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.
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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'
when g and h are composable germs and y = t(h)
,
This is precisely the formule of Thurston for the codbillon-Vey invariant.
5. Differentiable cohomology with coefficient in E. As in 3,
E denotes the smooth rM-sheaf
vector bundle on M on which
r
acts smoothly, determined by a representation
M
of G in the fiber V.
Definition. The differentiable cohmno~ogtlHi(rM;
the compler
c:(rM
;
of smooth sections of a
E) is
the cohomotogy of
e).
In 111, 3 we gave an example of a smooth cocycle for which the bundle E was the trivial line bundle.
gn be
the L i e algebra of formal vector fields on R" which vanish a t 3 ; it can be considered as theLie algebra of G = l&m Gk (cf. Let
I,8). We denote by C*(gn ; 3,; V) the 3 -basic complex of Gelfand-Fuchs n cochains (i.e. depending only on finite order jets) on with value in Yn the G-module V (the fiber of E). I
The analogue of Van Est'theorem is
;
lheorem 8;(rM
E) ie isomorphfc
Let p : PM/On
+
to H*(Ynfln
; V).
M be the natural projection. L et
E~(~) be the
sheaf of germs of differential forms along the fibers of p, considered as the sheaf on M of germs of smooth sections of the vector bundle .Qq(p) -1 whose fiber above x is the vector space of q-forms on p ( x ) . We can consider
.L?q(p)
as a sheaf of 0 -modules, where 0 =)M('?L. is the sheaf of germs of M M smooth functions on M. We have a differential d : flq(p) + .L?P+l(p) which is smooth, hence passing to smooth sections, we get a differential
-
d :~
~
+
(fll(p) ~ 1which -isgM-linear.
- ...
'he sequence 3
+
o*+ Ij) (p)
+
d(p)
+
is exact, because the fiber is contractible by a smooth deformation. In fact one has a smooth homotopy operator h : nq
which is a
smooth morphism of vector bundle and such that hd + dh = id. Tensoring with
E over OM, we
get an exact sequence
0+ E +L?(p)~&+$(p)@~+....
Consider the double complex C*(r d M i n -*(p)
BE)
.
First stee : The inclusion :
ci(rM
;
E) c*d(r M'- n*(p) @g) +
induces an isomorphism in cohomology (for the associated simple complexes). As in 11, 3, this follows from the fact that the sequence €
o c: (rM -+
; E)
(rM
cdp
QE)
;
+
c:
(rM
; cl(p)
+
is exact for each p. (An homotopy operator is constructed using the h above). Let (.Q*(p) OJ~)~' of c*(p)
be the complex of sections of the subsheaf
@E of elements invariant by
rM.
Second step : The inclusion E2
: (n*(p)
@ ElrM
+
Cz(rM i c*(p)
induces an isomorptism in cohomology.
The proof of this fact is formally the same as the proof of lemma 2 in I I , 3 .
A proposition analogous to the one of 1,8 shows that (Q*(p)
, On
is isomorphic to C*(
@E)
; V), hence the result.
%n Remark This result gives an interpretation of the group H*(y,On ; V) as a group of differentiable cohomology of r in a sheaf associated to the -module M V, and gives a new insight in the computations of Koszul [ 341.
8
In his paper, I[oszul computes I!1(y
4( l-mod~leV.
, On
; V) for a semi-simple
There are essentially three cases.
In the first case, V is the trivial module R. A generator of El1 is represented by the 1-cocycle associating to a formal vector fieldon Rn without constant term the trace of its linear part. By the isomorphism above, it correspond to the cocycle in example 111, 3. n In the second case, V is the ~ l - m o d u l1e of ~ linear maps of R in gln=yl. A generator of H 1(g,On,~) corresponds to the Atiyah-Molino class which sits in Hd1 (rM; where%t is the sheaf of g e n s of sAooth sections of the associated'bundle with fiber ~ ( c f .1411 et 1421 ). n In the last case, V is the module dual to R ; the generator of
HI( lJ
, On
; V) corresponds also to an Atiyah class which represents the
obstruction for an invariant connection in the bundle of n-forms on M.
6. Conclusion Recall that by definition, we have a natural morphism of the differentiable cohomology H*(r ; R) in the real cohomology H*(r ; R) (which d can be interpreted as the cohomology of the classifying space Br). It is the homotopical and cohomological properties of
~r which
are
important to know to solve geometric problems. The differentiable cohomology is just a way of constructing interesting cohomology classes in Br, so that one can detect non trivial homology classes of Br by evaluating on them smooth cohomology classes. But there is a little more, namely some qualitative properties of differentiable cohomology also reflect qualitative properties of the cohomology. So if one can prove something on one side, cjne can try to prove the analogous property on the other side.
To illustrate that, we mention some results to compare the smooth cohomology B*(r ; R) with the real cohomology H*(rn; d n
R)uH*(B~;
R).
For n = 1, it'is known that BT is 2-connected (Mather 137]), and 1 that n (Brl) surjects on R'(Thurston i431. This last result is obtained by 3 evaluating the Godbillon-Vey invariant (i.e. the generator of H3(r ,R)) on d 1 suitable examples of foliations. No other invariant is known. For n > 1, the differentiable cohomology of Tn is 2n-connected (cf. 4 and I.8),
but it is only known that
BTn
is (n+l)-connected (Thurston
1441, Mather 1383). Many homology classes in dimension 2n+l of B? n are known to be non zero (by checking that the evaluation of differentiable cohomology classes are non zero), and even vary continuously (Thurston, Heitsch 1271). .One is timpte'd to conjecture that BT A step n is also in-connected, toward this conjecture (which would imply that n2n+1(~Fn) = H2n+l(Brn)) would be to construct examples of 7-structures on S2n+i for which the differentiable n cohomology classes of dimension 2n+1 vary independently. The polynomialsin Pontrjagin classes of degree 2n don't vanish in the differentiable cohomology of rn(this is just the bound given by the vanishing theorem of Bott C41) ; so it is natural to conjecture that they don't vanish in the cohomology of Br but very little-isknown about that P' in the range between n+1 and 2n. The more general conjecture is that the map Hz(rn; R)
+
~ ~ ( R) 7 + ~ H* ;(% or IV, 2).
H*(rn;R)
=
H*(Brn;R)
is injective, as well as the map
; R) (this would be the analogue gf the theorem of I I , 4
Some partial results are knowit in that direction (Thurston,
Bott-Haefliger,
amber-~o'ndeur
131) Aeitsch [27], etc,)
According to our definition, H*(r ;R) and H*(?; R) are iqomorphic, d n d n n wherk n denotes the groupoid 6f germs of analytic local diffeomorphisms of R So one can ark if the natural map H*(r ;R) + H*(< ;R) is an isomorphim. n
A remarkable example of .the parallelism between differentiable cohomology and cohomology is the analogous of the theorem of Thurston-Mather
-
1441 explained by Mather in his lectures, relating the homology of BDiffM and the homoldgy of the space of sections of a suitable bundle. The analogous theorem for the differentiable cohomology was called the Bott Conjecture,and has been proved by several people (cf. 1261).
.
As a last example, the differentiable cohomology of
r0(resp.
m
(cf. III,l,Ex.3)
is the cohomology of the Lie algebra? of formal con&,act 2n+ 1 vector fields on R (resp. the Lie algebra's of formal symplectic vector 2n fields on R ). Note that?$ can be interpreted as the Lie algebra of formal power series on R~~ with the Poisson bracket (cf. Vey 1491). It is proved in [491 that Ii*(+)
is Zn-connected ; this is also true for
H*(FO; R) = H * ( B ~ ~ ;R) (cf. 1241). Also according to [491, H*(s ) is, in degreej;Zn, a polynomial algebra in a generator of degree 2 ; on the other hand in degree 2n,Bi;n
i s a n Eilenberg-MacLane complex K(R ; 2), The Hochshild-
Serre spectral sequence relating H*(+
) and H*
Serre spectral sequence of the fibration
(S )
BFu BTn ' +
correspond to the
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- Cohomology
of groups. Algebraic 'number theory edited
by Cassels and Frzlich
- Academic Press (1967), 94-115.
[2] I.N. Bernstein and B.I..Rosenfeld- On Characteric classes of foliations
- Functional
Analysis and applications, 6 (1972), 68-69
-
[3] A. Bore1
Compact Clifford-Klein Forms of Symmetric Spaces, Topology 2 (1963), 111-122.
-A
141 R. Bott
topological obstruction to integrability
- Proc.
Symp.
Pure Math. Vol XVI (1970) 127-131.
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[5] R. B0t.t
on Characteristic Classes and Foliations
-
Springer Lectures Notes No 279 (1972). 161 R. Bott and A. Haefliger
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Bull. AMS (1972), 1039-1044.
- On the Chern-Weil homomorphism and the continuous
[71 K. Bott
cohomology of Lie groups. Advances in Math 11 (19731, p. 289-303. [81 J.P. Buffet-J.C. Lor
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assez large de r-structures
universe1 pour une classe
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Acad. Sc. Paris 270
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[lo] 'Dupont - Simplicia1 de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976) 233-245. [ 11 1 C. Ehresmam. Les connexions in£initgsimales dans un fibr6 dif f6rentiable
Colloque de Topologie de Bruxelles 1950. CBRM, 29-55. 112; Van Est
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Nedel Akad.
Wetensch. Proc. Ser. A 58 (1955) I, 225-233 ; 11, 286-294. [13] Van Est
-
Une application d'une m6thode de Cartan-Leray, Indag. Math. 18 (1955), p. 542-544.
1141 I.M. Gelfand-D.B. Fuchs. The cohomologies of the Lie algebra of the
vector fields on the circle. Funct. Analiz. 3 No 4 (1968) 92-93. [15] I.M. Gelfand-D.B. Fuchs. The cohomology of the ~ i algebra e of formal
vector fields, 1zvestia An. CCCR
- 3i (1970),
322-337.
[I61 I.M. Gelfand-D.B. Fuchs. The cohomology of the Lie algebra of tangent
vector fields on a smooth manifold, I and 11 Analysis
- Voi.. 3 No 3 (1969),
- Functional
32-52 and vol. 4 No 2 (1970)
23-32. 1171 I.M. Gelfand-B.L.. Feigin-D.B. Fuchs. Cohomologies of the Lie Algebra
of Formal vector fields with coefficients in its adjoint space and variations of characteristic classes of foliations. Funct. Analysis-8 (No 2) (1974) 13-29. [ 181 G. Godbillon-J.Vey.
Un invariant des feuilletages de codimension un, C.R. Acad. Sc. Paris Juin 1971.
1191
G, Godbillon. Cohomologie d'algsbre de Lie de champs de vecteurs formels, S6minaire Bourbaki 1972-1973, expos6 421.
[20] R. Godement. Thi5orie des faisceaux, Hermann Paris (1964). [21] A. Grotlendieck. Sur quelques points d'algsbre homologique. Tohoku Math.
Journ. 9 (1957), 119-183. 1221 V. Guillemin-S. Sternberg. Deformation Theory of Pseudogroup structures,
Memoires AMS No 64 (1966). [231 V. Guillemin.Cohomology of vector fields on Manifolds, Advance in Math 10, 192-220 (1973). [241 A. Haefliger. Homotopy and Integrability
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No 197
Springer-Verlag (1971, 133-163.[2i] A. Haefliger. Sur les classes caractgristiques des feuilletages
-
SCminaire Bourbaki annee 1971-1972, No 412. [261 A. Haefliger. Sur la cohomologie de Gelfand-Fuchs.Differentia1 topology
and geometry No 484.
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[27] J. Heitsch -'Variation of Secondary classes (to appear). 1281 G. Hochschild
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[291 G. Hochshild-G.D.
Mostow
Holden-Day Inc. 1965.
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(1962) 367-401. [30] F. Kamber-P. Tondeur. Non trivial Characteristic Invariants of homogeneous foliated ~undles,Ann. Ec. Norm. Sup. 8. (1975) 433-486. [31] F. Kamber-P. Tondeur. On the linear independence of certain cohomology classes of Br. To appear. [32] Kobayashi-Nomizu. Foundations of differential geometry I and I1
-
Interscience Tracts in Pure and Applied Math. (1963). I331 J.L. Koszul. Homologie et Cohomologie des algebres de Lie. Bull. Soc. Math. de France 78 1950, 65-127. 1341 J.L. Koszul. Homologie des formes differentielles dlordre supIrieur. Ann. Sc. Ec. Norm. Sup. 7 (1974) 139-153. [35] M.V. Losik. Cohomology of the Lie algebra of vector fields with coefficients in a trivial unitary representation
-
Functional Analysis 6 (1972), 24-36. [361 S. MacLane. Homology. Grundlehren der mathematishen Wiss, Springer 1967. [371 J. Mather. On Haefliger's cla.ssifying Space I. Bull. AMS 77 (1971) 1111-1115. 1381 J. Mather. ~ntegrability in codimension I, Co~mn,Math, Helv. 48 (1973) 195-233. C391 J. Mather. Commutators of diffeomorphisms I and 11, Conrm. Mith. Helv. 49, (1974), p. 512-528 and 50 (1975) 33-40 1401
G.D.MOS~OW.
Cohomology of Topological groups and ~olvmanifolds. Ann. of Math. (1961), 20-48.
1411 P. Molino. Classes caractIristiques et obstructions dlAtiyah pour les fibres principaux feuilletgs. C.R, Acad. Sc. Paris 272 (1971). [421 C. Roger. MIthodes homotopiques et cohomologiques en thIorie des feuilletages. ThSse (to appear).
1431 W . Thurston. Non Cobordant Foliations of S3 , Bull. AMS 78 (1972), 511-514. [44] W . Thur'ston. Foliations and groups of diffeomorphisms, Bull. AMS 81 (1975). 1451 W . Thurston. The theory of foliations in codimension greater than one, Comm. Math. Helv. 49 (1974) 214-231. [46] W . Thurston. Existence of codimension one foliations, Ann. of Math. (to appear). 1471 G . Segal. Classifying spaces and spectral sequences. IHES mathktiques No 34 (L
- Publications
105-112.
1481 N. Steenrod. The Topology of Fiber Bundles. Princeton University Press 1951. 1491 J . Vey, Sur la cohomologie des.champs de vecteurs symplectiqnes formels C..R. Acad. Sc. Paris 280.(1975) 805.
Universitg de.Gen&ve Section de mgthhatiques 2-4, rue du Lievre Case Postale 124 CH-1211 GENEVE 24
CENTRO I N TERNAZIONALE MATEMATICO E S T I W (c.I.M.E.
1
ON THE HOMOLOGY OF HAEFLIGERIS CLASSIFYING SPACE
JOHN N. MATHER
C o r s o tenuto a Varenna dal 25 agosto al 4 Settembre 1976
On the Homology of Haefliger's Classifying Space Course Given at Varenna 1976
John N. Mather Princeton University Princeton, New Jersey
1. Haefliger Structures ( ~ einition). f Let Z be a topological space. we let s(y),
If y is a germ of a diffeomorphism of Z,
t(y) E Z denote the source and target of y, respectively.
By a groupoid of germs of homearnorphisms of Z is meant a set
r
having
the following properties:
1) Each y a
r
is a germ of a harneomorphism of Z.
2) For each z a Z, idz a
3) If y a 4)
r,
1f y, y' a
then y-l a .1;
r
and s(yt) = t(y),
5) ($he& Property) If y g : U -+ V of y gZ
E F
r.
E
r,
then y'y E
r.
then there is a representative
(where U, V are open in 2) such that
foreach z a U .
