On Some F o r m u l a s f o r the C h a r a c t e r i s t i c C l a s s e s
of G r o u p - a c t i o n s
Raoul Bott
1.
Introduction.
t
T h i s is an account only of the m a t e r i a l of my l a s t two l e c t u r e s of the
Rio conferenc% a s the e a r l i e r l e c t u r e s dealt with well known m a t t e r s .
My e m p h a s i s h e r e then i s the study of o u r foliation i n v a r i a n t s in the context of group
a c t i o n s on a manifold~ and I will s t a r t by showing you the naturality of o u r b a s i c c o n s t r u c t i o n
now p a y s off by extending directly to the equivariant situation.
Recall then the main c o n c l u s i o n s of o u r e a r l i e r d i s c u s s i o n s .
E s s e n t i a l l y they
amounted to t h i s (see a l s o [ i ] , [ 2 ] for details):
If we let
IRn
to
JM
M ~ then
cechain a l g e b r a
denote the space of j e t s - based at
JM
C*(an)
0 E IRn - of d i f f e o m o r p h i s m s of
c a r r i e s an a l g e b r a of n a t u r a l f o r m s which is i s o m o r p h i c to the
where
an
is the Lie a l g e b r a of f o r m a l v e c t o r - f i e l d s on
IRn .
In s h o r t t h e r e is a n a t u r a l a r r o w :
(1. 1)
of
C*(0n)-->
C*(= n)
onto the a l g e b r a of
although (t. 1) induces the
once we divide
JM
Inv
Diff M
~*JM
Dill(M) - invariaut f o r m s
0 - homomorphism
by the n a t u r a l action of
0
on
JM.
F u r t h e r we saw that
in cohomology, it induces an i n t e r e s t i n g one
n
= the orthogonal g r o u p of IRn .
Indeed then (1. 1) induces a " b a s i c " h o m o m o r p h i s m :
(1.2)
C*(an;On) - - >
Inv f~*(JM/Ou)
Diff M
and t h i s m a p certainly r e c a p t u r e s all the u s u a l c h a r a c t e r i s t i c c l a s s e s of
M.
#The a u t h o r gratefully acknowledges with thanks p a r t i a l s u p p o r t f r o m the NSF under
G r a n t MPS 74-11896.
26
M o r e p r e c i s e l y we s a w that :
(1.3)
H*(Q n, O n) ~ H*(WOn)
where
WO
( i . 4)
n
i s the d i f f e r e n t i a l a l g e b r a given b y :
W.% = IR[Cl,'",
chI®
k
E(hl, h3,""
h k)
,
= ci
odd and = n or
n-1
with d i f f e r e n t i a l :
dCl®l
d(l®hi)
denoting the quotient of the indicated p o l y n o m i a l r i n g by t h e e l e m e n t s of
and
the
= 0
c,
I
havingdimension
d i m > 2n ,
2i.
F u r t h e r we s a w - and t h i s i s of c o u r s e quite s t a n d a r d - that
H*(f~*(JM/On)) ~ H*(M)
so t h a t (1.2) i n d u c e s a n a t u r a l m a p
H*(an, On) - - >
and finally we identified t h e i m a g e of
of
c2i
H*(M)
,
u n d e r t h i s a r r o w with t h e Pontryagin c l a s s e s
M.
T h e m a i n v i r t u e of t h i s point of view i s then, that we s e e that a manifold d e t e r m i n e s
i t s own Pontryagin f o r m s n a t u r a l l y on t h e s p a c e
:
JM/O
w h i c h I t h i n k of a s a n a t u r a l l y t h i c k e n e d v e r s i o n of
,
M,
and that f u r t h e r m o r e t h e s e f o r m s
27
a r e invariant under any diffeomorphism of
M.
Now let us put this construction to use when an a b s t r a c t (L e . , discrete) group
M via diffeomorpbism.
acts on
(~. 5)
F
These data naturally define two semisimplicial manifolds
MF: M~- F x M E F x F x M
and
(1.6)
BF : •
~
r
E
rxr
whose geometric realizations then respectively correspond to, the
induced by the
F
action over the classifying space
]BF ] of
F,
M - bundle ~
and
[BF
IMr I
itself.
Now if
(1. 7)
X : X0 ~ X 1 ~ - X 2 " "
is any semisimplicial manifold the cohomology of its geometric realization
computed in various ways.
F i r s t of all the double complex
(1.8)
O**X
~**X
~ * * X : D * ( X 0) 6 > ~ * ( X l )
obtained from (1. 7) by applying the de Rham functor
~*
6 >...
to each
(Xi)
the sum of the de Rham differential and the differential operator
the simplicial structure, computes
(i.9)
X 1 can be
and then giving
6
derived from
H*( I X I) thus:
H*(I x I) = H{O**(X)}
O n the other hand one also has the complex of "compatible forms" on the realization
of
X :
(1.10)
I x I = x 0 U X 1 ×51 u x 2 x A2 U - ' '
28
in t h e s e n s e of W h i t n e y - T h o m - S u l l t v a n .
T h i s c o m p l e x i s denoted by
¢2" [ X I , and once
a g a i n one h a s :
(1.11)
H*([Xt)
= H*(~*IXt)
•
F o r a fine a c c o u n t of all t h i s I r e f e r you to Dupont,
[ 5 ].
T h e 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 t h e n , we can c o m p u t e the c o h o m o l o g y of
(I.
12)
~**h~? : ~*(M)
f r o m the double c o m p l e x
6 > ~*(M X F)-->
F u r t h e r m o r e a s we will s e e l a t e r , on t h e f i r s t c o n s t i t u t e n t
(1.13)
MF
6o0 I M x j
= ~®1
O*M
> ,~** MF
6
takes the form
-j%.~®l
Hence t h e 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 t h e
F - invariant forms
on
M.
In view of t h 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 e p l a c e
Then the arrows
(1. 15)
by
MF.
(1.2) a n d (1.14) c o m b i n e to yield a chain m a p
c*on, On) - - > ~*(~) - - > ~ * * ~
On t h e o t h e r hand the homotopy e q u i v a l e n c e of
and
MF
~IF .
M
with
M
e a s i l y i m p l i e s that of
IvIF
Hence in h o m o l o g y (1. 15) i n d u c e s 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 that t h e i n v a r i a n c e of o u r c o n s t r u c t i o n i m m e d i a t e l y y i e l d s " e q u i v a r i a n t
characteristic classes" for a
F - manifold in
I MF I .
And h e r e of c o u r s e , a s t h e c o h o m o l o g y
29
of
Mr
h a s no a p r i o r i bound~ all the c l a s s e s of
Finally if
M
i s a compact
c h a r a c t e r i s t i c n u m b e r s of
M
WC
potentially c o m e into play.
n
F - orientable manifold
we can define the equivariant
by following (1. 16) with integration o v e r the fiber in the
fibering
MF
BF
.
