Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
640 Johan L. Dupont
Curvature and Characteristic Classes...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
640 Johan L. Dupont
Curvature and Characteristic Classes
SpringerVerlag Berlin Heidelberg New York 1978
Author Johan L. Dupont Matematisk Institut Ny Munkegade DK8000 Aarhus C/Denmark
AMS Subject Classifications (1970): 53C05, 55F40, 57D20, 58A10, 55J10 ISBN 3540086633 SpringerVerlag Berlin Heidelberg NewYork ISBN 038?086633 SpringerVerlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by SpringerVerlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140543210
INTRODUCTION
These
notes
Mathematics year
are b a s e d on a series
Institute,
of the lectures
to the c l a s s i c a l real
homology
ChernWeil
coefficients
differentiable
during
the a c a d e m i c
theory
for
theory
only basic
and Lie groups
of the c l a s s i c a l M
a compact
0
k = 0.
(1.4)
× U ~ U
g(s,x)
For
j
I IR, Proof.
let
=
(1s)x,
g*~
s 6 [ 0, I],
6 Ak([0,1]
× U)
x 6 U.
is u n i q u e l y
expressible
(The
(k1)form
as g ~ = ds ^
where
~
and
is u s u a l l y
B
are
~ + 8
forms
denoted
not
involving
i ~ (g*~).)
Then
ds.
define
~s
= ~
hk(~)
1
o~
s=0 which
means
respect
that w e
integrate
to the v a r i a b l e
s.
the c o e f f i c i e n t s In o r d e r
of
to p r o v e
~
(1.3)
with notice
that
g*d~ where dx@
= d(g*~)
we h a v e
only
+ ds ^ ~ s B +...
written
terms
= da  ds A ~ e .
k =
0
the
involving
ds,
and w h e r e
Hence 1 = is=0
hk+1(d~) For
= ds A dx~
clearly
@ = 0
~s~
 dx~.
so
I
h1(d~) (x) = ~ s = 0 For
k > 0 ,
~~ w ( s e + ( 1  s ) x )
= ~(e)
 ~(x),
x £ U.
B1 0
×
U =
(id)*~
B11
× U = g~
=
=
0,
g1(x)
= e,
x
6 U.
Hence I
hk+1(dm)
= m
 d I
e = m
 dhk(m)'
s=0 which
proves
The gives
de
theorem
we
nsimplex
determined
nform
First
Actually
standard
on
by
An
which
is
interpretation
manifold.
forms.
is
Rham
a geometric
general of
(1.3).
we
An . the
is
of
need
shall
a
only
The
nform
by
f = j
r
[jtj
An ~
c
~n
5)
ei
2.
: A n1
e
(1.6)
the
Rham
integrate
about
nforms
on
An
d t n.
this
cohomology
remarks
expressible
f(tl
An
or
chapter of
over
the
rather
Explicitly
Vn
every
dt n
o dt • tn)dt I ' " "' "" n
{(tl ''" . ' tn)
£ ~nl
t.i => 0,
Stoke's
theorem:
i
that
dt I ^
.^ d t ""
Show
the
~ An , i =
(t0,...,tn_
~ £ A n1 (A n ) .
IA n
I n
~.
following
0,...,n,
I)
=
=
"
case be
the
of
face
de
=
n
map
(t o , .... t i _ 1 , 0 , t i , . . . , t n _ 1 ) 
Then
[ (I) i i=0
jA n  I (ei) *m
a
integration
as
I ^...^
A n0 =
set
Show
[ JAn Exercise
Let
is
I.
•
Let
de
of
1},
Exercise
(1
few
object
definition
j An where
the
dt I ^...^
uniquely
main
orientation
= f(tl,...,tn)dt
and
the
(Hint: I =
First
show
[0,1],
(1.6)
by
a similar
(see e . g . M .
using
=
Exercise of
the
therefore A k ( A n)
~
3.
An
n +
First
Stoke's
Now
I
theorem
that
homology
and
cohomology.
and
maps
we
shall
recall
which
spaces
only
use
starshaped
1,2 .... , for
any
(nl)
(equation
us
topological
In c ~n, Then
deduce
by
i = 0,1,...,n.
the
let
C~
clearly
(1)nh
show
cube
818]~
given
corresponding
that
[ m = j An
(Hint:
is
k =
Show
p.
: In ~ A n
ei,
Ak1(An),
i = 0,...,n.
[29,
the
,s 1 . . . s n _ I (1s n) ,s I .... s n) .)
vertices
have
(1.7)
g
for
( ( 1  s 1 ) , s I (1s 2 ) , s l s 2 ( 1  s 3) ....
• ..
each
Spivak
the map
g ( s I .... ,s n)
formula
satisfying
o...o
the
on
elementary We
field
of
h(i ) :
on
with
e = ei,
An
the
right
and
then
the
maps
satisfies use
about
case
analogous
real
1.2 w e
(1.3)
facts
consider
continuous
to
h(o) (~) (en).
above)
is c o m p l e t e l y and
~
respect
lemma
operators
operator
the
By
nform
(1.6)
with
singular
of
C~
to t h e
usually
numbers
induction.)
manifolds
case
of
considered.
~
Also
as c o e f f i c i e n t
ring. Let M
is a
M
be
C
map
simplex.
Let
simplices
in
i = 0 ,...,n, Ei
: S~(M)
a
C~ o
S~(M) n M.
As
be
the
~ S ~n _ l ( M ) ,
si(o)
manifold.
: A n ~ M, denote
A
in e x e r c i s e
set
i = 0 .... ,n,
= o o e i,
of
2 above on
s i .... n@ular
An
where the
inclusion
C~
the by
a 6 S~(M). n
is t h e all
C~
let
ith
ei
face.
nsimplex
in
standard
n
singular
n
: A n1 Define
~ An ,
Notice
that
(1.8)
e i o ej
The
group
the
free
space
of
of
C~
have
(1.8)
vector
space
finite
formal
homology
Dually
the
cients
is
with
: Hn(C group
have
the
Explicitly
numbers,
(1.9)
Again
nth
coefficients Hn(M)
If
f
This f~
: Cn(M)
n
~ Cn_I(M)
C~
singular
~ Cn_I(M))/3Cn_I with
and
(M).
real
coeffi
(M) ,JR)
6 =
3"
c = {c
o
: Cn(M) c
},
~ cn+1(M). : S~(M) n
6 S
~ ~
or
of
real
(M),
n
by
n+1 [ (1)icE.y, i=0 i
C~
~ Cn_1(M)
nth
ncochains
is
maps
: Cn(M)
the
~
the vector
The
: Cn(M)
have
is a f u n c t i o n
singular
= Hn(C~(M))
map
clearly :
we
sin@ular
is g i v e n
• o.
in
oo
T 6 Sn+I(M).
cohomology
group
with
real
is
: M ~ N
induced
: ker(~
coboundary
(~c) T 
the
and
= Hom(C
i.e.
~ = [i(1)lsi
coefficients
C~
coefficients
a ei
real
of
@
to
j.
(M),
n
[o6S[(M)
extend
a collection
and
S
on
0
(M))
i
0,
H0(Pt)
= ]R
Hi(pt)
=
0,
i > 0,
H0(pt)
= ]R.
(Homotopy
property).
homotopic,
i.e.,
such
FIM
that
f1~ si
,
are
f19#
there
 f049
= si1
F
there
are
: M ×
[0,1]
Then
f0#
× I = f1"
i.e.,
such
: M ~ N
map
C ~
FIM
homot0pic,
~ Ci+I(N)
f0,fl
is a
x 0 = f0'
chain
: Ci(M)
Suppose
~ N and
are
homomorphisms
} 6Z
is an o p e n
that
0 ~ +
~ o s i.
In p a r t i c u l a r
f1*
= f0*
f~ : f~
(1.12)
(Excision
: H.(M)
: H*(N)
~ H,(N),
~ H*(M).
property).
covering
of
and
let
singular
nsimplices
of
o ( A n)
U
be
the
~
M
Suppose
for
some
corresponding
"with
support
in
i,
: C,(U)
~ C,(M),
be
the
natural
U")
maps
U
S~(U) n M, e.
denote
the
set
: A n ~ M,
such
Let
(C*(U),%)
and
or
and
let
induced
{U
o
chain
i*
=
cochain
: C*(M)
by
the
complexes
~ C*(U
incluslon
of that (C*(U),B) (called
10
I : S~(U)
c S ~n ( M ) .
equivalences,
now
~ H(C,(M)),
define
a natural
the
f
they
I*
are
induce
chain
isomorphisms
~ H(C*(U)).
map
~ C n(M)
formula
(1.13) I
and
H(C*(M))
I : A n(M)
by
l,
in p a r t i c u l a r
H(C,(U))
We
Then
I(~)o
is c l e a r l y : M ~ N
where
f*
induced
=
IA n °*~'
a natural C~
is a
: A*(N)
~
~ 6 An(M) ,
transformation
map,
then
I o f*
= f•
A*(M)
and
0 6 S ~n (M) . of
functors,
that
is,
are
the
o 7,
f4~
: C*(N)
~ C*(M)
maps.
Lemma
1.14.
I
is a c h a i n
map,
i.e.
I o d = 6 o I.
In p a r t i c u l a r
induces
a map
I : H(A*(M))
This
Proof.
!
/(de) T
simply
on h o m o l o g y
~ H ( C * (M)) .
follows
using
exercise
2 above:
f An+1
~*(dco)
= JAn+ I dT*~
=
n+l ~ (I)i I i=0 An
=
n÷ I [ (1) i l ( ~ ) e i ( T ) = 6(I(~o)) T, i=O
(~i)*T*m
=
n+1 I ~ (I) i i=0 An
co
6 An(M),
T 6 S n + 1 (M) .
(Si(T))*~
if
11
Theorem
1.15.
an i s o m o r p h i s m
for a n y
First
notice:
Lemma
1.16.
a star
shaped
set
consider and
C
set
1.15
in
shaped with
the h o m o t o p y
g(,0)
= e
g
(1.11)
the
in s i n g u l a r together
cohomology, Lemma
respect
to
= sx +
{e}
~
I
is an o p e n
covering
empty
finite
intersection
diffeomorphic
onto
U) .
g(,1)
1.2
= id
statement
follows
~
H (C* (U))
)
H(C*
from
(1.10)
diagram
(e))
IR
there
to a s t a r
Choose
of
U
C~
manifold
U = {U
open
a Riemannian
(i.e.,
of a s t a r
In p a r t i c u l a r ,
} £ Z,
U 0~...N
shaped
a neighbourhood
diffeomorphism
in L e m m a
II
For any
point
an
an isomorphism
the c o m m u t a t i v e
I
1.17.
every
As
with
induces
IR
has
to
(1s)e.
II
point
diffeomorphic
M = U c~n
e 6 U. ~ U
U
so t h e
1.2 a n d
H(A*(e))
Proof.
M
given by
H (A* (U))
Lemma
for
to c o n s i d e r
: U x [0,1]
inclusion
with
is
M.
is t r u e
enough
g(x,s)
By
~ H(C*(M))
~n.
It is c l e a r l y
star
I : H*(A*(M))
manifold
Theorem
open
Proof. open
(de R h a m ) .
U
for e v e r y
U
of d i m e n s i o n
such
s0,
set of
metric
which
shaped
U p,
M
on
that
...,~p
n
every
non
6 ~,
is
~n.
M.
is n o r m a l q 6 U,
neighbourhood
is g e o d e s i c a l l y
Then with
eXpq of
every
respect
to
is a 0 6 Tq(M)
convex,
that
is,
12
for e v e r y Segment U.
pair
in
of p o i n t s
M
joining
(For a p r o o f
6.4).
Now
sets.
Then
p,q p
and
see e . g . S .
choose any
6 U q
and
Helgason
a covering
nonempty
there
is a u n i q u e
this
is c o n t a i n e d
[14,
Chapter
U = {U }~6 E
finite
geodesic
with
intersection
in
I Lemma
such open U
n...N
U
~0 is a g a i n
geodesically
neighbourhood clearly
of
In v i e w
using
are
algebraic
with
only
=
modules
use
R = ~). d
a double
PI, qI
d'
It is t h e r e f o r e
shaped
region
in
two
as
it is o b v i o u s
in L e m m a
about over
1.17.
double
~n
(via
is a
together
d"
R
is a
n £ ~,
~
with
: C p'q ~ Cp+1'q
ring
C~
: C n ~ C n+1,
What
that we inductive is n e e d e d
complexes:
a fixed
A complex
complex
Cp'q ,
lemmas
1.15 b y s o m e k i n d of f o r m a l
facts
a differential
Similarly, C~,~
last
a covering
We consider shall
~k
is a n o r m a l
points.
to a s t a r
Theorem
argument some
so
map).
of t h e
to p r o v e
and
e a c h of its
diffeomorphic
the e x p o n e n t i a l
want
convex
(actually ~graded
such
× ~graded
that
we
module d d = 0.
module
two d i f f e r e n t i a l s
: C p'q ~ C p'q+1
satisfying (1.18)
We
shall
d'd'
=
0,
actually
complex, Associated
that
is,
to
d"d"
assume Cp'q =
=
0,
d"d'
+
that
C~, •
0
if e i t h e r
( C ~ , ~ , d ' , d '')
is the
d'd"
~ I p+q=n
CP'q,
d = d'
0.
is a I. q u a d r a n t p < 0
total
where
cn =_
=
+ d".
double
or
q < 0.
complex
(C*,d)
13
For
fixed
to
d'.
q This
Now
gives
and
respecting clearly
f
induces
take
I C*, ~
suppose
gives
Lemma
bigraded
and
and
f~
1. q u a d r a n t
are
commuting map
two d o u b l e
double
complexes
d'
and
associated
d".
total
Also
Then
complexes
clearly
f
We n o w have:
f : IC*, • ~ 2C~, •
complexes
Then
respect
is a h o m o m o r p h i s m
with
of the
with
E~ 'q = H P ( c * ' q , d ' ) .
: H(IC*,d ) ~ H(2C*,d).
Suppose
isomorphism.
C ~,q
module
2 C~' *
: IE~ 'q ~ 2E~ 'q"
1.19.
of
f : IC*, • ~ 2C*, •
a chain
induces fl
the h o m o l o g y
another
the g r a d i n g
and h e n c e
is an
can
suppose
as above,
of
we
also
and f,
is a h o m o m o r p h i s m
suppose
fl
: H(IC~)
: IE~ '* ~ 2E~ '~
~ H(2C~)
is an
isomorphism.
Proof.
For
a double
(C~,d)
complex
define
complex
(C*,~,d', d'')
the s u b c o m p l e x e s
with
total
F q~ cC ~, q 6 ZZ, by
F* = [ I C * ' k q k__>q ~ ~l. 4 q 5 ~ Then
clearly ...
and
d
: F~ ~ F~ q q
isomorphic a map fl
to
that
f
from
q1
Notice
complexes
~ 2E ~ ' q
: IF~/1F~+1
in h o m o l o g y .
F ~
(ce,q,d').
of d o u b l e
~P,q : I~I
•
m
=
is an
m
F *
=
m
q =
that
Therefore the
0 ~
* I F*q+r / IF q+r+1 ~f
0 ~ 2 F q + r / 2 F ~ +r+1
...
for
~/ F*q+r+1 2 F q'2
f
: I c~'~
induces
~ 2C ~ ' ~
~ F q~+ r I F q/1
to s a y i n g
an i s o m o r p h i s m it f o l l o w s
complexes ~
is
that
r = 1,2,..•
of c h a i n
I F q~ / I F q~+ r + 1
* * (Fq/Fq+ I ,d)
is e q u i v a l e n t
q 6 ~,
%f ~
for
isomorphism,
diagram ~
m

