From calculus to cohomology: de Rham cohomology and characteristic classes

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From Calculus to Cohomology

From Calculus to Cohomology .

de Rham cohomology and characteristic classes

lb Madsen and J�rgen Tornehave University r�f'Aarhus

CAMBRIDGE

·:� " UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pin Building, Trumpington Street, Cambridge CB2 I RP, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge, CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia ©I. Madsen & .T. Tomhave 1997 This book is in copyright. Subject to statutory exception and to lbe provisions of relevant collective licensing agreements, no reproduction of any part may take place wit110ut lbe written pennission of Cambridge University Pre�s First puhlished 1997 Printed in t.be United Kingdom at the University Press, Cambridge A catalogue recordfor this book is avail£Jblefrom the British Library

Library ofCongress Cataloguing in P�th!ication data Madsen, I. H. (Ib Henning), 1942From calculus to cohomology: de Rham cohomology and chracteristic classes I lb Madsen and .T111rgen Tornehave. em. (1. Includes bibliographical references and index. ISBN 0-521-58059-5 (he).-- ISBN 0-521-58956-8 (pbk). 1. Homology theory. 2. Differential forms. 3. Characteri�tic classes. I. Tomehave, .T9Srgen. II. Title. QA612.3.M33 1996 514'.2--dc20 96-28589 CJP

ISBN ISBN

0 521 58059 5 headback 0 521 58956 8 paperback

v

Contents

.

Preface

. vii

Chapter

1

Introduction . . . . . . ..

. I

Chapter

2

The Alternating Algebra.

. 7

Chapter

3

Chapter

4

Chapter

5

The Mayer-Vietoris Sequence

Chapter

6

Homotopy ................

Chapter

7

Applications of de Rham Cohomology

47

Chapter

8

Smooth Manifolds . .. .........

57

Chapter

9

Differential Forms on Smoth Manifolds .

65

de Rham Cohomology . .

15 25

Chain Complexes and their Cohomology .

.

.

. . . ..

.

.

33 39

.

. .

83

Chapter

10

Integration on Manifolds

Chapter

11

Degree, Linking Numbers and Index of Vector Fields.

Chapter

12

The Poincare-Hopf Theorem .. . .

Chapter

13

Poincare Duality .

Chapter

14

The Complex Projective Space

cpn .

139

Chapter

15

Fiber Bundles and Vector Bundles ..

147

Chapter

16

Operations on Vector Bundles and their Sections

157

Chapter

17

Connections and Curvature ... . .. .. . . . . .

167

Chapter

18

Characteristic Classes of Complex Vector Bundles

181

Chapter

19

The Euler Class.

Chapter

20

Cohomology of Projective and Grassmannian Bundles

Chapter

21

Thorn Isomorphism and the General Gauss-Bonnet Formula

.

.

.

.

. . .

.

Smooth Partition of Unity ..

Appendix B

Invariant Polynomials . .

Appendix C

Proof of Lemmas

Appendix D

Exercises

97 113

.

. . . ..

. .. . .. . .

Appendix A

.

.

.

.

127

.

. . . ..

.

193

.

199 .

21I

. . . .

. 22 1

. . . . . .

. 227

12.12 and 12 . 1 3 .

. 233 . 243

References

. 281

Index ....

. 283

vii PREFACE This text offers a self-contained exposition of the cohomology of differential forms, de Rham cohomology, and of its application to characteristic classes defined in terms of the curvature tensor.

The only formal prerequisites are

knowledge of standard calculus and linear algebra, but for the later part of the book some prior knowledge of the geometry of surfaces, Gaussian curvature, will not hurt the reader. The first seven chapters present the cohomology of open sets in Euclidean spaces and give the standard applications usually covered in a first course in algebraic topology, such as Brouwer's fixed point theorem, the topological invariance of domains and the Jordan-Brouwer separation theorem.

The next four chapters

extend the definition of cohomology to smooth manifolds, present Stokes' the­ orem and give a treatment of degree and index of vector fields, from both the cohomological and geometric point of view. The last ten chapters give the more advanced part of cohomology:

the

Poincare-Hopf theorem, Poincare duality, Chern classes, the Euler class, and finally the general Gauss-Bonnet formula.

As a novel point we prove the so

called splitting principles for both complex and real oriented vector bundles. The text grew out of numerous versions of lecture notes for the beginning course

in topology at Aarhus University. The inspiration to use de Rham cohomology as a first introduction to topology comes in part from a course given by G.Segal at Oxford many years ago, and the first few chapters owe a lot to his presentation of the subject.It is our hope that the text can also serve as an introduction to the modern theory of smooth four-manifolds and gauge theory. The text has been used for third and fourth year students with no prior exposure to the concepts of homology or algebraic topology. We have striven to present

a11 arguments and constructions in detail. Finally we sincerely thank the many

students who have been subjected to earlier versions of this book.Their comments

have substantially changed the presentation in many places. Aarhus, January 1996

INTRODUCTION

1.

It is well-known that a continuous real function, that is defined on an open set of

R

has a primitive function. How about multivariable functions,? For the sake of

coo-)

simplicity we restrict ourselves to smooth (or

functions, i.e. functions that

have continuous partial derivatives of all orders. We begin with functions of two variables. Let

f: U

defined on an open set of R2. Is there a smooth function F: U ---+

Question 1.1

&F

(1)

- =

&x1

h

Since

&F and - =h,

&x2

---+

R,

R2

be a smooth function

such that

where f = (h,h)?

&2F

&2F

OX20X1

0X}0X 2

we must have &JI

(2)

&x2

=

&h

axl

0

The correct question is therefore whether F exists, assuming

(2).

Is condition

also sufficient?

(2)

Consider the function

Example 1.2

f: R2

---+

R2

f = (h,h) satisfies

given by

It is easy to show that (2) is satisfied. However, there is n o function F:

R

that satisfies (1). Assume there were; then

{21r o

J

R2- { 0} ---+

d

F(cosO, sin O)dO = F(l, 0)- F(l, 0) = 0. dO

On the other hand the chain rule gives F ·cosO ·(-sinO)+ f) &y =- fi(cosO, sinO) · sinO+ h(cosO, sinO)· cosO= 1.

!!:_F(cosO. sinO)= dO ·

dF

dx

This contradiction can only be explained by the non-existence of F.

I.

2

INTRODUCTION

Definition 1.3 A subset X c Rn is said to be EX if the line segment + (1point for all x E X.

xo

{txo

star-shaped with respect to the E [0, 1]} is contained in X

t)xit

IR2

be an open star-shaped set. For any smooth function Let U C that satisfies (2), Question l . l has a solution. (!I, h): U --)

Theorem 1.4

IR2

Proof. For the sake of simplicity we assume that x0 = 0 E function F : U --) R,

Then one has ()F :tt, x2 =

(

)

IR2. Consider the

f (t:r:1, txz) tx2 (txt, tx2)] dt {l [ ( tx1, t:c2) t:t1 &xt &1:1 ./0 fl

+

{)

t

+

{)f2

&x1 and 8fl (tx1, tx2) txz � 8!1 (tx > tx2). -ddt tfl (tx1, tx2)= h (tx1, tx2) tx1 � UXj UX2 l +

+

Substituting this result into the formula, we get

��(xi, uXt x2)

=

f t [dtd tfi(txt, tx2) + tx2 (�VXh (tx1. tx2)- ��1 UX2 (txt, tx2))] dt

}0

!th(tx1,tx2)]Z==o= fl(x1,x2). Analogously, g� = h(xi, x2). =

I

··Example 1.2 and Theorem 1.4 suggest that the answer to Question 1.1 depends on the "shape" or "topology" of U. Instead of searching for further examples or counterexamples of sets U and functions we define an invariant of U, which te1ls assuming us whether or not the question has an affirmative answer (for all the necessary condition (2). Given the open set U dim V. Indeed, let e1, . . . , en be a k basis of V, and let w E Al t (V). Using multilinearity, w(6, ... , �k) = w

(L Ai,lei, ... , L Ai,kei) = L AJw(eil, .

.

. , ejk)

with AJ = .\h,l . . . Aj� n, there must be at least one repetition among the elements ej1, . . . , ejk. Hence w ( CiJ, . . . , eJk) 0. =

The symmetric group of permutations of the set {1, . . . , k} is denoted by S(k). We remind the reader that any permutation can be written as a composition of transpositions. The transposition that interchanges i and j will be denoted by ( i, j). Furtherniore, and this fact will be used below, any permutation can be written as a composition of transpositions of the type (i, i+l), (i, i+l)o(i+l, i+2)o(i, i+l) = (i, i + 2) and so forth. The sign of a permutation: sign: S(k) - { ± 1 },

(I)

is a homomorphism, sign(cr or) = sign(cr) o sign(r), which maps every transpo­ sition to -1. Thus the sign of cr E S(k) is -1 precisely if cr decomposes into a product consisting of an odd number of transpositions.

Lemma 2.2 If w E Altk(V) and cr E

w(�.,.(l)• ... , � 0, and H0 (U) = R. Proof. We may assume U to be star-shaped with respect to the origin and wish to construct a linear operator

0 E Rn,

such that dSp + Sp+td = id when p > 0 and S1d = id e, where e(w) = w(O) for w E S1°(U). Such an operator immediately implies our theorem, since dSp(w) = w for a closed p-form, p > 0, and hence [w] = 0. If p = 0 we have w - w(O) = S1dw = 0, and w must be constant. -

First we construct

Every w E SlP(U

x

IR) can be written in the form

w= where I =

(ib . . , ip) .

L fr(x, t)dx1 + L 9J(x, t)dt and J =

Sp(w) = Then we have that

(11 "" ( + �

dSp(w) + Sp+ld(w) = L...t J,t. A

A

0

(j 1 ,

.

.

.

I:: (fol

9J(x,t)dt)dxJ.

og,(x, t) 0 . dt) dxi A dx1 x,

)

� ( ofi(x,t) dt dxr ot

(fol ofr�:' t) dt) dxr

( {1

7,: lo

= L fr(x, 1)dxl - L fi(x, O)dxJ. We apply this result to cf>*(w), where

¢: U

{

x

R - U, cf>(x, t)

=

'ifJ(t)x

and 'ifJ(t) is a smooth function for which

'ifJ(t) = 0 1/J(t) =

1

dxJ

, Jp- 1). We define

� lo

=L

!\

0 � 1/J(t) � 1

if t � 0 if t � 1 otherwise.

09J

OXi

)

dt dxi (\ dx,

3.

24

DE RHAM COHOMOLOGY

Define Sp(w) Sp(c/>*(w)) with Sp: 0.P(U x R) -+ QP- 1 (U) as above. Assume that w = L_ hi (x)dXJ. From Example 3.14 we have =

In the notation used above we then get that

This implies that

dsP (w) + Sp+l d(w·) -

_

{ 'L(x1h1(x)dx1 w ) - w(O)

=w

p>O p = 0.

0

25

4.

CHAIN COMPLEXES AND THEIR HOMOLOGY

In this chapter we present some general algebraic definitions and viewpoints, which should illuminate some of the constructions of Chapter 3. The algebraic results will be applied later to de Rham cohomology in Chapters 5 and 6. A sequence of vector spaces and linear maps (1)

is said to be exact when Im f

= Ker g, where as above

Ker g = { b

E Blg(b) = 0} (the kernel of g) lm f = {f(a)la E A} (the image of f).

Note that A .L B - 0 is exact precisely when f is surjective and that 0 is exact precisely when g is injective. A sequence A* = {Ai ,

di},

B l!... C

(2)

of vector spaces and linear maps is called a chain complex provided di+1 o di for all i. It is exact if

=0

Ker i = Im d -1

i

for all i. An exact sequence of the form

o - A L s i!... c - o

(3)

is called short exact. This is equivalent to requiring that

f is injective, g is surjective and Im f = Ker g.

The cokemel of a linear -map f: A

-

B is

Cok(f) = B jim f. For a short exact sequence,

g induces an isomorphism ""

g: Cok(f) � C. Every (long) exact sequence, i be used to calculate A )

as

in (2), induces short exact sequences (which can

26

4.

CHAIN COMPLEXES AND THEIR HOMOLOGY

Furthermore the isomorphisms

Ai-l /lm £f-2 � Ai-l /Ker di -l

d:_:1 Im i- 1 Ol

are frequently applied in concrete calculations. The

If

direct sum of vector spaces A

{ ai }

{bj} A EB B.

and

basis of

and

B

is the vector space

A EB B = {(a,b)i a E A, b E B} >.(a, b) = (>.a, >.b), ). E R (al , b1 ) + (a2, bz) = (al + a2, b1 + b2).

are bases of

A

and

B,

respectively, then

{(ai,O), (O, bj)}

is a

In particular dim(A E9

B) = dim A + dim B.

Suppose 0 � A L B � C � 0 is a short exact sequence of vector spaces. Then B is finite-dimensional if both A and C are, and B � A EB C. Lemma 4.1

{ ai } of A and { Cj} of C. Since g is surjective there exist bj E B with g(bj) = Cj· Then {f(ai),bj} is a basis of B: For b E B we have g(b) = 2:::::: AjCj. Hence b - L Ajbj E Ker g. Since Ker g = Im f, b - I: Ajbj = f(a), so Proof.

Choose a basis

(

)

b - L Ajbj = ! L J.Liai = L J.Ld(ai)· This shows that

b

can be written as a linear combination of { bj} and

is left to the reader to show that

{ bj, f (ai)}

{! ( ai)}.

are linearly independent.

Definition 4.2 For a chain complex A* = { . . .

--+

AP-1

we define the p-th cohomology vector space to be

dp- 1 --+

dP

AP --+ AP+l

--+

.

It

0

. .}

flP(A*) = Ker dPjim dP- 1 .

The elements of Ker dP are called

p-cycles (or are said to be closed) and the p-boundaries (or said to be exact). The elements of HP(A*) are called cohomology classes. A chain map f: A* -+ B* between chain complexes consists of a family fP: AP --+ BP of linear maps, satisfying d� o fP = JP+ l o d�. A chain map is illustrated elements of Im

dP- I

are called

as the commutative diagram

4.

27

CHAIN COMPLEXES AND TH.EIR HOMOLOGY

Lemma 4.3 A chain map

f: A* � B* J* = H* (J): HP(A*)

induces a Linear map �

HP(B*),

for all p.

a E AP be a cycle (dPa = 0) and [a] = a + Im dp- l its corresponding cohomology class in HP(A*). We define !*([a]) = [JP(a)]. Two remarks are needed. First, we have d� JP(a) = JP+ l d� (a) = JP+ l (O) = 0. Hence JP(a) is a cycle. Second, [JP(a)] is independent of which cycle a we choose in the class [a]. If [a 1 ] = [a2] then a1 - a2 E Im d�- 1 , and fP(a1 - a2) = fP d�- 1 (x) = d�-1 JP-1 (x). Hence JP(a1 ) - JP(a2) E Im d�- 1 , and JP(a1), JP(a2) define the Proof. Let

same cohomology class.

0

A category C consists of "objects" and "morphisms" between them, such that

"composition" is defined. If f: Ct � C2 and g: C2 � C3 are morphisms, then there exists a morphism g o f: C1 � C3. Furthermore it is to be assumed that ide: C � C is a morphism for every object C of C. The concept is best illustrated by examples: The category of open sets in Euclidean spaces, where the morphisms are the smooth maps. The category of vector spaces, where the morphisms are the linear maps. The category of abelian groups, where the morphisms are homomor­ phisms. The category of chain complexes, where the morphisms are the chain maps. A category with just one object is the same as a semigroup, namely the semigroup of morphisms of the object.

Every partially ordered set is a category with one morphism from d, when c � d.

c

to

contravariant functor F: C � V between two categories maps every object C E obC to an object F(C) E ob V, and every morphism f: C1 � C2 in C to a morphism F(J): F(C2) � F(C1) in V, such that

A

F(g o f) = F(f) o F(g) , F(idc) = idF(C)· A covariantfunctor F:C � V is an assignment in which F(J): F(C ) l

and

F(g o f) = F(g) o F(f),

F(idc) = idF(C)·

�

F(C2) ,

4.

