply the restriction of the R−fundamental class to an open cover.

The converse holds only if M is compact. To prove it, we first need the vanishing theorem:

Theorem 1.2.2. Let M be an n−manifold. For any coefficient group π H_i(M ; π) = 0 if i > n

H˜n(M ; π) = 0 if M is connected and not compact

We proceed now proving that an R−orientation determines an R−fundamental class for M at K, with K a compact subset.

This will be later extended to the whole M given that M is compact.

Suppose that K is contained in a coordinate chart U ∼=RⁿBy excision and exact- ness:

Hi(M, M\ K; π) ∼= Hi(U, U \ K; π) ∼= ˜Hi−1(U\ K; π) Since U\ K is open, ˜Hi−1(U \ K; π) = 0 for i > n.

We have the Mayer-Vietoris sequence, for i = n:

Hn(M, M \ K ∪ L)−→ H^{Ψ n}(M, M \ K) ⊕ Hⁿ(M, M\ L)−→ H^{φ n}(M, M\ K ∩ L) → Applying the vanishing theorem, it follows:

→ 0−→ H^{Ψ n}(M, M \ K) ⊕ Hⁿ(M, M \ L)−→ H^{φ n}(M, M\ K ∩ L) →

Taking zK ∈ Hⁿ(M, M\ K) and z^L∈ Hⁿ(M, M\ L) R−fundamental classes given by our R−orientation, and noticing that they both map to z^K∩L,

φ(zK, zL) = zK∩L− z^K∩L = 0

1.2 Orientability in Z² and the Poincar´e Duality theorem 7 and therefore there exists a unique zK∪L such that:

Ψ(z_K∪L) = (zK, zL) Where zK∪L is an R−fundamental class of M at K ∪ L

Corollary 1.2.2.1. Let M be a connected, compact n−manifold.

If M is non orientable, then Hn(M ;Z) = 0.

If M is orientable, then

Hn(M,Z) → Hⁿ(M, M\ x; Z) ∼=Z is an isomorphism.

Proof. Since M \ x is connected but not compact, we have Hn(M \ x; π) = 0 for any group π. This leads to the following being a monomorphism:

Hn(M ; π)→ Hⁿ(M, M\ x; π) ∼= π Applying the Universal Coefficient Theorem, we have that

H_n(M ;Z) ⊗ Zq → Hn(M, M\ x; Z) ⊗ Zq ∼=Zq

is a monomorphism too, for q∈ Z, q > 0.

The fact that it has to be a monomorphism for every q, leaves us with no other choice than (if Hn(M ;Z) 6= 0),

Hn(M ;Z) ∼=Z

Then the generator of Hn(M ;Z) has to be sent into ± the generator of Hⁿ(M, M\ x;Z), in order to preserve the injectivity.

Therefore, the isomorphism has been proven.

The last important step, is to prove that

Theorem 1.2.3. Every closed, connected n−manifold M is orientable in Z² Proof. As proved in the beginning of this chapter, we have

Hn(M, M\ x; Z2) ∼=Z²

Since we have a connected manifold, defining an orientation is the same as choosing a generator of Hn(M, M\ x; Z²).

it is trivial that any open cover of M will determine an R−fundamental class at each open set, since Z² has a single generator.

By excision, for any two points x and y contained in a ball B,

H_n(M, M \ x) = Hn(B, B\ x) = Hn(B, B\ y) ∼= Hn(M, M\ y)

and therefore the two fundamental classes are mapped into the same element, making it an R−orientation.

We then have the Poincar´e duality theorem (the proof can be found in [3], it goes beyond the scope of this thesis):

Theorem 1.2.4. Let M be an R−oriented, n−manifold.

Then for an R−module π,

H^k(M ; π) ∼= H_n−k(M ; π)

Since Stiefel-Whitney classes will use coefficients in Z² and every manifold is Z² orientable, this will always hold throughout the next chapters.

Chapter 2

Stiefel-Whitney Classes

2.1 Axioms

The aim of this chapter is to describe the axioms that define the Stiefel-Whitney classes, and to introduce the total class.

Let B denote a topological space (when regarding vector bundles, it is referred to as base space).

For all the Stiefel-Whitney classes part, the homology and cohomology is going to be with coefficients in Z². This because every closed, connected n−manifold is orientable in Z², by Theorem 1.2.3.

There are four axioms that characterise the Stiefel-Whitney (cohomology) classes.

• Rank

To each vector bundle ξ, there corresponds a sequence of cohomology classes:

w_i(ξ) ∈ Hⁱ(B(ξ);Z2) such that:

– w0 = 1 ∈ H⁰(B(ξ);Z²)

– if ξ is an n-plane bundle, wi = 0 ∀ i > n

• Naturality

for each bundle map (let f : B(ξ)→ B(η) be the corresponding map between the base spaces)

wi(ξ) = f^∗wi(η) 9

• Whitney product theorem

Given η and ξ vector bundles over B,

wk(ξ⊕ η) = Xk

i=0

wi(ξ)wk−i(η)

• Normalisation

On the line bundle over P¹(R) (i.e. the twisted line bundle γ1¹ over the circle P¹), the class w1 is non-zero.

