Differential Topology (C.I.M.E. Summer Schools, 73)

V. Villani ( E d.)

Differential Topology Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, August 25 - September 4, 1976

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11101-3 e-ISBN: 978-3-642-11102-0 DOI:10.1007/978-3-642-11102-0 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Liguori, Napoli 1979 With kind permission of C.I.M.E.

Printed on acid-free paper

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CENTRO INTEIW AZIONALE MATEMATICO ESTIVO (c.I.M.E.

I11 Ciclo

- Varenna d a l

)

25 agosto a 1

D I F F E R E N T I A L Coordinatore:

A.

Banyaga

G.

A.

P.

Michor

Fredricks

V. Poenaru F.

Sergeraert

G.

Wallet

Prof.

4 settembre 1976

TOPOLOGIY

Vinicio VILLANI

: On t h e group of diffeomorphisms preseE ving an exact symplectic : Some remarks on Cauchy-Riemann S t r u c t z res : D i f f e r e n t i a b l e Cohomology : On -the homology of Haef l i g e r l s Classifyng space : Manifolds of d i f f e r e n t i a b l e maps : Some remarks on low-dimensional t o p o l o gy and- immersion theory : La c l a s s e d e cobordisme d e s f e u i l l e t a g e s de reeb d e s3 e s t n u l l e : I n v a r i a n t de bodbillon-vey e t diffeomorphismes cornmutants

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (c.I.M.E.)

ON THE GROUP OF DIFFEOMORPHIS PRESERVING AN EXACT SYMPLECTIC FORM

k. BANYAGA

Corao tenuto a Varenna d a l 25 agosto a1 4 settembre 1976

ON THE GROUP OF DiFFEWRPHISMS PRESERVING AN EXACT SYWLECTlC FORM

Statement of the result w

Let M be a smooth paracompact connected manifold and let DiffK (M) w be the group of all C -diffeomorphisms of M, supported in a compact subset w

m

K of M, equipped with the C -topology. Denote by Diff (M) the group ~iffi(~)with the direct limit topology. A symplectic manifold is a couple (M,S2) where M is a smooth manifold

of dimension even 2n and S? is a closed 2-form such that fin= fi

A,..

.

fi

is everywhere non zero. Let (M,Q) be a symplectic manifold. We shall denote by ~ i f f i(M) the subgroup of ~iff~(t1)whose elements are those diffeomorphisms h such that h*n =

n.

Denote by Gn(M) its identity Component.

In this note we prove a theorem on the structure of the group Gn(M) in the case where the symplectic form f2 is exacf. It is known that,this condition requires M to be non compact. The compact case has been studied in [I]. We have shown that if (M,fi) is a closed and connected symplectic manifold, the commutator subgroup [GQ(M),Gn(M)]

is a simple group, and that

the abelianised group GR (M)/[Gn(M) ,Gn(M)] of GRfl.f)is isomorphic to a quotient of the first de Rham cohomology group of Mt Recall that a group is said to be simple if it contains no non trivial normal subgroup. If G is a group, the abelianised group of G, G/[G,G] will to denoted by HI(G). The result is the following : Theorem

Let (M,n) be a connected symplectic manifold. Suppose that fl is an exact form. If the dimension of M is 2 4, then : (i)

[Gn(~),Gn(M)1

is a simple group 1 is isomorphic to the direct product H~(M,R) @ R

(ii) HI(Gn(M)) 1

shere H~(M,P) is the first de Rhan cohomology group of M, with compact support.

I would like to thank Professor Haefliger for helpful conversations.

11. Sketch of proof

We construct a surjective homomorphism

cP a @

: Ga(M)

+

1 Ec(M,R)

and we show that its kernel, Ker 2, is a simple group. As Ker 2 contains [Gn(n) ,Gn(M)]as

a normal subgroup, Ker

= [G~(M),Gn(M)

I.

2.1. The homomomorg~sm = dl. For.any h e G (M), one shows n 1 that the cohomology class [h'*A-Ale Hc (M,R) of the closed 1-form Let A be a 1-form such that

h*A-A is idependent of the choice of A and that the correspondance h

I+

[h*A

- A]

is a continuous and surjective homomorphism

a: Gn(M)

+

1 Hc(M,R )

which coincides with the homomorphisms S considered in ti]. 2.2.

The homomorphism P For any h in the kernel of a, Ker C$

there exists a unique

fonction f(h,X) with compact support such that : h*A-A = df(h,h]. the dimension of M is 2n, denobby [f(h,A) nnj E R

.

integral of the 2n-form

If

H~~(M,P) c the

f (h,l)nn over M. One shows that[f (h,A)nn1

is independent of the choice of A and that the correspondance

h

I+

[f (H,x)Q~]~s a surjective homomorphism p : Ker a * R

which coincides with the homomorphism R considered in [I]. 2.3.

he homomorphism.P For any he C (M) , denote by [h* A

n

6

hA Q

n-ll

en

-

w2"(n,4 c

)

the integral of the 2n-form h* h* Xh nn^'over M. One shows thiir if n-l1 is independent of the choice of A, arr3 that n 2 2,[h*h r Q

the correspondance h

I+

[h*Xh X

nn-']

is a surjective

homomorphism

-

P : GS1(M)

-+

which extends the homomorphisms p above.

2.5. End of the proof We'have constructed a surjective homomorphism

r Since

= u (36 :

cn(n)

+

1 H~(M,R) @R

extends p, Ker C = Ker p.

If (M,S1) is any connected non compact symplectic manifold, reapeting the arguments of [I], one can see that the kernel of the homomorphism R mentionned above, is a simple group. Since coincides with R, Ker R = Ker p = Ker C is a simple group.

References [ 1]A. Banyaga : "Thesis, University of Geneva 1976".

Universite de Geneve Section de math6matiques 2-4, rue du Lievre Case Postale 124

CBNTRO I N TERNAZIONALE MATEMATICO ESTIVO

(c.I.M.E.1

SOME REMARKS ON CAUCHY-RIRQWN STRUCTURES

GREGORY A. FREDRICKS

Corso tenuto a Varenna d a l 25 a g o s t o a1 4 settembre 1976

SOME REMARKS OM CAUCHY-RIEMANN STKDCTURES

Gregory A. Fredricks

In t h i s l e c t u r e we consider r e a l a n a l y t i c Cauchy-Riemann structures.

We formulate r e s u l t s about t h e i r complexifications,

t h e i r l o c a l and global embeddability and t h e i r p a r t i a l ordering. The main reference f o r t h i s m a t e r i a l i s Andreotti C13, where t h e study of Cauchy-Riemann s t r u c t u r e s i s carefully developed from t h e elementary d e f i n i t i o n s 80 t h e l o c a l and global embeddability questions. Let

U

be an open s e t i n lRm

1 complex-valued

C?

v e c t o r f i e l d s on O ,

which i s i n v o l u t i v e , i.e. [P,,PJ

Definition: on -

U

$-

@

and consider a system of

f o r which

=

m k=1

k

dij

Pk with

h EC'(U). i3

-

A l o c a l Cauchy-Riemann (c-R) s t m c t u r e of type

~ystem (*)

f o r which

l i n e a r l y &@dependen$(over

C)

- P ,,...,Pg,P1, ...,PA

a t each p o i n t of U.

are

.t

The l o c a l C-R s t r u c t u r e i s oalled r e a l analytic i f the fbctions c a r e r e a l analytic. ij If M i s an m-dimensional r e a l manifold we define a l o c a l C-R structure of type A on a chart ( x , ~ ) of M exactly a s above. Definition: A 2-g s t r u c t u r e of tgpe I an M is a csllection of l o c a l C-R structures of type A on an open cover {u,) of I charts of M, say U ; . . B P ~such that the c ' ( u ~ ~ u ~ ) i is equal t o the span of PI, ,P: r e s t r i c t e d t o U nU

...

C0(UiAU

)

span of

P3I , .

..,PL3

i

3

r e s M c t e d t o VinU

j 3' Equivalently, a C-R s t m e t a r e of type 2 on M (see Greenfield C61) i s an A-dimensional complex subbundle H(M) of T(N)Q)C d c h t h a t (i) H(M)~= = 10) (pier0 section) '. ( i i ) H(M) is involtative, i.e. CP,Q3 is a , section of H(M) whenever P and Q a r e sections of H(M). In the case A = 0 the zero section defines a C-R struc-

t u r e of type 0 wbich i s called the t o t a l l y r e a l structure of M. Real analytic C-R structures a r e defined on r e a l analytio manifolds i n the obvious way. C-R structures often a r i s e i n the following way. Let X be a complex manifold of complex dimension n and suppose t h a t M i s a l o c a l l y closed submanifold of X of r e a l dimen-

-

sion m. Letting F(M) denote the sheaf of germs of C* f h c t i o n s on X which vanish on M, the holomor~hictangent space t o M at p&M i s defined by n

B T ( M )=~

(P =

where

,...,zn

z1

i= 1

a a p ) :~ ( f ) ( ~ = )o f o r e v e q Q azi a r e l o c a l holomorphic coordinates a t

Setting l ( p ) = d+

RP(M)~ one can shew t h a t

fr.~(~)~), peX.

i s an

upper semicontinuous Function of

Now if

p

along M

i s constant and equal t o

C-R s t r u c t u r e of type

and t h a t

1, then

HT(I~T) i s a

1 on M a s ( i ) i s c l e a r and (ii)

follows from t h e f a c t t h a t a section

P

of

KT(&?)i s char-

acterized by being holomorphic and s a t i s f y i n g

In t h i s setting M

inherits 5

t h e complex s t r u c t u r e of For example, R~ e

n

-

s t r u c t u r e of type

from

X.

cn

and a r e a l hypersurface i n type

2-g

inherits the totally r e a l structure cn i n h e r i t s a C-B s t r u c t u r e of

1.

i s a r e a l a n a l y t i c submanifold of X, then t h e inherited C-R s t r u c t u r e i s r e a l analytic. Definition:. A l o c a l l y closed submanifold M of X ( a s above) is generic i f n 2 m and =m n ( i . e . R(p) i s conIf

M

-

.

stan t and minimal) We now s t a t e the Local Embeddabilits Question:

Given a l o c a l

C-2 s t r u c t u r e V of p in U

(*)

and pcU, does there e x i s t a neighborhood such t h a t t h e and a l o c a l l s closed embedding a : V -, m-L ---~ i v e nC-g s t r u c t u r e and the inherited 2-2 s t r u c t u r e aLqge? That i s , -such t h a t -(a)

O(V)

is a

generic submanifold

& cm",

(b) where every

u,(H(v)) = H T ( ~ ( v ) ) , A( V) i s the space spanned b~ P, (q)

,...,P~(q) for

~EV.

An e l emen t a r y proof shows t h e f 011 owing

Theorem:

The

S n c s Embeddabilitg Question can be answered

affirmativei~r_i f the given

2-2

s t r u c t u r e i s r e a l analytic.

L. Nirenberg t8j has given an example of a C-R structure of type 1 on a neighborgood of the origin i n E? f o r which the answer t o the Local Embeddability Question is negative. Our main i n t e r e s t here is a consideration of the Global Embeddability Question: If M has a C-R structure of type 1 which i s l o c a l l g embeddable, does there e x i s t 2 splex manifold X of complex dimension m-1 and a l o c a l l x closed embeddinp; a : M - + X such t h a t the given 2-2 structure and the inherited $-,R structure ?mee? A t t h i s point we can consider the Global Embeddability Question only f o r r e a l analytic C-R s t r u c e e s . We therefore assume, f o r the remainder of t h i s l e c t u r e , t h a t I i s a r e a l analytic manifold of r e a l dimension m and that H(M) is 'a r e a l analytic C-R s t r u c t u r e of type 1 an M. Definition: A p a i r (X,a) i e a c o ~ p l e x i i i c a t i o nof XI i f X is a complex manifold of complex dimension m-1 and a : M 4 X is i s a r e a l analytic l o c a l l y closed embedding such that dl!) a generic submanifold of X and U,H(M) = HT( a ( l ) 1. llhe following theorem s t a t e s t h a t the answer t o the Qobal Bnbeddability Question is affirmative f o r real analytic C-R s t r u c t u r e s and also that the germ of the complexification of M along M i s unique i n t h i s case. Theorem 1: (a) There exist complexifications (x,u) of NI. (b) If ( ~ , a ) and ( Y , ~ ) a r e two complexifications of M, then there e x i s t neighborhoods U af U(I) 2 X and V of p( M) in Y and a biholmorphic h : U -,V such t h a t the diamam

----

--

--

---

commutes. Moreover, h ts unilq.lelg defined i e n t l y s m a l l and connected-with o(M).

if

U

suffic-

The t o t a l l y r e a l case, i . e .

A = 0, of theorem

1 was proven

independently by Shutrick 193, by Haefliger 171 and by Bruhat and Whitney C31. The proof of theorem 1 (see ~ 4 3 )proceeds along the same l i n e a s the proof given by Bruhat and Whitney. (see a l s o Andreotti and Holm t23.) By a theorem of Grauert t5J we know t h a t there i s a complexification (x,u) of the t o t a l l y r e a l structure of a r e a l analytic manifold M wtth X Stein. Since Stein i s the same a s 0-complete we a r e lead t o the following Conjecture: There exis't complexifications (x,u) of M for 1 which X 2 1-complete. A s a p a r t i a l r e s u l t we have Theorem 2: conjecture holds i f M & compact. The proof of theorem 2 ( see C4]) i s modelled on the proof given by Grauert i n the t o t a l l y r e a l cage, i.e. 1 = 0.