If U and V are open subsets of Z and g : U-rV is a horneomorphism, then g will be said to be a r-homeomorphism if gZ E for each z E U. g : U -C V is a
We will provide
r
r
with the sheaf topology. If
r - homeomorphism, then
: z E U)
w i l l be an open set,
in the sheaf topology.
The sheaf topology will be the smallest topology
compatible with this propc2-t;~. Let space 2,
r
be a groupoid of g e m s of homeamorphisms of a topological
A l-cocycle on X
and let X be a second topological space.
with values in I. consists of an open cover two indices U,
B,
{ua)
a continuous mappbg yap : Ua
of X,
n
and for any
Up + r.
In order to be
a cocycle this collection of data must satisfy the cocycle condition: for
,
any indices a, 8 8,
we have
In particular, the left side is defined, that is s(yap) = t(yp
a).
From the cocycle condition, it follows that yaa(x) is the germ at -1 sane point of the identity and -yap (x) = -pB a (x) , for any x E X. TWO
yap ) I and
cocycles
-
{u&, Y&B}J
are said to be equivalent
if for each a E I and p e J there exists a continuous mapping :
ua n u;
r
such that
on u a n d 8 n u 1
'ap,
Yae6pp'
where
IL'
% S E I, w * J
on U& n u,$ n u1 where a E I, BYp E J. 4xpYbp' 'ap k7 It is easily verified that this is an equivalence relation.
--
An equivalence class ,ofl-cokycles is called a Haefliger . .
If f : X ;re may define a
Y is a continuous and o is a
r
structure
a l-cocycle representing
represented by
* f o
on X,
o. We define
{f-ba, -yap, f).
If X is a subspace of Y and i : X
*
r - structure on
as follows.
* f w
r - structure.
Let
{ua,
Y,
-ya
be
to be the r-structure
-
Y denotes the inclusion
mapping, we let olx = i w.
If
0,
and
y
are t w o
r - structures on
X,
we say
o0 and
'5
are hmotopic and. write uO such that
nl x x o =
-Y
r - struct.ure
I2 on X x I
nl x x I = q.
w and 0
The Classifying Space for r
$2.
if there is a
- structures.
Haefliger has defined .[l,2]
a classifying space B r
for
r - struc-
tures. This represents a functor, and indeed one way to define Haefliger's classifying space is to let it be the space representing this functor [2], which is known to exist by Brown's representation theorem.
r - structures on X, and let h1(xJ) denote the set of hanotopy classes of r - structures on X. Then B r represents hl( . , r) in the sense that there is a natural equivalence Let
$(x,F)
denote the set of
of functors
(1)
hl(*,
on the category of finite polyhedre.
r)= r a SB
~ I
Here [x,Y~denotes the set of homo-
topy classes of mappings of X into Y. We do not require that there be a natural equivalence (1)on the category of all topological spaces. Consequently B r
may not be uniquely
defined. However, its homology and homotopy groups are uniquely defbied. We will adopt the following definition. If X is any topological space, a second topological space Y will be said to be a model for X if there is a natural equivalence of functors [
, X]
= [
, Y]
on the
category of finite polyhedra.
$3. The Normal Bundle. Let p = If r
ri
-
r rn, the gmupoid of germs of
2 1, the mapping
G 4 (n, R).
y
4
d y (s(y))
r C diffeomorphisms of lRn.
is a homomorphism of groupoids
Consequently, we have a rapping V :
B ~ -+BGc(~,~). Z
This generalizes for the case r = 0 to a mapping 0 B top,. v : B rn -+
The latter must be defined in a different way, however. If w is a
rn0
structure on X, the graph construction of Haefliger [2] associates a microbundle on X, wbich induces a mapping of X + B topn. Applying this 0 to X = Brn, we get the mapping we wished to define.
We let
BT:
r = fiber (v:Br,-+
BGt(n,
a)),r>l
top,),
r = 0:
0 = fiber (v:B~,'+B
Here, and throughout this paper, "fiber" will mean homotopy theoretic fiber. If w is a
structure on X, then there is associated to it a
vector bundle, by the homomorphism
<
+
G$ (n, R), described above. The
associated vector bundle is called the normal bundle of w
In the case
we have only a normal microbundle, as described above.
r= 0
$4.
.
The Classifying Theorem. Haefligerlsclassirylng space is in a sense a classiFying space for
r'oliations. M a n y geometric questions concerning existence and extension of foliations would be solved if it were possible to understand the hmotopy and homology of Haefliger's classifying space. However, the homotopy and homol o w is still a mystery. Here is the basic theorem relating foliations and Haefligerlsclassifying space. Classifying Theorem (~aefliger-Thurston). &J fold
-9
X,
X be a smooth mani-
& K a closed subset of X, 3 a foliation of a neighborhood of K
and
v a sub-bundle of T X
in some neighborhood of K.
such that v
in
is the normal bundle of
r>1 Suppose cod 3
X
-K
has no relative-
ly compact components. ?hen the followg two conditions are equivalent.
on
(1) There exists a foliation
~(9 ) - V rel. nbd. the normal bundle of 5' in X. neighborhood of K
(2)
and
X, such that
==in a
$ r C
Here ~ (' ) rdenotes
of K.
In the diagram below, we can find a mapping
5'
making it
hmotopy commutative.
Here $ and v -
:
/d denotes the mapping canonically associated with the foliation .y
X -,B G $(n, R) the mapping canonically associated to the bundle v.
This theorem is discussed in Thurston's lectures. Here is an example of what we would like. -r Conjecture 1. Brn is 2'-connected, all r 2 0.
-
If this were true, we would have:
5
Conjecture 2. If 1 d i m X + 5, 1 then 2
and
is a sub-bundle of T X
is homotopic to an integrable sub-bundle of X.
For, if v denotes the complementary buadle to
v
:
fiber dim. f
X -,BGt(n, R) (n = fiber dim. V)
in T X ,
then
can be lifted to a mapping into
B rz by standard obstruction theory and the assertion in Conjecture 1. The main reason for making Conjecture 1 is that it is known that
BT:
is not (2n+l)-connected, if r _> 2. On the positive side it is
known that
BF:
is (~1)-connected, at least if r
#
ntl.
The proof of
this result relies on Thurston's theorem, which we will prove. fact that
~i;:
From the
is (n+l)-connected and the classifSlng theorem, it follows
t h a t w e r y plane f i e l d is hanotopic t o an integrable plane field.
On t h e other hand, conjecture 2 does not imply conjecture 1. 95.
BZ.
Model f o r Let
G be a topological group.
contractible, i.e.,
We w i l l suppose t h a t
t h a t . t h e r e i s a neighborhood
W
of
is locally
G
1 fn G which i s
contractible t o 1 i n G. We l e t G6
denote t h e under*
d i s c r e t e group, and s e t
-G = f i b e r (0'
G),
where "fibern means homotopy theoretic fiber. the path-space
PG
Then
5=
maps i n t o G by the endpoint map.
6 G
xG PG
Thus,
-G
where is a
topological group, and we have a hcmotopy theoretic f i b r a t i o n
BE + B G ~ + B G . Now we replace G with the geametric r e a l i z a t i o n of i t s singular complex
l ~ i n gG
1.
ThIsCis a group object i n the category of compactly genera-
ted topological spaces, and G8 of
l ~ i n gG
1.
i s a subgroup, consisting of t h e v e r t i c e s
Then we have a well lmown f i b r a t i o n
GI
Since B , I ~ i n g
ha8 t h e homotopy type of
B G,
comparison with the previous
homotopy theoretic f i b r a t i o n shows we have a homotop equivalence BE =
1s-
o1/G6
The rig& side provides a simplicial model f o r
$6.
B5
.
Thurston's theorem.
if< M
If
M
is a smooth manifold, and K
denote the group o f
C'
i s a compact subset of M, l e t
diffeomorphisms of M
which have r u ~ p o r t
in
if< M
Let
K.
support.
denote the group of diffeomorphisms of
Clearly ~ i f Mc
f
1 9 K
if$ My
where the l i m i t i s taken over a l l compact subsets of
if< M
M with compact
with t h e Let
T
M
cr
if< M
topology and
M.
We provide
with t h e d i r e c t l h u i t topology.
M and also its classifying
denote the tangent bundle of
mapping T*
:
M
+
B ~ & ( n R). ,
-
By replacing t h e spaces involved with homotopy equivalent spaces, we m a y
assume, without l o s s of generality, t h a t t h e mapping V : B rz i s a Serre fibration.
Thus, we obtain a Serre f i b r a t i o n
Theorem (Thurston).
Suppose B
a~
=
.
it
r
M
V,
B G.f,(n, Xt )
with base M.
There e x i s t s a mapping
M +rc(~>)
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 -< ... <_ u9 5
O<
5
Clearly t h i s homotom begins a t t h e identity. Aq be a face inclusion.
It i s e a s i l y cbecked t h a t
the following diagram comutes:
Consequently, i f
... -<
1 1, so H
K i s a simplicia1 complex, we may define a homotopy
H :M x K x I
-
M x K by defining it as above on each simplex.
Let v(A~) be the set of (tl, integer, 1 5 i 5 q.
Let V(K)
..., tQ)
t
flq such that Nti is an
denote the union of all v(aq),
union is taken over all simplices of K.
where the
Let I, = H~(Mx V(K) ) C M x K.
To summarize, we have associated to functions
2, ..., AN
on M,
a homotnpy H, and a subset L of M x K, provided hi _> 0, C Xi
It is easily seen that if the hi's
<_ 1.
are-piecewise linear with respect
to scane triangulation of M, then L is an n-dimensional subcomplex of
$ 12. Proof of the First Deformation lemma. We consider continuous mappings f, g which make the following diagram commute:
Here K is a finite simplicia1 canplex and KO is a subcomplex. Both of the vertical arrows are the inclusion mappings.
We will construct homotopies
F, G which make the following diagram caanrmte:
diag. 12.1
such that Fo = f, Go = g, and G1 : K
4
rc
factors through
rc.
This will
be enough to prove the first defonnation lennna. For t
t
K, x
E
M, we write g(t,x)
for g(t)(x),
so we may think
of
g as a mapping of
of the s e t of
x e. M
i s easily seen t h a t
K
x M into
such t h a t supp g
The hanotopies
We l e t
E.
g(t,x) f *(x)
supp g denote the closure f o r a t l e a s t one t a K..
i s compact.
F and G w i l l be constructed i n three stages.
*
f i r s t i s based on t h e assumption t h a t the base section
i s good.
means t h a t there i s a f i b e r preserving homotopy h : E 4 E ho =
ia
It
The
This
such t h a t
.
$'(*(M)) i s a neighborhood of *(MI, and ht(*(n)) = *(HI, . f o r 1111 t e I. We l e t F ; ( ~ ) ( x ) = h 8 ( f ( t ) ( x ) ) ~ i ( t ) ( x = ) h8(g(t)(x)), teK,
X E M ,
S E I .
It i s e a s i l y seen t h a t i f
t
in K
such t h a t i f
t e K, then there i s a neighborhood U of
t *l U then supp ~ i ( t)' C supp f ( t ) .
But
supp f ( t )
i s i n the union of a t most n b a l l s of radius
~ 2 - " for
some n. It follows t h a t f o r a suitable subdivision of
%
has the property t h a t
supp
most n open b a l l s of radius
E 2 'n
31
(~il U)
K,
each simplex
i s contained i n t h e union a t
f o r some n.
We provide
K with t h i s
subdivision from now on. We l e t
q = dim K.
We triangulate M and we l e t
piecewise l i n e a r f u k t i o n s on M a neighborhood of
supp g in M,
such t h a t each
hi
and each supp hi
radius ~ 2 ' ~ W . elet H : M x K x I
-+
M x K
_>
0
1,...,
be
and C hi = 1 i n
i s I n an open b a l l of
and L C M x K b e t h e
homotopy and subcomplex defined in t h e previous section i n terns of
1,
* - - 9
$ Since dim L = dim M. = n and the f i b e r of n : E *M
tea, there i s a homotopy G" : M X K ~ " ( x ,t, s ) = *(x)
if
X
g ( x , t ) = *(x),
is
n-connec-
I d E such t h a t G: = G i , f o r x e M, t e K, s e I,
and
G;(L) C *(H), where
*
denotes the base section, and the following diagram
commutes, f o r all t E I: M X K
+
The proof of the existence of
G: -
E
a
i s not d i f f i c u l t , and w i l l be carried out
G"
i n the next section. We l e t
G ':
= G;
We claim t h a t
"
We l e t
I-$.
has t h e required properties.
G
there e x i s t s
F
such t h a t diagram 12.1 colmnutes.
there e x i s t s
N
such t h a t
G(K0
O
G ~ ( K ~ ) C for I ' ~ , ~ IJ
be a simplex of
follows t h a t all
t
E
I.
supp (G;
n
M
- H&(L
IU)
and
It i s enough t o show t h a t
It i s obvious t h a t
by the construction of supp
supp (G;'
( ~I ri)
€ 2 'n
I r)
G' and
-
G1
: K
Let
G".
i s contained i n
f o r some n,
it
liave t h e same property, f o r
This completes t h e proof of the existence of
-* rc f a c t o r s through
M x K has t h e property t h a t f o r each
t
E K,
F.
' rE'
But
we have t h a t
n M x t ) i s in t h e union of a t most q open b a l l s of radius
€ 2 " where q = dim K. we have
rEYr
open b a l l s of radius
Second, we show t h a t :M x K
I) C
From the f a c t t h a t
I$,.
the union of a t most
X
F i r s t , we show t h a t
G"'
1st
Since G ~ ( L=)
has support in a t most
*,
we obtain t h a t f o r each t E K,
q open b a l l s of radius
E2' q.
This completes the proof of the f i r s t deformation lemma, except for
the proof of the existence of G", which will be carried out in the next section.
Q 13. Existence of G". We will prwe the existence of G", which was used in the previous section. Let
R :M X
as a section of R*E Gi (x,t) = *(x))
K
I -M
X
We may regard G;
denote the projection.
aver M x K .x 0. We have that {(x,t)
is a neighborhood of {(x,t)
E
aM x K :
M x K : g(x,t) = *(x)),
by
the construction of G1. Hence there is a polyhedron V such that {(x,t) e M
-
Let
K: g(x,t)
X
= *(x))
be a section of R*E
C
r M
X
K: Gi(x,t) = *(x)}.
w e r (M x K x 0) U (V x I) defined by
the fiber is n-connected and dim L
*E
{(x,t)
if s = 0, and qx,t,s) = *;(x) if (x,t) E V.
~(x,t,s)= Gi(x,t)
R
vC
< n,
there exists a section
w e r (M x K x 0) U (V x I) u (L x I) extending
S(x,t,l) = *(x)
if (x,t)
E
L. Finally,
8
Since of
such that
extends to a section G",
having all the required properties. 1
Second Deformation Lemma. Let M be a smooth n-manifold, X a topological space, and w a
structure on X XxM
-r
X.
X
<-
M, transverse to the fibers of the projection
We suppose n = dim M.
We will say w is horizontal if w is
the.inverse image of the differentiable structure on X x M under the projection X x M +M.
If t r X, x r M y we will say w is locally
horizontal at (t,x) if there is an open neighborhood U x N of (t,x) in
X x M such that over Y, -
w
1U x
IV is horizontal.
