Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
640 Johan L. Dupont
Curvature and Characteristic Classes
Springer-Verlag Berlin Heidelberg New York 1978
Author Johan L. Dupont Matematisk Institut Ny Munkegade DK-8000 Aarhus C/Denmark
AMS Subject Classifications (1970): 53C05, 55F40, 57D20, 58A10, 55J10 ISBN 3-540-08663-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-08663-3 Springer-Verlag 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, re-use 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 Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
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
Chern-Weil
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
<
knowledge
is the G a u s s i a n
of
elementary
a compact
manifold,
d i f f e r e n t i a l form
Pontrjagin
forms,
M.
I ~K
see chapter 4 e x a m p l e s and the i n t e g r a t i o n
chain
in
In this way
class
(e.g.
the Euler
which
turns
out to be a d i f f e r e n t i a l
that it depends
considered
there
class
Thus a r e p e a t i n g quantities
M
is by a
or one of the
a singular
or one of the P o n t r j a g i n
vector
is a
is r e p l a c e d
is done over
is d e f i n e d
~ priori
global
[ ~ JM where
I and 3) a s s o c i a t e d
topological
a singular cohomology classes)
invariant
bundle
to
of
in
M
bundle.
theme of this theory
which
are a c t u a l l y
(I)
only on the t a n g e n t
as a t o p o l o g i c a l
states
in 3-space
dimensions
the P f a f f i a n
tensor
the sense
g
in
the c u r v a t u r e M.
theorem which
In p a r t i c u l a r
In h i g h e r
(e.g.
to h i g h e r
= 2(I-g)
curvature.
of
Riemannian
generalization
of genus
IM K
topologicalinvar±ant
geo m e t r y
classes
t o g e t h e r with
Gauss-Bonnet
surface
I 2--~
certain
an i n t r o d u c t i o n
of c h a r a c t e r i s t i c
is the p r o p e r
(1)
closed
to give
theory.
dimensions
where
was
presupposihg
manifolds
Chern-Weil
that
of Aarhus,
g i v e n at the
1976-77. The p u r p o s e
with
University
of lectures
is to show that
d e p e n d on the local
topological
invariants.
differential Fundamental
IV
in this c o n t e x t is of c o u r s e the de R h a m t h e o r e m w h i c h says t h a t e v e r y real c o h o m o l o g y
class of a m a n i f o l d
p r e s e n t e d by i n t e g r a t i n g a c l o s e d and on the o t h e r h a n d singular chains exact.
I we give an e l e m e n t a r y p r o o f of this t h e o r e m
tools u s e d s e v e r a l
times
[34]) w h i c h d e p e n d s
t h r o u g h the lectures:
g r a t i o n o p e r a t o r of the P o i n c a r ~ (iii)
form over
the z e r o - c o c y c l e t h e n the f o r m is
( e s s e n t i a l l y due to A. W e i l
covering,
can be re-
form over singular chains
if i n t e g r a t i o n of a c l o s e d
represents
In c h a p t e r
M
the c o m p a r i s o n
(I have d e l i b e r a t e l y
lemma,
(ii)
on 3 b a s i c (i) the inte-
the n e r v e of a
t h e o r e m for d o u b l e c o m p l e x e s
a v o i d e d all
m e n t i o n i n g of s p e c t r a l
sequences).
In c h a p t e r 2 we show that the de R h a m i s o m o r p h i s m r e s p e c t s products
and for the p r o o f we use the o p p o r t u n i t y
another basic differential
tool:
(iv)
differential topology,
sets.
as we call
it,
the c a l c u l u s o f
forms to the c o m b i n a t o r i a l m e t h o d s
its a p p l i c a b i l i t y
occuring
Chapter and c u r v a t u r e
3
of t h e s e
of a l g e b r a i c lectures
geometry.
an a c c o u n t of the t h e o r y of c o n n e c t i o n
in a p r i n c i p a l
G-bundle
(G
a Lie-group)
ly f o l l o w i n g the e x p o s i t i o n of K o b a y a s h i and N o m i z u c h a p t e r ends w i t h explaining
some r a t h e r
theory of an a f f i n e c o n n e c t i o n
struction
long e x e r c i s e s
the r e l a t i o n of the g e n e r a l
Eventually,
in c h a p t e r
@
[17]. T h e 7 and 8)
theory to the c l a s s i c a l
4 we get to the C h e r n - W e i l
and c u r v a t u r e
manifold mentioned above
(nos.
essential-
in a R i e m a n n i a n m a n i f o l d .
in the c a s e of a p r i n c i p a l
a connection
is to
in the theory of c h a r a c t e r i s t i c
in d i f f e r e n t i a l contains
t h e o r y of
The r e s u l t i n g s i m p l ~
connects
and one of the m a i n p u r p o s e s
demonstrate classes
the W h i t n e y - T h o m - S u l l i v a n
forms on s i m p l i c i a l
cial de R h a m complex,
to i n t r o d u c e
~
G = O(n)
G-bundle
~: E ~ M
conwith
(in the case of a R i e m a n n i a n and
E
is the b u n d l e of
V
orthonormal associated
tangent
~
invariant
proving
principal
notion
ment behaves
istic
classes
classifyin~
WE(P)
Chern-Weil
in order
construction
to define
p(~k)
on
on
M
a topological in chapter
5
for t o p o l o g i c a l
an a s s i g n m e n t such that
to b u n d l e
maps.
of a c o h o m o l o g y the assignThe m a i n
that the ring of c h a r a c t e r ring of the
the c h a r a c t e r i s t i c it suffices
for the u n i v e r s a l
of a s i m p l i c i a l
a simplicial
set w h e r e
Therefore
de R h a m complex
BG
G-bundle
class
to make EG
manifold,
that
in chapter
manifolds,
construction
carries
In this way we get a u n i v e r s a l
it
is, roughly
the set of p - s i m p l i c e s
we g e n e r a l i z e
the
over
is not a m a n i f o l d
to s i m p l i c i a l
turns out that the C h e r n - W e i l bundle.
class
G-bundle
is that a l t h o u g h
a manifold.
universal
we discuss
to the c o h o m o l o g y
any t o p o l o g i c a l
is the r e a l i z a t i o n
simplicial
states
P
BG.
Now the p o i n t
speaking,
form
G-bundle
respect
is
polynomial
is a c t u a l l y
G-bundle
with
there
WE(P) EH2k(M,]R).
this class
is i s o m o r p h i c
E
class
space of every
space
for
stitute
differential
of the chapter
Therefore,
homogeneous
By this we mean
naturally
(5.5)
situation
of a c h a r a c t e r i s t i c
G-bundles.
in the base
theorem
that
of the p r i n c i p a l
the g e n e r a l
BG.
a closed
in turn a c o h o m o l o g y
Before
class
In this
to every G - i n v a r i a n t
the Lie a l g e b r a defining
frames).
con6 the
and it over
to the
Chern-Weil
homomorphism w: I*(G)
where
I~(G)
denotes
~ H*(BG,IR)
the ring of G - i n v a r i a n t
polynomials
on
the Lie algebra In chapter groups
obtaining
7 we s p e c i a l i z e in this way
the c o n s t r u c t i o n
the C h e r n
to the c l a s s i c a l
and P o n t r j a g i n g
classes
VJ
w i t h real c o e f f i c i e n t s .
We also c o n s i d e r the E u l e r class de-
fined by the P f a f f i a n p o l y n o m i a l the G a u s s - B o n n e t Chapter
formula
for
G
that
to the p r o o f of the t h e o r e m
w:
I~(G)
a c o m p a c t Lie group.
t h e o r e m that of
in all e v e n d i m e n s i o n s .
8 is d e v o t e d
due to H. C a r t a n
H~(BG,~)
H~(BT,~)
is an i s o m o r p h i s m
is i s o m o r p h i c group
to the i n v a r i a n t p a r t
W
of a m a x i m a l
torus
T.
r e s u l t for the ring of i n v a r i a n t p o l y n o m i a l s
(due to C. Chevalley) is r a t h e r
~ H~(BG,]R)
(8.1)
A t the s a m e time we p r o v e A . B o r e l ' s
u n d e r the W e y l
The c o r r e s p o n d i n g
and in an e x e r c i s e we s h o w
d e p e n d s on some Lie g r o u p
theory which
far f r o m the m a i n topic of these notes,
t h e r e f o r e p l a c e d the p r o o f
and I h a v e
in an a p p e n d i x at the end of the
chapter. T h e final characteristic
c h a p t e r 9 deals classes
equivalently with compact
with
the s p e c i a l p r o p e r t i e s of
for G - b u n d l e s w i t h a flat c o n n e c t i o n or
constant transition
it f o l l o w s
functions.
from the above m e n t i o n e d
If
G
is
t h e o r e m 8.1 t h a t
e v e r y c h a r a c t e r i s t i c class w i t h real c o e f f i c i e n t s
is in the
image of the C h e r n - W e i l h o m o m o r p h i s m and t h e r e f o r e m u s t vanish. In g e n e r a l
for
K ~ G
a m a x i m a l c o m p a c t s u b g r o u p we d e r i v e
f o r m u l a for the c h a r a c t e r i s t i c c l a s s e s over certain singular
s i m p l i c e s of
p r o v e the t h e o r e m of J. M i l n o r flat
Sl(2,~)-bundle
v a l u e less t h a n
[20]
involving
G/K.
integration
As an a p p l i c a t i o n we
t h a t the E u l e r n u m b e r of a
on a s u r f a c e of genus
h
has n u m e r i c a l
h.
I h a v e tried to m a k e the notes as s e l f c o n t a i n e d giving otherwise proper references S i n c e our s u b j e c t and e s p e c i a l l y
a
is c l a s s i c a l ,
to w e l l - k n o w n
as p o s s i b l e
text-books.
the l i t e r a t u r e is q u i t e
in r e c e n t y e a r s has g r o w n rapidly,
no a t t e m p t to m a k e the b i b l i o g r a p h y c o m p l e t e .
large,
so I h a v e m a d e
Vll
It should be noted the m a i n
text and also
exercise. weekly
that m a n y of the e x e r c i s e s some details
In the course
exercise
session
from w h i c h played
in this
Johanne
Poul
Lund C h r i s t i a n s e n ,
Lune N i e l s e n Finally
for their v a l u a b l e I would
Aarhus,
December
notes
course,
Klausen,
1977.
the notes
the
I am g r a t e f u l
especially
to
Laitinen
and S # r e n
and s u g g e s t i o n s .
Albrecht
for a careful
Dold
in this
in
left as an
derived
role.
Erkki
criticism
and prof.
for i n c l u d i n g
15,
these
like to thank Lissi D a b e r
typing of the m a n u s c r i p t Springer-Verlag
in the text are
an e s s e n t i a l
to the active p a r t i c i p a n t s
are used
and the series.
CONTENTS
page
Chapter
I.
Differential
2.
Multiplicativity.
3.
Connections
4.
The C h e r n - W e i l
5.
Topological
bundles
6.
Simplicial
manifolds.
for
forms
I
and c o h o m o l o g y The
simplicial
in p r i n c i p a l
de Rham c o m p l e x
20 38
bundles
61
homomorphism and c l a s s i f y i n g
71
spaces
The C h e r n - W e i l
homomorphism 89
BG
7.
Characteristic
classes
for some classical
8.
The C h e r n - W e i l
homomorphism
9.
Applications
to flat b u n d l e s
for c o m p a c t
groups
groups
97 114 144
References
165
List of symbols
168
Subject
170
index
CURVATURE AND C H A R A C T E R I S T I C
i.
Differential
forms and co homology
First let us recall the basic
facts of the calculus
differential
forms on a d i f f e r e n t i a b l e
differential
form
vector
fields
~ ( X l , . . . , X k)
~
of degree
XI,...,X k
k
manifold
associates
a real valued
C
M. to
depends only on
and such that it is m u l t i l i n e a r For an 1-form
~I
and a k-form
k
C~
function
X1p, .... Xkp
(i.e.
for all
and alternatin~ ~2
of
A
such that it has the "tensor property"
~(XI, .... Xk) p
the
CLASSES
in
the product
p 6 M) XI,...,X k-
~I ^ ~2
is
(k+l)-form defined by
ml ^ ~2(XI ' .... Xk+l)
=
I (k+l) ~ o s i g n ( ~ ) ~ 1 (Xd(1) ..... X~(1))'~2(Xo(I+I) ..... Xq(l+k))
= where
o
product
runs through all permutations is associative
of
1,...,k+l.
and graded commutative,
This
i.e.
~I ^ ~2 = (-I)ki~2 ^ ~I" Furthermore k-form
~
there is an exterior differential associates
a
d IIXl' ... 'xk+11 +
(k+1)-form
=
1 rk+l
(-I)
d~ i+Ix
d
i
(m(X1
'Xi' ........
properties:
fields,
d
Xk+1
))
'Xk+1 )]
where the "hat" means that the term is left out. of the vector
to any
defined by
[ (-1)i+J~([Xi'Xj]'Xl '''" 'Xi ' . . . . 'Xj' i<j ..
is the L i e - b r a c k e t
which
Here
[Xi,X j]
has the following
(i) (ii)
d
is
dd
=
linear 0
d(mlAm 2
(iii)
For
(iv)
a
= (dm 1) A m2 + ( - 1 ) k m l C
(df) (X)
(v)
d
is
~IU
In
a
local
unique
over
function
f
and
A dm 2
X
for
a vector
local,
that
d~!U
coordinate
is,
for
any
open
= 0.
system
~ I~ii
a. ll.--i k Suppose F a k-form
M
such
where
F~
( u , u l , . . . , u n)
any
k-form
~
has
refer
is
on
N. for
is
with
set
Rham
of
any
the
of
U.
M
that
Then
naturally understand
there
is
vector
A...A
at
du
U.
m a D~
of
C
manifolds
a unique fields
differential
k-forms
an
a manifold.
suppose
C ~
a
k
on
ik
^ du
induced
of
F
let
k-form
XI,...,X k
F.
and
on
F
M
Vq
6 M,
preserves
A
d.
al~ebra)
U ~
and
is Then
(A~(M),A,d)
For
functions
: M ~ N
= dF
to
(or d e
C~
i2
a . . du 11 .... ,1 k
(~) (X I .... ,Xk) q = W F ( q ) ( F ~ X l q .... , F ~ X k q ) ,
commutes The
are
that
F
U
U,
presentation
be
and
field
set
iI
on
a k-form
= X(f)
= 0 ~
=
where
ml
on or
of
every
just
denoted
A~(M)
subset
suppose point
any
point
identified
with
a collection
is
Ak(M)
as
the
de
and Rham
we
shall
complex
M.
open Now
M
A~(U) U ~
of
U
q 6 U T ~
q
q
of
M
is
is
a
the
(M) .
is
By
clearly a closed
limit
tangent a k-form
k-linear
defined
of
since
subset
of
interior
space on
alternating
points
Tq(U) U
we
M
is
shall
forms
on
a
3
Tq(M), M U
which
(it is e n o u g h by a "bump
the U
q 6 U,
that
is d e t e r m i n e d
have
d
on
by
U.
Example consider
I.
A n,
vectors
e.
Notice
~ Ak+1(U)
that
of the
The
f o r m o n all o f
neighbourhood
let
Ak(u)
interior
is w e l l - d e f i n e d
following
standard
This
form on
of
and we
observation
U. again is
example:
n-simplex
hull
of
denote
a differential
to the
(A~(U),A,d).
the c o n v e x
=
Again
its r e s t r i c t i o n
: Ak(u)
because
to a n o p e n
argument).
a de Rham complex
important
to a d i f f e r e n t i a l
it e x t e n d s
function"
s e t of k - f o r m s
Therefore
extends
of t h e
(0,0,...,I,0,...,0)
A n.
In
~n+1
s e t of c a n o n i c a l
with
1
on the
basis
i-th place,
l
i = 0,1,...,n.
That
A n = {t =
is
(t o ..... tn) Iti 2
0, i =
0 ..... n,
[jtj
= I}
t 2 ~ t1
~
0 Vn = {t 6 ~ n + l l ~ j t j = 1}
Thehyperplane and
An ~
Vn
V n.
So it m a k e s
barycentric have on
their Vn
is c l e a r l y sense
the
differentials An )
of its
to t a l k a b o u t
coordinates
(or
closure
i =
is e x p r e s s i b l e
interior
Ak(An).
(t o ..... t n) d t i,
is c l e a r l y a manifold Considering
as f u n c t i o n s
0,...,n,
in the
form
dt. ik
where
points
and
on
every
in the
Vn
we
k-form
co
ai 0~i0<...
on
.i k d t i 0""
Vn
tn = I
(or
An).
implies
{ d t l , . . . , d t n}
^...^ 0
Notice
dt 0 + . . . +
generates
that dt n =
A~(An).
a. 10'''ik
are
the r e l a t i o n 0,
so a c t u a l l y
the
C
Now
return
manifold d~
and dd
e
the
= 0
real
A
is
M
or
closed
on
U
e = de' form
The
k-th
is
de
subset
is
for
of
called
some
a
C
closed
if
(k-1)-form
~'.
closed.
Rham
cohomology
group
of
U
vectorspace
= ker(d
: Ak(u)
k = 0 , 1 , 2 ....
Example is
open e
if
exact
1.1.
Hk(A~(U))
1-form
an
k-form
exact
every
Definition is
U ~
as b e f o r e .
= 0
Since
to
of
2.
For
the
form
~
Ak+I(u))/dAk-I(u)
(A-I(u)
M = ~2 e =
= 0).
with
fdx
coordinates
+ gdy
and
de
(x,y) = 0
is
any just
the
requirement ~f _ by Now
take
U = ~2~{0}
It
is
easily
seen
and I
-
~g ~x "
x2+y2 that
consider
(xdy-ydx)
e
is
the
l-form
"
closed
but
S
~ =
2~
so
~
is
SI not
exact.
It the
Hence
is
classically of
M.
respect
to
e 6 U,
segment
from
Lemma with
1.2.
respect : Ak(u)
(1.3)
e
x
example that is
(Poincar~'s
to ~
FOr
to
e 6 U.
Ak-I(u),
hk+1(de)
% 0.
wellknown
geometry
with
hk
HI(A~(U))
= S ]-e
let
is,
U ~ ~n
for
all in
lemma).
Let
there
1,2 .... ,
- dhk(e)'
[e(e)
H~(A~(M))
contained
Then
k =
that
- e,
are such
be
x 6 U U.
Then
U ~ ~n
is
related
to
star-shaped the
whole
line
we
have:
be
star-shaped
operators that
k ~ 0 k = 0.
for
any
e 6 Ak(u),
In p a r t i c u l a r Hk,A,,U~ ~ ~ ,,
(1.4)
The g
Clearly
operators
: [0,1]
O,
k
any
hk
follows
are
from
defined
(1.3).
as follows:
be the m a p
= se +
~ £ Ak(u),
> 0
k = 0.
(1.4)
× U ~ U
g(s,x)
For
j
I IR, Proof.
let
=
(1-s)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
(k-1)-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
n-simplex
determined
n-form
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
n-form
by
f = j
r
[jtj
An ~
c
~n
5)
ei
2.
: A n-1
e
(1.6)
the
Rham
integrate
about
n-forms
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 n-1 (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
star-shaped
1,2 .... , for
any
(n-l)
(equation
us
topological
In c ~n, Then
deduce
by
i = 0,1,...,n.
the
let
C~
clearly
(-1)nh
show
cube
8-18]~
given
corresponding
that
[ m = j An
(Hint:
is
k =
Show
p.
: In ~ A n
ei,
Ak-1(An),
i = 0,...,n.
[29,
the
,s 1 . . . s n _ I (1-s n) ,s I .... s n) .)
vertices
have
(1.7)
g
for
( ( 1 - s 1 ) , s I (1-s 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
n-form
(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
i-th
ei
face.
n-simplex
in
standard
n-
singular
n-
: A n-1 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
n-th
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)
n-th
n-cochains
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 <
(M)
n
8~ =
an n - c o c h a i n
equivalently
if
with
operator
Cn(M)
we
0 ei
n-chains
sums
clearly
that
group
Hn(M)
and
C
the boundary
implies
ej-1
singular
e i , i = 0 ..... n, we
=
C~(N)
contravariant
is a S(f)
= k e r (6 : C n ( M )
C~
map
: S~(M)
extends ~ C~(M). functors
to
of
Ca
~ C n+1 ( M ) ) / 6 C n-1 (M) .
manifolds
~ S.(N)
defined
f~ : C . ( M )
~ C~(N)
Obviously
C~
respectively.
and Also
we
by
S(f) (0)
and C~
an
induces
covariant and
get
= f o ~.
dually
are f~
clearly
~
and are
chain-maps,
i.e.
f#
Therefore
Let
(1.11)
we
have
f~
: H~(M)
us
(1 .10)
o ~ =
~ o f~
induced
recall
6 o f%#
= f%#
o 6.
maps
~ H~(N),
f~
the
following
: H~(N)
~ H~(M).
wellknown
facts:
Hi(Pt)
=
0,
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
= si-1
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
n-simplices
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
(1-s)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
non-empty
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
bi-graded
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~ c---C ~, 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
q-1
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)
"edge-homomorphism").
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 ~ Ap-1'q
Sp+ I o 6 + 6 o Sp
this
just
~e ~ Us,
choose
Ve
6 [,
a partition and
(Sp~) (~0 ..... ~p-1 ) =
id.
of u n i t y
to v e r i f y
that
with
define
(-1)P [ ~e~(~O
'~)
~6~
'''''~p-1
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
n-simplices
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))
cup-product
® 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 : ~q-1
(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
k-form
~
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 p-I
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
i-th
we get a k-form
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 k-cochain
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)
~ Ak-1(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 k-1 (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)
(j-l) (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,
p-simplex
As
: Ak (A p) ~ A k-1 (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
a k-l-form that
i.
on
For
o...o
the o p e r a t o r
= k~ IIl=k [ ~I
show
contain
we
s
k + I
zero)
~i ~
associated
Ap
^ .. .^ dt
dt i
and
as
= e
Also
on
a k-cochain
Sk(~) ° = is c l e a r l y
^...^
~I
I below) :
as f o l l o w s :
(2.25)
form"
~ : ck(s)
a k-form
interpreted
is d e f i n e d
Let
(c)
k + 1 = p.
h(i0)
by
~
"'" > Jl ~ 0
in E x e r c i s e
E(c)q
which
which
o...0
now d e f i n e
(2.24)
and
= 0) and
the d e g r e e
c =
I
the k-
andi similarlY31 w e
P ~ Jl >
where
of
Corresponding onto
{ei0 .... ,eik}
"elementary
is g i v e n
For
: A k ~ Ap
ejl to
"di1
I@I = -I).
I
p h 0.
integers
The
Explicitly,
h I = h(ik) which
put
by
of
i k ~ p.
k [ (-I )st i dti0 s=0 s
put
integer
: Sp ~ S k.
sequence
there
(2.23)
I = ~
spanned
a face m a p
complementary
<...<
inclusion
have
a fixed
be a s e q u e n c e
0 ~ i0 < i I
is
to
Consider
(~)
sk
the :
6 Ak(s)
II{~ I ^ hi(~
Ak
sum
(S) ~
and
is of A k-1
~ 6 Sp
(S) put
)
A p. satisfies I =
Definition
( i 0 , . . . , i k)
we have
2.8
(ii) :
does
not
is < 1 < ±s+1"
and
26 we put
I' = ( i 0 , . . . , i s , i s + 1 - 1 , . . . , i k - 1 )
(£i)~
[(c)o = k~
: k~ since
it is easy
I' =
[ ~I' KIl=k;l{I [ ~I' JI'[=k
to see that
(10,...,ik)
runs over
0 :< i~ < ... < i{ =< p - I, [(c)
w h i c h was
to s a t i s f y
2.8
N o w let us p r o v e identity
of
(2.17)
us c o n c e n t r a t e 6 S
d[(c)o
On the other (2.27)
= k~
= pI, (elO).
Similarly
(i) u s i n g
(2.21)
the i d e n t i t i e s exactly one:
Now s i n c e
satisfying above
(2.25)
equals
is s h o w n
above.
(2.17):
The first
as L e m m a
1.14,
For
c E ck(s)
so let and
dtio
A...A dtik
(k+l) ' [ Ii l=k+1
~I
(6c)
= (k+1) ' ~ Ill=k+1
mI
=
(j0,...,jk) , 0 ~ J0 <'''<
involving
Cpj (~)
in
(2.28)
(k+1) !
cpj(o)
(2.27)
=
in
(2.27).
(J0 ..... Jk )"
therefore
~I (0) k+1 ( [ (-I)ic i:0 elU I (0)) " Jk ~ p
we shall
Now
el~ I = ~j
The c o e f f i c i e n t
find the iff of
is
k+1 ~ (-I) 1 ~ (-1)st. at. ^ . . . ^ a t ^...Adt. il[(j 0 .... ,Jk ) s=0 i s 10 is lk+ I
(10 .... 'ik+1) NOW
• cpi(O ) •
hand
[(6c)c
J =
k ( [ (-1)Sdt. A d t i 0 A . . . A d [ i A-..Adtik)C~i(o ) s=0 is s
[ IIl=k
(k+1) ~ IIl:k[
(i0 ..... il ..... ik+1)
where
c~ I (SlO)
the last e x p r e s s i o n
is p r o v e d
this n o t a t i o n
cpi(o)
all s e q u e n c e s
on the second
=
terms
With
we have
P
(2.26)
For
~i(o)
to be proved.
Definition
.
(2.28)
=
equals
(J0 ..... J l - 1 ' i ' 3 1 ..... Jk ) "
27
(k+1) ~
~ i~(Jo,...,jk)
[ ~ (-1)s+it. dt~ h...hdt, h . . . h d t ^dt.hdt. ^.. s
• •. h d t j k + t i d t jO h" • .hdt3k
[-1)s+l-lt. S~I
= (k+1) ~
+
=
dt 3S
+
^...hdt. 30
31_1
^dt.hdt. h...hdt, h...hdt. ] i 31 3S 3k
~ [tidt. h...hdt. + i((j 0 ..... Jk ) 30 3k
k ~ - t dt. h . . . h d t . ^dt.hdt ^...hdt. ] s=O 3S 30 3S_I l 3S+I 3k
(k+1) ~[
[ t.dt. ^...hdt. + i~(j 0 ..... jk ) 1 30 3k
k + [ ~ -t. dt~ h...hat. ^dt.hdt. ^.,. hdt. 3k s=O i~(J0,...,jk) 3s 30 3s_i l 3s+i = (k+1) ~[
~ t.dt. h...hdt. + i~(j 0 ..... jk ) i 30 3k
k k p + [ t. dt h...hdt. - [ t. dt. ^...hdt. h (I dti)Adt" ^--hdtjk] s=O 3S Jo 3k s=O 3S 30 3S-I i=O 3S+I P = (k+1) I [ t.dt. h...hdt. = i=0 ~ 30 3k P [ dt. = 0 i=O 1
since
and
(k+1) ~dt. ^...^dt. 30 3k
P ~ t i = 1. i=O
Hence
| . h...hdt3k. E(6c) ~ = (k+1) '. [j~=kdt30 by
(2.26)
which
To prove C = (Co), (2.24)
proves
the second
the first
equation
O 6 Sk
E(c)
and we shall
is the k - f o r m
on
" c~j(O)
identity of
show Ak
(2.18) that
of
= dE(c) (2.17).
consider
a k-cochain
l(E(c)) a = ca"
given by
By
28 k £(c)o = k'.c j~0 (-1)Jtjdt0A...Ad£jA...Adt k
= k%co[t0dtiA...Adt k + k k + j=1[(-I) 3tj (-s=1 [ dts)^dtIA'''^dt'A3 "''^dtk]
= k'.c [t0dtiA...Adtk+
k [ (-I) J-ltjdtjAdtiA...Adt ^ . . . A d t k] j=1 3
= k~codtiA...^dt k. Therefore
I(E(c))o : k~co~AkdtIA...Adt k = c o by Exercise
I of Chapter
I.
For the proof of the second equation of observe that an iterated application of
(2.18) first
(2.20) yields the
following Lemma 2.29.
Let
~ 6 Ak(AP) , k > 0,
I = (i0,...,ir),
0 < r < p,
Suppose
Then
k > r.
with
and consider
0 < i 0 <...< i r __< p.
f[j=O (1)Jh(io ..... ~j ..... ir) (~)-(-1)rdhi(~)
, k > r
h I (de) = k - [ (-1)Jh j=0 (i0 ..... fj ..... ik) (~)+(-1)kh(i0 ..... ik_ I) (~) (eik), k = r. Now let
~ 6 Ak(s)
and
~ 6 Sp.
there is nothing to prove).
Assume
By (2.29)
p ~ k
(otherwise
29 (2.30)
Sk+l(d~) ~ =
= ii1=k [ k~ I
[ 1 I I ~ I ^ hi(d~ ~) 0~lli~k ((-I) kh(i0,-..,ik_ . 1 ) (~) (eik)) Ill
-
[
i I i ~ I A ( ~ (-1)Jh
0~Iil~k -
[
j=0
ij (i0 . . . . . . . . . . iIIl) (~g))
IIIL~ I ^ ((-I) I Ildhi(~e)).
0~IIl
IIl~d~ I ^ hi(~o)+(-1)IIi II1~I^dhi(~c).
d(Sk~) ~ = 0~III
By (2.22) (-1)kh (i 0 ..... ik_1 ) (&oq) (eik) = IAk(~I)* &o = I Akq0 i(~)
= ~(~)~i(~ ) •
Therefore adding (2.30) and (2.31) we obtain (2.32)
Sk+1(d~) ~ + d(Sk~) a = 111 [ IIl[~i^( [ (-1)Jh ~j = E(I(~))o-~o-0
'ilIi) (~))
[ II1~d~ I ^ h i ( ~ ) . 0~III
However the last two sums in (2.32) cancel by exactly the same calculations as in the proof that (2.26) equals
(2.27) above.
This proves the second equation of (2.18) and ends the proof of Theorem 2.16. We now return to the proof of Theorem 2.4. in the commutative diagram
Notice that
30
A*(M)
i
~ A, (S~(M))
C*(M) all maps
induce
i : A*(M)
~ A*(S~(M))
2.4 t h e r e f o r e Theorem diagram
isomorphism
follows
Also
multiplicative.