Definition 2.1.1. H^Π(B;Z²) is the ring of formal infinite series a = a₀+ a₁+ a₂+ . . .

where ai ∈ Hⁱ(B,Z²) with the product as:

(a0+ a1+ . . .)(b0+ b1+ . . .) = (a0b0) + (a1b0+ a0b1) + (a2b0+ a1b1+ a0b2) + . . . And the sum as:

(a₀+ a₁+ . . .) + (b₀+ b₁ + . . .) = (a₀+ b₀) + (a₁+ b₁) + (a₂+ b₂) + . . . This allows us to define the following:

Definition 2.1.2. The total Stiefel-Whitney class of ξ over B is defined as:

w(ξ) = 1 + w1(ξ) + w2(ξ) + . . . + wn(ξ) + 0 + 0 + ....

A first usage of the total classes is to simplify the Whitney product theorem.

In fact, it follows immediately:

w(ξ⊕ η) = w(ξ)w(η)

2.2 Stiefel-Whitney numbers 11

2.2 Stiefel-Whitney numbers

Using the Stiefel-Whitney classes, this chapter defines the Stiefel-Whitney num- bers.

These numbers are an important tool, that will be used to define a correspondence to the cobordism group, through Thom’s theorem.

Let M be a closed smooth n-manifold.

We have the fundamental homology class (unique!) µM ∈ Hⁿ(M ;Z²)

Therefore, for v∈ Hⁿ(M ;Z²), is defined the Kronecker index

< v, µM >∈ Z2

by the evaluation

Hⁿ(M ;Z²)⊗ Hⁿ(M ;Z²)→ Z²

For (r1, . . . , rn) non-negative integers such that r1+ 2r2 + . . . + nrn = n (in this way the degree of the monomial is going to be n), we have the following monomial, for any vector bundle ξ, (we will now consider the tangent bundle):

w1(τM)^r1 · . . . · wn(τM)^rn ∈ Hⁿ(M ;Z²) Definition 2.2.1. We define the Stiefel-Whitney numbers as:

< w1(τM)^r1 · . . . · wⁿ(τM)^rn, µM >

Note that two different manifolds M and M⁰ have the same Stiefel-Whitney numbers if: w₁^r1. . . w_n^rn[M ] = w^r₁¹. . . w_n^rn[M⁰] for every monomial w^r₁¹. . . w_n^rn of dimension n.

Theorem 2.2.1 (Pontrjagin). Let B be a smooth compact (n+1)-manifold with boundary M . Then, the Stiefel-Whitney numbers of M are all zero.

Proof. First, note that the natural homomorphism

∂ : Hn+1(B, M )→ Hⁿ(M ) maps µB into µM.

For any class v∈ Hⁿ(M ), we have the following (where δ : Hⁿ(M )→ Hⁿ⁺¹(B, M ) is the natural homomorphism)

< v, ∂µB >=< δv, µB >

Now, let us consider the tangent bundle. Choosing an Euclidean metric on τB, we have a unique outward normal vector field on M. This vector field spans a trivial line bundle ¹. We therefore have:

τ_B|M = τ_M ⊕ ¹

This means that the Stiefel-Whitney classes of τB|^M are the same as the Stiefel- Whitney classes wj of τM.

The last step is done using the exact sequence

Hⁿ(B)→ Hⁿ(M )→ Hⁿ⁺¹(B, M ) Since the left map is a restriction, we must have:

δ(w₁^r1. . . w_n^rn) = 0 Using the equality above, we then find:

< w^r₁¹. . . w^r_nⁿ, ∂µB >=< δ(w^r₁¹. . . w_n^rn), µB >= 0

Theorem 2.2.2. Thom

If all of the Stiefel-Whitney numbers of M are zero, then M can be realised as the boundary of some smooth compact manifold.

Proof. Due to the length and complexity of the proof, the proof will be treated as a separate chapter.

Definition 2.2.2. Two smooth n-manifolds belong to the same unoriented cobor- dism class if and only if their disjoint union is the boundary of a smooth compact (n + 1)−manifold.

The two theorems above give us the following

Corollary 2.2.2.1. Two smooth closed n-manifolds belong to the same cobordism class if and only if all of their Stiefel-Whitney numbers are equal.

Proof. The proof is immediate, once it is pointed out that we have coefficients in Z².

2.3 Grassmann manifolds 13

2.3 Grassmann manifolds

In order to be able to prove the existence and uniqueness of the Stiefel-Whitney classes, we need a basic notion of Grassmann manifolds. This chapter will briefly describe the Grassmann manifolds and the infinite Grassmann manifold.

Let R^∞ be the space of the infinite sequences of real numbers x = (x₀, x₁, x₂, . . .)

where only a finite number of xi is non zero.

Definition 2.3.1. The Grassmanian manifold Gn(R^n+k) is the set of all the n- dimensional planes through the origin in R^n+k.

It can be equipped with a topology, as a quotient space.

An n-frame is a n-tuple of linearly independent vectors in R^n+k.

Definition 2.3.2. The Stiefel manifold Vn(R^n+k) is the set of all the orthonormal n−frames in R^n+k.

Note that Vn(R^n+k) is an open subset of R^n+k× . . . × R^n+k .