T h e r e r e s u l t s an additive h o m o m o r p h i s m
(1.17)
H*(an;O n) - - >
H*(BF)
,
and my a i m in the next s e c t i o n s will be to d e r i v e s o m e explicit r e c i p e s f o r ( 1 . 1 7 ) , and
to r e v i e w s o m e of T h u r s t o n ' s e x a m p l e s in t h i s f r a m e w o r k .
2.
F o r m u l a s for the Godbillon-Vey C l a s s in H*(BF).
Note that (1.16) can also be thought
of in t h i s m a n n e r .
The d i s c r e t e n e s s of
F
naturally defines a folitation
v e r s a l to the f i b e r s in the projection
c l a s s e s of
~F
MF ~ BF,
gF
on
and of codimeasion
integrated o v e r the f i b e r induce ( 1 . 1 6 ) .
MF
n.
which i s t r a n s The c h a r a c t e r i s t i c
With this i n t e r p r e t a t i o n it s u g g e s t s
itself that the c o n s t r u c t i o n s which a r e known to r e p r e s e n t the c h a r a c t e r i s t i c c l a s s e s of
folitations should extend to the s e m i s i m p l i c i a l situation provided only that one h a s a suitable
de Rham t h e o r y at hand.
Now the double complex
muItiplication i s not artticommutative.
Q**X
does not fit the bill b e c a u s e i t s
On the o t h e r hand the compatible c o m p l e x of Dupont
i s p e r f e c t I y suitable and t h e r e f o r e y i e l d s explicit r e c i p e s quite readily.
30
, itself. This i n v a r i a n t - correspon-
L e t m e s t a r t with t h e Godbillon-Vey c l a s s
ding to a g e n e r a t o r of
of c o d i m e n s i o n one,
H3(a I ' O1) - i s defined on o r i e n t e d f o l i a t i o n s
a n d c a n b e c o m p u t e d a c c o r d i n g to t h e a l g o r i t h m :
Let
~t be d e s c r i b e d a s t h e k e r n e l of a n o n - d e g e n e r a t e
T h e n i n t e g r a b i l i t y i m p l i e s that t h e r e e ~ s t s a
d~0 = ~ A ~
(2. 1)
1 - form
1 - form
~
.~.
with
.
Now s e t
=
Then
d~o=0.
,r'l A d ~
F u r t h e r the c o h o m o l o g y c l a s s of
t h e c h o i c e s involved, and r e p r e s e n t s
~
is independent of
~0(3).
T h e e x t e n s i o n 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~
a foliation
on
X
X IE
i s s i m p l y a foliation
X2 0 " "
3k
on e a c h
Xk ,
s u c h that, all t h e s t r u c t u r e
maps:
X(~) : Xk - >
a r e t r a n s v e r s a l to
~k'
Xk,
and induce i s o m o r p h i s m s
~(rv) : ~k----:> X(oO" 1 gk,
Such data then define a foliation
in the following m a n n e r :
]31
on t h e g e o m e t r i c r e a l i z a t i o n
IxI
of
x,
31
On
X k × Ak ~ t h e n a t u r a l p r o j e c t i o n
~rL
X k < - - X k ×A k
(2. 2)
i n d u c e s t h e foliation
which assemble the
" F a t r e a l i z a t i o n " of
-1
,ITL o ~k
Xk x Ak
and t h e s e f o l i a t i o n s a r e c o m p a t i b l e u n d e r t h e i d e n t i f i c a t i o n s
to f o r m
IX 1.
P r e c i s e l y we h a v e in mind h e r e the s o - c a l l e d
IX I , w h i c h t h e r e f o r e only i d e n t i f i e s by t h e b o u n d a r y m a p s .
T h u s if
i s s u c h a b o u n d a r y map, and
:
t h e c o r r e s p o n d i n g m a p of t h e
cation c o r r e s p o n d i n g to
a
Ak - 1
simplex
k - I
>
5k
Ak - 1
onto a f a c e of
Ak
t h e n t h e identifi-
i s d e s c r i b e d by t h e d i a g r a m :
X k x Ak ' l lEA(a)
Ak
............ > Xk ×
(2.3)
X(a) X 1 ]
Xk_ 1 X Ak-1
That i s , t h e two i m a g e s of a point
v e r t i c a l a r r o w s a r e to be identified in
It is clear then that the
on
~r
(p,q) E Xk x Ak-1
IX I •
o ~k
do define a compatible collection of foliations
IX I , and that is precisely what one means by a foliation on
Similarly, one defines the de R h a m complex
IXt
under the horizontal and
in t e r m s of t h e d i a g r a m (2. 3 ) :
Thus:
IX l-
~* IX I of "compatible forms" on
32
A
on
q - form~o
on
I XI
isa collection -~{¢Ok~ of q - forms
such that
X kxA k
(2.4)
{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 " i n t e g r a t i o n o v e r the s i m p l e x e s "
L
~ ~ ~ ~.
(2. 5)
where
~rL
is the projection (2.2) and
lr L
denotes integration o v e r the f i b e r of
Note that the s u m h a s only q nonzero t e r m s , when
~
is of dimension
~rL .
q.
Now then, with all t h i s understood o u r f i r s t r e m a r k is :
PROPOSITION 2.6.
Let
compact simplicial manifold
X.
;
he an oriented codimension
Then
w (5) E H*(~ [ X ])
1
foliation on the p a r a -
can be computed by the Oodbfllon-
V.ey...algorithm (2.1) provided only we i n t e r p r e t " f o r m " to mean "compatible f o r m " on
lX I •
The p r o o f of t h i s i s quite s t r a i g h t f o r w a r d , so let m e only s t a r t the a r g u m e n t and in
the p r o c e s s derive an explicit a l g o r i t h m f o r
C o n s i d e r then an
r e p r e s e n t e d a s kernel of
~
t~(~)
in t e r m s of the s t r u c t u r e m a p s of
a s envisaged in the proposition, and let
~0 with
d~
=
~
.
~0
on
X0
be
X.
33
W e next try to extend
the notation let
0
and
1
~
and
~
to
X I x AI
be the vertices of
(70 : AO- - > Al ,
to
0
and let
x0
and
xI
T o fix
be the
A I . Also let
corresponding barycentric coordinates on
be the inclusions sending A0
A1
in a compatible manner.
and
and c~1 : AO - > A 1
i
respectively, and let
frO and (71 be the
corresponding maps :
Xo ~
Now by our hypothesis on
;Y the f o r m s
cr~W
both represent
/~i
on
X1
Using
51
Xl
and
in the s a m e orientation.
cr~¢~
Hence there exists a smooth positive function
such that
it 1
we now c o n s t r u c t the f o r m
x
(2.7)
1rL
1
~01 = ( 7 ~ . /ZI
H e r e we have identified the f o r m s on
1
and ~1
is defined by
1
on
XI × A
X 1 w/th t h e i r pullback t o
x 1 (p, a) = ~l(p)Xl(a)
X 1 × A1 under
34
Hence
~1
r e s t r i c t s to
the k e r n e l of
O1
(y~¢~ on
clearly represents
Next we wish to extend
(2. 7 ) .
