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

Table of contents :
CONTENTS
Preface
I EXPONENTS IN HOMOTOPY THEORY
II THE EXPONENT OF A MOORE SPACE
III THE SPACE OF MAPS OF MOORE SPACES INTO SPHERES
IV THE ADAMS SPECTRAL SEQUENCE OF Ω^2S^3 AND BROWN-GITLER SPECTRA
V HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES
VI MAPPING TELESCOPES AND K*-LOCALIZATION
VII THE GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION
VIII EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS
IX THE ROLE OF THE STEENROD ALGEBRA IN THE MOD 2 COHOMOLOGY OF A FINITE H-SPACE
X MAPS BETWEEN CLASSIFYING SPACES
XI GENERIC ALGEBRAS AND CW COMPLEXES
XII DEFORMATION THEORY AND THE LITTLE CONSTRUCTIONS OF CARTAN AND MOORE
XIII FREE (Z2)^3 - ACTIONS ON FINITE COMPLEXES
XIV EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA
XV A DECOMPOSITION OF THE SPACE OF GENERALIZED MORSE FUNCTIONS
XVI ALGEBRAIC K-THEORY OF SPACES, CONCORDANCE AND STABLE HOMOTOPY THEORY
XVII THE MAP BSG → A(*) → QS^0
XVIII VECTOR BUNDLES, PROJECTIVE MODULES AND THE K-THEORY OF SPHERES
XIX LIMITS OF INFINITESMAL GROUP COHOMOLOGY
XX ALGEBRAIC K-THEORY OF GROUP SCHEME ACTIONS

Citation preview

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.



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.



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



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.



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 ^

0

V ^

C(a).

such that *s

nullhomotopic. The result then follows using Corollary 1.2 . Suppose Then where

H.(V ^ X ) = 0 iv n' v = 6 Let

Since

H.(X } = 0 lv n y

if

m =5

for

p =2 ifp = 2

H^(V ^

for

i < -b n

i where

certain constant depending only on resolution for

V ^ ^n+k

H*(V ^ ^n+k) ; then the

if p > 2 .

the vanishing line

results of Anderson and Davis [5] (p = 2) show that

i > t + r, n

if p > 2 .

*s ^ree over

Wilkerson [13] (p > 2)

i > t . n

p.

c

is a

Build an Adams

us*nS a minimal resolution for s ^ cover

ES

is

(ms - (c +

- 1 )-connected. Now at least

1 ^

v ^X -> V ^ X , has Adams filtration n n+k fk fk, so itlifts to E , and hence isnull

provided that

:

162

DAVIS, MAHOWALD, AND MILLER

(*)

tn + v < mfk - (c + bn + k ).

5 > 0

By assumption there exists i > 0

there exists

j > i

such that for all

for which

b. (**) v J Pick

i > n

=

j > i

§3.

< mf - p sib

f/^^

|tr

pb/^ ^ pb If

t = °°,

7rf

filtration exactly 4,

is our map

tf>. Since

does too.

spectral sequence shows that in p

Kq

— 16— > p

\/: p b+8

appears as

16 IP^

^as

The Atiyah-Hirzebruch the diagram

163

MAPPING TELESCOPES AND K^-LOCALIZATION 16

2 oo

'

ie\

so

200

/

^

^ is an isomorphism. If

t < °°,

we consider the canonical cofibration kb

b

Smashing with an Adams tower

-»p^ i • t+1 {S}

for the sphere

spectrum gives a sequence of Adams towers which is a cofibration sequence at each stage. map

?rf

lifts to a map

The filtration four

g : ^k+g

^b ^

t+8

claim that the composite map

anc^ we ^ S

in the

diagram Pb+I “ £ — * Pb +8 - S— i \

I P* ~ S ^ b

is null-homotopic.

i

) P, ~ S -C^ b

-> P,. , ~ S t+1

This is the outcome of an easy lifting

argument, again using the Adams edge. g

Pt+ 1 ~ S

» Pb ~ S

results, and the composite

The dotted lifting

t+8 t 7rg : P. 0 -» P. D+o D

is our

filtration 3 map . By construction we have a commutative diagram pt+8 b +8

7Tg

t b

p —El b +8

»p b

*

164

DAVIS, MAHOWALD, AND MILLER

We saw that in

Kq,

7rf

induces an isomorphism.

vertical arrows induce monomorphisms, so But

t+g ^o^b+8

so

§4.

?rg

The

does too.

£ anC^

are

SrouPs of equal order,

is an isomorphism.

THE "TELESCOPE CONJECTURE" We expand somewhat on Ravenel’s description of the

construction of a K-theory self-equivalence of a finite p-torsion spectrum.

Let

identity map has order extends to a map

X p

i*

for some

ji ' X ^ M^

introduction, there is a for suitable

be such a spectrum.

k;

X.

As

Its

r > 1 , and hence noted in the

K -equivalence x

A :2 M

M

r

r

so we may form the composite

: 2kX — =-- > S^X ^ M r

A -> X ^ M--— ^--> X . r

We claim this is a K^-equivalence. To see this, notice that if e ^ 1 : M^ Since

A

e : S

K ^ M^ is a

K

is the K-theory unit map then

i s a generator of

K^-equivalence,

is a generator of

A(e^l)=(e^l)oA

;Z/pr) . But if

is a power of Bott periodicity, then is also a generator of altering

A

K^(Mr ;Z/pr) = Z/p1*.

j3 ■ K ^

(p ^ 1) o 2

K^(2^Mf ;Z/pr) = Z/pF ;so,

k

K (e ^ 1) by

if need be, we may assume they are equal.

Hence their K-module extensions

K^ S

^ M

r

-»K^M

r

are

MAPPING TELESCOPES AND K^-LOCALIZATION

165

We now form the following commutative diagram. t K (Sk - X) -- £-» K (S

- M

3

- X)

P * U A*

and the result follows. The map

(f> clearly has Adams filtration at least

equal to that of

A, which we may take to be

k/2(p-l).