We clearly have the function that maps each n-frame to the n-plane it spans:

q : Vn(R^n+k)→ Gn(R^n+k) Gn(R^n+k) is now given the quotient topology as:

U ⊂ Gn(R^n+k) is open ⇐⇒ q⁻¹(U ) ⊂ Vn(R^n+k) is open.

Definition 2.3.3. The infinite Grassmann manifold Gn = Gn(R^∞) is the set of all n-dimensional linear subspaces of R^∞. The topology is given as the limit of

G_n(Rⁿ)⊂ Gn(Rⁿ⁺¹)⊂ . . .

The space Gn(R^m) can be given a cell-subdivision, making it a CW−complex (see Appendix 4.5). Let X be a subspace of Rⁿ.

We clearly have:

0≤ dim X ∩ R¹

≤ . . . ≤ dim (X ∩ R^m) = n

Definition 2.3.4. A Schubert symbol σ = σ1. . . σn is a sequence of integers such that:

1≤ σ1 < σ2. . . < σn ≤ m For each Schubert symbol σ, we can define

e(σ)⊂ Gⁿ(R^m) the set of all n-planes X such that, for i = 1 . . . n:

dim(X∩ R^σi) = i dim(X∩ R^σi−1) = i− 1 X ⊂ Gⁿ(R^m) belongs to one and only one sets e(σ).

Note that e(σ) is an open cell of dimension

d(σ) = (σ1− 1) + (σ²− 2) + . . . + (σⁿ− n)

Then, one can verify that this leads to a well defined CW structure for both the Grassmannian manifold and the infinite one.

Definition 2.3.5. The infinite projective space P^∞ = G1(R^∞) Is defined as the limit of

P¹ ⊂ P² ⊂ P³ ⊂ . . .

Definition 2.3.6. A canonical universal bundle γⁿ over Gn is constructed as follows.

Let

E(γⁿ)⊂ Gn× R^∞

be the set of all the pairs of an n-plane in R^∞ and a vector in such plane.

We have the projection

π : E(γⁿ)→ Gn

that is defined by

π(X, x) = X

Definition 2.3.7. A topological space is paracompact if it is Hausdorff and if every open cover has a locally finite open refinement.

2.3 Grassmann manifolds 15 First of all, let us note that every compact space is trivially paracompact.

A paracompact space is of fundamental importance to us because it gives local compactness.

Theorem 2.3.1. Any Rⁿ-bundle ξ over a paracompact space admits a bundle map ξ → γⁿ

2.4 Uniqueness of Stiefel-Whitney classes

Theorem 2.4.1. The cohomology ring H^∗(Gn) is a polynomial algebra over Z², freely generated by w₁(γⁿ) . . . w_n(γⁿ).

To prove that, we will use the following lemma:

Lemma 2.4.2. There is no polynomial relation (with coefficients in Z2 ), among wi(γⁿ).

Proof. Suppose there is a polynomial relation p (a polynomial in n variables with coefficients inZ²).

For any n-plane bundle over a paracompact space, there exists a bundle map g : ξ→ γⁿ

We then have:

wi(ξ) = g^∗(wi(γⁿ)) Therefore, the polynomial relation

p(w1(ξ), . . . , wn(ξ)) = 0 becomes

g^∗p(w₁(γⁿ), . . . , w_n(γⁿ)) = 0

This means that our task is now to find an n-plane bundle so that there are no polynomial relations between w₁(ξ), . . . , w_n(ξ).

Let us now consider γ¹ over P^∞.

H^∗(P^∞,Z²) is a polynomial algebra with a single, one-dimensional generator a.

More, w(γ¹) = 1 + a.

Then, H^∗(P^∞× . . . × P^∞;Z²) is a polynomial algebra on n one-dimensional gen- erators a1. . . an. ai := π₁^∗(a), where the map is induced by the projection map πi : P^∞× . . . × P^∞→ P^∞.

Let ξ be an n-plane bundle over P^∞× . . . × P^∞

ξ = γ¹× . . . × γ¹ ∼= (π₁^∗γ¹)⊕ . . . ⊕ (πn^∗γ¹) The total Stiefel-Whitney class

w(ξ) = w(γ¹)× . . . × w(γ¹) = π^∗₁(w(γ¹)) . . . π_n^∗(w(γ¹))

2.4 Uniqueness of Stiefel-Whitney classes 17 is equal to the n-fold product

(1 + a)× . . . × (1 + a) = (1 + a¹)(1 + a2) . . . (1 + an) That means

w1(ξ) = a1+ a2+ . . . + an

w2(ξ) =P

i6=j

aiaj

...

wn(ξ) = a1a2. . . an

Since wk = k-th elementary symmetric function of a1. . . an, and the elementary symmetric functions share no polynomial relation, we are done.

We can now prove the theorem:

Proof. Denoting by C^r the group of mod 2 r-cochains, and |^r ⊂ B^r the corre- sponding cocycle and coboundary group, we have that the number of r-cells is equal to:

rk(C^r)≥ rk(Z^r)≥ rk(Z^r/B^r) = rk(H^r)

The number of r-cells in the CW-complex Gn is the number of partitions of r into at most n integers.