~(X 1 x A1)
rl
and is compatible.
Furthermore
rtLl~Yl. Indeed any n o n z e r o multiple of
compatibly to
X 1 x A1 .
¢~eO would.
F o r this p u r p o s e d i f f e r e n t i a t e
One obtains
(2. 8)
d ~ l = (log ~1 " d x l + x l d log b~t + c r ~ ) ^ ¢Pl
Now the t e r m in t h e b r a c k e t i s not compatible with
~
a s it stands.
However, it can be
modified to b e c o m e so in the following m a n n e r :
By differentiating (2.6) we obtain
d u : G = (TI~ . O':e = {d log ~i1 + c r0* ~ lJ ~*¢~
1
whence
(2.9)
Hence we may r e p l a c e
dlOg~l in (2.8)by
d~o I
=
cr:~-Cr~ toobtain:
{iog~ i dx I + x I ¢T~ I + x 0 a ~ } A V I
and this t i m e the t e r m
~]I = {log ~iI dx I + x i ( ~
(2. 10)
i s c l e a r l y compatible with
~.
This c o n s t r u c t i o n now extends to all of
collection
¢Pk and
~k
+ x0 ~ }
on
X k x Ak .
i X i to yield the followingcompatible
35
For each
the inclusion of
k,
we let
AO into
Then we define
o'0,0"1, ° - ' , o"k
Ak
as the
k+l
bLi, i = l , ' " ,
k,
be the m a p s of
X k - X0
c o r r e s p o n d i n g to
vertexes.
by
and finally set
(2. 11)
and c o r r e s p o n d i n g l y set
k
(2. 12)
The f o r m s
xi
~k A d ~ k = ¢ek(l~)
an algorith m f o r computing
¢0(~)
a r e t h e r e f o r e again cgmPatible~ closed, and give
in
H*( 1 X I )
At this stage one m a y of course return to the m o r e economical complex
i n t e g r a t i n g o v e r the s i m p l i c e s .
•
"
i
u~~' ] 6 D (Xj)
Then
and u s i n g (2. 13) one obtains explicit formulae.
(2. 14)
while
by
~ .L ~( ~ ) = ~o3~0 + ~02~ 1 + wl~ 2 + co0, 3
(2. 13)
with
~**(X)
Fox example :
03,0 = ~A~0,
J,
1
is obtained by i n t e g r a t i n g
component i s given by :
HI A d ~ l
o v e r the
I - simplex
41 . Thus t h i s
36
~ ; ~ + x 1 (r~q + log DI dxl) A (dxl(cr~ - Cr;~) + d log ~i dxl
fl(X
+x
Only the t e r m s involving a
dx
0 do~
+xld(~)
survive and hence with a little a l g e b r a one
at
arrives
(2. 15)
0 2 , 1 = Cr~ A cf;'~ - d { l o g ~ll(
To obtain the next t e r m we p r o c e e d s i m i l a r l y with
2
~2 A d~2.
)}
The r e s u l t is
01, 2 = i,j=O
2 f2 log ~i (dlog ~j - cr;~) dx i d d
0 < i
< 2
-- ' j --
(2. 16)
E
(- 1)i- j +I
O<i<j<2
Finally
w0, 3
logui(dlog~lj- CT~~
')
v a n i s h e s b e c a u s e t h e r e i s no t e r m involving t h r e e
So f a r I have d i s c u s s e d only the original Godbillon-Vey c l a s s
codimension
1.
If
i s g e n e r a l i z e d to
F,
~
this t i m e of dimension
is orientable
F
in
~3 ^ d~3 "
u~ for foliations of
but has h i g h e r codimension say
by again d e s c r i b i n g
q.
dx's
q
- their class
a s the kernel of a d e c o m p o s a b l e f o r m
~p ,
Then integrability again i m p l i e s that
d~ = ~ A ~
but now it is the form
oo(~) = ~ A (d~) q
which i s an invartant of
In
WO
q
a
and thus g i v e s the extended Oodbillon-Vey c l a s s in
this c l a s s then r e p r e s e n t s
h1 clq.
H2q+I(M).
u)
37
In t h e s e m i s i m p l i c i a l s i t u a t i o n t h i s e x t e n s i o n works: equally well and (2. 12) a g a i n
y i e l d s an explicit a l g o r i t h m f o r t h e c o m p u t a t i o n s of
codimension
q ,
co(J).
Note by the way that in
only c o m p o n e n t s of t h e type:
~rL co(g) = o~2 q + l ' 0 + . . . . + coq, q+l
(2. 17)
survive because
~k A d ~ k q
c a n contain no m o r e t h a n
(q + i)
dx' s
f o r arty
k
F u r t h e r m o r e t h e e x t r e m a l t e r m s a r e e a s y to c o m p u t e f r o m (2. 12) :
¢o2 q + l ' 0 = ~ A (d~) q
(2. 18)
~ q , q+l
= f
i
i
(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 f o r t h e o t h e r c l a s s e s in
H*(WO n)
but t h e y a r e
h a r d e r to w r i t e down e x p l i c i t l y , b e c a u s % a s in the e a s e of j u s t a single foliation
:
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 a n d R i e m a r m i a n c o n n e c t i o n s .
H o w e v e r t h e c o m p a t i b l e c o m p l e x a g a i n e n a b l e s one to extend the s i n g l e - s p a c e r e c i p e s
to t h e 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 t h a n d i s c u s s i n g
t h e s e l e t m e now s p e c i a l i z e o u r f o r m u l a e to t h e c a s e of a g r o u p action.
recall briefly how
MF
and
Recall first, that if C
BF a r e c o n s t r u c t e d and why
is a
a
MF
We t h e r e f o r e h a v e to
c a r r i e s a n a t u r a l foliation.
c a t e g o r y , t h e n it d e f i n e s a n a t u r a l s e m i s i m p l i c i a l
object
BC : O(C) ,~--~I(C) ~ ~,2(C)
(2. 19)
s t a r t i n g with t h e o b j e c t s
m o r p h i s m s in
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
.~-'~.~"'~
(2. 20)
the
y ' s
being a r r o w s in
C.
.....
Yk
~--~.
The boundary m a p s in
BC
a r e then given by sending
such a d i a g r a m into one with e i t h e r 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
When applied to the c a t e g o r y
FM
determined by the action of
F
on
F
yields
M , it yields
BF o
MF:
Thus
MF --- BFM
w h e r e the category
with Y 6 F ,
m6 M.
FM
h a s a s objects the manifold
M ,
and a s m o r p h i s m s p a i r s
(y,m)
Finally the composition
(2.22)
( 7 1 , m i ) o (72, m2) = (71 o Y2,m2)
e x i s t s only if
72(m2) = m 1
and is then given by the RHS of (2. 22) .
Pictorially t h e s e a r e then b e s t d e s c r i b e d by a r r o w s of the f o r m
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.
to
a
on
M.
