Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZCirich
36 Armand Borel The Institute for Advanced Study Princeton, New Jersey, USA
Topics in the Homology Theory of Fibre Bundles Lectures given at the University of Chicago, 1954 Notes by Edward Halpern
1967
Springer-Verlag. Berlin. Heidelberg. New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. @ by Springer-Verlag Berlin 9 Heidelberg 1967. Library of Congress Catalog Card Number 67-28561 Printed in Germany. Title No. 7356.
INTRODUCTION This fascicle consists of the Notes of a course given at the University of Chicago in 1954, the purpose of which was to discuss some then recent developments in the homology theory of fibre bundles, pertaining to H-spaces, classifying spaces and characteristic
classes.
spectral sequences,
Since then, of course, alternative
approaches to some of these topics have been introduced, and new results have been obtained, which make; these Notes outdated in several respects. because)
In spite (or maybe
of this, it was recently suggested that they be included in this series.
No changes have been made in the original version, written by E. Halpern, to whom I am glad to express my hearty thanks. However, which, without aiming at completeness,
some comments have been added at the end,
point out further results and give more recent
references. A. Borel
TABLE OF CONTENTS Chapter
I.
The homologlcal
properties
of H-spaces.
Algebraic preliminaries. Topological preliminaries. Structure theorem for Hopf algebras. Primitive elements, Samelson's theorem. The PontrJagin product. Homology of H-spaces. Spaces on which an H-space operates. Bibliography Chapter II. .
9. io. ii. 12. 13. 14. 15.
Spectral
Chapter 21. 22. 23. 24. 25.
2. 4. ii. 14. 17. 19. 25.
sequence of a fibre bundle.
Differential and filtered modules. Notion of a spectral sequence. Spectral sequence of a differential filtered module. Systems of local coefficients. Fibre bundles. Spectral sequences of a fibre bundle. Some simple applications. Pairing of the spectral sequence of a principal bundle with the homology of the structural group. Bibliography.
Chapter III. Universal bundles 16. 17. 18. 19. 20.
i.
and classifying
Classifying
spaces of the classical
notes and comments.
52. 55. 56. 57. 64. 70.
groups.
Unitary groups. Orthogonal groups, cohomology mod 2. Orthogonal groups, cohomology mod p ~ 2. Integral cohomology of Bo(n) and BSO(n ) . Stlefel-Whltney classes, PontrJagin classes. Bibliography.
Bibliographical
46. 51.
spaces.
Universal bundles and classifying spaces. p(U,G) : three fiberings involving classifying spaces. Some results on universal spectral sequences. Proof of one theorem on universal spectral sequences. Invariants of the Weyl group and classifying spaces. The Hirsch formula. Bibliography. IV.
26. 28. 29. 31. 53. 37. 41.
71. 75. 82. 84. 87. 92. 93.
26
C H A P T E R II SPECTRAL
SEQL~NCE
In this chapter we shall define of the spectral
sequence
OF A FIBRE B U N D L E
spectral
of a fibre bundle.
the reader to the f o l l o w i n g
sequences
and give the m a i n p r o p e r t i e s
For the proofs
of these
statements
we r e f e r
literature:
(i)
the Cech a p p r o a c h
(ii)
the singular
[1],
approach
[5],
[9],
[ll],
(iii) in the case where the base
[12], space is a locally
finite p o l y h e d r o n
[6],
[7],
[io]. F i n a l l y we shall make
8.
Differential
and filtered m o d u l e s
Differential
module:
that dd = O. R e l a t i v e H(M)
= d-l(o)/dM.
m e a n i n g dM i C graded H(M)
some simple a p p l i c a t i o n s .
an A-module M t o g e t h e r with an e n d o m o r p h i s m
to the d i f f e r e n t i a l
d we define the d e r i v e d m o d u l e
If M is graded then we require
Mi+ r where r is an i n t e g e r
in a n a t u r a l way.
is then n a t u r a l l y Differential
Similarly
d:M ~ M such
that d be h o m o g e n e o u s
independent
if M is b i g r a d e d
of degree r,
of i. It follows that H(M) we require
is t h e n
that d be b l h o m o g e n e o u s ;
bigraded.
al~ebra:
a differential
m o d u l e M w i t h an a u t o m o r p h i s m
w satisfying
dw + wd = 0 d(x.y) If M is graded are d of degree
(bigraded)
-1 (homology),
Filtered modules: g i v e n by an i n c r e a s i n g homology
(cohomology)
Corresponding on M by setting f(x) Clearly valued
f(O)
filtration
is a graded
or d of degree
an i n c r e a s i n g
(decreasing)
(bigraded)
(decreasing)
sequence
to a d e c r e a s i n g
~ min(f(x),f(y)),
f(x) on M w h i c h
satisfies
integer
(f(x)
of an A - m o d u l e M is
M i such that M = u M i. F o r
(decreasing)
i (possibly
and f(ax)
~ f(x).
these p r o p e r t i e s
~ i }.
cases
= (-1) d~
case.
of M we define a f i l t r a t i o n
for M by s e t t i n g M i = {x
The s t a n d a r d
and w(x)
filtration
of s u b m o d u l e s
filtration
equal to the m a x i m u m
algebra.
+l (cohomology),
we u s u a l l y c o n s i d e r the i n c r e a s i n g
= ~, f(x+y)
function
then H(M)
= dx.y + w(x).dy.
|
function
f
such that x e M i.
Conversely
given an integer
we can define a d e c r e a s i n g
27
Similarly
an i n c r e a s i n g
filtration
can be c h a r a c t e r i z e d
W i t h each f i l t e r e d m o d u l e M is a s s o c i a t e d GrM = ~ Mi/Mi+l,
(decreasing
case),
GrM = ~ Mi/Mi_l,
(increasing
case).
A decreasing above
filtration
An i n c r e a s i n g
for i small e n o u g h and limited above
c u s s i o n holds
filtration
is limited below if M i = 0
if M i = M for i big enough.
in the i n c r e a s i n g
L e m m a 8.1.
of f i l t e r e d m o d u l e s
is a h o m o m o r p h i s m
there
(a) If fw is inJective
If fw is surJective
case,
an analogous
dis-
case.
f(M i) c M' i. Given such a h o m o m o r p h i s m
(b)
a graded module:
follows we shall only c o n s i d e r the d e c r e a s i n g
A homomorphism
function.
is limited b e l o w if M i = M for i small enough and limited
if M i = 0 for i big enough.
In what
by a f i l t r a t i o n
f : M § M' such that
is induced a h o m o m o r p h i s m fw:GrM + GrM'.
and if ~ M i = 0 t h e n f is also inJective.
and the f i l t r a t i o n
of M' is limited
above then f is sur-
Jective. Proof.
(a) If x ~ M, x ~ O,
x ~ Mj+ 1. T h e n the p r o j e c t i o n in t u r n implies (b) M~ C f(M).
f(x)
~ O. Thus
Let x ~ M'. Suppose
x'-f(x)
Filtered further require multiplication
of M' is limited above,
J i s ' s u c h that Mj+ 1 C f(M);
alsebra:
there
fw(~)
~ 0 which
and
r M'j+l,
(b) follows
if M is an A - a l g e b r a
y ~ Mj have x and y as p r o j e c t i o n s
in detail,
if ~ is its p r o j e c t i o n
and hence there
is a y ~ M w i t h
by induction. w h i c h is f i l t e r e d
that M i . M j C Mi+ j. For a f i l t e r e d induced naturally;
for i big enough
let x' ~ M~ and let x' be its p r o j e c t i o n
is an x E Mj such that
= x'. T h e n x'-f(x)
= f(y). Thus x' ~ f(M)
J such that x ~ Mj but
f is injective.
Since the f i l t r a t i o n
M j / M j + 1 we have fw(X)
an integer
~ ~ Mj/Mj+ 1 of x is not zero. T h e r e f o r e
in Mj/Mj+ I. Since fw is surJective in
there exists
algebra,
as an A - m o d u l e
GrM is an a l g e b r a w i t h
if x ~ Mi/Mi+l,
y c Mj/Mj+ l, and x ~ Mi,
then ~.~ E M i + j / M i + j + 1 is the p r o j e c t i o n
x.y E Mi+ j. The c o n d i t i o n M i . M j E Mi+ j insures
we
that x.y is independent
of
of choice
of
x and y. Filtered filtration
~raded module:
M = v M i that
if M is graded by M = [ i M
then we further
it satisfies M i = [ M i ~ JM. It t h e n follows
require
of a
that GrM is hi-
graded naturally. Differential many a p p l i c a t i o n s
f i l t e r e d module:
a filtered
A - m o d u l e M with a d i f f e r e n t i a l
we have dM i C M i but we do not require
it here).
d. (In
28
If M is a d i f f e r e n t i a l
filtered module
then H(M)
is f i l t e r e d n a t u r a l l y
by the
submodules: J.
: {x
I x contains
E H(M)
1
a cycle of Mi}.
Examples: (I)
A,B:
differential
graded modules,
M : A @ B, d the usual total d i f f e r e n t i a l , M i : jli- JA ~ B
(ii)
X : a compact M
(or M i : JZi~A | JB). complex
: set of e x t e r i o r
d : exterior
analytic manifold,
differential
dimension
n,
forms on X,
differentiation,
M i : ~ A p'q where the A p'q are the m o d u l e s of forms of type (p,q) d e f i n e d p~i Let (Zl,...,Zn) be a local c o o r d i n a t e system on J; then each element of
as follows.
A p'q is e x p r e s s e d
locally
By de Rham's
in terms of p of the dz i and q of the dz i.
t h e o r e m H(M)
is the cohomology
(iii) Let L be a finite d i m e n s i o n a l a l g e b r a M is the quotient xy-yx-~x,y].
ring of X .
Lie a l g e b r a over K. Its u n i v e r s a l
enveloping
of its tensor a l g e b r a by the ideal g e n e r a t e d by the elements
An i n c r e a s i n g
filtration
is o b t a i n e d by t a k i n g as M i the p r o j e c t i o n
in
i
M of
~0 ~ j L. The B i r k h o f f - W i t t ~
t h e o r e m may be e x p r e s s e d
by saying that GrM is iso-
J morphic 9.
to the symmetric
Definition
algebra
of a spectral
A spectral
sequence
over L.
sequence.
consists
of a sequence
of d i f f e r e n t i a l
modules(Er,dr) ,
r 2 ro,(r o may be - ~),such that Er+ 1 : H(E r) with respect to d r . r 1 be the p r o j e c t i o n Let kr+ projection
onto E s of the e l e m e n t s
onto Er+ 1 of the cycles of E r. Let k rS ~ (r < s), be the in E r which are cycles
in the d o m a i n of k rs if drX = O, d r +_kr+~x , r •
for dk, r i k < s, (i.e.,
x is
: O, "''' ds-l~s .s-l.s-2. ~s-1 "'k~+lX = 0). Thus we
may write k rs = k ss-l's-2 r Ks-l" ..k r+l for r<s. We put krr = identity. A cycle x E E r is a p e r m a n e n t The limit E~ a sequence and
of a spectral
cycle if dik~x
sequence
(x i) where x i is a p e r m a n e n t
(yi) are i d e n t i f i e d
= 0 for all i z r.
is d e f i n e d as follows.
An element
i cycle of E i and xi+ 1 = ki+lX i. Two elements
if x i = Yi for i big enough.
We make E~ into an A - m o d u l e
obvious way by d e f i n i n g (x i) + (yi) The same d e f i n i t i o n s
of E~
: (x i + yi ) ;
apply to spectral
~(x i) : (~xi) , a ~ A. sequences
of graded
(bigraded)
is (xl)
in the
29
differential
modules
and differential
algebras.
and in the case of algebras E| becomes (x i)
E| is then naturally
an A - a l g e b r a
graded
(blgraded)
under the definition
9 (yi) = (xi.Yl). In the standard
cases E r is graded
and d r is homogeneous
of degree
p+r +_r(i.e., drE ~ C E r- ). A canonical
spectral
sequence
(a)
E r is a differential
(b)
d r has bldegree
for homology
module
bigraded
(Er,dr), r by Ep,q,
(rZro),
has the properties:
and Ep,q r = 0 for p < 0 or
q < O;
A canonical
spectral
(-r,r-1).
sequence
(a)
E r is a differential
(b)
d r has bldegree
(c)
the a u t o m o r p h l s m
for cohomology
algebra blgraded
(Er,dr) , (rZro) , has the properties
by E~ 'q, and E~ 'q = 0 for p< 0 or
q < O; (r,l-r); w r of E r is given by Wr(X) where lo.
x is a homogeneous Spectral
sequence
element
= (-1) t~
d~
with respect
of a differential
objective
is only to illustrate
to the total degree.
filtered
We shall not make use of the material
9 x
module.
in this
to some extent
section
the origin
in the sequel.
and purpose
Our
of a spectral
sequence. Let M be a d i f f e r e n t i a l d. Then the associated filtered
filtered
graded module
module
(decreasing
is GrM = [ Mi/Mi+l.
filtration)
with differential
The derived module
H(M)
is
by Ji = { x r H(M)
and its associated
graded module
is GrH(M)
I x contains
= [ Ji/Ji+l.
C p = { x I x is a cycle
a cycle
in M i } ,
If we set
in M i } ,
D p = dM ~ M p, then Ji/Ji+l The spectral
sequence
we shall now describe.
connects
= cP/c p+I + D p. GrM and GrH(M)
by "successive
Let cPr = { x lx ~ Mp, dx ~ Mp+ r }, DPr ~ dCPr-r ;
approximations"
as
50
then we have inclusions
CrP D ~p+l and D p C cPr (as well as others). -r-I r EPr : cP / C p+l r-1 + Dpr-l' Er = ~ E P
then E r is a graded module.
We define
;
We define a d i f f e r e n t i a l
cp+lr-1+ Dr_l)
d r as follows.
We have
~p+r _p+r+l +
Co §247
(vr ' r-i
p+r Dr-1)
hence we have an induced m a p Ep = cpr / ~p+l + DP_l vr-1
dr §
_p+r+l + p+r ~+r CP+r / Ur-1 Dr-1 = "
We define d r = ~ drP. Clearly drd r = O. As an e x e r c i s e the reader may show that Er+ 1 = H(E r) w i t h respect to d r . Thus we have a spectral
sequence
(Er,d r) a s s o c i a t e d
with M. p+l1 + DP_l tends to C p / C p+l + D p : Ji/Ji+l. As r § +| one sees that E p = CPr / Cr_ Thus GrH(M) C E ; with suitable Similarly assumptions
assumptions
we obtain GrH(M)
= E|
p+l1 + DPr,1 tends to Mp/Mp+ 1 as r § -~ EPr = CrP / Cr_
we get E_| = GrM.
Usually
one assumes
that d M i ~
and u n d e r suitable
M i, in w h i c h case we have
CrP = Mp for r _> O; then
cPo / cp+lo § ~ Therefore
(under suitable
sequence
:Mp I p§
assumptions)
E o : OrM.
the spectral
GrM = E o , E I , E 2 , . . . , E
If M is g r a d e d by submodules
of M may be viewed as a
= GrH(M).
iM we define
C P'q r = cPr ~ P+qM, then E r is b i g r a d e d
sequence
D p'q = DPr ~ P+qM;
by submodules EP,q :
CP,q / cP+l,q-i ~r-i
+ ~P,q ~r-l"
Now assume (m)
Mi = M
(B)
MIniM
for i ! O, = 0
if i > J.
(These imply that JM = 0 for J < 0.) We assert that E~ 'q = 0 if p < 0 or q < O. Suppose first that p < O, then by (~), Mp = M = Mp+ 1. T h e r e f o r e E 'q = O. Suppose
q < O, t h e n by (8) we have Mp ~
C~ 'q = Vr-l~P+l'q-1 and hence
P + q M = O. T h e r e f o r e
: O and C P'q r
hence also E~ 'q = O. Assume further that d(JM) C J+EM. (r,E-r).
Then it follows easily that d r has b l d e g r e e
If M is also an a l g e b r a with a u t o m o r p h i s m
easily that w induces
an a u t o m o r p h i s m
w such that w(M i) C M i it follows
w r on E r s a t i s f y i n g
31
drW r + Wrd r = O, dr(x.y) Thus E r is a bigraded w(x)
= (-1)d~
differential
it follows
that
= (drX).y
algebra.
(Er,dr)
If we assume
is a canonical
For r ~ p or r ~ q+l one sees readily follows
that E| H(M)
is blgraded
with the successive
by JP'q
quotients
An alternative filtered module
= Jp~
procedure
is described
homomorphisms
sequence
for cohomology.
= E~ 'q, and it
fr
the spectral
filtered
modules
: E| §
E~. Note that
by f is the r e s t r i c t i o n
0
sequence
of a differential
of f~
and f : M § M' a homomorphism.
which commute
r
with the kr+ 1. It follows
if we regard
. We have the following
If f : M § M' and the filtrations
(or if M and M' are graded
~
lll~.
E'r
induced
lo.1.
f~
..-DJk'~
= j i , k - ~ / ji+l,k-i-1
: Er
a homomorphism
and for fixed k we have the filtration
= jo,k D j l , k - l D
for defining
in
induces
sequences)
spectral
that E~ 'q = E p'q .... r+l
HP+q(M),
E~,k-i
Let M, M' be differential
Theorem
(a,B) as well as E = +l and
= GrH(M).
Hk(M)
f induces
+ Wr(X)dr(Y).
that f
DGrH(M)
then fw
fundamental
theorem.
of M and M' are limited
in such a way that they determine
and if fro is an i s o m o r p h i s m
E|
for some r o then f~
Then
canonical : H(M)
above
spectral
~ H(M')
is an
isomorphism. The proof are isomorphisms ll.
S~stems
is easy.
Clearly
and the t h e o r e m
of local
ring, module,
follows
by lemma
for all r - r o. Hence
f~ and fw
8.1.
coefficients.
Let X be a t o p o l o g i c a l group,
fr is an i s o m o r p h i s m
space and let G be some algebraic
or algebra).
A w
of local
coefficients
structure (G,~)
(e.g. a
in X consists
of
the following: (a) To each P r X corresponds (b) To each path homotopy ~(mpQ)
: Gp § GQ such that This
implies
that
an i s o m o r p h i s m
class
~pQ Joining
: Gp § G,
P to Q corresponds
an i s o m o r p h i s m
if apR = aQR~pQ then ~(~pR ) = @(aQR)~(apQ).
if ~pp is the class of the trivial
identity map and also
O(ap~l) = O(apQ)-l.
A homomorphism
f : (G,~) § (G',~')
of a family
Wp
of homomorphisms
If each fp is an i s o m o r p h i s m
fp
of systems
path at P then~(app)
of local
: Gp § G~ which commute
coefficients
with the @(mpQ)
then we say f is an isomorphism.
is the
in X consists
and $'(~pQ).
If each fp is a
mono-
32
morphism then (G;r
may be identified with a sub-system of (G',r
the latter notion
being defined in the obvious way. Each system (G,r
in X clearly induces a homomorphlsm Cp : ~l(X,P) § Aut(Gp).
If X is arcwise connected then Cp completely determines
(G,r
as follows. Consider the
set of all pairs (g,~pQ) where g ~ Gp and apQ is a homotopy class of paths from P to Q. -1 Identify (g,epQ) = (h,SpQ) if h = Cp(8pQepQ).g, and let GQ be the set so obtained. We define r
in the obvious way, namely,
The system (G,r
r
= (g,~pQ).
is said to be simple if for all P,Q c X the map WQr
is
the identity map of G. When this is the case the isomorphisms Wp provide a canonical identification of the Gp with G. If X is arcwise connected then (G,r only if the induced homomorphism Cp : ~l(X,P) § Aut(Gp) is trivial.
is simple if and In particular (G,r
is simple if X is simply connected. In general the set G ~l(x) of elements on which ~l(X) acts trivially determines
a subsystem (G,r ~l(x) of (G,r
maximal simple subsystem. We shall usually write G f for G ~l(X), (G,r f for
which is actually the G for (G,r
and G f or
(G,r ~l(x)
If X is a locally finite polyhedron then we can give a simpler definition for
(0,r (a)
To each vertex P corresponds an isomorphism ~p : G F § G,
(b)
If P and Q are vertices of a simplex then there is an isomorphism
cpQ : Gp + GQ such that if P,Q,R are in a simplex then CPR = CQR'r This is essentially equivalent to the original definition.
For if P and Q are
vertices of X then to each equivalence class of routes apQ Joining P to Q we define r
as the composition of the isomorphisms corresponding to adjacent vertices of a
representative route in spQ. If P and Q are not vertices and ~PQ is a path homotopy class Joining them then we can define r
by making use of a slmplicial
approximation. We shall assume hereafter that G is abelian. Cohomology with local coefficients: cP(x,G) consists of functions f defined on (p+l)-tuples (Po,...,Pp) of vertices in X with values f(Po,...,Pp) 6f(Po,...~Pp+l)
r Gpo. We define the coboundary 6f by the formula
= CPiPof(P1,...,Pp+l ) + [ (-1)if(Po,.-.,Pi,..,Pp+l).
In the usual way one shows 66 = O, and we define HP(x,G) :
6-i(0) / 6CP-I(x,G).
33
If X is connected then H~
-- G f. For if f is a o-cocycle then
~f(PoPl ) : CPiPof(Pl)-f(Po ) : O. Thus ~l(X) acts trlvally on f(P), and conversely. By a pairing ~: GI,G 2 ~ G 3 we mean a family of pairings ~p : Gip,G2p ~ G3p which commute with the maps r
• r
and r
Given such a pairing we define a
pairing ~
~' : cP(x,GI),cq(x,G2)§
+
C p q(X,G3)
by ~'(fg) (Po'" '''Pp+q) : ~'Po (f(P~
CPpPo g(PP'''''PP+q))"
As usual the coboundary formula 6fg = ~f.g + (-l)Pf-6f is satisfied. Thus we can define a pairing
u* : H P ( x , G 1 ) ,
Hq(X,G 2) + HP+q(X,G3)
in the usual way. If G 1 = G 2 ~ G 3 are simple u* reduces to the ordinary cup product. Homqlogy with local coefflcients:
Cp(X,G) c o n s i s t s
of finite
linear
combinations
of p+l-tuples
of vertices
of X
such that (Po,...,Pp) has a coefficient in GPo. The boundary is defined by the formula ~g (Po,...,Pp) : CPiPog.(Pl,...,Pp) + [ (-llig'(Po,---,Pi,.--,Pp)-
As u s u a l
we h a v e a~ = 0 so t h a t
we c a n d e f i n e
Hp(X,G)
= ~-i(o)
/ ~Cp§
If X is connected one verifies readily that
Ho(X,G) = a / [ g - y g ] , where y is an automorphism of G defined by ~l(X) and [g-yg] is generated by elements of the form g-~g. 12.
Fibre bundles. A (locally trivlal) fibre bundle consists of three topological spaces E,B,F and
a continuous surJective map ~ : E ~ B subject to the following condition: for each b g B there exists a neighborhood V b of b and a homeomorphism Vb: ~-l(Vb) ~ Vb • F such that Wb(w-l(b)) = b • F. We shall denote the fibre bundle by (E,B,F,~) or (E,B,F): the spaces E,B,F are called respectively the total space, base space, and standard fibre and ~ is called the projection map. Sometimes we shall refer to E as the fibre bundle. For b ~ B, ~-l(b) is called the fibre over b : it is evidently homeomorphic to the standard fibre F. A representation (also homom0rphism) of (E,B,F,~) in (E',B',F',~') is a map
34
: E + E' w h i c h m a p s other words morphism
r induces
(E,B,F,~)
to
in
conditions G if the
given
(E',B',F
following
gii
(b)
gijgjkgki
(c)
The h o m e o m o r p h i s m
by
(b,f)
= id,
gijgji
can write
hold
(a) and
of the
s : B § E which
A fibre
bundle
induced ~(b')
(E,B,F,~)
bundle map
(E',B',F,~') induced
on B'. Let
of
(E,B,F,~) of
continuity has
structural
group
U i and a c o r r e s p o n d i n g
the properties:
in the
that
Uj))
(in the above
compact
with
defined
notations)
is
is equivalent
E' C
B'
s
for r is equivalent bundle
By a local
with
by
~.s = identity.
cross-section point.
a trivial
(b',e)
of pairs
§ b' and
(locally
(b',e)
trivial)
local
bundle
for
with
+ e respectively. fibre
a topological
~ is a r as the
such that
bundles
group
Then
every
(E',B',F,~'). group
a map
bundle.
in (E~B,F,~) (b',e)
We shall
we m e a n
Clearly
r : B' + B. An i n d u c e d
subset
structural
on E as follows:
topology
sets
of every
is a r e p r e s e n t a t i o n
for ~. For to
continuity
compact-open
: B + E such that
to B x F is called
x E as the
bundle
F and s a t i s f y i n g
the r e q u i r e d
by the
we
set of F.
and let
there
~' and ~ be d e f i n e d
E Aut
fibre b u n d l e
I g(C) C V, g-l(c) C V}
in a n e i g h b o r h o o d
for w h i c h
that
trivial
and Aut F is given the
is a map
bundle
gij(b)
further
the t o p o l o g y
(E,B,F)
be a fibre
Define
on the right
case of a local
+ (b,gij(b)f)
It can be shown
i.e.,
which
E is a p r i n c i p a l operates
bundle
identity.
the b u n d l e sets
In
r is a h o m e o -
is a r e p r e s e n t a t i o n
F (with the usual
of B by open
of s as a c r o s s - s e c t i o n .
is an induced
bundle
if there
of B'.
Uj ~ U k ,
= {g E Aut F
(E',B',F,~')
= ~(e).
A fibre
r : B § B is the
true
is a c r o s s - s e c t i o n exist.
fibre
on
§ G with
set and V is an open
image
cross-sections
Let
map
Wj (~-l(u i ~
(b,f)
(b).
to g and g-l,
C is a compact
speak
(E',B',F',~').
Wi v~l of
if F is locally
A cross-section also
If in a d d i t i o n
We say that
is a c o v e r i n g
in U i ~
always
(C,V) where
7'r =
r
o v e r points
+ (b,gij(b)f).
conditions
w i t h respect
induced
Uj
fibres
and fibre)
operating
: UI ~
of B into
= id,
= id
~iw; 1 in the f o r m
conditions
to
effectively).
giJ
(a)
the
group
there
maps
such that
(same base
such that
holds:
It is of course
also
,~')
necessarily
of c o n t i n u o u s
~ B'
is isomorphic
be a t o p o l o g i c a l
but not
over the points
r : B
(E,B,F,~)
(E',B,F,~')
Let G
fibres
a map
t h e n we say
is e q u i v a l e n t
family
the
G if G
35
(a)
(e.g).g'
: e.gg',
e.1
(b)
e.g : e
for some
e
(c)
On the
this unique
e its orbit
are closed
(c) can be r e a d i l y as the f o l l o w i n g
Euclidean through
plane
the
operates
on E by right
evidently
dense
A.M.
Gleason,
fibre pact
bundle
that
the The
is locally
in g e n e r a l
with
(E,B,~)
closed
subgroup,
subgroup
the d e f i n i t i o n S0(3)
and hence
(E,B,~)
The (i) lations.
general
linear (ii)'
G the
E the
space
group
E the
orthogonal
locally
there
group
imply
this
that
then
is not true
coefficients.
(a,b)
then
group.
The
hold,
orbits
group
orbit
topology
(E,B,~)
in the
The
clearly
w i t h the n a t u r a l the
Let L be a line
a 1-parameter
compact.
of the
through
(c) must and let
e is
fail. ~:E § B
of E. By a t h e o r e m
is a (locally
case w h e r e (E,B,~)
following
G which
of
trivial)
G is l o c a l l y
com-
is a (locally
example
shows
however
trivial: many
copies
corresponding a local
it is clear
But
of S0(3). infinite
Let H C S0(3)
product
cross-section that
it is k n o w n
of p r i n c i p a l
group,
G a closed
of left
this
that
for
would
no such
be a
of H. T h e n (E,B,~).
imply
G is a
Then
from
that
cross-sections
exist,
bundles: subgroup
operating
on E by right
trans-
cosets.
on a d i f f e r e n t i a b l e
n. The base
of o r t h o n o r m a l
O(n).
