Algebraic Topology and Algebraic K-Theory (AM-113), Volume 113: Proceedings of a Symposium in Honor of John C. Moore. (AM-113) 9781400882113

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Annals of M athematics Studies Number 113

ALGEBRAIC TOPOLOGY AND ALGEBRAIC K-THEORY

E D I T E D BY

WILLIAM BROWDER

Proceedings o f a conference O ctober 24-28, 1983, at Princeton University, dedicated to John C. M oore on his 60th birthday

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

1987

C op y rig h t © 1987 by P rinceton U niversity Press A LL RIGHTS RESERVED

T he A nnals o f M athem atics Studies are edited by W illiam B row der, R obert P. L anglands, John M ilnor, and Elias M. Stein C orresponding editors: Stefan H ild eb ran d t, H . B laine L aw son, Louis N irenberg, and D avid V ogan

C lo thbound editions o f Princeton U niversity Press books are printed on acid-free paper, and binding m aterials are chosen for strength and durability. P a­ p erb ack s, w hile satisfactory for personal collections, are not usually suitable fo r library rebinding

ISB N 0-691-08415-7 (cloth) IS B N 0 -6 9 1-08426-2 (paper)

Printed in the U nited States o f A m erica by P rinceton U niversity Press, 41 W illiam Street P rin ceto n , N ew Jersey

☆

L ibrary o f C ongress C ataloging in Publication data will be found on the last printed page of this book

DEDICATION TO JOHN MOORE John Moore, in his mathematical career and his more than 30 years at Princeton, has made a lasting impact on the subject of topology, both in his research and in his influence on the many graduate students who worked with him. This conference and this volume are dedicated to him in recognition and warm appreciation by his ex-students, by other coworkers in topology, and by his many friends, numbered in both groups.

Students of John Moore

Richard Swan William Browder James Stasheff Allan Clark Stephen Weingram Paul Baum Wu-yi Hsiang J. Peter May William Singer Harsh V. Pittie Fred William Roush Harold Hastings Joseph Neisendorfer Brian Smith Philip Trauber James Lin Haynes Miller Leo Chouinard Robert Thomason Paul Selick John Long

v

1957 1958 1961 1961 1962 1962 1964 1964 1967 1970 1972 1972 1972 1972 1973 1974 1975 1975 1977 1977 1979

LIST OF SPEAKERS AT THE CONFERENCE

J . F . Adams D. Anick E. H. Brown G. Carlsson F . Cohen R . Cohen W. Dwyer E. Friedlander S. Halperin D. Kan J. Lin M. Mahowald H. Miller J. C. Moore J. Neisendorfer F. P. Peterson D . Ravene1 P. Selick J.-P. Serre W. Singer C. Soule J. Stasheff R . Swan R. Thomason F. Waldhaussen A. Zabrodsky

CONTENTS ix

Preface I

II

III

IV

V

VI

EXPONENTS IN HOMOTOPY THEORY by F. R. Cohen, J. C. Moore, and J. A. Neisendorfer

3

THE EXPONENT OF A MOORE SPACE by Joseph A. Neisendorfer

35

THE SPACE OF MAPS OF MOORE SPACES INTO SPHERES by H. E. A. Campbell, F. R. Cohen, F. P. Peterson, and P. S. Selick

72

THE ADAMS SPECTRAL SEQUENCE OF Q2S3 AND BROWN- GITLER SPECTRA by Edgar H. Brown and Ralph L. Cohen

101

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES by Donald M. Davis and Mark Mahowald

126

MAPPING TELESCOPES AND K^-LOCALIZATION

152

by Donald M. Davis, Mark Mahowald and Haynes Miller VII

VIII

IX

X

XI

THE GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION by Douglas C. Ravenel

168

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS by W. G. Dwyer and D. M. Kan

180

THE ROLE OF THE STEENROD ALGEBRA IN THE MOD 2 COHOMOLOGY OF A FINITE H-SPACE by James P. Lin

206

MAPS BETWEEN CLASSIFYING SPACES by A. Zabrodsky

228

GENERIC ALGEBRAS AND CW COMPLEXES by David J. Anick

247

vii

viii XII

XIII

CONTENTS DEFORMATION THEORY AND THE LITTLE CONSTRUCTIONS OF CARTAN AND MOORE by James Stasheff

322

FREE (Z2 )3 - ACTIONS ON FINITE COMPLEXES

332

by Gunnar CarIsson XIV

XV

XVI

XVII

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA by J. P. May

345

A DECOMPOSITION OF THE SPACE OF GENERALIZED MORSE FUNCTIONS by Ralph L. Cohen

365

ALGEBRAIC K-THEORY OF SPACES, CONCORDANCE AND STABLE HOMOTOPY THEORY by Friedhelm Waldhausen

392

THE MAP

418

BSG -> A(*) -» QS°

by Marcel Bokstedt and Friedhelm Waldhausen XVIII

XVIX

XX

VECTOR BUNDLES, PROJECTIVE MODULES AND THE K-THEORY OF SPHERES by Richard G. Swan

432

LIMITS OF INFINITESMAL GROUP COHOMOLOGY by Eric M. Friedlander and Brian J. Parshall

523

ALGEBRAIC K-THEORY OF GROUP SCHEME ACTIONS by R. W. Thomason

539

PREFACE From October 24 to October 28, 1983, a conference entitled 'Algebraic Topology and Algebraic K-theory’ was held at Princeton University, and was dedicated to John Moore on his 60th birthday. It was planned by an organizing committee consisting of William Browder, chairman, Franklin P. Peterson, James Stasheff and Richard Swan. It was the intention of the organizers to organize a conference that emphasized homotopy theory, algebraic topology more generally and some more topologically related aspects of algebraic K-theory, and to create an interesting conference with a relatively narrow focus, without trying to cover all aspects of topology, or even of algebraic topology. Neither was it the intention of the committee to try to reflect explicitly the interests of Moore, or to emphasize explicitly his role and that of his students in so many of the modern developments. In the event, these things were, in fact, well displayed during the conference. The conference was part of a special year in Algebraic Topology and K-theory at Princeton University during the 1983-84 academic year, which was supported by the National Science Foundation. Special thanks are due to Nancy Schlesinger who handled most of the day to day organization of the conference, as well as to a large number of Princeton graduate students who pitched in to help with refreshments, and other aspects of hospitality for the visitors.

W. Browder

ix

Algebraic Topology and Algebraic K-Theory

I EXPONENTS IN HOMOTOPY THEORY F. R. Cohen, J. C. Moore, and J. A. Neisendorfer

1

INTRODUCTION An abelian group positive integer) if that

N

A a

has exponent in

A

N

(where

implies that

is the least such integer.

N

Na = 0

is a and

The first interesting

result concerning exponents in homotopy theory was proved by I.M. James [4] in the middle fifties. his result may be stated as follows:

The main part of

suppose

q

and

n

are strictly positive integers, then the 2-primary component of

7rq[®^n+^] ^ias exPOnent dividing

2^n .

Shortly thereafter it was proved by H. Toda [13] that if and

n

are as above and

p

is an odd prime, then the

p-primary component of p

2n

„

has exponent dividing

^ r

. One important fact about these results is that they

do not depend on

q.

Moreover, it has been known for many

years now that they do depend sharply on are unstable.

1,

Supported in part by NSF grants.

3

n,

i.e., they

q

4

COHEN, MOORE, AND NEISENDORFER After considering the preceding results of James and

Toda, Michael Barratt conjectured that for an odd prime one should be able to replace shown to be the case for

p

n = 1

2

by

p.

p

In 1978 this was

by P. Selick [11].

A

short time later our work led to a proof that this was indeed the case for all

n

[1,2,8].

The techniques used

were at least superficially quite different from those employed by Selick. Aside from answering some questions, the recent work on exponents has posed a number of new questions. Classically one looked at exponent questions in topology by supposing that one had some functor from spaces and maps or pointed spaces and maps to abelian groups, and then asking which spaces go into abelian groups having exponent dividing

N

when this functor is applied.

Typical

examples of such functors are classical homology, homotopy, or for a given prime these functors.

p,

the p-primary component of one of

The homotopy group functor is particularly

pertinent since it is often difficult to obtain information concerning homotopy groups.

However, the proofs of the

more recent results mentioned earlier lead to the belief that one should look for more geometric results which would imply the desired results concerning the functors. Over any pointed category the notion of cogroup makes sense.

Furthermore, the morphisms in the original category

5

EXPONENTS IN HOMOTOPY THEORY from a cogroup to an object form a group.

Fixing the

cogroup, one obtains a functor from the category to that of groups.

Clearly such a functor is limit preserving.

Moreoever if the fixed cogroup is abelian the functor takes values in abelian groups. In homotopy theory the basic pointed category is the one such that an object is a pointed space with a compactly generated topology having the homotopy type of a

CW

complex, and the morphisms are pointed homotopy classes of pointed maps. co-H

The cogroups over this category are the

spaces such that the comultiplication is homotopy

associative and has a homotopy

inverse.

Any

the original category gives rise to a cogroup reduced suspension of

X.

objectX 2X,

in

the

The typical example of an

abelian cogroup is the suspension of a cogroup. 0, the homotopy functor of degree

q

For

q >

is that where the

^ fixed cogroup is the

q-sphere

Over any pointed category sense.

Sq = the notion of

groupmakes

The morphisms in the original category from an

object to a group form a group.

The morphisms from a

cogroup to a group form an abelian group whose operation is determined by either the cogroup structure of the domain or the group structure of the range. When dealing with the basic pointed category of homotopy theory, any object

X

gives rise to a group

OX,

6

COHEN, MOORE, AND NEISENDORFER

the loop space of

X.

The group objects are the

H

spaces

such that the multiplication is homotopy associative and has a homotopy inverse.

The typical example of an abelian

group is the loop space of an

H

space.

Over any pointed category, suppose that one has a group object

Y

and the identity of

observe that for any cogroup morphisms from is an odd prime.

X

to

Y

X

has order

N, then

the abelian group of

has exponent

Localize the

Y

2n+l

N.

Suppose that

sphere at

p,

p

and

then either take the universal covering space of the 2n-fold loop space of the localized sphere or else equivalently first take the

2n+l

connective cover of the

localized sphere and then take the 2n-fold loop space of the connective cover.

The order of the identity of the

group so obtained is

pn . This implies the stated result

on

p-primary homotopy of spheres assuming the known result

of B. Gray [3] that there are elements of order homotopy of

pn

in the

s^n+*.

Results of the type just described could not have been obtained a couple of decades ago because they use localization in an essential though straightforward way. However, it is possible that their implications concerning homotopy groups could have been proved via C-theory [12]. Nevertheless results of this type encourage the attempt to characterize those simply connected finite complexes such

7

EXPONENTS IN HOMOTOPY THEORY

that some iterated loop space has an identity map of finite order, or an identity map which becomes of finite order after localization at a

prime.

We will exhibit a few.

There is a current conjecture of Michael Barratt to the effect that if

p

is a prime,

Q222X

is a pointed space,

2 2 X

and the order of the identify of order of the identity of

X

is

xi p ,

is

then the

pn+1.

When we started writing this paper there were no known examples of this conjecture.

The principal result of the

paper is to prove that if in addition

p

is odd,

X

is

simply connected and has a single nonvanishing homology group, then the order of the identity of i.e.,

2 2

0 2 X

has an exponent.

n22 2x

is finite,

We indeed give a bound for

the exponent, but it is a crude bound. Recently the third author has been able to proceed beyond the techniques of this paper and show that under the hypotheses of the preceding paragraph the conjecture of Barratt is valid. One of the principal difficulties with proceeding further in the direction of study outlined above is that we have no appropriate extension theorem. X'

X

Q222X '' exponent? those of

X' '

i s a cofibration sequence,

have exponents, then does If

n2^

n2s2x 1 and

For example, when 2 2 , Q 2 X ’ and

have an

has an exponent, is it determined by n2^ ' ■

together with some

reasonable invariant of the cofibration sequence?

8

COHEN, MOORE, AND NEISENDORFER If

7r

is an abelian group of exponent

exponent of the Eileriberg-MacLane space K(7r,n) = n^K(7r,n+2), and the cogroup finite exponent if

n > 1.

pn ,

K(7r,n)

2^K(7T,n)

then the is

pn ,

has no

Thus no naive dual of Barratt’s

conjecture is close to being true.

§1.

THE WEAK PRODUCT DECOMPOSITIONS OF THE SPACES

nPm (pr)

As indicated in the introduction we shall work in the context of compactly generated spaces with nondegenerate basepoint having the homotopy type of a pointed complex.

CW

Sometimes the category will be that where the

maps are pointed maps, and sometimes it will be the corresponding homotopy category. An object in any category is decomposable if it is isomorphic with a nontrivial product, and otherwise it is indecomposable.

In the categories where we will work there

is also the notion of infinite weak product.

Sometimes we

shall be able to show that some of the objects we are considering admit infinite weak product decompositions into indecomposable factors. Now as in our earlier work [1,2], let space obtained by attaching an by a map of degree

pF

for

be the

m-cell to the (m-l)-sphere

m > 2.

For

m > 3, these

spaces are suspensions, and hence cogroups. the order of the identity is

m r P (p )

For

pr . Further for

p m > 3

odd and

EXPONENTS IN HOMOTOPY THEORY in

p odd the spaces

r

OP (p )

9

admit infinite weak product

decompositions with indecomposable factors.

Looping once

more and looking at the factors, it will follow that the order of the identity of

fi^Pm (pr) dividesp^r+^.

as in our earlier work, let

S^n+^{pr}

theoretic fibre of a map of degree itself for

pr

Also

be the homotopy of

into

n > 1.

The basic decomposition result for the spaces Qp2n+2(pr)

a irea(jy been proved [1], though it does not

give rise to a decomposition into indecomposable factors.

PROPOSITION 1.1.

If p

Is an odd prime and

in the homotopy category the spaces ^

s2n+^{pr} x

0P^n+2(pr)

aru^

n > 1, then

are isomorphic.

The principal task of this section is to obtain a somewhat analogous decomposition of the spaces

0p2n+^(pr)

and to obtain some hold on the factors.

l

DEFINITION 1.2

Let

connected spaces.

X

i Y -» Z

be a fibration sequence of

It is split if either of

the following

equivalent conditions is satisfied. i)

the map

i'

X Y

ii)

the map

j:

Y -» Z

has a left inverse (retraction). has a right inverse (section) such

that the resulting map extension to

X x Z.

tlf:

X^Z-»Y

admits an

10

COHEN, MOORE, AND NEISENDORFER Note that the usual situation is that the map

i

is a

cofibration and in this case the existence of a left homotopy inverse for

implies the existence of a left

i

inverse. Note also that a map fibration

j:

Y

Z

j: Y -> Z

may be replaced by a

if and only if

(j):

is surjective (assuming of course that fibrations are surjective), and in this case the natural map a homotopy equivalence. retract of

Indeed

Y

is

is a deformation

Y.

CONVENTION 1.3.

f:

If

X -> Y

Is a map, f

denote the map obtained by replacing and if

surjective,

map obtained by making If

g: Y -> Y

l 7r F-»E-»B

f

let

f:

let

f:

X

Y

by a cofibration, X

Y

denote the

into a fibration.

is a fibration sequence with

E

connected, there is a canonical shift of this sequence to

the fibration sequence

~ i QB -» F -» E.

If all spaces are

simply connected one can replace the third canonical shift Ql

by the fibration sequence If

L j A-^B-^C

OF

Q tt

0E

-» QB.

i s a cofibration sequence, there is a

canonical shift of this sequence to the cofibration j sequence

B

v C -» 2A.

The third canonical shift can be 2j

replaced by the cofibration sequence

2A

2B

2C.

11

EXPONENTS IN HOMOTOPY THEORY The definition of

m r P (p )

canonical cofibration sequence

implies that there is a Sm

Pm+^(pr) -» Sm+^ , the

right hand map being called the pinch map since it pinches the m-cell of

to a point.

In the shift of this

rjn+1, r. ^m+1 ^m+l sequence to P (p ) -» S -» S ,

the right hand map is

of degree

Pm *^(pr)

.

pF .

Note that one has the commutative diagram Pm+1(pr)

-»

sm+1

-»

sm+1

where the top edge is part of the preceding cofibration sequence.

Making all maps in this diagram into fibrations,

one obtains for

m = 2n

X.2H+1 f r. , E {p }

-,2 n+l f r, F {p }

the commutative diagram

-.20 + 1 , r. , 02n+l f r, — »P (p ) —* S {p }

_,2 n+l, r> — » P (p )

I

~ 2 n+l — >S

I

ns2n+i

— >

I

*

— > s2n+1

with rows and columns fibration sequences [2,10]. that

F^n+^{pr}

4.x. • -u the pinch map

was defined to be the homotopy fibre of

r»2n+l ( r*. 02n+l P (p ) -+ S

, „2n+l r r. and E \P /

homotopy fibre of the natural map [1,2]. X*

Recall

In the preceding diagram if

the

P^n+^(pF) -» S^n+^{pr} X

is a space, then

is a space canonically homotopy equivalent with

The right hand column is the defining sequence of

X.

12

COHEN, MOORE, AND NEISENDORFER

S^n+l{pr},

and therows are equivalent to the

defining

fibration sequences of their left entries. In our earlier papers a bouquet of spaces (with p now denoted

mod p

odd) proved useful [1,2,8].

n P(n,p ) = \MP (p ),

r

Moore

This bouquet,

is such that

n^ > 4n,

and only a finite number of indices occur in any degree. r 2n+1 r P(n,p ) -+ F {p }

There is a map

fibration with fibre

W^n+*{pr}

which if made into a

gives rise on shifting the

fibration sequence twice to a split fibration squence rm/ T\ ™2n+l f r. T1T2n+l r r-, fiP(n,p ) ■+ QF (p } W {p } where the left hand map is equivalent to a loop map [1,8]. It was also shown that the space

W^n+^{pr}

homotopy type of

k n *{pP+*}

S^n * x

has the [1, Theorem

12.1, 8 ] . t*

The map

P(n,p )

factor through

F

9

E^n+*{pr}

n

+1

T*

(p }

could be chosen to

and by the same procedure gives

rise to a split fibration sequence OP(n,pr) -» QE2n+1{pr} -» V2n+1{pr} the left hand map being equivalent to a loop map, and y2n+ l{prj. having the homotopy type of

C(n) x

k ^k>l^P n

where

the double suspension

3.2].

C(n)

is the homotopy fibre of

2^-* S^n *

Q^S^n+*

[2, Theorem

EXPONENTS IN HOMOTOPY THEORY r P(n,p ) -> P

Let

2n+1r (p )

the composite of maps P^n+l^prj

be an admissible map, i.e.,

P(n,pF)

E^n+*{pr}

as described above. T2"* V >

13

and

E^n+*{pr}

Let

P^p')

P2" V )

be the resulting fibration sequence, thus defining the space

T^n+^{pr} .

We wish to know

sequence obtained by shifting this split fibration sequence.

that the fibration sequence twice is a

This will follow from the next

results.

LEMMA 1.4.

If

h:

simply connected p

X mod p

Y is a map between bouquets of

r

Moore spaces which induces a h

homology monomorphism, then

mod

has a left homotopy

inverse.

Proof:

The r-th Bockstein

homologies

H^(X;Z/pZ)

and

differential modules over

makes the reduced

pT

H^(Y;Z/pZ) Z/pZ,

considered as a submodule of

and

into acyclic H^(X;Z/pZ)

H^(Y;Z/pZ).

may be

Since over a

field acyclic differential modules are the injectives in the category of differential modules, there is an acyclic differential submodule natural map isomorphism.

C

of

H^(Y;Z/pZ)

H^(X;Z/pZ) © C -»H^(Y;Z/pZ) Since in this situation the

such that the is an mod p

Hurewicz

map induces an epimorphism of all terms in the Bockstein

14

COHEN, MOORE, AND NEISENDORFER

spectral sequence, it follows that there is a map Y

where

that C.

g

X’

is a bouquet of

mod p

J*

Moore spaces such

induces an isomorphism between

This implies that

h 1 g:

LEMMA 1.5.

X

If

simply connected

H^(X';Z/pZ)

X ^ X' -» Y

equivalence, and the lemma follows.

g: X' -»

and

i s a homotopy

^

has the homotopy type of a bouquet of mod p

r

p

Moore spaces with

r

^2,

then

so does 2QX.

Proof: then

A result of Milnor implies that if

2Q2A

has the homotopy type of

denotes the j-fold smash of may assume

that

isomorphic with

A

X = 2A. P

s+1

A

is a space,

2A^A

with itself [6].

where A^A Now we

s r t r Noting thatP(p ) ^ P (p ) is

r s+t—1 r (p ) ^ P (p )

in the homotopy

category, and taking account of the behavior of smash products vis-a-vis coproducts, the lemma follows.^

LEMMA 1.6

If

k:

QX

Z

is a map, then

homotopy inverse if and only if

2k:

2QX

k

has a left 2Z

has a left

homotopy inverse.

Proof: We cofibration.

may

suppose without loss thatk

Then so also is 2k.

isa

15

EXPONENTS IN HOMOTOPY THEORY Let

tr*.