Corollary C, asserting that the mapping telescope

is

K^-local, now follows from Theorem B. We end with some further comments on Ravenel’s conjecture ([14], (10.5)).

First, notice that the

Bousfield class of

M = C(p|S^),

spectra

X

which are

nontrivial.

spectrum has

K -local, and

K^~equivalence of a finite p-torsion K -local mapping telescope.

false, however.

For instance,

and let

H^(P;Z/p) = 0

P :V

let

—6

2

V

V

p = 2) . Let

K^-equivalence with

K^Y ^ 0

is not contractible,

evidently a counterexample.

be a space with

p > 2,

a ’ Y -» 2

and

a0 V

This is clearly

be a non-nilpotent map with

(e.g., Smith [16] when

Mahowald [8] when

P

p-torsion,

is the class of

The conjecture thus seems to assert that any

HZ/p^-trivial

K^V = 0

K ^ M,

Davis and be a

H^(a;Z/p) = 0. :Y V V

Since

2 ^eY V V

is

166

DAVIS, MAHOWALD, AND MILLER We note also that Ravenel offers a proof of the

telescope conjecture for K-theory (Theorem 10.12).

The

proof actually addresses the more natural question of the existence of appropriate self-maps, but even as such appears incomplete —

why is

X

a module over

in the

last paragraph?

REFERENCES [1]

J. F. Adams, On Chern characters and the structure of the unitary group, Proc. Camb. Phil. Soc. 57(1961), 189-199.

[2]

J. F. Adams, A periodicity theorem in homological algebra, Proc. Camb. Phil. Soc. 62(1966), 365-377.

[3]

J. F. Adams, On the groups J(X). (1966), 21-71.

[4]

IV, Topology 5

J. F. Adams, Operations of the n ^ kind of K-theory, 00

and what we don’t know about RP , London Math. Soc. Lecture Notes Series 11(1974), 1-9. [5]

D. W. Anderson and D. M. Davis, A vanishing theorem in homological algebra, Comm. Math. Helv. 48(1973), 318-327.

[6]

A. K. Bousfield, The localization of spectra with respect to homology, Topology 18(1979), 257-281.

[7]

M. C. Crabb and K. Knapp, Adams periodicity in stable homotopy, Topology 24(1985) 475-486.

[8]

D. M. Davis and M. E. Mahowald, v^- and r^-periodicity in stable homotopy theory, Am. J. Math. 103(1981), 615-659.

[9]

D. M. Davis, Odd primary bo-resolutions and K-theory localizations, 111. Jour. Math 30(1986) 79-100.

MAPPING TELESCOPES AND K^-LOCALIZATION

167

[10] K. Y. Lam, K0-equivalences and existence of nonsingular bilinear maps, Pac. J. Math. 82(1979), 145-153. [11] M. E. Mahowald, The image of J in the EHP sequence, Ann. of Math. 116(1982), 65-112. [12] H. R. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure and Appl. Alg. 20(1981), 287-312. [13] H. R. Miller and C. W. Wilkerson, Vanishing lines for modules over the Steenrod algebra, J. Pure and Appl. Alg. 22(1981), 293-307. [14] D. C. Ravenel, Localization with respect to certain periodic homology theories, Am. J. Math. 106(1984), 351-414. [15] L. Schwartz, K-theorie des corps finis et homotopie stable du classifiant d ’un groupe de Lie, J. Pure and Appl. Alg. 34(1984), 291-300. [16] L. Smith, On realizing complex cobordism modules, Am. J. Math. 92(1970), 793-856.

Donald M. Davis Lehigh University Bethlehem, PA 18015

Mark Mahowald NorthwesternUniversity Evanston, IL 60201 Haynes Miller University of Washington Seattle, WA 98195

VII THE GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION Douglas C. Ravenel^

The chromatic resolution is a long exact sequence 0 -» BP^ -» M° -» M1 -> ... of BP^(BP)-comodules (to be specified below) introduced in [3] to study periodic families in the stable homotopy groups of spheres. abbreviate BP^(X)

Given such a comodule

Ext^p ^pp^(BP^.M) by

for a suitable spectrum

spectrum, then

Ext(M)

M,

we

If

M

Ext(M). X,

is

such as the sphere

is the E^-term of the Adams-Novikov

spectral sequence (ANSS) converging to the p-component of

7r^(X) . Using classical homological algebra one derives from the resolution above the chromatic spectral sequence converging to

Ext(BP^) with E*’S = ExtS(Mn ).

This object is interesting for two reasons. the resulting filtration of families’ for various

n,

Ext(BP^) is by

First,

'v^-periodic

so the spectral sequence behaves

^Partially supported by the National Science Foundation.

168

GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION like a spectrum in the astronomical sense. is closely related to the continuous a certain pro - p - group

S . S^

169

Second, Ext(Mn )

mod(p) cohomology of is contained in the

group of units of a certain division algebra over the p-adic numbers.

It is known to contain the p-Sylow

subgroup of the group of units of the ring of integers of any n-th degree extension of cohomological dimension periodic cohomology if

n

Q.

It is also known to have

if (p - 1)|n and to have

(p - 1)|n.

This construction is entirely algebraic and raises the question of an underlying geometric phenomenon.

The object

of this paper is to show that there are spectra

M^

satisfying

BP (M ) = Mn n

and that the homomorphisms in the

resolution are realized by maps between these spectra. Unfortunately this is not a complete solution to the problem as we are unable to show that the ANSS for converges. paper.

M^

We will discuss this question at the end of the

We will show that convergence follows from a

strengthened version of the smashing conjecture (10.6 of [5], which says

LrX = X ^ LnS0).

The problem of realizing the chromatic resolution was discussed extensively in [5], and we shall use results from that paper freely.

