From calculus to cohomology: de Rham cohomology and characteristic classes

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)• ... , � ... , �u(p))(w2 (\ W3) (�u(p+l)• ... , �u(p+q+r)) uES(p,q+r)

L

sign(O")

uES(p,q+r) =

I;

sign(r)[wl(�u(l)•

. .

·•�u(p))

rES(p,q,r)

W2(�ur(p+l), · · · '�ur(p+q))w3(�ur(p+q+l), · · · , �ur(p+q+,·))]

L [si gn(�L)WJ (�tt(l)• ·

·

·, �u(p))w2(�u(p+l)> · · ·, �tt(p+q))

uES(p,q,r) W3(�1L(p+q+ 1)> · · · > �u(p+q+r))]

where the last equality follows from the first equation in (2). Quite analogously one can calculate [(w1 1\ w2) 1\ w3j (6 , . , �p+q+r ) employing the second equation in (2). 0 .

Remark 2.10

.

,

In other textbooks on alternating functions one can often see the

definition

w1i\w2(6, ... , � p+q)

1

p!q!

2::

uES(p+q)

sign( (J)w1 (�u(l),

. . . •

�u(p))w2 (�u(p+ 1)>... , �u(p+q)).

2.

Note that in this formula

THE ALTERNATING ALGEBRA

II

{ a(1), . . ., a(p) } and { a(p + 1), ... , a(p + q )} are not S(p) x S(q) ways to come from an ordered set to the

ordered. There are exactly

arbitrary sequence above; this causes the factor

, p-;;h .q.

so

WIAw2 = WI

w2.

/\

A consists of a vector space over R and a bilinear map w Ax A A ft(a,/-l(b,c)) = 1-l(l-l(a,b),c) for every a,b,c E A. The algebra is called unitary if there exists a unit element for 1-l· 1-l(1, a) = 1-l( a,1) a for all a E A.

An R-algebra

---*

which is associative,

=

Definition 2.11 (i) A graded R-algebra and bilinear maps

A* is a sequence Ak+t 1-l: Ak x Az ----+

of vector spaces

Ak, k

=

0, 1. . .,

which are associative.

(ii) The algebra A* is called connected if there exists a unit element

1E Ao Ao, given by t: ( r ) = r 1, is an isomorphism. The algebra A* is called (graded) commutative (or anti-commutative), if k 1-l(a,b) = (-1) 1!-l(b,a) for aE Ak and bE Az. and if t:: R

(iii)

The elements in

----+

·

Ak are

said to have degree k. The set

Alt k(V)

is a vector space

over R in the usual manner:

(wi + wz )(6, . . ,�k) = wi(6, ..., �k) + wz(6, ... , �k) (,\w)(6, ... ,�k) =,\w(6, ...,�k), ,\E R .

AltF(V) x Altq(V) to R and expand the product to Alt0(V) x AltP(V)

The product from Definition 2.5 is a bilinear map from

Altp+q(V).

We set

Alt0(V)

=

by using the vector space structure. The basic formal properties of the alternating forms can now be summarized in

Theorem 2.12

Alt*(V)

Alt*(V)

is an anti-commutative and connected graded algebra.O

is called the exterior or alternating algebra associated to

Lemma 2.13 For 1-forms

V.

WI, ... ,wp E Alti(V), WI(�p) wz(�p)

)

(

. . . Wp �p) Proof. The case to Definition 2.5,

p

=

2

is obvious. We proceed by induction on

p.

According

12

THE ALTERNATING ALGEBRA

2.

WI 1\ (w2 1\ ... 1\ Wp)(�l, · ..• �p)

"+1 =� L.) - 1)1 w1(�j)(w21\ ... l\wp) 6 , . . . ,�i····•�P j=l (6, . . . ,�j, ... , �p) denotes the (p- I)-tuple where �j has been omitted.

(

where

)

A

The lemma follows by expanding the detenninant by the first row. Note, from Lemma 2.13, that if the 1-fonns

w1 , ... ,wp E Alt1 (V)

0

are linearly

w1 1\ ... 1\ wp =/: 0. Indeed, we can choose elements �i E Wi(�j) = 0 for i -:/: j and Wj(�j)= 1, so that det(wi(�j)) == 1. Conversely, if WI, ..., wp are linearly dependent, we can express one of them, say wp. as a linear combination of the others. If wp= "L ���riwi, then p-I · WI 1\ · · 1\Wp- 1 1\Wp = 2:: TiWI 1\ · · · 1\ Wp-I 1\Wi = 0, i=I as the determinant in Lemma 2.13 has two equal rows. We have proved independent then

V

with

Lemma 2.14 For 1-fonns w1 . . are linearly independent.

.

.

, Wp on

V, WI 1\... 1\ wp -:/: 0 if and only if they 0

Theorem 2.15 Let e 1, . . . , en be a basis of V and Alt1 (V). Then

E!,

.. . , En the dual basis of

{ Eo-(1) 1\ Eo-(2) 1\... 1\€o-(p)} o-ES(p,n-p ) is a basis of AltP(V). In particular

dimAltP(V) Proof. Since

Ei(e1)

=

(

;V ) .

= 0 when i-:/: j, and Ei(ei) = 1, Lemma if

(3) Here

di

if u

is the pennutation

w

=

u(ik)

L

aES(p,n- p)

= jk.

From Lemma 2.2 and (3) we get

€o-(l) 1\ ... 1\ Ea(p)

generates the vector space

independence follows from (3), since a relation

L aES(p,n- p)

evaluated on

{il,·· .,ip} f.: {jl,···,jp} {i l,· .. ,ip} = {jl,· .. ,jp}

w(eo-(1)• ... ,eo-(p))€a(l) 1\ ... 1\ Eo-(p)

for any alternating p-fonn. Thus

AitP(V). Linear

2.13 gives

Ao- o-( 1) 1\... 1\ Eu(p) = 0, E

(e n. A basis of Altn(V) is given by �:1/\ . .. I\ £11• In particular every alternating n-fonn on Rn is proportional to the form in Example 2.3.

0

A linear map f: V

�

W induces

the linear map

(4)

by setting AltP(J)(w)(6, . . . , �p) = w(J(6), . .. , f(�p)). For the composition of maps we have AltP(g of) = AltP(J) oAltP(g), and AitP(id) = id. These two properties are summarized by saying that AltP (-) is a contravariant functor. If dim V = n and f: V � V is a linear map then is a linear endomorphism of a !-dimensional vector space and thus multiplication by a number d. From Theorem 2.18 below it follows that d = det(f). We shall also be using the other maps AltP(J)� AltP(V)

-4

AltP(V).

Let tr(g) denote the trace of a linear endomorphism Theorem 2.16

g.

The characteristic polynomial of a linear endomorphism f: V -4 V

is given by

det(f- t) = where n

=

dimV.

Proof. Choose a basis vectors of f,

n

L (-1)itr (Altn-i(f)) ti,

i=O

e1, . . . , en

of

V. Assume first

f(ei) = >.iei, Let

and

E1,

that

e1,

. .. , en are eigen­

i = 1, . . . , n.

. . . ,En be the dual basis of Alt1(V). Then AJtP(J) (Eu(l)

1\ · · · 1\ Eu(p))

tr AltP(J)

=

=

Au(l) ... Au(p)Ea(l) 1\ · · · 1\ Eu(p)

L Au(l). · · Au(p)· aES(p,n-p)

On the other hand det(J- t) =

IT(>.i- t) = L (-1t- p(L>.a(l) ... Aa(p))tn-p.

This proves the formula when f is diagonal.

14

2.

THE ALTERNATING ALGEBRA

If f is replaced by gf9-1, with 9 an isomorphism on V, then both sides of the equation of Theorem 2 . 1 6 remain unchanged. This is obvious for the left-hand side and follows for the right-hand side since

by the functor property. Hence tr A ltP (9 of o 9-1) = tr AltP(f). Consider the set = {9f9-1lf diagonal, 9 E GL(V)}.

D

If V is a vector space over C and all maps are complex linear, then is dense in the set of linear endomorphisms on V. We shall not give a formal proof of this, but it fo1lows since every matrix with complex entries can be approximated arbitrarily closely by a matrix for which all roots of the characteristic polynomial are distinct. Since eigenvectors belonging to different eigenvalues are linearly independent, V has a basis consisting of eigenvectors for such a matrix, which then belongs to D

D

For general f E End(V) we can choose a sequence dn E D with dn � f (i.e. the (i,j)-th element in dn converges to the (i,j)-th element in f). Since both sides in the equation we want to prove are continuous, and since the equation holds for 0 dn, it follows for f.

It is not true that the set of diagonalizable matrices over R is dense in the set of matrices over R - a matrix with imaginary eigenvalues cannot be approximated by a matrix of the form 9f9-1, with f a real diagonal matrix. Therefore in the proof of Theorem 2.16 we . must pass to complex linear maps, even if we are mainly interested in real ones.

15

3.

DE RHAM COHOMOLOGY

U will denote an open set in Rn, { e1, ..., en} the standard basis c1, ... , en} the dual basis of Alt 1(Rn). Definition 3.1 A differential p-fonn on U is a smooth map w: U AltP(Rn).

In this chapter and {

The vector space of all such maps is denoted by OP(U).

---+

If p = 0 then l = and D.O is just the vector space of all smooth real-valued functions on = c=(u, is denoted The usual derivative of a smooth map U ---+ and its value at x by It is the linear map

A t0(Rn) R (U) U, Q0(U) R). w: AltP(Rn) Dxw. Dxw: Rn AltP( Rn), ow (x). (Dxw)(ei) = -dd w(x + tei)t::;O �

Dw

---+

with

.

.

=

t

UXi

In AltP (!Rn) we have the basis €£ = €i1 1\ .. . 1\ £ip• where I runs over all sequences can be written in the with 1 � i1 < i2 < ... < ip � n. Hence every E = with smooth real-valued functions of E U. form The differential is the linear map

w flP(U) x w(x) 'L'::wr(x)t1, w1(x) Dxw OW[ . (1) DxW(€j) = """"' � ox . (X)€I , J = 1, . .. , n. J I The function x Dxw is a smooth map from U to the vector space of linear maps from Rn to AltP(Rn). Definition 3.2 The exterior differential d: QP(U) f2P+l(U) is the linear op­ �

---+

erator

[::;}

It follows from Lemma 2.7 that

p+l """"'

dxw E Altp+l (IRn). Indeed, if �i = �i+l· then

Dxw(�t)(�l, · · · , ��.- · · · '�p+l) =( -1)i-1 Dxw(�i)(6, ... , �i, �p+l) + ( -1)iDxw(�i+I)(6, ..., ��+1, ... , �p+l) 1-1

� (-1 ) 1=1

· · · ,

=0

\6

DE RHAM COHOMOLOGY

3.

because (6, ... , �i, ... , �p+ l )

Example 3.3 Let Xi: U

constant map dxi: X (1) shows that

(6, ... , �i+l, ... , �p+l)·

be the i-th projection. Then dxi E 01(U ) is the Ci· This follows from (1). In general, for f E n°(U), --+

--+

dxf(()

(2)

=

R

=

f (x)( 1 � UX]

+

··· +

f (x)(n � UXn

with (( 1,... , (n) = (. In other words: df = L,. -3-te:i

Lemma 3.4

Proof.

=

L,. 3-fdxi.

If w(x) = f(x )tJ then dxw = dxf 1\ cJ.

By (1) we have Dxw (()

===

(Dxf)(()tJ

(

OXn

·

1

and Definition 3.2 gives dxw(6, . . . , �p+l)

)

&f &f = - (1 + .. + - ( n cf = dxf(()tJ &x

l

p+l

(

)

= '2::: ( - ) k- ldxf(�k)ti 6, .. , �k, · · · �p+l k=l

·

= [dxf 1\ cJ](6, ... , � p+l)·

0

Note for cJ E AltP(Rn) that tkl\t[

=

{�-lre:J

if k E I if k r/. I

with r the number determined by ir < k < ir+l and J = (il> ... , ir, k, . . . , ip). For p � identically zero.