Then
a
Let
F
be a foliation on
extends to a foliation
In p a r t i c u l a r the foliation of
Note of c o u r s e that
as a
simplicial manifold
aM
M
aM ,
MF.
aM
o_..~n M F .
MF
matte.rs simplify to s o m e extent a s now
cO(aM) o v e r the f i b e r in the projection
q - dimensional
only
>BF
0~q,q+l
e n t e r s into the computation.
m a n n e r w e obtain the following algorithm for
t7, a)(a M)
PROPOSITION 2.25.
•
The c h a r a c t e r i s t i c " n u m b e r "
y . o~(~M) E Hq+I(BF)
r e p r e s e n t e d by the cocycle :
(2.26) ~0(~i,... , ~q+l) = ifq+I×M (log~li dxi) (d log ~j dxJ)q
w h e r e the
Di
a r e functions on
Choose a volume
(2.27)
considered
K we f u r t h e r m o r e follow o u r r e c i p e , in the c a s e when
MF
being
actstransversally
in the obvious s e n s e .
is oriented and compact, and integrate
M
F
then c o r r e s p o n d s to the foliation by points of
can c l e a r l y be taken to be zero.
then -
on
such that
by points induces a n a t u r a l foliation
When we specialize o u r f o r m u l a e to
M
FF
M,
~0 on
M,
determined in the following m a n n e r :
M , and then define
( Yi Yi+l • -"
~i
y q*.i)*~ =~i $
by:
is
In t h i s
40
Remark.
(i)
T h u r s t o n h a s b e e n a w a r e of t h i s f o r m u l a all a l o n g a l t h o u g h h i s c o n -
v e n t i o n s and h i s s e t t i n g might have differed s o m e w h a t .
i n s p i r e d it w a s h o w e v e r not so obvious.
my remarks
F o 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
On t h e o t h e r hand - and t h i s i s the m a i n point of
- in the c o n t e x t of t h e c o m p a t i b l e c o m p l e x the f o r m u l a does b e c o m e e a s i l y
accessible.
(2)
The following g e o m e t r i c a l i n t e r p r e t a t i o n of 2. 25 w a s pointed out to m e by
T h u r s t o n quite r e c e n t l y .
If
Yl ' "" " ' Yq+l
a r e c o o r d i n a t e s in
•q+l
then the form
/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 t h e identity
d g~ = n v o l u m e in IRn .
On t h e 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 i v e s t h e i n t e g r a l of
pulled back to
M
under the map
fF : M
> IRq+l
defined by
(2. 29)
i
y o f F = l°g(/gi)
Thus our formula for the Eilenberg MacLane cocycle
(2.30)
¢o(~) can be written;
¢~(~) (~1 ' " ' " ' Yq+l ) = v o l u m e e n c l o s e d by fF M in 1Rq+l .
41
3.
The Infinitesimal Case.
When the action of
F
by r e s t r i c t i o n of an action of a connected Lie group
c l a s s e s of the
F - action
on a foliation
G
containing
can be d e s c r i b e d in L i e - a l g e b r a t e r m s .
over
M
comes
F , the characteristic
Such a description is
often m o r e e a s y to handle, and o u r a i m in t h i s section is t h e r e f o r e to d e s c r i b e an infinit e s i m a l analogue of (2. 18) in this f r a m e w o r k .
to
(G/K) x M , with
K
The technique involves the lifting of
a m a x i m a l compact subgroup of
a
to
G ~ and is thus typical of the
V a n - E s t t h e o r e m s and t h e o r e m s on continuous cohomology, as well a s of the whole a p p r o a c h
of K a m b e r and Tondeur to fotiated bundles.
See [ 7 ] .
However I will not p u r s u e t h e s e a s p e c t s , but r a t h e r head a s quickly as p o s s i b l e t o w a r d s
the infinitesimal f o r m u l a f o r
~(~).
Recall then, f i r s t of all, that
(3. 1)
G/K
i s contractible:
G / K = pt
Hence if we c o n s i d e r the natural
F
action on
equivalence of the quotient of G / K by the action of
(3.2)
then (3. 1) extends to a homology
F (denoted [G/K]F)wittl
BF ,"
[G/K]F ~
Thus
in
G/K
[G/K ] F
is a thickened v e r s i o n of
BF .
Now c o n s i d e r the edge h o m o m o r p h i s m
[G/K ] P ,
(3. a)
inv ~.*(G/K) - ~**( [ G / K ] F)
F
Because
F c G,
the left hand side certainly contains the subcomplex of
f o r m s on
G / K , which is finally identified with the r e l a t i v e Lie a l g e b r a c o m p l e x
~*(E; K) .
Thus we have
G - invariant
42
(3.4)
•*(g; K) =- Inv f~*(G/K) c Inv ~ * ( G / K )
G
F
combining (3.4) with (3.3) and (3.2) gives r i s e to an a r r o w
(3.5)
I., : H*(g;K) - H*(BF)
t
,
and o u r f i r s t aim is then to show that:
PROPOSITION 3.6.
action lift naturally to
In the situation envisaged the c h a r a c t e r i s t i c c l a s s e s of the
H*(g; K)
To s e e this r e s u l t one f i r s t h a s to u n d e r s t a n d the following d i a g r a m of
(3.7)
G X M t > (G/K) X M fr> M
K
where
fr
is the n a t u r a l product projection, and the action of
product action.
F
On the o t h e r hand
G
a c t s on
G x M
K
G
spaces:
,
G
on
(G/K) x M
purely on the Ieft~ and
is the
G x M
K
is
--
defined a s the quotient
(3. 8)
with
G x M = (G x M)/K
K
K
acting by
(3. 9)
(g, m ) . k = (gk, k ' l m )
Finally the twist map
(3. 10)
t
is induced by
'~: G x M - G x M
sending
(3.11)
(g,m)
to
(g,g.m)
In [ 5 ] and also independently in a r e c e n t p a p e r of Shulman and T i s c h l e r , this a r r o w is
explicitly d e s c r i b e d on the chain level.
43
It i s then c l e a r that
into the
K
action
t
i s a d i f f e o m o r p h i s m which s e n d s the
(g,m)~ (gk,m)
and so induces the equivalence
v i r t u e of t h i s t w i s t ~ g map
t,
the left t r a n s l a t i o n of
It follows that the
G.
identified with the f o r m s on
i s of c o u r s e that the
G x M~
K
G - action
G - invariant
on
of ( 3 . 7 ) .
O x M
K
f o r m s on
The
i s given by
G x M
K
can be
which a r e
(1)
invariant u n d e r left t r a n s l a t i o n of
(2)
K = b a s i c u n d e r the action (3.8) of
G
by
K
G,
on
an_._dd
Gx M.
Thus:
Inv ~*(G x M) ~ K basic f o r m s in f~*(~[) ® ~*M
G
K
(3. 12)
t
action (3.9)
.