A crucial ingredient of our proof,

unavailable when [5] was written, is the existence of

170

RAVENEL

certain finite complexes recently constructed by S. Mitchell [4]. To be more specific, the chromatic spectral sequence is derived from certain short exact sequences of BP^-modules, 0 -> N definedinductively 2

for

M

N

0

n > 0 by

= BP^and

Mn

=

yi

v^ N , where spectra

N

v^ = p. and

M

The question is whether there are with

BP (N ) = Nn

and

BP fM ) =

Mn , and cofibrations realizing the above short exact sequences. N~ = S° 0

We will show this can be done inductively with

and

M = L N , where n n n

localization (see[1] of homology theory compute BP^(L X)

THEOREM 1.

denotes Bousfield

§1 of [5]) with respect to the

v ^BP . To make this work we need to n « in terms of BP^(X). We have

For any spectrum

particular , iff

L n

X,

v \ b P ^ X = pt, n-1

BP ^ L^X ~ X ^ then

BP a L X = X a

n

v_1BP. n

This is conjecture 10.7 of [5].

It is shown there

that the two statements are actually equivalent. described in §6 of [5]. the case

X = N , so n

^BP

The second statement applies to

BP 1M ) = Mn * v n'

as desired,

is

GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION

171

The language of Bousfield equivalence is especially convenient for our purposes (see [12] or 1.19-1.24 of [5]). Briefly, spectra (denoted by

E

and

E ~ F)

if

F

are Bousfield equivalent

E ^ X = pt. iff F

X = pt., or

equivalently if the corresponding homology theories give the same localization functor. is denoted by . >

if

the class of of

E

These classes are partially ordered by

E ^ X = pt. implies

F ^ X = pt., i.e., if

E^-acyclic spectra is contained in the class

F^-acyclic spectra.

and

The equivalence class of

Hence

is the biggest class

is the smallest.

The classes also admit wedges and smash product.

In

some (but not all) cases there is a complementary class C =

satisfying

E v F ~

and

E ^ F ~ pt.

The

collection of such classes is a Boolean algebra denoted by BA.

Bousfield [2] shows that any wedge of finite spectra,

represents a class in

BA, while

BP

and

H (the integral

Eilenberg-MacLane spectrum) are known not to ([5], 2.2 and 3.1).

THEOREM 2. n > 0

(Mitchell [4]).

For each prime

there is a finite p-local spectrum

v_1,BP fX ) = 0 n-1 nJ

and

v_1BP fX ) * 0. n nJ

p X^

and each

with

RAVENEL

172

Mitchell constructs such complexes explicitly as stable retracts of the homogeneous spaces

U(pn)/(Z/(p))n

using the Steinberg idempotent, but the only property of X n

that we need here is that stated in the theorem.

V(n - 1)

of Smith [6]

The

and Toda [7] have the same

property, but are known to exist only for small values of

In [5] we made some conjectures (10.4 and 10.8) concerning the Structure of

FBA, the subalgebra of

BA

generated by classes of finite spectra and their complements, namely that each p-torsion finite complex is Bousfield equivalent to some

and that these classes

generate the ‘p-component* of that C n

FBA.

We also conjectured

is < ^ S/(q) v v_1BP > qj*p n

where the wedge summation is over all primes other than Mitchell’s theorem shows that

FBA

is at least as big as

expected. To prove Theorem 1, let localization map

X -»L X, n

C^X

be the fibre of the

Consider the two fibrations

C BP a L X ----- » BP a L X n n n

» L BP A L X n n

L BP a C X ----- > L BP a X n n n

> L BP a L X. n n

If we can show

C BP A L X n n

and

L BP A C X n n

are

contractible, then the result will follow since BP a L X n and

X a L BP n

would both be equivalent to

L X a L BP. n n

p

GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION

173

The following was essentially proved in [5]; details will be given below.

LEMMA 3.

L BP - v_1BP n n

P(n + 1 )

is the

and

C BP ~ P(n + 1), n v J

where

BP-module spectrum satisfying

T7gp(n + 1) = BPx/(p,v1,v2 ... vn).

Now

v ^BP (C X) = 0 n “ n

implies that shows

L BP a C X n n

C BP a L X n n

P(n + 1)

is

is

by definition so the lemma is contractible,

iff P(n + 1) /s L X is. v J n

v^RP^-acyclic, we know

In other words the

L X -cohomology of n **

trivial, but we need to know the P(n + 1) vanishes.

The lemma also Since

[P(n + l),LnX] = 0. P(n + 1 )

is

L X -homology of n ^

The difficulty is that a vanishing

generalized cohomology group on an infinite complex such as P(n + 1 )

does not give a vanishing homology group.

We

need Mitchell’s cpmplexes to surmount this obstacle.

LEMMA 4.

P(n) - X ^ P(n).

To finishthe proof of Theorem 1 we have n+1

^ L X) = [DX 1 ,L X] n J L n+1 n J

v *BP -acyclic. n * J

Rut

[2] and the latter is

DX ,1~ n+1

which vanishes if DX

. n+1

is

X ,1by Proposition2.10 n+1

of

v BP -acyclic by Theorem 2. n “

Hence

174

RAVENEL pt.

L X ^ P(n + 1) n v '

n+1

X

. A L X

n+1

n

^ Pfn + 1) a L X ^ C BP a L X v ' n n n and Theorem 1 follows. We still need to prove Lemmas 3 and 4, and we do the latter first.

P(m)^(Xn )

can be computed with the

Atiyah-Hirzebruch spectral sequence, which will collapse for large enough

m

since

X^

is finite.

P(m)

PW *(X n) = ^

Hence we have

® H.. Xn ;Z/(p)

which is therefore a free

7r P(m) -module for

A standard argument shows

P(m) ^ X^

suspensions of

P(m)

m

large.

is a wedge of

so Xn - P(m) ~ P(m).

By 2.1 (c) of [5], the

P(n) ~

V K(i) v P(m), n L X

AS

n

B p ( s ).

