Cohomology Operations (AM-50), Volume 50: Lectures by N.E. Steenrod. (AM-50) 9781400881673

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Annals of Mathematics Studies Number 50

COHOMOLOGY OPERATIONS LECTURES BY

N. E. Steenrod W RITTEN AND REVISED BY

D. B. A. Epstein

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1962

Copyright © 1962, by Princeton University Press All Rights Reserved L.C. Card 62-19961 ISBN 0-691-07924-2 Fourth Printing, 1974 Sixth Printing, 1996

Printed in the United States of America

PREFACE

Speaking roughly, cohomology operations are algebraic operations on the cohomology groups of spaces which commute with the homomorphisms in­ duced by continuous mappings.

They are used to decide questions about the

existence of continuous mappings which cannot be settled

by examining

cohomology groups alone. For example, the extension problem is basic in topology. X,Y,

a subspace A C X,

and a mapping

problem is to decide whether

h

h:

A -- > Y

If spaces

are given, then the

is extendable to a mapping

f : X -— > Y.

The problem can be represented by the diagram X^ \ f

fg

-

h

"A

A ----------- > Y h where

g

is the inclusion mapping.

Passing to cohomology yields an alge­

braic problem H* (X)

/

^

g/

g q> = h

H* (A)
2 ),

and applications of these.

contains a proof that the squares and

Chapter VI contains Chapter VIII

p"^ powers are characterized by some

of the axioms assumed in I and VI. The method of constructing the reduced powers, given in VII, is new and, we believe, more perspicuous.

The derivation of Adem's relations in

VIII is considerably simpler than the published version. proof of VIII is also simpler.

The uniqueness

In spite of these improvements, the con­

struction of the reduced powers and proofs of properties constitute a lengthy and heavy piece of work.

For this reason, we have adopted the axio­

matic approach so that the reader will arrive quickly at the easier and more interesting parts. The appendix, due to Epstein, presents purely algebraic proofs of propositions whose proofs, in the text, are mixed algebraic and geometric. The reader should regard these lectures as an introduction to co­ homology operations.

There are a number of important topics which we have

not included and which the reader might well study next.

First, there is

an alternate approach to cohomology operations based on the complexes K(ir,n)

of Eilenberg-MacLane [Ann. of Math., 58 (1953), 5 5 -1 0 6 ; 60 (19511 ),

^9-139; 60 0 95*0, 51 3-5553 . This approach has been developed extensively by H. Cartan [Seminar i95*+/55h

A very important application of the squares

has been made by J. F. Adams to the computation of the stable homotopy groups of spheres [Comment. Math. Helv. 32 (1 9 5 8 ), 1 8 0 -2 1 U]. do not consider secondary cohomology operations.

Finally, we

J. F. Adams has used these

most successfully in settling the question of existence of mappings of spheres of Hopf invariant 1 [Ann. -of Math., 7? (i960 ), 20-ioU; and Seminar, H. Cartan 1958/59]. Princeton, New Jersey, May, 1 9 6 2 .

N. E. S. D. B. A. E.

CONTENTS

P RE FA CE............................................................

v

CHAPTER I.

Axiomatic Development of the Steenrod Algebra

i

CHAPTER II.

The Dual of the Algebra

Gt(2). . .

&(2)......................... 16

CHARTER III. Embeddings of Spaces in Spheres....................... 30 CHAPTER IV.

The Cohomology of Classical Groups and StiefelManifolds.

CHAPTER V.

Equivariant Cohomology................................ 58

CHAPTER VI.

Axiomatic Development of the Algebra

CHAPTER VII.

Construction of the Reduced Powers..................... 97

37

Cb(p)........... 76

CHAPTER VIII. Relations of Adem and the Uniqueness Theorem.......... 1 1 5 APPENDIX:

Algebraic Derivations of Certain Properties of the Steenrod Algebra

13 3

COHOMOLOGY OPERATIONS

CHAPTER I.

Axiomatic Development of the Steenrod Algebra

In § i, axioms are given for Steenrod squares.

( S t (2)

(The existence and

uniqueness theorems are postponed to the final chapters.)

In § 2 , the ef­

fect of squares In projective spaces is discussed, and it is proved that any suspension of a Hopf map is essential. Q(2)

Is defined and the vector space basis of Adem h] and Cartan [2 ] is

obtained. algebra

In §1, it is shown that the indecomposable elements of the 2i 0t(2) are represented by elements of the form Sq . Some geo­

metric applications of this fact are given. maps

In §3, the algebra of the squares

S2n_1 -- > Sn

Hopf invariant when

is defined. n

In § 5 , the Hopf invariant of

The existence theorem for maps of even

is even, and some non-existence theorems, are given.

Unless otherwise stated, all homology and cohomology groups In this chapter will have coefficients

Z-2 .

§ 1 . Axioms. We now give axioms for the squares

Sq1 . The existence and unique­

ness theorems will be postponed to the final chapters. 1) For all integers

i > 0

and

q > 0,

there is a natural transformation

of functors which is a homomorphism Sq1 : Hn(X,A)--- > Hn+1 (X,A) 2 ) Sq° =

,

n> 0.

1.

5 ) If

dim x

U) If

i > dim x,

=

n,

Sqnx Sq^-x

= x2. =

0.

5 ) Cartan formula

Z

k Sq"Sc . S q ^ y i=0

.

2

I.

AXIOMS FOR THE ADOEBRA

We recall that if

x e H^(X,A)

and

a ( 2)

y e H^-(X,B),

then

xy e

H^+^-(X,AuB) . This is true in general in simplicial cohomology, but some condition of niceness on the subspaces

A

and

B

is necessary in singu­

p

of the coefficient sequence

lar cohomology. 6)

Sq1

is the Bockstein homomorphism

0 — > z 2 ---- > 7)

Adem relations. If

Zk

o < a < 2b,

SqaSqb

— >

Z2

---- > 0 .

then

= Y ^

( taj'O

Sqa+b-jsq^ .

The binomial coefficient is, of course, taken mod 2 . The first five axioms imply the last two, as will be proved in the final chapter 1.1

LEMMA.

The following two forms of the Cartan formula are

equivalent in the presence of Axiom 1 ):

PROOF. jections.

Let

Sq1 (xy)

=

Sqjx . Sqi_Jy

Sqi(x x y)

=

Zj Sq^x x Sq1_^y

p:

X x Y -- > X

and

q:

.

X x Y -- > Y

be the pro­

If the first formula holds, then Sq1(x x y) = Sq^Cx x 1) . (1 x y)) = Zj SqJ'(x x l) . Sq1"^ (1 x y) = Zj Sq^ (p*x) _

*

i

. Sq1-,*(q*y) *

-? — i

= Z 5 P Sq x . q Sq1 Jy = Z^ (Sq^x x 1) . (1 x Sq^~^y) = Z^ Sq^x x S q ^ y Let

d:

X—

> X x X

be the diagonal.

. If the second formula

holds, then Sq1 (xy) =

Sq^d*(x x y)

=

d*Sq1(x x y)

- d*Z. Sq'Sc x Sq1" ^ 1 . 2 LEMMA.

If

S:

.

Axioms 1 ), 2 ) and 5) imply: H?(A)

> H?+ 1 (X,A) BSq1

PROOF.

= Zj SqSt . S q ^ y

We will show that

product with a 1 -dimensional class.

8

=

is the coboundary map, then Sqi 8 .

is essentially equivalent to a

x~

Then the Cartan formula applies to give

§2.

the desired result.

PROJECTIVE SPACES

(This method can be used for any cohomology operation

whose behaviour under x-products is known.) Let with

Y

be theunion of

(0} x A.

and let

X

and

B = [1 / 2, 1 ]xA

Let

A ’ = {1 } x A

and

I x A,

C Y

and

A" = B n Z.

with

A C X

identified

Z = X U [0,i/2]xA

C Y,

We then have the following commu­

tative diagram. H^-(A)

> H ^ d x A)

V

V

H^-+1 (X,A) — > ^

EfyA'uZ) ----- > H^A'uA")

> eP(A') Hq+1(X,A') 0.

dim Sq1 = i.

We write

be

We denote the gen­

a (2)

( a - 2 j 0 Sqa+]:)"^ 0 Sq^

is the quotient of

Sq° = 1

in

when

is called the length

a < 2b .

Ct( 2) .

Given a sequence of non-negative integers

by

M

by relations of the form Sqa ® Sqb -

I

Let

of

I.

We write

m(I) = Zg_! sig. A sequence k > s > 2> and Sq1

-

I

k = 4(1).

I = (i1,i2,...,i^),

We define the moment of

is called admissible if both

i^ > 1.

k

We write ... Sqlk.

8

I .

If

AXIOMS FOR THE ALGEBRA

Iis admissible, we call

sible.

admissible.

We shall also speak of the moment of 3.1.

for

Sq^

THEOREM.

Ct( 2)

We also call

Sq°

admis­

Sq1 .

The admissible monomials form a vector space basis

Ct(2). PROOF.

We first show that any inadmissible monomial is the sum of

monomials of smaller moment, and hence that the admissible monomials span Ct(2).

Let

For some

r,

I = (i1,..'. ,i^)be an inadmissible n = ip < 2ir+1 Sq1

where

e Zg.

=

and

2m.So, by the Adem relations,

SqNSqnSqmSqM

It is easy to

smaller moment than j = o

=

Sq1

sequence with no zeros.

=

Zj >.jSqNSqn+ln'^Sq;3SqM

verify that each monomial on the right has

(separate arguments are needed for the cases

j > o). By induction on the moment,

it follows that every

monomial is a sum of admissible monomials. We still need to show that the admissible monomials are linearly independent.

Let

P

be

is the polynomial ring Cartesian product of

P

oo-dimensional real projective space. Zgfu],

where

with

dim u = i.

itself. Let

Let

w = ux u

Pn

Then

H*(P)

be the n-fold

x u € H ^ P 11).

The

following proposition will complete the proof of 3 .1 . 3 .2 .

evaluation on

PROPOSITION. w,

The mapping

Ct(2 ) — -> H*(Pn ), defined by

sends the admissible monomials of degree

< n

into

linearly independent elements. PROOF.

The proposition is proved by induction on

n.

For

n = 1,

it follows from 2 .U. Suppose monomials aj = 0 4(1).

Sq1

for each

Z a^Sq^w

0 ,where thesum

=

of a fixed degree q, I.

Suppose that

is taken over admissible

where q < n. We wish to prove that

This is done by a decreasing induction on the length a^ = 0

for

4(1) > m.

The above relation takes the

form (1 )

^4(I)=m aiSclIw + ^4 (I)< m aq Sq W =

The Hinneth theorem asserts that H^+ n (Pn )

„

Zg HS(P) ® Hq"'n"s(Pn''1)

0

•

§3 . DEFINITIONS. Let

g

THE BASIS OF ADMISSIBLE MONOMIALS

w f € Hn~1 (Pn~1).

u x w*, where (2 )

Sq^w

where

J< I

=

1 0 2 ,2 )

case

^j 2. 1.5.

THEOREM.

If

H*(X;'Z2)

polynomial ring on a generator Vq = 2 for some k. PROOF.

x

is a polynomial ring or a truncated

of dimension

q,

and

x2 4 0 ,

then

Since

H*(X) is a polynomial ring, H^+2 (X) = 0 for i . k 0 < 2* < q. Therefore Sq2 x = 0 for 0 < 2i < q. By Sq2x 4 ° k k for some k such that 0 < 2 < q. So q = 2 . REMARKS. for

k

J. F. Adams has shown

0 ,1 ,2 ,3 .

are

[3 ] that the only possible values

His methods entail a much deeper analysis of the

algebra Ct(2). Examples of spaces which satisfy the hypotheses of the theorem are i)

any dimension, with

q = 1;

Complex projective spaceof any dimension, with

q = 2;

Real projective space of

ii) iii)

Q,uaternionic projective space of any dimension, with

iv)

The Cayley projective plane with, q = 8.

1.6. that

THEOREM.

H^(M) = 0

power of

q =

for

Let

M

be a connected compact 2n-manifold, such

1 < q < n,

and with

H^M) = Z2. Then

n

is a

2.

PROOF. of IhPcm) ,

u2

H2n~q-(M) = 0

for

1 < q < n,

and, if

u

is the generator

is the generator of H2n(M) . We now apply 1 .5 .

I.

AXIOMS FOR THE AUOEBRA §5.

Let

f:

The Hopf Invariant.

S2n“1 — — > Sn (n > i) . Let

obtained by attaching a 2n-cell

e2n

EpfXjZ)

Z,

«

Z

G(2)

and

H2n(X;Z)

«

to

Sn

X

be the adjunction space

by the mapping

f . Then

while for other positive dimensions

the cohomology groups are zero. Let x e Hn (X;Z) and y e H2n(X;Z) be o generators. Then x = h(f) .y for some integer h(f) called the Hopf in­ variant of .f.

It is defined up to sign.

homotopy type of

X

A homotopy of

f

leaves the

unchanged, and so the Hopf invariant is an invariant

of the homotopy class of

f.

Sometimes the double covering

S

-- > S

is assigned the Hopf

invariant 1 . In this case, the adjunction space is the projective plane. 5 .1 .

THEOREM.

Hopf invariant, then PROOF.

If there exists a map

n

Let

is a power of

Tj:

( t\ x )

Hence of

2

ny, since

=

be the map induced by the

> Z2* This map is a ring homomorphism. h(f)

1

=

(mod 2 ).

By U.5 , n

is a power

2.

REMARKS, for then 2)

Z

of odd

2.

H*(X;Z) -— -> H*(X;Z2)

coefficient homomorphism

f:S2n-1 *— > Sn

x

2

i)

=

-x

We easily see that, if

2

2x

and so

2

= 0

n

h(f) = 0

is odd, then

(integer coefficients).

The following are the standard maps of Hopf invariant one, and their

adjunction space: ■ 3

7

S

— —> —

>

1 tr

f:

PROOF. We

say

f

has degree

3,

where

bd(E1 x E 2)

E^ =

quaternionic projective plane;

S

Cayley projective plane. If

even

S1 ,S2 and

has degree (a,ft)

of the choice of Let

with any

Let

complex projective plane;

S

(Hopf [5 ]).

THEOREM.

S2n~1 -— > Sn

S

4 p

S J -- > 5-2.

2

S

n

is even, there are maps

Hopf invariant. be

if

(n-i)-spheres and f |S1 x p2

f:

has degree

(p1,p2) € S1 x S2 . The degree of

f

S1 x S2 --- >; a

and

is independent

(p1,p2) . be

the

n- cell such that

(E1 x S2) U

(S1 x E2)

is a

= bd E^

(i = 1 ,2 ) . Now

(2n-i)-sphere and

f|p1

§5 .

(E

x S2)n (S1 x E2) =

S'

=

E+ u E_

where

(E1

E+

in

S2) U (S1 x E2)

we extend

> E+ U E .

f

=

=

Then

S.

to a mapping

S'

C(f)(S1 x E2) C E_.

C(f)

is

will follow from two lemmas.

5.3. LEMMA. PROOF.

Let

S 1 x S 2 -- > S,

E + n E_

S.

S2* ' 1---- > Sn . 5 .2

C(f)

E_ are n-cells and

C(f)(E1 x S2) C E+ and

such a way that

a map

Let

and

f:

X

13

S 1 x S2. Let S ’ be the suspension of

Given a mapping C (f) :

THE HOPF INVARIANT

X

h(C(f))

= ae.

Throughout this proof integral coefficients will be used.

be the adjunction space gives rise to a map

x

be a generator of

verse images of

x

(E1 x E2)

(E1 x E g,E1 x S2 ,S1 x E2)

g:

Hn (X;Z) . We define

under the isomorphisms

Hn (X,E+) -- > Hn (X)

S ’. The attaching map

respectively.

x+

> (X,E+ ,E_).

and

x_

to be the in­

Hn (X,E_) -- > Hn(X)

and

Now we have a map (X,0,0) — > (X,E+ ,E_) .

This gives rise to a commutative diagram H11®

--------- > H 2n (X)

® Hn (X)

t

T > H 2n (X,S’) .

Hn(X,E+) ® Hn (X,E_) The vertical maps are isomorphisms. has image

x2

under the map

Therefore the cup-product

x+ u x_

H 2n (X,S’) -- > H 2n (X) . We have the following

commutative diagram *

H11®

H ^ x

'I

Eg,S 1 x Eg)

*

I

*

v

> Hn (E+,S) -S— > Hn (E,x p2,S,x pg) A

A

81 "

Hn"l(S) — £— > Hn'1(S x p )

U

Z By the diagram

g*x+

=

where

By a similar diagram, we see that Hn (E1 x E ^ x

S2) .

---— w+

g*x_

>

generates = 3w_, where

Z Hn (E-jX w_

E2) .

generates

I.

14

Let x^

p^:

€Hn (Ei,Si)

E 1 x E 2 -- > Ei by p*x1

w+ u w_ Hence

AXIOMS FOR THE ALGEBRA

=

w+

1 ,2 ). We define the generators

(i =

and

p*x2

=

(x1 x i) u (1 x x2)

= p*x1 u p2x2

g*x+ u g*x__ =

C t( 2 )

a^(x1x x2)

=

and

w_ . Now =

(x1 x

x2) .

(x1 x x2) generates

H2n(E, x E2,E, x S2 , S, x Eg) . Now

g:

(E1 x E2,E1 x S2 ^ S1 x E 2) —

homeomorphism and therefore induces an

> (X,Sf)

is a relative

isomorphism of cohomology groups.

So we have the isomorphisms H2n(X) 0 ).

In §k the canonical anti­

of a Hopf algebra is briefly discussed.

constructions with modules over the algebra

In §3 it

g —

0t(2) = C t ) ,

\|r of generators > Ct ® Ct . We have

ker a>.

We have a map of modules a: given by

a(u (8) v)

=

is an isomorphism.

u x v.

Let

X = Pn = P x...x P.

H*(X) (8) H*(Y) —

P

be

> H*(X x Y)

By the Kunneth relations for a field, this 00-dimensional real projective space.

Then, using the notation of Chapter I §3, the evalua­

tion map on w, w: Ct -— > H*(X) , is a monomorphism I 3.3. Therefore the map w ® w: phism in degrees

Let

Ct ®

in degrees

Ct—— > H*(X)® H*(X)

< n. 16

< n by

is a monomor­

§1.

THE ALGEBRA

ft ( 2)

I S A HOPF ALGEBRA

17

We have the diagram ft h *(X) ® H* (X) — |— > H* (X x X) A ~ w X w

a ---------------- £----------------> a -X*

We now prove that this diagram is commutative. module and hence, using the map

jr,

phism a,

thisgives

the structure of an

H* (X xX)

hasits usual structure as an

H*(X x X)

is an

H (X) 0 H (X)

ft -module.

is an Ct 0 6,

Using the isomor­ g

g -module via

-module.However, cd.

These

two

g -modules are identical, for (a>Sqk)(u x v)

= Z S q ^ x Sq^^v = a( (Z Sq1 0 Sqk"'1) (u

0 v))

= a(tSqk . u 0 v). Since the two

& -modules are identical, the diagram above is commutative.

Now, if

meg,

gram is commutative and

deg m < n, w 0 w

and

com = o,

then, since the dia­

is a monomorphism in dimensions < n,

tym = 0 .

This completes the proof of the theorem. Let

A

H with a unit. 1)

be an augmented graded algebra over a commutative ring We say

A

is a Hopf algebra if:

There is a "diagonal map" of algebras >|r: A--- > A 0 A;

2)

The compositions

s A 0 A

A i

H 0 A are both the identity. We say

t

is associative if the diagram A- ----- 4---- >

A ® A

Lfi_4

A ® A

> a ® A ® A

II.

18

is commutative.

THE DUAL OP THE ALGEBRA.

We say

t

G ( 2)

is commutative if the diagram

A is commutative.

(See I §3

1.2.

for the definition of

THEOREM.

is a Hopf algebra, with the commutative and

of i.1 .

associative diagonal map t PROOF.

0 ( 2)

T.)

The map

is a map of algebras by 1 .1 .

is connected, we have the unique augmentation c

e:

0,(2)

—

Since

0(2)

•> Z . In the

diagram

a

all the maps are homomorphisms of algebras.

The compositions are both the

identity on the generators of O , and they are therefore the identity on all of

O .

Using the fact that

Tjr iscommutative and

t

is an algebra homomorphism, we see that

associative by checking on the generators.

This com­

pletes the proof. Let Let

M

A

be an

defines an

be a Hopf algebra with diagonal map q: A-module.

if

m

M ® M

A-module structure on

multiplication in A,

Then

M.

We say that

is a homomorphism of 1.3.

is an

M ® M. M

A -— -> A ® A.

A A -module.

Let

m:

The map

M g> M — — > M

t

be a

is an algebra over the Hopf algebra

A-modules.

PROPOSITION.

If

X

is any space,

H*(X;Z2)

is an algebra

over the Hopf algebra 0 ( 2 ) . PROOF. y

This results immediately from the Cartan formula, since

is a homomorphism of algebras. Let

X

be a graded

A-module, where

A

is a Hopf algebra over a

§ 2 . THE STRUCTURE OF THE DUAL ALGEBRA ground ring

R,

with an associative diagonal map

tensor algebra of m:

X

over

r(X) ® r(X) -- > r(X)

R.

Let

r(X)

be the

It is obvious that the usual multiplication

is an

algebra over the Hopf algebra

ty.

19

A-homomorphism.

Therefore

r(X) is an

A.

§2 . The Structure of the Dual Algebra If

X is a graded module over a field

finite type if X* of

X,

Xn

R,

we say that

is finite dimensional for each

to be the graded module with

X*

=

n.

If

A

and diagonal

(f ® g)(a b)

Hom(Xn ,R) . If

= (-i)pqfa ® gb, p = deg a,

ijr, we easily verify that and diagonal

For k > 0 ,

let

A

respect to

1

Sq1, where I

x1

LetPn

x x2 I 3 .3 ,

1 ^. e (2*

m ik

is admissible and m k has degree 2 - 1 .

=

M^.

x(I)

x(I)

I

=x d ^

2 .1 .THEOREM.

=

P.

In

x €H 1 (P;Z2)

Hn (Pn ;Z2) we have

will enable us to

findthe structureof (2*.

=

n,we shall define

haslength less than

a c (2,

is finite, since

have the same dimension.)

x(I) € H*(Pn)

n.Now. suppose

x x(i2,... ,in) and If

Thefollowingtheorem,

and

§(I) € Q*

is a sequence of non-negative integers. If 2i x and |(I) = ^ . Suppose x(I) and

a(x1 x...xxn ) (The summation

P x P ...x

Let

=

|(I)

*

I

§(I)

= (i^...,^)

1(1^ |(i2,... ,In) .

then < |(I),a > x(I) .

< |(I),a>

=

0

^

has

eachx^ = x.

I = (i1 ,...,in)

are defined when We put

be the dual of

x...x xn , where

By induction on

I = (i) ,we put

cp

(2 k- 1 ,2 k~2 ,...,2 ,1 ).

=

P be 00-dimensional real projective space.

together with

where

q = deg g.

the basis of admissible monomials in G. Then< l^'^k

the generator.

the element

Y

%

cp*.

=

and < > = 0 if k 2 - 1 and therefore Let

be

and

is also a Hopf algebra, with

is an admissible monomial in Ct. Let

degree

X

(X ® Y)*

is a Hopf algebra of finite type, with multiplication

multiplication

with

is of

We define the dual

are of finite type, then we have a canonical isomorphism X* Y* defined by

X

unless

|(I)

and

a

20

II.

PROOF.

THE DUAL OF THE ALGEBRA

We prove the theorem by induction on n. If a is ad2^ ax = o unless a = Mk, and M^x = x by I 2 .7 .

missible, then

The formula is therefore true for each element in true for

® (2 )

n = i,

when

a

is admissible.

Since

Ct is the sum of admissible monomials, the theorem is

n = 1. We now assume the theorem is true for integers less than

\|ra

=

n.

Let

By 1 .3 and i .1 .

Z. a! ® a!.

a(x1 x...x xn)

=

a

=

a^(x2 x...x xn)

x

^i,I < ^ ^ "I) ^ai > < 3-
X U)

® x(I)

=

Zj < i(i.,) 0 |(i2,.. .,in) ,>ira > x(I)

=

^(I)=n
Gfc* C

. ****

PROOF.

Let

-

Zi=o *k-i ®

•

a,3 € 'Cfc . We have to show that

*

< cp ! k , a 3 >

=

.

That is, we have to show
< l-pf* > •

We shall prove this by using 2.1 . Let

x

be the generator of

H^PjZp).

Let

d:

P

>Pn be the

22

II.

diagonal, where

THE DUAL OF THE ALGEBRA

n = 21 . Then x

pi 2 ^-

So

o:*x

where mute

n(I) = 2 1+...+ 2 n I,

G ( 2)

if

we do not alter

*

=

d (x1 x ...x x ) .

=

a*d (x1 x...x x ) .

=

d*a(x1 x...x x )

=

d*(2f(I)=n < 1(1) ,a > x(I))

=

Z f(I)=n < g(I)'a > xIl(I)

*

I = (i],..., i ).

< |(I),a > xn^

.