In any c a s e , the plan of p r o c e d u r e is now s u g g e s t e d b y t h e diagram:
Inv O* (G x M) -* f~**(G x MF) ~ ~**(MP)
G
K
K
(3. ~3)
Inv ~ * ( G / K )
G
~ i'~* (G/KF)
w h e r e the l o w e r line induces ( 3 . 5 ) , and
~r,
~ f~**(BlT)
denotes integration o v e r the fiber.
The
homotopy equivalence in the u p p e r Hne is of c o u r s e again a consequence of the contractibility
of
G/K.
c l a s s e s of
In view of (3. i3) our lifting p r o b l e m c l e a r l y a m o u n t s to r e a l i z i n g the c h a r a c t e r i s t i c
~
lifted to
G x M,
PROPOSITION 3. 14.
by
G - invariant
forms.
The c h a r a c t e r i s t i c c I a s s e s of
Thus we need the following.
g
admit
a n a t u r a l l i f t i n g to
the complex ( 3 . 1 2 ) .
Let me c a r r y out the proof~ but again only f o r o u r Godbillon-Vey c l a s s
then the pull-hack
(3. 15)
~
of
~
to
Gx M,
GxM
~o~
u n d e r the map
> M
.
o~(F).
Consider
44
To d e s c r i b e
g$ TmM,
~
let u s identify the t a n g e n t s p a c e of
u s i n g the left i n v a r i a n t v e c t o r - f i e l d s on
G
using the projections for the direct sum decomposition.
the Lie a l g e b r a of
G ,
let
induced by t h e action of
G
P r e c i s e l y if
et x
:~ E F(TM)
on
G X M,
at
to identify
Gg
A l s o ff
x Eg
:~
m
with
with
g ~ and
i s a n e l e m e n t of
be the c o r r e s p o n d i n g i n f i n i t e s i m a l m o t i o n on
M
M.
i s t h e o n e - p a r a m e t e r s u b g r o u p g e n e r a t e d by
(3. 16)
(g,m)
= t a n g e n t of e ~ m
With t h i s u n d e r s t o o d , t h e k e r n e l of
x + y E T(g, m) (G x M)
(3. 17)
~
at m
x E g,
then
.
i s d e s c r i b e d by:
i s in "~ if and only if
:~m + Ym E
Indeed the c u r v e
t a n g e n t at
t=0
(ge tx, m)
g o e s to
g*(Xm + Ym ) E g .
But
g*~m"
G
g o e s o v e r into
getXm
On t h e o t h e r hand
y
u n d e r o u r map, and h e n c e i t s
g o e s to
g*Ym"
Hence
p r e s e r v e s t h e foliation so that (3. 17) follows.
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.
decomposable
q
form
decomposable form
this complex ~
¢0.
Let t h e foliation
Then
3
~
a d m i t s a n a t u r a l r e p r e s e n t a t i o n a s t h e k e r n e l of a
~0 E f~*(_g) ® Q*(M) .
F u r t h e r m o r e ~ in the n a t u r a l double g r a d i n ~ of
has components:
,~ = , ~ q , O + . . . .
with:
,~0, q = •
.
be d e s c r i b e d a s the k e r n e l of the
~0, q
+co
45
~l,q-I
(3. 19)
where
xa
r u n s o v e r a b a s i s of
Proof.
g_, a n d
x~
o v e r a dual b a s e in
This is purely a linear algebra matter.
If
e
g* .
i s a 1 - f o r m with
its kernel, then
(3. 20)
will h a v e "~ in i t s k e r n e l .
~(x + y ) e
Indeed f o r a n y
x +y
s u b j e c t to (3. 17) we t h e n h a v e
= e(y) + e(~) = o .
H e n c e if
e 1A °''A0 q =~
locally, then O 1 A . . .
AO q
describes
"~ locally, and
e x p a n d i n g t h i s p r o d u c t c l e a r l y y i e l d s (3. 1 9 ) .
To p r o c e e d f u r t h e r we need to c o m p u t e
complex
do
and e x p r e s s it a s
~ A
in t h e double
~*(g) ® o*(M) .
F o r t h i s p u r p o s e , let u s s e t
and u s e t h e double index s u m m a t i o n convention, so that
(3.21)
V = ~+x~
A~¢~÷""
d e s c r i b e s t h e "beginning" of ~ .
A l s o , let u s a s s u m e t h a t on
(3. 22)
where
M
t h e i n t e g r a b i l i t y of
d~ = ~ A
D
i s a global
1 - form.
T h e n I c l a i m that:
~
i s e x p r e s s e d by
46
PROPOSITION 3. 23.
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
such that
F u r t h e r the
is given by
= ~-x~(u(~)
(3. 24)
Proof.
- ,(~)}
where ~ ( x ) i s
a e ~ i . e d by ( 3 . Z 7 ) .
Differentiating (3.21) yields
(3. 25)
d ~ = d ~ - x~x A d %
+""
Further
d% = d~(~ )~ = £ ( ~ ) ~ - ~(~)dv.
(3. 26)
where
£(~)
is the Lie derivative in the direction
~, £(~), x ~ g
(3. 27)
This
must p r e s e r v e
Because
G
preserves
¢0 up to multiples, whence
£(~)~0 = ~(x)¢0 , x ¢ g ,
/~(x)
:~.
with
/~(x) E ~°(M)
is, of course, the infinitesimal 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 o r d e r
(3. 2s)
> 2
in the
g
A ( ~ ( ~ ) ~ - '~ s, % ) + . . .
direction, is given by
47
But a s
fixes
~
e x i s t s and c l e a r l y i s t h e s u m of f o r m s of type
~.
(1, 0)
(0, 1) this equation
and
Q.E.D.
To a s s e m b l e t h e p i e c e s , we s h a l l h a v e to d e t e r m i n e w h e t h e r
~
and
~
are
K
However, by a v e r a g i n g o v e r
K,
b a s i c in o u r complex.
In g e n e r a l t h i s will, of c o u r s e , not be the c a s e .
we c a n a r r a n g e it that both t h e
action of
K,
~
and t h e
~
of o u r d i s c u s s i o n a r e i n v a r i a n t u n d e r t h e
L e., that infinitesimally
(3.29)
£(~)¢0 = 0
;
£(~)~ = 0
for
xEk
and u n d e r t h i s h y p o t h e s i s we h a v e t h e following.
PROPOSITION 3.30.
r e l a t i v e to t h e action of
T h e condition (3.29) i m p l i e s that
K
on
~
and ~
are
K
basic
GxM.
T h e p r o o f of t h i s fact i s a s t r a i g h t f o r w a r d c h e c k (though n o t quite t r i v i a l ) which I
will t a k e up in g r e a t e r g e n e r a l i t y at a n o t h e r t i m e .
At t h i s s t a g e , we a r e r e a d y to give an i n f i n i t e s i m a l r e c i p e for
to(J).