Then we have equivalences X

an

Lli p ( s ) AS L BP -=-» L X AN B p ( s ) AS L BP

n

^

X

n

AS

gp(s)

A L

n

BP

n

s:

n

L X

n

A

BP*S)

AS

BP

where the last equivalence is given by Theorem 1. gives the desired

This

L^BP^-equivalence

X a L B P (s+1) ^ L X a ^ n n

1).

i

REFERENCES [1]

A. K. Bousfield, The localization of spectra with respect to homology, Topology 18(1979), 257-281.

GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION

179

[2]

A. K. Bousfield, The Boolean algebra of spectra, Comment. Math. Helv. 54(1979), 368-377.

[3]

H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic Phenomena in the Adams-Novikov spectral sequence, Ann. of Math. 106(1977), 469-516.

[4]

S. A. Mitchell, Finite complexes with A(n)-free cohomology, Topology 24(1985), 227-248.

[5]

D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106(1984), 351-414.

[6]

L. Smith, On realizing complex bordism modules, Amer. J. Math. 92(1970), 793-856.

[7]

H. Toda, On realizing exterior parts of the Steenrod algebra, Topology 10(1971), 53-65.

Douglas C. Ravenel University of Washington Seattle, WA 98195

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

§0.

BACKGROUND AND MOTIVATION We start with a brief explanation for our interest in

diagrams of spaces (where by ’’space” we mean simplicial set or topological space).

0.1

Diagrams of spaces. Let

S

be the category of spaces and let

small category.

has as objects the functors

A

transformations between them. an

S

A

or indexed by

defined as a map

object

A € A,

f: X

the map

fA:

the category which

An object of

A.

be a

and as maps the natural

A-diagram of spaces or a diagram

shape of A S^

A S^

Then we denote by

A

S^

is called

of spaces with the

With a weak equivalence in Y €

A

such that, for every

XA YA € S

is a weak

*This research was in part supported by the National Science Foundation.

180

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS (homotopy) equivalence of spaces, the category

S^

181

has a

rich and interesting homotopy theory ([1,p.314],[7]), with suitably defined homotopy classes of maps, function complexes, fibration and cofibration sequences, etc., which generalize the usual homotopy-theoretical constructions in S. It is sometimes useful (although never necessary [5]) to allow

A

to be simplicial, i.e., enriched over

simplicial sets, and to consider the resulting diagram category

0.2

A S^

of simplicial functors

A -* S.

Restricted diagrams. In many contexts it is natural to single out a full

subcategory

A

1,

Xa^: Xn^ -» Xl^ € S

(homotopy) equivalence

irn (Xl ) u **

A $ S ^ ’ , where

Xn^

A X € S^:

the product of

gives rise to a weak

(Xl^)n € S,

and

the resulting abelian monoid structure on

is actually an abelian group structure, Loop spaces [8,10],

k-fold loop spaces [15],

k-fold suspensions [14], etc.

0.5

Usefulness of diagrams. Not only are diagrams everywhere in homotopy theory,

but they are also useful, as a general result in diagram

184

DWYER AND KAN

theory can have many different and seemingly unrelated specializations.

For instance, [4] contains a general

classification theorem for diagrams of spaces which specializes to (i)

the usual classification of bundles [9],

(ii)

a general classification theorem for equivariant

homotopy theory [5], (iii)

a (so far unexplored) classification result for

infinite loop space structures on a space (iv) for all

X,

a classification of spaces of the same n-type n [11,13],

and many other results of the same sort.

These

specializations are in some sense automatic, though from a practical point of view there is, in each special case, some work to be done to rewrite the general classification formula in a useful and suitable form.

§1.

INTRODUCTION.

1.1

Summary. Let

S

denote the category of simplicial sets (or, if

the reader prefers, topological spaces), small category, and let can consider the category

U C A A

let

A

be a subcategory. of

be a Then one

A-diagrams of simplicial

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS sets (which has as objects the functors

A

S

185

and as maps

the natural transformations between them), as well as the A U A S ^ ’^ C S^

full subcategory (i.e., functors

A

S

of

U-restricted

A-diagrams

which send all maps of

(homotopy) equivalences in

U

to weak

S). These categories come up

naturally in homotopy theory (0.3) and an obvious question to ask is when a functor

f: (A,U)

(B,V)

between two

pairs of categories induces an equivalence of homotopy theories (1.3)

f*:

S^'H.

The main aim of this note

is to show that this question has a surprisingly nice answer, namely that the induced functor

f*:

is an equivalence of homotopy theories iff weak equivalence

L f : L(A,U) -> L(B,V)

simplicial localizations (1.3).

S^ ’H f

induces a

between the

Actually we prove a

simplicial version of this result.

1.2

Organization of the paper. After fixing some notation and terminology (1.3), we

state our main result (2.1 and 2.2) and derive a few of its immediate consequences, the most interesting of which are that (2.4) the homotopy theory of infinitely homotopy commutative

A-diagrams is equivalent to the homotopy

theory of (strictly commutative) A-diagrams and that (2.5

186

DWYER AND KAN

and 2.6) every simplicial category is weakly equivalent to the simplicial localization of a "discrete" category. Next we show (in §3) that a functor

f: A

B

between

two small categories gives rise to a pair of adjoint functors

with many nice properties.

And finally (§4) we use some of

these properties (3.4, 3.8 and 3.9) to prove our main result. 1.3

Notation, Terminology, etc. We will freely use the following notation, terminology

and results, most of which can be found in [2, 3, and 6]. (i)

Simplicial categories. As usual simplicial categories will be assumed to

have the same objects in all dimensions.

If

A

is a

simplicial category, then the simplicial set of maps between two objects A honwfAj.A^) (ii)

or

A ^ , A^

€ A

will be denoted by

homfA^.A^).