assumption
N o w by i n d u c t i o n
commutative
q+1
the c o m p l e x
~ 2 F q'2 ~/ F q+1' •
the
F •
~ 0
4f ~ 2 F ~ / 2 F ~ +r ~ 0
14
and
the
induces r =
five
lemma
that
f
IF*/~ I
an isomorphism
1,2, . . . .
C*,*
F*
~
q+r
in h o m o l o g y
However,
for all
q 6 ~
for a I. q u a d r a n t
double
and
r > n
and complex
we have
n
FO
so the
lemma
similar
Cn
=
Notice
that
follows
from
denoted
d"
E PI' q
(1.18)
e
for
that
d"
q by
double
induces
: (E~'*,d")
we get
complex
C*'*
a differential p.
: C 0'q ~ C 1'q)
of chain
C p'q
a
H q ( C P ' * , d '')
for e a c h
inclusion
in
it
also
In p a r t i c u l a r ,
~ C 0'q
since
~ Cq
complexes
(C*,d)
"edgehomomorphism").
Corollary
I 20 •
induces
and
replaced
: E ~ 'q ~ E ~ 'q+1
a natural
the
p
for a 1. q u a d r a n t
E ~ 'q = ker(d'
(called
0
r
Interchanging
lemma with
we have
Fn =
follows.
Remark.
Suppose •
Rp'q =
0
for
p > 0.
Then
I
an isomorphism
: H(E~'*,d")
Proof Lemma
2 F q* / 2 F q* + r
1.19
Ep'q for
is a d o u b l e
the n a t u r a l
H(C*,d).
complex
inclusion
with
d' = 0 .
E ~ 'q ~ C p ' q
Apply
e
15 Note. G. B r e d o n ~§ 3 and
For m o r e
information
[7, a p p e n d i x ]
on d o u b l e
or S. M a c L a n e
complexes
[18, C h a p t e r
see e.g. 11,
6].
W e now t u r n to
P r o o f of T h e o r e m of
M
as in L e m m a
complex
where
1.17.
as follows:
~ 'q
=
1.15.
Choose
Associated
Given
is o v e r
n...n
where
d
exterior
differential
For
Sp
(p+1)tuples % ~.
The
(s0,...,s p)
"vertical"
AP,q +I
U p ) ~ Aq+1 (U 0 n...n operator.
Ap, q
is g i v e n
U
) ~p
all o r d e r e d
: AP,q ~
: Aq (Us0 n...A
u
50
~. 6 ~ s u c h t h a t U N...n i S0 d i f f e r e n t i a l is g i v e n by (1)Pd
consider
Aq(U
with
U = {Us} 6 E
to this w e get a d o u b l e
P,q 2 0
H (s 0 , .... ep)
the p r o d u c t
a covering
The
UsP )
horlzontal
is the differential
.p+1,q
as follows: ~ =
(~(s 0 ..... ~p))
Aq(u 0 N . . . N U
)
6
the c o m p o n e n t
~'q
of
6~
in
is g i v e n by
~p+ 1 p+ I (1.21)
(6~) (a 0 ..... eP+1 ) =
It is e a s i l y double
seen that
66 = 0
(i) i i=O
and
e(e0 ..... ~i ..... Sp+l) 6d = d6
so
complex.
Now notice
that there
Aq(M ) c
is a n a t u r a l
H A q ( u s 0 ) = 4 'q . s0
inclusion
A~ 'q
is a
18
Lemma
1.22.
For
each
q
the s e q u e n c e
0,q 1,q ~ AU ~ AU ~
0 ~ Aq(M)
...
is exact. Proof.
In fact
,q = Aq(M) AUI
putting
we
can construct
homomorphisms Sp
such
that
(1.23)
TO do supp
: Ap'q ~ Ap1'q
Sp+ I o 6 + 6 o Sp
this
just
~e ~ Us,
choose
Ve
6 [,
a partition and
(Sp~) (~0 ..... ~p1 ) =
id.
of u n i t y
to v e r i f y
that
with
define
(1)P [ ~e~(~O
'~)
~6~
'''''~p1
w It is easy
{~}~£Z
s
'
6 A~ 'q
is w e l l  d e f i n e d
and
that
(1.23)
P is s a t i s f i e d . It f o l l o w s
that
= f 0,
p > 0
EP,q A q (M), Together
with
Lemma there
Corollary
1.24.
is a n a t u r a l
eA which
induces
Let
1.20
A U*
p = 0. this
be
proves
the total
complex
chain map
: A*(M)
~ A U
an i s o m o r p h i s m
in h o m o l o g y .
of
*'* . AU
Then
17
We now want the
singular
to d o
cochain
the
same
functor
thing with
C ~.
As
A*
before
replaced
we get
by
a double
complex
C~ 'q =
H cq(u (s 0 .... ,ep) e0
U
where
the
"vertical"
the coboundary the
in t h e c o m p l e x
"horizontal"
(1.21)
above.
Again
Lemma
we have
: C*(M)
by
N...N
a natural ~0,*
)
is g i v e n
s0 is g i v e n
~ ~U
u ep
C*(U
differential
ec and we want
differential
n...n
by
(I) p
times
U
) and where P t h e s a m e f o r m u l a as
map of chain
complexes
,
=c C U
to p r o v e
1.25.
eC
: C*(M)
* ~ CU
induces
an isomorphism
in
homology.
Suppose finish
for t h e
the proof
For
U ~ M
moment that Lemma
of T h e o r e m we have
of double
by
(I .13)
above.
Therefore
I : A P ' q ~ Cp ' q
a commutative
+e A
A*(M)
and
l e t us
this.
~ C*(U)
complexes
and we have
is t r u e
a chain map
I : A*(U)
as d e f i n e d
1.15 u s i n g
1.25
diagram
+e c
"~ C*(M)
we
clearly
get
a map
18 By
(1.24)
and
(1.25)
in h o m o l o g y . map
It r e m a i n s
induces
following
the v e r t i c a l to s h o w
an i s o m o r p h i s m
Lemma
1.19
to e a c h
of
it s u f f i c e s
U
with
A*
replaced
this
D...n
with
support
let
cq(u)
Then
C*.
However,
C~ 'q
cq(u)
and
0 ~
is exact.
In f a c t w e
as
follows:
o(~q)
cq(u)
~ Us(o),
an e a s y
s
It f o l l o w s e C = ec map
as
in
if w e
defined there
is a n a t u r a l
p
Lemma
1.16
applied
that L e m m a restrict Thus
1.22
holds
to c o c h a i n s as in
(1.12)
on s i m p l i c e s is a
restriction
U
with
map
~ CZ ' q ~ C~ ' q . . . . construct
each
~ 6
homomoprhisms (Cu 1'q = cq(u)),
S~(U) q
choose
s(o)
6 ~
such
that
and d e f i n e
calculation
that
the c h a i n where
(1.12)
(~) =
shows
o d + ~ 0 s
p+1
0 I*,
true
S~(U) q
o 6
Sp(C) (s 0 , . . . , s p _ 1 ) Then
for e a c h
it is true.
: C~ 'q ~ C~ I
For
that
the s e q u e n c e
(1.26)
Sp
U
the q  c o c h a i n s
there
the r e m a r k
is e x a c t l y
It is not
o 6 S~(U), i.e. for each q 0(A q) ~ U s.
N o w by
U
in the c o v e r i n g denote
horizontal
~p
1.25. by
the u p p e r
to see
s0 of L e m m a
isomorphisms
~ H(C~'*)
However
the sets
Proof
that
induce
in h o m o l o g y .
I : H(A~'*) is an i s o m o r p h i s m .
maps
I*
and w h e r e
p map
(~)Pc(s0,...,~p_1,s(~))
(~)"
that
=id.
eC
: C*(M)
: C*(M) ~
the e d g e
~ C~
factors
C*(U) is the n a t u r a l homomorphism
into chain
19
ec
induces
an i s o m o r p h i s m
exactness
of
in h o m o l o g y
(1.26). by
Exercise
4.
For
H(C~(M))
(Hint:
and
S~(M)
ends
simplices
agree with therefore
cohomology
groups
5.
(1.11)
Note.
The
[34].
the proof
1.20 and the
an i s o m o r p h i s m
of Lemma
1.25 and
space
X
let
st°P(x) n
nsimplices
of
be the c o r r e s p o n d i n g
Show that
~ st°P(M)
for a
C~
induces
the usual
from T h e o r e m
M in h o m o l o g y
as in Lemma
b a s e d on
singular
chain
~ H(C*(M)) .
for a c o v e r i n g
and c o h o m o l o g y
X,
manifold
isomorphisms
H (Cto p(M))
complexes
It follows
property
C~ (X) top
the h o m o l o g y
Exercise
induces
singular
~ H(ct°P(M)),
Use d o u b l e
Hence
also
a topological
complexes.
the i n c l u s i o n
by C o r o l l a r y
1.15.
c~°P(x)
and c o c h a i n
A. Weil
this
~
the set of c o n t i n u o u s
and let
~ C~
in h o m o l o g y Since
(1.12)
also of T h e o r e m
denote
: C*(U)
homology
1.15 that
C~
1.17).
singular
and cohomology.
the de Rham
are t o p o l o g i c a l invariants.
Show d ~ r e c t l y
the a n a l o g u e
of the h o m o t o p y
for the de Rham complex.
above proof It c o n t a i n s
of de Rham's the germs
For an e x p o s i t i o n
of de Rham's
F. W. W a r n e r
chapter
[33,
5].
theorem
goes back
of the theory
theorem
to
of sheaves.
in this c o n t e x t
see e.g.
2.
Multiplicativity.
In C h a p t e r M
The s i m p l i c i a l
I we s h o w e d
the de R h a m c o h o m o l o g y
invariants
of
M.
(2.1)
makes
A~(M)
induces
that for a d i f f e r e n t i a b l e
groups
As m e n t i o n e d
A : Ak(M)
an a l g e b r a
de R h a m c o m p l e x
Hk(A~(M)) above
® AI(M)
manifold
are t o p o l o g i c a l
the w e d g e  p r o d u c t
~ Ak+I(M)
and it is easy to see t h a t
(2.1)
a multiplication
(2.2)
^ : Hk(A~(M))
In this c h a p t e r w e shall invariant.
More
(2.3)
be the u s u a l
show that
precisely,
V
: Hk(c*(M))
cupproduct
® HI(A~(M))
~ Hk+I(A~(M)) .
(2.2)
is also a t o p o l o g i c a l
let ® HI(c~(M))
in s i n g u l a r
~ Hk+I(c*(M))
cohomology;
then we shall
prove
Theorem
2.4.
For any d i f f e r e n t i a b l e
manifold
M
the
diagram Hk(A,(M))
® HI(A,(M)) +I ®
Hk(c*(M))
A
~ Hk+I(A,(M))
I
~I
® HI(c~(M))
~
~ Hk+I(c~(M))
commutes.
For
the p r o o f
it is c o n v e n i e n t
de R h a m c o m p l e x w h i c h closely h a n d has
related
is a p u r e l y
to the c o c h a i n
the s a m e
to i n t r o d u c e
combinatorial
complex
formal properties
C*
the s i m p l i c i a l construction
b u t on the o t h e r
as the de R h a m c o m p l e x
A ~.
21 We shall d e f i n e
it for a g e n e r a l
Definition S = {Sq},
2.5.
A simplicial
q = 0,1,2,...,
of sets
e i : Sq ~ Sq_1 . i. =. 0, .
,q,
(ii)
S
is a s e q u e n c e
together with
which
satisfy
gig j = £ j _ i s i ,
i < j,
Ninj
= Nj+INi ,
i ~ j,
nj_lei,
i < j,
f
(iii)
set
set:
and d e g e n e r a c y
H i : Sq ~ Sq+ I, i = 0,...,q,
(i)
simplicial
ein j = J i d ,
face o p e r a t o r s
operators the i d e n t i t i e s
i = j, i = j+1,
I
(~jEi_1, Example Sq = S ~q(M)
I.
We s h a l l m a i n l y
or
i = 0,...,q,
i > j + I.
st°P~Mjq ,, . where
ei
Here : ~q1
(2.6)
el(t0 , .... tq_ I) =
Analogously,
the d e g e n e r a c y
Hi(o)
consider
as in C h a p t e r ~ Aq
I,
where
ei(~ ) = ~ 0
ei
is d e f i n e d by
(to, .... t i _ 1 , 0 , t i .... ,tq_1).
operators
= o 0 n i , i = 0, .... q,
the e x a m p l e ,
Hi
are d e f i n e d
i : Aq +I ~ Aq
where
by
is d e f i n e d
by
(2.7)
H i ( t 0 , .... tq+ I) =
We l e a v e
it to the r e a d e r
A m a p of s i m p l i c i a l commuting S~
and
(t O .... , t i _ l , t i + t i + 1 , t i + 2 ..... tq+l).
with S top
manifolds s implicial
to v e r i f y
sets
the a b o v e
is c l e a r l y
the face a n d d e g e n e r a c y become
functors
(respectively sets.
identities.
a sequence
operators.
f r o m the c a t e g o r y
topological
spaces)
of m a p s Obviously
of
C~
to the c a t e g o r y
of
F
22
Definition A differential of k  f o r m s (i)
2.8.
Let
kform
~
such ~o
S = {S } q on
S
be a simplicial
is a f a m i l y
set.
~ = {~ }, o 6 ~ S p P
that
is a k  f o r m
on
the s t a n d a r d
simplex
Ap
for
o 6 S P (ii)
~e.o
=
(el)~o
' i = 0,...,p,
o 6 Sp,
p =
1,2,...
1
where
e i : A pI
Example if
~
2.
The Ak(s) . A ~
~
If
we have
for
M
a
as d e f i n e d
C~
by
(2.6).
manifold.
~ = {~o}
on
Then S~(M)
o 6 S~(M) . P
on a simplicial ~ 6 AI(s)
set
we have
S
again
is d e n o t e d the w e d g e  p r o d u c t
by
= ~
^ ¢o'
the exterior
(d~)~ = d~o,
commutative
that
and
~ £ Sp,
differential
p = 0,1,...
d
: Ak(s)
~ Ak+I(s)
d
associative
and graded
satisfies
= d~ ^ % +
(A*(S),^,d)
complex
then clearly
is a g a i n
p = 0,1,2,...
and
^ ~)
call
o 6 Sp,
^
that
dd = 0 d(~
de Rham
(M)
face map
by
It is o b v i o u s
shall
ith
we get a kform
for
(~ ^ ~ ) ~
(2.11)
M
~ 6 Ak(s),
(2.10)
(2.12)
on
= ~
defined
defined
S = S
s e t of k  f o r m s
(2.9)
Also,
is the
Let
is a k  f o r m
by putting
We
~ Ap
of
we get
S. f~
(f*%0) o = ~fo,
If
(1)k~
^ d~,
the simplicial f
: S ~ S'
: A * ( S ') ~ A*(S)
~ 6 Ak(s'),
~ 6 Ak(s),
de Rham
algebra
is a s i m p l i c i a l defined
o 6 Sp,
~ 6 AI(s).
map
by
p = 0,1 ....
or
23
and
thus
A*
is a c o n t r a v a r i a n t
functor. oo
Remark manifold
I.
M
Notice
that by Example
a natural
(2.13)
: A*(M)
which
is c l e a r l y
injective,
forms
on
as s o m e
We now want simplicial
set
so w e
The chain
vector
on
and
k ~ i=0
(0) =
Dually C*(S) c =
the cochain
complex
= Hom(C,(S),JR), (c),
~ 6 Sk,
(2.14)
Again
~
with
we have
a natural
M.
real
where
Ck(S)
is the
is g i v e n
free
by
o 6 Sk
coefficients
a kcochain
is
is a f a m i l y is g i v e n
by
T £ Sk+1 "
map
~ ck(s)
by
(2.1 5)
and we
with
~ C k+1 (S)
on
for a n y
C,(S)
k+1 [ (1)ic ~. T ' i=0 1
I : Ak(s)
defined
forms
~ C k _ I (S)
real
: ck(s)
(6c) ° =
of
(1)ie i (0) t
so a g a i n
and
of s i m p l i c i a l
theorem"
complex
~ : Ck(S)
think kind
complex
is of c o u r s e the Sk
can
a "de R h a m
coefficients space
C
~ A~(S~(M))
generalized
to p r o v e S.
for a n y
transformation
i
S~(M)
2 we have
~ (4) o = IAk ~0'
can
o 6 S k,
now state
Theorem chain map
~0 6 A k ( s ) ,
2.16
inducing
(H. W h i t n e y ) . an isomorphism
I : A*(S)
~ C*(S)
in h o m o l o g y .
is a
In f a c t
there
24 is a n a t u r a l
chain map sk
homotopies