Theorem
from
For any s i m p l i c i a l
set
S
the f o l l o w i n g
commutes H(A*(S))
® H(A*(S)) +I
H(C*(S))
where
is o b v i o u s l y
immidiately 2.33.
in homology.
the upper
of s i m p l i c i a l
@
®
~
H(A*(S))
I
+I
H(C*(S))
horizontal
forms
^
, H(C*(S))
map
is induced
by the w e d g e - p r o d u c t
and the lower h o r i z o n t a l
map
is the cup-
product.
Before
proving
of the c u p - p r o d u c t Consider sets
this in
C,
is a natural
equal
to
of simplicial
and c h a i n maps
~) .
(as
An approximation
to
transformation
: C,(S)
(in p a r t i c u l a r
the d e f i n i t i o n
from the c a t e g o r y
of c h a i n - c o m p l e x e s
take c o e f f i c i e n t s
the d i a g o n a l
let us recall
H(C*(S)).
the functor
to the c a t e g o r y
usual we
theorem
a chain map)
~ C,(S)
such
® C,(S)
that in d i m e n s i o n
zero
is g i v e n by ¢(o)
It follows
using
it is unique Chapter
=
o
®
acyclic m o d e l s
o,
that
up to chain h o m o t o p y
6, § 11, E x e r c i s e
4].
o
6
S 0.
there exists
(see e . g . A .
The c u p - p r o d u c t
some
Dold
~
and
[10,
is now simply
31
induced
by the
~
: C~(S)
An explicit defined
mapping
® C~(S)
choice
~ Hom(C~(S)
for
~
AW(~)
this
choice
of
¢
is r e p r e s e n t e d
a v b
(2.35)
a =
the
then
(avb)o
Proof
(a)
= a
It is t h e r e f o r e
cup-product 6 cP(s) by
2.32.
in the
to s h o w
that
(p+q)-cochain
I (E(a) ^ E(b))
of
a
H(C~(S)).
and
b
be d e f i n e d
by
(2.36)
in
#~(a ® b)
claim
inducing
that
there
(2.36).
Put
= p~
[ I =
ip ~ n,
has
one
suppose
let
#~
and
J
have
exactlv
b £ cq(s)
® C~(S)
~ C~(S)
b C cq(s).
to the d i a g o n a l formula Then
= q~
[
for on
(2.36) : An ,
b
iJl= q ~j(~)~I' and
J =
(J0 ..... Jq)
Jq ~ n.
in common. then
~ A~(S) .
the c u p - p r o d u c t
: C~(S)
~ £ S n.
0 ~ J0 <'''<
in c o m m o n
simplicial
E • C ~(s)
a 6 cP(s),
E(b)~
( i 0 , . . . , i p)
o 6 Sp+q.
(~),
a 6 cP(s),
an e x p l i c i t
and c o n s i d e r
integer
integers I
of
represents
a
0 ~ i 0 <...<
two
us f i n d
given
6 cq(s),
every
iil= p ~ i ( o ) ~ I '
as u s u a l
at l e a s t
for
is an a p p r o x i m a t i o n
n = p + q
E(a)~
So
2.16
image
= l(E(a) ^ E ( b ) ) ,
Let
(b m)
• b ~(p,...,p+q)
By T h e o r e m
to a form
enough
b =
n"
the c o c h a i n
the
than
AW
~ 6 S
is e x p l i c i t l y
and
(0 ..... p) (~)
of T h e o r e m
is c o h o m o l o g o u s
where
map
n [ ~ (~) ® ~ (~) p=0 (0, .... p) (p ..... n) '
=
Let
We
~ C~(S).
is the A l e x a n d e r - W h i t n e y
as f o l l o w s :
form
® C~(S),]R)
by
(2.34)
With
composed
If
I
obviously two
Then and ~I
integers
I J
satisfy and
J
has m o r e
^ ~J = 0. in common,
Now say
32 isl = Jr I < i s2 = jr 2 sl+r I miA~j=(--1)
+(-1)
ti
s2+r2t
Then
t. dt. A...Adt. ^...Adt. Adt. ^...^dr. A..Adt. 10 l l 30 3q S I 3r 2 sI P ]r 2 t. dt, ^..^dt, ^..Adt, Adt. A,.Adt. ^..^dt, is2 ip 30 ]q is 2 3r I l 0 3r I
and it is easy to see that these two terms are equal with opposite signs so and
J
mI ^ ~J = 0
also in this case.
have exactly one integer in common,
miA~j=(-1)s+rt,
+
Finally suppose say
is = Jr'
I then
t. dt. ^...^dt. A...Adt. Adt. ^...adt. A , . . A d t iS ]r 10 is ip 30 ]r 3q
[ (-1)s+kt. t. dt. A...Adt. a...Adt. A d t A..Adt. A..^dt. k$r i s 3k l 0 is ip 30 ]k ]q
+ [ (-1)r+it. t dt, A...Adt i A...Adt i Adtj0A..Adtjr^..Adtjq, I%S ll 3r 10 i p n
Using
dt I = 0
we get
~=0 WiA~j=[(-1)s+rt
t. + [ (-1)s+k+r+kt. t + [ (-1)r+l+l+st. t ] is 3r k%r is 3k l%s ii 3r
• dt. A.,.Adt. A...Adt. ^dt. A...Adt. A . . . A d t 10 is ip 30 3r 3q = (-1)s+rt, dt. A...Adt. A...Adt. Adt. A...Adt. A...Adt. is l 0 is Ip 30 3r 3q It follows that
(£(a) AF(b))
=p'q' Iil=p IJi=q
aui(a)buj(o) "(-1)r+st. dti0^...^dt. ^.. is is ..^dt. ^ d t ^...Adt. ^ . . . ^ d t ip 30 3r 3q
where the sum is taken over
I
and
J
such that for some
s
33 and
r
is
sgn(I,J)
Jr
and no o t h e r
be the sign
permutation
taking
(_~)p-s+r
(0,...,n)
^
integers
.
are common.
. times
the s i g n of the
into
.
^
(i0 ..... is ..... i p ' i s = 3 r ' J 0 ..... Jr ..... Jq) ;
sgn(I,J)~ An
N o w let
then
(-1)r+st. d t i 0 ^ . . . A d t ^...^dt. ^dt. ^...^dt. ^...Adt. is is ip 30 3r 3q
= IAnt0dtiA...Adt
- [ n-J{tl+'''+tn=<1'ti~0}
= I
dt 0. {to+ . . . + t n < 1 , t i > 0 }
.dt "
= I n
(1-(tl+...+tn))dtldt2...dt n
dtiAdt2^..
Adtn+ I = I/(n+I)'
An+1
Hence (2.37)
~(a
® b)~ = I(E(a)
^ E(b)) °
P'q" [ sgn(I,J) a (a)b j (p+q+1) ' II[=p PI (~) [Jl=q where
again
Therefore
I
and
J
if we d e f i n e
have
exactly
on~ i n t e g e r
in common.
the m a p
: C,(S)
~ C,(S)
® c,(s)
by (2.38)
then
#(~)
~
:
[ -P~q: (n+1)~ p+q=n
g i v e n by
(2.36)
is an a p p r o x i m a t i o n
[ sgn(I,J) pi(a) [I[=p [J[=q
is the dual map.
to the diagonal:
® pj(~) ' ~ 6 S n
We w a n t
Clearly
to show that
~
is n a t u r a l
and %(o) It r e m a i n s
= o ® a
to show that
it is e n o u g h
to see that
#
for
~ 6 SO .
is a c h a i n map. ¢*
However,
is a c h a i n map w h i c h
for this
is easy:
.
34
~(6(a®b))
= #~(da®b
+ (-I) p a ® 6b)
= l(E(6a) hE(b))
+ (-I)PI(E(a) h E ( ~ b ) )
= l(dE(a) h E(b) +
(-I)PE(a) h d E ( d b ) )
= l(d(E(a) hE(b))
=
BI(E(a)
hE(b))
= ~(a®b).
This ends the proof.
Remark.
Notice that the term in
I = (0,...,p), J =
(p,...,p+q)
Whitney cup-product
(2.35).
(2.37) c o r r e s p o n d i n g to
gives exactly the A l e x a n d e r -
Thus
(2.37)
is an average of the
A l e x a n d e r - W h i t n e y c u p - p r o d u c t over the p e r m u t a t i o n s given by (I,J)
in order to m a k ~ the p r o d u c t ~raded c o m m u t a t i v e on the
c o c h a i n level.
In fact the A - W - p r o d u c t is not graded
c o m m u t a t i v e on the cochain level as since
h
is graded commutative.
~
clearly must be
On the other hand the A-W-
p r o d u c t is a s s o c i a t i v e on the cochain level w h i c h not.
In order to achieve both properties
to replace the functor
C•
~
is
it seems n e c e s s a r y
by the chain e q u i v a l e n t functor
A•. Exercise with
I.
C o n s i d e r for
0 ~ i 0 <...< i k ~ p
k < p
and let
a sequence A~ ~ A P ~ ~ p + 1
I =
(i0,...,i k)
be the
set
A~ = {(t0,...,tp) Isome tis>0} = A p - {ti0=ti1=...=tik=0}, (i.e. we s u b t r a c t a nI
: A~ ~ A k
p-k-l-dimensional
face).
be the p r o j e c t i o n 1
~i(t0,...,tp)
a)
Show that on
= ~tis
(ti0'''''tik)"
~
~(dtlh...hdtk)
=
(~t i )-(k+1)~ I s s
Let
35 where
~I
b)
is g i v e n Show
the
(i)
(2.23).
following
properties
(~I)*m I = dt I ^ . . . ^
(ii)
the
by
(~J)~I
c)
Conclude
form
E(c) °
that on
of
dt k
= 0
if
IJl = k,
for
c =
(c o )
Ap
satisfy:
~I:
J # I.
a k-cochain
For
any
I =
and
~ 6 Sp,
( i o , . . . , i k)
as a b o v e
(~I)*E(c) o = k:c ~i(o)dtl
2.39)
d)
Observe
that
for
~ 6 Sk
E(c) o = k~codtl is the of
simplest
(2.18).
Show
condition E(c) 0
(2.39)
for
of s i m p l i c i a l
a)
Let
sets.
Show
f
f~ 0 E = E 0 f ~
i = O,...,q, (i)
(ii)
to s a t i s f y choice
simplicial each
eoho
the
for
requirement
s k 0 f~ = f* o s k,
such
Ak
first
o £ Sk for
identity the
the c h o i c e
of
maps q
be a s i m p l i c i a l
fo,fl
there
are
k = 1,2,... : S ~ S' functions
that = fo'
eih j =
~
map
that
(ii)
if for
on
dtk
: S ~ S'
I o f~ = f ~ o I
Two
dtk"
p > 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,
thjei-1'
= 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 hi-1 ,
if
set.
A k-form
if it f u r t h e r m o r e
i = 0,...,p,
is the i-th 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
k-forms
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)
k-cochains
[~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 k-1 (S) : AN(S ) , A N
of normal cochains,
.T = 0 1
i = O,...,k-1.
: 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<...
ai0...i k
are p o l y n o m i a l s
in
t0...t n
be a simplicial
set.
with rational
coefficients. N o w let S
S
is c a l l e d
Ak(s,
~) a)
which
rational
denote
rational
Let
values.
~)
is a r a t i o n a l
the e x t e r i o r
~)
denote
the c o m p l e x
~)
(ii)
E : C*(S,
~) ~ A*(S,
~)
sk
simplicial
: Ak(s,
~) ~ A k - I ( s ,
t h a t the T h e o r e m s r e p l a c e d by
Formulate
Note.
Let
vector d
space and e x t e r i o r
of c o c h a i n s w i t h
Show that ~) ~ C*(S,
(see E x e r c i s e
o 6 Sp.
differential
[ : A*(S,
C*(S)
for
on
k-forms.
(i)
and c o n c l u d e
c)
6 Ak(A p, ~)
~ = {~o}
^. C*(S,
(iii)
and
A*(S,
is c l o s e d u n d e r
b)
~
the set of r a t i o n a l
Show that
multiplication
if
A k-form
A*(S,
and p r o v e
~)
2.16 and
2.33 h o l d w i t h
~)
C*(S,
and
a normal
version
A*(S)
~). of q u e s t i o n
b)
3).
For a s i m p l i c i a l de R h a m c o m p l e x
complex
goes b a c k
the c o n s t r u c t i o n to H. W h i t n e y
of the
[35, C h a p t e r
7].
3.
Connections
The
theory
"parallel
one
M;
i.e.
V
is
y
to
= q
and
t 6
.
an
6 M
is
[a,b],
be
P
with
a basis
v a = v.
or
let
data
frame
~
an
p
~ M
However,
the q,
concept that
curve
from
vector;
then
denote
the
= p
a
course for
is,
y(a)
1
!
a differentiable
It is of
P
impossible
to
{ V l , . . . , v n}
: F(V)
v 6 V
vector
weaker:
from
over
a vector
to be
a given
to t h e s e
V
: Vp ~ V q .
T
something
of
for
bundle
"parallel"
seems
a curve
concept So
and
is a d i f f e r e n t i a b l e v 6 V
Therefore
p,q
this
the
a differentiable
vector
isomorphism
bundle
associate
translate
T M of
corresponding
along
let
will
v t 6 Vy(t),
V
the
~ M
bundle
points
from
manifold.
a real
is p o s s i b l e
: [a,b]
"connection"
space
of
translation
suppose
parallel
Given
require
What
parallel
y(b)
we
tangent
a trivial
requirement. of
originates
generally
n.
a concept
6 Vq,
unless
the
or more
dimension
wants
T(v)
connections
consider
manifold of
of
bundles
t r a n s l a t l o' n " in a R i e m a n n i a n
motivation
M
in p r i n c i p a l
family
enough
to
the
vector
frame
bundle
P over of
M,
all
i.e.
bases
the
bundle
(frames)
whose
for
V
fibre .
over
Then
a
p
is e q u a l
"connection"
to
the
set
simply
P associates
to
any
a lift
Y
through
~(a)
of
= e
defines
and
curve
y
: [a,b]
e,
that
is,
z o ~ = y.
Now
let
a tangent
vector
X 6 T
~ M
and
a curve q
(M)
~
tend and
any
: [a,b]
to
~
point
p~
e £ F(V)y(a) ~ F(V)
then
defines
with
y
a tangent
P vector a
X
6 Te(F(V))
"connection"
defines
mapping
isomorphically
that
actually
is
below.
Notice
such
z~X
"horizontal"
onto
how we
that
a
that
are
F(V)
T
(e) (M)
going is
the
= X.
So
infinitessimally
subspace for
to d e f i n e principal
every
H e ~ Te(F(V)) e 6 F(V).
a connection
And
formally
Gl(n,~)-bundle
39
associated about C~
to
V.
principal
So
first
G-bundles
let
for
us
any
recall Lie
the
group
fundamental G.
Let
facts
M
be
a
manifold.
Definition mapping
~
3.1.
: E ~ M
differentiable
(i)
A principal of
right
For
differentiable
G-action
every
G-bundle
p £ M
E
E
is
manifolds
x G ~ E
= ~
a differentiable
-I
together
with
a
satisfying
(p)
is
an
orbit.
P (ii)
(Local
neighbourhood such
triviality)
U
and
Every
point
a diffeomorphism
of
~
M
: z
has
-I
(U)
an
~ U
open x G,
that (a)
the
diagram -I
(U)
~ U
x G
--... / U
commutes, (b)
~
is
equivariant, ~(e-g)
where
G
= ~(e)'g,
acts
translation
E E
= ~
is -I
called
(p)
is
the
the
at
e 6 ~-1(U),
trivially
on
total
fibre
i.e.
on
U
g 6 G,
and
by
right
G.
space,
M
the
base
p.
Notice
that
so
induces
space
by
(i)
and ~
is
onto
P and
by
(ii)
it
is
of
the
orbit
space
G
on
E
free
given
by
often
refer
space
E.
is
an
open
mapping
E/G
to
(i.e.,
xg
M.
~
Also
= x ~ g =
a homeomorphism
observe
that
1)
and
the
for
every
the
action
mapping
of
G ~ E P
g ~
eg
is
a diffeomorphism
e
£ E
.
We
P to
a principal
G-bundle
by
just
writing
its
total
shall
40
Example bundle.
I.
Then
Suppose
the
V ~ M
bundle
is
F(V)
~ M
be
two
an
n-dimensional
of
n-frames
is
vector a principal
Gl(n,~)-bundle.
Let
E ~ M
isomorphis
m
morphism. an
~ M
~
Now
x G
: ~
~
in
gBs
are
called
and
they
(3.2)
the
n UB
x G -~ U c~ N UB
to b e
=
~ G
the
given
functions
quotient
(p,a)
C
6 Ue
x G
x G
E
p 6 U
This with
N UB
system
respect
6 U s nu~nu
vp
U =
(3.2) as
{U
one
}
the
with
6 U s h UB,
and
can
follows:
with
identified Vp
× G
{gBs } to
U
Y
I.
G-bundle ~ U
a
condition
= gy~(p),
satisfying
of
choose
consider
a 6 G,
for
cocycle
=
and
trivializations
function•
a covering
principal space
a
The
form
(p,gBs(p)"a),
the
• gBs(p)
hand
the
and
trivialization.
D UB % ~
functions
satisfy
gyB(p)
of
is
transition
clearly
corresponding
with
: U
the
transition
together U
G-bundle
: E ~ M
if
: U s DUB
other
~
Then
gs~
On
G-bundle
x G.
seen
M
a local
U
-I K0B o ~ p (p,a)
where
called
an diffeo-
a trivialization.
6Z
easily
of
called
Then
preserving
principal
{Us}
(U)
is
is
G-bundles. fibre
trivial
is
a principal
-I %08 o ~ e
which
a
x G
above
principal
a G-equivariant
course
: E ~ M
(ii)
U =
-I
is
is o f
consider
covering
F ~ M
: E ~ F
isomorphism
mapping
~
and
a system
construct the
total
of
a space
is
identifications
(p,gBs(p)-a)
a 6 G.
6 UB x G
41
Again f
let
: N ~ M
be
f~
; f*E
~ N
f~E
~ N
~
: E ~ M
is
map.
the
principal
=
{ (q,e) If(q)
G-bundle
The
and
let
"pull-back"
G-bundle
with
total
space
x E
projection
onto
a principal
a differentiable
f*E
and
be
the
f*z
first
give s an
given
factor.
equivariant
by
The
map
f
= ~(e)}
the
restriction
of
projection
onto
: f*E
covering
~ E
the
the
projection
second f,
factor
i.e.
the
diagram
T
f* (E)
, E
N
, M
commutes.
Exercise transition U = for
{U
} 6~
then
Let
map
is
map
and
f
f.
Show
that
a pair
: M ~ E b)
z*E
is c)
to
any
2.
a)
iff
it has
such Let
z
that
if
is
be
where
to
the
covering
an
{gas}
relative
E ~ M
bundle the
E
the
is
and
that
0 f}
(f,f),
: F ~ E
f*E
trivial
for
F ~ N,
Exercise
s
Show
{g~
relative
b)
is
a)
functions
f*E
: F ~
I.
set
bundle
that
i.e.
the
set
transition {f-Iu
is
functions
} 6 Z. A bundle
a differentiable
differentiable into map
of
covering
G-bundles.
a principal
a section, ~ 0 s =
=
: N ~ M
factorizes
canonical
Show
of
f-lu
equivariant map
the
principal f
is
an
f*(E)
map
covering
isomorphism ~ E
G-bundle
as
~
a differentiable
above.
: E ~ M map
id.
: E ~ M
be
a principal
G-bundle.
: E ~ M
be
a principal
G-bundle
Show
that
trivial. Let
~
and
let
H ~
G
42
be
a closed
(Hint: using
subgroup.
First the
construct
let
N
associated ~N
: EN
under and
Let
~ M
the
bundle
where
with
: E ~ M
be
a principal
a
is
induced
that
EN
is
a manifold
the
sense
trivial
in
U
a diffeomorphism
by
that
left
fibre
(e,x) "g =
~N
with
the
×G N
where
Show
a principal
of
with
EN = E
G-action
is
sections
z
a manifold
fibre
E ~ E/H
bundle
H-bundle.
G ~
G/H
map).
3. be
that
local
exponential
Exercise and
Show
G-action
N
is
is
the
the
projection
and
that point
%0 : ~ I
(U)
space
e 6 E,
the
E
fibre
of
M X
of g
The
× N 6 G,
followed
bundle
has
is
by
that
the
diagram -I
%0
~N (U)
~ U × N
U co~utes.
In particular
Now be
let
and
a homomorphism
principal
Then
we ~
of
H-bundle
suppose
to
H
there
will or,
relative "relative
is
~N
G
be
Lie
and
to
Suppose
a differentiable
Ep,
%0(x • h)
= ~(x)
that
E
(when ~").
it
is
map
%0 : F ~ E
" ~(h) ,
Vx
is
6 F,
: H ~ G is
a
G-bundle
and
satisfying
h 6 H.
F
to
G
relative
a reduction
of
E
to
what
of
~
: F ~ M
a principal
and
clear
let
is
Vp 6 M,
F
and ~
an extension
that is
and differentiable.
Lie-groups
: E ~ M
%0(Fp) ~
say
~
two
open
groups. ~
equivalently,
to
is
e
is
we
will
~.
locally
a neighbourhood
such
N
E
x 6 N,
on
the
U
× N ~ N.
mapping
orbit
(eg,g-lx),
every
G
G-bundle
omit
H
43
Example
2.
An n-dimensional
principal
Gl(N,~)-bundle
act on the
left on
with
fibre
~n
F(V)
~n
and vector
bundles.
let
Fo(V)
fibre.
F(V)
Then
Fo(V)
~ M
inclusion
versely
a reduction metric
Exercise consider g 6 G. ZG
G
with
Show
b)
of
of
F(V)
O(n)
a) L e t the
z
: F ~ M
is a G - e x t e n s i o n
of
~
that a principal
{~ 0 h ~ y }
H
relative
with
{hsy}
(3.2)
Before
we
introduce
it is c o n v e n i e n t
coefficients
: Uy
the
there
is a
defines
a
O(n).
frames
In f a c t
in e a c h
orthogonal
bundle
clearly
gives
Con-
rise
to a
by
h
H-bundle
- g = a(h)g,
bundle
with
: F ~ M,
and
fibre
and h £ H,
G,
show that
an
k
~ ( X l , . . . , X k)
with
~
: E ~ M
iff t h e r e
a
is a c o v e r i n g
functions functions
has
for
E
of t h e
form
satisfying
fl U 8 ~ H).
notion
space.
dimensional
of degree
e
to c o n s i d e r
in a v e c t o r
a finite
to
a set o f
(hsy
function
bundle
the r e d u c t i o n .
G-bundle
a n d a set of t r a n s i t i o n
M
V group
given
fibre
U = {Uy}
on
fibre
is u n i q u e .
Show
V
Gl(n,~)
be a principal
left H-action
to
and
the
V.
reduction
bundle
on
defines
to
that
has
Gl(n,~)-bundles
the o r t h o n o r m a l
~ F(V)
V ~ M
Hence
principal metric
that the associated
: FG ~ M
extension
4.
bundle.
is t h e c o r r e s p o n d i n g
Fo(V)
on
Notice
to t h e o r t h o g o n a l
consist
and the
Riemannian
A Riemannian ~ M
~ F(V)
between
bundle
the associated
is j u s t t h e v e c t o r
correspondance
of
~ M.
and that
one-to-one
reduction
vector
of a connection differential
So l e t
M
vectorspace. values
: M ~ V
in
to e v e r y
V
be a
in a p r i n c i p a l
forms with C~
manifold
A differential associates set of
C~
a
form C~
vector
44
fields
XI,...,X k
nating
and has
a basis
forms.
+...+ Let
in
defined chain
the
V,W
has as
If we
the
choose
form
is a set of u s u a l
set of k - f o r m s
on
an e x t e r i o r
in C h a p t e r
dd = 0).
@ AI(M,W)
This
M
k-
with
differential
1 and
time,
~ Ak+I(M,V
two v e c t o r s p a c e s . define
A~(M,V)
however,
d is a
the w e d g e -
~I
In f a c t
® W)
for
~1
^ ~2 6 A k + I ( M , V
6 Ak(M,V)
® W)
and
by
^ ~2(XI ..... Xk+l) I (k+l) ~ ~q s i g n ( ~ ) ~ 1 (X~(1) ..... X~(k))
where
as u s u a l
Again
we h a v e
(3.4)
q
runs
through
d(w1 ^ ~2 ) =
Similarly
for
induced
a linear commuting
(d~1)
let
as u s u a l
the
can
be
element
all p e r m u t a t i o n s
: M ~ N
it c l e a r l y d G
and be
set of
a
For
C~
(-I)k~I
of
1,...,k+l.
maps
the let
P
vector
Ad(g)
if
P
we have
: V ~ W
is
~ A~(M,W)
as above.
Lie
tangent
manifolds
: A~(M,V)
F~
The
left-invariant
g 6 G
C~ Also
a map
group.
with
of
~ A~(M,V).
induced
^ d~2'
~2 6 A I ( M , W ) .
map
induces
a Lie
identified
1 6 G.
^ ~2 +
6 Ak(M,V),
F ~ : A~(N,V)
with
Now
also
F
map
map
® m 2 ( X o ( k + 1 ) ..... X~(k+l))
the f o r m u l a
~I
an
is of
and a l t e r -
is a m a p
~2 6 AI(M,W)
-
the
A~(M,V)
is,
~
( W l , . . . , ~ n)
formula
(that
multilinear
as b e f o r e .
then
denote
same
Ak(M,V)
~I
V
where
Again
by the
is a g a i n
property"
for
~nen
V.
~
"tensor
Ak(M,V)
complex
product
for
M~
{e I .... ,e n}
= ~iei
values
on
algebra fields
space
of
:22 ~ ~ / I
~
of
G
on
G.
This
G be
is
at the u n i t the a d j o i n t
45
representation, x ~
gxg
let
map
vx
: ~
with
~
differential
: E ~ M
G ~ E
given
~ Tx(E)
and
(x) (M) .
That
T
(3.5)
The
the
at
I
of
the
map
-I
Now the
i.e.,
0
,~
vectors
in
by
a principal
g ~
the
x •g
quotient
is,
x
the
be
we
image
an
~ T
of
u
induces space
have
Tx(E )
G-bundle. an
is
exact
x
6 E
injection
naturally
identified
sequence
(x)(M)
are
For
, 0.
called
vertical
and
we
want
x
to
single
i.e., is
out
we
want
equivalent
a complement
in
to
split
exact
to
a linear
(3.6)
It
is
be
a
8
therefore l-form we
consider
the on
want
left the
trivial
E
: M
i.e. action
given
for on
Lemma
E = M
horizontal
sequence
(3.5).
: Tx(E)
~ ~
vectors, This
such
A d ( g -I)
0
course
that
a connection that
(3.6)
condition E = M
on
in
holds
8.
To
x G ~ M
and
E
simply
for
all
motivate let
8
=
(Lg-1
is
0 ~2),,
the
g.
Now
the
action
x 6 M,
to
x 6 E. this be
the
projection for
of
x G,
by
the
For
e
defined
g g
6 G on
right
and let
the
g 6 G,
L -I : G ~ G is g R : E ~ E denote g
principal
action
on
G
(3.7)
we
have
G-bundle
and
M.
3.8.
of
id
such
by
R*8 g where
of
by
x G ~ G
by
(E)
define
bundle
given
translation map
to
x
0x
=
x
a further
8(x,g)
72
o u
x
natural
(3.7)
where
map
8 6 AI(E,~)
However,
l-form
the
T
= Ad(g-1)
: A I (E,~)
0 @,
by
Vg
~ A I (E,~)
6 G,
is
induced
by
the
E,
trivial
46
A d ( g -I)
:~
~
Proof.
•
Since
to c o n s i d e r
8
is i n d u c e d
M = pt.
That
is,
via
8
72
from
is the
G
l-form
it is e n o u g h
on
G
defined
by 8y =
(L - I ) * 7
: Ty (G) ~ T I (G)
=~
.
Then
(R*8)
= 8
=
With
: E ~ M
6x
3.9.
o ux
R*8 g
=
on
Remark vectors,
R
vanish check from
o
i.e.
= A d ( g -I)
o e
. Y
0 @,
: E ~ E
g
in a p r i n c i p a l
ux
Vg
G-bundle
satisfying:
: ~/~
Tx(E)
is
the
g ,~ xg. 6 G,
is g i v e n
H x c= T x ( E )
If
by
H x = ker
8x,
is the
then
Vx 6 E,
(ii)
the
action
of
g
implies
(ii) ' and
on h o r i z o n t a l
vectors
(granted
on v e r t i c a l
(i) and L e m m a
3.8.
vectors
subspace
of h o r i z o n t a l
is e q u i v a l e n t
to
Vg 6 G.
clearly
(ii)
(Rg),
(Rg),
E.
I.
(ii)
where
of the m a p
(ii) ' Rg,H x = Hxg,
In f a c t
(L _i), 7
-I )* o
8 6 AI(E,~)
id
= A d ( g -I)
where
(L-I
A connection
differential (ii)
0
=
we h a v e
is a l - f o r m
(i)
(Rg),
(L _i), g
this m o t i v a t i o n
Definition
o
in w h i c h
since
both
sides
of
(ii) ') it is e n o u g h case
(ii)
(ii) to
is o b v i o u s
47
Remark has
2.
By L e m m a
a connection
connection that has
if
or ~
given
the
by
: F ~ E
is an @
every
from
connection
called
the
The
flat
is
again
3.10.
Any
functions
in on
connection
~ M
in
3.11.
Any
manifold
Proof.
By
M
Remark
In g e n e r a l
connections
0a.
a partition
follows
from
Exercise
x G.
G-bundles
bundle.
the
Notice
and
if
in
a connection
given
E F.
induced
This
combination
precisely:
=
let I.
is
also
trivialization.
of
Let
Then
connections
0 1 ,...,0 k
11,...,Ik
in
has
be
be
realvalued
@ = ~ili0i
is
again
a
E IU of
every
n
trivial
: E ~ M
bundle
trivializations
for
unity
a)
G-bundle
on
a
a connection.
local
Proposition
5.
principal
2 above
connection.
choose
M
flat
is o b v i o u s .
and
~i~i
the
E.
Corollary paracompact
with
by
of
x G ~ M
a connection
product
convex
More
: E ~ M
of
has
induced
proposition
a connection.
connections
the
M
is c a l l e d
defines
bundle
in
bundle
connection
~0
connection
Proposition
This
isomorphism
trivial
following
product
(3.7).
then
In p a r t i c u l a r flat
the
Maurer-Cartan
a connection
the
3.8
{I
3.10
Suppose
{U
} 6~
}
and
that
9
has
define
a covering put
@ = ~
a flat
flat of
%
.