If we cyclically per­

Since

n = 21 ,

the number

of different sequences, obtainable by cyclic permutation from one particu­ lar sequence

I,

is some power of

2, say

2^-^.

if

j(I) > 0,

terms in the summation corresponding to cyclic permutations of cancel out mod 2 . So we are left with terms for which m = i1 = i2 = ... = i . For such sequences n(I)

=

|(I)

will

j(I) = 0 . That is =

21

and

2m+i. Therefore pi

pi

ax Now

I,

I

the

Zk
x

= ^

2k

pm+i

< V>a > x

=

x

pi

pi

^i,m < ^m >a > < Equating coefficients of




x

2k

-

2.1

2m+i

> x

, we see that

A + i=k < *£>


which proves our theorem.

§3. Let nalAn ideal If

M

A M

be a Hopf algebra

Ideals of finite type over a field, with diago

is called a Hopf ideal, if

\|r(M) C M ® A + A g>M.

is a Hopf Ideal, then A/M has an induced Hopfalgebra structure

(assuming i k M) . If A* is the dual of A and M+ the dual of A/M, + * then M is the Hopf subalgebra of A which annihilates M. Conversely t * + If M is a Hopf subalgebra of A , then the dual algebra to M Is the

§3quotient of

A by the

IDEALS

23

Hopf ideal M which annihilates

Mf

In the algebra

Cfc(2)*,' let ,...,jk,...) be the ideal genn ^k erated by the elements ik , where n = 2 (k=i,2,...). 3.1

. LEMMA.

If

Jk_1
H^(X,X - U) give a representation of PROOF. X,

let

Y

HQ-(X,X - U)

Suppose first that

denote its closure and

joint topological union of the spaces

as a direct sum. I

Y

of the spaces

Then(V,W)

is finite. its boundary. U^,

and let

is a compact pair. 30

For any subspace Let

V

W C V

Y

of

be the dis­ be the union

The following

diagram

§1.

THOM’S THEOREM

31

is commutative

(tJ±,

> (v,w)

I

1

(X,X - Ui) /

Cechtheory, taking limits over 1.3. LEMMA.

If

finite subsets

of

e is any cohomology

I.

operation of onevariable,

such that

then

e:

H^(X)---- > Hn (X)

e:

H?(Sn ,Y) -- > H ^ S ^ Y )

only axiom

e

spaces,

need not be a homomorphism.)

e

let

P

is zero.

(Note that the

needs to satisfy is naturality with respect to mappings of

PROOF. dimension,

(o < q < n)

For any cohomology operation

e(o) = o.

be a point.

e,

The proof is as follows.

with image in a positive Let

X

be any space, and

Then we have the commutative diagram i f i ( P ) ------ > H*(X)

je

I e

Hn(P)------ > Hn (X) induced by the map 0,

where

X -- > P.

Since

n > o,

Hn (P)

=

o,

and so

e (o) =

0 e H^-(X) . So

1.2 shows that we have only to prove

Hn (Sn /Sn - U.)

9: pP(Sn ,Sn - U^) -- >

is zero, in order to prove our lemma.

Now, we have the

32

III.

EMBEDDINGS OF SPACES IN SPHERES

commutative diagram Hq (Sn ,Sn - Ui) — 5— >

-U±)

I* P

V

0 Since and

th

=

V

^---- >Hn (Sn)

—

}f{S n)

* 0

is connected, we have by Alexander duality, Hn~1(Sn - U^)

pf^S3"1 - Ph)

=

o.

Therefore the vertical map on

diagram is an isomorphism. Let subcomplex such that

U

K

be any neighbourhood of

KC U

and

Y.

Y

the right of the

Y C K - L.

Then there is a connected

n-manifold with boundary

We can construct

Sn , by taking a fine subdivision.

connected, since

is connected.

K

H*(Y)

The set of such manifolds

is the direct limit of the groups

Let

F

be a field.

L,

from thesimplicial

We can assume

inclusion maps between them form an inverse system with limit fore

0

This proves the lemma.

ofSn , which is a compact

structure of

=

K

is

K, and the Y.

There­

H*(K).

We have the cup-product pairing

HP (K,L;F) ® H ^ C K s F ) -- > EpCKjLjF)

«

F.

Lefschetz duality tells us that the induced map a: is an isomorphism.

HP (K;F)--- ~> Horn (Hn 'p (K,L;F) ,F)

Let

x € H^(K;Z2).

Hn”q”i (K,L;Z2) — by the formula such that

y — — > Sqxj

a(Q1x)

w x.

> Hn (K,L;Z2) Let

is a homomorphism

(See II §P

PROOF.

0° = c(Sq°). definition,

Q1

For any

=

Z2 Hq+i(K;Z2)

Then y w Q^ x. > Hq+i(K;Z2) .

cCSq1)

as a homomorphism

Hq (K;Z2) — ->

c.)

coefficients throughout this proof.

by induction on x e H^K)

-

be the element of

for the definition of

We shall use

The proof is

=

Q1 : Hq (K;Z2)

i.P. PROPOSITION. Hq+i(K;Z2).

Q^x

isthe homomorphism. Sq1y w x

Q1

We define a homomorphism

and

i. Obviously Q° = 1.

Therefore

y € Hn“q“i(K,L) , we have, by

§1.

THOM'S THEOREM

y u (Q1 + Ij"] Q J" Sq1'0" + Sq1)x = S q W o x + I1'} =

Sq°y u Sq1'0! + y u Sq1x

Sq1(y u x) € Hn (K,L)

.

We have the commutative diagram Hn_i(K,L) ------- ^

>

Hn (K,L)

T

T

Hn'1(Sn ,Sn - Int K) — § 1 — > T p ( S n , S n - Int K) . The vertical maps are excision isomorphisms. Sq1 (y

ux)

Q1

=

= o if

i>

o. Therefore, from

= - Zj c(Sq^) . Sq1”"5 ~ Sq1

by our induction hypothesis

= cCSq1)

by the definition of

THEOREM.

then, for each

If the compact space

Y

c .

can be embedded in

Sn ,

i > o, we have that Q1 : Hn"2i(Y)

and

the computation above,

- Ej Q j . Sq1"^ - Sq1

1.5.

iszero.

By 1.3, we have

> Hn"i (Y)

Equivalently, ifa compact space

Y

is such that, for some

r

i > o, Q1 : Hr (Y)

is not zero, then PROOF.

Y

> Hr+i(Y)

is not embeddable in

Suppose

Y

21

S

can be embedded in

Sn . We construct a mani­

fold K as described above. Let y e ff^CS^S11 - Int K) . Then Sq1y p y = o by I § 1Axiom 3 and 1 .i . We have thecommutativediagram H^fKjL) --------

>

=

H2i(K,L)

T

T - Int K) —

— > H2i(Sn ,Sn - Int K) .

The vertical maps are excision isomorphisms. map is zero, so is the upper one.

Since the lower horizontal

3it

III.

Let

EMBEDDINGS OF SPACES IN SPHERES

x € H01'21^)

and

y s H^(K,L) . Then

->!•x By duality

Qxx

=

0,

Q^x

LEMMA.

Q*Sc = 2h = x'

0

PROOF. k = o.

If

unless

m:

Themiddle pm

2m+1

=

if

Pn

h x“ # 0 .

x

0 . y.

be a i-dimensional cohomology class mod 2.

has the form

x

=

=

> H*(X)

2k - 1 ;

j_f ^

k.

_

2^ -

i }

then

It is obvious for

Q'Sc ♦ Q k- 1x 2 .

be the cup-product, and let

be the diagonal.

Then

c(Sqk"1)x2

= m[ ^(cSq^1) . x ® x]

by II 1.3

= m[ (c x c)T^ Sqk"1

by II § b

. x ® x]

=

m[Z cSq^ x cSqk“i_1 . x ® x]

=

Z f l l ^ x . Q ^ 1" ^

.

out in pairs (mod 2), exceptfor the

term occurs when pm

i = k - i - 1,

middle term,

if

and, by induction, is

v>n

ir_ i

p

.x

if i = 2 - i and is zero otherwise. So Q x m 1 k= 2 - i and is zero otherwise. This proves the lemma.

1 .7 .

space

Let

Etc

> Ct(2 ) 0 Ct(2)

The summationcancels

equal to

=

This is proved by induction on

Q ^ x 2

x

k

H*(X) ® H*(X) —

any.

u x

k > o, we have o

Let

Sqxj

since the above equation is true for all

1.6. Then

=

THEOREM.

If

1 < 2k < n < 2il+1,

then real projective n-

cannot be embedded in a sphere of dimension less than PROOF. Let

x

be the generator of

i H (PMll ;Z0) .Then d

2^+1 . p^~ i Q, x

=

By 1.5, the theorem follows. 1. 7

was

first proved for regular differentiable embeddings by

using Stiefel-Whitney classes.

Let

M

§2.

HOPF'S THEOREM

§ 2.

Hopf *s Theorem.

be a closed

(n-i)-manifold embedded in

Alexander duality with coefficients we find that

M

closed subset of duality to

A

A

and

B

and

then to

B, *

MCA

and

j:

separates

Sn

Z,

into two open

A u B = Sn . By duality no proper and

so

A n B=

M.Applying

we see that

Hr(B)

for any ring of coefficients. 2.1. THEOREM.

Sn ,

Sn . Applying

and then with coefficients M

such that

M can separate

Hr (A)

i:

Z2

is orientable and that

sets with closures

35

=

0

(r > n-i)

We have the following theorem due to Hopf.

Under the above hypotheses, the inclusion maps

M C B

induce a representation of

eP(M)

as a direct

sum H^(M) Here

■X*

i

and

-K*

j

=

I*#(A) + j*H?(B)

are monomorphisms.

and the identification

Hn~1 (M)

=

for

Using

F,

o < q < n-i.

a field of coefficients

cup-products in

M

F,

give an isomor­

phism i*H^(A) PROOF.

or

B

(B) ,F)

for 0 < q < n-i .

The first statement followsimmediately

Vietoris sequence. A

=» Horn (

Since

Hn”1(A)

=

with values in dimension

Hn“1(B)

(n-i)

=

o,

are zero.

from the Mayercup-products in

The rest of the

theorem follows by Poincare duality. 2.2. COROLLARY. be embedded in

=

n > 2,

then real projective n-space cannot

Sn+1.

2.3. LEMMA. Sq^St

If

Let

x £

and let

r + k = n - i,

then

0. PROOF.

Let

x

=

i*a + j*b,

The lemma follows by naturality since Let

s

c(Sq^)

as in

action of the Steenrod algebra

where H21"1(A)

§1.

0,(2)

is determined by the following theorem.

Let on

a e Hr (A) =

H31"1(B)

and =

x c Hr (A;Z2).

H*(B;Z2)

is known.

b c Hr (B). o. Suppose the Then

Sq^

36

III. 2A .

EMBEDDINGS OP SPACES IN SPHERES

THEOREM.

x e Hr (A;Z2);

Let

s = n -1

and let

ye HS(B;Z2),

then i*Sq^x u

PROOF. for

- r - k

The

j*y =

i*x u j ' V V .

theorem is proved by induction on

k.

It is obvious

k ~ o. By 2.3,

S q ^ ^ x u j*y)=

0

if

k > o.

So by the Cartan

formula 0

=

Sqmi*x U Sqk-m J*y

=

i*x o Qm Sqk- V y

♦ 1*Sq^X u f y

by our induction hypothesis =

i*x u j^Q^y + i*Sqlcx u j*y by the definition of

Therefore

i*x u j*Q^y

=

in

II §^.

i*Sq*Sc u j*y.

BIBLIOGRAPHY Thom, R .,mEspaces fibres en spheres et carres de Steenrod," Ann. Sci. Ecole Normale Sup. 69 (1952), 1 0 9 -1 8 2 .

CHAPTER IV. The Cohomology of Classical Groups and Stiefel Manifolds.

In this chapter, we find

the

and quaternionic Stiefel manifolds. We

cohomology rings of the real,complex

also obtain the Pontrjagin rings

of the orthogonal, unitary and symplectic groups and of the special orthogo­ nal and special unitary groups.

The method is to obtain a cellular de­

composition of the Stiefel manifolds (following [i] and [2]) . We then find the action of the Steenrod algebra real Stiefel manifolds.

Cfc(2)in the cohomology rings of the

Using this information, we obtain an upper bound

on the possible number of linearly independent vector fields on a sphere.

. Definitions.

Let

F =

d = 1 , 2 or h.

according as over

F,

Let

zero elsewhere.

u^

Let

be the n-dimensional vector space

be the column vector with x

=

x^.

< x,y1 > + < x,y2 >;

< x,y >

V = Fn

Z^ u^x^ e V,

We define the scalar product

is the conjugate of =

Let

consisting of column vectors with entries in

on the right.

Zi u^y^.

be the real or complex numbers or the quaternions,

=

< x,y >

Then < x,yx > < x 1 + x2,y >

< y,x > . We embed Fn

where

= =

in

1

F.

x^ e F =

We write scalars

in the

i^*1 row and

and let

=

Z^ xiyi , where

< x,y >x

if

x e F;

< x p y > + < x2,y >; Fn+1

y

< x,y1+ y 2> and

by putting the last coordi­

nate equal to zero. Let

G(n)

scalar products. for all

x,y € V.

be the group of transformations of That is, If

A

A e G(n)

if and only if

is represented by the

tiplying column vectors on the left, then t^k

-

I.

G(n)

V

which preserve

< Ax,Ay >

=

n x n-matrix

A € G(n)

< x,y > mul­

if and only If

is the orthogonal, unitary or symplectic group according 37

38

as

IV .

d = 1 , 2 or ^

embedding

COHOMOLOGY OP CLASSICAL GROUPS

We have an embedding

Fn C Fn+1 . The matrix

G(n) C G(n + i)

A ,e G(n)

induced by the

corresponds to the matrix

€ G(n + 1 )

We write

G(o)

=

I.

The Stiefel manifold

G(n,k)

is the manifold of left cosets

G(n)/G(k). Let G l .(n,k) be the manifold of (n-k)-frames in n-space. mapping

G(n) -- > G*(n,k),

matrix as the G*(n,k)

(n-k) vectors of an

which is obviously onto.

have the same last

(n-k)-frame,

A

A “1B e G(k).

and

B

G(n,k) — in

>

G(n)

Therefore the map

is a homeomorphism, and we can identify the two spaces.

G ’(n,n-1) is the manifold of unit vectors in 1.1.

(n-k) columns of a

induces a map

If two matrices

(n-k) columns, then

G(n,k) -- > G'(n,k) Wow

which selects the last

The

G(n,n-i)

is homeomorphic to

V.

Snd~1 C V

Therefore =

F31 by the map

which selects the last column of a matrix. 1.2. in

Definition of

V = Fn . Then

S^”1

cp. Let

Sn^~1 be the sphere of unit vectors

is the sphere of scalars of unit norm in

F.

We

construct a mapping

xx. That is

x(x-i) < x,y > + y

or in matrix notation.

If inclusion

m < n F131 —

we have an inclusion

> Fn . This induces ,md-i

x S'

smd"1 _ >

a further inclusion ,nd-i x -> SJ

The following diagram is obviously commutative

cp v

G(m)

snd~1 ,

V

->

G(n) .

induced by the

§1. DEFINITIONS 1 .3 .

Sd~1

Definition of

induced by

cp.

the identifications (y,i).

in

be the quotient space of

It is the set of pairs (x,x)

=

(xv ,v~1xv)

s1^ " 1 x

(x,x) e Snd~1 x Sd-1 where

v e S^1

and

under (x,i)

=

That these are the only identifications is easily seen by looking

at the fixed point set of Q,0

Q^. Let

39

cp(x,x) . Let

by sending

Q0

Q,q

Qh

(n > i)

If

n > m > i, we have an embedding

be a single point.

We embed

to the equivalence class of

(x,i).

induced by

Smd‘1-- x S^”1 -> S ^ " 1 x Sd"1 . By the commutativity of the previous dia­ gram, we have another commutative

diagram

V V G(m) --- > G(n) whenever

n > m > o. ^

is a compact Hausdorff space and so the vertical maps are em­

beddings.

We identify

with its embedded image in

Under the identification

Q0

becomes the identity of

G(n)

(n > m ).

G(n) .

Let

£(n“1)d

be the ball consisting of all vectors x€ S ^ ”1 C V

- Fn , with

xn real

and

Let

xn > o.

Then

xn

is determined by

fn :E^11”1)d------> snd~1 te the inclusion map.

—— >

(Sd“1,i) be the usual relative homeomorphism V

Elld'1

>

Let

g:

x ],... ,x

.j.

(Ed“1,Sd~2)

(S_1 =

0).

Let

(n > 1)

be the composition giid-i

Let

Snd”2

be the boundary of

i.1*. 1^:

= E (n-i)d x Ed -1 _ £ n _ L L > s1**'1 x sd ~1

LEMMA..

o-cell

The map

Q0

.

End'1 . defines a relative homeomorphism

(End_1 ,Snd-2)----> (Qjj, Qn_1)

plex with a

>

if

and with an

n > 1.

Therefore

^

(md-i)-cell for each

is

a CM com-

m such that

i < m < n. PROOF.

(f

Snd_1 x Sd~1 where

x g)Snd“2 xn = o

consists of points of the form or

x = 1 . Therefore

(x,x) e

h(Snd~2) C Qn_, .

In any equivalence class{(x,x)} e Snd-1 we can choose a representative (x,x)

so that

xn

is real

and

xn > o.

Moreover, if

x4 1

and xn >°,

ko

COHOMOLOGY OF CLASSICAL GROUPS

IV.

this representative is unique. (Qn^^n-i)

This proves that

(E3”^ ” 1 ,Snd~2) — ->

3^:

is a rela1:ive homeomorphism.

The rest of the lemma follows by induction on 1.5.

DEFINITION.

rr: G(n) -— > G(n,k)

11

Let

n.

be the multiplication in G(n). Let

be the standard projection.

A normal cellof

G(n,k)

is a map(or the image ofa map) of the form E1^ 1 x...x Elr>d 1

x...x 0. --- £H_> G(n,k) r We denotesuch a cell by (i1 ,...,ip |n,k)

■

h > Q.

X1

where

n > i 1 > ± 2 >...> ip > k.

or simply by

(i^...,! )

if this will cause noconfusion.

The cells

of

(other than Q0), described in i.U, may be identified with the normal cells

(m|n,o)

where

By u on

We denote such a cell of

we shall also denote the action of

by

(m) .

G(n) by left translation

> k > 0 ).

G(n,k) (n

§2.

n > m > o.

The Cellular Structure of the Stiefel Manifolds.

In this section we shall prove the following pivotal theorem. 2 .1 .

THEOREM.

G(n,k)

cells (see 1.5) and the

cellular

o-cell

*(I). The map

Qh x G(n- 1 ,k) —

n: is

is a CW complex, whose cells are the normal

and induces

> G(n,k)

(k < n)

an epimorphism of chain complexes.

Before proving the theorem,.we state and prove a corollary. 2 .2 .

G(m,I) —

COROLLARY.

> G(n,k)

(i1 ,...,ir |m,£)

If m < n

is cellular.

if

d

= 1 and

& < k,then the induced map

This map sends the normal cell

to the normal cell

(i1,... ,ip_T |n,k)

and

(i k > ip

=

if

ip > k; to

i> i =

0;

and

degenerately otherwise. This follows immediately from 2 . 1

PROOF.

and the definition 1 . 5

of normal cells. We now begin the proof of 2 .1 . Let us denote by (On,On.,) —

a:

(0^0^-,) —

> (Snd~ 1 ,un ) the composition

> (G(n,n - i),G(n - i ,n - 0) -- >

§2.

THE CELLULAR STRUCTURE OF THE STIEFEL MANIFOLDS

k

where the map on the right is the homeomorphism of i.1 . 2 .3 .

LEMMA.

(Qn ,Qn_1)--- > (Snd~1,un)

The map a:

is a relative

homeomorphism. PROOF,

for if

^

cc:

> Snd_1

x(x - i)xn + un

=

un ,then

Suppose we are given must show thatthere is exactly xn

sends

((x,x)} (x e Snd_1,X e Sd_1)

by 1 .2 . The inverse image of

x(x - i)icn + un

real andxn > o,

and

x

under

= i

or

xn

y e snci~1 C V =

Fn

such that

one element

x j i,

(x,X)

such that

In the real d-dimensional space F,

x(x

(y

(y

-i) ^ o.

dimensional space (y

- 1),

for

can solve uniquely

real,

xn > °,

xn , x^is determined uniquely =

y - un , xn real,

o. y ^

We

Dxn

-

- i)

=y - ^

lies in - 1 ),

(x

.

the closed

where |x| = i.

So, projecting from the origin inthe real d-

F, we

xn

=

to

is

e Snd~1 x S^”1with

ball bounded by the sphere of scalars of the form Moreover,

a

|x|

for

= iand

We have to check that

|x| =

x^(x - 1 )

the equation

x ^ 1 . Knowing

i and

i < i < n-i.We now

x

and

havex(x - i)xn

x ^ i.

< x,x > = i.

On evaluating thescalar pro­

duct of each side of the above equation with itself, we find < x,x > (2 - x - x)x^ Also, weknow

that

x^(x - i)

< X ,X >

Since

|x| =

we deduce that

and

-

X - x)x£

(2 -

=

X

X)x^ .

-

(2 - x - x) ^

o.

Since also,

xn ^ 0

< x,x > = 1. If

n > k > 0,

then

(Q^ x G(n - 1,k) ,Qn _1 x G(n - i,k))----- >(G(n,k),G(n - i,k))

is a relative homeomorphism and maps PROOF.

A e Q ^ B c G(n - 1)

A € G(n - i).

Qh n G(n m.

1)

x G(n - i,k)

The inverse image of G(n -

To see this, let Then

2 - yn - yn .

s (yn -i) . Hence

x ^ i, we have

PROPOSITION.

2.k.

n:

1

(2

=

=

and suppose

On projecting into

Qn _] . Therefore

is one-to-oneon

i,k)

is

onto

Qn-1 x G(n - i,k).

ABG(k)

G(n,n - i),

G(n,k).

C G(n - 1 ).

we see by 2.3 that

A € Qn_1.

(Q^ - QT1_1) x G(n - i,k).

To see this, let

42

I V . COHOMOLOGY OF CLASSICAL

A, C c

-^ n - i'

CDG(k).

Then

811(1 l e t

AG(n - l)

we see by 2.3 th a t n maps

^

B >-D €

A = C.

Ce

th e re i s

anelem ent

D € G(n - 1 )

2.5.

LEMMA.

€Sd_1 C F be a

such

L et

such th a t

x € Snd‘ 1 C V

f ix e d i f

< A x ,y >

=

0

A e G (n ). By 2 . 3 ,

= AG(n- 1 ) .

T h erefo re

A = CD.

=

Fn

be a u n it v e c t o r , l e t

u n it s c a la r and l e t A € G(n) .

By d e f i n i t i o n ,

ABG(k) G (n,n - i ) ,

= DG(k) .

th a t CG(n - i)

(See 1 . 2 f o r d e f i n i t i o n o f PROOF.

y

BG(k)

x G (n -i,k ) onto G ( n ,k ) . To see t h i s , l e t an elem ent

cp(Ax, x ) .

811(1 suPP°se tha1:

On p r o je c t in g in t o

T h erefo re

th e re i s

x

^

= CG(n - i ) .

GROUPS

Then

A cp(x,x)A” 1

=

cp.)

cp(Ax,x)

i s the tra n sfo rm a tio n w hich keeps

and w hich sends

Ax

to

Axx.

The lemma f o l ­

low s. 2 . 6.

PROPOSITION,

PROOF. where and

m = 2.

M(Qm x Q ^ .,) C G(m)

fo r

m > 1.

m i s a r b i t r a r y to the case

We th e r e fo r e b e g in by ch eckin g the p r o p o s itio n f o r

m= 1

m = 2.

=1

ij(Q 1

=

We s h a ll reduce the case where

I f m = 1, n

x Qjh)

then

Q1

=

G ( i ) . We see t h is from 2.4 by p u t tin g

and k = 0 .The p r o p o s itio n fo llo w s x

C G( i ) If

p u t tin g

( r e c a l l from 1.3 th a t

m = 2,

n = 2

then

and

v io u s p aragraph.

n(Q2 x

k = 0,

s in c e Q0

G (i)

=

ji(Q1 x Qq) C

i s the i d e n t i t y elem e n t).

= G(2)»

We see t h i s from 2 .4 ? by

and r e c a l l i n g th a t

Q1

= G(1)

from th e p r e ­

So G(2)

=

M.(Q2 x

C fi(Q2 x Q2) C G (2) .

The p r o p o s itio n fo llo w s . Wow l e t tra ry

elem en ts.

x , y e Smd" 1 C V Then

We must show th a t L et

cp(x,x)

=

and

F111 and l e t q>(y,v)

e Sd_1 C F

be a r b i-

a re a r b i t r a r y elem ents o f

Qm.

cp(x,x)cp(y,v) c

W be a 2-d im en sion al sub space o f

U sing the in c lu s io n

x,v

Fr C Fr+1 0

c

V

c o n ta in in g

o f § 1 , we have the sequence

W n F1 ...

cW nF ^

= W

x

and

y.

§2.

THE CELLULAR STRUCTURE OF THE STIEFEL MANIFOLDS

o f v e c to r

sp aces over

We choose

an in t e g e r

a l.

A € F(m)

Ax

L et =

x’

e F2

F, r,

such th a t

map

and

in c r e a s in g by

W onto

Ay

=

1
1,

has

has degree

has degree -1

[i 1 [ i ]

and, by 2 .6 ,

1,

by 1.^

= Qi Qi „ 1 ,

then by 2 . 7 ,

o f dim ension

( i) (i) C Q1 ,

i s con tain ed

(2 i - i) d - 2

which i s l e s s than

i s zero f o r dim en sional rea so n s, i f w hich has dim ension

d - 1.