28

CHAIN COMPLEXES AND TIIEIR HOMOLOGY

thus are the "structure-preserving" assignments between categories. The contravariant ones change the direction of the arrows, the covariant ones preserve directions. We give a few examples:

Functors

A be a vector space and F (C) Hom(C, A), the linear maps from C to A. For ¢: C1 � C2, Hom(¢, A): Hom(C2, A) � Hom(C1 , A) is given by Horn(¢, A)('�/�) = 'lj; o ¢>. This is a contravariant functor from the category of vector spaces to itself.

Let

=

F(C) = H om(A, C), F(¢): '1/1 f-t o '1/1. This is a covariant functor from the category of vector spaces to itself.

Let U be the category of open sets in Euclidean spaces and smooth maps, and Vect the category of vector spaces. The vector space of differential p-forms on U E U defines a contravariant functor 0.P: U --+ Vect.

Let Vect* be the category of chain complexes. defines a contravariant functor D*: U Vect*.

The

de Rham

complex

--+

For every p the homology HP: Vect*

--+

Vcct is a covariant functor.

The composition of the two functors above is exactly the de Rham Vect. It is contravariant. cohomology functor HP: U --+

A short exact sequence of chain complexes 0

-

A*

L B* � C*

consists of chain maps f and g such that 0 for every p.

�

-

AP

0

..!... BP

� CP -+ 0 is exact

Lemma 4.4 For a short exact sequence of chain complexes the sequence

is exact.

Proof. Since gP o fP =

0 we have

g* o f* ( [ a ] ) = g* ( [fP (a) ] ) = [gP CfP(a) )] = 0

every cohomology class [a] E HP(A* ). Conversely, assume for [bJ E HP(B) that g*[bJ 0. Then gP(b) = dF 1 (c) . Since gP-1 is surjective, there exists

for

=

4.

CHAIN COMPLEXF.S AND THEIR HOMOLOGY

29

b1 E BP-l with gP- 1 (b!) = c. It follows that gP (b - d�- 1 (b1 )) = 0. Hence there exist a E AP with JP(a) = b - d�- 1 (b ). We will show that a is a p-cycle. Since 1 JP+l is injective, it is sufficient to note that JP+ 1 (d�(a)) = 0. But + jP 1 (� (a)) = d�(JP(a)) = d�(b - d�- 1 (b1 )) = 0 since b is a p-cycle and dP o dP-1 = 0. We have thus found a cohomology class 0 [a] E HP(A), and f* [ a ] = [b - d�- 1 (b1 )] = ( b ]. One might expect that the sequence of Lemma 4.4 could be extended to a short exact sequence, but this is not so. The problem is that, even though gP: BP - CP is surjective, the pre-image (gP) - 1 (c) of a p-cycle with c E CP need not contain a cycle. We shall measure when this is the case by introducing Definition 4.5 For a short exact sequence of chain complexes C* - 0 we define

0 - A* L B* .!!...

&*: HP(C* ) - HP+1(A* ) to be the linear map given by

/)* ( ( C J)

=

[ (JP+l) -1 (d� ((gP)-l (c)))] .

There are several things to be noted. The definition expresses that for every b E (gP)- 1 (c) we have d�(b) E Im(JP+l), and that the uniquely determined a E AP+ l with p+1(a) d�(b) is a (p + ! )-cycle. Finally it is postulated that =

[ a ] E HP+l (A*)

is independent of the choice of

b E (gP) -l (c).

In order to prove these assertions it is convenient to write the given short exact sequence in a diagram:

0

0

0

-J_,

J_

r-• B 1

(i)

J_

C 1-

ld�-1 Jd�-1 ld�-1 - AP BP CP J ��cp+Jdc - AP+d l r+1 BP+ 1 l 1 1 of&*. l p

__1_

The slanted arrow indicates the definition assertions which, when combined, make (ii) (iii)

9,_, p

0

__f!___

0

gP+1

0

We shall now prove the necessary well-defined. Namely:

&* If gP(b) = c and d�(c) = 0 then d�(b) E ImjP+ 1• If JP+ l (a) = d�(b) then d�+1 (a) = 0 . If gP(b1) = gP(b2 ) = c and JP+ l (ai) = d�(bi) then [ad HP+ l (A*).

[ a2 ] E

30

4.

CHAIN COMPLEXES AND THEIR HOMOLOGY

gP+ 1 d�(b) = �(c) = 0, and Ker gP+ l = Im JP+l; (ii) uses the injectivity of jP+2 and that jP+2d�+l (a) = d�+I JP+l(a) = d�+ 1 d�(b) = 0; (iii) follows since b1 - bz = fP(a) so that d1f:J(b1 ) - d�(bz) = 1 1 d�jP(a) = jP+ld�(a), and therefore (Jp+lr (d�(bl )) = (JP+lr (d�(bz)) + d�(a). The first assertion follows, because

Example 4.6 Here is a short exact sequence of chain complexes (the dots indicate that the chain groups are zero) with &* i= 0:

1 1 ! 1 "d lid 1 1 l 1 "d

o- o - R � R - o o- R�R- o -o

One can easily verify that

&*: R ---+ R is an

Lemma 4.7 The sequence

HP(B*) g• HP(C*) a· HP+l(A*) lS. exact. ---+

isomorphism. 4

Proof. We have &*g*([b]) = &*[gP(b)] = [(JP+lr \dB(b))] = o. Conversely assume that &* [ c ] = 0. Choose b E BP with gP(b) = c and a E AP, such that

d� (b) = jP+l (d� a) . Now we have d�(b - JP(a)) = 0 and gP(b - fP(a)) = c. [ c] . Lemma 4.8 The sequence

Hence

g*[b - fP(a)] = D

r HP+ 1 (B*) is exact. HP(C*) a· HP+1(A*) ---+ ---+

j*&*([ c]) = [d�(b)) = 0, where gP(b) = c. Conversely assume f*[ a] = 0, i.e. JP+l (a) = di:J(b) . Then d�(gP(b)) = gP+l JP+ 1 (a) = 0, and 0 &*(gP(b)] = [ a ] .

Proof. We have that

We can sum up Lemmas

4.4, 4.7

and

4.8

in the important

Theorem 4.9 (Long exact homology sequence). Let 0 ---+ A* L B* � be a short exact sequence of chain complexes. Then the sequence · ·

·

---+

is exact.

HP(A*) � HP(B*) � HP(C* ) � HP+l(A* ) !_: HP+l(B* )

C*

---+

·

---+

0

· ·

0

4.

CHAIN COMPLEXES AND THEIR HOMOLOGY

Definition 4.10 Two chain maps j, g: A*

s: AP

if there exist linear maps

--t

sp-l

31

B* are said to be chain-homotopic, satisfying -+

for every p. In the form of a diagram, a chain homotopy is given by the slanted arrows.

The name chain homotopy will be explained in Chapter 6. Lemma 4.11 For two chain-homotopic chain maps f, g: A*

r

=

--t

B*

we have that

g*: HP(A*) - HP(B*).

Proof. If [ ] E HP(A*) then

a

(!* - g*) [ a ]

=

[fP(a) - gP(a)] =

[d�-1s(a) + sd�(a)] = [d�-1s(a)] = 0.

0

Remark 4.12 In the proof of the Poincare lemma in Chapter 3 we constructed

linear maps SP: 0P(U) - nP- 1 (U)

dP-1SP SP+ldP

+ = id for p > 0. This is a chain homotopy between id and with 0 (for p > 0), such that id* = 0*: HP(U) - HP(U), However id*

p > 0.

=

id and 0*

=

0. Hence id

=

p > 0.

0 on HP(U), and HP(U) = 0 when

Lemma 4.13 If A* and B* are chain complexes then

HP(A* $ B*)

= HP(A*) $ HP(B*).

Proof. It is obvious that

(d�ELlB) = Ker d� $ Ker d� Im(d��1s) = Im d�-l El1 Im d�- l

Ker

and the lemma follows.

0

33

5.

THE MAYER-VIETORIS SEQUENCE

This chapter introduces a fundamental calculational technique for de Rham cohomology, namely the so-called Mayer-Vietoris sequence, which calculates

H*(Ul u Uz) as a "function" of H*(Ut), H*(Uz) and H*(U1 n Uz). Here U1 and Uz are open sets in Rn. By iteration we get a calculation of H* ( U1 U · · · U Un) as a "function" of H*(Ua), where a runs over the subsets of { 1 , . . . , n} and Ua = Ui1 n · · · n Ui" when a = { i 1 , . . . , ir}. Combined with the Poincare lemma, this yields a principal calculation of H*(U) for quite general open sets in Rn . If, for instance, U can be covered by a finite number of convex open sets Ui, then every Ua will also be convex and H* (Ua) thus known from the Poincare lemma. Theorem 5.1 Let U1 and v = 1 , 2, let

iv: Uv

Then the sequence

-+

Uz be open sets of Rn with union U = U1 u Uz. For U and )v: U1 n Uz Uv be the corresponding inclusions. -+

Proof. For a smooth map ¢: V -+ W and a p-form w = Efidx1 E nP(W),

In particular, if ¢ is an inclusion of open sets in

Rn, i.e. ¢i (x) = Xi, then

Hence

¢*(w)

(1)

=

E(Jr o ¢)dxJ.

i11, j11, v = 1, 2. It follows from (1) that JP 0 then ii(w) = 0 = i2(w), and

This will be used for ¢ = If namely

JP(w)

=

if and only if fi imply that fi

o

iv

=

0

for all I. However fi

= 0 on all of U, since U1 and U2

Ker JP = Im JP. First,

JP o JP(w)

o

i1 =

is injective.

0 and fi o

iz = 0

cover U. Similarly we show that

= J2i2(w) - jiii(w) = j* (w) - j*(w) = 0,

34

5.

THE MAYER-VIETORIS SEQUENCE

where j: UI n u2 � u is the inclusion. Hence Im JP s; Ker JP. To show the converse inclusion we start with two p-forms Wv E QP ( Uv), Since JP(w1 , w2) = 0 we have that ii (w! ) = ii(w2), which by ( 1 ) translates into !I o j1 = 9I o )2 or fi(x) = 9I(x) for x E U1 n U2. We define a smooth function hi: U - Rn by hi(x) =

/J(x) { 9I(x), ,

X E U1 x E U2.

Then JP(L_ hidxI ) = (w1, w2). Finally we show that JP is surjective. To this end we use a partition of unity {Pl,P2} with support in {UJ. U2}, i.e. smooth functions Pv: U --t [ 0, 1 ] ,

1/ =

1, 2

for which SUPPu(Pv) C Uv, and such that PI(x) + P2(x) = 1 for x E U (cf. Appendix A). Let f: U1 n U2 �R be a smooth function . We use {PI, P2} to extend j to U1 and u2. Since SUPP u (PI) n u2 c ul n u2 we can define a smooth function by

h(x) =

{�

Analogously we define

!I (x) =

j(x)p1(x)

{�

(x )p2 (x)

if x E U1 n U2 if x E U2 - suppu(Pl)·

if x E U1 n U2 if x E U1 - SUPPu(P2) ·

Note that fi(x) - h(x) = f(x) when x E U1 n U2, because PI(x) + P2 ( x) = 1 . For a differential form w E rv'(U1 n U2) , w = L_ fidxJ, we can apply the above to each of the functions fi: U1 n U2 - R. This yields the functions !J,v: Uv --+ R, and thus the differential forms Wv = L_ fi,vdXI E QP(Uv)· With this choice D JP(w1 ,w2) = w. It is clear that n*(Ut) EB n*(U2) J: D*(UI) EB D*(U2) � D*(UI n U2)

I: D*(U)

�

are chain maps, so that Theorem 5.1 yields a short exact sequence of chain complexes. From Theorem 4.9 one thus obtains a long exact sequence of cohomology vector spaces. Finally Lemma 4.13 tells us that

We have proved:

5.

Theorem 5.2 (Mayer-Vietoris) Let U1 and U2 be open sets in Rn and U = There exists an exact sequence of cohomology vector spaces · ··

-

35

THE MAYER-VIETORIS SEQUENCE

U1 UU2.

r r a· HP+1 (U) - · · · HP(U) HP(Ul ) EB HP(U2) HP(U1 n U2 ) -

Here I*([ w]) = (ii[ w ] , i2[ w ] ) and notation of Theorem 5 . 1 .

Corollary 5.3 If U1 and

J* ( [ wl ], [ w2 ]) = Ji[ wl ] - }2[ w2 ]

in the 0

U2 are disjoint open sets in Rn then

is an isomorphism.

Proof. It follows from Theorem 5 . 1 that

is an isomorphism, and Lemma 4.13 gives that the corresponding map on coho­ mology is also an isomorphism. 0 Example 5.4 We use Theorem 5.2 to calculate the de Rham cohomology vector

spaces of the punctured plane

R2 - {0}. Let

= R2 - {(x1,x2) I Xt � 0, x2 = 0} u2 = R2 - {(xt,X2 ) I Xt � 0, X2 = 0}. These are star-shaped open sets, such that HP(U1 ) = HP(U2) = 0 for p > 0 and H0(U1) = H0(U2) = R. Their intersection U1 n U2 R2 - R = R� u n: U1

l

=

is the disjoint union of the open half-planes (2)

x2 > 0 and x2 < 0. Hence if p > 0 if p = 0

by the Poincare lemma and Corollary 5.3. From the Mayer-Vietoris sequence we have ·

For

p > 0,

·

·

r cr - HP(U1) Ee HP (U2) HP(Ut n U2 ) HP+l (R2 -{0}) � HP+1(Ut) EB HP+l(U2) - . . .

S.

36

THE MAYER·VIETORIS SEQUENCE

is exact, i.e. ()* is an isomorphism, and Hq (R2 - {0}) = 0 for q � 2 according to (2). If p = 0, one gets the exact sequence

H0 (R2 - {0}) £: H0(U ) EB H0(Uz) � 1 2 � H0(U1 n Uz) H1 (R - {0}) � H1(U1) EB H1 (Uz).

H-1 (U1 n Uz)

(3)

---+

Since H-1(U) = 0 for all open sets, and in particular H-1 (Uv) = 0, 1° is injective. Since H 1 (Uv) = 0, ()* is surjective, and the sequence (3) reduces to the exact sequence

R EB R

II

R EB R

II

0- H0(R2 - {0}) L. H0(U1) EB H0(Uz) L. H0(U1 n U2 ) .£:. H1 (R2 - {0}) -- 0. 0

0

However, R2 - {0} is connected. Hence H0(R2 - {0}) � R, and since 1° is injective we must have that Irn J0 � R. Exactness gives KerJ0 � R, so that J0 has rank 1 . Therefore IrnJ0 � R and, once again, by exactness

Since H0(U

1

n

Uz) / IrnJ0 � R, we have shown W(R -

{0}) =

{:

if p � 2 ifp = 1 if p = 0.

In the proof above we could alternatively have calculated

by using Lemma 3.9: H0(U) consists of locally constant functions. constant function on ui. then

° J0 (!I ) = iiiUtn U2 and 1 Uz) =

-

If

h is a

fzw1 nu2

so that J0(a, b) = a - b. Theorem

ul , .

.

.

5.5 Assume that the open set U is covered by convex open sets Then HP(U) is finitely generated.

' Ur.

r = 1 the assertion follows from the Poincare lemma. Assume the assertion is proved for r - 1 and

Proof. We use induction on the number of open sets. If

let V

=

U1 U

· · · u U1·- 1 ,

such that U

the exact sequence

which by Lemma

37

THE MAYER-VIETORIS SEQUENCE

5.

=

V U Ur·· From Theorem

5.2 we have

4.1 yields HP(U) :::! Im8* EB Iml*.

Now both V and V n Ur convex open sets.