Since
space
trivial.
of frames
of order
space
in g e n e r a l
it can be shown that
existed
topology
locally
space
G is not
Moreover,
of i n f i n i t e l y
E a topological
that
structural
are examples
B is t h e n the (ii)
group.
a cross-section.
following
integer
F b are p r e c i s e l y
and let G be the
is not
but
it as the quotient
and G is a Lie group
is not
of the product has
(a,b,c)
to G. If G is compact
Then L d e t e r m i n e s
of orbits
fibres
G as
of E. Suppose
~ SO(3)/H
(a,b),
with
closed.
contractible
Let E be the product 1-parameter
Note
is not
G as s t r u c t u r a l
bundle
of
and r e g a r d
slope.
space
if E is r e g u l a r
fibre
g ~ G. C o n d i t i o n s
are h o m e o m o r p h i c
of points
translations.
projection.
and B = E/G
trivial)
lattice
in E and hence
with
set of all e.g,
torus
irrational
Let B = E/G denote be the n a t u r a l
e.g = e'
shows:
the
with
is a g such that
of e and e'
of E w h i c h
2-dimensional
modulo
g : l,
for w h i c h there
to be a c o n s e q u e n c e
example
origin
is the
that
(e,e')
function
subspaces
shown
Let E be the
implies
of pairs
g is a c o n t i n u o u s
For fixed the orbits
subspace
: e,
space
frames
manifold
M of d i m e n s i o n
n, G the
B is t h e n M itself. on a R i e m a n n i a n
manifold
of d i m e n s i o n
n,
36
Let
(E,B,G,~),
(E',B',G',~')
r : E § E' is a r e p r e s e n t a t i o n is a h o m o m o r p h l s m fibers
and
(or a h o m o m o r p h i s m )
u : G § G' such that
induces
isomorphism
be two p r i n c i p a l
a map
~ : B § B' such that
if r is a h o m e o m o r p h i s m .
B • G,
(r b e i n g
a well
known
the
identity),
theorem,
r (e.g)
the
a principal
We say latter
bundle
bundles.
of
(E,B,G)
= r
into
Then
~.~
= ~'.r
(E,B,G,~)
being
We say that
(E',B',G')
r maps
if there
fibers
representation
is t r i v i a l
acted
is t r i v i a l
The
a map
into is an
if it is i s o m o r p h i c
upon by G in the n a t u r a l
if and only
way.
to By
if it has a cross-
section. Let E be a p r i n c i p a l on w h i c h valent
G operates
if there
bundle
on the right.
with
We c o n s i d e r
is a g E G such that
(E,F) G. We define
~
structural
choose
the u n l a b e l l e d
for
a fixed
some
moreover, ~-l(b)
point
•
, E
1
l
bundle.
is in 1-1 c o r r e s p o n d e n c e
associated trivial) the above
bundles
fibre
((E,F)G,B,F,~)
Let El,
trivial
of
with
projections.
example
(e,f)
Denote
and
(e',f')
the q u o t i e n t
Then
any p a i r
(e,f)
(E,F) G contains it can c o n t a i n
bundles
with
a structural
F o r the details
E 2 be p r i n c i p a l
EI~--~-"
B I,
with
of the
space
equispace
by
bundle
bundles.
is o b t a i n a b l e
the r e a d e r the
same
E1 ~ E2
(EI,E2)~ ~2
to
group
~ E2
, B2
fibre
This
(eo,fg -1)
shows
that
details
to the reader.
of E.
If the p r i n c i p a l
Moreover,
that
every
from a p r i n c i p a l [13]
is F
(eo,f) , and
It can be shown
groups.
that
to a p a i r
form
such pair.
the r e m a i n i n g
structural
we r e f e r
bundles
a pair
an a s s o c i a t e d
group
standard
is e q u i v a l e n t
no other
w i t h F. We leave is called
It is easy to verify
to show that the
so are its a s s o c i a t e d
are fibre
bundle
process.
For
e ~ in ~-l(b).
each element
E is l o c a l l y
f' = f.g.
~ be a t o p o l o g i c a l
' B
are the n a t u r a l
it is easy to see that
The bundle bundle
maps
is a fibre
g; hence
E x F and define
e' = e.g and
(E,F) G
(~,F)G,B,F,~)
G. Let
: (E,F) G § B by the d i a g r a m
E
where
group
(section
G. By m e a n s
8) or
the
(locally bundle
by
[4].
of the d i a g r a m
137
in which the u n l a b e l l e d
maps are natural
above a pair of associated
projections,
constructions
Let
(E,B,F,w)
sufficiently H~(F,A) bundle
((EIE2)G,B2EI,#2)-
are due to C. Ehresmann.)
be a fibre bundle with B a locally
fine slmpllcial
subdivision
= (H~(F,A),@ *) as follows. is trivial.
as
bundles,
((EI,E2)G,B2,EI,~I), (The above
we obtain by the same process
we construct
If P is a vertex
finite
polyhedron.
Assuming &
a system of local coefficients
of B then on StP,
(star of P), the
Thus we have a h o m e o m o r p h l s m Yp : ~-l(stP)
§ StP x F,
and hence a h o m e o m o r p h l s m ~p : Fp § P x F where Fp is the fibre over P. This defines W~ : H~(F,A) If P,Q are vertices define
fpQ = Wp.
an i s o m o r p h i s m §
H~(Fp,A).
of a simplex then UpQ = S t P ~ ; this determines
StQ is not empty.
On UpQ x
F we
an i s o m o r p h i s m
~pQ
: H~(F,A)
~pQ
: H*(Fp,A)
§ H~(F,A).
We define § H*(FQ,A)
by ~PQ = -~ ~ Q f -~ Q ( W ~ ) - l "
It is easily
Similarly
a system Ha(F,A) , and if F is simple,
homotopy 13.
we can define
seen that we have a system of local coefficients. systems
consisting
of
groups.
Spectral Theorem
sequences i~.i.
finite polyhedra. filtration
for fibre bundles
Let
(E,B,F,~)
be a fibre bundle with E,B,
Let L be a principal
of H,(E,A)
and a canonical
and F connected
ideal ring and A an L-module. spectral
sequence
locally
Then there
for homology,
is a
(Er,dr),r
Z 2,
such that E" = GrH,(E,A), E p2 , q
= Hp (B,Hq(F,A)) '
where Hq~,A) d e n o t e s t h e s y s t e m ( H . ( F b , A ) , ~ . ) . Note that E r is an L-module. Theorem
13.2.
finite polyhedra. filtration
Let
(E,B,F,~)
be a fibre bundle
Let L be a principal
of H~(E,A)
and a canonical
with E,B,
and F connected
ideal ring and A an L-algebra. spectral
sequence
locally
Then there
for cohomology,
is a
(Er,dr) , r ~ 2,
38
such that E| = GrH*(E,A), E~ 'q = HP(B,H~q(F,A)), where Hq(F,A) denotes the system (H*(Fb,A),$*). Moreover, if A is commutative then E r is an L-algebra which is anticommutative with respect to the total degree, and the isomorphism E 2 = H*(B,H*(F,A)) is multipllcative. We remark that the theorems have been proved under the following other assumptions: v
(i)
E,B,F locally compact, Cech cohomology with compact carriers
([11,[9]).
(ll)
E,B,F locally compact, E a principal bundle with connected compact V
structural group, Cech cohomology with compact carrlers[9]. (ill) E,B,F arbitrary, homology (and cohomology)
(E,B,F) a "fibre space" in the sense of Serre, singular
[12]. v
(iv) carriers
B paracompact, F compact, Cech cohomology with arbitrary closed
[5].
(v)
E,B,F locally finite polyhedra ([6],[71,[lo]).
The last is a special case of (Ill). We note further that for real cohomology of dlfferentlable bundles the spectral sequence may be constructed using exterior differential forms (provided F is compact). We llst the main properties of the spectral sequence for cohomology.
In the
applications it is these rather than the actual definitions which are most useful. (13.3)
If H~(F,A) is a simple system, as is the case if B is simply connected
or the structural group is connected, then we may canonically identify H~(Fb,A) with H~(F,A) by the isomorphism w~. Then since the cohomology groups are finitely generated we have, by the KGnneth rule, 0 § HP(B,A) | Hq(F,A) § E~ 'q + Tot (HP+l(B,A),Hq(F,A)) § O where the tensor and torsion products are taken over L. Thus in particular field K we have E~ 'q = HP(B,K) | Hq(F,K). Similarly, if A = Z and if B or F has no torsion then E~ 'q = HP(B,Z) e Hq(F,Z). L
if A is a
39
(13.4)
is surJectlve
Clearly drEW'~ = O; hence r ~,o _p~o > kr+ 1 : E § mr+l, r - 2 so that
is also surJectlve.
Recall that E~,O : jp,o C HP(E,A), E~ '~ = HP(B,H~
= HP(B,A).
Then (13.4) asserts that ~* is identical with the composed map HP(B,A) (13.5) d2-cocycles
2 = E~,o kp+l
E~,o
,
= jp,o C HP(E,A).
From E~ 'q = 0
for p < 0 it follows that ~P'q -r+l is the submodule of E~ 'q. We therefore have E~
: E~
"
q
C ~o,q,... ~
of
C E ~ 'q
-q+l-
"
Also E2o,q = HO(B,Hq(F,A)) (recall Hq(Fb,A) f consists of the elements
: Hq(FD,A)f ~ Hq(Fb,A)
fixed under the action of ~l(B).) We have
E~ 'q = Hq(E,A)/J l'q-1. Then (13.5) asserts that the map i b given by the inclusion F b C E is identical with the composed map Hq(E,A) The kernel of i~ is J1
9
§ E~ 'q C E~ 'q = Hq(Fb,A)f C Hq(Fb,A).
the image of i~b is the submodule
fixed under the action of ~l(B). The isomorphlsms canonical
identification
of permanent
Wb* : H~(F'A)
cocycles which are
§ H$(Fb,A)
give a
of Hq(Fb,A)f with the "fixed" submodule Hq(F,A) f C Hq(F,A).
Thus we may identify all i~ with i*:
(13.6) the submodule
Hq(E,A)
§ Hq(F,A) f C
Hq(F,A),
i ~ : (T~)-li~.
The map dq+ 1 : E~149 § Eq+lq+l'O is called the transgression. of Hq(F,A)
which corresponds
Let Tq(F,A) be
under the isomorphism E~ 'q = Hq(F,A) f to
E ~q i We saw above that E q+l'O is isomorphic to a factor module of Hq+I(B,A) q+l submodule which we shall denote by M q+l. The diagram Eo,q q+l
Tq(F,A) defines the map
dq+l
~
by some
-q+l,O ' ~q+l
, Hq+I(B,A)/M q+l
9 which we also call the transgression.
The elements
of Tq(F,A)
are
4o
said to be transgresslve. There are various other definitions
of the transgression,
see
[2]; the following
one is very useful: The maps i b : F b C E and ~ : E § B induce cochain maps i'
: Cq(E,A)
An element x ~ Hq(F,A) e c Cq(E,A)
§ Cq(F,A),
~' : Cq(B,A)
is said to be transgressive
§ Cq(E,A).
if there exists a cochain
such that
(a)
i'e is a cocycle in x,
(b)
6e = w'(b), where b r cq+l(B,A).
Since ~' is inJective this implies that b is a cocycle. b then we write
If y is the cohomology class of
~(x) = y. This defines the t r a n s g r e s s i o n
~. This definition can be
proved to be equivalent to the p r e c e d i n g one. Let (E,B,F,w) and (E', B',F',~')
(13.7) H~(~,A)
and H~(F ',A) respectively.
Let E
X
B
be a representation;
be fibre bundles with simple systems
'~
E'
~B'
then there is induced a h o m o m o r p h i s m of spectral sequences,
X~: (E'r) § (Er). The induced map
X~ : H*(E',A)
+ H=(E,A)
satisfies
x* (J'P'q) C JP'q,
and hence defines a map X* : GrH~(E',A) w
~ GrH~(E,A).
Let ~b be the r e s t r i c t i o n of ~ to the fibre Fb; then ~b together with ~ induces a map HP(B,,Hq(F,,A))
§ HP(B,Hq(F,A)).
Applying the isomorphism E~ 'q = HP(B,Hq(F,A))
w~ obtain maps
X*r : E'P'qr § E rp'q (r -> 2), and eventually a map above map ~
X~: E'~ 'q § E~ 'q. The maps
x~ and the
are identical.
(13.8) (E',B',F,~')
x; : E'~ 'q § E~ 'q ,
Let F be a finite polyhedron of dimension n and let (E,B,F,~) and be fibre bundles over F wi~h simple local systems H*(F,A).
If X is a
representation E B
such that X is a h o m e o m o r p h i s m then the induced h o m o m o r p h l s m
X
W
"
E'
~ B'
on each fibre and ~
is an isomorphism up to dimension k
X~: (E'r) § (Er) of the spectral sequences and the
41
induced map
~
: H~(E',A)
Proof.
~induces
§ H*(E,A)
are i s o m o r p h i s m s
an i s o m o r p h i s m
k~ : H e ( B ' , H ~ ( F , A ) ) up to d i m e n s i o n
up to total degree k-n-l.
§ H~(B,H~(F,A))
k. In view of the Isomorphisms E' 2 = H ~ ( B ' , H ~ ( F , A ) ) ,
E 2 = H~(B,HS(F,A))
we see that d 2 : E' 2 + E 2 is an i s o m o r p h i s m
up to total degree k. S i m i l a r l y
is an i s o m o r p h i s m up to total degree k-l, and so on. H o w e v e r follows
d r ~ 0 for r ~ 2 so that E'n+2
isomorphism 14.
up to total degree k-n-l,
= El and En+ 2 = E. and
(13.8)
d3:E'3 § E 3
since F has d i m e n s i o n n it
. Then
~:
E' § E is an
follows readily.
Some simple applications. We call the r e a d e r ' s
Exposg X in
Poincar6 p o l y n o m i a l s
of the spectral
sequence
and E u l e r c h a r a c t e r i s t i c s
to
[9],
[121 , Exposg
IX in
[1], and
are Ps(Er)
= dim SEt, r ! |
will be t a k e n with respect
The
to the total
explicitly, P(Er,t) The fibre
induced map i ~ Theorem
: [ Ps ts, X(E r) : P(Er,-I).
F is said to be totally
: H~(E,A) 14.1.
§ H~(F,A)
non-homolosous
(b)
H~(F,A)
is simple and E 2 = E., assume
the image of i* is Hq(F,K).
(b). R e f e r r i n g
This proves
(a). Again r e f e r r i n g ~Hq(E,K)
Since by a s s u m p t i o n moreover,
Hq(F,K)
(explicitly,
d r = O, r ~ 2).
to (13.5) we see at once that E 2 = E| is simple it follows
that
(a). to
(13.5) we see that
: E~ 'q C Hq(F,K) f C Hq(F,K).
i a is s u r J e c t i v e
consists
are equivalent:
to O,
that the image of i~ is Hq(F,K) f. Then since H~(F,K)
Now assume
(relative to A) if the
If A = K then the f o l l o w i n g two conditions
F is totally n o n - h o m o l o g o u s
First
to zero
is surJective.
(a)
Proof. implies
particularly
[5].
The Betti numbers
degree;
attention
equality
of permanent
Also observe that we always have dr(E~ '~
holds.
Therefore
cocycles ~ T h e r e f o r e = O. To complete
Ha(F,K)
dr(E~'q)
is simple and,
= 0 for r ~ 2.
(b) it remains
to show that
d r ~ 0 for r ~ 2. Since A = K is a field we have by (13.3) E~ 'q = HP(B,K) We may thus r e g a r d an element b c HP(B,K),f
e Hq(F,K).
| Hq(F,K).
x e E~ 'q as a finite
Since the above
linear
isomorphism
combination
is m u l t i p l i c a t i v e
of elements
b | f,
we may write each
42
b | f : (b | i)
9 (I @ f)
and regard b | i in E p ,o and 1 | f in E o,q 2 . But we have observed above that d2(EP'~
= 0 and d2(E~'q)
= 0; hence it follows that d 2 (b @ f) = O. Thus we have shown
that d 2 -- O. Similarly we can show by induction that d r = 0 for r > 2, and it follows that E 2 = E|
This completes the proof of (b).
Theorem 14.2.
by ~
If A = K and F is totally non-homologous
(a)
~* is InJective,
(b)
H*(F,K)
H+(B,K), (c)
to zero then
is isomorphic to a factor module of H~(E,K) by the ideal generated
where H~(B,K)
~ HI(B,K), i 1 = PK(B,t).PK(F,t).
PK(E,t)
:
Proof (a) Referring to (13.4) we see that ~mHP(B,K)
= jP,~ C HP(E,K).
But jp,o = EP,O which by assumption is equal to E p'~ = HP(B,K). (b)
Thus wXis inJective.
i~ is surJectlve and by (13.5) has kernel Jl" Consider ~HP(B,K).Hm(E,K)
This implies that w*H+~(B,K) ( J l remains to show that Jl C Consider j p - l , l ~
= JP'~
so that the ideal
o g Jp C Jl-
(~'H~(B,K))
is contained in Jl" It
(~*H+~(B,K)). jp,o = ~,HP(B,K).
We have
jp-l,1/jp,o : E -l,1 : E -l,l by our assumption.
Therefore jp-l,I/jp,o
= HP-I(B,K)
| HI(F,K).
Let x ~ Jp-l; then we may write its p r o j e c t i o n x c jp-l,i/jp,o
as a finite linear com-
bination ~ = X bi | fi' where 5 i r HP-l(B,K)
Therefore there are
elements b i c HP-l(B,K), clearly in the ideal
fi c Hl(F,K)
(~-H~(B,K)),
and ~i r HI(F'K)"
such that x - X bl.f i ~ jp,o. Since the sum is
(as is also jp,o), it follows that x belongs to the
ideal. Thus we have shown that jp-l,l
C (~H~(B,K))
for p-i > l,(If p = 0 there is nothing to prove).
Similarly by induction we may show
that ji,p-i
((~*H~(B,K)),
For i = 1 we have
jl,p-i C (~H+~(S,K))
i ~ i.
43
from which it follows that Jl is contained in the ideal ( ~ H 2 ( B , K ) ) .
This completes the
proof of (b). (c) PK(E,t)
We have E. = E 2 by (14.1); then (c) follows at once from the fact that
= PK(E~,t)
= PK(E2,t).
Theorem 14.3.
If H~(F,K)
is simple then
(a)
the Betti numbers satisfy Pk(E) ! Pk(B • F),
(b)
the Euler characteristics
satisfy •
= •
9 •
Proof (a) We have E 2 = H~(B,K)
~ H~(F,K).
This implies that Pk(E2) Since E 3 is a sub-quotient
=i+~=k dim E~ 'j :i+~=k PI(B)'pj(F)
: pk(B • F).
of E 2 we have
Pk(E3 ) ~ Pk(E2)
= Pk(B ~ F),
Pk(Er) ! Pk(E2)
= Pk(B • F),
and more generally
Therefore Pk(E.) ~ Pk(B • F). Since Pk(E) (b)
for r Z 2.
= Pk(E|
(a) is proved.
From the above isomorphism it follows that x(E 2) = •
9 x(F).
Since d r
increases the total degree by 1 it follows from a general theorem that with respect to the total degree • Since x(E) = x(E|
(2) is proved.
Theorem 14.4. H*(F,K)
2) : x(E 3) :...: x(E|
If the Betti numbers satisfy Pk(E) ~ Pk(B • F) for all k then
is simple, i~ is surJectlve, Proof.
and Pk(E)
= Pk(B • F).
Clearly H (F,K) = K is a simple system. Now assume that Hk(F,K)
simple system for i s k; we shall prove that Hk+I(F,K) have observed in the proof of (14.3) Pk+l (E) = Pk+l (E~) ~ Pk+l (E2)" By the inductive assumption we have E~ 'j = HI(B,K) ~ HJ(F,K), hence
J ~ k;
is a
is also a simple system. As we
44
Pk+l(E2)
:
[ dim E~ 'j = =[k+iPi(B)pj(F) i+J=k+l i+J
=
[ Pi(B)pj(F)i+J=k+l
_o,k+l + dim ~2
dim Hk+l(F,K)+dim -o,k+l m2
= Pk+l(B
x F)- dim Hk+l(F,K)
+ dim E~ 'k+l.
Pk+l(B
x F)- dim Hk+l(F,K)
+ dim E~ 'k+l,
Therefore Pk+l (E) < -
Pk+l(E)-dim Hk+l(F,K)
_o,k+l + dim ~2
This implies that dim Hk+l(F,K)
~ dim E~ 'k+1 = dim Hk+l(F,K) f,
and hence equality must hold. But Hk+l(F,K) f ~
Hk+l(F,K);
therefore Hk+l(F,K)f=Hk+l(F,K)
so that H~k+l(F,K) is a simple system. Thus we have proved that H~(F,K)
is a simple
system. Then E 2 = Ha(B,K) ~ H~(F,K) and Pk(E2)
= Pk(B ~ F) for all k, whence
Pk(E) = Pk(E| so t h a t
equality
holds throughout.
! Pk(E2) = Pk(B x F) ! P k ( E ) , This p r o v e s t h a t Pk(E) = Pk(B ~ F) and Pk(E=) = Pk(E2)
for all k. Hence E| = E 2. Applying theorem Theorem 14.5.
Let Hi(F,K)
(14.2) it follows that i~ is surJectlve.
= 0 for i > s and let ~l(B) act trivially on HS(F,K).
If there exists a non-zero element x c HS(F,K) which is in the image of in then W~ is InJective. Proof. Referring to (13.4) we see that ~ p,o (drE r ) ~ E r = 0 for all r a 2. Suppose that ~
is inJectlve if and only if is not inJectlve.
Then for some r Z 2
we can find an element b c H~(B,K), b ~ O, and an element y ~ O, y E E r such that drY = k~b. Assume this is the first such r; we shall produce a contradiction. Let x r HS(F,K) be a non-zero element in the image of i~, and consider y.k~x. We have
d r (y.k x) = (dry Ck x) , Y.dr(k x).
o Since x is in the image of i9 it is a permanent vanishes.
(a)
cocycle so that the second term
Therefore
d r ( Y . k ~ x ) = ( k ~ b ) ( k ~ b ) ( k r 2X )
= k~(b.x).
Since drY = k~b the element y has bidegree
(r,r-1) and y.k~x is in E p-r'r-l+s r
45
But r-l+s>s on Hs(F)
for r_2,>9 hence Y-krX2 = O. On the other hand since
Under this isomorphism is a permanent then k~(b.x)
cocycle,
~ O, which by (~) contradicts
or H~(F)
| HS(F,K).
b.x corresponds
and having maximal
The above theorems
to b @ x and is thus not O. Clearly b.x
fibre degree y.k~x
it cannot
has no torsion
the theorems
then by t h e o r e m
remain valid (14.2)
is false as is shown by the following
be a coboundary.
But
= O.
were proved under the assumption
If i~ is surJective however
acts trivially
we have E~ ,s = HP(B,K)
H*(B)
~l(B)
~
that A was a field K. If
if we take A = Z.
is inJectlve.
example
The converse
(due to G. Hirsch)
which we shall
outline briefly: Consider
the Hopf fibering
E 2 is thus d e t e r m i n e d
S 7 § S 4 with fibre S 3. We have E 2 = HS(S4)
by generators
d4Y = x. By a w e l l - k n o w n
result
there
f : S2 of degree
i. Consider
y and x of degrees
the bundle
•
In this bundle
$2
x
. It follows
E
that
# P(S 2 x $2 x S3,t).
(E,S2,S 2 • $3, ~".~')
S2
induced by f,
, $4
d 4 ~ 0 so that E 2 ~ E|
a new bundle
S2 § S4
s7
P(E,t) Now consider
with
is a map
(E,S 2 x $2,$3)
E
S2
4 and 3, respectively,
| H~(S3) ;
determined
by
S7
x
$2---~ S 4
S2
where
7" is the p r o j e c t i o n
which has a cross-sectlon (~"-~')*
is clearly
map ia cannot
on one of the factors, and is therefore
inJective
be surJective P(E,t)
homeomorphic
(and in fact there
in this bundle
The fiber
is a principal
to S 2 x $3. On the other hand
is a cross
section),
since
# P(S 2 • S2,t).P(S3,t)
bundle
= P(S 2 • S 2 • S3,t).
but the injection
46
Problem f o l l o w that
(Koszul).
if ~
If the structure
is inJective
We shall later prove a result principal 15.
then i ~ is surJective?
structural
of the spectral
In general the answer is unknown.
of Kudo that the i m p l i c a t i o n
a commutative
sequence
of a p r i n c i p a l
a principal
in the case of
bundle w i t h
bundle E with structural
in w h i c h
r defines
by
+ b.
group X. C o r r e s p o n d i n g l y
E
x
X
of the
the o p e r a t i o n s
space on w h i c h an H-space
B
x
p
1
, B
of X on E, p is a fixed point
X operates
has fixed points,
freely,
identity
operation
quotient
space B (relative to the e q u i v a l e n c e
is a fibre map
(say in Serre's
(b) for a p r i n c i p a l
bundle.
sense). Apart
situation.
from p r i n c i p a l
Let E be a topo-
, of E onto its
satisfying
bundles,
of X)
conditions
the s t a n d a r d
of a given t o p o l o g i c a l
(a)
case is where
space and X is
of paths d e f i n i n g the operations.
type as X. This
follows we assume always
is d ~ f i n e d
by w h i c h we m e a n that only the
Thus we get a bundle
We shall call this a q u a s i - p r i n c i p a l
~
r e l a t i o n d e f i n e d by the o p e r a t i o n s
the space of loops at the point with c o m p o s i t i o n are of the same h o m o t o p y
of B, and
and assume that the p r o j e c t i o n
E is the space of paths ending at a fixed point
In what
we have
r ,E
At the same time we w i s h to c o n s i d e r a more general
the fibres
the h o m o l o g y
diagram
Jl
and
holds
group.
Consider
logical
Lie group does it
bundles.
Pairing
(b,p)
group is a c o n n e c t e d
case is studied
in
Then
[3].
bundle.
a coefficient
field K w h i c h we shall omit
writing. L e m m a 15.1.
Let
(Ei,Bi,Fi,~i), (I:I,2), be two fiberings
with H*(F i) simple,
and
consider the f l b e r i n g (E,B,F,~) Denote the r e s p e c t i v e
(cohomology)
there is an i s o m o r p h i s m Proof.
= (E 1 x E2,B 1 x B2,F 1 x F2,~ 1 x ~2). spectral
sequences
by
(iEr),
E r = iEr e 2Er w h i c h is compatible
C o n s i d e r the c o m m u t a t i v e
diagram
(i=l,2),
and
(Er). Then
with all structures.
47
~i E1
Sl , E1 x E2
, E1
~l BI
B2 , BI x B2
. BI
where ~i and S I are given by e I § (el,e~) and (el,e 2) § el, respectively,
and al, and i 1
are the induced maps. As in (13.7) el,B1 induce maps on the spectral sequences, 8~ : (iEr) § (Er), a~ : (E r) § (iEr) such that ~18~ is the identity map. Then it follows that iEr is isomorphic under 8~ t o a subalgebra of E r. Similarly ~=
2Er is isomorphic to a subalgebra of E r. We define
B~ | 8~ : 2Er | i Er § E r
by 8*(a | b) = 8~(a)'8~(b).
This defines
8*as a multiplicative
compatible with d r and the total differential
homomorphism which is
on 2Er | 1Er . For r = 2 we have
E r = H~(B 1 • B 2) | H~(F1 • F2) , and from section 13 and the details of the KGnneth rule we see that E 2 = H~(B1) @ H*(B 2) | H~(F l) | H~(F 2) = 1E2 | 2E2 9 It follows trivially that E r = 1Er | 2Er (2~r~|
and the lemma is proved.
A similar lemma can be proved for the homology spectral sequences. We apply the lemma (15.1) to
the two fiberings which appear in the initial
diagram of this section. The second fiberlng X + p is of course trivial; we have [0
if p
>
0
2E~ 'q =[Hq(X),q d r ~ O. Combining the isomorphism of the lemma with the induced map ~
gives a map,
which we also denote by ~ ~
: E r + E r | H~(X);
~
: E~ 'q §
more precisely, [ E~ 'q-i | Hi(x). i>O In a similar fashion using the anal~gous lemma for homology and the induced map ~
we obtain a map #~ : E r | H . ( X )
+ Er
which has the following properties: (a) denote
~(a
~, is a h o m o m o r p h i s m Erp,q | Hi(X) + Erp,q+l" If a a E rp,q and u a Hi(X) we | u) by a v u.
48
(b)
If we identify Ep,q2 = Hp(B) 9 Hq(X) then (b 9 u) v v = b 9 (u v v); b r Hp(B), u e Hq(X), v e HI(X),
where the second v denotes the PontrJagln product in H,(X). (c)
(a v u) v v = a v (u v v); a c E p,q' r
u,v c H,(X),
where the second v denotes the PontrJagln product in H~(X). (d)
r $, commutes with d r and kr+ 1.
(e)
If Ji is the i th level in the filtration of H~(E) then JIvHs(X) C Ji' and
moreover the induced operation of H~(X) on GrH~(E) is given by the induced homomorphism E | e H,(X) ~ E | The map #~ is the analogue of the PontrJagin product which we defined in section 7. Continuing the analogy we now define a "cap" product E r 9 Hs(X) §
E r as follows. Let
a r Er; then $~ (a) is in E r 9 H~(X), and we can write $~(a) = [ a i @ x i ' a i r E p'q-i r
'
x i r Hi(X).
Set a v u = [ ai
; then the following properties hold: (a)
a v u pairs E~ 'q, HI(X) to E~ 'q-i.
(b)
If we identify E~ 'q = HP(B) e Hq(x) then (b 9 x) v u = b 9 (x v u), b r HP(B), x r Hq(X), u r Hi(X),
where the second v denotes the Pontrjagin "cap" product in X. (c)
(a v u) v v = a (u v v), a a E~ 'q,
u,v r H~(X),
where the second v denotes the Pontrjagln product in X. (d)
r v commutes with d r and kr+ I.
