Curvature and Characteristic Classes [640] 9783540086635

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Lecture Notes in Mathematics Edited by A Dold and B. Eckmann

640 Johan L. Dupont

Curvature and Characteristic Classes

Springer-Verlag Berlin Heidelberg New York 1978

Author Johan L. Dupont Matematisk Institut Ny Munkegade DK-BOOO Aarhus C/Denmark

AMS Subject Classifications (1970): 53C05, 55F40, 57D20, 58AlO, 55J10 ISBN 3-540-08663-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08663-3 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 64 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-643210

INTRODUCTION

These notes are based on a series of lectures given at the Mathematics Institute, University of Aarhus, during the academic year 1976-77. The purpose of the lectures was to give an introduction to the classical Chern-Wei 1 theory of characteristic classes with real coefficients presupposihg only basic knowledge of differentiable manifolds and Lie groups together with elementary homology theory. Chern-Wei 1 theory is the proper generalization to higher dimensions of the classical Gauss-Bonnet theorem which states that for

M

a compact surface of genus

( 1)

where

g

in 3-space

2 (1-g)

K

is the Gaussian curvature. In particular

topological invariant of

M.

JM

K

In higher dimensions where

a compact Riemannian manifold,

K

in (1)

is a M

is

is replaced by a

closed differential form (e.g. the Pfaffian or one of the Pontrjagin forms, see chapter

examples 1 and 3)

associated to

the curvature tensor and the integration is done over a singular chain in

M.

In this way there is defined a singular cohomology

class (e.g. the Euler class or one of the Pontrjagin classes) which turns out to be a differential topological invariant in the sense that it depends only on the tangent bundle of

M

considered as a topological vector bundle. Thus a repeating theme of this theory is to show that certain quantities which

a

priori depend on the local differential

geometry are actually global topological invariants. Fundamental

IV

in this context is of course the de Rham theorem which says that every real cohomology class of a manifold

M

can be re-

presented by integrating a closed form over singular chains and on the other hand if integration of a closed form over singular chains represents the zero­cocycle then the form is exact. In chapter 1 we give an elementary proof of this theorem (essentially due to A. Weil [34]) which depends on 3 basic tools used several times through the lectures: gration operator of the Poincare lemma, covering,

(i) the inte-

(ii) the nerve of a

(iii) the comparison theorem for double complexes

(I have deliberately avoided all mentioning of spectral sequences) . In chapter 2 we show that the de Rham isomorphism respects products and for the proof we use the opportunity to introduce another basic tool:

(iv) the Whitney­Thorn­Sullivan theory of

differential forms on simplicial sets. The resulting simplicial de Rham compleK, as we call it, connects the calculus of differential forms to the combinatorial methods of algebraic topology, and one of the main purposes of these lectures is to demonstrate its applicability in the theory of characteristic classes occuring in differential geometry. Chapter 3

contains an account of the theory of connection

and curvature in a principal G­bundle

(G

a Lie­group) essential-

ly following the exposition of Kobayashi and Nomizu [17>]. The chapter ends with some rather long exercises (nos. 7 and 8) explaining the relation of the general theory to the classical theory of an affine connection in a Riemannian manifold. Eventually, in chapter 4 we get to the Chern­Wei I construction in the case of a principal G­bundle a connection

e

and curvature

manifold mentioned above

G = O(n)

IT:

E

M with

(in the case of a Riemannian and

E

is the bundle of

v orthonormal tangent frames).

In this situation there is

associated to every G-invariant homogeneous polynomial the Lie algebra

p(n k)

a closed differential form

defining in turn a cohomology class

WE(P)

EH

2k

P

on

on

M

(M,lR).

Before proving that this class is actually a topological invariant of the principal G-bundle we discuss in chapter 5 the general notion of a characteristic class for topological principal G-bundles. By this we mean an assignment of a cohomology class in the base space of every G-bundle such that the assignment behaves naturally with respect to bundle maps. The main theorem (5.5) of the chapter states that the ring of characteristic classes is isomorphic to the cohomology ring of the classifying space

BG.

Therefore, in order to define the characteristic class WE(P)

for

E

any topological G-bundle it suffices to make the

Chern-Weil construction for the universal G-bundle BG.

Now the point is that although

BG

EG

over

is not a manifold it

is the realization of a simplicial manifold,

that is, roughly

speaking, a simplicial set where the set of p-simplices constitute a manifold. Therefore we generalize in chapter 6 the simplicial de Rham complex to simplicial manifolds, and it turns out that the Chern-Weil construction carries over to the universal bundle. In this way we get a universal Chern-Weil homomorphism W:

where

I*(G)

r*(G) ... H*(BG,lR)

denotes the ring of G-invariant polynomials on

the Lie algebra In chapter 7 we specialize the construction to the classical groups obtaining in this way the Chern and Pontrjaging classes

VI

with real coefficients. We also consider the Euler class defined by the Pfaffian polynomial and in an exercise we show the Gauss­Bonnet formula in all even dimensions. Chapter 8 is devoted to the proof of the theorem (8.1) due to H. Cartan that for

G

H*(BG,JR)

is an isomorphism

a compact Lie group. At the same time we prove A.Borel's

theorem that of

w: I*(G)

H*(BG,JR)

H*(BT,JR)

is isomorphic to the invariant part

under the Weyl group

W of a maximal torus

T.

The corresponding result for the ring of invariant polynomials (due to C. Chevalley) depends on some Lie group theory which is rather far from the main topic of these notes, and I have therefore placed the proof in an appendix at the end of the chapter. The final chapter 9 deals with the special properties of characteristic classes for G­bundles with a flat connection or equivalently with constant transition functions. If

G

is

compact it follows from the above mentioned theorem 8.1 that every characteristic class with real coefficients is in the image of the Chern­Weil homomorphism and therefore must vanish. In general for

K

G

a maximal compact subgroup we derive a

formula for the characteristic classes involving integration over certain singular simplices of

G/K.

As an application we

prove the theorem of J. Milnor [20] that the Euler number of a flat

Sl(2,JR)­bundle on a surface of genus

value less than

h

has numerical

h.

I have tried to make the notes as selfcontained as possible giving otherwise proper references to well­known text­books. Since our subject is classical, the literature is quite large, and especially in recent years has grown rapidly, so I have made no attempt to make the bibliography complete.

VII

It should be noted that many of the exercises are used in the main text and also some details in the text are left as an exercise. In the course from which these notes derived the weekly exercise session played an essential role. I am grateful to the active participants in this

course, especially to

Johanne Lund Christiansen, Poul Klausen, Erkki Laitinen and S¢ren Lune Nielsen for their valuable criticism and suggestions. Finally I would like to thank Lissi Daber for a careful typing of the manuscript and prof. Albrecht Dold and the Springer-Verlag for including the notes in this series.

Aarhus, December" 15, 1977.

CONTENTS Chapter

PClge

1.

Differential forms and cohomology

2.

Multiplicativity.

3.

Connections in principal bundles

38

4.

The Chern-Weil homomorphism

61

5.

Topological bundles and classifying spaces

71

6.

Simplicial manifolds. for BG

89

The simplicial de Rham complex

20

The Chern-Weil homomorphism

7.

Characteristic classes for some classical groups

97

8.

The Chern-Weil homomorphism for compact groups

114

9.

Applications to flat bundles

144

References

165

List of symbols

168

Subject index

170

CURVATURE AND CHARACTERISTIC CLASSES

1.

Differential forms and cohomology

First let us recall the basic facts of the calculus of differential forms on a differentiable manifold differential form

w

vector fields

•• 'X

w(X 1 , .•. ,X

k)

w(X 1 , ••. ,X k)p

X1"

of degree

k

associates to

a real valued

k

M.

COO

A k

function

such that it has the "tensor property" depends only on

X1p""'Xk p

For an l-form

and a k-form

(i.e. p E M)

for all

x1 , ... ,X k.

and such that it is multilinear and alternating in

the

COO

is

the product

(k+l)-form defined by

where

runs through all permutations of

0

1, ... ,k+l.

This

product is associative and graded commutative, i.e.

Furthermore there is an exterior differential k-form

w

associates a

(k+1)-form

dw

d

defined by

1 [k+1 i+' (do ) (X1, ... ,X k+ 1) = k+1 J1(-1) Xi(w(X"

+

L

i

M,

cc- (U) ,3)

such that and

rc- (U), 0)

be the corresponding chain or co chain complexes (called "with support in

un) and let C* (M)

->

C* (U )

be the natural maps induced by the inclusion

10

1

c SOO (M). n

SOO (U)

:

n

Then

1

*

and

1

*

are chain

equivalences, in particular they induce isomorphisms H(C*(M)),

H(C*(U))

H(C*(M))

H(C*(U)).

We now define a natural map

by the formula (1.13)

I

I (w) a =

a E Soo{M).

a*w,

n

is clearly a natural transformation of functors, that is, if COO

is a

map,

I where

f*:

A*(N)

then

f* = f#

0

A*(M)

and

0

I,

f#

C*(N)

C*{M)

are the

induced maps. Lemma 1.14.

I

is a chain map, i.e. I

In particular

o

d

0

l(dw)T

I.

induces a map on homology

I

H(C*

1: H(A*(M)) Proof.

0

(M)) •

This simply follows using exercise 2 above:

r n+1

I

oJ

n-e l i=O W

. (_1)1

dT*w

.

{E1)*T*W

{_1)1

i=O

I

r

+1 T* (dw)

I(w)

E An(M),

T

()= Ei

E

o(l(w))T'

T

n+l (M) .

SOO

11

Theorem 1.15.

(de Rham).

an isomorphism for any

COO

I : H*(A*(M»

manifold

H(C*(M)

is

M.

First notice: Lemma 1.16.

Theorem 1.15 is true for

M diffeomorphic to

a star shaped open set in llin . Proof.

It is clearly enough to consider

open set star shaped with respect to consider the homotopy and

g(-,O) = e

g: U

x

e

E U.

U

[0,1]

M = U clli

n

an

As in Lemma 1.2

with

g(-,l)

=

id

given by g(x,s) = sx + (l-s)e.

By (1.11) the inclusion

{e}

U

induces an isomorphism

in singular cohomology, so the statement follows from (1.10) together with Lemma 1.2 and the commutative diagram H (A* (U»

-

1

H(A* (e)

I

H (C* (U»

I

H (C*

II

(e )

II ]R

JR

Lemma 1.17.

For any

there is an open covering

COO

manifold

U = {Ua}aEL'

M of dimension

such that every non-

U n... n U , aO, •.. ,a p E aO ap diffeomorphic to a star shaped open set of ]Rn. empty finite intersection

Proof.

Choose a Riemannian metric on

point has a neighbourhood every point of

U

U

M.

U).

(i.e., for every

In particular,

U

I,

is

Then every

which is normal with respect to q E U,

eXP q

diffeomorphism of a star shaped neighbourhood of onto

n

is a

oE

T

q

(M)

is geodesically convex, that is,

12

for every pair of points segment in U.

M

joining

p,q E U p

and

there is a unique geodesic

q

and this is contained in

[14, Chapter I Lemma

(For a proof see e.g. S. Helgason

= {Ua}aEZ

6.4).

Now choose a covering

sets.

Then any non-empty finite intersection

U

with such open

n ... n u a k

is again geodesically convex and so is a normal neighbourhood of each of its points.

It is therefore

clearly diffeomorphic to a star shaped region in

(via

the exponential map).

In view of the last two lemmas it is obvious that we want to prove Theorem 1.15 by some kind of formal inductive argument using a covering as in Lemma 1.17.

What is needed

are some algebraic facts about double complexes: We consider modules over a fixed ring shall only use

A complex

R =

with a differential

C

-+

n+1

Similarly, a double complex is a

ip,ql

cp,q,

,

z::

is a

nEZ::, x

(actually we module

such that

dd = O.

Z::-graded module

together with two differentials ... CP

d'

C*

R

+1, q ,

satisfying (1. 18)

d'd'

0,

d"d"

d"d' + d'd"

0,

We shall actually assume that complex, that is, Associated to

cp,q = 0

(C*'*,d' ,d")

C*,*

O.

is a 1. quadrant double or

q < O.

is the total complex

(C* ,d)

if either

p < 0

where d

d'

+ d".

13

For fixed to

d'.

q

we can take the homology of

C*,q

with respect

This gives another bi-graded module

Now suppose

1c*,*

as above, and suppose

and

2C*,*

are two double complexes

f

respecting the grading and commuting with clearly

f

d'

and

d".

Then

gives a chain map of the associated total complexes

and hence induces

Also clearly

f*: H(1c*,d) .... H(2c*,d).

.... EP,q f 1 .. 1EP,q 1 2 1

induces

HP(C*,q,d').

=

Lemma 1.19.

Suppose

We now have: f : 1C*,* .... 2C*,*

of 1. quadrant double complexes and suppose is an isomorphism.

f

Then also

is a homomorphism f 1 : 1E1'* .... 2E1'*

f* : H(1C*) .... H(2C*)

is an

isomorphism. Proof. complex

For a double complex

(C*,d)

(C*,*,d' ,d") F*

define the subcomplexes

q

with total

= C*,

C

q E:

7l,

by

F* q

Then clearly ::>

and

d : F* .... F*. q

isomorphic to

q

F* q-l

::>

F* q

::>

Notice that the complex

(C*,q,d').

Therefore for

is f

1

C*,* ....

2

C*,*

a map of double complexes the assumption that f

1

:

EP'q....

1 1

that

EP,q

2 1

f:

is an isomorphism, is equivalent to saying ....

in homology.

q E: 7l,

Now by induction for

induces an isomorphism

r = 1,2, ...

it follows

from the commutative diagram of chain complexes 0

....

0

....

1F*q+r / 1F*q+r+1 H

2F q+r / 2 F*q+r+l

.... 1F*/ .... 1F*/ q 1F*q+r q 1F*q+r+1 H

H

....

2F*/ q 2F*q+r+1

.... 0

....

2 F*/ q 2 F*q+r

.... 0

14

and the five lemma that f:

F*/ F*

F*/ F*

1 q 1 q+r

2 q 2 q+r

q E

induces an isomorphism in homology for all r

=

C*, *

1,2, . . . .

and

However, for a 1. quadrant double complex

we have

o

for

r > n

so the lemma follows. Remark.

Interchanging

similar lemma with

p

and

replaced by

q

in H

cp,q

q (C p ,

we get a

* , d ")

•

Notice that for a 1. quadrant double complex follows from (1.18) that denoted

d"

d" :

it

induces a differential also for each

E O,q 1

C*,*

In particular, since

p.

ker(d'

we have a natural inclusion of chain complexes e

(Ef'* ,d")

(C* ,d)

(called the "edge-homomorphism"). Corollary 1.20.

o

Suppose

for

p >

o.

Then

induces an isomorphism H(Ef'*,d") Proof.

q

H(C*,d).

is a double complex with

Lemma 1.19 for the natural inclusion

d'

= 0

cp,q.

Apply

e

15

Note.

For more information on double complexes see e.g.

G. Bredon [7, appendix) or S. Mac Lane [18, Chapter 11, 3 and 6). We now turn to Proof of Theorem 1.15. of

M as in Lemma 1.17.

complex as follows:

Choose a covering

p,q

n

0

Aq(U

(a O ' ••• ,a )

a

p

{Ua}aEE

0

consider

n ... n U ) a p

where the product is over all ordered (p+l)-tuples such that

=

Associated to this we get a double

Given

=

U

n... n

U

Ua p

aO

differential is given by

P,q AU

-+

* @.

(ao, ... ,a p )

The "vertical"

AP,q+l U

n... n U ) -+ Aq+ l (U n... n Ua is the aO ap aO p exterior differential operator. The "horizontal" differential

where

d: Aq(U

AP,q U

-+

AP+l,q U

is given as follows: For Aq(u

Cl O

(1.21)

w =

(w

n••• o u

(a O ' ••• ,a p)

Clp + l

(ow)

(Cl

)

O,

) E

,q

the component of

is given by p+l

L

· · · , ap + l )

It is easily seen that

00

=

i=O 0

and

double complex. Now notice that there is a natural inclusion

ow

in

16

Lemma 1.22.

For each

q

the sequence -+

•••

is exact. Proof.

A-1 , q

In fact putting

we can construct

U

homomorphisms

such that (1 .23)

id.

To do this just choose a partition of unity

with

and define (s w)

p

(U

O

, · · · , up _ 1 )

aEL

U

(U

O , · · · ,a p _ 1 ,a) ,

w E It is easy to verify that

s

p

is well-defined and that (1.23)

is satisfied. It follows that p > 0 p

O.

Together with Corollary 1.20 this proves Lemma 1.24.

Let

A*

U

be the total complex of

there is a natural chain map eA

: A*(M) ...

which induces an isomorphism in homology.

Then

17

A*

We now want to do the same thing with the singular cochain functor

replaced by

As before we get a double

complex

n

Cq(U

(aO, .•• ,a ) p

a

0

n... n

U ) a p

where the "vertical" differential is given by

c*(u

the coboundary in the complex

(-1)P

times

n... n

U ) and where a p the "horizontal" differential is given by the same formula as (1.21) above.

"o

Again we have a natural map of chain complexes

and we want to prove

C*(M)

Lemma 1.25.

induces an isomorphism in

homology. Suppose for the moment that Lemma 1.25 is true and let us finish the proof of Theorem 1.15 using this. For

U c M

we have a chain map

A*{U)

I

C*(U)

as defined by (1.13) above.

Therefore we clearly get a map

of double complexes I and we have a commutative diagram

U

u

teA

t e C

A* (M)

C* (M)

A* ----+. C

18

By (1.24) and (1.25) the vertical maps induce isomorphisms in homology.

It remains to show that the upper horizontal

map induces an isomorphism in homology.

