Differential characters and geometric invariants

DIFFERENTIAL
CHARACTERS
Jeff
AND
GEOMETRIC
INVARIANTS
Cheeger*
and
James
State
University
Stony
Simons**
of N e w Y o r k at S t o n y B r o o k
Brook, NY 11794
Abstract
This
were
paper
first
distributed
Geometry,
remains)
held
the
version.
at S t a n f o r d
authors'
But,
the original
notes.
recently
work
of C h e e g e r
first
In t h i s
lectures
proper
subgroup
determines
both vanish
class
general
with
H*
~,
but
M.
is the
mod
A
contains
more
invariants
Year).
For
ring
of s m o o t h
A reduction
a class
f
of
reduction
information
R/A
than,
we
form.
assigns
if
A c R
to a
is a
(mod A)
k-cycles
that
f
cocycle
out
R/A,
closed)
uniquely
whose
It t u r n s
is a
to
(necessarily
seen
of a real
of t h e
reasons
character
of s o m e
~.
in the
the r i n g of
u 6 Hk+I(M,A)
is an
and
original
which
singular
It is e a s i l y
for
we discussed
these
speaking,
a differential
requests
the s u b j e c t
in t h e i r
H*(M),
(and
detailed
in p h y s i c s
(which w a s
Roughly
to the de R a h m c l a s s
if
the m a i n
the W e i l
classes
and
interest
homomorphism
a consequence,
*
mod
a more
study of a functor
graded
on
it has b e e n
to r e c e i v e
e.g.
albeit
notes which
real
image
that
~
and
the c o h o m o l o g y
class.
Thus,
in
and
forms
A-cohomology
A-periods.
Perhaps
that
the
if a n d o n l y
of which
Special
~ 6 Ak+I(M).
not only
is c o h o m o l o g o u s
u
is the
form
secondary
contexts,
"collapse"
from the group
coboundary
differential
sketch
of the r e a l s ,
f
the
then
lecture
on D i f f e r e n t i a l
available
continued
the notes,
a certain
characters"
homomorphism
whose
M
on
Since
we
of
Institute
to m a k e
time,
at t h e
publish
paper we
smooth manifold
"differential
in 1973.
in s o m e n e w
and Gromov
to f i n a l l y
Summer
Moreover,
arisen
author's
in a c o l l e c t i o n
intention
in the m e a n
have
decided
appeared
at the A.M.S.
we obtain
** P a r t i a l l y
supported
construction
a refinement
characteristic
Partially supported
GP 3 1 3 5 9 X - I .
of o u r
can be naturally
forms.
comes
factored
of the t h e o r y
In a p p r o p r i a t e
by A l f r e d
P. S l o a n
by N.S.F.
Grant
from the
through
PO 29743002.
As
of characteristic
contexts,
Foundation
fact
H*.
this
and N.S.F.
gives
Grant
51
rise
to o b s t r u c t i o n s
as w e l l
and
as
R/A
foliations.
draw
some
to c o n f o r m a l
characteristic
Moreover,
conclusions
immersion
cohomology
the calculus
f r o m the r e c e n t
of R i e m a n n i a n
classes
we develop,
"geometric
manifolds
for f l a t b u n d l e s
may be used
index
to
theorem"
of
Atiyah-Patodi-Singer.
We
should mention
differential
with
forms
connection.
In fact,
playing
results
in this
in
H*
was
The
format
develop
how
These
the p r e s e n t
in the b a s e
the general
resulting
In p a r t i c u l a r
previous
results
Mn
foliations.
distinguished
range
family
the a s s o c i a t e
consequence
with
these
of Bott's
4*.
relate
characters
IRn+k.
are
theorem,
are
classes.
are
their values
Finally,
(mod Q)
of t h e
to t h e v o l u m e s
in S e c t i o n
discretized
explicitly
9, w e
reformulate
computations
structural
to d e r i v e
some
with
of our
In
invariants
from
and
R/Z
We
relate
o n the s p h e r e .
index
theorem
invariants,
special
a
a n d as a
group.
resolution
the g e o m e t r i c
in t e r m s
mani-
t a k e up
classes.
to c o m e
simplicities
an
In a s u i t a b l e
case our
shown
in t h e b a r
of g e o d e s i c
of A t i y a h - P a t o d i - S i n g e r
our previous
of
classes
7 we
cohomology
cohomology
of
classes.
our
of connection
R/Z
TP(@).
and give
6 we apply
by Bott.
become
classes
forms
consideration
is e q u i p p e d
in w h i c h
These
to the
for a R i e m a n n i a n
independent
show
the
show how these
In S e c t i o n
defined
2, w e
and study
intrinsically
conditions
in
1 we
and Pontrjagin
In S e c t i o n
to f l a t b u n d l e s
these
The multiplication
We
Chern
specialize
cohomology
Earlier
H*
them
8 we
construct
TP(@).
In S e c t i o n
Section
Borel
[9].
objects
detailed
of a f o l i a t i o n
vanishing
in
In S e c t i o n
connection.
and
of c o n n e c t i o n s
characters
to the
bundle
to d e f i n e
[17].
through
with more
necessary
bundle
and Simons
follows:
to the E u l e r ,
conformally
The normal
in
of the r i n g
sum formula.
to g i v e
to i m m e r s e
by C h e r n
to t h a t o f the
be as
concerned
we construct
of the W h i t n e y
related
[7].
connection
corresponding
closely
of a p r i n c i p l e
of t h e a t t e m p t
c a n be f a c t o r e d
of bundles
5 are
are
space
formulated
in
properties
analogue
fold
were
paper will
change with
the c h a r a c t e r s
out
analogous
developed
homomorphism
3, 4 a n d
total
considered
arose
a role
invariants
invariants
Sections
were
work
of t h i s
invariants
on t h e
direction
already
the W e l l
that our
TP(O)
results
and use
in the c a s e
flat bundles.
We are very happy
conversation.
several
We
important
throughout
to t h a n k A.
are especially
insights
the development
Haefliger
grateful
and made
many
of this work.
and W.
to J o h n
Thurston
Millson
stimulating
for h e l p f u l
who
provided
suggestions
52
§I.
Differential
Let
ential
M
Characters
be a
C~
manifold
on
M.
Let
forms
and
let
A*
C k ~ Zk ~ B k
denote
denote
the
ring
the g r o u p s
of d i f f e r of n o r m a l i z e d
smooth
s i n g u l a r cubic chains, c y c l e s and b o u n d a r i e s , a n d
3 : C k ÷ Ck_ 1
6 : C k ÷ C k+l
be the u s u a l b o u n d a r y and c o b o u n d a r y o p e r a t o r s .
If
and
A c R
is a p r o p e r
k-forms
with
homomorphism.
real
If
cochain
the v a l u e s
mod
A.
then
~
only
w ÷ ~
6 ck+I(M,R/A)
Let
via
A k0
we w r i t e
R~
R/A
for
integration,
we may
regard
R/A-cochain o b t a i n e d
for the
the c l o s e d
be the n a t u r a l
~
as a
by r e d u c i n g
A.
a non-vanishing
lying
~k+l
in
~ ( Ak
~
that
of the reals,
lying
and w r i t e
of
Observe
subring
periods
in a p r o p e r
subring
differential
A c R.
is an i n j e c t i o n ,
form
never
Therefore,
and we m a y
takes
values
the m a p
regard
~ ck+I(M,R/A) .
Definition.
Hk(M,R/A)
=
{f 6 H o m ( Z k , R / A ) If o ~ ( A k + l } .
The m o s t
interesting
A smooth
map
Hk(MI,R/A)
^-i
H
(M,A)
objects
module
with
we w i l l
will
,
(M R/A)
call
sequences.
r
A = Q,Z,0.
a homomorphism
~* : H k ( M 2 , R / A )
properties.
= • Hk(M,R/A),
We
is a g r a d e d
characters.
A-module
A ring
÷
set
whose
s t r u c t u r e on this
presently.
size
of
H
by i n s e r t i n g
it in some
exact
Set
Rk(M,A)
Here
the
be
functorial
differential
be i n t r o d u c e d
We can m e a s u r e
will
induces
the o b v i o u s
~,
= A.
cases
¢ :M 1 + M 2
= {(e,u)E
i k0 × H k ( M , A ) J r ( u )
is the n a t u r a l
de R a h m
class
of
(u,~)
(v,~)
=
~.
Theorem
i.i.
There
R(M)
r :Hk(M,A
[~]}
and
÷ Hk(M,R)
has an o b v l o u s
ring
~ Rk(M,A)=
and
[~]
structure
(u U v , w A ¢ ) .
0 ÷ Hk,M,R_A)(
/
0 ÷ Ak(M)
map
=
are n a t u r a l
÷ H
~_k.M,R_A.
(
/
Ak(s)
0
sequences
~i ~ A ~ + l ( m ) ÷
÷ Hk(M , R/A)
0 ÷ Hk(M,R)/r(H(M,A)
exact
2+ H k + I ( M , A )
2)
÷ ^
Hk(M,R/A) (61'6
- - ÷
0
÷ 0
R
k+l
(M,A)
÷ 0.
R*(M,A).
is the
53
In p a r t i c u l a r
if
Hk(M,R)
=
0,
then
f
is
determined
uniquely
by
~ l ( f ) ,~2 (f).
Proof.
T
Let
with
6 A k+l
6e
f E ~k.
TI z k =
- 6c
and
- ~c.
differential
s
and
~
- ~'
[c']
= u.
61,62
are
£ Hk+I(A)
an
~.
there
the
0
= c'
Set
~'
[~]
T
with
6T = ~
R/A
an
R/A
cohomology
represented
by
character
f.
Finally,
e
= ~
for
some
z
Thus
the
0
~
of
= ~.
Again,
~ Ak(s)
By
by
with
~ ÷ ~IZk
sends
third
sequence
follows
Ak
an
f
de
Rahm
=
ker
by
and
62
T"
an
class
~l(f)
c = ~e
theorem,
0
Thus
defines
a differential
so
we
that
theorem,
ziZ k.
with
f
s
with
Rahm
such
so
cohomology
defines
- c
find
6 R/A.
so
of
exists
can
E ~k
R/A
choice
immediately
w = ~'
cochain
f i B k -- 0
is
onto
c + 6d.
that
we
a real
= T + 6s,
de
lift,
E ck(M,A)
there
u
as
T/Z k =
¢IZ k =
~-
E A k0+ l
6 (T-e-0)
the
d
follows
given
= ~
the
Then,
it
~,u
another
some
+ 6d =
that
that
= u.
s IZ k
6T
for
that
the
=
subgroup
follows
is
if
~
v
then
then
map
k
A0.
clearly
The
form
if
it
T'
Then
so
a proper
fact
exact
= T + 6s
s
in
claim
above
exists
0 = 62T
We
given
is
Then
= ~-c,
= 6T
cochain
there
= r(u) .
Conversely
T'
~ Ak(M) .
E zk(M,R).
a closed
fact
independent
=
as
a real
a nonvanishing
only
62(f)
6T = -c,
~ (T-e)
e
and
Similarly
62(f)
In
= 6T'
- c.
cocycle
is
Then
for
some
some
if
E ck(M,A).
d%
class.
f
T.
~ - c
co-cycle.
= ~-c.
6T
[~]
and
= r(u).
Then
61(f) = ~,
62(f) = u.
If
f E k e r 61
then
defines
In
= u.
lying
is
assumption
= T + d + 6s
- c + 6d
~
6T
Since
- c'
=
there
by
mentioned,
and
of
[c]
exists
0.
T'
surjective.
that
values
=
that
61(f)
f o ~,
have
choice
Then
with
Let
= 6c
so
E ck-I(M,R).
Therefore
such
E Hk+I(M,A)
of
divisible,
6T =
as w e
takes
dw
= u
T - T'Iz k =
and
u
never
conclude
~ A k+l,
[c]
0
are independent
then
is
6T =
Since,
form
we
R
Since
c 6 ck+I(M,A)
= d~
A c R,
Since
f.
there
=
for
0.
some
have
also
T -e
-9 =z
exists
So
T I Z k = % + ¢ + e.
and
its
combining
kernel
the
is
first
two.
q .e od.
Corollary
associated
1.2.
to
Let
the
i)
62 IHk (M,R/A)
2)
~±IAk/A~
=
B :Hk(M,R/A)
coefficient
=
d.
-B.
+ Hk+I(M,A)
sequence
denote
the
O + A + R ÷ R/A
Bockstein
÷ O.
Then
54
Proof.
This
We w i l l
in
often
write
to c h e c k
dl(f)
= mf
Let
0 ÷ A 1 ~--~A2 + R.
Let
A i.
The
induces
H k ( M , R / A 2)
Corollary
inclusion
as w e l l
1.3.
0 ÷ ker
i
as
We h a v e
= Hn(M,R/A)
^k
H (M,R/A)
= 0
ters
Example
connection
1 ~
2--~
simple
in g e o m e t r y .
1.5.
Let
e.
curve
let
H(y)
X
the
=
y
and
periods
i :Hk(M,R/AI)+
÷ Hk(m,R/A2 ) .
¢
1-~Hk(M,R/A 2)
k+l~ k+l
+ A 2 /A 1
example
illustrates
In m a n y
ways
+ 0.
be a c i r c l e
denote
class,
Euler
how differential
it t y p i f i e s
real
be h o l o n o m y
1-cycles
a chain
X(y)
1
+ ~-~-~(y).
seen
that
X 6 HI(M,R/Z).
one
forms w i t h
map
sequence
~ E A2(M)
to all
curve
It is e a s i l y
then
closed
an o b v i o u s
+ E ~--~ M
H(y) 6 $0(2)
2~i~
(y).
X(x)
Thus
denote
= uf.
its
the g e n e r a l
bundle
curvature
1 ~ E A2
2~z
0-
around
y,
charac-
over
M
form.
For
case.
with
Since
7
a closed
and d e f i n e
2(Y)
E R/Z
= e
Extend
closed
S0(2)
Let
represents
~2(f)
above.
k > n = d i m M.
following
arise
the a r g u m e n t s
~-- C ~(M,S I)
Hn(M,R/A)
The
Ai
the e x a c t
H0(M,R/Z)
from
and
i, : H k ( M , R / A I)
i, ÷ H k ( M , R / A I )
1.4.
Example
by
is s t r a i g h t f o r w a r d
X
as follows.
y
E C2
is w e l l
If w e
let
Let
so t h a t
x 6 Z1
and
x = Y + ~y.
choose
Set
defined
and
clearly
X o ~ =
denote
the
integral
Euler
X
class
can c h e c k
1
~i (~)
=
carries
vanish
As
ring.
Let
and
2-~ ~'
more
when
information
X
already
does
~ : C, +
~
C,
not,
mentioned,
To d e f i n e
let
=
62 (~)
X.
than
~
e.g.
be the
X
standard
homotopy
together,
since
both may
M = S I.
the d i f f e r e n t i a l
the m u l t i p l i c a t i o n
be its c h a i n
and
characters
we m u s t
introduce
subdivision
to
1
(see
map
form
in c u b i c a l
[ii]).
a graded
subdivision.
I.e.
theory,
a
55
1 - A
Since
on
~
~
+ ~3.
is n a t u r a l ,
Thus
the
Consequently,
if
and
~.
=
ant u n d e r
if
is a
then
differential
If
8, ~ E A*
may
thus
cup
Kervaire
has
allows
we m a y
@
one
~
of
differential
= f(x)
8,w,
and get
and
E Ck+ 1
operates
= f(x)
8 A w
real
is zero.
on e v e r y t h i n g ,
characters.
product
another
is s u p p o r t e d
as c o c h a i n s )
- f(~x)
A
~(~)
~(~)
A
(regarded
to c o n n e c t
regard
and
volume
then
~ o 9 = 0.
forms
So are
A(f) (x) = f(Ax)
Subdivision
k-simplex
(k+l)-dimensional
subdivision.
then
q
~ E A k+l
in p a r t i c u l a r
x E Zk
(1.6)
are
In fact
- wf(~x)
and
U
as real
cochain
invariif
= f(x).
product.
cochains.
@ U ~.
In
We
[12]
shown
lim An(@ Uw)
=
8
~.
A
1.7)
n-~oo
It is b e c a u s e
Let
~i
of this
formula
W l , ~ 2 E A ZI,
A w 2 - e I U~2
A ~2
exact
that we use
be closed.
cubical
Using
in a c a n o n i c a l
theory.
(1.7)
way.
w e can m a k e
Define
£i+~2-i
E ( W l , ~ 2)
( C
(M,R)
by
0o
E ( w I , ~ 2) (x) = -
A straightforward
is d o m i n a t e d
estimate
by a g e o m e t r i c
is then o b v i o u s
1.8)
[ ~i U ~2(~£ix)
i=0
shows
series
that
the
and h e n c e
right
hand
converges.
side of
Moreover,
1.8)
it
that
lim E ( W l , ~ 2) (Anx)
=
0.
(1.9)
n-~oo
Now
co
6 E ( W l , W 2) (x) = - [
i=O
.
co
w I U~2(}AI3x)
.
= - [ w I Uw2(@~Alx)
i=O
co
.
= - [ ~i U w 2 ( ( I - A - ~ ) A I x )
i=0
,
= lim [ w I Uw2((l-A)Alx)
n÷co
i=0
= lim - ~i U w 2 ( ( l - A n + l ) x )
n÷co
=
where
we have
since
the
w.
1
used
are
(w I A ~ 2 - W l
(1.6),
closed.
U ~ 2 ) (x)
(1.7)
and
Hence
the
fact
that
6(w I U w 2)
=
O,
58
8E(e 1,e 2)
=
e 1A e 2 - e 1U e 2.
The m a i n p o i n t
a sequence
fact,
if
in the above
of natural
8
chain
n
All e x p r e s s i o n s
homotopies
of c e r t a i n
will
differ
as a finite
expressions
between
1
{[i=0
and
the p r o p e r t y
~AI}
An+l.
is
In
that
E(el,~2) (x) = l i m - e I U e2(SnX).
universally
sum of terms
in
k
Tg
with
is that
then we can take
so o b t a i n e d
One such can be w r i t t e n
x
computation
is any such s e q u e n c e
lim - e I U e2(0n x) exists,
over
(i.i0)
by exact
involving
cochains.
integrals
e l, e 2.
~k 2
k1
N O W let
f 6 H I(M,R/A), g E
(M,R/A)
and choose
k2
~
=
( C
(M,R)
with
TfiZkl = f, TgiZk2
g"
Tf
6 C
(M,R),
Definition.
f * g = Tf U e g
Theorem
kI ~
ef U T g
~
- Tf U S T g + E(ef,eg) IZkl+k2+ I.
^kl+k2+l
f * g 6 H
(M,R/A)
i.ii.
the c h o i c e s
- (-i)
of
Tf, T .
g
is well
defined
independent
of
Moreover,
i)
(f *g)
* h = f , (g ,h)
2)
f , g =
3)
ef,g = efA Wg
and
Uf,g = uf U Ug ^ i.e.
81
and__ 82
rinq h o m o m o r p h i s m s as is
(81,82):H(M) ÷ R(M) .
4)
If
(kl+l) (k2+l)
Proof.
g , f
~ :M ÷ N
Let
is a
8Tf = ef - cf,
To see that
computes
(-i)
f,g
map,
C ~
then
¢*(f*g)
8Tg - eg - Cg
is a d i f f e r e n t
with
character
=
~*(f),~*(g).
[cf] = uf,
such that
are
[Cg] = Ug.
3) holds,
one
that
k1
8(Tf Ueg - (-i)
=
=
(ef-cf)
formal
choose
U e g + e f U (eg-Cg) - ( e f - c f )
ef A e g
3) follows
choices
of
is the mod
Tf, Tg
That
the d e f i n i t i o n
is s t r a i g h t f o r w a r d .
and 4) is trivial.
as above.
A
U (eg -c g ) + e f
Ae g - e f U e g
cf Uc g .
immediately.
argument
Th
ef U T g - T g U d T g + E ( e f , e g ) )
A direct
reduction
of
is i n d e p e n d e n t
of the
2) can be p r o v e d^ by a simple
To see i) , let
computation
shows
h 6 H k3
that
and
(f*g*h-f*(g*h)
57
k1
E(wf,Wg) U e h + E ( w _ZA w _g, ~ h) + (-i
f
U E(Wg,W h) - E ( w f , W g A ~ h) ,
and that the c o b o u n d a r y of this e x p r e s s i o n is zero.
similar estimates show that the limit of
a cocycle with zero periods,
Note that if
A
(1.9)
(1.12)
and
(1.12) under subdivision,
and i) follows,
is discrete,
e.g.
is
q.e.d.
A = Z,
then by use of
(1.7)
we have
f,g
=
lim A n ~
k l ~ g
(T 1 U ~ g -(-l)
- T f U~Tg) IZkl+k2+ I.
Two special cases are important and follow easily from the definition.
kl+l
f,g =
(-i)
f,g =
(-i)
k2
uf U g
g E H
~f A g
g E A
kl+l
k2
(M,R/A)
(1.14)
k2
/A 0 .
(1.15)
T h e o r e m i.ii may be p a r a p h r a s e d as saying that
from m a n i f o l d s to rings and
tion of functors.
property.
The
*
(61,62) :H* ÷ R*
H*
is a functor
is a natural transforma-
product is p r o b a b l y c h a r a c t e r i z e d by this
It is also p o s s i b l e to represent d i f f e r e n t i a l characters b y
d i f f e r e n t i a l forms w i t h singularities
respect to this representation,
(although not canonically).
With
there is a nice formula for the p r o d u c t
w h i c h generalizes that of Example 1.16 below.
(For more details see
[7]).
Example 1.16.
M = S I, f,g E H0(SI,R/Z)
be r e p r e s e n t e d by functions
G(x+2~)
= G(x) + n 2
HI (SI,R/Z) ,
f,g(S I)
§2.
with
F,G : R + R
nl,n 2 E Z.
= C~(SI,R/Z).
so that
Now
f
F(x+2~)
and
g
= F(x)
f*g E HI(sI,R/Z)
may
+ n I,
=
and
=
f2z
nlG(0 ) - J
FG'.
0
A Lift of Weil H o m o m o r p h i s m
Let
G
be a Lie group w i t h finitely many components,
c l a s s i f y i n g space and
Let
space
e = {E,M,%}
M
I*(G)
0.
Let
G - b u n d l e w i t h total space
e(G)
e ÷ I*(G), H*(BG,R),
E,
G.
base
be the category of these ob-
jects w i t h m o r p h i s m s being c o n n e c t i o n p r e s e r v i n g bundle maps.
have the functors
its
the ring of invariant p o l y n o m i a l s on
be a p r i n c i p l e
and c o n n e c t i o n
BG
H*(BG,A),
Then we
H*(M,A), H*(M),
58
A~I(M)
(= closed
forms).
(In the first
three
cases,
to any m o r p h i s m
we a s s i g n
the i d e n t i t y map).
The Weil h o m o m o r p h i s m c o n s t r u c t s a
homomorphism
w : I k (G) + H 2k (BG,R)
and a natural t r a n s f o r m a t i o n
W : Ik(G)
+ A2k(M)
formations
such that the f o l l o w i n g
I*(G)
W
dR
(2 .i)
H* ( M , A ) .
are p r o v i d e d
by the theory
is the de Rham h o m o m o r p h i s m .
of c h a r a c t e r i s t i c
If
P ~ Ik(G),
~ is the c u r v a t u r e form of
e E e then e x p l i c i t l y ,
k
P ( ~ ) ,
and
CA(U) = u(e),
the c h a r a c t e r i s t i c class.
K2k (G,A)
{ (P,u)
K*(:G,A) = @ K 2 k ( G , A )
(2.1)
phrased
x H2k(BG,A)
[ w(p)
W(P)
=
Set
: r(u)}.
forms
a graded ring in an o b v i o u s way.
MoreWxc A
K*(G,A)
~ R*(M,A).
Our result may be para-
induces
as saying
(Ik(G)
classes
u E H*(BG,A)
and
over
trans-
H* (BG,A)
< r
H*(M,R)
CA, C R
dR
of natural
[C R
A~I(M)
and
r
~ H*(BG,R)
W
Here,
diagram
commutes
that
there
exists
a unique
natural
transformation
^
S : K*(G,A)
+ H*(M,A)
such that the d i a g r a m
H* (M, R/A)
- WXCA*
K*(G,A)
commutes.
Theorem
R* (M,A)
In more detail:
2.2.
a unique
Let
Sp,u
(P,u)
6 K2k(G,A).
H2k-i (M, R/A)
i)
~l(Sp,u(a))
= P(~).
2)
62(Sp,u(e))
= u(~).
3)
If
8 6 e(G)
and
For each
~ ~ e(G)
there
exists
satisfying
~ :e ÷ ~
is a m o r p h i s m
then
~*(Sp,u(B))
=
Sp, u (a) •
Proof.
An o b j e c t
8N =
(EN,AN,P N)
6 £(G)
is called
N-classifying
if
59
any
(E,M,8)
= ~ E e(G)
with
for any two such morphisms,
smoothly
homotopic.
objects
exist.
dim M < N
the corresponding
It is well known that
N-classifying
large.
to Theorem
isomorphism
and the theorem
N-classifying
objects
follow in general
to
BN,,N'
~i :AiN ÷ AN'
above,
i.i,
(61,62)
follows
by setting
= 0
trivially
=
>> N,
such that
8~
be the corresponding
~(Sp,u(SN,))
= Sp,u(B~).
are
[15] such
and since
for
N-sufficiently
+ R2k(AN)
in the category
There
is an
It will
is an
admit morphisms
it suffices
8N
of such
(@i'62)-I(P(~))"
of
maps of
N-classi-
to
maps of base spaces.
Therefore,
and
= 0
:~2k-l(~)
Sp,u(BN)
BN
fl,f2 ÷ M
H°dd(BG,R)
H2k-I(AN,R)
f$ (Sp,u(B~ )) = fl*(Sp,u(B~)).
then
fying object
Let
maps
to
if we can show that if F0,FI
are morphisms
0 8N1 with
f.1 :M ÷ A N± the corresponding
BN'
N-classifying
base spaces,
a morphism
By a theorem of Narasimhan-Ramanan
is topologically
Referring
admits
BN,.
By the
to check that
(%0 o f0)*(Sp,u(BN' )) = (%1 o fl )*(SP, u(BN,)).
Let Gt be a homotopy
between
~0 o f0 and
%1 o fl" Further, choose
G t to be constant
near
t = 0,
forms of
t = i.
BN,
and
Let
z E Z2k_l(M)
G[(SN,),
and
~, ~
be the curvature
(the latter being a bundle over
M ×I).
Since
(¢lOfl)*(Sp,u(BN,))
- (¢0of0)*(Sp,u(SN,))
(z) = Sp,u(~Gt(zxI))
= ]
P (~q)
Gt (z×I)
we must show that
t = 0, t = i,
t
fGt(zxi)P(~)
the induced
near these points.
we obtain
denote
E A.
connection
By identifying
the characteristic
p(~)
JGt (z×I) P (~)
z x S1
2.3.
=
]z×I
is a cycle and
The map
Gt
is constant
G~(EN,)
over
near
is independent
G~(EN,) IM x 0
form for this bundle.
t
Corollary
on
a bundle with smooth connection
=
Since,
Since
with
M × S I.
of
G~(EN,) IM x i,
Let
P(~)
Clearly
F
]
P(~) .
z×S 1
E A° ,
P(~)
S :K*(G,A)
the theorem
~ H*(M,R/A)
follows.
is a ring homomorphism.
i.e.
SpQ,uUv(e)
This
follows
uniqueness
1.2 we see
=
Sp,u(~) * SQ,v(C~) •
immediately
statement
from the properties
in the theorem.
of
*
From Theorem
product
and the
i.i and Corollary
60
Corollary
2.4.
Suppose
P(~)
i)
Sp, u (e) 6 H 2k-I(M,R/A)
2)
B(Sp,u(e))
Example
reasons
Suppose
If
and
Then
= -u(e).
2.5.
oriented.
= 0.
(P,u)
S
~ = {E,M2k-I,@}
6 K2k(G,A)
(a)
then
P(Q)
6 H2k-I(M,R/A).
P,u
c y c l e we get the c h a r a c t e r i s t i c
Sp,u(~) (M 2k-l)
where
M 2k-I = 3 M
and that
be any e x t e n s i o n
Let
~
~ = {M,E,8}
Sp,u(~) (M 2k-l)
It m i g h t
appear
E
extends
of
we h a v e the m o r p h i s m
Since
=
~l(Sp,u(~))
l
J
since
to
8
E,
a principal
G-
to a c o n n e c t i o n
~-~ ~.
Thus
in
Sp,u(~)
E.
I
= P(~)
P(~).
f r o m this
b u t this is false
for d i m e n s i o n
on the f u n d a m e n t a l
6 R/A.
M.
Setting
and
number
bundle
M 2k-I = Sp, u(e).
is c o m p a c t
vanishes
Evaluating
NOW suppOse
over
M 2k-I
(2.6)
formula
that these numbers
d e p e n d o n l y on
P,
whose boundary
only extends over a manifold
2k-i
is a f i n i t e u n i o n of c o p i e s of
M
, a n d the c h o i c e
of
a rational
u
removes
In
[9] the a u t h o r s
TP(8)
where
=
forms,
P(~)
one m a y
Proposition
2.8.
This m a k e s
E
If
in
reduced
: E ÷ M,
when
ambiguity.
considered
= t~ + ~(t2-t) [@,@],
2
Ct
E
the forms
TP(8)
defined
in
E
by
tl
~-l)d t
k } P(8 A
2O
=
dTP(6)
These
in g e n e r a l
E.
m o d A,
(2.7)
are the lifts of the
Sp,u(~).
Letting
show
~*(Sp,u(e))
the c h a r a c t e r s
has a g l o b a l
e0,e I
and s h o w e d
=
TP(e)
representable
I Z2k_I(E).
by s p e c i f i c
differential
forms
cross-section.
are c o n n e c t i o n s
on
E
set
~i = {E'M'0i}"
Then
6 2 ( S p , u ( e l ) - Sp,u(~0)) = U(el) - u(e 0) = 0. T h u s by (i.i)
the
d i f f e r e n c e of the c h a r a c t e r s m u s t be the r e d u c t i o n of a form.
Let
be a s m o o t h c u r v e of c o n n e c t i o n s
joining
80
to
81 ,
let
~t
@t
be the
61
curvature
at time
Pr°p°siti°n
t,
2"9 •
! =
and set
Sp,u(~l)
d/dt(et).
@t
- S p , u (S 0)
This m a k e s
sense s i n c e
@'
vanishes
t
is the lift of a form on
M.
grand
A bundle
is c a l l e d
flat
if
that in this case the h o l o n o m y
connected.
this
{E,M}
is a l w a y s
is i n d u c e d by a m a p
: B H ÷ B G,
and for
are s o m e t i m e s
called
see
[2].
We r e c a l l
all torsion,
Proposition
If
e
2.10•
inclusion
we get
is f i n i t e
This
which
a product.
{EH,M},
H _c G
~*(u) 6
classes
[i] shows
t o t a l l y dis-
H-bundle
and
induces
H2k(BH,A).
These
of the r e p r e s e n t a t i o n ,
its i n t e g r a l
If
e
cohomology
is
Sp, u (a) 6 H 2 k - I ( M , R / A ) .
is flat then all
flat then all
S
(e) = 0.
If
P,u
=
theorem
is a r c w i s e
to an
The
u 6 H2k(BG,A),
H
and the inte-
flat if it is t r i v i a l l y
reducible
the c h a r a c t e r i s t i c
(and is c o n s e q u e n t l y
Sp,u(~)
§3.
The h o l o n o m y
H _c G
p : M + B H.
that if
vectors,
H2k-I(BH,R/Z) ~ H2k-I(BH,Q/Z)--~ H2k(BH,Z) .
and
is g l o b a l l y
-holonomy
~ = 0.
[9] we h a v e
I~ p (@LA~ k - 1 ) d t I Z2k_I(M) •
k
on v e r t i c a l
group
It is c a l l e d g l o b a l l y
The G - b u n d l e
=
As in
flat)
p*(B-l(~*(u))
f o r m u l a was p o i n t e d
is s t r a i g h t f o r w a r d
and
~
has f i n i t e
--
A = Z
then
~ H2k-I(M,Q/Z).
out to us by J o h n M i l l s o n ;
appears
in his d i s s e r t a t i o n
its proof,
[13].
The E u l e r C h a r a c t e r
It is p o s s i b l e
character
X.
bundle over
SV ~-~ M
to g i v e a m o r e
Let
M,
with
V
(s2n-l)
*
> H2n_I(M)
of the E u l e r
2n-dim Riemannian
covariant
s p h e r e bundle.
÷ H2n_I(SV)
construction
be a real
denoting
be the a s s o c i a t e d
H2n_l
intrinsic
V 2n = {V,M,q}
differentiation.
vector
Let
We have the h o m o l o g y
sequence
(3.1)
÷ 0.
be the i n t e g r a l E u l e r class and let
PX 6
Let
X 6 H 2n (Bso(2n) Z)
In(So(2n))
be the u n i q u e p o l y n o m i a l w i t h
w ( P X) = X.
(Px
is u n i q u e
since
G
bundle
of
P
becomes
(D)
X
form
Q
satisfies
is c o m p a c t ) .
V,
(see
Let
F(V)
with connection
exact
[8]) on
in
SV,
SV
@
6 c(SO(2n))
and c u r v a t u r e
and in fact t h e r e
which
be the o r t h o n o r m a l
is n a t u r a l
Q.
frame
The E u l e r f o r m
is a c a n o n i c a l
in the c a t e g o r y
(2n-l)
and w h i c h
82
~*(Px(9))
=
dQ
and
]
Q = 1.
S
w
Let
z 6 Z2n_I(M).
6 C2n(M ) with
z
=
~,(y)
We d e f i n e
=
cult
and
In the
then
compact
(3.1)
42(X(V))
=
special
choose
x(V)
easily
that
and
(H2n-I(M,R/Z),
(3.3)
shows
x(V)
@I(X(V))
= X(V) •
case
to be w e l l
= PX(~),
Since
Q
and
defined.
It
it is not d i f f i -
is n a t u r a l
X
is n a t u r a l ,
(3.4)
that
a global
{V,M,V}
Let
V =
over
one
on
V • W,
get
the n e w
d i m M = 2n - i ,
cross-section
(3.3)
simplifies.
% : M ÷ SV,
and
if
M
We
is
%*(Q) (M).
M.
3.6.
Proof.
Since
classifying
and
product
letting
V @ V'
V ~ W
may
§4.
Characters.
G = GI(n,C).
A
be an
polynomial,
1
det(ll -2-~
X(V)
=
n
[
k=O
from
vector
naturally
the n a t u r a l
a bundle
it s u f f i c e s
map
from
W
complex
C k 6 Ik(Gl(n,C))
A)
W
V • W = {V @ W ,
has v a n i s h i n g
The W e i l
and
induces
connection
M,
we
? ~?'}.
, x(W).
follows
n × n
be two R i e m a n n i a n
V
denote
be i n d u c e d
again
theorem
on
bundle
by n a t u r a l i t y
a product
and the
Let
vector
=
dimensions
Chern
W = {W,M,V'}
inner
X(V ~W)
spaces,
Such
(3.5)
and
The
Riemannian
Theorem
Chern
(Z2n_l(SV)
and o r i e n t e d
bundles
Let
y
Spx,x (F(V)) .
=
kernel.
find
2.2
x(V) (M)
there.
we can
+ PX(~) (w).
(3.3)
so by T h e o r e m
x(V)
may
of
from
to s h o w
character,
Q(y)
An analysis
is i m m e d i a t e
(3.1)
+ ~w.
the E u l e r
X(V) (z)
By
(3.2)
2n-i
to c h e c k
real
Theorem
is onto,
matrix
over
a product
the
cohomology
i.i
theorem
in odd
and T h e o r e m
b u t has
and d e f i n e
of
2.2.
a large
the
k th
by
[C k(A) + i D k(A) ]In-k.
(4.1)
63
Letting
(Ck,C k)
ck
(
Let
denote
the
k th
integral
C h e r n class,
w(C k) = c k,
and
K2k(GI(n,C),Z).
En_k+ 1
be the S t i e f e l m a n i f o l d
We do n o t r e q u i r e
i < 2k - i .
H 2 k _ l ( E n _ k + I) ~ Z
U(n)/U(k-l)
c En_k+ 1
V = {Vn,M,?}
En-k+l
of
n - k + 1
t h e s e to be o r t h o n o r m a l .
and the i m a g e of
gives a generator,
bases
H.(E~ n _ k + l ) = 0
be the S t i e f e l
C n.
S 2k-I = U ( k ) / U ( k - l )
h2k_l,
of this group.
be a c o m p l e x v e c t o r b u n d l e w i t h c o n n e c t i o n .
÷ Vn
~
n-k+l
in
for
c
Let
Let
M
b u n d l e of
n - k +l
d i m b a s e s of
V n.
Analogous
to
(3.1) we h a v e
Z ~ H 2 k _ l ( E n _ k + I) ÷ H 2k-i (V n-k+l
n
)
an e x a c t s e q u e n c e .
Letting
E(V)
(4.2)
~ H 2 k - i (M) ÷ 0
be the
GL(n,C)
basis bundle
of
V, w i t h c o n n e c t i o n
@ and c u r v a t u r e
~,
~*(Ci(~))
is e x a c t in
Vn
n-k+l"
In fact, there is a f a m i l y of c a n o n i c a l
(2k-l)
forms,
Q2k-l'
n-k+l'
Vn
in
the c a t e g o r y
defined modulo
and w h i c h
exact
forms w h i c h
is n a t u r a l
in
satisfies
f
=
dQ2k_l
Let
~*(Ck(~)) ,
z ~ Z2k_I(M).
w (C2k(M)
+ ~w.
We d e f i n e
the
Ck(V) (z)
=
X
k th
(4.2) we can find
~k(V)
it is e a s i l y
=
W e set
c(V)
If
V, W
their W h i t n e y
3.6
Chern
Q2k_l(y)
61(Ck(V)) = Ck(~),
is n a t u r a l , T h e o r e m
Theorem
By
Q 2 k - i = i.
(4.3)
n
y ( Z 2 k _ l ( V n _ k + I)
and
with
z = ~,(y)
As w i t h
;~h2k_l
character,
Ck(V)
~ H2k_I(M,R/Z)
+ Ck(~) (w) .
s h o w n that
and
62(Ck(V))
2.2 shows
(4.4)
Ck
is w e l l d e f i n e d
= Ck(V) .
and t h a t
S i n c e the f a m i l y
S C k , C k (E (V)) .
= 1 + ~I(V)
Q2k-I
(4.5)
(V) ( H ° d d ( M , R / Z ) .
n
are two c o m p l e x v e c t o r b u n d l e s wit]] c o n n e c t i o n s
sum as in
by
+...+ ~
§3 and u s i n g
(4.5)
and T h e o r e m
we form
2.2 show as in
84
Theorem
4.6.
c(V @ W )
The W h i t n e y
ful.
Let
let
~
=
c(V)
* c(W).
sum c o n n e c t i o n
be a n o t h e r
on
V ~ W
connection
on
is not
V ~ W,
R
:V
• W + V
• W
be its c u r v a t u r e
x,y
m
m
m
m
d i r e c t sum p r o j e c t i o n
~
induces connections
W.
We
l)
iv
2)
Rx,y(Vm)
Using
call
~
=
compatible
v,
~
=
Theorem
4.7.
Let
The m o s t
situation.
flat
=
other.
4.8.
c(V)
If
* c(V -I)
V
has
fibers
then
vector
bundle.
n
(V)
the
V -I
character,
ponding
The
~v
V
and
~w
on
on
of
if t h e r e
e.g.
immersed
bundle
4.7 o c c u r s
V
following
a globally
the c o m p l e x i f i e d
in
is not
to
in the
exists
Rn
are
tangent
inverses
of e a c h
unique.
V
real
hermitian
bundle,
relation
V R,
inner
product
on
its
is a
2n - d i m
Riemannian
holds
(4.9)
useful
ch
to w o r k m o d u l o
( H e v e n (BGI(n,c),Q)
polynomial.
So
Sp
and
V ~ W.
-x(V R)
Let
m
shows
V @ W.
constant
expected
Pch
=
By
if
of T h e o r e m
is i n v e r s e
a covariant
and let
ch(V)
transformation.
= i.
It is s o m e t i m e s
i.e • in R/Q.
on
inverse
the u n d e r l y i n g
=
one
an i n v e r s e
of a m a n i f o l d
If
use-
x, y 6 T(M)
* c(W).
instance
connection
bundles
In g e n e r a l
Corollary
$(V)
is c a l l e d
compatible
and n o r m a l
(2.9)
be c o m p a t i b l e
important
W
the m o s t
~ W m.
formula
~
&({VeW,M,?})
V ~ V'
for
v'
~ V m, Rx,y(Wm)
the v a r i a t i o n
with
always
and
= P c h ( C l ..... Cn)
W(Pch)
,ch(E(V))
= ch,
elements
of
finite
order,
be the t o p o l o g i c a l
6 I(Gl(n,c))
and we
Chern
be the c o r r e s -
set
6 Hodd(M,R/Q)
ch
ch
work
is just
in
R/Q
the u s u a l
because
formula
for
ch,
of d e n o m i n a t o r s ,
with
*
e.g.
if
replacing
V
U.
is a line
We
bundle.
85
n
Sh(V)
=
&l*&l
2
+"
1 + Cl +
- -
Given
V, W
there
as in T h e o r e m
Theorem
§ 5.
let
connection
=
ch(V)
+ ch(W)
ch(V ® W )
=
ch(V)
* ch(W) .
V • W,
be a real v e c t o r b u n d l e w i t h
be its c o m p l e x i f i c a t i o n .
=
and one shows
connection,
and
We set
(-i) k C2k(VC) .
we h a v e
Pk ( I 2 k ( G l ( n ' R ) )
=
(5.1)
the P o n t r j a g i n
with
the b a s i s b u n d l e
Pk(V)
on
(4 .i0)
"'"
Characters
V = {Vn,M,V}
BGI(n,R )
denote
+
c h ( V ~W)
Pontrjagin
Pk(V)
In
is a s t a n d a r d
n!
3.6.
4.11.
Let
c
V
"+ C l * ' ' ' * C l
"
W(Pk)
of
V
= Pk"
class
p
Letting
one e a s i l y
and the p o l y n o m i a l
E(V)
( E (GI(n,R))
shows
S P k , p k(E(V)) .
We d e f i n e
~
(5.2
and c o m p a t i b l e
connection
as in the c o m p l e x
2 elements
in
case,
and o b t a i n
P(V ~ W )
= P(V) * p ( W )
p(V~W,M,~)
=
? [n/2]
1 + Lk=l
In o r d e r
c^ik
=
Pk^± =
Defining
= p(V egW)
+ order
V
compatible
H°dd(M,R/Z)
where
(5.3
(5.4
Pk"
to get a p r o p e r
inverse
f o r m u l a we i n d u c t i v e l y
define
-Ck - Ck_
A 1 * c^±I -.. .- Cl , Ck_
^± 1
Ck (Vc)
inverse
as in the c o m p l e x
~k(v -1) : ~(v).
case we see
(5.5)
86
§6.
Applications
Let
M
be a R i e m a n n i a n
connection,
and
set
V,
Pk(M)
Theorem
6.1.
= Pk(M,g)
=
Theorem
sion
Euclidean
connection
Setting
V,
N(M)
by
on
on
T(M)
in
and
T(M)
Mn
for
i >
Riemannian
= {T(M),M,V},
metrics
on
as
in
M.
Then
~
see
with
implies
the
immersion
N(M)
Proposition
2.8
show
forms
is i s o m e t r i c .
flat
V'
is an i n v e r s e
[~]
the
immer-
the R i e m a n n i a n
a connection
i >
that
characters.
a conformal
for
= Pi(N(Mn) ) : 0
Pk
these
be the g l o b a l l y
induces
that
[9] for
[9] that
the
admit
[
induces
that
in
[~]
assume
let
• N(M).
we
same
so are
that
and
and also
together
R n+k
thus
we can
bundle;
T(M)
PC± (Mn)
theorem,
immersion
theorem
= {N(M),M,V'}
(5.5)
This
condition
is the
It i ~ p r o v e d
and
^±(M n) = 0
Pi
is that
be the n o r m a l
connection,
g,
Set
equivalent
(2.9),
forms.
invariant,
By the p r e v i o u s
N ( M n)
Thus
TP(8)
A necessary
R n+k
Proof.
Let
be c o n f o r m a l l y
formula,
of the
conformally
6.2.
in
T(M).
: pk(T(M)) .
g, g
The d i f f e r e n c e
are
with metric,
bundle
pk (M,g) .
the d i f f e r e n c e
forms
Geometry
manifold,
in the t a n g e n t
Let
pk (M,g)
Proof.
to R i e m a n n i a n
on
N(M).
of
T(M) .
.
TP I. (8)
that
in the
conformal
frame
l
bundle
are c l o s e d
result
of
The
next
doctoral
Theorem
Then
6.3
Proof.
(Millson) .
In the
the
flat
F = {vn+I,M,~}
where
Since
the t r i v i a l
F
M
is
shown that
to J o h n
This
~ TP
Millson,
be a c o m p a c t
its
(8)
is a m a i n
is
integral.
and are p a r t of his
nonnegative
space
connection
a n d we a g a i n
on
may
over
group
line
induced
~
M
2.10.
form.
by
~
Thus
by
use P r o p o s i t i o n
and
acts
to the
easily
is the R i e m a n n i a n
with
(5.3)
2.10.
V ~ V',
and
holo-
freely.
By t o p o l o g y
and one
finite
In the p o s i t i v e
associated
F.
bundle,
is c o m p a t i b l e
L'
is flat w i t h
is f i n i t e
flat b u n d l e
T(M)
is flat
bundle
from P r o p o s i t i o n
fundamental
is a t r i v i a l
on
tangent
F ~ 0(n+l)
be the
of
the
follows
where
L
hhe c o n n e c t i o n
pi(F),
it
are due
Let
case
theorem
representation
V~
fact
cohomology.
(H4i-I(M,Q/Z).
M = sn/F,
V n+l,
in
two t h e o r e m s
Pi(M)
and
case
where
integral
dissertation.
all
nomy
[9],
and r e p r e s e n t
(5.4)
Let
inclusion
T(M)
~ L'
sees
that
connection,
where
Pi(M)
V'
=
is
67
Example
6.4
(Millson).
Let
M 4k-I = L
be the lens s p a c e
P;ql'''''q2k
by the c y c l i c g r o u p of o r d e r
p generated
o b t a i n e d by d i v i d i n g
S 4k-I
2~iq I
2~iq2k
by
(e
P
P
,...,e
t i v e l y prime.
),
As in Ex.
where
,
^ . 4k-i
Pk (M
)
where
ok
is the
1 m o d p.
e.g.
Theorem
dim
6.4
p
are p a i r w i s e
rela-
give numbers,
2
elementary
q' (l+q2)
P
-
Z.
symmetric
notation,
~
functions
the
and
qlql -=
3 - d i m lens space
(Millson).
mod
w i t h the n o n - i m m e r s i o n
For e a c h
smoothly
Z.
P
these calculations
lens spaces
immersible
and
and
-=
Coupling
k th
qi
ql .... 'q2k )
mod
P
in the s t a n d a r d
Lp,q = L p;l,q'
Pl(Lp,q)
( 2
q{'''q2k°k
=
the
2.5 the top c h a r a c t e r s
k
theorem
shows
there are i n f i n i t e l y m a n y
immersible
in
R 4k
(4k-l)
b u t not c o n f o r m a l l [
R6k-1.
in
The n o n n e g a t i v e
space
forms
themselves
m a y be u s e d as t a r g e t
manifolds.
Theorem
6.5.
A necessary
in a n o n n e g a t i v e
for
i >
Proof.
N
space
m o d Q,
Reduce
of
M
that
Mn
is that
curvature
that on
~
=
T
*
as an
class.
i(N) = 0
R/Q
of
vector
is c o n s t a n t ,
(5.3)
p(N)
6.3 shows
=
bundles
Pi
M
the c o n n e c t i o n
as R i e m a n n i a n
a n d by
p(T)
But T h e o r e m
TO v a n i s h
tangent
t e n s o r of
T ~ N,
p(T)
and r e g a r d all
together with
t i o n we r e g a r d
mology
~n+k
be c o n f o r m a l l y i m m e r s i b l e
( H 4 i - i (Mn,Q/Z)
p ~ ( M n)
[~].
be the R i e m a n n i a n
bundle
condition
form
and
( H4i-I(M'R/Q)"
and
M,
over
its c o n n e c t i o n
T,
M.
M.
By r e s t r i c Because
is c o m p a t i b l e
the
with
(5.4)
m o d Q.
p(T)
= 1
m o d Q,
a n d so
i>k
character
is e q u i v a l e n t
T,
and the f o r m a l
i n d u c e d by
bundle
Let
to b e i n g a
Q/Z
coho-
68
The
Since
but
case
the
of c o n s t a n t
characteristic
it seems
folds
highly
are all of the
form
discontinuously
Letting
< , >
be
n,l
H n = {x I []XIln,1 = -i},
is p o s i t i v e
of
bundle,
definite
M n,
F,
^
forms
unlikely
property
group
negative
all vanish,
that
the L o r e n z
with
M.
As
the
Pi
where
F
in
acts
induced
n +i
metric
hypersurface.
This
are
gives
in the p r o o f
freely
from
mani-
and
space
H n.
we m a y i d e n t i f y
< ' >n,l'
F
is the
a flat
0(n,l)
of T h e o r e m
classes,
These
space,
Now
deeper.
R/Z
on the h y p e r b o l i c
metric
the
is c o n s i d e r a b l y
are all r a t i o n a l .
isometries
F c 0(n,l).
over
they
M n = Hn/F,
as
on this
and
curvature
6.3 one
which
fundamental
vector
shows
^
P i ( M n)
= Pi(F).
the h o l o n o m y
7.
However,
group,
we get no r a t i o n a l i t y
F,
is not
conclusion
because
finite.
Foliations
Let
G = GI(n,R),
and set
I (G) = ker w.
Then
I = ZI k
is
o
trA2k_ 1
o
o
by the p o l y n o m i a l s
. Taking
A = {0}
we
the ideal
generated
see t h a t
Q <-> (Q,0)
is an i s o m o r p h i s m
between
Ik
and
K2k(G,{0}).
O
If
~ 6 ~(GI(n,R))
and
Q
~ Ik
set
O
Q(~)
Let
N(F)
=
SQ,0(~)
F
be
E H2k-I(M,R).
be a f o l i a t i o n
its n o r m a l
connections
in
of c o - d i m
bundle.
N(F),
In
all of w h i c h
curvature transformations,
k
"~ ~..~.~--~ = 0 if
k > n,
shows
that
certain
to c o n s t r u c t
R
in a m a n i f o l d
have
vanish
x,y
and thus
Pontrjagin
secondary
n
[4] B o t t
defined
if
x, y
vanish,
M,
and
a family
the p r o p e r t y
P(~) = 0
classes
cohomology
has
that
their
if
6 F . This
m
P 6 Ik(G).
and
it also
invariants.
Let
N(F)
let
of
guarantees
This
leads
one
{N(F),M,?}
=
^
where
V
is such
Pi(N(F)).
Bott's
Q(F)
a Bott
connection
curvature
and
vanishing
6 H2k-I(M,R)
Q
set
Q(F)
theorem
6 I k,
= Q(N(F)),
Pi(F)
=
shows:
k > n
(7.1)
O
Pi(F)
A simple
6 H4i-I(M,R/Z)
application
Proposition
ently
7.3.
of c h o i c e
The
of P r o p o s i t i o n
classes
of B o t t
It is s t r a i g h t f o r w a r d
are n a t u r a l
under
2i > n.
smooth
to s h o w
maps
2.9 y i e l d s
Q(F)
connection,
(7.2)
and
Pi(F)
and are
that
the
transverse
to
thus
are
classes
F,
defined
invariants
and
Q(F)
that
independof
F.
and
Pi(F)
they
are
69
cobordism
space
invariants.
Thus,
for f o l i a t i o n s ,
see
letting
[5],
6 H2k-I(B~n,R)
example
Q
have been defined
treatment
are n o n - v a n i s h i n g :
Theorem
7.4.
Let
independently
classifying
by others.
is the G o d b i l l o n - V e y
of the
classes
B
denote Haefliger
2i > n.
(trA) 2 ( H 3 ( B F I , R )
extensive
n
k > n
p. ( H 4 i - I ( B F ,R/Z)
1
n
The c l a s s e s
BF
we get
Q
c l a s s e s m a y be f o u n d in
9 : B F n ÷ BGI(n,R)
For
invariant.
An
[6].
be the n a t u r a l map.
The
Then
Pi
letting
denote Bockstein
B(Pi)
=
Corollary
-~*(pi).
7.5.
Pi ~ 0.
The p r o o f of the t h e o r e m
corollary
from Corollary
then f o l l o w s f r o m e x a m p l e s of B o t t - H e i t s c h
.th
n,
the
l
integral Pontrjagin classes
of c o - d i m
bundles
is i m m e d i a t e
do not v a n i s h
for
i > n.
This
shows
2.4.
The
[6] of f o l i a t i o n s
of w h o s e n o r m a l
~*(pi ) ~ 0
and thus
~i/o.
§8.
Flat Bundles
Let
p : ~I(M)
G
G-bundle
Sp,u(P)
be a Lie g r o u p w i t h
÷ G
Ep.
For
= Sp,u(Ep) .
S p , u (p)
(P,u)
to
we set
u(p)
E K2k(G,A)
Corollary
components,
Associated
p
and let
we get a flat
= U(Ep)
and
2.4 shows
(H2k-I(M,R/A)
B(Sp,u(p))
Let
finitely many
be a r e p r e s e n t a t i o n .
% :N ÷ M
(8.1)
(8.2)
= -u(p).
be smooth.
Then
p o ~ : ~I(N)
÷ G,
and T h e o r e m
2.2
shows
Sp,u( p o %)
Let
G
=
denote
¢*(Sp,u(p)) .
G
equipped with
d e n o t eo its c l a s s i f y i n g
BG
o
(8.3)
space.
the d i s c r e t e
t o p o l o g y , a n d let
G o J--~ G is
The i d e n t i t y m a p
70
continuous
u = j*(u)
Sp,u
and i n d u c e s
BG 3 ~
BG"
For
( H2 k (BG ,i).
o
F ro o m
(8.1)
and
u 6 H2k(BG,A),
(8.3)
we g e t
one shows
( H 2 k - l ( B G ,R/A)
(8.4)
O
B (Sp
)
= -u.
(8.5)
rU
Po
Any representation
G
O
i ÷ G.
Since
Oo
p : ~I(M)
÷ G
is continuous
can be f a c t o r e d
it
induces
as
~I(M)
Po : M ÷ B G
,
÷
and
O
p*(S_o~,u)
Proposition
=
8.7.
(R/i)to r ~ R/A
SpQ,uUv
(8.6)
Sp,u(P)"
Let
(P,u)
denote
=
6 K
2kl( G
the t o r s i o n
u O SQ, v 6 H * ( B G
,A), (Q,v)
( K
2k 2
(G,A),
and let
subgroup.
, (R/A)tor) .
o
Proof.
F o r any
SpQ,uUv(p)
is flat,
p : ~I(M)
= Sp,u(p)
u(p)
= u(E
÷ G,
* SQ,v(0)
Corollary
= u(p)
) ( H 2k (M,A)
p
tor
2.3 and
(1.14)
U SQ,v(p) •
and
show
Moreover,
H[or(M,i)
since
U H*(M,R/A)
EQ
c
H* (M,R/A) tor ) .
In p a r t i c u l a r
we see
( H*(B G ,Q/Z)
SpQ,uUv
i = z
(8.8)
O
SpQ,uUv
If
smooth
p
t
if,
=
0
: ~I(M)
A = Q.
÷ G
for e a c h
Using Proposition
is a f a m i l y of r e p r e s e n t a t i o n s
h 6 ~I(M),
P r o p o s i t i o n 8.10.
I_~f
0
pl)
S p , u ( P ) = Sp,u(
•
As we w i l l
dition
k ~ 2
sitions
will
p
t
: Zl(M)
is a s m o o t h
question
÷ G
is smooth,
as c h a r a c t e r s
outside
of
k ~ 2
k ~ 2,
X (HI(Bs0(2)
w i l l be the v a l u e s
8.7 and 8.10 show that for
values
and
in this t h e o r e m ,
they are r e g a r d e d
not p r o d u c e
c u r v e in
it
G.
shows
see b e l o w in the case of
is n e c e s s a r y
A dominating
classes when
pt(h)
2.9 one e a s i l y
we call
,R/Z) ; the cono
t a k e n by the
on
Sp, u
H2k_I(BGo ) .
elementary
(R/A)to r.
then
Propo-
constructions
Moreover,
Proposition
71
8.10
The
seems
Euler
to i n d i c a t e
Character.
corresponding
sphere
on
so
SV.
Let
Let
and
let
shows
and
=
agrees
arbitrary
such
fibre.
a
and
be the
P
be the a s s o c i a t e d
SV ÷ M
via
let
and
exists
then
V
f o r m on
S n-l,
the c o n n e c t i o n
defines
y 6 Hn_I(SV )
y
We
is c o u n t a b l e .
be the v o l u m e
choose
that
h o m o l o g y class in the
Hn-I(M,R/Z)
by
This
Let
/ n-i ~ = i,
and e x t e n d e d
S
Since
SV
is flat,
de = o
(3.1)
(p) (z)
bundle.
of v a l u e s
÷ S0(n),
~ ( An-Isv
z 6 Hn_I(M) ,
Sequence
the r a n g e
p :~l(M)
flat v e c t o r
bundle,
normalized
~,
that
define
[~] 6 Hn-I(sv,R).
so that
and
~,(y)
is u n i q u e
X(p
to a form,
= z.
up to a
6 Hom(Hn_I(M),R/Z)
[m] (y).
=
(8.11)
with
the o r i g i n a l
S0(2n)
bundles,
definition
and
extends
of
X
it,
gzven
in the
in §3
flat
for
case,
to all
S0(n).
Proposition
8.12.
i)
X(p)
2)
X(p)
3)
X(Pl~P2
Proof.
could
and
has o r d e r
( H
n-i
)
Let
use
this
S0(2),
that
finite
be the a n t i p o d a l
(see Ex.
p(1)
perturbed
of
y
follows
in
map.
(8.11).
Since
For
from P r o p o s i t i o n
n
~Q A = ~,
odd,
2.10,
and
A*(~)
we
to each
1.5).
are d e a l i n g
closed
curve
In p a r t i c u l a r
= e 2~i~,
and
so that
is n e c e s s a r y
p(1)
with
in
a flat
M
if
takes
M = S1
8.10.
One
bundle,
and
p
in
S0(2),
Now
easily
The h i g h e r
tion,
and
dimensional
and o r i e n t e d ,
let
SV
angle
of
p :~l(S) 1 ÷
Since
any v a l u e
in P r o p o s i t i o n
circle
may
be
we
see
HI(Bs0(2)O )
shows
: HI(Bs0(2)o)-~ R/Z.
compact
= -~,
3) f o l l o w s
its a s s o c i a t e d
X(Q) (S I) = e.
S 0 ( 2 ) o / [ S 0 ( 2 ) o ,S0(2) o] ~ S0(2) o ~ R/Z.
be
we
(1.14).
n = 2,
assigns
k ~ 2
2)
3.6 and
then
smoothly
p(~I(M))
instead
i).
case
odd
X(p l) U X(P2 ) 6 H * ( M , Q / Z ) .
A,(y)
X(Q)
holonomy
(M,Q/Z)
=
shows
In the
n
A : SV ÷ SV
from Theorem
and
2
(8.13)
cases
let
are m o r e
p : Z l ( M 2n-l)
be the a s s o c i a t e d
flat
interesting.
÷ S0(2n)
(2n-l)
dim
Let
M 2n-I
be a r e p r e s e n t a sphere
bundle.
72
Let
For
ml,...,m N
each
be the v e r t i c e s
vertex
choose
top d i m e n s i o n a l
of a s i m p l i c i a l
v. ( SV
If
3
mj
let
b I denote
simplex
subdivision
of
M 2n-l.
o. = (m. ,...,m.
) is a
i
10
12n_l
its b a r y c e n t e r ,
and let
w. ,...,w.
6 SVb.
be the v e c t o r s o b t a i n e d by p a r a l l e l t r a n s l a t i n g
io
12n-i
i
• ,...,v.
a l o n g c u r v e s in
~..
N o t e that since
SV
is flat the
vl o
12n_ 1
i
{w. } do not d e p e n d on the c h o i c e s of these curves.
We call
1.
3
are
v I, • .. ,v N
g e n e r i c if for e a c h
~i. the v e c t o r s
W l.o ,...,w.12n-i
linearly
independent.
N-tuples
Vl,...,v N
In the g e n e r i c
It is e a s i l y
form an o p e n
case
let
seen
that
dense
Z . ~ S 2n-I
the
subset
denote
set of g e n e r i c
of
SVml
the u n i q u e
x...×
SVmN
convex
oriented
1
~eodesic simplex spanned
its o r i e n t e d volume,
Theorem
8.14.
fundamental
volume.
Let
cycle
by
w i , .... w
,
o
i2n-i
Vl'''''V2n-iN
of
be generiC2n_l and
M
Let
S
and
let
VoI(z ~)
1
let
°l
+'''+ ~k
be n o r m a l i z e d
to h a v e
denote
be a
unit
Then
(p) (M 2n-l)
=
Z VoI(z .) •
1
This
the bar
tuples
theorem
suggests
resolution
of g r o u p
of
a direct
S0(2n)o.
elements
(go .... ,gk ) ~ (hg ° .... ,hgk),
g i , . . . , g k ) . The h o m o l o g y and
of
go(e) ..... g2n_l(e)
are
f o r m an o p e n
generic
let
spanned
by
unit
Z (J)
Vol(o)
and call
linearly
dense
~ S 2n-I
and
=
as a c o c y c l e
are
in
(k+l)
the e q u i v a l e n c e
Vol(Z(o))
independent
subset
of
Let
in
~S0(2n
R2n.
x...×
oriented,
S 2n-I
The
generic
S0(~).
For
geodesic
be n o r m a l i z e d
if
generic
simplex
to h a v e
( R/Z.
V o l ( ( h g ° ..... hg2n_l))
cochain
= V o l ( ( g o, .... g 2 n _ l ) ) ,
on the g e n e r i c
(go ..... g2n )
are g e n e r i c ,
-~
(6 Vol) (y)
(go , . . . , g 2 n _ l )
set
(2n-l)
Y =
J =
be the convex,
Since
Let
under
X
k-simplices
and
~ (go ..... gk ) = Z ~ = 0 ( - l ) i ( g o ' ....
c o h o m o l o g y of this c o m p l e x are i s o m o r p h i c
go(e) ..... g2n_l(e) .
volume,
faces
of
that
B
G
Let us fix o e ( s2n_l,
simplices
recall
(go ..... gk )'
^
to that
definition
We
and
be a
let
f--~
= Vol(o7)
Yi
=
Vol
defines
simplexes.
2n
simplex,
denote
its
2n
Vol( [ (-l)i
i=0
all of w h o s e
(2n-l)
.th
i
face.
Then
J
Z(yi) 0 = 0
an
73
since
and
z2n
i=0(-l) i E (7i)
(2n-l)
so by our n o r m a l i z a t i o n
cycle
on its d o m a i n s
where
a Borel
R/Z.
It is shown
cocycle
Thus
map
in
[14]
theorem
Theorem
sense
8.15.
one
X =
singular
cycle
~on
volume.
Thus
Vol
and c l e a r l y
of
that
such
and all
such
,R/Z)
[14]
defines,
from
S0(2n)
a cochain
s2n_l,
is a co-
almost
×...×
are
to a
cohomologous.
defined.
of the c h o i c e
every-
S0(2n)
can be e x t e n d e d
extensions
is w e l l
o
is i n d e p e n d e n t
[Vol]
dim
integral
of d e f i n i t i o n
( H 2n-I (Bs0(2n)
that
previous
has
(in the
on all chains,
[Vol]
shown
and
is a
It is e a s i l y
of
e.
By u s i n g
the
shows
[Vol]
thus we o b t a i n
Corollary
8.16.
X (H2n-l(Bs0(2n)
,R/Z)
is a B o r e l
cohomology
class.
O
Let
Range
(8.13)
X
be the
this
is of
Proposition
8.17.
Proof.
Let
Let
Range
field
bundle
to
mod
Let
has
Z.
n = 2,
and
For
this
=
shows
VoI(Z)
the
determines
1
~
that
this
=
unlikely
rational
is p r o b a b l y
acting
S0(2n)
on
S 2n-l.
The
associa-
defined
by the n o r m a l
shows
X(E) (s2n-I/F)
(8.11)
choose
freely
bundle.
F ~ Z k.
g o e d e s i c simplex.
Let
~x.
denote
.th
jth
1,j
l
and
face.
The set
Z
up to c o n g r u e n c e ,
Sn
to h a v e
in
S2
and we
volume
call
i.
Then
for
simplexes
false.
have
rational
volume.
In p a r t i c u l a r
f ( x l , 2 , . . . , X n , n + I)
transcendental
that
b u t we do not k n o w
Thurston.
By
(Xl,2+Xl,3+x2,3-2),
is a n o n - e l e m e n t a r y
W.
we m a y
n-dim
between
it in
x.
6 Q.
Normalize
1,j
the G a u s s - B o n n e t t h e o r e m g i v e s
n ~ 3
highly
k
subgroup
flat
if all
VoI(Z)
) ÷ R/Z.
o
cross-section
Using
any
be an
angle
(Xl,2,...,Xn,n+l)
rational
R 2n.
For
X : H2n_I(Bs0(2n)
n ~ 2.
be a f i n i t e
a canonical
in
Z c Sn
the d i h e d r a l
for
be the a s s o c i a t e d
S 2n-I
i/ord(F)
of the m a p
only
X ~ Q/Z.
F c S0(2n)
E ÷ s2n-I/F
sphere
image
interest
f
takes
function
rational
a counterexample.
(see
values
The
[i0],
at all
following
[16]).
rational
theorem
It seems
points,
is due
to
74
Theorem 8.18
simplexes
(Thurston).
E ~ S 3,
m VoI(Z)
Thus
X
For all but a finite number of rational
there exists an integer
H3(Bs0(3)O )
finite number of rational geodesic
3-simplices
The proof of this theorem depends
unpublished
construction
associating
to a given rational
constant negative
X(p)
We should emphasize
Z,
large,
~ m Vol(~)
the denominators
Similarly Theorem
8.18 implies
8.19.
Let
erated by the volumes
of Theorem
V
associated,
on the values of
be the vector
that
form
of the
Let
is
Since
i.
En_k+ 1
Vn_k+ 1
gives a
For example,
(all but finitely many)
Then
homomorphism
rational
p :~I(M)
÷ U(n)
Let
and let
Vn_k+ 1 ÷ M
~eodesic
Let
Vp
En_k+ I.
~2k-i
~2k-i
be the
be the flat
We recall
be the unique harmonic
whose value on the generator
is flat,
gen-
Rank H3(Bs) (4)o) ~ dim V - i.
H2k_l(En_k+ I) ~ Z.
on
X
space over the rationals
Stiefel bundle with fibe~ the Stiefel manifold
(4.2)
of
p :~I(M 3) ÷ $0(4).
H2n_l(S0(2n)).
flat, hermitian bundle.
(2k-l)
of whose
M3
and hence is not finitely generated.
8.18.
The Chern Characters.
of
mod Q.
8.17 shows that this group has a nontrivial
H ~ Q/Z
consists
a compact manifold
that information
onto some group,
from
8.14 and a recent and
This construction
simplex
lower bound for the homology group
simplices
so that
have rational volume.
curvature and a representation
He then shows that
Corollary
r 6 Q
unless all but a
on Theorem
of Thurston.
angles are sufficiently
Proposition
and
+ r E Range X.
takes irrational values on
dihedral
m ~ 0
3-
U(k)/U(k-l)
defines a closed form on
Vn_k+ I,
and we denote its cohomology
Let
Sequence
z 6 H2k_I(M),
and choose
y ~ H2k_l(Vn_k+ I) with
~,(z)
(4.2) shows that such a z exists and is unique up to a
multiple
of the generator
H2k-I(M,R/Z)
~ Hom
~k(p) (z)
=
This definition
given in
Since
shows
of
class by
H2k_l(En_k+l).
(H2k_I(M),R/Z)
[e2k_l ] H2k-I(Vn_k+l,R).
We define
Ck(P)
by
[~2k_l ] (y).
agrees,
(8.19)
for flat bundles,
with the general definition
(4.4).
U(n)o 5 $0(2n) o,
= y.
we may also consider
X(p),
and
(4.9)
75
Cn(P)
=
-X. (P)
(8.20)
•
For any space
X with
~ E HZ(X,R/Z),
* ~ = -B(~) U ~ ~ Hk+Z+I(x,R/Z).
Note
B(e "8)
-- B(~)
,
8.21.
product,
Proof.
(8.21)
Ck
Cl,...,Cn
cn
is Borel
flavor of
using
[Vol],
H*Borel
~k
of
from
(8.19),
(8.20)
however,
similar
forms
under
of the ring.
and Corollary
are Borel one needs
U(n)o,
(B G ,R/Z)
o
Moreover,
set of generators
follows
to prove the lower
may be derived
G,
E HBoreI(Bu(n)o,R/Z).
in the bar resolution
Cl'
Lie group,
the~ are a complete
That
However,
set
U B(B).
It is easily seen that for any
a ring under
, product,
Theorem
8 E Hk(x,R/Z),
(8.16).
a formula
to that for
for
X.
This
it does not have the canonical
and we omit the details.*
The simple
exception
is
and
cl(go'gl )
Let
1 log
)
2~i
det(golgl "
-
j :Bu(n)
÷ BU(n)
be the natural
map, and let
o
j :H*(Bu(n),Z)
[18] Wigner
+ H*(Bu(n)o,Z)
Im j*
B(Ck)
Since
the
map.