Lemma 3.5

0

the composition QP(U)

--+

QP+1(U)

--+

QP+2(U) is

Proof. Let w = ft1. Then

of of OX OX 1 n Now use ti 1\ ti = 0 and ti 1\ tj = -tj 1\ ti to obtain that n &2 f 2 = 8 · . ti 1\ (tj 1\ t J) d w XtoxJ t,.J.=l dw

= df 1\ cJ = -q 1\ cJ + · · · + -tn 1\ cJ.

'2:::

=

. . (aXt I: t<J

()�1

X .J

1\ Ej 1\ f.J = - 0°2£ X1 Xt. ) €i .

0.

0

3.

17

DERHAM COHOMOLOGY

The exterior product in Alt* (Rn) induces an exterior product on D.*(U) upon defining (w1 1\ w2)(x) = w1(x) 1\ w2(x). The exterior product of a differential p-form and a differential q-form is a differential (p + q)-form, so we get a bilinear map

For a smooth function J

E

C00(U, R), we have that

This just expresses the bilinearity of the product in Alt* (Rn). Also note that f 1\ w = fw when f E D.0(U) and w E D.P(U). Lemma 3.6 For

w1 E D.P(U)

and w2

E D,Q (U),

Proof. It is sufficient to show the formula when

then w1 1\ w2 = fg

cJ

1\ EJ, and

w1 = fcJ and w2 = 9€J. But

d(wl 1\ w2) = d(J g) 1\ cJ 1\ EJ = ((df)g + fdg) 1\ EJ 1\ f.J = dj9 1\ E[ 1\ EJ + jdg 1\ f.[ 1\ f.J = dj 1\ EJI\ 9EJ + (-l?fcJ 1\ dg 1\ tJ = dw1 1\ w2 + (-l)Pwl 1\ dw2.

0

Summing up, we have introduced an anti-commutative algebra D.*(U) with a ·

differential,

d: D.*(U)--+ n*+1(U),

dod= 0

and d is a derivation (satisfies Lemma 3.6): (D.*(U), d) is a commutative DGA (differential graded algebra). It is called the de Rham complex of U. Theorem 3.7 There is precisely one linear operator d: OP(U) 0, 1, . , such that

..

--+ f2P+1(U),

D.0(U), df = *ftcl + · · · + -§t en (ii) d 0 d = 0 (iii) d(w1/\w2) = : U1 - U2 a smooth map. The induced morphism D.P(¢>): DP(U2) - DP(UI) is defined by

Definition

f2P(if>)(w)x = AltP(Dxif>) o w( if>(x)),

D0(¢>)(w)x = wcf>(x)·

Frequently one writes ¢>* instead of f2P(¢>). We note that the analogue of (4) is satisfied. Indeed, ¢>*(w)x( 6, · ·, �p) = Wcf>(x)(Dx¢>(6), · · · , Dxif>(�p)), and using the chain rule Dx(7/J o ¢>) = Dcf>(x)(7J;) o Dx(¢>), for ¢>: U 1 U2, 1/J: U2 - U3, it is easy to see that f2P( 7/J 0 ) = f2P( if>) 0 f2P ( 7/J), f2P(idu) = idnP(U) . ·

It should be noted that f2P(i)(w) = w 0 i when i: Ut then Dxi id. =

�

u2 is an inclusion, since

Example 3.11 For the constant 1-forrn €i E D1(U2) we have that

*( €i ) = with if>i

8¢i � � -€k = dif>i

OXk k=l

the i-th coordinate function. To see this, let ( E R11• Then ¢• ( i Ln ( = L -€1 (() = dif>i( () . OXI OX[ I

0

1=1

1=1

Theorem 3.12 With Definition 3 .I 0 we have the relations

(i) if>*(w 1\ T) = if>*(w) 1\ if>*(T ) (ii) ¢>*(!) = J o ¢ if J E D0(U2) (iii) d¢>*(w) = ¢>*(dw).

Conversely, if¢': D*(U2) - D* (Ut) is a linear operator that satisfies the three conditions, then if>' = (x)(Dx¢>(6), ... , Dxif>(�p+q))

[

= L sign( cr) Wcf>(x) (Dx(�cr(p)))

Tcf>(x) (Dx¢>(�cr{p+l)) ,

= L sign(O')if>* (w)x(�cr(l)>.

0

.

]

, Dx¢>(�cr(p+q))) ) ( , �cr(p))¢*(T)x �cr(p+l)> · · ., � cr(p+q)

=(if>*(w)x 1\ *(T)x)(6, · · ·, �p+q)·

o

•

•

DE RHAM COHOMOLOGY

3.

21

This shows (i) when p > 0 and q > 0. If p = 0 or q = 0 the proof is quite analogous, but easier. Property (ii) is contained in the definition of ¢* for degree 0. So we are left with (iii). We shall first show that d¢* ( ) = *(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 and H0 ( U) = R.

---+

U is a 0

HP(U) = 0 when

Proof. Let F: U x [0, 1] -+ U be a homotopy from fo = idu to a constant map h with value xo E U. For x E U, F(x , t) defines a continuous curve in U,

which connects x to xo. Hence U is connected and H0(U) = R by Lemma 3.9. If p > 0 then Dl(ft): Dl(U) - DP(U) is the zero map. Hence by Theorem 6.8.(i) we get that ids,(u) =

and thus HP(U)

= 0.

fo = fi = o

0

In the proposition below, Rn is identified with the subspace Rn x {0} of Rn+l and R · 1 denotes the !-dimensional subspace consisting of constant functions.

6.

HOMOTOPY

Proposition 6.11 For an arbitrary closed subset

isomorphisms

43

A of R with A

i=

Rn we have

HP+ l (Rn+l - A) � HP(Rn - A) for p � 1 1 +l H (Rn - A) � H0(Rn - A)/R 1 H0(Rn+ l - A) � R. ·

Rn+ 1 = Rn x R, U1 = Rn x ( O,oo) U (Rn - A) x (-1, oo) u2 = Rn X ( -oo, 0) u (Rn - A) X (-oo, 1).

Proof. Define open subsets of

Then ul u u2 = Rn+l - A and ul n u2 = (Rn - A) X ( - 1 , 1). Let ¢: ul -+ ul be given by adding 1 to the (n + 1)-st coordinate. For x E U1 , U1 contains the line segments from x to ¢(x) and from ¢(x) to a fixed point in Rn x (O,oo). As in Example 6.5 we get homotopies from idu1 to ¢ and from ¢ to a constant map. It follows that ul is contractible. Analogously u2 is contractible, and HP(Uv) is described in Corollary 6.10. Let pr be the projection of U1 n U2 = (Rn - A) x ( -1, 1) on Rn - A. Define i : Rn - A -+ U1 n U2 by i(y) = (y, 0). We have pr o i = idR"-A and i o pr � idu1nu2• From Theorem 6.8.(iii) we conclude that is an isomorphism for every

p. Theorem 5.2 gives isomorphisms

8* : HP(Ul

n

U2 )

-+

HP+l (Rn+l - A)

for p � 1 . By composition with pr* one obtains the first part of Proposition 6.1 1 . Consider the exact sequence

0 -+ H0 (Rn+l - A) J..: H0(U1) E9 H0(U2) !; H0(U1 n U2 ) � H1(Rn+ l - A) -+ 0. An element of H0(UI) ffi H0(U2) is given by a pair of constant functions on U1 and U2 with values a1 and a2. Their image under J* is by Theorem 5.2 the constant function on ul n u2 with the value a l - a2. This shows that Ker 8* = Im J*

=R·1,

and we obtain the isomorphisms

H1 (Rn+l - A) � H0(U1 n U2)/R-1 � H0(Rn - A)/R · 1 .

We also have that dim(Im (!*)) = dim(Ker (J*)) = 1 so H0(Rn+l - A ) � R.D ,

44

6.

Addendum 6.12

HOMOTOPY

In the situation of Proposition 6. 1 I we have a diffeomorphism

R: Rn+ l - A

�

Rn+ l - A

defined by R(x1 , . . . , Xn , Xn+l) = (x1, . . . , Xn , - Xn+I). The induced linear map

is multiplication by -1 for p � 0. Proof.

In the notation of the proof above we have commutative diagrams, which the horizontal diffeomorphisms are restrictions of R:

m

In the proof of Proposition 6 . 1 I we saw that

is surjective. Therefore it is sufficient to show that R* o 8*([w]) = -8*([w]) for an arbitrary closed p-form w on U1 n U2. Using Theorem 5 . 1 we can find Wv E 0P(Uv), v = 0, 1, with w = jj(w1 ) -j2(w2)· The definition of 8* (see Definition 4.5) shows that 8*([w]) = [T] where T E QP+ l (Rn+ 1 - A) is determined by i� (T ) = d;,;.;v for v = 1, 2. Furthermore we get

-R{jw = R0 jz(w2) - R{j ji(w1) = ji(Riw2) - J2( R2w 1 ) ii(R*T) = Rj(i2T) = Rj(dw2) = d(Riw2) i2(R*T) = R;(ijT) = R2(dwl ) = d(R2wi). o

o

These equations and the definition of 8* give 8*(-[R0w]) (2)

8* R(i([wJ) = -RQ o

For the projection pr: U1 n U2 the composition

�

o

= [R*T]. Hence

8* ([w]).

Rn - A we have that pro Ro = pr and therefore

is identical with pr* . Since pr* is an isomorphism, RQ is forced to be the identity map on HP(U1 n U2), and the left-hand side in (2) is 8*[wj. This completes the proof. 0

6.

HOMOTOPY

45

Theorem 6.13 For n � 2 we have the isomorphisms

HP (Rn

_

{ O}) �

{ 0R

zfp =

�,

n

-1

otherwzse.

Proof. The case n = 2 was shown in Example 5.4. The general case follows from induction on n, via Proposition 6. 1 1 . 0

An invertible real n a diffeomorphism

x

n matrix A defines a linear isomorphism Rn - Rn, and

For each n � 2, the induced map fA.: Hn-1 (Rn - {0}) Hn-1(Rn - {0}) operates by multiplication by det A/Jdet AI E { ±1 }.

Lemma 6.14

Proof. Let B be obtained from A by replacing the T-th row by the sum of the T-th row and c times the s-th row, where T =f- s and c E R,

B = (I + cEr,s)A,

where I is the identity matrix and Er,s is the matrix with entry 1 in its T-th row and s-th column and zeros elsewhere. A homotopy between fA and fB is defined by the matrices (I + tcEr,s) A,

0 � t � l.

From Theorem 6.8 it follows that fA. = f'B· Furthermore detA = detB. By a sequence of elementary operations of this kind, A can be changed to diag ( l, . . , l, d), where d = det A. Hence it suffices to prove the assertion for diagonal matrices. The matrices

.

(

. JdJtd dtag l, . . . , l, ldl

),

yield a homotopy, which reduces the problem to the two cases A = diag(l, . . . , l, ±l), so fA is either the identity or the map R from Adden­ dum 6.12. This proves the assertion. 0 From topological invariance (see Corollary 6.9) and the calculation in Theorem 6.13, supplemented with if p = 0 if p =!= 0 we get

6.

46

Proposition 6.15 If n i=

HOMOTOPY

m then Rn and Rm are not homeomorphic.

Proof. A possible homeomorphism Rn

Rm may be assumed to map 0 to 0, and would induce a homeomorphism between Rn - {0} and Rm - {0}. Hence

for all

�

D

p, in conflict with our calculations.

Remark 6.16 We offer the following more conceptual proof of Addendum 6.12.

