The Homology of Hopf Spaces

The Homology of HopfSpaces Richard M. KANE University ofWestern Ontario Ontario, Canada

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Library of Congress Cataloging-in-Publlcatlon Data

Kane, Richard M., 1944The homology of Hopf spaces. (North-Holland mathematical library v. 40) Bibliography: p. Includes index. 1. H-spaces. 2. Homology theory. I. Title. II. Series. QA612.77 .K36 1988 514' .24 88-16458 ISBN 0-444-70464-7 (U.S.)

PRINTED IN THE NETHERLANDS

IN MEMORY OF ALEX ZABRODSKY

vii

INTRODUCTION

This book is an exposition of the theory of finite Hopf spaces detailing how the subject has developed over the past thirty years. Our main theme is the homology of such spaces. But before beginning we will make a change in terminology. A Hopf space is more commonly referred to by topologists as a H-space. The use of the term "Hopf space" in the cover ti t l e was primarily designed to make the subject matter of the book as clear as possible to non topologists. From now on we will adopt the more conventional term of "He-space' in referring to our objects of study.

A H-space is a generalization of a topological group, A H-space is a (X,~) where X is a topological space with basepoint and ~: X x X ~ X

pair

is a continuous map such that the basepoint acts as a two sided homotopy unit. In other words, the following diagram is homotopy commutative.

X LL

1

Xx L

where LL and

L

R

~

x-x

1~

X

R are the standard inclusions. We will also always assume

that X has the homotopy type of a connected CW complex with finite skeletons. A H-space is finite if X has the homotopy type of a finite CW complex.

Fini te H-space theory is an outgrowth of the homotopy theory of Lie groups. With the advent of algebraic topology in the 1930's and 1940's mathematicians, quite naturally, began to study the homology and cohomology of Lie groups. It became apparent that some of the results obtained did not really depend on the entire Lie group structure but rather only on the much more limited structure captured in the finite H-space concept. The Hopf

[1]

structure theorem for

the rational

cohomology of Lie

groups was the first example of such a fact. Given a Lie group G, the Hspace structure on G induces. what is now called,a Hopf algebra structure

The Homology of Hopf Spaces

viii

on H* (X;~). The fact that H* (X;~) is a finite Hopf algebra then forces H* (X;~) to be an exterior algebra E(x •...• x where r) l §2) .

IXi I

= 2n - l i

(see

The explicit concept of a H-space is due to Serre [5]. Because of his interest in the path space fibration OX --+ EX --+ X he was lead to consider loop spaces. The usual mul tipl ication of loops is not associative (see §6). So he was lead to introduce the idea of a H-space to describe the mul tiplication on OX. As we wi 11 see,

fini te loop spaces have been an

object of intensive study. They are closely related to compact Lie groups (see §8). Throughout

the 1950's and early 1960's mathematicians continued

to

analyse the homology of the Lie groups. The homology was calculated for the semi-simple compact Lie groups See Appendix A for a summary of these calculations. In addition, a number of interesting general properties were obtained. Notably, we have Borel's mod p version of the Hopf decomposition (see §2) as well as his study of the cohomology of classifying spaces (see §20-1), Bott's proof that H*(nG) is torsion free in the I-connected case and Hodgkin's proof that K*(G) is torsion free in the same case. We also cannot fail

to mention Schereer's theorem that I-connected compact Lie

groups all have distinct homotopy types as well as the fact that, throughout this period. the only known examples of connected finite H-spaces were products of Lie groups, S7 (= the units in the Cayley numbers) and ffip7. After this period the focus in homotopy theory shifted away from Lie groups in themselves to the more general category of finite H-spaces. So it is at this point that our book begins its tale. We should note, however, that Lie groups have continued to provide a major stimulant to the development of finite H-space theory. As we will see. it has been mainly concerned with understanding the above Lie properties in the more general context of finite H-space theory. So Lie groups might be described as the experimental data of H-space theory. Our exposi tion is roughly chronological in order. From this point of view finite H-space theory has occured in two waves. The first wave consists of the work in the 1960's of Adams, Browder. Clark, Hubbuck, Milnor, Moore.

Stasheff and Thomas among others.

This material

is covered in

§§1-19. The second wave consists of the work in the 1970's and 1980's of

Introduction Adams. Harper. Kane. described in §§20-51.

ix

Lin. Wilkerson and Zabrodsky among others.

It is

("H-spaces from A to Z"?). A convenient marker

dividing the two periods is the Hilton-Roitberg [2] "criminal" (see §1O). It is probably more useful to break down the book on a thematic basis. From this point of view the book can be conveniently divided under three main headings. (a) The study of finite H-spaces with torsion free integral homology The study of such H-spaces has focused on their allowable type. Since H*(X) is torsion free one can write H*(X) = E(x 2n i - 1. Provided n

~ ... ~

n

then (2n

1..... 2n

,x where IX I = 1•··· r) i is called the type of

1 r C r-1) X. The study has focused on the type of finite loop spaces (However, one

should also consul t §18 and §19 for arguments involving the projective plane which apply to any finite H-space). The exposition of the work on finite loop spaces is contained in §§5-8 and §§20-27. The main result is a generalization to all finite loop spaces of structure theorems obtained by Borel in the Lie group case. More exactly. one studies the classifying space associated to the loop space and deduces structure theorems for the cohomology of the classifying space. (b) The study of finite H-spaces with homology torsion The study of torsion has centred on the study. for each prime p,

of

H*(X;W ) as a Hopf algebra over the mod p Steenrod algebra A*(p). Unlike p

the torsion free study this study does not generally demand extra structure for X but rather works as much as possible with an arbitrary finite H-space. As a consequence the arguments tend to be long and complex. The main tools of this study are Bockstein spectral sequences and secondary operations. The results obtained are described in §§11-17 and §§28-47. The main results of this study. besides a detailed knowledge of how A*(p) acts on H*(X;W ). are the proofs that H*(OX) and K*(X) are torsion free for any p

simply connected finite H-space. These. of course, generalize the previous mentioned results of Bott and Hodgkin. (c) The construction of finite H-spaces A natural question to ask about H-spaces is what examples of finite Hspaces can be produced over and above the classical examples. The answer

The Homology of Hopf Spaces

x

is qui te a few. Beginning wi th the Hi 1ton-Roi tberg [2] example many new finite H-spaces have been produced in the 1970's and 1980's. In §§9-10, §24 and §§47-51 we describe various constructions of new finite H-spaces. Localization plays a major role in all these constructions. Indeed. once we introduce localization in §9. we will work with localized spaces for the rest of the book. Correspondingly, we will cease to consider finite H-spaces and rather consider mod p finite H-spaces from that point on. We emphasize again that the main focus of this book is the homology and cohomology of finite H-spaces. Aside from the construction of finite Hspaces indicated in part (c) above and the study of H-space multiplications in §5 and §6 we do not work at the space level. Probably the major omission of the book is the topic of maps between classifying spaces. The author can only say that he regrets the omission of this important topic. There have been a number of other treatments of Lie groups and finite H-spaces. In the 1950' s there were two surveys of the homology of Lie groups. namely Samelson [2] and Borel [3]. In the 1970's four treatments of H-space theory appeared. Two were the brief surveys Curtis [2] and Rector-Stasheff [1]. The other two were book length expositions, namely, Stasheff [4] and Zabrodsky [15]. In addition the conference proceedings of the 1970 Neuchatel H-space conference (Springer-Verlag LNM # 196) might also be considered as another treatment of finite H-spaces. One should also mention Steenrod's 1968 collection of Math Reviews in Topology. The relevant sections provide a good survey of the early history of H-spaces. Regarding background

this book assumes a working knowledge of

basics of Algebraic Topology.

the

The material covered in Spanier [2] or

Whitehead [1] is a reasonable approximation of the level of knowledge being assumed by this book. This book has gradually taken shape over a number of years. The original impetus was a graduate course in H-space theory which I taught at the University of Western Ontario in 1980-1981. The book took final shape during a six month visi t to the Centre de Recerca Matematica, Insti tut d'Estudis Catalans. Barcelona during the first half of 1985.

I am very

grateful to the Centre for its financial support during that period.

I

Introduction

would also like to thank NSERC,

xi

the National Science and Engineering

Research Council of Canada, for its continued financial support over the years. I also owe a debt of thanks to various people who offered comments on a preliminary version of the book. In particular. I thank Steve Halperin. John Harper. John Klippenstein and Haynes Miller. Finally I would like to thank the typist who did such a painstaking job of typing the manuscript, namely myself. In turn I would like to thank Jo-Ann Kane for pretending that having a husband who typed endlessly for an entire academic year was a normal way to live one's life.

PART I: HOPF ALGEBRAS

We begin our study of finite H-spaces by considering Hopf algebras. Hopf algebra structures will playa pervasive role throughout this book. For the W and ~ cohomology of a H-space possesses a Hopf algebra strucp ture. In §1 and §2. following the exposition of Milnor and Moore [1], we develop some basic facts about Hopf algebras. Notably we prove the Hopf and Borel decomposition theorems. In §3 and §4 we demonstrate what might be called elementary applications of Hopf algebras. These are applications

to the cohomology of finite H-spaces not involving cohomology operations. They were among the first results obtained in the theory of H-spaces.In §3 we describe Browder's study of the covering spaces of finite H-spaces. In particular we demonstrate the usefulness of Hopf algebra structures in spectral

sequence arguments.

In §4 we describe Browder and Spanier's

demonstration that fini te H-spaces are Poincare complexes and also self dual. These results are attained by using the fact that the Borel and Hopf decompositions hold for the cohomology of finite H-spaces.

3

§1

HOPF ALGEBRAS

In this opening section as well as in §2 we will discuss some basic facts about graded Hopf algebras. The main reference in the literature for such Hopf algebras is Milnor-Moore [1]. We are distilling what. for our purposes. are the essentials of that paper. §1-1: Definitions and Examples

We will work in the category of graded connected modules of finite type over a ground field K. K will always be assumed to be perfect. As usual

Ixl

denotes the degree of an element. All tensor products. unless other-

wise indicated. are over K. Given a module M. then

L Mi. i~l

Mdenotes

the submodule

We will also use M to denote the identity map on M.

An algebra is a pair (A.
@

A ----> A is a map sa.t t sf'y tng the

following diagram: A

LL A L

R

1

@

t

A

~

A...:£......... A :;:;.

(Here LL and L are the left and right inclusion maps). A coalgebra is a R pair (C.A) where A: C ----> C @ C is a map satisfying the following diagram:

(Here PL and PR are the left and right projection maps). The diagram asserts that. for any c € C. we have A{c) where 0

<

I. Ic~1 < 1 1

Ic:

= c@l

+ l@c +

L

c:@c~ 1

1

Icl. A Hopf algebra is a triple (H.'I'.A) where: (i) (H.
The Homology of Hopf Spaces

4

(iii) A is a map of algebras (Anti!)commutativity and associativity of algebras can be expressed by the usual diagrams. By reversing the arrows we get diagrams expressing the cocommutativity and coassociativity of coalgebras. It should be noted that. unlike Milnor-Moore [1]. we do not include associativity and coassociativity as part of our definition of a Hopf algebra. When we wish to use these properties we wi 11 always specify them. Our defini tion of a Hopf algebra corresponds to what is called a quasi-Hopf algebra in Milnor-Moore [1]. In the non associative case we will adopt the convention that we always multiply from the right. So, a1~"'~

= ( ... «a1a2)~)"')~'

Finally. we have the concept of a Hopf ideal. This is an algebra ideal I such that A(I) C I 0 H + H 0 I. This property ensures that the quotient H/ I inherits a Hopf algebra structure from H. not just an algebra structure. EXAMPLES:

(1) The example which motivates our interest in Hopf algebras. in terms of this book.

is the K homology and cohomology of an H-space (X.Il). Given

such a X then (H* (X;K).A* .Il* ) and (H*(X;K).Il*.A*) are Hopf algebras.

In

the latter case Il* is called the Pontryagin product. The Hopf algebras H* (X;K) and H*(X;K) are dual in the sense to be discussed in §1-2 (2) The other standard example of a Hopf algebra which we will be constantly encountering is A*(p). the mod p Steenrod algebra (see Appendix B). Here K = IF • 'I' = mul tipl ication whi Ie A is derived from the Cartan p

formula. For p odd A(p n )

= L

p i 0Pj

i+j=n

A(o) = 001 + 100

For p

2 A(Sqn) = L Sqi@Sqj i+j=n

§1-2: Duali tv One of the great advantage of Hopf algebras is that their duals are also Hopf algebras.

More precisly, given a Hopf algebra (A.'I',A)

then

(A*.A*./") is also a Hopf algebra. The algebra structure '1': A 0 A -

A

I: Hopf Algebras

5

dualizes to give a coalgebra structure ~*: A* ~ A* @ A* while the coalgebra structure A: A ~ A @ A dualizes to the algebra structure A*: A* @ A*

~ A*. The Steenrod algebra illustrates the usefulness of dualizing. To describe A* (p) it is equivalent to describe the dual A*(p). And it is also much easier to describe A*(p). First of all, A*(p) is a free commutative and associative algebra.

p odd

where ITil = 2pi_1 I[i

I=

2pi_2

Secondly, to obtain the coalgebra structure of A*(p) we need only describe the action of ~*: A*(p) ~ A*(p) @ A*(p) on the generators. We have

*

~ ([k)

pj

= L hi

@[j

i+j=k

*

j

~ (T k) = L [~ @T j + Tk@1 i+j=k Given a Hopf algebra A. if we denote the Kronecker pairing < , ): A* @ A ~ F then the products and coproducts for A and A* satisfy the following useful relations <x·y,z)

<~(x@y).z) = <x@y,~*(z»

for any x,y E A* and a,~ E A. In particular, the reader can consult Milnor [1] to see how Milnor uses these to pass back and forth between A*(p) and A*(p) . For any H-space

(X.~).

H* (X;K) and H*(X;K) are dual Hopf algebras. The

Kronecker pairing will always be denoted in this case as < , ): H* (X;K) @ H*(X;K) We will constantly use the duality relation
~

K

6

The Homology of Hopf Spaces

where f: X

~

Y. x € H* (Y;K) and a € H*(X;K). We also recall that when K

=

then H*(X;~ ) and H (X;f ) are dual as Steenrod modules. See §31-2 for p p * p further details on this point.

f

§1-3: Hopf Algebras with Single Generators We now restrict our attention to monogenic Hopf algebras. These are Hopf algebras which. as algebras. are generated by a single element. We will only deal with the case of commutative (i.e. anti-commutative) Hopf algebras. It is easy to classify such Hopf algebras. There are fewer possibilities than one might. at first. suspect.

(1) char k - 0 The monogenic Hopf algebras are of the form H

= E(x)

Ixl odd

H

= K[x]

Ixl even

= K[X]/(Xn)

In other words. Hopf algebras of the form H

where Ixl is even

do not occur. To see this. observe that the generator x must satisfy A(x) = X®1 + 1®X Therefore

If we attempt to substitute x

n

=0

but xi ~ 0 for


we obtain the con-

tradiction n

.

.

o = L (~)xl®Xn-l i=O

We can rephrase this argument. An element y A(y)

~

0

€

H is said to be primitive if

1

y@1 + 10)'. A necessary and sufficient condition for K[X]/(Xn) to

have a Hopf algebra structure is that x (*) shows that x (2) char K - p

n

n

be primitive. However. equation

is never primitive.

>0

There are more examples of monogenic Hopf algebras than in char For now. equation (*) shows that x

n

is primi tive when n

based on the following binomial identity.

= pk.

=

O.

This is

I: Hopf Algebras

7

Lemma: Given p-adic expansions n mod p. In particular.

k

(~1 )

=0 mod p

< i < pk.

if 0

Thus the examples of monogenic Hopf algebras are: (i) P odd

(ii) P

=2

H=

E{x)

Ixl odd

H

K[x]

Ixl even

H

K[x]j

H

K[x]

H

Ixl even

k

(xp )

Ixl odd or even

= K[x]j (x2k )

Ixl odd

or even

The above monogenic Hopf algebras wi 11 playa basic role in the rest of this chapter as well as in §2. In particular.

the Hopf-Borel structure

theorem (see §2-l) shows that any commutative. associate Hopf algebra is isomorphic. as an algebra. to a tensor product

~

Ai where each Ai is one

1

of the above monogenic Hopf algebras. Moreover. many of the arguments of this chapter proceed by induction on the number of elements needed to generate the Hopf algebra. The monogenic Hopf algebras are the initial case of such inductions. Even at this introductory stage we can observe that our resul ts on monogenic Hopf algebras are useful. By looking at H* (S 2n ;~) we conclude: PROPOSITION: S2n is not a H-space for any n Motivated by the monogenic Hopf algebras we also introduce the concept of height. Given an algebra A then

o and

x E A has height n if xn - l ~ 0 while xn =

has height ~ if xn ¢ 0 for all n. As the monogenic case illustrates.

the possible heights of generators of Hopf algebras are restricted. Remark: If we drop our commutative (i.e. anti-commutative) hypothesis then further examples of monogenic Hopf algebras occur. For example. as H H* (OS2n ;~) demonstrates. we can have H §1-4: The Decomposition B - A

~

= K[x]

where

=

Ixl is odd.

C

We now prove a key technical fact about Hopf algebras. Let A C B be

8

The Homology of Hopf Spaces

= K 0 AB. We will show that B ~ A 0 C as left A modules. The left A module structures on B and on A 0 C

associative Hopf algebras over K and let C are defined by the maps

A0B~B0B-L..B

Actually we will give two proofs of the isomorphism B

~

A 0 C. Both ver-

sions will be useful. In the first version the isomorphism is established using the product map

~:

B 0 B

~

B. In the second version the isomor-

phism is established using the coproduct map A: B

~

B 0 B. A0f

A

LEMMA A: Let f : C -+ B be any map such that ,.-f = C. Then f: A 0 C - +

A 0 B ~ B is an isomorphism of A modules. Proof: The following commutative diagram asserts that f is a map of A modules. (Notice that we use the associativity of B) A 0 A 0 C

1

A 0 C

---_o,

A0f

Since f is an A module map

it

B

will be surjective if K 0

A

K0f (A 0 C) - +

K 0 B is surjective. But K0f is just the identity map C -+ C. A To show f is injective it suffices to show that A 0 C

.J:.... B -!... B

0 C

is injective. Filter A 0 C and B 0 C by the rule

LA

i~q

0 C. 1

L B 0 C. i~q 1 We claim that Af(

LK 0

C.) C F (B 0 C). First of all. since Af is a map 1 q

i~q

of A modules it suffices to show that Af(L K 0 C.) c F (B 0 C). Secondly, i~q 1 q A

it is easy to see that Af(K 0 C C Fs(B 0 C) for any s ~ 1. s) If we pass to the associated graded modules we now have a well defined map EO(Af): EO(A 0 C) -+ EO(B 0 C) of bigraded modules. The injectivity of EO(Af) is a consequence of the following commutative diagram

9

I: Hopf Algebras

°

EO (A 0 C) p,q

A

E (A£) • EO (B ® C) p,q

II

II

i0C

--=-"'---->.

A 0 C p q

B 0 C p q

Q.E.D.

We now extend the isomorphism of Lemma A. Observe that the isomorphism enables us to pick a map g: B

~

A of A modules such that gi

= A.

We now

prove

LEMMA B: Let g: B

~: B ~ B

~

A be a map of A modules such that gi

~ A0

0 B

= A.

Then

C is an isomorphism of left A modules

Proof: As in Lemma A the appropriate commuting diagram tells us that g is a map of left A modules. To show that g is an isomorphism we consider the composite gf: A 0 C

~

A 0 C

where f is the map in Lemma A. If we filter A 0 C as in Lemma A then we have a well defined map

°

°

E (gf): E (A 0 C)

°

Moreover. E (gf) AA

= the

AA

~

°

E (A 0 C)

identity map. Q.E.D.

Before closing §l-4 let us deduce a simple corollary of the preceeding proofs. Given associative Hopf algebras A C B we say that A is normal in B (A 4 B) if A.B

= B.A.

Normal i ty is exactly the condi tion r-equf r'ed to en-

sure that C = K 0 AB = B/A.B is a Hopf algebra (i.e. A.B is a Hopf ideal). When A 4 B we will think of C as the quotient Hopf algebra and write it B //A'

COROLLARY: Given associative Hopf algebras A 4 B 4 C then B//A is a sub Hopf algebra of

C

B

//A (i.e. the induced map //A

~

C

//A is injec-

tive) Proof: The isomorphism of Lemma A enables us to pick a map g: C modules such that gi = A. By restriction we have a map g': B modules such that g'i

= A.

~ ~

By Lemma B we have a commutative diagram

A of A A of A

10

The Homology of Hopf Spaces

g

B -

A @ B// A

-

A @ C// A

1

C

!

g

So, B

~

C injective implies that

B//A~ C//A is injective as well.Q.E.D.

§1-5: Primitives and Indecomposables Let A be a Hopf algebra. We define the module of indecomposables by Q(A) =

A/

A-A

= cok{ '/':

A@A~ A }

and the module of primitives by peA)

{a

E

A

jA(a)

= a@l

+ l@a }

ker{ A: A ~ A @ A } Indecomposables are a standard concept to consider. Primitives will turn out to be equally useful. We might note that primitives arise quite naturally in the topological situation. For. given a H-space (X.j.l) then x

E

Hn(X:C) is primitive if and only if the representing map X ~ K(C;n) is an H-map (see §6-4 for the definition of a H-map). There are two basic relations which exist between primi tives and indecomposables. The first relation is self evident. If we pass from A to its dual A* then the quotient module Q(A) dualizes to the submodule P(A*). Similarly, peA) dualizes to Q(A*). In other words THEOREM A: Q(At = P(A*) and peAt = Q(A*)

For the second relation we will only deal with commutative, associative Hopf algebras. If K is of char p t 0 we can define the Frobenius pth power map (: A

~A

((a)

= aP

Then ( is a map of Hopf algebras and (A)

Image ( is a sub Hopf algebra

of A. THEOREM B: Let A be a commutative associative Hopf algebra

I: Hopf Algebras

(i) If char K (ii) If char K

11

= 0 then peA) ~ Q(A) is injective = p > 0 then the following sequence o ~ P(f(A» ~ P(A) ~Q(A)

is exact

To illustrate Theorem B again consider the monogenic Hopf algebras as discussed in §1-3. char K - 0: For A

= K[a]

or E(a) we have P(A)

char K - P > 0: (i) For A (ii) For A

(i ii) For A

= K[a]

we have P(A)

= K[a]/ =



k

we have P(A)

(aP ) E(a) we have P(A)

k-l

=
>

=

Before proving Theorem B let us note some useful corollaries. It is an obvious consequence of the exact sequence in part (ii) of Theorem B that P(A)

~Q(A)

is injective in many degrees even when char K

~

O. We need

only check the degrees in which P(CA) is trivial. COROLLARY A: Let A be a commutative associative Hopf algebra. (i) If char K

=2

then podd(A) ~ QOdd(A) is injective

(ii) If char K

=p

> 2 then pn(A) ~ Qn(A) is injective provided

n Moreover.

;I;

0 mod 2p

i f A is bicommutative and biassociative the injections become

isomorphisms. COROLLARY B: Let A be a bicommutative biassociative Hopf algebra. (i) If char K

=0

then P(A)

(ii) If char K

=2

then

(iii) If char K

=p

> 2 then pn(A) ~ Qn(A) provided n

~

Q(A)

podd(A) ~ QOdd(A) ;I;

0 mod 2p

Proof: Corollary A applies to both A and A*.the dual of A. Moreover. pn(A*) ~ Qn(A*)

injective is equivalent to pn(A) ~ Qn(A) surjective.

Q.E.D. A Hopf algebra A is said to be primitively generated i f peA)

~

Q(A) is

surjective COROLLARY C: Let A be a Hopf algebra. (i) If char K

=0

then a is primitively generated if and only if

12

The Homology of Hopf Spaces

A* is commutative and associative (ii) If char K

=p > 0

then A is primitively generated if and only

if A* is commutative, associative and f: This

result

follows easily from Theorem B since

A*

~

A*

is trivial

the surjectivity of

peA) ~ Q(A} is equivalent to the injectivity of P(A*} ~Q(A*}. We now begin the proof of Theorem B. We need one lemma before we begin the proof proper. LEMMA: If B is an associative Hopf algebra and i:A C B is a normal sub Hopf algebra then the following diagram commutes P(i} P(1r} B peA} ------> PCB) ------> P( / / A}

o~

111

Q(A) ------> Q(B} ------> Q(B/ / A) Q(i} Q(1r)

~0

and also has exact rows Proof: (i) exactness of top row A.

A

~

Let x € PCB} where 1r(x) = O. Let g: B - + B 0 B =::..., A 0 isomorphism of A modules from Lemma 1-48. Since g(x}

= x01

B

//A be the

it follows that

x € A. (ii) exactness of bottom row

B

cok

= cok = cok

//A 0

-

-

B

//A ~

{ B 0 B ~

B/IB} B)

{ (A) + (B 0

cok

B//A

~ B }

A ~ B/BoB }

A

B

cok { /AoA ~ /BoB} cok { Q(A} ~ Q(B) } Proof of Theorem B (i) The case char K

Q.E.D.

=0

We want to prove that peA)

~

Q(A) is injective. We can reduce to the

= ~ Aa where Aa are finitely generated. We will proceed by induction on the number of elements need-

finitely generated case since. in general, A

ed to generate A (i.e. on rank Q(A}). If A is monogenic then peA)

~

Q(A)

I: Hopf Algebras

13

(see the discussion following Theorem B) If A is generated by n elements we can find a sub Hopf algebra A' generated by n-l elements where

A

//A' is

monogenic, By the above lemma we have a commutative diagram 0---> P(A') -----+ peA) -----+ p(A//A')

111

o -> Q(A')

-----+ Q(A) -----+ Q(A/ / A') ---> 0

with exact rows. We have

since A//A' is monogenic. By the induction hypothesis we also have peA') ->

Q(A') injective. It follows that peA)

(ii) The case char K - p Let A'

= rCA).

->

Q(A) is injective as well.

>0

Again, using the above lemma we have the diagram 0---> peA') ---> peA) ---> p(A//A')

111

Q(A') ---> Q(A) ---> Q(A/ / A')

A

with exact rows. As in case (i) P( //A') lows that peA')

->

peA)

->

->

A

Q( //A') is injective. It fol-

Q(A) is exact. Q.E.D.

There is one further relation between primitives and indecomposables which we will have need of when we prove the Borel-Hopf structure theorem in §2. Let

be the iterated Frobenius map. PROPOSITION: Let A be a commutative associative Hopf algebra over K of char p t O. Suppose ff: Then peA)

=0

in deg

A-> A is

trivial and Q(A)

=0

in deg

> N.

> pfN

Proof: We again reduce to the finitely generated case and proceed by induction on the number of elements needed to generate A. The monogenic case was discussed following Theorem B

In general. if A is generated by n ele-

ments we can find a sub Hopf algebra A' generated by n-l elements where A

//A' is monogenic. We have an exact sequence

14

TheHomQrogyofHopfSpac~

Since the proposition is true by induction for A and AII for A. Q.E.D.

A,

it is also true

§1-6: Differential Hopf Algebras We will make great use of the concept of a differential Hopf algebra. It is a pair (A.d) where A is a Hopf algebra and d: A tial ~:

(d

2

A0 A

= 0) ~

of degree ±l

A and A: A

~

such that

~

A is a differen-

the product and

coproduct maps

A 0 A are maps of differential modules. In other

words, the following diagram commutes

A~A0A...:£...A

1

1

dId d A--r A 0 A---;pA (A 0 A has the product differential d(x0y)

= d(x)0y

+ (_I)lx1x@d(y»

Given a Hopf algebra (A,d) the homology H(A) is also a Hopf algebra. Moreover, many properties of A are inherited by H(A). In particular, associativity,

coassociativity,

commutativity

and

cocommutativity

are

hereditary properties. Most of the spectral sequences which we deal wi th wi 11 be spectral sequences of differential Hopf algebras. The presence of the Hopf algebra structure severely limits how the differential can act. In this section we give a few indications of the restrictions which can be obtained. These results will suffice until we reach §13 when more detailed structure theorems for differential Hopf algebras will be developed. The restrictions of this section are based on the following simple, but extraordinarily effective, lemma about the action of d on a differential Hopf algebra A.

o

DHA LEMMA: Let (A.d) be a differential Hopf algebra. If d i


on Ai for

then: (i) deAn) C peA) (ii) d annihilates the decomposables of An

Proof: If x E A and A(x} n

+ 10dx +

L [dx:0x~ 1 1

+

= x01

+ 10x +

L x:0x~ 1 1

(-I}lxilx:0dx~J = dx01 1 1

then A(dx)

= dA(x)

dx01

+ 10dx. Q.E.D

Despite its simplicity this lemma will be used over and over again.

In

particular,when we combine it with the restrictions on primitives obtained

I: Hopf Algebras

15

in §I-S then the action of d on A is severely limited. Our next proposition is an illustration of this fact. PROPOSITION: If (A.d) is an exterior algebra on generators of odd degree

Proof:

then H(A)

= A (i.e.

Observe

that peven(A)

d

= 0) =

Qeven(A)

=

peven(A) we use Corollary I-SA). We prove d

<

Suppose d = 0 in degree

N. If d : ~ ---+

O.

=o

~+ 1

(For

the

triviali ty of

by induction on degree.

is non trivial it follows

from the DHA Lemma that QN(A) t 0 and ~+l(A) t O. This contradicts our first statement. Q.E.D. The usefulness of the proposition in future applications is due to the following COROLLARY: If {E is a spectral sequence of Hopf algebras and ~ is an r} exterior algebra on generators of odd degree then ~ = E"" For explicit applications of the above lemma and proposition in topological si tuations the reader should consul t the proof of Theorem B in §3 (Serre spectral sequence). most of the results in §11 and§12. (Bockstein spectral sequence) and

the proof of Theorem A in §29 (Eilenberg-Moore

spectral sequence). As we have already mentioned. we will return to the study of differential Hopf algebras in §13. As a last comment, we might mention duality as discussed in §1-2 extends to differential Hopf algebras. Given (A.d) and its dual (A*.d*) then d and d* are related by

<x.d(a» for any x € A* and a € A.

16

§2: DEroMPaSITIONS OF HOPF ALGEBRAS

In this chapter we discuss the Borel-Hopf decompositions of commutative associative Hopf algebras into a tensor product of the monogenic Hopf algebras discussed in §1-3. §2-1: The Borel-Hopf Decompositions The main result of the chapter is THEOREM A: (Borel,Hopf) Let A be a commutative associative Hopf algebra. Then A

~

@ A. as algebras where each Ai is a Hopf algebra on a i

1

single generator. We will prove this theorem in §2-2 In the rest of this section we deduce some consequences. The theorem is due in char 0 to _Hopf (see Hopf [1]) and in char p # 0 to Borel (see Borel [1])

Recall that in §1-3 we

classified the monogenic Hopf algebras. So the possible choices for the Ai factors in the above theorem are very limited. In particular, when

(x.~)

is a H-space then H*(X;~) and H*(X;r ) are commutative associative Hopf p

algebras. So the above theorem implies strong restrictions on the cohomology algebra structure. COROLLARY A: (Hopf) If (X.~) is a finite H-space then H*(X;~) = @ Ai where i

COROLLARY B:(Borel) If (X.~) is a finite H-space then H*(X;r ) where A '" E(x i) or

r [x.]/ p

1

p

(x~ 1

= @ A.1

h.

1).

The only point to observe is that finiteness removes the possibility of polynomial algebras occuring among the factors from Theorem A. The decomposi tion H* (X;Gl) write H*(X;Gl)

= E(x 1, ... '''Ie)

= E(x1 •.... x r)

is called the Hopf decomposi tion. If we

where Ixil

= 2n i - 1 and

n l ~ n2 ~"'~ n r then

17

I: Hopf Algebras

(2n l-l ..... 2n

r-l)

is called the type of X and X is said to have rank r.

The decomposition H*(X:W ) p

=0 i

A. is called a Borel decomposition and the 1

elements {x.} are called Borel generators of H*(X;W ).

P

1

The Leray-Samelson theorem asserts that. under certain conditions. the Borel-Hopf decomposition can be dualized. We will use this theorem in §7-3. THEOREM B:(Leray-Samelson) Let A

= 2n i-l.

bra where Ixil then

A*

= E(yl •.... yr)

= E(xl ..... x r)

be an exterior Hopf alge-

If the dual Hopf algebra A* is associative where IYi l

= 2n i-l.

Proof It suffices to show that A is primitively generated. For the surjectivity of P(A) tive. Thus A

=0

Q(A) means that we can choose the {Xi} to be primi-

~

E(x

i)

as Hopf algebras. Dualizing. A*

=0

[E(x

i)]

*

as

Hopf algebras. By Corollary l-SC it suffices to show that A* is commutative associative and has trivial pth powers. By hypothesis A* is associative. Regarding the other two properties observe: (i) P(A*) ~Q(A*) is surjective

(For. dually. P(A) ~ Q(A) is injective by Theorem l-5B and Qeven(A) Thus it suffices to show x p elements belong to peven(A*)

= yp = O.

= [x.y]

= 0)

* Such 0 for all x.y € Podd (A).

Q.E.D.

As wi th the Borel-Hopf theorem the Leray-Samelson theorem applies to compact Lie groups and.

indeed. originated from a study of that case.

Given a compact Lie group G we know. by Corollary A that A isfies the hypothesis of the theorem. So

H*(G:~)

= H*(G:~)

= E(yl •...• yr).

sat-

Moreover,

if H*(G) has no p torsion then. by our next proposition, A = H*(G;W ) satisfies the hypothesis of the theorem. So H (G;W ) case as well.

*

P

= E(yl •...• y

p

r

) in that

We now use Corollaries A and B to deduce one further consequence. It is a strengthened version of Corollary B. tions in §8-2 and §11-2.

It will find important applica-

The Homology of Hopf Spaces

18

PROPOSITION:(Borel) If (X.~) is a finite H-space and H*(X;~) H*(X)

E(x1.···.x r) over ~ then

= E(x1 ..... x r )

IT or ® Wp

. N such that H* (X;W ) particular. there eXIsts p

>N *

Proof: Let us begin with H (X)/ H*(X X X)

H*(X)

I Tor =

-

the embedding all x € H (X) I

H*(X)

I Tor ®

Tor

when p

p

H*(X)

I Tor C H*(X:~)

.

.

I Tor' It IS an exterIor algebra since

= E(x1 •...• x r )

implies that x

2

=0

for

Tor'

* (X)I

bras is obtained by the maps

H* (X)I H*(X)

= E(x 1 ....• x r )

® W . It is a Hopf algebra since

The correspondence between H

If

over W .In p

I Tor® Wp

= E(Yl'"

Tor

Tor

® W and H*(X;~) as exterior algep

®W p

·.y s )

E(Yl,····Y s) (over Z(p))' So H* (X:~) ~) Q.E.D.

=

§2-2: Proof of the Borel-Hopf Decomposition The general case is the direct limit of the finitely generated case. We will prove the finitely generated case by induction on the number of elements needed to generate A. If A is generated by n elements let A' be a sub Hopf algebra generated by n-l elements such that We have the canonical map

If we pick f:

A

II

A,

~

r:

A such that wf A' ® AliA'

= AII A,

~A

® A

A

II A, is monogenic.

then. by Lemma l-4A. the map

...J!..... A

is an isomorphism of A' modules. If f were mul tiplicative we would be done. (We can apply the induction hypothesis to A') Now i and

~

are alge-

I: Hopf Algebras

bra maps. For

~

we use the fact that

~

19

is commutative So we want to show

t ha t f can b e c h osen to b e an a 1ge b ra map. I n ot h er wor d s , I f AllA' generated by the element

a we want

to find a € A such that

=a

'IT(a}

= height

height a This is only a problem when char K

char K = p

~

a

A

II A, is a truncated polynomial

0 and

~

algebra. So assume 0

a = pf

height If we let B

.

1S

(f(A') then we can form the diagram

r

A'II

----7

B

i

q

r

All ~ All'

A' - A

T

Rather than lifting

a directly

II

'IT

A

A

lilA'

T

j

B

B

B

to A we will first lift a through 'IT': AilE

~ AliA' and then through q: A ~ AI I B. Let us first observe a few facts about diagram (*). We have Q(A'} = Q(A}

Q(A'II

B}

= Q(AI I B}

Also, by Corollary 1-4, A/ I B is a sub Hopf algebra of AI I B. And we have the usual isomorphism

So, in particular, 'IT' is surjective. (i)

Lifting for 'IT: Pick b

€

AliB

AII A,

----7

AI I B such that 'IT'(b} A(b}

= a.

= b0l

A'

I( liB}' Since ( f

A(bP )

= bP

We want to show that bP

f

= O.

+ 10b + c f

f

=0 01 +

on

A'liB it follows that

We have

The Homology of Hopf Spaces

20

Consider the exact sequence

= N it

If Ibl

A'

A'

f

A

=0

degree) p N. (For Q{ //B) and Q{ //A') both p

f-l

A'

//oB and

N. Thus b

A

=0 Cf = 0

follows from Propostion 1-5 that P( //B) and P{ //A')

pf

A//B')

in degree) n and A

It follows from (**) that P( //B)

in on

0 in degree)

= O.

(ii) Lifting for q: A ~ A// B f

It suffices to show that a P € B. For

Pick a € A such that q{a) = b. f f then. if a P # 0 we have a P =

ab

f

for ~ € A'. So we merely replace a by

a-aO' By Lemma 1-4A we can pick a B module map g: A

~

B such that gj

= B.

f

f

By Lemma 1-4B the map ~ A g: A -----> A

@

A

~

=04

B I!IJ

~

f

A

is an isomorphism of B modules. However. g(aP ) f

a P @1 + l@aP

f

+ dP

f

//B

= g{aP

)@l. (For A{aP )

f

where d € I(A') @ I(A')) SO. a P € B.

§2-3: Extensions of the Borel-Hopf Theorem There are various possible extensions of the Borel-Hopf decompositions. First of all. if we relax our definition of Hopf algebras so that the ground field K need only be a ring we can ask if Borel-Hopf type decomposi tions hold for such Hopf algebras. The case K = il(p) has received a great deal of attention. See Husemoller [1].Ravenel-Wilson [2] and Hubbuck [14]. There are many results about the choice of generators of a Borel-Hopf decompositions. The choice of generators is far from unique. For example. if A

= E{x 1 •..• x r)

then we can replace each x.1 by x.I =I x. +I d. where d.1 is

decomposable and we again have A = E{x ....• x When A is commutative. 1 r). associative and primitively generated the Borel-Hopf theorem can be strengthened to assert that we can choose the generators of the decomposition to be primitive. In other words. A

~ ~

Ai as Hopf algebras (not just

1

as algebras) where each Ai is a Hopf algebra on a single generator.

I: Hopf Algebras

21

More generally, if we drop our commutativity hypothesis and assume only that A is associative and primitively generated we can use peA) to define a generalization of the Borel-Hopf theorem. The Lie bracket map [ , ]: peA) 0 peA) [x,y]

= xy

~

peA)

- (_l)lxllylyx

gives peA) the structure of a Lie algebra. If char K = 0 then A = UP(A).

the universal enveloping Hopf algebra

associated to peA). (The construction of UP(A) is a simplification of

«( = 0)

that given below for VP(A». The Bott-Samelson theorem is a special

case of this result. It is the case

A = H.. (OX;~) peA)

= II.. (OX)@(Q

where X is arbitrary and the bracket product on II*(DX)@(Q is defined by the Samelson product. (see Bott-Samelson [1]) If char K

=p >0

we also have an identity A

= VP(A)

for A associative

and primi tively generated. Here, VP(A) is the universal enveloping Hopf algebra of the restricted Lie algebra P(A). The restricted Lie algebra structure consists of the bracket product [

]: peA) 0 peA)

~

peA) plus

the Frobenius pth power map (: peA) ~ peA). To define VP(A) let TP(A)

= the

tensor algebra generated by peA)

and let VP(A)

= TP(A)/I

where I is the ideal generated by { [x.y] - xy + (_l)lx l Iylyx and {(x) - xP

o

I x,y

I

x.y€P(A) }

€ peA)}. In particular, the abelian case (i.e.[x,y]

for all x,y € P(A» of the identities A

= UP(A)

=

or A = VP(A) is the

Borel-Hopf decomposition. Regarding the primitivi ty hypothesis, we might remark that it is an open question as to whether H"(X;~) is always primi tively generated for some choice of the multiplication

~:

Xx X

~

X on each finite H-space X.

Another question about Borel decompositions concerns Steenrod operations. If (X,~) is a H-space then H*(X;W ) is not only a Hopf algebra but p

also a Steenrod module. One can ask about the relation between the two structures. A natural question to ask is whether a Borel decomposi tion

o

A. can be chosen to be compatible with the Steenrod module structure of 1

H*(X;W ). Asking that each factor A. is invariant under the action of p

1

22

The Homotoqy of Hopf Spaces

A*(p) is too much. Rather the appropriate question would seem to be if H* (X;W p)

= U(M)

where M is an unstable Steenrod module. For the definition

of an U(M) algebra as well as a study of the question H*(X;W ) §47.

p

= U(M)

see

23

§3 COVERING SPACES

In this chapter we study the covering spaces of finite H-spaces. Our main tool is the Serre spectral sequence. Notably, we will demonstrate the usefulness of introducing a Hopf algebra structure into this spectral sequence.

fhe reader should observe how the DHA Lemma of §1-6 is used to

control the action of differentials. Our arguments are based on the work of Browder [lJ. §3-1:Main Results It is a fact of elementary topology that the covering spaces of X correspond to the subgroups of "l(X), Given H covering space p:X ~ X satisfies "l(X)

"l(X) then the corresponding

~

= H.

See, for example, Massey [lJ.

Regarding "1 of a H-space we have: THEOREM A: For any H-space

(X,~),

"l(X) is abelian.

Proof: Given a,~: Sl~ X we need a homotopy of the form

We do this in two stages

The inner diamond is obtained from the map 8 outer triangles are based on the fact that ~(xO.~(t».

1

x 8

aCt)

1

~ X x X ~ X. The

~ ~(a(t).xO)

and

~(t) ~

Q.E.D.

In the "1 abelian case we can obtain the covering space (up to homotopy

24

The Homology of Hopf Spaces

type) as the fibres of Eilenberg-Maclane spaces. Namely, given He ITI(X) the

corresponding

covering

space

is

the

fibre

of

the

map

X-->

K(IT1(X)/H,l). In particular, the universal covering space is simply connected and is the fibre of the map X --> K(IT1(X),l). We want to show: THEOREM B: The covering spaces of finite H-spaces are also finite H-spaces The proof is a Serre spectral sequence argument and will be done in the next section. §3-2:Proof of Theorem B

To prove Theorem B we use H-fibrations i. e.

fibrations F.!..... E

i..... B

where F.E,B are H-spaces and i.f are H-maps (see §6-4 for the definition of a H-map). We have the following basic facts regarding such fibrations LEMMA A: Given a fibration F .!..... E

i..... B where

E,B are H-spaces and f is a

H-map then F is a H-space and i is a H-map. LEMMA B: If

(X,~)

is a H-space then f:X --> K(G,n) is a H-map if and only

if [f] E Hn(X;G) is primitive So. in our case, the map X --> K(IT1(X)/H,l) will be H-maps and the corresponding covering space will be a H-space. We are left wi th demonstrating the fini teness of the covering space. First of all, we can reduce to the case IT

1(X)

finite.

PROPOSITION: For any H-space (X,~) X '" Sl x .. x

s'

fini te Proof: If rank IT

r we define Xby a fibration of the form

1(X)

-

X -+ X --> S

(recall S

1

K(1,1»

x X where IT 1 (X) is

1

X ..•

x S

1

We then use the multiplication on X to define a map

s'

X ••• x

Sl x X --> Xx ... x X

---->

X

which is obviously an equivalence in IT*( ). Q.E.D. To prove the finiteness of our covering space in the IT l finite case we use the Serre spectral sequence with mod p coefficients. Given a H-fibration F --> E --> B the resulting spectral sequence

25

I: Hopf Algebras

H*(F;W ) 0 H*(B;W ) p

p

==>

H*(E;W ) P

is a spectral sequence of commutative associative Hopf algebras. This Hopf algebra structure imposes strong restrictions on the action of differentials in the spectral sequence. Notably, we have the DHA Lemma of §1-6 Since we have reduced to the case of ITl(X) finite (and abelian) we need only demonstrate for H-fibrations of the form

that X is finite whenever X is finite (up to homotopy type). Let us deal with the case p odd until further notice. The Serre spectral sequence is of the form E 2 where lal

=1

= H*(X;Wp )

=2

and Ibl

==> H*(X;Wp )

0 E(a) 0 W [b)

p

(see Appendix B for the cohomology of Eilenberg-

Maclane spaces). Since H*(X;W ) is finite there must be a non trivial difp

ferential in the spectral sequence. Suppose d.

=0

d

"# 0

1

We know dra

= drb = O.

r


when i

So the action of d

effect on H*(X;W We will use the p).

r

is completely determined by its

DHA Lemma of

§1-6

to show that d

r

first acts non trivially in a transgressive manner i.e. in the following

manner H*(X;W )

d~ r

E(a) 0 W[b) p

Moreover, this transgressive action completely determines the action of d on E

r.

(i) r

Regarding the transgressive action we have:

= 2ps

for some s ~ 1

(ii) there exists indecomposable x €

~s-l(X;W

To see this pick the smallest i such that d Lemma d (Hi(X;W r

r

p

»

p

) such that d i -

r

"# 0 on H (X;W

2ps

p).

(x)

By the DHA

C peE ). By Lemma 1-5 we have an exact sequence r

= bPS

26

The Homology of Hopf Spaces

PH*{X;F } ~ peE } ~ P{E(a) @ F [b]) p

p

r

Obviously d Hi(X;F } n Hi(X;IF} = O. Also P(E(a) I» IF [b]) has basis r p p p 2 {a,b,bP,bP , ... } (see the discussion following Theorem 1-5B). Next, the relation d d

2ps

on E 2ps

(iii) E2ps

s

= bP

(x)

2p More precisly,

*= H (X;lFp)//E(x)

s

completely determines the action of

where d 2p s (x )

@ E{a) I» E{x) @ IFp[b]

= bP

s

This identity holds as differential Hopf algebras. The verification of (iii) uses strongly the fact that p is odd.

We can certainly choose such

a decomposition as Hopf algebras. (since p is odd x

2

= 0).

Regarding the

action of d2ps we have d In other words, d E2p s

/

2ps

_ = H* (X)

s

(bP )

s

C bP oE 2ps(E2ps) 2ps

acts trivially on the quotient Hopf algebra IF [b]

I» E(a) @ p

/

s. We establish the fact by an argu-

(bP )

ment similar to the previous one. Next. choose a Borel decomposition of E2ps

including {x,a,b} as generators and with the remaining generators

from ker d2ps. We place y by height of

Y= y

can

always do so. For, if d2ps{y)

- xz. (Observe that since x

2

=0

= bP

s

z T- 0 we

can

re-

=

the

the height of y

y)

It follows from (iii) that (iv) E2ps+ l

*-

=H

(X;lFp)//E(X) I» E(a) @ IFp[b]/(bps)

Finally

To prove this we argue as before that if d. T- 0 for i act non

trivially in

a

1

trangressive manner

IF [b] s E{a) I» p /(bP ) ' However,

> 2ps

and hit a

it would first primi t i ve

of

there are no primitives in the appropriate

degree. The identity

*-

H (X;IF )// p

E(x)

@ E( ) I» IF [b]/

a

p

s (b P)

27

I: Hopf Algebras

now establishes that H*(X;W ) is a W vector space of finite rank. p

p

This concludes the proof of Theorem B for p odd. When p =2 we have the extra

complication

of

odd

degree

elements

having

possible

non

zero

squares. But we can repeat the above argument if we alter the algebraic structure. Namely, assume that odd degree elements have zero squares by introducing new indecomposables to replace the non zero square of any odd degree indecomposable. §3-3:Examples The above proof enables one to explicitly calculate the cohomology of covering spaces. First, consider rational cohomology. Suppose H*(X;Ill) = E(x

1,

.. ,x

r).

Splitting off copies of S1 from X amounts to removing the ex-

terior algebra generators of degree 1. Fibering X using K(l/pk,1) does not affect H*(X;Ill). Next,

o

consider W cohomology. p

Pick a Borel decomposi tion H*(X;W ) = p

A,. As we split off copies of S1 we remove exterior algebras E(a) where 1

lal = 1 from H*(X;W ). When we fibre X using K(l/pk,1) we replace factors p

of the form E(a) 0 Wp[b]/(bp s) where lal = 1 and Ibl = 2 by an exterior algebra E(x) where Ixl = 2ps-1. Examples (1)(Baum-Browder[1]) Let X 1I1 (X) (p ) =

PU(n). Write n

l/l

*

H (X;Wp) = E(x 1,x3,·· .,x2pk+1,·· .,x2n+1) 0

W[b] p

If X is given by the fibration X ~ X ~ K(l/pk,1) then

(2)(Ishitoya-Kono-Mimura [1]) Let X = 1I 1(X)

1, We have

pkm where (p.m)

= r/z

Ad(~).

Then

k

/(bP)

28

The Homology of Hopf Spaces

Then X

= E7

is given by the fibration X --+ X --+ K(Z/2, 1) and

This example illustrates the fact that the Serre spectral sequence does not give complete information about the algebra structure of H*(X:f ) when p

p

= 2.

If we follow our recipe as outlined above for calculating H*(X;f

from H*(X;JF we would replace JF2 [ X1]/(xi) by E(x However, 2) 3). also a non trivial algebra extension obtained by f 2[x3]/

4. This extension is not picked up by

(x 3 )

replacing E(x

2)

there is 3,x6)

by

the Serre spectral se-

quence. Observe. also. that this type of extension can not happen in the p odd case.

29

§4 POINCARE AND S-DUALITY

In this chapter we prove that a fini te H-space is a Poincare dual ty space. We also prove that if we work in a stable category then X+ (= X plus a disjoint basepoint) is S-dual to itself. Both of these results are consequences of the existence of Borel decomposi tions for H*(X;IF ). The p

type of non singular pairing between homology and cohomology which is necessary for such dualities easily follow from the canonical structure of H*(X;IF ). The reference in the literature for Poincare and S duality are p

Browder [1] and Browder-Spanier [1]. §4-1: The Cap Product and Slant Product In order to define Poincare duality we need the notion of the cap product. In order to define S-duality we need the notion of the slant product. The Cap Product

(c*,a)

Let us begin with the cap product. Let chains and cochains of X. Then

a and

(X,aa)

and (C* ,0) be the singular

0 are related by the rule

= (ox,a)

The diagonal map A: X ...... X x X induces a map A*: C* Pick a € C

n

such that aa

o.

C*


C*

Suppose

A*(a) = L ai0ai Then we define n a: C* ...... C* as follows

n a: Cq ...... Cn-q x n a = L (x,a:)a~ 1 1 This map induces a well defined map

n a:

Hq(X;G) ...... H (X;G) n-q

for any G. It suffices to do the case G

= ~.

We prove

30

The Homology of Hopf Spaces

LEMMA A: For x

Cq, 8(x n a)

€

This lemma implies that

= (-1)q+1(ox) n

n a: C*

a

C* is a "chain map" in the sense that

-+

"'* ,0) '" where if we replace (C*',0) by (C

2I ~ then

n a: CA*

-+

=


= (-1)q+1 0

on

cn-q

C* is a legitimate chain map.

Proof of Lemma A 8(x

n a) = 8[L <x.ai>aiJ = L <x,aj>8ai (-l)q

L (-1) lail<x,ai>8ai

Regarding the last equality observe that <x,ai>

o

unless lai

I

q.

Our

second series of equalities is (ox) n a

= (ox) =

na - x n

(Ba)

L ai - L <x,8ai>ai

= -L

(-1)

= 0) - L (_1)lail<x,a:>8a~ 1 1 (Ba

lail<x,a:>8a~ 1 1

For the last equality we use identity (*) Observe that when G = F

p

Q.E.D.

(and. indeed. any field) the cap product can

be defined directly at the homology level. We do not need to pass to the chain level. Given a € H (X;IF) let A (a) = L a:@ a'". Then x n a = n p * 1 1 L <x,ai>ai. It follows that the cap product is directly related to the cup product in cohomology when G = IF p' LEMMA B: Given x.y € H*(X;IFp ) and -Proof:

<xy,a>

a

€

H~(X;Fp ) then <xy, ~

<x,y

n a>

=

= <x@y,A*(a» =

L <x,aj> ai>

=

Q.E.D.

The Slant Product To define the slant product pick x € (C*@ C*)n such that 8x define xl: C* -+ C* as follows

O. Then we

I: Hopf Algebras

x./': C --+ q <x/a.~>

for any a E

Cq'~

31

cfl- q

= <x.a0~>

E C n_q' This induces a well defined map x/: H (X;G) --+ Hn-q(X;G) q

because of the lemma

Again. when G

Wp' we can define the slant product directly at the homo-

logy level.

LEMMA D: For any a,~ E H (X;W ) and x E H*(X x X;W ) we have

*

<x/a.~>

p

p

= <x.a0~>

§4-2: Poincare Duality A finite CW complex X is a Poincare complex of dimension n if (i) Hi(X)

=0

for i

(ii) Hn(X)

~ ~

generated by [X]

(iii)

n

>n

[X]: ~(X) --+ H (X) is an isomorphism for 0 ~ q ~ n n-q

The rest of this section will be devoted to proving THEOREM:(Browder) A finite H-space

(X.~)

is a Poincare complex

Condition (i) and (ii) Let

m

= max

Hi(X;~)

n

= max

Hi(X;W ) # 0 }

Since H*(X;~)

= the

# 0 }

p

= E(x1 ....• x r)

= the

rational dimension of X mod p dimension of X

it follows that

Hm(X;~)

=~

generated by x 1x2 ... x Similarly, the Borel decomposition H*(X;W r' p)

=0

Ai

implies that Hn(X;IF' ) p

= FP

Thus to verify properties (i) and (ii) for any H-space X it suffices to show that mod p dimension of X

rational dimension of X

32

The Homology of Hopf Spaces

Case I: X simply connected. Obviously m

~

n. All the generators {ail of

the Borel decomposition 0 Ai have degree

~

2. Thus. besides Hn(X:Wp )

n-l we also have H (X;W) p

= O.

= wp

n

By the universal coefficient theorem H (X)

=

Z. Hence ~(X;~) # 0 and n ~ m.

Case II: "1 (X) finite. Let Xbe the universal covering space of X. By case I

mod p dimension of X

= rational

dimension of X

By the arguments in §3

= mod p dimension of X = rational dimension

mod p dimension of X rational dimension of

I Case III: general X. By Lemma 3-2C X = X x S

fini teo

X .•• x

X of X S

1

where 11 ( X) is 1

Condi tion (11 i) First of all, we can reduce to considering the mod p cap product n[X]: H* (X;Wp ) ->

H~(X:W ~

p

). For, by Lemma 4-1A we can regard n[X]: C*-> C* as a

"chain map". And, given a chain map f:A -> B between free chain complexes, then f*: H(A) -> H(B) is an isomorphism if and only i f f*: H(A 0 Wp) = the mapping cone of f

H(B 0 W ) is an isomorphism. To see this, let C p

C

n

= An

+ B

n

d: Cn -> Cn - 1

d(a.b)

= (d'(a),d"(b)

+ f(a))

The short exact sequence 0 -> A ~ B -> C -> 0 induces an exact triangle

"

H(A @ G)

f*

~

H(B @ G)

./

H*(C @ G)

By the universal coefficient theorem, H(C)

0 if and only if H(C @ W ) p

=

O. Secondly, the mod p cap product is an isomorphism. This follows from the Borel decomposition of H*(X:W ). For, by Lemma 4-1B. it suffices to show that the bilinear form

p

I: Hopf Algebras

33

is non singular. If we write [~ 1

then [X]

is dual

to f

=

h ] E(a ) ] @ [@ Fp [b.]/ J (bP j ) i j

p l

"i ... asb

hI

-1

j

ht

... b

p-I t

Moreover. a basis of

Hq(X;F and Hn-q(X;F can be chosen from the monomials in {ail U {b p) p) j}. Thus given each basis element x € hq(X;F ) there exists an unique basis p

element y E hn-q(X;F such that <xy,[X]> # O. Just let y p)

= fix'

§4-3: S-Duali ty

To define S-duality we must stabilize. Our stable category will be of the most elementary type. The objects will still be CW complexes. But the morphisms will be stable maps. In other words, {A.B}

=

lim[~nA,~~]. Thus

n""'"

we are considering a sub category of the usual category of CW spectra. (see Adams [7]). Two finite CW complexes A and Bare S-dual if there exists ~ E {A A B,SN} such that

is an isomorphism. Examples (1) Si and sj are S-duals (2) If M is a compact manifold and normal bundle

T(v) is the Thorn space of the stable

v of M then M+ and T(v) are S-duals (see Atiyah [lJ)

In general. any fini te CW complex X has as-dual D(X). However,

it

should be noted that D(X) is not defined as an explicit CW complex. Rather it is only determined up to homotopy type. The original proceedure for obtaining S-duals worked only for finite CW complexes.

(see Spanier [1]).

Now S-duality can be easily defined for the full stable category of CW spectra. When X is finite the mirror image isomorphism given above for ordinary homology and cohomology actually holds for any homology and cohomology theory E*( ) and E* ( ) defined by a CW spectra E (see §5 of Part III of Adams [7]). The main result regarding S-duality is: THEOREM:(Browder-Spanier) If (X,~) is a finite H-space then X+ is its own S-dual

34

The Homology of Hopf Spaces

Before proving the theorem let us note the need for adding a disjoint basepoint to X. If Y is its own S-dual then H*(Y;W ) is symmetric. More p

precisly, let~: YAY ~ SN be the S-dual map. Then H.(Y;W ) ~

1

for all i. It follows that

for all i. Such a symmetry holds for ~* H (X+ ;W ) p

P

= H* (X;Wp ).

::'::N-i

H'

~

(Y;W) p

This follows

from the arguments in §4-2. For, if X is a Poincare complex of dimension n then the non singular form ( , ): Hi(X;W ) 0 Hn-i(X;W ) ~ W p

p

p

defined in §4-2 establishes the isomorphisms

On the other hand,

i f we work wi th X rather

.

than X+ then this symmetry

= 0 whIle H (X;Wp) = Wp' p) The above discussion also gives a very strong hint regarding how to ~O

would be destroyed since,now, H (X;W prove the theorem. Suppose

(X,~)

~n

is a Poincare complex of dimension n.

Then our duality is defined as the composite

The last map q:X+ ~

Thus, if [X]

= q*(LN )

gN

is defined by the property

generates ~(X), we want to prove

LEMMA: ~*([X])/: H (X) ~ ~-q(X) is an isomorphism for q

0 ~ q ~ N

Proof: The proof is analogous to the argument in §4-2 proving condition (iii) for Poincare duality. First of all, it suffices to replace the coefficient Z by W in the lemma. Secondly, by Lemma 4-lD, it suffices to p show that the bilinear form ( , ): Hq(X;Wp) 0 Hn_q(X;Wp) ~ Wp = < [X],xy>

(x,y)

is non singular. Thirdly, if we have a Borel decomposition for H (X;W ) we

*

p

can continue to imitate the argument in §4-2 and show that ( , ) is non singular. In general, we have no such decomposition since H*(X;W

p)

is nei-

I: Hopf Algebras

35

ther commutative nor associative. However. since H.. (X;lF is coassociap) tive we can filter H.. (X;IFp ) so that the associated object OE(H.. (X;IFp » is a commutative associative Hopf algebra. We will delay the actual description of this filtration until §13 when we will give a systematic treatment of it. The fact that (

) is non singular for OE(H (X;IF » implies that .. P

( • ) is non singular for H.. (X;lF as well. p) §4-4: H-spaces and Manifolds

The above results raise the question about the relation between Hspaces and manifolds. In particular. the existence of Poincare duality suggests that every finite H-space could have the homotopy type of a compact manifold. Such a result is very appealing. particularly if one wants to think of fini te H-spaces as generalizations of compact Lie groups

How-

ever the question is still unanswered. though partial results are availc

able.

In Browder [6] it is demonstrated that a simply connected finite

H-space has the homotopy type of a compact topological manifold and that simply connected finite H-spaces of dimension 4k are actually smooth manifolds. In Pederson [I] it is shown that many non classical finite H-spaces are smooth manifolds. Cappell-Weinberger [1] have similar results.

PART II: a..ASSIFYING SPACES

The next spaces.

four chapters are devoted

to

a

treatment

of

classifying

In §5 and §6 we discuss the various approaches to classifying

spaces developed in the late 1950's and early 1960's. We sununarize the work of Milnor, Dyer-Lashof, Milgram and Stasheff. The main theme of these approaches is to obtain a construction of Ex which works under increasingly weaker hypothesis on X. As we will see it is the degree of associativity possessed by the multiplication quirement.

Jl:

X x X --> X which is the key re-

In §7 and §B we describe the structure theorems which were

obtained during the same period of time by Borel and Clark. for the cohomology of classifying spaces

In §7 we present the bar and cobar spectral

sequences and use them to derive Borel's result for H*(Ex:W In §B we p)' is willing to assume

describe Clark's astonishingly easy [prov tded one Dirichlet's Theorem) restrictions on H* (Ex;~).

39

§5: LOOP SPACES AND CLASSIFYING SPACES

In this chapter and §6 we will determine necessary and sufficient conditions for a H-space

(X,~)

to be a loop space. As we have mentioned. the

degree of associativity of the mul tipl ication

~:

X x X -> X is the key

requirement. Most of the discussion will be centred around the construc-

Ex

tion of a classifying space for X. namely, a space such that OB ~ X. X In §5 we consider constructions due to Milnor. Dold-Lashoff, and Milgram for the cases of topological groups and associative H-spaces.

In §6 we

further analyse the use of associativi ty in the construction of

Ex.

We

consider the concept of a An space as defined by Stasheff. The An structures measure the degree of associativity possessed by the multiplication u: X x X -> X. Notably.

ACIJ spaces possess enough associativi ty to have

classifying spaces. We can crudely summarize the results of §5 and §6 by saying that. in the homotopy category of connected CW complexes we have the following identities: loop spaces

= topological

groups

= associative

H-spaces

= ACIJ

spaces

The main references in the literature for the results of this chapter are Milnor [lJ [2J. Dold Lashof [lJ Stasheff [4J [7J and Milgram [lJ. §5-1: Lie Groups and Classifying Spaces Classifying spaces were first defined for compact Lie groups. The concept of a classifying space arose out of the theory of characteristic classes as developed in the 1930's and 1940·s.

The first classifying

spaces were the Grassmanian manifolds. Let n k G (lR + ) k

the space of k dimensional linear subspaces

0

f IRn + k

It was recognized that k sphere bundles over a finite complex X correspond to homotopy classes of maps

40

The Homology of Hopf Spaces

for n

» o.

To establish the correspondence one constructs a canonical Sk

bundle over Gk+I(IR"') and then uses the map f : X ---- Gk+1(IR"') to pull it back to X. See Whitney [IJ, Pontryagin [lJ and Steenrod [IJ for the results. By the late 1940' s classifying spaces associated to any compact Lie group had been constructed. We can consider G as a subgroup of O(k) for some k. The universal bundle is G ---- En ---- Bn

where

E _ O(n+k)/ n O(n) x I

k B _ O(n+k)/ n O(n) x G

The universal bundle is

G----E----B where E

=

B

= lim

lim E n n-+ oo

B n n-+'"

The total space En can be identified with the Steifel manifold Vn+k,n the set of k orthogonal frames in IR i


n+k.

it follows that IT. (O(n+k)/ ) 1 O(n)

=0

= ITi(O(n+k»

Since ITi(O(n» for i

< n.

=

for

So

En is (n-l) connected E is contractible

For a Hni te complex X the homotopy classes [X,BJ classify principal G bundles over X. The correspondence is established using the map f: X ---- B to "pull back" the bundle G E B over X. This concept of a classifying space includes the previously noted one of ~+1 (IR"') as classifying space for Sk bundles over X.

For when G

O(k+l) we have the identities B _ O(n+k+l)/ n O(n) x O(k+l) B

n+k+l

= Gk+1 (IR

)

co

= ~+l(IR

)

We can use O(k+1) ---- E ---- ~+ 1(IR"') to construct the universal Sk bundle

=

41

1/: Classifying Spaces

Sk

~+I(IR<Xl)

E'

II k E xO(k+l)S ExSk

E

Sk

Here /O(k+l) x /~ where (xog,y) ~ (x. goy) for any x € E. y € k <Xl k S . g € O(k+l). We can use this bundle to relate maps [X.G )] to S k+1(1R bundles over X. §5-2: Topological Groups Milnor

extended classifying spaces

from

Lie groups

to

topological

groups. Given a topological group G. Milnor constructed a contractible fibre bundle

Moreover. if the group operations are cellular then E and B are CW comG G plexes. Observe that we must then have flBG '" G For B CW implies that flB is CW as well. Also. the long exact sequence in G G IT*( ) plus the fact that E is contractible implies IT*(flBG) ~ IT*(G). G Remark: As with Lie groups the classifying space B acquires its name from G the fact that it classifies principal G bundles over finite X. As with Lie groups such bundles are in 1-1 correspondence wi th the homotopy classes

(i) Join construction To construct contractible fibre bundles we require the join construction. Define the q simplex Aq C IRq+1 by q A

={ (t 1.···.tq+ 1)1

def

t. ~ 0, 1

L

t l.

I}

Then. given spaces {X •.... X we have the join 1 q}

={

t 1x l+ ... +t x I (t ... ,t ) € 1, q def q q q terms for which t 0 } i

x

Aq- 1 and we omi t

=

For example pt

*

X

= ex.

the unreduced cone on X

~2

*

X

= LX.

the unreduced suspension of X

42

The Homology of Hopf Spaces

*

In general. one can interpret X

Y in terms of straight lines running

from X to Y. When X and Yare connected we have a canonical homeomorphism

X*

y ~

!(X

/I

Y)

We can extend the join to the infinite case by

*

lim Xl def q ... co

*

Xq

(ii) The Milnor Construction The following example illustrates and also motivates the general Milnor construction. One can use the join construction to obtain the fibration n co 7U2 --+ S --+ RP

7U2

1 Sn 1n

Rp Letting n

---+

7U2

1An (n+l 7U2 1

7U2

*... *

7U2

*

*

7U2j

fold diagonal) (n+l copies)

7U2

co we have 7U2

t

= 7U2 7U2

*

7U2

*...

7U2

*

7U2

*"'j

S

R~co

7U2

For an arbi trary topological group one can define maps Pn : En..... B and n p: E --+ B in an analogous manner G G G

1 E t

G

G

* ... *

G

1 = G * G * ... 1

(n+l copies)

EG

B

BG

n

LEMMA A: G --+ E --+ B is a fibre bundle G G Proof: Let t. : G 1 be the

.th

1

*

G

* ...

-->

co-ordinate function. Let Vi=pt i

We have equivalences

-1

ro.n

[0.1]

G

*

G

* .. 'I

G

43

1/: Classifyinil Spaces

where

~i

is defined by ~i(P(tlg1+···+tqgq).g)

and Pi is defined by Q.E.D. Observe that the full group structure of C is required to prove Lemma A. Also. we have ignored the problem of topologizing any of the objects defined above. Some care must be shown in treating this problem. The other properties of E and B are easily established. C C

Proof: Since E = lim En C n~CXl

it

suffices to show En is n-r l connected. This

follows from the standard homeomorphism Y LEMMA C: E

C

and B

C

*

Z

~

2Y A Z Q.E.D.

are CIi complexes

Proof: Since E is a CIi complex the identity Y * Z structure on En

= En_ 1 *

~

IY A Z imposes a CIi

C and. hence. on E as well. The fact that C is a C

cellular group implies that C acts cellularly on E So EC I C' as well.

Q.E.D.

C

is cellular

(iii) Simplicial Croups This process of replacing a topological group by a loop space can be reversed. If X

~

OY where Y has the homotopy type of a connected countable

CIi complex then one can find a CIi topological group C such that X

~

C. The

idea. which is due to Milnor. is to replace OY by a simplicial loop space. A CIi complex has the homotopy type of a simplicial complex. So we can assume that Y is simplicial. A simplicial loop in Y is a sequence (Yn'Y n- l" ..• yO) of points in Y such that each pair Yi'Y lies in a common simplex i- l of Y and Y = Yo = basepoint. Introduce an equivalence relation by the n rule

if Yi

= Yi + 1

or Yi

= Yi- 1.

Then

C(X)

= simpicial

100Ps/~

Multiplication in C(X) is by juxtaposition. Observe that the inverse of

The Homology of Hopf Spaces

44

[yn"·· ,yO] is [yo,··· 'Yn]' One can show that G(X) is a CW complex, the group operations are cellular and X ~ G(X). See Milnor [1] for details. §5-3: Associative H-spaces We next describe the construction of classifying spaces for associative H-spaces. This broadens Milnor's construction in that inverses are no longer needed. The basic construction is due to Dold and Lashof. However, we will also describe Milgram's variation of the construction. For we are

Ex

interested in the CW structure of

and his model has a cellular struc-

ture directly related to that of X. Rather than constructing a fibre bundle as in §5-2 the Dold-Lashof approach concentrates on constructing a principal quasi-fibration. says that f:X -> Y is a quasi-fibration i f 1I.. (Y) f-

1(pt).

=

one

1I.. (X,F) where F

A quasi-fibration possesses the important property of inducing a

long exact sequence ... ->

lIn(F) -> lIn(X) -> lIn(Y) -> IIn + 1(F) - ...

(i) Dold-Lashof Construction Given a principal X bundle p' E _

B with action

~,

Xx E -

E one can

define another principal X bundle DL{E}

= {X x

DL(B}

= CE Up B

1

with action

CE} UE

X x DL (E)

~:

{x,t,y}

1

~

x'{x,t,y}

{t,p{y}} ----+

DL(E}

= (x'x,t,y)

The usual topology on DL{E} must be altered to ensure the continuity of

~.

This construction can be iterated to give DLn{E} -> DLn{E}. In our case We start off with the principal X bundle X _ ->

X_

pt with action

X and define

In particular, observe that

}J(

= DL1 Cpt}

h: X -

fiX -

C B So we have a map X'

nBx

~:

Xx X

1/: Classifving Spaces

One shows that

EX

-+

Bx

45

is a quasi-fibration and that

EX

is contractible.

It then follows that h*: 1l*(X) -+ 1l*(nRx) is an isomorphism (ii) The Milgram Construction Milgram defines the quasi-fibration

EX -+ Bx by (x, t .v)

1

( t.y)

Bx

The identifications for

are of the form

::: [( .........• ~i •.. '::iXi+l.···)

The identifications for

EX

x ·

• t _ i 1+t i

(

)

i

if

t

i

if xi

= 0

= pt

are of the same type. Just ignore the extra

copy of X. It is an observation of Stasheff that the Dold-Lashof are obtained if we ignore the identification for xi

= pt.

EX

and

Bx

Equivalently. if

we take the Dold-Lashof construction and replace the ordinary cone by the reduced cone then we obtain the Milgram construction. If the"multiplication on X is cellular then B has a very explicit cellular structure. The

X

n

cells are A x e

1

x ... x en where n

~

0

and e

ranges over the cells of X.

i

(iii) Moore Loop Spaces We and

can

partially extend the Milnor correspondence between loop spaces

topological groups

to one between loop spaces and associative H-

spaces. We have already shown how to replace associative H-spaces with cellular multiplication by loop spaces. Conversely. given a loop space X

=

OY there is no problem in selecting an associative H-space of the same homotopy type. Just replace OY = (a: [0.1] -+ Y) by the Moore loop space

n'Y

= (a:

[O.r] -+ Y

I

r ~ O. a(O)

= a(r) = pt}

There is an associative multiplication on n'Y. Namely. given a: [O,r] -+ Y

(3: [0,

a

* *

a

*

then the product loop a

s]

-+ Y

(3 is defined by (3: [O.r+s] -+ Y (3(t)

= [~~g ~ ~

t

t

~ r ~

r+s

46

The Homology of Hopf Spaces

The homotopy equivalence between OX and {'l'X is established by using the inclusion map OX C {'l'X and the normalizing map {'l'X

~

OX.

Observe, however, that working with {'l'X forces us to abandon CW properties. Even if {'l'X has a CW structure we do not know that the multiplication is cellular, We can, of course, use the cellular approximation theorem to choose a multiplication which is cellular. But,

inevitably,

the

strict associativity of the multiplication will not survive such perturbations.

So,

cellulari ty is at odds wi th strict associativity. On the

other hand, the cellular multiplication property is important. It provides

Bx with a

CW structure. In turn this fact is used to prove

0Bx

~

X. All of

this suggests that we should look for a more general construction of classifying spaces which will enable us to perturb the multiplication up to homotopy. In §6 we describe such a homotopy invariant approach.

47

§6: A -SPACES n

The structures imposed on X in §5 to produce classifying spaces are not homotopy invariant. This creates gaps in our theory of classifying spaces. For example. if X has the homotopy type of a topological group or an associative H-space it does not follow that X is itself a topological group or an associative H-space. It would seem plausible that X should have a classifying space but we cannot produce one by our previous constructions. Again, the previous constructions do not allow us to perturb multiplications up to homotopy. We have already discussed this problem at the end of §5. In this chapter we discuss a construction of classifying spaces which relies on homotopy invariant properties. Such approaches have been provided by Stasheff and Sugawara. We will only discuss Stasheff's work. References in the literature for his approach are Stasheff [1] [2] [4] and [7]. See Sugawara [1] for Sugawara's approach. Stasheff's approach is based on the concept of a An form. The An form measures the degree of associativity possessed by the multiplication J.L:

X x X ---. X. In particular, Aoo spaces possess enough associativi ty to

have classifying spaces. The homotopy invariant approach has many benefi ts. The types of problems discussed earlier disappear. In particular, cellular properties are now always available. From the purely theoretical point of view the homotopy invariant approach is satisfying in that one obtains necessary and sufficient condi tions for a space to have the homotopy type of a

loop

space. Moreover, Stasheff's classifying space construction can be applied directly to any such space. The mentality of Stasheff's approach leads to the idea of an obstruction theory for loop spaces. The An forms are a series of intermediate stages between H-spaces (= A spaces) and loop spaces (= Aoo spaces). More2 over, this obstruction theory can be used in an effective way. Namely, by using projective planes one obtains necessary conditions in cohomology for spaces to possess a An form. We will discuss this briefly in §6-3. Many later chapters are devoted entirely to this theme. See, in particular, §7. §S and §§15.16.17.

48

The Homology of Hopf Spaces

As a final conunent,

the classifying space

space X but also on the multiplication

~:

Bx

depends not only on the

Xx X

~

X. A given space may

possess many different multiplications giving rise

to many different

classifying spaces. Equivalently, there may exist many different spaces B (up to homotopy type) such that

nBx

X

~

X. See Slifker [1] for the study of

S3. We will not discuss this problem. §6-I: ~ ~

Let

(X,~)

be a H-space. By using

~:

X x X ~ X one can produce a number

n

of maps X ~ X for each n ~ 2. Consider all the possible homotopies which can arise between these maps. The situation can be represented by a cell

Kn . For example K2 =

* (xy)z

x(yz)

o~----oo

.r> . (xy)(zw)

«xy)z)w

x(y(zw) ) ~(X(YZ»w x( (yz)w) Thus the O-cells (= vertices) represent maps, the I-cells (= edges) represent homotopies between the maps, the 2-cells represent homotopies between homotopies, and so on. To envisage general K we reason as follows. Each n n of the maps X ~ X corresponds to introducing n-2 sets of brackets. A homotopy between two of these maps corresponds to introducing only n-3 sets of brackets, namely, the n-3 sets of brackets the two maps possess in common. For example, in K 4 x(y(zw»

(xy)(zw)

Or-----~O

(xy)(zw)

«xy)z)w

o------~o

corresponds to the bracket xy(zw) corresponds to the bracket (xy)zw

A homotopy between homotopies corresponds

to

introducing n-4 sets of

brackets and so on. Thus K will be a n-2 dimensional cell and in partin cular. the faces of K correspond to introducing one set of brackets n xlx2"'(~"'~+s_1)xk+s ... xn' We define Kn inductively by

49

1/: Classifying Spaces

Kn

U (Kn- s+ l x Ks)k

cone on 8Kn

2~s~n-l l~k~n-s+l

The

face

corresponds

(Kn - s +1 x Ks)k

xl" .(~. "~+s-1)" .xn·

to

introducing

the

brackets

n-2 and, indeed, In all cases Kn is a subset of I

is homeomorphic to In. See page 284 of Stasheff [lJ for pictures of K , 2,K3 K , K from this point of view. We will say that (X,~) is a An space if it 4 5 possesses all the homotopies respected by K . Let us be more precis. X is n said to admit a ~ form (2 ~ n ~ 00) if there exists maps

M : K x Xi ~ X for 2 ~ i

where

i

~ n

2 2: X ~ X is a H-space multiplication

(i) M

(ii) the M are compatible in that M 1 . x Xi can be defined in terms i i 8K of M for j j

< i.

1

Namely, the map i

(K i - s + l x Ks)k x X

can be defined by

An forms are designed to be homotopy invariants in the following sense. If X possesses an An form and f: Y '" X then we use f to induce a An form on Y. The details of the construction of the A form on Yare, however, rathn

er complicated. In particular. it is not obtained merely by composition with the An form on X. Pasting and gluing techniques are also required. We will shortly indicate how to perform this construction when X is a strictly associative H-space. If X admi ts an An form then X is called an ~ ~ . An A space is 2 just a H-space. An A space is called a homotopy associative H-space. We 3 are particularly interested in Aoo spaces. Such structures characterize loop spaces. Let us begin by showing that everything studied so far falls

under the heading of Aoo spaces. Examples of Aoospaces 1) Any associative H-space. We can denote the n fold multiplication unambiguously as n

X

~X

(xl'" ,xn) ~ x lx2·· .xn

50

The Homology of Hopf Spaces

Then define M : K x Xn ~ X n

n

(2) Any space X having the same homotopy type as an associative H-space. f

Let (Y,~) be the associative H-space. Let X ~ Y be the homotopy equig

valences. Define a multiplication on X by the rule

(*)

~'

= ~(f

x f)

n Choose a homotopy ~: fg ~ l y indexed by I. Consider any of the maps h: X ~

X which can be obtained from

~'.

Then there is a family of deformations

of this map indexed by I n-2.

One uses I

n-2

~

to obtain these deformations. Intuitively, each copy of I C

corresponds to removing a set of brackets. For. if we rewrite the map

n h: X ~ X using (*) then n-2 copies of fg appears in our expression. We then use ~ to remove each copy of fg. The maps I n-2x Xn ~ X associated to the various maps h:X

n

~ X fit together nicely to give a map

which is an An form on X. The following pictures suggest the pattern xyz

(xy)z

x(yz)

O~----'o'----o

II (xy)(zw) xy(zw) x(y(zw»

12 (xy)zw

o~~

«xy)z)w

J

(x(yz»w

J

J (xyz)w

~

x(yzw1'o~(yz)W x((yz)w)

(3) Any loop space. This is a special case of (2). For, given OX. let Q'X be the strictly associative Moore loop space as defined in §4-3. Let f: OX inclusion map and g:Q'X

~

~

Q'X be the

OX the normalizing map. The loop mul tiplication

on OX and Q'X are related as in (*) above. So we can use the argument from (2) to show OX is an Aoo space. §6-2:

Aoo Spaces and Classifying Spaces

II: Classifying Spaces

51

Every Aoo space possesses a classifying space B The Aoo structure enX' ~ B The construction is analo-

Ex

ables one to define a quasi-fibration

X'

gous to the previous ones for groups and monoids.

'" U K. 2 1+

i£O

X

i +l

X

/

(a.xl·····xn+ l)

~

1

i

Bx =i£O U Ki 2 x X /:: + :: The identification

~

can be expressed as follows

Bx '" ----+ lim Bn

and

n

where

Ex '" ----+ lim En n

B n

n n and. for (Kn-s+ 1 x Ks)k x X C 8K x X we have the identities n+2 whi Ie

E

E

n

n-l

U

K

8K x Xn+ 1 n+2 n+2

x

Xn + 1

n 1 n 1 and. for (Kn- s+ x Ks)k x X + C 8K x X + we have the identities 1 n+2 (a.~.xl···,~·Ms(~·~+I··.. xk+s),xk+s+l.··· ,xn+ 1 )

This

construction

is

a

direct

generalization

of

the

Dold-Lashof-

Milgram model for the classifying space of associative H-spaces. Indeed, Stasheff demonstrates in Stasheff [1] that their model agrees with his in the case when X is an associative H-space and we use the trivial An forms on X in constructing

Bx.

The Stasheff classifying space satisfies properties analogous to the previous models. In particular. when the multiplication is cellular (and we can always alter the multiplication to be cellular while preserving the Aoo property) a cellular structure can be imposed on the B spaces hased on n the fact that B = B - U CE l' So is a CW complex as well. Observe n n 1 Pn-l nLX = B1 C BX' One can then show that is contractible and thus that the canonical map h: X ~ nLX C nBx is a homotopy equivalence.

Bx Ex

§6-3: Projective Planes The preceeding study of loop spaces and classifying spaces can be car-

52

The Homology of Hopf Spaces

ried further. One can break down the previous classifying spaces into a series of "projective planes"

The definition of Pn(X) are as follows (topological groups) Pn(X)

=G*

(associative H-space) Pn(X)

= U As

G

*... * s

x X/

s~O

(Am spaces) Pn(X)

=U K

GIG

;:: s

s~O s+

2 x XI

;::

The term "projective plane" arises from the fact that. for the topological groups SO.S1 and S3. we have the identities P (SO) n p (S1) n p (S3) n

= IRpn = lCpn = Hpn

The theory of projective planes blends nicely with Stasheff's theory of An structures. He gives an

independent characterization of the projective

plane and shows that X has an An structure if and only if Pn(X) exists. As we suggested in the introduction to §6 the theory of An forms can be viewed as an obstruction theory. Projective planes and. more specifically, their cohomology provide the major means of effectively computing these obstructions. The spheres are the canonical examples of this fact. (i) H-spaces

We lmow that SO.S1 . , S3 and S7 are H-spaces. Are any other spheres Hspaces? The answer is "no". Suppose that the sphere Sn-1 is a H-space. Then P2(S

n-1

n-1 ) exists. One can show (see §15) that P2(S ) H*(P (Sn-1)·W ) - W2[x]/ 3 2 ' 2 (x )

n Thus Sq (x)

= x2

where deg x

n 2n = SUe

and

=n

. i "I- 0. On the other hand. S1llce H (P

°

n-1 );W2) = for n < 2(S s. i < 2n. It follows that Sqn cannot factor. So n = 2 A further analysis

using secondary operations or K-theory (see §18) shows that n are the only possibilities. (ii) Homotopy Associative H-spaces

= 1.2.4,8

1/: Classifying Spaces

53

°

So, among the spheres. only S ,S 1 .S 3 ,S 7 possess A forms. What about An 2 forms for n ~ 3? The topological groups SO,Sl and S7 are, of course, Aoo spaces. On the other hand, S7 is not even a A space. For suppose S7 is 3 homotopy associative. Then P3(S7) exists. One e

24

can

7

show P ) 3(S

8 16 = SUe

U

and

Thus p

1p3

(x )

H* (P 3 (S7 );F3 )

= F3 [xJ /(x4)

= p4(x) = x3

t 0. But P3(x)

where deg x €

20(P3(S7);F H

=8 3)

=°

The above type of argument, as well as the obstruction theory of An forms, will achieve greater lucidity once we introduce localization in §9. For the above arguments are really local in character. The H-space argument in (i) is totally mod 2. What it shows is that the localized sphere n-l S(2) is not a H-space unless n

= 1,2,4,8.

2n-1

In fact S(p)

is a H-space for n

1 when p is odd. (see, for example, Adams [3J). SO. the obstruction to

~

Sn-1 being a H-space is totally mod 2. Again, the homotopy associative H7 space argument in (ii) is really mod 3 and shows that 8(3) is not homotopy 7

associative. A mod 2 argument can also be given to show that 8(2) is not homotopy associative. (see James [2J or Goncalves [lJ) On the other hand. 7

S(p) is homotopy associative for p L 5. 80 the homotopy associativity obstruction is really mod 2 and mod 3. Mod 2 arguments involving the projective plane P (X) will be used in 2 later chapters. Most of this book is devoted to the theme of obtaining necessary conditions for the existence of a H-space structure on a complex X. H* (P (X) ;F ) is one of

the major tools which have been successfully 2 2 exploited to obtain such results. We treat the cohomology of the projective plane in §§15-19. §6-4:

~Maps

Besides An forms there is also the concept of maps which preserve An forms. Given two An spaces X and Y we can define the concept of an An map f: X

~Y,

namely a map which respects the An structure of X and Y in an

appropriate sense. We will only discuss the cases which we will need in later chapters. namely

~.~.

and Aoo maps.

The Homology of Hopf Spaces

54

A2

(i)

Maps

(=

H-maps)

Here X and Yare H-spaces and an A map (or H map) is a pair (f .H) 2

where

r.

X --> Y

H: X x X --> py

H: f(xy)

~

f(x)f(y)

In other words, the following diagram is homotopy commutative

XxX~Yxy

1-.i..-- 1 y

X

Given any map f: X --> Y we can measure its H-space deviation D We need f. to first recall the concept of an algebraic loop. Given a H-space

(Z,~)

then. for any space M. the homotopy class of maps

[M.Z] is an algebraic loop. In other words. if we define addition by the rule gs-h: M ~ M x M ~ Z x Z .E.... Z then. for any g.h E [M.Z]. there exists a unique D(g.h) E [M.Z] such that g

=h

+ D(g,h)

If Z is homotopy associative then [M.Z] is a group and D(g,h)

g - h (see

Stasheff [4] Zabrodsky [15]). If we return to our map f: X --4 Y then D E [X x X,Y] is defined by f D

f

+ f~

= ~(fxf)

Although D is defined as a map D X x X --> Y it always factors as f f:

Df

XxX--4¥

!/

X" X So it can be regarded as a map D X " X --> Y. The H-deviation obeys a f: composition law. Given X

-1 Y ~ Z

In particular, given a diagram

where X.¥.Z are H-spaces then

II: Classifying Spaces

55

where X,K,E are H-spaces and q is a H map then

(ii)

A3 maps

Let X and Y be homotopy associative H-spaces where the homotopy associativity is given by a: X x X x X --+ PX a: (xy)z ~ x(yz)

fJ: Y x Y x Y --+ PY fJ: (xy)z ~ x(yz)

An A3 map is a triple (f,H.A) where (f,H): X --+ Y is a H-map A: X x X x X ~ P(PY) A: f(xy)z)

f(31)f(z)

0

0

J.-----Q--l

(x(yz»

o

f(x)f(yz)

[f(x)f(y)Jf(z)

f(x)[f(y)f(z)J

Given any H map we can measure its failure to be a A map. If we compare 3 the two ways around the above diagram then we obtain A : X x X x X --+ OY f

As with D we can factor A and regard it as a map A : X A X A X --+ OY. f

Given X

-1 Y ~

f

f

Z where X,Y,Z are homotopy associative H-spaces and f and

g are H maps then

(iii)

Am

maps (- loop maps)

For a detailed discussion of Aoo maps the reader should consult Stasheff [lJ [2J. We only want to observe that Aoo maps correspond to loop maps. Given a Aoo space X and Y an Aoo map f: X --+ Y induces a map Bf: Ex --+ By. If we replace X and Y by

to the map OOf: 0Ex - -

nBx and

DEy.

OBy then the Aoo map f: X --+ Y corresponds So, Aoo maps are just loop maps.

§6-5: The Classifying Spaces of Loop Spaces B and [) are inverse constructions. We have shown that OOx Aoo space X. We now want to show that PROPOSITION: BOX

~

X for any CW complex X.

~

X for any

56

The Homology of Hopf Spaces

Actually we wi 11 only suggest the main idea of the proof. We wi 11 1imi t ourselves to the trivial case of the Aoo construction. We replace OX by the associative Moore loop space O'X and use Milgram's model for BO'X' We will construct a diagram

=

O'X

1

O'X

1

EO'X --iE'X

1

BO'X

1

---->

X

By the 5-lemma we then have U*(BO'X) ~ U*(X) and so BO'X ~ X provided that X and BO'X are CWo In constructing the map BO'X

~

X we will use the iden-

tity

from §5-3. It suffices to construct maps An x (O'X)n ~ X which are compatible with this identity. We have the evaluation map Al x O'X ~ X (t,A) ----> A(tr)

where A: [O,r]

~

X

The required maps are higher dimensional analogues. Again, to simplify, we will only do the cases n

= 2,3.

They suffice to indicate the general pat-

tern. Given AI: [O,r ] l A [0, r ] 2 2:

2 we want a map A x {AI) x {A 2} diagram

~

x.

~

X

~

X

It is determined by the following

Given [O,r [0.r [0.r

l] 2] 3]

3 we want a map A x {AI} x {A x {A 3} 2} diagram is

~X ~

X

~X

~

X. This time the representing

57

1/: Classifying Spaces

o

The map EO'X

~

r

1+r2

X is defined analogously. We have EO'X

= U O'X n~O

x An x (O'X)~

~

and we define the maps O'X x An x O'X x ... x O'X ~ EX by ignoring the first factor of O'X and generalizing the evaluation map e: Al x O'X ~ X e(t,A)(S) A(str)

=

where A: [O,r]

~

X

58

§7: THE COHOMOLOGY OF CLASSIFYING SPACES

In this chapter we describe a spectral sequence which computes the cohomology of a classifying space. We will use i t to show that,

in many

cases, H* (Bx:W ) is a polynomial algebra. References in the literature for p this chapter are Milnor [2J and Clark [2]. §7-l: The Bar Construction The bar construction is closely related to the constructions of classifying spaces in §5 and §6. Indeed, as we will soon see, under the appropriate hypothesis, classifying spaces produce geometrical realizations of the bar construction. Given a differential graded augmented algebra A over a ground ring R with differential d of degree -lone can construct a differential bigraded coalgebra (BA,a as follows. BA is defined by: T) (BA)n

=0

(BA)O

=R

(BA)

=

n

for n

n @

i=l

<0

(the ground ring)

A

An element of (BA)n will be denoted [all ... lan ]. If lall+ ... +Ian 1= k then [all ... Ian] €

(BA)n,k' We call n the external degree and k the internal

degree. The coalgebra structure of BA is defined by: [all ... Ian]

~ [all ... Ia i]

----->

@

[ai+ll ... Ian]

1

Letting a

= dI

+ dE

i

(_l)lai la i then the differential aT:BA where d I: (BA)n,k

----->

~

BA is defined by

(BA)n,k_l

dI([all ... [an ]) =.L [all ... la.1- llda.l la.+ll ... [a ] I n and

1

aT

59

1/: Classifying Spaces

dE: (BA)n,k ~ (BA)n_1,k a dE([ a 1 1... la]) 1 ... [an ] n= .[[ 1 1... la.a. 1 1+ 1 1

In turn, the differential coalgebra (BA,8

A

Tor (K,K) This definition extends

_

T)

is used to define

H(BA,8 T)

=

def

the usual definition of Tor.

(see Hilton-

Stambach [1] or Cartan-Eilenberg [1]). That definition is for algebras, not differential algebras (i. e, d

= 0).

In appl ications that is all we

will have to deal with. However, the extended definition for differential algebras is still the key to these applications. The applications arise in topology. For the bar construction is closely related to the classifying constructions of §5. If X has an associative mul tiplication then we can apply the bar construction to C*(X) and form BC*(X). In turn. we can con.

C (X) 0 R

SIder Tor *

(R;R)

= H(BC*(X)

0 R). We have the following basic fact

THEOREM; If X is an associative H-space then H*(BX;R)

= TorC*(X)

0 R(R;R)

In the above B is the Dold-Lashof-Milgram model for B It is 'of the form X X' Ex

= u /;n n~O

n

x X/ . So, we have maps

;:;

n

® C*(X)

i=1

n)

= C*(X

-----+ C*(Ex)

This gives a chain map BC*(X) -----+ C*(B which turns out to be an equivaX) lence. For if we filter C*(BX) by the projective planes then

So the theorem holds. In particular, when X is a CW associative H-space and the multipl ication is cellular then Milgram's construction gives an equivalence BC*(X)

~

C*(Ex) of cellular chain complexes. So Milgram's mod-

el is, in effect, a geometric realization of the bar construction. A different version of the above theorem is due to John Moore. If X O'Y, the Moore loop space, then

§7-2: The Cobar Construction

=

The Homology of Hopf Spaces

60

We can dualize all of the above. Given a differential graded augmented coalgebra Cover R with differential d of degree +1 one can construct a differential graded algebra (OC,a (OC)n

T)

=0

as follows. OC is defined by: for n

<0

(OC)O = R n

= @C

(OC)

i=l

n

for n ) 0

The algebra structure is defined by juxtaposition of elements.

The differential aT: OC

~

OC is defined as aT

d I: (OC)n,k dI([cll ... lc n])

and

= L [cll ... ICi_lldcilci+ll ... ICn] (OC)n+l.k

L

dE([cll ... [c ]) =

= c@l

+ dE where

(OC)n,k+l

dE: (OC)n,k -

where A(c)

= dI

n

[cll ... lc'.!c':I ... [c ] J

J

+ l@c + Lcj@cj. Then (OC,a

and we define CotorC(R;R)

n

T)

is a differential algebra

- H(OC;a T) def

The analogue to Theorem 7-1 is THEOREM:If X is an associative H-space then H* (Bx;R)

= CotorC*(B

)@R(R;R).

X

§7-3: The Bar and Cobar Spectral Sequence If we filter BA and OC by external degree then we obtain spectral sequences. We have a spectral sequence {Er(A)} of coalgebras where EO

= BA

El

= BH(A)

E 2

= TorH(A)(R;R)

Eoo

= EO(Tor A(R;R»

And we have the spectral sequence {Er(C)} of coalgebras where EO = OC E

l

= OH(C)

61

1/: Classifying Spaces

2 E

= CotorH(C)(R;R)

'" = E°(CotorC(R;R))

E By letting A

= C*(X).

C

= C*(X)

and using Theorems 7-1 and 7-2 we can ob-

tain the following topological spectral sequences. THEOREM A: Let

(X.~)

be an associative H-space and let K be a field. Then

r r} (a) there exists a first quadrant spectral sequence {E .d of coalgebras where:

co

(ii) E = EO(H*(BX;K)) r (iii) d has bidegree (-r.r+1) (b) there exists a first quadrant spectral sequence

{Er.d

algebras where: (i) E2 (ii) E", (iii) d

r

r}

of

= CotorH*(X;K)(K;K) = EO(H*(BX;K)) has bidegree (r,-r+1)

These spectral sequences hold for any loop space. The A", structure defined on

ax

in §6-2 makes the inclusion OX C

duced map BOX

~

lences enable us to replace THEOREM B: Let

a'x

an A", map. So we have an in-

Ba,X which is also a homotopy equivalence. These equiva-

(X.~)

a'x

by

ax

in the above theorem. We have

be a loop space and let K be a field. Then

r r} (a) there exists a first quadrant spectral sequence {E .d of coalgebras where: (i) E2 (ii) E'" (iii) d

r

= TorH*(X;K)(K;K)

= EO(H*(BX;K)) has bidegree (-r.r+1)

(b) there exists a first quadrant spectral sequence {Er.d

r}

of

algebras where: (i) E2 (ii) E", (iii) d Alternatively.

r

these

= CotorH*(X;K)(K;K)

= EO(H*(Bx;K)) has bidegree (r,-r+1) spectral

sequences arise

from

the

filtration of

62

The Homology of Hopf Spaces

H*(Bx;K} and H* (Bx;K) given by the geometric filtration {PnX} of Bx.(see Milnor [2] and Rothenberg-Steenrod [I]}. This approach has the advantage of working for such homology theories as BP theory or K theory. provided that H*(X) is torsion free. This hypothesis is invoked in order to ensure that we have a Kunneth isomorphism h* ()(l) theory h*( ) in question.

=0n

i=1

h* (X) for the cohomology

In §47 we will use the BP variation of this

spectral sequence. §7-4: The Borel Structure Theorem Borel determined when H* (Bx;f THEOREM:(Borel} If

.

(X,~) IS

p)

is a polynomial algebra.

a loop space and H* (X;f

p)

= E(x

l,

.... x r) where

Ixil = 2n i-1 then H*(Bx;fp) = f p[yl.··. 'Yr] where IYil = 2n i· Proof: We use the spectral sequence from §7-3. We also need the following hasic facts about Cotor. (i) If C = E(x) then CotorC(K;K) = K[sx] where sx has bidegree (I, Ixl) (ii) If C = C10 C then CotorC(K;K) 2

= Cotorc

1

(K;K) 0 Cotor c (K;K) 2

By Theorem 2-IB H (X;f ) = 0 E(x.} as Hopf algebras. It follows from (i)

*

P

i

1

and (ii) that CotorH*(X;fp)(fp;fp} = fp[sxl·····sx r] Since all non trivial elements have even total degree it follows that E2 = E",. There is no extension problem in passing from E", E0* H (BX;fp) to H*
63

1/: Classifying Spaces

Remark 2:

Once we have described the Eilenberg-Moore spectral sequence

(see §29)

H* (X;lFp )

the converse of =

the above

E(x I•··· .x r) where IX i

IFp[yl'····yrJ where

IYi l = 2n.1

I =

theorem will

easily follow.

So.

* (!3x:lFp ) =

2n - 1 if and only if H

i

In general the cohomology of a classifying space !3x can he quite complicated if X has torsion. It is not even known if H* (!3x;lF ) is finitely p

generated as an algebra. Wilkerson [12J has verified that Cotor H*(X'1F )(lFp:lFp ) • p spaces. Moreover,

is a

if

finitely generated algebra

for

finite

loop

is the cobar spectral sequence converging to

H* (!3x;lFp ) ' then it has been verified that, for p odd and G a simply connected compact Lie group, Eo(H* (B G:lFp »

= Eoo = E2 = CotorH*(G;lF

) (lFp :lFp)' p On the other hand, such a collapse result is not true when p = 2. even for Lie groups. The groups G

= Spin(2 s+1)

provide counterexamples. See §28 for

further discussion. That chapter will concentrate on the topic of classifying spaces with torsion.

64

§S: THE RATIONAL roHOMOLOGY OF LOOP SPACES

In this chapter we describe an argument due to Allan Clark which puts strong restrictions on the rational type of a finite loop space. The reference in the literature for the argument is Clark [1]. It should be emphasized that the arguments of this chapter. unlike any other argument we have presented or will present. are not local in character. In the next chapter. we will introduce localization After that. we will deal exclusively with local arguments.

In this chapter. however. we need to know

that we are dealing with a space which is a loop space for all primes or. at least. all sufficiently large primes. not just for a single prime. §S-l: The Main Results Recall that for any H-space (X.~) the rational cohomology H*(X;~) is an exterior algebra E(xl •.... x

on odd degree generators. All of §S will be r) centred around the following result: THEOREM:(Clark) Let

(X.~)

be a finite loop space. If the algebra generator

of highest degree in H*{X;~) occurs in degree 2m-1 then H*{X;~) has generators in all degrees 2k+1 where 0

< k < m and

(k.m)

=1

The exceptional Lie groups provide examples of the power of this theorem. Examples

= G2 has type (3.11). Thus m = 6 and k = 1.5. In other words. the presence of a generator in degree 11 forces a generator in degree 3

(1) X

= E7 has type (3.11.15.19.23.27.35). Thus m = IS and k = 1.5.7.11. 13.17. So. this time. the generator of degree 35 forces generators in

(2) X

degree 3.11.15.23.27.

=

ES has type (3.15.23.27.35.39.47.59). Thus m = 30 and k = 1.7.11. 13.17.19.23.29. So the generator of degree 59 forces all the other gen-

(3) X

era tors Hence the theorem contains an enormous amount of information about the

65

1/: Classifying Spaces

rational cohomology of a finite loop space.

In particular. we have the

following striking consequence COROLLARY: Let Then IT

(X,~)

3(X)

be a finite. connected, simply connected loop space.

0 ~ t 0 provided H*(X;~) t O.

Proof: Since H*(X;~) t 0 we can apply the theorem to X. Observe that k is always an admissible value in the

theorem.

So

t

H3(X;~)

=I

O. Since

H*(X;~) is an exterior algebra on odd degree generators, HI(X;~)

0

forces H2(X;~) = O. Thus we have the Hurewicz isomorphism IT 0 ~ 3(X) Q.E.D.

H3(X;~),

Remark: When we study the Bockstein spectral sequence we will see that the hypothesis H*(X;~) t 0 can be weakened to X is not contractible. See, for example. Proposition 13-3. One other consequence of the theorem is a "symmetry" in the degrees of the exterior algebras of H*(X;~). For. if (k.m)

=1

then (m-k.m)

=1

as well.

So H*(X;~) must have generators of degree 2k+I and of degree 2(m-k)+I. In the example X

= ES

the eight generators are paired off 3 - 59. 15 -

47.

23 - 39, 27 - 35. Degree 31 is the centre of symmetry. The symmetry can be

compared to the much weaker symmetry obtained in §4. It was shown there that Hn(X;K) ~ HN-n(X;K) for all 0 ~ n ~ N where N

= dimension

X and K

=~

or IF . p

§S-2: Proof of Theorem S-I Write H*(X;~)

= E(x I .... ,xr)

where IxII ~ ... ~ Ixrl

= 2m-I.

For all suf-

ficiently large primes p. H*(X) has no p torsion. So. for all primes p

>N

it follows from Proposition 2-1 that 2m-I In turn. it follows from Proposition 7-4 that where Iy. I = [x, 1 1

I

+ 1

< k < m where (k.m) = 1, it suffices to show H* (Bx;lFp) has a polynomial generator of degree 2(m-k)+2 when p is an appropriate large

Thus. given 1

prime. To choose the prime we use

66

The Homology of Hopf Spaces

=1

DIRICHLET'S THEOREM: If (k.m)

then the series {k.k+m.k+2m .... } con-

tains an infinite number of primes

>

We then choose our prime p

N so that p - k (mod m).

In the algebra

H* (~-:W let -x p ) = Wp [yl •...• y] r I

= the

ideal (y1 •... 'yr-l)

We will consider the action of the Steenrod operation P 1 on H* (!3xWp)' Case 1: I is invariant under pI This cannot happen. If it does then H*(!3x:Wp)/I

= Wp [yr ]

has a well de-

fined action of pl. Thus

o

# y

p

Il m = Pm (y r ) = -,(P ) (y r ) m.

r

As a consequence we must have the divisibility relation mlp-1 For the identity pl(y ) r

2sm. In other words. p-l Since p

= qysr (mod = (s-l)m.

I) forces the identity 2m + 2(p-l)

=k (mod m) we also have the divisibility relation mlk-l

Such a relation contradicts our hypothesis. Case 2: I is not invariant under pI There must exist a generator Ys where

= qyrt

I

p (ys)

(mod I) and q # 0

It follows that

Since p

=k

Iy s

I

+ 2(p-l)

= 2tm

(mod m) the identities can be written as Iy s

I =-2k+2

(mod m)

Since 0 ~ IYsl ~ 2m we must have Iy

s

I = 2m-2k+2

Q.E.D.

§8-3: Further Conjectures for Loop Spaces The preceeding resul ts suggest that fini te loop spaces bear a close

67

II: Classifying Spaces

resemblence to compact Lie groups. In particular. the examples immediately following Theorem 8-1 give evidence of a common pattern. A number of conjectures have been made about the relationship between loop spaces and Lie groups. The first conjecture arises directly from the work of Clark. Conjecture A: If (2n

1-1

(X.~)

•...• 2n

is a finite loop space then its rational type

r-1)

is that of a Lie group.

By extending Clark's argument. Ewing verified the conjecture for finite loop spaces of rank

~

5. The structure theorem of Adams and Wilkerson (see

§25) provide the strongest known restrictions on the type of finite loop spaces. But

they provide a specific example of a

type which cannot be

eliminated by their techniques (see Adams-Wilkerson [1]). One can make a conjecture involving the full machinery of mod p cohomology which is analogous to the above. Namely Conjecture B: If

(X,~)

is a finite loop space then there exists a Lie

group G where H*(X:W ) ~ H*(G:W ) as Hopf algebras over A*(p) for each prime p

p

p

We can also make a similar conjecture at the classifying space level. Conjecture C: If

(X,~)

is a finite loop space then there exists a Lie

group G where H* (Bx:W p) prime p

~

H* (BG:W

p)

as algebras over A* (p) for each

When one passes from cohomology to the space level then the possible conjectures about finite loop spaces are limited. The strongest possible conjecture which can be made is that any finite loop space has the homotopy type of a compact Lie group. Zabrodsky's theory of "mixing" homotopy types (see §9-4) easily provides counterexamples to this conjecture. Likewise. not every finite loop space has the genus (see §9-S) of a compact Lie group. Specifically. Zabrodsky [21] gives a space X where X '" G2 x Sp(2) (2) X '" S3 x Sp(3) for p odd. (p) One possible conjecture about the relation between loop groups and Lie groups concerns the existence of maximal tori. Rector [2] has extended the idea of possessing a maximal torus from Lie theory to loop space theory. However, not every loop space need possess a maximal torus.

Indeed, it

68

The Homology of HopfSpaces

might be possible that: Conjecture D: The classifying space Ex of a finite loop space X has the homotopy type of B where G is a Lie group if and only if X has a G maximal torus Wilkerson [4] has shown that if X has a maximal torus T and the "Weyl group" W(G) of the maximal torus satisfies the Lie like property that:

H*(Ex;~) then (Ex)l

N

= (BG)l where N

N

=

= H*(Br;~)W(G)

IW(G)I.

For a discussion of this conjecture in the case of H* (Ex) Ixl

=4

see Rector [1] and

= Z[x]

where

McGibbon [1] [6].

As a final remark we emphasize that none of the above conjectures are valid if we localize. We must have a global loop space for such patterns to exist. When we work with mod p loop spaces then many new non Lie ex2n-1

amples arise. For example S(p)

is a loop space if n lp-L. However.

it

should also be noted that the pattern obtained (so far!) for mod p loop spaces still suggests a strong relationship between such loop spaces and Lie groups. Lie groups are classified by the Weyl groups = reflection groups defined over Z. The results of §§20 - 27 extend this relation by demonstrating a close resemblance between mod p loop spaces and p-ad i c reflection groups.

PART III: LOCALIZATION

In the next two chapters we introduce localization and describe some of its applications in finite H-space theory. Principally we explain how it can be used to construct many new (non- Lie) finite H-spaces. Thus far, all the results obtained have tended to reinforce the impression that finite H-spaces behave like Lie groups. In particular, the only examples of fini te H-spaces which we have referred to, so far. are the compact Lie groups and S7.

In §9 we describe the general theory of localization and

mixing. In §10 we describe the many new non classical examples of finite H-spaces which were obtained via localization theory. The work described in these two chapters span the late 1960's and early 1970's. Since that time the local viewpoint has tended to dominate finite H-space theory. In line with that change chapters 9 and 10 also mark an important change in focus in this book. After §10 we will cease to deal with homotopy types and H-spaces. Rather we will deal with mod p homotopy types and mod pH-spaces. Localized spaces wi 11 be used throughout the remainder of this book. There are a number of reasons for making such a switch. They are discussed in §9-1. The basic point is that most of the phenomena we now study are local in nature. So far,

there has been no

pressing reason to introduce localization. Indeed. there has been a major reason for not introducing it. We refer to the results of Allan Clark in §8. These results are not local. However, from this point onwards,

the

local approach will dominate. Moreover, aside from the Clark result, all of our previous results pass to local spaces with no problem. For example, in §4 we could talk about mod p Poincare spaces or mod p S duals. And so on.

71

§9: LOCALIZATION

In this chapter we will develop the idea that homotopy theory can be divided into two parts: (1) first, the study of the mod p homotopy type of the space (2) secondly, the study of how the mod p homotopy types unite to form the integral homotopy type This programme constitutes the theory of localization. The mixing theory arising from part (2) of the programne will be used in §10 to obtain new examples of finite H-spaces. In particular, a given H-space often gives rise to new ones simply by leaving the mod p homotopy types alone but altering the mixing data. §9-1:Localization The idea of localization arose out of Serre's concept of mod C homotopy theory in Serre [1]. An even more direct antecedent is the study of spheres in Adams [3]. The concept of localization in homotopy theory was first explicitly formulated by Sullivan [1]. In the early 1970's localization underwent intensive development. In particular, it found many useful applications in H-space theory. It is these applications which we summarize in §9 and §10. The standard reference for these results is HiltonMislin-Roitberg [5]. However, see Bousfield-Kan [1] for another approach to localization. We should also remark that an alternative but essentially equivalent approach is that of mod p equivalence theory. See chapter IV of Zabrodsky [15] for a brief summary. Notably Zabrodsky uses this approach in his arguments. So, in this chapter and in §10, any result of Zabrodsky referred to is actually stated and proved in terms of mod p equivalence theory. Let IT

= the

set of primes. Given any subset P C IT one can localze the

integers with respect to P.

~P As usual

= { a/bla,b

~(p)

€

~ and b is prime to the elements of P }

will denote the integers localized at the prime p.

~(p)

= { a/bla,b

€

~ and (b,p)

=1

}

72

The Homology of Hopf Spaces

We will also let Zo

= ~.

So Zo denotes what should be denoted

nected CW complex is nilpotent if IT tently on ITn(X) for n

~

Z~.

A con-

is nilpotent and operates nilpo-

1(X) 2. In particular, connected H-spaces as well as 1-

connected spaces are nilpotent. There is a functorial construction which assigns to any nilpotent CW complex X its localization Xp for each P C IT. The localization is well defined up to homotopy type. There exists a canonical map X

~

Xp which satisfies the rules

H* (Xp)

~

H* (X) @Z Zp

IT*(X p) ~ IT*(X) @Z Zp In addition, localization respects all of the usual constructions such as fibrations. cofibrations. products and coproducts. For example. if X

~

Y

Z is a fibration then Xp ~ Yp ~ Zp is also a fibration. Since the construction is functorial we have canonical maps Xp ~ X when P C Q. In Q line with our previous conventions X{p) wi II denote X localized at the

~

prime p and X will denote the rationalization of X.

o

We can use localization to define the concept of mod P homotopy type. Given P C IT we will say that X and Yare mod P equivalent (X

~

P

Y) if Xp

~

Yp' Likewise X and Yare mod p equivalent (X(;)Y) if X(p) ~ Yep)' We will say that X is a mod pH-space if X(p) is a H-space. Also X is mod p finite ~ a finite complex or, equivalently, H*(X;IF ) is a finite rank IF p p (p) vector space.

if X

There are a number of reasons for dealing with localized spaces rather than ordinary spaces. (1) Localized spaces often have a simpler structure than non localized spaces. The Serre splitting ("",4n-1) ) S2n-1 ( "",2n ,~ (p) ~ (p) x,~ (p) for p odd is the original example of such a phenomena. In §9-2 we will give many examples of Lie groups splitting when localized. At a more theoretical level we have Wilkerson's result (see Wilkerson [6]) that a simply connected H-space of finite type splits into irreducible factors X

= ITi

X. 1

in an unique way (up to order). As we will see in §9-5,in our discussion of genus, this contrasts sharply with the case of non localized spaces. (2) Conversely, localized spaces often have extra structure which is not present in the non local ized version. Notably, spaces often possess H-

III: Localization

73

space structures when localized. For example,

the splitting (OS2n)(p) ce

2n-1 4n-l 2n-l (S )(p) x (OS )(p) shows that (S )(p) is a H-space. (It is a retraction of a H-space) In subsequent chapters we will produce many examples of localized H-spaces. (see §24 and §51) (3) All of our future arguments and structure theorems will be mod p in nature. So localized H-spaces provide the natural setting for such arguments and results. As an example, we have already commented in §6-3 how the study of spheres as An spaces is clarified when we localize. The study of mod p finite loop spaces in §§20-28 is another clear example of natural structure theorems emerging when we localize. §9-2: Decomposition of Localized Lie Groups In this section we use Lie groups to illustrate the fact that spaces often decompose when localized. Our first example actually holds for any finite H-space. Given a finite H-space

(X,~)

the Hopf decomposition

where Ix.1 can be rephrased as

r X

'"

I = 2n.-1 1

2n.-1

IT S

1

(0) i=1

The mod p version of this resul t has been extensively studied. Given a ~

prime p we say that X is p regular if X

r

2n.-1

IT S l I t was shown by (p) Le I '"

Serre (classical case) and Kumpel (exceptional case) that a simply connected compact Lie group is p regular if and only if p ~ max {n (see i} Serre [1] Kumpel [1]). This was extended to simply connected finite loop spaces by Wilkerson and Underwood (see Wilkerson [1] Underwood [1J ). More over. Wilkerson observed that any simply connected fini te H-space is p ~ max {n i}. The related concept of p quasi-regular is treated in Mimura-Toda [1J

regular if p-2

Harper [7] and Wilkerson [7]. Now the factors are not only spheres but also the spaces Bn{p). The mod p cohomology of Bn(p) is of the form

H*{B (p);W ) n

p

= E(x,y)

Ixl where

Iyl

2n+1

= 2n+1+2(p-l)

pl{x)

=y

The mod p cohomology characterizes the mod p homotopy type of Bn(p). As a representative resul t obtained for p quasi-regularity we quote Harper's

74

The Homology of Hopf Spaces

result that any simply connected finite H-space with H*{X;~) primitively generated is p quasi regular if 2p-1 l max {n

(see Harper [4J,[8J) i}. Another approach to splitting localized spaces uses self maps of class-

ifying spaces. This approach began with Sullivan's construction of "Adam's operations" for BU(n). Nishida used these maps to split U(n) into factors. Wilkerson extended the techniques and results to any torsion free compact connected semi-simple Lie group G. In Nishida [1J it is shown that if H*{G) has no p torsion then G and H* (~;~) mod p-1.

= E(x 1, ... ,x i)

~

p-1 IT ~ where ~ is indecomposable (mod p)

(p) k=l

has type

(2k1~1, ...

,2ki-1) where each k

i

=k

Various results about splitting Lie groups with torsion have also been obtained. Harris (see Harris [2J) demonstrated that E 6

~

(3)

F

4

x

E 6/ F

4

Wilkerson (see Wilkerson [5J) reobtained this result using the above approach of self maps of classifying spaces as well as a splitting of E S A x B where

~

(5)

~

On the other hand. Goncalves [2J has shown that the factor A is mod 5 indecomposable. Harper (see Harper [8J) has shown that F 4

~

(3)

C x D where

H* (C;1F3)

H* (D;1F 3) He has also shown that the factor B further splits as B

~

(3)

Ex F where

§9-3: Reassembling The Global Picture The most subtle point of localization theory is reassembling the local information into a global picture. The local information does not of itself determine the global picture. Notably, the fact that X ~ Y for all (p)

11/: Localization

primes p does not mean that X

~

75

Y. This is the genus phenomena (see §9-5).

To reconstruct X one must also be given coherence data about the localizations {Xp}. It suffices to be given the canonical map Xp ~ X for each o space Xp ' We reassemble X using the pullback construction. Given f:X ~ Z and g:Y

~

Z we define the pullback P and weak pullback WP as follows

PCXxY P

= {(x,y) I

f(x)

WPCXxEZxY

= g(y}}

WP

= {(x.a.Y)la(O) = f(x}.

a(l)

= g(y)}

See 7-1 of Hi! ton-Mislin-Roitberg [5J for a further discussion of pullbacks and weak pullbacks. Given a partition IT

=

P U Q then X is obtained from the pullback dia-

gram X -----> Xp

1

1

X -----+ X o Q

This process can be extended to any finite partition IT

= PI

U... U Pk. One

defines spaces Xl' .... ~ inductively by the requirement that X -----> X _

n l

n

1

1

Xp -----> X n

is a pullback diagram. And X

= ~'

o

!fis proceedure cannot be extended to

infinite partitions. For, suppose one is given a partition IT = PI U P2 U..

and maps Xp C X One can construct a tower of spaces O' n PI P2 Pn Xl +-X2 +-",+-Xn +-...

by the requirement that X is obtained from X by a pullback construcn n_ l tion as in (*). One then wants the identity X = ~ X to be valid. Hown ever. by [Bousfield-KanJ. ~ X satisfies an exact sequence n

o~

limlIT + l(Xn } k

+--

~ ITk(lim X } ~ +-lim ITk(Xn } ~ 0 +--n

(k fixed). The lim l term creates problems. Notably. in the case X = Sk+l. +if one is given the rational and mod p components {XO.X{2}.X(3}.X(5} •... } # ~ X n' For the ~l term in the above X # O. The steps in the argument are as n} follows. I am grateful to Rob Seymour for the details. Write the primes IT

and the maps X(p}

~ Xo then X

exact sequence forces

ITk{~

{Pn}' First of all (l)

76

The Homology of Hopf Spaces

It now follows from the pullback (*) that we have an exact sequence

(2)

Let Qn

= {Pl ..... Pn}.

(3) ITk+l(Xn)

= ~Q

n

One can apply an inductive argument to (2) and show

plus (Pn-l)# in (2) is the canonical map induced by the

inclusion Qn-l C Qn, With this identity we can show (4) Recall that. given a tower of abelian groups

there is an exact sequence IT A ~ limlA n-n

0 - lim A _

-n

0

{a n- anan+ So. (4) is equivalent to showing l}. IT ~Q


n~l

1 {p'

- - IT ~O n~l

= {a n-


n

"n

a n+l}

1 tl - , ... ,-, ... } € II ~Q does not belong to 1m
2 P3

a

n+l

=

.!.

Pn+l

we must have + ... +

The divergence of the series

1 Pn+l

L.!.

contradicts a finite. l Pn It should be noted that, despite the above example. X can still be re-

covered from its rational and mod p homotopy types. It is demonstrated in Hilton-Mislin-Roitberg [5] that there is a pullback diagram of the form

§9-4: Mixing and Twisting Homotopy Types The idea of mixing homotopy types is due to Zabrodsky. It is an exten-

III: Localization

77

sion of the ideas in §9-3. Suppose we are given

X

f

~Z

Y~

where f and g are rational equivalences (i.e. f and g induce isomorphisms in rational cohomology). Given a partition IT

=P

U Q we can define a space

W by the pullback construction

Since Xp

~

p

Zo and YQ ~ Zo it follows that the space W satisfies Q

W~XandW~Y

Q

P

So W is obtained by "mixing" the homotopy types of X and Y. To apply the construction we need only know that X and Y have the same rational cohomology.

Notably.

if

X and

E(x •...• x) where Ix. 1 r 1

Yare

I = 2n.-l 1

H-spaces

let Z

and

H*(X;~)

r

=LeIT I K(~.2n.-l) 1

and let f and g be .

maps representing the generators {xi}.This case of the construction will be used in §10-2 to produce examples of non classical fini te H-spaces. A concept associated to mixing fomotopy types is that of "twisting homotopy types". It is also due to Zabrodsky. Given a complex X and a partition IT

=P

U Q we can. of course. recover X as the pullback of the maps

Now. however. choose a rational equivalence

~;

pullback diagram

X

o

~

X Define O'

X~

by the

------>'X p

1

~Xo ~Xo ~

This construction gives a "twisted" copy of X. The construction does not affect mod p homotopy types. Namely, X

~ X for all primes p. On the (p) other hand. the integral homotopy type of X is quite often affected. Nota-

Q

bly. Zabrodsky has demonstrated that. for a finite H-space. X, this construction produces the entire genus of X. See Zabrodsky [12].[14] [15].

78

The Homology of Hopf Spaces

§9-5: Genus We have touched on the concept of genus at several points in the preceeding discussion. Genus is a generalization of the concept of homotopy type. Given a space X let [X] denote its homotopy type. The genus of X is the set G(X) = {[Y]

I Y of

finite type and X ~ Y for all primes p} (p)

The restriction to finite type is important. Otherwise. for example. G(Sl) is not finite.

(see III-1 of Hilton-Mislin-Roitberg [5]. On the other

hand. Wilkerson has shown that. with the above definition. G(X) is finite for all finite complexes. (see Wilkerson [9]) In §10-3 we will analyse in detail an example of a non trivial genus. Namely. Sp(2) and E have dis5 tinct homotopy types even though Sp(2) ~ £5 for all p. In fact G(Sp(2)) {Sp(2) .E5}. IT

(p)

As another example Zabrodsky has

shown that

[C(SU(n))

I

~

~(~)! where ~ is the Euler function. (see Zabrodsky [12]). He approach-

k~n-1

es genus via miXing. In particular. as we mentioned in §9-4. he has shown that every element in G(X) can be produced by twisting the homotopy type

of X. Genus is connected wi th the failure of unique factorization. We have already noted that unique factorization exists at the local level. In contrast. we have the follOWing result. PROPOSITION:(Mislin-Wilkerson-Zabrodsky) Let X and Y be finite H-spaces of type (2n ..... 2n Then the following are equivalent: 1-1 r-l). (i) X and Yare of the same genus k

k

(ii) There exists k>O such that IT X ~ IT Y i=l i=l 2n -1 2n -1 2n -1 2n -1 (iii) X x S 1 x ... x S r ~ Y x S l x ... x S r For the proof consult Mislin [3]. Wilkerson [8] and Zabodsky [12]. §9-6: Completion Besides localizing one can also complete a space. Consult Sullivan [1] [2] or Bousfield-Kan [1] for discussions of completion In particular. one can form the p adic completion X of a space. If p tegers then

H*(X ) p

A

IT*(X ) p

~p

denotes the p adic in-

= IT* (X)®~

I£.

A

~

P

III: Localization

79

One can recover the p localization of a space from its p-adic completion via the following pullback square.

x

1(p)

Xo

----+

--->.

X ( (Xp)o

We will have need of the p-adic completion in §24 where we use p-adic reflection groups to produce spaces realizing certain mod p polynomial algebras W [x1 •.... x ]. The above fibre square will enable us to replace the p

n

constructed space by a localized space.

80

sro

LOCALIZATION AND H-SPACES

Thus far, all the results obtained have tended to reinforce the impression that finite H-spaces behave like Lie groups. In particular. the only examples of finite H-spaces which we have referred to. so far, are the compact Lie groups and S7. I n this chapter we will use localization theory to produce many new non classical examples of finite H-spaces. This chapter is a

summary of

the work

in,

among others.

Hilton-Roitberg [2],

Stasheff [3] and Zabrodsky [10]. §10-1: Localization and H-spaces Localization has found important applications in H-space theory. For. the study of H-space structures on a fini te complex X can be reduced to the study of H-space structures on its localizations {Xp}' Namely THEOREM: (Zabrodsky) Let X be a finite complex. Given a partition IT PI U P U... then X is a H-space if and only if Xp. is a H-space 2 1

for each i. Consequently, we have

COROLLARY: Let X be a finite complex. Then X is a H-space if and only if X(p) is a H-space for each prime p. One implication in the theorem is obvious. The functorial nature of localization means that the mul tiplication tions

~:

X x X -+ X induces mul tiplica-

Xp x Xp -+ Xp for all P C IT. Regarding the converse, assume that we have a partition IT = PI U P U... and that Xp. is a H-space for each i. 2 ~p:

1

We want to show X is a H-space. We can assume (i) The partition is finite i.e. IT

= PI

U... U P k

This follows from two facts. First of all. since X(p) is a H-space H* (X;~) H*(Xp. :~) N

>0

1

= E(x l,··· .xr)

such that

where Ixil

= 2n i-l.

Consequently, there exists

81

11/: Localization

r X '"

2n.-l

II S

1

Q i=1

where Q

= {pip>

N}. This is the p regularity of X as discussed in §9-2.

2n.-l

Secondly, as noted in §6-3 or §9-1, Sp

1

is a H-space when P

= the

set

of odd primes. Next, we will assume

For the argument which we give for this case easily iterates to cover the finite partition case. Consider the pullback diagram X-X 2

1

1f 2

Xl - Xo f 1

= Xp

Here Xl

and X2 1

= Xp

2

. The spaces (Xl'~I) and (X2'~2) are assumed to

be H-spaces. In addition. X o abelian group. If f 1 and f

r

= II

K(~,2n.-l) is also a H-space. indeed an i=l 1 are H-space maps then X inherits a H-space

2 structure from (*). Hence we want to show that the H-space structures on

X and X can always be altered so as to make f and f H-maps. o 2 1,X2 1 Since X As in §6-4 we have the H-deviation D Xi A Xi ~ X of f o o i: i. II K(~.2n.-l) and f. is a rational equivalence D. factors uniquely J 1 1

D.

Xi fiAf i

A

....!..... Xo

Xi

1 ~W

Xo A Xo

i

In each case if we replace the multiplication map f

i

~o

on X by o

~o

+ Wi then the

is a H-space map. More crudely, D is non trivial because H* (Xo;~) i

is primitively generated while H*(X.;~) is not. So we alter the coproduct 1

on H* (XO;~) to match that of H* (Xi;~)' To simultaneously make f maps we obviously need WI

W Let 2. u

= W1-

~

and f 2 H-

W2

We will show by induction on degree that u* degree

l

=0

in rational cohomology in

n. To make this argument we can assume

LEMMA A: X has a finite Postnikov system Proof: If ... ~ X n

~

X n- 1

~ ... ~

X is the Postnikov system pick N o

»

0

82

The Homology of Hopf Spaces

such that (f N) # : rri{X) ~ rri{~) is isomorphism for i ~ 2m where m = dim ~ is a H-space then the multiplication ~ x ~ ~ X N induces a map

X. If

XNm x XNm ~ XNm and.

hence. X x X in our argument. Q.E.D.

~

X. Consequently. we can replace X by

~

It now follows

LEMMA B: There exists maps gi: Xo ~ Xo such that fig i al PI

'\:

"'Pk

~

Ail

XO

where Ai

and Ai € Pi'

Proof: It follows from Lemma A that the Postnikov factorization of f ~

Xi i: X is finite. If we pick Ai large enough to annihilate the k invariants

o

then the map Ai: X

o ~ Xo

lifts through the Postnikov system. Since f i is

a mod Pi equivalence the k invariants

all have order m

'\:

a

= PI l ... Pk

where

Pi € Pi' Q.E.D.

fiAf i aiu gi vi: X A X Xo A Xo ~ Xo ~ Xi Use vi to alter the multiplication

Since u*

=0

~i'

Let

in degree ~ n it follows that v~

If a multiplication

~

o

in degree

~

n as well.

is altered by W then

whenever

Consequently.

*

~i

*

~i

in degree

~

n

~*1' = ~~1 + v~1 in degree n+l If we calculate the H-space deviation for the map f i: (Xi '~i) ~ (XO'~o)

then we obtain a diagram

83

III: Localization

D.

f.Af. 1

1

where

A*

Consequently u*

o

1 X.1 A X.1 - X O

1

W.1

Xo A X

o

*

Wi

Wi in degree

W~1

W~1 + a.A.u* in degree n+1 1 1

in degree

~

~

n

n+1 Q.E.D.

§10-2: Sphere Bundles We now apply the preceeding ideas and exhibi t some new examples of finite H-spaces. Suppose we are given a bundle of the form H

----> G ---->

S2n+1

where H C G are topological groups and G/ we can construct a new bundle H

----> ~ ---->

S2n+1. Then. for each k € ~.

H

S2n+ 1

via the pullback construction. The two bundles are related via the usual commuting diagram H ----> ~ ----> ~n+ 1 II

H

11k

----> G ---->

S2n+ 1

Bundles of this form are classified by IT = U2n+ 1(BH} = [S2n+1. BH]. If 2n(H) a € U (H) classifies the original bundle then ka classifies ~. So the 2n

number of distinct bundles obtained equals the order of a. Moreover. we are only interested in the total space

~

of the bundle. So we must also

keep in mind that

This follows from the 5 Lemma. For we have a commutative diagram H ----> E-k --+ S2n+ II

1

H --+ ~

1

1

-1

--+ S2n+ 1

Now apply IT*( ). We will consider (G.H) for the canonical bundles

84

The Homology of Hopf Spaces

SU(n) -> SU(n+1) -> S2n+l Sp(n) -> Sp(n+1) -> S4n+3 The first case of interest are fibrings over the 7 sphere Example #1: (Curtis-Mislin [2]) The fibre bundle SU(3) -> SU(4) -> S7 is classified by a generator w € IT = Z/6. Since w has order 6 the 6(SU(3» above onstruction gives rise to 6 distinct bundles. If {~} are the total spaces then there are four distinct homotopy types {E For the O.E1.E2.E3}. identity ~ ~ E_ reduces us to these four cases. (For example. E ~ E_2 ~ k 4

=

E And IT1(~) Z/6/ then distinguishes these four. 2). kZl6 Two of the four are well known. Namely EO

S7 x SU(3)

E1

SU(4)

The remaining two are "twisted" versions of these. Namely E 2 E 2 E 3 E 3

~

S7 x SU(3)

(2) ~

(p) ~

SU(4) ifp#-2 S7 x SU(3)

(3) ~

(p)

SU(4) if p #- 3

Observe that by Theorem 10-1, the last two examples are also H-spaces. So we have two non classical H-spaces. Example #2: (Hi I ton-Roitberg [2] Stasheff [3] Zabrodsky [5]) Historically, this case gave rise to the first non classical finite H-space -

the so

called Hil ton-Roi tberg "criminal". It is the space E in what follows. The 5 fibre bundle Sp(1) -> Sp(2) -> S

{~}

3

IT 1» = IT ) = Z/12. The total 6(S 6(Sp( of our associated bundles have 7 distinct homotopy types

is classified by a generator w € spaces

7

{E •...• E As before we can reduce to at most these cases. This time, 6}. O Z/12 however, the fundamental group IT (~) = /kZl12 does not distinguish 1 all the cases (e.g. k = 1 and k = 5). We must work a little harder to show

III: Localization

85

{EO.··· .E6} have distinct homotopoy types. Suppose show k

= ±m.

E1c '"

We can write

E . We want m

to

Our homotopy equivalence can be extended to a homotopy equivalence of pairs h:

(E1c.s3 )

by h

'" (E Now consider the commutative diagram induced m,S3). h# 3 3 IT7(~'S ) -----+ 1I7(Em·S )

a1

1a

IT 6(S3) 3 IT7 ( Em, S )

1I 6(s3)

= h#t k

= ±tm

Moreover, by standard bundle theory

So ko =

±mw.

This time {EO,E1,E3,E4,ES} are H-spaces. For

E

S3 x S7

E1

Sp(2)

o

E 3 £4

'" (3)

S3 x S7 while

£3

'"

S3 x S7 while

£4

(2) £S

'" Sp(2) (p)

'"

Sp(2)

P f. 3

'"

Sp(2)

P f. 2

(p) (p)

for all p

On the other hand, £2 and £6 are not H-spaces. This was originally shown

in Zabrodsky [S].

For an alternative proof employing K theory consul t

Sigrist-Sutter [1]. Observe that £S and Sp(2) have distinct homotopy types even though E S

'" Sp(2) for all p. This is an example of two distinct (p) spaces having the same genus. See §9-5 for our discussion of genus.

Example #3: [Zabr-odsky [4] [S] [10]) The bundles arising from the fibre bundles

86

The Homology of Hopf Spaces

SU(n) --+ SU(n+1) --+ S2n+1

n ~ 4

Sp(n) --+ Sp(n+1) --+ S4n+3

n ~ 2

have been studied by Zabrodsky . In the complex case we have 1I2n(SU(n)) = Zln!. The spaces only if k

{~)

= ±m

have

n :tdistinct l

possibil i ties. That is.

~

'" Em i f and

mod n!. Moreover ~

is a H-space

<=>

k is odd

The symplectic case has similar results. This time we have

1I4n+2 (Sp ( n »

=[

ZI(2n+1)!

n even

Zl2(2n+l)!

n odd

The arguments to establish (*) are general. First of all. i f k is even then trouble develops in H* (~;f2)' It agrees with that of SU(n) or Sp(n) as algebras but the cohomology operations act in a different (and unacceptable) manner. We have H* (~;f2)

= E(xl.x3.···.x2n_l.x2n+l)

or

Once we begin to study the action of cohomology operations on the mod 2 cohomology of finite H-spaces it will become apparent that the above generators must be linked to each other by cohomology operations if be a H-space. Zabrodsky [5J showed that degree generator X

= x 2n+l

~

~

is to

a H-space forces the highest

or x

to be linked to the lower degree gen4n+3 i_l). i_l) erators via primary (Ixl # 2 or secondary operations (Ixl = 2 The primary operation case is treated in later chapters .• If one is willing to assume that H* (~;f2) is primitively generated then the reader should consult Theorem 15-1. For an argument without the primitivity assumption the reader should consult §41 and §42. To show that

~

is a H-space for k odd we use the following general

resul t. THEOREM:(Zabrodsky) Let He G be topological groups where G/ ~

be defined by the pullback diagram

H

= S2n+1.

Let

87

III: Localization H ~ G ~ S2n+1

T

II

Tk

H ~ ~ --+ S2n+1 where k is odd. Then

is a H-space.

~

By Theorem 10-1 i t suffices to show that primes p. For p

=2

(~)(p)

there is no problem since

is a H-space for all

SU(n) or Sp(n). For (2) odd primes we explicitly construct a multiplication on (~)(p)' The data (~)

~

required is as follows. Consider the pullback diagram

w ------>

1

X

1g

S ------> S f

where X and S are H-spaces and (i) f is a H-map (ii)

X acts

on

S.

That

is,

there

exists a

map 1-'0: X x S --+ S

satisfying the following homotopy commutative diagram I-'

X x X ------> X 1xg g X x S ------> S

1

1

1-'0

Since S is a H-space [X A X,S] is an algebraic loop (see §6-4) and we can define the difference element w: X A S --+ S of the maps 1-'0 and I-'S(gx1). That is (iii) I-'O(x,s)

~

w(x,s)·g(x)·s

Finally, suppose that we have a homotopy commutative diagram w X A S ------> S

(iv)

1Af

1

1f

X A S ------> S w

Then we can construct a multiplication on W C X x ES x S. Let us give a simplified version of the multiplication. Assume that all of the above diagrams are strictly commutative and take the strict pullback W C X x S. The multiplication is defined by (x,s)·(X,S)

This is well defined. For

88

The Homology of Hopf Spaces

g(xox) = 1l 0(x,gx)

by (ii)

= w(x,gx)ogxogX = w(x,fs)ofs'fs

by (iii) since gX = fs gx = fs

fw(x,s)ofsofs

by (iv)

f[w(x,s)osos]

by (i)

In the general case we use the weak pullback and we replace all the above strict identities by homotopies. We can apply the above construction to produce the desired multiplication on

~.

For, if we take X

= G(p)

and S

= S~)l

then all of the above

properties are satisfied. Property (ii) holds even without localizing. As for (i)

LEMMA: 8

2n 1 + is a homotopy commutative H-space (p)

Proof: We know that

a2x

is homotopy commutative for all Y. And, by the

suspension isomorphism for homotopy groups we have the identity of groups

Q.E.D. It follows from the lemma that the k power map satisfies (i) and we can construct was in (iii). Regarding property (iv), if we write S

= LS

then

we can identify (iv) with the diagram I(G " 8) k'

1

I(G " 8)

--!!...

S

1k

---> 8

w

Here k' is k times the identity obtained using the suspension structure. This diagram commutes since the two group operations in [I(G " 8),8] obtained from the suspension I(G "

8)

and the H-space 8 agree.

Remark: For further discussion including a complete summary about the Hspaces which can be constructed via pullbacks of classical Lie group fibrations the reader should consult Hemmi [2]. §10-3: Mixing Homotopy Types The idea of mixing homotopy types was discussed in §9-4. This construc-

89

III: Localization

tion can be used to produce many non classical finite H-spaces. For suppose W is obtained by mixing the mod P homotopy types of X with the mod 0 homotopy types of Y. Then we have the following two results. PROPOSITION A: Let W be I-connected. Then W is finite if and only if (i) X is mod p finite for p € P (ii) Y is mod p finite for p € 0 PROPOSITION B: If W is I-connected and finite then W is a H-space if and only if Xp and YO are H-spaces. The second proposi tion is obtained by arguing as in §10-1. As a simple example of the uses of the above we have Example:(Mislin [2]) Let X = Sp(6),

Y = F x S7 x S19,p = {2},O = 4 {pip ~ 3}. This gives an example of an H-space W which has 3 torsion but

no 2 torsion. This is of interest since. for the compact Lie groups, odd torsion only occurs when 2 torsion is present as well This type of example can be extensively generalized. The most general result is probably the following Example: (Zabrodsky [15]) let P be a set of odd primes. Let X be a complex such that Xp is a H-space. Then there always exists a product S of odd dimensional spheres such that H*(X x S:~)

=

= IT

2m -1 S i

H*(G:~) for some Lie

group G. So we can produce a finite H-space W such that

w'" X x P

W

where 0

= IT

- P.

'"0

S

G

A concept associated to mixing homotopy types is that of twisting homotopy types. It also was discussed in §9-4. We again note Zabrodsky's result that, for a finite H-space. X. this construction produces the entire genus of X. See Zabrodsky [12].[14].[15].

PART IV: THE BOCKSfEIN SPECTRAL SEQUENCE

We now introduce one of the major themes of this book, namely the study of torsion in H-spaces. The next four chapters are concerned wi th the Bockstein spectral sequence. In §§11-13 we describe the results obtained by Browder in the early 1960's. In §11 we outline the basic facts about the Bockstein spectral sequence. In §12 we derive Browder's "implication" theorems for the Bockstein spectral sequence of a finite H-space. In §13 we use the machinery of differential Hopf algebras to deduce further facts about the Bockstein spectral sequence of finite H-spaces. The full impact of Browder's results will not be seen until we discuss secondary operations in §§29-42. We will see that they form the initial step in an inductive, and highly effective, study of the mod p cohomology of fini te H-spaces. In the last of the four chapters, §14, we suggest how Browder's structure theorems might be extended to the Bockstein spectral sequences arising from Morava K-theory. We will return to these conjectures in §40. They will play an important role in the proof of the loop space theorem

93

§ 11:

TIlE BOCKSfEIN SPECfRAL SEQUENCE

In this chapter we begin our study of torsion in H-spaces. Our topic is the Bockstein spectral sequence. We describe some of the basic facts about the Bockstein spectral sequence. In §12 and §13 we then explain the structure theorems obtained by Browder for this spectral sequence in the case of finite H-spaces. The machinery of differential Hopf algebras plays a leading role in proving these structure theorems. The reference in the literature for the work of this chapter is Browder [3]. §11-1: The Bockstein Spectral Sequence Let C* be the singular chain of X. The short exact sequence

P

x·p

a ---+ c

---+ C

---+ C ~

* * * of chain maps induces the exact triangle

IF P

---+

a

x·p H*{X}

----+

<,

H*{X) P

,/

H*{X;lFp)

This exact couple induces a spectral sequence called the homology Bockr r}. stein spectral sequence {B ,d This construction yields the following properties. Properties 1 (1) B

= H (X;IF ) * p

and d

(2) The map p:H (X)

*

If we let

1

= O.

~ H~(X;IF ~

the Bockstein p

) induces maps

Pr :H*(X) ---+ Br

then Ker Pr

= pH*{X)

1m Pr

= Pr{T r)

+ Tr_ 1

The Homology of Hopf Spaces

94

In particular property (2) asserts that d

r

detects direct summands l/pr c

H*(X) (3) Boo (4) {Br.d

r}

satisfies the multiplicative formula Br(X x Y) ~ Br(X) 0 Br(y) d r ( x0y ) = d r(x)0y + (_I)lx 1x0dr(y)

Only (4) requires any extra justification. It is an extension of the fact that H*(

;~p)

satisfies the Kunneth formula.

We can also define a cohomology Bockstein spectral sequence {Br.d r}. It satisfies the obvious parallels of the above properties. When

(X.~)

is a finite H-space the Bockstein spectral sequences satisfy

the following additional property. PROPOSITION: If (X.~) is a H-space then {B .d } and {Br.d r

r}

r

are spectral

sequences of Hopf algebras. Moreover they are dual to each other. The Hopf algebra structures in the proposi tion are induced by the maps X

~ X x X ~X. In particular Br is a commutative associative Hopf al-

gebra. The duality in the proposition refers to the fact that. for each r

> 1. -

{Br .d r ) and {Br,d

r}

are dual as differential Hopf algebras.

§11-2: The Bockstein Spectral Sequence and Finite H-spaces We now restrict our attention to the case of finite H-spaces and concentrate on the cohomology spectral sequence. Let p be a fixed prime and let {B be the cohomology Bockstein spectral sequence for p torsion. r} First of all. Proposi tion 2-1 tells us that Boo is an exterior algebra. Moreover. it is the same exterior algebra as that given to us by H*(X;~). So we have PROPOSITION A: Let (X.~) be a finite H-space. Let H* (X;~) Then Boo

=~

p

= E~(xl""

.x r).

(x 1.···.xr)

In addition. by Proposition 1-6, the spectral sequence {B collapses as r} is exterior.

soon as B

r

I V; The Bockstein Spectral Sequence

PROPOSITION B: Let

(X,~)

95

= Boo

be a finite H-space. Then B r

if and only if

Br is an exterior algebra on odd degree generators One consequence of Proposition B is that it is very easy to determine when H*(X) has no p torsion. COROLLARY: Let (X,~) be a finite H-space. Then H*(X) has no p torsion if and only if H*(X:W ) is an exterior algebra on odd degree generap

tors.

To illustrate the above assertions consider the following examples. Example #1 Let X

= the p

exceptional Lie group G

2.

Then

=2

p odd For p odd B l

= Boo Bl

Hence

~

and H* (G2) has no odd torsion. For p 2 E(x3) ® E(x5) ® E(x 3)

and B2 = Boo· Observe also that H* (G2;~) for each of these spectral sequences.

and

d

1(x5)

= E(x3,x11)

which agrees with Boo

Example #2 Let X

= the

exceptional Lie group F , Then

4

and and

For p

=2

Boo

For p

=3

Boo

For p :?: 5

Boo

= B2 = E(X3,x15,x23)

2

® E({x5x3})

2 = B2 = E(x3,xll,x15) ® E({~xS}) = B1 = E(~.xll,x15,x23)

= 2 we have = x 32

96

§l2: IMPLICATIONS AND THE BOCKSTEIN SPECTRAL SEQUENCE

In this chapter we describe the structure theorems obtained in Browder [3] for the Bockstein spectral sequence of a finite H-space. §l2-1: Browder's Structure Theorems Browder proved a number of restrictions on how differentials can act in the Bockstein spectral sequence of a fini te H-space.

In particular he

proved that, for a finite H-space, r} (i) {B has no primitive boundaries of even degree {B has no primitive boundaries of even degree i f X satisfies r} certain commutativity and associativity conditions

(ii)

These resul ts are the foundation for the study of torsion in fini te Hspaces. Some of their consequences will be described in this chapter. However, it is only when these results are linked with other techniques that their full power becomes apparent. In particular, the reader can consult our treatment of secondary operations in §§30-42 for important appl ications In the rest of §l2-l we describe the basic theorems which are used to prove (i) and (ii). In §l2-2 we deduce, among other results. (i). In §l2-3 we deduce, among other results, (ii). Browder's arguments use the concept of an implication. Let A be a Hopf algebra over IF

p

and let A* be the dual Hopf algebra. Then x € A is of

implication" i f there exists 0 7- x. (i)

X

1

o =x

(ii) ei ther xi+l

€

A . for each i 2pln

x~1 or there exists x

€

~

"00

0 where

A* such that

<x,x/ 1- 0

Thus, each Xi

.

IS

either a p

th

::P ,x + > 7- 0 <x i l

power or the dual of a p

th

power. We should

97

I V: The Bockstein Spectral Sequence

remark again that if A is not associative we will define pth powers by the

= (... «xx)x)x ... )x.

convention that x P

Browder's structure theorems assert that, in the Bockstein spectral sequence of a H-space. primitive boundaries of even degree often have ro implication. In the homology Bockstein spectral sequence this is true without restriction. THEOREM A: Let (X.~) be a H-space. If 0 # x € Image d has ro implication

r

n Peven (Br) then x

In the cohomology Bockstein spectral sequence primitive boundaries of even degree produce ro implications only under certain hypotheses. THEOREM B: Let (X.~) be a H-space. Then 0 # x € Image d

r

n peven(B r )

has ro

implication if (i) P

=2

and X is homotopy commutative

(ii) p is odd and X is both homotopy commutative and homotopy associative These are the two basic theorems. Theorem A was proved in Browder [3] while Theorem B was proved in Browder [5J. In other papers variants of Theorem B have been proved. Browder and Namioka [1] showed that 0 # x € Image d

r

n

peven(B

r)

has ro implication if r ~ 2 and H*(X;W is commutap)

tive and associative. Browder [8] showed that if 0 # x E peven(B

rather r). then x has a k implication. In

than being hi t by dr' survives to B + r k other words. the previously defined co implication exists at least until the k

th

stage.

We. however. will focus on the results stated as Theorem A and Theorem B. Before proving them we first describe their consequences for finite Hspaces. §12-2: Consequences of Theorem 12-1A for Finite H-spaces Theorem A imposes restrictions on the action of differentials in the Bockstein spectral sequence of finite H-spaces. The point is that ro implications are not allowed for finite H-spaces. COROLLARY A: Let

(X.~)

(i) Image d

r

be a mod p finite H-space.Then

n peven (Br)

0

98

TheHomomgyofHopfSp~~

Proof: Fact (i) is a trivial consequence of Theorem A. Regarding (ii), it follows from (i) that d

r:

Podd(B

r)

->

Peven(Br)

is trivial. Dually, we know that d : Qeven(B ) - > QOdd(B ) r r r is trivial. We have a commutative diagram

Moreover, by Corollary l-5A, the right vertical map is injective. It follows that

is trivial as well.

Q.E.D.

In proving Corollary A we have also established the following restrictions on the action of d COROLLARY B: Let

r

and dr.

(X,~)

be a mod p finite H-space. Then the following maps

are trivial r: r) r) (i) d Podd(B -> P even (B (ii) d r : Qeven(B ) - > QOdd(B ) r r > podd(B ) peven(B ) d : (iii) r r r These technical results contain a great deal of information. They will be used constantly throughout the rest of this book. Even now we can use them to deduce some consequences. First of all, the differential restrictions can be used to impose restrictions on the Pontryagin product in H*(X;W These restrictions will p)' be of great importance both in projective plane arguments and in secondary operation arguments COROLLARY C: Let

(X,~)

be a mod p finite H-space. Then

2 (i) for any a € Podd(H*(X;Wp » a = 0 (ii) for any a,~ € Podd(HM(X;Wp» [a,~]

o

99

I V: The Bockstein Spectral Sequence

2 P Suppose a f. O. By odd(H*(X;lFp))' 2 Corollary B(i) a is a permanent cycle in {B }. Hence a is a permanent Proof:

For statement (i) pick a

E

cycle as well. By Corollary A(i) a

2

r

is not a boundary. It follows that

P (Boo) f. O. Dually, Qeven(Boo) f. O. But this contradicts Proposition even 11-2A which asserts that Boo is an exterior algebra on odd degree generators. Statement (ii) is proved in an analogous manner. Q.E.D. Lastly,

we deduce what is certainly the most quotable result of

this

chapter COROLLARY D: Let

(X,~)

be a mod p finite H-space. Then when we localize at

p the first non vanishing homotopy group occurs in odd degree. In particular. U2 (X) ( p )

= o.

Proof: The first part is equivalent to asserting that the first non trivial group of H*(X;lF occurs in odd degree. So. suppose that H =0 p) i(X;lFp) for i < 2n while H (X;IF ) f. O. Just as in Corollary C we deduce that 2n p P f. O. Thus Q2n(Boo) f. 0 which. as before, contradicts Proposition 2n(BOO) 11-2A. The U statement follows from the first part if X is simply connected. 2 In general. U ~ U (X) where 2(X) 2

X is the universal covering space. Q.E.D.

§l2-3: Consequences of Theorem l2-lB for Finite H-spaces Since'" implications cannot occur for finite H-spaces we can use Theorem B to restrict the action of the differentials in {B We have the r}. following obvious consequences of Theorem B. (i) If p

=2

Image d

and

(X.~)

n P(B r) r

(ii) If p is odd and

is a homotopy commutative mod 2 finite H-space then

=0 (X,~)

is a homotopy commutative, homotopy associative

mod p finite H-space then Image d

r

n P(B r) = 0

However, these restrictions have a much simpler form. For. if we combine these restrictions with Corollary l2-2A then we have Image d

r

n P(B r) = 0

It follows from the DHA Lemma of §1-6 that d Therefore

r

100

The Homology of Hopf Spaces

COROLLARY A: Let

be a homotopy commutative mod 2 finite H-space.

(X,~)

Then H*(X) has no 2 torsion. COROLLARY B: Let

be a homotopy commutative, homotopy associative.

(X.~)

mod p finite H-space. Then H*(X) has no p torsion. These results can be substantially generalized. Hubbuck has proved that if X is a homotopy commutative mod 2 finite H-space then X ~ Sl X ... x Sl. (2) The proof uses Corollary A however. We will discuss the proof in S19. S12-4: Proof of Theorem A We will construct an

d

r

n

P

2n

where s

00

implication by induction. Suppose 0

r

(B ) . We will show that there exists 0 ~ x' E Image d

=r

~

S

x E Image

n

P

2~

(Bs)

or r+l. Moreover, the construction will show that x' is relat-

ed to x by implication. So. if we iterate the construction we have an

00

implication as desired. First of all, if xP ~ 0 just let x' are done. So suppose x P

=0

= xP = O.

E Image d

r

n

r P (B ) and we 2 pn

and pick y E B~+l such that dr(y)

Then dr(xp-Iy)

= xP

=x

So we have {xp-1y} E Br+ 1 We will prove 2pn+l'

PROPOSITION A: {xp-1y} is primitive

If we let

x·

= dr+1{xp-1y} r

r). P (B Our proof 2pn of Proposition B will also show that x and x' are related via an implicathen. by the above propositions. 0 ~ x' E Image d

tion. Proof of Proposition A We will only do p

= 2.

We have A*(x)

x01 + 1®X

Suppose A (y)

*

= y@l

+ l@y +

L y:@y~ 1 1

n

101

I V: The Bockstein Spectral Sequence

Then (*)

A*(xy)

= xy@1

+ 1@xy + x@y + ~ + (>&11 + 10x)(L yi0yi)

We have dr(y) = x and dr(L y:0y~) = O. Hence 1

1

r d ( y0y ) = x@y + ~ d

r(Y@1

=

+ 10y)(L y:0y~) 1 1

(X@1 + 10x)(L y:0y~) 1 1

Proof of Proposition B To prove Proposi tion B we wi 11 dualize and work wi th the cohomology Bockstein spectral sequence {B }. Pick x € B2n such that r

r

<x,x> t 0 Let

y = d r (x)

2n+ 1

B

€

r

We can use x and y to define classes {xP- 1-y } except when r

and p

= 2.

B2pn+1 r+l

€

In that case we have the class

- 2n {x·y + Sq (x)} For, when p is odd or p

O. When p

=2

and r

= 1.

=2

and r

1 - d (xoy)

~

€

4n+l B 2

::p-1-

2, we have dr(x

= -2 y

may be non zero.

= SqlSq2n(y) = d 1[Sq2n(y)]. Hence dl[xoy + Sq2n(y)]

= (p-l)x::p-2-2 y = -2 2n+l But y = Sq (y) y)

= O.

LEMMA A: <{xP-1y},{xp-1y}> to for p odd or for p = 2 and r ~ 2 -2nLEMMA B: <{x·y + Sq (y)},{xy}> t 0 when p Proof: We will only do the case p

en

r

=2

and r

=1

= 2.

>2

It suffices to show <xoy,xy> t O. We have <xoy,xy>

<X"0Y,A*(xy» <X"0Y,x@y>

t 0 Regarding the last equality,we gave the expansion of A*(xy) in equation

102

The Homology of Hopf Spaces

(*) during the proof of Proposition A. For reasons of degree

X0Y

annihi-

lates all terms in this expansion except for x0y and yi0xyi where IYi l = Ixl and IXYil = Iyl. We can eliminate yi0yi. For

"»

- r (XYi = = <x,d

<X,a>

=

a

(ii)r-l

a.

This time we must show that (Sq2n cY),xy> =

We will use the dual

right action of A* (2) on H*(X;W Consult Appendix II for details on this 2). action. First of all, we have the duality relation 2n 2n <Sq (y),xy> = Secondly, by the Cartan formula,

2 (xy)Sq n =

i '

L xSq .ySqJ i+j=2n

Thirdly, if we dualize the fact that A*(2) acts unstably on H*(X;W then 2) xSqi

=a

for i 2 n

ySqj

=a

for j 2n+l

and

The fact that xSqn = P2n(H*(X;~2»'

n

Q

* (H (X;W » 2

a

is especially delicate. We need the fact that x E

For although Sqn ; H* (x;W ~ H* (X;W ) is not trivial, Sq n ; 2) 2 2n * n ~ Q (H (X;W is trivial. Dually, Sq: H*(X;W ~ 2) 2»

H*(X;W is not trivial but Sqn; P » ~ Pn(H*(X;W2 » is trivial. 2) 2n(H*(X;W2

Q.E.D.

Since d (xP) r LEMMA C: d

LEMMA D: d

r+

= a we have l{xP}

-2 } 2{x

2pn the classes {xP} E B . r+l

= {xP- 1y}

-= {x'y +

Sq

if p is odd or p = 2 and r 2 2

2n(y) if P = 2 and r

=1

To prove the lemma we reduce to the universal example X = K(~/pr,2n) with

x = L2 n

and

y = d r (L 2 n ).

It follows from Cartan's calculation of tor-

sion in H*(K(~pr,2n» (see Cartan [lJ) that the lemma holds in that case. Proposition B is a simple consequence of the above lemmas. We have

103

IV: The Bockstein Spectral Sequence

r 1{xP}.{xp 1y}>

or

f- 0

Observe that the non trivial pairing also ensures that x and x

are re-

lated by an implication. §12-5: Proof of Theorem B The proof is analogous to that of Theorem A. We assume that we have 0 fx € d

s

Image d

r

n

p

2n

{B ) and we show that

there exists 0 f- x'

r

E

Image

n p2pn {Bs ) where s = r or r+1 and x is related to x' by an implication.

= xP.

If xP f- 0 let x'

have {xP- 1y}

If xP

=0

pick

y

E B;n+1 where dr{Y)

= X.

We

B2pn- 1 We show r+1 .

E

PROPOSITION A: {xP-

1y

}

is primitive in B r.

= d r + 1{xP-1y} .

Then we let x

The proof of Proposition A is as before. The proof of Proposition B is also analogous with one significant difference. The case p

=2

is easiest

to describe. So let us concentrate on that case. First of all we dualize to homology. Pick x € B~n such that <x.x> f- 0 Let y ~

= dr{x)

2 we can define the class

E B2n- 1 for r

r +1 { xy } € B4n-1 r

Observe that B Thus

r

must be commutative in order to ensure d (xy)

the commutativity hypothesis for X is employed here.

weakened form. For r

=1

= x 2 = O.

though in a

we have a class

The operation Q is a homology operation which is defined as follows. Since

X is homotopy commutative there exists a

~2

equivariant map

~: S1 x X x X ~ X where ~2 acts on S1 x X x X by the rule (t.x.y) ~ (I-t.y.x) while ~2 acts trivially on X. The map

~

is defined by

The Homology of Hopf Spaces

104


where H: I x X x X

~

-_ [H(2t,X'y) H(1-2t,x,y)

for for

X is the map expressing the homotopy commutativity

of X. If we quotient out the

action then we have a map

~2

1

F: mp x X ~ S

1

~2(X

x X)

~

X

Define Q: Hn(X;~2) ~ Hn+l(X;~2) Q(x)

= F*(g@x)

1 1 where 0 F g € H1(mp ;~2)' The fact that d Q(x)

=0

that d1{xy + Q(x)

= Sq 1Q(x) = x 2

and, hence,

is the most elementary Nishida relation. As before,

we prove F 0 for r ~ 2

LEMMA A: «x·y). {xy}

<{x'y},{xy + Q(x)}> F 0 for r LEMMA B: d

r+l

2 {y}

1

2

d {y }

= {xy}

= {xy

for r

2

~

+ Q(x)} for r

and then deduce
r+l - 2 {x·y}.{y}> F 0

It should be noted that in the proof of Lemma A the fact that <x'y,Q(x»

o

requires a detailed analysis of Q (which we have omitted).

§12-6: Examples The proof of the previous two sections demonstrate that the infinite implications of Theorems 12-1A and 12-1B occur in a very precise way. Consider

Theorem

12-1A.

( i) 0 F x s € P2p s n (B

In each case r

s+ 1

r

r

s

s

)

The

n

infini te

Image d

or r

s+l.

r

implication

forced

by

0 F x €

s

If x

s+ 1

is linked to

X

s

via a pth power in

I V: The Bockstein Spectral Sequence

= xP) s

homology (i.e.x s+ 1

then r s+ 1

=r s .

If x s+ 1 is linked to x s via a pth where (x,x > i 0 and (x,x >

power in cohomology (i.e. there exists x E B

r

i 0) then r s+ 1

r

s

105

+1

s

s

s

Example #1 Let X

= S3(3).

the 3 connected covering space of the 3 sphere S3. It is

defined by the fibration

where [f] generates

rr3(K(~.3»

~ ~.

One can show deg x = 2p

IF [x] @ E(y)

where

p

[

deg y o(x)

= 2p+1 =Y

The cohomology Bockstein spectral sequence has the following form deg -. = 2pr B

Here x

r

r

= IFp [x r ]

pr {x } and y

r

=

where

@ E(y ) r

[

deg Yr dr(x r)

= 2pr+1

= Yr

r 1 {xp - y} The argument used to prove Theorem 12-IA

forces this pattern. Since

is non zero we can dualize to obtain a non zero map 0: P2p+1H*(S3(3>:lFp)

~ P2pH*(S3(3>:lFp)

o(a) E P2pH*(S3 (3);lFp) n Image 0

As in Theorem 12-1A any non zero element

forces an infinite implication {a ) The non trivial pth power linking a s

and a

~

s

1 must. in each case, be in cohomology. For H*(S3(3>;1F ) is primip

tively generated So all pth powers are trivial in H*(S3(3>;lFp) and. hence. in each Br. By our previous remarks. we must have

oi

s) as E P (B 2ps

n

Image d

S

If we take these non zero differentials and dualize to cohomology then we can deduce the pattern for {B r} example. (*) forces

described above in (*) and (-). For

IF [x'] p

@

E(y')

The Homology of Hopf Spaces

106

where

= {x P}

x The fact that

o#

a2 €

2 P 2(B) 2p

and y'

n

Image d

2 then forces

= y'

d (x ' ) 2

And so on.

= {xp-1y}

Example #2 Similar remarks apply to {Br} and Theorem 12-1B. Let X can

=

08

3(3).

One

show tha t deg a = 2p-l where

deg [

Theorem 12-1B forces an infinite implication {x

<x r . {~p

o

# x

r

€

r

r

= E(ar )

0 Wp [~ r ]

=a

}> # 0 r

If we dualize these non trivial differentials we

B

{j{~)

= 2p

in {B where r}

r}

2 r p P (B ) n Image d

r}. {B We obtain

~

r can

use them to calculate

107

§13: DIFFERENTIAL HOPF ALGEBRAS

We briefly discussed differential Hopf algebras in §1-6. The main problem in the theory of differential Hopf algebras is to develop general techniques for computing R(A) when we know (A,d). In this chapter we examine some spectral sequence techniques which were used by Browder [7] for the case of the Bockstein spectral sequence. §13-1: Examples

We will restrict our attention to Hopf algebras over f p ' Let us begin by listing some simple cases of non trivial differential Hopf algebras plus their resulting homology. A

A

A

= E(a)

R(A)

d

0 f [b] p

h = E(a) o f p [b]/ (b P)

d(a)

= bP

d(a)

= bP

s

s

=

R(A)

R(A)

f [b]/

p

s (b P )

s = E(c) o f P[b]/ (bP) c

A

= E(a)

0 f [b] P

A

= E(a)

0

IF [b] s P /(bP)

deb)

=a

R(A)

deb)

=a

R(A)

where h-s

= {ab P

-I}

= E(c) 0 WP Ed] where 1} and d = {bP} c = {ab P= E(c) 0 f P Ed]/(dPh-l )where 1) c = {ab Pand d = {bP}

We also have the usual multiplicative relation PROPOSITION: If (A,d)

~

0 (Ai,d

i)

as differential Ropf algebras then R(A)

108

The Homology of Hopf Spaces

~ ~ 1

H(A as Hopf algebras i)

Consequently. we can now calculate the homology of a large family of differential Hopf algebras. Of course, (A,d) will not always be a tensor product of the above simple Hopf algebras. For example, consider A

= E(a)

@ E(b) @ W [c] p

where

deal d(c)

Moreover. in the p

=2

= cP =b

case, the above simple differential Hopf algebras

do not even exhaust all possibilities in the 2 generator case. Consider where

= b2

deal

s

The technique in more complicated situations is to filter A as a differential Hopf algebra so that EO(A) does split as a tensor product of simple factors and then study the spectral sequence of Hopf algebras converging from H(EOA) to H(A). As usual. one can lose a great deal of information in the process. But one obtains, at least, some hold on H(A). §13-2: Primitive and Biprimitive Forms of A Assume that A is associative. The augmentation filtration {F A} is deq

fined by the rule FOA

=A

F~ = Image { '1':

In other words, F A n

= nn.

A @ Fn_1A

--+

A }

the n fold decomposables of A. The associated

graded object EOA has a Hopf algebra structure induced from that of A. To prove this we use the identity EO(A @ A)

~

EOA @ EOA. We will call EOA the

primitive form of A. Observe that EOA is a bigraded Hopf algebra with the bigrading being given by (E A)s.t

o

The Hopf algebra structure of EOA is determined by the following two results. PROPOSITION A: If A is commutative then EOA

~

A as algebras

Proof: By the Borel decomposition (see §2-1) A

~

@ A. as algebras where A.

i

l

l

is monogenic. Obviously EOA ~ Ai as algebras. Moreover EO(~ Ai) ~ ~ EoA i i 1 1

I V: The Bockstein Spectral Sequence

109

PROPOSITION B: EoA is primitively generated

E~A

= F1A/F

A ~ Q(A) generates EOA as an algebra. Moreover. the bi2 grading of EOA ensures that the elements of EOA are primitive. Q.E.D. Proof:

It follows from the above proposition that the spectral sequence associated to the augmentation filtration is a spectral sequence of primitively generated Hopf algebras. We will call this spectral sequence the primitive spectral sequence. We have E

1

= EOA,

the primitive form of A

The circumflex "A" is used to indicate that the fil tration induced on H(A) need not be the augmentation filtration. So the associated graded Hopf algebra need not be the primitive form of H(A). It will be primitively generated but the bigrading may differ from that of the primitive form, If A is coassociative we can fil ter A by

dualizing the fil tration

{F A*} of A*. Thus q

FqA

= the

annihilator of F

q+

1A*

This gives rise to a graded object OEA which is the dual Hopf algebra of EOA* . We can use Corollary 1-5C to dualize Proposition B and obtain PROPOSITION C: OEA is commutative, associate and has only trivial pth powers. As before, there is a spectral sequence associated with this filtration. Finally, if A is associative, we can combine the two filtrations to obtain a graded object OEEOA called the biprimitive form of A. It is primitively generated and any Borel decomposition consists entirely of exterior algebras and polynomial algebras with the generators truncated at height p. In particular. if A is commutative and has a Borel decomposition

then OEEOA i ~ 0 OEEO\ And we can determine OEEOA

i

by the following identities

The Homology of Hopf Spaces

110

h __ ~ IFp[Yk]/ ~ EE IF [x]/ ( p) ~ ( ) O 0 P X' k=1 k where xP

i

is a representative of Y So the objects OEEOA is obtained from i.

~

ph by IFp[Yk ] / p . (x) k=1 (Yk) By splicing together the spectral sequence associated with each of the

A by replacing tensor factors IFp[x]/

two fil trations used to define OEEOA we obtain the biprimi tive spectral sequence {E where r} E1 = OEEOA. the biprimitive form of A Em

= OEEOH(A).

the biprimitive form of H(A)

In the next two sections we will give some applications of these spectral sequences. §13-3: The Type of a Finite H-space Recall that a finite H-space if H* (X;~)

= E(x 1 •...• x r)

(X.~)

has rank r and type (2n - l •...• 2n 1 r-1)

where deg Xi

= 2n i - l .

In this section we use the

biprimitive spectral sequence of §l3-2 to show how the rank and type of X are reflected in the mod p cohomology of X. We will apply it to the terms of the cohomology Bockstein spectral sequence {Br}.Our main result is PROPOSITION: Let

(X.~)

be a mod p finite H-space. Then. for each r

~

(i) rank QOdd(B ) = rank Qodd (B 1) (as IF vector spaces) r r+ p odd (ii) If a basis of Q (B has degrees (2m1-1 ..... 2mr-l) then a r) k

k

basis of QOdd(B 1) has degree (2p 1m •...• 2p r m some r+ 1-l r-1)for kl· .. ··kr~O.

= Br + 1

(iii) B r

if and only if k

.. _ k

1

r

=0

It easily follows from the proposition that COROLLARY: Let (2n

1-1

(X.~)

..... 2n

be a mod p finite H-space of rank r and type r-l).

Then

(i) rank Qodd H* (X;IF ) p

=r

(ii) H*(X) has no p torsion if (p.rrn.) i

1

=1

To illustrate this corollary consider the examples G and F discussed in 2 4 §11-2. The types of G2 and F are (3.11) and (3.11.15.23). So the corol4

IV: The Bockstein Spectral Sequence lary says that H* (G

2)

and H* (F

4)

111

have at most 2 torsion and 3 torsion. In

fact, H* (G has 2 torsion while H* (F ) has both 2 torsion and 3 torsion. 2) 4 Usually the corollary does not give such effective answers. For example, H*(SU(n»

is torsion free but has type (3,5, ... 2n-l). So the corollary

only limits SU(n) to p torsion where p

En].

~

Proof of Proposition A Let (A,d)

=

(Br,dr)and let {E be the biprimitive spectral sequence r} described in §13-2. By our results there we have QOdd(A) ~ QOdd(OEEOA) QoddH(A) ~ QOdd(OEEOA)

= Qodd(E 1)

= QOdd(Eoo)

Since E is primitively generated and has all elements truncated at height r p it is easy to decompose E in the form r E

r

~

0 A.

1

where each Ai has one of the following forms Ai

= E(a)

dr(a)

0

A.

= IFp[a]/(aP )

dr(a)

=0

A.

= E(a) o IFP [b]/ (b P s )

dr(a)

=b

A.

= E(a)

dr(b)

a

1

1

1

0

IF [b] P

/(bP )

In the latter two cases where c

= Cabp-l }

Thus, every stage of computing the homology of H(A) consists of replacing algebras of the form Ai deg c

= 2pn-l

if deg a

= E(a) 0 IFp[b]/(bP) by algebras H(Ai) = E(c) where = 2n-l. Lastly, there are only a finite number of

stages to consider. For since A is fini te dimensional the spectral sequence collapses after a finite number of terms. Q.E.D. Remark:

The fini teness of X is essential

Consider X

= K(l/p,l).

in the preceeding argument.

Then deg x = 1 [

It follows that B 2

odd O. So the rank of Q (B

r)

deg y o(x)

=2 =y

is not a constant.

112

The Homology of Hopf Spaces

This failure can also be seen in the previous argument. When we calculate B2 = H{Bt ) by using the biprimitive spectral sequence {E for B then the r} l replacement process described above occurs an infinite number of times. So rank QOdd(E r)

=1

for all r. But the degree of the generator of QOdd(E r) odd tends to infinity. Consequently, Q (Eoo) = O.

§13-4: A Collapse Result One of the main facts to be proved about torsion in finite H-spaces is the lack of higher torsion in H*{X). In other words. all the torsion in H*{X) consists of direct summands of Z/p. This is equivalent to asserting that B2

= Boo'

The following proposition will be relevant in proving such a

resul t.

PROPOSITION: Let (X,~) be a mod p finite H-space. If 0: QoddH*(X'W , p) Ir ) Qeven H* (X·'~p

•

IS

• surJective t h en B2

= B'00

By 11-2B it suffices to show that B = H(H*(X;Wp);O) is· an exterior alge2 bra on odd degree generators. To simplify notation let A d

We will use the primitive spectral sequence {E defined in §13-2. so r} E 1

= EOA,

the primitive form of A

See §13-2 for the significance of the circumflex LEMMA A: E 2

= H(EOA)

"~".

We will show that

is an exterior algebra on odd degree generators.

By Corollary 1-6 the spectral sequence {E then collapses and Eoo r}

= E2 =

EOH{A) is an exterior algebra on odd degree generators. It follows that H(A) is also an exterior algebra on odd degree generators. For H(A)

=

EOH{A) as W modules means that that the H(A) has a Poincare series of the p 2n -1 form IT(1+t i ). On the other hand, if we choose a Borel decomposition H(A)

i

= A'

@ A" where A'

= @ E(a.) i

1

2k.-1 1

113

I V: The Bockstein Spectral Sequence

A"

... IF [b.]/

=~ J

P

J

h.

deg b

(b~ J)

J

2m.

j

J

h.

2m p J 1-t j ) .1I(......:....e::-,.--h j j 1-tP

2k -1

then H(A) has a Poincare series

i

II( 1+t i

Proof of Lemma for p odd It suffices to show that EaA = 0 Ai as a

differential Hopf algebra

where each factor Ai is one of the following

Ai

= E(a) = E(b)

A

- E(b) @ IF [c]/

Ai

@ IFp[c]

P

i -

s

(cp )

d(a)

=a

deb)

c

deb)

=c

For.by the calculations in §13-1. H(A is an exterior algebra on an odd i) degree generator in each of the above cases. So the same is true for H(A)

=~

H(\)

I

Choose any Borel decomposition of A. Let {c

i}

be the even degree gener-

ators. They project to a basis of Qeven(A). Choose odd degree generators

U {b which project to a basis of QOdd(A) and satisfy the following j} i} identities in Q(A).

{a

(*)

d(a )

a

d(b )

c

j

i

The elements S

i

= {a j} U {b i} U {c i}

define a Borel decomposition of A. We

are merely replacing the exterior algebra generators of our initial Borel decomposition by the elements {a.l U {b.}. (The p odd hypothesis is being J'

I

invoked here to ensure that the squares of odd degree elements are zero). The elements of S determine elements in EaA under the map

The algebra isomorphism EaA

~

A of 13-2A tells us that these elements are

generators of a Borel decomposition of EaA. Proof of Lemma for p - 2 We now have the added complication that the squares of odd degree ele.

2n+1

ments may be non zero. However, gIven x € A

= ..?n+1 H(X;1F 2)

= Sq 1Sq2n (x) = 6Sq2n (x)

we have

The Homology of Hopf Spaces

114

The fact that the squares of odd degree elements lie in the image of 0 enables us to easily modify our previous proof. Crudely put, we we reduce to the previous proof by replacing the squares of odd degree elements by new indecomposables. Q.E.D. We should also note one further fact which arises out of the above proof. LEMMA B : Let p be odd and let (X.~) be a H-space where 0: QoddH*(X:~p) ~

Qeve~*(X;~ ) is surjective. Then every element in p

odd * Q H (X;W ) p

Proof: Let A

n Ker

= H* (X;Wp)

* 0 has a representative in H (X;W ) p

and d

= o.

n

Ker 0

By the proof of the above lemma the

odd * elements Q H (X;W n Ker 0 C EOA are permenent cycles in the spectral p) sequence {E and determine elements in Eoo r} position. Q.E.D.

= EOH(A).

This implies the pro-

115

§14:MORAVA K-THEORY

Both the Bockstein spectral sequence and the type of structure theorems obtained by Browder and described in §12 can be generalized. In this chapter we outline the possibilities. These generalizations will constitute a major theme in our study of torsion in H-spaces. We will return to their study in §40. §14-1: The Bockstein Spectral Sequence for Morava K-theory Morava K-theory is the simplest form of BP theory (see Appendix C). For each n ~ 1 k(n)*(X) is a module over k(n)*

= k(n)*(Pt) = Wp [vn ]

where

Unlike more complicated BP theories the structure of k(n)*(X) .1s easy to envisage. For n ~ 1 we can decompose k(n)*(X) as a W [v ] module as folp

n

lows k(n)*(X) = III M.1

where

= Wp [v n ]

It is useful to let k(O) * (X)

= H* (X)(p)

For then the above decomposition of k(n)*(X) corresponds to the decomposition of k(O)*(X) into copies of Z{p) and the cyclic groups Zips. So the copies of W [v ] are the "torsion free" part of k(n)*(X) while the copies p n of Wp[Vn]/{Vs) are the "torsion" of k(n)*(X). As in the classical n = 0 n

case,

torsion in k(n)*{X) can be analyzed via a Bockstein spectral se-

quence. Associated with each Morava K-theory is the exact couple

The Homology of Hopf Spaces

116

x v k(n)*(X) ~ k(n)*(X)

~ Pn

""'"H*(X'IF) 'p

where P is reduction "mod v The exact couple induces a Bockstein specn n". tral sequence {Br.d

which analyzes v

r}

n

torsion in k(n)*(X). We have

H*(X;IF ) p

ken) * (X)/

Tor

IF IF [v ] P

@

n

p

The differentials raises degree by 2(pn- l)+1. In particular, we have the identity

= Qn'

dl

the Milnor element in A* (p)

This spectral sequence is actually the Atiyah-Hirzebruch spectral sequence {E r}

in disguise.

To make the identification define a map from E2 =

H* (X;lFp) @ IFp[vn] to Bl

= H* (X;lFp)

by sending v

n

= vry n

in Er+ 1 corresponds to d r (x) = y in Br . The theories ken) are multiplicative. For p

to 1. In general dr+l(x)

=2

the multiplication is

associative but not commutative. For p odd the multiplication is both associative and commutative. See Wurgler [1]. By Dold [1] {Er} is therefore multiplicative. It follows that {B is also multiplicative. Moreover, the r} multiplication in {B is commutative and associative even for p = 2. For, r}

=

in B H* (X;lF we have the usual multiplication. l p) The differentials detect torsion in a manner analogous to the classical case. So d

r

detects torsion of order v r i.e. d n

r

detects direct summands

IFp[Vn]/(Vr) C k(n)*(X). We also have the following fact about cycles and n

boundaries in {B r}. LEMMA A: x € H*(X;lF

p)

is a permanent cycle if and only if x E Image Pn .

LEMMA B: x € H*(X;IF ) survives {B } non trivially if and only if x p

r

Pn(y)

where y € k(n)*(X) generates a free summand of k(n)*(X). §14-2: Two Conjectures Based on our description in §12 of Browder's study of the classical

117

I V; The Bockstein Spectral Sequence

Bockstein spectral sequence we offer two conjectures. Let

(X.~)

be a mod p

finite H-space and let {B be the Bockstein spectral sequence associated r} with v

torsion in k(n)*(X).

n

OONlliCTURE I: The even degree algebra generators of H*(X;F ) can be chosen p

to be permanent cycles in {B r}

OON.TECTURE II: In degree ~ 2pn the even degree generators can be chosen to be boundaries in {B r} In the classical (n

= 0)

case these conjectures are consequences of the

structure theorems obtained in §12. Conjecture I is a reformulation of the r} fact that. in the case of mod p finite H-spaces. {B has no primitive boundaries of even degree. For this fact forces d Beven C d D r

where D

dr

--->

r

r

the decomposables of B In other words. the map Q(B r) r.

B

rid B

is trivial

= BriD

in even degrees. To see this just dualize. We

r r

. a map d r : (Br I d D)*----> peven(B) ob ta rn r ' S'rnce Im d r n peven(B r ) -_ 0 this r

map is trivial. Conjecture I now follows. For given an indecomposable x € even . , B = Heven (X:Fp) we can use (*) to rewrIte x. uSIng decomposables. so 1 as to make x a permanent cycle in {B (The mod p finiteness of X ensures r}. that {B collapses after a finite number of stages. So the rewriting need r} only be done a finite number of times.) Regarding Conjecture II it is essentially Hopf's theorem in disguise in the classical case. For H*(X;~) that Boo

= E(x 1 •...• x r)

= H*(X) I Tor ® Fp = E(x 1, .... x r)

where Ixil

where Ixil

2n

= 2n i-l

i-l.

implies

(see Proposi-

tion 12-2A). Now consider an indecomposable x € B = H*(X;F of even p) 1 degree. By I x is a permanent cycle in {B So we have {x} E Boo. COnjecr}. ture II is equivalent to {x} = 0 in Boo. For if {x} # 0 then {x} is decomposable. Write {x} {x}

= L {Yi}·{zi}'

If we replace x by

x = x - L Yizi

then

= O. These conjectures contain an enormous amount of information about clas-

sical p torsion in H*(X)(p)' As already mentioned. the first differential of these spectral sequences are the operations

{~}.

These operations are

The Homoloqy of Hopf Spaces

118

built up out of the Bockstein 0 by the recursive formula

n

Q

n+1

n

=PPQ

n

-QPP

n

There are no such simple relations between the higher differentials. However, Johnson-Wilson [1] shows that v

n

torsion in k(n)*(X) forces v. tor1

sion in k(i)*(X) for 0 ~ i ~ n. So the different torsions are related. In §40 we will prove special cases of conjecture II. Our study of these special cases will also bring out the connection between torsion in Morava K-theory and classical p torsion. Besides the above conjectures one might also conjecture a means

of

proof, namely, an "implication" argument. We defined and extensively used the mod p version of this concept in §12. A characteristic zero version of this concept was briefly seen in §1-3. The elements {x

n}

of K[x] satis-

fies the coproduct formula n

.

.

[ (n)x1@Xn-l Le I i

A sequence of elements tied together via the coproduct as in (*) will be called an implication. Equation (*) was the key to proving Hopf's theorem. For as the argument in §1-3 illustrates, given an even degree primitive algebra generator x E H*(X:~), then (*) forces x

n

# 0 for all n

~ 1.This

contradicts the finiteness of X. In this manner we conclude that H*(X;~) is an exterior algebra on odd degree generators. As we have already mentioned Conjecture II in the classical case is just Hopf's theorem. This suggests that we might attempt to prove Conjecture II by an impl ication argument in Morava K-theory analogous to the above one for Hopf's theorem. Given an algebra generator x E Heven(X;W ), p

one might hope to demonstrate that there exists a sequence of elements of k(n)*(X) where (i) Pn(x O)

=x

(ii) ~(x 1) s+

v x 0 ... @X n s s

in k(n)*(X A... A X)

(X A.. A X is the p fold smash product) If x is not a boundary in {B then r} by (i)

X

o

is a torsion free element in k(n) * (X). We can then use (ii) to

prove, by induction, that { x

... } are also torsion free and, in par1,x2' ticular, non zero. So the elements {x contradict the finiteness of X. s}

IV: The Bockstein Spectral Sequence

119

Unfortunately. a sequence such as above can only be constructed in special circumstances and only so as to obey weaker properties than (i) and (ii). As a result the implication arguments in Morava K-theory tend to be much more laborious than suggested above. We will return to implications in §40 when we prove the mod 2 loop space theorem.

PART V: THE PROJECfIVE PLANE

In the next five chapters we study the mod 2 cohomology of finite Hspaces via the projective plane. The method is to convert information about the coalgebra structure of H* (X:W

2)

into information about the alge-

bra structure of H* (P One then analyses this algebra structure 2(X);W2). using cohomology operations. The restrictions obtained can be used to impose restrictions on H* (X;W as well. 2) In §15 we introduce the projective plane and explain how the coalgebra H* (X;W

2)

is related to the algebra H* (P2(X);W

2),

In §16 and §17 we use the

ideas of Thomas to analyse H* (X;W and H* (P via Steenrod opera2) 2(X);W 2) tions. In §18 and §19 we use the ideas of Hubbuck to analyse H* (X;W2) and

H* (P

2(X);W2)

via K-theory operations.

123

§15: THE PROJECTIVE PLANE

In §6-3 we defined. for each n ~ 2. the concept of the plane

Pn(X).

H-space can

(X.~).

be

nth

projective

In

used

particular. the projective plane. P exists for any 2(X). In the next five chapters we will demonstrate that P 2(X) to impose strong restrictions on the cohomology of mod 2

finite H-spaces. We will concentrate on the cases of mod 2 cohomology

and

of K-theory. As we will see. the projective plane is a device for converting

information

the coalgebra structure of H* (X;~2) or K* (X) into

about

information about the algebra structure of H* (P2(X);~2) or K* (P 2(X».

can

then

One

analyse this algebra structure using cohomology operations. The

restrictions obtained translate into restrictions for H* (X:~2) or K* (X). §15-1: Mod 2 Cohomology of the Projective Plane Let the

(X.~)

be a mod 2 finite H-space. Then. as in §6-3.

projective

we can

define

plane.

P of X. However. one can also define P2(X) 2(X). without everl introducing any of the machinery from §6. Namely P 2(X)

- cofibre of L(X A X) def

~

~

LX

The mod 2 cohomology of P often has a very simple form. First of all. 2(X) observe that P IX U C(2X A X) is of category three and. so. all 3 2(X) fold products are trivial in H* (P2(X);~2)' Next. we have a long exact sequence A n+l L n ¢ n A n+2 L .. ~ H (P2(X);~2) ~ H (P2(X);~2) ~ H (X A X;~2) ~ H (P2(X);~2) ~ .. arising from the definition of P The maps in this sequence satisfy 2(X). the following properties: (i)

¢ = M*.

(ii) Image

the reduced coproduct defined by M*(x) L

= P(H* (X;~2»

= ~*(x)

- 1®X - x01

124

The Homology of Hopf Spaces

(iii) Given x 1,x2 E P(H*(X;~2» xi then YIY2 = A(x l 0x2)

pick YI'Y2 E H*(P2(X);~2) where t(y i)

The first two properties are straightforward. For a

proof

of

the

third

property see Thomas [1]. We can use the above long exact sequence plus our knowledge

H* (X;F 2)

of

=

to

calculate

H*(P

2{X);f2).

For example. suppose

H* (X;~2) E(x1 •...• x r) is primitively generated as a Hopf algebra. We can x .. assume { x 1., ..• x r } are primitive. Pick { Y1,··· 'Yr } where t(Y i) 1 Then it is easy to deduce from properties (i),(ii) and (iii) that

~ ) H* ( P2 (X) ;1'2

= ~2[YI""'Yr]/D3

m S '"

Here 3 D

= the

3 fold decomposables of

~2[YI'"

.• y r]

and if D denotes the decomposables of H* (X;~2) then

Observe

that S is invariant under A*(2). Moreover. S is an algebra ideal.

We have

For it is a general fact that, given f: X

~

Y. if we consider

the

exact

triangle

~ * ~ then the image of H (X;~2) in H (Cf;~2) annililates all of H (Cf;~2)' (See

the appendix to Browder-Thomas [1].) To summarize. we have PROPOSITION A: Let

(X.~)

be a mod 2 finite H-space where H* (X;~2) is a

primitively generated exterior algebra. Write H* (X;~2) E{xl.···.x r) where {Xi} is a basis of H* (P2(X);~2) where t{y i)

= Xi'

PH* (X;~2)'

. PIck

Then

H*(P2(X);~2) = ~2[YI""'Yr]/D3 III where I is an algebra ideal invariant under A*(2).

= {Yi} in

125

V: The Projective Plane

The

existence of such structure theorems for P enables us to put res2(X) trictions on the cohomology of X. In particular. one can factor out I and

obtain

a

well

defined

unstable

action

of

A*(2)

Example: Is it posible to have a mod 2 finite H-space

= E(xS'Xg)' dition

on

(X.~)

The above structure theorem for P2(X) gives a

for E(xS'xg) to be realizable. Namely. W2[Y6'Yl OJ/(

an unstable action of A*(2). This is impossible.

For

the

algebra

where H* (X;W

2) con-

necessary Y6·YlO

consider

)3 admits the

Adem

relation

We have Sq

Consequently,

structure

10

(y 10)

theorems

such as Proposition A are extremely

useful. We would like such theorems to be as general as possible.

We

now

set about removing the hypothesis that H*(X;W is an exterior algebra. 2) This is not entirely straightforward. For the most obvious generalization of Proposition A does not work. That would be to choose a basis {xl""x s} of

PH*(X;W

2)

and hope that H*(P

= W2[yl···· 'YsJ/( Y

2(X);W2)

l,

.... y

)3 ffi I s

where L(Y = Xi and I is an A*(2) invariant algebra ideal. But there i) no obvious choice of I. In particular. we cannot. as before. let I X(D @ H* (X;W

2)

*

+ H (X;W

is

=

2)

@ D).

For,

Consequently. I and W2[y l··· "YsJ/(

in

general. D n PeVenH* (X;W 2) to.

Yl'··· ·y s

)3 would not be

disjoint

for

odd * such a choice. The solution is to concentrate attention on P H (X;W2). PROPOSITION B:(Browder-Thomas) Let

(X.~)

be a mod 2 finite H-space where

H* (X;W

is primitively generated. Let {xl"" ,x r} be a basis of 2) odd * . . * Xi' P H (X;W2)· PIck {Yl'.·· ,Yr} In H (P 2(X);W2) where L(Y i) Then

=

The Homology of Hopf Spaces

126

H*(P

2(X);W2)

= W2[yl'··· 'Yr J/D3 ~

I

where I is an algebra ideal invariant under the action of A*(2) The ideal I in the above theorem is ~ H*(X;'-2 ~) + H*(X ;'-2 ~ ) "" ~ D + peve~'*(X"~2) ~ H*(x·,r I = ~(D A "" 11."" .- 2)

*

+ H (X;W

2)

@ Peven.ex 11 (X;W » 2

It follows from Browder's results for the Bockstein spectral sequence that the ideal I is invariant. For. by Corollary 12-2B(iii), Sq1peve~*(X;W2)

O. Consequently,

pevenH*(X;W2) is invariant under A*(2). As noted by Lin

this use of the finiteness of X is essential and the more asserted

by

Browder-Thomas

[1J

general

result

is incorrect. They assert that one only

requires pOddH*(X;W to be finite dimensional. But consider S3<3>. the 2) 3. connected cover of 8 As we stated in §12-6

where

Sq1(x

4)

=

= xs .

3

Consequently, peve~*(S3<3>;W2) and, hence, I are not

invariant under A*(2). Proposition B is the key to structure theorems for

H* (X;W2).

Proposi-

=

* H*(P (X);W ) the A (2) algebra A 2 2 II' By 2) forcing restrictions on the structure of A one thereby forces restrictions tion

B associates

to

* H (X;W

on the structure of H* (X;W in §16 and §17.

2)

as well. This programme will be

carried

out

§lS-2: The K-Theory of the Projective Plane

Most

of

the

above discussion applies to any cohomology theory h*( ).

However, the resulting structure theorem for H* (P any

practical

significance.

In

2(X»

the case of H*(P

that we can analyse non zero squares via Steenrod the

is not

usually

of

it is the fact 2{X);W2) operations which make

structure so important. K-theory is another case where such an analy-

sis can be made. This time, Adams operations

{,h are

used.

It

is

con-

venient to localize. So we will consider K*( )(2) = K* ( ) @71. 71.(2)' Most of the ideas of §lS-l translate directly into K-theory. Since

V: The Projective Plane

127

. K* (P P2(X) is of category 3 all 3 fold products In are trivial. We 2(X»(2) have a long exact sequence

Provided K* (X)(2) is a

primitively

generated

exterior

algebra

we can

deduce a structure theorem similar to Proposition 15-1A PROPOSITION: Let

(X.~)

be a mod 2 finite H-space where K* (X)(2) is a Write K* (X)(2)

primitively generated exterior algebra.

odd * . E(x 1.···.xr) where {xi} is a basis of P K (X)(2)' PIck {Yi} in

o

K (P 2(X»(2) where L(Yi) KO(P 2(X»(2)

= xi'

Then

= ~(2)[Y1' ····yr J/ D3 $

I

where I is an algebra ideal invariant under the action of the Adams operations {,pk}. In

particular.

. H* (X)(2) is torsion free and H* (X;~) is primitively If

generated then K* (X)(2) satisfies the hypothesis of the proposition. by

Proposition

2-1.

H* (X)(2)

= E(x 1 ..... x r).

free. the Atiyah-Hirzebruch spectral sequence E(x ..... x r). 1

Since H* (X)(2) is torsion

collapses

and

K* (X)(2)

For. by an argument analogous to that used to prove Theorem

2-1B. we can show that the primitivity of K* (X)(2) or K* (X;~) lent to the associativity of K*(X)(2» iative. And this implies K*(X)(2) C Remark:

For,

is

equiva-

or K*(X;~). Thus, K*(X;~) is assoc-

K*(X;~)

is associative.

If H* (X)(2) is torsion free but H* (X;~) is not necessarily assoc-

iative then one can show that K(P

2(X»(2)

rest of the structure theorem for K(P be important in §19

is torsion free even though

2(X»(2)

the

may not hold. This fact will

128

§16: STEENROD SQUARES AND TRUNCAlED POLYNOMIAL ALGEBRAS

An algebra of the form A

= F2 [ a 1 · · · · ,a n J / (

a l , · · · ,an

)3 with an unstable

action of A*(2) is called a finitely generated truncated polynomial algebra over A*(2). Our motivation for considering such objects is the discussion of the mod 2 cohomology of the projective plane in §15. In this chapter we obtain structure theorems for the action of A*(2) on such algebras. In the next chapter we deduce consequences for the mod 2 cohomology of finite H-spaces. This chapter. as well as §17, are based on the work of Thomas [3J and [5J. §16-1: Main Results We will adopt the notation that 2

means that 2 (n

l

k

>.... > n t).

k

f. n

is missing from the 2-adic expansion n

+ ... +

This chapter is devoted to proving

THEOREM: (Thomas) Let A

= F2[al····,ar J/(

a l · · · · .a r

)3 be a finitely generat-

ed truncated polynomial algebra over A* (2). Let Q

= Q(A).If

2 k f. n

then (i) Qn+l

= Sq2

k

(ii) Sq2 Qn+l

k

k

Qn-2 +1

(provided 2

k+l

< n)

=0

For example, in low degree, the theorem gives the following action of Sql and Sq2 on Q.

k

Since the operations {Sq2 } generate A*(2) the theorem gives complete in-

V: The Projective Plane

129

formation about the action of A*(2) on Q. By the results of §15-1, theorem also tells us about the action of A*{2) on H*(X;W

2)

the

when (X.~) is

a mod 2 finite H-space and H* (X;W is primitively generated. These conse2) quences will be discussed in §17. The rest of §16 is devoted to the proof of Theorem 16-1. Our proof will consist of an analysis of A*(2) on A.

We might remark that. apart from

the fact that A*(2) acts unstably, we will only need the relation

A

§16-2: Action of Sq n on A As a preliminary to proving theorem 16-1 we first prove PROPOSITION: Let A be a finitely truncated polynomial algebra over A*(2). Let Q

= Q(A).

Then

Q2n+l

= Sq

A

s

s Q2n-2 +2

if 2n+1

> 2 s-1

Before proving the proposition we note a special property of indecomposables. Let D be the decomposables of A. An indecomposable a € A is irreducible if for every subspace Y satisfying Dey C have a

2

A and

W ·a ~ Y p

= A we

-

f. Y·A.

LEMMA A: Every indecomposable is irreducible Proof: If A -----

= W2[al,··· ,ar]/{ a

)3 then, without loss of generality,

1····,ar a and a •... ,a € Y. The point is that any set of ele-

we can assume a = r 1 2 ments giving a basis of Q can serve as {a maps onto Q we can choose {a

l,

... ,a

r}

.... a And since Wp.a III Y r}. 1, as above. Q.E.D.

Our proof of Proposition 16-2 proceeds by induction on s.

CASE s - 1 We wi 11 prove

So if Q2n+1

4 Image

Sq1 then Q4n+1

4 Image

Sq1 as well. We could iterate

130

The Homology of Hopf Spaces

this fact an infinite number of times to produce a contradiction to the finiteness of A. Therefore Q2n+1 C Image Sq1. Proof of Lemma B The fact that a t D implies that a is irreducible. Pick Y where

A

IF ·a 1Il Y p

< s and consider the

Assume Proposition 16-2 is true for s' GENERAL CASE s

>2

A s-l By induction we have Q2n+1 = Sq s-l Q2n-2 +2 So we prove LEMMA

C:

If

Sq2n-2

s-l

a

A

Sq s-l(b)

E

A2n+1

and

A

+1(b) E Sq SA + D provided 2n+1

a

t

sures

ISq2n-2

s-I

+1(b)1

> 2n+l.

(Observe that

Proof of Lemma C Again a is irreducible. Pick Y where A

C

IF ·a 1Il Y P

So a

2

A

=A

s-l A +l(b) E Sq SA + D then t Y·A. But if Sq2n-s a2

= Sq2n+l(a)

= Sq2n+lSq = Sq = Sq E

and produce a

Ibl = 2n+l > 2 s_1

So the new element has higher degree.)

SqsA+D

Sq

A

s-l(b)

A

s-l

A

s-l

sSq2n-2 sSq2n-2

A

"n

A

C Sq s(A).A

+l(b) +l(b)

then

> 2 s-1.

Again we can iterate this lemma an infinite number of times . . 2n+1 As contradictIon If Q q Image Sq

D

en-

131

V: The Projective Plane

c yo7\. The third equality is relation (*) from §16-1. The fourth equality uses the fact that A*(2) is acting unstably. Namely. Sq2n+l{b)

=0

<

since Ibl

2n+l.

Q.E.D.

§16-3: Contractions of A

Let A be a finitely generated truncated polynomial algebra over A*(2). We now describe how to form another such algebra R{A) called the contraction of A. The construction of R{A). as an algebra. is rather simple. Let

Since B is concentrated in even degree we can "contract" it and form R{A) by the rule

It is straightforward to see that R{A) is a finitely generated truncated polynomial algebra. It is not so straightforward to see that R{A) inherits from A an unstable action of A*(2). First of all odd) LEMMA A: The ideal (A is invariant under A*(2). odd) Proof: The ideal (A is invariant under Sql. For. by Proposition 16-2. 1

we can choose odd degree algebra generators for A from Image Sq . Thus Sq annihilates

the generators and

. so.

by

the Cartan formula.

1 odd Sq A

1

C

odd). (A odd Since A is mapped to itself by Sq2n (n

L 0)

odd) the invariance of (A

under all Sqn follows from the Cartan formula. Q.E.D. Thus there is an induced action of A*(2) on B. Since B is concentrated in

*

even degree this action factors to give an action of A (2)/(Sql) on this B LEMMA B: There exists a Hopf algebra isomorphism A: A*(2)/{Sql) defined by A{Sq2n)

= Sqn

~ A*(2)

132

The Homology of Hopf Spaces

Proof: The dual of A* (2) is W

2[(I.(2 .... ]. 2 dual sub Hopf algebra W 2[(I.(2 .... ]. The squaring operation defines an the

2

2[(I.(2 .... ] onto W2 [(i.(; ....

isomorphism of W A. Q.E.D.

].

The dual of this map is

We now define the Steenrod module structure of R(A). A*(2)

The action of

*

I(SqI) on B induces an action of A (2) on R(A) by the rule Sq

n

= p(Sq2n (x»

p(x)

for any x € A. Here we are using p to denote the contracting map p: A A I(Aodd)

~

~

R(A). By Lemma B this action is well defined.

§16-4: Proof of Theorem 16-1 We will prove the following statement by induction on n. Statement Sen): For every finitely generated truncated polynomial algebra over A*(2) and for every integer k (i) Qn+l

= sq2

k

(ii) Sq2 Qn+l

k

k

Qn-2 +1

~ a

if 2

k

( n

t~en

. k+l (provIded 2
=0

Suppose S(n) is true for n' < n Case n

= 2m-l

We are assuming that 2

k+ 1

< 2m-I and that 2

k

( 2m-I. (Thus k ~ 1) By

. I ating . 2 -ad iIC expanSIons one can conc I u d e that 2 k < m-l and 2 k- 1 rnanlpu

m-l. By the induction hypothesis we have Qm(R(A» Sq2 It follows that

Case n = 2m

k-l

= Sqm-2 Qm(R(A»

k-l

(R(A»

=0

a~

133

V: The Projective Plane

We are assuming that 2 k+ 1

= Sq1Q2m

that k ~ 1. For Q2m+1 2m

< 2m and that 2k ~ 2m. We can also assume

n

=2

1

by Proposition 16-2. Write n

+... + 2

h

(i) 1 ~ k < n1=l By Proposition 16-2 Q2m+1

= SqAk+1Q2m-2k+1+2

Furthermore, one can show. by manipulating 2-adic expansions. that

By the induction hypothesis 2m_2k+1+2

Q

= Sq

2k 2m_2k+1_2k+2 Q

k

k+1 Sq2 Q2m-2 +2

A 1 2 Since Sq k+ Sq

k

= Sq2

k

=0

A 1 Sq k + (see relation 16-1(*» k

it follows that

k

Q2m+1 = Sq2 Q2m-2 +1 k

Sq2 Q2m+1

=0

= n 1-1 Again. by 16-2

(ii) k

Q2m+1

= Sq

A k+1 k+1Q2m-2 +2

First of all we have

A

The first identity follows from the definition of Sq k+1 2k 2m- 2 k+ 1+2 second identity, Sq Q

=0

* since A (2) acts unstably.

By arguing as above. we can also show k A k+1 Sq2 Sq k+1Q2m-2 +2 A k k+1 = Sq k+1Sq2 Q2m-2 +2

=0

Regarding the

134

§17: STEENROD SQUARES AND PRIMITIVELY GENERATED CDHOMOLOGY

In this chapter we use the results of §16 to deduce structure theorems for the mod 2 cohomology of finite H-spaces. §17-l: Structure Theorems for H* (X:W21 It follows from the results of §16 that THEOREM: (Thomas) Let

(X.~)

be a mod 2 finite H-space such that H* (X;W is 2)

primitively generated. The action of A*(2) on P

= PH*(X;W2)

sat-

isfies the following: If 2k ( 2n+l then (i) p2n+l k

= Sq2

(ii) Sq2 p2n+ l

k

k pn-2 +1

(provided 2k+ l

< 2n+l)

=0

For. by Proposition l5-1B. given X as above. then A

) = H*(P2 (X);W 2/ 1

is a

finitely generated truncated polynomial algebra over A*(2) and oddH*(X;W

;:: Q(A) (degree shift of +1). So Theorem 16-1 implies the 2) above theorem. p

k As an illustration of the theorem the action of the operations {Sq2 }

on p

odd

in low degrees can be represented as follows

Thomas's theorem is an extremely important result. It describes how elements

0 fp

odd

Ied togeth er via S teenrod operatIons. . For t h e actIon . are tIe

0f

k

A*(2) is determined by the action of the operations {Sq2 }. Moreover. a knowledge of how A*(2) acts on pOddH*(X:W A* (2) acts on all of H* (X;W

2)

is equivalent to knowing how

* 2). For H (X;W 2) is generated. as an algebra.

135

V: The Projective Plane

odd * * odd * So we can extend the action of A (2) from P H (X;W to by P H (X;W 2). 2) * ~d* all of H (X;W via the Cartan formula. The fact that P H (X;W gener2) 2) ates H* (X;W2) as an algebra is a highly non trivial result. Since

* 2) - + QH* (X;W2) is surjective an equivalent assertion is that even . the prImItIvely . . . H*( X:W ) = O. One can prove that QevenH* (X;W = 0 In Q 2 2) generated case by extending the argument used to prove Theorem 16-1. Since PH (X;W

we are going to prove the same resul t for an arbi trary simply connected mod 2 finite H-space (see §40) we will omit a proof in this chapter. The above

theorem is Theorem 1.1 of Thomas [5]. Thomas states his

theorem for all Sqn

rather

than for

the

indecomposables

{Sq

k 2

}.

His

k . . 2n+I-2 cond i t i on that ( k) mod 2 is equivalent to the condi tion that 2 k+ 1 k 2 ( 2n+l and 2 (2n+1. The equivalence is obtained via the binomial identity given by Lemma 1-3. We should also note that Theorem 1.1 of Thomas [5] is incorrect as stated. The error originates from the one by Browder and Thomas regarding the structure of H* (P hypothesis that p

2(X) oddH*(X:W

;W discussed in §15-1. 2)

2)

One must replace the

is finite dimensional by the stronger hypo-

thesis that H*(X:W

is finite dimensional. Again the space S3(3) provides 2) a counterexample to the weaker hypothesis. For Theorem 16-1 cannot apply to H* (S 3 (3);W

= W2[x4] @ E(x5) since x 5 ( Image Sq2 . 2) Thomas's formulation of Theorem 16-1 enables one,

in many cases,

give an elegant description of the action of A*(2) on p

odd.

to

We will note

but not prove be a mod 2 finite H-space such that H* (X;W is 2) primitively generated. Suppose that all basis elements {x of 2 i+ 1}

COROLLARY A: Let

(X,~)

podd have distinct degrees. If we adopt the convention that x

o when

p2i+l

=0

2 i+ l

then 2i

Sq

Corollary A appl ies to X

(x2 i+ 1)

2j+l

= ( 2i

= SO(n) , U(n)

)x2(i+j)+1

and Sp(n). We close this section

with two more corollaries illustrating the very strong restrictions imposed on H* (X:W Our first is an easy consequence 2).

136

The Homology of Hopf Spaces

be a mod 2 finite H-space such that H* (X;W ) is 2 odd primitively generated. Then p is generated. as an A*(2) module.

COROLLARY B: Let

(X.~)

by elements in degree 2

n-1

where n ~ 1.

By further analysis involving secondary operations (see Thomas [2]) Thomas shows that the first such generator must occur in degree 1.3.7 or 15. The second restriction we offer is COROLLARY C: Let

(X.~)

be a mod 2 finite H-space such that H* (X;W ) is 2 s

2 2n+ 1 ~ 0 for all primitively generated. If p +1 ~ 0 then p

Proof: Let 2

s+1

-2n+1

2

sl

+ ... + 2

sh

(sl

>... > sh)'

0

<

2 +1 Pick x € P n . Using

s sl si · d e 1ements Xi € p2 -(s + .. +2 )+1 where Theorem 17 - 1 one can f In x

Sq

xl

xh - 1 In particular. Ixhl

= 2n+l.

Sq

2

2

= Sq2

s1

(xl) s2

(x 2) sh

(x h )

Q.E.D.

§17-2: Generalizing Thomas's Results Thomas's structure theorems cannot be extended in their

totality to

arbitrary mod 2 finite H-spaces. As he himself pointed out. when X

=

the

exceptional Lie groups E6'~ or E one has p9 ~ 0 but p7 = O. Thus p9 ~ 8 p7. Sq2 Nevertheless. Thomas's work has been very influential. Many of the subsequent results obtained about the cohomology of finite H-spaces can be viewed as generalizations of his work. We will close §17 by briefly discussing some possible generalizations.

= QH* (X;W2 ) rather generate H* (X;W ) while 2

First of all. it is better to talk about Q

than P

=

PH* (X;W ) . For. in general. P does not Q does. 2 Thus Q is a more canonical object. Notably. results about Q extend to all of H* (X;W ) via the Cartan formula. 2

V: The Projective Plane

137

The negative aspect of Thomas's structure theorem should extend without restriction to any finite H-space. k

Conjecture A: Sq2 Qn

=0

_ 0 mod 2) if 2k ( n (i.e.if (n k) 2

In particular Corollary 12-2B verifies that the conjecture is true for Sql (i.e. SqlQeven

= 0).

Let us now turn to Thomas's results about the non trivial action of A*(2)

As we have already pointed out the exceptional Lie groups E

6,E7

and

ES demonstrate that they cannot totally extend, So we will adopt restricted versions. Given n = 2

nl

+ ... + 2

~

define

(n

l

>... > nh)

let w(n) = h.

If we

F Q q

then it is a consequence of Conjecture A that {FqQ} filters Q as a Steenrod module. Observe that the counterexample associated with E

6,E7

and E

S

concern a non trivial action which would change filtration. The non triv-

ial actions which do not change filtration should extend to any finite Hspace. Conjecture B: If 2k ( n and 2k + 1 E n then Qn

= Sq2

k

Qn-2

k

Notably, conjecture B implies that Q is generated over A*(2) by

n_l [ Q2 n~l

and also that Qeven

= O.

This last fact, at first sight, does not concern

the action of A* (2). However, the unstable action of A* (2) means that Q2n

o

can be restated as Q2n

= SqnQn.

We will attack conjectures A and B in §§3S-42. We will only prove special cases. However,

these special cases wi 11 have important conse-

even quences. In particular. we show Q

= O.

. * SInce H (X;W

will no longer 2) be considered to be primitively generated the projective plane is not the appropriate tool for this study. Rather we will use a series of secondary operations. The use of secondary operations also reverses the order of the proofs. Thomas's arguments concentrate on proving Conjecture B. Conjecture . 1 2n A then follows quite easIly. For example, he proves that Sq Q

= Sq 1Q2n-l

and. as a consequence. he has SqlQ2n = SqlSqlQ2n-l = 0 as well. Future arguments will tend to reverse this order of implication. Notably. knowing

138

The Homology of Hopf Spaces

that Sql acts trivially in certain cases enables us to define certain secondary operations which are then used to obtain cases of Conjecture B. In particular. we obtain Sql Qodd

= Qeven.

See §33 for more discussion of

this point. All of the above conjectures and comments have p odd analogues. However, we must restrict our attention to Qeven. For any product of odd dimensional spheres is a mod odd H-space. So the type of restrictions obtained by Thomas cannot hold for QOdd. See §§35-37 for the restrictions which have been obtained for Qeven. See also §51 for a further discussion . . of Qodd . For, despIte the above comment, results about the actIon of A* (p) on QOdd are still possible.

139

§18: K-TIIEORY AND TORSION FREE H-SPACES

The restrictions obtained in §17 were obtained by studying the relations between primary operations. One can also use higher order operations to analyse the relations between primary operations. One might expect such use of higher operations to give further restrictions over and above those obtained in §l7. This is the case. Consider the question of which odd dimensional spheres are mod 2 H-spaces. The results of §l7 restrict S2n-l to the cases n

= 2k

for k

~ O. (see Corollary l7-lB). By using secondary

operations Adams eliminated all these cases as well except. of course, for Sl. S3 and S7. (see Adams [2]). Adams' argument has been difficult to generalize. Higher order operations are much more difficul t to handle than primary operations. Zabrodsky developed an approach by which secondary operations can be used to obtain systematic results about Steenrod operations and H-spaces. We will intensively study these techniques ·in §30-42. Now. however. we focus on another profitable approach. One can pass from ordinary cohomology to other cohomology theories and use the primary operations associated with these theories. These operations often contain information which is only available in terms of higher order operations when we deal with mod 2 cohomology. The Adams' operations associated with K-theory provide the outstanding example of

such operations

Notably.

Adams and Atiyah used K-theory to reprove the above resul ts concerning which spheres are mod 2 H-spaces.(see Adams-Atiyah [1]) In the next two chapters we will study the K-theory of projective planes. In this chapter we will describe restrictions obtained for torsion free mod 2 finite Hspaces via K-theory.

In particular. we will describe

the Adams-Atiyah

treatment of the spheres. In §19 we will use K-theory to classify homotopy commutative mod 2 fini te H-spaces.

The main resul ts are due

to John

Hubbuck. The main references in the literature for the work of these chapters are Hubbuck [2] and [3]. The use of K-theory in finite H-space theory began with the K-theory proof by Adams and Atiyah that Sl,S3 and S7 are the only spheres which are H-spaces.

140

The Homology of Hopf Spaces

r u A'. (Adams) S2n-1 IS . a mod 2 H-space on 1y if n THEonNU~

~

1.2 .4

Proof: (Adams-Atiyah) Suppose S2n-l is a mod 2 H-space. We are dealing with the simplest case of Proposition 15-2. We have

We have 2

nx

2

+ ax

where a

=1 mod 2

(These identities are based on the fact that ~k is multiplication by k ~*

K (S

2n

2 ) and -IJ (x)

~

2

2

x -2" (x)

=x

2

n

on

mod 2.} If we substi tute the above

identities into the equality

then we obtain n3nx n nfJ 2 2I1;}nx + (a3 + fJ22n}x 2 ~ 2 + (2 + 3na}x The coefficients of x

2

give 3n(3n_ l}a ~ 2 n(2n-l}fJ

So 2

nI3n_l.

This is only possible if n

= 1,2,4. Q.E.D.

This argument can be extended to more complicated cases, But all the basic principles are already evident (indeed more evident) in this simple case. There is no new theoretical input. Rather the computations become increasingly complex. There are more and more divisibi 1i ty relations to keep track of. For a straightforward extension of the argument in Theorem A to the rank 2 case see Douglas-Sigrist [1]. A more subtle generalization of the argument in Theorem A has been produced by John Hubbuck in Hubbuck [2]. In order to impose more structure on the divisibility arguments Hubbuck replaces the Adams' operations by a family of operations Sn: H*(X}(2}

~ H*(X}(2} and the relations

-IJk-IJE

= ~E~k

by a series of rela-

tions between the {Sn}. With this formulation the divisibility arguments are easier to organize. Hubbuck is able to prove

THEOREM B:(Hubbuck} Suppose that

(X,~)

is a mod 2 finite H-space where

H* (X}(2) is torsion free and H* (X:~) is primitively generated. If rank X

~

5 then the type of X is a union of the types of:

141

V: The Projective Plane

Sl.S7. SU{n}.Sp{n} {n

~

5} i.e. (1}.{3}.{7}.{3.5}.{3.5.7}.{3.5.7.9).

{3.5.7.9.ll}.{3.7}.{3.7.l1}.{3.7.ll.l5)and {3.7.11.15.19}. Remark: The hypotheses on X are. of course. to enable one to make use of Proposition 15-2.

Also.

as we remarked in §2-3.

the hypothesis

that

H*{X;~} is primitively generated may be superfluous. Thus Theorem B suggests that torsion free mod 2 finite H-spaces resemble the Lie groups plus S7 at the level of cohomology. Based on the methods of his arguments Hubbuck has suggested some more modest conjectures stressing the analogy between mod 2 finite H-spaces and Lie groups. Conjecture A: Let

(X.~)

be a mod 2 finite H-space where H* (X)(2) is tor-

sion free. If H*{X;~} has generators above degree 7 then H*{X;~) has generators below degree 7 as well. In particular. if X is 3 connected then X ~ IT S7 while if X is 7 connected then X ~ * {2} Conjecture B: Let

(X.~)

be a simply connected mod 2 finite H-space where

H* (X}(2) is torsion free. If X has type {2n

<... < n r

•... 2n r-l} where n l l-l then X has the type of SUer} or Sperl i.e.the type is

{3.5.7 •.... 2r+1} or {3.7.ll ..... 4r-l}

One can also ask whether Conjecture A holds without the assumption of H* {X}{2} being torsion free {§24-4 is relevant to this question}.

142

§19: HOMOTOPY COMMlITATIVE H-SPACES

The most dramatic result obtained from applying K-theory to finite Hspaces is Hubbuck's classification of homotopy commutative mod 2 finite H-spaces. TIIEOREM: (Hubbuck) Let (X,J.l) be a homotopy conunutative mod 2 fini te Hn

space. Then X ~ IT Sl (2) i=1 Hubbuck's proof depends on previous work of Browder and James. As we have already seen (see Corollary 12-3A) Browder proved that H* (X){2) is torsion free. Also James [3J constructed an "involution" f: P2(X)

---> P as fol2(X) lows. Asserting that (X,J.l) is homotopy commutative means that the follow-

ing diagram conunutes. J.l XxX----+X

T

x Here T is the twist map T(x,y)

1

1/

x X----+X J.l (y,x). We can use (*) to induce a diagram

L(X

LJ.l X) ----+ LX

"1 1h L(X X) ----+LX " LJ.l

(**)

g

For, i f we write

L(X " X)

then the vertical maps in (**) are

where r{t)

I-t. Diagram (**) induces a map f: P 2(X)

--->

P which fits 2(X)

143

V: The Projective Plane

into the diagram h

LX

---+

P2{X)

LX

---+

P2(X)

1

1f

LX

---+ LX "

where T is the switch map. We now turn to Hubbuck's K-theory argument. As in §18 we will only do the case of odd dimensional spheres in detail. Proof of Theorem for X - S2n-1 Diagram (***) gives a K-theory diagram ~ 2n K{S )(2)

g*

~ K(S2n " S2n) (2)

i

T*

2n K{S )(2) ~

+-

i

K(S2n " S2n) (2)

Also

~K{P2 (S2n-1» (2)

= Z(2)[x]/(x3)

h f i 1 tration x were

= 2n

We have

~2(x)

= 2nx

f*(x)

= -x

+ ax

+ (3x

2

where a

=1 mod 2

2

If we substitute these identities into the equality

(f*-1*)~2(x) = ~2(f*-1*)(X) then we obtain L.H.S.

{f *-1 * )(2n x + ax2 )

n 2 2 n 2 {2 (-x + (3x ) + ax } - {2 x + ax }

= _ 2n + 1x

=0 mod 4 R.H.S.

= ~2{_ =-

+ 2n{3x2

provided n

~

2

x - (3x2 - x)

2(2nx + ax 2 ) _ (3~2(x2)

=- 2ax2 mod 4 ;l;Omod4 So S2n-1 is homotopy commutative only if n

= 1.

Q.E.D.

The general proof is clearly modelled on the above proof. In particular by Corollary 12-3A H*(X)(2) must be torsion free if X is homotopy commu-

144

The Homology of Hopf Spaces

tative. This enables one to deduce that K(P

2(X»(2)

is torsion free. On

the other hand. there is no reason to assume that K* (X)(2) is primitively generated. So K* (P

2(X»(2)

does not necessarily satisfy a structure theo-

rem like Proposition 15-2. But K(P

2(X»(2)

is well behaved enough to per-

form divisibility arguments similar to those outlined for X

= S2n-1.

The above theorem does not extend to the p odd case. Recall that in Lemma 10-2 we showed that. given n

~

2n-1 1. S(p) is a homotopy commutative

H-space for all odd primes p. This result has been extensively generalized. Iriye-Kono [1] (see also McGibbon [8]) have shown that. for p odd. any mod pH-space possesses a multiplication which is homotopy commutative. (McGibbon [8] also studies the homotopy commutativity properties of the Lie groups with their standard multiplication).

PART VI: REFLECrION GROUPS AND CLASSIFYING SPACES

The next nine sections will explore the relation between reflection groups.

invariant theory and the cohomology of classifying spaces. Our

study will be centred around the following problem which was emphasized by Norman Steenrod. STEENROD'S PROBLEM: Determine the graded polynomial rings

Wp[x1..... xn] which can be realized as the mod p cohomology of a space.

Steenrpd's problem amounts to determining the mod p cohomology of the classifying space of mod p loop spaces which have no integral p torsion. For. given a mod p finite loop space X with classifying space

Bx.

the

following are equivalent: (i) H*(X)(p) has no p torsion and

H*(X:(Q)

where [x,1 I

2d.-l 1

(ii) H*(X:Wp)

= E(x l.· ... x n)

(iii) H* (R.:W -x P )

where Ixil

= 2d i-1

= Wp [yl •...• y] where Iy. 1= 2d. n I l

The equivalence of (i) and (ii) was established in §ll-2. The equivalence of (ii) and (iii) was discussed in §7-3. We have already obtained a major restriction in the case of mod 2 polynomial rings. We are refering to the restrictions on the action of the Steenrod squares. {Sqn}. obtained in §l6. We now obtain major restrictions in the case of large primes. We will demonstrate that. for such primes. if H* (Bx:Wp) = Wp[xl •.... xn] then Wp[x1 •...• xnJ is the ring of invariants of a W reflection group. We will also demonstrate that the group in question p is actually a p adic reflection group and then use this fact to determine what polynomial algebras can occur as the mod p cohomology of a space. In §20.§21.§22 and §23 we discuss reflection groups and. in particular. their invariant theory. The ideas of these sections lead naturally into §24.§25. §26.§27 and §28. There. we describe the arguments of Clark-Ewing. Adams-Wilkerson and others concerning the mod p cohomology of classifying spaces

147

§20 REFLECfION GROUPS AND INVARIANT THEORY

This chapter is concerned with some of the basic facts about invariant theory. General references for the material are Benson-Groves [1], Bourbaki [1] and Hiller [IJ. We will cite other. more specialized, references during our discussion. §20-1: Lie Groups and Reflection Groups The work of Borel and Chevalley in the early 50's brought out the connection between Lie groups,

reflection groups,

and

invariant

theory.

Chevalley [IJ used invariant theory to calculate the rational cohomology of the exceptional Lie groups G and E Borel [IJ then reform2,F4,E6,E7, 8. ulated this invariant theory in terms of classifying spaces. Borel's main result is as follows. Let G be a semi-simple compact Lie Group and let T C G be a maximal torus. Let

be the Weyl group of G. By the results in §7-3 we know

H*(~:~} H* (BG:~)

= ~[tl •...• tnJ = ~[xl,····xnJ

Itil Ixil

=2 = 2d i

The action of W(G} on T induces an action of W(G} on B and, hence, on T H* (BT;~)' Then

H*(BG;~}

H*(BT;~}W(G}, the ring of invariants

and IW(G} Moreover, if (p,lTd.) i

1

I = lTd. i 1

=1

then all of the above holds with

~

replaced by W

P

Chevalley then gave, at the algebraic level, a very solid reason for the above relation. Namely, the Weyl group is a reflection group and rings of invariants of reflection groups are polynomial algebras. The rest of §20 as well as §2I,§22 and §23 are devoted to reflection groups and their invariant theory. In §24 §25,§26 §27 and §28 we will return to topology and use invariant theory to study classifying spaces. In particular, we

The Homology of Hopf Spaces

148

solve a major portion of Steenrod's problem which was described in the introduction to Part VI. In the rest of this chapter we define reflection groups and explain how they can be used to produce polynomial algebras. §20-2: Reflections and Reflection Groups We begin by defining the concept of a reflection and of a ref lection group. n: DEF Let V be a vector space of finite rank over a field k. a reflection (over k) is a linear transformation of V which is of finite order and leaves a hyperplane invariant. Reflections are also referred to as "pseudo-reflections" and "general ized reflections". Wi th this terminology "reflection" is reserved for the case of reflections of order two. Provided p and the order of a reflection are prime then the reflection is diagonalizable and has all its eigenvalues, with one exception, equal to 1. The remaining eigenvalue will be a nth root of unity, some n

L

rn = e

27ri n

for

1. The allowable values of n will vary with the field. For dif-

ferent fields contain different roots of unity. For example,

= ±l

k

IR

n

k

c

n arbi trary

k

=~,

DEF

n:

the p adic numbers

nlp-1

Let k and V be as above A reflection group (over k)

is a

subgroup of GL(V) generated by

reflections. Reflections in the real and complex case have a canonical form. Consider the case of real reflections.Given a finite group G C GL (lR) we can n n choose a positive definite form (x,y) on V = IR which is invariant under the action of G. Just choose any positive definite form (x,y)' and average to obtain (x,y)

[ (gx,gy) , Suppose ~ gtG

t

G is a reflection. Let

L

the hyperplane of V left pointwise invariant by

a

a vector on which

~

is multiplication by -1 i.e.~(a)

= -a

~

149

VI: Reflection Groups and Classifying Spaces

= O.

Then (n.L)

For. given x

-(a.x). It follows that

~

L. we have (a.x)

t

(~(a) .~(x))

(a.-x)

is of the form

=x

~(x)

-

2(a.x)

(a.a) a

for all x e Y. We need only observe that this formula gives the right action on L and a. In the complex case every reflection of order n is of the form

~(x) = x

- (1-(

where

Cn = e

n

)~ a

n (a.a)

211"i

These formula show that a real or complex reflection is

completely determined by its order and its reflecting hyperplane. See Theorem C for an application of this observation. We will shortly discuss examples of reflection groups for the cases k ffi and

~

=

as well as other fields.This will be done in §2l.§22 and §23. In

the rest of this chapter we examine the fundamental relation between reflection groups and polynomial algebras. §20-3: Reflection Groups and Invariant Theory Given any group G acting on a vector space Y we

can

define an action of

G on the polynomial algebra S[Y]

= k[tl •..• t n]

Here. S[Y] is the symmetric algebra on Y*. the dual of Y. In particular. y* C S[Y]

and any basis {t •... t of Y* gives n} l

the identity S[Y]

k[t ....• t The action of G on Y induces an action on Y* by the rule l n]. gof(x) for any f

t

Y*. g

t

G and x

t

= f(g-lx)

Y. This action extends multiplicatively to

S[Y]. We can now define the ring of invariants of the action. DEF

n:

S[y]G

= {x

Observe that S[Y]

€ S[Y]

I gx

= k[tl ..... t n]

a a mial tll ... t n its degree n

x for all g € G}

= L a.). 1

is a graded ring.(assign to every monoThe action of G respects this grading.

So S[y]G is also a graded ring. The first basic fact about rings of invariants is

150

The Homology of Hopf Spaces

THEOREM A: (Hilbert) S[yJG is a finitely generated algebra

However.

in general. S[yJG is not a polynomial algebra. There usually

exist relations among the generators of S[YJ G. Consider

the following

example studied by Richard Stanley

~and G = V4 generated by a . The action of G on Y is

EXAMPLE: Let Y =

determined by a

= [_~ ~ ].

If S[YJ

= [[X,Y,ZJ/

2~. 2 (Z -X-Y+4Y )

generated by:

= [[t 1,t2J

then one can show S[yJG is

Since f2

3

S[VJ G

where

[~:~ ~ deg Z

2 4 4

So S[yJG is not free

The failure of S[yJG being a polymonial algebra in the above example is due to G C GL(Y) not being a reflection group.

For we have the following

fundamental result. THEOREM B:(Chevalley-Shephard-Todd-Bourbaki) Let Y be a vector space of finite rank over a field k. Let G C GL(Y) be a finite group. (i) If S[yJG is a polynomial algebra then G is a reflection group

=0

(ii) If char k

or char k

=p > 0

and IGI is prime to p then S[yJG

is a polynomial algebra if and only if G is a reflection group (iii) If S[yJG

= k[x 1, .... x n J

where Ix.1

1= d.1

then IGI

= Ud1.

The numbers {d

... ,d are called the degrees of G. The fact that reflec1, n} tion groups produce polynomial algebras as invariants was shown by

Chevalley [2J (He only did the case provided char k

=0

or char k

k

=p >0

= ffi.

However, his proof is valid

and IGI is prime to pl. The fact

that polynomial rings of invariants must come from reflection groups was first observed in the case k

=[

by Shephard-Todd [IJ. The fact that it

VI: Reflection Groups and Classifying Spaces

151

holds for arbi trary k is hidden away as an exercise in Chapter V of Bourbaki [1] The distinction between (i) and (ii) in Theorem B is necessary. For if p divides

IG I

then S[V]G is not necessarily a polynomial algebra even

though G is a reflection group. The Weyl group W = W(F ) of the exceptio4 nal Lie group F provides an example of a W reflection group whose ring 4 3 of invariants is not a polynomial ring. By Tocla [3] we have

The correspondence between reflection groups and polynomial algebras given by Theorem B can be extended. Namely. if k[t , ...• tn]G = k[x •.... x then 1 l n] we know more than just that G is a reflection group. We can determine the structure of G as a reflection group. at least in the cases k = ffi or k[t 1 , ...• t

n]

~.

If

G

= k[x , .... x the Jacobean is defined by 1 n] J = det

ax.1 ] __ [ at j

THEOREM C:(Coxeter-Steinberg) Let char k = O. Let G C GLn(k) be a finite

= k[t 1 •..•• t n].

reflection group. Then. in S[V]

r.-l 1 J=cIIL.

R

1

where 0 "# c € k

= the set of reflections L. = the linear functional 1 R

hyperplane of ri

the order of

{~i}

in G

in V* giving the reflecting

~i ~i

This result was observed by Coxeter on an empirical basis for real reflection groups. The general fact for char k

=0

was then proved by Steinberg

[1]. As we have already observed. in the case k

= ffi or

~.

a reflection

~

is completely determined by its order and its reflecting hyperplane. Consequently. in these cases, Theorem C tells us that a reflection group is completely determined by its ring of invariants. In closing let us note the following fact about the extension S[V]G C S[V] for arbitrary G.

152

TheHomo0gyofHopfSpac~

PROPOSITION: Let G C GL(V) be a finite group. Then (i) S[V] is a finitely generated S[V]G module (ii) S[V] is a free S[V]G module if and only if S[V]G is polynomial.

153

§21 REAL REFLECfION GROUPS

In a number of cases when char k

=0

a complete classification of re-

flection groups has been obtained. In this section we will describe the results which have been obtained for real reflection groups. These results are not really necessary for any future topological arguments. However, they are a good introduction to our discussion in §22 of complex and padic reflection groups. And those groups will have topological applications.

Some general references for real ref lection groups are Benson-

Groves [1], Bourbaki [1] and Hiller [1]. §21-1: Coxeter Groups Coxeter groups provide a characterization of real reflection groups. A Coxeter group is a "generalized dihedral group". Recall that the dihedral group D is usually described as a semi-direct product: m

Here 1/2 acts on 1/m by sending elements to their inverses. Alternatively, we can write

The correspondence between the two descriptions is provided by sl s2

= xy

=x

and

where x generates 1/2 and y generates 1/m. The second description

can be used to show that D has a representation as a reflection group. m Let D act on V m

= ffin

by the rule

x

any reflection in ffi2

y

a rotation through an angle of 2IT

m

Then it is easy to see that sl and s2 are both reflections. Moreover, the reflecting hyperplanes of s1 and s2 are separated by an angle of

~.

Alter-

natively, the vectors orthogonal to the hyperplanes are separated by an angle of IT -

~. So, all the relations si = s~ = (s1s2)m = 1 are mirrored

in the geometry of the reflection group.

154

The Homology of Hopf Spaces

The concept of a Coxeter group is simply the extension of this type of explicit correspondence between algebra and geometry to an arbitrary real reflection group. n: A Coxeter group is any group of the form DEF

I

W = <si € S

(siSj)

m ij

1>

where S is a set and (i) m ii (ii) m.. € {2.3 .... } U {<xl} IJ

The presentation

i f i;fj

(W.S) is referred to as a Coxeter system. One can show

that all the elements of S are non zero in W and that the order of each SiSj is exactly m So W is generated by involutions and mi j measures the i j. lack of commutativity between the {si}' (Observe that SiSj = SjSi if and

only if mi j = 2). The fundamental result about Coxeter groups is

THEOREM:(Coxeter) Let G be a finite group. Then G is a Coxeter group if and only if G has a representation as a real reflection group. As in the dihedral group case. the structure of a Coxeter group mirrors the structure of the corresponding reflection group. (i) Given a reflection group G we can find reflections S G

= <so1

€

S

I

m..

(s.s.) IJ 1

J

{s) such that

= 1>

Moreover. the integers {m.. } are determined by the fact that the vectors IJ

orthogonal to the reflecting hyperplanes of s. and s. are separated by an J

1

angle of IT _

IT.

m ••

IJ

(ii) Conversely. given a Coxeter group. we can construct a representation (the Tits representation)

of G as a reflection group. Choose vectors {a •...• a in n} 1

= cos(IT

(a .. a.) 1

J

For each a. define a reflection s 1

s

a

i

(x)

ai

IT - -) m ij

IT = -cos(-) m

by the rule

_ x _ (ai·a j) a (ai.a i i)

ij

mn

such that

VI: Reflection Groups and Classifying Spaces

This is a reflection. For sa.(a )

= {x I

(x,a i)

i

1

=

= -a i

=1

while sa.

155

on the hyperplane Hi

1

O} Our representation is defined by s

a.

1

We can describe Coxeter groups via Coxeter graphs DEF

n:

A Coxeter graph is a graph with each edge labelled by an integer L 3. (The integer will be omitted when it equals 3)

We can assign a Coxeter graph to any Coxeter system (W.S). Let the vertices

= S.

Let there be an edge between s. and s. if m.. ) 3 The edge is 1

J

IJ -

then labelled by the integer m... Such a representation of (W.S) has the IJ

very pleasent property that (W.S)

is irreducible as a Coxeter system if

and only if the graph is connected. When we classify irreducible finite Coxeter groups we obtain the following list. (An) (Bn

= Cn)

(D n) (E 6) (E (E

7) 8)

0-0- ............ -0-0

(n

L 1)

o-o-............ -oio

(n

L 2) L 4)

0-0- ........ 0-0-0-0-0

_0<0 0

(n

I

0 0-0-0-0-0-0

I

0 0-0-0-0-0-0-0

I

0

(G ) 2

o-oio-o ofio

(H 3)

o~-o

(H 4)

o~o-o-o

(I 2(m»

o!!!o m = 5 or m L 7

(F

4)

Here. the groups A •... F agree with the Weyl groups to be disn.Bn 4.G2 cussed in §21-2. They are distinguished among Coxeter groups = real reflection groups by the fact that they are crystallographic. This means that there is a lattice L

= zn

c

mn

which is invariant under G. Alterna-

tively. they are characterized by the fact that m ij i , j.

= 1.2.3.4

or 6 for all

The Homology of Hopf Spaces

156

§21-2: Weyl Groups The Wey1 groups are the most important examples of reflection groups. Every semi-simple compact Lie group G gives rise to such a group. Given such a Lie group G let T C G

T

Slx ... xS 1 (n factors)

be a maximal torus of G. Let

where N(T) is the normalizer of T in G. There is a well defined action of W on T. In turn. this induces an action of W on t

= the

tangent space of T

at the identity. One can show that W acts as a reflection group on t

= mn .

Since any two maximal tori of G are conjugate the above is independent of our choice of T. It also only depends on the local isomorphism class of G.

So. we can reduce to G being simply connected. In that case G decomposes into a product of simple Lie groups and the Weyl group of G exhibi ts a corresponding decomposition. 'The simple Lie groups fall .i n t;o 4 infini te families An.Bn,C and D (n 2 1) and 5 exceptional cases G , E . E and 2,F 4 6 7 n n E . These groups all appear on the list of Coxeter groups in §21-1. 8 The Weyl groups of the infinite families are as follows:

= ~n+1

(i)~}

Let

{e

Decompose

1.e2

.... ,e

n+ 1}

be

the

standard

orthonormal

basis

of

mn + 1 .

where

v = { L x.e. I L x.1 = 0 1 1 W is generated by e

}

1+e2+·

.. +e

n+ 1

n acts on m + 1 by permuting factors then it respects the above de-

If 2 n+ 1 composition. We claim that the induced action of 2 reflection group .. It is well known that 2 positions {si where a

= (i.i+1)1

= e i-e i+ 1.

on V is that of a n+ 1 is generated by the trans-

n+ 1 1 ~ i ~ n) A basis of V is given by

{a1 , · · · ,an}

And si is the reflection with respect to a i. In other i words. si sends a to -a and leaves the hyperplane orthogonal to a i i i pointwise invariant. These properties will follow if si satisfies the formula

157

VI: Reflection Groups and Classifying Spaces

=x

2(x,a ) i

-

a

(ai,a i )

i

for all x c V. To verify this formula we just verify it for the basis

{a1 •. . . . a n) of V.

z

The ring of invariants S[V] n+l is easy to calculate. The decomposition 1 rnn+

=V ~

W of

~n+ 1 modules gives rise to identities

= S(V)

~

~

n+10 SeW) n+1

mn+1 ) ~n+ 1 Moreover, it is well known that s( m the symmetric polynomials and ~

n+1) S(rn n+l

= m[ m t •... ,t ] 1 n

~n+1

. consists

0

f

= rn[o1""'On+1]

where Ok

= the k t h elementary symmetric polynomial = E t 1. . . . t.ik where 1 ~ i l <... < i k ~ n 1

Since

~n+1

acts as the identity on W we have

z

= SeW) = rn[u]

SeW) n+1 where u

= e 1+ ... +e n+l.

In particular, lui ~

= 1.

Consequently.

= m[x2 , · .. ,xn +1 ]

S[V] n+1 where IX i I i .

(ii)~) = W(Cll~ ~ (V2)n Let V ~

n

= mn

with standard orthonormal basis {el •... ,e n). The action of

~ (V2)n on V is determined by ~n

= permutation

group of {e

1,·

.. ,en}

(V2)n changes the sign of {e 1 •...• e n} Again. this action is that of a reflection group. The group is generated by {s1 •.... sn_1} U {sn} where si permutes e i and e i+ 1 (1 ~ i ~ n-1) and sn changes the sign of en' All these elements are reflections. The ring of invariants of ~

n

~

S[V] n

~ (V2)n

= the

~ (V2)n acting on rn[t •...• t ] is 1 n

2 2 symmetric polynomials in {t 1 .... ,t n}

The Homology of Hopf Spaces

158

where

(iii)

W(Dn~ ~ (Y21 n - 1

Let I

n

~ (Y2)n act on V as in (ii). Then I

where (Y2)

n-l

n

~ (Y2)n-l is the subgroup

changes the sign of an even number of elements from {e l ...

en} The ring of invariants of In ~ (Y2)n-l acting on ffi[t ..... t is n] l I ~ (Y2)n-l S[V] n where {x .... x are as in (ii) and 2 2k_2}

ffi[x ..... x 2 2k_2.x]

159

§22: ruMPLEX AND P-ADIC REFLECfION GROUPS

We first describe the Shephard-Todd classification of the finite complex reflection groups. This classification is used, in turn, to obtain the Clark-Ewing classification of the finite p-adic reflection groups. §22-i:Complex Reflection Groups The finite complex reflection groups have no simple description as do the real reflection groups. Likewise, the classification of finite complex reflection groups is a much more complicated affair than in the real case. The irreducible complex reflection groups divide into 37 cases. They consist of 3 infinite families

{l/m} , {Ln} , {G(m,p,n)}

and 34 exceptional

cases. The classification was first achieved by Shephard-Todd [iJ. A more modern treatment is given in Cohen [lJ. In dimension i we obviously must have G ment for dimension

L2

= l/m.

The classification argu-

falls into 3 cases.

(i) Imprimitive Reflection Groups A group G C GL(V) is said to be imprimitive if V = VI ffi.•• ffi Vk (k L 2) and the {V.} are permuted transitively by G. If p jm we can define the 1

reflection group G(m,p,n) where

L P< A(m,p,n) n

the permutation matrices in

Ln

o

A(m,p,n)

o

GLn(~)

w.m 1

i}

w

n

This construction includes many Coxeter groups as special cases. G(l,i,n)

= Ln = WeAn)

G(2,i,n)

= Ln

G(2,2,n)

L

n

ffi trivial representation

= WeBn ) = W{Cn )

P<

(l/2)n

P<

(Zl2)n-i

= WeDn )

160

The Homology of Hopf Spaces

The cases n

2

2 of G(m,p,n) give all irreducible imprimitive reflection

groups. (ii) Primitive Reflection Groups of Dimension 2 This case produces a large number (19!) of the exceptional cases. We want G C GL We can decompose 2(lC). GL (l[; ) 2

= l[;

x SL (l[; ) 2

where l[; is the scalar matrices. The conjugacy classes of finite subgroups of SL (l[;) are known. We have 2

<

Zlm

~ < [~ ~]. [.~!:~!]>

"2m T

= the

tetrahedral group

o

the octahedral group

I

the icosahedral group

= 3,4,5

The last three are the cases k

= 1>.

[.~!:~m'] >

of the group <x,y

x

2

3

Y

Given subgroups

where

we can define, for each k

2

I, a group

as the pullback in the diagram G~K

1

1

In other words. G arises as an extension

o -+Yk

-+G -+K-iO

All the subgroups of GL (l[; ) arise in this manner. Moreover, 19 of these 2 cases give finite primitive reflection groups The cases K = T,O and I give

161

VI: Reflection Groups and Classifying Spaces

rise to 4.8 and 7 reflection groups. respectively. (iii) Primitive Reflection Groups of Dimension

~3

This case can be handled by techniques similar to the real case. We describe the irreducible reflection groups via root graphs (or.

in one

case. by what is called a neat extension of a root graph). Root Graph in ~n n} with the property that (a .. a.) = 1 if and only if a. = a .. There is an

The vertices consist of n linearly independent vectors {al ..... a 1

J

J

1

J

1

O. Both vertices and edges are is labelled by an integer w(a L 2. The edges i)

edge between a. and a. provided (a .. a.) 1

labelled. The vertex a

J

~

i between a. and a. are labelled by the numbers (a .. a.). We

o#

1

can

J

associate

a

reflection

a € ~ and an integer n

s

a.n

L 2. (x)

group

to

1

J

each

such

graph.

Given

define x -

(l-r

)~ a

n (a.a)

=

=

=e the nth root of unity. Since sa.n(a) rna and sa.n 1 n on the hyperplane H {xl(x.a) = O} we see that s is a reflection of a a.n order n. Given a root graph with vertices {a we associate to it the i} reflection group generated by {s } ai·w(a i) . Conversely. every primitive irreducible reflection group of dimension L

where r

n

3 (with the one exception already noted) can be produced by such a root graph. The resulting classification produces one infinite family L and 15 n exceptional cases. We should note that the Coxeter graphs and their corresponding groups fit into the above pattern. To make the transition label each vertice by 2 and label an edge by "- cos(!.)" rather than "m". For example. the Coxeter m

2 -cos(!') 2 graph of D I is O~O. Its root graph is 0 n O. m 2(m) We might ask why all complex reflection groups cannot be treated in

=

terms of root graphs. The obstruction is that root graphs only produce n dimensional reflection groups which are also generated by n reflections. Some n dimensional complex ref lection groups require n+ 1 reflections to generate them. For our purposes. the degrees of the various complex reflection groups are their most important property. These were explicitly determined by Shephard and Todd.

In order to avoid repetition. we will delay actually

listing the degrees of the 37 classes of fini te irreducible complex re-

162

The Homology of Hopf Spaces

flection groups until the next section. For, as we will see, the classification of the p-adic reflection groups is really just the same as the above one for complex groups, only with some extra embellishments. §22-2: p-adic Reflection Groups The p-adic reflection groups were studied and classified by Clark-Ewing [1J. The p-adre reflection groups are always complex reflection groups. For suppose G is a p-adic reflection group via the representation

There exists a representation

with the same character. Moreover, a is actually defined over K C

~

where

K/~ < 00. (Let K be generated by the entries in the matrices of a(G».

deg

If we let L be a common extension of p,a:

G

---->

~

and K then the representations

GLn(L)

are conjugate since they have the same charac ter . So a must al so be a reflection group. However. a given complex reflection group is not a p-adic reflection group for all primes p. The key obstruction is the character field of the representation. DEFn: Let o: G _

GL (~) be a representation with character ~: G n

The character field ~(~) is the extension of ~ generated by I~ Obviously, in order to have p: G _ ~(~)

GLn(~)

defined over K

~*.

c ~*

we must have

C K. This condition is usually not sufficient. An arbitrary represen-

tation p: G -

is usually not defined over

~(~)

But, if peG) C

GLn(~)

con-

tains a reflection then the representation is defined over

~(~).

(see

GLn(~)

with character

~

itself, only over a finite extension of

~(~).

Clark-Ewing [1]). So, for such representations, p is defined over K if and only if

~(~)

C K. The above discussion

gives the following equivalence.

THEOREM:(Clark-Ewing) Let G be a finite group. Then G has a representation as a (irreducible) p-adic reflection group if and only if (i) G has a representation as a (irreducible) complex reflection group o:

(ii)

G-

~(~)

C

GLn(~) ~

where

~

is the character of p

VI: Reflection Groups and Classifying Spaces

163

We now have a programme for classifying p- adic reflection groups. Take each of the 37 irreducible complex reflection groups and then determine for which primes p one has

~(~)

C

This programme was carried out by

~.

Clark-Ewing [I). First of all, a case by case study of the complex reflection groups yields that, in every case, 21Ti ving (n

=e

(n+(_n,~,V-2,v5 and

n

~(~)

v-7.

is an extension of

~

invol-

Regarding roots of unity, it is

we11 known tha t

<=>

(n e. ~

n

2 or nlp-1

It is also a fact that

Here ~(n) is the Euler function. Observe that the case pin and ~(n) ~ 2 consists of five possibilities: n

= 2,4,6

To deduce the equal i ty observe that ~«(n)

~

for p

=2

and n

al «( +( ) 'P n -n

= 3,6

for p

= 3.

impl ies that III

C

'P

is, at most, a quadratic extension. The degrees of the cyclotomic

extensions ~ C ~«(n) are well known (see §4 of Chapter IV of Serre [6)). The quadratic extensions are the cases given in the RHS of the equality. Conversely, for these cases, the Galois group (= map (

is generated by the

-1

. So (+( € al in these cases. n -n 'P As for ~,V-2,v5 and we have n

~

(

~2)

n

v-7

Va €

~ <=>

a is a quadratic residue mod p

<=> We can establish when (~) p

(~)

p

=1

=1

for a

where (~) is the Legendre symbol p

= ±2,5,-7

by using the following facts

about quadratic reciprocity (- ~) p

(2)

<=>

p '= 1 mod 4

(~)

<=>

p '= ±1 mod 8

p

p

(-1)

p-1 g=!. 2 2 where p and q are odd primes

5

7

In particular, the last relation enables us to pass from (-) or (-) to p p or

(¥)

n

(L 5)

which are easy to solve. A standard reference for all of the above

164

The Homology of Hopf Spaces

facts on elementary number theory is Niven-Zuckerman [1]. The classification of p-adic reflection groups now follows. For. given a complex reflection group p: G

~

GLn(C) with character

above discussion to find integers m. al ..... ~ such that only if p

=

~.

we can use the

~(~)

C

A

~

if and

a 1.a2 ....• or ~ (mod m). The following table lists Clark and Ewing's results for the 37 types of

irreducible complex reflection groups. The numbering of the groups as well as the calculation of the degrees is due to Shephard and Todd (see §22-1). The Weyl groups are distinguished on this list by the fact that they are p-adic reflection groups for all primes.

VI: Reflection Groups and Classifying Spaces

Number Rank

Order

1

n

2a

n

n-1 qm n!

2b

2

2m

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 6 7 8

(n+1)!

=e

m

.i

Primes

Del!:rees

CQ

all

2,3, ... ,n-1

CQ(O CQ«(+(-l)

p :: 1 mod m

m.rn+1, .... {n-1)m.qn

p :: ±1 mod m (m~3) p = 2 (m = 4.6) p = 3 (m = 3.6) p :: 1 mod m p::lmod3 P :: I mod 3 p :: 1 mod 12 p :: 1 mod 12 P == 1 mod 4 p :: 1 mod 8 p :: 1 mod 12 p :: 1 mod 24 p :: 1.3 mod 8 P == 1 mod 8 p :: 1, 19 mod 24 p :: 1 mod 24 p :: 1 mod 5 p :: 1 mod 20 p :: 1 mod 15 p == 1 mod 60 p:: 1,4 mod 15 p:: 1.49 mod 60 p:: 1.9 mod 20 p::1.4mod5 p:: 1.2.4 mod 7 p :: 1 mod 3 p :: 1 mod 3 p:: 1.4 mod 15 all p :: 1 mod 4 p :: 1.4 mod 5 p::1mod4 p :: 1 mod 3 p::1mod3 p == 1 mod 3 all all all

2.m

m CQ«) 24 CQ(w) 72 CQ(i) 48 CQ(i.w) 144 CQ(i.w) 96 CQ(i) 192 lIl(i,v2) 288 III ( i .w) 576 lIl(c,w) 48 lIl{vI-~ 96 1Il{ i, ) 144 CQ{w, vi-I) 288 CQ{i,w,v2) 600 CQ{TJ) 1200 lIl(i,TJ) 1800 lIl(w.TJ) 3600 lIl(w.i.TJ) 360 lIl(w.~ 720 lIl(i .w. ) 240 III( t , vS) 120 lIl(vt) 336 lIl(vI-7) 648 lIl(w) 1296 lIl(w) 2160 lIl(w.vS) 1152 III 7680 lIl(i) 14.400 lIl(vS) 64-6! CQ(i) 216'6! lIl(w) 72-6! CQ(w) 108'9! CQ(w) 72'6! III 8'9! III 192'1O! III 2~i

Here (

Character Field

2~i

2~i

= m.

m 4.6 6.12 4.12 12.12 8.12 8.24 12.24 24.24 6.8 8,12 6,24 12.24 20.30 20.60 30,60 60,60 12.30 12.60 12.20 2.6.10 4.6.14 6.9.12 6.12.18 6.12.30 2.6.8.12 4.8.12.20 2.12.20.30 8,12,20,24 12,18,24.30 4,6,10,12.18 6.12.18.24,30.42 2.5.6.8.9.12 2.6.8.10.12.14.18 2.8.12.14.18.20.24.30

2~i

S= vI-1.w = e3""""""5 .TJ = e .f'. = e

and m are related by the rule pq

165

Moreover. in case 2a. p.q

166

§23: MOD P REFLECfION GROUPS

As of yet no one has obtained a classification for W reflection groups p

analogous to those obtained in §21 and §22 for k provided one assumes that

Icl

= m,~

and

~

However,

is prime to p then Clark and Ewing obtained

strong restrictions by relating W reflection groups to the already disp

cussed p-adic reflection groups. In fact,

the Clark-Ewing classification

for p-adic reflection groups simply carries over to W reflection groups. p In this chapter we outline their results. §23-1:Main Results Clark and Ewing treat W reflection groups by reducing to the results p

for

~

reflection groups described in §22.

THEOREM (Clark-Ewing) Let C be a finite group of order prime to p. Then C has a representation as a W reflection group if and only if C has p a representation as a

~

reflection group.

The correspondence is established by using Zp representations. Let C have order prime to p. One shows that, given a representation p: C there exists a representation p: G duces a

~

~

~

GL (W ), n

p

GL (Z ) inducing p. In turn, p inn

p

representation. As we will see,

reflection representations to

~

this construction sends W

p

reflection representations. Moreover we

obtain, in this manner, a one to one map between W reflection represenp tations and

~

reflection representations.

This correspondence between reflection groups carries over to their invariant theory. This is not surprising. For, since

Ici

is prime to p,

C c CL(V) being a reflection group is equivalent to its ring of invariants S[V]G being polynomial. We will show PROPOSITION: Let C be a finite group of order prime to p. Given a repre-

167

VI: Reflection Groups and Classifying Spaces

sentation p: G

GL (~ ) then the following are equivalent: n p G (1) ~p[tl.···,tn] = ~p[xI"",xn] where Ixil di •

~

A

A

= Wp[x1 ....• Xn]

where Ixil

~[tl'···.tn]G = ~[Xl.···.Xn]

where Ixil

(ii) Wp[t l,·· .,tn]G (iii)

d

i

= di pl. It follows

(The action of G in (ii) and (iii) is that induced from

from the proposition that we can determine the polynomial ring of invariants which arise from W reflection groups of order prime to p. The answer p

is the same as for \

reflection groups. The table of degrees given in

§22-2 for irreducible p-adic reflection groups applies to irreducible W p reflection groups as well. One must only add the extra restriction that the degrees {d ITd i

=

... ,d are prime to p. This restriction is forced because l, n} IGI t 0 mod p.

To summarize.

the W reflection groups of order prime to p have. as p

invariants. polynomial algebras W [x •..•• x ] of type {d ••..• d } where: p

(i) ITd

1

n

1

n

is prime to p

i

(ii) {d •..• d

is an union of the sets of degrees appearing in the last n} 1 column of the table in §22-2

All the above only applies when IGI is prime to p. There is very little known about the modular case Le. when p divides IGI. Certainly. the

~

reflection groups do not give the answer. There are W reflection groups p which are not p-adic reflection groups. The example GL (W ) is discussed n

p

in §23-4. §23-2: Representation Theory

As already stated. the link between W and \ p through

~p'

is obtained by passing

The following two lemmas provide a 1-1 correspondence (provid-

ed the group has order prime to p). In all that follows we will always work with free -

~

p

modules.

LEMMA A: Suppose G has order prime to p. Then (i) any representation p: G

~

GL (W ) lifts to a representation p: G n

p

168

TheHomowgvofHopfSpac~ ~

(ii) if P1,P2:G ~ GLn(Wp) have liftings P1,P2:G ~ GLn(Zp) and PI ~

P2 then PI

=P2· ~

Proof:(i) Given P we want to find P so that GL (Z )

.i-:

G

n

<.

commutes. Since Z

p

= ~p lim Z/ r

1

GL (W )

P

Ps : G

p

p

n

we have GL (Z ) np

GLn (Z/p r) we want to find Ps+ 1: G

~

P

~

=< lim GL (Z/ r). --np

So, given

GLn (Z/ p r+1) so that

1 GL (Z/ r+1)

~n

G

p

1

0s G Ln (Z/p r) We have a commutative diagram 1

1

K

K

1

1

1

G'

1

1

c

GL (Z/ r+l) n p

c

GL (Z/ r) n p

"Y

ps(G)

1

1

1

where

1

1

K

= Ker

...

G' = ...- l(p (G» s

Now K is a abelian p group and ps(G) has order prime to p. Thus 2(P H

= 0 and the short exact sequence 1 ~ K ~ G' ~ ps(G) ~ 1 splits s(G);K) as a semi-direct product G' = ps(G) ~ K. Consequently, we can define Ps + 1" ~

(ii) Think of PI and P2 as Wp[G] modules V and W and PI and P2 as Zp[G] "

A

A.

"

modules V and W. So V and Ware the mod p reductions of V and W. Given a W [G] module isomorphism 0: V p

isomorphism isomorphism

n:

V ~ W. If

=W we can certainly

~

lift it to a Z

p

module

we average 0 over G we obtain a Z [G] module p

169

VI: Reflection Groups and Classifying Spaces

covering O. Q.E.D. We also have a relation between

representations and lp representa-

~

tions. Before establishing the relation let us observe some consequences of Lemma A. It tells us that direct sum splittings of f

Zp [G] modules.

respond to direct sum splittings of

p

[G] modules cor-

Since (IGI.p)

=I

it is

a standard fact that every fp[G] module decomposes into irreducible modules (i.e. f

vector spaces V such that only {O} and V are invariant under

p

G). It follows that l p [G] modules likewise decompose where an irreducible is a free l

p

module such that the only G invariant submodules are {O} and

pSM for s ~ O. LEMMA B: Suppose that G has order prime to p. (i) Any representation representation (ii) If

° 1,°2 :

A

0:

0:

G

G

~ GLn(~)

is conjugate to a

A

A

GL (l ) C GL (ID )

~

n

n

p

'p

° 1,°2 :

G ~ GLn(~) are conjugate to A

~

G ~ GLn{lp) and

A

~

01 ~ 02 as ~[G] modules then 01 ~ 02 as lp[G] modules.

Proof: (i) Pick any "lattice" L

= (Zp )n

C

(~)n. Now consider

= Ell

L'

gL gEG

We claim that L'

is an invariant lattice. Obviously, it is invariant. We

must show that L'

= (lp )n . A

1 = l p [-] P

A.A

Since ID

P

i t follows that any lp module

is of the form F Ell T where F is free and T consists of p torsion. The inclusion L' C (~)n tells us that L' is a free lp module. Since L'0 ~ =' A nAn

we must have L'

(~)

sentation

= (lp)

. The invariant lattice L' defines the repre-

0.

( i i) We can reduce to the case of irreducible l [G] modules. We have G p

A

A

invariant lattices M and N such that M 0 ~ _ N 0 ~ as ~[G] modules. Then M C M 0

~ ~

N0

~

is a lattice of N 0

A

~.

S

So p MeN for some s

~

O. But every non trivial l [G] submodule of N is of the form ptN. So M ~ N p

as l [G] modules. Q.E.D. p

§23-3: Proof of Theorem 23-1 and Proposition 23-1

170

The Homoloqv of Hopf Spaces

lei

Since

is prime to p, G being a reflection group is equivalent to

its ring of invariants being polynomial. Hence, Theorem 23-1 follows from Proposition 23-1. The proposition is easy to prove. Proof of Proposition: (i

il

(it il

<=>

Obviously. (ii) ator Av(x) image.

==>

= ~ L gx gEG

Consequently.

(iii). Regarding (iii)

==>

(ii) the averaging oper-

is a projection operator with the invariants as its the

ring

a

direct

summand of

~

The averaging operator can be used to show that 7L [t ..... t J p l n

G

-->

§23-4: The Dickson Invariants As we mentioned in §23-1 groups and

~

the correspondence between IF p

reflection groups does not hold when p divides the order of

the group. GL (IF ) provides an example of a group which is a IF n

p

group but. in most cases. has no representation as a The IF

reflection

~

p

reflection

reflection group.

reflection group property is seen from the fact that the invariants

p of GL (IF ) form a polynomial algebra. n

p

Dn where Iq. I 1

= pn

= IFp[tl •...• tnJ

GL (IF ) n

p

= IFp[ql •.... ~_lJ

- pi-I. This algebra is called the Dickson invariants It

has played a major role in algebraic topology in recent years. See Wilkerson [13J for a convenient summary of the properties and uses of the Dickson invariants

171

§24:RINGS OF INVARIANTS AS THE roHOMOLOGY OF CLASSIFYING SPACES

In this section we construct. for each finite p-adic reflection group G of order prime to p. a space X such that H* (XG;W is a polynomial algep) G bra consisting of the invariants of G. In the previous four chapters we regarded S[V]

= k[tI ..... t n]

a a mial tII ... t n its degree n

L

as a graded ring by assigning to every monoa .. We now change the grading convention re1

garding S[V]. For in order to identify S[V] with the cohomology of a space we must assume that S[V] is concentrated in even degrees. So we will assume that deg V - 2 All of the results of the last four chapters are valid for this new convention if we make the obvious modifications. References in the literature for the results of this section are Clark-Ewing [1]. Sullivan [1] [2] and Holzsager [1]. §24-I:Realizing Rings of Invariants The construction is quite simple. Given our group G. we will take an appropriate G space X and pass to XI

G, The following proposition gives the

basic relation between such spaces and invariant theory.

PROPOSITION: Let G be a finite group and let X be a space on which G acts freely. If char k H*(X/ G: k)

= p>O assume

also that

IGI

is prime to p. Then

= H*(X:k)G

Proof: The finite covering G ----> X ----> XIG gives rise to a transfer map t: 2

coX

I G ----> 2coX. Given x E H* ( XIG;k) then

(*) Given y E H*(X:k) then

t*p*(x) = x

172

TheHomowgyofHopfSpac~

In particular, given z € H* (X;k) G we have

Since p: X ~ X/

G

is G equivariant (X/

. map p* : H*X well defIned ( /C;k)

~

c

has a trivial C action) we have a

... H* (X;k) C By (*) G is InJectIve. By (***)

p* is surjective.

Q.E.D.

§24-2:The Sullivan-Holzsager Theorem Sullivan [1]) was the first to use the above to realize polynomial algebras. (See however the work of Holzsager [1] for a parallel treatment). He observed that if nlp-1 then Zln C Zlp-1 Maclane space

K(~

p

~ * = (~) p

,2) (= the p-adic completion of

acts on the EilenbergK(~.2)

as discussed in

§9-6) and if we let

X _-

K(71 .2)/ P Zln

then

H* (X;W p )

where

= Wp [t] Zln = Wp [x] It! = 2 [x ] = 2n

The converse is also true for p odd. Namely, we have THEOREM A: (Sullivan-Holzsager) Let p be an odd prime. There exists a space

X such that H*(X;W ) p

= Wp [x]

Proof: We want to show that H*(X;W ) First of all, we can assume

p

where Ixl

= Wp [x]

= 2n

if and only if nlp-1

where Ixl

= 2n

forces nip-I.,

n=Omodp pi * t (t ~ 1) then P = 0 on H (X;W for i < t while pP (x) = x P p) # 0 The question as to whether such a situation can exist is very similar For. if n

=p

t

to the Hopf invariant one question. And it can be elimated by similar techniques. (see §18). Secondly, n ",0 mod p forces the factorization

x P and p

n- 1(x)

= qx S

for some 1 ~ s ~ p-1and 0 # q €

rp

we

173

VI: Reflection Groups and Classifying Spaces

have the identity pn The p

=2

= 2sn

+ 2(p-l). So (p-s)n

= p-l

and thus nlp-l.Q.E.D.

analogue of Theorem A is more delicate. We have

THEOREM B: There exists a space X such that H*(X;W if and only if n

1,2,4

2)

= W2[x]

=n

where Ixl

The spaces are provided by X = JRp"',a:;p'" and HP"'. The elimination of the cases n

~

1,2,4 has already been discussed in §18.

§24-3:The Clark-Ewing Construction Clark and EWing observed that the cyclic groups Zln where n!p-l are the one

dimensional

p-adic reflection groups and

that an analogue of

the

Sullivan-Holzinger construction could be applied to any p-adic reflection group of order prime to p to produce a mod p cohomology polynomial algebra. As we have seen in §22 and §23 they classified the p-adic reflection groups and, using this classification, they determined which mod p polynomial algebras can be realized by their construction. We now describe their results in more detail. A

Regarding the construction let G C GL (Z ) be a finite group. There is n

p

an induced right action of G on the Eilenberg-Maclane space K(i the p-adic completion of K(zll,2)

= IT K(Z,2).

p

n,2)

i.e.

To ensure the action is free

l~i~n

we consider the action of G on

n

X = K(l p .2) x EG A

(x,y)og

= (xog,g-ly)

Let

If G has order prime to p then, by Proposition 24-lA plus the fact that EG is contractible we have

*

H (XG;W ) Since H* (K(lpn ;2);W A

H* (XG;W

p)

p

= Wp[t l'

G = H* (K(ZAn ,2);W) p p ... , t where I t I i

n]

= 2 we are reduced

to

... ,t ] G. By the results of §23 we obtain a

calculating ) = W [t p p l, n polynomial algebra precisly when G is a p-adic reflection group via the A

composition G C GLn(lp) C

A

GLn(~)'

Conversely.

applied to any p-adic reflection group.

For,

the construction can be as we observed in Lemma

174

The Homology of Hopf Spaces

23-2B. every p-adic reflection representation G representation G

~

is induced by a

~ GLn(~)

GL (Z ) n

p

The table in §23-2 now tells us precisly which polynomial algebras can be obtained by

the Clark-Ewing construction.

Namely.

polynomial algebras F [x1, ... ,x] of type {2d •.... 2d } p n n 1

one can

obtain

wher~

(i) Hd is prime to p i (ii) {d .... ,d is a union of the sets of degrees appearing in the l n} last column of the table in §22-2

Remark 1: Each p-adic reflection group G C

GLn(~}

gives rise to a unique

space via the above construction. Factor the inclusion as G C GL (Z ) C n

GLn(~} and apply the construction to G C GLn(Zp}' Since IGI

$

p

0 mod p the

isomorphism class of G C GL (Z ) only depends on its image in GL (ID ) (see n

p

n

'p

Lemma 23-2B). It follows that the homotopy type of X only depends on the G image of G in GLn(~)' Remark 2: The Clark-Ewing construction is a best possible result. The next three sections are devoted to showing that if F [Xl' .... X ] is realizable p

n

and has type {2d ..... 2d where Hd is prime to p then it must be one of l n} i the polynomial algebras realized by Clark and Ewing This result is due to Adams and Wilkerson They demonstrate that any such Fp[x1 .... x form S[V]

G

n]

is of the

for some G C GL(V).By the results of §23 G must then be a p-

adic reflection group. Consequently. S[V]G can be realized by the ClarkEwing construction. Recent work of Dwyer.

Miller and Wilkerson has carried this resul t

further. They show that. for p odd,

~

p-adically completed space real-

izing F [xl ..... x ] must agree with the Clark-Ewing space. p

n

Remark 3: The spaces X are not simply connected and do not have the mod p G homotopy type of a CW complex of fini te type. For p-adic completion is used in the construction. However. one can use the fibre square of §9-6 to replace X by a local space having the above properties. In view of the G above mentioned p-adic uniqueness theorem of Dwyer. Miller and Wilkerson one should also mention a recent result of McGibbon demonstrating the lack of uniqueness at the local level. He shows that. in most Lie group cases. there are uncountably many p local spaces realizing a given F [xl .... x ]. p

n

One should also mention that Smith [13] has a construction exhibiting

VI: Reflection Groups and Classifying Spaces

175

X as a actual CW complex of finite type and. in addition. possessing a G x ... x BU(~) inducing a surjective map in lF cohomomap f: X -. BU(n l) G p logy. His construction only works for certain G. however. §24-4: Small Primes The Clark-Ewing construction runs into problems for small primes. Little is known about systematically realizing polynomial algebras which have any generators in degrees divisible by p. The Dwyer-Miller-Wilkerson arguments referred to above also show that. for p odd. if H*(X:lF ) is a poly-

*

p

nomial algebra then H (X:lF ) p

= S[V] G

for some G C Gl

clear how to produce spaces satisfying

H* (X;IF

p

)

A

n

(~

p

). But it is not

= S[V]G

. when p d1vides

IGI. The Clark-Ewing construction does not seem to be the appropriate vehicle. One

should

also

be

aware

that

the

correspondence

between

p-adic

ref lection groups and polynomial algebras breaks down for small primes. For many small primes, given a p-adic reflection group with degrees {d .d

. l, there cannot exist a space realizing a polynomial algebra of type

n}. {2d ....• 2d One uses cohomology operations of various kinds to elimin}. l nate the possibility of such a space. Because of this problem it is better to think of a connection existing between finite p-adic reflection groups having degrees {d {2d

....• 2d

... ,d and mod p fini te loop spaces of rational type l, n} . The Clark-Ewing construction then concerns the case of

l-I n-l} realizing torsion free mod p finite loop spaces (i.e. those whose classi-

fying space realizes a polynomial algebra as its mod p cohomology). If one consults the list of irreducible p-adic reflection groups in

§22-2 then the groups which fail to realize polynomial algebras via the Clark-Ewing construction are given in the chart below. In the last column we have indicated whether. for some other reason. a loop space (possibly wi th torsion!) of the appropriate type exists. Among these loop spaces those wi th torsion come from the Lie group cases. Among the remaining possibilities offered by

the above

table only

the mod 2

loop

space

corresponding to group #24 could have torsion. If it existed. its mod 2 cohomology would have to be of the form

Lin has announced a proof that there cannot even exist an H-space with such mod 2 cohomology. The only known examples of mod 2 fini te H-spaces

176

The Homology of Hopf Spaces

continue to be products of classical H-spaces i.e. the Lie groups. S7 and Rp7. Other Construction #1

pl{n+1)!

#2a

pin!

Quillen [3]

#12

p=3

Zab [21]

#24

p=2

Lie group

#28

p=2.3

#29

p=5

#31

p=5

#34

p=7

#35

p=2.3.5

Lie group

#36

p=2.3.5.7

Lie group

#37

n=2 3 5 7

Lie "Toun

Lie group Zab [21]

177

§25: ALGEBRAIC CLOSURES

The next three sections are devoted to expanding the Borel structure theorem of §20-1 beyond the case of classifying spaces of Lie groups. We will describe the Adams-Wilkerson programme for demonstrating that,

in

many cases, if H*(X;W ) is a polynomial algebra then H*(X:W ) is the ring p

of

p

invariants S[Vf of some group GCGL(V).This result is more than a

general

structure

theorem. It opens

the way to specific classification

resul ts about which polynomial algebras can occur as the cohomology of a space. For, by Theorem 20-2B, such a G must be a W reflection group. So, p we can appeal to the classification resul ts of §23 to severely limit G and, hence, S[VJ G. As might be suspected, such a result about H* (X;W ) bep

ing a ring of invariants was motivated by the Clark-Ewing construction of §24. The main classification result will be to show that, for many primes p,

the Clark-Ewing construction gives all possible polynomial algebras

which can be realized as the mod p cohomology of a space. §25-1: Discussion of the Programme

The proof of the identity H*(X;W ) p

= S[VJ G follows

a programme develop-

ed in Wilkerson [10J and [llJ and brought to fruition by Adams-Wilkerson [lJ. One works in a suitable category of W algebras over A*(p) and develp

ops in this category an appropriate analogue of Galois theory. There are two main steps in the programme: (i) Show that H*(X;W ) has an algebraic closure and identify this p

algebraic closure with S[VJ (ii) Show that the extension H*(X;W ) C S[VJ is Galois and that H*(X;W ) S[VJ

G

p

p

where G is the Galois group of the extension.

In carrying out this programme one uses the A*(p) action to

impose

restrictions on the algebraic structure. In particular, much use is made

The Homology of Hopf Spaces

178

of the operations {Qs} where QS

A

=P

when s ~

s

and QO = mul tiplica tion by d in degree 2d Given an algebra Hover A*{p) these operations are intrinsically related to the algebra structure of H For they are derivations and. as such. can be used in any of the roles assigned to derivations in classical field theory. Notably. we will use them to study algebraically independent elements (see §25-4) and inseparable extensions (see §26). We.emphasize that this approach simply bypasses the entire question as to how elements in H*(X;IF ) are tied together via Steenrod operations. (This,

p

for example. was the approach used in §I6 for mod 2 polynomial

algebras). Rather it uses the Steenrod algebra as a control device for Galois theory arguments. It is conceptual and structural in nature rather than computational. Of course. the identity H*{X;IF ) p

= S[V]G

gives a

=

complete description (in theory!) of the action of A*{p) on S[V] IFp[tI ....• t

n].

For the action of A* (p) on IFp[tI •...• t pI{t.)

tl?

pn(t

0

1

i)

n]

is forced. Namely,

1

if n ~ 2

So. we can compute how A* (p) acts on the invariants IF [t1, ... t ]G. Howp

n

ever. for most G, both the G invariants and the action of A*(p) on these invariants is very complicated. For a good illustration of the advantages of the group theoretic approach over the approach of analyzing Steenrod operations the reader should compare the treatment of the algebra IFp[x4.x2p+2] in Steenrod [5] and Wilkerson [2]. Of course, some detailed arguments involving the Steenrod algebra are necessary. However. as a general rule, we will ignore the more involved of these. The arguments given in §16 are indicative of what is required. Phrases such as "one can show" or "it can be shown" usually refer to such arguments involving Steenrod operations.

The arguments are present

in

Adams-Wilkerson [1] We have chosen to omit such computations and concentrate attention on the more conceptual aspects of the proof. In particular, we will focus attention on the operations {Qs} §25-2:AIgebraic Closures

VI: Reflection Groups and Classifying Spaces

179

There are two categories in which we will work.

, = the

category of graded IFp integral domains admi tting an unstable

action of A*(p)

~

the category of graded IF

fields admitting an action of A*(p)

p

In ~ we require that that the action of A*(p) satisfy the Cartan formula and the Adem relations be satisfied. In ,

we are also demanding that,

2n given x € H , then

= xP = 0 for

pn(x) pi(x) As a consequence, elements of , degree O. (The equation xP

i

>n

must be connected and equal to IF

= pO(x) = x

in

can only have p solutions ) On the

other hand. it can be shown that elements of kO[u.u

p

~

are always of the form k

-1

] where k is the elements of degree O. Observe also that when p O is odd the elements of , and ~ must be concentrated in even degrees. There are functors

U

where F(H)

the field of fractions of H

and U(k)

= the

unstable elements of k

Regarding FC) the action of A*(p) on H extends to F(H), For the total Steenrod power p: H - - H [ n

P(x)

=x

+ pl(x)( + ... + pn(x)(

is a ring monomorphism and extends to a map

p: F(H) P(x/ ) y

----4

F(H)[[(]]

= P(x)/P(y)

(This uses the fact that the constant term of P(x) is nonzero) Regarding UC) one can show that unstable elements are invariant under A*(p). So there is a well defined action of A*(p) on U(k). The functor U(_) enables

180

The Homology of Hopf Spaces

us to "push" results from

'!J

into .9'.It's existence is the first indication

of the added power obtained by imposing an action of A*(p) on our algebras In this chapter we will construct and study algebraic closures for the categories .9' and

'!J.

As we will see, algebraic closures in .9' and

'!J

corres-

pond under the map U. Indeed the algebraic closure of H E .9' will be defined to be U(k) where k is the algebraic closure of F(H) in

'!J

We will obtain

the closures by pursuing graded analogues of classical field and ring theory. A reference for this material is. for example. Lang [1]. In the rest of this section we describe how to define a theory of a1gebric extensions and closures for our categories .9' and itions can be applied to either .9' or

All of the following defin-

'!J.

'!J.

Given an embedding H C K then x E ~ is algebraic over H if f(x)

=0

for some homogeneous polynomial f(X) E H[X] where the variable is assumed to have degree 2d. We say K is an algebraic extension of H if every element of K is algebraic over H. L is algebraically closed if the following universal property is satisfied. Any diagram H

~K

~L

where K is an algebraic extension of H can be completed to a commutative diagram

K is the algebraic closure of H if K is algebraically closed amd also an algebraic extension of H. Any algebraic extension can be broken down into separable amd inseparable extensions An extension is separable if. every x E K. the defining polynomial f(x)

=0

algebraic extension H C K is purely inseparable if, for any for some s

~

for

also satisfies f'(x) # O. An X

s E K, x P E H

O. An algebraic extension H C K can always be decomposed H C

S C K where H C S is separable and S C K is purely inseparable.An extension He K is Galois if it is separable and H

= KG

where G are the group

of automorphisms of K which fix H. In general one has an inclusion. H C KG

but not equality.

§25-3: Algebraic Closures in '!J First of all, given a graded field k, one can construct a graded ana-

181

VI: Reflection Groups and Classifying Spaces

logue k C

k of the classical algebraic closure. One follows the classical

construction except for the obvious restrictions imposed by grading. Notably, x E

k

does not have a pth root in

k unless

Ixl

=0

mod 2p.

The algebraic closure k C t

for the category ~ is a subalgebra of k.

It is the maximal subalgebra of

k to which the action of A*(p) can be

extended. Some restrictions are required. In particular, pth roots which exist in

k must

our field.

sometimes be excluded when we impose an action of A*(p) on

For, given an A*(p) action,

then the differentials {Qs} of

§25-1 must act on k. And, if x = yP, we must have QS(x) = QS(yp) = O. So, given x E k2pn where QS(x) i 0 for some s

~

1, we can no longer allow x to

k C k to a smaller extension k C t so as

have a pth root We will restrict to avoid this problem. Decompose the extension k C

k into two extensions k

esc k

where (i) k C s is separable (ii) s C

k is purely inseparable

The extension k C s We can extend the action of A*(p) from k to s with no problem. Let x E s have degree 2d. Find the homogeneous polynomial f(X) E k[X] where degree X=2dand

=0

(*)

f(x)

(**)

f' (x) i 0

We determine the action of {pn} on x by induction on n. Because of (*) we have the identity pnf(x) = O. But the Cartan formula also gives the identity pnf(x)

,n 1 n-1 f (x)P (x) + g(x,P (x), ... ,P (x»

So, by (**), we have pn(x) = _

1 n-1 g(x,P ex), ... ,P (x» f' (x)

Alternatively, (***) says that if xEs is defined by the equation

The Homology of Hopf Spaces

182

over k then the total Steenrod power P(x) €

seer]] is chosen to satisfy

the equation

over k[[r]]. So.

the extended action of the operations {pn} is chosen to

behave mul tiplicatively. In particular it satisfies the Cartan formula. The extended action also satisfies the Adem relations. For if one takes the free Hopf algebra F generated by {pn} wi th the coproduct induced by the Cartan formula then the Adem relations generate a Hopf ideal I. Consequently, the element

a

E I of minimal degree which acts nontrivially on s

must act as a derivation. However. with defining equation f(x) we have ax

= o.

The extension s C

= 0,

a acts trivially on k. = af(x) = f'(x)ax.

then 0

So, given xEs Since f'(x) to

k

When we attempt to extend the action from s to

k

we encounter the dif-

ficulty with pth roots described previously. So we limit the extension to avoid this problem. We only take pth roots for

n QS.

We use the following

s~O

construction

DE~: Let m C k be a subalgebra admitting an action of A*(p} R(m)

= {x

E

k where ~ = y for some y E m such that QS(y)=o for s ~ O}

Obviously m C R(m). One can show that R(m) is a subalgebra of k. Moreover, R(m) admits an action of A*(p) extending the action on m. The action of pR on x E R(m) is determined by the action of ppR on y

n Ker

QS implies that ppR(y) E

s~l

n Ker

= xP

E m. For y E

QS (There exists a relation of the

s~l

R form QSpP

R. s. R 1Q 1) So pP (y)

=L p

= zP

for some z E R(m) and we then let

i

pR(x)

= z.

We can use the R construction to extend the action of A*(p)

beyond s in a canonical manner. Define the sequence s(O) C s(l) C.... by s(O) s(n+1)

=s = R(s(n»

Let t

=U n~O

sen)

VI: Reflection Groups and Classifying Spaces

183

Then t is the algebraic closure of k. Remark: Observe that, by the above construction, x € and only if QS(x)

=0

for s ~ 0 s

cation. Given x € n Ker Q

t is a pth root if

We need only comment on the "if" impli-

0 then Q (x)

=0

means that x has degree

s~O

=0 mod

2p So the above root construction applies to x and we can find a pth root

for x in t.

(not just in k) This p th power criterion wi 11 playa major

role in §25-4. §25-4: Algebraic Closures in 1 It is a simple consequence of §25-3 that THEOREM A: Every element of 1 has an algebraic closure Given H € 1 we construct the algebraic closure H eTas follows. Let F(H) C t be

the algebraic closure of F(H) as constructed in §25-3. Let T

=

U(t). For T to be an algebraic closure we must be able to complete diagrams in 1 as follows H

/~ I

~~

f

T

The property of t being an algebraic closure in

~

enables us to obtain a

map

Since g maps unstable elements to unstable elements it must map Q to T U(t). So f

= glQ'

If H has finite Krull dimension then its algebraic closure in 1 is of a very simple form. The Krull dimension of H is the maximal number of algebraically independent elements in H (over W In the rest of this section p)'

we wi 11 prove

THEOREM B: Given H € 1 of finite Krull dimension then the algebraic closure of H is of the form S[V] We might remark that finite Krull dimension is an obvious restriction to impose on H. For if V has rank n as a W vector space then S[V] has Krull p

184

The Homology of Hopf Spaces

dimension n and any subalgebra H C S[V] has Krull dimension

~

n.

The

embedding H C S[V] to be constructed in Theorem B will have the property that rank V

the Krull dimension of H.

The rest of this section is devoted to the proof of Theorem B. Let H be a unstable integral domain of fini te Krull dimension algebraic closure of H in

~

Let H C T be the

as constructed above. Then T also has finite

Krull dimension The point is that F(T) is algebraic over F(H) So.

they

have the same transcendence degree (= maximal number of algebraically independent elements). We have the standard relation that the linear independence of derivations {a · .... a } in T corresponds to the algebraic independence of eleI k ments {t in T. Let {Qs} be the derivations on T defined in §23-I. I·· ... t k} One can show that there exists n ~ 1 such that

(*)

sl s any n operations {Q ..... Q n} are linearly independent in T sl sn+l . any n+l operatIons {Q ..... Q ) are linearly dependent on T

This type of resul t is based on the fact that the operations {Qs} are s

s

related to each other via the recursive formula QS+I = QSpP - pP QS for s ~

1. By (**) we have a relation n

on T. By

.

I a.QI = 0

(R)

i=O

(*}. a i

# 0 for 0

~

~

1

n. Consider the homogeneous polynomial n

A(X) =

I a.XP

i=O

i

1

where X has degree 2. To make the identity T = S[V] we must first locate elements in T of degree 2. The operations {Qs} act on elements of degree 2 by the rule xP

s

for s ~ 0

2. Consequently. for any y € T relation (R) forces A(y) = O. So. we will consider the roots of the equation A(X) = O. Observe that A'(X) = a implies that A(X) is separable. So. the roots of A(X) the extension t C F(H)

is purely inseparable. (recall

extensions F(H) esc t C F(H) and s C F(H)

# 0 O 0 belong to t. For

that we have

the

is purely inseparable). One

can also show that the roots of A(X) are unstable. Consequently.

they

185

VI: Reflection Groups and Classifying Spaces

actually belong to T

= U(t).

And the separability of A(X) means that the

roots of A(X) are all distinct. In other words. the pn roots of A(X) form a vector space VCr of rank n. We thus obtain a map S[V]

\]I:

~

T

of A*(p) modules. We are left with shOWing that {tl ....• t (a)

n}

\]I

is an isomorphism. Let

be a basis of V.

is injective

\]I

One shows that A

i

det(t~) t 0. This implies both that

det(Qi(t.» J

J

the derivations {QO ....• Qn-l) are linearly independent and that the elements {tl ..... t

are algebraically independent in T.

n} i

where (al ..... a zero a

° it

i

= IT (alt l+ ranges through all non zero tuples where the

To show det(t~ ) t

suffices to show det(t~ )

n) is 1. By row operations one shows that each altl+ ... +ant i n, i

and

i

hence their product. divides det(t~ ). Moreover. det(t~ ) and IT (a t .. l l+ J n-l .+a t ) have the same degree and the monomial t P ... t P occur in both n n n lt 2 with the same coefficient (= 1). So. they are equal. (b)

\]I

is surjective

It suffices to show that any y € T can be expanded (*)

y

= [yi t I I

where YI E T

= (il •... i n)

I t

I

and

°

~

is ~ p-l

i1 in t ... t n l

For. by induction on degree we can assume YI (and hence yi) belong to S[V]

= Wp [tl •...• t n ].

Thus y E S[V] as well.

Consider the operations E(j) QJ

= QO(Qo_l}(QO_2) ... (QO_j+l)

= E(jo)(Ql)~~ .. (Qn-l)jn-l

where J

= (jO.jl'"

,jn-l) and

° ~ js ~ p-l

Rather than proving (*) we prove the more general fact that. letting I and J be as above. then (**) For each J we have QJ(y}

=[ I

yiQJ{t

I)

186

The Homology of Hopf Spaces

Observe that (*) is obtained from (**) by letting QI= the identity. The advantage in considering (**) rather than (*) is that we can use linear algebra. Given y, we have pn equations [ xIQJtI = QJ(y) I

in pn unknowns {xI}' First of all, the equations have an unique solution. For one can show that det(QJ(t

I»,

the determinant of the coefficient

matrix, is non zero by an extension of the argument used to show A = det(Qi(t.» t- O. Secondly, J

the solutions are actually pth powers. One

= 0 for s ~ O. It then follows that xI = yi I) for some YI € T. For the remark made at the end of §23-3 obviously applies

shows that, for each I. QS(x

not only to t but to T = U(t) as well. Namely, we have the following: ~

th

Power Criterion: Given x € T then x = y

This concludes the proof that T

P for some y

€

T if and only if

S[V]. We will further analyze the proof

in §26-3. Remark 1: We have isolated the pth power criterion because of its importance in the above proof. Given any unstable domain H closed with respect to separable extensions then all of the above proof applies except for the final assertion concerning the pth power criterion. In other words, we can construct an injective map S[V]

~

H. However. the map is not surjective

unless the pth power criterion is satisfied. So the pth power criterion actually characterizes S[V] among such domains. Remark 2: Smith and Switzer [1] obtain the embedding H C S[V] by constructing a "spli tting field" of the polynomial A(X). They also make use of the Dickson invariants in their arguments.

187

§26 INSEPARABLE EXlENSIONS

Given an unstable domain H of finite Krull dimension we now know that we have an embedding H C S[VJ. Our next goal is to determine when such an extension is Galois i.e. H

= S[VJ G

for some G C GL(V). This involves the

study of the usual Galois properties: normality. separability. and finiteness. In this chapter we study the separability property. The results obtained are based on Wilkerson [14J. §26-1: Main Result Let H be an unstable domain.

In §25-3 the algebraic closure of the

field of fractions F(H) (in the category

~)

was constructed in two stages

F(H) esc t The first extension is separable while the second is purely inseparable. Consequently, the algebraic closure of H in 1 can be broken down into two extensions

H esc T by letting S

= U(s).

Again, the two extensions are separable and purely

inseparable. respectively. In §25-4 we demonstrated that T

= S[VJ

when H

has finite Krull dimension. We now demonstrate that S also has a canonical form under the same hypothesis. Since the extension S C S[VJ is purely e

inseparable we have S[VJ P esc S[VJ for some e ~ O. We will prove e

THEOREM: Let H be an unstable domain where S[VJ P C H C S[VJ for some Wp vector space V of finite rank. Then H decomposition V

= Vo m... mVs

= Wp[Vo.Vi ..... V~

s

J for some

In particular. this result applies to S. The rest of §26 will be devoted to the proof of this theorem. §26-2: Integral Extensions

188

TheHomorogyofHopfSpac~

Integrality will playa major role in the arguments of §26 and §27.We now summarize a few basic facts. Given graded W domains A C B recall that p

b E B is integral over A if there exists a monic polynomial

= Xn

f(X)

= O.

where f(b)

n-I + an_IX +

+ a1X + a O

(a i € A)

As usual X has the same degree as b and f(X) is homogen-

eous. If every element of B is integral over A we say that B is integral over A. We have the following standard fact. PPOPOSITION:Given graded W domains A C B where B is finitely generated p

then

(i) B is finite over A if and only if B is integral over A (ii) If B is finite/integral over A then A is finitely generated Given an integral domain H we say that H is integrally closed if the only elements of F(H) which are integral over H consist of H itself. Any UFD is integrally closed. For, given a,b E F(H), any integral relation

can be converted to the relation a

n

n-l n-l n b +... +aIab + aOb + an_Ia

=

°

in H. So, any factor in b must be present in a. Thus bla and a/b E H C F(H).

In

particular,

polynomial

algebras

k[x

closed. §26-3: The Lie Algebra

l,

... ,x n]

are

integrally

~

We want to make some further comments about the proof of Theorem B in §25-4. Given a A*(p) algebra H define

~

~

C Der(H)

~

= {L hiQil

hiEH}

is a restricted Lie algebra. Regarding the Lie bracket [x,y]

(-1) Ixllylyx. we have

= 0 if s , t [QO,QO] = o s s [Q ,Q ] = s(p-l)Q

[Qs,Qt]

°

~

if s ~

xy -

189

VI: Reflection Groups and Classifying Spaces

Regarding pth powers. we have (QO)p

= QO

°

(Qs)p = We will apply

~

if s ~ 1

in the next section. In the rest of this section we note

the two important properties of

~

which we will require. The properties

arise from the argument employing QS operations in §25-4. They apply to

* e any A (p) domain satisfying S[V]p C H C S[V] for some e ~ 0. The first property is the restriction to H C S[V] of the pth power criterion given at the end of §25-4 for S[V]. We have the following weakened version of the pth power criterion. LEMMA A: The constant field of

~

in H (= the elements of H annihilated by

The second, considerably deeper. property is LEMMA B: Given

a€

~

then

a acts

trivially on H if

a acts

trivially on H

in degree 2. The proof of Lemma B mimics the argument used to prove Theorem B of §25-4. As we remarked at the end of §25 all of that argument. except for the p

th

power criterion given at the very end is valid. not only for S[V]. but also for H C S[V]. (Instead of the pth power criterion we have the weaker version given in Lemma A) First of all.

~

is a free H module of finite

rank (say n) with basis {QO •... Qn-l} (see (*) and (**) of §25-4). Thus we want to show that every non trivial linear combination of {QO •... ,Qn-l} acts non trivially on~. It suffices to find n linearly independent (over 2 . F ) elements {x1 ....• x } in H . For. as in §25-4. det(Q1x.) n

p

° and

J

= det(x.J P

i

) #

this suffices to force the derivations {QO •.... Qn-l} to be indepen-

dent. To locate the elements {x1 •... ,x we proceed as in §25-4. We have a n} n . 2 n i relation L a.Q1 = on H. For any y € H it forces L a.yP = 0. So we i=O

1

°

consider the separable polynomial A(X)

i=O

n

= L a.XP i=O

i

1

where X has degree 2. H

1

is closed under separable extensions. So the pn solutions of A(X)

= ° all

190

The Homology of Hopf Spaces

lie in H. They give a f a basis of W.

p

vector space We

ff2

of rank n. Let {x1 .... x } be n

§26-4: Proof of Theorem 26-1

Assume that we have

=0

induction on e. The e al e

~

S[vt

e C H C

S[V]. The proof of the theorem is by

case forces H

= S[V]

and we are done. For gener-

1 we consider two cases.

(i} H C S[V1P

In this case let H'= the pth power roots in S[V] of elements of H. Then the root construction in §25-3 shows that H' is an unstable domain. Moree-l over. S[V]p C H' C S[V]. So, the induction hypothesis gives the desired structure for H'. And H the form H

= fp[V~.V~

2

=

(H')P inherits the structure. Indeed, H is of s

....• V~ ] (i.e. Vo

= o}

(ii) He S[V]p Fi rs t of all. let H" = H

n S[V]P.

e

Then S[V]p C H" C S[V]p and. by case

s

s

ff2).

(i), H" = IFp[V~ ..... V~ ]. Let Vo = H n V (= We have IFp[VO'V~, ...• V~ ] C H. We want to show equality. We will use the following result. e

PROPOSITION: Given unstable domains S[V]p C B cAe S[V] where: {i) B is integrally closed (ii) A

n S[V]p = B n S[V]p

2 2 (iii) A = B

Then A = B The rest of

this section is devoted to

the proof of

Assume that we are given A and B as in the proposition. LEMMA A: The following are equivalent: (i) A = B

(ii) F(A)

= F(B)

(iii) F(A) C F(B) is separable (iv) DerF(A)F(B)

=0

the proposi tion.

VI: Reflection Groups and Classifying Spaces

Proof: The equivalence (iii)

191

(iv) is a standard algebraic fact. (see,

(=)

for example, Lang [1] X S7) The other two equivalences require the special hypotheses that we have imposed on A and B (i) (-) (ii): To deduce that F(A)

= F(B)

implies A

=B

we use the fact

that B C A is finite (and hence integral) and that B is integrally closed (ii) (-) (iii): This equivalence follows from the fact that F(A) C F(B) is a purely inseparable extension Let

C Der(F(A»

~

Q.E.D.

be the Lie algebra discussed in S26-3.

LEMMA B: DerF(Bl(A) C ~ Proof:

Let C

=A n

S[V]p

=

B

n S[V]P. The inclusion

C C B C A gives

inclusions F(C) C F(B) C F(A) DerF(B)F(A) C DerF(C)F(A) Next, we appeal to a result of Jacobson [1] §4-8.

(*) There is a 1-1 correspondence between subalgebras F(A)P eKe F(A) and restricted Lie subalgebras DC Der(F(A». The correspondence is given by

K = constant field of D

By the results of S26-4 F(C) DerF(C)F(A)

= ~.

=

the constant field of

LEMMA C: DerF(B)F(A)

2. = 0 on A Since B2 F(B).

So,

by (*),

=0

Proof: By the results of §26-4, given

a o on

~.

Q.E.D.

Q.E.D.

aE

= A2 it follows that

D,

a =0

a =0

on F(A) if and only if

on F(A) if and only if

a=

192

§27: GALOIS THEORY

In this chapter we focus on the second stage of the programme discussed in §25-1. We will demonstrate that, in many cases, an unstable polynomial algebra is a ring of invariants S[V]G for some V and G C GL(V). The resuIts of this chapter are based on the work of Adams and Wilkerson. §27-l: The Adams-Wilkerson Theorem In this chapter we will prove THEOREM: (Adams-Wilkerson) Let Fp[xl, ... ,x be a polynomial algebra n] admitting an unstable action of A*(p). If p is prime to ITd. where i

Ixil

= 2d.1

then F [xl, ... ,x ] P

n

= S[V]G

1

for some G C GL(V) of order ITd i i

The proof of the theorem is an extension of the arguments in §25 and §26. By the resul ts of §25 we have an embedding Fp[x

... ,x C S[V]. We now l' n] further examine it and extract its Galois properties. We might note that the hypothesis of p being prime to ITd

is important. For example, the i P] embedding F [t C F [t] of A*(p) algebras ( It 1 = 2) is not Galois. There p

p

is no G such that F [t P] p

= FP[t]G.

The Adams-Wilkerson resul t

implies,

in turn, a major classification

result for the cohomology of classifying spaces. First of all, we know exactly what polynomial algebras can arise as rings of invariants S[V]G when IGI is prime to p , By the resul ts of §23 G is a p-adic reflection group and is a product of the irreducible p-adic reflection groups appearing in the list from §22-2. Moreover, the ring of invariants, S[V]G,

is a

polynomial algebra of type {2d

is a

~

2d

~ ... ~

2d

where {dl, .... d

n} n} l 2 union of the sets of degrees appearing on the same list. Secondly. by the results of §24, all such polynomial algebras are realizable as the cohomology of a space. Summarizing, we have obtained the following converse to the Clark-EWing construction of §24.

193

VI: Reflection Groups and Classifying Spaces

COROLLARY: If f [Xl' ...• X ] admits an unstable action of A* (p) and p is p

n

I = 2d

prime to ITd. where Ix.

then f [XI .... X] is the ring of

i I I p

n

invariants of a p-adic reflection group and is one of the polynomial algebras realized by the Clark-Ewing construction of §24. So.

the Steenrod problem discussed in the introduction to Part VI is

settled in an important case. One can go further. Recent results of Dwyer. Miller and Wilkerson include: (a) An Adams-Wilkerson type theorem for the modular case. If p is odd and H*(X:fp) = fp[xI ..... x then fp[xI, .... x n] n] reflection group.

= S[V]G

The hypothesis of H*(X;f ) p

where G is a p-adic

being realizable is.

of

course. much stronger than merely assuming that f [xI .... x ] admits an p

n

unstable action of A*(p). However. some such hypothesis is required in the modular case. For. as we pointed out in §23. in the modular case there are mod p reflection groups G C GL (f ) which are not p-adic ref lection n

p

groups. In most cases the resulting invariant rings S[V]G are not realizable. (b) The uniqueness of the Clark-EWing construction in realizing polynomial algebras in the non modular case. If p is odd and X is a p-adically completed space where H* (X:f ) = IF [xI ..... x ] then X has the homotopy type p

p

n

of a Clark-Ewing space X where G is a p-adic reflection group. G The rest of §27 will be spent in proving the Adams-Wilkerson result. We must analyze the embedding f [xI •... x ] C S[V] wi th respect to the usual p

n

Galois properties of normality. separability and finiteness. §27-2: Normal! ty The normal i ty property always holds.

Indeed.

let H be an unstable

domain and let F(H) esc t C F(H)

U H

C

U T

be the extensions constructed in §25. In particular. F(H) is the graded analogue of the classical algebraic closure of F(H). PROPOSITION: Any automorphism a: F(H)

~

F(H) which fixes F(H) maps each

194

The Homology of Hopf Spaces

of s and t to itself Moreover. the resulting maps respect the A*(p) actions Proof: First of all, a maps s to itself. For. as usual. the roots of any separable polynomial f(X) € H[X] are permuted among themselves by a. Moreover. a: s -> s is a map of A*(p) algebras. For the action of A*(p) , as obtained in §25-3. is the same for each root. Secondly. a maps t to itself and preserves the A*(p) structure. To see this. posi tion t

= U s(n) n~O

already treated s

Since

a:

take the decom-

from §25-3 and work by induction on s(n). We have

= s(O).

Q.E.D.

F(H) -> F(H) also preserves the unstable elements of t we have

CX>ROLLARY: Any automorphism a: F(H) -> F(H) which fixes F(H) maps T to itself and. also. respects the A*(p) action on T §27-3: Finiteness Unlike normali ty, finiteness and separabil ity do not always hold for the algebraic closure He T.The embeddings ideal (t

I)

C Wp[tI,t2]

W [t P] C W [t] p p provide counterexamples. The first is neither finite nor separable. The second is not separable. In this section we investigate finiteness under the assumption of finite Krull dimension. This hypothesis enables us to use the resul ts of §25-4. In particular. T

= S[V].

Adams and Wilkerson

obtain the following equivalence PROPOSITION:If H is an unstable domain of finite Krull dimension then the following are equivalent: (i) H is finitely generated (i.e. noetherian) (ii) He S[V] is finite (i.e. integral) One of the implications is standard. Since S[V] is fini tely generated. Proposition 26-2 shows that He S[V] finite forces H to be finitely generated. The reverse implication demands much more work. The embedding

195

VI: Reflection Groups and Classifying Spaces

provides food for thought. Observe that even assuming that H is a polynomial algebra of the same rank as S[V] does not guarantee that H C S[V] is finite. The A*(p} action must be invoked in an essential way. For the rest of this section assume that H is a fini tely generated unstable domain and let H C S[V] be the algebraic closure as constructed in §25-4. First of all, any algebraic extension A C B can be decomposed in the form A

esc B where A C S is separable and S

C B is purely inseparable.

Such a dcomposition was studied for He S[V] in §26 We now show that H C S[V] has the remarkable property that it can also be decomposed in the reverse order. Let I

I

r

= {x

€ S[V]

I xP

r

€ H}

=U I

r~O r

By construction I satisfies the pth power criterion

given at the end of

§25-4. (For S[V] satisfies it) Also, He I is purely inseparable. We now show LEMMA A: I C S[V] is separable Proof: Given x € S[V] we know that x is algebraic over H. So, for some r

~

r

0, xP is separable over H and, hence, over I as well. We are reduced to showing that, for y Let z

= yp.

€

S[V], yP separable over I forces y to be separable.

Choose a non-zero polynomial

of minimal degree such that fez} pth power. We can arrange a Since QS(z}

m

= o.

We want to assume that each a

to be a pth power. Apply QS to fez}

= QS(yp} = 0 = QS(a m}

i

is a

= O.

we obtain

Since feZ} has minimal degree we conclude

Since the above holds for each s

~ 0 we conclude from the p th power

196

The Homology of Hopf Spaces

criterion that a.

1

we have g(y)p

= b~1

for each i. Letting

= f(z) = 0

= O.

and. so. g(y)

Since the roots of f(Z) are

distinct the roots of g(Y) are also distinct.

In other words. z·

= yP

separable forces y to be separable. Q.E.D. It is a consequence of H being finitely generated that LEMMA B: For each r 2 0 I

is finitely generated

r

r

Proof: The map x ~ x P gives embeddings frH C I generated

these

extensions

are

finite.

r

C H. Since H is finitely

Hence.

generated. Q.E.D.

I

is

r

also

finitely

Let

~r

{[ a.QiJa. E I } . 1 1 r 1

In particular. since 1 = H we have ~O = ~ where 0 of Der(H) discussed previously in §26-3 LEMMA C:

~r

is finitely generated as a I

r

~

is the' sub Lie algebra

module n

Proof: If {x1,···,xn } generates I r then the map ~r ~i~lIr given by imbeds {ax1.···.ax n}

~r

as a submodule of a Noetherian module. Q.E.D.

It follows from Lemma A that S[V] is separable over I S[V]

= Wp[t1 •.... t n ]

I

It follows from Lemma C that

r)·

. •Qm} for some m

a~

r

for r

»

0 (Write

and choose r such that {t ..... t are separable over n} I

> O.

~r

is generated as a I

r

o

module by {Q ., .

So. we have a relation

in I

Since I C S[V] is separable the relation holds on S[V] as well. r r. (see Lemma 26-4A) Applying this relation to each tiE S[V] = Wp[t ,t 1,··· n] we have

t~ 1

m+l +

Multiplying exponents by pr we obtain

o

VI: Reflection Groups and Classifying Spaces

t~

m+r+1

1

r

+ L a~ t~ O~j~mJ 1

j+r

197

o

r

is integral over H. Thus S[V] = i F [t •...• t ] is integral over H. By Proposition 26-2 S[V] is finite over 1 p n where now a~ € H. Consequently. each t

H.

§27-4: The Adams-Wilkerson Theorem Suppose that H

= Fp [xl ..... x n ] where Ix. 1= 2d. I l

and d.

1

*0

mod p for

each i.We want to show that F [x ..... x ] = 8[V]G for some G C GL(V). We p 1 n begin with a standard fact about field extensions. LEMMA A: Let k C s C t be graded F field extensions where p (i) each extension is finite

( ii) k C s is separable (i 11) s C t is inseparable (iv) k C t is normal Then k

s

n

t

G

where G is the group of automorphisms of t fixing k

8ee. for example.Proposition 12 of Lang [1] VII 7. We can deduce a version of this lemma at the level of unstable domains LEMMA B: Let H be an unstable domain which is finitely generated and integrally closed in its field of fractions. Let He 8[V]

be the

algebraic closure of H. Let H C 8 C 8[V] be the separable closure of H as in §26-1. Then H morphisms of 8[Y]

=8

n 8[V]G where G

= the

group of auto-

which fix H

Proof: If we pass to fraction fields then F(H) C F(8) C F(S[V]) satisfies the hypothesis of Lemma A.The point is that the extensions H C 8 C S[V] satisfy the equivalent properties (For normality and finiteness see §27-2 and §27-3) 80, Lemma A tells us that H C 8

n 8[V]G have the same fraction

fields However. this extension is also integral (i.e.finite) Thus. since H is integrally closed. H In particular.

= S n S[V]G.

any polynomial

Q.E.D.

algebra F [xl, ... ,x] p

n

with an unstable

action of A*(p) satisfies the hypothesis of Lemma B and. hence, is of the form F [xl, .. ,x ] p

n

=8

n S[V]G. The final step in the argument is to use

198

The Homology of Hopf Spaces

the assumption that the generators {xi} have degrees prime to p to force S G

= S[V]. By the above.we will then have Wp[xl ...• xn] = S[V] . First of all.

LEMMA C: S[V] is a free H module of degree lId

i

Proof: A sys tern of parame ters for a Wp graded algebra A is a se t of algebraically independent elements {wl •... wn } such that W·p [wl ..... wn ] C A is finite. The property of being free over W [wI"" p

is independent of

,W ]

n

the choice of parameters. (see, for example. Theorem 2 on page IV-20 of Serre [4]). Since W [xl •.... x ] C S[V] is finite it follows that S[V] is p

n

free over Wp [xl •...• x n ]. Any

S[V]

set

of

elements

in

II", [

] generates p xl·· .. ·xn Poincare series we have

S[V]

projecting

S[V]

as

a

Wp [Xl' .. ,x] n

a

Wp

Wp[x .... ,xn] l

If

P(S[V]II

to

) _ P(S[V])I -

basis

module.

of

Taking

P(Wp [xl" ..• xn ])

II (l-tfl II (l_tdi)-l

d.

(l-t 1)

II

(l-t)

n

l1(l+t+.,.+t i=l Letting t = 1 we conclude that S[V]II vector space. Q.E.D.

d.-l 1

) n

has rank lId. as a W Wp[xl····,Xn] i=11 p

By extending the above type of argument we can deduce that

LEMMA D: S = S[V]. The decomposi t i on W [Xl •.... x ] eSc S[V] into separable and purely inp

n

separable extensions forces the relation lId. = degS[V]II", [ ] = deg(S[V]lls)deg(SII", [ . x]) 1 If If p Xl' ., n p Xl' .... x n $ 0 mod p means that deg(S[V]ll ) s i 0 mod p. The second possibility cannot happen. Because. by Theorem

This. in turn forces S = S[V]. For lId 1 or

$

26-1. deg(

S[V]

liS) = p

r

for some r

~

O.

199

§28: CLASSIFYING SPACES WIlli TORSION

The resul ts of §27 demonstrate that we have reasonable control of H* (!3x;Wp) when it is a polynomial algebra. We now concern ourselves with the structure of H*{!3x;W in the non polynomial case. As we explained in p) the discussion of Steenrod's problem in the introduction to Part VI. asserting that H*(R.;W ) "X p

= Wp [Xl' .... Xr ]

where

Ix.1 I = 2d.1

is equivalent to

asserting that H* (X)(p) is torsion free. So we are now dealing with the case of finite loop spaces with torsion. References in the literature for the work of this chapter are Rector [4] and Lam [1]. §2S-1: Classifying Spaces of Lie Groups The study of the classifying space of loop spaces wi th torsion has proved to be a very intractable problem. At the moment the only known examples of such loop spaces are the Lie groups and the calculation of H* (BX;W

p)

is not even complete for them.

The simply connected cases of Lie groups where H* (G)(p) has torsion are p

2

G = Spin{n) for n

p

3

G = F4• E6· E7· ES

p = 5

~

7. G2· F4• E6· E7· ES

G = ES

As we mentioned at the end of §7 the main approach has been to study the cobar spectral sequence {E } converging to H* (BG;W ) and attempt to prove that E 2

= Eoo = CotorH*(G'W

'p

are:

p

r

)(Wp;Wp)' A list of calculations involving {Er}

G= G

for p = 2 2: Borel [1]

G= F

4:

Borel [1] for p = 2 Toda [4], Kono Mimura Shimada [1] for p

G = E Kono-Mimura [2] for p = 2 6: Kono-Mimura [6] for p =3

3

The Homorogy of Hopf Spaces

200 G

G

E Kono Mimura Shimada [2] for p 7: Mimura-Sambe [1] for p = 3

= E8:

Mimura-Sambe [2] for p

3

Mimura-Sambe [3] for p

5

2

The reader should consult §46 for an illustration of how one calculates E 2 = CotorH*{G:1F )(lFp;lFp) from a Imowledge of the coalgebra structure of p

H*{G;IF ). Typically the spectral sequence only gives the module structure p

of H* (B For there are usually extension problems in passsing from Eoo G;lFp)'

= EoH* (BG;IFp )

to H* (B ). G;IFp There are two cases in which the collapse E = Eoo has not been proved. 2 namely, (G.p) = (Spin{n),2) and (E In the case (E it is unlmown 8,2) 8,2). what happens. In the case (Spin{n),2) it is known that {E does not alr} ways collapse. For. on the one hand, Quillen [1] has demonstrated that H* (BSpin{n);1F

2)

is not a polynomial algebra if n

~

10. On the other hand.

s+1 is Imown to be primitively generated if n = 2 (see May2) Zabrodsky [1]). Consequently, CotorH*{SPin{2s+1);1F2)(1F2;1F2) is a polyno-

H*(Spin(n);1F

mial algebra. The collapse result can.of course, be studied for any mod p finite loop space. But even less is known in that case. §28-2:Mod p Cohomology Even if one knows H* (Bx;W in all cases the problem remains of obtainp) ing general structure theorems to explain the data. At the moment. eVen the most basic questions about H* (B remain unanswered. For example. X;lFp) is H* (Bx;lF finitely generated for all mod p finite loop spaces? This p) result is Imown to be true for Lie groups (see Quillen [2]) and for Cotor H*{X'1F )(lF (see Wilkerson [12]). So the collapse of the above p;lFp) , p cobar spectral sequence would imply the result, The noetherian question is pivotal to an understanding of H* (BX;lFp)'

Lam and Rector, working independently, have demonstrated how one can obtain structure theorems for H* (B if one works under a noetherian X;lFp) hypothesis. They use the machinary developed by Adams and Wilkerson which has been described in the last few chapters. Notably,

they work in the

VI: Reflection Groups and Classifying Spaces

category The

gj

of graded IF

Lam-Rector

p

201

algebras admitting an unstable action of l'(p).

structure

theorem

is

based

on

Quillen's

approach

to

H* (BC:lFp ) as developed in Quillen [2]. The idea is to view H* (BX:lFp ) as a number of polynomial algebras "glued together" in an appropriate manner. We will follow the approach from Rector [4]. One associates to each H E a category

of polynomial algebras over

~(H)

H and shows that H =

gj

~ ~(H)

modulo nilpotent elements. The Category

~(H)

(i)Objects As in previous chapters S[V] is the IF polynomial algebra generated by p V where the elements of V are assumed to have degree 2. For each invariant prime ideal P C H one can construct a polynomial algebra integral over H. One has H -+

H

I p C S[V]

where the inclusion Hlp C S[V] is the algebraic closure of the integral domain Hlp as constructed in §25. There is a one-one correspondence between such objects and the collection of invariant prime ideals of H. The collection of

invariant prime

ideals

is

finite.

So

~(H)

is a

finite

category. (ii) Morphisms Morphisms in

~(H)

are maps over H Le. maps such that the following

diagram commutes S[V]

-

S[V']

~/ H

Mor~(H) (S[V] ,S[V' ]) "#


i f and only i f PCP'

where P and P' are the

corresponding invariant primes. The necessi ty of PCP'

is obvious. Re-

garding sufficiency, PCP' forces a map Hlp -+ Hlp,. In turn, we have a map S[V] -+ S[V'] between their algebraic closures. The fact that S[V] is integral over H forces all the rnorphisms to be epirnorphisms. We now describe the results of Lam and Rector. A morphism f: A -+ B of algebras over IF (i) (ii)

p

is a purely inseparable isogeny if

Ker f consists of nilpotent elements n

For each z E B, zP E Im f for some n l

202

The Homology of Hopf Spaces

In particular coker f also consists of nilpotent elements.

There is a

canonical map H

-+ !!!!!. '€(H).

THEOREM:(Lam-Rector) For each H E

the map H

~

-+!!!!!. '€(H)

is a purely

inseparable isogeny. For proofs of the theorem see Lam [1] and Rector [4]. This result can be viewed as a generalization of the Adams-Wilkerson theorem that H for H E I. For if H is an integral domain then '€(H)

= S[V]W

consists of one ob-

ject, namely the algebraic closure H C S[V] as constructed in §25. Moreover,

Mor~(H)(S[Y],S[V]) = W and !!!!!. '€(H)

= S[y]W

The theorem is also a generalization of the structure theorem for compact Lie groups deduced in Quillen [2]. The elementary abelian p groups {V} of G form a category with the morphisms being given by inclusions and

conjugation. The inclusions Y C G induce a map H* (BG;W ) p

~

lim H* (By;W }

+--

Y

P

Quillen proved that this map is an inseparable isogeny. This result can be subsumed under the Lam-Rector theorem. (i) One can reduce to conjugacy classes of elementary abelian p groups. There is a one-one correspondence between such conjugacy classes and the invariant prime ideals of H* (BG;W

P

= Ker

given by

p}

H*(B 'W )

G'

p

-+ H* (By;Wp)/NIl.

where Nil is the nilpotent elements of H* (BG;W p)' (ii) One can replace H* (By;W p) identifying V with

b

y

H* (By:W )/ p Nil

S[Y] .

Here we are

~(By;Wp)'

The above structure theorems are, of course, only an approximation to H* (Bx;W As an example of the phenomena which they are attempting to p), describe consider the calculation of H* (BF ) in Toda [4]. We have 4;W3 H*(BF 4;1F3)

= 1F3[x4·x8'X9,x20,x21,~,x26'X36,x48]/R

where R is the ideal generated by

VI: Reflection Groups and Classifying Spaces

203

2

x 4x9, xSx g, xg , x4x21' XSX2 1 , x4x25+xSx21' x 20x21, x 20x25, 2 3 3 3 2 2 x25 ' X9X20-x4x25+xSx2I' x 20 -x4 x4S+xS X36+X4XS x 20 ' x26x4+x21Xg' x 26xS+x25Xg, x26x20-x2Ix25 If one ignores nilpotent elements then one It(BF4;1F3 )

= 1F3[x26,x36,x48]

B

= 1F3[x36,x48]

write

=A U C B

where A

can

c = 1F3[x4·Xg·x20·x36·x4S]/

3 3 3 2 2 (x20 -x4 x 20-xS ~6+x4x8 x20 )

As this calculation illustrates there is a significant gap between known results and the exact structure of H* (!3x;lF

p)'

which the purely inseparable isogeny H* (~_;IF )

-x

There are several ways in ~

p

lim S[V] seems capable of

--

inprovement. On the one hand, Adams and Stong have raised the question of when the map H* (BG;IF ) --> lim H* (~-;IF ) is injective. Information about p

"v

--

p

the nilpotent elements of H* (BG;lF would be captured in this case. On the p) H*(~_'IF ) other hand, one can obviously replace S[V] by -x' p /p C S[V] and consider

the purely inseparable

*

isogeny H (!3x;lF

p)

--> ~

H*(~-'IF ) -X' p /p'

It

gives a better approximation of H* (!3x:lF Provided one can obtain strong p)' restrictions H* (!3x;lFp)

on

the

would be

S[V]SL 3(1F3), C GL (IF ) S[V] 2 3

factors

*

H (BX;lFp)/p

considerably advanced.

= S[V]W(F4)

= 1F3[xI2,xI6]

then In

the

understanding

the above

example A

of

=

while B is the purely inseparable extension of given by x36

= x l 23 ,x48 = x l 63 .

§28-3: p-Adic Reflection Groups This chapter arose from the fact that the correspondence between p-adic reflection groups and mod p finite loop spaces given by Clark-Ewing-AdamsWilkerson cannot hold in mod p cohomology when p is a small prime. We close out the chapter by pointing out that there is another approach to fini te loop spaces designed to extend the correspondence to all primes.

204

One

The Homology of Hopf Spaces

passes

from mod p

cohomology and considers

classifying spaces. The objective to show that

the K-theory of

K* (BX)(p)

where G is a p-adic reflection group. This time,

should hold for all mod p finite loop spaces not just for sidered in previous chapters. The motivation for K*(X)(p) is torsion free (see §44).

=

however.

their

[K* (Br)(p)] G the identity the ones con-

this approach is

that

It follows that K*(Ex)(p) is torsion

free. The question is whether K* (Ex)(p) is open to an Adams-Wilkerson type analysis (using A operations rather than Steenrod operations) yielding the

. K* (BX)(p) identIty

= [K* (Br)(p)] G .

rhe difficulties involved in this

approach are discussed by Adams [12].

PART VII: SEalNDARY OPERATIONS

The next fourteen chapters are essentially devoted to the theory of secondary operations.

We will use secondary operations

to analyse the

action of the Steenrod algebra on the mod p cohomology of finite H-spaces. In the next four chapters, as a preliminary to that study, we will develop the basic

theory of secondary operations and examine the relation of

secondary operations to the mod p cohomology of H-spaces. In particular, we will develop techniques for analysing ll*¢<x) where Q> is a secondary operation and ll*:H*(X;1F ) ----> H*(X;r ) 0 H*(X;IF) is the coproduct map. p p p In §29 we introduce the Eilenberg-Moore spectral sequence. It enables us to systematically handle the relationship between H*(X;IF ) and p

H*{OX;r). In §30 and §32 we develop the theory of secondary operations p

and H-spaces. The main result is a coproduct formula for ll*¢<x). In §31 we give some simple applications of secondary operations. The applications are simple in that we deal with H-spaces where H*(X;r) is primi tively generated.

This hypothesis enables us

p

to easily handle

the coproduct

ll*~(x). Most of the difficulties encountered in the applications in chapters §§33-42 can be traced to the fact that H*(X;1F ) will not be assumed to be primitive

p

207

§29: TIIE ElLENBERG-MOORE SPECfRAL SEQUENCE

In this chapter we will describe the Eilenberg-Moore spectral sequence converging to H*(OX:K). Some references in the literature for the work of this chapter are Clark [2], Kraines [2], Rector [1] and Smith [4] [5]. §29-1: The Eilenberg-Moore Spectral Sequence In §7 we used the fact that H*(!\;K) = H(BC*(X) 0 K ) when X is an associative Ii-space to construct a spectral sequence {Er} converging to H*(!\:K). We now deal with a cohomology version of this spectral sequence. We must slightly alter the bar construction with respect to grading. Given a K algebra A we let

= 0 for n > 0 (BA)O = K Inl (BA) = 0 A for n < 0 (BA)n

i=l

n

Thus. the non trivial part of BA now occurs when n

~

0 rather than n

~

0

as in §7. So BA considered as a bigraded object now lives in the second quadrant rather than the first. Modulo this change we define

dy:

BA

~

BA

and TorA(K;K) as before. Adams showed that H* (OX)

= H(BC* (X) ;dy)

for any space X. By an argument

similar to that of §7 it follows that THEOREM A: For any space X and field K there exists a second quadrant spectral sequence {Er,d

r}

of coalgebras where

= TorH*(X;K)(K;K) Eoo = EoH* (OX;K)

(i) E2 (ii)

(iii) d

r

has bidegree (-r,r+1)

We will call this spectral sequence an Eilenberg-Moore spectral sequence. Eilenberg and Moore deal with a general ization of the above si tuation. Namely, given a fibre square

208

The Homology of Hopf Spaces

there exists a second quadrant spectral sequence {E where r} (i) E2 (ii) E""

= TorH*(B;K)(H* (BO;K);H* (E;K»

= EOH*(EO;K)

= X. E = PX, BO = {pt). EO = OX. The spectral sequence has some addi tional properties which wi 11 be

Theorem A is the case B

utilized at various times. First of all. the spectral sequence is multiplicative and natural. When X is a H-space one can use the maps X

~Xx X

~ X to show

THEOREM B: Given a H-space

(X.~) then {Er.d is a spectral sequence of r} bicommutative biassociative Hopf algebras.

For example see the argument in Clark [2] This spectral sequence gives complete information about the coalgebra structure of H'*(QX;IF ). P

shown in Kane [1] that Eco ~ H*(OX;IFp ) as coalgebras. When K

= IFp

It is one can

also introduce an action of A*(p) into the spectral sequence. Observe that A*(p) acts on BH*(X;IF ) via the Cartan formula. So there is an induced p

action of A* (p) on TorH*(X'IF )(IFp;IFp)' With respect to this action we have , p THEOREM C: When K

= IFp

{Er,d

r}

is a spectral sequence of Steenrod modules

For a proof of this fact see Rector [1] or Smith [4]. §29-2: Calculating the E Term 2 We are principally interested in the Eilenberg-Moore spectral sequence in the case of H-spaces. We now want to record how one can read off the from any Borel decomposition H*(X;IF ) p

So we need only consider the individual factors Ai of the Borel decomposi-

VII: Secondary Operations

209

tion. The following table summarizes what can appear

E(a) IF [a]/

p

f(sa)

= (-1.

where bidegree sa

E(sa) 0 f(ta)where bidegree sa

(aPs) where p odd or p = 2 and s ~ 2

IF [a]

E(sa)

p

=

where bidegree sa

jail (-I.lal)

(-1. jail

The elements {sa} are called suspension elements. The element sa is defined via the bar construction by the element [a] €

A.

The elements {tal

are called transpotence elements. The element ta is defined via tha bar s. . construction by CaP -1Ia 1 ] €

assuming that A

A 0 A for

any 1

s

i ~ pS_I.

(Here we are

= IFp[ a ] / (a Ps ».

) is the one suggested The Hopf algebra structure on Tor *(X:1F )(lF p;lFp H p

by the above notation. In particular. the suspension elements {sa} and the transpotence

elements

{tal

give a

basis

of

P(Tor H*(X' 1F )(lFp:lFp • p

»'

It

should also be noted that the suspension elements induce an isomorphism

§29-3: The Differential d

r

Clark has shown that the Hopf algebra structure in {E } severely limits 2 the action of the differentials. THEOREM A: (Clark) (i) d.

(ii) E pr_ 1

=0

1

=0

unless i

= pr_ 1

or i

= 2pr_1

A. as a differential Hopf algebra where either A. is a 1

1

trivial differential Hopf algebra or Ai

= [(sail

0 E(sb i) and

~ (sa.) = sb. pr_ 1 pr 1 1 (iii) E = @ A. as a differential Hopf algebra where either Ai is 2pr-1 1 a trivial differential Hopf algebra or Ai = retail 0 E(sb i) and sb.

d

1

The homology of the non trivial factor of E r

p -1

or E given in part 2pr-1

210

The Homology of Hopf Spaces

(ii) or (iii) is easy to calculate. LEMMA: Given f(x)

~

E(y) where

k(x)

=y

k ~ p }

or

d~

then H(f(x)

E(y»

~

p

f(sa)1

Once a factor

Iq

{~q(sa)

f( tal / {~

q

(ta)

arises in this manner its elements are permanent cycles in the spectral spectral. For, by checking external degrees, one can see that such elements are mapped into the first quadrant (which is trivial). Factors of the form E(sa) are similarly permanent cycles. Thus the differentials d can only act non trivially on factors of the form f(sa)

or f(ta).

i By

making use of the DHA lemma from §1-6 one can deduce that 2pr-1 and that, in these cases, d. acts as in the theorem. 1

and d in pS_1 2ps_1 terms of cohomology operations. We will only need d _ It was also charKraines [2] has characterized the differentials d

p 1'

acterized by Moore and Smith. THEOREM B: For p odd d

p-

1~ (sx) p

= sopn(x)

when degree x

2n+1

For our use of this result see the next section. §29-4: The Suspension Map The suspension map a * : QH*(X;O' ) --> PH*(OX;f ) can be defined in a p

number of ways.

(i) It is the map induced by the canonical map

p

~

--> X which is adjoint

to the identi ty (ii) If x € H*(X;f ) is represented by the map f: X --> K(l/p,n) then a*(x) p

is represented by the map Of: OX --> K(l/p,n-1). Because of this definition the suspension map is also called the loop map. (!!) (iii) Finally, the suspension map can be defined in terms of the Eilenberg -Moore

sequence.

spectral

We have already observed

that QH*(X;f) p

~

-1,* -1,* TorH*(X'f )(0' ;f ) and that the elements of TorH*(X'O' )(f ;f ) are primi'p

p

p

tive and permanent cycles. So we have a map

'p

p

p

211

VII: Seconasry Operations

This defini tion justifies the term "suspension map" It also enables one to use the restrictions on {Er.d

r}

obtained in

§29-3 to impose restrictions on a*. THEOREM:(Clark) Let (X.Il) be a H-space. Then (i) a*: Q~*(X;W )

-4

p

(ii) a*: Qn+1H*(X;W ) p

Proof: d

Regarding

(i)

pn-1H*(QX;W ) is injective unless n p

P~*(QX;W ) is surjective unless

-4

p

a*(x)

=

0 only

if

sx

=

d r

= 2pm+2 n = 2pm-2

~ r(sy) or sx

p -1 p

r ~ r(ty). Suppose the former is true. We know that 2p -1 p bidegree sy = (-1.2q) for some q ~ 1

So bidegree

r(sy)

~

p

and bidegree d

~ (sy) pr_ 1 pr

= (-1.2(q-1)pr+2)

The latter possibility is handled similarly. Regarding (ii) the transpotence elements have total degrees of the form 2pm-2. So Image a

)(

=

P(E oo)

except in degrees 2pm-2. Q.E.D. Theorem A has the following consequences COROLLARY:

Let (X.Il) be a H-space. Then

* (i) a:

-4

peve~*(QX·W ) is injective

* (ii) a:

-4

poddH*(OX'W ) is surjective • p

'p

§29-5: The Primitives PH* COX;Wpl

A knowledge of the primitives PH*(QX;W ) will play an important role in §46 The transpotence elements {tx} of E2

p

= TorH*(X'W )(W ;W ) • p p p

were defined

in §29-2. As observed in §29-3 they are permanent cycles in {Er}. Moreover. any primitive element of E- 2.* has a primitive representative in 00

H* (OX;Wp)' So if we let

212

The Homology of Hopf Spaces

= Image a *

S

*

then the transpotence elements define a subset T C PH (OX;Wp)/s. Actually, we have the identity _ PH* (OX;W )/ T p S For the argument of Clark used to prove Theorem 29-4 shows that any primitive element in E",

= EoH* (OX;Wp )

and, hence,

(;W ), arises from

in

p

either a suspension or a transpotence element. Thus we have a short exact sequence

o -+ S

-+

PH*(OX;W ) P

-+

T

-+

0

Thus the Eilenberg-Moore spectral sequence gives us a very tight hold on the primitives. Notably, we can use it to understand the A*(p) structure of PH*(OX;W ). The above short exact sequence presents PH*(OX;W ) as an p

p

extension of two Steenrod modules S and T. The A*(p) structure of S is the one induced from

the surjective map a*: QH*(X;W ) p

-+

S.

The action of

A*(p) on T can also be determined from H*(X;W ). We extend the action of p

A*(p) from H*(X;W ) to BH*(X;W ) via the Cartan formula. In particular, we p

can then determine how

p

* A (p)

acts on tx

= [xpS_i Ix i J.

This fact will play

an important role in §46 §29-6: The Loop Space Conjecture Bott [IJ [3J showed via Morse theory that the loop space of a I-connected compact Lie group has no integral torsion. His result motivated the following conjecture Loop Space Conjecture: If (X,/l) is a I-connected mod p finite H-space then H*(OX) has no p torsion. Most of the work in the next ten chapters is devoted to proving this conjecture. At the moment we only want to observe that the loop space conjecture can be reduced to a statement about the mod p cohomology of X. More exactly, it can be reduced to a conjecture about the module of indecomposabIes QH*(X;W ). p

VII: Seconoary Operations

*

~ ~)

Conjecture: If even.ex

then (i) Q

H

(X.~)

(X;W

213

is a I-connected mod p finite H-space

Oifp=2

2) (ii) Qeve~*(X'W } , p

L 6pnQ2n+IH*(X;W

n~1

} if P is odd

p

This conjecture can be related to the loop space conjecture via a series of equivalences. LEMMA A: H*(flX} has no p torsion if and only if Hodd(flX;W}

0

p

Proof: H* (X;al) = E(x l . · · · .X r } implies TorH*(X;al} (al;al) = f(sx l .. ·· ,sx r}. Since the {xi} are of odd degree it follows that all non trivial elements of f(sx l .... sx have even total degree. So. the Eilenberg-Moore spectral r} sequence converging to H*(OX;al} collapses and Hodd(flX;al} = 0 Then H*(flX}(p} torsion free implies that HOdd(flX}(p} = HOdd(OX;W

p}

= 0 as well.

Conversely. if Hodd(OX;W } = 0 then the Bockstein spectral sequence colp

lapses. Q.E.D.

LEMMA B: Hodd(flX;Wp }

---

Proof:

Only

one

of

the

implications

needs

any

comment.

Suppose

odd * odd * P H (flX;W ) = O. By Corollary I-5B we have Q H (flX;W ) p

p

= O. So H*(flX;W } is concentrated in even degree. Q.E.D. p

Next. it follows from Corollary 29-4 that

These equivalences lead us from the loop space conjecture to the QH* (X;W p) conjecture. The QH*(X;W } conjecture easily implies the loop space conjecp

. ..?n+l ture. For. gIven x E ~ (X;W). then p

o*6pn(x} = 6p no*(x} = 6[o*(x}]P= 0 Actually. the two conjectures are equivalent. But we have not yet developed enough machinery to prove the reverse implication. The basic idea is to show that when H*(flX} has no p torsion then E = Em in the Eilenbergp

214

The Homology of Hopf Spaces

Moore spectral sequence converging to H*{OX;W ). The definition of a* in p

terms of the Eilenberg-Moore spectral sequence plus the characterization of

d

from p_ 1 even.je Q "(X;W) p

29-3C

=

then

forces

n 2n+ 1 * L oP Q H (XW

'"

n~l

p

)

even..* Q "(X;W 2) for p odd. The

0

for

. relatIon

p

=

2

and

between the two

conjectures of this section is explored in more detail in Kane [IJ and

[6J.

215

§30: SECONDARY COHOMOLOGY OPERATIONS

In this chapter we begin our discussion of secondary operations by describing the operations in their most naive version. In the next chapter we wi 11 present some representative applications of

the operations.

More

complex operations and applications will appear in subsequent chapters. A reference for the work of this chapter is Zabrodsky [7J. §30-1: Preliminary Discussion It might be helpful to begin with a brief summary of the philosophy underlying our applications of secondary operations in H-space theory. This discussion is meant to be a sketch. Any unexplained terms will be dealt with in detail in subsequent chapters. Given any relation

I a.b. = 0 in A*(p) it is well known that one can 1 1

associate a secondary operation

(see §30-2). It is also well known that secondary operations are, in general. almost impossible to handle. If one has complete information about the action of A*(p) on H*(X;f ) one can determine n Ker b. i.e. the elep

1

ments on which ~ is defined. But how can one determine Im lar, how does one know when ~ ~ 0 mod

¢q

In particu-

I Im a.? The action of A*(p) tells 1

us nothing about these questions. Zabrodsky made the key observation that.

in the case of a H-space

(X.IL). one can use the coproduct IL*: H*(X;f ) ---+ H*(X;f ) @ H*(X;f) to p p p extract information about rm ~ from our knowledge of the Steenrod module structure of H*(X;f ). His result is perhaps clearest if stated in terms p

of the iterated coproducts ()_

n=*

IL n : H (X;~ ) ---+ @ H (X;f ) p

They are defined recursively by the rule

Le I

P

216

The Homology of Hopf Spaces

~(2)

~. the reduced coproduct

~(n)

(1 0 ~(n-1»~

If one is working in degree 2n and has a relation of the form

L a.b. = opn 1 1 then the associated secondary operation

~(p)¢{x) =

¢ will

x0 ... 0x mod

satisfy a coproduct formula

L 1m

a

i

in 0p H* (X;W ) whenever x E P2~* (X;W ) and b.(x) = 0 for all i. (Actually, i=1 p P 1 we do not need x to be primitive. But when x is non primitive the coproduct becomes more involved. See §32) This coproduct formula makes

¢ palatable.

If we have information about

* * the action of A (p) on H (X;W ) and. hence, on

n * 0 H (X;W ). we can deteri-I P mine whether x0 ... 0x € L Im a .. If x0 ... 0x f L Im a. then the R.H.S. of p

i

l

l

L 1m a i as well. In our applications secondary operations serve the role·of a middleman.

the above coproduct must be non zero. So ¢(x) ~ 0 mod

We are primarily interested in the action of A*(p) on H*(X;W ). Given a p

certain amount of knowledge of the Steenrod module structure of H*(X;W ) P

we use secondary operations to obtain more information. To take a very simple example consider a mod pH-space where H*(X;W ) p

=

Wp[x]/(xp) and deg x = 2n. Such a H-space does not exist. We have already presented projective plane arguments which can eliminate it in the p = 2 case. (see §15,§16 and §17) Let us now see how secondary operations can eliminate it for all primes. Obviously x E PH* (X;W and P i (x) = 0 for i ~ p) O. Consider the relation (o)(pn) = opn. In other words consider the relation ab = op

n

where a = 0 and b = pn.

let

¢

be the secondary operation

associated to this relation. We have pn(x) = O. So ¢(x) is defined and

~(p)¢{x) = x0 ... 0x mod 1m 0 But Hi(X;W ) = 0 in degree

~ 2pn. So ¢(x) = 0 and. hence. ~(p)¢{x)

well. Therefore x0 ... 0x €

Irn o.

p

This forces 0

~

p

o as

*

0 in 0 H (X;W ) and, i=1 p

hence. in H*(X;W ) as well. Since o = 0 in H*(X;W ) we have a contradicp

tioll and X cannot exist.

p

VII: Seconaery Operations

217

Generally. however. we will be dealing with H-spaces that do exist. We use secondary operations to obtain structure theorems about the action of A*(p) on mod p cohomology. They are used inductively to extend our given knowledge about the A*(p) action to obtain new knowledge. As an example suppose we have

and a relation

Suppose we also know enough about the action of A* (p) to deduce that bi(x)

=0

¢ associated

for all i. Then the secondary operation

to our relation

is defined on x and the coproduct formula is valid

~(p)¢{x)

= x0 ... ®X

mod I 1m a. .

1

1

Suppose we also have enough information about the coproduct ~(p) to know that x0 ... ®X cannot appear as a non trivial summand in ~(p)(y) for any y E

H* (X;Wp)' Then the coproduct formula forces x0 ... ®x E

L 1m a.1

A careful analysis (using the Cartan formula) enables us to deduce that x E 1m t for some t E A*(p). For example. if p

=2

and

x@x

E 1m Sq2 we must

have either X E 1m Sq2 or x E 1m Sql. So. we have obtained new information about the action of A*(p). With the introduction of secondary operations Browder' s restrictions from §12 concerning the action of the Bockstein

(j

acquire a new impor-

tance. They are the initial step in an inductive study of the action of A*(p). We can use the fact that operations and obtain more

(j

=0

in many cases to define secondary

restrictions on the action of A*(p).

These

restrictions. in turn. are used to define other secondary operations. And so on. See. in particular. the proof of theorem 31-2 for an illustration of this technique. In general our arguments are limited only by our ingenuity in finding relations which give information about a particular X E H*(X;W ). Again. it should be emp':asized that. in such arguments. we never p

really look at the secondary operation

¢.

Rather. we spend all our time

studying the action of A*(p). (first to show that bi(x) in analysing the assertion that x0 ... ®X E

=0

and. secondly.

I 1m a i in order to determine

The Homology of Hopf Spaces

218

for what ~ € A*(p) we must have x E 1m ~) §30-2: Unstable Secondary Operations We begin by treating unstable secondary operations without introducing H-space structures. That will be done in §30-3. Suppose that in degree n we have a relation in A*(p) of the form la i

L a.b. = 1 1

lb.1

0

I >0 I >0

lail + Ibil

=N

(fixed!)

(i) We can realize the relation by the follOWing commutative diagram K(Yp,n)

w

K(Yp,n)

1

1

K -->

K(7Up,n+N)

r

where K W* (tn+lb.l) 1

r*(t n+N)

= IT i

K(Yp,n+lb.l) 1

= bi(t n)

=

7ai(tn+lbil)

and the right hand vertical map is the trivial map. (ii) We can extend (*) to form the commutative diagram QK

j

1

~ K(7Up,n+N-I)

1

h E - - - + K{7Up,n) x K(7Up,n+N-I)

K{7Up,n) w

1

1

= K(7Up,n)

1

K ~ K(7Up,n+N) Here the vertical arrows form fibration sequences while h and Or are the induced maps between the fibration sequences. (iii) If we let

v

= h* (t n+N- I)

€

H* (E;Wp )

219

VII: Secondary Operations then v defines a

secondary operation associated to

L

a b i i

O. This

operation is a map of the form


->

Ifl+N-1(X'1F ) • P /'C' L.

i

n Ker b.1 be represented by f:

1m a

i

X -> K(Yp.n). Then we

have the following diagram

OK f

1j

E

/lq f

X ---+K(Yp,n} w K ~

n Ker b.) Observe that f has a whole range of

(f exists since x €

possibil i ties.

But

1

i

any

1

two choices differ by a map factoring

through

j: OK -> E. Let

Since

it follows that
= (Or)*(t n+N- 1) = Li a.(t N 1) 1 n+well defined modulo L 1m a .. 1

§30-3: Secondary Operations and H-spaces We now restrict our attention to H-spaces and introduce a coalgebra into our secondary operations THEOREM: Let la i I > O. Ib i I > 0 la I + Ib I = 2n(p-l} + I i i be a relation holding in degree 2n+l. Let (X,~) be a H-space. We

L

aib i

pIl

=s

can define a secondary operation

€

n {n Ker b.}. then 1

p

~
p-l

=L

~p) xi@xP-imodLlmal'

i=1 p i

L a.b. = lipn 1 1

L a.b. = 0 in 1 1 fact that L aib i

~ 0 in degree 2n+l we have

degree 2n. So the discussion in §30-2 applies to
The Homology of Hopf Spaces

220

opn i 0 in degree 2n+l is very important. The term op n is the key to proving the coproduct formula for ¢{x) (see Lemma A). Consequently, the above theorem only applies to very special relations, namely decompositions of

Remark 2: In the case p

=2

the theorem can be strengthened. There is no

need to restrict our attention to elements of even degree.

For any n ,

given a decomposition Sqn+l = L a.b. we have a secondary operation 1

1


in

degree n where M*¢{x)

= x0x

mod

L Im a. i

1

This strengthened version will be used in §41 and §42. If we apply the iterated coproduct ~(p)

to ¢{x) then the theorem gives

the coproduct formula

~(p)¢(x) = x0 ... 0x + LIm a i discussed in §30-1. It is this weakened version which we will find easiest to use in applications. This version makes clear that


is the dual of a

pth power modulo indeterminancy. For if we dualize ~(p) we have the map P

° i=1

Jl(p): Jl(p)(a

10

~(X;lFp)

H*(X;lFp) -

.. . 0a )

p

= a 1a2 ... a p

(Pontryagin product)

So, given a € H*(X;lF where <x,a> i 0, we have p) 0a) >

<¢(x) ,Jl(p) (a0 <Jl(p)¢{x) ,a0 <x0 ... 0x +

<x,a>P + i 0 mod

L

L

0a>

L Im a i
,a0

i,a0
0a> 0a>

Formula like the above will be used over and over in our arguments. We now set about proving the theorem. Our first result explains how op produces the coproduct formula. Consider the fibration 6pn EO --+ K(Vp,2n+l) K(Vp,2pn+2) Looping, we have the fibration sequence

n

221

VII: Seconasry Operations 6pn

K(Yp.2pn) -> flEO -> K(Yp.2n) Since 6p

n

=0

---->

K(Yp.2pn+1)

in degree 2n it follows that flEO '" K(Yp.2n) x K(Yp.2pn)

and so H* (flEO;Wp)

= H* (K(Yp,2n);Wp)

0 H* (K(Yp.2pn);W p)

this splitting of H* (flEo;W is only with respect to the algebra p) structure. Regarding the coalgebra structure we have

However.

-

LEMMA A: M (L 2 pn ) -Proof: p

= 2:

p-1

L ~~)(L

i=l P

1

.

2n

)P-l 0 (

L2n

)i

Consider the spectral sequence E2 = TorH*(EO;W2)(W2;W2) ===> H* (flEO;W2)

We want to calculate E in low degree. Now H*(K(lo2.2n+2);W is a polyno2) 2 mial algebra with generators {SqR(L 2n+1)

I

R = (r

1.r2

•... ) and

L

r i ~ 2n}.

) = E(L (see Appendix II) Thus. in degree ~ 4n+2, H* (E 2n+1) 0 polynoO;W2 mial algebra. (Use the Serre spectral sequence associated with the fibra-

tion (*) plus the fact that 6Pn = Sq2n+1.IS the squar i. ng map in degree 2n+1). So. in total degree

~

2n

TorH*(EO;W2)(W2;W2)

= f(sL 2n+ l)

0 exterior algera

By Clark's restrictions on the action of the differentials (see §29-3) 'Y survives {E We then choose a 2(sL 2n+l) r}. K(Y2.4n) such that L4n = 'Y2(Sl2n)' p odd: We again conclude that

map flEo

---->

K(Y2.2n)

x

TorH*(EO;Wp)(Wp;Wp) = f(sL 2n+1) 0 .... n

Also. by §29-3. d p- 1'Yp (SL 2n+ 1) = s6P (l2n+1) permanent cycle in {E Q.E.D. r}.

=

is a

Now consider any relation in degree 2n+1

L a.b. = 6pn 1 1

la i

I > O.

I >0 = 2(p-1)n

Ib i

Jail + Ibil

We can realize this relation by the following diagram

+

222

The Homology of Hopf Spaces

K(Z/p.2n+l)

1

w'

=

K(Z/p.2n+l)

1

orP

K - - - - + l K(Z/p.2pn+2) r'

where K

= IT i

K(Z/p,n+l+lb.

1

I)

(W·)*(L 2n+ 1+lb.l) = b i(L 2n+ 1 ) 1

~

(r')*(l2pn+2) =

1

a i(l2n+I+lb.l) 1

By looping we obtain the following fibre square n

2 ,

_?_.~

irK ~

1 h E ---> 1

j q

K(Z/p.2n) w

1

= Ow'

K(Z/p,2pn)

1

K(Z/p,2n) x K(Z/p,2pn)

1

= K(Z/p,2n)

OK

define v € ~pn(E;W ) by p

v = h* (l2pn )

Then 1 * =<~) q (L

.

(a) ~ (v)

= p-l L

(b) j*(v)

= L. a.(L 1 2n+ Ib . I)

-K

i=l p

1

1

2n

*

)p-l 0 q (L

2n

)

l'

1

Part (a) follows from Lemma A. Part (b) follows from the construction. As in §30-2 we can use v to define a secondary operation

¢.

The next lemma

suffices to prove the theorem. LEMMA B: If (X.Il) is a H-space and x € ~*(X;r ) n {n Ker b.} then p

i

1

Proof: We use the concept of a H-map deviation as discussed in §6-4. The map f; X

~

K(Z/p.2n) representing x is a H-space map because x is prim-

itive. A lifting

223

VII: Seconaery Operations

o~

1j f /E ./ lq

X - > K(iVp,2n) f

is not necessarily a H-map. However. the H-deviation Of: X x X

~

E of f

at least factors through ~ i.e.

XXX~E

1

f

q

K(iVp.2n)

This factorization is equivalent to showing that qDf identities qOf

= Df = O.

= O.

And we have the

(The first identity is Lemma 6-4A) It follows

from this factorization that -*.......* ~ f (v)

*'-* = (fIV....... x f) ~ (v)

p-1

L ~~)

i=1 P p-1

L

'* '*

+ D j (v)

x p- i @ xi + D*j*(v)

1

~~) x p- i @ xi mod

i=1 p

1

L 1m a.

1

The last equality is a consequence of identities (a) and (b) above. Q.E.D. §30-4: More Secondary Operations The operations in §30-3 can be generalized by using more compl icated relations than <5pn =

L a.b. 1 1

L

a. b.. We can extend relations of the form <5pn = 1

1

in at least two ways. First of all. following Lin. we can consider

relations of the form <5pn where aic

i

= O.

= L ai(b.-c.) 1 1

In other words, we can add on null terms to the existing

decomposition of <5pn. Such a proceedure is useful. For if bi(x) t 0 then the operation associated to <5pn

= L aib i

can find elements {xi) such that bi(x) ation associated to <5pn

= L a.(b.-c 1 1 i).

is not defined on x. But. if we ci(x

i)

then we can use the oper-

Such operations playa key role in

the study of mod 2 cohomology.

Secondly, following Kane. we can consider relations of the form

224

The Homology of Hopf Spaces

At the simplest. we can take a relation of the form opn = [ multiply on the right to obtain a decomposition opn~ type of extension can be useful. For. if

bi~

=0

=[

aib

i

and

a.b.~. Even this 1

1

then the term

aibi~

dis-

appears from the decomposition of opn~. Thus the indeterminancy created by a i can be ignored. See the proofs of 35-2C and 35-2D for a key application of this idea. More generally. however. there are decompositions of bpn~ which do not arise from decompositions of bpn. In particular. consider the relation opn

s

= (Qs+ 1)(opn-p).

The operation associated with this relation

will playa key role in §46.

All of the above can be summarized in the following theorem. THEOREM: Let OP~

=[

a.(b.-c.) be a relation holding in degree 2n+1-1~1 1

1

1

where la i

I > 0,

(X.~)

I >0 + 1

=0

aic i Let

Ib i

= 2n(p-1)

lail + Ibil

be a H-space. Given

where bi(x) = ci(x i) then we can define a secondary operation ¢(x.x ..... x such that 1 n) __ _ p-1 1 p . . ~ ¢(x,x .,X ) = [ :{.) ~(X)p-l @ ~(X)l mod [ 1m a. 1'" n i=1 p 1 1 Proof: The proof is an extension of that of §30-3. Consider the diagram realizing the relation opn~

= [a.(bi-c.) 1 1 g'

K O w'l

) K(Zlp,2n+1)

1opn

K

) K(Zlp,2pn+2) r

where KO

= K(Zlp,2n+1-1~1)

x IT. K(Zlp,2n+1-1~1+lb. 1 1+lc.1 1

I)

225

VII: Secondary Operations

= IT K(Z/p.2n+l-I~I+lb. I)

K

i

1

(g')*(t 2n+ 1 )

= ~(t2n+l-I~I)

(w'>*(t2n+l_I~I_lb.l) = b i(t 2n+ 1- 1 1/J 1) - c i(t 2n+ l-II/JI+lb.I+lc.l) I I I (r')*(t2pn+2)

= ~ ai(t2n+l-I~I_lb. 1

1

,)

If we loop then we obtain the following diagram Or

ifK - + K(Z/p,2pn) j

!

!

h

---->, K(Z/p,2n) x K(Z/p,2pn) ! ! g

E q

aK

w!

o - - - > K(Z/p,2n) ! r

OK - - - > K(Z/p,2pn+l) We use v

= h*(t 2pn )

§30-3 we have -*

(i) ~ (v)

H*(E;Wp ) to define the operation ¢. In analogue to

p * p-i * i ~i) q l/J(t 2n+ll/Jl) ® q ~(t2n-ll/Jl)

~ 1

=L

(ii) t(v) =

€

~ 1

a i(t2n+la.I+lb.l) 1

The theorem follows. Q.E.D.

1

226

§3l: H-SPACES Willi PRIMITIVELY GENERATED COHOMOL<X;Y

We now illustrate how the secondary operations defined in §30 can be used to study the action of A*(p) on H*(X;IF ). We will only deal with p

finite H-spaces where H*(X;IF ) is primitively generated. In later chapters p

we will treat the non primi tive case by using more sophisticated operations. §31-1: Torsion in H-spaces Our first resul t

is designed to illustrate how effective secondary

operations can be when properly applied. With really a minimum of effort we show THEOREM: Let (X.~) be a mod p finite H-space where H*(X;IF ) is primitively p

generated. Then H* (X)(p) has only elementary torsion i.e. px

=0

for all x € Tor H* (X)(p)' This result was originally proven by Browder [8]. Our proof using secondary operations follows the arguments of Lin [9]. We want to show that B 2

= Boo

where {B is the Bockstein spectral ser}

quence analysing p torsion in H* (X)(p)' By §13-4 it suffices to show that en..*.. . * ) .IS surJec. -H (X;IF ) ~ Qe v H (X;IF ) IS surJective. SInce PH (X;IF o.. Qodd__* p p p tive it suffices to show peve~*(X'IF ) C 1m 0 • p

Let f

odiL* sub Hopf algebra generated by OP ~ (X;IFp)

= the

Observe that f is concentrated in even degrees. We can choose a Borel decomposition

f ~ 0 f .

1

i

where f.

1

= IFp[a i]/

h.

( a P 1)

and the Borel generators

i

{ail span opoddH*(X;IF Every element of P(f) satisfies (*). For P(f) is p)'

227

VII: Seconaery Operations

k

k

= obi

spanned by {a~ # O}. And. if a.

then a~ So. to prove (*) it suffices to show 1

1

k

= o(bia~

).

P(f) -- peve~*(X'F • p) By the exact sequence (**)

* PH*(X;F -+ p(H (X;Fp)ll ) p) f (see Theorem l-SB) it suffices to show 0-+ p(n

-+

We proceed by downward induction on degree. Assume that (***). and hence (*). are true in degree> 20. Pick x p(H* (X;Fp)ll ). r

in

€

P2nH*(X;F ) but suppose that x # 0 P

We will produce a non trivial pth power in H*(X;F

p)

This. as we will see. will contradict the primitivity of H*(X;W ). p

We can assume that pn(x)

= xp

€

opoddH*(X'W ) 'p

The point is that we are free to rewrite x by any element of.p(r) {PH*(X;W

*

p(H (X;Wp)ll ) } . By our induction hypothesis xP

-+

p)

f

ker

€

p2p~*(X;W ) = P2pn ( f) . So xP can be expanded in terms of the elements p

{a

pj

i

). If j

1 and a

~

pj

i

appears as a non trivial summand with coefficient

~ then replace x by x - ~~

j-1

We now apply the secondary operation associated to the relation op o(pn_o). Pick y such that o(y)

= xP = pn(x).

n

=

Then. by Theorem 30-4. ¢{x.y)

is defined and

~p)¢{x.y) = x0 ... 0x for some z

€

+ o(z)

*

p

0 H (X;W ). To see that this is impossible we pass to the i=l p

quotient Hopf algebra Since H*(X;W

p)

H*(X'W ) . p Il

r.

is primitively generated it follows that H* (X;Wp)ll r is

also primitively generated. Also. since H*(X;W ) is primitively generated p

and since opeve~*(X;W )

*

p

on H (X;WP )11f' Hence. in

=0 p

0

1=1

(see Corollary 12-2B) it follows that 0

*

=0

H (X;F P )11f' our coproduct formula becomes

228

The Homology of Hopf Spaces

~(p)¢{x,y) = x0 ... 0x If we pick a €

(

H*(X;IF) p Il

*

r)

where <x,a> # 0 then

<¢{x,y) '~(p) (a0

00:»

<~(p)¢(x,y),a0

00:>

<x0 ... 0x,a0 ... 00:>

*

However a P # 0 contradicts H (X;lFp)ll

r being primitively generated. For

Corollary §1-5C implies that the pth power is trivial on H*(X;lF p)' §3l-2: The Module of Indecomposables Our next result is more representative of future results.

It is de-

signed to serve as an introduction to the arguments of the next few chapters. Both the result obtained and the techniques used to prove it illustrate how we will be using secondary operations. The arguments of the next few chapters are organized around proving the loop space conjecture of §29-6.

In §29-6 we showed that the loop space conjecture reduces to a

statement about the action of A*(p) on QH*(X;IF ). The next eleven chapters p

are devoted to the study of QH*(X;IF ) and, in particular, to the proof of p

that statement. We will

even(X;1F look at Qeve~*(X;IF). By Corollary 12-2B /5: H ) p

p

-+

QoddH*(X;1F ) is trivial. So the action of A*(p) on QH*(X;IF ) induces an p

p

action on Qeve~*(X;IF). In the next eleven chapters we will prove many p

results about this action of A*(p) on Qeve~*(X;1F ). They are all analop

gous to the patterns obtained by Thomas and described in §17. Namely, the action of A*(p) is determined by the p-adic expansion of the degree in which we are working. Let s I

,.(s) = ~ p-1

So '"1(0)

o while

'"I(s)

ps-1+ ... +p+1 for s ~ 1. We now prove

THEOREM: Let p be odd and let

(X.~)

be a mod p finite H-space where

229

VII: Secondary Operations

is primitively generated. Then QevenH*(X;W ) is generp

ated. as an A*(p) module. by

L Q2~(s)H*(X;W

s~O

p

)

The restriction to p odd in the theorem is not accidental. There is a major gap separating the study of QH*(X;W ) for p odd from the study of p

QH* (X;W The theorem is true when p = 2 but it requires a more compli2). cated argument to establish it. Mod 2 arguments are much more subtle. The difference is easy to state. A mod 2 obstruction arises from the fact that n the Bockstein 6(=Sql) appears in the Cartan formula for p ( =Sq2n ) . We will discuss

the mod 2 obstruction in more detail after

the proof of

the

theorem. First of all.

it

is convenient

to dualize and

think in

terms of

PH (X;W ) rather than QH* (X;W ). We have already observed that PH*(X:W ) * p p p and QH*(X: W ) are dual. Moreover. p

the duali ty extends to being one of

Steenrod modules. The left action of A*(p) on H*(X;W ) dualizes to give a p

right action of A*(p) on H*(X;W The two actions are equivalent and are p)' related by the rule <~(x).a>

= <x.a~>

for any x E H*(X:W ). a E H (X;W ) and ~ E A*(p). Observe that the action

*

p

p

of A*(p) on H (X:W ) lowers degree.

*

p

So. the theorem is the same as asserting that. for any a E P 2nH*(X;Wp) where n # ~(s) for some s ~ O. we have ap

q

# 0 for some q

> O.

prove the theorem by contradiction. Given 0 # a E P 2nH*(X;Wp)

~(s) and

ap

q

=0

for q

>0

We will

where n #

we will use secondary operations to prove a P #

O. This contradicts H*(X;W ) being primitively generated. For. by Corolp

lary l-SC the pth power map is trivial on H*(X;W Pick x E P2n..* tl (X:W where p)

<x.a>

p)'

# O. The existence of x follows from

the duality between PH*(X;W and QH*(X;W plus the surjectivity of the p) p) map PH*(X;W ) p

----->

QH*(X;W ). p

Let {Qs} be the Milnor elements in A*(p) defined recursively by the rule s s pP Q - Q pP for s ~ s

s

230

The Homology of Hopf Spaces

Let ~ be the secondary operation in degree 2n associated with the relation 6pn

=L

(_1)s pn-7(s)Q

s~O

This operation is defined on x.

s

For, by Corollary 12-2B. the ideal (6)

acts trivially on peve~*(X;W ). So Q (x) p

S

=0

for s ~ O. Moreover, ¢(x)

satisfies the coproduct formula of Theorem 30-3.

~(p)¢{x) = x0 ... ®X + y where y

(*) in

p

€

LIm pn-7(s) s~o

*

0 H (X;W ). We can use (*) to show i=l P

We have the identities <¢(x),aP)

<¢(x)'~(p)(a0

<~(p)¢{x) .a0 <x0 ... ®X.a0

0<%» 0<%) 0<%> +
In the fourth equality we use the fact that
O. For. i f y

pn-7(s)(z) then

=0

We have now shown that a P t 0 which is the desired contradiction. Q.E.D. In §33 we will analyse in more detail just how the primitivity hypothesis was used in the above proof. At the moment, we only want to point out the necessity of assuming p is odd in the above proof. When we attempt to follow through the above argument for P

=2

we run into an obstruction.

even(2) Let a and x be as before. We are assuming that A acts trivially on a and we want

to

show that a 2 t

O. We work with the operation ¢(x)

associated to the relation Sq2n+l

= L Sq2n-(2 s + ... +22 +2)Q

We have ~¢(x)

= X@X

s

s 2 + L Im Sq2n-(2 + ... +2 +2)

s 2 So. a 2 t 0 provided a0a € ker Sq2n-(2 +... +2 +2) for each s ~ 1. However,

231

VII: Seconaary Operations

even(2) the fact that A acts trivially on a is not sufficient grounds to even(2) conclude that A acts trivially on a@a. For Sql appears in the Cartan formula for Sq ;f.

o.

2n

. For example. if aSq

2

=0

and aSq

1

;f.

0 then (a@a)Sq

2

232

§32: SECDNDARY OPERATIONS FOR QUOTIENTS

In this chapter we extend the operations from §30. We will describe how to define secondary operations on quotient Hopf algebras. The main reference in the literature for the work of this chapter is Zabrodsky [6J and Lin [3J.

§32-1: Qperations on Quotient Hopf Algebras The secondary operations from §30 only satisfy the coproduct formula p-l I 'r L

~p.) _\

Le I p

.

xP - l

"" '"

.

Xl

+

'r L

Im a

1

i

. prImItIve. ... I n or d er to wh en x IS

apply secondary operations in a systematic way to the cohomology of finite H-spaces it is important, and indeed necessary, to eliminate the primitivity hypothesis. For. given a finite H-space

(X.~).

many indecomposables of

H*{X;W ) will not be primitive. There is a simple algebraic way of reducp

ing to the primitive case. Namely. one passes to a quotient Hopf algebra. Even allowing for the action of done.

As we will see at

the Steenrod algebra this can still be

the end of §33. given an indecomposable x E

H*(X;W it is a simple matter to choose a sub Hopf algebra B C H*(X;W p), p)' * invariant under A (p). so that 0

~

x E P(

we obtain primitivity by passing to assume

that

we can

define

H*(X'W ) 'p //B)' However, although

H*(X'W ) . p //B'

secondary

there is no reason to

operations

on

H* (X;W )// p

B'

For

H* (X;W )// is a purely algebraic object. It is not the cohomology of any p B space. Even more. it would be nice to be able to define secondary operations on quotient Hopf algebras

H*(X'W ) . p //B which satisfy a coproduct

formula simi lar to the one discussed in §30. decomposition lip

n

=

L a.b. and x 1 1

E P(

In

0

ther words.

given a

H*(X'W ) 'p //B) where b (x) = 0 in i

H* (X;W )// we would like a secondary operation ~defined on x such that p B

233

VII: Secondary Operations

p-l

L ~~)

j1*¢<x) .

In

H* (X;W )//

B

p

~ ~

i=l P

I

H* (X;W )//

P

B.

!f(!!) such operations exist

then we could begin to study H*(X;W) P

where X is an arbtrary H-space by passing to quotient Hopf algebras H* (X;Wp)//B" Such operations do exist, at least provided a few additional assumptions are made. Zabrodsky [6] explained how to construct them. A detailed exposition of the construction is provided by Lin [3]. The main resul t is THEOREM:(Zabrodsky-Lin) Let 8pn~ = 2n+I-I>fI1 where la i

L aib i

I > 0,

Ib i

be a relation holding in degree

I >0

la.l+lb.1 = 2n(p-l)+I+I~1 I 1

Let (X,M) be a H-space. Given x € H2n- I>fII(X;W ) and a sub Hopf p

algebra B C H*(X;W ) invariant under A*(p) where p

(i) j1*(x) € B @ B (ii) bi(x) €

BoB

then we can define a secondary operation

¢ A

on x

€

H*(X'W ) ' P //B such

that j1*¢<x) = .

In

H* (X;W )// p

B

p-l

L ~~)

i=l p

@

[>fI(x)] i

~ H* (X;W )//

P

~

¢ on H* (X;Wp)//B A

So to define

[>fI(x)]p-i

I

B.

we must assume hypothesis (i) and (ii). They

*

are strenghtenings of the hypotheses that x € p(H (X;Wp)//B) and bi(x)

=0

in H* (X;Wp)//B' In §33 we will discuss the problem of verifying hypothesis (i) and (ii) for a given x € H*(X;W ). P

The rest of §32 will be devoted to the proof of the above theorem. We will first attempt to define

o

in H* (X;W )// p

B'

¢ on

x € p(H* (X;Wp)//B) assuming that bi(x)

=

The difficul ties of making such a construction soon

234

The Homology of Hopf Spaces

become apparent. We then demonstrate that the construction can be completed by making the stronger assumption that hypothesis (i) and (ii) hold. In all that follows we will concentrate on the case rem. As in §30 the case

~

=1

of the theo-

t 1 is a straightforward extension of the

~

~

=1

case.

§32-2: Twisted Multiplications As a preliminary to our construction of the secondary operations we discuss the concept of "twisted multiplications" spaces

(X,~)

and

(Y,~')

and a map w: X A X

~

on H-spaces. Given H-

Y we can define a twisted

multiplication on X x Y by the rule (x,y)*(x'y')

= (x·x' ,w(x,x')·y·y')

Notably, since Eilenberg-Maclane spaces classify cohomology it is easy to construct tailor made twisted multiplications on products of EilenbergMaclane spaces. For example, we can give K(Zlp,3) x K(Zlp,7) a multiplication where

by using the twisting map

w: K(Zlp.3) A K(Zlp,3)

~

K(Zlp.7)

We will use such twisted multiplication to construct H-maps X

~

K where K

is a product of Eilenberg-Maclane spaces. Let B C H*(X;W ) be a sub Hopf p

algebra invariant under A*(p). Let (b.} be a set of algebra generators of I

B. Choose maps

r I.. realizing the elements {b

i}.

f = IT f

i:

X~K(Zlp,lb.l) I

The map X ~ K = IT K(Zlp,lbil)

can then be made an H-space map by giving K the appropriate twisted multiplication. The coproduct in H*(K;W ) mirrors that in B. p

§32-3: An Attempt to Construct Operations for Quotients

In §30 we defined the operation associated to (E.v) where E is defined by the fibration

E a.b. I

I

lipn in terms of

235

VII: Secondary Operations j

w

q

OK ---+ E ---+ A ---+ K

A = K(Vp.2n) K = IT K{Vp.2n+lb. i

1

I)

PEl ~~) q*(t )p-i 0 *( )i i=l P 1 2n q t 2n

;t(v)

Let x E H2n(X'~ • p ) be an indecomposable. We will assume that we have an invariant sub Hopf algebra B C H*(X;~ ) where p

x E

p(H* (X;Wp)II

and

B)

bi(x) = 0 in H* {X;Wp)II

B.

We will attempt to define a secondary operation on x E

H'* (X;~P)IIB associ-

ated to 6pn = [ a.b. satisfying a coproduct formula. Two major problems arise.

We

solve

1

1

these problems

stronger assumptions that ;t(x) E

in subsequent

B 0 B and

sections by making

that b.(x) E 1

the

BoB.

Now secondary operations are defined on the cohomology of a space. On the other hand. logical

H*(X'W ) . P liB is an algebraic construction and has no topo-

interpretation.

However.

we can pick a

product of Ei lenberg-

Maclane spaces X and a H-space map

o

g : X ---+ X

o

satisfying

1m g*

=B

In order to make g a H-map we must put a twisted multiplication on Xo as explained in §32-3. If we let X = the fibre of g then a Serre spectral sequence argument shows

236

The Homology of Hopf Spaces

Proof: The point is that differentials in the mod p Serre spectral sequence are compatible with the action of A*(p). Suppose that in the Serre spectral sequence H*(B;W ) @ H*(F;W ) p p where

Then d (x) n

~(x)

==>

H*(E;W ) we have x € ~(F;W ) p p

. Hn+l (B;W). GIven W € A* (p) where W raises degree by k then

€

p

n+k € H (X;W).

The

p

compatibility

of

Steenrod

operations

with

the

differentials means that, in the above circumstances, we must also have

< n+k

drW(x)

0 for r

dn+kW(x)

= Wdn(x)

Consequently, given a fibration determined by d (L ) n

n

€

K(~p,n) ~

n 1(B;W H + ). For d

can act non trivially on

p

L

n

•

r

E

~

B, all differentials are

is the only differential which

Moreover H*(K(~p,n);W ) is primitively genp

era ted and PH*(K(~p.n);Wp) is generated as a Steenrod module by Ln' Similar remarks apply to the fibration OX

~ X ~ X since OX is a proO O duct of Eilenberg-Maclane spaces. In particular, the lemma follows. Q.E.D.

The lemma enables us to pass from

H*(X'W ) ' p liB to the cohomology of a

space. In particular, we can define the operation ~ on x € PH*(X;W ) n (n Ker b.) in the usual fashion. Namely, consider the diagram i

p

1

OK

1j go 1 1q X-A gIl w f

X ---> E

Xo

and let

K

¢g~(x) = r*(v). This operation, however, faces other difficulies *~

before we can apply it. The difficulties arise from the fact that H (X;Wp)

= H*(X'W . P ) liB the summand

ffi ? where? is highly non trivial. We are only interested in

H*(X'W ) ' p liB' This summand is invariant under primary opera-

tions but not necessarily under secondary operations.

237

VII: Secondary Operetions

PROBLEM I: we want
*

want a coproduct formula. In H (X;IF ) 'V

~

p

:*

f'V

H (X;IF ) we have the identity p

p-l

. L ~~)g *(X)p-l ~ *()i n:*() i=l p l O g O X + f v 1 * . * i =<~)go (X)p-l ~ go (x) + i=l p 1

=p-l L

L ai(Yi) *

~P~RO~B~L~E~M~I~I:we want the elements {Y

*

to belong to H (X;lFp)//B~ H (X;lFp)//B

i}

For then our coproduct formula holds for H* (X;lFp)//B C H*(X;lF not just p) for the much larger Hopf algebra H*~ (X;IF ). p

We will show that the above problems can be solved in the affirmative if we assume p*(x) E B ~ B and b.(x) E 1

B·B.

To solve I we will construct a

secondary "operation" ¢'(x) C ~n(X;lFp) such that gO*¢'(x) c ·
factors through X A X

go A go l

*¢(x).

O

Regarding II we will show

Df

X A X.

§32-4: Toda Brackets We will solve Problems I and II in §32-6. In order to solve Problem II we reformulate our operations in terms of Toda brackets. In this section we show that we can reformulate the operations from §30 in terms of Toda ~ brackets. Given liPn = 'a,b. and yEP 2nH* (Y;IF ) n (n Ker b.) consider the I I p

sequence

where w: A

~

K are as before and K = K(Vp.2pn+l) 1 w1*(t2pn+l)

=~ 1

a i(t 2n+lb. ,) 1

i

1

238

The Homology of Hopf Spaces

Pick homotopies expressing the fact that wf and w1w are null homotopic. 2

1:

Y ---+ PK

2 1: wf '"

Define

2 A ---+ PK 1 2:

*

*

22: w1w '"

u(2 Y ---+ OK 1 1,22}: u(i , 2 } = (Pw - 22 f 1 2 1}Zl

q,+(y)

= {u{2 1,22) I

2

allowed to vary}

1,2 2

If ¢{y) is the operation constructed in §3Q then we have a proper inclusion ¢{y} C ¢+(y). For

= (OW)**[Y,OK]

indeterminancy ¢+(y)

=

+ f ** [A,OK 1]

L a.H2pn-lai'(y;W

p

1

) +

~(y)

~

where

2pn(p} € A

Because of the factor ~(y) the indeterminancy of ¢+(y) is bigger than ¢{y}.

However,

the bigger

indeterminancy makes

no difference.

For y

primitive implies that ~(y) is primitive as well. So ¢+(y) satisfies the same coproduct formula as ¢{y}. Namely -th+

~ ~ (y)

v 1 P = -~, I~y} =p-1 L :{.)

i=1 p

.

i

yP-l @ Y +

1

L 1m a i

Therefore there is no harm, for our purposes, in using ¢+(y) instead of ¢{y}. All our arguments depend on ji*¢{y} not on ¢{y). In particular, we can replace ¢ by ¢+ in §32-3 and it suffices to solve Problems I and II for th+ ~ (go * (x»

rather than

* (x».

,Iv

~go

§32-5: Ladder Toda Brackets Toda brackets can be generalized to what Zabrodsky calls ladder Toda brackets. Given a homotopy commutative diagram f X-Y g

gl and homotopies

1

f

1h

O X o - Yo

1

1

f hI 1 Xl - Y 1

VII: Secondary Operations k l:

239

feft '" hf

k2: flg l '" hlf O Define u(e

l,e2,kl,k2

): X

u(e l,e2,kl,k2) We can represent u(e

~

= Pfle l

OY

I

+ k~ + Phlk

l - e2f

by

l,e2,kl,k2)

--+* We will now set up a homotopy commutative diagram X g

1

X

(*)

o

gl

f

'A f

O

1

1w

'K

1WI

f l XO"X l -------. Kl

where X,XO.A,K,Kl,f,g,w,w are as before, Then we will apply a slightly l modified version of the ladder Toda bracket to this diagram and define an operation ¢'(x) C ~pn(X;W ). p

The Map

go

Since f * w* (L 2n + lb . I) 1

such that Image f

*

O

b. (x) € 1

BoB

and Image g*

C the decomposables of

The Space Xl and the lola

B we can choose f O

H* (X;Wp)'

-: X -----. X "Xl

We begin by picking generators of Ker g*. Without loss of generality assume g* is surjective. Since H* (XO;W is a free commutative algebra we p) can write where

= E(a i )

or W [a.]

= E(aj )

or Wp[aj]

p

[b.]/l . h or Wp (bI? 1) 1

1

240

The Homology of Hopf Spaces

h. (Choose b.,then a .. and finally a.) Ker g*' is generated by { a P1. l} U { a } . 1 1 J j Let hi subker g*' = the algebra generated by (ar } U {a } j

Pick Xl' a product of Eilenberg-Maclane spaces; and g: X a Image gl*'

~

Xl such that

=:

subker g*'

A

[decomposables of H*'(X;F and, in p)]

Define gl to be the composite

By construction Image gl*'

Ker g*

C

odd degree. we have equality.

By construction fa *'WI *'(L 2pn + l )

E

Ker g *'

n

[decomposables of H*'(X;F p)]'

So. by the above. fa*'"i * (L E Image f l*' Therefore. we can choose 2pn+ 1) f 1 .. X A Xl ~Kl completing the diagram. a The Operation

ell'

We now use the diagram (*') to define an operation ¢'. Pick homotopies k l:

fif '"

El: *' '" gIg

wf

k 2 : flg l '" wlfO

E2: *' '" w1w

We can use E to define the homotopy l

il

Then we let

= (g

A El)A:

* '" gIg

¢, (x) Remark: The replacement of E by i in the above reduces the possible inl l determinancy. §32-6: Proof of Theorem 32-1 To prove the theorem we must solve Problem I and II of §32-3. Actually. we will only solve Problem II in its stated form. The desired factorization of

Of

follows from

LEMMA A: Consider a fibre square

241

VII: Secondary Operations

f

X ---+ E

1 1

go

f

1q 1w

X ---+ A g

f

o o ---->

X

K

where A and K are general ized Ei 1enberg-Mac lane spaces wi th an abelian multiplication and w is a group homomorphism. If D f factors as gAg then

Of

F

X A X ---+ X

o

factors as XAX

gOAgO

A X

o

---+ A

D

j

lXAX---->l?K--->E

We then know that ¢gO*(x) has the right coproduct formula Next we replace ¢ by ¢+ as defined in §32-4. Since ~~x) still have the desired coproduct formula.

Moreover,

~¢+(x) we

letting ¢,

be

the

the operation described by

the

operation defined in §32-5. we can prove

So,

letting ¢ex)

=

gO*¢'(x),

we have

theorem. We now briefly sketch how to prove Lemmas A and B. Proof of Lemma A Define maps X - I- 4 X x A - -w4 K by

o

I(x) w(x,a)

(g(x).f(x» fO(x)

-1

w(a)

Then choose a homotopy E: woI", *. Next, use F to put a "twisted" H-space structure on X x A.

o

(x,a)o(x' ,a')

= (x-x ' .Ffx .x ' )oaoa')

Then I is a H-map. Now w is not necessarily a H-map. But

Dw factors

as

242

The Homology of Hopf Spaces

and there exists a homotopy

e: ~

F for F

1,F2 : Xo A Xo

-4

F(gAg)

~

= mO(F l ~

*. We can write ~

x F2)A

K. Also. there exist homotopies

We use the above homotopies to define D: X A X Dxl'~}

*

ax

-4

by the rule that

is the following loop

----+

W'f(x

) ----+ W(f(X )} l,x2 l),f(x2 II

W· f(x l) 'W' f(x2}

1 ~ w· f(x l) ·w· f(x 2}' F(g(x l} ,g(x2» II

~

~

w'f(xl)'W'f(x2)'Fl(g(xl),g(x2»'F(g(xl),g(x2) The identity

Of

=

jD(gO A gO) is proven by imbedding the above loop in a

larger homotopy diagram. Since the proof is long and tedious we omit it. See Lin [3] for details. Proof of Lemma B Consider the diagram

X~E go

1

1

X~A

1

g

f

o 1w

Xo-K

gl Xo plus

1

A

f 1 wI 1 Xl -> KO

the homotopies kl,k2,el,e2,fl defined before.

Consider also

homotopies

el

eo: *

~

ggo

pfOe O + klgO:

*

~ f~O ~

*

Lemma B then follows from the following homotopy identities ~

(i) u(fl,e l,kl,k2)go ~ - Oflu(eO,fl) + u(e 1,e2) Again, we omit the details.

the

PART VIII: TIIE MODUlE OF INDECOMPOSABLES QH*(X:IF ) P ODD p

The next four chapters are concerned wi th structure theorems for the A*(p) module QH*(X:IF ) when p is odd and (X.J1.) is a mod p finite II-space. p

In §33. as an introduction. we will outline the general framework of our arguments involving secondary operations. p

=2

These remarks also apply to the

case which will be discussed in Part IX.

In §§ 34.35 and 36 we

deduce our main results for QH*(X:IF ) in the p odd case. Notably, in §35. we prove that QevenH* (X:IF ) = p

p

" n L oP Q2n+l H* (X:IF ). As we explained in

TIll

P

§29-6 this resul t is equivalent to asserting that H* (QX) (p) is torsion free.

245

§33: THE USE OF SECONDARY OPERATIONS

The purpose of this chapter is to explain just how we will use secondary operations in studying the cohomology of finite H-spaces. We will concentrate attention on the p odd case. First we discuss the most obvious question. On what elements can a particular secondary operation be defined? Then we discuss. in a general way, how we will Use secondary operations to analyse cohomology. We have already worked through one argument which we described as being a good introduction to future arguments involving secondary operations. This is the proof of Theorem 31-2. Essentially, this chapter is a discussion of how to generalize that argument. §33-1: Defining Secondary Operations The operations described in Theorem 32-1 are particularly well sui ted to the study of indecomposables. Let sition. 6p X

n

= [a.b .• 1 1

(X,~)

be a H-space. Given" a decompo-

the associated secondary operation

2n(X;IF) E H if we can find a A*(p)

can

be applied to

invariant sub Hopf algebra B C

P

H*(X;IF ) where p

(a) x # 0 in Q(H* (X;lFp)//B)

M*(x)

(b)

E B 0 B

(c) b.(x) E 1

Conditions (b) and (c) are,

BoB

of course,

explicit hypotheses in Theorem

32-1. But. as a pratical consideration we also want condition (a). It is really only indecomposables which are amenable to systematic study via secondary operations. Notably, the use of duality, as explained in §33-3, is available for indecomposables. Given an indecomposable

xE

Q~*(X;lF ) there is no problem finding a p

2n(X;lF representative x E H ) satisfying (a) and (b). Let p

B(q)

= the

sub Hopf algebra of H* (X;IF ) which is invariant under A* (p) p

246

The Homology of Hopf Spaces

and generated by

L

Hi(X;~)

i~q

P

Observe that ~B(q+1) C B(q) 0 B(q). The filtration 0= B(O) C B(l) C ... C B(q) C B(q+1) C... C H*(X:~ ) p

induces a filtration C F QH*(X:~ ) C... C QH*(X:~ ) q

p

p

Given an indecomposable -x € QH* (X:~ ) choose N so that -x

-X

p

FN+ 1QH* (X:~p)' Choose a representative x then (a) and (b) are satisfied. €

~

FNQH* (X;~ ) but p

B(N+1). If we let B

€

= B(N)

Condi tion (c) is the more subtle restriction. Verifying it requires more analysis. Indeed, it will only hold in special circumstances. We will restrict our attention to the case p odd. The p = 2 case requires even more careful arguments and we will delay our discussion until §37. We know from (b) that ~b. (x) € 1

criterion for showing that bi(x) €

B 0 B. The following is our basic

BoB.

PROPOSITION: Let p be odd. Let B C A be commutative associative Hopf algebras over

~

p

. Given a

€

A where

(i) lal ~ 0 mod 2p (ii)

I(a)

(iii) a

=0

€ B 0 B

in Q(A)

BoB

Then a €

Thus, in many cases, we need only show that b.(x) 1

=0

in QH*(X:~ ). This p

condition is easy to verify. In particular, we already know from Browder's structure theorems that, for finite H-spaces Q : Qeve~*(X:~ ) ~ Qod~*(X:~ ) is trivial s

for each s opn

=L

~

p

p

O. (see Corollary 12-2B) Hence, the operation associated to

(_l)spn-~(s)Q can always be used. This operation was used in the s

proof of Theorem 31-2. It is now a simple matter to extend Theorem 31-2 to any finite H-space. We will do so in §34-1. The next two lemmas play a crucial role in the proof of the proposition. Let D denote the decomposables. It follows from Theorem 1-5B

that

LEMMA A: Let p be odd and A a commutative associative Hopf algebra over ~

p

. Then peA) n DA

=0

in degrees

~

0 mod 2p.

247

VIII: The Module of lnaecompossbles OH* (X;IFp) p ODD

We also have LEMMA B: Let B C A be commutative associative Hopf algebras. Given a € A

=0

where A(a) € B ~ B then a € B if and only if a Proof: By Lemma 1-4B we can choose g: A

~

B such that

~: A ~ A ~ A ~ B ~ A//

B

o in A//

is an isomorphism of left B modules. If A(a) € B ~ B and a then g(a)

= g(a)~l.

B

So a € B. Q.E.D.

Proof of Proposition First of all we want to establish a € B. By Lemma B it suffices to show a

=0

By condition (ii) of the proposition we have a in A// B.

€

p(A// B).

D(A// By Lemma A a = O. B). Secondly, to show a is decomposable in B let B' be the sub Hopf algebra

By condition (iii) we have a

€

of B generated by elements of degree

B

<

lal. We want to show a

A

We have a € P( //B') C P( //B')' By Lemma B, a

= O.

=0

in B//B'

Q.E.D.

Remark: When p = 2 only a weakened version of the proposi tion holds. A difficulty arises from the fact that odd degree elements can have non zero squares. As a result Lemma A and the proposition are only true in degrees t 0 mod 2 rather than t 0 mod 4.

§33-2: The Case

~

In §37-2 we will return to this problem.

t

It is much more difficult to apply Theorem 32-1 when we are dealing

= L a.b. where ~ 1 1 = 1 case (notably,

with secondary operations associated with relations Dpn~

t 1. Even what were easy properties to verify in the

~

properties (a) and (b) of §33-1) become much more difficult to handle. For now, given x €

~-I~I(X;1Fp ),

we want a sub Hopf algebra B C H*(X;IF ) p

where ;:rex) € B ~ B and ~(x) f. B. As before there is no difficul ty in finding B C H*(X;IF ) so that ;:rex) € B ~ B and x f. B. But there is no guarantee that

p

~(x)

f. B. Consider the example of the mod 3 cohomology of

the exceptional Lie group E The structure of H* (E as a Hopf algebra S;1F3) S' over A*(3) will be analysed in §47-5, We have

248

The Homology of Hopf Spaces

Let x = x 15 and

=

~

-

= Q1' Since ~ (x

=

=

3

15)

s

Q1(x 15) x 20 P (xS) € B. There is one si tua t t on where the case

~(x)

dealing with a relation of the form opn~ that ;;:*(x) E B

@

we must let B = B(S). But

= ~ @X

~

~

I is palatable. If we are

= L a.(b.~) 1 1

then the requirement

B can be weakened to ;;:*~(x) E B ~ B. So we can apply

Theorem 32-1. For, as before, we can choose B where ;;:*~(x) € B ~(x)

@

B and

f. B. This is an important qualification. Lemmas 35-2C and 35-2D are

two cases where this weakened hypothesis comes into play. §33-3: The Study of QH*~): Induction and Duality Our major use of secondary operations will be to study the A*(p) structure of QH*(X;W ). In general, given a H-space (X.~), only the elements of p

H*(X:W ) which are primitive or can be made primitive by passing to a quop

tient Hopf algebra H* (X;Wp)//B (where B is A*(p) invariant!) are amenable to study by secondary operations. As we demonstrated in §33-1 the indecomposables of H*(X:W ) are such elements. p

(a) Induction Our study of QH*(X;W ) will usually involve induction, often several p

inductions at the same time. We will usually work by decreasing induction on degree t .e . we will assume that the result to be proved is true in degree> n and then prove it in degree n. Notice that the finiteness of H*(X;W ) gives us a place to ground such inductions! In degree n we will p

often use a second induction. As already remarked, the sub Hopf algebra 0

= B(O)

C B(l) C ... C B(q) C ... c H*(X:W ) induces a filtration p

0= FOQH* (X:Wp) C FIQH* (X;W C... C FqQH* (X:Wp) C... C QH* (X:Wp) p) We

WI'11

nH*(X· Ir assume t ha t t h e resu 1t to b e prove d is true f or FQ q '~p ) and

then prove it for F 1QnH*(X;IF). q+ p Actually, the entire framework of our approach to QH*(X;Wp ) could be described

as

being

inductive

in nature.

Our

structure

theorems

for

249

VIII: The Mooule of lnaecomposables QH* (X;IFp) p ODD

QH*(X:W ) are obtained in a series of increments where we take the given p

knowledge about the action of A*(p) on QH*(X;W ) and use secondary operap

tions to extend it. In this proceedure. Browder's results about the action of the Bockstein 6 on H*(X:W ) are the starting point. We have already p

noted in the discussion following Proposi tlon 33-1 that 60

evenH*(X;W p

)

=0

to the relation 6p n

the condi tion

enables us to define the secondary operation associated

= L (_1)spn-~(s)O s

on representatives for Q2~*(X:W ). p

By judicious USe of these operations we can accumulate a large amount of information about how the elements of Qeve~*(X;W ) are linked to each p

other via Steenrod operations. The work of Thomas from §17 plays a major role in this study. The results obtained are patterned on those of Thomas. Notably. as in his work, the action of A*(p) on QevenH*(X:W ) is related p

to the p-adic expansion of the degree in which we are working. All of the above represents the first stage in our study of QH*(X:W ) p

namely. how the elements of OH*(X:W ) are linked to each other via Steenrod operations.

p

In the second stage we use this information {p l us more

secondary operations) to determine how the elements of OoddH*(X:W ) are p

linked to those of OevenH*(X:W ) via Steenrod operations. notably via the p

elements {Os}. The structure theorems which imply the loop space conjecture are of this type. This conjecture is the final goal towards which all our inductive arguments are directed. When p involve other

=2

this second stage will

tools besides secondary operations. Morava K-theory will

also playa major role. Admittedly we will be using what amounts to secondary operations in Morava K-theory. (b) Duali ty Our study of QH*(X;W ) will also involve duality. Rather than considp

ering

QH* (X:W ) p

of A* (p) on

we will dualize and consider PH*(X:W ). The right action p

P~(X:Wp)

is related to the left action of A* (p) on QH*(X;Wp )

by the rule <~(x).a>

= <x,a~>

for any x € QH*(X;W ). a € PH (X;W ) and ~ € A*(p). It is convenient to p

*

P

work with PH*(X:W because our secondary operations are duals of pth p) powers. Namely «!Xx).aP> = <x,a>P modulo indeterminancy. So. secondary

250

The Homology of Hopf Spaces

operations wi 11 produce non zero p th powers unless the p th powers are "killed" by secondary operations.

This dichotomy gives us

a means

of

forcing A*(p) to act non trivially on PH~(X;W ). When H*(X;W ) is primip

~

p

tively generated (as in §31) this is straightforward. We are not allowed to have any non trivial pth powers in H*(X;W (see Corollary l-SC). So. p).

the identities

o = <¢(x) .aP> = <x0... 0x.a0 ... 00:>

+ [ i 1

(see §31-2) forces <1m a . a0 ... 00:> to for some i in order to cancel out i <x0... 0x.a0 ... 00:> to. Thus (a0 ... 00:)a to. By the Cartan formula. a~ t 0 i for some ~ as well. A detailed(!) analysis tells us just what ~ acts non trivially on a. We also want to deal with the situation where H*(X;W ) is not primip

tively generated. There is now no a priori reason why a P

=

0 for a €

PH*(X;W We must be more subtle in discovering reasons why pth powers p)' are trivial. Generally. we invoke our induction hypotheses. We use proofs by contradiction. Given a € P (X;W ). if "too many" elements of A*{p) 2nH* p act trivially on a then we can find a secondary operation so that <¢(x).aP> t

o. Hence. a P t o. By the Cartan formula "many" elements of

A*(p) must act trivially on a P , Our induction hypothesis now becomes involved. For we are assuming that certain specified Steenrod operations must act non trivially on PH (X;W ) in degree> 2n. And we have produced

*

p

an element on which they act trivially. This contradiction leads us to

assume that more elements of A*(p) than originally assumed act non trivially on a.

In particular a detailed (!) analysis yields

that certain

Steenrod operations must act non trivially on a. Nota Bene: In closing we should make a comment about the operation the relation bpn nancy 1m p

n.

= [ (_l)spn-~(s)Q s .

For. given a

€

¢ associated

We can always ignore the indetermi-

P H (X;W ) then we have 2n * p n <1m p . a0 ... 00:>

=0

This follows from the fact that A*(p) acts unstably on H*{X;W

=0

2· .-* on Q IH (X;W

p)

to

q

for q ~ i we have p

particular. we have (a0 ... 00:)p

n

= o.

=0

p).

on P for i 2 iH*(X;Wp)

Since pq ~

pq. In

251

§34: TIlE STRUCfURE OF Qever~t(X:1F }: PART I p

The next three chapters study the mod p cohomology of finite H-spaces for the p odd case.

In §34 we study how elements of QevenH*(X;1F ) are p

1inked via Steenrod operations.

In §35 we determine how the elements of

. Qodd H* (X;IF ) are lInked to those of Qeve~.* H (X:IF ) via Steenrod operations, p

p

notably via tha operations {Qs}' The loop space theorem for p odd arises out of these results. In §36 we return to the study of Qevel){*(X:1F ) and p

prove more extensive structure theorems. References for the work of this chapter are Kane [3J and Lin [5J. §34-1: Main Results Let (X,~) be a mod p finite H-space. Since OOevenH*(X:1Fp ) ~ 0. the action of A*(p) on Q leaves Qevel){*(X:1F ) invariant. In this chapter we p

prove some structure theorems about the Steenrod module structure of QevenH*(X:1F ). Our results are akin to those obtained by Thomas which p where described in §17. Namely, the action of A*(p) will be determined by the p-adic expansion of the degrees in which we are working. We will make s 1

1

extensive use of the -r(s) (~ Pp=1 ~ ps- +... +p+1)

function defined in

§31-2. We have the following positive result about the action of A*(p). THEOREM A: Let p be odd and let n

> -r(s)

if n

=-r(s)

(X,~)

be a mod p finite H-space. Given

mod pS and n

$

-r(s+1) mod ps+1 then

We can deduce from Theorem A the following negative result about the action of A*(p). THEOREM B: Let p be odd and let

(X.~)

be a mod p finite H-space. If n -

252

The Homology of Hopf Spaces

~(s)

mod p

2n.... k then Q tl (X;W ) n 1m P

s

p

=0

_ s unless k = 0 mod p

The relation between Theorems A and B is perhaps clearer i f we observe that the conclusion of Theorem B i

pP = 0 for

can

be reformulated as Q2~"(X;W ) p

n

1m

0 ~ i ~ s-l.

These two results give extensive information about the action of A"(p) on Qeve~"(X;W). Theorem 31-2.

In particular.

p

COROLLARY A: Le t p be odd and let Qeve~"(X'W ) 0

. P

we have

(X.~)

I

W A"(p) P

the

following

extension of

be a mod p finite H-space. Then

Q2~(s).

slO

The fact that every element of QevenH"(X'W • p ) can be traced back. via Steenrod operations.

to an element of degree

is a consequence of

2~(s)

Theorem A. The fact that the elements of degree

2~(s)

cannot be further

traced back is a consequence of Theorem B. As we explained in §33. we are very interested in knowing that a P

0

for elements of PH.. (X;W We have the following fact p)' COROLLARY B: If a E P2 2H (X;W ) then a P pn+" p

= O.

For, by Theorem B. Q2pn+~"(X;W ) n Im pI = O. Dually. apl P

By the Car tan formula. Theorem

A

tells

us

that

(aP)pl

2 Q2p n+2PH*(X;W ) p

=

=

0 if a E

O. On the other hand.

= plQ2p

2

+~*(X;W). p

P2p2n+2pH*(x;Wp) ~ P2p2+2H.. (X;Wp) is injective. Consequently, a P In §36 we will again study the Steenrod module structure of

So

pI:

= O.

Qeve~*(X;W). We will demonstrate that much more comprehensive results p

can be obtained about the action of A*(p) on Qeve~*(X;W ). We have isop

lated the above facts because they are all that we need to know about the Steenrod module structure of Qeve~"(X;W ) in order to prove the general p

structure theorem Qeve~*(X;W ) p

proved in q35.

§34-2: The Action of pI

=I

nll

~pnQ2n+lH*(X;W ). This result will be p

253

VIII: The Moaule of tnaecomposebles QH* (X;IFp) p ODD

Before proving the theorems of §34-1 we first study the action of pIon

Qeve~*(X;Wp ). The arguments of this section are a prototype for the proof of the theorems.

PROPOSITION A: Let p be odd and let $

p

mod p then Q2n* H (X;W ) p

be a mod p finite H-space. If n

= plQ2n-2p+~*(X;Wp ).

1 mod p then Q2~*(X;W )

PROPOSITION B: Let p be odd and

(X.~)

(X.~)

n 1m pI

be a mod p finite H-space. If n

=

= O.

Proof of Proposition A We prove the proposition by double induction. (I) We use downward induction on degree. Assume the proposition is true in degree

> 2n and that n

$ 1

mod p

(II) Consider the filtration (F QH*(X;W )} as in §33. Assume that the q

p

proposition is true in degree 2n for elements of FNQH* (X;Wp)' So. we want to prove the proposition for elements of F + H*(X;Wp)' N IQ2ll 2n H* (X;W Let B = B(N). As in §33-2 any -x E F has a representative x ~ N+ 1Q p) B(N+l) C H*(X;W ) and p

x i 0 in Q(H* (X;Wp)II

B)

;i.*(x) € B ~ B

Qs(x) €

BoB

for s i 0

It follows by the last two conditions that the operation ~ associated. as in Theorem 32-1. to the decomposition opn

=

on x and satisfies

in

p

~

H* (X;W )11

i=1

p

B'

We want to use (*) to force

II.

x( 1m pI

in Q( H

*(X ' ~ ) .Ir

x€

P II) B'

1m pl. If Dua I iZIng. '

such that <x.a> i 0

x(

1m pI then. by Assumption

we f Ind a

€

P(

H*(X W) * ; p liB)

The Hamology af Hopf Spaces

254

The latter restriction forces aP

k

I I k-l = ~P P =0

Cartan formula we then have (a0 ... 0a)pk n

$

I mod p means

n-~(s)

~

0 mod p for s

(a0 ... 0a)pn-~(s)

= 0 if k > O. It

=0

if s

. If k

$

0 mod p. By the

0 mod p. In particular,

$

follows that

>0

We also have n

=0

(a0 ... 0a)p since A*(p) acts unstably on H*(X;W

p)'

(see 333). These restrictions force

a P t O. For we have the identities

< cjl(x),aP >

< cjl(x)./-l(p)(a0 < /-l(p)cjl(x) .a0

0a) 0a

>

L 1m pn-~(s),a@ ... 0a > >

< x0

®X

< x0

®X,a0 •.. 0a

+

>

¢ 0 On

the other hand,

by Assumption

P H*(X;IF) n Ker pl. p 2pn contradiction.

So

1,

we must have a P = O. For a P €

the assumption that apl

=

0 has produced a

Proof of Proposition B This

is a

simple consequence of Proposi tion A.

For,

if

0 t

x

€

=

pI(x for xl € Q2P(k-1)+'1f*(X;lF then we can p) I) apply the proposition to obtain X. E Q2p(k-i)+2(i+I)H*(X;1F ) for 1 < ~ P 1 p1Ip Ip whe~e x. = P (x. 1)' However, x = (P ) (x ) contradicts the relation (P ) 1 1+ P =0

Q2pk+~*(X;lFp) and x

334-3: Proof of Theorems 34-IA and 34-IB The proof is by induction on s . When s = 0 Theorem A is Proposi tion 34-2A while Theorem B is vacuous. Now consider general s

>0

(i) Theorem B For Theorem B we must show that Q~*(X;IF ) The cases i

< s-1

p

n

1m pP

i

=0

for i ~ s-l.

follow directly from the induction hypothesis. When i

=

s-1 we deduce Theorem B just as we deduced Proposition B from Proposition

255

VIII: The Module of lnaecornposebtes QH* (X;lFp) p ODD

=a

A in §34-2. However, rather than using (pl)p

s-l p s-2 (pI pP", .. P ) and use the relation (P p )p

we work modulo the ideal

=a

mod (pl.pp, .... Pp

s-l

).

(ii) Theorem A Regarding Theorem A. suppose n

=

mod p

~(s)

s

but n

= ~(s+l)

mod p

~l

We proceed as we did in the proof of Proposition A in §34-2. We use the secondary operation

¢

associated to the relation opn

= L (_l)spn-~(s)Q

. s We assume Theorem A holds in the case s in degree> 2n as well as for the FNQ2nH*(X;lFp)

elements

C

Q2~*(X;1F ).

Let

p

* FN+lQH (X;lFp) we choose a representative x defined and

~<jl(x) in

*

p 0 H (X;lFp)//B'

If

i=l such that

x f. apq

= x0 ... 0x Irn pP

s

+

L 1m

€

a

~

i

Given

B(N) .

B

x

E

H*(X;IF ) such that <jl(x) is p

pn-~(s)

*

then there exists a E p(H (X;lFp)//B>"

< x.a > f. a = a unless q =a

mod ps+l

The second equality is the same as asserting that aP P cases

s~O

i

a

for i

~

s. The

s-l are a consequence of Theorem B being true in the case

~

s-l. The case i

=s

s

is a consequence of x f. 1m pP .

It now follows that

< /l(p)<jl(x).a0 ... 0a

< x0 ... 0x

However, a P f. q

a

= a mod pS+l.

is impossible. For a P In particular, aPpl

+

€

L

>

Im pn-~(s) .a0 ... 0a

P H (X;IF ) and aPpq 2pn * p

>

=a

O. By Proposition 34-2A. pI:

unless

256

§35: THE STRUCfURE OF QH* (X; IF ) p

In this chapter we study the operations Q : QoddH*(X;1F ) ~ s

p

Qeve~*(X;1F } for finite H-spaces. In particular. we prove that the loop p

space of a I-connected mod p finite H-space has no integral torsion for p odd. Our proof is a simplification of the proof given in Lin [5J. §35: The Loop Space Theorem This chapter will be devoted to proving THEOREM: (Lin) Let p be odd and let

(X.~)

be a I-connected mod p finite

H-space. Then Qeve~*(X;1F ) p

=[

n~1

lipnQ2n+l H*(X'IF) • p

As we explained in §29-6 this theorem implies COROLLARY A: Let p be odd and let

(X.~)

be a I-connected mod p finite

H-space. Then H*(QX} has no p torsion. This theorem implies that Qeve~*(X;1F ) = OQoddH*(X;IF). So i t follows from Proposition 13-4 that

p

p

COROLLARY B: H*(X) has only elementary p torsion. Further consequences of the theorem will be discussed in §§43.44.45.46 and 47. Notably. in §44 we will demonstrate that the theorem also implies that K*(X} has no p torsion.

In general. III (X)

= HI (X)

can have torsion of

arbitrary order. (e.g. H = Z/pk when X = PU(pk }}. So. the hypothesis 1(X) of X being I-connected is essential in Corollary B. On the other hand. the theorem holds in degree

~

3 when X is non simply connected. And Corollary

A also holds in the non simply connected case if

ax is replaced by its

identity component QOX. We now verify the theorem modulo the proof of two key propositions. Let

257

VIII: The Mooule of tnaecomposebles QH* (X,-IFp) p ODD

us make precise just what is to be proved. Let M be an unstable A*(p) module. Define A: M -----+ M

A(X)

=

[oPn(X) pn(x) = ~

if x E

i2n + 1 2n

ifxEM

That A(M) is invariant under the action of A*(p) follows from

=0

(i) OA(X)

=0

(ii) pqA(X)

unless q

By Corollary 34-1A Qeven is generated by to prove QevenH* (X;W ) p

prove

s~O

p

= op~(s)Q2~(s)+lH*(X;W

p

~

Q2~(s)H*(X:W). So, in order

[

p '" n 2n+ 1 * . . = ~ oP Q H (X;W ), It suffIces. by the above. to

Q2~(S+1)H*(X;W )

for each s

=0 mod p

)

1. This equality can be rephrased as Q2~(s+1)H*(X;W )

(*)

p

= Q Q2~(s)+lH*(X;W s

To derive this formula one uses the formula 6pn the fact

p

that Q2~(s+1) H* (X;W

34-1A).

p)

n 1m Pq

=

P

)

= [(_1)spn-~(s)Q s~O

0 for q

~

s

plus

1 (see Corollary

One proves (*) by decreasing induction on s. For large s the statement must be vacuous since we are dealing with finite H-spaces. So assume

To prove Q2~(s+1)H*(X;W ) p

PROPOSITION A;

= Q Q2~(s)+1H*(X;W s

Q2~(S+1)H*(X'W ) n • p

p

) we will show

ps Ker p C 1m Q

s

s+l PROPOSITION B: Q2p +~(s)H*(X;W) C 1m Q p s+1 They easily imply the desired result. Pick

=0

in Q then, by Proposition A. s

that pP (x) t O. By Proposition B

x E Q2~(s+1)H*(X;Wp ).

x E 1m Qs'

s

If pP (x)

Suppose. on the other hand,

258

The Homology of Hopf Spaces

s pP (x)

= Qs+l{y)

for some y

s

S

pP Q (y) - Q pP (y) s

= pP s

(In the last equality p P (y)

s

Then pP (Xl)

=

s Q

s

s

(y)

= 0 since Iyl < 2ps).Let x' = x - Qs (y)

0 and. so. as we have already shown,

x = Q s (y)

x'

= Q (Y'). Hence

s

+ Q (y')

s

The next section will be devoted to proving Proposition A and B. §35-2: Proof of Proposition A and B All

secondary operations will be of the type discussed in §34.

Our

first lemma contains Proposition A as a special case. 2 k+2 'Y {s )

LEMMA A: Q p

*

H (X;IF ) p

n Ker

k pP C Image Q for k ~ s ~ s

Proof: Fix k and s. Filter Q2 pk+2'Y{s)H* (X;IF ), as in §33-3. by the rule p

k

k

~ Q2p +2'Y(s)H*{X;lF » I mage { Q2p +2'Y{s){B{q » ~ p

Assume the lemma is true for FqQ2l+2'Y(S)H*{X;lFp) and pick k

n Ker pP

.

xE

Pick a representative x E B{q+1). Let B

;:i*{x) E B @ B

Let

¢ be

the operation associated to

liPP

k

+'Y{s)

= L

k k (_l)t pP +'Y{s)-r{t) + Q pP

O~t~s

s

This operation is defined on x. That is Qs(x) E k

pP (x) E

BoB BoB

To verify these we use Proposition 33-1. We have already observed in §33

VIII: The Module of Inaecomposables QH* (X;lFp) p ODD

that we have Qs (x) E

k that IpP (x) I

=

BoB.

k A similar argument gives pP (x) E

k 0 mod 2p and pP (x)

=0

259

BoB.

(Observe

in Q by hypothesis.)

We have the coproduct formula

~(p)qxx) If

k

= x0 ... @X

xe

+ Elm pP +~(s)+~(t) t<s

1m Q we can pick a E p«H* (X;lF p )/ / B)*) such s

< x,a )

;t

0

By Theorem 34-IB

< Image pq.a )

=0

unless q

=0 mod pS

Then

< /l(p)qxx) .a®

0a )

< x0 .•. @x.a®

But a P

;t

0a )

0 contradicts Corollary 34-IB.

Q.E.D.

We now set about proving Proposition B. The proof involves three lemmas. 2 k+2~(s) * H (X;IF ) In the first two lemmas we show Q p k

of all. we know from Theorem 34-IB that Q2p for i

< s.

=0

for k

Z s+2.

+2~(s)H*(X'1F , p) n

First .

p1 1m p

=0

Now we prove k

LEMMA B: Q2p +2~(s)H*(X'1F ) , p Proof:

p

We will

n

s

Image pP

k show Q2p +2~(s)

-

=0

2(p

s+1

for k

Z s+2

s -p )H*(X'IF ) • p

O. To simpify

notation let m

= p k-l -p s

Then we want to show Q2pm+~(s+I)H*(X;1F ) degree 2pm + of degree

p

~(s+l)

2~(t+l)

= O.

By Theorem 34-IB elements of

can be traced back via Steenrod operations to elements where t ) s . By the induction hypothesis from §35-1

260

The Homology of Hopf Spaces

invariant under A*(p) (see again §35-1) it follows that

opm+~(s)

= L

(_I)spm+~(s)-~(t)Q

t~O

plus the fact that

Pm+~ ( s ) -~ ( t )

€

t

1

P

P ,P , ... ,P

(

pS

) for all t

~

0

imply that

However. by Theorem 34-1B

Q2pm+2~(s+I)H*(X;Wp) n k

Q2p +2~(s)H*(X;W ) p

=0

Image ( p 1,pP , ... ,P

for k

ps)

=0

Q.E.D.

> s+2

-

2 k+s+2~(s) * Proof: By Theorem 34-1A Q p H (X;W ) P

= pP

s+k-l P s+k-2 p

pS 2~(s+l) * P Q H (X;W ). So Lemma C is equivalent to the assertion that pP

s+1

p

s pP Q2~(s+I)H*(X;W ) p

= O.

Pick the minimal k such that

s+k Ps+k-l s 2~(s+l) * p ... Pp Q H (X;W pP contradiction. By Lemma A

= O.

p)

We will assume k ~ 2 and produce a

k+s Q2p +2~(s)H*(X'W) C Image Q

s

, p

Pick x Qs(Y)

Qs(Y)

E

k+s Q2p +2~(S)H*(X;Wp)'

Pick representatives y and x =

in H*(X;W ) and a sub Hopf algebra B p

C

H*(X;W ) invariant under p

A*(p) such that

x (. B

~(x) E B @ B Let ~ be the operation associated to k+s +~(s)Q oPP

s

= L t;ts

This operation is defined on y. For

as in §33-1. So

k+s pP +~(s)-~(t)Q Q t

s

261

VIII: The MOdule of Indecomposables QH* (X,IFp) p ODD

~(p)¢{y)

= ~ ... 0x

k+s Image pP -~(s)-~{t)

L

+

t;.!s H*{X'IF) * Pick a € P{{ 'p liB) ) such that 34-1B plus Lemma B that, for k

< Image

< x,a > ;.! O. It follows from Theorem

2,

~

> = 0 unless

pq,a

q

=0

mod ps+1

Therefore

< Image

k+s

pP

+~(s)-~{t),a0 ... 0a

>=0

and so

< ¢{y) ,aP >

< ~(p)¢{y) ,a0 < ~ .. .0x,a0 < x,a

0a 0a

>

>

>p

;.! 0 But a P ;.! 0 contradicts Corollary 34-1B.

Q.E.D.

Remark: The reader should consult the end of §33-2 for a discussion of one of the subtler points of the above proof. The same remark applies to the next lemma. s+1 LEMMA 0: Q2p +2~{s)H*{X'IF) C Image Q ' p s Proof: It follows from Lemmas A and C that

Pick x

s+1 +~(s)H*{X'IF) C Image Q Q2p , p s s+l Q2p +2~{s)H*{X;IF). Pick representatives y and x

Q (y) € s

p

Qs{Y) in H*{X;IFp)' Pick B such that xl£

~(x) €

Let

¢ be oPP

B B I» B

the relation associated to the relation s+1

+~{s)Q

s

= L pP

s+1

This operation is defined on y pP

s+l

+~(s)-~{t){Q Q ) + (Q )(pp t s s+1 €

*

H {X;fp)II

s

pP

-p Q (y)

s

=

s+1

-p

s

B.

s -P Q ) s

In particular, observe that

{x,>

s+1 s s pP -p pP (z)

=0

s+1

The HomowgyofHopfSpaces

262

The second equality follows from Theorem 34-1A. The last equality follows from Theorem 34-1B plus the fact that pP

*

Pick a € p«H (X:Wp)//B)*) such that

< Image

pq.a

s+l

5-1 s s 1 P -P pP €(P.P ..... PP ).

< x.a > to.

> = 0 unless q

If x ( Image Q then we can assume that s < Image Qs.a

By Theorem 34-1A

=0 mod p

S

> =0

as well. We can then conclude using the coproduct formula

~(p)¢{x) that

= x0

< ¢{x),aP > = < x0

0x +

L Image

0x,a0 ... 0a

contradicts Corollary 34-1B.

Q.E.D.

pP

s+l

+~(s)-~(t)

> to.

+ Image

Q s

But the fact that a P t 0 then

263

§36: THE STRUcrURE OF Qevel1f*(X;1F ): PART II p

Theorems A and B of §35 have consequences for

the Steenrod module

structure of Qevel1f*(X;1F ). In this brief chapter we will show p

.' 2~.* THEOREM'(l) Q tl (X;lF p) ~

k

=0

unless n

i

(ii) Q2(p + ... +p + ... +P+l)H*(X;1F ) p

= pk +pk-l + ... +p i + ... +p+l = pP

i

~

k

Q2(p + ... +p

i+l

+ ... +p+l)H*(X;1F ) p

This theorem completely determines the action of A*(p) on QevenH*(X;1F ). p

The Steenrod module structure of Qevel1f*(X;1F ) described by the theorem p

can be represented as follows 2

pP

r--;l p +1)

(p+l

pP

21 (p +p+l

3

pP

pP

3

1

r--;l 2 p +p +1)

p +p+l

pP

2

1

(p 3 +p2+p+l

4

1

3

p +p +p+l ...

The lines indicate the non trivial action of A*(p) linking elements in the various degrees 2n in which Q2~.* tl (X;IF ) may be non zero. Thus the action p

of A*(p) on Qeven H*(X;1F ) is determined by the p-adic expansion of the dep

grees in which Qevel1f*(X;1F ) is non zero. Observe that the above result p

generalizes (and simplifies) Theorems A and B of §34. Given an integer n ~ 1 wi th p-adic expansion n = binary (with respect to p) if n p

sl

+p

s2

+ ... +p

sk

(sl

>

s2

s

>... >

=0

or

L n s pS

we say n is

for each s. In other words. n

=

sk)' Note that the integers in the above

theorem are binary. To prove the above theorem it suffices to prove PROPOSITION: Q2n H* (X;lF p)

=0

unless n is binary and n

=1 mod p

Observe, first of all. that part (ii) of the theorem follows from part (i) via Theorem A of §35-1. In turn, part (i) follows from the proposition. For part (i) of the theorem asserts that Q2n

=0

unless n is binary, n

=1

264

TheHomowgyofHopfSpac~

mod p. and n has at most one "gap" in its p-adic expansion. For example. p6+p4+ 1 is considered to have 4 gaps in its p-adre expansion while p3+ 1 has 2 gaps. The proposition gives the first two conditions. Regarding the third condition, suppose n is binary and n

=1 mod p but n has

two or more

gaps in its p-adic expansion. Then. by repeated use of Theorem 34-1A we k

can use Steenrod operations to link Q2nH*(X;1F ) with Q2p +2'T(s)H*(X;1F ) for some k and s where k

~

p

s+2. For example.

p

363 pP Q2(p +p +l)H*(X'1F ) • p

32<) 6 2 pP pP Q' lP +p +1)H*(X'1F ) , p

pp3pp2ppQ2(p6+p+1)H*(X;1F ) p

k

However. by Lemma 35-2C. Q2p +2'T(s)H*(X'1F ) • p

o

if k

~

s+2 .

It remains to prove the proposition. Proof of Proposition

< nand

Assume the proposition holds for Q2i H*(X'1F , p ) where i Q2n H* (X;lF p)

for some m

~

O. By Theorem 35-1 we have

1. So 2n

~

= 2pm+2 and

is no harm in assuming that m Q2nH*(X;1F ) p

~

=

it suffices to prove m is binary. There

'T(t) for any t

~

1. We can rewrite (*) as

[pm-'T(s)Q Q2m+1 H*(X;1F ) s p

s~1

Pick r such that pm-'T(r)Q Q2rn+1 H*(X;1F ) ~ O. Let q q Q2 H*(X;1F ) ~ 0 and q p

r

< n.

p

= r+pr.

then

By the induction hypothesis q is binary. Hence,

it suffices to show LEMMA: q

='T(r+1)

mod pr+1

Proof: We will prove, by induction, that. for s The fact that q

= 1 mod p

~

r. q

is given by Theorem 35-1. For any 0

have the series of implications q

='T(r+1)

='T(r+1)

mod pS

=> =>

='T(r+1) n ='T(r+1) m

mod pS mod ps+1

~

mod p s

~

s+1

q we

265

VIII: The Module of tnaecomposebles QH* (X;IFp) p ODD

q

=~(r+l)

mod ps+l

use

Theorem

34-lB

=> For

the

last

implication

we

Q2n H* (X;lF ) and Q2q H* (X;lF ) are p

p

linked by

Steenrod operation (namely pm-~(q)).

plus

the

fact

that

the non trivial action of a Q.E.D.

PART IX: TIlE MODULE OF INDECDMPOSABLES QII*(X:1F 2)

In the next few chapters we will be dealing with mod 2 finite H-spaces (X,).!) and deducing structure theorems for QH* (X:1F Much of what we do is 2). analogous to our study in previous sections of QH*(X:IF ) for p odd. As p

before our main goal is to prove the loop space theorem. By §29-6 the loop space

. even.,« theorem reduces to showfng Q 11 (X: IF2) = O.

The proof of

thi s

result takes up §38.§39 and §40. As before. the proof conveniently breaks up into two parts, First.

in §38 and §39. we study how the elements of

Qeve~*(X:1F2) are

to

linked

each

other

via Steenrod

operations.

resul ts obtained are modelled on those of Thomas from §17, Secondly. even..*

§40. we study how the elements of Q

11

The in

(X:1F are linked to those of 2)

odd * , Ai . Q H (X;1F V1a the Milnor elements {Sq }. Th1S corresponds to the study 2) odd * eveTLJ* . of the maps Q : Q H (X;IF ) ~ Q --H (X;IF ) for p odd 1n §35. However. s p p we introduce a new tool which was not used in the p odd case. We use Morava K-theory and.

in particular. we introduce the idea of "secondary

operations" in Morava K-theory.

even x

0, we turn to another topic (X;1F 2) which has no p odd analogue. In §41 and §42 we discuss Lin's analysis of Hav Ing demonstrated that Q

H

odd * H (X:1F are linked via Steenrod operations. The 2) resul ts are again patterned on Thomas. The techniques are secondary and

how elements

of

Q

even tertiary operations.

269

§37: THE USE OF SECONDARY OPERATIONS: p

2

In this chapter we discuss how we will be using secondary operations in the p = 2 case. All of our previous remarks in §33 apply. However. a number of extra comments are now in order. §37-1: ~1 Problem Many of our previous arguments for p odd have no p = 2 analogues. The obstruction to using the p odd arguments arises from the fact that. when p = 2. the Bockstein 0

Sql appears in the Gartan formula for pn

Sq2n. We

have already illustrated. at the end of §31-2. the existence of such a problem. We now consider another simple example of this obstruction. EXAMPLE: Suppose we want to show that there is no H-space

For

p odd we would

use

the

(X;~)

where

secondary operation associated to op

2

p2 0 + p1Ql' If such a H-space exists then we have

it¢(y) in H*(X;W ) 0 H*(X;W ). Since the operations {pn} act trivially on p p H*(X:W ) they act trivially on H*(X;W ) 0 H*(X;W ) as well. Thus p p p _

~ ¢(y) =

p-l 1

L ::
i=l P

.. yP-l 0 yl

0

7-

1

4p(X;W However. ¢(y) 7- 0 contradicts the fact that H ) = O. This argument fails when p 1

=

211

P

2. For since o(x) = Sql(x) 7- 0 we have

P (x0x) = Sq (x0x) = Sq (x) 0 Sq (x) = y0y 7- O. Consequently. the identi ty

270

The Homology of Hopf Spaces

could collapse to ~¢{y)

= O.

So. no contradiction.

It is easy to get around the problem of Sq1 in the above case (use the relation op

2

= p2 0

+ Q1P1)

However, if we attempt to deduce more detailed

results about the A*(2) structure of Qeve~*(X;IF2) the above type of problem is more intractable and much greater analysis is required. Hence the

=2

length of the proof of the loop space theorem for p

as opposed for p

odd. We will use many different techniques to overcome the problem associated with Sq1. We will make much more use of secondary operations as sociated to relations of the form Sq2n+l~

= L a.b. 1 1

instead of using the operation associated to Sq

¢

~ Sq2n+l-~(i)SqAi

2n+1

L

iLl

we

= L (Sq2n+1-~(i)Sql)SqAi

It is iLl is defined on a given x. But if we can

use the operation associated to Sq2n+1Sq1 now much harder to verify that

where ~ ~ 1. For example.

verify that ¢{x) is defined then the type of indeterminancy problem encountered in our above example will disappear. The difficulty in using operations associated to the relation Sq2n+1

= L Sq2n+1-~(i)SqAi

iLl the indeterminancy is too big. We have to worry about x0x

Sq

2n+1-~(i)

is that

=

even * odd * (a0b) where a b € Q H (X;IF or avb € Q H (X;IF If we 2). 2) i

use the relations Sq2n+1Sql =

L (Sq2n+1-~(i)SqI)SqAi

iLl effect.

ignore the possibility of

considering the possibility of

a b € i

Qodd H* (X;IF 2).

then we can.

in

For we are now

x0x = Sq2n+l-~(i)Sq1(a'0b')

where a'

€

. Qodd H* (X;IF and b' € Qeven..* tl (X;f (or VIce versa). By Corollary 12-2B. 2) 2) 1 even * odd * . . 1 , Sq : Q H (X;f -> Q H (X;f IS t r tv ta l , So Sq (b ) = 0 and we can 2) 2) reduce to the considering the possibility of x0x

=

Sq2n+1-~(i)(Sq1(a')0b'). The Sq1 problem also forces us to pay much closer attention to

the

algebra structure of H*(X;IF As with the p odd case we will usually 2). dualize and consider PH*(X;IF

2)

rather than QH* (X;f

2).

However, manyargu-

IX: The Moaute af lndecompossoles QH* (X;IF2)

271

ments are much more subtle than than they were in the p odd case. Consider the proof of Proposition 34-2A. We proceeded by downward induction on degree. We assumed that. in degree> 2n. pI: PH~(X;W ) ~ PH~(X;~ ) is inp

~

jective in degrees ap

l

= O.

$

2 mod 2p. Consequently. given a

we know from the Cartan formula that (aP)pl

p

~

€

P2nH*(X;~p)

= O.

Hence a P

where

= O.

So. when we used secondary operations to force a P 1- 0 we had a contradiction to the induction hypothesis. When p show that api 1

=0

implies (a2)pl

= O.

=2

we must work harder to

We have (a2)Sq2

= (aSq2 )a

+ a( aSq2)

I

+

(aSq )(aSq ) =0. For the last equality we use the hypothesis that aSq

o

plus the fact that [13."lJ = 0 for 13."1

I2-2C) So.

the algebra structure of

€

PoddH*(X;W2).

H*(X;~2)

important facts about the action of A* (2) on

2

(see Corollary

can be used

to

tell us

H*(X;~2)'

§37-2: Canonical Generators Another annoying,

though less crucial. problem in the p = 2 case is

that odd degree elements can have non zero squares. As a resul t., some of the Hopf algebra techniques used in the p odd case are not valid when p

=

2. Proposition 33-1 is among such results. On the other hand it is clear that if we hope to imitate any of the p odd arguments then we must have. if not Proposi tion 33-1.

the p

=2

then some equally effec tive version holding in

case. We can obtain such a version by passing from the module of

indecomposables QH* (X;~2) to generalized module of indecomposables. Let

) . where Dn+l are the n+l fold decomposables of H* X;~2( . In partIcular Q l

QH* (X;~2)' We will principally be working with Q2' Given x wi 11 use

{x) ~

for any s,t Qeven

~*

to

denote ~*

0 H

its

class

in Q2'

H

(X;~2)

~

1. We define Q C Q as follows 2 {x} € Q 2 { {x}

€

Q 2

(X;~2)

The

€

H* (X;~2) we

reduced coproduc t

induces maps

~{x} E QevenH*(X;~2) 0 QevenH*(X;~2) }

~{x}

€

=

QOddH*(X;~2) 0 Qeve~*(X;~2) }

--* I.l :

272

The Homology of Hopf Spaces

The action of A*(2) on H*(X;W

induces an action on Q and Q is invariant 2

2)

under this action. (We use the fact that Sql Qeven

= 0)

lies in the fact that every element of QH*(X;W

has a representative x E

*

H (X;W

2)

such that {x} E Q. In other words ~

PROPOSITION A: The canonical map Q

= Bodd

Proof: Pick a basis B of QH* (X;W

QH* (X;W 2)

2)

The usefulness of Q

2) ~

~

QH* (X;W

QH* (X;W

2).

2)

* (X;W

~QH

is surjective.

2)

U Beven of QH* (X;W

Given {x} E

in terms of B

~

G2

expand

Then B

2).

Jt{x}

~

B is a basis

E

B.

(i) x has even degree odd Given u,v E B then (*) u~ does not appear in (**) u~v appears in

J!{x}

J!{x} if and only if ~ appears in J!{x}

These restrictions follow from the fact that a a,~ E

2

= ~2

=[a,~]

=0

for all

PoddH*{X;W2). (see Corollary 12-2C) We can use them to rewrite x so

-*{x} E Qeve~.* that ~ H (X;W 2) appears in

~

QH* (X;W

2).

For, if

(and hence

u~

v~)

Jt{x} replace x by x - u'v' where u' and v' are any represen-

tatives in H* (X;W for u and v. 2) (ii) x has odd degree Given u E Beven and v E Bodd if u~ appears in

J!{x} replace x by x -

u'v' where u' and v' are representatives in H* (X;W

-* odd * fashion we can reduce to ~ {x} E Q H (X;W 2)

~

for u and v. In this 2) even * Q H (X;W2) Q.E.D.

We will use Q to verify the hypothesis of Theorem 32-1. In other words, given x E H*(X;W

2)

and a relation Sq2n+l

= L aib i,

we are interested in

finding an invariant sub Hopf algebra B C H* (X;W where 2) (a) x

(b)

t 0 in Q(H* (X;W 2)//B)

J!{x}

E B ~ B

273

IX: The Module of lnaecomposebtes QH* (X;IF 2)

As in §33-1 it is easy to find B

= B(q)

satisfying (a) and (b). We use the

following to determine when (c) can be verified. Given a commutative. associative Hopf algebra A we can define Q

= Q(A)

just as above.

PROPOSITION B: Let B C A be commutative associative Hopf algebras over W

2.

Given a € A where {a) € Q and (i) lal (ii)

0 mod 4

$

rea)

€

B @B

(iii) {a) E Qeven.Qeven in Q Then a E

B·B

The proof is analogous to that of Proposition 33-1. Indeed. in odd degree. we can use the same proof. In degree 4k+2. however. we replace the use of Lemmna 33-1A by the following lemma. Let 1 be the ideal generated by 2k elements of degree ~ 2k. It follows from Theorem 1-5B that a primitive of degree 4k+2 is either indecomposable or the square of an indecomposable of degree 2k+1. So we have 4k+2A LEMMA: Given a commutative associate Hopf algebra then P n

I2~ = O.

Proposition B is the key to defining secondary operations. It implies ~.~ PROPOSITION C: Let x E H* (X;W where {x) E Q. Let B C H (X;W be an 2) 2)

invariant sub Hopf algebra such that ~(x) E B @ B. Given ~ E A*(2) where (i) {~(x» (ii) deg then ~(x) €

E Qeven.Qeven

~(x) $

0 mod 4

B·B

The next two results are the key to the application of Proposition C. PROPOSITION D: Sq1Q2n

=0 ~odd

Proof: It is an easy coalgebra argument to deduce that Q

~

odd. Q IS in-

1 even.,» jective. So the lemma follows from the fact that Sq : Q tl (X;W 2)

~

274

The Homology of Hopf Spaces

Proof: First of all. we can reduce to considering Sq2(d) where d

Q40 is

€

decomposable. For. by our hypothesis. any x € H4o (X;W ) can be written 2

x

= Sq2 (y)

+ d

where y is indecomposable and d is decomposable. By Proposition A we can 2 ~ assume that y. and hence Sq (y). € Q. Moreover Sq2(x) = Sq2Sq2(y) + Sq2(d)

= Sql Sq2Sql(y)

+ Sq2(d)

= Sq2(d) For the last equality we use Proposition D. Secondly, it is an easy coalgebra argument to deduce that ~{d} eve~.*

Q

tl

eve~.*

(X;W2) ® Q .

1

tl

(X;W

even.,»

ments. SInce Sq : Q

tl

2)

forces d to be a product of ·even degree ele-

(X;W 2) 2

€

~

odd__* Q -H (X;W

~4k+2

the Cartan formula that Sq (d) € Q

.

2)

.

is triVIal it follows via

IS also a product of even degree

elements. Q.E.D. In S38 our first task will be to demonstrate that Q40* H (X;W Sq

2 40-2._*

2)

=

Q -H (X;1F actually holds for all n ~ 1. Consequently. the conclu2) sion of Proposition E always holds. Having established this fact we will

return in S38-3 to the discussion of secondary operations and amplify the remarks made here. S37-3: The Extended Module Of Indecomposables

In this last section of S37 we consider a refinement of Q C

G2.

It will

reappear in §39. Let

~ By Proposi tion 37-2D the action of A* (2) on Q induces an action on Q. So Q

has a natural Steenrod module structure. This Steenrod module structure

275

IX: The Module of Indecomposables QH* (X/IF2)

determines that of QH* (X;W map Q A

~

~ 2). For the map Q C

QH* (X;W2) induces a

~ ~

QH* (X;W2) which, by Proposition 37-2A, is surjective. More gener-

ally, we have the following exact sequence which explains our passing from Q to Q.

PROPOSITION A: The sequence 0

~

Q«(H* (X:W 2

»

~

exact in degrees t 0 mod 4.

Proof: Choose a Borel decomposition H* (X;W ) 2 Then {a i} U {aia j ¢ 0) is a hasis of to deduce that {a i} U {aia j ¢

01

Q.E.D. Exactness

Q

* H (X;W

2

)

fails

= O.

A

i}

U {a

2 ¢

01

in degrees ;: 0 mod 4. A4n

It follows that Q

=0

0 is

l

l

It is an easy coalgebra argument

~.

i

~

@ A. with generators {a.}.

i

deg ai' deg a j even} U {a i

is a basis of Q C Q2. Thus {a

4n

Q ~ QH* (X;W 2)

2

deg a

i

¢

01

deg a i odd}

odd} is a basis of Q.

For, in §38,

we will

prove

as well.

Our use of Q is a refinement of our use of Q as explained in §37-2. We use Q rather than Q because it is possible to write down more systematic structure theorems for the Steenrod module structure of Q rather than Q (the "junk" has been eliminated). Notably, we have the exact sequence from Proposition A. Although structure theorems for Q give structure theorems for QH* (X;W2)

= Q/Q«(H*(X;W » 2

it does not seem possible to study

QH* (X;W ) by itself. Rather we must work with the more general structure 2

*

of Q. As with QH (X;W ) we study Q via secondary operations and we use 2 inductive arguments on degree. These arguments would not work well if we A

attempted to study only QH*(X;W

2).

At each stage of the induction we need

to know that the inductive hypothesis holds for all of Q not just for

QH* (X;W ) . Otherwise, we do not have enough information about the action 2

of A* (2) on H* (X;W ) to handle the next series of secondary operations 2 which are used in the argument. In defining secondary operations the following Q version of Proposition B in §37-2 will be used

276

The Homology of Hopf Spaces

PROPOSITION B: Given x € H* (X;F where {x} € Q let B C H* (X;F be an 2) 2) A

invariant sub Hopf algebra where ;i*(x) € B ~ B. Let

I[f

€

A*(2) where (i) {\[I(x)}

=0

(ii) deg \[I(x) Then \[I(x) €

in Q

=0

mod 4

BoB

The definition of Q given here replaces the one given in Kane [20]. We have chosen this new definition only because it fits in more easily with the exposition of the book. In Kane [20] we defined

If we now denote this object by Q' then we have an inclusion Q' C Q. This A

induces a map Q'

A

~

Q. One can show that Q'

~

Q in degrees

We can also use Q to deduce results about PH* (X;F mutative diagram

with exact rows in degrees

=0

2).

=2 mod 4.

For we have a com-

mod 4. The exactness is a consequence of

Theorem 1-5A and Proposition A of this section. Theorem I-5A also tells us that p(rH* (X;F follows that

2

»

~

Q(rH * (X;F

PROPOSITION C: P(H*(X;F

2

2

» ~Q

» is injective i.n degrees

is injective in degrees

=0

=0

mod 4. It

mod 4

Consequently, any restriction on the Steenrod module structure of Q imposes restrictions on P(H*(X;F » as well. We will use Proposition C in 2

§40-5.

277

The next two chapters are devoted to studying how elements of QevenH* (X;1F

2)

. are l Inked to each other via Steenrod operations.

chapter we study QH* (X;1F

* QH (X;1F

2)

In this

proper. In §39 we' study the extended module of

~

indecomposables as defined in §37-3. This chapter is an expo2) sition of the work of Hubbuck-Kane [1] and Kane [10]. §38-1: Main Results In §38 we concentrate on the study of QH* (X;1F We prove THEOREM:Let The

rest

in degrees - 0 mod 4.

be a mod 2 finite H-space. Then Q4n* H (X;1F

(X.~)

of

2)

this

chapter

is

devoted

to

the proof

of

2)

=.0 for n>O

this

theorem.

Throughout this chapter we will make the following Assumption:

is a I-connected mod 2 finite H-space

(X.~)

Regarding the I-connectedness i t follows from Q4nH*(X;1F

2)

~ Q4nH*(X;1F

2)

the argument in §3 that

is injective when X is the universal covering

space of X. So, it suffices to prove the theorem for X.

§38-2: Action of Sq2 As an important preliminary to the theorem we prove PROPOSITION: Let 4n__*

Q-~

(X;1F

(X.~)

2)

be a I-connected mod 2 finite H-space. Then

2 4n-?-_* = Sq Q -H (X;1F

2)

for n > O.

The importance of this proposition is explained by Proposition E of §37-2. We use it to ensure that certain secondary operations are defined. These secondary operations are used in the proof of the main theorem.

278

The Homology of Hopf Spaces

Our proof will be inductive. We will proceed by downward induction on degree. Since QH* (X;W is a finite dimensional F vector space the pro2 2) position is trivially satisfied in large degrees. So we have a place to begin the induction. Assume that the proposition is true in degree> 4n. We can now apply the conclusion of Proposition 37-2£ in degree> 4n. We will use this fact plus Proposi t I on 37-2D in defining secondary operations. We will also be using the restrictions of Corollaries B and C of §12-2 in controlling the indeterminancy of the secondary operation. Case n

>2

Consider

the secondary operation in degree 4n associated with

the

factorization

Given an indecomposable FqQH* (X;W

2).

x€

Q~*(X;F2) suppose

x€

x f.

F + QH* (X;W ) , 2 q I

Assume that we have verified the proposition for FqQH* (X;F2).

Pick a representative x € B(q+l) and let B

= B(q).

Then ~(x)

€

B 0 B and

by the results of §37-2 I

--

Sq (x) € B'B Sq2Sq4n-2(x)

B'B

€

(Here we use n l 2!) It now follows that there is a secondary operation

¢

defined on x such that

~¢(x) = x0x + Im Sq4n + Im Sq2Sql in H* (X;W 2 ) / / B 0 H* (X;W2)//B' Hence

= x0x

~¢(x)

+ Im Sq2Sql

in Q(H* (X;W2)//B) 0 Q(H* (X;F2)//B)' (As explained at the end of §33-3 we can ignore 1m Sq4n since A*(2) acts unstably). It now follows that

x

€

1m

plus the fact that Sql: PoddH*(X;F 2)

~

Sq2 in QH*(X;F Otherwise. we could find a € P 4nH*(X;F2) where 2).

< x,a > f. < 1m

Sq2. a

0

>=0

The second restriction on a implies

< 1m For the hypothesis that aSq2

Sq2Sql. a0a

=0

>=0

279

IX: The Module of Indecomposables OH * (X;IF2)

PevenH*(X;W2) is

[(aSq2)~

+

trivial

(see Corollary

12-2B)

forces

(a0a)Sq2SqI

=

= o.

a0(aSq2)]SqI

It now follows that

< ¢ex) ,a2 >

(*)

< jl*¢ex),a0a > < x0x + Im Sq2Sql,a0a > < x0x,a0a > T- 0

We now have

< 1m Sq2,a 2 > = 0

(**)

This follows from the fact that (a2)Sq2

= (aS q l )2

= O.

= (aSq2)a

+ (aSql)(aSql) + a(aSq2)

For the last equality we are using the fact that a

oddH*(X;W2). (see Corollary C of §12-2) 80...* It follows from (*) and (**) that Q tl (X;W

2

= 0

for

a € P

is a contradiction to our induction hypothesis. Case n

2)

2 8n-2.-_*

T- Sq Q

-H (X;W

2). This

=1

This case is handled by the projective plane. Since H* (X;W2). is primitively generated in low degree, the structure of H*(P is well 2(X);1F2) behaved in low degree. We use the long exact sequence described in §15.

... ->

* H (X;W

2)

~ * ~ H (XIIX;1F

2)

X L * ~ ~ H*(P ;1F ~ H (X;1F 2) ~ ... 2(X) 2)

4(X;1F Given x € H where L(Y)

= x.

we know that x must be primitive. Pick y 2) We have

€

~(P2(X);1F2)

y2 = Sq5(y) = Sq4SqI(y) + Sq2SqlSq2(y) However, as we will see below. the following maps are trivial 6(P (i) Sql:~(P2(X);1F2) -> H 2(X);1F2) 7(p (ii) SqI:H

2(X);1F2)

It follows that y2

->

= O.

8(P H

2(X);1F2)

But this is impossible. For y2

= X(x0x).

So, by

= -~ (z) for some Z € H8 (X;1F2). As in the previous case this forces the existence of a € P so that 4H*(X;1F2)

exactness, x0x

< x,a > T- 0

< z.a2 > T-

0

280

The Homology of Hopf Spaces

a 2 € Ker Sq2

Thus z ( 1m Sq2 contradicting our induction hypothesis. Regarding the proof of (i) and (ii) we will only verify (ii). The proof of (i) is analogous but simpler. Before proving (ii) we record two facts which wi 11 be needed for the proof. We use Browder's resul ts from §12. Since X is I-connected it follows from Corollary 12-20 that X(2) is actually 2-connected. It follows that Hi(XAX)(2) = 0 for i < 6 [ H6(XAX)(2) is torsion free

(*)

(The second part follows from the first part via the universal coefficient theorem). It follows from (*) that for any coefficient group G i L: H ( p

(**)

2(X);G)

~ Hi-1(X;G) is injective for i ~ 7

To prove (ii) we will show p: H7(P Given x €

2(X);:l)

~ H7(P2(X);1F2) is surjective.

6 * H7 (P let y = L(X) € PH (X;1F 2(X);1F2) 2).

Bockstein spectral sequence analysing torsion in

Let

H* (X)(2)'.

{B r}

be

the

The discussion

in §14-2 can be easily modified to show that any even degree primitive of H* (X;1F

2)

eventually becomes a boundary in this spectral sequence. So, y

p(z) where z €

- (z) = O. So there exists Tor H6 (X)(2)" By (*) IL

7

= z.

H (P 2(X»(2) where L(W)

W

€

The commutative diagram L

*

--> H

(X)(2)

1p tells us that pew) and x both map to y. By (**) pew)

= x.

§38-3: The Use of Secondary Operations (Again) In §37-2 we discussed techniques for handling secondary operations. HavIng proved Proposi tion 38-2 we are now in a posi tion to make some further comments. We list the facts which will be used in the proof of Theorem 38-1. Let Q be as defined in §37-2. PROPOSITION A: Given x

€

~(X;1F2) where {x}

€

Q and an invariant sub Hopf

IX: The Module of tnaecomposebles QH* (XJF2)

algebra B C H* (X;W

4n(X;W

PROPOSITION B: Given x € H algebra B C H* (X;W 18

~ ~ (x) €

where

2)

2)

2)

-~

where

B·~

281

- if R t 2S B then SqR (x) €-B·B

where {x} € Q and an invariant sub Hopf (x) € B

~

R€ - if R t B then Sq2(x) B·B

The point is that SqR belongs to the ideal (Sq1) if R t 2S and Sq2R belongs to the ideal (Sq1,Sq2) if R t 18. And in §37-2 we treated the cases Sq1 and Sq2. We will use Propositions A and B to define secondary operations. The following facts extend those obtained in §12-2 and will be used to control indeterminancy. R

PROPOSITION C: (i) Sq

=0

(11 ) Sq2R

PROPOSITION D: (i) a (ii) a

=0

on P2k+ 1H*(X;W2) if R t 2S on P + if 2R t 18 4k 2H*(X;W2)

2

= ~2

[a,~]

2

=~

[a,~]

2

=0 =0

if

a,~

€

PoddH*(X;W2)

if a € P4 i + 2H*(X;W2)

and.~ €

P4 j +2H*(X;W2) In each proposition case (i) is Browder's result from §12-2. Case (ii) is based on Proposi tion 38-2. If we dualize Proposi tion 38-2 asserts that Sq2: P4kH*(X;W2) P4k+ 2H*(X:W2)

~ P4k_~*(X;W2)

~ P4kH*(X;W2)

is

is

injective.

trivial. (lise

It the

follows relation

that

Sq2:

Sq2Sq2

plus the fact that Sql: P + ~ P4k + 1H*(X;W2) is 4k 2H*(X;W2) trivial) This establishes case (ii) of Proposition C. Regarding (ii) of

Sq1Sq2Sq1

Proposition D consider, for example, a 2 . By Proposition 38-2 we must have (a2)Sq2 t 0 if a 2 t O.

But (a2)Sq2

O. (For the last equality aSq2

=0

=

(aSq2)a + (aSq1)(aSq1) + a{ aSq2)

by Proposition C(ii) while (aSq1)2

=0

by Proposition D(i» §38-4: Proof of Theorem 38-1

We now prove that

Q~*(X;W2) = 0 for n l 1 for any mod 2 finite H-

space. The proof is fairly complicated. In particular,

it involves the

consideration of a number of special cases. However, the proof is moti-

The Hornoloqy of Hopf Spaces

282

vated by a simple idea. The idea is to try to force Q~*(X;W2) to be in A the image of Sq t where t grows larger and larger. Eventually, we then

4n * At have Q H (X;W being hit, under Sq , by elements of negative degree. So 2) Q4n* H (X;W = O. 2) Unfortunately, this programme does not quite work out and we are forced to make modifications which considerably complicate the argument. Where do the difficulties come from? To make the above programme work we have to use operations associated to relations of the form Sq4n+1~

= L a.b. 1 1

where

t 1. We have already discussed in §33 the type of difficulties which can arise when we attempt to deal wi th the ~ t 1 case. Notably, given x E ~

H4n (X;W and y E H4n-I~1 (X;W where 2) 2)

~(y)

= x,

the problem is to find an

invariant sub Hopf algebra B where x ( B but ~(y) E B 0 B. To overcome our difficulties we end up introducing secondary operations based on more complicated factorizations. We consider the form Sq4n+1~

= L a.b. 1 1

where ~

=

factorizations of

A 2A 2A At l,Sq t,Sq s or Sq s Sq . We end up

*

A

seeing very little of Q~ (X;W being hit by Sq t where t ~ 00. Rather 2) 4n* we choose the minimal n such that P (and hence Q H (X;W » t O. 2 4nH*(X;W2)

2q

Given 0 t a E P we prove a to for all q ~ 1. This, of course, 4nH*(X;W2) contradicts the finiteness of X. The observant reader, however. will detect many traces of the initially suggested approach in this new proof. So pick a as above and assume w

= a2

q

t O. We want to show that w2 t O.

Part I: Choice of B and x 2 To prove w t 0 we use secondary operations. As usual we need to choose an invariant sub Hopf algebra B C H* (X;W as well as x E H4n (X;W on 2) 2) which the operation is defined. In the cases when x = ~(y) the choice must be made so that ~(y) E B 0 B but x ( B. First we observe LEMMA A: Any element of A*(2)

which acts non trivially on w can be

written as a sum of the elements {Sq Proof; We prove by induction on q that w

2As

a

2

q

Sq

At

}.

satisfies

283

IX: The Module of lnaecomposebles QH* (X;IF2)

(*)

wSqR

=0

> 0 and IRI

if IRI

=0 mod 4

The case q = 1 follows from the fact that P = 0 for i 4 iH*(X;1F2) Suppose that the result is true for w IRI

=

= a2

q

Pick R such that IRI

2)SqR 0 mod 4. By the Cartan formula (w

be written

can

< n.

>0

and

as a sum of

the elements

R

R

[wSq 1,wSq 2]

where R + R 2 1

(wSqS)2

where R

= 28

By the induction hypothesis IR I ~ 0 mod 4 and 1

R

R

Proposition 38-3D, [wSq 1,wSq 2]

=R IR21 ~ 0 mod 4. Then, by

= (wSqS)2 = O.

Next we show (**) any element of A*(2) which acts non trivially on PevenH*(X;1F2) can

be written as a sum of the elements {Sq2RSq Ai} . It suffices to consider the hasis elements {SqR}. Supopose the statement is true for elements of degree Otherwise, we are done. Suppose r SqR

= Sq

i

=1 mod 2.

< d.

IRI

d and

We can assume R ~ 28.

Then

A. R-A. R. A. lSq 1 = I Sq JSq J R. A. Suppose Sq JSq J ~ 0 on PevenH*(X;1F2)'

By

R.

Proposition 38-3C(i) IR.I is even. By the induction hypothesis Sq J cml be J

written as a sum I Sq

2~

.

Finally, we have (***)

If wSq2R ~ 0 then R

= Aj

for some j

The proof is illlalogous to (**). By (*) R 2R wSq

2Ai 2R-2A. Sq J

= wSq

~

28. Suppose r

=I

2R. 2A. wSq JSq J

i

- 1 mod 2. Then

2R. 2A j Suppose wSq JSq ~ O. By Proposition 38-3C IRjl is even. By (*) IR.I

o

2R. and so Sq J

J

1. Q.E.D.

Fix s and t by the rule that Sq

2A A SSq t is an element of maximal degree

284

The Homology of Hopf Spaces

such that wSq

2A A SSq t t O. Moreover, make the choice so that t is the maxi-

mal possible integer. It then follows that wSq

2A A SSq t

= wSq

A 2A tSq s

(a) Choice of B Let

m

= 4n

- 2 s+ 1 - 2

t

+ 2

B = B(m) Then Lemma B:

< B,w > = 0 < u,w > t 0 we must have u indecomposable. But then where v has degree ~ m. So < u,w > = < ~(v),w > = < v,w~ > = O.

Proof:lf u E B(m) and u

= ~(v)

The last equali ty follows from the fact that

has degree greater than

~

2A A Sq SSq t. Q.E.D. (b) Choice of x and y We have already mentioned that we will only be using secondary opera-

I a.b. where ~ 1 1

tions coming from relations of the form Sq4n+l~ =

A 2A A Sq t or Sq SSq t. In each case we need to choose x

2A s

4n

E H

(X;W on 2) m+l (X;W ) where secondary operation. Choose z E H 2

which to define our 2A

= l,Sq

A

~(y)

< z,wSq SSq t > t O. Let Sq Then

< x,w > t

choice of

~.

O. So, by Lemma B, x

~

At

Sq

2As

(z)

B. The choice of y depends on our

The following table gives the possibilities

In each case x

~

Sq

~

Sq

~

= Sq

= ~(y).

2A A SSq t 2A s At

Y

z

Y

A Sq t(z)

y

= Sq

2A s(y)

And ~(y) E B @ B!

Part II: Comments about Secondary Operations For each relation Sq4n+l~

=I

a.b. which we employ in Part III during 1

1

the proof of the theorem it will follow from Propositions A and B of §38-3

285

IX: The Module of Indecomposables QH* (X,'IF 2)

B·B.

that bi(y) €

So. in each application of a secondary operation, only

our treatment of the indeterminancy

L 1m a i

il*
+

in the coproduct formula

L

1m a

i

requires further comment. In each application we will show (W®W)a Thus ( 1m ai,W®W

>=0

and so

(
i

O.

(il*

>

( x0x + L 1m a. ,W®W ( x0x,W®W

>

1

>

¢ 0

2 Therefore we have forced w ¢ 0 as desired. We also remark that, in each

= O.

application, we always have (W®W)Sq4n

H* (X;f2 ) forces wSql.

=0

if

For A*(2) acting unstably on

~ 2n.

Part III: Proof of Theorem Our strategy is to divide and conquer. Define the integers k and R by 2A 2A. wSq k ¢ 0 while wSq 1

wSq Obviously, 1 (a)

t

(

Ae

A.

=0

=0

¢ 0 while wSq 1

for i for i

>k

>e

~ t ~ R. We want to show w2 ¢ O. We

~ s ~ k and

can assume

R-1

First we must have R

~

2. for the secondary operation associated to

Sq4n+1 = Sq4n(SqI) + SqOI(Sq4n-2) can be used to force w2

¢

0 when w €

Ker Sq0l. Secondly,

the operation

associated to

A

Sq4n+1Sq t 2

= Sq4n(Sq1Sq

can be used to show w ¢

A

A

t

t) + Sq t+1(Sq2Sq4n-2 +1)

° when 2 ~ t ~ R.

It follows from (a) that (b) R-I

<s

- k

First of all, Sq

Ae

e-1

~

s. For s

~

e-2

(plus t

~

e-1

from (a»

implies that

2A A is of larger degree than Sq SSq t. But any element from A*(2) of

degree larger than Sq

2As At Sq acts trivially on w. Secondly, s

k (plus the now established fact that t

~

s) implies that Sq

= k.

2A k

For s (

is of lar-

286

The Homology of Hopf Spaces

2A A than Sq SSq t

ger degree

This again contradicts

the maximal i ty

of

2A A Sq SSq t We can strengthen (a) and (b) to the inequalities (c) 2


1 - s - k

First of all. i-I

= s.

For. if i-I

<s

we can use the operation associ-

ated to

(which holds in degree 4o_2 s+1_1) to deduce w2 # 0. Observe that since s+1

> k,i we have A

A

(ufu)Sq s+2 (ufu)Sq

A

[wSq s+2 0 w] + [w 0 wSq s+2]

2A A A s+1 = [wSq s+1 0 w] + [wSq s+1

o wSq

=°

A 2A s+1] + [w 0 wSq s+l]

=

°

(d) As our final step we eliminate the case given in {c}. We use the secondary operation associated to the relation

= Sq4o(SqlSq

2A A SSq t)

+ Sq

2A 2A s+1 t -2 -2) s+ISq t(SqlSq0lSq40-2

+ Sq

2A 2A s+1 t -2 -2) s+ISq1(Sq tSq01 Sq4o-2

A s+1 A -3Sq t) + Sq s+2Sq01(Sq40-2 (which holds in degree 4o_2 s+1_2t+5). The restrictions in (c) force (ufu)Sq

2A 2A s+1Sq t

A A 2A (wSq 8+1 0 wSq s+1)Sq t

= wSq

(since s+1

> k)

A 2A A s+ISq t @ wSq s+1 A + wSq s+1

@

A 2A wSq s+ISq t

=°

A

The last equality follows from the fact that Sq s+1Sq

2A

(by 38-3C)

t has larger degree

2A At than Sq SSq . Finally (ufu)Sq

2A s+1Sql

2A

= (wSq

s+1 0 w + w @ wSq

2A s+1)Sql

(by 38-3C)

287

IX: The Module of Inaecomposables QH* (X;IF2)

=

and

°

(since s+1 ) k)

A

(wSq s+2 0

=° 2

We conclude that w t 0.

A

w + w 0 wSq s+2)Sq01 (since s+2 ) R)

288

In order to continue our study of QH*(X;W for mod 2 finite H-spaces 2) ~

we work with the more general structure Q

= QH* (X;W2). ~

the extended module

of indecomposables as described in §37-3. The structure theorems obtained for Q will also apply to QH* (X;W work

2).

However. our arguments force us to

wi th Q. We have indicated our reasons in §37-3. The work in this

section is based on Kane [20] as well as Lin [8]. One major alteration has been introduced. As we explained in §37-3 the definition of Q differs from that given in Kane [20]. §39-1: Main Results . . . even

In this chapter we determine how the elements of Q .

*'

H (X;W

2)

and

hence.

the elements of QevenH*(X;W are linked together via Steenrod 2). operations. The resul ts of this chapter correspond to those obtained in §34 for Qeve~*(X;W ) when p is odd. As there our results are modelled on p

those of Thomas from §17. The action of A*(2) is related to the 2-adic expansion of the degree in which we are working. More particularly. the action of A*(2) in a given degree n is related to the "gaps" in the 2-adic expansion of n. We have two main results: one positive and one negative. First we have our positive result. THEOREM A: Let

(X.~)

be a mod 2 finite H-space. Given k.s

~2s+1k+2s_2

Q

C

Im Sq

2s

+ Im Sq

~

1 then

2 sk

We can then deduce from Theorem A the following negative result. THEOREM B: Let

(X.~)

be a mod 2 finite H-space. Given k.s Sq

2s~2s+1k+2s_2 Q

=0

~

1 then

289

IX: The Module of tndecomposebles QH* (X;1F2)

These theorems have consequences which we will need in the next chapter. It is a simple consequence of Theorem A that COROLLARY A: Let (X.~) be a mod 2 finite H-space. Then Qeven is generated.

*

as an A (2) module, by

i_2

L QA2

.

i~2

The next result is just a reformulation of Theorem B. COROLLARY B: Let (X,~) be a mod 2 finite H-space. Let deg SqR be even.

For Theorem B is equivalent to asserting that sk+2 s_2 i Q2 n 1m Sq2

=0

Since Sq1 Qeven

=0

~

for 1

1

I

2

$ 0 mod 2 s for some r in R i i As a consequence of Corollary B we have

r

COROLLARY C: Let 2 s +1

~Q

s-1

sk+2s_2 we also know that Q2 n 1m (Sq1)2

lary now follows. For the ideal (Sq .Sq •.... Sq ments {SqR

~

i

(X.~)

2 s- 1

(r

= O.

The corol-

) has as basis the ele-

1,r2 · · · .)}.

Q2s-1H*(X;~2)

be a mod 2 finite H-space. Then Sqa S

z..

--H*(X;~2) is trivial.

a

Sq

One uses the relation Sq s B plus the fact that Sq

2 s_1

2s_1

+

L Sq

2s_2i

Sq

a.

1

and applies Corollary

1~i<s

2 s_1 * : Q H (X;~2)

~

2s+1_~_* Q -H (X;~2) is trivial.

The above results can be applied to PH* (X;~2)' For. as we demonstrated

in Proposition B of §37-3. PH* (X;~2)

~

A Q is injective in degrees

$

0 mod

4. Notably. Theorem B holds for PH* (X;~2)' COROLLARY D: Let

(X.~)

be a mod 2 finite H-space. Given k.s s

Sq2 p2

s+1

s

k+2 -~*(X;~2)

~

1 then

=0

The above resul ts are all that we wi 11 need in proving the loop space theorem in §40. Consequently. Theorems A and B are all that we will prove.

290

The Homology of Hopf Spaces

However. stronger results can be obtained. In Kane [20J it is proved that TIIEOREM C:

Let (X,Il) be a mod 2 finite H-space.

(i) If 2 (ii) If 2

s (2n s and 2

< 2n

then Q2n

s ( 2n 2 SA2n then Sq Q

Sq

2 SA2n_2s Q

=0

§39-2: The Use of Secondary Operations (Once Again) We have already discussed the use of secondary operations in §37-2 and §38-3. We now state some further facts which will be required in proving Theorems A and B. The theorems will be proved simultaneously. For they are interdependent. Not only does Theorem A imply Theorem B but, also. Theorem B is required in the proof of Theorem A. We use

it to handle certain

secondary operations which are used in the proof of Theorem A. It is used to define secondary operations and ,in its reformulation as Corollary B. it is used to analyse the indeterminancy of the secondary operations. We will proceed by induction on s. The case s established. For since

Q~*(X;W2)

=0

=

1 has already been

(see §38) it follows that Q4n

=0

as well (see the exact sequence Of §37-3). The proof for the general case s will, of course, involve more secondary operations. (A) Relations in A*~ In making our arguments we will have need of a number of relations in A*(2). For any s,t ~ 1 we have Sq2t+l

(R-l)

L

Sq2t-2

i+2SqA

i + SqASSq2t-2s+2

l~i~s-1

In particular, if we choose s large then the final term is trivial (Sqi

o

for

< 0)

and, so, we have

Sq 2 t+ 1

(R-2)

i 2 A. Sq2 t-2 + Sq 1

L i~1

. Slnce Sq

2 s+ 1_1

Sq

2 s+ 1k_2s

0 we can modify (D) in the case 2t

= 2 S+1k+2 s-2

to obtain (R-3)

L Sq2s+1k+2s_2iSqAi

+ (SqAs+ 1 + Sq2s+1_1)Sq2s+1k_2s

l~i~s

There are some cruder relations which we wi 11 also have need of.

Let

IX: The Module of Indecomposables QH* (X,-IF2)

291

(~l""~m) be the two sided ideal of A*(2) generated by ~1' ···.~m· (R-4)

s s s-l i Sq2 Sq2 k == 0 mod L (Sq2)2

for k odd

i=O

(R-5)

s s s s s-1 i Sq2 Sq2 k == Sq2 k+2 mod E (Sq2)2

for k even

i=O

(B) Defining Secondary Operations To define the secondary operations we wi 11 use extensions of the results from §38-3. Since we are assuming that Theorem B holds for s

<s

we

have the following analogue of Propositions A and B of §38-3. __2 S+ 1k+2s_2 (X;W PROPOSITION A: Given x E H-

*

~

2)

where {x} E Q and an invariant

sub Hopf algebra B C H (X;W where 2) E

B'B

_

~

R

(x) E B @ B then Sq (x)

if

(i) deg SqR(x) t 0 mod 4 (ii) IRI

to

mod 2

s

We will also be forced to deal with cases where deg SqR(x) == 0 mod 4. These cases will be dealt with in a more ad hoc manner. In all such cases.

*

given Be H (X;W

2)

where

_

~

The difficulty is to force SqR(x) E

=0

R

(x) € B @ B. it is easy to force Sq (x) E B.

B·B.

The solution will be to show Q(B)

for the degrees in question.

(C) The Indeterminancy of Secondary Operations Regarding the indeterminancy of our secondary operations we wi 11 use the following restrictions on the A*(2) action to control indeterminancy. Since we are assuming that Corollary B holds for s'

<s

we have

PROPOSITION B: Given a € P H*(X;W and SqR of even degree then 2) 2 sk+2s_2 s aSqR = 0 unless r == 0 mod 2 for each r in R = (r 1.r2 .... ). i i We should also mention that Theorem 38-1 dualizes to assert that

PROPOSITION C: a 2

=0

for all a € PevenH~(X;W2) ~

292

The Homology of Hopf Spaces

This means that a force a

2

2

¢ 0 is an automatic contradiction. We no longer have to

q

¢ 0 for all q

I (as in the proof of Theorem 38-1) before ob-

~

taining a contradiction. §39-3: Divisions in the Proof of Theorem A Suppose Theorems A and B hold for 1

s. The proof of theorem A in the case k' three cases.

A s+l s E Q2 k+2 -2.

Pick {x}

<s

s'

~

= k,

and for k' ) k when s

s

We will

=s

will be divided into

consider

the

following

possibi l t ties: Case I

~Q}

{x} E 1m { Q(rH*(X;F2»

2 and {x} ¢ 0 in QH* (X;F

2)

and {x} ¢ 0 in QH* (X;F

2)

Case II

k

~

Case III

k

=1

By the exact sequence 0

~

Q(rH* (X;F 2»

~

A Q

~

QH* (X;IF2)

~

0 of §37-3

these cases include all possibilities. §39-4: Proof of Theorem A: Case I Given {x} E 1m { Q(rH* (X;F 2» 2 A2sk+2s- I _1 {y} where {y} E Q . So

~

Q } of degree 2 s+l k+2 s -2 we have {x}

Ass s-l (Sq s + Sq2 -1)Sq2 (k-1)+2 {y}

The second identity is relation (R-3) of §39-2. The third identity is obtained by applying Corollary B (for s' s

Sq2 k+s

s-l

i

A

-2 Sq i{y}

and then applying Corollary B (for s'

= [Sq l~i~s-l

2 s_2 i

Sq

Ai

to show

< s) to show

=0

(1 ~ i ~ s-2)

A 2s-1 < s) plus the identity Sq s + Sq

IX: The Module of Indecomposables QH* (X;IF2)

Ass s-l (Sq s + Sq2 -1}Sq2 (k-1}+2 {y}

293

=0

§39-5: Proof of Theorem A: Case II Given an indecomposable

-x

~

-

x E Q2s+1k+2s_~.* -H (X;W 2)

2)

Fq+IQH* (X;W2 ) . Assume also that Theorem A holds for FqQH* (X;W2) in the

case of our fixed s. Pick a representative x

~

_ * suppose x E FqQH (X;W

(x)

E

B @ B. We will also assume that {x}

B(q+1) and let B

E

Q ~

E

C ~.

= B(q).

So

The secondary opera-

tion associated to relation (R-3) of §39-2 is defined on x. In other words

A.

BoB

LEMMA A: (i) Sq lex) E (11) Sq

2 s+ 1k_2s

(x) E

BoB

Proof: Only (ii) needs comment. We want to show

..

( ) (Here (Sq (for s'

2

i

i

) is the ideal generated by Sq 2 ) For it follows from Theorem B

< s) plus the fact that Sq1Qeven

i

=0

that the elements of (Sq2 )2

acts trivially on {x} E Q. By relation (R-4) of §39-2 we have mod

s-l

L

i

(Sq2)2

i=O

(Here we use k

~

Theorem B (for s'

2 s+ 1 (k-1)

2!). Observe Sq

=s

and k' Sq

2s

Sq

> k)

{x}

E

Q~2s+ 1 (2k-1 )+2 s_2

So. by

we have

2 s + 1 (k_1)

{x}

=0

Q.E.D.

So ¢{x) is defined and we have the coproduct formula ;t+¢{x) =

x@x

+

L

Im Sq 2

s +1k+2 s _2 i

A 1 2 s+ 1_1 + Im (Sq s+ = Sq )

l~i~s

H*(X'W) * Pick a E P« '2 //B) ) C PH.. (X;W that. for any such a. we have aSq2

s

2)

where

< x,a > #

# 0 or aSq

2

sk #

o.

opposite and use the above coproduct formula to force a tion to 39-2C. It suffices to show

O. We want to show We will assume the 2

# O. a contradic-

294

The Homology of Hopf Spaces

A s+l s+l + Sq2 -l,a0a

LEMMA B:

< 1m Sq

LEMMA C:

< 1m Sq2

s+l

s i k+2 -2 ,a0a

>

0

> = 0 for 1 ~ i ~ s.

For then, by the above coproduct formula. we have

< ¢(x) .a2 > < ¢(x) ,il*(a0a) >

< jt¢(x).a0a > < x0x.a0a > "F-

0

Proof of Lemma B First of all. by relation (R-3) of §39-2. we have A s+l (a0a)(Sq s+l + Sq2 -1)

= L

(a0a)Sq2

s+l

-2

i

l~i~s

On

the other hand, for each 1 (a0a)Sq2

s+l

~

~

i A. -2 Sq 1

s. we have

L 2u+2v

(aSq2u

@

aSq2v)SqAi

= 2 s+ 1_2i

o

The first identi ty follows from the Cartan formula plus the fact

that

A.

Sq 1: PoddH*(X;~2) ~ PevenH*(X;~2) is trivial. The second identity follows from LEMMA D: aSq2i

=0

unless 2i

=0 mod 2s+ 1

This lemma is a consequence of Theorem B (for s that aSq

2

s

< s) plus our assumption

=0.

Proof of Lemma C Expand

L u+v

aSqU

@

aSqV

2 s+ 1k+2s_2i

(i) u and v even By Lemma D we can reduce to u we must have i

= s,

u

= 2 s+ 1k'

=v =0 and v

mod 2 s+ 1. Since u+v

= 2 s+ 1k"

where k

A*(2) acts unstably we can eliminate all cases except k This case can be eliminated since we are assuming aSq 2

sk

= 2 s+ 1k+2s_2i

= k'

= 21, = O.

+ k". Since k'

= k" = 1.

295

IX; The Module of tnaecompossbles OH* (XJF 2)

(ti) u and v odd

Since u+v ~ 2 Suppose u

s +1

k and aSq

2 sk

o

it follows that either u or v

> 2 sk.

> 2 s k and write u

= 2 sk

+ q

where

0

< q < 2 s_1

To derive the upper bound on q we use the fact that A*(2) acts unstably on

*

H (X;f

(a) q

2).

. showaSqu We wIll

=0

for each value of q.

= 2 r_1

We have the relation

By Lemma D plus the assumption that aSq

2 s + 1k

= 0 it follows

that every

term in the right hand side is trivial. In particular, we must expand

to make use of Lemma D. (b) q

= 2r -1

The argument is similar to the above. We use relation (R-2) of §39-2 plus Lemma D. §39-6:Proof of Theorem A: Case III As before.

-x

* (X;f

F q+1QH * FqQH (X;f 2) €

given an

indecomposable

( FqQH* (X;f Assume that Theorem A holds for 2). .i n degree 2 s+l +2 s -2. Pick a representative x € B(q+1) where 2).

-x

{x) € Q C Q2' This time, we let

B

= the

sub Hopf algebra generated by B(q) and

CH* (X;f2)

.. * --* H*(X;f ) So B is InvarIant under A (2) and M (x) € P( 2 lIB)' To prove x € PH*(X;f2) where

1m Sq

2

s

we again dualize. Pick a

€

< x,a > ¢ O. We want to show that, for

p«H* (X;f 2) ~

B) * ) C

such a. we have

296

aSq

o

The Homology of Hopf Spaces

2

s # O. It suffices to showaSq

2i

Z 1.

# 0 for some i

for some j Z, 1. And we can eliminate j

>

Sq

and use the secondary operation

2 s+ 1+2s_1

= L

#

s since A*(2) acts unstably

Our proof will be by contradiction. We will assume that aSq2i

Z1

j

< s by Theorem B (for s' < s).

while we can eliminate j

all i

2

For then aSq

Sq2

¢ associated

=0

for

to the relation

s i A, s 2A 2 s 2A 1 +2 -2 (Sq 1) + Sq2 Sq1(Sq s) + Sq Sq s (Sq)

s+1

l~i~s

< ¢{x),a2 > #

to demonstrate that

O. Observe that a

2

# 0 contradicts Pro-

position C of §39-2. To prove

<

2 ¢{x) .a

>

#

0 we must

first demonstrate

that ¢{x)

is

defined. In other words

A.

BoB

Proposition A:(i) Sq lex) € (ii) Sq

2A sex)

€

BoB

We must then show that a0a annihilates the indeterminancy in the coproduct formula

In other words Proposition B:(i) (a0a)Sq

2 s+ 1+2s_2i s

(ii) (a0a)Sq2 Sq1 s

(iii) (a0a)Sq2 Sq

=0

for 1

~

~

s

=0

2A s

=0

We will verify these propositions in reverse order. Proof of Proposition B By an argument similar to that used to prove Lemma 38-4A we can show LEMMA A: If deg Sq

R

>0

and aSq

R

# 0 then deg R

Since A*(2) acts unstably on H*(X:W LEMMA B: aSq

2 i_l

o

for

Z s+l.

2)

i

= 2 -1 and aSq

we also have

R

2 = aSq

i-l

.

IX: The Module of lnaecomposeotes OH* (X;IF2)

Proposi tion B follows easily from

these restrictions.

a

(a0a) SqR ¢ 0 unless deg R = 2 _1 + 2b - 1 where 1

297

For

they

imply

~ a.b ~ s.

Proof of Proposition A Only (ii) needs comment. First of all. we have 2.1 Sq sex) E B -* 2.1 For since ~ Sq sex) E B 0 B it suffices. by Lemma 33-1B. to show that

Sq

2.1" 2.1." sex) 0 in H (X;W2)//B' We have Sq sex) E p41(H (X;W2)//B)' By

=

Theorem 1-5B p(H" (X;W2)//B) ~ Q(H* (X;W2)//B) is injective (CH"(X;W . .. Bl) and. by Theorem 38-1. Q41(H (X;W2)//B)

.

= o.

2.1 So Sq sex)

=0

2)

C

in

H (X;W2 ) / / B '

The rest of the proof consists of showing that s+2

Q2

s

+2 -4H* (X;W 2)

In view of (.. ) this forces Sq

2.1 sex)

€

=0

B·B.

SO we will be done. First of

all.

Proof:It follows from Lemmas A and B that the B(q) which appears in B can even .. H (X;W

, '" IS sur jec t rve

~

degree ~ 2 s+ 1_2.

By our induction hypothesis Theorem A and Corollary A

hold

<

in

degree

2

s+ 1 even -2. So Q (B)

be chosen with q

2 s+ 1+2s_2.

Thus

~

the

Q

2)

surjectivity

even .. Q H (X;W extends up to the required degree. 2)

of

Qeven(B)

Q.E.D.

It follows easily from Lemma C that 4' 4'" LEMMA D: Q I(CB) ~ Q I(CH (X:W

2

»

is surjective for 4i ~ 2

Now consider the commutative diagram

s +2 s +1 +2 -8.

III

~

298

The Homology of Hopf Spaces

The rows are exact by Lemma 1-5. In particular, by Lemma D, g is injective in degrees 4i ~ 2 s+2+2 s+ 1_8 Since

B //(H*(X;lF

--.

H*(X;lF ) 2 //(H*(X;lF

2) exterior Hopf algebras we also have

2)

is an

injective map between

h is injective in all degrees

For suppose E --. F is an injective map between exterior Hopf algebras. Given an indecomposable x E E let E' generated by the elements of degree

be the sub Hopf algebra of E

<

E

Ixl. thus 0 t x E P( liE') C

F F F P( liE')' By Theorem 1-5B P( liE') --. Q( liE') is injective. So 0 t x E

Q{F/I E,). Thus Q(E) --. Q(F) is injective. It follows from (*) and (**) that f is injective in degree 4i 2 s+2+2 s+ 1_8 . Hence. by Theorem 38-1. Q4i(B) s+2 s particular, Q2 +2 -4(B)

= O.

=

~

0 for 4i ~ 2 s+2+2 s+ 1_8 . In

(5 ~ 2!).

§39-7:Proof of Theorem B Theorem B is a simple consequence of Theorem A. At the current stage of our inductive argument we are allowed to assume that Theorem A holds for s'

~

s and Theorem B for s'

< s.

~2s+1k+2s_2

Given (x) E Q

A we can write

where

A2 s k+2 s _2

{y) E Q

A2 s k- 2

{z} E Q

By relation R-4 of §39-2 we have

then, by Theorem

299

IX: The Module of Indecomposables OH* (X;IF2) s

s

In establishing the final equality we use the fact that Sq2 k+2 {y} since A*(2) acts unstably as well as the fact that the elements of (Sq for

< s.

0

s:

i

s:

Q.E.D.

=0 2

1

)

2

s-1 act trivially on {y} and {z} since Theorem B holds for s·

300

§40: IMPLICATIONS IN MORAVA K-THEORY

In this chapter we use Morava K-theory to prove H*(OX) has no 2 torsion when X is a I-connected finite H-space. We return to the conjectures discussed in §I4-2 and prove special cases. The loop space theorem follows as a simple consequence of

these special cases. Actually. because of the

length of the proof. we only sketch many details of the argument. Total details appear in Kane [21]. §40-I:Main Results This chapter will be devoted to proving THEOREM: Let

(X.~)

be a I-connected mod 2 finite H-space. Then

Qeve~*(X;W2)

=0

It then follows from the discussion in §I3-4 that COROLLARY A: Let (X.M) be a I-connected mod 2 finite H-space. Then H*(OX) has no 2 torsion. It follows from Proposition 14-4 that COROLLARY B: Let (X,M) be a I-connected mod 2 finite H-space. Then H*(X) has only elementary 2 torsion. In §44 we will also use the above theorem to prove K*(X) has no 2 torsion. Lin gave a proof of Theorem A using the hypothesis that H*(X;W

is assoc2) iative. His techniques are secondary and tertiary operations. We will dis-

cuss some of his techniques in §4I and §42. Our approach is via Morava Ktheory. The connection arises quite naturally.

By Corollary 38-IA the

theorem is equivalent to asserting

By Corollary 39-1C. the statement in Reduction I is equivalent to shoWing Reduction II: Sq

A. 1

IX: The Module of lnaecomposebles QH* (X;IF2)

To attack this problem in degree 2 quence {B } analysing v r

n

n+2_

301

2 we use the Bockstein spectral se-

torsion in k(n)*(X) which we described in §14.

The statement in Reduction II is equivalent to asserting Reduction III: The algebra generators in B 1 can be chosen so that:

= H*(X;W2)

of degree 2

n+2_2

(i) they are permanent cycles in {B r}. (ii) they eventually become boundaries in {B r}. For, suppose III holds. Since d

r

raises degree by r(2

n+1-2)+1

and since X

is I-connected it follows that the algebra generators of degree 2 must lie in 1m d

=>

III is easy.

1.

Since d

1

= Sq

n+2_2

A n+l, condition II holds. The converse II

Properties (i) and (ii) of Reduction III were discussed in §14-2. We also suggested in §14-2 a way of proving them; namely, by constructing an "implication"

in k(n)*(X).

We

will

use

a

modified

version

of

that

approach. §40-2: Implications in Morava K-Theory We can verify properties (i) and (ii) from Reduction III by following the approach suggested in §14-2. Given an algebraic generator x E H2

n+2

-2(X;W

2)

we first convince ourselves that

This amounts to verifying property (i) from Reduction III. Suppose we can write x

=

Pn(~)'

To verify property (ii) from Reduction III we construct

an implication as described in §14-2. We have to extend {~

= ~O'~l'~2""} (** )

~

to a collection

where

;:t(~ s+l ) = v n ~ s 18hs

For suppose that x is not a boundary. This amounts to saying that

~O

gen-

erates a free W summand of ken) * (X). We can then use (**) to force 2[vn] all the elements {~O'~l '''(2' ... } to be torsion free as well as lying in distinct W summands. Thus the elements 2[vn] have a contradiction to the finiteness of X.

{~s}

are all distinct and we

Unfortunately, we must introduce modifications in order to obtain

(*)

and even a weakened version of (**). Notably, we must work with skeletons

302

The Homology of Hopf Spaces

of X. Passing to skeltons easily gives property (*). Given x E 2n+2_2 2n+2 H -2{X;W replace X by X . Then we can choose ~ E 2) n 2_2 n+2 2 _2 2 + __2n +2_2 _2n +2 _2 ken) (X ) such that Pn{'Y) = x E }1 (x;W ments

2).

{'Yl'~2""}

The ele-

will also be constructed on appropriate skeletons of X.

We will have

and we will obtain property (**) for the multiplication (X x X) 2

n +s + l 2n+2_4 2 n+s+ 1 2n+2_4 + ~ X +

Actually we will only obtain a seriously weakened version of property (**). We will construct "secondary operations" in Morava K-theory which

satisfy a coproduct formula which looks like (**) modulo indeterminancy. The indeterminancy will require extensive analysis. Our main result in this vein is THEOREM: Let (X.~) be a H-space. Suppose that in degree 2m+l-I~1 we have the relation

A _ Sq n+ 1 Sq 2 m+l.r. 'I! J( + "L a i b i

where

lail + Ibil = I~I + 2m + 1 la i

I.

Ib i

I >0

Suppose also that we have x E P2m-I~1 H* (X;W ) where bi{x) = O. 2 Then there exists a E k{n)4m{X4m), P E k{n)4m{{XAX)4m) where (i) (ii)

;tea)

= vnP

Pn{a) = J({X)

(iii) Pn{P)

= ~(x)

~ ~(x) +

1: ai{Yi ~ zi)

for some Yi,zi E

H* (X;W2) 2n+2_2 Given x E H (X;W

we will apply this theorem to x and construct the 2) element 'Y s by induction. The theorem demands that x be primitive. If x is

not primi tive then we "primi tivize" x by constructing a fibration f

~

K so that

g * (x)

have constructed

~

s

E

~ PH* (X;W

and

2).

Regarding the elements {'Y

s}'

X !L. X

suppose we

303

IX: The Moaule of traecompossbtes QH* (X;IF2)

Then we apply the theorem to x using the following relation which holds in degree 2n+2_ l Sq2

(So '1!

= Sq

to obtain

n+s+l 2n+l_l 2 + Sq

2n + lA s and

~s+l

J(

Sq

n+l

A

S

2n+2 A n+l sSq2). We will apply the above theorem

where

Pn (~s+l ) = J( () x = Sq

2n+2 A n+l 2 n+lA sSq2 (x) __ Sq s+l(x)

(The last equality arises from the structure theorems of §39. See. in particular. Theorem 39-lB) Moreover. the theorem tells us that

~s

and

~s+l

are related by a coproduct formula analogous to the desired property (**). We have in ken) 2

ll+ l_4 n +s +l+2n+ 1_ 2 ((X A X) 2n+s+l +2 )

where A satisfies A

Pn(A) E 1m Sq n+s+

1

+ [ 1m Sq

2

n+2

A

2n+2 2 i

SSq

-

i~n

A careful (and prolonged!) analysis of A enables us to deduce that generates a free summand if

~s

~s+l

does. We will discuss the analysis in

§40-4. We close this section by sketching a proof of our theorem concerning secondary operations in Morava K-theory. Proof of Theorem As in §15 let P ( X) be the projective plane of X and let 2

be the associated long exact sequence. Since x _.2m-I'1!I+l y E H(P2(X);F

2) such that

t(y)

A

LEMMA: Sq n+lJ«x)

= ~['1!(x)

E

Ker ~ we can choose

=x

0 '1!(x) + [ai(Yi 0 zi)J for some Yi zi E

304

The Homology of Hopf Spaces

.

Here we use the relatIon Sq

2m+l

~(y)

2

~

= Sq An + 1~

= A(~(X)

+ [ aib

i

plus the fact that

@ ~(x»

For the first identity see §15. For the second identity we use bi(x)

0

Next. we pass to skeltons of X and X A X. We also work stably. Let Y

4m

= the

suspension spectrum of X

Z = the suspension spectrum of (X A X)4m The multiplication

X x X ~ X induces a stable map

~:

ii:Y ~ Z Let

the cofibre of

P

ii:

Y ~ Z

We have a diagram Y~Z---7P---+Y

1 ~(y) k(n)

where

the horizontal

~

v

k(n)

--+

HZ/2

~

Pn

n

maps are

k(n)

Tn

tha co fibre

sequences.

We can

form

a

commutative diagram P ---> Y

~(y)

1

HZ/2

113

--+ T

where

k(n)

n

P (13) = ~(x)~(x) + [ a.(y.@z.) n I l

1

This follows from the lemma plus the fact that

i (since H (X;W ) commutes

2

=0

for i

> 4m+2)

plus the fact that the following diagram T

HV2 ~k(n)

A.............. 1

Sq n+l ~

1 Pn HV2

IX: The Module of tnaecomposebles QH* rX;IF2)

305

cofibre sequences. So we can extend (**) to form

Stably, fibre sequences the diagram

y ...l!:....-., z ----> p -----> y

!

(***)

J3

! a

!

!

\[I(y)

J3

ken) --+ ken) --+ H7l 2 --+ ken) V Pn Tn n for some a. By the commutativity of the left and middle square we have

Pn(a)

= \[I(x)

~(a)

= vn J3

Q.E.D.

§40-3:The Sub Module Tor(n) The techniques described in §40-2 demand that we work with skeletons of X. On the other hand. it is more pleasant to work with the cohomology of X rather with that of skeletons of X. In particular, H* (X;F

has a Hopf 2) algebra structure. We now reformulate our approach so as to be able to work wi th X as much as possible. Tor ken) *(X)

denotes v

n

torsion in

k(n)*(X). Let Tor(n) It follows

= 1m {Tor

Pn H* (X;F ken) * (X) --+ 2)}

from the multiplicative properties of the Bockstein spectral

sequence {B associated to v torsion that r} n LEMMA A:(i)Tor(n) is a subalgebra (ii)Tor(n) is a coalgebra ideal of H* (X:F 2) i.e.

--* ~ Tor(n)

C Tor(n) @ H* (X:F

2)

+ H* (X;F

2)

@ Tor(n)

Moreover. proving that x E H* (X;F is a permanent cycle in {B and also r} 2) becomes a boundary is equivalent to proving that x E Tor(n). So, by the discussion in §40-1, the proof of Theorem 40-1 reduces to proving Reduction IV: The algebra generators of H*(X;F chosen from Tor(n) C H* (X;F

2)

of degree 2

n+2_2

can be

2).

Our implication argument can also be done in terms of Tor(n). The property that x t Tor(n) is a weakened version of the property that x

= Pn(Y)

where

y E ken) * (X) generates a free F summand. Namely, x t Tor(n) means 2[vn]

306

TheHomowgyofHopfSpac~

q that when we restrict to an appropriate skeletons X C X then x where y € k(n)*(X

q)

following two facts.

=

Pn(Y)

generates a free W J summand. This is based on the 2[vn

Rather than showing that the failure of property (*) forces a sequence of elements {~s) in k(n)*(X), each generating a free W J summand, we will 2[vn show that the failure of (*) forces a sequence of elements {x in s} H* (X;W where X ( Tor(n) for s 2) s to the finiteness of X.

~

O. We will still have a contradiction

§40-4:The Primitive Case In this section we prove our theorem under a special hypothesis. Our proof is designed to serve as an introduction to the proof of the general case of the theorem. We will prove THEOREM:Let

(X,~)

degree

<

be a I-connected mod 2 finite H-space. Suppose that. in

2n+2·_2. H*(X;W is an exterior algebra on odd degree 2)

generators. Then Q~~--H*(X;W

2)

= O.

We should remark that our indexing is chosen with the proof in mind. We want to think of ourselves as working in degree 2

n+2_

2 because our proof

will then require the use of ken) theory and the module Tor(n) C H* (X;W 2). We will assume for the rest of this section that X is a H-space satisfying the hypothesis of the theorem. The hypothesis implies

Proof:Let {B and {B be the Bockstein spectral sequences analysing 2 r} r'} torsion in H*(X) and H*(X A X) ,respectively. By the discussion in §14-2 each even degree indecomposable x € B = H* (X;W can be chosen so as to 1 2) be a permanent cycle and an eventual boundary in {B So {x} = 0 in Boo· r}. Hence ~{x}

=0

in Boo'.

IX: The Module of IndecomposablesQH* rX;IF2)

307

Since H*(X;f is an exterior algebra in degree < 2 n+2_2 it follows, as 2) in Proposition 1-6, that B = B00 in degree < 2n +2_2. Since Br ' = Br @ Br 1 we also have B1 '

= Boo'·

Thus ~(x)

=0

in B 1'

= H*(X A X;f2 ) ,

Q.E.D.

In view of the discussion in §40-3 the proof of our theorem reduces to showing p

Given x €

p

2

n+2

2

n+2

-~*(X;f2) C Tor(n)

-~*(X;f2) let xo

=x

x

= Sq

s

2n + 1A sex)

We will show x

~ Tor(n) forces x ~ Tor(n) for s ~ 1. In particular, x s ~ s 1 contradicts the finiteness of X. Our proof that x s ~ Tor(n) will be by induction. Fix s ~ 0 and assume

o

for s

~

2n + 2 _2--* Induction Hypothesis:For all x € P -H (X;f Tor(n) .

2),

if x

By making use of our structure theorems for QB+(X;f

~

2)

Tor(n) then x

from §39 we

s

~

can

considerably strengthen this induction hypothesis. Let B(q) C H* (X;f2 ) be

as in §31-1. Let B

= B(2 n+1+2n _2 )

I

= the

ideal of H* (X;f ) generated by B + Tor(n) 2

*

By Lemma 40-3A I is a Hopf ideal. In other words, H (X;f2) / I is a Hopf algebra. We can extend the induction hypothesis to PROPOSITION B: For all x €

H* (X;f2)/

n+2

p2

-~*(X;f2) if x ~ Tor(n) then x s

# 0 in

r:

We will prove Proposition B in the next section. Let us continue with the proof of the theorem. To prove x

~ Tor(n) we use the secondary operas+ 1 tion in Morava K-theory associated wi th relation R from §40-2. This s secondary operation is defined on x. Namely, we have

A.

PROPOSITION C:(i) Sq lex)

=0

for i ~ 1

308

The Homology of Hopf Spaces

2n+ 1A n+1 sSq2 (x)

(ii) Sq

=0

(See Corollary 12-2 as well as Corollary 38-1D) So. combining Proposition C with Theorem 40-2. we have n+ s +2 2n+1_2

a € ken) 2

+

n+s+2

~ € k(n)2

(X

2n+s+2 2 n+2_ 4 +

)

n+2 2n+s+2 2n+2_4 +2 -4«x A X) + )

where (i)

= vn~

;:i*(a)

= Sq

(ii) Pn(a)

n 2A

2 +

s

2n + 1

Sq

+

..

(x)

L Sq

2 n+2 A n+2 i A sSq2 -2 (a.®b.) + Sq n+s+l(c®d) 1

i~2n+1

1

for some ai.bi.c. d in H (X;W ) . We can deduce from Proposition B that 2 PROPOSITION D: Pn(~) ~ Im;:i* + Tor(n).

..

.

.

Proof: Suppose Pn(~) E 1m ;:i* + Tor(n). Then (.. )

xs®X

s

€ 1m

{;:i*: H (X;W 2)/I ~ H (X;W 2)/I ® H (X;W 2)/ }

..

.

For, because of the degrees involved a @ b i i

= c0d = 0

r

in

H (X;W2)/I ® H (X;W2)/I' However. (.. ) is not possible. There are two cases to consider. By Theorem 1-5B either

=

is indecomposable or X y2 where s s y is indecomposable. We will show that. in either case. (.. ) leads to a X

contradiction about the algebra structure of H.. (X;W ) . Write xs®X 2 s H"(X'W ) ' 2 II'

for some z € (i)

~s

= -* ~ (z)

indecomposable

Choose a

€

Peven «

H*(X;W)

..

2 II) ) C PH..{X;F2 ) where

< z.a2 > < z.~..(a®a) > < ;:i*(z).a®a > <x s ®X s ,a®a >

# 0

< xs.a > #

O. Then

309

IX: The Module of tnaecomposebles QH* (X;IF2)

Hence a (ii)

2

# 0 which contradicts Proposition 39-2C.

~s~_:i

Choose a € Podd«H* (X;W2)/r)*) and ~ € (H* (X;W2)/r)* where

< y.a

) f.

A.*({3)

= a0a

< y.~

) f. 0

0

We must justify the existence of such an a and

Observe. first of all.

~.

*

that if we have a Borel decomposition H (X;W2)/r of the Borel generators then we can obtain a and algebra decomposition H* (X;W2)/r

~

A. with y being one

~

i

1

~

by dualizing. For the

Ai dualizes

to give a coalgebra

i

decomposition (H* (X;W 2)/r)*

=~ i

C rf y and hence i.

just let a and {3 be the dual elements in

X

y2 belongs to A k

s

Unfortunately, there is no

~.

reason to assume that y is a generator of any Borel decomposi tion of H* (X;W 2)/r

However. if we pass to

Q

2

(H* (X;W2)/r' we can choose a "Borel

decomposition" of ~(H* (X;W2)/r) which includes y among its generators. H*(X'W) * H*(X;W) * , 2 /r» C ( 2/ r) then, as in (i).

Dualizing. we have the desired a,{3 € (Q2( Granted the existence of a and

~

< z.{32

) f. 0

On the other hand

< z.[a.{3]a >

< z,~*([a,{3]~) )

< ~(z).[a.~]~ ) <

( [a.{3].x

=0

) )( a.x

[a,f3]~.xs~s

s

For the last equality, observe that deg

X

s

s

)

) deg y

from the above that f32 f. [a.f3]a. But we must have primitive and

= deg f32

a.

Lt follows

[a.f3]a. For a

A.*(~) = a0a forces ~ - [a.f3]a € PH*(X;W2). Moreover, we are

working in degree 4k. So. by Theorem 38-1. P * (X;W 2) 4kH

= O.

Q.E.D.

The Homology of Hopf Spaces

310

Finally we have PROPOSITION E: Pn(a) ( Tor(n).

Y = ~

Z

= (X A X)q

We have a commutative diagram Ifl(Z;1F 2)

I

II

Hq(X;1F

2)

--*

.a, Hq(Y;1F2)

III

~ Ifl(X A X;1F

2)

The vertical isomorphisms depend on choosing the cell structures of X and XAX so that the cells are in 1-1 correspondence with bases of H* (X;1F2) and H* (X A X;1F

2)

respectively. So Proposition D implies Pn(~)

( 1m

* (Z;1F

--* :H {~

2)

~

H* (Y;1F2)} + Tor(n)

Since k(n)q(Z) ~ Hq(Z;1F and k(n)Q(y) ~ Ifl(Y;1F it follows that 2) 2) (*) The relation M*(a)

~ ( Im --*: ken) * (Z) / ~

= vn~

Tor

~ ken) * (Y) /

Tor

now forces a to generate a free 1F2[vn] summand in

k(n)*(Z) (So p (a) ( Tor(n) as desired). First of all ~ t 0 in

..

..

n

ken) (Y)/ which forces a t 0 in ken) (Z)/ . Secondly, if a does not Tor Tor generate a free summand then a

("). Q.E.D.

= vn a i n

..

k(n) (Z) /

Tor'

This contradicts

§40-5: The Sub Hopf Algebra B Before proving Proposition B of §40-4 we first establish some facts A

about The Hopf algebra B. We can define Q(B) just as we did Q We have two main results to prove about B. PROPOSITION A: For each s

~

1 we have a commutative diagram

*

= QH A

(X;1F 2)·

IX: The MOdule of tndecomposebtes QH* (X;IF 2)

311

where (i) the horizontal maps are isomorphisms (ii) the vertical maps are surjective PROPOSITION B: p2

n+2

-2(B) C Tor(n)

In Proposition B we are, of course, considering both PCB) and Tor(n) as submodules of H* (X;W of

2).

The rest of this section is devoted to the proof

the above propositions. We begin by considering Q(B). By the same

argument as in Proposition 37-3A we have an exact sequence

o --+ in degrees

=0

Q«(B) --+ Q(B) --+ Q(B)

mod 4. The key to all the results of this section is that A

many of the structure theorems obtained in §39 for Q

=

A

*

QH (X;W

2) also

apply to Q(B). In particular. the negative results given by Theorem 39-1B plus Corollaries 39-1B and 39-1C all apply. This follows from A

A

*

LEMMA A:The canonical map Q(B) --+ QH (X;W2) is injective in degrees

=0

mod 4. Proof:We have a commutative diagram A

Q«(B)

I

Q(B) _

Q(B)

111

Q«(H'*(X;W 2)

--+

A,* QH (X;W 2)

--+

with exact rows. Pick 0 t {x} € Q(B) of degree

QH* (X;W2)

=0

mod 4.

(i) If {x} t 0 in Q(B) (i.e. x is indecomposable) let B' be the sub Hopf algebra generated by the elements from B of degree

<

[x I. Then 0 t x €

p(BI I B,) C p(H* (X;W 2)II By Theorem 1-5B either x is indecomposable in B,). H*(X"W ) 2 liB or x case {x}

to

= y2 *

where y is indecomposable in

in QH (X;W A

2).

H*(X'W ) • 2 liB' In either

312

TheHomomgyofHopfSp~~

(ii) Suppose that {x} € Q«(B). We can assume {x}

=

. od,L.. odd Q (B). By the above argument {y} # 0 In Q ~ (X;W

in

* QH (X;W A

2

{y}

2

where {y}

= {y}

2). So {x}

0 in

~

2

# 0

) . Q.E.D.

Since SqlQeven = 0 it follows from Lemma A that SqlQeven(B) = O. So ..

A

A

A (2) leaves Qeven(B) invariant. It is possible to show that Qeven(B) is .. A2n+2_2 generated, as a A (2) module. by Q (B). We will only need the following weaker result about the A"(p) structure of Qeven(B). A

even

LEMMA B:Q

*

(B) is generated. over A (p). by

=0

Proof: First of all. Q2i(B)

A2i

~

Q (B). 2n+2_2~2i~2n+2+2n-4 L

< 2n+2_2. For. in these degrees,

in degree

H* (X;W ) and. hence. B is an exterior algebra on odd degree generators. 2 This gives the lower bound of our lemma. Regarding the upper bound. pick {x} E Qeven(B). We can write (Of)

{x}

= SqR{w}

where deg {w} ~ 2n+ 1+2n_2. For the image of {x} in Q(B) this is obvious.

= {y} 2 = Sq2n+l {y}

So we can assume that {x} E Q«(B). Thus {x} A2n+l Aodd Q (B). Since Q (B) Hence {x}

~

odd Q (B). {y}

= Sq2n+ 1SqR' {w}.

= Sq

R'

(x) where w

.

IS

where

{y} E

as above.

Expanding Sq2n+lSqR' in terms of the Milnor

basis {SqR} of A* (2) we have (*). Next we reduce to {x}

(**)

For, in degree

< 2n ~-2,

. 8

= Sq

~i

Q (B)

odd degree. Thus we can write 8qR

Sq

= O. = Sq

,\

{y}

80 {y} and. hence. Sq

A

iSqR'

=I

R'

R in (*) have

A

Sq j 8q j

Lastly, we can assume deg Sq

(***)

It suffices to show that k 2k 2 i

I Sq i~k-l

A.

~

~ {y}

~

n+l. If k

~

n+2 we have the identity Sq

Sq l{y}. This is relation R-2 of §39-2 with Sq

2k-l

Ak

{y}

{y} removed

IX: The Module of tnaecomposeotes QH* (X;1F2)

313

since A*(2) acts unstably. Consequently. we can replace Sq 11. Sq 1 where i ~ k-l.

,\

in (**) by

Q.E.D.

One explicit case of Lemma B is the following.

LEMMA C: For each s L surjective. Proof: We begin by showing

(*) A2n+s+ 1 2n- 1 2 n+2 Pick {x} € Q + - (B). By Lemma B. {x} = ~({y}) where 2 -2 ~ deg y ~ 2n+2+2n+ 1_4. By Corollary 39-1B we must have deg y

=0 mod 2n+1_2.

So deg y = 2n+ 1_2. To prove

~

is of the required form we will prove by induction on k

that. for each 0

~

k

s,

~

In particular. when k

s we must have m = 2 =0. So

is of the required

~

form. k = 1: We can assume

~

is a monomial in the operations {Sq

m 2 Sq 2 ... Sq 2 . Since A* (2) acts unstably we must have 2 39-lB we must have deg Sq

general case: Suppose have deg Sq

2

2n + 111

2

Sq

k

2

2

Sq

~

=0

{y}

mod 2

~

2

i

}. Write

~

=

n+l. By Corollary

n+ 1 -2. So 2 = n+l.

2m 22 2 ... Sq Sq

n+l

11k

By Corollary 39-1B we must

_ n+l n+l * {y} = 0 mod 2 -2. So 2 L 2 . Since A (2) acts

unstably we must have 2n+ 1 ~ 2 ~ 2n+k+ 1. We can expand 2 2n+ 111 2 k n+l . Sq2 Sq k _ I a Sq(2 -2 i ..... 0.2 -1.0 .... ) -

i

By Corollary 39-1B we must have i

=0 mod 2n+1.

n 1_i Hence. since 2 + LOwe

have i = 0 or 2n + 1. And we can eliminate i = 0 since A*(2) acts unstably.

314

The Homology of Hopf Spaces

~2n+2_2

LEMMA D: Q

(B) ~2n+2

Proof: Given {x} € Q

2 - (B) then. arguing as in Lemma B. we can show {x}

s

where deg {y} "'even

Q

Sq

R

(B)

= 1).

=0

.

In

2

n+1

n +2 -2 and k

degree

< 2n+2_2

A

SqRSq k{y} ~

A n+l. Now Sq k {y} € A

~even

Q

we must have deg Sq k{y}

(B). Because

= 2n+1_2.

Because of the degrees of {x} and {y} we must have k

(and

= n+ l.

Q.E.D. n+2 ~2n+2 2 LEMMA E: p2 -2(B) =0 Q - (B). LEMMA F: p2

n+2

A -2(B) C 1m Sq n+1

Observe. first of all. that we have inclusions A 1 2n+1_l 2 n+2 2 ~2n+2 2 Sq n+ B C P - (B) C Q - (B) Regarding the first inclusion in degree

< 2n+2_2

on odd degree generators.

from the DHA Lemma of §1-6 that

It follows

A Sq n+1 acts trivially on elements of degree

2 p

n+2

<

B is an exterior algebra A n 2n+1_1 and Sq n+1B2 -1 C

-2(B). The second inclusion follows from Proposition 37-3C.

It now follows from Lemma D that the above inclusions are equivalences. Proof of Propositions A and B Regarding Proposition A the bottom map is an isomorphism (by Lemma E). the top map is injective (by Proposition 37-3C) and the right vertical map is surjective (by Lemma C). This is enough to force all the maps in the diagram to be isomorphisms. Proposition B follows from Lemma F. §40-6: Proof of Proposition 40-4B We now apply the properties obtained in §40-5 to prove Proposition B of

IX: The Module of tnaecornposebles QH* (X;IF2) 2 n + 1A

s

Sq

315

~

(x) € Tor{n). This contradicts the induction hypothesis of §40-4.

Thus no such x exists. To prove the above implication we need some facts about simple systems of generators of B and H*{X;W Given a commutative associative Hopf al2). gebra A over W2' if we ignore the possibility of non zero squares in A then we can think of A as being an exterior algebra. Any set of elements, S, such that A = E(S) is called a simple system of generators. We will obtain three facts of ascending order of difficul ty concerning simple system of generators. (a) The HOof Algebra B In degrees

$

0 mod 4 the exact sequence 0

~

Q{CB)

~

Q{B)

~

Q{B)

~

0

tells us that any set of elements representing a basis of Q{B) can be chosen as a simple system of generators for that degree. The isomorphism n +s +1 2n+l_2 1 1 A2n+s+ 1 2n+1_2 P2 + (B) ~ Q + (B) tells us that, in degree 2n+s+ +2n+ +2 we can choose the simple system of generators from PCB). (b) The HOof Algebra H* {X:Wzl First of all, we can choose a simple system of generators SUS' where S projects to a simple system of generators of H* (X:W2)/I and S' C 1. Secondly, since B + Tor{n) generate lone can deduce from (a) that, in degree 2n+s+l+2n+l_2, we can choose the simple system of generators of H* (X:W

2)

from P(B),Tor(n) and S.

(c) The Hoof Algebra B r Let {Br} be the Bockstein spectral sequence analysing v

n

torsion in

k{n)*{X). Let T

r

the image of Tor C k{n)*(X) under the map

p

n,r

:k{n)*(X) ~ B

r

l) which survive to B Pr = the elements of PCB) C H*(X:W (= B r 2) In particular, T

1

= Tor{n)

and PI

= PCB).

There is a well defined map Br

~ H* (X;W2)/I' (For if {x}

= {y} in Br then x-y € Tor{n) C I). By an inductive argument, begining with the results in (b), one can show that, in degree 2n+s+l+2n+l_2, the simple system of generators for H*(X:W

2)

can be

316

The Homology of Hopf Spaces

chosen from Pr,T r and Sr where Sr maps to a subset of a simple system of

generators for H* (X;~2)/I' This result demands a fairly involved differential Hopf algebra argument.

It certainly is not obvious. So we are

omitting some major details in avoiding this argument. Granted this result for {B we can easily prove our desired contrar} 2n+2 ?_-

diction, Suppose we have 0 ~ x € P

o LEMMA:

X

X

s

in H* (X;~2)/I' We can assume

p + t where p

s

-JH*(X;~2) where x ( Tor(n) but

PCB) and t

€

Tor(n),

€

Proof: We will find p € PCB) such that

X

s

+ p € Tor(n). In other words

(i) X + P is a permanent cycle in {B s r} (ii) {X s + p} = 0 in Boo Regarding (i) suppose

X

+ p has survived

s

{x + p} s

where p

p=

However, {x

p + p',

s

+

p}

+

s

€

p} =

=0

Since dr(t}

r

s

+

p}

€

t

Thus

+ d

we have

dr{x s + p} peven(B ) forces dr{x

while

p' + t + d

Pr,t € T and d is decomposable. r

€

{x

where

=

to B By (c) we can write r,

= dr(d)

pOdd(B C QOdd(B r) r)

dr(d) + p survives to Br + l, O. So x dr{x s + p} dr(d) s Regarding (ii) suppose {x + p} t- 0 in Boo By fact (c) established s above plus the fact that Too = 0 we can write

Thus

{x

s

+ p}

= p'

+ d

where p' € Poo and d is not primitive. This is impossible unless {x s + p} + p = d = O. Thus we can alter p to obtain {x + p} = 0 .. Q.E.D. s By Proposition A we can write

IX: The Module of lndecomposebtes QH* (X;IF2)

where p

€

p

2

n+2

-2(B). By Proposition B. 2 n + 1 !J.

Then x f. Tor(n) but Sq

Sex)

p€

t €

317

Tor(n). Consider x

So.

Tor(n).

=x

- p.

x contradicts the

induction hypothesis of §40-4. §40-7: The General Case The general case is handled by. in effect. reducing to the primitive case. To be more exact. we obtain as close an approximation of the primitive case as we can manage. So. the philosophy of our argument is exactly as before. However. many extra complications are present. We assume (X.il)

is a

. even * provIng Q H (X;f ~

=

I-connected mod 2 finite H-space.

2 i _ 9__* 0 amounts to proving Q iH (X;f

2) 3. We proceed by induction. We can assume 9__*

Q-H (X;f For.

by

Corollary

12-20.

2-connected. Thus Q~*(X;f2)

=0

2)

for each i

= Q6 H* (X;f2) = 0

2)

X

As before.

I-connected

= O.

means

that

X

is

actually

And. since X is 2-connected. "H*(X;f

an exterior algebra on odd degree generators in degree

~

is 2) 6. So. we can

apply the argument of the primitive case to prove Q6 H* (X;f = O. Thus. 2) fix n ~ 2 and assume Q2i H*(X;f

2)

=0

for 2i

< 2 n+2_2

2n + 2 2...

We "primitivize" the elements of Q

--H*(X;f

by passing from X to an

2)

associated space X. Let A

= the

sub Hopf algebra of (H* (X;f

2).

. under invarIant

generated by the elements from (H* (X;f The action of

2)

of degree

A*(2). and

< 2n+2 -2

A*(2) on A induces an action on Q(A). It is easy to deduce

*

from our structure theorems for QH (X;f ) that Q(A) is generated .over 2 A

A*(2). by

L

S-2(A).

Q2 3SsSn+l

For each 3

f : X~ K S

S

= ITi

~

s

S n+l

K(Z 2.2

s-2(A). be a map realizing a basis of Q2 Let K

=

IT KS

we let

s-2)

318

The Homology of Hopf Spaces

f=l1f:X-+K s

where 3 ~ s ~ n+l. Then 1m f* = A. We can put a twisted multiplication on K so that f is a H-space map. (see §32-2}. Let

x=

the fibre of f

So. we have a fibration sequence f

~ g

X-+X-+K *~

of H-spaces and H-maps. We now pass from X to X and work in H (X;W One 2). can

show that each element of QH* (X;W

so that

. x E H* (X;W2) has a representatIve

2)

2n+2_~_* ~

*

t l (X;W )

g (x) E P

2

g*(x} f. Tor(n} We again consider the elements {x X

x

o s

where

s}

g*(x} 2 n + 1A

= Sq

Sg*(x}

We want to show. by induction on s. that X

f. Tor(n}. However. the argus ment given in §40-4 for the primitive case is not sufficient to handle the

elements {x

in the general case. The reason is not hard to discover. The

s}

* * ~ * ~ -+ H (X;W induces a map g*: H (X;W2 }//A ----+ H (X;W 2) 2) 2) which is injective. Denote the quotient Hopf algebra of this inclusion by * * map g : H (X;W

r. One would like to be able to split H*(X;W2) = H* (X;W2}//A 0 r as Hopf algebras over A*(2). However. there is no reason to assume that this can be done.

In other words. we have extension problems. The coalgebra and

Steenrod module extension problems wreck havoc with our previous argument. In particular.

the key step in proving that

Tor(n} is to show x

s0

X

s

+

L Sq

2n+1A n+2 i sSq2 -2 (a 1m

i0

-*

{~

b

i)

X

s

f. Tor(n} forces x

A + Sq n+s+l(c0d}

*~

: H (X;W

f.

f.

*~

2)

s+ 1

*~

-+ H (X;W 0 H (X;W2}} 2)

Our extension problem means that new arguments must be devised. The solution (surprise!) is to use a more complicated relation than before. One must also make heroic efforts to control the above extension

IX: The Moaule of Inaecampasables QH* (X,'IF2)

problems. There is an

important fact which enables us

Steenrod module structure of ~-

2).

Namely, Of

~

This means that when we consider the action of

O~.

H*(x;r2)//A C H*(X;W -->

H*(X;r

*

~

H (OX;r

2)

319

to control

*. Therefore OX A* (2)

on

H*(X;r

~

the OX x

2) then

~ splits as a direct summand modulo ker {a* : H* (X;W

2)

We also have more information about QH* (X;W ) and, hence, 2)}. 2

*

Q(H (X;r2)//A) than was required for the primitive case. Namely, Theorem 39-1C can be used to control the Steenrod module structure of H*(X;W2) as well.

320

In this chapter, as well as in the next, we will describe obtained

the

results

Lin regarding the A* (2) structure of QoddH* (X;W ) when 2

by

(X,~)

is a mod 2 finite H-space. In this chapter we study QOddH*(X;W directly. 2) In the next chapter we pass to H* (OX;W ) and study QoddH* (X;W ) using the 2 2 * odd H* (X;W ) ~ PevenH* (OX;W ) . The results of these chapimbedding a:Q 2 2 ters, as with much of our previous results, can be viewed as an effort to generalize

the

structure

theorems

of

Thomas from §17. A number of new

techniques will be introduced in these chapters. In spend

most

that

sense

we

of our time discussing the framework of our proofs as opposed

to the technical aspects of the proofs. The techniques we discuss in chapter

will this

are centred around the use of Cartan formulae in secondary opera-

tions. References for the results of this chapter are Lin [11] and [13]. §41-1: Secondary Operations First of all, let us note that the secondary operations which were only applied in previous chapters to indecomposables of even degree can be plied

=2

in the case p

in §30-3 that, for any n and any factorization Sqn+1 fine

ap-

to indecomposables of any degree. For we observed

a secondary operation

¢ in

=L

degree n where ~¢<x)

a b . , we can i

l

= x@x

1

de-

+ L Image a .. 1

This is also true of the more sophisticated operations constructed in §32. We will use these operations in odd degree without

any

further

comment.

Our

throughout

the

rest

of

§41

comments will be more concerned with

various refinements which enable us to use these operations

with

maximal

effectiveness. There

are

two

major

problems

encountered

when we attempt to apply

secondary operations in odd degrees. They are the same type of problems we have faced before in dealing with secondary operations in even degrees. Problem I: defining secondary operations on a given x

IX: The MOdule ot lndecompossbles OH* (X;IF2)

321

Problem II: analysing the coproduct ~~(x) Most of §4l and §42 is devoted to explaining how one handles I and II.

As

we will see, the solution to these problems are much more involved for the odd

degree

case

than

for

the even degree case. In this chapter §4l-2,

§41-3 and §4l-4 will basically be devoted

to

treating

Problem

I

while

§4l-5 and §41-6 will be more concerned with Problem II. In the rest of this chapter we will be making the following 2 4k+1 * Assumption: Sq : Q H (X;f We

will

delay

2)

~

. . -H {X;f is trIvIal for k 2}

4k+~_*

Q

1

~

our proof of this fact until §42-5. For although it logi-

cally preceeds the results of this chapter, from the expositional point of view, it is better to delay the proof until we have introduced the niques

of

this

will enable us to define certain secondary operations. Just result

that

tech-

chapter. This assumption is connected with Problem I. It

=0

SqlQevenH*{X:f 2}

gation of QevenH*(X;f

2}

as

Browder's

was the starting point of our investi-

so the above assumption will be the starting point

of our investigation of QOddH*(X:f

2}.

§41-2: The Associativity Hypothesis Throughout §4l and §42 we will work with the hypothesis is

that

H*{X;f

2} associative. This associativity hypothesis enables us to use secondary

operations in a more effective manner. When H*{X:f elements

of

odd * Q H (X;f 2)

-* : H* {X;f coproduct map ~ 2}

have ~

is associative the 2} . * . representatives In H (X;f on WhIch the 2)

H* (X;f2) 0 H* {X;f2} is

Notably, we can define the submodule

n={x

€

Hodd {X;f 2}

n

I -* ~ (x)

odd(X;f C H €

2}

very

well

behaved.

by

Hodd (X;f2}0fH* (X;f2)

and use the argument of Baum-Browder [1] to show

(In

order

*

to apply their argument and deduce this result we need to know

that H (X;f2 } / r

is an exterior algebra on odd 2) This follows from Theorem 40-l). H*(X;f

degree

generators.

322

The Homology of Hopf Spaces

The

use of the elements of 0 as canonical representatives for the ele-

ments of QoddH*(X;W

2) operations.

secondary

will play an important role in our

used to handle both of the problems listed in the rest

of

this

applications

of

Actually. there are two important roles. For it is

section

last

section.

In

the

we will discuss the relation of associativity to

Problem I. We will delay our discussion of II until §41-5.

The

relevance

of 0 to Problem I arises from even(2) Lemma: (i) A maps 0 to itself ..

odd

(11) A

Proof:

* (2) maps 0 to rH (X;W

~*(x) =

Write

2).

We know that rH* (X;W is invariant under 2)

L y.~z.2. 1 1

A*(2). A Hopf algebra argument establishes that (just

-*

c 0

1J. (0)

~

rH* (X;W 2)

compare (~*~1)~* with (1~*)~*). For (i) we can reduce to Sq2n

L

~*(Sq2n(x)

-*

s+t=n

Sq2S(Yi)~Sq2t(Zi2). For (ii) we can reduce to

= L Sq1(Yi)~zi2.

1J. (Sq 1 (x»

Sq1

And And

Q.E.D.

Defining Secondary Operations We can use the lemma to define secondary operations. For suppose that x €

O.

Case (i): a € Aevenf2l

a(x)

Then

€

0

~

Qodd H* (X;W So 2).

~(x)

odd * = 0 if a(x) = 0 in Q H (X;W 2).

Notably we will apply our assumption that Sq 2Q4k+1 H* (X;W over again via this fact.

2)

=

0

over

and

Odd Case (ii): a € A f2l

a(x)

Then

in rH* (X;W 1.)

€

rH* (X;W

2).

(Qeve~.* tl (X;W

If deg a(x)

=0

mod 4 then a(x) is decomposable

= 0 forces Q4k (rH * (X;W2 » = 0 for

each k ~ 2). 2) This is useful because for any x € 0 we can choose a sub Hopf algebra

Be H* (X;W rH* (X;W

2)

2).

invariant under A* (2) such that x

(

B.

-* (x) 1J.

€

~B

c B. To obtain B let

B(n) = the sub Hopf algebra generated over A*(2) by rH* (X;W 2) and pick N where x ( B(N) but x

€

B(N+1).

L Hi(X;W2)

i
and

and

323

IX: The Module of Indecomposables QH* (X'U=2)

Simplified Decompositions We

can use the above ideas to simplify our use of secondary operations

in the study of indecomposables of degree 4n+l. The

secondary

operations

used to study such elements are based on decompositions of Sq4n+2. We need only

be

where

lal

concerned

with

For such terms are irrelevant in terms

treatment of the secondary operation of Sq4n+2

Sq4n+2 modulo terms of the form a~

decomposing

= I~I =3 mod 4.

¢ which

of

our

arises from our decomposition

Consider both problem I and II mentioned above. First of all, ~

* € (H* (X;W 2)o(H (X;W 2) Secondly, a can be ignored when we study the in-

creates no problem in defining ¢. For, as above,~(x) I~(x)

since

I

=0 mod 4.

= x0x

determinancy of ~¢(x) But

Ia I

+ ?

odd and Sq 1QevenH* (X:f

2)

We will use the notation

We are only concerned if x0x

=0

Sq4n+2 to indicate that Sq4n+2

= L aib i

§4l-3: The Action of Sa 2

s

As an

illustration

+

THEOREM A: Let

Image

a.

eliminates such a possibility.

=L a ib , 1

L aj~j

1

where

laj l

I~.I J

=3

mod 4.

of the ideas discussed in the previous section we

will prove, or at least suggest how to prove, analogues of theorems obtained for

€

QH* (X;W ~

2)

the

structure

in §39.

be a mod 2 finite H-space such that H*(X;W2) is associative. Then for each s ~ 2 (X,~)

s+l s (i) Q2 k+2 -3H*(X;W s+l s (ii) Q2 +2 -3H*(X:W As a

companion

to

2)

2)

this

C C

Image Sq Image Sq

2

2

s

sk

+ Image Sq

+ Image Sq

2

2

s

if k

~

2

s-2

theorem we have the following extension of our

. . §4l-l that Sq 2Q4k+l H* (X;W = O. assumptIon In 2) THEOREM B: Let

(X.~)

be a mod 2 finite H-space such that H*(X;W2) is ~ 2

associative. Then for s

s s+l s Sq2 Q2 k+2 -~*(X;W2)

0

324

The Homology of Hopf Spaces

Remark: The first part of Theorem A can be strengthened to assert that 2s+1k+2s_~_* -H (X;F

Q

C Image Sq

2)

2 s+ 1k

(see Lin [10J).

The proof of the theorems follows a familiar pattern. consult

the

arguments

of

The

reader

argument. We prove the two results simultaneously by induction prove Theorem B from Theorem A we expand Sq Sq

2s

Sq

sion

2 s_2

s

2 Sq2

s

on

s.

acts

trivially

or

in the degree in question either because A*(2) acts

< s.

The proof of Theorem A is based on secondary Sq

2 s+ 1k+2s_2

for

k

~

operations

derived

analogues of the decompositions of Sq

=s

from

1. One uses the modified factorizations

discussed at the end of the previous section. The factorizations used

When t

To

using the Adem rules and then show that each term in the expan-

unstably or because Theorem B holds for s'

factoring

should

§39 for detailed versions of the same type of

2 s+ 1k+2s_1

are

obtained in §39-2. We have

we can modify (*). using the fact that Sq

2 s+ 1_2

Sq

2 s+ 1k_2s

O.

to obtain (**) Sq

2 s+ 1k+2 s_2

_

s-1

L Sq

2s+1k 2s_2j+1 2A. 2A 2s+1_2 2s+1k_2s + Sq J + (Sq s +Sq )Sq

j=O We will concentrate on the proof of part (i) of Theorem A. It is (**)

relation

which is used to prove part (i). There are three separate inductions

involved in the proof. (a) We assume that Theorem A (and hence Theorem B) are true for s' (b) We assume that Theorem A is true for the case s in degree> 2 (c) We assume that Theorem A is true in degree 2

Q(N) = 0

s+ 1k+2s_2

<s

s+ 1k+2s_2

for elements

of

n Ker ~(N).

x € O(N+l). defined on x. For

Consider

The

operation

~

2A. J(x)

0

Sq

by assumption 41-1. And

associated to relation (**) is

325

IX: The Module of Indecomposables QH* (X;IF2)

since the theorem holds for s'

<s

and for case s in degree> 2

An analysis of the indeterminancy of

~Qxx}

= x0x

s+ lk+2s_2.

+ ? using our

tion hypothesis on Theorems A and B then yields that x € Image Sq In

particular.

having Sq

2As

2

2 s+ l_2 +Sq appear in relation

inducs

+ Image

C**} in-

2A stead of just Sq s enables us to ignore the indeterminancy created by -* 2A 2s+l_2 in ~ Qxx}. One rewrites Sq s+Sq

_

it

2A L SqRSq i and applies the induc-

tion hypothesis. §4l-4: Augmented Relations In

§30-4

we discussed

given factorization

o

the idea of adding "null terms" a c i i

to a

thereby obtaining a factorization of the form

We will describe such a relation as an augmented relation. The of

such an extension occurs when bi(x}

ated to Sqn

= L a.b. 1 1

{Xi} such that bi(x}

this

more

usefulness

and thus the operation associ-

is not defined on x. We may be able to find

= ci(x i}.

ation associated to Sqn use

to

= L a.(b.-c.} 1 1 1

complicated

elements

Hence, as in §30-4. we can define the operon {x,x

operation

l'

... ,x

k}.

We then attempt to

to analyse the A*(2} structure of

H* (X;1F ) exactly as we have used the operation associated to the first 2 relation As we wi 11 see in the next section this analysis is a non trivial matter.

We

have not so much eliminated a problem as exchanged it for an-

other. This new problem will be discussed below under the topic of formulae.

Cartan

At the moment we only wish to discuss some important null terms

which we will be adding to relations, Good examples of null terms which can be added are 2A 2A Sq SSq s

o

A

A

o

Sq SSq s

and

A (s ~ 2). For example, if Sq s~ appears in a factorization

A

A

of Sqn then we can expand the factorization to include Sq s(~_Sq s}

as

a

326

The Homology of Hopf Spaces

term. The idea of adding null terms can be extended in various directions. (a)

First

of

all

we can add relations rather than null terms. However,

this usually forces us into an inductive argument. In particular

we have

the relations Sq which

can

be

2

k Sq

2

used

k

+

L

Sq

2

O~j~k-l

j

a. =0 J

in this fashion. We add the relations (*) with k de-

creasing until we reach the null term SqlSql 2

inated

the

121 + Sq (Sq Sq )

= O.

For example. we have the

= Sq 2 and bi(x) i 2 2 121 . = Sq2 (y) for some y. If we augment (*) by Sq Sq + Sq Sq Sq we have elImrelation Sq Sq

2

= O.

Suppose that in (*) a

problem of bi(x) # 0 only to be faced with the new problem of

2 1 .21 possibly Sq Sq (y) # O. However. ~ Sq Sq (y) can augment by the null term SqlSql (b)

=0

= Sq 1 (z)

for some z then

we

to eliminate this problem as well.

The idea of adding null terms can be extended to involve higher order

operations. Let relation

with

~(r)

be the r

th

order Bockstein. We can add Sqlp(r)

no problem. For example. the relation Sq2n+l

= Sq

to

1Sq2n

a can

be expanded to Sq2n+1 = SqI(Sq2n - ~(r» The universal example associated to this relation

is

(E.v)

where

E

is

defined by the fibration sequence

nx.J...ELA!....K where

A

= K(Zl2.2n)

x K(Zl2

r.4n-I)

K = K(Zl2,4n) w* (L 4n )

= Sq2n (L2n)

4n(E;Zl2) and v € H satisfies j*(v)

- p(r)(L 4n- 1)

= Sql(L 4n_ 1).

r.n);W L € H*(Zl2 as defining an operation Per) 2) n nCX;W which is defined on x € H ) provided P(i)CX) = 0 for i ~ r. In other (c)

We

can

think

of

2

words

x

is

in

the image of the mod 2 reduction map Per): H* (X;Zl2 r ) ~

=

H* (X;W We are only interested in the case r 2 and in adding 2). to a relation. For example. the augmented relation

1

Sq P(2)

327

IX: The Module of lndecomposebles QH* (X;IF 2)

Sq2n+l gives

rise

to

the

= Sq

universal

1

(Sq

2n

example

-P(2» (E,v)

where E is defined by the

fibration sequence

where A

= K{Y2.2n)

K

= K{Y2,4n)

w* (L 4n )

x K{Y4.4n)

= Sq2n (L 2n )

and v € H4n (E;W2) satisfies J.*(v)

=

Sq 1 (L

- L

4n

_ ). 4n 1

The above examples, plus more complicated ones. will be used QoddH*{X;W

2)

to

study

.

§41-5: Cartan Maps The

above discussion of augmented relations leads into a discussion of

Cartan maps. For in adding null terms we have avoided one problem only run

into a new problem. The old problem was. of course, that

to Sqn Sqn

= L aib i

was not defined on x. Now

= L a.{hi-c.) 1 1

¢ = ¢(x.x1 •.... xk )

is at least defined. The new problem is to

to

¢ associated

associated to analyse

the

coproduct ~¢(x.xl"" ,~). It is at this point that the concept of a Cartan map becomes relevant. By our previous arguments we can select a Hopf algebra B C H* (X;W

2).

in-

variant under A*(2). where

~(x)

€ B 0 B

-* Jl (Xi) € B 0 B

Cartan maps are used to show that we can write

Moreover ,prOVided

we know ~(x) and ~(x.), we can actually compute just 1

what elements of B 0 H* (X;W

2)

+ H* (X;W

2)

0 Bare present in

~¢(x,Xl""~)' In our case this criterion is meet. For, because of associativity

hypothesis. we know

-* Jl (x)

and

-* Jl (Xi)

the

or, at least, are able

to impose strong restrictions. So we will be able to control the

indeter-

The Homology of Hopf Spaces

328

minancy created by B @ H* (X;W ) + H* (X;W ) @ B. 2 2

-""

. ~ ~x'XI"" ,xk) is necessary. This new approach to indeterminancy In The approach in previous chapters was to completely ignore the indeterminancy

created

by

H* (X;W ) 2

@ B + B @ H* (X;W ) . We merely passed to the

2

H*(X'W ) quotient Hopf algebra ' 2 lIB' But to make this approach work one viously

needs to choose B so that x # 0 in H* (X;W 2)II

B.

to arrange this fact restricted the range of

ob-

(Indeed, the need

applications

of

our

argu-

ments) Now we cannot guarantee this property. For with the introduction of augmented

relations

and

the

accompanying

elements {xi} we must make B

large enough so that ~(x.) € B @ B. We are forced to admit that

possibly

I

x € B. Hence the need of a new approach. This leads to Emery Thomas's concept of a Cartan map and Lin's application of it to the theory of secondary operations. A map m: Al x A 2 called a Cartan map if there are maps n 1 .. K x A 1 2

~K

WI: Al

~

n

~K

w A 2: 2

~K2

x K 2: Al 2

w:

~.A

is

K1

A~K

where

k,K are products of Eilenberg-Maclane spaces and the following 1,K2 diagram commutes m

Al x A2

'A

w

K

T

Kx K

AA

x A 1 2 Al x A2 x A1x A2

n Xll 1 2 x Al x K2 ' K1 x A 2

T

w 1xlxlxw2

We can use such maps to induce a Cartan formula for secondary operations (i) The Map

e

Let E E

i

the fibre of w: A

~

the fibre of Wi: Ai

K

~

K i

Then, given a Cartan map m: Al x A ~ A there exists a map 2 E fitting into the following fibre square

e:

E

1

x E2

329

IX; The Module of lndecornposebles QH* (X;IF2)

• OK

OK x OK 1 2

jl x j2

To define

e

e

.E

1

rn

.A

~:

=

2,A2»

P

(rn(al,a2),~(nl(Al,a2),n2(al ,A2»

K x K ~ K induces A

PK x PK -----+. PK

~:

= ~(A(t),A'(t»

~(A,A')(t)

and n

1

consider E C A x PK and E. C Ai x PK.. Then 1 1

e«a 1,A1 ) , (a where

1j

1

E x E 2 1 PIXP2 Al x~

K x A2 l: l

~

K induces

= n l(A l(t),a2)

~

n l(A 1,a2)(t) (ii) Calculating e*{yl In

our

applications

we define a secondary operation by choosing v E

H* (£:f2 ) in the stable range i.e. in degrees where quence holds. Given such a v we

the

Serre

exact

se-

compute e*(v) in the following manner.

can

The maps lllP2

Kl

II

£2

----->

£1

A

K2

----->

nl Kl A A2 - - K

PI Al

induce maps

as

before.

Moreover,

the

n

Al

A

2

K2 - - K

k l: OK I A £2

-->

OK

k

-->

OK

2:

£1

OK 2

A

map n.* 1

determines

fundamental class of any factor of K. Then

LEMMA: If n * (t) l If n2*(t) Also. the maps k

=[ =[ *

l following diagram

ai@b

k. * 1

For

let

t

be the

i

ci@di

and k

*

2

suffice to determine

e* .

This is because of the

The Homology of Hopf Spaces

330

flK

(*)

1

j 1,.1 E

,. OK2

k ~ ,. E - fll K + - - E l 3

l

l,.

j

E 2

1

1

1"j2

e-- E

E

2

,. E2

l

plus the fact that (**) If u € H* (E

l ,. E2;F2 ) is in the stable range for both PI and P2 then u € Image PI * @P2* (=) u € Ker (jl,.l) * and u € Ker (1,.j2) *

To see how k

l*

and k

2*

determine 2* suppose that j

Then

* (v) = ~~

k l * j * (v)

k 2* j * (v)

= jl*('iJ.1 ')

If we write a.' 1

* (L

~a

k

)

= E ~kl ** a (L k ) = E a i'@p2*(bi')

all

effectively

€

Image jl *

where d. 1

€

Image j2*

1

= E pt(Ci')@di'

and d.' == j2*('iJ.") then 1

1

It should be emphasized that the classes are

where a.

Hi '},{b i '}.H i " }

computable. To do so we need to know k

and

{d

i"}

* * . l (l.e.n l ).

k 2* (i.e. n * ) and the operations ~ appearing in j * (v). In our case when 2 we apply Cartan maps to secondary operations all of these are known. The above classes can be characterised in terms of primary and secondary operations acting on H* (X;F In what follows one particular secondary opera2). tion will arise. Let ~s be the secondary operation associated to the relation

Sq

jl*¢<x)

2A 2A sSq s

= O.

Not surprisingly. ~s will

2A 2A when the null term Sq SSq s

cp.

=0

arise

in

the

coproduct

appears in the relation defining

As a final point it should be emphasized

that

the

use

of

augmented

relations and Cartan formulae is geared to the simpler operations discussed

in §30 not to the sophisticated operations defined in §32 and which we

have used constantly up to this point. Hence. if we deal with tion

cp arising

ted

relation) we must show that bi(x)

from a relation

the

opera-

E aib i = Sqn (or a more complicated augmen-

=0

not just that that bi(x) is de-

331

IX: The Module of tndecomposebles QH* (X;IF2)

composable as before. We note. however. that the type of simplified relations Sq4n+2 discussed at the end of §41-2 are compatible with relations a~ where

dary

and

lal

= I~I =3 mod 4 are

operation

Pcreates

use

of

Cartan formulae. Given a decomposition of Sq

¢

arising

from

=[

aib i augmented

4n+2

still irrelevant in terms of

the terms

the

secon-

the factorization. Any such term can be

written as a sum of terms of the form asql (i)

the

no problem in defining

¢.

p where Ipl

=3 mod 4.

We will only apply

¢

to elements x

E 0 of degree 4n+l. If bi(x) # 0 we at least know (from our discussion

§41-2) that bi(x) E CH* (X;~2)' Since Sq 1CH* (X;~2) *

=0

in

we know that

* CH (X;~2)

*

C

we replace

asql~ by the augmented expression asql(~-P(2»

Image {P(2):H (X;Z/4) ~H (X;~2)}· Thus bi(x) E 1m P(2). If then

we have

added the precise term to cancel off bi(x). (ii) asql and before

-aSq 1

~P(2) contribute nothing to the indeterminancy of M*¢(x). As . contributes nothing sInce Sq 1 : QevenH* (X;~2)

is trivial. We must also consider

~-P(2)

~

Q'odd H*( X;~2)

since in the expansion

given by the above Cartan formula analysis

~-P(2)

contributes

terms

to

B @ H* (X;~2) + H* (X;~2) @ B. However. a (tedious!!) analysis of the Cartan formula argument establishes that nothing involving iativity

x@x

arises. The assoc-

hypothesis comes into play here. For as we explained before. the

coproduct of P(x) influences just what elements of H* (X;~2) @ B +

B @ H* (X;~2) are contributed by duct of

~(x)

~-P(2)'

Hence restrictions on

the

copro-

imply restrictions on the resulting elements.

§41-6: Action of Sq

2A dd * s~o H (X;~21

In this section we apply the ideas of the last two sections to show THEOREM: Let

(X.~)

be a I-connected mod 2 finite H-space. Given s

then. for each 2

~

s+2.

2 s+1 Q2 +2 -~*(X;~2)

= Sq

2A 2 s Q2 -lH*(X;~2)

~

2

332

The Homology of Hopf Spaces

For

a fixed s. one proves the theorem by a decreasing induction on de-

gree (i.e. on 2). The proof is based on secondary operations derived factoring Sq

2 2+2s+ 1_2

. By the argument at the end of §41-S we need only be

concerned with factoring Sq

= deg

a

~

=3

from

2 2+2s+ 1_2

modulo terms of the form

where deg

a~

mod 4. As in the arguments from §41-3 our key relation is

still s-l

_ L

Sq

22+2s+1_2i+1

Sq

2A.

1

+ Sq

2A 22 sSq

i=l All other relations will be generated out of this one. We note the following restriction which will be used to control the indeterminancy

of

some

of our operations.

o (The cases 2

i

~

~

if i =0 or 2

~

i

~

s

s follow from §41-3 while s = 0 follows from §12-2). We

now proceed by induction on s. The Case s = 2 We want to prove that 2

Q2 +SH*(X;IF Fix

2

and

2)

assume that (I) is true for 2'

>2

(for large

e

the result is

vacuous). One begins by showing

To prove the proposition we use the operation ¢(x.x augmented relation

2 Sq2 +6

Here deg x

= 2 2 +S.

deg xl

1)

associated

to

the

=Sq2 2+~2 + =Sq2 2+4Sq2 + 2 2+1_1 and. by induction on 2. we have Sq

2

e

(x)

= Sq02(x1). A (tedious!) analysis of the indeterminancy of ~¢(x.x1) then yields the proposition. Since this is the simplest situation we will counter have

en-

we will take the opportunity to expand on this last statement. We

IX: The MODule of Inaecomposables QH* (X;IF2)

As usual we know that x0x does not appear as

a

non

333

trivial

summand

in

~QXx,x1)' Our problem is to ensure that x0x does not lie in the indeter-

~QXx'XI)

minancy of forces

x0x

created by B 0 H* (X;W ) + H* (X;W ) 2 2

0

B. This

then

2

lie in the indeterminancy created by 1m Sq2 +4 + 1m Sq 02 .

to

The theorem then easily follows. Let

~2

be the secondary operation associated to the relation Sq02Sq02

O. In particular the indeterminancy from B 0 H* (X;W ) + H* (X;W ) 0 2

2

B

in-

volves the elements ~2(xi')0~2(xi") where ~(x1) = L x i'0xi". We Can E assume, by some detailed Hopf algebra arguments, that x ." E!1 and x . 1

rH* (X;W2 ) ·

This eliminates ~2(xi' )0x i"· It also eliminates xi since we have the following general result.

!s LEMMA: Let Sq

2A

~s

'0~2(xi")

be the secondary operation associated to the relation 2A

SSq

C 1m Sq

1

s

2A

s

2

by

2 2-1 of all, by Adem relations, Q2 +5H*(X;W n 1m Sq2 +2 C 1m Sq2 2) then can use the operation associated to the relation

We

Having established Proposition A we finish the proof of the case s showing

2

2

PROPOSITION B: Q2 +5H*(X;W n 1m Sq2 +5 C 1m Sq02. 2) First

2 2 (Sq2 +4)Sq1(Sq2Sq 1) + Sq02Sq2 (Sq2) 2 2 :; Sq2 +4Sq1(Sq2Sq1_ p(2)) + Sq02Sq2 Sq1_Sq02) to prove Proposition B. Previously, the fact that x0x could belong allowed the possibility of x E 1m Sq

2 2- 1+2

to

1m

. Now. however, x0x Can-

2

not belong to 1m Sq2 +4Sq1 The General Case The proof for general s is analogous. We proceed by downward on

2.

induction

The argument breaks down into the same two steps represented above

by Propositions A and B. First of all we use the relation

The Homology of Hopf Spaces

334

Sq2~+2s+1_2

=L

Sq2~+2s+1_2i+lSq24i + Sq24s(Sq2~_Sq24s)

l~i~s-1

2~+2s+1_l_* 2~-1+2s_2 24 to force Q -H (X;W C 1m Sq + 1m Sq s We must use a 2) detailed Cartan formula analysis. In particular we must use the ~s lemma.

Secondly, we use the relation s-2

2~ 2 s+ 1_2 i+ 1

_ L Sq +

Sq

24 i

Sq

24 2~ 24 1 24 -1 S + Sq SSq Sq s-

i=l 2 s+l ~ s+l 24 -4Sql)(Sq2Sq1) + Sq2 +2 -~2(Sq s-1_ Sq2) _ (Sq2 +2 s-2 2~ 2 s+ 1 2 i+ 1 24. 24 1 L Sq + Sq lSq si=2 24 ~ 24 24 + Sq s(Sq2 _Sq s)Sq s-l +

2-1 s 24 -3H* ( X;W ) n Im Sq2 +2 -2 C Im Sq s 2 with three comments about this part of the argument ~

to show that Q2 +2

s+l

Let us finish

(1) In the last relation we added the null term Sq2Sq2 + Sq1Sq2 Sq1

O.

(2) To apply this last relation we use the facts that ~

Q2 +2

(i) (ii) (iii)

s+l

-3H*(X;W 2)

n

1m Sq

+

C 1m Sq

24

2 2- 1+2s_2

One

the

fact

that

2 s+l Q2 +2 -3H*(X;W 2)

~s

C

uses the relation (*) and the restrictions (**) from

above. Fact (iii) follows from induction on 2. (3) The

s-l

2 s+l ~ s+l Q2 +2 -3H*(X;W n 1m Sq2 +2 -2 C 1m Sq2 2) ~ ~ s+l 24 Sq2 Q2 +2 -3H*(X;W C 1m Sq s 2)

Both (i) and (ii) are deduced from 1m Sq

2~ 2s+1_2

lemma is used to control the indeterminancy.

335

. section we study Qod~.* and using In thIS -H (X;W by passing to H* (OX;W 2) 2) the c-invariant. The results of this chapter are due to Lin. We refer the reader to Lin [11],[13] and [16] for any omitted details of this chapter.

§42-1: The c-Invariant The c-invariant measures the extent to which a H-map preserves homotopy commutativity. Given two homotopy commutative H-spaces X and Y and a H-map h: X --+ Y there is an invariant

c(h) € [X A X.UY] measuring the extent

to which h preserves homotopy conunutativi ty.

We

define c(h) to be the loop

We have c(h)

=0

if and only if the fibre of h wi th the usual induced

multiplication is homotopy commutative. The invariant satisfies the usual type of composition law. LEMMA A: Given X ~ Y ~ Z where X,Y,Z are homotopy commutative H-spaces and f,g are H-maps then c(gf)

Ogc(f) + C(g)(fAf)

We will apply the invariant to loop maps Df: OX --+ DY where X and Yare H-spaces. Let D be the H-map deviation of f. The questions if c(Of) = 0 f and if D = 0 are closely related. f LEMMA B: Given f:X --+ Y where X and Yare H-spaces then c(Df): OX

ify is adjoint to mx A zex

~ X A X ~ Y.

A

OX --+

336

The Homology of Hopf Spaces

COROLLARY A: If Y = K(Y2.n) and Df

= L xi'@Xi"

E H*(X A X;1F2) then c(Of)

= L a*(x.I ')~*(x."). I It then follows from the loop space theorem that

COROLLARY B: If

(X.~)

is a I-connected mod 2 finite H-space and Y

K(Y2.n) then c(Of)

= O.

These last properties explain why the c-invariant is so useful in the case of finite H-spaces. As we will see in §42-2 our arguments in H* (OX;W in2) volving the c-invariant are analogous to our arguments in H* (X;W in2) volving secondary operations. In particular. the c-invariant plays the role formerly played by the H-map deviation. Since c(Of)

=0

our arguments

involving the c-invariant correspond to the simplest arguments involving secondary operations. namely.

the primitive case. So passing to OX and

working with the c-invariant gives rise to considerable simplifications. §42-2: The c-Invariant and Finite H-spaces As we have just suggested the c-invariant gives rise to a coproduct formula similar to the coproduct formula ~*¢{x) = X@X + L 1m a satisfied i by secondary operations. Moreover. the formula appears under less stringent hypothesis than we required in the case of secondary operations. We have already suggested the lack of need of any primi tivi ty restriction. Given a decomposition Sqn+l

= L aib i

we also required that bi(x)

H* (X;W or, in (very!) special circumstances. that bi(x) 2) QH* (X;1F

2)

Now we will demonstrate that whenever bi(x)

=0

=0

=0

in

in Q H* (X;W 2 2)

H*(X'IF ) * . 2/ then we obtain a coproduct formula in H (OX;1F 2). D3 . t h e approach use d"In prevIous argumen t s. GI'ven Sqn+l Let us summarIze

L a.b. I I

and x E P~*(X;1F2) where b.(x) I

=0

we dealt with a diagram

in

IX: The Moaule at tndecomposebles QH* (X;IF2)

337

nK

1j

f

x where f * {t 2 } n

=x

and w

./E

Ip

~

'K{iV2.n}

f

= ITi

1w

K

b . A secondary operation cD is defined by i

2n

choosing v € H {E;W } satisfying 2

-* {v} = p * {tn}@P* {t } n

~

t{v}

L

€

1m a

The relation ~ ~

i

*-*

*

{fAf} ~ {v} + {Of} {v}

then becomes

~*¢(x}

= X@X

+

L 1m a.1

In particular {Of}*{v} € L 1m a is a consequence of the primitivity of x. i For pOf = Df = O. So Of factors through j: nK ~ E. Now suppose only that bi{x} € n3 . the 3 fold decomposables. We have a diagram

OK

1

~E P2{fIX}

~

X

f

1 1w

~ K{iV2.n}

K

As in §6 the projective plane P includes into BOX 2{fIX}

= X.

The lifting f

exists because all 3 fold cup products vanish in H* {P {As noted 2{fIX};W2}. in §15-1 P is of category 3}. If we loop the above diagram then. 2{fIX} since fIX is a retract of OP we obtain 2{OX}.

338

The Homology of Hopf Spaces

fA<

1OJ

~

ox ----+

OE ~ K(V2.2n-1)

.Y lOp

OP2(OX) ----+ K(V2,n-1)

lnw

Of where h is defined by h* (t since ~(v)

OK

= P*(tn )@p*(tn ) c(h)

= 0* (v).

2n-1)

It is easy to calculate c(h). For

it follows from Corollary 42-1A that

= (Op) * (t n_ 1)@(Op)* (tn-I)

The map g is a H-map. For OX is a H-retract of OP So we can also 2(OX). consider the c-invariant of g. Since c(Op) = c(Of) = 0 it follows from Lemma 42-IA

fA<

->

that

(Op)c(g)

O~. Since (02j)*(rlv)

=0. €

1:

Consequently c(g)

factors

through 02 j :

1m a. we have

c(g) * (n* (v»

1

<:" L

€

1m a

i

Remark: Lin-Williams [1] demonstrate that c(g)*(O*(v»

€

1:

a.(K@K) where K 1

3 ~* -> @ H (OX;F }. As we will indicate 2) 2) i=l later (see §42-6) this is a very powerful restriction.

=

(3) ~* Ker {~ :H (OX;F

Now consider the c invariant of OX ~ DE ~ K(V2,2n-1). We have

o = c(hg) = Ohc(g) These

two

equalities

respectively.

Applying

follow

from

these maps

obtain

+ C(h)(gAg)

Corollary to

t

2n-1

€

42-1B __?n-1

It--

and

Lemma

(K(V2,2n-1);F

42-1A 2)

we

Remark: All of the above discussion can be extended to augmented relations in the various versions discussed in §41-4. §42-3: The Secondary Operation

~

One operation in particular will playa leading role in the arguments of the next two sections. That is the secondary operation the relation Sq2Sq2

=

~

associated to

0 on V4 cohomology classes. We define ~ by

universal example (E,v) where E is defined by the fibration sequence

the

339

IX: The Module of tnaecomposebles QH* (X/IF2)

K(V2,n+l}

l.... E E.... K(V4,n}

~ K(V2,n+2}

where

n+3 is chosen to satisfy and v € H (E;W 2) j * (v) = Sq2 (L + ) n l Since Im Sq2 = Im 1f!(0} we can view If! as a general ization of Sq2. Since SqlQevenH*(X;W2}

=0

we also have the relation Sq2Sq2

= SqlSq2Sql = 0

on

QH* (X;W The rest of this section is devoted to proving the following 2). generalization of this fact. PROPOSITION; Let (X,~) be a mod 2 finite H-space. Then Sq2 I m If! QOddH*(X;W

=0

in

2)

To prove the proposition we consider another universal example (EO'v l,v2). We define EO by the fibration sequence flK where

O

jo PO Wo ---. EO ---. A ---. K O O

A O

= K(V4,n}

K O

= K(V2,n+2}

x K(V4,n+4} x K(V2,n+4}

* 2 o (L n+2 ) = Sq (L n) * 4 W (L + ) = Sq (L ) o n 4 n W

- Ln+4

2 2 1 1 * Since Sq (L n+2 ) and Sq Sq (L + ) + Sq (L +4 ) € Ker W it follows from the n 2 n o Serre exact sequence that there exist elements n+3 vI € H (EO;W2) v

2

€

n+4

H

(EO;W2)

.* 2 .* 21 1 where J O (vI) = Sq (Ln+l) and J O (v 2) = Sq Sq (L n+l) + Sq (Ln+l). One can then deduce from the Serre exact sequence that 2 1 * Sq (vI) = Sq (v 2) modulo 1m Po 1m PO* ~ W2 generated by PO*I3(2)(L n+4}. identity implies that In degree n+5,

Sq2(V l) € 1m Sql + 1m 13(2} The space EO and the above facts are used to prove LEMMA: Given a space X and x

€

Ker Sql

n Ker

Sq2 then

So the above

340

The Homology of Hopf Spaces

2 I Sq ~(x) C 1m Sq + 1m ~(2).

Proof: First of all, since any two elements in w{x) differ by an element in 1m Sq2 and since Sq2Sq2

= SqlSq2Sql

we need only demonstrate that w{x)

2 contains one element y such that Sq (y)

1m Sq

E

I

+ 1m ~(2)'

Secondly, we have a commutative diagram flKO -----. EO --+ AO

I

hI

K O

--->

I

I

K{V2,n+l) ----+ E ----+ K{V4,n) ----+ K{V2,n+2)

= VI'

where h*(v)

Thirdly, if we think of x as being represented by f: X f can be factored X Sq ISq4{x)

f

OA

~

= Sq4Sql{x)

a map f': X

~

O

~

~

K{V4,n) then

K{V4,n) such that f O lifts to EO' For

= 0. Thus Sq4{x) = fxf'. Q.E.D. O

+ Sq0ISq2{x)

K{V4,n+4). Let f

The proposi tion follows from the lemma. For

can be represented by

it was demonstrated in

§12-2 that 1m Sql + 1m ~(2) C decomposables of H*(X;F

2)

in odd degree.

§42-4: Desuspension Results At this point the Achi lIes heel of c-invariant arguments appears. We can use the relation a*(x)@a*(x) E for some specific

~.

L 1m

a. to conclude that a*{x) E 1m ~ 1

But so what? We want to prove that x E 1m

~

and there

is no obvious way to desuspend our result. Put another way, if a*{x)

=

~(y) there is no guarantee that we can choose y E 1m a*. This problem can sometimes be handled. There are two stages. By Hopf algebra arguments one reduces to the case where y is primitive. One then obtains desuspension results in the particular case of primitive elements. The process works well for ~

= Sq2.

In this section we discuss the desus-

pension result for Sq2. As in §42-3 let w be the secondary operation associated to the relation Sq2Sq2

= ° on V4

cohomology classes.

PROPOSITION: Given a I-connected mod 2 finite H-space (X.M)and x E 4k-~_*

P

2

-H (UX;F then Sq (x) E 2)

n* (1m ~).

So we can desuspend Sq2 only at the price of introducing the higher order

341

IX: The Module of tnaecomposebtes QH* (X;IF2)

operation '1J. We divide the proof of the proposition into a number of cases. First of all, i f x E Im a* there is no problem. For i f x

a* (Sq 2 (x)).

So suppose that x ( 1m

(see §29-S). Write x . SInce QevenH* (X;f2)

then Sq 2 (x)

=

Then x is a transpotence element.

where y is an algebra generator of H* (X;f2).

= t(y)

=0

a* .

= a* (x)

(see §40-1) we could actually assume that y is of

odd degree. However, we will do the even degree case as well. For the odd

degree case is handled,

in large part. by reducing to the even degree

case. Case (i): deg y

= 2~ = 2s .

Assume that height y

= y 2S-4Sq4~(y 2) .

z We have

2 4~ 2 Sq Sq (y)

= y4 .

Let

2 So Sq (z)

s

= y 2 = O.

Consequently '1J(z) is de-

fined. We will show that Sq2(x) E a*'1J(z). Let (E.v) be the universal example for '1J considered in §42-3. We have a commutative diagram K(V2,2 s + 12-1) j1

1

E PI

h

1

1

g

K(V2,22) WI

1

K(V2, 2 s + 1e) where

WI* (l s 1) 2 + 2

* (l

W

2

s+1 ) 2

*

g (l s+l ) 2 2-2 Since Ow

~

*

= (l22) 2

s

= Sq2 (l 2 s+l 2-2 ) 2 s _4- 42

=Y

~

2 (y)

there is a splitting nE ~ K(V2,2~-1) x K(V2,2 s+ l

Recall that v E H* (E;f satisfies j * (v) 2) Then

12-2)

= Sq2 (l

1 ). Let VI 2 s + 2-1

342

The Homology of Hopf Spaces

for some a € A*(2). Observe that deg a is odd. Now nE

l

is the universal example for the transpotence element t(y).

Namely. given f: X E

l

K(Z/2.2e) where f * (L

and

-* (l0L (Df)

We also have f*(V a

~

=0

1 ) 2 s + e-2

= y.

2e)

on H* (DX;f2 ) . So the identity (*) gives Sq2(t(y»

€

~

= t(y).

E ~(z). And since HOdd (DX ;f

l)

then f lifts to -f: X

2)

o

(see §40-l) we have

a*(~(z»

Case (ii): deg y - 2e+l and height y - 4 We use the same argument as above. However. y having odd degree results in a having even degree. So we cannot automatically ignore a when we pass to H* (DX;f ) as above. To eliminate a we invoke the hypothesis of height y 2 4. This forces a to be of degree 6e+4. Moreover. by the results of

=

§42-3. we have Sq2(v

l)

E 1m Sql. This forces Sq2a(L

this forces a to be of the form a VI - Sq2 ~p* (L

= Sq2~

+

2e

»

E 1m Sql

In turn

Sql~. If we replace VI by

VI =

Ze- l ) then we have

* -

= Sq 1~(L2e)01

a (VI)

2 + 10Sq (LSe+2)

Passing to H* (DX;f ) we again have Sq2 (t(y» Z Case (iii): deg y

= 2e+l

and height y

= 2s

E a* (~(z». (s

> 3)

We handle these cases by reducing to the even degree case treated in (i). we have a commutative diagram K(Z/Z.Zs+l e+2s-1) j2

1

q

E 2

P2

1

r

K(Z/2.2e+l)

1

W2 K(Z/Z.zs+le+2 S )

where the right hand column is the fibration from (i) and

*

r (L 4e+2 )

= (L Ze+ 1 )2

343

IX: The Module of Indecomposables QH* (X;IF2)

We have a splitting nE ~ K(Zl2.2e) x K(Zl2.2 s+ le+2 2

s-2).

If we let v

2

q* (VI) then we can deduce from the results of (i) that

o* (v 2) = a(Or) * (L 4e

2)01 + 10Sq2 (L s I ) + 2 + e+2 s-2

Since a has odd degree we can again ignore it when we pass to H* (nx;~2)' So we again obtain Sq2 (t(y» §42-5: The Action of

€

a *~(z)

*

Sq 2~ (X:~21

We are now in a position to deduce some results about the action of Sq2 on QoddH*(X;~2) for mod 2 finite H-spaces. Our main result is PROPOSITION: Let

(X.~)

be a I-connected mod 2 finite H-space. Then

Q4n+IH*(X:~2) c 1m ~ + 1m Sq2n This proposition has consequences for the action of Sq2 n

=I

First of all when

we have

COROLLARY A: Let

(X.~)

be a I-connected mod 2 finite H-space. Then 5 * Q H (X;W 2)

:'1-_* (X;W = Sq2Q-H 2).

For. by Corollary 12-2D. X is actually 2-connected. Consequently. 1m

~(O)

~

1m Sq2 in degree 5. Secondly. we have

COROLLARY B:Let n

(X.~)

be a I-connected mod 2 finite H-space. Then for each

l

2 4n+1 We have already shown that Sq ~H (X;W Sq2Sq2nQ2n+IH*(X;W2)

= (Sq2n+2

2)

=0

. _•.* ln ~ (X;W 2). We also have

+ SqISq2nSqI)Q2n+I H*(X:W

The second equality is due to the fact that Sq2n+2

=0

2)

=0

in degree 2n+1 plus

the fact that Sq I Qeven..* tl (X;W

= 0 by Corollary 12-2B 2) This last corollary was the foundation in §41 for the study of

QoddH*(X;W via secondary operations. So we have now filled the final gap 2)

344

The Homology of Hopf Spaces

in the arguments of §41. The rest of this section is devoted to the proof of the proposition. We will proceed by passing back and forth between H*(X:W The and H*(OX:W 2}. 2} c-invariant as discussed in §42-1 and §42-2 is our means of passing from H* (X:W

* 2) to H (OX:W 2). The desuspension results of §42-4 enable us to pass

back to H* (X:W

2). We will actually prove a more precise result than the one stated in the

proposition. Write 4n+1

= 2 sm+1

where m is odd. We will show

Assume that (*) holds in degree 4n'+1 for n

> n.

Pick

xE

Q4n+1H*(x:W Consider the relation 2}. (*) Sq4n+2 = Sq2~ where ~ = Sq4n + Sq1Sq4n-2Sq1

o

in Q8n+1 H*(X:IF then we can apply the secondary operation 2}

defined by (*) to prove that

x~

1m Sq2 (= 1m ~(O}). So we would be done.

- i 0 In . Q8n+1 H* (X;IF . Therefore assume that ~(x) We WIll augment relation 2). (*) in order to cancel the term ~(x). We can assume that

Proof; Since 8n+1

= 2 s+ 1m+1

we can assume by induction that

~(x) C 1m IlJ +

L

Im Sq

2s

~

2 s-1

... Sq

O~i~s

1m ~ + Sq4n(lm IlJ + L 1m Sq2 15;i5;s-1

2 s- i

s-1

~

m... Sq2

s-i

~)

For the purposes of proving (*). we can obviously rewrite x using elements

L

of 1m IlJ +

1m Sq2

1~i~s-1

we have

s-1

4n+1 * H (X:IF 2)

~

m... Sq2

s-i

~. Moreover. since Sq1QevenH*(X:W 2}

* = Sq 4n~4n+1 ~ H (X:IF2).

= O.

Q.E.D.

Motivated by Lemma A we now construct the main operation used to prove (*). In view of Lemma A we would like to take the relation Sq

4n+2

and add on the term Sq2~. Unfortunately, Sq21lJ is not a null term. However. we know from §42-3 that 1m Sq21lJ C 1m Sq1. We can use that fact to add on a

345

IX: The Module of Indecomposables QH* (X;IF 2)

further term to cancel off the excess. Let (EO'v l,v2) be the universal example considered in §42-3 We can think of Was being defined by VI' Let w' be the operation determined by v

2

H* (E ;W ) . We will use the augmented relation O 2

€

Sq

4n+2

= Sq2 (.,,-W)

1

,

+ Sq ('lJ -P(2» A

A

It gives rise to the universal example E where E is defined by

the

fibration A

A

W

A

A

flK-+E-+A-+K where A

= EO

K

= K(~2.Sn+l)

x

K(~2.4n+l)

A* W (LSn+ l ) A* W

By a

careful

VI

L

Hopf algebra argument one

(the coassociativity of

can find a

as well as elements y,z

2)

K(~4.Sn+2)

K(~2.Sn+2)

= "'(LSn+ l ) = v 2 - Sn+2

(L Sn +2 )

playing a major role)

QH* (X;W

x

x

(H* (X;W

€

.,,(x)

= 'lJ(y)

W· (y)

=Z

mod D3 . Also. since Sq 1(H* (X;W 2)

representative x

=0

fl for

x€

satisfying

2)

we have

€

Z €

1m P(2)' Now, bY f a 11ow-

ing the c-invariant argument given in §42-2. we can deduce that

a* (x)@a* (x) in H* (flX;W

2).

. Hodd (flX;W SInce

2)

1m Sq 1 + 1m Sq2

€

=0

a* (x)

we can ignore Sq 1 and deduce that €

1m Sq 2

By some further Hopf algebra arguments we have LEMMA B; a*(x)

Proof;Let ~ L

k~O

€

I

O~i~s-l

i

where Yi

€

Im'lJ.

1

r be the sub Hopf algebra generated by I P4k+2-.* ~ (flX;W 2) and

2 4k+?_.* -H (flX;W

Sq P

a* (y.) 2

k~O

2).

Observe that

ant under Sq2. Moreover we have

r is primitively generated and invari-

346

The Homology of Hopf Spaces

the isomorphisms p4 i+ 2 ( f) ~ p4i+~*{OX;1F2) ~ Q4i+~*(OX;1F2) tell us

For

*

that H {OX;1F2)ll

f

= 0 except in degrees == 0 mod 4. Thus Sq2 = 0 in

H* (flX;1F 2)ll . The exact sequence 0 --+ P(f) --> PH*(nv ."'; IF2 ) --> f

*

p(H (OX:1F2)ll ) (see Lemma 1-5) then gives the above inclusion. f

Next, let f' C f be the sub Hopf algebra generated by '" L

Sq2 P4k+?_-* IH (OX;1F

k~O

2).

We claim that

P(f') = P(f) in degree == 0 mod 4 4k+?__* 2 For, given x € P -H (OX;1F then x € P{f'). This follows from the 2), identities x 2 = Sq4k+2(x) = (Sq2Sq 4k + Sq4k+1Sq1)(x) = Sq2Sq4k(x) € 28~~~

Sq P

-H (flX;1F

tors of degree

2).

f

Consequently,

=2 mod 4.

I l , is an exterior algebra on generaf

Thus p(fl l f,) is restricted to the same degrees. f

The exact sequence 0 --> P(f') --> P{f) --> P( I l , ) (see Theorem 1-5) then f

yields the above identity.

s '" 2 4k+~-* L Sq P -H (OX;IF2) then the . set { xi2 T- 0 }

If {xi} is a basis of

k~O

spans P(f'). Moreover, by Proposition 42-4, each x.

1

€

a*(Im >1'). Conse-

quently we can write a*(x) € P(f') as a * (x)

=

L

O~j~s-l

a * (y.) 2

* QOddH*( X;1F ) since a: 2 desuspend Lemma B and obtain

La st 1y,

J

~ ~

j

where YJ'

€

peven..*(nV',1F H ''''

Q.E.D,

rm >1'

2)

is

injective

we can

§42-6: Exterior H-Spaces The main difficul ty faced in the above argument was stated at

the

beginning of §42-4. We can use the c-invariant to deduce results about the action of A* (2) on PH* (OX;1F PH* (OX;1F

2)

2).

However, the suspension map a * :QH* (X;1F2) -->

is not, in general. an isomorphism. So we cannot automatically

desuspend

the results

QH* (X;1F

The limited results of §42-4 and §42-5 were obtained by care-

2).

to obtain facts about

the action of A*(2)

on

347

IX: The Module of Indecomposables QH* (X;IF2)

fully analysing the cokernal of a * (Le.

the transpotence elements) in

certain cases. In this section we discuss the more systematic resul ts which can be obtained when a* is an isomorphism. Under this assuption one can use the c-invariant in a way which exactly mirrors the use of secondary operations as practiced in §§30-39. One uses an inductive procedure analogous to that described in §33-3. Given certain restrictions on the action of A*(2) on

*

QH (X;IF

2)

one uses them to ensure that a c-invariant is "defined" on a

n certain x € H (X;IF In other words. we have a relation Sqn+1 2).

and bi(x)

=0

=I

a.b 1

i

= H* (X;IF2 ) / n3.

in G2H*(X;IF

This enables one to perform the 2) c-invariant arguments described in §42-2 and thereby deduce fu ther res-

=

trictions about the action of A* (2) on QH* (X;IF PH* (ITX;IF With these 2). 2) new facts we repeat the cycle until we have relatively complete information about QH* (X;IF 2). We will work under the hypothesis that H*(X;IF is an exterior algebra 2) on odd degree elements. Equivalently (see Corollary 11-3) we. can assume that H* (X)(2) is torsion free. By the discussion in §29 this hypothesis ensures that a * :QH* (X;IF 2) the following results THEOREM: Let

(X.~)

~

PH'*(ITX;IF

2)

is an isomorphism. Lin [16J proved

be a mod 2 finite H-space such that H* (X;IF is an 2)

exterior algebra on odd degree generators. Then (i) Q4k+l H*(X;IF

2)

= Sq2 Q4k-1 H*(X;IF2)

r+1 r (ii) Q2 k+2 -l H*(X;IF COROLLARY: Let

(X.~)

2)

= Sq2

r

+ Sq2kSq2Q2k-1 H*(X;IF 2)

r+1 Q2 k-1 H*(X;IF

2)

for k ~ 1.r ~ 2

be a mod 2 finite H-space such that H* (X;IF2) is an

exterior algebra on odd degree generators. Then for k.r r

r+1 r Sq2 Q2 k+2 -l H*(X;IF 2)

~

1

=0

These results are based on Thomas's structure theorems from §17. As such they are strongly analogous to various results obtained using secondary operations. See for example §39-1. As with the secondary operation results the theorem and the corollary are proved together by an inductive argument. Only instead of using the secondary operation associated to a relation we use the c-invariant. The corollary (for case r) enables us to

348

The Homology of Hopf Spaces

define and use c-invariants to deduce the theorem (for case r+1). PROOF OF THEOREM

Actually we will only sketch Lin's argument. We begin with the Case r - 1 The proof that Q4k+1H*(X;1F

= Sq2Q4k-1H*(X;f

2)

2)

+ Sq2kSq2 Q2k-1 H*(X;f 2)

is analogous to (but simpler than!) the proof in §42-5 that Q4k+1H*(X;f C 1m

~

+ 1m Sq

2k

2)

. We proceed by downward induction on degree. In degree

4k+l we begin by considering the relation Sq4k+2 = Sq2,

where, = Sq 4k + Sq 4k-1Sq1

4k+1 * Given x E Q H (X;f then, by induction, we know 2) 4k22,(x) Sq Sq (Y1) + Sq (Y2) for some Y1'Y2 E

QH* (X;f

the case

=

2).

Replacing x- by x- - Sq 2 (Y l) we have reduced to

-

2 -

4>(x) = Sq (y)

Using Hopf algebra arguments we can choose representatives'x,y E H* (X;f such that ,(x)

= Sq2(y)

3

mod n

and Sql(y)

=0

3

2)

mod n . So, if we augment

the above relation to form

we can use the c-invariant argument from §42-2 to deduce that *

2 1 2 + Im Sq Sq

o (x) E Im Sq

Odd(OX;f Since H 2) §42-2 we have

o

we can ignore 1m Sq1Sq2. By using the remark from

u*(x) E Sq2(primitives) + Sq2(2-fold products of primitives) By the same type of argument used to deduce Lemma B in §42-5 we can show that a* (x) Hence,

desuspending,

= Sq2 (1m a* ) we

have

+ Sq2k-~2 (1m a* )

Q4k+1 H*(X;f

2)

Sq2 Q4k-1 H*(X;f

2)

+

Sq2kSq2Q2k-1H*(X;f2)' General Case The general case is similar but more complicated than the above. Working in degree 2 r+ 1k+2r_1 we start off with a relation of the form

349

IX: The Module of Indecomposables QH* (X;IF 2)

Sq

2 r+ 1k+2r_l

L Sq

2i

induction. that

a

i.

r+l r Given x € Q2 k+2 -lH*(X:1F we can assume, by 2}

for some xi' In order to feed this fact into our argument we augment our previous relation by

and form the relation

By induction we also have aij(x i)

= Sq2

j

-

(x i j)

for some x

Again. using (*), we can augment our relation to accommodate i j. these identities. We repeat the process again and again. Each time we aug-

ment the relation using (*), i decreases until the final relation added is Sq1Sql

O.

We then choose representatives for x,xi,x

i j"

.. so that the above idenr

2 tities hold mod n3 and use the c-invariant to force a*(x) € 1m Sq . Conr +1k+2r _ l * sequently Q2 H (X;1I'2)

= Sq2

r

r+l Q2 k-l H*(X;1F

2}.

PART X: K-TIIEORY

The next two chapters are devoted to proving that the K-theory of a fini te H-space is torsion free. We approach K-theory via Brown-Peterson theory. For the Conner-Floyd isomorphism (see §44-2) tells us that K*(X)(p) is determined by BP*(X). We also approach BP*(X) via BP*(OX). For the presence of torsion in H (X)( ) means that BP (X) can be quite compli-

*

p

*

cated. On the other hand. since H*(OX)(p) is torsion free BP*(OX) has a simpler structure. In §43 we study the algebra structures of H*(X;Wp) and BP*(OX). In §44 we use these results to study BP*(X) and. in particular. deduce that K*(X)(p) is torsion free. The reader should note that Appendix C is devoted to an introductory discussion of BP theory.

353

§43: TIIE HOMOLOGY OF ex

In this chapter we collect some facts about the homology algebras H*(flX;lFp) and BP*(flX) where X is a mod p finite H-space. We will use these facts in §44 and §47.

If X is an H-space then OX is a homotopy commutative H-space. (See the proof in §3 that ITI(X) is abelian for H-spaces) Since flX is also homotopy associative (see §7) it follows that H (OX:IF ) is a bicommutative bias soc-

*

iative

p

Hopf algebra. When X is a mod p finite H-space. the algebra struc-

ture of H (OX;IF ) is particularly simple. This is based on the following result.

*

p

THEOREM: Let

be a H-space where H (X;IF ) is commutative and associa-

(X,~)

tive and 0 then a P

*

=0

p

on H (X;IF ). Given a E

*

= O.

p

.

H~(X;IF ~

p

) of finite height

By the loop space theorem (see §35 and §40)the hypothesis and, hence. the conclusion of the theorem applies to the loop space of I-connected mod p fini te H-spaces.

By some algebraic manipulations we can extend the

theorem to the following useful form. COROLLARY: Let

(X,~)

be a I-connected mod p finite H-space. Then H*(OX;lFp)

contains a sub Hopf algebra T C H (OX;IF ) which is invariant

*

p

under A*(p) and satisfies: (i) a P (ii)

=0

for all a E T

~(flX;lFp) ~

T @ P as algebras where P is a polynomial algebra

This corollary is the basic structure theorem for H*(flX;lF

We also want p)' to record a further fact about T which will be needed in §44 in our study

of the K-theory of finite H-spaces. If we dualize T C H*(flX;lFp) then we obtain the quotient Hopf algebra H*(flX;1F ) ~ T* which is also invariant p

The Homology of Hopf Spaces

354

under A*(p). Since T is comutative, associative and has only trivial pth powers it follows that T* is primitively generated (see Corollary l-SC). The Hopf algebra T* satisfies

* PROPOSITION: Q2n(T)

=0

unless n

=1 mod p

We will prove the above results in §43-3.§43-4 and §43-5. First we explain the consequences for BP*(nx) of the above results. §43-2: The Algebra BP*(nx) We now consider the Brown-Peterson theory of OX where X is a I-connected mod p finite H-space. By Corollary 43-1 we can write

~(nx;fp) = f p[X 1]/I where I is the ideal generated by {x free BP*(OX) is a free BP* T: BP*(OX)

~

p

I

@f

p[X2]

x € Xl}' Since H*(nx)(p)is torsion

= ~(P)[vI,v2""]

module and the Thorn map

H (OX;f ) is surjective. Let

*

p

= ~1 U ~2 = representatives in BP*(OX) for the elements D = the set of non zero monomials in the elements of ~ of ~

X

= Xl

weight

U X2

L2

which do not include the pth power of any element from ~1' Then

~

U D is a BP* basis of BP*(nx). In fact. by our structure theorems

for H*(OX;f p)' we have PROPOSITION A: If

(X.~)

is a I-connected mod p finite H-space then BP*(nx)

= BP*[~]/J

as an algebra where J is the ideal generated by

{ RIjI

~l } and each RIjI is of the form

I IjI

€

RIjI

= tp -

L ~iljli - L wjd j

(P.v l,v2 , ... ) C BP*,ljIi

E~.

d j € D.

Therefore J defines the relation by which monomials in powers of elements from if one works mod (P,v

~l

l.v2

can be written

in terms of

~

~

involving p

th

U D. Futhermore.

2

.... ) • the only such monomials which cannot be

expanded entirely in terms of ~ are the monomials {

tp I

IjI € ~l }.

If we dualize to cohomology then we can deduce the following useful

fact about representatives in BP theory, Given a

€

PH*(OX;f ) and A P

€

X: K-Theorv

=a

BP*(OX) such that T(A)

355

then. in general. A is only known to be primi-

tive mod (P.v 1.v2 •... ). For Ker T = the ideal (P.v •... ). The following 1.v2 fact will be used in the proof of Theorem A of §43-1. PROPOSITION B: If 2~.*

P

tl

(X.~)

is a I-connected mod p finite H-space and a €

(OX;f ) where 2n P

where

(i) T(A)

$

*

0 or 2 mod 2p then there exists A € BP (OX)

=a

(ii) A is primitive mod (P.v

1.v2

.... )2

Proof: Dualizing the above basis llJ U D of BP*(OX). we have a basis of BP*(OX). Let

n be

the duals of the elements of llJ. Since llJ projects to a f

p basis of QH (OX;f ) it follows that the elements of 0 project to a basis of

*

PH.. (OX;f). p

p

(see Theorem A of

§1-5).

So

they are

primitive mod

(P.v 1.v2 .... ). An element of n is primitive if and only if its dual in llJ is not required to expand any monomial of weight

~

And, as already observed. if we work mod (P.v

, ... )2 then the only

1,v2

tp I ~ € llJ 1 }. Iwl = Itpl - Ivsl

2 in the elements of llJ.

monomials we need consider are {

So, if w € 0 is not primi-

tive mod (P.v 1.v .... )2 then 2 In particular. I~I 0 or 2 mod 2p. Q.E.D.

for some

=

~

€

llJ and s

1

~ O.

§43-3: Proof of Theorem 43-1 We want to show that there cannot exist a € H (X;f ) where a P ¢ 0 but * P s an = 0 for some n > p. Pick such an a of minimal degree. Suppose a P ¢ 0 s+1 but a P

O. We will use secondary operations to force a P

s+1

¢ O. The

contradiction then eliminates a. Let ~

s

= aP

. We want to show ~p ¢ O. The theorem holds in degree < lal.

We begin by deducing two facts about

~.

k k 0 for each k ~ Write d*(a) = [ ai 0 ai. then A*(a P ) = [ (a:)p 1 s+l s+l s+1 = 0 or (a'.')p 1. So aP = 0 implies (a:)p = 0 for each i. By our 1

1

minimal degree assumption on a we must then have (a:)p 1

=0

or (a~)p 1

= O.

356

The Homology of Hopf Spaces

(b) 8P

=0

for all ~ € A*(p) when I~I k

= pP

It suffices to consider ~

l

= (aPs )p

{:lp

l

. By the Cartan formula = [

First we show m

>0

0 k-s s k < s (apP )p k L s

0 for all m L O.

(apP)p

The Cartan formula gives the identities

for any m.r L

o.

So, a P m

all m L O. Thus (apP )p

s+l

m s+1 implies (apP )p

=0

=0

(aP

s+1

)Pp

m+s+1

since the theorem holds in degrees

=0

for

< lal.

We now use (a) and (b) to show {:lP 7- 0 Suppose (:l

€

P

H*(X;IF). Pick an indecomposable x P

< x,{:l > 7- O. Let 2:

€

~~

¢ be

2p sn(X;1F H ) where . P

s

the operation associated to oPP n

s (-1) tpP n-rr ( t } Qt' As in §33-1 we can choose a

sub Hopf algebra B C

H*(X;IF ), invariant under A*(p), where x E B but ~(z) € B @ B Since 0 p

=0

on H*(X;lF we have Qs(z) p)

~(p)¢(x)

= x0 ... 0x

for s L

+

o.

=0

So. ¢(x) is defined and

2: 1m pPs n--y(t) in

p

*

)

@ H (X;lFp //B'

i=1

Since

< x,{:l > 7< 1m pP

we can deduce

< ¢(x).{:lP > 7-

s

0

n--y(t),{:l

>=0

O. See. for example, the proof of Proposition

34-2A for the details of this argument.

Q.E.D.

§43-4: Proof of Corollary 43-1

Let I

={ a

€

H (OX;IF )

*

p

I

aP

=0

}. Then I is a Hopf ideal of H (OX;IF )

*

p

over the Steenrod algebra. It is obviously an ideal. For the remaining properties we appeal to the arguments of the last section. For the Hopf

X: K·Theory

357

ideal property see the argument used in step (i) of the last section. For invariance under the action of the Steenrod algebra see the formula at the beginning of step (ii) of the last section. Let P -_ H*(OX;IFp )/ I Dualizing. p* C H*(OX;IF ) is a sub Hopf algebra invariant under A*(p). Let T* =

H* (OX;lFp)//p*.

p

Finally let T = T-.

§43-5: Proof of Proposition 43-1 We use secondary operations. The required secondary operations are defined on an element x E T* by picking a representative of x from H*(OX;IF ) P

and defining the secondary operation on the representative. The projection H*(OX;IF ) ~ T* enables one to assume that the secondary operation takes p

values in T*. Except in one case which we will mention, the fact that the Bockstein D acts trivially on H*(OX;IF ) ensures that the required operap

tions are always defined on the representatives in H*(OX;IF ). Let p

p

= 2.

We first use the secondary operation ¢1 associated with the

Adem relation

to deduce that (*)

Q4n(T*)

= Sq2 Q4n-2(T*)

for all n ~ 1

~ associated wi th In particular, Q4 (T* ) = O. We then use the operation '+'2

the relation SqlSq4n to deduce that

In this case, however. it is not automatic that ¢2 is defined. We must be sure that there exists representatives on which Sq2Sq4(n-1) acts trivially.

Now. any element

of Q4n(T*) has a primitive representative

in

H*(OX;IF ). For P(T*) ~ Q(T*) is surjective and. dualizing Corollary 43-1. p

H*(OX;IF ) ~ T* p

@

p* as coalgebras. But Sq2Sq4(n-1) acts trivially on any

4n primitive element x E H (OX;IF). For. by Theorem 1-5B. Sq2Sq4(n-l)(x) E p

8n-?-.* P -H (OX;IF ) is either indecomposable or a perfect square. But p

358

The Homology of Hopf Spaces

Sq2Sq4(n-1) (x) cannot be indecomposable because of (*) and Sq2Sq2 = Sq lSq2Sql. And Sq2Sq4(n-l) (x) =

i

the relation

is not possible since

i

Sq1Sq4n-2(y). For

p odd we make analogous arguments.

As in §34-2 we first

secondary operations associated with the Adem relations '"" s n-..,(t) Q L (-1) P

t

use

6PP =

to deduce Q2n(T*)

= p 1Q2n- 2p+2 (T* )

if n

= 1 mod p

So it suffices to show 2pn * Q (T)

=0

for n

~

1

pn To do so we use the secondary operation associated with the relation 6P

= PpnQO -

1 p-1 p(n-l) Q P and argue as in the p 1(P)

=2

case.

359

§44: K-THEORY

In this chapter we will prove that the K-theory of a finite H-space is torsion free. The results of this chapter are based on Kane [8]. In turn Kane [8] is a generalization of the work of Petrie [1]. §44-1: Main Results In this chapter we will prove THEOREM: Let

(X.~)

be a I-connected mod p finite H-space. Then K*(X)(p) is

torsion free. The possibility of such results was first demonstrated in Hodgkins [1] for the case of Lie groups. His proof was a case by case calculation using IFp cohomology operations and spectral sequence arguments. When p is odd Hodgkins approach is based on the operation QI' It follows from a spectral sequence argument that K*(X)(p) has no p torsion if QI: QOddH*(X;lFp) . ah- H'i r-zeb ruch Qeven H* (X'. IFp ) t s sur j ec t t. ve , Th ere are two steps. An At iy spectral sequence argument tells us that K*(X;IF ) is an exterior algebra p

on odd degree generators. And a Bockstein spectral sequence argument then

. algebra on odd degree generators as tells us that K* (X)(p) is an exterlor well. Lin [5J has verified that Hodgkins approach can be used to eliminate odd torsion in the K-theory of all I-connected finite H-spaces. He demonstrates that the above Q condition holds. The arguments are an extension I of those in §35. Unfortunately, the Q condition is not true for p = 2. I The exceptional Lie groups E and E provide counterexamples. The failure 8 7 of this Q condition provided a significant obstruction for Hodgkins to I overcome in his arguments. In general. the above approch does not seem viable as a way of eliminating 2 torsion in K* (X)(2)' We will adopt an alternative approach to K*(X)(p) which was pioneered by Petrie [1] and then extended to a general argument in Kane [8]. We employ BP theory. For the Conner-Floyd isomorphism tells us that the Ktheory of a space X is determined by its BP theory. We will work with BP

360

The Homology of Hopf Spaces

and K homology rather than cohomology. This enables us to approach X via its loop space OX. There is an Eilenberg-Moore type spectral sequence {E r} BP.. {OX) converging to BP.. (X) with E = Tor (BP.. ,BP.. ). And because H.. (OX)(p) 2 BP.. (OX) has no p torsion, BP.. {OX) and Tor (BP.. ,BP.. ) have relatively pleasant structures. Notably, Tor vI' This tells us that

BP.. {OX)

(BP.. , BP.. ) has no p torsion when we invert

BP,,(X)[~I]

has no p torsion. Since inverting vI is

compatible wi th the Conner-Floyd isomorphism we deduce that K.. (X) (p) has no p torsion. We might also remark that the arguments and results of this chapter are not limited to finite H-spaces. The arguments of Kane [8J show that, for

K.. (X)(p) is torsion free if

any I-connected H-space of finite type,

H.. (OX)(p) has no p torsion. All tensor products, unless otherwise specified. are over

~.

§44-2: The Conner-Floyd Isomorphism The Conner-Floyd isomorphism enables us to use BP theory to study Ktheory. It is defined using the Todd genus map. This is a map Td : MU.. --+

~

characterized by the fact that

=I

Td[lCpnJ

If we localize at p and restrict Td to BP.. Td(v )

=I

Td(v n)

=0

l

= ~(p)[vI,v2, ... J

c

MU.. (p) then

for n l 2

This is easily established using the identities v

n

m n

pm n

L

i+j=n

~

n+1 (Work by induction on n). The Conner-Floyd isomorphism tells us that

.

K..(X) = MU..(X) @MU K.. (X)(p)

= BP.. (X)

~

@BP..~(p)

Observe that we can extend Td to a map Td: BP.. Conner-Floyd isomorphism then tells us that

[~l]

--+

~(p)'

The extended

X: K-Theory

K.. (X) (p)

= BP.. (X)

[&J

361

[1 ]71(P) BP.. -v

@

1

We also know slightly more about the relationship between BP and K theory. The inclusion 7l(p) C 7l(p)[v I,v2 •... ] imbeds ~(X)(p) as a direct summand

K..(X) (p) CBP.. (X) [&1] So. to prove K.. (X)(p) is torsion free. it suffices to show Reduction I: BP.. (X)

[&1]

is free of p torsion.

§44-3: Rothenberg-Steenrod Spectral Sequence There is a spectral sequence relating the BP homology of

nx

to that of

X. It was first established by Milnor [2] for the case of ordinary homol-

nx'

ogy. It arises from the fact that X has the same homotopy type as B classifying space of

quence {P } where P is the n fold projective space of n n

= BP.. (Bnx )

induces a filtration of BP.. (X) 4

th

the

nx. The space Bnx is filtered by an increasing se-

nx

(see §6} This

and. hence. a bigraded 1

st

and

quadrant spectral sequence {E where r} (.. )

E

2

(**) Eoo

= TorBP.. (OX){BP... BP.. )

= EOBP.. (X)

I t should be noted that (.. ) is only valid because H.. (OX)(p) is torsion free. {E

r

(See §7-2) We can localize {E r}

and obtain a

spectral

sequence

[l]} where VI

..

( ),

(**)"

E2

BP {nX} [-1 ] 1 1 VI (BP [-] .BP [-])

[&J = Tor"

Eoo[&J

= EOBP.. (X)

[l]

To show that BP (X) .. VI

Reduction II: Tor

BP.. (nX)

.. VI

.. VI

[&J

is torsion free it suffices to show

[lVI ](BP

[1] [1] - .BP - ) is torsion free .

.. VI

.. VI

[l]} are

For the differentials in the spectral sequences {E and {E r} r VI

tor-

The HomorogyofHopfSpaces

362

sion valued. Consequently.

E2[~J

=

Eoo[~J

= EOBP*(X)

[~J

is also torsion

free. In turn, BP*(X) [~J is torsion free.

To calculate E2 and E2 [~J we use our knowledge of BP*(OX) as detailed in §43-2. If we write BP*(OX) as in Proposition 43-2A then Tor can be calculated as follows. Let E r

= =

@ E(sw)

= BP*[~]/J

BP.. (OX)

(BP*,BP*) and its vI localization

where sw has bidegree (1. Iwl)

¥-'¥

@ r(tW)

where tw has bidegree (2.2p!wl)

¥-'¥1

For each W € ~1' given R w let

~-

L Ai~i - L Wjdj as in Proposition 43-2A.

Q = L Aiswi w Define a differential d on E @ r by the rule

d(sw)

=0

d(~i(tW»

= Q~~i_l(t~)

(ii) Tor

BP*(OX)

(BP*,BP*)

= H(E @r

~

for ~

~1

€

= d(x)y

It is extended to products by the rule d(xy) LEMMA: (i) Tor

for W €

+ (-I)!x 1xd(y).

@BP.. )

BP.. (OX) [-1 ] 1 1 vI (BP [- ].BP [-]) * vI * vI

1

= H{E@r@BP*[-]). vI BP*(flX)

(BP.. ,BP.. ) = H(T @BP*(OX)BP*) where T is any BP*(OX) free resolution of BP* Let T Proof: We will only do (i). By definition Tor

BP*(OX) @E @ r where d acts by the rule d(s~)

=w

for

~ €

~

A

d(~i(t~»

= Q~~i-l(t~)

for ~

€ ~1

, -1

where Q'iJ

363

X: K·Theory

s'iJ + Q'iJ =

, -1 [ s'iJ +

"is'iJ i· Then T 0BP*(f2X}BP* is the differential algebra considered above. Moreover T is acyclic. To see this

filter BP*(OX) by the skeleton filtration i.e. F BP*(f2X} q

= the

BP module generated by

*

[

i~q

BP.(f2X} 1

The filtration on BP*(f2X} induces a filtration on T. Moreover EO(T 0 W } p

H*(OX;Wp} 0 E 0 f where. if we write H*(f2X;Wp) §45-1 then

=x

d(sx}

=0

Then EO(T 0 Wp} A.

0 Wp[xi]/

1

(x. p )

A.I = E(sx.} 0 W [x.] I p 1 In each case H(A Q.E.D.

W' p

= Wp

i}

for x

€

~

for x

€

~I

0

Wp[~2] as in

Ai where each factor Ai is of the form

= E(sx.}

1

= Wp[~l]/I

0 f(tx

i}

d(sx . }

= X.1

d(sx i)

= X.1

1

1

(of degree O). So H(EO(T 0 Wp})

We now turn to calculating the homology of E 0 r 0

= Wp

BP*[~J.

and H(T)

Notably, we

will investigate the relation {R'iJ } in BP*(f2X} and the corresponding elements {Q'iJ} C E H(E

((I

r 0 BP

*

[1

vI

[1

[1]

[1

[1

r 0 BP* vI . To prove Tor BP*(f2X} "r J(BP* vI J,BP* vI J

((I

J)

is torsion free it suffices to show

Reduction III: {QJ} represents part of a BP* [~J basis of Q(E((IBP*(f2X) [~J} For H(E 0 r 0 BP*[l ]} is then a free BP

*

vI

{Q'iJ} to a

BP*[~J

then H(E 0 r 0

basis of Q(E

((I

[1] module. vI

BP*[~J)

Indeed, if we expand

using elements A = {DI,··

.D s }

BP*[~J) = E(A} ((I BP*[~J

§44-5: The Algebra BP*(f2Xl

in

In this section and the next we put restrictions on the relations {R'iJ} and BP (f2X) 0 W . They, in turn, will be used in §44-7 to put

BP~(OX) ~

*

p

restrictions on {Q'iJ}' The restrictions on {R'iJ} are of interest for their

364

The Homology of Hopf Spaces

own sake since they demonstrate that the algebra structure of BP*(OX) refleets not just the algebra structure of H (OX;W ) but also its structure

*

p

as a coalgebra and as a Steenrod module. As before, define a filtration {Fn} on BP*(OX) by Fq = the BP* submodule generated by n

=

This induces a filtration on BP*( IT OX )

i=l

coproduct map

[BP.(OX)

i~q

1

n

0 BP*(OX). Define the reduced

i=l

n

(flA)n: BP*(OX) - - @ BP*(OX) i=l by the recursive formula the reduced coproduct (OA)* (flA)n_10 (OA)* The algebra generators

= ~1

~

U

for n

~

2

of BP*(OX) are far from unique. This

~2

section will be devoted to proving that we can choose

t

PROPOSITION: Given

€

such that deg

~1

t

~

~1

to satisfy

0 mod 2p then there exists Y

€

BP*(OX) where

=t 0 ... 0 t mod F1 =~ - pY + v 1r 1(Y) + vld mod F2 where d

(i) (flA)p(Y) (ii) R t

First of all, the elements

~

is decomposable.

map to the algebra generators

We can assume that the elements of

~

~

of H*(OX;Wp)' ~ 0 mod 2p.

are primitive in degrees

For, by Theorem 1-5B we have LEMMA A: Pn H« (OX;Wp )

---

~

0'11* H (OX;WP ) if n

Secondly, we can assume that, in degrees

0 mod 2p

$ ~

0 mod 2p, the elements of

primitive mod Fl' In other words, each element of

~

~

are

can be lifted to an

element in H*(OX)(p) which is primitive. This follows from LEMMA B: P

---

n

H~(OX)( ~

p

)

~

P H (OX;W ) is surjective if n n

*

p

$

0 mod 2p.

Proof: First of all (*)

QnH*(OX)(p) is torsion free if n

~

0 mod 2p

For, in analogue to our description of BP*(OX) , we can wri te H*(OX) (p)

Z(p)[~]/J as an algebra where J is the ideal generated by {R = ~ t [ \ t i - [ wjd j I t € ~1}' It is easy to deduce that QH*(OX)(p) is the free Z(p) module generated by {t} modulo the relations induced by {R }· t

X: K·Theory

365

=0 mod 2p.

But all such relations lie in degrees

Obviously PH*(OX)(p) C H*(OX)(p) is torsion free. It follows from (*) and Corollary l-SB(i) that (**) rank PnH*(OX)(p) = rank ~H*(OX)(p) if n

;I;

0 mod 2p

Lastly. consider the commutative diagram IF

@

Both

L

p

p ------>

PH*(OX; IFp )

and p are injective (For

isomorphism in degree

E

>fi

l

0 mod 2p.

where deg 'iJ

PH*(OX)(p)' Since p(xP )

=0

p

is an

=x

@••• @

mod 2p.

By Lemma B. x = T('iJ l)

E

in H*(OX;lFp) we can define y E H*(OX)(p) by xP y

(b) x P

Lp

Hence p is an isomorphism in these

;I; 0

In particular. y satisfies (a) (0,1,) (y)

use Lemma A). By (*) and (**)

Q.E.D.

degrees as we11 . Now pick 'iJ

;I;

L

p

x

= py

Pick Y E BP*(OX) where T(Y)

= y.

We want to show that Y satisfies (i) and

(ii) of the proposition. Property (i) follows from (a). Regarding property (ii) observe that (b) implies (c)

,pP = pY

+ VIZ mod F

2 for some Z E BP*(OX)

So. in QBP*(OX). we have pY + VIZ Quillen operation r have (d) prl(Y) + pZ

=

0 mod F Taking the image under the 2. and using the fact that rl(v = p and r l(F2) C F l we l l)

=0 mod FI

in QBP*(OX). Since QBP*(OX) mod F I agrees with QH*(OX)(p) and since QH*(OX)(p) is torsion free in degrees ;I; 0 mod 2p (see (*) in the proof of Lemma B) we actually have (e)

rl(Y) + Z

=0 mod FI

in QBP*(OX). Consequently. in BP*(OX). we have (f)

Z

=-rl(Y)

+ d mod F

I

Property (ii) of Y follows by combining (c) and (f).

366

The Homology of Hopf Spaces

§44-6: The Algebra BP*fQX) 0 W

p

When we reduce mod p we can prove a strengthened version of Proposition 44-4.

We first need to be more precise in our choice of elements in

BP*(OX). (I) The Elements XsLll We begin by making a very specific choice of the algebra generators We begin with the algebra generators

~1

~1'

of T C H (QX;W ). As observed in

*

p

§43-1 the dual Hopf algebra T* is primitively generated. Choose a Borel decomposition T*

=0

A where each A is generated by a single primitive s sES s h

element as' Suppose as has height p s t

A

gives a basis of

= { asP I

The set

< hs

s E S. 0 ~ t

P(T*). Dualizing T*

=0

A we obtain a Hopf algebra sES s

decomposition of T. In particular. if xs{t)

= the

t

dual of asP

Then

= { xs(t) I

~1

s E S, 0 ~ t

< hs

give a basis of Q(T), If we let Xs{t) Then. we

can

=a

assume

representative of xs{t) in BP*{QX) ~1

Xs(t)

s

E

S.

The above choice guarantees that the elements of degrees

*0

mod 2p.

For the elements of

~1

0 ~1

~ t < h s }. are primitive in

are either primitive (namely

{xs(O)} or divided pth powers of primitives (namely xs(t) for t

> 0)

latter have degrees ;: 0 mod 2p, This primi tivi ty restriction on enough to ensure that Proposition 44-4 applies to

~l

The is

~1'

(II) The Elements YsLll We now pick another group of indecomposables {Ys(t)} in BP*(OX). Again we begin with T and T* . The algebra isomorphism

~(QX;Wp) ~

T 0 P of

Corollary 43-1 dualizes to give a coalgebra isomorphism H*(QX;W ) ~ T*0P*. p

Under this (non unique) isomorphism we can consider T* as lying in H*(QX;W For each as E T* we have deg as p), Let

= 2pn+2

by Proposition 43-1.

367

X: K·Theory

b So. pl(b

s}

= asp. t

s

= ppn(as }

It follows. by induction on t. that

t

a P

LEMMA A: pP (b P ) s

t+l

s

B

t

= {b s P I

s E S. 0 ~ t

< hs }

LEMMA B: B is a linear independent set Proof: In view of Lemma A we need to know that the set A+ h

s

t

I

0 ~ t ~

is linearly independent. We have already examined the set A C A+. We

s}

are left with A+-A a

{a p

=

(o)(p

If a P= 0

2n. n)

= {asp

hs

}. Let a

= asP

=0

and x

then the secondary operation

is defined on a and satisfies

fact that xP

h s -1

for x E T.

<

¢ex} .xp

= xs(hs-l}. ¢

Suppose deg

associated to opn

> tf.

=

0 contradicting the

Q.E.D. t

Pick representatives {B in BP (OX) for the elements {b }. Then B p s} * s s is a representative for each 0 ~ t < h Pick elements {Ys(t}} which have s' t

a Kronecker pairing with the elements {B p }. In other words s

t

Since the elements {B p } project to a linearly independent set in s

PH (OX;W ) it follows that the set {Y (t}) project to a linearly indepen-

*

p

dent set in

s

QH~(OX;W ~

p

}. In particular. the elements {Ys (t}) are indecom-

posable in BP*(OX}. (III) Relations between {Xs(t)} and {Ysilll If we reduce mod P we can obtain a non singular pairing involving

{Xs(t}} and {Ys(t}}. Expand {Ys(t}} to a set jects to a basis of

~(OX;Wp)'

~

C BP*(OX} 0 Wp which prothe set of non zero

As in §43-2 let D

=

monomials in the elements of W of weight l 2 which do not include the p powers of any element from WI'

th

368

The Homology of Hopf Spaces

Proof:

In both cases 4> or If/ project to a basis of QH*(OX;IFp) while D

projects to a basis of the decomposable elements in H (nx;IF).

*

We want to expand our relations R~ basis

~

U D. We

can

L Wjdj

-

in terms of the

use Proposition 44-4 to deduce the following

PROPOSITION: Given ~ € 1f/

= XsCt)

= ~ - L ~i~i

Q.E.D.

p

1

then the coefficient of Ys(t) in ~ is VIP

t

if ~

and 0 if ~ ~ Xs(t) and deg ~ ~ deg XsCt).

As we will see in the next section our main theorem follows easily from this result. To prove the proposition we dualize and work in BP*(OX) @ IF . p

We have already chosen the representatives {B for {b Choose represens}' s} tatives {As} for the elements {as}' To prove the proposition it suffices to show mod F t P +1

Perhaps a comment is in order about such dualizing. In general, some care must be shown when dualizing between BP*COX) @ IF and BP*(OX) @ IF ' p p Notably. if we dualize the basis ~ U D of BP*COX) @ IF

then Bs P p

t

is not

t

necessarily the element dual to Ys(t). For B P may not be non zero when s evaluated on elements of D. It is a question of what appears in the coprot

duct

(OM)*(B p). s

Lemma D eliminates any

such problems

in degree

~

[xs Ct)P I· Consequently. duali ty works in the desired manner in these

degrees and we obtain the proposition from Lemma D. To prove Lemma D it suffices to prove (OM)*(B ) s

p-l

= VI L ~~) i=1

A p-i @ A i mod F 2 S

pIS

t

For since we are working mod p we have (OMt(B p ) s (*) is equivalent to asserting (*)' if d is a monomial of weight

o mod F2

then d

= Xs (O)p

and

~

2 in the elements If/ and

< Bs.d > $

< Bs .Xs (O)p > = VI'

Since If/ U D is a basis of BP*(OX) we can certainly choose B to satisfy s

369

X: K-Theory

Moreover, as observed in the discussion following Proposition 43-2A, the only monomials of weight are {~

I

~

2 in

~

~l}' Given ~

€

which cannot be expanded in terms of D

~

~l we have

€

< Bs'~ > ~

the same degree as Xs(O). In particular deg Proposition 44-5 to such

< Bs'~

=v l < Bs,rl(Y» =v l < bs,pl(y) 1

= vl< P

remark that,

< Bs,vld

)

(by 44-5)

>

)

in the second equality above,

< Bs,vld

)

mod F2

= v l < as0···0as'~--* (y) ) < a s 0 ... 0a s ,~(y) ) to if and

) because

~ has apply

mod F2

(bs)'y

= v l < asP,y By 44-5 again

can

We have

~.

= < Bs.vlrl(Y) > + < Bs.vld

)

0 mod F unless 2

0 mod 2p. We

~ ~

= v l < Bs,d

)

~

only if ~

= Xs (0).

We should

we were able to eliminate

= O.

v l < bs,T(d) )

The last

fact is due to b s being primitive and T(d) being decomposable. §44-7: Proof of Main Theorem By the reductions performed in §44-2.§44-3 and §44-4 we can reduce to showing

{Q~}

represents part of a

reduce mod p and show

{O~}

BP*[~J

basis of Q(E

represents part of a

@

BP*[~J)'

BP*[~J

@

We will

IFp basis of

Q(E 0 BP*(OX) [~J 0 IFp)' Define a map s: BP*(OX) s(Y) where Y

=[

~

1

1

1

=[

1

J J

~

s(~) If Y

Q(E 0 BP*)

a.sX.

a.X. + [w.d. is the expression of Y in terms of the BP~ basis

U D. In particular

~

=[

= Q.p

aiY i + [ Wjd is the expansion of Y in terms of ~ U D then s(Y) j

=

[ais(Y i)· If we reduce mod p or invert VI then all of the above is valid. Index the elements {Q.p} as {Ql'~"" ,Qk} where IQll

s

I~I ~ ...

s

lOki. Index

the elements {Ys(t)} as {Y l,Y2,· .. ,Yk} where Ys(t) = Yi if 0i = Q.p and.p = X (t). Let A = (a. ) be the k x k matrix where a .. is the coefficient of s

s(Y

1

i)

j

IJ

in Qj' It follows from Proposition 44-6 that if we reduce mod p then

370

TheHomorogyofHopfSpac~

A

o

sk

vI

If we invert v I as well then A is invertible. Since sl[J is a BP* [~J basis of Q(E

@

BP*[~J

@

IFp) and since A is invertible over

BP*[~J

@

IFp

0 IFp

it follows that we can replace the elements {s~i} in sl[J by the elements {sQi} and obtain another BP

*

[1.] vI

0 IF

p

basis of Q(E 0 BP

*

[1.] vI

0 IF ).

P

PART XI: TIIE HOPF ALGEBRA H*(X;IF ) p

In the next three sections we study the structure of H*(X;IF ) as a Hopf p

algebra over A*(p). We have three structures to consider: algebra. coalgebra and Steenrod module. These structures are not independent.

Special

assumptions about one of them inevitably forces restrictions on the others.

We have already seen a good example of such interaction in the case

of Thomas' structure theorems in §17. There. the assumption that H*(X;IF ) p

is primitively generated forced strong restrictions on the algebra and A*(2) structures. We now undertake a similar programme in the case of p

. deduce structure theorems for H* (X;IF under a primitively odd. We wIll p) generated hypothesis. Conversely, we will study how H*(X;IF ) fails to be p

primitively generated when these structure theorems are violated. In each of the three chapters we will use a different technique to study the Hopf algebra structure of H*(X;IF ). In §45 we will use the structure theorems p

from §35. In §46 we will use secondary operations. In §47 we use BP operations.

373

§45: TIlE ALGEBRA H*(X:IF ) p

Let p be odd. This chapter is really an extension of §35. We will use the structure theorems for Qeve nu* (X:1F ) obtained there to deduce strucp

ture theorems for H*(X;IF ) as a Hopf algebra over A*(p). The results of p

this chapter are due to Lin. The reference in the literature is Lin [5]. Throughout this chapter we will make the following ASSUMPTION: p is odd and

(X.~)

is a I-connected mod p finite H-space

§45-1: The Sub Hopf Algebra f Let

A = L op~2n+l(X:1F ) p

n~l

f

= the

algebra generated by A

Obviously. r C H*(X;IF ) is concentrated in even degrees. Our interest in f p

arises from the following properties. TIlEOREM A: (Lin) f is a Hopf algebra invariant under A*(p) THEOREM B:(Lin) Q(f) ~ Qeve~*(X;1F ) p

Proof of Theorem A We begin by noting that the Frobenius pth power map (:: H*(X:IF ) -> p

H*(X:IF ) satisfies p

(a)

Cf

= CH*~l

For p odd means that we have a factorization H* (X:lF p)

(:

----+

H* (X p)

~ * r:lFr:

H (X;lFp)//E

374

The Homology of Hopf Spaces

where E C H*(X;W)

is

p

the

sub algebra generated by

the odd degree

generators in any Borel decomposition of H*(X;W ). And Q(f) ~ p

-4Q(H* (X;Wp)//E)

Qeve~*(X;Wp)

are surjective maps.

(b) f is invariant under A*W It suffices to show that A is mapped to f under A*(p). As we observed at the begining of 935, if we define

~: H*(X;W ) ~ H*(X;W ) P P ~(x)

then ~*(X;W) is

[

OPn (x )

if x € H2n+ 1(X;W )

xP

if x € H2n (X;W )

p p

invariant under A*(p).

p

inclusion

=

A C

~*(X;W )

C

p

Moreover,

we also have

the

f

To establish the second we use (a) plus the fact that ~*(X;W ) cAe p

(c) f is a coalgebra It suff ices

2q+1 (X;W H

to show ~A C f @ f.

Suppose

2).

~(y) where zi ,Zj deg Zj

So choose x

€

= 2n j+1

=L w,@zi i

L z.0w j

+

j

1

J

odd H (X;Wp) and wi ,wj € Heven(X;Wp ).

2n.+1 and 1

then

~(x)

= L wi P@ i

n.

OP l(zi) +

n ,

L oP j

J(z.) @ w P J

j

Using (a) we conclude that ~(x) € f@f. Proof of Theorem B It follows from Theorem 35-1 that Q(f) -4 Qeve~*(X;W ) is surjective. p

Regarding injectivity observe that A and, hence Q(f), is concentrated in degrees

=2 mod 2p.

We now apply the following

LEMMA: Given A C B commutative associative Hopf algebras over Wp then Q(A) ~ Q(B) is injective in degrees t 0 mod 2p.

375

XI: The Hopf Algebra H' (X;IFp)

Proof:

Given

x€

Qn(A} where n ~ 0 mod 2p. pick a representative x € A.

L Ai.

Let A' = the sub Hopf algebra generated by

o ;t.

A x € P( I I

Then

i
A,)

A B By Corollary 1-4. II A, ellA' is an inclusion. Consequently 0 ;t. x n B

€

n B

P (II A,). By Corollary 1-5B. O;t. x € Q (II Consequently, O;t. x € A,). Qn(B}. Q.E.D.

§45-2: The pth Power Map

Ww now demonstrate how the restrictions on Qeve~*(X;f ) obtained in §36 p

imply strong restrictions on the algebra structure of H*(X;f ). For any p

Borel decomposition H*(X;f } = 0 A. with generators {ail we have p

THEOREM:(i} If la

i

I ;t.

2(ps+

(ii) If lail = 2(ps+

1

+p+l) then a t = 0 +p+1} then a. p 1

2

= 0

We might remark that there are no known examples of I-connected mod p finite H-spaces for p odd where xp ;t. 0 for any x

€

~*

H (X;f ). On the other p

hand. when p = 2 or when p is odd and X is not simply connected then there are examples. namely, X= Spin(n) and X = PU(n). respectively. In §47 we will demonstrate, using BP operations. that

xP

= 0 for all x € H*(X;f ) p

provided H*(X;f } = U(M). P

Proof of Part (i) of Theorem k We will show a P ;t. 0 forces lail = 2(ps+ ... +p+1}. Let D denote the k i fold decomposables of H*(X;f }. As usual. we wri te D2 = D. Since A-> p

Qeve~*(X;f } is surjective we can write p

where x

€

A and d E D. Then xP

=a. p ~ 0 mod nP+ 1 1

Regarding the second inequality observe that the non zero monomials in the Borel generators {a

j}

of weight ~ p+1 give a basis of ~1. We can write

376

The Homology of Hopf Spaces

Ixl = 2pn+2

for some n ~ 1. Let

= p 1ppn+ 1(x) = ppn+1(x)

Furthermore p 1(y)

= 2p2n+2.

Then Iyl lows that

Y # 0 in QH*(X;f ) p

For x

r

€

implies y

=2

~1. For Iyl

€

r.

€

Thus y

=0

in Q(r)

=Qeve~*(X;fP )

implies that y

mod 2p and r is generated by elements of degree

mod 2p. But then we must also have x P

=

=2

pl(y) € DP+ 1 contradicting (*).

2p2 n+2-_* -H (X;f ) By the binary theorem in §36. Q

~ 1. So (**) forces Ixl

.. + p2+ 1 for some s

= 0 unless p2n+1 = p s+ 1 = 2pn+2 = 2(ps+ ... +p+1). P

+

Proof of Part (ii) of the Theorem The proof is analogous to the above. Assume la. a

i

=x

+ d for some x

A. d

€

1

D. We have

€

1= 2(ps+ ... +p+1).

Write

2 2 2 1 xP = a. p mod if + 1

Choose y € r where Iyl

= 2(ps+1+ ... +p2+1)

2 xP . So it suffices to show that yP y

=L j

=0

and pl(y)

= xp .

Then Pp(yp)

2 mod D2p +1. We can write

a. + J

d

where a j are Borel generators and d is decomposable. By part (i) yP = So we want to show 45-1 CH*(X;f2) ments of degree Hence

(iP

€

(iP

= cr.

2

=0 mod tiP +1.

As observed in the proof of Theorem

So we can assume d € r. But r is generated by ele-

=2 mod 2p.

2+ tiP p.

(iP

So d € D and Idl

=2 mod 2p

forces d € tiP+

1.

Q.E.D.

§45-3: The Algebra H (X;f 1

*

p

Next we consider the coalgebra structure of H*(X;f ). It is more conp

venient to dualize and study the homology algebra H (X;f ). In addition to

*

p

the assumptions made at the beginning of §45 we will also make the additional

377

XI: The Hopf Algebra H* (X;IFp)

ASSUMPTION: H*(X;W ) is associative p

We will study

the commutator map [ , ]: H*(X;Wp) @ H*(X;Wp) [a,/3] = a/3 - (_l)la l l/3lf3a

~

H*(X;Wp)

Actually we will restrict our attention to the primitives. The commutator map restricts to give a well defined map

The bracket product satisfies the usual two relations: (i) [a,/3] (ii)

= -(-1) lal 1/31[/3.a]

[a.[/3.'Y]

= (-1) la

(anticommutativity)

l l/3I[/3.[a.'Y]

+

[[a./3],'Y] (Jacobi identity

We also have the Frobenius pth power map (: PH (X;W )

*

p

~ PH~(X;W ~

p

)

The above maps give PH (X;W ) the structure of a restricted Lie algebra.

*

p

In all known cases. these maps completely determine the algebra' structure of H (X;W ). For. in all known cases. x P

*

p

=0

~*

for all x E H (X;W ). Conp

sequently. H (X;W ) is primitively generated. (see Corollary 1-5C) But, as

*

p

noted in §2-3. this means that H (X;W )

*

p

= VPH

*

(X;W ). the universal envelp

oping Hopf algebra of the restricted Lie algebra PH*(X;W p

if examples are produced in which x bilities are

so

limited

p)'

Moreover, even

~*

0 for some x E H (X;Wp) the possi§45-2) that the inclusion VPH*(X;Wp) C

(see

~

H (X;W ) will have a very small cokernel.

*

p

The structure maps of PH (X;W ) satisfy

*

p

2 LEMMA A: Given a./3 E P ddH (X;W ) then a o * p

LEMMA B: Given a,/3 E P H (X;W ) then a P even * p

= /32 = [a,/3] = 0 = /3P = [a./3] = 0

For Lemma A see Corollary 12-2C. Lemma B follows from the results of §36. For.

by

H (X; W ) and Qeve~*(X; W ). we have even * p p 0 only if n = 2 mod 2p. So we-can assume deg a = deg /3 = 2

the duali ty between P

P H (X;W ) 2n * p

~

mod 2p. But then deg a P

= deg

/3P

=0

mod 2p and deg [a./3]

=4

mod 2p.

Consequently, when we study PH (X;W ) we can ignore ( and we need only study

*

p

[ , ]: PoddH* (X;Wp ) @ PevenH~(X;W) ~ p ~ P0 ddH~(X;W ~ p)

378

The Homology of Hopf Spaces

Moreover, by using rules (i) and (ii) for the Lie bracket product, all the non zero commutators in PH (X;W ) reduce to ones of the form

*

p

[ .... [[a'~lJ'~2J··'~kJ where a € PoddH*(X;W p) ~l""~k € PevenH*(X;Wp) and the order of the {~i} is immaterial. So we have reduced our study of H (X;W ) to the question of these commutators.

*

In §46 and

p

§47 we will determine families of non

trivial commutators in PH*(X;Wp)' §45-4: The Sub Hopf Algebra

In closing our discussion of H*(X;W ) let us note one further property p

of the sub Hopf algebra r C H*(X;W ) defined in §45-1. p

PROPOSITION: If H (X;W ) is associative then r and H*(X:Wp)//r are primi-

*

p

tively generated Hopf algebras. So the Hopf algebra H*(X;W ) is an extension p

W p

->

r

->

H*(X;W )

P

->

H* (X;Wp)//r

->

W

P

of two primi tively generated Hopf algebras. Since p is odd and Q(f)

~

even * . H*(X;W ) H (X;Wp) 1 t follows that p / / r is an exterior algebra on odd

Q

degree generators. Consequently, the extension splits as algebras. We have a (non canonical) isomorphism H*(X:W p)

~

r @ H*(X;Wp)//r

of algebras. On the other hand, the above isomorphism does not extend to the coalgebra structure of H*(X;W

p)'

*

For primitive elements of H (X;Wp)//r

may have only non primitive representatives in H*(X;W ). P

This non trivial extension problem is the dual of the one arrived at in §45-3. The question of commutators in PH (X;W ) being non trivial is equi-

*

p

valent to one of the algebra generators of H*(X;W ) possessing non trivial terms in its coproduct. For example,

p

~

= [a3,a4 ]

# 0 in P

7H*(X;Wp) would

be equivalent to H* (X;W ) having an algebra generator x- E H7 (X;W ) where p {P

* p) @ H (X;Wp) See Kane [22] for a more detailed discussion of the above correspondence. It is more

--* ~ (~)

= x3@X4

+ possibly other terms in H* (X;W

XI: The Hopf Algebra H* (X/UV

379

convenient to work in terms of commutators in H*(X;W and we will do so. p) Proof of Proposition

*

First of all, H (X;Wp)//f is primitively generated.

As already ob-

served. H* (X;Wp)//f is an exterior algebra. Since H* (X;Wp)//f is also coassociative it follows from the argument in §7-3 that H* (X;Wp)//f is primitively generated. Regarding f, we must use the structure theorem for Q(f) ~ Qeve~*(X;W ) p

given in §36 to deduce primitivity. It follows from §36 that Q(f) is cons

... i

centrated in the degrees 2(p + ... +p +... +p+l) where s

>

i

~

1.

It then

follows from Theorem 1-5B and Theorem 45-2 that P(f) is also concentrated in these degrees. We know f* is associative and we want to show i t is commutative and has trivial pth powers. Consider commutativity. If [a,~] is a non trivial commutator of the lowest possible degree then a and must be indecomposable and

[a,~]

must be primitive. However,

~

these re-

quirements are incompatible with our restrictions on the degrees of P(f*) and Q(f*). The Frobenius pth power map is handled in a similar manner.

380

§46: HOMOTOPY ASSOCIATIVE H-SPACES

In this chapter we again use secondary operations to study the module of indecomposables Qeve~*(X:f ). However. p

this time we impose a homotopy

associativity hypothesis on our spaces. With this added structure both our arguments and our results acquire a simplicity and elegance which is perhaps missing from previous applications of secondary operations. The resu I ts of this chapter originate from the work in Zabrodsky [1] and the extensions of that work obtained in Kane [4]. §46-1: Primitively Generated Cohomology In

the case of homotopy associative fini te H-spaces a commutativi ty

hypothesis on

H~(X:W ~

Notably. we have

p

) imposes srong restrictions on the algebra H* (X:Wp ).

THEOREM:(Zabrodsky) Let p be odd and

(X.~)

a I-connected homotopy associ-

ative mod p finite H-space. Then H*(X;W is commutative if and p) only if Qeve~*(X;f ) p

= O.

A few remarks concerning this theorem might be in order. (a) Observe that by Corollary 11-2. Qeve~*(X:f ) p

=0

if and only if H*(X)

has no p torsion. So the conclusion of the theorem could be restated as: H*(X:fp) is commutative if and only if H*(X) has no p torsion. (b) This theorem extends a previous result obtained by Browder [9] for the case of fini te loop spaces. Moreover. Zabrodsky actually proved a much stronger theorem than we have stated. He showed that. given an odd prime p ands any homotopy associative H-space (X.~). i f H*(X:W ) is primi tively p

generated then H*(X:f ) is a free algebra. In the case of finite H-spaces p

we can eliminate polynomial factors. So H*(X;f ) must be an exterior algebra on odd degree generators.

p

XI: The Hopf Algebra H* (X__IFp)

(c)

381

In the next chapter we will study just how H* (X;lFp)

fails

to be

primitive when Qeve~*(X;1F ) # O. p

We now prove the theorem modulo the proof of one proposition. It will be handled in §46-3. One implication in the theorem follows by purely

=0

algebraic arguments. Assume Qeve~*(X;1F ) algebra

on

odd

degree

generators.

p

Then

i.e. H*(X;IF ) is an exterior H (X;IF)

*

p

p

associative

forces

H*{X;lF to be an exterior algebra on odd degree generators. (See Theorem p) 2-IB) In particular. H (X;IF ) is commutative.

* *

p

Conversely, assume H (X;IF ) is commutative. We will use secondary operp

ations to show Qeve~*(X;1F ) p

= O.

We will use the secondary operation

degree 2n associated wi th the factorization op

n

(0) (p

¢ in

n). The point of

assuming homotopy associativity is that we can feed it in the operation to restrict the indeterminancy. In §46-3 we will prove the following variation of our previous structure theorems for secondary operations. PROPOSITION: Let

(X.~)

be a homotopy associative H-space. If x €

= xp = 0

p2nH*(X:1F ) satisfies pn(x) p

then the secondary operation

associated to the factorization

satisfies

it¢<x)

=

p-I

L ~J:» ~-i

i=l P

0 xi + o(y)

1

where Y € Ker(it 0 1 - 1 0

it)

Granted this result and. in particular. the restriction on y, we can easily prove Qeve~*(X;1F ) P

= O.

For the purposes of the next section we also want to locate the precise point. in our argument where the use of the commutativi ty hypothesis on is used in an essential way. So. until further notice, we will H*(X;lF p) only use the fact that (X,~) is a I-connected homotopy associative mod p finite H-space. In particular H (X:IF ) is associative.

*

p

Suppose Q2n H* (X;IF ) # O. Let f C H* (X;IF ) be the sub Hopf algebra dep

. . §45-I. The maps P2n (f) f1ned 1n

~

Q2n (f)

p

~

Q2n..* tl (X;IF ) are surjective. p

(See Theorem 45-lB and Proposition 45-4). So pick 0 # x €

2nH*(X;lF P p)

382

The Homology of Hopf Spaces

where x is indecomposable. We can assume

For suppose x P t- O. First of all.

Ixl = 2(ps+ ... +p+l}. For. by Theorem

45-2, (H*(X:IF ) is generated by elements of degree 2(ps+ ... +p+l) and of p

height p. Consequently. by Theorem 1-5B, P«(H*(X;IF }} is concentrated in p

= pP

degree 2(ps+ ... +p+l). Secondly, we can write xP s

plpP +···+p(x). Let x' €

p2(p

s-l

degrees

s

= pP

s

+ ... +p+l(x)

=

+···+p(x). Then x' is indecomposable since x

2 +... +p +l)H*(X:1F ) and PH (X;IF ) -+ OH (X;IF) is injective in p

* 0 mod 2p.

p

',*

Also (x'}p

=0

~'*

p

by our first fact. So replacing x by x'

we obtain an element satisfying (*). By (*) the secondary operation

¢ of

the above proposition is defined on

x and we have

~¢Cx)

(**)

where y €

p-l

= L

~~) x p- i @ xi + a(y)

i=l p

1

Ker(~@l - l~). We can assume that equation (**) holds in

Cotor * (IF ;IF ). For. if we define Cotor by the cobar construction as H (X'IF) P P , P in

§7

then 2 * Cotor *' (IF ;IF ) H (X'IF) p P

Ker (~@l - l~-*)/

-* Im JL

'P

In our case

p-l

L

~~) x p-i @ xi

i=l p

€

1

Ker (~@1 - l~-*) since x is primitive.

And a(y) € Ker (~@l - l~-*) by the above proposition. Consequently, we have the equation { a(y) } in Cotor. Moreover (****) x p- i @ xi that

= ~(y)

< x,a > t- O.

then X@ ... @X Then

< y.aP > < y,JL(p)(a@ < JL(p)(y),a@

@a) @a

> >

= JL(p)(y).

Pick a

€

XI: The Hopf Algebra H* (X;I'P)

< x0 ... 0x, < x,a >p

a@ ••• 0a

383

>

f. 0

AndaP To 0 contradicts Lemma 45-3B. So no such y exists and (****) is established On the other hand, i f H (X:IF ) is commutative we can force

= O.

{o(y)}

*

For let a be as above and let

p ~

= aO.

Let 0 C H (X:IF ) be the

sub Hopf algebra. invariant under O. generated by a and

o = IFp[a]/(aP) The relations a P lation

[a.~]

~P

=0

*

Then

~.

p

@ E(~)

follows from Lemmas 45-3A and 45-3B. The re-

= 0 follows from

the fact that H*(X:lF is commutative. p)

Dualizing, we have a quotient Hopf algebra H*(X;IF ) ~ 0* and a map Cotor ~

Cotorn*(1F ,IF) of 0 algebras. "

p

p

p

Both (***) and

(****)

are valid in

Cotorn*(1F ,IF ). On the other hand "

p

p

{o(y)}

(*****)

To see this observe that 0*

= IFp [a*]/«a*}p)

over. one can compute that if A CotorA(1F ,IF ) p p

= E(sa*)

CotorB(1Fp .IFp )

= IFp [s~*]

= o{y} = 0

@ IF

p

= IFp[a*]/«a*)P).

[ta*]

i=1

*

(a)p

-i

@

*

i

= O.

= E(~*) th~n

where s~* has bidegree (1,2n-1)

(a ) }. Since Cotor 0*

2.2pn-1 conclude Cotor O*

B

where [ sa* has bidegree (1,2n) * ta has bidegree (2,2pn)

The respective classes on the cobar level are sa*

{ p-1 L

*

@ E(~ ) as coalgebras. More-

Thus {y}

= {a},

s~*

= {~},

ta*

= CotorA @ B = CotorA @ CotorB

we

= O.

§46-2: Commutators in H (XilF 1 IE P We can refine the arguments from §46-1 and obtain a more precise hold on the relation between the algebra structures of H*(X:IF ) and of H (X;IF ). As in §46-1 assume that

*

p

p

(X,~)

is a I-connected mod p fini te

homotopy associative H-space. Suppose also that Q2~*(X;1F ) To O. Then, by p

Theorem 46-1, H*(X;lF cannot be commutative. In many cases, by re-examp) ining the proof of Theorem 46-1 we can discover precisely how H*(X;lF p) fails to be commutative.

384

The Homology of Hopf Spaces

THEOREM A: Let p be odd and

a I-connected homotopy associative mod p

(X.~)

finite H-space. Suppose x € P~*(X;~ } is indecomposable and pn(x}

= xp = O.

p

Then, for any a

< x,a

P2nH*(X;~p} satisfying

€

) f. 0

we can find non zero elements {~s}l~s~p-l in PH*(X;~p} defined recursively by the rule

= ao = [~s_l,a]

~l ~s

for s ~ 2

We might remark that the homotopy associativi ty hypothesis ensures that

peve~*(X'~ } ~ Qeve~*(X'~ } is onto. (See 45-4A) Also, the requirement 'p

that x P

=0

'p

is only required as an extra hypothesis in certain degrees

(See Theorem 45-2) So Theorem A appl ies qui te generally.

Notably.

it

applies to compact Lie groups with odd torsion. We will discuss some examples at the end of the section. Proof of Theorem € P H (X;~ ) where < x,a > f. O. Let ~ = ao and let 2n * p . be the 0 invariant sub Hopf algebra generated by a and ~. As

As in §46-1 pick a

o

C

H*(X;~p}

in §46-1 we have (***) and (****). Consequently, there exists

o such that o{y}

o

2.2pn-1

f. {y} € Cotor n* ..

p-l

= - { L ~~} i=l p

is associative, a P

~p

~N

p- i

p

0 xi

f. O.

p

)

On the other hand, we know

1

=0

commutator formed from a and theorem. Suppose

x

(~ ,~

and. by §45-3. the only possible non zero ~

f. 0 and ~N+l

are

= O.

where

{~s}

~s

is defined as in the

We want to show N

~

p-l. To do so we

use the May spectral sequence. Filter 0* by the augmentation filtration and let EOO* be the associated graded Hopf algebra.(We emphasize that, in

. what follows. we only consider EOO* as a graded object, not as a bigraded object}. The filtration induces a spectral sequence {E where r} E2 Eoo

= CotorEon*(~p .~P } = EoCotorn*(Fp,Fp}

The structure of EOn* and Cotor fi 1ter

n

O*(~

Eo

p

,~

p

} is easy to determine. If we

by the dual of the augmentation fil tration and let Om be the

associated graded Hopf algebra then

385

XI: The Hopf Algebra H* (X;IFp)

OEn

= E(~l)

0

E(~2)

E(~v)

0 ... 0

0 Wp[a]/(aP)

as Hopf algebras. Consequently * * * W [a*] EOn = E(~I) 0 ... 0 E(~N) 0 p /«a*)p) So. arguing as in §46-1, we have CotorE n*(W .W ) 0" p p

* = Wp [s~I]

* 0 E(sa* ) 0 W eta* ] 0 ... 0 W [~N] p p

where bidegree s~~

= (l.l~kl) = (1,2kn-l)

bidegree sa* = (1. la bidegree ta*

=

I)

(2.plal)

2,2pn-l So Cotor E n* (Wp.Wp) t 0 only if N

o

~

= (1, 2n)

= (2.2pn)

p-l. (The only possible non zero

element of that bidegree is (sa* )(s~* 1»' p-

Q.E.D.

All of this can be extended. The previous theorem and its proof was based

= (6)(pn ) .

on the secondary operation associated to the relation 6p n

For k

lOwe have the (unstable) relation

Oapn +p

k

n)

~ = (-Qk+l)(QOP

in degree 2n+2. This is a factorization of the form op~

= n+p

k

and

~

= Qk'

PROPOSITION: Let

= E aib i

where m

In analogue to Proposition 46-1 we have

(X.~)

be a homotopy associative H-space. If x

p2n + 1H* (X'• Wp ) satisfies QOPn(x)

=0

€

then the secondary operation

associated to the factorization k

n p QOp + Qk

n = (Q - k+l )(Q0 p )

satisfies

jt¢<x)

p-l 1 P p-i i L =<.) Qk(x) 0 Qk(x) + Qk+1 (y) i=1 P 1

where y € Ker (jt0l - 1@jt). This proposition is proved by an argument analogous to the one given in the next section for Proposition 46-1. Moreover. we can apply the proposition by an argument analogous to the one above and establish THEOREM B: Let p be odd and

(X,~)

a I-connected homotopy associative mod p

386

The Homology of Hopf Spaces

. n finite H-space. Suppose x € P2n+l H* (X;W p) satIsfies QOP (x)

y

= Qk(x)

isfying

= O.

If

is indecomposable then, for any a E P lH*(X;W ) sat2n+2p p

< y,a > t- 0 we can find non zero primitive elements defined recursively by the rule ~l Ps

= aQk+l = [~s-l,a]

for

5 ~

{~s}l~s~p-l

in PH*(X;Wp)

2

Let us see some examples of Theorems A and B from among the Lie groups. The only I-connected compact simple Lie groups with odd torsion are the exceptional Lie groups F

4,E6

,E7 and ES '

Example 1: The exceptional Lie group F has 3 torsion. Equivalently, 4 even * Q H (F 4;W3 ) t- O. We have * W [x ] H (F 4;W3) = E(x3,~,x11,x15) @ 3 S /(xS3) Observe that Theorem 35-1 forces particular,

x3'~

and

X

s

= P1 (x3)

and Xs are primitive. On the other ~

--

=

o(~)

= Q1(x 3).

In

hand Theorem B forces

XII and x 15 to be non primitive. We must have ~ (XII)

= x3

(x 15 )

=~

~

For, let dual to

=

= (as)o and a 3 (as)o be the homology primitives which are and x3. Applying Theorem A to X we have s

as'~ xS'~

s @ Xs

@X

a 15

= [~,aS]

t- 0

Applying Theorem B to ~ in the case of the relation Q 4Q oP O we have

Example 2: The exceptional Lie group E has 5 torsion which generates a S pattern in mod 5 cohomology similar to our first example. We have

*

H (ES;W5)

= E(x3,x11,x15,x23,x27,x35,x39,x47)

W [x

@ 5

] 12/(x125)

1(x The elements x3,x11 = P and x 12 = o(x = Q1(x 3) are primitive. The 3) 11) remaining generators are non primi tive. If we dual ize and consider the primitives a 12,a11

= (a12 ) o

and a

3

= (a 12)Ql

then Theorem A applied to x 12

387

XI: The Hopf Algebra H* (X;IFp)

gives a

23

u35 u 47

= [a l l,aI2] ~ 0 = [[uII,uI2],uI2] ~ 0 = [[[all,aI2].aI2],aI2]

~ 0

Likewise, Theorem B applied to XII for the relation OOP600 gives al5

a 27 a 39

Moreover. a

= (all)P 1

3

= [a3,a I 2 ]

0

= [[a3,aI2],aI2] ~ 0 = [[[a3,aI2],aI2],uI2]

~ 0

forces l5

= (u 23 ) P3

u27

= (~)P3

u39

= (u 47 ) P3

a

~

The two above examples illustrate part (ii) of the following consequence of Theorems A and B. COROLLARY: Let (X.~) be a I-connected homotopy associative mod p' finite H-space. (i) If rank H*(X;W )

< 2(p-l) then H*(X) has no p torsion

(ii)If rank H*(X;Wp )

< 2(p-l)

p

and H*(X) has p torsion then H*(X;Wp )

UPH*(X;Wp)' the universal enveloping Hopf algebra of PH*(X;Wp) where PH*(X;W has a basis {a} U {~s} U {~t} (1 ~ s,t ~ p-l} where p) a E P 2H (X;W ) and {~ } U {~ } are defined recursively by the 2p+ * pst relations ~l ~l

= aQO'

= aQI'

Proof: We can find y E (*)

QO(Y) ~ 0

~s ~s

= [~s-l,a] = [~s_l'u]

for s ~ 2 for s ~ 2

p2pk+I H* (X;W ) where

but

p

QOPPk(y)

=0

For, by Theorem ~-1, the first degree in which Qeve~*(X;f ) is non trivp

ial is of the form 2pn+2 and Q2pn+~*(X;f ) p

= 6pnQ2n+I H*(X;fp ).

45-4B H*(X;f ) is primitively generated in degrees p

n H (X;fp) such that 6P (w) ~ O. Let y

~+l*

w E P

QOPpn(y) ~ 0 replace y by y'

= ppn(y).

replacement process until you obtain

< 2pn+2.

= Pn (w).

By Theorem

So we can find

So QO(Y) ~ O. If

Again, OO(y') ~ O. Continue this

an element satisfying (*).

388

The Homology of Hopf Spaces

If we let x

= OO(y)

€

p2pk+~(X;Wp) then

x is indecomposable and xP

(**)

=0

The first fact follows from the fact that PH*(X:W ) ~ QH*(X:W ) is injecP P tive in degree $ 0 mod 2p. The second statement follows from (*) and the fact that A*(p) acts unstably. For we have the series of identities xP

= ppk+l(x)

= ppk+lOO(y) = QOPPk+1(y)

= Q1 Ppk (y) = (p100

+ Q Pk(y) 1P

(A* (p) acts unstably)

- QOP 1)ppk(y)

= p 1QOPPk(y)

- OOPPk+\y)

=0

The last equality uses (*) plus the fact that A*(p) acts unstably. We can now apply Theorem A to x and Theorem B (in the case of the relaPk » to y and obtain non zero primitives {a} U OP {~s}1~s~p-1U {'Ys}1~s~p-1 where a € P2pk+2H*(X;lFp) satisfies < x,a > -;! 0 while {~s} and {'Y } are defined from a as in the statement of the theorem. s It is easy to see that the elements {~s} U {'Y all have distinct degrees. s} odd * . Consequently, rank Q H (X:lFp) = rank PoddH*(X;lFp) L 2(p-1). ThIS p;oves part (i) of the theorem. Regarding part (ii) rank Podd~(X;Wp) 2(p-1) tion QOPPk+1 QO

=

(-Q1)(Q

=

implies

} U {'Y } is a basis of P ddH (X;W ), We are left with showing k s s 0 * p = 1. Since {~ s } U {'Ys ) is a basis of P0 ddH* (X;1FP ) we have aQO = ~l ¢ 0 and aQ1 = 'Y 1 -;! 0 but a~ = 0 for any ~ € Aodd (p) where I~ I > 2p-1. However, the {~

structure aQOP

k

-;!

theorem Q2pk+~*(X;IF) p

O. Consequently k

= 1.

=

Q kQ2k+1H*(X:IF) p Op

dualizes

to

give

Q.E.D.

Let us do one more example. We want to use the exceptional Lie group E 8 to illustrate that Theorems A and B fit very closely with the cohomology of H-spaces. They seem to apply just when we want them to apply. Example 3: The mod 3 cohomology of the exceptional Lie group E is of the S following form.

*

H (ES:1F3)

= E(x3,~,x15,x19,x27,x35,x39,x47)

Theorem 35-1 gives

0

II' [x ,x

3

S

] 20/(xs3,x203)

389

XI: The Hopf Algebra H* (X;IFp)

1 s = liP (x3 ) 3 x 20 = lip ( "7 } X

So we also have

1

"7 = P (~)

x 19 x

20

3

=P =P

1

("7) (x

s)

Then x 3'''7,xS,x19 and x are all primitive. If we apply Theorem A to X s 20 then we obtain the non zero commutators 20

and x

a 15 a

39

t 0 t 0 I9,a20]

[~,aS]

[a

We can apply theorem B for QOP 10 Q = -QI (QOP9 ) to x and obtain l9 O a 15

a On

35

(a20}QI t 0 [a I5,a20] t 0

the other hand, we cannot apply Theorem B in the case of the relation 4

3

3

QOP QO = (-QI}OaP to "7' For QOP ("7) t 0 and so the associated secondary operation is not defined on Xl' This is fortunate since P I IH*(ES;W3)

= QI IH*(ES;W3} = 0

while if Theorem B did apply we would have [a 3,a7 ] t O.

Again we cannot apply Theorem B in the case of the relation QOPlOQ I (~)(OaP7) to x 15 . This time QoP7 (x 15} 0 but x is not primitive. l5 Again this failure is compatible with the structure of H*(ES;W3}, For P23H*(X;Wp} = 0 while Theorem B would force

[~,a20]

t 0 in this case.

Consequently, Theorems A and B are very sensitive to the structure of

390

§47: U(M) ALGEBRAS

Given a H-space (X.~) a Borel decomposition H*(X;F ) ~ ~ A. is far from p

1

unique. As we discussed at the end of §2 a natural question to ask is whether we can choose the Borel decomposi tion to be compatible wi th the action of A*(p) on H*(X:F ). Is H*(X:F ) the enveloping algebra of an unp

p

n, stable A*(p) module? A Steenrod module is unstable if. for all x € M k P (x)

=0

when 2k

>n

k and 6P (x)

=0

when 2k

~

n. Given such a M we define

U(M) as follows. Let SCM) I U(M)

the graded symmetric algebra generated by M the ideal of SCM) generated by M

= S(M)/I

In all cases, U(M) has a Borel decomposition. We are only interested in the case p odd and M finite. In that case U(M)

=~

Ai as an algebra where

the factors Ai are of the type Ai

E(a i)

A

= Fp[a i]

i

lail odd /(a. p

n

la . l

I

even and n ~ 1

1

where a. € M. Thus if H*(X:F ) 1 P

= U(M)

we are asserting that we can choose

a Borel decomposition of H*(X;F ) which is compatible with the action of p

A*(p)

in the sense that the Borel generators plus

their iterated p

th

powers are invariant. (Observe that they span M C U(M». If H*(X;F ) adp

mits such a structure then it is called a U(M) algebra. In this chapter we will study necessary conditions for the existence of a U(M) structure on

*

H (X;F

p)'

If H*(X;F ) is primitively generated then H*{X:F ) is a U(M) algebra. p

p

Just let M = PH*{X;F ). So. in particular. as in §46. we are studying p

necessary conditions for H*(X:F ) to be primitively generated. As in §46 p

XI: The Hopf Algebra H* (X;IFp)

391

we will also study how H*(X;W ) fails to be primitively generated when the p

necessary conditions are violated. The results of this chapter are treated in the papers Kane [12].[14].[15].[18] and [22]. S47-1: Main Results

Necessary conditions for H*(X;W ) to be a U(M) algebra are as follows p

THEOREM A: Let p be odd and let space such that H*(X;W ) p

(X,~)

= U(M).

THEOREM B: Let p be odd and let space such that H*(X;W ) p

be a I-connected mod p finite H-

(X.~)

then xP

=0

for all x

€

~(X;W ). P

be a I-connected mod p finite H-

= U(M).

Then Q2~*(X;W ) p

=0

unless 2n

=

2(ps+ ... +p+l). Again we emphasize that primi tively generated.

the above

theorems apply when H*(X:W) p

is

In Theorem A the hypothesis that p is odd and X

I-connected are both necessary. Counterexamples are provided by H*(Spin(n);W

and H*(PU(n);f On the other hand. it is not clear. at 2) p)' the moment whether the U(M) hypothesis is really necessary. As we have

. . .*

...*

the Frobenius map C: H (X;W ) -+ H (X;W ) is trivial for all

remarked,

known cases when p is odd and

p

(X.~)

p

is a I-connected mod p finite H-space.

In Theorem B the hypothesis of p odd and X I-connected probably can be eliminated. However. this time. the U(M) hypothesis is essential. Notably. the exceptional Lie group £8 at the prime p

=3

2n..* shows that Q tl (X;Wp) can

be non trivial in degrees other than 2(ps+ ... +p+l). For Q20H*(£8;W to. 3) So H*(X;f ) being a U(M) algebra forces added restrictions on p

QevenH*(X;W ). p

Theorem B can be compared to previous restrictions obtained on QevenH*(X;W ) when p is odd and (X.~) is a mod p finite H-space. In §36 it P

was demonstrated that Q2~*(X;W ) # p

° only

in a restricted set of degrees

which include the degrees 2(ps+ ... +p+l) but other degrees as well. In §46 it was demonstrated that. with enough added assumptions.

(namely

(X,~)

homotopy associative and H*(X:r ) primitively generated) QevenH*(X;W ) is p

p

trivial in all degrees. Thus. Theorem B is an intermediary result inter-

392

The Homology of Hopf Spaces

polated between these results. Under appropriate circumstances we can also study the converse of Theo2n..* s * rem B. If Q tl (X;W ) to where n t p + .. +p+I then. by Theorem B. H (X;W ) mus t fail

p

to be primitively generated. As in §46 we can describe

p

the

failure of primitivi ty in terms of a family of non zero commutators in PH*(X;W

p}'

First recall that asserting that QevenH*(X;W is non trivial p}

in degrees other than 2(ps+ ... +p+I} is equivalent to asserting that A*(p} acts non trivially on Qeve~*(X;W }. For. by Corollary 34-IA. p

QevenH*(X;W } is generated. as an A*(p} module. by the elements of degree p

2(ps+ ... +p+I}. The next result describes how H*(X;W ) fails to be primip

tively generated when A*(p} acts non trivially on Qeve~*(X;W }. We will p

use the notation ad(x}

=[

THEOREM C: Let p be odd and

= ad(x)o ... oad(x)

.x] and adi(x) (X.~)

be a I-connected mod p finite H-space

such that H (X;W ) is associative and x

*

Given

x,y

(i times).

p

E Qeve~*(X;W ) such that pP

t

p

p

=0

~*

for all x E H (X;W ). P

(x) = y then, for any

a.~ E

PevenH~(X;W} ~ p satisfying

< x.a > t 0

< Y.f3 > t we have non zero elements defined by

{~ .. }

IJ

0

in PH (X;W ) for 1 * P

~

i+j

~

p-2

or ~

..

IJ

The equivalence of the two definitions of

~ij

follows from the identity

(The last equality uses Lemma 45-3B which tells us that

[a.~]

= 0) As we

have said. the above pattern tells us how H*(X;W ) fails to be primitive. p

The commutators in Theorem C dualize to indecomposables in H*(X;W ) with p

non trivial coproducts. However. the coproducts are qui te messy. So the description in terms of commutators in H*(X;W ) is more palatable. Theorem C complements p

results

on

p

} ~ P H (X;IF ) obtained in §46. The commutators produced in §46 are "unstable".

*

the

[ • ]: H (X;W ) 0

*

p

H~(X;W ~

393

XI: The Hopf Algebra H* rX:IFp)

They require

(X,~)

to be homotopy associative and they tend to disappear

when we perturb the multiplication. For example, in Example I of §46-2 we produced the non zero commutators [a other hand. F4

~

(3)

Xl x

~

3,aS] and

where

[~,aS]

in H*F4;W3). On the

W [x ]

E(x3'~) @ 3

S /(xS3)

E(xll,x I S)

=

With the product multiplication on F Xl x X H* (F is primitively 2 4;W3) 4 is commutative and [a = [~,aS] = O. 3,aS] 4;W3) The commutators produced by Theorem C are "stable". They are non zero

generated. So H*(F

no matter what multiplication is chosen for X. For they are forced by the Steenrod module structure of H*(X;W ). In §47-S we will have more to say p

about Theorem C. Notably, we will discuss the mod 3 cohomology of the exceptional Lie group E in terms of Theorem C. S In §47-2 and §47-3 we will prove Theorem A. We will only briefly discuss the proofs of Theorems B and C in §47-4. For they are analogous to the proof of Theorem A. However, detailed proofs would require the introduction of bu theory and a careful analysis of its operations. §47-2: Proof of Theorem 47-IA We prove Theorem A by using Brown-Peterson theory. There are two points about the proof which should be emphasized. First of all. in the proof we pass from X to the loop space OX. Our discussion in §29 of the EilenbergMoore spectral sequence demonstrated that the algebra structure and the Steenrod module structure of H*(X;W ) are reflected in the coalgebra and p

the Steenrod module structure of H*(OX;IF ). See, in particular, the disp

cussion of the short exact sequence

0-+ S -+ PH*(rlX;W ) -+ T -+ 0 P of Steenrod modules in §29-S. Secondly, since H* (rlX)(p) has no p torsion, the BP theory of OX is easy to determine and one can use BP cohomology operations to effectively analyse the action of A*(p) on H*(rlX;W ). Notably. the freeness of BP*(OX) p

will enable us to make divisibility arguments. Throughout this section we will make the following Assumption: (X,~) is a I-connected mod p finite H-space where H*(X;W ) p

394

The Homology of Hopf Spaces

U(M} for some unstable A*(p} module M.

~: We begin by showing that the non triviality of the pth power in H*(X;Wp } can be reinterpreted in terms of the Steenrod module structure of PH*(lIX;W} (We wi 11 be assuming and using the ideas of §29). Choose a p Borel decomposition H* (X;W )

=

p

~

A. with generators {a 1

Suppose aiP # O. By Theorem 45-2 we know deg a ~

1. So

i

i}

where a.

= 2(ps+ ... +p+l}

1

E

M.

for some s

ps+ ... +P+l{ } a. p -_ p a. 1

1

= plpp

S

+",+p(a.} 1

= pl(a.} J

pS+ ... +p s+l 2 (a Observe that deg a = 2(p + ... +p +l} So, by where a j = P i). j Theorem 45-2. a j P = O. As in §29-5. the elements a and a determine j i transpotence elements ta

i,

ta . J

E

Tor -2. * H*(X;W

p}

(W.W). If we define Tor via p p

~*

the bar construction BH (X;W ) then p

[a i

i

p

2

-s]

[a j sl a j p-s ]

ta j Moreover, since pl(a

=

sla

j}

= aiP,

for any

~

s

~

p2 -1

for any

~

s

~

p-l

we have. by the Cartan formula, pp[ajlajP-l]

[a.Pla.p{p-l}]. So 1

1

As in §29-5 the elements tai,ta

j

pass through the Eilenberg-Moore spectral

sequence {E comverging to H*(lIX;W and determine elements tai,ta E T r} j p}

*

PH (lIX;fp}/s also satisfying pp(ta

j)

=

= ta i.

Step II: Next. we use Brown-Peterson operations to analyse the action of A* (p) on PH* (lIX;Wp)' We want to demonstrate that Pp( ta j } # 0 in T

*

PH (lIX;fp}/s is not possible. By construction ta

j

=

has total degree 2(ps+l+

.. +p3+p) _ 2. Let a be an arbitrary primitive element of degree 2(ps+l+ ... +p3+p}_2 in PH*(lIX;W }. We will prove pp(a) E S. p

395

XI: The Hopf Algebra H* (X,'IFp)

Since OX has no integral p torsion, BP*(OX) is a free BP*

=

l(p)[v 1.v2 •... ] module and the Thorn map T: BP*(OX) ~ H*(OX:W is surjecp) tive. (The reader should observe how the freeness of BP*(OX) is used throughout the following proof to do divisibility arguments.

It is to

obtain this freeness we pass from X to OX.) Choose a representative A E BP*(OX) for a E H*(OX;W ).(i.e. T(A) p

= a).

Since the operation r

p

covers

~(Pp) = - pP i.e. the following diagram commutes r

p

- pP

we

can

analyse

Using

the

relation

up to an unit in l(p)' Now consider r p_2(A). By Proposition assume A is primitive mod (P.v PROPOSITION B: rp_1(A)

=B

1,v2

43~2B

we can

.... )2. Using this fact one can show

+ pB' + v1B" mod (p2.pvl'v12,v2.v3' ... ) where

(i) BE Im{a*: QBP*(X) ~ PBP*(OX)} (ii) T(B'). T(B") E S Moreover, using the property that H*(X:W ) p

= U(M)

one can show that

These last two propositions will be proved in the next section. They force the theorem. First of all. we have

Proof: Apply r 2 to the expansion of r in Proposition B. We have p_2(A) (*)

r

2rp_2(A)

== r

2(B)

+ pr

2(B')

+ pr

2(B")

mod (p2,v

1.v2

... )

Moreover. each term in the right hand side of (*) can be written pD where T(D) E S. Proposition C verifies this for the term r

2(B).

Regarding the

396

The Homology of Hopf Spaces

other terms we need only show that T[r and T{B")

€

2{B')]

and T[r

2{B")]

€

S. But T{B')

S and S is invariant under A*(p) So we can appeal to the

commutative diagram

Q.E.D. Secondly, we have

Proof: Combining Propositions A and D we have prp{A) Since BP* (OX)/

2 (p ,v

1,v2 , · · · )

rp{A)

=pD mod (p2,v 1,v2, ... ) = H* (OX;Z/p2 )

is a free Z/p2 module we have

=D mod (P,v 1,v2' ... ) =-

It follows from Lemma B that pp{a)

T[r (A)] p

Q.E.D.

= T{D)

€

S.,

§47-3: Proof of Proposi tions 47-28 and 47-2C In order to complete the proof of Theorem

A we are left with proving

Propositions 47-2B and 47-2C. Actually the two propositions are not valid without passing to skeletons. Let 2n

= deg

p P (a)

= 2{p s+l +... +p2 )

- 2

Replace X and OX by y

= {OX)2n

Z

= {X)2n+l

and the suspension map a* by the map induced by 2{OX)2n+1

(20X)2n+l _

2n+1. X The purpose of such a replacement is to obtain

LEMMA A: For i ~ 2n - 4(p-1) + I, x T: BP* (Z) -

€ Hi{X;W

H* (Z;Wp ) if QO(X)

p)

lies in the image of

= Q1(x) = O.

Proof: We can factor T into the maps BP

= BP(ro>

_ .... _

BP(n) _

BP(n-1> _ ... _

BP(O) -

BP(-l>

= HW

p

397

XI: The Hopf Algebra H* (X;IFp)

Each

map BP*{Z) ~ BP*{Z)

can be

analysed via

the

spectral

sequence {Br} associated to the exact couple BP*{Z)

x v

n , BP*(Z)

<, /

Pn

BP*(Z)

= BP*(Z)

Namely. B1

and x E BP*(Z) belongs to Image P if and n only if x is a permanent cycle in {Br}'

Casen-O This is the Bockstein spectral sequence asa studied in §ll,§l2 and §l3. If QO(x)

then x € H* (Z;~p)

=0

For, by Corollary 35-1B,

= BP<-I> * (Z) * BP (Z) has only

= BP * (Z). . torsIon So BP2

lifts to H* (Z)(p) elementary p

= BPoo Case n In degree

~

2n+1-4(p-I). if OO(x)

= QI(x) = 0

to BP*(Z). First of all, as above. GQ(x) H* (X)(p)' Secondly Ql(x)

=0

=0

then x E H* (X;~p) lifts

means that x lifts to y €

forces y to be a permanent cycle in the spec-

tral sequence {Br}' For since Hi(Z)(p)

=0

in degrees> 2n+1 the only

differential which can act non trivially in {BPr} in degree

~

2n+l-

4(p-l) is d . Moreover. since the differentials are torsion valued and i

since Bl

= BP*(Z)

only if dl(y)

=0

has only elementary p torsion, dl(y) # 0 if and

mod p. Lastly. dl(y) mod p agrees with QI(x),

Case n > 2 In degree ~ 2n+1-4{p-l) Bpi(Z) ~

= BPi(Z).

2n+l. Q.E.D.

For Hi(X;~ ) p

o

in degree

It follows from Lemma A that we have

Convention: To simplify notation, as well as our argument, we will assume that Lemmas A and B hold not just for Z C X but for X itself. We then avoid having to constantly pass back and forth between X or ax and Y or Z. Proof of Proposition 47-2B

398

The Homology of Hopf Spaces

Proof: We use the commutative diagram

a*

~

a

Let

b

= T[r p-2(A)]

PH* (f1X;1F ) p

PH* (f1X;1Fp )

€

By Theorem 29-4 and the hypothesis that H*(X;IF ) p e € M such that

U(M) we can choose

Moreover Qo(e)

= Q1(e) = 0

For deg QO(e) == 4 mod 2p while deg Ql(e) == 2p2_2P+2 mod 2p2. But %(e). even. even Q1(e) € M And M is trivial in degree $ 0 or 2 mod 2p by the Proposition from §36. By Lemma A e

for some E

= a*(E).

BP*(f1X). Let B

€

== B mod Ker T. And Ker T

=

= T(E)

(P.v

1.v2

By the commutativity of diagram (*) A

.... )

Q.E.D.

..

LEMMA D: rp(A) = B + pB' + vlB" mod (p2. v12. v2 .... ) where T(B·).T(B")

€

PH (f1X;1Fp)' == B + pB' + vlB" mod (p2. v12. v2 ... ) p_2(A) for some B' .B". Consider B'. It follows from Lemma C above plus Pr opc-:

Proof: We can certainly expand r

s I t i on 43-2 that pB' + vlB"

= r p_2(A)

2

- B is primitive mod (P.v 1.v2 ... ) .

Thus pB' is primitive mod (p2. v1. v2 .... ). Since BP*(OX)/

2 (p .v 1·v2····)

BP* (OX x f1X)/ 2

= H*(f1X; 7lIp2) = H*(f1X

x OX; 7lIp2)

(p .v l·v2 · · · · )

as (P.v

free 7lIp2 modules we can l.v2

divide by p.

So B'

.... ). The case of B" is analogous. Q.E.D.

is primitive mod

399

XI: The Hopf Algebra H* (X;IFp)

Finally, to complete the proof of the proposition, we need only observe that T(B') and T(B") E S since T

= PH* (OX;lFp )/ s = 0

in degree

;I'

-2 mod 2p

while deg B' == 2 mod 2p and deg B" == 0 mod 2p. Remark: The use of the U(M) hypothesis in the proof of Lemmma C is not really essential. By harder work we could have shown QO(e) = Q (e) l without invoking H*(X;IF ) p

= U(M).

=0

The essential use of the U(M) hypothesis

occurs in the proof of Proposition 47-2C. Proof of Proposition 47-2C As above let b

pp-2(a) E PH*(OX;IF ) and choose e E M C

= T[r p-2(A)]

P

H*(X;IF ), E E BP*(X) such that p

u*(E) u*(e) T(E)

=B =b =e

The essential use of the U(M) hypothesis arises in proving

Proof: First of all, P2 (e)

in QoddH* (X;IF ). For o * : Qodd H* (X;IF ) --> p p evenH*(OX;1F p ) is injective and u*~(e) = p2u*(e ) = ~(b) = p2pp-2(a) = O.

=0

p

. Secondly, SInce H* (X;IF ) p

Q.E.D.

= U(M)

and Modd

By Lemmas B and E we can write r sequently, r

2(B)

= u*r 2(B) = po* (F).

2(E)

~

Qodd H* (X;IF ) we have P2 (e) p

= pF

= O.

2n+l for some F E BP (X). Con-

§47-4: Proof of Theorems 47-1B and 47-1C Actually this section will explain why we will not prove Theorems B and C. The propf of Theorem B is analogous to that of Theorem A but considerably more involved. Given indecomposables x,y linked by ~ € A*(p) one can use the structure theorems for Qeve~*(X;1F ) from §36 to reduce to case 'lJ

= l«(PP

t

). So y

Their transpotences

= l«(PP

t

p

A

)(x) where deg x

the

= 2(ps +... +p t+l +... +p+l).

400

The Homology of Hopf Spaces

= [xp-1Ix] = [yp-1!y]

tx ty satisfy x(pP

t+1

)(tx)

= ty

# 0

*

in T = PH (OX;IFp) / S' We want to show that this si tuation is impossible. . SInce deg tx

= 2(p s+l + ... p t+2 +p t+1 + ... +p)-

2 it suffices to show

t+1

)(a) E S for all a E PH* (OX;IF ) of degree 2(p M1 + ... +p M2+ ... p)-2. P t+1 p ) i.e. the following diagram commutes Since r t+1 covers X(P

X(pP

p

r t+l p

BP*(OX) T

I

1

BP*(OX)

1

T

H*(OX;IF ) p

p One might try to analyse X(P

I

p x(P

t+l

t+l

H*(OX;IF ) p

)

) via BP operations in a manner analogous

to our proof of Theorem A. If one examines the proof of Theorem A in §47-2 then it is the analogue of Proposition C which creates the problem. One can

find a relation pr t+l(A) p

=L ai~i(A)

mod (p2,v 1,v2'···)

for BP operations a .. ~. E BP*BP. And one can expand 1

~i(A)

1

= Bi

where

+

L

j21

vjB i j mod (P,v l,v2'···)

B. E 1m 1

T(B The difficulty is to show

i j)

ai(B i)

€

2

c,* S

= pCi

for some C. E 1m a*. This is the result corresponding to Proposition C of 1

§47-2.

If one

examines the proof of Proposi tion C in §47-3 then it is

obvious that a control of 1m {T: BP*(X) -> H*(X;IF )} is essential. It is p

this control which is missing in our now more general situation. The proof of Theorem B can only be completed via BP operations if one assumes that

XI: The Hopf Algebra H* (X;IFp)

n Ker

Qs'

401

This assumption has so far not

s~O

been verified.

An alternative proof is available using the machinery of connective Ktheory operations. Although these operations are more complicated to use, one does know that 1m bu * (X)(p) ~ H* (X;W = Ker QO n Ker Q1 for finite p) H-spaces and p odd. (see Kane [17]). One can this fact to push through the proof of Theorem B. (see Kane [IS). Crudely, one argues as above only replacing BP operations by bu operations. The proof of Theorem C is, not surprisingly, the converse of the proof of Theorem B. Suppose H*(X;W ) possesses even degree indecomposables x and p

t

Y where ~(Pp )(x)

= y.

By Theorem B, H*(X;W ) is not a U(M) algebra. If p

one examines the proof of Theorem B one can locate the essential point at which the U(M) hypothesis was invoked. Further, one can determine exactly how H*(X;W ) must fail to be U(M) in our present circumstances. This, in p

turn, tells us how H*(X;W ) fails to be primitively generated. The homop

logy generators listed in Theorem C arehe result of all this

analysis.

Since the argument again relies on bu operations we will not present any details. §47-5: The Mod 3 Cohomology of E S In this section we give a further discussion of Theorem

C

of §47-1.

Theorem C explains how the algebra structure of H*(X;W ) forces results p

about the coalgebra structure. There is only one example of Theorem C in action. To apply Theorem C we must have two even degree generators linked by a Steenrod operation. The mod 3 cohomology of the exceptional Lie group

ES provides this For

*

H (E S ;W3 )

= E(x3,~,xI5,xI9,x27,x35,x39,x47) .

0

W[x ,x ] 3

S

By Theorem 34-1 we can assume

20 /

(x

3

S

3 ,x ) 20

By Theorem 35-1 we have

In particular, x

3

primitive forces

~,x8,x19

and x

20

to be primitive. On

402

The Homology of Hopf Spaces

the other hand, the remaining generators {xI5,x2T"3s'''39,x47} are not primitive. This follows from Theorem C. As in Theorem C we dualize and consider the Lie algebra PH*(ES;W3). It has a basis {a3,~,aS,aI5,aI9,a20' 3 a2Ta35,a39,a4T}' The fact that P (xS) = x 20 forces the following commutators in PH*(E S;W3) a l5

[~,aS]

a 27

[~,a20]

~

[a

a

I5,a20 ]

[a

39,aS

39

]

= a l 9 , a S] = [a s,a27 ]

= [a 27,a20]

Observe that, by the sparseness of PH*(E plus the restrictions on S;W3) commutators given in §45-3, these are all possible non zero commutators in PH*(ES;W3). So Theorem C asserts that every commutator in PH*(ES:W3) which can be non zero is indeed non zero. This example suggests that, to some extent anyway, Theorem C is a "best possible" resul t. §47-6: Some Questions We close this chapter with some questions raised by our discussion, in the last three chapters, of the structure of H*(X;W ) as a Hopf algebra p

over A*(p), In all that follows let

(X,~)

be a I-connected mod p finite H-

space. First of all, the algebra structure of H*(X;W ) forces restrictions on p

its coalgebra and Steenrod module structures. The discussion of the Lie groups F and E in §46-2 and §47-5 are good illustrations of this fact. 4 S It has been conjectured that, when X is a compact Lie group, the algebra structure of H*(X:W ) should completely determine the structure of p

H*(X;W ) as a Hopf algebra over A*(p). It seems reasonable to raise the p

same conjecture

for X just being a loop space. However, some restriction

on X is necessary. For such a conjecture cannot hold in general. example, H*(X;W 3) I

E(x3'~) occurs both for X = Sp(2) (in which case . = S3x 7 S (In

I

which case P (x3) = 0). In another direction one can ask about the relation between the coalge-

P ("3)

= ~)

=

For

and for X

bra structure and the Steenrod module structure of H*(X;W ). We have obp

served that H*(X;W ) primitively generated implies "*(X;W ) p

let M

= PH*(X;Wp ».

p

= U(M).

(Just

What about the converse of this relation? If H*(X;Wp )

403

XI: The Hopf Algebra H* (X;I'P J

= U(M)

does X have a multiplication ~:X x X ~X so that H*(X;F ) is primp

itively generated?

Finally, one can ask about the pth power map on H*(X;F ). Is it true p

that x

p

=0

~*

for all x E H (X;F )? Notably, we have the strong restrictions p

imposed by Theorem 45-2 and Theorem 47-1A.

PART XII: POWER SPACES

The next four chapters are centred around the concept of a power space. We will use them to construct families of mod p finite H-spaces for p odd. For our purposes. power spaces are useful as technical devices in altering the given H-space structure on a space so as to obtain one with more pleasant properties. Such alterations can be used in conjunction with Postnikov systems. Given a finite complex X and its Postnikov system

... ~

~+1

X ~ ... ~ Xl ~ X then X wi 11 be a H-space i f each X is a H-space o N N the k invariants ~: ~ ~ K(I1 + are H-maps. We proceed by N 1(X).N) induction. If ~ is a H-space we alter the multiplication on X to make ~ N a H-map So X + fibre k must also be a H-space. It is in this argument N 1 N that power spaces make their appearance. Actually we will work with primi-

~

and

=

tive Postnikov systems rather than Postnikov systems. But aside from that we proceed as above. In §48 we will discuss power spaces. In §49 we will discuss the alteration of H-space structures. In §50 and §51 we will construct families of H-spaces. Throughout these four chapters we will make the following Assumption: p is odd and h is a positive integer which generates W * W - {a} multiplicatively.

p

p

The results of these four chapters are mostly due to Harper and Zabrodsky. either jointly or separately. Some references in the literature are Harper [8]. Cooke-Harper-Zabrodsky [1] and Harper-Zabrodsky [1].

407

H8: POWER SPACES

In this chapter we introduce the concept of a power space and discuss some of its main properties. §48-l: Power Spaces A power space is a pair

(X.~)

where

~:

X

~

X is a self map inducing

multiplication by A on QH*(X;F ). Observe that if (X.~) is a power. space p

then (X.~) is also a power space. The canonical example of a power space is the A power map on a H-space (X.M) defined by ~A:

A X X Axl X --. x -

X

X

x Xx

-. . .

Axlx .. xl

... --' X x X x X f! xl

'lX x

.... x

X

MXlx.. xl ,

J

A factors , X x X J!:...., X

However. there are many other examples. For instance. any wedge

v i

~.

of

1

suspensions is a power space. So we can regard a power space as a weakened version of a H-space. Despite this weakening. power spaces possess enough structure to still be useful. One of the major advantages of power spaces over H-spaces is that they are better behaved wi th respect to liftings. This will be discussed in the next section. As the first sign of the structure possessed by power spaces we have the following fact. PROPOSITION: Let (X.~) be a power space with H*(X;F ) finitely generated p

as an algebra. Then we can choose algebra generators {xi} of

*

r

*

H (X;F ) so that (~ ) (xi) p

Proof:

By induction on degree.

QH*(X;F ) and ~*(x.) p

1

= Ax.1

= Ax.1

Suppose {x) projects

in degree

< n.

* (x j ) = Ax j

~

for some iterate ~

r

of~.

to a

basis of

Given x. € Hn(X;F ) we can write J P + d

where d is decomposable. By our induction hypothesis

408

The Homology of Hopf Spaces

= d 1+· .. +dp_ 1

d where.p* (d

k k) = A ~. For each 1

= xj

xj then

sk

.p* (x

j

)

~

p-l. if we replace x

L

Ak_A + d

Using induction on t we can establish (.p*)t(x.)

* P (x(.p)

In particular. Ax

i

in degree

~

J

j

Axj"

)

= At-x j

by

dk

- ktl

= Ax- j

j

+ tA

l

t-l

d

l

If we replace .p by .p

n. Q.E.D.

.pP then ~*(x.) 1

Remark: The type of argument given in the above proof is canonical.

It

will occur many times in this chapter. It follows from the proposition that the cohomology of a power space (X,.p) has a decomposition H*(X:W) P

= H(l)

ffi H(2) ffi••• ffi H(p-l)

k where .p* is multiplication by A on H(k). More generally

*

n

H ( II X:W ) LeI P

Even if H*(X;W ) is not finitely generated these decompositions are still p valid. For, if we work by induction on degree. we can reduce to finitely generated sub algebras of H*(X;W). These decompositions will be extenp

sively used when we alter H-space structures. The guiding principle behind our alterations is to make the coproduct ~: H*(X;W ) ~ P

H*(X;W ) I8l H*(X;W ) compatible (in an appropriate sense) with the above p

decompositions.

p

Besides power spaces we also have the concept of a power space map. Namely. f:

(X •.p)

~

(Y. cj»

is a power space map if,

for some r l 0,

the

following diagram is homotopy commutative.

X f

.pP

r I

1

Y

1 f

Y

cj>P

r

I

Y

The need to allow iterates of .p and cj> in the above definition is explained

409

X /I: Power Spaces

by results such as the previous proposition. We are often forced to replace power maps by their iterates in order to obtain an algebraic property. Again. the canonical example of a power space map is an H-space map f:(X,~) ~ (Y.~')

between H-spaces.

Finally. we have the concept of a power H-space

(X.~.t).

This is a

space X which is both a H-space and a power space Moreover, the structures are compatible space map.

in the sense that the H-space map

In other words.

for some r

O.

~

~:

Xx X

~

X is a power

the following diagram is

homotopy commutative r

X ~

xX

.pPx.pP

'X

1 .pP

In particular. if

~:

X x X

~

Notably,

xX

1

X

is a power H-space.

r

~

'x

r

X is associative and abelian then

(X.~.t~)

this occurs for Eilenberg-Maclane spaces

K(Yp.n). We also have the concept of a power H-space map. We omit its defini tion. §48-2: Power Spaces and Fibrations Power spaces are particularly well suited for Postnikov systems as well as for other fibrations. First. let us note that fibrations present problems for power spaces which are not present for H-spaces. The fibre of a H-space map has a naturally induced H-space structure. This is not true for power spaces. Given a power space map

(X.~)

~ (Y,~)

there is an ob-

vious way to induce a self map of the fibre F. Namely. choose r

~

0 so

that the diagram

Y -----> Y ~p

r

commutes. Then there is an induced map t: F volves choices and. hence.

~

F. This map obviously in-

indeterminancy. A more relevant objection is

that. in some cases, none of the choices make F a power space. Consider the fibration

410

The Homology of Hopf Spaces

where

L

oo) is a generator of 11 (!Cp 2

oo

= 71..

give multiplication by X in ~(S2;F ) p

If the power maps on S2 and !Cp

= ~{lCpoo;Fp } = Fp

then any choice of

the induced map on S3 gives multiplication by X2 in If3(S3;1F }. We will p

shortly demonstrate that this type of problem does not arise when we deal with power H-spaces rather than power spaces. First, however, we need to consider the positive side of power spaces. One of the main advantages of power spaces over H-spaces is that they are much better behaved with respect

to

liftings.

Suppose we have a

lifting

f/ 1p E

X-B f

The whole theory of secondary operations as developed in §30 and §32 is a painful lesson in the fact that when we are dealing with H-spaces f need not be a H-space map even though f is a H-space map. (see. for example, the end of §30-3). When we deal with power spaces the situation is different. Notably. we have PROPOSITION A: Given a diagram of power spaces and power maps

k

f

(X,,p) where (i) K = K(71. p,n} (ii) E

~

B

~

{B.
(K,c)

K is a fibration

If f lifts then a lifting can be chosen which is a power map. Proof: Given d: X

~

OK

g: X

~E

we will use d*g to denote the map dxg : X _

A

dxg

Xx X_

OK x E -

J.L

E

where J.L is the principal action. As usual, if g,h: X ent liftings of f then h

= d*g

~

E are two differ-

for some d. We can assume that

mute with f,k,p. So, given a lifting g: X

~

E,

,p,
there exists w: X

com-

~

OK

such that (*)

~g

= w*(g>JI} = O. It

We want to alter g so that w

is easy to see how w varies as we

XII: Power Spaces

vary g. Given d: X

~

411

OK let gd wd

Then

d*g (Oc)d +

W - d~

= O.

So we want to choose d such that w d

If we consider

W

as a cohomology

class of H*(X;W ) then by the discussion following Proposition 48-1 we can p

wri te w

where",* (w )

= w1+w2+ ... +wp_ 1

= Ak wk '

k

(In degree

~

is finitely generated as an algebra). We can choose d so that

Iw.I H*(X;Wp )

Let Then (Oc)d d~

= AWl

+ [

i¢1

= Ad Ai

---.--- w. ,,1_

A

1

(ii)~d=O

For any w, formula (*) can be extended, for each t ~tg

t-l

= [

.. [(Oc)t-1~1]

i=O In the case w = WI and t = P we obtain

~Pg

p-l

=[[

i=O

1 AP- w ] 1

For the last equality we use AP- 1

*

~

1, to

[~t]

* [ ~] = 0 * [~]

=1 mod p.

Q.E.D,

The following consequence of Proposition A will be useful COROLLARY: Let (E,~) ~ (B,~) ~ (K.c) be a fibration of power spaces where K

= K(Z/p,n). (i)

(B,~)

If

is a power H-space (with respect to

~)

(ii) E is a H-space (iii) p is a H-map Then we can choose a new H-space structure on E so that E is a power H-space (with respect to

~)

and p is a power H-space map.

Proof: The given multiplication on E is a lifting of E x E ~ B x B ~ B. Applying Proposition A we obtain a power space lifting. Q.E.D.

412

The Homology of Hopf Spaces

Let us now return to considering the type of difficul ty wi th power spaces illustrated by the fibration S3 ~ S2 ~ ~pro. If we deal with power H-spaces rather than just with power spaces then. in the following important case, the problem associated with inducing a power space structure on the fibre of a map disappears. PROPOSITION B: Let k: p: E

~

(B,~.~) ~ (K.~'

,c) be a map of power H-spaces. Let

B be the fibre of k. Then (i) E is a power H-space (ii) p: E

~

B is a map of power H-spaces.

The rest of this chapter will be devoted to the proof of this proposition. As usual. E has a naturally induced H-space structure Choose r

~

~":E

x E

~

E:

0 so that the following diagram commutes ~P

r

K - - -.... 'K

Let

cP

~:

E

~

r

E be the self map induced by (*). We will show that

(E,~)

is a

power space. If so then. by the above corollary, (E.~" .~p) is a power Hspace and p : (E.~" .~p)

(K,~"l) is a map of power H-spaces.

--+

Consider the Serre spectral sequence {E } associated to the fibration r 11K ~ E ~ B. Because of the presence of H-space structures {E } is a r spectral sequence of Hopf algebras. We have E

2

Any multiplication on 11K ated.

= H* (B;Wp )

= K(Vp,n-l)

0 H* (I1K;W ) p

makes H*(I1K;W ) primitively generp

(For all the algebra generators can be traced back to the funda-

n-l I € H (K;W) via Steenrod operations). By differential np Hopf algebra arguments similar to those in §3 we can deduce that the mental class

L

differentials in {E each r

~

2.

r)

act in a transgressive manner. We deduce that. for

E where B

r

C H*(I1K;W)

action of ~* on

p

r

= H* (B;Wp)//A r

0 B r

is a primitively generated sub Hopf algebra. The

XII: Power Spaces

EoH*(E ;Wp )

can be written

= EO) = H* (B;Wp)//A

413

~ B

'"

In passing to H*(E;W ) we must solve an extension problem. We have a commutative diagram

p

QH*(E;W) ~ Q(B)

C

p

1

(,/r>* Q{H* (B;Wp)//A)

1 ~* C

1

(cpr >*

QH*{E;W ) ~ Q(B) p

with exact rows. We also have LEMMA: If the commutative diagram

A~B.....L.C

A.....,...+B-:->C 1

J

of Wp modules and maps has rows exact and a,

~

are multiplication

by X then f3P is multiplication by X. Proof: Let b € B. then f3(b)

Setting t

= P yields

= Xb +

f3 t(b)

the lemma.

=

tea) for some a E A. By induction on t

tXt-It(a) + Xtb Q.E.D.

It follows from the lemma plus diagram (**) that (E,~P) is a power space.

414

§49: ALTERATIONS OF H-SPACE SfRUCfURES

In §49 we will develop techniques for altering H-space structures. The purpose of the a I tera t ions wi 11 be to make the coproduc t Ii*: H*(X; IF ) ---> P

H*{X;IF ) @ H*{X;IF ) as simple as possible. A reference in the literature p

p

for the work in this chapter is Harper-Zabrodsky [1]. §49-1: General Facts Given a H-space (X,Ii) and w: X A X ---> X we can define a new multiplication li :

w

liw

= Ii

+ w:

X x X ---> X

liXW,XxX~X

X x X AX x X ,XxXxXxX

(Here we consider w as the composite w: X x X ---> X A X ---> X) The coproducts are related in the following fashion.

So. if x € Hn{X;IF ) and w*(x) p

(*)

Ii *(x) W

= X@1

+

1@X

=0

for degree

+ }": x ,

1

'@X." 1


then

+ w*{x)

Our method will be to work by induction on H*{X;IF ) and attempt to choose p

a multiplication so that H*(X:IF ) is primitively generated. We will assume that this is true in degree X] so that w*

=0

in degree

p

< n.

We will then attempt to pick w € [X A X.

< nand

w*(x)

= }":

x.

1

'@X .... 1

By (*) we will have

made H*{X;IF ) primitively generated in degree n as well. p

Our alterations will be done via the difference construction. Recall (see §6-4) that. given a H-space (X.Ii).

then [Y.X] is an algebraic loop

for any complex Y. So, for any f.g € [Y,X]. there exists unique D{f.g) € [Y,X] satisfying f Given f .s

=g

+ D{f,g).

k II X ---> X which are homotopic on the fat wedge

i=l

{1. e.

the

415

XII: Power Spaces

k k IT X which contain the basepoint in any factor} then D(f,g}

elements of

i=l

factors through X(k} (= X A.•. A X) to give a map D LEMMA B: If f*

= g*

= D(f ,g}:
for degree

X(k} ---+ X

then D*

=0

in degree


while D*

=

f *-g* in degree n. §49-2: Alteration of H-spaces I Given a H-space

(X.~)

then

(X'~A)

is a power space where

~A

is the A

power map. By the discussion following Proposition 48-lA we have decompositions

= H(l}

H*(X;W} p

o H* (X:W

n

1=1

P

=

)

$ .... $ H(p-l)

$ [H(r

l} 0 ... 0 H(r n}]

l~ri~p-l

If H*(X:W } is primitively generated then we would have ~(r) C p

H(i} 0 H(j} for each r .

$

In a t t empt Ing

to prove that H*(X:W} p

is

1+j'=r mod p-l primitively generated in degree n this is the first property at which we aim. Throughout an alteration the power map

~A

and the resul ting decompo-

sition H(l} @... $ H(p-l} remain fixed. They are the ones induced by the initial

multiplication~:

X x X ---+X.

Let

x = {ail.

the A characteristic generators of H*(X:Wp}

chosen in Proposition 47-lA

xn = the

elements from X of degree n

PROPOSITION: Let (X.~) be a H-space with H*(X;F ) primitively generated in degree

< n. There is a multiplication (i) ~* (ii)

v*x

= v* n

C

in degree $

i+j == If

(X,~)

p

v such that


H(i} 0 H(j}

mod p-l

were a power H-space then property (ii) would be automatic. We

The Homology of Hopf Spaces

416

will measure the failure of

to be a power H-space and use the re-

(X.~)

sulting deviation maps to alter

The new multiplication will then sat-

~.

isfy (i) and (ii). n Given x € X we can write

~*(x)

= X@l

p-l

E

+ 10x +

r=l

where

Y r

H(a)

~

€

Yr

H(b)

~

a+b =: r mod p-l

We want to eliminate {Y 2,Y 3, •... ,Y p - 1}. Let t

= t A be

the A power map induced by~. Let t

Since H*(X;W ) is primitively generated in degree degree


p

(titlH(a)

I

i


= to ... ot

(i times)

it follows that in

ai

=A

( tix t i )* H(a)~H(b) = A(a+b ) i

~*H(r) C

~

H(a) ~ H(b)

a+b =: r mod p-l

If we let

ii = D(ti ~.~(t x t ):

Wi

XAX

-4

X

then it follows from the above, plus Lemma 49-1B. that

=0

w.* ].

i

The (p-2) x (p-2) matrix A ~

B

= CD

in degree

(A -A

ir


)1~i~p-2 is invertible. For we have 2~r~p-l

where

B

(Air_A(i+l)r) A_A2

c

o

= o

A

XII: Power Spaces

D

417

"},.

"},.p-3

Ai

Ai(P-3)

"},.p-2

"},.(p-2)(p-3)

Obviously det C i 0 while det D i 0 by the Vandermonde determinant xl······· xl det

x.

X.

X

X

1

n-l n-l

II (x.-x.) 1 J

1

n

n

i(j n-l

So B and, hence, A also have inverses. Using A-I we can reverse (**) to obtain an identity p-l

L

r=2 Now replace

~

y

r

=

by M = D(M,W) where w w

p-2

L

i=l

p-2

L

Le I

a.w.*(y) 1

a.w 1

1

i

(Recall that D(M.W)* = M* - w*) Remark: The proposition can be restated in a slightly stronger form. Namely, given a H-space (X,M) with H*(X;W ) primitively generated in degree ( n,

p

then. without loss of generality. one can assume property (ii) holds. ~A and the resulting decomposition H* (X;W = H(l) $ ... p) induced by the derived multiplication v agrees with that induced

For the power map

mH(p-l)

by the original multiplication M in degree

~

n. So,

in property (ii) of

the proposition we can consider the decomposition H*(X;W ) = H(l) $

H(p-l) to be induced by v rather than M.

P

$ ...

§49-3: Alterations of H-spaces II Ideally, one would like to continue the alteration process begun in the previous section and, finally. arrive at a mul tiplication for which all the algebra generators of degree n are primitive. This can only be done under special circumstances. We will treat those cases in §49-5. In this section we extend Proposition 49-2 by obtaining some further restrictions

418

The Homology of Hopf Spaces

on the coproduct. They will be the key to the primitivity results. As in §49-2 let ~ C H(l) be the X characteristic generators of H*(X:f ). p

PROPOSITION: Let (X,~) be a H-space where H*(X:f ) is primitively generp

ated in degree


C H(l) such that

and ~~n C $ H(i) 0 H(j). Given a subspace W i+j == I mod p-l

~~n C W@ H(p-l) +

$ H(i) @ H(j) i+j :; I mod p-l i ~ 2

then there is a multiplication v such that (i) ~*

= v*

in degree

(ii) ~~n C I

@


H*(X:f ) where I is the ideal of H*(X:f ) p

generated by W.

p

Part I: The Projection Maps pea) Before making modifications in our multiplication (so as (ii»

to obtain

we must define and study some "projection" maps. As always let 'iJ =

the decomposition induced by

~.

*

.

Let H (X;fp) = H(l) ~ ... $ H{p-l) be Consequently, we have projection maps

'iJX be the X power map induced by

~.

Pea): H*(X:f ) ~ H(a) p

for each I ~ a ~ p-l. Since H*(X:f ) is primitively generated in degree p

n we have LEMMA A: A*[P(I) @ P(a-l)]~*

= aP(a)

in degree


<

for 2 ~ a ~ p-l.

Proof: Let A

=

the sub Hopf algebra of H*(X;f ) generated by p

L Hi(X:f ) i
Then A is primitively generated. A basis of peA) gives a "simple system of generators" for A. More exactly, if {xi} is a basis of podd(A) and {Yi} is a basis of peven(A) then we can think of A as being of the form

(*)

f [yl····yt]/ A -- E( xl'" ,x ) ~ s '" P

P P (yI , .... y t )

Here we are ignoring the possibil i ty of non zero pth powers. If we take the non zero monomials allowed by the identity (*) then we obtain a basis of A.

X 1/: Power Spaces

B

={x

E F

y

= xl

el

... x s

es

YI

f

l

"'Yt

f t

I

419

0 ~ e i ~ I, 0 ~ f j ~ p-I }

The coproduct on xEyF is *

E F

~ (x y )

(**)

EI F1 E2 F2 y 0 x Y E I+E2 = E F l+F 2 = F

=

L

x

= H(l)

~ ... ~ H(p-l). Namely the monomials of weight a mod (p-l) gives a basis of H(a). In particular. it follows from (*) that

The basis B respects the splitting H*(X;f )

=

p

~*H(r) C ~

(***)

a+b

=r

H(a)

0 H(b)

mod (p-l)

It follows from (***) that A*[P(l) 0 P(a-l)]~*

=0

on H(r) unless r

= a.

Regarding the action of A*[P(I) 0 P(a-l)]~* on H(a) we know that

(l+~)k

= (l+~)a mod p

So, taking mod p coefficients of

L (

~.

k

if k

=a

mod (p-l)

we have

i~O s+i(p-l)

) _ (a)

-

s

However, given a monomial xEyF of weight d where d easy to see from

L ( .k

i20 l+l(p-l)

(**)

=a

mod (p-l), it is

that A*[P(l) 0 P(a-l)]it is multiplication by

) on xEyF. So, by the binomial identity (****).

A*[P(l) 0 P(a-l)]it is multiplied by (~)

=a

if k

=a

Remark: The above lemma does not extend to the case a

mod (p-l). Q.E.D.

= 1.

The operations

A*[P(l) 0 P(p-l)]it obviously cannot satisfy A*[P(l) 0 P(p-l)]it = P(l). For

it

annihilates peA) while pel) is the identity on primitives.

Part II: Realizing the Projection Maps pea) There is no reason to believe that the projection maps pea) can be realized by a map. However, maps

more complicated expressions involving the

pea) can be realized. And these expressions wi 11 suffice for our

purposes. We now show LEMMA B: For each 2 ~ a ~ p-l there exists a map ~ : X(3) ~ X satisfying a

(i) ~ *: a (11) ~ * a

=0

in degree

= pel)


0 P(a-l) 0 P(p-a)[it0 1 - 1 0

it]it

in degree n

420

The Homology of Hopf Spaces

To obtain

~a

we alter the map A

= D(~(~l),~(l~»:

By Lemma 49-1B

*

A

Let ~ C HOm(Hn(X:F X where f*

=0

k = a~l

l

k ~2

X

~a:

2

k ~3

3

[0

-

(~

--

@1 -

1~)~

in degree


in degree n

n(x(3):F » be the map realized by maps f: X(3) ~ p),H p

in degree

polynomial algebra power map

=

X(3) ~ X

< n.

So A* E ~. We can define an action of the

Fp[~1'~2'~3]

~

X just as

on

~. For each a E F we can define the a p was defined in §48-1. For each monomial m

~A

let

One uses addition and subtraction in [X(3),X] to realize the corresponding operations in F [A ,A ,A One has the relations p I 2 3].

m°(fl * + f 2 * ) (m

1

mo f

+ m ) of * 2

*

+ mo f

1

2

*

ml·r* + m2·r*

Remark: The need to obtain the second identity explains why we restrict our attention to the rather special set

~,

This identity is based on the

fact that addition on both sides can be induced by addition in [X(3) ,X]. Linear addition in Hom(H*(X;F ),H*(X(3);F » is not generally induced by p P the addition in [X(3),X]. One must restrict to the set ~ to obtain this fact. Now consider A*

E

~. Observe that, for 1 ~ a ~ p-1 [

0 mod p for

=a

1 mod p for

f- a

Consequently, the operations I - (~ _ Aa )p- 1 (k k

= 1,2,3),

A* , satisfy

I - (~1 -

(*)

Aa )p- 1A*

- (~2 - Aa )p- 1A* 1 - (~3 - Aa )p- 1A*

(P(a) @ 1 ~ I)A* (1

~

pea) 0 I)A*

(1 0 1 0 P(a»A*

when applied to

XII: Power Spaces

421

This is based on the fact that A* = (~@l - l~-*)~ in degree n and that

L

~~n

H(i) ~ H(j). It follows from (*)

that

i+j=l pel) @ P(a-l) @ P(p-a)(~ @ I - I @ ~)~ € ~. This completes the proof of Lemma B. Part III: Proof of the Proposition We now set about modifying 11 so as to obtain property (ii) of the proposition. Let

and, for each 2

~

a

p-l, define

~

We have wa * = 0 in deg have,by Lemma B above


while, given a generator x €

~

of degree n, we

(i) wa *(x) = (A*@I)[P(I) @ P(a-l) @ P(p-a)][~ @l - l@ ~]~(x) It follows from Lemma A above that (ii) (A*@I)[P(I) @ P(a-l) @ P(p-a)](~@I)~(X) = a[P(a) @ p(p-a)]~(x) Also. given any W C H(l) such that

~(x)

€

W@ H(p-l) +

m H(i) @ H(j) i+j=l mod (p-l) i

it is easy to see that

~

2

(iii) (A*@I)[P(I) @ P(a-l) @ P(p-a)](~@I)~(X) € I @ H*(X;W ) where I is p

the ideal generated by W. As a consequence of (i),(ii) and (iii) we have -* (x) mod (I @ H* (X;W » LEMMA C: wa * (x) = a[P(a) @ P(p-a)]l1 p So. if we let p-l w=L:.!.w a=2 a a then it follows that l1 w

D(l1,W) satisfies ~ *(x) € I @ H*(X;W ). (Recall

that D(l1,W)* = 11* - w*) §49-4: Alteration of Power H-Spaces

w

p

422

The Homology of Hopf Spaces

The alteration of H-space structures carried out in §49-2 and §49-3 can be modified so as to be compatible with power H-space structures. (X.~.~)

Let

be a power H-space. Let H(l)

~ ... ~

H(p-l)

be the decomposition obtained from the power space map alterations

with

respect

to

obtained from the X power map

~

~.

this

decomposition

rather

= ~X

as in §49-2 and §49-3.

We can do our than

the

one

Suppose that H*(X;lF is primitively generated in degree < n. First. p) observe that the power H-space structure automatically gives the restrictions obtained in §49-2. For we have

~H(r) C Moreover. by Proposition 49-1.

~ H(i) 0 H(j) i+ J:r mod (p-l) the algebra generator J( of H*(X:IF ) are p

contained in H(l).

When we make the al terations from §49-3 we must impose an addi tional hypothesis on our decomposition H*(X:IF ) p

p-l

=

m H(i). Namely. we must assume

i=l

that PH*(X;IF ) C H(l) p

in degree

< n.

This property was used in the proof of Lemma 49-3A. It is

automatic for the decomposition H*(X;IF ) power map

~

= ~X

p

induced by

~.

p-l

= mH(i) Le I

obtained from the X

However. if we use a different power map

and. hence. a different decomposition it might not be true. Granted this hypothesis. then the alteration from §49-3 can be carried out. Observe that if

(X.~.~)

is a power H-space then

power H-space where v is the new multiplication. For

(X.v.~) ~

the maps used in constructing the perturbation w: X A X ly.

~

is still a

commutes with all ~

X. Consequent-

also commutes with v. If we let J( C H(l) be the X characteristic

vectors then we obtain PROPOSITION: Let

(X.~.~)

be a power H-space such that.

in degree

<

n,

H*(X;IF ) is primitively generated and PH*(X:IF ) C H(l). Given a p

p

subspace We H(l) such that v*J(n C W0 H(p-l) +

m H(i) 0 H(j) i+j=l mod (p-l) ~

there is a multiplication v such that

2

XII: Power Spaces

(i)

(X,v,~)

(ii) ~*

= v*

423

is a power H-space


in degree

n (iii) V*X C I @ H*(X:W ) where I is the ideal of H*(X:W ) p

generated by W.

p

§49-5: Primitively Generated Cohomology In certain circumstances the types of restrictions obtained in §49-2 and §49-3 tell us that X has a multiplication

Xx X

~:

~

X so that

H*(X:W ) is primitively generated in low degrees. We will derive two such p

results. They are expressly designed to be used in §50 and §5l during our construction of families of H-spaces. Our first result is fairly obvious. PROPOSITION A: Let

(X,~)

be a H-space where in degree


(i) H*(X:W ) is primitively generated p

(ii) H*(X:W ) is an exterior algebra on ~ p-2 generators. p

Then there is a multiplication v for which H*(X:W ) is primitively generated in degree

p

n.

~

For the fact that H*(X;W ) is an exterior algebra on ~ p-2 generators in degree


p

=0

means that H(p-l)

in degree

< n.

Consequently, in Propo-

sition 49-3 we may choose W = O. Thus I = I(W) = 0 as well and V*(x) = 0 for each algebra generator of degree n. If we deal with a power H-space H*(X;W ) p

(X,~,~)

and let

= H(l) m...mH(p-l)

be the decomposition obtained from

~

then, using Proposi tion 49--4,

we

obtain PROPOSITION B: Let

(X,~)

be a power H-space where in degree

(i) PH*(X;W ) P


= H(l)

(ii) H* (X;W is an exterior algebra on ~ p-2 generators p) Then there exists a multiplication v such that (X,v.~) is a power H-space and PH* (X;W p)

= H(l)

Note that the condition PH*(X;W ) P

in degree

= H(l)

~

n.

strengthens the condition that

H*(X:W ) is primitively generated. For the algebra generators of H*(X:W ) p

p

424

TheHomo0gyofHopfSpac~

are always contained in H(l). As our second result we obtain PROPOSITION C: Let

(X.~)

be a H-space where. in degree

< n.

(i) H*(X;W ) is primitively generated P (ii) H*(X;W ) p

If n

> 2p2+2p+2

then there exists a multiplication v such that

H*(X;W ) is primitively generated in degree ~ n. p

Proof: Let M(k) = the subspace of H*(X;W ) generated by the monomials in p

{x3,x2p+l,x2p+2} of weight = k So, in degree

< n.

H(a) =

~ M(a) Observe that M(k) = 0 for k k=a mod (p-l) p+l. By Proposition 49-2 we have a multiplication v such that

v*(x) E

>

~ M(i) 0 M(j) + ~ M(i) 0 M(j) i+j=p i+j=2p-l

We can eliminate terms from M(i) 0 M(p-i). For such terms have degree p(2p+2)

2P2+2p while n

> 2p2+2p+2. Since M(k) =

0 for k

~

> p+l we are

reduced to v*(x) E

~

i+j=2p+l

M(i) 0 M(j)

p-2~i.j~p+l

First. applying condition (iii) of Proposition 49-3 we can reduce to v*(x) E M(p) 0 M(p-l) + M(p+l) 0 M(p-2) If

we apply

multiplication

the

twist map T:

X x X

~

X x X to

v

we obtain a

where v*(x) E M(p-l) 0 M(p) + M(p-2) 0 M(p+l)

Finally again applying Proposition 49-3 (observe that we can now choose W = 0) we have a multiplication v where v*(x) = O. Q.E.D.

425

§50: H-SPACES WITH TORSION

In §50 and §51 we will construct families of mod p finite H-spaces. In §50 the H-spaces will have integral p torsion. In §51 the H-spaces will be torsion free. In both cases the machinery of power spaces as developed in §48 and §49 will playa major role. §50-1: H-spaces with Torsion In §50 we will prove THEOREM: (Harper) For each odd prime p there exists a mod pH-space X(p) where H*(X(p);W ) p

= E{x3.x2p+ 1)

~ Wp[~2J/( x

2p+2

p)

and A*(p) acts by the rule

Observe that the non trivial action of the Bockstein 0 means that H*{X(p);W ) has p torsion. Observe also how well H*(X(p);W ) fits into the p

p

structure theorems for Qeve~*(X;W ) from §36. Given a I-connected mod p 2iH*(X;W finite H-space then Q ) p

p

=0

for 2i

P. X(p) provides an example of a H-space

< 2p+2.

So. for each odd prime

where QevenH*(X;W ) is non trivp

ial in the first allowable degree. In higher degree there remains a gap between the structure theorems of §36 and examples satisfying them. The 2 2+?__* exceptional Lie group E demonstrates that Q p ~ (X;W can be non zero. p) 8 at least for p = 3. Beyond that the structure theorems of §36 are.so far. vacuous. The above theorem was originally proved in Harper [8J. A modified and simpler proof was presented in Harper-Zabrodsky [IJ. Our argument is based on that paper.

426

The Homology of Hopf Spaces

§50-2: Primitive Postnikov Systems Our approach to H-space structures will be via Postnikov systems. It is a standard fact that X is a H-space if and only if the Postnikov system of X consists of H-spaces and H-maps. More generally, given f: X

~

Y where Y

is a H-space then X is a H-space and f is a H-map if and only if the Postnikov decomposition of f consists of H-spaces and H-maps. The approach of this chapter is a variant of these ideas. For reasons to be explained we will deal with primitive Postnikov systems rather than Postnikov systems. Given a rational equivalence f: X

~

X between spaces localized at p

o

then 1I*{fibre f) consists of finite p groups, So if we localize at the prime p then the usual Postnikov system of f can be further refined to obtain a primitive Postnikov system. It is a tower

where (i) Kn K(Zlp,sn) (ii) X + 1 ~ X ~ K is a fibration n n n (iii) (f n )# : lIi(X) --4 ITi(X is an isomorphism for n) (iv) Sn--4 00

< sn

The maps {k

are called the k-invariants of the system. We will work with n} rational equivalences f: X ~ X So we are free to work wi th primi tive

o

Postnikov systems rather than Postnikov systems. There is a definite advantage in that the k invariants are in cohomology with W coefficients p f .e . k

n

€

[X,K(Zlp,s )J n

s

= H n(X n ;Wp )

When we introduce power space structures into the primi tive Postnikov system the k invariants will turn out to be compatible with the power map in a way that forces significant restrictions. The primitive Postnikov system of f: X theory to X being a H-space and f: X PROPOSITION A: Suppose f: X

... ~ X n

~

~ X provides an obstruction o X being a H-map.

o

~

X is a rational equivalence and

~

X n_ 1

o

~ ... ~

Xl

~

X

o

427

XII: Power Spaces

is the primitive Postnikov system of f. If the,spaces {Xn} have H-space structures such that the maps {k are H-maps then X is a n} H-space and f is a H-map. Proof: We wi 11 work wi th skeletons of X. For each m

1 we can produce

~

maps

as follows. For N

»

x"

m 2m x X --+ X

0 (f N)# : ITi(X}(p} ~ ITi(~}(P} is an isomorphism for

i ~ 2m. Consequently, we can'choose a CW structure on ~ such that (X N}2m 2m 2m. m = X The multiplication ~ x ~ -+ ~ induces a map X x x" ~ X The above maps obviously satisfy the following diagram x

m

m x X

1

(*)

1

xm+ l

) X2m+2

Xm x Xm

) X2m

xm+l x and

(**)

X2m

J

fxf

1

1

Xo x Xo

f

• Xo

By (*) we can use the maps to produce a H-space multiplication

xx

X

---0..+)

II

II m

U (X

x

U X2m

m )

X

m~O

m~O

By (**) f : X

~

X

o

X

is a H-space map.

Q.E.D.

If X is a finite complex then our requirements can be weakened. Namely PROPOSITION B: Let X be a finite CW complex, f: X ~ X a rational

o

equivalence and

... ~ X -+ X - ~ ••. ~ X n l o n the primitive Postnikov system of f. If the spaces {Xn} are Hspaces then X is a H-space.

Proof: Let dim X

= m.

Pick N

is an isomorphism for i cation XN x ~

X. Q.E.D.

~ ~ ~

.

»

0 such that (f N)# : ITi(X)(p} ~ ITi(~}(p}

~ 2m. So we can assume (~}2m = X The multipli-

I nduce s a map

(~)

m

x

(~)

m

-+ (~)

2m

and. so, X x X

428

The Homology of Hopf Spaces

The freedom from requiring the H-space structures on {X to be comn} patible (via the maps Pn: X ~ X - ) allows much greater choice. So the n 1 n chances of X being a H-space are significantly strengthened. Our approach will be to choose H-space structures on the spaces X so that the k invarn iants are primitive. Suppose we have a multiplication on X such that the n k invariant is primitive i.e. k

n

€

s * P ~ (X ;F ) n

p

k

p

X ~ K(Zlp.sn) is·a H-map. Since X + ~ X ~ K(l P.sn) is a n: n n n 1 fibration we have a H-space structure induced on X such that Pn is a n+1 H-map. It is at this point that the machinery of H-space al terations as

Then k

s 1 € H n+1(X l;F) is not n+ n+ p primitive then we alter the H-space structure on X + and make k primn+1 n 1 itive. This alteration may. of course, destroy the property of Pn+l being

developed in §49 enters our argument.

If k

a H-map. All of

the previous results are valid for power H-spaces not

just

H-spaces. In §51 we will need the following extended version of Proposition B. PROPOSITION C: Let X be a fini te CW complex and let

r.

X

~

rational equivalence. Let ... ~ X ~ X ~ ... ~ Xl ~ X o n n- 1 be a primitive Postnikov system of f. If the spaces spaces then

(X.~)

is a H-power space.

X be a

o

{Xn } are power

§50-3: Construction of H-spaces with Torsion We now construct the H-spaces X(p) of Theorem 50-1. We will assume p 5. This is a useful assumption for our argument. However. the p

=3

~

case

can be obtained by alternative methods. In §9-2 it was pointed out (though not proved) that F 4

'" X x Y where H* (X;F

(3)

3)

Since X is a retract of the H-space F it also is a H-space. 4 For p ~ 5 we will obtain X(p) from the Eilenberg-Maclane space K(l.3). We have H*(K(l.3);F ) p

where

E(~.X2p+1·X 2 .... ) 2p +1

0 W [x2p+2' X 2 P

2p +2

•... ]

X II: Power Spaces s pP (x

2ps+l

x

429

)

o(x s ) 2p +1

2ps+2

Since x 3 is primitive it follows that all the generators are primitive. So. in low degrees. H*(K(~.3):W ) looks like the mod p cohomology of our p

intended space X(p). We obtain X(p) by simply killing all the excess cohomology of

K(~.3).

Let A

~ f

= E( x.y )

~

p

[z]/

(zp)

where Ixl

= 3.

and

Iyl

= 2p+l.

Izl

= 2p+2

pl(x) = y

=z

c'i(y)

Observe that dimension A

= 2p2+2P+2.

Our construction of X(p) can be con-

veniently broken up into two steps so that H* (XO;f p)

(I) We will construct a H-space X

o

2p2+2p+2 and H*(XO:lll)

=

0 for i

~ 2p2+2p+2. So.

2p2+2P+2 homology approximation of X is a rational eqUivalence.

o

=

A in degrees

if we let X(p)

then W(X(p):lF p)

=A

s

be the

and X(p) C Xo

(II) By using the al teration techniques of §49 we wi 11 demonstrate that

each space in the primitive Postnikov system of X(p)

is a H-space. By

Theorem 5O-2B X(p) will then be a H-space. Part I: The Space X

o

(a)The Space X To begin. let K

= K(~.3) X_

Since x

x

and consider the fibration

x 2 K 2p +1, K(Vp. 2p2_ l)

is a H-map (i.e. x 2 € PH* (K;W » X is a H-space. Indeed 2p2+ l 2p +1 P is an infinite loop map and X is an infinite loop space. It is easy

2 2p +1 to apply the Serre spectral sequence to the fibration

2 j K(Vp.2p ) ---+ X ---+ K and deduce

430

The Homology of Hopi Spaces

H*(X;IF ) as an algebra over elements e

l

and e

2

A* (p)

p

=A ~

r

where. in degree

~

.* J (e ) j

= P 1(t

2p

* (e 2) = GbQI(t

Regarding the coalgebra structure we have LEMMA A: (i) x.y,z,e

Proof: Since x

2

r consists of two

of degree 2p2+2p_2 and 2p2+2p. respectively. satisfying l

(ii) If e

2p 2 +2p+2.

2)

2p

2)

are primitive

l

is not primitive then M*(e

p-l

L ~1:') zp-i ~ z i . i=l P 1

2)

2 is an iterated loop space H (X;IF ) is a commutative * p 2p +1

and associative algebra. Consequently, the failure of H*(X;IF ) to be primp

itively generated is measured by the non triviality of the Frobenius pth power map f: H*(X;lF p)

-+

H*(X;lF (see Corollary l-5C) Obviously. p)

there

cannot be non trivial pth powers in the degrees of x,y.z or " i So the first non trivial pth power must occur in the degree of e If e that

2

2.

Q.E.D.

is not primi tive then we can al ter the H-space structure on X so

LEMMA B: e

2

is primitive.

Proof: By Lemma A we must eliminate

p-l I

L =<1:')

i=l p

..

ZP-l 0 Zl

_

from ~ (e

1

2).

Alter

the multiplication using d

2

j

w: X x X - - K(ll p.2p ) - - X where

d

p-2

L ~1:') zj 0 xyzp-j-2 + xyzp-3 ~ z j=l P 1

= w*(y) = w*(z) = O. w*(e = 0 as well. So. l)

Observe that w*(x) it follows that

=

Since j*(e w*

=0

l)

= pl(l)

in degree

and pl(w)

=0

< 2p2+2p and. as

in §49-l.

i z . So . by Lemma A

XII: Power Spaces

O.

431

Q.E.D.

The spaces X and X " X have only elementary p torsion in degree

~

2p2+2p+2. It suffices to consider X. Let {B be the Bockstein spectral r} sequence analysing torsion in H* (X}(p)'

= H*(X;F p) = A @ f

description of B 1

It is easy to deduce from our

that. in degree ~ 2p2+2p+2. Boo

E@f

where E is an exterior algebra on {x} and {yzp-l}. It now follows. by differential Hopf algebra ar-gument.s , that B 2 (see Corollary 1-6) The above argument also shows that e

2

= Boo

in degree

s

2p2+2p+2.

is a permanent cycle in {B In r}.

other words. e 2 is the mod p reduction of some e E H* (X)(p)' Moreover 2 LEMMA C: e

is primitive

2

.-?n2+2 Proof: We have ~ (e2) E H-r p(x" X)(p)' As observed above _

2 22

_A

H P + p(x " X) mod p. So.

A

~

_

A

consists of elementary p torsion. By Lemma B ~ (e

(p)

= O.

(e 2)

2)

0

Q.E.D.

(b) The Space X

o

We define the space X by the fibration

o

Xo ---> X By Lemmas A and C. e

e 1x e 2

1x

e

2

2 2 , K(Vp.2p +2p-2) x K(Z.2p +2p}

is a H-map. So. X is a H-space. Moreover. it is

o

easy to calculate. from a Serre spectral sequence argument. that H* (X 2;Fp)

=A

in degree ~ 2p2+2p+2. Q.E.D.

Part II: The Primitive Postnikov System X(p) C X

o

(a) The Space X(p) Let X(p)

= the

2p2+2p+2 homology approximation of X

o

LEMMA D: X(p) C X is a rational equivalence.

o

Proof:By construction X(p) C X

o

is an isomorphism in rational homology in

degree ~ 2p2+2p+2. Moreover. Hi(X(p);~) Q.E.D.

= Hi(X2;~) = 0

for i

> 2p2+2p+2.

432

The Homology of Hopf Spaces

(b) The Primitive Postnikov System of

xep) c Xo

Given the primi tive Postnikov system .. --- ~+l --- ~ ---... --- Xl --- Xo of X(p) C Xo suppose that ~ is a H-space. Write H* (XN;Wp)

=Ae

IN

where IN consists of the elements of degree # k

€

> 2p2+2p+2. Pick an element 0

IN of minimal degree. By Proposition 49-5C we can alter the multi-

N plication on

~

so as to ensure

~

--- K(Z!P,sN) is a H-space as well.

is primitive. Then

~+l

= fibre

~:

XN

433

§51: H-SPACFS OF LOW RANK

This chapter is concerned with constructing torsion free finite H-spaces of low rank. It is based on the results of Cooke-Harper-Zabrodsky [1]. §51-1: H-spaces of Low Rank We will work with spaces localized at the prime p. This chapter will be devoted to proving THEOREM (Cooke-Harper-Zabrodsky) Let H* (X)(p) plication

= E(x 1 ....• x r) ~:

Xx X

~

(X.~)

be a power space such that

where r (p-l. Then there exists a multi-

X so that

(X.~.~)

is a power H-space.

We should make some comments on this theorem. First of all. some restrictions aside from H* (X)(p)

= E(x 1 ..... x r) is necessary in order for X to be a H-space. For. as Zabrodsky [13] has demonstrated . given a €

(S2n+l ) of odd ord er , t h ere exists on I y one H-space 0 f t h e f orm X " 2m s2n+1Uae2m+1uJ3e2m+2n+2. On the other hand. there usually exists many choices

of

the attaching map 13:

S2m+2n+l ~ s2n+1 Ua e2m+l

such

that

H* (X)(p) = E(x2n+l.x2m+1)' Secondly. once one reaches rank p-l the theorem fails. There are power spaces X where H* (X)(p)

= E(x1 •...• x r) but X is UQ1 a H-space. Such examples are discussed in Harper-Zabrodsky [2]. The theorem is very useful in constructing mod p finite H-spaces of low rank on a rank by rank basis. It can be used to show that. for rank

~

p-2.

there are an enormous number of finite H-spaces in the torsion free case. To be more precise. suppose that we have a mod p finite torsion free Hspace

(X.~)

of rank r where r

Y(X.a) by the pullback diagram

< p-2. Pick a

€

"

2n (X) . We can define Y

=

434

The Homology of Hopf Spaces

x !

Here the right hand is the Hopf fibration obtained from applying the Hopf construction to the multiplication (X.~)

~:

X x X

X. Given

~

~

such H-space

of rank r and a € U then 2m(X)

PROPOSITION: Y

= Y(X,a)

is a H-space of rank r+1.

To prove the proposition we need to verify that (i) H* (X)(p) is an exterior algebra on r+1 generators. (ii) There exists a self

Y

map~:

~y

so that

(Y,~)

is a power space.

The first fact is easy. Consider the Serre spectral sequence for X

~

Y

~

1 S2n+1 . We ha ve E00 = E = H*(X)'" (p) '" H*(S2n+ ) (p) For differentials are 2 torsion valued while E is torsion free. Regarding the second fact. let ~ 2 = ~A be the A power map on X induced by ~. By the theorem we can assume that

(X.~.~)

is a power H-space. So. for some r

~ x

~r

1

l~

However.

the

fibration

r

, X

Xx X commutes.

O. the diagram

IX

Xx X r

~

r

(X.~ )

~

r (X * X.~ *

r

~)

J!Wl..

r

(XX.l~ ) is a fibration of power spaces. The power map ~ has the property that it induces multiplication by A on IT*(X). So. the map p r) (S2m+1. A _ is a map of power spaces. (AP map .,,: Y

~

r

(X.~r)

is the degree map) Diagram (*) gives a self

Y. Since we have a short exact sequence

o ~ QH*(X:IF

) ~ QH*(Y;IF ) ~ QH*(S2n+1;lF ~ 0 p) P Lemma 48-2 that ." induces multiplication by p

it

follows

from

A on

QH* (Y;lFp)'

An alternative (possibly less general) approach to the construction of

torsion free finite H-spaces of low rank is provided by Cohen-Neisendorfer [lJ. We might also remark that. given a torsion free H-space Y of rank

<

435

X /I: Power Spaces

p-l. i t is easy to construct a map Y

~

S2m+l so that the fibre X is a

H-space of rank r-l. So the above method provides all torsion free finite

< p-l.

H-spaces of rank

However, the process may yield the same homotopy

type in different ways. So we do not have a classification result. §5l-2: Proof of Theorem 51-1 Our proof can be broken up into two stages. (I) We will construct a rational equivalence f: X

X where X is a o o

~

power H-space. (II) We will construct a primitive Postnikov system

... ~ X ~ X N l N ... ~ Xl ~ X of f and demonstrate (by induction) that each ~ is o a power H-space.

By Proposition 5O-2B X will then be a power H-space. So assume that

(X,~)

= E{xl ..... x r).

is a power H-space and H*(X){p) Par t I: The Space X o

Let

r

IT K{Z{ ),2n.-l)

i=l where deg Xi

= 2n i - l .

p

1

If we take the canonical abelian group multiplicak

tion ~ on Xo and the A power map ~ = ~A induced by ~ then (Xo'~'~ ) is a power H-space for each k ~ O. We want to construct a map f: X

~

X

o

which is a rational equivalence and also a power map. The map is the obvious choice. Namely. we choose a map realizing the algebra generators

=

{xi} of H* (X)(p) E(x ... ,x So f is a rational equivalence. The prol, r}. perty of being a power map demands more work. The algebra generators are. of course. far from unique. We would like to show that we can choose {Xi} so that ~*(Xi) ~:

X

~

= Ax i

for each 1 ~ i ~ r. To do this we replace the given

X by a new power map

~:

X

~

X and replace

~:

Xo

~

Xo by an

k

appropriate iterate ~ . We will prove LEMMA: There exists a self map

~:

X -> X plus a choice of the algebra

generators {Xi} of H* (X){p) where

436

The Homology of Hopf Spaces

(a) Algebra Generators for H*(X:Vl!sl We begin by working mod pS. We can prove. by induction on s. that. for the given map "'.

there exists algebra generators {x.} of H*{X:Vps) I

E(X1.··· .Xr) satisfying

*

'" (xi) for some k

»

= -,...pk xi

O. The initial case s

Suppose that H* (X;Vp s ) sentatives {Yi}

=1

follows from Proposition 48-1A.

and '"* (xi)

= E(x1 ..... x r)

. H* (X;Vp s+1 ) In

= -,. .p

k

xi' Choose repre-

for {Xi}' Then. for each y

*

'" (y)

=

= -,...Pk y

= Yi'

+ pd

for some d € H*(X;Vps+l). To eliminate d we argue exactly as we did for Proposition 48-1A. We write d eliminate {d ..... d

2

p_ 1}

= d1

+ ... + d

by replacing y by

p_ 1

*

where", (d

p-1 y=y-pL . 2 k. k J= (-,...p)J _ "AP We eliminate d (b) The Map

<jl

1

j

)

by replacing", by ",p.

and Canonical Generators for H*~(p)

* (x.) = -,...pk x. I I

<jl

+ pSd. where k.s I

{psd.} we alter the map I

(Hence

<jl

We

d. J

By part (a) we can choose generators {Xi} of H* (X){p) where

= {-,...pk )J.d j .

<jl.

»

= E{xl •.... x r)

O. To eliminate the error terms

We will find a map

<jl

so that

is also a power map) and

r

Let f: X --+ Xo

= II

K(Z{ ).lx.I) be the map realizing {Xi}' Suppose we can i=l p I construct a map ~: X x X --+ X covering the multiplication of Xo in the

o

sense that the following diagram commutes.

437

X II: Power Spaces

TI

Xx X

(*)

f x

>/J

1

IX

o

1

f

XOX Xo

' Xo

~o

where>/J is the map (_ps._ps ..... _ps). Let g: X ~ X be the map realizing

o

{d

i}.

If we then define (**)

~* then ~ (xi)

X

~:

k

= AP xi

~

by

A

~xg

--+

Xx X-

TI X x Xo ----> X

as desired.

Unfortunately. we can only construct an approximation of the map TI. But it will suffice for our purposes. Let ... ~

~ ~ ~-l ~ ... ~

the primitive Postnikov system of f. For each m map TIm: X x X

o ~ Xm fitting x ./

x

into a commutative diagram

11

Xo

1

-----="---------->,

pX "0

X x Xo where p is the map (P

Xo be

lone can construct a

~

1 x

"0

f

---"" Xo

x Xo

11 2

'I'

---->

Xo

p). The maps TIm are constructed by induction.

Given TIm then Tlm+l exists provided the composition lxP

TI

k

o -2!!.-. Xm ....!!!.....,. Km =

X x X ----> X x X

o

K(Vp.s ) m

is trivial (k is the k invariant). Triviality follows from two facts. The m map -p : H* (XO;F ~ H* (XO:F is trivial. Also we can choose TIm so that p) p) t TI k X ~ X x X ....!!!... X .n, K(Vp.s ) is trivial. O m m We can assume that X is a finite complex. Suppose X is of dimension d. d So X

= X.

Pick s

»

0 so that (f s )# : ITi(X)(p)

~

ITi(Xs)(p) is an isomor-

phism for i ~ 2d. Choose a CW structure on X so that (X )2d map TIs: X x X

o~

X induces a map s

Thus TIs satisfies an analogue of and produce the desired

~.

(*). We

s

s

= X.

Then the

can use it as in (**) to alter

~

438

The Homology of Hopf Spaces

Part II: The Primitive Postnikov System Let f: X --. Xo be the map realizing {xi} and let .. --' ~ --. ~-l --....

--. Xo be the primitive Postnikov system of X. We will prove by induction on N that

(i)

is a power H-space

~

(il) f

N:

X --.

~

is a map of power spaces

Part I establishes these properties for X and fO' We will assume that (i)

o and (ii) hold for ~ and f Let ~: ~ --. KN N.

= K(Zlp,sN}

be the Nth k

invariant. We want to show k

(Here 48-2B,

KN

N

is a map of power spaces.

has its canonical power H-space structure) For, by Proposi tion

~+l

is then a power H-space and

spaces. By Proposition 48-2A f map of power spaces. First of all,

~

N

~+l

lifts to f

--.

N+1:

~

X --.

is a map of power H~+l

is a map of power spaces. For since f

map of power spaces it follows that, in degree sN' ker f under "'N* where "'N:

~

--.

~

where f

N+ l

is a

N:

X --. X is a N

*

is invariant

N

is the power map on XN' Moreover, we have a

short exact sequence (*)

0 --. Ker f

* sN * f * sN --. Q H (~;lFp) ~ Q H*(X;lF --.0 N p}

(H*(X;lF is an exterior algebra and f H*(~;lFp} ~ H*(X;lFp} in degree p) N*:

<

sN'} Since "'N* is multiplication by A in QH* (~;lFp) it is also multiplication by A on ker f * and, hence, on 1m N

* C Ker f * . In other words N

~

1m ~* C H(1} The map

~:

= H*(X;IFp }

X --.

KN

is also a H-space map. For, in degree

is an exterior algebra on

<

p-2 generators. So,

Proposition 49-2B, we can deduce

PH*(~;lFp}

(......) in degree

~

sN' Combining (.... ) and (...... )

< sN' H* (XN;lFp)

H(l}

by applying

439

APPENDIX A: MOD P COHOMOLOGY OF LIE GROUPS

The Cartan-Ki 11ing classification of

locally isomorphic compact Lie

groups gives 7 classes of simple Lie groups: An (n

1). Bn (n ~ 2). C (n n Each class contains a simply con~

~ 3). D (n ~ 4). G F E ~ and E n 8, 2• 4• 6• nected representative unique up to global isomorphism. The other groups in

the class are quotients of the former by subgroups of its centre. Group An B n Cn D n

= SU{n+l) = Spin(2n+l) = Sp{n) = Spin(2n)

Centre

~

Vn+l

(3.5.7 ..... 2n+1)

V2 V2 V2 Ell V2 n even V4 n odd

(3.7.11 .... ,4n-l) (3. 7 .11. ...• 4n-l) (3.7.11 .... 4n-5,2n-l)

G 2 F 4 £6

(3.9.11.15.17.23)

£7

(3.11.15,19,23,27.35)

V3 V2

£8

(3.15.23.27.35.39.47.59)

0

0

(3.11)

0

(3.11.15.23)

Let G be simply connected and let reG be a subgroup of the centre of G. In certain cases G/

has a standard name. The quotient of G by its full r centre is denoted PC or Ad{G). In the Spin(k) case if we quotient by the

appropriate V2 we obtain SOCk). For k

= 2n+l

or k

there is no abigui ty about the V2 factor. For k

=

= 2n

where n is odd

2n when n is even.

Spin(2n) contains three central subgroups of order 2. One of these gives SO(2n) as a quotient. The other two give isomorphic copies of SS(2n). the semi-spinor group. as the quotient. We will describe the mod p cohomology of the simple compact Lie groups. We will not attempt to give any historical account of the results, Notably the papers we list are not necessarily the original source of a given calculation. The reader might consult Kacs [1] and Kono [3] for a brief historical summary of the calculations listed in this appendix.

440

The Homology of Hopf Spaces

(1) The An Groups (Borel [2J. Baum-Browder [IJ) The simply connected case is given by

where

the generators

are all

primitive and

the action of

A*(p)

is

determined by

(~)

x 2 j+2 i(p-l)+1

In the p = 2 case one replaces pi by Sq2i. To interpret this rule one adopts the convention that Xi

=0

unless i

= 3.5 ..... 2n+l.

Regarding the non simply connected case choose k where kin. The mod p . the same as above unless p Ik. In that case cohomology of SU(n+l) /ZIk IS write

n

= prn,

where (n' .p)

k

= pSk'

where (k' ,p)

Then

W [zJ

~

E(x3.x5.···.x •.... x 2 +1) @ E(y) @ p 2pr-1 n . where jyl unless p

=2

and s

= 1.

In that case z

= y2

= 1.

/

r (zp )

Izi

=2

The action of A*(p) is given

for pi as in the simply connected case while 5 acts by the rule o(x ) r-1 2p -1

When s

1 we also have D(Y)

= z.

= zp

r-l

The coproduct is non trivial. For

~*(X2i+1) = °rsY@z +j~i (~)

x 2 j+ 1@zi-

j

where Drs is the Kronecker product. (2) The G Groups (Borel [2J. Baum-Browder [IJ) n The simply connected case is given by

where all the generators are primitive and A*(p) acts by the same rule as in the An case. (This time the convention becomes that Xi 3,7 ....• 411-1)

The non simply connected case is the same except when p

=0

unless i

= 2.

Write

=

441

Appendices

n

= 2 rn'

where n' is odd

Then

*

W [y]

= E(x3'~""

H (PSp(n);W2)

.x2r+l_l •...• x4n-l) @ 2

/(y2r+1)

where Iyl

= Sq2i

The action of pi

=

is again as in the A case while Sql is determined n

by 1

Sq (y) = y

2

1

and Sq (x

2 r_l

) = y

2

r

The coproduct is given by -*(

~

(3)

~

x4 i+ 3

)

= ~L

(i.) J

~.4i-4j

X4j+3~J

and Dn Groups (Baum-Browder [1]. May-Zabrodsky [1]. IshitoyaKono-Toda [1])

We can describe the B and the D cases of the groups G = Spin(n). n n SO(n), PSpin(n) simutaneously. For p odd the B cases have the same mod p n cohomology as Sp(n) (actually the same mod p homotopy type). In the D n case we have, for p odd.

H* (Spin(n);W2)

= H* (SO(n);W 2) = H* (PSpin(n);W2) = E(x3'~""

'4n-5) @ E(y)

where Iyl

= 2n-l

The generators are primitive. The action of pi on the generators {xi} is as in the A case while pi(y) n

As regards p V(x

' I

=2

x. ) ' t

=0

for i ~ 1.

let

=

the commutative associative algebra over

E

E

= 0.1

(i) { (Xi ) 1... (x. ) tiE. ' t 1 1

(ii) ( Xi ) 2 q

(x, )2 1

= x 2i q =0

l'f 2,' q

i

s

~2

} is a !lasis

for some 1

~

s

otherwise

q

The easiest case to describe is SO(n).

where all the generators are primitive and A*(2) acts by the rule Sqi(x j )

= (i)x i+ j

such that

~

t while

442

The Homology of Hopf Spaces

The two other groups are modifications of this. Write n = 2

rn'

where n

is

odd.

*

~

A

H (PSpin(n);~2) = Y(x1.x2.···.x2r_1.··· ,xn- 1) @ 2

[y]

l(y2r)

where Iyl The action of A*(2) is as above plus we also have 1

Sq (x r 1 ) 2 - -1 when r

~

3 and j

=x

2r - 1

1 + Y 2r -

1. The coproduct is given by -*

~

i (x.) = L (j) x.@Y i-j j
1

As for Spin(n) choose s where 2 s

J

< n ~ 2 5+ 1. We have

H*(SPin(n);~2) = Y({ xil3 ~ i ~ n-1 and i t 2 j })

@ E(z)

where Izi

= 2 5 + 1_1

The action of A*(2) is as before plus we also have

The coproduct is given by i!*(z)

L

X2 i @X2 j _1

i+j=2 s-1

(All other generators are primitive). As for the group SS(2n) (n even) its mod p cohomology for p odd agrees wi th the other D groups. Regarding p = 2 choose r and s where 2n = 2n r'odd s-1 2 and 2 < 2n ~ 2 s. We have

t 2

j

}) @ ~2[Y]1

where Iyl The A*(2) action on {xi} is as before while

L

Sq1(z) =

i+j=22 The coproduct is given by

s-l X2i X2j +

r

(i )

@ E(z)

1 and Izi = 2

s+ 1_1

443

Appendices

-*

" 2"J _ L (~j)Y 0x i 2 j

~ (xi)

+ ioxi@y

for

¢ 2

r-I

i<j
~ 1...

~ (z)

i+j+k=2s-I

i+j 2i (j)Y x 2 '~k

+

J

L

i+j=2 s- I

2 Y x2ix2 j @Y

i<j

(4) The Exceptional Groups (Kana [3J,Mimura-Toda [IJ, Thomas [3J) First of all we have

H* (G2;Wp )

= E(x 3 , x I I)

H* (F 4;Wp) = E(x3,xII,xIS,x23) H*(E6;Wp)

= E(x 3,x9,xI I ,xIS,xI7,x23)

for p

~

3

for p

~

S

for p

~

S

for p

~

S

for p

~

7

All generators are primitive. We will not write down the action of A*(p). It amounts to the action of pI and is described in Mimura-Toda [IJ. Basically pI acts non trivially as often as it can. For p = 2 we have

*

H (G2 ;W2 ) = E(x S) ~

W [x J 2

3/(x34)

*

= E(x S'xIS'x23)

*

= E(xS,X9,XIS,XI7,x23)

H (F 4;W2) H (E6;W2)

H*(E7;W2) = H*(E8;W2)

W [x 3/(x34) J

@ 2

E(xIS,xI7'~3,x27)

= E(xI7,x23'~7,x29)

W [x

@ 2

J

3/(x34)

@ W2[X3,XS,XgJ/(X34,XS4,Xg4)

W2[~'XS'X9'XlS/(X3I6'X58'Xg4'XIS4)

0

The action of A*(2) is given by 2 Sq (x ) 3

4

Sq (x ) S 8 Sq (x

I5 )

= Xs

2 Sq (XIS) = x I7

= xg

4 Sq (x

= x 23

23 )

8

Sq (xg )

= x 27 = x I7

444

The Homology of Hopf Spaces

groups we have

= E6 G =E 7

for G for

Using this coproduct plus the fact that x is primitive we can deduce all 3 coproducts by applying Steenrod operations. In the non simply connected case we have

The A*(2) action and the coproduct structure for H*(P~;f2) are obtained from that of H*(E ; f ) via replacing x 3 7 2 For p = 3 we have

*

H (F4;f3)

= E(x3,~,xll,x15)

f

@ 3

2

by x

6.

[x ] S /(xS3)

* _ f [x ] H (£6;f3) - E(x3,~,X9,XIl,X15,x17) @ 3 S /(xs3)

*

H (E7;f3)

*

.

= E(x3,~,xll,x15,x19,x23,x27,x35)

f [x ] @ 3 S /(x 3)

S

_

f [x ,x ] H (ES,f3) - E(x3,~,X15,X19,X27,X35,X39,X47) @ 3 S 20/(xS3,x203)

The A* (3) action and the coalgebra structure of H* (F

and H* (£S;f3 ) 4;1I'3) were described in §46 and §47. The inclusions F C £6 C £7C E force simiS 4 lar results for £6 and E The only new fact to be added is that, in 7.

-* (x ) H* (E6;1I'3)' xg is primitive while ~ 17

= XS@Xg'

In the non simply connected case we have

. The A* (3) actlon.can be deduced from that of H* (E ; f By using the A*( 3) 6 3). action the coproduct structure can be deduced from

Finally, for p

=5

we have

445

Appendices

The structure of H* (ES:W as a Hopf algebra over A* (5) was described in S) §46.

APPENDIX B: STEENROD OPERATIONS We will give a brief summary of some important facts about Steenrod operations which are used in various parts of the book. We will only discuss A*(p) for p odd. The p = 2 case is analogous and. moreover, is treated in more detail in the book. We tend to use, almost exclusively. the Milnor [3] approach to A*(p). At times,

however,

the Steenrod [3]

approach wi 11 be useful. The Milnor elements {Qs} are defined recursively by the rule Q = O. the Bockstein O s

Q = pP Q - Q pP s+1 s s

s

for s ~

a

They generate an exterior subalgebra of A*(p). In §1-2 we described A*(p). the dual Hopf algebra of A*(p). Let QE

e eO e l k Q Q O l .. ·Qk

pF

the dual of the monomial

( Here e

i

the dual of the monomial TO fF = f

eO e l T I

e "

.T

k

k

f

f f 1 2 ... f e f 2 e l

. I ar p(n,O.O .... ) = pn. = 01 . wun 1 e fi > _ 0) . I n partlcu

the usual

Steenrod operation. A*(p) has a W basis { qEpF } where E= (eO.e .... ) and p l F = (f I' f 2' ... ) run through all sequences of non negative numbers wi th only finitely many non zero terms. Let As denote the sequence (O .... O.I.O ... ) where I occurs in the sth position. Let pSF denote the sequence where each term in F is multiplied by pS. For any F and any s ~ 0 there is the relation

445

Appendices

The structure of H* (ES:W as a Hopf algebra over A* (5) was described in S) §46.

APPENDIX B: STEENROD OPERATIONS We will give a brief summary of some important facts about Steenrod operations which are used in various parts of the book. We will only discuss A*(p) for p odd. The p = 2 case is analogous and. moreover, is treated in more detail in the book. We tend to use, almost exclusively. the Milnor [3] approach to A*(p). At times,

however,

the Steenrod [3]

approach wi 11 be useful. The Milnor elements {Qs} are defined recursively by the rule Q = O. the Bockstein O s

Q = pP Q - Q pP s+1 s s

s

for s ~

a

They generate an exterior subalgebra of A*(p). In §1-2 we described A*(p). the dual Hopf algebra of A*(p). Let QE

e eO e l k Q Q O l .. ·Qk

pF

the dual of the monomial

( Here e

i

the dual of the monomial TO fF = f

eO e l T I

e "

.T

k

k

f

f f 1 2 ... f e f 2 e l

. I ar p(n,O.O .... ) = pn. = 01 . wun 1 e fi > _ 0) . I n partlcu

the usual

Steenrod operation. A*(p) has a W basis { qEpF } where E= (eO.e .... ) and p l F = (f I' f 2' ... ) run through all sequences of non negative numbers wi th only finitely many non zero terms. Let As denote the sequence (O .... O.I.O ... ) where I occurs in the sth position. Let pSF denote the sequence where each term in F is multiplied by pS. For any F and any s ~ 0 there is the relation

The Homology of Hopf Spaces

446

If

we let

F

E (_1)spn-,(s)Q

= s

(n.O.O .... )

then we can deduce

the

n relation GDp

used throughout §34.§35 and §36.

An important restriction used throughout

this book is the fact that

A*(p) acts unstably on mod p cohomology. In other words pn(x} The

~

=

P

[x

o

Ixl = 2n

if

i f Ixl

6pn(x)

> 2n

=0

if Ixl

> 2n+1

of an operation is the minimal degree in which it can act non

trivially. The test case in degree n is the universal example X

=

K(~p.n}. Kraines [1] showed that the excess of an element of A*(p} can be defined by the rule

=E

eX(QEpF}

e. + E 2f. J

1

The mod p cohomology of K(~p,n} can be described in terms of A*(p}. First of all, H*(K(~p.n};F } is a primitively generated Hopf algebra. So p

the map PH*(K(~p.n};F } ~ QH*(K(~p.n};F } is surjective. Both p

p

PH* (K(~p.n};Fp) and QH* (K(~p.n);Fp) are cyclic Steenrod modules generated by the fundamental class r

L

€ Hn(K(~p,n);F

n

= { QEpF

p

) ~ F . The set p

lEe. + E 2f. ~ n } 1

J

is a basis of PH*(K(~p.n};F } while the set p

r+

= { QEpF

lEe. + E 2f. ~ n-l } 1

J

is a basis of QH*(K(~p.n};Fp)' The set r - r+ consists of the iterated pth powers of the elements of r+ A*(p} possesses a canonical antiautomorphism X: A*(p) ~ A*(p) defined by the rule X(6}

= -6

X(pn)

=-

E

pi XCp j)

i+j=n x(ab)

= (-1) la11bll(b))(a)

There are both left and right actions of A*(p) on H*(X:Fp)' The right action is obtained from the usual left action of A*(p} on H*CX:F } by the p

rule

Appendices

447

< = (-I) Ixll for any x € H*{X;W ), Y € H (X;W ) and 'I' € A*{p). The left action is re-

*

p

lated to the right by the rule
p

= (x)J({
where )( is the canonical anti-automorphism. We will have need of both of these actions. We will use the right action in our treatment of secondary operations. The left action arises in our use of BP theory.

APPENDIX C: BROWN-PE1ERSON TIIEORY

Brown-Peterson theory is a localized version of complex cobordism. The standard introductions to Brown-Peterson theory are Adams [7] and Wilson [3]. Brown-Peterson homology BP*{X) is a module over

while Brown-Peterson cohomology is a module over

If we identify the coefficient rings via the rule (*) Bpi have a bilinear pairing

BP_ then we i

BP*{X) 0 BP*{X) ~ l{p)[v l,v2' ... ] When H*{X){p)

is torsion free the Atiyah-Hirzebruch spectral sequences

converging from H*{X;BP*) to BP*{X) and from H* (X;BP* ) to BP* (X) collapse. The collapse implies that BP*{X) and BP* (X) are dual free l{p)[v l.v2 .... ] modules. The duality is in the sense of (*). So BP*{X) is connected and locally finite while BP*{X) need not be either. BP homology is related to ordinary homology by Thom maps T: BP*{X) ~ H*(X)(p) T: BP*{X)

~

H*(X;Wp)

(We use the same symbols in both cases). When H*{X){p) is torsion free the Thorn maps are surjective and

Appendices

447

< = (-I) Ixll for any x € H*{X;W ), Y € H (X;W ) and 'I' € A*{p). The left action is re-

*

p

lated to the right by the rule
p

= (x)J({
where )( is the canonical anti-automorphism. We will have need of both of these actions. We will use the right action in our treatment of secondary operations. The left action arises in our use of BP theory.

APPENDIX C: BROWN-PE1ERSON TIIEORY

Brown-Peterson theory is a localized version of complex cobordism. The standard introductions to Brown-Peterson theory are Adams [7] and Wilson [3]. Brown-Peterson homology BP*{X) is a module over

while Brown-Peterson cohomology is a module over

If we identify the coefficient rings via the rule (*) Bpi have a bilinear pairing

BP_ then we i

BP*{X) 0 BP*{X) ~ l{p)[v l,v2' ... ] When H*{X){p)

is torsion free the Atiyah-Hirzebruch spectral sequences

converging from H*{X;BP*) to BP*{X) and from H* (X;BP* ) to BP* (X) collapse. The collapse implies that BP*{X) and BP* (X) are dual free l{p)[v l.v2 .... ] modules. The duality is in the sense of (*). So BP*{X) is connected and locally finite while BP*{X) need not be either. BP homology is related to ordinary homology by Thom maps T: BP*{X) ~ H*(X)(p) T: BP*{X)

~

H*(X;Wp)

(We use the same symbols in both cases). When H*{X){p) is torsion free the Thorn maps are surjective and

448

The Homology of Hopf Spaces

(P.v 1,v2, .. )

= Ker

T: BP.. (X)

-4

H.. (X;Wp)

The Thorn maps exist in cohomology as well. Moreover. the pairings between homology and cohomology for the various theories are respected by the Thorn maps. The algebra BP*BP of

Brown-Peterson operations

is generated as a

module (provided one allows infinite sums) by elements {r E} .... ) ranges through all sequences of non negative intewhere E = (e 1.e2 gers with only finitely many non zero terms. The element r is of degree E ~(p)[v1,v2"']

lEI

= L 2e.(pi-1). 1

There is a natural left action of BP"BP on BP.. (X) obtained by interpreting them as homotopy classes of maps.

[BP.BP] and

.

~.. (BPAX).

respec-

tively. This action is related to the left action of A (p) on H.. (X;Wp)

discussed in Appendix B. Namely. for each E.

the following diagram com-

mutes r

_........:::E----+l BP.. (X)

where X is the canonical anti-automorphism. The algebra BP*BP acts on BP..

= ~..(BP)

and on H.. (BP). H.. (BP) is also a

polynomial algebra. H.. (BP) The Hurewicz map h: ~.. (BP)

= ~(p)[m1,m2' ... ] ~.. (BP)

-4

H.. (BP)

is a monomorphism.

Considering

C H.. (BP) we have the identities

- L

v. i+j=n 1 i, j
pj

m,

J

In particular h is an isomorphism when we tensor by Ill. The morphism h respects the action of BP*BP. Given F F rE(m ) =

[~

= (f 1.f2 .... )

F let m

= m1

f I

... ~

if E = F

if E;tFand lEI L IFI In particular we have r p. 1(v1) All of the preceeding holds with homology replaced by cohomology.

=

fk

449

REFERENCES

The reader is referred to James [4] for a previous bibliography on H-spaces. We have listed papers from that collection only when they seemed particularly relevant to the topics in this book. Our listing of post 1970 papers reflect those same prejudices. ADAMS J.F.

[1]

On the Cobar Construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956).

[2]

On the Nonexistence of El.ements of Hapf Invariant One, Ann. Math. 72

409-412 (1960), 20-104 [3J

The Sphere. Considered as a H-space Mod p. Quart. J, Math. Oxford 12

[4]

H-spaces with Few Cells, Topology

[5]

Lectures on Lie Groups, W.A. Benjamin 1969

[6J

Lectures on Generalized Cohomology, Springer Verlag LNM # 99 (1969).

[7]

Stable Homotopy and Generalized Homology, University of Chicago

[8]

Primitive Elements in the K-Theory of BSU. Quart. J, Math. Oxford 27

(1961). 52-60 (1962) , 67-72

1-138 Press 1974

(1976). 253-262

[9] Maps Between Classifying Spaces, Springer Verlag LNM

# 658 (1978).

1-9 [10J

Maps Between Classifying Spaces, L'Enseigement Math, XXIV (1978),

79-85 [11]

Maps Between Classifying Spaces II, Inventiones Math. 49 (1978),

[12J

Finite H-spaces and Lie Groups, J. Pure Appl. Algebra 19(1980), 1-8

1-65

ADAMS J,F. and ATIYAH M.F.

[IJ

K-Theoryand the Hopf Invariant, Quart. J. Math. Oxford 17 (1966),

31-38 ADAMS J,F, and HILTON P.J.

[1]

On the Chain Algebra of a Loop Space, Comment. Math. Helv. 30 (1956). 305-330

450

The Homology of Hopf Spaces

ADAMS J. F. and MAHMUD Z [lJ

Maps between Classifyina Spaces, Invent. Math 35 (1976), 1-41

[2J

Maps between Classifying Spaces III. London Math. Soc. LNM # 86

(1986), 136--153 ADAMS J.F. and PRIDDY S.D. [lJ

Unigueness of BSO, Math. Proc. Carob. Phil, Soc. 80 (1976), 475-509

ADAMS,J.F. and WILKERSON C. W, [1]

Finite H-spaces and Algebras over the Steenrod Algebra, Ann. Math.

111 (1980), 95-143 AGUADE J.

[1]

The Realizability of Certain Algebras as Cohomology Rings, Rev.

[2]

Univ. Santander 2 (1979). 463-464 Fibrings of Spheres by Spheres Mod p. Quart. J. Math. Oxford 31

[3]

(1980), 129-137 On the Space of Free Loops of an Odd Sphere Pub. Mat. U.A.B. 25

[4]

A Note on Realizing Polynomial Algebras, Israel J. Math. 38 (1981),

(1981). 87-90 95-99 [5]

Cohomology Algebras with Two Generators, Math. Z. 177 (1981).

[6]

Realizability of Cohomology Algebras: a Survey, Pub. Mat. U.A.B. 26

[7]

Invariants of Modular Representations and Polynomial Algebras over

289-296 (1982). 25-68 the Steenrod Algebra, Duke J. Math. 52 (1985). 315-328

AGUADE J. and SMITH L. [1] Modular Cohomology Algebras, Amer. J. Math. 107 (1985), 507-530 [2J

On the Mod p Torus Theorem of John Hubbuck. Math. Zeit. 191 (1986). 325-326

ARAKI S.

[1]

DiFFerential HopF Algebras and the Cohomology Mod 3 of the

[2]

Exceptional Lie Groups E and EW- Ann. Math. 73 (1961), 43-65 7 HopF Structures Attached to K-Theory: Hodgkin's Theorem, Ann. Math. 85 (1967), 508-525

[3]

On a Type of DiFFerential HopF Algebra, Springer Verlag LNM

# 196

(1970), 74-84 ARAKI S. and YOSIMURA Z.

[1]

On a Certain Type of Hopf Algebra. Pr-oc, Japan. Acad. 46 (1970),

451

References

332-336 ARKOWITZ M. [1]

Whitehead Products as Images of Pontryagin Products. Trans. Amer.

Math. Soc. 158 {1970}. 453-463 [2]

The Generalized lnbrodsky Theorem. Springer Verlag LNM # 418. {1974}. 6-12

ARKOWITZ M. EWING J. and SCHIFFMAN S. [1]

H-structures on LocaLized and CompLeted Spheres. Quart.

J.

Math.

Oxford 26 {1975}. 295-307 ATIYAH M.F. [1]

Thom CompLexes. Proc. London Math. Soc. 11 (1961). 291-310

[2]

On the K-Theory of Compact Lie Groups, Topology 4 (1965) , 95-99

ATIYAH M.F. and HIRZEBRUaI F. [1]

Vector BundLes and Homogeneous Spaces, Proc. Sym.Pure Math. III Amer. Math. Soc. (1961), 7-38

BAliN P.F. and BROWDER W. [1]

The CohomoLogy of Quotients of CLassicaL Groups, Topo!ogy.3 (1965). 305-336

BENSON C. T. and GROVES L. C. [1] Finite ReFLection Groups. Spinger Verlag GTM # 99, 1983 BODY R.A. and DOUGLAS R.R. [1]

Homotopy Types within a RationaL Homotopy Type. Topology 13 {1974}.

209-214..... BOREL A. [1]

Sur La CohomoLogie des Espaces Fibres Prtncipaux tl des Espaces

[2]

Sur L'HomoLogie et La CohomoLogie des Groupes de Lie Compact

[3]

Connexe. Amer. J. Math. 76 (1954), 273-342 TopoLogy of Lie Groups and Characteristic CLasses. Bull. Amer. Math.

Homogenes de Groupes de Lie Compacts, Ann. Math. 57 (1953). 115-207

Soc. 61 (1955), 397-432 [4]

Sous-Groupes CommutatiFs et Torsion des Groupes Compact Connexes. Tohoku J. Math. 13 (1961), 216-240

BOREL A. and CHEVALLEY C [1]

The Betti Numbers of the ExceptionaL Groups. Memoirs Amer. Math. Soc. 14 (1955). 1-9

BOIT R.

452

[1]

The Homology of Hopf Spaces

On Torsion in Lie Groups. Proc. Nat. Acad. Sci. U.S.A. 40 (1954). 586-588

[2]

An AppLication of Morse Theory to the TopoLogy of Lie Groups, Bull. Amer. Math. Soc. 84 (1956). 251-281

[3]

The Space of Loops on a Lie Group, Michigan Math. J. 5 (1958). 36-61

BOTT R. and SAMELSON H. [1]

On the Pontrjagin Product in Spaces of Paths, Comm. Math. Helv. 27 (1953). 320-337

BOURBAKI N. [1]

Groupes et ALgebres de Lie. Chapters 4.5.6 Hermann. 1968

BOUSFIELD A.K. and KAN D.M. [1]

Homotopy Limits, CompLetions and LocaLization. Springer-Verlag LNM # 304, 1972

BROWDER W. [1]

The CohomoLogy of Covering Spaces of H-spaces, Bull. Amer. Math.

Soc. 65 (1959). 140-141 [2]

HomoLogy Operations and Loop spaces, Illinois J. Math. 4 (1960). 316-319

[3] [4]

Torsion in H-spaces. Ann. Math. 74 (1961). 24-51 Fibrings of Spheres and H-spaces Which are RationaL HomoLogy Spheres, Bull. Amer. Math. Soc. 68 (1962). 202-203

[5]

Homotopy Commutative H-spaces, Ann. Math. 75 (1962). 283-311

[6]

Homotopy Type of DiFFerentiabLe ManiFoLds. Proc. Aarhus Symp 1962

[7]

On DiFFerentiaL HopF ALgebras, Trans. Amer. Math. Soc. 107 (1963). 153-178

[8] Higher Torsion in H-spaces. Trans. Amer. Math. Soc. 108 (1963). 353-375 [9]

HomoLogy Rings of Groups, Amer. J. Math. 90 (1968), 318-333

BROWDER W. and NAMIOKA I. [1]

H-spaces with Commutative HomoLogy Rings, Ann. Math. 75 (1962). 449-451

BROWDER W. and SPANIER E. [1]

H-spaces and DuaLity, Pacific J. Math. 12 (1962). 411-414

BROWDER W. and THOMAS E. [1]

On the Projective PLane of a H-space, Illinois J. Math. 7 (1963). 492-502

CAPPELL S. and WEINBERGER S.

References

[IJ

453

Parallelizability of Finite H-spaces, Comm. Helv. Math. 60 (1985), 628-629

CARTAN H.

[IJ

Algebres d'Eilenberg-Maclane et Homotopie, Seminaire Henri Car tan , 1954-55, Paris, 1955

CARTAN H. and ElLENBERG S.

[IJ

Homological Algebra. Princeton University Press 1956

CHEVALLEY C.

[IJ

The Betti Numbers of the Exceptional Lie Groups, Proc. Cong. Math.,

Cambridge, 1950, [2J

Invariants of Finite Groups Generated by Reflections, Amer. J. Math. 77 (1955), 778-782

CLARK A. [2J

On IT3 of Finite Dimensional H-spaces, Ann. Math.78 (1963), 193-196 Homotopy Commutativity and the Moore Spectral Seguence. Pacific J.

[3J

Hopf Algebras over Dedehind Domains and Torsion in H-spaces, Pacific

[4J

Categorical Constructions in Algebraic Topology, Springer Verlag

[IJ

Math. 15 (1965), 65-74 J. Math. 15 (1965), 1,19-1,26 LNM

**

196 (1970), 68-73

CLARK A. and EWING J. R. [IJ

The Realization of Polynomial Algebras as Cohomology Rings, Pacific

J. Math. 50 (1974) 425-434 CLARKE F. [IJ

On the K-Theory of a Loop Space of a Lie Group, Proc. Camb. Phil. Soc. 57 (1974), 1-20

COHEN A.

[IJ

Finite Complex Reflection Groups. Ann. Sci. Ec. Norm. Sup. 9 (1976). 379-436

COHEN F.C., MOORE J.C. and NEISENDORFER J.A.

[IJ

Decompositions of Loop Spaces and Applications to Exponents,

Springer Verlag LNM [2J

**

763 (1979). 1-12

Torsion in Homotopy, Ann. Math. 109 (1979), 121-168

COHEN F.C. and NEISENDORFER J.A.

[IJ

A Construction of p-Local H-spaces, Springer Verlag LNM ** 1051 (1984), 350-359

454

The Homology 0 f Hopf Spaces

COOKE G.E. [1]

RepLacing Homotopy Actions by TopoLogical Actions, Trans. Amer. Math. Soc. 237 (1978). 391-406

[2J

Constructing Spaces with Interesting Zip Cohomology via c Actions on Loop Spaces. Amer. J. Math. 103 (1979). 515-542

COOKE G.E.. HARPER J. R. and ZABRODSKY A. [IJ

Torsion Free Mod pH-spaces of Low Rank. Topology 18 (l979). 349-359

COOKE G.E. and SMITH L. [IJ

Decomposition of Co-H spaces and Applications. Math. Z. 157 (1977),

[2J

On Realizing Modules over the Steenrod ALgebra. J. Pure Appl.

155-157 Algebra 13 (1978). 71-100 (lJRJEL C. R. and DOUGLAS R. R. [IJ

On H-spaces of Finite Dimension. Topology 10 (1971). 385-390

(lJRTIS M. [IJ

Finite DimenslonaL H-spaces. Bull. Amer. Math. Soc. 77 (1971). 1-12

[IJ

H-spaces Mod p II. Springer Verlag LNM # 196 (1970) .. 11-19

(lJRTIS M. and MISLIN G. [1]

[2J

Two New H-spaces, Bull. Amer. Math. Soc. 76 (1970). 851-852

H-spaces Which are BundLes over S7~ J. Pure Appl. Algebra 1 (1971), 27-40

CURTIS M.• WIEDERHOLD A. and WILLIAMS B. [IJ

Normalizers of Maximal Tori. Springer Verlag LNM # 418 (1974). 31-47

OOLD A. [IJ

On General Cohomology. Aarhus LNM (1968)

OOLD A. AND LASHOF R. [1]

Principal Quqsi-fibrations and Fibre Homotopy Equivalence of Bundles. Illinois J. Math. 3 (1959). 285-305

OOUGLAS R.R. and SIGRIST F. [IJ

Sphere Bundles over Spheres and H-spaces. Topology 8 (1969). 115-118

DROR E. and ZABRODSKY A. [IJ

Unipotency and Nilpotency in Homotopy Eguivalences. Topology 18

(1979). 187-197 DUCKWORTH P.W. [IJ

The K-Theory Pontriagin Ring for the Loop Spaces on the Exceptional

455

References

Lie Groups, Quart. J. Math. 35 (1984). 253-262 ELLENBERG S. and MOORE J.C.

[1]

Homology and Fibrations I, Comm. Math. Helv. 40 (1960) , 199-236

EWING J. R.

[1]

Sphere Bundles over Sphere as Loop Spaces Mod p

Duke Math. J. 40

(1973). 157-162 FESHBACH M.

[1]

The Transfer and Characteristic Classes. Springer Verlag LNM # 657.

[2]

The Transfer and Compact Lie Groups, Trans Amer. Math. Soc. 251

156-162 (1979), 215-244 [3]

The Image of H*(BG;Z) in H*CBT;Z) for G a Compact Lie Group with Maximal Torus T. Topology 20 (1981), 93-95

FRIEDLANDER E.M.

[1]

Exceptional Isogenies and the Classifying Spaces of Simple Lie

Groups. Ann. Math. 101 (1975) , 510-520 FRIEDLANDER E. M. and MISLIN G.

[1]

Cohomology of Classifying Spaces of Complex Lie Groups and Related

Discrete

Groups. Comm, Math. Helv. 59 (1984). 347-361

GANEA T.

[1]

Cogroups and Suspensions, Invent. Math. 9 (1970). 185-197

GILBERT W.J.

[1]

Homotopical Nilpotence of the Seven Sphere. Proc. Amer. Math. Soc.

32 (1972). 621-622 GONCALVES D.C.

[1]

Mod 2 Homotopy Associative H-spaces. Springer Verlag LNM # 657

[2]

The Mod 5 Splitting of the Compact Exceptional Lie Group E Publ. 8, RIMS Kyoto Univ. 19 (1983) , 1-6

(1978). 198-216

HARPER J.R.

[1]

On Certain Non Abelian Loop Spaces, Bol. Soc. Mat. Mex. 15 (1970), 30-32

[2]

Homotopy Groups of H-spaces. Comm. Math. Helv. 47 (1972). 311-331

[3]

The Mod 3 Homotopy Type of

F4~

Springer Verlag LNM # 418 (1974),

58-67 [4]

Mod p Decompositions of Finite H-spaces. Springer Verlag #428.

456

The Homology of Hopf Spaces

(1974), 44-51 [5]

On the Construction of Mod pH-spaces, Proc. Sym. Pure Math. XXXII

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On the Construction of Mod pH-spaces II, Springer Verlag LNM # 741.

(1978). 207-214 548-556 [7]

Torsion Free Mod pH-spaces. Springer Verlag LNM # 657, (1978).

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H-spaces with Torsion. Memoirs Amer. Math. Soc. # 223 (1979)

144-149 [9] [10]

Regularity of Finite H-spaces. Illinois

J. Math. 23 (1979). 330-333

Rank 2 Mod 3 H-spaces, Canadian Math. Soc. Conference Lecture Note

Series # 2 (1982), 207-214 HARPER J. R. and SCHOCHET C. [1]

Coalgebra Extensions in Two Stage Postnikov Systems, Math. Scand. 29 (1971). 232-236

HARPER J. R. and ZABRODSKY A. [1]

Alteration of H-Structures, Amer. Math. Soc. Contemporary Math. Series # 19 (1983). 71-84

[2]

H-spaces and Self Maps, Springer Verlag LNM # 1051 (1984), 371-373

HARRIS B. [1]

On the Homotopy Groups of the Classical Groups,

Ann. Math. 74

(1961), 407-413 [2J

Suspensions and Characteristic Maps for Symmetric Spaces, Ann. Math. 76 (1962) , 295-305

HARRISON J. and SCHEERER H. [1]

Zur Homotopietheorie von Abbildungen in Homotopieassoziative H-raum, Archiv der Mathematik 23 (1972). 319-323

HARRISON J. and STASHEFF J. D . [1]

Families of H-spaces. Quart. J. Math. Oxford 22 (1971), 347-351

HEMMI Y. [1]

Mod p H spaces and p-Regularities. Hiroshima Math. J. 12 (1982).

399-419

[2] On Finite H-spaces Given by Sphere Extensions of Classical Groups, Hiroshima Math. J. 14 (1984), 451-470 [3]

On 3-Connected Finite H-spaces, Hiroshima Math. J. 15 (1985), 55-56

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HILLER H. [1J

Geometry of Coxeter Groups. Pitman Research Notes in Mathematics #

54. 1982 [2]

On the Cohomology of Loop Spaces of Compact Lie Groups. Glasgow Math. J. 26 (1985). 91-99

HILTON P.J. [1J

Localization of Nilpotent Spaces. Spinger verlag LNM # 428 (1974). 18-43

HILTON P.. MISLIN G. and ROITBERG J. [1]

Sphere Bundles over Spheres and Non-Gancellation Phenomena. Springer

[2]

H-spaces of Rank 2 and Non-Cancellation Phenomena. Inventiones Math.

Verlag LNM # 249 (1971). 34-46 16 (1972). 325-334 [3]

Topological Localization and Nilpotent Groups. Bull. Amer. Math.

Soc. 78 (1972). 1060-1063 [4J

Homotopical Localization. Proc. London Math. Soc. 26 (1973). 693-706

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HILTON P. J. and ROITBERG J. [1J

Note on Principal

g3

Bundles. Bull. Amer. Math. Soc. 74 (1969).

957-959 [2J [3J

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On Principal Bundles ~ Spheres. Ann. Math. 90 (1969). 91-107 On the Classification Problem For H-spaces of Rank Two. Comm. Math. Helv. 45 (1970). 506-516

[4]

[5J

On the Classification of Torsion Free H-spaces of Rank Two. Springer Verlag LNM # 168 (1970). 67-74 On the Classification Problem for H-spaces of Rank Two. Comm. Math. Helv. 46 (1971). 506-516

HILTON P.J. and STAMBACH U. [lJ

A Course in Homological Algebra. Springer Verlag GTM # 4. 1974

HODGKINS.L. [1]

On the K-Theory of Lie Groups, Topology 6 (1966). 1-36

HOFFMAN P. [1]

Cohomology Realisations of Q[xl. Quart. J. Math. Oxford 24 (1973).

251-253

458

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HOFFMAN P. and ZABRODSKY A. [1]

Thin Lens Spaces. Canadian Math. Bull. 21 (1978). 31-35

HOLZSAGER R. [1]

H-spaces of Category

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Topology 9 (1970). 211-216

HOPF H [1]

Uber die Topologie der Gruppen-Hannigfaltigkeiten lind Ihre

Verallgemeinerungen. Ann. Math. 13 (1940). 22-52 HOSLI H. [1]

Sur les H-Espaces a Deux Generateurs Cohomologigues. C.R. Acad. Sci. Paris 270 (1970). 746-749

HUBBUCK.J. R.

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Some Results in the Theory of H-spaces. Bull. Amer. Math. Soc. 74

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On Homotopy Commutative H-spaces,Topo1ogy 8 (1969). 119-126

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Amer. Math. Soc. 141 (1969). 335-360 [4]

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HUBBUCK J. R. and KANE R. [1]

On U3 of a Finite H-space, Trans. Amer. Math. Soc. 213 (1975),

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HUBBUCK J. R. and MAHMUD Z. [1]

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The Structure of the Hopf Algebra H*(BUl over a Z(p) Algebra, Amer.

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ISHITOYA K. ,KONO A. and TODA H. [1]

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Protective Dimension and Brown-Peterson Homology, Topology 12

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455-467 (1979) 557-587 [14J

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On Cohomology Mod 2 of the Classifying Spaces of Non-Simply Connected Classical Lie Groups, J. Math. Soc Japan 27 (1975), 281-288

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A Secondary Operation of Exceptional Lie Groups, J. Math. Kyoto Univ. 17 (1977), 23-26 On the Cohomologu Mod 2 of E8, J. Math. Kyoto U. 24 (1984). 275-280

On the Cohomology of the 2-Connected Cover of the Loop Space of

Simple Lie Group, Publ. RIMS 22 (1986). 537-541 KONO A. and KOZIMA K.

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The Space of Loops on a Symplectic Group, Japan J. Math. 4 (1978),

461-480 KONO A. and MIMURA M. [1J

On the Cohomology of the Classifying Spaces of PSU(4n+2) and PO(4n+2), Publ. RIMS Kyoto Univ. 10 (1975), 691-720

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Cohomology Operations and the Hopf Algebra Structures of the Compact

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475

INDEX

algebra 3 differential 58 Hopf 3 Lie 21 monogenic 6 restricted Lie 21 universal enveloping 21 algebraic 180 closure 180 extension 180 loop 54 A

n

form 49

An map 53 An space 49 anti automorphism 446 augmentation filtration 108 augmented relations 325

bar construction 58 spectral sequence 60 binary integer 263 biprimi tive form 109 spectral sequence 110 Borel decomposition 17 Brown Peterson theory 447 Car tan maps 328 character field 162 classifying space 38 Aoo spaces 51 associative H-space 44 Lie group 38 topological group 41

476

The Homology of Hopf Spaces

c-invariant 335 coalgebra 3 cobar construction 60 spectral sequence 60 completion (p-adic) 78 contraction 131 coproduct 216 Cotor 60 Coxeter graph 155 group 154 system 154 degree external 58 internal 58 of a reflection group 150 duality Hopf algebra 4 Poincare 31 primitives and indecomposables 10 S-duality 31 Steenrod module 229 excess 446 extended module of indecomposables 274 Frobenius pth power map 10 Galois 180 graph Coxeter 155 root 161 genus 78 group crystallographic 155 Coxeter 154 dihedral 153 imprimitive 159 reflection 148 Weyl 156 height 7

Index

Hopf algebra 33 differential algebra 14 decomposition 16 ideal 4 H-fibration 24 H-space vii finite vii homotopy associative 49 homotopy commutative 142 mod p 72 impl ication 96 indeterminancy of a secondary cohomology operation 217 integral 187 integrally closed 188 join construction 41 k invariant 428 Krull dimension 183 K-theory 137 Lie

algebra 21 group 439 lifting 219 localization 72 loop map 55

space 45 space conjecture 212 mixing of homotopy type 77 mod p equivalent 72 H-space 72 fini te 72 mod P equivalent 72 module extended 274 of indecomposables 10 of primitives 10 Steenrod 445

477

478

The Homology of Hopf Spaces

unstable Steenrod 390 Morava K-theory 115 nilpotent CW complex 72 normal 9 operations Adams 140 Bockstein 92 Brown Peterson 447 secondary 215 Steenrod 445 Poincare complex 31 polynomial algebra truncated 128 unstable 179 Postnikov system 426 primitive system 426 power H-space 409 H-space map 409 map 406 space 406 space map 408 p regular 73 p quasi-regular 73 primi tive 6 form 108 spectral sequence 109 generated 11 Postnikov system 426 product cap 29 Pontryagin 4 slant 30 projective plane 52 pullback 75 weak 75 purely inseparable 180 purely inseparable isogeny 201

Index

quasi-fibration 44 reflection transformation 148 group 148 separable 180 simple system 315 spectral sequence Atiyah-Hirzebruch 447 bar 60 Bockstein 92 cobar 61 Eilenberg Moore 206 Rothenberg-Steenrod 361 Serre 25 Steenrod algebra 445 operation 445 suspension elements 209 map 210 Thorn map 447 Toda bracket 237 ladder bracket 238 Tor 59 torsion ordinary homology 92 Morava K-theory 115 transpotence elements 209 transgressive 25 twisting of homotopy type 76 type 17 U(M) algebra 390 universal covering space 22 enveloping algebra 21 unstable polynomial algebra 179 Steenrod module 390

479