= (U1 n Ur) U · · · U (Ur-1 n Ur )

are unions of (r - 1 )

Therefore Theorem 5.5 holds for H*(V n Ur ) , H*(V) and

H*(Ur), and hence also for H*(U).

0

39

6.

HOMOTOPY

In this chapter we show that de Rham cohomology is functorial on the cate­ gory of continuous maps between open sets in Euclidean spaces and calculate

H*('Rn - {0} ). Definition 6.1 Two continuous maps fv: X � Y, v = 0, 1 between topological spaces are said to be homotopic, if there exists a continuous map

F: X x [0, I] such that

_,

Y

F(x, v) = fv(x) for v = 0, 1 and all x

E

X.

This is denoted by fo � h, and F is called a homotopy from fo to fi. It is convenient to think of F as a family of continuous maps ft : X _, Y (0 � t � 1), given by ft(x) = F(x, t), which defonn fo to fi . Lemma 6.2 Homotopy is an equivalence relation. Proof. If F is a homotopy from

fo to fi, a homotopy from h to fo is defined by G(x, t) = F(x, 1 - t). If fo � h via F and h :::= h via G, then fo :::= h via H( x, t) = .

Finally we have that f

F(x, 2t) { G(x,

O�t��

2t - 1)

! � t � l.

Lemma 6.3 Let X, Y and Z be topological spaces and Let fv: X continuous maps for v = 0, 1. If fo � !1 and 90

9v: Y _, Z be 9o fo :::= 91 o

°

0

� f via F(x, t) = f(x).

fi.

Y and � 91 then �

Proof. Given homotopies

F from fo to h and G from 9o to 91· the homotopy H from 90 fo to 91 h can be defined by H(x, t) = G(F(x, t), t). 0 o

o

Definition 6.4 A continuous map f: X

Y

is called a homotopy equivalence, if there exists a continuous map g: Y -+ X, such that 9 o f � idx and f o 9 � idy. Such a map g is said to be a homotopy inverse to f. -+

Two topological spaces X and Y are called homotopy equivalent if there exists a homotopy equivalence between them. We say that X is contractible, when X is homotopy equivalent to a single-point space. This is the same as saying that idx is homotopic to a constant map. The equivalence classes of topological spaces defined by the relation homotopy equivalence are called homotopy types.

40

6.

HOMOTOPY

Example 6.5 Let Y � Rm have the topology induced by Rm. If, for the con­ tinuous maps fv: X ---+ Y, 11 = 0, 1, the line segment in Rm from fo(x) to ft(x) is contained in from

fo

Y for all x E X,

we can define a homotopy

to !I by

F: X x [0, 1]

---+

Y

F(x, t) = (1 - t)fo(x) + t h (x). In particular this shows that a star-shaped set in Rm is contractible.

Lemma 6.6 If U, V are open sets in Euclidean spaces, then (i) Every continuous map h: U

---+

V is homotopic to a smooth map.

(ii) If two smooth maps fv: U ---+ V, 11 = 0, 1 are homotopic, then there exists a smooth map F: U x R ---+ V with F(x, 11 ) = fv(x) for v = 0, 1 and all x E U (F is called a smooth homotopy from fo to fi). Proof. We use Lemma A.9 to approximate h by a smooth map f: U -+ V. We can choose f such that V contains the line segment from h(x) to f(x) for every x E U. Then h ::::= f by Example 6.5.

Let G be a homotopy from fo to ft. Use a continuous function '1/J: R with '1/J(t) = 0 for t � and 'lj;(t) = 1 for t 2: � to construct

!

-+

[0, 1]

H: U x R ---+ V; H(x, t) = G(x, 'lj;(t)).

k

Since H(x, t ) = fo(x) for t � and H(x, t) ft (x) for t 2: �. H is smooth on U x ( -oo, !) u U x ( � , oo ). Lemma A.9 allows us to approximate H by a smooth map F: U x R ---+ V such that F and H have the same restriction on 11) H(x, 11) = fv(x). 0 U x {0, 1}. For 11 = 0, 1 and x E U we have that =

F(x,

Theorem 6.7 If j, g: U -+ V are smooth maps and f maps

::::=

=

g then the induced chain

J* , 9*:n*(v) - n* (u) are chain-homotopic (see Definition 4.10). Proof. Recall, from the proof of Theorem 3.15, that every p-form w on can be written as

If ¢: U -+ U

x

R is the inclusion map

¢(x) = ¢o(x) = (x, 0), then

¢* (w) = L fi (x, O) d¢1 = L fi(x, O)dx.f.

U

x

R

HOMOTOPY

6.

41

Indeed, 0 such that J l f(y) - f (xo )ll < 2, provided it is additionally assumed that E is flat in IRn, that is, there exists a 8 > 0 and a continuous injective map ¢: sn-l x ( -6, 6) - Rn with E = ¢( sn-l x {0}).

A result by M. Brown from

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

Example 7.18 One can also calculate the cohomology of

55

"Rn with m holes",

i.e. the cohomology of

V = Rn -

( U Kj ) . m

j=l

The "holes" Kj in Rn are disjoint compact sets with boundary Ej , homeomorphic to sn- l . Hence the interiors Kj = Kj - Ej are exactly the interior domains of I:j. One has (3)

HP(V)

:::e

{ �m

if p = 0 if p = n otherwise.

1

We use induction on m. The case n = 1 follows from Proposition 7 . 1 6. Assume the assertion is true for V1

m-1

=

(

)

Rn - U Kj . j= l

Let V2 = Rn - Km . Then V1 u V2 = Rn and V1 n V2 = V. For p 2: the exact Mayer-Vietoris sequence

0 we have

If p = 0 then H0(Rn) � R and I* is injective. We get H0(VI) � R by induction and H0(V2) � R from Proposition 7 . 1 6. The exact sequence yields HO(V) � R. If p > 0 then HP(Rn) = 0 and the exact sequence gives the isomorphism HP(Vl ) EfJ HP(Vz) � HP(V). Now (3) follows by induction.

57

8. SMOOTH MANIFOLDS A topological space

X

has a countable topological base, when there exists a

countable system of open sets written in the form

Uie/ ui.

V

=

{Ui

I i E N}, such that every open set can be

I � N.

where

For instance Rn has a countable basis for the topology given by

V = {Dn (x;�;) where

b

(x; �;)

I

x = (XI, . . , xn ), Xi E Q; �; E Q, .

is the open ball with center at

A topological space

X

x

and radius

t:.

t: > O }

is a Hausdorff space when, for arbitrary distinct

Ux

Ux II Uy = 0.

x, y E X,

and

Uy

with

Definition 8.1 A topological manifold

M

is a topological Hausdorff space that

there exist open neighborhoods

has a countable basis for its topology and that is locally homeomorphic to The number n is called the dimension of

M.

Rn.

Remark 8.2 Every open ball bn(o, t:) in Rn is diffeomorphic to Rn via the map

q> given by

?f.. (

'K y) = (Smoothness of q> and

{ tan(7r llvll/2�;) · Y/IIYII 0

q>-1

at

0 can be shown

# 0, if y = 0. if y

IIYII < f

by means of the Taylor series at

0

for tan and Arctan.) Thus in Definition 8.1 it does not matter whether we require that

Mn

is locally homeomorphic to Rn or to an open set in Rn.

Definition 8.3

(U, h) on an n-dimensional manifold is a homeomorphism h: U --+ U', where U is an open set in M and U' is an open set in Rn. A system A = {hi: ui --+ Uf I i E J} of charts is called an atlas, provided {Ui I i E J} covers M.

(i) A chart (ii)

(iii) An atlas is smooth when all of the maps

hJ;· · - hJ· o hi-1 · h;· ( i II U·) J --+ hJ· ( U f � n U·) ; J .

are smooth. They are called chart transformations (or transition functions) for the given atlas.

Note in Definition 8.3.(iii) that Two smooth atlases

A 1 , A2

hi(Ui n Uj)

is open in Rn.

are smoothly equivalent if

A1 U A2 is a smooth atlas.

M. is an equivalence class A of smooth atlases on M.

This defines an equivalence relation on the set of atlases on on

M

A smooth structure

8.

58

SMOOTH MANIFOLDS

Definition 8.4 A smooth manifold is a pair ( M, A) consisting of a topological

manifold M and a smooth structure

A on M.

Usually A is suppressed from the notation and we write M instead of (M, A).

{ x E Rn+ l l ll x ll = 1 } is an n­ dimensional smooth manifold. We define an atlas with 2(n + 1) charts (U±i , h±i) Example 8.5 The n-dimensional sphere

sn

=

where

u i = {X E sn I Xi < 0} and h±i: u±i - iJn is the map given by h±i(X) = (xi, . . . ' Xi, . . . ) Xn+l )· The circumflex over Xi denotes that Xi is omitted. The inverse map is u+i = {X E sn I Xi > 0},

.. )

(

h±� (u) = u1 , . . . , Ui- 1, ±V1 - llull2, 1Li, . , un

It is. left to the reader to prove that the chart transformations are smooth. Example 8.6 (The projective space

relation:

X

"'

y

¢>

RPn) On sn we define an equivalence

X=

y

or X =

-y.

The equivalence classes [xJ = {x, -x} define the set lllPn. Alternatively one can consider RPn as all lines in Rn+l through 0. Let 1r be the canonical projection We give RPn the quotient topology, i.e.

U � RPn open ¢> n-1(U) � sn open.

With the conventions of Example 8.5, n(U-i) 1r(U+i). We define Ui n(U±i) � RPn, and note that 7r- 1 (Ui) = U+i U U-i with U+i n U_i = 0. An equivalence class [x] E Ui has exactly one representative in U+i and one representative in U-i· Hence 11': U+i ---+ Ui is a homeomorphism. We define =

·i =

1,

. . . , n.

=

hi = h+i 0 Jr- 1 : ui - iJn , This gives a smooth atlas on RPn.

Definition 8.7 Consider smooth manifolds M1 and M2 and a continuous map f: M1 - M2. The map f is called smooth at x E M1 if there exist charts h1: U1 - U{ and h2: U2 - U2 on Mt and M2 with x E Ut and j(x) E U2,

such that 1 1 h2 o j o h1 : h1 (f- (U2))

- U2

is smooth in a neighborhood of h1(x). If f is smooth at all points of M1 then f is said to be smooth.

8.

SMOOTH MANIFOLDS

59

Since chart transformations (by Definition 8.3.(iii)) are smooth, we have that Definition 8.7 is independent of the choice of charts in the given atlases for M1 and M2. A composition of two smooth maps is smooth. A diffeomorphism f: M1 � Mz between smooth manifolds is a smooth map that has a smooth inverse. In particular a diffeomorphism is a homeomorphism.

As soon as we have chosen an atlas A on a manifold M, we know which functions on are smooth. In particular we know when a homeomorphism f: V � V' between an open set V c M and an open set V' p(q) { 7f;o(h(q)) 0 M - ¢; 1 (0) U, 8.11 (M p M M

can now be defined by

hp

if E V otherwise.

h

the function coincides with and On the open neighborhood = therefore maps diffeomorphically onto Choose '1/Jo E C00(Rn, R) with compact support suppR (?/Jo) � and > and let ..

if otherwise.

=

Since

�

0

the final assertion holds.

choose ¢p and /p as in compact). For every E Proof of Theorem Lemma 8.12. By compactness can be covered by a finite number of the sets M - ¢; 1(0). After a change of notation we have smooth functions

¢i : M --+ R,

fr M --+ Rn

satisfying the following conditions:

(i) The open sets

(ii)

fj\Ui

maps

Uj

Uj

4>-;t

(0)

(1 ::; j ::; d)

Mcover Af. diffeomorphically onto an open set =

Uj � Rn.

8.

We define a smooth map

f: M

61

SMOOTII MANIFOLDS

._

Rnd+d by setting

f(q) = (h (q) , · · · , fd (q ) , rf>I (q) , · · · , cPd(q )) . f(ql) = j (q2) , we can by (i) choose j such that q1 E Uj. ¢j (q2) = cPj (q1 ) =I= 0, q2 E Uj, and by (ii), q1 = Q2 · Hence f is injective.

Assuming

M

is compact, f is a homeomorphism from

M

to

f(M).

Then Since

Let

n coordinates and the last n(d - 1) + d coordinates, respectively. By (ii) 1r1 o f = h is a diffeomorphism from U1 to U�. In particular 1r1 maps f(UI) bijectively onto Uf. Hence f(U1) is the graph of the smooth map be the projections on the first

g

1 ·. U1'

._

Define a diffeomorphism

Rn(d-l) +d '. h1

from

1 1r1 (Un

h1(x, y ) = (x , y - 9t (x)),

to itself by the formula

x E Uf, y

E

Rn(d- l)+n .

h1 maps f(U1) bijectively onto U{ x {0}. Since f(U1) is open in f(A1), f(UI) = f(M) n W1 for an open set W1 � Rnd+d, which can be chosen 1 to be contained in ?T1 (U{). The restriction h11w1 is a diffeomorphism from W1 onto an open set W{, and it maps f(M) n W1 bijectively onto W{ n Rn, as required by Definition 8.8. The remaining f (Uj) are treated analogously. Hence f(M) is a smooth submanifold of Rnd+d . Note also that fiU1 : U1 - f(U1) is a diffeomorphism, namely the composite of h 1u1 : U1 - U{ and the inverse to the diffeomorphism j(U1) - Uf induced by 1r1. The remaining U; are treated analogously. Hence f: M - f(M) is a diffeomorphism. 0 We see that

Remark 8.13

The general case of Theorem 8 . 1 1 is shown in standard text books

on differential topology.

To get

one uses Theorem 1 1 .6 below.

k = n+1

(Whitney' s embedding theorem)

In the proofs above one can change "smooth

manifold" to "topological manifold", "smooth map" to "continuous map" and "diffeomorphism" to "homeomorphism". This will lead to the theorem below, where the concept (locally flat) topological submanifold is defined in analogy to Definition 8.8, but with a homeomorphism instead of the diffeomorphism

h.

Theorem 8.14 Every compact topological n-dimensional manifold is homeomor­ phic to a (locally flat) topological submanifold of a Euclidean space Rn+k . 0 On a topological manifold functions jll[

._

Mn

C0(M, R) of continuous A on Mn gives a subalgebra

we have the R-algebra

R. A smooth structure

· c= (( M, A), R) � C0 (M, R )

8.

62

SMOO'fH MANIFOLDS

consisting of the maps M � R, that are smooth in the structure A on (and the standard structure on R). Usually A is suppressed from the notation, and the R-algebra of smooth real-valued functions on M is denoted by C00(M, R). This subalgebra of R) uniquely determines the smooth structure on M. This is a consequence of the following Proposition 8.15 applied to the identity maps idM

M

C0(M,

Proposition 8.15 If g: N � M is a continuous map between smooth manifolds N and M, then g is smooth if and only if the homomorphism

given by g*('lj;)

=

't/J

o

g maps C00(M, R) to C00(N, R).

Proof. "Only if'' follows because a composition of two smooth maps is smooth. Conversely if the condition on g" is satisfied, Lemma 8.12 applied to p :::: g (q) yields a smooth map f: Mn � Rn and an open neighborhood V of p in M, such that the restriction fw is a diffeomorphism of V onto an open subset of Rn. For the j-th coordinate function jj E C00(M, R) we have fi o g = g*(jj) E C00(N, R), so that j o g: N ---4 Rn is smooth. Using the chart fw on M, g is seen to be smooth at q. 0 Remark 8.16 There is a quite elaborate theory which attempts to classify n­ dimensional smooth and topological manifolds up to diffeomorphism and home­ omorphism. Every connected !-dimensional smooth or topological manifold is diffeomorphic or homeomorphic to R or S1 . For n = 2 there is a complete classification of the compact connected surfaces. There are two infinite families of them: Orientable surfaces:

Non-orientable surfaces: RP2 , Klein's bottle, etc. See e.g. [Hirsch], (Massey]. In dimension 3, one meets a famous open problem: the Poincare conjecture, which asserts that every compact topological 3-manifold that is homotopy equivalent to S3 is homeomorphic to S3. It is known that every topological 3-manifold has a smooth structure A and that two homeomorphic smooth 3-manifolds also

M3

8.