Let

0

A* -L. B* ____g.:._ C* - 0

o· l 0 - A. -

1�·

1�·

gi C* f -1 -0 *1 i B*1 ---be a commutative diagram of chain complexes with exact rows. It is not hard to prove that the diagram �

HP( C*) __L._ HP+ 1 ( A* )

· 17

o! ·

HP ( Ci ) � HP+l ( Ai ) is commutative. In the situation of Addendum 6. 1 2 consider the diagram

o 0

-

D*(U) -L 0:"(U, ) 9 D*(U2 )

R ! ! .Q*(U) -L n*(Ul) E17 .Q"'(U2)

-

' R

-

with R(w1,w2) addendum.

=

-

D*(U1

n

U2 )

D*(Ul

n

U2)

1-Ro

-

-

0 0

(Riw2, R2w1 ). This gives equation (2) of the proof of

the

47

7.

APPLICATIONS OF

DE

RHAM COHOMOLOGY

Let us introduce the standard notation

Dn = { X E

sn-

1

Rn

l llxll $ 1 } l llxll = 1 }

= {x E Rn

A fixed point for a map

f: X

--t

(the n-ball) (the ( n - 1)-sphere)

X is a point x E X, such that f(x) = x.

Theorem 7.1 (Brouwer's fixed point theorem, 1912) Every continuous map J : vn ---7 vn has a .fixed point.

Assume that j(x) � X for all X E vn. For every X E the point g(x) E sn-l as the point of intersection between from j(x) through x.

vn

sn-1

Proof.

o"

we can define and the half-l.ine

f(x)

X

g(x)

We have that

g(:c) = x + tu,

where

t = -x -u + ·

u=

���=��:�11• and

J1 - llxll2 + (x · u)2 .

Here x · u denotes the usual inner product. The expression for g(.r,) is obtained by solving the equation (x + 't'u) - (x + tu,) = L There are two solutions since the line determined by f (x) and X intersects sn- l in two points. We are interested in the solution with t 2:: 0. Since g is continuous with 9iSn-1 = idsn- 1 , the theorem 0 follows from the lemma below. Lemma 7.2 There is no continuous map g: Dn

--t

sn-1 ,

with .9!S"-'

=

idsn- 1 .

Proof. We may assume that n 2:: 2. For the map T : Rn - {0} --t Rn - {O},T(x) = x/111:11. we get that idR"-{O } � ·r, because Rn - {0} always contains the line segment between x and T(x) (see Example 6.5). If g is of the indicated type, then g(t · T(x)), 0 :::; t ::; 1 defines a homotopy from a constant map to 1'. This shows

48

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

that Rn - {0} is contractible. Corollary which contradicts Theorem 6.13.

6. 10

asserts that Hn-l(Rn - {0})

=

0,

0

The tangent space of sn in the point X E sn is Txsn = {X} 1.. ' the orthogonal complement in Rn+l to the position vector. A tangent vector field on sn is a continuous map v: sn -+ Rn+l such that v(x) E Txsn for every X E sn. Theorem 7.3 The sphere S" has X E 5n if and only if n is odd. Proof.

Such a vector field by setting

v

a tangent vector field v with v(x)

-:fi

0 for all

can be extended to a vector field w on Rn - {0}

We have that w(x) # 0 and w(x) · x = 0. The expression F(x, t) = (cosn t)x + (sin 1r t)w(x)

defines a homotopy from fo = idcw•+' -{o}) to the antipodal map h , h (x) = -x. Theorem 6.8.(i) shows that fi is the identity on Hn(Rn+l - {0} ) , which by Theorem 6.13 is I -dimensional. On the other hand Lemma 6 . 1 4 evaluates !{ to be multiplication with (-l)n+ I . Hence n is odd. Conversely, for n = 2m - 1, we can define a vector field v with

In I 962 J. F. Adams solved the so-called "vector field problem": find the maximal number of linearly independent tangent vector fields one may have on sn. (Tangent vector fields v 1 , . , vd on sn are called linearly independent if for every X E 5n the vectors VI (X), . , Vd(X) are linearly independent.) .

.

.

.

For n = 2rn - 1, let 2m = (2c + 1)24a+b, where 0 :5 b :5 3. The maximal number of linearly independent tangent vector fields on sn is equal to 2b + 8a - l. Adams' theorem

(Urysohn-Tietze) If A � Rn is closed and J: A -+ Rm continuous, then there exists a continuous map g: Rn Rm with 91 A = f.

Lemma 7.4

-+

Proof.

We denote Euclidean distance in Rn by d(x , y) and for x E Rn we define

d(x, A) = yEA in£ d(x, y).

7.

APPLICATIONS OF DE RHAM COHOMOLOGY

49

For p E Rn - A we have an open neighborhood Up � Rn - A of p given by

Up =

{ x E Rn I d(x, p)


p(x)J(a(p))

pe

if X E A

if X E Rn - A

where for p E Rn - A, a(p) E A is chosen such that

d(p, a(p)) < 2d(p, A ) . Since the sum is locally finite on Rn - A, g is smooth on Rn - A. The only remaining problem is the continuity of g at a point x0 on the boundary of A. If x E Up then

1 d(x0, p) :5 d(xo, x) + d(x, p) < d(xo, x) + 2 d(p, A)

:5

1 d(xo, x) + 2 d(p, xo).

Hence d(xo,p) < 2d(xo, x) for x E Up. Since d(p, a(p)) < 2d(p, A) :5 2d(xo,p) wegetforx E Up that d(xo, a(p)) :5 d(xo,p)+d(p,a(p)) < 3d(xo,P) < 6d(xo,x). For x E Rn - A we have

g(x) - g(xo) =

L

pER"-A

p(x)(f(a(p)) - f(xo))

and (I )

JJg (x) - g(xo)!l :5

L p(x) Jlf(a (p)) - f(xo)JJ, p

where we sum over the points p with x E Up. For an arbitrary 0 choose 8 > 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 <J?h is a bijection. The linear structure on TpM is well-defined. This can be seen from the following commutative diagram, where F = h o h,-1 , q = h(p)

Nn be a smooth map and p E M. (i) There is a linear map Dpf: TpM ---+ Tf(p) N given in terms of representing

Lemma 9.3

Let f: Mm

---+

curves by

Dp!([aJ) = [f o a]. (ii) If (U, h) is a chart around p in M and (V, g) a chart around f(p) in N, then we have the following commutative diagram:

Tf(p)M

l�Rm�h Dh(p) (gofoh-1) �1Rn�9

= g o f o h-1, defined on the open set h(U n j-1(V)), shows that the bottom map in the diagram is linear and given

Proof. Remark 9 . l .(ii) applied to F

on representing curves as stated there. Since h and 9 are linear isomorphisms, there exists a linear map Dpf making the diagram commutative. The formula in 0 (i) follows by chasing around the diagram. ·

Note that the linear isomorphism <J?h in Definition 9.2 can be identified with Dph through the identification discussed in Remark 9.l .(ii). From now on we write Dph: TpM ---+ Rm for this linear isomorphism, and similarly Dh (p) h-1: Rm TpM for its inverse. ---+

9.

67

DIFFERENTIAL l''ORMS ON SMOOTH MANIFOLDS

Suppose Mm � R1 is a smooth submanifold with inclusion map i: M ---... R1 . Definition 8.8 implies that Dpi: TpMm ---+ TpRl "' Rl is injective. In this case we usually identify TpM with the image in R1, consisting of all vectors a'(O) where a: I ---+ M � R1 is a smooth parametrized curve with a(O) = p. For a composite rp o f of smooth maps f: M ---+ N, rp: N ---+ we have the chain rule, immediately from Lemma 9.3(i),

P and a point p E M

Dp (rp o f ) = Df(p)'P o Dp'P·

(3)

Remark 9.4 Given a smooth chart (U, h) around the point p E Mm we obtain a basis for TpM

(8�;

where )p is the image under Dh (p) h- 1 : Rm ---+ TpM of the i-th standard basis vector ei = (0, . . . 1, . . . E Rm. A tangent vector Xp E TpM can be written uniquely as

, 0)

(4 ) where a = (a1, . . . , a ) E Rm. If Xp m a representing smooth curve, we have a =

=

[a],

where

a: I ---+ U

with

a(O) = p

is

(h o a)'(O).

Given f E C00(M, R) we have the tangent map (5) The directional derivative Xpf E R is defined to be the image in R of Xp under 1 (5), i.e. Xpf = (f o a)'(O). In terms of f o h- we have by the chain rule

d Xpf = dt (fh-1

In particular

(6)

o

m Bfh-1 (h(p))ai. ha(t) ) lt=O = 2:=

(�) f OXj p

i::=l

=

ox·t

8fh-1 ( (p) . h ) OXj

Under the assumptions of Lemma 9.3.(ii) there is a similar basis (8/ByJ f(p)' 1 � j � n, for Tf(p) N and the matrix of Dpf with respect to our bases for TpM

9.

68

VlFFERENTIAL FORMS ON SMOOTH MANIFOLDS

and Tf(p)N is the Jacobian matrix at h(p) of g o f o h- 1 . Specializing this to the case N = M, f = idM we find that

(�)

(7 )

OXi p

=

f. ocpi (h(p)) (_!___)

j=1

OXi

OYj p

.

This holds when (U, h) and (V, g) are two charts around p with transition function (cpi . . . . ' 'Pm) = cp = g o h-

1

expressing the Yj-coordinates in terms of the Xi-coordinates. Suppose X is a function that to each p E M assigns a tangent vector Xp E TpM. Given the chart (U, h), formula (4) holds for p E U with certain coefficient functions ai: U - R. If these are smooth in a neighborhood of p E U, X is said to be smooth at p. From (7), this condition is independent of the choice of smooth chart around p. If X is smooth at every point p E M, X is a a smooth vector .field on M. Let us next consider families w = {wp } pE M of alternating k-forms on TpM, where wp E Altk (TpM). We need a notion of w being smooth as function of p. Let g: W --+ M be a local parametrization, i.e. the inverse of a smooth chart, where W is an open set in Rm. For x E W,

is an isomorphism, and induces an isomorphism

We define g*(w): W

--+

Altk(Rm) to be the function whose value at x is

Definition 9.5 A family w = { wp hE M of alternating k-fonns on TpM is said

to be smooth if g*(w) is a smooth function for every local parametrization. The set of such smooth families is a vector space D,k (M). In particular, fl0(M) = C00(M,R). Lemma 9.6 Let gi: Wi --+ N be a family of local parametrizations with N = U 9i(Wi)· If gi(w) is smooth for all i, then w is smooth.

N be any local parametrization. and z E W. We show that g*(w) is smooth close to z. Choose an index i with g(z) E 9i(Wi). Close to z

Proof. Let

g: W

--+

69

DD'F .ERENTIAL FORMS ON SMOOTH MANIFOLDS

9.

we can write g = 9i o gi1 o g = 9i o h , where h = gi1 o g: g-1(gi(Wi)) is a smooth map between open sets in Euclidean space. Thus

---t Wi

g"'(w) == (gi o h)*(w) = h"'(gi(w))

in a neighborhood of z, and the right-hand side is

a

smooth k-form by assumption.

0

The exterior differential

d: Qlc (M) ---t Qlc+l (M ) can be defined via local parametrizations g: W is a smooth k-form on M then

---t

M as follows. If w = {wp }pEM

(8) It is not immediately obvious that dpw is independent of the choice of local parametrization, but this is indeed the case: Given a local parametrization g, then any other locally has the form g o ¢, with ¢: U - W a diffeomorphism. Let 6, . . . , elc + l E TpM. We choose VI , . . . , vk+ l E Rn so that Dx(go ¢)(vi) = ei· We must show that

dyg* (w) (wb . . . , wk+l) = dx(9 o ¢) "'(w )(v1 , . . . , V!c+I)

where ¢(x) = y and Dx¢(vi)

Wi ·

=

This follows from the equations

(g o ¢)*(w) == ¢* (g* (w)) d¢"'(r) = ¢"'d(r),

where r = g*(w); see Theorem have defined a chain complex ·

We have

· ·

D,k(M) = 0

• •

---t

• .