2Z

C = 2Z/2QX

has a left homotopy inverse n/

orvTT 2Z --- > 2QX ^ C

evaluation map adjoint

Z

be the natural map. g, then the composite

is a homotopy equivalence.

2QX

QX

X

If

extends to a map

k

2Z -» 2Z

Hence the

2Z

X

is a left homotopy inverse for

whose k.

The opposite implication being clear, the lemma follows.

f§

PROPOSITION 1.7. connected QY

mod p

If r

X

and

f

are bouquets of simply p

Moore spaces with

is a map which induces a

then,

Y

mod p

r

^ 2,

and

f'-

QX -»

homology monomorphism,

has a left homotopy inuerse.

The proposition above follows at once from the preceding lemmas.

PROPOSITION 1.8. nnr r\ QP(n,p )

There is a commutative diagram ™2n+l f r, »QE {p }

,r2n+l, r. , » V {p }

i=

i

i

nnr r\ OP(n,p )

v or»2 n+l, r. > QP (p )

^ n + l r T, > T {p }

^ *

r^2n+l f r, »OS {p }

=

rto2n+lr r. > OS {p }

such that the rows are split fibration sequences, and the inclusion map of the right hand column is equivalent with the natural map

V^n+^{pr} -*T^n+*{pr}.

16

COHEN, MOORE, AND NEISENDORFER

Proof:

There is a commutative diagram with rows and

columns fibration sequences y2 n+i{pr} ,

T^n+l f r, , T {P }

nr T\* P(n,p )

i-2n+lr r, » E {p }

I

I

rtf P(n,p )

Tj2 n+1 f r. » P (p )

i

i

*

» S2n+1{pr}

no2n+lf r, OS {p }

where the right hand column is a defining fibration sequence for

E^n+^{pr}, and the rows are equivalent with

defining fibration sequences for their left entries. Shifting appropriately a diagram of the desired form is obtained.

Since it has already been observed that the

upper row is a split fibration sequence, and the preceding proposition implies that the middle row is also, the proof is complete.

f|

COROLLARY 1.9.

If

p

If

T^n+*{pr}

p

is an odd prime and has exponent dividing

in

x OP(n,pr) and

are isomorphic.

COROLLARY 1.10. the space

prime and n > 2, then

the spacesT^n+^{pr}

the homotopycategory OP^n+l(pr)

Is an odd

n > 2, then p^r+*.

The first of these corollaries follows at once from the fact that the central row of the diagram of the

EXPONENTS IN HOMOTOPY THEORY preceding proposition splits.

The space

17

Q2S2n+^{pr}

has

•p

exponent space p

r+1

p

[9], and the product decomposition of the

V2n+V }

.

Now if

has exponent

implies that F

EB

s

and B

exponent dividing

st.

fiV2n+^{pr}

has exponent

is a multiplicative fibration, has exponent t,

then

E

F

has

Applying this fact to the loop of

the right hand column of the preceding proposition, the second corollary follows. We remark without proof or reference that Proposition 1.1

fails if

pr = 2 and often fails if

greater than one.

It would imply that

pr = 2r

S2n+*{2r}

space and this is usually not the case.

with

r

is an H

Since Sections 1

and 2 depend heavily on Proposition 1.1, the results here are for the most part restricted to odd primes.

§2.

BOUNDS FOR THE EXPONENT OF THE SPACES RELATED SPACES FOR AN ODD PRIME

n2Pm (pr)

AND

p

The work of the preceding section gives rise to an almost immediately obtainable bound on the exponent of the spaces

fi2pm (Pr). This of course shows that these spaces

do have an exponent.

PROPOSITION 2.1.

connected spaces such that if then

Z

£

Suppose that Z

has the homotopy type of

and

3.

in r QP (p )

have exponents for

p

an odd prime

In this section we will show that the spaces

do not have an exponent for any prime

p.

This

will be obtained from some elementary considerations concerning Hopf algebras, and hence is a special case of a much more general result.

CONVENTIONS CONCERNING HOPF ALGEBRAS AND For the purposes of this section

H

H SPACES 3.1.

space will mean

space having a homotopy associative multiplication. is a field, Hopf algebra over algebra over

k

k

will mean connected Hopf

These have sometimes been called

homology Hopf algebras [7].

over

If

having a commutative diagonal or

comultiplication.

is a connected

k

H

H

With these conventions, if

space, then

H^(G;k)

G

is a Hopf algebra

k

DEFINITIONS AND OBSERVATIONS 3.2. Hopf algebra over diagonal of

k.

A, and

multiplication of

A

A (1)

A(n): A

Let

(n) * • ®nA -» A n

where and

(1)

integer.

Thus

A,

is the diagonal of

A(2)

Suppose that ®nA

A

is a

be the n-fold

the n-fold

is a strictly positive are both the identity of

A, and

(2)

is the

22

COHEN, MOORE, AND NEISENDORFER A.

multiplication of

Note

Hopf algebras, and that(n)

that A(n) is

is a morphism of

a morphism of coalgebras

but not usually of algebras unless the multiplication of A

is commutative. The n-th power map of p(n) = (n)A(n): A

of coalgebras A

has finite exponent if n.

coalgebras for some n

then

If n

is the exponent

PROPOSITION 3.3. 0(n): G

G

H^(0(n);k)

If G

its

p(n)

A.

is the morphism

The Hopf algebra

is the trivial morphism of is the

least such integer,

of A.

H

is a connected

n-th

A

power

space, and

map, then fork

a field

is the n-th power map of the Hopf algebra

H*(G;k).

The proposition above follows at once from the definitions and standard considerations.

COROLLARY 3.4.

If

exponent dividing

G

is a connected

H

space with

n, then the n-th power map of

H^(G;k)

is trivial.

PROPOSITION 3.5. i)

for

m

map

p(m)

k

If A

a strictlypositive integer,

k, then

the m-th power

is an isomorphism if the characteristic of

is zero or if

k, and

is a Hopf algebra over

m

is prime to the characteristic of

23

EXPONENTS IN HOMOTOPY THEORY 11)

m

for

and

n

strictly positive integers

p(mn) =

P ( m) p ( n ) •

Proof:

If

x

is a primitive element of

Thus since p(m)

A,

p(m)x = mx.

is a morphism of coalgebras the conditions

of part (i) imply it is injective.

However, since

A

is a

filtering colimit of Hopf algebras of finite type, then when

p(m)

is a monomorphism it is an isomorphism.

Now p(m)p(n) = (m)A(m)p(n).

Since

p(n)

is a Now

morphism of coalgebras ®mp(n) = ®m(n)®mA(n), ®mA(n)A(m) = A(mn), 1.

If

p

is cm odd prime, then

1) S2n+1{pr}

is atomic if

n > 1.

it) T2n+*{pr}

is atomic if

n > 2.

3 lit) T {p}

p is

is not atomic, but

3 r T {p }

is atomic if r > 1.

27

EXPONENTS IN HOMOTOPY THEORY The proof occupies the rest of this paper.

The least degree nonvanishing homotopy groups of s2n+l{prj.

T^n+*{pr}

are both

Z/prZ.

To show that

these spaces are atomic, it suffices to show that any self map inducing an isomorphism on these groups also induces an isomorphism of mod For algebra

n > 1 =

p

homology.

let

B

denote the differential Hopf

n

S(x,y) == the primitively generated symmetric

algebra on

x

and

y, with degree

2n+l, and differential

d

specified by

homology Bockstein differential H^(S^n+^{pr};Z/pZ)

x = 2n,

0

j*

degree y =

dy = x.

The r-th

makes

into a differential coalgebra which is

isomorphic to the underlying differential coalgebra of

B .

(This follows from the fact that the fibration sequence Qg^n+l

g2n+l^r^

zero mod

p.)

g2n+l

tota^ y n0nhomologous to

Since any self map of

S^n+*{pr}

induces an

endomorphism of this differential coalgebra, 4.1 (i) follows from:

LEMMA 4.2.

f: B

If

Since

-» B

n

is a morphism of differential

f(x) ^ 0, then

coalgebras and

Proof:

n

f

is a monomorphism.

f

Is an isomorphism.

is a self map it suffices to show that it For this it suffices to show that it is

28

COHEN, MOORE, AND NEISENDORFER

a monomorphism on primitives.

Since

has rank at most

one in each degree it suffices to show that on a basis of the primitives, i.e., that f |xP j

Assume that

f(y) ? 0

since

f jxP j / 0

for

Af |xP

diagonal. k .. xP V

f(x),

f(y).

Hence

f £zP

df(y) = f(dy) =f(x) j* i < k. Then

*yj = (f®f)A|xP ^yj

*yj ^ 0

and

f JxP

where

0.

*yj ^ 0 A is the

is a nonzero multiple of

r k ^ > k d[xP XyJ = xP ,

Since

multiple of

rk-i f |xP J

is a nonzero

xP .

Now we shall show that or

is nonzero

are al1 nonzero.

Clearly

since

f

T^n+*{pr}

is atomic if

n > 2

r > 1. Recall that

H^|oP^n+*(pr);Z/pzj

is the primitively

generated tensor algebra

T(u,v)

degree

r-th Bockstein differential

v = 2n,

specified by

and the

i* 0 v = u.

with degree

u = 2n-l,

Consider the elements

k t,

= adP

(v)(u)

and

p 1 ^,pk -

[ad5 ^vjfu),

Since the fibration sequence ^,2 n+l^r^

adP

£ 1 (v)(u)j.

DP(n,pr) -» QP^n+*(pr)

. , it follows from [ 1 ,

split by proposition 1 8

j u,

v,

map to a basis

u,

section 12] that the elements

k ,

t ^,

vP

with

j j > 0

and

k > 1

v,

k , t ^,

v

EXPONENTS IN HOMOTOPY THEORY of the primitives

|l^+ ^{pr}; Z / p z j . Observe that

of

these primitives have rank r P

Bockstein =0

T

be the natural map.

commutes with colimits, respect to

P

p

and

h^

H^(T;Z/pZ)

Suppose that the kernel

h^

but,

if

n = 1,

K

K

of

If

h^

n > 2,

then the image of

k /3rvP =1^*

^

is an

is nonzero.

If

is then so is under the

u ®

is acyclic with respect to

Since

g^

is injective.

reduced diagonal is contained in that

Since homology

Hence,

is the first nonvanishing degree, then contained in the primitives.

and

is acyclic with

is surjective.

isomorphism if and only if

gn

® u.

Note

p

P .

cannot be generated by

k vP .

30

COHEN, MOORE, AND NEISENDORFER

Suppose that

generates

acyclic, there is n > 2, then 1, let Then

to

to

in

image

to^

Since

such that

K

is

r P w =

pj

is primitive, which is impossible.

to be the image of to =

K^.

+ co^

where

to to^

in

If

If

n =

H^|oP^n+^(pr);Z/pzj.

has degree 3 in

of to^ is primitive.

.

u

Hence, to^ = 0

and the if

j >1 p

and P

to^ = At ^

r

for a scalar

r + P to^ with p

of

P to^

A

r p to^ = 0.

if

j = 1.

Suppose that primitive, it is

generates a multiple of

Since

h

by

the fibre.

dimensions

vP -

is in

a fibration T

Note that

F

K„ 1 . £+1

On +1

p

{p } -» T

3

short exact

r {p } = mod

Hurewicz map of Hence, for and the

mod p

classes

t

and

p F

tt^T,

and let

is (2-1) connected.

This follows from:

1, the

mod p

QS^n+^

k shows that

is also in it. k tlP

for the transgression Since

j ^ 0.

p^

only if

H ^ k_^(fl^S^n+* ;Z/pZ).

we must have

k J*

Proposition 14.5],

in

Hence, the same is true

is in the

r = 1. As in [1, mod p

n = k = 1.

On the other hand, suppose that generator

Hurewicz image

a

of

tt^ ( Q S

3

;Z/pZ)

in the image of an element

a'

Consider the fibration sequence

n = k = r = 1.

has order in

p

A

and hence is

3 7r2 p ( ® (p}:Z/pZ).

V^{p}'

T^{p} -> fiS^{p}

32 of

COHEN, MOORE, AND NEISENDORFER proposition 1.8.

a'1

in

the

mod p

If

ir2p(T3 {p};Z/pZ),

da' = 0 But

then

Hurewicz map.

in

is the image of an element

a'

a* ' maps to

vP

under

Hence, it suffices to show that

tt^ 1(V3 {p},;Z/pZ).

V^p}

has the homotopy type of

C(l) x

k ^k>l^P

^{P^}

C{ 1)

has the type of the 3-fold loop

space of the 3-connected cover of Z /p z j = 0 , Z/pZ).

3 S . Since

lies in the summand

da'

The composite

^/pZ) -*

H2 p_i(S2p_1{p2 } ;Z/pZ) -» H2p_1(T3 {p};Z/pZ) I^)»

da' = 0.

being injective f|

3 T {p}

We conclude the proof of 4.1 by showing that is not atomic. t 2p +1{p

This is done by exhibiting maps

} -» T3 {p}

composite Z/pZ).

ba

Since

and

b: T3{p} -* T ^ +^{p}

induces an automorphism of T ^ +^{p} X{p}

is atomic,

f

^gp-l^^

ir2 p - l ^ ^ >

(with image generated by

^p-l

ba

equivalence.

If

replacing

by a fibration, then since

a:

such that the HQ ( T ^ + ^{p};

is a homotopy

is the fibre of the result of 3

b

space, it follows that 3 T {p} x X{p}. Let

Hence,

3 T {p}

a''

is the composite

a'

a's, then

H

is not atomic.

a " : P2p(p) -» T3 {p}

extends to a map

is an

has the homotopy type of

3 T {p}

be as in the proof of 4.3.

T {p}

and

Since of

S: T2p+1{p} -» fiP2p+1(p) 3 T {p} is an H space,

fi2P^(p) = fiP^+ *(p).

a^v = v^.

If

a

33

EXPONENTS IN HOMOTOPY THEORY To construct; invariant

b,

recall that the p-th James-Hopf

h^: fiP^(p) -» fl2(P^(p)^)

® ... 0 v, p-times.

Since

satisfies

Pm (p) ^ En (p)

ha-s the homotopy

type of Pm+n(p) ^ Pm+n *(p), there is a map P^P (p)• Consider the composite Q2P^(p).

Since

obtain a map

T^n+*{p)

P^(p)^P

fiP^(p) -» Q2(P^(p) P )

-*

is a retract of

b: T^{p} -* T^P+*{p)

with

we

h^vP = v.

Remark: It is not hard to show that the space atomic.

=

X{p}

is

Its universal cover has the homotopy type of

REFERENCES [1]

F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsion in homotopy groups, Ann. of Math. 109 (1979), 121-168.

[2]

The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. 110 (1979), 549-565.

[3]

B. Gray, On the sphere of origin of infinite families in the homotopy groups of spheres, Topology 8 (1969), 219-232.

[4]

I. M. James, On the suspension sequence, Ann. of Math. 65 (1957), 74-107.

[5]

I. Kaplansky, Infinite Abelian Groups, Univ. of Michigan Press, Ann Arbor, 1971.

[6]

J. W. Milnor, On the construction FK, in Algebraic Topology — A Student’s Guide by J. F. Adams, Cambridge Univ. Press, 1972.

34

COHEN, MOORE, AND NEISENDORFER

[7]

J. C. Moore and L. Smith, Hopf algebras and multiplicative fibrations I, Amer. J. Math. 90 (1978), 752-780.

[8]

J. A. Neisendorfer, 3-primary exponents, Math. Proc. Camb. Phil. Soc. 90 (1981), 63-83.

[9]

J. A. Neisendorfer, Properties of certain H-spaces, Quart. J. Math. Oxford (2), 34 (1983), 201-209.

[10] J. A. Neisendorfer, The exponent of a Moore space, appearing in this volume. 3 [11] P. S. Selick, Odd primary torsion in

}> Topology

17 (1978), 407-412. [12] J-P. Serre, Groupes d ’homotopie et classes de groupes abeliens, Ann. of Math. 54 (1951), 425-505. 2 [13] H. Toda, On the double suspension E , J. Inst. Polytech. Osaka City Univ. Ser. A. 7 (1956), 103-145.

F . R . Cohen University of Kentucky Lexington, KY 40506

J. C. Moore Princeton University Fine Hall Princeton, NJ 08540

J. A. Neisendorfer University of Rochester Rochester, NY 14620

II THE EXPONENT OF A MOORE SPACE Joseph A. Neisendorfer^

INTRODUCTION Throughout this paper, let Pm (pr )

be the

mod

pr

p

be an odd prime and let

Moore space

Sm ^ U

em . The Pr

object of this paper is to prove the following theorem.

THEOREM 0.1. fi2pm (pr)

m > 3, then the double loop space

If

iias a nun

homo topic

pF+^ - st power map.

And hence*-

COROLLARY 0.2.

If

m > 3,

then

pr+1irx Jpm (pr )J = 0.

We note that Theorem 0.1 is best possible in three different ways. One,

m

must be

>3.

An elementary geometric

exercise shows that the universal cover of same homotopy type as a bouquet of

p

r

- 1

2 r P (p )

copies of

^This work was supported in part by an NSF grant.

35

has the 2 S .

36

NEISENDORFER

The Hi 1ton-Milnor theorem shows that this bouquet has no exponent for its homotopy groups, m r QP (p )

Two,

has no null homotopic power maps

whatsoever [4]. Three, of order

p

7r^|pm (pr)j r+ 1

contains infinitely many elements

[2,6].

The method for proving 0.1 is as follows.

We have

from [2] and [4] two product decompositions as below.

PROPOSITION 0.3.

(a)

0P2n+2(pr)

has the same homotopy

00

02n+lf r.

S

is the homotopy theoretic fibre of the degree

p

g2n+l

S

2yi+1

of

\P }

r

r p :

map

g2n+l (b)

T

{p } x

w „4n+2kn+3r rA V P (P ) k=0

02n+lf r,

, where

type as

QP2n+^(pr)

r r {p } x QP(n,p )

mod p

r

has the same homotopy type as where

Moore spaces and

r P(n,p ) T

2n+1

is a certain bouquet

r {p }

is a space to be

described in some detail later (c.f. section 8). From 0.3, the Hi 1ton-Milnor theorem, and P^(pr) = pk+^(pr) V pk+^ *(pr),

k r P (p ) A

it follows that

fiPm (pr)

has the same homotopy type as an infinite weak product of spaces

S2k+1{pr}

and

are repeated only finitely often.

£

and

T2^+ ^{pF}

where the indices Hence,

k

fi2Pm (pr)

has the same homotopy type as an infinite weak product of nS2k+1{pr}

and

QT2£+1{pr}

is multiplicative.

and the homotopy equivalence

THE EXPONENT OF A MOORE SPACE But we know that homotopic

S

2 k+ j

j* {p }

r p ~th power map [7].

37

is an H-space with a null Hence,

OS

2k+1 r {p }

has a

i*

null homo topic

p -th power map.

Theorem 0.1 by proving that p

r+l

§1.

OT

Thus, we can prove

2£+l

r {p }

has null homotopic

-st power map.

THE PROOF IN MINIATURE Let

pr : Sm -» sm

be a degree

pr

map.

Then, up to

homotopy, we have the commutative diagram below in which the rows and columns are fibration sequences.

s2n+l{pS} I

--- > S2n+1{pr+S}

=

» S2n+1{pr}

i

02n+l , s, S \P }

--- »

02n+l S

I pS 02n+l — — »S

i r+s I p

i r ip

^2n+1

We will use the shorthand notation product

00 IT S k=l

k

^

^

IT r

s^n+1

for the weak

p

{p }.

The top row of the above diagram

implies that, if we take products, we get fibration sequences up to homotopy

IT^ -» ^r+s

• We call these

the canonical fibration sequences. A standing convention for us will be that all spaces are localized at

p.

In section 8, we shall show that, up

38

NEISENDORFER

to homotopy, there is a commutative diagram in which the rows and columns are fibration sequences. nr

\

T2n+1

tt

c (n ) * nr+1

-- » T

r

r,

{p }

i

»

002n+l r r .

{p }

I =

s2n_1 x n r+ 1

—

I

» T2n+1{pr}

I

—

ns2n+1

»

I

n2 s2n+1

—

i

»

x —

0 p r

os2n+1

»

In this diagram, called the main diagram, the left column is the evident product of

with the fibration

2 sequence

s^n+* — -- >

C(n)

of the double

suspension. The main work of this paper is the construction of a sort of semi-splitting map.

PROPOSITION 1.1. such that, if

Namely,

There exists a map

t' E^+^

T^n+*{pr}

0: T 2 n+ 1 {pr} -* IT is the restriction of

the map in the main diagram, then the composition 21

has homotopy theoretic fibre

6^’ Er+^

E^.

Note that we do not state that this fibration sequence 2L -» IT . 1 r+1

IT r

is the canonical fibration sequence.

Note also that, since

E

r

is 2pn - 3

must have the following property. diagram gives maps composition with

C(n) 0

connected,

0

Restriction of the main

S^n * -» T^n+^{pr}.

is null homotopic.

The

39

the: exponent of a moore space

If Proposition 1.1 is granted, we can prove*.