M.
Now
It
is a c o n n e c t i o n .
we
have
a bundle
then
f*0
defines
map
of
principal
G-bundles F
N
If
E
has
a connection
-* E
f
~ M.
8
a connection
in
F.
48
b)
If
E ~ M
is a t r i v i a l
G-bundle
then
there
is a b u n d l e
map
and
the
flat connection
Maurer-Cartan
G
M~
pt.
is j u s t
connection
Now consider
E~
induced
in the G - b u n d l e
a principal
X £ Tx(E)
the
@.
For
the
t e r m v_~ertical
for
X 6 im Ux,
for
X 6 H x = ker
8 x.
Now
with
coefficients
in s o m e v e c t o r s p a c e
is h o r i z o n t a l vectors
if
R~ g
= g
the
are
G
exactly
the
In f a c t
suppose
define
~ £ Ak(M)
choose
x 6 z
~ . X i = Xi'
-I
= 0
If
image
w
For
X l , . . . , X k 6 Tx(E)
i = 1,...,k
w
the
(left)
is the
trivial
invariant.
Notice
coefficients
: A~(M)
and
p £ M
that
is e q u i v a r i a n t V
~
is h o r i z o n t a l
say
is a
with
of
introduced
is a k - f o r m
V
if
E
connection
j u s t o n e of
that
on
the
and horizontal
We will
is c a l l e d
forms
as f o l l o w s : and
~ Tx(E) ,
V.
say
with
already
whenever
form
in the
~ 6 A*(E)
(p)
:~
In p a r t i c u l a r
horizontal forms
: E ~ M
~ £ A*(E,V)
then we will
an e q u i v a r i a n t
invariant
z
we have
ux
of
G ~ pt.
is v e r t i c a l .
~, Vg 6 G.
representation that
~(Xl,...,Xk)
of
-I
vector
suppose
X l , . . . , X k 6 Tx(E)
representation if
a tangent
G-bundle
connection
in
~ A~(E) •
invariant;
then we
and
Xl,...,Xk
such
that
£ Tp(M)
and put
~ ( X 1 ..... X k ) = ~ ( X I ..... Xk). This
is t h e n
Furthermore on
M
independent if
we can by
Xl,...,Xk local
the c h o i c e s are
of
~Xi
~(Xl,...,Xk)
so
extended
triviality
a neighbourhood = Xi'
x
of
to
C~
of
vector is
C~
E
of
x
and
to
C~
extend
fields
X l , . . . , X k.
vector
fields
Xl,...,X k
in
satisfying
in a n e i ~ h b o u r h o o d
of
x.
49
Now that
consider
8
the c o n n e c t i o n
is an e q u i v a r i a n t
the
adjOint
the
image
induced
action
of
of
8 A 8
Proposi£ion connection
and
Let let
~
a)
: E ~ M
:~
Let
coefficients
[0,8]
®~
~
in ~
6 A2(E,~)
A2(E,~
®~) .
with
denote
~ A2(E,~)
T h e n w e have:
E = M x G
be a p r i n c i p a l
~ £ A2(E,~)
(3.14)
with
the
flat
be
G-bundle
the c u r v a t u r e
with
connection
form defined
by
de = -½[0,0] + ~
(the s t r u c t u r a l c)
equation).
Furthermore
~
(3.15)
Then
Proof.
d~
a)
induced
from
= 0
because b)
both
observe
that
To see any
(3.16)
In o r d e r
X,Y
and
clearly that
~
£ Tx(E)
(dO) (X,Y)
to show
that [8,0] Ad(g)
since
equivariant.
~
with
by E x e r c i s e G ~ pt
and
is e q u i v a r i a n t
are e q u i v a r i a n t : ~
X
vectors.
5
8
is
therefore
by b).
we must
vertical
= -½[@,e] (X,Y)
since
(for the
preserves
is h o r i z o n t a l
(3.16)
and
identity
of h o r i z o n t a l
G-bundle
it is h o r i z o n t a l
de
the B i a n c h i
on sets
f r o m b)
the p r i n c i p a l
It is o b v i o u s
is h o r i z o n t a l
[~,8].
vanishes
follows
n
satisfies
an =
In p a r t i c u l a r
for
let
Observe
d0 = - ½ [ O , O ] .
b)
hence
with
the m a p
[-,-]
e 6 AI(E,~).
Then
(3.13)
O
Also
under
3.12.
8.
l-form
G.
by the b r a c k e t
from
0
and
second
one
the L i e b r a c k e t ) .
show
for
x 6 E
and
that
= -½[@ (X) ,% (Y) ].
it is e n o u g h
to c o n s i d e r
I)
Y
vertical
50
and
2)
Y
I)
horizontal. First
associated where
the
{Rgt}, see
C~
ux
that
notice
: ~
that
vector
t 6 ~
that
as
,
for
where
A,B
gt
by
trivial is
local
left
Therefore
6 ~
E
defined
induced
by
group tA,
of
there by
g ~
is
A~ = x xg.
.
Also
it
u
(A)
x
Observe
diffeomorphisms
t 6 ~
an
is
is
easy
to
=
it
x G
invariant is
[A~,B~].
is
enough
in w h i c h
vector
immidiate
prove
case
field from
to
on
the
this
A~ = G
0 @
for
A
where
associated
definition
of
a
to
the
A.
Lie
in ~ .
bracket
NOW, enough
to p r o v e
(3.16)
for
X
and
Y
vertical
it
is
clearly
to p r o v e
(dS) ( A ~ , B ~)
Rut
is
= exp
triviality E = M
(3.17)
on
A
6 ~
G-bundle the
vector
A•
usual
[A,B]~
fact
any
l-parameter
(3.17)
In
field
~ Tx(E)
associated
for
since
8 ( A ~)
= -½[ 8 ( A ~ ) , 8 ( B ~ ) ] ,
= A,
8 ( B ~)
(d@) ( A ~ , B ~)
= B
= -%8(
are
2)
Again
A 6~.
Also
extend for
Y
denoted
vector
field
also
vector
field
Z
Since
Y
So we
must
(3.18)
is
X
and
to
horizontal
the
field
extend Y
put
we
.
conclude
= -%8 ( [ A , B ] ~)
a vector
by
6~
= -½18(A~),8(B~)].
horizontal
then
constants
[ A~,B*])
= -½[A,B]
A,B
it
(first Yy
right
of
to
the
form
a horizontal
extend
Y
to
- Vy
o 8y(Zy) , y
hand
side
of
(3.16)
show
(dS) (A~,Y)
=
0
for
A
£ ~
,
Y
a horizontal
vector
field.
C~
any
Zy
:
A ~,
C~
£ E).
vanishes.
51
Now
since
8 (A ~) = A
is c o n s t a n t
(as) (A~,Y)
As r e m a r k e d
in I) the
Rg t' gt = exp
tA,
t 6 ]R.
gt Yx =
(Rg t) • (Y
= 0
group
associated
to
A~
is
Therefore
x
lim l ( Y g t - Y x) t~0
=
-I ) " xg t
Since
0 (Yxg t) = A d ( g tI- ) 0 0 (Y
we
e(Y)
= -%@ ([A~,Y]) .
l-parameter
[A~,Y]
where
and
-I) xg t
= 0
and
8 (Yx)
= 0,
conclude
@([A~,Y] x) which
proves c)
(3.18)
and h e n c e
Differentiating
since
= d~ -
[de,el
=
[~,G]
-
[[8,9],8]
= 0
proves
(3.14)
0 = d~ - ½[de,@]
d~
= 0
by
b).
we get
+ ½[@,d6] = d~ -
[~,@]
the J a c o b i
+ ½[[8,8],8]
identity.
This
proves
the
proposition.
Remark. by
Let
X, Y
be h o r i z o n t a l
vector
fields
on
E.
Then
(3.14)
(3.19)
which
n(X,Y)
gives
another
Definition is c a l l e d
flat
way
3.20.
= -½%([X,Y])
of d e f i n i n g
A connection
if the c u r v a t u r e
~.
e
in a p r i n c i p a l
form vanishes,
that
is,
G-bundle ~ = 0.
52
Theorem : E ~ M
3.21. is
flat
neighbourhood
U
restriction
Proof. =:
This
of
the
Remark
any
leaf
of Now
: Hx
T p (M)
U
of
and
: ~
let
T(E)). by
subspace = Hyg ,
the
the
is
a
that
the
be
the
flat
let
a). c= T x (E)
Hx
X 6 Hx on
E
iff
(i.e.
this
is
[29,
Chapter
through
3.9
that
onto
a differen-
6])
some
integrable defines
such
It
: E ~
g
= 0.
theorem
x~ R
@(X)
an
integrability
leaf
Definition
is
and
choose
Since an
x 6 ~
Tx(F)
that
follows E,
g
(possibly
we
V
the
x G
of
be in
the U
Ty.g(E), 8
and
~
Then y
8'
EIU~
: U
H
x
from
6 G,
different)
6 V,
g
defines
is
in
hence
EIU obvious
£ G, the
F
such s
is same
(U)
g
induced that
that
: U ~ V exercise
is g i v e n
q 6 U,
the
a neighbourhood
by
-I
consider
since
find
× G ~ ~
in
it
and
and
inverse
= s(q)-g,
connection
x G.
x
trivialization
where
(p)
can
The
of
-I
= Hx
isomorphism
a section fact
~ U
8'
so
there such
3.12
(3.19)
Spivak
to
x.
In
(U)
in
By
Frobenius'
e.g.M.
defines
connection
M
EIU
from
i.e.
a distribution
~(q,g)
Now
of
6 E
a diffeomorphism.
trivial. -I
x
a neighbourhood
is
therefore
of
Proposition
vectors,
p 6 M
~x
: V ~ U
point
G-bundle
foliation.
let
p
a principal
induced
diffeomorphically
through
-I
For
0.
space
F
is
~ =
of
(see
the
is
by
hence
leaf
every
obvious
I following
maps
leaf
is
tangent
in
x G.
subbundle
a foliation is
EIU
defines
distribution
e
a trivialization
horizontal
clearly
tiable
U
~
around
to
Suppose
subspace
iff
e
in
connection
and
of
connection
A
2
EIU
by is
defined
by
6 G.
from the
flat
horizontal
(Rg).(Ty(V)) horizontal
the
= Rg~Hy subspaces
=
53
and
therefore
must
Corollary The
3.22.
following
ge6
are
I)
E
2)
There
a s e t of
agree.
has
E
has
Then
2) ~
is c o n s t a n t be
2) a n d Let
3)
to
are
~s
: ~
with
Let
connection
be the
connection bundle
in
U
be a principal
G-bundle.
{ge6}
for
sets E
and
{ Ue}s6Z
such
that
for
all
e,6
6 E.
G
with
the d i s c r e t e
topology.
G d.
equivalent -I
U
in
Now
curvature.
by open
U
the constant
x G.
s
vanishing M
the g r o u p
a reduction
I):
of
functions
trivializations e
with
is a c o v e r i n g
Gd
Proof.
: E ~ M
a connection
: Us n u 6 ~ G Let
~
equivalent:
transition
3)
Let
x G,
s 6 Z,
transition
E IU
there
by Exercise
induced
4. be
the
functions from
the
is a c o m m u t a t i v e
gs6" flat
diagram
of
maps -I ~oc~ o ~oB U
OB x G
' Us
n uB x G 2
L gsB
G
G
and
let
80
definition
be 80
the M a u r e r - C a r t a n
connection
is l e f t
and
invariant
(~s 0 ~ ; I ) * ~ 8 0
or equivalently we
can define
@s
on
E I U s.
has
for
all
8
and
a global Clearly e.
88
has
G ~ pt.
By
therefore
= z~6 0
agree
connection 8
in
on 8
E I U s N U 6. in
vanishing
E
which
curvature
Therefore agree with since
8s
54
I) =
2):
8 3.21
from
the
we ~
can
cover
: U ~ U
flat connection
in M
E by
with open
sets
× G
such
that
U
× G.
Now
in
vanishing {U }
81~-Iu fix
~,B
6 Z
let
~o = kOc~ o ~oB
Again
let
clearly
80
be
i.e.,
particular
~(U
and
it f o l l o w s
the
let
Show
is t h e
(x,g0g)
induced
the f r a m e
projection.
be
e
(U
~ go
N U B)
of
N UB)
in
such
m a p of L i e
bundle The
geB
for
Then
the h o r i z o n t a l x g, g 6 G. some
g o 6 G,
the
: ~n
H-bundle
that
with
~ T
equal
to
to
G
go"
homomorphism
connection
~'8 E = ~,
tangent group Since
o
G.
be a Lie group
be a manifold
structure ~n)
6
then
0 8 F,
8 F.
there where
coefficients
in
(M)
~n
and
bundle,
is
Gl(n,~)
x 6 -1(p),
there
defined
let z
F(M)
: F(M) with P 6 M,
is a l - f o r m
w
= F(TM) ~ M
Lie
the
algebra
is an on
F(M)
by
-I L0x
a)
Show
l-form,
where
that
~
Gl(n,~)
on
is ~,
P with
In
algebras.
M
of
g
is c o n s t a n t l y
is the e x t e n s i o n
E
Let
N UB,
Vx 6 U
: H ~ G
: F ~ E
7.
x
leaves (U
a principal
= Hom(~n
isomorphism
the
U s n U B × G.
the f o r m
function
Let
8E
Exercise
~(n,~)
=
~
a connection
of
in
x G.
that
6.
if
permutes
N U B x I) =
F ~ M
that
connection
~
sets
transition
Exercise and
so
the
~ ( x ,g)
Hence
N U B x G -~ Uc~ N UB
: U
the f l a t
~'8 0 = 8 0
foliation,
be
be a connection
trivializations
is i n d u c e d and
let
By Theorem
curvature. and find
Now
X
F(M) acts
on
O
Z.
•
is a h o r i z o n t a l ~n
equivariant
by t h e u s u a l
action.
55
b)
For
M = ]Rn
and for
F ( IRn)
defined
connection
in
TIR n
x ]Rn,
~ ]Rn
0 6 AI(F(M)),~n,]R))
by the n a t u r a l
trivialization
show that de = - 0
where
the
the w e d g e - p r o d u c t
denotes
A e
the c o m p o s i t e
A 1 (F(M),J(n,]R)) ® A I (F(M),JR n)
^ ,
map
A2(F(M),J(n,]R) /
A2(F(M)
@ J R n)
,IR n) . 2
(Hint:
Notice
coordinates a real
that
y =
(yl,...,yn)
n x n-matrix.
For
M
F ( ~ n) = ~ n
Then
(3.23)
c)
0 = X-Idx and
with
X = {xij}i,j=1,..., n and
@
e = x-ldy).
a connection
@ £ A2(F(M),~n)
defined
in
F(M)
by
and h o r i z o n t a l .
With respect
I
e ,..., I
O
n
to the c a n o n i c a l
are u s u a l
l-forms
I 81 el .......... n
=
n @n 81 .......... n Then
and
c ]Rn x ]Rn
de = -@ ^ e + @
is e q u i v a r i a n t
where
6 ~n
a general manifold
s h o w that the t o r s i o n - f o r m
x GI(n,]R)
(3.23)
takes
the f o r m
on
basis
of
F(M) .
~n
we write
Similarly we write
56
(3.23)'
d~i
d)
Show
the
form
on
F(M).
that
every
~ = [ifi ~i,
e) @
= -I @~ ^ mj + 8i, j 3
Now
and Show
@
be d e f i n e d
that
(3.23)
(3.23)
still
show and
that
write
Conclude
framebundle Levi-Civita
Fo(M)
Again
for
then
every
Notice
connection
8.
Let
vector
M
bundle.
Show
= Hom(~n, that
takes
for
first
that
[j~j fij
torsion
free
= 0
determined (ei) and
is a
if w e
M
the
connection
t h a t by E x e r c i s e
6
(the
this
extends
F(M).
and
z
i.e.
the b u n d l e
: F ~ M
is the Lie
V ~ M be the
an nassociated
of n - f r a m e s algebra
^ % +
denotes
above.
~ =
manifold
the f o r m
the w e d g e - p r o d u c t
if
^ ~J
8 6 AI(F,~(n,~)),
d0 = -0
F(M)
Let
= fji )"
Let
~n)
let
Fo(M).
is u n i q u e l y
Riemannian
in
and
Fo(M)
8
be a m a n i f o l d
Gl(n,~)-bundle,
(3.14)
where
as in d),
bundle
functions
= I .... n.
satisfying
connection).
~(n,~) a)
Show
is of
C~
metric
as for
on
then
a unique
Exercise
principal
0
has
to a w e l l - d e f i n e d
dimensional
i,j
(Hint:
that
that
F(M)
valued
frame
exactly
and
8 =
l-forms
~j = [ i f i j m i f)
if
on
a Riemannian
Fo(M)
holds
(3.24).
r o w of h o r i z o n t a l
(3.25)
on
e
are r e a l
is g i v e n
0~ = -0~, 3
Furthermore
F,
M
fi
l-form
in the o r t h o g o n a l
(3.24)
by
where
suppose
be a c o n n e c t i o n
horizontal
i = 0,... ,n.
the c o m p o s i t e
8
Of
in
V.
Gl(n,~)
a connection
in
O
57
AI ( F , ~ ( n ) )
®.~AI(F,#(n))
A
, A2(F,/(n)
® /(n))
e' n, l of
maps
of
canonical
composition
]R n
into
basis
of
1Rn ) .
Furthermore,
~(n,lR),
e
11 1
and
with S% a r e
respect given
6) 1 . . . . . . . . . . e 1 n
to
the
by matrices
n
I
n 01 . . . . . . . . . .
of
I- a n d Show
2-forms that
correspondence where
of
V
sections
on
~n
show that
correspondence
with in
to e v e r y v e c t o r
of
C~
V
are
in
I-I
functions
of
F
The
into
~n
set of
C~
F(V) . C~
sections
equivariant
~n.
i,j = 1 , . . . , n .
l
in the u s u a l way.
is d e n o t e d
Similarly
coefficients
C~
equivariant
acts
to
~Y eki ^ ~k] + ~i]
that
with
Gl(n,~)
sections
is e q u i v a l e n t
d S -, = ] Observe
\ ~ ..........~n
respectively.
(3.25)
(3.25) '
b)
@n n
T~M ® V
horizontal
Alternatively
X p 6 Tp(M)
of
l-forms
are on
I 6 F ( T * M ® V)
an e l e m e n t
Zx
6 Vp
in F
I-I with
associates
such
that
P (i)
(ii)
1 x +y = 1x + ly , P P P P if
X
p ~
1X
is a is a P
C~
llx
vector C~
=
ll x
P field
section
,
~ 6 ~,
P
of
on V.
M
then
the
function
58
c)
Let
define
again
V(s)
8
be
6 AI(F,~n)
ds
(here
s
that
is
V(s)
considered is
£ F(T~M d)
as
horizontal
: -8-s
c)
is
called
and
?
s 6 F(V)
and
Show
that
let
a function and
F
equivariant,
and
X
6 T
any
the
F(V)
into
~n).
hence
Show
defines
(M)
let
covariant
V(s)
6
F(T~M
® V)
as s
defined
b) .
This
in
the
differential
in
direction
X
corresponding
P
to
satisfies:
V x + y (s) P P
= VX
(s)
+ Vy
P
(s),
VIX
P
(s)
:
IV X
P
If
X
is
a
function
C
p ~
vector Vx
(s)
field is
a
on
C
(s), P
s 6 F(M),
(ii)
s 6
P
Vx
called ?
(i)
of
(s) = V(s) x 6 Vp P P covariant derivative of
the is
For
+ V(s)
P in
F.
® V).
For
as
in
by
(3.26)
?(s)
a connection
M
I 6
then
section
IR.
the
of
V.
P This
(iii)
is
denoted
Vx(fS)
= X(f)~x(~)
+ fVx(S)
a
real
function
C~
directional
e) that is
As
for
y
a unique o ~
Notice
= y, that
translation
Vx(S) .
before
derivative
let
: [a,b] liftet
valued
8
~ M
be a
curve
such
that
the
this
lift
defines
along
y")
of
M
: [a,b]
an
and
f
X(f)
the
f.
curve
tangents
Ty(t)
on
s 6 F(V),
a connection
C~ ~
for
and
~ F of
in
are
isomorphism
: Vy(a)
: F ~ M.
x 6 ~-I (y(a)) with
~
~
~ Vy(t),
~(a) all (the
Show there
= x,
horizontal. "parallel
t 6
[a,b].
8.
59
f)
For
X
6 T P
(M)
let
y
: [-e,e]
~ M,
e >
0
be
a
P
oo
C
curve
with
be
parallel
y(0)
p,
=
translation
(3.27)
Vx
along
(S)
=
lim t~0
p (Hint:
Observe
a section lift the
of
that
v
of
y.
components
in
FIU
Now
y.
o
Show
such
that
and
Let
Tt
that
for
: Vp
-+
y(t)
V
s 6 r (v)
-I T t S (y (t) ) -s (p) t
neighbourhood
s =
v
= Xp
some
write
of
y' (0)
v
U
o y
of
defines
p
there
is
a horizontal
[ aiv i where ( V l , . . . , v n) are i a. : U ~ ~ , i = 1,...,n, are C~ 1
functions). g) 8.
Now
Show
function
let
that of
~ 6 A2(F,W(n,~))
for F
any
that
section
X E
~(X,Y)
h)
Now
in Exercise form
for
~(X,Y)
(3.29)
@.
let 7.
that
is g i v e n
=
be
that
for
TM,
that
is
(Hint:
defined Notice
is,
fields
of
as
an
equivariant
on
M
~
defines
Show
that
- Vy
o V x - V [ x , y ]) (s),
let
~
be
a connection X,
form
@ A V(S) •
o Vy
and
curvature
interpreted
vector
½(V x
e
Y
a new
the in
l-form F(M)
a
Vs
6 r(v).
considered
with
vector
fields
on
M
vector
field
@(X,Y)
torsion @
defines and
show
by
@(X,Y)
V
• s -
Y
the
have
r(Hom(V,V)).
Let
Observe
(3.30)
we
V = TM
of
this
= ~
and
(s)
a section
where
~n,
dV(s)
Notice
s £ F(V),
into
(3.28)
be
=
½(Vx(Y)
- Vy(X)
-
[X,Y])
in d ) . first
that
for
any
vector
field
~
on
F(M)
60
which Z*Xx
is a lift = Xzx'
Vx 6 F(M))
the e q u i v a r i a n t
Note. follows [17,
Our
closely
Chapter
of a v e c t o r the
function
treatment the
I and
field
function
corresponding
of p r i n c i p a l
exposition II].
X
on ~(~) to
M
(that : F(M) X
bundles
by S. K o b a y a s h i
is, ~ ~n
as in b)
and
is above.
connections
and K.
Nomizu
4.
The Chern-Weil
We n o w c o m e to c o n s t r u c t G-bundles
characteristic
V
let
of
real
Equivalently
acting
on
o
defined
under V.
these
cohomology
real
in
the a c t i o n
There
vector
space.
of
symmetric
variables P
the
For
on
V.
: V ®...®
symmetric
V ~
group
is a p r o d u c t
: s k ( v ~) ® S I ( v *) ~ s I + k ( v ~)
by
(4.1)
P o Q(v I .... ,Vk+ I) = _
I
(k+l) ! [oP(vq1 ..... Vok) where
~
runs
S * ( V ~) = 1[ k~0 algebra.
[x I .... ,xn]k in s o m e
through
s k ( v ~)
Exercise
k
for p r i n c i p a l notation:
map
of
namely
some
space k
is a l i n e a r
lectures,
classes
First
the v e c t o r
functions
P 6 s k ( v ~)
V ®...®
of
dimensional
denote
valued
is i n v a r i a n t
object
a connection.
be a f i n i t e s k ( v ~)
multilinear
which
to the m a i n
by m e a n s
Let k ~ I
homomorphis m
I.
all p e r m u t a t i o n s
(S0(V * ) = ~)
Let be
variables
the
;
{e I .... ,e n}
" Q ( V q ( k + 1 ) ..... Vo(k+l) of
then
be
Show
S ~ ( V ~)
a basis
set of h o m o g e n e o u s X l , . . . , x n.
I .... ,k+l.
for
is a g r a d e d
V
polynomials
that
Let
and
of d e g r e e
the m a p p i n g
: s k ( v ~) ~ ~ [x I .... ,xn]k
defined
by
~ ( x I .... ,x n) = P(V ..... v),
let
v = [ixiei ,
82
for
P 6 sk(v~) ,
is an i s o m o r p h i s m
: S*(V ~) ~ ~ [x I .... ,x n] shows
that
given
by
P
and that
is an a l g e b r a
is d e t e r m i n e d
v ~ P(v,...,v).
isomorphism.
by the p o l ~ n o m i a l
The inverse
of
This
function
on
V
is called
polarization. Now
let
the a d j o i n t
G
be a Lie g r o u p w i t h
representation
for every
induces
Lie a l g e b r a
an action
of
~
Then
G
on
sk(~)
k:
(gP) (v I, .... v k) = P ( A d ( g - 1 ) v I ..... A d ( g - 1 ) V k ) , vl,...,v k 6~ Let
Ik(G)
be the G - i n v a r i a n t
the m u l t i p l i c a t i o n (4.2)
(4.1)
Ik(G)
polynomials
on
~
E
with
manifold
curvature
P 6 Ik(G)
form
2k-form.
gives
Theorem Let
and
P
by
Notice
that
~ Ik+l(G).
is c a l l e d
the a l g e b r a
G-bundle
~ : E ~ M
and suppose
~ E A2(E,~).
to a 2 k - f o r m
p(~k)
invariant
p(~k)
of i n v a r i a n t
8
Then
p(~k)
is a c o n n e c t i o n for
= A2k(E,~
p(gk)
is h o r i z o n t a l ,
on a
k ~ I
we have
®k)
6 A2k(E). and since
is an i n v a r i a n t
is the lift of a 2k-form
in
on
Since ~
is
horizontal M
w h i c h we
p(~k).
4.3.
WE(P)
M,
rise
also
Hence
also d e n o t e
sk(~*).
a multiplication
{ A2k(E,~®...®~)
~
is h o r i z o n t a l equivariant
I*(G)
a principal
~k = ~ A . . . A
SO
I
g 6 G.
.
Now c o n s i d e r differentiable
induces
® II(G)
In v i e w of E x e r c i s e
part of
,
a)
p(gk)
6 H2k(A~(M))
6 A2k(M)
is a c l o s e d
be the c o r r e s p o n d i n g
form. cohomology
63
class.
Then
b)
WE(P)
does
and
in p a r t i c u l a r
of
E. c)
wE
d)
For
does
: I~(G) f
not d e p e n d only
on the c h o i c e
depend
~ H(A~(M))
: N ~ M
on
the
isomorphism
is an a l g e b r a
a differentiable
of c o n n e c t i o n class
homomorphism.
map
wf~ E = f~ o w E. Remark.
The m a p
phism.
Sometimes
bundle
in q u e s t i o n
WE(P)
is c a l l e d
to
injective since
(4.4)
by
is c a l l e d
shall
the C h e r n - W e i l
just denote
is c l e a r
from
the
the c h a r a c t e r i s t i c
it by
w
context. class
of
homomor-
when For
E
the
P 6 I~(G)
corresponding
P.
Proof
Now
we
wE
(3.15).
of T h e o r e m it is e n o u g h P
4.3.
a)
to s h o w
is s y m m e t r i c
d P ( ~ k)
Since
= kP(d~
On the o t h e r
that
and
~
^ n k-l)
hand
z~
: A*(M)
dP(~ k)
= 0
~ A~(E) in
is
A~(E).
a 2-form = kP([~,~]
since
^ ~k-1)
P £ sk(~
~)
is i n v a r i a n t
we have
(4.5)
P ( A d ( g t ) Y 1 ..... A d ( g t ) Y k ) gt = exptY0'
Differentiating
(4.5)
at
= P(YI ..... Yk ) '
Y0,YI,...,Yk
t = 0
6 ~
, t 6 JR.
we get
k P(YI ..... [Y0'Yi ] ..... Yk ) = 0 i=I or e q u i v a l e n t l y k A [ P ( [ Y 0 ' Y i ] ' Y I ..... Yi ..... Yk ) = 0, i=I
Y0''" "'Yk 6 ~
.
64
F r o m this it f o l l o w s together with b)
For
Chapter
(4.4)
P([0,~]
^ ~ A...^
this we n e e d the f o l l o w i n g
4.6.
5 or L e m m a
Let
be the o p e r a t o r
~) = 0
which
ends the p r o o f of a).
I, E x e r c i s e
Lemma
that
h
lemma
(compare
1.2):
: Ak(M
sending
easy
x [0,1])
~ Ak-I(M),
~ = ds ^ e + B
k = 0,I,...,
to
I
h(~)
= I
~
(h~ = 0
for
~ E A0).
s=0 Then
(4.7)
dh(~)
where
io(p)
=
(p,O),
Now suppose forms
principal
G-bundle
and
~0
and
obvious p(~k) f o r e by
that
(p,1),
01 ~I
3.10
=
p £ M.
are two c o n n e c t i o n s respectively.
in
Consider and
(1-s)00x + Selx'
~
is a c o n n e c t i o n
form of
i~
= ~0
is a c l o s e d
~.
Since
and
i~
2 k - f o r m on
E
with
the
let
i~
= ~I"
(x,s) in
E x [0,1].
= 80' N o w for
E x [0,1]
6 E × [0,1].
i~
= @I
Let it is
P 6 Ik(G),
by a) above.
There-
(4.7) d(h(p(~k)))
= i { p ( ~ k) - i ~ p ( ~ k) = P(~)
and h e n c e in
x [0,1])
be the f o r m g i v e n by
~(x,s)
be the c u r v a t u r e
=
E x [0,1] ~ M x [0,1]
x [0,1])
By P r o p o s i t i o n
6 A*(M
= ii*~ - i ~ ,
ii(p)
00
curvature
£ AI(E
+ h(dw)
P(~)
H2k(A*(M)) .
and
P(~)
This
shows
- P(Q~)
represent that
WE(P)
the same c o h o m o l o g y
class
d o e s not d e p e n d on the
65
choice
of c o n n e c t i q n .