We also have that the distinct (linearly independent mod 2) monomials w1(γⁿ)^r1. . . wn(γⁿ)^rn ∈ H^r(Gn)

have also size equal to s the number of partitions of r into at most n integers.

To each sequence of non-negative integers r1, . . . , rn such that r₁+ 2r₂ + . . . + nr_n= r

it can be linked the following partition of r:

rn, rn+ rn−1, . . . , rn+ . . . + r1

By the linear independence of the monomials we have that all the ranks above have to be equal. H^r(Gn) is a polynomial algebra over Z², and the monomials

w1(γⁿ)^r1. . . wn(γⁿ)^rn form a base for it.

As last step, it follows that the natural homomorphism g^∗ : H^∗(Gn)→ H^∗(P^∞× . . . × P^∞)

is a isomorphism into the algebra of all polynomials in a1. . . anwhich are invariants under the permutations of 1 . . . n.

Theorem 2.4.3. Let ξ be a vector bundle over a paracompact base space.

There exists at most one correspondence ξ → w(ξ) such that w are Stiefel-Whitney classes.

Proof. Suppose there exist two correspondences, to w and ˜w.

By axioms 1 and 4, we have:

w(γ₁¹) = ˜w(γ₁¹) = 1 + a Embedding in P^∞, we get (from axioms 1 and 2)

w(γ¹) = ˜w(γ¹) = 1 + a We now pass to the n-fold Cartesian product

ξ = γ¹× . . . × γ¹ = π₁^∗γ¹⊕ . . . ⊕ πn^∗γ¹ And therefore, by axioms 2 and 3,

w(ξ) = ˜w(ξ) = (1 + a1) . . . (1 + an)

Apply now the bundle map ξ → γⁿ, also noticing that H^∗(Gn) injects monomor- phically into H^∗(P^∞×. . .×P^∞) (follows directly from the proof of Theorem 2.4.1).

We now have

w(γⁿ) = ˜w(γⁿ)

The last step, is to choose, for any n-plane bundle η over a paracompact base space, a bundle map f : η→ γⁿ.

We finally have:

w(η) = f^∗w(γⁿ) = f^∗w(γ˜ ⁿ) = ˜w(η)

2.5 Existence of Stiefel-Whitney classes 19

2.5 Existence of Stiefel-Whitney classes

Let ξ be an n−plane bundle.

Using the previous notation (E total space, B base space, π : E → B the projection map), let E0 be the set of all non-zero elements of E, and F0 the set of all non-zero elements of the fiber F = π⁻¹(b). Trivially, F0 = F ∩ E⁰.

The following holds:

Hⁱ(F, F0;Z²) =

0 i6= n Z2 i = n Hⁱ(E, E0;Z²) =

 0 i < n

Hⁱ⁻ⁿ(B;Z²) i≥ n

Theorem 2.5.1. Hⁱ(E, E0) = 0 for i < n, and Hⁿ(E, E0) contains a (unique) class u such that, for each fiber F = π⁻¹(b), u|(F,F0) is the (unique) non zero class in Hⁿ(F, F0)

Note that there is an isomorphism H^k(E)→ H^k+n(E, E₀) for every k, induced by x→ x ^ u.

Definition 2.5.1. Such u is the fundamental cohomology class.

Theorem 2.5.2. The Thom isomorphism is

φ : H^k(B)→ H^k+n(E, E0)

that is defined as the following composition of two isomorphisms (the first is in- duced by a section of the projection, that sends B into a deformation retract of E that is isomorphic to B).

H^k(B)→ H^k(E)→ H^k+n(E, E0)

Definition 2.5.2. We now need the Steenrod operations, that are described by the following properties:

1. ∀ Y ⊂ X, ∀ n, i ≥ 0 there is the following homomorphism:

Sqⁱ : Hⁿ(X, Y )→ Hⁿ⁺ⁱ(X, Y ) 2. f : (X, Y )→ (X⁰, Y⁰)⇒ Sqⁱ◦ f^∗ = f^∗◦ Sqⁱ

3. a∈ Hⁿ(X, Y )⇒

• Sq⁰(a) = a

• Sqⁿ(a) = a ^ a

• Sqⁱ(a) = 0 ∀ i > n 4. Sq^k(a ^ b) = P

i+j=k

Sqⁱ(a) ^ Sq^j(b)

We now can define the Stiefel-Whitney classes as:

wi(ξ) = φ⁻¹Sqⁱφ(1) that is equivalent to:

φ(w_i(ξ)) = Sqⁱφ(1) We further have:

π^∗wi(ξ) ^ u = φ(wi(ξ)) = Sqⁱφ(1) = Sqⁱ(u)

In order to simplify things, we can introduce the total squaring operation Sq(a) = a + Sq¹(a) + . . . + Sqⁿ(a)

that satisfies:

Sq(a ^ b) = Sq(a) ^ Sq(b) Sq(a× b) = Sq(a) × Sq(b) This because we can now write:

w(ξ) = φ⁻¹Sq φ(1) = φ⁻¹Sq(u) We shall now verify that the axioms are satisfied.

1. Rank

The first axiom follows directly from the properties 1 and 3 of the Steenrod operations.