If Y C X, the support of w
denoted supp w, will mean the closure in M of the set of Y x E M for which there is at least one t E Y such that w 1 Y x M not
locally horizontal at (t,x).
suppy.w C M and w 1 Y x (M-sup% w)
Thus,
is horizontal. Now we suppose that M
is prodded with a Riemannian metric satisf'y-
ing the conditions stated at the beginning of Q 10. Let K be a finite semi-simplicial complex, without degeneracies. Let
w
be a
projection
<
structure on
1141 x M 141. +
Ildl X My
transverse to the fiber of the
We w i l l say w is
&
- regular (for the given
triangulation of K) if for each simplex a in K, there exists n
2
such that s u m w is contained in at most n open balls of radius
E2 'n.
LenmLs.Let acies, and let
.%
K be a finite semi-simplicial complex, without degener-
-
let
be a subcomplex.
w be a
transverse to the fiber of the projdction
(I[o(I
supqldl w is compact. Suppose w 1
xM
transverse to the fiber of the projection
n I c(IKJI
M
18
&
-C
IIdI.
/Id1x My
Suppose
- regular and
r 2 1.
1141 X
and a <-structure. a
x 0 is the given triangulati&,
w,
X
rz-structure on
x I,, whose restriction to
Then there d s t s a triangulation of
a 1 lldl x o =
1
x I) U
1141 x I x M 1141 x +
(lldl x 111
xM
2
E-
I X My
Iy such that regular, with
respect to the given triangulation. Thus, the laau provides a h-tom
and an g - regular one, rel.
%,
between the given $-structure
provided r
2 1.
This 1em.s will be proved in the next section. § 15.
Proof of the second deformation lemma. First, we choose a family
such that AS:
0, C .A
1 1,
'1, ..., \
C hi =
of smooth functions on M y
and supp Ai is contained-inan dpen ball of radius 2-&, This is possible, since su
Pglrdl "
Sl dl lldl.
1 in a neighborhood of sup
q = dim
is compact by assumption.
Next, we choose a piecewise linear subdivision
L of K.
We tri-
lldl x
@ate
I so that for each simplex
complex, so that
IIKII
x [0,
1
lldl
and
x
of K, 1
IF,
u
X
I is a
sub-
11 are subcomplexes, so that
1
the triangulation of x 5 is L, so that for each simplex U of L, 1 x [?, 11 is a subcomplex, and finally so that for each cell u of N-th rectilinear subdivision of L, we have that u x 1 is a subcomplex. The N-th rectilinear subdivision is defined as follows. Let A"
{(tl,
..., tI)
For any integers 0
: 0
5
5 tl 5 ... 5 tI 5 11
il
5
... 5 iI 2
...,tI) E:'A
ui = {(tl,
I?-1,
be the standard 1-simplex.
let
i < M t < i + 1 for ,j = l,...,~}
3-
3-
3
th These ui and their faces are defined to be the cells of the EFI rectilinear subdivision of A This subdivision is compatible with face th inclusions, so we have a notion of the I?rectilinear subdivision of the
.
fat realization of any semi-simplicial complex without degeneracies. If ' A M x' A
e denotes the standard I-simplex, we let H : M x A x I
denote the homotopy defined in § 11, and depending on
-
5, ...,AN'
Since this homotopy is compatible with face inclusions, it defines a homotopy H:M
X
(141X I'M
X
(IdI.
1141 = Ildl.
Since L is a subdivision of K, we have homotopy
3 : M x lldl x I
-+
M
X
We define a
Illdl by
%(xY~) ' XY
x EM
X
IIdI,
1
0S t 5 2
= ~(~,2t-1)
We let R =
%* w .
We assert that the triangulation and
Q
have the required properties,
provided the subdivision L of K is chosen sufficiently fine. If 'a
is an I simplex of I, and t : ' A
standard map, we define wu = (t
X
* idM) o.
4
' U
denotes the
We may arrange for wu to be
as nearly horizontal as we wish (independently of c) by taking L to be
Since the following diagram commutes:
a fine enough subdivision.
diag. 15.1
it follows t h a t i f
wu is sufficiently near t o being horizontal then
H IM x t x s i s transverse t o w,
f o r each t E A
If
i s a simplex of
U
then H~(Mx
K,
diag. 15.1 commutes, it follows t h a t
n 1 IIKJI
,
lldl
i s transverse t o the f i b e r of the projection
Q
I
.Q
s E I.
x IxM
-C
I n t h i s case,
lid1 x
x I) C M x u.
suppaWCQ C s u p p e
I.
Since
Hence,
x I i s E-re@ar.
From the definition of H, it follows e a s i l y t h a t i f . u i s any c e l l of t h e q-t h rectilinear subdivision of L, then supp Q C supp hi(llU.. .U supp hi ( llfor same il, open ball of radius
E
...,iI
2q '
and
0x1
with X 1
-
1 < q.
(Id(x Q 1 (Id( x 1
i s a subcomplex of
respect t o the given triangulation, it follows t h a t E
i s i n an
Since each supp hi
1 with
is
- regular.
§ 16.
Spme of Sections.
*
We will pruvide a model f o r the space re(% v ) Throughout t h i s section,
introduced i n § 6.
M w i l l be a smooth n-dimensional manifold.
F i r s t , we consider the semi-simplicial complex U wfiose q-cells are gems a t
x d i e . M of
fibers of the projection
nl
:
M
X
&d(Aq)x
-
x M x M transverse t o the
on
: fq x M x
M
nq x M. Here
denotes t h e projection on the f i r s t factor.
M -M
Lema 1. -
Proof. -
<-structures
*
U i s a model f o r r ( r M v ) .
It i s easily seen t h a t
* v)
r(r
Y
represents the functor which
assigns to a finite simplicial complex K the set of homotopy classes of gems at K x diag. M of ri-structures on K fibers of the projection
id(^) x
:
K
X
M
X
X
M
X
M transverse to the
M
-+
K
X
M.
The fact that
also represents this fbctor may be proved by the method of §
Iu~
6.
Let uq denote a q-cell in this complex. We w i l l say that uq is horizontal at x E M
if there is a neighborhood V of x in M
the restriction of dl to projection
X
x V x V
such that
is the ri-structure given by the
V x V + V, on the last factor. By the support of cqy we
mean the set of x E My where
a
is not horizontal. Note that supp uq
is a closed subset. Let Uc be the semi-simplicial subcomplex of U consisting of cells having cmpact support.
*
Lemma 2. Uc is a model for rC(rM" ) Proof. This is similar to the proof of Lemma 1. -
The functor which
is represented by the two spaces assigns to a finite simplicia1 complex K the set of homotopy classes of germ at K x diag. M
of c-structures on
K x M x M transverse to the fibers of the projection
id(^) x nl
:
K
X
M
X
M + K x M, and horizo~taloutside a compact subset of M. Let UE denote the semi-simplicial complex consisting of cells having support in at most q open balls of radius ~ Lemma 3. Proof. -
'for ~ some q
2
_> 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
<-structures
projection
We define a subcomplex
-
the given q-cell, has support in Nl
we l e t
of
<-structures.
as follows. ' A q-cell of
'
E
the support of such a q-cell we mean the union o f t h e supports of
all the constituent
-
2'N
5,w ~ill be
an N-tuple
-
of germs a t Aq
x diag.
M
on Aq x M x My each one trasverse t o the f i b e r s of the
q
id(Aq) x
: Aq x M
xM
.
x My
and ha*g
support in a
E gN
b a l l of radius
By the support of such a q-cell, we mean the union of the supports of the constituent germs of of
-]kYN,as follows.
2-Nl E
A q-cell of.
We define a subcomplex w i l l be a q-cell of
ATM,N
A?
if
MYN
balls of radius
. = . S y N/ ATMyA'
We l e t &,N
-a
W e have a mapping product of
f :
]k
we get a mapping on
-]k,*
the given q-cell bas support in Al
< N,
f o r some N1
<-structures.
]kYNand
N
T
$,,
Tn
~ k , n + b , a obtained
by taking the Cartesian
with i t s e l f and r e s t r i c t i n g .Passing t o the qudiient ,
XMYN
,a,N.Moreover, the symmetric group
N
N
:
-
, :
+
nN 'acts
by permutation of the' cmponents of an A-tuple. This
action leaves the base q-cell fixed, and i s f r e e on t h e remaining q-cells. W e have obvious identifications
Moreover, R
commutes with
FN,
syN InN agrees with the mapping N
rr
inaucea by f :
and the induced mapping
N: % , N / ~ N
+
fN : % , N / % , ~ ~ . + % , N ~ ~ , I u - ~
l k , +~ Xl,f
Definition.
Let
u :A
-+
B be a continuous-mapping (or a semi-
simplicia1 mapping of semi-simplicial sets).
We say u is
j-acyclic
if
i <_ j,
H~(M(u),A) = 0 for
where ~ ( u )denotes t h e mappix cyc1indt:r of
u. Lama. If f~
FN :
ji(,N*%,N
j-acyclic then so i s
: %,N/%,N-1 %,N/%,N-1-
FTOO~. Let -
cone.
Clearly,
freely on
c
C(FN) = M ( Y ~/)
n
a c t s on
N
-*
*
( ~ ( f)), P+q N immediately.
H
20.
(%
U (* x I)) denote t h e mapping
Y
c(H ),
leaving the base point fixed and a c t i n g
N
c
Clearly,
i s a s p e c t r a l sequence with
5
is -
/ TN =
= (n (c(%))) converging t o P Y ~P N Y 9 denotes reduced homology. The l e m a follows
where
Construction of
n
E~
A
%,H'
%,N'
=*
We wish t o compare t h e homology of and the homology of
It follows t h a t there
( f ) .
N
%,a
x i t h t h a t of
" %,N 5ln B
with t h a t of
.
B
Dlf< b
To do t h i s we must con-
s t r u c t some more auxilary semi-simplicial sets. Recall t h a t
i s chosen so t h a t b a l l s of radius
&
geodesically convex.
u = (uly E2
-"-admissible
h
s e t s such t h a t
A
ij'
be a q-cell of exists
MI
Consider a q-cell (u, B
ij
A
A
< N and
Lenrma. -
)
of
(res~.A % , ~ ) h
)k
,N
(resp. A Y M , ~ i)f f o r each j = 0, N1
open b a l l s of radius
The natural mappings
(B ) 13
~
n
(resp. g Y N ) . This w i l l
h
A $,M
matrix
supp ul C B
We w i l l ale0 define a subcomplex A$ ,N (resp. s , ~ ) .
are
and it i s geodescially convex.
SyN (resp. w i l l consist of a q - c e l l ..., %) of 3,. (resp. k Y N ) and an N x (q + 1 )
A q-cell of
of
E,
in M
i s E-admissible i f it
We w i l l say a subset B of M
i s open, it i s i n at1 open b a l l of radius
E
2
..., q,
there
~ union ~ contains whose
ls,Nl,lagyN 1 la&yN
l$,Nl
1
are all homotopy equivalences. Proof. -
It i s enough t o show t h a t inverses images of points are Let x be i n the i n t e r i o r of a q-cell
contractible. Let e 2-"
... BN )
S denote the s e t of all N-tuples ( B ~ , admissible and
supp
Ui
C Bi
= (cl,
such that
... IYN). Bi
is
(1n the case of the second and fourth
mapping, ye add the .condition ,that there e x i s t N of radius E 2N '1 whose union contains U
.
N, A
< N and Let
)
j.=l
semi-simplicial s e t whose p-cells are mappings of
A
S* denote the
...,
(0,
open balls
N,
The following sub-lama clearly implies .the lemma, since
into
S.
1 ~ * 1 ~ +i s~
contractible. Sublemma.
Proof.
.
The inverse image of
Let
?T :
x
2 1s*lq+'.
1 ~ +1 I B ~ denote t h e mapping i n question.
a
Let
denote the semi-simplicial s e t whose n-cells are the weakly order preserving mappings
[o,n]
t o the q-cell 7
: LX
s%*'
-, [0,q]. (J
..., q.
verified t h a t morphism int
:
i n the standard way.
+E
be given by
a+ B
denote the mapping associated
Note t h a t
Then
T
Let
..., Vq) = (~(cp),@~,. .., qq),
~(cp,@~,
-
x IS* lq+l =
r :
aq x ls*lq+l
I p x S+~+'I A
if
h
IX x YI
A
(E (
induces a homo-
?T-jint 1 ~ 1 ) . Here we have used Milnorls
=
1x1 x 1 ~ 1i f
semi-simplicial s e t s (cf. 163).
-
T o ~ o l o gof~ X D , ~ , Y D Y l p ~ etc. W e let D
aq.
i s a semi-simplicial mapping, and it i s easily
theorem which asserts t h a t
1
=
i s weakly order preseming, and $i : [o,n] -, S f o r
cp : [o,n] -, [0,q] i = 0,
We l e t
denote the unit disk,' D".
X
and Y
are
lifts f : k,N-s,N
The mapping A
defined by fN(ui,
Bid) = (TN(ui), Bij).
For any topological space X, sion of
t
€
let
xn
X.
point i n
A
and
snX
We have
X.
X €
representations a s f o r some i f j.
2N-tuples
(t,x) =
Lemma. -
*
denote t h e
X
if t =
(snxlN (tl,
For 19 = 1, l e t
the p a i r of topological spaces
we l e t
can be represented as
denote t h e subset of
v#
a
naturally t o
or
suspen-
(t,x), where x =
*.
For N > 1,
consisting of points having
..., tN,xN)
Xl,
plpX
*
n-t h
*.
5
Let
such t h a t
=
aenote
V#
v).
((snx)',
ti
There e x i s t s a homotopy commutative diagram
-
where the r i g h t v e r t i c a l arrow i s the arrow functorially associated t o t h e
-r Pn
mapping B Dif f
SLn B 7 defined i n
3
n
17,
and the horizontal arrows
induce isomorphisms i n i n t e g r a l homology. h -r n N - ( s n s ~ i f f C a) 'D,N r n -wNBDiffc PI both induce isomorphism i n i n t e g r a l homology, and
The l a s t phrase means t h a t t h e mappings
ma
similarly f o r t h e lower horizontal arrow. The proof w i l l be carried out i n
$3
22-25.
From t h e definitions, it follows t h a t we have equality of semisimplicia1 sets: A
I k , l ( ~2-N) if
(c, BO,
(rather than
and
= $,1(~ 2-*)
..., B9)
(€2
-N)N
and
CPN- t , 1 ( ~ 2 - N ) 1 9 . Here
are defined l i k e
i s a q-cell we require t h a t
h
)k,l Bi
A
and YM,l be
except t h a t
~ 2 - N admissible-
~-ar3-xissible). Obviously t h e problerc of describing the topolo-
gy of
<
l ( ~ 2 - N ) i s the same a s t h a t of -describing the topology of
we have only t o replace
by E 2",
E
To describe the topology of
4,I( ~ 2 - ~ and )
L be a sequence
..., BQ)
(B0,
of
~
2 - 9 , we intro-
$
9,
as follows.
duce three s e m i - s h p l i c i a l s e t s J, K, and L, of
%,I ;
throughout.
2
We l e t a q-ceU
~adrmssible ' s e t s i n M.
Such
...
B n 0 q " i s non-void (resp. has non-Void intersection with a ~ ) .
a q - c e l l w i l l be a q-cell of
K (resp. J )
It is e a s i l y seen t h a t
if
L i s contractible.
Moreover,
K has t h e
since it i s the nerve of a Coveri'ng of
hcanotopy: type of My
M,
with t h e
pr'operby t h a t the intersection of any f d l y of s e t s In the cover is contracti b l e o r void.