T E D,k(N);

A:

1\

. . . , 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 be a constant such that (4) Here fi is the j-th coordinate function off, and II II denotes the Euclidean norm. Let n T = IT !ti,ti + aJ i=l

be a cube such that K � T, and let t > 0. Since the functions {}fJj{}xi are uniformly continuous on L, there exists a ti > 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

<J? (x, t) =

n

L XicPi(tx ) i= l

,

and in particular that has a smooth extension to an open set W with U0 x [0, 1]

W � Uo

x

R.

For each t E [0, 1]. 0 such that pDn � Uo. The homotopy sn- 1 X [0, 1] - sn-1 given by Xt(Px ) / IIXt(px)i! ,

and Corollary

1 1 .2

0 :::; t � 1,

shows that

Since A: Rn - Rn is linear we have (A- 1 ) * Y = yields the commutative diagram

A-1 o Y o A: Rn - Rn.

This

Rn - { 0 } (A-t).Y Rn - {0}

lA

Rn - { 0 }

--L

lA

Rn - { 0}

Now use the functor Hn- l and apply Lemma 1 1 .17 to both Y and get 1-((A-1) * Y;O) = 1-(Y;O). Hence 1-(F;O) = 1-(Y;O).

(A- 1 )* Y

to

0

Definition 11.19 Let X be a smooth tangent vector field on the manifold Mn, n 2: 2 � it p0 E M as an isolated zero. The local index 1-(X; Po ) E Z of X is defined by

where (U, h) is an arbitrary smooth chart around Po with h(po) =

0.

We note that Lemma 1 1 . 1 8 shows that the local index does not depend on the choice of (U, h). One can picture vector fields in the plane by drawing their integral curves, e.g.

JJ.

DEGREE, LINKING NUMBERS AND INDEX OF VECTOR FIELDS

'=

Let X be a smooth

-1

Figure 4

'=

tangent vector field on Mn

109

+2

and let Po

E

Mn be a zero. Let

for a chart (U, h) with h(po) = 0. If DoF: Rn ---+ Rn is an isomorphism, then Po is said to be a non-degenerate singularity or zero. Note that by the inverse function theorem F is a local diffeomorphism around 0, such that 0 is an isolated zero for F'. Hence p0 E Mn is also an isolated zero for X . Lemma 11.20

If Po is a non-degenerate singularity, then t.(X,po) = sign(det DoF) E {±1}.

Proof.

B y shrinking U we may assume that h maps U diffeomorphically onto an open set U0 � R'\ which is star-shaped around 0, and that F is a diffeomorphism from Uo to an open set. As in the proof of Lemma 1 1. 18 we can define a homotopy

G : Uo

x

[0, 1]

---+

Rn ;

G(x, t ) =

{

DoF F(tx )j t

if t = 0 if t # 0,

where G can be extended smoothly to an open set W in Uo x R that contains Uo X [0, 1]. Choose p > 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<EA subordinate to (Ucr) QEA • and

define a smooth tangent vector field on M by X=

L Pcr Xcr,

0<E A

where PaXa is taken to be 0 outside Ua · If p E M is not a critical point for f then, for every a E A with p E Ucr and q = ha(P), we have Indeed, there is at least one C¥ with Pa(P) > 0 and

We see that X satisfies condition (i) in Definition 12.7.

120

12.

THE POINCARt-HOPF THEOREM

If p is a critical point of f then there exists a unique a E A with p E Ua. It follows from (a) that X coincides with Xa on a neighborhood of p, and condition (b) above shows that assertion (ii) in Definition 12.7 is satisfied. D The next lemma relates the index of a Morse function to the local index of vector fields as defined in Chapter 1 1 .

Lemma 12.9 Let f be a Morse function on M and X a smooth tangent vector

field such that dpf(X(p)) > 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 m, ui n Wm- 1 i= 0} -

m-1

u lj

j=l

If Im- 1 is finite, then Wm- 1 is relatively compact and (ii) implies that Wm_1 only intersects finitely many of the sets Ui. This shows inductively that Im is indeed finite for all m. Moreover, if m 2: 2 does not belong to any Ij with j < m then it certainly, by definition of Im, belongs to Im. Thus N is the disjoint union of the finite sets Im. Since we already know that finite unions are in U we have

lm = 0, Wm = 0 E U). Similarly, (7) shows that Wm n Wm+ 1 =

U

Wm E U (if

Ui n Uj E U.

(i,j)Elm X Im+l

Note also from (8) that Wm n Wk = 0 if k 2: m + 2. Indeed, if the intersection were non-empty, then there would exist ·i E lk with Wm n Ui i= 0, and by (8) i E lj for some j s m + 1. One now uses condition (iv) of Theorem 13.9 to see that the following three sets are in U:

W(O)

=

00

U

j=l

W j, W( l ) 2

=

U W2j-1 , W(o) n W(l) = U (Wm n Wm+d· j=l j=l 00

00

13.

POINCARE DUALITY

But then Theorem 1 3.9.(iii) implies that the union

135

U� 1 Ui = W(O) U W ( l ) E U. D

Proof of Theorem 13.9. Consider first the special case where an open subset. We consider the maximum norm on Rn

M = W � Rn is

llxlloc = l�t� max lxi j n

whose open balls are the cubes IIi= I (ai, bi). The argument of Proposition gives us a sequence of open II 11 00-balls Uj such that (i) w = Uf=l Uj = Uf=l Uj. (ii) (Uj )j EN is locally finite. (iii) Each u1 is contained in at least one

A.6

V13.

A finite intersection Uj1 n . . . n U1m is either empty or of the form [1� 1 (ai, bi ), so is diffeomorphic to Rn. Thus we can conclude from Lemma 13.10 that W E U. In the general case, consider first an open coordinate patch (U, h) of M, h: U W a diffeomorphism onto an open subset W � Rn. We apply the above special case to W with cover (h- 1 (V13)) /3EB ' and with uh the open sets of W whose images by h- 1 belong to the given U. The conclusion is that U E U for each coordinate �

patch. If M is compact then we apply the argument of Lemma 13.10 to a finite cover of M by coordinate patches to conclude that M E U. In the non-compact case we make use of a locally finite cover of M by a sequence of relatively compact coordinate patches; cf. Theorem 9. 1 1 . (One may alternatively imitate the proof of Proposition A.6 using relatively compact coordinate patches instead of discs to construct the desired cover). D

Proof of Theorem 13.5. Let

U = { U � Mn I U open, Df1 an isomorphism for all p}

and let V = (V13) /3EB be the trivial cover consisting only of M. Corollary 13.8 D tells us that the assumptions in Theorem 13.9 are satisfied, so M E U. We close the chapter with an exact sequence associated to a smooth compact manifold pair (N, M), i.e. a smooth compact submanifold M of a smooth compact manifold N. Let U be the complement U = N - M, and let

i: U - N,

j: M - N

be the inclusions. Then we have

Proposition 13.11 There is a long exact sequence

. . . - Hq-l(M) � Hg(U) !..; Hq(N) L Hq(M) - . . . The proof of this result is based upon the following:

13.

136

POINCARE DUAUTY

Lemma 13.12 (i) j"' = nq(j): n,q(N) � nq(M) is an epimorphism. (ii) If w E nq(M) is closed, there exists a q-form T E nq(N) such that j "' (T) == w and such that dT is identically zero on some open set in N containing M. (iii) If T E nq(N) has SUPPN (dT) n M = 0 and j"'(T) is exact, there exists a form u E n,q-l ( N) such that T - du is identically zero on some open set in N containing M. that N is a smooth submanifold of Rk . gives us tubular neighborhoods in Rk with corresponding smooth

Proof. We can assume by Theorem Theorem

9.23

8. 1 1

inclusions and retractions (VN , i N , rN), (VM, iM,rM) for N and M respectively; we may arrange that VM ; VN. Let 0} - #{i I Ui


lzol 2 - ! z1 l 2 ) .

(3)

Hopf discovered (in 1931) that

11

is not homotopic to a constant map.

The Riemann surface of Example 14. 1 is holomorphically equivalent to CP1 . Indeed, by (2), the compositions

are

h0 1 '1/J- (z, t) = [1, z/(1 + t)), h11 '1/J+ (z, t) = [z/( 1 - t), 1]. These expressions agree when t E ( 1 1 ) , so we can define a homeomorphism o

o

-

\11 (z, t)

(4)

,

=

{ [1

+

t, z] , (z,t) # (0, -1) [z, 1 - t], (z, t) # (0, 1).

Actually w is holomorphic with holomorphic inverse. The complex projective plane CP 1 is often identified with C U { 00} by letting [zo , ZI ) E CP1 correspond to z0 1 Z!t and assigning oo to z0 = 0. In this identification \11 : S2 -t C u { oo} becomes the stereographic projection '1/J- from the south pole p-, extended by mapping P- to oo. One may generalize the Hopf fibration to the map (5)

Its fiber 7r- 1 (p) is the unit circle in the complex line p E CPn. For n = 1 this is nothing but the Hopf fibration of Example 1 4. 1 . Indeed with the notation of (3), (4)

w- 1 ( [l , zQ' 1 zl ] ) = '!/': 1 (zQ'1zl ) = (2zozl , lzol2 - lzi i2 ). The unit circle S1 � C acts on the sphere sZn+l � cn+l by coordinatewise w -1 ([zo, zl])

=

multiplication, (6)

The action is free, and the set of orbits of this action is precisely cpn in the sense that the orbit space s2n+l I S1 is homeomorphic to cPn . .

14.

TilE COMPLEX PROJECTIVE SPACE

cpn

141

Theorem 14.2 The cohomology of CPn is

H2i(CPn) = R Hk(CPn) = 0 Proof. The embedding en

c cn+l

for 0 � j � n otherwise.

induces an embedding of

cpn-1

into

CPn,

(CPn, cpn- 1 ). We can assume the and that n � 2, since the cohomology of CP 1 = S2 was given

and we can use Proposition 1 3. 1 1 on the pair result for

cpn-1

in Example

9.29.

The complement

cpn - cpn-l = U n is by ( 1 ) and

(2) homeomorphic with R2n.

The exact cohomology sequence takes

the form

We know from Lemma

13.2

that

Hl(R2n)

is non-zero only for

is a copy of R, and the result follows easily.

q = 2n,

---+

---+

would be the identity, but this is impossible as Since

0

14.2 that the general Hopf map 1r: S2n+1 cpn cannot s2n+l : if s existed with 1f 0 s = id, then s: cpn H2(cPn ) � H2 ( s2n+1 ) £ H2(cPn )

It follows from Theorem have a section

when it

H2n(cPn)

=

R

H2(CPn) =!=

0 and H2(S2n+l) = 0.

we know from Exercise 10.4 that

CPn

is an oriented

manifold, and hence from Theorem 13.5 that the bilinear map

given by the wedge product (followed by the integration isomorphism) is a dual

(non-singular) pairing. In particular, the generator of H2P(CPn) and the generator of

H2n-2P(CPn)

has non-trivial product.

Theorem 14.3 The cohomology algebra H*(CPn) is a truncated polynomial

algebra

H*(CPn )

= R[c]/(cn+ l)

where c is a non-zero class in degree 2, and (cn+1 ) the ideal generated by cn+l . Proof. We use induction over n, so suppose the theorem proved for cpn- 1 . The inclusion

14.

