On Some F o r m u l a s for the Charac te r i s t i c C la s ses

of Group-ac t ions

t Raoul Bott

1. Introduction. This is an account only of the ma te r i a l of my las t two l ec tu res of the

Rio conferenc% a s the e a r l i e r l e c tu re s dealt with well known ma t t e r s .

My emphas i s he re then i s the study of our foliation invar ian ts in the context of group

act ions on a manifold~ and I will s t a r t by showing you the natural i ty of our basic const ruct ion

now pays off by extending direct ly to the equivariant situation.

Recall then the main conclus ions of our e a r l i e r d i scuss ions . Essent ia l ly they

amounted to th is (see a lso [ i ] , [ 2 ] for detai ls) :

If we let JM denote the space of je t s - based at 0 E IR n - of d i f feomorphisms of

IR n to M ~ then JM c a r r i e s an a lgebra of natural f o r m s which is i somorphic to the

cechain a lgebra C*(an) where a n is the Lie a lgebra of formal vec to r - f i e lds on IR n .

In shor t there is a na tura l a r r o w :

(1. 1) C*(0n)--> Inv ~*JM Diff M

of C*(= n) onto the a lgebra of Dill(M) - invariaut f o r m s on J M . F u r t h e r we saw that

although ( t . 1) induces the 0 - h o m o m o r p h i s m in cohomology, it induces an in te res t ing one

once we divide JM by the natural action of 0 = the orthogonal group of IR n . n

Indeed then (1. 1) induces a "bas ic" h o m o m o r p h i s m :

(1.2) C*(an;On) - - > Inv f~*(JM/Ou) Diff M

and th i s map cer tainly r e c a p t u r e s all the usual cha rac t e r i s t i c c l a s s e s of M .

#The author gratefully acknowledges with thanks par t ia l support f rom the NSF under Grant MPS 74-11896.

(1 .3)

w h e r e

( i . 4)

More p r e c i s e l y we saw tha t :

WO n

26

H*(Q n, O n) ~ H*(WOn)

i s the d i f fe ren t ia l a lgeb ra given by:

W.% = I R [ C l , ' " , c h I ® E(h l , h 3 , " " h k)

with d i f f e ren t i a l :

and

the

k odd a n d = n o r n - 1

dCl®l = 0 , d(l®hi) = c i

denot ing the quotient of the indicated polynomia l r ing by the e l e m e n t s of

c, having dimension 2 i . I

F u r t h e r we saw - and th i s i s of c o u r s e qui te s t anda rd - that

so tha t (1 .2)

d im > 2n ,

H*(f~*(JM/On)) ~ H*(M)

induces a n a t u r a l m a p

H*(an, On) - - > H*(M) ,

and f ina l ly we ident i f ied the i m a g e of c2i u n d e r t h i s a r r o w with the Pontryagin c l a s s e s

of M .

The ma in v i r t u e of th i s point of view i s then, that we s ee tha t a manifold d e t e r m i n e s

i t s own Pont ryagin f o r m s n a t u r a l l y on the space

: J M / O ,

which I th ink of a s a n a t u r a l l y th ickened v e r s i o n of M , and tha t f u r t h e r m o r e t h e s e f o r m s

27

are invariant under any diffeomorphism of M .

Now let us put th is construction to use when an abs t rac t (L e . , d iscre te) group

acts on M

(~. 5)

and

( 1 . 6 ) BF : • ~ r E r x r

whose geometric real izat ions then respect ively correspond to, the M - bundle ~ I Mr I

action over the classifying space ]BF ] of F , and [BF itself. induced by the F

Now if

F

via diffeomorpbism. These data natural ly define two semis impl ic ia l manifolds

MF: M~- F x M E F x F x M

(1. 7) X : X 0 ~ X 1 ~ - X 2 " "

is any semisimplic ia l manifold the cohomology of i t s geometr ic real izat ion X 1 can be

computed in var ious ways. F i r s t of all the double complex ~**X

(1.8) ~ * * X : D * ( X 0) 6 > ~ * ( X l ) 6 > . . .

obtained from (1. 7) by applying the de Rham functor ~* to each (Xi) and then giving

O**X the sum of the de Rham differential and the differential operator 6 derived from

the s implicial s t ructure , computes H*( I X I) thus:

(i. 9) H*(I x I) = H{O**(X)}

On the other hand one also has the complex of "compatible forms" on the realization

of X :

(1.10) IxI = x 0UX 1 ×51 u x 2 x A 2 U - ' '

in the s e n s e of Whi tney-Thom-Sul l tvan .

aga in one h a s :

28

T h i s complex i s denoted by ¢2" [ X I , and once

(1 .11) H * ( [ X t ) = H * ( ~ * I X t ) •

F o r a f ine accoun t of a l l t h i s I r e f e r you to Dupont, [ 5 ] .

The f o r m u l a (1 .9) i s a l s o to be found in [ 4 ]o

In p a r t i c u l a r then , we can compute the cohomology of MF f r o m the double c o m p l e x

( I . 12) ~**h~? : ~*(M) 6 > ~*(M X F ) - - >

F u r t h e r m o r e a s we wil l s e e l a t e r , on the f i r s t cons t i tu ten t 6 t a k e s the f o r m

(1 .13)

Hence the n a t u r a l l i n e a r m a p

(i. i4)

b e c o m e s a c o c h a i n m a p only on the

6o0 I M x j = ~ ® 1 - j % . ~ ® l

O*M > ,~** MF

F - i nva r i an t f o r m s on M .

In v iew of th i s s t a t e of a f f a i r s it s u g g e s t s i t s e l f to r ep l ace MF by M F .

Then the a r r o w s (1.2) and (1.14) combine to y ie ld a chain map

(1. 15) c*on, On) - - > ~*(~) - - > ~ * * ~

On the o the r hand the homotopy equ iva lence of M with M eas i ly imp l i e s tha t of IvIF

and ~IF . Hence in homology (1. 15) induces a h o m o m o r p h i s m

( L 16) H*(an, O n) - - > H*(t MF t)

T h u s we s e e tha t the i nva r i ance of our c o n s t r u c t i o n i m m e d i a t e l y y ie lds "equ iva r i an t

c h a r a c t e r i s t i c c l a s s e s " for a F - manifold in I MF I . And h e r e of c o u r s e , a s the cohomology

29

of Mr has no a p r io r i bound~ all the c l a s s e s of WC potentially come into play. n

Finally if M is a compact F - or ientable manifold we can define the equivariant

cha rac te r i s t i c number s of M by following (1. 16) with integrat ion over the f iber in the

f ibering

There r e s u l t s an additive homomorph i sm

MF

BF .

(1.17) H*(an;O n) - - > H*(BF) ,

and my a im in the next sec t ions will be to der ive some explicit r ec ipes for ( 1 . 1 7 ) , and

to rev iew some of T h u r s t o n ' s examples in th is f ramework .

2. F o r m u l a s for the Godbillon-Vey Class in H*(BF). Note that (1.16) can also be thought

of in th is manner .