Weak equivalences between simplicial categories. A functor

f: A

B

between two simplicial

categories is called a weak equivalence if (a) induced map

for every two objects homfA^.A^)

hom(f A^ ,fA^)

(homotopy) equivalence, and

A ^ , A^ € A, is a weak

the

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS (b) 7Tq B A

is

187

every object in the "category of components" equivalent to an object in the image ofir^f.

If

and Bare "discrete", such a weak equivalence is just

an equivalence of categories. Two simplicial categories will be called weakly equivalent whenever they can be connected by a finite string of weak equivalences. (iii)

Weak r-equivalences. A functor

f:

A

B

between two simplicial

categories will be called a weak r-equivalence if (a) induced map

for every two objects homfA^.A^)

hom(fA^,fA^)

A ^ , A^ € A, the is a weak

equivalence, and (b) 7Tq B

is

every object in the "category of components" a retract of an object in the image of

Clearly, every weak equivalence (ii) is a weak r-equivalence and a weak r-equivalence which is "onto on objects" is a weak equivalence. (iv)

The standard resolution of a category. The free category on a category

category

FA

which has the same objects as

A

is the free A

and which

has exactly one generator for every non-identity map of

A.

188

DWYER AND KAN

The standard resolution of a category category

F^A

which in dimension

(k+l)-fold free category

F^+ ^A

k

A

consists of the

and which has the obvious

[2, 2.5] face and degeneracy functors. functor

F A -» A /V/

/V

is the simplicial

The canonical

is a weak equivalence and so is for every

simplicial category

B, the corresponding functor

diag F B -» B. (v)

The simplicial localization. The simplicial localization of a small simplicial

category

A

with respect to a subcategory

simplicial category

L(A,U)

defined by

U C A

is the

L(A,U) = diag

F A[F U *], i.e., the simplicial category obtained from diag

F^A

by ’’formally inverting” all maps that are in

F U.

The smallness of

complexes in

L(A,U)

A

/X /

insures that the function

are small, i.e., that

L(A,U)

is

indeed a simplicial category. (vi)

category

Simplicial diagrams of simplicial sets. Let

A

si

of

be a small simplicial category.

Then the

A-diagrams of simplicial sets (i.e.,

simplicial functors

A

S

and natural transformations

between them) admits a closed simplicial model category

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS

189

structure in which the function complexes are the obvious ones and a map

A € S^

x*.

is a weak equivalence or a

fibration whenever

for every object

xA: X^A

is so [6, 2.2].

X^A € S

For every object the obvious map isomorphism.

A € A

and every diagram

hom(hom(A,-) ,X) -» XA € S

U € A

is

r: A°^

A X € S^,,

is an

Moreover the correspondence

gives rise to a full embedding If

A € A, the map

A -» hom(A,-) S^. A U A S ^ ’^ CS^

a simplicial subcategory, the

will denote the full subcategory of the

U-restricted

A-diagrams (1.1). (vii)

Equivalences of homotopy theories. Given two pairs of small simplicial categories

(A,U)

and

(B,V),

a simplicial functor

k:

S

which preserves weak equivalences will be called an equivalence of homotopy theories if (a) fibrant object

for every cofibrant object XQ €

equivalences cof ibrant and homfY^.Y^) ^ S

A U

A U

and

and every pair of weak

-» kX^, kX^ Y^

X^ €

B V Y^ € with Y^

fibrant, the induced map

is a weak equivalence, and

homfX^.X^)

190

DWYER AND KAN (b)

B V

every object of

to an object in the image of

is weakly equivalent

k.

This definition is justified by the fact that [3, §4] one has:

If

k:

A U

B V

is an equivalence of homotopy

theories, then the full subcategories of

A U

and

B, V

spanned by the objects which are both fibrant and cofibrant are weakly equivalent.

§2.

THE MAIN RESULT Our main result is (in the notation and terminology of

1.3)

2.1

THEOREM.

Let

f : A -» B

simplicial categories. A

f

be a functor between small

Then the induced functor

* B f : ->

is an equivalence of homotopy theories iff the function itself is a weak r-equivalence

and more generally

2.2

THEOREM.

Let

f: (A.U)

(B,V)

pairs of small simplicial categories. functor

f*:

be a functor between Then the induced

is an equivalence of homotopy

theories iff the simplicial localization L(B,V)

is a weak

r-equivalence.

L f : L(A,U) -»

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS

191

In view of [1, Ch.XI, 8.1 and Ch. XII, 2.6] theorem 2.2 implies

2.3.

A COFINALITY THEOREM.

Let

f: (A,U) -> (B,V)

be a

functor between pairs of small categories such that its simplicial localization

L f : L(A,U) -> L(B,V)

Is a weak

r-equivalence. Then (1)

for every diagram holinuf*Y

B V Y € S^’^,

the induced map

holinwY € S [1, Ch. XII]

is a weak

equivalence, and (it) for every fibrant diagram induced map

B V Y € S^ ’^ (1.3(v))

the

holinuY -» holin£f*Y e S [l,Ch. XI] ~

Is a weak equivalence.

Other immediate consequences are

2.4 A REALIZATION THEOREM.

If an infinitely homotopy

commutative diagram of simplicial sets indexed by a small category

A

is defined as an

F^A-diagram (1.3(iii)), then

theorem 2.1 implies that every infinitely homotopy commutative A-diagram of simplicial sets is weakly equivalent to a diagram induced from an actual A-diagram by the canonical functor

F^A -> A.

192 2.5

DWYER AND KAN A DELOCALIZATION THEOREM.

Every small simplicial

category is, in a natural manner, weakly equivalent to the simplicial localization of a small category with respect to a suitable subcategory.

2.6

Remark.

This result also holds in the not necessarily

small case, if one interprets "simplicial localization" and "weakly equivalent" as in [2, §3].

Proof.