: Ak(s)
(2.17)
I o d=
(2.18)
I o E = id,
E : C*(S)
~ A~(S)
~ Ak1(S),
~ o I,
and n a t u r a l
k = 1,2 .... ,
such
chain that
E 0 ~ = d o E E o I  id = Sk+ I o d + d o s k, k =0,1,...
For the p r o o f we f i r s t usual
Ap c ~p+1
is the s t a n d a r d
=
canonical
basis
coordinates respect
{e0, .... ep}
(t0,...,tp).
to e a c h v e r t e x
have operators each
j
h(j)~
= 0
h(j)
as d e f i n e d for
k = 0,1,2,..., (i) (2.20)
(ii)
For
in the p r o o f
The o p e r a t o r s
=
=
(2.22)
Lhe f o l l o w i n g
~,
k > 0
w(ej)e,
k = 0
0,...,p
(Ei) ",
i > j
o
i < j
= (el) ~,
e 6 Ak(~ k)
IA k e =
(1)kh(k_1)
o...o
for
lemma
I):
: Ak(A p) ~ A k1 (A p) ,
{
o
we
Also put
~ 6 Ak(g p)
(ei) * 0 h(j)
For
of
1.2.
3 of C h a p t e r
h(j)
(jl) (iii)
of L e m m a
The p r o o f
{~(j) (2.21)
and t h e r e f o r e
satisfy
i,j
by the
is star s h a p e d w i t h
(of. E x e r c i s e
(j)d~ + d h ( j ) ~
For
Ap
j = 0 ..... p,
~ 6 A0(&P) .
2.19.
spanned
and w e use the b a r y c e n t r i c
Now
ej,
psimplex
As
: Ak (A p) ~ A k1 (AD), k = 1,2 .....
is l e f t as an e x e r c i s e
Lemma
need some p r e p a r a t i o n s .
h ( o ) (~) (e k) .
25
Next
some
notation:
Let
I =
(i0,...,ik)
satisfying I
IIl = k
I
we have
dimensional
(for the
face
and
~I = ejl o...o
to
I
is the
~I =
(for
~I
I = ~
lowers We
can
motivation
~
is c l e a r l y
course
First
1 6 {0,...,p}
and
Then
•
(for
I = @
~ Ak(s)
is the
defined
by
ik
~ =
on
suppose s
put
h~ = id).
as f o l l o w s
~ £ S
(a
put P
c i (~) (if
p < k
Similarly
(2.24)
for some
e
: A~(A p) ~ A~(A p)
Ap
[ 0~111 k.
(i)
(iii)
homotopic
this
is a n e c e s s a r y
2.
the k  f o r m
^'''^
in o r d e r
that with
~ 6 Sp,
Exercise
b)
choice
^'''^
eq+lhq
h j _ l e i,
thjei1'
= fl
if
i < j,
if
i > j+1,
ej+lhj+ I = ej+lhj'
are hi
called : Sq
S q+1' i
36
= ~ hj+INi' (iii)
Show that c) and b)
Nihj
f~,f~ Let
f0,fl
= {~o}
3.
on
i
on
Let
S
(iii)
(2.7).
f$,f~
: Ap+I
i > j. are c h a i n h o m o t o p i c .
be h o m o t o p i c . ~ A~(S)
chain homotopies
S
normal
S h o w that a) are c h a i n h o m o t o p i c .
in c) .
be a s i m p l i c i a l
( l).~o,
~ AP
k AN(S)
Let
if
: A*(S')
is c a l l e d
~nio =
i < j,
~ C*(S)
: S ~ S'
Find explicit
Exercise
where
: C*(S')
imply that
d)
~ h J hi1 ,
if
set.
A kform
if it f u r t h e r m o r e
i = 0,...,p,
is the ith d e g e n e r a c y
o £ Sp, p = 0 , I , 2 , . . map defined
be the s u b s e t of n o r m a l
~ Ak(s)
satisfies
by
kforms
S. a)
Show that
f : S ~ S' normal b)
d
and
^
preserve
is a s i m p l i c i a l m a p
then
f*
forms and if
also preserves
forms. Show
k = 0,1,...,
t h a t the o p e r a t o r s j = 0,...,p,
h(j)
h(i)D j
(ii)
kcochains
[~h(i_l
h ( i ) h ( i ) = 0, k CN(S) ~ ck(s)
Let
c =
(c o )
i < j ),
i > j
i = 0 ..... p.
be the
such t h a t
c
set
Show that
(i)
I : A~(s)
~ c~(s)
(ii)
£ : C~(S)
~ A~(S)
(iii)
sk
k k1 (S) : AN(S ) , A N
of normal cochains,
.T = 0 1
i = O,...,k1.
: Ak(A p) ~ A k  I ( A P ) ,
satisfy
* =~D3h(i), (i)
c)
normal
VT £ Sk_ 1,
i.e.,
37 and conclude Hence
that
~ : A~(S)
s i n c e the i n c l u s i o n
equivalence
(see e . g . S .
the i n c l u s i o n
A~(S)
Exercise
4.
r
D)
C~(S) MacLane
* A*(S)
is a c h a i n e q u i v a l e n c e .
~ C*(S)
is a c h a i n
[18, C h a p t e r
7, § 6] a l s o
is a c h a i n e q u i v a l e n c e .
(D. S u l l i v a n ) .
set of p o l y n o m i a l 6 Ak(A n
* C~(S)
forms w i t h
Let
Ak(A n, ~)
rational
is the r e s t r i c t i o n
denote
coefficients,
of a k  f o r m
in
the
i.e.
~n+l
of
the f o r m
L0 =
a. . dt. ^...^dt. 10'''l k 10 ik
i0 nO
and
suppose
we
have
defined
an
invariant
X
74
open
set
Let
p
Un_ I ~ EG(nI)
: An
x G n+1
observe
that
W ~
x
DA n
since h'
is
: W'~
G
Shrinking
Now
consider
Clearly
can
an
~
W'
W"
a ~
is and
an
Now
p1(Un_1) U'
= W"
Un
h"
= Un_ I
clearly h
n
U =
: U
n W"'
n
We
can
II
for
h =
now
open
assume
and
an
Let
principal
correspondence.
is
an
The
the
map
Gbundles
h"
to
a map
neighbourhood defined
of
on
W'.
6 W'}.
notice
that
On
the
An
x G n+1
other
W ~ W" hand
such
Ginvariant
: U'
we
that
~ G
G
and
and
This
ends
main
result
set
h
the
associating element
by
is
n
this
to
EG(n)
of
the
so
let
proposition.
chapter:
a characteristic
c(E(G))
and
extension
inductively,
proof
of
equivariant.
in
an e q u i v a r i a n t n
subset.
..... g n g 0I) )'go"
invariant
U
the
~
open
: W ~
open
Then
= p1(Un_1)
defines
U h . n n
and
W"'
define
construct
state
5.5.
x G n+1)
and
by
set
subset
: Un_ I ~G.
extends
h'
Ginvariant.
x G n+1
hn_ 1
This
and
Theorem c
EG(nI).
an o p e n
defined
hn_ I 0 p is
and
~ G.
U U n
is
can
hn_ I
projection
: W ~ G
,gn) ) = h, ( t , ( 1 , g l g 0 1
p(U')
h" n
~A n
extends
U
is
N (3A n
~
h.(t, (g0, "'" Clearly
we
map
W = p1(Un_1).
Ginvariant
W
a Ginvariant
let
x G n+1
x G n+1
open
hence
W"'
since
An
into
hn_ I o p
little
An
natural
subset
map
~
the
x G n+1
closed
the
equivariant
{ (t, (g 0 ..... gn)) I (t, ( 1 , g ~ g 0 1 ..... g n g 0 1 ) )
n
find
ANR
an
be
DA n
the
W'
W"
since
be
where
W.
=
~ EG(n)
maps
~+I
G
W"
p
and
£ H*(BG)
class is
a
75
For
the proof
"simplicial" Let suppose face
we
point
shall
study
EG
and
BG
from
a
of view:
X = {Xq}, that each
q = 0,1,..., X
and degeneracy
be
a simplicial
is a t o p o l o g i c a l
q
operators
called
a simplicial
space
called
fat realization,
are
space
the
space
li x tl =
~
such
continuous.
and associated
t h a t all
Then
to this
[l X ii g i v e n
An
set and
X
is
is the so
by
× Xn/~
n>0 with
the
identifications
(5.6)
(£1t,x)
~
t £ A nl,
(t,£ix) ,
x 6 Xn,
i = 0,...,n, n = 1,2,...
Remark
(5.7)
I.
It is c o m m o n
(nit,x)
~
furthermore
(t,~ix),
t 6 A n+1,
to r e q u i r e
x 6 X n,
i = 0,...,n, n = 0,1,...
The resulting denoted
by
space IXi.
is a h o m o t o p y
Remark
Example consider The name from
this
X
One
can
equivalence
2.
I.
Notice
If
the ~ e o m e t r i c
show
that
under
X = {Xq}
realization
the n a t u r a l
suitable
that both
as a s i m p l i c i a l
"geometric
II'II
realization"
for
is
il X hi ~ IXi
conditions.
and
1I
is a s i m p l i c i a l space with
map
and
are
set
then we
the discrete
the
space
iXi
functors.
can
topology. originates
case.
Example the
is c a l l e d
2.
simplicial
Let
X
space with
be a topological NX
q
= X
and
space
all
face
and
let
NX
be
and degeneracy
76
operators
equal
to the
I N X IL =
IL N(pt)
identity.
II x X,
II N(pt)
with
the
apropriate
Example
3. group)
spaces
and
(Here
NG
NG(0)
In are
NG
given
INXl
= X
Anu
...
identifications.
G
and
be
a Lie
consider
group
the
(or m o r e
following
two
simplicial
NG(q)
= G .... x G
(q+1times),
NG(q)
= G x...x
(qtimes).
consists
of o n e
e i : NG(q)
G
element,
~ NG(q1)
namely
the
empty
and
H i : NG(q)
=
(go ..... gi ..... gq)
~ i ( g 0 ' .... gq)
=
(go ..... g i  1 ' g i ' g i ' ' ' ' ' g q
in
NG
ei
c i ( g 1 ' .... gq)
: NG(q)
~ NG(qI)
)'
is g i v e n
= ~(g1'
igi+1'''''gq
L(g I , : NG(q)
~ NG(q+~)
Hi(g1 ..... gq) By d e f i n i t i o n map
y
0tuple
~ NG(q+I)
=
EG =
: NG ~ NG
by
i =
)'
I,...,qI
i = q
,gq_1 ), by
(gl ..... g i  1 ' 1 ' g i ' ~ ' ' ' ' g q II N G II a n d
given
i = 0, .... q.
i = 0
I
~i
any
by
(g2''''i~q)'
and
generally
NG:
E i ( g 0 ..... gq)
Similarly
and
where
II = A 0 U A I U . . . U
Let
topological
Then
by
if w e
consider
)' the
i = 0 ..... q. simplicial
!).
77
(5.8)
Y{g0
it is e a s y
gq) = (g0g~1
. . . . .
to see t h a t t h e r e
is a c o m m u t a t i v e
EG  
I IL y II
BG
~ II NG It
s u c h t h a t the b o t t o m h o r i z o n t a l therefore
identify
The simplicial
diagram
il N G [I
YG i
will
gq_~g~1)
. . . . .
BG
spaces
map
with
NG
is a h o m e o m o r p h i s m . IING II
and
NG
and
above
YG
We
with
11 y II.
are s p e c i a l
cases
of the f o l l o w i n g :
Example "small"
4.
Let
category
C
be a t o p o l o g i c a l
c__ategory,
such that the set of o b j e c t s
set of m o r p h i s m s
Mot(C)
are t o p o l o g i c a l
i.e.
0b(C)
a
and the
spaces
and such
Mor(C)
~ 0b(c)
that (i)
The " s o u r c e "
and
"target"
maps
are
continuous. (ii)
"Composition": where
M0a(C) ° c Mar(C) =
pairs
of c o m p o s a b l e
(f,f') Associated n e r v e of NC(2)
to C
there
where
morphisms
= M0r(c) °,
= 0b(C),
consists
space
NC(1)
(f')).
NC
= Mor(C),
and g e n e r a l l y
c__ Mot(C)
x...x
is the s u b s e t of c o m p o s a b l e fl
f2
Mot(C)
of the
(i.e.
(f) = t a r g e t
is a s i m p l i c i a l
NC(0)
is c o n t i n u o u s
x M0r(C)
6 MOA(C) O ~ s o u r c e
C
NC(n)
MoA(C) 0 ~ Mot(C)
(n
strings f
n
times)
called
the
78
That
is,
(fl,f2 .... ,fn ) 6 NC(n)
i = I,...,nI.
Here
iff
e. : NC(n) l
~
source
~ NC(n1)
(fi)
= target
is g i v e n
by
i = 0
(f2 ..... fn ) '
e i ( f 1 ' f 2 ' .... fn ) = 1 ( f ] '
(fi+1) ,
'fi o fi+1 .... 'fn )'
0 < i < n
'fn1 )'
i = n
! 0}.
that for
I, ~ 6 {,
from the identity
(k) [ (D_l)kix, (D_~)iy],
x,y E ~ ,
k=0,I,2,..,
i=0 which is proved by induction on Now let let
~
T ~ G
= ~®~
be a maximal
~ ~ ~
connected Lie group. ad(t) metric.
: ~
~7
k.
This proves
torus with Lie algebra
and let
Tff ~ G~
Every element
t £ ~
~
,
be the corresponding is semisimple
since
is skewadjoint with respect to a Ginvariant
Therefore
every element of
~
and we have the root space decomposition Chapter III,
the lemma.
§ 4]) =
7~
$
/~(E $ c~E~'e~e~ '
is semisimple
as well
(see e.g. Helgason
[14,
137
where
~ : ~ ¢ ~ ~,
onedimensional
e 6 #,
subspaces
[t,x ] = a(t) Furthermore
let
Then both
~
~+
t 6 ~,
x
6~
are
.
be a choice of positive
~
and
i.e. ~ e
and
" x ,
¢+ ~ ~
=
are the roots,
t
:
roots and let
"
are subalgebras
of
~
since
18A31 Also let
B ~ G~
be the group with Lie algebra
~.
With this
notation we now have Lemma
~
.
8.A.4.
a) ~ ¢
Furthermore
every element of
every element of b)
is a maximal
~
+
~
is semisimple
v 67¢
with
more,
then the semisimple
to
G
The inclusion and
NT ~ NT¢
G~, respectively, W = NT/T
d)
If
there exists
Proof. of
of
and
n 6~ +
g 6 G{
and
such that
[t,n] = 0.
part of
v
Further
is conjugate
t. c)
in
t 6 4,
there is
Ad(g)v = t+n 6 6 v 6 ~ +,
subalgebra
is nilpotent.
For every element
if
abelian
v.
If
s 6~
induces
For
[v, ~ ]
such that
v £ f~ = 0
of
T
and
T~
an isomorphism
~ N T c / T C
and if for some
w 6 NT¢
a)
of normalizers
let
g 6 G~,
[Rev ,~]
 O,
6~
then
Ad(w) s = Ad(g)s.
v
be the complex conjugate
then clearly also
both the real and imaginary part
Ad(g)s
Rev
and
[Imv,~
[U, ~ ] Imv ] = 0
= 0
satisfy
so
138
SO by m a x i m a l i t y is a m a x i m a l already
abelian
proved
h)
and
By the
Chapter
of ~
VI,
v = Rev subalgebra.
the
last
Iwasawa
Theorem
G@
The
clearly
= 0.
This
second
follows
decomposition
6.3])
(8.A.5)
in p a r t i c u l a r
+ i Imv
shows
statement from
(see e.g.
that
~
is
(8.A.3). Helgason
[14,
we have
=
G
and
B ~ G : T
" exp~ +
• exp(i~)
the
inclusion
G ~ G~
induces
a diffeomorphism
G/T ~ G { / B
so the E u l e r
characteristic
(cf.
Adams
[1, p r o o f
fore
conclude
an e l e m e n t
g 6 G~
such
group
Ad(g1)v
4.2]]).
fixed
that
For
point
from
v 6 ~
theorem
gB E G{/B
: G~/B
= exp(rv)xB,
g
Hence
is d i f f e r e n t
we
that
is f i x e d
zero there
there
under
is
the
of d i f f e o m o r p h i s m s
hr
hr(XB)
G{/B
of T h e o r e m
by L e f s c h e t z '
oneparameter
where
of
I
66.
~ G~/B,
r [ IR,
exp(rv)g
We
can
r £ ~,
that
£ B,
is,
Vr 6 ~R.
therefore
suppose
v 6 ~
,
and
we write
~+ X
v = t +
N O W we
claim
of
so t h a t
B
is a m i n i m a l
t h a t we x
root
can
% 0
t
{,
change only
v
for
so that b o t h
~
~
by c o n j u g a t i o n ~(t)
x
x
% 0
= 0. but
•
by e l e m e n t s
In fact ~(t)
suppose
% 0.
Then
139
I/__ x ))(v) A d ( e x p (   ~ t) x ))v = E x p ( a d (e(t) co
where
e'
> e
means
that
=
v
:
t