The d i s c r e t e n e s s of F natural ly defines a folitation gF on MF which i s t r a n s -

v e r s a l to the f ibe r s in the project ion MF ~ BF, and of codimeasion n . The charac te r i s t i c

c l a s s e s of ~F integrated over the f iber induce (1 .16 ) . With this in terpre ta t ion it sugges t s

i tself that the cons t ruc t ions which a re known to r e p r e s e n t the cha rac t e r i s t i c c l a s s e s of

foli tations should extend to the semis impl ic ia l situation provided only that one has a suitable

de Rham theory at hand. Now the double complex Q**X does not fit the bill because i t s

muItipl icat ion i s not artticommutative. On the o ther hand the compatible complex of Dupont

i s perfec t Iy suitable and the re fo re yields explicit r ec ipes quite readily.

30

Let m e s t a r t with the Godbi l lon-Vey c l a s s

ding to a g e n e r a t o r of H3(a I ' O1) - i s defined on o r i en ted fo l ia t ions

and can be computed a c c o r d i n g to the a l g o r i t h m :

, itself. This i nva r i an t - correspon-

of c o d i m e n s i o n one,

Let ~t be d e s c r i b e d a s the ke rne l of a n o n - d e g e n e r a t e 1 - f o r m .~.

Then in tegrab i l i ty i m p l i e s tha t t h e r e e ~ s t s a 1 - f o r m ~ with

(2. 1) Now se t

a fol ia t ion

m a p s :

d~0 = ~A~ .

Then d ~ o = 0 .

= ,r'l A d ~

F u r t h e r the cohomology c l a s s of ~ is independent of

the c h o i c e s involved, and r e p r e s e n t s ~0(3).

The ex tens ion of (2. 1) to the s e m i s i m p l i c i a l c a s e i s now i m m e d i a t e .

Given a s . s . manifold

X : X 0 ~ X IE X 2 0 " "

on X i s s i m p l y a fol ia t ion 3k on each X k , such that , a l l t he s t r u c t u r e

a r e t r a n s v e r s a l to

X(~) : X k - > X k,

~k' and induce i s o m o r p h i s m s

~(rv) : ~k-- - - :> X(oO" 1 gk,

]31 on the g e o m e t r i c r ea l i za t ion Such data then define a foliation

in the following manner :

IxI of x,

On X k × A k ~ the n a t u r a l p ro jec t ion

(2. 2)

-1 i nduces the fol iat ion ,IT L o ~k

which a s s e m b l e the X k x A k

" F a t r ea l i za t ion" of IX I ,

i s s u c h a boundary map, and

the c o r r e s p o n d i n g ma p of t h e k - I

31

~r L X k < - - X k ×A k

and t h e s e fo l i a t ions a r e compat ib le unde r the ident i f ica t ions

to f o r m I X 1 . P r e c i s e l y we have in mind h e r e the so - ca l l ed

which t h e r e f o r e only iden t i f i es by the boundary maps . Thus if

A k - 1 5 k : >

s i m p l e x A k- 1 onto a f ace of A k then the ident i f i -

ca t ion c o r r e s p o n d i n g to a i s d e s c r i b e d by the d i a g r a m :

X k x A k ' l lEA(a) Ak ............ > X k ×

(2 .3) X(a) X 1 ]

Xk_ 1 X Ak-1

That i s , the two i m a g e s of a point (p ,q) E X k x A k-1

v e r t i c a l a r r o w s a r e to be identif ied in I X I •

on

under the horizontal and

It is clear then that the ~r o ~k do define a compatible collection of foliations

I X I , and that is precisely what one means by a foliation on I X l-

Similarly, one defines the de Rham complex ~* I X I of "compatible forms" on

I X t in t e r m s of the d i a g r a m (2. 3 ) : T h u s :

(2.4)

A q - form~o on I XI

on X kxA k such that

32

isa collection - ~{¢Ok~ of q- forms

{i x A(a)}% k : {x(a) x i}*%_z

Finally one has the Dupont extension of the Whitney-Thorn-Sullivan Theorem to the effect

that

H*(~X) ~ H(~**X)

the i s o m o r p h i s m being induced by " integrat ion over the s implexes"

L (2. 5) ~ ~ ~ ~.

where ~r L is the project ion (2.2) and lr L denotes integrat ion over the f iber of ~r L .

Note that the sum has only q nonzero t e r m s , when ~ is of dimension q .

Now then, with all th is understood our f i r s t r e m a r k is :

PROPOSITION 2.6. Let ; he an oriented codimension 1 foliation on the pa ra -

compact s implicial manifold X . Then w (5) E H*(~ [ X ]) can be computed by the Oodbfllon-

V.ey...algorithm (2.1) provided only we in te rp re t " fo rm" to mean "compatible fo rm" on l X I •

The proof of th i s i s quite s t ra ight forward , so let me only s t a r t the a rgument and in

the p r o c e s s derive an explicit a lgor i thm for t~(~) in t e r m s of the s t ruc tu re maps of X .

Consider then an ~ as envisaged in the proposi t ion, and let ~0 on X 0 be

represen ted as kernel of ~0 with

d ~ = ~ .

33

We next try to extend ~ and ~ to X I x A I in a compatible manner. To fix

the notation let 0 and 1 be the vertices of A 1 and let x 0 and x I be the

A I corresponding barycentric coordinates on . Also let

0 AO Al 1 (7 : - -> , and c~ : A O - > A 1

be the inclusions sending A 0 to 0 and i respectively, and let frO and (71 be the

cor respond ing maps :

Xo ~ Xl

Now by our hypothes is on ;Y the f o r m s

cr~W and cr~¢~

both represent 51 in the same orientation. Hence there exists a smooth positive function

/~i on X 1 such that

Using it 1 we now cons t ruc t the fo rm

1 x 1

(2.7) ~01 = (7~ . /Z I on X I × A

A 1 Here we have identified the f o r m s on X 1 w/th t he i r pullback to X 1 × under 1

1r L and ~1 is defined by

x 1 ~l(p)Xl(a) (p, a) =

34

Hence ~1 r e s t r i c t s to (y~¢~ on ~(X 1 x A 1) and is compatible . F u r t h e r m o r e

the ke rne l of O1 c lea r ly r e p r e s e n t s rtLl~Yl. Indeed any nonzero mult iple of ¢~eO would.

Next we wish to extend rl compatibly to X 1 x A 1 . F o r th is purpose d i f fe ren t ia te

(2. 7 ) . One obtains

(2. 8) d ~ l = (log ~1 "dx l + x l d log b~ t + c r ~ ) ^ ¢Pl

Now the t e r m in the b racke t i s not compat ib le with ~ a s i t s tands . However , i t can be

modified to become so in the following manner :

By di f ferent ia t ing (2.6) we obtain

= . = +c r*~ l ~*¢~ du:G (TI~ O':e {d log ~i 1 0 J 1

dlOg~l in (2.8)by cr:~-Cr~ toobtain:

d~o I = {iog~ i dx I +x I ¢T~ I +x 0 a~} AV I

whence

(2.9)

Hence we may rep lace

and th is t ime the t e r m

~]I = {log ~i I dx I + x i (~ + x 0 ~} (2. 10)

i s c l ea r ly compat ib le with ~ .