Now by the remark

following Lemma 1.19 it suffices to see that for each

is an isomorphism.

However this is exactly Lemma 1.16 applied

n ... n u cx p

to each of the sets Proof of Lemma 1.25. with

A*

replaced by

It is not true that Lemma 1.22 holds

C*.

However, if we restrict to cochains

U

with support in the covering let

Cq(U)

o(6

q

C

it is true.

Thus as in (1.12)

denote the q-cochains defined on simplices

a E Soo(U), i.e. for each q)

p

U cx.

a E SOO(U)

there is a

q

u

with

Then there is a natural restriction map

cq(U)

and the sequence

(1 .26)

-of'

is exact.

•

&.

In fact we construct homomoprhisms

Cp-1 u as follows:

For each

(C- 1 , q U

a E Soo(U)

choose

q

cx(o)

E

L

such that

Then an easy calculation shows that

It follows that the chain map e

C

=

e

C

0 1*,

where

e

1*: C*(M)

c

: C*(M) C*(U)

u

C

factors into

is the natural chain

map as in (1.12) and where the edge homomorphism

19

induces an isomorphism in homology by Corollary 1.20 and the exactness of (1.26).

Since

t*

also induces an isomorphism

in homology by (1.12) this ends the proof of Lemma 1.25 and also of Theorem 1.15. Exercise 4.

For a topological space

X

denote the set of continuous singular n-simplices of and let

and

and cochain complexes. the inclusion

S;(M)

X,

be the corresponding chain Show that for a S;oP(M)

COO

manifold

M

induces isomorphisms in homology H(C*(M)).

(Hint:

Use double complexes for a covering as in Lemma 1 .17).

Hence the homology and cohomology based on

COO

singular

simplices agree with the usual singular homology and cohomology. It follows therefore from Theorem 1.15 that the de Rham cohomology groups are topological invariants. Exercise 5.

Show directly the analogue of the homotopy

property (1.11) for the de Rham complex. Note.

The above proof of de Rham's theorem goes back to

A. Weil [34].

It contains the germs of the theory of sheaves.

For an exposition of de Rham's theorem in this context see e.g. F. W. Warner [33, chapter 5].

2.

Multiplicativity.

The simplicial de Rham complex

In Chapter 1 we showed that for a differentiable manifold M

the de Rham cohomology groups

invariants of

makes

A*(M)

M.

Hk(A*(M»

are topological

As mentioned above the wedge-product

an algebra and it is easy to see that (2.1)

induces a multiplication (2.2) In this chapter we shall show that (2.2) is also a topological invariant.

More precisely, let

be the usual cup-product in singular cohomology; then we shall prove Theorem 2.4.

For any differentiable manifold

M

the

diagram Hk(A*(M»

@ Hl(A*(M»

n Hk(C*(M»

(i9

Hk + l

/\

(A* (M) )

n

I

Hk+l (C* (M) )

@ Hl(C*(M»

commutes. For the proof it is convenient to introduce the simplicial de Rham complex which is a purely combinatorial construction closely related to the cochain complex

C*

but on the other

hand has the same formal properties as the de Rham complex

A*.

21

We shall define it for a general simplicial set: Definition 2.5.

A simplicial set

S = {Sq}' q = 0,1,2, •.. ,

S

is a sequence

of sets together with face operators

E:.

Sq

->

S q- l' i

O, ... ,q,

and degeneracy operators

ni

S

->

Sq+1' i

O, .•• ,q,

which satisfy the identities

1.

q

(i) (ii)

E: _ j 1E: i,

i < j,

n j + 1n i ,

i

f

i < j,

j

=

(iii)

Example 1. S

SOO (M) q

q

i

=

i ' id, "

i

n E: _ , j i 1

i > j + 1.

j+1,

j, i

We shall mainly consider the example, where StoP(M) .

or

0, ... ,q,

j,

q

where

E:

i

Here as in Chapter 1,

nq -

1

->

nq

E:i(O) = 0

0

E: i ,

is defined by

(2.6)

Analogously, the degeneracy operators

ni

(0 )

=0

0

ni, .1. = 0 , ... ,q,

h were

are defined by

nq + 1

->

nq

is defined

by (2.7)

We leave i t to the reader to verify the above identities. A map of simplicial sets is clearly a sequence of maps commuting with the face and degeneracy operators. SOO

and

stop

become functors from the category of

Obviously COO

manifolds (respectively topological spaces) to the category of simplicial sets.

22 Definition 2.8.

Let (jJ

A differential k-form

{S } q

S on

be a simplicial set. (jJ='{(jJa}' a EllS p P

is a family

S

of k-forms such that (i)

(jJa

is a k-form on the standard simplex

i *(jJa' (jJE.a =' (E)

Ei

where

w

is the i-th face map as defined by (2.6). co

Let

for

S =' S (M)

is a k-form on

by putting

O, ••• ,p, a ESp' P =' 1,2, ....

a,

Example 2. if

C

a E

for

(jJ A

If

(jJ E Ak(S),

co

manifold.

(jJ =' {(jJa}

we get a k-form

M

(jJa =' a*w

a

M

Then

on

S'" (M) •

P

The set of k-forms on a simplicial set Ak(S).

for

E Al(S)

S

is denoted

we have again the wedge-product

defined by 0,1, ...

(2.9)

Also, we have the exterior differential

d

defined by

a ESp'

(2.10)

It is obvious that

A

commutative and that (2.11)

dd =' 0

d

satisfies

=' d(jJ A (A*(S) ,A,d)

de Rham complex of then clearly we get (2.12)

is again associative and graded

and

d«(jJ A We shall call

P = 0,1,2, ...

S.

If

+ (_1)k(jJ A

E Al(S).

the simplicial de Rham algebra or f: S

f* : A*(5'} (jJfa'

(jJ E Ak(s),

5' A*(S}

is a simplicial map defined by 0,1, .••

23 and thus

A*

is a contravariant functor.

Remark 1. manifold

M

Notice that by Example 2 we have for any

COO

a natural transformation i

(2.13)

: A*(M)

-->

A*(Soo(M»

which is clearly injective, so we can think of simplicial forms on

Soo(M)

as some generalized kind of forms on

M.

We now want to prove a "de Rham theorem" for any simplicial set

8. The chain complex

with real

C*(8)

coefficients is of course the complex where vector space on

8

a

and

k

C k(8) k

a (a)

'i

(-1)

i=O

-->

C k(8)

is given by

Ck- 1 (8)

i E (a), i

is the free

a E Sk'

Dually the cochain complex with real coefficients is C* (8) = Hom(C* (8) ,JR),

so again a k-cochain is a family

c = (c

0 : Ck(8)

a),

a E 8

k,

and

(2.14 )

(DC)a

=

k+1

'i

i=O

Ck + 1 (8)

-->

is given by

.

(-1)

l

c

EiT

,

T

E 8 k + 1•

Again we have a natural map

defined by (2.15)

I(j)a

(j)

k

E A (8), a E 8

k,

and we can now state Theorem 2.16 (H. Whitney).

I : A*(8}

-->

C*(8)

chain map inducing an isomorphism in homology.

is a

In fact there

24

E : C*(S)

is a natural chain map

k A - 1 (S), k

k

homotopies

sk : A (S)

(2.17)

1

0

d

o

(2.18)

1

0

E

id,

I,

0

A*(S) and natural chain

E

0

E

0

0

1 , 2 , ... ,

suc h th a t

doE

- id

sk+1

0

d + d

= 0,1 , ...

k

For the proof we first need some preparations. usual

sk'

0

As

is the standard p-simplex spanned by the

canonical basis coordinates

{eo, •.• ,e

(to, •.. ,t

p)' respect to each vertex e

have operators each

j

h(j)

and we use the barycentric

p}

Now j,

j

is star shaped with and therefore we

0, ... ,p,

for

:

as defined in the proof of Lemma 1.2.

o

w E

for

Also put

The prooL of Lhe following lemma

is left as an exercise (cf. Exercise 3 of Chapter 1): Lemma 2.19. k

k

The operators

P

k-1

0,1,2, ... , satisfy

k > 0

(2.20)

k (ii)

(2.21 )

(iii)

(2.22)

For

I ,j

0

0, ... ,p i

> j

i

< j

p

25

Next some notation: Consider a fixed integer Let

I

(iO, •.. ,i

satisfying I

is

to

be a sequence of integers

k)

< i < ... < i p, The "d.Li.reris Lon" of k 1 O (for 1=0 put 101 -1). Corresponding

we have the inclusion

dimensional face spanned by have a face map

: Sp

and

E.

=

E.

0 ••• 0

J1

{e.

I

onto the k-

, ... ,e. }

and similarly we

(2.23) (for

WI =

k

L

s=O

j1 >

> jl

and

k + 1

s

p.

A ••• A

E O ••• O

is the

0

A ••• A

dt.

0) and the operator

We can now define

E

(for

k + 1

which lowers the degree by

c = (c

(2.24)

T

E{c)

o

L

=

W

III=k I

which is clearly a k-form on

is defined as follows: (2.25)

=

.

h

0

= id) .

=

L

P

put

c

p < k

the sum is of k- 1{S) sk: Ak{S) A

Similarly

For

put

as follows (a

o E S

(if

course interpreted as zero).

0

below) :

a k-cochain and

)

I =

Ak{S)

Ck{S)

:

motivation is given in Exercise

E Ak{S) III !W I

and

[j E sP

put

A

which is clearly a k-1-form on First we show that (2.24) satisfies Definition 2.8 (ii): Let

1 E {O, ... ,p}

contain

j 1

defined by

on dt.

E

Also associated

A

(-1) t. dt.

I

For

j 1

p

there is the "elementary form"

I

I

u

where

J1

:

Explicitly,

Sk.

complementary sequence to to

o.

0 < i

III = k

I

p >

1.

and suppose

Then for some

s

I = (i

.. ,i does not k) O" and we have is < 1 < i + s 1

26

k .'

x:

\'

L

I I' I=k

since it is easy to see that I' = (ib, ... ,i

o

k)

ib < ••• < i

• c

wI'

L

III=k;lU

llI(a)

wI'

llI(a) = llI' (Ela).

Now since

runs over all sequences satisfying

k

p - 1,

the last expression above equals

E(c)

which was to be proved. Similarly (2.25) is shown Ela to satisfy Definition 2.8 (i) using (2.21) above. Now let us prove the identities (2.17):

The first

identity of (2.17) is proved exactly as Lemma 1.14, so let us concentrate on the second one:

a E S

p

c E Ck(S)

For

and

we have

k:

(2.26)

L

tII=k

(k+1):

L

III=k

dt. 10

dt. 1k

A ••• A

On the other hand (2.27)

(k+1) :

L

(Dc) II

III=k+1 (k+1) :

L

III=k+1 For

J

= (jo'· ··,jk)' 0

terms involving

W I

•

() in (2.27). llJ a (i o , · · · , i l,···,ik+ 1) = (jo,···,jk)·

c

llJ (a)

(a)

k+1

( L

1=0

jo < •.• < jk

c

I

1 (-1)

P

cE

lllI

we shall find the

Now

in (2.27) therefore is

(i O,···,i k+ 1) = (jo, ... ,jl-1,i,jl, ..• ,jk). Now (2.28) equals

iff

The coefficient of

(2.28)

where

(a)) •

27

L

(k+1):

[ L

.... (.JO, ••• ,J. ) s< 1 k

1.-.

(_1)s+lt. d t , A..• Adt. A..• Adt. Adt.Adt. A •. Js J0 Js J 1-1 1. J1

•• Adt. +t.dt. A ••• Adt. + Jk 1. JO Jk +

L.\'

(_1)s+1-1 t. dt. A ...A d t. d] A d t.A d t. A •••A t. A .•• At. JS JO J l- 1 1. Jl JS Jk

(k-e l l !

'lE('J " O

1.

.) ' " Jk

[t.dt. A..• xd t; . + 1. J0 Jk

k

L

+

+

L

S=O

-to dt. A ... Adt. Adt.Adt. A .•• Adt.] JS JO J s- 1 1. J s+ 1 Jk

k

L

L

S=O ilE(jO, .•• ,jk)

L

(k+1):[

-to dt. A..• xd t . Adt.Adt. A ••. Adt.] JS JO J s- 1 1. J s+1 Jk

ilE(jo, ... ,jk)

+

t.dt. A •.. xd t . + 1. JO Jk

k

k p dt. A ••• Adt. - L t. dt. A.•. xd t , AlL d t S=O J S JO Jk S=O J S JO J s- 1 i=O

L t.

j

1.

l xd t , A•. Adt.] J s+1 Jk

p (k+1):

L t.dt.

i=O

I

since

A ••. Adt. = JO Jk

J.

o

dt.

i=O

1.

E(oc)a = (k+1): by

(k-r l I Ld t , A.•. xd t . JO Jk

p

and

L

IJ I=k

L t.

i=O

1.

Hence

1.

dt. A... xd t, . • c () Jo Jk].lJ a

= dE(c)

a

(2.26) which proves the second identity of (2.17). To prove the first equation of

c = (c a), a E Sk (2.24)

E(c)o

(2.18)

and we shall show that

is the k-form on

consider a k-cochain

I(E(c))a = ca'

given by

By

28 k

k c

.

I (-1 ) J t

a j=O

.d to A... Adt . A.•. Adt

J

J

k

k.c [t ..• Adt + I (-1) k j=1 a odt 1A I

. 1 J-

k

t.dt.Adt J

J

A

1A

... Adt.A ... Adt J

Therefore

I (E (c)) a by Exercise 1 of Chapter 1. For the proof of the second equation of (2.18) first observe that an iterated application of (2.20) yields the following Lemma 2.29. I

=

(i

Let

, 0 < r ' ••• , i O r)

Suppose

Now let

k

r.

k w E A (f1,P) , k p,

with

and consider

0,

0 < i

O

< ••• < i

< p. r =

p

(otherwise

Then

E Ak(S)

and

a E S .

there is nothing to prove).

p

Assume

By (2.29)

k

k]

29

(2.30)

sk+1 (do) a

-

ill

L I! I

A

(

'

L (-1)J h ( 1, j=O

-:' ,)(\p» a 0 , ••• , 1 j , ••• , 1 1I I

Also

By (2.22) (-1)

k

h(,

1

' ) (.p ) (e.

0 , · · · , l k _1

a

1

k

)

Therefore adding (2.30) and (2.31) we obtain

E(I(\p»

a

-\p -

a

L 0< I I I

II I

'

L (-1)J h ( 1. j=O

0;-

0 , •• , 1 j , ••

,)(\p» a 1I I

,1

However the last two sums in (2.32) cancel by exactly the same calculations as in the proof that (2.26) equals (2.27) above. This proves the second equation of (2.18) and ends the proof of Theorem 2.16. We now return to the proof of Theorem 2.4. in the commutative diagram

Notice that

30 i

A* (M)

C* (M) all maps induce isomorphism in homology. i

: A*(M)

A*(Soo(M))

Also

is obviously multiplicative.

Theorem

2.4 therefore immidiately follows from Theorem 2.33.

For any simplicial set

S

the following

diagram commutes H (A

* (S))

H (A

+I

H(C*(S))

* (S) ) I

H(C*(S))

where the upper horizontal map is induced by the wedge-product of simplicial forms and the lower horizontal map is the cupproduct. Before proving this theorem let us recall the definition of the cup-product in

H(C*(S)).

Consider the functor

C*

from the category of simplicial

sets to the category of chain-complexes and chain maps (as usual we take coefficients equal to

An approximation to

the diagonal is a natural transformation

(in particular a chain map) such that in dimension zero is given by

It follows using acyclic models that there exists some

and

it is unique up to chain homotopy (see e.g. A. Dold [10, Chapter 6,

§

11, Exercise 4].

The cup-product is now simply

31

induced by the composed mapping *

C* (8 )

C* (8 ) ... Hom (C* (8 )

An explicit choice for

C* (8) , lR) ... C* (8) .

is the Alexander-Whitney map

AW

defined by (2.34)

n I u (a) p=o (O, •.. ,p)

AW(a)

With this choice of

u

(p, •.. ,n

a = (a ) E CP(8) a

then

is represented by the cochain

(2.35)

(avb)a=a

u ( 0 , ••• , p)

Proof of Theorem 2.32.

and

(a)·b

T

() u (p, ... , p+q ) a ,

By Theorem 2.16 every simplicial

form is cohomologous to a form in the image of It is therefore enough to show that for the (p+q) -cochain

I (E (a)

of

H(C*(8).

a

and

b

in

n•

q(8); b = (b ) E C

Let

b

E 8

a

the cup-product is explicitly given

as follows: a

) (a),

A

E (b) )

E: C*(8) ... A*(8).

a E CP(8), b E C

q(8)

represents the cup-product

80 let

* : C*(8)

C*(8) ... C*(8)

E(b»,

a E CP ( 8 ) , b E C

be defined by (2.36)

*{a

b)

I (E(a)

A

q(8).

We claim that there is an approximation to the diagonal inducing (2.36). Put E(a)

i

and consider

a E 8

n.

i O < ••• < p

I = (i ' o

s

n,

0

... ' i p)

and

n. If

I

than two integers in common then obviously I

6

(jo' ••. , jq)

J

jo < ••• < jq

has at least one integer in common.

suppose

Then on

n

,

I a ()w I I I =p III a I'

=

where as usual 0

Let us find an explicit formula for (2.36):

n = p + q

a

and

J

Then and wI

I

and

J

has more

J A

satisfy

W

J

= O.

Now

have exactly two integers in common, say

32

Then

s +r

+ ( -1) 2

2t . 2

t . d t . A.. Adt . Jr 1

s2

A•. Adt. Adt. A.. Adt . A•. Adt . JO Jr Jq P 1

and it is easy to see that these two terms are equal with opposite signs so and

wI A

W

= 0

also in this case.