-

The f i n a l topic of this l e c t u r e i s the p a r t i a l ordering of r e a l analytic C-R structures on I. Definition: If H(M) and H(M)' a r e r e a l analytic C-R structures on I of types 1 and R' respectively, then dominates H(M) i f H(M)c H(M)' ( s o 1 5 1').

H(N)

'

mote that the t o t a l l y r e a l structure of

M i s dominated by every other r e a l analytic C-R structure on I. From 141 we have Theorem 3 : If H(M)' dominates H(M) a s above and a : M 4 X 1 An n-dimensional complex manif016 X i s 1-complete if there e x i s t s a C* flrnction p : X -.R such t h a t ( 1) {XEX : P(X) c C) i s r e l a t i v e l y compact f o r every coR. (2) A t each point xO€X the Levi form of P T

has a t l e a s t n-f

p?sitive eigenvalues.

and p (M,H(M)

of

:M

4

'1

Y

are complexificationa of

(M,H(x))

and

r e s p e c t i v e l y , then t h e r e e x i s t open neighborhoods

X and V V s u r j ec t i v e map h : U

U

o(l)

3

af p ( ~ ) Y and a holomorphic af maximal. rank such t h a t t h e eMoreover, h 2 uniquely defined U

--+

m

commutes. i s s u f f i c i e n t l y small and connected w i t h (**)

-

u(M).

A. ATJDREOTTI. Introduzione All'analisi Complessa. Contrib u t i d e l Centro Linceo I n t e r d i s c i p l i n a r e d i Scienze W t e matiche e Loro Applicazioni, IT* 24, Accademia Nazionale d e i Lincei, Rome, 1976. A. ANDREOTTI and P. HOLM.

Quasi-analytic and parametric. spaces. To appear i n t h e Proceedings of the Nordic Summer School i n Mathematics, Oslo, 1976.

F. BRURAT and H. WHITETEY.

Quelques p r o p r i i t 6 s fondamenComment, Math. Helv. t a l e s des ensembles analytiqnes-re'els. 33 (1959), 132-160. G. A. FREDRICKS. Higher Order D i f f e r e n t i a l Geometry and Some Related Questions. Ph.D. t h e s i s , Oregon S t a t e Univ.,, 1976. GRAUERT. On L e v i t s problem and t h e imbedding of r e a l a n a l y t i c manifolds. Ann. of Math. 68 ( 19581, 460-472. H.

S. J. GREENFIELD. Canchy-Riemann equations i n s e v e r a l varia b l e s . Ann. Scuola Rorm. Sup. P i s a (3) 22 ( 1968), 275-314. A. HAEFLIGER. S t r u c t u r e s f e u i l l e t 6 e s e t cohomologie & valeur dans un f a i s c e a u de groupoldes. Comment. Math. Helv.

32 (1958)q 248-329.

L. NIREKF$ERG. Lectures on Linear P a r t i a l D i f f e r e n t i a l Equations.' Regional Conf. S e r i e s i n Math., No. 17, Ameri c a n IvIath. Society, Providence, R.I., 1973. 9.

H. B. SHUTRICK. Complex extensions. Oxford Ser. ( 2 ) 9 (19581, 181-201.

Quart:J.

Math.

CEN TRO I?ITERNAZIONALE MATEJ4ATICO ESTIVO (c.I.M.E.

1

DIFF'ERENTIAL COHOMOLOGY

Am HAEFLIGER

C o r s o tenuto a V a r e n h a dal 25 agosto a1 4 settembre 1976

DIFFERENTIABLE COHOMOLOGY by Andr6 HAEFLIGER

This is a slightly expanded version of two lectures given at the Institute for Advanced Study of Princeton in the fall 1972. Some of this material was supposed to be included in a joint paper with R. Bott on smooth cohomology. These notes do not contain any concrete new result. We just try to explain the pkitosophy of differentiable cohomology. Namely one would like to compute the cohomology of classifying spaces Br of some topological groupoids

which plays an important role in

the problem of constructing foliations (cf. [45]et[24]

). Unfortunetely this

seems to be extremely difficult. Now the classifying space Br(or rather the groupoid

r)

carry also

a softer topology with some kind of differentiable structure. In the complex of cochains giving the cohomology of Br, one can single out the subcomplex of smooth cochains whose cohomology, called the differentiable or smooth cohomology of Br, is computed in many important cases, because it is isomorphic ao the cohomology of some Lie algebra associated to

r. This is proved

as the theorem of Van Est (cf. el21 ) relating the differentiable cohomology of a Lie group to the cohomology of its Lie algebra. The titel of this course could also have been : "variations on a theorem of Van

st".

By its very definition, the smooth cohomology of Br maps into the cohomology of Br, and almost all informations we have on the last one are obtained by ewluating differentiable cohomology classes on some homology classes of Br.

Throughout these notes, instead of using the notation H*(Br;A) familiar to topologist, we use the notation H*(r

; A), following the al-

gebraist.

The differentiable cohomology will appear in many cases as the cohomology of some complex of invariant forms. So in chapter I, we consider several cases useful later. In chapter 11, we explain the theorem of Van Est, and this will be a model for the more general cases. In chapter 111, an abstract cohomology theory for ficient in a l'-sheaf

r

with coef-

is sketched, introducing the definition.of differentiable

cohomology given in chapter IV. We-then state the analogous of Van Est theorem. Of course many very important points concerning differentiable cohomology are left out here. Let us mention the study of deformation of structures and the variations of smooth cohomology classes, the fact that the map from the cohomology of Br with the soft topology into the cohomology of Br factorizes through the differentiable cohomology. Also we don't consider the Cheeger-Simons theory of differential characters, the multiplicative structures, the complex analytic case, etc..

.

We have tried to avoid too much homological algebra, although it is clear that the final stage of the theory should be expressed in those terms (like in Iloct~stiild -Hostow [ 29 ]for Lie groups).

Chapter I : Invariant forms

1. ~asicformulas. ( ~ f 1321) . Let M be a differenti&le manifold and let

$1'

M

b c ? t l l c a vector space

of differential forms of degree p.

A smooth vector field X on M defines two operations : i) The Lie derivative

4(

:

Qi

ii) The interior product ix : (iXa)(X2

given by

+

,...,XP)

+

-

a(X,X2

,...,XP).

One has also the exterior differential d : Cartan formula iii)

5

-

+

and the

di + i d X X

The formula for d is

The cohomology of the complex d differential forms on M

will be denoted by H*(M)

or H*(M ; R).

If A is'a finite d-hensional vector space over R,L1 P (?$;A) will denote the differential p-forms on M with value in A (this means that

x k a (XI,.

..,XP) (x)

is a smooth map of M in A. In this more general case,

one has the same formulsas above and the cohomology of the complex R*(M;A) will be denoted by H*(M;A).

2. Cartan Theorems

Let G be a Lie group acting smoothly on M and let us assume also that a smooth linear representation of G as automorphisms of A is given. We shall say that A is a smooth G-module. Then G acts on R(M;A)

,...,xp)(x)

(@)(xi

by

= g [a(g,

-1

-1 Xp)(g -1x)

Xl,..-,g,

where g*X is the image of X under the differential of x

+

I

gx.

This action commutes with the differential; so we :get an action of G on H*(M

; A).

;A) G .and H*(M ; A) G the subspaces of elements G left invariant by G. Of course R*(M ;A) is a subcomplex of O*(M;A), and we G shall denote its cohomology by H*(Q*(M;A) ). Let us denote by R*(M

Theorem

(E. Cartan)

Suppose G.i s compact. Thensthe, :n&turaZ in&Zusion o*(M;A)~ -+ R*(M;A)

induces an isomorphism H*(Q(M

; A ) ~ ) + H*(M

;A)~

Corollary

Asswne G compact connected and the action of G on A t r i v i a l . Then H*(M;A) { s isomorphic t o the cohomoZogy of the compZex of Ginvariant forms.

This is because G acts trivially on H*(M;A). is not trivial, then H*(M;A)~ = H*(M)@

If the action of G

A~.

To prove injectivity in the theorem, one constructs a retraction

defined by

where dg is an invariant measure on G of volume I and ga is considered as a form depending on the parameter g. The proof of surjectivity can be obtained along the same line, using a partition of unity on G. Symmetric space. M is a symmetric space under G if i) G acts transitively on M ii) There is a m o t h involution s on M such that sgs(x)

= g'(x)

for some g' in G and all x e M.

iii) s has a fixed point such that the differential of s at x is the central symmetry on the tangent space T M. X

Theorem (E'. Cartan) If M is symnetric under G, then every ~ i n v a k a n form t on M

is closed (the coefficients are trivial). Proof. -

P Then s*a(x) = (-1) P a(x). Let a € QM.

and is equal to (-1)'

But s*a is also G-invariant

a at x, so everywhere on M, because G acts transi-

tively. tifferentiating this equality, we get s*da = (-llPda, but as d is invariant of degree p+1, we also have s*da=(-1)~"da.

So da = 0.

If we combine the two theorems of Cartan, we get Corollary

For a synanetric space under G compact connected, then B*(M) is iscmorphic t o the algebra of Ginvcrriant forms on M. This corollary is very useful for computing the cohomology of symmetric spaces. Examples of symmetric spaces are ,the spheres, the grassmannians, etc. Also any Lie group is a symmetric space under thb action -1 1 of E x G, the action of (gl,g2) on x being g 1 x g2 and the symmetry sx = x

-

3 . cohomology of Lie algebras (cf. [ g

Let?

1 et [331

)

be the Lie algebra of G (left invariant vector fields on 6).

Let A be a smooth G-module. For any X E Y and a c A , we can define Xa = velocity vector at t = 0 of the curve exp tX.a. With this operation, A

a

is a 3-module, namely a vector space with an actionr'x linear and such that [X,Y] a = XY a

- PX a.

A

.+

A

which is bi-

G acts on itself by left translations : Lg(x) = gx. The complex

of left invariants f e r a Q*(G;A) G is isomorphic to the complex C*(y ;A) of cochains on% with value in A. A p-cochain c ii a p-linear alternate form on lawith value in A. For c E

cP(

a

;A), then dc is defined by

The isomorphism between S2(G;A) triction to left invariant vector fields.

; A) is obtained by res-

.

Definition in the

3

m e cohomoZogw H*( A) of the Lie algebra A w i t h coeffir~icnt moduZe A {s the cohomoZogy of the complex ;A).

7;

c*(Y

fiis definition makes sense of course for any abstract Lie algebra over a field and a

7.-module A.

Note that G also acts on itself by right translations ; as this action commutes with left translations, we get an action on called the

a.

adjoint representation. The derivative of this action on R(G;A)

is the

Lie derivative L given by formula 1,i. We also have the interior product X iX : :A) + C'-'(~;A) defined for x 6 by formula 1,ii). Formula

~'(3,

7

ii5) is also valid.

Consider R as a trivialr module. The algebra of left and right (biinvariant) invariant forms on G with value in R is isomorphic to the subalgebra C*(g

;APof elements df C*(a;A)

annihilated by all

5.If G

is compact, then by Cartan theorem, E*g;R) -is isomorphic,,to-thealgebra

. This is also true if7 is reductive, i.e. if there is a compact group 5 whose Lie algebra 3 has the same complexification as . In that 7 case, H* (9 ;R) E*($ ;R) = ~ * ( ;R) 8.

c(~;R)

=.

For instance, the Lie algebra gln(R) of (n,n),real

matrices

and the Lie algebra u of the unitary group U have the same complexification n n Hence

namely gln(C).

n(gln(~)

;R) = a(un ; R)

and this is an exterior algebra E(hl,

...,hn

)

in generators h of degree 2i-1. i

4. K-basic forms and relative cohomology of Lie algebras Let K be a Lie group acting smoothly on a manifold E. Any element X of the Lie algebra X of K defines a vector field on E, still denoted by X, defined by X(x)

=

d/dt (exptx)xl

t=o'

The K-basic elements of R(E;A)

are the forms invariant by K and

annihilated by all i X E K. They form a subcomplex of R(E;A) denoted by X' R(E;A)K. Note that if K is connected then R(E;A) is also the subcomplex K of Porms annihilsted by all i and LX, X E g. X

Suppose that E is a principal K-bundle, i.e. K acts freely on the right on E and the space of orbits B = .E/K is a differentiable manifold, the projection p : E

+

E/K being a submersion. Then R(E;AIK i s the image of

Q(B;A) under the map p* :Q(B ;A)

+

R (E ;A).

So the basic forms come from

the base space. In the particular case where K is closed subgroup of a Lie group G, the action of K being by right translations, then R ( G ; A ) ~ ~R(G/K;A).

Assume that A is a smooth G-module. The above isomorphism is compatible with the action of G on itself (by left translations) and on G/K. Passing to G-invariant forms, we get the isomorphism

By the isomorphism $G;A)~

+

C * q ;A),

a

ments is mapped onto the subspace C(%,K;A)

tke subspace of K-basic ele-

of 4 7 ; ~ )formed by the elements

c such that i) iX c = 0 for all X G E ii) c is K-invariant, K acting on and on A.

7 by the adjoint representation

If K is connected, ii) is equivalent to LX c = 0 for a11

ii)' Definition

X.5.

H*(Y

The relative cohomology ,K ;A) i s the cohomology of the subcomplex ,K ;A) which i s isomorphic t o the comptex of G-invariants

c*(T

forms on G/K. As an example, H*(gln,On ; R) .is isomorphic to an exterior algebra in generators h1, h3 , - , with degree hi = 2i-1, i odd and isn.