This cons t ruc t ion now extends to all of

¢Pk and ~k on X k x A k . col lect ion

i X i to yield the following compatible

For each

the inclusion of

Then we define

and finally set

(2. 11)

and co r re spond ing ly se t

(2. 12)

35

k , we le t o'0,0"1, ° - ' , o" k be the m a p s of

AO into A k as the k + l ve r t exes .

bLi, i = l , ' " , k , by

X k - X 0 co r r e spond ing to

k xi

The f o r m s ~k A d~k = ¢ek(l~ ) a r e t h e r e f o r e again cgmPatible~ c losed, and give

an a lgor i th m fo r comput ing ¢0(~) in H*( 1 X I )

At this stage one may of course return to the more economical complex

i n t eg ra t ing ove r the s impl i ces . Then

~**(X) by

L ~ ~o3~ 0 + ~02~ 1 + wl~ 2 + co0, 3 (2. 13) ~ . ~( ) =

• " i with u~ ~' ] 6 D (Xj) and us ing (2. 13) one obta ins expl ic i t fo rmulae . Fox example :

(2. 14) 0 3 , 0 = ~ A ~ 0 ,

while J , 1 i s obtained by in t eg ra t ing HI A d ~ l ove r the I - s implex 41 . Thus t h i s

component i s given by :

36

f l ( X ~ ;~ + x 1 (r~q + log DI dxl) A (dxl (cr~ - Cr;~) + d log ~i dxl

0 + x d o ~ + x l d ( ~ )

Only the t e r m s involving a dx survive and hence with a l i t t le a lgebra one

arrives at

(2. 15) 02 , 1 = Cr~ A cf;'~ - d{ log ~ll( 2 )}

To obtain the next t e r m we proceed s imi l a r ly with

(2. 16)

~2 A d~2.

01, 2 2 f2 (dlog - cr;~) dx idd 0 < i j < 2 = i,j=O log ~i ~j -- ' --

E (- 1) i - j +I log ui(d log ~lj- CT~'~ ) O<i<j<2

The resu l t is

Final ly w0, 3 van i shes because the re i s no t e r m involving t h r e e dx ' s in ~3 ^ d~3 "

So f a r I have d i scussed only the or iginal Godbil lon-Vey c l a s s u~ for fol ia t ions of

codimension 1 . If ~ is or ientable but has h igher codimension say q - t he i r c l a s s u)

i s genera l i zed to F , by again desc r ib ing F as the kerne l of a decomposable fo rm ~p ,

th i s t ime of d imension q . Then in tegrabi l i ty again impl ies that

d~ = ~A~

but now it is the form

oo(~) = ~ A (d~) q

which i s an invar tant of a and thus g ives the extended Oodbil lon-Vey c l a s s in H2q+I(M).

In WO th is c l a s s then r e p r e s e n t s h 1 c l q . q

37

In the s e m i s i m p l i c i a l s i tua t ion t h i s ex t ens ion works: equal ly wel l and (2. 12) aga in

y ie lds an explici t a l g o r i t h m for the computa t ions of co(J). Note by the way that in

cod imens ion q , only c o m p o n e n t s of t h e type:

(2. 17) ~r L co(g) = o~ 2q+l ' 0 + . . . . + co q, q+l

s u r v i v e b e c a u s e ~k A d ~ k q can conta in no m o r e than (q + i ) dx ' s fo r arty k

F u r t h e r m o r e the e x t r e m a l t e r m s a r e e a s y to compute f r o m (2. 12) :

¢o 2 q + l ' 0 = ~ A (d~) q

(2. 18) ~ q , q+l i i

= f (log/~i dx ) {(d log/~j - c~;~) dx~) q }

Aq

T h e r e a r e c o r r e s p o n d i n g f o r m u l a e for the o t h e r c l a s s e s in H*(WO n) but they a r e

h a r d e r to wr i t e down expl ic i t ly , becaus% a s in the e a s e of j u s t a s ingle fol iat ion : on M ,

one h a s to b r i n g in a c o m p a r i s o n of t o r s i o n - f r e e and Riemarmian connec t ions .

However the compat ib le complex aga in enab le s one to extend the s i n g l e - s p a c e r e c i p e s

to the s e m i s i m p l i c i a l c a s e in a m o r e o r l e s s s t r a i g h t f o r w a r d m a n n e r , but r a t h e r than d i s c u s s i n g

t h e s e le t m e now spec ia l i ze o u r f o r m u l a e to the c a s e of a g roup act ion. We t h e r e f o r e have to

recall briefly how MF and

Recall first, that if C

object

(2. 19)

s t a r t i n g wi th the ob jec t s

m o r p h i s m s in C .

BF a r e c o n s t r u c t e d and why MF c a r r i e s a na tu ra l fol iat ion.

i s a a ca tegory , then it de f ines a na tu r a l s e m i s i m p l i c i a l

BC : O(C) ,~-- ~I(C) ~ ~,2(C)

O(C) of C , and hi(C ) beingthe i-tuples of composable

Thus a typical element in r0,k(C) is given by the diagram:

38

71 7"2 Yk

(2. 20) . ~ - ' ~ . ~ " ' ~ . . . . . ~ - - ~ .

the y ' s being a r r o w s in C . The boundary maps in BC a re then given by sending

such a d iagram into one with e i ther an end a r r o w deleted, o r two consecutive one composed.

F o r example

~0(71,72 ) = 72

(2. 21) ~i(71,72 ) = 71 o 72

~2(71,72 ) = 71

This construction applied to the category determined by the group F yields BF o

F M determined by the action of F on M , it yields MF: When applied to the category

Thus

MF --- BF M

where the category F M has a s objects the manifold M , and as m o r p h i s m s p a i r s (y ,m)

with Y 6 F , m 6 M . Finally the composi t ion

(2.22) (71 ,mi ) o (72, m2) = (71 o Y2,m2)

ex is t s only if 72(m2) = m 1 and is then given by the RHS of (2. 22) .

Pictorially these a re then bes t descr ibed by a r r o w s of the fo rm

7

(2. 23) 7(m) m

In any case, all of this granted it is easy to see that the following proposition holds.

39

PROPOSITION 2.24. Let F be a foliation on M , such that F a c t s t r a n s v e r s a l l y

to a on M . Then a extends to a foliation FF on MF.

In par t i cu la r the foliation of M by points induces a natura l foliation a M o_..~n MF.

Note of course that a M then co r r e sponds to the foliation by points of MF considered

a s a simplicial manifold in the obvious sense.