In his
thesis,
shows
B :H*BoreI(Bu(n)o,R/Z)-~
But
be t h e a s s o c i a t e d
Im j*.
is the ring generated
=
by
j*(c I) ..... j*(Cn)
and by
(8.5)
-j*(ck).
Ck
are Borel,
and
B
We need not be restricted
to
maps
,
products
into cup products
we are done.
following
(4.5),
(4.10),
Ck(P)
H2k-i (M,R/Z)
~h(P)
H°dd(M,R/Q).
and
U(n).
If
P :~I(M)
÷ Gl(n,c),
(8.1) we define
* We wish to thank John Millson for acquainting us with Wigner's
and for suggesting that the X, c k might be Borel.
thesis
76
Theorem
8.22.
i)
Ck(P 1 ~ p 2 ) : ck(Pl ) + Ck(P2 ) +
2)
&h(p)
3)
ch(Pl @ p 2 ) = &h(Pl)
4)
ch(Pl ~ p 2 ) = nlch(P2)
Proof.
= n +
n
[
i=l
i) follows
k-i
[ ci(P I) U ~k_i(P2 )
i=l
i-I
(-i)
&i(p )
(i-l)'
mod Q
+ &h(P2)
+ n2~h(Pl)
immediately
- nln 2.
from Theorem
4.6 and
(i.14) .
ch is
^
only defined
mod Q,
since
is torsion.
ci(P)
and
see 4.10)
all product
immediate
from Theorem
Let
~I(M),
R(~I(M))
and
4.11,
shows
unitary
ch
(e.g.
3) is
representation
ideal of virtual
ring of
representa-
ch extends to R(zI(M))
and defines a
÷ H°dd(M,R/Q)
as Q-modules.
4) of the
ch(Pl.~p 2) = 0
if
Pl' Q2 6 I(~I(M))"
This
÷ H°dd(M,R/Q)
Q-module
homomorphism.
If we suppose
finitely generated group whose classifying
dimensional manifold we get
ch : I(L)/I2(L)
= 0 mod Q
for
and so is 4) by virtue of 2).
the augmentation
ch : I(~I(M))/I2(nI(M))
is a well defined
U cj(p)
formula
and 2) is what remains.
the rational
tions of dim 0. Clearly
homomorphism
ch : R(~I(M))
above theorem
, cj (p) = ci(P)
in the general
terms vanish,
denote
I(~I(M))
ci(P)
Thus
space,
BL,
L
to be a
is a finite
÷ H°dd(BL,R/Q).
At this point we have no information
as to the kernel
and range of this
map.
We remark that by constructions
in this section,
it is possible
the real continuous
mology,
which corresponds
compact)
9.
invariant
to those given
cocycles
Lie groups.
polynomials
come as a special
on
g/k
case of the
representing
This coho(k-maximal
~k.
Index Theorem of Atiyah-Patodi-Singer
L k = Lk(P I, .... pk)~ H4k(BGI(n,R),Q)
L-class
analogous
to give explicit
of noncompact
to invariant
does not in general
The Geometric
Let
sal
cohomology
completely
and let
polynomial.
PLk = Lk(PI,...,Pk)
If
V = {Vn,M,V}
denote
denote
the
k th
univer-
the corresponding
is a real vector bundle with
77
c o n n e c t i o n w e let
L(V)
= 1 + L I ( V n ) + . . . + L [ k / 2 ] (V) ( H * ( M , Q )
PL(V)
i(V)
denote
= 1 + PLI(9)
= 1 + il(V)
+ . . . + PL[k/2] (~)
+ . . . + {'[k/2] (V) ( H * ( M , R / Q )
the c o r r e s p o n d i n g
differential
rational
character.
The L.
Pi
and
*
il
=
If
{M,g}
product,
form,
can of c o u r s e be w r i t t e n
and
in terms of
e.g.
Y Pl
45
is a R i e m a n n i a n
be the class,
form,
tangent bundle.
class,
characteristic
1
A
the
class,
manifold
and c h a r a c t e r
we let
In s p i t e of the fact that
it is i m p o s s i b l e
which maps naturally
to r e f i n e
under
L(M) , P L ( M , g ) ,
corresponding
L(M,g)
isometries.
L(M)
is an i n t e g r a l
to get an
The
L(M,g)
to the R i e m a n n i a n
R/Q
R/Z
character
character,
L(M,g)
is of c o u r s e n a t u r a l .
Let
{M,g}
V = {V,M,?}
be c o m p a c t ,
oriented,
be a c o m p l e x H e r m i t i a n
and odd dimensional,
v e c t o r bundle.
denote
V-valued
k-forms.
T h e c o n n e c t i o n on
d : A k (M,V) ÷ i k + l ( M ,V) , a n d the m e t r i c on
M
An-k(M,V).
V
Let
allows
defines
and let
ik(M,V)
one to d e f i n e
, : A k (M,V) ÷
Define
T : Z ~ A2P(M,V)
÷ ~ • A2P(M,V)
by
T
=
*d +
T
=
i(*d + (-l)Pd,)
In
(-1) p d*
dim M
=
4k - 1
dim M
=
4k + i .
[3], A t i y a h - P a t o d i - S i n g e r
It has d i s c r e t e
spectrum with infinite
Letting
{li},
spectrum
they f o r m the
Nv(s)
=
{yi }
denote
~ ~7 s i=l
i
NCV)
=
positive
~
elliptic
and n e g a t i v e
operator.
range.
and s t r i c t l y
f u n c t i o n of a c o m p l e x v a r i a b l e
negative
s,
(-~i)-s
i~l
to a m e r o m o r p h i c
T h e y also s h o w that
NV(O).
symmetric
its s t r i c t l y p o s i t i v e
and s h o w this to be c o n t i n u a b l e
e n t i r e plane.
s t u d y this
N(0)
function
in the
is real and finite.
Set
78
Now suppose that
Let
g
M = ~M
be any metric on
and that
M
V
extends to
w h i c h induces
g
product metric in a collar n e i g h b o r h o o d of
T h e o r e m 9.1
~ = {V,M,V}.
M,
and w h i c h is
M.
(Atiyah-Patodi-Singer)
(-i) k+l~ (V)
where
on
=
N(M,M,V)
I~ Pch(V)
A PL(M,g)
+ N(M,M,V)
is the index of a certain boundary value p r o b l e m
a s s o c i a t e d to the data and is therefore an integer.
The left side of this equation is clearly an intrinsic function of
the odd d i m e n s i o n a l R i e m a n n i a n manifold,
bundle
ever,
{V,M,V}.
T h e r e f o r e of course,
is, defined only when
the interior.
M
M,
and the H e r m i t i a n vector
so is the right side.
is a b o u n d a r y and when
V
It, how-
extends over
One can avoid this r e s t r i c t i o n and get an intrinsic
right hand side w h i c h is always defined by working
mod A.
Some
topology is lost, but one gains n a t u r a l i t y and some c o m p u t a t i o n a l
facility.
T h e o r e m 9.2.
V
over
For all complex, Hermitian,
R i e m a n n i a n vector bundles
{M,g]
(-l)k+l~(v)
Proof.
H
(ch(V)
* L(M,g)) (M)
mod Q.
It is always the case that one can find an integer
compact manifold,
over
M,
M,
where
the c o n n e c t i o n to
so that
ZV
V
is
on
V
V,
~M = ZM
and so that
on each c o m p o n e n t of
and choosing a metric
£V
and a
extends to
iM.
g
£,
Extending
on
M,
product
near the boundary, we get
(-ik+l~(~V)
Clearly
=
iM Pch(V)
~(IV) = Z~(V) ,
(_i) k+in (V)
=
A PL(M,g)
and w o r k i n g
+ integer.
mod Q,
1 r
£ IM Pch(~) A PL(M,g)
=
!i [d 1 (ch(V)) ^ 61(£(M,g))] (M)
=
Z
=
!
1
[$h(IV),£(T(M)
[ IM)] (£M)
The a s s u m p t i o n of product metric means that
T(M) I £M = T(iM)
m L,
79
where
i
is a t r i v i a l
(-ik+l~(v)
This f o r m u l a
Riemannian
= }[ch(ZV)
line b u n d l e .
*~.(£M,g)] (ZM) =
seems of i n t e r e s t
Thus
[ch(V) * L ( M , g ) ] (M).
for flat b u n d l e s .
Let
p :Zl(M)
÷ U(n),
and set
n(p)
=
n(v
Using Theorem
Corollary
P
).
8.22 and
9.3.
If
(1.14) we see
d i m M = 4k + l
(-l)k+in(p)
1
i=0
T] (Pl ® P2 )
If
then
(2k-2i)'
Li(M)
U C2(k_l)+l(p)
mod Q
(M)
m o d Q.
n 2 n ( p I) + nl~(p 2)
d i m M = 4k - i
then
k-i
(_ik+l~(p)
-_-nLk(M-g ) i=0
~(pl ® p 2 )
Let
That
in
~(p)
- n2D(Pl)
- n([ n)
depends
clarify
this
under
independence
Example
9.4.
reversing
~(p)
By
(8.20)
ated,
b u n d l e of d i m e n s i o n
o n l y on
p
was p r o v e d
c h a n g e of m e t r i c
and c a l c u l a t e
in terms of c h a r a c t e r i s t i c
d i m M = 4k + i ,
complex
classes
of
p
and
The above
_ ~([n)
M.
n = d i m p.
in [3] by s h o w i n g
is zero.
~(p)
mod Q
m o d Q.
+ nln(p 2) - nln21.k(M,g)
be the t r i v i a l
t h a t the d e r i v a t i v e
(~)(M)
(2k-2i-i) ! L i(M) U C2(k-1)
m o d Q,
formulae
explicitly
(Note t h a t in case
n(i n) = 0).
Let
M
isometry.
be a c o m p a c t
Then
LI(M,g)
- -c(p) (M) = X(p) (M)
and T h e o r e m
3 - m a n i f o l d w i t h an o r i e n t a t i o n
= 0
and
m o d Q.
8.14 we see that the s e r i e s
up to a r a t i o n a l ,
as a sum of s i m p l e x
volumes
n(P)
on
m a y be e v a l u S 3.
80
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II:
Characters