5. Bundle with discrete structural group and characteristic homomorphism. 6

Let E(G) be a principal: G-bundle as above, with a reduction to G the Lie group G with the discrete topology.

,

This situation can be described as follows. One gives on the base space B an open covering%=

{ui)

and continuous maps y ij : Uic Uj B is locally connected, the y . . ( x ) are locally constant) such that 1J

+

& (as

yik(x) ykj (XI = yij(XI

for

XE

uin u. n uk J

.

yP (y . ) over the covering U. Two

Such a datum is a G'-cocycle

lJ

6

such cocycles are equivalent if there is a G -cocycle over the covering union of the two given one whose restrictions to each of them are the given cocycles. m e n E(G) is obtained from the disjoint union of the Ui x G by the identification of (x g.)€U. x G and (x.,g.)~U.x G if xiE x = x and i 1 J J J j gj = yji (x) gi. Two equivalent eocycles define isomorphic bundles (cf, [4$). More generally, suppose G acts smoothly on a manifold M. The

associated bundle E(M) with fiber M is the quotient of the disjoint union

M by the equivalence relation identifying (x mi) and i' (x.,m.) if x = x = x and m. = y..(x)mi. J J i j J 31 6 The bundle E(G ), where the fiber is G with the discrete topology,

of the Ui

x

6 G -covering. Such a covering is characterized by a repre-

is a principal

sentation H of the fundamental group

r

E(G

6

called the holonortg of the bundik CI

space of B and so that B =

r

r\%,

of B in G, defined up to conjugacy, ).If $ is the udiversaL covering

is represented as the covering translations group of B 6 N then E(G ) = Bx G, quotient of 7 x O by the identification

r

for any y E I' (cf, [ Illet R81).

of (x,g) with (YX, H(y)g)

The characteristic homomorphism. Recall that for any Lie group with a finite number of connected components, there is a maximal compact subgroup K, unique up to conjugacy, 6

and that G/K is a manifold which' is contractible by a smooth deformation to a point (cf. 128

I).

For instance 0 is a maximal compact subgroup of n Gln(R) and Gln/O is the space of scalar products on

un.

n

For a bundle with discrete structural group G ~ , given by a 6

G -cocycle (y. .) on 1J

{u.):1

Y*: H(% ,K;R)

we want t~ define an homomorp11ism +

H(B;R)

depending only on the eq~~ivalcnce class of y and which is functorial with respect to pull back of bundles.

Let E(G/K) be the associated bundle with fibre G/K. Let a be a G-invariant form on G/K ; let a. = p; a be the form on 1

Ui x G, where p

: U. x G/K +G/K is the projection. As a is G-invariant

i 1 and the y.. locally constant, the forms ai which are defined on the 1J

Ui

X

'G/K, identified to open sets of E(G/K) ,match together, and define

a global fo m

y* a

on E(G/K)

.

The homomorphism

-y*

: S~(G/K)~+ $2 (E(G/K)

commutes with d. This map could as well be described using the flat connection on E ( G / K ) . As the bundle E(G/K) smooth section s : B

+

+

B has a contractible fiber, there is a

E(G/K) and two such sections are homotopic. So

after passing to cohomology, the map s* Y* : H(TK)

+

'G+ gives

a well defined homomorphism

H(B)

called the characteristic homomorphism More generally, for a smooth G-module A, o w h a s a homomorphism Y* : H*(~,K;A)

+

H*(B,A~).

Here AYmeans the local system of groups on B obtained by taking the associated bundle E(A) = AY ; H*(B,A'~

can be described as the complex

N

of forms on B with values in A invariant by the action of

r,

6. Main example

Let K be a maximal compact subgroup of a Lie group G. Suppose is a discrete subgroup of G such that r\G is compact. Assume also that

r

r is

without torsion so that 'I acts also on G/K without fixed point (because the intersection of I' with any conjugate of K is the identity). Such a subgroup always exists for G semi-simple (cf. Bore1 [ 33). Then r \ G/K is the base space of a principal G -bur?dle, namely E(G ) = G/K xrG, quotient of G/K x G by the identification (gK,h) = (ygK,yh)

6

6

for all y~

r.

The associated bundle E(G/K) section s : G/K

E(G/K),

+

is CIK

x,,

namely s (gK) = class of (gK,gK). It is easy to

see that the characteristic map : n(G/K)

GIK. There is a canonical

; A ) ~+

Q(r\ GIK,*Y )

=

r

~1((;/~;i)

is just the natural i n c l . u s i o n . The following property is familiar to people like Matsushima or Bore1 working on discrete subgroups of Lie group. Proposition

The naturat in&Zusinn R(G/K

;

+

B(GIK;A)

r

induces an injection in cohomo7.ogy . One can check that H*(~,K;A) is linked to A*(% ,K;A) by a Poincar6 duality (cf. [ 331) and that the generator of ;K,R) is mapped on a volume fo m of

r \ G/K

(n = dim G/K)

~"(7

.

The following more direct proof was communicated to me by

W. Thurston. To prove the proposition, it is sufficient to construct a linear

aqeb that i) p is a retraction (i.e. if a is G-invariant, then pcr = a). ii) p commutes with d iii) p conslutes with the right action of K and also with the action of the Lie algebra of K by interior product.

-lr\G

To construct p, let dg be the biinvariant measure on G such that for the induced measure d z on G, we have d g = I. Anyleft invariant r\ vector field X on C gives n vector field X on ,\G. Then define (, by the formula (XI..

...

=

1

Properties i ) and iii) i;e that, for a smooth function f on

r\G,

.(x,,...A)(i)6.

immediate ; ii) follows from the fact then

Exercise Suppose G acts smoothly on a manifold M. Consider on G/K the invariant forms under the diagonal action. First express this algebra as a double complex C*( 3 , K ; Q*(M)), bracket in

the first differential coming from the

2 and 6the second one from6the exterior differentiation.

Let E(G ) be a principal G -bundle over B and let Em) be the associated bundle with fiber M. After choice of a section s : B

+

E(G/K),

construct a characteristic morphism

For the example above, show that this morphism induces an injection in cohomology.

7. Invariant forms on groups of diffeomorphisms For a manifold M and a compact set C, denote by Diff.CM the group of diffeomorphisms of M with Support inc, with the topology of uniform convergence of all derivatives. Let Diff M be the group of diffeomorphism C

of M with compact support, with the topology of the direct limit of the Diff p . We want to study the left invariant forms on the "differentiable manifold" DiffCM. To avoid artificial considerations on infinite dimensional manifold, one can proceed as follows. For a manifold X, a smooth map f : X such that the associated map F : X x M

+

+

Diff M is a continuous map C

M, defined by F(x,m)

=

f(x)(m),

is

smooth. A smooth form a on Diff M is a rule associating to each such f a C

smooth form a on X such that, if u :. Y X is a smooth map, then f a! = u*(af). The differential is defined by ( d ~ ?= ~daf. fou A left invariant form on Diff M is a form a such that, for each C g s Diff M, then af = a , where L is the left translation of Diff M bg g. C f 0 Lg I3 C +

One can check that the complex of invariant forms on DiffcM is

canonically isomorphic to C*(v M), the complex defined by Gelfand-Fuchs C

(cf. 861) of multilinear alternate continuous forms on the Lie algebra vCM of smooth vector fields on M with compact support, with value in the trivial module R

.

To define continuity, on puts the topology of uniform convergence of all derivatives on vCM = vector fields with support in the compact C.

A multilinear form on v M is continuous if its restriction to v M is conC C tinuous for every compact C. In C*(v M), the differential is given by C

Suppose M compact and let K be a compact Lie group acting smoothly on M. Then K acts also on vector fields, and also on C*( vM). Let C*(v M, K) C

be the subalgebra of K-basic elements. Let E be a bundle over E with fiber M and structural group K. Assume there is a foliation on E coqlementary to the fibers. Then there is a characteristic morphism

which generalizes the one given in 5 (cf. 125 3 ).

8. Invariant forms on frames bundle For a manifold M'of dimension n, a frame r of order k 'at y& at 0 of a local diffemrphism f of a nbhd of 0 in R" n on a nbhd of y = f ( 0 ) . If yl,. ,y are local coordinates around y, then is the k-jit jof'

..

the numbers

up to order k, are the local coordinates of r.

k The frames of order k on M form a differentiable manifold PM which k k is naturally a principal G -bundle over M, h e r e G is the Lie group of

n k-jets at 0 of local diffeomorphisms of R preserving 0. The projection k p : P + M,associates the point y to a frame at y. The action of jk h on f t k 0 r = jof is jo(f Oh). One has natural projections

The bundle P of infinite frames is by definition the inverse M k limit of the P

H'

k The orthogonal group 0 sits in G = lim G as infinite jets of rotan tions. One can consider.the quotient PEI/Onof P by the right action of On.

M

By definition, a differential form on P is locally the direct M k limit of forms on the P

M'

The pseudogroup of local diffeomorphisms of M operates on the left k .k on each PM, by g.r = jO (gof). We want to investigate the invariant forms on P ' under this action. M Gelfand-Puchs cochains on formal vector fields be the Lie algebra of formal vector fields on R ~ .A formal i i i vector field is a . expression of the form Z v (x) a/a x , where the v (x) Let%

.

are formal pover series in xis.. ,xu '; it can be considered as an infinite jet of vectors fields at 0. The bracket is defined by the usual formula.

A continuous cochain on yl is a multi'linear alternate form c o n an which depends only on finite order jets of the arguments. Those continuous cochains form an algebra C*(U

) with a differential as before (the action. n on the coefficient R is trivial) (cf. [IS] and 1191 )

.

n The o r t h o g o ~ lgroup 0 operates on R , so on formal vector fields, n hence on C*(U n). The Lie'algebra of On is also contained in&, as jet of orthogonal vector fields. We shall denote by C*(qn,On) of C*(an) of On-basic elements. (cf. 119 1).

the subalgebra

Proposition The differential algebra of invariant forms on PM (resp. P#/o,)

is isomorphic to the GeZfand-Fuchs algebra A natural map J : C*(%)

-+

(resp C* (fin,on) )

C*

.

n*(P M ) is described as follows.

n Let h be a 1-parameter family of local diffeomorphisms of R ; t 00 then v = j (d/dt htlt ,O) is a formal vector field. It gives on PM an 0

invariant vector field whose value at r = j 0f is v(r) k For c E C (&,), then J(c) is defined by J(C)(V~(~)~.

..,vk(r))

=

c(vl,.

= jo(d/dt(f

o

1

ht) tPo).

..,vk1.

It is easy to see that J is an isomorphism of C*(Rn)

on the

invariant forms on PM, commutidg with d. Insteadof giving a formal proof (cf. [22D, we indicate how any 1-form on P is expressed in terms of invariant M foms. A basis for C1((n ) is given by the w i 6here 16 i s n and n a a = (ao,. ,an) is a multiindex,. ai integers-20 and

..

Then in the local coordinates y

where 6 + y = Results of

G, and

i

a

/3k is the sequence (b1..

., b t 1,..,..,b n).

elfa and-~uchs

We mention them without proof (cf. 1'i7let 1 199).

...,

c 1 be a polynomial algebra in the variablgc. of 1'n 1 degree 2i, and let R[ci, ] , c be fie quotient by the ideal of elements (Let R[c

...,

of total degree> 2n. Let E(hl,...,h

) be the exterior algebra in the variables hi of n degree 2i-1 (everything on the reals).

On the algebra Wn = E(hl,.

..,hn)

@

-

..

R [c1,. .

is defined by d(hi

@

1) = 1 @ ci and d(1

@

ci) = 0.

,C ]

n

a differential d

We also consider the differential subalgebra WOn

E

E (hl,hg. .) @ B[c1,.

..,cn 1

(all the h. with i even are dropped). 1

.

Theorem of Gelfand-Fuchs (cf [I51 1191).

There is a ce-mmtative diagram of rnorphisms ef differential

where the horizontaZ arrows induce an isomorphism in cohomology. It follows that El*(%)

and H*(%,On)

are finite dimensional,

and that the first one is Z~connected. For n = I, H*((Xa)

= H*(tXI,O1) is an algebra with one generator

1 1 in dimension 3, represented by the cocycle o A dwl

.

CHAPTER I 1 --

- DIFFERENTIABLE COHOMDLWY OF LIE GROUPS

-

1. Cohomology of discrete' groups (see for instance [l] ). a) The category of G-modules Let G be an abstract group. A G-module A is an abelian group with an action of G as automorphisms of A. A G-morphism f : A

+

B of G-modules is a homrmorph.ism of groups

commuting with the action, namely f(ga) = gf (a) for all a € A and g R . The group of G-morphisms 6f A in B is denoted by HomG(A,B). For a G-module A,:.the subgroup of elements left fixed by all g~ G will be denoted by AG It is isomorphic to %(Z,A). where Z are the

.

integers considered as a Ctmodule with the trivial action. b) The standard resolution Denote by F~(G,A) the abelian group of.maps of Gn+l in A. This is a G-modu1e.b~the action (gf)(gO,

*.**,

gn) = g Ef(g

-1

-1

gos*:.sg

gn)l

We have an exact sequence of G-mdules

where (€8) (g)

-

a LL

Definition

The n-th cohcnrotogy group #(G;A) the cohamotogy group of the canptex

of G with coefficient i n A is

F*(G;AE~.

For a geometric iaterpretation, the sequences (go,.