When we special ize our formulae to a M , matte.rs simplify to some extent as now

can c lear ly be taken to be zero. K we f u r t h e r m o r e follow our recipe, in the case when

M is oriented and compact , and integrate cO(aM) over the f iber in the project ion

MF >BF

then - M being q - dimensional only 0~q, q+l en te r s into the computation. In th is

manner we obtain the following algorithm for

t7, a)(a M) •

PROPOSITION 2.25. The charac te r i s t i c "number" y . o~(~ M) E Hq+I(BF) i s

represen ted by the cocycle :

... if q+ (log ~l i dx i) (d log ~j dxJ) q (2.26) ~0(~i, , ~q+l ) = I×M

where the Di a re functions on M , determined in the following manne r :

Choose a volume ~0 on M , and then define ~i by:

• -" y .i)*~ =~i $ (2.27) ( Yi Yi+l q*

40

R e m a r k . ( i ) T h u r s t o n h a s been a w a r e of t h i s f o r m u l a a l l a long a l though h i s con-

v e n t i o n s and h i s se t t ing might have di f fered somewha t . Fo r t h o s e of u s l e s s g e o m e t r i c a l l y

i n s p i r e d it was however not so obvious. On the o the r hand - and t h i s i s the ma in point of

m y r e m a r k s - in the context of the compat ib le complex the fo rmu la does b e c o m e eas i ly

a c c e s s i b l e .

(2) The following g e o m e t r i c a l i n t e rp re t a t i on of 2. 25 was pointed out to m e by

T h u r s t o n qui te r ecen t ly . If Yl ' "" " ' Yq+l a r e c o o r d i n a t e s in •q+l then the f o r m

/K (2. 28) • = E ( - I) i Yi dYl A . . . . dy i A .** dy n

c l e a r l y s a t i s f i e s the ident i ty

d g~ = n v o l u m e in IR n .

On the o t h e r hand, up to a u n i v e r s a l c o n s t a n t (2 .26) c l e a r l y g ive s the i n t eg ra l of

pul led back to M u n d e r the m a p

fF : M > IR q+l

def ined by

i (2. 29) y o f F = l°g(/gi)

Thus our formula for the Eilenberg MacLane cocycle ¢o(~)

(2.30)

can be wri t ten;

¢~(~) (~1 ' " ' " ' Yq+l ) = volume enclosed by fF M in 1R q+l .

41

3. The Infini tesimal Case. When the action of F on a foliation

by res t r i c t ion of an action of a connected Lie group G containing

c l a s s e s of the F - action can be descr ibed in L ie -a lgebra t e r m s .

over M comes

F , the characteristic

Such a description is

often m o r e easy to handle, and our a im in th i s section is the re fore to desc r ibe an infini-

t e s ima l analogue of (2. 18) in this f ramework . The technique involves the lifting of a to

to (G/K) x M , with K a maximal compact subgroup of G ~ and is thus typical of the

Van-Est t h e o r e m s and t h e o r e m s on continuous cohomology, as well as of the whole approach

of Kamber and Tondeur to fotiated bundles. See [ 7 ] .

However I will not pursue these aspects , but r a the r head as quickly as poss ible towards

the inf ini tes imal formula for ~ ( ~ ) .

Recall then, f i r s t of all , that G/K is contractible:

(3. 1) G/K = pt

Hence if we cons ider the natural F action on

equivalence of the quotient of G/K by the action of

(3.2)

Thus

in [G /K ] P ,

(3. a)

Because

f o r m s on G/K ,

~*(E; K) .

[ G / K ] F ~

G/K then (3. 1) extends to a homology

F (denoted [ G / K ] F ) w i t t l BF , "

[G/K ] F is a thickened ve r s ion of BF . Now cons ider the edge h o m o m o r p h i s m

inv ~.*(G/K) - ~**( [G /K ] F) F

F c G , the left hand side cer tainly contains the subcomplex of G - invariant

which is finally identified with the re la t ive Lie a lgebra complex

Thus we have

42

(3 .4 ) •*(g; K) =- Inv f~*(G/K) c Inv ~*(G/K) G F

t combining (3.4) with (3.3) and (3.2) gives r i s e to an a r row

(3.5) I., : H*(g;K) - H*(BF) ,

and our f i r s t a im is then to show that:

PROPOSITION 3.6. In the situation envisaged the cha rac te r i s t i c c l a s s e s of the F

action lift natural ly to H*(g; K)

To see this r esu l t one f i r s t has to unders tand the following d iagram of G spaces:

( 3 . 7 ) G X M t > (G/K) X M fr> M , K

where fr is the natural product projection, and the action of G on (G/K) x M is the

product action. On the other hand G ac ts on G x M purely on the Ieft~ and G x M is K K - -

defined a s the quotient

(3. 8) G x M = (G x M)/K K

with K acting by

(3. 9) (g, m ) . k = (gk, k ' l m )

Finally the twist map t is induced by

(3. 10) '~: G x M - G x M

sending

(3.11) (g ,m) to ( g , g . m )

In [ 5 ] and also independently in a recent paper of Shulman and Tischler , this a r r o w is explicitly descr ibed on the chain level.

43

It i s then c lea r that t i s a d i f feomorphism which sends the K action (3.9)

into the K action ( g , m ) ~ ( g k , m ) and so induces the equivalence t of ( 3 . 7 ) . The

v i r tue of th i s t w i s t ~ g map

the left t r ans la t ion of G .

identified with the f o r m s on

t , i s of cour se that the G - action on O x M is given by K

It follows that the G - invar iant f o r m s on G x M can be K

G x M~ which a re

(1) invariant under left t rans la t ion of G by G , an_._dd

(2) K = bas ic under the action (3.8) of K on G x M . Thus:

(3. 12) Inv ~*(G x M) ~ K basic f o r m s in f~*(~[) ® ~*M . G K

In any case , the plan of p rocedure is now s u g g e s t e d b y t h e diagram:

(3. ~3)

Inv O* (G x M) -* f~**(G x MF) ~ ~**(MP) G K K

Inv ~*(G/K) ~ i'~* (G/KF) ~ f~**(BlT) G

where the lower line induces ( 3 . 5 ) , and ~r, denotes integrat ion over the f iber. The

homotopy equivalence in the upper Hne is of cou r se again a consequence of the contractibil i ty

of G / K . In view of (3. i3) our lifting prob lem clear ly amounts to real iz ing the charac te r i s t i c

c l a s s e s of ~ lifted to G x M , by G - invariant fo rms . Thus we need the following.

PROPOSITION 3. 14. The cha rac t e r i s t i c c I a s s e s of g admit a natural l i f t ing to

the complex (3 .12 ) .

Let me c a r r y out the proof~ but again only for our Godbillon-Vey c l a s s o~(F). Consider

then the pull-hack ~ of ~ to G x M , under the map

~o~ (3. 15) G x M > M .

44

To d e s c r i b e ~ let u s ident ify the tangent space of G X M , at ( g ,m) with

g $ T m M , u s i n g the le f t i nva r i an t v e c t o r - f i e l d s on G to identify Gg with g ~ and

u s i n g the p ro jec t ions for the d i r ec t s u m decompos i t ion . A l so ff x E g i s an e l e m e n t of

the Lie a lgeb ra of G , le t :~ E F(TM) be the c o r r e s p o n d i n g in f in i t e s ima l mot ion on M

induced by the act ion of G on M .