SMOOTH

MANIFOLDS

63

are diffeomorphic. In the mid 1 950s J. Milnor discovered smooth ?-manifolds that are homeomorphic to S1' but not diffeomorphic to S7. In collaboration with M. Kervaire he classified such exotic n-spheres. For example they showed that there are exactly 28 oriented diffeomorphism classes of exotic ?-spheres. In 1960 Kervaire described a topological 1 0-manifold that has no smooth structure. During the 1960s the so-called "surgery" technique was developed, which in principle classifies all manifolds of a specified homotopy type, but only for n � 5. In the early 1 980s M. Freedman completely classified the simply-connected com­ pact topological 4-manifolds; see [Freedman-Quinn]. At the same time S. Don­ aldson proved some very surprising results about smooth compact 4-manifolds, which showed that there is a tremendous difference between smooth and topolog­ ical 4-manifolds. Donaldson used methods originating in mathematical physics (Yang-Mills theory). This has led to a wealth of new results on smooth 4manifolds; see [Donaldson-Kronheimer]. One of the most bizarre conclusions of the work of Donaldson and Freedman is that there exists a smooth structure on R4 such that the resulting smooth 4-manifold "R4 " is not diffeomorphic to the usual R4 . It was proved earlier by S. Smale that every smooth structure on Rn for n =!= 4 is diffeomorphic to the standard Rn.

65

9. DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

In this chapter we define the de Rham complex

Q*(M) of a smooth manifold Mm

and generalize the material of earlier chapters to the manifold case.

For a given point p E Mm we shall construct an m-dimensional real vector space

TpM

M N Tf(p)M known as the tangent map of f at p.

called the tangent space at p. Moreover we want a smooth map f:

to induce a linear map Dpf; TpM

._

-t

Remarks 9.1 (i) In the case p E tangent space to

U U

� Rm,

where

at p with

Rm.

U

is open, one usually identifies the

Better suited for generalization is the

following description: Consider the set of smooth parametrized curves TI

-t

U

with

'Y(O) =

p, defined on open intervals around 0.

equivalence relation on this set is given by the condition

'Yi (0)

There is a 1-1 correspondence between equivalence classes and

An

'Yz (O).

Rm, which

')'1(0) E nm. Consider a further open set V £; Rn and a smooth map F: U V. The Jacobi matrix of F evaluated at p E U defines a linear map DpF: Rm Rn. For 1: I U, 'Y(O) = p as in (i) the chain rule implies that DpF(r1(0)) = ( F o 1)1(0). Interpreting tangent spaces as given by ['Y]

to the class (ii)

of 'Y associates the velocity vector

=

._

-t

-t

equivalence classes of curves we have

(I) In particular the equivalence class of F o

1

depends only on

['Y] ·

(U, h) be a smooth chart around p E Mm. On the set of smooth curves ex: I ._ M with cx(O) = p defined on open intervals around 0 we have an

Let

equivalence relation

(2) This equivalence relation is independent of the choice of is another smooth chart around p, one finds that

(h o cxl ) (0) I

=

(h o cx2 ) (0) I

{:}

·

(h o a1)'(0) �

=

(U, h).

In fact, if

(U, h)

(h o a2) I (0) �

by applying the last statement of Remark 9.l .(ii) to the transition diffeomorphism

F

=

h o h,- l

and its inverse.

66

9.

DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS

Definition 9.2 The tangent space

TpMm is the set of equivalence classes with M, a(O) = p.

respect to (2) of smooth curves a: I

---+

We give TpM the structure of an m-dimensional vector space defined by the following condition: if (U, h) is a smooth chart in M with p E U, then

is a linear isomorphism; here [a]

E

TpM is the equivalence class of a.

By definition . . . , Xn-1, t) = j(x 1 , . . . , Xn-1. t) -

n-l

L j= l

!:l

ug ·

J (x1, . . . , Xn-l)P(t) axJ

= f(xl> . . . , Xn-b t) - g(x1, . . . , Xn-dP(t). Finally from (9) it follows that

/00 = 1: -

oo

h(x1 . . .

·.

,

Xn-1, t)dt

j(x1, . . . , Xn-1, t)dt - g(x1, . . . , Xn-d

= 0.

1:

p(t)dt 0

Lemma 10.16 Let (Ua)aeA be an open cover of the connected manifold M, and

let p, q E

M.

There exist indices a1 , . . . , ak such tho.t

(i) p E Ua1 and q E Uak (ii) Ua; n Uai+l =j:. 0 when 1

s; i s;

k - 1.

Proof. For a fixed p we define V to be the set of q E M, for which there exists a finite sequence of indices a1, . . . , ak from A, such that (i) and (ii) are satisfied. It is obvious that V is both open and c.losed in M and that V contains p. Since M is connected, we must have V = M. 0

be an open set diffeomorphic to Rn and let W � U be non-empty and open. For every w E n�(M) with suppw s;;; U, there exists a "' E n�- 1 (JV!) such that supp "' � U and supp (w - d"') � W. Lemma 10.17 Let

U�

M

Proof. It suffices to prove the lemma when M =

U,

in variance it is enough to consider the case where M =

and by diffeomorphism

U = Rn.

10.

94

INTEGRATION ON MANIFOLDS

fRn WI = 1. Then where a = r w. }Rn

Choose WI E n�(Rn) with suppwl � w and

r }R� (w - awl) = 0,

By Lemma 10.15 we can find a "' E n�-1(Rn) with

w - aw1 = d"'.

Hence w - d"' = aw1 has its support contained in W.

0

Lemma 10.18 Assume that Mn is connected and let W � M be non-empty and open. For every w E S1�(M) there exists a "' E n�-1(M) with supp(w - d"') � w. Proof. Suppose that suppw � U1 for some open set U1 � M diffeomorphic to Rn. We apply Lemma 10.16 to find open sets U2, . . . , Ub diffeomorphic to Rn, such that Ui 1 n Ui #- 0 for 2 .:::; i ::; k and Uk � W. We use Lemma 10.17 to successively choose "'I' "'2 ' . . . , "'k -I in n�- 1 (M) such that

( - t d! 0 such that (5)

lx-

Yll � ti ::::;.

lar ar 1 x : ( ) a ox� x

(y) �

(1 $ i,j $ n and x, y E L).

€,

We subdivide T into a union of Nn closed small cubes Tz with side length and choose N so that (6)

x

N•

For a small cube Tt with T1 n K =I= 0 we pick E Tt n K. If y E T1 the mean value theorem yields points �j on the line segment between x and y for which

(7) Since �j

E T1 �

L,

the Cauchy-Schwarz inequality and (4) give

l ()()

11.

DEGREE. UNKJNG NUMBERS AND INDEX OF VECTOR FIELDS

and by (6),

llf ( y) - f (x) ll

(8)

�

c diam(T1) Cvn

==

anC

N

.

Formula (7) can be rewritten as

J(y) = f(x) + Dxf(y - x) + z,

(9)

where z = ( zl> . . . , z ) is given by n

By (6), ll�i - x ll S 8, so that (5) gives l zil ::; (10)

l l z ll

S f.

n;n

a

m

N.

Hence

.

Since the image of Dxf is a proper subspace of Rn , we may choose an affine hyperplane H � Rn with

f(x)

+ Im(Dxf) � H.

By (9) and ( 1 0) the distance from f(y) to H is Jess than f.an#. Then (8) implies that f(Tt) is contained in the set Dt consisting of all points q E Rn whose orthogonal projection pr(q) on H lies in the closed ball in H with radius and centre f(x) and llq - pr(q)JI ::; we have J.1,n(Dt) =

7f

e.anp. For the Lebesgue measure J.1,n on Rn

( c)

r,;; anv •• an 2c � N

n-1

Vol(Dn-1 )

=E

c

Nn

,

where c = 2 annn+icn- lVo1(Dn-l). For every small cube Tt with Tt n K ::/= 0 we now have J.1,n (j(T1)) ::; E Jn . Since there are at most Nn such small cubes Tt, J.1,n (J(K)) ::; EC. This holds for every c > 0 and proves the assertion. 0 Lemma 11.8 Let p E Mn be a regular value for the smooth map f: Nn -+ Mn, with Nn compact. Then J-1 (p) consists offinitely many points Ql , . . . , qk. Moreover, there exist disjoint open neighborhoods Vi of qi in Nn, and an open neighborhood U of p in Mn, such that

( i) J- 1(U) =

U:::t Vi

(ii) fi maps Vi diffeomorphically onto U for 1 ::; i $ k.

11. DEGREE, LINKING NUMBERS AND INDEX OF VECI'OR FIELDS

f-1(p), D9f: f-1(p). .. f f( i) p ( f (Wi)) f (

101

T9N ----t TqM is an isomorphism. From the E inverse function theorem we know that f is a local diffeomorphism around q. In particular q is an isolated point in Compactness of N implies that consists of finitely many points q1 , . , Qk· We can choose mutually disjoint open neighborhoods Wi of Qi in N, such that maps diffeomorphically onto an open neighborhood W of in M. Let k k N-U U= n Proof. For each q

f-1(p)

Wi

i=l

i=l

Wi) .

Since N - u�l wi is. closed in N and therefore compact, N - Uf=l is also compact. Hence U is an open neighborhood of in M. We then set Vi = wi n J- 1 (U). o

f(

p

f:

Wi)

Consider a smooth map Nn - Mn between compact n-dimensional oriented manifolds, with M connected. For a regular value E M and q E define the local index n ; T9N ----t TpM preserves orientation 1 if I (ll) q = -1 otherwtse.

d(f ) {

p

f-1(p),

D9f.:

Theorem 11.9 In the situation above, and for every regular value

deg(J) = In particular

deg(f) is an integer.

L qEJ-l(p)

Proof. Let Qi , Vi, and

Ind(f; q).

p,

U be as in Lemma 1 1 .8. We may assume that U and Vi U is positively or negatively hence Vi connected. The diffeomorphism oriented, depending on whether lnd(f; Qi) is 1 or - 1 Let E nn(M) be an n-forrn with

Jrv.: -

w

.

suppM(w) k U, JM w = 1 . V1 Then suppN(f*(w)) k f - 1(U) Vk> and we can write k f* (w) = L Wi , i=l where Wi E nn(N) and supp(w ) £:; "\ti. Here Wij V; Ujv;) *(wiU) . The fonnula =

U ... U

=

i

is a consequence of the following calculation:

k

k

* (wiU) 1 f deg(f) deg(f) 1M w = 1N f* (w) = L lN Wi = L ( , v ;) i =

k

k

=

i=l

= l Vi

nd (f qi). L Ind(f;. Qi) 1 wiU = I i=l i=l

u

)

;

0

Jl. DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

102

f-1(p) = 0 the theorem shows f*(w) = 0). Thus we have

In the special case where proof above we get

Corollary 11.10

that deg(f)

= 0 (in the

If deg(J) =/= 0, then f is surjective.

0

Proposition 11.11 Let F: pn+ l -t Mn be a smooth map between oriented smooth manifolds, with Mn compact and connected. Let X � P be a compact domain with smooth boundary Nn = oX, and suppose N is the disjoint union of submanifolds Nf, . . . , NJ:. If fi = FjN,, then

k

L deg(fi) = 0. i=l

Proof.

Let

f =

FJ N

so that

k

deg(J) = L deg(fi). i=l

On the other hand, if

g

de (J )

w E nn(M)

=

has

JM w = 1,

then

f f*(w) = f dF*(w) = f F*(dw) = 0 JN lx lx

where the second equation is from Theorem

0

1 0.8.

We shall give two applications of degree. We first consider linking numbers, and then treat indices of vector fields.

Definition 11.12 Let Jd

and

K1 be

two disjoint compact oriented connected

smooth submanifolds of Rn+ l , whose dimensions d 2: 1 , l 2: 1

Their

linking number

lk(J, K)

w(x, y) = JXK

d + l = n.

= deg ( w J,I 0 with

l l lxll

S p} � U

and such that sn - 1 by

0 is the only

zero for F in

pDn.

Define a smooth map

=

{x E

Fp(x)

=

and Theorem

Definition 11.16

Fp: sn-l

--+

F(px) I I F(px))i

Fp is independent of the choice of p , I 1 .9, degFp E Z is independent of p.

The homotopy class of

1 1 .2

2, and let

F is also called a

Rn

pDn

2:

The degree of Fp is called

and b y Corollary

local index of F at 0, and is denoted

t,(F; 0). Lemma 11.17

Suppose F E C00(Rn, Rn) has the origin as its only zero. Then

induces multiplication by t(F; 0) on Hn- 1(Rn - {0}) � R.

11.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

107

Let i: sn- l Rn - {0} be the inclusion map and r: Rn- l - {0} --+ sn-l the retraction r(x) = x/llxl l · We have t.,(F; 0) = degFt, where F1 = r o F o i. The lemma follows from the commutative diagram below, where Hn-1 (i) and Hn-1 ( r) are inverse isomorphisms: Proof.

--+

Hn-1 (Rn -

l

{0 } ) Hn- l (F ) Hn-1 (R - {0})

l

Hn-l(r ) H'' - I (i) Hn-1 (1·) 1 Hn- l (sn - 1) H"- (Fl ) Hn- I (sn - 1)

0

Given a diffeomorphism ¢: U V to an open set V � Rn and a vector field on U, we can define the direct image ¢*F E c=(v, Rn) by --+

If F E c=(u, Rn) has 0 as an isolated singularity and ¢: U --+ V is a diffeomorphism to an open set V � Rn with ¢(0) = 0, then

J_,emma 11.18

By shrinking U and V we can restrict ourselves to considering the case where 0 is the only zero for F in U, and where there exists a diffeomorphism 1/J: V Rn. The assertion about ¢ will follow from the corresponding assertions about 1/J and 1/J o ¢, since

Proof.

--+

1/;*(¢* F)

= (1/; o ¢) *F.

Thus it suffices to treat the case where ¢: U Rn is a diffeomorphism and where Y = ¢*F E c=(Rn, Rn) has the origin as its only singularity. Let U0 � U be open and star-shaped around 0. We define a homotopy --+

:

Uo X [0, 1]

--+

Rn ;

t(X)

{

(Do¢)x = (x, t) = c/J(t x)/t

For x E Uo,

where cPi

E

c=(Uo, Rn) is given by cPi (x) =

1 1 8¢ (tx) dt. �

0 UXi

if t = 0 if t # 0.

108

11.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

It follows that

0 so that pDn � Uo. We get a homotopy G: sn- l X [0, 1] -

sn- 1

'

G(x, t) = G(px, t) / l!G(px, t)IJ,

between the map Fp in Defi nition 1 1 . 1 6 and the analogous map Ap with A = DoF. It follows from Corollary 1 1 .2 that t.(X ; Po ) = [,(F; 0) = deg(Fp) = degAp = t(A ; 0). {0} ---+ Rn - { 0} induced by A operates on Hn-l (Rn - {0}) by multiplication by t(X;po); cf. Lemma 1 1 . 17. The result now follows from Lemma 6.14. 0

The map fA: Rn

-

110

11.

DEGREE, LJNKING NUMBERS AND INDEX OF VECTOR FIELDS

Definition 11.21 Let X be a smooth vector field on Mn , with only isolated singularities. For a compact set R to be

R £; M

we define the total index of X over

Index (X ; R ) = I >(X;p),

where the summation runs over the finite number of zeros p E compact we write Index (X) instead of Index (X; M).

R for X . If M

is

Theorem 11.22 Let F E C00(U, Rn) be a vector field on an open set U £; Rn, with only isolated zeros. Let R £; U be a compact domain with smooth boundary oR, and assume that F(p) =/= 0 for p E aR. Then Index(F; R)

where f: aR

-t

= F(x)

sn-I is the map j(x )

Proof. Let p1 , . . . , Pk be the zeros in

D1 £; R - aR,

= deg j,

R for F,

with centers Pi · Define

fj : aDj

-t

I II F(x )ll. and choose disjoint closed balls

fj (X) = F(x) I IIF(x) l l ·

sn-I ;

We apply Proposition 1 1 . 1 1 with X = R- uj Dj. The boundary ax is the disjoint union of aR and the (n - !)-spheres aD1 , . . . , aDk· Here aDj, considered as boundary component of X, has the opposite orientation to the one induced from Dj. Thus

k

deg(f) +

L

-deg(fi)

i=l FinaJly deg(Ji)

= 0.

= t( F; Pi) by the definition of local index and Corollary 1 J .3.

0

Corollary 1 1.23 In the situation of Theorem 1 1.22, Index(F; R) depends only on the restriction of F to aR. 0 Corollary 1 1.24 In the situation of Theorem 1 1.22, suppose for every p E aR that the vector F(p) points outward. Let g: aR -t sn-I be the Gauss map which to p E oR associates the outward pointing unit normal vector to aR. Then Index(F; R)

= deg g.

Proof. By Corollary 1 1 .2 it sufficies to show that f and g are homotopic. Since j(p) and g(p) belong to the same open half-space of Rn, the desired homotopy can be defined by

(1 - t)j(p) + tg(p) 11( 1 - t )f(p) + tg (p) [l

(0 � t � 1).

0

11. DEGREE, LINKING NUMBERS AND INDEX OF VECI'OR FIELDS

III

Lemma 11.25 Suppose F E C00(Rn, Rn) has the origin as its only zero. Then there exists an F E C00(Rn, Rn), with only non-degenerate zeros, that coincides

with F outside a compact set.

Proof.

We choose a function ¢ E C00(Rn, [0, 1]) with

{

1 if llxll � 1 ¢(x) - 0 if llxll � 2. _

We want to define F(x) = F(x) - ¢(x)w for a suitable w E we have F(x) F(x). Set

IRn.

For llxll > 2

=

c=

inf IIF(x)fl > 0 l �llxll�2

and choose w with llwll < c. For 1 � llxll � 2, IIF(x)ll � c - llwll > 0. Thus all zeros ofF belong to the open unit ball bn. Since F coincides with F -w on bn, We can pick w as a regular value of F with llwll < c by Sard's theorem. Then = DpF will be invertible for all p E .F- 1 (0) , and F has the desired 0 properties. DpF

Note, by Corollary 1 1.23, that (14)

t(F;O) =

L t(.F,p)

peF-1(p)

Here is a picture of F and F in a simple case: Figure 5

The zero for F of index -2 has been replaced by two non-degenerate zeros for F, both of index -1.

112

11.

DEGREE, LINKING NU�mERS AND INDEX OF VECTOR FIELDS

Coronary 1 1.26 Let X be a smooth vector field on the compact manifold Mn with isolated singularities. Then there exists a smooth vectorfield X on M having only non-degenerate zeros and with Index(X) = Index(X).

Proof. We choose disjoint coordinate patches which are diffeomorphic to Rn around the finitely many zeros of X, and apply Lemma 1 1 .25 on the interior of each of them to obtain X. The formula then follows from ( 1 4).

Let Mn � Rn+k be a compact smooth submanifold and let N€ be a tubular neighborhood of radius E > 0 around M. Denote by g: 8Ne. -t sn+k- l the outward pointing Gauss map. If X is an arbitrary smooth vector field on Mn with isolated singularities, then Theorem 1 1.27

Index(X)

=

deg g.

Proof. By Corollary 1 1 .26 one may assume that X only has non-degenerate zeros. From the construction of the tubular neighborhood we have a smooth projection 1r: N -t M from an open tubular neighborhood N with NE � N � Rn+k, and can define a smooth vector field F on N by F( q) = X(1r (q) ) + ( q - 1r( q)) .

(1 5)

Since the two summands are orthogonal, F(q) = 0 if and only if q E M and X(q) = 0. For q E oN, , q - 1r(q) is a vector normal to TqfJN< pointing outwards. Hence X(1r(q) ) E Tq8Nf., and F(q) points outwards. By Corollary 1 1 .24

lndex(F; N€) = deg g.

and it suffices to show that �(X;p) :::::: �(F;p) for an arbitrary zero of X. In local coordinates around p in M, with p corresponding to 0 E R'l, X can be written in the form ( 1 6) where fi ( 0)

(17)

By

X=

=

a 2:: ]i(x)�, x�

n

i::: l 0, and by Lemma 1 1 .20 t( X; p) is the sign of det

( ![; (0)) .

differentiating (16) and substituting 0 one gets

a x (o) = x aj

t

8 /i (o)�. 8xj axi i=l lt follows from (15) that DpF: Rn+k -t Rn+k is the identity on TpMJ., and by ( J 8) DpF maps TT>M into itself by the linear map with matrix (8f:,/OXj (0)) (with respect to the basis (8j ax i)0). It follows that p is a non-degenerate zero for F and that detDpF has the same sign as the Jacobian in (17). 0

( 1 8)

113

THE POINCAR.t-HOPF THEOREM

12.

..... In the following, Mn � Rn+k will denote a fixed_pn60th submanifold. If the cohomology of Mn is finite-dimensional (e.g. -when Mn is compact), the i-th Betti number is given by ./

( 1)

The

Euler characteristic

of Mn is defined to be x(Mn) =

(2)

n

L ( -l) i bi( M). i=O

This chapter's main result is: Theorem 12.1 (Poincare-Hopf) Let X be a smooth vector field on a compact manifold M. If X has only isolated zeros then

lndex(X) = x(M) . By the final result of Chapter 1 1 it is sufficient to show the formula for just one such vector field X on M. We shall do so by making use of a Morse function on Mn. Given f E C00(M, R), a point p E M is a critical point for f if dpf = 0. Proposition 12.2 Suppose that p E M is a critical point for f E C00(M, R).

(i)

(ii)

There exists a quadratic form d�f on TpM characterized by the equation

where a: ( -15, 15) - M is any smooth curve with a(O) = p. Let h: U - Rn be a chart around p and let q == h(p). Then the composition

is the quadratic form associated to the symmetric matrix

I 14

Proof. Set h

12.

o a(t) =

THE POINCARi-HOPF THEOREM

'Y(t)

calculation yields

(! o a)'(t)

=

- ('Y1(t), . . , 'Yn (t)) .

(¢> o 'Y) ' (t) =

Since p is critical, (8¢>/oxi)('Y(O)) = substituting t = 0, we get

(! o a)" (O) This is the value at 'Y'(O) Both (i) and (ii) follow.

n

=

n

0.

and ¢> = f

t :: .

i=l

o

h-1. A direct

('Y(t))'Y�(t).

1

By differentiating once again and

82 ¢>

L L axl·ax . (q)'Y�(o)'Yj(o). J

. 1 ]= . 1

t=

= Dph(a'(O))

E

Rn

of the quadratic form from (ii).

0

Consider another chart h: [J --... Rn around p with q = h(p) and let F = h o h - l defined in a neighborhood of q. The last formula i n the proof above can be compared with

where ¢ = f o Tt-1 and i = h o o.. Let J denote the Jacobi matrix associated with F in q. Then i'(O) = J'Y'(O) for the column vectors i'(O) and 'Y'(O). By substituting this and comparing, one obtains the matrix identity

(3)

Definition 12.3 A critical point p

E M

of f E C00(M, R) is said to be non­ degenerate, if the m�trix in Proposition 12.2.(ii) is invertible. We call f a Morse function, if all critical points of f are non-degenerate. The index of a non­ degenerate critical point p is the maximal dimension of a subspace V s;:; TpM for which the restriction of d�f to V is negative definite. For smooth submanifolds Mn �

Rn+k

Theorem 12.4 For almost all Po

E Rn+k the function f: M

one can get Morse functions by:

1 f (p) = 2 I IP - Poll 2 is a Morse function.

--...

R defined by

12.

THE POINCAR.E-HOPF THEOREM

115

Proof. Let g: Rn --+ M be a local parametrization and Yj: Rn --+ nn+k, j = 1 , . . . , k smooth maps, such that Yi (x), . . . , Yk (x) is a basis of Tg(x) M.1 for all x E Rn. By Lemma 9.21 we know that M can be covered by at most countably many coordinate patches g(U) of this type. Therefore it suffices to prove the assertion for g( U) instead of M.

We define : Rn+k

(4)

-t

Rn+k by

k (x, t) = g(x) + I: tj Yj(x)

j= I

(x E Rn , t E Rk ) .

By Sard's theorem it suffices to prove that if Po is a regular value of , then f becomes a Morse function on g(U). Set k = f o g: Rn --+ R; we can show instead that k becomes a Morse function on Rn. We have

k (x) =

(5)

1 2

(g (x) - Po , g(x) - Po),

where ( ' ) denotes the usual inner product on nn+k . Since (agjaxi (x), Yv(x))

= 0, it follows by differentiation with respect to Xj that

(6) From (5) we have

(7) and therefore (8) Furthermore, by (4),

(9)

k a ag I: aYv = + tv ax·J ax·J v=l axJ· )

-

-

Assume that p0 is a regular value of and let x be a critical point of k. It follows .l from (7) that g(x) - Po E T_g(x)M . Hence there exists a unique t E Rk with

( 1 0)

g(x) - PO =

k

- I: tv Yv(x), v=l

1 16

12.

THE POINCARE-HOPF THEOREM

and (x, t) E -1(po). The n + k vectors in (9) are linearly independent at the point (x,t). At this point, the equations (8), ( 1 0), (6) and (9) give

Let A denote the invertible (n + k) x (n + k) matrix with the vectors from (9) as rows. Then ADxg takes the following form: o2k

&2k OXjbX I

�

o2k bx,.8x1 0

82k � 0

0

0

Since Dxg has rank n, so does ADxg. Hence the n x n matrix

is invertible. This shows that x is a non-degenerate critical point. Example 12.5

Let

f: Rn - R be

0

the function

2 2 + . . . Xn, f(X) = C - x12 - x22 - . . . - X,\2 + X>.+ l

where c E

R, A E 7L

and 0

:::; A :::; n.

Since

gradx(f) = 2( -Xt, . . . , -X>., X>.+1 , . . . , Xn), 0

is the only critical point of f. We find that

( [J2j ) oxilxj

(0)

. -2, . . . , -2, 2, . . . , 2) = dtag(

with exactly A diagonal entries equal to -2. Thus the origin is non-degenerate of index A. We note that the vector field grad(!) has the origin as its only zero and that it is non-degenerate of index ( -1) >. .

12.

THE POINCAR.t-HOPF THEOREM

1 17

Theorem 12.6 Let p E Mn be a non-degenerate critical pointfor f E C00(M, R). There exists a C00-chart h: U - h(U) � Rn with p E U and h(p) = 0 such that

n

f

o h - 1 (x) = f(p) + L OiXl , x E h(U), i=l

where Oi = ± 1 (1 :::; i ::; n) (By an additional pennutation of coordinates we can put f into the standard jonn given in Example 12.5.)

Proof. After replacing f with f - f(p) we may assume that f(p)

= 0.

Since the problem is local and diffeomorphism invariant, we may also assume that f E C00(W, R), where W is an open convex neighborhood of 0 in Rn and that 0 is the considered non-degenerate critical point with f(O) = 0.

We write f in the form

n

J(x ) = L Xi 9i (x) ; i:=l

Since 9i(O)

=

-3f(O)

=

0,

we may repeat to get

n

( ) 9i (X) = L Xj 9ij X ; 9tJ0 · (X) j=l

11 agi(sx) !:1 o

ux1·

d

8.

On W we now have that n

n

J(x) = L L Xi Xj 9ij(x), i:::l j=l

where 9ij E C00(W, IR). If we introduce hij = !(9ii + 9ii) then symmetric n x n matrix of smooth functions on W, and n

(11)

f(x)

(hij)

becomes a

n

= L L Xi Xj hij(x). i=l j=l

By differentiating (I I ) twice and substituting 0, we get

a2f 0Xi 0Xj

( 0)

= 2hij(O).

(hij(O)) is invertible. Let us return to the original f E C00(M, R). By induction on k, we attempt to show that the C00-chart h from the theorem can be choosen such that f 0 h- 1 In particular the matrix

is given by ( 1 1 ) with

1 18

12.

THE POJNCARt-IIOPF THEOREM

with D a (k - 1) x (k - 1) matrix of the form diag(±1, . . . , ±1), and E some symmetric (n - k + 1 ) x (n - k + 1 ) matrix of smooth functions. So suppose inductively that k- 1 (12) f(x) = L t5ixf + L L Xi Xj hij(x), n

i=l

n

i=k j=.:k

for x in a neighborhood W of the origin. We know that the minor E is invertible at 0. To start off we can perform a Jinear change of variables in Xk , . . . , Xn , so that our new variables satisfy (12) with hkk(O) # 0. By continuity we may assume that hkk (x) has constant sign 6k = ±1 on the entire W. Set q=

and introduce new variables:

� E C00 (W, R),

(

t ��)

Yk

= q(x)

Yj

= Xj for j # k, 1

xk

+

i=k+l

ik Xi z

::;

j

kk

::;

n.

The Jacobi determinant for y as function of x is easily seen to be 8yk foxk(O) q( O) # 0. The change of variables thus defines a local diffeomorphism w around 0. In a neighborhood around 0 we have for y = 'll(x): =

f

0

w-1(y) = f(x) n n k-1 n = OiXJ + x�hkk(x) + 2xk XiXjhij(x) Xj hjk (x) + i=l i=k+l j=k+l j=k+ l n k-l h. ( ) 2 OiXJ + hkk(x) Xk + = Xj 1 i=l j=k+l hkk ( ) n n n hj - hkk (x) Xj X;Xjhij(x) + kk (x) j=k+ i=k+l j=k+l n n k = + iY xixj hij(x) s l i=l i=k+l j=k+l n n k = S;y} + YiYj hij o w-1(y), i;:k+l j=k+J i=l

L

L

(L

(

:L

:L :L

:L

:L :L

L

L

))2

) �:

L L

L L

where hij E C00(W, R). This completes the induction step.

0

We point out that with the assumptions of Theorem 12.6 p is the only critical point in U. If M is compact and f E C00(M,R) is a Morse function then f has only finitely many critical points. Among them there will always be at least one local minimum (index ). = 0) and at least one local maximum (index ). = n).

·

12.

THE POINCARt-HOPF THEOREM

119

Definition 12.7 Let f E

C00(M, R) be a Morse function. A smooth tangent vector field X on M is said to be gradient-like for f, if the following conditions are satisfied: (i) For every non-critical point p E M, dpf (X(p)) > 0. (ii) If p E Mn is a critical point of f then there exists a C00-chart h : U --+ h(U) � Rn with p E U and h(p) = 0 such that f o h-1(x) = f(p) - xi - · · · - xl + x� + l + and h.Xtu = grad ( ! o h-1).

A

smooth parametrized curve a: I � M is an a'(t)

· · ·

+

x;,

integral curve

x E h(U),

for X, if

= X(a(t)) for t E I.

one gets (f o C¥)1(t) = da(t) f(X(a(t))). If a(I) does not contain any critical points, then f o C¥: I - R is a monotone increasing function by condition (i). Hence

Lemma 12.8 Every Morse function on

M admits a gradient-like vector field.

Proof. We can find a C00-atlas (Ua , ha)aEA for M which satisfies the following two conditions: (i) Every critical point of f belongs to just one of the coordinate patches Ua . (ii) For any a E A either f has no critical point in Ua or f has precisely one critical point p in Ua, ha(P) = 0, and f o h;;1 has the form listed in Example 12.5.