..,g9 are )

the n-cells of a simplicial set with boundary' operator

and on which G acts diagonally. Its geometric realization is a contractible cell complex EG on which G acts freely. So the quotient BG = EG/G has G is fondamental'group G .and all other houmtopy groups zero. Then F*(G,A) the complex of cochains on BG with value in twisted coefficients A. The space BG is a classifying space for principal G-bundle, G discrete. G is called the comptex of homogencm8 cochains. The complex F*(G,A)

n G As an abelian group, F (G,A) is isomorphic to the group c"(G;A) n o of G to A (by -.onventionC (G ; A) = A).. G To T E F~ (G,A) correspond fdn(~,A) given by

...A

f(gl,

-

= f(e.glg2,...,gl--.g

n nTo ~EC~(G;A)correspond f given by -1 1 f(gor..-sgn) gof(gO g1,*",Sn 8,).

of maps

1.

-

By this identification, the boundary 6 becomes 6 : c~(G;A) + C~+'(G;A)

The complex C*(G;A) Note that H'(G;A)

is called the compZa of non homogenous cochains. 1 . = AG and E (G;A) = Horn(G,A) if A is a trivial

The characteristic homomorphism

C)

Any principal G-bundle (G discrete) over a topological space X can be given by an open covering%'ij

: U i n Uj

-+

{ui}

and a G-cocycle y = (yij), where

G is a continuous map verifying the cocycle condition of I,5.

If A is a G-module, then y defines on X a -locally trivial system of coefficient ,'A

namely the associated bundle with fiber A(cf. 1 .S)

The Cech cochains cP(U,AY) sequence (i) = (io, continuous map f

(i)

are the maps f associating to each

...,iP) such that U (i) = Ui n....n : U(i)+

U. is not empty, a 1 P

A.

The boundary map

6 : cP(%

, AY

)

+

cP+'(q~

k +

k >o

1

f.lo*.

,AY) is defined by

..*-i , y .

.**lp+l(XI for

n... n u i

xr U. 1

0

P+1

We have a natural homomorphism C*(G;A)

+

C*(s;A Y ) mapping

f on the cochain

Passing to cohomology, and to the limit over the open coverings of X, we get the characteristic homomorphism, depending only on the equivalence class of y, y* : H*(G;A)

J +

H*(x;A~)

V

where H denotes the Cech cohomology (cf. [201 TI,5 3.

2. Continu~us and aifferentiable cohomology of Lie groups Let G be a Lie group. The cohomology of G as a discrete group is in general very complicated. For instance if G'is the group of reals and the coefficients are the reals with the trivial action, then H 1(R ;R) is the group of all homomorphisms (discontinuous or not) of R in R. fiorc n generally H (R ; R) is the group of 2-multilinear alternate maps of R.in R depending on n arguments.. For n > 1, they are all discontinuous (except the 0).

It is natural to consider more restrictive cochains on G. This led Van Est 112 1 to the following definition. As G-modules A, consider to simplify finite dimensional vector spaces A over R on which G acts through a continuous (hence smooth) linear representation. For more general G-modules, see [ 2 9 3 . Let :C

(G;A) be the continuous u p s of G~ to A and G(:C

;A)- the

smooth maps of G" to A. The boundary operator is defined by the formula *) above in 1,b). Definition : The continuous fresp. differentiable or smooth) cohomology of G

with value in A denoted by Hc(G;A) ( resp. by H~(G;A)) is the c o h o l o g y of the complex of continuous cochains Cc(G;A) (resp. the complex of smooth cochains Cd (G;A) )

F*C (G,A)

G

.

One can also consider the continuous or smooth homogenous cochains * G or Fd(G,A) which are isomorphic to CZ(G;A) or C$ (G;A) as in 1,b).

Theorem 1 : (cf

. f401). The inotusion C*d(G;A)

isomorphism in cohomotogy, same for a Lie group G.

80

+

Cg(.G;A) induces an

that continuous and differentiablecohomology are the

Proof. Let $ be a mooth function on G with compact support such that

.

where dg is a left invariant measure on

G.

G F" (G;A)~-+ :F (G,A) by r C, -1 (sf)(go,. .gn) = I n+l,f(hop .hn) ) ( g ; ho). G Define also H : :F (G,A>~ F:-'(G,A)~ Define

..

:

+

H f , .

..g

IG+dg = 1

=

-

1

.f (gl,.

J-'&

.((go

-1 h )dh o....

dh,.

by

..,gj,gjllj,. ..gnhn)$(hj). ..$(h n)dhj..

dhn

A straightforward computation shows that 6Hf + ~ 6 =f 2 f

-

(

2 sf)

which implies the result because H maps smooth cochains in smooth cochains. 3. Van Est theorem 'Suppose G has a finite number of connected components and let K be a maximal compact subgroup of G. Let A be a finite dimensional vector space which is a smooth G-module Theorem of Van Est : The differentiable cohomology Hi(G;A)

is isomorphic

to H* ( LJ,KIA), i. e. the cohomo~ogy of Dinvariant forms on G/X with vatue in A, We give the original proof of Van Est El33. Consider the double complex

cpsq = :c

(G ; Q~(G/Ic,A))

An element a of

cPpq is a

differentkbla q-form a(gl,..,g

with value in A, depending smoothly on the p parameters gl,...,g

) on GIK P in G

P (this means that in local coordinates, the coefficients of a are smooth

functions of gl,...,g

).

P The first differential -Farator 6 : cP,q + C~+l,q

is given by formula *) of l,b), and the second one d : CP,q + C~,q+l is the exterior differentiation. One has d6= 6d

The associated simple complex Kn= @ cpsq p+q-= n has the differential Da = 6a + (-I)~da, for ae: cPpq. There is a natural inclusion of complexes El : C$ (G ; A) where c:(G;A) :C

(G ; QO(G/R,A))

-+

K*

is identified by

to the subspace of cPs0=

given by the inclusion of A in QO(G/~,A).

-

There is also a natural inclusion of complexes E

2

: w(G/K,A)~

K*

induced by the inclusion Qq(G/K,A)

G -+

nq(G/K,A) =

Lemma 1 : For each p h o , the foZZaring sequence i s exact d 0-C: (G

coyq,

....

Proof. There is a smooth deformation of G/K to a point 0. This induces, by the classical homotopy formula for differential forms, a smooth map h :

Qq(G/R,A)

-+

Q~-~(G/K.A) such that hda + dha = a for q 2 - 1.

. cPYq

We get an homotopy operator H 1' with h, so that dH a + H da = a for q z l . 1 1

+

cpyq"

by composition

It is obvious that Im El = Ker d. Remark. This lemma is also true if we replace the smooth cochains by. the non necessarily continuous cochains.

emm ma

2

. For each q 20, - the foZZaring sequence is exact

Proof. It is clear that Im

E~

= Ker 6.

An homotopy operator H2 is defined in terms of homogenous cochains by

where dk is an invariant measure on K of total volume 1.

It is straightforward to verify that H26 + 6H2 = id. By a standard argument on double complexes (see for instance

151, p.211, lemma '1 implies that el induces an isomorphism in cohomology, and lenrma 2 that the same is true for c2, hence the theorem. Remarkl. If a € Q9(G/K,A) G is a .closed form, the image of its cohomology class by the Van Est isomorphism is (up to sign) the class of the cocy~le(6H Iqa 1 which is in c~(G;A). d Remark 2. For G semi-simple, see !lo] for an equivariant map of R*(G/K,A) in F*(G;A) inducing an isomorphism in cohomology~whenpassing to the G-invariants elements. Remark 3. For a treatment of continuous and differentiable cohomology in the framework of relative homological algebra, see Hochschild-Mostow 1291. 6

Remark 4. Given a principal G -bundle on a manifold B, we have constructed

Y

a

in

m p : H*(Y * K;A) + H*(B,A ).This map is the same as the composition of the isomorphism with the characteristic map

van- st

described in l,c, i.e. the following diagram commutes

H*( UJ .,K;A)

1 H* (G ;A)

I*e*(x.J)

3 pv. a* ( x,A')

To check it, use the double Cech-de Rham complex on B (see for instance 151,

§

3).

4. Comparison between cohomology and differentiable cohomology Theorem. Suppose 'I is a di8crete subgroup of G without torsion and such that

r\

G

is compact. Then the natwviZ map

is injective.

Proof. This follows from the proposition of 1,6 and the preceding remark applied to the example of I,6. As noticed by Borel, the theorem is still valid for a

r

with

torsion.

6 .6 Remark. Let BG and BG be the classifying spaces for principal G -bundles and G-bundles respectively (d6 =.G with the discrete topology, G = G with its topology of Lie group).

6 + BG. The map Bi* : H*(BG ;R) + H*(BG 6 ;R) induced in cohomotagy factorizes through differentiabte cohomo~ogg (cf 1251). The identity map i : G~ + G induces a map Bi : BG

.

One has the commutafive diagram

I (K)

B*(BG;R)

I(K) is the algebra of invariant polynomialson the Lie algebra of K (cf. [32]).

The cohomology H*(BG;R)

is isomorphic to I(K).

The map I(K)

-+

is isomorphic to H*(BK;R),

which

H*(Y ,K;R) is the Chern-Weil homo-

morphism given by a K-connection in the algebra C*(y ,R). The map H*(BG;R)

+

Hi(G,R)

can be defined using differential

forms on the nerve of G like in Bott [ 73

.

5. Case of the group of diffeomorphisms For the group Diff M of diffeomorphisms with compact support of M, c we can define the notion of differentiable cohomology as in 2(cf. 1,7 for the definition of differentiable functions on DiffcM). Van Est theorem can be genxalized if there is a compact subgroup

K such that Diff M/K is smoothly contractible. This is not true in general 1

but in soms particular cases like Diff R and Diff S C

.

If we restrict to the identity component ofethe identity (i.e.

+

to orientat,ion preserving diffeomorphisms) then Diffc R is contractible + 1 and the quotient of Diff S by the rotation group SO2 is also smoothly contractible. In those cases, the proof of Van Est theorem works and we get Theorem.

the if:f

R) is isomorphic to H*(vcR)

According to Gelfand-Fuchs 1141, the first group is a polynomial algebra in one generator of degree 2. The second one is an algebra .with two generators of degree 2, the Euler class e and a class u (often called the Godbillon-Vey invariant) with the relation u.e. = 0 (cf. [25].

1. Topological groupoids and homomorphisms. Examples- (cf

A groupoid

r

. 124 1).

is a category whose morphisms are all invertible and

r

form a set. The set U of objects of

is considered as the set of units

of r. One has two projections s : r + U (source) and t associating to each morphism its right and left unit.

:

'l

-+

U (target)

A topological groupoid 'I is a groupoid with a topology such that composition and passage to inlerse is continuous. The space of units U is whikh the induced topology and s and r are continuous.

A homomorphism of topological groupoids h : r map such that h (y y f = h(yl)h(y2). 1 2 of U in the space U' of units of Example 0.Gracpofd

+

a continuous

F!is

~ o t ethat h induces a continuous map h

r'

and hs = sh, ht = th.

rU'associated to an open covering U = {ui

3

0Of

a

topoZogicaZ space X. The space of units U is the disjoint union of the U.. The 1

space

rU

is the disjoint union of the family U.. = U.n U.. For a point 13 1 J denote by x.. the same point considered as a point of U.'. ; X E X in U.nU 1 j7 13 1J if xe U.n U. n Uk, then the composite x x is defined and equal to xik. . 3 . ij jk A continuous homombrphism of r in a topological groupoid r is

U

also called a r-cocycle over U(cf

1241) ; it is a family of c~ntinuousmaps

r

such that y.. (x) y. (x) = yik(x) for ICE Ui (I U. n Uk. 1J ~k J is another covering of X, two r co~yclesOn and (UC are equivalent 2f they are the restriction of a r-cocycle defined over the

: Ui n U. y .ij J If

+

w

union of the two coverings. An equivalence class of r-cocycles is called a r-structure. TWO r-structures are homotopic if there is a r-structure on

X

x

[0,13 inducing on X = X

x 0

and X = X

x

1 the given structures.

For each r, one can construct a ctassifying space Br with a universal r-stpcture on it such that the set of homotopy classes of mclps of x in ~r are is in bijection with the set of homotopy classes of r-structures on X (for more details, see [ 8 3 and *(The reader can skip 4 and 6).

124

3

).

Example 1. The groupo?d GM. Let G be a topological group acting contiliuously on a space M. Then G will denote the topological groupoid G x M with space M of units M, s and t being defined by s(g,m) = m, t(g,m) = gut, and the composition by (g,m) (g' ,m') = (gg',ml),

assuming of course m = g'm'

.

A GM-cocycle over a covering U of X gives a principal bundle together with a continuous section of the associated bundle with fiber M. The classifying space BG

M is the bundle with fiber M associated to the principal G-bundle with base BG. For Cnstance, if K is a subgroup o f G, and M = G/K, then a GMcocycle gives a principal G-bundle with a reduction of its structural group to K (cf. [ 481. For another important particular case, consider a topological 6 6 group G and denote by G this group with the discrete topology. Then G. acts on G by left translation, sa we can consider.the groupoid 'G = G 6 A -cocycle gives a principal G -bundle and a trivialization as a

5.

principal G-bundle. The classifying space can be interpreted as the 6 homotopic fiber of. the map BG. -+ BG induced by the identity+'G G. Example 2. The groupoid I' (or M

rn). aD

'I M is the groupoid of germs of C -diffeomorphisms of the differentiable manifold M, with the sheaf topology.

An element y of I'M is the germ at x e M of a diffeomorphism g of a nbhd U of x on some open set of M. A basis for the fopen set in

rM

is

obtained by taking the g e m ' of g at the points of U. The space of units is the manifold H. When M is R ~ ,then TM will be noted

rn.

is an open covering of a manifold X, a rM-cocycle over %

If-

such that the yii : U.