PROPOSITION 1.2. homo topic

Proof:

The space

0T2 n+ 1 {pr}

has a null

pr+^ -- st power map.

Apply the loop functor to the top row of the main

diagram and to Proposition 1.1 to get the diagram below.

QC(n) x 021^

1 1 QC(n) X QCTr+ 1

2 n+lr r-L

-S-> QT

In the above diagram, sequence,

h

f n2 e2 n+lr r — » fi S \P }

{pr}

f

and

g

is the composition of

form a fibration Q0

and

g,

h

and

restrict to the loop of the fibration sequence of 1 .1 , and i

is the evident product map. The loop multiplication allows us to multiply maps.

We shall write this additively. identity map of

OT

2 n+ 1

r {p },

Thus, if

then

r p 1

1

is the

indicates the

p -th power map. We shall use the fact that loop space) has a null homotopic fact that

2 £+l

S

r {p }

(and hence any

pr-th power [7] and the

C(n) has a null homotopic p-th power.

latter is the main result of [3].

(See [6 ] for

The p = 3.)

i

40

NEISENDORFER Since f(prl) = prf(l) ^

nT2 n+ 1 {pr} -* QC(n)

and

tj:

prl ^ g(e,n)

where

a-

QT2 n+ 1 {p1'} ^ mir+1.

By the second paragraph after the statement of 1.1, he

^

*

and hence

* ^ pr |o0 (l)J = (Q0 )(prl) ^ (Q0 )g(e,Tj) =

hfe.r?) = he + hr? ^ hq.

Hence,

(&,77) ^ i(a,rj)

17:

where

0T2 n+ 1 {pr} -* U 1 . Thus, p

r+i

j» — — 1 = p(p 1) ^ pg(e,17) ^ pgi(e,rj) ^ gi (pe.pq)

This proves Theroem 0.1.

The rest of this paper is

devoted to the proof of Proposition 1.1 and to the construction of the fibration diagrams in the first and third paragraphs of this section.

§2.

pr

SUSPENSIONS OF FIBRES OF DEGREE We show that the suspension

homotopy type of a bouquet of

2 S

2£+l

mod p

MAPS r {p }

has the

Moore spaces.

An

important technical point is that this decomposition is compatible with the suspension of the fibration sequence 02 £+lf s.

S

02 £+lr r+s,

{p } -» S

{p

02 £+lf r,

}

S

The decomposition of

2 S

\P }

2 £+l

, r. , . defined m

r {p }

.. 1 section 1.

is based upon the

following well known lemma.

LEMMA 2.1.

There exists a natural homotopy equivalence 2(X x Y)

2X V 2Y V 2(X A Y) .

41

THE EXPONENT OF A MOORE SPACE Proof:

Since

XX V XY -* X(X x Y)

cofibration sequence split.

XX V XY

admits a retraction, the

X(X x Y) -* X(X A Y)

Hence, if we suspend the natural maps

X x Y -» Y,

and

structure of equivalence

X x Y

X A Y,

X(X x Y)

is

X x Y

X,

we can use the co-H-space

to add them and get a homotopy

2(X x Y)

2X V 2Y V 2(X A Y ) .

1

Iterating 2.1 gives*.

COROLLARY 2.2.

There is a natural injection

X(X^ A ... A X^)

2 (X^ x ... x X^)

onto a summand of a

bouquet decomposit ion.

PROPOSITION 2.3. v

There is a homotopy equivalence p2 «+2 k£+2 ^prj

2 S25+1{pr}.

k =0

Proof: Let compute

H^

denote

mod p

homology.

First, we

H ^ ( S ^ +*{pF} ) .

The first left translate of the fibration sequence 02 £+l f r-.

S

02 £+l

{p } -* S

p 02 £+l — — »S

. , . . 1 r.i . is the principal fibration

^ 2^+1 02 £+l f r-. 02 ^+l t OS -» S {p } -» S .In

sequence

a

the

mod p

homology Serre spectral sequence of the latter, Hence,

H ^ ( S ^ +^{pr})

H ^ ( S ^ +*)

as an

is isomorphic to

H^ffiS^**)

module.

E

2

00

= E .

H^(fiS^+ *) ®

Since

H ^ ( Q S ^ +*)

42

NEISENDORFER

is a polynomial algebra 2 £+l

S(u,v)

S(u)

where

u

has degree

22,

r {P })

is a free (graded) commutative algebra

as an

S(u)

Let

module where

denote the t-th

v

mod p

Hw (S2 ^+ 1 {pr}).

differential for

has degree

If

22+1.

Bockstein t < r,

0

then

»(r) ^ o(r) k k +1 P y Ju = 0 , P K Ju v = u

j and

Jjl. - S c2^->^OSnc2 ^ 1 u be a generator ofr

T + Let

such that the Hurewicz I . Let

image of preduces

. ^22+1,rA ^22+1 rr-. v- P (p ) -» S \P } r r {p };Z/p Z)

22+1

7r2 ^+ i(S

image of

v

is

For each

{rnc u S 2 ^ + L)

mod

p to

u.

, r be a generator of

such that the

mod p

r

Hurewicz

v.

k > 0, use the maps

^ 2^+1 \!S

p action of

tt^

J22+1 f re S \P }

on

and

jjl

v

and the

­ to form maps

J22 J22 r.22+1, r, J22+1, r. S x ... x S xP (p ) -> S {p} where 2 .2

there are

k factors of S

22

.Suspendthese

and use

to form the compositions •

gk

a a

I

c 2 ^

a

T>2 ^ + 1 r

r>l

*''

)J

,L 2 2 02 2 j.22+1, r j v 02 £+l, r. ;S (p } -» 2 S {p } . x . . . x S xP

»t

. Note that

C 2 A A ... AA S

2 S

ri22+2k.2+2, rA T P (p). Let

u^.

\TI22+2'k2+2r r. HjP (P } k A 2 u v

where

A ^ 0.

and

^

,

, ( r ) y k

Hence,

TD2 AA P

^+1 ( p r )]

=

, be generators of =

V

T h e n

Sk*Uk =

(rl

g k * V k

=

k 2 u v =

43

THE EXPONENT OF A MOORE SPACE Therefore, the maps g^

define a map

00

v/ ,,2£+2k£+2,(pr.} -» 2v S 02£+l, r. ,. , . , { p } whichinduces a

g: V P k =0

homology isomorphism. p, g

Since these spaces are localized at

is a homotopyequivalence.

Let

17: Pm (pr+S) -» Pm (pr)

the natural epimorphism let

||

be the map which induces

Z/pF+SZ -» ZpFZ

f : Pm (pS) -» Pm (pr+S)

natural monomorphism

,

mod p

on

and

be the map which induces the

Z/pSZ -» Z/pr+SZ.

g2 ^+l^r+Sj _j7_^ g2 ^+l^r^

;Z)

If

S ^ + *{pS} —

denotes the fibration sequence

defined in section 1 , then the maps

17,

17,

f,

and

f

are related by:

PROPOSITION 2.4.

If we suspend the homotopy equivalences

of 2.3, we get homotopy commutative diagrams .. Tt2£+2k£+2rr+Sx V 17 w T,2£+2k£+2/- r^ V P (P ) ---- 1-- > V P (p ) k =0 k =0

1 2S25 + 1 {pr+S} 00

1 ...S .V k r—

2S25 + 1 {pr}

> 00

w Ti2£+2k£+2,s. V p C w Ti2£+2'k£+2f r+s, V P (p ) — ~— *— > V P (P )

44

NEISENDORFER This is a consequence of the naturality condition

Proof:

in 2 . 2 and of the use of the principal action in the proof T

O

O

of 2.3.

t £*

•

If

rjoi2£+l

p: OS

0 2 £ + l

xS

f

t-»

0 2 £ + l

{p } -» S

,

t-.

principal action, then we have the formulas p(x,qy)

and

.

\P }

,

is the

rjp(x,y) ^

fp(x,y) ^ p jfi(pr)x,fyj . These formulas

follow from the vertical fibration sequences below and from the maps between them. s2 ^+ 1 {ps} ---£-- » S2 m

{pr+s} -- 2 -- » s2 ^+ 1 {pr}

1

g2 fi+l

ps

g2 «+l

pr

s2^+l

1

g2«+l

s2£+1

I s P

s 2e+l

K

The middle and right column and the maps between them are already in the defining diagram at the beginning of section 1.

For the first and middle column, the reader should do

exercise 8.5.

iff

Recall the canonical fibration sequences 27 r+s

— HU— >21 r

defined in Section 1.

II^

^ — »

If we let

n (pr)

be the least dimensional Moore space in the bouquet decomposition of

p k 2 S ^ n {pF}> then 2.1 and 2.3

give the

following technical result which will be used in the construction of the semi-splitting in section 5.

THE EXPONENT OF A MOORE SPACE COROLLARY 2.5. Z(n,pr )

2 U

and

There are bouquets of Z^(n,pr )

for

k > 0

mod p

45 j*

Moore spaces

such that:

(a)

Z(n,pr) = P2** n (pr) V Zj^n.p1")

(b)

there are homotopy equivalences

a

(c)

and there are maps

such that the

Z(nfpr) -+

T and

following diagrams are homotopy commutative.

?Vf

k

P2 p k n ( p s ) V Z j^ (n ,p S )

T?Ve:

P2 p n ( p r + S ) V Z j ^ n . p 1"*8 ) ------ ^

k p 2 p n ( p r ) V Z j ^ n . p 1-)

|a r+s

^

211

3.

21

>

|a r

^

r+s

)

A COFIBRATION SEQUENCE In the diagram below, we begin with the upper left

square of maps between spheres and expand it to the homotopy commutative diagram in which the rows and columns are cofibration sequences up to homotopy. Sm_1

1 s'* - 1

1

Pm{pr+s)

— 2^—

»

r+s

Sm_1

> Pm (pr)

| r+s jp

y

s"1-1

1

| r+s | p * pn,(p r )

—

.

1

- pr— , pm(pr+s) — :!— * pm(Pr ) v p ^ V )

21

r

46

NEISENDORFER

In this diagram, we require a suspension. Since if

p

then

is odd,

s > 0

pt

m > 3

so that every space is

indicates

pr+S: Pm (pr)

pt

times the identity.

Pm (pr)

is null homotopic

[5].

It follows that the right column is a cofibration sequence.

Hence, the bottom row, which is the subject of

this section, is a cofibration sequence also. A suspension if

k

2X

is said to have additive exponent

times the identity map of

2X

k

is null homotopic.

Since the suspension of a map is a co-H map, we have*

LEMMA 3.1. 2X

Let

f: Pm (pr+S) -» X r

p ,

has additive exponent f

where

be a map with then

2f

Pm+*(pr) V Pm+^(pr)

is a map

m > 3.

factors into 2X.

It is easy to check that:

LEMMA 3.2. The composition

✓

^

(project) Is the map Z/p

r+s

Z

.

rJftr

J-P(p

t+ s

^

„m, r .

rjn +l, r A

) -» P (p ) V P

^m, r ,

(p)^P(p)

rf which induces the natural epimorphism r Z/p Z

in integral homology.

™ 2 n+2 A r. 02 n+l f r. QP (p ) — S {p } x

t ■« In the splitting 00

q

v k =0

p^n+2kn+3^rj

^

pr0p 0 Siti0n 0 .3 (a), the map

If — fj

THE EXPONENT OF A MOORE SPACE

47

CO

~ w ~4n+2kn+3, r> ™2n+2, rx fi V P (p ) -* OP (p ) k=0

. , r is the loop of a map O r i+

T*

which we will indicate by

k

: Y(2n+2,p ) -» P

9

T*

(p ),

00

Y(2n+2,pr) =

LEMMA 3.3.

V P4n+2kn+3(pr). k=0

p2n+2(prj a(f)

and

k > 4,

If

homo topic to a sum, (3(f)

f' P^(pF+S)

then any map

f^a( f ) + P(f),

where

are compositions as indicated below:

a(f): Pk (Pr+S) - L Pk (pr) V Pk+1(pr) - S L m , P2n+2(pr) P(f): Pk (pr+S)

Proof:

> Y(2n+2,pr) - ^ P 2n+2(pr).

The homotopy theoretic fibre of

P2n+2(pr)

is

S^n+^{pr}.

QY(2n+2,p1 )

has a section Let off.

power

The related fibration sequence

QP2n+2(pr)

► S2n+1{pr}

f: P^ ^(pF+S) -* fiP2n+2(pr)

2n+1 S {p } —

map, pfp

r

i.e.,

pa ^ 1.

be the right adjoint

p f : P^*(pF+S)

-»S2n+^{pr}.

r is an H-space with null homotopic is null homotopic. Hence, T*

through the cof ibre of ^

r ' Y(2n+2,p )

a: S2n+*{pr} -» 0P2n+2(pr),

Consider the map

Since

k

p ,

—

i.e.,

—

-

pf

r»k-l, rA w —k, r> f' 02n+l f r, P *P (p ) V P (p ) — — > S \P } for some map Since popf ^ p f ,

f - crpf ^ (Qfc)g’

i': Pk_1(pr+S) -» nY(2n+2,pr).

factors

]/—1

pf = f 'j : P

r p -th

T *+ S

(p

)

7.

f .

for some map

48

NEISENDORFER Now,

a( f),

a'(f),

/3(£),

respective left adjoints of and

g '.

If

and

crpf,

P'(f)

f',

are the

(Qfc)g',

§§

f=a(f)+j3(f)

is the equation of right adjoints

which corresponds to 3.3, we note that

pf = pa(f)

and

v i w n * *.

§4.

SUSPENSIONS OF PIECES OF THE LOOPS ON A MOORE SPACE The object of this section is to prove Corollary 4.8.

We begin with:

PROPOSITION 4.1.

k

type of a bouquet

2T2 n+ 1 {pr}

The space VP a

cl

r (p }

Proof: Since

T^n+^{pr}

is a retract.

Since

p

k

where the

positive integers greater than

has the homotopy

homotopy type of a bouquet

run over some

2 n.

is a factor of is odd,

a

2 QP

^r r V P (p )

fiP^n+^(pr),

2 n+ 1

has the

where

P some positive integers greater than

r (p )

it

run over

P 2n

[4, Lemma 1.9].

The next lemma completes the proof of 4.1.

49

THE EXPONENT OF A MOORE SPACE LEMMA 4.2.

If

X

is a retract of V P

homotopy type of a bouquet

^r

Y

and

r

(p ),

Y

has the

then

X

has the

P k VP a

homotopy type of a bouquet

The mod

Proof'X

(r 1 Pv '

Let 1.

i:

X

va

=

k a

Y

Let v and a

u a

r: Y

generate

We can pick maps

ua *

Hence,

H^(X;Z/pZ)

denote the degree of

and

^ct*v ~(v •Then a J ) =* i^x a f

Y.

x

kct X* H (P (p );Z/pZ) «

—

f

:

P a (p T)

Y

f (u a)J= i^(P^r^x a).J a*v

p,

COROLLARY 4.3. exponent

Proof'

rf

X

such that

and

VP a

where such that

rf

is a

r (p )

cl

is a homotopy equivalence.

The space

22T 2 n+ 1 {pr}

mod

are §|

has additive

pr .

Proposition 4.2 implies that

homotopy type of m > 3,

Since

ri =

Adding& up^ the

k phomology isomorphism.

a.

X be maps such that

k cl r gives a map f: V P (p ) -> Y a

localized at

with

is acyclic and it must have a basis

fr) 6 V yx . Let a

x , a

r (p ).

homology Bochstein spectral sequence of

is a retract of that of

differential

P

p

cl

in

j*

P (p )

k +1 cl r V P (p ) a

2^T^n+^{pr}

as a co-H-space.

has additive exponent

j*

««

p . H

has the

But, if

50

NEISENDORFER Restricting the map

C(n) x Ur+^

T^n+^{pr}

main diagram (c.f. section 1 ) gives a map T^n+l{pr|

of the

t• *

By Corollary 2.5, there are homotopy

equivalences

aT+ \ : Z(n,p

r+i

)

217r+^

where

Z(n,p

r+1

)

is

p

a bouquet of

mod p

PROPOSITION 4.4. into

Moore spaces.

Hence, 3.1 yields:

(2^t)(2ar+^)

The composition

factors

Ij: 2Z(n,pr+1) -» 2Z(n,pr) V 22 Z(n,pr) -» 22 T2 n+ 1 {pr} .

PROPOSITION 4.5.

There is a bouquet

X

spaces and a map

f : X -» 2^T^n+*{pr}

such that adding

and

— i

gives a homotopy equivalence

of

mod pr

Moore f

r

g: X V 2Z(n,p ) V

JZrj, r. ^2 ^ 20 + 1 , r, 2 Z(n,p } -» 2 T {p }.

Proof: By the proof of [4, Lemma 1.4], it is sufficient to show that 8.3,

i

the map

induces a i

mod p

induces a

With differential equal to

homology monomorphism.

mod p P

fr)

,

By

homology monomorphism. the following lemma

proves 4.5.

LEMMA 4.6.

Let

complexes with

C — -— » C' — ^— » C' ' ji

monic.

isomorphic to the injection then

j

is an injection.

If C

bemaps of chain

d(C) = 0 and C

11

i

2C where

is d2x = x

51

THE EXPONENT OF A MOORE SPACE It is trivial.

Proof'-

We can desuspend 4.5 in the following weak sense.

COROLLARY 4.7.

Y

There exists a bouquet 2Y = X

Moore spaces such that

mod pF

of

and a homotopy equivalence

h: Y V Z(n,pr) V 2Z(n,pr) -» 2T2 n+ 1 {pr} such that

2h

For a bouquet of

Proof:

mod p

Hurewicz map is surjective.

oo

Z(n,p

Z(n,pr) =

V

VP

r (p )

)

with 9

Let

2T

n^ < n ^+ ^* n. {p1*} -> P X(pr)

1

be the composition of

projection on these summands of p^

h

—1

with the

r r Z(n,p ) V 2Z(n,p ).

Let

be the composition r^ir r+ 1 ^ P (p )

rjf r+1 ^ ar+l w Z(n,p ) ------ »

COROLLARY 4.8. of the map (b) homo topic.

(a)

If

i < j,

2l t +1

The composition

pr : P ^(pr+^)

r

T *+ l

(p

1=1

n.+l i

mod p

n

V P i=l

n. P 1 (pr).

homology map as

Moore spaces, the

i

) = 00

Then

t

J*

f|

r*+ 1

Write

mod p

induces the same

X^P^

*

^ 2 n+l r r^ {p

ts the cofibre

P *(pr+*)-

then the composition

XjP^

null

52

NEISENDORFER If

Proof: Pm (pr+*)

m < n, to

n

> 4,

and

Pn (pr),

then

f

and

g

are maps from

is homotopic to g

only if they induce the same

§5.

f

mod pr

if and

homology maps.

f§

CONSTRUCTING THE SEMI-SPLITTING 0: T2 n+*{pr} -» H^

We shall construct a map properties listed in 5.2 Let

H' r

denote

is a natural map

below.

the product

Hr

AT

with the

00 k H S2p n ^{pr}.There k=l

from the weak product to the

product.

LEMMA 5.1.

is a weak equivalence.

II^

In any fixed degree

Proof’ sum.

The map

£,

is a finite

direct

|

Since

T^n+^{pr} has the

complex, the homotopy

homotopy type of a

classes[T2 n+ 1 {pr },I7r]

bijectively to the homotopy classes 7r^: 17^

construct

S2p n {pr} 0,

CW

map

[T2 n+^{pr},2T].

denote the projections,

k > 1.

Let To

it is sufficient to construct the maps

0^.

= 7r. 0 . k Our notation is: cells in

S2p n ^{pt},

9 k _i P p n (p )

l

denotes the bottom two

denotes the inclusion

THE EXPONENT OF A MOORE SPACE

53

2 T^n + ^{pr} f

,

IT , r+1

ITi — ^— > IT . — -— » IT 1 r+1 r

and

denotes the inclusion

n ^{pr+^}

is the canonical

fibration sequence.

PROPOSITION 5.2.

For all

k

T

r r. 02p n-1 , T, {p } -» S ^ (p }

k > 1,

there exist maps

, x ^ such that 2

(1)

(11)

® i LL]z

restr^cts to

If

^V

nu^

(111)

0^.:

n

V: P

r+1 (P

) “*

homo topic.

null homotopic.

In the proof of 3.3, there are maps nP2£+2(pr)

> S2*+ V }

T*

Ok”

QY(2^+2,p ) ---- »

-> Y (2 £ + 2 ,p r ) - £ - > P2e+ 2 (p r )

where any two successive maps form a fibration sequence up to homotopy. oo+o

QP

There is also a section

r

(p ),

2p^n

r

(p )

r {p }

per ~ 1.

We set

2T? + 2 = 2p n

and use

0^: T^n+^{pr) -+

which has the properties listed in 5.3 below.

Then if we set

0^ =

The left adjoint of 2 kn r -» P n (p ).

2^+1

\r

i.e.,

this in the construction of a map QP

o o'"- S

2.5 proves 5.2. 0^

is denoted by

nr^*- 2T^n+^{pr}

The homotopy equivalence of 2.5 is denoted by

54

NEISENDORFER

PROPOSITION 5.3.

