An Introduction to the Chern Conjecture

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

An Introduction to

the Chern Conjecture

Candidato:

Relatore:

Andrea Clini

Prof. Roberto Frigerio

Contents

Introduction . . . iii

1 Getting Started 1

1.1 Connections and Affine Manifolds . . . 1 1.2 The Euler Class . . . 9 1.3 Characteristic Classes and Classifying Spaces . . . 17

2 Flat Manifolds and the Chern Conjecture in Dimension 2 33

2.1 Flat Bundles . . . 33 2.2 Milnor’s Inequality and the Chern Conjecture in Dimension 2 . . . 37 2.3 The Counterexample of Smillie . . . 47

3 Chern-Weil Theory and Complete Affine Manifolds 53

3.1 Chern-Weil Theory . . . 53 3.2 Chern-Weil Theory in GLnR . . . 70 3.3 The Chern Conjecture for Complete Affine Manifolds . . . 82

Introduction

The Chern conjecture predicts the vanishing of the Euler characteristic of affine mani-folds, a particular kind of smooth manifolds carrying a rather rigid differentiable struc-ture. It is therefore a statement that links together topological and geometric concepts. One of the first classical results providing a bridge between topology and geometry is of course the Gauss-Bonnet Theorem. This theorem furnishes a formula for the Euler characteristic of a closed oriented surface Σ:

1 2π

Z

Σ

κ dA =

χ

(Σ). (1)

Here κ is the gaussian curvature of Σ associated to some riemannian metric and dA is the volume form determined by the metric and the orientation.

The higher dimensional extension of this formula had to wait until 1945, when Chern [Che45] proved his generalized version, the so called Chern-Gauss-Bonnet Theorem. Here M is a closed oriented 2m-dimensional riemannian manifold and 1

2πκ in (1) is replaced

by a closed differential form associated to the Riemann curvature tensor K. Explicitly we have: 1 (2π)m Z M Pf(K) =

χ

(M ) (2)

where Pf is a particular homogeneus polynomial called the Pfaffian. In particular, the corresponding cohomology class  1

(2π)m Pf(K), called the Euler class of (the tangent bundle of) M , turns out to be indipendent of the metric. This is in fact what is called a differential topological invariant: it depends only on the tangent bundle of M considered as a topological vector bundle and not on the differentiable structure. Using different methods, the Euler class can be equivalently defined for any topological oriented rank n vector bundle ξ = (E, B, p) as an element e(ξ) ∈ Hn_{(B) in}

the cohomology of the base space. In case B is a closed oriented smooth manifold and ξ = T B is its tangent bundle, evaluating the Euler class on the fundamental class of the manifold yields:

he(T B), [B]i =

χ

(B).

The vanishing of the Euler characteristic of a smooth manifold is therefore the same as the vanishing of the Euler class of its tangent bundle.

An affine manifold is a smooth manifold with an affine structure, that is an atlas of charts whose changes of coordinates are (locally) affine transformations. The existence of such a structure is equivalent to the existence of a flat torsion free connection on the tangent bundle.

Around 1950 Chern tried to understand which topological constraints an affine struc-ture imposes on a closed smooth manifold. He apparently proposed the following: Conjecture (Chern, 1955). Let M be a closed affine manifold, then

χ

(M ) = 0. Since the Euler characteristic is multiplicative under passage to a finite covering space, in the conjecture one can assume that M is oriented; also, as a closed odd dimensional manifold has zero Euler characteristic by Poincaré duality, one can assume that M has even dimension. As a reason for the conjecture, notice that if the flat torsion free con-nection corresponding to the affine structure happens to be the Levi-Civita concon-nection of some riemannian metric on M , then the curvature form vanishes and hence (2) immedi-ately gives

χ

(M ) = 0. However not every flat torsion free connection is the Levi-Civita connection of some metric (see [Sch73]), and hence Chern-Gauss-Bonnet Theorem does not solve the conjecture.

In the late ’40s, in the wake of ideas following this theorem, Chern and Weil started developing a new theory, which was an important step forward in the study of charac-teristic classes. Chern-Weil theory computes topological invariants of fibre bundles on a smooth manifold by evaluating particular homogeneus polynomials on the curvature form of some suitable connection on the bundle.

For an oriented vector bundle ξ, most of these invariants can be computed using any connection on ξ. Hence, in the case of the tangent bundle of an affine manifold, using the associated flat torsion free connection, one immediately sees that they must vanish. This is not however the case for the Euler class: this must be computed using a metric connection, i.e. a connection compatible with some riemannian metric on the bundle ξ. In particular the Chern conjecture is not trivially solved by this construction, and is not a general statement on flat vector bundles.

One could nevertheless ask whether it is a statement on flat, not necessarily torsion free, connections on tangent bundles. In [Ben55] Benzécri proved the Chern conjecture for closed 2-manifolds: among them only tori admit affine structures. In [Mil58] Milnor proved his celebrated inequality:

Theorem (Milnor). A rank 2 oriented vector bundle E over the closed oriented surface Σg of positive genus g admits a flat connection if and only if |he(E), [Σg]i| < g.

This implies, in particular, the following stronger version of Benzécri’s result: the tangent bundle of a closed oriented surface Σ admits a flat, not necessarily torsion free, connection if and only if

χ

(Σ) = 0. Milnor asked if this result could be generalized in all dimensions. However in [Smi77] Smillie showed the following:

Theorem (Smillie). For any n ≥ 2 there exist closed 2n-dimensional manifolds with non-vanishing Euler characteristic whose tangent bundle admits a flat connection (with nonzero torsion).

Hence the Chern conjecture is really a question on affine structures, and not on flat connections on tangent bundles.

The simplest examples of affine manifolds are the complete ones, namely, the ones that are geodesically complete with respect to the associated connection. Equivalently they can be expressed as a quotient Rm

/Γ for some discrete subgroup Γ ⊆ Aff(Rm_{) with free}

and properly discontinuous action. Using a Chern-Weil theory based strategy, Kostant and Sullivan [KS75] managed to prove the conjecture for this class of affine manifolds. Theorem (Kostant-Sullivan). If M is a closed affine complete manifold, then

χ

(M ) = 0.

Other partial results were then achieved by Hirsch, Thurston and Goldman (see [HT75] and [GH81]) and also recently by Bucher and Gelander [BG11]. Finally, in 2015 Klingler [Kli17] proved the conjecture for special affine manifolds, that is affine manifolds admitting a parallel volume form. Markus conjectured in 1960 that a closed affine man-ifold is special if and only if it is complete (see [HT75]). This conjecture is still largely open.

The aim of this thesis is to give an introduction to the Chern conjecture through the theorems stated above, as well as to give an exposition of the necessary background about characteristic classes, with particular focus on Chern-Weil theory.

The work is structured as follows. In Chapter 1 we set the context for our problem. We recall the main facts about connections on vector bundles and introduce the reader to affine manifolds. Next we focus on the Euler class: we give a topological definition, which will suffice for the first two chapters, and study some of its properties. The remaining section is devoted to three major topics: principal bundles, characteristic classes and classifying spaces.

Chapter 2 starts with the study of flat vector bundles: we give a complete characteri-zation of these bundles in order to set the conjecture in the right context. In the following section we prove Milnor’s inequality and then easily deduce the solution of the conjecture in dimension 2. In Section 2.3 we finally present Smillie’s construction of manifolds with flat tangent bundle and nonzero Euler characteristic.

Chapter 3 is entirely devoted to Chern-Weil theory. We extend to principal bundles the concepts of connections and curvature and work out all the details of the general theory. In the next section we specialize to the case of vector bundles; we give a differential

definition of the Euler class and prove its equivalence with the topological one we gave before. We also prove the Chern-Gauss-Bonnet Theorem. Finally, we present Kostant and Sullivan’s solution of the conjecture for affine complete manifolds.

Chapter 1

Getting Started

1.1

Connections and Affine Manifolds

Connections on Vector Bundles

Affine manifolds will be the main object of this work. We shall see that for a manifold M the condition of being affine is equivalent to having a connection on its tangent bundle T M which satisfies certain properties. Therefore, let us recall some definitions and facts about connections on an n-vector bundle p : E → M .

Definition 1.1.1. A connection in a vector bundle p : E → M over a smooth manifold M is bilinear map

∇ : X(M ) × Γ(E) → Γ(E) satisfying the following conditions:

(1) ∇f Xs = f ∇Xs,

(2) ∇X(f s) = f ∇Xs + X(f )s,

where f ∈ C∞(M ), X ∈ X(M ), s ∈ Γ(E). That is, in the sense of Definition 3.1.1, a connection is a linear map

∇ : Γ(E) → Ω1_{(M, E).}

We call ∇Xs the covariant derivative of s relative to X.

Also, we define its curvature:

Definition 1.1.2. Let ∇ be a connection in a vector bundle p : E → M . The map defined by

R(X, Y )(s) = (∇X∇Y − ∇Y∇X− ∇[X,Y ])(s), s ∈ Γ(E), X, Y ∈ X(M )

is called the curvature of the connection.