[ ~(ad(~(~ i=2 "
is a p o s i t i v e
b 6 B
Ad(b)v = t +
+
x )
root.
Iterating
such that
[ + z . ~(t)=0
T h e r e f o r e w e put [t,n]
= 0;
Notice in
~¢+ ~ z
n =
hence
Ad(b)v
that c o n j u g a t i o n
~
b 6 B
in the d e c o m p o s i t i o n
statement c)
and we c l e a r l y
6 ~+
= t + n
by
have
is the J o r d a n d e c o m p o s i t i o n . does not c h a n g e
(8°A.6)
which proves
the c o m p o n e n t the s e c o n d
in b). Clearly
NT/T ~ NT~/T~
NT ~ N T ~
and since
is i n j e c t i v e .
leftmultiplication
by
has a f i x e d p o i n t
N o w for
T~ D G = T g 6 T
the m a p
a regular
element,
g
Lg
: G ~ / B ~ G~/B
for e v e r y
element
in
NT~/NT~n
B.
Therefore
the c o m p o s i t e
N T / T ~ N T { / T { ~ N T { / NT~ D B
is a b i j e c t i o n however,
is t r i v i a l
of the f o r m the p r o o f d) Consider
so it r e m a i n s
to s h o w that
T~ = N T ~ n B.
f r o m the f a c t that e v e r y e l e m e n t
a  exp(n)
with
a £ T~
and
n 6 ~ +.
of This
of c). Let
s 6 ~
and
the L i e a l g e b r a
g 6 G~
i
Y~,
+
e'  ~
this p r o c e d u r e w e can f i n d
(~[V,Xc]
with
Ad(g)s
= t 6 4.
This, B
is ends
(v)
140
J= and let
D c= G{
Then clearly
{v 6 ~ {
J Iv,t] = 0}
be the associated
~
c__J
and also
connected Ad(g) ~
subgroup of
__c J
G~.
since for
[Ad(g) (x),t] = [x,s] = 0. Also
~
and hence
Ad(g) ~ {
are Cartan subalgebras
nilpotent
algebra with itself as normalizer).
conjugacy
theorem
Th~or~me
(see e . g . J . P .
2]) there exists a
d 6 D
Ad(g) / ~ dlg 6 NT~
Hence
and
Serre
(i.e. a
Hence by the
[25, Chapitre
III,
such that
= Ad(d) ~ .
Ad(dlg)s
= Ad(d)t = t.
This ends the
proof of the lemma. After these preparations differentiability
of
8.3.
~
Recall that
Lie group
G
polynomial
of degree
P'
: ~
we now return to the proof of the ~ ~
in the proof of Proposition
is the Lie algebra of a compact connected
with maximal k
torus
T
and
P
on the Lie algebra
is a homogeneous ~
of
T.
P'
:~
is defined by the formula P' (v) = P(ad(g)v) We shall show that !
PC
on
P'
where
Ad(g)v 6 ~
extends
for some
to a complex analytic
g 6 G. function
~.
Since
G
is compact
is the center and [14, Chapter
~'
~
= ~
@ ~'
is a semisimple
II, Proposition
6.6]).
where
ideal
(see Helgason
Furthermore,
if
Z ~ G
141
is the center of
G
then
!
~
the Lie algebra of the group
is n a t u r a l l y identified with G' = G/Z.
r e p r e s e n t a t i o n factors through
G'
Ad I.
coefffcients C,NF.
x 6 F
using
by W h i t e h e a d ' s
integral
now define
representing
into
skeletons
is c o n t r a c t i b l e
the h o m o l o g y to the h o m o l o g y
and we c l a i m z 6 C2(NF)
that
defined
by
z =
(Xl,X 2) +
I (XlX2,X I ) + . . . +
I I I (XlX2Xl x 2 . . . X 2 h , X 2 h _ 1 )
+
(Xl,X~I). +