P r e c i s e l y if e tx i s the o n e - p a r a m e t e r subgroup g e n e r a t e d by x E g , then

(3. 16) :~ = tangent of e ~ m at m . m

With th i s unde r s tood , the ke rne l of ~ i s d e s c r i bed by:

x + y E T(g, m) (G x M) i s in "~ if and only if

(3. 17) :~m + Ym E

Indeed the c u r v e (ge tx, m) goes ove r into getXm unde r our map, and hence i t s

t angent at t = 0 g o e s to g * ~ m " On the o the r hand y goes to g*Ym" Hence

g*(Xm + Ym ) E g . But G p r e s e r v e s the fol iat ion so tha t (3. 17) fol lows.

An i m m e d i a t e c o r o l l a r y of (3. 17) i s the following:.

PROPOSITION 3. 18. Let the fol iat ion ~ be desc r ibed a s the ke rne l of the

d e c o m p o s a b l e q f o r m ¢0. Then 3 a d m i t s a na tu ra l r e p r e s e n t a t i o n a s the ke rne l of a

d e c o m p o s a b l e f o r m ~0 E f~*(_g) ® Q*(M) . F u r t h e r m o r e~ in the na tu ra l double g rad in ~ of

t h i s c o m p l e x ~ h a s componen t s :

~ 0 , q ,~ = , ~ q , O + . . . . +co

with:

,~0, q = • .

(3. 19)

w h e r e

45

~l,q-I

x a r u n s o v e r a b a s i s of g_, a n d x ~ o v e r a dual b a s e in g* .

Proof. T h i s i s pu re ly a l i n e a r a l g e b r a m a t t e r . If e i s a 1 - f o r m with

i t s ke rne l , then

(3. 20)

wil l have "~ in i t s ke rne l .

~(x + y ) e = e(y) + e(~) = o .

Hence if e 1A ° ' ' A 0 q = ~ local ly , then

expanding t h i s p roduc t c l e a r l y y ie lds (3. 19 ) .

To p roceed f u r t h e r we need to compute d o

complex ~*(g) ® o*(M) .

F o r t h i s p u r p o s e , le t u s se t

Indeed for any x + y sub jec t to (3. 17) we then have

O 1 A . . . AO q d e s c r i b e s "~ local ly , and

and u s e the double index s u m m a t i o n convent ion, so tha t

(3.21) V = ~ + x ~ A ~ ¢ ~ ÷ " "

d e s c r i b e s the "beginning" of ~ .

and e x p r e s s it a s ~ A

Also , le t u s a s s u m e tha t on M the in tegrab i l i ty of ~ i s e x p r e s s e d by

(3. 22) d~ = ~ A

w h e r e D i s a global 1 - f o r m . Then I c l a i m that:

in the double

PROPOSITION 3. 23.

such that

(3. 24)

(3. 25)

(3. 26)

Fur the r the

Proof.

Fu r the r

where

~, £(~), x ~ g

(3. 27)

46

The ~ of ( 3 . 2 2 ) l i f t s n a t u r a l l y t 0 o n e ~ in f~*g@f~*M

is given by

= ~ - x ~ ( u ( ~ ) - , ( ~ ) } where ~ ( x ) i s a e ~ i . e d by ( 3 . Z 7 ) .

Differentiating (3.21) yields

d ~ = d~ - x~x A d % + " "

d% = d~(~ )~ = £ ( ~ ) ~ - ~(~)dv.

£ ( ~ ) is the Lie derivat ive in the direct ion :~. Because

must p rese rve ¢0 up to multiples, whence

G p rese rves

£(~)~0 = ~(x)¢0 , x ¢ g , with /~(x) E ~°(M)

This /~(x) is , of course, the infini tesimal analogue of the /~((r) in Section 2 .

In any case, combining (3.25), (3.26), (3.27) with (3.22) one obtains the formula

which, up to t e r m s of order

(3. 2s)

+ % A ( ~ ( ~ ) ~ - '~ s, % ) + . . .

> 2 in the g direction, is given by

47

But a s ~ e x i s t s and c l ea r ly i s the s u m of f o r m s of type (1, 0) and

f i x e s ~ . Q . E . D .

To a s s e m b l e the p i e c e s , we sha l l have to d e t e r m i n e whe the r ~ and ~ a r e K

b a s i c in our complex .

In g e n e r a l t h i s wil l , of c o u r s e , not be the c a s e . However , by ave rag ing ove r K ,

we can a r r a n g e it tha t both the ~ and the ~ of o u r d i s c u s s i o n a r e i nva r i an t u n d e r the

ac t ion of K , L e . , tha t i n f in i t e s ima l ly

(3.29) £(~)¢0 = 0 ; £(~)~ = 0 for x E k

and u n d e r t h i s h y p o t h e s i s we have the following.

PROPOSITION 3.30. The condi t ion (3.29) i m p l i e s that ~ and ~ a r e K ba s i c

r e l a t i v e to the ac t ion of K on G x M .

T h e proof of t h i s fac t i s a s t r a i g h t f o r w a r d check (though not qui te t r iv ia l ) which I

will t ake up in g r e a t e r gene ra l i t y at a n o t h e r t ime .

At t h i s s tage , we a r e r eady to give an i n f in i t e s ima l r e c ipe for to (J ) .

Indeed, expanding ~ ( d ~ ) q , will have al l p o s s ib l e type of components :

o~(~) = o ~ 2 q + l ' 0 + . . - + 0 , 2 q + l ,

of which the simplest are given by:

(3. 31) 0 , 2q+l . d~ q = ~

(3.32) wq+l,q = - x' A "'" x' u du A ' " dya ~1 %+1 ~i ¢¢2 q+l

w h e r e we have now se t

t The K b a s i c f o r m s a r e t h o s e annih i la ted by the v e c t o r f i e lds a long the orbi t of the ac t ion and inva r i an t u n d e r tha t act ion.

(0, 1) this equation

(3.33)

and have abbrev ia ted

If we think of

va lues ill ~* M )

n°(M):

48

v(x) = ~(x) - ~(:~) x ~ g

v(~) to v .

f~*(g )®a*(M) as the complex a*(_g;O*(M)) of f o r m s on g with

then (3.33) can be thought of a s a 1 - f o r m on g with va lues in

u E ~ l ( g ; ~ ° ( M ) ) ,

and t h e r e (3.32) t akes the fo rm

. .o 1 " . . . d v ~ x i ) . . , dV(Xq+l) (3.34) o~ q+ l ' q(xl , , Xq+ 1) = ~ E (- 1) 1 U(xi) dv(x 1)

Hence we get the fo l lo~ing co ro l l a ry , wtRch i s an in f in i t e s ima l analogue of 3 . 2 5 .