142

THE COMPLEX PROJECTIVE SPACE

cpn

induces the map

The proof of Theorem 14.2 shows that j* is an isomorphism for i Hence cn-l i= 0 in H2n-2(CPn), and the pairing

H2(CPn) X H2n-2(CPn)

---t


: E(� EB �) --+ R such that ¢>: Fb(�) EB Fb(O --+ R is an inner product on

each fiber Fb(�).

Proposition Proof.

15.13 Every vector bundle over a compact B has an inner product.

Choose local trivializations

hi = ui

X

Rn

7ri1(Ui) where · U1 , . . . , Ur cover B, and choose a partition of unity { a:i};=l with supp(a:i) c Ui. The usual inner product in Rn induces an inner product in Ui x Rn and hence, via hi, an inner product o f are continuous. In particular note that � � X x V and 77 � X x W give Horn(�, 77) � X x Hom(V, W). All of the constructions in (9) produce smooth vector bundles when � and 77 are smooth.

For (smooth) vector bundles� and 77 there are isomorphisms � � e* and e 0 11 � Hom (� , 77). Lemma 16.9

Proof. For finite-dimensional vector spaces there are natural isomorphisms

V ---+ V**,

V* 0 W - Hom(V, W)

defined without any· reference to basis. This gives maps of vector bundles (over the identity)

which are isomorphisms on each fiber, and we can apply Lemma 15. 10.

0

For finite-dimensional vector spaces V � V*, since the dimensions agree. How­ ever, it is not true in general that � � C; cf. Properties l8. l l below. The above proof breaks down because there is no isomorphism from V to V* defined in­ dependently of choice of basis. As a result one cannot define a homomorphism from � to C.

16.

OPERATIONS ON VECTOR BUNDLES AND THEIR SECTIONS

163

The isomorphisms of Theorem 16.7.(iii) are again defined without reference to a basis, and so give isomorphisms (1 1) for any vector bundles

�.

We have in Theorem 1 6.7, Addendum 16.8, Lemma 16.9, and (1 1) concentrated on real vector spaces and vector bundles, and leave the reader to formulate and prove the corresponding results for the complex cases. Every complex vector bundle� gives rise to a real vector bundle �R upon forgetting part of the structure. On the other hand, every real vector bundle induces a complex vector bundle 'f/c upon complexifying each fiber, 'f/c = 'fJ ®R t:b where t:t denotes the trivial complex line bundle. There are the following relations between these operations. Lemma

16.10

(i) For a real vector bundle fl, (rlc)R � rt EB fl. (ii) For a complex vector bundle �. (�R)c � � EB C .

Proof. We leave (i) to the reader and prove (ii). Given a complex vector space V we get a new one V ®R C where scalar multiplication is in the second factor. The map ) + ( - 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((A(l + tE?j)) where Eij is the basic matrix with 1 in the ( i, j)-th entry and zero elsewhere. Now (6) yields the relation

(7) and using (5) and the Bianchi identity we get

-dP(F9) = Tr(P'(F9) 1\ F9 1\ A - P'(F9) 1\ A 1\ F9) = Tr(F9 1\ (P'(F9) 1\ A) - (P'(F9) 1\ A) 1\ F9)

=

0.

0

18.

I K3

CHARACfERJSTIC CLASSES OF COMPLEX VECl'OR BUNDLES

18.2. Let \7o, 'VI be two connections on �. and 1r: M x R - M the projection onto the first factor. Let V1, = 1r*(\711) be the induced connections on 1r*(O; cf. Lemma 17. LO. Define a new connection on 1r*(O by

Proof of Lemma

v(s)(p. t) = ( 1 - t)vo(s)(p, t) + tvl (s)(p, t)

where (p, t) E M x R. Apply Lemma 1 7 . 1 0 to see that i()(v)

=

\7o ,

=

it(v)

'V1

lvf x IR are the two inclusions in respectively top and bottom. where i11: lvf From ( 1 7.2 1 ) it follows that _,

i0(F") = F"0,

�

ij ( F")

=

F"1

and hence i�(P(F9))�= P(F9"), 11 � 0, 1. Since ·io we have that i(i({P(F") ] ) = i j ( [P(F"')]).

-:::

i1

and P(F") is closed,

0

Note that isomorphic vectorbundles define identical cohomology classes [P(F'V)], since a smooth fiberwise isomorphism ]: � - ( induces isomorphisms between section spaces, and since we can choose connections to make the diagram no (�) _sz_ nl(�)

!

1

j.

j_

n°(t) _y_f2 1 (e)

commute. Thus the matrices for F" and p-v' are identical with respect to corresponding frames for � and (, and P(F9) P(F"''). In particular, if � is a trivial vector bundle, then [P (� ) ] [P(c-c ) ] = 0. Indeed, we just use the flat connection 'Vo on cc·

=

Definition

=

18.3

(i) The

k-rh Chern class

q, (O = (ii) The

of the complex vector bundle � is

1 [ ( Z1r-A F")] CJ1;;

k-th Chern character class

chk(� ) =

2k E H (1vf; C) .

is

:! [sk (27r� F")] E H2k(M; C).

Here \7 is any complex connection on �· If k cho(�) = dim�.

=

0 then co(O

=

1 and

184

18.

CHARACJ'ERISTIC CLASSES Ot' COMPLEX VECTOR BUNDLES

The reader may check that Definition l 8.3.(ii) agrees with Definition 1 7 . 1 5. We shall prove some properties of these classes. First note that they determine each other, since

for certain polynomials Pk and Qk; cf. Appendix B. For example we have

The integration homomorphism

is an isomorphism by Corollary 10. 14, and the inclusion j: an isomorphism

by Theorem 14.3. We now chose

property that

c

E H2(CPn; C)

I(j*(c))

(8)

It follows from Example 14. 10 that

of CP 1

=

- 1r c

CP1

c o�n. induces

once and for all with the

- 1.

is the cohomology class of the volume

with the Fubini-Study metric (cf. Theorem 1 4.8) and if we identify form 1 2 S with Cll� via \fJ then -4r.c corresponds to the volume form of 82 in its natural metric as the unit sphere and with its complex orientation. Let

Hn

be the canonical line bundle on

E(HnJ Then

.i*(Hn)

=

H1 is the

=

cpn with total space

{(L, u) E CPn x cn+l ! v. E L}.

canonical line bundle

of Example 15.2.

Theorem 18.4 The integration homomorphism maps c1 (H1 ) to - 1 . Proof.

Apply the two positively oriented stereographic charts 'lj;_ and "if; on + = CP1 from Example 14. 1 . In Example 17.9 we calculated the pre-image of the curvature form F'V under g = ('lj;_ ) to be

S2

-l

g * ( F" ) =

2·i

d

( - ?:"' X 1\ dy. 1 + lzl )

18.

CHARACTElUSTIC CLASSES OF COMJ'LEX VECTOR BUNOLES

We integrate this form by changing to polar coordinates (:�:, y) = ( 'rcos Since dx = cos Bdr· - rsin Bel() and dlJ = sin Bdr· + nos fJd() we see that

rl�c

1\ dy =

r dr 1\ cf(l,

185

e. ·r

sin fJ).

and

This calculation implies that

Indeed we can apply a partition of unity 1 = Po + Pl with supppo an arbitrarily large f.-1-sphere in the chart g and suppp 1 a correspondingly small f.-sphere in the other chart. In the limit c ---4 0 the integral of g*(F'il) (over all of IR2 ) is equal to the integral of F'il over CP1. 0 Theorem 18.5 Let J: N

-;

lvf

be a smooth map and � a complex vector bundle on M. For every invariant polynomial we have f*[P(O] = [P(f*(�))]. Proof. We give J* ( �) the connection f* ( \7) of Lemma 1 7. 1 0. By formula ( 17 . 1 3), 0 r ( F9) = pF(9l. Hence .f* (P(F9)) = P(Fr(9l).

For a line bundle L, D2 (Hom(L, L)) = D2 (A'f; C) so that F9

E D2(M: C), and

This gives

(9) so that chk (L) becomes the k-th term in the power series Theorem 18.6 For a swn of complex vee/or bundles,

(i) ch�c(�o $ (1) = chk((o) + cbd(t) k (ii ) ck(�o 8 �I ) 2: (.A�o)ck-v(�I). v=O =

exp( c1 ( L)).

186

CHARACfERISTIC CLASSES OF COMPLEX VECTOR BUNDLES

.18.

Proof. Choose complex connections

Oi(c;o ) $ Qi(6)�

then

on �v· We identify Oi(�o EB �1) with

'Vv

\7o EB 'VI: 0°(�o 61 6) is a connection on �0

EB �1

-

01(�o ffi 6)

with curvature

For direct sum of matrices

Ao ffi A1

=

(� 11 ) 0

E M +m( C) n

formula (3) of Appendix B gives the equations

sk(Ao EB A1 )

k

sk(Ao)+sk(AI)

=

and

Cik(Ao EB A 1 ) = L CTv(Ao)·CTk-v(AI), v=O

which prove the assertions. Theorem 18.7

0

For a tensor product of complex vector bundles, k chk (�o 0 6) L ch v (�o)chk-v (�1) =

where cho(�v) = dimc �v·

v=O

The tensor product of linear maps, applied fiberwise, defines a map of vector bundles

Proof.

Hom(�o, �o) 0 Hom(6, 6 ) -+ Hom(�o 18> 6 , �o 18> 6 ) and thus a product

For connections

(17. 15):

'\i'o , '\71

on �o, 6 , we have the connection '\7 on �o 18> 6 from

?(so 0 si)

= 'Vo(so) 1\ 1 + so s

1\ 'V1(s1).

The corresponding curvature form becomes F"' = F"'0 1\ id + id 1\ F"'1 where id E Q0(Hom(�v, �v) ) is the section that maps p E M to id: �P -+ �p· It follows that

,

18.

CHARACTERISTIC CLASSES OF COMPI,EX VECTOR BUNDl,ES

187

There is a commutative diagram: Di(Hom(�o, �o)) ® Di (Hom(6, �1)) 2- ni+i(Hom(�o ® 6 , �o ® 6))

lTr®Tr

(11)

1\ -

Di (M ; C) ® Di(M; C )

lTr

From ( 10) and ( 1 1 ) we get

which is equivalent to the statement of the theorem.

D

Let H2*(M; C) denote the graded algebra

H2*(M ; C) = EB H 2i (M; C). i�O For a complex smooth vector bundle �, we define the Chern character by

This defines a homomorphism ch: Vectc(M) .._. H2*(M; C), which by Theorem 1 8.6.(ii) and the universal property of the Grothendieck construction can be extended to a homomorphism

ch: K(M)

--t

H2*(M; C).

An application of Theorem 18.7 shows that ch is a multiplicative map, when the product in K(M) is defined by

([�o] - [rJo])([6] - [m]) = [�o ® 6 ] + [rJo ® 111] - [�o ® m] - [rJo ® 6]. Without proof we state: Theorem 18.8

The Chern character induces an isomorphism of algebras ch: K(M) ®z C

---t

H 2*(M; C).

There exists precisely one set of cohomology classes ck(() k H2 (M; C), depending only on the isomorphism class of(, and such that (i) I(c1(Hi ) ) = -1, ck (Hn) = 0 when k > 1, and eo (Hn) = 1 ���) f* ck (� ) = ck (f* �()) (111) ck(�o EB 6) = Li=O ci (�o ) ck-i (6 ) .

Theorem 18.9

0 E

188

18.

CHARACTERISTIC CLASSES OF COMPLEX VECTOR BUNDI.ES

The uniqueness part of Theorem 18.9 rests on whose proof is deferred to Chapter 20.

the

so-called splitting principle,

Theorem 18.10 (Splitting principle) For any complex vector bundle � on M there exists a manifold T = T(�) and a proper smooth map f: T --t M such that

(i) j*: Hk (M) --t H k (T) is injective (ii) J*(€) � "Yl EB · · · ffi "Yn

for certain complex line bundles

11, . . .