(1,1)

(1,1)
(x2,x~1)z + . . . +
(1,1)
+ 
I (X2h_1,X2h_1)
155
which
is e a s i l y
checked
is the s u m of a l l shown
the
in t h e a b o v e Now
a map
any
B~
to b e (4h2)
figure
flat
<e(E),z>
Now
it is e a s y
(x,x I) the
contribute
trace
matrix). terms This
to s e e
of
proves
the f i l l i n g
: F ~ Si(2,~) It f o l l o w s
~
by geodesic
in t h i s
case
hand
side
of
the
form is
and a symmetric
consists
contribute
the
integrand
with
of
4h2
less
than
for
remark I
It is s t r a i g h t f o r w a r d so w e
can
I/4.
simplices.
G
semisimple
Theorem
First
l e t us r e d u c e
~
=~
with maximal
@/
we have
= z 0 exp
:~
following
Proposition
: G/K ~ G
For
9.12
that using the
group
~ G/K
9.11.
is an e m b e d d i n g
such
Therefore that
i , G
P 6 II(K)
compact
the d i f f e o m o r p h i s m
Then we have
9.20.
to c h e c k
apply
G/K
Lemma
is a
variables:
G/K
commutes.
by
that
that a simplex
a skewsymmetric
9.15.
decomposition
o ~
of
is s e m i  s i m p l e
In g e n e r a l
I = exp
(since
numerically
of i n t e g r a t i o n
as in t h e
(9.16)
the r i g h t
of T h e o r e m
and Cartan
is i n d u c e d
the corollary.
G = Sp(2n,~)
number
~
simplices.
: E ~ BF
I below).
from
the p r o d u c t
e a c h of w h i c h
Proof
where
triangulation
= .
zero
Therefore
~
f,z 6 C , ( X h)
in the
some degenerate
Sl(2,~)bundle
(see E x e r c i s e
In f a c t
2simplices
plus
: BF ~ B S I ( 2 ~ ) d
homomorphism
a cycle.
and
gl,g2
6 G,
the diagram
K
156
(9.21)
r = j
J(P(~K )) (gl,g2)
l*P(e K) P (gl 'g2 )
where
P(g1'g2 )
glg20
(that is, Proof.
in fact
is the geodesic
curve in
G/K
p(gl,g2) (s) = g1~0(s~01(g20)),
P(DK )
P([SK,SK])
considered = 0
as a form on
since
P
P(~K ) = d(P(SK))
G
from
to
s 6 [0,1]). is actually
is Kinvariant, on
g10
hence by
exact, (3.14)
G
and so (9.22)
P(~K ) = d(l*P(SK))
Now by
(9.8) the geodesic
on
2simplex
G/K. a(gl,g 2) : A 2 ~ G/K
is given
by
(9.23) where
d(gl,g 2) (t0,tl,t 2) = ht1+t2(glht2/(t1+t2) hs(X) = %0(s~01(x)) , x E G/K,
s 6 [0,1].
(g20))
Notice that
OF
vanishes
on the tangent fields along any curve of the form exp(sv), i s £ [0,1], and since I o o(gl,g2 ) o e , i = 1,2, is of this
form we conclude
from
(9.22)
J(P(~K)) (g1'g2)
that
= I A 2 d(g1'g2)*d(l*P(SK))
=I
AI
which is just NOW for =~(2n,]R)
(9.21). G = Sp(2n,IR) is contained
~(2n,]R) The Lie algebra ;(n)
(O(gl g2 ) 0 e0)*I*P(SK )
~
c_ GI(2n,]R), in
= {X = K = U(n)
C>jtc = C '
is the subspace tA = A}
157
with
complement
in
~
~(n)
~(2n,~)
:
= {A = ItA  A,
the v e c t o r s p a c e
(as in E x a m p l e
class
to
X = A + iC.
c I 6 H2(BU(n),~)
the l i n e a r
form
P £ I1(U(n)) I tr(X) =2z~
is i d e n t i f i e d w i t h
U : G/K ~ G l ( 2 n , ~ )
[23, p.
i : G/K ~ G
l(p)
Also
if
let
p
along
p = p(s), denote
tr(C)
I tr(JX) ' =  4~
G N P(2n,~)
= g
20]).
g,
X 6
~(n) .
v i a the m a p
g 6 G
Under
above
= p½,
t
this i d e n t i f i c a t i o n
the
is g i v e n by
p 6 G N P(2n,m)
s 6 [0,1],
is a c u r v e
the d e r i v a t i v e ,
i.e.
.
in
G fl P ( 2 n ; ~ )
the t a n g e n t v e c t o r
field
P. Notice t h a t the p r o j e c t i o n
= <J, (I + e x p ( 
= . z k (ad ~) (J) = zkJ,
Now it is easy to see that
hence
;
tr(jr1~)
=  = tr(J(1 + exp(Z))IQ) = tr(J(1 + p  1 )  I p ( 0 )  I p ( 0 ) ) .
Finally
let
t gl 0 = gl gl
p = p(s),
s 6 [0,1],
to
t t = glg2 g2 g1'
( t s t = gl g2 g2 ) g1'
p(s) Then
p(0)
glg20
= gl
log
(g2tg2)
be the g e o d e s i c that
s6
[0,1].
and we c o n c l u d e
= tr(Jg11 [1+tgl I (g2tg2)
s
= t r ( j [ t g l g I + (g2tg2)s] I t
9.12
in
I gl J = Jgl
Theorem
together
with
Remark.
It w o u l d
(9.16)
is b o u n d e d
Exercise
I.
and s e m i  l o c a l l y
group
X
: X ~ X Fcovering)
X
space
and let
Suppose
also
for
G
is a p r i n c i p a l
tg11
log
log
(g2tg2)tg I)
(g2tg2)
(g2tg2))
9.15 now c l e a r l y
follows
from Theorem
(9.25). to know
if the e x p r e s s i o n
n > I.
topological
z : X ~ X.
locally
pathconnected
space
so that it has a
F
be the f u n d a m e n t a l
Let
be any Lie group.
e : F ~ G
and that
log
be a c o n n e c t e d
lconnected
covering
of
and
 ]I t  I gl I gl
be i n t e r e s t i n g
Let
universal
a)
(9.21)
from
is,
tr(jTIT) (S) = tr(J[1+tg11 (g2tg2)Sg~ I]I
since
curve
is a h o m o m o r p h i s m . Fbundle
the a s s o c i a t e d
(therefore extension
Show called
that a principal
to a p r i n c i p a l
G
160
bundle
~
b) Show
: E
~ X
Suppose
that
F = {I]
every
Show
:
Gdbundle
of
~
2.
Let
X = X
is s i m p l y
is t r i v i a l .
(Hint:
is a c o v e r i n g
t h a t in g e n e r a l
the e x t e n s i o n ~
so t h a t
flat Gbundle
the c o r r e s p o n d i n g c)
is a f l a t G  b u n d l e .
every
: X ~ X
to
flat
G
Observe
space
of
Gbundle
relative
connected. that
X). on
to s o m e
X
is
homomorphism
r ~ G .
Exercise components group.
and
Let
For defines
a
K ~ G
: F ~ G
be
Let n
Exercise bundle
~ : E
la)
with
and
is j u s t b) subgroup
m
and
torsion
flat right
J~(~) ( E )
principal
Fcovering
: M
the a s s o c i a t e d
G/K.
xFG/K ~ M
Show
that
class
back e z
that
J~(~)
let
xg = g  l x
£ H~(MF,~)
lift
is the
~
the
to
for
~
: E
be
is r e p r e s e n t e d
Faction
F
space
is d i s c r e t e
the a s s o c i a t e d on
G/K
g 6 F).
to a
Show
in
A ~ ( M F)
by
~.
(Hint:
Observe
that
the unique
^
form
~
whose
the d i a g o n a l ~
: F \ (G/K
lift
to
G/K ~ G/K
x G/K)
G/K × G/K
~ MF).
is j u s t induces
in
of a d i s c r e t e
is the c o v e r i n g
~ MF
xF G/K,~)
M × G / K ~ G/K.
provided
x 6 G/K,
6 H~(M
x G/K
the c a s e
left
fibre
is r e p r e s e n t e d
inclusion
= MF
(see
an i s o m o r p h i s m
(E))
6 H~(M,~)
: G/K ~ F\G/K
(first change
~(J~(~)
the p r o j e c t i o n
: F ~ G
Gbundle
induces
form whose
under
Again
be
~
the p u l l  b a c k
the u n i q u e
free).
by
Gbundles.
be a d i f f e r e n t i a b l e
(this is a c t u a l l y
Gbundle action
for f l a t
6 H~(BGd,~),
~
that
of a m a n i f o l d
J~
let
suppose
such
subgroup.
flat
pulled
Now
a discrete
the c o r r e s p o n d i n g
of the c h a r a c t e r i s t i c by
from
be
and
x F G/K)
many
~ M
fibre
in c o h o m o l o g y
compact
the e l e m e n t
class
: M ~ M
finitely
be a homomorphism
a maximal
a characteristic
let
A~(M
be a Lie group with
~ 6 InVGA~(G/K),
a) and
let
G
a section
of
that
the b u n d l e
161
c) for
Again consider
P 6 Ii(K),
by the form connection
G, F
w(P) ( E )
P(~)^
G
for
K
as in b) and show that
£ H21(MF,~)
where
~K
given in step I.
direct proof by observing to
and
of the principal
is represented
is the curvature (Hint:
that
z
Kbundle
in
A21(M F)
form of the
Either use b) or give a : E
~ MF
is the extension
F \G ~ F\G/K).
In particular,
dim G/K = 2k,
(9.27)
r [MF]> = ]
<w(P) ( E ) ,
P(~),
P 6 Ik(K).
for all
MF d) where
Let £I
and
dimensional and
ze2
~I
: F1 ~ G F2
and
: E~2 ~ M2
MI
and
proportionality
be homomorphisms
groups of two M2
be the corresponding
There is a real constant (9.28)
: F2 ~ G
are the fundamental
compact m a n i f o l d s
the H i r z e b r u c h
~2
and let flat
: E~I Show
principle: c(~1,e 2)
such that
<w(P) (E i) ,[MI]> = c(~1,~2)<w(P) (E 2) ,[M2]>
Furthermore,
if
FI
and
F2
are discrete
and
induced from a left invariant metric on
G/K
i = 1,2, as in b) above then
Riemannian metrics (which exists since
where
~
Now cohsider
subgroups
are given the
has an inner product which
is invariant
K).
G = PSl(2,~)
on the Poincar~
of
c(~1,e 2) =
MF''I i = 1,2,
under the adjoint action by
by isometries
P £ Ik(K).
G
= vol(MF1)/vol(MF2)
e)
z~1
Gbundles.
for all
M i = MF., l
2k
= S i ( 2 , ~ ) / {±1}.
upper halfplane
H = {z = x + iy 6 C i y > 0} with Riemannian metric 12(dx ~ dx + dy ® dy). Y
G
acts
M I
162
The action is given by
z
(az + b)/(cz + d),
:
z 6
for ~) 6 S i ( 2 , ~ ) .
The isotropy s u b g r o u p at G/K
with
H.
i
is
,
and let where
so we identify
Here the Lie algebras are
=#(2,~)
Let
K = SO(2)/{±I}
&
be
the
P 6 II(K) v : SO(2)
= {\c

I a'b'c 6 ~ }
projection
:
X
be the p o l y n o m i a l such that
~ K
is the p r o j e c t i o n
and
v~P = Pf
Pf 6 II(so(2))
is
the Pfaffian. i)
Show that I
(9.29)
where
p..(~K) : ~ v
is the volume form on
u
H.
It is w e l l  k n o w n from n o n  E u c l i d e a n g e o m e t r y C.L.
Siegel
F ~ G
[27, Chapter 3])
F
that there exist d i s c r e t e subgroups
acting d i s c o n t i n u o u s l y on
surface of genus,
say
h.
triangle
G/K
F~H
a
In fact the fundamental d o m a i n of
AABC
is
~ L A  LB F~H
4h
sides.
 LC,
that the Euler
is
X (F\H) (Hint:
with q u o t i e n t
Check using the fact that the area of a n o n  E u c l i d e a n
c h a r a c t e r i s t i c of
of
H
is a n o n  E u c l i d e a n polygon w i t h ii)
(see e.g.
= 2(Ih) .
O b s e r v e first that the principal is the e x t e n s i o n to
SO(2)
S O ( 2 )  t a n g e n t bundle
of the p r i n c i p a l
Kbundle
163
G ~ G/K
relative
subspace
~
iii) iv) above
and
inverse flat
to the a d j o i n t
= ker(
the
of
a surface be the
Let
In this
as o b s t r u c t i o n
(9.17)
F\ H
Show
9.2 u s i n g
In g e n e r a l
F
the
<e(E
of C o r o l l a r y
and
(2,JR)).
: F~c~SI(2,~)
(9.30)
such
=c ~
bundle.
Exercise
representation
the
Ginvariant
in the
image
cochain
~64
i)
is a cocycle,
s(~)
s(~)
hence defines a class
6 Hq(BGd,~/~).
^
ii) choice of iii)
does not depend on the choice of q  f i l l i n g or
s(w) e
in the de Rham c o h o m o l o g y class.
Suppose
generator.
If
Hq(F,~)
~ ~
and that
B : Hq(BGd,~/~)
h o m o m o r p h i s m then
~(s(~))
Let
G = Gl(n,~).
is the B o c k s t e i n
is the o b s t r u c t i o n to
over the q + 1  s k e l e t o n of c)
represents a
~ Hq+I(BGd,~)
e x i s t e n c e of a section of the u n i v e r s a l F
w
the
G d  b U n d l e with fibre
BG d. For
G  b u n d l e the kth Chern class
YG : EG ~ BG
the u n i v e r s a l
ck 6 H2k(BG,~)
is the
o b s t r u c t i o n to the e x i s t e n c e of a section of the a s s o c i a t e d fibre bundle w i t h fibre 2k2connected
and
F = Gl(n,~)/Gl(k1,~).
H2kI(F,~)
c l o s e d complex valued
form
= ~.
In fact
F
is
Show that there is a
~k £ InvG(A2kI(F'C))
representing
the image of the g e n e r a t o r in the de Rham c o h o m o l o g y w i t h complex coefficients. f i c a t i o n of
(Hint: U(n)
Observe that
Gl(n,~)
is the c o m p l e x i 
and n o t i c e that any c o h o m o l o g y class of
H~(U(n)/U(kI),~)
can be r e p r e s e n t e d by a
real valued form).
C o n c l u d e that if
U(n)invariant
j : Gl(n,f) d ~ Gl(n,~)
is the natural map then
(9.32) where
Bj*c k = ~(~(Wk) ) S(~k ) 6 H 2 k  1 ( B G l ( n , ~ ) d , C / ~ )
particular
Bj*c k
C o r o l l a r y 9.2.
maps to zero in
(The classes
S(~k )
studied by J. Cheeger and J. Simons
is given by H2k(BGd,~)
(9.31).
In
w h i c h proves
have been introduced and (to appear)).
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R. Bott,
[6]
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 Heidelberg
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W. A. Benjamin,
G. Bredon,
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McGrawHill,
IVVI,
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 London,
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La t r a n s g r e s s i o n
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S. S. C h e r n
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fibr~ principal,
fibr@s),
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in: C o l l o q u e
5771,
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de t o p o l o g i e
George Thone,
Characteristic Ann.
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Dold,
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Berlin
Math.
 Heidelberg