The
second
statement
is o b v i o u s
from
this. c)
For
to v e r i f y
P 6 II(G)
(P o Q) (~k+l)
from which d) then
c)
If
trivially 8
f~8
form
forward
in
E ~ M in
with f~E
curvature
~ N
form
with
since
= p(~,~)k
follows.
Remark.
Let
G-invariant E
follows.
Therefore
~,p(~k)
bundle
it is s t r a i g h t
^ Q(~k)
is a c o n n e c t i o n
f*~.
(4.9)
clearly
= p(~l)
is a c o n n e c t i o n
clearly
curvature
Wail
Q 6 Ik(G)
that
(4.8)
d)
and
I~(G)
polynomials with
be
the a l g e b r a
on ~
connection
Then 8
of
for
we get
complex
valued
any p r i n c i p a l
a similar
complex
GChern-
homomorphism
(4.10)
I~(G)
L e t us polynomials exhibit
~ H(A~(M,~))
~ H~(M,C).
some
end
this
chapter
for
some
classical
the p o l y n o m i a l
with
examples
groups.
function
of
In all
the
v ~ P(v,...,v),
invariant examples
v 6~
,
we
for
P £ Ik(G).
Example matrices. the L i e For
I.
The
G = Gl(n,~), Lie
algebra
g 6 G,
algebra
~
of all m a t r i c e s
Ad(g) (A) = gAg
a positive
integer
degree
which
k
the g r o u p
let
Pk/2
I,
=~ with
(n,~)
=
Hom(~n,~
Lie b r a c k e t
for all be
of n o n - s i n g u l a r
A 6
(n,~).
the h o m o g e n e o u s
is the c o e f f i c i e n t
of
I n-k
[A,B]
n x n n)
is
= AB - BA. For
k
polynomial
of
in the p o l y n o m i a l
66 in
1
(4.11)
d e t ( l ' 1 - 2~A)
Clearly
= [PI,~(A ..... A)~ n-k k K/z
Pk/2 £ I k ( G l ( n ' ~ ) ) ;
Pontrjagin
polynomial,
the Pontrjagin Example satisfying
t
det(ll
is zero.
t
where
O(n)
is
g
the subgroup of matrices
is the transpose of
~'(n) ~ ~ ( n , ~ )
= det(ll
to
k
are non-zero on Example
3.
The
of skew-symmetric
odd the restriction
Notice O(n),
odd is zero for any
g.
+ 2~A)
Therefore we only consider
a reduction
images are called
A 622"(n)
- 1A)
that for
1 = 0,I,...,[~].
matrices
and the Chern-Weil
Since for
it follows
is called the k/2-th
G = O(n) ~ G I ( N , ~ ) ,
g = I
Lie algebra of matrices.
.
classes.
2. g
Pk/2
A 6~n,~)
Pk/2
to
A/'(n)
Pl 6 I21(O(n)),
that since every
the Chern-Weil Gl(n,~)-bundle
of
Gl(n,~)-bundle
image of although
Pk/2
has
for
k
the polynomials
~(n,]R). G = SO(n) ~ O(n),
satisfying
det(g)
= I.
so again we have the Pontrjagin
the subgroup of orthogonal The Lie algebra
polynomials
~(n)
= 4F (n)
P1 6 I21(SO(n)
,
1 = 0,1 ..... [~]. Now suppose homogeneous
n
is even,
polynomial
Pf
n = 2m,
and consider
(for Pfaffian)
the
of degree
m
glven
by
(4.12) where where
Pf(A, .... A)
_
I !(sgn ~ ) a .a 22m mm ~ I~2"" (2m-1)o(2m)
the sum is over all permutations A = {aij}
satisfies
aij =
of
-aji-
1,2,...,2m,
and
67 In o r d e r t h a t if
to see
g = {xij}
that
Pf
6 SO(n)
is i n v a r i a n t
first notice
then
gAg -I = gA tg = A'
where
A' : {a]~} ±J
is g i v e n by
a ~13 .
[k I , k 2 X i k l a k l k 2 X j k 2
=
so
Pf(A', .... A')
=
[ k2maklk2...ak2m_ik2m • kl,...,
[ s g n ( ° ) X o I k l X o 2 k 2 " ' ' x o ( 2 m - 1)k2m_1
xo(2m)k2m The c o e f f i c i e n t matrix
of
{Xik
}. 3 is a p e r m u t a t i o n
the p e r m u t a t i o n = Pf(A,...,A) if
det{xij}
aklk2...a k2m_Ik2 m
This determinant of
I...2m
since so
Pf
= -I
Pf
later this
det{xij}
is the c o n t e n t
= Hom(~n,~n). polynomials polynomial
= I.
Hence
Pf(A',...,A')
polynomial.
=
Notice
that
= -Pf(A,...,A)
polynomial
of the c l a s s i c a l
G = GI(n,C)
which
for
i m a g e of
has L i e
Here we consider Ck
(k I .... ,k2m)
then
s h o w t h a t the C h e r n - W e i l
4.
of the
case it is the sign of
is an i n v a r i a n t
is not an i n v a r i a n t
Example
is zero u n l e s s
in w h i c h
P f ( g A g - 1 , . . . , g A g -I)
so
is the d e t e r m i n a n t
O(n).
Pf
is the E u l e r
Gauss-Bonnet
algebra
to
class;
theorem.
/n,~)
the c o m p l e x v a l u e d
are the c o e f f i c i e n t s
We shall
invariant
I n-k
in the
68
(4.13)
det(l'1
where
A
is an
The C h e r n - W e i l classes
with
classes.
I
n × n image
Notice
matrix
of t h e s e
complex
: [ Ck(A, ,A) I n'k "" " k
A)
2~i
of c o m p l e x polynomials
coefficients
that
numbers
and
give
they
the r e s t r i c t i o n
of
and
characteristic
are c a l l e d Ck
i = /:~.
to
the C h e r n
,;/(n,~)
7 satisfy
(4.14)
i k C k ( A ..... A)
It f o l l o w s is
1
(-I)
that
the
times
l-th
the
extension
g
such
The
Lie
that
5.
g t~ is
matrices,
Chern
class
A 6 ~(n,m).
of a of
Gl(n,~)-bundle
the c o m p l e x i f i c a t i o n .
Gl(n,~)-bundle
is the
Gl(n,~)).
= I ~
class
a principal
G = U(n)
algebra
hermitian
of
to the g r o u p
Example
Pontrjagin
21-th
(The c o m p l e x i f i c a t i o n
= P k / 2 ( A ..... A),
~ Gl(n,f) (g
is the
is the c o m p l e x
(n) ~ ( n , ~ ) ,
that
is,
the
A 6 ~(n)
subgroup conjugate
subalgebra satisfy
of m a t r i c e s of
g).
of skew-
A = _t~.
Therefore det(l'1
- ~
1
A)
= det(l'1
+ ~
when
the p o l y n o m i a l s restricted
naturally
to
in r e a l
Exercise
2.
Ck
~(n).
defined The
cohomology
Let
V
by
(4.13)
Chern-Weil
tA)--
I A), 2~i
= det(l.1 hence
I
are
image
real
A 6 ~(n)
valued
therefore
again.
be a f i n i t e
dimensional
vector
Let
T*(V)
be
the
tensor
algebra
of
=
V,
lies
_[]_ V ®k k~O i.e.
the g r a d e d
algebra
with
space.
69
Tk(v)
= V ®...@
V
(k
Tk(v)
The
symmetric
where
I
v ® w - w ® v. S k(v)
and
a) then space
V
ideal
generated
that
T*(V)/
image
of
V*
: E ~ M a)
(2k-1)-form
that
for
TP(@)
on
(4.15)
(Hint: (where flat
dTP(@)
Observe ~
that
: ~*E ~ E
connection
such
vectorspace
of
in
sk(v*), k
V
the v e c t o r -
variables.
and J.
®
S j (W) .
Simons
with
[9]).
Let
connection
there
8.
is a " c a n o n i c a l "
that
= p(~k) .
z*E
has
two c o n n e c t i o n s :
is the m a p of
induced
is d e n o t e d V.
to
form
V, W
P 6 Ik(G) E
power
the
of
~ si(v) i+j=k
G-bundle
of
S *(V)
symmetric
forms
Chern
be a principal Show
in
for v e c t o r s p a c e s
(S.-S.
elements
isomorphic
S k ( V ® W) ~
3.
Tk(v)
multilinear
that
Exercise
by all
is the d u a l
is n a t u r a l l y
Show
product
I
the k - t h
if
the n a t u r a l
is the q u o t i e n t
=
of s y m m e t r i c b)
Tk+I(v).
~
S*(V)
The
S k(v*)
TI(v)
of
is c a l l e d
Show
and w i t h
®
algebra
is the
factors)
from
total
spaces)
the c a n o n i c a l
8 1 = ~*@ and
@0
trivialization
the of
~*E). b)
Suppose
(4.16)
c)
f
: N ~ M
TP(f*e)
Show
that
TP(8)
is
covered
by
f
E
by
= f*TP(8).
is g i v e n
on
:
f*E
~ E.
Then
70
1 (4.17)
TP(e)
= k
k-1 ^ Us )
P(e s=0
where
U s = s~
Exercise and
let a)
defined
e~
+
½ ( s 2 - s) [ 8 , 8 ] .
4.
Let
: ?~
Show
~
that
e
: H ~ G
be e~
the
be
a Lie
associated
induces
a map
group
Lie e•
algebra
: I~(G)
Suppose
~ : E ~ M.
Show
(4.18)
~
: F ~ M
that
for
is
an
H-bundle
,
with
P E Ik(G). G-extension
P E I~(G)
mF(a*p ) : mE(P ) .
Note. one
~ I~(H)
= P ( e ~ v I ..... e , v k) Vl,...,v k E~.
the
homomorphism.
by e ~ P ( v I , .... v k)
b)
homomorphism
by
Our S.
exposition
Kobayashi
and
of K.
the
Chern-Weil
Nomizu
[17,
construction Chapter
XII].
follows
5.
Topological
In this notion
of
section
space
the w o r d s
G
X
denotes
by
this
the f o l l o w i n g
"continuous"
homomorphism G-bundles
principal
"differentiable"
defines
and
and
only
the u n d e r l y i n g
section point
we
shall
of v i e w .
coefficients principal
study
domain
Definition
5.1.
A
to e v e r y
principal
G-bundles
~
the Chern-Weil
of
topological
such
for e v e r y
continuous
classes
In this
cohomology
is a s s u m e d take
with
to be a
A = ~).
class
f
chapter,
G-bundle.
c
class
a cohomology map
of
denotes
isomorphism
: E ~ X
of
from a general
(we s h a l l m a i n l y
associates
are
classes
which
A characteristic
3.1,
purpose
in the p r e v i o u s
topological
ring
G-bundles
that
The
that
classes
H*
on a
as in D e f i n i t i o n
is to s h o w
as d e f i n e d
The
: E ~ X
"homeomorphism".
In the f o l l o w i n g
ideal
n
"diffeomorphism"
characteristic
in a f i x e d
as b e f o r e .
the c h a r a c t e r i s t i c
bundles,
on
exactly
characteristic
differentiable depend
G-bundle
and
section
in p a r t i c u l a r ,
spaces
a Lie group
is d e f i n e d
replaced and
and classifying
a topological
topological only
bundles
for principal of
topological
class
: Y ~ x
c(E)
a n d for
6 H*(X), ~
: E ~ X
a G-bundle
(5.2)
c(f*(E))
We shall called classes in
show
that
the c l a s s i f y i n g are
in
H*(BG). As u s u a l
I-I
The
there space
is a t o p o l o g i c a l for
correspondence
construction
An ~
= f*c(E).
~n+1
G
such
with
that
space
BG,
the c h a r a c t e r i s t i c
the c o h o m o l o g y
classes
is as f o l l o w s :
is the
standard
n-simplex
with
bary-
72
centric (n+1
coordinates times)
and
t =
(t0,...,tn)
.
Let
G n+1
= G
x...x
G
let
EG
=
~
An
x Gn+I/~
n~0 with
the
following
idenfitications:
(£1t' (g0 ..... g n ))
~
(t' (g0 ..... gi ..... g n )) ' t 6 A n-1 , g 0 , . . . , g n 6 G,
Now
G
acts
on
the
right
on
( t , ( g 0 ..... g n ) ) g
and
we
let
BG
= EG/G
Proposition
Proof. free
(i.e.
xg
furthermore
let in
Let
F
be
F~ ~
F
x F
the
defined
same
if
T
easy
(compare
It enough
~
: EG
~ BG
~ BG
that
the
action
I),
and
it
strongly
is
of
the
in
a natural action
is
The
Chapter
3):
be
a space
with
is
a
projection.
G-bundle.
G
on
EG
to
see
that
the
sense:
x G ~ F
with
x
map
T
said
to
following
and
and
: F~ ~ be
y G
strongly
lemma
is
free
G-
a strongly
trivial
is
following F
(x,y)
of
: F ~ F/G
easy
G-action
continuous.
F
of
is
pairs is
the
a principal
free
a free
and
2
action
G-bundle
iff
~
Proposition
5.3
it
has
section.
follows
that
construct
Equivalently,
is
the
: EG
there
x T(x,y)
Let
YG
set
Then
by
0,...,n.
(t,(g0g,...,gng))
with
the
Exercise
Then
to
be
5.4.
a continuous
is
a space
: F~ ~ G
Lemma
g =
action
y =
free
action.
notice
orbit.
by
YG
= x ~
the
=
with
5.3.
First
EG
i =
for
in o r d e r local
any
to
show
sections
point
x 6 EG
of
YG we
: EG
shall
is
~ BG. find
a G-invariant
73
open
neighbourhood
which
is
(then
equivariant
the map For
U
y ~
this
we
of with
shall
is a c l o s e d
subspace
continuous,
there
in
M ~
~q
with as F
X.
any manifold
such
that
there
and
f
Now required EG(n)
is
: A ~ M
~ M
let above
~ EG,
by
G
space of
of
i.e. X
f
and we
the
shall
N
there
the
no
be
smallest
A n0
x G n0+1
is
A ~ X is
M
an
in
~q A ~
then
U
and
the
of
space
extension
neighbourhood
image
of
whenever
is
and
successively
is
it
: A ~ G
of
Hence
construct
EG(n)
f
G
: U ~ U/G).
in a E u c l i d e a n
theorem)
to
on
to a n e i g h b o u r h o o d
(rim = id) °
f
¥G
: U ~ G
whenever
and
embeds
extension
h
is a m a n i f o l d
(ANR),
M
constructing
where
right
a section
is c o n t i n u o u s ,
extends
x 6 EG
G-action
the
a neighbourhood
: N ~ M
(by T i t z e ' s
: F-I(N)
to
since
a normal
fact
r
map
retract
In
: X ~ ~q
r o F
that
is a n e x t e n s i o n
a retraction
above
of
a continuous
defines
use
neighbourhood
and
respect
yh(y) -1
an absolute
A
x
F-I(N) . h
as
restrictions
~ Ak k
x G k+1
to
in
EG. First
let
represented
in
x = Then
all
V
integer
such
that
x
is
by
((t o ..... tn0), (g 0 ..... g n 0 ) ) .
t O , .... tn0
neighbourhood
the
> 0
of
and
we
can
(t 0 , . . . , t n 0 )
clearly such
that
find
an o p e n
V c= i n t ( A n 0 ) .
Define n0+1 Un0
and
let
h
: U nO
coordinate Now
of let
= V
x G
~ G no no+1
be
~ E G ( n 0)
the map
which
project
onto
the
first
G n > nO
and
suppose
we
have
defined
an
invariant
X
74
open
set
Let
p
Un_ I ~ EG(n-I)
: 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
p-1(Un_1) U'
= W"
Un
h"
= Un_ I
clearly h
n
U =
: U
n W"'
n
We
can
I-I
for
h =
now
open
assume
and
an
Let
principal
correspondence.
is
an
The
the
map
G-bundles
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
G-invariant
: 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 0-I) )'go"
invariant
U
the
~
open
: W ~
open
Then
= p-1(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'
G-invariant.
x G n+1
hn_ 1
This
and
Theorem c
EG(n-I).
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 = p-1(Un_1).
G-invariant
W
a G-invariant
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 n-l,
(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
1-I
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+1-times),
NG(q)
= G x...x
(q-times).
consists
of o n e
e i : NG(q)
G
element,
~ NG(q-1)
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(q-I)
)'
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
0-tuple
~ NG(q+I)
=
EG =
: NG ~ NG
by
i =
)'
I,...,q-I
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,...,n-I.
Here
iff
e. : NC(n) l
~
source
~ NC(n-1)
(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
'fn-1 )'
i = n
! <(f1' and
Bi
: NC(n)
~ NC(n+I)
~i(fl,...,fn) Remark of
I.
category
categories
2.
with
Observe
in E x a m p l e
Example
4.
Example
3 is e x a c t l y
follows:
y y
: G ~ G
study
U = {U
(x,U
0) ~
the
and
MoAG
= go
and
(g0,gl)
5.
of
= G × G,
the
functor
The
following
an o p e n
covering
category
(x,U) i)
O b ( x u) =
with
of
X.
NG
defined G
=
and
M o r ( x u)
=
(g0,gl)
called
as
= g1' Finally y)
space with
is a u n i q u e
U i
That
is,
n u
(~0,~i) S0
in
and U
there
An o b j e c t
there
u
in
4 is u s e f u l
as f o l l o w s :
Jl
in
defined
(g0,g2) .
(also
as
as d e f i n e d
Associated
defined
x = y 6 Ua0D ,
NG
be a t o p o l o g i c a l
x 6 U
iff II u
XH
G
c a s e of E x a m p l e X
is a t o p o l o g i c a l
source
(gl,g2)
by
Let
of
the c a t e g o r y
0
category
continuous
that
space
-I Y(g0'gl ) = go gl
of G - b u n d l e s .
(y,U
it f o l l o w s
simplicial of
are
group
the n e r v e
= G
is a t o p o l o g i c a l is a p a i r
and
the
spaces.
a topological
nerve
given
}d6 Z
simplicial
3 is e x a c t l y
is the n e r v e
Example the
of
i = 0 .... ,n.
from
the m o r p h i s m s
the
0b(G) (g0,gl)
is a f u n c t o r
(where
that
Furthermore
: NG ~ NG
N
just o n e o b j e c t
defined
target
that
to the c a t e g o r y
Remark
by
( f l , . . . , f i _ 1 , i d , f i .... ,fn ) ,
Notice
topological
functors)
=
is g i v e n
~I
morphism
79 where
the d i s j o i n t
with
Ua0D
union
Ual ~ @.
n-simplices
is t a k e n o v e r
all p a i r s
In the s i m p l i c i a l
space
=
~
U
N
...
A
(e 0 ..... a n) a0 where again
the d i s j o i n t
(e0,...,an)
with
that w h e n
U = {X}
in E x a m p l e
2.
N o w let
already then
(3.2) define
gab
: U
~e
: ~
a continuous
-I
(U)
where
Notice
~ U
x G
also
space considered
principal
G-bundle of
and t r a n s i t i o n
t h a t the c o c y c l e
t h a t the t r a n s i t i o n
of t o p o l o g i c a l
O[/(E)
that this
be an o p e n c o v e r i n g
Notice
by s a y i n g
functor
I.
is the s i m p l i c i a l
A U 8 ~ G.
can be e x p r e s s e d
Notice
in C h a p t e r
U = {U }
(n+1)-tuples
The face and d e g e n e r a c y
be a t o p o l o g i c a l
and let
trivializations
NX
U
an
inclusions.
appeared
z : E ~ X
a Lie group)
functions
the set of
is t a k e n o v e r all
are g i v e n by n a t u r a l space
with
union
Ua0 D ... n Uan % @.
simplicial
(G
NX u
is
NXu(n)
operators
(a0,a I)
condition functions
categories
: X U -, G
Su(E) IUa0 N Ual
the c o v e r i n g
. S i m i l a r l y let V {V } be = ga0al = -I of the total s p a c e E by V = ~ (U). Then
the t r i v i a l i z a t i o n s
{~a}
defines
~U (E)
a functor
: EV ~
where ~u(E) IVa0n Va I = (here
7 2 : Va0 Q Val
factor). continuous
Finally functor
x G ~ G
is the p r o j e c t i o n
the p r o j e c t i o n ~U
(~2 0 ~ a 0 ' ~ 2 ~ a l )
: E~ ~ X u
~ : E ~ X
on the s e c o n d
induces
a
s u c h t h a t the d i a g r a m
80
~U
EV
,
(5.10)
~u XU commutes.
Also we have
, G
the
commutative
E
~
EV
X
~
XU
diagram
(5.11)
where
the h o r i z o n t a l
nerves
maps
and realizations
commutative
are
induced
we get
from
by
the
(5.10)
inclusions.
and
(5.11)
Taking
a
diagram
E =
INEI
~U
II N E V II
, II N G
II =
EG
, II N G
II =
BG
I (5.12)
II ~U II1
eU X =
where
fu =
is i n d u c e d
4'
INXl 4.
II ~ u ( E ) I I , by
fu
II NX U II
fu
= II ~ u ( E ) I I
the p r o j e c t i o n
on the
and
second
cU
: II N X u II ~ X
factor
in
1~ A n x NXu(n) . N o t i c e t h a t the u p p e r h o r i z o n t a l m a p s in (5.12) n a r e e q u i v a r i a n t a n d u s i n g L e m m a 5.4 it is e a s i l y s e e n t h a t t h e map
II z U II
in the m i d d l e
(5.13)
~E
For (5.12)
the p r o o f
of T h e o r e m
in c o h o m o l o g y .
cohomology
of t h e
the r e m a i n d e r notation: denotes
is a p r i n c i p a l
the
For
5.5 w e
shall
More generally
chapter
a topological
s e t of c o n t i n u o u s
Therefore
= fiE(G).
fat realization
of t h i s
G-bundle.
we
study
l e t us
study
of a s i m p l i c i a l shall
space
X,
singular
use S
q
the d i a g r a m
the
the space.
In
following
(X) = s t ° P ( x ) q
q-simplices,
and
for
A
81
a fixed
ring
cochains
cq(x)
with
= cq(X,A)
coefficients
Hq(X)
Now consider denote
= Hq(x,A)
coboundary
= Hq(c*(x,A)).
X = {Xp}
6'
6"
in C h a p t e r
is g i v e n
I
Example then C~ 'q C~
6.
If
the d o u b l e
manifold
: X ~ X'
induces
C~
that (that
a map
and
(-I) p
CP'q(x)
the
total
(Xp+1 complex
is a c o v e r i n g
C P ' q ( N X U) that
Cq ~
in C h a p t e r singular
a simplicial
map
f = {fp}
of d o u b l e
)
"
of
{CP'q(x)}.
the d o u b l e
I we considered
C~
f
the
of a s p a c e
is e x a c t l y
denoted
is,
times
the h o r i z o n t a l
: cq(XP )
U = {U } 6 Z
I (except
and
let
by
denotes
complex
of C h a p t e r
Notice f
C~(X)
is
C~(Xp)
6' = p+1 [ (-I) is ~ i=O 1 AS
and
= cq(Xp) .
differential
in the c o m p l e x
differential
and
space
CP'q(c)
the v e r t i c a l
A
the set of s i n g u l a r
complex
(5.14)
Here
in
a simplicial
the d o u b l e
denotes
X complex a
cochains).
of
simplicial
spaces
: X ~ X' is c o n t i n u o u s for all p) P P P complexes f~ : C P ' q ( x ') ~ CP'q(x).
We n o w h a v e
Proposition
5.15.
Let
X = {Xp}
be a s i m p l i c i a l
space.
Then
H*([l X H) ~ H(C*(X)). Furthermore
this
is a s i m p l i c i a l
isomorphism map
of
is n a t u r a l ,
simplicial
spaces
f
: X~
i.e.
if
then
the d i a g r a m
X'
82 H*([[ X' I])
H(C*(x')
I [I f [[*
~f*
H*(II X [[) commutes,
where
Sketch
f*
Proof.
is a C . W . - c o m p l e x the g r o u p forward
is i n d u c e d
First with
of c e l l u l a r
to c h e c k
that
p-chains
complex
cohomology
of the c o m p l e x
see A.
even
and
the
the n a t u r a l
6"
identity
Vq,
o(t)
6 Y).
isomorphism
C
(X)
P boundary
ei(o)
Dold
Therefore it is s t r a i g h t -
is g i v e n
o 6 X
by
. P
[10,
Chapter
V,
isomorphic On
§§
I and
with
the o t h e r
the
hand
for
hence
= Hom(Cp{X),A),
q
. P and
JR X l[
Vq,
~ CP'q+1(X)
is zero
Therefore
by C o r o l l a r y
odd.
for
q 1.20
p (p
= cP'0(X)
c cP(x)
on h o m o l o g y .
This
proves
the p r o p o s i t i o n
case.
In p a r t i c u l a r
cohomology
i
: CP'q(x)
for
an i s o m o r p h i s m
map
just
x 6 X
Then
inclusion
in the d i s c r e t e
natural
each
Hom(C,(X),A).
cP(x)
induces
is d e s c r e t e .
is n a t u r a l l y
= Xp,
the d i f f e r e n t i a l
is
: C * ( X ') ~ C*(X).
'
CP'q(x)
and
X for
(-I)
H * ( H X [L)
Sq(Xp)
f#
the c e l l u l a r
that
discrete
by
a p-cell
It f o l l o w s
X
H(C*(X))
assume
2(0) = [ i (For the c e l l u l a r
~
if
Y
is a t o p o l o g i c a l
: 11S(Y) II ~ Y is d e f i n e d
Notice
that
in h o m o l o g y
by
induces sending
by a s i m i l a r with
A
space
then
an i s o m o r p h i s m (t,o) argument
coefficients.
£ Ap p
× S
the in P
(Y)
induces
to an
6].
83
Now
for a g e n e r a l
the d o u b l e face
simplicial
simplicial set
S(X)
space
X = {Xp}
= {Sq(Xp)},
consider
that
is we h a v e
operators
e~l : Sq PX ~ Sq p-1'X i = 0,...,p,
j = 0,...,q,
~'!3 : Sq(Xp)
such
~ Sq_ I (Xp)
that
~! o ~'[ = ~'~ 0 ~ l 3 3 i and
similarly
simplicial
for
the d e g e n e r a c y
set w e h a v e
II S X II =
with and
suitable the
set of n - c e l l s Again
the o t h e r
realization
of
is a n a t u r a l is d e f i n e d isomorphism
double
× Aq
x S
(Xp)/~ q
Again
in
I-I
checks
this
is a C . W ° - c o m p l e x
correspondence
that
with
H * ( H S(X) ]J)
is i s o m o r p h i c
hand
the
LI S(X) II
simplicial
simplicial
above
and,
map
is h o m o e m o r p h i c
space
in h o m o l o g y .
{ll S(Xp)li }.
P = {Dp}
as r e m a r k e d The
with
where
there,
pp
induces
proposition
now
Now
the
fat
there
: IJ S(Xp) 11 ~ Xp an
follows
from
following
Lemma simplicial isomorphism Then
~P
are
one
this
H(C~(x)) . On
the
~ p,q~0
For
fat r e a l i z a t i o n
identifications.
Sq(Xp). p+q=n with
the
operators.
5.16.
Let
f
spaces
such
that
in h o m o l o g y
IJ f i[ : J1X II ~
in h o m o l o g y
as w e l l
: X ~ X'
with II x' II
be a s i m p l i c i a l
map
of
: X ~ X' i n d u c e s an P P P c o e f f i c i e n t s in A for all f
also
induces
as in c o h o m o l o g y
with
P.
an i s o m o r p h i s m coefficients
in
A.
84
Proof. Then the
Let
II X II (n) filtration,
[I X II (n) c l] X il b e the i m a g e = is a f i l t r a t i o n of i[ X II and that
it is e a s y
to see
(A n
induces
× Xn,
induces
that
~A n
an i s o m o r p h i s m
Jl f IL :
x Xn)
II x il ~
]i X' II
induces
coefficient
thus
the p r o o f
Corollary
p
there
are continuous
il f o i l Proof.
f0 let
*
=
theorem
Now
II f l
and
ii (n) , ii X'[[
iterated
of
hi
]i*
: H*(ll
In f a c t c o n s i d e r
: cp'q xl
S p + I : c P + I , q (X')
the
X'
il f II :
in h o m o l o g y . now
: X ~ X'
follows,
are
spaces X'p+1'
and
ll)
for e a c h
i = 0, "'" ,p,
~ H*(ll
2).
X ll).
maps
cp'q<x
P ih#% [ (-I) i i=O
By
simplicially
(i.e.,
of C h a p t e r
induced
~ C P , q (X)
the
5.15.
: Xp 2b)
u s e of
therefore
simplicial
maps
(n-l))
II X' [i (n), n = 1 , 2 , . . . ,
the r e s u l t
f0,fl
of E x e r c i s e
Sp+ I =
(ll X'
of P r o p o s i t i o n
maps
i) - iii)
by a s s u m p t i o n
an i s o m o r p h i s m
Suppose
simplicial
satisfying
and
5.17.
~
in h o m o l o g y
the U n i v e r s a l finishes
hence
[I X II ( n - l ) )
II f II : i] X II (n) ~
an i s o m o r p h i s m also
map
(II X II (n),
in h o m o l o g y .
that
induces
homotopic
~
(][ X 11 (n) , ]i X [; ( n - l ) )
shows
-~ I] X' [I (n).
the n a t u r a l
in h o m o l o g y ,
an i s o m o r p h i s m
five-lemma
~ £k × Xkk
is,
II f il : II X [[ (n)
Now
of
be defined
by
Then
85 Then
Sp+l as in E x e r c i s e
o 6' + 6'
2 of C h a p t e r
0 Sp
2.
f
- f
o
Furthermore
Sp+ I 0 ~" + ~" o Sp+ I = 0 since
f~
h~l
and
induce
are
f~ the
and
covering ~e
: ~
let
~ of
~ C'iX)
that
X
there
5.5.
that
also
is the
let
there
homotopic
c
are
and c o n s i d e r
is a c o m m u t a t i v e
H*(X) eC
chain
be a p r i n c i p a l
such × G
First
H*(II NX u II)
where
are
It f o l l o w s
that
and hence
in h o m o l o g y .
: E ~ X
(U s) ~ U
C * ( X p!+ I) ~ C*(Xp).
maps
of T h e o r e m
U
-I
Notice
: C*(X') same m a p
Proof class
chain
~
eC
be a c h a r a c t e r i s t i c G-bundle.
Choose
a
trivializations
the d i a g r a m
(5.12)
above.
diagram
H(C*(NXu))
, H(C~)
isomorphism
of L e m m a
1.25,
so t h a t
e~
is
an i s o m o r p h i s m . N o w by n a t u r a l i t y
(5.18)
c~(c(E))
of
c
= f~(c(EG))
and
since
c~
is an i s o m o r p h i s m
by
c(EG)
and
equation
On
the o t h e r
principal
(5.19)
G-bundle
hand
is u n i q u e l y
determined
(5.18). let
the c l a s s
e ~(c(E))
c(E)
c o 6 H*(BG) c(E)
= f~(c0).
by
and d e f i n e
for a
88 we must show that Now if of
X
is well defined:
U = {U }a6 Z
then
and clearly
c(E)
W = {U
and
U' = {U;}B6Z,
D U;}(~,B)6ZxZ,
there is a commutative
are two coverings
is also a covering of
X
diagram
IINX W II
/l
(5.20)
II NXu,e U ' ~ ~ X ~ IE[ W
Also let ~W
fw : It NXwIi
~ BG
II
UIL
be the realization
is given by the transition
trivializations
x
~ IU D U;.
functions
of
N~W
corresponding
Then clearly
where to the
there is a commutative
diagram
NX W II
fw
(5.21)
-'-"~BG
NX U II
~
From the diagram (5.20) and (5.21) i t follows t h a t i t is enough to show that for any covering f~(c 0) 6 H*(Ii NXu I ) of trivializations So let associated
{~ } to
corresponding
in cohomology. 1
: U
the element
does not depend on the particular
and
continuous
{~'}
be two sets of trivializations
and let functors.
~,~'
: XU ~ G
induce the same map
Now the family of continuous maps defined by
be the
We want to show that the
fu' fu : II N X U il ~ BG
~ G, ~ 6 ~,
choices
{~ }.
U = {U s}
associated maps
U
87
<0' (~ 0
=
(X,l
(x)'g)
(x,g)
t
6 U
x G,
satisfy
4 ~ 0 ,~1)(x) Hence
• I 1(x)
I = {I }e6 ~
1 : ~ ~ ~' therefore
= I 0(x)
is just a c o n t i n u o u s
of the f u n c t o r s follows
• 4(~0,~i)(x),
4
and
from Corollary
natural
4'.
That
x £ U 0N U i transformation
f~ = f6*
5.17 and the f o l l o w i n g
general
lemma:
Lemma
5.20.
of t o p o l o g i c a l continuous
satisfying simplex
in
4,4'
categories
natural
simplicially
Proof.
Let
C, D.
We shall i) - iii)
be two c o n t i n u o u s
If
transformation
homotopic
NC
: C ~ D
1 : 4 ~ 4'
then
simplicial
construct
: NC ~ ND
1
: NC(p)
2b)
~ ND(p+I) , i = 0,...,p,
in C h a p t e r
f
f2 Ale
i = 0,...,p,
simplex
in
A2~
P
......
ND
h. 1
associates
to this
A0,...,A p 60b
Ap,
string
6 M0r(C).
the
(p+1)-
g i v e n by the s t r i n g
4' (f I) @, (A0) ~
N o w a p-
2.
f0,...,fp For
are
is a s t r i n g
fl A0 ~
is
maps.
h
of E x e r c i s e
N4,N4'
functors
4' (f i) 4, (At) ~. . . . .
•
4, (Ai) 4
IA. 1
4(fi+1 ) 4(Ai)~"
4(fp)
h i : NC(p)
~ ND(p+I)
is c l e a r l y
forward
to c h e c k
the i d e n t i t i e s
Chapter
2.
proves
This
. • • ~
4 (Ap)
continuous
and
i) - iii)
the lemma.
.
it is s t r a i g h t -
of E x e r c i s e
2b)
in
(C),
88
It follows and it is easily condition
Note. is due the one
(5.2).
that
c(E)
checked This
that
ends
The o r i g i n a l
to J. M i l n o r
[20].
in G. Segal
[24].
defined
by
c(E)
the proof
construction Our
(5.19)
satisfies
is well
the n a t u r a l i t y
of T h e o r e m
5.5.
of a c l a s s i f y i n g
exposition
defined
follows
space
essentially
6.
Simplicial
In t h i s
manifolds.
chapter
coefficients. Chern-Weil
H•
homomorphism
and
is a s i m p l i c i a l
C
is n o t
: I~(G)
a manifold.
a n d all
cohomology
However, That
but
BG = is,
BG
real
G
the
the t r o u b l e
II N ( G ) I f ,
X = {Xq}
manifold
and degeneracy
for
with
for a L i e g r o u p
a simplicial
face
homomorphism
~ H~(BG) ;
manifold.
s e t is c a l l e d
manifolds
denotes
to d e f i n e
w
BG
s implicial
again
We now want
is t h a t NG
The Chern-Weil
a
if all
operators
Xq
are
are C
maps.
Example simplicial
I.
Again
manifold
a simplicial
with
all
X
set
X = {Xq}
considered
q
as
is a
zero dimensional
manifolds.
Example space
NM
equal
2.
with
to the
Example and
NG
U = {U } 6 E manifold. G-bundle
with
M
= M
and
3.
G
For
C~
all
simplicial
M
For
a
the
simplicial ~
C~
the corresponding simplicial
operators
manifold.
simplicial
and
manifold
space : E ~ M
differentiable of
manifolds
simplicial
and degeneracy
a simplicial
the
the
~
spaces
: NG ~ NG
NG
is a
map.
4.
if
manifold
face
a Lie group
simplicial
the nerves
differentiable
is a
is a g a i n
Finally,
taking
obtain
NM(q)
are also
Example
if
identity
differentiable
then
Also
NM U
with
is a l s o
the d i a g r a m s
maps.
of
(5.10)
~
: ~
and
simplicial
covering
a simplicial
is a d i f f e r e n t i a b l e
trivializations
diagrams
an o p e n
-I
principal U s ~U
(5.11)
manifolds
we and
xG
9O Now let us study the c o h o m o l o g i c a l p r o p e r t i e s of a s i m p l i c i a l manifold,
in p a r t i c u l a r we want a de Rham theorem.
Again in this chapter for
M
a manifold
C*(M)
c o c h a i n complex with real coefficients based on
denotes the C
singular
simplices. Now c o n s i d e r a simplicial m a n i f o l d Chapter
5 we have the double complex
that by Lemma 1.19 and E x e r c i s e
X = {Xp}.
CP'q(x)
4 of Chapter
As in
= cq(Xp) .
Notice
I the natural map
C ~ o p ( X p) ~ cq(Xp) induces an i s o m o r p h i s m on h o m o l o g y of the total complexes. We also have the double complex the v e r t i c a l d i f f e r e n t i a l differential 6'
: Aq(xp)
in
A*(Xp)
d"
is
AP'q(x)
(-I) p
= Aq(Xp).
Here
times the exterior
and the h o r i z o n t a l d i f f e r e n t i a l
~ Aq(Xp+ I)
is d e f i n e d by
6' =
p+1 [ (-l)Zc~. i=O
F u r t h e r m o r e we have an i n t e g r a t i o n map
I X = AP,q(x)
~ CP'q(x )
w h i c h is clearly a map of double complexes. Lemma
1.19 we easily obtain
Proposition Then
By T h e o r e m 1.15 and
6.1.
I x : AP'q(x)
Let
~ CP'q(x)
H(A*(x))
X = {Xp}
be a simplicial manifold.
induces a natural i s o m o r p h i s m
~ H(C*(x))
~ H*(II x I I ) .
N o w there is even another double complex a s s o c i a t e d to a simplicial m a n i f o l d w h i c h g e n e r a l i z e s c o m p l e x of Chapter
2:
the simplicial de Rham
gl
Definition manifold on
6.2.
X = {Xp}
A p x Xp,
(6.3)
A simplicial
n-form
is a s e q u e n c e
such
~ =
~
on
{~(P)}
simplicial
of n - f o r m s
~(P)
that
( i x id)~
(p) =
(id x e i ) ~ ( P - 1 )
on
A p-I
i = 0,...,p,
Remark.
the
Notice
that
~ = {~(P)}
x Xp,
p = 0,1,2,...
defines
an n - f o r m
on
Jl A p x X and t h a t (6.3) is the n a t u r a l c o n d i t i o n for a f o r m p-J0 P on JJ X JJ in v i e w of the i d e n t i f i c a t i o n s (5.6). In the following
the r e s t r i c t i o n
~(P)
of
~
to
Ap x X
is a l s o P
denoted agrees
~. with
Let Again
Notice
also
Definition
An(X)
differential
d
for
X
discrete
Definition
6.2
2.8.
denote
the e x t e r i o r
that
the
set of s i m p l i c i a l
differential
: An(x)
on
~ An+I(x)
n-forms
on
X.
Ap x X and
defines a P a l s o w e h a v e the e x t e r i o r
multiplication ^
satisfying
: An(x)
the u s u a l
® Am(x)
identities.
The
complex
(A~(X),d)
a double
complex
(Ak'l(x),d',d").
Ak'I(x),
k+l
iff
= n
~ An+m(x)
is a c t u a l l y Here
~JA p x X
the
total
an n - f o r m
is l o c a l l y
of
the
complex ~
of
lies
in
form
P = [ ai where Ap
I"
..
ik'J1"
(t0,...,tp)
and
{xj }
are
.
. "Jl d t l I A .
as u s u a l local
are
the b a r y c e n t r i c
coordinates
that An(x)
and
that
=
~ k+l=n
A d t i k A d X j A. . . . I.
A k ' l (X)
in
Xp .
.^dx 31 coordinates
It is easy
in
to see
92
d = d'
where
d'
is
barycentric
over
exterior
coordinates
derivative Now
the
with
restricting
Ak
yields
a
is n o w
Theorem
there
and
are
chain
such
a map
(k,l)-form
(-I) k
times
to
Ak
: Ak'I(x)
~ Ak'l(x)
of
complexes.
double
For
and
each
× Xk
to the
and
natural
The
generalization
1
the
( A * ' I ( x ) , 6 ')
the exterior
integrating
two
are
chain
chain
of
following Theorem
2.16:
complexes
equivalent.
In
fact
maps
IA
: Ak'l(x)
~ Ak'l(x)
: E
sk
: Ak'I(x)
~ Ak-I,I(x)
6'
o I,
homotopies
that
(6.5)
I A o d'
(6.6)
d'
o E = E o 6 ',
(6.7)
I
o E = id
(6.8)
E
A
In p a r t i c u l a r
morphism (6.9)
respect
x-variables.
a strightforward
6.4.
( A * ' I ( x ) , d ')
is
with
a map
is c l e a r l y
theorem
d"
to t h e
IA
which
derivative
and
respect
+ d"
on
o
IA
=
-
I A o d" E o d"
id = S k + I o d'
I/X : A k ' l ( x ) the homology
H(A*(X))
~
the
total
H(A*(x))
= d"
+ d'
.-* A k ' I ( x )
of
= d"
N
o IA o E
0 Sk,
s k o d"
induces complexes
H*(ll X ll).
+ d"
a natural
0 s k = 0.
iso-
93
Also
l e t us
the f o l l o w i n g
Theorem where
state without
generalization
6.10.
the product
and where
As fold
on
the p r o d u c t
with
simp]icial
the n a t u r a l
by
~P
these
A...n
Corollary
U
the n a t u r a l
map
6.4
l e t us c o n s i d e r and
~ U
let
the natural
For
A~(M)
N...~
U
~0
~ A~(NM)
6.11.
A-product
NM U
of
the
a mani-
be
the
category
M U-
projections
induce
A~(M)
by the
to t h e n e r v e
ep
also
is m u l t i p l i c a t i v e
: ]INMuII ~ M
~0 and that
[11])
map
the n a t u r a l
× U
Dupont
is t h e c u p - p r o d u c t .
U = {U }
associated
eU
is i n d u c e d
is i n d u c e d
of T h e o r e m
L.
2.33:
(6.9)
the right
a covering
manifold
that
left
on
(see J.
of T h e o r e m
isomorphism
the
an application
M
Notice
The
proof
c M ~p =
map
~ A~(NMu) .
U = {U s}
~ A~(NMu)
an open
induces
covering
of
M
an i s o m o r p h i s m
in
homology.
Proof.
In f a c t
A~(M)
is t h e m a p
Now folds.
A
sequence
of Lemma
turn
z
: E
M = {Mp}
1.24.
G-bundle
~ M P
I A .~ A ~ ( N M U) = A~
to C h e r n - W e i l
simplicial P
E = {Ep},
....A ~ ( N M u )
eA
let us
the c o m p o s i t e
n
theory : E ~ M
of differentiable
for
simplicial
is of c o u r s e G-bundles
mania
where
P are
simplicial
manifolds,
x
: E ~ M
is
94
a simplicial by
g 6 G, R : E ~ M
differentiable : E ~ E,
g
map
6.2 above)
restricted
to
also right
is s i m p l i c i a l .
is t h e n a l - f o r m
Definition
and
with
Ap x E
0
on
E
coefficients
is a c o n n e c t i o n
multiplication
A connection (in the in ~
in
sense such
of
that
in the u s u a l
0
sense
in
P the bundle
Ap × E
~ ~P
× M
P Again for
we have
P 6 Ik(G)
representing
the c u r v a t u r e
we get
that Theorem
: NG
~ NG.
this bundle Let
There
%0
be
onto
the
i-th factor
Then
0
qi
is s i m p l y
: &p in
sense
(6.3).
Theorem
6.13.
3.10, and
for
~ G
be
in
over
&P
x NG(p)
+...+
are
and
let
%1
qi60 "
by
tp0p
the b a r y c e n t r i c
01A p x NG(p)
coordinates
is c l e a r l y from
in
a connection
(6.12)
that
@
summarize:
a) T h e r e
: I*(G)
P E Ik(G),
in the b u n d l e
the p r o j e c t i o n
i = 0,...,p,
it is a l s o o b v i o u s
We now
w
G-bundle
connection
connection
G p+I
(t0,...,tp)
By Proposition
that
form
simplicial
a canonical
x NG(p)
% = t000
in the usual
such
and
as f o l l o w s :
given
(6.12)
satisfies
a closed
3.14
H2k(ll M ll)
the
the M a u r e r - C a r t a n
let
as u s u a l
~
consider
is a c t u a l l y
constructed
Also
~P.
6 H2k(A*(M))
l e t us
G ~ pt.
where
6 A2k(M)
by
4.3 h o l d s .
In p a r t i c u l a r y
p(~k)
defined
form
a class
WE(P)
such
. P
is a c a n o n i c a l
homomorphism
~ H*(BG)
w(P)
is r e p r e s e n t e d
in
A2k(NG)
by
95
p(~k)
where
defined
by
b)
~
is t h e c u r v a t u r e
f o r m of
the c o n n e c t i o n
@
(6.12).
Let
for
P £ Ik(G),
characteristic
class.
differentiable
G-bundle
w(P) (')
Then
if
we
have
~
be the
: E ~ M
corresponding
is an o r d i n a r y
w(P) (E) = WE(P)
where
~*
wE
: I~(G)
c)
w
d)
Let
: I*(G)
~ H~(M)
: I~(G) ~
is the u s u a l C h e r n - W e i l
~ H*(BG)
: H ~ G
~ I*(H)
be
is an a l g e b r a
be the
a Lie group induced
homomorphism.
homomorphism.
homomorphism
map.
Then
and
let
the d i a g r a m
lw
I*(G)
-~ I*(H)
H* (BG)
, H* (BH)
commutes.
Proof. b)
a)
Choose
trivializations
is a d e f i n i t i o n . an o p e n c o v e r i n g of
of d i f f e r e n t i a b l e
E
U = {U s}
so t h a t w e h a v e
simplicial
NE
of
M
and
a commutative
diagram
bundles:
NE V Nz U
NM
By the p r o o f II N M U N in
of T h e o r e m
is g i v e n
H ( A ~ ( N M U)
by
NG
NM U
~
by
5.5
the p u l l
f~(w(P))
the C h e r n - W e i l
.
back
which image
of
clearly of
P
w(P) (E)
to
is r e p r e s e n t e d for
the
simplicial
96
G-bundle
NE V ~ NM U
connection in
E ~ M
8
defined
induces
the p u l l - b a c k
of
(6.12).
another
@'
in
induced
On the other
connection
@"
H(A*(NMu))
image of
by the a r g u m e n t
is i n d e p e n d e n t
connection
by
WE(P)
by the C h e r n - W e i l However,
with
P
using
of T h e o r e m
of the choice
in
from
hand
of connection,
and
represented
the c o n n e c t i o n b)
a connection
NE V ~ NM U
is clearly
4.3,
the
e"
the C h e r n - W e i l which
proves
image
that
£~(w(P) (E)) = e~(WE(P)),
where
e U : tl N M U II ~ M
Since
eU
proof
induces
is the natural
an i s o m o r p h i s m
in c o h o m o l o g y
this
above.
ends
the
of b). c)
follows
again
4.3 c) and T h e o r e m d)
the s i m p l i c i a l
Notice
and the proof
that by a), A*(NG)
w(P)
and
The c o n s t r u c t i o n
due to H. S h u l m a n
[26] g e n e r a l i z i n g
manifolds
[4],
and
follows
of T h e o r e m
[5]).
of
[11].
to the reader.
in
by c a n o n i c a l l y w(P)
in
a construction
The e x p o s i t i o n
J. L. D u p o n t
is left
is also r e p r e s e n t e d
C~(NG)
elements.
[2],
analogue
6.10.
the total c o m p l e x e s defined
from
is s t r a i g h t f o r w a r d
Note.
(see
map c o n s i d e r e d
in terms
A~(NG)
is
by R. Bott of simplicial
7.
Characteristic
We
shall
classes
now study
defined
Chern
For
valued
defined
by
we
define
(4.13).
represented
by
every
For
= W E ( C k)
form
then
since
Ck(E)
6.13 we
a real
(represented
inclusion
d)
to that
has
£ H2k(B
Again ~(n)
: BU(n)
in
the C h e r n
(C O = I), ~
: E ~ M
classes
where
Ck(~k), ~
: E ~ M.
a Hermitian lies
in the
by
Notice
metric, image
that
i.e.
a
of the
4 of C h a p t e r
the d e f i n i t i o n
4).
of the C h e r n
first
defining
Gl(n,~),~)
ck
is a real
is a real
by a r e a l
valued
c Gl(n,~)
that
Bj
called
the r e s t r i c t i o n
j : U(n)
4 the
Gl(n,~)-bundle
(cf. E x e r c i s e
extend
4 Example
k = 0,1,...,n
Gl(n,~)-bundle
5.5.
6.13
it f o l l o w s
4.
2k-forms
actually
can
from Theorem
natural
bundle
t_~opological
restricted
class
valued
c H2k(M,~)
use T h e o r e m
Ck
the c h a r a c t e r i s t i c
k = 0,1 ..... n,
of a c o n n e c t i o n
U(n),
to any
classes
£ H2k(M,~),
vector
H2k(M,~)
Ck,
a differentiable
c k = w(C k)
and
groups
in C h a p t e r
polynomials
the c o m p l e x
By T h e o r e m classes
of
of C h a p t e r
considered
complex
to
inclusion
we
characteristic
is the c u r v a t u r e
reduction
classical
the p r o p e r t i e s
invariant
Ck(E)
since
some
in the e x a m p l e s
G = Gl(n,f)
(7.1)
for
classes.
complex
thus
classes
~ B Gl(n,~)
class.
polynomial of
ck
form),
to
In f a c t
it f o l l o w s BU(n)
is
and s i n c e
the
is a h o m o t o p y
equivalence
98
induces
an i s o m o r p h i s m
Proposition two Lie groups (coefficients induces
7.2.
which
be a homomorphism
: H ~ G
an i s o m o r p h i s m
A) .
Then
in h o m o l o g y
By K~nneth's
follows
Before
e
also
we
have
of
in h o m o l o g y Be
: BH ~ B G
as
in c o h o m o l o g y
as w e l l
A).
an i s o m o r p h i s m
therefore
Let
in a P . I . D .
(with c o e f f i c i e n t s
induces
In g e n e r a l
induces
an i s o m o r p h i s m
Proof.
in c o h o m o l o g y .
formula
in h o m o l o g y
by Lemma
continuing
No(p) for
: NH(p)
each
~ NG(p)
p.
The
proposition
5.16.
the
study
of
the C h e r n
classes
we make
X
a principal
a few definitions: Suppose
we
consider
Gl(n,~)-bundle Then
~ : E ~ X
the Whitney
scribed
in t e r m s
First
a topological
sum of
and
(~ @ ~)
with
Gl(m,~}-bundle
~ : F ~ X.
: E @ F ~ X
is m o s t
functions
as f o l l o w s :
transition
easily
de-
let
: Gl(n,f)
be
a
space
the h o m o m o r p h i s m
taking
x Gl(m,~)
~ Gl(n+m,~)
a p a i r of m a t r i c e s
(A,B)
to the
matrix
Now
choose
a covering
and
F
have
and
{h 8}
and
F
with
transition
and
F
U = {Ue}d6 Z
trivializations be
over
the corresponding
respectively.
Then
functions
are differentiable
of
X
Ue,
such
e 6 Z,
transition
that both and
functions
~ S ~ : E S F ~ X {g~6 S h e s } . then
also
Notice
E @ F
let
is.
E
{ges} for
E
is the b u n d l e that
if
E
99
Notice group
that
of n o n - z e r o
GI(I,~)
= ~* = C ~
complex
numbers.
correspondence
with
called
line bundles).
complex
canonical
l-dimensional
line bundle
Here
~pn
under
the a c t i o n
{0},
Gl(1,~)-bundles complex
An
as
of
~*
(z0,z I .... ,z n)
example
space
of
to see
that
~n
• I =
is a p r i n c i p a l
{n+1
¢*-bundle.
~
The
is b y d e f i n i t i o n
the c a n o n i c a l
denoted
for H. H o p f ) .
We
H* n
(H
Chern
7.3. class
c(E)
is the
space C n+1
~pn. TM
{0}
I 6 ~*.
projection
{0} ~ ~ p n
associated
complex
line bundle.
The
line bundle
total
space
For
a
b e the
= c0(E)
Gl(n,¢)-bundle
~
: E ~ X
let t h e
sum
+ c I (E) + . . . +
Cn(E)
6 H*(X,~).
Then
a)
b) : E ~ X
(7.4)
(also
can now prove
Theorem total
I-I
(z0"l ..... Zn'l),
the natural
:
in
by
z 0 , . . . , z n 6 ~,
It is e a s y
are
bundles
projective
the q u o t i e n t given
vector
important
on the complex
is d e f i n e d
the m u l t i p l i c a t i v e
ci(E)
6 H2i(x,~),
c0(E)
= I
and
(Naturality). a
i = 0,1,...
ci(E) If
Gl(n,~)-bundle
c(f*E)
f
= 0
for
: Y ~ X then
= f*(c(E)).
i > n.
is c o n t i n u o u s
and
is
100
c)
(Whitney
Gl(n,{)-bundle
duality
and
[ : F ~ X
(7.5)
or
formula).
c(E
@ F)
~ : E ~ X
Gl(m,Q)-bundle
: c(E)
is a
then
• c(F)
equivalently
(7.5) '
Ck(E
d) line
@ F)
=
[ i+j=k
(Normalization).
bundle.
h
6 H2 ({Pn, ZZ)
n
Proof.
sum
a)
follows
c)
Let
E @ F
induce
by
the
Pl
: Gn
B(G
cj(F) ,
~n
k = 0,1 ..... n+m.
: H*n ~ { p n
be
the
canonical
n
I - hn
canonical
has
generator.
definition. 5.5.
= Gl(n,f)
is c l e a r l y
for
short.
a homomorphism a reduction
The
and
to
Gn
map
the Whitney x
G
m
.
projections
×
G
m
in t h e
Gn'
~
P2
:
G
n
x
G
m
~
G
m
diagram B
x G ) m
n
by
=
, BGn+ m
I Bpl x BP2
(7.7) BG
(7.5)'
shall
G
definition
with
the
Theorem
us w r i t e
the maps
(7.8)
is
trivial
from
x Gm ~ Gn+m
together
and
is
b)
Gn
~
Then
c(H~)
where
:
ci(E)
Let
(7.6)
We
a
If
n
× BG
will
m
clearly
follow
if w e
(B 8 ) * c k = i + j[= k (BPl) *ci ~ prove
this
by proving
the
can
prove
the
formula
(BP2) *cj .
.k .= .0 I.
corresponding
formula
.. n + m .
on
the
101
level of d i f f e r e n t i a l
forms
N(G n
in the d i a g r a m N
Gm)
x
-* N G n + m
(7.9) NG
NG
n
m
For this w e f i r s t n e e d s o m e n o t a t i o n . algebra
of
and let (6.]2)
G
e(n ) with
(i.e.
n
M
is
n
the
be the c a n o n i c a l ~(n)
the
Lie
M
connection
denote
n
algebra
corresponding
i I : M n ~ M n + m, be the i n c l u s i o n s
Let
of
in
n x n
NU n
curvature
the Lie
form.
matrices)
defined Also
by let
i 2 : M m ~ Mn+ m
g i v e n by
T h e n it is easy to see t h a t
(7.10)
(N~)*O(n+m)
and since
the Lie b r a c k e t
is zero it f o l l o w s
(7.1]) N o w for
=
n
of the two forms
(NP2)*(i 2 0 0(m ) ) on the r i g h t of
(7.10)
that
(N~)*~(n+m) A 6 M
(NPl)*(i I o 0(n ) ) +
=
and
(NPl)*(i I o ~(n))
B 6 M
2~i
<
11 -
= det
= det(ll
from which we conclude
(it(A)
+ i2(B)))
I
2---~A
0
I o
(NP2)*(i 2 0 ~(m)).
m
I
det(11
+
11
B/
-
2zil A ) d e t ( l l
- ~I
B)
I02
(7.12)
Ck(iiA
+ i2B,...,iiA
+ i2B)
C i ( A ..... A) "Cj(B ..... B) .
[
=
i+j=k Therefore
by
(7.13)
( N @ ) ~ C k ( ~ k) =
which
(7.11) we h a v e
clearly d)
implies
[ N p ~ C i ( ~ i) ^ Np~Cj(~J),. i+j =k
(7.8)
The restriction
morphism
and
hn
maps
and ends
map
to
the p r o o f
H2(~pn,~
hI,
which
k = 0,1 ..... n+m,
of c).
) ~ H 2 ( ~ P I ,ZZ ) by d e f i n i t i o n
is an iso-
is the class
such t h a t
where
~p1
2-form
is g i v e n
dx ^ dy
coordinate dinates
the c a n o n i c a l
where
it is c l e a r l y
the p r i n c i p a l nl
is c l e a r l y
coordinates
(7.14)
in
is the c o m p l e x
with
homogeneous
coor-
it is e a s i l y
enough
to p r o v e
bundle.
and c o n s i d e r
Let
complex that
the c u r v a t u r e
for
@
(z0,z I)
n = I,
be the
the c o m p l e x v a l u e d
(~0dz0 + ~ i d Z l ) / ( I z 0 12 +
checked
(7.6)
: f2 ~ {0} ~ ~pl
~2 \ {0}
the b a r d e n o t e s
conjugation
l-form
[zl ]2)
and
is a c o n n e c t i o n
[z] 2 = z~. and since
Then ~
is
f o r m is g i v e n by
(7.15)
N o w let
~p1
by the
~*-bundle
a differentiable
8 =
abelian
sphere
determined
(z0,zl) .
so w e c o n s i d e r
where
orientation
z = x + iy = Zl/Z 0
in the R i e m a n n
By n a t u r a l i t y
which
= 1
~ = d0.
U = ~PI~{ (0,I)}
coordinate
z = Zl/Z 0.
= { (z0,zl) Iz 0 % 0} Then
z I = z0z
and
and use the local dz I = zdz 0 + z0dz.
103
Hence = [z0dz0 + z 0 z ( z d z 0 + z 0 d z ) ] / i z 0 1 2 ( 1 dz0 (--~-0 +
=
Z dz0 )/( Iz'2 z0 + ~ d z I +
+
Lzl 2)
) dz0 Izl2. = z0 + --i+iz12 dz.
Therefore z
= d@ = d(
dz)
It f o l l o w s
t h a t in
U
c1(n) Therefore
Ci(~) I 2~i
=
(cf. E x e r c i s e
Now put
H
~
)
j
is g i v e n by dz ^ dz (i+izi2)2"
2 b) below)
~p1
z = r e 2~it
dz
(I+Iz12)2
r
dz^
-
1+zz
Then
I f ] 2~i ~
C I (~)
d z ^ dz (i+iz12)2
dz ^ dz = 4 ~ i r d r I
^ dt~
Hence
oo
dr d t 0 0
(I+r2) 2 dr
0 This proves
(7.6)
Remark. characterizes methods
and ends the p r o o f
the C h e r n
classes
under
It f o l l o w s
the n a t u r a l
Pontrjagin
For
7.3
By t o p o l o g i c a l ck ~ H2k(BGI(n,~),~)
classes
satisfy
t h a t t h e s e m a p to our C h e r n c l a s s e s ~ ~ ~.
classes.
invariant
For
see t h a t T h e o r e m
classes
characteristic
m a p i n d u c e d by
G = Gl(n,~)
realvalued (4.11).
uniquely.
o n e can s h o w t h a t t h e r e e x i s t
7.2.
of the t h e o r e m .
In the n e x t c h a p t e r we shall
such t h a t the c o r r e s p o n d i n g Theorem
I.
(1+r) 2
we c o n s i d e r e d polynomials
z : E ~ M
in C h a p t e r
Pk/2'
4 Example
k = 0,...,n,
a differentiable
Gl(n,~)
I the
defined
- b u n d l e we
by
104
defined
the P o n t r j a g i n
(7.16)
Pk/2(E)
represented form
= WE(Pk/2)
by the
= 0
for
topological
Pk/2
P k / 2 ( ~ k) ,
As n o t i c e d
k
odd.
Again
Gl(n,IR)-bundles
= W(Pk/2)
using
£ H2k(M,IR),
2k-forms
of a c o n n e c t i o n .