Finally suppose

have exactly one integer in common, say

J

+

'\ s+k L (-1) t. t. dt. k*r Jk

n

Using

J

L dt;\

;\=0

o

is = jr;

I then

A... xd t , A... xd t.. Adt. A.. xd t . A.. xd t . JO Jk Jq A

we get

It follows that

(E(a)AE(b»

a

=p:q:

'\ r+s L a ()b ()'(-1) t. dt. A... xd t , A.. I I I =p ]JI a ]JJ a A

IJI=q

.. Adt. Adt. A•.. Adt. A... Adt . JO Jr Jq where the sum is taken over

I

and

J

such that for some

s

33

and

r

i

sgn(I,J)

jr and no other integers are common. Now let s be the sign (_1)p-s+r. times the sign of the

permutation taking

(O, •.. ,n)

into

(io,···,is,···,ip,is=jr,jo' " " ; r , ... ,jq);

r

sgn(I,J)J

t-,n

(-1)

r+s

t. dt. A... xd t,

then

A... Adt. xd t . A... Adt. A•.• xd t . JO Jr Jq A

is

1/ (n+1): • Hence (2.37)

I(E(a)

p:q:

(p+q+1):

where again

I

and

l:

III=p IJI=q J

A

E(b»(}

sgn(I,J)a

()b ()

() ()

have exactly one integer in common.

Therefore if we define the map

by 4> (o )

(2.38)

then 4>

4>*

l:

(

p+q=n

n+

1)

\" I L . I I I =p IJI=q

given by (2.36) is the dual map.

is an approximation to the diagonal:

() E Sn

@

We want to show that

Clearly

4>

is natural

and 4>(})

() @ ()

It remains to show that it is enough to see that

for

() E SO'

is a chain map.

4> 4>*

However, for this

is a chain map which is easy:

34

q,*(o(a@b))

q,*(oa@b + (-1)P a@ob) I(E(oa) "E(b)) +

(-1)

P l ( E ( a ) "E(ob))

l(dE(a) "E(b) + (-1)PE(a) "dE(ob) l(d(E(a) "E(b)) = ol(E(a) "[Cb)) = oq,*(a@b). This ends the proof. Remark.

Notice that the term in (2.37) corresponding to

I = (O, ... ,p), J = (p, •.. ,p+q) Whitney cup-product (2.35).

gives exactly the Alexander-

Thus (2.37) is an average of the

Alexander-Whitney cup-product over the permutations given by (I,J)

in order to make the product graded commutative on the

cochain level.

In fact the A-W-product is not graded

commutative on the cochain level as since

"

is graded commutative.

q,*

clearly must be

On the other hand the A-W-

product is associative on the cochain level which not.

q,*

is

In order to achieve both properties it seems necessary

to replace the functor

C*

by the chain equivalent functor

A*. Exercise 1. with

0

Consider for

i O < ••• < i k

k < P

and let

P

a sequence c

I = (iO, ••• ,ik) be the

set = {( to' ..• , t ) I some t. >O} = p

(i.e. we subtract a TIl :

- {t.. =t. = ... =t. =O},

p-k-1-dimensional face).

be the projection 1 s a)

Show that on

I

( t . , ... ,t.

).

Let

35

where

is given by (2.23). b)

Show the following properties of

c)

(i)

(IlI)*W

(ii)

(IlJ)*W

r

dt

I

a

E(c)o

on

c

dt

k

IJI = k, J 'f I.

if

Conclude that for

the form

A ••• A

1

wI:

(co)

=

satisfy:

a k-cochain and

For any

I

a ESp'

(iO, ... ,i

=

k)

as above (2.39) d)

a E Sk

Observe that for

the k-form on

is the simplest choice in order to satisfy the first identity of (2.18).

Show that with this choice for

a E Sk

the

condition (2.39) is a necessary requirement for the choice of for

a ESp' P

Exercise 2.

a)

Let

of simplicial sets. (L)

b)

1

=

f

: S

f*

0

E

(iii)

sk

0

f*

=

f*

sk'

0

Two simplicial maps

(i)

(ii)

be a simplicial map

f*

0

(ii)

O, •.. ,q,

S'

Show that

homotopic if for each i

k.

>

q

1,2, ..•

k

fO,f,

S

S'

there are functions

such that Oh O

=

fa,

q+1 hq

{h.J- 1c 1., , e h . = 1. J h j i-1'

=

f1

if

i < j,

if

i > j+1,

i

j+1 hj+1

=

j+1 hj'

are called hi

36

(iii)

Show that c) and b)

C*(S')

Let

f

O,f 1

imply that

d)

: S

i

> j.

are chain homotopic.

be homotopic.

Let

S

S

->

A*(S)

Show that a) are chain homotopic.

be a simplicial set.

ni

A k-form

is called normal if it furthermore satisfies

(iii)

on

if

Find explicit chain homotopies in c) .

on

(2.7).

< j,

fO,f, : A*(S')

Exercise 3.

where

i

C*(S)

->

S'

->

if

=

i

6 P+ 1

6P

->

Let

O, ••• ,p,

0

E S , P p

= 0,1,2, ..

is the i-th degeneracy map defined by

c Ak(S)

be the subset of normal k-forms

S. a)

f : S

->

Show that S'

d

and

preserve

A

is a simplicial map then

normal forms and if

f*

also preserves

normal forms. b) k

Show that the operators

0,1, ... , j

fl '!'h ( ')'

h(i)h(i) = 0,

(ii) Let

k-cochains

satisfy

J h (L) n·* J { n * h ( i _ 1 )' j

(i)

c)

O, ••• ,p,

=

Ck(S)

c

=

v

i = 0, ••• , k-1 . (L)

(ii) (iii)

h(j)

E

->

sk :

->

i

> j

be the set of normal cochains, i.e., c

niT

Show that ->

j

i = O, ... ,p.

such that

I

i

Ak- 1 (S) N

0

VT E Sk-1'

37

and conclude that

I :

is a chain equivalence.

Hence since the inclusion

C*(S)

is a chain

equivalence (see e.g. S. MacLane [18, Chapter 7, the inclusion

A*(S)

Exercise 4.

§

6] also

is a chain equivalence.

(D. Sullivan).

Let

m)

denote the

set of polynomial forms with rational coefficients, i.e.

w E

is the restriction of a k-form in

lli

n+1

of

the form

where

a.

.

are polynomials in

lO·· .lk

to ... t

n

with rational

coefficients. Now let S

S

be a simplicial set.

is called rational if

Ak(S, W) a)

E

A k-form m)

for

0

E S. p

Show that

multiplication

is a rational vector space

Let

C*(S, m)

(i)

(ii)

d

and exterior

A.

rational values.

denote the complex of cochains with

Show that

I E

C*(S,

A*(S,

and conclude that the Theorems 2.16 and 2.33 hold with and

C*(S) c)

Let

denote the set of rational k-forms.

which is closed under the exterior differential

b)

on

=

replaced by

A*(S)

and

Formulate and prove a normal version of question b)

(see Exercise 3). Note.

For a simplicial complex the construction of the

simplicial de Rham complex goes back to H. Whitney [35, Chapter 7].

3.

Connections in principal bundles

The theory of connections originates from the concept of "parallel translation" in a Riemannian manifold.

So for

motivation consider the tangent bundle TM of a differentiable manifold

M; or more generally a real vector bundle

M of dimension

over

p,q E M and a vector

Given points

n.

V

v E V P

one wants a concept of the corresponding "parallel" vector

E V

T(V)

i.e. we require an isomorphism T : V Vq. However, p q, V is a trivial bundle this seems to be an impossible

unless

requirement.

What is possible is something weaker: the concept

of parallel translation along a curve from suppose to

[a,b]

y

y(b)

q

p

to

q,

that is,

M is a differentiable curve from v E V

and let

y(a)

p

be a given vector; then a

p

\

i

"connection" will associate to these data a differentiable family with

v t E Vy(t)' t E l a b l , j

v

a

parallel translate a basis or frame

V.

space

Therefore let

P

over

11

{v ' ••• ,v

for the vector n} M denote the frame bundle 1

F(V)

:

M, i.e. the bundle whose fibre over

of all bases (frames) for associates to any curve a lift of rCa)

It is of course enough to

= v.

=

e

y and

through 11

0

Y

X E Te(F(V))

[a,b]

M and any point

e, that is, a curve

=

y.

is equal to the set

Then a "connection" simply

p

y

defines a tangent vector vector

V.

p

Now let

q

X E T (M) p

such that

11*X

y : [a,b]

tend to and =

y

X.

T

11(e)

(M)

p;

F(V)

then

with

y

defines a tangent So infinitessimally

a "connection" defines a "horizontal" subspace mapping isomorphically onto

e E F(V)y(a)

for every

He

Te(F(V)) e E F(V).

And

that is actually how we are going to define a connection formally below.

Notice that

F(V)

is the principal

Gl(n,JR )-bundle

39

associated to

V.

So first let us recall the fundamental facts

about principal G-bundles for any Lie group

COO

G.

Let

M

be a

manifold. Definition 3.1.

mapping

TI

:

E

M

A principal G-bundle is a differentiable of differentiable manifolds together with a

differentiable right G-action (i)

For every

P E M E

E x G

(Local triviality)

(ii)

neighbourhood

U

TI

P

-1

E

satisfying is an orbit.

(p)

Every point of

and a diffeomorphism

:

M TI

-1

has an open (U)

U x G,

such that (a)

the diagram TI-

1 (U)

U

x

G

U

commutes, (b)

is equivariant, i.e. .g,

=

where

G

acts trivially on

translation on E

e E

TI-

1

(U),

U

g E G,

and by right

G.

is called the total space,

M

the base space and

1 TI is onto TI(p) is the fibre at p. Notice that by (i) P and by (ii) it is an open mapping so TI induces a homeomorphism

E

to

M.

is free (i.e., xg

= x

of the orbit space G

on

E

given by

g

eg

E/G

Also observe that the action of g

=

1)

and the mapping

is a diffeomorphism for every

e E E

p.

G

E.

P

We shall

often refer to a principal G-bundle by just writing its total space

E

40

Example 1. bundle.

Suppose

V

Then the bundle

M

F(V)

is an n-dimensional vector M of n-frames is a principal

Gl (n, lR ) -bundle. Let

E

isomorphism morphism.

M

M

and

F

(j)

E

F

x

an isomorphism mapping

(j)

G

be two principal G-bundles. Then an

M

is a G-equivariant fibre preserving diffeo-

is of course a trivial principal G-bundle and

(j)

E

M

x

is called a trivialization.

G

in (ii) above is called a local trivialization.

Now consider a principal G-bundle covering (j)

a

The

U

= {Ua}aEL

1

: n- (U ) a (j)S

of

U x G. a 0

n

E

:

M

and choose a

together with trivializations

M

Then if

*¢

U n Us a

consider

-1

(j)a

which is easily seen to be of the form (j)S

0

-1

(j)a (p,a)

gSa: U n Us G is a COO function. This system {gsa} a are called the transition functions for E with respect to U

where

and they clearly satisfy the cocycle condition (3.2)

gSa (p) g

=

aa

gya (p),

=

1.

On the other hand given a covering

=

and a system of a transition functions satisfying (3.2) one can construct a U

{U }

corresponding principal G-bundle as follows: the total space is the quotient space of (p,a) E Ua

liU a a x

G

x

G

with the identifications with

(p,gSa(p) ·a) E Us

Vp E Ua n US' a E G.

x

G

41

Again let f

: N

M be a principal G-bundle and let

M be a differentiable map. f*E

f*n

n: E

N

The "pull-back"

is the principal G-bundle with total space

and

1T (e) }

{{q,e) If (q )

f*E f*1T

given by the restriction of the projection

onto the first factor.

The projection onto the second factor

giveg an equivariant map

f : f*E

l

E

covering

f, i.e. the

f*{E) f* 1T

f

N -----'''---.. M

commutes. Exercise 1.

a)

Show that if

transition functions for U = {Pa}aEl: for

f*E b)

then

f.

0

f}

Let

F

N, E

M

(f,f),

f: F

E

where

is the set of

relative to the covering is the set of transition functions

relative to the covering

map is a pair map and

{gaB

E

{gaB}

-1 f-1 U = {f Ua}aEl:.

be principal G-bundles. f

: N

M

A bundle

is a differentiable

is an equivariant differentiable map covering

Show that any bundle map factorizes into an isomorphism : F

f*E

and the canonical bundle map

Exercise 2.

a)

f*{E)

E

Show that a principal G-bundle

as above. 1T : E

M

is trivial iff it has a section, i.e. a differentiable map s : M b) 1T*E

E

such that Let

1T

E

1T

0

sid.

M be a principal G-bundle.

Show that

is trivial. c)

Let

n : E

M be a principal G-bundle and let

H

G

42

be a closed sUbgroup.

Show that

E

E/H

is a principal H-bundle.

(Hint: First construct local sections of the bundle

G

G/H

using the exponential map). Let

Exercise 3. and let

N

: E

M

be a manifold with a left G-action

associated fibre bundle with fibre : EN

M where

EN

under the G-action and where Show that

be a principal G-bundle

= E

x

(e,x) 0g

G

N

N

N.

is the orbit space of

N

The

E x N

= (eg,g-1 x), e E E, x E N, g E G,

is induced by the projection on EN

x

is the mapping

followed by

E

is a manifold and that the fibre bundle is locally

trivial in the sense that every point of

u

G

with a diffeomorphism

:

-1

(U)

M

U

x

N

has a neighbourhood such that the

diagram -1

(U)

U x N

U

commutes.

In particular

Now let

Hand

G

is open and differentiable. be two Lie-groups and let

be a homomorphism of Lie groups. principal H-bundle and

E

Suppose M

p

= Ep'

Vp EM,

=

'a(h),

Then we will say that to

a

E

"relative to

a

F

M

is a

E

satisfying

and "Ix E F, hE H. to

G

relative

is a reduction of

E

to

(when it is clear what a").

F

is an extension of

or, equivalently, that

relative to

: F

G

is a principal G-bundle and

suppose there is a differentiable map (F ) c

a: H

a

F

is we will omit

H

43

Example 2. principal

An n-dimensional vector bundle

Gl (N, lR) - bundle n lR

act on the left on n lR

with fibre

F (V)

M.

FO(V)

is just the vector bundle. Hence there is a

F(V) F(V)

fibre. Then

Gl (n, lR) - bundles

g E G. 'IT

G

:

G

O(n).

In fact

M is the corresponding orthogonal bundle FO(V)

Riemannian metric on

consider

defines a

consist of the orthonormal frames in each

F(V)

versely a reduction of

Exercise 4.

V

M to the orthogonal group

FO(V)

and the inclusion

Gl (n, lR)

and that the associated fibre bundle

and vector bundles. A Riemannian metric on

let

M has the

Notice that

one-to-one correspondance between principal

reduction of

V

F(V)

defines the reduction.

to

O(n)

clearly gives rise to a

F

M be a principal H-bundle and

V.

a) Let

'IT

:

with the left H-action given by

.

h

g

=

a(h)g, h E H,

Show that the associated fibre bundle with fibre

FG

Con-

M is a G-extension of

'IT

:

and show that an

M,

F

G,

extension is unique. b)

Show that a principal G-bundle

reduction to

u = {u} y {a

0

h

H

relative to

a

'IT

:

E

iff there is a covering

and a set of transition functions for

} with SY

{h

M has a

E

of the form

} a set of functions satisfying SY

(3.2)

Before we introduce the notion of a connection in a principal bundle it is convenient to consider differential forms with coefficients in a vector space. So let and

w

V on

M be a

COO

manifold

a finite dimensional vectorspace. A differential form M of degree

function

k

w(X1' ••. 'X k)

with values in : M

V

V

associates a

to every set of

COO

COO

vector

44

fields

Xl, ••. ,X k

on

w

M;

is again multilinear and alter-

nating and has the "tensor property" as before. a basis

{e 1, ... ,e

n}

w = w1 e 1 + ... + wne n forms.

Let

values in

V

then

where

Ak{M,V) V.

for

Again

w

(w , •• • ,w l

If we choose

is of the form is a set of usual k-

n)

denote the set of k­forms on A*{M,V)

M with

has an exterior differential

defined by the same formula as in Chapter 1 and chain complex (that is, dd = 0).

A*{M,V)

d

is a

This time, however, the wedge­

product is a map

for w

2

V,W

two vectorspaces.

E Al{M,W)

define

where as usual

w

1

In fact for

A W

2

w

l

E Ak+l(M,V 0 W)

E Ak(M,V)

by

runs through all permutations of

a

and

1, ... ,k+l.

Again we have the formula (3.4)

(dw 1) w

Similarly for

F: M

an induced map

w2

k

+ (-1) w1

k

1

N

E A (M,V), w

a

A*(N,V)

F*

A

COO

Now let

d

map of A*{M,V).

and induced maps

E A (M,W). 00

C

manifolds we have

P: V

Also if

P: A*{M,V) F*

W is

A*{M,W)

as above. G

as usual the set of left­invariant vector fields on

G.

This

can also be identified with the tangent space of

at the unit

element

1 E G.

For

g E G

let

The Lie algebra

is

of

G

be a Lie group.

dw 2 '

1

2

a linear map it clearly induces a map commuting with

A

Ad (g)

:,

1

G

be the adjoint

45

representation, i.e., the differential at x .... gxg

,

Now let

with

E .... M

:

IT

G .... E

the map

vx

of the map

-1

given by

be a principal G-bundle. 9

....

For

induces an injection

x'g

and the quotient space is naturally identified

.... Tx(E)

That is, we have an exact sequence

TlT(x) (M).

v

IT*

2...,

(3.5)

--->

The vectors in the image of

v

T

lT (x) (M)

--->

O.

are called vertical and we want

x

to single out a complement in

Tx(E)

of horizontal vectors,

i.e., we want to split the exact sequence (3.5).