Definition 1.1.3. A connection ∇ is called flat if its curvature R is identically zero. Let us recall the usual properties of the curvature. The easy proof is left to the reader. Lemma 1.1.4. The curvature R has the following properties. For any X, Y ∈ X(M ), f, g, h ∈ C∞(M ) and s ∈ Γ(E), we have

(i) R is multilinear. (i) R(Y, X) = −R(X, Y ).

(ii) R(f X, gY )(hs) = f ghR(X, Y )(s).

That is, in the sense of Definition 3.1.1, it defines a linear map R : Γ(E) → Ω2_{(M, E).}

Remark 1.1.5. Let ∇ be a connection in p : E → M and R its curvature. Given an open set U ⊆ M and a local frame s1, . . . , sn ∈ Γ(E|U), for any vector field X ∈ X(U )

we may write ∇X(sj) = n X i=1 θij(X)si

where, by the properties of ∇, each θij is an ordinary 1-form on U , i.e. θij∈ Ω1(U ). So,

denoting them collectively as θ = (θij), we get a 1-form that takes its values in M (n, R)

and we write θ ∈ Ω1_{(U, M (n, R)). Notice that the form θ completely determines ∇ over} U . This is called the connection form of ∇ on U relative to the frame s1, . . . , sn.

Now we do the same for the curvature. For any vector fields X, Y ∈ X(U ) we may write R(X, Y )(sj) = n X i=1 Θ(X, Y )ijsi,

and by Lemma 1.1.4 we have

Θij(X, Y ) = −Θij(Y, X), Θij(f X, gY ) = f gΘij(X, Y )

for every i, j. Hence Θ = (Θij) defines a 2-form with values in M (n, R), i.e. Θ ∈

Ω2_{(U, M (n)), called the curvature form of R on U relative to the frame s}

1, . . . , sn.

In the case of the tangent bundle T M of a riemannian manifold M , let ϕ : M ⊇ U → Rnbe a chart with coordinates (x1, . . . , xn) and consider the induced frame _∂x∂

1, . . . ,

∂ ∂xn on T M . Then, for the Levi-Civita connection ∇ of M , we have that θ(_∂x∂

i)jh= Γ

h ij are

the Christoffel symbols and Θ(_∂x∂ i,

∂

∂xj)kl = K

l

ijk are the components of the Riemann

curvature tensor K.

A connection and its curvature are related by the following structural equation. Theorem 1.1.6. Let E → M be a vector bundle with connection ∇ and curvature R. The connection and curvature forms on an open set U ⊆ M relative to some frame of E are related by dθ = −θ ∧ θ + Θ. (1.1) Componentwise we have dθij = − n X k=1 θik∧ θkj+ Θij. (1.2)

Proof. Take an open subset U ⊆ M , vector fields X, Y ∈ X(M ) and a local frame s1, . . . , sn. A straightforward computation gives

R(X, Y )(sj) = ∇X∇Ysj− ∇Y∇Xsj− ∇[X,Y ]sj

=X(θij(Y ))si− Y (θij(X))si− θij([X, Y ])si+ θik(X)θkj(Y )si− θik(Y )θkj(X)si,

where repeated indices are summed over. Now if we substitute the identities dθij(X, Y ) = X(θij(Y )) − Y (θij(X)) − θij([X, Y ]), θik∧ θkj(X, Y ) = θik(X)θkj(Y ) − θik(Y )θkj(X), we get R(X, Y )(sj) = dθ(X, Y )ijsi+ θik∧ θkj(X, Y )si. Since we have R(X, Y )(sj) = Θij(X, Y )si

Finally, we see how the connection and curvature forms change when we use two different local trivializations of the vector bundle.

Proposition 1.1.7. Let p : E → M be a vector bundle with a connection ∇. Given two open subsets Uα, Uβ ⊆ M with local trivializations ϕα : p−1(Uα) → Uα× Rn and

ϕβ : p−1(Uβ) → Uβ× Rn, let gαβ : Uα∩ Uβ → GLnR be the corresponding transition function. Denote by θα, Θαand θβ, Θβ the connection and curvature forms on the open

sets Uα and Uβ relative to the frames induced by the trivializations. Then we have

θβ= g−1αβθαgαβ+ gαβ−1dgαβ, (1.3)

Θβ = g−1αβΘαgαβ. (1.4)

Proof. Let sα1, . . . , sαn and s β 1, . . . , s

β

n be the local frames induced on Uα and Uβ by ϕα

and ϕβ respectively. For simplicity, let us write g = gαβfor the corresponding transition

function and recall that repeated indices are summed over. Then, on Uα∩ Uβ, we have

sβ_j = gijsαi.

Applying ∇X on both sides we get

∇Xs β j = (θβ)kj(X)s β k = (θβ)kj(X)giksαi = ∇X(gijsαi) = X(gij)siα+ gij(θα)ki(X)sαk,

and we conclude that

gik(θβ)kj(X) = dgij(X) + (θα)ki(X)gij.

Since this holds for arbitrary X and indices i, j, we conclude that gαβθβ= dgαβ+ θαgαβ,

and multiplying on the left by g_αβ−1 we get (1.3).

Now we deal with the second equation. From Theorem 1.1.6 we have Θβ= dθβ+ θβ∧ θβ.

Now we take the exterior derivative on both sides of (1.3). Here functions and 1-forms appear as matrices, but their exterior derivative can be easily handled by the usual rules. For instance, if we write g for gαβ for simplicity, then exterior differentiation of

g−1g = I, where I is the identity matrix, gives dg−1g + g−1dg = 0, from which we get dg−1= −g−1dgg−1. Now we can compute as follows:

Θβ= dθβ+ θβ∧ θβ

= −g−1dgg−1∧ θαg + g−1dθαg − g−1θα∧ dg − g−1dgg−1∧ dg

+ (g−1θαg + g−1dg) ∧ (g−1θαg + g−1dg)

= g−1(dθα+ θα∧ θα)g = g−1Θαg.

Thanks to Remark 1.1.5 and Proposition 1.1.7, it is straightforward to prove the following:

Corollary 1.1.8. Let p : E → M be a vector bundle. Let U = { Uα}α be an open

covering of M such that we have local trivializations ϕα: EUα→ Uα×R

n _{with transition}

functions gαβ. There is a 1-1 correspondence between connections on E and collections

a connection ∇ to the collection { θα}_α of its local connection forms relative to the

trivializations ϕα.

Remark 1.1.9. The covariant derivative ∇Xs(p) at the point p ∈ M depends only on

the value of X(p) and the restriction of s along a curve tangent to X at p. To see this, take an open neighbourhood U ⊆ M of p and a local frame s1, . . . , snof the vector bundle

E → M , so that ∇ has connection form θ and moreover we can write

s =

n

X

i=1

αisi

for some α1, . . . , αn ∈ C∞(U ). Then, using properties of ∇, we get

∇Xs(p) = n X i=1 X(αi)(p)si(p) + n X i=1 αi(p)θki(X(p))sk(p),

and this expression depends only on X(p) and on the values of the terms X(αi)(p), which

in turn depend only on the restriction of s to a curve tangent to X in p.

A connection on the vector bundle π : E → M gives us a well-defined way to derive sections along a curve c : I → M . That is, maps s : I → E such that s(t) ∈ Ec(t) for all

t ∈ I.

Proposition 1.1.10. Let π : E → M be a vector bundle over the manifold M , endowed with a connection ∇, and let c : I → M be a differentiable curve. There exists a unique correspondence which associates to a section S along c another section DS_dt along c, called the covariant derivative of S along c, such that:

(a) _dtD(S + S0) = DS_dt +DS_dt0.

(b) _dtD(f S) = df_dtS + fDS_{dt, where f : I → R is a differentiable function.} (c) If S is induced by a section ¯S ∈ Γ(E), i.e. S(t) = ¯S(c(t)), then DS_dt = ∇dc

dt ¯ S. Proof. Let us suppose initially that there exists a correspondence satisfying (a), (b) and (c). Let ϕ : E|U → U × Rn be a local trivialization over the open subset U ⊆ M such

that U ∩ c(I) 6= ∅. Let s1, . . . , sn be the induced local frame. Then we can write the

section S locally as

S(t) =X

j

aj(t)sj(c(t))

for some differentiable functions a1(t), . . . , an(t). By conditions (a) and (b), we have

DS dt = X j daj dt sj+ X j aj Dsj dt . By condition (c) we have Dsj

dt = ∇dc/dtsj, and hence we get

DS dt = X j daj dt sj+ X i,j ajθij(dc/dt)si (1.5)

where θ is the local connection form of ∇ relative to the trivialization ϕ. Then expression (1.5) shows that if there is a correspondence satisfying the conditions above, then it is unique.

To show existence, define DS_dt over U by formula (1.5). It is easy to verify that (1.5) possessed the required properties. If W ⊆ M is another open subset with a local trivialization E|W ∼= W × Rn and we define DSdt over W by (1.5), the definitions agree on

U ∩ W , by uniqueness of DS_dt on U . It follows that the definition can be extended over all of M , and this concludes the proof.

Definition 1.1.11. Let π : E → M be a vector bundle over the manifold M , endowed with a connection ∇. A section S along a curve c : I → M is called parallel if DS_dt = 0 for all t ∈ I.