Let
Similarly,
(US Bo,
a~ .
has the hamotopy type of
J
..., Bq )
be a q-cell of
,
..., B9 )
The mapping which associates
(B~,
(nsp.
to
2-'))
...; B9 )
(o, Bo,
-IJ
n
is a
'
semi-shp1iciP-l napping of semi-simplicial s e t s S , ~ ( E ~) -+L (resp.
Let
(B0,
..., B9)
denote a q-cell of
point of t h i s c e l l i s
f i b e r is.one point.
..., BP
Y ~ n.. o J,
then
Bo
n
but not i n J,
. nBq Bo fl
i s h a o t o p y equivalent t o
y ~ n o
B
... n Bq
a~
meets
-
... n B~
ism i n homology whenever
Y ~ on..
and
5o,,... ..n
Bi 6
..., B9 )
(B~,
5
n end
E 2-'
(resp.
(resp.
% , L 1 XBO
(B0,
b a l l s , we have
5). I f t h e c e l l i s i n
, ... nBq
....
then t h e
then since
=<xn
i s homotopically equivalent t o SZnB
There a r e
... n Bq = 9,
are geodescially convex and contained i n
y ) i s contractible t o a' point. B4. Notice t h a t t h e inclusion (resp.
... fl Bq )
n
B 'O
i.e.,
I f the c e l l i s i n K,
50".. n 89
that
... n B~ (resp.
I f the c e l l i s not i n K,
three cases.
B ,
n
%e. f i b e r over an i n t e r i o r
L.
) Bq
n
B '0
n
... f l
... ,-,Bi f l ... "q '
n
induces an isomorph-
..., Biy ...., Eq ) n
ax.e
.
both i n
and not i n J.
K
-
n
Thus, we see t h a t the semi-simplicial mapping (reap.
2-3
§ 23.
I)
Construction of
5%
We l e t identifications
S%
-
(t,xl )
if
*
x = i
*
The mapping
and
(tl,
(6%lN
3%
-+
2%
induces a mapping
-
S%
il
f o r some i
5, ..., t 5, ..., tj, xi, ..., tN,%),
X
(t,
"Y$[
in
vp
+
If
X.
s"X
X
i s to
-
i
-Y$[).
*
for
For, i f
I$$.
( s % ) ~ i s the same as
v$
CW ccnnplex, t h e mapping
2M-tuples
or x =
j,
whatever
ti = t3'
is a
#
tj
5
in
i s a homotopy
+
equivalence, since the f i b e r s are all products of copies of
0 2 = ( (s?)',
E
consisting of
ti = t
particular, we can talre Again, i f
x, x'
3% has t h e same h w t o p y
such t h a t
then t h e image of
t h e image of
CW- complex),
denote the subset of
4) ..., tN,%)
some i.
t e a D n = Sn-'
modulo the
t o a point.
We l e t (tl,
t E Dn, x e X,
(t,x),
since t h e e f f e c t of t h e natural mapping
identify D~ x
(R,J).
v -2.
i s a reasonable s p a c e (e.g. a type as
-
2-') Y
i s a r e l a t i v e homology f i b r a t i o n over
be the s e t of pairs
(t,x)
]k
Dn.
We l e t
We have a hcmotopy equivalence of pairs
V$.
Construction of
§ 24.
In
*
v X. N
$22, we associated three complexes L, K, and J
In t h i s section, we take M = D
n
.
We l e t
sn*x =
t o a manifold
(151 x *) U
IKI
X
M.
X
( t , x ) = ( t , x 9 ) i f t 6 1 ~ 1 , x, x v e ' X. Since 1are contractible, Sn* X has the same homotopy type as
modulo t h e identifications
IL~
and
'
1 ~
1 ~ x1 X
modulo t h e same identifications.
type of
D"
end
IJ 1
the homotopy type of
But since Sn-l
,
1 ~ 1has
t h e homotow
it follows t h a t
W e let1
n :
Let
sn*x
GX denote the svbset of 4
~ L I denote
the projection.
L~ consisting of a l l simplices for
i = 0,
..., q,
2 - N l ~ such that
U
*
W e let
=
n*
((s
ij
(sn*x
IJ
(sn*x x
N X)
Lemma 1. If -
CW
< B and
j=l,
..., N
such t h a t
open balls of radius
Nl
n* ( I L ~X * X S X
We define
... sn*X) I L ~ x * x ... x sn*x, u ...
sn*x x
X
X
... x . 1 ~ x1 *>.
cccrmplex we have homotopy equivalence of
pairs:
-
*
VNX
Proof. -
Let T be the subccmplex of
..., q;
i=o,
, C*L I ~ X , ) . is
X
x
defined as follows.
l i e s i n t h e i r union.
/TI u
u
)
Bl
... U BiN
= ($)-I
%X
* vNx
(B
there exists
Bil
(sn*x)"
" v#.
V#
W e have constructed the l a s t homotogy equivalence in the
previous section. W e triangulate
D~
with a very fine triaagulation.
we associate the open b a l l of radius
~2-',
To each vertex,
centered a t t h a t vertex.
The
balls centered a t the vertices of a simplex form the vertices of a simplex of
K,
i f t h e original triangulation was f i n e enough.
simplicial mapping 1 : D~ i x id :
D%
seen t h a t
-c
K.
Note t h a t
i(sn")
Thus, we have a
C J. . Thus, the mapping
x -,K x X induces a mapping T :'3% -,sn*x. * T'I~) C %X. We w i l l show t h a t
-
P : ((s%)", i s a homotopy equivalence of pairs. already done that
yN :
n*
El
((s X)
9
It is easily
*
PBX)
I n fact, it i s obvious from what we have
( s ~ x),~(S"*X)~ i s a homotogy equivalence.
For the proof t h a t
xN :
introduce the following spaces.
-
*
P X
N
i s a homotogy eguivalewb, we
For any ~ a z l t i t i o n p = PI,
..., '~1 of
..., N),
(1,
simplices
we l e t
T be the subcomplex of P
..., q ; $=I, ..., N
(~ij)i=o,
condition t h a t there e x i s t N b a l l s 1 that
Bij C
B;,
j e Pk.
if
We l e t
..., B1N 1 of radius
2-IPlg
S W ~
where the union i s taken
T1 = U Tp,
elements.
N
8.
It i s e a s i l y seen that i f we substitute T 0 * * i n the formula defining %X , we get the same space. Let uNXl
We l e t for T
To = T
il
be the space obtained by substituting
*
Clearly TICTOy
PNX.
*
Sublemma. Proof.
i n Dn
B1 U
i n the formula defining
*
Nl
*
P X.
N
(Dn)"
consisting of those open b a l l s of radius
N1
l i e s i n t h e i r union.
defined as follows.
No
Let
A s e t i n Uo
open b a l l s of radius
It i s e a s i l y seen t h a t
Uo
w i l l be
~ 2 . ~ whose 1
i s an open cover
and any intersection of a f i n i t e n h b e r of members Of
No,
UO be
i s an ~ 2 - ~ - a d m i s s i b l subset e
Bi
and Nl
N1 C N
... U %
of
< N an8
... x BAY where each
and there e x i s t s
union contains
T
CYpX."
..., tN
t
the collection of subsets of
of' D"
1
be the open subset
No
such t h a t
of the form Bl x
for
T
i s a deformation r e t r a c t of
pNXl
Let
*
so $X1
..., tN such t h a t there e x i s t s
2-q s
of
and satisfy
which are i n
B;,
over all p a r t i t i o n s having fewer than
tl,
KN consisting of a l l
U
0
is
contractible o r void. Given a p a r t i t i o n
p = {ply
..., PN,
of
.. , N we define (tl, ..., t N ) such t h a t
1
-L
Np
t o be the open subset of
there e x i s t N l
t. E B1 i f
open b a l l s
(Dn)'
B;,
consisting of
..., B1N1 of radius
E 2-N1
such t h a t
P Let U be the collection of subsets of N defined 3' P P as follows. A s e t i n U w i l l be of the form Bl x X BN, where each P Bi . i s an C2-N admissible subset of D' and there e x i s t Nl open b a l l s 1
B y
3
i
E
..., BN1 of
easily seen t h a t
...
radius U P
E 2-N1
such t h a t
U
P
j
i c P
j'
It i s
and any intersection of a P' i s contractible o r void.
i s an open cover of
f i n i t e number of members of
Bi C Bv i f N
Since N = U N To i s the nerve of Uo, Tp i s the nerve of U 0 P' P P and T1 = u Tp, it follows t h a t T and T1 both have the hmotopy type 0 P of N~ ; i n particular T1 i s a deformation r e t r a c t of To. Applying t h e same argument, and using t h e f a c t t h a t any intersection of members of or
P
x Dn,
Dn X a D n X
...
X
...,
D ~ ,
t h e deformation retraction of J
K
X
0 a D n x Dn x
has contractible br void intersection with the s e t s
U
X
X
K, K
X
J
X
K,
*
P X
H
into
can be chosen so as t o preserve
.-..
Such a' deformation r e t r a c t i o n of retraction of
...
and a l l t h e i r intersections, we see t h a t
To i n t o T1 X
U
'TO
induces a deformation
i n t o TI
*
wNX1.
Proof of Lemma 1 (continued). For any p a r t i t i o n A
P that
denote t h e subset of
of
6
Pk f o r some k = 1,
denote the projection.
The mapping
The union of the spaces appearing on the l e f t is spaces appearing on the r i g h t i s
rN
that
..., IT},we l e t
G,
..., t N )
(D~)* e o n s b t i n g of N-tupa (tly
ti = t j whenever i, j
S"X +lIn
p :
..., P"1)
p = (P~,
* %XI.
..., HI.
such
Let
?-induces
the union of t h e
?$,
Therefore, it i s ehough t o show
induces a homotopy equivalence between any intersection of t h e
spaces occuring on the l e f t and the corresponding intersection of spaces on the r i g h t , i n order t o complete t h e proof of Lemma 1. The f a c t t h a t any intersection of t h e spaces.
,S.,~
'SnX x..
.X Dn
m...X
Xmaps v i a a hoaotopy equivalence t o the corresponding intersection of
spaces on the right side i s obvious from the f a c t t h a t the mapping
snx sn*x 4
Let
i s a homotopy equivalence. p = (pl,
..., %)
be any sequepce of p a r t i t i o n s of (1,
..., NI.
n ... n A
~ e tA = A
1t is clear that A = A
where p is the P P partition p1 ~ e tT ~ = Tn . . . n ~ ~ ~N, ~ - N ~ ~ . . . ~ N Pl Pky UP = U n nU It is easily seen that T is the nerve of the Pl %' P cover U of N that N is a convex neighborhood of- A and that the P P' P P intersection of any finite family of members of U is contractible or void. P For any p as above and any subset A of (1, , N) we define to be the intersection of (pN )- 1(A l t l -A ) and all the EA, 9 % Znx x.. x Dn xW.. .x fi where the - D~ x * appears in the i-th position
%'
='l.
p
... .
...
...
...
.
with i N -1 (T )
E
(IT
A.
Likewise, we define F
n...n
A, P
1)
T
and all sn.*x
PIS
to 'be-the. intersection of
...x 1 ~ 1 ...xsn*x, X*
where L X *
th appears in the i-- position.with i e A. To finish the proof of Lemma 1, it is enbugh to show that
is always a homotopy equivalence, where we can assume that p is a non-empty sequence of partitions. For any subset B of {I.,
..., N),
we let A denote the set of P,B it,, , tN} in AP such that ti E aDn whenever i e B. We let TPYB be the subcomplex of T consisting of all simplices (B. ) P i=o,. ,q ; 3 4 , in, Tp such that N BOj n aDn f @, if j E B.
...
u
...,
". . "B
sj
We let M be the nkber of elements 'in (1, identie
..., N)
..
not in A.
We
]r with the subset of $ consisting of (5,..., %) such
that x =' * .if i e A. i The space E is A x ?? with the identifications (t, %, ,%) A,P P (t, , if t E A and xi = x1 whenever i f B. Likewise PYB i the space is X with the identifications (t, 5, %) F ~P, (t, si) if t E 1 and xi = x il whenever i f B . PYB
-
...
5, ... 5)
IT,~
5, ...,
9
IT
...,
-
It follows t h a t , i n order t o prove t h a t t h e mapping
yN .
E
~ - C, F ~ , p
i s a homotopy equivalence, it i s enough t o show t h a t
inducei a homotopy equivalence, f o r each subset
B
of
(1,
..., N).
...,
W e let
B be t h e s e t of (tl, t N ) i n Np such t h a t PYB ti r if i B clearly N i s a neighborhood of A and it i s PYB P,B e a s i l y seen t h a t there i s a r e t r a c t i o n of N to A Since U is a P,B P,B' P cover of N it i s a l s o a cover of N and T i s the nerve of U P PYB PYB P when it i s considered a s a cover of N i , e . when a p simplex i s a PIB' sequence of p elements i n U whose i n t e r s e c t i o n i n t e r s e c t s B It P P,B' follows t h a t iB: A i s a homotopy equivalence. This completes PYB t h e 'proof of Lemma 1. We will also need: Lemma 2. The inclusion of (nB) -1 2 %X* i s & homotopy
son
- IT^,^
I
IT[
-
equivalence. Proof. -
We l e t Zi
where
1 ~ x1 *
-
(='I-1
I L ~XIX...X
IT1 fl (sn*x x...x
occurs i n t h e
1% place.
We have
where U means union over a l l p a r t i t i o n s with fewer than P. proof of the sublemma shows t h a t
u
($I-' P i s a homotopy equivalence.
IT,~
U 2,
u...u
sn*x),
-2 ,
R elements.
The
( 4 1 - l It1
We consider the mappings
I f we can show t h a t these mappings, and all mappings 'induced on intersections are homotopy equivalences, it w i l l then follow t h a t
u...u zi u..
u (2)-I
. L)
D
is a homotopy equivalence, and then the Lemma will follow immediately. N -1 For an intersection which involves one of the (n ) 1 ~ ~ 1the , corresponding mapping is the identity. Therefore it is enough to consider the mapping of an intersection of the Zi into the corresponding intersection of n* *X. the S X x...x xW...X n* We will write Yi = S X x...x xh(...x sn*x where (I)x * th appears in the iposition. If A B (1, , N) , we write 'A = A:i
IL~
~LI
...
Y = nY A PA i' diagram:
..., N) -
(4, ..., %)
We consider the commutative
where (xi)isAc A, and M denotes the nmber of elements in AC.
Here the right arrow maps AC = (1,
IT 1.
Clearly, ZA = YA I7 l')?(
into
To complete the proof of Lemma 2, it is enough to show that
z A C YA vertical
is a homotopy equivalence.
~~~~~~~are homotopy
-
We will do this by showing that both
equivalences. In fact, it is obvious that the
right arrow is a homotopy equivalence. To prove that ZA the ccmrmutative diagram:
(sn*xlM
is a homotopy equivalence, we consider
with the obvious arrows.
IT^
The mapping
semi-simplicial mapp&
T
-+
M L
.
-r
lLIM i s the realization of a
Using t h e f a c t t h a t
M
< IB
( i . e. A
# g),
On the
one sees e a s i l y t h a t the f i b e r s of t h i s mapping are contractible. other hand
z
=
I
n* M (S X )
x M
. is a f i b e r product,
(LIM a r e contractible and ZA
Since the f i b e r s of
it follows easily t h a t ZA
+
over t h e skeletons of
~ LMI.
x
= (
we uiu l e t
( s ~ * x ) ~i s a homotopy equivalence, by induction
vrmx = ((s X-R
T
n*
m, % x).