This of course such that

is equivalent to a linear map (3.6)

id

It is therefore natural to define a connection in 1 8 E: A

be a 1-form

E = M

consider the trivial bundle 1-form on

E

8 (x,g)

Z

left translation by

action on

E

= M

To motivate this

8

g.

Now for

G,

x

9 E: G 9

let

L R

9 9

:

-1 :

E

be the

For

by the right action on

R*8 9

Ad (g-1)

0

:

8

G

A 1 (E'j)

0

8,

....

E

Vg E G,

1 .... A (E'1)

is

denote E,

and the trivial

defined by (3.7) we have

Ad (g -1)

G .... G

on the principal G-bundle

M.

Lemma 3.8.

where

x E: E.

x E: M, 9 E: G,

is the projection and

the map given by the action of i.e. for

8.

G .... M and let

x

OlT z ) * '

=(L g-l

M x G .... G

:

simply to

given by

(3.7)

lT

E

such that (3.6) holds for all

However, we want a further condition on

where

x E: E

is induced by

46

Ad(g-l ) Proof.

e

Since

to consider

= pt.

M

from

is induced via That is,

e

G

it is enough

is the 1-form on

G

defined

by

ey Then (R*e) g

y

With this motivation we have Definition 3.9. n

E

M ( L)

A connection in a principal G-bundle

is a 1-form ex

0

V

x

=

1 e E A

satisfying: is the

where

id

differential of the map (El

R*e g

= Ad(g-l)

where on

g

xg.

Vg E G, is given by the action of

g

E. If

vectors, i.e. R *H

e, E

Remark 1.

(E)'

0

g

H

H x

x

x

Tx(E)

is the subspace of horizontal

then (ii) Vx E E,

In fact (ii) clearly implies (ii)

is equivalent to

Vg E G.

I

and since both sides of (ii)

vanish on horizontal vectors (granted (ii) ')

it is enough to

check (ii) on vertical vectors in which case (ii) from (i) and Lemma 3.8.

is obvious

47

Remark 2.

By Lemma 3.8 the product bundle

has a connection given by (3.7).

E

F

M

M

x

G.

Notice

is an isomorphism of G-bundles and if

8

has a connection

G

x

This is called the flat

connection or the Maurer-Cartan connection of that if

M

then

E

defines a connection in

In particular every trivial bundle has a connection induced from the flat connection in the product bundle.

This is also

called the flat connection induced by the given trivialization. The following proposition is obvious. Proposition 3.10.

Any convex combination of connections

is again a connection.

More precisely:

connections in

E

M

M with

LiA

functions on

TI

connection in

:

paracompact manifold

connection.

i

=

1.

A , ••• ,A 1

Then

8

=

8

k

1,

. . . ,8

k

be

be realvalued

Li Ai 8 i

is again a

E.

Corollary 3.11.

Proof.

and let

Let

Any principal G-bundle M

TI

E

M

on a

has a connection.

By Remark 2 above every trivial bundle has a flat In general local trivializations define

flat

a covering of M. a in EIU a for {Ua}aEZ choose a partition of unity {A } and put 8 = LAa 8 a • It a a follows from Proposition 3.10 that 8 is a connection. connections

8

Exercise 5.

a)

Now

Suppose we have a bundle map of principal

G-bundles

I

F ----+ E

1

If

E

has a connection

1

8

then

I*8

defines a connection in

F.

48

b)

If

E

M is a trivial G-bundle then there is a bundle

map

and the flat connection is just the induced connection of the Maurer-Cartan connection in the G-bundle

G

Now consider a principal G-bundle

e.

For

X E Tx(E)

X

E H

x

X

ex

ker

=

E

:'1

im v x' v x

Now suppose

w

with coefficients in some vectors pace is horizontal if

: E

M with connection

a tangent vector we have already introduced

the term vertical for for

pt.

w(X

1,

... ,X

= 0

k)

A* (E,V)

E

and horizontal

Tx(E),

is a k-form

We will say that

V.

w

whenever just one of the

is vertical. I f V is a (left) X1,··· ,X k E Tx(E) representation of G then we will say that w is equivariant -1 if R*w w, Vg E G. In particular i f V is the trivial g = g

vectors

representation an equivariant form is called invariant. that the invariant horizontal forms on

E

with coefficients in

are exactly the forms in the image of In fact suppose define

w

E

choose

x

E

=

Xi' i

w E A*(E)

Ak(M) -1

=

(p)

1, ... ,k

: A*(M)

For

X1 , ... ,X k

p E

E

M and

Tx(E)

Furthermore if

COO

X 1,.",Xk

are extended to

=

Xi'

so

to

COO

w(X 1 , ... ,Xk )

and

P

X1, ... ,X k.

vector fields

X1"."Xk vector fields satisfying

M we can by local triviality of x

T (M)

E

and put

x

a neighbourhood of

X1 , · · · 'X k

such that

This is then independent of the choices of

on

A*(E).

is horizontal and invariant; then we

as follows: and

Notice

is

COO

E

extend

in a neighbourhood of

in

x.

49

Now consider the connection from that

8

is an equivariant 1-form with coefficients in

the adjoint action of the image of

8

A

8

G.

[-, -]

Proposition 3.12. connection

8.

E

Let

: E Q E

M

x

G

with the flat

be the curvature form defined by d8

(the structural equation). Furthermore

(3.15)

Q

dQ

In particular Proof.

drl

Then

Q

is horizontal and equivariant.

satisfies the Bianchi identity [rl ,8] .

vanishes on sets of horizontal vectors.

a) follows from b) since by Exercise 5

induced from the principal G-bundle

= 0

Then we have:

M be a principal G-bundle with connection

(3.14 )

Q

1

d8

and let

c)

2(E" A

:..., ®

Let

denote

Then

(3.13 )

8

a)

with

[8,8] E

Also let

under the map

induced by the bracket

b)

Observe

G

pt

e

is

and therefore

because it is horizontal by b) . b)

It is obvious that

hence both

d8

and

observe that clearly To see that for any (3.16)

rI

X,Y E Tx(E) (d8) (X,Y)

(e,8]

rI

is equivariant since

and

are equivariant (for the second one

1-

Ad (g)

preserves the Lie bracket) .

is horizontal we must show for with

8

X

x E E

and

vertical that

-\(8,8](X,Y)

-\(8

(X),8 (Y)].

In order to show (3.16) it is enough to consider 1)

Y

vertical

50 and 2)

Y

1)

horizontal.

First notice that for any vector

COO

associated

vector field

where

A*

on

E

A E

JJjr

there is an

= Vx(A)

defined by

as usual is induced by

g

xg.

Observe

that the associated 1-parameter group of diffeomorphisms is {R

},tElR,

gt see that for

where A,B

gt

= exp tA, t

Also it is easy to

E lR.

E'1 [A,B]*

(3.17)

[A*,B*].

In fact by local triviality it is enough to prove this for a trivial G-bundle A

= M

E

x

G

in which case

is the left invariant vector field on

Therefore (3.17) bracket in

1.

A* G

A

= 0

where

associated to

A.

is immidiate from the definition of the Lie

Now, to prove (3.16)

for

X

and

Y

vertical it is clearly

enough to prove (d8) (A*,B*) But since

8 (A*)

A,

(d8) (A* ,B*)

8 (A*),8 (B*)], 8 (B*)

B

A,B E

1- .

are constants we conclude

-'-,8 ( [A*,B*]) = -'-,8 ( [A,B]*) -'-,[A,B] = -'-, [8(A*),8 (B*)].

2)

Again extend Also for

Y

X

to a vector field of the form

horizontal extend it to a horizontal

vector field also denoted by vector field Since

Y

Z

and then put

Y

Y

(first extend

Y

Y

=

Z

Y

- v

y

0

to any

00

C

00

C

8 (Z ), Y E E). Y

Y

is horizontal the right hand side of (3.16) vanishes.

So we must show (3.18)

A*,

(d8) (A*,Y)

o

for

Y

a horizontal

vector field.

51

Now since

e (A*)

A

e(y) = 0

is constant and (de)(A*,Y) =

As remarked in 1)

the 1-parameter group associated to

A*

is

Therefore g

lim 2. (Y t_ y ) t .... O t x x

where

Since g

e (Y x t)

1

Ad (g

)

0

e (Y

xg

0,

-1 ) t

we conclude

which proves (3.18) and hence proves b). c)

Differentiating (3.14) we get

o

+

-

[ de, e] =

-

e] +

[ [e ,e] , e]

-

since

[ [e , e] , e]

o

by the Jacobi identity.

This proves the

proposition. Remark.

Let

X, Y

be horizontal vector fields on

E.

Then

by (3.14) (3.19 )

O(X,Y) =

which gives another way of defining Definition 3.20.

A connection

e

in a principal G-bundle

is called flat if the curvature form vanishes, that is,

= O.

52

Theorem 3.21. IT

:

E

M

A connection

in a principal G-bundle

is flat iff around every point of

neighbourhood

U

restriction of

and a trivialization of to

6

connection in

U

Proof.

EIU

M

EIU

there is a such that the

is induced from the flat

G.

x

is obvious by Proposition 3.12 a).

n

Suppose

q:

6

O.

=

For

x E E

let

subspace of horizontal vectors, i.e.

tiable subbundle of

T(E».

x

X E H

x

E

This clearly defines a distribution on

Tx(E)

H

iff

be the

6 (X)

=

O.

(i.e. a differen-

By (3.19) this is an integrable

distribution hence by Frobenius' integrability theorem defines a foliation (see e.g. M. Spivak [29, Chapter 6]) such that is the tangent space to the leaf through Remark

following Definition 3.9 that

x. R

It follows from E

g

Hx

E, g E G,

maps any leaf diffeomorphically onto some (possibly different) leaf of the foliation. Now let leaf IT

through

Hx

x

U IT

F

of :

p E M

Tp(M) p

x.

Since

x E

Tx(F)

V

is a diffeomorphism.

:

IT-

H X

of

x

Elu;

U

x

G

where

:

U x G

be the connection in

6'

connection in

so

and consider the

and since

in

F

such that s : U

V

Elu

In fact the trivialization is given by

1 (U)

subspace in

(p)

hence by exercise 2

s(q)·g, Now let

-1

The inverse

therefore defines a section of is trivial.

IT

is an isomorphism we can find a neighbourhood

and a neighbourhood U

V

and choose

U

x

G.

and

6'

-1

(U)

is defined by

qEU,gEG.

Elu

induced from the flat

Then it is obvious that the horizontal

T (E), Y E V, g E G, y.g 6

IT

is

(R )*(T (V») g Y

=

R *H g Y

defines the same horizontal subspaces

53

and therefore must agree. Corollary 3.22.

Let

IT

:

E

M be a principal G-bundle.

The following are equivalent: 1)

E

has a connection with vanishing curvature.

2)

There is a covering of

a set of transition functions gaS

Ua n Us 3)

Then

E

{gaS}

is constant for all

Let

G be the group G d has a reduction to G d,

Proof, 2)

G

M by open sets for

E

{Ua}aEL

such that

a,S E L.

with the discrete topology.

2) and 3) are equivalent by Exercise 4. be the

Let

=> 1):

trivializations with the constant transition functions Let

and

gaS'

8

be the connection in E.I Va induced from the flat a connection in U x G. Now there is a commutative diagram of a bundle maps

F

• U n Us a

G

IT

1

8

definition

z

be the Maurer-Cartan connection in

0 8

0

G

• G

G

and let

x

G

pt.

By

is left invariant and therefJre

Therefore and 8 agree on EIU n US' a a S we can define a global connection 8 in E which agree with or equivalently

8 a.

on

Clearly

EIU a·

has for all

8

a..

8

has vanishing curvature since

8

a

54

1)

2):

curvature.

Now let

8

be a connection in

By Theorem 3.21 we can cover

and find trivializations

U .... U

M

is induced from the flat connection in

with vanishing

{Ua} 1U such that Slna x G. Now fix a,S E L

x G

a

E

U a

by open sets

and let

Again let

8

U n Us a

be the flat connection in

0

x

Then

G.

(jJ*SO = 8 0 so (jJ permutes the leaves of the horizontal foliation, i.e. , the sets of the form In (U n US) x g, g E G. a

clearly

(jJ(U n Us a and it follows that

particular

1) = (U

x

a

n

US)

Hence the transition function Exercise 6. and let

Let

x

go

go E G,

for some

is constantly equal to

a : H .... G

be a Lie yroup homomorphism

F .... M be a principal H-bundle with connection

Show that if

(jJ

F .... E

is the extension to

a connection

8E

in

such that

E

(jJ*8

E

G

= a*

0

SF'

then 8

F,

is where

a*

is the induced map of Lie algebras. Exercise 7.

Let

M be a manifold and let

be the frame bundle of the tangent bundle, projection.

...1 (n,m) -r

=

The structure group is n Hom(m n ,m)

isomorphism

Since

x : m n .... T (M) p

with coefficients in

JRn

x

Show that

1-form, where

w on

GI(n,m)

-1

x E

1T

F(M) .... M

n

the

with Lie algebra

(p), P E M,

there is a 1-form

w

is an on

F(M)

defined by W

a)

Gl(n,m)

F(M) = F(TM)

x

-1

F(M) acts on

0

11*.

is a horizontal equivariant m n

by the usual action.

55

b)

For

M=lR

n

F (lR n)

connection in

and for

8EA

1(F(M)),#n,lR))

the

defined by the natural trivialization

show that dw

-8

II

w

where the wedge-product denotes the composite map

(Hint:

Notice that

.coordinates a real

n

x

For

F (lR n) = lRn

x

Gl (n , lR)

lRn

x

JRn

2

wi th

n

y= (Y1""'Yn) E lR n-matrix.

and X={x·'}"1 lJ l,J= , ... ,n 1dX and w = x- 1 d y ) . 8 = X-

Then

M a general manifold and

show that the torsion-form (3.23)

8

a connection in

S E A2(F(M) ,lRn) -8

dw

II

F(M)

defined by

w+ S

is equivariant and horizontal. c)

where

With respect to the canonical basis of

1

w , ••• ,w

e

n

are usual 1-forms on

·· ··· ·'n

='

8

1,

... .. ..

on

••••••••. en

Then (3.23) takes the form

F(M).

([)

lRn

we write

Similarly we write

56

(3.23)

d)

Show that every horizontal 1-form

the form on

i Lifi w

Ci =

,

where

on

Ci

00

are real valued

f.

1

C

functions

M is given a Riemannian metric and let

Now suppose

be a connection in the orthogonal frame bundle

wand

S

be defined on

FO(M)

exactly as for

Show that (3.23) still holds and that on (3.24)

i,j

J

Furthermore show that if by (3.23) and (3.24).

S

=

F(M)

write

Ci

j

f)

=

Lifijwi

FO(M)

e

is uniquely determined

(Hint: Show first that if

Ci =

and if we

a)

Let

F(M).

M be a manifold and Let

K

:

V .... M

an n-

F .... M be the associated

= Hom(mn, m n)

Show that for

e

is the Lie algebra of

E A1 (F,,t(n,ml),

e

(3.14) takes the form

(3.25)

M the

Gl(n,m) -bundle, i.e. the bundle of n-frames in

,l(n, m)

is a

Notice that by Exercise 6 this extends

dimensional vector bundle.

Again

J

has a unique torsion free connection (the

to a well-defined connection in

principal

(Ci.)

as in d), then

Levi-Civi ta connection).

Exercise 8.

above.

FO(M)

Conclude that for every Riemannian manifold

framebundle

Let

1, ... , n.

then

0

FO(M).

row of horizontal 1-forms satisfying

F,

is of

F(H)

F(H) . e)

e

0, ... ,n.

i

I

d8

-8

II

e

+

n

where the wedge-product denotes the composite

V.

GI (n, m)

a connection in

57

{here "(n, JR) of maps of

@

JRn

l'

(n, JR) ... "(n, lR) JRn).

into

canonical basis of

is given by composition

Furthermore, with respect to the

e

?(n,JR),

Si

and

are given by matrices

e11 •••••••••• en1

Si 1··········Sin

en1·

Si 1··········Sin

n en

1

1

n

n

of 1- and 2-forms respectively. Show that (3.25) is equivalent to (3. 25)

I

-1:

]

b)

i, j

k

Observe that

COO

sections of

correspondence with equivariant where

Gl (n, JR)

sections of

acts on

V

C

00

V

1, •..

are in 1-1

Similarly show that

into

n lR

The set of

COO

functions of

JRn in the usual way.

is denoted

,n.

F

r{V). C

00

T*M @ V

sections of

are in 1-1

correspondence with equivariant horizontal 1-forms on coefficients in

JRn.

Alternatively

to every vector

t E r{T*M

an element

E V P

(i) (ii)

@

V)

F

with

associates

such that

A E JR,

if

X

is a

p

tx

P

is a

COO

vector field on

COO

section of

V.

M

then the function

58

c)

Let again

6

be a connection in

V(s) E A (F , lli n ) 1

define

that

is considered as a function of

V(s)

s E r(V)

-6-s + V(s)

ds s

For any

by

(3.26)

(here

F.

F

into

lli

n).

Show

is horizontal and equivariant, hence defines

V(s) E r(T*M ® V). d)

s E r (V)

For

and

X

P

V (s) = V(s)x x

as in c) and let

P

P

V

V(s) E r(T*M ® V)

as defined in b) _

E V P

P

is called the covariant derivative of and

let

E T (M)

s

in the direction

V

P

V (s) ilXp

P

ilV

=

sEr(M), (ii)

If

X

is a

function

p

....

V x

a

C

(s )

P

is a

s

y : [a,b]

6 M a

is a unique liftet curve

Y

=

Y,

(s), p

ilElli.

then the

section of

for

real valued function on

As before let

M

x

V.