2. Naturality

The bundle map f : ξ→ ξ⁰ induces g : (E, E0)→ (E⁰, E₀⁰).

u⁰ ∈ Hⁿ(E⁰, E₀⁰)⇒ g^∗(u⁰) = u And therefore

g^∗◦ φ = φ ◦ f^∗ composing by φ⁻¹ from left, we find

wi(ξ) = f^∗wi(ξ⁰)

2.5 Existence of Stiefel-Whitney classes 21 3. Whitney product theorem

Let u∈ Hⁿ(E, E0), u⁰ ∈ Hⁿ(E⁰, E₀⁰) be the respective fundamental classes.

We consider the Cartesian product

ξ⁰⁰= ξ× ξ⁰ and the following projection map:

π× π⁰ : E× E⁰ → B × B⁰ We then have:

u× u⁰ ∈ H^m+n(E× E⁰, (E× E0⁰) ^ (E0× E⁰)) well defined.

Let E₀⁰⁰= (E × E0⁰) ^ (E0× E⁰), clearly a subset of E⁰⁰.

We will now prove that E₀⁰⁰ is the set of non-zero vectors in E⁰⁰ (in other words, it is the fundamental class u⁰⁰ ∈ H^m+n(E⁰⁰, E₀⁰⁰)).

We proceed noticing that

u× u⁰|(F⁰⁰,F₀⁰⁰) = u|(F,F0)× u⁰|(F⁰,F₀⁰) 6= 0

We just proved that the restriction is non-zero, and therefore it is the fun- damental class.

Now, given a = π^∗(a), b = π^0∗(b), we have that

(a× b) ^ (u × u⁰) = (a ^ u)× (b ^ u⁰) From which immediately follows:

φ⁰⁰(a× b) = φ(a) × φ⁰(b)

We can finally compute the total Stiefel-Whitney classes:

φ⁰⁰(w(ξ⁰⁰)) = Sq(u⁰⁰) = Sq(u×u⁰) = Sq(u)×Sq(u⁰) = φ(w(u))×φ⁰(w(u⁰)) = φ⁰⁰(w(ξ)× w(ξ⁰)) And thus

w(ξ× ξ⁰) = w(ξ)× w(ξ⁰)

If the base space is the same for ξ and ξ⁰, we have (lifting to B):

w(ξ⊕ ξ⁰) = w(ξ) ^ w(ξ⁰)

4. Normalisation

Let γ₁¹ be the line bundle over P¹.

Quite intuitively, the space of vectors with length ≥ 1 ⊂ E(γ1¹) is a M¨obius band M , whose boundary is a circle S¹.

M is a deformation retract of E, and S¹ is a deformation retract of E₀. Therefore, it follows that

H^∗(M, S¹) = H^∗(E, E0) By Excision, we have (since P²− D² ∼= M )

H^∗(M, M₀) ∼= H^∗(P², D²) Therefore, we have the following isomorphisms:

Hⁱ(E, E0)→ Hⁱ(M, M0)← Hⁱ(P², D²)→ Hⁱ(P²)

Since u∈ H¹(E, E0) cannot be zero, it must correspond to the generator of H¹(P²). Therefore,

u ^ u = Sq¹(u) = Sq¹(a) = a ^ a Since a ^ a6= 0 (lemma),

w1(γ₁¹) = φ⁻¹Sq¹(u)6= 0

Chapter 3 Cobordism

3.1 Cobordism groups of smooth closed mani- folds

Definition 3.1.1. We say that two smooth closed manifolds M and N are cobor- dant if there exists a smooth compact manifold W such that

∂W = M q N

Definition 3.1.2. Nⁿis the set of cobordism classes of smooth closed n−manifolds.

Nⁿ can be given a group structure as follows:

• ∅ ∈ Nⁿ ∀n

• The addition is the disjoint union of manifolds, and trivially the disjoint union is commutative and associative.

Now, we clearly have, ∀M

∂(M × I) = M q M = M q M q ∅ = (M q M) q ∅ That means that M q M is cobordant to ∅, and thus M = −M.

This means that Nⁿ is an infinite dimensional vector space overZ². The Cartesian product of manifolds

Nⁿ× N^m → N^n+m

trivially defines a multiplication (whose identity is a point considered as a zero- dimensional manifold), that allows us to turn Nⁿ into a graded algebra overZ².

23

3.2 h-cobordism

Taking M as a smooth, closed (n + 1)−manifold, suppose we have

∂M = M1q M²

with M1, M2 smooth closed n−manifolds. Then we say that M is an h−cobordism if the inclusion maps M1 ,−→ M, M² ,−→ M are homotopy equivalences.

There is an important theorem, called h−cobodism theorem.

Theorem 3.2.1. For n≥ 5, M a compact (n + 1)−manifold as above, such that M, M1, M2 are simply connected.

Then, M is diffeomorphic to M1 × [0, 1].

This theorem is quite important since it implies that cobordant manifolds in dimension ≥ 5 are also homotopy equivalent. In general, the converse is not true.

3.3 Surgery

Cobordism comes out naturally from topology, when dealing with surgery.

Let M₁ be an (p + q)−manifold.