X)
)(-Ic
gy m a s
1 and 2, we have homotopy equivalences of pairs
5
25.
End of the Proof of t h e Lemrma i n
Since
i n t Dn
B ~ i f f f ( i n on) t
i s diffeoanorphic t o
with B C
aq x
Dn
have a mod91 f o r iln aloG D"
-aq
nq x
diag.
x Dn,
nn
we may identify
lRn
w$b. Thus we have
r f o l i a t i o n on Aq x Dn
q - c e l l consists of a the projection
§ 21.
-aq
-r
i n t Dn.
Likewise, we
q-cell consists of a g e m of a
transverse t o the projection
and having support i n . i n t Dn(cf.
.. a
transverse t o the f i b e r of
and having support i n
~ 7 :: a
n
a node1 f o r B Diffc IR
id(Aq)
X
C'
foliation
3 : Aq
x Dn
X
em ma 2 i n § 16).
We have natural mappings
These a r e described a s follows. A
xDY1(s 2'*)
(or
A
YDY1(E2-W))
Let
.
(u, Boy
..., Bq.)
be a q-cell i n
I t s image i s described as
(u, BOY
.., Bq).
There a r e three cases which must be considered t o see t h a t these mappings a r e well defined.
(1) I f
..., BP)
(B~,
i d a c e l l of L, not of Et,
then
supp u =
fly
i. e.
,
cr i s horizontal and
..., B9 )
((J, BOY
..., ~ i i)s a c e l l of K not i n J, -ff ~ g .'.., B ) i s a c e l l of I K ~x B ~ i f Bn 9 the case may be. (3) Finally, if ( B ~ ,..., $1 (2)
i s in
ILI x
then supp cr C i n t D~
(B~,
or
(0,
I K ~x nn
i s in. J,
B%,
(Boy
*.
so
as
..., Bs'
uniquely determines a c e l l , since the other factor has been identified t o a point.
-
The above mappings give r i s e t o mapping
a,, A
CI
Y,,
(sn* B ~ i f . f l R ~ ) ~
.+(sn* rP B ? ) ~ .
It i s e a s i l y seen t h a t these induce mappings
I n shor-t, we have a conmutative diagram:
To complete the proof it is enough t o show t h a t the top and bottopn arroys a r e homology isomorphisms, by the lemma i n
3 24.
gain, we mean hmology
isomorphism on each member of the pair). A
To show t h a t the mapping XD .. ,N
n* -r n N - (S B Diffc B ) i s a hmology -+
isomorphism, it i s enough t o ahmr tp&t Y
homology isomorphism.
l !)'-2~
Let
is a
1~1,
It i s enough t o show t h a t t h e mapping
induced on each f i b e r i s a homology isomorphism.
here.
- lRn
sn*B ,if<
Each of these spaces has a natural mapping i n t o
so we get a commutative diagram.
1 ~ -1 1 ~ 1i s
-+
The f i b e r over a point of
one point f o r both spaces, so we have a homology isomorphism
..., B4)
(B~,
be a p-cell of
IKI.
The f i b e r of
over an i n t e r i o r p o i ~ tof BO
n
... n B
fla D n =
..., 391
fBo,
is
..,, B )
\ n ... n eq-
In the case
i s not i n J )
t h i s fiber is
$ (i.e. ( B ~ ,
9 4 The f i b e r of the model we have adopted f o r B Dif f (B0 ll ... n B ) . 9 sn"B lRn i s B Dif<(int D ~ ) . The mapping between the fibers
=<
is
...
and it i s easily seen fl n B t--) i n t Dn, 0 9 t h a t t h i s induces isomorphism i n homology. I n the case B n.. .n B n a D n 0 9 induced by the inclusion
+
9
then
XBg
n...
n
.B
sn*B
=
(s~*BDiff lRn)'
induces
i s contractible and the f i b e r of
8,
w e r an i n t e r i o r point of
..., Bq ) .i s a single point.
(B0,
-
h
Thus, we have proved t h a t thd lapping
%
9
The proof that
isomorphism in integral homology.
'
h
%
-
induces isomorphism i n integral homology i s , similar. Let nl : projections.
5,N
-
Since A
and n2 : ( s n * ~ -1 = ( IT^) and
1 1 1 '
5 h
CB ;ayN- %
9
9
D i f g X?
homology isomorphism on each f i b e r above T A n phism. The proof t h a t A Yg ,N yl 52 B% i s analogous.
5 26.
A Property of the Functor
*
(sn*
nn
BF~)'
111' denote the
= xi1(
w -
argument given abwe shows t h a t the mapping A
-
-+
B Dif
<
IT I), lRn
the
induces
and hence i s a homology isomor-
induces is-*ism
in hmology
vN.
F i r s t , tre extend the definition of an acyclic mapping ($19) t o a
-
mapping of pairs. Definition. uO : AO - B b
Consider a mapping u
be the r e s t r i c t i o n of
H~(M(U),A U M ( u ~ ) = ) 0 ' for cylinder of
5
j,
W e say u
is
Let
j-acyclic i f
where ~ ( u )denotes the mapping
u.
Lemma 1. Let arcwise connected. acyclic.
i
u.
( A , A ~ ) (B,BO) of pairs.
f :X
Then
'Y
j-acyclic,
vN(f) : v~(x)+ (
and suppose X
Y
2
(j + n + 2N
Y
- 2) -
are
x
Proof. and Y.
We l e t
We l e t
X
# Y
denote
R
/
= X X Y
~ Y N generalized diagonal, i.e. {(tl,
sn #
X
v
denote the smash product of
Y
... # S"
..., tN) : ti
(N factors) = t
5
moddo the
for some i f j). ~t is
e a s i l y checked t h a t
#
( S ? I ~ / ~ An,,
X
#
X
#- .# X
(where = denotes homeomorphism and there are N check t h i s , it i s helpful t o note t h a t if (tl,
a point of
(snx)"
For,
v#.
(tly
and
.
5. .-.
4+1~.--, tN~
3)= *
(ti,
tNy
where j
f
X's on the right).
q, ..., tNy
f o r some 1, then
(tl,
TY...Y~~-~Y
= (tls
tj.
TO
represents
..., %) *
Y
ti+ly
i.
I n order t o complete the proof we need: Lemma 2.
n
is -
%,n
+N
- 2 acyclic.
Bid of the proof of Lemma 1. By what we have j u s t done, it i s enough t o check t h a t
%,N.#x#.*.#x-%,N
is
(j
+
n
+
2N
- 2)-acyclic.
#Y#
...# Y
But t h i s follows f'rcuu Lenrma 2 and the f a c t
that
X H is
(j
+ #N
+ N-1)-acyclic, by the assumption t h a t f
and the assumption t h a t
By definition,
a,,
where X = ( n,N show t h a t X
t
is
~ Y N
..., t
=
sn #....#
): t
(n + N
= t
sn(~ factors)/% 3
YN f o r some i = j . It is enough t o
- 3)-acyclic.
More generally, i f a i s a s e t of pairs of members of we l e t that
%,a = { (tl,..
%,a.
is
(n + N
j-acyclic;
and Y a r e arcwise connected.
X
Proof of Lemma 2.
is
: X-+Y
., tN: ti = t3
- ))-acyclic
elements i n a. We have t h a t
X2
f o r some ( i ,j )
L
(1,
..., N),
a}. We w i l l show
by induction on N and the number of
i s an n-sphere and so it is (n-1)-connect-
More generally, i f a
ed.
has just one e l w e n t then
sphere, so it i s n(N-1)-connected,
and hence (n
t h e inductive step, we consider f3 = a ff ( i , j ).
'
and %,a %, ( i , j ) * %-1,a' $,(iyj) a under a subjective mapping of (1,
sends i and
%,B - 'n,a
We have
onto
(1,
u
..., N-1) which
j t o the same number.
It follows e a s i l y t h a t
%,a are i s (n + N
Xlg
- 3)-connected.
+N
For
a' be t h e image of
where we l e t
..., N}
acyclic by induction hypothesis. (n
- 3)-connected.
But %,(i, j ) and 3)-acyclic by induction hypothesis, and %-l,al
-
(n + N
+ EJ
i s an n(N-1)
54
,B
- 4)-
is
Proof of the Corollary of Thurston's Theorem.
;5 27.
I n t h i s section we will prow the Corollary of Thurston's T h e o r q In
Q
denote t h e
28,
we w i l l deduce Thurston's Theorem from it.
n disk DL'.
f :%
We consider t h e homotopy equivalence Since the inclusions
(5
it follows t h a t
18),
We l e t D
f :
XD,
%,-(-,
%,m
'%,-
+r)
defined i n § 17.
Y a r e hmotopy equivalences
i s a homotopy equivalence.
l e t ~ ( f )denote the mapping cylinder of the l a t t e r mapping. ) ~t h e mapping cyclinder of ~ ( f )by l e t t i n g ~ ( f be We l e t ( ~ (),f
We
We f i l t e r
f : %,N+,%,N.
r
-
{ E ~ , ~ denote ) homology spectral sequence associated t o the p a i r
% m) 9
of f i l t e r e d spaces.
From the f a c t t h a t
f :
)[D,m
-C
YD,m
i s a homotopy equivalence, we obtain immediately: L-a 1. -
E~ = 0. P*9 We l e t { E ~(x)) and PYQ nssociated t o the f i l t r a t i ' o n s
{ E ~(Y)) denote the spectral sequences P,Q 1 of GbYN} of and {Y D,N The long exact sequence of hanology gives an exact sequence
...
-
we w i l l pro&:
1 E (x) P*Q
5,-
4
1 E p , q ( ~ )4E:,~
+
1 Ep,q-l(~)
+
...
If
Lemma2.
ma -
p
2 1.
1 El = O 99
for
q s j
then
1 E = O P99
for
q ~ p + j - l ,
Assuming t h i s Lemma, we can complete the proof of Thurston's Theorem, as follows.
r
F i r s t , from Lemma 2, it follows t h a t let
qo be the smallest
q
such t h a t
E = 0 f o r a l l p,q,r. For, P,9 f 0, assuming such exists. All
El1
,9 E1 end a t a group which i s 0 by Lemma 2. 1,401 1 1 Hence E which.is 0 by Lenrma 1. Hence E l,q(0) ~Y,q(o)' l , q ( 0 ) = O' contradicting the choice of q Hence = 0 f o r all q. From Lemma 2, 0' 1,q d i f f e r e n t i a l s beginning a t
-
it then follows t h a t
El
Psq
= 0 f o r all p,q.
Hence Er = 0 f o r all P%q
P Y ~ Y ~ .
it follows t h a t
Since E l = 0 PYP By definition
~ ; , q ( x ) = %+q(]b,p'
%,PI)
1 Ep,q(y) = h ( Y g , p '
%,pl).
E? (x) P99
+
2P,'2 (Y)
i s an isomorphism,
We a s s e r t t h a t there are homology equivalences
This follows f r m the Lemma in § 21 together the homotopy equitralences'
- ;a,,,
-
bYl
A
YD,l, which follow as a special case of the homotopy
19-20, and, i n any case, are e a s i l y seen direct-
equivalences obtained i n ly.
*
= *. Thus, from the f a c t t h a t 590 + $ y q ( ~ ) i s an isomorphism, it follows t h a t
Moreover
1 (x) *PA
%,0 =
and
-
r n H,(B Diffc 1~ )
4
n -r ~ , ( nBG)
i s an isomorphism, which i s the Corollary t o Thurston's Theorem. The conclusion of Lemma 2 is equivalent t o the
Proof of Lemma 2. assertion t h a t
-
$ : %.pl%,p14
k,p/%,p-l
i s (2p + j
- 1)-acyclic. N
'P, : %,P
-,YM,p definition of
By the Lemma i n $ 19, it i s enough t o prove t h a t
i s (2p + j
- 1)-acyclic.
By t h e Lenrma i n § 20, and the A
N
A
A
it i s enough t o prove t h a t f M,p : ( l k y p yA %,p) In,pY L P ' i s (2p + j 1)-acyclic. By the Lemma i n § 21, it i s
-
enough t o p r w e t h a t
i s (2p
+j
B
JRn
wft
- 1)-acyclic. 0%
-+
%
is
By Lenana 1 i n
$26 it i s enough t o prove t h a t
(3- n + 1)-acyclic.
show t h a t t h i s i s the same a s
I?? = 0 l,q
But the homqlogy equivalence (*)
for
q
5
j.
Proof of Thurston's Theorem.
$ 28.
According t o $ 17,
Y
:
% -+& i s
the mapping which appears i n the
statement of Thurston's Theorem. 'We will show t h a t it induces an isomorphism i n integer homology. The spectral sequence associated t o t h e f i l t r a t i o n of
5 (resp. 5 ) ($ 18).
converges t o the homology of prove t h a t
N-~ 5,N /%,N-1$ . I , N ~ ~ M , induces
homology, f o r each N prove t h a t
XM ( r e s ~ .
Thus, it i s enough t o
isamorphism i n integer
_> 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
for
i s a neighbourhood base f o r
8)-topology,.
c o n s i s t i n g of open s e t s ,
= ( y n ) r u n s through a l l sequences of continuous
s t r i c t l y p o s i t i v e f u n c t i o n s on X with. (supp f i n i t e . The m e t r i c s dn.
y,/
locally
dj -topology i s indepcnuent of the choice of t h e
(c) The system of s e t s of t h e form
R ( L , U ) = [ f e L-(x,Y):
. ~ " ~ ( X - L ~ O ) Sf oUr ~a l l n )
i s a base of open s e t s f o r t h e $ -topology on C-(x,Y),
where L = (Ln) r u n s trough a l l sequences of compact s e t s LnS X such t h a t (x-LnO) i s l o c a l l y f i n i t e and U = (Un) runs through a l l sequences of open s e t s U,s
a -topology
4.
J~(x,Y).
on ~ Y ( x , Y )i s f i n e r t h a n t h e % h i t n e y - c -
topology. It i s e x a c t l y t h e topology who proves t h a t ?(x,Y)
twof
i s a B a i r e space i n t h i s topology.
it was mistoken t o be t h e Whitney-topology by LESLIE Ve now l i s t some p r o p e r t i e s of t h e ( a ) A sequence ifn,
,
WORLET i n [ 2 3
.
(53
3 .-topology:
& (P(x,Y) converges t o f i f and only
if
t h e r e e x i s t s a compact s e t K C X such t h a t a l l b u t f i n i t e l y k k riang of t h e f n t s e q u a l f off K & j f,j f "uniformly" on K -
f o r a l l k. So convergence of sequences i s t h e same f o r
t h e ilhitney-topology and f o r t h e
3 -topology.
See
['/I .
(bj ~f T i s a connected m e t r i z a b l e compact t o p o l o g i c a l space and
f : T 4 C@(X,Y) i s any continuous mapping ( f o r t h e
3 -topolog~),
then t h e r e i s a compact s e t K S X such t h a t t ~ f ( t ) ( x )2 consfant on T
for
x e X-K.
Proof: Any t c T has a neighbourhood Vt i n T such t h a t t h e s t a t e d p r o p e r t y h o l d s on V t :
tn + t
i f n o t one may f i n d a sequence
i n T such t h a t t h e sequence f ( t n ) does n o t s a t i s f y
t h e c o n d i t i o n i n ( a ) . Now use t h a t T i s compact and connected. ( c ) For each k Z O t h e ma2 continuous f o r t h e
b
jk: CP(X,Y)-F(X,$(X,Y))
- t o p o l o m . See [ 71
is
a r e smooth f i n i t e dimensional manifolds t h e n
( d ) ~ fX,Y,Z
00
composition C (Y,z) x c ~ ~ ~ ( x -+ , Yc*(x,z), )
, is
( f ,g) -fog
-
-
continuous i n t h e
g i v e n by
3 -topology,
where
I
00
(X,Y) i s t h e space of a l l smooth p r o p e r maps f : X--+ C ~ r o ~ i . e . f - ' ( ~ ) i s compact i f K i s compact. See C73.