(e)
If Ji is the i th level in the filtration of H~(E) then Ji v H (X) C Ji'
and the induced operation on GrH~(E) coincides with the induced map E= v H~(X) § E| (f)
If u is a homogeneous element of H~(X) and is orthogonal to the
decomposable elements in H*(X) and al,a 2 are homogeneous elements of E r then (ala 2) v u = al.(a 2 v u) + (-I ) d~ The verifications of properties
(a I v u).a 2.
(a,b,c,d) for the homology pairing and
(a,b,c,d,f) for the cohomology pairing are similar to those of section 7 making use also of lemma (15.1) and its homology analog. The proof of (e) requires explicit knowledge of the filtration, see [3]. Theorem 15.2.
(Kudo [7]). Let (E,B,X,~) be a quasi-princlpal bundle. If ~
inJective then i ~ is surJectlve (and E 2 = E| Proof.
Since the fibres are connected H~(X) is a simple system. Thus
is
49
E~ 'q = HP(B) Assume that i* is not surJectlve;
e Hq(x).
t h e n for some first index s,d s ~ O. We have
E~ 'q = E~ 'q = HP(B) Since d s ~ 0 there
e Hq(x).
is an i n t e g e r q and an element
x ~ Hq(x)
such that ds(l | x) ~ 0. We
can write
ds(l e x) = ~ b i | xi, where the b i are n o n - z e r o elements
of Hq-S+l(x).
elements
of HS(B)
Let u I ~ H~(X)
and the x i are l i n e a r l y
independent
be dual to Xl; t h e n
ds(X V u I) = (dsX)
V u I = (~ b i 9 x i) v u I = ~ b i 9 (x i v Ul),
= [ b i @ 1 <xi,ul 9 = b I e 1. Thus b I | 1 is a c o b o u n d a r y
s
so that ks+lb I = O. It follows that the p r o j e c t i o n
E| is zero and E 2s,o § Es,o | is not inJective. which contradicts Theorem
the h y p o t h e s i s .
i~.3.
Let
Proof.
be a q u a s i - p r i n c i p a l
element
We have E o,q 2 = Hq(X).
x r Hq(x)
An element
for r ~ q. On the other hand x is n o n - p r i m i t i v e such that x v u # 0 where two c o n d i t i o n s
in d i m e n s i o n
is t r a n s g r e s s i v e
m e a n s there
v u) = (drk~X)
= 0 for
is also primitive.
d~
= 0
< d~
We shall show these
of the theorem.
V u = 0
if drk~X
is an u c H,(X),
"cap" product.
u n d e r the h y p o t h e s i s
drk~(x
bundle w i t h Hq(E)
x c Hq(X)
v denotes the P o n t r J a g i n
are i n c o m p a t i b l e
is not inJective
Thus i~ must be surJective.
(E,B,X,~)
q 9 O. T h e n every t r a n s g r e s s i v e
But t h e n ~
of b I in
We have
for r ~ q.
Therefore drk~(x~ v u) = 0 but then by a general p r o p e r t y
of the spectral
2 drkr(X so that x v u is a p e r m a n e n t follows that Hm(E) transgressive Remark.
v u) = 0
cocycle.
for r ~ d~
v u) + l;
sequence for r 9 d~
If we set d~
v u) + 1
v u) = m we have E ~
~ O. Since m 9 0 this c o n t r a d i c t s
the h y p o t h e s i s .
Thus if u is
it must be primitive. One can d e v e l o p
similar
considerations
in other cases w h e r e a group
operates,
for example
in the f o l l o w i n g
case first
(E,B,F,~)
be a b u n d l e
(or fibre space)
and X be an H - s p a c e
operations
~ O, and it
c o n s i d e r e d by J. Leray
c o m m u t i n g with 7. T h e n we have a c o m m u t a t i v e
K8]. Let
o p e r a t i n g on E and B, the
diagram
s
50
E • X'---* E
B • The spectral
sequence
2E~ '~ = HP(x);
Id
of X
, B
X iS of course given by 2E~ 'q = 0 (q > 0),
O, and using lemma 15.1 we get homomorphlsms
dr
@~ : E~ ' q ~
[ E~ - l ' q
|
HI(X),
and in homology, %
r e Hs(X) : Ep,q
These again give rise to PontrJagin products base-degree
but otherwise have properties
§ E p+s,q r .
and "cap" products,
which now alter the
similar to those above. For instance,
in
cohomology we have (a')
a v u pairs E p'q r
(b')
(b |
x)
HI(X) to E~ -i'q, 9
V u = (b v u )
|
x,
and (c) to (f) as before. A further generalization
would be the case where there is a homomorphism
f of X
onto another H space Y operating on Y, such that E • X---+ E
B • Y--~B is commutative.
We conclude the chapter with the following digression.
Theorem 1~.4.
If ~n denotes the space of loops on the sphere Sn, n~2, then H@(~n,Z)
= Z[u], d~
= n-l,
the product being the PontrJagin product. Proof. acycllc
Let E be the space of paths on S n starting at a fixed point; then E is
and the map w : E § S n defined by projecting each path to its end point is a
fibre map in the sense of Serre with ~n as fibre. spectral
In the corresponding
(homology)
sequence we have E 2 = H~(Sn,Z)
| H~(nn,Z),
E~, O = Z, E~,q = 0 One shows readily that d r vanishes
for p+q>O.
except for
n d n : E~,k(n_l ) § Eo,(k+l)(n_l) ,k = o,i,2,..., which are isomorphisms.
It follows that
51
= [ Z Hi(an,Z) For the details see e k r Hk(n_l)(~n,Z)
[0
for i = k(n-1), otherwise.
[12 I. Now let x be a generator for Hn(Sn,Z). We can choose such that dn(X m l) = el,
dn(X @ ek_ l) = e k.
We shall prove by induction on k that e k = e~ (= e I v---V e I , k times). Assuming the result up to k then ek+ 1 = dn(X | e~) = dn((X | e
) v e I ) = (dn(X | e k+l = dn(X | ek_ I) v e I = e k v e I = e I ,
el,
and the induction is completed. Applying the converse of (5.3) H*(nn,Z) is a twisted polynomial ring. A result in [3] generalizes
(15.4) to the case of finitely many copies of S n Joined at a common
point.
BIBLIOGRAPHY [i I
A. Borel, s~mlnaire de Topologie alg~brique de ~'E.P.F., Zurich (1951).
[21
A. Borel, Sur la cohomologle des espaces fibres princlpaux et des espaces homog~nes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115-2o7.
[31
R. Bott and H. Samelson, On the PontrJagin product in spaces of paths, Comment. Math. Helv. 27 (1953), 32o-337.
[41
H. Caftan, S~minalre de Topologie alg~brlque de I'E.N.S. II, Paris (1949-5o).
[51
H. Cartan, idem III, (195o-51).
[61
S. Eilenberg, Idem III, (195o-51).
[71
T. Kudo, Homological structure of fibre bundles, Jour. Inst. of Polyt. Osaka City Univ. 2 (1952), lol-14o.
[81
J. Leray, Espaces o~ op~re un groupe de Lie compact connexe, Comptes rendus de l'Acad~mle des Sciences, Paris, t. 228 (1949), pp. 1545-47. - -, Applications continues commutant avec les ~l~ments d'un groupe de Lie, ibid., pp. 1784-86. -
[9]
J. Leray, L'homologle d'un espace fibr6 dont la fibre est connexe, Jour.Math. Pures Appl. IXs. 29 (195o), 169-213.
[!o1
S.D. Liao, On the theory of obstructions of fibre bundles, Ann. of Math. (1954), 146-191.
[11] [12]
S. Mac Lane, Spectral sequences and homotopy, U. of Chicago (mimeographed note).
[13]
(2) 60
J.P. Serre, Homologie singuli~re des espaces flbr@s. Applications, Ann. of Math. 54 (1951), 425-5o5. N. Steenrod, The topology of fibre bundles, Princeton (1951).
CHAPTER THE H O M O L O G I C A L 1.
Algebraic
I
PROPERTIES
OF H - S P A C E S
preliminaries.
We i n t r o d u c e
the
following
notations,
terminology,
and c o n v e n t i o n s
which
will be
used t h r o u g h o u t . Z: 7
the r i n g of integers, :
the r i n g of i n t e g e r s
9
the
m
Z
field
modulo
of r a t i o n a l
m,
numbers,
O
K, K
:
a field
G:
the t o r s i o n
of c h a r a c t e r i s t i c
p.
P
Tors elements
of G w h i c h
Torsp
G:
are of finite
the p - p r i m a r y
of G w h o s e
order
is a prime
and Torsp
that
subgroup
are a p o w e r
of an a b e l i a n
group
G. This
consists
of the
order.
component.
For p a prime
this
of p; for p = O Tors o G c o n s i s t s
G = 0 we say that
G has no p - t o r s i o n ;
consists of the
of the e l e m e n t s
zero element.
it is also
convenient
If p
to say
G has no o-torsion. A:
a ring
(with unit
M:
an A - m o d u l e
i).
(1 induces
the
identity
on M).
It is said to be g r a d e d
M = ~ M i (weak direct sum) w h e r e the M i are A-modules. In all cases i we shall assume M i = 0 for i < O, and that M i is f i n i t e l y g e n e r a t e d field the
Poincarg
polynomial
(or P o i n c a r g P(M,
The degree
d~
of a non
t)
series
if M is i n f i n i t e
of b i d e g r e e
zero element
then
is the
it is a graded
(if any) must I
A-module
and
satisfies
If M is a graded if for h o m o g e n e o u s
smallest
A-algebra
elements
of odd d e g r e e
A-algebra
t h e n be in M ~
t
!
M i ' J - M i 'Jc M i+i
dimensional)
A is a is
integer
J such that
is said to be
then
if it is g r a d e d
Similarly,
as an A - m o d u l e
if it is b i g r a d e d
as an
!
'J+J
we
say it is a b i g r a d e d
it is a n t i - c o m m u t a t l v e
(also
A-algebra.
skew commutative)
x a n d y we have xy
If A = K 2 t h e n this
o v e r A. W h e n
(i,J).
If M is an A - a l g e b r a and M i M J c M i+j . A unit
we c o n s i d e r
= ~ dim Mi.t i.
x E ~ Mi If x e M i, we say that it is h o m o g e n e o u s of d e g r e e i. i:O S i m i l a r l y M is b i g r a d e d if M = ~ M i'j. An element x c M i'j bihomogeneous
which
if
is simply
we have x 2 = O.
=
(-1)d~176
commutativity.
If A : Kp w i t h
p ~ 2 then
for an element
x
If M1, M 2 are graded A-modules tensor product defining
bigraded
then their tensor product
by M 1 s M . We also have a total
the homogeneous
elements
of total degree
If M1, M 2 are graded A-algebras A-algebra
under the m u l t i p l i c a t i o n
readily
that
to the total
degree.
AP:
the Grassmann
algebra
and has
graded
xix j = -xjx i implies An A - a l g e b r a (weak)
direct
monomials generators
A(Xl,
A(Xl,
(b)
A[x]
(c)
A[x]/(x s) has a simple
has
(x, x 2,
case
it as an
if p ~ 2 then
generated
(Xl, x2,
...) if it is the
by 1 (if M has a unit l) and the
..., x m) is a simple
system of
examples:
(x, x 2 ,
...) as a simple
system of generators
system of generators; if and only if s is of the form
..., x 2k-1 ) is a simple only associative
system.
algebras.
preliminaries.
of the time we shall
only t o p o l o g i c a l
consider
spaces X which are arcwise
X to be a finite
For more general
theories.
spaces
polyhedron
we shall consider
It is to be understood
connected.
and all homology
Most
theories
either the singular
once and for all that X satisfies
or ~ech one of the
conditions:
(1)
X is a finite polyhedron,
(2)
X is a space with finitely
also has finitely
(and hence
If we regard
..., Xm). We cite the following
..., x 2k,
we shall consider
We shall consider
(3)
it by A(Xl,...,Xm).
dimensional
..., Xm) ;
Throughout
following
P. If P is finite
s2stem of generators A-modules
(a)
homology
then M 1 s M 2 is anti-commuta-
space
with i I < i2...
for M we write M --
will coincide.
s (x2Y2).
is odd for all i.
sum of the m o n o g e n i c
Topological
~ M1i e MJ2. i+J=n is a bigraded
A-modules)
then we assume the x i to be homogeneous;
M has a simple
2 k (k > O) in which
2.
algebra
that d~
XilXi2...Xik
(XlYl)
of a vector
(Xl,...,x m) as basis then we denote
anti-commutative
of
by
by
if M1, M 2 are a n t i - c o m m u t a t i v e
tive with respect
of M 1 ~ M 2 obtained
n to be the elements
(x I | x2)(y I | Y2 ) = (_l)d~176 One verifies
grading
then M 1 @ M 2 (as graded
defined
M 1 ~ M 2 is the usual
generated
X is a compact
singular
generated
cohomology
singular
generated
groups
(and henc~
groups),
space and has finitely
also has finitely
homology
generated
Cech homology
Cech cohomology
groups).
groups
As usual we denote the i th dimensional with coefficients
The latter
in the ring A by H i (X,A)
is the cohomology
associative
homology
and cohomology
and Hi(X,A)
respectively,
H,(X,A)
: ~ Hi(X,A),
H*(X,A)
= ~ Hi(X,A).
ring of X (under cup products)
ring with a unit element,
and moreover
groups
of X
and we set
and is known to be an
is anti-commutatlve
in case A is
commutative.
spaces
If A = Kp then the homology
and cohomology
groups
are finite
and the Poincarg
of X (Poincarg
series
in cases
polynomial
dimensional
(2) and
vector
(3)) is
defined by = [ dim Hi(X,Kp)t i.
Pp(X,t) We recall the KGnneth then the following 0 ~
The map
~ Hi(X,A) i+J =n
k is given by
X x y. Note that factor H'(X
sequence
is torsion
x Y,A).
rule.
If A is a module
e HJ(Y,A)
~ Hn(X
where
x Y,A)§
~ Tor(Hi(X,A),HJ(Y,A)) i+J =n+l
#l and ~2 are the natural
~l and #~ are m o n o m o r p h i s m s
In that
then Tor vanishes
(f)
in cases
or H*(Y,Z) homology
(f)
of
so that
k identifies
H~(X,A)
e H*(Y,A)
with
(d~
> 0),
for y § (Xo,Y). further
comment
(1) and (3), and also in case
is torsion
if we replace
the universal
projections
case the map induced by x § (x,y o) is given by
We shall use without holds
§ O.
If A = K, or if A = Z and either
x ~ 1 + ~ x i e Yi ~ x, and similarly
ideal domain L
is exact:
k = #l e ~
free,
over a principal
free.
In case
the fact that the KGnneth rule always (2) when A = K, or when A = Z and H~(X,Z)
(2) we also have the above
n+l by n-1 in the last sum.
coefficient
t h e o r e m which asserts
exact
In these various
the following
sequence
cases we also have
sequences
0 § Hi(X,L)
~ A § Hi(X,A)
§ Tor(Hi+I(x,L),A)
~ O,
0 § Hi(X,L)
~ A + Hi(X,A)
§ Tor(Hi_I(X,L),A)
§ O.
for
are exact:
We shall use the following terminology. A map f:A § B is inJective if a ~ a' implies f(a) ~ f(a'); it is surJectlve if f(A) = B; it is bi~ective if it is inJectlve and surJective. An injectlve (resp. surjective, biJective) h o m o m o r p h l s m is a m o n o m o r p h i s m (resp. epimorphism, isomorphism).
3.
The structure
of Hopf
A Hopf algebra negative
degrees)
such that
algebras.
consists
w i t h a unit
of an a n t i - c o m m u t a t i v e element
if x is a h o m o g e n e o u s
element.with
h(x) where d~
u i a n d v i are h o m o g e n e o u s > O, d~
> O, the
We shall An e q u i v a l e n t
denote
definition
where once
p and
from that
satisfies
the
observation first
A monogenic d~
integer
exists
description
algebra
of x is the
t h e n we define
of m o n o g e n l c
Theorem
3.1.
h:H § H | H
> 0 then
such that
instead
d~
= d~
: ~(x)
(H,h)
that
+ d ~ i and
of H: the e q u i v a l e n c e if h satisfies
the
would
be m o r e
precise.
h satisfy
~ i + 1 ~ a(x)
Hopf
is g e n e r a t e d
+ X ui ~ vi of the d e f i n i t i o n s
second
condition
integer
by 1 and a h o m o g e n e o u s
s satisfying
the height
s = |
The
then
follows
at
(p-l|
Let H be a m o n o g e n i c
Hopf
is odd t h e n H = A(x).
(b)
If p ~ 2 and d~
is e v e n then
(c)
If p = 2 then This
(b)
Since
x with
~ 0, x s = 0. If no such
following
algebra
If p ~ 2 and d~
(a)
x s-I
element
theorem
gives
a complete
algebras.
(a)
Proof.
by non-
condition. Hopf
> 0. The height
that
H ~ and a h o m o m o r p h i s m
by H a l t h o u g h
is to r e q u i r e
a are a u t o m o r p h i s m s
d~
H (graded
finite.
algebra
h(x)
spans
algebra
= x s 1 + 1 e x + X ui | vi
elements
summation a Hopf
1 which
graded
over a field
of c h a r a c t e r i s t i c
p.
s = pk or |
s = 2 k or ~. is i m m e d i a t e d~
since
x 2 = O.
is e v e n x is in the
center
of H. T h e n
h(x s) : (x @ 1 + 1 | x) s : ~ (~)x i | x s-i F r o m the d e f i n i t i o n we can write Thus
s = p k m where
in the above
But t h e n
of s we k n o w x i and x s-i are not 0. If the a s s e r t i o n
x s cannot (c)
(m,p)
= 1 and m > I. It follows
sum for h(x s) there be 0 w h i c h Since
is not true
is a n o n - z e r o
contradicts
e a s i l y that (~k) ~ m (mod p). k k t e r m mx p | x s-p so that h(x s) ~ O.
the d e f i n i n g
p = 2, H is c o m m u t a t i v e
property
and the
same
of s. argument
as in
(b)
applies. A s~stem (Xl, x2,
of g e n e r a t o r s
...) w i t h the
of type
following
(M) is a sequence
properties:
of h o m o g e n e o u s
elements
(1)
(Xl, x2,
(2)
d~
(3)
if P = P (Xl,
element
...) is a m i n i m a l
~ d~
for i ~ J,
of degree d~
existence
..., Xk_l)
is any p o l y n o m i a l
then the height
s k ~ height One proves r e a d i l y
system of generators,
of (x k + P).
that every Hopf a l g e b r a has a system of type
of the h o m o m o r p h l s m
(M) (in fact the
h is not required).
then it is called perfect.
that
Note that
We shall refer to graded algebras as algebras
a homogeneous
of x k satisfies
If Kp is a field with the p r o p e r t y elements
which represents
it contains if p = 0
a p-th root of each of its
Kp is perfect.
such that H i is finitely
generated
for all i
of finite type.
The m a i n result
of the section
is the f o l l o w i n g
structure
t h e o r e m for Hopf
algebras: Theorem is i s o m o r p h i c
3.2.
If H is a Hopf a l g e b r a
(as an algebra)
of finite type over a perfect
to the t e n s o r product
In v i e w of (3.1) this is an immediate T h e o r e m 3.3. Let H be a Hopf a l g e b r a let(x i) be a system of g e n e r a t o r s (i) The m o n o m i a l s except
of type
consequence
Hopf algebras.
of the f o l l o w i n g theorem.
of finite type over a perfect
(2) If s i is the height
field Kp and
(M). We conclude:
x rl I x r2 2 ...x~ m, where 0 -< r i < s i (i = 1,2,..,m)
for a finite n u m b e r of indices,
form a v e c t o r basis
with r i = 0
for H.
of x i and
(a)
p = 2 then s i = 2ki or s i = |
(b)
p ~ 2, d~
(c)
p = O, d~ i even,
(d)
p ~ 0,2, d~ i even,
From
of m o n o g e n i c
field then it
odd, then s i = 2, then s i = | then s i = pkl or s i = ~.
(2) it is clear that each x i g e n e r a t e s
a monogenic
Hopf a l g e b r a H i under
hi(x)
= x | 1 + 1 | x. T h e n | H i is a Hopf a l g e b r a under | hi, and a p p l y i n g (1) we i i o b t a i n the p r e c e d i n g theorem. In general h and | h i are unrelated; this is the m e a n i n g i of the p a r e n t h e t i c a l r e m a r k in t h e o r e m (3.2). Note that
generated
by e l e m e n t s
even degree. generated
if p = O, H is isomorphic of odd degree
and a r i n g
If in a d d i t i o n H is finite
by e l e m e n t s
of odd degree.
to the t e n s o r product of p o l y n o m i a l s
dimensional
This
of an e x t e r i o r
generated
then H is an e x t e r i o r
is the o r i g i n a l
Hopf t h e o r e m
algebra
by elements algebra
[4].
of
Although theorems
are
we are
probably
We b e g i n in
the
true
proof
interested
more
in H o p f
algebras
of f i n i t e
type
the
generally.
of t h e o r e m
3.3 w i t h
some
preliminaries
(the
proof
appears
[i]). If d~
mean
that If
the
primarily
ideal
hence,
is odd
x ~ = 0 when
p = 2 or if p ~ 2 t h a t
d~
(x i) is a g i v e n
of g e n e r a t o r s
(Xl,...,Xk)
h being
h(x
system
is even.
s H in H s H. T h e n
) ~
r ~ Xk s Q +
Q = Q (Xl,...,Xk_l)
of type
case (M)
x is the
for H we
~ xk s 1 + 1 s xk
(mod
Ik_l),
h(x i)
~ 1
(mod
Ik_ l)
s xi
r
> 1 will
center
let
of H.
I k stand
for
for
i ~ k-1
:
if r > 1
~ i + 0~
xrk-2 rl rk-1 k-2 "''Xl ) ~ 1 e Xk_ 1
r h(XkQ)
In e i t h e r
x r # 0 with
we can w r i t e
~ 1 + 1 ~ Xk )r ~
-
p ~ 2; h e n c e
h(x k)
a homomorphism,
h(x~k{1
where
and r > 1 t h e n
~
xk
rk-2 rl Xk_ 2 ...x I
r i r-i ~ (i)(xk ~ x k )(1 s Q) O~i
(mod
Ik_l) ,
(mod
Ik_l) ,
(mod
Ik_l) ,
is a p o l y n o m i a l .
A monomial rk-1 r1 a = x~ k Xk_ 1 ...x I ; 0 < r k < Sk, is c a l l e d then
normal.
a ~ b is said
generated that
normal
> 1).
degree Note
monomials.
monomials
of a n o r m a l For
degree
of a we m e a n
to be a n o r m a l
by n o r m a l
the
degree
n(n
By the
monomial
Thus
are
linearly
n = 1 this
that
this
means
that
identical; a i and
P = P (Xk,...,Xl)
non-zero
coefficients
the
bj
be any and
than
are
n are
less
linear
such
that
Assume
any two
in p a r t i c u l a r ,
let
of t h e o r e m
independent.
is t r i v i a l .
are
of the
(1)
3.3
We p r o v e
it r e m a i n s
this
by
to s h o w
induction
on the
monomial.
less
degrees
~ r~d~ If a,b are n o r m a l m o n o m l a l s l~i~k ~ • in H m H. It is e a s y to s h o w that H is
to p r o v e
a l , a 2 , . . . , a j of d e g r e e s formally
0 ~ r i < s i (I ~ i < k)
linear
equal the
than
it is t r u e
n are
P = 0. We
combinations
as e l e m e n t s
normal
combination
for d e g r e e
of H
monomlals linearly
of n o r m a l shall
of n o r m a l if a n d
a i | bj independent
monomlals
produce
less
than monomials
only
such
if t h e y
that
the
in H | H. N o w of d e g r e e
a contradiction.
n with
(a)
We assert
that
P can be w r i t t e n = x rk + R ( X k ,...,x 1 )
P(Xk,...,Xl) where
the e x p o n e n t Proof.
of x k in the p o l y n o m i a l
Clearly
R is less t h a n r.
we can write r
P(Xk,...,Xl) (everything exponent
written
in l e x i c o g r a p h i c
of x k in R. Suppose
formula
for h(x~Q)
we
a priori
see that
it contains
but
then
contradicts
the
fact that
Q the a s s e r t i o n (B)
If the
also
monomial
cannot
characteristic
of b o t h
also that
characteristic
of the
in Q. Observe
monomials
Q cannot
formula
that
in h(R)
preceeding
have
that
~rom the above form x~ @ ha where by the r e s t r i c t i o n
which
can cancel
(e).
h(P)
positive
Since
this,
the
~ O. But this
degree.
for h(x r) we see that
r-i e xk , 0 ~ i
by any linear
By d i v i d i n g
it contains
at least
factors
h(P)
are
in a p o s i t i o n
contradicts
to prove
the
linearly
(y)
If the
This
follows
(6)
in h(R).
this
P = O; hence
term
r must
(1) of the m a i n t h e o r e m
Since
p = 0 and
is n o n - z e r o ,
and
be 1.
(3.3)
w h e n the
p = O. We have
n are
coefficients
of terms
less t h a n n, we k n o w that
~ O. But this
contradicts
of degree
r,
combination
P(Xk,...,Xl) this
monomial
on the
p = 0 t h e n r = 1.
f r o m the above
be c a n c e l l e d
We are now
But
t h e n r d O X k < n.
n and k ~ O, it follows
P = O. T h e r e f o r e
condition
of the f o r m
the d e g r e e s hence
a normal
specified
in H ~ H by the r e m a r k
(1)x which
degree;
+ R (Xk,...,x l)
is proved.
If r ~ 1 t h e n one t e r m
w i t h the
are no n o r m a l
r of x k and a are less t h a n
degrees
out
normal
of x k in R there
formally,
order)
Q has p o s i t i v e
~ 0 and a(~ O) is the g r e a t e s t on the e x p o n e n t
= x k Q (Xk_l,...,Xl)
z is a m o n o m i a l The p r o o f
that
p ~ 0 then
by an a r g u m e n t in the p r o o f
normal
(x i) is a m i n i m a l
system.
Thus
the normal
monomials
independent.
characteristic
recalled Every
fact
= x k + R ( X k , . . . , x l) = O.
monomial
(not n e c e s s a r i l y is by i n d u c t i o n
similar
to
of t h e o r e m
r >
1 and is a power
(B)
of p.
from the p r o p e r t y
of b i n o m i a l
3.1.
in P can be w r i t t e n
in the f o r m z r
where
~e Kp and
normal). on the d e c r e a s i n g
lexicographic
order
in P. Note that
the first
r in P is x k which
normal monomial
is of this form.
Assume
that we can write
P
in the form P = x rk + S + x~ U(xj_ I, . .,x .l ) . + .V (.x j , where by i n d u c t i v e Since the field
,x I)
a s s u m p t i o n S = [ u i z [ and t h e e x p o n e n t o f xj i n V i s l e s s t h a n t .
is perfect
and r is a power of p we may write
S = ~ (~izi)r
= ~
-r where ~ = ~i' ~ = [ ~izl :
hence we have p = (x k + ~)r + x] U(Xj_l,...,Xl)
+ V(xj,...,Xl).
We may write h(x k + ~) = (x k + ~) ~ 1 + 1 e (x k + ~) + [ cia i @ bi, a i | b i being independent
normal m o n o m i a l s
with d~
> O, d~ i > O:
therefore h((x k + ~)r) and the non-zero
terms
r r = (x k + ~)r | 1 + 1 | (x k + ~)r + ~ ~ ci a i e
of the last sum are
(up to coefficients)
independent
r bl, normal
monomlals. Assume greatest
first that d ~
normal m o n o m i a l
cannot
be cancelled
exists
an i such that x
> O; then we consider
in U. As in (a) we see that h(x~U)
in h(x~U + V). Hence,
familiar cannot
3 d~
argument
It follows
implies
that h(~x
be cancelled readily
We now prove
always
by h(~x
contains
~x] ~ b which
using the remark p r e c e d i n g
r whence = • a[, b = • bl,
Assume now that U is a constant by r since d~
the term ~x~b, where b is the
x ~ b = (clalbi)r
~. If t is a power
(s), there
with c ri = ~ 1.
of p then it is divisible
t Z r. If t is not a power of p we see by the now
) contains
a term
+ V) and must
~x ~ s xjt-s , (0 < s < t, ~ # 0), which
therefore
r be equal to one term cla ri m b i.
that x~ = (cialbl) r. This completes (1) of t h e o r e m
the proof of (6).
3.5 in case p # O. By (6) we can write
P in the
form P = x kr + [ wIz~ ' Ul ~ Kp, where
z i is a m o n o m l a l
which does not contain x k. Thus
P = (Xk + i ~i zl )r = (Xk + ~)r = (x k + ~(Xl,...,Xk_l))r. Since P = 0 we have x k + ~ (Xl,...,Xk_l) (x k + ~) 5 r < height
x k. But this contradicts
(M). Thus the induction Part
= O, and hence,
to degree
(2) Is proved
n is complete
similarly
to theorem
we have height
the fact that and
(x i) is a system of type
(I) is proved.