Pk/2(E)
and
classes
Theorem
is a h o m o t o p y
6 H2k(B
5.5.
where
in C h a p t e r we
extend
~
is the c u r v a t u r e
4 Example
2,
the d e f i n i t i o n
to all
by d e f i n i n g
GI(n,]R) ,~),
This
equivalence
k = 0,1 ..... n,
time
hence
the
k = 0,1 ..... n,
inclusion
by P r o p o s i t i o n
j : O(n) 7.2
~ GI(n,]R)
induces
an
isomorphism
(Bj)*
and is
since zero
: H*(B
for
k
odd
it f o l l o w s
The
proof
Theorem total
Gl(n,•)
of
Pontrjagin
Pk/2
that the
7.17.
,~)
~ H*(BO(n) ,JR) ,
restricted
Pk/2
= 0
following
For
class
a be
p(E)
= P0(E)
a)
Pi(E)
6 H4i(x,~)
P0(E)
= I
to the Lie
for
k
theorem
the
xr(n)
odd.
is left
Gl(n,]R)-bundle
algebra
to the reader.
~ : E ~ X
let
the
sum
+ pl (E) + . . . +
p[n/2] (E) 6 H*(X,IR)
.
Then
that
b)
Let
is,
the e x t e n s i o n
(7 • 18) C)
~
and
(Naturality).
=
i = 0,1 ....
Pi(E)
: E{ * X
Pi(E)
,
to
be
= 0
for
the c o m p l e x i f i c a t i o n
Gl(n,@).
f
: Y ~ X
: E ~ X,
of
Then
(-I) i c2i(E~) , If
i > n/2.
i = 0,~,... is c o n t i n u o u s
and
105
: E ~
X
is
a
Gl(n,]R)-bundle
(7.19)
p(f~E)
d)
(Whitney
Gl(n,]R)-bundle
duality
and
(7.20)
or
then
~
= f~p(E).
formula).
: F ~ X
p(E
8 F)
If
a
~
: E ~ X
Gl(m,]R)-bundle
= p(E)
is
a
then
• p(F),
equivalently
(7.20)'
Pk(E
8 F)
Pi(E)
=
~
pj(F),
k =
0 , 1 , 2 ..... [ ( n + m ) / 2 ] .
i+j=k The
Euler
Finally defined For
the
class.
consider
G = SO(2m).
invariant
polynomial
a differentiable
Euler
SO(2m)
e(E)
Again
we
extend
using
the
Theorem
Theorem
7.22.
6 H2m(X,]R) a)
a
by
bundle
= WE(Pf)
definition
z
the
4 Example Equation
: E ~ M
we
3 we
(4.12). define
the
by
putting
5.5.
We
For
topological
bundles
6 H2m(BSO(2m),]R)
then
{
6 H2m(M,]R).
to
e = w(Pf)
e(E)
Pf
Chapter
class
(7.21)
and
In
have
: E ~ X
a
SO(2m)-bundle
satisfies
(Naturality).
For
f
: Y ~ X
continuous
and
~
: E ~ X
SO(2m)-bundle
(7.23)
b) bundle
e(f*E)
(Whitney and
~
duality
: F ~ X
a
= f~e(E).
formula).
For
SO(21)-bundle
~
: E ~ X
a
SO(2m)-
106
(7.24)
e(E
c) be
the
For
= ~
U(m) @ i~
= e(E) ~
a
U(m)-bundle
: E ~ X
realification,
inclusion ~m
~
@ F)
i.e.
c SO(2m)
@ ~
@ i~
~...@
For
~
: E ~ X
an
(7.26)
b)
a)
First
A
and
see
this
consider
trivial
observe
B
are
notice
A
and
that
the
(where
the
identifying
= ~2m) .
Then
= Pm(E).
Theorem
for
since
of
by
SO(2m)
~ X
= Cm(E) .
5.5.
A 6 4~(m)
skew-symmetric
B
0.
Pf
and
B £ ~(1)
(that
matrices)
is
invariant
it
is e n o u g h
to
form
0
-b I
".
0.
B =
0 -a m •
0
Then
2
by
that
to
: E~
I0al1 I0bl1 -a I
A =
is
~
SO(2m)-bundle
e(E)
Proof.
To
i~
e(E]R )
d)
let
extension
is d e f i n e d
(7.25)
is,
the
e(F) .
0
.
am 0
0
"0
b1
-b I
0
clearly al-..a m - (2~) m '
Pf(A,...,A)
P f ( B .... ,B)
-
bl..-b 1 1 (27)
and Pf[
so
that
(7.27)
(7.27) exactly
B0> ..... < O 0 > ]
is o b v i o u s as
in t h e
= a1"''amb1"''bl (27) m + l
in t h i s proof
of
case.
Now
Theorem
(7.24)
7.3
c).
follows
from
107
c)
The inclusion
map of Lie algebras Hermitian
r : U(m) c SO(2m)
r~ : ~(m)
m x m-matrix
~ 4~(2m)
correspond
to the
which sends the skew-
X = {ast + ibst}
into the
2m x 2m-
matrix
al I
-bl I
al 2
-bl 2
alto
-blm ~
blm
alm
amm
-bmm
° ° . o .
b11
a11
am1
-bml
bml
aml
b12
a12
r.(X) =
I
..................
which is clearly skew-symmetric.
Now
bm m
amm/
(7.25) follows from
Theorem 6.13 d) and the following identity of polynomials: (7.28)
Pf(r,(X) ..... r,(X))
= Cm(X ..... X),
Since both sides are invariant polynomials again assume that
X =
X
is diagonalized,
i11
"..
0 1,
on
~(m)
X 6~(m). we can
that is,
b 1...b m 6 ]R.
•ib m Then Pf(r~(X) ..... r~(X))
: (-I) m b1"''bm (2z) m
whereas I Cm(X ..... X) = det(- ~ which proves d)
(7.28) and hence
X) = (-I
(7.25).
clearly follows from the identity
)m b1"''bm m (2~)
108
(7.29)
which
P f ( A ..... A) 2 = P m ( A ..... A)
is p r o v e d
Remark. defined see
in t h e
Usually
differently
in the n e x t
properties
But
class
as w e
determined
by
is
shall
the
7.22.
This
exercise
definition
of
H~
cohomology
with
GI(n,IR) + ~
A 6 ~(m),
the E u l e r
I below).
it is u n i q u e l y
topological denotes
2 ~ A),
(7.27).
Topology
(see E x e r c i s e
chapter
I.
as
in A l g e b r a i c
of T h e o r e m
Exercise
same way
= det(-
Gl(n,~)
be
deals
the E u l e r
with
class.
coefficients
the
the a l g e b r a i c
subgroup
In t h e
in
~.
following Let
of m a t r i c e s
with
positive
determinant a) space
Show X
n,
a preferred for
every
preserving Now
point ~
let
Theorem
9.1])
class
E.
of
of
an o r i e n t a t i o n
X
such
that
isomorphism {0})
and
suppose
: E ~ X
with
x £ X,
such
is t h e
U
denotes
a n d J.
and
a
orientation). the
zero
Stasheff
[19,
U 6 H n ( E , E 0)
x 6 X onto
of
is o r i e n t a t i o n
X x 0
class
for e v e r y
n ~
~
bundles
the c a n o n i c a l
Milnor
is a u n i q u e
preserving
i~U 6 Hn(~n,~
where
(see e . g . J .
there
bundles
which
is g i v e n
(X x 0),
Recall
E)
(~n
vector
is a n e i g h b o u r h o o d
~ U x ~n
fibre
on a topological
E x = ~-1(x)
fibre
there
: n-1(U)
that
vector
of e v e r y
E0 = E ~
of
to o r i e n t e d
n-dimensional
on e v e r y
section
class
i.e.
Gl(n,~) +-bundles
bijectively
orientation
trivialization
Thom
topological
correspond
dimension
that
that
and for
the
canonical
fibre
(the ix
Ex,
: ~n~E the
generator.
X
NOW s(x)
% 0
(7.30)
let for
Y ~ X all
x £ Y.
e (E,s)
= s~U
Define
s : X ~ E
is a s e c t i o n
the relative
6 Hn(x,Y).
Euler
class
with
109
b)
Show
particular
that
for
£(E,s)
£(E)
is i n d e p e n d e n t
of
s
show
class
as o r i e n t e d
c)
Observe
= £(E,s)
that
that
£(E)
X =
for
is j u s t t h e c a n o n i c a l
d)
~ ~n
Let
X = M
differentiabel oriented such
s
Now choose
Ai ~i
to
0
as
and
=
on
M.
~n~{0},
=
~
: E ~ M
Suppose
Clearly
IndexA
the
=
U. 1
X
x
of x £ mn.
n-dimensional be
an n - d i m e n s i o n a l is a s e c t i o n
s e t of p o i n t s of ~i
~n
~ ZZ
s : M ~ E
orientation
and w e d e f i n e
isomorphism
E
(x,~(x)),
oriented
at a finite
with
section).
the
the d e g r e e
diffeomorphisms
~ U i x ~n.
(7.32)
s(x)
neighbourhoods
together
in c)
times
let
only
preserving and
depends
, y
~n
where
on
vanishes
: ~-1(Ui)
section
bundle
disjoint
orientation
In
bundle.
be a compact
manifold
vector
that
sIX-Y.
s = zero
6 Hn(]Rn,]R n ~{0})
generator
- {0},
choose
only
vector
£(E,s)
: ~n~{o}
on
6 Hn(X)
(so w e c a n
Furthermore, E
not depend
Y =
(7.31)
of
does
A. l
A I , . . . , A N.
together
: Ui ~ ~n
preserving
with
taking trivializations
s 1 = ~i 0 s o ~ I
defines
integer
index)
(the l o c a l
a
(s) = d e g ( ~ i ) 1
Show
that
~°i' ~i'
IndexA. (s) 1 and show the
is i n d e p e n d e n t following
of
formula
the c h o i c e s
of
U i,
of H. H o p f :
N
(7.33)
~ IndexA. (s) = <e(E) ,[S]>. i=I 1
In p a r t i c u l a r For show
that
the
left hand
the t a n g e D t
bundle
s i d e of TM
(7.33)
: TM ~ M
is i n d e p e n d e n t one
can use
of
(7.33)
s. to
110
n
(7.34)
the
<e(TM) ,[M]>
Euler-Poincar6
=
x(M)
vector
field
of
have
the
sum
local
indices
[21,
Theorem e)
of
a Morse
M.
function
equal
to
In
fact
is
easily
x(M)
e.g.
the
seen
(see
J.
to
Milnor
5.2]).
Show
oriented
[ ( - 1 ) i d i m ~ H . (M ~) i=0 ~ i ' '
characteristic
gradient
of
=
that
vector
e(E)
bundle
£ Hn(x),
of
defined
dimension
n,
for
has
n
the
: E ~
X
an
following
properties: i) an
(Naturality).
oriented
vector
for
£(f*E)
e
defines
principal ii)
(Whitney
dimensional
vector vector
be
the
z
vector
•
: E ~ X
bundle
~
Notice
~)
with
: E ~
isomorphism
of
that E
and
an
X
class
with
For
~
~
: F ~ X
the and
=
e(~)
~
~
: E ~ X
~-coefficients
: E ~ an
X
an
oriented
oriented m-
oriented
the
e(~).
vector
opposite
£
Hn(X)
antipodal E
bundle
orientation.
let
~
: E-
Then
= -e(E).
n-dimensional
e(E)
(Hint:
and
bundle
e(E-)
For
continuous
f*e(E) ,
formula).
bundle
(7.37)
iv)
X
+ -bundles.
e(~
For
=
duality
(7.36)
iii)
: Y ~
a characteristic
Gl(n,~)
n-dimensional
f
bundle
(7.35)
hence
For
).
has
map
with
n
order
on
each
odd
2.
fibre
defines
an
~
X
111
v)
For
considered (coming
nn
: H* ~ CP n n
as a p l a n e
from
the c a n o n i c a l
bundle
the u s u a l
g(H n)
where
h n 6 H 2 ( ~ P n)
(Hint:
Use
(7.33)
Remark. determines Theorem
is the
image
of
line bundle
orientation
C = ~
• i~
= ~2 )
= -h n
canonical
for the b u n d l e
In the n e x t
the
the i n d u c e d
identification
(7.38)
~I
chapter g(E)
generator.
we
: H~ ~ ~p1).
shall
in r e a l
show
that
cohomology.
i) - v) Hence,
Dy
7.22,
£(E)
(7.39)
for a n y
SO(2m)-bundle
Exercise
2.
differentiable
preserving
=
~
Let
e(E)
M
be The
the u n i q u e
map
n-chain
representing
a)
Hn(M;~)
As usual
contained
such
that
image
~ H n ( M , M ~ { x } ;~ ) . [M]
let
and denote
An c ~n+1 =
in t h e h y p e r p l a n e
be
[M] E H n ( M , ~ )
and
for
V n = {t =
x = ~0(0)
~ Hn(M,M~{x},~)
of
[M]
Choose
a
it a l s o b y the
oriented
for a n y o r i e n t a t i o n
~ U c M
to the
compact
class
~ Hn(U,U~{x},ZZ)
generator
natural
fundamental
~0 : IRn
diffeomorphism
canonical
an n - d i m e n s i o n a l
class
qO. : Hn(]Rn,I~n~{0},ZZ) the
6 H2m(X,]R)
: E ~ X.
manifold.
is b y d e f i n i t i o n
takes
with
complex
under C~
the
singular
[M].
standard
n-simplex
(t0,...,tn) I~t i = 0}. 1
Consider
a
C~
singular
to a n o r i e n t a t i o n of
An
image
in of
associated
Vn
to
preserving
onto
intA n
and ~.
n-simplex
an o p e n let
Show
o
: An ~ M
diffeomorphism set of
M.
[~] 6 Cn(M) that
in
C
n
(M)
which
of a
Let denote
extends
neighbourhood
U ~ M
b e the
the n - c h a i n
112
[M]
for
some
c 6 C n + I (M)
b)
For
and
co 6 An(M)
-
d
[o]
=
Zc
+ d
6 Cn(M-U).
recall
that
the
integral
I co
is
M
defined
as
for
with
M
follows: supp
diffeomorphisms
Choose
a finite
partition
c= U
together
with
I
~
: ]Rn
~ U
.
Then
~
= !
[ .
{I
}
preserving
~*(I
co)
]19n
"
that
(7.40)
< I(w) , [ M ] >
(Hint: in
unity
orientation
Mco Show
of
First
assume
co
has
r ]MCO"
=
support
in
a set
U
as
considered
: E ~ M
be
a
a) ) . c)
Now
suppose
differentiable form
~.
n =
2m
and
SO(2m)-bundle
Show
using
with
(7.39)
~
connection
8
and
of
M
curvature
that
<e(E) , [ M ] >
(7.41)
let
[
=
pf(~m) .
JM In p a r t i c u l a r proves
the
for
E = TM
Gauss-Bonnet
×(M)
this
form
d) induced that T
from
if
the
to W.
Consider
~ Sn
For sub-matrix
in an
{x 6
bundle
of
: SO(n+1)
(n+1) where
by
x
the
[J
bundle
this
and
C.
(the g ~ S n.
=
gN
I}
pole)
is
the
Consider
and
Allendoerfer
SO(n+1)
north
~ Sn
row
B.
11xl
that
(n+1)-matric last
pf (~m) M
~n+1
Observe
given
•
=
Fenchel
(0,0,...,0,1)
tangent
connection
Sn =
~n+1.
N =
: SO(n+1)
for
due
tangent
formula
(7.42)
(in
the
defined A
let
column
with
the
metric
acts
on
Sn
then
the
map
principal
furthermore as
- A.
Weil).
and
SO(n)-bundle the
follows: A
have
denote been
the
n
cancelled.
x n
113
Now
consider
of
(n+1)
x
SO(n+1)
(n+1)-matrices
0 =
on
SO(n+1) Now
where (Hint: of
is
form
_
the volume
Observe
S0(2m+I)
volume
show
pf(~m)
u
that so
has
orthonormal
basis).
Since
X ( S 2m)
the
(tx = t r a n s p o s e
,
the
set
l-form
of
X)
that
form
associated
sides
enough
definition
= 2
V 0 Z ( S 2m)
are
the
conclude
-
with
invariant
to e v a l u a t e
22m+I (7.44)
that
M(n+1,~)
(2m) ! 22mmm ~ u
both
i t is by
Show
of
a connection.
n = 2m
(7.43)
X.
(tXdX) ^
defines
for
as a s u b m a n i f o l d
value
that m
~m.
(2m)!
,
at
the metric. under N.
I/(2m) !
the
Obs: on
an
action The
8.
The Chern-Weil
In this
chapter
coefficients.
8.1.
Then
w
Remark. compact
We
I~(G)
computes
[8]).
~ H~(BG)
shall
see below
for
G
3.1])
cohomology
Let
the
following
be
a compact
G
(Proposition computable.
8.3)
K
G/K
Bj ~ : H *(BG)
Lie
and
G
number
has
a
is d i f f e o m o r p h i c
Hochschild j : K ~ G
[15,
to
Chapter
induces
by Proposition
G
also
a finite
in t h a t c a s e
hence
real
that for
This
In f a c t
so the i n c l u s i o n
with
is a n i s o m o r p h i s m .
(see e . g . G .
in h o m o l o g y ~
~roups
any L i e g r o u p w i t h
subgroup space
compact
is to p r o v e
: I~(G)
components.
compact
morphism
object
is in p r i n c i p l e
some Euclidean Theorem
for
again means
(H. C a r t a n
H~(BG)
of c o n n e c t e d maximal
H~(-)
The main
Theorem group.
homomorphism
15
an iso-
7.2,
~ H ~(BK)
is an i s o m o r p h i s m .
In t h e f o l l o w i n g the
identity
G
component,
which
the group
of c o m p o n e n t s .
following
we
polynomial
shall
4 Exercise on
G0
on
I ~ ( G 0)
4)
so
4 Exercise I~
so w e h a v e
P 6 I~(G)
of
study
with
we get
G/G 0
the
s e t of
is n o w w h a t before
In t h e invariant by
(cf. C h a p t e r since
(right-)
By d e f i n i t i o n
GO and
be
G/G 0
we denoted
an i n d u c e d
I * ( G 0)
GO
with
I~(G) :
so in p a r t i c u l a r
on
Let
subgroup
As mentioned
g ~ Ad(g) ~.
an a c t i o n
let us
I~(G)
is a f u n c t o r
by conjugation by
First
I.
Lie group.
is a n o r m a l
identify
functions,
in C h a p t e r
is a c o m p a c t
action
acts
G
acts of
G
trivially,
also by definition
115
(8.2)
the
I ~(G)
invariant Now
torus NT
part
suppose
T c G
of G
I ~(G0)
tori
Chapter
4]).
and consider
be I ~(T) on
the L i e
i
= S ~(~)
and
T
of
group Weyl
G
see
the
and
the action
and
8.3.
Let
i : T ~ G
group
W.
the
Then
(8.4)
i
a maximal where
properties
e.g.J.F. and
let
respectively.
W
on
~
of
Adams
inclusion T
of
If
then clearly action
by
i~
induces
W,
[I, ~
and
Clearly an a c t i o n
(cf. A d a m s
that
i*
to
~
polynomial
is i n v a r i a n t
under
on the
i~P = 0.
abelian
subalgebra
[I, C o r o l l a r y 6 ~
.
Every
4.23])
element and
since
there
v 6 ~ all
such
is a
Hence
= P(Ad(g)v)
= 0,
P = 0. surjective:
function
of d e g r e e
W.
v 6 ~7~
For
torus with
an i s o m o r p h i s m
is an i n v a r i a n t
Ad(g) (v)
P(v)
is,
a maximal
Lie
~- I n V w ( I ~ ( T ) ) .
Suppose
in a m a x i m a l
conjugate
of
connected
i~P 6 I n V w ( I ~ ( T ) ) .
injective:
such
induces
P £ I ~(G)
so
be a compact
inclusion
the restriction
is c o n t a i n e d
g 6 G
G
: I~(G)
i~
Proof.
that
be
G/G 0 •
W = NT/T,
(for the b a s i c
: T ~ G
by
I ~ (T) .
Proposition
are
action
Then we choose
Lie groups
algebras
the
the Weyl group
of
in c o m p a c t Let
under
is c o n n e c t e d .
is t h e n o r m a l i z e r
maximal
= I n V G / G 0 ( I ~ ( G O ))
Suppose k
choose
on
~
P and
g 6 G
is a h o m o g e n e o u s suppose such
that
P
polynomial
is i n v a r i a n t Ad(g)v
6 ~
under
and
118
define
the
function
P'
: ~
(8.5)
P'
~ ~
by
P' (v) = P ( A d ( g ) v ) .
is w e l l - d e f i n e d .
In fact
suppose
t I = Ad(gl)v , both
lie in
n 6 N(T) hence
~
.
~
.
such
Then
that
t 2 = Ad(g2g~1)t I
t 2 = Ad(n) t I
P(t2)
= P ( t I)
We want
to s h o w
By d e f i n i t i o n
t 2 = Ad(g2)v
since that P'
and
then
(cf. A d a m s
is an
there
[I, L e m m a
4.33]) ~
P 6 InVw(I*(T)). P'
is an i n v a r i a n t
is an i n v a r i a n t
polynomial
function
on
~
,
on that
w
is,
(8.6)
and
P' (Ad(g)v)
also
P'
is c l e a r l y
(8.7)
P' (~v)
In an a p p e n d i x
to this
C
function
is a c t u a l l y following
on
~
chapter
a homogeneous
v 6 ~,
of d e g r e e
Vv 6 ~ ,
we
(a s u r p r i s i n g l y
8.8.
Suppose of d e g r e e
(8.9)
f(Ix)
shall
show
k,
that
is,
I 6 ~.
that
non-trivial
polynomial
Let
Differentiating rule
and
f
: ~n~
k,
of d e g r e e
x = (8.9)
phtting
P'
is a
fact).
Then
k
due
P'
to the
C~
function
which
is s a t i s f i e s
polynomial
be
times with ~ = 0
is a
Vx 6 ~ n ,
(Xl,...,Xn) k
~
that
= Ikf(x),
is a h o m o g e n e o u s
Proof.
chain
homogeneous
= ~kP' (v),
is h o m o g e n e o u s
f
Vg £ G,
lemma:
Lemma
Then
= P' (v) ,
yields
I 6 ~.
of d e g r e e
k.
the c o o r d i n a t e s respect
to
~
in
using
~n. the
117
(8.10)
ii Sn = k~f(x), x I ...x n
a i1+...+in= k
x 6 ]Rn,
il-''i n
where ~kf a.
11 This
proves
the
In v i e w 8.1
for
(0).
=
•.
iI in ~x I ...~x n
-i n
lemma
and
ends
of P r o p o s i t i o n
G = Tn
8.3 w e
T I = U(1)
identify
T n = ~n/~n
via
(n
Then
the Lie
bracket,
so
algebra I * ( T n)
the p o l y n o m i a l ~I x ~
ring
is i d e n t i f i e d 2zix
and
in one v a r i a b l e
x
Proposition variable on
~
I
w(x)
group
is
,
#n
~
8.11.
~ •
I*(T I)
Chern-Weil
where
By P r o p o s i t i o n
as the h o m o l o g y AP'q(NT I )
zero
6 ~n.
Lie with
For
n =
w(x)
I
the m a p
is the p o l y n o m i a l
is a p o l y n o m i a l x
. , . Xn)
identified
under
image
ring
= -c I 6 H 2 ( B T I ) .
ring
is the i d e n t i t y
of
the
total
6.1,
H * ( B T I)
complex
in the
polynomial
of
can be c a l c u l a t e d
the d o u b l e
complex
with
A P , q ( N T I) = A q ( N T I (p))
above
shall
= ~.
Proof.
AS
We
X l , . . . , x n.
(I) = i ~
that
with
is a c t u a l l y
H ~ ( B T I)
£ H 2 ( B T I)
~.
(Xl,.
= ~n
in the v a r i a b l e s
with
in
2ZiXn)
= S ~ ( ( ~ n ) ~)
it f o l l o w s
i.e.
, .... e Tn
with
Theorem
the m a p
(e of
prove
times)
circle
2~ix I e x p ( x l , . . . , x n) =
first
torus,
TI
is the u n i t
of the p r o p o s i t i o n .
shall
the n - d i m e n s i o n a l
T n = T I x...x
where
the p r o o f
identify
T p = IRP/zz p
with
= A q ( T p)
coordinates
(Xl,...Xp).
118 Now consider
the d o u b l e
complex
(8.12)
A p'q c A P ' q ( N T I) 0 =
where
A 'q
I =< Jl
<'''<
AP'q = 0 0
jq =< p
for
vanishes induces
is the v e c t o r s p a c e
on
AP,q -0 "
and that It is easy
by all
dx31 ,
~= A q ( ~ P ) ) "
dx jq
A...A
Notice
that
the v e r t i c a l
differential
to see that
the i n c l u s i o n
d" (8.12)
an i s o m o r p h i s m Ap,q
H e n c e by L e m m a on h o m o l o g y
1.19
calculate
~ Tp'
~
Hq(AP,*(NTI)).
=
(8.12)
induces
It follows
an i s o m o r p h i s m that
~I HP(A~ 'q) p+q=n
H p'A*'q) tA 0
i = 0,...,p,
I ei(Xl, .... Xp+1)
~
complexes.
Hn(BTI)
so we shall : T p+I
-
the i n c l u s i o n
of the total
(8.13)
ei
A p'q -0
(so
p < q
spanned
for each
q.
Here
is g i v e n by
(X2, .... Xp+1) ,
i = 0,
"(XI .... ,Xi+Xi+ I .... ,Xp+1),
i = I .... ,p,
(X 1,...,xp) ,
i = p + I,
(Xl, .... Xp+ I) 6 ]RP+I/zz p+I By a s t r a i g h t f o r w a r d
calculation p+1
.
it is seen that Ap,q
.p+1 ,q
i=0 is g i v e n
(8.14)
by
~'(dxj3A'''AdXjq)
31 = (i=0 ~ (-1)i)dXjl+lA'''AdXjq+l
J2 +
( ~ (-1)i)dx31 Adx.32+I A. " .Adx.3q+I +. "" + i=j I
p+l + ( ~ (-1)i)dx. i=j~
A...A dX~q. 31
,
119
Now
define
maps AP-l,q-1 0
R for
q =
1,2,...,
÷ Ap, q 0
p = q,q+1,...,
(8.15)
R(~)
(8.16)
T(e
by
= ~ ^ dXp,
^ dx
P
w 6 A p-1'q-1
+ 8) = ~,
~ ^ ~,
Then
it
is e a s i l y
between
the
checked
complexes
On
that
R
A0-1'q-1
(8.17)
: T
8
and
dx do
T
+
B 6 A p'q
not
contain
give
A~,q
and
P
~0
'
chain
dx
P
maps
clearly
and
T 0 R = id.
the
other
hand
if w e
let
: T p-I ~ T p, p = 1,2,..., be
s P
induced
by
(Xl, .... X p _ I ,0) ,
S p ( X I .... ,Xp_ I) = then
i t is
easy
(-I)P6
(8.18)
on
to c h e c k
A ~ 'q.
The
6 ~p-1
that
o s P~ +
details
(x I ,... ,xD_1) _
are
(-I) p+I S p + I o 6 = id - R o T
left
as a n
exercise.
It follows
that
q,
P%
p = q. Also
the
dx I ^...^
generator dXp
is r e p r e s e n t e d
for
p = q > 0.
Hn(BTI)
and we want
to s h o w
that
is r e p r e s e n t e d
in
connection
given
e
By
= {~
NT I
by
the by
I
for
(8.13)
n
odd
n
even
w(x) p # 0.
A 2 ( N T 1) in
by
For
we
p = q = 0 now
this
curvature
have
notice
form
and
~
that of
the
w(x)
120 P [ tidY i i=0
@ = where
N T I (p)
(Y0 .... ,yp) .
= T p+I
= ]RP+I/2z p+I
has
=
dt i A d Y i
=
that
~P
on
is t h e
lift
dtp
^
Therefore
This
ends
the
now
: E ~ X
Notice
e(E)
= ¢(E)
To
7
from
prove
e
(dyp
- dy 0) ,
in
SO(2) and
i =
I ..... p,
A P ' P ( N T I) ±
= TI
e
are
by
by
I A(~ p)
the
generator
and
that
both
6 H 2 ( B T I)
and
Whitney
dXp
(5.8). =
in c o h o m o l o g y .
proposition.
-w(x)
(7.24)
the in
the
that
with
is
^ dx I A . . . ^
represents
of
in C h a p t e r
follows
^...^
dx i = d Y i _ I - dy i,
which
proof
H2(BSO(2),~)
dtp
is r e p r e s e n t e d
dXp
Remark. defined
since
w(x) p
= ±dx I ^ . . . A
( d Y l - d Y 0)
of
~P = ± p ~ d t I ^ . . . ^
Ap x NTI(p)
(dy i - dy 0) "
x N T I (p)
~P = +-p'.dtI ^ . . . ^
on
coordinates
~ dt i ^ i=I
i=0
which
the
Now
= de
It f o l l o w s
x NT ] (p)
Ap
on
(7.36)
sum
of
by
that m
the
classes
identified (7.25)
when
and
in (7.38).
It
a SO(2m)-bundle
SO(2)-bundles
then
H2m(x,~).
Theorem
8.1
for
G = Tn
we
now
any
group
need
the
following
proposition:
Proposition
8.20.
is c o n t r a c t i b l e . of h o m o t o p y
(8.21)
a)
For
In p a r t i c u l a r
Lie
there
is a n a t u r a l
groups
~i(BG)
~ ~i_1(G),
i =
G,
1,2 ....
the
space
isomorphism
EG
121
b)
For
G
and
BPl
induced a weak
by
the
any
x BP2
: B(G
a)
By
groups
x H)
Pl
equivalence,
in c o h o m o l o g y
Proof.
Lie
projections
homotopy
morphism
H
the
~ BG
natural
x BH
: G x H ~ G,
in p a r t i c u l a r
with
any
definition
map
P2
: G x H ~ H,
it i n d u c e s
an
is
iso-
coefficients.
EG
is
a quotient
space
of
co
Ap
x NG(p).