Given a small simplicial category A,

let bA

be

its flattening (i.e., [5,§7] the category which has as objects the pairs n

(A,n), where

is an integer

> 0

A € A is an object and

and which has as maps

(A^,n^) -» (A^.n^)the pairs

(a,q)

where

simplicial operator from dimension and

a

is a map

a: A 1 X

A9 € A

the subcategory consisting is an identity map. L(bA,W)

Z yyH p

n^ )

q

is a

to dimension

anddenote by

of the maps

n^

W€

rv

bA

(a,q) for which

(V

a

Then the simplicial localization

is, in a natural manner, weakly equivalent to A.

To see this denote by as objects the pairs

(A,n)

A

the simplicial category with as above and with function

complexes given by the formula

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS

193

hon5 ^(Aj,n1),(A2 ,n2 ) hom |^A[n^] ,hom(Aj ,A2 )J x horn |A[n2] ,A[n^]j and observe that there are and

j': A -» A, "

obvious functors jL : bA

A

»v

with the latter a weak equivalence.

/VI /V

Moreover it is not difficult to construct a functor k:

A

->

that b*.

A

which preserves weak equivalences and is such

k = id, A

bA -» S ^

while

of [5, §7].

k

is the flattening functor

The desired result now follows by

combining the results of [1, §5] with theorems 2.1 and 2.2. We end with observing

that, in view of [4, 6.5and

6.15], theorem 2.2 applies to

2.7 an

L-COFINAL AND R-COFINAL FUNCTORS. L-cofinal (resp.

the nerve of

f :A

B

be

R-cofinal) functor between small

categories (i.e., for every object (i)

Let

f ^B

B € B

is contractible, and

(ii) the inclusion functor

f *B -» fIB

cofinal (resp., the inclusion functor

is right

f ^B -» Blf

is left

cof inal)). and let all maps

U C A

denote the subcategory which consists of

a € A

such that

fa € B

Then the simplicial localization equivalent to

B.

is on identity map.

L(A,U)

is weakly

194 §3.

DWYER AND KAN A PAIR OF ADJOINT FUNCTORS In preparation for the proofs (in §4) of theorems 2.1

and 2.2, we discuss here a pair of adjoint functors that can be associated with a functor simplicial categories.

3.1

f : A -» B

between small

The first of these is

A HOMOTOPY PUSH DOWN FUNCTOR,

f : S^-» S®. -»* ~ ~

This is

the functor which assigns to a simplicial diagram

X €

A

g the simplicial diagram the bisimplicial (i)

f X €

B-diagram

(fX)n ,xB = ^

which is the diagonal of (fX)

given by the formula

XAq x hom(A0>A 1) x ... x

hom(An_^,An ) x hom(fAn>B) wherethe disjoint union (Aq ,...,An )

of

is taken over all (n+l)-tuples

objects of A

or equivalently given by the

formula [1, Ch.XII] (ii)

(fX)M nB = holim -> ’ -»

where by

X

X :A n ^n and

-» (sets) C ^

j: A^iB

A^

that, for every diagram

Ar,iB * j Xr

S

denotes the functor determined

is the forgetful functor. X €

A

and every object

there is an obvious map i : XA -» (fxX){fA) = ( f % X ) A € S which is natural in

X,

but not in

A.

Note A 6 A,

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS 3.2

195

EXAMPLES. (i)

functor

If f

A

and

B

are both "discrete” , then the

coincides with the homotopy push down functor

of [4, 9.8]. (ii) If, in addition, XII]

fK = holing:

B

is trivial, then [1, Ch.

-» S. ~

~

A key property of the functor for every object

A € A, the

A f : S^

B-diagram

strong deformation retract of the

B S^

is that

hom(fA,-)

B-diagram

is a

f^hom(A,-).

In fact the arguments of [4, 6.4] yield

3.3 € A,

PROPOSITION.

For every diagram hom(fA,-)

the B-diagrams

and

X € S^

and object

f^hom(A,-)

are

cofibrant and can be connected by an obvious pair of maps hom(fA,-) — g S

f^hom(A,-)

and

f^hom(A,-) — ^ hom(fA,-) C

such that (1)

the map

t

(li)

ts = id, and

is natural in

(Hi) st ~ id re I. homffA,-).

A,

A

196 3 .4

DWYER AND KAN COROLLARY.

The diagram ( 1 . 3 ( v i ) ) Aop — E

,

SA

t°p{

IL

Bo p _ j _ ^

SB

commutes up to a natural weak equivalence.

The other functor is

3.5

A HOMOTOPY PULL BACK FUNCTOR

f*: S? -> SA . This is

g the functor which assigns to a diagram f*Y €

such that, for every object

Y €

the diagram

A € A,

(f*Y)A = hom(f^hom(A,-),Y) Its name is justified by the following proposition which is an easy consequence of 3.3.

3.6

PROPOSITION.

The natural transformation

which assigns to every diagram

Y €

f* -> f^

and object

A € A

the map (f*Y)A = hom(hom(fA,-),Y) induced by the map equivalence.

t

hom(fwhom(A,-),Y) = (f*Y)A

of 3.3 Is a natural

Moreover for every object

weak

A € A, this map A/

(f Y)A -* (f Y)A (f Y)A

has as a left inverse the map

induced by the map

s

of 3.3.

(f Y)A -»

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS

197

As expected, one has 3.7

PROPOSITION.

f: A

Let

B

small simplicial categories.

be a functor between

Then

A b * the functors f • S^ «—> S^ :f are -T ~ ~ -*

(i)

’’simplicially” adjoint in the sense that, for every pair of objects

A X € S^

isomorphism of (it)

and

B Y € S^, /V

simplicial sets

there is a natural hom(f^X,Y)

£ hom(X,f Y ) ,

the functor f

preserves weak equivalences

and

the functor f -»

preserves weak equivalences

and

fibrations, (lit)

cofibrations, and (in)

* B f :

the functor

A S^

Is an equivalence of

homotopy theories iff for every diagram every diagram

g Y 6 S^,

X €

and for

the adjunction maps

X -> f*f*X -» -»

and

f^f*Y -* Y

are rneah equivalences.