, In·

Proof of Theorem 18.9. The Chern classes of Definition 18.3 satisfy the three conditions, so it remains to consider the uniqueness part. From (i) it follows that c1 (H1) == c in the notation of (8). Let L be an arbitrary line bundle and L.l a complement to L, with L EB L.l = M X cn+l . We can define

where proh: M

X

M - CP71; X 1-+ proj2(Lx) cn+ l - cn+l . There is an obvious diagram L __:fr.._. H.., 7r:

1

1

M � CP71 with -ft'p an isomorphim for every p E M. Hence 1r*(H..,) follows that

C>{

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., >.- 1 with say ,\ � 1. Then ,\ depends continuously on A. The case ,\ = 1 occurs only for the identity matrix I. If A i= I we can write

where P is the orthogonal projection on the (1-dimensional) >.-eigenspace of A. Here P depends continuously on A E Q - {I}. We define F on (Q - {I}) x [0, 1] by F(A;t) = ,>.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 · · · , A?r(n) ) ·

In particular we have

for all 1r E En· Theorem B.3 gives that P(d(AI, . . . , An)) = p(o-1, . . . , Un),

where Ui = ui(AI, . . . , An), and p is a certain polynomial. Hence

P(A) for all

= p(u1(A), . . . , o-n( A))

A E Dn, and hence for all A E

D

Mn(e).

We finally consider the Pfaffian Pf(A). This is a homogeneous polynomial in n(2n - 1) real variables, or alternatively a polynomial in the skew-symmetric 2n x 2n matrices A = (Aj)· It has degree n. We can consider Pf(A) as a map

Pf: so2n

-+

R,

from the space of skew-symmetric matrices . For an A E .so2n• we let w(A) =

and define

Pf (A)

where vol = e1

L Aij ei 1\ ej E A2(R2n) i<j

by the equation

1\ e 1\ 2

w(A) 1\ . . . 1\ w(A) = n!Pf(A)vol, .

..

1\ e2n· For the block matrix

a simple calculation gives

(4)

w(A) = a1 e1 1\ e2 + a2 e3 1\ e4 + · · · + an e2n-1 1\ ezn,

and one sees that

w(A) 1\ . . . 1\ w(A) It follows that

Pf(A)

= a1 . . . an and

=

n!(a1 . . . an) vol.

Pf (A)2 = det(A)

in this case.

B. INVARIANT POLYNOMIALS

231

Theorem B.S lf A E so 2n and B is an arbitrary matrix then

2 (i) Pf(A) = det(A) (ii) Pf(BABt) = Pf(A)det(B).

Proof. Since A is a real matrix and AAt = At A, A is normal when considered as element in M2n(C). The spectral theorem ensures the existence of an orthonormal basis of eigenvectors ei, . . . , e2n in C2n and eigenvalues Ai E C, A ei = Ai ei. By conjugation we see that the complex conjugate vector ei is an eigenvector with eigenvalue Xi . We claim that the basis can be chosen so that e2j = e2j -I (1 � j � n) . This is easy if all eigenvalues are zero, i.e. A = 0, so we may assume ei to be an eigenvector with non-zero eigenvalue AI. By skew-symmetry

AI = (Ae�, ei) = ( - e1 . Ae1 ) = -XI,

so A1 is purely imaginary. Pick e2 = e1, and note that e2 has eigenvalue A2 = XI f:. A1 . Hence e2 is orthogonal to e1 in C2n. Note that the orthogonal complement Span(ei. e1 )j_ is invariant under A and also under complex conjugation. This makes is possible to repeat the process in this subspace. Having arranged that e2j = e2j 1> we obtain an orthonormal basis for R2n

-

consisting of the vectors

Moreover for ai E R given by A2j-l = Hai we have Avj Awj = ajVj· This proves the existence of g E 02n such that

(

gAg-1 = diag ( -�

I

�1 ) . . . . , (-� n

= -ajWj

and

�n ) ) .

In particular

I 2 2 1 P£ (gAg - ) = (a1 . . . an) = det(gAg- ) = det(A) .

2 Since g-1 = gl, assertion (ii) implies that Pf(A) = det(A) for all A. In order to prove (ii) we consider the elements fi = Bei E R2n. fi = I: Bviev we have that T = L Aij /i 1\ /j

Since

= L Bvi Aij BJ.�-j ev 1\ e�-' = L (BABt) v�-' ev 1\ e�-',

so that T = w(BABt). Hence

w(BABt) 1\ . . . 1\ w(BABt)

= T 1\ . . . 1\ T = n! Pf(A)JI 1\ . . . 1\ hn ·

By Theorem 2.18 the map

. A2n (B) : A2n (R2n )

-t

A2n ( R2n)

232

B.

INVARIANT POLYNOMIALS

is multiplication by det (B), so that h 1\ . . . 1\ hn == det(B)e1 1\ . This gives (ii).

..

1\ e2n· 0

Let sun C Mn(C) denote the subset of skew-hennitian matrices. The realification map from Mn(C) to M2n(R) induces a map from sun to S02n. denoted A .- AR, and we have

Since a skew-hennitian matrix has an orthononnal basis of eigenvectors, we may assume A is diagonal, A == diag(Ha1, . . . , .;=Tan) with ai E R. Multiplication by ;=Tai on C corresponds to the matrix

Proof.

so by (4) Pf(AR)

=

(

and the result follows.

(0ai -a·0 ) -1ta 1 ... an.

t

E

so2

On the other hand

0

C. PROOF OF LEMMAS 12.12 AND 12.13

C.

233

PROOF OF LEMMAS 12.12 AND 12.13

In the proof of Lemma 1 2 . 1 3 the diffeomorphism of (iv) is obtained by applying Lemma 1 2.12 to a new Morse function F: M ._ IR, which coincides with f outside small neighborhoods of the critical points Pi· The sets Ui and Vi from (ii) and (iii) together with F in a neighborhood of Pi are constructed by means of the following lemma applied to f o ki 1 , where ki is a C00-chart around Pi given by Theorem 1 2.6. Without loss of generality we may assume f o k,; 1 to be in the standard form of Example 12.5. Lemma C.1 Let W � Rn be an open neighborhood of the origin in Rn and let

f: W

-t

R be the function ).

n

f(x) = a - :Lxf + L i=>-+1 i::::l

where a

E R, A E 7L and 0 :::; A :::; n. E

=

{

x

E Rn

Choose c

> 0 such that W contains the set

I txl + 2 t xf :::; 2 } c

i""l

Then there exists a Morsefunction F: W V � Rn-A+l that satisfy:

(i) (ii) (iii) (iv) (v)

xl, .

i=A+ 1

._

R and contractible open sets U � Rn,

F(x) = j(x) when x E W - E. The only critical point of F in W is 0 and F(O) < a - c. F-1 (( -oo, a + c)) = f -1 (( -oo, a, + c)). F- 1 ((-oo, a - c)) = j- 1 ((-oo, a - c)) U U. feomorphic with sA- l X V. f- 1 (( - oo , a - c )) n u is dif

Proof. We introduce the notation � =

2:�1 x7 and T/ = L�>.+ 1x7.

Then

(1)

and we define F E C00(W, R ) to be (2)

where J.t E C00(R, R) is chosen to have the properties:

(a) -1 < p!(t) :::; 0 for all t E R. (b) J.t(t) = 0 when t � 2t:. (c) f.L is constant on an open interval around 0 with value J.t(O) > c.

234

C. PROOF OF LEMMAS 12.12 AND 12.13

;

\

\ \

, , I I t I I

�·

- -,. €

"'·

f

l

I

-, 'j.l '

I

Figure 1

If x E W - E, then � + 2rt > 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 -- l

and 1J
0 we

by

q to q =

2 2 min(s - t, t - s /2)}.

=

(s, xA+b ... , xn) E V, then V ( s, 0, . .. , 0) and from q to ( so, 0, . .. , 0),

Define finally a diffeomorphism

B;

'l!(y, s, XA+l> . . . , Xn)

= (s . y, XA+1> . , Xn), ..

D

Some of the sets introduced in the proof above are indicated in Figure 2. We note that Figure 2 only displays a quarter of the constructed sets; it should be reflected in both the Vf. axis and the Jfi axis for n = 2, and rotated correspondingly for

236 n

> 2. In particular

C. PROO.t" OF LEMMAS 12.12 AND 12.13

E is represented by the quarter ellipse between Jf_ and

J2€.

E: m

U:

l;ssj

8: �

Figure 2 (U hatched;

r 1 ( (-oo, a - e)) n U

double-hatched)

Proof of Lemma 12.13. By Theorem 12.6 we can choose a smooth chart hi: vVi ---. Wi � IRn with Pi E lVi, hi(P) = 0 and such that .>.,

n

f o h,j 1(x) = a - 'L,:rJ + L xJ for x E Wi. j= l

j= >-.+ 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

.. .t:1 and show that the general case follows from the special case.)

2

2.12.

Let V be an n-dimensional vector space. Show for a linear map f: V -+ V the existence of a number d(j) such that

Altn(f)(w) = d(f)w for

w E Altn(V).

Verify the product rule

d(g 0 f )

= d(g ) d(f)

for linear maps f, g: V -+ V using the functoriality of Altn. Prove that d(f) = det(f). (Hint: Pick a basis e 1 , . . . , e for V, let q , . . . , En be the dual basis for n Altn(V) and evaluate Altn(f)(t:I 1\ . . . 1\ fn) on (e1, . . . , e ) in terms of n the matrix for f . with respect to the chosen basis.)

D. EXERCISES

246

3 . 1 . Show for

open set in R2 that the de Rham complex

an

is isomorphic to the the complex 0 --+ C00(U, R) �d C00 (U, R2 )

·� C00(U, R) --+ 0.

Analogously, show that for an open set in R3 the de Rham complex is isomorphic to

0

-t

C00(U, R)

g� C00(U, R3) � C00(U, R3 ) � C00(U, R) -+

0

defined in Chapter 1 . dxn the usual constant 1-fonns 3.2. Let U � Rn be an open set and dx1, (dxi = €i)o Let vol = dx1 1\ . . . 1\ dxn E nn(U). p Use *: AitP(Rn) Altn- (Rn) (from Exercise 2.9) to define a linear operator (Hodge's star operator) .

. 0

,

-t

and show that *(dxl 1\ 1\ dxp) = dxp l 1\ + ( n-p)_ w- 1 (U) by D -lt efine d*: QP(U) ( 0

0

0

0

0

0

1\ dx

n

and

*

o*

-t

d* (w) = ( - ltp+n- l * o d o * (w) .

Show that d* o d* = 0. Verify the formula

and more generally for 1 � i1 < i2 < d*(fdxi1 1\

o

o

.

1\

dxip) =

t (-It:�

v= l

0

0

0

< ip �

n

1\ . . . 1\

d;iv 1\

dXi1

v

that . . .

1\

3.3. With the notation of Exercise 3.2, the Laplace operator �: W(U)

is defined by

� = d 0 d* + d* 0 d.

dxiv· -t

QP(U)

Let f E n°(U). Show that � (fdx1 1\ . . . 1\ dxp) = �(f)dx1 1\ . . . 1\ dxp where

D.

247

EXERCISES

(Hint: Try the case p = 1, n = 2 first. What can one say about D. (! · dx1) where I = (i1 , . , ip )?) A p-form w E W(U) is said to be harmonic if ..!l(w) = 0. Show that .

.

*:

3.4.

DP(U) � nn -p(U)

maps harmonic forms into harmonic forms. Let AitP(Rm, C) be the C-vector space of alternating R-multilinear maps (p factors). Note that

w:

Rn

w

X

· •

•

X

Rn - C

can be written uniquely

w = Re w + i Im w,

where Rew E AitP(Rm), Imw E AIV'(Rn). Extend the wedge product to a C-bilinear map AitP(Rn, C) x Altq ( Rn , C) � Altp+q(Rn, C) 3.5.

and show that we obtain a graded anti-commutative C-algebra Alt* (Rn, C). Introduce C-valued differential p-forms on an open set U � Rn by setting (see Exercise 3.4) stP(U, C)

=

C00(U, AltP(Rn, C)).