PROPOSITION 3.35. Suppose Z C M i s a cycle of d imens ion q on M . Then

(3.36) / ' / c o ( a ) E Hq+l(g ; K)

Z

i s r e p r e s e n t e d by the cocycle

( 3 . 3 7 ) COZ(a) ( X l , ' - " , Xq) = f V(Xl)dV(x2) " ' " dV(Xq+l)

Z

whe re V(x) i s defined by

(3.38) {~(x) - ~(~) }~ = v (x )~

4. On the Examples of Thurs ton and Hei tsch.

va ry in g c l a s s e s co(a) i s roughly a s follows.

The h i s t o ry of example s of fo l ia t ions wi th

49

In the complex ana ly t ic case~ I had o b s e r v e d a l r eady be fore 1970 that the foliation:

(,. l) ~ = {~iZl ~ + ~2z2 -~2 )

¢ 2 on - 0 , had for i t s g . v . invar ian t :

and thus varied continuously with k.

k 2 + - - - 2 } ,

k I

At tha t t i m e , I thought tha t the c o r r e s p o n d i n g r e a l i nva r i an t would a lways van i sh .

However~ soon t h e r e a f t e r in 1971, the p a p e r of Godbi l lon-Vey appea red wi th the R o u s s a r i e

e x a m p l e of the fol iat ion ~ on F~SL(2 ; IR) induced by the Lie a l g e b r a of t r i a n g u l a r m a t r i c e s

in SL(2 ; IR) , F be ing a d i s c r e t e subgroup with c o m pac t quot ient space .

T h e r e a f t e r , T h u r s t o n p roduced h i s e x a m p l e s . In p a r t i c u l a r he cons t ruc t ed e x a m p l e s

S 1 of a f ami ly of ac t ions of a group F ac t ing on , such tha t the c o r r e s p o n d i n g g .v . n u m b e r

E H 2 ( B F ) v a r i e d cont inuous ly . In an appendix, Rober t Brooks h a s wr i t t en up the de ta i l s

t r e a t i n g t h i s e x a m p l e with the f o r m u l a (2.25) m u c h l ike Dupont in [ 5 ] t r ea t ed the Eu le r

c l a s s of f lat bundles . Here le t m e j u s t out l ine and c o m m e n t on the plan of th i s v e r y ingen ious

example .

We s t a r t by o b s e r v i n g tha t SL(2 ; IR)

az + b (4. I) z - - >

cz + d '

a c t s on S 1 in the c l a s s i c a l m a n n e r

1 [ab~ [zl = '\cd/ESL(2'IR)

S 1 Hence for any F ~ SL(2, IR) t h e r e i s a n a t u r a l ac t ion of F on .

W h e r e I~SL(2, lR) i s compac t , H2(F ; IR) # 0 and the g .v . c l a s s of the act ion will a l so be

nonzero . On the o t h e r hand, a s we le t F v a r y in a con t inuous f ami ly of such subg roups ,

50

the c o r r e s p o n d i n g c h a r a c t e r i s t i c c l a s s does not v a r y .

do not f u r n i s h varying e x a m p l e s .

T h u r s t o n t h e r e f o r e twis ted t h e s e h o m o g e n e o u s ac t ions on

rnanner.

Consider the double cover

S I ' ,¢t > S 1

given by send ing z to z2 ; I z I = 1 . Then e v e r y d i f f e o m o r p h i s m

p r e c i s e l y two l i f t ings "~ r e l a t i v e to ~ . T h u s ,

' ~ : S 1 >S 1

is a diffeomorphism of S 1 , with

~.of = fo~

On the double cove r Diff (2) S 1 of Diff S 1 the funct ion

s ingle va lued and de f ines a h o m o m o r p h i s m :

s 1 ~'2 (4. 2) Diff (2) > Diff (S I)

Note tha t if f i s l i f ted to an " e q u i v a r i a n t m a p " :

t hen f i s r e p r e s e n t e d by:

_T(x) =

f : IN > I R

Thus, the moduli of Riemann surfaces

S 1 in the following

f of S 1 admits

f -->'~ now becomes

_f(x + 2~) =_f(x) + 2~,

1 / 2 f ( 2 x ) o r 1 / 2 { f ( 2 x ) + ¢ t }

51

Hence, in pa r t i cu la r , if f i s a ro ta t ion by a and hence r e p r e s e n t e d by

x >x+o~

then f is represented by:

(4.3) x - - > x + ~ / 2 , and x - - > x + ~ + r r

It follows that if the map

SL(2, JR) > PSL(2, IR)

given by (4. 1) , i s l i f ted to a map of SL(2, tR) to

t h e r e r e s u l t s a homomorph i sm of

(4. 4) SL(2, IR) * SL(2, IR) so(2)

> D i l l S 1

Di.ff(2)(S 1 )

1% > Diff (S 1)

and then followed by ~2 '

where on the lef t we have in mind the f r ee product amalgamated along the ro ta t ions accord ing

to (4. 3) , and it i s th i s action which g ives r i s e to T h u r s t o n ' s example. More p rec i se ly , he

takes the F r e p r e s e n t e d by g e n e r a t o r s

{X,V,Z,W}

and with the re la t ion

(4.5) [ X , Y ] = [ Z , W ] ,

chooses X,Y 6SL(2; IR) so that [ X , Y ] i s a ro ta t ion by

detai ls) , and a lso Z ' , W ' 6 SL(2,1R) so that

Z' and W' and applying ~2 ' one obtains

re la t ion (4. 5 ) .

0 < ~ < ?r , ( see Appendix for

[ Z ' , W ' ] i s a ro ta t ion by 2 5 . Then l i f t i ng

Z, W in Diff S 1 , which c l ea r ly sa t i s fy the

52

Now v a r y i n g ~ , T h u r s t o n obta ins h i s example . Note tha t a s th i s example i s

obtained by a m a l g a m a t i o n of two i n f i n i t e s i m a l s i t ua t i ons , i t cannot be d i r ec t ly t r ea t ed by

our i n f i n i t e s i m a l me thod , a l though, a s Bob Brooks point out our global and in f in i t e s ima l

c o c y c l e s a g r e e w h e r e they should.

More r e c e n t l y T h u r s t o n h a s found e x a m p l e s of v a r y i n g the h i g h e r g . v . c l a s s e s by

a c t i o n s of c e r t a i n F ' s on the s p h e r e s . T h u s , h i s e x a m p l e s ac tua l ly v a r y the c h a r a c t e r i s t i c

n u m b e r s of ce r t a in F ac t ions .

Dur ing the Rio con fe rence , J a m e s He i t sch s u g g es t ed a d i f fe rent approach to t he se

e x a m p l e s , which qui te r e c e n t l y h a s enabled h i m to v a r y a l a r g e n u m b e r of c h a r a c t e r i s t i c

c l a s s e s independent ly ( see [ 6 ] ) .

In our t e rmino logy , Hei t sch p a s s e s f r o m the s p h e r e , w h e r e T h u r s t o n worked, to a

IR n fol ia t ion ~ . on - 0 of the type I u sed in the c o m p l e x c a s e , but h e r e he s t a r t s out

with a suf f ic ien t ly spec ia l t so tha t ~ l a d m i t s a l a rge group of a u t o m o r p h i s m . Let m e

conclude by taking up the f i r s t i n s t a n c e of h i s c o n s t r u c t i on to v a r y h l C l ~- E H4(BF).