Given an embedding

φ : S^p× D^q → M we have the manifold

M2 = (M1− int im φ) ∪φ|_Sp×Sq−1 (D^p+1× S^q−1)

that is nothing but M1 where we performed the following surgery: first we cut out S^p× D^q, then we glued D^p+1× S^q−1.

It relates to cobordism as follows:

M = (M1× I) ∪S^p×D^q×{1}(D^p+1× D^q) is an (p + q)−manifold with boundary

∂M = M1q M²

3.3 Surgery 25 In fact, we started from M1 × I and attached a handle D^p+1× D^q.

An example is the following: we start from M1 = S¹ × D⁰ and we do a surgery cutting out S⁰ × D¹. Basically, we start from a circle and cut out two disjoint segments. If we now glue D¹×S⁰ we can obtain two copies of the circles, therefore M₂ = S⁰× S¹.

The edges that come out naturally from the surgery can always be smoothened.

M then will be a cylinder to which we attach a 1−handle (it is attached as follows:

one extremity of the handle in each of the two disjoint circles that are the boundary of the cylinder S¹× I: ∂(S¹× I) = S¹q S¹).

Chapter 4

Thom’s Theorem

This chapter will go through the proof of Thom’s theorem, and therefore com- pletes proving the relation that exists among the Stiefel-Whitney numbers and the cobordism group: two manifolds are cobordant if and only if they have the same Stiefel-Whitney numbers. If two manifolds M1, M2 have the same Stiefel-Whitney numbers, then trivially the manifold M = M₁q M2 that is obtained by a disjoint union of them has Stiefel-Whitney numbers zero, and what we are going to prove is that if this is the case, then M can be realised as the boundary of some smooth, compact manifold.

Theorem 4.0.1. Thom

If all of the Stiefel-Whitney numbers of M are zero, then M can be realised as the boundary of some smooth compact manifold.

We are going to start with some definitions needed in order to prove the theo- rem.

Definition 4.0.1. Given an n−plane bundle ξ, define Sph(E) as the one-point compactification of each fiber of ξ, obtaining spherical fibers of dimension n.

Definition 4.0.2. The Thom space is defined to be T (ξ) = Sph(E) /B

constructed as compactifying all the fibers, and then identifying all the points at

∞ of every fiber.

27

We define Gr,n to be the space of unoriented r−planes in R^n+r, and γ_n^r the r−plane bundle over it, that consists of the pairs:

{r−plane in R^n+r; point in that plane} From here, we can define the space

BO(r) = lim

n→∞Gr,n

and the universal plane bundle is constructed in the same way:

γ^r = lim

n→∞γ_n^r

Finally, we define T O as to be the Thom space of this bundle.

Denote

fn: Bn→ BOⁿ a fibration (see appendix 4.5).

Let ξ be a vector bundle over a space X

X Bn

X× I BO_n

ξ˜

X×{0} fn

H

By the definition of fibration, one can easily see that the outer square of the diagram above commutes.

Let

i : M → R^n+r be an embedding.

This defines a map of the vector bundles from the vectors normal to M into γ_n^r. Including it in γ^r, we have a bundle map of the normal bundle to i(M ) into γ^r. We can now define the induced map on the base spaces:

ν(i) : M → BO^r

Definition 4.0.3. We call a (Bn, fn) structure on an n−dimensional vector bundle ξ over X, classified by ξ : X → BOn, is a homotopy class of liftings of ξ to Bn. In other words, it is an equivalence class of maps

ξ : X˜ → Bⁿ s.t. fn◦ ˜ξ = ξ

29 The equivalence is defined by ˜ξ1 ∼ ˜ξ2 ⇐⇒ they are homotopic by the homotopy H:

fn◦ H(x, t) = ξ(x) ∀(x, t)

Suppose that our bundle ξ is induced from the two bundles η : Y → BO^r

and

g : X → Y Then, the bundle map

g∗ η = ξ → η gives rise to

T g : T ξ → T η The inclusion:

Jr : BOr → BO^r+1

induces a vector bundle j^∗(γ^r+1) over BOr defined by the Whitney sum of γ^r and the trivial line bundle.

Considering T j_r^∗(γ^r+1) as ΣT γ^r, we have the commutative diagram ΣT Br T Br+1

ΣT BOr T BOr+1 T gr

ΣT fr T fr+1

T jr

and the homomorphism

T gr◦ Σ : πn+r(T Br,∞) → πn+r+1(T Br+1,∞) Denote by  the trivial line bundle.

The Thom space T (γq)⊕  has a canonical homorphism to the suspension ΣT O(q) Also note that the bundle map

γ^q⊕  → γq+1

induces a map:

σq: ΣT O(q) → T O(q + 1)

T O(q) and σq define a prespectrum T O, whose homotopy group is π_n(T O) = colim π_n+kT (k)

and BO is defined to be the base space of T O.