Define
the
homotopy
h
p=0 by
h s ( ( t 0 , t I ..... tp), (g 0 ..... gp))
=
This
is e a s i l y
to t h e from
: E G ~ EG,
s 6
[0,1],
s
point
the
fibre
seen
((1-s,st0,...,Stp)
to b e w e l l - d e f i n e d
((I),(I))
homotopy
=
6 A
sequence
0
x G.
for
.
a contraction
of
and
(8.21)
the
, (1,g0,...,gp))
now
clearly
fibration
EG
follows
EG ~ BG
with
principal
G
G. b)
Pl
and
P2
clearly
induce
a map
of
x H-
bundles
Since
both
induces second
an
total
[28,
Corollary That
is,
x H)
~ EG
x EH
B(G
x
"* B G
×
spaces
isomorphism
statement
Spanier
E(G
are on
follows
Chapter
is
w
BH.
contractible
homotopy from
the map
groups
by
Whitehead's
7 § 5, T h e o r e m
8.22.
H * ( B T n)
H)
: I * ( T n)
a polynomial
B(G
(8.21),
x H) and
~ BG
the
theorem
(see e . g . E .
is
isomorphism.
9]).
~ H * ( B T n) ring
an
in t h e v a r i a b l e s
x BH
122
W ( X i)
6 H2(BTn),
generators
of
Proof. gether
I ~ ( T n)
with
the
we
K~nneth
dimensional
manifold.
shall
called
use
alon~
from
the
U
a coordinate
(Xl,...,Xk).
respect
we
x N),
does
N.
p~
extends
be
canonical
and
Theorem Let
a compact
the
8.20
to-
8.1 M
we
be
need
an
oriented
projection
p
nk-
: M
x N ~ M.
p~
not
reader
will
lemma
(Lemma
1.2).
has
of
contain
6 AI-k(M)
we
is
the
First
suppose
Then
we
B
x U,
where
coordinates
coordinates
are
k-form
with
can write
on
M
and
6
a degree
M
local
a positive
dXl,...,dx k to
the
inside with
dx k ^ ~ +
integrate
on
N
that
N.
= IudXl
We
is w e l l - d e f i n e d forms
in
recognize
x U,
only
less
involves
than
k.
Then
to b e
X l , . . . , x k.
to a l l
support
dx k
dxl,...,dx k
that
of
~ A~-k(M)
furthermore
p~
functions
of
The
1 ~ k,
orientation
containing
means
8.11
nature:
N
N)
×
dx I ^...^
(8.24)
that
let
~ = dx I ^...A
~
proof
neighbourhood
that
to t h e
define
which
the
Poincar~
Suppose
(8.23)
where
the
Ix I ..... Xn].
Propositions
Consider
inte@ration
~ 6 AI(M
terms
and
: A~(M
such
are
l
a homomorphism
that
chosen
= ~
a technical
p~
technique
is
with
of
manifold
the
x
theorem.
proceed
preparations
where
= S ~ ( ( ~ n ) ~)
from
dimensional
We
1,...,n,
Obvious
Before a few
i =
M
A...^
the leave and
x N
dx k A ~,
coefficients it as
that using
the
of
e
an e x e r c i s e definition
a partition
as to v e r i f y of
of
p~
unity.
123
Also
the
following
Lemma b)
lemma
8.25.
For
a)
is
p,(dm)
m £ A*(M
choose
of T h e o r e m
a maximal
inclusion
=
x N)
p,(m
Proof
left
and consider
(-1)kdp,~,
and
^ p'v)
8.1.
compact
as an e x e r c i s e :
=
(p,~)
T.
and
the
fact
image
for
x ~ g
-I
g £ G
xg.
homotopic induces w
Bi*
1 g
to t h e
: G ~ G
~ H*(BT)
commutative
5.20
identity
identity
be
the
acts
H*(BT)
diagram
W = NT/T
is c o n t a i n e d
let
: T ~ G
H* (BT) .
=
functoriality
Bi*
i
and
I
Then by Lemma
the
: I*(T)
the
of
connected
, I*(T)
H*(BG)
that by
G
Let
commutative
(8.26)
notice
^ V.
assume
subgroup
I*(G)
First
× N).
U 6 Am(M)
First
the
~ 6 A*(M
in the
be NI
g
the
on cohomology. is a m a p
of
invariant
inner
: NG ~ NG
so b y C o r o l l a r y Also
on
part.
In
conjugation is s i m p l i c i a l l y
5.]7,
B1
g
: BG ~ BG
by functoriality
W-modules,
hence
(8.26)
yields
diagram
I*(G)
, InVw(I* (T))
(8.27)
Bi*
H*(BG)
where
the u p p e r
horizontal
morphisms
by Proposition
Therefore
it is e n o u g h
(8.28)
Bi*
map
8.3
, InVw(H*(BT))
and
right vertical
and Corollary
8.22,
map
are
respectively.
to s h o w
: H*(BG)
~ H*(BT)
iso-
is i n j e c t i v e .
124
To p r o v e
this w e c o n s i d e r
the c o m m u t a t i v e
ET/T
) EG/T
BT
)
and o b s e r v e
that
equivalence
by P r o p o s i t i o n
to s h o w
8.20
EG/T
cohomology.
This
is the
simplicial
manifolds
: NG ~ NG
manifold
map
given
i'
by
BG
horizontal
the m a p
y
that
the u p p e r
a).
~ BG
map
is a w e a k
Hence
(8.28)
induces
realization
: NG/T
(5.8).
diagram
~ NG
Here
is e q u i v a l e n t
an i n j e c t i v e of
the m a p
induced NG/T
homotopy
by
is the
map
in
of
the m a p simplicial
with
(NG/T) (p) : N G ( p ) / T ,
where
T
G x...x the
G
acts
by
(p+1
times).
simplicial
the d i a g o n a l This,
manifold
action however,
N(G;G/T)
on the r i g h t can be
e. : N(G~G/T) (p) ~ l
given
by
£i(gl ..... gp,gT)
I
(NG/T) (p-l),
(g2'''''gp'gT)
= ~(gl ' ~(g1'
In f a c t
is g i v e n
the m a p
with
x G/T is
i = 0,
.,gigi+ I ,. . .,gp,gT) , i
I ,. . .,p-1
"'gp-1 ,gpgT) ,
p.
N(G;G/T)
by
identified
i = 0,...,p,
,
the i d e n t i f i c a t i o n
(8.29)
NG(p)
where
N(G;G/T) (p) = NG(p) and
of
~ NG/T
i
=
125
(gl,...,gp,g-T) Under
this
identification
to the m a p first
~
the m a p
i : N(G,G/T)
factor
in
NG(p)
(gl...gp-g,...,gp-g,g)T.
~ NG
i'
given
x G/T.
: NG/T by
(8.28)
~ NG
corresponds
the p r o j e c t i o n
is t h e r e f o r e
on
the
equivalent
to
(8.28)'
which
11 ~ N*
is p r o v e d We
of e v e n
shall
: H~(IL N G ll) ~ H~( I N(G;G/T)II
)
is i n j e e t i v e ,
as f o l l o w s :
see b e l o w
dimension,
say
that 2m,
G/T and
is an o r i e n t a b l e
(8.24)
therefore
manifold
produces
a
map ~,
for e a c h
:
p.
A~(~ p x NG(p)
It is easy
requirements
x G/T)
to see
in D e f i n i t i o n
6.2
that
suppose
Lemma
we h a v e
8.30.
proved
There
these
so w e g e t
A~(N(G,G/T))
Now
~ A ~ - 2 m ( A p x NG(p))
maps
preserve
the
a map
~ A*-2m(NG).
the f o l l o w i n g
is a 2 m - f o r m
~ 6 A2m(N(G;G/T))
such
that (i) (ii)
d~ = 0 The
restriction
satisfies
We
can
then
~0
to
G/T
= 40
x
N ( G , G / T ) ) (0
[ ~0 % 0. J G/T
define
a map
: A*(N(G,G/T))
~ A*(NG)
by T(%9) = ~ , ( ~
^ ~),
~0 E A * ( N ( G , G / T ) ) .
126
By L e m m a hence
8.25
a)
induces
and Lemma
a map
T~
By Lemma
It f o l l o w s
6 A0(NG)
= ~(~)
is c l o s e d ,
that
T o ~*
constant.
This
shows
(8.28).
Proof
is i n v a r i a n t G
under
the
8.30.
, that
is,
and f i n d
split
7
Ad(exp(t)),
t £ ~
into
acts
a constant,
T. o
injectivity to p r o v e
Choose
that
II ~ II*:
an i n n e r of
a root
of
product G.
~:
on ~
(This
space
is p o s s i b l e
for
bundle
of
sum
,~such
that
the matrix
COS
2 ~ I (t)
-sin
2T~O~I (t)
sin
2~e I (t)
cos
2~e I (t)
0
Ad(exp(t))
details
which
decomposition
direct
{el,...,e2m}
on ~ b y
and hence
existence
an o r t h o g o n a l
basis
II ~ II~
(8.31)
where
is,
by a non-zero
of
the
action
Now make
an o r t h o n o r r a a l
hence
so a l s o
the a d j o i n t
is c o m p a c t ) .
~ 6 A~(NG).
is m u l t i p l i c a t i o n
It r e m a i n s
of L e m m a
- ~,
= ~ G / T ~0 % 0. and
H~(II N G II) ~ H~(II N G ]I)
~
is a c h a i n m a p ,
: H~(II N(G; G/T) Jl ) ~ H~(II NGII).
~*(~)
since
T
8.25 b),
~(T)
proves
(i),
in c o h o m o l o g y
0 ~(~)
Here
8.30
=
/ ~. : ~ l
I
G/T
2ze
i = I ..... m,
Adams can be
[1,
Chapter
identified
(t)
-sin
2~em(t)
cos
2zero(t)
m sin
~ ~,
s e e e.g. of
" cos 0
are
4]). with
linear
2~m(t)
forms
Notice the
on
that
/ ~
the
2m-dimensional
(for
tangent vector
127
bundle
: G X T ~
which
is c l e a r l y
by the b a s i s Now @
be
an o r i e n t e d
bundle
G/T
with
the o r i e n t a t i o n
given
{el,...,e2m}.
let
<
:~
be
the c a n o n i c a l
the o r t h o g o n a l
connection
in
NG
projection
given
by
and
(6.12).
let Then
clearly @T = K o @ defines let
a connection
aT
given
be
by
the
in the p r i n c i p a l
curvature
the p o l y n o m i a l
form.
on
NG/T
under
and w e
the
is just G ~ G/T
~
It r e m a i n s
the C h e r n - W e i l with
the
v 6~
P(~)
and
image
of
@T
P
given
(L _i),, g
Unfortunately
f o r m on
so c l e a r l y (ii).
.
is a c l o s e d
the c o r r e s p o n d i n g
to p r o v e
connection
a T = d@ T.
P 6 Im(T)
2m-form
(8.29),
(@T)g = < 0 and
consider
m n e. (v), i=I ±
(-I) m
be
identification
satisfied.
=
theory let
NG ~ NG/T
function
P(v ..... v)
T h e n by C h e r n - W e i l
Also
T-bundle
d~ = 0
Now
N(G;G/T) is 6 A 2 m (G/T)
~0 = p ( ~ )
in the p r i n c i p a l
form
T-bundle
by
g 6 G,
it is n o t
so easy
to C a l c u l a t e
I
P{~) directly. H o w e v e r , as n o t i c e d a b o v e the e x t e n s i o n G/T of the b u n d l e G ~ G/T to the g r o u p SO(2m) v i a the a d j o i n t representation it is easy
to see
On the o t h e r Whitney
on ~
hand
s u m of
is just
that
P(~)
it f o l l o w s
SO(2)-bundles.
the
tangent
bundle
is e x a c t l y from
(8.31)
Therefore,
of
G/T
the P f a f f i a n that
and form.
the b u n d l e
as r e m a r k e d
is a
after
128
Proposition
8.11
(8.32) NOW
the r i g h t for
regular
hand
element
bundle
field
v0
local
well-known
s(gT)
of
s
fact
a neighbourhood
hood
of
{T}
in
7),
can c o m p u t e
~i(v0) the
given
= 0
iff
s
the
formula
Choose
a
every
root
of
the v e c t o r
by
g 6 G
o Ad(g-1)v0),
W = N T / T ~ G/T. is
exponential
of
0 6 ~
projection. so
s
For
map
diffeomorphic
so w e g e t
a local
the
recall
:~Z~ onto
at the
that
this w e
exp
Since
vanishes
Now we claim
+I.
the
G/Tr
for
section
g 6 NT
gT 6 W
by
form:
% 0
is the o r t h o g o n a l
at
that
we
the f o l l o w i n g
(g, (id-<)
~ /
maps
gT
=
set of p o i n t s index
of
(i.e.
~ G/T
:~
is r e g u l a r
finite
(8.32)
and c o n s i d e r
s(gT)
<
of
v0 6 ~
~ : G XT~
again
2 of C h a p t e r
= <£(T(G/T)),[G/T]>.
side
a vector
i = 1,...,m)
where
Exercise
IG/TP(~)
(7.33)
ei'
(cf.
the
G ~ G/T a neighbour-
trivialization
near
by
(g expx,v)
It is t h e r e f o r e
enough
~ v,
to see
x £~
that
near
the m a p
zero,
~
:~
v 6~4~.
~44~
given
by
~(x)
=
(id-<) ( A d ( e x p ( - x ) ) V g ) ,
is an o r i e n t a t i o n
preserving
differential
at
x 6 44~ .
s~
0
Differentiating
Vg = A d ( g - 1 ) v ,
diffeomorphism
is g i v e n (8.31)
by
and
~(x) taking
near
= -[X,Vg]
m
Vg)
=
The = ad(vg) (x),
the d e t e r m i n a n t
gives
det(ad
0.
x 6/,~ ,
(27r)m ~ ei(Vg) 2 > 0 i=I
now
129
so the
local
index
of
s
at
G/T which
proves
for
G
is +I.
It f o l l o w s
that
G/T 8.30
and
finishes
the p r o o f
of T h e o r e m
8.1
connected.
For to
Lemma
gT
G
a general
compact
group
we g e t
a diagram
similar
(8 .27) : % I*(G)
,
1
(8.33)
l
H*(BG) where and
again
connected. the
Again
:
map
map
to s h o w
Bi*:
this
H*(BG)
~ H * ( B G 0)
is e q u i v a l e n t
is d e f i n e d
on
the
factor.
T :
is s i m p l y
if
i
GO
is
: GO ~ G
is
given
This
A*N(G;
is i n j e c t i v e .
il )
NG(p)
N ( G ; G / G 0 ) (p) ~ NG(p)
first
since
(8.2)
that is i n j e c t i v e ,
where
as f o l l o w s :
N ( G ; G / G 0) (p) =
:
that
to s h o w i n g
: H*(li N G ll) ~ H*(]LN(G;G/G0)
~
isomorphism
is an i s o m o r p h i s m
it s u f f i c e s
N ( G ; G / G 0) ~ N G
and
is the
then
(8.34)
II ~Li*
horizontal
vertical
inclusion
As b e f o r e ,
., I n V G / G 0 ( H * ( B G 0))
the u p p e r
the r i g h t
InVG/G0(I*(G0))
x G/G 0
is g i v e n
by
the p r o j e c t i o n
time
(G/G0)))
-~ A*(NG)
by
Sg~,
6 A*(N(G,G/G 0)
g6G/G 0 where
Sg
: Ap
x NG(p)
~ Ap
×
N ( G ; G / G 0 ) (p)
i s g i v e n by p u t t i n g
130
gG 0 6 G/G 0
on the last c o o r d i n a t e
a simplicial easily
map but still
c h e c k e d that
T
by
]G/G 0
multiplication
by
IG/G 0
i n d u c e d by
of T h e o r e m
8.1.
Corollary
8.35.
connected
Lie group,
a maximal
torus.
N(G;G/G0)
Obvious
Corollary are u n i q u e l y
8.36.
This
(A. B o r e l a n d let Bi
H*(BG) Proof.
also
is not
Again
it is
T 0 ~*
T, O II ~ II*
is
is
where
T.
Then
is w e l l - d e f i n e d ) .
Hence
: H*(I
is the m a p
Sg
is a c h a i n m a p and that
multiplication
T,
T
(notice that
II ) ~ H * ( I I NG 11)
shows
[3]).
(8.34)
Let
i : T ~ G
: BT ~ BG
G
and ends
the p r o o f
be a c o m p a c t
be the i n c l u s i o n
induces
of
an i s o m o r p h i s m
,InvwH*(BT) . f r o m the d i a g r a m
(i) The C h e r n
determined
(8.27).
classes
by the p r o p e r t i e s
of
Gl(n,C)-bundles
a) - d) of T h e o r e m
7.3. (ii) F u r t h e r m o r e
H~(BGI(n,~)) is a p o l y n o m i a l
ring w i t h
~ H*(BU(n)) the C h e r n
~ ]R [c I .... ,c n]
classes
c I ,...,c n
as
generators.
Proof. U(n)-bundles.
As n o t i c e d N o w let
in C h a p t e r
7 it is e n o u g h
to c o n s i d e r
i : T n ~ U(n)
be the n a t u r a l
I
0
X I 12
.
inclusion
1 A I ..... A n 6 U(1)
i(X I ..... Xn ) = 0
An
131
and
let
qj
: T n ~ U(1)
j =
1,...,n.
That
Hence
by
This,
however,
after
Proposition
yj =
Bi •
: H~(BU(n))
is,
the C h e r n
U(n)-bundles (7.5)
(ii)
they
which
first
on
Tn
with
Chern
8.11.
by p e r m u t i n g
8.22
der W a e r d e n
H*(BTn)
W
Hence
§ 29]).
proves
8.37.
SO(2m)-bundles
formula
and v) (7.39)
(ii)
on
and
the
U(1)-bundles.
U(1)-bundles. immediately
W
symmetric
where
c I 6 H2(BU(1))
symmetric
I n v w ( B T n)
acts
group on
is
acting
H * ( B T n)
is a p o l y n o m i a l
by
ring
polynomials
Yl ..... Yn by
by
(see e . g . B . L ,
van
(7.5)
k = I ..... n,
the c o r o l l a r y .
Corollary
ii),
so
(i) .
i.e.
However,
of
as r e m a r k e d
is the
in
sums
= ~ [YI' .... Yn ]
ok(Yl, .... yn ) = i~ck,
i),
factor,
torus
are d e t e r m i n e d
cI
(7.6)
proves
the e l e m e n t a r y
[32,
by
factors,
ok(Yl , .... yn ) , k = I, .... n,
for
j-th
~ H ~ ( B T n)
j = I, .... n Now
the
y l , . . . , y n.
by
This
class.
generators
which
the
is a m a x i m a l
are W h i t n e y
is d e t e r m i n e d
By C o r o l l a r y
permuting
Tn
classes
are d e t e r m i n e d
(Bqj)*c I 6 H 2 ( B T n ) ,
the
that
onto
8.35
is i n j e c t i v e . on
the p r o j e c t i o n
It is w e l l - k n o w n
by C o r o l l a r y
values
be
(i) T h e
Euler
is u n i q u e l y
of E x e r c i s e
class
with
determined
I e) of C h a p t e r
by 7.
real
the p r o p e r t i e s In p a r t i c u l a r
holds.
Furthermore H~(BGI(2m,~)
+)
coefficients
~ H*(BSO(2m)) ~ [Pl .... 'Pm-1 'el
132
is a p o l y n o m i a l classes
ring w i t h g e n e r a t o r s
pl,...,pm_ I
(iii)
and the E u l e r
the f i r s t class
m-1
Pontrjagin
e.
Finally H*(BGI(2m,~))
~ H~(BO(2m)) ~ [Pl ..... Pm ]
is a p o l y n o m i a l
Proof.
in the P o n t r j a g i n
The m a x i m a l
be the set
I
ring
Tm
cos
2~x I
sin
2~x I
torus
of m a t r i c e s
-sin
in
classes
SO(2m)
p l , . . . , p m.
is w e l l - k n o w n
of the form
2zx I
cos 2~x I
cos
2~x
0
-sin
m
sin 2nx m (x I ..... x m) inclusion
6 ~m/~m.
and let
projection
remark
of
5.17)
following
Proposition H~(BTm)
t h a t the
Weyl group
the y j ' s and c h a n g i n g to d e t e r m i n e
A = Invw(~ First notice changing
that
A
the sign of
cos
i : T m ~ SO(2m)
: T m ~ SO(2),
(Bqj)~e E H2(BTm) .
We w a n t
let
2~x
2zx m /
be the
(i) f o l l o w s
~ H ~ ( B T m)
m
be the
j = I ..... m,
As b e f o r e
i ~ : H~(BSO(m))
(ii) A g a i n yj =
qj
Again
on the 3-th factor.
injectivity
to
together
from the
with
the
8,11.
~ ~ [Yl .... 'Ym ] It is e a s i l y W
where
seen
acts on
(cf. A d a m s
H ~ ( B T m)
Example
by p e r m u t i n g
the sign on an e v e n n u m b e r
of the yj's.
the s u b r i n g
[Yl .... 'Ym ]) ~ ~ [Yl .... 'Ym ]" has an i n v o l u t i o n Yl'
say.
T : A ~ A
Then clearly
g i v e n by
A = A+ • A_,
133
where
A+
and
A_
fl
are the
eigen
spaces
for
T.
Notice
that
A+ = Invw,(3R [ Y I ' ' ' ' ' Y m ]) where
W'
is the g r o u p g e n e r a t e d
yj ! s t o g e t h e r w i t h of any n u m b e r
by the p e r m u t a t i o n s
the t r a n s f o r m a t i o n s
of the
yj's.
which
changes
It is n o w e a s i l y
of the the sign
seen that
A+ = ~ [o I ..... Om ] where
oj = o j ( y ~ ..... y~)
polynomial
in
2 2 y l , . . . , y m.
seen to be d i v i s i b l e
is the j-th e l e m e n t a r y Now every
e l e m e n t of
symmetric A_
is e a s i l y
by the p o l y n o m i a l
e = Yl "" "Ym" Hence
A = A+ @ A+g. Now
£ 2 = Om(Yl2 ..... ym2 ) £ A+;
hence
A = ]R [ O l , . . . , O m _ 1 , g ] . Here e =
oj = (Bi)~e (iii)
(Bi)~pj, by
j = I, .... m, by
(7.24).
By T h e o r e m
H*(BO(2m))
This p r o v e s 8.1
and
(7.20)
and
(7.26),
and
(ii) .
(8.2)
~ I~(O(2m)) I n v o ( 2 m ) / S O ( 2 m ) (I ~(SO(2m))) .
Here
O(2m)/SO(2m)
action clearly
~ ZZ/2
of an o r i e n t a t i o n fixes
acts on reversing
the P o n t r j a g i n
I~(SO(2m)) orthogonal
polynomials
using matrix.
and c h a n g e s
the a d j o i n t This the sign
134
of the P f a f f i a n p o l y n o m i a l
(see Chapter
Hence the i n v a r i a n t part of ring in the v a r i & b l e s
4, Example
I~(SO(2m))
PI,...,Pm_I
I and 3).
is the p o l y n o m i a l
and
pf2 = Pm"
This proves
the corollary.
In a similar way one proves
Corollary
(i)
8.38.
H~(BGI(2m+I) +) Z H*(BSO 2m+I))
~ ~ [Pl ..... Pm ]
is a p o l y n o m i a l ring in the P o n t r agin classes.
(ii) H~(BGI(2m+I))
~ H*(BO(2m+I))
~ H~(BSO(2m+I))
~ [Pl ..... Pm ]" Remark.
In all the cases c o n s i d e r e d above
= Invw(S~(~)) In fact if
V
is a p o l y n o m i a l ring.
is any real v e c t o r s p a c e of d i m e n s i o n
InVw(S~(V*))
N. B o u r b a k i
is a p o l y n o m i a l ring in
[6, C h a p i t r e V,
=
This is no coincidence.
is a finite group g e n e r a t e d by reflections then
H~(BG)
1
and
W
in h y p e r p l a n e s of 1
generators
V,
(cf.
§ 5, th~or~me 3]).
APPENDIX
We will in this a p p e n d i x give a proof of the d i f f e r e n t i a b i l i t y of the f u n c t i o n
P'
8.3 by the formula
:~
~ ~
(8.5).
d e f i n e d in the proof of P r o p o s i t i o n First we recall some rather standard
facts from the theory of Lie groups. In the following suppose
G
simple Lie group w i t h o u t center. ~
=~
®~
•
is a compact c o n n e c t e d semiLet
~
the c o m p l e x i f i c a t i o n of ~
complex analytic Lie group
G~
be the Lie algebra and Then there is a
(the c o m p l e x i f i c a t i o n of
G)
135
and an injection algebra
of
~ ~
G~
i?
without
j : G ~ G~ and
Ad
= {ad(v)
group with
inclusion G
is
and the image with Lie algebra
by
Iv 6 I
' ad(v)(x)
= Iv,x],
c= G I ( ~ { )
Lie algebra
~ ad(~{)
that since
is injective
G~ = I n t ( ~ f )
complex
is the Lie
is the natural
Int(~ ) ~ GI(?)
defined
We can then take
ad : ? ~
: G ~ GI(~)
subgroup
~ End(~)
ad(~)
~ ~
~{
To see this notice
is the connected ad(~)
j~ : ~
= ~.
center
such that
ad(~{)
is an isomorphism
x 6I
}.
the complex =c E n d ( ~ { ) .
and
analytic
Here again
j : G ~ G{
is given
by the composite Ad
G
, Int(~)
In the following
, Int(?{).
we shall
identify
G
with
We also need the Jordan-decomposition For a complex
vector
space
V
the image of elements
a linear map
in
G{.
of
~{:
A £ End(V)
has a unique Jordan-decomposition A = S + N, with for
S S)
semi-simple and
particular ad(v)
N
(i.e.
nilpotent
for
v 6 ~
£ End(~¢) Lemma
SN = NS v
has a basis of eigenvectors
(i.e.
Nk = 0
for some
k ~ 0).
we have a Jordan-decomposition
In
of
and we have
8.A.I.
For
v 6 ~¢
there
is a unique
Jordan-
m
decomposition is nilpotent Proof.
v = s + n and
such that
adv
ad n
is semi-simple,
[s,n] = 0
We must show that the semi-simple
(and hence also the nilpotent
part)
part of
lies again in
adv
136 ad(~)
c_ E n d ( ~ )
.
Since
~
is semi-simple
the Lie algebra of derivations [14, Chapter lies in
II,
Proposition
ad(~f) D[x,y]
D
That
S
Helgason
D 6 End(f~)
x,y 6 f f .
,
So let
then also the semi-
D = S + N
be the Jordan
Then there is a direct sum decomposition
I,
(~)I
is the eigenspace
S
of
with
that is
= {v 6 ~
I (D-I) k v = 0
is a derivation
This, however, =
for some
simply means
easily follows k
(D-l-~)k[x,y]
that is,
is a derivation
such that
~I
(see e . g . S .
6.4]),
= [Dx,y] + [x,Dy]
simple part is a derivation.
eigenvalue
~
is
iff
We must show that if
decomposition.
of
ad(~)
k > 0}.
that for
I, ~ 6 {,
from the identity
(k) [ (D_l)k-ix, (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 semi-simple
since
is skew-adjoint with respect to a G-invariant
Therefore
every element of
~
and we have the root space decomposition Chapter III,
the lemma.
§ 4]) =
7~
$
/~(E $ c~E~'e~e~ '
is semi-simple
as well
(see e.g. Helgason
[14,
137
where
~ : ~ ¢ ~ ~,
one-dimensional
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 semi-simple
v 67¢
with
more,
then the semi-simple
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(g-1)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 '
one-parameter
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 .
left-multiplication
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) / ~ d-lg 6 NT~
Hence
and
Serre
(i.e. a
Hence by the
[25, Chapitre
III,
such that
= Ad(d) ~ .
Ad(d-lg)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 semi-simple
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
Clearly the adjoint
and
z C}
, v C~',
g ~ G,
w
where and
g' = gZ 6 G'.
~=
~
~
Notice that center.
~ D ~' G'
Also
T' = T/Z
where
/D
is a maximal
~'
torus in
G'
is the Lie algebra of
T'
is a compact s e m i - s i m p l e Lie group w i t h o u t
T h e r e f o r e we shall r e s t r i c t to the case w h e r e
s e m i - s i m p l e w i t h o u t center.
G
is
The reader will have no d i f f i c u l t i e s
in e x t e n d i n g the arguments to the general case. The h o m o g e n e o u s p o l y n o m i a l
P :~
a complex h o m o g e n e o u s p o l y n o m i a l P~
P~
~ ~ : ~
clearly extends to ~ ~
is invariant under the adjoint action of
8.A.4 c) and the i n v a r i a n c e of Now define For
P~
: ~
v 6~
~ ~
P
and o b v i o u s l y NT~
by Lemma
under the action by
W
on
~
.
as follows:
choose
g 6 G~
such that
I
Ad(g)v = t + n
as in Lamina 8.A.4 b), and put
P~(v)
= P~(t).
Then this is clearly w e l l - d e f i n e d by the u n i q u e n e s s of the Jordand e c o m p o s i t i o n and Lemma 8.A.4 d) . F i r s t we show that let
~ : ~ =
~
~
~ ~ +
P~' : ~
Clearly also ~ ~
then we can w r i t e
= t + n',
with
For this
decomposition
Ad(g)v = t + n
g = u • b, u 6 G, b 6 B
A d ( u ) v = Ad(b -I) (t+n)
= P'.
is continuous:
be the p r o j e c t i o n in the and notice that if
P~L
by
as above
(8.A.5)
n' 6 ~ + .
and then
142
It follows that
(8.A.7)
P~(v)
= P~(z(Ad(u)v))
and by the second part of Lemma 8.A.4 b) this equation holds for any
u 6 G
such that
Ad(u)v 6~.
P~'
To show that w h e n e v e r a sequence
is continuous
it suffices to show that
{Vk}, k = 1,2,...,
there is a s u b s e q u e n c e
{Vk }
converges
such that
uk 6 G
such that
Hence
Ad(Uk)V k 6~
Ad(Uk)V k ~ Ad(u)v
P&
.
Since uk
G
is compact
converges
to
u,
~ P~(~(Ad(u)v) ) = P~(v) .
is a c t u a l l y complex a n a l y t i c it suffices
by the Riemann r e m o v a b l e singularity and H. Rossi
Now
and so
P~(v k) = P~(~(Ad(Uk)Vk)) To see that
then
1
we can assume by taking a s u b s e q u e n c e that say.
v,
P~(Vk ) ~ P~(v) .
1
choose
to
[13, Chapter
theorem
(cf. R. C. Gunning
I, § C, T h e o r e m 3]) to show that it
is complex analytic o u t s i d e a closed algebraic
set
S ~ ~.
For this c o n s i d e r the complex analytic m a p p i n g
d e f i n e d by
F(g,t)
and notice that
!
Pc(F(g,t))
analytic near points at
(g,t).
only if ad(t)
t
: 3~
1 = dim C ~ C
= Ad(g)t,
t 6~C,
= Pc(t).
v = Ad(g)t
g 6 G~,
It follows
for which
Now it is easy to see that
F
F
that
is strictly bigger
is singular at
and let
S ~ ~f
i
than ~ C "
be the set
is
is n o n - s i n g u l a r
is singular in the sense that the kernel of ~ ~
!
P f
Now let
(g,t)
143
S = {v 6 ~ C
i the semi-simple satisfies
Notice that if actually near
v 6~
- S
semi-simple
v.
different
It remains from
a0(v)
~C:
dim(ker
s
of
ad(s))
v
> 1 }
then by Lemma 8.A.4 b),
so by the above
P~
to show that
is an algebraic
S
S = {v 6 ~ C
polynomial
I a0(v)
of
elements
U ker ~ ~6~ outside
the complex analyticity
of
subset
n = dim~ 7~,
Then clearly
= 0}
set and since
% ~ S.
P~
adv.
=...= al(v)
a closed algebraic
//~ D S = there exist
is
is complex analytic
+ a1(v) l +...+ an(n) In = det(ad(v)-ll),
which is obviously
v
For this let
be the c h a r a c t e r i s t i c
8.3.
part
This finishes
the proof of
and ends the proof of Proposition
9.
Applications
Again
let
components.
G
showed
s e t of
called
G-bundle
flat
if t h e c u r v a t u r e
has
for
of
be
with
corresponding
proposition
Proposition
9.1.
(i)
I* (G)
(ii)
I~ (G)
Corollary
9.2.
trivial
with
w
w
a)
For
this
take
this
as the
in a r i n g of the
composite
classes
A
5.5 are
H*(BGd,A). identity
from Theorem
Bj*
spaces.
6.13
maps
map)
d) :
are
zero
~ H *(BGd,~), , H* (BGd,e) .
with
real
coefficients
zero. with
geometric
However,
so easy.
i.e.
also makes
of c l a s s i f y i n g
• H* (BG,~)
a
real
coefficients
of
flat
zero.
a differential
topological
functions,
elements
Bj* --
The Chern
to h a v i n g
Then by Theorem
, H*(BG,~)
classes
the u s u a l
shall
and
the underlying
(actually
following
The Pontrjagin
remarks.
map
: BG d ~ BG
are
are
the
is o b v i o u s
The
in a p r i n c i p a l
of c o u r s e
coefficients
with
Gl(n,~)-bundles
Gl(n,~)-bundles
From
Bj
Gd,
in g e n e r a l .
the n a t u r a l map
so w e
many
form vanishes
transition
last condition
correspondence
j : Gd ~ G
following
classes
finitely
is e q u i v a l e n t
to the g r o u p
of a f l a t G - b u n d l e
Let
this
G-bundles,
with
a connection
constant
This
topological
in o n e - t o - o n e
b)
with
G.
the c h a r a c t e r i s t i c
of f l a t
that
a reduction
group
definition
The
3.22)
trivializations
discrete
Lie group
3 we
(Corollary
the bundle
sense
b e an a r b i t r a r y
In C h a p t e r
differentiable we
to f l a t b u n d l e s
a direct
definitions
as w e l l
point
as for
of v i e w
proof
of C h e r n
these
are
of C o r o l l a r y classes
the g e n e r a l
just
9.2 f r o m
is r e a l l y
subject
of
this
not
145
chapter 4]
see F. W. K a m b e r
and also
[16a].
ly d i f f e r e n t Notice conclude for
G
that
that
shown
G
W
: I~(G)
that there exist (see J. M i l n o r
othe r
hand we shall where
K ~ G we
H~(BGd,~)
any d i s c r e t e
zero.
[22],
or E x er c i s e
group
Since
algebraic
the n e r v e
NH
set and by P r o p o s i t i o n
5.15,
complex
a q-cochain
(q
C*(NH)
factors
(9.3)
of
where H)
J. M i l n o r with
non-zero
2 below).
Bj ~ only
subgroup. Gd
not
On the
depends
on
In the
is a d i s c r e t e
group
description.
In fact
is a d i s c r e t e
simplicial
H~(B~,~)
and w h e r e
However,
is in general
Sl(2,~)-bundles
K.
8.1 we
is zero.
flat
of
for a c o m p l e t e -
For example
compact
Chapter
[12]).
~ H~(BG,~)
is a m a x i m a l
has an e x p l i c i t
and
then by T h e o r e m
see that the image of
fix a choice
especially
3 below,
~ H*(BGd,~)
Bj * need not be
class
following
is c o m p a c t
Bj~ : H ~ ( B G , ~ )
and
[16,
of view A. G r o t h e n d i e c k
if
Euler
G/K
(See also E x e r c i s e
non-compact
surjective has
point
and Ph. T o n d e u r
is the h o m o l o g y
is a f u n c t i o n
the c o b o u n d a r y
c
6
for
of the
: H x...x
H ~
is g i v e n by
6(c) (x I .... ,Xq+ I) = c(x 2 .... ,Xq+ I) +
The h o m o l o g y group
+
q [ (-1)ic i=I
+
(-1)q+Ic(xl,...,Xq) ,
of this
cohomology
of
giving
an e x p l i c i t
(9.4)
I~(K)
(Xl'''''xiXi+l'''''Xq+l)
complex ~.
description
~ H~(BK,~)
X l , . . . , X q + 1 6 H.
is k n o w n
In this
+
as the E i l e n b e r g - M a c L a n e
chapter
we shall
of the c o m p o s i t e
study
Bj ~ by
map
~ H~(BG,~)B-~J~H~(BGd,~)
= H(C~NGd).
=
This
is done
Step
I.
in two steps:
By C h e r n - W e i l
theory
P 6 II(K)
defines
a closed
146
G-invariant
21-form on
Step II.
G/K.
Using the contractibility
any closed G - i n v a r i a n t Step I .
Let ~
respectively.
q-form on and
~
a l-form
projection eK 6 A I ( G , ~ )
the principal curvature a closed
hence
P(~)
on
and
in ~
and let
.
on
Let
G/K.
of
which
K :~
~K
C~NG d. G
and
be the <
defines
a connection
be the associated
theory
P 6 II(K)
Notice that since
where again
also
G
defines eK
by
~K
and
acts on the left
For this we introduce 9.5.
A filling of
the following G/K
is a family of
C~
simplices
p = 0
such that for
: A p ~ G/K, a(@)
= 0
gl ..... gp 6 G,
p = 0,1,2 ....
is some "base point",
usually
0 = {K})
p = 1,2,...,
Lg I o o(g2,...,gp), (g1'''''gp)
0 ei = I
i = 0,
(g1'''''gigi+1' .... gp) , a(gl,...,gp_1) '
(Here
in
G/K.
o(gl,...,gp)
(9.6)
K,
is invariant
~ ~
the left G-action
G-invariant,
in
By left-translation
G ~ G/K.
P(~}
are
Definition
(so for
~
is invariant under
Step II .
singular
K,
Then by Chern-Weil
21-form
we define for
be the Lie algebras
which clearly defines
K-bundle
form.
definition
G
onto
G/K
a q-cocycle
Choose an inner product
under the adjoint action of orthogonal
G/K
of
Lg I : G/K ~ G/K Lemma 9.7.
as usually
is given by
There exist explicit fillings
0
< i < p,
i = p, Lg1(gK)
of.
G/K.
= glgK).
147
Proof. h
Let
0 = {K}
: G/K ~ G/K,
s
0,
that
is,
explicitly
h0(x)
given
that
near
by C~
0
=
o(~)
the
and
equal
with
of
h I = id
exponential
map,
0
6(I)
= I
G/K
cf.
s
to
can be the
8.1).
for
6 : [0,1]
let
(this
Theorem
to
where
function
and
contraction
following
h6(s),
point
We can
near
~
[0,1]
and
6(s)
zero is a
= 0
zero.
Now we define p
using
the b a s e
C~
Vx 6 G/K
is c o n s t a n t l y
hs
non-decreasing s
= 0
be
be a
in the r e m a r k
hs
by r e p l a c i n g
for
[0,1]
constructed
reference assume
s £
6 G/K
= 0
o(gl,...,gp)
and
for
inductively
p = I
as
follows:
: A I ~ G/K
o(gl)
For
is g i v e n
by
o(g I) (t0,t I) = htl (gl °)" For
p > I
consider
{e I, .... ep} ~ ~ p + 1 . that
face must
this m a p
as
Then
be g i v e n
to the c o n e
(9.8)
Ap
the c o n e
on the
the r e s t r i c t i o n
by
face
of
the c o n t r a c t i o n
o(gl ..... gp) (t o ..... tp)
by
o(g I ,...,gp)
Lg I 0 o ( g 2 , . . . , g p ) ,
using
spanned
hs.
and we
to
extend
Explicitly
=
= h 1 _ t 0 [ g l o ( g 2 ..... gp) (tl/(1-t0~ ..... t p / ( 1 - t 0)) ]. It is n o w
straightforward
The m e r i t to c o n s t r u c t
of a f i l l i n g
explicit
InVG(A~(G/K))
consisting
of G - i n v a r i a n t
the
left
G-action
J
by
~
of
(9.6)
on
of
inductively.
G/K
Eilenberg-MacLane
subcomplex
by
to c h e c k
is that cochains:
the de R h a m
forms G/K).
: InVG(A*(G/K))
it e n a b l e s
complex
(where
the G - a c t i o n
Define
the m a p
~ C ~ ( N G d)
Consider
us the
A~(G/K) is i n d u c e d
148
(9.9)
J(~) (gl ..... gP) : I AP o(gl ..... g p ) ~ ' gl,...,g p 6 G, ~ 6 AP(G/K) , p = 0,1,2,.. Proposition b)
9.10.
a)
J
is a chain map.
The induced map on homology J~
: H(InVGA~(G/K))
is i n d e p e n d e n t Proof.
~ H(C~(NGd))
= H~(BGd,~)
of the choice of filling.
a)
By Stoke's
J(dw) (gl '''''gp+1)
theorem
= I Ap+1
P~
i=I
(9.6)
o(gl ' "" "'gp+1 )~d~
IAp[Lgl +
and
o o(g2' .... gp+1 ) ] ~
iI
(-1)
Ap
~Cg
+
.
I ' "''gigi+1 '''''gp+1
)*~
+ (-I) P+I I Ap o(g I '" - .,gp)*~ = 6(J(~)) (gl ..... gp+1) since
L* m = w. gl b)
We give an a l t e r n a t i v e
map of simplicial
description
of
J~:
Consider
the
manifolds : N(Gd;G/K)
~ NG d
where N(Gd,G/K) (p) = NGd(p) and the face o p e r a t o r s
ei(g1'''''gp'gK)
× G/K
are given by
=
(g2 .... ,gp,gK) ,,
i = 0,
(g1'"
,gigi+ I ...,gp,gK),
0 < i < p,
(gl'"
'gp-1 ,gpgK)
i = p.
I
+
149
is just g i v e n by the p r o j e c t i o n the p r o o f of T h e o r e m bundle with fibre t h a t if
~
L
8.1.
G/K
the f i r s t factor.
The realization
associated
is a f i l l i n g
-I
onto
of
0 ~(gl,...,gp)
G/K
to
of
~
(Cf.
is the f i b r e
7G d : E G d ~ BG d.
Notice
then the f a m i l y
: A p ~ G/K, g l , . . . , g p 6 G, p = 0 , 1 , 2 , . . . ,
(gl.-.gp) defines
a s e c t i o n of
N o w if
e £ Aq(G/K)
f a m i l y of forms on the p r o j e c t i o n s Clearly :
which
onto
d~ = d--~,
explains
is an i n v a r i a n t
A p x NGd(P) G/K,
x G/K,
defines
f o r m t h e n the c o r r e s p o n d i n g
p = 0,1,...,
an e l e m e n t
~ H(A*(N(Gd,G/K))).
i n d u c e d by
~ 6 Aq(N(Gd;G/K)) •
On the o t h e r hand,
since
is c o n t r a c t i b l e
: N(Gd;G/K) induces
an i s o m o r p h i s m
Theorem
6.4.
Hence
~ NG d
in de R h a m c o h o m o l o g y
defined
(I.e. w i t h o u t
we c l a i m t h a t this is just
J~
g e t an e x p l i c i t
~*
inverse
~* ~
5.16 and
, H(A ~(N(Gd,G/K)))
H(A*(NG d)) is c a n o n i c a l l y
by L e m m a
the c o m p o s i t e m a p
H ( I n v G A ~(G/K)
where
the d e f i n i t i o n ) .
so we h a v e an i n d u c e d m a p on h o m o l o g y
H ( I n v G A~(G/K))
G/K
II ~ N
to
: A*(N(Gd,G/K))
: A p x NGd(P)
I
, tt(C*NG d)
a choice
In f a c t g i v e n
of filling) a filling
and ~
we
~ A~(NGd)
~ A p x NGd(P)
x G/K,
p = 0,I,2,...,
is
g i v e n by
(t,(g0, .... gp))
=
(t, (go .... 'gp) ' (gl "''gp)-1~(gl ..... gp) (t)) t 6 A p,
g l , . . . , g p 6 G, p = 0 , 1 , 2 , . . .
150
Then obviously
I (~)
for
~ 6 InvGAP(G/K)
(g1' .... gp)
= IAp[ L
_
.gp)-1 (gl
4
o o(g1'
"''gp) ] ~
i
¢
lap o(gl '''''gp) ~ This proves
the proposition.
Remark. hs
In the proof of Lemma 9.7 we replaced
by the c o n t r a c t i o n
h6(s)
where
in order to be able to define the of
A p.
6(s) = 0
C~
map
On the other hand the inductive
the original
contraction makes
corresponding the integral
(9.9).
3, § 7])
: G ~ G/K
is the projection
and
The curves
s ~ hs(X)
corresponding
construction
=~
~
on all
(9.8)
exp
:~
:y
~ G
and the
h
explicit-
s
~ G/K
[14, Chapter
(where
6, Theorem
1.1]).
defined by x 6 G/K,
s 6 [0,1]. to a G-invariant
and we shall therefore refer
filling defined
Then
the exponential
are geodesics with respect G/K
using
(see Helgason
~ = z o exp
h (x) = ~(s~-1(x)) , s
Riemannian metric on
by geodesic
~
(see Helgason
Therefore we get a contraction (9.11)
near zero
is semisimple with finite center:
and the map
is a d i f f e o m o r p h i s m
s
o(gl,...,gp)
let us describe
we can choose a Caftan d e c o m p o s i t i o n [14, Chapter
for
does not affect the value of
In particular G
the contraction
sense on the open simplex
change of parameter
ly for the case where
map)
= ] (~) (g1'''''gp) "
inductively
by
(9.8)
to the
as the filling
simplices.
We can now describe
the composite map
Theorem 9.12.
P 6 If(K)
For
Bj~ : H ~ ( B G , ~ )
~ H~(BGd,~)
is represented
in
of
H 2 1 ( C ~ ( N G d ))
(9.4):
the image under
w(P)
6 H21(BK,~)
~ H21(BG,~)
by the E i l e n b e r g - M a c L a n e
cochain
151
1
3(P(~K)),
P(~K1 ) 6 invG (A21 (G/K))
where
above and (9.13)
J
is given by (9.9).
Bj* (w(P)) (gl ..... g21)
where
is a filling Proof.
Let
commutative
of
~
That is,
= I A2I ~ (gl , - - - , g 2 1 ) * P ( ~ )
be the inclusion
of simplicial
N(Gd,G/K)
in step I
G/K.
i : K c-* G
diagram
is defined
and consider
the
manifolds
, NG/K < NI
NK/K
(9.14) N(Gd ) where
Nj
~ : NGd(P)
, NG+
morphisms
(9.14)
all maps
~* 0 (NY) *-I < :~
and let
8
(6.12).
Then
bundle
~
above.
and
Therefore
connection
of
8K
~K and
For
P 6 II(K),
represented
by the form
(Ni)*-lw(p) P(~)
~* o (NY)*-Iw(p) by the form
induce
iso-
we shall calculate
projection
in
is a connection
and we let
Nj
H(A*(N(Gd~G/K))).
NG ~ NG
~K
to
as in step I given by
in the principal
be the curvature
the connection and curvature
is represented
~
be the orthogonal
8K = < 0 8
that the restriction obviously
except
: H(A*(NK))~
be the canonical
NG ~ NG/K
is given by
= (gl...gpg ..... gpg,g)K.
in de Rham cohomology.
For this let
NK
x G/K ~ NG(p)/K
~(gl,...,gp,gK) In the diagram
Ni
NG(0)
= G
forms defined
6 H21(A*(NG/K))
6 A21(NG/K)
form.
KNotice
are in step I is clearly
It follows
that
£ H21(A*(N(Gd;G/K))) P(~ )
where
now
P(~K1 ) 6 A21 (G/K)
152
denotes the G - i n v a r i a n t form d e f i n e d in step I and w h e r e 6 A~(N(Gd,G/K))
for
~ 6 InVGA~(G/K)
is the a s s o c i a t e d
s i m p l i c i a l form as in the proof of P r o p o s i t i o n 9.10 b). fore it follows
from the d i a g r a m
~*(Nj)~(Ni) ~-I (w(P))
(9.14)
that
1 P(~K)
represents
6 H(A~(N(Gd,G/K)))
and the theorem follows from the d e s c r i p t i o n of the proof of P r o p o s i t i o n
There-
J,
given in
9.10 b).
As an example we shall now study T h e o r e m 9.12 in the case G = Sp(2n,]R),
the real symplectic group.
of n o n - s i n g u l a r m a t r i c e s t
where
g
g 6 Gl(2n,]R)
is the transpose of
J =
I
g
g tg
= I)
(equivalently
U(n) =c Sp(2n,]R) J).
first C h e r n - c l a s s Let
m a t r i c e s and
Let
S(2n,IR)
(g 6 0 ( 2 n ) U(n)
is the subgroup of elements
M(2n,]R)
exp
be the set of all
2n x 2n
the set of symmetric matrices.
: S(2n,]R)
d i f f e o m o r p h i s m with inverse log.
The image
First some notation:
be the set of p o s i t i v e d e f i n i t e
=c M(2n,IR)
T h e n the e x p o n e n t i a l map
T h e o r e m 9.15.
K = G D O(2n)
The first class to study is therefore the
=m GI(2n,]R)
s y m m e t r i c matrices.
is the m a t r i x
to the unitary group
c I 6 H2(BU(n),JR).
P(2n,]R)
J
tgjg = j
0]"
w h i c h is isomorphic
commuting with
such that
and
Here the m a x i m a l compact s u b g r o u p is iff
This is the subgroup
~ P(2n,]R)
is a
We then have
(Bj)~c I 6 H2(BSp(2n,]R) d , ~ )
of
the first Chern class is r e p r e s e n t e d by the cochain
(9.16) where
(Bj~c I) (gl,g2) tr
means
trace.
I [Itr (j[ tglgl + [g2tg2) -s ]- 11og g2tg2) ds = - -4~ J0
153
Remark. definite
Notice
symmetric
that
matrix
tglg I + hence
(g2tg2)-s
invertible,
is a p o s i t i v e so the r i g h t h a n d
side is w e l l - d e f i n e d .
Before proving n = I.
Then
matrices
Theorem
G = Sp(2,~)
of d e t e r m i n a n t
the E u l e r
class
9.15
= Si(2,~)
I.
Here
e £ H2(BSO(2),~)
0>
g2tg2 = k-l< y
let us s p e c i a l i z e
y-1
k,
the g r o u p of
K = SO(2) .
to the c a s e
For
and
2 × 2
cI
gl,g 2 6 G
equals write
y > 0, k 6 SO(2) ,
and k-1
tglglk =
d '
It is e a s y to see t h a t
(9.16)
(Bj~e) (gl,g2)
(and e q u a l
to zero for
b = ~
ad-
a,d > 0.
= I,
then r e d u c e s
to
[I log y ds I0 dy'S+ayS+2
= ~[Arc
tan<~>
b = 0).
Notice
_
b
[Y
dt
27 J0 a t 2 + 2 t + d
- Arc
t a n /~ ~
h /]
t h a t the n u m e r i c a l
value
satisfies I
(9.17)
(This i n e q u a l i t y
can a l s o be d e d u c e d
see E x e r c i s e
2 below).
to J. M i l n o r
[22]:
Corollary
9.18.
o v e r an o r i e n t e d class
(9.19)
~
I
I (Bj~e) (g1'g2) I < 2--~ " 2 - 4 "
e(E)
This has
Let
surface
directly
the f o l l o w i n g
~ : E ~ Xh Xh
of g e n u s
satisfies
I<e(E),[Xh]>l
< h.
from Theorem consequence
be a flat h > I.
9.12; due
Sl(2,~)-bundle Then
the E u l e r
154
Proof. topology with
of
We
first
need
surfaces.
pairwise
Xh
some w e l l - k n o w n
facts
can be c o n s t r u c t e d
identifications
of
the
sides
about
as a
4h-polygon
x.Nx! 1
the
as on the
1
figure
x~
Here
the
sides
F
with
group
Xl,...,X2h the
...-
, ," /
single
give
generators
of
the
fundamental
relation
-I -I -I -I X l X 2 X I x 2 . . . X 2 h _ i X 2 h = I. Furthermore
the u n i v e r s a l
covering
in E x e r c i s e
2 e)
We can
f : BF ~ X h generators
below).
as f o l l o w s : Xl,...,X2h
For
the c o r r e s p o n d i n g
curve
over
the
Br
groups
z i ( X h) = 0
equivalence with of
of
the c o m p l e x
for
the g e n e r a t o r
the
a word
and m a p
of
Xh
Hence
fact
Clearly
theorem.
is r e p r e s e n t e d
choose x
by
the c h a i n
the m a p
the h o m o t o p y
is a h o m o t o p y
is i s o m o r p h i c ~ ~
in the
extend
In p a r t i c u l a r
H2(C,N?)
map
AI x x ~ AI x F
Now
that f
(see r e f e r e n c e
a continuous
in the p o l y g o n .
i > 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 (4h-2)
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
4h-2
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
semi-simple
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 skew-symmetric
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
2-simplices
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~0-1(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 K-invariant, 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
2-simplex
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~0-1(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 =
= {X = (A
of
M(2n,]R)
the Lie algebra as the set of matrices
B t C = C, tB = B} . _tA> K = U(n)
-C>jtc = C '
is the subspace tA = -A}
157
with
complement
in
~
~(n)
~(2n,~)
:
= {A =
is i d e n t i f i e d w i t h
complex matrices
_AB>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 <(X)
For
i m a g e of
g i v e n by
~(gK)
embedding
the f i r s t C h e r n
g i v e n by
=-~ I
Now
(see G. M o s t o w
A/
is g i v e n by the C h e r n - W e i l
P(X)
n × n
4) by l e t t i n g
In this n o t a t i o n
(9.24) G/K
of H e r m i t i a n
5 of C h a p t e r
X = correspond
tB = B} .
P £ I1(U(n))
therefore s 6 [0,1],
takes in
< : ~(2n)
= ½(X - tx) ,
g i v e n by
(9.24)
the f o l l o w i n g G N P(2n,~) :
~(n)
is g i v e n by
X 6~(2n).
above
the f o r m
form along a curve
~*P(0 K) P = p (s) ,
158
1
I*P(eK)(P) But
tr(jt(~-l~))
=-8--~ t r ( J ( ~ - 1 ~
= tr(T-ITt7)
-
t -1 (
= - tr(jI-ll)
I tr(jT-1{), I*P(@K) (P) = -4--~
(9.25)
p
Now suppose
p(s)
is a g e o d e s i c
= ~0 e x p ( s Y ) ~ 0 '
~)))'
in
p½.
so
T = p½
G D P(2n,~),
s 6 [0,1],
~ =
that is,
Y 6~
,
T O 6 G A P(2n,IR). Then
p-l~
(9.26)
is a c o n s t a n t = exp(Z(s)), Chapter
in y Z(s)
= T01Y~ 0 = 9(0)-16(0)
= Q
.
if w e w r i t e
p(s)
(see H e l g a s o n
[14,
6y
II, T h e o r e m
On the o t h e r hand, , s 6 [0,1],
then
=
1.7]):
-I. 1-exp (-ad Z) P P = adZ
({)
Z
where
again
T =
=
1 - e x p ( - a d 7)) (2) Z (I + e x p ( - a d ~)) ( ad~2
=
Z T-I (I + e x p ( - a d ~)) ( T) ,
p½ =
z
exp ~.
tr(jT-1~)
Now since of
M(2n,~)
Z 6 S(2n,~), with
Therefore
= tr(J(1
ad Z
respect
H e n c e by
(9.26)
Z
+ e x p ( - a d ~])
is a self a d j o i n t
to the inner p r o d u c t
= tr(tAB)
-I (Q)) .
= tr(AtB) .
transformation
159
tr(jT-1~)
a d ~Z) ) -I (Q)>
= -<J, (I + e x p ( -
= <(I + exp(- ad2))-1(J) ,Q>. z k (ad ~) (J) = zkJ,
Now it is easy to see that
hence
;
tr(jr-1~)
= -<(I + e x p ( - Z ) ) - I J , Q > = 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 F-covering)
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
path-connected
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
l-connected
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(jT-IT) (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 . F-bundle
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
:
Gd-bundle
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 G-bundle
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
G-bundle
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
F-covering
: 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
F-action
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
G-bundle
induces
form whose
under
Again
be
~
the p u l l - b a c k
the u n i q u e
free).
by
G-bundles.
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
G-bundle 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
K-bundle
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
G-bundles.
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(I-h) .
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
K-bundle
163
G ~ G/K
relative
subspace
~
iii) iv) above
and
inverse flat
to the a d j o i n t
= ker(<) Show
that
Let
F ~ G
let
e
image
of
SI(2,~)
classes
inequality
with
F ~ G.
3.
let
a differentiable a ~-filling
of
~(gl,...,gp)
that a)
(9.6) Show
that
G
(9.29).
inclusion
h
as
of the
be
the a s s o c i a t e d
shall
Steenrod and
C~
of C h e r n [30,
F
G x F ~ F. of
a refinement
definition
a Lie g r o u p
to be a f a m i l y
make
§ 41]).
a manifold
For
q ~ 0
singular
with
define
simplices
g l , . . . , g p 6 G, p = 0 , I , 2 , . . . , q ,
is s a t i s f i e d
for
that q-fillings
p ~ q.
exist
if
are h o m o t o p i c
F
is
(q-1)-connected
(in the o b v i o u s
sense)
if
is q - c o n n e c t e d . b)
and form
s(~)
Now
let
suppose
F
~ 6 InVG(Aq(F,~))
representing
of the
is
(q-1)-connected be a closed
an i n t e g r a l
inclusion
Hq(F,~)
6 Cq(NGd,~/~)
by
(9.31)
s(~) (gl ..... gq)
class
= I
show
with
complex
(i.e.
c Hq(F,~)).
q-filling valued
a class
Define
~(gl ..... g q ) * W Aq
and
we
(see N.
G-action
two q - f i l l i n g s
from
of g e n u s
~ F\ H
topological
: A p ~ F,
on the
= h - I.
exercise
be
follows
K
that
classes
left F
: E
) ~ [ F \ H]>
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
G-invariant
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 k-th 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 2k-2-connected
and
F = Gl(n,~)/Gl(k-1,~).
H2k-I(F,~)
c l o s e d complex valued
form
= ~.
In fact
F
is
Show that there is a
~k £ InvG(A2k-I(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(k-I),~)
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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W. A. Benjamin,
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15
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166
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Classes
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Tracts
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Berlin
Berlin-Heidelberg-New
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215-305,
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Cliffs,
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and Ph.
Geometry,
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functions
Characteristic 493, [17]
Geometry
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schemas,
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Prentice-Hall,
San F r a n c i s c o [16]
des
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and
New York
Wissensch.
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114),
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[20]
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1974.
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of L o c a l l y
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78),
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Symmetric
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[24]
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J. P. Serre,
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H. Shulmann,
Semi-Simple
On Characteristic
C. L. Siegel,
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Series
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F. W. Warner,
Interscience
(Princeton Press,
Princeton,
1973, pp.
of Tokyo Press,
Algebra
I,
37-49, Tokyo,
ed. 1975.
(Grundlehren Math. Berlin - G~ttingen
-
1960.
Scott,
(1952), pp.
Series 1957.
25),
forms and the topology of mani-
33), Springer-Verlag,
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]
105-112.
1971.
Differential
A. Hattori,
[33]
and Abelian
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[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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of California, [27]
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and Co.,
de de Rham,
Manifolds
Glenview,
Comment.
and
1971.
Math.
Helv.
119-145.
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
Alexander-Whitney approximation
barycentric base
bundle -
the
diagonal
of principal
3 G-bundle
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,
Chern-Weil
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
Chern-Weil
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
G-bundle
functor
87
transformation
derivative
58
differential
58 20,30
cup-product 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
Euler-Poincar~
15,17
14
map
-
a covering
83
set
edge-homomorphism
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 G-bundle
differential
map
-
9
property
7,21
E. 1
75
realization
fibre
42
bundle
-
of p r i n c i p a l
39
G-bundle
filling
146,163
flat
144
bundle
-
47,51
connection
52
foliation frame free
bundle
38
G-action
72
fundamental
Gauss-Bonnet 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
"pull-back")
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
Jordan-decomposition
local
56
connection
index
-
of v e c t o r
field
connection
maximal
torus
natural
109 40,42
trivialization
Maurer-Cartan
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
k-form
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
Levi-Civita
4,
48
form
37
function
62
classes polynomials root G-bundle F-covering
66,103 66 137 39 159
174
page rational
differential
37
form
106
realification reduction regular
of
42
a G-bundle
128
element
relative
Euler
108
class
137
root root
space
semi-simple simplicial
126 ,136
decomposition
135
element chain
23
complex
-
cochain
complex
23
-
de
complex
20 ,22,91
-
form
22 ,91
-
G-bundle
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
-
skew-hermitian standard
algebra
68
matrix
3
simplex
star-shaped Stoke's
in a L i e
42
-
4
set
6
theorem
strongly structural symmetric -
symplectic
free
72
G-action
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 G-bundle
55
torsion-form
117
torus total
Chern
-
complex Pontrjagin
-
space
transition trivial
vertical
12
of principal functions
duality sum
G-bundle
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