Proposition 1.1.12. Let π : E → M be a vector bundle over the manifold M , endowed with a connection ∇. Let c : I → M be a differentiable curve and let S0be a point in the

fibre over c(t0), for some t0∈ I, that is S0 ∈ Ec(t0). Then there exists a unique parallel section S along c such that S(t0) = S0. The section S is called the parallel transport of

S0 along c.

Proof. Suppose that the theorem was proved for the case in which c(I) is contained in an open subset U ⊆ M such that E|U ∼= U × Rn is trivial. By compactness, for any t ∈ I,

the segment c([t0, t]) ⊆ M can be covered by a finite number of trivializing open subsets,

in each of which S is defined, by hypothesis. From uniqueness, the definitions coincide when the intersections are not empty, thus allowing the definition of S along all of [t0, t].

Therefore we only have to prove the theorem when c(I) is entirely contained in a trivializing open subset U ⊆ M . Let s1, . . . , sn be a local frame of E over U and let

S0=P aj0sj(c(t0). Suppose that there exists a parallel section S along c with S(t0) = S0.

We can write S =P

ja j_(t)s

j(c(t)) for some differentiable functions a1, . . . , an, and then

we get: 0 = DS dt = X j daj dt + X i aiθji(dc/dt)  sj

where θ is the local connection form of ∇ relative to this trivialization. The system of n differential equations in aj(t), 0 = da j dt + X i aiθji(dc/dt), j = 1, . . . , n, (1.6)

possesses a unique solution satisfying the initial conditions aj_(t

0) = aj0. It follows that if S

exists, then it is unique. Moreover, since the system is linear, any solution is defined for all t ∈ I, which then proves existence (and uniqueness) of S with the desired properties. Corollary 1.1.13. Let π : E → M be a vector bundle over the manifold M , endowed with a connection ∇. Let c : I → M be a differentiable curve and let t0, t1 ∈ I. The

mapping

P = Pc,t0,t1 : Ec(t0)→ Ec(t1)

which sends a point S0∈ Ec(t0) to S(t1) ∈ Ec(t1), where S is the parallel transport of S along c, is a linear isomorphism.

Proof. The linearity properties of the covariant derivative show that the map P is linear. The existence and uniqueness of parallel transport show that P must be bijective.

We remark that in general the parallel transport depends on the curve c, and not only on its extremal points. Actually, parallel transport along different curves joining the same points may differ even if the curves are homotopic.

Affine Manifolds

Let us now introduce the main object of our investigation.

Definition 1.1.14. An affine structure on a manifold is an atlas such that all transition functions are locally affine and it is maximal with this property. An affine manifold is a manifold together with an affine structure. A map between affine manifolds is affine if it is an affine trasformation when it is read in charts.

Explicitly we have the following. An atlas { ϕα: Uα→ Vα⊆ Rn}_α of M is affine iff

for every α and β the change of coordinates ϕα◦ ϕ−1_β : ϕβ(Uα∩ Uβ) → ϕα(Uα∩ Uβ) has

locally the form

ϕα◦ ϕ−1_β (x) = Ax + B, A ∈ GLnR, B ∈ Rn, that is, its differential d(ϕα◦ ϕ−1_β ) is locally constant.

Equivalently, if we look at the induced trivializations ¯ϕα: T M|Uα → Uα× R

n _{of T M ,}

given by

¯

ϕα(p, v) = (p, d(ϕα)p(v)), p ∈ Uα, v ∈ TpM,

this is the same as requiring the corresponding transition functions gαβ : Uα∩ Uβ →

GLnR, which are given by gαβ(p) = d(ϕα◦ ϕ−1β )ϕβ(p), to be locally constant. Let us look at a few examples.

Example 1.1.15. (1) The torus Tm := Rm

/Zm has a natural affine structure for which the projection map π : Rm→ Tm_{is an affine local diffeomophism.}

(2) Similarly, let us fix a real number λ > 1 and consider the action Z y Rm_{−{0} given}

by k · x := λk_{x. Since the action is free and proper, the quotient is a manifold; and}

as Z acts by affine trasformations, the quotient is an affine manifold. Topologically, these manifolds are diffeomorphic to Sm−1_{× S}1 _{for m ≥ 1.}

(3) In general, when a subgroup G < Aff(M ) of the affine automorphisms of M acts freely and properly, the quotient M/G is again an affine manifold.

Here is a useful characterization of affine structures.

Proposition 1.1.16. Let M be a manifold. There is a natural bijective correspondence between affine structures on M and flat torsion free connections on T M .

Proof. Let { ϕα: Uα→ Vα⊆ Rn}α be an affine structure on M . We want to uniquely

define a connection ∇ by requiring that, in the induced local trivializations ˜ϕα of T M ,

its local connection forms θα vanish. We have to show that this local definitions patch

together and give a well defined global connection on M . The forms θαshould transform

as

θβ= gαβ−1θαgαβ+ g−1αβdgαβ. (1.7)

Since the atlas { ϕα}_α is affine, the transition functions of the induced trivializations,

i.e. gαβ = d(ϕα◦ ϕ−1_β ), are locally constant, and hence we have dgαβ = 0. Thus we

can coherently ask θα = 0 for every α and get a well defined connection ∇ on all of

M . Flatness and symmetry of ∇ now follow from θα ≡ 0 and the structural equation

(Theorem 1.1.6).

Conversely, suppose that, given a flat torsion free connection, the following claim holds:

Claim. There exists an atlas { ϕα: Uα→ Vα⊆ Rn}α of M such that, in the

corre-sponding trivializations ˜ϕαof T M , the local connection forms θα of ∇ vanish.

Then, since formula (1.7) must hold and since we have θα= θβ= 0, we must as well

have dgαβ= 0. It follows that the functions gαβ= d(ϕα◦ ϕ−1β ) are locally constant, and

hence the atlas { ϕα}α is an affine atlas of M .

We now prove the claim. For any point p ∈ M we shall find a chart ϕ : U → V ⊂ Rn_,

with p ∈ U , such that the corresponding local connection form θ vanishes. We start by taking a chart ψ : V → (−ε, ε)n _{⊆ R}n _{such that p ∈ V and ψ(p) = 0. We have an}

induced isomorphism between the tangent bundles ˜

ψ : T M|V → T (−ε, ε)n
= (−ε, ε)n× Rn,

and we still denote by ∇ the induced connection on T (−ε, ε)n
. We use coordinates (x1, . . . , xn) on (−ε, ε)n. We now extend the basis_∂x∂₁, . . . ,_∂x∂_n at 0 to a frame X1, . . . , Xn

x1, then along x2, and so until xn. At the i-th step the frame is defined only on the slice

Fi = { xi+1= · · · = xn= 0 } of the cube, and at the end it is defined everywhere. It is

smooth because parallel transport depends smoothly on the initial data. By construction we have:

∇ ∂

∂xiXk = 0 on Fi ∀k. We now prove that in fact

∇ ∂

∂xjXk = 0 on Fi ∀k, ∀j ≤ i.

We show this by induction on i. The case i = 1 is done, so we suppose that it holds for i and prove it for i + 1. We already know that ∇∂/∂xi+1Xk = 0 on Fi+1. If j ≤ i by induction we have that ∇∂/∂xjXk = 0 on the hyperplane Fi. To conclude it suffices to check that ∇∂/∂xi+1(∇∂/∂xjXk) = 0 on Fi+1. Then it follows that ∇∂/∂xjXk is parallel along all the lines parallel to the xi+1-axis, and hence, being zero on Fi, it is zero an all

of Fi+1. The coordinate fields _∂x∂₁, . . . ,_∂x∂_n commute, hence flatness gives

∇∂/∂xi+1(∇∂/∂xjXk) = ∇∂/∂xj(∇∂/∂xi+1Xk) = 0. The inductive proof is completed and when i = n it shows that

∇∂/∂xjXk= 0 ∀k, j everywhere on the cube. Since ∇ is torsion free we get

[Xi, Xj] = ∇XiXj− ∇XjXi= 0.

It follows that the vector fields X1, . . . , Xn commute and hence there exists a chart

ϕ0 : U0 → W ⊆ Rn _{with 0 ∈ U}0 _{⊆ (−ε, ε)}n _{that straightens these vector fields, i.e. it}

transports Xi to _∂x∂_i for all i. Now it is straightforward to check that the composition

ϕ = ϕ0◦ ψ : ψ−1(U0) | {z }

=U

→ W

is the chart we were looking for.

Remark 1.1.17. If M is affine and π : fM → M is a covering map, there exists a unique smooth structure on fM such that π is smooth. This structure supports an affine atlas that makes π affine. In particular, if fM is the orientable 2-cover of a closed affine manifold M , then it is affine and satisfies χ( fM ) = 2

χ

(M ). Therefore, in our work on the Chern conjecture, we can and will always suppose that M is oriented.

Being affine is a rather strict condition. For example we have:

Lemma 1.1.18. Let M(m) _{and N}(n)_{be connected affine manifolds and f, g : M → N}

be affine immersions. If f and g coincide on a nonempty open subset of M then they are equal.

Proof. Let us set

C = { p ∈ M | f ≡ g in an open neighbourhood of p } .

By construction C is open, and by hypothesis it is non-empty. If we show that it is also closed, then we are done by connectedness. Let p ∈ M − C, there are two cases. If f (p) 6= g(p), then by continuity there exists an open neighbourhood V of p such that f (q) 6= g(q) for all q ∈ V . Then V ∩ C = ∅. Now suppose f (p) = g(p). Let us take affine charts ϕ : M ⊇ U → Rm

and ψ : N ⊇ V → Rn _{for some open neighbourhoods of p}

charts the maps f and g looks like:

x 7→ Ax and x 7→ Bx.

Since p /∈ C, we must have A 6= B and thus there exists y ∈ Rm_{such that (A − B)y 6= 0.}

Then, for every q = ϕ−1(x) ∈ U we have either f (q) 6= g(q), or f (q) = g(q) and ψ(f (ϕ−1(x + ty))) = Ax + tAy = Bx + tAy 6= Bx + tBy = ψ(g(ϕ−1(q))) for every t > 0, and we can conclude that q /∈ C. It follows that U ∩ C = ∅. We conclude that C is clopen and hence C = M .

We would like to have a third description of affine manifolds in terms of a globally defined map with values in Rm. First we need the following:

Proposition 1.1.19. Let M be an affine manifold of dimension m and let G be the group Aff(Rm) seen as a discrete group. There is a natural principal G-bundle π : τ (M ) → M (see Definition 1.3.2) such that sections of π are in natural bijective correspondence with affine immersions from M to Rm_.

Proof. For each p ∈ M we define

Cp= { ϕ : U → V ⊆ Rm| ϕ is an affine diffeomorphism and p ∈ U } .

There is an equivalence relation ∼ on Cp given by declaring that ϕ ∼ ϕ0 if and only if

there exists an open neighbourhood W ⊆ U ∩ U0 of p such that ϕ|W = ϕ0|W. Let us

denote by Lp the set of equivalence classes of Cp and set

τ (M ) =G

p

Lp.

There is a natural map π : τ (M ) → M that sends Lpto p. The Lie group G acts on the

right on each set Lp by composition:

ϕ · g = g−1◦ ϕ,

and this action is free and transitive. Moreover, an affine chart ϕ : U → V ⊆ Rminduces a natural identification:

ˆ

ϕ : U × Aff(Rm) → π−1(U )(p, g) 7→ [g−1◦ ϕ]p,

where [g−1◦ϕ]pdenotes the class of g−1◦ϕ in Lp. There is a unique topology on τ (M ) such

that ˆϕ is a homeomorphism for all affine diffeomorphisms ϕ. The map π : τ (M ) → M is a principal G-bundle with respect to this topology. Let σ be a section of π. Then we define ˜_{σ : M → R}m_by:

˜

σ(p) := σ(p)(p).

By construction, the map ˜σ is an affine immersion. Conversely, an affine immersion f : M → Rmdefines a section σf given by:

σf(p) := [f ]p.

Corollary 1.1.20. Let M(m) be a simply connected affine manifold. Any affine chart φ : U → V ⊆ Rm extends uniquely to an affine immersion from M to Rm.

Proof. Consider the principal bundle π : τ (M ) → M constructed in the above proposi-tion. In particular, this is a fibre bundle with discrete fibre Aff(Rm) and hence π is a covering map. Since M is simply connected this covering map is trivial and any local section extends uniquely to a global one.

Corollary 1.1.21. If M is an affine manifold with finite fundamental group then M is not compact.

Proof. If M is compact then so it is its universal cover ˜M which is simply connected and therefore admits an immersion to Rm_{. This is impossible.}

Finally, here is the description we were looking for.

Proposition 1.1.22. Let M(m)be a manifold and fM be its universal cover. Fix a point x0 ∈ M and consider the usual left action of π1(M, x0) on ˜M , as explained in Example

1.3.3.(5). Then we have a bijective correspondence between affine structures on M and equivalence classes of pairs (h, D), where h : π1(M, x0) → Aff(Rm) is a homomorphism

and D : ˜_{M → R}m _{is a local diffeomorphism, satisfying}

D(γ · p) = h(γ) ◦ D(p) (1.8)

for all γ ∈ π1(M, x0) and p ∈ ˜M . Such a pair (h, D) is called holonomy representation

and developing map. Here two pairs (h, D) and (h0, D0) are considered equivalent if there exists L ∈ Aff(Rm_{) such that}

D = L ◦ D0 and h(γ) = L ◦ h0(γ) ◦ L−1, for all γ ∈ π1(M, x0).

Proof. Take an affine structure on M and endow ˜M with the induced affine structure as explained in Remark 1.1.17. Then the action π1(M, x0) y ˜M is by affine transformations.

By Corollary 1.1.20, taking any chart of ˜M , we get an affine diffeomorphism D : ˜M → Rm. By Lemma 1.1.18, for every γ ∈ π1(M, x0) there exists a unique map h(γ) ∈ Aff(Rm)

such that formula (1.8) holds, and it is easy to show that h : π1(M, x0) → Aff(Rm) is a

homomorphism.

Conversely, suppose we are given h and D as above. Then, for every x ∈ M we take an evenly covered open subset U ⊆ M and a corresponding open set ˜U ⊆ ˜M such that π : ˜U → U is a homeomorphism and D_{| ˜U} : ˜_{U → R}m_{is a diffeomorphism onto its image.}

Then we get a chart ϕ : U → Rm_{by setting ϕ = D ◦ π}−1_{. In this way we construct an}

atlas for M . Since formula (1.8) holds, we conclude that the transition functions of this atlas are actually given by the affine transformations h(γ) for γ ∈ π1(M, x0). Hence this

atlas is affine. Details are left to the reader.

1.2

The Euler Class

In this section we introduce the reader to the so called Euler class, which is basically a (cohomological) extension to general vector bundles of the Euler characteristic and will be of major importance throughout this work.

First we need the following very useful result by Thom, which investigates the coho-mology of the total space of an oriented vector bundle p : E → B. We shall work with singular cohomology and coefficients are in Z when omitted. If F is a fibre of E, let us denote by F0 the set of its non-zero elements, and similarly let us denote by E0 the set

of non-zero elements of the total space. Recall that an orientation of E is basically a coherent orientation of its fibres F ' Rn, that is, cohomologically speaking, a coherent choice of a generator of Hn_{(F, F}

0; Z) ' Hn(Rn, Rn0; Z) ' Z for every fibre F .

Theorem 1.2.1 (Thom’s isomorphism). Let ξ = (E, B, p) be an oriented n-vector bun-dle. Then the cohomology group Hi_{(E, E}

0; Z) is zero for i < n and Hn(E, E0; Z) contains

a unique class u = u(ξ), called the Thom class, such that its restriction i∗u ∈ Hn_{(F, F} 0; Z)

is equal to the preferred generator uF, determined by the orientation, for any fiber F

of ξ. Furthermore, the correspondence y 7→ y ∪ u maps Hk

(E; Z) isomorphically onto Hn+k_{(E, E}

As π : E → B is a homotopy equivalence, the composition Hj_{(B)−→ H}π∗ j_(E)−−→∪u

Hj+n_{(E, E}

0) gives an isomorphism φ : Hj(B) → Hj+n(E, E0) which is called Thom’s

isomorphism.

Proof. We shall divide the proof into three steps.

Step 1. Suppose ξ = (E = B × Rn_{, B, p) is a trivial vector bundle. Let µ be a}

fixed preferred generator for the group Hn

(Rn

, Rn

0) ' Z. By the relative version of the

Kunneth formula we have that the map

γ : H0(B) → Hn_{(B × R}n_{, B × R}n₀)

given by α 7→ α × µ is an isomorphism. For any point b ∈ B let Fb be the fibre over b and consider the following commutative diagram

Hn_{(E, E} 0) i∗ F b // H_n (Fb_{, F}b 0) H0_(B) γ OO i∗ b // H₀ ({ b }) γb OO

where γbis the map obtained using the Kunneth theorem for { b } and (Rn, Rn0). Observe

that the preferred generator for Fb _{is u}

Fb = 1 × µ and that, by the commutativity of the previous diagram, the element 1 × µ ∈ Hn_{(E, E}

0) is such that i∗Fb(1 × µ) = uFb. Furthermore, since 1 ∈ H0_{(B) is the only element which restricts to 1 for each point}

b ∈ B, by the commutativity of the diagram we conclude that 1 × µ is the only element in Hn_{(E, E}

0) which satisfies the required property.

Applying again the relative version of the Kunneth formula we have that the map Hj(B) ⊗ Hn_{(R, R}n₀) → Hj+n_{(B × R}n_{, B × R}n₀) | y ⊗ µ 7→ y × µ

is an isomorphism. Then, any element in Hj+n_{(B × R}n_{, B × R}n₀) can be written uniquely as y × µ for some y ∈ Hj(B). Since by the Kunneth theorem we also have that any element in Hj_{(B × R}n) can be written uniquely as y × 1 with y ∈ Hj(B), we can conclude that the assignment

y × 1 7→ (y × 1) × (1 × µ) = y × µ is an isomorphism. Thus the theorem is proved in the trivial case.

Step 2. Suppose that B is the union of two open sets U and V such that the theorem holds for E0 = p−1(U ), E00 = p−1(V ) and E∩ = p−1(U ∩ V ). Consider the following Mayer-Vietoris sequence: · · · → Hi−1(E∩, E₀∩) → Hi(E, E0) → Hi(E0, E00) ⊕ H i_(E00_{, E}00 0) → H i_(E∩_{, E}∩ 0) → . . . .

By assumption, there exist fundamental cohomology classes u0 and u00 for ξU and ξV

respectively. By the uniqueness of the fundamental cohomology class for ξU ∩V, the

cohomology classes u0 and u00 have the same image in Hn_(E∩_{, E}∩

0). By exactness of the

sequence above, there exists a cohomology class u ∈ Hn_{(E, E}

0) that gives u0 and u00

when restricted to U and V respectively. Clearly the restriction of u on each fiber gives us the orientation of the vector bundle, as u0 and u00do so. Furthermore this cohomology class is unique since we have Hn−1_(E∩_{, E}∩

0) = 0 by assumption. Now consider the

Mayer-Vietoris sequence

· · · → Hj−1_(E∩_{) → H}j_{(E) → H}j_(E0_{) ⊕ H}j_(E00_{) → H}j_(E∩_{) → . . .}

with i = n + j. Mapping this sequence to the previuous Mayer-Vietoris sequence via the correspondence y 7→ y ∪ u and applying the Five Lemma, we obtain that Hj(E) ∼=

Hj+n_{(E, E} 0).

Step 3. Suppose that B has a finite open covering U1, . . . , Um such that the vector

bundles ξUi are trivial for each i. We will prove that the result holds also in this case, by induction on m. Clearly the result holds for m = 1 by our Step 1. Assume now that the result holds for m − 1. Then we have that the theorem is true for the vector bundles ξU1∪···∪Um−1, by inductive hypothesis, and for ξUm and ξ(U1∪···∪Um−1)∩Um, since this bundles are trivial and by Step 1. Applying Step 2 we see that the theorem holds for ξ as well.

In particular, Step 3 shows that the theorem holds for bundles over compact spaces, and this will suffice for our scopes. For the general case just a little more effort is required, and we recommend the reader to consult [MS74, Chapter 10].

Remark 1.2.2. (1) The above theorem and its consequences hold, with the same proofs, for coefficients in an arbitrary ring Λ.

(2) When the base space is compact, Thom’s isomorphim can be described in terms of compactly supported singular cohomology. Recall that for a space X this is defined as

H_c∗(X) = lim

K compactH

∗_{(X, X − K),}

where the limit is over the directed set { H∗(X, X − K) | K ⊆ X compact } and the maps are induced by inclusions. Now fix a riemannian metric on E and set Er:= { v ∈ E | kvk > r } for every r ≥ 0. As M is compact, E − Eris also compact

and moreover the set { H∗(E, Er) }r≥0is cofinal in { H∗(E, E − K) | K compact }.

Since (E, Er) ,→ (E, E0) is a homotopy equivalence, we have that

Hc∗(E) = lim K compactH ∗_{(E, E − K) = lim} r→∞H ∗_{(E, E} r) ' H∗(E, E0),

and the following diagram commutes

H_c∗(E) λ // H∗(E) H∗(E, E0) ' i∗ OO j∗ 88

where λ is the natural morphism from compactly supported cohomology to the usual one. In particular, the Thom class u(E) may be interpreted as a generator of Hcn(E; Z) ' H0(E; Z) ' Z.

Let us now state a few properties of the Thom class. Proposition 1.2.3. The Thom class satisfies the following:

(i) if the orientation of ξ is reversed, then the Thom class u(ξ) changes sign;

(ii) if ( ˜f , f ) : (E0B0) → (E, B) is a morphism of vector bundles, then ˜f∗u(E) = u(E0); (iii) the Thom class of a Whitney sum is given by u(ξ ⊕ ξ0) = u(ξ) ∪ u(ξ0), and the

Thom class of a cartesian product is given by u(ξ × ξ0) = u(ξ) × u(ξ0).

Proof. Property (i) is obvious. We turn to (ii). The mapping ˜f is an orientation pre-serving linear isomorphism when restricted to fibers. Since for every b ∈ B0 the diagram

E_b0_{ _}  ˜ f _{// E} f (b) _  E0 f˜ // E

commutes, it follows that the class ˜f∗(u(E)) satisfies the property defining the Thom class, and hence it coincides with u(E0) by uniqueness. We now prove (iii). Let the fiber dimensions be m and n respectively. Using the relative version of the cohomology Kunneth formula and the commutativity of the diagram

Hm(E, E0) ⊗ Hn(E0, E00) × _// i∗⊗i∗  Hm+n(E × E0, (E × E0)0) i∗  Hm_{(F, F} 0) ⊗ Hn(F0, F00) × _{// Hm+n} (F × F0, (F × F0)0)

for every pair of fibres F ⊆ E and F0 ⊆ E0_{, we see that the Thom class of ξ × ξ}0 _{is given}

by

u(ξ × ξ0) = u(ξ) × u(ξ0).

Now suppose ξ and ξ0 have the same base space B. Pulling both sides of this equation back to Hm+n(E ⊕ E0, (E ⊕ E0)0) via the bundle map E ⊕ E0 → E × E0 associated to

the diagonal embedding B → B × B | b 7→ (b, b), we obtain the formula u(ξ ⊕ ξ0) = u(ξ) ∪ u(ξ0).

We are now ready to define the Euler class of an oriented vector bundle p : E → B. Let j : E → (E, E0) be the natural inclusion and let s0: B → E be the zero-section.

Definition 1.2.4. The (topological) Euler class of an oriented n-vector bundle ξ = (E, B, p) is the cohomology class e(ξ) = s∗₀j∗u(ξ) ∈ Hn

(B; Z).

A bunch of properties of the Euler class follow readily from its definition and Propo-sition 1.2.3.

Proposition 1.2.5. The Euler class satisfies the following:

(i) if the orientation of ξ is reversed, then the Euler class e(ξ) changes sign;

(ii) (naturality) if ( ˜f , f ) : (E0, B0) → (E, B) is an orientation preserving map of oriented vector bundles, then f∗e(E) = e(E0);

(iii) the Euler class of a Whitney sum is given by e(ξ ⊕ ξ0) = e(ξ) ∪ e(ξ0), and the Euler class of a cartesian product is given by e(ξ × ξ0) = e(ξ) × e(ξ0_).

Remark 1.2.6. In particular, thanks to naturality, two isomorphic oriented vector bun-dles over the same base B have the same Euler class. Similarly, since every trivial vector bundle over some space B is the pullback of the bundle Rn _{→ ∗ over a point, the Euler}

class of any trivial vector bundle is zero.

Corollary 1.2.7. If the oriented vector bundle ξ = (E, B, p) possesses a nowhere van-ishing section, then its Euler class must be zero.

Proof. Let  be the trivial line bundle spanned by the section above. Take any riemannian metric on ξ and consider its orthogonal complement ⊥. Then we have e(ξ) = e()∪e(⊥). Since e() = 0 the result follows.

Corollary 1.2.8. The Euler class of an oriented n-vector bundle η with n odd satisfies e(η) = −e(η).

Proof. For any vector bundle η = (E, B, p) the map defined as f : (b, v) 7→ (b, −v) is a bundle isomorphism. If n is odd then this map reverses the orientation and thus f∗(e(η)) = −e(η). On the other hand, since the base space map is the identity, we must have e(η) = −e(η).

The Euler class allows us to relate the cohomology of the base space to that of the total space through a long exact sequence.

Theorem 1.2.9 (Gysin sequence). For an oriented n-vector bundle ξ = (E, B, p) we have the following long exact sequence

· · · → Hi(B)−−−→ H∪e(ξ) i+n(B) p ∗

−→ Hi+n(E0) Γ

−→ Hi+1(B) → . . . . Here Γ is given by the composition Hi+n_(E

0) δ − → Hi+n+1_{(E, E} 0) φ−1 −−→ Hi+1_{(B), where δ}

is the boundary operator of the cohomology long exact sequence of (E, E0) and φ is the

Thom isomorphism.

Proof. We have the following commutative diagram with the top row exact // Hi+n_{(E, E}

0)

j∗ _{// Hi+n}

(E) i

∗

// Hi+n_(E) δ // Hi+n+1_{(E, E}

0) // // Hi_(B) φ OO // Hi+n_(B) p∗ OO p∗ _{// Hi+n} (E0) Id OO Γ // Hi+1_(B) φ OO //

where j : E → (E, E0) and i : E0 → E are the inclusion maps. Since all the vertical

maps are isomorphisms the bottom row is exact as well. Moreover, for the unlabelled arrow in the bottom left corner we have the following computation

(p∗)−1j∗φ(a)

=(p∗)−1j∗(p∗(a) ∪ u(ξ)) =(p∗)−1(p∗(a) ∪ j∗u(ξ)) =a ∪ (p∗)−1j∗(u(ξ)) = a ∪ e(ξ). This concludes the proof.

We now give an intereseting example about the Euler class that we shall need later. Recall that the complex projective space PnC (resp. P∞C) has a usual CW-structure with one 2h-dimensional cell for every 0 ≤ h ≤ n (resp. h ∈ N) and no other cells (see [Hat09, p. 7]). Using a cellular cohomology argument we have that the cohomology groups of the complex projective space PnC (resp. P∞C) are the following:

Hk_(PnC; Z) = (

Z if k = 0, 2, . . . , 2n (resp.k ∈ N even)

0 else .

Nevertheless, with our available tools we can refine this result one step further. Set En

= { (x, u) ∈ PnC × Cn+1| u ∈ x } and similarly E = { (x, u) ∈ P∞C × C∞| u ∈ x }. Recall that the projection onto the first factor gives a complex 1-dimensional vector bundle γn,C₁ = (En_{, P}nC, p), called the canonical (complex) line bundle over PnC, and similarly for γC

1 = (E, P∞C, p). The next lemma is a well known fact (see [MS74]) and is left to the reader.

Lemma 1.2.10. If ξ is a complex n-dimensional vector bundle over a base space B, then the underlying real 2n-dimensional vector bundle ξrhas a canonical preferred orientation.

In particular the Euler class e(ξr) ∈ H2n(B; Z) is well defined.

Now let (γ₁n,C)r denote the underlying real vector bundle of the canonical complex

line bundle over PnC. Using the Gysin sequence, we can prove the following:

Theorem 1.2.11. The cohomology ring H∗_(PnC; Z) is a truncated polynomial ring terminating in dimension 2n and is generated by the Euler calss e((γ₁n,C)r) ∈ H2(PnC). Proof. Let us consider the Gysin sequence of the oriented real vector bundle (γ₁n,C)r=

(En_{, P}nC, p), that is

. . . // Hi+1(E₀n) Γ // Hi_(PnC) ∪e // Hi+2(PnC)

p∗ _{// Hi+2}

All the points of the space En

0 are of the form (L = line through the origin, u) with u ∈ L

non-zero, thus En

0 can be identified with Cn+1−{ 0 } and has the same homotopy type as

S2n+1. From the Gysin exact sequence we then obtain short exact sequences of the form 0 → Hi_(PnC)−∪e−→ Hi+2(PnC) → 0

for 0 ≤ i ≤ 2n − 2. From these short exact sequences we obtain that H0_(PnC)∼= H2(PnC)∼= . . . ∼= H2n(PnC) and that H2j(PnC) is generated by e(γ1n,C)

j_{for 0 ≤ j ≤ n. This concludes the proof.}

Corollary 1.2.12. The cohomology ring H∗_(P∞C; Z) is the polynomial ring generated by the Euler calss e((γC

1)r) ∈ H2(P∞C).

Proof. Let n ∈ N be any natural number. We have an oriented bundle map (˜i, i) : (En

, PnC) → (E, P∞C) where both the top and the bottom map are induced by the inclusion Cn+1

,→ C∞, so that we have i∗e((γC

1)r)) = e((γn,C1 )r). Also, considering the

usual CW-structure on the complex projective spaces, it is clear that the map i : PnC ,→ P∞C induces a chain map ϕ• between the respective cellular cochain complexes, such that ϕi _{= Id for 0 ≤ i ≤ 2n and ϕ}i _{= 0 otherwise. Thus, i}∗ _{: H}i

(P∞C) → Hi(PnC) is an isomorphism for 0 ≤ i ≤ 2n, which implies that H2j_(P∞C) is generated by e((γC1)r))j

for 0 ≤ j ≤ n. Letting n tend to infinity we get the stated result.

Finally, we would like to relate the Euler class of the line bundle γC

1 to another

’famous’ cohomology class. We recall that there is a construction, sometimes called the homotopy construction of cohomology, that allows us describe cohomology groups in terms of homotopy classes of maps with values in suitable Eilenberg-MacLane spaces. We recommend the reader to consult [Hat09, pp. 393-404] for an excellent exposition. Explicitly, we have the following result:

Theorem 1.2.13. For every n ≥ 1 and every abelian group G, there are natural equiv-alences

T : [X, K(G, n)] → Hn(X; G) between the functors [−, K(G, n)] and Hn_{(−; G) from CW}

h, the homotopy category of

CW-complexes, to Set. Explicitely, such a T has the form T ([f ]) = f∗(α)

for a certain distinguished class α ∈ Hn_{(K(G, n); G). A class α satisfying this property}

is called a fundamental class.

Let us see why we are interested in this theorem. It is a well known fact that P∞C is a K(Z, 2) space. We now take e(γC

1) as a generator of H2(P∞C; Z) so that we have an identification Z ' H2_(P

∞C) given by k 7→ ke(γ1C). In this case a map f : P∞C → P∞C induces a map f∗: H2_(P

∞C) → H2(P∞C) that has the form k 7→ f∗(k) = dfk for some

fixed integer df ∈ Z. A fundamental class α ∈ H2(P∞C; Z) has the form α = he(γ1C) for

some h ∈ Z, and since we have an isomorphism [P∞C, P∞C]∼= H2(P∞C) via [f ] 7→ f∗α, there must be a map f such that f∗α = dfhe(γ1C) = e(γC1). In particular, we must have

h = ±1, that is α = ±e(γC

1). Since there exists a map g ∈ [P∞C, P∞C] such that dg= −1,

namely the one which gives T ([g]) = g∗α = −α under the natural equivalence above, we conclude that we can take α = e(γC

1).

Proposition 1.2.14. The Euler class e(γC

1) of the canonical complex line bundle over

P∞C is a fundamental class for P∞C = K(Z, 2). In particular the assignment

[X, P∞C] → H2(X; Z) [f ] 7→ f∗(e(γC

is a natural equivalence of functors CWh→ Set.

Let us go back to general oriented vector bundles. If M is a closed oriented m-dimensional manifold, then Poincaré duality gives Hm

(M ; Z) ∼= Z via the map β 7→ hβ, [M ]i. Therefore, for an oriented m-vector bundle π : E → M , the number he(E), [M ]i completely determines the Euler class e(E) ∈ Hm_{(M ).}

Definition 1.2.15. Let M be a closed oriented m-dimensional manifold and let π : E → M be an oriented m-dimensional vector bundle. The number

D(E) := he(E), [M ]i is called the Euler number of the bundle.

Finally, we prove the following celebrated result, which relates the Euler class to its original definition in terms of obstruction to the existence of a section in a vector bundle. Theorem 1.2.16 (Hopf Index Theorem). Let M be a closed oriented m-dimensional manifold and let π : E → M be an oriented m-dimensional vector bundle. Let s : M → E be a section with only isolated zeros, then

X

zeros of s

index(s, p) = he(E), [M ]i

where index(s, p) denotes the index of the section s at the point p. First we need the following:

Lemma 1.2.17. Let µ ∈ Hn

(Rn

, Rn

0) be the generator given by the canonical orientation

of Rn and let us consider the trivial bundle E = Rn× Rn

→ Rn

. Let s : Rn → E be a section of the form s(x) = (x, σ(x)) with σ(x) 6= 0 for all x 6= 0. Then s∗u(E) ∈ Hn_(Rn_{, R}n0) is the generator µ times the degree of the map σ : Rn− { 0 } → Rn− { 0 }.

Proof. Let µ be as above and let 1 ∈ H0_(Rn) be the unique element that restricts to 1 for each point b ∈ Rn. Then we know that the Thom class of E is given by u = 1 × µ = π∗₁1 ∪ π∗₂µ. It follows that s∗u = s∗π∗₁1 ∪ s∗π∗₂µ = 1 ∪ σ∗µ = deg(σ)µ as desired.

Proof of the Hopf Index Theorem. Let p1, . . . , pkbe the zeros of s and let us take disjoint

open neighbourhoods Ui 3 pi together with orientation preserving charts ϕ : Ui →

Rn such that ϕi(pi) = 0. Also, let us suppose that E|Ui is trivial, so that we have isomorphisms of oriented vector bundles

E_|Ui ψi // π  Rn× Rn  Ui ϕi // Rn.

Finally, let us denote by the same name ij both the inclusions Uj ,→ M and E|Uj ,→ E, so that we have other morphisms of oriented vector bundles

E|Uj ij // π  E π  Uj ij // M .

We have the following commutative diagram Hn(E, E0) (i∗_j)j _// s∗  L jH n_(E Uj, (EUj)0) ⊕js∗_|Uj  L jH n (Rn× Rn , Rn× Rn 0) ⊕jψ∗j oo ⊕js∗j  Hn_{(M, M − ∪} j{ pj}) (i∗_j)j // L_jHn_(U j, Uj− { pj}) LjH n (Rn , Rn 0) ⊕jϕ∗j oo

where the maps sjare defined as sj= ψj◦ s ◦ ϕ−1j . Since all the maps above are induced

by morphisms of oriented vector bundles, we see that in the previous diagram the Thom class u(E) ∈ Hn(E, E0), thanks to its naturality, is mapped as follows:

u(E) (i∗ j)j_// s∗  u(EUj)
j ⊕js∗_|Uj  u(Rn_{× R}n₎
j ⊕jψj∗ oo ⊕js∗j  s∗_u(E) (i ∗ j)j // s∗_u(E Uj)
j s ∗ ju(Rn× Rn)
j . ⊕jϕ∗j oo (1.9)

Moreover, by Lemma 1.2.17, we have s∗_ju(Rn× Rn_{) = deg(s}

j)µ and hence s∗u(EUj) = deg(sj)µj, where µj is the generator of Hn(Uj, Uj− { pj}) ' Z given by the orientation

of M .

Now consider the inclusion J : M → (M, M − ∪j{ pj}). Notice that by definition of

the fundamental class [M ], the mapping Hn(M ) J∗ −→ Hn(M, M − ∪j{ pj}) P jij∗ ←−−−− ' ⊕jHn(Uj, Uj− pj)

sends [M ] to (1Uj)j ∈ ⊕jHn(Uj, Uj − pj), where 1Uj ∈ Hn(Uj, Uj − pj) ' Z is the generator given by the orientation of M . Finally, we have the following commutative diagram Hn_{(M )} ∩[M ]  Hn_{(M, M − ∪} j{ pj}) J∗ oo (i∗_j)j ' // ∩J∗([M ])  L jH n_(U j, Uj− { pj}) ⊗j∩1Uj  Zoo _Id Z P LjZ jxj←[(xj)j oo

where ∩ denotes the cap product. By definition of the Euler class we have J∗s∗u(E) = e(E), and by diagram (1.9) we have (i∗_j)j s∗u(E)
= deg(sj)µj

j. In conclusion we get he(E), [M ]i =X j hdeg(sj)µj, 1Uji = X j deg(sj) = X j index(s, pj) ,

and the theorem is proved.

Remark 1.2.18. It is a well known fact from Morse Theory (see [Mil63]) that a compact manifold always admits a vector field with isolated zeros whose index sum equals the Euler characteristic of the space. In view of the theorem above we thus have

he(T M ), [M ]i =

χ

(M ).

1.3

Characteristic Classes and Classifying Spaces

Principal Bundles

We have seen that the Euler class is an invariant for isomorphism classes of oriented vector bundles. We would like to do more and completely characterize, up to isomorphism, a general fibre bundle. Here we shall work with continuous maps and topological spaces, but we remark that almost every definition and result translates into the differentiable setting with obvious modifications.

Remark 1.3.1. Given a base space B with an open covering U = { Uα}α, another

space F and a set of functions gαβ: Uα∩ Uβ→ Homeo(F ) satisfying the so called cocycle

condition gαβ◦ gβγ= gαγ on Uα∩ Uβ∩ Uγ, we know that

E := G

α

Uα× F

.

∼ with (α, b, p) ∼ (β, c, q) iff b = c and gαβ(c)(q) = p

is a fibre bundle over B with local trivializations E|Uα' Uα× F and transition functions the maps gαβ above. Viceversa, any F -fibre bundle E0, with local trivializations E_|U0 _α'

Uα× F and transition functions the same maps gαβ, is bundle-isomorphic to the bundle

E just constructed.

This means that all the information about a fibre bundle is carried by its transition functions and this suggests to take a closer look at the group G < Homeo(F ) generated by the maps gαβ. The next definition should now seem more motivated.

Definition 1.3.2. Given a topological group G, a principal G-bundle is a continuous mapping π : P → B together with a continuous right G-action P × G → P satisfying:

(i) for every b ∈ B, the fibre Pb= π−1(b) is an orbit;

(ii) (local triviality) every point of B has an open neighbourhood U and a homeomor-phism ϕ : π−1(U ) → U × G such that

(a) the diagram

π−1(U )  ϕ _{// U × G} yy U commutes; (b) ϕ is equivariant, i.e. ϕ(p · g) = ϕ(p) · g, p ∈ π−1(U ), g ∈ G, where G acts trivially on U and by right translation on G.

In particular notice that π is an open mapping and induces a homeomorphism P/G ' B. Also observe that the action of G on P is free and for any p ∈ P the mapping G → P given by g 7→ pg is a homeomorphism onto the fibre Pπ(p) = π−1(π(p)). Finally notice

that for every subset B0 ⊆ B the restriction π : π−1_(B0_{) → B}0 _{is again a principal}

G-bundle.

Example 1.3.3. (1) For any topological group G the projection onto the first factor π : B × G → B gives a principal G-bundle, called the product bundle.

(2) For (V, B, p) an n-dimensional vector bundle, the associated frame bundle (F (V ), B, ˜p) is a principal GLnR-bundle.

(3) If V is endowed with a riemannian metric, the bundle of its orthonormal frames (FO(V ), B, ˜p) is a principal O(n)-bundle.

(4) Again, if V is oriented and we only consider oriented frames then (F+_{(V ), B, ˜p) is}

a principal GL+

nR-bundle; and similarly, if we only consider oriented orthonormal

frames, we get a principal SO(n)-bundle.

(5) Let M be a manifold, x0 ∈ M a point and p : ˜M → M the universal cover. We

identify ˜M ' S

y∈Mπ(M, x0, y), where π(M, x0, y) denotes the set of homotopy

classes relative to the endpoints of paths from x0 to y in M , and take the right

action of π1(M, x0) by monodromy

α · g := g−1∗ α, α ∈ ˜M , g ∈ π1(M, x0),

where g−1∗ α denotes the path following g−1 _{and then α. Then ( ˜M , M, p) is a}

principal π1(M, x0)-bundle.

Definition 1.3.4. A morphism of principal G-bundles P → X, Q → Y over the map f : X → Y is a commuting square P  ˜ f // Q  X f // Y

where the map ˜f is required to be G-equivariant. Given a map f : B → X and a principal G-bundle P −→ X, we define its pullback fπ ∗P = { (b, p) ∈ B × P | f (b) = π(p) } in the usual set-categorical sense and it is the unique principal G-bundle up to isomorphism making the following diagram commute:

f∗P πB  πP // P π  B f // X .

We say that a principal bundle is trivial if it is isomorphic to the product bundle B × G. In Example 1.3.3.(3) and (4) above one sees that the associated principal O(n)-bundle (resp. SO(n)-bundle) does not depend, up to isomorphism of principal bundles, on the chosen riemannian metric. This is a consequence of the following well known result, whose proof can be found in [Mar17].

Proposition 1.3.5. Let p : V → M be a vector bundle equipped with two arbitrary riemannian metrics g and g0. There exists a bundle map (ϕ, IdM) : (V, M ) → (V, M )

which is an isometry between the two metrics. Furthermore, there exists a bundle map isotopy (ϕt, IdM) over the identity of M , between (ϕ0= IdV, IdM) and (ϕ1= ϕ, IdM).

Being a principal G-bundle is a far stronger condition than being just a local product with fibre G. Here are two striking facts that illustrate this claim, which can be easily proved using the right action of G on the bundle.

Proposition 1.3.6. (i) Any morphism of principal G-bundles over the identity map is an isomorphism.

(ii) A principal G-bundle is trivial iff it admits a section.

The difference between a principal G-bundle and a common local product with fi-bre G can be illustrated further in terms of transition functions. Comparing two local trivializations of a G-fibre bundle π : E → B over two open sets U, V leads to a homemo-rphism (U ∩ V ) × G → (U ∩ V ) × G of the form (b, g) 7→ (b, φ(b)(g)), where the transition function φ is a map φ : U ∩ V → Homeo(G). In a principal G-bundle local trivializations

are compatible with the right action of the group G. Thus in the above homeomorphism we have

(b, g) = (b, e) · g 7→ (b, φ(b)(g)) = (b, φ(b)(e)) · g = (b, φ(b)(e) · g), and hence φ(b)(g) = φ(b)(e)g for all g ∈ G, b ∈ U ∩ V .

Proposition 1.3.7. In a principal G-bundle each φ(b) is a left translation by an element of G and φ : U ∩ V → G is continuous.

Remark 1.3.8. Recall that any left G-action on a space X can be converted to a right action, and vice-versa, by setting xg = g−1x, x ∈ X. If W is a right G-space and X is a left G-space, the balanced product W ×G X is the quotient space W × X/ ∼,

where (wg, x) ∼ (w, gx). Equivalently, it is the orbit space under the diagonal action (w, x)g := (wg, g−1x). The following facts should be noted:

(i) If X = ∗ is a point, then W ×G∗ = W/G.

(ii) Let G, H be topological groups and X a (G, H)-space, that is a left G-space and an H-right space such that the two actions commute: (gx)h = g(xh). Then W ×GX

receives a natural right H-action. A symmetric result follows if W is a (K, G)-space. (iii) If X = G with the left and right translation actions, then W ×GG is a right G-space

and we have a natural G-equivariant homeomorphism W ×GG ' W .

The following results are straightforward and the proofs are left to the reader. Proposition 1.3.9. The balanced product is associative up to natural isomorphism: let X be a right G-space, Y a (G, H)-space and Z a left H-space, then there is a natural homeomorphism

(X ×GY ) ×HZ ' X ×G(Y ×HZ).