0. k

There exists

such that: r+ 2

2 k (I )

Tk ^ L^ar+l restr^cts to

2 kn

rj: P

(p

) -»

r (p ) on the first summand.

P

p+ 1

(II) Tk ^ L^ar+l restr^cts to a 2pkn

P p

r

(p )

on the second summand which factors through k

k .wo k r> ^2 p n, Tx - Y(2p n,p ) -» P H (p ).

Proof: Recall from section 4 that °° n i r+1 V P (p ) i=l

with

< n.. + 1. 1

ni P^(pr+^)

such that p2 p n^r+lj

Z(n,p Let

p+l

i^

) = be the integer

1

is the distinguished summand

shall use the maps

of 4 .8 in

andp^

\

our construction. We claim that, for 2T^n+*{pr}

k p^P n (pr)

Suppose that j

i^,

j < i.

then

i > 1,

j = i^, then

*s nu^

^p^

homo topic.

h: Y V Z(n,pr) V 2Z(n,pr ) -> 2T2 n+ 1 {pr} equivalence of 4.7 and that

^

r

P (p )

the decomposition of the domain with is not the summand ^k i+lk

nr.k, 1.h.

t0

^

nr^

:

with the following properties.

If

^k ipj

there exist maps

is

17.

Suppose that

is the homotopy

is any Moore space in £
0.

It is clear that

has the

required properties. If

i < i^,

f . (project)

let

y^ ^

be the composition

k k . ™ 2 n+lf r. _ ~2 p n, r. .. ^ p n+ 1 , r + K : 2T {p } -» P ^ (p ) V P K (p ) -»

x *0

o k p2p n (Pr). Let

i > i^

Tk i-1

and suppose that we have constructed required properties.

Let

comultiplication of a suspension and let be the map given by 3.3. composition

Set

y^ ^

v

be the

a' = a '(nrjc ^

equal to the

1 [-a'])(l V \^)v: 2T^n+*{pr} -»

(nr^

2T2 n+ 1 {pr} V 2T2 n+ 1 {pr} -» 2T2 n+ 1 {pr} V P ^ f p 1-) V p" 1 1 (pr) 2 p^n

P

r (p ).

"Y, . 1 k, l-l

-a'

is the map which is on the second and

f§

IDENTIFYING A FIBRE Let

0: T^n+^{pr}

section 5.

II

be the map constructed in

To prove 1.1, we must show that the composition

0 c:

^as homotopy theoretic fibre Let

ni

1 [-a'])

on the first summand and

third. §6 .

Here,

F

— * ffr+l

11^.

be the homotopy theoretic fibre of Ur

0t.

If

is the canonical fibration sequence,

56

NEISENDORFER

then 5.2(iii) implies that

f

factors through

F.

Hence,

we get the vertical maps of the horizontal fibration sequences below.

MI

Qrf

JL

17

MI

r+1

n .1 — 2 r+1

1 0c

JimiUnB Q27 r+1 r

We need to know that

IT

r+1

is a homotopy equivalence.

p

By the five lemma, it is sufficient to check that

is a

a

homotopy equivalence. The

mod p

product of the

homology of mod p

027^

is the infinite tensor

homologies of

The fibration sequence

OS

.

is an

commutative.

k > 1.

0 ^ S ^ +* -+ Q S ^ + ^{pt} -+ QS"^+ *

is clearly totally nonhomologous to zero 027?+l r t. S {p }

k 2pn X t {p },

T T r-ry H-space [7],

■ »

t {p }

OS

It follows easily that

Hx (fi2 S2 ^+ 1 ;Z/pZ) ® HM (fiS25 + 1 ;Z/pZ)

mod p.

H^(QS

Since

.

. is homotopy

2 £+l

t {p };Z/pZ) =

as a Hopf algebra.

The

first factor is the free commutative algebra

. 2£p1- l ’ 2 £pJ-2 .1 * 0 , j> 0

and the second factor is the degrees.

S(c^).

The subscripts indicate

The generators are all primitive and we have

the following formulas for Bocksteins and Steenrod operations [i]=

THE EXPONENT OF A MOORE SPACE

22

p

a

22-1

"

.

= b

2£px-l

’" ' W - i

.

2£p1-2

,

i > 1

= ?*hzep-z = P-c2 « = °

P^b i 0 = -fb ... ] , x 2 fip - 2 I 25 p i-l_2J

Notice that 5.2(i) implies that a2£-l

k ^ = P n“l*

^°r

hold when a 2 ^_i•

QU

57

k > 1.

replaces

i >

|fl(0 i)j) 1, then the

commutative diagram 22+1

p

-22+1

'

r- 1

-

22+1

-22+1

yields the map of fibration sequences no2 fi+l, rx OS {p }

02 S25+1

_ nc2fi+1 -» OS Op

no2^+ 1{p} r i OS

02 S2* +1

Hence, it is impossible if Thus, §7.

and

a^

r- 1

2 2+1

- ,

-» OS

r > 1.

are isomorphisms.

f§

APPENDIX ON CUBICAL DIAGRAMS OF FIBRATION SEQUENCES In order to avoid drawing high dimensional diagrams,

we introduce some convenient language. A partially ordered set can be regarded as a category in which the objects are the elements of the set. and

b

are elements with

regarded as a map from

a

If

a

a > b, then this relation is to

b.

60

NEISENDORFER If

C

and

D

are two partially ordered sets, then so

is the cartesian product and only if

c > c'

C

and

x D

where

(c,d) > (c',d')

if

d > d'.

We are especially interested in the partially ordered [£] = {0 ,1 ,...,2-1,2}

sets powers

and their n-th cartesian

[£]n = [£] x ... x [£].

DEFINITION 7.1.

Ann-dimensional cubewith side length F

is a covariant functor

[£]n

from the category

2

to the

category of pointed topological spaces and continuous maps.

Cubes have sections which are also cubes. defined as follows. j , < n n“k '

and

Given

n - k constants

k-dimensional section of of

F

(x. d. . Jn-k

F

indices

in m

n

1
(b.,...,b ), v 1 nJ ~ v 1 n'

F(a1,...,a ) > F(b1,...,b ). v 1 n' “ v 1 nJ sections of

F.We say that

G

Let

w

of

G

G

and

dominates

and if, for all noninitial vertices noninitial vertex

then we write

with

v

H H

of

w > v.

be two if

H,

iG > iH there is a

62

NEISENDORFER If

G

dominates H, then there is a commutative

diagram of maps iG ----- > iH

i

i >m

m and hence a map

fG

Suppose that

fH.

F

is a totally fibred n-dimensional 1.

cube with side length of

F

We now define an extension

to an n-dimensional cube with side length

Given any point

(a^.... a^)

in

[2]n ,

F

2.

let

d. ,...,d. J1 Jn-k

denote the coordinates which are either 0 or

1.

s(a^

F

Denote by

a^)

the k-dimensional section of

defined by restriction of

[l]n

with objects

F

(x.,...,x )

to the full subcategory of where

1

x. H

= d . ,...,x. H Jn-k

= d. Jn-k If

(a1

dominates

an ) > (bj...... bn ), then

s(b^,...,b ).

s(ax.... aR )

Hence, there is a map

fs(a1f...,a ) -» fs(b1,...,b ). v 1 n' v 1 n' If

(a^.... a )

s(a.,...,a ) v 1 nJ

is a point in

is the vertex

Hence, an extension

F

an ) = fs(a1

aR )

F(a 1 The cube

F

[1 ]n , then

Ffa.,...,a ). v 1 n' of

F

for

is defined by (aj

aR )

is called the fibre extension of

In what follows, we also write

in

[2 ]n .

F.

sF(a1,...,a ) v 1 n

for

63

THE EXPONENT OF A MOORE SPACE LEMMA 7.3.

A section of a fibre extension is a fibre

extension.

Proof:

Let

F

be the fibre extension of the totally

fibred cube

F.

then let

be the restriction of

H

If

H

enough to show that

is a k-dimensional section of

iH -> 2E

~ H

to

k [1] .

F,

It is

is a fibration and that

fH =

iH. Since

F

is a fibre extension,

isiH = isiH, Let

L

there is a map

isiH

and is parallel to

noninitial vertices of

Note that

L.

H,

isv

H.

siH

Since

siH As

which v

runs

runs over the

There are maps

£H

be the dimension of the section

v siH

isv. of

is the (m-k)-dimensional section of

which passes through w

fsiH.

iH -» isiH.

over the noninitial vertices of

m

is

be the k-dimensional section of

passes through

Let

iH

isiH

and is perpendicular to

is a noninitial vertex of

siH, define

£H -» w

F. siH H.

If

to be

the trivial map. The vertices

isv

and

the noninitial vertices of paragraphs define a map

£H

w

above are cofinal among all

siH.

The two preceding

£siH.

commutative diagram iH ----- » isiH

i

i_

m

----- > «siH

This gives a

64

NEISENDORFER Since this is a cartesian square (pullback), the

result follows.

LEMMA 7.4.

ff§

Up to homotopy equivalence, any n-dimensional 1

cube of side length

is equivalent to a totally fibred

cube.

Proof: Let If (a^

F a )

be an n-dimensional cube of side length is a point in [1]n ,

be the section of

F

then let

1.

S(a^,...,an )

defined by restriction to those

(xi ,...,x ) < (a. ,...,a ) . Note that y I nJ v 1 nJ

iS(a. v 1

a ) = nJ

F(a1 ,...,a ). Recall that any map = gi • 'X

Y

where

g

f: X

Y

can be factored into

is a fibration and

cofibration and a homotopy equivalence. by

g

in a diagram, then we say that

i If

f

f

is a f

is replaced

is replaced by a

f ibration. Linearly order the

(a^a^)

in a manner

compatible with the original partial ordering new order, replace each

< .

In this

F(a^,...,an ) -» £S(a^.... a^)

by a

fibration. We must show that, if S(a 1 ,...,a )with

v 1

-»

nJ

is a section of

iH = F(a. ,.

is a fibration.

F(a^,...,an )

H

v1

..,a ) ,then

nJ

Ffa,. ,. ..,a )

v 1

There is a factorization

£S(a^,...,an )

£E.

If

K

is another

n'

65

THE EXPONENT OF A MOORE SPACE section of K

S(a.,...,a ) v 1 n'

is contained in

£S(a^,...,an ) that, if

K

H,

with

iK = F(a.,...,a ) 1 n'

then there is a factorization

iE -» SK.

Hence, it is sufficient to show

is a codimension 1 section of

F(a^, ... .a^),

then

and if

£H -» iK

H

is a fibration.

with

iK =

By

induction, we may suppose that this is done for all vertices which are Let

< (a.,...,a ). v 1 n'

J be the dimension 1 section of

perpendicular to

K

be the codimension and has

Since

§8 .

il = £J„

il

£1

H

which

F(a^..... a^).

and hasij = 1section ofH

is

Let

whichis parallel

I to

K

There is a cartesian square

is

m

-------» il

m

» ei

a fibration,

sois

£H

£K.

f|

APPENDIX ON CERTAIN FIBRATION SEQUENCES Recall the pinch map

the bottom cell of

P^n+*(pr)

P^n+^(pr)

■□2 n+l, r> 02 n+l pr 02 n+l P (p ) -» S — - »S up to homotopy.

s^n+*

to a point.

which pinches

The sequence

. is a cofibration sequence

Hence the diagram

p2 n+l(pr) ----- > g2 n+l

66

NEISENDORFER

is homotopy commutative. fibration

Replace

q: s^n+* -* g^n+l

pr : g^n+l

g^n+l

Then use the homotopy lifting

property to make the diagram strictly commutative. n = 2

7.2 with

by a

Apply

to get a homotopy commutative diagram in

which the rows and columns are fibration sequences up to homotopy.

^ n + l , r,

E

^ n + l , r,

{p }

>P

>S

r£n+l,(prA}

^2n+lf r-»

F

v 02n+l f r,

(p )

{p }

%

>P

»

{p }

02n+l

S

K ns;2 n+ l

»

*

>

g2 n+l

Applying the loop functor to this diagram yields another homotopy commutative diagram in which the rows and columns are fibration sequences up to homotopy. In the introduction, we mentioned a bouquet of

mod p

r

Moore spaces.

QE2n+l{pr} j [3,6]. classifying map of

Let

r

fiP(n,p ) -» QP p2n+l^r^

Qpij. I

2n+l

QP(n,p ) -> QE

compositions yield loop maps T»

t*

(p )

r QP(n,p )

There is a loop map P(n,pr) -» E2 n+ 1 {p1'} r

{p }.

r

QP(n,p ) -> QF

and classifying maps

P(n,pr) ->P^n+^(pr).

r P(n,p )

be the

The obvious 2 n +1

r {p }

and

t*

P(n,p )

Thus, we get the

homotopy commutative diagram below in which the rows are fibration sequences up to homotopy.

67

THE EXPONENT OF A MOORE SPACE

n p ( n ,p r ) ----------> nE2n+1{pr } --------- » V2n + 1{pr } ----------> P ( n ,p r )

1=

i

I

1

> E2n+

-

OP(n,pr) ----- >nF2n+1 {pr} ----- ► W2n+1 {pr} -----► P(n.pr) 1=

1

1

f i P ( n . p r ) ----------- ►0P2 n + 1 ( p r )

PROPOSITION 8.1.

> F2n+

I'

> T 2 n + 1{p r }

1

»P ( n , p r )

*p2n+

The above fibration sequences up to

homotopy with respective bases T^n+^{pr}

1

V^n+*{pr},

J^n+^{pr}9

and

are split, i.e., there are compatible homotopy

equivalences nE2n+1{pr} ----- ^ V2n+l{pr} x np(n ,pr ) QF2 n+ 1 {pr} ----- » l 2 n+ 1 {pr} x nP(n,pr) np2 n+l(pr) Proof:

-------------------->

T2 n+ 1 {pr}

X

f2P(n.pr). r OP(n,p )

In [4, section 1], it is shown that

retract of

Since

OP^n+*(pr).

S^n+*{pr}

This is clearly sufficient.

is the fibre of

q,

is a §|

there is a

commutative diagram

P2n+1(pr )

» S2n+1 N s s2 n+ 1 {pr}

^

lq ,2 n+l

Replace

P^n+^{pT) -4S^n+^{pr}

~ 2 n+l, rx 4-1 r P(n,p ) -» E {p }.

strictly commutative diagram

'. q

. is

~ 2 n+l r r^ E {p }

Hence, there is a

68

NEISENDORFER

Apply the case

n = 3

of 7.2 to the above cube.

One of

the resulting faces yields:

PROPOSITION 8.2.

There exists the homotopy commutative

diagram below in which the rows and columns are fibration sequences up to homotopy.

v2n+1{pr > ------- »T2n+1{pr } ------- » ns2 n+V }

I

=

i

1

W 2 n+l{pr} ----- „ x2 n+ l{pr} ----- , ^ n + l

I

I

n2 g2 n+l

Remark 8.3. 0 p2 n+l(prj

p,

----- >

^

i , ns2n+1

Since the fibration sequence ns2n+1

QF2n+^{pr} -»

is totally nonhomologous to zero

mod

the same is true for the middle row above. If we take another face and extend it by first left

translates of fibration sequences, then we get the homotopy commutative diagram below in which the rows and columns are again fibration sequences up to homotopy.

THE EXPONENT OF A MOORE SPACE E2 n+ 1 {pr}

» V2 n+ 1 {pr}

I

I

1+ If

ri

!

»W

! Tif

r^

-p2n+1

{p } ----- > P(n,p )

i

1 n2g2n+l

» E 2 n+ 1 {pr}

» P(n,pr)

T„2n+1 r

{p }

69

»F

1

— ---> n2 S2n+1

>

1 ----- > 0S2n+1

x

V2 n+^{p1'} -» W2 n+^{pr} ,

In order to describe the map

we recall from [3] and [6 ] a description of the left-hand column. Let

C(n)

be the homotopy theoretic fibre of the

2

1 : S'2n 1

double suspension ^p2 n+l^r^

-> n2s 2n+1.

representsa generator of

if

s2n 1

7T2 n_^(QF^n+*{pr}),

then this map fits into the homotopy commutative diagram below which is a map of fibration sequences up to homotopy. » nE2 n+ 1 {pr}

C(n)

I

1 02 n-l

S

nr2 n+lr r^ ----- » QF {p }

!

1

n2s 2n+i —=— » n2s 2n+1 In [3] and [6 ], we showed the existence of a certain 00

map

^E^n+^{pr}

where

By composition, we get a map We have maps composition, a map

^r+-^

fiF2 n+ 1 {pr}.

0P(n,pr) -» QE2 n+^{pr} r

QP(n,p )

k

IIr+^ = 21 S ^ n ^{pr+*}. k=l

QF

2n+ 1 r

{p }.

and, by

70

NEISENDORFER In the obvious way, form the fibration sequence up to

homotopy 2 2 n +1

-+ Q S

T* Qri—1 t* C(n) x ffr+j x 0P(n,p ) -+ S x 2Ir+^ x 0P(n,p ) . Multiply maps to get the map below of fibration

sequences up to homotopy. X 0P(n,pr) ----- » QE2 n+ 1 {pr}

C(n) X n

'i 02 n+l

S

I or,2 n+lf r. »QF {p }

r. x QP(n,p )

x IT

ri

i > Q2 S2n+1

n2g2n+l Then [3] and [6 ]

assert that the horizontal maps in

the diagram above are homotopy equivalences localized at p.

Restricting to the first two factors in the left column

and recalling 8 . 1 gives:

PROPOSITION 8.4.

In the homotopy commutative diagram

below, the horizontal maps are homotopy equivalences localized at

p. C(n) x 2Ir+ 1

» V2 n+ 1 {pr}

I ^

i

1 * ffr+l ----- " w2 n+ 1 {pr}

I n2s 2n+i

EXERCISE 8.5.

i —=— „ n2s 2n+1

Apply 7.2 to the cube below to get two

. , eqmvalent definitions of the map in section 1 .

02 n+lr s.

S

02 n+lr r+s.

{p } -> S

(p

}

THE EXPONENT OF A MOORE SPACE

Remarh 8 .6 . fibration

71

Putting 8.4 together with 8.2 gives the second

diagram of section 1 .

REFERENCES [1]

F. R. Cohen, T. J. Lada, and J. P. May, The Homology of Iterated Loop Spaces, Springer-Verlag, 1976.

[2]

F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsiox in homotopy groups, Ann. of Math. 109 (1979), 121-168. F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups oi spheres, Ann. of Math. 110 (1969), 549-565.

[3]

[4]

F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Exponents in homotopy theory, appearing in this volume.

[5]

J. A. Neisendorfer, Primary Homotopy Theory, Mem. A.M.S., No. 232, 1980.

[6 ]

J. A. Neisendorfer, 3-primary exponents, Math. Proc. Caml Phil. Soc. 90 (1981), 63-83.

[7]

J. A. Neisendorfer, Properties of certain H-spaces, Quarl J. Math, Oxford (2), 34 (1983), 201-209.

[8 ] P. S. Selick, A reformulation of the Arf invariant one me p problem and applications to atomic spaces, Pacific Joui Math, 108 (1983), 431-450.

J. A. Neisendorfer University of Rochester Rochester, NY 14620

Ill

THE SPACE OF MAPS OF MOORE SPACES INTO SPHERES H. E. A. Campbell, F. R. Cohen, F. P. Peterson, and P. S. Selick^

§INTRODUCTION Let

Map^(X,Y)

denote the function space of

continuous based maps from and

X

X

to

Y.

When

Y

is a sphere

a finite complex, the homological properties of

this space were studied by J. C. Moore [7] who calculated H^|Map^(X,Sn )J S

s—1

s e .

U

for a range of dimensions.

Let

PS(2r) =

In this paper we study the homological and

2r homotopical properties of

Is r xiI Map^ P (2 ),S .

Our main interest is in determining whether or not

[Ps(2r ),Sn I when

s

is small.

Map^ |p^(2 ),S^j where

can decompose as a product of other spaces

3 S

F. R. Cohen [4] has shown that

is homotopy equivalent to is the 3-connected cover of

the fibre of the double suspension

Q^S^ x W^, S

3

E^: S^n ^

and

W

n

QS^n+^.

is F.

R. Cohen and P. S. Selick [5] have similarly decomposed Mapx [p4 (2),S9].

^The authors were partially supported by the National Science Foundation and the Natural Sciences and Engineering Research Council of Canada.

72

SPACE OF MAPS OF MOORE SPACES INTO SPHERES o 7 ic OS ^ S xfiS

Clearly, the splittings S^xOS^ 4 or 8 .

give decompositions

4

and

for Map^ |p^(2r) ,Snj

OS

^

for

n =

Our main theorem 3.2 shows that M a p ^ P (2),Snj n ^ 4, 5, 6 , 8 , 9, 16, or 17

does not decompose if n > 4.

73

For these values of

theorem fails.

If

when

n, the proof of our main

n = 4, 5, 6 , 8 , and 9, the failure of

our proof can be traced to the existence of elements of s ir^

Hopf invariant one in n = 16 and 17

is due

Arf invariant one in

and

s

7T^.