4.1 Outline of the proof

To prove the theorem, we will prove that this diagram is well defined and commutes:

Hⁿ(BO)⊗ Nn Hⁿ(BO)⊗ πn(T O) Hⁿ(BO)⊗ Hn(T O)

Z² Hⁿ(BO)⊗ Hn(BO)

id⊗α

#

id⊗h

id⊗Φ

<,>

If the Stiefel-Whitney numbers of M are zero, then w#[M ] = 0 ∀ w ∈ Hⁿ(BO)

Since we supposed that the diagram was commutative, it is the same as

< w, Φ◦ h ◦ α([M]) >= 0

And, considering that <, > is nothing but an evaluation of dual vector spaces (chains and cochains), it is equivalent to

Φ◦ h ◦ α([M]) = 0

The last step, is that Φ and α are isomorphism, and h is a monomorhism therefore it boils down to [M ] = 0 and therefore M is a boundary.

We will now proceed describing the morphisms above, and proving that the di- agram is commutative (note that we can prove the first line without considering the tensor product Hⁿ⊗).

4.2 N

→ π

(T O) is an isomorphism

• Definition of the homomorphism

First of all, we have to define the map α :Nⁿ→ lim_r

→∞πn+r(T Br,∞)

The proof uses the famous Pontryagin-Thom construction, also called Thom space.

We can summarise the Thom space as follows:

4.2 Nn → πn(T O) is an isomorphism 31 – Embed M into R^k+n. For r sufficiently large, any two embeddings of

M are homotopic.

– We take a tubular neighbourhood¹ N of M . – We make a map

R^k+n→ R^k∪ {∞} ∼= S^k by sending what is out of N to ∞.

We can now go through the construction in a rigorous way.

Let M ∈ Nn. Let N denote the total space of the normal bundle of M , that can be thought as a subspace of R^n+r × R^n+r.

The map

e :R^n+r × R^n+r → R^n+r : (a, b)→ a + b maps it differentiably to R^n+r.

If we consider M = M×0, the map above restricts trivially to the embedding of M into R^n+r.

Denoting N the subspace of N of vectors of length less than  we have that, for a sufficiently small  > 0, N is embedded by e|^N.

We now define a map

S^n+r → T fr^∗(γ^r) We consider S^n+r asR^n+r ∪ ∞ and the map

c : S^n+r → N^

∂N

that collpases all points of S^n+r outside ∂N to a point.

We can now define another map:

N

∂N → T N induced by the multiplication by ⁻¹.

We define the map:

n× (ν ◦ π) : N → γ^r× Br

where n is n applied to the inclusion γ_n^r → γ^r

1For more insight about the tubular neighbourhood, see [1] p. 115.

and n is the bundle map

n : Br → γn^r

π is the projection

π : N → M and

ν : M → Br

is a lifting of the normal bundle:

f ν(i) = ν(i)

Such map is a bundle map into f_r^∗(γ^r), and induces the map:

T (n× (ν ◦ π)) : T N → T Br

Applying ⁻¹ and c, we get the map of pairs:

θ := T (n× (ν ◦ π)) ◦ ⁻¹◦ c : (S^n+r,∞) → (T Br,∞) The map:

i : M → R^n+r ⊂ R^n+r+1 gives rise to

α := T gr◦ Σθ : Nn → lim

r→∞πn+r(T Br,∞) that defines an element of

rlim→∞πn+r(T Br,∞)

and passing to the colimit we define the map from the cobordism group to the homotopy group.

One now has to check that α is well defined, invariant, it is an isomorphism and then solve the homotopy problem, i.e. that two homotopic manifolds have the same Stiefel-Whitney numbers.

• Invariance

If we take a smaller , the two maps ⁻¹◦ c will be homotopic and therefore they will not change the homotopy class of α.

If we use a different lifting ν, we will obtain a homotopy of T (n× (ν ◦ π)).

4.2 Nn → πn(T O) is an isomorphism 33

• The class of α is independent of the embedding of M

Let W be a (B, f ) manifold, and

j : M + ∂W → R^n+r an embedding with a lift

˜

ν : M + ∂W → Br

such that it gives the same (Br, fr) structure on M . Let

H : M× I → R^n+r

be a homotopy of i and j|M (that exists since any two embeddings are homo- topy equivalent if the space in which they are embedded is sufficiently big) such that

H(x, t) =

 i(x) if t < δ1

j(x) if t < 1− δ2

Let

k : W → R^n+r × (0, 1]

be j×1 on ∂W and embedding a tubular neighbourhood of ∂W orthogonally along j(∂W )× 1.

We can define an embedding on a close neighbourhood of the boundary (we chose δ1, δ2 appositely for this) as follows

(H × π2) + k : M× I + W → R^n+r× I

We can use a homotopy fixed on that neighbourhood to obtain the embed- ding.

F : M× I + W → R^n+r× I

Notice that F|M×I is a regular homotopy and it gives us a covering map A : M× I → Br that is ν on M× 0. Since the normal map is constant near M × 1, we can make A agree with ˜ν on M × 1.

Now we focus on W . Since the (B, f ) structure on ∂W is induced by the one of W , we can find a lift that agrees with ˜ν on ∂W .

From here, the homotopy is preserved up until the homotopy group. In fact, we have

M × I + W −→ R^{F n+r}× I → S^n+r× I −→ N^{c }

∂N

⁻¹

−−→ T N and thus any two embeddings are sent into the same homotopy class.

• α is a homomorphism

Take M1 and M2 two classes in Nⁿ, with i1, i2 the embeddings in R^n+r. We embed them such that the last coordinate is positive for i1 and negative for i2.