We will u s e ou r i n f in i t e s ima l r e c i p e for t h i s p u r p o s e , so r e c a l l f i r s t of a l l tha t the

h o m o m o r p h i s m

(4. 6) H*(g; K) ............. > H*(BF)

i s in jec t ive for any s e m i - s i m p l e Lie group and any d i s c r e t e subgroup F with F \G compac t .

Indeed F will then have a subgroup of f ini te index F' c F such that F' a c t s f r e e ly on

G / K . The n a t u r a l m a p

(4. 7) H*(_g; K) ...... > H * ( F ' \ G/K)

Pomcare i s then in jec t ive in the top d i m e n s i o n and both s i d e s s a t i s fy " ' duali ty. Hence (4.7) i s

53

in jec t ive , but F ' \ G / K ~ BF' , and BF' and BF have equal ra t iona l cohomology. Q . E . D .

F ina l ly r eca l l that by a t h e o r e m of Bore l ' s any s e m i - s i m p l e G a d m i t s a F with

F \G c o m p a a . Thus for our p u r p o s e s , v a r y i n g the i n f in i t e s ima l c l a s s with a s e m i - s i m p l e

group G a l so v a r i e s the c l a s s in s o m e BF.

(4. 8)

With t h e s e r e m a r k s we a r e r eady to take up the Hei t sch example .

Le t then ~k be g e n e r a t e d by X k in IR 4 - 0 w h e r e

= Z l i xi + Yi Xk i=l

Then the na tura l ac t ion of SL(2;IR) on the (x i , yi) space de f ines an ac t ion of

O = SL(2 , IR) X SL(2, IR)

Oil ]R 4 which obviously p r e s e r v e s X k and hence ~k

IR 4 The in f in i t e s ima l ac t ion of g on i s t h e r e f o r e gene ra t ed by

(4.9) ui = xi ~ i - Yi --~Yi ; vi = xi --~Yi + Yi ~x---~1

and

= - - - Yi 5x-~ hi xi bYi 1

T h u s the h. g e n e r a t e the act ion of K , and a c l a s s in 1

va lue on u 11% v 1 A u 2/% v 2 .

Let u s now apply the in f in i t e s ima l r ec ipe to ~k "

H4(g; K) i s de t e rmined by i t s

For ¢0 we may choose

(4o10) ~ = c(Xk)v , v = dx 1 dy 1 dx 2 dy 2

Then

Hence an admissible

(4. 11)

w h e r e we have set

(4. 12)

Indeed

54

d m = £(Xk)v = 2(X t + k2)v .

i s given by

= (x 1 + ~2 ) . _ _ ~X~ r2i , i = 1,2

2 2 2 r i = xi + Yi

is c l ea r ly invar iant under K , and sa t i s f i e s the re la t ion

do = ~ A ~ •

Further note that ~ is invariant under all of G, so that the v

ent i re ly given by the formula

Now by d i rec t computation

(4. 13)

hence

u(x) = n(x)

u i ' r 2 = i 2(x~ - y~)

2 v i . r i = 4 x i Yi

of (3.33) is

55

(4. 14)

Y(ui) = - ~(ui)= ( k l + k 2 ) 2 k i ( x ~ " Y~)

G k2i r~

Y(Vi) = - ~3(Vi) : (kl + k2) 24Xi 2 xi Yi

~" k i r i

Now (3.37) together with (2.28) yield the formula:

¢0(~t ) (u 1' Vl' U2' v2) volume enclosed by the map S 3

~ b _ z the four functions

v(u 1), V(v I), ~(u 2), ~(v 2)

o__nn S 3 c R 4 .

> R 4

Clearly this map is homogeneous of degree zero, hence we may change coordinates

from x. to and setting 1 Xi/ki '

(4. 15) zi = xi + ~r:-~ Yi

we see that the volume enclosed by the map in question will be proportional to

(k 1 + k2)4/(klk2 )2 times the volume enclosed by the unit sphere under the map,

2 22} 1 t2 2 (4.16) {z t , z 2} ...... > { z 1, i z 1 + I z 2 I ,

which is easily seen to be nonzero. Hence

f o~(~k) (Ul,Vl, U2,V2) = const. k + k2) 4

(klk2) 2

and therefore varies with X •

56

In h igher even d imens ions th is method of He i t sch ' s works equally well and leads

to an independent var ia t ion of all the hlC~ c l a s se s . In odd d imens ions the argument i s a

l i t t le m o r e subtle. However~ h e can a lso t rea t th i s case by construct ion which - on the

sphe re - goes back to Thurs ton.

All in all then Thurs ton and Heitsch have se t u s well on the way of showing that all

the potential c l a s s e s of H*(an;On) a re non- t r iv ia l , independent and, in the appropr ia te

c a s e s , var iable . F o r c l a s s e s involving many h ' s , cor responding independence t heo rems

were f i r s t obtained by Kamber and Tondeur [ 7 ] .

57

Appendix

by

Robert Brooks

In th is appendix, we will evaluate the GodbiHon-Vey c l a s s in the case of some

specific act ions of g roups on the circle . We wilt then show how these caIcutaVions lead to

an example due to Thurs ton showing how one can va ry the Godbillon-Vey " n u m b e r . "

Given an action of G on S 1 , reca l l that on BG, a~ [s given by the 2 - c o c y c l e

oo(g, f) = f log(~f) d log(.g~) S 1

If G is a d i sc re te subgroup of PSL(2, IR) , then we have a natura l action of G

on S 1 , which we view as the boundary of the upper half plane, given by the l inear fract ional

t r an s fo rma t ions -

(; cz÷daz ÷b

F o r ease in computation, we can conjugate th is action by the l inear fract ional

z - i t r ans fo rmat ion z -* ~ taking the upper half plane into the disk I z I <__ 1 - we will

want to p a s s back and for th freely between the two p ic tures .