Y,
-
5. Theorem: Zet X , Y be smooth manifolds.
Then
C?(X,Y)
3 -to~olopy.
Z a i r e space with t h e
This was proved by FIORLET C 2 ] . We give h e r e a q u i t e d i f f e r e n t
3 -topology. sequence of 3 -open-
proof u s i n g t h e e x p l i c i t d e s c r i p t i o n of t h e Proof: Let U1, -
U2,
... be a countable
dense s u b s e t s of c*(x,Y).
8 -dense.
We have t o show t h a t
i s again
QU,
Choose m e t r i c s dn on Jn(x,Y), n = 0 , .
. , compatible
with t h e t o p o l o g i e s , such t h a t each Jn(X,Y) becomes a complete
Let be f o r ~ ( X , Y )and V ( f o ) be any
m e t r i c space w i t h d,.
'P
neighbourhood of f o a s i n 3 ( b ) . It s u f f i c e s t o show t h a t Vy(fo)" let
? Ui
# O
? y = = (*%),
Vit(fo),
=
[
. Let t h e n f o e V 4 y ( f o ) ~ ' Y f y ( f o ) ~ ~ y ( ,f owhere )
g~ cW(x,y) : f y,(x)
x b X and f o r a l l nB 0 Viy(fo) n
ou,
. It
dn(jnf ( x ) , jng(x)
C1
for a l l
c l e a r l y s u f f i c e s t o show t h a t
# 0.
To do t h i s we c h o o s e ' i n d u c t i v e l y a sequence of f u n c t i o n s ( f i )
i n C-(x,Y);
a sequence ( y ( i l )
o f f a n i l i e s a s i n 3 ( b ) such
t h a t t h e following holds :
Choose fl& V
i~( f o ) n
U1
which i s p o s s i b l e , s i n c e Ul
i s Oense.
So (A1 ). holds.. U1 i s open and f l
Ui
-
, then
such t h a t V2*m(f1) 5 U1
r(4)
r
s o we can f i n d a family
V ,,,,(fl
t U1 so (BI) holds.
)
i s empty. Now assume i n d u c t i v e l y t h a t t h e d a t a i s chosen
(CI)
f o r a l l j ~ i - I .We w i l l choose f i s a t i s f y i n g ( d i ) and (Ci) and not using any (Cj),, j < i ,and then we can e a s i l y f i n d Yc%uch t h a t (Bi) holds. Consider t h e open s e t V (f 7 i-I) where 7 = , ( 0 , 2 ~ ;2i,0,0 with i - t i m e s z i .
,...
Let
Ei =
and fi,l
f Y (f ,)
V
fi
n
Ei by (Ai,l
by d e n s i t y of ui.
,...)
i-l
n V ,t~,(f ,)
i=c
) ,' SO Ei # f8 and we may p i c k
t'i y ( f o ) nP~,+,*(f
d s ( j fi-,,(x),
f i e B i n Ui
Furthermore we have f o r l g s ~ i i s o (Ci) i s 1 / 2 by t h e form of 7
.) o Ui.
J
a4
S
Ei i s ' open
hen c l e a r l y (Ai) holds s i n c e we have
i-q
E
, then
o V (f 3 i-1
jsfiix))
<
,
s a t i s f i e d . F i n a l l y f i e Ui, such t h a t VZy~o(fi) C Ui,
Ui
so
i s open, s o t h e r e i s a family cy
- -
'''
Vy cn(fi) 6 Ui and (Bi) halds too. ,
liow we use t h i s d a t a t o prove t h e theorem. Define g S ( r ) = T* i m jsfi(xj .I ..o
t J'(x,Y).
ks
l i m i t e x i s t s since f o r
each s ds i s a complete metric on J'(x,Y) sequence ,j?fi(x) jofi(x)
and f o r each x t h e
i s a Cauchy-sequence by (C)
= (x,fi(x)),
. Since
t h e graph of f i , we can d e f i n e g:X*Y
by gO{x) = (x,g(x)).
We claim t h a t g i s smooth. Yhis i s a l o c a l
question ana i n a chart-neighbourhood we see t h a t a l l p.artia1 derivative's of f i converge uniformly by ( C ) ,
s o g i s smooth
by a c l a s s i c a l theorem of llubini. How f i e Vty(fo) b j (Ai), i . e . y n ( x ) dn(,jnfo(x), j n f i ( x ) ) < 2 f o r a l l x c a and n > 0. Since jnfi(x) --+jnglxj we conclude t h a t so g
6
7%). t.r
yn(x) dnijnf0(x), --
f o r a l l x and
n
jng(x)) C 2 f o r a l l x and n ,
By (Bi) Y G ' ~ chosen a~
SO
that
d
v y,.ir
(fi)
c Ui
.
and by (Ai) we have t h a t f s a vyca(fij f o r a l l s > i , i .e
<
( x ) ( f i x ) ~ . ~ f , ( x ) ) 1 f o r a l l x and n. Since jnfs(xg 9 jng(x) f o r a l l x A d n we conclude t h a t y'''(x) dn(jnfi(x), jnglx)) Q 1 f o r a l l x and n , i . e . n g e ~(j. r h i s h o l d s f o r a l l i. So by (B) wehave Y" 00 qed. g € Yfy ( f o l o flVym(fi) S Vi ( f o ) n Ui. i- 4 ~'f e 84
--
-
-n
6, Examples: I n h i s l e c t u r e Plather introduce6. a topology on ~ i f f z k, t h e space of c r - d i f f e o m ~ r ~ h i s r n swith compact s u p p o r t of a smooth manifold K , by t h e formula uiffE1.l = 1 3
iff;^ , K
compact i n M. I f r =
i s e x a c t l y t h e topology induced from t h e
caD(~,N), if
, then
3 -topology
this on
r < o = t h e n it i s t h e topol.ogy induced from t h e
'iihitney-~r-topology. The same topology was used by Banyaga i n h i s t a l k on t h e space .of smooth s y m p l e c t i c diffeomorphisms with compact support.
7. We now i n t r o d u c e a refinement of t h e f f - t o p o l o g y on C-(x,Y)
which i s needed f o r t h e manifold s t r u c t u r e l a t e r
on. It i s c a l l e d t h e'0
-topology i n
171,
not a very good
name, It i s g i v e n by t h e f o l l o w i n g p r o c e s s : If ~ , ~ € c ' . ( x , Y ) and t h e s e t j x c X: f ( x ) # g(x))
has
compact closure' i n X we c a l l f e q u i v a l e n t t o g
(f-g).
his
i s an e q u i v a l e n c e r e l a t i o n . The $w-topology
i s now
t h e c o a r s e s t among a l l t o p o l o g i e s on C?(X,Y), which a r e f i n e r than the
8
-topology and f o r which a l l equivalence
c l a s s e s of t h e above r e l a t i o n a r e open. Another d e s c r i p t i o n i s : e q u i p each e q u i v a l e n c e c l a s s w i t h t h e t r a c e o f t h e
a -topology
and t a k e t h e i r d i s j o i n t union.
The i n t r i n s i c d e s c r i p t i o n s of s e c t i o n 3 a r e s t i l l v a l i d with a l t e r a t i o n s , j u s t add f-g
t o t h e d e f i n i t i o n of V ( f ) i n ( b )
1
and i n t e r s e c t M(m,U) r e s p . Mt(L,U) with equivalence c l a s s e s . The p r o p e r t i e s 4 ( a )
- 4'(d)
remain v a l i d f o r t h e $w-topolo@J
t o o , s i n c e t h e maps and c o n s t r u c t i o n s used t h e r e a r e compatibel with t h e equivalence r e l a t i o n . 0
C'O(X,Y) i s no l o n g e r a Baire space with t h e a - t o p o l o g y s i n c e
i t looks l o c a l l y l i k e t h e model space I ~ ( ~ * T Y as) we s h a l l s e e i n t h e n e x t s e c t i o n and f u n c t i o n a l a n a l y s i s t e l l s u s , t h a t t h i s i s no Baire space. But i t i s a Lindel8f space i f X i s second
countable, s o cOO(X,Y) i s normal and paracompact with t h e $--topology. 8.
Iie now d e s c r i b e t h e manifold s t r u c t u r e on C-(x,Y).
be a smooth map such t h a t f o r each y C Y t h e map
? : TY-Y
'ty:T Y + Y 9
Let
i s a diffeomorphism onto an open neighbourhood
of y i n Y. Such a map may be c o n s t r u c t e d by u s i n g a f i b r e r e s p e c t i n g diffeomorphism from TY onto an open neighbourhood of t h e z e r o - s e c t i o n i n TY followed by an a p p r o p r i a t e e x p o n e n t i a l map.
If f
r ~ ( x , Y )c o n s i d e r t h e pullbach ~ * T Y .which i s a
vectorbundle over X, and t h e space
8 (~'TY) of
a l l smooth
s e c t i o n s with1compact support of t h i s bundle, equipped with the
aW- ( o r 3 -)
topology.
Let
yf: )( ~ ' T Y ) --r C?(X,Y) be t h e mapping yf ( s ) ( x ) = tf( x ) s ( x ) 4 Y. Denote t h e image
of yl
by U f .
u L X ~ rf
(Tf cx)Y)) i s an open neighbourhood of ( X 2 = X (in fact t h e graph i ( x , f ( x ) ) , x e ~ ) of f i n X s Y = J'(x,Y) a t u b u l a r neighbourhood), and Uf c o n s i s t s of a l l g k u"(x,Y) such t h a t t h e graph of g i s contained i n Zf and g ~ f s,o
yf
Uf i s open i n the $--topology.
has a continuous i n v e r s e
tf f :
i s continuous by 4 i d ) and
Uf + a \ f * i ~ ) ,
giverrby
y f ( g ) ( x ) = ~ ? ; ~ ) ( g j x , ) ,a s i s e a s i l y checked up. ve use
yf
a s c o o r d i n a t e map. xiow l e t us check t h e form of
t h e coordinate c h a n ~ e : l e t f , g C C ( X , Y ) with U f o Ug # 0. -
>,or s e yi.(Uf 0 u g j we have
y g y,f ( s ) ( x )
=
2-p;*)(yflS~(x)) =
f f ( x l ( ~ ( x , ) , s o t h e map
=
yf
yge
?
(Uf o Ug) E
3( ~ T Y* ) 3(g*r~)
i s given by
z f j * , by pushing forward s e c t i o n s by a f i b e r bundle diffepmorphism t - l *tf * t h i s i s c l e a r l y continuous. i 3 00 s o we have constructed on C (X,T) a s t r u c t u r e of a t o p o l o g i c a l
(Z-
.
manifold, where each f e C-(x, r ) h a s a coordinate neighbourhood
uf homeomorphic t o a whole space
3 ( f * 1 ~ ) of
compact support of t h e v e c t o r bundle
s e c t i o n s with
~ * Y Y over
X.
Ithe c o n s t r u c t i o n we have given here i s a s i m p l i f i e d v e r s i o n of t h e one given i n C73.
9.
Yo make c ~ ( x , Y )i n t o a d i f f e r e n t i a b l e manifold we j u s t have
t o t a k e a s u i t a b l e n o t i o n of 'C t h e coordinate change ( t of c l a s s
fin t h e
- mappings and t o show t h a t
-' Vf)
Q
i s.'c
we remark t h a t i t i s
sense of E43, a r a t h e r str& n o t i o n , a s i s
shown i n 171, and probably of c l a s s coof o r any n o t i o n of d i f f e r e n t i a b i l i t y t h a t h a s appeared i n t h e l i t e r a t u r e u n t i l now. The tangent space a t f
bf (X,TY) = $(f'rf)
0
?(x,Y)
t u r n s out t o be
and t h e whole t a n g e n t bundle i s
t h e space of a l l smooth maps A+TY
3 (X,TY) ,
which d i f f e r from zero only 00
on a compact s e t . lt i s a v e c t o r bundle over C (x,Y) ( i . e , l o c a l l y t r i v i a l ) i n t h e manifold s t r u c t u r e it i n h e r i t s fsam C ~ ( X , T Ya)s an open s u b s e t . T h i s t a n g e n t bundle seems t o be independent of t h e n o t i o n of d i f f e r e n t i a t i o n a p p l i e d . See [?J.
10. The i n v e r s e function theorem I presented i n my t a l k i s
wrong due t o . d i f f i c u l t i e s with chain-rule f o r t h e notion of d i f f e r e n t i a b i l i t y applied. The statement t h a t remains t r u e i s t o o s p e c i a l t o be of any i n t e r e s t .
References [I]
C. BESSAGA, A . PEY;CYl'?SKY: S l e c t e d t o p i c s i n i n f i n i t e
diraensional topology, P o l i s h s c i e n t i f i c P u b l i s h e r s 1975.
121 H. CARTAB: ~BminairtiE.N.S.
1961/62 : Topologie d i f feren-
t i e l l e , 6xpos6 by NQRI;ET.
[33
PI.
GOLUEITSkT, V. GUILWHIN: S t a b l e mappings and. t h e i r
s i n g u l a r i t i e s , Graduate t e x t s i n mathematics 1 4 , Springer. [4]
H.H.
KELLER: D s f f e r e n t i a l c a l c u l u s i n l o c a l l y convex
spaces, Springer l e c t u r e Cotes 417 151
J.A.
(1974).
LESLIE: On a d i f f e r e n t i a b l e s t r u c t u r e f o r t h e group
of d i f f eomorphisms , Topology 6 (1967), 263-27'i [6]
.
J. KATHER: S t a b i l i t y of C -mappings 11: i n f i n i t e s i m a l
s t a b i l i t y i n p l i e s s t a b i l i t y . Ann. Math. 89 ( l 9 6 9 ) , 254-291.
[7]
P. MICHOR: FIanifolds of smootb maps, t o appear i n Cahiers
de topologie e t g6ometriB d i f f e r e n t i e l l e . P. Kicbor, Rathenatisches I n s t i t u t d e r U n i v e r s i t z t , Strudlhofgasse 4, A-1090 Wien, Austria.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (c.I..M.E.)
SOME REMARKS ON LOW-DIMEN SIONAL TOPOLOGY AND IMMERSION THEORY
V.
POENARU
Corso tenuto a Varenna d a l 2 5 agosto a1 4 settembre 1976
Some remarks on low-dimensional topology and immersion theory (*) by V. POENARU University P a r i s XI Orsay Department of Mathematics 91405 Orsay France
-
-
-
This is the text of a lecture which I gave at the 3rd CIME session of 1976 and describes some technical ingredients (in very simplified versions)
and open problems, connected with a certain research
program in low-
dimensional topology. Although the various items described here might seem unrelated to each other, this is actually not the case. A rather long paper which I am preparing binds all these technical threads together and will (hopefully) contain the proof of some of the conjectures stated below.
-I-
I will s t a r t by recalling the following (see [ I
THEOREM 1 .
1) :
Let Z3 be a s m t h homotopy 3-sphere.
I'
h hen the smooth
open 4-manifold C3 x R contains 2 open subsets U1 and U2 such that : 1)
Z3 x R
2) U
1
= U, U'U2
.
and U2 a r e diffeomorphic to R~
.