The failure for

to the existence of an element such that

i

0 of

rj6 = 0.

In order to prove this theorem, we need to know

(Map^(P s(2r),Sxi) I

as a module over the Steenrod algebra

for small values of

F. R. Cohen and L. R. Taylor [6 ]

s.

have shown how to calculate

H^|Map^(PS(2r),Sn )j

dimensions as an algebra if

s+1 < n,

but their results do

not give the Steenrod

operations.

give the structure of

H^ |Map^(PS(2r),Sn )j

over the Steenrod algebra

in all

Our theorems 2.1 and2.2

in many cases.

as a module Unfortunately,

we have not been able to use these methods to determine the structure of

H^|Map^(X,Sn )j

algebra for more general

§2.

as a module over the Steenrod

X.

HOMOLOGICAL RESULTS Using the cofibration,

the fibration

Ss *

PS(2r) -» Ss ,

we obtain

74

CAMPBELL, COHEN, PETERSON AND SELICK

(*)

Q SSn - i* Mapx [pS(2r ),Sn] -JU Op_1Sn .

Alternatively,

the cofibration

SS

--- » SS

» PS (2r )

Map^ |pS (2r ) ,Snj — > fiS *Sn

gives the fibration

(In the notation of [4],

MaP)( (^PS(2r),Snj = (nS_1Sn){2r}.)

We will study the fibration (*). Let

g: Sn

the fibration

QSn (*}

be stabilization.

Applying

we get the fibration

g

to

g(*)

n sQSn = Q(Sn_s) -> Map^ jjPs(2r } ,QSnj -> fiS_1QSn = QSn_S+1. Cohen and Taylor [6] show that if spectral sequence in

mod 2

s+1 < n, the Serre

homology for

(*)

collapses

by comparing it to the Serre spectral sequence for Furthermore,

they prove that

|Map^(PS (2r ),Sn )j

g(* ) . is

generated as an algebra by elements in the image of the homology suspension provided and cohomology groups have

5 < s+2 < n. Z/2Z

(All homology

coefficients unless

otherwise indicated.) We now give names to elements in Let

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

i . < i .,. . .1 " .1+1 H

Let

x

n-s

Map (Ps (2r),Sn ) n-s I v J

I s r n I H lMap^(P (2 ),S ) .

be an admissible sequence,

that is,

denote the generator of & = H

n-s

U SSn

and

x

J.1 n-s+1

denote the

generator of Hn_s+1 [topw (PS(2r),Sn )] = Hn_s+1(nS_1Sn ) . Q t (x ) = i^QT(x ), Iv n-sy n-s' denote an element of

with

i. < s-1. k ”

|Map^(PS (2r),Sn )j If

8+2 ^ n>

Let

Let

Q Tx I n-s+1

such that then

¥ n - s +l

75

SPACE OF MAPS OF MOORE SPACES INTO SPHERES

may be chosen to be in the image of the homology suspension and thus may be chosen to be primitive by [6 ]. s+1 ,

then it may not be possible to choose

If

n =

Qjxn_s+i

to

be primitive. Map^|pS(2r),S n j

Since we choose

is an

Q T(x (1) = Q T(x ,.) Iv n-s+ly Iv n-s+1 '

Similarly,

if

we choose i

j

f

t

s

i, < s-3. k “ with

i.
3.

(2

),S

)J

as a

However, it only

partially determines the Steenrod algebra structure, namely squaring operations can be computed on elements which are actual Dyer-Lashof operations of other elements using the Nishida formulae.

We will need more complete information

and nearly complete information is provided by the theorems below.

By our choices above, we need only compute

Sc$ ( s - 2 .... s-2)(xn-s+ l)' we denote

by

Sq^ s - 2 (xn-s+l)

Qg_2 *

If

1 = (s"2 .... S"2 ) ’ a times>

^ne wa^

ls to comPute

result does this in most cases.

computing

SxQ 3 _2 (xn_s+1).

Our first

(We assume throughout that

n > s+2.)

THEOREM 2.1.

(I)

- Q? * « T and

n £ 3(4) ,

r' 1

SPACE OF MAPS OF MOORE SPACES INTO SPHERES

(tt>

g*«s-2 + others

r = 1

if

"others” = 0

if

and

n £ 3(4) with

a = 1 , and

S*Qs-2 {xn-s+ l> = ‘S - A - s + l *

(“ O

77

^

r > 1

f°r ®*«

n.

If

n = 3(4),

then

Sq^Q^(x 0 ) « l n-z

is computed in some

cases.

THEOREM 2.2.

Let

n = -l(2l)

with

s < 2t

and

r = 1.

Then (O

Sq^Q^(x ) ** 1 n— s

can be computed by assuming that

g^ l (xn-S) = Q l K - s ^ (ii)

SqiQ ) ** s o(x z n-sj.1 +1

tf

t 1 2 ' ^

cari ke computed by assuming that

gA - 2 < xn-s+l> = Qs-2( V s+P-

Clearly

is a monomorphism, as we have remarked.

However, our proof of theorem 2.1 gives the following corollary.

COROLLARY 2.3.

If

r > 1

and

n = 0(2), then

S*: H*[Map*(P3 (2 r ),S n );Z { 2 )] - H* [Map*(P3 (2 r ) ,QSn ) ;Z( 2 ) ] is not a monomorphism.

78

CAMPBELL, COHEN, PETERSON AND SELICK

§3.

ATOMICITY RESULTS We first recall some notions from [2].

DEFINITION. at

p

1f

f: Y -» Y

(r-1 )-connected space

An

H^fY.-Z/pZ) = Z/pZ

isomorphism, then

and given any self-map

f^:Hr(Y;Z/pZ)

such that

is called atomic

Y

Hr(Y;Z/pZ) is

f : H^(Y;Z/pZ) -^H^(Y;Z/pZ)

an

is an

isomorphism. DEFINITION.

An

H-atomic at

p

E-maps

(r-1)-connected H-space

Y

is called

if the above condition is true for all

f.

We note the following relationship between these two notions.

PROPOSITION 3.1.

If

1-connected, then

Y

QY is H-atomic at isatomic at

p

and Y

is

p.

We are interested in spaces of the form Map^ PS(2r),Sn|

and we note that

Map^ (PS+^(2r ),Snj .

0 Map^ |pS(2r),Sn] =

We will state our theorems in the form

that a certain space is H-atomic at reader to apply proposition 3.1.

2

and leave it to the

79

SPACE OF MAPS OF MOORE SPACES INTO SPHERES THEOREM 3.2. 2

n > 6 . Map^ |p4 (2) ,Snj

Let

is H-atomic at

n / 6 , 8 , 9, 16, or 17.

if

THEOREM 3.3. H-atomic at

If

that

2

if

f : Y -» Y

f^: H^(Y)

n > 6

Let

HJY)

n

r > 1.

and

Map^|p4 (2P),Snj

is

2^ .

i s a self-H-map, then to show is an isomorphism, it is enough to show

f^: QH^(Y) -> QH^(Y)

is an isomorphism, where

denotes the module of indecomposables.

QHX (Y)

In order to prove

theorems 3.2 and 3.3, there are various cases to consider depending on

n.

The following theorem gives a result

which can be applied to the various cases.

THEOREM 3.4. n > 5, on

H. J

Let

f

be a self-map of

which is an H-map. for

j < 2n-4, then

If f *

f

Map^jp4 (2r),Snj ,

induces an isomorphism

is an isomorphism.

The following theorem will be used in [4].

THEOREM 3.5.

Let

f : X -► Mapx |p3 (2),S2n+1j

be a map which

induces an isomorphism on the module of primitives in dimensions homology of

2n-2 X

and

4n-3

for

n > 2.

is isomorphic to that of

Map^|h*(2},S^n

as a coalgebra over the Steenrod algebra, then isomorphism.

mod-2

If the

f

is an

80

CAMPBELL, COHEN, PETERSON AND SELICK Let

Wn

denote the homotopy theoretic fibre of the

, ,, . double suspension

THEOREM 3.6.

If

,-,2 . 02 n-l E : S

0202 n + 1

12 S

n > 1,

Is atomic

(at 2).

Theorems 3.2 to 3.5 will be proven in sections 5 and 6.

Theorem 3.6 will be proven in section 7.

§4.

PROOFS OF THE HOMOLOGICAL RESULTS We first prove the formula in theorem 2.1(iii) which

states that

= Qs-2 (yn-s+l)

g ^ ^ V s * ^

Here notice that in case ^s-2^n-s+l^ + z

r = 1,

If

r > 1

as given in section 2.

and

trivial map because

s+2 < n,

But then slnCe

then

H^Mapx (SS+*,QSn )

Thus

p^z = 0

S*P* =

is t^ie is a primitively

generated Hopf algebra and the algebra map annihilates primitives.

T > 1 '

g^Qa 0 (x .1 ) = s-2 v n-s+ 1 J

g*P*Qs_2 (xn-s+1) = p X - 2 {yn-s+ l} + P*Z Pxg*.

if

j i r- 1 **

and theorem

2 .1 (iii) follows.

Next observe that where

A

g*Qg_2 (*n_s+1) = ^ ^ n - s + l 5 + A

is in the image of

linear combination of elements 0 < i-1 _ < ... < i,k “ = Qt(yn-2 ) + < ? ' V yn-3> s= 1.

is

where

o

if n *

3

is the homology

3114

r

81

SPACE OF MAPS OF MOORE SPACES INTO SPHERES

g*Qg_2 (xn-s+i) = Qs-2 ^yn - s + P

suspension, It follows that a- 1 + Q fy ) + others. s- 2 sw n-s'

Furthermore the degree of

is greater than that of

Qs-2^n-s+l^

’"others” = 0

and theorem 2.3(ii) follows.

if

In case

a = 1

s = 3,

we have

by degree considerations. a

a

= 1

if

n £ 3(4)

a

= 1

for

a > 1:

The only elements in

and

r = 1.

Apply

= 1 Sq* “

H^Map^

Since if

(*) to obtain

p 3(2).s "]

„ith

Q0Q“ ’(y„_2)

^(xn-2 ^ + e^2 ^Xn-3 ^ *

(For a check of degrees here, see lemma 5.1(ii).)

Sq* 1.

and then deduce

to

as a term in its g^-image are e = 0,1.

3^

Furthermore, we claim that

Assume the result that a

n > s - Hence

S * ^ Xn-2 > = Qt(yn-2 ) + “ a ^ V

(* )

4 y n-s

Thus

it follows that

= ^o^l" ^ Xn-2^

if

e = 0

a > 1

and

a = a. . a 1 We now compute case

n = 0(2).

a^

for

Note that

torsion and hence

s = 3.

We first consider the

tt^|jlap^(P^(2r),Sn )j

Map^(P^(2r),Sn );zj

is all

is all torsion.

Thus the Bockstein spectral sequence for H*[Mapx (P3 (2 r),Sn )J + ^

3 (7 ^ 3 )*

is zero at

e“ .

The Bocksteins of interest are

Sqi(xn-2 * Xn-3] =

’ SqiQl(xn-3} = Q0(xn-3)'

Pi V Xn-2> = Xn-2 * Xn-3 + V

xn-3>

for SOme hlgher

Pi

82 and

CAMPBELL, COHEN, PETERSON AND SELICK

p.Q^Cx 0) = Q0(x 0) for some j lv

SqiQ3 ^ n - 3 ] = Q2(xn-3} and

and only if

j.

2 V n- 3 '

n - 2 J

Because

SqiQl(xn-2} = °'

a1 = 1 because

g

Pj = Sql

is a monomorphism.

lf

(Note

p. may be a higher Bockstein because g

that

is not J * necessarily a monomorphism on higher terms of the Bockstein spectral sequence.)

Thus, to compute

, we must find

the order of the element represented by Q q (x H2 n_4 |Map^(P (2r),Sn);zJ.

We have a fibration

Map^|p^(2r),Sn)j ---» Q^Sn — H-space

Q^Sn, where

2r-th power map.

generator

y.

Hurewicz map,

For some

~ Z 0^0,

0y

h(fi CLn ,cn^ =

multiplication by

r

2

in

with

is in the image of the

Since

on 7r^,

is the

induces j*

JrX(y) = 2 y.

An easy

check of the Serre spectral sequence for this fibration

zj

gives that

|Map^(P^(2r),Sn);

that

a1 = 1

that

g^

if

r = 1 and

o' = 0

= Z/2rZ. if

Hence we see

r > 1.

We note

is not a monomorphism on integral homology when

r > 1. We now let

n = 1(4).

Write

n = 4k+l.

F. R. Cohen

[3] and M. G. Barratt (unpublished) prove that the second Hopf invariant

h^:

factors through

Map^ |p ^(2) ,S^+^j ; there is a homotopy commutative diagram

SPACE OF MAPS OF MOORE SPACES INTO SPHERES 2

-

2

-

h*Q-|^X 2k-l^ = ^l(x 4 k-l^’ Apply

and

2-

-

to obtain

h^O = S q ^ f x ^ ^ ) .

Q l{y4k-1} + “ lQ3 (y4k-2t

Also

^q^ to this equation -

g^Qjfx^p =

ThuS

g*Sq*Q1 (x4k_1) = ( 1 + Thus

a^=l

if

r = 1.

if

We first take

gJ3 0x = Q y + aQ y * s- 2 n-s+1 s- 2 n-s+1 sn-s Sq^

2*

Sq^

=0

a = 1.

as before.

We will

of the right-hand side of this equation is

independent of as

r = 1.

This finishes the proof of 2.1.

We now prove theorem 2.2.

show that

83

It is enough to calculate

a. for

i > t

2* Sq^ , i < t,

by dimensional reasons (and

n = -l(2 t)): n-s+s- 2 Sq2 Q syn-s

^s-2 * ^n-s

0i

n- 2 0i

Q s-2 i yn-s

r2 t«+l + ... + 2 t 1 - 2 1 Q s-2 i yn-s

= 0 Thus the term 2 *Sqw Q

and

qX .. s- 2 n-s+1

s = 3.

.

aQ y sn-s

makes no contribution to

Similar considerations hold for

a > 1

84

CAMPBELL, COHEN, PETERSON AND SELIOC

§5.

PROOF OF 3.k AND 3.5 To Drove 3.4 and 3.3 from 3.4. we need an explicit I 4 r n1 H lMap^(P (2 ),S L

description of the indecomposables of n > 5.

This is given by lemma 5.2 below.

immediately from 5.1.

5.2 follows

The proof of 5.1 is straightforward

and we leave the details to the reader.

LEMMA 5.1.

If

3 n PH^(Q S ).

n > 5,

PH^|Mapx (P3 (2r),Sn )j = PHK (n2Sn ) ©

Furthermore, exactly one of the following

holds' • (i)

PH^|Map^(P^(2r),Sn )j

is of dimension at

most one

with basis'-

(it)

(a)

(QiQ2xn_3)2J’ j - 0,

(h)

x ^ 3.

{c>

xn-2’

(d)

Qjxn_2 - a £ 1,

b -°’

j> 0.

PH^jMap^(P3 (2r),Snj

or is of dimension a+1

- a

(Qixn_2)

basis:

a ^ 1,

and

(^2

2 5>^

Xn-3^

a,j 2 0.

LEMMA 5.2. (t)

Let

If

n > 5.

q = 1(2),

QHq |Mapx (P4 (2r).Sn )] =

PHq |Mapx (P4 (2r),Sn )J (a)

Q ^ Q bxn_4 ,

(b)

Qjxn_4 > J > 1.

and has a basis:

j > 0.

with

a > 1.

’

85

SPACE OF MAPS OF MOORE SPACES INTO SPHERES

(ii)

(c )

xn_4

(d )

Q i + 1 Q2Xn - 3

If

n = 1(2).

if

q = 0(2),

xn_3

n = 0(2),

if

a *J ^ ° -

QHq jMap^(P4 (2r),Sn )j

is at most

one dimensional with basis'(a )

x n_4

(b>

^2Xn-3’

(c)

*^Q3xn- 4 ’

n = 0(2),

if

a

* 1-

xr_3

if

n=l(2),

b * °-

Before we start the proof of 3.4, we give two preparatory lemmas.

LEMMA 5.3 n > 5,

Let

(11) v J (iii)

IK

A * i n-4

Is an H-map.

for

f^

is an

Then

= Q.x A + others i n-4

fJ3.x q = Q.x 0 + others * l n-3 l n-3

if i = 0,1,2,3, if

i = 0,1,

fxQ2xn_3 = Q 2 Xn_3 + others,

In (i) with

i =0,1,

primitive in that degree. follows by applying

Sq^

the others to the reader. AQ3xn 4 + BQ1xn_3 AQ_x . and so 2 n-4

by 5.2.

A = 1

n j* 6.

there is at most one

In other cases, the result to an equation to evaluate the

undetermined coefficients.

degree.

Assume that

j < 2n-4.

where others Is zero In (111) If

Proof *

Map,w jp4 (2r),Sn],

be a self-map of f

such that

isomorphism on (O v J

£

as

We give an example and leave Assume Apply f *

n = 1(2). Sq*

to get

^*^3Xn-4 = f*Q2Xn-4 =

is an isomorphism in this

86

CAMPBELL, COHEN, PETERSON AND SELICK

LEMMA5.4.

Let

£

be a self-map of

n > 5,

which is an H-map.

on

for

Hj

j < 2n-4,

-2

If

induces an isomorphism

then

= Q 1^2Xn-3’

-2

Proof: ^ ^ 2 x n _ 3 = ^ 2 Xn-3 + scluares

1-2

-

Sq = V n - 4 ,

Sq Sq Q 0x ~ = Q nx ., ** **2 n —o u n-^±

Since

P = 1,

Sq2QOx n-3 =

2

Hence

a

S% Q 3Xn-4 = Q 2Xn - 4 ’

We know

a = 1.

j3Q^x

= Q 2xn _4 ,

*S t^10 o n ^

= “V n - 3 -

the following

=

and all

an H-map, and since

x

1.We have

Sq*Q 2Xn - 4 = V n - 4 '

1 Sq Q. x .= Q nx ., l n _ t; u n-“Ti Since

r =

is defined. In bur case,

Sq^x o = x A so we must be able to choose a ^ n-3 n-4

relation

92

CAMPBELL, COHEN, PETERSON AND SELICK

with no

b i = Sq*.

Since n_

n = 1(4), and

^

find a relation

Sq

= 2 a.b..

|a.[ = 3(4) J

a. = 2u»v, with J

M

so

n-1 / 2 t, we can

i If some

ii

|v|

b . = Sq ,

then

J

> 0 . Hence

Sq11 *

= 2 u*(vSq*) + others. Let

n = 2(8),

r = 1.

If we can show 1 Sq^

preserved, the rest are preserved by

and

*S

2 Sq^.

By

4 — 4-

2.1,

Sq 0ox 0ox 00 == CL.x CLx .,A, and Sq andasas the theonly onlyelement element in that * 2 n-3 0 n-4

dimension, Let

is preserved.*S Preserve 2t+*. Consider

a self-map ofMap^jp^ (2),Snj. Sq

Q2Xn-3

and thus

Q

By 2.1, x 2 - 2 n-2>l

suspension we see that

is

^2Xn-3

is preserved. Let

n = 6(8),

we must show then

n > 6,

Q2Xn-3

r = 1.

As

*S Preservec*-

CLx A € Ker f also. 1 n-4 *

^

However,

in the above case, 2Xn-3 € ^er Q 1x . 1 n-4

is detected

by an unstable secondary cohomology operation coming from

n_2

factoring Sq as in the case n = 1(4), n ^ 2 +1. 2 Sqfactors as long as n > 6, with thesame properties

^

as

in the case Let

1(4),

n = 1(4),

n = 2*4-1,

n

t > 5,

/ 2^+1. r = 1.

As

we need to show ^2Xn-4 ^ ^er **'

€ Ker f^.

Let

M

in the case of

n =

assume ^2Xn-4

be the mapping telescope of

f,

SPACE OF MAPS OF MOORE SPACES INTO SPHERES f: Map^jp4 (2),SnJ -» M,

Map^ |p4 (2) ,Snj . Then

1: F and

and let

H ^ qCFiZ) = Z/2Z.

F = fibre

F

Is

g: P

(2) -+ F

mod 2 .

g = ig: P^n ^(2) -» Map^ jp^(2) ,S n j . Consider

Let

g*(y2n-5}- Sqig^(y2n-5> = Q2Xn-4 30

with

(2n-7)-connected

Thus there exists

which is a monomorphism on homology

f,

93

311(1

^ ( y 2 n-5) = Q3Xn-4 + Q lXn-3’ Let -> n3 Sn . Then

7r: Map^|p4 (2),Sn

Since

g ' = g|S

0 = irg' : S2n 6 -» 03 Sn .

(^g)v,(y0 K ) = Q -x Q , we see that ^ zn-o l n o

invariant one [9].

Clearly

p2 n-5 (2 ) — 2

= 0

2 n _0

27Tg ^ *,

^ p2n-5^2 ^

0

has Arf

so we have a diagram trg

Q -

s2n-s where

c

is the collapsing map.

is divisible by 2.