Vx(s) .

directional derivative of

that for

C

00

X(f);x(S) + fVx(s)

Vx(fs) 00

vector field on

COO

This is denoted (iii)

6.

satisfies:

Vx +y (s)

(i)

IT 0

Xp

is called the covariant differential corresponding to

Show that

e)

This

s E r (V) M

curve and

y : [a,b]

such that the tangents of

X(f)

and

the

f.

be a connection in COO

f

I

F

IT

:

x E IT-

1

F

M.

(y(a))

Show there

= x,

with

yare all horizontal.

Notice that this lift defines an isomorphism (the "parallel translation along

y")

t E

[a,b] .

59

f) C

00

For

X

E T

p

p

(M)

let

y (0) = p,

curve with

y

y' (0)

be parallel translation along

(3.27)

T

'V X (s)

t

M,

P

Let

T

lift of

v y.

of

Flu

.... V Y(t)

s E r (V)

s(y(t»-s(p)

such that

v

P

t .... O

l

i a.

and

v

0

L a.v.

s =

Now write

the components of

V

t

Show that for

{Hint: Observe that in some neighbourhood a section

be a

s > 0

lim - = - - - : t - - - - -

=

p

X

y. -1

....

[-s,d

:

l

.... lR,

of

p

there is

defines a horizontal

y

where

l

U

U

are

(v 1'··· ,v n)

are

i = 1, ... ,n,

C

00

functions) . g) 8.

2(F'7(n,lR» rl E A

Now let

s E r(V),

Show that for any

function of

F

into

(3.28)

X

and

s

Y

-

rl (X, Y)

(3.29) h)

9.

V

TM

=

'V

(s) •

M

rl

defines a

Show that

Let

8

and let

TM,

w

be the 1-form considered

be a connection in

Observe that for

a section of

A

(s)

Now let

in Exercise 7.

8

vector fields on

rl(X,Y) E r(Hom(V,V».

section

interpreted as an equivariant

we have

rl •

d'V(s)

Notice that for

form

lRn,

be the curvature form of

X, Y

F(M)

with torsion

vector fields on

that is, a new vector field

M

'9(X,Y)

e

defines and show

that this is given by (3.30) where

8(X,Y) 'V

-

'Vy(X)

-

[X,Y])

is defined in d) •

(Hint: Notice first that for any vector field

X on

F(M)

60

which is a lift of a vector field

TI*Xx = XTIX , Vx E F(M))

X

the function

on

(that is,

w(X)

the equivariant function corresponding to Note.

M

is X

as in b)

above.

Our treatment of principal bundles and connections

follows closely the exposition by S. Kobayashi and K. Nomizu [17, Chapter I and II].

4.

The Chern-Weil homomorphism

We now come to the main object of these lectures, namely to construct characteristic cohomology classes for principal G-bundles by means of a connection. Let k

V

let

First some notation:

be a finite dimensional real vector space. Sk{v*)

denote the vector space of symmetric

multilinear real valued functions in Equivalently

For

P E Sk{v*)

k

variables on

is a linear map

V.

P : V ®... ® V

lli

which is invariant under the action of the symmetric group V ®... @ V.

acting on

There is a product

o

defined by

P

(4.1)

Q{v 1 , ... ,v + l ) = k

0

1l)

= (k+ a

where S* (V*)

LoP (v 01'

.•. ,v o k)

Q{v O {k + 1 ) ' " .,vo{k+l))

1, ... ,k+l.

runs through all permutations of

II

(SO (V*) = lli);

Sk (V*)

then

S* (V*)

Let

is a graded

algebra. Exercise 1. lli [x ' ••• ,x 1 n]

k

k

Let

{e

1,

... ,e

n}

be a basis for

V

and let

be the set of homogeneous polynomials of degree

in some variables

Show that the mapping

defined by P{v, ... ,v),

v

62

for

P E Sk(v*), s*(v*)

-->

shows that given by

JR[x , ••• ,x 1

P v

is an isomorphism and that is an algebra isomorphism.

n]

is determined by the P(v, ... ,v).

This

function on

V

is called

The inverse of

polarization. Now let

be a Lie group with Lie algebra

G

the adjoint representation induces an action of for every

k: P (Ad (g

-1

) v 1 ' ••• , Ad (g

-1

v 1,···,vk E1' Let

Ik(G)

G

) v k) ,

E

g

G.

be the G-invariant part of

Notice that

the multiplication (4.1) induces a multiplication (4.2)

I*(G)

In view of Exercise

is called the algebra of invariant

polynomials on Now consider a principal G-bundle differentiable manifold E

with curvature form

M,

and suppose

Q E A2 (E'1)'

llk =llA ... I\QE 2k A (E'1@" so Q

P E Ik(G)

gives rise to a 2k-form

is horizontal also

equivariant and

P

2k-form.

p(Qk)

Hence

also denote by

p(Qk)

invariant

8

-->

M

on a

is a connection in

Then for

we have

k

2k(E'1®k) = A p(Qk) E A2k(E) .

is horizontal, and since p(Qk)

Since Q

is

is an invariant horizontal

is the lift of a 2k-form on

M which we

p(Qk).

Theorem 4.3. Let

.@')

E

n

a)

p(Qk) E A2k(M)

2k(A*(M)) WE(P) E H

is a closed form.

be the corresponding cohomology

63

class. b}

Then WE{P}

does not depend on the choice of connection

and in particular does only depend on the isomorphism class of

E. c}

WE: I*{G}

H{A*{M}}

d}

For

M a differentiable map

Remark. phism.

f

: N

The map

WE

is an algebra homomorphism.

is called the Chern-Wei 1 homomor-

Sometimes we shall just denote it by

W when the

bundle in question is clear from the context. WE{P} to

is called the characteristic class of

P E I*{G}

For E

corresponding

P. Proof of Theorem 4.3.

a}

Since

TI* : A*{M}

= 0

injective it is enough to show that Now since

P

is symmetric and

A*(E) in

is

A*{E}.

a 2-form

{4.4} by {3.15}.

On the other hand since

P E

is invariant

we have {4.5}

P {Ad (g t) y 1

gt

=

I

••• I

Ad (g t) y k } = P (Y 1

exptyo'

Differentiating (4.5) at

t

YO'Y1""

o

I

••• I

'Y k E

1'

we get

o or equivalently 0,

Yk)

I

t E JR.

64

From this it follows that

A

o

A ••• A

which

together with (4.4) ends the proof of a). b)

For this we need the following easy lemma (compare

Chapter 1, Exercise 5 or Lemma 1.2); Lemma 4.6.

Let

h : Ak(M

w

be the operator sending

J1

=

h(w)

s=O

a

--+

Ak- 1 (M),

k

0,1 , ... ,

S to

A

a +

(hw

o

ds

=

[0,1])

x

for

w E A0) .

Then

w E A*(M

dh(w) + h(dw) = iiw - iOw,

(4.7)

where

i 0 (p)

= (p, 0),

eO

Now suppose

= (p, 1 ),

L 1 (p)

and

curvature forms

e1

and

principal G-bundle e E A1 (E x [0,1])

are two connections in

--+

M

x

[ 0,1 ]

and let

be the form given by

By Proposition 3.10

e

be the curvature form of

P(D

k)

with

E

Consider the

(x,s)

obvious that

[0,1])

P EM.

respectively.

E x [0,1]

x

is a connection in

e.

Since

o =

and iiD 0 is a closed 2k-form on E i*D

i(;e Q1.

x

[0,1]

=

E E

x

[0,1].

E x [0,1].

eO'

Now for by a)

i*e = 1

Let

e1

D

it is

PErk (G) , above.

There-

fore by (4.7)

and hence in

H

2k(A*(M)).

and

represent the same cohomology class

This shows that

WE(P)

does not depend on the

65

choice of connectiQn.

The second statement is obvious from

this. c)

For

P E Il(G)

and

Q E Ik(G)

it is straight forward

to verify that (4.8)

from which c) d)

If

trivially follows. e

is a connection in

then clearly

I*e

curvature form

E

is a connection in

M with curvature form

f*E

N

with

Therefore since

(4.9)

d) clearly follows. Remark.

Let

be the algebra of complex valued

G-invariant polynomials on bundle

E

with connection

Then for any principal Ge

we get a similar complex Chern-

Weil homomorphism (G)

(4.10)

H(A* (M,lC»

!O
0

and we can clearly find an open nO (to, ... ,t ) such that V c int(tJ ).

of

nO

Define

and let

h

nO

coordinate of Now let

TI G

nO nO+ 1

n > nO

G

be the map which project onto the first

and suppose we have defined an invariant

74

open set

U EG(n-1} and an equivariant map h U ... G. n- 1 n- 1 n- 1 n n 1 p : 6 x G + ... EG(n} be the natural projection and

Let

observe that W

d6 n

h' W.

G

Shrinking

Now consider

since

be the closed subset

where

W"

w"

Un

h

n- 1 1 6 n x Gn +

W'

W'

6

n

Gn + 1

x

(U n 1)

Now let

U·

Clearly

h"

and

h"

: Un ... G. and

and def-ine

W'"

extends

clearly

U = U Un n

extends to a map

h'

W'.

defined on

W"

On the other hand we

W"' c 6 -1

W

n

n 1

x G

+

such that

(U -

n 1)

is an open G-invariant subset.

= W" n

p(U'}

n

Then

defined by

W is G-invariant.

c

Un = Un- 1 U

h

(Un- 1).

is an open G-invariant set and notice that and hence

-1

-1 -

W ... G

a little we can assume c

Let

is an open neighbourhood of

P

P

P

W p

0

can find a G-invariant open subset

since

EG(n-1}.

into

is an ANR the map

: W' ... G

Clearly

maps

ef+1

x

since

p

h

n- 1

0

h" : U' ... G

p : W ... G

and is equivariant.

is an open invariant set in h

n- 1

EG(n}

and

defines an equivariant extension

This construct h = U h n. n

by

Un

and

h

n

inductively, so let

This ends the proof of the proposition.

We can now state the main result of this chapter: Theorem 5.5. c

The map associating to a characteristic class

for principal G-bundles the element

1-1 correspondence.

c(E(G}} E H*(BG}

is a

75

For the proof we shall study

EG

and

BG

from a

"simplicial" point of view: Let

X = {X }, q = 0,1, ... , q

suppose that each

X

be a simplicial set and

is a topological space such that all

q

face and degeneracy operators are continuous.

Then

X

is

called a simplicial space and associated to this is the socalled fat realization, the space

II X" given by

II X II

with the identifications ( E i t ,x )

(5.6)

(t , E i X) ,

tEALJ n-l, x E X n'

n

Remark 1.

0, ...

1..

=

,n,

1,2, ...

It is common furthermore to require 0, ...

(5.7)

n

=

,n,

0,1, ...

The resulting space is called the geometric realization and is I X I.

denoted by

One can show that the natural map

II X II .... I X I

is a homotopy equivalence under suitable conditions. Remark 2. Example 1. consider

X

Notice that both If

X

=

{X}

II' II

and

I· I

are functors.

is a simplicial set then we can

q

as a simplicial space with the discrete topology.

The name "geometric realization" for the space

Ixi

originates

from this case. Example 2.

Let

X

the simplicial space with

be a topological space and let NX

q

=

X

NX

be

and all face and degeneracy

76

operators equal to the identity. II NX II = II N (pt) II

Then

x

INXI

and

where

x X,

with the apropriate identifications. Example 3.

Let

G

be a Lie group (or more generally any

topological group) and consider the following two simplicial spaces

NG

(Here

and

NG(O) In

NG: NG(q)

G

x ••• x

G

(q+l-times),

NG(q)

G

x ••• x

G

(q-times).

consists of one element, namely the empty a-tuple !).

NG

NG(q)

NG(q-l )

NG(q)

and

->

NG(q+l)

are given by

0, .•. ,q.

i

Similarly in

NG

£i (gl""

£i : NG(q)

,gq) =

->

NG(q-l)

is given by

(g 2 ' ... ,gq) ,

i

o

(gl"" ,gi gi+l"" ,gq)'

i

1, •.. ,q-l

(g l' ... ,gq-l) ,

i

q

{

and

n

i

: NG(q)

->

n· (gl,···,g ) .i, q By definition map

y

: NG

->

EG

NG

NG(q+l)

by

= (gl,···,g· 1,1,g., ,... ,g q ),

i

= O, ... ,q.

= II NG [I and if we consider the simplicial given by

77

(5.8) it is easy to see that there is a commutative diagram II NG II

111 y

II

- -......·IINGII such that the bottom horizontal map is a homeomorphism. will therefore identify

BG with

The simplicial spaces

NG

and

with

and

II NG II NG

We II y II.

above are special cases

of the following: Example 4.

Let

C

be a topological category, i.e. a

"small" category such that the set of objects set of morphisms

Ob (C)

and the

are topological spaces and such

that (i)

Ob(C)

The "source" and "target" maps

are

continuous. (ii)

"Composition":

->

is continuous

where

x

consists of the

pairs of composable morphisms (i.e. (f,f') E Associated to nerve of NC(2)

=

C

C where

source (f)

=

target (f'».

there is a simplicial space NC(O) = Ob(C),

NC

NC(1) =

and generally NC(n) c MM(C)

x ••• x

MM(C)

is the subset of compos able strings

(n

times)

called the

78

That is, i = l , ... ,n-l.

NC(n)

Here (f

(f , f , ... ,f = 1 2 n)

E:

i

2

(f {

n

NC(n)

i:

->

1,

•.. ,f

i

f

0

i

(f , •.. ,f -

NC(n+l)

is given by

, ..• ,f ) , n

n 1

1

and

NC(n-l)

->

i+ 1,

... ,f

0

0 < i < n

n),

) ,

i

=

n

is given by 0, ...

Remark 1.

Notice that

N

,n.

is a functor from the category

of topological categories (where the morphisms are continuous functors) to the category of simplicial spaces. Remark 2.

Observe that a topological group is a topological

category with just one object and it follows that

NG

as

defined in Example 3 is exactly the nerve of

G

as defined in

Example 4.

NG

defined in

Furthermore the simplicial space

Example 3 is exactly the nerve of the category

Ob(G) = G

follows:

target (go,g,) = go y

NG

y

G

NG G

and

= G

and

(go,g,)

Let

is a topological category is a pair )

O

Now i t is easy to see that the natural map

induces an isomorphism in homology, hence by assumption II f II :

(II X II (n) , Ii X II (n-1))

->

induces an isomorphism in homology. five-lemma shows that

II f II :

(II X'II (TI) ,ii X'II (n-1)) Now iterated use of the

II X II (n )

II x'II (n), n

-->

induces an isomorphism in homology and therefore

II X II

->

II X' II

=

1,2, ... ,

II f II :

also induces an isomorphism in homology.

By

the Universal coefficient theorem the result now follows, and thus finishes the proof of Proposition 5.15. Corollary 5.17.

Suppose

f

O

,f

1

: X

->

X'

are simplicially

homotopic simplicial maps of simplicial spaces p

there are continuous maps

satisfying i) -

and let

i

-->

=

O, ... ,p,

iii) of Exercise 2b) of Chapter 2).

II fOIl* Proof.

hi : X p

(i.e., for each

=

II f

1

11* : H*(II X'II)

-->

H*(II X II).

In fact consider the induced maps

c p + 1 ,q (X') p

L

i=O

->

cp,q (X) . #

(-1)lh.

1

be defined by

Then

85

Then

as in Exercise 2 of Chapter 2.

s since ff

f::

6" + 8"

0

p+l

are chain maps

and

C* (X')

....

Furthermore S

0

0

p+l

C*(X' ) .... C*(X ). p+l p

C*(X)

It follows that

are chain homotopic and hence

induce the same map in homology. Proof of Theorem 5.5.

n

class and let

U of

covering : n

-1

E .... X

(U

a

X

) .... U

a

First let

c

be a characteristic

be a principal G-bundle.

Choose a

such that there are trivializations

x G

and consider the diagram (5.12) above.

Notice that there is a commutative diagram

---='---...... where

U)

H (C

is the isomorphism of Lemma 1.25, so that

E* U

is

also an isomorphism. Now by naturality of (5.18)

E

U

U(C (E»

c

= f U(c (EG) )

and since

E

by

and equation (5.18).

c(EG)

is an isomorphism

On the other hand let principal G-bundle the class (5.19 )

E

U(c (E»

Co

c(E)

H*(BG) c(E)

by

is uniquely determined

and define for a

86

we must show that Now if of

X

then

c(E)

is well defined:

=

U

{ua}aEL

W

{Ua nuS} (a,S)ELXL'

and

U'

{US}SEL'

are two coverings

is also a covering of

X

and clearly there is a commutative diagram

(5.20)

Also let ljJW

f W : II NXW"

-->

BG

be the realization of

NljJW

where

is given by the transition functions corresponding to the

US.

trivializations

Then clearly there is a commutative

diagram

(5.21 ) IINXull/ From the diagram (5.20) and (5.21) it follows that it is enough to show that for any covering

q; (cO)

E H* (II NX U II)

U

the element

does not depend on the particular choices

of trivializations So let

and

be two sets of trivializations

} and let ljJ,ljJ' : Xu --> G be the a corresponding continuous functors. We want to show that the associated to

associated maps in cohomology.

Aa : Ua

-->

U =

{U

f U' f U : II NX II U

-->

BG

induce the same map

Now the family of continuous maps

G, a E L,

defined by

87

(x, \1

(x,g) E U

(X) 'g) ,

X

Ct

G,

satisfy A

Ct

Hence A :

A

W

=

W·

(x)

1

is just a continuous natural transformation of the functors

and

f Q= f U *

That

therefore follows from Corollary 5.17 and the following general lemma: Lemma 5.20.