Using the map

d : S^n+r → S^n+r∨ S^n+r

that just collapses the equator to a point, and we denote α1 = α([M1]) and α₂ = α([M₂]) we clearly have

α([M1] + [M2]) = (α1∨ α²)◦ d

We can do this as long as we choose  small enough such that when passing to N we still remain in the half sphere.

• α is an isomorphism

We now only need to prove that α is injective and surjective. To be able to prove them one must use the transversality theorem, and it’s beyond the scope of this thesis. Proofs can be found in [2] and [3].

4.3 π

(T O) → H

(T O) is a monomorphism

We start from the Hurewicz homomorphism:

Theorem 4.3.1. For any space X, ∀k ∈ Z, there exists a homomorphism h : πn(T ) → Hⁿ(T ;Z)

Then, we compose it with the map

Hn(T ;Z) → Hⁿ(T ; R) and we considerZ2, so we have the final map

h : πn(T )→ Hⁿ(T ;Z²)

4.4 Hn(T O)→ Hn(BO) is an isomorphism 35

4.4 H

(T O) → H

(BO) is an isomorphism

Using Theorem 2.5.2, we can prove the following version of the Thom isomor- phism:

Theorem 4.4.1. Given an n−plane bundle ξ : E → B we have the isomorphism

φ : H^k(B; R)→ ˜H^k+n(T ξ; R) given by

φ(x) = x ^ u where u is the fundmanetal class:

u∈ ˜Hⁿ(T ξ; R) for the bundle ξ.

Proof. Theorem 2.5.2 states that the following is an isomorphism:

φ : H^k(B)→ H^k+n(E, E0) On the right side, we have:

H˜^∗(T ξ) ∼= H^∗(Sph(E), B) ∼= H^∗(Sph(E), Sph(E)0) ∼= H^∗(E, E0)

Where E0 and Sph(E)0 are the subspaces of E and Sph(E) obtained by deleting {0} from each fiber.

We therefore have the isomorphism:

Φr : Hⁿ(BO(r))→ ˜H^n+r(T O(r))

From here, the following diagram commutes since ∆ (the diagonal map) is natural.

The definition of ∧ can be found in Appendix 4.5.

ΣT O(r) BO(r)+∧ ΣT O(r)

ΣT O(r + 1) BO(r + 1)₊∧ T O(r + 1)

∆

σq ir∧σr

∆

By means of some diagram chasing, one can prove that the following diagram is also commutative:

Hⁿ(BO(r + 1)) Hⁿ(BO(r))

H˜^n+r+1(T O(r + 1)) H˜^n+r+1(ΣT O(r)) H˜^n+r(T O(r))

i^∗_r

Φr+1 Φr

σ_r^∗ id⊗h

That yields a stable isomorphism

Φ : Hⁿ(BO)→ Hⁿ(T O)

4.5 The diagram commutes

The final step, is to prove the commutativity of the diagram:

Hⁿ(BO)⊗ Nn Hⁿ(BO)⊗ πn(T O) Hⁿ(BO)⊗ Hn(T O)

Z2 Hⁿ(BO)⊗ Hn(BO)

id⊗α

#

id⊗h

id⊗Φ

<,>

Proof. First, we embed M in R^n+r. Let ν be the normal bundle and

f : M → BO(r) classify ν.

The class of M : α([M ]) is given by

S^{n+r t}−→ T ν −→ T O(r)^{T f} Passing to homology:

H˜n+r(S^n+r) H˜n+r(T ν) H˜n+r(T O(r))

H_n(M ) H_n(BO(r))

t_∗ T f_∗

Φ Φ

f_∗

4.5 The diagram commutes 37

Call in+r ∈ ˜Hn+r(S^n+r) the fundamental class. By the diagram, we have:

f_∗◦ Φ ◦ t∗ = Φ◦ T f ◦ t∗

Applying it to the fundamental class, we have

(Φ◦ T f ◦ t∗)(in+r) = (Φ◦ h ◦ α)([M])

If z = (Φ ◦ t∗)(in+r) ∈ Hⁿ(M ) is the fundamental class², we have (for w ∈ Hⁿ(BO(r)))

w#[M ] =< w(ν), z >=< f^∗w(γr), (Φ◦ t∗)(in+r) >

that is equivalent to (remind that we are evaluating chains on cochains)

< f^∗w(γr), (Φ◦ t∗)(in+r) >=< w(γr), (f_∗◦ Φ ◦ t∗)(in+r) >

and finally, using the equality above:

< w(γr), (f_∗◦ Φ ◦ t∗)(in+r) >=< w(γr), (Φ◦ h ◦ α)([M]) >

Finally, we have

w#[M ] =< w(γ_r), (Φ◦ h ◦ α)([M]) >

And thus the commutativity of the diagram is proved.

Thom’s theorem is therefore proved and we have a way to compute the cobordism class of manifolds, since two manifolds are cobordant if and only if they have the same Stiefel-Whitney numbers.

2To prove this one can show that z maps to a generator of Hn(M, M\ {x}) for every x ∈ M.

However, it requires some machinery and it is beyond the scope of this paper. The proof can be found in: