The equality of two approaches to Chern classes ofcomplex vector bundles

Reinier Kramer6051405

rkramer@science.uva.nl

19 July 2012

σ0,n //

σ1,n−1

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Bachelor thesis

Supervisor: prof. dr. E. M. Opdam

KdV Institute for mathematicsFaculty of Science

Abstract

In this paper, two important views of the Chern classes, characteristic classes ofcomplex vector bundles, will be related to each other. To do this, chapter 1 willdevelop De Rham theory to a sucient level, including the Poincaré lemmasand the Mayer-Vietoris principle.In chapter 2, the basic theory of ber bundles is explained, with a focus onvector bundles. Some constructions of new vector bundles will be be given, andthe cohomology of these bundles will be calculated.In chapter 3, the notion of connections and the related concept of curvature areoutlined. In line with the constructions in chapter 2, induced connections forsuch vector bundles are dened.Finally, chapter 4 will give two denitions of the Chern class, one via the Eulerclass of an associated sphere bundle and the projectivisation of the vector bun-dle, and a second via invariant polynomials of the curvature of a connection. Inthe second part of this chapter the equality of these two denitions is shown.

DataTitle: The equality of two approaches to Chern classes of complex vector bundlesAuthor: Reinier Kramer, rkramer@science.uva.nl, 6051405Supervisor: prof. dr. E. M. OpdamSecond reviewer: dr. B. J. J. MoonenEnd date: 20-07-2012

Korteweg de Vries Institute for mathematicsUniversity of AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.science.uva.nl/math

Contents

Introduction 4

1 Cohomology 5

1.1 De Rham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1 De Rham cohomology of vector spaces . . . . . . . . . . . . . . . . 51.1.2 De Rham cohomology of manifolds . . . . . . . . . . . . . . . . . . 91.1.3 The compact De Rham cohomology . . . . . . . . . . . . . . . . . 11

1.2 Integration and Poincaré lemmas . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Poincaré lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 The Mayer-Vietoris principle . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.1 The Mayer-Vietoris sequence . . . . . . . . . . . . . . . . . . . . . 191.3.2 The Mayer-Vietoris argument . . . . . . . . . . . . . . . . . . . . . 221.3.3 The generalized Mayer-Vietoris principle . . . . . . . . . . . . . . . 25

2 Fiber bundles 29

2.1 Introduction and constructions of ber bundles . . . . . . . . . . . . . . . 292.1.1 Fiber bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.2 Constructing new vector bundles . . . . . . . . . . . . . . . . . . . 33

2.2 The Leray-Hirsch theorem and the Künneth formula . . . . . . . . . . . . 36

3 Connections 39

3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Curvature and induced connections . . . . . . . . . . . . . . . . . . . . . . 40

4 Chern classes 44

4.1 Chern classes via the Euler class of sphere bundles . . . . . . . . . . . . . 444.1.1 The Euler class of a sphere bundle . . . . . . . . . . . . . . . . . . 444.1.2 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Characteristic classes via the curvature of a connection . . . . . . . . . . . 47

2

4.3 Equivalence of the denitions . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.1 Equivalence for line bundles . . . . . . . . . . . . . . . . . . . . . . 504.3.2 The splitting principle and the Whitney product formula . . . . . 51

Populaire samenvatting 54

Bibliography 56

3

Introduction

The subject of this thesis, characteristic classes, is a subject in algebraic topology, specif-ically cohomology theory. I rst encountered this eld of mathematics in my second yearproject and in the course `analyse op variëteiten', and it directly caught my attention.The idea of classifying manifold, which are very moldable structures, by using rigidvector bundles via the abstract manipulation of functions appealed to me greatly, andespecially since it had something to do with theoretical physics, albeit remotely. How-ever, the introduction in the subject was quite sparce, and since I was quite fascinated,I decided to do my bachelor thesis about this subject. I therefore went to prof. dr. EricOpdam, who agreed to be my supervisor and recommended the subject of characteristicclasses.

This subject arose in 1935, when Eduard Stiefel and Hassler Whitney separately beganstudying homology classes associated to tangent bundles (Stiefel) and sphere bundles(Whitney). The Stiefel-Whitney classes have been named after them. A little laterWhitney developed the theory in the clearer language of cohomology theory. In 1942,Lev Pontrjagin started developing homology of Grasmann manifolds, via which he founda new sort of characteristic classes for real vector bundles, now called Pontrjagin classes.In 1946 Shing-Shen Chern dened a similar sort of characteristic classes for complexvector bundles, which turned out to be much neater than the real versions.[1] TheseChern classes are the subject of this paper.

The cohomology theory of choice is the De Rham cohomology, which was developedby Georges de Rham, who showed it to be a topological invariant theory in 1931.

One way used to reach the Chern classes is by use of connections, a notion which cameabout around 1900, but the used form is that of Jean-Louis Koszul, who developed theconnection as a dierential operator in an algebraic framework in 1950.

My main references are Bott & Tu[2] for information on De Rham cohomology andthe Euler class method, and Bradlow[3] for the study of ber bundles and connections.

Finally, I would like to thank my supervisor prof. dr. Eric Opdam for his guidanceduring the project.

4

Chapter 1

Cohomology

For any introduction in cohomology, one should be aware of the notions of cochaincomplexes and graded algebras.

Denition 1.1. A cochain complex C is a set of vector spaces Cii∈Z together with aset of homomorphisms (called the dierential operator) di : Ci → Ci+1i∈Z such thatdi di−1 = 0. The subscript i in the dierential operator is often omitted.

A graded algebra is an algebra A which can be written as a direct sum A =⊕n∈NAn

such that if a ∈ Ai and b ∈ Aj, then a · b ∈ Ai+j.A dierential graded algebra (DGA) is a graded algebra A, equipped with a map d

of degree either 1 or −1, which satises both d2 = 0 and d(a ·b) = da ·b+(−1)degaa ·db.

Remark. Note that a DGA is also a cochain complex if degd = −1.

Since d2 = 0 in a cochain complex, imdi ⊆ kerdi+1, and it makes sense to look attheir quotient:

Denition 1.2. Given a cochain complex C with dierential operator d, its cohomologyis given by H(C) =

⊕i∈ZH

i(C), where

Hi(C) B (kerd ∩ Ci)/(imd ∩ Ci)

1.1 De Rham cohomology

1.1.1 De Rham cohomology of vector spaces

The De Rham cohomology is, as we shall see, an important dieomorphism invariant ofmanifolds. But to dene it, we must rst dene this cohomology on Rn, in order to laterextend this denition to manifolds using charts.

The De Rham cohomology is an analytic cohomology, in that it is made up from(equivalence classes of) forms, which are in turn built from functions, as follows:

5

Denition 1.3. Consider Rn with standard coordinates x1, . . . , xn, and dene elementsdx1, . . . ,dxn with multiplicative conditions dxi ∧ dxj = −dxj ∧ dxi. Often the notationdxI B dxi1 ∧ · · ·∧ dxiq is used.

A linear combination of elements dxi1 ∧ · · ·∧ dxiq is said to be of degree q.

The space Ωq is dened as the vector space over R with the basis dxI|I|=q, theelementary elements of degree q.

DeneΩ∗ B⊕nq=0Ω

q. Observe thatΩ∗ has a dened multiplication, by extendingthe multiplication on the basis, and it is a graded algebra on Rn.

The C∞ dierential q-forms on Rn are the elements of Ωq(Rn) B C∞(Rn)⊗Ωq.These again piece together to form Ω∗(Rn) B

⊕nq=0Ω

q(Rn) = C∞(Rn)⊗Ω∗. There is a dierential operator d : Ωq(Rn)→ Ωq+1(Rn) dened by:

df =∑

∂f∂xidxi if f ∈ Ω0(Rn)

dω =∑dfI ∧ dxI if ω =

∑fIdxI

Remark. First note that dxi ∧ dxi = −dxi ∧ dxi, so dx2i = 0.

Second, Ω∗(Rm) is again an algebra with multiplication∑I fIdxI ∧

∑I gJdxJ =∑

I,J fIgJdxI ∧ dxJ.Note that τ ∧ ω = (−1)degωdegτω ∧ τ, since switching dxI anddxJ requires degω · deg τ switches of neighbors. An algebra in which multiplicationis commutative or anticommutative depending on the degrees of the elements is calledgraded-commutative or supercommutative.

This also means that, since functions are by denition 0-forms, fω = f∧ω = ω∧ f =

ωf.

Now we have dened this d, we should check some of its properties. For example,it is a linear operator which is based on the derivative, so we would like to know if ananalogue of the Leibniz rule holds:

Proposition 1.4 (Graded Leibniz rule). d is an antiderivation:

d(ω∧ τ) = dω∧ τ+ (−1)degωω∧ dτ

Proof. Since the multiplication is dened by an extension from the basis elements, it

6

suces to prove the claim for monomials ω = fdxI, τ = gdxJ:

d(ω∧ τ) = d(fdxI ∧ gdxJ) = d(fgdxI ∧ dxJ) = d(fg)∧ dxI ∧ dxJ

=∑i

∂fg

∂xidxi ∧ dxI ∧ dxJ =

∑i

(∂f

∂xig+ f

∂g

∂xi)dxi ∧ dxI ∧ dxJ

= dfg∧ dxI ∧ dxJ + fdg∧ dxI ∧ dxJ

= df∧ dxI ∧ gdxJ + (−1)degdxI degdgfdxI ∧ dg∧ dxJ

= dω∧ τ+ (−1)degωω∧ dτ

since degdxI = degω and degdg = 1+ deg g = 1.

In order to dene a cohomology on the De Rham complex Ω∗(Rn) = Ωq(Rn)q∈Z,where Ωq(Rn) = 0 if q < 0, ...,n, we must rst check wether this sequence is in fact acochain complex. This amounts to proving the following proposition.

Proposition 1.5. d2 = 0

Proof. We start by checking this on functions:

d2f = d(∑i

∂f

∂xidxi)=∑i

∑j

∂2f

∂xj∂xidxj ∧ dxi

=∑i<j

∂2f

∂xj∂xidxj ∧ dxi +

∂2f

∂xi∂xjdxi ∧ dxj = 0

Now, let ω =∑I fIdxI:

d2ω = d2(∑I

fIdxI)=∑I

d(dfI ∧ dxI) =∑I

d2fI ∧ dxI − dfI ∧ d(dxI) = 0

by proposition 1.4.

Corollary 1.6. Ω∗(Rn) is a DGA.

The dierential forms in the kernel of d are called closed forms, and those in its imageexact forms. Thus, proposition 1.5 is equivalent to saying that all exact forms are closed.

Now we can dene the cohomology of the De Rham complex:

Denition 1.7. The q-th De Rham cohomology HqDR(Rn) of Rn is dened to be

H∗DR(Rn) B Hq(Ω∗(Rn)).

The (total) De Rham cohomology H∗DR(Rn) of Rn is then H∗DR(R

n) B H(Ω∗(Rn)) =⊕q∈ZH

qDR(R

n).The cohomology class of a form ω is denoted [ω].

If there is no chance of confusion, the subscriptDR is often omitted, and the distinctionbetween a form and its class is also often neglected.

7

Functoriality of Ω∗

Now we have associated to every Euclidean space Rn a graded algebra Ω∗(Rn), we wouldlike to now how the algebras of dierent spaces relate to each other, for otherwise wecannot hope to take seriously the claim that the De Rham cohomology is a dieomorphicinvariant.

Look at smooth functions f : Rm → Rn. We would like to dene a pullback functionf∗ : Ω∗(Rn) → Ω∗(Rm), to relate these algebra's. On the space of smooth functions(C∞ = Ω0(Rn) ), we have a pullback f∗ : Ω0(Rn) → Ω0(Rm) : g 7→ g f. This inspiresthe following denition:

Denition 1.8. Let x1, . . . , xm and y1, . . . ,yn be the standard coordinates on Rm

and Rn respectively, and let f : Rm → Rn be a C∞ function. Then dene the pullbackof f, f∗ by:

f∗ : Ω∗(Rn)→ Ω∗(Rm) :∑I

gIdyi1 ∧ · · ·∧ dyiq 7→∑I

(gI f)d(yi1 f)∧ · · ·∧ d(yiq f)

Proposition 1.9. f∗ commutes with d, i.e. df∗ = f∗d.

Proof. By linearity of both d and f∗, it suces to show the claim for monomials.

df∗(gIdyi1 ∧ · · ·∧ dyiq) = d((gI f)d(yi1 f)∧ · · ·∧ d(yiq f))= d(gI f)∧ d(yi1 f)∧ · · ·∧ d(yiq f)

f∗d(gIdyi1 ∧ · · ·∧ dyiq) = f∗(∑

i

∂gI

∂yidyi ∧ dyi1 ∧ · · ·∧ dyiq

)

=

(∑i

(∂gI

∂yi f)d(yi f)

)∧ d(yi1 f)∧ · · ·∧ d(yiq f)

= d(gI f)∧ d(yi1 f)∧ · · ·∧ d(yiq f)

So we have a way to associate to the Euclidean spaces graded algebras and to smoothmaps between these spaces homomorphisms of the algebras in a nice way. It turns outthat Ω∗ is a functor:

Proposition 1.10. Let f : Rm → Rn be a smooth function. Dene Ω∗(f) = f∗. ThenΩ∗ is a contravariant functor from the category of Euclidean spaces and smoothmaps to the category of DGAs and homomorphisms.

Proof. We should rst show that the categories in the proposition are indeed categories:For the category of Euclidean spaces and smooth maps: if f : Rm → Rn and g : Rn → Rpare smooth, then clearly gf : Rm → Rp is too. Also IdRn is clearly smooth. Associativityis also clear.

8

For the category of DGAs and homomorphisms (homomorphisms in this category shouldboth respect grades and commute with the dierential): if f : A → B and g : B → C

are homomorphisms such that dBf = fdA, dCg = gdB, deg f(a) = dega and deg g(b) =

deg b, then clearly g f is a homomorphism, dCgf = gdBf = gfdA and deg g(f(a)) =

deg f(a) = dega. Clearly any identity is a homomorphism which respects degrees, andit commutes with the dierential by denition. Associativity works in the same way.Now the proof of the proposition itself. Clearly, Ω∗(Rn) is a graded dierential algebra,and if f ∈ Hom(Rm,Rn) then Ω∗(f) = f∗ is by denition a degree-respecting homomor-phism. By proposition 1.9, it also commutes with d, so Ω∗(f) ∈ Hom(Ω∗(Rn),Ω∗(Rm)).Now suppose f ∈ Hom(Rm,Rn) and g ∈ Hom(Rn,Rp), and ω = ωIdxI ∈ Ω∗(Rp).

Ω∗(g f)(ω) = (g f)∗(ωIdxi1 ∧ · · ·∧ dxiq)= (ωI g f)d(xi1 g f)∧ · · ·∧ d(xiq g f)

Ω∗(f) Ω∗(g)(ω) = f∗ g∗(ωIdxi1 ∧ · · ·∧ dxiq)= f∗((ωI g)d(xi1 g)∧ · · ·∧ d(xiq g))= (ωI g f)d(xi1 g f)∧ · · ·∧ d(xiq g f)

Finally, let IdRn ∈ Hom(Rn,Rn) be the identity.

Ω∗(IdRn)(ω) = Id∗Rn(ωIdxi1 ∧ · · ·∧ dxiq)= (ωI IdRn)d(xi1 IdRn)∧ · · ·∧ d(xiq IdRn)= ωIdxi1 ∧ · · ·∧ dxiq) = ω = IdΩ∗(Rn)ω

Thus, Ω∗ satises the axioms of a contravariant functor.

1.1.2 De Rham cohomology of manifolds

In order to generalize the denitions from the former section, we would rst like to knowif the exterior derivative is dependent of the choice of coordinates on Rn. This turns outnot to be the case:

Proposition 1.11. d is independent of the choice of coordinate system on Rn, i.e.,if x1, . . . xn are the standard coordinates on Rn, and y1, . . . ,yn is another system,then for all forms ω =

∑I gIdxI =

∑J hJdyJ:∑

I

n∑i=1

∂fI

∂xidxi ∧ dxI =

∑J

n∑j=1

∂gJ

∂yjdyj ∧ dyJ

Proof. There is a dieomorphism f : Rn → Rn such that yi = xi f = f∗(xi), sodyi =

∑nj=1

∂yi∂xjdxj. Dene d as the exterior derivative in the standard coordinate

9

system and take τ = hIdyI. Then,

dτ = d(hIdyI) = d(hIdf∗(xI)) = dhI ∧ df

∗(xI) + (−1)|I|hId(df∗(xI))

=

n∑i=1

∂hI

∂xidxi ∧ df

∗(xI) =

n∑i=1

n∑j=1

∂hI

∂yj

∂yj

∂xidxi ∧ df

∗(xI) =

n∑j=1

∂hI

∂yjdyj ∧ dyI

Thus, the denition of d does not depend on the choice of coordinates for monomials.By linearity, it is thus also unambiguous on polynomials.

Now, recall the denition of a manifold:

Denition 1.12. A (smooth) manifold is a second countable Hausdor space M to-gether with a dierentiable structure F , with consists of:

An open cover, or atlas U B Uαα∈A which is maximal with respect to inclusions;

And homeomorphisms, called trivialisations ϕα : Uα ∼→ Rn, such that the transi-tion functions gαβ = ϕαϕ−1

β : ϕβ(Uα∩Uβ)→ ϕa(Uα∩Uβ) are dieomorphisms.

Any trivialisation ϕα of Uα gives rise to a coordinate system by xi = ri ϕα, whereri

ni=1 is the standard basis on Rn.

Denition 1.13. A manifold with boundary is a manifold in which the trivialisationsgive homeomorphisms with either Rn or Hn = (x1, . . . , xn)|xn > 0. The boundary ∂Mof M is itself a manifold of codimension 1.

Remark. Although most theorems and propositions are stated and proved for Rn andmanifolds without boundary, they also hold for Hn and manifolds with boundary, exceptwhere noted.

A function f on M is said to be dierentiable on Uα if f ϕα is dierentiable, withits derivative given by:

∂f

∂xi(p) =

∂(f ϕ−1α )

∂ri(ϕα(p))

A dierential form on a manifold M is now a set of forms ωα ∈ Ω∗(Uα) denedon Uα such that, if i : Uα ∩ Uβ → Uα and j : Uα ∩ Uβ → Uβ are inclusions, theni∗ωα = j∗ωβ. Since Ω∗ is a functor, the exterior derivative, the wedge product and thepullback all extend naturally to manifolds, so the denition of Ω∗(M) as the DGA of allforms on M makes sense.

Recall moreover the idea of a partition of unity : a collection of functions ρaα∈I suchthat

∑ρα = 1, and this is a nite sum in a neighborhood of every point. Given an open

cover of a manifold, one can nd a partition of unity subordinate to it. In general, Mwill always denote a manifold with U = Uα as a trivialising open cover, trivialisationsgiven by ϕα and ρα will always be a partition of unity subordinate to U.

10

1.1.3 The compact De Rham cohomology

Analogously to the De Rham cohomology, we can dene the compact De Rham coho-mology of Rn:

Denition 1.14. The De Rham complex with compact support Ω∗c(Rn) is dened as

follows:Ω∗c(R

n) B C∞ functions on Rn with compact support⊗R Ω∗

Remark. Although, trivially, Ω∗c(Rn) ⊆ Ω∗(Rn), it is often not the case that H∗c(R

n) ⊆H∗(Rn).

If we try to dene the functorΩ∗c analogously toΩ∗, however, our approach fails. This

is because the pullback of a map on manifolds need not preserve compactness. Thus Ω∗cis not a contravariant functor with respect to manifolds and smooth maps. However, Ω∗ccan be seen as a covariant functor with respect to inclusions of open sets, by extendingforms to zero outside the original set: if j : U → V is an inclusion, its image under Ω∗c,j∗ = Ω

∗c(j), is given by:

j∗ : Ω∗c(U)→ Ω∗c(V) : ω 7→

(j∗ω : x 7→

ω(x) if x ∈ U,0 if x < U.

)It is obvious that if i : V →W is another inclusion, (ij)∗ = i∗j∗, and that (IdU)∗ = IdU,since it is the map extending a form on U to a form on U by setting it zero on U\U = ∅.This covariant notion of Ω∗c is the way in which it will be used in this text. Sincetrivialisations give embeddings of Rn in manifolds, the extension of Ω∗c to manifoldsmakes sense.

Since we still have that d2 = 0, and d preserves compactness, we can dene

Denition 1.15. The q-th De Rham cohomology with compact support Hqc (M) of amanifold M is dened to be Hqc (M) B Hq(Ω∗c(M)).

The (total) De Rham cohomology with compact support H∗c(M) is then H∗c(M) B

H(Ω∗c(M)) =⊕q∈ZH

qc (M).

1.2 Integration and Poincaré lemmas

1.2.1 Integration

On the Euclidean space Rn with standard coordinates x1, . . . , xn, there is for any smoothfunction f with compact support the notion of its (Riemann) integral

∫Rnfdx1 · · ·dxn.

We would like to extend this notion to an arbitrary manifold M. As suggested by thenotation, dierential forms are the natural tools for this.

We will start with a denition for the integral of forms on Rn:

11

Denition 1.16. Let ω = fdx1 ∧ · · ·∧ dxn ∈ Ωnc (Rn). Then we dene its integral as:∫Rnω =

∫Rnfdx1 · · ·dxn

Remark. There are several things to be noted in this denition:

The integral is only dened for compactly supported top forms, i.e., elements ofΩnc (R

n). Dierential forms have wedges between their dxi, whereas the usual Riemann in-tegral notation doesn't. In order to better distinguish between the two, I will usedots in the Riemann integral, when appropriate.

Although the order of dxi is of no importance in the Riemann integral, it doesmatter in the dierential form. The elements should be ordered in the standardorder for the integration.

In order to generalize this denition to manifolds, however, we need this denition tobe coordinate-free. We thus want to know how our current denition transforms undercoordinate transformations:

Proposition 1.17. Let ω = fdx1 ∧ · · · ∧ dxn be a top-form on Rn with the stan-dard coordinates xi

ni=1. If yj

nj=1 is another coordinate system such that xi =

Ti(y1, . . . ,yn), ∫Rnω = sgn J(T)

∫RnT∗ω

where J(T) is the Jacobian determinant of the dieomorphism T .

Proof. First we look how the dierential elements transform:

T∗dx1 ∧ · · ·∧ dxn = dT1 ∧ · · ·∧ dTn =

(n∑i1=1

∂T1

∂yi1dyi1

)∧ · · ·∧

(n∑in=1

∂Tn

∂yindyin

)

=

n∑i1,...,in=1

n∏j=1

(∂Tj

∂xij

)dyi1 ∧ · · ·∧ dyin

=∑σ∈Sn

n∏j=1

(∂Tj

∂yσ(j)

)dyσ(1) ∧ · · ·∧ dyσ(n)

=∑σ∈Sn

ε(σ)

n∏j=1

(∂Tj

∂yσ(j)

)dy1 ∧ · · ·∧ dyn

= det

(∂Ti

∂yj

)i,j

dy1 ∧ · · ·∧ dyn = J(T)dy1 ∧ · · ·∧ dyn

12

where we can restrict the sum to the symmetric group in the fourth line because dy2i = 0.

Now we can calculate the integrals of both ω and T∗ω (which is ω in the other base):∫RnT∗ω =

∫Rn(f T)dT1 ∧ · · ·∧ dTn =

∫Rn(f T)J(T)dy1 ∧ · · ·∧ dyn∫

Rnω =

∫RNfdx1 ∧ · · ·∧ dxn =

∫Rnfdx1 · · · · · dxn

=

∫Rn(f T) |J(T)|dy1 · · · · · dyn =

∫Rn(f T) sgn(J(T)) J(T)dy1 ∧ · · ·∧ dyn

which proves the proposition.

Thus, not all dieomorphisms preserve the integral. Any dieomorphism that does,is called orientation-preserving. By the proposition, a dieomorphism is orientation-preserving if and only if its Jacobian determinant is positive.

To dene an integral of a form ω on a manifoldM, we seek to take the integral of thepullback on all open sets in the cover, and sum them in an appropriate manner. To dothis unambiguously, we want the integrals of dierent functions to agree on overlaps ofopen sets. More precisely, we require that in the dierentiable structure Uα,ϕαα∈A,all gαβ = ϕα ϕ−1

β be orientation-preserving. If this is the case, we say that the atlas isoriented. A manifold which allows an oriented atlas is called orientable. Without furthernotice, if a manifold is orientable, its trivialisations will be assumed to be oriented.

For a manifold with boundary, an oriented atlas induces an oriented atlas for itsboundary:

Proposition 1.18. Let T : Hn ∼→ Hn be orientation-preserving. Then T induces anorientation-preserving dieomorphism T : ∂Hn ∼→ ∂Hn.

Proof. According to the inverse function theorem, T−1((Hn)) = (Hn), so T(∂Hn) =

∂Hn.T is then dened by Ti(y) = Ti(y, 0) ∀i ∈ 0, . . . ,n− 1.T is orientation-preserving, so for all y ∈ ∂Hn,

J(T)(y, 0) =

∣∣∣∣∣∣∣∂T1∂y1

(y, 0) · · · ∂T1∂yn

(y, 0)...

. . ....

∂Tn∂y1

(y, 0) · · · ∂Tn∂yn

(y, 0)

∣∣∣∣∣∣∣ > 0

Because Tn(y, 0) = 0∀y ∈ Hn, ∂Tn∂yi

(y, 0) = 0∀i < n, and ∂Tn∂yn

> 0, by denition of T .From this it follows that

J(T)(y) =

∣∣∣∣∣∣∣∂T1∂y1

(y, 0) · · · ∂T1∂yn−1

(y, 0)...

. . ....

∂Tn−1∂y1

(y, 0) · · · ∂Tn−1∂yn−1

(y, 0)

∣∣∣∣∣∣∣ > 0

13

A useful criterion for nding out whether a manifold is orientable, is the followingproposition:

Proposition 1.19. A manifold M is orientable if and only if is has a global nowherevanishing top form.

Proof. (⇐ ) Supposeω is a global nowhere-vanishing top form onM. For any coordinatemap ϕa : Uα ∼→ Rn, ϕ∗αdx1∧ · · ·∧dxn = fαω, where fα is a nowhere-vanishing functionon Uα. Thus, fα is either positive or negative. Without loss of generality, we can assumefα to be positive, for otherwise we compose ϕα with the map changing the sign of therst coordinate. Therefore g∗αβ(dx1∧· · ·∧dxn) = (ϕ−1

α )∗ϕ∗β(dx1∧· · ·∧dxn) is a positivemultiple of itself, and the atlas (Uα,ϕα) is oriented.

( ⇒ ) Let (Uα,ϕα) be an oriented atlas for M, and let λαβ be the positive functionsuch that

(ϕαϕ−1β )∗(dx1 ∧ · · ·∧ dxn) = λαβdx1 ∧ · · ·∧ dxn

from which it follows that

ϕ∗αdx1 ∧ · · ·∧ dxn = (ϕ∗βλαβ)ϕ∗βdx1 ∧ · · ·∧ dxn

Dene ωα B ϕ∗αdx1 ∧ · · ·∧dxn, which is nonzero. Also, ωβ = (ϕ∗αλαβ)ωα is a positivemultiple of ωα. Take ρα as a partition of unity subordinate to Uα. Then ω =∑ραωα. This is nowhere-vanishing, since the ρα are positive and

∑ρα = 1.

We thus see that an orientation of a manifold is given by specifying a global nowhere-vanishing top form. Two top forms ω, τ give rise to the same orientation if and only ifthere is a positive function f such that ω = f · τ. This leads to the following denition:

Denition 1.20. Two top forms ω, τ on a manifoldM are said to be equivalent if theydier by a positive function. An orientation of M, written as [M], is an equivalenceclass of top forms.

If Hn is given the standard orientation [dx1 ∧ . . .∧ dxn], the induced orientation onthe boundary ∂Hn is given by [(−1)ndx1 ∧ . . .∧ dxn−1].

If M is an oriented manifold with boundary and orientation [M], the induced orien-tation [∂M] on ∂M is the orientation such that for all orientation-preserving dieomor-phisms ϕ :M ⊇ U → Hn, ϕ∗[∂Hn] = [∂M]|∂U.

With these tools at hand, we can dene the integral of a top form on a manifold:

Denition 1.21. Let [M] be an orientation of the n-dimensional manifold M, withorientation-preserving cover (Uα,ϕa) and let ω ∈ Ωnc (M). Dene the integral of τ by:∫

[M]

τ B∑α

∫Uα

ρατ =∑α

∫Rn(ϕ−1α )∗(ρατ)

14

Proposition 1.22. The denition of the integral is independent of the oriented atlasand the partition of unity.

Proof. Let (Uα,ϕα) and (Vβ,ψβ) be oriented atlases ofM, with respective partitionsof unity ρα and σβ subordinate to them. Then∑

α

∫Uα

ρατ =∑α,β

∫Uα

ρασβτ =∑α,β

∫Vβ

ρασβτ =∑β

∫Vβ

σβτ

since supp(ρασβτ) ⊆ Uα ∩ Vβ.

An important theorem about integration, which generalizes the theorems of Gauss,Green and Stokes in three dimensions, is the following.

Theorem 1.23 (Stokes). Let M be an oriented manifold, with ∂M given the inducedorientation. If ω ∈ Ωn−1

c (M), then∫M

dω =

∫∂M

ω

Proof. The proof goes in three parts: rst the case Rn, then Hn, and the third partgeneralizes this to all manifolds.

Part I By linearity of both the integral and the dierential operator, we can takeω = fdx2 ∧ · · · ∧ dxn, so that dω = ∂f

∂x1dx1 ∧ · · · ∧ dxn. By Fubini's theorem and the

compactness of ω,∫Rndω =

∫ (∫∞−∞

∂f

∂x1dx1

)dx2 · · · · · dxn

=

∫(f(∞, x2, . . . , xn) − f(−∞, x2, . . . , xn))dx2 · · · · · dxn = 0

Since ∂Rn = ∅, this concludes the proof for Rn.Part IIWrite ω =

∑ni=1 fidx1∧· · ·∧dxi∧. . . · · ·dxn, where the hat denotes omission

of that element. Then dω =∑ni=1(−1)

i−1 ∂fi∂xidx1 ∧ · · ·∧ dxn. For 1 6 i 6 n− 1,∫

Hn

∂fi

∂xidx1 ∧ · · ·∧ dxn =

∫ (∫∞−∞

∂fi

∂xidxi

)dx1 · · · · · dxi · · · · · dxn = 0

by the same argument as in part I. Hence,∫Hndω =

∫Hn(−1)n−1 ∂fn

∂xndx1 · · · · · dxn = (−1)n−1

∫ (∫∞0

∂fn

∂xndxn

)dx1 · · · · · dxn−1

= (−1)n−1

∫(fn(x1, . . . , xn−1,∞) − f(x1, . . . , xn−1, 0))dx1 · · · · · dxn−1

=

∫fn(x1, . . . , xn−1, 0)(−1)

ndx1 · · · · · dxn−1

=

∫∂Hn

f(x1, . . . , xn−1, 0)dx1 ∧ · · ·∧ dxn−1 =

∫∂Hn

ω

15

Where the last equality holds since dxn restricted to ∂Hn is 0.Part III Since ω =

∑α ραω and the theorem is linear, we only need to prove the

assertion for ραω. But supp(ρaω) ⊆ supp ρα ∩ suppω ⊂ Uα, so the support is aclosed subset of a compact set and thus compact, and it is contained in Uα. since Uα isdieomorphic to either Rn or Hn, the proof of the theorem extends to Uα. So∫

M

dραω =

∫Uα

dραω =

∫∂Uα

ραω =

∫∂M

ραω

which is the conclusion of the proof.

1.2.2 Poincaré lemmas

So far, we have only seen the denition of the De Rham cohomologies, but we have yetto compute any. In this section, I will compute both the normal and the compact DeRham cohomologies for the case of Rn. The results are called Poincaré lemmas.

Since the only possible forms on R0 = 0 are constant 0-forms, and these forms areclearly compactly supported, it follows that

Hq(R0) = Hqc (R0) =

R q = 0

0 q > 0

In this section, all integrals will be understood to be with respect to dt, where t is thecoordinate on R. Thus,

∫f B∫fdt.

The ordinary case

The argument for higher dimensions goes by induction on the dimension: it will beshown that H∗(M× R) is isomorphic to H∗(M).

Let π :M×R1 →M be the projection on the rst factor, and let s :M→M×R1 bethe zero section. It will be proved that these induce inverse isomorphisms in cohomology.

It is clear that s∗ π∗ = Id, since π s = Id.The other way around, however, is not that easy, since s π , Id, and therefore

π∗ s∗ , Id for forms. However, it is sucient that they are equal in cohomology, so ifId−π∗ s∗ can be shown to be equal to an operator which induces 0 in cohomology, theassertion is proved. It will be shown that there is a so-called homotopy operator K suchthat Id− π∗ s∗ = ±(dK± Kd), which indeed induces zero.

Proposition 1.24. H∗(M× R) H∗(M), with mutually inverse isomorphisms givenby s∗ and π∗.

Proof. If x are the (local) coordinates on (an open subset of)M, and t is the coordinateon R, then every form on M× R can be written as a linear combination of π∗ϕ∧ f and

16

π∗ϕ ∧ f dt, where ϕ ∈ Ω∗(M), and f is a function on M × R. K is then dened as anorder to integrate dt:

K : Ω∗(M× R)→ Ω∗−1(M× R) :

π∗ϕ∧ f 7→ 0

π∗ϕ∧ f dt 7→ π∗ϕ∫t0f

Now, we would like to check if indeed Id − π∗ s∗ = ±(dK − Kd). For the rst case,ω = π∗ϕ∧ f(x, t), this goes as follows:

(Id− π∗ s∗)ω = π∗ϕ∧ f(x, t) − π∗ϕ∧ f(x, 0)

(dK− Kd)ω = −Kπ∗(dϕ)∧ f+ (−1)qπ∗ϕ∧

(∂f

∂x(x, t)dx+

∂f

∂t(x, t)dt

)= (−1)q+1π∗ϕ∧

∫ t0

∂f

∂t(x, t) = (−1)q+1π∗ϕ∧ (f(x, t) − f(x, 0))

And for the second case, ω = π∗ϕ∧ f(x, t)dt:

(Id− π∗ s∗)ω = ω− π∗(ϕ∧ f(x, 0)d(s∗t)) = ω

Kdω = K

(π∗(dϕ)∧ f dt+ (−1)qπ∗ϕ∧

∂f

∂xdx∧ dt

)= π∗(dϕ)∧

∫ t0

f+ (−1)q−1π∗ϕ∧

∫ t0

∂f

∂xdx

dKω = π∗(dϕ)∧

∫ t0

f+ (−1)q−1π∗ϕ∧

(∫ t0

∂f

∂xdx+ f dt

)(dK− Kd)ω = (−1)q−1ω

Where q = degω in both cases, so in both cases (−1)q+1(dK−Kd)ω = (Id−π∗s∗)ω.

Lemma 1.25 (Poincaré).

Hq(Rn) =

R q = 0

0 q > 0

Proof. By repeated use of proposition 1.24, Hq(Rn) = Hq(0), which we have alreadycomputed, and is indeed given as in the proposition.

Corollary 1.26 (Homotopy Axiom). The pullbacks of homotopic maps are equal.

Proof. If f ' g : M → N, then there is a map F : M × I → N such that f(·) = F(·, 0)and g(·) = F(·, 1). We can extend F to M × R by setting F(·, t) = f(·) ∀t < 0, F(·, t) =g(·) ∀t > 1. If s0, s1 :M→M× R are the 0- and 1-section, then f = F s0, g = F s1.

Since both sections induce the inverse of π∗, they must be equal. Thus

f∗ = (F s0)∗ = s∗0 F∗ = s∗1 F∗ = (F s1)∗ = g∗

17

The compact case

For the compact case, the argument again goes by induction, but now an isomorphismwill be established between H∗+1

c (M × R) and H∗c(M). Since the pullback of any ofnonzero form onM toM×R has non-compact support, π∗ is not an operator on compactforms. However, we can dene an chain map, called integration along the ber π∗ :

Ω∗c(M× R)→ Ω∗−1c (M).

As before, any compactly supported form onM×R is a linear combination of π∗ϕ∧ f

and π∗ϕ∧ f dt. However, f now must be compactly supported. Then

π∗ : Ω∗c(M× R)→ Ω∗−1

c (M) :

π∗ϕ∧ f 7→ 0

π∗ϕ∧ f dt 7→ ϕ∫Rf

Let e ∈ Ω1c(R) with integral 1, and dene the chain map e∗ : Ω∗c(M) → Ω∗+1

c (M × R) :ϕ 7→ (π∗ϕ)∧e. Then obviously π∗ e∗ = 1. For e∗ π∗ to be the identity in cohomology,we will nd a homotopy operator K such that Id− e∗π∗ = ±(dK− Kd).

Proposition 1.27. H∗c(M×R) H∗−1c (M), with mutually inverse isomorphisms given

by π∗ and e∗.

Proof. K will now be dened as:

K : Ω∗(M× R)→ Ω∗−1(M× R) :

π∗ϕ∧ f 7→ 0

π∗ϕ∧ f dt 7→ π∗ϕ∧(∫t

−∞ f−A(t) ∫R f)where A(t) =

∫t−∞ e dt.

Now, we would like to check if indeed Id−e∗π∗ = (−1)q−1(dK−Kd), where q = degω

and ω is again either of the cases mentioned. For the rst case, this goes as follows:

(Id− e∗ π∗)π∗ϕ∧ f = π∗ϕ∧ f

(dK− Kd)π∗ϕ∧ f = −K

(π∗(dϕ)∧ f+ (−1)qπ∗ϕ∧

(∂f

∂xdx+

∂f

∂tdt

))= (−1)q−1π∗ϕ∧

(∫ t−∞

∂f

∂t−A(t)

∫R

∂f

∂t

)= (−1)q−1π∗ϕ∧ f

And for the second case:

(Id− e∗ π∗)π∗ϕ∧ f dt = π∗ϕ∧ f dt− π∗(ϕ

∫R

f

)∧ e

18

dK(π∗ϕ∧ f dt) = d

(π∗ϕ∧

(∫ t−∞ f−A(t)

∫R

f

))= π∗(dϕ)∧

(∫ t−∞ f−A(t)

∫R

f

)+ (−1)q−1π∗ϕ∧ fdt

+ (−1)q−1π∗ϕ∧

(∫ t−∞

∂f

∂xdx− e

∫R

f−A(t)

∫R

∂f

∂xdx

)Kd(π∗ϕ∧ f dt) = K

(π∗(dϕ)∧ f dt+ (−1)q−1π∗ϕ

∂f

∂xdt∧ dx

)= π∗(dϕ)∧

(∫ t−∞ f−A(t)

∫R

f

)+ (−1)q−1π∗ϕ∧

(∫ t−∞

∂f

∂xdx−A(t)

∫R

∂f

∂xdx

)(dK− Kd)π∗ϕ∧ f dt = (−1)q−1π∗ϕ∧

(fdt− e

∫R

f

)Therefore, in both cases, (−1)q−1(dK− Kd)ω = (Id− π∗ s∗)ω.

Lemma 1.28 (Poincaré for compact supports).

Hqc (Rn) =

R q = n

0 q < n

Proof. By repeated use of proposition 1.27, Hqc (Rn) = Hq−nc (0), which we have already

computed, and is indeed given as in the proposition.

1.3 The Mayer-Vietoris principle

Now that we have computed the De Rham cohomology of Rn, we would like to computecohomologies of more complicated manifolds. The Mayer-Vietoris sequence and its gen-eralisation, the Mayer-Vietoris principle, are useful for this purpose. They are also verystrong tools to prove certain theorems concerning cohomology.

1.3.1 The Mayer-Vietoris sequence

The Mayer-Vietoris sequence is a manner of computing the cohomology of a manifoldM from the cohomologies of two of its open subsets. More specically, let M = U ∪ V,where U and V are open in M. There is a sequence of inclusions

M U∐V

ioo U ∩ V∂V

oo∂Uoo (1.1)

19

where ∂U and ∂V are the inclusions in U and V, respectively.This induces a sequence of restrictions in the De Rham complex:

Ω∗(M)i∗ //Ω∗(U)⊕Ω∗(V)

∂∗U //

∂∗V

//Ω∗(U ∩ V)

By taking the dierence of the last two maps, we obtain:

Denition 1.29. The Mayer-Vietoris sequence of a manifold M = U ∪ V is given by:

0 //Ω∗(M) //Ω∗(U)⊕Ω∗(V) δ //Ω∗(U ∩ V) // 0

(ω, τ) // ∂∗Uω− ∂∗Vτ

Remark. The notation of restrictions is often omitted, i.e., if j is an inclusion, j∗ω isoften just written ω. The domain will be clear from the context.

Proposition 1.30. The Mayer-Vietoris sequence is exact.

Proof. The exactness at the rst two steps is clear. At the last step, the assertion isthat the dierence operator δ be surjective. Proving the proposition thus amounts toshowing that any form on U∩V is the dierence on two forms on U and V respectively.Let ω ∈ Ω∗(U ∩ V), and let ρU, ρV be a partition of unity subordinate to U,V.Then ρVω ∈ Ω∗(U) and −ρUω ∈ Ω∗(V), and since ρVω − (−ρUω) = ω, this δ issurjective.

The Mayer-Vietoris sequence is a sequence of exact complexes, so it can be writtenas:

0 //Ωq+1(M)d

OO

//Ωq+1(U)⊕Ωq+1(V)d

OO

//Ωq+1(U ∩ V)d

OO

// 0

0 //Ωq(M)

d

OO

//Ωq(U)⊕Ωq(V)d

OO

//Ωq(U ∩ V)d

OO

// 0dOO

dOO

dOO

In this way, it induces an eponymous long exact sequence in cohomology:

Denition 1.31. The cohomology-sequence induced by the Mayer-Vietoris sequence isgiven by

· · ·

Hq+1(M) // Hq+1(U)⊕Hq+1(V) // Hq+1(U ∩ V)

oo

Hq(M) // Hq(U)⊕Hq(V) // Hq(U ∩ V)d∗

..

· · ·

//

20

where d∗ is given by:

d∗[ω] = [(i∗)−1 d (δ∗)−1(ω)] =

[−d(ρV)∧ω] on U

[d(ρU)∧ω] on V

since ω is closed. This sequence is also called the Mayer-Vietoris sequence.

Example 1.32 (The cohomology of the circle). As an example of the Mayer-Vietorissequence, we can compute the De Rham cohomology of the circle, S1. We can coverS1 with two open sets U and V, which are the circle minus the north or south polerespectively. Thus, U V R, and U

∐V U ∩ V R

∐R. The cohomology of these

last two is therefore given by

Hq(U∐

V) = Hq(U ∩ V) =

R⊕ R q = 0

0 q > 0

The Mayer-Vietoris sequence now gives a short exact sequence:

0 // H0(S1) // H0(U∐V)

δ // H0(U ∩ V) d∗// H1(S1) // H1(U

∐V)

which is:

0 // H0(S1) // R⊕ R δ // R⊕ R d∗ // H1(S1) // 0

(ω, τ) // (τ−ω, τ−ω)

Clearly, dim im δ = 1, so H0(S1) = ker δ = R, and H1(S1) = coker δ = R, and the otherdegrees are zero.

Example 1.33 (The cohomology of Sn). The cohomology of the n-sphere can be calcu-lated by induction. Suppose we have found that

Hq(Sn−1) =

R q ∈ 0,n− 1

0 0 < q < n− 1

Then, let Sn = U ∪ V, where again U and V are the sphere without north and southpole respectively. Then U

∐V Rn

∐Rn and U ∩ V Sn−1. We get two short exact

sequences

0 // H0(Sn) // H0(U∐V)

δ // H0(U ∩ V) d∗ // H1(Sn) // H1(U∐V)

Hn−1(U∐V) // Hn−1(U ∩ V) d

∗// Hn(Sn) // Hn(U

∐V)

which are

0 // H0(Sn) // R⊕ R δ // Rd∗ // H1(Sn) // 0

0 // Rd∗ // Hn(Sn) // 0

Since the image of δ in the rst sequence is clearly one-dimensional, H0(Sn) = R

and H1(Sn) = 0. The second sequence clearly gives Hn(Sn) = R. By using the former

21

example and induction, we now have for all n:

Hq(Sn) =

R q ∈ 0,n

0 0 < q < n

The Mayer-Vietoris sequence for compact support

The same argument gives the Mayer-Vietoris sequence for compact supports. The sameequation:

M U∐V

ioo U ∩ V∂V

oo∂Uoo (1.2)

yields a sequence of forms in the compactly supported De Rham complex:

Denition 1.34. The compactly supported Mayer-Vietoris sequence of a manifoldM = U ∪ V is given by:

0 Ω∗c(M)oo Ω∗c(U)⊕Ω∗c(V)+oo Ω∗c(U ∩ V)

δoo 0oo

(−j∗ω, j∗ω) ωoo

Proposition 1.35. The compactly supported Mayer-Vietoris sequence is exact.

Proof. Trivial.

Denition 1.36. The eponymous cohomology-sequence induced by the compactly sup-ported Mayer-Vietoris sequence is given by

· · ·

Hq+1c (M)

//

Hq+1c (U)⊕Hq+1

c (V)oo Hq+1c (U ∩ V)oo

Hqc (M)d∗

pp

Hqc (U)⊕Hqc (V)oo Hqc (U ∩ V)oo

· · ·

oo

where, again because ω is closed,

d∗[ω] = [(δ∗)−1 d (+∗)−1(ω)] = [−d(ρU)∧ω] = [d(ρV)∧ω]

1.3.2 The Mayer-Vietoris argument

The Mayer-Vietoris argument is a method of proof by induction on the cardinality of anite good cover. The idea is to prove an assertion for Rn rst, and then by using theFive Lemma on the Mayer-Vietoris sequence, give an inductive step.

Denition 1.37. A good cover is an open cover U such that any nite non-emptyintersection Uα0···αm B Uα0

∩ . . . ∩Uαm Rn.A manifold is of nite type if it has a nite good cover.

22

If U = Uαα∈I and V = Vββ∈J are two open covers, if there is a map ϕ : J→ I suchthat Vβ ⊆ Uϕ(β), then V is said to be a renement of U, written U < V.

Theorem 1.38. Every open cover on a manifold has a renement which is a goodcover. Every compact manifold is of nite type.

Proof. The proof of this theorem is outside the scope of this text. See e.g. Bott & Tu[2]

The other important tool in the Mayer-Vietoris argument is the following:

Lemma 1.39 (Five Lemma). If in the commutative diagram

· · · // Af1 //

α

Bf2 //

β

Cf3 //

γ

Df4 //

δ

E //

ε

· · ·

· · · // A ′g1 // B ′

g2 // C ′g3 // D ′

g4 // E ′ // · · ·the vertices are Abelian groups, the arrows are group homomorphisms, the rows areexact and α, β, δ and ε are isomorphisms, then so is γ.

Proof. First, let c ∈ kerγ. Then f3(c) ∈ ker δ = 0 by commutativity, so by exactnessc = f2(b) for some b ∈ B. By commutativity, g2 β(b) = 0, so there is an a ′ ∈ A ′ suchthat g1(a ′) = β(b). There is an a ∈ A such that α(a) = a ′, so f1(a) = b, since theyboth map to the same element in B ′. By exactness, c = 0.

Secondly, let c ′ ∈ C ′. There is a d ∈ D with δ(d) = g3(c′). By commutativity and

exactness, f4(d) = 0. Hence, there is a c ∈ C which maps to d. Then c0 = c ′−γ(c) mapsto 0 in D ′. Thus, there is a b ′ ∈ B ′ such that g2(b ′) = c0. This element comes from ab ∈ B, and γ f2(b) = c0, so f2(b) + c maps to c ′. Thus, γ is an isomorphism.[4]

Finite-dimensionality of De Rham Cohomology

Proposition 1.40. IfM has a nite good cover, then its cohomology and its compactcohomology are nite-dimensional.

Proof. For Rn, this is obvious from the Poincaré lemma.By the Mayer-Vietoris sequence

· · · // Hq−1(U ∩ V) d∗ // Hq(U ∪ V) r // Hq(U)⊕Hq(V) // · · ·

we obtain that Hq(U ∪ V) im r⊕ imd∗. If M has a good open cover U0, . . . ,Up, letV = U0 ∪ · · · ∪ Up−1. Then both Up and V have open covers of at least p elements.On Up ∩ V we can take a good open cover with p elements as U0p, . . . ,Up−1,p. Byinduction hypothesis, the cohomologies of Up, V and Up ∩ V are nite-dimensional, sothe cohomology of M = Up ∪ V is too.

The argument for compact cohomology is similar.

23

Poincaré duality

Poincaré duality is a way of pairing compact cohomology and normal cohomology in anon-degenerate way. By Stokes' Theorem, integration descends to cohomology. We canthen dene on an n-dimensional oriented manifoldM a pairing between the compact DeRham cohomology and the ordinary De Rham cohomology:∫

: Hq(M)⊗Hn−qc (M)→ R : ω⊗ τ 7→∫M

ω∧ τ

Theorem 1.41 (Poincaré duality). The pairing dened above is non-degenerate if Mis of nite type. In other words,

Hq(M) = (Hn−qc (M))∗

In order to prove this, we rst need the following lemma:

Lemma 1.42. The following diagram, combining Mayer-Vietoris sequences, is com-mutative up to sign:

· · · // Hq(U ∪ V) i∗ // Hq(U)⊕Hq(V) δ∗ // Hq(U ∩ V) d∗ // Hq+1(U ∪ V) // · · ·⊗ ⊗ ⊗ ⊗

· · · Hn−qc (U ∪ V)oo ∫U∩V

Hn−qc (U)⊕Hqc (V)+oo ∫

U+∫V

Hn−qc (U ∩ V)oo ∫U∩V

Hn−q−1c (U ∪ V)d∗oo ∫

U∪V

· · ·oo

R R R R

Proof. The rst two squares are clearly commutative. The third square then remains tobe checked.

Let ω ∈ Hq(U ∩ V) and τ ∈ Hn−q−1c (U ∪ V). By the denition of both d∗ and d∗,

and since suppd∗ω ⊆ U ∩ V),∫U∪V

d∗ω∧ τ =

∫U∩V

dρU ∧ω∧ τ = (−1)degω∫U∩V

ω∧ dρU ∧ τ

= (−1)degω+1

∫U∩V

ω∧ d∗τ

which proves that the nal square is commutative up to sign.

Now we can do the proof of Poincaré duality:

Proof of theorem 1.41. This is, as indicated, a proof by induction on the cardinalityof the good cover. The basis step (n = 1) follows from the Poincaré lemmas for bothcohomologies.

Now suppose Poincaré duality holds for all manifolds with an open cover of at mostp subsets, and let M have a good cover U0, . . . ,Up. We can take a good open coverwith n elements on (U0 ∪ · · · ∪Up−1) ∩Up as U0p, . . . ,Up−1,p. Then Poincaré duality

24

holds for (U0 ∪ · · · ∪Up−1)∩Up, and U0 ∪ · · · ∪Up−1, so by the Five Lemma it holds forM = U0 ∪ · · · ∪Up as well.

Actually, Hq(M) = (Hn−qc (M))∗ always holds. However, Hqc (M) = (Hn−q(M))∗ doesnot if M has no nite good cover.

1.3.3 The generalized Mayer-Vietoris principle

The Mayer-Vietoris sequence can be reformulated in order to generalize it to any count-able cover.

Dene

Kp,q = Cp(U,Ωq) =∏

α0,α1,...,αp

Ωp(Uα0α1...αp)

K∗,∗ = C∗(U,Ω∗) =⊕p,q>0

Kp,q

If U = U,V, then this gives the Mayer-Vietoris sequence with two dierential operators,d and δ, in which case the δ-sequence of course terminates after two groups. Since d andδ are independent, they commute.

In order to extend this notion, we take U = Uαα∈J, where J is required to be countableand ordered. Since any manifold is second countable, this is not a severe restriction. Thenthere is a sequence of inclusions:

M∐α0Uα0

oo∐α0<α1

Uα0α1

∂0oo∂1

oo∐α0<α1<α2

Uα0α1α2oooooo · · ·oooo

oooo

Where ∂i includes U...αi−1αiαi+1... in U...αi−1αi+1... (i.e., it omits the i-th subscript). Thissequence induces a sequence of forms

Ω∗(M)r //∏α0Ω∗(Uα0

)∂∗0 //∂∗1

//∏α0<α1

Ω∗(Uα0α1)

// ////∏α0<α1<α2

Ω∗(Uα0α1α2)

// ////// · · ·

where ∂∗i : Ω∗(U...αi−1αi...)→

∏αΩ

∗(U...αi−1,ααi...).

Denition 1.43. The dierence operator δ is dened as follows: ifω ∈∏Ωq(Uα0...αp),

where ωα0...αp ∈ Ω∗(Uα0...αp), then

(δω)α0...αp+1 =

p+1∑i=0

(−1)iωα0...αi...αp+1

where restrictions ∂∗i have been omitted and hats denote omissions of that particularsubscript.

Proposition 1.44. δ2 = 0.

25

Proof.

(δ2ω)α0...αp+2 =∑

(−1)i(δω)α0...αi...αp+2

=∑j<i

(−1)j(−1)iωα0...αj...αi...αp+2 + (−1)j−1(−1)iωα0...αi...αj...αp+2 = 0

Denition 1.45. The sequence

Ω∗(M)r //∏Ω∗(Uα0

)δ //∏Ω∗(Uα0α1

)δ //∏Ω∗(Uα0α1α2

)δ // · · ·

is called the generalized Mayer-Vietoris sequence.

Proposition 1.46. The generalized Mayer-Vietoris sequence is exact.

Proof. Let ω ∈∏Ω∗(Uα0...αp be a cocycle (δω = 0 ).

(δω)αα0...αp = ωα0...αp +∑i

(−1)i+1ωαα0...αi...αp = 0

Dene a cochain τ by

τα0...αp−1 =∑α

ραωαα0...αp−1

Then its dierence is given by:

(δτ)α0...αp =∑i

(−1)iτα0...αi...αp =∑i,α

(−1)iραωαα0...αi...αp

=∑α

ρα∑i

(−1)iωαα0...αi...αp =∑α

ραωα0...αp = ωα0...αp

Thus, every cocycle is a coboundary.

Denition 1.47. The double complex C∗(U,Ω∗) with dierential operators d and δ iscalled the ech-De Rham complex.

The ech-De Rham complex can be made into a singly graded complex by taking thegroups Kn =

⊕p+q=n K

p,q and the dierential operator D = δ+ (−1)pd.

Remark. Observe that D2 = d2 + δ2 + (−1)pδd+ (−1)p+1dδ = 0, so the denition doesindeed give a dierential complex.

26

The (augmented) ech-De Rham complex can conveniently be represented as a table:

q

0 //Ω2(M)r // K0,2 K1,2 K2,2

0 //Ω1(M)

d

OO

r // K0,1 K1,1 K2,1

0 //Ω0(M)

d

OO

r // K0,0 K1,0 K2,0

//

OO

C0(U,R)iOO

δ // C1(U,R)iOO

δ // C2(U,R)iOO

p

0

OO

0

OO

0

OO

Here the rst column is the De Rham complex, and analogously the bottom row, C∗(U,R)is the ech complex of the cover U. Cp(U,R) consists of the locally constant functionson p+ 1-fold intersections of elements of U. Its cohomology, H∗(U,R), is called the echcohomology. We can use this double complex to compute the De Rham cohomology byusing the following theorem:

Theorem 1.48 (Generalized Mayer-Vietoris principle). If all rows of an augmenteddouble complex are exact, then the D-cohomology is isomorphic to the cohomologyof the initial column.

Proof. Using the terminology for the ech-De Rham complex, let ϕ be a D-cocycleof degree n. Then ϕ consist of elements ϕi of the n + 1st anti-diagonal, such thatdϕ0 = δϕn = (−1)idϕi+δϕi−1 = 0. Then, by exactness, ϕn = δψn, so [ϕ] = [ϕ−Dψn].By iterating this, ϕ can be assumed to be nonzero only in K0,n. Because δϕ = 0,it comes from an element ψ ∈ Ωn(M) by r. Because dϕ = 0, ψ is a cocycle, andr∗[ψ] = [r(ψ)] = [ϕ]. Hence, r∗ : H∗DR(M)→ HDC

∗(U,Ω∗) is surjective.If r(ω) = Dϕ ∈ K0,n, then still δϕn−1 = (−1)idϕi + δϕi−1 = 0, so we can assume

ϕ ∈ K0,n−1. Then δϕ = 0, so ϕ comes from a τ ∈ Ωn−1(M), and r(ω − dτ) =

r(ω) − r(dτ) = Dϕ − dr(τ) = dϕ − dϕ = 0. By injectivity of r (a function which iszero on a cover is zero on the total set), ω = dτ. Therefore, in cohomology, r∗([ω]) = 0

implies [ω] = 0, so r∗ is also injective.

Corollary 1.49. The restriction map r : Ω∗(M) → C∗(U,Ω∗) induces an isomor-phism in cohomology

r∗ : H∗DR(M) ∼→ HDC∗(U,Ω∗)

Theorem 1.50. If U is a good cover of M, then

H∗DR(M) H∗(U,R)

27

Proof. Because U is a good cover, each Uα0...αp is contractible, and by the Poincarélemma, their cohomologies all vanish for p > 0. Therefore, all columns of the ech-De Rham complex are exact. By the generalized Mayer-Vietoris principle, H∗(U,R) HDC

∗(U,Ω∗), which is in turn isomorphic to HDR(M).

Corollary 1.51. The ech cohomology H∗(U,R) does not depend on the choice ofgood cover U.

Proof. Indeed, the De Rham cohomology does not depend on the cover.

Denition 1.52. The ech cohomology of a manifold M is dened as

H∗(M,R) B limUH∗(U,R) =

∐U

H∗(U,R)/ ∼

Where [fU] ∼ [fV] if and only if for some W < U,V, ρUW([fU]) = ρUV([fV]), and the ρ denote

restrictions.

Proposition 1.53. The ech cohomology of a manifold M is isomorphic to its DeRham cohomology, i.e.,

H∗(M,R) H∗DR(M)

Proof. Since the good covers are conal in all covers, We can restrict the limit to onlythe good covers. By theorem 1.50, the isomorphism holds under the limit, and it iscompatible with renements. Hence, the isomorphism hold in the limit.

28

Chapter 2

Fiber bundles

2.1 Introduction and constructions of ber bundles

2.1.1 Fiber bundles

While the former chapter developed cohomology theory for manifolds, the rest of thispaper will mainly focus on ber bundles:

Denition 2.1. A ber bundle is a quadruple (E,B, F,π), such that the total space E,the base space B and the ber F are smooth manifolds and the projection π : E → B

is a map such that for all b ∈ B, π−1(b) F and there is an open U 3 b such thatE|U B π−1(U) U× F, in a way that preserves bers.

If F is a vector space, and E|U U × F in a linear way, the bundle is called a vectorbundle. In this case, F may be a complex vector space.

If F is a Lie group with a free, ber-preserving, smooth right action on E, the bundleis called a principal F-bundle

A bundle (E,B, F,π) is often denoted π : E → B, and in the rest of this paper, π, Eand B will mean the projection, total space and base space of a ber bundle respectively,and F will denote its ber.

Denition 2.2. A section is a function s : B→ E such that π s = IdB.The set of all sections of π : E→ B will be denoted either C∞(B,E) or Ω0(B,E).

Example 2.3. A simple example is given by the trivial bundle: E = B × F with π :

B × F → B the projection on the rst coordinate. A section is then given by s : B →B × F : b 7→ (b,σ(b)), where σ is any smooth function from B to F. The idea of berbundles is actually to generalize this notion.

Example 2.4. To each point x on a manifold M of dimension n we can associate atangent space TxM, which is the vector space of germs of paths through x. Then we can

29

dene the tangent bundle:

TM B∐x∈M

TxM =⋃x∈M

x× TxM

Then π : TM → M : (x, v) 7→ x is a bundle. Locally, a basis of TM is given byπ∗xi,π

∗ ∂∂xi

ni=1.

Example 2.5. In this context, with x ∈ U, dene T∗xM B (TxM)∗, and let ( ∂∂xi

)∗ =

dxi. Then we can extend this to∧q

T∗xM and∧∗T∗xM =

⋃nq=0

∧qT∗xM by setting

ω∧ τ(X, Y) = ω(X)τ(Y) − τ(X)ω(Y). This gives us the bundles∧q T∗M B

∐x∈M

∧q T∗xM =

⋃x∈M

x×∧q T∗xM∧

∗ T∗M B∐x∈M

∧∗ T∗xM =

⋃x∈M

x×∧∗ T∗xM

Then it is easily seen that∧q

T∗xM Ωq,∧∗T∗xM Ω∗, and

Ωq(M) = C∞ (M,∧q T∗M

)Ω∗(M) = C∞ (M,

∧∗ T∗M

)Note in particular the case q = 0 :

∧0T∗xM B R, so that Ω0(M) = C∞(M,R).

Denition 2.6. A bundle map between to bundles π1 : E1 → B1 and π2 : E2 → B2 is apair of maps (u, f) such that the following diagram commutes:

E1

π1

u // E2

π2

B1 f// B2

On a bundle, we have again the notion of trivialisations and transition functions:

Denition 2.7. Let U = Uαα∈A be a cover of B such that E|Uα Uα×F, with ψα givingthis dieomorphism. Then ψα is said to be a trivialisation, and the coordinate mapstogether form a local frame : a set of local sections whose restrictions freely generate theber. If gαβ = ψβ ψ−1

α : Uα,β × F → Uαβ × F, we call it a transition function. Sincegαβ is invertible, it can be regarded as a function gαβ : Uαβ → End(F). End(F) is saidto be the structure group of E. The notations ψα and gαβ will always be used forthe trivialisations and transition functions of π : E→ B, unless stated otherwise.

Remark. Note that, although these denitions are very similar to the denitions of triv-ialisations and transition functions of a manifold, they are not equal. The trivialisationsof the manifold E are not the trivialisations of the bundle π : E → B. From now on Uwill be assumed to trivialise both the ber bundle π : E→ B and the manifold B.

30

As is clear from the denition, the transition functions satisfy the following conditions:

1. (Identity) gαα = Id ∈ End(F)

2. (Inverse) gαβgβα = Id

3. (Cocycle) gαβgβγgγα = Id

In fact, given a manifold B with open cover Uα and functions gαβ : Uα,β → End(F),which satisfy those three conditions, the converse holds too:

Theorem 2.8. Let B be a manifold of rank m with open cover Uαα∈A. If there arefunctions gαβ : Uα,β → End(F), one for every non-empty Uαβ, such that they satisfythe above three conditions, then there is a vector bundle π : E→ B which has gαβ

as its transition functions. Moreover, this bundle is unique up to isomorphism.

Proof. First dene Z =∐α∈AUα × F, and dene a relation:

(x, v)α ∼ (x ′, v ′)β i x = x ′ and v ′ = gβα(x)(v)

By condition 1, (x, v)α ∼ (x,gααv)α = (x, v)α, so ∼ is reexive.By condition 2, (x, v)α ∼ (x ′, v ′)β i x = x ′ and v ′ = gβα(x)(v) i x ′ = x and v =

gαβ(x)(v′) i (x ′, v ′)β ∼ (x, v)β, so ∼ is also symmetric.

By conditions 2 and 3, (x, v)α ∼ (x ′, v ′)β and (x ′, v ′)β ∼ (x ′′, v ′′)γ i x = x ′ =

x ′′ and v ′′ = gγβ(x)(v′) = gγβ(x)gβα(x)(v) = gγα(x)(v) i (x, v)α ∼ (x ′′, v ′′)γ, so ∼

is transitive, and hence an equivalence relation.Now, dene E = Z/ ∼. The quotient map Z → E is open, since ∼ identies open

subsets of Z. Also, ∼⊂ Z × Z is closed: if ((x, v)α, (y,w)β) ∈ Z × Z\ ∼, we have one ofthe following:

x , y, so we can nd open disjoint neighborhoods U 3 x and V 3 y, which meansthat the open neighborhood (U× F)× (V × F) ⊂ Z× Z\ ∼.

x = y and w , gβαv, so we can nd open neighborhoods W 3 w and V 3 v suchthat W ∩ gβα(V) = ∅. Then, in the same way, (Uα × V)× (Uβ ×W) ⊂ Z× Z\ ∼.

Then, by theorem 7.8 and corollary 7.10 of Tu [5], E is a manifold.We can rene the cover Uα of B such that it is maximal and admits trivialisations

ϕα : Uα → Rm. Let Vα = [Uα × F] ⊆ E. Then take ϕα : Vα → Rm × F : [(x, v)α] →(ϕα(x), v). On Vα then, we have that the projection π : E→ B is given by π = π1 ϕα:

Vαπ

ϕα

!!Uα Uα × F

π1oo

31

Hence, π is smooth. We also have that, if x ∈ Uα,

ϕ−1α π−1

1 (x) = [(x, v)α]|v ∈ F ⊂ π−1(x)

and if x ∈ Uβ too, then [(x, v)β] = [(x,gαβ(x)(v))α], so π−1(x) = [(x, v)α]|v ∈ F F, sothe bers are indeed F.

By the same argument, π−1(Uα) = [(x, v)α]|x ∈ Uα, v ∈ F Uα × F, so π : E→ B islocally trivial.

Finally, these trivialisations of π : E→ B are given by ψα : E|Uα → Uα×F : [(x, v)α] 7→(x, v), Hence, g ′αβ(x)(v) = ψα ψ−1

β (x, v) = ψα([(x, v)β]) = ψα([(x,gαβ(x)(v))α]) =

gαβ(x)(v).Taking this all together, the given construction does indeed give a manifold E, a

projection map π : E→ B with bers F and transition functions gαβ.The isomorphism between two such bundles is given by the identity on the trivialised

spaces Uα × F.

Remark. Note that in the case of a rank n vector bundle, End(F) = GL(F) = GL(n,R),and in the case of a principal G-bundle, End(G) = G.

We would also like to know when two sets of transition functions gαβ and hαβ givethe same bundle. This is given by the following lemma:

Lemma 2.9. Two sets of transition functions gαβ and hαβ dene the same berbundle if and only if there exists maps λα : Uα → End(F) for which

gαβ = λαhαβλ−1β

Proof. (⇒ ) If ψα and ϕα are the trivialisations belonging to gαβ and hαβ re-spectively, then there must be maps λα : Uα → End(F) such that ψα = λαϕα.Hence,

gαβ = ψαψ−1β = λαϕαϕ

−1β λ

−1β = λαhαβλ

−1β

(⇐ ) Let ϕα be the trivialisations of hαβ, and dene ψα = λαϕα. Then these arealso trivialisations of the bundle, and clearly

gαβ = λαhαβλ−1β = λαϕαϕ

−1β λ

−1β = ψαψ

−1β

Hence, both sets are transition functions for the same bundle.

Denition 2.10. Two sets of transition functions as in the former lemma are calledequivalent.

If there is a set of transition functions on a bundle, such that they take their valuesin a subgroup G of End(F), the structure group is said to be reducible to G.

32

Denition 2.11. A real vector bundle is called orientable if its structure group isreducible to GL+(n,R).

Proposition 2.12. The structure group of any real vector bundle can be reduced toO(n). If the bundle is orientable, it can be reduced to SO(n).

Proof. Dene a metric on E by rst pulling back the orthonormal basis on Rn to theUα via the the trivialisations to obtain 〈·, ·〉α and then dene 〈·, ·〉 =

∑α ρα〈·, ·〉α. By

Gram-Schmidt, the trivialisations can be orthonormalised, so that gαβ sends orthonormalframes to orthonormal frames, and thus takes values in O(n).

If π : E→ B is orientable, then gαβ(x) ∈ GL+(n,R) ∩O(n) = SO(n).

Now we know when Uα and gαβ piece together to form a bundle, we would like tohave have the same idea for sections.

Let E be given byE =

(∐Uα × F

)/gαβ

with trivialisations ψα : E|Uα → Uα × F. If s : B → E is a section, clearly its restrictionto Uα can be given as sα B ψα s : Uα → Uα × F : b 7→ (b,σα(b)). Hence, s induces aset of `local sections' σα : Uα → F.

Suppose now a collection σα : Uα → F is given. When can we nd a global sections : B→ E with s|Uα = σα? If they piece together, the following diagram must commuteon Uαβ:

E|Uαβψβ

yy

ψα

%%Uαβ × F Uαβσβ

oo

σ

OO

σα// Uαβ × F

Hence, it follows that σβ = gβασα is a sucient and necessary condition to form a globalform, which is then dened at any Uα by s = ψ−1

α σα, and globally can be written ass =∑α ραψ

−1α σα, where ρα is a partition of unity subordinate to Uα.

2.1.2 Constructing new vector bundles

Given some vector bundles, it is possible to dene new vector bundles in a way analogousto vector space constructions:

Denition 2.13. Let E = (∐Uα × Rn) /gαβ be a vector bundle. Its dual bundle is

dened asE∗ =

(∐Uα × Rn

)/(g−1

αβ)>

It is clear that the transition functions satisfy the required conditions. However, it isnot obvious that this is indeed the dual bundle:

33

Proposition 2.14. E∗ is the dual bundle to E, i.e., (E∗)b = (Eb)∗.

Proof. Suppose λ|b : Eb → R, and let ψα be the trivialisations of E. Then, we get anassociated set λα = λ ψ−1

α of dual vectors on the Uα. On Uαβ, we get

λα = λ ψ−1α = λ (gαβ ψβ)−1 = λ ψ−1

β g−1αβ = λβ g−1

αβ

If ei is a basis for Eb, the dual basis is given by e∗i . This identies the 1 × n matrixλ with its transpose vector λ>. Hence λ>α = (λb g−1

αβ)> = (g−1

αβ)> λ>b . Therefore the

transition functions of the dual bundle are indeed (g−1αβ)>.

Some other common constructions are given in the following denition:

Denition 2.15. Suppose E = (∐Uα × Rn) /gαβ and F = (

∐Vγ × Rm) /hγδ are

vector bundles over a base space B. The direct sum bundle, the tensor product bundleand the homomorphism bundle are dened respectively as

E⊕ F B(∐

(Uα ∩ Vγ)× Rn+m)/gαβ ⊕ hγδ

E⊗ F B(∐

(Uα ∩ Vγ)× Rn·m)/gαβ ⊗ hγδ

Hom(E, F) B E∗ ⊗ F

where gαβ ⊕ hγδ and gαβ ⊗ hγδ are made by the usual direct sum and tensor productof matrices.

Again, it is clear that these transition functions satisfy the identity, inverse and cocycleconditions, and (E⊕F)|b = E|b⊕F|b, (E⊗F)|b = E|b⊗F|b and Hom(E, F)|b = Hom(E|b, F|b).

The pullback of a vector bundle

Suppose we have the following diagram:

E

π

Xf // B

Then there is a unique pullback bundle f∗E → X which commutatively completes thesquare:

f∗Ef //

p

E

π

X

f// B

This is dened as follows:

34

Denition 2.16. Suppose F = Rn and f : X → B a smooth map. Then the pullbackbundle of E to X by f is dened as p : f∗E→ X, where

f∗E = (x, e) ∈ X× E|f(x) = π(e)p(x, e) = x

Proposition 2.17. p : f∗E→ X is a vector bundle with ber equal to the ber of theoriginal bundle, and with transition functions gfαβ(x) = gαβ(f(x)).

Proof. The ber is p−1(x) = (x, e)|f(x) = π(e) = x × π−1(f(x)) Rn, so it is indeedequal.

Let U ⊆ B be an open set over which E can be trivialised: E|U π×ψ U × Rn, anddene V = f−1(U). Then

f∗E|V = (x, e)|π(e) = f(x) ∈ U ⊂ V × E|U V ×U× Rn

Since every point x ∈ V corresponds to exactly one point f(x) ∈ U, and hence also tof(x)× Rn ⊆ U× Rn, we have that f∗E|V V × Rn by

Id×ψ : V × E|U → V × Rn : (x, e) 7→ (x,ψ(e))

If Vα and Uα = f(Vα) are trivialising open covers of their respective bundles, we havethat gfαβ : Vαβ → GL(n,R) : x 7→ (ψβ(e) 7→ ψα(e)), and since π(e) = f(x), we have thatgαβ(f(x))(ψβ(e)) = ψα(e) and therefore, gfαβ(x) = gαβ(f(x)).

The following proposition shows that pullback constructions are essentially unique:

Proposition 2.18. Suppose p : H→ X and π : E → B are two vector bundles with abundle map (f : X → B,u : H → E) such that u is an isomorphism when restrictedto bers. Then H f∗E.

Proof. We have that, ber wise, H|x E|f(x) f∗E|x. Hence f−1 u : H ∼→ f∗E gives thefollowing commutative diagram:

H

p

f−1u //f∗E

π ′

u−1foo

X

which directly proves the proposition.

Corollary 2.19. If p1 : H1 → X and p2 : H2 → X are two ber bundles with bundlemaps (f : X → B,u1 : H1 → E) and (f : X → B,u2 : H2 → E), which are bothisomorphisms on bers, then H1 H2.

35

2.2 The Leray-Hirsch theorem and the Künneth for-

mula

In order to link the concepts of De Rham theory and ber bundles, it is useful to knowhow to compute the cohomology of a ber bundle. A useful tool for this is the followingtheorem.

Theorem 2.20 (Leray-Hirsch). Suppose B has nite good cover. If there are globalcohomology classes which, restricted to a ber, freely generate the cohomology ofthe ber, then for all n

Hn(E) Hn(B)⊗Hn(F) Bn⊕p=0

Hp(B)⊗Hn−p(F)

Proof. The projections π and ρ = p ψα : E|Uα → Uα × F→ F piece together to a mapof forms ε : ω⊗ τ 7→ π∗ω∧ ρ∗τ, the defenestration map. Since

εd(ω⊗ τ) = ε(dω⊗ dτ) = π∗dω∧ ρ∗dτ = dπ∗ω∧ dρ∗τ = d(π∗ω∧ dρ∗τ)

and

dε(ω⊗ τ) = d(π∗ω∧ ρ∗τ) = π∗dω∧ ρ∗τ+ (−1)degτπ∗ω∧ ρ∗dτ

we have that ε preserves both exactness and closedness, and hence induces a map ε∗ incohomology.

To prove: ε∗ : H∗(B)⊗H∗(F)→ H∗(E) is an isomorphism.If B = Rm and E = B× F, this is the Poincaré lemma.Now, let U = Uα be an open good cover of B over which E can be trivialised. The

Mayer-Vietoris sequence

· · · → Hp(U ∪ V)→ Hp(U)⊕Hp(V)→ Hp(U ∩ V)→ · · ·

gives an exact sequence by tensoring with Hn−p(F):

· · · → Hp(U ∪ V)⊗Hn−p(F)→ (Hp(U)⊗Hn−p(F))⊕ (Hp(V)⊗Hn−p(F))→ Hp(U ∩ V)⊗Hn−p(F)→ · · ·

Now summing over all p yields the sequence

· · · →⊕p

Hp(U ∪ V)⊗Hn−p(F)→⊕p

(Hp(U)⊗Hn−p(F))⊕ (Hp(V)⊗Hn−p(F))

→⊕p

Hp(U ∩ V)⊗Hn−p(F)→ · · ·

This sequence can be mapped by ε∗ to the sequence

· · · → Hn((U ∪ V)× F)→ Hn(U× F)⊕Hn(V × F)→ Hn((U ∩ V)× F)

36

The restriction maps clearly commute with ε∗. Furthermore, we have, by denition 1.31,

d∗ε∗(ω⊗ τ) = d∗(π∗ω∧ ρ∗τ) = d((π∗ρUπ∗ω∧ ρ∗τ)

= (dπ∗(ρUω)∧ ρ∗τ = π∗(d∗ω)∧ ρ∗τ = ε∗(d∗ω⊗ τ)

where ρU, ρV is a partition of unity. Hence, ε∗ commutes with all maps in the Mayer-Vietoris sequence.

By the Five Lemma, the theorem now follows by induction on the cover U.

Corollary 2.21 (Künneth formula). For any two manifolds B and F, where B is ofnite type,

H∗(B× F) H∗(B)⊗H∗(F)

Proof. Interpret B× F as the trivial bundle, and take as free generators the extension ofthe generators of F, which are constant on B. Then the Leray-Hirsch theorem yields theresult.

Since the cohomology of Rn is trivial, cohomology cannot distinguish between a basespace and its vector bundles. Therefore, we need to nd another way to classify thesespace. We will come back to this in chapter 4.

In order to come to a usable result for compact cohomology, we rst need the followingresult on orientations for vector bundles:

Lemma 2.22. Suppose π : E → B is an orientable vector bundle and B is an ori-entable manifold. Then E is an orientable manifold.

Proof. Let ϕα : Uα → Rm be oriented trivialisations of B with transition functionshαβ, and ψα : E|Uα → Uα × Rn oriented trivialisations of the bundle π : E → B

with transition functions gαβ. Then (ϕα× Id) ψα are trivialisations of E, and theirtransition functions are given by

(ϕα × Id) ψαψ−1β (ϕβ × Id)−1 : Rm+n → Rm+n : (x,y) 7→ (hαβ(x),gαβ(ϕ

−1α (x))y)

And its Jacobian determinant is clearly given by

J(T) =

∣∣∣∣∣D(hαβ) ∗0 gαβ(ϕ

−1α (x))

∣∣∣∣∣ = detD(hαβ)detgαβ(ϕ−1α (x)) > 0

Hence, E is orientable.

Now, we can use Poincaré duality to nd the compact cohomology of a vector bundle:

Corollary 2.23. If π : E → B is an orientable vector bundle and B is an orientablemanifold with nite good cover, then H∗c(E) H

∗−nc (B).

37

Proof. By Poincaré duality on E and B, we have that:

H∗c(E) (Hm+n−∗(E))∗ (Hm+n−∗(B))∗ H∗−nc (B)

which proves the corollary.

38

Chapter 3

Connections

3.1 Properties

Recall that sections on trivial bundles π : B × Rn → B are in one-one correspondencewith functions B → Rn. As such, the exterior derivative is dened on such sections.Unfortunately, on arbitrary vector bundles, this is not always well-dened. Therefore,there is need for an appropriate generalisation, which locally behaves like d:

Denition 3.1. A connection is a linear map

D : Ω0(B,E)→ Ω1(B,E) B Ω0(B, T∗B⊗ E)

which satises a Leibniz rule: D(fs) = df ⊗ s + fDs for all functions f : B → R andsections s : B→ E.

Proposition 3.2. Any connection can locally be written as D = d+A, where A is amatrix of one-forms on Uα.

Proof. If ei is a local frame on Uα, then locally a section s is writable as s =∑i siei.

Hence

Ds = D(∑i

siei

)=∑i

D(siei) =∑i

dsi ⊗ ei + siDei

=∑i

dsi ⊗ ei + si∑j

Aji ⊗ ej =∑i,j

(dsi +Aijsj)⊗ ei = (d+A)∑i

siei

since Dei is a local section of T∗U⊗ E, which must have the form∑jAji ⊗ ej.

In order for local descriptions to be compatible, we need the Aα to be compatible inthe right way:

Deαi =∑j

Aαji ⊗ eαj

39

but also

Deαi = D∑j

gαβji eβj =∑j

dgαβji ⊗ eβj + gαβji De

βj

=∑j

dgαβji ⊗ eβj + gαβji

∑k

Aβkj ⊗ eβk =∑j,k

(dgαβji + gαβki Aβjk)⊗ e

βj

=∑j,k,l

(dgαβji gβαlj + gαβki A

βjkg

βαlj )⊗ eαl =

[gβαdgαβ + gβαAβgαβ

]ji⊗ eαj

Therefore, Aα = gβαdgαβ + gβαAβgαβ.This means that D = d only is a well-dened connection if dgαβ ≡ 0.

Denition 3.3. If gαβ can be chosen so that dgαβ ≡ 0, then π : E→ B is called a atbundle.

Proposition 3.4. Let D1 be a connection. Any other map D2 : Ω0(B,E)→ Ω1(B,E)

is a connection if and only if D1 −D2 ∈ Ω1(B, End(E))

Proof. (⇒) Since Ω1(B, End(E)) = Ω0(B, T∗B ⊗ End(E)) = Ω0(B, T∗B ⊗ E ⊗ E∗) =

Ω0(B, Hom(E, T∗B⊗E), the only requirement is that D1−D2 be linear with respectto `local scalars', or functions:

(D1−D2)(fs) = D1(fs)−D2(fs) = (df⊗ s+ fD1s)− (df⊗ s+ fD2s) = f(D1−D2)s

(⇐) Write D2 = D1 +A. It is linear since both D1 and A are. Furthermore,

D2(fs) = D1(fs) +A(fs) = df⊗ s+ fD1s+ fAs = df⊗ s+ fD2s

since A is linear in functions.

Corollary 3.5. The space of all connectives, A(E), is an ane space based onΩ1(B, End(E)): for any connection D0, A(E) = D0 +Ω

1(B, End(E)).

3.2 Curvature and induced connections

Until now, connections where only dened on sections. However, as we see them asa generalisation of the exterior derivative, we would like to extend this denition. Inparticular, we would like to know if this extension also denes a complex.

Proposition-denition 3.6. Given a connection D, there exists a unique linear exten-sion D : Ω∗(B,E) → Ω∗+1(B,E) such that D(ω ∧ σ) = dω ∧ σ + (−1)qω ∧ Dσ forall ω ∈ Ωq(B) and σ ∈ Ωp(B,E) B Ω0(B,

∧pT∗B ⊗ E). This extension is called the

covariant derivative.

40

Proof. If σ ∈ Ωq(B,E), we can write it as σ =∑ωi ⊗ si, where ωi ∈ Ωq(B) and

si ∈ Ω0(B,E). Then, by the required properties, D must be given by

D(σ) =∑

D(ωi ⊗ si) =∑

dωi ⊗ si + (−1)pωi ∧Dsi

(recall that ⊗ = ∧ in these cases). This is clearly a consistent denition, by the propertiesof the wedge product, the exterior derivative and the connection.

Denition 3.7. The curvature of a connection D is dened as FD = D2 : Ω0(B,E) →Ω2(B,E).

Proposition 3.8. Locally, if D = d+A, then FD = dA+A∧A.

Proof. If s =∑i siei, where ei is a local frame, we have

FD(s) = D(Ds) = D((d+A)s

)= D

(∑i

(dsi ⊗ ei +

∑j

siAjiej))

= D(∑i

(dsi +

∑j

sjAij)⊗ ei

)=∑i

d(dsi +

∑j

sjAij)⊗ ei −

(dsi +

∑j

sjAij)∧∑k

Akiek

=∑ij

d(sjAi,j)⊗ ei − dsj ∧Aij ⊗ ei −∑k

sjAkj ∧Aik ⊗ ei

=∑i,j

(dsj ∧Aij + sjdAij − dsj ∧Aij + sj

∑k

Aik ∧Akj)⊗ ei

=∑i,j

(sjdAij + sj(A∧A)ij

)⊗ ei = (dA+A∧A)

∑i

si ⊗ ei = (dA+A∧A)s

So the curvature FD measure the failure of D to yield a complex. If FD = 0, however,we can take D = d, so then π : E → B is at. Therefore, a connection is said to be atif its curvature is zero.

Proposition 3.9. The curvature is a section of∧2T∗B ⊗ End(E), in other words:

FD ∈ Ω2(B, End(E)).

Proof. We have that FD = D2 : Ω0(B,E) → Ω2(B,E) = Ω0(B,∧2T∗B ⊗ E) is linear on

bers. Furthermore,

FD(fs) = D2(fs) = D(df⊗ s+ fDs) = D(df∧ s) +D(f∧Ds)

= d2f∧ s− df∧Ds+ df∧Ds+ fD2s = fFDs

So FD gives a homomorphism on bers, and we have FD ∈ Ω0(B, Hom(E,∧2T∗B⊗E)) =

Ω0(B,E∗ ⊗∧2T∗B⊗ E) = Ω0(B,

∧2T∗B⊗ End(E)) = Ω2(B, End(E)).

41

This way, it is interesting to know if D(FD) can be given a meaning, and see what itwill be. In order to do this, we need to induce a connection on End(E) from the givenconnection D on E.

Proposition-denition 3.10. Let π1 : E1 → B and π2 : E2 → B be two vector bundleswith connections D1 and D2. Then the following are the induced connections :

On E1 ⊕ E2, dene D⊕ = D1 ⊕D2 : s1 ⊕ s2 → D1s1 ⊕D2s2.

On E1 ⊗ E2, dene D⊗ = D1 ⊗ Id+ Id⊗D2 : s1 ⊗ s2 → D1s1 ⊗ s2 + s1 ⊗D2s2.

On E∗, dene locally D∗ = d−A>.

If f : X → B is a map, and D = d + Aα on Uα, then f∗(D) = d + f∗(Aα) locallydenes the pullback connection on E|f∗Uα, with respect to the pullback frame.

Proof. Clearly, all these connections are linear. Then we have:

D⊕(f · (s1 ⊕ s2)) = D⊕(fs1 ⊕ fs2) = D1(fs1)⊕D2(fs2)

= (df⊗ s1 + fD1s1)⊕ (df⊗ s2 + fD2s2) = df⊗ (s1 ⊕ s2) + fD⊕(s1 ⊕ s2)D⊗(f · (s1 ⊗ s2)) = D⊗((fs1)⊗ s2) = D1(fs1)⊗ s2 + fs1 ⊗D2s2

= (df⊗ s1 + fD1s1)⊗ s2 + fs1 ⊗D2s2 = df⊗ (s1 ⊗ s2) + fD⊗(s1 ⊗ s2)

So the rst two are connections.For the third part,

−A>α = −(gβαAβgαβ + gβαdgαβ)> = −(gβαAβgαβ)

> − (gβαdgαβ)>

= −g>αβA>βg>βα − dg>αβg

>βα = −(g>βα)

−1A>β(g>αβ)

−1 + g>αβdg>βα

= g∗βα(−A>β)g

∗αβ + g∗βαdg

∗αβ

where I used that 0 = d Id = d(gαβgβα) = dgαβgβα + gαβdgβα.For the last point, we have for a vector eld Y ∈ Ω0(X, TX).

f∗(D)(f∗s)(Y) = df∗s(Y) + f∗(Aα)(f∗s)(Y) = f∗(ds(f∗Y) + f∗(Aα(f∗Y))(s) = f

∗(Ds(f∗Y))

This is a connection, and since local frames are pullbacks, the denition makes sense.

Proposition 3.11. With this denition of D∗, the function b 7→ sb(tb) is smooth forall s ∈ Ω0(B,E∗), t ∈ Ω0(B,E).

Proof. The requirement for being smooth is that ds(t) = (D∗s)(t) + s(Dt). Checkingthis on basis elements, we have:

e∗i (b)(ej(b)) = δij

0 = d(e∗i (ej)) =∑k

A∗kie∗k(ej) + e

∗i

(∑k

Akjek

)= A∗ji +Aij

42

which is exactly the condition A∗ = −A>. Then by linearity and the Leibniz rule, itworks for all sections s ∈ Ω0(B,E∗), t ∈ Ω0(B,E).

On Hom(E, F) = E∗ ⊗ F with basis eij = e∗j ⊗ fi, we have locally

D(h) = D(∑ij

hije∗j ⊗ fi

)=∑ij

D(hije∗j ⊗ fi)

=∑ij

dhije∗j ⊗ fi + hij

(∑k

−(AE)>kje∗k ⊗ fi +

∑k

e∗j ⊗AFkifk)

= dh+∑ijk

(hkjAFik − hikA

Ekj)e

∗j ⊗ fi

= dh+∑ij

(AFh− hAE)ije∗j ⊗ fi = dh+AFh− hAE

Therefore, on End(E), D(h) = dh+Ah+ hA = dh+ [A,h].

Proposition 3.12 (Bianchi identity). D(FD) = D(D D) = 0.

Proof. By using the covariant derivative on Ω2(B, End(E)), we have locally

D(FD) = dFD + [A, FD] = d(dA+A∧A) + [A,dA+A∧A]

= dA∧A−A∧ dA+A∧ (dA+A∧A) − (dA+A∧A)∧A

= dA∧A−A∧ dA+A∧ dA+A∧A∧A− dA∧A−A∧A∧A = 0

43

Chapter 4

Chern classes

According to the Leray-Hirsch theorem (theorem 2.20), cohomology cannot distinguishbetween a manifold and any vector bundle over it, since the cohomology of its bersis trivial. However, all is not lost, since although the cohomology of the bundle itselfdoes not contain any information, a vector bundle does correspond to certain classes inthe cohomology of the base space, which are preserved under isomorphisms of vectorbundles. For complex bundles, these characteristic classes are called Chern classes, andthey will be approached in two dierent ways in the following chapter: via the Eulerclass on associated sphere bundles and via the curvature of a connection.

4.1 Chern classes via the Euler class of sphere bundles

4.1.1 The Euler class of a sphere bundle

In order to dene the Euler class, a sphere bundle needs to be orientable:

Denition 4.1. Let π : E → B be a sphere bundle, i.e., it has ber Sn, n > 1. Thisbundle is orientable if there is an open cover Uα of B and generators [σα] ∈ Hn(E|Uα)such that [σα] = [σβ] ∈ Hn(E|Uαβ). Such a set [σα] is called an orientation.

Remark. Recall that, by the Künneth formula, Hn(E|Uα) Hn(Uα × Sn) Hn(Uα) ×

Hn(Sn) Hn(Sn) R, so this cohomology does have a generator.

To any vector bundle, we can associate a sphere bundle in the following way: letπ : E→ B be a rank n+ 1 bundle, whose structure group is reduced to O(n+ 1). Thenthe associated (unit) sphere bundle ~π : S(E) → B is the sphere bundle with ber theunit sphere of the ber of E, and the same transition functions.

Proposition 4.2. A vector bundle is orientable if and only if its sphere bundle isorientable.

44

Proof. (⇒ ) Let σ be a generator of Sn and gαβ : Uαβ → SO(n+1) transition functionsfor the bundle E. Then ∫

Sng∗αβσ =

∫gαβ(Sn)

σ =

∫Snσ

Hence, [σ] = g∗αβ[σ] ∈ Hn(Sn). If we dene [σα] = ϕ∗α[σ], where projections areomitted, then on Hn(E|Uαβ) we have [σα] = ϕ∗α[σ] = ϕ∗αg

∗βα[σ] = ϕ∗β[σ] = [σβ],

and S(E) is orientable.

(⇐ ) If (Uα, [σα]) is an orientation of S(E), and σ is a generator of Sn, choose metric-preserving trivialisations ϕα such that [σα] = ϕ∗α[σ]. Then g

∗αβ[σ] = [σ], so gαβ ∈

SO(n+ 1). Then the gαβ give an orientation on E.

The construction of the Euler class

To dene the Euler class of a sphere bundle π : E→ B with ber Sn, we rst take a goodcover U = Uα of B and an orientation [σα] of E. This orientation can be viewed as anσ0,n ∈ C0(π−1U,Ωn).

We want to extend this class to a class σ = σ0,n + σ1,n−1 + · · · + σn,0 such thatDσ = δσn,0. Since [σα] is a cohomology class in Hn(Uα), dσ = 0.

By orientability, (δσ0,n)αβ = σβ − σα is exact so we can nd a σ1,n−1 with dσ1,n−1 =

δσ0,n. Hence, D(σ0,n + σ1,n−1) = δσ1,n−1.We can repeat this procedure since dδσj,n−j = δdσj,n−j = −δδσj−1,n−j+1 = 0, and

Hq(Sn) is trivial for 0 < q < n, whence the column is exact at those nodes.

σ0,n //

σ1,n−1

OO

σn,0 // −i∗ε

−ε

OO

Finally, we obtain σ = σ0,n + σ1,n−1 + · · ·+ σn,0 such that Dσ = δσn,0. Again, dδσn,0 =0, so we have an ε ∈ Cn+1(U,R) π∗(ε) ∈ Cn+1(π−1U,R) such that Dσ = i∗(−ε)

(for simplicity, π∗ is omitted). Since δε = 0, it denes a cohomology class e(E) ∈Hn+1(U,R) Hn+1(B). This class is called the Euler class of the oriented spherebundle.

For an oriented vector bundle, its Euler class is dened to be that of its sphere bundle.

45

The Euler class is well-dened

A priori, the Euler class dened this way seems to depend on both the choice of goodcover and the choice of the elements σj,n−j. However, this turns out not to be the case:

Proposition 4.3. For a given orientation [σα] of the sphere bundle, the Euler classdoes not depend on the choice of the σj,n−j in the above construction.

Proof. Suppose σ0,n and σ0,n both represent [σα] in C0(π−1U,Ωn). Then σ0,n− σ0,n =

dτn−1. Hence, d(δτn−1) = d(σ1,n−1 − σ1,n−1), and δτn−1 − (σ1,n−1 − σ1,n−1) = τn−2.This goes on until δτ0 − (σ0,n − σ0,n) = iτ. Therefore, taking δ of this equation,

ε− ε = δτ

whence they represent the same cohomology class.

Proposition 4.4. The Euler class does not depend on the good cover chosen in theconstruction.

Proof. If V is a renement of U, then let σ0,nV be the restriction of σ0,n

U . Then εV is therestriction of εU, so they dene the same class in Hn+1(M,R). For arbitrary covers U,V, let W be a common renement. Then εU = εW = εV ∈ Hn+1(M,R).

If E = B× F, then clearly e(E) = 0. But the Euler class can be zero in other instancestoo:

Proposition 4.5. If an oriented sphere bundle has a section, then its Euler class iszero.

Proof. Let s be a section. By construction, −π∗ε = Dσ. Hence,

−ε = −s∗π∗ε = Ds∗σ

so e = 0 ∈ Hn+1(M).

4.1.2 Chern classes

The Chern classes are characteristic classes of a complex vector bundle, bundles withber Cn. All general results obtained until now also hold for complex bundles, bymaking the appropriate changes. In particular, a modication of proposition 2.12 givesthat the structure group of any complex vector bundle may be reduced to U(n). Bythe isomorphism between U(1) and SO(2), there is a bijective correspondence betweencomplex line bundles and oriented real plane bundles.

Denition 4.6. The rst Chern class of a complex line bundle π : L → M is denedas c1(L) = e(LR) ∈ H2(M), where LR is the realisation of the bundle.

46

On S(LR) we have one coordinate, the angle. Thus, the σ0,1 is a collection 1-forms ofangles dϑα, and their dierence is given by δσ0,1 = dσ1,0. Therefore σ1,0 = ϑα − ϑβ.Since the transition functions are given by gαβ = ei(ϑα−ϑβ), we have that σ0,1 = log gαβ,up to constants. Then, e(LR) ∝ log gαβ, so we have that c1(L ⊗ L ′) = e(LR ⊗ L ′R) =

e(LR) + e(L′R) = c1(L) + c1(L

′).Since the identity map is a section of End(L) = L⊗ L∗, its Euler class is zero. Hence,

c1(L) + c1(L∗) = c1(L⊗ L∗) = 0, and

c1(L∗) = −c1(L)

Now let p : E → B be a complex vector bundle with gαβ : Uαβ → GL(n,C). LetPGL(n,C) = GL(n,C)/CId. Then the projectivisation π : P(E)→ B of E is the bundlewith ber P(Eb) and transition matrices ~gαβ : P(Uαβ)→ PGL(n,C) induced by gαβ.

On P(E), there are some special bundles: the pullback bundle π∗E, whose ber at lpis Ep, the universal subbundle S = (lp, v) ∈ π∗E|v ∈ lp, and the universal quotientbundle Q, dened by the tautological exact sequence

0→ S→ π∗E→ Q→ 0

Now, let x = c1(S∗) ∈ H2(P(E)). Since at every ber, −x = c1(~S), where ~S is the universalsubbundle of P(Ep), the cohomology classes 1, x, . . . , xn−1 freely generate the cohomologyof the ber by an argument of spectral sequence theory, see e.g. Bott & Tu [2]. Therefore,the Leray-Hirsch theorem 2.20 states that H∗(P(E)) is a free module over H∗(B) withbasis 1, x, . . . , xn−1.

Denition 4.7. The Chern classes of a complex vector bundle are the coecientsci(E) ∈ H2i(B) of xn in this basis:

xn + c1(E)xn−1 + · · ·+ cn(E) = 0

The total Chern class is then

c(E) = 1+ c1(E) + · · ·+ cn(E)

Since for line bundles this equation is x + c1(L) = 0, this agrees with the originaldenition of the rst Chern class.

If E = B×F, H∗(P(E)) = H∗(M×P(V)) = H∗(M)⊗Rn/xn = H∗(M)/xn, so xn = 0.The Chern classes can thus be seen as a measure of the twisting of the bundle.

4.2 Characteristic classes via the curvature of a connec-

tion

Another way to get to the Chern classes is by looking at polynomials P of the curvatureof a connection D. Since curvatures are form-valued, the output of such a polynomial

47

would be a form too. If P(FD) is closed, this gives a class [P(FD)] in cohomology. Inorder for it to be characteristic, it should be independent of the chosen curvature andthe frame. The notation P(D) = P(FD) is often used.

Denition 4.8. A polynomial P : Matn → C is said to be GL(n,C)-invariant ifP(ABA−1) = P(B) for all A ∈ GL(n,C).

Clearly this denition ensures the same condition for form-valued matrices B.It turns out that any GL(n,C)-invariant polynomial already satises the other two

conditions, calculating a closed form independent of the chosen connection.

Proposition 4.9. If P is a GL(n,C)-invariant polynomial and D is a connection,then dP(D) = 0.

Proof. We can assume P to be homogeneous of degree m. Then P(A) = P(A, . . . ,A),where P(A1, . . . ,Am) is an m-linear and symmetric map, dened as 1

m!times the coe-

cient of t1 · · · tm in the expansion of P(∑mi=1 tiAi). This is true, because by denition,

it is linear in each Ai, symmetric, and

P(A, . . . ,A) =1

m!·(coef. of

∏ti in P

((∑ti)A))

=1

m!#Sm · P(A) = P(A)

Therefore we have, since dF = −[A, F] = [F,A] by the Bianchi identity,

dP(F) = dP(F, . . . , F) = P(dF, F, . . . , F) + · · ·+ P(F, . . . , f,dF)= mP(dF, F, . . . , F) = mP([F,A], F, . . . , F)

Thus if we can nd, for every x, a frame such that A(x) = 0, the proposition is proved.Suppose A(x) =

∑Aidxi , 0 in a frame where x = O. If we take the coordinate

transformation g = Id−Aixi, then g(x) = Id, dg(x) = −A(x), so in the new frame

A ′(x) = g−1(x)A(x)g(x) + g−1(x)dg(x) = IdA(x)Id+ Id(−A(x)) = 0

So by invariance dP(F)(x) = mP([A, F], F, . . . , F)(x) = 0 for all x.

Proposition 4.10. If P is a GL(n,C)-invariant polynomial and D0 and D1 are twoconnections on π : E → B, then there exists a form TP(D0,D1) such that P(D1) −

P(D0) = dTP(D0,D1).

Proof. Dene Dt = tD1 +(1− t)D0 and ϑ = D1 −D0. Since we can always pick a framesuch that At0(x) = 0, and therefore dFt0(x) = 0 too,

_Ft = d _At + _At ∧At +At ∧ _At = d _At = d( ddt

(A0 + tϑ))= dϑ

48

So we have that

P(D1) − P(D0) =

∫ 10

_P(Ft, . . . , Ft)dt =

∫ 10

mP( _Ft, Ft, . . . , Ft)dt =

∫ 10

mP(dϑ, Ft, . . . , Ft)dt

= m

∫ 10

dP(ϑ, Ft, . . . , Ft) − (m− 1)P(ϑ,dFt, . . . , Ft)dt

= d(m

∫ 10

P(ϑ, Ft, . . . , Ft)dt)= dTP(D0,D1)

Corollary 4.11. The class [P(D)] ∈ H∗(B) is independent of the chosen connectionD. We can therefore write [P(E)].

If we take a connection and a frame such that d〈s, t〉 = 〈Ds, t〉 + 〈s,Dt〉 (such aconnection is called unitary), then

0 = d〈ei, ej〉 = 〈Dei, ej〉+ 〈ei,Dej〉 =∑k

Aki〈ek, ej〉+∑k

A∗kj〈ei, ek〉 = Aji +A∗ij

Therefore also FD + F∗D = 0, FDF∗D = F∗DFD, so FD is diagonalisable. So P(FD) onlydepends on the eigenvalues of FD, and must in fact be a symmetric function of them.