3.1 using the following
lemma.
The
details
are left to the reader. Lemma.
Let
(x i) be a system of g e n e r a t o r s
of type
(M). If x k is in the center
of H, s is not a p o w e r of p, and x ks-i ~ 0 then x ks ~ O. Corresponding
to each x i we can write a Poincar6 p o l y n o m i a l
F i + t d~
+ t 2d~
series
+ ... + t (si-l)d~
if s i < |
Pp(Hi,t) (1-td~
(In the
-1
if s i = |
latter case we m e a n of course the infinite
given by Pp(Hi,1) ; hence dim H = s I 9 s2... Proposition a simple
Therefore
The d i m e n s i o n
of H i is
if H has finite d i m e n s i o n we have
sm. 3.4. Every Hopf a l g e b r a of finite type over a perfect
field K 2 has
system of generators.
Let elements
s i = dim H i .
series.)
(x i) c o n s t i t u t e
2J xi ,
a system of g e n e r a t o r s
1 ! 2 j < si ' form a simple
Proposition
H = A (Xl,...,x m) w i t h d~
(b)
(Xl,... , x m) is a simple
(c)
dim H = 2m.
Hopf a l g e b r a over a perfect
system for H,
to show
(c) + (a). Let
(M) for H and let Sl,...,s m be the r e s p e c t i v e
Proposition
s i = 2 for all i. This proves
3.6.
field Kp and the Polncar6 p o l y n o m i a l = (i + t kl)
(Xl,...,x m) be a
heights.
Then by (c)
(a).
If H is a finite d i m e n s i o n a l
P(H,t)
field
odd,
(a) ~ (b) § (c). It remains
Sl...s m = 2 m so that
(3.3) the
are equivalent:
(a)
system of type
(M). Then by
system of generators.
3.5. If H is a finite d i m e n s i o n a l
Kp, p ~ 2, t h e n the f o l l o w i n g
Clearly
of type
Hopf a l g e b r a over a perfect
has the form (i + t k2)
...
(1 + t km) w i t h k i odd
then H = A (Xl,...,x m) w i t h d~ For p ~ 2 this reduces consists
of m elements
constructed and height
to 3~
of d e g r e e s
odd.
Let p = 2. Any simple
kl,k2,...,k m respectively.
in the p r o o f of 3.4 there are odd d e g r e e s 2. This proves the proposition.
system of g e n e r a t o r s But
in the simple
only if the x i have odd
of H
system degrees
lo
Clearly
our d e f i n i t i o n
of Hopf a l g e b r a
(over Kp) may be e x t e n d e d
by c o n s i d e r i n g
Z (or any ring) algebras. where
in place of K . Little is known of the structure of such Hopf P E v e n without t o r s i o n H may be complicated. As an example we cite H~(~n+l,
Cn+l is the loop space of an odd d i m e n s i o n a l
H~(~n+l,
Z) is a t w i s t e d p o l y n o m i a l
ring;
sphere Sn+ 1. It is known that
explicitly,
H i ( ~ n + l , Z) = { 0
if i ~ kn
Z with multiplication
Z)
if i = kn, g e n e r a t o r
e k.
given by ej e k = (Jjk)ej+ k.
Theorem
3.7.
If H is a Hopf a l g e b r a over Z of finite rank with no t o r s i o n t h e n H = A (Xl,...,Xm),
Proof. decomposable
d~
Let D i be the group of d e c o m p o s a b l e elements
in Hi
|
integers mi~
D i. We r e g a r d H | H | Zo,in w h i c h
in H i and D i P
(Yi,l'''''Yi's
~ 0 the elements milYil
case D i generates
(Hopf)
(all i).
elements
Zp. We c a n c h o o s e a b a s i s
for H i such that for suitable
H @ Zo, and we know f r o m the
odd
structure t
i'
the space of
Xi,l''''xi,ti
form a basis for
Di'o Since H has finite rank so has
t h e o r e m that !
H ~ Z o = A (xi,...,Xm),
t
O
d x i odd,
t
where x i = x i | 1. Thus we can write the P o i n c a r g p o l y n o m i a l o I o t P(H @ Z o , t ) = II(1 + t d x i ) , d x i o d d . For p r 0 we know H | Zp -- H / p H w i t h D i m a p p e d onto D i.p T h e n H | Zp is a Hopf a l g e b r a u n d e r the h o m o m o r p h i s m
induced by h. Clearly
dim H i ~ Z and hence
p
o
= rank H i ,
it follows that P(H | Zp,t)
Therefore
: dim H i | Z
by p r o p o s i t i o n
= P(H | Zo,t).
3.6 it follows that H | Zp = A (xpl,... ,Xpm) , d~
= d o x i. '
This implies that dim Dpi = dim D oi = rank D i, and hence we have Thus mij
= • 1 and D i is a direct
summand,
and the t h e o r e m
(mij,P)
= 1 for any p.
is proved.
If H is a Hopf a l g e b r a over Z then H/Tots H is a Hopf a l g e b r a u n d e r morphism naturally Corollary
induced by h. Hence we have the f o l l o w i n g
3.8.
corollary.
If H is a Hopf a l g e b r a over Z w h i c h is finitely H/Tors H = A (Xl,...,Xm),
the homo-
d~ i odd.
generated then
)
ill
4.
Primitive
elements~
Consider and h d e f i n e d
the
Samelson's
following
theorem.
example.
Let H =A (x~x 2) over K 2 w i t h d~ I = l, d~
= 2,
by h(x l) = x I @ 1 + 1 ~ Xl, h(x 2) = x 2 @ 1 + 1 ~ x 2 + x I ~ x 1.
T h e n H is i s o m o r p h i c
to H 1 | H 2 where H 1 = A (x l) w i t h
hl(X l) = x I ~ 1 + 1 9 Xl,
H 2 = A (x 2) w i t h h 2 ( x 2) = x 2 ~ 1 + 1 9 x 2. However Thus
there
an a l g e b r a Let
h(x)
is c l e a r l y can have
of a system
particular We
essentially
x r H be a h o m o g e n e o u s
= x | 1 + 1 | x. One
existence
no i s o m o r p h i s m
case where
sees
distinct
element
readily
of p r i m i t i v e
homomorphlsms
w i t h d~
that
h onto h I | h 2.
h.
> O. We say x is
the u n i q u e n e s s
generators
we can o b t a i n
say h is a s s o c i a t i v e
of H on H 1 9 H 2 c a r r y i n g
of t y p e
primitive
of h is equivalent
if w i t h the
(M). We shall now discuss
a
such a system.
if the
following
(H | H) | H
9
diagram
is c o m m u t a t i v e
, H e (H e H)
h | id~
~ d
| h
H | H § H
where
id d e n o t e s
the
If h is a s s o c i a t i v e shows
that
not
identity
map
we say that
every
and the h o r i z o n t a l
H is an a s s o c i a t i v e
h is a s s o c i a t i v e :
d~ 3 = 3, x I and x 2 p r i m i t i v e ,
H = A (Xl,
map Hopf
is the
canonical
algebra.
x2, x 3) w i t h
The
isomorphism.
following
d~ I = l, d~
example
= l,
and
h(x 3) = x 3 | 1 + 1 | x 3 + XlX 2 | x 1. Theorem
4.1.
Let H be a H o p f
algebra
H = A ( X l , . . . , x m) w i t h d~ i odd t h e n there d~
= d~
such that
This The p r o o f follows
is the
given
a proof Proof.
exist
primitive
associative elements
h.
If
ui,...,Um
wlth
(5.8)
(6.6).
H = A (ui,...,Um).
algebraic
formulation
by H. S a m e l s o n by J. Leray Since
over Kp w i t h
of S a m e l s o n ' s
[8] is r e s t r i c t e d
[6] w h i c h
x I is p r i m i t i v e
holds we take
theorem;
see
to c h a r a c t e r i s t i c
and
O. The p r o o f
we give
for any p. u I = x 1. Assume
there
exist
primitive
12
elements Ul,U2,..,u k with d~
: d~
and such that H = ^ (Ul,...,Uk,
We propose to find a primitive
element Uk+ 1 of the form
Xk+l,...,Xm).
Uk+ 1 = Xk+ 1 + P(Ul,...,Uk) , where P(Ul,...,Uk)
is a polynomial
of degree d~
. This will prove the theorem.
We can write
h(Xk+l)
= Xk+ 1 ~ 1 + 1 e Xk+ 1 + ~
9
9
ci I ..ij ; ij+ 1 ..i s ui I
9
~
" u i j uij+ 1
ee.
ui s
where (i) il<-..
= d~ s
(lii) The summation
is taken over all such Uil ,...,Uis and all J such that
I ~ J < s. o Note that since d Xk+ 1 is odd, condition The coefficients subscripts
Cil...ij
il<...
il,...,i s by anti-symmetry
(ii) implies that s ~ 3.
; ij+l...i s are here only defined for the i s . We extend the definition
in each batch of indices.
special
to all subscripts
We assert the following
properties : (~)
Cil'" "~-j , ij+l...is ~l-''ij
(~) (~)
-- 0 if the two batches
; ij+l---i s
have an integer in common.
Cil--.ijlj+ 1 ; ij+2--~ s,
ci .... ij ; ij +i ,..i s =-Cl i . "'ij-lij+l ; ijij+2 l if kl...k s is any permutation of il...i s then
sgn (il.-.i j ; ij+l'''is)
ci 1...ij
(s > J+l),
"'is'
(s > J+l),
; lj+ 1...i s = sgn (kl...k m ; km+l...k s) Ckl-..km;km§
Assuming
(6) proved we can complete the proof of the theorem as follows.
First note that if Vl,...,v s are primitive h(Vl'--v s) = Vl...v s
1 + 1
elements then
Vl-..v s
+ l ~ < s S g n (kl'''k m ; km+ l'''ks) Now let Cil ...ij h(Xk+l).
s.
; ij+l...isuil'''uij|
Vkl'''Vkme
Vkm+ l'''v s 9
uij+l...u s be a non zero fixed element
By (~) the indices are pairwise different
; therefore
in
we may write
sgn (il,. "" ,ij ; ij+ 1 '''" ,i s ) Uil 9 "'Uis = u ~l ...u Us, ~l,...,~s
being the permutation
that the summation
corresponding
of the ij with ~l(...<Us . Using this and ( 6 ) to
we see
13
h(sgn(il...ij;lj+l..-is)cil...lj;ij+l...isUil...Uis l~<ssgn(il'''ij;ij+l'''is)Cil
is
...ij;ij+ 1..issgn(kl'''km;km+l'''ks)ukl'''ukm | Ukm+l'''uks
= l!~<sCkl ...km,km+l...k s "
Ukl'''Ukm| Ukm+ l'''uks
where the summation is extended over the permutations of il,...,i s with kl<.-.
> O, d~
> 0), than h(Xk+l).
The construction of Uk+ 1 then follows easily by an inductive argument. It remains to prove (~,~,y,6). (~)
Suppose (ll,...,i j) and (ij+l,...,i s) have an integer in common. Using
antlsymmetry if necessary, we may assume that ij is the common integer. Hence h(Xk+ l) contains a term
Cil...ij;ij+l...isUi l" following term to the image of h | id:
..
uij
|
uij+ 1
...
ui s
which contributes the
Cil'''lj;ij+l" 9 "is(Uil'''uij _i | uij) | u i J+l ..-u i s and, moreover, is the only one which does so. By the associativity of h, Cil.. "ij;lj+l'''isUil ' 9 "uij-I | (uij | uij+l ...u i s ) is then in id | h. Thus
| ... must be in the image of Cil...ij;lj+l...i s uij uij+ 1 ul s h(P(Ul...Uk)) 9 Since uij appears on both sides of | we see that this impossible.
Hence ci I ...ij ;ij+ l" ..i s = O. (8) As above it suffices to prove the assertion for the coefficients of the term in h(Xk+l).
If Cil ...ij;lj+l...isuil...uij
| uij+l'''Uis
is in h(Xk+ l) then it
contributes the following term to id | h: ci I. 9 .ij ;ij+l. " "isUil" " "uij | By assoclatlvity
i
j+l
~ ui
J+2
) 9 9 "Uis 9
of h we can see that the following term is in h | id: ci l" . ij;ij+ . . . l-
i s (ul I ... ulj @ uij+ l) | uij+ 2... ui s
But then it must be contributed by the following term which proves (B): Cil...ljij+l;ij+2...isUil'''ulj+l
@ uij+2"''Uis"
14
(y)
This follows easily
from
(B).
(6)
This follows
from
(8) and
easily
(y).
The same p r o o f applies to the f o l l o w i n g theorem. Theorem associative
4.2.
h.
Let H be a Hopf a l g e b r a over Z, without
If H = A (Xl,...,x m) with d~
U l , . . . , u m with d~ Proposition generated
= d~ 4.3.
torsion,
and w i t h
odd then there exist p r i m i t i v e
elements
such that H = A (ul,...,Um). If H is a H o p f a l g e b r a over Z w i t h a s s o c i a t i v e
then H / Tots H = A ( u l , . . . , U m ) , the u i p r i m i t i v e
with respect
h and f i n i t e l y to the map
induced by h. 5.
The P o n t r ~ a s i n product
([2],[3],[5],[7]).
Let H ~ denote a Hopf a l g e b r a
(of finite type)
dual space to H i and define H~ = [ H i . The duality
over a field Kp. Let H i be the
b e t w e e n H ~ and H ~ is e x p r e s s e d
as
usual by <x,u>
This induces
: u(x),
x ~ H~ ,
u
i~.
~
a duality b e t w e e n H*e H ~ and Hs @ H~ by < a @ b,u @ v> :
Corresponding
.
to H ~ and H, we have the h o m o m o r p h l s m s H~
~ HA
H A @ H~
H*
where h * is the Hopf h o m o m o r p h i s m
HA
of H ~ and h, is the t r a n s p o s e
of h e . E x p l i c i t l y
h~
is d e f i n e d by <x,h~(u
| v)>
(Note that H A | H ~ is the tensor product tensor product
of graded modules).
:
, v>.
of g r a d e d algebras
The element
but H ~ ~ H m is only the
h~(u | v) is called the P o n t r ~ a g l n
product
of u and v and will be d e n o t e d by u | v. The f o l l o w i n g p r o p e r t i e s
readily
seen: (a) There
is a unit element
(b) dO( u v v) = d~
in each variable,
(d) if h ~ is a s s o c i a t i v e (d) as follows.
in H~ w h i c h spans Ho,
+ d~
(c) it is d i s t r i b u t i v e
We prove
of v are
then v is associative.
We have <X,(U V V) V W> = <(h~| id).h~(x),
u | v m w>,
15
<x,u
v (v v w)>
= <(id e h * ) . h * ( x ) ,
u ~ v ~ w>.
If (h#~ Id)-h~(x)
= ~ a i o (b i o c i)
then <x,(U which proves
V v)
V W>
= I
= <x,u
v(v v w)>
(d).
Hereafter
we assume
h ~ is a s s o c i a t i v e .
We shall prove
some r e l a t i o n s
between
H ~ and H,. Proposition
~.l.
to the d e c o m p o s a b l e Proof.
An element
elements
Suppose
d~
, d~
V v~
the
if it is o r t h o g o n a l
u ~ v>
suppose
= <x | 1 + 1 ~ x, u ~ v> = 0
x is o r t h o g o n a l
to the d e c o m p o s a b l e
elements.
t h e n we can write
h~(X) there
if and only
We have
=
> O. Conversely,
If x is not p r i m i t i v e
primitive
of H,.
x is primitive.
<x,u since
x a H~is
= X | 1 + 1 | x + [ ai | bi
a i are l i n e a r l y
independent
and s i m i l a r l y
= 1 =
;
the b i. Choose
= 0 =
u,v
a H ~ such that
for i z 2.
T h e n we have <x,u v v> which
contradicts
the
=
fact that
u | v> = 1
x is o r t h o g o n a l
to all d e c o m p o s a b l e
elements.
Therefore
x is p r i m i t i v e . We r e m a r k (or a p r i n c i p l e Let
that
the
same
ideal r i n g
e be the a u t o m o r p h i s m
h * is s ~ m m e t r i c
Proposition
5.2.
F o r the p r o o f Lemma.
= (-1) d~ if e h*
d~
(b | a),
H ~ a Hopf algebra
over
of 5.2
see
by a,b h o m o g e n e o u s .
= h e.
H ~ is a n t i - c o m m u t a t l v e
If x is a p r i m i t i v e
be p r o v e d
if we c o n s i d e r
of H ~ | H ~ d e f i n e d
if and only
if h 9 is symmetric.
[2]. element
have
This may
is v a l i d
L) w i t h no torsion.
e (a | b) We say that
argument
<xn,un>
= n!(<x,u>) n
easily
by induction.
in the
center
of H ~ t h e n
for any u a H ~ we
Z
16
Proposition 5.3.
If H ~ : Z[x] with d~
ring in "one" variable u with d~
= d~
even then H~ is the twisted polynomial
More precisely; if d~
= n then
: I 0 if i ~ kn, Hi
[
Z if i = kn, generator Uk,
and multiplication u i 9 uj = (i+J i ) ui+j" The converse is also true. Proof.
Define the uj by <xi,uj? = 6ij. By the lemma <xi,uli> = iI(<X,Ul>)i = i! ;
hence u i = uli/i!
. Therefore u i 9 uj = (uli/il)(UlJ/J!) - uli+J/ilJ!
fi+J = ~ i ) ui+j"
The converse, as well as the following proposition, may be proved similarly. Proposition 5.4.
If H ~ is a twisted polynomial ring in "one" variable x of
even degree and is generated by primitive elements then H, = Z[u I with d~ Proposition 5.5.
= d~
If H* = Zp[Xl, p ~ O, and x in the center, then H, is
commutative and H. = H 1 | H 2 | ... where H i is generated by 1 and u i with uiP = 0 and d~
= pid~ Proof.
We choose the generators u i as follows. Let u I be dual to x. By the
lemma (uln,xn~ = nl so that
Now choose u 2 so that = l; then
Continuing we obtain infinitely many
generators u i with the asserted properties, as is easily seen. Since x is primitive it is orthogonal to the decomposable elements; hence H, = H 1 | H 2 | .-- where H i is generated by 1 and u i. Since x is in the center commutativlty of H~ is clear. Proposition 5.7.
H * = A(Xl,...,x m) with x i primitive if and only if
H, = A(Ui,...,Um) where (ui,...,Um) and (xi,...,x m) are dual bases and d~
= d~
.
Proof. Assume first that H ~ = A(Ui,...,u m). Let (Xil...i k) be a dual basis to the basis
(UilV..VUik), il<.-.
elements x i are orthogonal to the decomposable elements of H~ and hence are primitive. It follows that <XilXi2,UilV
Ui2~ = I , (il<
<XilXi2,UjlV...vujk~
= 0
i2),
otherwise.
Using induction on k one sees that the elements Xil...Xik , (il<...(ik),
(1 ~ k ~ m),
form a basis for H ~ which is dual to (ui v...VUik). Thus (Xil-..Xik) is a simple I system of primitive generators for H ~ .
17
The c o n v e r s e Combining version
is p r o v e d analogously;
(5.7) w i t h Samelson's
of Samelson's Theorem
5.8.
see[5].
theorem
(4.1) we o b t a i n a second algebraic
theorem: If H * = A ( X l , . . . x m) w i t h d~
odd then H ~ = A ( U l , . . . , u m) and
H~ ~ H* (and hence H, is a n t i - c o m m u t a t i v e . ) A similar p r o o f gives the same result torsion.
C o m b i n i n g this with
Proposition
5.9.
(3.7) we get
If H* is a f i n i t e l y
H,/ Tots H, = A(Ul,...,Um) , d~ 6.
Homology
Let x o be a fixed point
p~ and ~
space and h:X x X ~ X, h(x,y)
p~ and a* of H*(X,A).
are clearly
The s t a n d a r d
examples
there
It is obvious
6.1.
an H-space
X • X + X with a homotopy
Proof. H ~ ( X • X,Kp)
product
there exists
(2) stated in t h e o r e m
h then
product
unit;
by the map k (see section
and
for any A, and
automorphlsm. H*(X,Kp)
is a Hopf a l g e b r a
a system of g e n e r a t o r s
of type
(M) w h i c h
(3.3).
Using the K G n n e t h rule we may identify H~(X,Kp)
of H*(X,Kp)
A topo-
to e.
has an A - e s s e n t i a l
If X has a K p - e s s e n t i a l
(i) and
are H-spaces.
e E X such that e 9 e = e and the maps x § x 9 e
under the induced map h*. M o r e o v e r have p r o p e r t i e s
for X if x 9 x o and
products
to the identity map of X r e l a t i v e
then that
map.
of x o.
if we take x o = e then p~ and a ~ are the identity
Theorem
element
product
of the choice
if it has a product
is an element
x § e 9 x are h o m o t o p i c
moreover
independent
= x 9 y, a c o n t i n u o u s
Since X is a s s u m e d to be arcwise
of spaces w i t h A - e s s e n t i a l
space X is an H-space
explicitly,
Hopf a l g e b r a over Z then
odd.
in X; then h is an A - e s s e n t i a l
x o 9 x induce a u t o m o r p h i s m s
logical
generated
of H-spaces.
Let X be a t o p o l o g i c a l
connected
for a Hopf a l g e b r a H ~ over Z without
| H*(X,Kp)
2). One sees readily that
and
if x is a h o m o g e n e o u s
then h~(x)
w h e r e u i and v i have p o s i t i v e
: p*
degrees
(x) ~ i + 1 ~ ~
less than d~
(x) + ~ u i ~ v i
and d~
+ d~ i = d~
Thus h ~ is a H o p f h o m o m o r p h i s m . U s i n g the u n i v e r s a l H*(X,Zp)
| Kp.
It then follows
of h ~. But Zp is a perfect generators
coefficient
of type
t h e o r e m we may i d e n t i f y
that H*(X,Zp)
field
H*(X,Kp)with
is a Hopf a l g e b r a u n d e r the r e s t r i c t i o n
so we may apply t h e o r e m 3.3 to H (X,Zp).
(M) for H~(X,Kp)
will t h e r e f o r e
A system of
have the d e s i r e d properties.
18
As immediate corollaries we have: Proposition 6.2.
If p = 2 then the Hopf algebra H~(X, K 2) (under h~) has a
simple system of generators. Proposition 6.3.
If p # 2 and X is a finite polyhedron with a
Kp-essential
product then H~(X,Kp) has a simple s~stem of generators if and only if
H*(X,Kp) = A(Xl,..,Xm) , d~ Making use of some previous results
odd ( a l l i ) .
(section 3) we can consider the case where
X has a Z-essential product. Proposition 6.4.
If X is a finite polyhedron with no p-torsion and X has a
Z-essentlal product then H*(X,Kp) Proof.
= A(Xl,...,Xm) , d~
odd (all i).
The Z-essential product h is clearly also a Zo-essentlal product. Since
X is finite dimensional it follows from the Hopf theorem that H~(X,Zo) = A(Xl,...,Xm) , d~ i odd (all i). Moreover since X has no p-torsion h will also be a Zp-essential product and Po(x,t) = Pp(x,t). Applying (3.6) we get that H~(X,Zp)
= A(Xl,...,Xm) , d~
odd (all i).
Identifying H~(X,Kp) with H~(X,Zp) | Kp the proposition then follows. In view of (3.7) and(3.8) we have the following: Proposition 6.5. Let X have a Z-essential product h. (a)
If H ~ (X,Z) is free and finitely generated then He(X,Z)
(b)
= A(Xl,...,Xm)~ ~ d~
odd (all i).
If H ~ (X,Z) is finitely generated then H*(X,Z) / Tots H*~X,Z) = ^(Xl,...,Xm),d~ i odd (all i),
where the quotient is regarded as a Hopf algebra under the homomorphlsm naturally induced by h*. Let X be an H-space with an associative product h. Then one sees easily that h * is associative as defined in section 4. Using the KGnneth rule we identify H~(X • X,Kp) with H~(X,Kp) | H~(X,Kp); hence h induces a map he: H~(X,Kp) | Ha(X,Kp) § Ha(X , Kp). If u, v z H,(X,Kp) then h~(u | v) is called the (topological) PontrJagln product of u and v. Since h~Is associative and h~ and h ~ are mutually transposes it follows that
19
h,(u | v) is p r e c i s e l y
the
(algebraic)
same is true of the t o p o l o g i c a l H,(X,Z)
and algebraic
(6 4) and
(6.5) with
(5.7,
(a) with p = 0 is the t o p o l o g i c a l T h e o r e m 6.6.
associative (a)
5.8,
products
in section
naturally
5. The
defined
in
5.9) we get the f o l l o w i n g t h e o r e m
formulation
Let X be a finite p o l y h e d r o n
product
of S a m e l s o n ' s
which
p-torslon
is an H-space with an
then
H*(X,Kp)
= A(Xl,...,Xm) , d~
H,(X,Kp)
= A(Ul,...,Um) ,
is the P o n t r J a g i n
ring.
The elements
u i are a dual basis to the x i w i t h d~
= d~
odd,
x i are p r i m i t i v e
If X has no t o r s i o n then in (a) we may replace
(c)
If X has no t o r s i o n then
Kp) by the q u o t i e n t s
H~(X,Z)
(a) is still valid
/ Tots H*(X,Z)
The x i are then p r i m i t i v e
under h z, and the
.
(b)
respectively.
theorem.
h.
If X has no
where H~(X,Kp)
H,(X,
Pontrjagin
as d e f i n e d
/ Tors H,(X,Z). Combining
where
P o n t r J a g l n product
relative
Kp by Z.
if we replace H~(X,Kp)
and H~(X,Z)
and
/ Tors H,(X,Z)
to the h o m o m o r p h i s m
naturally
induced
by h *. 7.
Spaces
on which an H - s p a c e
We assume t h r o u g h o u t
operates.
that
our H - s p a c e s
have an a s s o c i a t i v e
product. \
An H - s p a c e continuous
map
X operates
(on the right)
r : E x X ~ E such t h a t ( w r i t i n g
(1)
a(x.y)
where e is the unit We c o n s i d e r
homology
= ax)
up to a given homotopy,
(and cohomology)
with the
|
H.(X)
i n d u c e d map r r
a dual p a i r i n g
gives
: Ha(E)
We call this p a i r i n g the g e n e r a l i z e d
In a similar
with coefficients
in a
field K (which we
Then the K G n n e t h rule gives an i s o m o r p h i s m
H,(E)
follows.
is a
of X.
shall omit writing).
We define
(a,x)
space E if there
= (ax)-y,
(ii) ae = a
which together
on a t o p o l o g i c a l
§ H,(E
• X)
a map
| H~(X)
PontrJagin
§ H~(E). product
(thus a g e n e r a l i z e d
and write
"PontrJagin
fashion to above we obtain a map
r
(a ~ x) = a v x.
cap" product)
as
20
$~: H~(E) For a r H*(E),
u ~ H,(X)
we define
§ H*(E)
a v u r
H~(E)
by
= , v ~ H ~ ( E ) ,
where the second v denotes the g e n e r a l i z e d r
| H*(X).
P o n t r J a g i n product.
= [ a i e Yi E H~(E)
In detail;
if
| H=(X)
then we set a v u = [ ai. This clearly defines
a p a i r i n g Hk(E),
(a)
a v i = a, where
(b)
a v (u v v) : (a v u) v v,
(c)
if X o p e r a t e s
Hs(X) § Hk-S(E)
w i t h the properties:
1 is the unit of H,(X),
on E and E', and f : E § E' commutes w i t h the o p e r a t i o n s
of
X then f*(a'
(d)
if u e H,(X)
v u)
is o r t h o g o n a l
= (Z'a')
v u, a' ~ H * ( E ' ) ,
to all d e c o m p o s a b l e
elements
(a 9 b) v u = a-(b v u) + (-i) d~176 where
. m e a n s the o r d i n a r y T r o m the e x i s t e n c e
it follows
H~(X)
of ~ ( X )
then
9 (a v u)- b
cup product. of a (homotopy)
unit and the remarks
at the end of section
that $~(a)
whence p r o p e r t y Property
u E
(a). Property
= a e i + [ a i | Yi' 0 < d~
(b) follows
from the c o n d i t i o n
< d a ,
a(x 9 y) = (a 9 x)y.
(c) is the dual of f4,(a v u) = ( ~ a )
v u, a e H.(E),
u r H,(X)
and is obvious. To prove
(d) choose an a d d i t i v e <~,u>
basis
(u, U l , . . . , U m , ~ l , . . . , ~ n) such that
= i, <~l,U>
and such that the ~i span the d e c o m p o s a b l e
: O, <~l,U>
elements.
: 0
Relative
to this basis we can
write ~*(a)
= a 9 1 + a o 9 R + [ a i | ui + ~ ai 9 ~i'
r
= b 9 i + b o | u + [ b i | ui + Z bi | ~i"
Therefore r where each t e r m in R has a right ui" A p p l y i n g the d e f i n i t i o n s
: ab 9 I + (ab o + ( - l ) d ~ 1 7 6 factor w h i c h is d e c o m p o s a b l e
of a v u, b v u,
(ab) v u,
b) e u + R,
or involves
(d) follows.
some of the
2
21
As an example let E = X and let r be the H-space product h : X • X § X. If h~(x)
= ~ xi ~ Yi then x v u = ~ x i < Yi,U >. Suppose now H*(X)
x i prlmltive.
Then we know by a previous theorem that H,(X)
= A(Xl,...,x m) with the
=A(Ul,...,u m) with the
(ul) dual to (xi). Clearly the u i are orthogonal to the decomposable
elements of H~(X).
Since the monomials Xil...Xik , il<...
I sgn (ll,...,Iki s) Xil" ~Is'-if i : i = [ "" xi k s' 0 if i = il,...,Ik,
where ^ means as usual that the variable is to be omitted.
To see this, note that
since the elements x i are primitive we can write h~(Xil'''Xik)
= [ sgn (Pil'''~is;P4~s+l'''"k) Xpl.'.Xps 9 Xps+l--.Xpk
where the summation is taken over the permutations t e r m s l xi I -" "xi k and xi I "-" xl k Xll "'" Xik
i (for s = o,k). Then
V ui
: ~ sgn(~il "''~Is'Pis+l . . . .
If i # il,...,i k then <Xps+l-..X~k,Ui> all <Xps+l..-Xpk,Ui>
= 0 except
described earlier with the additional
: O. If i Is among il,...,ik,
<xpj,ui>
Let H*(X,Kp)
say i : ij, then
= i. Therefore the sum reduces to
Xil" 9 -Xlk V u i = sgn (~l,...,pk,Pj) Lemma 7.1.
~ik)Xp~.-x Ps <x Ps+l . - - x ~k ,u->. I
= A(Xl,...,Xm)
x l" ..x ~J 9 "'Xpk 9
with x i primitive
. If P is the
subspace addit~vely generated by the x i then any base of F is a simple system of generators. If p ~ 2 then we know by a previous theorem that H~(X,Kp) the lemma is trivial.
= A(Xl,...,Xm) , and
If p = 2 the lemma may be proved as follows.
Let yl,...,y m be any
base of P, then we can write each x i as a linear combination of Yi" This means that each generating monomial Xil. 9 "Xik (i I <'''< i k) corresponds to a polynomial in the rI rk suffices to prove that any such monomial is a monomlals Yil 9 " "Yi k" It therefore linear combination of generating monomials k = 1 this is trivial.
of the form Y J l ' ' ' Y J k
(Jl <'''< Jk )" For
We then proceed by induction on the number of factors r I +'-'+r k.
Since p = 2 it follows that the square of the primitive element Yl is also primitive rl rk and hence is in P. Therefore, if Yl appears as a factor in Y i l ' ' ' Y i k with exponent greater than 1 we can replace y~ by a linear combination of the yj
and so obtain a
22
smaller n u m b e r of factors hypothesis
in each term w h i c h arises.
and the lemma follows.
Theorem
7.2.
Let X,E be finite p o l y h e d r a with X an H - s p a c e w h i c h operates
on E. Regard X as o p e r a t i n g
on itself
commutes w i t h the operations primitive
(on the right),
of X on E and itself.
then the image of H~(E,K)
Proof.
We have p r e v i o u s l y
and let f:X + E be a map w h i c h
If H~(X,K)
= A(Xl,...,x m) w i t h x i
u n d e r f~ is g e n e r a t e d by p r i m i t i v e
proved that H~(X,K)
are dual to the x i. Since f commutes
=
we have
v ul, a E H~(E,K).
shows that the image of f ~ is stable under the
v-product
with any u i. The t h e o r e m
will then follow if we show that any s u b a l g e b r a A c H*(X) w h i c h v-product
with all u i is g e n e r a t e d by p r i m i t i v e
Let P denote the subspace basis
additively
so that the first k basis elements
guarantees
A
m
is stable u n d e r
elements.
generated
additively
by the xi; if we change the
generate
that the n e w basis is still a simple system.
Xl,...,x k generate
elements.
A ( U i , . . . , u m) where the u i
with the operations
f~(a v u i) : (f~a) This
Then we can apply the i n d u c t i v e
A ~ P then lemma 7.1
Hence we may assume that
P. Let a E A, then we can write a =
. . . . . x i + other terms, Cil isXil" s where xi I ~ "xi s is the longest m o n o m i a l a p p e a r i n g in a. By the above example we have that Xil 9 - -Xis v (UisV... vui2) Since A is stable we have that Xil ~ A ~ P. Similarly longest m o n o m i a l
of
a
is g e n e r a t e d
show that every m o n o m l a l that A is g e n e r a t e d
in a
by the p r i m i t i v e
xi2'''''Xis
by basis elements
is g e n e r a t e d basis
by basis elements
= Xil . ~ P' and hence
the
of A n P. By i n d u c t i o n we may
elements
of A m P. It follows
Xl,...,Xk,
and the t h e o r e m
is
proved. Theorem
7.3. Let X,E be finite p o l y h e d r a
Regard X as o p e r a t i n g commutes
on itself
with the o p e r a t i o n s
(on the right),
annihilates
Proof.
If p # 2 and
= A(Xl,...,Xm) , d~
= A ~ B, where A and B are s u b a l g e b r a s
A and is inJective By Samelson's
on E.
and let f:X + E be a map w h i c h
of X on E and itself. H~(X,Kp)
then H~(E,Kp)
and X an H - s p a c e w h i c h operates
odd,
of H~(E,Kp)
such that f~
on B.
t h e o r e m we may assume the elements
x i to be primitive.
23
Then H,(X,Kp)
: A(Ul,...,Um)
with (ui) dual to (xi). By the preceding theorem we have
that f*H~(E,Kp)
is generated by primitive
Thus f~H*(E,Kp)
=
(m)
elements which we may assume to be Xl,...,x s.
A(Xl,...,Xs).
We assert there exist elements Yi ~ H~(E'Kp ) with the properties: f*(yi ) : x i , 1 ~ i ~ s, Yi v uj= ~iJ' 1 ! i, J ! s.
Since f commutes with the operations
of X there exist elements Yi c H~(E,Kp)
such that f~(~i ) : x i , I ~ i ~ s, Yi v uj : ~iJ' 1 ~ i ! j ! m. Such Yi can always be constructed Yi from the Yi by induction. properties;
by property
(c) of v. We now construct
the elements
Set Yl = Yl " Assume yl,...,y k chosen with the required
then we construct
Yk+l also satisfying these properties
as follows.
Setting Yk+l,k = Yk+l - (Yk+l v u I v-.-v u k) yk--.yl, we get f~(Yk+l,k ) : f~(Yk+l ) - f~(Yk+l ) v (u I v-..~ u k) f~(yk)---f~(yl ), = Xk+ 1 - Xk+ 1 v (u I v...v u k) Xk.-.x I = Xk+ I. We also have Yk+l,k v uj : Yk+l v uj - Yk+l v (u I v...v u k) yk...yl = 6k+l, j
v uj,
for k + 1 ! J ~ s.
Then Yk+l,k V(Ul V--.V u k) : Yk+l V(Ul V---V u k) - [Yk+l V(Ul V...V uk)Yk-..Yl]V(UlV-..V
= Yk+l V(Ul v--.v u k) - Yk+l This implies that Yk+l,k is annihilated
V(Ul v---V u k) = O.
by Ak(ul,...,Uk ).
Now assume that we have constructed
an element Yk+l,t with t ! s with the
following properties: f~(Yk+l,t ) = Xk+ I, Yk+l,t
v uj: 6k+l, j
Jk+l,t We then construct
an element Yk+l,t-1 Yk+l,t-i
: Yk+l,t-
for k+l ~ J ~ s,
is annihilated
by
At(ul,...,Uk ).
satisfying the same properties
by setting
. . t)Y~t . . ~ Yk+l,t v (UUl v . .vu ~l<'''<wt~k
YuI"
u k)
24
Thus by induction we see that there is an element Yk+l,1 having the properties: f*(Yk+l,1 ) = xk+ l, Yk+l,1 v uj= 6k+l, j
for 1 ! J ! s,
Yk+l,1 is annihilated by ^' (Ul,...,Uk) , or equivalently by each u i. We therefore define Yk+l = Yk+l,l' and (0) is proved. Now let A to be the subalgebra annihilated by ^ (Ul,...,us), and let B denote the subalgebra generated by yl,...,y s. (8)
f* is inJective on B: The degree of Yi is odd since x i has odd degree. Since p # 2 it follows
that B = A (yl,...,ys). Since f*(yi ) = xi, f'is clearly inJective. (y)
We assert A 9 B = H*(E,Kp): Let x be an element annihilated by A'(Ul,...,Us), then x e A. Assume that
if x is annihilated by A j (Ul,...,us),J
> i, then x ~ A 9 B. Suppose y is annihilated
by AJ+l(ul,...,Us), and consider y' = y -if<. ! 9<Jigs (y v (UilV. "" vulj)) Yij .. "Yi 1 9 A straightforward computation shows that y' is annihilated by Aj (Ul,...,Us). Similarly
y v (UilV-..vuij)
is annihilated by A' (Ul,...,u s) so that
y v (Uil v...v uij) r A. Clearly then the sum is in A 9 B. By the inductive assumption y' ~ A 9 B, and hence so is y. (6)
A and B are linearly disjoint: Given an element h
:
[
il<...
...ij
9
Yil'" YIj
:
o
with ail...i j a A, we have to show that ail...i j = O. Let al...nYl..-y n be the longest monomial in h. Then h V (u n V...V u I) : al, ' .. ,nYl.-.yn V (u n v...v u I ) : a I, ... ,n" Since h is O, we have a I . ,
that all ail...ij
..,n
= O. By induction on the length of monomials we have
= O.
Combining (y) and (6) yields the desired result. We mention two applications: (i)
Let X be a compact connected Lie group G, let E = G/U be the left coset
space of G modulo a closed subgroup U, and let f:G * G/U be the canonical projection. Then theorem 7.3
generalizes a theorem of Leray[7].
25
(ll) Let X be a compact connected Lie group G. Let E be a compact connected Lie group G having a closed subgroup
isomorphic
to
G. Let f:G § ~ be the injection
map. Then theorem 7.3 generalizes a theorem of Samelson
[8].
BIBLIOGRAPHY
[1]
A. Borel, Sur la cohomologie des espaces flbr~s principaux et des espaces homog~nes de groupes de Lie compacts, Ann. of Math. 57 (1953),115-2o7.
[2]
A. Borel, Sur l'homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math. 76 (1954), 273-342.
[3]
R. Bott and H. Samelson, On the PontrJagin product in spaces of paths, Comm.Math. Helv. 27 (1953), 32o-337.
[4]
H. Hopf, 0bet die Topologie der Gruppenmannigfaltlgkelten und lhrer Verallgemelnerungen, Ann. of Math. 42 (1941), 22-52.
[5]
T. Kudo, Homologlcal structure of fibre bundles, Jour. Inst. of Polyt. Osaka City Univ. 2 (1952), lol-14o.
[6]
J. Leray, Sur la forme topologlque des espaces et sur les points fixes des repr~sentatlons, J. Math. Pures Applic. 54 (1945), 95-167.
[7]
J. Leray, Espaces o0 op~re un groupe de Lie compact connexe, Comptes rendus de l'Acad6mle des Science, 228 (1949), 1545-1547; - - - ~ p p l i c a t l o n s continues commutant avec les 61~ments d'un groupe de Lie, ibid., 1784-1786.
[8]
H. Samelson, Beltr~ge zur Topologie der Gruppen-Mannigfaltlgkelten, (1941), lo91-1137.
Ann.of Math.42
52
CHAPTER III UNIVERSAL BUNDLES AND CLASSIFYING
16.
Universal bundles and classifying Let G be a compact Lie group.
E(n,G) whose structural space B(n,G)
spaces.
An n-universal
the following classification
classes of principal correspondence
bundle for G is a principal bundle
group is G and whose first n homotopy groups vanish. The base
is said to be an n-classifying
Theorem 16.1.
SPACES
classifying
space for G, as is suggested by
theorem.
Let B be an n-dimensional bundles with structural
with the homotopy
polyhedron.
Then the equivalence
group G and base space B are in 1 - 1
classes of maps B + B (n,G), the bundle corresponding
to a given map being the induced bundle. For the proof see section 19 compact
space).
The existence
[6], (and
of n-universal
bundles)
for any n > o and any compact
Briefly,
if G = O(m) we take for E(n,O(m))
Wm+n+l, m = O(m+n+l)
/ O(n+l);
Gm+n+l, m = O(m+n+l)
/ O(n+l)
bundle E(n,G) n-unlversal
representation
bundles
(in fact, compact differentiable
Lie group G is well known
(see
[6],
[5]).
the Stiefel manifold
then B(n,O(m))
is the Grassmann manifold
• O(m). Since for any closed subgroup
is also an n-universal
bundles
[5] in the case where B is a locally
bundle E(n,U),
U C G an n-universal
it follows that there exist
for any closed subgroup of O(m). Making use of the Peter-Weyl
theorem we then have the existence
of n-universal bundles for any
compact Lie group. Proposition corresponding
16.2.
Let E1,E 2 be n-universal
n-classifying
morphism Hi(B1,A)
= Hi(B2,A)
spaces.
bundles
for G with B1,B 2 the
Then for any coefficient
ring A there is an iso-
compatible with the product up to n,(i.e.,
Consider the commutative
i ~ n).
diagram fl
f2
E1 '
E1 • E2
' E2
B I ~"
(EI,E2) G
~ B2
where fl,f2 are the natural projections
and fl,f2 the induced maps. Then rl,f 2 are
fibre maps with E2,E 1 as fibres respectively.
By the following
lemma ~l ~ and r2 ~ are
53
isomorphisms canonical
up to dimension n. This proves
16.2.
(Note that the isomorphisms
are
in the sense of transitivity.)
Lemma. ~*: Hi(B,A)
Let (E,B,F,I)
§ Hi(E,A)
Proof.
be a fibre bundle with HJ(F,A)
= 0 for 0 < J ! n. Then
is an isomorphism up to n.
We have [ Hi(B,A)
if J = 0
E~ 'j 0 Thus an element that Hi(E,A) boundary.
if 0 < J ~ n.
of E 2 of total degree i+J ! n must be zero if J ~ O, and it follows
= E i'~174 Moreover every element
Thus for i ~ n, ~
= _i,o 52 = . .9=
"pass to the limit n § |
n-universal
A more concrete way of defining H~(BG,A)
limit of
bundles E i (n I < n 2 <...) together with injections
fi : Ei § Ei+l" The existence as E i Stiefel manifolds.
and
is to introduce a universal bundle E G for G, (that is, an
bundle for all n.) The bundle E G may be defined as the inductive
a sequence of ni-universal
dimensions,
Ei,o | = H i (E,A).
16.2 allows us to define a graded algebra H~(BG, A) which for each
n is isomorphic up to n wlth H~(B(n,G),A). to
and none is a d 2-
is given by Hi(B'A)
Proposition
of E~ '~ is a d2-cycle
of such sequences
follows from the fact that we may take
Then E G is a principal bundle which is acyclic
and its base space BG, determined
up to homotopy type,
in all
is a classifying
space for G for all n. Let E be a principal bundle for G with base space B, then by the classification theorem it is induced by a map @ : B § B G. The map @~ : H$(BG,A) uniquely determined by the bundle structure Its image is called the characteristic morphism of the spectral
sequences spectral
map.
rin~ of E. Of course @ also defines a homo-
sequence of (EG,BG,G)
into
that of (E,B,G).
13.8 we see that when G is connected
isomorphism up to total degree n-dim G between the spectral
of two n-universal
bundles
sequence for G as a spectral
for G. We may therefore sequence
nothing other than the spectral in singular or Alexander-Spanier
define a universal
(Er)G, r ~ 2, which for each n is iso-
morphic up to n-dlm G with the spectral sequence
bundle
is thus
of E and is called the characteristic
Using the diagram in 16.2 and property there is a canonical
§ H~(B,A)
of (E(n,G),B(n,G),G).
sequence of a universal bundle cohomology.
(E,B,G) then defines a homomorphism
Clearly it is
(EG,BG,G) taken either
A map # : B § B G which induces a given
of (Er) G into the spectral
sequence of the
54
bundle. Proposition Suppose
16.3.
Let
E and F are p r i n c i p a l
(E,B,F,~) bundles
w i t h the a d m i s s i b l e maps b e t w e e n bundle.
T h e n we can define
be a f i b e r l n g w i t h group a compact
for G such that its o p e r a t i o n s
the fibres F b and the s t a n d a r d
a new f i b e r i n g
i : F/G + E/G be the n a t u r a l map.
(E/G,B,
We assert
i*: H*(E/G,A) contains
the c h a r a c t e r i s t i c
Proof.
Consider
F/G,~)
Lie group G.
on E and F commute
fibre F of the original
with ~ d e f i n e d naturally.
Let
that the image of § H~(F/G,A)
ring of (F,F/G,G).
the d i a g r a m i F
rE
I
F/G where
r induces
(F,F/G,G)
the p r i n c i p a l
is equivalent
16.4.
then F/G is t o t a l l y
~ BG
structure
of E. Since the p r i n c i p a l of the map r
non-homologous
to 0
in any bundle
a universal
an a s s o c i a t e d
associated
((EG,F)G,BG, F,~) where E G is a u n i v e r s a l acyclic.
Corresponding
The induced maps characteristic spectral Note that
rings
sequence
to any bundle
~* and
for
r
for the bundle.
if F = G/U then
at
bundle bundle
(E,B,F,~)
ring of (F,F/G,G),
bundle
space
F on w h i c h G acts
((E,F)G,B,F,~)
for the pair
with group
:(EG, F) G
B
'
(on the
(G,F) as a bundle
for G. Note h o w e v e r that
E
with base
space F on w h i c h G o p e r a t e s
(EG,F) G is not
G we can write a d i a g r a m
BG
are c h a r a c t e r i s t i c .maps and the c o r r e s p o n d i n g
(G,~;
follows
(E/B,B,F/G,~).
bundle E w i t h group G and a t o p o l o g i c a l
we have e a r l i e r d e f i n e d
we define
the a s s e r t i o n
is the c h a r a c t e r i s t i c
space B = E/G. Now given a group G and a t o p o l o g i c a l right)
bundle
theorem.
If A = K, and HS(F/G,K)
Given a p r i n c i p a l (on the right)
01
,E/G
to the induced bundle
once from the c l a s s i f i c a t i o n Corollary
bundle
~, E G
The spectral
sequence
it is m a p p e d h o m o m o r p h i c a l l y
rings
of (EG, F) G is the u n i v e r s a l by ~
(E,F) G may be i d e n t i f i e d with E/U.
into that of (E,B,F).
55
17.
p(U,G)
: 3 fiberings
If U is a closed
involving
subgroup
gives rise to the fibre map We can g e n e r a l i z e on G by Tu(g)
we obtain a p r i n c i p a l
three
special
fiberlng
on itself by right
((Eu,G)u,Bu,G).
translations,
U C G then
(Bu,BG,G/U,p(U,G)).
Let f : U + G be a h o m o m o r p h i s m
We can then form the bundle
and let U operate Since the o p e r a t i o n s
it is easily
seen that
bundle with group G. It is induced by a map of B U into BG,
p(f), w h i c h g e n e r a l i z e s
Beginning
: B U § B G of the n a t u r a l
p(U,G) as follows.
= f(u).g.
spaces.
of G then E G is also an E U. The i n c l u s i o n
p(U,G)
of T u commute with G o p e r a t i n g
d e n o t e d by
classifying
with a f i b e r l n g
cases listed
0(U,G).
(E,B,F,#)
we may derive
a new f i b e r i n g
(E',B',F',~')
in
in the f o l l o w i n g table:
Case I.
Principal
II.
U closed subgroup
III. U closed and invariant
~'
i'
=y(1) BU
X
X
#
G/U U
BU
G/U
p(U,G)
•
G/U
=BG(1)BG/U
BU
p(#)
p(U,G)
B
i~
X
Y
U
G G
bundle of G
F'
E
E'
#
U
B'
BG
(• is the c l a s s i f y i n g map of the given bundle.) We r e m a r k that U is a compact may
p o s s i b l e be finite.
sense of group theory. is u s e f u l
In the latter case H~(B U) is the c o h o m o l o g y We have already
considered
We consider two fiberlngs
latter the fiber E U is acyclic, the derived
fibering
((Eu,X)u,Bu,X)
connected
and
of U in the
The d e r i v e d
fibering
and ((Eu,X)u,Y,Eu).
in I. To show that the fibre map induces
in III
fibering
In the
corresponds
to
the c l a s s i f y i n g map of the
bundle we c o n s i d e r the d i a g r a m g
f EU x X
BU +
that
II above.
so (Eu,X) U = Y. Thus the first
EU~
where
is not n e c e s s a r i l y
in c o n n e c t i o n with group extensions.
Case I.
original
Lie group w h i c h
(E U x X ) U _ _ + y
f,g are the n a t u r a l p r o j e c t i o n s . (r~)-l~ x is the c h a r a c t e r i s t i c
and define
The map r * is an isomorphism.
map r
~ : X ~ E U x X by x § (~(x),x).
that
(~,)-l~,
= r
(I)
=X m e a n s a space whose h o m o l o g y those of X.
~X
of (X,Y,U).
Let ~
We have to show
: X + E U correspond
Then g~ = ~ and f~ = identity.
It is also easy to see that or h o m o t o p y
to r
It follows
i' = #. groups
are c a n o n i c a l l y
isomorphic
to
56
Case III.
Consider the fib,ring (EG/U = Bu,BG,G/U) of II. Since U is invarlant
this is a principal bundle with fibre G/U. Then E = (EG/U,EG/u)G/U has two fib,rings (E,BG,EG/U)and(E,BG/u,Bu);the fibre of the first being acycllc we see that E = B G from which it follows that the second flbering gives the derived fiberlng in III. We leave the details to 18.
the reader.
Some results on universal spectral sequences. We have previously defined the universal spectral sequence (Er) G for a compact
connected Lie group G and observed that if (E,B,G) is induced by r
: B § B G then
r
defines a homomorphlsm of (Er) G into the spectral sequence of (E,B,G). It follows that properties of the former have a "universal" character in the sense that they are valid in any principal bundle. For instance if x c H~(G,A) is transgressive in E G then it will be transgressive in any principal bundle for G. We call such elements universall2 transgressive. The study of (Er) G is an important problem which has not yet been fully investigated. We list below some results obtained so far. A part of these results will be proved in the next section. Theorem 18.1. (a)
Let H~(G,Kp)
= A(Xl,...,Xm) , d~
H*(G,Kp) = A(x~,...,X'm) d~
odd, then
= d~
where the x' are universally transgressive; i (b) H*(BG,K p) = Kp[Yl,...,ym] , d~
= I + d~
the Yi being images of the x i by transgression. A similar statement is valid for integer coefficients. As a converse to (b) we have the following: Theorem 18.2. H*(G,Kp)
H*(BG,Kp) = Kp[Yl,...,ym] , d~
If
even, then
= A(Xl,...,Xm) , d~ i = d~
where the x i are universally transgressive elements whose images are the Yi' and similarly with Z in place of Kp. Theorem 18.3.
If Ha(G,K 2) = A(Xl,...,x m) where the x i are unlversally trans-
gressive then H~(BG,K2)
= K2[Yl,...,ym]
where the Yi are images of the x i by trans-
gresslon, and conversely. For the proofs we refer the reader to [1] and [2]. As a consequence of (18.1) we have the following: Proposition 18.4. p-torslon.
If H~(G,Kp)
= A(Xl,...,x m) d~
odd, then G and B G have no
57
Proof.
From (18.1b) we see that H~(BG,K p) has only even degrees; hence it has no
p-torslon, and Pp(BG,t) = Po(BG,T).
By Hopf's theorem the assumption of 18.1 always
holds in characteristic zero; thus we see that H2(BG,K p) and H~(BG,K o) are both rings of polynomials over elements which, because of the equality of the Poincar~ polynomials, must have the same degrees. Then (18.2) gives PpiG,t) = Po(G,t) so G has no p-torslon. A comparison of Polncar6 polynomials and application of (18.4) then gives the following: Proposition 18.5. (b)
(a) If G has no torsion then B G has no torsion.
G has no p-torsion if and only if H*(G, Kp) = A(Xl,...,Xm) , d~
19.
odd.
Proof of one theorem on universal spectral sequences. We shall prove the following theorem: Theorem 19.1.
If H~(G,Kp) has a simple system of universally transgressive
elements Xl,...,x m then H~(BG,K p) = Kp[Yl,...,ym] where Yl is an image of x i by transgression. Notice that for p ~ 2 our assumption implies that H*(G,Kp)
= A(Xl,...,Xm), d~ i
odd, (proposition 3.5), so that the Yl have even degrees and the statement is in agreement with anti-commutatlvlty. The theorem includes 18.3, and for p ~ 2 is weaker than 18.1 since it gives (b) if we assume (a). Although a more "conceptual" proof can be given, we follow here the method of [1] since it is also used to prove theorem 18.1. We begin with some algebraic preliminaries. Let B be an algebra over a field K, graded by B~ (i ~ 0), antl-commutative, and with B ~ spanned by the unit element 1. Let al,...,a m be homogeneous elements of B; we shall say they are ann-free (annlhilator-free) up to k if the annihilator of the projection of a i in B/(al,...,ai_ l) is zero up to degree k-dOai for all i = 1,2,...,m, a I in B vanishes up to degree k - d~ independence.
If k ~ d~
(for i = 1 we mean the annihilator of , (1 ~ i ~ m), this implies linear
If the a i are ann-free up to k for all k we simply say they are ann-free.
One shows readily by induction that if the a i are ann-free they are algebraically independent. It will be apparent by 19.2 (and could also be seen directly) that if the a i are ann-free up to k, (k ~ d~
then so are the elements of any basis of the
vector space Q spanned by the a i. Thus any permutation of the a i are ann-free up to k. We may also speak of Q as being ann-free. In what follows we will have to consider not
58 only an algebra
A(Xl,...,x m) but also certain of its subspaces
x i and their products Xil... multiplicative
Xik,
spanned by some of the
(i I <...< ik). Such subspaces retain a part of the
structure of A(Xl,...,Xm) , (for example,
if x i and xj are in the
subspace and i < J, their product xix j is also in the subspace), but in general they are not subalgebras.
In order to give them a more precise formal description we shall
denote by At(Xl,...,Xm)
a graded vector space which
(a) is a module over the exterior
algebra ^(Xl,...,Xm) , (b) has a basis composed of an element 1 of degree 0 and its transforms
Xil . . . Xik .
(i),
(il<.-.
Xil...Xlk.
The grading in A~(Xl~...,Xm)kis
d~177''.•X•
the latter elements will also be written 8;iven by
=j:1 [ d~
There is an obvious isomorphism AT(Xl,...,x k) s
AT(Xk+l,...,Xm ) = AT(Xl,...,Xm )
which maps 1 ~ 1 to i, x i s 1 to xi, 1 s xj to xj, x i s xj to xixj, etc. Ebidently every
A(Xl,...,x m) may be regarded as a AT(Xl,...,Xm ). Of course we could identify
AT(Xl,...,x m) with A (Xl,...,Xm) , but for the reasons m e n t i o n e d above we wish to avoid it and rather regard a product
as an external law.
Let J = B | AT(Xl,...,Xm).
Then J is a vector space graded in a natural way, and
we denote by DB, DF, and D respectively, (d ~ in AT(Xl,..,Xm)),
the base degree
and the total degree
(d ~ in B), the fibre degree
(D = DB + DF). In addition J is graded by
subspaces B @(e)AT(Xl,...,x m), spanned by the generators b @ Xil-..Xie, exterior degree. We may consider J as a modale over B @ In particular,
if h = b |
Xll
(a @ l)h
=
9
"'xi e
ab |
(1 | Xl)h= b
and call e the
A(Xl,...,x m) in an obvious way.
we have 9 , Xil "'xi e
| Xil-..Xie
We shall hereafter assume that d~
= d~
if i I _< 2). where ai,...,a m are elements of
degree s in the center of B. If p ~ 2 this implies that s is even and d~ We define a differential
d on J by
dh = d(b s Xil-..Xle) (^ indicates
ommission).
= [ (-1)J-lbaij
s Xil
... xij ^
We have clearly
d((a | l)h) = (a | l)dh, d((l | Xl)h)=
odd.
(al| l)h - (i | Xl)dh
(for i I Z 2).
.-. xi e '
59
Then d is homogeneous
with respect
DF by s-l,
D by l, and decreasing
increasing
with respect
to d (coefficients
in particular
we note Ho(J)
Lemma 19.2.
al,...,a m are ann-free
(2)
H p'q = 0
Proof.
conditions
degree by 1. The homology
graded
by H(J)
=
of J
~ HP'q; p,q,e e
are equivalent:
up to k,
for p < k - s,
(e 9 1),
-
H~ 'q = 0
the exterior
DB by s, decreasing
= B/(al,...,am).
The following
e
of J, increasing
in K) is then naturally
(1)
(3)
to all gradings
-
for p ~ k - s.
For m = 1 the lemma is trivial.
Assume the lemma true for m-1.
Let
J1 = B 9 At(x2,...,Xm ), J~ : B/(al) and let k : Jl § J'l be the natural map. a (B e A(X2,...,Xm))
submodule
a differential
~ Xil.-.Xik ) : ~
(b' a B', a~j = k(aij)) , and then d'k for J~, and it remains (1)~(2)
Evidently
to infer
to show it is a coboundary.
of J, in fact
d' on J~ by
(-I) j-lb'a~j
^ ~ Xil . . . Xij ...
=kd. By our inductive
Xik ,
assumption
the lemma holds
it for J.
: The al,...,a m being ann-free
a~,..., ma' and so (2) holds
Jl is a subspace
stable under d. The kernel of k is clearly
(a I) ~ At(x2,...,Xm ). We define d(b'
@ At(x2,...,Xm ),
up to k, the same is then true for
for J~. Let h ~ J, DBh _< k-s, DFh 9 O, be a cocycle.
We have
Write h = (1 ~ Xl)h I + h2; hl,h 2 c Jl'
with DBh I = DBh 2 = DBh and DFh 2 = DFh > O. Then dh = (a I | 1)h I + (-1) s-1 the first
and last terms
on the right
side being in Jl" Since dh = 0 we have
(a I | 1)h I + dh 2 = O, Projecting
into J~ we see that d'h~
h i ~ J~ with DBh~
= k-2s
(1 @ Xl)dh I + dh2,
= O. Therefore
(1 9 Xl)dh I = O. in J~ we have h~ = d'h% for some
: hence h 2 = dh 4 + (a I ~ 1)h4,
DBh 3 = DBh 4 = k-2s,
with h3,h 4 r Jl" Substituting, h =(1 9 Xl)h I + dh 3 + (a I 9 1)h 4, dh = (a I 9 l)(h I + dh 4) + (-1)s-l(1 Thus
(a I | 1)(h I + dh4)
annihilator
= O. The annihilator
of a I in B) ~ 1 which vanishes
9 Xl)dh I = O.
of (a I e l) is identical
up to DB ~ k-s.
with
(the
Since DB(h I + dh4) ~ k-s,
6o
we h a v e
h I + dh 4 : O. T h e r e f o r e h : dh 3 + (a I | l)h 4 - (i ~ X l ) d h 4, = d(h 3 + (I | Xl)h4) ,
so h is a c o b o u n d a r y . (2)--~(3):
trivial.
(3)
first
cocycle
~(1):
in J~ w i t h
we
exterior
show
that
(3)
is also
fulfilled
in J~.
Let
h' be a
l; we h a v e to s h o w that h' is a c o b o u n d a r y . We w r i t e m m [ b i'a i' | 1 = 0 . h' : [ b~ | xi, d'h' = i=2 i=2 m m b i'a i' = O, and h e n c e i X 2biai = - b l a l f o r some b I ~ B. T h e r e f o r e h = i ~ i b l | x i is a
m ~ i=2 c o c y c l e in J of e x t e r i o r
Thus
where
shall
degree
degree
u = (1 | X l ) h l + h 2 , w i t h
1. S i n c e
(3)
h l , h 2 r Jl" h = du
= -(1
is a s s u m e d
in J it f o l l o w s
that
h = du,
Thus | X l ) d h I + (a I @ 1)h I + dh2,
and h e n c e m
Projecting
into
By our annihilator
J' we get
inductive
of the
I = 2,3,...,m.
h'
assumption
projection
To c o m p l e t e
of a I in B is zero
b i | xi = (a I | 1)h I + dh 2. I=2 = d ' h ' 2 , and the a s s e r t i o n is p r o v e d . a n d the
fact
Just
proved
it f o l l o w s
of a i in B / ( a l , . . . , a i _ I) v a n i s h e s
the proof
up to d e g r e e
k-s.
of
(1)
it r e m a i n s
We
shall
do t h i s
only
that
the
up to d e g r e e
to s h o w
by i n d u c t i o n
that
the
on the
k-s
for
annihilator
degree
t
of B t . For t = 0 the ba I = O. T h e n
assertion
is t r i v i a l :
h = b | x I is a c o c y c l e . h = du,
where
hl,h 2 r B | At(x2,...,Xm)
respectively.
h'2
which
b | x I = -(1
= d'h'3
(3) we
it for t-1
and
let b r B t be
such
degrees
2
can w r i t e
u = (1 | X l ) h I + h2,
, D B h i = D B h 2 = t-s,
and e x t e r i o r
1 and
Then du
from
By
Assume
for
some
= (a I | 1)h I -(1
| X l ) d h I and
h' 3. T h e r e f o r e
| X l ) d h I + dh2,
(a I | 1)h I + dh 2 = O. T h e n
d'h'2
= 0 so t h a t
we can w r i t e
h 2 = dh 3 + (a I | l)h 4 for
some
h4,
and
hence (a I | l)h I + (a I | 1 ) d h 4 = (a I ~ 1 ) ( h I + dh 4)
By the
inductive
assumption
on t - 1 we h a v e
h I = -dh 4
: hence
= O.
that
61
b = • and
(i) is proved. Remark:
This completes
By i n d u c t i o n
t h e n B is g e n e r a t e d B = K[al,...,am]
I = •
4 = 0
the lemma.
on the degree
(as an algebra)
it is clear that if B = (al,...,a m) + B ~
by 1 and the a i. Hence
in order to show that
it suffices to prove that the a i are a n n - f r e e
and that B
B = (al,...,a m) + B ~ To i l l u s t r a t e m-dlmenslonal universally
we shall now prove t h e o r e m 19.1 in the special
torus T. We have H*(T)
transgressive.
= A(Xl,...,Xm)
where d~
case where G is an
= 1 and the x i are
As J we take E 2 = H*(BT)
@ A(Xl,...,Xm);
then d 2 is given by d 2 ( H ~ ( B T) | i) = 0 and d2(l @ x i) = (Yi @ i). We first prove by i n d u c t i o n that the Yl are ann-free. T h e n by (3) in lemma d2-coboundary
Suppose they are ann-free
(19.2) t h e r e must be a d 2 - c o c y c l e
h r E 2 w h i c h is not a
and is of the form h = ~ b i | xl, d~
Since DFh = d~
= l, h is a permanent
s > 2 and an element
which contradicts H~ 'q = 0
up to k but not to k+l.
cocycle.
u r E 2 such that drk u = 0 DBu = DBh-s
= k-l-s
DFu = l+s-1
= s>2,
(2) of the lemma.
= k+l-2
= k-l.
Since E| is t r i v i a l there must be an for r < s and dsk ~ u = k 2h s ~ O. But < k-4,
Thus the Yi are ann-free.
We t h e r e f o r e
have
for e > 0 and so E 3 = H * ( B T , K ) / ( y l , . . . , y m) | i, E| = E 3.
Since E| is t r i v i a l we have H*(BT,K) r e m a r k we then have the d e s i r e d
= H~
+ (yl,...,ym).
In view of the above
result,
HZ(BT,K ) = K[Yl,...,ym]. The g e n e r a l result
of 19.1 will be o b t a i n e d
h o w e v e r we shall prove a p r e l i m i n a r y
as a c o n s e q u e n c e
(below).
First
sequence
(Er) , r ~ 2, over a field K w i t h E r a n t l - c o m m u t a t i v e
degree.)
We assume
gresslon;
Consider
a canonical
(with respect
19.4 spectral
to total
further E 2 = B 9 A(Xl,X2,...) ,
(of finite type)
lemma.
of t h e o r e m
and the x i t r a n s g r e s s i v e .
explicitly,
E~ 'q = B p 9 (A(Xl,X2,...)) q, Let Yi r B be an image of x i by trans-
2 1 = d s + i k s2+ l Y i. We introduce if s = d~ i then ks+
the f o l l o w i n g
62
notation: P
: the vector
Fr
: the v e c t o r space spanned by the x i of degree r,
Q
: the vector
space spanned by the Yi'
: the vector
space spanned by the Yi of degree r,
Qr
space spanned by the xi,
J~r Br
J~r
2 = kr(B | l)
:
p,o [ Er . P
%-r-I Note that B r | A ~ has a n a t u r a l
differential
d'
r
d e f i n e d by
d ' r ( B r | i) : 0, d ' r ( 1 | x i) = 0
if x i E p r
d,r( 1 | xl ) = krY 2 i | i The e l e m e n t s defined
if x i r pr-1 .
of B | AtP~r-1 are d i - c o c y c l e s
and is a d r - i n v a r i a n t L e m m a 19.3.
subspace
(2 r)
(3 r) Proof.
of E r.
Assume B 1 = E l'O = 0, d r is an i s o m o r p h i s m
and that Q 2 ...,Qi are a n n - f r e e (I r)
r-i for i < r; hence T r = k2(B eA t P, ) is well
up to k. Then for 2 < r < i we have
T_r-l, H(B r | a ~ ) C Er+ 1 for DB <_ k+l, T_r-1, /(Qr) | %pr H(B r | A r~ ) = Br A ~ , % r -- Br+ 1 | A P~ = Er+ 1 t-r-1 Br+ 1 | A r~ = Tr+ 1 for DB < k+l. We p r o c e e d by i n d u c t i o n
(12)
Clear.
(22 )
We can write
of pr-1 onto k2Q r for r_
for DB _< k-r,
on r.
2 E 2 = B | AtF 1 : B | Atpl @ A t P~, and by the K G n n e t h rule t 2 E 3 : H(B | atp I) | A P,. The d e s i r e d result (32 )
now follows r e a d i l y
We have
T 3 : k (B ~ A t ~ ) whence
: (B | A t P $ ) I ( B
t2
~ A P$) ~
d2E 2,
(32). N o w assume
induces
from lemma 19.2.
a map fr
(1 r . 1),(2 . r 1),(3r . . 1 ) for r < i. The map k 2r : B | At pr-l~ § Tr : Br | a t_r-1 ~ + T r w h i c h is surJectlve,
and frd'r
= drf r. We assert
63
(~) that
Tr~
drE r = T r A
drT r
is to say, if a cocycle
for DB ~ k,
of T r of DB i k+l bounds
h ~ Tr, DBh ~ k+l, h = drU , then DBu ~ k-r+l. DB ~ k-r+l;
hence
in E r it bounds
coboundary
u e T r.
fr(h)
= drU; then u E T r and DBu ! k-r+l.
in B r ~ AtP r-l~ . Therefore
(2 r) The first (m) and
(2r_l),and
subspace
of E
For if
But by (2r_ l) and (3r_l) , E r = T r for
(lr) f~r: H(Br ~ Atpr -1 ) + Er+l 9 t_r-1 If h is a cocycle in B r m A ~ with DBh ! k+l then fr(h) in E r. Suppose
in T r.
equality
f~ is InJective
follows
the third from
from lemma
(2r_l)
is a cocycle
in T r and hence
By (3r_ l) h is then a
up to DB _< k+l and (1 r) is proved.
19.2, the second
from 19.2 using also
using the fact that Er+ 1 is a quotient
of a
r
(3 r) We have
Tr§ Using
= krr§
(e) we have the desired Theorem
19.4.
Let
r
= k §
: kr§
AT r
result.
(Er) , r ~ 2, be a canonical
with E r antl-commutatlve.
r .
spectral
sequence
over a field K
Suppose E 2 = B | A%(Xl,...,xl,...)
where the x i are transgressive, only finitely E~ '~
many
have odd degree
if the characterlstle
of the x i have the same degree.
= K,E p'q = 0
if p+q
> 0). Let Yl be an image
Assume
further
of K is not 2, and
that E
is trivial,
of x i by transgression.
Then
B = K[Yl,...,yl,...]. As a corollary section 9
we have at once theorem
19.1 stated at the b e g i n n i n g
Serre has also made use of the t h e o r e m
berg-MacLane
on the Eilen-
19.4.
We have B 1 = E~ '0 = E l'O| = O. Moreover
ErO,r-1 is E~
= E~
isomorphically
- 0. Thus d r is InJective
for r _> 2 the kernel on E ~
D
of d r on
pr-1 and so ~ maps pr-1
onto Qr.
(8) We assert
that the Yi are ann-free 9
k and that o = Q1,Q2,...,QI Q1,...,Qi+l
computations
groups.
Proof of theorem (a)
in certain
of the
are ann-free
are ann-free
that the Yi are ann-free
up to
up to k+l for some i > 2. We shall prove that
up to k+l. Suppose H(BI |
Assume
this
is not true.
%p l-l) ~ Ei+l
By (ll) we have
for DB ! k+l
64
U s i n g the K G n n e t h rule and the d e f i n i t i o n H(B i , Atp~ -I)
of B i we can write
: H ( B / Q I + . . . + Q i-l) ~ Atp~ -1) ~ Atp~
to w h i c h we can then apply lemma 19,2. H a v i n g a s s u m e d Q I , . . , , Q I + I
not a n n - f r e e
k+l it follows
h E El+ 1 w h i c h is not
from
a di+l-CObOundary
(3) in the lemma that there
and is such that DBh = k+l-i,
degree h is a p e r m a n e n t element
is a d i + l - c o c y c l e
cocyole.
Since E
u r E 2 such that drk u = 0 DBu = DBh-s
DFh = i-l. B e c a u s e
is t r i v i a l there
of its low fiber
is an s > i + 1 and an
for r < s and risk u =
= k+l-i-s
up to
s
< k-s+l
hence by (2s_ I) in lemma 19.3, we have .
But also DFu ~ D F h + s - i hence k~u is not d e c o m p o s a b l e
= i-l+s-i
= i+s-2<2(s-l);
and
e Bs | Since d s v a n i s h e s
on t r a n s g r e s s i v e
elements w i t h degree
g r e a t e r than s-1 and d s k ~ U # 0
we must have k2 pS-1 su E B s @ Thus s-1 = i+s-2, the induction,
i m p l y i n g i = 1 in c o n t r a d i c t i o n
and the Q1 Q2
Having established
of the fact that i > 2. This completes
... are ann-free.
(~) and (8) we can now apply lemma 19.3 for all k. Thus for
any k we have Bk = B k / ( Q l + . . . + Q k) ~ B k k+l from w h i c h we see that B = B ~ + (Q1 + Q2+...). d e s i r e d result B = K[y I .... ,Ym,'''] 2o.
Invariants Let
closed
of the Weyl group~
(E,B,G)
connected
be a p r i n c i p a l
subgroup
now N is a subgroup
by the r e m a r k f o l l o w i n g and c ~ a s s i f y l n g
spaces.
The H i r s c h formula.
f i b e r i n g w i t h G a compact
is c o n t a i n e d
u § nun -1 and,
Lie group.
x U of E and o p e r a t e s
on the f i b e r l n g
of E, the c o h o m o l o g y
of E/U,
in the n o r m a l i z e r
since nU = U n , ( n
(E,E/U,U).
and the spectral
Therefore
sequence
we have the
19.2.
of G: then we also have a p r i n c i p a l
of G w h i c h
by the inner a u t o m o r p h l s m s
Since the Yl are a n n - f r e e
fibering
Let U be a (E,E/U,U).
of U t h e n N operates
E N), it p e r m u t e s N operates
of (E,E/U,U).
If on U
the cosets
on the c o h o m o l o g y We have the
f o l l o w i n g properties: (a)
N/N ~ U acts on H~(E,A)
and H ~ ( E / U , A ) ,
the o p e r a t i o n s
being compatible
with
65
the f i l t r a t i o n (b)
and the map induced by the fiber map.
the operations
of N on E 2 = H~CEIU,
commute with the canonical
H~(U,A))
isomorphisms
E~ ,~ : H P ( E / U , A ) , In p a r t i c u l a r (c)
N ~ U operates t r i v i a l l y
the operations
on H*(E,A).
E~ 'q : Hq(U,A).
of N/N ~ U on E | agree with those
In p a r t i c u l a r
b e t w e e n H*(E,K)
and E
so that N/N ~ U operates
if A is a field K t h e n there
w h i c h commutes with the
We can also consider the f i b e r i n g N operates
on G/U by right t r a n s l a t i o n s ,
the fibering,
and t r l v l a l l y
since U is connected,
N/N ~ U operates
induced by the operations
is a vector
space i s o m o r p h i s m
operations.
(E/U,B,G/U),
(G/U the left coset
on E/U, the o p e r a t i o n s
on B. T h e r e f o r e
on Er, r ~ 2.
N operates
on H$(G/U,A).
space).
being compatible
on the c o h o m o l o g y
Then with
of G/U, and,
In a d d i t i o n we have the f o l l o w i n g
properties: (a')
N/N ~ U operates
being compatible (b')
with the f i l t r a t i o n
Assuming
(E/U,E/G,G/U),
on H*(E/U,A)
and t r i v i a l l y
the operations
and the map induced by fiber map.
G c o n n e c t e d then N operates
the o p e r a t i o n s
on H~(B,A),
on the spectral
sequence
(E r) of
on E 2 = H~(B,H*(G/U,A))
c o m m u t i n g with the canonical
Isomorphisms E~ '~ = HP(B,A),
Since N ~ U operates trivially, (c') on H * ( E / U , A ) .
The operations
on E
E~ 'q = Hq(G/U,A).
N/N ~ U operates
on (Er), r ~ 2.
agree with t h o s e
induced by the o p e r a t i o n s
of N/N ~ U
If A = K then there is a v e c t o r space i s o m o r p h i s m b e t w e e n H*(E/U,K)
and E|
w h i c h commutes with the operations. Let G be a compact of these are conjugate
Lie group then it contains m a x i m a l
under an inner a u t o m o r p h i s m
called the rank of G. Let T be a m a x i m a l It is k n o w n that T has finite
total
subgroups.
Any two
of G, t h e i r common d i m e n s i o n
torus and let N T be the n o r m a l l z e r
index in N T so that W(G)
is
of T in G.
= NT/T is a finite group w h i c h
is called the Weyl group of G. The map t ~ ntn -1, n c N T, t E T, is an a u t o m o r p h i s m T which depends
only on the coset mod T in w h i c h t lies and induces
of
a map W(G) § Aut(T)
w h i c h is faithful when G is connected. In the p r e c e d i n g d i s c u s s i o n we take U = T and N = N T and c o n s i d e r the p r i n c i p a l
66
flbering
(E G = ET, BT, T). T h e n W(G)
Now T = S 1 x ... x S1 ' (m times),
operates
has no
in a c c o r d a n c e
torsion
E 2 = H~(BT,Z)
with
(a),(b),(c)
above.
so that
~ H*(T,Z).
Using the a c y c l i c i t y
of E G and the fact that d r ~ O, r ~ 3, it follows that the
transgression
HI(T,Z)
T maps
Isomorphically H~(BT,Z)
where v i is an image of a g e n e r a t o r H~(BT,Z)
with p o l y n o m i a l s
operations
onto H2(BT,Z).
= Z[Vl,...,Vm],
of Hl(T,Z)
over Hl(T,Z)
by
By t h e o r e m 19.1
v i ~ H2(BT,Z),
by t r a n s g r e s s i o n .
T is easily
The i d e n t i f i c a t i o n
of
seen to commute with the
of W(G).
The ring of invariants kx e I G and w e W(G) that x = w(x).
I G of W(G)
then kx = w ( k x
in H~(BT,Z)
= kw(x),
is a direct
and since H*(BT,Z)
summand.
For if
has no t o r s i o n
it follows
Then I G ~ Zp C
H~(BT,Z)
We shall now study the second f l b e r l n g Proposition t h e n G/T has no
2o.1.
torsion,
If G is a compact
~ Zp = H ~ ( B T , Z p ) -
(BT,BG,G/T).
connected
its odd d i m e n s i o n a l
Lie group and T a m a x i m a l
B e t t l numbers
are zero, and
torus
x(G/T)
is
equal to the order of W(G). For the p r o o f of 2o.1 in the case of the classical in the g e n e r a l
case
Proposition is e q u i v a l e n t
and G2,F4,
see
[i] and
E3],E4]. 2o.2.
(Leray).
The n a t u r a l r e p r e s e n t a t i o n
rw of W(G)
in H*(G/T,R)
to the r e g u l a r r e p r e s e n t a t i o n .
Proof. a fixed point
groups
Let T n be the h o m e o m o r p h i s m then n e T. Let w e W(G), L(w)
of G/T d e f i n e d by g § g.n.
then for the Lefschetz
= ~ (-1)itrrw
Clearly
number
if T n has
L(w) we have
in Hi(G/T,R),
= ~ trr w in H I ( G / T , R ) , = If w ~ e then L(w)
trr w in H~(G/T,R).
= O; hence trF w = O. On the other hand if w = e then
Trr w = dim H*(G/T,R)
= order of W(G).
This
is p r e c i s e l y
the character
of the r e g u l a r
representation. As a corollary we see that the t r i v i a l r e p r e s e n t a t i o n H~
Since W(G)
operates
on H ~ ( G / T , Z o) and since
H ~ ( G / T , K o) =-H*(G/T,Z o) | K o, this applies
also to H ~ ( G / T , K o ) .
occurs
exactly once
(in
67
Theorem 2o.3.
Let G be a compact
connected
Lie group with no p-torslon and let
T be a maximal torus. Then (a)
p~(T,G)
: H*(BG,K p) + Ha(BT,K p) is an isomorphism
(b)
H~(G/T,Kp)
is the characteristic
ring of the flbering of G by T and + : H*(BT,Kp)/(I G ~ Kp),
H*(G/T,Kp) (I~
onto I~ ~ Kp;
is generated by elements
of positive degree).
Remark.
A similar statement
holds over Z when G has no torsion.
Proof.
We consider the fibering
(BT,BG,G/T) ; we have
E 2 ~ H*(BG,K p) ~ H~(G/T,Kp), E~ : GrH*(BT,Kp). The second factor in E 2 has only even dimensions
as a consequence
of proposition
2o.1.
Since G has no p-torslon we have H~(G,Kp) Therefore
by the theorem on transgression
Thus E 2 has only even dimensions p~(T,G)
is inJectlve. Case p = o.
of the section). H~
= A(Xl,...,Xm) , d~
(19.1), H~(BG,K p) has only even dimensions.
and it follows that E 2 = E . This proves
It remains to determine
the image ~t(T,G) H~(BG,K p ) C
We know W(G) operates trivially
on H~(BG,Kp),(see
By (2o.2) it acts as the regular representation
o) as the only trivial representation.
is the set of all invariants
(b) and that H~(BT,Kp).
(a') at beginning
on H~(G/T,K o) with
This implies that
H~(BG,K o) | H~
~(T,G).
odd.
)
of W(G) in E 2. Its projection
We have already proved that E 2 = E|
in E| is then the image of
Using the vector space isomorphism
in (c')
we can identify E| = H*(BT,Ko) , and hence also H*(BG,K o) | H~ Thus the image of pt(T,G) Case p ~ O.
) = I G | K o.
is the full set of invariants
Since I G is a direct summand,
dim (~G | Kp )i is equal to the rank
of (IG)i , and hence by the preceding to dim Hi(BG,Ko). no p-torsion
Since G has no p-torsion,
(see 18.4, 18.5); thus dim Hi(BG,Kp)
= dim Hi(BG,Ko).
Moreover
H~(BG,Kp) ~ H~(BG,Z) ~ Kp, whence
I G | K o.
B G has
68
and the image Q of p~(T,G) is contained in I G 9 Kp. By (a') and the p r e c e d i n g remarks, Q ~ Hi(BT,K p) and (I G ~ Kp) i have the same dimension and so must be equal. Proposition 2o.4.
If G is a compact Lie group and T a maximal torus then p~(T,G)
maps H ~ ( B G , K o) isomorphically Proof.
onto I G @ K o.
Let G O be the connected component of e, then T is maximal torus in G o .
We have natural projections EG/T § EG/G o § EG/G, that is, BT p(T,Go) with p(T,G)
= k-p(T,Go).
BG ~
~
~ BG '
The map k is clearly a covering map with covering group G/G o .
Clearly W(G o) is an invariant W(G)/W(G o) = G/G o . We have
subgroup of W(G) and one sees readily that W(G)IW(G
#(T,G)
=
~(T,Go).~.
It is known that
the invariants of the covering group in H~(BGo , Ko) as image.
k* is inJectlve with
If we now apply theorem
2o.3 to p*(T,Go) the p r o p o s i t i o n is proved. Remark. the
If p is prime to the order of G/G o it is easily seen that I G | Kp is
set of all invarlants of G/G o in I G
p~(T,G) maps H~(BG,Kp)
e K . Using this one proves as before that o P onto I G | Kp when G o has no p - t o r s i o n and p is
isomorphically
prime to the order of G/G ~ . Theorem 2o.5. torus.
Let G be a compact connected Lie group and let T be a maximal
If 2rl-1,...,2rm-1
are the degrees of primitive
elements in H*(G,K o) then m is
also the rank of G, and P(G/T,t)
Moreover,
the order of W(G)
Proof.
= ( 1 - t 2 r l ) . . . ( 1 - t 2rm) (l_t2) m
= Hr i and the dimension of G = 2 ~ ri-m.
We have seen earlier that H~(BT,Ko)
= Ko[Vl,...,Vm] ,
d~
= 2;
hence p (BT,t)
:
i (l-t2) TM
By Hopf's t h e o r e m H(G,Ko)
: A(Xl,...,Xm) , d~
odd.
Applying 18.1 we have H~(BG,K o) : Ko[Yl,...,ym, ], hence
d~
: 2ri;
69
P(BG,t) We have already
:
(l_t2rl)...(l-t2rm ')
noted that in the spectral
since all degrees
sequence
of (BT,BG,G/T)
we have E 2 = E
are even. Therefore P(G/T,t)
= P(BT,t)/P(BG,t), = (l-t2rl)...(l-t 2rm') (l-t2)
We assert series.
that m = m'.
But this
If m'
contradicts
On the other hand if m'
< m then on dividing
P(G/T,1)
•
= P(G/T,-I)
Betti numbers
P(G/T,t)
contradicts
(2o.1)
t = 1 in m = H (l+t2+--.+ i=l
the fact that
the last equality
t 2ri-2)
P(G/T,1)
we have that the order
= P(G/T,I),
are present.
of W(G)
holding
m' H (l-t 2rl) i=m+l
> O; hence m' = m. is equal to
since
m H (l+t2+..-+t 2ri-2) I t=l i:l Since G/T is an orientable m a n i f o l d we have m dim G/T = ~ (2ri-2) = 2[rl-2m. i=l Thus dim G-m = 2~ri-2m , and we have the desired result.
closed
2o.6.
connected
primitive generators
W(G)
=
Let G be a compact
subgroup
generators
hence m'~m.
only even dimensional
Therefore
order of
Theorem
is an infinite
> m then putting
= O. This
By p r o p o s i t i o n
we have that
the fact that H * ( G / T , K o) has finite dimension;
P(G/T,t)
gives
TM
connected
of the same rank.
m
=
H r 1. i=l
Lie group of rank m. Let U be a
Let 2rl-l,...,2rm-i
of H*(G,K o) and 2Sl-1,...,2Sm-1
the degrees
be the degrees
of the
of the primitive
of H*(U,Ko).
(a)
If U has no p - t o r s i o n
(conjectured
by Hirsch
then G/U has no p-torsion
and we have the
formula
for p = 0): (l_t2rl)...(1-t2rm) P (G/U,t) P
(b) subrlng,
:
If G and U have no p-torslon
(l_t2Sl)...(l-t2Sm) then H~(G/U,Kp)
is equal to its characteristic
and H~(G/U,Kp)
Proof.
(a) Consider
G D U D T. We have
the spectral
= (I U ~ Kp) sequence
+ / (I G e Kp).
of (G/T,G/U,U/T,~)
defined
by
7o
E 2 = H*(G/U,Kp) E
~ H~(U/T,Kp),
= GrH*(G/T,Kp).
Since U has no p-torslon then by (2o.3) H*(U/T,Kp)iS By corollary Therefore
(16.4) U/T is totally non-homologous
E2 = E
(2o.5) the Hirsch formula follows.
get the same P(G/U,t), (b)
to 0 in (G/T,G/U,U/T,~)
ring.
(mod p).
and Pp(G/U,t).Pp(U/T,t)
Applying
equal to the characteristic
= Pp(G/T,t).
Noting that in characteristics
o and p we
we see that G/U has no p-torslon.
Consider the spectral
sequence
of the fibering
(Bu,BG,G/U,p(U,G)):
we have
E 2 : H*(BG,K p) ~ H*(G/U,Kp), E| : GrH*(B U Kp) By(19.1)
and the Hirsch formula the factors
that E 2 = E.. Therefore Jective.
G/U is totally non-homologous
But i~ is the characteristic
H*(G/U,Kp)
is the characteristic
and it follows
to 0 which means that i~ is sur-
map as is shown in section 17. This proves that
ring. If we also apply theorem 14.2 we get
H~(G/U,Kp) Applying ~ ( T , U ) ,
in E 2 have only even degrees
which by 2o.3(a)
= H*(Bu,Kp)
is inJective,
HS(G/U,Kp)
/ (~(U,G)H+(BG,Kp)).
we get
= I U ~ Kp/ (p~(T,U)p~(U,G)
: I U ~ Kp/ (~(T,G)
H+(BG,Kp)),
H+(BG,Kp)),
= I U 9 Kp/ (I~ s Kp), the last equality holding by 2o.3(a). Remark:
We have a similar statement
over Z:
if G and U have no torsion and G/U
has no torsion then H*(G/U,Z)
= IU/(I~).
BIBLIOGRAPHY [1]
A. Borel,
Ann. of Math.
[2]
A. Borel, Amer. J. Math.
[3]
A. Borel, K~hlerlan coset spaces of semlsimple Vol. 4o, No. 12, (1954), 1147-1151.
[4]
R. Bott, On torsion in Lie groups,Proc.Nat.Acad.Scl.,Vol.4o,No.7
[5]
H. Cartan,
[6]
N. Steenrod,
Sgmlnalre,
57(1953),
76 (1954),
Paris
The topology
115-2o7. 273-342. Lie groups,
Proc.Nat.Acad.Sci.,
(1949-5o).
of fibre bundles,
Princeton
(1951).
(1954),586-588.
71
CHAPTER CLASSIFYING
21.
Unitary
SPACES
S(al,...,a r)
~i
the elements
Let U(n)
denote the group
of orthonormal
s-frames
Wn, 1 = S2n_l and Wn, n = U(n).
then the natural
map
(Wn,s,Wn,t,Wn_t,s_t).
~s,t
Hermitian
: Wn,s
+ Wn,t
space.
Wn, s = U ( n ) / U ( n - s ) i n More generally
Evidently
the usual way and let
we may consider
is a fibre map for the fibering
Moreover
if u ~ t ~ s then clearly ~s,u = ~t,u'~s,t" 9 (s) (s) (s) H~Wn,s,Z) = ^(X2n_l,X2n_3,-.-,X2(n_s)+l) where the subscripts
of the elements. *
Moreover
_(t)
,
~s,t(~2J+l j =
the elements
x(S)
2J+l
(n-t
'
< J < n-l)
-
-
of ~s,t"
For s = 1 the p r o p o s i t i o n
sider the fibering
is clear.
(Wn,s,Wn,s_l,Wn_s+l,l,~s,s_l). E 2 : H*(Wn,s_I,Z)
then, using the inductive
assumption E2
One shows readily
and Wn, s the complex Stlefel
t ~ s;
the degrees
Proof.
function.
in n - d i m e n s l o n a l
21.1.
the image
in S(al,...,ar),
matrices
Proposition
generate
in al,...,ar,
of n • n unitary
We identify
U(n-t) D U(n-s),
degree
symmetric
+ Wn, s denote the natural map.
U(n) ~
denote
functions
of positive
: the i th elementary
: U(n)
GROUPS
notation:
: the ring of symmetric
S+(al,...,ar):
~n,s
OF THE CLASSICAL
groups.
We shall use the following
manifold
IV
by an argument
Assume
(s>l),
and con-
We have
~ h~*(Wn_s+l,l,Z);
and Wn_s+l, 1 = S2(n_s+l)_l,
A(X (s-l) x(S-1) 2n-1 ''''' 2(n-s+l)
=
it for s-l,
on the fibre degrees
+l
) ~ Atx (1) ~ 2(n-s)+l )"
of elements
in E r that d r ~ 0 for
r > 2; hence E 2 = E| = GrH*(Wn,e,Z). By the analogous
statements
to 14.1.
and 14.2
(a) for integer
coefficients
we have
I
that
i* is surJectlve
and an element
and ~* inJective Consider the elements s,s-i ~,s, s_l( X2(n_i)+l (s-l) (s) ) : X2(n_i)+l , i : O,l,...,s-l,
x~ s) (n-s)+l
such that
i~ x(S) 2(n-s)+l
: x (I) 2(n-s)+l"
72
These are clearly linearly independent odd degrees and H*(Wn,s,Z)
and generate H*(Wn,s,Z).
is torsion free it follows that
(s)
H~(Wn,s,Z) This also
Since they also have
proves the second statement
(s)
: A(X2n_l,...,X2(n_s)+l). in the theorem in the case where t = s-1. The
general statement then follows easily by induction. Proposition 21.2. are universally Proof.
,
transgressive. (f) If s=l, Wn, s is a sphere so the generator x (1)2n_l is clearly universally
transgressive. transgressive
(s)
(s)
The elements X2n_l,...,X2(n_s)+l
Assume the proposition
for s-l, (s > i); then v(s-l) ~2n-i '''" ,x(S-l) 2(n-s+l)+l are
in the fibering (Eu(n)/U(n-s+l),Bu(n),Wn,s_l).
The natural map : Eu(n)/U(n-s) defines a representation that
§ EU(n)/U(n-s+I)
of (Eu(n)/U(n-s),Bu(n),Wn,s)
in the preceding fibering such
~ : BU(n) § BU(n) is the identity map. If we canonically
fibres Wn, s and Wn,s_ 1 with the various fibres in Eu(n)/U(n-s) respectively,
identify the standard and Eu(n)/U(n-s+l)
then ~ restricted to a fibre is precisely the map ~s,s-1
: Wn,s§ Wn,s-l"
It follows from this that the elements ~s) = i tx(S-1)~ x n-i ~s,s-I < 2n-i "' are transgressive
in (Eu(n)/U(n-s),Bu(n),Wn,s).
i = 1,3
'''"
,2s-3
The element x ~ _ s ) + l
' having lowest
possible degree is clearly also transgressive. Theorem 21.3.
BU(n) has no torsion and H~(Bu(n),Z)
: Z[y2,Y4,...,Y2n]
(n) by transgression. where the y's are images of the X2n_i This follows from 18.5(a) and the analogue of 18.1 for integers. The unitary group U(n) clearly has rank n; a maximal torus T n consists of the diagonal matrices of the form
0
(f)
e 2~ixn
In analogy with the definition of section 18 we say in the case of associated bundles that x E H'(F) is universally transgressive if it is transgressive in the bundle ((EG,F)G,BG,F) introduced in section 17.
73 The Weyl group W(U(n))
is then the group of permutations
shown in Chapter III that H~(BTn,Z) of polynomials fication.
over Hl(Tn,Z),
may be identified
the operations
being compatible
= Z[Vl,...,Vn] , d~
The Weyl group is then the permutation Theorem 21.4.
p*(T n, U(n))
with the ring of with the identi-
Given a flbering image by transgression determined
extension
(E,B,Wn,n_i+l)
isomorphlcally
= S(Vl,...,Vn).
onto S(Vl,...,Vn).
for integers.
we define the Chern class C2i r H2i( B,Z) as the _(n-i+l) x2i_l
of the generator
in .2i-i.. n lWn,n_i+l, Z) . It is uniquely
d r of the spectral
sequence
are
zero for 2 ~ r < 2i. (Usually C2i is defined as the obstruction
of a cross-sectlon
definition
of 2o.3(a)
because by 21.2 the differentials
identically
= 2.
group of (Vl,...,v n) and Iu(n)
maps H ~(BU(n),z)
This follows from the analogue
universal
by transgression
Thus H~(BTn,Z)
gression
W(U(n))
of the diagonal terms. We have
in the given bundle,
but this is equivalent
(see 37.16 in [7]) 9 The image C2ir H 21(BU(n) ,Z) of
in the fibering Chern class.
If r : B § Bu(n) the universal
(Eu(n)/U(i-l),Bu(n),Wn,n_i+l)
Chern class C2i under
to the above
~2i-IY(n-i+l)by trans-
= (Bu(i.l),Bu(n),Wn,n_i+l)
Thus theorem 21.3 asserts H~(Bu(n),Z)
is a classifying map for a flberlng
to
= Z[C2,C4,...,C2n].
(E,B,Wn,n_i+ I) then the image of
clearly the corresponding Chern class in the
r
latter fibering. Proposition ~*(U(i-l),U(n)) Proof.
21.5. The universal
in dimension
Chern class C2i generates
the kernel of
2i.
In the spectral
sequence of (Bu(i_l),Bu(n),Wn,n_i+ I) we have
E 2 = H~(Bu(n),Z)
~ H~(Wn,n_i+l,Z), . (n-i+l) .(n-i+l) = Z[y2,Y4,...,Y2n ] e A[X2n_l ,...,x2i_l );
E|
: :
GrH~(Bu(i_l),Z) : arZ[g2,~4,...,~21_2], Z[Y2' ~ 4 ' 'Y2i-2]
since there are only even degrees. Z[y2,Y4,...,Y2i_2 ] isomorphically
Clearly then p~ (U(i-l), U(n))maps onto H~(Bu(i_I),Z).
H2i(Bu(n),Z) where D 2i consists p*(U(i-l),U(n))
of the decomposable
in dimension
Proposition
We can write
= O 2i + Z[Y2i ]
elements
of degree 21. Thus the kernel of
2i is Z[Y2i ] = Z[C2i I.
21.6. ~ ( T n , u ( n ) )
is the
maps C21 onto •
74
We first make a general remark on o~(U,G) where U is a closed connected subgroup of G. Suppose we choose maximal tori in U and G respectively such that T s C T n. Correspondingly we have a commutative diagram H*(BTs,A):
H*(BTn,A)
H~(Bu,A ) ~
H~(BG,A )
where the indicated maps are induced by the corresponding ~ maps. Suppose now that the vertical maps are inJective as will be the case in the proposition we wish to prove. Then instead of ~ ( U , G )
we may consider p~(TS,T n) restricted to the image of H~(BG,A)
in H~(BTn,A). Proof of 21.6.
If we regard U(s) C U(n) in the usual way then we have a natural
embedding i : T s C T n. The matrices of T n are diagonal matrices whose diagonal elements are e2~IXl,...,e2~iXn.
Then the diagonal elements of the matrices of T s are 1,...
1,...,1,e2~i~n-s+l,...,e 2~i~n. One sees readily that we may regard the xj and ~k as generators of Hl(Tn,Z) and Hl(TS,z) respectively; hence the map i* : Hl(Tn,Z) ~ Hl(TS,z), is given by i*xj =
0 _ xj
if J <_ n-s, if J > n-s.
We may write H*(BTn,Z)
= Z[Vl,...,Vn] , vj = T(xj),
H*(BTs,Z ) = Z[~n_s+l,...,~n],
~j = ~(xj),
in which case the map p~TS,T n) is given by
P*(TS'Tn)(vJ)
=
0 vj
if J ~ n-s, if j > n-s.
We want to prove p~(Tn,u(n)) maps C2(s+l) onto ~Cs+l for any s. By the preceding proposition we know that C2(s+l) generates the kernel of ~ ( U ( s ) , U ( n ) )
in dimension
2(s+l). On applying the preceding general remarks here we see that p~(Tn,u(n))(C2(s+l)) is a symmetric function of the vj of degree s+l which vanishes when Vl,...,v s are set equal to zero; hence (by a theorem on elementary symmetric functions) it follows that p~(Tn,u(n))(C2(s+l))
= kv I -.. Vs+ 1.
Since C 2 ,... ,C2(s+l) generate H 2(s+l) (Bu(n),Z) we must have k = •
75
Remarks.
We shall not discuss
which are necessary
for so called
the various
"duality"
sign conventions
formulas.
Briefly,
for the Chern classes
these
are such that
p~(Tn,u(n))(C2i ) = o i for suitable
bundle
v i permuted
by W(U(n)).
We also recall
the "duality"
Given two principal
bundles
(E,B,U(nl)
over B by taking the bundle
• U(n2))
(E(i),B,U(ni)),
(i = 1,2),
formulas:
over B we construct
induced
on the diagonal
a
of B x B
by the bundle (E (I) x E (2), B x B, U(n I) x U(n2)) . Extending
the structural
(E(3),B,U(n)) associated
which may be called
bundle
of the original direct
group to U(n),
with
fibres
sum" of the original
with fibre C n is obtained C nl and cn2 respectively
two bundles.
from associated
An
bundles
by taking as fibre the
sum of the fibres. Now let ~(i) ~2J ' (i = 1,2,5)
the
the "Whitney
of the sum bundle
bundles
n = n I + n2, we then have a bundle
"duality"
formulas
be the c o r r e s p o n d i n g
col)
:
Chern classes.
then we have
[4],
<21.7) As is well known
jth - Chern Classes;
(and readily
For consider
seen)
a+8=J 2a "u28 this is in fact a relation
the standard
inclusion
among the universal
U(n l) x U(n2 ) C U(n);
this induces
a
map r
H~(Bu(n),Z)§
(using the KOnneth rule and the absence (21.8)
H~(BU(nl),Z)
of torsion).
e ~Bu(n2),Z)
Thus duality means:
r
Now a maximal
(C(3) C(1) ~(2) " 2J ) = ~ 2~ ~ u2B " ~+B:J torus T n of U(n) may be regarded as a product T nl • T n2 where T ni is a
maximal
of U(ni).
torus
of the elementary
Using the preceding
we see that
(21.8)
is a t r a n s l a t i o n
identity:
oj(Yl,...,Ynl,Zl,...,Zn2)
(21.9)
results
=
aa(Yl,..-,Ynl)'~B(Zl,...,Zn2
)"
~+S:J 22.
Orthogonal Let O(n)
matrices
groups,
and SO(n)
respectively,
in n - d l m e n s i o n a l
cohomolo~y denote
mod
2.
the groups
and let Vn, s denote
Euclidean
space.
of n x n orthogonal the Stlefel m a n i f o l d
and special
of orthonormal
Clearly
Vn, 1 = Sn_ 1 , Vn,n_ 1 = Vn, n = SO(n). We embed O(n-s) C O(n)
in the usual way and identify
orthogonal s-frames
76
Vn, s : O(n)/O(n-s) If O(n) D flbering
: SO(n)/SO(n-s).
O(n-t) D O(n-s) then the natural map ~s,t (Vn,s,Vn,t,Vn_t,s_t).
: Vn,s § Vn,t is a fibre map for the
If u ~ t ~ s then ~s,u = ~t,u"
~s,t" We assume the
following well known results: Lemma 22.1.
(a)
For n even, H*(Vn,2,Z)
. (2) ~(2)) = A ~Xn_l,~n_2 9
(b)
For n odd,
= H2n-3(Vn,2,Z)
H~
Hn-I(Vn,2,Z2)
= Z, = Z2,
Hi(Vn,2,Z 2) = 0 Corollary.
otherwise.
For n odd:
(a)
H*(Vn,2,Zp)
(b)
(x (2) x(2)~ H*(Vn,2,Z 2) : A . n-l' n-2"
(c)
V3, 2 : S0(3)
if p # 2.
= H *(S2n_3,zp)
if n > 3.
= P3; hence
H*(V3,2,Z 2) : Z2[x]/(x4). Proposition
22.2.
, (S)
(S)
(S)),
(a)
H*(Vn, s'z2) : A ~ X n - l ' X n - 2 ' ' ' ' ' X n - k
(b)
~
Proof.
t(x~t!)
= Xn_ (S)i
dOx!S) 3
= j.
for 1 < i < t < s < n-1
For any n, if s = 1 then Vn, 1 = Sn_l; hence the proposition
is true for
s = i. Assume it true for n-i and any s and also for n and any s-l; then we prove it for n and s as follows.
The inclusions
(Vn,s,Sn_l,Vn_l,s_l).
O(n) D
O(n-l) D O(n-s)
define a natural fibering
In its spectral sequence we have E 2 -- H~(Sn_I,Z2)
s H*(Vn_I,s_I,Z2),
,-(i)% ~(s-l) ~(s-l)% = A ~Xn_ I, s A( n-3 ''''' n-s "' by inductive assumption.
_o-(s-l) a Xn_ i
= n-i,
We assert that Hn-S(Vn,s,Z 2) # O. For s = 2 this follows from
lemma 22.1. If s > 2 one sees readily that x (s-l) is a non-trlvlal permanent n-s
cocycle and
hence the assertion follows. Now consider the spectral sequence of the fibering which is given by the inclusions
O(n) D
O(n-s+l) ~
(Vn,s,Vn,s_l,Sn_s,~s,s_l)
O(n-s). We have
E 2 = H~(Vn,s_l,Z 2) ~ H*(Sn_k,Z2), x(S-l) (s-l) .. x(S-l)) = A ( n-i ' Xn-2 '" ' n-s+l
. (i)) ~ A[Xn-s "
Clearly d r - 0 if r ~ n-s+l and we assert that dn_s+ 1 -- O. If dn_s+ 1 ~ 0 then it must not vanish on x (1) since n-s
77
n-s E
n-k+l =
n-sE2
has one generator. But then n-sE
n-k+2
o,n-s . (i) = E2 = A[Xn-s) = 0 which in turn implies that
n-sE~ = 0 which
contradicts the assertion Hn-S(Vn,s,Z 2) ~ 0 proved above. Therefore dn_s+ 1 ~ O, and hence E 2 = E|
We define (S) : ~,s, s_l(_(s-l) Xn-i Xn_ i )
for i ~ s - i,
and x (s) as any element such that n-s
i*(X (s)) : X (1) "
n-s-
n-s"
It follows readily from elementary results on the tensor product of simple systems that H~(Vn,s,Z2 ) = A[Xn_l,...,Xn_ - (s) _(s)) s 9 This also proves (b) in the case where t = s-l; the general case follows easily by induction. Fropositlon 22.3.
The Xn_ i(s), (i _< s) are universally transgressive
The proof is analogous to the proof of (21.2).
For i 2 2 we define the it h
universal Stlefel-Whitney class mod 2 to be the element w i which is the image by trans(n'i+l) in the bundle gression of x i-1 (Eo(n)/O(i-1),Bo(n),O(n)/O(i-1))
= (Bo(i_l),Bo(n),Vn,n_i+l),
or equivalently in the bundle, (Eo(n)/SO(i-1),Bso(n)
SO(n)/S0(i-1))
= (Bs0(i_l),Bso(n),Vn,n_i+l).
Thus w i denotes an element of Hi(B0(n),Z2 ) or of HI(Bs0(n),Z2); however no confusion will arise since the map that
p~(SO(n),O(n)) obviously maps the first onto the second.
(Note
p(SO(n),0(n)) is actually a covering since it has a discrete fibre Z2.) Finally we
define the Stiefel-Whitney class mod 2, w I to be the non-zero element of HI(Bo(n),Z2 ). Actually we could also define it using transgression, but in a fibering with fiber Z 2. Proposition 22.4. H~(Bso(n),Z 2) : Z 2 ~2,...,Wn],
d~
: i.
This is an immediate consequence of theorem 18.3. Our next objectives are to determine H*(Bo(n),Z2) and to interpret the w i as elementary symmetric functions. To do this we proceed analogously with the unitary case. The i th Stiefel-Whitney class mod 2 of a principal bundle (E,B,O(n))or (E,B,SO(n)) will then be defined as the image of the characteristic map of the corresponding universal class, or equivalently as the image by transgression of x(n-i+l) i-i in the bundle (E/O(i-l),B,Vn,n_i+l) or (E/SO(i-l),B,Vn,n_i+l).
78
Let E be a principal bundle with fibre a finite group N, let Q be Abelian, Q C N, and let E be simply connected. Then N operates on the fiberlng (E,E/Q,Q) by xQ § x n Q, and hence N/Q operates on E/Q which is in fact a principal bundle (E/Q,E/N,N/Q). For y a N/Q let Ty : Q § Q be defined by q § y-lq Y and let Y~ : ~l(E/Q) § ~l(E/Q) be the map induced by right translation of E/Q by y. Then there exists a canonical isomorphism r : Q § ~I(E/Q)
such that the following diagram is commutative:
Q T
r
,~I(E/Q)
Y Q
'~I(E/Q)
We define r as follows. Let x be a fixed point of E/Q and let ~ be any point of E over it. If q E Q we define Cx(q ) to be the element of ~l(E/Q) whose representative loops are the projections of the paths in E which Join ~ to ~-q. Note that Cx is independent of choice of ~ since Q is Abellan. Let Q(n) C 0(n) be the subgroup of diagonal matrices whose diagonal elements are •
and let SQ(n) denote the corresponding subgroup of S0(n). Evidently Q(n) = (z2)n , SQ(n) = (z2)n-1.
Let NQ(n) be the normalizer of Q(n) in O(n) and define
X(O(n))
= NQ(n)/Q(n). Then
X(O(n)) operating on Q(n) by inner automorphisms acts as ~n, the symmetric group in n-varlables.
In what follows Q(n) and X(0(n)) play the role of T n and the Weyl group in
the unitary case. In the preceding discussion we now take E = E0(n) ' Q = Q(n), and N = NQ(n) ; then Eo(n)/Q(n)
= BQ(n) and X(O(n)) operates on BQ(n) naturally and on ~l(BQ(n)) by maps
induced by right translation. One sees readily that H*(BQ(n),Z2)
= Z2[Ul,...,Un] , d~
= 1.
In view of Hl(BQ(n),Z2 ) = Hom (~l(BQ(n)) ,Z2 ) and the above isomorphism r it follows that X(O(n)) operates on H*(BQ(n),Z 2) by permuting the generators u i.
79
The fibre map p(Q(n),0(n))
: BQ(n)
and its induced map p~(Q(n),0(n)) clearly operates trivially
+ B0(n)
commute with the operations
of X(O(n)).
Since X(O(n))
on H*(Bo(n),Z 2) it follows that
p~(Q(n),
O(n))H~(Bo(n),Z 2) ~ S(u!,...,Un).
Similar remarks apply if consider S0(n) and SQ(n) in place of O(n) and Q(n). In particular,
we note H~(BSQ(n),Z 2) : Z2[Ul,...,Un_l] ,
d~
= i.
Note that the fibre in both cases is the same since 0(n)/Q(n)
= SO(n)/SQ(n).
We shall
denote it by F n. Lemma 22.5. Proof.
dim HI(Fn,Z2 ) Z n-l.
In the spectral
sequence
of (BSQ(n),Bso(n),F n) we have
E 2 = H*(Bs0(n),Z 2) ~ H*(Fn,Z2). Since BS0(n )
is simply connected
E~ '0 = O; hence
IE 2 : E~ ,I = HI(Fn,Z2 ), IE| = HI(BsQ(n),Z 2) = (Z2)n-I Then dim HI(Fn,Z2 ) = dim IE 2 ~ dim 1E| : n-l. Proposition P2(Fn,t) Proof.
22.6.
H*(Fn,Z 2) is generated by elements
of degree l, and
= (1-t2)(1-t3)...(1-tn)(l_t)l-n We proceed by induction
on N. For n = 2,
F 2 = SO(2)/Z 2 ~ TI/z2 ~ T 1 = S 1 and hence the proposition
holds.
Assume the proposition
for n-l,
(n > 2), and consider
the inclusions O(n) D where Z 2 • O(n-1)
Z 2 x O(n-1) D
Z 2 x Q(n-1)
and Z 2 x Q(n-1) are embedded (Fn,O(n)/Z 2 x O(n_l),Fn_l)
where Pn-1 is n-1 dimensional
= (Fn,Pn_l,Fn_1)
real projective
E 2 = H~(Pn_l,~*(Fn_l,Z2))
in the usual way. This defines a fibering
space.
In its ~pectral
sequence we have
,
E~ '0 = Z2, E~ 'I = HI(Fn_I,Z2 )f Then dim IE 2 : 1 + dim HI(Fn_I,Z2 )f ! 1 + dim H I(Fn_I,Z2).
8o
Using the inductive assumption the formula for P2(Fn_i,t) gives dim HI(Fn_I,Z 2) = n-2; hence dim 1E 2 ~ n-1. On the other hand, dim 1E 2 Z dim 1E| = dim Hl(Fn,Z 2) _> n-1 by the preceding lemma, and hence n-1 ~ 1 + dim Hl(Fn_l,Z2 )f _< 1 + dim Hl(F n_l,z2) _< n-1. Thus Hl(Fn_l,Z2 )f = Hl(Fn_l,Z2) which shows that all the elements of E~ 'l = Hl(Fn_l,Z2 ) are permanent cocycles and that ~*(Fn_l,Z 2) is simple in dimension 1. Since by our inductive assumption H*(Fn_I,Z 2) is generated by elements of degree 1 we may conclude that ~*(Fn_l,Z 2) is a simple system and all the elements of H~(Fn_I,Z 2) are permanent cocycles. Thus E 2 = HX(Pn_l,Z 2) e HX(Fn_I,Z2), dr ~ 0 for r ~ 2, and E 2 = E . Moreover since H*(Pn_I,Z 2) : Z2[x]/(x n) we have l_t n P2(Fn 't) = l-t
P2(Fn-i 't)
which proves the desired formula. Note that each factor in E| = E 2 is generated by elements of degree 1. Then by an elementary result on filtered rings the result follows for H*(Fn,Z2) , and the proposition is proved. Theorem 22.7.
p*(Q(n),O(n))maps
H~(Bo(n),Z 2) isomorphically onto S(Ul,...,u n)
and maps w i onto a i. Proof.
Consider the spectral sequence of the fibering (BQ(n),Bo(n),Fn): E 2 = H*(Bo(n) ,~*(FnZ 2)), E~ ,0 = HI(B0(n),Z2) ,
E~ 'I = Hl(Fn,Z2 )f,
E| : GrH*(BQ(n),Z2). Then dim IE 2 > dim IE| = dim H I (BQ(n),Z 2) : n; _
,
dim IE 2 = dim E~ '0 + dim E~ 'I, 1 + dim HI(Fn,Z2 )f , l+dim HI(Fn,Z2 ) ~ 1 + (n-l) = n. Therefore dim IE 2 = n and ' n = 1 + dim HI(Fn,Z2 )f ~ 1 + dim HI(Fn, Z 2) = n,
81
from which it follows that HI(Fn,Z2 )f = HI(Fn,Z2). Thus E~ 'I = E O'l~ so that all the elements of E~ 'l z Hl(Fn,Z2 ) are permanent cocycles and ~ ( F n , Z 2) is simple in dimension 1. In view of proposition 22.6 it follows that ~ ( F n , Z 2) is simple and H~(Fn,Z2 ) consists of permanent cocycles. Then E 2 = H~(Bo(n),Z 2) e H~(Fn,Z2), d r ~ 0 for r ~ 2, and so E 2 ~ E|
By theorems 14.1 and 14.2(a) we see that ~(Q(n),O(n))
is inJective. By 14.2(c) we have P2(Bo(n),t).P2(Fn,t)
= P2(BQ(n),t),
and hence P2(Bo(n) 't) "
(1-t2).--(1-t n) _ ! )n-1 n ' (l-t (l-t)
P2(Bo(n),t ) :
1 (1-t)(l-t2)...(l-t n)
But this is precisely P2(S(Ul,...,Un),t)
which shows that p~(Q(n),O(n)) maps
Ha(Bo(n),Z 2) onto S(Ul,...,Un). The proof of the second statement is analogous to the proof of proposition 21.6 and we leave it to the reader. Corollar F 22.8. (a)
H~(Fn,Z 2) = Z2[Ul,...,Un]/(S+(Ul,...,Un)).
(b)
H~(O(n)/Q(n),Z 2) = H~(SO(n)/SQ(n),Z 2) is equal to its characteristic ring.
Proposition 22.9.
~(SO(n),O(n))
is surJective and has (w I) as kernel.
Proof. Consider the fibering (Bo(n),Bo(n)/SO(n),Bso(n),~). O(n)/SO(n)
Note that
= Z 2. We have E 2 = H~(Bz2,~(Bso(n),Z2)) , E| = GrH~(Bo(n),Z2).
We also have P2(Bz2,t)
= l/l-t,
e2(Bo(n),t) Pa(Bso(n),t)
= 1/(l-t)(1-t2)...(1-tn), = i/(l-t2)...(l-tn);
the last formula is a consequence of proposition 22.4. Therefore P2(Bz2,t)
9 P2(Bso(n),t)
= P2(Bo(n),t),
and applying theorem 14.4 we have that ~@(Bso(n),Z2) case III in section 17
is simple and i~ is surJective. By
we may identify i~ with p~(SO(n),O(n)).
It remains to determine
82
the kernel. By 14.2(b) we have that i
identifies
H~(B0(n),Z2)/(~H~(Bz2,Z2))
: H~(Bso(n),Z2), =
Z2[w2,...,Wn].
By an argument on dimension the left side is then readily seen to Remarks.
be Z2[Wl,...,Wn]l(Wl~
(a) In H*(BQ(n),Z 2) the Steenrod squares are given by
Sq~ i = u l ,
sqlui
and by Cartan's product formula.
= u~
It follows that
2 ..uj Sqi(aj(Ul,...,Un )) : ~ u~'''UlUi+ I" where the right side denotes the symmetric function with the given summand as a typical term. Thus the determination
of Sqiwj is reduced to the problem of expressing
this symmetric function as a polynomial
in the elementary
symmetric functions with
coefficients mod 2. We shall not give the solution, due to Wu Wen Tsgn
[8](see also
[1]), but shall merely derive a formula to be used later. We have sql(~j(Ul,...,Un )) : ~ u~u2.., uj : ~l-aj-(J+l)~j+l; therefore sqlwj = WlW j + (J+l)wj+ 1. In H*(Bso(n),Z2)
we then have sqlwj = (J+l)Wj+l,
(b)
The identification
(2 i J ~ n).
of wj with elementary
derive a duality formula from the identity 23.