Indeed, e x p a n d i n g ~ ( d ~ ) q , will h a v e all p o s s i b l e type of c o m p o n e n t s :
o~(~) = o ~ 2 q + l ' 0 + . . -
+
0,2q+l
,
of which the simplest are given by:
(3. 31)
0 , 2q+l
(3.32)
wq+l,q = - x' A "'" x'
u du A ' " dya
~1
%+1 ~i ¢¢2
q+l
= ~
. d~q
w h e r e we h a v e now s e t
t T h e K b a s i c f o r m s a r e t h o s e a n n i h i l a t e d by t h e v e c t o r f i e l d s a l o n g the orbit of the
a c t i o n and i n v a r i a n t u n d e r t h a t action.
48
(3.33)
v(x) = ~(x) - ~(:~)
and have abbreviated
v(~)
to v .
If we think of f ~ * ( g ) ® a * ( M )
v a l u e s ill
x ~g
as the complex
~ * M ) then (3.33) can be thought of a s a
a*(_g;O*(M))
1 - f o r m on
of f o r m s on
g
g
with
with v a l u e s in
n°(M):
u E ~l(g;~°(M))
,
and t h e r e (3.32) takes the f o r m
o~q + l ' q(xl, ..o , Xq+ 1) = ~ 1
(3.34)
E (- 1) 1" U(xi) dv(x 1) . . . d v ~ x i ) . . , dV(Xq+l)
Hence we get the follo~ing c o r o l l a r y , wtRch i s an i n f i n i t e s i m a l analogue of 3 . 2 5 .
PROPOSITION 3.35.
(3.36)
Suppose
ZC M
i s a cycle of d i m e n s i o n
q
on
M.
Then
/ ' / c o ( a ) E Hq+l(g ; K)
Z
is r e p r e s e n t e d by the cocycle
(3.37)
COZ(a) ( X l , ' - " , Xq) = f
V(Xl)dV(x2) " ' " dV(Xq+l)
Z
where
V(x)
i s defined by
(3.38)
4.
{~(x) - ~(~) }~ = v ( x ) ~
On the E x a m p l e s of T h u r s t o n and Heitsch.
varying classes
co(a)
i s roughly a s follows.
The h i s t o r y of e x a m p l e s of foliations with
49
In the c o m p l e x a n a l y t i c case~ I had o b s e r v e d a l r e a d y b e f o r e 1970 that the foliation:
~ = {~iZl~
(,. l)
on
¢2
- 0,
+
~2z2-~2)
had f o r i t s g . v . i n v a r i a n t :
k2
+-kI
-
2}
,
and thus varied continuouslywith k.
At that t i m e , I thought t h a t t h e c o r r e s p o n d i n g r e a l i n v a r i a n t would a l w a y s v a n i s h .
However~ soon t h e r e a f t e r in 1971, t h e p a p e r of Godbillon-Vey a p p e a r e d with t h e R o u s s a r i e
e x a m p l e of t h e foliation
in
SL(2 ; IR),
F
~
on
F~SL(2 ; IR)
induced by t h e L i e 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
b e i n g a d i s c r e t e s u b g r o u p with c o m p a c t quotient space.
Thereafter, Thurston produced his examples.
of a f a m i l y of a c t i o n s of a g r o u p
E H2(BF)
varied continuously.
F
S1 ,
a c t i n g on
In p a r t i c u l a r he c o n s t r u c t e d e x a m p l e s
such that the corresponding g.v. number
In an appendix, Robert Brooks h a s w r i t t e n up t h e d e t a i l s
t r e a t i n g t h i s e x a m p l e with t h e f o r m u l a (2.25) m u c h like Dupont in [ 5 ] t r e a t e d the E u l e r
c l a s s of flat b u n d l e s .
H e r e let m e j u s t outline and c o m m e n t on t h e plan of t h i s v e r y i n g e n i o u s
example.
We s t a r t by o b s e r v i n g that
(4. I)
z-->
Hence for any
Where
nonzero.
F ~ SL(2, IR)
I~SL(2, lR)
SL(2 ; IR)
az + b
cz + d
'
a c t s on
S1
[ab~
[zl = 1 '\cd/ESL(2'IR)
t h e r e i s a n a t u r a l a c t i o n of
is compact,
H2(F ; IR) # 0
On t h e o t h e r h a n d , a s we let
F
in t h e c l a s s i c a l m a n n e r
F
on
S1 .
and t h e g . v . c l a s s of t h e action will a l s o be
v a r y in a c o n t i n u o u s f a m i l y of s u c h s u b g r o u p s ,
50
t h e 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 d o e s not v a r y .
Thus, the moduli of Riemann surfaces
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 t w i s t e d t h e s e h o m o g e n e o u s a c t i o n s on
S1
in the following
rnanner.
Consider the double cover
S I ' ,¢t > S 1
g i v e n by s e n d i n g
z
to
z2 ; I z I = 1 .
p r e c i s e l y two l i f t i n g s "~ r e l a t i v e to
~.
'~:
is a diffeomorphism of
Then every diffeomorphism
f
of
S1
Thus,
S1
>S 1
S 1 , with
~.of
On t h e double c o v e r
Diff (2) S 1
= fo~
of
Diff S 1
t h e function
f -->'~
now becomes
single v a l u e d and d e f i n e s a h o m o m o r p h i s m :
Diff(2) s 1
(4. 2)
Note t h a t if
f
f
~'2
> Diff (S I)
i s lifted to a n " e q u i v a r i a n t m a p " : _f(x + 2~) =_f(x)+ 2~,
f : IN
then
admits
>IR
i s r e p r e s e n t e d by:
_T(x) = 1 / 2 f ( 2 x )
or
1/2{f(2x)+¢t}
51
Hence, in p a r t i c u l a r , if
then
f
f
i s a rotation by
a
and hence r e p r e s e n t e d by
x
>x+o~
is represented by:
(4.3)
x-->x+~/2,
and
x-->x+~+rr
It follows that if the map
SL(2, JR)
> DillS 1
> PSL(2, IR)
given by (4. 1) , is lifted to a map of
SL(2, tR)
to
Di.ff(2)(S 1)
and then followed by
~2 '
t h e r e r e s u l t s a h o m o m o r p h i s m of
1%
(4. 4)
> Diff (S 1)
SL(2, IR) * SL(2, IR)
so(2)
w h e r e on the left we have in mind the f r e e product a m a l g a m a t e d along the rotations a c c o r d i n g
to (4. 3) , and it i s t h i s action which gives r i s e to T h u r s t o n ' s example.
t a k e s the
F
More p r e c i s e l y , he
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 relation
(4.5)
chooses
[X,Y] = [Z,W]
X,Y 6 S L ( 2 ; I R )
details), and also
Z'
and
W'
Z ' , W ' 6 SL(2,1R)
and applying
r e l a t i o n (4. 5 ) .
so that
[X,Y]
,
i s a rotation by
so that
~2 ' one obtains
[Z',W']
Z, W
0 < ~ < ?r , (see Appendix for
i s a rotation by
in
25.
Thenlifting
Diff S 1 , which c l e a r l y satisfy the
52
Now v a r y i n g
~,
Thurston obtains his example.