Note that w E stP(U, C) can be written uniquely w

= Re w + i Im w,

where Rew E DP(U). Extend d to a C-linear operator d: stP(U, C) � stP+1(U, C)

3.6.

and show that Theorem 3.7 holds for C-valued differential forms. Generalize Theorem 3.12 to the case of C-valued differential forms Take U = C - {0} = R2 - {0} in Exercise 3.5 and let z E D.0(U, C) be the inclusion map U � C . Write x = Rez, y = Imz. Show that Re (z-1dz) = dlogr,

where

r:

U � R is defined by r(z)

lzl

=

Jx2 + y2• Show that

X -y dx dy. + 2 2 X + y2 +y (Observe that this is the 1-form corresponding to the vector field of Example 1.2.) Prove for the complex exponential map exp: C � C* that

Im (z-1 dz) =

3.7.

=

2 X

dz exp = exp(z)dz and exp*(z-1 dz)

=

dz.

D. EXERCISES

248 4. 1 .

Consider a commutative diagram of vector spaces and linear maps with exact rows A 1 - A2 - A3 - A4 - As

lh

!h

lh

. lh

lh

!h

lh

4.2.

4.3.

!h

B1 - B2 - B3 - B4 - Bs Suppose that !4 is injective. h is surjective and h is injective. Show that /3 is injective. Suppose that h is surjective, /4 is surjective and J5 is injective. Show that h is surjective. In particular we have that if /I , /2, f4 and J5 are isomorphisms, then h is an isomorphism. (This assertion is called the 5-lemma.) Consider the following commutative diagram O - A1 - A2 - A3 - 0 O - B� - B2 - B3 - 0 where the rows are exact sequences. Show that there exists a exact sequence 0 -+ Ker h -+ Ker h -+ Ker h -+ -+ Cokfi -+ Cokh -+ Cok /3 -+ 0. (Hint: Try the long exact cohomology sequence). In the commutative diagram

0

0

!

0 _ Ao,o

__

!

0

!

0

A1,0 _ A2,0

__

!

A3,0 _ . . .

!

!

!

!

!

!

!

l

l

l

l

!

l

!

!

!

0 - AO,l - Al,l -- A2,1 -- A3,1 -- . . . o_

Ao,2 _ A1,2 _ A2,2 _ A3,2

__ .

.

.

0 _ Ao,3 _ A1,3 _ A2,3 -- A3,3 _ . . .

the horizontal (A*·q) and the vertical (AP•*) are chain complexes where Ap,q = 0 if either p < 0 or q < 0. Suppose that HP( A*·q) = 0 for q :fo.O and all p Hq(AP•*) = 0 for p :fo.O and all q.

Construct isomorphisms HP( A*•0)

-+

HP( A0•*) for aJl p.

D.

4.4.

do

dl

EXERCISES

249

dn- 1

Let 0 ---+ A0 ---+ A1 ---+ · · · ---+ An ---+ 0 be a chain complex and assume that dimR Ai < oo. The Euler characteristic is defined by n x(A*) = L ( - 1) i dim Ai. i=O Show that x(A*) = 0 if A* is exact. Show that the sequence

0 ---+ Hi( A*) ---+ Ai /Im di-1 ! Im di ---+ 0 is exact and conclude that

4.5.

5.1.

dimR Ai - dimR Im di-1 = dimR Hi(A*) + dimR Im di. . n Show that x (A* ) = L: ( -1)t dimR Hi(A*) . i=O Associate to two composable linear maps an exact sequence 0 ---+ Ker(f) ---+ Ker(g o f) ---+ Ker(g) ---+ ---+ Cok(f) ---+ Cok(g o f) ---+ Cok(g) ---+ 0. Adopt the notation of Example 5.4. A point (x, y) E U1 can be uniquely described in terms of polar coordinates (r, O) E (O,oo) x (0,211'). Let arg1 E 0°(U1 ) be the function mapping (x, y) into 0 E (0, 27r) (why is arg1 smooth?). Define similarly arg2 E 0° ( U2) using polar coordinates with 0 E ( -1r, 1r) and prove the existence of a closed 1-form r E 01 (R2 - {0}) such that

(v = 1 , 2) . . 1luv = i�(r) = d arg11 Show that the connecting homomorphism &0: H0(U1 n U2) ---+ H1(R2 - {0}) carries the locally constant function with values {0, 27T} on the upper and lower half-planes respectively into [r]. . ' 5.2. Show that the 1-forms T E n 1 (R2 - {0}) of Exercise 5 . 1 and Im(z-1dz) of Exercise 3.6 are the same. 5.3. Can R2 be written as R2 = U U V where U, V are open connected sets such that U n V is disconnected? 5.4. (Phragmen-Brouwer property of Rn) Suppose p # q in Rn. A closed set A £; Rn is said to separate p from q, when p and q belong to two different connected components of Rn - A. Let A and B be two disjoint closed subsets of Rn. Given two distinct points p and q in Rn - (A U B). Show that if neither A nor B separates p from q, then A U B does not separate p from q. (Apply Theorem 5.2 to U1 = Rn - A, U2 = Rn - B.)

250

D. EXERCISES

6. 1 . Show that "homotopy equivalence" is an equivalence relation in the class of topological spaces. 6.2. 6.3.

Show that all continuous maps f: U --+ V that are homotopic to a constant map induce the 0-map f*: JfP(V ) --+ JfP(U) for p > 0.

{

Let Pb . . . , Pk be k different points in

Hd(Rn - {PI. · · · , Pk }) �

Rn, n � 2.

IRk R 0

Show that

for d = n for d = O otherwise.

1

6.4. Suppose f, g: X --+ sn are two continuous maps, such that f(x) and g(x) are never antipodal. Show that f � g. Show that every non-surjective continuous map f: X --+ sn is homotopic to a constant map. 6.5.

Show that sn-l is homotopy equivalent to continuous maps

Rn - {0}.

are homotopic if and only if their restrictions to 6.6. 7. 1 . 7.2.

7.4.

are homotopic.

sn- 1 is not contractible. Show that Rn does not contain a subset homeomorphic to Dm when m > n. Let E � Rn be homeomorpic to Sk (1 s; k s; n - 2). Show that Show that

JfP (Rn _ E) � 7.3.

sn- 1

Show that two

{R 0

for p =: 0, n - k - 1, n otherwise.

-1

--+ sn- l with g lsn- 1 � idsn-1. Let f: Dn --+ Rn be a continuous map, and let r E (0, 1) be given. Suppose for all X E sn-l that llf(x) - xll � 1 - r. Show that Im f(Dn) contains Show that there is no continuous map g: Dn

the closed disc with radius r and center 0. (Hint: Modify the proof of Brouwer's fixed point theorem and use Exer­ cises 6.4 and 7.3.)

7 .5. Assume given two injective continuous maps ex , /3:

ex( 0) = ( - 1 , 0) , /3(0) = (0, - 1) ,

[0, 1)

--+

D2

such that

ex(1) = (1, 0) /3(1) = (0, 1).

Prove that the curves ex and /3 intersect (apply both parts of Theorem 7.10). 8.1. Fill in the details of Remark 8.2.

8.2. Let cp: N --+ M be a continuous map from a smooth manifold N to a smooth submanifold M of Rk. Let i: M --+ Rk be the inclusion. Show that cp is smooth if and only if i o cp is smooth.

D. EXERCISES

8.3.

8.4.

8.5 .

251

Suppose that M � Rk (with the induced topology from Rk) is an n­ dimensional topological manifold. Include M in Rk+n. Show that M is . locally flat in Rk+n

Set Tn = Rnj7Ln, i.e. the set of cosets for the subgroup zn of Rn with respect to vector addition. Let 1r : Rn -+ Tn be the canonical map and equip Tn with the quotient topology (i.e. W � Tn is open if and only if 1r - 1 (W) is open in Rn). Show that Tn is a compact topological manifold of dimension n (the n­ dimensional torus). Construct a differentiable structure on Tn, such that 1r becomes smooth and every p E Rn has an open neighborhood that is mapped diffeomorphically onto an open set in Tn by 1r . Prove that T1 is diffeomorphic to s 1 . Define A: R2

-+ R2

by

A(x,y) = (x + �. - y)

smooth map A: T2 -+ T2 satisfying A o 7f = 8.4.) Show that A is a diffeomorphism, that A for all q E T2 .

.

Show that there exists a

A. = A- 1 1r o

(Consult Exercise and that A(q) =I= q

Let K2 be the set of pairs {q,A(q)}, q E T2. Show that K2 with the quotient topology from T2 is a 2-dimensional topological manifold (Klein's bottle). 2 Construct a differentiable structure on K . 8.6.

9.1.

9.2.

Let Po E sn be the "north pole" Po = (0, . . . , 0, 1). Show that sn - {po} is diffeomorphic to Rn under stereographic projection, i.e. the map sn {Po} -t Rn that carries p E sn into the point of intersection between the line through Po and p and the equatorial hyperplane Rn � Rn+ l .

M � Rl be a differentiable submanifold and assume the points p E R1 and Po E M are such that l i P - Po ll � liP - qll for all q E M. Show that p - Po E Tp0M.l.. A smooth map t.p: Mm -+ Nn between smooth manifolds is called immer­ sive at p E M, when Let

injective. Show that there exist smooth charts (U, h) in M h(p) = 0, and (V, k) in N with q E V, k(q) = 0, such that

is

with

p

E U,

in a neighborhood of 0. (Hint: Reduce the problem to the case where t.p: W -+ Rn is smooth on an open neighborhood W in Rm of 0 with

=

( llx iJ)

0 is given by log r (r) 1 _ 2-n _ 2_n r =

{

if n = 2 1'f n > 3 . _

Show for the closed form w defined by equation ( 19) of Example 9.18 that *dP = w, and conclude that p is harmonic (i.e. D.p = 0). Apply the last identity of Exercise 10.14 to f = p, g E Cgo(Rn, R) and Re = {x E Rn I f. � llxll � a}, where f. > 0 and a is suitably large, to obtain

{ pD.g JR"-eDn

=

1eSn-l

p 1\ * dg

- 1eSn-l g 1\

w,

where EDn is the open disc of radius t around 0 and tsn- l is its boundary sphere. Show that lim

e-+0+

1

eS"-1

p 1\ *dg = 0 and

lim

�-+0+

1

eS"-1

-

g 1\ w = Vol ( sn 1 )g(O).

(Hint: The pull-back of w to c:sn- 1 is c: 1 -n Volesn-L) Conclude that

g(O)

= (��- 1. Vol

1)

pD.g d n p,

for every g E c;o (Rn, C), where the right hand side is the Lebesgue integral.

D.

EXERCISES

259

1 1 . I . Generalize the concept of degree to the case of continuous maps in such a way that Corollary 1 1.2 holds for f continuous. Show that Corollary 1 1.3, the statement deg (f) E l of Theorem 1 1 .9, Corollary 1 1 . 1 0 and Proposition 1 1 . 1 1 generalize to the case of continuous maps. 1 1 .2. Prove the formula of Theorem 1 1.9 under the following assumptions: (i) f is continuous. (ii) f is smooth outside a closed set A � (iii) p E M - f(A) is a regular value of f restricted to A. (Hint: Lemma 1 1.8 holds with U � M - f(A), A. Construct a homotopy F from f to a smooth map such that Ft (0 :::; t :::; 1) is the identity near the points qi E j-1(p).)

N. Vi � N-N -

Nn

Nn n

be an oriented closed (i.e. compact) manifold. Show that every 1 1.3. Let � 5 integer occurs as the degree of some smooth map (Use Exercise 1 1.2.) 1 1 .4.