Viewing f and g a s acting on I z l = 1 , we may now wri te

z ' ~ 1 . z ' ~ 2 . f(z) = 8 t ( ~ ) ; g o f(z) = 82(~2--~-~_ I)

with 1 0 1 1 = t 0 2 1 = 1 , t~1I , l~21<l and so viewing S I as IR/z, then f and g become

58

e 2 ~ x e2~ r ix . i - ~i I log(e~( ^ . 2 ))

f(x) = ~ i l ° g ( e l ( ~ . l e 2 ~ i x - 1)) ; g o f(x) = ~ -~ 2 "~2 eZmX " 1

C h o o s i n g t h e s t a n d a r d v o l u m e on S 1 , b;f = f ' a n d b;g = g ' , w e c a n now e a s i l y c o m p u t e

by r e s i d u e s to g e t

S 1

i = log(1 - e-2Z~i%l)(1 e2~rlX~l)[ 2~ri ] (

0

i - ~2~i = 2m l o g ( ~ )

- 21fix e ~2 ~ 2

- 2 ~ i x 21 f ix_ ) dx 1 - e ~2 1 - e v] 2

We can give this number a somewhat more geometric flavor by now passing to the

a-i b-i upper half plane - writing ~i =a+i ' ~2 =b+i ' we have

. ,a+i,,b-i, .~- b) ,,~(f, g) = 2 ~ l o g ~ - - ~ _ i) t~-$-'[) t~ a "

ie Now in the u p p e r h a l f p l a n e , a g e o d e s i c i s of t h e f o r m r e + k ,

ie I i8 2 e arerealo So given z I= re +k, z 2 =re +k, the expression

H e n c e w e l l - k n o w n f o r m u l a s in n o n - E u c l i d e a n g e o m e t r y g ive u s

(*) ~(g, g) = (- 41r 2) area (A) ,

where r,k and z -~

l o g ( - i 1 2 ) = i ( e l _ e 2 ) . z 2 - z 1

where A is the geodesic triangle with vertices i, a, and b. Here, of course, a and

b satisfy f(a)=i, (go f) (b)=i .

T h e f o r m u l a ( . ) r e l a t e s w e l l to the f o r m u l a s o f [ 5 ] in t he fo l l owing m a n n e r : If

a L i e g r o u p G o p e r a t e s on a m a n i f o l d M of d i m e n s i o n q , t h e n w e ge t a m a p f r o m

g = L i e (G) to t h e L i e a l g e b r a of v e c t o r f i e l d s on M in t h e fo l l owing way : g i v e n X ~ g ,

59

then it de f ines a o n e - p a r a m e t e r group ft of d t f f e o m o r p h i s m s of M , and the

i n f i n i t e s i m a l g e n e r a t o r of t h i s flow i s a v e c t o r field on M .

Le t K be a m a x i m a l c o m p a c t subgroup of G , wi th Lie a l g e b r a ~ , and choose

a K - i n v a r i a n t v o l u m e @ on M . Then £X(e) = ~X 0 e de f ines a funct ion /~X on

M , which we m a y th ink of a s the " d i v e r g e n c e " of X . One m a y define an " in f in i t e s ima l "

Godbi l lon-Vey c l a s s by the f o r m u l a

~(X0, XI,'" , Xq) = f ,Xo d~x I A "" A d~x q

and one sees easily that this defines an element of Hq+l(g/~) , independent of e.

In [5] is constructed, given a discrete subgroup F of G , a map

H*(g/]~) H*(BF). In the specific case of our standard PSL(2,1R) action on S l - . , o n e

can check eas i ly tha t t h i s " in f in i t e s ima l " Godbil lon-Vey c l a s s a g r e e s with our "global"

Godhi l lon-Vey c l a s s .

U s i n g the f o r m u l a ( , ) , we can c o n s t r u c t fo l ia ted c i r c l e bund le s hav ing a r b i t r a r y

Godbi l lon-Vey n u m b e r , a cco rd ing to the following s c h e m e due to Thu r s ton . Let a and b

be any two po in t s in the uppe r ha l f p lane. Then t h e r e i s a unique p a r a l l e l o g r a m (in t he s e n s e

tha t oppos i te s i d e s a r e of equal length) hav ing a s t h r e e v e r t i c e s i , a, and b . Label l ing

the four th v e r t e x c , then t h e r e a r e unique e l e m e n t s f and g s a t i s fy ing

f(c) = a f(b) = i

g(c) = b g~a) = i

Then f g f - l g - l ( i ) = i and so fgf - I -I , g is a rotation. A little non-Euclidean geometry

shows us that the rotation is the non-Euclidean area of the parallelogram. Since the

parallelogram may have an arbitrary area, we have shown

60

LEMMA. F o r any 0 < a < 2~r, t h e r e a r e f, g C PSL(2, IR) whose c o m m u t a t o r

i s a ro ta t ion th rough angle a .

Now le t G be the fundamen ta l group of the two-holed t o m s . G i s g iven as

{X,Y, Z , W : X Y X ' I y -1 = Z W Z - I w -1} , and BG h a s a fundamen ta l 2 - cyc le

[BG] = (X,Y) - (XYX " l , x ) - ( X Y X - 1 y - 1 , Y ) - ( Z , W ) + ( Z W Z - 1 , Z ) + ( Z W Z - I W " l , w )

and th i s i s the object on which, for ca re fu l c h o i c e s of X,Y,Z~ and W , w e w a n t t o

eva lua te our cocycle 0~.

F i x i n g a pos i t ive i n t e g e r n , and s o m e a < 21t , choose f and g with n

c o m m u t a t o r ~ , and f' and g' with c o m m u t a t o r n a .

Set X = g , and Y = f o We se t f ' = ( x ) = l f ' ( n x ) , and s i m i l a r l y for ~ - we

m a y t h i n k o f ~ and ~ as l i f t ings of f ' and g' to an n - fold cove r ing of the c i r c l e .

Of c o u r s e i t i s c l e a r tha t the c o m m u t a t o r of "~ and ~ i s now ~ .

Now ~ ( g , f ) - ¢o(gfg " l , g ) i s s imp ly (- 41r 2) . a , a s can be s e e n by applying (*)

to the p a r a l l e l o g r a m with v e r t i c e s i, a, b, and c . o~(gfg-lf " l , g ) = 0 , s ince

g f g - l f - t i s a rota t ion. S imi la r ly , o~(g' ,g ' ) - ~ ( g . f , g , - 1 , g , ) = (_ 41r2) . h a , and i t i s e a s y

to check by the or ig ina l in t eg ra l f o r m u l a tha t p a s s i n g to an n - fold cove r ing s imp ly

m u l t i p l i e s the r e s u l t by n . Hence choos ing X = g , Y = f , Z = g , W = ' ~ , we ge t

o~([BG ]) = 4~2(~ 2 - 1)(~.

2v Varying c~ between 0 and ~-, and taking n arbitrarily large, we see tha t

6o may take any real value°

61

Bibliography

1. R. Bott, M. Mostow, J. Perchik, Gelfand-~uks cohomology and foliations, Proceedings of the Eleventh Symposium New Mexico State University, 1973.

2. R. Bott, Some aspects of Invariant theory, C.I.M.E. Lectures delivered at Varenna, August 1975, to be published.

3. R. Bott, Lectures on characteristic classes and foliations (Notes by Lawrence Conlon), Lecture Notes in Mathematics vol. 279, 1-94. Springer-Verlag, New Y o ' r ' k , . . . . . . . .

4. R. Bott, H. Shulman, J. Stasheff, On the de Rham theory of certain classifying spaces, to be published in Advances of Math.

5. J. L. Dupont, Semisimplicial de Rham cohomology and characteristic classes of flat bundles, to be published in Top.

6. J. Heitsch, Independant variation of secondary classes~ to be published.

7. F. W. Kamber, P. Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Mathematics vol. 493, Springer-Verlag, New York.

8. W. Thurston, Foliations and groups of diffeomorphisms, Bull. AM~, 80, 304-312.