Let

C

of topological categories

D

C, D.

be two continuous functors

If

W

A :

continuous natural transformation then

W'

NW,NW'

is

Q

NC

ND

are

simplicially homotopic simplicial maps. Proof.

We shall construct

hi : NC(p)

ND(p+l), i

satisfying i) - iii) of Exercise 2b) in Chapter 2. simplex in A

For

o

...

f

NC

f

1

ND

O, ... ,p,

Now a p-

is a string

i = O, ... ,p,

simplex in

=

hi

P

A

O'

••• ,A

E

MOJL (C) •

associates to this string the

(p+l)-

p

E Ob (C) ,

given by the string

w'

(f.) l

1jJ'(A.)< l

AA. l

W(f ) P

hi : NC(p)

ND(p+l)

is clearly continuous and it is straight-

forward to check the identities i) - iii) of Exercise 2b) in Chapter 2.

This proves the lemma.

88

It follows that

c(E)

defined by (5.19) is well defined

and it is easily checked that condition (5.2). Note.

c(E)

satisfies the naturality

This ends the proof of Theorem 5.5.

The original construction of a classifying space

is due to J. Milnor [20]. the one in G. Segal [24).

Our exposition follows essentially

6.

Simplicial manifolds. In this chapter

coefficients.

H*

The Chern-Wei I homomorphism for again denotes cohomology with real

We now want to define for a Lie group

w : I*{G)

Chern-Weil homomorphism is that

BG

and

is a simplicial manifold.

NG

is not a manifold.

H*{BG);

However,

the

but the trouble

BG

That is,

G

II N{G) II,

X

a

{X }

q

simplicial set is called a simplicial manifold if all COO

BG

X

manifolds and all face and degeneracy operators are

are

q

COO

maps. Example 1.

X = {X is a q} considered as zero dimensional

Again a simplicial set

simplicial manifold with all

X

q

manifolds. Example 2. space

NM

with

Also if NM{q)

=

M is a

00

C

manifold the simplicial

M and all face and degeneracy operators

equal to the identity is again a simplicial manifold. Example 3. and

NG

For

G

a Lie group the simplicial spaces

are also simplicial manifolds and

y : NG

NG

NG

is a

differentiable simplicial map. Example 4.

U

=

{Ua}aEL

manifold.

For

MaC

00

manifold with an open covering

the simplicial space Finally, if

E

NM U

is also a simplicial

M is a differentiable principal

G-bundle with differentiable trivializations

: U a a a then taking the nerves of the diagrams (5.10) and (5.11) we

obtain the corresponding diagrams of simplicial manifolds and differentiable simplicial maps.

x

G

90

Now let us study the cohomological properties of a simplicial manifold, in particular we want a de Rham theorem. Again in this chapter for

M a manifold

C*(M)

denotes the

cochain complex with real coefficients based on

COO

singular

simplices. Now consider a simplicial manifold Chapter 5 we have the double complex

X = {X}. p

cp,q(X)

=

As in

Cq(X ). p

Notice

that by Lemma 1.19 and Exercise 4 of Chapter 1 the natural map

Cq (X) ... top p induces an isomorphism on homology of the total complexes. Here

We also have the double complex the vertical differential differential in 8'

Aq{X

p)'"

A*{X) p

Aq{X + p 1) 8'

d"

is

(-1)P

times the exterior

and the horizontal differential is defined by p+1

L

i=O

. {-1)lS'!'. l

Furthermore we have an integration map

which is clearly a map of double complexes.

By Theorem 1.15 and

Lemma 1.19 we easily obtain Proposition 6.1. Then

Let

X

Ix: AP,q(X) ... cp,q{X)

H(A*(x»

=

{X} p

be a simplicial manifold.

induces a natural isomorphism

H(C*(X»

H* (II XII).

Now there is even another double complex associated to a simplicial manifold which generalizes the simplicial de Rham complex of Chapter 2:

91

Definition 6.2. manifold

X = {X} p

A simplicial n-form is a sequence

on the simplicial

=

of n-forms

such that

on

(id

(6.3)

x

O, ... ,p,

i

Remark.

Notice that

L'l p - 1 x X ,

on

=

P

P

=

0,1,2, ...

defines an n-form on

II L'l P x X and that (6.3) is the natural condition for a form p=O p on II X II in view of the iden tif ica tions (5.6). In the

following the restriction denoted

of

Notice also that for

is also

to X

discrete Definition 6.2

agrees with Definition 2.8. Let

An(X)

denote the set of simplicial n-forms on L'l P

Again the exterior differential on differential

An+ 1 (X)

d: An(X)

x

X

P

X.

defines a

and also we have the exterior

multiplication 1\

satisfying the usual identities. The complex

(A*(X) ,d)

is actually the total complex of

a double complex

(Ak,l(X) ,d' ,d").

Ak,l(X), k+l = n

iff

x

X

p

Here an n-form

lies in

is locally of the form

=L where L'l P

(to, ... ,t p)

and

{x.} J

as usual are the barycentric coordinates in

are local coordinates in

that

II

k+l=n and that

Ak,l (X)

X. p

It is easy to see

92

d where

d'

d' + d"

is the exterior derivative with respect to the

barycentric coordinates and

d"

is

(_l)k

times the exterior

derivative with respect to the x-variables. Now restricting a over

6

k

6k

(k,l)-form to

x

X k

and integrating

yields a map

which is clearly a map of double complexes.

The following

theorem is now a strightforward generalization of Theorem 2.16: 6.4.

(A*,l(X),d')

and

For each

the two chain complexes

1

(A*,l(X) ,0')

are chain equivalent.

In fact

there are natural maps E

and chain homotopies

such that (6.5)

1

(6.6)

d

6 I

0

d' = 0'

0

E

E

0

0

1,

E

0 '

0

d"

=

d"

0

E

id

(6.7)

o.

(6.8)

In particular

1

6:

Ak,l(X)

Ak,l(X)

induces a natural iso-

morphism on the homology of the total complexes (6.9)

H(A*(X))

H(A* (X))

H* (II X II).

93

Also let us state without proof (see J. L. Dupont [11]) the following generalization of Theorem 2.33: Theorem 6.10.

The isomorphism (6.9) is multiplicative

where the product on the left is induced by the

A-product

and where the product on the right is the cup-product. As an application of Theorem 6.4 let us consider a manifold

M with a covering

U =

{U}

a an:

and let

NM U

be the

simplicial manifold associated to the nerve of the category

MU.

Notice that the natural map

is induced by the natural projections

and that these also induce the natural map A*(M) Corollary 6.11. the natural map

A*(NM ) .

A*(NM) For

A*(M)

U

U=

{U}

a

A*(NM )

an open covering of

M

induces an isomorphism in

U

homology. Proof.

In fact the composite

is the map

eA

of Lemma 1.24.

Now let us turn to Chern-Wei I theory for simplicial manifolds.

A

sequence E =

{E

P

simplicial G-bundle n

p

}, M =

: E

P

{M }

P

M

p

n : E

M

is of course a

of differentiable G-bundles where

are simplicial manifolds,

n: E

M

is

94

a simplicial differentiable map and also right multiplication g E G, R : E .... E, g

by 'IT

:

E

A connection in

is simplicial.

.... M is then a 1-form

on

8

(in the sense of

E

1

Definition 6.2 above) with coefficients in restricted to

t-,P

x E

p

for

8

is a connection in the usual sense in

E .... liP x M P P Again we have the curvature form liP

the bundle

such that

x

P E Ik(G)

we get

p(n k ) E A

n

2k(M)

defined by 3.14 and a closed form

representing a class

such that Theorem 4.3 holds. In particular let us consider the simplicial G-bundle y : NG

.... NG.

There is actually a canonical connection in

this bundle constructed as follows: Let G .... pt.

80

be the Maurer-Cartan connection in the bundle

Also let

qi : liP

8

is simply given over

(6.12 )

liP

NG(p)

x

t

(to, ... ,t

By Proposition 3.10,

p)

be the projection

O, ... ,p,

=

8

where as usual liP.

NG(p) .... G

GP+ 1, i

onto the i-th factor in Then

x

and let

8.

1

by

8

P P

are the barycentric coordinates in 81l1 P x NG(p)

is clearly a connection

in the usual sense and it is also obvious from (6.12) that satisfies (6.3). Theorem 6.13.

We now summarize: a) There is a canonical homomorphism W

such that for

8

I*(G) .... H*(BG)

P E Ik(G),

w(P)

is represented in

2k(NG) A

by

95

p{n k )

n

where

is the curvature form of the connection

e

defined by (6.12). b)

P E Ik{G),

Let for

characteristic class.

w{P) (.)

Then if

TI:

be the corresponding

E

M

is an ordinary

differentiable G-bundle we have W{P) (E)

where

a*

WE : I* (G)

c)

W:

d)

Let

I*{G)

H* (M)

I*{G)

H*{BG)

a: H I*{H)

is the usual Chern-Weil homomorphism.

G

is an algebra homomorphism.

be a Lie group homomorphism and let

be the induced map. a*

I*{G)

-1

H* (BG)

Then the diagram

• I* (H)

1-

Ba*

• H* (BH)

commutes. Proof.

a)

is a definition.

b) Choose an open covering trivializations of

E

U

=

{Ua}

of

M

and

so that we have a commutative diagram

of differentiable simplicial bundles:

}n

NE ... '---

F-u---+' 1

NM+'--- NM U -----'>. NG . By the proof of Theorem 5.5 the pull back of

II NM U II in

is given by

H{A*{NM U)

fii

(w

(P) )

w{P) (E)

to

which clearly is represented

by the Chern-Weil image of

P

for the simplicial

96 G-bundle

NE

e

connection in

E

M

NM

V

U

e'

with connection

defined by (6.12).

On the other hand a connection

induces another connection

the pull-back of

WE(P)

in

by the Chern-Weil image of

H(A*(NM P

induced from the

en

in

NE

NM

V

U

and

is clearly represented

U)

using the connection

However, by the argument of Theorem 4.3, b)

en.

the Chern-Wei 1 image

is independent of the choice of connection, which proves that

where

IOU:

Since

IOU

II NM

U

II

M

is the natural map considered above.

induces an isomorphism in cohomology this ends the

proof of b). c) follows again from the simplicial analogue of Theorem 4.3 c) and Theorem 6.10. d)

is straightforward and the proof is left to the reader.

Note.

Notice that by a),

the total complexes defined elements.

A*(NG)

and

w(P)

is also represented in

C*(NG)

The construction of

by canonically w(P)

in

A*(NG)

is

due to H. Shulman [26] generalizing a construction by R. Bott (see [2], [4], and [5]).

The exposition in terms of simplicial

manifolds follows J. L. Dupont [11].

7.

Characteristic classes for some classical groups

We shall now study the properties of the characteristic classes defined in the examples of Chapter 4. Chern classes. For

G

we considered in Chapter 4 Example 4 the

=

complex valued invariant polynomials defined by (4.13).

C k k,

=

0,1, ... ,n

=

(CO

For a differentiable

TI

:

1),

E

we thus define characteristic classes called the Chern classes 0,1, ... ,n,

(7,1)

where

represented by the complex valued 2k-forms is the curvature form of a connection in

TI

:

E

M.

Q

Notice that

since every complex vector bundle has a Hermitian metric, i.e. a reduction to inclusion

U(n),

Ck(E)

2k(M,B) H

actually lies in the image of the

H2k(M,a:)

(c f . Exercise 4 of Chapter 4).

By Theorem 6.13 we can extend the definition of the Chern classes to any topological

and then use Theorem 5.5. since

C k

Gl(n,cr)-bundle by first defining

Again

restricted to

from Theorem 6.13 d)

c

is a real class. In fact k is a real polynomial it follows

to BU(n) is k a real class (represented by a real valued form), and since the natural inclusion

that the restriction of

j : U(n)

C

Gl(n,cr)

it follows that Bj

BU(n)

B Gl(n,a:)

c

is a homotopy equivalence

M

98 induces an isomorphism in cohomology. Proposition 7.2.

Let

a : H

In general we have

G

be a homomorphism of

two Lie groups which induces an isomorphism in homology (coefficients in a P.I.D.

h).

Then also

Ba : BH

BG

induces an isomorphism in homology as well as in cohomology (with coefficients Proof.

h).

By Runneth's formula

Na(p)

: NH(p)

induces an isomorphism in homology for each

NG(p)

p.

The proposition

therefore follows by Lemma 5.16. Before continuing the study of the Chern classes we make a few definitions: Suppose we consider a topological space

E Then the

X

s:

and a

: E e F

sum

with a principal

X

X

X.

F

is most easily de-

scribed in terms of transition functions as follows: First let

be the homomorphism taking a pair of matrices

(A,B)

to the

matrix A Now choose a covering and

F

and

{haS}

and

F

e

B

U =

have trivializations over

F

X

such that both

Ua' a

and let

E

{gaS}

be the corresponding transition functions for

respectively.

e

Then

with transition functions and

of

{gaS

: e

e

E

F

haS}'

are differentiable then also

E

X

is the bundle

Notice that if

e

F

is.

E

E

99

=

Notice that

=

the multiplicative

group of non-zero complex numbers.

are in 1-1

correspondence with l-dimensional complex vector bundles (also called complex line bundles).

An important example is the H* (II NG II) constant.

\1/

r*

is multiplication by a non-zero

This shows the injectivity of

proves (8.28).

II i 11*

Choose an inner product on

is invariant under the adjoint action of G

is compact).

that is, split

if

Ad(exp(t», t E./-

acts on

(8.31)

21TCi (t) 1

where

Ci

i

M'Z,

--> JR,

G/T

that

by the matrix

0

cos 21TCi (t). 1

° i = 1, ... ,m,

cos 21TCi (t) m

-sin 21TCi m(t)

sin 21TCi m(t)

cos 21TCim (t)

are linear forms on

details see e.g. Adams [1, Chapter 4]). bundle of

for

2m}

-sin 21TCi (t) 1

Sl.n 21TCi (t) 1

: "

(This is possible

into an orthogonal direct sum

1

C'

G.

which

Now make a root space decomposition of

and find an orthonormal basis {e , ••• ,e

Ad(exp(t) )

and hence

It remains to prove the existence of

Proof of Lemma 8.30.

since

II i II *:

0

,i

(for

Notice that the tangent

can be identified with the 2m-dimensional vector

127

bundle G

E;

x T ""'

.... G/T

which is clearly an oriented bundle with the orientation given by the basis

{e

Now let 8

K

1, :

1

,e

2rn}.

,j.

be the orthogonal projection and let

be the canonical connection in

NG

given by (6.12).

Then

clearly

defines a connection in the principal T-bundle

NG .... NG/T

let

P E Im(T)

n

T

be the curvature form.

Also consider

and

given by the polynomial function m

P(v, ... ,v) = (_l)m

1I a..(v),

i=l

Then by Chern-Weil theory the 2m-form on

NG/T

and we let

is a closed form

be the corresponding form on

under the identification (8.29), so clearly satisfied.

It remains to prove (ii).

is just the G .... G/T

v E)..

l

image of

with connection

8

T

P

Now

d'l'

=

N(G;G/T)

is m) = P(n E A2m (G/ T) 0

T

in the principal T-bundle

given by g E G,

and

f

nT

=

d8 T.

Unfortunately it is not so easy to calculate

directly. However, as noticed above the extension G/T of the bundle G .... G/T to the group SO{2m) via the adjoint representation on

is just the tangent bundle of

it is easy to see that

G/T

and

is exactly the Pfaffian form.

On the other hand it follows from (8.31) that the bundle is a Whitney sum of

SO(2)-bundles.

Therefore, as remarked after

128 Proposition 8.11

(cf. Exercise 2 of Chapter 7),

I

(8.32)

G/T

P(o*) = .

Now the right hand side of (8.32) we can compute by the formula (7.33) for a vector field of the following form: Choose a regular element ai' i

=

bundle

V

1, ... ,m) i;,: G x

o

E )

and consider the section 4ft ....

T

G/T

s(gT) where again va

given by

(g, (id-K)

0

Ad(g

-1

)v

0

s

at

g E NT

iff NT/T

W

gT E W

maps a neighbourhood of

gT

{T}

g E G

s(gT)

is

in

vanishes at the

+1.

For this we recall the exp

.... G .... G/T

diffeomorphic onto a neighbour-

0 E

G/T,

Since

Now we claim that the

G/T.

c

s

so

well-known fact that the exponential map

hood of

a) ,

is the orthogonal projection.

finite set of points local index of

of the vector

1 .. ;

K

is regular

s

so we get a local trivialization near

by (g exp x,v) ..... v,

x E

near zero, v E "4(.

It is therefore enough to see that the map

s

....

given

by i(x)

=

(id-K) (Ad(exp(-x))v

g),

v

g

Ad (g

is an orientation preserving diffeomorphism near differential x E

s*

at

0

is given by

i*(x)

-1

O.

) v, x E mt , The

= -[x,v g] = ad(v g) (x),

Differentiating (8.31) and taking the determinant now

gives det(ad v ) g

(2,r)

m.!!1..

Ila.(v) i=1 l g

2

>0

129

so the local index of

J

s

'ji0

at

gT

J

G/T

is +1.

PIn;)

=

It follows that

Iwl >

0

G/T

which proves Lemma 8.30 and finishes the proof of Theorem 8.1 for

G

connected.

For

G

a general compact group we get a diagram similar

to (8 .27) : I* (G)

1

(8.33)

Inv G/ G (I*(G O)) 0

---+

H* (BG)

1

Inv G/G (H* (BGO) )

---+

0

where again the upper horizontal map is the isomorphism (8.2) and the right vertical map is an isomorphism since connected.

Again it suffices to show that if

GO

i : GO

is G

is

the inclusion then (8.34 )

Bi*: H*(BG)

H*(BG

is injective.