Let Xa be a tangent vector field on Ucr determined by Xcr = (h;; 1 ) * (grad (f o h;;1 ) ) . Choose a smooth partition of unity (Pcr)O O for every p E M that is not a critical point for f. Let Po E M be a critical point for f of index A. If X (po) = 0, then t-(X ; Po) = ( -1) A .

Proof. We choose a gradient-like vector field X. By Definition 12.7.(ii) and Example 12.5,

Let U be an open neighborhood of Po that is diffeomorphic to Rn and chosen so small that Po is the only critical point in U. The inequalities

dpj(X(p)) > 0, valid for p E U - {Po}, show that X (p) and half-space in TpM. Thus

X (p)

( 1 - t)X(p) + tX (p) (0 � t defines a homotopy between JC and X considered Rn - {0}, and t(X; Po) = t-(X; Po).

belong to the same open

1) maps from U

- {Po}

to

D

Remark 12.10 Given a Riemannian metric on M and f E C00(M, R), one can

define the gradient vector field gradf by the equation

(gradp(f) , Xp) = dpf(Xp) for all Xp E TpM. Then Lemma 12.9 holds for grad(!).

Theorem 12.11 Let Mn be a compact differentiable manifold and X a smooth

tangent vector field on Mn with isolated singularities. Let f E C00(M, R) be a Morse function and cA the number of critical points of index A for f. Then we have that Index(X) =

n

L

A=O

(-l)A cA .

!

·J ·i

t(X;po) = (-1) A .

!

12. THE POINCARE-HOPF THEOREM

121

Proof. It is a consequence of Theorem 1 1 .27 that any two tangent vector fields with isolated singularities have the same index. Thus we may assume that X is gradient-like for f. The zeros for X are exactly the critical points of f, and the

0

claimed formula follows from Lemma 12.9.

It is a consequence of the above theorem that the sum ( 1 3)

is independent of the choice of Morse function J E C00(M, R). Given Theorem 1 2 . 1 1 , the Poincar6-Hopf theorem is the statement that the sum ( 1 3) is equal to the Euler characteristic; cf. (2). We will give a proof of this based on the two lemmas below, whose proofs in turn involve methods from dynamical systems and ordinary differential equations, and will be postponed to Appendix C. Let us fix a compact manifold Mn and a Morse function f on M. For a E R we set M (a) = {p E M I f(p)


0, and disjoint open neighbourhoods Ui of Pi. such that

(i) PI> . . . , Pr are the only critical points in f-1 ([a - E, a + E]). (ii) Ui is diffeomorphic to an open contractible subset of Rn. (iii) ui n M(a - €) is diffeomorphic to s).·- 1 X v;, where v; is an open contractible subset ofRn->.;+1 (in particular Ui nM(a - E) = 0 if Ai = 0). (iv) M ( a + E) is diffeomorphic to U1 u . . . u Ur u M(a - E). 0

Proposition 12.14 In the situation of Lemma 12.13 suppose that M( a - E) has finite-dimensional cohomology. Then the same will be true for M (a + E). and r

x( M( a + t:)) = x(M (a - t: )) + L ( -1)>.' . i=l

12. THE POINCARE-HOPF THEOREM

122

Proof. For U = U1 U . . .

u

Ur,

Lemma 12.1 3.(ii) and Corollary 6.10 imply that

HP(U) c:=

{

�f p # 0

0

W

If p = 0.

This gives x(U) = r. Condition (iii) of Lemma 12.13 shows that Ui n M(a is homotopy equivalent to S..\•- 1 , and Example 9.29 gives x ( Ui n M(a -

t) ) =

)

€

1 + ( - 1) ..\' -1.

Since U n M(a - €) is a disjoint union of the sets finite-dimensional de Rham cohomology, and

Ui

n

M(a

- €) ,

r

r

it has a

r

,\ x(U n M(a - t)) = 2:::: ( 1 + (-1) ' - 1) = r - L ( - 1 ),\' = x(U) - L (- 1 ),\'. i=1 i= 1 i= 1 The claimed formula now follows from Lemma 12.1 3.(iv) and the lemma below, 0 applied to U and V = M(a - t) . Lemma 12.15 Let U and V be open subsets of a smooth manifold. If U, V and UnV have.finite dimensional de Rham cohomology, x(U

the same is truefor Uu V, and

u V) = x(U) + x(V) - x(U n V).

Proof. We use the long exact Mayer-Vietoris sequence .

.

.

� HP- 1 (U n v)

�

HP(U u V)

� HP(U) G1 HP(V)

�

HP(U n V)

�

.

.

.

First we conclude that dim HP(U u V) < oo. Second, the altematirlg sum of the cf. Exercise dimensions of the vector spaces in an exact sequence is equal to 4.4. 0

zef; /_ -

Theorem 12.16 If f is a Morse function on the compact manif old Mn, then x(Mn)

n

= L ( - 1) ,\ C

..\=0

,\ ,

where c..\ denotes the number of critical points for f of index >.. < ak be the critical values. Choose real numbers bo < a1, bj E (aj, ai+d for 1 :S j :S k - 1 and bk > ak· Lemma 12.12 shows that the dimensions of Hd( M(bj)) are independent of the choice of bj from the relevant interval. If M(bj-1) has finite-dimensional de Rham cohomology, the same will be true for M(bj) according to Proposition 12.14, and Proof. Let

(14)

a1

< a2 < . . .
0 } - # { p E s I N (p) = P±' [( (p) < 0} . ===

Since P+ is a regular value for N, we have by Theorem 1 1 .9

1

# { p E N- 1 (P+ ) I K(p) > 0} - # { p E N (p+ ) I K(p) and analogously with P- instead of P+· It follows that

deg(N)

=

-

x(S)

(15)

=


. . . ' Xn ) E sn � Rn+ l , and real numbers ai. The differential

Example 12.20

(Morse function on i.e. to even functions

=

of f at

x

is given by

n dxf(vo, . . . , vn) = 2 L aiXiVi, i=O where v = (vo, VI, . . . , vn ) E TxSn so that n L XiVi = 0. i=O

x is a critical point for f if and only if the vectors x and (a0xo, a1 x1, . . . , anxn ) are linearly independent. If the coefficients ai are distinct, this occurs precisely for x = ±ei = (0, . . , ± 1, . . , 0), and f has ex­ actly 2n + 2 critical points. The induced smooth map j: RPn - R then has n + 1 critical points [ei]· In a neighborhood of ± eo E sn we have the charts h with Thus

.

.

h±1(u1, . . . , u,) = (±VI - 2:::1 u; , and in a neighborhood of 0

).

n

UJ. . . . , U

E Rn,

The matrix of the second-order partial derivatives for f o matrix

h;±1 (at 0) is the diagonal

diag(2(al - ao), 2(a2 - ao), . . . , 2(an - ao)). Hence ±eo are non-degenerate critical points for f; the index for each is equal to the number of indices i with 1 :::; i :::; n and � < ao. An analogous result holds for the other critical points ±ej. For simplicity, suppose that ao < a1 < a2 < . . . < an . Then the two critical points ±ej for f: sn R have index j. The induced function /: RPn - R is a Morse function with critical points [ej] of index j. We apply Theorem 12.16 to

j.

-

Since

cA

=

1

if n i s even if n is odd. This agrees with Example 9.3 1.

for 0 :::; >. :::; n, we get

127 13.

POINCARE DUALITY

Given a compact oriented smooth manifold is the statement that

Mn of dimension n, Poincare duality

(1) where Hn-p(M)* denotes the dual vector space of linear forms on Hn-P (M). The proof we give below is based upon induction over an open cover of M. Thus we need a generalization of ( I ) to oriented manifolds that are not necessarily compact. The general statement we will prove is that

(2) where the subscript

c

refers to de Rham cohomology with compact support.

For a smooth manifold M we let 0.�(M) be the subcomplex of the de Rham complex that in degree p consists of the vector space 0.� ( M) of p-forms with compact support. The cohomology groups of (0.�(M), d) are denoted H�(M), i.e.

Ker(d: 0.�(M) -+ n�+ 1 (M)) Im(d: n�- 1 (M) -+ n�(M)) · Note that when M is compact 0.�(M) = 0.*(M), so that H;(M) HP (M ) c

=

this case.

=

H*(M)

in

The vector spaces H�(M) are not in general a (contravariant) functor on the category of all smooth maps. However, if c.p: M -+ N is proper, i.e. if c.p-l ( K) is compact whenever K is, then the induced form c.p* (w) will have compact support when w has. Indeed

and c.p* becomes a chain map from 0.�(N) to 0.�(M). Hence by Lemma 4.3 there is an induced map

Hf (c.p): Hg(N) -+ Hf(M) and Hg(-) becomes a contravariant functor on the category of smooth manifolds and smooth proper maps. A diffeomorphism is proper, so Hf(M) � Hf(N) when M and N are diffeomorphic.

13.

l28

POINCARE DUALITY

Remarks 13.1

(i) The vector space H2(M) consists of the locally constant functions f: M - R with compact support. Such an f must be identically zero on every non-compact connected component of M. In particular H2(M) = 0 for non-compact connected M. In contrast H0(M) R for such a man­ ifold. (ii) If Mn is connected, oriented and n-dimensionaJ, then we have the iso­ morphism from Theorem 10.1 3, =

f

JM

: H�(M) � R

whereas by (i) and (2), Hn(M) Lemma 13.2

Hq(Rn) c

=

=

0 if M is non-compact.

{R

if q = n 0 otherwise.

The above remarks give the result for q = 0 and q = n, so we may assume that 0 < q < n. We identify Rn with sn - {Po}, e.g. by stereographic projection, and can thus instead prove that Proof.

Now the chain complex rl� ( sn - {Po}) is the subcomplex of rl* ( sn) consisting of differential forms which vanish in a neighborhood of p0. Let w E ng(sn - {Po}) be a closed form. Since Hq(Sn) 0, by Example 9.29, w is exact in rl*(Sn), so there is a T E nq-l(Sn) with &r w. We must show that T can be chosen to van.ish in a neighborhood of PO· Suppose W is an open neighborhood of {p0}, diffeomorphic to Rn, where w l w = 0. If q == 1, then T is a function on sn that is constant on W, say T l w = a. But then K = T - a E S1� (Sn - {Po}) and dK = W. If 2 :::; q < n then we use that Hq- l ( W) � Hq- 1(Rn) = 0, and that T l w is a closed form, to find a CJ E nq- 2 ( W) with dCJ = T l w. Now choose a smooth function .u1r

D.,�

T,p cpn

Multiplication by >. can be considered as an R-linear map cn+ I � cn+l, and l. Dv

0 such that every homomorphism g: � - 1J that satisfies llfb - 9bll < f for b E B is also an isomorphism.

Proof. If � and 1J are trivial, then after choice of frames,

f

and

g

are represented

ad { ] ) : B -+ GLn(R) and ad(g): B -+ Mn(R). Since B is compact and GLn(R) is open, some f-neighborhood of ad(})(B) in Mn(R) is still contained in GLn(R). But then ad(g)(B) C GLn(R) when g satisfies the condition of the

by maps

lemma. In general, we can cover B with a finite number of compact neighborhoods

152

15.

FmER BUNDLES AND VECTOR BUNDLES

over which the bundles are trivial, and take the minimum of the resulting epsilons.

0

Given two smooth vector bundles � and 17, one might wonder if there is any essential distinction between the notions of continuous and smooth isomorphism. The next result shows that this is not the case. Along the same lines one may ask if each isomorphism class of continuous vector bundles over a compact manifold contains a smooth representative. This is indeed the case (cf. Exercise 1 5 .8). Lemma 15.17 If two smooth vector bundles � and 17 over the compact manifold

B are isomorphic as continuous bundles, then they are smoothly isomorphic.

..

U1, . . . , ur of B and smooth local orthonormal frames = (ti, . . . , t�) for � and 1'}, over Ui .

Proof. We choose a cover

si =

(si, . , s�) and

ti

A continuous isomorphism

/: �

-+

17

gives continuous maps

Let Gi : Ui -+ GLn(R) be a smooth €-approximation with IIGi(x) - ad(]�) II < E for x E Ui . Construct a smooth homomorphism gi : 1ri1 (Ui ) -+ 7r;j 1 (Ui ) with ad(gi) = d by the formula

9t(L Aksk (b) ) = L tk (b) Glv(b) · Av. ·

k,v

Then ad(gi) = Gi, and llfb - 9t ll < E for partition of unity {ai};=l with supp(ai) c

b E Ui . We can then use a smooth

Ui r

to define

g: � -+ 1'}, 9b = L ai(b) g� i=l

.

Then II !� - 9bll = ll!b - Eai(b)9tll � Eai(b)llfb - 9�11 S Eai(b)E = €. With € as in Lemma 15.16, g becomes an isomorphism on every fiber and hence a smooth isomorphism by Lemma 15.10. 0

In Examples

15.5 and 15.6 we constructed two vector bundles associated to a submanifold Mn c Rn+k, namely the tangent bundle T and the normal bundle v. It is obvious that r El7 v is a trivial vector bundle. Indeed, by construction Tp $ 1/p = Rn+ k for every p E M, and there is a globally defined frame for r $ v. Hence r ED v � c: n+k where c:n+k is the trivial bundle over M of dimension n + k. We now give the general construction of complements of vector bundles.

,

l5o

153

FIBER BUNDLES AND VECTOR BUNDLES

Theorem 15.18 Every vector bundle � over a compact base space B has a complement rt, i.e. � EB rt � eN (jor a suitably large N). Proof.

Choose an open cover

U1 ,

0

0

, U'o

B

of

admitting trivializations

�I U•· and let {ai} be a partition of unity with supp(ai) c 0

Ui o

hi

Denote by

of

fi

the composite

and define

(2)

S: E(O -+ B x Rnro; S(v) = (n� (v ) , a1 ( n�(v) )l(v), 0 , o:r (n� (v)) /" ( v)) 0 fiberwise map and gives a homomorphism S: � e:nr 0

This is a

inclusion on each fiber. We give

Rnr-

.

-+

which is an

the usual inner product and let

It is easy to see that

1] =

(E(rt), B, Rnr-n, proh )

is a vector bundle (cf. Example If� in Theorem

1 506) and by definition ( Efl rt =

0

e:nro

15.18 is smooth then so is the constructed complement 17, provided

hi and ai are choosen smooth. The above proof uses that B is compact to ensure r < oo. The theorem is not true without the compactness condition; see however Exercise

1 5 . 1 0 when � is a smooth vector bundle - it is the finite-dimensionality

which counts. Let

Vectn(B)

denote the isomorphism classes of vector bundles over

dimension n. Direct sum induces a map

Vectn(B)

x

B

of

Vectm (B) � Vectn+m(B)

such that

Vect(B)

=

II Vectn(B) 00

n==O

becomes an abelian semigroupo The zero dimensional bundle the unit element.

e:0 = B

x

{0} is

To any abelian semigroup (V, +) one can associate an abelian group (K(V),

a - b. or pairs (a, b), subject to the relation (a + x) - (b + x) = a - b

defined as the formal differences

+)

154

15.

FIBER BUNDLES AND VECTOR BUNDLES

where x E V is arbitrary. The construction has the universal property that any homomorphism from V to an abelian group A factors over K(V), i.e. is induced from a homomorphism from K( V) to A. The construction V -+ K ( V) , often called the Grothendieck construction, corresponds to the way the integers are constructed from the natural numbers, except that we do not demand that cancellation "x + a = y + a ::::} x = y" holds in V. When B is compact, we define

KO(B)

(3)

=

K(Vect(B)).

By Theorem 15.18 every element of KO(B) has the form [�J - [skJ, where [�] denotes the isomorphism class of the vector bundle f�j. Indeed

if we choose

772

to be a complement to 6.