Thus

0rjc ^ *

and

0tj

The result will follow from the

following lemma.

g

LEMMA 6.1.

Let

0 € tt

be an element of Arf invariant

2 -2

1

with

Qr} = 2x.

t < 4.

Then

Proof: We consider the Adams spectral sequence for cohomology.

= Ext^(Z/2Z,Z/2Z). When

3-fold products of for the relations

2

ki^i+2 =

h *s

s = 3,

0

the

are linearly independent except

h.h.f1 h. = 0 , h?h . l0 = h ? - , l l+l j l i+2 l+l

see [1 ]*

mod-2

*s r e Pr e s e n ted by

2

and anc^ thus

94

CAMPBELL, COHEN, PETERSON AND SELICK

617

2

is represented by

= 2x.

2

Since

h^ht_^ ?£ 0

. Let

s = 2 ),

or

2

h^ht_^

or

2

h^h^_^

2

h^h^ ^ = h^

doesn’t happen because only h-jh^if h^

t > 5.

(element

The first case

is there and

•frt *

The second case doesn’t happen since

is not an infinite cycle when

t > 5.

The third case

doesn’t happen because there are no elements with of dimension

Let

n = 2*\

t > 5,

r = 1.

€ Ker f*.

As in the case of

^2Xn-3

*S Preservec*-

Then S q ^ x ^

= Q g X ^ e Ker f*.

By a mapping telescope argument, as in the case t > 5,

s = 1

2 t.

n = 0(4), we need to show that Assume

s = 1,

is twice an element with

is hit by a differential.

2

617

and assume

and is assumed to be an infinite

cycle, for this to happen, either with

t > 5,

we obtain

g ‘P^n ^(2) -* Map^|p ^(2) ,S n j

= V n - 3-

Adjoint

g

n = 2*4-1, such that

to get

g 1 : p2 n-3 {2 ) -► Map*[p3 (2),Sn] such that

g^(y2n_3 ) = Q ^ n_2 - Let

j : Map^ P3 ( 2 ) , Snj -» Mapx ( p 3 ( 2 ) , nsn+1 j = Map^ (p 4 ( 2 ) , Sn+1 j . j*g*{y2n_3) = Q ^ n_z + Q3 xn_3 .

Then

situation we had in the case

This is exactly the

n = 2*4*1,

t > 5,

and the

same argument applies to get a contradiction. We now prove 3.3. r > 1.

Let

n = 1(4)

or

n = 2(4),

All elements are easily seen to be preserved.

Let

SPACE OF MAPS OF MOORE SPACES INTO SPHERES n = 3(4),

r > 1.

One needs to use a secondary cohomology Q ^x n_3

operation to show that

*s preserved and then the

rest of the elements are preserved. n ^ 2^,

*s preserved. ^ 3x n_4

n = 2t(2t+*),

n > 2t+* . Consider

Mapx |p2 + 1 (2r),SnJ.

x 2

that

§7.

n = 0(4),

The rest will be also if we

can show that

Q

Let

r > 1 . Again a secondary cohomology operation ^ix n_4

shows

95

*s preserved.

Sq2 Q tx t = 2 n 2

is preserved by

Find

t

such that

^(f), a self-map of t

Qq x

and thus

n 2

2^-3 (0 f) . By suspension we see

n- 2 Q 3x n _4

*s preserved.

PROOF OF THEOREM 3.6 We first prove 3.6 when

fibration

0S^n * -*

monomorphism in

H^.

n > 2. W . n

Consider the The first map isa

Thus, as a coalgebra over the

Steenrod algebra and as a module over

H^(QS^n ^),

H^(n3 S2n+1) ^ H^(QS2n-1) ® H^(Wn ). This proves the following proposition.

PROPOSITION 7.1.

As a coalgebra over the Steenrod algebra,

H (W ) = Z/2Z QTQ2 X 2 n - 2 nJ

I a+b * 1] >

if

n>~ 2 -

96

CAMPBELL, COHEN, PETERSON AND SELICK Now let

f: W -» W n n

^lX 2 n-2 *

want to sh°w

{(QaQ 2x 2 n~2 ^ that rank

f*

^ a+^ -

PH^fW^) < 1

J ^ 0}*

a ^ix2n-2

*S Preservec*-

k,£.

Since

We now consider

Qf.

(nf)*Q lQ2x2n-3 = Q lQ2x2n-3’

,

this

2a_ 1

Sq^

= ^ i x2 n-2^2

Thus all primitives are preserved if

preserved.

•••

for SOme

^ i x2n-2^

2j

Slnce

Sq? Q lQ3x2n-3 = ( ^ ) ^ i ^ 3 x 2n- 3 =

There is only one primitive in this dimension

Q 1 Q 3 x 2

(0f)xQ^Q3 x2n_3 = «QjQ3 x2n_3 . But

(Qj"lQ3x2n_3 )2 ,

a = 1

so

which is true by applying Let

n = 1.

shown that if i < 2,

then

Sq^

=

(nf)MQ 3x 2n_3 = and

a.

= 03 S3 , and F. R. Cohen [4] has

Then

f: W^ -* W^ f

if

S c ^ J ^ x ^

induces an isomorphism on

is a homotopy equivalence at 2.

7r\ ,

We will

use his result and methods. Let

3 4 4 w- IRP -» Qq S

Whitehead and cell. 4

is

We want to show that

it is sufficient to prove

so

) 2

t*iere Is a Steenrod operation

Sq^(QaQ2 x2 n 2 ) 2

such that

Let us

= ^ l X2 n-2 ^

f*(Qix2n-2^

Sq^Sq^Qa+^X2n_2 = ^ ^ 2 X2n-2’ Sq^

PH^fW^) =

It is easy to check

... S q j s q ^ x 2n_2 = (QjX^

SqJ

shows that

preserves

in any given dimension.

assume that we can prove Since

f O ix 0 0 = * 1 Zn-Z

be such that

i: S

4

-> BS

denote the map constructed by 3

Thus there is a map

0 (i)*w;

g

the inclusion of the bottom 3 3 3 g: IRP -* QqS

is clearly non-trivial on

given by

t .

SPACE OF MAPS OF MOORE SPACES INTO SPHERES LEMMA 7.2. that

3 3 3 k: fi^S -> IRP

There does not exist a map

kg

97 such

ts a homotopy equivalence.

Proof: Apply composition

tt^

to the map

Z/4Z -+ 0 -» Z/4Z

kg

to get that the

is an isomorphism.

Thus

k

cannot exist.

We assume given x^

and

f^Q^x^ = 0 .

applies.)

(If

3 3

f^ = 0

1

a,b > 0

primitive in a given dimension. element for such that

a = 1.

(Q^[1])

2b

(Q1 [1])

2

€ Ker f^

€ Ker f . hence **

2k+1

E Ker f^

a given dimension. shows that

and there Let

(xk)^

x^

Q^Q^Cl] +

is at most one

be this latter

Cohen produces sequences

first prove by induction that

(Qi [1]) i

a > 0 , and

I 2b+^ Sq^x^ = ((^[1]) and

C Ker f^.

to be the

U

**

of the elements Q^[l]. for

f^(x^) =

3 3 PH^(Qq S )

on

3 3 PH (fiJS )

x 1 . Cohen computes

(Q^+b[l])(Q^[l] ) 2 ,

such that

f^Q^x^ ^ 0, then Cohen’s proof

We wish to show that

except for 2 i-powers

3 3 f: Q^S

I

and

J

T 2b Sq^xfe = (Q-^l]) . We

(Q^[l])

2b

€ Ker f^

by hypothesis.

and

x^

By induction

x, € Ker f , and so D **

as there is at most one primitive in

The equation € ^er

Let

(Q^+b[l])(Qb [l])2 ; (x1 b = xb ).

2iT

Sq^

(xb)

xa b =

2}

2 *+b

= +

There are sequences

K

"98

CAMPBELL, COHEN, PETERSON AND SELICK 1

such that = (Q^Cl] ) 2

^ = •

Then the equations

9 a+i-l

(x^

^

and

such that

Sq^ ^(xa

=

i Qi 9 i+a-l Sq~ L (Q*[l]r = (Q^l])

all primitives are in

Ker f^

* Sq^Q^[l]

except for

show that and

2

x^.

LEMMA 7.3. Under these hypotheses, ^ * 33 ^ 33 f : H (OqS ) -» H (OqS ) Is zero on all elements except r *a2 , andj x*^ , (x^)

Proof:

, ^3 . (x^)

x ^ x f (Xj) = Xj

computation.

^4

by hypothesis.

(x^)

=0

n f

By the above computation,

by an easy

is zero on all

the indecomposables of degree greater than one. careful in dimension 2 , and choose that

(Q^x^)

(Be

correctly so

f*((QoX;l)*) = 0 .)

We return to the proof of theorem 3.6 for X = mapping telescope of 3 3 h: QqS -» X

LEMMA 7.4.

f.

There is a map of

isomorphism on

3

Then

is an isomorphism on

H^(X) = H^(RP ), 7

and

P ’ X -* RP

.

Proof: Consider the composite

n = 1.

0

[RP3-* n3s 3 -» x

Let and

.

3 which induces an

SPACE OF MAPS OF MOORE SPACES INTO SPHERES

99 3

which induces a homology isomorphism. homotopy equivalent to

3 Since IRP

3 r: 0 2 IRP

there is a retraction the composite

2 X.

n 2 X * fi 2 IRP3

X

Thus

2 IRP

is

is an H-space,

3 IRP . The map

p

is

IRP3 .

To finish the proof of 3.6, observe that the composite Rp^

Oq S^

X

contradicts lemma 7.2.

RP^ Thus

is an isomorphism on f^Q^x^ = Qq x ^

. This

£Lnd we are

done.

REFERENCES [1]

J. F. Adams, "On the Non-Existence of Elements of Hopf Invariant One," Ann. of Math., 72(1960), 20-104.

[2]

H. E. A. Campbell, F. P. Peterson, and P. S. Selick, "Self-Maps of Loop Spaces, I," Trans. AMS, V293 (Jan. 1986), 1-39.

[3]

F. R. Cohen, "The Unstable Decomposition of Q^2^X its Applications," Math. Zeit., 2(1983), 553-568.

[4]

F. R. Cohen, "Two-Primary Analogues of Selick*s Theorem and the Kahn-Priddy Theorem for the 3-Sphere," Topology, 23(1984), 401-421.

[5]

F. R. Cohen and P. S. Selick, "Suspending Loop Spaces and the Fibre of the H-space Squaring map," preprint.

[6 ]

F. R. Cohen and L. R. Taylor, "The Homology of Function Spaces,'* Contemporary Math., 19(1983), 39-50.

[7]

J. C. Moore, "On a Theorem of Borsuk," Fund. Math. 43(1956), 195-201.

and

100

CAMPBELL, COHEN, PETERSON AND SELICK

[8 ]

F. P. Peterson, ’’Self-Maps of Loop Spaces of Spheres,” Contemporary Math., 12(1983), 287-288.

[9]

P. S. Selick, A Reformulation of the Arf Invariant One Mod p Problem and Applications to Atomic Spaces, Pac. J. Mathematics, 108(1983), 431-450.

H. E. A. Campbell Queens University Kingston, Ontario Canada

F. R. Cohen University of Kentucky Lexington, KY 40506

F. P. Peterson Mass. Inst, of Technology Cambridge, MA 02139

P. S. Selick University of Toronto Toronto, Ontario Canada

IV THE ADAMS SPECTRAL SEQUENCE OF AND BROWN GITLET SPECTRA

Q2 S3

Edgar H. Brown and Ralph L. Cohen^

§1.

INTRODUCTION The main results of this paper are computations of

some differentials in the Adams spectral sequence for s 2 3 7T^(Q S ).

As an application we derive directly from loop

space technology the existence and uniqueness properties of the Brown-Gitler spectra at the prime two.

(The same

analysis probably works for odd primes.) Recall, using a configuration space model of May [15] defined a filtraction, Fn (X) ^ f/V^X

= FjtXJ/Fj^jCX) = F(IRn ,k)+ < ^ X (k)

F(IRn ,k)+

is the space of ordered set of

distinct points in (k) Xv J

is the

... F^fX) C F^+^(X) C ...

with subquotients

d £(X)

where

QnSnX,

IRn

together with a disjoint basepoint,

k-fold smash product, and

symmetric group.

k

Furthermore, if

X

is the

is connected,

QnSnX

^Both authors were supported by NSF grants, and the second author by a fellowship from the A. P. Sloan Foundation.

101

102

BROWN AND COHEN

and

V k>l

D?J(X)

are stably homotopy equivalent by a theorem

R

of Snaith [17].

where

For

n = 2

and

X = S*,

d ^CS1)

= t(fk ) p

is the vector bundle

F(IR^,k)/2^,

and

k

F(IR ,k) x

IR

*s its Thom space.

Let

T(f^.)

denote the Thom spectrum, indexed so that the Thom class

2

has dimension zero.

Recall from [1] that

K(j5^,l),

is Artin’s braid group on k-strings.

where

The bundle

p^

F(IR ,k)/2^ =

is induced by a representation of

given by the composition

P^ -» 2^

P^.

0(k). The first map in

this composite associates to a braid the corresponding permutation of the endpoints of the strings, and the second map represents

2^

as permutation matrices.

The Brown-Gitler spectra

B^

([2]) have proved to be

useful because on the one hand, they were constructed by a Postnikov system and hence have a moderately well understood homotopy type, while on the other hand, T(^2 k)

are homology equivalent at the prime

2.

B^

and

Thus the

B ^ ’s connect up with vector bundles and loop spaces [4], [5], and [10].

In view of the equivalence of

T(f2 k)» one m ight simply define

B^

to be

B^

Tff^)

and and

thus eliminate the need for the rather complex arguments used to prove the existence of

B^

in [2].

The working

out of this alternative approach is the main aim of this

103

ADAMS SPECTRAL SEQUENCE OF n2S3 paper.

The main computationally useful feature of

Postnikov construction of the

’s

is that the

k-invariants have a very peculiar but useful property; namely, there is a Postnikov tower J spectra building

such that the

Pontrjagin dual spectrum of KfZ^.iJ's

([2], 5.1(iv)).

X n

X n- 1

(2 k+l)

til

of space of the

splits into a product of This has been reformulated in

various ways, for example, in the concept of adapted manifolds ([5]).

One can also easily translate this

condition into (l.l)(ii) below.

We first make a

def inition.

DEFINITION.

A spectrum

E

is said to have space-like

cohomology if it admits a map to the suspension spectrum of a

C.W.

mod 2

complex,

f: E -» X,

that induces an surjection in

cohomology.

Remark:

If

E

has space-like cohomology,

unstable module over the Steenrod algebra, for any

x € H^fEjZ^), Sqn (x) = 0

for

H (E;Z^)

is an

A.

That is,

n > q.

However,

having space-like cohomology is a strictly stronger condition than having unstable

A-module structure as can

be seen by the following example. U

where

p 1' .

Let

E

be the spectrum

is the stable map given by

V the square of the Hopf map

r/ €

s

tt^

0

(S ).

We leave it to

104

BROWN AND COHEN

the reader to verify that

H (E)

has an unstable A-module

structure (in fact its a trivial A-module) but it does not have space-like cohomology. THEOREM 1.1.

The spectrum

T(f^)

satisfies the following

properties: (i)

H* Tf^k)] ~ A^A{\(Sqi): i > k} where of

(li)

\

as

A-modules,

denotes the canonical antiautomorphism

A. E

Let

be any spectrum with space-like

cohomology.

Then the Adams spectral sequence of

^ E)

satisfies the following

properties ' • EX’ b.

= H (E),

The differentials

zero for r = 1

t = 2 k+l.

for

T-s < 2k+l

d : E ^ ,t: r r and r >

ES+r ’t+r * are r

1.

(The case

depends on the particular resolution we

choose.) Properties (i) and (11) above characterize the stable homotopy type of

Remark:

Let

H

and let

h: T(f^)

^(?2k^

at t^ie Pr^me

be the Eilenberg-MacLane spectrum H

K(Z^)

represent the Thom class.

Condition (ii) above implies that every element of E^ ,C1

2.

is an infinite cycle for

induced in generalized homology,

q < 2k+l.

^(E) =

Hence the map

ADAMS SPECTRAL SEQUENCE OF (1.2)

T(f2 k )q (E)

V

is surjective for

q < 2k+l.

105

Hq (E)

In [5] it was shown that

properties (1 .1 )(i) and (1 .2 ) characterize This paper is organized as follows.

B^. In section 2 we

prove a proposition that reduces the proof of Theorem (1.1) to showing that there exist certain cofibration sequences among the

T(fk )’s.

The proof of this proposition uses a

lambda algebra calculation that is postponed until section 4.

In section 3 we prove the existence of these

cofibration sequences, thus completing the proof of Theorem (1.1).

In this section we also show how various uniqueness

theorems for Brown-Gitler spectra that appear in the literature follow directly from theorem (1 .1 ). Throughout this paper all spaces and spectra will be localized at the prime 2 and all (co)-homology will be taken with

§2.

mod 2

coefficients.

THE MAIN PROPOSITION Let

be the

A-module = A/A{x(Sq1):

i > k}.

The goal of this section is to prove the following.

PROPOSITION 2.1. of spectra and

Suppose a-

(X^.: k = 0,1,2,...} aru^

maps satisfying the following properties:

^k-

is a family -»

are

106

BROWN AND COHEN

(1)

H (Xk ) = Mk . k

(11)

a o p:

^ ^[k/2]

nu^ homotopic,

and

(111)

In cohomology, a

Pinduce the exact

and

sequence

0 ----- > V

where

"k- 1 —

- 2- ~ " k

a* (1) = \(Sq^)(l)

Y

if

]

and

*°

P*(l) = 1-

Then,

Is any spectrum with space-like

cohomology, the differentials in the Adams spectral

^

sequence for

Y) ,

, . i—is*ti-,s+r,t+r-l d :E -* E r r are zero for

t-s < 2k+l

Proof : By (ii) and (iii),

which yields a map

Let

fibre of since Then

^

/(I)

/y

nr

give a cofibration

nr

a

Ak

1 and let lifts to

= -r*j3*(l) = 0.

and

a,/3

P

y

Ak - 1

H represent

h.The map

P

, y

[k/2 ]

h: Xk

and

r > 1.

nr making a cofibre sequence:

^V A

a

and

nr:

yky

A

Xk

___

[k/2 ]

Xk

bethe

2^

^k-1

LetE^’5(k) = E®,1:(Xk

define maps

E^ ’t(k-l) -» E^ ’t(k)

a*: E ^ ’t(k) -» e®*t_k([k/2 ]) V

E * ’t ( [ k / 2 ] )

-» E ^ + 1 ’

t+k(k-l) .

i

- Y).

107

ADAMS SPECTRAL SEQUENCE OF Q2S3 Recall the

X-algebra,

algebra with unit over

Let

is the associative graded

If

2i


f°r ©very

generatedby admissible

(i. ,...,i„) with

v 1

in = k, £

= (i^,...,i^)

% = s,

X^,

j.

Let

I =

\ i . = n. In L j

section

four we prove:

LEMMA 2.3.

An Adams spectral resolution for

a

Y)

may be chosen so that * 1 AW P

°

9 V

Y)

and d^(Xj ® u) = y x ^x j ® uSq*+^ where

uSqJ

denotes the

Hom( ,Z^)

dual of the cohomology

SqJ. Furthermore: (1 )

a : E^'^fk) -»Ei?,t ^([k/2 ]) ** 1 1 2k

and if

|X^ ® u| = 2 k

is zero for (I may = [ ])

u € H^pfY)# then a*(X2 I ® u) = Xj ® uSqP (^)

P~: E*f,t:(k-1) ->E^,t:(k) JL

I

® u) =

is given by ®

u

t-s < and

108

BROWN AND COHEN

(ill)

E ® ,t:[k/2] -» E®=,t+k(k-l)

is

y*(XI ® u) = XiXk-i ® u *^(kjJ = 0

dx

(iv)

t-s < 2 k+l.

for

We now prove (2.1), that Is, t-s = 2k+l

by

covered by (iv) than

r.

2k+l. via

The case r = 1

for is

Suppose it is true for values less

® u € E ^ ,t: = E^ ,t:(k) Xj. ® u

so that q = 2(k-l)+l.

and

q = |Aj ® u|
k K

Y

I = [ ].

be a product of

a map suchthat hence

or

KfZ^.iJ’s

f : H (K) -» H (Y)

f^: H (Y)

H^(K)

and

f : Y -» K

be

is an epimorphism and

is a monomorphism.

The fact that

has space-like cohomology is equivalent to the existence

of such a map. hypothesis, 2 k+l.

Sq^: HP (K)

E^ = E^

by the inductive

finduces a monomorphism on

Hence we

SqP : H ^

Then, since

may assume

(X) -»

Y =K.

s t E^’ ,

t-s
0 }

satisfies the hypotheses of prop. (2 .1 ).

That is, we prove the following.

LEMMA 3.1.

There exist maps

* - « 2 [k/2 ]>•

^

a' T(f2k)

T « 2 k-2 > ^ T « 2 k>

ttal satisfy the

following properties.

(t) (1 1 ) (Ill)

H*[T(f2k)] * 1^. a op

Is nullhomotopic. a

In cohomology,

and

p

induce the exact

sequence a

* M[k/2] where

Remark:

M

P

“k " ^ — ^“k -l --------->

a*( 1 ) = x(Sq^)(l)

and

P*(l) = I.

A proof of (3.1) already exists in the literature.

Part (i) was proved by Mahowald in [14].

Parts (ii) and

(iii) were proved by F. Cohen, M. Mahowald, and R. J.

110

BROWN AND COHEN

Milgram in [9].

For the sake of

outline of a proof of this

completeness we include an

lemma here.

To prove (i) it will be easier to work in homology

Proof'-

than in cohomology, so we begin by identifying the dual vector space

M*

Let

€ A

2 1-!,

and let

as a subspace of

be the Milnor generator of dimension t. = x~(C-}* 1

antiautomorphism

^

\.

1

Thus

Here A

weight to monomials in the wt(l) = 0

b.

wt(t^) = 2 *

c.

wt(xy) = wt(x) + wt(y).

M^ C__> A R

monomials

t

\ ^

= Z^ft

is the dual of the t •••]■

We give a

t^’s by the rules

a.

LEMMA 3.2.

A*.

j

= 2^[t^,t^...]

is spanned by those

* C A

with

wt(t ) < 2k.

Proof:

Recall from [3] that

< 2 k},

where the excess of a cohomology operation

e(b) if

M^. = {a € A: dim(a) + e(\(a))

is defined to be the smallest integer i

( t q) *

6 Hq K ( Z g . q ) j

q

b =

such that

is the fundamental class, then

b

0.

Consider the Milnor basis, monomial

R ^ f C A

A

=

we define e(rR ) = e(SqR )

*^2* ’'‘-J‘

^OT> a

ADAMS SPECTRAL SEQUENCE OF Q2S3 where for

Sq A.

t^

is the class in the corresponding dual basis It is now easy to verify that for every monomial

in the

t .* s , 1

wt(tR ) = dim fR + e(fR )

(3.2)

111

.

|

now follows.

Now recall that the disjoint union

11 F(IR —2,k)/X

JLL k > 0

is homotopy equivalent to a C^-space in the sense of May [15], as is every 2-fold loop space.

This in particular

implies that there exist "cup-1 " pairings q: S 1 xz where

FflR^k)/^ x FfD^.k)/^ -» F(IR2 ,2k)/22k ,

acts antipodal ly on

description of the map

q

. The combinatorial

is given in [15], but since both

the source and target space of

q

described group theoretically.

If

then

q

are j3

K(7T,1 )’s

q

can be

is the braid group,

is induced by the homomorphism z X Pk X Pk - P2k

defined by associating to a triple 2k

(n.b^.b^)

strings defined by twisting the braid

braid

b^

by

n

The "cup-1" operation formula

b^

the braid on around the

half-twists. pairings

q

define an Araki-Kudo

Qj : Hr JV(IR2 .k)/^) -» H2 r+ 1 |^F(IR2 ,2k)/22kj

by the

112

BROWN AND COHEN

Q x(x ) = % ( e1 ® x ® x). F. Cohen [8 ].

We recall the following calculations of LEMMA 3.3.

Let

B/3^ = lim B/3k = lim FflR^k)/^. k

k

= ^2 ^-X l ,X2 ’**'^

(i)

where

(ii)

x^

is the image of

^F(IR^,2 )/2 ^J = H^(S^),

the nontrivial class in and where

is defined inductively to be the

image of

€ H .

The inclusion

|f(IR2 ,k)/2j -*

^2 ^-X l ’‘ '-1

Then

^(IR2 ^ 1)/^

j. =

is a monomorphism, with image spanned x

by monomials

R

above, wt(x )

R wt(x ) < k.

with

Here, like

is defined by the rules

(a)

wt(1 ) = 0

(b)

wt(x.) = 2 1

(c)

wt(xy) = wt(x) + wt(y)

Observe that (3.2) and (3.3) imply that H* |f (IR2 ,2k)/sJ = H^^T(f2k)j vector spaces.

and

M*

are isomorphic as

To prove (3.1)(i) it is therefore

sufficient to show that if u: T(f2k) -» H represents the Thom class in X u* : HxT (^2 k^ A has image

f2k^] ’ t*ien -X Mk ‘

To prove this, we first recall that in [7] F. Cohen proved that the c^structure of

11 FfER^.k)/^ k

and that of

ADAMS SPECTRAL SEQUENCE OF fi2S3 Z x BO

113

correspond under the classifying maps of the

bundles

^

Thom spectrum level this implies that

the following diagram homotopy commutes:

T(?4k) where Z^, q.

H

classifies the generator of

6

and where

Tq

Observe that ^ 2 r+l*

a^,

H a H

is the map of Thom spectra induced by 6

induces a Araki-Kudo operation

which, by the commutativity of the diagram

is compatible, via the Thom isomorphism, with the operation Q1

Hence the fact that the image of

on

u* :

A*

follows from (3.2), (3.3) and the

following result, which is a straightforward calculation in A*.

LEMMA 3.4.

Remark.

In

A*,

Q 1 (t. 1) = t., lv i-l' i

for all

i.

The above calculations were modelled after

calculations done at odd primes in [1 2 ].

We now sketch how parts (ii) and (iii) of (3.1) were proved by F. Cohen, M. Mahowald, and R. J. Milgram in [9].

BROWN AND COHEN

114

Consider the natural inclusion of braid groups, ^2 k-l C~ _* ^2 k'

£*2k-2 ^

F(IR^,2 k-2 )/22^._2

This induces a map

2

F (ER ^ k ) / ^ ^ , which in turn induces a map of Thom spectra ^

T « 2 k-2 > ^ T « 2 k> '

The fact that in cohomology, since

p

p (1) = 1

is immediate,

preserves Thom classes.

The construction of the map is somewhat more complicated,

^^^2[k/2]^

a•

is constructed by use of

a

the James-Hopf map h: QS3 -» OS5 . More specifically,

2 3 Qh:fiS

2 5 S

is used in the

following manner. 2 5 2 2 3 Q S = Q 2 S

The Snaith decomposition of

is given

by fi2 s5

*

V k

where

t(3f^) In [9]

D 2 (S3 ) =V k

t(3f )

is the Thom space of three times the bundle it was shown that

has order 2.

Thus

t(3fk ) * 22 k t(fk ) = 23 kT(fk ). T(f

) -»

is defined to be the composition

a: 22 kT(f2k) = tff^)

The fact that the maps (iii)

V

t(f .) ~s Q2 S3

*

V

t(3fj) -» t(3fk ) = 2 3 kT(fk ).

a

and

P

n2 S5

satisfy (3.1)(ii) and

was proved in a straightforward manner in [9].

main component of their proof was a calculation of the

The

ADAMS SPECTRAL SEQUENCE OF 02S3 (fih)^: H^(Q2 S3 ) -*H^(n2 S5 ).

homomorphism

115

We refer the

reader to [9] for details. This completes our outline of the proof of (3.1). Thus modulo Lemma 2.3, the proof of Theorem 1.1 is complete. We end this section by describing some uniqueness properties of Brown-Gitler spectra, all of which are easy consequences of Theorem 1.1. Define

to be

anc* ^et

^n

t*ie

n-Spanier-Whitehead dual spectrum to

Thus

^ ( G^)

05 Hn_q(B [n/2 ]'

LEMMA 3.5.

Recall that

Proof: > n}

has space-like cohomology.

H

*

is the quotient of

operations

a € A

= M [n/2]^ = A/A(x(Sq1): 2i

A

by the ideal generated by

such that

a(U

) = 0

for every

Mn n-manifold

M , where

U

is the Thom class of the Mn

stable normal bundle ^[n/2 ]

*S a

of

v Mn

(see [2]).

Since

dimensional vector space, there exists

a finite set of operations a basis for

Mn

a^,...,a^ € A

^ n / 2 ] ' and so that for each

an n-manifold disjoint union

Ma

with

a.(U

j* 0.

that project to j

Define

there exists Nn

to be

116

BROWN AND COHEN

*n =11Mn j Then the homomorphism

A -» H

J defined by

(Td

a -> a(U^)

factors through a monomorphism Mr [n/2 ]

H*(Tu

) . Nn'

Now consider the fundamental class (1.1)

this class represents an infinite cycle in the Adams 7r^(®^n/2 ] ^ ^+) *

spectral sequence for ^ N^)

be a class that is represented by

E ^ ’11.

By duality we get a map a: Tr (Nn )

so that in cohomology -x a

{Nn } € Hn (Nn ). By

Br t

a (1) = U^.

is a monomorphism.

Let

a e 7rn ^ [ n / 2 ]

{Nn } € Hn (Nn ) =

1 By the above remarks,

Apply duality again, we get a

stable map a- G

-> Nn

that induces an epimorphism in cohomology.

Thus

G

has

space-like cohomology. The following characterization of Brown-Gitler spectra is essentially the same as that given by Miller in [16].

THEOREM 3.6.

A spectrum

the prime 2) to

B^

is homotopy equivalent (at

if and only if it satisfies the

following properties (i)

H (Y^) = M^

as

A-modules, and

ADAMS SPECTRAL SEQUENCE OF fi2S3 (1 1 )

If Z

Is the

(2k+l)-S-dual of

Y^,

117 then

Z

has space-like cohomology.

Proof:

By (3.5) we know that

properties.

satisfies these

So conversely, assume

satisfying (i) and {ii).

is any spectrum

Let

3 e " a w i ' 2 ) = H° ‘Yk> = z 2 represent the generator.

By Theorem (1.1)

j

is an

infinite cycle in the Adams spectral sequence for Z). Let

g € ^k+l^k A ^

0 2 k+l

j € E^’

tt^ (B^ a

a c lass represented by

. Taking duals we get a map

so that in cohomology equivalence.

—x — g (1 ) = 1 . g

is clearly a

mod 2

f|

Remarks: 1.

In [16] Miller observed that the excess conditions that define

^[-^2 ]

(see the proof of (3.2))

translate via S-duality to the statement that J

H G * n

is projective in the category of unstable, right A-modules.

This fact together with CarIsson’s

calculations in [6 ] concerning how X 00 ® H (RP ) n

H*jG(2 *n)j

and

are related, formed the leaping off point

for Miller’s proof of the Sullivan conjecture [16]. 2.

In [16] Miller conjectured that

is a stable wedge

BROWN AND COHEN

118

summand of the suspension spectrum of a space.

(This

property is strictly stronger than that of

having

space-like cohomology.)

A proof of this conjecture

was recently announced by Lannes and Zarati [13].

The following characterization of Brown-Gitler spectra was proven in [ 1 1 ].

THEOREM 3.7.

Let

(Y^.: k > 0}

be a family of spectra

satisfying the following pro p e r t i e s .

(i) (ii)

H (Y^) =

as a A - m o d u l e s , and

There exist pairings

Yk - Y r ^ Yk+r : S 1 a_

1

+

and

Y . ^ Y . -> Y

Z2

21

21

2

that induce nontrivial homomorphisms in cohom o l o g y .

Then each

2-equivalent

to

Is homotopy

B^..

An examination of the proof of 3.7 given in [11] shows that these pairings were used to inductively show that the appropriate duals of the

Y ^ ’s

have space-like cohomology.

We leave for the reader the exercise of doing this directly.

119

ADAMS SPECTRAL SEQUENCE OF Q2S3 §4.

PROOF OF LEMMA 2.3 We recall some results from [3].

Let

A^(h) = Hom(A®(k),Z2 ) Cs(k) = A ® A*(k) Let

d: ^ ^ ( k )

(4.1)

^ s(^)

.

defined by

d(u ® A 1) = ^ A I(A>j)x(Sq'j+1) ® AJ

where

{A*}

is the basis dual to j = -1

the sum ranges over e' Cq -» M^.

by

and

e(a ® A ^ ) = a.

-»cs(k} - L c ^ f k )

A/Js>

J q = {Aj € A|

J

admissible.

Let

In [3] it is shown that

M^.

last entry

For any integer n,

admissible} and

- ... c0 (k) - ^ M k

is a free acyclic resolution of Let

{Aj| I

let

I < s}.

n = [n/2].

Note

AS =

Carlsson

proved 4.4 below in [6 ].

LEMMA 4.2.

J

s

(4-3) Furthermore, if

is a left ideal of

A

V l C J s+ |I|

■

i £ |l| < 2 i^

where

and

ii^ = last entry

I,

then (4.4)

Proof:

Vi 21 = AIAi+|l| mod Ji+|l| *

When a non admissible word

A^

is transformed into

a sum of admissible words using (2 .2 ), the last entries can only decrease.

Hence

Jg

is a left ideal.

120

BROWN AND COHEN We prove (4.3) and (4.4) by induction on

if

By 2,2

2.

2 i < j,

A.A. = eA. - A . - mod J. i J J-J i+J i+J where

e = 0 or 1

4.3 and 4.4 for Let

2-1.

and if

j

is even,

e = 1.

This proves

Suppose 4.3 and 4.4 are true for

2=1.

I' = (i^,i^,...,i^_j). Then C J 4 1

A. t+|r | U

C J t+|i|

•

Also X iX2I 6 (X I'X i+|r | + Ji+|l' P X21^ C X IX i+|I'| + Ji+|l| i+ |l'| = i +111 -

since

Let C(k) Cs+i(k),

< 2 i^ - i£ < 2 1 ^ .

be chain complex defined by

s > 0

with the same differential

— Cg(k)

is a free acyclic resolution of

fibre

X^. -»H.)

Let

C g(k) = d.

x _ H (X^)

nr*: C(k-l) -» C(k)

|

be the

Note

—

(X^ = A

map given by t (Av

for all admissible

LEMMA 4.5.

J) = A

if l = k-1

=0

if i > k - 1

(I,i).

C(k) C C(k-l)

One easily proves:

is a subcomplex, and

induces an isomorphism of chain complexes

nr*

linear

121

ADAMS SPECTRAL SEQUENCE OF n2S3 C(k-1)/C(k) £ C(k)

.

_

^

nr

Note 4.5 implies -» C(k-l)

by

is a chain map.

pfX1) =

dp + pd = 0 mod C(k) ct*: C(k) -» C(k)

Define

p*- C(k)

Lemma 2.5 shows that and hence we may define a chain map

by

a* = dp + pd.

Let

P*: C(k) -> C(k-l)

be the inclusion.

LEMMA 4.6.

a*: C(k) ->C(k),

C(k-l) -» C(k) H^(Xk_^)

lift the maps

P*: C(k) ^ ( 2 * ^ ) -»

-» H*(2k ^X)

and

C(k-l)

n r*:

and

, H*(Xk ) -»

respectively.

Proof: a*(A^) = (dp + pd)A^ = dX^ 1 = X (Sqk )X[] pV

]) = x C]

^ ( X 1) = x H =0

Recall, at the cohomology of

K(Sqk), P*(l) = 1 if

if

i = k- 1

if

i > k- 1 .

level,

and we must verify that

i = k-1, and is zero otherwise.

For

a (1 ) =

nr*(x(Sq*)) = 1

a € A,

^ ( a ) = t J I ) = (a* ) _1 (a) where

nra

is the functional cohomology operation of

a

122

BROWN AND COHEN

and

a : H (2 Xg) -» H (X^.).Calculating

homomorphism

a

from the

Mg -» M^, we see that a*(x(Sq^)) = 1

if i = k

=0

if i > k.

This completes the proof of 4.5.

Proof of 2.3. 7T^(X^ ^ Y)

action as

To form the Adams spectral sequence for

we take

C(k)0 H (y)

a resolution

with

the diagonal

ofH (X^ ^ Y ) . Hence

= HomA (Cs(k) ® HX (Y),Z2 ) = AS(k) ® H ^ Y ) The verification that

d^

immediate from 2.6.

u € A(k) ® H (Y) = E,

1

and

.

is as in (2.3) is an exercise in

dual vector spaces using 4.1. are

A

Parts (ii) and (iii) of

We next prove (iv). Suppose

2.3 A^

III + lul = 2 k+l.

d 1 (AJ ® u) = J A A j ® u SqJ+1 . Note

e Jj+i+ |j| an |u|, uSq^+ ^ = 0.

satisfying|u|/l = j+1 >

|u| = 2k+l - [I|. Hence there

are no values

giving nonzero terms in the above sum.

Finally we prove (2.3)(ii). Suppose and

V € HomA (C(k) ® H.Zg),

|I| + |v| = |V| - k .

A 1 e Ck ,

v e H*(Y)

ADAMS SPECTRAL SEQUENCE OF n2S3

123

a*V(A^ 0 v) = V((pd + dp)A^ ® v) = v[[ I A I(AJXJ)x(SqJ+1 )AJ , k ~1

+ XI’k_1(XjXJ)x(Sqj+1)XJ] = V [ I AI,k_1(Xj.XJ)XJ ® SqJ+1vj The last step utilizes

T k -1

V(A *

® x) = 0

. and

V((x(Sqt)u) ® V) = V( I x(SqS)(u » Sqt_Sv) = x(SqS)V( Y u ® Sqt_sv) = V(u If S q ^ ^ v ^ 0, then j+1

® SqV) . =|v|

and

ifX^ ,k+'*'(X .X.) J

?£ 0

then

j+1 +

|J | = k,

since

J

A A j €Jj+1^j C JL+1+|k |.

Thus if a*V ? 0, |V| - |v| - j- 1 = |J| > 2 |J | > 2 (k - j-1 ) > 2 k - j- 1 - |v | |v| >2 k.

Hence j+1.

Thus

a*(V) = 0

Suppose V = Xgj ® i.

= 2k - 1.

Xgj 6

for |v|

= 2k.

|v| =

A^, u €

H*{V)

and|u|

Then

a*(X2I ® u) = I ^Jk~ 1 (Ak_|I |_1 A2 I)AJ ® uSqk . By 4.6 since

(k-

|l| - 1) + |l| = k-1 < i^,

\-|l|-lX2I = X l\- 1 mod Jk-1 Thus

a^fXgj®

complete.

u) = Xj ® u Sqk ^ and theproofof 2.3 is

124

BROWN AND COHEN REFERENCES

[1]

J. Birman, Braids, Links, and Mapping Class Groups, Annals of Math. Studies #82, 1974.

[2]

E. H. Brown and S. Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283-295.

[3]

E. H. Brown and F. P. Peterson, Relations among characteristic classes I, Topology 3 (1964), 39-52.

[4]

E. H. Brown and F. P. Peterson, On immersions of n-manifolds, Adv. in Math. 24 (1977), 74-77.

[5]

E. H. Brown and F. P. Peterson, On the stable 9

r+ 9

decomposition of Q ST 287-298. [6 ]

, Trans. A.M.S. 243 (1978),

G. Carlsson, G. B. Segal’s Burnside ring conjecture for (Z/2)k , Topology 22 (1983) 83-103.

[7]

F. R. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), 99-110.

[8 ]

F. R. Cohen, T. Lada, and J. P. May, The homology of iterated loop spaces, Springer Lecture Notes #533, 1976.

[9]

F. Cohen, M. Mahowald and R. J. Milgram, The stable decomposition of the double loop space of a sphere, A.M.S. Proc. Symp. Pure Math. 32 (1978), 225-228.

[10] R. L. Cohen, The geometry of Q^S^ and braid orientations, Invent. Math. 54 (1979), 53-67. [11] R. L. Cohen, Representations of Brown-Gitler spectra, Proc. Top. Symp. at Siegen, 1979, Springer Lecture Notes #788 (1980) 399-417. [12] R. L. Cohen, Odd primary infinite families in stable homotopy theory, Memoirs of A.M.S. 242 (1981). [13] J. Lannes and S. Zarati, Derives de la destabilisation, invariants de Hopf d ’order superieur, et suite spectral d*Adams, preprint 1983.

125

ADAMS SPECTRAL SEQUENCE OF 02S3

[14] M. Mahowald, A new infinite family in

2 17** Topology

16 (1977), 249-256. [15] J. P. May, The geometry of iterated loop spaces, Springer Lecture Notes #271, 1972. [16] H. Miller, The Sullivan conjecture on maps from classifying spaces, to appear. [17] V. P. Snaith, A stable decomposition for Lond. Math. Soc. 2 (1974), 577-583.

Edgar H. Brown Brandeis University Waltham, MA 02154

QnS*X, J.

Ralph L. Cohen Stanford University Stanford, CA 94305

V HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES Donald M. Davis and Mark Mahowald^

§1.

INTRODUCTION If

b

is an integer and

there is a spectrum

P^

t>b

which when

is an integer or b >0

spectrum of stunted real projective space

is the suspension RPt/RP^)

will prove in Section 2 that for all odd integers even or infinite integers

t

00,

We b

and

there are maps

e rpt __ » pt-8 b, t b b- 8 of Adams filtration 4, nontrivial on the bottom cell.

DEFINITION 1.1.

P^

is the mapping telescope of the

sequence „t b

V t

r t- 8 * b- 8

^b-8 ,t- 8

pt-16 ^b-16, t-16 „t-24 * b-16 * b-24

The main purpose of this paper is to calculate the homotopy groups

tt

^

(pJ^). D

In [11] (unpublished) this calculation was

used to prove that

P^

is the K-theory localization of

^The authors were supported by N.S.F. research grants and S.R.C. research grants.