1972. Simplicial
de Rham c o h o m o l o g y
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Topology
15
and c h a r a c t e r i s t i c
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233245.
166
[12]
A° Grothendieck, lin~aires sur
Classes
la c o h o m o l o g i e
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R. C. Gunning
S. Helgason,
Press,
G. Hochschild,
F. W. K a m b e r Notes
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 London
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Applied
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 Sydney,
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Homology,
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in Pure
Publ.,
Math.
 G~ttingen
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Tracts
Interscience
London
SpringerVerlag,
Bundles Notes
Foundations
(Interscience
15),
Berlin
BerlinHeidelbergNew
and K. Nomizu, III,
(Lecture
1968.
Tondeur,
S. K o b a y a s h i
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Classes,
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Spaces,
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 New York,
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 Amsterdam,
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215305,
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Cliffs,
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and Ph.
Geometry,
[18]
functions
Characteristic 493, [17]
Geometry
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1968.
Englewood
New York  London,
in Math.
Kamber
exp. VIII,
Amsterdam,
Analytic
The S t r u c t u r e
Heidelberg
in: Dix exposes
schemas,
Co.,
PrenticeHall,
San F r a n c i s c o [16]
des
Publ.
Differential
Academic [15]
discrets,
and H. Rossi,
variables, [14]
de Che~n et r e p r e s e n t a t i o n s
des groupes
and
New York
Wissensch.