Example 15.19 The normal bundle to the unit sphere S2 � R3 is trivial, since the outward directed unit normal vector defines a global frame. We also know that rsz $ vsz = s3, such that

in Vect(S2 ) . However, [rsz] =!= [s2] in Vect ( S2) . Indeed, if [rsz] were equal to [s2], then there would exist a section s E f(rsz) with s(x) =!= 0 for all x E S2 . However, Theorem 7.3 implies that rsz does not have a non-zero section. We see that cancellation does not hold in Vect(S2). Definition 15.20 Let f: X -+ B be a continuous (smooth) map and � a (smooth) vector bundle over B. The pre-image or pull-back !*(0 is the vector bundle over X given by

E(f*(O) = { (x , v) E X x E(Oif(x) = 1r�(v) } ,

1l"J·(�) = proh.

We note the homomorphism (!, ]): !*(�) -+ � given by ](x, v) = v. It is obvious that the pull-backs of isomorphic bundles are isomorphic, so f* induces homomorphisms J*: Vect(B)

-+

and (g o f) * = f* o g*, id* functors.

Vect(X) and j*: KO(B)

=

-+

KO(X),

id. Thus Vect(B) and KO(B) become contravariant

15.

FIBER BUNDLES

AND VECTOR BUNDLES

155

Theorem 15.21 If fo and h are homotopic maps, then j0(0 and fi(�) are isomorphic. Proof. Let F: X x I --+ B, I = [0, 1] , be a homotopy between fo and JI, fo(x) = F(x, 0) and fl(x) = F(x, 1). When t E I we get [it*(OJ E Vect(X). It is sufficient to see that the function t --+ (ft(�)) is locally constant, and thus constant. Fix t and consider the bundles

(

==

proji ft (O

and

over X x I. Since F = ft o proh on choose a fiberwise isomorphism

X

x

7J = F"(� ) {t}, (

= 11 on X x

{t}. We can

___!!:_ E ( 7J)

E(()

"'-.

X

X

/

{t}

The first step is to extend h to a homomorphism of vector bundles on X x [t t:, t + t:] for some t: > 0. This can be done as follows. Since X is compact, there exists a finite cover of X with Ei > 0 such that both ( and 1J are trivial on Ui x [t - Ei, t + ei J . We can extend h to

U1, . . . , Ur

E (()

�X

Ui Let

0:1: . . .

, O:r

/

[t - Ei, t + fi]

be a partition of unity on X with

E(()

E(1J)

�

/

supp(o:i) C Ui.

We define

E(7J)

X x [t - t + eJ where

f.

e,

= min (Ei ) by setting

k(v) = L o:i(proj1 o 1r((v)) hi(v) . ·

Since hi(v) = h(v) when 1r((v) E X x {t}, and since I: O:i(x) = 1 , we have k(v) = h(v) on X x {t}. In particular k is an isomorphism on X x {t}. We finally show that k is an isomorphism in a neighborhood X x [t - e1, t + e1 ] of X x { t}. Since X is compact, it suffices to show that k is an isomorphism on

15.

156

FmER BUNDLES AND VECTOR BUNDLES

a neighborhood V(x, t) of any point (x, t) E X x {t}. Let e and s be frames of ( and r7 in a neighborhood W of (x, t), and ad(k): W ---+ !Vln(R) the resulting map, cf. ( I ). Since GLn(R) C Mn(R) is open and ad (k)(x, t) E GLn(R), there exists a neighborhood V(x, t) where ad (k ) E GLn(R), and k is an isomorphism. 0 The above theorem expresses that Vect(X) (and hence also KO(X)) is a homo­ -:::: g: X ---+ Y induce the same map

topy functor: homotopic maps f j* Corollary 15.22

=

g*: Vect(Y)

f(B) =

{b}.

Vect(X).

Every vector bundle over a contractible base space is trivial.

Proof. With our assumption idB w��

---+

Hence /*(0

�

f , where f is the constant map with value �· But /*(�) is trivial by construction when f is -::::

0

In the above we have concentrated on real vector bundles. There is a completely analogous notion of complex (or even quaternion) vector bundles. In Definition 15.4 one simply requires V and 1r-1(x) to be complex vector spaces and h(x , - ) to be a complex isomorphism. The direct sum of complex vector bundles is a complex vector bundle. A hennitian inner product on a complex vector bundle is a map ¢ as in Definition 15.15 but such that it induces a hennitian inner product in each fiber. Proposition 15.13 and Theorem 15.18 and 15.21 have obvious analogues for complex vector bundles. The isomorphism class of complex vector bundles over B of complex dimension n is denoted Vect�(B). These sets give rise to a semigroup whose corresponding group (for compact B) is traditionally denoted

(4)

K(B)

=

K(Vectc(B)).

It is a contravariant homotopy functor of B, often somewhat easier to calculate than its real analogue KO(B).

157

16.

OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS

The main operations to be considered are tensor products and exterior products. We begin with a description of these operations on vector spaces, then apply them fiberwise to vector bundles, and end with the relation between the constructions on bundles and their equivalent constructions on spaces of sections. Let R be a unital commutative ring and let V and W be R-modules. In the simplest applications R = R or C and V and W are R-vector spaces, but we present the definitions in the general setting. Denote by R[V x W} the free R-module with basis the set V x W, i.e. the space of maps from the set V x W to R that are zero except for a finite number of points in V x W. In R[V x W], we consider the submodule R(V, W) which is generated (via finite linear combinations) by elements of the form

(1)

where

Vi

E V, Wi

(v1 + v2, w) - (vi, w) - (v2 , w) (v,w1 + w2) - (v,wi) - (v,w2) (rv, w) - r(v, w) (v,rw) - r(v,w) E lV and r E R.

tensor product V 0 R W of two R-modules is the quotient W]j R(V, W).

Definition 16.1 The

module R[V x

Let ?T : R[V x W] ---+ V 0R W be the canonical projection and write v0RW for the image of (v,w) E R[V x W]. It is clear from ( 1 ) that ?T : V x W - V 0R W is R-bilinear. Moreover, it is universal with this property in the following sense:

Let V, W and U be R-modules, and let f: V x W U be any R-bilinear map. Then there exists a unique R-linear map j: V 0R W ---+ U, with f J 0 ?f.

Lemma 16.2

-+

=

W is a basis for the R-module R[V x W], f extends to an R-linear map ]: R[V x W] ---+ U. The bilinearity of f implies that 0, so that j induces a map 1 from the quotient V 0R W to j(R(V, W)) U. By construction f J o 1r, J is R-linear and since ( V x W) generates the 0 R-module V ®R W, 1 is uniquely determined by f.

Proof. Since the set V =

x

=

?T

It is immediate from Lemma 16.2 that tensor product is a functor. Indeed, if ) + ( - 1 )i ( s) dV')

E ni(Hom(�, ry)).

178

17.

where d9 corresponds to

CONNECI'IONS ANl> CURVATURE

v

= Ve, Vr" Ve- and vHom(e .e·)• respectively.

0

The definitions above may appear somewhat abstract, so let us state in local coordinates the case of most importance for our later use. Let \7 = \7e be a connection on �. and let e = (et, . . . , ek) be a frame over U. This defines isomorphisms Hom{�, �)lu � U x i\tfk(R) and induces

The connections \7 = become

They

are

(17)

\le

and

�=

'VHom({

L. From (ii) it

Ct(L) :::: 1r• ( c) .

Since ck(H...) = 0 when k > 1, the same holds for any line bundle. Therefore (i) and (ii) determine the Chern classes of an arbitrary line bundle. Inductive application of (iii) shows that for a sum of line bundles, ck (L1 Efl . . EB L..,) is determined by c1 (LI) , . . . , ct(L..,). Finally we can apply Theorem 18.10 to see that Ck(0 is uniquely determined for every complex vector bundle. 0 .

The graded class, called the total Chern class, (12)

is exponential by Theorem l8.9.(iii), and c(L) = 1 + c1(L) for a line bundle. Hence k c(L1 Ef) Ef) Lk) = IT (1 + Ct(Lv)) ::= L O"i(CI(Lt), . . . , Ct(Lk ) ) v=l and it follows that Ci(Lt EB . . EB Lk) = o-i(c1 (Ll ), . . . , Ct(Lk )). We have addi­ tional calculational rules for Chern classes: . • .

.

18.

189

CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDLES

Properties 18.11

(a) ck (O = 0 if k > dim � k k (b) ck ( C ) = ( - 1 ) ck(� ) . chk(C) = ( -l) chk(s ) (c) C2k+I(7Jc) = 0 and ch2k+1(7Jc) = 0 for a real vector bundle

7].

Proof. For a line bundle. (a) follows from assertions (i) and (ii) of Theorem 18.9. because every line bundle � has the form 7r*(Hn)· If s = 'Yl $ . . . $ 'Yn is a sum

of line bundles. then

c(�)

=

n ( 1 + cl(Tj))

and it follows that ck(�) = 0 when k > n. For an arbitrary � we can apply Theorem 18.10. The proof of (b) is analogous: if dime� = 1 then C ® � = Hom(�,�) is trivial and Theorem 18.7 gives that ch1(�*) +ch1(�) = 0, hence c1(�*) = -c1(0. For a sum of line bundles, (13) This shows that ck (�*) = ( -l) k ck(�), and the splitting principle implies (b) in general. For a real vector bundle 7], 77* � 7], as we can choose a metric ( , ) on ry and use the isomorphism a : ry - Hom(ry , R) ;

a( v) =

(u , -).

Then (nc)* = (n*)c so that Ck(77�) = c�c(nc ). Now (c) follows from (b). Note that (c) implies that ch(7Jc) is a graded cohomology class, which can only be non-zero in the dimensions 4k. One defines Pontryagin classes and Pontryagin character classes for real vector bundles by the equations: (14) We leave to the reader to check that the total Pontryagin class p(TJ) is exponential.

= 1+p1 (77)+· · ·

Remark 18.12 Definition 18.3 gives cohomology classes in H*(M; C), but

actually all classes lie in real cohomology. This follows from Theorem 18.9.(i) for Hn, and for a sum of line bundles from (ii) and (iii). The general case is a consequence of Theorem 18. 10. Theorem 18.8 actually gives isomorphisms ch: K(M) ®z R � H2*(M) ph: J s2) where {s1. s2) is the function that maps p E j\1 to {s1 (p), s2(p)) and w1,w2 E S1*(M). < , ): !"i(�) 0 ni (�)

�

=

Definition 19.1 A connection \J on (�,

{ , ) ) is said to be metric or orthogonal if

A associated to an orthonormal frame. Let E S1°(0 be sections over U, so that e1 (p), . . . , ek(P) forms an orthonormal basis of� for p E U. Let A be the associated We express this condition locally in terms of the connection form

e1. . . . , ek

connection form,

For every pair

(i, k)

metric one gets

0

= =

\J(ei) = L Aij ® ei . we have (ei, ek ) = 8ik (on U),

so

d(ei, ek)

=

If \J is

0.

(EAij 0 ej, ek) + (ei, EAkj 0 ej) "'£i Akj(ei, ej) = Aik + Aki· "'£i Aij(ej, ek) +

Thus the connection matrix with respect to an orthonormal frame is

symmetric. frame, then Let

F9 E

S1

If conversely

\J

is metric.

A

skew­

is skew-symmetric with respect to an orthonormal

2( Hom (C �)) be the curvature form associated to a metric connection.

After choice of an orthonormal frame for �IU , S12 ( Hom (�, Ow)

�

Mzk(S12 (U)).

In ( 17 .10) the corresponding matrix of 2-forms

F9(e) where

=

F9 (e)

was calculated to

be

dA - A 1\ A

A is the connection form associated to e.

In particular,

F9 (e)

is skew­

symmetric, and we can apply the Pfaffian polynomial from Appendix B to to get

(1)

F9 (e)

194

19.

In another orthonormal frame

e'

THE EULER CLASS

over

U

(2) where

Bp

is the orthogonal transisition matrix between e(p) and e'(p).

Now suppose further that the vector bundle � is oriented. and that are oriented orthonormal bases for �P• p

E U.

Then Bp

e(p) and e'(p)

E S0 2b and by Theorem

B.5, (3 ) It follows that of Lemma

Pf(Fv)

18.1

becomes a well-defined global 2k-form on

shows that

Pf(Fv)

M.

The proof

is a closed 2k-form.

We must verify that its cohomology class is independent of the choice of metric on � and of the metric connection. First note that connections can be glued together

(V'a) aEA is a family of connections on e and (Pa)aEA M, then \Js = E Per \Jas defines a connection if each \}a is a metric for g = ( , } then \} = E Pa\}a is

by a partition of unity: if

is a smooth partition of unity on on (. Furthermore, also metric.

Indeed, if

(4) then

(\JSt, s2} + (s1, vsz} = L (POt.\Jo:Sl,sz} + L (sl , Pa'Vasz )

L P01.( (V01.sb sz) + (sr, \101.sz)) = L .Oad(sl, sz) d(s1, sz). =

==

In this calculation we have only used and not neccesarily on all of

M.

(4) over open sets that contain suppM(Po:).

This will be used

in the proof of Lemma 19.2

below. Consider the maps

with iv(x)

i�(�) =

=

(x, v)

� for 11 =

Lemma

and 1r(x,

0, 1

t) = x,

and we have:

and let �

=

1r*(�) over

M

x

R.

Then

19.2 For any choice of inner products and metric connections g11, \111 ( 11 = Q, 1) on the smooth real vec!, _or bundle � over M, there is an inner product g on � and a metric connection \7 compatible with g such that i�(9) = g11 and

i�(v) = 'Vv·

19.

THE EULER CLASS

195

Proof. We can pull back by n* the metric 9v and the metric connections \Jv

( Let {Po, Pl} be a partition of unity on M x R subordinate to the cover M x ( -oo, 3/4) and M x (1/4, oo). Then g = pon*(go) + p1n*(g1) is a metric on { which agrees with n* (go ) over M x ( -oo, 1/4) and with n*(g1) on M x (3/4, oo) . In particular i�('g) = 9v· to

Let \J be any metric connection on � compatible with g. We have connections ?r*(\Jo), v and 1r*(71) compatible with g over M x ( -oo x 1/4), M x (1/8, 7/8) and M x (3/4, oo) respectively. We use a partition of unity, subordinate_!o this cover, to glue together the three connections to const�ct a connection \7 over 0 M x R. This is metric w.r.t. g, and by construction, i� \7 = \7v · -

-

Corollary 19.3 The cohomology class [Pf(F9)] E the metric and the compatible metric connection.

H2k(M)

is independent of

Proof. Let (g0, 'Vo) and (g1 , 'VI ) be two different choices an_:! let (g, v) be

the metric and connection of the previous lemma. Then i� (Fv) = Fv and _ hence i�Pf (Fv) = Pf(F'1v ). The maps io and i1 are homotopic, so i0 = ii: Hn(M x R) ._ Hn (M). Thus the cohomology classes of Pf (Fv0) and Pf (Fv1 ) agree. 0 v,

Definition 19.4 The cohomology class

is called the Euler class of the oriented real 2k-dimensional vector bundle �. Example 19.5 Suppose

M is an oriented surface with Riemannian metric and

that � = T* � TM is the cotangent bundle. Let e1, e2 be an oriented orthonormal frame for D0(T1u) = D1(U), such that e1 1\ e2 = vol on U. Let a1, a2 be the smooth functions on U determined by

and let A12 = a1e1 + a2e2. We give

A=

T!U the connection with connection form

0 (-A12

A12 0

)

so that '\J(e1) = A12®e2 and 7(e2) = -A12®e1. This is the so-called Levi-Civita connection; cf. Exercise 19.6. By (17. 10)

Fv

·

=

dA

-

A 1\ A =

(-dA12 0

dA12 0

)

196

THE EULER CLASS

19.

since A12 1\ A1 2 = 0. In this case Pf(FV') = dA12 is called the Gauss-Bonnet form, and the Gaussian curvature "' E S1°(M) is defined by the formula -J.tp + ,\- t (I - P) = ,\-t I + (>.t - ,\-t)P. Observing that each matrix entry in (>.t - ,>.-t)p has numerical value at most ,\ - ,>.- 1 , we see that F extends continuously to Q x [0, 1] by F (I , t) = I. 0 Proposition 20.6

The map

= cpm-1 - RPm- 1 , and consider the smooth map (cf. (?)) 4>: V2(Rm ) X

GLt(R) ----+ 7r- 1 (Wm );

(x, y, (: �))

=

(ax + by) - i(cx + dy).