126

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES P.*', b

and some related results.

127

A more elementary, and less

computational, proof of these localization results was achieved by H. Miller and will appear in [12].

The present

paper is culled from [1 1 ], for it is felt that the method of calculation of tt (P-5) X D Let

bo

denote the spectrum for connective real

K-theory localized at from

bo

is of independent interest.

2,

4 2 bsp

the spectrum obtained

by killing the first 3 homotopy groups, and

the fiber of

\/^-l:bo -» 2^bsp ([15],[17]).

J

The following

result is proved in Section 3 using bo-resolutions.

THEOREM 1.2.

tt (P^)

The Hurewicz homomorphism

^

b

J^(P^) ^

b

Is an isomorphism.

In Section 2, we will calculate 1.2

TTw(P^) 7K D .

J~(P^)» und hence by ^

b

A novel feature of this calculation will be

the use of Adams-type homotopy charts with negative (as well as positive) filtrations. In Section 4 some

K^-localization results in addition

to those of [12] are discussed. that

2P^n 1

is the

For example, it is proved

K -localization of the ^

spectrum, and that the K^-localization of cofibration with

P^

S^

Moore

fits into a

and the rational Moore spectrum.

In Section 5, we show that if K -localization map

mod 2^n

— { 00 S /2

— P., ,

^ bu

is applied to the

then the cofibre is

128

DAVIS AND MAHOWALD

essentially a Brown-Comenetz dual ([16]) of

3

2 bu.

We

generalize this argument and obtain the following universal coefficient theorem.

THEOREM 1.3.

If

G is any divisible torsion abelian 00

group, such as

Z/p

or Q/Z,

X

and

is any spectrum,

there is an exact sequence -skun (X;G) -» KUn (X;G) -» Hom(kun 2 (X) ,G) -> kun+ 1 (X;G) -> .

We would like to express our gratitude to the University of Warwick and especially John Jones, who organized a seminar in Autumn, 1982, to study [19], out of which many of these ideas originated.

The first author

thanks the Institute for Advanced Study, where [11] was written, and Haynes Miller.

§2.

CALCULATING

J (r S by

The spectrum T(bft_^),

where

can be defined as the Thom spectrum b is possibly negative and

Is the

^ Hopf bundle over

RP

* Alternatively it may be defined

using James periodicity as in [3]. A map has (Adams) filtration written as a composite of Z^-cohomology. map

Sn

X

We use

Z^

has filtration

s

> s

if it can be

maps, each trivial in

and s

Z/2

interchangeably.

if it is detected in

A

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES S X Ext^(H X, from s.

X

Let

^s^ X

by killing

Ext

•

denote the spectrum obtained classes of filtration less than

00

We often write

129

P^ as

P^.

The following result is

similar to one used by Lam in [13].

PROPOSITION 2.1.

For all odd integers

t, there are filtration-k maps

infinite integers

e b,t

Proof (H . Miller). P^

t

(P^_g)

(P^_g)^^-

Hence

where

L

16c

lifts

By Adams’ edge ([2]),

pt b- 8

f

rpt

.

1 b-8 ;

factors as c

t rb

7

fpt ., irb-8 j

as always, denotes the collapse map. t = °°,

S-dual of

i f

pt b- 8

If

K .

and

is (b-1 )-connected and so the composite

is trivial.

c,

K

Since it has filtration 4,

pb-1 b- 8

where

, pt- 8 b- 8

b

which induce isomorphisms in

to a map

ft and even or

P^

we are done. (Here

Otherwise,

T = L - b + 7

is highly 2-divisible.)

let

and

T P^ 0 o—o

B = L - t + 7 ,

Then the argument of the

preceding paragraph gives a filtration-4 map

T f T P^ -- » ^B-8 *

whose S-dual is a filtration-4 map can be lifted to

P* ~ b- 8

(P^ v b-8 '

be an

which The connectivity

argument of the previous paragraph can be applied to factor this map through

DAVIS AND MAHOWALD

130

The diagram T f T P -i-* P B B -8 •16

\ / T P B -8 shows

f

7*

is injective in

surjective in

K^( ).

K n( ), and hence

—i

(Df)

is

Now the diagram P1 b- 8 c

shows

^ £

Df

P t J L p t_8 b b- 8 0

is an isomorphism in

K ( ).

It is clear from 1.1 that there are equivalences pt __ , pt-8 i

b

That

P^

b -8 i '

is independent of the choice of maps

satisfying 2 . 1

is not so clear, but it follows from the

uniqueness of

K^-localization.

We use homotopy charts

See [11] or [12].

s t E ’ (X)

with the usual

(t-s,s)-coordinates (e.g., [15; pp. 93-95], [8 ; p.41], [10; p. 149]).

For a sequence of filtration-4 maps such as

those in 1 . 1

xo - * x i - *x2 - . . .

,

we form a chart for the mapping telescope r s ,t T r S - 4 i E r (X) = lim Er

i

,t-4irv . (X.)

X

by .

131

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES For example, all homomorphisms in the sequence for tt

(P1 ^ bo) = ko_(P1)

following chart for

are injective, yielding the

E^ = E^.

(See [7], [10], or [15].)

t-s =

Thus

ko^fP^)

TL/2

i = 3(4)

Z/2

i = 1 ,2 (8 )

0

otherwise

The negative filtrations are due to

our reindexing; they

seem essential to the utilization of charts for mapping telescopes. Charts of

ko (P, ) “ D

for other odd

b

are constructed

^^(P^ ^ S^bsp). A chart for

similarly, as are

^(P^

is formed with -,s— 1 ,t ,5 (P, ~ S^sp) E“ *"(Pb ~ J) = E 2 S ’t{Vh ~ b o ) © Eg v*b and for

r > 1,

d

r

towers in dimensions where not

) Ext

s,t r

„s+r,t+r-l

t-s

satisfying

:E

denotes the exponent of 2.

charts, but by [15; 7.1]

is nonzero on u^ft-s+l) = r+1 , Such charts are

they do correspond to a

DAVIS AND MAHOWALD

132

direct limit of charts derived from resolutions of t-8 i ^b-8 i ^ 7.7].

differentials were established in [15; This yields as part of the chart for

):

The homotopy groups are read off as follows.

THEOREM 2.2. Uj+1)+1

Z/2

j = 3(4),

00

j,(p4i±1)

j ^ -1

Z/2

j =

- 1 or - 2

Z/2

j =

4i or 4i+2

Z/2 © Z/2

j =

4i + 1 (8 )

0

otherwise

Next we calculate

*

b

)

when

t

(8 )

is finite.

The

first step is to determine

ko^(P^), which can be found

from charts of

as in [7; §3].

ko fP^ ?*) * b~oi

Typical are the

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES

133

following charts: 11

/ I

/I

V

FIRST TYPE

• /

-1

'1

,r ■

ko^ P3 4 >

/

/

/

SECOND TYPE /

All

'

ko^(P^)-charts are of one of these two types.

b = 4B ± 1

and

t = 4T -1 ± 1.

on towers in dimension

= 4T-1 (8 ),

are on towers in dimensions

for all

= 4B + 3 (8 ).

t = 3 {4 )

chart has the second type;

If

j

and tops

a chart of the first type is obtained; Z/2 (t_b+l)/2

^

Bottoms

Write

If

are

^ B = T(2),

ko^(P^) ~ B S T{2),

then the

134

DAVIS AND MAHOWALD _t ^

f Z/2 ^t - b + 3 ^/ 2

if

i = 4T - 1(8)

{Z/2 ^t - b _ 1 ^/ 2

if

i = 4T + 3(8)

A chart for of

kox fP^), v hJ

Jw(P^) “ D

is obtained by summing two copies

one unshifted and the other shifted one unit

to the left and two units down, and inserting differentials by the same rule as was used in establishing 2.2. example, a portion of the chart for

Jw (p J^)

is as below.

Then one can easily write out results such as

PROPOSITION 2.3.

If

e, A € {0,1}, then ' Z/2 © 1/2 © TL/2

T (p8n i ^ 1-2A ;

i = 0,1(8)

2 /2 m ^ >

i = 3,6(8)

TL/2 © TL/23m (i)

i = 2,7(8)

0

For

i = 4,5(8)

135

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES m(i) = min(4n-e+A, 1 + maxfi^Ci+1), v^(±+2))).

where

§3.

PROOF OF THEOREM 1.2. The proof that

^(P^)

injective uses

bo-resolutions. That it is surjective is proved by constructing homotopy classes.

We begin with the

injectivity. After possibly reindexing, it suffices to show that if a ' ■ Sn

P.5 b

.. the composite is trivial.

becomes trivial in

P^ ^ J, b

0n a nt ^b,t S -- > P, > ... b

then for some

^b-8k+8 ,t-8k+8

k

„t-8 k »P, b- 8k

By duality it is equivalent to show that if

f : 2n+^P_^_j -» S^

becomes trivial in

J

then for some

k

the composite o0 ,f 55
tt^(D^X /\ bo), where

D^X

is a

f stable

N-dual

of

X.

If

s > 2,

and

X —

Is

has

s+i

s+1 »I

f Adams filtration such that

p

> 2, then there is a map

.p f - P f . The same is true if s - l s s+1 s- 1 s

has Adams filtration true if

X

s = 1

1

dim X < 5s.

and

f s

The same is also

and the first component of the horizontal

composite 2 _1bo f1

X — -

2 ^bo.

factors through

Now suppose implies

^ »I^ bo ---- » 2 bsp x/ W

>I

f

is asin (3.1).

that it lifts to amap f^

Its triviality in satisfying the

hypothesis in the last sentence of 3.2.

Similarly to [8 ;

4.2], the first hypothesis of 3.2 is satisfied. 2^

has Adams filtration 4,

, .p lifts to a map

vn+1 2

3.2 implies that

b+8K— 1

J

Since each £2^ ... 2^.

t2K+ 1 * ■ Choose

„ K =

Then further liftings satisfy the hypothesis dim X < a map

5s

of 3.2.

vn+l0-b+8k-l 2 P_t+ 8 k_1

Hence f ^

... \ # K+1 •*• \

T2K+4(k-K)+l _ I • Choose

lifts

, ^ , 0 k = n-b-2.

to

137

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES

Then „ (dimension of and hence

^n+l^-b+Sk-K . , . . « P_t+gk_j) ^ (connectivity of

2

f£^ ... ^

factors through a trivial map.

The proof that

^P^

is surJective follows,

for the most part, as in [15; 7.14 and 7.18], with the case

The

T4k-2K+1 I ).

We begin

t = 00.

Z ’s

easily handled.

in

i = 0, 1, 2(4)

Let

e = ±1

and

d = 8 , 9, or 10.

The

top arrow in the diagram below is surjective by [16; Tables 8.2,

8.4, 8 .6 , 8 .8 ]. 7r4k+dP4k+e

J4k+dP4k+e

7r4k+dP4k+e±8i

J4k+dP4k+e±8i

Therefore, so is the bottom one. When

i = 3(4), our work is directed toward proving

THEOREM 3.3.

^ ( N ) = 4e + a > 2

Suppose

with

1 < a < 4.

For 7 9 d =

11

let

f(d)

13 Let

b = N - 8 e - 8y - d

ko^_^(P^) f(d) - e.

with

y > 0.

Then

^^(P^)

maps onto all elements of filtration

> 4y +

DAVIS AND MAHOWALD

138

The surjectivity of

^P^

b

^ ° ^ ows from the

observation that, in the notation of 3.3, ko^_i(P^) in 3.3.

is injective with image exactly that described Sur jectivi ty of

for perhaps an isolated m i b-8k

J-Pi m i b-8k

also covered

-> J^P^

it

i > b)

when

i < b

(and

requires surjectivity of k > 0 , but this is

for appropriate

of course by 3.3 and the sentence preceding

this one.

Proof of 3.3.

By use of the filtration 4 maps

suffices to prove 3.3 when

y = 0.

2

^

(x),

it

The argument is

exactly that of [15; 7.14, 7.15], which we review. There is a commutative diagram M4n+5

* B4n+2

i P4n+3

i ~

* P4n+2

I Q4n+3

.

f, l

i f3

P4n-1

’ Q4n+2

such that i)

the vertical maps are cofibrations which define the spectra

Q

(called

..x I, 04n+5 ii)M. - = S J 4n+5 „ 04n+2 B. 0 = S 4n+2

C

in [15; 7.15];

,, 4n+6 , LL e and 2 .. 4n+4 .. 4n+5; U e LL e 77 2

^Except for the case e=4, d=7, where we need to use y=l, and this case is implied by the case e=4, y=0, d=13.

139

HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES iii)

the maps

iv)

the maps

f

and

have Adams filtration 1;

induce monomorphisms in

ko^(

).

Remark 3.5. a)

b)

asArmodulesare

H*Q4n+3and

H*Q4n+2

and

a a) of [7;3.6],

^4n+\

Theproof of

the existence and properties of (3.4) is

quite clearly presented in [15; p. 104] and so is omitted here.

PROPOSITION 3.6. nontrivial in

If

H (

u (N) = 4e+e > 2,

there is a map

; 7L^)

S"_ 1 - V s e - 4 - e

-

1

s" _1 - P h ^ . ^ 2 .

2

" "
0,

is the suspension spectrum of the stunted projective space IRpVlRpk ^ . By consideration of Thom spectra, or by James periodicity [4], the integers negative.

b < t

may be allowed to be

The usual properties, and the canonical collapse

and inclusion maps (which we denote

uniformly

byc),all

extend easily.

telescopes

wemay allow

By forming mapping

t = 00 as we 1 1 .

PROPOSITION D.

For any

b < t < 00 with

b

even, there exists a factorization of the map obtained by smashing P^, as b

16 €

tt^(S^)

odd and

t

1 6 |P^,

with the identity map of

155

MAPPING TELESCOPES AND K^-LOCALIZATION pt+8 __£ b +8

where

K^-equivalence of Adams filtration at least

is a

3 (and exactly 4 if

t = °°) .

Special cases of this have been considered by K. Y. Lam [10]. [9]).

There is an evident odd-primary analogue (cf.

The second author has shown that in fact

may be

chosen with filtration (exactly) 4.

COROLLARY E. n

■j . let

Let

b

be odd and „t-8 n P, _ b- 8n

, ^t-Sfn-l) k/(p2 - p - 1 )

The existence of such exactly

k/2 (p-l) (for

k = 2(p-l)p

r—1

if

p > 2.

A , with Adams filtration r

k =8

if

p =2

r < 2,

and

and

otherwise) is standard, using the Adams

conjecture (see, e.g., [8 ], Prop. 2.3).

Results of this

type have been obtained independently by M. C. Crabb and K. Knapp [7] and L. Schwartz [15] and apparently also by S. Oka.

§1.

FORMULATION OF KNOWN RESULTS. Our starting point is essentially a special case of

Theorem B, which we treat as given for the purposes of this paper.

We hope that this formulation recommends itself to

other workers. When

p > 2,

C(r])

where

We deal with a specific finite spectrum Y. Y = M = C(p|S^); but when

rj : S^ -» S^

expression of the theorem,

p = 2,

is the nontrivial map. B

Sq

if

if

and by

THEOREM 1.0.

Qq = p

and

There exists a map

that (i)

H C(a)

(it)

a^Y

is free over is

K^-iocai.

B.

In the

will denote the sub Hopf

algebra of the Steenrod algebra generated by p = 2,

Y = M ^

= [P*,j3]

a '

1

^ Y -» Y

and p > 2.

such

Sq

2

MAPPING TELESCOPES AND K^-LOCALIZATION

157

We make some remarks on existing proofs of this result.

When

p > 2, Bousfield [6 ] showed how it follows

from Miller’s calculation of the Adams conjecture. out for [11]

A similar deduction may be carried

p = 2, starting from Theorem 1.0 of [11].

To explain this, let

be the spectrum of connective orthogonal

localized at

p,

let

ko[4]

1 : ko -» ko,

chosen in the

usual way.

j

^([6 ],[11]).

The unit map

j.

Y, a

' ko

-1

K-theory

be its 3-connected cover, and

k ko[4] lift \p -

let

S

But in

Mahowald actually proves more, allowing one to avoid

reference to the Adams conjecture. ko

in [12], by use of

ir^(a

Let S

where k

denote the fiber of

ko

lifts uniquely to

In [11] it is proved that for suitable

with Y ^ j

H^C(a)

B,

is an equivalence.

lemma, that

LEMMA 1.1.

free over

a *Y

is

the map

e: 2

a ■ 2 Y

1 ^ e • ’a

-»

It then follows, by the next

is K^-local.

A p-local spectrum

X

is

K^-Iocal if

1 ^ e :

X -» X ^ j is an equivalence.

We prove this lemma at the end of the section. When

p > 2, [22 ^P_ 1 L,Y] = Z/pZ,

and any two

nontrivial maps differ by an automorphism of the source (or target).

When

p = 2,

however, more variations exist, and

we pause to comment on them.

158

DAVIS, MAHOWALD, AND MILLER There are eight maps

is free over

B.

2

a • 2 Y

Y

such that

x H C(a)

Each is (consequently) a K^-equivalence.

They fall into four orbits under the action of distinguished by

Aut Y,

Sq |H C(a). The paper [11] deals with

one having special properties; but our central result, Theorem B, clearly implies that (1.0) holds for any such a. One may prefer to deal with one of the 32 K^g

equivalences

A : 2 M

M;

cf. [3,8].

For example, M. C.

Crabb and K. Knapp [7] have recently shown that (for any p) there exists such an ’’Adams map”

A

for which

A ^M

is

K^-local, by a method (explored inconclusively in 1977 by M. G. Barratt) avoiding the Adams spectral sequence technology of [11] and [12].

We make two remarks about

this result. First, an easy Adams spectral sequence calculation g

([8 ], p. 634) shows that all maps filtration at least 2.

2 M -» M

have Adams

If we grant Theorem B, therefore,

all their mapping telescopes are

K^-local, since

8 < 10.

g

If

A :2 M

M

A ^M

is a

M

isa

K^-equivalence, it then follows that

K - localization.

Thus our work, together

with 1 .0 , implies their result. Second, the result as formulated by Crabb and Knapp can probably be made to yield (1.0).

Those authors do not

control the Adams filtration of their map

A.

However, it

MAPPING TELESCOPES AND K^-LOCALIZATION

159

seems likely that a direct calculation would show that for any

K^-equivalence

Adams filtration A

A :M

4k

2 ^M, A^ : M

for some

k > 0.

we may as well assume that

A

equivalences of Adams filtration 4.

2

a : 2 Y -> Y, 28m

is one of the two

, 2 8y

The easy argument of A,

and any

----- > Y

| A

----- » 22M

where the horizontal arrows are the canonical maps. a ^Y = C (77) ^ A ^M,

is nilpotent it follows that only if

A

K -

, 2 10 m

|

telescopes, we find that

In­

there is a commutative diagram

| A

M

has

Thus in considering

[8 ], top of p. 621, shows that for such equivalence

2

a ^Y

is

Taking

and since

rj

K^-local if and

is.

Following the argument of Bousfield ([16], Thm. 4.8) we note that (1.0) leads to a characterization of

K - local

spectra:

COROLLARY 1.2. only if

A p-local spectrum

C(a) ^ X = *,

where

X

is

2

a : 2 Y

Y

K -local if and satisfies

( 1 . 0 ).

We now return to a proof of Lemma 1.1. will be localized at

p

without mention.

All spectra

160

KO

DAVIS, MAHOWALD, AND MILLER denote

k map - 1 :

Let

J

KO.

Then Adams and Baird, and Ravenel [14] (see

[6 ]), prove that

the fiber of the stable

is the fiber of a map

which is an isomorphism in j -» J

lifts to

0 : j

Q 0

S^.

J

also

S

Thus the natural map

The localization map

S -»

can be chosen as the composite

It is easy to check that the homotopy of the cofiber S

j

is bounded below and p-torsion,

homotopy of the cofiber

D of

j

iv

C

of

and that the is bounded above

and p-torsion. Now assume that X ^ C = *

so

H^(X) = 0 . tower of

X

X ^ j

H^(X) Hx (C) = 0 .

is an equivalence. But

Then

H^(C) * 0,

so

It follows by induction over the Postnikov E

that

X ^ E = *

for any

p-torsion spectrum

E

with only finitely many nonzero homotopy groups.

D

is a mapping telescope of such spectra,

follows that

X -» X ^

Since

X ^ D = *.

is an equivalence; i.e. , X

It is

K - local. On the other hand, the results of = 0; S,, K

so the same arguments show that is a

diagram

[1] show that D ^ K = *,

H^(K)

and

K -equivalence. Now consider the commutative X

j

MAPPING TELESCOPES AND K^-LOCALIZATION If

X

is

161

K^-local, then the remaining horizontal arrows

are equivalences, and it follows that

X -» X ^ j

is an

equivalence.

§2.

PROOF OF THEOREM B Let

a

be as in Section 1, and write

We show that for all

n > 0 there exists

k 1 ^