If

i > 1 d >1 ,

i s zero f o r d im en sional rea so n s.

d = 1, I and

then by 3.2 th e re are two

-I.

( 1 ) i s the O - c e ll

-I.

O - c e l ls , namely the 1 x i T h erefo re [ i ] [ i ]

=

.

In ord er to com plete the p ro o f o f 3 .5 , i t on ly rem ains to be th a t the c la s s e s form a fr e e

0

(-1)^ .

the normal c e l l

in th e sk e le to n o f

If

U(n)

k l

and

b a s is f o r

Ci 1 ] [ i 2 3 . . . f i p ],

H*(G(n) ;R ) .

where

n >

i 1>

shown

i 2 *** > i r > 0

T h is fo llo w s from 3.U, sin ce

the d e f i ­

n it i o n l . 5 o f normal c e l l s shows th a t [ i , ] [ i 2] . . . The map

11 :

[ip ]

=

t i , , . . . ,ip|n,o] .

G(n) x G (n,k) ---- > G (n,k)

s tr u c tu r e o f a module over Pontr ja g in r in g 3 .6 .

THEOREM.

s in g le g e n e ra to r

H*(G(n,k) ;R)

the

H*(G(n) ;R) .

H*(G(n,k) ;R)

(n > k > i) .

g iv e s

i s a module over

H*(G(n)jR)

The d e fin in g r e la t io n s f o r t h i s module

are [i]

L = °

[i]

I

[1] 1 The

= = 1

normal c la s s

PROOF.

0

if k > i >0

i f d ^ i

if k > i >1

if d = 1

if d = 1 . C i1 , i 2, . . . , i

|n,k]

=

[i 1 3[

] ...

T h is fo llo w s im m ediately from 3.^ and 3.5

H . The Cohomology Rings

[ i ] i_ .

.

H* (G(n.k) ;R) .

We b e g in t h i s s e c tio n by rem inding the rea d e r o f our assum ption 3. 1 on

R. We s h a l l compute the cohomology r in g o f

and by use o f th e monomorphism (see 3.*0

G (n,k) by in d u c tio n on

n

on a

48

IV .

n*: If

d = 1,

if

d = 4,

d

-

H*(G(n,k) ;R) — >

we w r it e 0 (n ,k )

NOTATION.

if

d = 2,

and

G (n,k)

if

where

=i^

i f d = i.

if

i

> n

L et

k, where

{ i^ ,...,^ }

f o r some

i < s < t < r.

prod uct o f the normal c la s s o f

G (n,k) ;

s

( R e c a ll th a t

w hich i s odd i f d = 2 or 4, w h ile

if

1 < s < r,

{ i^ ...,!^ }

(i ) if

L et

be the ze ro cohomolo­

such th a t

O therw ise l e t

k > o

G (n,k) ob tain ed on perm uting

and the s ig n o f the perm u tation . ( i pd - 1 ) ,

n > k. =

We extend the n o ta tio n o f 3*3 as fo llo w s .

k> i

gy c la s s o f

where

U (n,k)

.

he a s e t o f in t e g e r s a l l g r e a te r than

2 or 4,

i

x G(n - i ,k) ;R)

= G (n,k) ;

Sp(n,k) = G (n,k)

4 .1 . i.j,...,i

COHOMOLOGY OF CLASSICAL GROUPS

or

denote the i^ ,...,^ ,

i s a c e l l o f dim ension

d = 1 , our r in g i s

Z2 .

T h erefo re t h i s n o ta tio n i s c o n s is te n t w ith the u su a l con ven tion f o r s ig n ch an gin g.)

C u rly b ra c k e ts w ith a space between them

p reted as T.

We a ls o use th e symbols

(b) and T,

denote the images o f th ese c la s s e s under the map 4 .2 .

LEMMA, a)

L et

n > k >i .

H * (0 (n ,k );Z 2) — >

n*:

{ } , should be i n t e r ­

where

0 < b < n to

H*(G(n) ;R) - — > H ^Q^R) .

Under th e monomorphism

E*(Q^

x 0(n -

(se e 3.4)

i , k ) ; Z 2) ,

we have, in the n o ta tio n o f 4.1 , n*Cbv ,

=

T x Cb1 , . . . , b r ) + £1) x Cb1 , . . . , b r ) +

b)

L et n > k > 0. n*:

L et

d = 2

or 4.

H *(G (n,k);R)------->

K*

x tbi>--->bi-i>bi+i>--->VUnder the monomorphism x G(n - 1 ,k) ;R) ,

we have, in the n o ta tio n o f 4 . 1 , n * C b ,,...,b r }

=

T x (b 1 , . . . , b r ) +

x PROOF. > bp > k,

x

{b1 , . . . , b i _ i,b i + l , . . . , b r ) .

W ithout lo s s o f g e n e r a li t y , we may assume th a t

n > b^ > . . .

s in c e in te rch a n g in g two a d ja c e n t b Ts m u lt ip lie s b o th s id e s o f

the eq u ation by

-1,

eq u a tio n are ze ro .

and i f two o f the b ’ s a re e q u a l, b oth s id e s o f the

§U .

THE COHOMOLOGY RINGS

H * ( G ( n ,k ) ;R)

^9

We prove the theorem by e v a lu a tin g b oth s id e s o f the eq u a tio n on c e lls of

x G(n - l , k ) .

The v a lu e o f th e le ft- h a n d sid e i s c a lc u la te d

w ith th e h elp o f 3.5 and 3 .6 . use th e s ig n change where

< c 1 x c 2,h-, x h2 >

c , € HP (X ),

h, € H^(X),

i.3 .

LEMMA.

c la s s e s in

a re z e ro .

and {a} (b)

When e v a lu a tin g the r ig h t hand s id e , we must

=

For d = 2 or

a l cohomology. S in ce

d = 1,

k,

and

h

h2 e Hq (Y) .

, cup p rod u cts o f p o s i t i v e dim en sional

n > a >b >

o,

then TCa} = o

When

1 .i

shows th a t

has

on ly odd dim ension­

the prod u ct o f two odd d im en sional c la s s e s

the lemma fo llo w s fo r d = 2 or

{a)

d = 2 or

and

{a + b - 1 } .

PROOF.

S ince

( - i ) pq < c 1 , j 1 > < c 2 ,h 2 >,

c 2 € H^Y)

If

If

=

d = 1,

h.

i s th e d i s j o i n t union o f

i s the g e n e ra to r o f

( a + b - 1} fo llo w s .

H3” 1 (P11” 1 ;Z 2) ,

The u n it elem ent i n

t h i s a s s ig n s the v a lu e

1

i s z e ro ,

to each

(T + {1}) {a}

H (Q ^ Z ^

{a}

and

P11" 1

the form ula

o - c e l l in =

Q0

by 3 .2 .

CaHb)

—

is

T + {1} ,

sin ce

T h erefo re

«

Ca} { i } .

The lemma fo llo w s . LEMMA.

k.k.

L et

n > k > 1.

{a} u [ b v . . . , b r )

=

In

H*(0(n,k) ;Z2) ,

C a,b1 , . . . , b r 3 + + 2i = i Cbi

where

a > k

and

PROOF.

bi > k

fo r a l l

h

. 2 a) and L et

k

A(n,k)

tio n s

Cb) Cb) =

{2b - 1}

1 .5

H*(0(m,£) ;Z 2) and

For n > k , i t fo llo w s by in d u c tio n on

b- i , if

THEOREM. k < £,

where

2b - 1 H*(0(n,k) ;Z2) .

Cb) -— > Cb]

n,

be the commutative a s s o c i a t iv e a lg e b ra on th e g e n e r­

Cb3 o f dim ension

and

n = k, s in c e then a l l term s in

.3 .

a to r s

n < m

+ a '

i.

The theorem i s tru e f o r

the form ula a re zero (see ^ .1 ) . u sin g

we have

if

b
b > k, and

n>k >

0 (n ,k )

1.

{bH b} Then

—— > 0(m,£)

Under t h i s map,

{b}

s u b je c t to the r e l a ­ =

o

i f 2b-i > n.

H *(0(n,k) ;Z2) ® A(n,k) . which in d uces a map -> 0

if b> n

If

50

IV .

PROOF.

COHOMOLOGY OF CLASSICAL GROUPS

From 3.J+, we see th a t

H *(0(n,k) ;Z 2)

b a s is c o n s is tin g o f th e normal c la s s e s > br > k.

shows by in d u c tio n on

k.k

n > k, g e n e ra te

H ( 0 ( n ,k ) ;Z 2) . (b) w {b}

(b 1 , . . . ,b } ,

w ith

n > b1 > ...

r , th a t the normal c la s s e s

A lso from i . i , *

has a v e c to r space

{b} , w ith

we see th a t

Cb,b} + { 2b - 1 } .

R e fe r r in g to our n o ta tio n 4 . 1 , we see th a t th e re i s an epimorphism A(n,k) -- > H *(0(n,k) ;Z2) . Suppose we have an elem ent i s zero. (bp ) ,

Q,

Q, e a ,

whose image in

H *(0(n,k) ;Z 2)

can be ex p ressed as the sum o f terms o f the form

where n > b 1 > . . . > br > k.. By

in d u c tio n on

{ b ^ fb g } ...

r , we see from

k.k

th a t (b 1 3 Cb2} . . . {bp } w ith

s < r.

In

=

{b1 , . . . , b p } +* terms l i k e

Q, i f we c o l l e c t the terms where

a p p ly t h i s form u la, we see th a t i.6 .

LEMMA.

a > k,

and

PROOF.

k,

then in H*(G(n,k)

u (b 1 , . . . ,t>p )

b^ > k

For

i s g r e a t e s t , and

fo r a l l

;R ),

=

,

i.

For n = k, the theorem i s tr u e

the eq u a tio n a re z e r o .

r

Q, = o.

I f d = 2 or

( -1 ) r Ca) where

(a 1 ? . . . , a s }

s in c e then b o th s id e s o f

n > k, i t fo llo w s by in d u c tio n u s in g

k

. 2 b)

and i . 3 . L et

r ( n ,k )

(b)o f dim ension h .

be the e x t e r i o r a lg e b r a over

bd - 1 THEOREM.

7.

n < m and

k
H*(G(n,k) ;R ) . {b)

— > {b}

if

PROOF.

§5 . S0(n)

k ) ,

R, gen erated by elem ents

w ith n > b > k. r ( n ,k )

fo r

G (n,k) ----> G(m,£)

Under t h i s map

d = 2 or

k.

If

which in d u ces a map

{b} —— > 0

if

b > n

and

b < n. T h is fo llo w s from 4.6 in the same way

The P ontr.jagin Rings o f and

SU(n)

SQ(n)

a re the subgroups o f

c o n s is tin g o f m a tric e s w ith determ inant

1.

th a t 4 . 5 fo llo w s

and SU(n) . 0(n) and U(n) r e s p e c t i v e l y ,

The com positions

§5.

THE PONTRJAGIN RINGS OF

SO(n) - i - > 0(n) — > 0 ( n ,i) a re b oth homeomorphisms.

SO(n)

and

AND

SG (n)

51

SU(n) — > U(n) — > U (n ,l)

The reason f o r t h i s i s th a t b o th in r e a l and in

complex n -sp a ce , th e re i s e x a c t ly one way o f com pletin g an (n -1)-fra m e to an n-fram e w ith determ in an t i . SU(n)

w ith

We s h a l l i d e n t i f y

THEOREM.

The P o n trja g in r in g

a lg e b ra on the normal c la s s e s PROOF.

Cb]

of

The normal c e l l

determ inant

space o f

w ith

H *(S0(n);Z2)

H *(0 (n , 1 ) ;Z2) (see 3 .3 ) .

( i . j , . . . , i ) of

( - i ) r (see i . t ,

i s the e x t e r io r

0(n)

c o n s is ts o f m a tric e s

1 . 5 and 3 .2 ) . T h erefo re

0(n) ,c o n s is ts o f the normal c e l l s

S 0 (n ),

( i 1 , . . . , i 2r)

. . . [ i 2 r 3 [i3 .

T h erefo re by 3 .5 ,

Cb3C i3 becomes

5 .2 . mal c la s s

THEOREM.

(n > b > i ) .

a lg e b ra over

R

PROOF. H1 (SU(n);R)

L et

=

0.

From

T h erefo re

*

= 0

if

i* { b ) = 0.

The P o n tr ja g in r in g

we know th a t

fie s

i^. Cb3

By

H*(SU(n);R)

H*(SU(n) ;R)

Cb 3

=

be a n o r­

i s the e x t e r io r

o f dim ension

Ej (TJ(n,l) ;R)

U . 7 , we know th a t

k

where

i

[b 3 where

**Cb} =

n> b > 1.

.6 and in d u c tio n on

The d u al map =

Mapping in to

2b - 1 .

0.

T h erefo re

H*(U(n) ;R) -£ — > H*(SU(n) ;R)

(b)

bi = ifo r some

{b 1 , . . . , b r 3.

b > 1.

The com position

i s th e i d e n t i t y .

i* (i]

[ i 1 ] [ 1 3[ i g 3C1 ]

by 3 .6 .

[b3 e H*(U(n, 1 ) ;R) =

H *(U (n ,l);R )

so

[b3,

[b 3 [ 1 3 w ith

gen erated by th e c la s s e s By 3 .6 ,

n > i1 >

the image o f H*(SO(n) ;Z 2) - — > H*(0(n) ;Z2)

the e x t e r io r a lg e b ra on the elem ents

H *(0 (n , 1 ) ;Z2) ,

as a sub­

where

. . . > i 2r > 0 . By 3 .5 , the normal c la s s o f such a c e l l i s

is

0 ( n ,i) and

U ( n ,i) .

5.1.

of

SO(n)

Cb}

Moreover

where

H1 (SU(n);R)

r , t h i s shows th a t

and th a t o th erw ise

n > b> 1. =

0,

i * ( b 1 , . . . ,b p}

i * ( b 1 , . . . ,b p 3

i * : H#(SCJ(n);R)

> H*(U(n) ;R) th e r e fo r e s a t i s ­

n > b > 1.

i*

S in ce

i s a monomorphism o f

P o n trja g in r in g s , the theorem fo llo w s . We now i n v e s t ig a t e th e embedding n io n ic n -sp ace and l e t n io n

Sp(n) C

W be complex 2 n -sp a ce.

q = q 1 + i q 1 + j q 3 +kqu ,

where

(q 1 + i q 2) + j ( q 3 - i q ^ ) • Then a

q 1 ,q 2 ,q 3

column v e c to r

U(2n) .

L et V

L et us w r ite and

qk

be q u a te r-

ev e ry q u a te r­

a re r e a l , as

( x 1 , . . . ,x R) e V

becomes

IV .

52

a column v e c t o r

COHOMOLOGY OF CLASSICAL GROUPS

( y 1 , . . . , y 2n) € W,

g iv e s an i d e n t i f i c a t i o n o f

V

and

by w r it in g

x.^ = jy 2i-i + y 2 i

W as complex v e c to r sp aces.

*

T h is

The id e n ­

t i f i c a t i o n p r e s e rv e s the s c a la r product o f a v e c to r w ith i t s e l f (bu t not w ith another v e c t o r ) . p rod u cts in maps

¥,

T h erefo re e v e ry elem ent o f

and we have an embedding

Sp(n) ---- > V

and

Sp(n)

p r e s e rv e s s c a la r

Sp(n) -— -> U(2n) .

We a ls o have

U(2n) —— > W ob tain ed by ta k in g the l a s t column

o f a m a trix , o r, e q u iv a le n t ly , by ta k in g th e image o f the v e c to r ( o , . . . , o , i ) under an elem ent o f

Sp(n)

or

U (2n). Sp(n) —

The diagram — •> V

i

I

U( 2n) ------— > ¥ i s com mutative.

On i d e n t i f y in g u n it v e c t o r s in

V

and

¥

w ith

S^11” 1 ,

we o b ta in th e fo llo w in g commutative diagram 5 *3*

Sp(n) — —— > S p (n ,n -t)

I

g ln -1

V

I

U(2n) — — -> U (2n ,2n -i)

5 .1 . phism — > o,

THEOREM.

The embedding

H*(U(2n);R) ---- > H*(Sp(n);R) where n > b > o.

Sp(n) ----> U(2n) giv en by

( R e c a ll th a t

(b)

ind u ces an epimor-

(2b) ---- > Cb) and has dim ension

lb - i , a cc o rd in g as i t den otes a normal c la s s o f PROOF.

.

The p ro o f i s by in d u c tio n on

n.

U(2n)

or

I f we tak e

2b - i

Sp(n - 1) _— -> u(2n - 2)

I Sp(n)

I ------ — -> U(2n)

T h erefo re th e diagram H*(Sp(n - i)

By our

2n~2.

in d u c tio n h y p o th e sis and th e com m utativity o f the diagram , th e theorem i s 2b —1 tru e f o r c la s s e s {b} e H (U (2n )), where b < 2n-2, and we have o n ly to check the theorem f o r {2n -l} € H^n ~3(U(2n))

(2 n -i)

and

(2 n ).

For dim en sional re a so n s,

goes in to the su b alg eb ra o f

elem ents o f the form

Cm}

H14'111'’ 1 (S p (n ))

e

where

0 < m < n. But by A . 7, * H ( S p ( n - 1 ) ) . By the commuta­

t h is su b alg eb ra i s sen t m onom orphically in t o t i v i t y o f th e above diagram , we see th a t To fin d the image o f use i . 7 and the diagram 5 .3 . H^ n - i(u ( 2n)) i s sen t to

0 ( n ,i) ,

{ 2n} €

(u( 2n))

Since b o th

Cn} €

in

( S p (n )) and

§6.

Cohomology O peration s in S t i e f e l M an ifo ld s.

k

U(n)

. 5 and

and

By in d u c tio n on

i . 7 , we need o n ly know t h e i r a c tio n in

Sp(n) .

them in has

tio n s in

>

x 0 ( n - i , l ) i Z 2)

and t h e ir behaviour under c ro s s p ro d u c ts.

By 3.2

SCPn~1 v S1 ,so we need on ly know

and t h e ir behaviour under

c ro s s

if

d = 2,

the op era­

p rod u cts (se e I 2 . 1 ) .

To fin d the o p era tio n s in Sp(n) we u se 5 A t h e ir a c t io n in

.

we can determ ine cohomology o p e ra tio n s , i f we know

the homotopy type o f CPn~1

S0(n)

We have the monomorphisms

H*(0(n, 1 ) ;Z2)

n,

{2n}

cohomology o p e ra tio n s in the S t i e f e l m an ifold s as

n*: H * (U (n );R )---------- > H * ^ x U(n-1 ,1 ) ;R)

and

we

{2n} e

(S^31" 1 ) ,

and the theorem i s proved.

From

H *(S p(n )).

( S p (n )) ,

{n}

n*:

Qh

{2 n -i) goes in t o zero in

a re images o f the fundam ental c la s s in

We can compute f o llo w s .

H*(Sp(n)) gen erated by

and our knowledge o f

U (2n).

The on ly e x p l i c i t com putation o f o p era tio n s w hich we s h a l l c a r ry o u t, i s

the e f f e c t o f Sq1 U sing the n o ta tio n 6.1 .

THEOREM.

(n > b > k > 1 ) .

on k

H *(0(n,k) ;Z2)

fo r k >

1.

. 1 , we have

Sqbb)

= ( b 71 )cb

+ i) in

H *(0(n,k) ;Z2)

The C artan form ula then g iv e s the a c t io n on the oth er

cohomology c la s s e s .

5i

I V . COHOMOLOGY OF CLASSICAL GROUPS

PROOF.

Under the monomorphism

n*: the image o f

Cb)

By I if

i > o.

H *(0(n,k) ;Z2) is

2 . Aand

>H * ^ x 0 (n -l ,k) ;Z 2) ,

T x Cb) + Cb) x [ ) 3 .2 , Sq1 tb)

=

(see

( b©

k

.2 a )).

Cb + i - 1) and

Sq3!'

=o

The theorem f o llo w s .

We s h a ll o b ta in another d e s c r ip tio n o f th e Cb ( 2 ) -module s tr u c tu r e in terms o f

the d e f i n i t i o n s in

II §5 .

A stu n ted p r o t e c t iv e space p r o je c t iv e

n -space

We have a map

Pn

p£

is

the space ob tain ed from

by c o lla p s in g the

Pn -— -> P^,

( r - i ) - s k e le to n Pr ~1

the r e a l

to

a p o in t.

w hich in d u ces a monomorphism

H*(P£;Z2) ---- > H*(Pn ;Z 2) . L et

wg be

the n on -zero elem ent o f HS(P^;Z2)

n a t u r a lit y and

ws

=

( O ws +i

if

s +i < n

Sq1 wg

=

0

if

s + i > n

d

Qn/Qk =

^

and

Qh

> 0 (n ,k)

if

= 1,

Pk” 1

(n > k > 1 ) .

pk’’

-

> 0(n'k)

map i s a homeomorphism in t o .

d u ctio n on

i s tru e f o r

same image in Qn ~ Qh.-] •

It

0 ( n ,k ) .

n = k.

if

If

n < 2k,

Suppose

THEOREM. I f

a lg e b ra generated by PROOF.

,±r

|n,k)

x ,y €

have th e. x e

has dim ension g r e a te r than

T h erefo re the 2 k -s k e le to n o f

the n -s k e le to n o f

6 .2 .

L et us

A(n,k) T h is

0 (n ,k)

k >

1

tak e

is

, then

H *(P^'1 ;Z 2) .

d e fin e d j u s t b e fo re 1 . 5 .

(n > b > k ) .

• We prove t h i s cla im by in ­

Our claim then fo llo w s by 2 . 3 .

r > 2.

morphic to

The map

By our in d u c tio n h y p o th esis we can assume

By 2 . 1 , a normal c e l l ( i 1 , . . . 2k

.

in d u ces a map

We claim th a t t h i s n.

by

2 . k,

I

By 3 . 2 ;

fo r r < s (it,A,K)

be a be

Then the image

H*(K;A) ---- > H*(K;A) i s p o in tw ise in v a r ia n t under the automorphism PROOF. determ ined by

L et y.

f:

(p,A,K) ---- > (p,A ,K )

* g . be th e in n er automorphism

Then by 1 .2 , th e fo llo w in g diagram i s commutative

V.

60

EQUIVARIAWT COHOMOLOGY H*(K;A)

H*(K;A)

V

*

V

H*(K;A) — S--- > H*(K;A) F u rth e r , 1 .4 shows th a t

f* = i .

Cohomology o f G roups. A r e g u la r c e l l complex

K

i s a c e l l complex w ith the p ro p e rty th a t

the c lo s u re o f each c e l l i s a f i n i t e subcomplex homeomorphic to a c lo se d b a ll.

If

K

i s i n f i n i t e , we g iv e i t the weak to p o lo g y

— th a t i s , a s e t

i s open i f and on ly i f i t s i n t e r s e c t io n w ith ev e ry f i n i t e subcomplex i s open, ( i . e . ,

K

i s a CW com plex).

c a r r ie r from

K

to

a subcomplex

C(t)

of

of

K

and

i s a fu n c tio n

C

L

fa c e o f

such th a t a

Let

p

and

jt

L

be c e l l com plexes.

w hich a s s ig n s t o each c e l l

An a c y c l i c c a r r i e r i s one such th a t

C(t).

t € K.

L

L et

i s a c y c l i c f o r each

C(t)

and

L

( c o n s is t e n t ly w ith t h e i r c e l l s t r u c t u r e s ) , and l e t

h:

morphism.

C(orr)

fo r a l l $

An e q u iv a r ia n t c a r r i e r i s one such th a t a € p

and

i s c a r r ie d hy 2. 1 .

te K.

C if

L et

L

K x L p le x o f

If

r e g u la r complex.

and

CW com plexes.

CW to p o lo g y .

K

and

L

L et

p-subcomplex o f a

we have an e q u iv a r ia n t ch ain map a c y c li c c a r r i e r from2 .2 .

LEMMA.

K— > L

c a r r ie d by cpQ and

cp1 .

K

L

K

a c t s on

We g iv e

KT—

C.

If

I L.

is a K xL

K x L .)

Suppose we have an e q u iv a ria n t L

between

f ix e d and a c t in g as b e fo re

§2.

on

61

COHOMOLOGY OF GROUPS

K .) PROOF.

We arran ge a

o f in c r e a s in g dim ension. C (t)

p -b a s is f o r th e c e l l s o f

We must d e fin e

i s a c y c li c f o r each

t,

cp

I x K

and we can d e fin e a c a r r i e r from

j e c t i n g onto

K

and then a p p ly in g

2 .3 .

LEMMA.

a c y c l i c s im p l i c i a l complex PROOF.

We g iv e

cpd

-

is a

I x K

in order

dcp.

we can do t h i s in d u c t iv e ly .

o f th e lemma fo llo w s from the f i r s t , s in c e ( see 2 . 1 ) ,

so th a t

K - K!

S ince

The second p a r t

p - fr e e complex

to

L

by f i r s t p ro -

C.

G iven a group

*,

we can alw ays c o n s tru c t a

W. *the d is c r e t e to p o lo g y and form

the i n f i n i t e

rep e a ted jo in W T h is rep e ated jo in w ith a

=

Jt* Jt* IT ... .

i s a s i m p li c i a l complex.

p o in t g iv e s us a c o n t r a c tib le sp ace.

a f i n i t e rep e a ted jo in T h erefo re

W’ .

Taking the jo in Any c y c le in

o f a complex

W

must l i e in

Such a c y c le i s homologous to zero in

W’ * Jt.

W i s a c y c lic .

We make on each f a c t o r

jt jt

a c t on

W as fo llo w s :

jt

a c t s by l e f t m u lt ip lic a t io n

o f th e j o in and we extend the a c t io n l i n e a r l y .

T h is

a c t io n i s obviou s f r e e and the lemma i s proved . Suppose wehave tt-fre e

a homomorphism

complex and V an a c y c li c

v a r ia n t a c y c li c c a r r i e r from C (t )

= V.

n -----> p

p - fr e e complex.

W to

V:

and

W i s an a c y c li c

Then we have an e q u i-

f o r each c e l l

t

By 2.2 we can fin d an e q u iv a r ia n t ch ain map

a l l such ch ain maps a re e q u iv a r ia n t ly T h erefo re a map o f p a ir s a map o f a lg e b r a ic t r i p l e s

f:

e

W,

we d e fin e

W -— > V,

and

hom otopic. (*,A )

> ( p ,B)

( jt,A,W) ---- > (p,B ,V )

e q u iv a r ia n t homotopy o f th e chain map W

> V.

a s in 1. i le a d s to

w hich i s determ ined up to By 1.2 we o b ta in a w e l l -

d e fin e d induced homomorphism f* : In the c la s s o f

H*(V;B)

> H* (W;A) .

tt-fre e a c y c li c com plexes, any two complexes a re

e q u iv a r ia n t ly homotopy e q u iv a le n t, and any two e q u iv a ria n t ch ain maps go in g from one such complex to another a re e q u iv a r ia n t ly hom otopic.

T h erefo re the

Tt-free

62

V.

groups

H*(W;A), as

EQUIVARIANT COHOMOLOGY

W v a r ie s over the c l a s s , a re a l l isom orph ic to each

oth er and the isomorphisms a re unique and t r a n s i t i v e . i d e n t i f y a l l th e se cohomology groups and w r it e LEMMA.

2,k.

H*(*;A)

We can th e r e fo r e

H*(*;A)

in s te a d o f

H*(W;A) .

i s a c o n tra v a ria n t fu n c to r from th e c a te g o ry

o f p a ir s ( see 1 . 1 ) .

§3.

Proper maps.

Suppose we have a continuou s map p le x e s .

A c a r r ie r

C

fo r

f( x )

CC (t)

r ie r

w hich a s s ig n s to each

ta in in g f on

fo r a l l c e l l s

f(x ).

x

K

i s a c a r r ie r K.

e

x c K

f

> L

from

The minimal

c e ll

E very c a r r i e r o f

PPQPe r

K

betw een two

to

L

c a r r ie r o f

CWcom­

such th a t f

i s the c a r ­

the s m a lle s t sub complex o f

c o n ta in s the minimal c a r r i e r .

i f the minimal c a r r i e r i s c y c l i c .

h: jt----- > p

L,

f

f :

i s a homomorphism and

If

a c t s on

L

con­

We say

K,

p a cts

f i s e q u iv a r ia n t, then the

minimal c a r r i e r i s a ls o e q u iv a r ia n t . 3 .1 . L et

it a c t

and l e t

LEMMA.

on

K

L et

and

p

K -----> L

f :

K

and

a c t on

L L,

be f i n i t e le t

r e g u la r c e l l com plexes.

h:

it

— > p

be a homomorphism

be a continuous e q u iv a r ia n t map.

Then

f

can be

fa c to r e d in t o prop er e q u iv a r ia n t maps K — L_> k ' -- > l ' — where

K*

and

L*

PROOF.

> L

a re b a r y c e n tr ic s u b d iv is io n s o f

K

and

L.

The f i r s t b a r y c e n tr ic s u b d iv is io n o f a r e g u la r c e l l complex

i s a s i m p li c i a l c e l l complex, a s we see by in d u c tio n on the dim ension. L*

be the n ^ b a r y c e n tr ic s u b d iv is io n o f

open s t a r o f th e L. p le x

l ^ 1

v e rtex

x^

of

L

L f . Then

We can choose a b a r y c e n tr ic s u b d iv is io n t

of

Kf

mal c a r r ie r o f

i s con tain ed in a s e t o f the x

fo r

n > 1.

{U^}

K1

of

T h erefo re

f :

a re o b v io u sly p ro p er.

K f ---- > L 1

i s p ro p er.

be the

i s an open c o v e rin g o f K

such th a t each sim­

form f 1 ^ ) .

c o n s is ts o f sim plexes a l l o f w hich have

L* -— > L

Let

The i d e n t i t y maps

Then the m in i­ x^

as a v e r t e x .

K —— > K f

The maps a re a l l e q u iv a r ia n t .

L*

and

T his p ro ves

the lemma. Note th a t we were a b le to choose

L et

to be any b a r y c e n tr ic

§3.

su b d iv is io n o f

L.

PROPER MAPS

63

Note a ls o t h a t any s u b d iv is io n f i n e r than

e q u a lly w e ll i n the p la c e o f

K1.

K1

would do

Lemma 3.1 i s tr u e b u t more d i f f i c u l t to

prove i f the words " f i n i t e " and " b a r y c e n tr ic " a re d e le te d from i t s statem en t. 3.2.

DEFINITION.

The c a te g o ry o f geom etric t r i p l e s i s d efin ed in

the same way as the c a te g o ry o f a lg e b r a ic t r i p l e s (see *1 . 1 ) , ex c e p t th a t we r e p la c e the ch ain complex v a r ia n t chain maps

f^

> L

by e q u iv a r ia n t continuous maps.

K

and e q u i­

We say a map o f

i s prop er i f the continuous map

i s p ro p er. L et

W

by a f i n i t e r e g u la r c e l l complex

(*,A ,K ) ---- > (p ,B ,L )

geom etric t r i p l e s K

K

be a

f:

(ir,A,K) ----- > (p ,B ,L )

ir-free acyclic complex and

exist by 2 .3 ).

where the a c t io n o f =

If a c y c li c

p

f

a

p-free acyclic complex (these

*

H*P (V x L;B) ---- > H*(W x K;A) Jt on

Wx K

(aw,ak) f o r a l l

the a c tio n o f

V

L et

We wish to construct a map f* :

a(w ,k)

be a map o f geom etric t r i p l e s .

on

i s the d ia g o n a l a c t io n — th a t i s

a € it,

w € W and

k € K — and s im ila r ly f o r

V x L,

i s p ro p e r, l e t i t s minimal c a r r i e r be

e q u iv a ria n t c a r r i e r

from

Wx K

to any c e l l o f the form

w x t € W x K.

chain map

f^ :

> V L

homotopy.

If

W (tc,A,K1 ) -- > (p,B,l' ) — -> (p,B,L) and d e fin e

f^

a s the com position o f th r e e ch ain maps W® K

> W ® K' ---- > V ® L 1----- > V ® L.

For a proper map o f geom etric t r i p l e s

(ir,A,K) ---- > (p ,B ,L )

have two d if f e r e n t c o n s tru c tio n s o f an e q u iv a r ia n t ch ain map V ® L.

W®K

>

The f i r s t i s ob tain ed d i r e c t l y and the second i s obtained by f a c ­

t o r i z i n g in t o th re e maps. homotopy.

The r e s u l t s d i f f e r by a t most an e q u iv a r ia n t

I t i s ea sy to see th a t th e d e f i n i t i o n o f

up to e q u iv a r ia n t in 3 . i .

we now

f^

does n ot depend,

homotopy, on th e number o f tim es wesu b d ivid e

I t e a s i l y fo llo w s th a t i f (*,A ,K ) - L >

(p ,B ,L ) -£ -> (a,C,M)

K

and

L

V.

6b

EQUIVARIANT COHOMOLOGY

are maps o f geom etric t r i p l e s and i f then

g $ f|:

W0 K

t in g

L = I x K,

> U 0 M

U

i s e q u iv a r ia n t ly homotopic to

i t fo llo w s th a t i f

homotopic a s continuous maps, then as ch ain maps from

¥ 0 K

i s an a c y c li c e q u iv a r ia n t a-com plex,

h ,k : h^

and

k^

a re e q u iv a r ia n t ly homotopic

to U 0 M.

to a map o f a lg e b r a ic t r i p l e s we show th a t

2A

L e t­

K ----> M a re e q u iv a r ia n t ly

T h erefo re a map o f geom etric t r i p l e s r is e

(gf)^»

•X*

H^(W 0 KjA)

(it,A,K) — > (p ,B ,L ) g iv e s

(*,A,W 0 K) ---- > (p ,B ,V 0 L ) .

does not depend on th e c h o ic e o f

As in

W.

We

have th e r e fo r e proved 3.3

LEMMA.

H*(W x .K;A)

i s a c o n tra v a ria n t fu n c to r from the c a t e ­

go ry o f geom etric t r i p l e s .

Induced maps a re independent o f e q u iv a r ia n t

homotopies o f th e v a r ia b le

K.

If

3 A

and

K

jt

i s a normal subgroup o f

i s a f i n i t e r e g u la r c e l l complex on which

A

p,

p

i s a p-module

a c t s , then

p

a cts

on the geom etric t r i p l e

(*,A ,K )

by th e same form ulas as in 1 .3 .

p

by 3 .3 .

T h is a c tio n commutes w ith e q u iv a r ia n t maps

a c t s on

H*(W x KjA)

o f the v a r ia b le

K. has

* = p,

tr iv ia l

then

p

n - a c tio n ,

a c t s t r i v i a l l y on

by 1 1 .

If

H*(W x

K;A) can

by

(se e 1.3 ) to a map o f the a lg e b r a ic t r i p l e

7

K

If

be found by ex te n d in g the automorphism o f

Using the i d e n t i t y map on

K, t h i s g iv e s a map

which in d u ces the automorphism o f If duces to

K

then th e a c t io n o f

i s a p o in t then

o f (*,A,W

H*(W x K;A) y

e

p

( * ,A)

on induced

in t o i t s e l f . ® K) in t o

its e lf,

H*(W x KjA) . H*(W x K;A)

i s ju s t

H*(jt;A)

and

3.3 r e ­

2 A .

§b

L et

K

be a

jr-free

.

P ro d u c ts.

CW complex and

a c t s , and suppose we have a homomorphism v a r ia n t map

f :

K — > L.

the c e l l s w hich form a homotopy map.

(*,A,W)

T h erefo re

I x K — > L,

n

L

a

—> p

CW complex on which

p

and a continuous e q u i­

By an in c r e a s in g in d u c tio n on the dim ension o f

jt-b a s is

fo r

K,

we can c o n s tru c t an e q u iv a r ia n t

which s t a r t s by b ein g

f

and ends as a c e l l u l a r

T h is g iv e s r i s e to an e q u iv a r ia n t chain map f^ :

determ ined up to e q u iv a ria n t homotopy.

K — > L,

which i s

We can i n s i s t t h a t , d u rin g the

§u.

PRODUCTS

homotopy, the image o f each c e l l in f.

Then

f^

is

Now ch ain map

f

K/n

ind u ces a map ---- > L/p.

s t a r t s by b e in g

and l e t

f

K

s ta y s w ith in the minimal

c a r r ie r

of

c a r r ie d by the minimal c a r r i e r . g:

K/it

>L/p.

The map

f^

T h is ch ain map w i l l do f o r

v a r ia n t homotopy I x K

L et

K

65

> L

ind u ces a homotopy

in d u ces a

s in c e th e e q u iI xK/it

-— > L/p w hich

g and ends as a c e l l u l a r map. and

L

be p r o p e r .

way to th a t above.

be r e g u la r c e l l com plexes w ith group a c t io n a s above, Then we can choose

f^ :

K ----- > L

in a d iffe r e n t

We can sim ply a p p ly 2 . 2 u s in g the minimal c a r r i e r o f

However our p re v io u s ch o ice o f

f^

f.

was a ls o c a r r ie d by the m inim al c a r r i e r .

T h erefo re the two proced u res lea d to th e same r e s u l t (up to e q u iv a r ia n t hom otopy). L et a c t on

W be a

re-free r e g u la r c e l l complex and l e t

W x W by the d ia g o n a l a c t io n .

an e q u iv a r ia n t proper map. i.1.

LEMMA.

a ch ain map in

L.

If

L

The d ia g o n a l

d:

L - W/it.

By the d is c u s s io n above we have W ind uces

which i s homotopic to a d ia g o n a l approxim ation

W i s a c y c l i c , then any e q u iv a r ia n t chain map

w i l l induce a map

*

W •— ~> W x W i s

Any e q u iv a ria n t d ia g o n a l approxim ation i n

> L L

L et

L -— > L ® L

W — > W W

w hich i s homotopic to a d ia g o n a l a p p ro x i­

m ation. L et

(*,A,M)

we have a t r i p l e

and

(p,B,N)

be a lg e b r a ic t r i p l e s (se e i . i ) .

(it x p,A ® B,M ® N) .

Then

We have a map

C*(M;A) ® C*(N;B) ---- > C*Xp(M ® N;A ® B) d e fin e d in an obvious way.

T h is g iv e s us a c ro s s -p ro d u ct or e x te r n a l -

prod u ct p a ir in g H*(M;A) ® H*(N n p jB) ---- > H*3tXp (M ® H;A ® B) . L et

W be an a c y c l i c

p - fr e e complex. cro s s-p ro d u ct

L et

(*,A ,K )

it-fr e e complex and l e t and (p ,B ,L )

Then the

above g iv e s us a map

Now the a lg e b r a ic t r i p l e s p,

be an a c y c l i c

be geom etric t r i p l e s .

H*(W x KjA) ® H*(V x L;B) --- > H*71 p (W x Tv P

(« x

V

(it

A ® B , W® V < g ) K ® L )

K

x V x L;A ® B) .

x p, A ® B , W®KV(8>L) and a re isom orphic v i a the map

w hich in terch a n g es

V.

66

V

and

K

EQUTVARIAM1 COHOMOLOGY

(w ith a s ig n ch an ge ).

W0 K 0 V 0 L

Here the a c t io n

irxp

a € *, p e p on

p

on

i s g iv en by ( a , 3 ) (w 0 k 0 v 0 £)

fo r a l l

o f xr x

*

(aw0 ak 0 pv 0 p£)

, w € W, k e K, v

W0 V 0 K 0 L

€ Vand

£ e L.

The a c t io n o f

i s g iv e n by

(a, p) (w 0 v 0 k 0 £)

=

(aw0 pv 0 ak 0 p£) .

T h erefo re we have an isomorphism between H* (W x K x V x L;A ® B) and H* (W x V xK x L;A ® B ) . Sin ce W x V i s JfXp JTXp a (jtx p )-fr e e a c y c l i c com plex, we see t h a t , composing t h i s isomorphism w ith th e c ro s s-p r o d u c t above, we have in tro d u ced a c ro s s-p ro d u ct ( it. 2 )

H*(W x K;A) ® H*(V x L;B) ---- > H*^xp (W x V x K x L;A ® B) ** p

d e fin e d on the fu n c to r o f 3. 3.

The image o f

u 0 v

i s denoted by

u x v.

We have a d ia g o n a l map o f geom etric t r i p l e s d: where

*

(jt,A 0 B,K) ——> (xr x * ,A 0 B,K x K) ,

a c t s onA 0 B

in the f i r s t t r i p l e by the d ia g o n a l a c t io n .

Hence wehave a map (se e 3.3) d*t where

jcxjt

H*X]t(W x ¥ x K x K;A ® B) ---- > H*(W x K;A ® B)

a c t s on

Wx Wx K x K

by

(a ,P ) ( v 1 , v 2 , k 1 ,k 2) fo r a l l

a ,p c

product o f

v 1 ,v 2

k.2,

7(

W and

(OT1 ,e v 2 ,a k 1 ,e k 2)

k 1 #k 2 € K.

Combining

d

w ith the c r o s s -

we have th e cup-product p a ir in g H*(Wx K;A) ® H*(W x K;B) ---- > H*(W xK;A ® B) .

(i t. 3) If

€

=

i s the t r i v i a l group, t h i s i s th e u s u a l cup-product i n

K. I f

K

a p o in t, then t h i s i s the u su a l cup-product in th e cohomology o f a group. k.h.

prod u cts in W.

REMARK. L

L et

W and

L

T h is i s p a r t i c u l a r l y u s e f u l when §5* Let

be a s i n ^ . i .

We can compute cup-

by c o n s tr u c tin g an e q u iv a r ia n t d ia g o n a l approxim ation in L

i s n ot a r e g u la r c e l l complex.

The C y c lic Group.

W be the u n it sphere the space o f i n f i n i t e l y many complex

is

§5.

v a r ia b l e s . where

That i s , ev e ry p o in t in

z^z^

Z

THE CYCLIC GROUP

=

1.

We g iv e

be d escrib ed as the

67

W has the form

( z Q, z 1 , . . . , z p , 0 , . . . )

W th e weak to p o lo g y .

A lt e r n a t i v e ly

W may

CW complex ob tain ed by ta k in g th e union o f the sequence

s 1 c s 3 c s 5 c __ L et

n

be any in t e g e r g r e a te r than one.

L et

T:

W — —> W be the tr a n s ­

form ation d e fin e d by

T(z0 , z ^ , . . . ) where

x

T

=

o f order

it

=

(Xz0 ,xZl,...)

o b v io u s ly a c t s f r e e l y and g e n e ra tes a c y c l i c group

n.

We now c o n s tru c t an e q u iv a r ia n t c e l l decom position f o r makes

W a r e g u la r c e l l complex.

so as to g e t

n

o -c e lls

e 1 , T e 1 , . . . ,Tn_1e 1 . S

pyj , 1

=

S

1

* S

1

Let

w ith the s e t o f p o in ts

=

( T - i ) e Q.

(where

*

means j o i n ) .

S

2 r - c e lls o f

l e t the

( 2 r + i ) - c e l l s be o f

have

c e l l s in each dim ension. L et

S2r+1

S2r+1 2 r—1

the form

+ Tn ~1

and

ft = Z0 0

and letu € Hr (K;Zp ).

K ; Z ^ ) is invariant under 7 €p,

for some odd number

7

K

7 .)

H2i+r(W x

Since

Let

*-action.

r > 0

w 2i->1 x u

i = m(p-i)

is invariant if and only if PROOF.

with trivial

K ; Z ^ ) is invariant under 7 €

H^i+p(W x

invariant if and only if

Then

be

be

Z ^

The following two lemmas will be important in Chapter VII. be afinite regular cell complex

be its

T.

has trivial

H*(W x K;Zp)

p

act on

and T. For i.

is an even permutation.

*-module and so if

H*(W x K;z£q))

= Tk(i).

and is thus a power of

as follows.

sign of the permutation.

T,

S(p)-module which is

S(p) -module.

is a trivial

p

. Then

7”ioTo:~1 7i * Therefore

i+ k

permutation as we see by letting 7

is an odd

be the

Then

suppose a €

7

=

if and only

m, and w 2 i - 1 x u e H2d*r""!(W x KjZp^)

i * m(p-i) / 2

for some odd number

is an.odd permutation, the map

g:

m. (ir,Zp^)

-- >

V.

70

(jr,Z p ^ )

induced by

g2 :

Z W w hich i s

and l e t

k

g 1 - e q u iv a r ia n t map.

the fo llo w in g form u las.

i s odd and

i s even;

Tk .

W ith

g^e2 i

q

g$e 2 i + 1

=

a s an in t e g e r ,

k 1 Ek ~Q

L et

©2 i +i *

1 < k < p .)

We e a s i l y check th a t N

and

a

=K

and

g.,(A)

K

and l e t

g^

g 1 - e q u iv a r i­ ^In

We extend

g^

to

i s a ch ain map b y u sin g

be the elem ents o f

Z(jt)

d e sc rib e d

Then g 1 (N) L et

a

be an

t a t i v e f o r the c la s s

r -c e ll of

u e Hr (K ;Z p ).

g# (w2i x u)-(e2i x a)

= Tk - 1. u denote a coch ain r e p re se n ­

Then

= g2 [(w2i ’ s# e2 i ^ u

=

'

g 2 [ki(u • a )] Ic^u • a)

■ (-k ^ u

* cr)

if

q

i s even

if

q

i s odd.

T h erefo re gf (w2 i x u)

r k 1 (w2i x u)

if

q

i s even

L

if

q

i s odd.

-

x

u)

A lso g#(w2i+1 X u ) . ( e 2 i +1 X a)

=

( - i ) r g2 [(w2i+1

. S |e 2j_+1) (u • 5 )]

=

( - l ) p g 2 [w2±+1 . k 1 Z ^ T ^ 2 . + 1 ](u • a)

=

( - D r g 2( Z j : ’ ki ) ( u • a)

=

( - 1 ) P g 2 (ki + 1 ) ( u •

a)

( - 1 ) P ki +1

if q

i s even

t ( - 1 ) P+1 Ic1 * 1

if q

i s odd.

r

T h erefo re r

8# C*(K;A)

i* :

H*(K;A)

> H*(K;A)

C*(K;A)

> C*(K;A)

in d u cin g a map

We d e fin e the t r a n s f e r t:

as fo llo w s :

if

u € C*(K;A) ™

and

• 0

c

=

€ K,

then

Za € PA

' “' V

where

a ran ges over a s e t o f l e f t c o s e t r e p r e s e n ta tiv e s

U^o^jt

=

p

and

the d e f i n it i o n o f If

3€

p,

n

=

t i s independent

f o r any f ix e d 3,

s e n ta tiv e s f o r t

i / j.

We check im m ediately th a t

o f th e ch o ice o f c o s e t r e p r e s e n ta tiv e s .

then 3 ” 1 (tu * 3c)

s in c e ,

0i f

{a^} — th a t i s

it

in

i s a ch ain map.

=

Z 3 ~ 1c^u . a “ 13c

the s e t (3~1ai } p.

T h erefo re tu

T h erefo re t:

t

7.1.

H*(K;A)----> H*(K;A) Jt P

of

p-com plexes it

in

K.

p— th a t i s , the

Co^} .

LEMMA. The com position C*(K;A) - L > C*

i s m u lt ip lic a t io n by PROOF.

I t i s immediate th a t

p

in d u ces a map

[ p: jt]denote the in d ex

elem ents in the s e t

tu

i s a se t of l e f t coset rep re­

c C*(K;A) .

which i s n a tu r a l fo r e q u iv a r ia n t maps o f Let

=

If

tu * c

C*(K;A)

Cp: jt]. u e C *(K ;A ), P

7

= Zi ci^u . o ^ c

then =

Z^

u

. c

=

[p :jt] u* c .

number o f

V.

72

L et

j

EQUIVARIANT COHOMOLOGY

be a subgroup o f

t a t i v e s o f double c o s e ts ix

n ( z~1az)

and

a

„

z

range over a s e t o f r e p re s e n ­

a

and

it

in

zjtz~1 n cr.

o f the in n e r automorphism o f an isomorphism.

L et

of

oztx

=

z

p.

p

L et

ad„z

adz

We w r it e

it

=

be th e r e s t r i c t i o n to

induced by z .

We a ls o denote by

p.

Then

adz :

«----- >

z

is

az

th e homomorphism

C*jt(K;A) ---- > C* (K;A) z z g iv e n by

adzu . c

=

z(u • z~1 c)

where

The rem ainder o f t h i s s e c tio n i s

c £ K. n ot

re q u ire d elsew h ere in th e se

n o te s. 7. 2.

LEMMA.

The fo llo w in g diagram i s commutative

C*(K;A)

1_ > C*(K;A)

C*(K;A)

— L —>

f *z * V Zz C^n(K;A) PROOF. oz

in

a.

L et

By

w h ile keep in g

yz

2 ad * --------------------------------- C0^(K;A)

range over a s e t o f l e f t c o s e t r e p r e s e n ta tiv e s o f

and z

we s h a l l mean ta k in g unions or sums over

fix e d * tfzzjt

yz,

Now =

z(z”1az z) tc

=

z(z«)jt

=

zix.

T h erefo re a ztx

and so

-

Uy

p =

z z

U_

z

z it

a ztx -

=

Uy

z

U_ U y_ z

y

z

ztx

ztx

We e a s i l y check th a t th e l a s t i s a d i s j o i n t u n ion.

Hence the elem ents

ran ge over a s e t o f r e p r e s e n ta tiv e s o f l e f t c o s e ts o f Suppose u e C*(K;A) Zz Tz adz ^ z u) •

and c

c =

€

T h is proves the lemma.

in

K. Then

Zz hr yz tadz (1zu)

=

=

sr

•

?~zo]

J z z l x zn ( z ' ^ z c ) ]

Zz,7

yzx[u * (yz2)_1c]

p.

yzz

§7.

Nowtake

cr =

THE TRANSFER

it.Then

73

z * = * n z~1itzand

zitz~1 n it.

it =z

Let

m = [p: Jt3. 7.3.

LEMMA.

If

p

is a prime not dividing

m,

then the composi­

tion H*(K;A) -i-> H*(K;A) -i-> H*(K;A) is an isomorphism of the p-primary part of PROOF.

By 7.1, t I

Hp(K;A).

= m. Also multiplication by

m

is an isomor­

phism on the p-primary part of an abelian group. 7 A.

=

i It u z

PROOF. choose

mT

Let

for

all

and suppose

so that

ad_Z i _Ti„ u = iTt u for all z z m m ’ = i modulo ps. Then by 7 , 2 *

z

If

z € p .

Suppose that

ixu

where

u € H*(K;A)

psu = 0 .

adz i * ^ i^ u for all z € p, then u is the imageunder i of some v € H (K;A) z p such that psv = 0 . If uis the image of some v € H*(K;A) then ad„Z i 7t_ u z

LEMMA.

t

jt2

ad„ i ^ u z z ic

a

£

z

t

xz

i

z € p.

We

u ,

nz

the sum ranges over a set of representatives of double cosets

jczic in

p. From the first paragraph of the proof of 1 , 2 , we see that as

runs through a set

of left coset representativesof

runs through a set

of representatives of double cosets

y_z form a set of left coset representatives of z Then Z2 m 2 a m. Hence iTU

=

2

T

i z

=

Therefore, on putting

Zz mz u

it in

k

in it,

and

z

irzit,

the elements

p. Let

rnz » [it:itz ]

U z by 7.1

v = T m ’u, we obtain the first assertion of the lemma.

The second assertion follows directly from the definitions. proves the lemma.

y2

This

V.

7k-

7.5. LEMMA. prime to

[put],

X

H (K;A). If u of

Let

Then

7 .6 .

if and

H*(it;A)

H (it;A)

X

H^KjA),

Let

is

only if

ad_Z —u

K

m H^(p;A) = 0

H^(K;A) = 0

if

q > o.

it be a Sylow p-subgroup of

-X

part of

u

H (p;A)

such that

We note that =

0

|jt[

=

ps

(p finite). H*(p;A) -- >

i:

isomorphicallyonto

ad^Z i ^ u

=

i

and

m

=

in positive dimensions.

Tf

u

the sub­

for each z €

p.

Z

[p:ir]

So

¥e

is prime to

H^(jt;A)

p.

is a p-group

The rest of the lemma follows from 7.^ and 7.3.

7.8.

PROPOSITION.

jt be a Sylow p-subgroup of

Let p.

it be a cyclic group oforderp,

and let

Let

in

a

be the normalizer of k

Then the monomorphic images of the p-primary parts of (in positive dimensions) coincide. elements of

H*(it;A)

PROOF.

by 7.7

z e p.

The lemma follows.

p

a p-group in positive dimensions and

By 7 .6 , psH*(jt;A)

Therefore

for all

jt= 1 .

then

Z

q > o.

u

be a p-free acyclic complex, and let

Let

groupof those elements

for

=

q>0.

m = |p|,

maps the p-primary

PROOF.

be

it is in the image

for

If

7.7. LEMMA.

X

p

This is immediate from 7.3 and 7 . k .

apply 7.1 and use the fact that

Then

p, and let

is an isomorphism of the p-primary part of

ri

H*(K;A), JT

LEMMA.

PROOF.

Jt be a normal subgroup of

is in the p-primary part of

i:H*(K;A) -— > P PROOF.

EQUIVARIANT -COHOMOLOGY

Since

p,

and

H*(cr;A)

The image is the subgroup of those

which are invariant under |jc| =

H*(p;A)

p.

a.

we have

jt n z”1jtz =

1

if

z 4 a

jt n z “1jtz =

jt

if

z € a.

and

i

» i = o in positive dimensions, if z 4 a , Therefore z z the conditions for an element to be in the p-primary part of

X

Im(H (p;A)) part of

are the same as the conditions for it to be in the p-primary

Im(H*( cr;A)) .

If

z € a,

then

adz: H*(*;A) -- •> H*(jt;A) is the automorphism induced by

z~ 1 (see 1 . 3 and the definition of ad ).

§7.

THE TRANSFER

75

This proves the proposition. 7.9* elements of

LEMMA. A

H°(tt;A)

under

jr.

is isomorphic to the subgroup of invariant

This isomorphism is natural for maps of

(*,A)

(see 1 .1 ). PROOF.

This follows immediately from the definition of

since an acyclic

jr-free complex must be connected.

7 .1 0 .

duced map of

A

H*(jt;A),

COROLLARY.

H°(p;A) — -> E°(jc;A)

which are invariant under

If

it C p

and

A

is a p-module, then the in­

has an image consisting of those elements p.

BIBLIOGRAPHY [1 ] C. H. Dowker,

’’Topology of Metric Complexes,"

Am. Jour, of Math.,

71 (1952) pp. 555-577. (2 ] H. Cartan and S. Eilenberg, Finite groups,

"Homological Algebra,"

Princeton University Press (1 9 5 6 ).

Chapter 1 2 :

CHAPTER V I .

Axiomatic Development of the Algebra

In

§1

we give the axioms for the

Steenrod algebra

P1 .

Cfc(p).

In §2 we define the

Cfc(p) and show it is a Hopf algebra.

the structure of the dual Hopf algebra. those in the mod 2 case.

In §3 we obtain

The proofs are very similar to

In §5 ve obtain some results about the homotopy

groups of spheres and in §6 we derive the Wang sequence .

§1 .

Let

p

Axioms.

be an odd prime and let H^XjZp) -- > H?1 + 1 (X;Zp)

3:

be the Bockstein coboundary operator associated with the exact coefficient sequence ° _ _ > Zp _ > We assume as known that

&

y

—

> Zp —

> 0.

is natural for mappings of spaces, that

p £ = 0

and that P(xy)

(0x)y + (-i)q x( 3y)

=

where

q = dim x.

We have the following axioms 1 ) For all integers

i > 0

and

q > 0

there is a natural transformation

of functors which is a homomorphism P1 : H^CXsZp) ■— -> Hq+21(P~1) (X;Zp ) . 2 ) P° = 1 . 3 ) If dim x = 2 k,

then

k)

If

2k > dim x, then

5)

, Cartan formula.

P^Sc P^x

=

xp .

=

0.

Pk (xy)

=

Z1 Pdx •Pk-iy .

76

§2.

6)

Adem relations.

a < b

If

a < pb

77

a (p )

then

=

^[a/p] ^_^a+t ^(p-i) (b-t)-1 ^ pa+b-tpt

pa£Pb =

p[a/p] ^_^a+t Ap-i) (b-t)"\ & pa+b-tpt t=o a-pt J

papb If

DEFINITION AND PROPERTIES OF

then

^[(a-i)/p] ^_^a+t-i ( (p-1 ) (b-t)-A pa+b-t^ pt t=o a-pt-1 J

+

We shall prove the axioms in Chapters VII and VIII and we shall show that the

other axioms imply Axiom 6 ) . As in Chapter I, we can show

that, in the presence of Axiom i), the Cartan formula above Is equivalent to Pk(x X y) P1

We can also show that s:

=

I F^x X pk_1” y

commutes with suspension and with >Hi+ 1 (X,A;Zp)

H^AsZp)

as in I 1 . 2 and I 2 .1 . Similarly

38

=

-8 3

3s

and

=

-s3,

where

s

is the suspension.

§ 2 . Definition and Properties of

Ct(p).

We define the Steenrod algebra &(p) to be the graded associative P1

algebra generated by the elements gree 1 , subject to in

Ct(p)

3

2

2 i(p-i)

= 0 , the Adem relations and to

and

0

3

of de-

= 1 . A monomial

P

can be written in the form e 0 si ei

where

of degree

=

0,1

and

sp

sk ek ... P K3 ^

3

P

=

1 , 2 , 3 -----

p

Wedenotethis monomial by

P^,

where I A sequence

I

=

is called admissible if

The corresponding

P^,

and also

We define the moment of the degree of 2 .1 .

(e0 ,s1 ,e1 ,sa,...,s]c,ek ,o,o..O.

P^,

I to be

P°,

PROPOSITION.

for each

i > 1.

will be called admissible monomials.

£ i(si + e^). Let the degree

which we denote by

of admissible monomials.

> psi + 1 +

of

I

be

d(I) .

Each element of

Ct(p)

is a linear combination

V I.

78

AXIOMS FOR THE ALGEBRA

f t (p)

As inI 3 .1 , we see by a straightforwardcomputation

PROOF.

the Adem relations express any inadmissible monomial asthe of smaller moment.

The proposition then follows by induction on the moment.

We shall investigate lens spaces.

Gfc(p)

LEMMA.

Let

x

space such that dim x = 1 that

P^x

=0

and

PROOF.

y

be mod p cohomology classes in any

and dim y = 2 . Then

unless i = 0

pV

2) .

by letting it operate on a product of

We first prove some lemmas

2 .2 .

imply

that

sum ofmonomials

-

and

( i ) yk+i(.p-1} ■

k = 1 , the result follows

For

For k >1 , it follows

Axioms 2 ),3 ) ,1 ) and 5)

by induction on

from k

§1 Axioms M,3)

and

and the Cartan formula.

2 .3 .

that i .

LEMMA. If y is as in 2 . 2 then Axioms 2 ),3 ),!) and 5) imply k k v k+1 P (yp ) is yp if i = 0 ; zero if i £ °,P'; and yp if

Pk. PROOF.

This follows immediately from 2 . 2 and I 2.6.

Let

be a cohomology class of dimension

u

q.

Let

I

be a sequence

(eQ ,sQ,e 1,s.j,..., sr ,e ,0 ...) . Then we have the formulas

of the form

Let

6 (u

x v)

= |3uxv + (-1)^ u

Pk(u

x v)

= Z Pxu

PX (uv)

=

PI(u x v)

=

L

and

X

x 3v,

Pk_1v ,

£fc+j=x ( - D q 'd(J) P ^ i W v

,

PKu x PJv .

e ff^LjZp)

be as in V §5.

Let

X = Lx...xL = L2n.

Let un where

x

w1

= 2 .1 .

where

I

of degree

and

= y

y x x ’x y x *

=

PROPOSITION.

xyxx e H3n (L2n ;Zp)

-wg. The elements

ranges over all admissible sequences

are linearly independent, (£Q ,s1 ,e1 ,...,si ,ei ,o,...)

< n.

PROOF.

Let

J. = T* = Jv

(o,pk_ 1 ,o,pk“2 ,...,o,p1 ,o,p°,0 ,...) and ^.1 ,0,p — 1 rs \ ,0,p ,...,0,p ,1,0,...).

( r\ —,k~ 1 ^ „k- 2

(°,P

§2. 3x = y

Recall that unless I

=

I

and

with J! (0 ,0 ,...): P ^ x

unless I

DEFINITION AND PROPERTIES OF

is

3y = o

(see V 5 .2 ).

G (p)

79

Therefore, by 2 .3 , P^x = 0

a number of pairs of adjacent zeros inserted, or k = yp and P°x = x. Also by 2 .3 , p^y = 0

H

is Jv with a number of pairs of zeros inserted, or J k = (0 ,0 ,...): P S ’ = yP a^d P°y = y. We note that P^(xxy) = 0

If there is more than one non-zero

in

We prove the lemma by induction on since the only monomials of degree 2m+i.

The Kiinneth theorem asserts that H ^ C L 211)

- ZS)t HS(L) ® H^L) 0 flq+3n-s-t(L2n-2) _

Let

gm

be the projection onto the factor with

h^

bethe projection onto the factorwith (1 )

Let

I

P1^

=

PI(yxxxun_1)

be admissible. (2 )

and

t = 1 . Let

t = pm .

=Sj+K+L=I (-i)d(L)PJy x

P1^

x PLun _1 .

if

4(1) < 2m+i,

then

=

if

4(1) = 2m+i,

then I >

and

0,

and

( V y x y i ^ x P ^ v , ,

i * deg (I - J^). We also assert that (3 )

if

4(1) < 2m, then

gmPI uR

=

if

4(1) s 2m, then

I > Jm

and

=(-1 ) 1 yP®

^ P 1^ where

= 2

s

and

We assert that

. where

s = pm

0 and

x x x P1" ^ un „l;

i = deg (I - Jm ). To prove (2 ) and (3 ), we refer to the first paragraphof this proof.

We

note that a sequence obtained from

greater than2m+i,

has length greater than 2m.Therefore We can now apply(2 ) Since

a^ = 0

by inserting zeros has length

and a sequence obtained

for 4(1)

from

Jm

byinserting

(2 )and (3 ) follow from (i).

and (3 ) to our decreasing induction on

> 2m+i,we see by

zeros

4(1).

applying (2 ) to ourrelation that

V I.

80

AXIOMS FOR THE AlfiEBRA

C t(p )

( -D 1

=o.

As

I ranges over all

admissible sequences oflength (2m+i)

I-

ranges over all

admissible sequences of length < 2m

q - 2pm +i .

By our induction onn,

we have

and degree

q,

and degree 4(1) = 2 m+i

a^ = o when

.

Now applying (3) to our relation, we see that yP"1 x x x As

=

0.

admissible sequences of length 2m and degree q,

I ranges over all

I - J

=2m (-1)1 ajP1 JlD un_1

< 2m

ranges over all admissible sequences of length

q - 2pm + 2 . By our induction on

n, we have

a^ = o

and degree 4(1) = 2m.

when

This completes the proof of the proposition. Combining 2. i and

tion

on

2.5.

THEOREM.

2.6.

COROLLARY.

The admissible monomials form a basis for mapping

Ot(p)

in degrees

< n.

PROOF.

Any(k 4 p1) is decomposable. r»i and Pp ( i = 0 ,1 ,2 ,...).

3

by

By the Adem relations,

( ^ _1

^ 0 a + b

0 £

< p

and

(p-i)b - l

(5t(p) .

> H*(L2n) given byevalua­

THEOREM.

is generated

where

The

un , is a monomorphism 2.7.

and

we obtain

pa+t)

Therefore

j_s decomposable if

Cfc(p)

a < pb

moci p * = .k =

kQ + k,p1 + ... + l^p®

k^ 4 o .

L et

b = pm . Then

=

(pm- 1 ) + (p-2 )pm

=

(p-i)(i + p 1 + ... + p®”1) + (p-2 )pm

=

kQ + k,p’ + ... + km _,pm 1 + (k^- i)p®

Now a So by

=

k - b

12.6, ( 3 ) , then

where

k - i or 2 .

We have a commutative diagram C* (K;Z) 0 Z

--- *

>

C*(K;Z)

1

i

C*(K;Z) ® Zp

1---> C*(K;Zp)

where the vertical map on the right is the coefficient homomorphism, and the lower horizontal map makes the diagram commutative. By the universal coefficient theorem for

C (KsZ) ® Z , we have

an exact sequence * 0 ---> H?(K;Z) ® Zp —

Since

H*(K;Z)

> Tord # " 1 (K;Z) ,Zp)

> H^(K;Zp)

is free, the third term is zero.

> 0.

Therefore using the com­

mutative diagram above, we see that the coefficient homomorphism eP(K;Z)

-- > H^(K;Zp)

induces an isomorphism H^KjZ) ® Zp

*= H^(K;Zp) .

Since the coefficient homomorphism is a map of coefficient rings, this iso­ morphism gives an isomorphism of rings. * Therefore H (K;Zp) is a polynomial ring on one generator dimension

p = 3

that

x 2 / 0 , we see from I k .5 that 2.10

^0)

xp = 0 , where

(possibly truncated by

we see from 2 . 8 for

x5 ^ 0, Since

2k

=

THEOREM.

3 ® 1 + 1® 3

The map of generators

Therefore *(Pk)

extends to a map of algebras Ct(p) -- > G (p) ®

G (p) .

of

t > 3 )* Since

k = m31, where

k = 2 ^.

x

m = 1 or 2. k = 1 or 2.

=I P 1 ® P ^ 1

and

V I.

82

PROOF. tute

L

Pn

AXIOMS FOR THE ALGEBRA

C t(p )

II 1 .l. We merely substi-

The proof is the same as that of

for the n-fold Cartesian product of infinite dimensional real

projective space, and substitute 2.ii.

THEOREM.

G(p)

u^

for

w.

is a Hopf algebra with a commutative and

associative diagonal map. PROOF. As in II 1.2.

§3.

2 .5 .)

The Structure of the Dual Algebra.

Let

G(p)*

be the dual of

Then

G(p)*

is a commutative associative Hopf algebra with an as-

sociative diagonal map. the dual of

Mk

=

Let

P^k

ik

G(p).

( G (p)

be the dual of

is of finite type by

Mk = P K

and let

in the basis of admissible monomials.

f

be

(J^ and

Yr

Jk are as in 2.1.) Then ik has degree 2(p - i) k 2 2p - 1. Since has an odd degree, = °.

and

has degree

We define t

(0)

6(1)

=

x(0) y(i)

!0 =

= = =

1,

r(i)

for i > i,

,

for i > 0,

x,

x(i)

=

n1"1 3T

yP1

for i >

i,

for i > o, •K*

where I

Let

=

x

and

y

(ij,...,! )

g(I)

zeros in

are the classes in

H (L;Zp)

described before 2 . k .

be a sequence of non-negative integers. x(I)

=

Let

We define

*(!.,) ... r(im ) e a(p)*

5(1)

=

5(i,) ... 5(im) € a(p)*

x(I)

=

x(i1)x...x x(im ) c H*(Lm ;Zp)

y(I)

=

y(i,) x...xy(im ) £ H*(Lm ;Zp)

be the minimum number of transpositions needed to transfer all I

to the right of

I.

The following lemma will enable us to determine the structure of

3 •1 • LEMMA.

Let

a € Ofc(p).

a(x., x .. .x xn x y 1 x...xym )

=

Then

Ej^j(-1) g^I)< •t(I) I(J) , a > x(I) x y(J)

§3.

THE STRUCTURE OF THE DUAL ALGEBRA

where the summation ranges over terms where I length less

and

a

n

and

J has

have the same degree.)

PROOF.

We prove the formula by induction.

= (0 ,1 )

or

M1 k

length

(The summation is finite since we get a zero contribution un­

t (I)|(J)

(n,m) or

m.

has

83

(i,o),

since non-zero

It is true for

terms occur only when

a =

by 2 .2 , 2 . 3 and V 5 .2 . Now suppose the lemma is true for

(o,m-i).

Let

ycc

=

£s

a”.

By the Cartan formula «Cy, X...X ym)

=

Es a's y, x a^(y2 x...x ym)

< 5(^ > as > < where J = (j,Jf)

This proves the lemma for

=

Z< |(j) ®

® a” >y(J)

=

Z< 1 (3 )

*

Z< V*(£(j) ® |(J')) ,a >y(J)

=

Z< |(J),a >y(J)

S(J!),1ra>y(J)

(0 ,m) .

Suppose the lemma has been proved for formula

(n-i,m). By the Cartan

deg a 3

a(x, x...x xn x y, x...x yffl) = 2 s(-i)

where =

E g ,I ?J

5

mod 2

a”

= (i ,I ’)

6

and

7

=

>x(I) x y(J) deg a” + g( I ’)

® T (I ' ^ ( J ) >a s * “s >

=

deg

+ g(I') + deg

for the non-zero terms of the sum.

term of the sum is non-zero, and

I < 'r^

where We must compute

ag *•, x «g(x2 x...x xn x y, x...xym)

( - 1 ) 7 < f(i),a's > < t(I 1) |(J)

=

t (I')^J)

>^(J)

then

t(i)

and

have the same degree.

Since

x

(deg t(I')l(J))• Now, if a

have the same degree, and |(J)

has

evendegree, we

have mod 2 5

=deg t(I 1) + g(I’) + deg T(i)-deg t(I’)

Since the number of

non-zero terms in

I*

is congruent mod

2 todeg t(I!),

V I.

we see that if

AXIOMS FOR THE AIGEBRA

i = 0, 5

If

i4

O t(p)

othen

s

deg r(I') + g(I')

deg r(i)

=

l

and

s

=

=

g(I) .

g(I')

=

g(I). This proves

that the expression above is Z (-i)e(I)
X(I) x y(J)

=Z (-l)SC1) < T(I)e(J),a > x(I) x y(J) . This proves the lemma. Let

&'

denote the free, graded, commutative algebra over

generated bytq ,t1 ,. ..

and

^ , Z 2,....

As is well known,

Zp

Ctr is a ten­

sor product E('t0>'ri ,. •

® P( It , i s , ■ ■ ■)

of an exterior algebra and a polynomial algebra (recall that has odd P ■K' degree and so = o) . Since Ofc1 is free and Ot(p) is commutative the map of the generators of a homomorphism of algebras 3.2.

THEOREM.

PROOF. Suppose

Q, '

0t' —

into

Ot (p)*

> Ot(p)*.

The map 0t’ —

We first show that

< T(I)|(J),a >

0

=

extends in Just one way to

> Ot(p)*

is an isomorphism.

0t* •-- > Ot(p)

for all choices of

a(x1 x...x xn x y 1 x...x ym)

■ft

is an epimorphism. I

and

=

J.

By 3 *1 ,

0

m and n. But, 2 . 6 shows that in this case * Ofc1 -— > Ot (p) is an epimorphism.

for all choices of Therefore

We now show that the map Ot ’ — — > 6t(p)*

is an isomorphism, by

showing that in each dimension, the ranks of 0tT and spaces over and Ct(p)

Zp

are the same.

=

= ^0£° £-,ri

(e0 'r i,si,‘‘*>rk,ek'0'***^

monomials

Ot (p)*

as vector

We have only to show that the ranks of 0tT

are the same in each degree.

We write 1

a = 0.

g^,

and

which form a basis of

ei

=

Ot ’,

Tic£k »

where

or

r± > 0 . The

0

i,

correspond in a one-to-one way T t

with such sequences

I.

The admissible monomials

sequences of integers I ’ = for each

i,

and

=

0 or

P

(£q,s1 ,...,sk ,e^,o,...)

€ Ot where

correspond to si > psi +1 +

.1 . It remains to set up a one-to-one

§3-

THE STRUCTURE OF THE DUAL ALGEBRA

correspondence between the sequences

I

1 1,

and

85

preserving the degrees

of the corresponding monomials. Let

R^

be the sequence with zeros everywhere except for

the

2 k-th place.

the

(2k+i)-th place.

Let Qk

l

in

be the sequence with zeros everywhere except in

Let

R£

=

(°,pk~\°,pk'2 ,---,°,p\0 ,p0 ,°,---)

%

=

(o,pk_ 1 ,o)pk‘2 ,...,o,p1 ,o,p°,i,o,...) .

The map from sequences I

to sequences

1 1can now be defined by extending

the map already defined on

R^

to be additive (with respect to

and

coordinates). Then if } 9e 19 ***,I>kJ£k*^,

we have

^

^

=

j>**•>

>

**•) >

and si

Solving for

= r^

+ ei> + o,

.

I 1, we obtain a unique sequence

and vice versa.

A computation of

degrees shows that deg e1

deg P1 ’ =

=

Zk TjS(pj- 1) + Zk e^(2p^- 1).

This completes the proof of the theorem. 3.3.

THEOREM.

The diagonal map

by

PROOF.

q>*:

**5k

i = Ei=0 *k-i ® 5i

ft* 0 ft*

ft*

311(1

® 1 + Zk=0 |P_i ®

T1

.

a , 3 e ft . We have to show that

Let

< tp*ek ,a ® g >

=

Z < 6^ i ® I1 ,a ® g >

< q>*Tk >a ® P >

=

+

That is, we have to show (!)

is given-




=

T. < 5^i »“ > < S-pP >,

and

.

86

V I . AXIOMS FOR THE AIOEBRA

as in

G(p)

(2)




=

< Tk,a > < i Q ,e, > +. Z

.

Let

x and

y

have the same meaning as in 3 •i • In the same way

II 2 .3 , we prove that oyP

i

=

„i Za < &P

=

a &y

—.a+i > yP

Now k
yp

py 3 . 1

= a z± < ei ,3

>

yP1

■n1

-pvi+a

= Za,i < >a > < h ’ P > y y, we see that(1 ) holds.

Equating coefficients of powers of to prove (2).

It

remains

Now k

< t0 ,a3 > X + Ek < Tk ,“ P > yP

by 3 . 1

=

=

a[< l0 ,e > x + E± < ^ , 3 > yP1 ]

=< ! 0,P X

S0,or >x

+ Ea < l 0,p >< Ta ,« >yPa

_i +

i,a < Ti ’e >

Equating coefficients of powers of y,


a

^a+i > yP

we obtain (2 ).

This proves the theorem.

. Ideals. Let

be the ideal of Q,

•¥r

generated by

tPk EP k ' 1 EP E T E T i » 2 ' • • • > sk ’ 5k + i ' k + i 5 •■■’ 5k+i> k + i ' ' ' ’ ' Then

is a Hopf ideal by

algebra.

Its dual

3*3 • Therefore Ct*/M^

is a Hopf subalgebra

(with minor embellishments), we see that of

Ct^..

is a finite Hopf Arguing as in II 3 . 2

Ct , CO.. 1

3 ,P ,...,P-t;

are all elements

It follows that U.i

THEOREM.

Ct is the union of the sequence

Ct ^

of finite Hopf

subalgebras. If defined by

A Xx

is any commutative algebra over ~

xp ,

mutes with maps of map of Hopf algebras.

then x

and

is a map ofalgebras.

algebras. Hence

if -A

A -— -> A Moreover

is a Hopf algebra,

x x

is

com­ is a

§fc.

Then of

x

x:

ft (p)

*

-— > ft (p)

is the ideal generated by k . 2 . LEMMA.

in

g^gg,...

factor

#

87

multiplies degrees by

o

otherwise.

p

as a factor.)

then

xP-P1

(Notice that if

I

Without loss of generality, we can suppose and

---

of the form

p.

The kernel

Tq ,^,....

If x € ft* andP1 € «,

I = pJ, and xP.P1" = T J P nor P can contain PROOF.

IDEALS

Let

x^-P1

o

t ^,since

o

4 0: then

.

Therefore

x-PJ

=

pj,

if neither

x is amonomial

x x

=

can contain no has even dimension.

We have x^-P1

=

\|r*(x 0 . .. x) -P1 (x ® ... ® x) •tP1

=

Z (x ® ... x) (PJl . ..

where the summation is over all sequences J 1+... + J

=

= Z (xP^1) ... (x pJp)

some term of the sum, two of the

permutation gives ifxPp1

J.j,...,Jp

4

such that

I . So xp . P1

If, in

0 pJP)

0,

p

are not equal, then cyclic

equal terms of the sum.

I = pJ

and

.

These cancel out mod p.

=(x P^)p

=

x PJ .

So,

This proves

the lemma. Let Gfc! be the Hopf subalgebra of (j= 1 ,2 ...). U.3.

Let

x*: & (p) --->

PROPOSITION,

degrees by

p. Theimage

generated by

P1

and

x* of

generated by

be the map dual to

PJ x.

is a map of Hopf algebras, which divides x*

is

Ct’, and its kernel is

=

PJ

if

=

o

the ideal

3. x* P1 *

x p

nel of

ft(p)

ft(p)

T

I

=

otherwise

pj .

PROOF. Using k . 2 , we see that we have only to checkthat the ker* 1 x is contained in the ideal generated by 3 and P . Applying

the formulas for

x*

to a linear combination of admissible monomials, we k 1 see that we have only to prove that P is in the ideal generated by P ,

V I.

88

if

k

is not a multiple of

p’pb k P

Therefore of

AXIOMS FOR THE ALGEBRA

=

p.

a (p)

By the Adem relations

(-i)((p'l1)b~1) Pb+1

=

is in the ideal generated by

P

1

(b + i)Pb+1 . if

k

is not a multiple

p. This proposition has been used by Wall [2] and Novikov [i3. PROPOSITION.

If we abelianize

bra, which is the tensor product of and the divided polynomial ring on phpk

PROOF. Let

A

Let

I

= dfc/l.

_

E(3),

Ct(p) , we obtain a Hopf alge­ the exterior algebra on

P1,P2 . . . ,

( h H - k ) ph+k

[ a ,a i.

be the ideal generated by all commutators in

Then

A

and A ® A

3,

i.e.,

Ofc(p).

are commutative algebras. Consider

the composition Ct -i-> Q, -® & -- > A ® A . This is an algebra zero on I.

homomorphism into a

commutative algebra and is therefore

Therefore V(I) c Ct I + I ® Ct

and

I is a Hopf ideal.

all elements

x e Gfc*

Suppose that Let

J

=

form

Therefore such that

tx is symmetric.

Z aj|J e A*

and

g"

(aj e Zp) . Then

® g.,m ,where

calculation shows that these terms I,11®

g.,

tq£ i

m and

s1 2 T ^ 2 ...

Pr 2 Prk ... lk_1

Tk _^

m

Z (r^ + s^)p1”1 > Z r^,

ei

I > i

0 for

and

r±

=

n

A*

consists of

Therefore

'r0,£1e A*.

£ ajCp*£J

is symmetric.

in

are maximal.

of the A short

are

We note that =

is a Hopf algebra.

(eQ ,r1 ,e1,... ,rk ,ek ,o,...) . We collect terms I*

and

A

0 ?or

where

n

=

where m

=

+ ei)p1'1

v Z* r± .

and that we have equality only if i > 2*

§5.

In maximal. for

Z

HOMOTOPY GROUPS OF SPHERES

we select those terms for which

By symmetry, we must have

i > 2.

8^

=

0

8$

z!f (r^ +

for

i > 1 and

Such terms are in the algebra generated by

induction on

Z^ (r^ +

generated by

tq

therefore shows that

and

r^

and

tq

A*

is =

o

11 . An

is the subaglebra

|1.

Dualizing, we see that

A

has the structure described in the

proposition.

§5. If

G

Homotopy Groups of Spheres.

is an abelian group, we let

whose orders are powers of the prime G

p.

Gp If

G

be the subgroup of elements is finitely generated then

can be expressed as the direct sum G

where

F

is a free group.

of an element of 5.1.

=

In this case we can talk of the p-primary part

G, by which we mean the component in

THEOREM.

^ ( S 3)

is finite for

ni(S 3)p Let

f:

F + ZpGp

S2p -- > S3

= { ° Zp

represent

p-primary part and let

E

Gp .

i > 3.

if i < 2P , if 1 - 2p

an element of n2p(S3)

be a

with a non-zero

(2p+i)-cell. Then, if

L

= S3

E,

P1: H3(L;Zp) -- > H2p+1(L;Zp) is an isomorphism. 5.2. of

(When

COROLLARY.

f, and let

M

=

p = 2, Let

SnL.

g:

replace

P 1 by Sq2, see I 2 .3 ).

Sn+2p

> Sn+3

be the n-fold suspension

Then

P1: Htl+3(M;Zp) -- > Hn+2p+1 (M;Zp) is an isomorphism.

Therefore

W

s n+3)p * °-

PROOF of 5.2*. As P1 commutes with suspension, the first part follows. Since

M

is formed by attaching a

the second part follows by taking

(2p+n+i)-cell to f

Sn+3

with the map

to be the generator of

*2p(S^)p .

g,

V I.

0

AXIOMS FOR THE ALGEBRA

G(p)

In fact the following stronger result can be deduced from 5.1 by using [3 ] Chapter XI, Theorem 8 . 3 and Corollary 1 3 .3 . 5.3.

COROLLARY.

If

p

is an odd prime, then

/on+3% n+i' }p

"

f0 \ zp

if if

i < 2p , i = 2p .

The remainder of this section will be concerned with the proof of 5.1.

We shall rely heavily on Serre's mod 6 theory.

to [33 Chapter X

We refer the reader

or to [k 3.

We would like to compute the homotopy groups of the (mod 6 ) Hurewicz theorem.

S3

by applying

But the Hurewicz theorem in dimension

only applies to spaces that are

(n-i)-connected (mod 6).

is an obstacle to this program.

We therefore construct a space

has the same homotopy groups as

S3

apply the Hurewicz theorem to

X.

except that

^(X)

The definition of

=

So

0,

n

^ ( S 3) ~ Z X

which

and then

X, which is rather

long, follows. 5.1. an integer.

DEFINITION. K(*,n)

zero except for

*n

Let

%

n > 2 be

be an abelian group and let

will denote any space whose homotopy groups are all which is isomorphic to

jr.

Such a space is called

an Eilenberg-MacLane space. 5.5

THEOREM.

there exists a REMARK.

For any abelian group

CW-complex which is a

jt and any integer

n > 2,

K(«,n) .

We can easily show by obstruction theory that all such

CW-complexes are homotopy equivalent. PROOF. between the

Let

xi ,s.

Jt

be generated by elements

xi

with relations

We take a bouquet of n-spheres, one for each

r^

x^

For

each relation rj where each degree

oc^

a

=

«lj x i + a2j x2 + " - + “mj ^

is an integer, we map an n-sphere into the bouquet with on the n-sphere corresponding to

n-sphere corresponding to

x2

and so on.

bouquet for each relation

r^

with this map.

x 1, with degree

We attach one

on the

(n+i)-cell to the

We now kill successively the

§5.

HOMOTOPY GROUPS OF SPHERES

homotopy groups in dimensions sions

91

n+i, n+2, etc., by attaching cells of dimen­

n+2, n+3, etc. Computing the nth homology group by using the cell structure, we

see that

Kp

~

*. By the Hurewicz theorem we have constructed

5.6. If x,

let

in

K

PK

K

x.

K

starting at

K

to its end-point.

Note that

PK

sequence for a fibration,

d:

MacLane space of type

K(xc,n-1).

=

let

K(Z,3)

then

represents agenerator of

it^(K)=» Z.

induced by thestandard fibration X

Let

over

PK

K.

the loops

> K

obtained

(See [3 ]

ftK.

By the homotopy exact

this shows

be a CW-complex.

be

QK

The fibre is

« ^^(nK).

K(*,n),

K

and

is contractible.

MacLane space of type

Let

x

¥e have the standard fibration p:

by sending a path in Chapter III.)

K(*,n).

is a path-connected topological space with base-point

bethepaths in

based on

a

If

K

that &K

Let

is an Eilenbergis an Eilenberg-

S3 -- > K

p: X --- > S3

be a map which

be thefibration

We have thecommutative

diagram

----- > PK

|p Ip v K where the vertical maps are fibrations with fibre By the homotopy exact sequences of the fibrations, p*:

^(X) «

Jti (S3)

We now find

for H*(X)

nK,

which is a K(Z,2).

*3(X)

=

0

and

i 4 3. and apply the Hurewicz theorem(mod 6)

to find the first non-vanishing higher homotopy group (mod

6 )of

to

X

S3. The

usual method of finding the homology

of a fibre space isby using a spectral

sequence.

space a sphere), the spectral sequence

In this simple case (base

reduces to the Wang sequence. 5.7.

THEOREM.

Let

X -- > Sn

be a fibration with fibre

F.

Then

we have an exact sequence (Wang’s sequence) H ^ X jA) where

A

> H^FjA) —

> Hi-n+1(F;A)

is a commutative ring with a unit.

that is, if

x e H^(F;A), and

y € H3(F;A),

> Hi+1(X;A)

Moreover, then

G

isa derivation:

V I.

92

AXIOMS FOR THE ALGEBRA

e(xy)

=

6x - y +

Ot(p)

(-1)^n_1^

x-6y.

We refer the reader to [5 ] p. ^71

PROOF. sequences or to

for

a proof byspectral

the next section of this chapter for a proof not using

spectral sequences. 5.8. H^fXjZ)

LEMMA.. =

K(Z,2).

K(Z,2)

=

Zk

and

X---> S3

with fibre

UK

which is

Now complex projective spaceof infinite dimension is also a

u.

H (ftK)

is a polynomial ring on a two-dimensional

We have the exact sequence (see 5.7)

H^(X;Z) In order to find

> H^nKjZ) — H* (X;Z),

Now since 0u

H2k(X;Z)

We have the fibration

and therefore

generator

then

0.

PROOF. a

If k > 0,

X

> Hi-2(nK;Z) -- > H1+1(X;Z) .

we need only find the derivation

is 3-connected,

- 1 • Changing

the sign of

9is a derivation,

8un = H2k(X;Z)

H^CX)

0

-

for i < 3 . Hence

u, we can ensure that

n un“1 by induction on =

0

and

9.

n.

H2k+1(X;Z) = Zk

0u

=

1.

Since

Therefore .

Let us first consider the class 6 of abelian groups which are finitely generated.

By the (mod 6) Hurewicz Theorem, the homotopy groups

of simply connected finite complexes are finitely generated. is erated

finitely

for all

generated forall

i. Hence

H^(X;Z)

We

now take

the class

Jt^(X)

^ ( S 3)

Taking the class orders prime to

p,

C

and so

^(X) is finitely gen­

is finitely generated

the universal coefficient theorem

the lemma that

i,

Therefore

for all

i. By

we deduce the lemma. 6 of finite abelian groups and deducefrom

is finite for all

i > 3.

to consist of all finite abelian groups with

we deduce that it (S3) i' 'p

~ / 0 ” \ Zp

if if

i < 2p , i = 2p .

This proves the first part of 5.1. Let S3

with a

f : S2p -— > S3

be as in the statement of 5.1.

(2p+i)-cell adjoined with the map

f.

Let

L

be

We can extend the map

§5.

HOMOTOPY GROUPS OF SPHERES

S3 -- > K(Z,3 )

which we have been using to a map

jt2p (K(Z,3 ))

0.

=

fibration over

Let

Y

-> L

93

L

> K(Z,3)

since

be the fibration induced by the standard

K(Z,3). By the cell structure of *^(L,S3)

=

0

2p+ T(L,S3)

=

Z.

L,

we have

i < 2p+i

for

and

Moreover the boundary map

maps the generator of the group on the left onto the element of represented by

f.

By the homotopy exact sequence for

(L,S3)

«2^(S3) we deduce

that flj_(L)

«

*.(S3)

for

i < 2p,

*2p^L^p

=

°*

By the same reasoning which gave us the homotopy groups of those of

=

0

for

i < k,

^ ( S 3)

*

*2p B

Then we have the maps * Ep(B/b) > EP(E,F) L p u € H (ftK;Zp) is transgressive.

erating class for the pair

(Y,ftK)

and since

*3 (L,x)

Y

is

with fibreGK,

*3 (Y,«K) — —

> *2 (nK)

z

where the vertical maps are Hurewicz homomorphisms and

x

1—

gen-

3 -connected we have the diagram

«

H3(L,x)
H2(£5k;Z) is the base-point

9b

in

AXIOMS FOR THE AIGEBRA

V I .

L.

By the universal coefficient theorem,

gcessive.

Let it correspond to

transgressive.

Since

H^fYjZ^

s: is a monomorphism. zero.

su-^

u e H (ftK;Zp)

is trans-

v e H 3(L;Zp) . By 5.9,

P1!* -

=

the map

0

HSp(fiK;Zp)

Hence

C t(p )

=

for

o < i4 < 2p,

is

> H2p+1(Y,nK;Zp) p*P1v

is non-zero.

Hence

pV

is non­

This completes the proof of 5.1.

§6.

The Wang Sequence.

In this section we shall prove 5.7 without using spectral sequences. We restrict ourselves to fibrations which have the covering homotopy pro­ perty for all spaces (not just for triangulable spaces). (See [3 ] Chapter III.) 6.1

THEOREM.

the fibre space over tel.

Then

EQ

X

and

Let

p:

E —

> X x I

obtained by restricting E1

E

be a fibration. to

X x (t)

Let

where

are fibre homotopy equivalent fibre spaces over

X. PROOF.

the identity on

p x 1:

Let

EQ x {0 } , we obtain a map

a fibre-preserving map map

g:

EQ x I ---> X x I.

f : EQ — ->E 1

EQ x {1 } -- > E 1 . So we have

and similarly a fibre-preserving

E 1 -- > Eq . We must prove that

to the identity and similarly for

Lifting this homotopy to

gf

is fibre homotopy equivalent

fg.

We have the map p x rr x 1 : EQ x I x I — We lift this map to

E

> Xx(o}xI

= Xxl.

on

Eq x ({0} x l u I x {0 } u {1 } xl)

,

by the constant lifting on I x {0 ) and using the constructions described above on

{0 } x I

{ 1 } xI.

and

extend the lifting to

By the covering homotopy property we can

EQ x I x I.

The homotopy between

tity are found by looking at the lifting restricted to This proves the theorem.

gf

and the iden­

EQ x I x {1 }.

Et

be

§6. 6 .2 . COROLLARY.

ted to a point

x

El—

Let

f:

X f -- > X

by a homotopy keeping

have a fibration over tion

THE WANG SEQUENCE

X

with fibre

be a map which can be contrac­

f(p)

F

95

over

= x.

x

fixed.

Then the induced fibra-

> X ! is fibre homotopy equivalent to the trivial fibration

Xrx F

>

into

by a map which is homotopic to the identity.

F

X r. The fibre homotopy equivalence maps F, the fibre over p,

PROOF. sent to

Suppose we

x.

We have a map

Let

E

X ’ x I -- > X

such that

be the induced fibration over

X! x i u p x I

X r x I.

is

The corollary

follows from 6 .1 . Now suppose we have a

fibration X---- > Sn . By 6 . 2 if we restrict

the fibre space to any proper subspace of

Sn , we have a fibration which

is fibre homotopy equivalent to the trivial fibration. where

E+ n

E_ =

Sn_1. Let

be the

F

Let

Sn = E+ u E_

be the fibre over a base-point

Let

X+

E_.

Then we have the commutative diagram

part of the fibre space over

E+

and

X_

x

€ Sn ~1.

the partover

(E+ x F ,Sn ~ 1 x F) ----- > (X,XJ K x K

Let

u c H (K)

u x v € H*(K x L).

If

is the diagonal we define the cup-product

u u v

=

*. . d (u x v) .

The cup-product is called an internal operation since all the cohomology classes exist in a single space

K; the cross-product is called an external

97

V II.

98

operation.

CONSTRUCTION OF THE REDUCED POWERS

The advantage of the cross-product is that its definition re­

quires no choice, even on the cochain level. product requires a diagonal approximation

On the other hand, the cup-

d^:

K

> K ® K.

Many diffi­

culties experienced with the cup-product in the past arose from the great variety of choices of looking formulas.

d^,

any particular choice giving rise to artificial-

Moreover, the properties of the cup-product such as the

associative and commutative laws follow easily from the corresponding pro­ perties for the cross-product by applying the diagonal.

The properties for

the cross-products themselves are easy to prove. Similarly we shall obtain the (internal) reduced powers as images, under an analogue of the diagonal mapping, of a certain external operation P.

We shall prove many of the properties of the (internal) reduced powers

by proving the corresponding properties for the external operation. Let

W x^ Kn

=

(W x Kn)A

and let

j

be the composition (which

is an embedding) Kn -- > W x K11 -- > V xTt Kn

.

The map

W xTt Kn — —> W/* is a fibration with fibre gy class u on K, we have a class u x ... x u on conditions we can extend in the total space the variable

K,

=

K?1. Under

this class in one and only oneway

W x^ Kn Po

Kn . Given a cohomolo-

o

so

that

suitable

to a class Pu

Pis natural

withrespect

tomapsof

and *

j Pu

=

u x ... x u.

For the n 0^1 power in the sense of cup-products, we have un To define reduced by

Pu.

We replace

W x^ K

=

d*(u x ... x u).

nth powers, we replace Kn d:

K — 1

Now

=

W /it x K.

>

Kn

xn d: W x„I tK

—

by

W x* Kn and

u x ... x

> W x xi Kn .

Hence (1 Xjr d)*Pu

e H* (W/it X K) .

If we are working with a fieldof coefficients, we can expand in by the Kunneth theorem. which lie in

H*(K)

u

by

The coefficients of the expansion of

are the internal reduced n ^*1 powers.

H*(W/jt

(i x

it

x K)

d)*Pu

§2,

CONSTRUCTION

99

In subsequent sections we replace cohomology classes on by equivariant cohomology classes on

W x^ Kn

W x K?1.

§ 2 . Construction, Let q-cocycle

K

u

be a finite regular cell complex. on

K

with values in an abelian group Gp = 0

a complex with all components Then we have a chain map Gn (q)

be the

u:

Gn

of the factors of factors of

Gn

if

q

is odd.

If

with no sign change.

Then

variant chain map which lowers degrees by Let > Kn

e:

Gn

q

W -> Z

as

G 0 = G.

by

q.

Let

in dimension zero. a

We let

and the permutation

is even we let

a permute the

un : Kn — —> Gn (q)

is an.equi­

nq.

j

be the augmentation on W.

Then

e 1

: W if1

is an equivariant chain map (using the diagonal action on

Therefore

G

It is zero in non-zero

by the product -of the sign of

Gn

We regard

which lowers degrees

S(n)-complex defined as follows.

act on

G.

except in dimension zero

K -- > G

dimensions and is the n-fold tensor product a € S(n)

Suppose we are given a

W ® Kn).

the composition w ® Kn

e ® 1 > Kn j L - > Gn (q)

is an equivariant chain map which lowers degrees by we have an equivariant

nq-cocycle on

nq.

In other words,

W Kn , which we denote by

Pu £ C ^ ( W ® Kn ; Gn (q)) . We now prove that if we vary

u

by a cohomology, then

Pu

varies

by an equivariant cohomology. 2 .1 .

such that

LEMMA.. There exists an equivariant map

h(o ® w)

PROOF,

h

=

on w

is equivariant on

variant acyclic carrier 2 .2 .

values in

LEMMA..

G,

and

then

If Pu

h(T ® w) o W

h:

=

Tn® w,

for all

and

T W.

We have the equi­

W In . The lemma follows from u

and

and

Pv

v

I ® W -- > In ® W w e W.

V 2 .2 .

are cohomologous q-cocycles on

are cohomologous

K

with

nq-cocycies in

C*(W ® Kn ; Gn (q))~ that is, they are equivariantly cohomologous. PROOF. of

u

into

v,

Now u cohomologous to v means that that is a chain map

D:

there is a chainhomotopy

I K -- > G lowering degrees

V II.

100

by

q,

CONSTRUCTION OF THE REDUCED POWERS

D (0 ®

such that

t

=

)

u(t)

and

D(T t )

=

v(t)

for all

r € K.

By 2 . 1 we have the following composition of equivariant chain maps > In ® W ® K11

I ® W ®

In ® K11 -s- uf"> (I ® K)n

Gn (q) .

The map shuf denotes the shuffling of the two sets of n factors with the usual sign convention. Pa

into

Pv

This composition gives the equivariant homotopy

of

which shows they are equivariantly cohomologous.

The lemma shows that

P

induces a map (not a homomorphism in gen­

eral) P: Let W ® K?1

w

pP-(K;G) ---> p£q (W ® Kn ;Gn (q)).

be a 0 -dimensional cell of

defined by

j(x)

=

w x

finite regular cell complex and

for all

f:

V 3 .3 the equivariant continuous map (fV:

W.

x € K?1. If

K -- > L f11:

We have a map L

K11 — — >

j:

is another

is a continuous map, then by

Kn -— > Ln

induces a map

H*(W x Ln ;Gn (q)) -- > H*(W x K?;Gn (q)) .

2 .3 .

LEMMA.

1)

j Pu

is the n-fold cross-product

u x .. .x u e

Hn^(K?1 jGn) . 2)

We have a commutative diagram H^(L;G) --

> H^g (W x Ln ;Gn (q))

f*

(fV

V

v

H^(KjG) ---------L _ : > H^q (W x Kn ;Gn (q)) . PROOF. 1 ) follows immediately by the definitions on the cochain level of

P

and of cross-products.

We reduce the proof of 2 ) to the case where V3.1 and V 3.3. Let K?1

to

Ln

C

be the minimal carrier of

which sends

o1 x . . . x a n

acyclic equivariant carrier for chain approximation to W®Kn _ >

w®

Ln .

f,

to

f f.

Then the carrier from

C(a1 ) x . . . x C(a )

f11. Therefore, if

we can use

is proper by using

1 ® (f|)n

f^*.

We have a commutative diagram

1 ® (fj)n V ®VLn

Ln

> L

is a

as our equivariant map

w ® K11 — £_®J— > K51 — £jfuL-..> Gn (q)

The lemma follows.

K

is an

> Gn (q).

§2. 2 .U.

S(p)

REMARK.

If

CONSTRUCTION

n - p

and

which permutes the factors of

101

G = Zp

K?

and

*

is the subgroup of

cyclically, then

ized by the properties in 2 . 3 and the fact that

Po = 0 .

P

is character-

(This can be proved

by the methods of VIII §3 .) Let

it C p C S(n)

and let

V

and

W

be respectively a p-free and

a *-free acyclic complex. 2 .5 .

LEMMA.

The following diagram is commutative . H?q (W x Kn ;Gn (q))

— ’----

f

_________ I P > H ^ C V x Kn ;Gn (q))

H^CKjG)

P

where the map on the right is induced as in V 3.3. It follows that independent of the choice of PROOF.

Let

P

is

W.

g^:-W --- > V

be an equivariant chain map.

The dia­

gram

W 8 K?1 S#®1 |

K11

Gn (q)

V ® Kn — ---is commutative. Let

The lemma follows.

u € H^(K;G)

and

regular cell complexes and Pu € Haq(W

X

Kn ;Gn (q))

and

G

v € Hr (L;F) and

F

where

K

and

are abelian groups.

Pv £ H ^ C W x Ln ;Fn (r)).

L

are finite

We have

By V k . S , we have a

cross-product Pu x Pv £ H^q^ ( W x W x Kn x Ln ; Gn (q) where

it x it acts on

W x W x K 11 x Ln

(«,P)(v,,v2 ,x,y) a , 3 € it,

for all

v -j>v 2 €

=

x e Kn

by the formula (ov,,Pv2 ,CK,Py) and

u x v £ Hq+r(K x L;G ® F) P(u xv) e where

V

8 Fn (r))

y € Ln . We also have and

x (K x L)n ;(G ® F)n (q + r))

is a it-free acyclic complex.

We have a map of geometric triples X:

(it,(G ® F)n (q + r),(K x L)")'----> (it x it,Gn (q) ® Fn (r),Kn x Ln)

V II.

1 02

CONSTRUCTION OF THE REDUCED POWERS

defined as follows: all

it -— > it x *

is such that

^(a)

=

(a,a)

for

a € it;

Gn (q) ® Fn (r)

\ 2:

> (G ® F)n (q + r)

is the obvious isomorphism which shuffles the two sets of n variables; the map

> K31 x Ln

(K x L)n —

unshuffles the two sets of n variables.

By

V 3 . 3 we have a map X*:

H*X„(W x W x K11 x Ln ;Gn (q) ® Fn (r)) ---> H*(V x(K x L)n ;(G ® F)n (q + r)).

2 .6 ,

LEMMA.

PROOF.

l*(Pu x Pv)

=

( - 1 ) n(n_1 )qr/2 P(u x v).

According to 2.5 we may take

acyclic complex.

Let

V

*

W x W

V

to be an arbitrary

with the diagonal action.

*r-free

We have the

commutative diagram of equivariant chain maps 1 (W ® W) ® (K ® L) £®1

jt n n £-> W W J T Ln

v

(u®v)nj,

e&e®l Gn (q) ® Fn (r) where

n is(_i)n (n_ 1 )r£l/ 2

thediagramgives

P(u x

times

v)

the inverse of

andthe right side

x2. The left

gives

side of

Pu xPv. Thisproves

the lemma.

§3. Now let phic to

5r

n = p,

Letit C s(p)

tation is it -module.

T (--1 )

=

G^(q)

is isomor­

acts on

.

be the cyclic group of order

which sends 1.

S(p)

G = Zp.Then

Z = G^(q) by the sign Jr is odd, and trivially if q is even. In the no-

q

V §6 , G^q)

permutation

a prime, and let

Z as an abelian group.

of the permutation if tation of

Cyclic Reduced Powers.

Since

i

to

(~1 ) P “ 1

(i + 1 ) mod p. =

1 (mod p),

p,generated by the The sign of this permu­ Z ^

is a trivial

§3.

3.1. action.

LEMMA.

CYCLIC REDUCED POWERS

Let K

be a finite regular

cell complex with no re­

Then H*(W x K;Zp )

*

H*(W/tt x K;Zp)

and this isomorphism is natural for maps of Let if

10 3

S(p)

d:

K — — > K?

acts on

K?

K.

be the diagonal map.

bypermuting its factors.

Then d

is equivariant,

By V 3 . 3

we have an in­

duced map d*: Since

Z ^

H*(W x Kp ,Z(q)) ---> H*(W x K;Z(q)) .

is a trivial

«-module, we can replace

Z ^

by

Zp .So, if

u e n f l ( K ; Z p ), we have by 3.1 and the Kunneth formula 3 .2 .

where

DEFINITION.

w^ e Hk(W/it;Zp)

d*Pu

=

A

wk * Dku

are the elements of V 5 .2 , and this defines

Dk : H^CKjZp) -- > H P ^ K h Z p ) . (Note that we have not yet shown that Let

f : K -- > L

Dk

is a homomorphism.)

be a continuous map between two finite regular

cell complexes with no group action. 3.3. PROOF.

LEMMA.

For each

We have

df

=

k, f^d.

f*Dk

= D^f*.

Hence the following diagramis com­

mutative (by V 3.3) H ^ t W x Lp ;Zp) —

-- > Hpq(W x L;Zp)

(fP )*

H ^ ( w xKp ;Zp)

| f*

--- > H ^ ( W x K;Zp)

Applying the commutative diagram of 2 . 3 to the left and theisomorphsims of 3.1

to the right of this diagram, the lemma follows. 3.1*. PROOF.

LEMMA. Let

D0 u = up . w

diagonal approximation.

be a

O-cell of

W.

Let

d# :

We have a commutative diagram

K -i— > W ® K d# i

I 'f #

KP _ ! _ > w ® Kp

K --- > Kp be a

10U

V II.

where

jx

=

CONSTRUCTION OF THE REDUCED POWERS

w ® x.

Now D0 u

3. 5. DjU

0

*

in t e g e r

LEMMA.

u n le ss

m.

If

( 2m + l) ( p - i ) -

1

H*(W x L ; Z ^ )

j q

PROOF.

is

V

3.3

J * ( S fc wk x Dku)

* *

=

j d Pu

=

dj

d *(u x . . .

=

up .

Let

odd,

x u)

u £ H ^K jZp)

D^.u

=

and l e t

2m ( p -l) - 1

or o

u n le ss

W ith n o ta tio n as in V § 6, l e t

y

p

a c ts .

as i n V Let

and V

3.k we.

# *(V

x Kp ;Z ( q ) ) - £ - » P

V

3.^ ,

2.

p >

I f even q is

fo r some n o n -n e g a tiv e

j m.

yep

2.3

by

fo r some n o n -n eg a tiv e in t e g e r

induced by

P

Pu

=

2m (p-l)

=

c e l l complex on w hich

2. 5,

=

( 2m + i)(p -i)

=

or

be th e automorphism o f

where

L

i s a f i n i t e r e g u la r

be a p -fr e e a c y c l i c complex.

By

have the commutative diagram H ^ V x K ;Z ( q ) ) - H l _ > P P

P

x K ;Z ( q ) ) P

P x Kp ;Z ( q ) ) - S — > H^(W x K ;Z ( q ) ) The lemma fo llo w s from V

6.1

and V

§U.

The T r a n s fe r .

tr a n s f e r .

the map induced by th e d ia g o n a l LEMMA.

4. 1

Then

PROOF.

Let

t

:

Hp q (W x K ;Z ( q ) ).

6. 2.

We have d e fin e d the t r a n s fe r i n V § 7. H* (W x K;Zp)

— y -------->

Let d:

H*(W ® Kp ;Zp ) --- >

d *:

H*(W x K?;Zp) — ->

K — ->

¥p.

H*(W ® Kp ;Zp ) denote the

d*r = o. We have a commutative diagram

H (W ® Kp ;Zp)— d .*

> H / W ® Kp ;Zp)

*

v

H*(W ® K;Z ) -i— > H* (W ® K;Z ) —

d

*

H*(W ® K;Z ) .

§i. Since

¥

is acyclic and

H^(W;Zp )

i*: is also onto. By V 7.1 i.2 .

If «

Let

> H° (W;Zp )

is onto,

u

The lemma follows.

is the group of cyclic permutations and

P: H^KjZp) --- > H ^ ( W x KpjZp) PROOF.

105

H^(W ®-K;Zp ) --- > Hn (¥ K;Zp ) Ti* = o.

LEMMA.

THE TRANSFER

and

v

then

d*P

is a homomorphism.

he q-cocycles on

K.

Then

P(v+u) - Pu - Pv

is given by the chain map w ® Kp -!SL_> Kp - C ^ .).p r..£ - .Yp > Zp .

According to U.i, we need only show that thiscocycle is in the image transfer.

It will be sufficient to show that

(u+v)p- u P - vp

of the

is in the

image of a cocycle under t

since

:

C*(Kp ;Zp) ---> C*(Kp ;Zp)

s 0 i is an equivariant map. Now

factors u

(u+v)P- u p - vP and

(p-k)

such factors freely. permutations under

is the sum ofall monomials which contain

factors

in

Kp

give each monomial exactly once.

v

Then

tz

=

Let

(u+v)*5- u^- v*\

since each monomial is a cocycle.

COROLLARY. For each

k.3.

Now jt permutes

Let us choose a basis consisting of monomials whose

of the monomials in the basis. a cocycle

where 1 < k < p-i.

v,

k

2

be the sum

Also

The lemma

z

is

follows.

k,

D k : H^-(K;Zp) ---> Hpq_k(K;Zp) is a homomorphism. 4.it. andD ( p __1 )^u u and

LEMMA. = a^u

If

u € H^CKsZ,.) then D,,u = 0 for k > (p-i)q; P K where a^ € Zp is a constant which is independent of

K. PROOF.

Let

Kp

be the q-skeleton of i*:

is a monomorphism for q-dimensional.

Let

r < q.

K. Then

H^K) ---> H^K^) By 3.3 we can therefore assume that

U q £ Efys^jZ-)

be the class dual to

S^.

K

is

There is a

V II.

1 06

map

f : k --- > sq

such that

map each q-cell of sentative for

CONSTRUCTION OF THE REDUCED POWERS

K

u.

into

f*uQ Sq

Then

j*

we let

By 3.3 we can assume that

D^u

q > 0 . Let

and

u:

If

s: -- > Sq

v, a cocycle repre­

K = Sq

k > (p-l)q,

to be non-zero and

j:

be a point and

with degree given by

second part of the lemma follows. ty remaining for

=

and

u = uQ . The

then the only possibili­

k > (p-i)q

is that

be the inclusion of a point

is an isomorphism in dimension zero and j *DpqU

= Dpqj * u

j*u

=

k s

=

pq

in

Sq .

o.

3'3

by

pq = o

by k . 3 .

This proves the lemma. U.5.

LEMMA.

Let

3

be the Bockstein operator associated with the

exact sequence

0 — > zp — > zp2 — > zp — > 0. Then

3 d*Pu

= o

p > 2

for

3 d*

PROOF. Since that on

K

gral

3Pu

or q

= d*3,

is in the image ofthe

represented

z

is an integral cochain on

3Pu

Kp

=

v be anintegral pz

where

3u e H^ * 1 (K;Zp).

z

cochain

is an inte­

The cochain

whose cohomology, class we denote by W ® Kp -- ->

Kp . Then

1 )*3 (vp) .

-X(e ® 1 ) , it will be sufficient to show that

t

commutes with

e(vp )

is in the image of 6VP

or

Let

5v

3(s l)*{vp } = (e

=

Since

is a mod p cocycle.

transfer.

represents

(vp) e Hpq(Kp ;Zp ). Let s ® 1 :

p-i

^.i shows that wehave only to prove

u € H^CKjZp). Then

(q+1)-cocycle, and

since either

even.

q

t

.

=

£P~J (~l)qS V S(6 V)VP_S " 1

-

p

-

p

=

p

C 5

h

^

v3

^ - 5-1 «(zvp-1)

t

(z v p ” 1 )

is even.

Since

v

is a mod p cocycle,

The above argument shows that

zvp_1

t(zvp_1) represents

vp

§U.

£(v^},

COROLLARY.

p > 2

If

” ■D2k-iu,^2k-lu PROOF.

=

FromV 5 .2 ,

6 w^

x

D

0

and

o ¥ 2k x ^ 2 k u

"

^k > o ¥ 2k+i

LEMMA.

Osk( u x v ) p

w k

=

1+.7.

= 2,

q = dim u

is even then

3D Qu

=

k u )

6 ^23 + 1

=

0

"¥ 2 j+2

=

^ > °)

Let

Hence

x ^D 2k+iu " ^k >1 ¥ 2k x D 2k-iu

comparing coefficients of u € H^KjZp)

and

■

Dk (u xv)

PROOF.

“0

w^.

v e Hs(L;Zp ).

If

p > 2

then

Zj,0 u 2j « x D sk. 2J v =

Ej_0 DjU x

D k_jV.

The map of geometric triples

x, used in 2.6, takes the

form X:

(it,Zp> (K x L)P ) -- > (« x it,Zp ,KP x LP),

¥e have a commutative diagram of maps of geometric triples (*,Zp ,(K x L)p )

> (* x jt,Zp ,Kp x Lp)

A

A

d1

d

d. (*,Zp ,K x L) ---- -— > (« x it,Zp ,K x L) where and

d

is

induced by the diagonal on

d ’ bycombining the diagonals on Let

free acyclic

0,

By 3.2 and i^.5

The lemma follows by

If

or

0‘

^ ( 2 k

^k >

10 7

and the proof of the lemma is complete. k.6.

^D 2ku

THE TRANSFER

W

K

K x L, and

be a re-free acyclic complex.

complex.

d

by the diagonal on

it

L. Then ¥ x ¥ is a

(« x *)-

From the above diagram and V 3.3 we have a commuta­

tive diagram H*(W x (K x L)p ;Zp)

.

(_l)P(P~->)(q-1)/2 aq_iai(u x v) .

)P(P-1)(Q.-1)/2

5/2 D (q_1} (p_, )U x

v

The lemma follows by induction.

In order to complete the determination of

I> ^ —^), we must find

a1 . This is done by appealing directly to the definition

in the case of

S1 . Suppose Then

K

Z^ w^ x D^u

is a finite regular cell complex and is represented by the composition

W x K where maps

d^

> W

X

Kp.-eft1....> Kp -lL.> zp ,

is a diagonal approximation.

W 0 K -— > kP ,

homotopic.

u e HfyKjZp).

By V 2,2 any two equivariant chain

carried by the diagonal carrier, are equivariantly

§5.

DETERMINATION OF

Hence, in order to find

D

Dq ( p _-,)

1 09

on a i-dimensional class, we need only

find an equivariant chain map W ® S1

tp:

carried by the diagonal carrier. breaking itinto two intervals bJ2

A - B.

=

Let

We make

J1

and

S1

J2

be the complex of V §5.

cp(e^ ® A)

=

S1

; 0(eQ ® B ) = 0

0. We need only

to an equivariant chain map > IP

and this will give a formula

J2

A - B

We define

q>: W ® I

J1

such that BJ1 «

cp is uniquely determined thus farby the carrier.

extend the definition of

where

into a regular complex by

Then the fundamental homology class of

W

0(eQ ® A) = Ap

In fact

> ( S V

W ® S1

by taking first

I.

We define A

is

the augmentation s 1 ,

s > 0 ).

We shall prove the following formulas a)

cp(e21+i ®

"

® I)=Scp ^(©2 i ® ^ ;

h)

cp(e2i

o)

cp(©j_ ® A) cp(e^ ® B)

Let

A

= 0 =

Scp ^(ei ® A)

if

i > 0

,

= 0 =

Scp S(ei ® B)

if

i > 0

.

T - i , where

-

Scp ^ e2i+i ® ^

T

is the element of

which sends

it

i

to

i + 1 (mod p). Then Scp d(e2i +1 ® ^

= Scp(A(e2i I)) SAcp(e2i 0 I)

= =

ii S

Z (A °IB

) (IA

IB ^...(IA IB 1> .

By (iii),terms with 3^ = 0 make no contribution. If 3 ^ > °, i ^i = Then by (iii) the above expression is equal to an 3n a. 3 or. 3» i! S Zg_ > Q (BA IB ) (IA IB ) ... (IA IB ) .

let

By (iv) this expression is equal to i; Zg_ >

0

(IA IB

)(IA IB )...(IA

This summation extends over allsequences p - 2i - 1

and

3i > 0 .

(or,3 )

IB 1)

such

. Z (on + 3 ^)

that

Therefore the expression is equal to 1 ,

then

5.2.

cpc

LEMMA,

PROOF.

is a chain in

W ®I

and

= Sep Be. cp is a chain map.

We prove this by induction on the dimension. It

ate in dimension 0.

is immedi­

In dimension i we have cpB(e1 ® A)

Also

cp.

Bcp(e1 ® A) = 0 .

=

*

0

.

Similarly cpB(e1 ® B)

a_ 2, then B 2. (-l)m mi

if

p > 2, a)

a

1

if

p = 2.

V II.

112

PROOF. and

o

on

CONSTRUCTION OF THEREDUCED POWERS

Let

u

Jg. Then

be the cocycle of u

generates

S1which has value

H 1(S1;Zp ).

(¥p_i ® Dp„iu) ^ep_i ®

1 on

J1

We have

=

^wp-i * ep - i ^ Dp-iu * ^J i “ J2 ^

-

a 1#

and (w;p__1 0 L)p_^u) If

p = 2

"" *^*2^

^^^p— 1 ® ^*^1 ""* ^ 2 ^ *

=

then 2,

a,

J,2- J22.

i. then (p-1)

is even and

•?(ep_.| ® (J, - J2)) Therefore

=

=

mi up . J,p

=

=

mJ (J,p - J£p).

(-i)P^P-1^/2 mJ

This proves the lemma.

Combining 5.1 and 5.3 we obtain 5.L a^u

where

THEOREM. aq

=

Let

q > 0

1

if

p = 2 and

aq

=

(-D^+O/S^jjq

§6. 6.1. u e

DEFINITION.

H^(K;Zp) . If

p

u € H^ K;Zp) . Then

lf

Let

K

=

r = i + m(q2 + q)/2.

Dq(p„i )u =

p > 2 _

The Reduced Powers

and1 Sq1 .

be a finite regular cell complex and let

> 2, let m = (p-i)/2.

Plu where

and let

tfe define

(mJ)q D (q-21)(p-l)U

^

If

p = 2,

Sq^-u

we define

= Dq-iu •

Restricting ourselves for the moment (in VIII §2 the restrictions are removed) to the absolute cohomology of finite regular cell complexes we have 6.2.THEOREM.

The

P1

satisfy all the axioms in VI §i (except

for the Adem relations which will be proved in Chapter VIII). We divide the proof into a number of lemmas. 6.3.

hEMMA.

(mi)2

s(-l)m+1 mod p.

§6. PROOF.

THE REDUCED POWERS

By Wilson’s Theorem, -1

a

(p-O!

(p—i) i

...

P1 =

AND

Sq1

113

-i. Therefore

(E^i) . (2^1)

...

=

1.2

(p-i)

s

1 .2 ... (!£.)[- U ^ l l ] ... (-2) (-1)

=

(m!)2 (-1)®

The lemma follows. 6.4.

LEMMA.

P° = 1 .

PROOF. Let dim u = q . Then by 6.1 P°u where

n * m(q2 + q)/2. 6.5.

LEMMA.

■

By 5.^

P u = u.

Cartan Formula. If Pk (u x v )

*

u € i f (K)

and

v € H^(L)

then

^s+t=k pSu x ptv •

PROOF. Psu x ? \ where

=

( - D n (m-)-r-^ D (q_2s)(p_1}u x D (r.2t)(p_1)v

n a s + t + m[q2 + q + r2 + rl/2.

^s+t=k pSu x p v

Therefore

“

(“T)n (m.’) r q Ss+t=k D (q-2s) (p-i)u x D (r-2t)(p-i)v

-

( - D m r q + n (m-’)'r-q D (r+q.sk)(p.l)(u x v) by i .7,

Now mrq + n -

k + m[(r + q ) 2 + (r + q)]/2 . The lemma follows by 6 .1 .

6 .6 . LEMMA.If dim u = 2k,

PROOF. where

r

and 3.5.

P^u

=

then

Pku = up .

(-l)r (m!)_2k D qu

= k +m(4k2 + 2k)/2

s k(m + i) mod 2 . By 6 . 3

(m!)-2k ^ (-i)k(m+1)

mod

p.

The lemma follows from 3 .U. Combining the lemmas we obtain 6 .2 . Restricting ourselves for the moment (in VIII §2 the restrictions are removed) to absolute cohomology of finite regular cell complexes we have 6.7.

THEOREM.

The

Sq1

satisfy the axioms of I §1 (except for the

1U

V II.

CONSTRUCTION OF THE REDUCED POWERS

Adem relations which we prove in Chapter VIII). PROOF.

The proof of Axioms i)-5) is very similar to the proof of

6 .2 , except that we do not have to worry about coefficients in

have only to prove that

6 = Sq1 .

= U 2q„.qu

Sq1u

Now if dim u = 2q

=

® 2 H*(W, x ^

x pKp2;Zp)

d2 : W 2 -— > W 2P .

The following diagram is commutative

HM (K;Zp) -----£------ > ^ ( W 2xpKp ;Zp) ----— ---> Hpq(W?/pxK;Zp ) P'

^^(W^^Xp^jZp)

I P

*

^

(d')*

and

P

t f ^ C V ^ f V g X p l P ) 5 ;^)— ^ > # S1(Vrix1t(Wa/pxK)p jZp) I d3*

V * H^^/itxWg/pxKsZp) 0 .

tuting in (1 ) we have that if (2 )

=

( i) -

nomial coefficient is equal to

then

H

'

(C^-2r)

Por 2 s > k

k < 2 s,2 c

and dim u =

Z^ (“j^r) Sq1C+°-r Sqru.

By I 2 . 6 this bi­

and q

r > 0 . Substi­ = 2S -1

+ c,

By VII 6 . 8 the theorem is

proved. 1 .6 .

THEOREM.

The

P^

defined in VII 6 . 1

satisfy the Adem rela­

tions. PROOF. and

v (q)

v(q) d*Pu

=

By VII 3.5, VII k . k and VII 6 .1 , we have, writing

2m * p -1

(m!)- k

By I 2.6 this binomial and

ps > k < pc

r > o.

and dim u = q =

then pC+k-r pru _

=

u

By VII 6.8 the lemma is proved. 1.9*

LEMMA.

PROOF.

The second Adem relation

Let

a

=

(pq - 2k)

and

b

is satisfied. =

(pq - 2 1) .

By 1.3, (2)

and (3) we have (5)

Let

((q; j i -i 1) pk-mq-i+* p p q

q

=

2ps + 2c

now arbitrary.

and let

=

q ( - D r+f+1 ((q- ^ )

+

zr ( - D r+f ((q; ^ - 1) pf-mq-r+k w ru

H

_ “

( (Q.-2r)m\

\ mq-k+r/

pr < k.

s,c

and

k

are

/(p-1)(l+...+ps_l) + (p-1)(c-i)\ \ i - c / 0 1

unless

The integers

.

Then

/(q-2i)m-1 \ V mq-£+i /

Now suppose that

c + mq.

-

w £'mq-r+k pru

"

if i ^ c if i = c .

_ /(q-2r)m\ \ k-pr J

k < pc.

_

by 12.6 and 1 .k

/(p-1) (ps+c-r)\ k - pr /

~

\

The binomial coefficient just examined is zero

Therefore it is zero unless

coefficient is equal to

for

r < c.

PS > k

By I 2.6 this binomial r > o.

have /(q-2r)m-i\ \ mq-k+r /

_ /(q-2r)m-i\ ~ \ k-pr-1 /

_ “

/(p-1)(ps+c-r)-i\ \ k - pr - 1 /

We also

V III.

122

RELATIONS OF ADEM, AND UNIQUENESS

This binomial coefficient is zero unless unless

c < r.

pr < k.

By I 2 .6 this binomial coefficient is equal to

0 P”k-pr-i^_1)

for

pS > k

Substituting in (5) we have that if 2ps + 2c

Therefore it is zero

^

r >°-

ps > k < pc and dim u = q =

then PkpP°u

=

Ep(-i)r+k ((p'kli°"r)) ppC+k'r pI>u

+

^ ( . , ^ 1

((p-O(c-r)-i) pc+k-r ppru

#

By VII 6.8the lemma is proved.

§2.

Extensions to Other Cohomology Theories.

We now extend the definitions of

P1

and

Sq1

so that they operate

on relative cohomology groups. 2.1.

THEOREM.

If

F

is a cohomology operation defined for abso­

lute cohomology groups, then there is one and only one cohomology operation defined on both absolute and relative cohomology groups, which coincides with

F

on absolute cohomology.

Furthermore these extensions to the rela­

tive groups of the reduced power operations axioms (see I § 1 PROOF.

and

If

Sq^

and

P*^

satisfy all the

VI §i).

a e K

we have a commutative diagram

0 -- > H^KjajG)

> H^KjG)

> H^ajG)

> 0

0 -- > ifd^ajG')

> H^KsG')

>jffasG')

> 0 .

By diagram chasing we obtain a unique definition

for F: H^CKjajG)

H^CK.ajG').The definition is natural for maps of pairs

---->

where the second

space is a point or is empty. Let A

attached.

(K,A)

be a pair of spaces.

Let

L be K

with the cone on

By excision we have an isomorphism H*(L,CA) ---> H* (K,A) .

Let

c

be the cone-point of

CA.

By the five lemma we have an isomorphism

§2.

EXTENSIONS TO OTHER COHOMOLOGY THEORIES

1 23

H*(L,CA) -- > H*(L,c).

pairs

These constructions and isomorphisms are natural for mappings of * (K,A) . Since we have defined F on H (L,c), we obtain F on

H*(k ,A). It is immediate to check that all the axioms listed in I §i VI §1

follows from the axioms for absolute cohomology.

and

This proves the

theorem. We now have (K,L)

where

K

Sq1

and

P1

defined on cohomology groups of pairs

is a finite regular cell complex and

2 .2 .

THEOREM,

a)

L

is a subcomplex.

There is a unique definition of

Sq1

and

on the singular cohomology groups of an arbitrary pair of spaces, which coincides with the definition on finite regular cell complexes, b)

There is a unique definition of

Sq*

and

P*

on the Cech cohomology

groups of an arbitrary pair of spaces, which coincides with the definition already given on finite regular cell complexes. The extensions in both a) and b) satisfy all the axioms in I §1 and VI § 1 . PROOF.

We shall leave the reader to check that the axioms are

satisfied whenever we extend the definitions of

Sq*

We first extend the definition to pairs infinite regular cell complex and

L

or

(K,L)

a subcomplex.

P*.

where

Now

L

is an

H^(K,L; zp)

is

naturally isomorphic to

Horn (H^(K,L) ,Zp). Therefore H*(K,L;Zp) is the * inverse limit of the groups H (Ka ,La ;Zp) where and La vary over the finite subcomplexes of

K

and

this gives a unique definition on

L. Since the reduced powers are natural * H (K,L;Zp) . A continuous map from one

pair of infinite complexes to another pair maps finite subcomplexes into subsets of finite subcomplexes.

It follows that

on the category of pairs

where

cell complex and

L

(K,L)

pair

L

(K,L)

and

P1

are natural

K is a (finite or infinite) regular

is a sub comp lex.

Now we extend the definition to pairs plex and

Sq1

a subcomplex.

According to

(K,L)

where

K

is a CW com­

J. H. C. Whitehead (see [2 ]), the

is homotopy equivalent to a pair of simplicial complexes.

This

P*

V III.

12k

RELATIONS OF ADEM, AND UNIQUENESS

obviously gives a unique and natural definition for P1

or

Sq1

on

H*(K,L). We now give the definition on of an arbitrary pair

X,Y.

Let

H*(X,Y),

SX

andSY

of the singular complexes of the spaces

X

have a singular homotopy equivalence h:

be the geometric and

(X,Y) .

of

and

P1

h* : H*(X,Y) —

Y

realisations

(see [2]).

Then we

(SX,SY) ---> (X,Y) . Moreover

this map is natural for maps of pairs CW complexes, we have defined

the singular cohomology

Since

Sq1

in

(SX,SY)

is a pair

H*(SX,SY).

Since

> H* (SX,SY)

is an isomorphism, this gives a unique and natural extension of Sq1

to singular cohomology groups.

P1

and

This proves the first part of the

theorem. We now extend Sq1 mology groups of a pair of

(X,Y)

and

(X,Y)

P1

to Cech cohomology.

The Cech coho­

are obtained by ordering the open coverings

according to whether one covering refines another, taking the

nerves of the coverings, and then taking the direct limit of the cohomology groups of the nerves.

Since we have introduced

Sq1

and

cohomology structure of the nerve of each covering, and natural, this defines Sq1 see that pairs

Sq1

(X,Y).

and

P1

and

P1

uniquely on

P1 Sq1

H*(X,Y).

into the and

P1

are

It is easy to

arenatural with respect to continuous maps of

This completes the proof of the theorem.

§3.

The Uniqueness Theorem.

In this section we shall prove that the

Sq1

and the

P1

are

uniquely determined by the axioms i)-5) in I §1 and i)-5) in VI §1 . We shall do this by investigating the cyclic product of spaces. Zp

We shall use

as coefficients throughout this section. 3.1.

LEMMA.

Let

K

be a chain complex over

Z . Then

homotopically equivalent to the chain complex which is isomorphic to

K

H^(K)

as a graded module and has zero boundary. PROOF. of

Kp

Let

Bp

be the boundaries in

which is complementary to the cycles.

complex which is

Hp(K) + BQ + Dp

K

and let

Then

in dimension

q,

K

Dp

be a subspace

is isomorphic to the

and whose boundary

is

§3.

operator is zero on Therefore B + D D,

K

THE UNIQUENESS THEOREM

H (K) +

and maps

125

isomorphically onto

is the direct sum of the chain complexes

has the contracting homotopy

H

and

B^_1 .

(B + D ) .

s which is defined to be a map into

and such that s: B -- > D , is the inverse SI Si 4+ the boundary. We extend s to K by letting s(H) = 0. Let

which is zero on

of p.:

K

The

[x\

> H

Let order

p

bethe projection and let

1

=

D

and K

\n

and

L

—

1

X:

H -— > K

by the homotopy

be chain complexes.

it act on

This proves the lemma.

Let

acting by cyclic permutations on

free acyclic complex and let

s.

K?

«

be the cyclic group of

and

W Kp

be theinjection.

Let

iP.

and

W ® Lp

W

be a Tt-

by the

diagonal action. 3.2.

LEMMA.

If

f ,g:

K

> L

are chain homotopic, then

1 ® fp ,1 ® gp : W ® KP--- > W ® Lp are equivariantly homotopic. PROOF. such that D:

By VII 2.1 there is an equivariant map

h(o* ®

w) =

® w and

h(T ® w)

=

I K -— > L be the chain homotopy between

h:

I W

® w. f

> Ip ® V

Let

and g.Then we

have

the equivariant chain maps I ® W ® Kp -M l _ > iP ® W ® KP -2-> W ® (I ® K)p - i ® L > W ® Lp . The composition is the required equivariant chain homotopy. 3.3.COROLLARY.

If

f:

K

> L

is a homotopy equivalence, then

i ® f-P is an equivariant homotopy equivalence. From 3.1 and 3.3, we see that variantly homotopy equivalent. equivalent to

W

Therefore

and

H o m ^ W ® Kp,Z )

each isomorphic to

Zp . Then H# (K)P =

B

=^

Thea c t io n o f

are equi­

is homotopy

Hom^CW H*(K)p ,Zp) .

We choose a direct sum splitting of

where

W ® H*(K)^

H^(K)

on A^p

Zi“ 1 A ^ .

into components

A^,

So

Z^_i A^p + Zp (it) ® B

< i2 < ... < i ^ it

=

H^(K)

< lp \

® ••• ®

i s by c y c l i c p erm u tation and

. on Z^( it) ® B

by

126

V III.

the usual action on

RELATIONS OF ADEM, AND UNIQUENESS

Zp(*)

and the identity on

B.

So, if

H*(K)

is of

finite type, Homjt(W ® H# (K)p ,Zp)

« Ei Ho m^ W ® A ^ Z p )

+ Horn (W ® Zp (*) ® B, Zp) .

¥e then obtain immediately 3.i. writing

LEMMA.

(¥ x Kp)/ir

d:

K

K

be a finite regular cell complex.

Then,

= ¥ x^ Kp ,

H*(¥ ® n Kp) Let

Let

¥/n x K

«

E± H*(W/« ® AiP) + H*(¥ ® Zp (jt) ® B)

be embedded in

¥ x# Kp

.

by the diagonal

map

> KP . 3.5.

H*(X,A)

REMARK.

For any pair of spaces

are modules over

H*(X)

(X,A), H*(X), H*(A)

in an obvious way.

and

Moreover it is easy

to see that the maps in the cohomology sequence are consistent

withthe

module structure. of

(X,A)

If we have a map X — > Y, then the cohomology sequence •Jfr -Xr gets an H (Y)structure via the induced map H (Y)— -> H (X) .

The cohomology sequence of via the projection

(¥ x^ Kp ,¥/rc x K)

¥ x^ K ? --> ¥/ jt. The u € H*(*)

is multiplication by on

u x 1P , where

=

is a module over

H*(jt)

action of a class

H* (¥/*) i

is the unit class (or augmentation)

K. ¥e have the maps H^-(K) U L > Hpq(¥ Xjr Kp) - U _ >

3.6.

PROPOSITION. * by the image of d P.

The image of

d*

Hpq(W/n x K) .

is the

H*(it)-module generated

PROOF. By VII ^.1 it will be sufficient to show that is the sum of erated by

Im t , where

Im P.

t

is the transfer, and the

H*(¥ n x K?;Zp )

H*(«)-module gen­

¥e see from 3.^ that we need only show

1) H* (¥ ® Zp (7t) B) C Im 2) H (¥/jc ® Pu± , where

t

and

is generated as an H*(*) -module by is dual to a generator of

A^.

the element

§ 3 . THE UNIQUENESS THEOREM 1 ). Let X:

PROOF of 1 e Zp all

to

1 €it.

Let

v:

other elements of

Zp

> Zp («)

Z (jt)

« to zero,

be the map which sends

> Zp x

i 2?

send

induces

1 € it to

i € Zp and

a map

X*: C*(¥ ® Zp (it) ® B;Zp ) --- > C*(¥ ® Zp ® B;Zp) . An equivariant cochain of •3f x . ¥e also have a map

¥ ® Zp (it) B is determined

v*: C*(W ® Zp ® B;Zp ) induced by

v.

Since v x

=

¥e must show that any the

transfer of

1,

>C*(W ® Zp («) ® B;Zp) x v

equivariant cocycle

u

a cocyclein¥ ® Zp (it) ® B. In orderto prove that

that

'=

* x *u )

image under

■¥r -K-

it follows that

¥ Zd (jt) ® B. x*(t v

x*.

by its image under

1. in ¥ ® Zp (jr) ® B

Nowv*x*u

Tv x u

=

u,

is

isa cocycle on

we need only show

x*u since an equivariant cochain is determined by its

From the definition of

t,

it follows that

x*tv

*

=

1.

This proves 1). PROOF The

of 2) . Let

H («) -structure on

v x ip

where

v e H*(it)

is a module over sider

A^

space to u.

o.

in

K^.

Now

Hom(Ai,Zp)

H (¥ x Kr)

generated by

u^generate

C^.

H*(¥ ® A^p )

%

H*(it) CLP

1 x u^p . Now as in 3.1 we can con­

for some

q.

Let

L^

be a complementary

as a cochain by insisting that

is defined by the composition (u.)p

W ® Kp whichis equal to

Therefore

¥e can represent

Pu.

andlet

is given by cup-products with elements

(see 3.5).

as a subspace of Ai

(L ) -

H*(it)

=

1 (ui)P *

Kp

Therefore

zp

1 ® u^p

=

Pu^ . Thisproves 2)

and the proposition follows. ¥e now define a graded module

S

-

(Sr]

where

Sr C Hp (¥/* x K)

defined by the formula sr Note that

j

3.7. maps

Sr

=

■

E0 < 3 < ( p - D r / p ® 3^

pj - (p-Oj < (p-i)(r-j).

LEMMA. 5:

Hr (¥/* x K) -- > Hr+1(¥Xif KP ,¥/*

monomorphically.

x K)

V III.

128

PROOF. Sr n Im i* erated

By the cohomology exact seuqence, we need only show that

=

0. By 3.6 we need only show that Sr n {H*(*) -module gen­

by Im d*P]

VII k ' k .

RELATIONS OF ADEM, AND UNIQUENESS

=0 . Now

=

Z ^ ^ " 1^ Wj x D^.u

by VII 3.2

and

By V 5.2 and VII 5.1)wk d*Pu

But

d*Pu

k+ q(p-i) >

=

wk+q(p-l) x aqu + other> terms.

(p-1)q which proves the lemma.

We now define a modified -transfer

such that the following

t1

diagram is commutative. H*(W x Kp) ---—

> H*(W x

Kp , ¥ x

K.)

/*

T X

' 3 J H*(W X

Kp) TT

Let

Kp

be subdivided so that the triangulation is invariant under

has the diagonal as a subcomplex (subdivide

it and

K to get a simplicial complex

and then take the Cartesian product of the triangulation as defined in r3 3 p. of

67) .Since W x Kp

a € Kp,

H*(W x

by

Kp)

e u

= H*(Kp), we can represent any cohomology class

where u

is

T !(e u)(w ® a)

If

a cocycle on

Kp.

If

w e W

and

we define

a€

then

oca

= a

t*(b ® u) Therefore

t t(s

s

Za€jf

s(ow) u (aa)

‘

Leit

s(w) u< >

for all a € it and so . (w ® a) =

p e(w) u(a)

= 0 .

® u) e C*(W x KP ,W x K^). This defines the modified trans­

fer. Let

h:

EP — 8:

Recall from 3.5that 3.8. If

P = 2,

> K

be the projection onto the first factor.

Let

H*(W/it x K) -— > H*(W Xjr KP ,W/it x K) . 5 isa homomorphism

LEMMA.~5(¥2i-l x 5(w^ x u)

of

H* (it)-modules.

= v 2i •.T,(1 x =

wi+1 .

t

T(1

x h*u)

• .

Let u e H^-(K) .

§3.

THE UNIQUENESS THEOREM

w^ . t ’ (1 x h*u)

PROOF,

129

is given by (see p. 67 for definition of N)

¥ ® Kp— >W ® Kp ® ¥ ® Kp

® Kp— M L _ > w ® Kp

>K Ji— >Zp.

The composition of the first two maps is equivariantly homotopic to the identity by V 2 .2 . Therefore w Let

h^u:

KP -— > Zp

K? - K^. u"N

Then

u"

v. ,

t

’( (1i

Xr

x h u)

Kp , _ b i _ P ^ a s M - > z . P

be written as

u ’ + u" where u 1 = 0 on

is invariant since

= 0 .Therefore

w. . t'(i

is fixed under

h\)

x

u” |Kd

=

u|K.

¥ ® nKp Now

= d 0 1

£

cocycle.

+ 1 0 £

Therefore

>

Therefore

=

& (hfu - u ’) 8 (Wj

¥e mustshow

= X

=

-8u’ since u)

that for

Zp.

d 0 i

since wu

is a

is represented by

J

8u"

Therefore

KpL 0 !..■■> Zp.

and we can leave out

(-i)q+J'(w. 0 uM)(i 0 b) Now

it.

= 0 on

8(w^ x u) is given by

iUlLh_> w

8(w^ x u)

and u"

is represented by

w ® Kp Now

is represented by

=

i

(Wj

X

even or

(-i)^(w. 0 su") . 0

u

is a cocycle.

8U’)(-1)J*+ 1

Therefore

.

p = 2 , -w^ 0 u'N

wi_1 0 8u! have the same class in H* (¥ x^ K?,¥/ir x K) .

and

It is sufficient

to

show that these two cocycles havethe same value on every relative cycle in (¥ x^ K?,¥/?t x K) . Such a relative cycle has the form

where e. ® n oq_j+i Therefore for

j

even or

p = 2

-e 0-1 ®* Scq-j+i+1 + wej-1 ®# °q-j+i e W ®* Kd Therefore N cq-j+i - 5 °q-j+i+i £ Kd Now since

u! = 0

on

KL

we have for

i

even or

p = 2

V III.

1 30

RELATIONS OP ADEM, AND UNIQUENESS

u N ) ( Z ^ ej

(w±

cq-j+i} =

=

u Koq

"

u Scq+1

( - D ^ s u )c q+i ® 5U ^ Ej=0 ej ® ° q . J ^

3.9.

THEOREM.

For a fixed odd prime

of VI §1 characterize the operations B1

(i = 0 ,1 ,2 ...)

(i = 0 ,1 ,2 ...).

the axioms 1 through 5 Precisely, if

is any sequence of cohomology operations satisfying

these axioms then, for each PROOF.

P1

p,

i,

B1

=

P1 .

From the axioms we deduce that

(1 )

&P1

=

Pis

(2 )

Plw 1

(3)

Cartan formula. ^k , „ v i r>k-i, ^i=0("1) W 2i(p-1) P

=

as in I 1.2. from VI 2 . 2 and so P1 (w1u)

0

*

=

w 1Plu

by the

vk / , \ i T -nk-i.u ^i=o^"1^ w 2i(p-i)+2 P yk~1 / vi pk-i-i ^ i ^ " 1' 2 (i+1 )(p-i) +2 p 1

By 3.7,

(k)

5:

Hr (W/it x K) —

> Hr+1(¥ x^ Kp ,¥/jr x K)

maps

Sr monomorphically. (5) By Let 6

7 = is an

3.8,

“&(w2i-i x

* v 2i 4 T ^ 1 x h*u ^ •

^ =0(“1 )iv2i(p-i )+i x pk""iu € H*(¥/jtx K) . ¥e recall that H* (jc)-homomorphism by 3 .5 . *7

¥e see that

■ 2L o (-1)1 W2i(p-1)

5(W 1pk_lu)-

= zto(-i)1 w2i(p-l) Pk-i(5(w1u))

by (1 )and (2)

= ^ . 0(-1)lt1w2i(p-i) pk"i(w 2- T '(1 x h*u ))^ = -w2Pk t 1(1 x h*u) If

q = dim u = 2s

t ’(i xh*u) =

q,

or

2s+i ,

we put

so

PkT*(i x h*u)

by (3)

k = s+i . =

0

(5)

and

Then

2k > q and dim 57

=

0.

§3.

Suppose 7*

(B1)

by replacing 5(Z|=0(-1 )±

The term

i

=

dim (7 - r 1)

P1

with

B1 . As above

+1

57’

is omitted since

P° - B°

2i(p -1) +1 + q + 2(s -

=

2(s + l)(p - 1) + q + l

i 3l

{P1}. Then we can define 0.

=

(PS~i+1- B S'i+1)u)

X

=

■ { Therefore

satisfy the same axioms as

W2 i ( p _ 1 )

s+ 1

THE UNIQUENESS THEOREM

-

6(7-7')

=

1 - 1

2(s + 1)p

if q = 2s + 1

2(s + 1 )p - 1

if q = 2s.

2 (s + 1) (p - 1 )

= 2s(p ~

2(3 + 0 ( p - 1 )

- (p - 1 )/p

(p - 1 ) dim (7 - r f)/p

Therefore

(7 - 7 1) € Sr

then

where

7 - 7 1 = 0.

3.7 shows that

2i >2k > dim u

3.10.

1 ) + 2(p - 1)

r = dim (7 - r 1)*

Therefore and

THEOREM.

=

W 2i(p-1)+i

5(7

sSq1 =

(2)

i o

P^u = B^u = 0. The theorem

6(1

Let

7

We recall that

= 0,

k.

If

is proved.

The axioms 1 through 5 characterize the operations R1

(i = 0 ,1 ,2 ...)

is any sequence of each

Sq^s

i,

R1

= Sq1 .

I 1 .2 .

as in

wi

=

£i=0 vl+1 S q ^ u +

=

w 1 Sq^U.

H^W /* x K) -- > Hr+1(¥ x^ Kp , W/* x K)

w.+a S q ^ - ’u

maps

morphically by 3.7. (i)

- 7 1)

0 < i
2i(p - 1 ) + 1 =

(i = 0 ,1 ,2 ...). Precisely if

(3)

o .Now

(p - 1 ) dim (7 - 7 r)/p

Therefore

Sq1

=

- 0.

i + 1 )(p - 1 )

= 2s(p - 1 )+ 2(p - 1) - (p

i > k

Therefore

xu)

=

w 1 t * (1 x h*u)

= Zf=0 w± x 5 is an 67

by

3.8.

Sq^”1 u € H* (W/* x K) .

* H (*) -homomorphism by 3.5. Therefore

=

£i=0 W± 8(1 x Sqk_1u)

=

Zk=Q

6Sqk-1(i x u)

Sr

mono-

V III.

132

If

RELATIONS OP ADEM, AND UNIQUENESS

q = dim u, we put Suppose

define y '

=

£^=0 wi S q ^ 1 5(1 x u)

=

Z f =0 w± S q ^ ^ T ' O

=

w 1 Sqkx ’(1 x h*u) .

k

= q+1 . Then

SqkT '(1 x h*u)

{R1} satisfy the same axioms as

by replacing

Sq1 with

Therefore If

i>k

( y - y 1) y - y x

then

=

= 2q + i - o

Sq'Si

and

by 3.7.

=

R^u =

by 1

&(r - r 1)

i < q < (2q+1)/2. Therefore 0.

=.0

(Sq1).

R1 . As above

6(Zi=o ¥i x Now dim

x h*u))

Sq1u

=

b y = 0.

Then we can =

0. Therefore

=

Hence Rxu

and so

0 . y- y 1

for

€ Sr .

o < i < k.

So the theorem is proved.

BIBLIOGRAPHY [1]

J. Adem, "The relations on Steenrod powers of cohomology classes," Algebraic Geometry and Topology, Princeton 1957, pp. 1 9 1 -2 3 8 .

[2]

J. Milnor, "On spaces having the homotopy type of a CW complex," Trans. A. M. S., 90 (1959), pp. 272-280.

[3 ]

S. Eilenberg and N. Princeton 1 9 5 2 .

E. Steenrod, "Foundations

of

Algebraic

APPENDIX

Algebraic Derivations of Certain Properties of The Steenrod Algebra Cfc , the Steenrod algebra mod p, has been defined in VI §2 (in I §2 for p = 2) in a purely algebraic manner as the free associative algebra Q,

over

Z

y

generated by the elements

degree 1 (for p = 2 by Sq1 P°- 1 (Sq°- l

if

P1

of degree

i)

of degree

2i(p-i)

and

3

of

modulo the ideal generated by

p = 2) and the Adem relations.

The theorem proved

in this appendix (Theorem 2) is purely algebraic both in hypothesis and con­ clusion.

It was proved in Chapters I, II

and

VI

operate on the cohomology groups of certain spaces. here will be purely algebraic.

by allowing

&

to

The proof to be given

The only new step is an identity between

binomial coefficients mod p which was proved by D. E. Cohen [i ] in a paper on the Adem relations. Let

^e t*1e polynomial algebra over 2(pi- 1 ) (of degree

ofdegree

2i-

1

p = 2 ).Let

if

on generators

of degree 2P1- 1 .

be

theexterior algebra over

Zp

Let

H =

p = 2 ) . We shall define a diagonal

^

P E

^

(H=P

^ ® ^

free to choose

which will make

-

H a Hopf algebra.

tyg on the generators

uniquely determined. *H«i

if

on generators

Zp

E(tq,t1,...)

and

In doing so

and then

tg

we are will be

Let

Ei l t - i ® 4

and

% Tk ■

Tk ® 1 + A

i t -i ® Ti

•

The following lemma is easily verified. LEMMA i. tg

is associative. H

algebra with an associative diagonal.

H

is a commutative associative Hopf is of finite type.

From now on we shall give no special discussion of the case since this can be obtained by replacing 133

P1

with

Sq1

p = 2,

and suppressing all

APPENDIX

arguments involving

3

or

.

We define a homomorphism of algebras 11:

>H*

a

by letting' tj(P1) be the dual of

and

n(p)

the dualof

tq

in the

basis of admissible monomials. THEOREM 2. sends

The map

induces

t\

an epimorphismft

> H

which

no non-zerosum of admissible monomials tozero. Theorem 2 has the following corollary. THEOREM 3 - a)

n

ft--- > H*.

induces an isomorphism

ft has a

basis consisting of the admissible monomials. b) i|r(Pi)

ft

is a Hopf algebra with diagonal given by

= Z. P^ ® Pi-J*. and c)

H

ft.

As in VI 2.1

3.

ft .

we see that the admissible Part a)

Part b) is proved by showing that

cpg ^(P1) = l(3) cp-^

3 ® 1+ 1 ®

They are linearly independent by Theorem 2.

of the theorem follows.

where

=

is the Hopf algebra dual to

PROOF of THEOREM 3monomials span

i|r(p)

Z n(PJ) © tjCP1^ ) and

=

n(3) ® 1

is the multiplication in

H.

+ 1 ® t)(3) ,

Part c)

is trivial.

We shall now prove Theorem 2. The first step is to show that 2 is zero on 3 and on the Adem relations. It is easy to see that tj(3

)

=

0,since

if

x

is a monomial in

< ti(32) ,x > byinspection

=

< n(3) x

of the formula for

H,

i\

then

tj( 3)

>

=

0

In order to seethat

r\ maps

each

Adem relation to zero we need a lemma. LEMMA k .

(X PROOF.

(Cohen M3)

)

3

if

o < c < pd then

Z. C - 0 C+J'( Ctd )((d-j);p-1)-1) mod p. J J c-pj

The formal power series in a variable

t

with coefficients

APPENDIX

in

Zp

form a commutative ring.

135

A power series whose constant coefficient

is non-zero has a unique inverse under multiplication.

Let

f

he the

element of the ring given by f(t)

=

(0 +

tp)0+d/0 + t)c(P'l) + 1 .

The lemma will be proved by expanding If

tain

in two different ways.

weapply the binomial theorem to the numerator of f(t)

Since

f(t)

=

( ctd )(-i)j tPJ(i ♦

Z.

J

'

c - pj < p(d - j)

only if j < d,

t c ~PJ

.

J

the expansion of

(1

+ t)

(P-1) ~1

tc

in

(_,)j ( c+d )((d- 3 H P - ’)-1)

On the other hand

will con­

in which case the coefficient of

/(d-j)(p-1)-i\ ^ Therefore the coefficient of \ c-pj / z

f(t) we obtain:

(i + t)p

(! + t ) ^ 1- tP

= =

i + tp

f(t)

tc_PJ is is

.

and so

1 - t o + t)P-1 .

Therefore f(t)

= (i - t(i + t)p_l)c+d/(i + t)c(p*l)+1 = Z . (-1)j ( °td )t3(i + J ' J

If we expand

.

(i + t) (3“°)(p~1)-1 we obtain a term of the form tc~^ only

if

j < c. Let x_ . be the coefficient of this term. —• cient of tc in f(t) is

2J, < - 7» VC-J, (\ I J* ) Now

xQ

=

k

1

and the lemma will follow if

x^

Thenthe coeffi-

• =

o

for

k > o.

(- (p-i)k - i) (-(p-i)k - 2) --- (-(p-i)k - i - k + i) =---------------------------------------------- ----k l

■ i

(■?)

=

if

0

k>0

PROPOSITION 5. relation to zero.

by 12.6.

The homomorphism

rj: ft--- > H

sends each Adem

136

APPENDIX

PROOF.

Suppose

x e H

is a monomial then

< PaPP,X > This is zero unless a+p

=

x

Then

R(a,b)

R(a,b)

=

0

< p“ ® PP,l|rHX > . and (for dimensional reasons)

Then

=

< p“ ® P^.d, ® s0 + i0 ® e1 )J

-

< p % ?! ^

=

(

=

-PaPb + X. (-i)a+i 1

< p“p e ,,|J|2 >

Let

^1^2

=

j + k(p + i).

=

(

i

v

® 6,)k >

+ p k ® ^ 3tk'm >

j \ V a-pk'

in

Ct if

a < pb.

zero on monomials of the form

1^2

(P‘1)-1) pa+b-ipi a-pi

Nov wbere

r\ R(a,b)

could only be non­

a + b

j + k(p+i)

=

and its

value on such monomials is /a+b-k(p+i)\ \ a-pk / By Lemma k vith

d

=

« , 1va+i/(b-i)(p-i)-i\/a+b-k(p+i)\ i ^ ' \ a-pi > \ a+b-i-pk '

b - k,

c

=

a-pk

and

j

=

i - k,

this expres­

sion is zero mod p. We nov have to show that if

a < pb,

then

tj

sends the following

expression to zero (1 )

- paw b + 1,«L ( - D a+i((bi l*"? \ ci P«L' 1b

p p ^ 13"1 p1

+ zk Let

Qe H* be

p ^ ' 1 p p1 -

dual to

(2)

t 1e

Then

n(Pa 6 - 6 Pa)

To see this we note that monomials of the form identity

(2)

onusing

then (i)

=

rj(Pa3) ,rj(£Pa) £aTQ

and

Q, t)(pa ^) and

|a”1Tl?

Q T|(Pa_1)

are zero except on

and on these monomials the

is easy to check.

If a H*.

As in VI h . 2 we can set up a one-to-one correspondence between sequences i

I

=

(eQ,i1,e1,...,i^,e^,o,...)

0 ,1 ,2 ,...

=

and admissible sequences

with I*

sr

=

*■ 0 or 1

and

(sQ ,i1,e1,...,ik ',ek ,o,...

by the equations !r

T1 Let P1 I I T Then g (1}

*^1

=

= ^ - P^-r+1 - 8r * ii £k 3 1 ... P k 3 and let i, e ' ik ek Ti 1 *k Tk £1

3u P en - T0 0 T' and P have the same degree.

LEMMA 7. PROOF.

< P1 ,|J > is zero for

The last non-zero element

sequence

I 1 with i£ =

ik

k = 1, M = * T’ J