3.8

COROLLARY.

Let

f: (A,U) -» (B,V)

be a functor

between pairs of small simplicial categories such that (i)

the functor

homotopy theories and

£** : S&* S^

is an equivalence of

198

DWYER AND KAN (It) the restriction

£ |U: U -» V

f* :

restriction

is onto.

Then the

is also an equivalence of

homotopy theories.

Proof of 3.7. (i)

The stated isomorphism clearly holds if

hom(A,-) x K

for some pair of objects

A € A

and

X = K € S

and the general case now follows from the observation that the functor

f

every diagram

respects difference cokernels and that A

X €

can be written as a difference

cokernal

11 XA

x hom(A,A* ) x hom(A' ,-) 5

11XA

x hom(A,-) -» X

where the disjoint unions are taken over all pairs of objects

A,

(ii)

A' € A

and all objects

Ay

The definition of

preserves fibrations.

f

A € A

Ay

respectively.

implies that

f

That it preserves weak equivalences

follows from 3.6. (iii)

Definition 3.1(i) implies that

weak equivalences.

f

preserves

That it also preserves cofibrations now

follows by a formal argument from (i) and (ii), since the cofibrations in a closed model category are the maps which have the left lifting property with respect to fibrations which are weak equivalences [7, I, 5.1].

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS (iv)

The "if” part is straightforward.

’’only if” part, one observes that

f

199

To prove the

induces an g

equivalence between the homotopy categories of A S^.

^ f

By 3.6, so does

functor induced by

f

and

and {(i), (ii), and (iii)) the

provides an inverse.

The desired

result now follows readily.

We end with

3.9 PROPOSITION.

Let

f : A -» B

be a functor between small

simplicial categories. Then the induced functor A S^

x B £ : S^ -»

is an equivalence of homotopy theories iff. (a)

for every diagram

the map (3.1)

A X € S^ A>

X

i' XA-*(f f^X)A € S

and object

A € A, A/

Is a weak equivalence,

and g

(b)

a map

y:

^ the induced map

Proof.



^

is a weak equivalence iff

^

f y: f Y^ -> f Y^ € S^

is so.

A straightforward calculation yields the existence

of a commutative diagram

200

DWYER AND KAN

in which the upward map is induced by the adjunction map and the vertical map is induced by the map

s

of 3.3.

Hence, (3.9) condition (a) is equivalent to:

For every

A X € S^, the adjunction map

A

diagram

weak equivalence.

** X -* f f X € -> -»*

is a

It is not difficult to verify that in

the presence of (a), condition (b) is equivalent to: every diagram

B Y € S^,

is a weak equivalence.

the adjunction map

For

^ f f Y -» Y €

B

The desired conclusion now follows

immediately from 3.7(iv).

§4.

PROOF OF THEOREM 2.1. (i)

The

"if" part.

This we prove by showing that

1.3(iii)(a) and 1.3(iii)(b) imply 3.9(a) and 3.9(b) respectively. For every diagram

X €

A

and object

A € A

consider

the commutative diagram (id*icy()A = (icyQA

XA

(f f„X)A in which

id: A -» A

denotes the identity functor and the

vertical map is induced by

f.

A straightforward

calculation using 3.1(ii) yields that the upward map is a

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS

201

weak equivalence and it follows readily from 1.3(iii)(a) and 3.1(i} that the vertical map is so too.

This proves

3.9(a). To prove 3.9(b), note that 1.3(iii)(b) implies that for every object j: B -» B' , tt^B

B € B, there exists a pair of maps

q: B ’

B € B

is the identity of

such that the image of B

and such that

B’

qj

in

is in the

image of the functor f. It follows that for every map y: g both horizontal compositions in the resulting commutative diagram

are weak equivalences. map

yB

map

yB'

is a weak equivalence of simplicial sets if the is so.

(ii)

The desired result is now immediate. The "only if” part.

implies 1.3(iii)(a). an object

A retract argument shows that the

B € B

To verify 1.3(iii)(b), consider for

the composite weak equivalence (3.6)

(fx f*hom(B,-))B -> (f^f*hom(B,-))B and let

(j: B -* fAQ ,

(f^f hom(B,-))B

q: fAQ -» B)

(hom(B,-))B = hom(B.B) be a vertex of

which goes to the component of the

identity of hom(B.B). B

Corollary 3.4 readily

is a retract of

Then it is not hard to verify

fAO

in

trOB,

that

i.e., 1.3(iii)(b) v y v / holds.

202

DWYER AND KAN

4.2.

Proof of theorem 2.2.

Consider the commutative

diagram:

SBV

XB.F V ___ oF*B[FV_1] _ _L(B.V)

sF*^F*H

»sF^[F^_1] = SL(^'H}

In view of theorem 2.1, it suffices to show that all horizontal functors are equivalences of homotopy theories. For the ones on the left, this is a consequence of 3.8. For the ones on the right, it suffices to functors

F A ^

F A[F U

check that the

and F B -» FB[F V

* ,v ,

/v.

satisfy

3.9(a), and this follows by a diagonal argument [2, 1.4(vii)] from 3.1(ii) and

4.3

LEMMA.

(C,W)

Let

be a pair of small categories

which are free and are such that every generator of

C. Then the functor

a generator of

induced by the localization functor

W

is

* cw crw"1!

q :

q-* C -* C[W ^],

is

an

equivalence of homotopy theories.