-1 n P(z) = zn + Q(z) = zn + 2.:

(The fundamental theorem of algebra) Suppose j=O

CLjZj

is a complex polynomial of degree n � 1 without any complex root. This leads to a contradiction as follows: Regarding 51 as the unit circle in C define for any r � 0 a smooth map

fr : 51

�

S1 ;

P(nu) fr(w) = IP(rw )l"

=

Show that deg(fr ) is independent of r and then (by taking r 0) that = 0. Prove for r chosen suitably large that the expression

deg(fr)

F( w, t)

1 1 .5.

+ tQ( r w) = i (rwt (rw t + tQ(rw) i ' = n.

0 :::; t :::; 1

defines a homotopy 51 x [0, 1] � 51 between fr and the map and conclude from this that deg(fr)

w

sn-l,

t-t

'Wn

Let E � W1 be a smooth submanifold diffeomorphic to n � 2, and let Ul> U2 be the open sets given by the Jordan-Brouwer separation theorem 7 . 1 0. For xo E Rn - E define

F: E

�

5n- 1 ;

Show that

deg(F)

={

F (X ) 0

if

±1 if

= II xo

xo

X - Xo

X - Xo

II "

E U2

E u1

where the sign depends only on the orientation of E.

260

1 1 .6.

D. EXERCISES

Prove the case n < m of Theorem 1 1 .5 (reduce to the case of equal dimensions). Apply this to show that every map f: Nn -t sm from a smooth man ifold of dimension n < m i s homotopic to a constant.

1 1 .7. Let U � Rn be a bounded open set and A a compact subset of U. Prove the existence of a compact domain with smooth boundary R such that

A 0, m > 0

that

Zjnl ®z ljml � ljdl

is the greatest common divisor of n and

m.

270

D.

EXERCISES

16.2. Prove the isomorphisms (4) listed above Lemma 16.4. 16.3. Let V be a finite-dimensional complex vector space with Hermitian inner product ( ) . Construct a hermitian inner product on i\k v satisfying ,

16.4. Let F be a covariant functor from the category of finite-dimensional real vector spaces to itself. Assume F to be smooth in the sense that the maps induced by

F HomR(V, W) - HomR(F(V), F(W))

are smooth. Construct for a given smooth real vector bundle � over B another smooth real vector bundle F(O over B with fibers F(Ob = F(�b), b E B. Show that this extends F to a covariant functor from the category of smooth real vector bundles over B and smooth homomorphisms over idB to itself. (Note: Many variations are possible: R can be replaced by C in the source and/or target category, F can be contravariant, and smooth can be changed to continuous.) 16.5. Prove that the construction of Exercise 1 6.4 is compatible with pull-back, i.e. construct for h: B' - B a smooth vector bundle isomorphism 1/J( F(h*(f.)) - h'"(F(f.)).

Show that pull-back by h extends to a functor h* from the category of vector bundles over B (see Exercise 16.4) to the corresponding category over B', and that the diagram F(h*(f.)) _i!s_ h*(F(�))

lF(h"(g))

1h.(F(g))

F(h* ( 1J) ) � h*(F (1J) )

commutes for every § : f. - 1J over idB. 16.6. Consider two smooth covariant functors F, G as in Exercise 16.4. As­ sume given (for each finite-dimensional vector space V) a linear map ¢>v: F(V) - G(V) such that the diagram F(V) ...!h:- G(V)

J F(h)

Jc(h)

F(W ) � G(W)

is commutative for every h E HomR(V, W). Construct for a given smooth real vector bundle over B a smooth bundle

D. EXERCISES

271

homomorphism ¢>( F(O ---+ C(�) over ids. and show that any smooth vector bundle homomorphism g: � ---+ 17 over ids leads to a commutative diagram F(O ___:h_ G(O

lF(ry) �G(ry)l F(g)

G(g)

Prove finally with the assumptions and notation of Exercise 1 6.5 that the diagrams

F(h*(�))

rf>h·(o

G(h*(0)

l�e h.(rt>e.) l�e

h*(F(O) commute.

h*(G(�))

16.7. Let V be a finite-dimensional real or complex vector space. Construct a linear map e: ®k V ---+ ® k V such that

e (v1 ® · · · ® Vk ) =

�! L

uES( k)

sign(o- )vu(I) 0 · · · ® Vu(k)·

Show that e o e = e, and that the quotient homomorphism ®kV ---+ AkV induces an isomorphism Im(e) � AkV. Apply Exercise 1 5 . 1 4 to give an alternative construction of the vector bundles Ak(O. 16.8. For finite-dimensional vector spaces V and W, construct a linear map carrying (v1

AkV 0 A1W ---+ Ak+1(V EB W) A . . . A vk

(0, w1 ) A . . . A (0, w1 ).

into (vh 0) A . . . A (vb 0) Show that these combine to give isomorphisms

)

®

(w1 A . . . A w1)

A

n

E9 Ak V 0 An-kw � An(v EB W). k=O

Extend this to vector bundle isomorphisms

n

EB Ak� ® An-k'l7 � An(� EB ry) . k=O

16.9. Prove Lemma 1 6. 1 0.(i). 16.10. Finish the proof of Theorem 16.1 3.(iv).

16. 1 1. Let � and 17 be smooth vector bundles over a compact smooth manifold M. Show that � and 17 are isomorphic if and only if n°(0 � n°(ry) as !1°(M)­ modules. Prove that every finitely generated projective !1°(M)-module P is isomorphic to n°(ry) for some smooth vector bundle 17 over M. (Hint: Apply Exercise 15.14 with � = €n.)

D.

272

16.12.

EXERCISES

Prove for any line bundle � (real or complex) that Hom(�, () is trivial. Prove for any real line bundle that �

16.13.

Let

T* M

manifold

®�

is trivial.

be the total space of the dual tangent bundle

Mn ,

T* M - M T1r(q)M) define

and

linear form on

-rr :

the projection. For

Show that this defines a differential 1-form

0

0

on

T* M

rM

q

E

of a smooth

T* M

(i.e. a

and moreover that

can be given in local coordinates by the expression

where

x1

, . . . , Xn

are the coordinate functions on a chart

t 1 , . . . , tn are linear dpXJ, . . . , dpXn· Show that

coordinates on

(T* M, dO)

U � M

and

r; M, p E U with respec t to the basis

is a symplectic manifold (see Exercise 1 4.7).

17 . 1 . A derivation on an R-algebra A is an R-linear map D: A - A that

satisfies

the identity

D (xy)

=

These form an R-vector space

(Dx)y + x(Dy). Der A.

Show for any smooth manifold

Mm

that there is a linear isomorphism

which to a vector field

X

assigns the derivation

Lx(f)

�

Xj

=

Lx

given by

dj (X ).

(Hint: Derivations are local operators.) Show that the commutator also belongs to

[D1 , Dz] = D1 oDz-DzoDl of D1 , Dz

DerA, and define the Lie bracket X, Y o n M by the condition

smooth vector fields

[X, Y]

E

E

DerA

D0( 'TM)

of

L[X,Y) = [Lx , Ly]. Prove that

D0 (TM )

is a Lie algebra, i.e. that the following conditions hold

(L I ) [ , } is bilinear. (L2) (X, X] = 0. (L3) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]J

=

0

(Jacobi identity).

D.

EXERCISES

Show for X, Y E r2°(rM) and f, g

E

273

r2°(M) that

[JX, Y] = f [X, Y] - (Yf )X [X, gY] = g[X, Y] + (Xg)Y.

Suppose X and Y Remark 9.4)

are

given on a chart U � M by the expressions (see

Y=

a L bi �· ux1 m

j= l

Show that in these coordinates [X, Y] is given on U by

[X,Y] = 17.2. Prove for w E

cku(Xl >

. . . l

( obk m oak ) L L ai� -L . bi� . k m

m

=1 �=1

uX,

J =l

uXJ

a

�· UXk

D,P(M) and Xj E r2°(rM ) , 1 � j � p + 1, the formula

Xp+l) =

p+l

L( i=l

-

w , ... ...

l )i - l xi ( (Xl

' xi ,

' Xp+I))

In particular

dw(X, Y) = X(w(Y)) - Y(wX) - w([X, Y]). for w E r2 1 (M), X, Y E r2°(rM) 17.3. Let M be a Riemannian manifold with metric ( , ) Prove the existence and uniqueness of a connection \1 on TM such that \1 x : n° ( r) 0° (r) satisfies the following two conditions for smooth vector fields X, Y, Z:

.

(a) X((Y, Z)) = (VxY, Z) + (Y, \1 xZ) (b) \lxY - \JyX = [X, YJ

Prove moreover that

2(\lxY, Z)

\7

-

satisfies the Koszul identity

= X(Y, Z) + Y(Z,X) - Z(X, Y)

- (X, [Y, Z]) + (Y, [Z, X]) + (Z, [X, Y]). (Hints: Let A(X, Y, Z) be the 6-term expression above. To prove unique­ ness, derive the Koszul identity by rewriting the six terms using the equa­ tions obtained from (a) and (b) by cyclic permutation of X, Y, Z. For existence verify first the identity A( X, Y, f Z) = fA(X, Y, Z), where f E D.0(M), and construct \7 x Y so that the Koszul identity holds.) This connection is the Levi-Civita connection on M.

274

1 7.4.

17.5.

� Rn+k

Prove for Mn

D.

EXERCISES

that the connection on 'TM constmcted in Example

17.2 is the same as the Levi-Civita connection of Exercise

is given the Riemannian metric induced from

1 7.3,

when j\tf

Let \7 be any connection on the vector bundle � over A1. Given two vector fields X, Y E 0°('TM) define the operator R(X, Y ) : 0°(0 R(X, Y)

= \lx o \Jy -

Verify that R(X, Y)

R(X, Y) =

is

F'j y ·

\Jy

o

\lx - '7[X,YJ·

-4

0°(0 by

a n°(M)-module homomorphism, and prove that

(Hint: Work loc�lJy using the formula

17 .6.

Rn+k .

that the two operators agree on

P-i-)

(1 7.4).

Show by direct computation

Let Mn be a Riemannian manifold and \7 the Levi-Civita connection on 'TM from Exercise 17.5.

17.3.

Define R(X, Y): 0°(,.,.M) -4 0°('TM) as in Exercise

Prove the Bianchi identity R(X,

for X, Y, Z

E

Y)Z + R(Y, Z)X + R(Z, X)Y = 0

D0('TM)·

Show that the value of R(X, Y)Z at p E M depends only on Xp, Yp , Zp E

Tp M. Hence

17.7.

R defines for Xp, Yp

TpM.

E Tp M a linear map R(Xp, Yp) : Tp M

h: U -4 U' � Rn be a chart in Mn and Oi U considered in Remark 1 7.4. The Christoffel

Let on

functions

rfi

on

U'

=

Prove the formula

(9ij)

the vector fields

L (rt h)ok. o

k

(

r ic = � � kl agjl where

0�,

symbols are the smooth

detem1ined by

vaJJi

�]

=

2

�g

f)xi

+

agil _ OXj

agii ). OXL

is the matrix of coefficients to the first fundamental form (see

below Definition 9.15) and where

k1 (g )

is the inverse matrix

(Hint: Apply the Koszul identity (cf. Exercise Define functions R':Jk on

U'

by

17.3)

to

(

)

(9ij ) - 1 .

Oi, Oj, 8�c.)

m Show that

---7

arm ki ar"!!Jt � rL ifm fl m Rm ijk = � -;-le jl - ji r kl . 1 ux1 uXk + �

.

D.

1 7.8.

EXERCISES

\7 be a connection on � and e1, . . . , ek on the same open set U (Aij ), A = (Aij ) of 1-forms on U Let

Let =

(