Note that a s t h i s e x a m p l e 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 u a t i o n s , it cannot be d i r e c t l y t r e a t e d by
o u r i n f i n i t e s i m a l m e t h o d , although, a s Bob Brooks point out o u r global and i n f i n i t e s i m a l
c o c y c l e s a g r e e w h e r e t h e y should.
M o r e 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 t h e 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
n u m b e r s of c e r t a i n
F's
F
on the s p h e r e s .
Thus, his examples
actually vary the characteristic
actions.
D u r i n g t h e Rio c o n f e r e n c e , J a m e s H e i t s c h s u g g e s t e d a d i f f e r e n t a p p r o a c h to t h e s e
e x a m p l e s , w h i c h q u i t e 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 independently
( s e e [ 6 ]).
In o u r t e r m i n o l o g y , H e i t s c h 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
foliation
~.
on
IRn - 0
with a s u f f i c i e n t l y s p e c i a l
of t h e type I u s e d in t h e c o m p l e x c a s e , but h e r e h e s t a r t s out
t
so that
~l
a d m i t s a l a r g e group of a u t o m o r p h i s m .
conclude by taking up t h e 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 o n to v a r y
Let m e
h l C l~- E H4(BF).
W e will u s e o u r i n f i n i t e s i m a l r e c i p e f o r 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 t h a t t h e
homomorphism
(4. 6)
H*(g; K)
.............
> H*(BF)
i s i n j e c t i v e f o r any s e m i - s i m p l e Lie g r o u p and a n y d i s c r e t e s u b g r o u p
Indeed
G/K.
(4. 7)
F
will then have a s u b g r o u p of finite index
F' c F
s u c h that
F
with
F'
F\G
compact.
a c t s f r e e l y on
The natural map
H*(_g; K)
......> H * ( F ' \ G / K )
i s t h e n i n j e c t i v e in t h e top d i m e n s i o n and both s i d e s s a t i s f y Pomcare
"
' duality.
Hence (4.7) i s
53
i n j e c t i v e , but
F'\G/K
~ BF' , and
BF'
and
BF
have equal r a t i o n a l cohomology.
F i n a l l y r e c a l l that by a t h e o r e m of B o r e l ' s any s e m i - s i m p l e
F\G compaa.
group
G
G
admits a
F
Q.E.D.
with
T h u s f o r o u r p u r p o s e s , v a r y i n g the i n f i n i t e s i m a l c l a s s with a s e m i - s i m p l e
a l s o v a r i e s t h e c l a s s in s o m e
BF.
With t h e s e r e m a r k s we a r e r e a d y to take up t h e H e i t s c h e x a m p l e .
L e t then
~k
be g e n e r a t e d by
(4. 8)
Xk
T h e n t h e n a t u r a l action of
Xk
in
IR4 - 0
Z
l i xi
i=l
=
SL(2;IR)
where
+ Yi
on the
(x i , yi)
space
d e f i n e s an action of
O = S L ( 2 , IR) X SL(2, IR)
Oil
]R4
w h i c h obviously p r e s e r v e s
T h e i n f i n i t e s i m a l action of
(4.9)
ui = xi ~ i
g
Xk
and h e n c e
on
IR4
- Yi --~Yi ;
~k
i s t h e r e f o r e g e n e r a t e d by
vi = xi --~Yi + Yi ~x---~1
and
- - Yi 5x-~
hi = xi -bYi
1
T h u s the
h.
v a l u e on
u 11% v 1 A u 2/% v 2 .
1
g e n e r a t e the action of
K , and a c l a s s in
Let u s now apply t h e i n f i n i t e s i m a l r e c i p e to
(4o10)
~ = c(Xk)v
,
H4(g; K)
~k " For
i s d e t e r m i n e d by i t s
¢0 we m a y choose
v = dx 1 dy 1 dx 2 dy 2
54
Then
d m = £(Xk)v = 2(X t + k2)v
Hence an admissible
.
is given by
(4. 11)
= (x 1 + ~2 ) . _~X~_ r2i
,
i = 1,2
w h e r e we have set
2
2
2
r i = xi + Yi
(4. 12)
Indeed
is c l e a r l y invariant under
K,
and s a t i s f i e s the relation
do = ~ A ~
Further note that ~
•
is invariant under all of G ,
e n t i r e l y given by the formula
u(x) = n(x)
Now by d i r e c t computation
ui'r2 =i
(4. 13)
hence
2(x~ - y~)
2
v i . r i = 4 x i Yi
so that the v
of (3.33) i s
55
Y(ui) = - ~(ui)= ( k l + k 2 ) 2 k i ( x ~ " Y~)
G k2i r~
(4. 14)
Y(Vi) = -
~3(Vi) : (kl + k2) 24Xi2 xi Yi
~" k i r i
Now (3.37) together with (2.28) yield the formula:
volume enclosed by the map S 3
¢0(~t ) (u 1' Vl' U2' v2)
~b_z
> R4
the four functions
v(u 1), V(v I), ~(u2), ~(v2)
o__nn S 3 c R 4 .
Clearly this map is homogeneous of degree zero, hence we may change coordinates
from
x.1 to
Xi/ki ' and setting
(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,
(4.16)
{zt, z 2}
......>{z 21, 22} 1 i z 1 t2 + I z 2I 2 ,
which is easily seen to be nonzero. Hence
k + k2)4
f
o~(~k) (Ul,Vl, U2,V2) = const.
(klk2) 2
and therefore varies with
X•
56
In h i g h e r even d i m e n s i o n s t h i s method of H e i t s c h ' s w o r k s equally well and leads
to an independent v a r i a t i o n of all the
little m o r e subtle.
However~ h e
hlC~
classes.
In odd d i m e n s i o n s the a r g u m e n t i s a
can a l s o t r e a t t h i s c a s e by c o n s t r u c t i o n which
- on the
s p h e r e - g o e s back to T h u r s t o n .
All in all then T h u r s t o n and Heitsch have s e t u s well on the way of showing that all
the potential c l a s s e s of
c a s e s , variable.
H*(an;On)
a r e n o n - t r i v i a l , independent and, in the a p p r o p r i a t e
F o r c l a s s e s involving many
h' s,
w e r e f i r s t obtained by K a m b e r and Tondeur [ 7 ] .
c o r r e s p o n d i n g independence t h e o r e m s
57
Appendix
by
Robert Brooks
In t h i s appendix, we will evaluate the GodbiHon-Vey c l a s s in the c a s e of s o m e
specific a c t i o n s of g r o u p s on the circle.