Note that if H is a subgroup of G, then G can be regarded as a (G, H)-space. Com-bining the third example above with the proposition, we find that the symbol ×GG (or

G×G) can be cancelled whenever it occurs.

Corollary 1.3.10. Suppose X is a right G-space, Y a left H-space, where H is a subgroup of G. Also consider X as a right H-space by restricting the G-action, then we have X ×GG ×HY ' X ×HY .

Taking Y to be a point and using the first example above, we get this important special case:

Corollary 1.3.11. Suppose X is a right G-space and H is a subgroup of G, then X ×GG/H ' X/H .

In the situation of this last corollary, suppose that X → X/G is a principal G-bundle. We can ask whether X → X/H is a principal H-bundle. In general it is not, even when X/G is a point: for example take X = G to be the additive group of real numbers acting on itself by translation and take H = Q. Then R → R/Q is not a principal Q-bundle: if it were locally trivial, then, since R/Q has the trivial topology, it would be globally trivial, which is clearly absurd.

We eliminate this pathology with a definition: we call a subgroup H of G admissibile if the quotient map G → G/H is a principal H-bundle. For example, any subgroup of a discrete group is admissibile.

Remark 1.3.12. Any closed subgroup H of a Lie group G is admissibile, that is G → G/H is a principal H-bundle. See [Mar17] for a proof.

Proposition 1.3.13. Suppose P → B is a principal G-bundle and let H be an admissible subgroup of G. Then the quotient map P → P/H is a principal H-bundle.

Proof. For any x ∈ P/G = B let us take a local trivialization ψ : P|W → W × G of the

bundle P → P/G for some open neighbourhood W ⊆ P/G of x. Since P|W ∼= W × G is

an open subset of P which is invariant under the right action of H, we just have to show the local triviality of P|W → P|W/H. Equivalently, we have to show the local triviality

of W × G → W × G/H. This follows from the fact that H ⊆ G is admissible.

We now go back to our usual fibre bundles and take into account the structure of the group generated by the transition functions.

Definition 1.3.14. Let G be a topological group and let F be a left G-space. We say that a fibre bundle π : E → B with fibre F admits a G-structure if there is an open covering U = { Uα}_α of B and local trivializations E|Uα

ϕα

−−→ Uα × F such that the

homeomorphisms ϕα◦ ϕ−1β : (Uα∩ Uβ) × F → (Uα∩ Uβ) × F have the form

ϕα◦ ϕ−1_β (b, f ) = (b, gαβ(b) · f )

for some continuous maps gαβ : Uα∩ Uβ → G satisfying the cocycle condition, which

are still called transition functions of the G-structure. Such a collection of trivializations { ϕα}αis called a G-atlas. A maximal G-atlas is also called a G-structure. We shall also

use the notation (E, B, p, F, G) for such a structure.

A morphism between fibre bundles p : E → B and p0 : E0 → B0 _{with fibre F and}

structure group G is a morphism of fibre bundles (˜h, h) : (E, B) → (E0, B0) such that for every b ∈ B there exist local trivializations ϕ : EU → U × F and ψ : E0V → V × F , in

the respective G-atlases, such that x ∈ U , h(U ) ⊆ V and ψ ◦ ˜h ◦ ϕ(b, f ) = (h(b), g(y) · f ) for some continuous map g : U → G.

Clearly any fibre bundle trivially has a Homeo(F )-structure.

Remark 1.3.15. Completely analogous results to those of Remark1.3.1 still hold for principal bundles and fibre bundles with structure group. Given an open covering U = { Uα}α of B and functions gαβ: Uα∩ Uβ → G satisfying the cocycle condition, we can

construct principal G-bundles and fibre bundles with structure group G in the same way as before. Also, two principal G-bundles (resp. fibre bundles with G-structure) that admit trivializations over U with the same transition functions gαβ are isomorphic as

principal G-bundles (resp. as fibre bundles with G-structure).

Now suppose we have a principal G-bundle P → B and a continuous left action G y F . In view of our previous considerations, we would expect this data to uniquely determine an F -fibre bundle with structure G.

Proposition 1.3.16. Let P −→ B be a principal G-bundle and let F be a left G-space.π Then P ×GF → B, induced by P × F

πP

−−→ P , is a fibre bundle with fibre F and structure group G.

Proof. The mapping P × F → P is G-equivariant and hence it descends to a mapping P ×GF → P/G. A trivialization ϕ = (ϕ(1), ϕ(2)) : P|U → U × G of the princiapl bundle

P induces a trivialization ˜ϕ : (P ×GF )|U→ (U × G) ×GF ' U × F given by ˜ϕ(p, f ) =

((ϕ(1)_{(p), ϕ}(2)_{(p)), f ) ' (ϕ}(1)_{(p), ϕ}(2)_{(p)·f ). Let us take an atlas { ϕ}

α: PUα → Uα× G }α

of P with transition functions gαβ : Uα∩ Uβ → G, which are given by gαβ(b) = ϕ (2) α ◦ ϕ−1_β (b, e) since we have ϕα◦ ϕ−1β (b, g) = ϕα◦ ϕ−1β (b, e) · g = (b, ϕ (2) α ◦ ϕ −1 β (b, e) · g).

Then we get an induced atlas { ˜ϕα}αof P ×GF with the same transition functions gαβ,

since we have ˜ ϕα◦ ˜ϕ−1β (b, f ) = ˜ϕα(ϕ−1β (b, e), f ) = ((b, ϕ (2) α ◦ ϕ−1β (b, e)), f ) ' (b, ϕ (2) α ◦ ϕ−1β (b, e) · f ).

Then this is a G-atlas for P ×GF .

Example 1.3.17. (1) Let (V, B, p) be an n-vector bundle and let (F (V ), B, ˜p) be its frame bundle. Consider the usual action GL(n, R) y Rn_{, then the associated fibre}

bundle (F (V )×_GL(n,R)Rn, B, ˜pRn) with fibre Rnis naturally isomorphic to (V, B, p).

(2) Consider the tangent bundle of a smooth n-manifold M and the associated principal GL(n, R)-bundle PT M → M . Then a k-dimensional distribution in M can be

interpreted as a section of the bundle PT M×GL(n,R)Gk(Rn) → M .

If P ×GF → B is a fibre bundle constructed as above and f : X → B is a continuous

map, the pullback f∗(P ×GF ) is again a fibre bundle over X with the same fibre and

structure group. To make sense of this, one has to know that the two ways of forming the pullback are really the same. We leave to the reader to check the following:

Proposition 1.3.18. Let π : P → B be a principal G-bundle, F a left G-space and let f : X → B be a map. Then there is a natural isomorphism of fibre bundles with G-structure f∗(P ×GF ) ∼= f∗(P ) ×GF .

As all the information about any bundle is contained in the transition functions, it should be possible to reverse the above construction.

Theorem 1.3.19. Let G be a topological group and F a left G-space. Given a space B, denote PG(B) and F(F,G)(B) the isomorphism classes of its principal G-bundles and

fibre bundles with fibre F and structure group G respectively. The functors PG, F(F,G):

Top → Set are naturally equivalent. Explicitely, for any open covering { Uα}α of B,

a principal bundle P and its corresponding fibre bundle E admit local trivializations { ϕα}α and { ˜ϕα}αwith the same transition functions { gαβ}_αβ.

Proof. Given a principal G-bundle P we can construct a fibre bundle P ×G F with

structure group G as in Proposition 1.3.16. Moreover, we have seen in the proof of the proposition that an atlas of P induces an atlas of P ×GF with the same transition

functions. If (¯h, h) : (P, B) → (P0, B0) is a morphism of principal bundles, then we have an induced morphism of fibre bundles (˜h, h) : (P ×GF, B) → (P0×GF, B0) simply by

setting ˜h (p, f )
= (¯h(p), f ). In particular, if ¯h is an isomorphism, then so is ˜h. This shows that the correspondence PG→ F(F,G)is well defined and natural.

Conversely, let us define a natural transformation F(F,G)→ PG. Given a fibre bundle

E with structure group G, we take a G-atlas { ψα: EUα→ Uα× F }αof E with transition

functions gαβ. We can construct a principal G-bundle PEwith induced trivializations ¯ψα

having the same transition functions gαβ, as we did in Remark 1.3.15, simply by setting

PE=

G

α

Uα× G_/∼ with (α, b, g) ∼ (β, c, h) iff b = c and gαβ(b)h = g.

This construction does not depend, up to isomorphism, on the trivializations ψα

con-sidered. To see this, take anothe G-atlas { ψ0_i: EU0 i → U

0

i × F }i of E with transition

functions g0_ij. If two open sets U_i0 and Uα overlap, we can consider the transition

func-tion hiα: Ui0∩ Uα→ G corresponding to the composition

ψ0_i◦ ψ−1_α : (U_i0∩ Uα) × F → (Ui0∩ Uα) × F .

We now consider the map G α Uα× G_/∼−→ G i U_i0× G_/∼

given by (α, b, g) 7→ (i, b, hiα(b)g), where i is any index such that b ∈ Uα∩ Ui0 and

hiα is defined as above. One readily checks that this map is well defined and it is an