Orthogonal groups~
symmetric functions allows us to
(21.9).
cohomology mod p # 2.
For n = 2m and n = 2m+l respectively
a maximal torus T m in S0(n) is given by the
diagonal matrices
where
10Dn
lllOID cos 2~x i
sin 2~xil
-sin 2wx i
cos 2~xi/
Di =
The x i are coordinates
in the universal covering of T m, and the preimage of the unit
element of T m is represented by points with integral coordinates. the x i with a basis of Hl(BTm,Z).
Hence we may identify
The normalizer of T m contains the permutations
Di; we may also replace Di by ADIA-1 where A =
(0 1),
of the
thereby changing xi into -xi .
83
Since A has d e t e r m i n a n t transformations.
For n odd we can always obtain
entry in the matrix. easily
seen,
These t r a n s f o r m a t i o n s
so the Weyl group W(SO(2m+l))
an a r b i t r a r y modulo
equal to -1, for n = 2m we must make an even number of such
n u m b e r of sign changes,
and a n a l o g o u s l y
x i c o m b i n e d with an a r b i t r a r y transgression;
least one odd exponent
T TM is also a m a x i m a l
torus
in O(n),
the group of p e r m u t a t i o n s
in sign.
of the
Let Yi be images of the x i by
by the elements
(1 ~ i ~ m - l ) ,
under W(SO(2m))
then it is d i v i s i b l e
23.1.
H*(SO(2m+l),Zp)
(b)
H~(SO(2m),Zp)
:
to show that
a monomial
by yl-..y m. This
and even numbers
For p ~ 2, SO(n)
(a)
sequence
of the x i m o d u l o
: Io(2m+l ) : I o ( 2 m ) : S ( y ~ , . . . , y ~ ) ,
is invarlant
of W(SO(2m))
Proposition
Proof.
as may be
is the group of p e r m u t a t i o n s
y l . . - y m. To prove the latter it suffices
P(yl,...,ym ) which
the o p e r a t i o n s
is the group of p e r m u t a t i o n s
n u m b e r of changes
~i(y~,...,y~),
nomial
the full n o r m a l i z e r
we see that W ( O ( n ) ) i s
is the r i n g g e n e r a t e d
and the product
by u s i n g -1 as first
then Iso(2m+l)
and Iso(2m)
generate
whereas W(SO(2m))
an even number of sign changes.
(n = 2m, 2m+l),
+l as d e t e r m i n a n t
if in a poly-
occurs w h i c h has at
can be easily
shown u s i n g
of sign changes.
has no p-torslon,
and
A(x3,xT,...,X4m_l),
= A(x3,x?, 9 9 9 X4m_5,X2m_l) 9
(a) may be p r o v e d r e a d i l y by i n d u c t i o n m a k i n g use of the spectral
of the f i b e r i n g SO(2m+l)/SO(2m-1)
= V2m+l,2
,
and 22.1(b). (b)
Consider
the spectral
sequence
SO(2m)/SO(2m-1)
= S2m_l
of the f i b e r l n g ;
we have E 2 = A(X2m_l)
~ A(x3,x7,...,X4m_5),
and it suffices to show that E 2 = E . Since the only n o n - t r i v i a l 2m-l,
only d 2 m _ l may not be i d e n t i c a l l y
other hand the x i are u n i v e r s a l l y whence d2m_l
= O, E 2 = E |
transgressive
shows f u r t h e r that SO(n)
= Po(SO(n),t), has no p-torsion.
(p ~ 2),
are 0 and
= E 2 and E2m = E . On the
and have odd degrees;
Since
Pp(SO(n),t) this
zero, and hence E2m_l
base degrees
hence d2m_iXi
= O,
84
T h e o r e m 23.2.
If p ~ 2 then
(a)
p~(Tm,S0(2m+l))
(b)
p*(Tm,S0(2m))
by S(y~,...,y~) (c)
maps H~(B
+ ,Zp) isomorphically SO(2m l)
maps H~(Bs0(2m),Zp)
isomorphlcally
onto S(y~
'"
..,y~)
'
onto the ring generated
and the product yl...ym .
p~(Tm,o(n)) maps H * ( B 0 ( n ) , Z p 0 i S o m o r p h i c a l l y
onto S(y~,...,y~)
for n = 2m,
2m+l. The theorem is a consequence
of 18.1, 2o.3, the remark following 2o.4, 23.1, and
the initial remarks of this section. Corollary 23.3. have no 24.
If p ~ 2 then ~ ( S O ( n ) , 0 ( n ) )
is inJective; BS0(n ) and B0(n)
p-torsion.
Integral cohomology of B0(n) and BS0(n ). Let X be a space with finitely generated integral cohomology groups. By the
universal coefficient theorem Hi(X,Z2 ) = HI(X,Z) Assume that the 2-primary component
~ Z 2 + Tor(HI+I(x,z),z2 ).
of Hi(x,z)
is a direct sum of ql cyclic groups,
and
that the dimension of Hi(X,Zo ) is Pi; then P2 (X't) : ~ (Pi + qi + qi+l )'ti : Po (X't) + (i + l/t)
X qi "ti"
As is well known, Sq I is the Bockstein h o m o m o r p h i s m attached to the exact sequence 0 --~ Z --* Z --* Z 2 --+ 0 followed by reduction Lemma 24.1.
mod 2. From this one deduces readily:
If X has finitely generated integral cohomology and A is the graded
subspace SqI(H~(X,Z2) ) then the 2-primary component
of H~(X,Z)
consists only of elements
of order 2 if and only if P2(X,t) Lemma 24.2.
- Po(X,t)
= (1 + l/t) P(A,t).
If X has finitely generated integral cohomology groups whose torsion
subgroups are direct sums of cyclic groups of order 2 then an element x ~ H~(X,Z)
is
completely determined by its images x 0 and x 2 in H~(X,Z o) and H~(X,Z 2) respectively. Proof.
We have to show that if x 0 = x 2 = 0 then x = O. If x 0 = 0 then clearly x
is a torsion element and therefore has order 2. But then x 2 = 0 implies x = O. We now consider a commutative graded algebra of finite type with H 0 = K. Let D be a derivation of H of degree +l and such that D 2 = 0, (for example, We introduce the following notation: A :
graded subspace stable under D,
Sq I on ~ X , Z 2 ) ) .
85
NA :
the kernel of D in A,
MA :
a supplementary
IA :
the image of A under D,
JA :
a supplementary
subspace to NA,
subspace to I A (in A).
Now let B be a second subspace stable under D and linearly disjoint K--i.e., the map a 9 b § a.b of A e B § H is inJectlve.
from A over
We denote the image of this map
by A.B. We propose to show that (24.3)
P(JA.B,t)
= P(JA,t).P(JB,t).
Since A = M A + I A + JA' and since D is an isomorphism of M A onto IA, we have P(A,t)
= (1 + 1/t)P(IA,t)
+ P(JA,t),
P(B,t)
= (i + I/t)P(IB,t)
+ P(JB,t),
and analogously, (24.4)
P(A.B,t)
= (i + I/t)P(IA.B,t)
+ P(JA.B,t).
Since A and B are linearly disjoint we have P(A.B,t)
= P(A,t).P(B,t),
from which we obtain (24.5)
\
P(A.B,t)
= (1 + I/t) 2 P(IA,t)P(IB,t) + (1 + I/t)(P(IA,t)P(JB,t) +
P(JA,t)
+ P(JA,t)P(IB,t))
P(JB,t).
On the other hand the image of D is spanned by IA(I B + JB ), IB(I A + JA ), and D(M A - MB). On MA-M B the d e r i v a t i o n D is clearly inJectlve; P(D(MA.MB),t)
hence
: tP(MA,t)P(MB,t) , : i/t P(IA,t)P(IB,t).
Making use of this one sees readily that
(l+i/t)
P(D(A.B),t)
is equal to the sum of the
first three terms on the right side of 24.5; hence 24.3 follows from 24.4. Theorem 24.6. Proof.
The torsion elements of H * ( B s o ( n ) , Z ) a r e
of order 2.
We have seen that BSO(n ) has no p-torsion for p ~ 2, and that H~(Bs0(n ), Z 2)
~ Z2[w2,...,Wnl
where sqlwl = (i-l)wi+ I. Therefore H~(Bso(2m+I),Z2)
= A 1 e-..e Am
where A i = Z2[w21,w21+l I and AI,...,A m are stable under Sq I and the cupproduct; H* (Bso(2m) ,Z 2) = A I e . . . e
Am
and
86
where AI,...,Am_ 1 are as before and A m = Z2[W2m ~ is annihilated by Sq I. In A i the image of Sq I is spanned by the elements w2i.w2i+l , st
(s _> O, s even, t > O ) : c o n s e q u e n t l y
we may
s t take the space spanned by the elements w21.w2i+l , (s odd , t > O) , as MAi . Similarly for jA i we may take the space spanned ~ by w21 s , (s even) P(
,t) : ~ (l-t4i)-l' JAi
Applying
L P(JAm't)
It follows that
if i ~ m or n ~ 2m,
= P(Am't)
= (1-t2m)-l'
if i = m, n = 2m.
(24.3) to m factors we get for n = 2m+l, P2(Bso(n),t)
: (I + l/t) P(SqI(H*(Bs0(n),Z2),t)
+ N~=l(l-t41)-l;
P2(Bs0(n),t)
: (i + l/t) P(Sql(H*(Bso(n),Z2),t)+(l-t)2m-ln~=l(1-t4i)-i
and for n : 2m,
Since in both cases the last term is Po(Bso(n),t)
by 23.2, the theorem follows from
lemma 24.1. Theorem 24. 7 . Proof.
The torsion elements of H~(Bo(n),Z)
are of order 2.
We have H*(B0(n),Z2)
= Z2[Wl,...,Wn] ,
sqlw i = WlWi + ( i - 1 ) w i + 1. We choose a new basis w I : Wl,W~i = w2i, w2i+ I = w2i+l + w21.Wl; then H~(B0(n),Z 2) -- Z 2[w~ .... ,Wn*],
sqlw . sqlw~
sql i§
2,
i- w2i+l'
: sqlw21§
§ w .sqlw21 § Wl(WlW21§ w21§
: 0
This gives H~(Bo(2m+I),Z 2) : A 0 ~ A I ~.-.~ A m where A 0 : Z2[w ~] , A i = Z2[w~i ,w2i+l ] , and Ao,...,A m are stable under Sq I ; and H~(B0(2m),Z 2) : A 0 , A 1 ,--. ~ Am_ 1 where A 0 = Z2[w~ * ] " A i = Z2[w~i ,w21+l * ] L 1 'w 2ml and AO,...,Am_ 1 are stable under Sq 1. As in the preceding theorem we have
87
P(JAi,t)
: (l-t4i) -I
(i > O)
For n odd it is clear that the elements
of strictly positive
of Sq I are W~s and that they span Sq I(Ao) ; hence P(JA,t)
degrees
in A O in the kernel
: i. We assert that this holds
also in case n is even. We have 1 s t Sq (WlW2m)
. . . . s+l t = k s ~ j w I W2m
which is zero if and only if s+t is even. But then for s > 0 it is equal to sql(w~-lw~m), get P(JAo,t)
and for s = 0 and t even, t > O, it is equal to sql(w~m I) from which we = i. Now the remainder
Corollary consists
24.8.
of the integral
This follows 25.
The kernel of Sq I in H~(Bso(n),Z2) , (and in H~(BO(n),Z2)) , cohomology reduced mod 2.
from 24.1,
Stiefel-Whltney
of the proof is the same as the proof in 24.6.
classes,
24.6,
24.7.
PontrJagin
classes.
In view of 24.2 and 24.8 we see that there is a unique element H21+I(Bso(n),Z),
(or in H2i+I(B0(n),Z)),
by W2i+ 1 9 Similarly W 2 is the element w~,
(for n ~ 3). W 2 and W2i+l
in the indicated
Stiefel-Whltney
classes;
coefficients
whose reduction mod 2 is w21+l ; we denote
of order 2 in H2(B0(n),Z)
(in the case of SO(n))
Whitney classes
of order 2 in
dimensions.
it
whose reduction mod 2 is
are the universal
integral Stiefel-
(In the case of O(n) we do not get the
these are defined with respect to "twisted"
integral
- we shall not discuss them.)
The element W2m which is the image by transgression H2m-l(S2m_l,Z)
of a generator
of
in the flberlng (Bso(2m-I),Bso(2m),S2m-I)
is called the universal
Euler-Poincarg
map of the unit tangent bundle r where
•
class.
We recall that if r is the classifying
of a differentiable
manifold B of dimension
: x(B).F
is the Euler characteristic
also called the 2m-Stiefel-Whitney mod 2 is the 2m-Stiefel-Whitney
and F the fundamental
class of B. r
class and in fact, by definition,
class mod 2. However,
map
classes of a bundle
of infinite
classes.
The
are defined as usual by means of the characteristic
of the bundle. Proposition
Then
is
its reduction
it is an element
order and not of order 2 as are the other integral Stlefel-Whitney Stiefel-Whitney
2m then
25.1.
Let T TM be the maximal torus of SO(2m)
described
in section
25.
88
p~(Tm,So(2m))(W2m ) = yl...y m. Proof.
Consider
the inclusion
U(m) ~ and i induces
SO(2m-l)
= U(m)/U(m-l)
BU(m-I)
BSO(2m)
BU(m)
~
all the maps are the c o r r e s p o n d i n g
p(U,G) maps.
(Bu(m_l),Bu(m),S2m_l)
H2m-l(S2m_l,Z)--in
that
Then
= SO(2m)/SO(2m-l).
~ defines
torus
a homomorphlsm
§ (Bso(2m_I),Bs0(2m),S2m_ l)
8~(W2m ) is the image by t r a n s g r e s s i o n
other words
T m is also a maximal
onto S2m_l
diagram
BSO(2m-l) "~
from which it follows
We have
= U(m-l),
the identity map of S2m_l
We also have a commutative
where
i : U(m) C SO(2m).
~ ( W 2 m ) = C2m by definition
of a generator
of the Chern classes.
Now
of U(m); hence
p~(Tm,So(2m))
= p*(Tm,u(m)).p~(U(m),SO(2m)), = ~ ( T m , u ( m ) ) . S ~,
and the p r o p o s i t i o n In general highest
follows
the integral
dimensional
25.2.
be their
then
Stiefel-Whitney
Let
(i = 1,2), be bundles
nl
one of the n i is odd,
from the identifications
anl and On2. Finally p*(SO(nl)
Let T be a m a x i m a l of SO(nl) particular
their respective
of section
torus
suppose
to consider rational
classes
(yl)
to be proved holds
coefficients.
say n I. Then the rational
reductions
If n is of both
If n I and n 2 are both even then the
of Wnl and Wn2 with the elementary
both n i are odd,
x SO(n2) ' SO(n))
of SO(n)
22 the formula
(Wn)
it is sufficient
symmetric
to prove
: O.
chosen as in section
25, and let T' be a maximal
x SO(n2) " We may choose T' so that T' C T with x I : O, p~(T',T)
and let
n2
W n and W n(I) , are zero so that the formula holds.
functions
but the
: w(1).w (2)
In view of the final remark
odd then n e c e s s a r i l y
follows
do not obey duality,
If W~I),w~ 2), and Wj denote
when reduced mod 2. By 24.2 it thus suffices
formula
21
classes
(E(1),B,SO(nl)),
sum bundle.
Wn Proof.
in section
ones do.
Proposition (E,B,SO(n))
from results
: 0 and hence
p (T',T)(Yl,...,y n) : ~ ( T ' , S O ( n ) ) ( W
n) : O.
(x i arbitrary).
torus
Then in
89
But p~(T',SO(n)) and p~(T',SO(nl)
: p~(T',SO(n l)
x SO(n2)).p~(SO(nl) ~O(n2),SO(n))
• SO(n2) ) is injective over the rationals,
so the assertion is proved.
As an application of 25.2 we cite the following theorem due to H. Samelson Proposition 25.5.
[6]:
If M is a compact orientable manifold of dimension 2m which
has a continuous field of odd dimensional plane elements then x(M) = O. The assumptions imply that the tangent bundle to M is a Whitney sum bundle of two bundles with groups
SO(2mi+l),
(i = 1,2), ml+m2+l = m. Therefore by 25.2, the 2m-
dimensional Stiefel-Whitney class of M is a product of the corresponding highest dimensional classes of the two bundles and hence must be an element of order 2. But as noted earlier it is also equal to • order we must have •
and since the fundamental class F has infinite
= O.
Let SO(n) C O(n) C U(n) be the natural inclusions. The image of the Chern class C2i under p~(SO(n),U(n)),
(or p~(O(n),U(n))), prefixed by (-1) i/2 when i is even, is
called the PontrJa~in class in dimension 2i; We denote it by P2i" Usually one considers only the classes P4i : the classes P4i+2 which we introduce here for convenience are elements of order 2 as we shall see. Let T n be a maximal torus in U(n) and Vl,...,v n a basis of H2(BTn,Z) as described in section 22. The maximal tori T m of SO(n) and O(n) have dimension m = [n/2]. We may without essential change replace O(n) by a conjugate subgroup in U(n), and it is convenient to choose it so that the corresponding tori T TM are now given by diagonal matrices of the form: Q
2~Ix I
i e2~ixl
e-2~ix 1 o
\
1(
0 'e2~iXm J '~ e-2~ixm \ ,. \
for n even and odd respectively.
e-2,~Xl "e2,,ixm e-2~iXm /, /
If we denote by yl,...,y m the basis of H2(BTm,Z) where
Yi is obtained from x i by transgression then p~(Tm,Tn)(vl ) = O, (n odd), P~(Tm'Tn)(v2i ) = -P~(Tm'Tn)(v2i+l ) = Yi' (n odd), P~(Tm'Tn)(v2i-1 ) = -P~(Tm'Tn)(v2i)
= Yi' (1 ~ i S m),
(n even).
9o
Therefore p~(Tm,Tn)~i(Vl,...,v n) = O, (i odd), p*(Tm,Tn)a2i(Vl,...,Vn ) : (-1)iai(y~,...,y~). Now making use of the identities p~(Tm,T n) p~(Tn,u(n))
= p~(Tm,0(n))
p~(Tm,T n) ~ ( T n , u ( n ) )
: p*(Tm,s0(n))
p*(0(n),U(n)), p~(SO(n),U(n)),
we obtain the following: Proposition
25.4.
With the above notations and for integer coefficients we have
p*(Tm,o(n))(P4i+2 ) : O, p~(Tm,o(n))(P41)
: ~i(y~,...,y~).
Clearly, the same formulas will hold for the reduction mod p of the PontrJagin classes, and will characterise
them for p ~ 2. Before considering the PontrJagin classes
reduced mod 2 we make the following observations.
If T n is a torus and Q(n) its sub-
group consisting of elements of order 2 then we recall H~(BTn,Z 2) = Z2[vl,...,Vnl, H*(BQ(n),Z 2) : Zm[Ul,...,Un] ,
(d~
= 2),
(d~
: i).
We now claim that (25.5)
p*(Q(n),Tn)(vi ) : u~ ,
(for the natural bases chosen as in sections 21 and 22). It clearly suffices to consider the case n = i. Then p~(Q(1),T I) is the projection map in the flbering (BQ(1),BTI,TI/Q(1))
whose fibre is S I. In the spectral sequence the term E 2 : Z2[Vl] | H*(SI,Z 2)
has the same Poincarg polynomial
as E| and hence E| = E 2. From this (25.5) follows
readily. Proposition
25.6. 2
(a) (b)
P4i+2 = (W2i+l) w~ = P2i
Proof.
9
mod 2 .
It is sufficient to prove the proposition in the case of O(n) -- it will
then follow for SO(n) upon application of p*(S0(n),0(n)). (disregarding
signs since we compute mod 2),
From the definition of P2i'
91
p~(Q(n),O(n))(P2i)
: p~(Q(n),U(n))(C2i) , = p~(Q(n),T n) p*(Tn,u(n))(C2i), = p~(Q(n),T n) Ol(Vl,...,Vn), = ~i(u~,...,u~), = o~(Q(n),O(n))(w~),
the last three steps following by 21.6, 25.5, and 22.7 respectively. Since p~(Q(n),O(n)) is inJective,
(as noted in the proof of 22.7), (b) follows. Moreover we
see that P4i+2 and W 22i+l are equal when reduced mod 2. From 25.4 it follows that the rational reduction of P4i+2 is zero and hence equal to the rational reduction of 2 (W2i+l) . Thus (a) follows because of 24.2. Remarks.
(a)
In view of the results of section 24 the integral Pontrjagin
classes are completely characterized by propositions 25.4 and 25.6. (b) The integral classes P4i by themselves do not obey duality (since obviously 2 they do their mod 2 reductions do not); however taken together with the classes W2i+l By 24.2 it suffices to prove this over the integers mod 2 and the rationals.
In both
cases the classes may be represented by elementary symmetric functions in the squares of certain variables; the duality relations then follow from an identity analogous to 21.9. (c) Let G be a connected compact Lie group. The Weyl group W(G) in operating on the fibering (BT,BG,G/T) of course acts trivially on BG; hence ~ ( T , G )
C I G. We recall
from section 2o. that the kernel of p~(T,G) consists of the torsion subgroup of H*(BG,Z). For the case G : U(n) the image of p~(T,G) is IG, and in fact the same is true for any group without torsion. The results of section 24 and propositions 25.1 and 25.4 show that it is also true for G = SO(n) and G = O(n). It is not known whether it is true in general. (d) The integral Stiefel-Whitney and PontrJagin classes may also be defined by transgression.
Consider a bundle (E,B,SO(n)ISO(m)), m = 2s-1 .
Then,
(see [2] section lo), the lowest dimensional non-vanlshing groups of the fibre are H~
H2S(Vn,n_m,Z)
: H4S-l(Vn,n_m,Z)
: Z
= Z2
Then W2s+l is the image by transgression of the generator of H2S(Vn,n_m,Z). A generator x E H4S-I is not in general transgressive; but 2x is always transgressive and its image is then the PontrJagin class.
92
BIBLIOGRAPHY I11
A. Borel, La cohomologie mod 2 de certains espaces homog~nes, (1953), 165-197.
[2]
A. Borel, Ann. of Math. 57 (1953), 115-2o7.
[3]
A. Borel, J.P. Serre, Am. J. Math. 75(1953),
[4]
S.S. Chern, On the characteristic classes of complex sphere bundles and algebraic varieties, Amer. J. Math. 75 (1953), 565-597,
[5]
L. PontrJagln, Characteristic 21, 63 (1947), 233-284.
[6]
H. Samelson, A theorem on dlfferentiable manifolds,
[7]
N. Steenrod, The topology of fibre bundles, Princeton U. Press (1951).
~
Comm. Math. Helv. 27
4o9-448.
cycles on differentiable manifolds, Mat. Sbornik N.S. Port. Math. lo (1951), 129-133.
Wu Wen TsGn, Les i-carr~s dans une vari~t~ grassmannienne, (195o), 918-92o.
C.R.Acad.Scl.
Paris 230
93
BIBLIOGRAPHICAL
NOTES AND CO~4ENTS
(added in 1967)
CHAPTER I The standard reference
for Hopf algebras
is now
[12].
CHAPTER II There are now a number of expositions for a treatment
geared to the discussion
of spectral
sequences,
of fibre bundles,
see e.g.
[4], or,
[7].
CHAPTER III For a more general construction transgression in reference berg-Moore
Theorem 18.1,
stated without proof here,
spectral
sequence
we refer to
[lo].The
can be proved more easily than
context.
[13]. See also an announcement[14]
where a similar spectral
Strictly
speaking,
sequence
and forthcoming papers
is introduced
in a more
these results do not include the purely algebraic
theorem of ~l}, but they cover the topological as well.
bundles,
[1] of Chapter III, to be quoted hereafter by {1}, by use of the Ellen-
by Rothenberg-Steenrod, geometric
of universal
In w 19, we have proved a special
applications,
and yield further results
case of 18.1 by a method similar to the
method of {1}, in order to give an idea of the general proof.
However,
this special
case can be handled more simply by means of the comparison theorem of spectral sequences
[16, 17]. The basic tool in the proof of Theorems
of a set of elements the terminology
in a ring which are "annlhilator-free"
of {1}), suggested
similar concept was introduced name of E-sequence,
in 1955 by Serre in the theory of local rings,
and this terminology has now become customary
to 19.2. For this, see
theorems mentioned
group G and a prime p : (1) (2)
no p-torsion.
In [1] it is checked,
A
under the
in homological
algebra.
If B G has no p-torsion
are also given.
in w 18 imply for a compact
If G has no p-torsion,
no p-torsion;
in
and the starting point is a lemma quite similar
[15], where further references
The transgression
assumption
(or "without relations"
in part by earlier work of Leray and of Koszul.
It occurs notably in dimension theory,
has no p-torsion.
18.1 and 19.1 is the notion
then its classifying
and H=(BG,K p) is a polynomial
connected Lie space B G has ring, then G has
case by case, that if B G has no p-torsion,
To prove this a priori,
using
then G
(2), one needs to know that the first
of (2) implies the second one. This implication
can be deduced from a result
94
of Serre's,
which states that if a polynomial
over a sub-algebra, that
algebra is a free module
then the latter is itself a polynomial
algebra.
B G has no p-torsion and let T be a maximal torus of G. Then,
facts about B G and G/T, the E2-term of the spectral flbering
(BT,BG,G/T)
E 2 = E| = H~(BG,Kp)
finite rank over H*(BG,Kp).
In fact, assume in view of known
sequence in cohomology mod p of the
contains non-zero terms only in even degrees, e H~(G/T,Kp)
of finite rank
hence
which implies that H~(BT,K p) is a free module of
Since the former is a polynomial
algebra,
Serre's result
implies that the same is true for H~(BG,Kp). The p's for which G has no p-torsion have been completely determined leads to interesting relationships partly
checked,
between various properties
partly proved a priori
Bott's paper
(see
[1],
and this
of G, which have been
[61; see also Theorem 4.2 of
[31 gives the proofs of the results announced
[4]).
in reference
[4] of
Chapter III. CHAPTER IV The characteristic
classes for the classical
framework of the cohomology
of classifying
This approach is convenient
to compute characteristic
bundles,
or of bundles
presupposes direct
associated
to bundles
a certain amount of machinery.
study of Grassmannians,
spaces,
groups are introduced here in the invariants
For a more geometric
description
over in algebraic geometry by Grothendieck
classes of homogeneous
of frames on a manifold
we refer to Milnor's
have also been given an axiomatic
of the Weyl group,
definitions
Appendix I]. Much of w16724, 25 has been incorporated
vector
[2]), but based on a
[ll I. Chern classes
[8], which has been carried
[51 . An exposition
in [9]. For a comparison between the different
treatment,
Lecture Notes
by Hirzebruch
(see
etc.
along those lines is given
of Chern classes,
in [2, Appendix
II].
see
[2,
95
REFERENCES [These are the references for the Appendix. Each chapter carries its own bibliography.] [i]
A. Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts \ connexes, Tohoku Math. J. 13 (1961), 216-24o.
[2]
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. Jour. Math. 8o (1958), 459-538, If, ibid. 81 (1959), 315-382, III, ibid. 82 (196o), 491-5o4.
[3]
R. Bott, An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France 84 (1956), 251-282.
[4]
H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, 1956.
[5]
A. Grothendieck, La th6orie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137-154.
[6]
B. Harris, Torsion in Lie groups and related spaces, Topology 5 (1966), 347-354.
[7]
P. J. Hilton and S. Wylie, Homology Theory, Cambridge University Press, Cambridge, 196o.
[8]
F. Hirzebruch, Topological Methods in Algebraic Geometry, 3rd ed., SpringerVerlag, Berlin, 1966.
[9]
D. Husemoller,
Fibre Bundles, McGraw-Hill, New York, 1966.
[l~
J. Milnor, Construction of universal bundles I, II, Annals of Math. (2) 63 (1956), 272, 43o-456.
[l~
J. Milnor, Lectures on characteristic classes, Princeton University 1967, mimeographed. (Notes by J. Stasheff.)
[I~
J. Milnor and J. C. Moore, On the structure of Hopf algebras, Annals of Math. (2) 81 (1965), 211-264.
[l~
J. Moore, Alg~bre homologique et homologie des espaces classifiants, S6m.E.N.S. 1959, Expos@ 7.
[14]
M. Rothenberg and N. Steenrod, The cohomology of classifying spaces of H-spaces, Bull. Amer. Math. Soc. 71 (1965), 872-875.
[15]
J. P. Serre, Alg~bre locale. Multiplicitgs, (rgdig@ par P. Gabriel), Lecture Notes in Mathematics ll (1965), Springer.
[i~
E. C. Zeeman, A proof of the comparison theorem for spectral sequences, Proc. Cambridge Phil. Soc. 53 (1957), 57-62.
[i~
E. C. Zeeman, A note on a theorem of Armand Borel, ibid. 396-398.