All symmetric polynomials are generated by the elementary symmetric functions:σi(x1, . . . , xn) =

∑τ∈Si

∏ij=1 xτ(j).

If we take

P(A) = det( i2πA+ I

)=

n∏k=1

(1+

iλk

2π

)=

n∑j=1

σj

(iλ12π

, . . . ,iλn

2π

)we have all these polynomials.

Denition 4.12. The Chern classes of a bundle p : E → B are dened by ck(E) =

[σk(i2πFD)] ∈ H2k(B;C), where σk(A) means the symmetric polynomial on the eigenval-

ues of A. The total Chern class is their sum, c(E) = [det(I+ i2πFD)] ∈ H∗(B;C).

Remark. We now have two denitions of the Chern classes. Until we prove that thesetwo coincide, I will denote the last-dened by bk(E) in stead of ck(E).

Corollary 4.13. The Chern class is real-valued, i.e., b(E) ∈ H∗(B;R).

Proof. Since we can take FD + F∗D = 0, i2πFD =

(i2πFD)∗, and therefore b(E) = det

(I +

i2πFD)= det

(I+ i

2πFD)= b(E).

The Chern class is characteristic in the sense that it is a natural transformation fromvector bundles to the cohomology over the base spaces:

Proposition 4.14. Given a vector bundle π : E→ B and a map f : X→ B, b(f∗E) =f∗b(E).

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Proof. If D = d+Aα is a connection on E, we have the connection f∗(D) = d+ f∗(Aα)

on f∗E, so f∗(F) is a curvature on f∗E. Therefore, b(f∗E) = [P(f∗(F))] = f∗[P(E)] =

f∗b(E).

4.3 Equivalence of the denitions

In order to prove the two denitions of the Chern classes agree, one rst needs to proveequality for line bundles. Then, by an induction argument, this result can be extended.

4.3.1 Equivalence for line bundles

Theorem 4.15. For complex line bundles π : L→ B the two denitions of the Chernclass agree, i.e., c(L) = b(L).

Proof. By denition, c0(L) = 1 = b0(L).If we interpret L as a real plane bundle, then we can take Aα = −i · dϑα, where ϑα

is the local angular coordinate. This denes a connection, since transition functions aregiven by gαβ = ei(ϑα−ϑβ), and therefore

Aβ = gβαAαgαβ + gβαdgαβ = ei(ϑβ−ϑα) ·−idϑa · ei(ϑα−ϑβ) + ei(ϑβ−ϑα)dei(ϑα−ϑβ)

= −idϑα + ei(ϑβ−ϑα)ei(ϑα−ϑβ)d(i(ϑα − ϑβ)) = −idϑβ

We then have∫S1Aα =

∫S1

−idϑα = −2πi, so [ i2πAα] ∈ H1(L|Uα) restricts to a generator

of the cohomology on each ber, and therefore σ0,1 B [ i2πAα] gives an orientation.

This orientation denes the Euler class, and under the isomorphisms (i∗)−1 r∗, i∗ :H∗(U,R) ∼→ HDC

∗(U,Ω∗), and r∗ : HDR∗(B) ∼→ HDC∗(U,Ω∗) respectively, we have

that

c1(L) = e(LR) [ε(LR)] [i∗(ε(LR)] =[i∗(ε(LR) +D

(σ0,1 + σ1,0

)]= [i∗(ε(LR) + dσ

0,1 − i∗(ε(LR)] = [dσ0,1] =[ i2πdAα

]=[ i2πFα][ i2πF]= b1(L)

Hence, c(L) = b(L).

Because of this result, naturality also holds for c(L), where L is a line bundle. We canthen extend this to

Proposition 4.16. Given a vector bundle π : E → B and a map f : X → B, c(f∗E) =f∗c(E).

Proof. If SE is the universal bundle over PE, we have that f∗PE = P(f∗E) and thereforef∗S∗E = S∗f∗E. If we again dene xE = c1(S

∗E), then, because S

∗E is a line bundle,

xf∗E = c1(Sf∗E) = c1(f∗S∗E) = f

∗c1(S∗E) = f

∗xE

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If we apply f∗ to the dening equation of the Chern classes we obtain the followingequation:

xnf∗E + f∗c1(E)x

n−1f∗E + · · ·+ f∗cn(E) = 0

Which implies that c(f∗E) = f∗c(E).

4.3.2 The splitting principle and the Whitney product formula

In order to prove the equivalence of the two denitions for arbitrary bundles, we need toshow that both denitions `add up' in the same way. This is done in two steps: rst, thesplitting principle is proved. This states that to prove a polynomial identity on vectorbundles, it is sucient to prove it for direct sums of line bundles. Second, the Whitneyproduct formula, c(E1 ⊕ E2) = c(E1) · c(E2) is proved for both denitions. Then, byinduction, the denitions agree.

The splitting principle

In order to prove the split principle, rst the existence of a split manifold must be proven:

Denition 4.17. A split manifold F(E) of a complex vector bundle π : E → B is amanifold, such that there is a map τ : F(E)→ B with τ∗E =

⊕ni=1 Li for Li complex line

bundles and H∗(B) is embedded in H∗(F(E)) by τ∗.

Proposition 4.18. A split manifold exists for any vector bundle.

Proof. For rank 1 bundles, this is trivial.For rank 2 bundles, there is the tautological exact sequence over P(E) given by 0 →

SE → τ∗E→ QE → 0, whence τ∗E = SE ⊕QE (this follows from the same fact for shortexact sequences of vector spaces). Also, since τ : P(E)→ B is a ber bundle, τ∗ embedsH∗(B) in H∗(P(E)). Therefore, P(E) is a split manifold for rank 2 bundles.

We can now use induction. Suppose the proposition is proved up to rank n − 1 andlet π : E→ B be a rank n bundle. Then over P(E), the bundle splits in τ∗1E = SE ⊕QE,where rankSE = 1 and rankQE = n − 1. By induction hypothesis, there is a manifoldF(QE) with map τ2 : F(QE)→ P(E) such that τ∗2QE =

⊕n−1

i=1 Li, so we have that

(τ1 τ2)∗E = τ∗2SE ⊕n−1⊕i=1

Li

Since τ∗1 embeds H∗(B) in H∗(P(E)) and τ∗2 embeds H∗(P(E)) in H∗(F(QE)), their com-position embeds H∗(B) in H∗(F(QE)). Therefore, F(E) = F(QE).

Corollary 4.19. For any nite number of vector bundles πi : Ei → B, there existsa manifold N with map τ : N → B which splits all bundles simultaneously. This isthe split manifold for E1, . . . ,Em.

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Proof. Iterate the former proof.

Proposition 4.20 (The splitting principle). To prove a polynomial identity for Chernclasses, it suces to prove it for direct sums of line bundles.

Proof. This proof, and hence the proposition, holds for both denitions of the Chernclasses, since it only depends on naturality.

Let P(c(Ei)) = 0 be a polynomial identity on the Chern classes of the bundles Ei. Bynaturality,

τ∗P(c(Ei)) = P(τ∗c(Ei)) = P(c(τ

∗Ei))

where τ∗Ei are direct sums of line bundles. By assumption, P(c(τ∗Ei)) = 0, whenceτ∗P(c(Ei)) = 0, so by injectivity P(c(Ei)) = 0, which proves the identity for arbitraryvector bundles.

The Whitney product formula

Now we have proved the splitting principle, the only thing left is to prove that the twodenitions of the Chern classes behave the same for the direct sum operations in orderto prove their equality. We rst need the following two lemmas:

Lemma 4.21 (Shrinking lemma). Let U = Uini=1 be an open cover of a normal space

X. Then there is an open cover V = Vini=1 with Vi ⊂ Ui.

Proof. Suppose we have Vim−1i=1 (m may be 1). Then dene Wm = (

⋃m−1

i=1 Vi ∪⋃nj=m+1Uj)

c. Then Wm and Ucm are disjoint and closed. By normality, they haveopen disjoint neighborhoods Vm ⊇ Wm and Ym ⊇ Ucm. Then Vm ⊂ Ycm ⊂ Um andVi

mi=1 ∪ Uj

nj=m+1 is still an open cover of X. This denes V.

Lemma 4.22. Let M be a manifold, U ⊆M open, and A ⊂ U closed. Then there isa C∞ function ρ with is zero on Uci and one on A.

Proof. The set U,Ac is an open cover ofM. If ρ,σ is a partition of unity subordinateto this cover, ρ satises the properties.

Proposition 4.23 (Whitney product formula). The Chern class of the direct sum oftwo bundles is the product of their Chern classes, i.e.,

c(E1 ⊕ E2) = c(E1) · c(E2) b(E1 ⊕ E2) = b(E1) · b(E2)

Proof. For the rst denition, we want to prove that, for E =⊕ni=1 Li,

c(E) =

n∏i=1

c(Li) =

n∏i=1

1+ c1(Li) =

n∑j=1

σj(c1(Li))

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where the σj are the symmetric polynomials, so cj(E) = σj(c1(Li)).If π : P(E) → B is the projectivisation of E, omit the π∗ in π∗Li, the pullbacks of

the Li. Since S is a line bundle, the projections si : S → Li are never all zero. Hence,Ui with Ui = x ∈ P(E)|si(x) , 0 is an open cover, and the bundle (S∗ ⊗ Li)|Ui has anowhere vanishing section si, which is a frame. So (S∗ ⊗ Li)|Ui is trivial.

If λi ∈ c1(S∗ ⊗ Li), then on Ui, λi|Ui = dωi. By the above two lemmas, since P(E)is normal, there are an open cover Vi with Vi ⊂ Ui and functions ρi which are zerooutside Ui and one on Vi.

Then λi − d(ρiωi) is a global form representing c1(S∗ ⊗ Li), which vanishes on Vi.Therefore,

∏ni=1 c1(S

∗⊗Li) = 0. Since S∗ and Li are line bundles, c1(S∗⊗Li) = c1(S∗)+c1(Li) = x+ c1(Li). Hence

n∏i=1

(x+ c1(Li)) = xn + σ1x

n−1 + · · ·+ σn = 0

And by denition, cj(E) = σj(c1(Li)).If E = E1 ⊕ E2, and E1 and E2 are direct sums of line bundles, this gives the Whitney

product formula. By the splitting principle, this works not only for direct sums of linebundles, but for all bundles.

For the second denition, if F1 and F2 are curvatures of E1 and E2 respectively, thenF⊕ =

(F1 00 F2

). Therefore,

b(E1 ⊕ E2) =[det(I+

i

2πF⊕)]

=

[det

(I1 +

i2πF1 0

0 I2 +i2πF2

)]=[det(I1 +

i

2πF1

)· det

(I2 +

i

2πF2

)]= b(E1) · b(E2)

Theorem 4.24. For all vector bundles π : E → B, c(E) = b(E). That is, the deni-tions of the Chern class agree.

Proof. By theorem 4.15, the theorem holds for line bundles. By the Whitney productformula (proposition 4.23), it is also true for direct sums of line bundles. According to thesplitting principle (proposition 4.20), it therefore hold for all complex vector bundles.

Because of this theorem, we now have two ways of looking at the Chern classes, whichboth can be used to gain information. For example, by the denition via the curvature,we see that the Chern classes are the only independent characteristic classes: any othersuch class must be a polynomial in the Chern classes.

However, the denition via the Euler class can also be very useful, since it behaveslike the Euler class, which is somewhat more easy to calculate.

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Populaire samenvatting

Een functie f : R → R kan worden voorgesteld door zijn graek, een lijn in de R2. Eengraek is dus een deelverzameling Γ(f) van R2, zo dat voor elk punt op de x-as er exactéén punt op de y-as is. Voor een functie g : Rm → Rn bestaat er op dezelfde manier eengraek, maar deze wordt natuurlijk wat moeilijker te tekenen.

Ook voor functies die een ander domein of bereik hebben dan Rn is zo'n graek tetekenen: als h : X → Y een functie is, dan is de graek van h een deelverzameling vanX × Y. Als je bijvoorbeeld de functie h : S1 → R : ϑ → sin2 ϑ bekijkt, dan heeft die degraek te zien in guur 4.1a.

Zelfs als een ruimte E niet het directe product is van twee andere ruimtes, maar eenvan dichtbij wel op lijkt (zo'n ruimte noemen we een vectorbundel) is dit mogelijk. Eendergelijke graek stelt dan een zogenaamde `sectie' voor. Een voorbeeld is de Möbius-band: dit lijkt op het product van een cirkel met een lijnstuk, maar is het net niet. Tochkan je er een graek op tekenen, zie guur 4.1b.

(a) De graek van h : S1 → R : ϑ 7→ sin2 ϑ. (b) Een sectie op de Möbiusband.

Figuur 4.1: Voorbeelden van graeken.

Op normale functies bestaat er een afgeleide, en dit kan je ook deniëren op sectiesvan vectorbundels. Een afgeleide, zoals d

dxwordt dan vervangen door een connectie D,

en daar blijken er erg veel van te zijn.Een connectie is belangrijk om de vectorbundel te bestuderen: je kan er bijvoorbeeld

de `kromming' van de bundel mee berekenen: in hoeverre de bundel niet lijkt op een

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direct product. Ook kan je er `karakteristieke klassen' mee deniëren, een soort specialefuncties op de basisruimte ( bij de Möbiusband de cirkel) die eigenschappen van devectorbundel weergeven.

Deze karakteristieke klassen zijn bijzonder omdat ze niet veranderen als je de vector-bundels op een gladde manier vervormt, en ze daardoor goed kunnen laten zien wanneertwee bundels verschillend zijn. Ook zijn de klassen die op deze manier gevonden wordende enige unieke klassen die bij een vectorbundel horen.

De karakteristieke klassen kunnen ook op een andere manier benaderd worden, via`bolbundels', en deze methodes geven hetzelfde resultaat. Ondanks dat geeft dit wel tweeverschillende handige manieren om tegen het onderwerp aan te kijken.

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Bibliography

[1] J. W. Milnor & J. D. Shashe, Characteristic Classes, 1st edition, Princeton Univer-sity Press, 1974.

[2] R. Bott & L. W. Tu, Dierential Forms in Algebraic Topology, 1st edition, Springer,1982.

[3] S. Bradlow, The geometry of vector bundles and an introduction to gauge theory,

class notes from math 433, http://cwillett.imathas.com/bundles/, 1998

[4] G. E. Bredon Topology and Geometry, 1st edition, Springer, 1993.

[5] L. W. Tu, An Introduction to Manifolds, 1st edition, Springer, 2008.

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