114),
 Heidelberg,
1963. [19]
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J. W. M i l n o r
and J. Stasheff,
Annals
of Math.
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Princeton,
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63
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Strong
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On the e x i s t e n c e
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(1956),
Morse
Princeton
Characteristic 76, P r i n c e t o n
1974.
Construction
of Math. [21]
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of Math.
32
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of L o c a l l y
Studies
Princeton,
1973.
78),
51,
1963.
of a c o n n e c t i o n
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of Math.
Studies
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Symmetric
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215223. Spaces,
University
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[24]
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Classifying
Hautes [25]
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J. P. Serre,
A~g~bre
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H. Shulmann,
SemiSimple
On Characteristic
C. L. Siegel,
Berkeley,
Topics
[31]
N. Steenrod,
Series
Differential
F. W. Warner,
Interscience
(Princeton Press,
Princeton,
1973, pp.
of Tokyo Press,
Algebra
I,
3749, Tokyo,
ed. 1975.
(Grundlehren Math. Berlin  G~ttingen

1960.
Scott,
(1952), pp.
Series 1957.
25),
forms and the topology of mani
33), SpringerVerlag,
of Differentiable
Foresman
Sur les th~or~mes
H. Whitney,
II, Auto
I, Publish or Perish,
 Tokyo,
University
Foundations
Lie groups,
/LD
Geometry
in: Manifolds
Heidelberg,
[35]
University
Inte@rals, (Interscience
14), Princeton University
B. L. van der Waerden,
26
Thesis,
The Topology of Fibre Bundles,
D. Sullivan,
A. Weil,
W. A. Benjamin,
1970.
Wissensch.
[34]
105112.
1971.
Differential
A. Hattori,
[33]
and Abelian
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folds,
[32]
Classes,
New York,
Math. 1951.
pp.
Inst.
1972.
Publ., M. Spivak,
(1968),
Complexes,
Tracts
Boston, [30]
34
in Complex Function Theory
morphic Functions
E28]
Math.
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1966.
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spaces and spectral
and Co.,
de de Rham,
Manifolds
Glenview,
Comment.
and
1971.
Math.
Helv.
119145.
Geometric
Integration
21), Princeton
Theory,
University
(Princeton Math.
Press,
Princeton,
LIST
A* (S)
page
OF S Y M B O L S
22
A n
V,V x
58 13 72
3
page
A* (X)

91
Ak ' l (X)

91
AN(S)

36
EP,q I EG
A ~ (S ,~)

37
E
25,92
A* (M)

eA
16
A* (M,V)

44
ec
17
AP,q

15
e(m)
105
Ap ' q (X)

90
e(E) ,e(E,s)
I08
Ad

44
ad

BG

C,(M) , C*(M)

C* (U)

CP'q,c n

12
G1 (n, JR) +

19
G1 (n, (E)

23
G(~
134
Ck

67
Gd
144
CP,q

17
Cp ' q (X)

81
(Z*

99

99
c(E)

71,99
r (v)
57
Ck(E)

97
YG
72
x(M)

110
C t°p (X) C,(S),
cpn
d
'
C* top (x)
C*(S)
2
t
i
135
D
6,21
• 1
,n i
21
F(V)
38
8
F O (V)
43
9
G1 (n, JR)
38
71
1,22,44,91
107 67
44 ~
n, JR) (n,¢)
54 67 134
H k (A* (M))
4
H n (M) , H n (M)
8

8,23
H~ n

8,23
H
99
161
I
I
I
~
0
I
I
"0
~
I
0"~
I
~
I
Oh
I
~ ~
I
(,,fl
I
"~ ~
I
0 ~,
I
"~ H,
I
I~
I
~ ~v'
I
~
I
~D H
I
1~0
I
E
!
I
O~
~ ~
I
i
~
0 ~
k
k ~D
I
~
I
~
~.
I
'J

r~
Z
I
!
I~
~ ~,1
I
.~
I
~ ~
I
I
Ol
o
~
I
,~
I
~ ~(~
I
~
I
~_~
I
0"1
I
c~
I
I
LD
i>
I
~0
I
I
bJ
~
I ~
I
I
I
L~
I
I
~8
0"~
I
i
I ~
v
0

I
I
I
(D
SUBJECT
INDEX
page absolute adjoint
neighbourhood
AlexanderWhitney approximation
barycentric base
bundle 
the
diagonal
of principal
3 Gbundle
isomorphism
40
map
40
connection line

94 99
bundle
orientation complex

of
~P
I
C

equivalence

homotopy
support
9 I0 9
map
9
characteristic
class
63,71 68,97
classes

polynomials,
ChernWeil
68
Ck
homomorphism


classifying
space
closed
63 for
BG

94 71
differential
form
cochain 
102 8,19,23
n with

Chern
39 49


30
identity
canonical
chain
73
31
map
to
(ANR)
44
coordinates
space
Bianchi
rectract
representation
4 8
complex 
Cn
8,19,23
with
support
9
cocycle
condition
complex
(of m o d u l e s )
12
complex
ChernWeil
65

line

projective
40
homomorphism
bundle
99 space
99
171
page
complexification 
of
a vector
of
a Lie
104
bundle
134
group
38,46
connection 
in
continuous
a simplicial

78
natural
covariant 
94
Gbundle
functor
87
transformation
derivative
58
differential
58 20,30
cupproduct curvature
degeneracy de
Rham 
49,94
form
operator
21
qi
4
cohomology
2
complex

11
's t h e o r e m for a simplicial set (= W h i t n e y ' s theorem) for
a simplicial
manifold
differentiable differential
simplicial
map
89
a chain
complex
12
in

's i n
differential
a double
12
complex
I
form


on
a simplicial


on
a simplicial

with
values
in
manifold
91
set
22
a vector space 43 52
distribution
12
complex


double
associated
simplicial
to
elementary equivariant
form
exact
25
~I
differential
form
48 39 05,108
class
EulerPoincar~
15,17
14
map

a covering
83
set
edgehomomorphism
Euler
92 36
derivation
double
23
characteristic
differential
form
10 4
172
page
excision extension exterior 
face
of
1,22,44,91
product,^
1,22,44,91
ci
6
operator
fat
42
a Gbundle
differential
map

9
property
7,21
E. 1
75
realization
fibre
42
bundle

of p r i n c i p a l
39
Gbundle
filling
146,163
flat
144
bundle

47,51
connection
52
foliation frame free
bundle
38
Gaction
72
fundamental
GaussBonnet geodesic
geometric
group
11
convex
75
commutative
I
cohomology
homotopy

150
realization
Hirzebruch
145
proportionality of C ~
principle
maps

of

property
Hopf
112
formula
simplex
geodesically
graded
111
class
9
simplicial
maps
99
's f o r m u l a

induced 
109
differential tangent
bundle
(=
form
vectors
"pullback")
differential
form
integration 
35 9
bundle
horizontal
161
48 38,46
41 2 6,112
along
a manifold
22
173 page
integration
10,23,92
map,/

operators,
invariant
h
differential

(i)
form
135
Jordandecomposition
local
56
connection
index

of v e c t o r
field
connection
maximal
torus
natural
109 40,42
trivialization
MaurerCartan
47 115
transformation
10
nerve
77

a covering,
79
of
nilpotent normal
NX U
135
element
35
cochain

neighbourhood
11

simplicial
36
oriented
vector
orthonormal
kform
bundle
frame
bundle
108 43
parallel
translation
38
Pfaffian
polynomial
66
Poincar~'s
lemma

upper
4 halfplane
Pontrjagin positive principal 
161 62
polarization polynomial
7,24
62
polynomial
LeviCivita
4,
48
form
37
function
62
classes polynomials root Gbundle Fcovering
66,103 66 137 39 159
174
page rational
differential
37
form
106
realification reduction regular
of
42
a Gbundle
128
element
relative
Euler
108
class
137
root root
space
semisimple simplicial
126 ,136
decomposition
135
element chain
23
complex

cochain
complex
23

de
complex
20 ,22,91

form
22 ,91

Gbundle
93
Rham

homotopy
35 ,84

manifold
89
map
21
set
21
space
75


singular
boundary
8
operator
8,19

chain

coboundary
operator
8

cochain
8,19

cohomology
8
element homology
8
simplex
7,19

skewhermitian standard
algebra
68
matrix
3
simplex
starshaped Stoke's
in a L i e
42

4
set
6
theorem
strongly structural symmetric 
symplectic
free
72
Gaction
49
equation
69
algebra multilinear power group
function
61 69 52
175
page tensor Thom
68
algebra
I
property
108
class
topological
category
topological
principal
77 Gbundle
55
torsionform
117
torus total
Chern

complex Pontrjagin

space
transition trivial
vertical
12
of principal functions
duality sum
Gbundle
39 40 40
bundle
tangent
104
class
vectors
45 115
group
Whitney 
99
class

Weyl
71
formula
100 , I 0 5 , 1 1 0 98