The proof of this theorem uses the notion of "Whitehead manifold" which I recall here : Consider an infinite sequence of smooth embeddings :
(*) Lecture held at the 3rd CIME Session 1976 (Varenna)
where : 1) Tk is a solid torus : Tk = n(k) # (S1 x D2). 2) Jk (Tk) c +k+l
-
3) Each map jk is null-homotopic. 3 Then, by definition, W = l i m Tk is a Whitehead manifold. This is clearly an open, contractible 3-manifold. It is very easy to see that W3 x R is diffeomorphic to R4
.
I will consider later the tlstandard Whiteheadtt manifold Wo3 which is defined by the following further specifications : 4) Each n(k) = 1
.
5) Each . jk is like in figure 1 below.
It is easy t o show that :W
can be (smoothly) embedded in R~ ; i t does 3 f 1). have, however, the same proper homotopy type as R 3 (since r - Wo Now, theorem 1 follows from the following lemma [ I ] :
" In the category of 3-manifolds
and smooth embeddings (into their inte-
riors) , any embedding of' a solid torus into a simply connected manifold, factors through a Whitehead manifold!' (This implies, in particular, that I?
is the union of two Whitehead manifolds).
- 11 -
Here are some open problems and conjectures connected with the
preceding paragraph. Problem 1 : If C3 is a smooth homotopy 3-sphere can one find two C3 such that UI U U2 = I:3 and U U a r e 1' 2 3 Whitehead manifolds which can be embedded in R ? open subsets UI ,U2
C
Warning : To the best of my knowledge, a positive answer does not necessarily lead to a proof of the Poincard Conjecture. I regard this problem rather a s a curiosity. Conjecture 2 : The following conjectural statement is much stronger then theorem 1 above : If C3 is a smooth homotopy 3-sphere, then (z3 4 morphic to R
- pt) x R
is diffeo-
.
Conjecture 2' : I will explain a slightly weaker conjecture. But, first, here is a definition : 4 Let V be a connected bounded smooth 4-manifold with connected boundary. 4 2 2 I shall define V # ( = # ( S x D ) ) a s being the following open, connected, smooth 4-manifold :
V
2
2
4
x D ) ) = V UM1UM2U 4 where : 1) MI is the cobordism BV x [0,1] # ( s 2 x D ~ ) 4 ( s2x D~ being attached to aV x 1 ). 4 4 2) a v 4 x 0 c BM1 isidentified to BV c V
...
.
3) M2 is the cobordism {a(v4 U M ~x ) [O, 1 ]I 4 ( s 2 x D ~ ,) and s o on.
..
Then, conjecture 2 implies the following conjecture 2' : Let a smooth homotopy 3-sphere and b3
C~
z3 be
- h3 .
Then : 2 2 ( b 3 x 1 ) # ( = J # ( s ~ x D ~=) )D ~ $ I ( ~ # (xsD ) ) . =
Note that this is a "stablev version of the Poincar6 Conjecture, since
f o r n finite a diffeomorphism of the type (63 x I) t (n 4 (s2 x ~
2 =)
D4 4 (n # ( s 2 x D ~ )) implies C3
=
S3
.
I don't know whether conjecture 2' implies (easily) conjecture 2.
Conjecture 3 : I will consider the Standard Whitehead manifold Wo3
f r o m the preceding paragraph. S o W3 = h i Tk , where Tk = S1 x D 2 and Jk Tk -+ Tk+l is like in figure 1. F o r convenience, I shall write : Wo3 = h l T k = T 1 U T 2 U
...
+
Then I can define the following non-compact smooth 4-manifold with boundary :
...
4 = ( 0~ ~ x [ - l , l ) ~ ~ ~ ~ ~ x ( - l( -, 1l, l)) )~U ) ~ ~ ? ' ~ x
X where
, for each
0
x E Ti, -1 < t < 1, the points (x, t) and (ji(x), t), a r e ,
by definition, equal. Note that
g4 = b4 ,
4 1 02 aX = S x D
.
4
Consider also an open smooth 4-manifold Y. obtained by adding a 2-handle D2 x b2 to x4, along
a x4.
I conjecture that there is noproper smooth function f : y4 + R with
exactly two non degenerate critical points, one of index 0 and one of index 2. 4 My conjecture means that 3X s i t s in a "wild positiontt. More exactly I conjecture that there is no smooth embedding *4
4
x4-
'P
D4 such that
*4 (in fact not even if one replaces D~ by some
cp(aX ) c 3 D ~ cp(X , ) =D 4
exotic A
, of rather doubtfull existance ...) .
Remark : Conjecture 3 is actually one of the difficulties which one can encounter trying to prove conjecture 2
.
This line of thoughts also leads to the following questions : Problem 4 : I shall consider a non-trivial h-cobordism ; in fact, to be more explicit, let
xn, Y"
be closed manifolds such that X f Y and
X x R = Y x R. Then there a r e two ways of putting a boundary to Z
=
X x ( 0 x 1 1, the standard one, leading to X x [0, 1 ) and a non-standard
one leading to an h-cobordisrn
T, in between X and Y. If one deletes
f r o m X x [O,1] . all of X x 0 except an n-disk smoothly embedded in X x 0 and f r o m T all of Y except an n-disk smoothly embedded in Y, a r e the resulting non-compact manifolds diffeomorphic ? Consider also a riemanian metric p on X x [0, 1 ] and a tubular neighborhood
YX[O,C]C
T (with Y X O E Y ) . Then y x ( O , c ]
smooth curves in X x (0,1]
are
. What can onesay about the p-length of these
curves ? I s i t always bounded ? Can i t be bounded for some y ' s and unbounded for others ?
...
- 111 -
I consider now two smooth compact manifolds X and Y, with n 2 d i m X < d i m Y andagenericimmersion f : X + Y . Let X x X - S X be the natural projection of the cartesian square of X to the symmetric cartesian square. I can consider the-s e t of double points of f in X, X2 or
s2x
: M2(f) c X, M2(f) c X x X, k2(f) c
s2x.
Note that while ~ ~ ( f )
6
has singularities, M (f) and %(f) a r e smooth manifolds (in fact, the map 2 2 M2(f) + M (f) induced by the projection of X x X on i t s first factor, is 2 the "resolution of singdarities" for M (f) )
.
The 2- sheeted covering
M2(f)
4'$(f)
turns out to be a very interesting object in a certain kind of geometric contexts. I shall write p (f) = 0 a s a shorthand f o r Itthe 2-.sheeted covering 2 above is trivial". In [2] , there is a "good" generalization o f p2(f) = 0 f o r the multiple points of f (of multiplicity > 2) : v3(f) = 0 . Very roughly speaking v3(f)
=
0 means (among other things) that if (xl..
..,xn )
n-tuple point of f , the double points : (x19x2), (x2,x3), (x3,x4)9
..., ( X , , - ~ , X ~ ) , ( X ~E, XM2(f) ~)
a r e not allowed to be in the same connected component of M2(f).
is an
Consider now a smooth homotopy 3-sphere C3 , two points x1 ,x2 E C3 and two'embedded 2-spheres S1,S" c C3 , S1 being concentrated around and S" around x From the Smale-Hirsch theory , i t follows that 1 2there exists a (generic) regular homotopy connecting S1 to Su in x
The following result was announced in [2] : THEOREM 2.
P2(9)
=
o
m d v3(cp) = 0 , then
c3
is h-cobordant to
s311.
Here is the sketch of the proof : the conditions ~ ~ ( (=00) and v (cp) imply the existance of a smooth isotopy 3
lifting cpt
s2 @t >. (z3 - {xl , x ~ I ) (see [2 ] , [3 1).
3 3 Let D c C be a smooth embedding and 3 3 3 ,C = A U D , 1 c a n . r e a d : C3 x D~ = A3 x D2
u D3 x DZ
R
= C3
= A 3 x D2
- b3.
From
+ (a certain 3-handle)
and the existance of Qt implies very e a s m that the 3-handle involved is trivial, which means that :
.
C3 x D~ = ( A x~ D2) # ( s 3 x D ~ )
The last equality tells me that there is a smooth embedding S3 c 3 (C3 x D2) which induces a homotopy equivalence S3
+
C3 x D2
. By
a simple argument, I can deduce the existance of a smooth embedding
3
S c
C3 x R, which is a homotopy equivalence, and this finishes the proof.
Remark : In the context of theorem 2 p2(rp) = 0
M
i(2(9)is orientable.
A better way to connect p2(f)-type questions and
x3
a will be included
in a paper in preparation.
- IV -
Here a r e some more conjectures. First, let me define the
llsingularllobject C(2n + 1). I consider 2 n
+1
disjoined embedded
2
and like-wise the D2i's, hence the a D2 I s . Let 5 ' be an oriented circle and )i : a D2i + s1 be diffeomorphisms such that I orient S
J,
,..,J,n+l are orientation-preservtng,
and )h+2,
orientation-reversing. By definition C(2 n (s2 So
-
2n+1 U D i) U s 1
s1c E(2 n + 1)
+ 1)
... J,2n+1 ?
is the quotient-space of
obtained by identifyingeach x E 3 D : is the singular locus of C(2 n
Conjecture 5 : Let O : C(2 n
are
+ 1) + s3 be a
to ) i ( ~ ) .
+ 1). smooth embedding. Then
@ ( S1) c S 3 is unknotted (or, at least, if one uses +(Sl) and the canonical Seifert trivialisation for i t s normal bundle, in order to add a 2-handle to 2 2 D ~ the , resulting manifold is S x D )
.
Remarks : a) Under the hypothesis above, the Alexander polynomial of ~E(s') is t
- I.
b) Conjecture 5 implies that a contractible manifold of the form : 4 D + (one 2-handle) + (one 3-handle) 4 is diffeomorphic to D
.
c) If 2 n
+ 1 = 3,
the conjecture is proved (Laudenbach) and
the same arguments seem to apply for 2 n + 1 = 5 (~audenbachand el ale)
References
.
[ 1] V POENARU , A remark on simply-connected 3-manifolds, BAMS, volume 80, no 6 (19741, p. 1203-1204. [2] V. POENARU, Some invariants of generic immersions and their geometric applications, BAMS , volum 81, No 6 ( 1975), p. 1079-1082.
.
.
[3
V POENARU , Homotopie r6gulihre et isotopie (to appear)
14 J
A. CASSON, Lectures P a r i s 1 9 7 4 ( t o appear). Paragraphs I and I1 have s t r o n g connections w i t h Casson's work..
.
CENTRO I N TERNAZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
LA CLASSE DE COBORDISME DES FEUILLETAGES DE REEB DE
s3 EST
NULLE
F a SERGERAERT
Corso tenuto a Varenna dal 25 agosto a1 4 settembre 1976
LA CLASSE DE COBORDISME DES FEUILLETAGES DE REEB DE
pour construire un feuilletage de Reeb de
g3 , on
s3
EST NULLE
peut proceder
cornme suit. .$ : LO, 1 [
Soit
cm
numerique de classe
, gl(x)>O
si Ogx41/2
lim $(x) = +m x+ 1 defini'e par : $(XI
. Soit
-+
R
une fonction #t(x) = 0
telle que si
1/2<x<1
,
et
+R
JI :]1/2,1]
I/.$' (x) si x
3
$(I) = 0 On demande aussi que
=
que JI(')(I)
o
Soit part
11)
1/2
0'
pour tout CJEN
a0
rotation autour de
cm
part les graphes de
(0) x R
, on
et -plate
en 1
.
le feuilletage de [0,1]
, d'autre
x R
soit
J,
1'
, c'est-l-dire
x ll dont lea feuilles sont d'une
x
-+
$(x)
+
obtient un feuilletage
X pour de
XsR D
2
.
Par
x R
, oii
. Ce feuilletage est compatible avec l'action 2 sur le second facteur de D x R . Par passage au quotient,
est le disque unite de R2
D~
canonique de
Z
on obtient un feuilletage $?. de D~ x s1 qui n'a qu'une feuille compacte 2 1 = S ,x s1 L'holonomie le long de T2 ne depend que du germe de 41 en 1.
.
T
On peut montrer qu'il en est de mgme de la classe de conjugaison de % Soit 0
. Soient
par
1
s
x
le groupe des germes de
pl
et
p2
(x0)
et
(x
L'holonomie
0
.
~ ~ - d i f f ~ o m o r ~ h i sde mes [o,+-[ en 2 les generateurs "standards" de rl(T ) representes 1 2 x S ' respectivement, oit xo = (1,o) 6 s c~
G+
.
de
$?,
le long de T2 est definie par
, et oii
oii e est 1'616ment neutre de G+
g
E G+
vdrifie
:
0 est seul point fixe
g(x) <x g
x>o
si
est m-tangentt 2 e
en 0
, i.e.
pour -tout qa Ei
La classe de conjugaison de §?, ne depend w e de g nous
%(g)
ce feuilletage
Comme s3 = L12 x
;
feuilletage 5
g1 et 92
&(gl) $(g1,42)
1 S
s1 U
et
aussi noterons
s3
x
D~
, on peut
g
.
recoller deux composan-
1 1 xs
a(g2) de
;
c'est la composante de Reeb associee 2
s ter de Reeb
,
;
1e long de l e u bord
T~ pour obtenir un
c'est le feuilletage de Reeb de .S3
associe
'
On note
Mizutani (2) avait considere le mSme feuilletage, mais ofi la seule feuille compacte T2 de metre T2 x I = T2 x
$(g1,g2)
feuilletage trivial de T2
etait remplacde par une famille 1 un para-
de feuilles diffdomorphes x
T2
. Notons
I dont les feuilles sont les T2
feuilletage de Mizutani x(g1,g2)
peut htre not6 come suit
x
(t}
%
le
. Le
:
On se propose de donner un resume de la demonstration du
:
THEOREME. La cZasse de cobordime de 6;(gl,g2) e s t nuZZe ; i . e . i Z e x i s t e une varidtg de dimension 4 v de bard av = s3 munie d'un c m - f i u i ~ ~ e t a g e de codimensbz I , t ~ m s v e r s e6 av , avec 1 av = (gl,g2)
9
q
a
.
Mizutani (2) avait d6moritr6 le mSme resultat pour son feuilletage N(glrg2) On donne d'abord le plan de la ddmonstration de Mizutani, explique ce qu'il faut y changer pour obtenir le th6orSme annonce.
p~1j.son
Mizutani construit d'abord, de faqon bl6mentaire,un cobordisme entre la s o m e disjointe de deux exemplaires de
m ( g 1 , g 2 ) d'une part, et un
feuilletage
d'autre part. Ici mie globale hEDiffZ(1)
2 T
3(h) est le feuilletage de
x
I
d6fini par l'holono-
2
.
s (T ) + DiffZ(1) qui envoie pl sur 'e = Id(1) et p2 sur 1 On a not6 ainsi le groupe des c"-diff60morphismes de . I = [0,1]
qui sont --tangents a l'identit6 en
0 et
1
.
Mizutani utilise alors le :
THEOREIB fkther
(I)). Le grarpe DiffW0,l (I) des c"-diffdomorphismes dgaux 2 Z'identitg duns un voisinage de o et de 1 'est parfait. pour construire une concordance entre et
8,(gl)#$
#
%(g2)
h
h
= x(g1,g2)
6#
'&(gl)#
$(h)#
de I
<# a(g2)
. Etait crucial le fait que
ofi. h'e Diffs (I) 0,l
;
6'est ainsi qu'intervenait le
resultat de Mather. Enfin un cobordisme trivial re le nul-cobordisme cherche pour
x(gl,g2) (gl,g2)
x
I permettait de construi-
, come
le schematise la
figure 1. Le m h e principe peut servir 6 construire un nul-cobordisme pour
$,(g1,g2)
. On devra alors construire u n e concordance entre
9 (gl)r#.J(h)#B$%(g2
et
9L(g1)#C#i
%g2)
.