?r - 1 (Wm ) has the form z = v - iw, where v and w are linearly independent vectors in Rm . The fiber - 1 (z ) is in 1-l correspondence with the orthonormal bases (x, y) of spanR (v, w) which determine the same orientation as (v, w). There is a global smooth section S of , constructed using the Gram-Schmidt orthonormalization process. The fibers of 4> are the orbits of the 30(2)-action, A point in

(x, y, B).Re

so

ci>

= ((x, y).Ro, R0 1 B),

induces a bijection from the set of orbits

This bijection commutes with the C* -actions, if we let C* act on the domain by right multiplication in GLt(R) and on ?r- 1 (Wm ) c em by scalar multiplication.

206

20. COHOMOLOGY OF PROJECTIVE AND GRASSMANNIAN BUNDLES

This gives a commutative diagram where the horizontal maps are bijections

V2 (Rm) X so(2) GLi(R)

- 1r-1(Wm)

l

1

(V2(Rm) X so(2) GLi(R))/C"'- W m � Lemma 20.4 gives a bijection Q GLi(R)/C*, so we may

identify the lower

left-hand corner of the diagram with the quotient space

X= for the S0(2)-action on

V2(Rm)

V2(Rm ) Xso(2} Q x

Q defined by

(x, y, A).Ro = ((x, y).Ro, R01ARo).

Altogether we obtain a bijection 1] (i E J), satisfying Vi, then there exist smooth functions (Pi: U (i) SUPPu(¢i) � Vi for all i E I. (ii) Every x E U has a neighborhood W on which only finitely many ¢i do not vanish. (iii) For every x E U we have I: tPi(x) 1

Theorem A.l

-+

iEl

::::=

We say that (¢i) iEI is a (smooth) partition of unity, which is subordinate to the cover V. A family of functions f/>i: U R that satisfy (ii) is called locally finite. Note that the sum l::ieJ ¢i in this case becomes a well-defined function U R. Moreover, it is smooth when all the ¢i are smooth. The proof of Theorem A. l requires some preparations. -+

Lemma A.2

The function w: R w(t) =

-+

R defined by 0 if t

{ exp(-1/t)

-+

o

is smooth. Proof. It is only smoothness at t = 0 which causes difficulties. It is sufficient to see that

.

hmt_.o+

for all for t

n

wPl,P2> . . . , such that

> 0

w(n)(t) = Pn (1/t) The result now follows because

exp(-1/t),

and

n

2: 0

.

limt-+O+ ( for k 2: 0.

(1/t)kexp(-1/t)) = limx-+oo expx ( k

x

)

=0 0

222

A. SMOOTH PARTITION OF UNITY

Corollary A.J

such that 'l{;(t) Proof.

For real numbers a < b there exists a smoothfunction 1/J: R = 0 for t � a and 'l{;(t) 1 for t 2: b.

-. [0, 1]

=

Set '!{;(t) = w (t

- a)f(w(t - a) + w(b - t)).

For x E R and t > 0 let Dt(x)

=

{y E Rn l ll y - x ll
0 there exists a smooth function ¢: Rn such that D£(x) = -1 ((0 , oo) ). [O,oo), Corollary

Proof.

._

Define by the formula

n 2 L (Yj - Xj) . 0, and thus A = ¢- 1 (0). 0 =

=

Lemma A.9 Suppose that A � Uo � U � Rn, where Uo and U are open in Rn and A is closed in U (in the induced topology from Rn). Let h: U - W be m a continuous map to an open set W � R with smooth restriction to U0. For any continuous function c U - (0, oo ) there exists a smooth map f: U � W that satisfies � ((x) for all x (ii) f(x) = h(x) for all x E A.

(i)

Proof.

llf (x) - h(x)ll

If W

E

U.

=j:: Rm then ((x) can be replaced by

t1 (x) = min (t (x) , !d(h(x), Rn - W) )

where d(y, Rn - W) inf{IJy - zll l z E Rm - W}. If f: U - Rm satisfies (i) with q instead of t, we will automatically get j(U) � W. Hence, without loss of generality, we may assume that W R m . Using continuity of h and we can find for each point p E U A an open set Up with p E Up � U - A, such that Jlh(x) - h(p)JI � e(x) for all x E Up. Apply =

=

e,

-

A. SMOOTH PARTITION OF UNITY

225

Theorem A. I to the open cover of U consisting of the sets Uo and Up. p E U - A. This yields smooth functions 2t: and (b) implies that J-t(� + 2rt) (i). A simple calculation gives: (3)

oF oxi

=

{

2Xi ( - 1 - J-t1 (� + 27])) 2xi(1 - 2p.'(� + 217))

if 1 � i � A if A + 1 ::; i s

"'-,

'·

_)

\.

-1

2E

= 0.

'•

2€

This gives

n.

By (a) we have

(4)

- 1 - J-t1(� + 2rt) < 0, 1 - 2J.£'(� + 2rt) > 0.

It follows from (3) and (4) that 0 is the only critical point of F. By (c), F coincides with f - J.£(0) on a neighborhood of 0 where J.£(0) > c. This shows that F is a Morse function and that (ii) is satisfied. Since J.£( t) 2:: 0 by (a) and (b), the fonnulas ( 1 ) and (2) show that F( x) � f (x) for all x E W. Hence

(5)

f- 1 (( - oo , a + t:)) � F- 1 (( - oo, a + t:)) f- 1 (( -oo, a - c)) � F- 1 (( -oo, a - F.)).

If (iii) were false, there would exist an x E vV with F(x) < a+t: and f(x) 2:: a+ c. By Equations ( 1 ), (2) and (b) we conclude that � + 27] < 2t:. This implies 1J < t:, which contradicts ( 1 ), because f(x) s a+17 < a+t:. This proves (iii). Analogously one sees that (iv) is satisfied for the open set U defined by

(6)

U = {x E W I � + 2rJ < 2t: and F(x) < a - c}.

We shall show that U is contractible. By (ii) we have 0 E U. Let i E U be a point of the fonn x = (x1, . . . , X,A, 0, . . . , 0) and let -.+ 1

The coordinate patches W1 (1 � i :S r) can be assumed to be mutually disjoint. Choose E > 0 such that a is the only critical value in [a- E , a+ c] and such that all Wi contain the closed ball J2ED11• We apply Lemma C. I to f o h.j 1 : Wi ---+ IR and this E. We obtain new Morse functions Fi : Wi ---. R with 1 :S i :S r, and an open set Ui � Wi such that hi(Ui) is the open contractible subset of Rn given by Lemma C. I . Thus we have satisfied Lemma J 2.1 3.(i) and (ii), and Lemma 1 2. 1 3.(iii) follows from assertion (v) of Lemma C. I . By assertion (i) we get a Morse function

F: M ---. R

if q E lVi if q � u�=l wi

C. PROOF OF LEM.'\-IAS 12.12 AND 12.13

237

and by Lemma C.l .(iii) and (iv), F- 1 (( - oo , a + c)) = M(a + c) F-1 (( - oo , a - c)) = M(a - E) u ul

u .. . u Ur.

We know from Lemma C . l .(ii) that F has the same critical points as f and furthermore that F(pi) < a - c (1 � i � ·r) . If p is one of the other critical points, then F(p) = J(p) � [a - c, a + cJ, and hence [a - c, a + c] does not contain any critical value of F. Hence assertion (iv) of Lemma 12.13 follows from Lemma 1 2 . 1 2 applied to F. 0 Lemma 1 2 . 1 2 is a consequence of the following theorem, which will be proved later in this appendix. Let Nn be a smooth manifold of dimension n 2: 1 and f: N - R a smooth function without any critical points. Let J be an open interval J � R with f(N) � J and such that f -1 ([a, b]) is compact for every bounded closed interval [a, b] c J. There exists a compact smooth (n - I)-dimensional manifold Qn-l and a diffeomorphism Theorem C.2

: Q X

such that f

o

: Q x J .-

J-N

J is the projection onto J.

Choose c1 < a1 and Cz > az, so that the open interval J = ( c , c2) does not contain any critical values of f. Since M is compact, 1 we can apply Theorem C.2 to N = f -1(J). We thus have a compact smooth manifold Q and a diffeomorphism : Q x J .- N such that f o (q, t) = t for q E Q, t E J. Consider a strictly increasing diffeomorphism p: J - J, which is the identity map outside of a closed bounded subinterval of J. Via p we can construct the diffeomorphism

Proof of Lemma 12.12.

Wp (p)

=

{: o (idQ

X p)

o q,- l (p)

if p E N if p � N.

a E J then wP maps JVJ (a) diffeomorphically onto M (p(a)). It suffices to choose p so that p(a1) = az. One may choose If

(7)

where

g

p(t) = t +

jtg(x)dx, Cl

E C�(R, R) satisfies the conditions:

supp(g) � J,

g(x)

> - 1 for

x

E R,

1°c 1 g(x)dx = 1

az - a1,

Jc1t2g(x)dx

= 0.

C. PROOF OF LEMMAS 12.12 AND 12.13

238

Now g can be constructed easily via Corollary A.4. See Figure

D

3 below.

,.1

I I

I

. \ i I

I

I

c,

·1

Figure 3

Remark C.3 In the proof above, we proved more than claimed in Lemma 1 2 . 1 2 . Indeed, we found a diffeomorphism IllP of

Ill

Let p5:

M

to itself for which

p ( M (al) ) = M (a2).

J --t J be the map Ps(t) = sp(t) + (1 - s)t, where s

E

[0, 1].

Then

W p, (P)

is smooth as a function of both s and p. Moreover, every IllP• is a diffeomorphism of

M

onto itself. This gives a so-called

isotopy from IllPo = idM

to Illp1 = Illp·

It remains to prove Theorem C.2. We first prove a few lemmas.

Lemma C.4 There exists a smooth tangent vector field X on Nn such that

dpf(X(p))

=

1 for all p E

M.

Proof. We use Lemma 12.8 to find a gradient-like vector field

p(p) = dpf(Y(p)) > 0

and we can choose

We shall investigate the integral curves open intervals

IcR

(8)

and satisfy

X(p) = p(p)- Y(p).

a: I

--t

N

for

X.

ftf a(t) = 1, o

Y

on

M.

Now

0

They are smooth on

o/(t) = X(a(t)).

By Lemma C.4 and (8) we have for a constant c.

1

which gives

f o a(t) = t + c

C.

PROOF OF LEMMAS

Lemma C.S Assume Po E N with

12.12

AND

12.13

239

f (Po) = to E J. Then

(i) /- 1 (to) is a compact (n - I)-dimensional smooth submanifold of N. (ii) There exists an open neighborhood Wo � f- 1 (to) of Po, a 6 > 0 with

(to -

6, to

+ 6) � J and a diffeomorphism

. :::; n) that either C.A-t = 0 or C.A = 0. Prove that bj = Cj for every j. 12.9. Let pr( Tn = Rn;zn ---+ R/Z be the i-th projection (see Exercise 8.4). Pick w E n1 (R/Z) representing a generator of H1 (R/Z) rv H1 (S1) � 1R and define Wi = pr;(w) E n1 (Tn). To an increasing sequence I: 1 :s; it < i2 < .. . < ip � n we associate the closed p-forrn

Prove that the resulting classes [wrJ E HP(Tn) are linearly independent. (Hint: Consider integrals of linear combinations L arwr over subtori.) I Prove with the help of Exercises 12.1 and 12.7 that

and conclude that there is an isomorphism

of graded algebras.

D.

263

EXERCISES

!VJn ---t Mn be a d-fold covering of closed manifolds M and M, i.e. Mn can be covered by open sets U with the property that 1r- 1 (U) is a disjoint union of d open sets UJ, . . , Ud such that 1r1u. : Ui ---t U is a diffeomorphism for 1 :::; i :::; d. Show that x.(M) = dx_(M). 1 2. 1 1 . Assume f and g are Morse functions on the closed manifolds ]'vf and N respectively. Show that a Morse function h on M x N can be defined by

1 2. 1 0. Let

1r:

.

h (p, q) = f(p) + g(q) . Describe the critical points and indices for h in tetms of similar data for f and g. Derive the product formula for Euler characteristics

x. (M X N) = x.(M)x.(N).

13.1.

A symplectic space (V,w) i s a real vector space equipped with an alter­ nating 2-form w E Alt2 ( V ) A linear subspace vV � V is said to be non-degenerate if for every e E W {0} we can find f E W such that w(e, f) =f:. 0. Assume from now on that V is non-degenerate of finite di­ mension. Let W � V be a non-degenerate subspace. Show that .

-

W_L = { x E Vlw(x, y) = 0 for every y E vV}

is a non-degenerate subspace with W $ Wj_ {e1 , JI , e2 , f2, . , en, J.,.. } such that .

= V.

Prove that V has a basis

.

This is called a symplectic basis. Note that dim V must be even. (Hint: Pick e1 =f:. 0 arbitrarily. Then find h and apply induction to W _L, where W is spanned by c 1 and h.) Let w1 , .TJ , w2, T2 , . . . , Wn, r.,.. be the dual basis of Alt 1 (V) = V* to a symplectic basis for V. Show that

n

w = LWj 1\ Tj. j=l

Let Mn be an oriented closed smooth manifold of dimension n = 2 (mod 4). Show that Poincare duality organizes Hnf2 (M) as a non­ degenerate symplectic space (see Exercise 1 3 . 1). Prove that x.(M) is even. 13.3. Consider a smooth map f: Nn ---t Mn between n-dimensional oriented smooth closed manifolds, where M is connected. Prove that if Hn(f) =f:. 0 then HP(j): HP(-M) ---t HP(N) is injective for every p.

1 3.2.

D.

264

EXERCISES

1 3.4. Let (S't, Mm) be a smooth compact manifold pair with 0 < m < n and

U

=

sn -

Afm.

Construct isomorphisms

Show that Hn(U)

=

0 and find short exact sequences

0 ---t Hn- 1 (U) ---t H1\M)* ---t R ---t 0 0 ---t R ---t H0(U) ---t Hn-1(M)* ---t 0. 1 3.5. Let 1r: M ---t M be the oriented double covering constructed in Exercise 9. 1 5 and define A: M ---t M as in Exercise 9.18. Find isomorphisms

1 3 .6.

1 3.7.

13.8.

1 3 .9.

where HZ(M)-t denotes the (-I)-eigenspace of A* on HZ(M). Compute Hn(Mn) for every smooth connected n-dimensional manifold Mn. (Hint: Use Exercise 13.5. The answer depends on whether !v1 is compact or not, and also whether M is orientable or not,) Prove that HZ ( M) for every smooth manifold and every q is at most countably generated. (Hint: Induction on open sets.) Show that any de Rham cohomology space HP(M) is either finite­ dimensional or isomorphic to a product n::::: l iR of countably many copies of IR. A compact set ]( � IRn is said to be cellular if K = n_i= 1 Dj. where each D1 � IRn is homeomorphic to Dn and Dj+l � b.i for every j. Show for K cellular that

HP(!Rn

_

K)

�

{R 0

?•

if p = n- 1 otherwtse.

1 3 . 10. For K � IRn compact denote by fi0(K, R) the vector ;pace of locally

constant functions

f{

---t

IR.

Construct an isomorphism

1 : H0(K, R)

---t

H.}(Rn - K)

such that 1 (f) = [d]] where j E Cgc'(Rn, R) is locally constant on an _ open set containing K and f extends f. Find an isomorphism :

Hn- t (Rn - K)

-+

H0(K, R)*

such that ([w]) for w E nn- 1 (Rn - K ) can be evaluated on f E fJO(K, IR) by the following procedure:

D.

EXERCISES

265

There are disjoint compact domains R 1 , . . . , RG