O)

As before, this is equivalent to showing that II ill*

i

: H*(II NG II)

N(G;G/G o )

NG

H*(IIN(G;G/G O) II)

is injective, whe r e

is defined as follows: NG(p)

and

i

x

G/G

O

is given by the projection

on the first factor. T

This time A*N(G;(G/G

o) ) )

A*(NG)

is simply given by T(lP) where

llP

x

NG(p)

L

gEG/G llP

s*lP, g

O

x

lP E A*(N(G;G/G o)

N(G;G/G

O)

(p)

is given by putting

130

gG

o

E GIGO

on the last coordinate (notice that

a simplicial map but still easily checked that

T

T

is well-defined).

I GIGO I.

multiplication by

IG/GOI

Hence also

of Theorem

0

T

r

II

0

i*

II *

is

is

where

T* : H*(II N(G;G/G T.

T*

is not

g

Again it is

is a chain map and that

multiplication by

is the map induced by

s

O)

II)

H*(II NG II)

-+

This shows (8.34) and ends the proof

8.'.

Corollary 8.35.

(A. Borel [3]).

connected Lie group, and let a maximal torus.

Then

Bi

i

T

Let -+

BG

BT

G

G

be a compact

be the inclusion of

induces an isomorphism

H* (BG) ------>. rnvwH* (BT) . Proof.

Obvious from the diagram (8.27).

Corollary 8.36.

(i) The Chern classes of

are uniquely determined by the properties a) - d)

of Theorem

7.3.

(ii) Furthermore H* (BU(n)) ';;; m. [c"

H* (BGl (n,(£))

is a polynomial ring with the Chern classes

... ,c

c, , ... ,c

n] n

as

generators. Proof. U(n)-bundles.

As noticed in Chapter 7 it is enough to consider Now let

i : Tn

(

o

-+

U(n)

be the natural inclusion

131

and let j

qj : Tn

1, ... ,n.

=

U(1)

be the projection onto the j-th factor,

It is well-known that

Tn

is a maximal torus so

by Corollary 8.35 Bi* is injective. values on

H*(BT n)

H*(BU(n))

That is, the Chern classes are determined by the

U(n)-bundles which are Whitney sums of

Hence by (7.5) they are determined by

c

on

1

U(1)-bundles.

U(1)-bundles.

This, however, is determined by (7.6) as remarked immediately after Proposition 8.11.

This proves (i).

(ii) By Corollary 8.22 Yj

=

n)

= JR[Y1""'Yn]

2(BTn), (Bqj)*C 1 E H j = 1, ... ,n

the first Chern class. on

H*(BT

Tn

Now

and

c

1

where 2(BU(1)) is E H

W is the symmetric group acting

by permuting the factors, i.e. n

Hence

Invw(BT )

W acts on

H*(BT

n)

by

is a polynomial ring

with generators the elementary symmetric polynomials 1, ... , n,

der Waerden [32,

§ 29]).

(see e.g. B.L. van

in

However, by (7.5) 1, ... In,

which proves the corollary. Corollary 8.37. for

(i) The Euler class with real coefficients

SO(2m)-bundles is uniquely determined by the properties

i), ii), and v)

of Exercise 1 e) of Chapter 7.

In particular

formula (7.39) holds. (ii) Furthermore H*(BG1(2m,JR)+)

H*(BSO(2m)) JR [P1'··· ,Pm-1 ,e]

132

is a polynomial ring with generators the first classes

and the Euler class

m-l

Pontrjagin

e.

(iii) Finally H* (BGl (2m,lR))

H* (BO (2m))

is a polynomial ring in the Pontrjagin classes Proof. be the set

The maximal torus in m T

SO(2m)

cos 21Tx

1

o

1

cos 21TX

o

1

sin 21TX

m. , ••• ,x ) E lRm/ ." IJJ

Again let

m

inclusion and let

-sin 21TX

m

cos 21TX

m

SO(2m)

i

SO (2), j

projection on the j-th factor. injectivityof

is well-known to

of matrices of the form -sin 21Tx

(x

Pl, ... ,Pm.

1, ... ,m,

=

m m

be the be the

As before (i) follows from the

i*: H*(BSO(m))

H*(BT

m)

together with the

remark following Proposition 8.11. m) H* (BT

(ii) Again Yj

=

5.17)

2

(Bq.) *e E H (BTm) . J

that the Weyl group

= lR [Yl'··· 'Ym]

where

It is easily seen (cf. Adams Example W acts on

m) H* (BT

by permuting

the Yj'S and changing the sign on an even number of the Yj's. We want to determine the subring

First notice that

A

changing the sign of

has an involution Y say. 1,

T

:

Then clearly

A A

A

given by

= A+

ffi

A_,

133

where

A+

and

A

are the

±1

eigen spaces for

Notice

W'

is the group generated by the permutations of the

T.

that

where

Yj's together with the transformations which changes the sign of any number of the

where

0.

Yj's.

is the j-th elementary symmetric

'=

]

It is now easily seen that

polynomial

Now every element of

A_

is easily

seen to be divisible by the polynomial £

'=

Hence

Now

Here £

'=

£

2

hence

0,

J

'=

(Bi)*e

{Bi)*p"

J

j

1, ... ,m, by (7.20) and (7.26), and

by (7.24).

This proves (ii).

(iii) By Theorem 8.1 and (8.2) H*(BO(2m))

r-J

I*

(O

(2m))

Inv O(2m)/SO{2m) (I*(SO{2m))). Here

0{2m)/SO{2m) ;;:: 'll./2

acts on

I*{SO{2m))

using the adjoint

action of an orientation reversing orthogonal matrix.

This

clearly fixes the Pontrjagin polynomials and changes the sign

134

of the Pfaffian polynomial (see Chapter 4, Example 1 and 3). Hence the invariant part of ring in the variables

I*(SO(2m))

P 1"",Pm-

is the polynomial

and

1

Pf2 = P. m

This proves

the corollary. In a similar way one proves Corollary 8.38. (L)

H*(BG1(2m+1)+)

H*(BSO(2m+l))

lR[Pl, ... ,Pm]

is a polynomial ring in the Pontrjagin classes. (ii) H*(BGI(2m+l))

H*(BO(2m+l))

H* (BSO(2m+l))

lR [p 1 ' ••• , Pm] . Remark.

In all the cases considered above

Inv w(S*(4*)) In fact if

V

is a polynomial ring.

H*(BG) =

This is no coincidence.

is any real vectorspace of dimension

I

and

W

is a finite group generated by reflections in hyperplanes of then

Invw(S*(V*))

is a polynomial ring in

N. Bourbaki [6, Chapitre V,

§

I

V,

generators (cf.

5, theoreme 3]).

APPENDIX We will in this appendix give a proof of the differentiability of the function

pI

8.3 by the formula (8.5).

defined in the proof of Proposition First we recall some rather standard

facts from the theory of Lie groups. In the following suppose

G

simple Lie group without center.

=

1 @lR

a:

is a compact connected semiLet

the complexification of

complex analytic Lie group

Ga:

be the Lie algebra and

1.

Then there is a

(the complexification of

G)

135

and an injection algebra of

1 .. 1

$

Ga

i1 =

j

:

and

1

without center

G .... Gn: j*

ad

(1)

= End

To see this notice that since

a;'

(1)

Ad: G .... Gl

(1)

is

G

is injective and the image c

Int(1) = Gl(1)

with Lie algebra

defined by

ad(1 ) = {ad (v) I v

E1 ,

i

;; Gl('1rr)

group with complex Lie algebra

1rr .... ad(

XE1

ad(v) (x ) = l v x ] ,

Ga; = Int(,

We can then take

ad :

is the Lie

is the natural inclusion

is the connected subgroup c

such that

(1 n:)

ad

c

is an isomorphism and

}.

the complex analytic End j

(1rr) .

:

Here again

G .... G rr

is given

by the composite

In the following we shall identify

G

with the image in

Grr.

We also need the Jordan-decomposition of elements of

v

For a complex vector space

a linear map

1e '

A E End(V)

has a unique Jordan-decomposition S + N,

A

with for

S 5)

semi-simple and

v E

ad (v) E End ( 1n:) Lemma8.A.1. decomposition

v

is nilpotent and Proof.

(i.e.

V

nilpotent (i.e.

N

particular for

SN

1 rr

NS

has a basis of eigenvectors Nk = 0

for some

k

In

0).

we have a Jordan-decomposition of

and we have For

v E

s + n [s,n]

1rr

there is a unique Jordan-

such that

=

ad v

is semi-simple,

ad n

0

We must show that the semi-simple part of

(and hence also the nilpotent part) lies again in

ad v

136

c:

ad(1 rr) = End(1 rr) ·

1rr

Since

1rr

the Lie algebra of derivations of [14, Chapter II,

Proposition 6.4]),

lies in

iff [ox,y] + [x,oy]

O[x,y]

We must show that if

0

ad(1 rr)

is semi-simple

(see e.g. S. Helgason

o E End (

that is,

,

x,y E

1rr .

So let

0

= S

be the Jordan

+ N

decomposition.

Then there is a direct sum decomposition

'1'rr=

such that

A

A eigenvalue

is the eigenspace of

(4jrr)A

S

S

with

that is

A,

o That

1rr)

is a derivation then also the semi-

simple part is a derivation.

E9

is

for some

k

is a derivation simply means that for

O}.

>

A,

E rr,

This, however, easily follows from the identity k

[x,y]

=

k

k L (.) [ (0-1..) k-i x ,

which is proved by induction on Now let let

k rr

=

T

4®JR

G II

connected Lie group. ad(t) metric.

i

i=O

k.

y],

x,y E

1rr'

k=O, 1,2, •.

This proves the lemma.

be a maximal torus with Lie algebra

irr

and let

Every element

4,

be the corresponding is semi-simple since

is skew-adjoint with respect to a G-invariant Therefore every element of

is semi-simple as well

and we have the root space decomposition (see e.g. Helgason [14, Chapter III, § 4])

r

137

where

et E ,

1

are the roots, i.e.

one-dimensional subspaces and

Furthermore let

t

(8.A.2)

+ c

.t and

Then both (8.A.3)

B

Go:

Ell = aE+ 1a:a

Ell etE+ G:et

t-+

1 er

are subalgebras of

[1a:et' 1a:s]

Also let

be a choice of positive roots and let

l' ' t- +

A. a: Ell

=

are

a:et

since

a,S E .

"da:(et+S)'

be the group with Lie

With this

notation we now have

i7o:'

Lemma 8.A.4.

b) Ad(g)v

+

= t+n E.Ir

is nilpotent.

with

v

is semi-simple and

v E l 0:

there is

tEA-a:' n EJ.+

and

g E Ga:

such that

[t,n]

O.

then the semi-simple part of

v

Further-

is conjugate

t. c) G

The inclusion and

d)

If

there exists

Proof.

v,

If

NT .... NTa:

of normalizers of

and

T

Ta:

Ga:' respectively, induces an isomorphism W

of

-f!r

For every element

more, if

in

is a maximal abelian subalgebra of

Furthermore every element of

every element of

to

a)

= NT/T ----=--. NTa:/Ta: .

s E 4a:

and i f for some

w E NTa:

a)

such that

For

[v,4a:J = 0

let

'1]

= 0,

Re v [Tm

A-rr

then

= Ad(g)s.

0e the complex conjugate

CiT, "a:J

then clearly also

both the real and imaginary part [Re v

Ad(w) s

v

Ad(g)s E

g E Go:,

rm v

and

J

=

0

=

0

satisfy

so

138

so by maximali ty of

j

v = Re v + i 1m v =

is a maximal abelian subalgebra.

o.

This shows that

4- II:

The second statement is

already proved and the last clearly follows from (B.A.3). b)

By the Iwasawa decomposition (see e.g. Helgason [14,

Chapter VI, Theorem 6.3]) we have G . e xp f L

(B.A.5) in particular

B n G

T

J) .

expp

and the inclusion

G

induces

a diffeomorphism

so the Euler characteristic of

is different from zero

(cf. Adams [1, proof of Theorem 4.21]).

For

v E

10::

we there-

fore conclude by Lefschetz' fixed point theorem that there is an element

g E

gB E

such that

is fixed under the

one­parameter group of diffeomorphisms

where

exp(rv)xB, r

hr(XB)

g Hence

-1

E,.t..

Ad (g -1) v

E JR,

that is,

exp(rv)g E B,

Vr E JR.

We can therefore suppose

VEt,

and

we write (B.A.6)

v

Now we claim that we can change of

B

so that

x

a

0

v

only for

is a minimal root so that both

by conjugation by elements a(t)

x

a

*0

=

O. but

In fact suppose a(t)

O.

Then

a

139

1

Exp(ad(a(t) x a)) (v) vt+ where

a'

> a

means that

a' - a

this procedure we can find

b E B t +

Ad(b)v

Therefore we put [t,n] = 0;

hence

4.11:

b E B

x ))i(v) a

a

Ya'

is a positive root.

Iterating

such that

L

n =

Notice that conjugation by in

L

a'>a

aE¢+ a(t)=O

L Z E aE¢+ ct Ad(b)v = t + n

L

a (1 [v,x + t) a]

z. ct

and we clearly have is the Jordan decomposition. does not change the component

in the decomposition (8.A.6) which proves the second

statement in b) . c)

Clearly

NT/T

NT

n G

and since

is injective.

left-multiplication by

Now for

gET

the map

T

a regular element,

g

has a fixed point for every element in

B.

Therefore

the composite

is a bijection so it remains to show that

T

a

=

n B.

however, is trivial from the fact that every element of of the form

a . exp(n)

with

a E Til:

and

n

Let

s E

4a

and

Consider the Lie algebra

g E Gil:

with

Ad(g)s

B

is

This ends

the proof of c). d)

This,

t E

ill:'

140

J= and let

0

Ga:

x E

E1a:

I

[v,t]

O}

be the associated connected sUbgroup of

j a:

Then clear ly

{v

J

Ad (g) ). a:

and also

Ga:'

since for

/.1£ 1. (f

Also

Ad (g) j..

and hence

1£

o.

l xvs l

[Ad (g) (x) , t]

are Cartan subalgebras (i. e. a

nilpotent algebra with itself as normalizer).

Hence by the

conjugacy theorem (see e.g. J. P. Serre [25, Chapitre III, Theoreme 2])

there exists a Ad (g)

Hence

d

-1

g E NTa:

and

d E 0

such that

Arr.

Ad(d

-1

Ad(d)t

g)s

t.

This ends the

proof of the lemma. After these preparations we now return to the proof of the

8.3.

Recall that

Lie group

G

m

pI

differentiability of

in the proof of Proposition

is the Lie algebra of a compact connected

with maximal torus

polynomial of degree

T

and

P

is a homogeneous

on the Lie algebra ), of

k

T.

pI

is defined by the formula pI (v)

P (ad (g) v)

We shall show that pI

a:

on

pI

where

lid (g) v E

4--

for some

g E G.

extends to a complex analytic function

11£.

Since

G

where

is compact

}= is the center and [14, Chapter II,

{v E

11

' is a l Proposition

[v,x]

=0

Vx E

7}

semi-simple ideal (see Helgason 6.6]).

Furthermore, if

Z c G

141

is the center of

'1'

then

G

is naturally identified with G' = G/Z.

the Lie algebra of the group representation factors through Ad(g) (z+v)

and

z + Ad(g' )v,

g' = gZ E G'.

where

A=

f

@

Notice that center.

G'

./.

G'

n

Also

i

Z

Clearly the adjoint

and E}

, v E

I"

g E G,

T' = T/Z

is a maximal torus in

G'

./ n

is the Lie algebra of

T' .

where

7'

is a compact semi-simple Lie group without

Therefore we shall restrict to the case where

semi-simple without center.

G

is

The reader will have no difficulties

in extending the arguments to the general case. The homogeneous polynomial

clearly extends to

P : , , / .... lR

a complex homogeneous polynomial

..

:

and obviously

is invariant under the adjoint action of 8.A.4 c) and the invariance of

For

I rr ....

pI

Now define

C :

v

under the action by

w on

A.

as follows:

0:

choose

Ell!

p

by Lemma

such that

g E

Ad(g)v

t + n

as in Lemma 8.A.4 b), and put

Then this is clearly well-defined by the uniqueness of the Jordandecomposition and Lemma 8.A.4 d) . First we show that let

1T

:

.t . .

j. I!

@

+

-t. = "-I! t

.•

....

is continuous:

be the projection in the and notice that if

then we can write Ad(u)v

P'

g

= Ad(b -1 )

=u

Ad(g)v

. b, u E G, b E B

(t+n)

=

Clearly also

t + n',

with

pl.

For this

decomposition

=

t + n

as above

by (8.A.5) and then n' E

..t +.

142

It follows that

=

(S.A.7)

and by the second part of Lemma S.A.4 b) any

u E: G

such that

To show that

Ad (u) v E:

P'

{v

there is a subsequence choose

uk E: G

fr.

is continuous it suffices to show that

a:

whenever a sequence

this equation holds for

k}, {v

such that

k

=

1,2, ... ,

converges to

v,

k .)

(v).

such that

k.}

(v

l

l

we can assume by taking a subsequence that say.

Hence

Ad(Uk)V

To see that

Ad(u)v

k

p'

uk

Now

is compact

G

Since

Ad(uk)v k E:£.

then

converges to

u,

and so

is actually complex analytic it suffices

c

by the Riemann removable singularity theorem (cf. R. C. Gunning and H. Rossi [13, Chapter 1,

§ C, Theorem

3]) to show that it

is complex analytic outside a closed algebraic set

S

*

For this consider the complex analytic mapping F

defined by F (g , t.)

= Ad (g) t,

= p((t).

and notice that analytic near points at

(g,t).

only if ad (t) 1

=

:

t

J(

= Ad(g)t

for which

Now it is easy to see that

F

F

is strictly bigger than

and let

be the set

is

is non-singular

is singular at

is singular in the sense that the kernel of

1

v

l}

then by Lemma 8.A.4 b),

actually semi-simple so by the above near

of

s

S

v

is

is complex analytic is an algebraic subset

For this let

aO(v) + a 1 (v)

x

+ •.. + an(n» ..

n

=

he the characteristic polynomial of

det(ad(v)-)"l), n ad v.

=

dimo:

10:'

Then clearly

o} which is obviously a closed algebraic set and since

/1 II n

S

=

U ker a. t a.E

there exist elements outside the complex analyticity of 8.3.

S. pI '

consider

{e 1 , ••• ,e

p}

0

JRP

and for

+' .

inductively as follows:

a(gl, ... ,gp) p = 1

6P

a(gl)

: 6'

G/K

For

is given by

as the cone on the face spanned by

Then the restriction of

a(g"

that face must be given by

L 0 a(g2, ... ,g ), g, p this map to the cone using the contraction h s'

... ,gp)

to

and we extend Explicitly

(9.8)

It is now straightforward to check (9.6) inductively. The merit of a filling

a

of

G/K

is that it enables us

to construct explicit Eilenberg-MacLane cochains:

Consider the

subcomplex

A*(G/K)

InvG(A*(G/K»

of the de Rham complex

consisting of G-invariant forms (where the G-action is induced by the left G-action on

by

G/K).

Define the map

148

(9.9) 0,1 Proposition 9.10. b)

J

a)

,2, ..

is a chain map.

The induced map on homology

is independent of the choice of filling. Proof.

a)

By Stoke's theorem and (9.6)

J (dw) (gl'··· ,gp+1)

J p+1

t>

J t>P

+

[L

0

gl

(gl'··· ,gp+1) *dw o(g2,···,g +l)]*w +

0

p

'J

PL (_1)l o(gl,···,g,g'+l,···,g +l)*w + i=l t>P l l P

+ (-1)P+1J P

o(gl, ... ,g )*w p

(w) ) (g 1 ' ... , gp+ 1 )

= w.

We give an alternative description of

J*:

Consider the

map of simplicial manifolds

where

and the face operators are given by i

o i

0, < i

p.

< p,

149

is just given

y

the projection onto the first factor.

the proof of Theorem 8.1. bundle with fibre

a

that if L

(g1 .•. gp)

G/K

associated to

is a filling of -1

II y II

w E Aq{G/K)

Now if

family of forms on

LIP

the projections onto

H{Inv G/K

=

dw G

dw,

x

Y G

d

G/K

is the fibre

y

: EG d

BG d.

Notice

then the family

LIP

a{g1,···,gp)

0

defines a section of

Clearly

The realization of

(Cf.

G/K, gl, ... ,gp E G, P = 0,1,2, ... ,

which explains the definition).

is an invariant form then the corresponding NGd{p)

G/K,

x

G/K, p

=

0,1, ..• ,

defines an element

induced by

w E Aq{N{Gd;G/K)).

so we have an induced map on homology

A*{G/K))

H{A*{N{Gd;G/K))).

On the other hand, since

is contractible

induces an isomorphism in de Rham cohomology by Lemma 5.16 and Theorem 6.4.

Hence the composite map

H{Inv

G

A*{G/K))

is canonically defined (i.e. without a choice of filling) we claim that this is just

J*

get an explicit inverse to

y*

where

a

In fact given a filling

and a

we

0,1,2, ... ,

is

given by

gl , ... ,gp E G, P

0,1,2, .•.

150

w E InvGAP{G/K)

Then obviously for

J

/),

p(L

(g1·· .gp)

_1

0

G{gl,···,gp)]*w

= J",pG(gl,···,gp)*w

J{w){gl,···,gp)·

This proves the proposition. Remark. h

In the proof of Lemma 9.7 we replaced the contraction

by the contraction

s

where

hots)

in order to be able to define the ",p.

of

o (s) = 0 00

map

C

for

near zero

s

on all

G(g l' ... ,gp)

On the other hand the inductive construction (9.8) using

the original contraction makes sense on the open simplex and the corresponding change of parameter does not affect the value of the integral (9. 9).

In particular let us describe

ly for the case where

G

§

TI

:

G

G/K

7])

explicit-

s

is semisimple with finite center:

'7

we can choose a Cartan decomposition (14, Chapter 3,

="

h

=

and the map

TI

0

Then

6l/ '

(see Helgason

exp

G/K

is the projection and

G

(where

the exponential

map) is a diffeomorphism {see Helgason (14, Chapter 6, Theorem 1.1]). Therefore we get a contraction defined by (9. 11 )

The curves

h

s

s

-1

=

(x)

hs{x)

Riemannian metric on

x E G/K,

(x l j ,

s E [0, 1 ] •

are geodesics with respect to a G-invariant G/K

and we shall therefore refer to the

corresponding filling defined inductively by (9.8) as the filling by geodesic simplices. We can now describe the composite map (9.4): Theorem 9.12. Bj*: H*{BG,lR) is represented in

For

Il(K)

P E

H*{BGd,lR)

of

H2 l{C*{NG)) d

the image under

W (P)

E H

2l (BK, lR)

H

2l

(BG, lR)

by the Eilenberg-MacLane co chain

151

1

1

where

above and

J

E InvG(A

21

is defined in step I

(G/K»

is given by (9.9).

That is,

(9.13 ) where

a

is a filling of

Proof.

Let

i

G/K.

:

G

be the inclusion and consider the

commutative diagram of simplicial manifolds N(Gd;G/K)

1y

(9.14 )

N(G where

----L

1

Nj

d)

j : NGd(p)

x

NT

NG/K ,

• NG

G/K

NK/K

1

Ni


0, k E SO(2),

and

b)d '

ad _ b2

1,

a,d>O.

It is easy to see that (9.16) then reduces to

1

r

t

0).

Notice that the numerical value

2nL Ar c (and equal to zero for

b

an

b

(1+a\] j - Arc tan,-S-j

satisfies rr

(9.17)

2"

1

4·

(This inequality can also be deduced directly from Theorem 9.12; see Exercise 2 below).

This has the following consequence due

to J. Milnor [22]: Corollary 9.18.

Let

over an oriented surface class (9.19 )

e(E)

: E X h

X be a flat Sl(2,IR)-bundle h of genus h > 1. Then the Euler

satisfies l1 < h.

154

Proof.

We first need some well-known facts about the

topology of surfaces.

X can be constructed as a 4h-polygon h with pairwise identifications of the sides as on the figure

/

/ / /

/ / /

/

.:..

. ...... Here the sides group

f

x

1,

..• ,x

give generators of the fundamental

2h

with the single relation 1•

Furthermore the universal covering is contractible (see reference in Exercise 2 e) below) .

,'Ie can now define a continuous map

f

For

:

Bf

-+

X h

generators

as follows: x 1 ' ... , x 2h

x

E

r

representing

choose a word in the x

and map

into the corresponding curve in the polygon. over the skeletons of groups

= a

for

Bf

i > 1.

with integral coefficients of C*Nf.

x

x c /',1

x

r

Now extend the map

using the fact that the homotopy Clearly

equivalence by Whitehead's theorem.

of the complex

/', 1

Hence

X h

f

is a homotopy

In particular the homology is isomorphic to the homology

H (C*Nr) :;:; 2Z 2

the generator is represented by the chain

and we claim that

z E C2(Nf)

defined

by z =

-1 -1 -1 -1 (x 1,x 2) + (x 1x 2,x 1) + ... + (xx 2x 1 x 2 ... x2h,x2h-1) +

+ (1,1) -

-1 (x 1,x1 ) + (1,1) -

-1 (x 2,x 2 ) + •.• + (1,1) -

-1 (x2h-1,x2h-1)

155

which is easily checked to be a cycle. is the sum of all the

(4h-2)

In fact

f*z E C*(X

h)

2-simplices in the triangulation

shown in the above figure plus some degenerate simplices. Now any flat a map

Be

:

Sf

S :

SI(2,:rn.)-bundle BSI(2lR)d

where

a : I'

homomorphism (see Exercise 1 below).

=

E

Bf

is induced by

SI(2,:rn.)

is a

It follows that

.

Now it is easy to see from (9.16) that a simplex of the form (x,x

-1

)

contribute zero (since in this case the integrand is

the trace of the product of a skew-symmetric and a symmetric matrix).

Therefore the right hand side consists of

4h-2

terms each of which numerically contribute with less than 1/4. This proves the corollary. Proof qf Theorem 9.15. G = Sp(2n,:rn.) the filling

It is straightforward to check that

is semi-simple so we can apply Theorem 9.12 using by geodesic simplices.

a

First let us reduce the

number of integration variables: In general for

G

semi-simple with maximal compact group

1 = -l J we exp :'1 G/K

and Cartan decomposition 'IT

0

ill

have the diffeomorphism

as in the remark following Proposition 9.11. 1

=

exp

0

-1

:

G/K

G

is an embedding such that the diagram

G/K 2-. G G/K commutes.

Then we have

Lemma 9.20.

For

Therefore

and

K

156

(9.21 ) where

P(gl,g2)

gl g2°

(that is,

P (gl ,g2)

1*P(8 K)

is the geodesic curve in

=

P (gl ,g2) (s)

Proof. in fact

J

=

] (P (nK) ) (g 1 ' g 2)

g 1tp (stp

-1

G/K

(g 2 0)

considered as a form on

P([8

K,8 K])

=

a

since

P

from

to

gl°

) ,

s E [0,1]) .

G

is actually exact;

is K-invariant, hence by (3.14)

and so (9.22) Now by (9.8) the geodesic 2-simplex

o(gl,g2)

6

2

G/K

is given

by (9.23) where

hs(x) = tp(stp

-1

(x)), x E G/K, s E [0,1].

Notice that

vanishes on the tangent fields along any curve of the form s E [0,1],

and since

10 o(gl,g2) 0 £i, i

= 1,2,

exp(sv),

is of this

form we conclude from (9.22) that

I I

6

2

0

1

(o(gl,g2)

(g l , g 2 ) *d ( l * P ( 8 K) )

6

0 £0)*1*P(8 K)

which is just (9.21). Now for

., =ft( 2n, lR)

G

=

Sp ( 2n, lR)

c

Gl (2n, lR) ,

is contained in

fy (2n, lR)

=

{x = C

= -"-(n)

The Lie algebra

=

M(2n, lR)

{x =

of

the Lie algebra as the set of matrices

C, t B K

=

U(n)

=

B}

is the subspace

157

wi th complement in

-f;r'( 2n, lR) : A,

(n)

tB =

B} .

is identified with the vectorspace of Hermitian

n

n

x

complex matrices (as in Example 5 of Chapter 4) by letting

=

x

X=

correspond to

A + iC.

In this notation the first Chern

2

E H (BU(n),lR) is given by the Chern-Wei 1 image of 1 the linear form P E 1 1 (U(n)) given by

class

c

=-

=-

(9.24)

P (X)

Now

is identified with

\l

:

G/K

G/K .... Gl ( 2n, lR)

tr (X)

1 2 'f[ tr (C)

=-

G n P(2n,lR)

1 4 'f[ tr (JX),

X E

.M(n).

via the map

given by g E G

(see G. Mostow [23, p. 20]). embedding

1 : G/K .... G 1 (p)

Also if let along

. p

p = p(s),

Under this identification the

above is given by

p,

pEG n P(2n,lR)

s E [0,1],

denote the derivative, i.e. the tangent vector field p,

Notice that the projection K(X)

For

G n P(2n;lR)

is a curve in

P E 1

1

(U(n))

-

t X) ,

K

:

1J-(2n) ....

X E

-

ff(2n).

given by (9.24) above the form

therefore takes the following form along a curve s E [0, 1 ] ,

in

is given by

Gnp ( 2n, lR) :

1*P(8 p

=

K)

p(s),

158

1 * P (8

K

) (p)

'" -

tr (J ( T -1

8n

t (T -1 i) ) ) , T '" P!;,

i -

so (9.25)

1*P(8 ) (f») '" K

Now suppose

p

1

-4"TI tr(JT

TO exp(SY)T

O'

T '" P

G n P(2n,JR),

is a geodesic in

p(s)

-1. T),

!;,

that is,

s E [0,1], YEt TO E G n P(2n,JR).

Then (9.26)

p

-1 .

-1

TO YT

p

is a constant in " . '" exp(2(s)), 2(s)

O

'" p(O)

-1

p(O)

Q

On the other hand, if we write

E 1 ' s E [0,1],

then

p(s)

(see Helgason [14,

Chapter II, Theorem 1.7]): p

-1 .

l-exp(-ad 2) ad 2

p

(1

+ exp(-ad

(1

where again

T '" P tr(JT

Now since of

-1

2

2'

2

2

(

2 ) 2

2)) (T

Hence by

. 2

-1 . T),

(9.26)

2 -1 t ) '" tr(J(l + exp(-ad 2)) (Q)).

2 E S (2n, JR),

M(2n, JR)

Therefore

'" exp

+ exp (-ad

(Z)

ad Z

is a self adjoint transformation

with respect to the inner product

159

Z -1 - < J, (1 + exp (- ad 2") ) (Q) > Z

«1+exp(-ad2"))

-1

(J),Q>.

Now it is easy to see that tr(JT

-1.

I

T) = -«1 +exp(-Z))

-1

J,Q>

tr (J (1 + exp (- Z) ) -1 Q} tr (J (1 + P-1 ) - 1 p (0) - 1 P(0) ) .

=

Finally let

p

t g1° = g1 g1

to

p(s), s E [0,1],

be the geodesic curve from

t t g1 g2 g2 g1'

g1 g2°

Then

that is,

and we conclude

t

since

g1J

Jg

-1

1.

Theorem 9.15 now clearly follows from Theorem

9.12 together with (9.21) and (9.25). Remark.

It would be interesting to know if the expression

in (9.16) is bounded also for Exercise 1.

Let

X

n > 1.

be a connected locally path-connected

and semi-locally 1-connected topological space so that it has a universal covering space group of a) n :

X

X

and let

Suppose X

G

a : r

is a principal

n :

X

X.

Let

r

be the fundamental

be any Lie group. G

is a homomorphism.

Show that

r-bundle (therefore called a principal

r-covering) and that the associated extension to a principal

G-

160

bundle

TI

b)

:

a.

E

X

a.

Suppose

is a flat G-bundle.

=

r

so that

{1}

=

X

X

Show that every flat G-bundle is trivial. the corresponding c)

Gd-bundle

a. :

r

-->

TI

:

X -->

X

(Hint:

Observe that

is a covering space of

Show that in general every flat

the extension of

is simply connected.

to

G-bundle

Let

G

group.

be a maximal compact subgroup.

Let

K

G

-->

w E InvGA*(G/K),

For

G

be a homomorphism from a discrete

J*w E H*(BGd,Th),

the element

defines a characteristic class for flat Let

TI

TI

Ea.

a.

M

:

-->

M

-->

Exercise 1a) and let bundle with fibre

M

be the corresponding flat Tla.: M xrG/K

G/K.

-->

Show that

of the characteristic class

is just b)

G/K) w

G-bundles.

be a differentiable principal

J*(w)

M

Tla.

in cohomology and that the pull-back

r

induces an isomorphism n*U*(w)(E)) E H*(M xrG/K,Th) a. a.

E H*(M,Th)

a. : r

-->

G

G/K

-->

r\G/K

TI

Again let

TI

= Mr

right action by

xg = g

J* (w) (Ea.) E H* (Mr, Th) whose lift to

the diagonal

G/K

-->

n a. : r \ (G/K x G/K)

-1

x

for

G/K x G/K -->

M . r)

M x G/K

-->

G/K.

x E G/K,

is just

is the covering space provided

r-action on

is represented in G/K

M x G/K

r

is discrete

be the associated

a.

flat G-bundle (first change the left

w

is represented in

is the inclusion of a discrete

of a manifold (this is actually the case and torsion free).

(see

be the associated fibre-

pulled back under the projection

Now suppose

r-covering

G-bundle

by the unique form whose lift to

subgroup such that

form

is

be a Lie group with finitely many

a. : r

A*(M x

X

relative to some homomorphism

G

components and let

and let

on

G.

Exercise 2.

a)

X).

w.

g E fl.

G/K

to a

Show that

A* (M r)

by the unique

(Hint:

Observe that

induces a section of the bundle

161

c) for

Again consider

P E Il(K),

by the form

G, rand

(E ) a where

w(P)

K

for

and show that A 21 (M )

r

is the curvature form of the (Hint:

direct proof by observing that

G of the principal

as in b)

is represented in

connection given in step I.

to

K

Either use b) or give a is the extension

11

a

r \G

K-bundle

r\G/K).

In particular,

dim G/K = 2k, = r

: r

G

1 r

1

J

for all

M

r

and

a

2

: r

G

2

be homomorphisms

are the fundamental groups of two

2

dimensional compact manifolds

M

1

and

M

2

and let

M be the corresponding flat a a 2 2 2 the Hirzebruch proportionality principle:

and

11

E

:

There is a real constant (9.28)

1

for all

P E Ik(K). G

and

Riemannian metrics induced from a left invariant metric on

G/K

Furthermore, if M. 1

=

=

Mr ' i i

vol(M

r1

=

f

r

and

1

1,2, as in b)

)/vol(M

r2)

where

are discrete subgroups of above then Mr.' i

=

c(a 1,a 2) =

1,2,

are given the

1

(which exists since

has an inner product which is invariant

under the adjoint action by e)

2

Now consider

K).

G = PSl(2,lR)

=

Sl(2,lR)/{±1}.

by isometries on the Poincare upper halfplane H

x + iy E iC I y > O}

{z

with Riemannian metric 1 Y

@ dx

+ dy @ dy) .

G

acts

162

The action is given by Z I----->

(az

+ b) /

+ d),

(c z

z E: