Symposium on Algebraic Topology (Lecture Notes in Mathematics, 249) 9783540057154, 3540057153

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich

249

Symposium on Algebraic Topology

Edited by Peter J. Hilton Battelle Seattle Research Center, Seattle, WA/USA

'fJ Springer-Verlag Berlin · Heidelberg . New York 1971

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich

249

Symposium on Algebraic Topology

Edited by Peter J. Hilton Battelle Seattle Research Center, Seattle, WA/USA

'fJ Springer-Verlag Berlin · Heidelberg . New York 1971

AMS Subject Classifications (1970): 55Bxx, 55Dxx, 55Fxx, 55Jxx, 57Dxx

ISBN 3-540-05715-3 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05715-3 Springer-Verlag New York· Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 79-185401. Printed in Germany. Offsetdruck:Julius Beltz, HemsbachlBergstr.

Dedicated to the memory of Tudor Ganea (1922-1971)

FOREWORD During the academic year 1970-1971 the University of Washington instituted a program of concentration in the area of algebraic topology in conjunction with the Battelle Seattle Research Center.

As part of that program the Center acted as host

to a symposium which took place during the week of February 22-26, 1971.

Several

topologists were invited from universities in the United States; and there were present, in addition to those invited, the regular members of the University of Washington mathematics faculty, the mathematicians associated with Battelle, the mathematicians visiting the University of Washington in conjunction with the year's activities in topology, and several other topologists who were interested to attend. Some of the talks given were of a very informal nature and, in those cases, the speakers preferred not to provide a manuscript.

On the other hand, in most cases,

the speaker did write up his talk subsequently so that this volume contains a fairly complete record of the scientific program. It is a pleasure to acknowledge the kindness of many people at the Battelle Seattle Research Center who helped to make the occasion such a very pleasant and productive one.

In particular, I would like to mention Mr. Louis M. Bonnefond, Miss

Kay Killingstad and Miss Penny Raines who made all of the necessary arrangements and insured that the symposium ran with the smoothness which one has come to associate with Battelle in Seattle.

Further, I would like to express my own appreciation to

Mrs. Lorraine Pritchett for having helped so very much in the final preparation of the manuscripts. A further and more somber duty devolves upon me.

The February symposium

was the last scientific meeting attended by my good friend and colleague, Tudor Ganea, before his death.

We topologists will all miss him very much indeed.

At the

symposium he was not able to give a talk but he did distribute a preprint containing

VI a list of unsolved problems in his particular area of interest.

I have therefore

included his catalog of probl81118 in the proceedings of the sYIIIPosiUII.

I have also

dedicated this volume to his melllOl'Y. a gesture which. I_sure. will cOlllllUlDd the assent of all of the participants.

Battelle Seattle Research Center. August. 1971

Peter Bilton

Contents D. W. Anderson: Chain Functors and Homology Theories. • • • • • • • • • • • • • •• E. Dror: A Generalization of the Whitehead Theorem. • • • • • • • • • • ••

13

T. Ganea: Some Problems on Numerical Homotopy Invariants. . . • • . . . . . .

23

S. Gitler and J. Milgram: Unst$ble Divisibility of the Chern Character. • • • • • • • • • • •

31

P. J. Hilton, G. Mislin, and J. Roitberg: Sphere Bundles Over Spheres and Non-Cancellation Phenomena. • • ••

34

A. Liulevicius: On the Algebra BP* (BP) • • • • • • • • • • • • • • • • • • • • ••

47

J. Milgram:

Surgery, BpL' BTOP' and the PL Bordism Rings. • • • • • . . . . •• G. Mislin: The Genus of an H-Space • • • • • • • • . • • • • • • • • • • • ••

54

75

J. C. Moore:

Bockstein Spectral Sequence, Modified Bockstein Spectral Sequences, and Hopf Algebras Over Certain Frobenius Rings. • • • • • • • • ••

84

J. C. Moore and F. P. Peterson:

Nearly Frobenius Algebras and Their Module Categories • • • • • ••

94

D. L. Rector: Loop Structures on the Homotopy Type of S3. • • • • • • • • • • ••

J. D. Stasheff: Sphere Bundles Over Spheres as H-Spaces Mod Addresses of Contributors • • • • • • •

2.

99 106 111

CHAIN FUNCTORS AND HOMOLOGY THEORIES

D, W, Anderson In his paper on homotopy everything H-spaces [7]# G, Segal showed that there was a relationship between a-spectra and certain types of' f'unctors f'rom the category of' f'inite basepointed sets to the category of' topological spaces, r -spaces,

These f'unctors he called special

We shall introduce the concept of' a chain functor below#

which is essentially the same notion as a special r-grouP# but our treatment of this concept will be entirely dif'f'erent f'rom Segal's, From our point of' view# the category of' f'inite basepointed sets will arise naturally. Chain functors seem to be a very convenient way to describe homology theories and their associated spectra.

Because spectra can

be constructed very explicitly f'rom chain functors# we get several new results,

For example# we obtain constructions of' the spectra f'or

the various connective

K-theories (including

Im

theory) which

J

lead to strictly associative multiplications on these spectra# as well as infinitely homotopy connnutative multiplications. As a second example of' a result of' this construction# we obtain an interesting spectral sequence# which I call the berg-Moore spectral sequence,

n-th order Eilen-

We shall define a functor

Torn**(A)

f'or a connnutative augmented graded algebra A over a f'ield n = 1#

Torn**(A) = Tor**A(K#K).

finite loop space# the has

=

If'

X = any#

where

n-th order Eilenberg-Moore spectral sequence

(H*(X;K»#

and has

=

(Y;K)# Here

is the ordinary homology of'

K#

X with coeff'icients in

usual Pontrjagin product structure. K[n]

If'

Y is an in-

graded group associated to a f'iltration of'

if' the group ring

K.

If'

rr

the

pth

given the

is abelian group, and

is considered to be concentrated in degree 0#

2

we obtain the relation H*(K(rr,n);K) above that

= Torn*, o(K[rr]).

(The condition

Y be an infinite loop space is not actually necessary.)

The theory of chain functors is made more useful by the theory of permutative categories.

Permutative categories arise in nature, and

give rise to all of the "geometrically" defined homology theories except for the bordism theories.

Also, every theory defined by a

permutative category has associated to it equivariant cohomology theories, in much the manner of equivariant

K­theory.

This will be dis-

cussed in a subsequent paper [1]. The theory of chain functors is adequate to describe all homology theories and all homology operations.

However, not all homology opera-

tions naturally present themselves as natural transformations of chain functors, but only as homotopy natural transformations. ­+

BGL(C)

Quillen's map

is one term in such a homotopy natural trans-

[5]

formation which is of some importance.

This will be discussed in

[2] •

1.

Chain functors A chain functor

is a zero preserving covariant functor from

the category of finite basepointed sets to the category of simplicial groups which satisfies the following relation. (1.1)

For any two basepointed sets ­+

x

X, Y,

the natural map

is a homotopy equivalence (of simplicial sets).

The assumption that

takes values in the category of simpli-

cial groups is made for technical convenience.

We could, with some

slight increase in effort, replace "group" by "monoid" or even "set". In the first instance l we obtain what we call semichain functors.

In

the second, we obtain Segal's "special r­spaces". The assumption that necessary.

takes values on finite sets is also un-

We could replace it by the assumption that

was com­

3

Up to

with direct limits.

one can use

direct limits to extend a chain functor from the category of finite sets to all sets.

Notice that (1.1) will again hold in this context

if we extend in this way. If

is a chain

and if

define a bisimplicial set of chains

(X)

of

(X)

by

X is a simplicial (X)

X defined by

i

(X 1 4.3

The Nilpotent Case We call a connected space frITnX = O.

such that

follows that if both so is 4.4

IT*f.

If

X nilpotent if for all

X is nilpotent, then

X and

n

0

= f:IT*X =

froIT*X

Yare nilpotent space and

H*f

there exists

fIT*X = O.

r

Thus, it

is an isomorphism, then

The nilpotent case is discussed in more detail in the Appendix.

The Complete Case The most general case in which

IT*X

For

the same proof will work in the next two examples 4.3 and 4.4.

+

IT*X

fro' r:

and

f

vanish, is when the map

(see 2.3) is an isomorphism, in which case we say that

So for a map

f: X + Y between two complete spaces, IT*f

X is (rr-)complete.

is isomorphic if

H*f

is.

As an example of a IT-complete space one can take the geometric realization of the pro-p-completion [3] of a free semi-simplicial group which is finitely generated in each dimension (Bousfield). 4.5

Perfect Fundamental Group If

fIT*X

is a perfect group, i.e., f 2IT 1X = IT1X (that is, H1X = 0) then f 2IT*X which means that f 2IT*X is IT1X-perfect (see 2.6). To see this

IT1X

= frorr*X =

let us recall that for any abelian IT-group

A, Ho(IT,A)

= A/r2A.

Hence it follows

17

from the exactness of the coefficient sequence

in which the leftmost group is trivial by the assumption on

= O.

which means that

4.6

Corollary

fA

Assume that

=

HIX

is an isomorphism, so is

= f2A.

= HlY

0

n , that

Further this implies

and H*f

Ho (11;f2A)

=0

f:A

and so one gets:

is an isomorphism.

if f1f*f

l1*f.

As a special case, one gets the following result. 4.7

Corollary

Let f: X + Y be a map of acycZic spaces is a homotopy equivaZenae if r1f*f is an isomorphism.

H*X

H*Y

0).

Then f

Corollary 4.7 suggests that in the category of acyclic spaces the functor r1f* plays a role similar to that played by the functor CW-complexes. 4.8

1f*

in the category of all

This idea is basic in the analysis of acyclic spaces given in [1].

The Case Where

f

is A Retraction

If the map f: X + Y is a retraction, i.e •• there exists a one-sided inverse

i: Y + X such that

and (ii) are satisfied. if clear.

To show that

osition 3.2.

r:1f*f

fi

idy. then it is easy to see that conditions 3.1 (i)

H*f

is an isomorphism.

is an epimorphism is

is a monomorphism, one needs to use. inductively, Prop-

In fact one just applies the five-lemma and 3.2 to the exact sequence

.

and epimorphic on n Thus, one gets the useful result

A map which is an isomorphism on isomorphism on 4.9

That

1f

1f

n

must be an

Corollary

Let

f

bijeative if rl1*f

be a retraction and assume that

H*f

is bijective.

Then l1*f is

is injective. 5.

AN ALGEBRAIC LEMMA

Proposition 3.2 is based on corresponding algebraic propositions for the abelian and non-abelian cases (see 2.2). Since the non-abelian case was proved by Stallings [4]. complete proofs will be given only for the abelian case.

18

5.1

Proposition (Stallings)

f

n' be a map of groups and assume that Hlf is bijective and Then the induced map n/f n n'/f n' is an isomorphism for aZZ r r (2.2 and 2.3).

Proof.

The proof is sketched after the proof of the following proposition.

Let

f: n

H2f is surjective.

r; thus sO is

5.2

Proposition

Let f: A A' be a map of n-moduZes. If Ho(n;f) is bijeative, and W surjeative, then f/f r : A/f r A' is an isomorphism for aZZ r 1 and thus so is t r (see 2.3). The crucial step in the proof of Proposition 5.2 is the following lemma. 5.3

Lemma

Let f: A A' be a map of into the submoduZe f2A'. Assume that and that f induces an isomorphism A/B A/f2B

whiah carries the submoduZe Then f

A'/B'.

B

f2A

is surjeative

HI(n;f):

induaes an isomorphism

A'/f2B'. Proof of 5.3. Hl(n;A) Since

Ho(n;B)

Consider the (twisted) coefficient exact sequence H1(n;A/B) B/f 2B and

Ho(n;A)

Ho(n;A/B)

(A/B)/f 2(A/B). one gets the map of

A/f2A

=

exact sequences H1(n;A)

HI (TI;A/B)

B/f 2B

HI(n;A')

HI (n;A' /B')

B'/f2B'

!

0

1

Hence. it follows from the assumptions and the five-lemma that Applying the five-lemma again to the sequence

0

B/f 2B

B/f2B

A/f2B

B'/f2B'.

A/B

O. Lemma 5.3

follows from the assumptions. Proof of Proposition 5.2. A/f A n

A/f2A

A'/f A'

=+

n

A'/f2A'.

Since

A = lim A/f A. it is enough to prove that +

n

for all n 1. In fact since Ho(n;f) is an isomorphism, so is Assuming the isomorphism, by induction, for n = r. one simply

n = r + 1. since fr+1A = f2frA. Q.E.D. Exactly the same method is used to prove Proposition 5.1. except that in-

applies Lemma 5.3 to get it for

stead of the coefficient sequence used in the proof of Lemma 5.3 one uses the lowdimensional homology exact sequence associated with the extension H2G for

H

f2G

H2(G/H)

a normal subgroup of

G.

(G/H)/f 2(G/H)

0

0

H

G/H

0:

19

This completes the algebraic preparation and we can now turn to the proof of the main theorem.

6. For

n

PROOF OF 3.2

= 1 the main homotopy-theoretical observation is that the canonical

map H2X

H2TI 1X is always surjective. This is due to Hopf [2]. Under the conditions of 3.2 this implies that the map H2TI 1X H2TI 1Y is surjective, and enables us to apply 5.1. In the higher dimensions the situation is slightly more complicated

since we do not have, in general, a surjection of

Hn+1X

onto

HI (TI1X,TInX).

Still

we have the following situation.

6.1

Lemma

Let

Y be a map of connected spaces such that

f: X

phism for 0 j jective. Then:

Assume that

n - 1.

TIjf is an isomor-

is bijective and that

Hnf

Hn+1f

is sur-

For n = 1, Hif is bijective for i = 1 and surjective for i = 2. For n> 1, Hi(TI1X,TInf) is bijective for i = 0 and surjective for

(i) (ii)

i = 1.

Proof: where

PiX

We use the spectral sequence of the fibration

denotes the i-th stage of the Postnikov tower of

E2

p,q

= HP (pn-l X,HqK(TI n ,n»

X.

K(TI

n We have

P n

n-l

X

H P X p+q n

X-module. Since H + K(TI,n) = 0 for all n-l n 1 TI and n > 1, we have for n > 1 only two non-zero groups in the E2 term with total degree n + 1, namely E2 and E2 It follows easily from the usual l,n n+l,O arguments that the classical Serre exact sequence can be extended to get:

where

H K(TI ,n) q

n

is regarded as a TI1P

Hn+2PnX

Hn+2Pn_ 1X HO(TI1X;TInX)

HI (TI1X; n X) HnX

Hn+1PnX

HnPn_1X

Hn+1Pn_1X

0

It follows from the last two terms of this exact sequence for the case Hn+1X too. f

Hn+1PnX

0

that if

Since we assume that

Hn+1f Pn_1X

is surjective Pn-1Y

Hn+1PnX

Hn+1PnY

is surjective,

is an equivalence, we may apply the map

to the extended Serre exact sequence above, and Lemma 6.1 easily follows from the

five-lemma.

Note that (i) follows from the last two terms of the sequence for

Proof of 3.2.

The conditions imply that

assumptions of Proposition 5.1 and 5.2. isomorphism.

TInf

f TI

con

TI

n

Hence, by those propositions

TI

n

f'TI

oon

0

= 2.

satisfies, by Lemma 6.1, the

Further, since one has the exact sequence of functors

o

n

; f n

is an

20

the assumptions imply that a monomorphism.

TInf

is an epimorphism.

o

fTI

n

TI

Since by assumption

fITnf

TInX/fTInX

is a monomorphism.

fact

TIny/rITnY

fnITnX

IT /fTI

n

n

0

n

This follows from the fact that

for some transfinite ordinal

n.

If

a

S + 1, then faG = f 2f

(b)

If

a

is a limit ordinal then

G.

Now since

n

SG.

n

f G

Sation A'" X ... (S3 U e 2p+l )

x

(S3 U e 2P+l )

in whiah X

has the homotopy type of a suspension? Next, the Hopf-Dold-Lashoff-Sugawara theorem on H-spaces suggests Problem 9.

Let B be a ao-H-spaae. u>ith

f a< 0

and Q

a
it

H*(G!PL,Z2)

Theorem 2.5:

q>*(k4i-2) = 0 unless

then q>*(k4i-2) 1

.s. r .s. 4i-3

H* i

is a power of 2

is the class which evaluates

, and evaluates zero on any other

Theorem 2.6:

It is possible to define

o unless i

is a power of 2, in which case

AlSO, note that the 5

is zero.

Theorem 2.7:

1

mod 2

Thus a new generator q>*(A,4)

evaluates

1

If

on the classes

i

is a power of 2 e4i-2-r

for

product.

mod 2

reductions of the

K4i , so

q> *(K4i)

reduction of the non-zero K-invariant in dimension 4 A,4 occurs in H , and we have finally on

e

l

el

the remaining class in this dimension not coming from

e

l

' and evaluates as zero on •

These three results, together with the remarks in §l, complete the proofs of Theorems A-D.

61

§,3.

AN INDEX INVARIANT FOR

*(K

As was indicated, to study

) , we must give a description of this class

4i

as a homomorphism of either unoriented bordism of into

Z2

([29]).

-SURGERY

GiFt into Z2 or of Z2-bordism

As the latter is easier, we describe the invariant in this latter

case. A

W

, so

f*(e

l)

reduced

Bordisms consist of bordisms

together with a homotopy class of maps

are

8 ,

f , f' •

Alternately, we can make f transverse regular to -1 0 f (*) ; then M - (I X N) is an oriented manifold

r

8

*

1

Let

C

W

with oriented boundary

N+N , together with a canonical Ft-homeomorphism of the two pieces. bordisms of

1

M to M' , together with a map F: W

W of

f

is the first Stiefel-Whitney class of M .

mod 2

M and M'

so the restrictions to

be

W,

is a manifold

W which give identical bordisms on the two ends,

N

l

Bordisms are and

•

At a first guess, in order to construct our invariant, we could take a surgery problem on the Z2-manifold M, and take

I(W) - I(W) •

rei N

would need

N

re:

M

M , open

M and

M

up as described above,

However, in order for this to be a suitable invariant, we

to be a homotopy equivalen::e. This is not true in general.

How-

ever, we have Lemma ,3.1 (Index Addition Lemma): N ; then

I(W U N( -w,) Let

Suppose

NJ

4n 4n W ,W'

have a common boundary component

I(W) - I(W') •

-4n-l

re : N

N are

is a bordism from

'"

Let

N4n-l be a degree 1 normal map of oriented manifolds.

simply connected, and

re

to a homotopy equivalence; then

of the bordism class of

re •

I(W)

is a

mod (8)

invariant

62

Applying 3.1, 3.2, we obtain the desired invariant in Corollary 3.3: there is a mod 2

Let

-4n 4n p : M -+ M be a degree 1

Z2 invariant

normal map of Z2-manifolds; then

depending only on the bordism class of p

T

1 [IeM) reduction of 8 - IeM)]

which is the

if M, -Mare orientable.

-4n 4n Open M -+ M up, obtaining

Proof:

(W,N) -+ (W,N) After doing surgery, we can assume attach two bordisms

and from 3.1, 3.2, by

8 .

C -+ N X I

(W,N), (W,N)

from

all simply connected, and then

to a homotopy equivalence

1C

2I(C) + leW) - I(W)

is well-defined mod 16 , and is divisible

Q.E.D. We now turn to the calculation of the invariant of 3.2.

4n-l has been made 2n-2 n: : -4n-l N -+ N

connected ([5]).

introducing any inde:x........across the bordism.

Suppose the map

This we can do without

Similarly, using the technique of §5 of

[13], it is easy to see that the torsion-free part of K2n-l(N)

can be killed with

zero index. We can now assume

1C

:

-4n-l 4n-l N -+ N

(finitely generated) torsion group. alence.

Note that

is

2n-2

connected, and

K2n-l(N)

is a

Now attach the bordism C to a homotopy equiv-

C is obtained from

N by attaching 2n

cells only.

Passing to

the exact sequence of kernels ([5, Chapter I])

3.4 we find duality.

K2n(C) , K2n(C,OC)

are torsion-free and canonically isomorphic via poincare

After identifying them with

Zen) , 3.4 becomes

63

with A symmetric and even.

From [28, §5], there is a quadratic form u

R2n-l(N} , taking values in Q/2Z.

The projection of u(x}

to Q/z

on

is the self-

linking number t(x,x} , and u(x+y) = u(x) + u(y) + 2t(x,y} • Again from [28, §5], let that

a(y}

=x

in 7.5i then yA-lyt

= u(x}

in

Q/2Z.

(Note

(y+zA}A-l(y+zA)t = yA-lyt + 2(z.yt) + zAzt , which is indeed equivalent to

yA-lyt

in

Q/2Z.)

us (at least

Next, note that it is exactly the index of A which interests

its residue class

L

Theorem 7.6: x where

IGI

Remark 7.7:

mod (8»

, and we have 11"1

edu(x) =

R2n-l(N)

IlL (N)lt -"2n-l

represents the number of elements in the set

G.

In a similar situation, E. Brown ([7]) defined the Arf invariant of a

quadratic form on a finite

space

L

-+ Z4]

[q : e

1Ii

in terms of the sum

q(x)

x

The proof of 7.6 is a fairly direct exercise in the analytic theory of quadratic forms.

The ingredients are the Poisson summation formula and the reciprocity

formula for generalized theta functions (used three times).

Indeed, the proof

requires only a slight extension of the techniques of [7] where a related formula is proved. 7.8:

The quadratic form

for the Arf invariant. is now

B ( PL

2n

admits a treatment analogous to that of [4], [7']

The cobordism theory in which the orientations are to be taken

} ' obtained from

2n th Wu class, i.e.,

u

B

PL

by killing the integral Bockstein of the

as the fiber in the map

Alternately, we can kill

V2n

by a map

V2n : The role played in the Browder-Brown treatment of the Kervaire invariant by is now played by K(Q/Z,2n-l) , and, aside fran some minor added canplexities, everything goes as before.

[5, Chapter 3, §5] to obtain a product formula

We can apply these results as in for our index.

This in turn gives

Theorem 3.9:

There are classes

tions of the

K 4i •

But if

M, then

K4i

f : M4n

4i

H

which are the

mod 2

restric-

G/ PL represents a surgery problem for the

..(M) '" (..jf*(K*) + Sqlp: visqlvif*(k*», [Ml} •

In particular, the Kervaire invariant and the index problem are not independent for surgery problems on. Remark 3.10: culation of

-manff'ol.da ;

These results can be regarded as a detailed exploration of Wall's calL*( Z, 1C)

where 1C : Z

is the non-trivial homomorphism, and 3.6 seems

to have further applications in studying the odd Wall groups. Remark 3.11:

SUllivan has given a purely geometric proof of 3.9, and has used these

techniques to canplete his discussion of PL-hanotopy types, using his characteristic variety theorem. Remark 3.12:

It is easy to check, using the product formula in 3.9, that the

are primitives for the Whitney sum pairing on H*

as required for §l.

G/PL.

This in turn determines

K

4i

6S

§4 THE SURGERY OBSTRUCTION FOR LOOP SUMS

Let nI

:

g: N -+ G represent tangential surgery problems

f:: M -+ G,

-

n: M -+ M ,

In this section, we evaluate the surgery problems f:or the "loop sum"

N -+ N.

problem; that is, the problem associated to the composite

Geometrically, the situation is easy to describe. 4.1:

Associated to !.( f:

3-sheeted covering identity,

p: -M X N U MX N U M X

I MX N

p

x g) , the surgery problem is obtained as the

is

n x

i, ,

and

We now identi1'y the kernel of: 4.2:

xe

K(p*)= K(n)

I

p

if

-+ M X N ,

MX

if

is

where

1 X

p

I

-M X N

is the

1('

p*.

H*(N) EEl H*(M) @ K(n') EEl H;(M X N) EEl

X N) :is identHied with (-x,x,O), and x e

X N) , where

X N) is identif:ied with

(-x,O,x) . Remark

The contribution of:

K(n)

\

H*(N) EEl H*(M)

K(n ')

is the number obtained as the obstruction to making the map homotopy equivalence. product in Let

MX if -+ M X N

into a

att this problem is associated to the ordinary composition

G. "i

be the non-zero class in

Hi

(1)1+1 ' the (i+l)st Stief:el-Whitney class.

,

Let

"

given as the suspension of: co

be the total class

co

X(SQ)" AlSO, the

to the surgery problem

"i

=

2

i=l

"i • 2-1

are primitive with respect to both loop sum

and they evaluate one on

L"i' i=l

*

and composition

(.) ,

66

Theorem 4.5: (M

X H,

X

Let

s* be the Kervaire invariant of the surgery problem

g»

if

M X H has dimension

4k-2, and suppose

s(.)

is the Kervaire

invariant of the usual composition; then

( -x,x,O) = (

e r 03>1 U SQ(x), [pi X pj)

,

in the Kervaire case, and

in the index case. Now note that the rmmber of where

a( r)

@

is non-zero is

1) , q>(l C8I e

j/ i/2 q>(e @ e- 2)

r.

2a(r)

Since this is

r , it is immediate that, in the Kervaire case, i+j

1+,1

2

Sqs(e r)

for which

is the number of ones in the dyadic expansion of

even for non-zero q>(e

s

2)

is also

are the only non-zero terms i f if

1

i

(b) is handled similarly.

is even.

i, j

are odd, and

(a) follows.

The quadratic form is evaluated in M(i,j) , and

again the above remark !llBkes it possible to ignore almost all middle dimensional classes. We now turn to the calculation of the

5.4: integers

Let

qi(k,t) =

1: ( k+;:l) mod t+2

l,j

£i,j' E

•

(2) • This is defined for all rational

k, t , and we have

qi (k,l) , Now, to check 2.5, we assume teristic numbers in

pi X pol

Ei,J

are given by

k

Example:

X

-+.f.

Now consider the diagram

Then

X is an H - space.

nx PX)il{ is a left

EY

Y is an H - space.

If X E..!!.!!!!. ,X an H- space and

and a integer

such that

k

Y be a rational equivalence.

Denote by f'i : X -+ Y a

Hence we can

If it's fiber is denoted by Z, then

Corollary.

lence for all but a finite number of primes

is monic.

PX exists since

Y ,... X X Z in the case that

Proof.

v(p)

since

Hence the cClllIPosite Y

Y E G{X) , then there is a

8

X an H- space.

Y is in addition an H - space, then

It'

The retraction

,I\j:x is finite.

and

!! .

i::Xll EY

find a map g : i:: Y

0L:y

!!

monic in

The Puppe sequence

X -+ Y -+ C f

,,\""1

H.

i::f

P1,""

Then

::.f .

1'1 is a p -equiva-

Pk -1' since

Pi _ 1- equivalence i

E8 is monic since

X XZ

u

0

=

X)

2 , .•• , k.

E8 ,... v

0

= H*(fj)

Consider

E8· u(p)

So the result follows by 1.5. 2 2 2 :: (Sp(2» • we get X Z :: (Sp(2» and

Sp(2)

w... -

(E by appl¥ing 1.6. (we can choose here k = 2). This implies that Z w 5)2 and W are rank 2 H- spaces of type (3,7) which, since 1T6z '; 1T6W '; 1T Sp(2) 6

are either

Sp(2)

or

E by the cJ.assification of those H- spaces [18]. 5w

Eithe.r

X

78

2

assumption leads to

2

(E :: (Sp(2)) 5w)

(This was also proved by Sieradski [16]).

§ 2 A Scbanue1 Type LeJmna 2.1 LeJmna:

Let

be fibra.tions over

X

over

prr 1

X which is a p - equivalence for all

pr(q)

Fl _ _---7'/ E/

Then

has no right inverse (section)}.

1 1 ;0 : F ... F

Denote by

F'" E X Fl.

l:Yis monic.

(pr) (p)

Yep)

as well as

Therefore again

l: Y are monic.

By 1.5

Yep)

and

is a pullback in

•

in

0

Hfin such that

111 pr 0 ex 0 f ::: 0

Now c;(-o ;010 h :: ;0 0 a :: f1 , where (-) E [W,

h. 0 (-)

FI-].

Y = (a that

E is an H- space,

p: E X

f :: c;(- o ?

X

quivalence. that

(B1)(p)

and therefore

E(p)

a(p)

Po

has

Y

Further

then, since

pr

0

;0 0 f ::

a

0

h :: f •

such that

so that

h. 0 (;010 h)

h. E [W , ax] on

we get a ccmnutative diagram in

Hfin •

Ii as constructed has to be unique up to homotopy, since

,c1)

in

(a, tp"j#, 0'" Tf/'f"__

H. -

It follows by applying B#> is exact; B

o ••

1 1 F ... E , which exists since X

pi

f: W ... E and

h :: ;0 0 f :: c;(-o f1

PI

: E

is a

It follows that

1 F ... F.

h: W...F

the map induced by an extension of 1 F ... F

1

;0

h. 0 (;010 h) :: ;ol(h. 0 h) , and a(h. 0 h) :: o o h •

0 h : W ... F

But the

.r

Y = (a, cp) :

(l:Y)(p) is monic; i f

denotes the result of the action of sane

By naturality

Ii = h.

Hence taking

0

E •

;0 and hence

Namely given

th,ere is an

F x

Consider

(l:Y)(p) are monic.

Y has a left inverse

I

p E Po = (q E P

Then

;0.

p E Po then

has a right inverse and hence, since

a left inverse. and

the map induced by

Namely if

p-equivalence; this implies that then

Suppose F ,E

Hfin •

are H - spaces and suppose that there is a

;0:-ir Proof.

with all spaces in

X

is an H-space.

(;o,ch-.i

1

EvF

Suppose

B

I

denotes

I

to

B:E

is epic; then

X

(p ,B)

induces an isanorphism in hanotopy and is therefore a homotopy e-

So it remains to show that is epic for all p. (BI) (p)

is epic.

If

which gives rise to a section

B

I

is epic.

If P E Po then p

I

Clearly it is enough to show B I E is a p -equivalence

Po then there is a section

sl= cp(p)

0

s :X(p) ... E(p)

s: X(p) ...

and, since

E(p) is

79

an H- space this implies that there is a retraction r: :m(p}

+ {a(p}}." a 2 '

lTn:m(p} , we JlI8¥ write it as Consider

{sh' a 2 } E TTnE{p}

+ {a(p}}." a 2 = sj Example:

$

1

TTJ'{p}'

(a(p)}"" a 2 = a.

+

E TTnX{p}

r(p}' and

Given

a E

a 2 E TTnr(p} •

Then {G{p)}'" {sn' a 2 } = { A 0 M-> M is an isomorphism of

morphism left

N = R 0 A N, and let f

Let

A-

modules. Let N -->

N be

a projective cover of

f : N - > M be a morphism such that

N over

R, and let

f is the morphism close above. Now

> A0M - > M is a morphism between free R-modules whose reduction mod p

A 0N

is an isomorphism. Proposition.

Thus this morphism is an isomorphism and the proposition follows.

If B is a connected Hopf aJ.gebras over R with cOllllllUtative multiplica-

tion, B is free as an R-module, and p is an odd prime, then either B is an exterior aJ.gebra on odd generators, or there is a surjective morphism of Hopf algebras 1(

:

B - > C such that C is free as an R-module and P(C)2n

Proof.

Let m < 0 be the least integer such that B

m

F O.

F0

for some n

If no such m exists

B = R and is the exterior aJ.gebra on no generators. Hence it may be supposed that m exists.

If m is even P(Em) = Em' and one may take the identity morphism of B

for the morphism :n: of the proposition.

If m is odd, let A be the exterior aJ.gebra

on B and let i : A --> B be the natural morphism of Hopf aJ.gebras. m,

i : A ->

E, and E is a free

free A module.

Let B(l) =

ptf'A,

Either one may take :n:(l) for with B(l).

A module.

Now

Thus by the preceding proposition B is a

and :n:(l) : B -> B(l) be the natural morphism. or the process carried out above may be repeated

1t

Thus proceding inductively the proposition follows readily.

Proposition.

If B is a connected Hopf aJ.gebra over R with cOllllllUtative multiplica-

tion, B is free as an R-module, and p

= 2,

then either B is an exterior aJ.gebra on

odd generators, or there is a surjective morphism of Hopf aJ.gebras such that C is free as an R-module and P(C)2n

F0

for some n

Let m > 0 be the least integer such that B m P(E ) = B. m m 2 2x F 0, 't2 Xl

F 0,

F O.

If m is even

2 If m is odd, suppose x B , 2x F 0, then x is primitive. I f m B2m is primitive and not zero. I f 2x 2 = 0, x2 = 2xl.' and if x2

and xl.

P(B ) . 2m

F0

Then the conditions of the proposition are satisfied by

the identity morphism unless x

2

= 0 for every x

B such that 2x m

F O.

In this

latter case l.et A be the exterior aJ.gebra on B and l.et i : A - > B be the m, natural morphism of Hopf algebras.

Now B is a free A-module.

Let B(l) = BfA,

91

and let 1£(1) : B -:> B(l) be the natural morphism.

Either one may take 1£(1) for

1£ or the process carried out above may be repeated with B(l).

If on the other hand

the process goes on indefinitely, one has that Q(B) is odd, and thus B is generated

A small conductive calculation now shows that if x is

by elements of odd degree.

of odd degree in B, P(B)2n = 0 for all n, and 2x

0, then x

2

= 0, and

't(li\e

pro-

position follows ,readily. Let S be a commutative ring, and n

E

> 0 and even.

lZ, n

Cartan's divided

polynomial algebra r(x,n) is the graded commutative S­algebra such that r(x,n)j= 0 for j

E

lZ, j

1=

0 mod n, r(x,n)kn is a free S­module with I­basis element

r ix)

rk(x), k ElZ, k :::: 0, ro(x) = 1, rl(x) = x, (i,j) is the binomial coefficient

fuJ2l

rj(x) = (i,j) r.+. [x) where J

J:

•

If S is Q the ring of rational integers, then rk(x) =

Further if

S = R, Cartan has &hewn that r(x,n) is generated by(rpt(x»)t > 0' indeed it de­ composes as a tensor product of the cyclic algebras with such generators, each being of height p,

Suppose k = r r

of k, then denoted by e(r

l, rk(x) = e(rl,.,r

•••

O r

+ rlP + ••• + rtpt, t is a unit R.

The inverse of this unit will be

.,r A simple calculation shows that over R, t). xro ••• rpt(x)rt• Also observe that t)

zero element of R whose reduction ili Proposition.

°:: r j < p in the p­adic expansion

Ris

is a non

zero.

If B is a connected Hopf algebra over R with commutative multiplica-

tion, B is free as an R­module, n s z, n

>

0,

0, and P(B)2'

n

In

=

°

j e lZ, j > 1, then there exists a morphism of algebras f : r(x,n) - > B such that

f :

f(x,n) ­ > B is a monomorphism of Hopf algebras.

Proof. x

Let Ii. denote the diagonal of B.

B2n is primitive.

L'!.(xP) = xP0 1 + 1 otherwise

rp (x)

E

xP

Choose x

Let ri(x) = xi/H. +

i?­ll

r,J: (x)

0

E

rp (x)

= xP.

:s :s

° and

Now 6(x) =:ll01 + 10 x + PYO' and

r . (x) •

Then x P

would be a non zero primitive in B • 2 pn

B2pn such that

B such that px 2n

0, but pxP =

° for

Choose an element

Now 6(rp (x») = r!!i=0

r.J: (x)

0

rp­c,. (x)

+ PY1'

Suppose r j (x) is defined for 1 j t, p r j (x) 0, j (x) = j p p r p r r l t = ri(x) 0 rpj_t(x) + PYj' where ri(x) = e(r1,.,rt ) x rp(x) ••• r i (x)

ii=o

o

92 for 1 = r

O

+ rlp + •••+ r pt, 0 t

.t(x)P) = 'Y t(x)p 01+10

:s r j

< p. Now

r t(x)P +

t+l 1 -

t+l,

'Y.(x) 0'Y t+l (x}, Observe (;E P-1 that in the Hopf algebra which is a quotient of B by the ideal generated by the p

p

, r t_l(x), the image of'Y t(x) is a non zero primitive. p P

image of x, r (x), P

'Y t(x)p

F 0,

P

p

Thus

p 'Y t(x)P = 0, and there exists 'Y t+l(x) such that p P t+l 'Y t+l(x) = 'Y t(x)p. Hence t+l(x» = r.(x) 0'Y t+l (x) + p-i P P P

Ei=o

+ PYt+l' Theorem.

The proof of the proposition If B is a connected Hopf

now readily be completed. over R with commutative multiplication,

B is free as an R-module, and the non-zero degrees of B are bounded above, then B is an exterior algebra on a finite number of odd generators. The theorem is an immediate corollary of the three preceding propositions. §4.

Homological application In this paragraph suppose that p is a prime.

'Dle ground ring R will be

the integers localized at p, Space X will mean compactly generated space.. X having the property that H (X;R) is a coproduct of cyclic R-module for n

n

q, q'

7L.

If

R, then Er(X,q,q') shall mean Er(C*(X),q,q') where C*(X) is the normalized

chain complex of X with coefficients in R (see §2). Proposition.

If X is a space, r

7L, r

1, and dr = 0 in the ordinary Bockstein

spectral sequence for X and p, then 1) R/p2R which is a free 2) is

a.

if r = 2t + 1, then Et+l(X,p2,l) is a coalgebra over

R/lR

module, and

if r = 2t, then Et(x,l,p) is a coalgebra over

R/lR

which

free R/p2R module. The proposition follows at once from the theorem of §2 and the Eilenberg-

Zilber theorem. Note Proposition.

R/lR = 7Lll7L. If X is a connected H-space, r

Bockstein spectral sequence for X and p, then

7L, r

1, and dr = 0 in the ordinary

93

1) ccmn:utative

if r • 2t + 1, Et +1(X,l,1) is a connected Hopf algebra with

canultiplication over R/lR which is free as an R/lR 2)

if r • 2t, Et(x,l,p) is a connected Hopf algebra with

commutative comultiplication over R/lR which is free as an Theorem (W. Browder).

moduJ.e, and

If X is a connected H-space N, r

2Z, r

moduJ.e.

2: 1, dr • 0 in the

ordinary Bockstein Spectral sequence for X and p, and H (X) • 0 for n n

d

S

•

> N, then

r(X,p,l) 0 for s 2: r, and E is an exterior algebra on odd generators. The condition that H (X) • 0 for n > N implies H (X) is finite type for all

n

n.

n

If r • 2t + 1, Et +1(X,l,1) is an exterior algebra on odd generators, or if

t(x,p2 r • 2t, E p) is an exterior algebra on odd generators as one sees by app1y:tng the last theorem of the preceding paragraph to the dual Hopf algebras.

Either of

the preceding conditions implies the desired resuJ.t at once.

REFERENCES [1]

W. Browder, "Torsion in H-spaces", Annals of Mathematics, Vol. 74, 1961, pp. 24-51.

[2]

W. Browder, "Higher torsion in H-spaces", Transactions A.M.S., Vol. 108, 1963, pp. 353-375.

[3]

H. Carlan, Seminaire ENS 1954/1955, "Algebres d'Eilenberg-MacLane et homotopie."

[4]

S. Eilenberg and J. C. Moore, "Limits and spectral sequences", Topology, Vol. 1, 1961, pp. 1-23.

[5]

J. W. Milnor and J. C. Moore, "On the structure of Hopf algebras", Annals of Mathematics, Vol. 81, pp. 211-264.

NEARLY FROBENIUS ALGEBRAS AND THEIR

CATEGORIES

John C. Moore and Franklin P. Peterson

51.

INTRODUCTION

Classically a Frobenius algebra A over a field k is an algebra such that in the category of left A-modules or the category of right A-modules. an object is projective if and only if it is injective.

Such algebras were once termed quasi-Frobenius. the

term Frobenius algebra being reserved for those algebras A having the additional property that the left regular representation is equivalent to the right regular representation. example of such an algebra is the group with coefficients in a field k.

The principal of a finite group

Here the equivalence of the left

and right regular representation is given by inversion in the group. A few years ago Sweedler remarked that this generalized directly to those Hopf algebras with commutative comultiplication which are finite dimensional vector spaces and have an involution. Recently. Adams and Margolis [1]. in studying the mod 2 Steenrod algebra. indicated that this graded algebra is injective as a module over itself.

However. they gave no appropriate setting

for this type of result and their approach did not indicate a reasonable class of graded algebras which have this property. In this note we propose to indicate a setting for studying graded algebras which are self-injective. and to show that there is a reasonable class of

having this property.

95

In section 2 we state our results and in section 3 we give some examples. §2.

STATEMENT OF RESULTS

Let R be a commutative ring.

Let A be a graded R-algebra

(possibly graded on all the integers), and let

AM

or MA denote the category of graded left or right A-modules with homomorphisms of degree zero respectively. DEFINITION 2.1.

A is a Frobenius algebra if, in AM and MA, an object is projective if and only if it is injective if and only if it

is flat.

DEFINITION 2.2.

A is a nearly Frobenius algebra if

i) the component of A in degree zero, AO' is a Frobenius algebra ii) every injective in AM or MA is flat, and iii) if 0 + X' + X + X" + 0 is a short exact sequence in AM or MA such that X' and X are flat, then X" is flat. Note that for any algebra A, if X" and X are flat, so is X' and if X' and X" are flat, so is X. DEFINITION 2.3.

Let A be a graded R-algebra and let I be a two-sided

ideal in A.

M is an object in AM, then M is projective at I if

If

i) A/I 8 A M is a projective A/I-module, and ii) (A/I, M) = 0 for n > O. If N is an object of AM, then N is complete at I if the filtration pPN • IP'N, for

p

£

Z, is a complete filtration.

The following proposition is one of the main steps in the proof of theorem 2.5. PROPOSITION 2.4.

Let A be a graded R-algebra and I a two-sided ideal

96

in A.

Let M and N be objects in AM such that M is projective at I

and N is complete at I. THEOREM 2.5.

Then

(M, N) • 0 for n

>

O.

Let A be a positively graded nearly Frobenius algebra.

Let X be an object of AM which is bounded below.

Then, if X is flat,

then X is both projective and injective. COROLLARY 2.6.

A positively graded nearly Frobenius algebra A is

self-injective. We now investigate how nearly Frobenius algebras behave under colimits. DEFINITION 2.7.

A coherent system of graded R-algebras consists of

a filtering ordered set I, that is, if

i O' i l

exists i 2 E: I such that i o i 2 and i l from I to graded R-algebras such that i f

E:

I, then there

i 2, and a functor A io

i l in I, then A(i l)

is flat as a left or right A(io)-module. A strongly coherent system of graded R-algebras is a coherent system such that A(i O il)O' the component of the morphism A(i o i l) in degree zero, is an isomorphism. THEOREM 2.8.

If I, A is a strongly coherent system of graded

R-algebras such that for each i

E:

I, A(i) is a nearly Frobenius

algebra, then B = colimI A is a nearly Frobenius algebra. S3.

EXAMPLES

Let A be a positively graded, finite dimensional, Hopf algebra over a field k with commutative comultiplication and involution.

As

remarked in Sl, A is a Frobenius algebra and hence is a nearly Frobenius algebra. of such for all i o

If A(i O i l) : A(i + A(i is a monomorphism o) 1) iI' then the system is coherent and we may

97

apply our theory. group.

Then

For example, let

0

be the infinite orthogonal

H.(Oj k) is a nearly Froben1us algebra.

Another example is the mod p Steenrod algebra,

a

a.

is a

union of finite Hopf subalgebras and hence is a nearly Frobenius algebra.

Let

a.. M' C a,M

which are bounded below.

be the full subcategory of

a -modules

In this category, every flat module is

a M'

free so we see that a module in

is free if and only if it is

projective if and only if it is injective if and only if it is flat.

elM,

In

there are injectives which are not projective.

The main result in [1] is a theorem giving conditions for S Let pS• Sq(O, ••• ,2 , ••• ) £ lZ, M £ 0.. M' to be free when p • 2. t where the 2s is in the tth place. Their main result is that Me

aM'

• 0 for all s < t , n 0 Let A' be the subalgebra geneTated by Pt, ••• ,P t-l and t Pgt• A' is an exterior algebra. Let A be the subalgebra generated t 2 0 t-l 0 t by A' and Pt. The relations in A are (P t) • pt·····p t • P2t '

o

is free if and only if H(M,

e

t

[P 2t, PtJ • O.

The main computational lemma in [lJ is the following

result. PROPOSITION 3.1.

Let

M £ AM'.

If M is A'-free, then M is A-free.

We give a short proof of proposition 3.1 using the structure of the extension qUite explicitly and using the methods developed in section 2.

Anderson [2] has given a more general version of

proposition 3.1.

We hope to be able to give a proof of

proposition 3.1 which will generalize to the mod p Steenrod algebra case, which we now describe. Let place.

• cP(O, ••• ,ps, ••• )

tl,

where the pS is in the tth o t-l 0 Let A' be the subalgebra generated by Pt, ••• ,P t and P2t• £

98

A' is a polynomial algebra with all generators truncated at height p. Let A be the subalgebra generated by A' and

The relations in

A are ( ptt )p • pOt • (pl)p-l t · 0.0· ( Ptt-l)p-l • (pO 2)p-l t.

CONJECTURE 3.2.

Let M £ AM'.

If M is A'-free, then M 1s A-free.

BIBLIOGRAPHY 1.

J. F. Adams and H. Margolis, "Modules over the Steenrod algebra", Topology, to appear.

2.

D. W. Anderson, to appear.

Loop Structures on the Homotopy Type of S3 David L. Rector 1 In order to understand the homotopy theoretic properties of compact Lie groups, it appears useful to study the purely homotopy theoretic category whose objects consist of a homotopy class of finite complexes G together with a complex BG such that OB G G. A map in this category is a homotopy class of maps f: BG B This category G,. has many of the homotopy theoretic properties of the Lie category (see e .g . , [3] and [6 J) • An important first step in proving classification and structure theorems in this category is to understand the group structures carried by the homotopy type of S3.

We provide here a partial list of such

structures by classifying those which are equivalent at each prime to Bg3 = HP B

P

I

.[;

0'

K(Q,4) ------------> K(Qp,4), where Qp denotes the p-adic numbers, is a fibre square.

It follows

that we need only construct a map of degree n on Bp . We give the construction in detail for p odd. Let T be a maximal 3 torus of the Lie group S , and let N be its normalizer. The inclusion of N in S3 induces a map B B with fibre RP 2. For p odd, this map N . H* . so induces an isomorphism H* (B;

=

(B )

N P

B

p

B may be constructed as follows, N reversal

N

p

be the p-adic integers, This map is equivariant so

where E is the universal K(Z ,2) associated to

the sign action on Zp'

lemma is

Let

K(Z ,2) the natural inclusion. P

we have a map B

Ep ... Bp .

act on K(Z,2) by sign

Then BN is the universal K(Z,2) bundle over BZ/ 2

associated to that action. K(Z, 2)

Let

p

Since B and E have isomorphic

N

,

cohomology,

A classical corollary to the "trivial case" of Hensel's

105

PROPOSITION.

=

If n

Zp' then E. has

square

in

iff (nip)

1.

Let a

2

= n.

Multiplication by a is an equivariant endomorphism -+

For p

=2

E.

*

We have H (K(Z ,2),Z ) p 2 P P degree a n.

a similar construction may be used.

A

A

Care must be taken,

however, at certain points.

B is obtained from B by successively N killing Z/2 classes in H +l(BN: Z) . This occasions no trouble except 4k for k = O. In order to make the appropriate arithmetic square a fibre square, each space must be made simply connected before completing. The key point to notice is that TIlE may be kd Ll.ed by attaching a cone on a subspace of the form (RP 2 f

2

invariant under the map f.

REFERENCES 1.

M.F. Atiyah, Power operations in K-theory, Quart. J. Math Oxford (2), 17 (1966), 163-93.

2.

A.K. Bousfield and D.M. Kan, Homotopy with respect to a ring. (Preprint) .

3.

W. Browder, Torsion in H-spaces. Ann. of Math. 74(1961), 24-51.

4.

G. Mislin, H-spaces mod p, I. (Preprint).

5.

M. Mimura, G. Nishida, and H. Toda, Localization of CW-complexes and its applications.

6.

D.L. Rector, Subgroups of finite dimensional topological groups. J. of Pure and Applied Algebra. (to appear).

7.

D. Sullivan, Geometric Topology I.

Notes. M.I.T. 1970.

Sphere bundles over spheres as H-spaces m:xl p>2 James Dillon The first H-space known to be a finite complex and not of the homotopy type of a

Lie

group or a product of a Lie group and 87 , s was the Hilton-

Reitberg "criminal" M7 , a 3-sphere bundle over 8 7 classified by 7 times the usual generator of 1Ts ( 8 3 ) . These examples and later ones due to myself, zabrodsky, fJarr>ison, and Curtis and Mislin had in commn the fact that they were for each prime p of the m:xi p homotopy type of products of Lie groups and 8 7 ,s .

On the other hand, Harris [3] and Mimura and Toda [4] observed that rrany classical indecomposable Lie groups were m:xi p equivalent to products,

the factors therefore being H-spaces.

The v.ork of Oka [S] and Mimura and

Toda called attention to certain sphere bundles B (p) over spheres and n showed that B (p) is a nod pH-space for n