We wilt then show how t h e s e caIcutaVions lead to
an e x a m p l e due to T h u r s t o n showing how one can v a r y the Godbillon-Vey " n u m b e r . "
Given an action of
G
on S 1 ,
r e c a 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~)
S1
If
on
S1 ,
G
i s a d i s c r e t e subgroup of
PSL(2, IR) ,
then we have a n a t u r a l action of
which we view a s the boundary of the u p p e r half plane, given by the l i n e a r fractional
transformations -
(;
cz÷daz
÷b
F o r e a s e in computation, we can conjugate t h i s action by the l i n e a r fractional
transformation
z-i
z -* ~
taking the u p p e r half plane into the disk
I z I <__1 - we will
want to p a s s back and f o r t h freely between the two p i c t u r e s .
Viewing
f
and
g
a s acting on
z'~
f(z) = 8 t ( ~ ) ;
with
G
1 0 1 1 = t 0 2 1 = 1 , t~1I ,
and so viewing S I
as
IR/z,
Izl = 1,
1
we may now w r i t e
. z'~ 2 .
g o f(z) = 82(~2--~-~_I)
l~21<l
then
f and
g
become
58
e2~x
i
e2~rix.
- ~i
f(x) = ~ i
l°g(el(~.le2~ix
I
log(e~(
- 1)) ; g o f(x) = ~ - ~
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
S1,
b;f = f'
and
^
.
2
))
2 "~2 eZmX " 1
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
S1
i
=
- 21fix
log(1- e-2Z~i%l)(1 e2~rlX~l)[2~ri] (
0
e
~2
-2~ix
1 - e
~2
~2
2 1 f i x _ ) dx
1 - e
v]2
i ~2~i
2ml o g ( ~ )
-
=
W e can give this n u m b e r a somewhat m o r e geometric flavor by now passing to the
upper half plane
- writing
a-i
~i = a + i
,a+i,,b-i,.~- ab)
.
,,~(f, g) = 2 ~ l o g ~ - - ~ _i) t~-$-'[) t~
b-i
' ~2 = b + i
' w e have
"
ie
e
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 is of t h e f o r m r e
+ k,
ie I
i82
the expression
arerealo So given z I = re
+k, z2 =re
+k,
where r,k
z -~
and
log(-i
12)=i(el_e2
z2 - z 1
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 i v e u s
(*)
~(g, g) = (- 41r2) area (A) ,
where A is the geodesic triangle with vertices i, a, and b .
b
Here, of course,
a
and
satisfy f ( a ) = i , (go f ) ( b ) = i .
The formula
a Lie group
g = L i e (G)
G
(.)
r e l a t e s w e l l to the f o r m u l a s of [ 5 ] in t h e f o l l o w i n g m a n n e r : If
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
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
q,
t h e n w e get a m a p f r o m
in t h e f o l l o w i n g w a y : g i v e n
X ~ g ,
).
59
t h e n it d e f i n e s 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
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
Let
a
M,
K
be a m a x i m a l c o m p a c t s u b g r o u p of
K - invariant
volume
@ on
M.
Then
and the
M.
G,
with L i e a l g e b r a ~
£X(e) = ~X 0 e
w h i c h we m a y t h i n k of a s the " d i v e r g e n c e " of
M,
X.
,
and c h o o s e
d e f i n e s a function
/~X on
One m a y define an " i n f i n i t e s i m a l "
G o d b i l l o n - V e y c l a s s by t h e f o r m u l a
~(X0, X I , ' " , Xq) =
f ,Xod ~ x I A " " A d ~ x q
and one sees easily that this defines an element of
Hq+l(g/~) , independent of
In [5] is constructed, given a discrete subgroup
H*(g/]~)- . H*(BF).
In the specific case of our standard
F
of
e.
G , a map
PSL(2,1R)
action on
Sl
,
one
can c h e c k e a s i l y that t h i s " i n f i n i t e s i m a l " Godbillon-Vey c l a s s a g r e e s with o u r "global"
Godhillon-Vey class.
Using the formula
(,),
we can c o n s t r u c t foliated c i r c l e b u n d l e s h a v i n g a r b i t r a r y
Godbillon-Vey n u m b e r , a c c o r d i n g to t h e following s c h e m e due to T h u r s t o n .
b e any two p o i n t s in t h e u p p e r half plane.
c , t h e n t h e r e a r e unique e l e m e n t s
f(c) = a
f(b) = i
g(c) = b
g~a) = i
Then f g f - l g - l ( i) = i , and so
a
and
b
T h e n t h e r e i s a unique p a r a l l e l o g r a m (in t h e s e n s e
t h a t opposite s i d e s a r e of equal length) h a v i n g a s t h r e e v e r t i c e s
the fourth vertex
Let
f g f - I g- I
f
is a rotation.
and
i, a,
g
and b .
satisfying
A little non-Euclidean geometry
shows us that the rotation is the non-Euclidean area of the parallelogram.
parallelogram may have an arbitrary area, we have shown
Labelling
Sincethe
60
LEMMA.
F o r any 0 < a < 2~r,
i s a rotation t h r o u g h angle
Now let
G
there are
f, g C PSL(2, IR)
a .
be the f u n d a m e n t a l group of the two-holed t o m s .
{ X , Y , Z , W : X Y X ' I y -1 = Z W Z - I w - 1 } ,
[ B G ] = (X,Y) - (XYX " l , x )
and
BG
evaluate our cocycle
Set
and
X=g,
maythinkof
~
- (XYX-1y-1,Y)- (Z,W)+(ZWZ -1,Z)+(ZWZ-IW "l,w)
f'
and
n,
and s o m e
g'
~
a s liftings of
~ ( g , f ) - ¢o(gfg " l , g )
f'
is simply
to t h e p a r a l l e l o g r a m with v e r t i c e s
gfg-lf-t
i s a rotation.
a<
and W ,
21t , c h o o s e
n
wewantto
and
i, a, b,
Similarly,
g'
n.
and
g
to an
and
~
i s now
(- 41r2) . a ,
and c .
n - fold
~
- we
c o v e r i n g of t h e c i r c l e .
~.
a s can be s e e n by applying
o~(gfg-lf " l , g ) = 0 ,
X= g, Y=f,
n - fold
Z=g,
(*)
since
o~(g',g') - ~ ( g . f , g , - 1 , g , ) = (_ 41r2) . h a ,
Hence c h o o s i n g
with
and s i m i l a r l y f o r
to c h e c k by t h e o r i g i n a l i n t e g r a l f o r m u l a that p a s s i n g to an
m u l t i p l i e s t h e r e s u l t by
f
n a.
f'=(x)=lf'(nx),
Of c o u r s e it i s c l e a r that t h e c o m m u t a t o r of "~
Now
X,Y,Z~
with c o m m u t a t o r
and Y = f o We s e t
and
i s given a s
0~.
Fixinga positive integer
~,
G
has a fundamental 2 - cycle
and t h i s i s t h e object on which, f o r c a r e f u l c h o i c e s of
commutator
whose commutator
and it i s e a s y
covering simply
W='~,
we g e t
o~([BG ]) = 4~2(~ 2 - 1)(~.
Varying
6o
c~ between
m a y take any real value°
0
and
2v
~-,
and taking
n
arbitrarily large, w e see that
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,
published.
Independant variation of secondary classes~ to be
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.