1

+

M are submersions gives on X a foliation of

codimension n. One can consider prolongation of I' to bundles on X. For instance, let

7M

M 1 be the groupoid whoje space of units is the bundle PM

of frames of order one (cf. I,8). The elements of FM are the germs 1 1 at points of PM of the prolongation of local diffeomorphisms of M to PM (cf. I,8). A ~M-cocycleis a rM-structure with a trivialization of its

B T (or ~ BT~) is the homotopic fiber to the morphism V : rn Gln mapping y

normal bundle. The classifying space of the map Brn + BGln associated on its derivative.

.

+

...

Example 3 Let w = dxo+ xi dx2 + + x2n-1dx2n be the contact form 2n+l 1 2 2n-1 2n 2n on R and R the symplectic form dx h dx +...+ dx A dx on R o 2n Let n : R ~ -+ ~R~~ +by the ~ natural projection (x , ,x2n) (xl ,x 1.

...

.

+

'...

Note that n* R = dw. Let

rW

and robe respect .[he

groupo?d of germs of local dif-

feomorphisms of R2" leaving invariant w and R reap. Any local diffeomorphism f of R ~ preserving ~ + ~ w is compatible with n and its projection gives a local diffeomorphism of R~~ preserving R. If this projection is the identity, then f is just a translation parallel to the xo-axis. Passing to germs, we get an exact sequence i)

1

+

rT+ ra+ % + 1

where rT is the groupoId of germs of translation of R. One can check that we get a fibration of classifying spaces ii)

BrW

+

BrR

the fiber being homotopically equivalent to BR6 , the classifying space of R with the discrete topology. By taking the prolongation of rw or rR to the bundle of contact or symplectic frames of order 1, one could define as above the groupoids

-rW

and

FR . One iii)

still get a fibration +

B T ~with

6 fiber BR

.

2. Abstract cohomology theory for

r

From now on, we restrict to the case where groupoid whose projections t and s :

r

+

r

is a topological

U are local homeomorphisms. Most

of the above examples satisfy this condition.

TWO extreme case are a) U is a point and b)

r

r

is a discrete group.

= U is just a topological space.

We want to describe an abstract cohomology theory for

r which

will give the usual one for a group in case a) and the sheaf cohomology in case b), We first describe the coefficients,

A r-sheaf 4 is a sheaf over U (namely a topological space with a continuous projection p : 4 + U which is a local homeomorphism (cf T201)) on which

-1(u)

acts continuously. This means that for each ag -AU= p

r

with u = ~(y), there is.an element ya in -A-t (u) depending continuously of y and' a. Of course y(yl) a = (w') a and ua = a. and y

E

If

A

is a sheaf of abelian groups,then y(a+al) = y(a) + y(al).

From now on, r-sheaf.wil1 mean r-sheaf of abelian groups. is an open covering of X, there is a bijection

Note that if'%

If (y. .) is a r-cocycle over % and 13 is a r-sheaf, one has on X a well defined sheaf AY obtained by taking

between sheaves on X and ra-sheaves. if

A

the d .b-joint union of the, induced sheaves a

E

Fii 9 with

(x,~..(x)a) 31

tii

= yti

11 on

U.1 and identifying

A.

€ 1 ~ ; ~

A r-morphism of r-sheaves f : A -+ 2 is a contipuous map commuting with the action~ofr and whose restriction to each fiber is a homomorphism. The group of l'-morphisw

ofA in

2 will

be denoted by Hoy(A,g).

One can check that the category of r-sheaves is an abelian category (cf. [ 201). Let Z be the constant sheaf with fiber the integers and the

r.

trivial action of

The basic functor associates to a r-sheaf

abelian group Hom (Z,&),

r

A

This is the composite of two functors :

the

r

the first one associates to 4 the subsheaf 4 of elements invariant by

r

(its fiber above u are the elements fixed by the Y with source and target the second consist of Theorem

r taking the continuous sections of A .

. (compare with

u);

[I])

n There are groups H (r;A) for each integer nso, unique up to

isomorphims, such that

ii) for each exact sequence of r-sheaves

o + - A'+

-A + -A" + 0

there i s a tong exact sequence

%

iii) for each r-sheafA,.Zet A be the r-sheaf of gems 'of maps f of

in

g

A

ldiicontinuou& not) such that f(g) EA+(g),for The action of YE r i s given by (yf) (g) =

=o

Then ) : ; ' l ( " ~

r -

T.

g~

yf(y-lg) .

for n > 0.

Morover m y morphism h : r 4 r' induces a morphism h* : H*(r ;A) + n*(rv; h*g) functoriaZ i n

r.

An indication of a proof is given in

4.

3. Standard =solutions (compare with I1,l). For a r-sheaf

A,

we shall denote by

cn(r,~)

...,yn'.of I'

the group of continuous

maps f associating to a sequence of n composable elements yl, an element f(yl,. sections of

A.

..,yn) of -At (y 1). By definition, cO(r;g)

The boundary 6 : cn(r;g)

+

cntl( r

is the group of

; ) is defined by

The complex C*(r cochains of

r with values

;A) is called

the o o m p h of non homogenous

in A.

r

It is isomorphic to the complex F*(r ;A) of homogenous cochains. n r An element f c F (I' ;A) is a continuous map associating to a sequence of of I' with the same target u an-element (n+l) elements go,...,g n such that, for any g of source u f(go, ...,g

The passage from homogenous to non homogenous cochains is given by the same formulas as in II,t,b)

Note that if %, is an open covering of X, then

e*(~~:A) is

just the complex of Cech cochains over % with value in the sheaf on X corresponding to b* (cf .r20], p. 203). More generally, one can interpret C*(r;A) of ucf. f 4 7 3 ) . Let Nk(r)

in terms of the nerve

be the space of sequences of k composable elements

...

yk) +t(yl), it is a sheaf (Y~, ,yk) of r. With the projection t : (y k on U. The elements of C (r;A) can be interpreted as continuous sections of the sheaf on N (r) inverse image of k

A by

t.

A continuous homomotphi'sm h :

r'

induces a morphism of complexes h* : C*(r;A) if y : I+%-+

r

.

+ -+

of topological groupoids C*(r'. ; h* 4). In particular,

is a cocycle over the covering 91 of X, the Sheaf h*& is -iden-

tified to a sheaf'A on X and h* is a map of C(r;&) Cf

U;

v H(%;

in the Cech complex

2).Passing to cohomology, and composing with the natural map v 2)+H(X ; c)(cf. 1203, p.223), we get the characteristic homomor-

phism

s*(c*(r ;A))

-

E*(x

;A Y ~

Example of a non trivial cocycle Let Example 2,i).

be the

rn-sheaf

of g e m of differential .p-fo-

on ~ ~ ( c f .

Let a c C1(l' log

I

y-l(x)

'1

,n 0) be the cochain associating to y the function -1 where (y ) '(x) is the jacobian of y at x. = t(y). This

"-7

is a 1-cocycle whose cohomology class is not trivial. To see that, consider a differentiable foliation of,codimensionn on .amanifold X given by the rn-cocycle f over the open covering U.. To 1 simplify, we assume the Y.. orientation preserving. The shaf fY is the sheaf 13

of germs of differentiable functions which are constant on pieces '("plaques1') of leaves. It is easy to s'ee that the image of the class of a by the above characteristic homomorphism is zero if and only if there is on X a transverse volume form, namely a closed n-form everwhere non zero and defining the foliation. In general, such a form does not exist

.

4. Construction of cohomology Given a sheaf A on U, there is a canonical resolution d i O+A+C'A+CA - - by "flabby sheaves" : (continuous or not) of If

1 A, C

9

a

.... is the sheaf of germs of local sections 0

the same for =/A,

etc. (cf. t 2 0

3).

is a r-sheaf, this is a resolution by r-sheaves.

Definition of cohomology

-

For a r -sheaf & , H*(I';A) &..the cohomology of the simple conrptez associated t o the double complez cPSq=cP(r;cqA) - where 6:.cP'% CP+' i s given by f o m 2 a *) in 3, and d : cPSq + c ~ i s ~obtained ~ ~ composition + ~ with d. All the properties mentionned in the theorem of 2 are easy to verify. Complete proofs will appear in the thesis of C. Roger [42].

H*(T;&)

should be the cohomology of the Milnor-Buffet-Lor

classifying space ~r(cf.[8!,[24])

W

with coefficient in the sheaf A

, where w

is the universal r-structure on Br. The advantage of the above definition is that it is functorial both in

A

and

r,

but it is not efficient for computations.

If the components Nk(r)

of the nerve are paracompact, one

can use a fine resolution instead of the flabby resolution above (cf.[20]). For an open covering % of X, then H*(rU

;A)

is the sheaf cohomo-

,

logy of X with coefficient in the sheaf A. So if y is a r-cocycle over% then it induces a characteristic homomorphism

Remark 1 If we compare with the homomorphism of 3, we get a commutative diagram

v H (X* ;

2)

H*(x,~)

-+

Remark 2 Suppose that

r acts

transitively on its space of unit$ (i.e. for

any u and v in U, there is a y such that s(y)

= u and t(y)

=

v). For an open

set U', let l" be the subgroupoid of elements of 'I with source and target in U'. Then there is a natural equivalence between the categories of r-sheaves and I"'-sheaves compatible with the functor Homr(Z, = H*(I";Af),

It_follows that H*(r;&)

where

A'

) and Hornrl(Z,

is the restriction o f 1 to U'.

For instance, for any n--manifold M, then H*(rM;R) example 2 o f 1.)

= H*(rn;R)

(cf.

5. Resolution using differential forms.

Assume that

r

is a differentiable groupoid, namely I- is a

differentiable manifold, and composition and the inverse map are differentiable. 6

-

For instance, this is the case for G \,rn, M' Let d l d

o+R+ n0(u)

-c

-n (u) +

etc.

...

be the resolution of the constant trivial f-sheaf R by the r-sheaves of germs

2

(U) of differential forms on U.

1.

One can consider the double complex

cP(r; gq(u))

with 6 as in 3

and d given by the exterior differentiation. If

is a paracompact manifold, then H*(r;R)-is

cohomology of the simple complex associated to C*(r,c*(U)).

isomorphic to the This is the case

6 n for G or for the groupoid of germs of analytic local diffeomorphisms of R M But

rn

.

is not ~ausdorf f In general, there is a morphism of the

simple complex associated to to

.

cP(r

; cq(u))

in the simple complex associated

cP(r;cq~). In the next chapter, the differentiable cohomology of

r will be

defined as a subcomplex of this complex. Example. Let 52 be the symplectic form on R~~ (cf 1, example 3). Then a represent a 2-cycle in cO(l' ; R2 (R2n). The image of its cohomology class

n -

can be interpreted as the characterist'ic clags of the fibration Br + Br with w a 6 fiber BR ( = K(R,l)). The characteristic homomomorphism Let X be a paracompact differentiable manifold with a given by a smooth cocycle y : U.n U. ij 1 J Let C*(%

; Q(X))

-t

rM-structure

M over%.

be the Cech-de-Rham double complex (cf. [5]~.17).

The cohomology.of the associated simple complex is isomorphic to H*(X ;R). We have a morphism of complex

Y*

: c~(~~,Q~(M)+

(pf)io,

cP!%

; aq(x))

given by

...,i (XI = yii*f (yi i (XI,.

.(XI ,yi P 0 1 p-lip Passing to cohomology, we get a commutative diagram

6. Formulation in the framework of relative homological algebra (cf. Mac Lane 1361).

An injective (resp. surjective) r-morphism f : A + !

of r-sheaves

is called s t r i c t if, for each L E U , there is a homomorphism of abelian groups

: B + A such that j U? iU is the identity of % (resp. iuo jU the ju 4 11 A identity of &) ; iu is the restriction of i to 1 1'

A morphism of r-sheaves f : A

+

1! is

strict if each morphism of the

canonical decomposition

is strict,

J is relatively injective if for each strict injection of r-sheaves arid r-morphism f : &-+ J, there is a morphism

A r-sheaf

i :4- 2

7 :B + J such that '?c,

i = f.

A reZativeZy injective resotution

of A is an exact sequence

of strict r-morphism of'r-sheaves

o+n

-

+ JO

+

-'J

-J 1 -...

+

i

where each J is relatively injective. If

-

0 +A

+A

-F

1 +

... is an exact sequence of r-sheaves with each r-morphisrn

strict, then there is a r-morphism of the complexz* in the complex A* inducing the identity on A ; this morphism is unique up to homotopy. "

& , there is a relatively injective r-sheaf A iii) and a strict injection 4 &. This imply that each A has a

For any r-sheaf (cf. 2, Th.

-+

relatively injective resolution. For instance, the simple complex associated to the double complex

F~(~,c~A) - is such a resolution, where $(r maps associating to a sequence Y

0'

....,y.

;A) is the r-sheaf of germs of

P

of elements of r,with the. same

target u an element of A

+.lo

The cohomology H*(r

; A) is obtained by applying the functor

) to a relatively injective resolution of

Homr(Z, logy of the complex obtained in this way. As Hoy(Z, Grothendieck 1211.

&

and taking the cohomo-

) is the composite of two functors, one could apply

CHAPTER I V

- DIFFERENTIABLE COHOMOLOGY OF GRWPOIDS W I T H TWO DIFFERENTIABLE STRUCTURES.

1. Examples of such groupords

A differentiable groupord is a groupoid r which is a differentiable manifold, U being a submanifold of r. The projections s and t : r -+ U are submersions ; composition is assumed to be smooth as well as passage to the inverse. In this chapter, we shall consider groupoids with two compatible differentiable structures, a fine one and a soft one. This generalizes the situation of chapter I1 where.the Lie group G was considered first 6 as a discrete group G (0-dimensional manifold) and also with its differentiable structure as a Lie group. G.

+

Exam le 1. Let G be a Lie group acting smoothly on a n-manifold M. Then GM = G x M (cf. II1,i) is a differentiable groupoid of dimension n. -. But we can also consider the differentiable groupoid GM= G X M,

wtge G is with its differentiable structure of Lie group, called the soft 6 one. The indentity map i : GM + GM is a morphism of differentiable groupoids, .in fact an immersion.

In this example, we could also replace G by the group DiffcM of m

diffeomorphisms with compact support with its C -topolo@ Example 2. The groupoid

rM

(cf. I,7).

of germs of local diffeomorphisms of M (cf.III,i)

can be considered as a differentiable groupoid, 'l M being a differentiable n-manifold, but not Hausdorff. On rM, one can consider another topology ; roughly two germs are closed together if their jets are close. For a precise definition, k

let J

rM

be the differentiable groupoid of k-jets of local diffeomorphisms

of M ; it is a differentiable manifold (compare with I,8) ; with the source We define JrM projection, it is a bundle with fiber isomorphic to pk M' k as the inverse limit of the J r ;it will be considered as some kind of M differentiable groupoid. We have a canonical morphism

associating to y its infinite jet. Note that the maps

rM -+

J

rM

are

immersiions for each k. In this example, we could replace of elements of a Lie pseudogroup (cf. [22]),

r.M

by any groupoid of germs

such as Tn,

rw7 etc.

@ other example is given by the groupoid n r(k)of germs along 0 x R of k-jets of diffeormorphisms of R x R of the n form h(x,t) = (ht(x) ,t). It plays a role .in the study of infinitesimal

deformations of foliations (cf. 1171). With the topology of germs,

,PI

is a n-manifold. But we can also consider the groupo~d~ ~ ( ~infinite ) ~ f n. jets of elements of r(k) which is as above an inverse limit of diffen rentiable manifolds. In the general situation of a groupoTd with two compat'ible differentiable structures as above, we wou1d.like to define the differentiable cohomology of

r with coefficient in a smooth r-sheaf A as the

cohomology of a complex of cochains for the fine topology which are smooth for the soft differentiable structure. There are certainly different ways of generalizing to this situation what has been done for Lie groups ([i3] 1291). The best would beb following the line of Hochschild-Mostow,to define a good category of smooth r-sheaves, define relative injectives and prove the existence of relatively injective resolutions. Here we shall only give examples of smooth resolution for the particular cases above.

6 2. Differentiable cohomology of G with real coefficient M Let

(resp.

a*) be the complex of' (resp. germs of) differential

--M

forms on M. According to 111, 4,5, the real cohomology H*(GM6 ;R) is the cohomology of the simple complex associated to the double complex q CP(G;,$) which is isomorphic to the double complex CP (G6;%I. It is then natural to define the differentiable cohomology

H*~(G~ ; R) as the cohomoZogy of the simple complef~associated t o tix: double c o q l s r :C (G'; .D;i) whose elemehts are forms depending s a w ~ ~ t h lan y p elements of G, as in II,3.

As c :(G~

;

G)is a subcomplex of CP(G6, G),we have a

morphism

Theorem. H$(GM;R) i s isomorphic t o the cohomoZogy of Ginvariant forms on G/K x M, where K .is a maiimai? compact subgroup of G, and G acts

diagonaZZy. (,II,3)

The proof is very similar to the proof of Van Est theorem

.

Also the example 1,6-shows, that i f . t h e r e i s a discrete

subgroup r without torsion such r \ G i s compact, then, the morphism 6 H$(GM ;R) 4 II?G~; R) i s injective. (cf the exercise in 1.6).

.

Remark 1. In general, we can express the differentiable cohomology as the cohomology of some invariants forms in several ways. Suppose for instance that G acts properly on M (i.e. for every compact C, the number of g e G such that CngC # 0 is finite), then the same kind of argixnent shows that Hi(% 6 ; R) = a*($).

BE ;R)+H*(Y

In particular 6 groupoId G described in 111,I, Ex. I). G

; R) where

a is the

i)- H*(+y,Ko

Suppose G acts transitively on M, and let K be a maximal compact subgroup of the isotrppy subgroup of G. Then H*(c6 d M i

;R).

2. One can also .define the differentiable cohomology of the groupoid

6 Diffc M

M as the cohomology of the simple complex associated to the

x

double complex C*(Diff M, * of forms depending smoothly on p elements d c %? of DiffcM in the C--topology. To generalize Van Est theorem, one hes the same difficulty as in 11,s. groupord

r

To avoid it, we can consider for instance the differential 6 = G x G/K defined by G = DiffcM acting on G/K, where K is

a compact Lie subgroup of G.Then the differentiable cohomology of I' is isomorphic to the relative cohomology H*(vC ; K ; R) (cf. I,7). In particular M Hi (B DiffcM ;R ) ,~*(vz,R). Similarly for

r= G6x

G/K

x

M, the differentiable cohomology is

isomorphic to the cohomology of the double complex C*(V;, C

K ; S2;)

K-basic continuous cochains on v with values in the complex M

of (this

complex plays an iaportant role in 1261). 3. Typical smooth rM-sheaves

We shall denote by E t h e rM-sheaf of germs of smooth sections of a vector bundle E on which rMacts smoothly. This means the following. To simplify, we restrict to the case where the fiber of E is a finite dimensional vector space (of course one should consider the

case where the fiber is a topological vector space). Let (sd, (si,

...,s;)

...,sk ) and

be smooth local sections of E giving local trivializations,

rM

and let h be a local section of

(i.e.a

local diffeomorphism). Then

.

h(x) si (s) = Z a. (j:h) 8 ; (x) J1 r where the a.. are smooth functions on J r for some r. 31 M In fact, such a vector bundle is given by a smooth linear r representation of the group G in the fiber V of E (cf. I,8), and E is k the bundle with fiber V associated to the principal bundle PM of frames of order k. For such an

:C

E,

we define the complex of smooth cochains

(TM ; g) as the subcomplex of

cP(rM; E)

of those smooth sections of E

depending smoothly on finite order jets of sequences yl...,Y

P

of

composable elements of I'

M'

For a more precise definition, let NkJrM = lim N ~ J be ~ the J ~ inverse limits of the manifolds N ~~r of sequences of r jets of k k M composable elements of Let t : NkJrM+ M be the map associating to M' k such a sequence the target of the first element. Then Cd(rM,E) is the vector space of smooth sections s of the bundle t*E, imerse image of E by t. In those tehs, the inclusion i : C*(r ;E) + C*(ri;~) is defined d M as follows. Let p : t*E + E be the natural projection ; let Yl, Ykbe a sequence of composable germs y. at x of local diffeomorphism g . .

...,

Let x = t(yl). Then

is)(^,,. ..,Y,)(x)

j

J

ps(jX gi,...,j 1

-

g$EX

"k

(when x varies in a nbhd, we get a smooth local section of X, so passing to the germ at x an element of

zt.

4. Differentiable cohomology of The sheaf

rM with real coefficient

$ of germs of p-forms on M is a smooth i.M-sheaf in the

preceding sense. Definition

R) is the cohomology of the s i q Z e c o q l e x associated to the double complex c;(rM;nM) which is a The differentiabte cohomology H$(rM;

subcomplex o p

cp(rM;

GI.

Recall that, in this double complex, the first differential 6 is defined by formula *) of 111, 3 and the second one d by the exterior differential of forms. The szme definition could be applied to groupoids coming from Lie

-

pseudogroups, like Fn, rQ, etc. For instance Ha(Tn; R) is the cohomology of

Functoriality P

0

Let G be the group Diff M with the C -topology and C

t the

same group

with the discrete topology. Consider the commutative diagram

where the horizontal arrows associate to (g,x) the germ (or jet) of g at x. It is clear that we have a morphism of complexes

hence a map Hf (rM; R)

-

H$(G~ ;R )

The analogue of Van Est theorem relates the smooth cohomology of M

to the Gelfand-Fuchs cohomology (cf. I,8). H$(rM

;R ) is isomorphic H*(en

Theorem

--

qvM;R )

is isomorphic A*(

; on)

an) .

As expected (compare. with 111,4, remark Z), 8*(rM;R) on the dimension n of M.

depends only

m e proof is very similar to the one of Van Est theorem and will be only sketched. Step 1).

The inclusion E1

: C:

(

rM

; $1

-

C:

induced by the inclusion

#

(

rM 4

;

cq

nq(PM / gn) induces an isom~rphism

of the cohomology of the associated simple complexes. Here c(PM/On) denotes the smooth sheaf on M whose sections above the open set U of M are the forms on PU/On' By a familiar argument (cf. [ p1, p.86), it is sufficient to prove that this itlclusion 2nduces an isolaorphism for the cohomology computed only with the differential d coming from the exterior differentiation (compare with lemma 1 of II,3).

M has a contratible fiber. Indeed G / 3; = a n / 0 is smoothly contractible (notation n of I,8), as well as Gk+'/Gk for k z 1 . So there is a smooth deformation of ?his follows from the fact that the bundle PMbn

+

1

PM/On onto a section of this bundle, preserving the projection. ?his first step is also valid if we drop the smoothness condition on the cochains. Step 2:

For each q, the following sequence is exact

r

Here Q ~ ( P ~ / ~ oM ~denotes ) the vector space of forms invariant by the action of r i.e. by local diffeomorphisms of M.

M' To prove exactness, one uses an homotopy operator defined exactly by the same 'formula as in II,3, lemma 2.

/3n)rM z s , C * ( a t 3 ), and we conclude from the M n n second step, as in II,3,that the inclusion BY I,8 n*(p

induces an isomomorphism in cohomology. For

M

the theorem is proved in the same way, PKbnbeing replaced

Remark 1 3ne can prove a similar theorem for any groupoid of germs of a transitive Lie pseudogroup. For instance the differentiable cohomology of

- is isomorphic to rn

the Gelfand-Fuchs cohomology of th'eLie algebra of 2n formal symplectic vector fields on R

.

Remark 2 Let (y) be a smooth rM-cocycle over a coveringw of a manifold X.

ca(r M ;

%en as in III,S, we get a homomorphism of

*

in

the Cech-de Rham complex C*(Q; R*), hence a characteristic homomorphism X

If we compose the isomorphism H*(an

; On)

+

Ha (

rM

; R) with

this characteristic homomorphism, we get the homomorphism described directly in [ 21 [ 61 [251. An example. For the computations, it is simpler to consider

as a sheaf 9, on which r acts on the right by wy y*w(instead of y.u = (y )*u). An M element u of cP(rM; $ associates to a composable Gequence yl.. ..',y an P element w(yl,. ..,y $, where - s(y 'lhe differential 6 is P P =

)E

x

).

We want to show that the formula given by Thurston for the GodbillonVey invariant in the complex C*(r ,R*R) can be proved following the preceding d 1 proof. J ' b simplify we restrict to the case of the groupoid

rl+ of germs

of orientation preserving diffeomorphisms of R. According to Gelfand-Fuch, H3 ( a ) has one generator represented 1

by the cocycle w hdwl, or in the local coordinates for PR given in I,8 1 by the invariant form 1 a = -3 dy dyl dy2 Y representing the Cbdbillon-Vey invariant.

+ 3 3 o can be considered as an element of cop3 = C'(T~;Q (93). (PI(). y2 dy dy1e~O' Hence 6p is a This form a is the differential of p = 3 Yl cocycle in cts2 cohomologous to a :

9.

--

where g~

r+I and

g',gl' are the first and second derivative of g at the

target y on g. But dy(g)(y)

-

d(-

If we define v e

g

cis'

log y1 dy) as the cochain u(g) =

- 811 log y dy, g 1

then

S\J is cohomologous up to sign to a, and we get

Gv(g,h) = d log

log h' (Y)

g')

which induces an isomorphism i n integer h o m l o q . Here

rC

denotes t h e sections which d i f f e r only on a cmpact s e t

from a pre-assigned base section.

The manifold s t r u c t M on M

is a

structure and hence it defines a homotopy c l a s s of mappings M + B r n cover

T

: M + B ~.f,(o,

We may take

any member

a), and

thus a homotopy class of sections M

T) =

,

Corollary.

-.

which

+

r

M

V.

of t h i s homotopy c l a s s t o be t h e base section.

The most important special case i s M , = Xtn. (

r-:

I n t h i s case

so we obtain There e x i s t s a mapping B

ZFFFJ?

-

nnsT:

which induces isomorphism in integer homology. The case was announced in

n = 1 had previously been proved by the author

131.

Thurston's theorem was announced i n 183.

141;

this

The proof we

will give here of Thurston's theorem i s based on t a l k s t h a t Thurston @;ave

i n a seminar a t Harvard 4 t h e spring of 1974.

The main ideas a r e due t o

Thurston, but many of the d e t d l s a r e due t o t h e author.

Thurston has found

three proofs of h i s theorem.

The f i r s t has never been published.

was outlined by the author in

151. The third, which we publish here, was

influenced by the ideas of

P. May on interated loop spaces.

give t h e proof on the case r

2

The second

We w i l l only

1. The second deformation lemma below

cannot be proved i n the same w a y i n the case r = 0, as i n the case

r

2 1.

Thurston has stated t h a t it can be proved by the methods t h a t Cernavskii o r Kirby used t o prove l o c a l c o n t r a c t i b i l i t y o f t h e group homeciuorphisms of lRn. Our original proof

(in t h e case n = 1 ) was an extension of work we

~ig 1R r onto F rl. C. Roger has written an exposition of t h i s proof in [TI. Having shown t h a t if? B if? R, if? B] maps onto F rlr , we

had done showing t h a t the comutator quotient group associated t o

maps

/

then attempted t o show t h a t this mapping i s injective.

We were able t o do

t h i s , and then extend t h e same method t o give t h e case n = 1 of Thurston's theorem.

r

$7.

- Btructures Transverse t o t h e Fibers. Let

If

z e Z,

r

then

identities i n yaa

be a groupoid of homeomorphisms of a topological space Z. id E

r.

If

r, SO we may identify Z ha, -yap 1 i s a 1-cocycle,

-

with the subspace of a , i l then -yaa : Ua

2.

a r e called the l o c a l projections associated t o t h i s 1-cocycle. A

rz- structure

on a smooth manifold M i s called a

cr

foliation

i f there i s a 1-cocycle representing it such t h a t t h e associated l o c a l projections are given

The

cr

ri- structure

submersions.

Then every 1-cocycle regresant;ing the

w i l l have t h i s property.

This definition coincides with the classical definition of foliation. Let n : E

+B

be a fibration whose fibers are smooth manifolds.

A

rr-structure u on E w i l l be said to be transverse to the fibers if its n restriction to each fiber is a cr foliation. In the case n : E d B is a smooth bundle, this is equivalent to the condition that w is a

cr

folia-

tion and its leaves are transverse to the fibers. We will be mainly interested in the case n = dim fiber, in which case the restriction of w to each fiber is the differentiable structure on that fiber. Local Model for B F

$8.

:.

In this and the next section, we will introduce two models for

B~E.

Each will be the realization of a semi-simplicial set. By a

FE -

structure on X we w n i mean a :p

normal bundle. We let hl(x,

Fz )

structure with framed

denote the set of hcratopy classes of

-

structures on X. For a semi-simplicial set L to be a model means n that there is a natural equivalence of functors hl(

-

, Fr) C , ILI I

on the category of finite polyhedra. We could equally well put the fa$ realization

11 L 1

the natural mapping

(degeneracies not identified) on the right side, since

11 L 11

-+

IL1

is a homotopy equivalence.

The local model L is defined as follows. A q-simplex of L is the on Aq x ?Rn transverse to the fibers of germ at Aq x 0 of a rr-structure n the projection Aq x ?Rn -+A~. Face and degeneracy operators are defined in the obvious way. Let K be a finite simplicial complex. A P i structure on K with framed normal bundle gives rise via the graph construction [2 ] to a germ at K x 0 of a :P projection K x ?Rn

structure on K x -+

K.

9,transverse to the fibers of the

The structure we obtain by restricting to a

q-simplex of

K defines a q-simplex i n

Thus, we associate t o any

L.

rrn

structure on K with framed normal bundle, a simplicial mapping K+L. It i s easy 'to check t h a t t h i s induces a natural transformation of functors

a:, Each q-simplex of tr-verse

x lRn

L corresponds t o a g e m of a

These f i t together, giving a

with framed normal bundle.

B(f) = f * ~i f Clearly

f :

rrn - structure

on

-structure on dq with

rz - structure

on

U

We define

P: 1 ,IILIII+~ by

r:

t o the f i b e r , and thus t o a

framed normal bundle.

IL~

,IIL~~I

1

1 -r (*,rn)

1 ~ 1 IL~. 4

B a = 1. The f a c t t h a t

a 19

t h e simplicial approximation theorem: i f f i n i t e simplicial complex i n t o

11 L 11

= 1 follows inmediately from

1 1

f : K

-+

11 L 11

is a mapping of a

then it i s homotopic t o a mapping which

simplicial with respect t o sane subdivision of

IK~.

To deduce t h i s simpli-

c i a l approximation theorem f r o m t h e ordinary one, note t h a t t h e second barycentric subdivision b2L simplicial mapping

IIb2L11 -r

$9. n i s k model f o r Let

Dn

i s a simplicial complex, and there i s a

11 L 11,

homotopic t o t h e identity.

BTf

denote t h e u n i t disk in

q-cell in t h e disk model i s a g e m along on dq x lRn,

lRn,

centered a t the origin.

fi X

D~

of a

r:-

A

structure

w

transverse t o the f i b e r of t h e projection dq x lRn-+ dq.

Again, face and degeneracy operators are defined as the obvious --back of

rnr

structures. Let

L'

denote the disk model and L t h e l o c a l model.

obvious mapping n : L9 -c L : i f i s the g e m of

A I

along

dq x 0.

w

There i s an

i s a q - c e l l i n the disk model,

n(u)

Lemma. -

n

:

IL'~

-,

IL~

is a homotopy equivalence.

11'1 is a model for

Clearly, this implies that Proof. -

BF~.

Suppose K is a finite simplicia1 camplex and K

0

subcomplex. Suppose f : K

+

is a

L and f' : KO -+L' are simplicia1 mappings

such that the following diagram commutes.

It is enough to show that there is a simplicialmapping F such that the diagram still commutes. From f' , we obtain a obtain a g e m at K X 0 of a

a

r:

structure

F:

structure u.

on

Kg x Dn. From

f, we

The commutativity of the

diagram implies the restriction to K 0 x Dn of w is the same as the g e m of w

0

at K x 0. It is easy to construct a 0

neighborhood U of

% X D~ U K

X 0 in

X

and such that the g e m of fl at K X 0 is

there is a mapping H : K of KO x is a

on u

K

x

X

Dn + U

G.

F nr

structure n in e

Dn such that

fll% x Dn=

It is also easy to see that

which is the identity on a neighborhodd

0, and is a diffeomorphism on each fiber. Then H*Q

-r :structure on

K x Dn.

It defines a mapping F : K +I'

which

makes the above diagram commute. 0

First Deformation Lama. Now we come to the first of the two deformation lemmas which

Thurston used to prove his theorem. We let M .be a smooth manifold, possibly with boundary, and let R :

wo,

E --r M be a locally trivial bundle.

We choose, once and for all, a

We will suppose that in a neighborhood of a M , the metric has the form ds2 + dt2, where ds2 is the metric on a M and t Riemann metric on M.

is a smoo+,b function in a collar neighborhood of a M

dM

#

and dt

such that t = 0 on

0. We may always choose a metric having this property. We

will also suppose the metric is complete, and that there is an E > 0 such that every €-ball is geodesically convex, i.e., two points x in y in such a ball BE can be joined in M by a unique minimal geodesic, which lies in BE.

If M is compact, these last two properties are automatically

satisfied. We suppose given a base section

*

:M

-+

E.

We will make the hypo-

thesis that the base section is good, i.e. there is a fiber preserving homot o w h of E such that ho = id, = *(M)

in E, and ht(*(M))

sl(*(~))

is a neighborhood of *(M)

for d l t.

By the support supp s of a section s : M the set of all x E M

such that s(x) f *(x).

+

E, we mean the closure of

We let rc(E)

denote the

space of compactly supported sections. Clearly, sets of M and

r,(~) = lim rK(E), where K ranges over all compact suba rK(E) denotes the space of sections with support in K. We

topologize rK(E)

with the compact open topology, and give rc(E)

the

direct limit topology.

Let

&

> 0 be so small that each ball of radius

convex. We let rS(E) n

2

2'n~.

&

is geodesically

be the set of sections s such that there exists

1 such that supp s lies in the union of n open ball's of radius For N

2 1,

we let

rEYN(E)

denote the space of those sections for

which the comsponding n can be taken

5 I(.

Thus

rE(~)=

rE,N(~).

U

E l We provide proride

rE

rE,H

with the topology induced by its Fnclusion in

with the direct limit topology:

rE =

1 9

rc

and

The inclusion

rE rc

mapping

-C

Lemma. -

i s c l e a r l y continuous.

I f , the f i b e r of

n

rE-+rCi s

t h e inclusion mappin$

: E

n-comnected (n = dim M),

-r M

,then

a weak hcanotopy equivalence.

This lenrma w i l l be proved i n $ § 11-13.

I n this'section, we construct a homotopy, which w i l l be used i n the proof of t h e deformation leamas. As in $10, we l e t M and l e t

on M,

n = dim M. and suppose

We l e t hi

be a smooth mautfold, possibly with boundary,

2, ...,

be continuous, real-valued flmctions

_> 0, Z hi 5 1. We l e t

q-simplex, by which we understand

{(tl,

denote t h e standard

..., t q ) : 0 ( tl 5 ... -< tq -< 1).

We w i l l construct a h~matopy H : M x A' x I -CM x AQ.

x e M, t = (tl,

Let

(x,~), where u r. = N t i 1

- Mi.

Mote t h a t

t

9

51

depends only on Xi

2

E

I. We define

i s defined as follows.

ai

( x , ti, s ).

0, it follows e a s i l y t h a t

i s a homotopy of M x Aq. Let

uq) e Aq

s

H(x,t,s) =

Let Mi = [I?ti]

Then

ui

and

- (q,...,

..., t p ) e A ,

: Aq-'+

Moreover, from 0 1 t

- q -< ... 1.

* rE($V )

is a model for (which was defined in $ 10. UG Similar to the two proofs just given.

§ 17. Construction of the Mapping

B

Diff: M

-+

rc(*k v).

Let M be a smooth manifold, possibly with boundary. We will suppose .M is provided with a Riemannian metric, satisfying the conditions stated at the beginning of § 10. Let n = dim M. We consider two semi-simplicial complex

%

and

5

defined as

A q-cell of

follows.

4

is a

r:-structure

i s the complement of a compact s e t i n M. for

uc

transverse t o the

and horizontal on Aq

f i b e r of the projection Aq x M ,Aq,

* rC(.rMV )

on Aq x M

5

The complex

X

U where

U

i s the node1

defined i n t h e previous section.

There are two cases of these constructions which w i l l be of use t o us. Case 1. -

n M = D , the unit disk i n

JRn.

4

I n t h i s case,

i s the

( 5 9).

disk model f o r B

@.I n this case,

Case 2. -

4

i s a model f o r

95

f a c t , it i s a special case of t h e model i n

for

BE,

B

D~P<M.

In

where

G = ~ i f f z ~ .

Now we define a mapping of semi-simplicial s e t s a q-cell

a

of

n, : M x M - M

%,

we l e t

f :

8,

f ( 8 ) = (id(Aq) x n,)'

we see t h a t

I'E-structure a t Aq x M X M,

x diag. M,

(id(Aq) x n2)*

8

Given

where Since uq

denotes projection on the second factor.

r 9 rn-structure on A x M,

% + 5.

is a

defines a

and it i s the germ of t h i s

$-structure

at

It i s e a s i l y verified t h a t

which we denote by f(Aq).

f

is

a semi-simplicial mapping.

In the case M = Dn,

B F ~and f

:

4 + I$ is t h e

I n t h e case and

8~

=

@,

and

]k

a r e two different models f o r

natural homo to.^ equivalence.

-

we have

= B D~P<M

md

5 = r c ( k*v ) ,

f ' i s the mapping which appears i n the statement of Thurston's theorem

($6). We w i l l show f induces isomorphism i n integer homology, t o prove Thurston's theorem.

9 18.

F i l t r a t i o n s of

5 "d %'

We. w i l l define f i l t r a t i o n s of

4

and

a mapping of' f i l t e r e d semi-simplicial sets.

5

so t h a t

f :

% -+ 5

is

I n $ 16 we have defined the support of a q-cell i n the support of a q - c e l l i n

6I

of

We say a q-cell

x u i s induced by the projection

$1 i s the s e t of points Let

a s follows.

x E M . i f there i s a neighborhood U of

i s horizontal a t that

%,

>0

&

3 (or 5

)

be small.

Let

positive integer p

5N

2-'~.

%,1

such t h a t

8 x

x u + U.

x E M a t which uq

of

%

in M

such

The support of

i s not horizontal.

N 'be a positive integei..

w i l l be said t o be i n

b a l l s of radius

5. We define

A q-cell

uq

(or K , ~ ) i f there exists a

%,N

supp $1 i s i n the union a t a t most

p

Thus we have ,increasing f i l t r a t i o n s

Cs,2 C

."

C $,NC

... c

=

u

lo>l

%,w

= U XIy2 C ... C y M Y N C... c 1 - 5,~ i s e a s i l y seen t h a t f : -+ 5 respects these f i l t r a t i o n s . Lemma. The inclusions sYw$ & andyO,- 5 a r e homotopy C

Y

It

equivdinces. proof.

The f i r s t assertion follows from the second deformation

lemma ($ 14).

The second assertion follows from the f i r s t deformation

lemma ($ l o )

$ 19.

and t h e identification of

Construction of

rY

N

%,N*

'M,N)

Throughout t h i s section, we l e t

5

N

as

*

rc(rM V ) and

as

be a positive integer, &d

M

a

smooth n-manifold provided with a Riemannian metric, satisfying the conditions s t a t e d a t t h e beginning of $10. We w i l l construct semi-simplicial s e t s denoted N

k,ny

A z ~AH^,^. ,~ A q-cell i n

%,N

w i l l be an

N-triple of

a c h one transverse t o the f i b e r of the projection

%,Ny

%,Ny

I':-structures

aq x M

'A~,

%,N

on AP x My and each

one hz:ir,g By

s q p o r t i n a b a l l of radius

'M,N

some Nl

< N,

.

jFd

ATM Y N

of

AT

i f for MYN balls of radius Z - ~ ( ~ ) E .

w i l l be a q-cell of

%,A

= %yN

-

9

A q-cell in

1. I n view of t h e Lemma i n $19, it i s enough t o

rH : GYN %,.

induces isomorphism i n i n t e g r a l homology.

N

+

In

view of the Lemma in $20, it i s enough t o prove t h a t

induces isomorphism i n i n t e g r a l hamology.

,

, -a a 5,1(&2, )

F i r s t we show

Since

GyN =

A

A

that

%, 1

Since

E

A

A

%,l-&,l

( 2-N) ~

-g

,1

and ( 2-I) ~

induces isomorphism i n i n t e g r a l homology. -N a = GS1(&2 )

gYN

it i s enough t o show

induces isomorphism i n i n t e g r a l homology.

i s an a r b i t r a r y small nmber it i s enough t o show t h a t induces isomorphism i n i n t e g r a l hamology.

According t o $22,

n 5

s,l A

and 9

1

a r e each equipped with a

A

semi-simplicial mapping into L, with which fN commutes. (Boy

..., B9)

Let A

denote a q-cell of L. The restriction of fN to a fiber

over an interior point'of this cell is f : XBo n.,.n Bq There are two cases. M e n Bo n

... n Bq = pi,

+

YBo n...n

Bq.

then both XBO n.. .n Bq

n...n B~ are one point. In the case Bo diffeomorphic to B , and f is the mapping B-D y%

fl

... fl Bq lRn

-

f, it is

"BP n which we

have shown to induce isomorphism in homology. h

Thus, the restriction of f to the fiber over the interior of any A

cell induces isomorphism in homology. A standard argument then shows that f

- AR,~

induces isomorphism in hcrmology. NOW we show

induces isomorphism in integral homology.

We let T C L~ be defined in the same way a6 in Cj 24; the only difference is

now our manifold M is arbitrary, whereas in

5 24 it was D ~ .

We have a canrmutative diagram

The fibers of the vertical arrows over an interior point of the cell of. T corresponding to (Bij) are X~(o,l> n.. .n ~(q,l)X - m - X%(o,N)

YB(O,N) n.. .n B(~,N)

YB(o,~) n.. .n ~(q,l)X...X

i?

It follows that the mapping induced by in homology, since either

%o,j are both one point (in the case

"

1 n.. .n n.

n.. .n B(~,N)

on each fiber is an isomorphism and Y ~(0~3 n. ) B(sYJ) B~~ = pi), or the mapping

~(q,j

..n

is the same as the mapping %(q,~) + 'B(O,J) n.. .n B(q,j) A B ~iffflRn+ F ~ (in B all~other cases). Hence f : "%,N + induces an isomorphism in homology.

'B(o,J)

-

.en

Haefliger, A., . Hmotopy and Integrability, Lecture Wtes in Math. , no. 1%' Springer-~erlag(191), pp. 133-163. Haefliger, A. , Feuilletages sur les vari&&

owertes, Topology 9

(1970), PP. 183-194. Mather, J., SOC.

~n

Haefliger's chss.i*ing

space,I ~ u l l .m. Math.

~7 (iml), pp. m - u 5 .

Mather, J.,

Integrability in Codimension 1, Comment..Math. Helv.

48 (19731, pp. 195-233. Mather, J.,

Loops and Foliations, Manifolds-Tokyo 1973, University

of Tokyo Press 1975, pp. 175-180. Milnor, J., The geometric realization of a semi-simplicial complex Ann. of Math.,

65 (1957),,pp. 357-362.

Roger, C., Etude des r-structures de codimension 1 sur la sphGre 2 Fourier, 23 (19'73), pp. 213-228. S , Annales de 'lvlnstitut Thurston, W.,

Foliations and groups of diffeombrphisms Bull.

Am. Math. Soc.

80 ,(1974'),pp. 304-307.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(c.I.M.E.

1

MANIFOLDS OF DIFFERENTIABLE MAPS

P. MICHOR

C o r s o tenuto a V a r e n n a dal 25 agosto a1 4 settembre 1976

Manifolds of d i f f e r e n t i a b l e maps

P e t e r Nichor Let X , Y be smooth f i n i t e dimensional manifolds, l e t

1.

C-(x,Y) be t h e s e t of smooth mappings from X t o Y ; f o r any non n e g a t i v e i n t e g e r n l e t

J ~ ( x , Y ) denote t h e f i b r e bundle

of n - j e t s of smooth maps from X t o Y, equipped with t h e can o n i c a l manifold s t r u c t u r e whlch makes i n t o a smooth s e c t i o n f o r each f

jnf : X

-+ Jn(x,y)

c ~ ( x , Y ) , where jnf(x) i s

t h e n - j e t of f a t x c X. Usually C-(x,Y)

i s equipped with t h e s o c a l l e d Whitney-C"-

topology: a b a s i s of open s e t s i s given by a l l s e t s of t h e M(u) = {f c ~ . ( x , Y ) : jnf (x).

form

U]

, where

U i s any open

s e t i n Jn(x,Y) and n-cN. See C3] hnd 1'63 f o r accounts of t h i s toMogy. We may d e s c r i b e i t i n t u i t i v e l y by t h e f o l l o w i n g words: i f you go t o i n f i n i t y on X you may c o n t r o l b e t t e r and b e t t e r

p a r t i a l d e r i v a t i v e s up t o a f i x e d o r d e r . 2.

Anyone f a m i l i a r with f u n c t i o n a l a n a l y s i s may have heard

t h e f o l l o a i n g words: i f you go t o i n f i n i t y (on X) you may c o n t r o l b e t t e r anii. b e t t e r more and more p a r t i a l d e r i v a t i v e s . This d e s c r i b e s t h e =

l i m a ( K ) , where --9

i n d u c t i v e l i m i t topology on

D(R)i s

B(R) =

t h e space of a l l smooth f u n c t i o n s

with compact s u p p o r t on IR and B ( K ) i s t h e subspace of those f u n c t i o n s . v ~ h i c hhave support contained i n some f i x e d compact K of X , equipped with t h e topology of uniform convergence i n

a l l partial derivatives. The topology induced by t h e khitney-Cm-topology

on

'%l @I>

could be described by t h e formula

8 (R)

= &m

( 12 f ( ~ ) ) , r 4 K where v r ( ~ )i s t h e space of a l l cr-functions on k with

support contained i n K.

This d i s c u s s i o n shows ( I hope) t h a t

t h e ~hitney-cW-topology i s n o t t h e most n a t u r a l topology on G"(X,Y).

3.

Ye now give an i n t r i n s i c d e s c r i p t i o n i n terms of j e t s

of t h e topology on

c~(x,Y)r e f e r r e d

t o i n 2. Ve c a l l i t t h e

t d e t a i l e d account of it Can 5e found i n [ 71

3 -topology.

'?here a r e t h r e e e q u i v a l e n t d e s c r i p t i o n s of t h e

.

3 -topology

on c@(x,Y~: (a) Fix a sequence K . = (Kn) of compact s e t s i n X such t h a t ii

0

=

13 , I$,-,,

t h e form

5

so, X

=

U Kn

. Then

t h e system of s e t s of

M(~,u= ) ~ ~ . L c * ( x , Y ) :j % f ( ~ - % ~ ) s Uf o~r a l l n]

i s a base of open s e t s f o r t h e

3 -topology

on C-(x,Y),

where

m =(rnn) r u n s through a l l sequences of non negative i n t e g e r s wd U = (Un j with Un open i n J~ (X,Y).

The

3 -topology

i?

independent of t h e choice of t h e sequence (K,). (b) F i x a sequence (dn) of m e t r i c s dn on J"(x,Y),

compatible

with t h e manifold t o p o i o g i e s . Then t h e system of s e t s of t n e form

v Y (r) =

r c ~ ( x , Y ) : Yn'(x) d n ( j n f ( x ) , jng(x))

a l l x i n X and f o r a l l n ) f e (P(x,Y) i n t h e where

i , i .e


x

g

sent %ales

,

i < n

si x > 0 .

f(")(~) = g(")(~) >

Alors, par le resultat de Takens,, f existe

--tangent d

$I

(f,g) tels que :

Id

en

0

f =

tel que

et

g

o

sont conjugu6s

f

'

:

= lim P+"

il

g g I-'

I1 est facile de trouver une solution "formelle" pour

Alors

;

:

de prendre

g

.

= lim

fP g-P

I?+" fP+l g-p = lim pfm

fP g-P+l = g g

.

#I ; il suffit

Que la suite

(fP ) ' g

soit convergente rdsulte essentiellement des

ieux observations suivantes. D'une part g'") (0)> 0 , la convergence de g-P(x) vers 0 sera d type de p-l/n-1 ; d'autre part, comme fg-1 est m-tangente 2 Id en 0

, la difference

"B d6croissance rapide" avec

p

entre fPg-P

et

fP+' g-P-'

est

.

On peut montrer que l'application

est suffisamment continue pour justifier l'argument de continuit6 indiqueS plus haut.

On trouvera les d6tails dans (3)

.

(1) J. MATHER.

Integrability in codimension one

;

Comm. Math. Helv., 1973,

vol. 48, pp. 195-233. (2) T. MIZUTANI. Foliated cobordisms of 4-manifolds (3' F. SERGERAERT.

;

and examples of foliated

Feuilletages et diffeomorphismes infiniment tangents

'11identit6; (4) S. STEWERG.

s3

Topology, 1974, vol. 13, pp. 353-362.

Local

parafhre.

cn

transformations of the real line

;

Duke Math. J

1957, vol. 24, pp. 97-102. (5) F. TAKENS.

Normal forms for certain singularities of vectorfields

Ann. Inst. Fourier, 1973, vol. 23, pp. 163-195.

Npartement de Mathematiques Faculte des Sciences F-86022 POITIERS CEDEX

;

CEN TRO INTERNAZIWALE MATEMTIC0 ESTIVO

(c.I.M.E.)

INVARIANT

DE GODBILLON-VEY. ET

DIFFEOMORPHISMES COMMUTANTS

G o WALLET

Corso tenuto a V a r e n n a dal 25 agosto a1 4 settembre 1976

INVARIANT DE GODBILLON-VEY ET DIFFEOMORPHISMES COMMUTANTS Guy WALLET

I. Definition de l'homomorphisme Soit G

et 6noncd du resultat principal.

GV

l'un des deux groupes discrets suivants

1) le groupe des difft5omorphismes de R

iff:^)

cm , not6

:

d support compact et de classe

;

2) le groupe des diffc5omorphismes de l'intervalle fern6 [o, 11 preservant

cm , note

l'orientation et de classe

.

~iff~[~,l]

Pour tout element c de HZ(G,2) orientee sans bord claasifiant BG

M

, et une

, telles que

par l'application induite de

, on associe

K(G,l)

rl(M) dans G

, il

existe une surface compacte

applicatio~continue h de M dans l'espace

c

soit l'image de la classe fondamentale de M

H2(M;I)

dans H (BG;Z) 2

d la classe d'homotopie de

. R6ciproquement1 la donnee d'un .

. Puisque

BG

est un

h un homomorphisme a de tel homomorphisme a permet de

definir un 616ment unique c de H2(G;2) K

[0,1]

d8signe l'ensemble R

. Soit

lorsque G = ~iff~[0,1]

considere l'action de G

sur

GXK

IY

Le quotient Ec = MxK/G structural discret G letage

Sc

, de

l'intervalle

le revetement universe1 de M

d6finie par

. On

:

est l'espace total d'un fibre a groupe

base M et de fibre K

gc.transverse aux

, et

lorsque G = ~iff:tR)

.

Ec

est muni d'un feuil-

fibres et d holonomie globale a

. Le feuilletage

etant trivial en dehors d'une variite compacte, on peut d6finir son inva-

riant de Godbillon-Vey GV( Sc) ER pondre GV ( S c )

[ 2 ]

D1aprBsdes resultats de W. THURSTON [8] l'homomorphisme

. L'application

, definit un homomorphisme note GV de GV

est surjectif.

qui I

c

fait corres-

H2 (G;Z) dans R

et de J.N. MATHER [4]

, on

.

sait que

DEFINITION. On appeZZe tore.de H2 ( ~ ; ntact dtdment de ce demier reprdsentd par un hoqmorphisme de nl (T) duns G 02 T ddsigne Za surface compacte orientde de genre 1. de

G

Un tore de H (G;Zf) est donc defini par un couple (f,g) d'elgments 2 qui commutent, images des generateurs de nl(T). On note le

tore associe B un tel couple.

THEOREME I .

L'imge par GV d!un t o r e de H~(G;z) gst nutte.

11. Reduction du theoreme 1 B un cas particulier.

On appelle diffeomorphisme local en

[ sur

intervalle [O,U

un intervalle [0,f3 [

0,un diffeomorphisme d'un

. Le resultat suivant enonce une

propriGt6 fondamentale des diff6omorphismes locaux qui commutent. 2

LEMME de N. KOPPEL [3] : Soit 4 un c -diffdomorphisme ZocaZ en 0 ne possddant 1 pas de point fixe diffdrent de o Si I/I est un c -diffdomorphisme ZocaZ en 0 cornmutant avec 4 et possddant un point fixe diffdrent de o, alors ~,.est dgaZ d Z'identitd.

.

soit un tore de lequel f

admet

N. KOPPEL, g

a et b

admet aussi

les restrictions de

,

de

f

et

H2(~iffD[a,b])

des feuilletages du type

le graphe de

f

et de

n2 (G;z?) , [a,b] un intervalld de

R

our

comme seuls points fixes. D'apres le lemme de

g

a

et

b

g

B

[a,b]

comme points fixes. Notons

. On peut donc

. Le feuilletage

#

$