(figure 2)-
Figure 1.
Figure 2,
Cette fois DiffI(1)
. Notre
3h
h
oe h'
theoreme sera donc demontr6 si on sait prouver le :
iff:(I)est parfait.
THEOREME.
Soit Gm le sous-groupe de G+ en 0
appartient seulement 5
constitue des germes --tangents
. Le theoreme precedent resulte immediatement du Le groupe
THEOREkE.
e
:
G~ e s t parfait.
-
En effet, si g € ~iffI(1) , on pourra alors trouver gl,hl, ..,gp, h € DiffZ(1) P
tels que gbl,hl].
..[gp,hd
t iff:,^ (1) .
E
Mais, par le theoreme de Mather d6jL cite, un tel element est un commutateur. On va maintenant donner des indications sur la demonstration d'un resultat plus precis.
Soit
THEOREME.
e G-
. AZors i Z existe
f , 4 e Gm
teZs que
On notera de la mSme f a ~ o nun element de
defini sur un voisinage de
.
[0, 11
dans [o,-]
.
G+ et un representant Soit donc T-, € Gw ' defini dans
[o, 11 Soit par ailleurs f1E G+ defini par f1 (x) = al x a1 est assez grand, alors filn n l apas de point fixe. Par aillews fyl et f-1 T-, ont meme serie de Taylor L l'origine, et le t e m e 1 -1 # 1 . I1 resulte de linedire de cette serie de Taylor est a-1 X avec al -1 Sternberg (4) que les diff6omorphismes fl m t fil sont conjugues. I1 un voisinage de
avec al>l
existe donc
. SI
C Gm
defini dans un voisinage de
[0,1]
On aurait tennine si fl etait un Blement de
tel que :
Gm
;
bien au contrair
re, il etait necessaire pour utiliser le theoreme de Sternberg que
f;(O) # 1.
Supposons qu'on sache trouver f n -1 -1 a) n = f n 4 , f n 4, b)
(fn
- ~d)
oii
(0) = 0
si
tels que :
i < n
i O<x
j
Alors
fn et $n convergeraient respectivement vers
f et
4
;
de mdme pour toutes les derivges. Par continuit6 on aurait encore
Enfin b) et c )
impliqueraient f
, +EG=. La perfection de
Gm
serait demontree. 11 reste donc B construire les suites
(f,)
et
(On) verifiant
a), b) , c ) , dl. Supposons construits fn et 'n+ 1 et
'n+l Soit' 8:
[o,+mL+
[0,1]
On , et
cherchons a construire
une fonction nudrique cm verifiant .:
Soit E>O assez petit. Prenons
.
(x) = x+B (x/E) (f (x)-x) n Un calcul facile montre que si E est pris assez petit, la condition d) pour £n+l est realisee ; ceci resulte pour l'essentiel de la n-platjtude de f n - Id.
1
,.
. fn+l = fn fn
Posons
et. cherchons
+n+l
SOUS
la forme 'n+l=
I
n
cornme on sait qua Bquivalente B
.
gn fil $ 2 , la relation
= fn
0 = fn+l
.
i1 n+l n+l 'n+l
n' 'n. est
On
Ainsi
apparait c o m e la solution d'un probleme de conjugaison On-1 fn +n Mais gn est -tangente d Id en 0 , et -1 donc les series de Taylor en 0 de fn Pn et fn $n fn gn sont egales. Ces entre
fn fn
et
.
A
fn
-
series deTaylor sont non triviales, car f ; (x) = n n
-1
~
+
...
a xn+l ~ ++~
a n+l
-
I
> 0
0
implique
>o.
de Takens (5) permet des lors d'affirmer que fn fn et -fn UnOn resultat . Pour satisfaisont bien conjugues, et ce.par un glement de
fn gn re d) pour
, il
. Remarquons
1) .
Sera proche de
Gm
..
faut encore pouvoir choisir que si 'E = 0
in
on peut dans ce cas choisir
in
avec
( f
Id
, et
= Id
, alors in
. Si
gn proche'de Id pour
= 1d
, et
fngn = fng;l?n@n
;
est positif rnais assez petit, -1 fnen proche de fn$n fn$n ; on peut donc E
A .
donc
prevoir, si on peut appliquer un argument de continuite, qu'il sera possible
Qn
de choisir
proche de
Soit donc a)
f
b)
f(0) = O = g ( O )
et
g
Dn
.
Id
l'ensemble des couples
sont des cW-diff~om~r~hismes definis dans .un voisinage de [0,1].
, et
f(x)
C)
les series de Taylor de
a)
f(i)(o) = g(i)(0) = 0 si
f
, g(x) et
> 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
#
$
i
et
considerer le tore
s'obtient en superposant
et des feuilletages triviaw.
le feuilletage
Le theoreme 1 decoule donc du resultat plus faible suivant THEOREM 2.
:
Soit un tore de H2(~iffW[0,l]) tet que f n'admette pas de point f k e sur 2 'intervaZZe ouvert loll[ AZors GV() est nul.
.
111. Centralisateur de certains diff~omorphismes.
f
satisfaisant aux hypoth6ses du theorOme 2, on designe par cr(f)
le groupe des elements de de la topologie
cr
.
iff^ [0, 11
qui commutent avec
f
. On munit cr(£1
Si, a m deux extre'mitds de Z'intervaZZe [or]l , te jet infini de f est diffe'rent de cetui de Zridentite', il existe un sous-groupe femne' F de R et un isomorphisme continu de F sur cm(f)
PROPOSITION I .
.
Si, d une des extre'mitds au moins de Z 'intervaZZe [o, 1 1 , Ze jet infini de f est e'gat~dcetui de Z'identitd; it existe un sous-groupe femne' F de R et un isomorphisme continu de F sur c1 (f) PROPOSITION 2.
.
Remarque sera que
On notera ft f1=f
:
l'image de
t t F par cet isomorphisme et on suppo-
.
La demonstration des propositions 1 et 2 resulte du lemme suivant. k L E ! m . It existe ua c -groupe d 1 pmamdtre de diffdomorphismes de t [o,l[,~fo3 , teZ sue :
a) fo1 est dgaZ d la restriction de f d [or 1 [ ; b ) Z'appZication t -+ ft (x) est strictement monotone pour tout x dif0 fe'rent de o ; c) si h e iff' [or 1 [ et conormte avec f , iz existe t a W tez que t
f = h ; 0
d) k=m
SOUS
les hypoth2ses de za proposition I ; sinon k=l
Demonstration du lemme et SERGEREART [5], l'intervalle
:
D'aprOs des travaux de STERNBERG
.
, TAKENS
on sait qu'il existe un ck-champs de vecteurs
[71
to SUr
dont l'integrale au temps 1 est Lgale 21 la restriction [0,1[ de f 2 [o, 1 [ . Soit Ift1 le groupe 2I 1-paramgtre engendre par Eo. 110 reste 2 verifier l'affirmation (c) du lemme. Soit h e Diff1[0,1[ tel que
f o h = h o f latifs n
I.
et
[ , il
Etant donn6 un nombre xo410,1
m
tels que
fn(xo) < h(xo) < fm(xo) ft (xo) = h(x )
.. Le
te doeun nombre reel
t
tel que
f-t o h
f
et possede un point fixe
commute avec
0
existe deux entiers re-
. Par continuit6, il exis-
;
1 C -diffeomorphisme
il est donc Lgal b l'i-
dentit6. DQmonstration des propositions $
l'application
La notation $IJ designe la restriction de t k J Soft {fl] le C -groupe b 1 paramstre
:
.
8 l'ensemble
.
1 donne par 1 'analogue du lemme precedent sui: 1 ' intervalle .]0,1] Si h € C ( f), t h 1[0,1[ est 6gale b un certain fo De m6me la restriction la restriction
.
. En v6=it6,
est Bgale b un certain ':f t <
9
< t'
t = t ' car si on suppse
on aboutit b la contradiction suivante :
oil b l'in6galit6 inverse. On d6finit : F = {tern Puis pour tout t ftl] 0.11 = f i et 1
de
I
to
l]oil[
= f:l]o,l~
.
appartenqnt B . F on d6finit 1; diffeomorphisme ft par t ft [o, 1 [ = fo application t -+ ft est un isomorphisme
.
1
.
F sur C (f) Soit (tn) une suite dq616ments de F convergeant vers le nombre reel t Pour tout x~]o,l[ et pour tout n , on a 1'Bgalite tn t t x0 = f (x), et donc L la limite fo(x) = fl (x) Le point t appartient 1 Dans b F , ce qui implique que ce dernier est un sous-group fen& de R t t s parale cas de la proposition 1 , {fo} et {fl} sont des ~ ~ - ~ r o u pbeun 1 m8tre d'oa lq6galit6 : C (f) = cm(f)
.
.
.
.
IV. D6monstration du theoreme 2 dans le cas dg la proposition 1. Nous aurons besoin du r6sultat' suivant :
L E .
a ) Pour tout f e G , .te tope e s t nut. b) Pour tout tore , pour tout couple d'ektiers
ZrdgaZCtk. : = nm
.
Donnons par exemple la demonstration du (b). Soit tion de cation
T = S'XS' s de
dans
BG
representant le-tore
S1xS1 dans lui meme par
(m,n)
, on
a
une applica-
. On d6finit l'appli-
s(9 1 ,B 2) = (nB1,mfl2)
. L'applicatian
o s represente le tore
+2 .=
[T]
. D'oii
designant la classe fondamentale de Soit maintenant
et
f
existe un nombre ratiomel
.
satisfaisant aux hypotheses de la proposition 1
fog = gof
ga~iff~[~,l] tel que
T
:
. Si g =
tel que
3'
F
est un groupe monogene, il
fg
. DvaprCs le lemme precedent
est donc nul. Si F est Bgal B R , l'application t GV est continue et s'annule sur l'enshble des nombres rationnels.
le tore t
+
L'idBe de d b n t r e r le th6orhe 2 par un argument de densitB pour la
cm
topologie
g 6td d o m B e par R. MOUSSU.
V. Une presentation simpliciale de l'homomorphisme
Dans ce paragraphe
G
est Bgal B
simplicia1 suivant de l'espace classifiant
GV.
.
~iffzFt) On considere le d e l e BG :
x ( n ) = G~ -1 -1 (ii) ao(gl,g2,-- ,gn) = (g2g1 r ---rgngl) (i)
...;g n (iii) oO(gl, ...,gn) =
..- .gi-l ,gi+l,- .- ,gn)
= (gl,
et ai(gl,
pour
l4icn
pour
ldia
(l,glr- -, gn)
et ni(gl,..-,g n
= (gl,..-,gi,gi~--,
gn)
W. THUXSTON a indiqu6 que, dans ce modele, l'homomorphisme
le 2-cocycle simplicial d6fini pa= tore
-
h
GV(u,v) =
log(ue)d(lcq(v'))
est represent8 par. le 2-cycle simplicial (f,gof)
-
est
GV
, [I]
(g,fog)
. Le . D'oii
VI. Fin de la demonstration du theoreme 2. 1. On considere d'abord le.cas oc le jet infini de de 18identitB en
0 et en 1
. Lorsque
f
est &gal b celui
m
C (f) est'un groupe monogene, la dB-
monstration du theoreme 2 est identique b ceile du paragraphe IV. D'apres la considerer l e c a s oii il existe un isomorphisme con1 1 sur C (f) (avec f =f) , tel que lvimage reciproque de
proposition 2, il reste tinu
t
+
ft
de IR
cm(f) par cet isomorphisme soit un sous-group dense l'application de
H
dans R
dBfinie par
H
de R
O (t) = GV
.
. Soit
O
LEW.
, le
Pour tout tore
de Diff~[0,1] coincide avec celui de l'identite ([3]
et en 1
.
se grotonge en une appZication continue de R dans R
@
jet in£ini de g
, page
169). On peut donc uti-
liser la formule de Thurston (paragraphe V) en integrant sur
. Pour tout
luer GV
tion par partie
:
-1
teH
@(t) =
[0,1]
pour eva-
, on obtient apres decomposition et integra-
1
(2)
en 0
1
loq(ft'of)d(lcq
t' log(£ )d(1og)f*o ft)
-
f')
Le membre de droite de 11dgalit6 (2) est defini meme lorsque ft 1 est seulement un C -difft5omorphisme. Donc cette dgalitd definit une application
8 de
dans R
continue car
{ft}
prolongeant la pr6c6dente. Cette application est
est un C1-groupe ZI 1 param6tre.
On ach6ve la demonstration du theor8me en montrant que la restric-
a l'ensemble des'rationnels est nulle. Pour tout element t de
@
tion de
,
H et pour tout entier n dgalite se prolonge a X
,
=
8(1) = 0
9
8(nt)
est &gal 5
n8(t)
. Par densite cette
. Finalement, pour tout nombre rationnel .
2. On suppose maintenant que le jet infini.de
f est dgal
l'identite en 0 et estdifferent de celui de l'identitc? en geDiffm[O,l~
tel que fog = go£
1
celui de
. Soit
. D'apres les resultats de STERNBERG [6]
.
de , il existe un champ de vecteurs C1 sur ] b,]l cm , engendrant un groupe 1 param6tre { $ t ~ de diffeomorphismes
et de TAKENS [7] classe de ]0,1] prolonge 2
El
en un champ de vecteurs
seulement, et infiniment plat en 2
et au temps t
-f
t et que $ = g
tel que $
et
'g
tels que
, on peut
cm
, s'annulant
ainsi prolonger
f et g
LO,^ , infiniment tangents a -- --
On
en des diffdomorphismes
l'identite en 0 et en
2
, et
. Come dans la premiere partie de ce paragraphe, on
peut utiliser la formule de Thurston pour montrer que et $
.
en 1 et en
. En prenant les intdgrales au temps 1
de
fog = go£
pour un certain t
sur ]0,2]
les restrictions de
--
-
f et
a [1,iJ I)
,-.
GV<~,T> = 0 soit
. I1 est elair que
?
:
-
GV = GV+ GV ..,# e On conclut en remarquant que GV = 0 car f et g sont les int6grales 8 different temps d'un champ de vecteur cWdefini sur tout l'interirailc [1,2].
BIBLIOGRAPHIE
R. B07.T.
On some formulas for the characteristic classes of groupes-
actions (preprint) C. GODBILLON et J. VEY.
Un invariant des feuilletages de codimension 1.
C.R. Acad. Sc. Paris 92, t. 273, 1971. N. KOPPEL.
Commuting Diffeomorphisms. Global Anal., Symp. Pure Math.
1970, 14. J.N. MATHER. Integrability in codimension 1
. Commentarii Mathematici
Helvetici. vol. 48, fasc. 2, 1973 (195-233). F. SERGERAERT. Feuilletages et diff6omorphismes infiniment tangents 3 11identit6 ( 8 paraitre). S. STERNBERG.
Local contractions and a theorem of Poincare. Amer. J.
Math. 79, 1957, (809-824). F. 'I'AKENS. Normal forms for certain singularites of vector fields. Annales de 1'Institut Fourier, t. XXIII, fasc. 2, 1973. W. THURSTON,
Non cobordant foliations of
Soc., vol. 78, number 4, 1972.
s3
. Bull.
of the Amer. Math.