Proof. diagram

3-2(1))

In view of 3.9, it suffices to show that for every C

W

Z € S^,’^

and object

C € C,

the map (3.1 and

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS i:

203

ZC -> (q*Z)C = holimq ^C j*Z

(where, as usual,

j denotes the forgetful functor)

is a

weak equivalence. The objects of C[W D^n

ending at and

£2n+l ’

qiC C.

can be considered as maps of

For every integer

n > 0

t*ie full subcategories of

qiC

denoted by are

spanned by the objects of the form -1 -1 c1w 1 ... cw 11 nn

and

respectively, where the Then the inclusions

D0

,

c^

^jZn+l

-1 -1 c1w i ... c w c ,. 11 n n n+1 are in

-+ D0 . ,J2n+l

C

and the

j Z.

in W.

are right cofinal

[4,9.4] and hence induce weak equivalences

holim~

wi

Moreover, the inclusions

^2n * holim j Z

D0 . -* D0 ^2n+l J2n+2

have a left inverse which is right cofinal, and this together with the homotopy invariance of homotopy direct limits [4,9.2], implies that the induced maps , i . i?2n+l , . ^2n+2 holim j Z + holim j Z

, , . , are also weak equivalences.

The desired result now follows immediately from the fact Do *

that

qic

is the union of the

D.

and that

holim~ j Z ~

204

DWYER AND KAN REFERENCES

[1]

A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304 (1972) Springer-Verlago.

[2]

W. G. Dwyer and D. M. Kan, Simplicial localizations of categories, J. Pure and Appl. Alg. 17, (1980) 267-284.

[3]

W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19, (1980), 427-440.

[4]

W. G. Dwyer and D. M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23, (1984) 139-155.

[5] W. G. Dwyer and D. M. Kan, Equivariant homotopy classification, J. Pure and Appl. Alg. 35 (1985), 269-285. [6] W. G. Dwyer and D. M. Kan, Singular functors and realization functions, Proc. Kon. Ned. Akad. van Wetensch. A87 = Ind. Math. 46 (1984), 147-153. [7]

D. G. Quillen, Homotopical algebra, Lecture Notes in Math, 43 (1967), Springer-Verlag.

[8]

G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312.

[9]

N. Steenrod, The topology of fibre bundles, Princeton Univ. Press (1951).

[10] R. W. Thomason, Uniqueness of delooping machines, Duke Math. J. 46 (1979), 217-252. [11] C. Wilkerson, Classification of spaces of the same n-type for all n, Proc. AMS 60 (1976), 279-286. [12] A. K. Bousfield and E. M. Friedlander, Homotopy theory of T-spaces, spectra and bisimplicial sets, Lecture Notes in Math. 658 (1977), Springer-Verlag, 80-130. [13] E. Dror, W. G. Dwyer and D. M. Kan, Self homotopy equivalences of Postnikov conjugates, Proc. AMS 74 (1979), 183-186.

EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS [14] M. J. Hopkins, Formulations of cocategory and the iterated suspension, Asterique 113-114 (1984), 212-226. [15] G. Dunn, Ph.D. thesis, Ohio State Univ. (1985).

W. G. Dwyer Univ. of Notre Dame Notre Dame, IN

D. M. Kan Mass. Inst, of Tech Cambridge, MA

205

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

In the first part of my talk I will develop some preliminary notions about finite H-spaces.

I will give a

bit of history and motivation and describe some of the early results in the field.

In the second part of my talk

I will describe a theorem proved by Emery Thomas in the 6 0 ’s for H-spaces with primitively generated cohomology.

mod 2

A new proof will be introduced and it will be

shown that it is possible to obtain results about finite H-spaces that do not have primitively generated mod 2 cohomology.

Finally, I will describe a number of unsolved

problems involving the mod 2 cohomology of finite H-spaces. To begin, an H-space is a pointed space with a continuous map

p: X x X -» X

X,*

together

such that the two

compositions X x x ----- > X x x — 1--- » X x x X ----- » X x X — ^-- > X are homotopic to the identity.

^"Partially supported by the National Science Foundation. 206

MOD 2 COHOMOLOGY OF A FINITE H-SPACE

207

H-spaces occur naturally in many contexts.

The loops

on any space

B

multiplication.

(denoted

QB)

is an H-space with the loop

This includes Eilenberg MacLane spaces,

and all strictly associative H-spaces by results of Milnor and Stasheff [16,18].

Another interesting class of

H-spaces are the Lie groups.

These objects are actually

manifolds with differentiable multiplication maps, a much stronger restriction than that imposed by the definition of an H-space.

A further characteristic of Lie groups that

distinguish them from other H-spaces is that they have the homotopy type of finite complexes. says that any Lie group group

K

G

is the product of a compact Lie

and a Euclidian space.

looks like

K

A theorem of Iwasawa

So up to homotopy

which is a finite complex.

G

This suggests

the definition of finite H-space, an H-space which has the homotopy type of a finite complex.

Hence all Lie groups

are finite H-spaces, but very few other loop spaces are finite H-spaces. Historically people studied the cohomology of Lie groups rationally and at different primes.

It was Hopf who

first observed that the cohomology rationally of a Lie group was the same as the rational cohomology of a product of odd dimensional spheres.

The interesting feature was

that this result did not depend on the differentiable manifold structure of the Lie group, but instead it only

208

LIN

depended on the finite H-space structure.

This motivated

other topologists to question the nature of the topology of Lie groups and to ask if there are other homological or homotopical properties of Lie groups that do not depend on the existence of a differentiable manifold structure.

The

correct framework for studying such questions seems to be the language of finite H-spaces. For -» X x X

k

a field, one observes that the diagonal map

and the H-structural map

X

ji’ X x X -» X yield maps

H*(X) ® H*(X) ----- » H*(X) H*(X) — --- » H*(X) ® H*(X) . The first map isthe cup product, the second map called the

A

coproduct. With respect to cup product,

is the

coproductis a map of algebras.These conditions define what is called a

Hopfalgebra.

The Universal Coefficient

Theorem tells us

that the homology

H^(X;k) = H^(X)

is

also a Hopf algebra and that the vector space dual of the cup product yields a coproduct in the homology and the vector space dual of

thecoproduct in cohomology yields a

product in homology.

We say that the cohomology and the

homology are dual Hopf algebras. Given an element Ax = x where

in

H (X)

x

Ax

has the form

® 1+ 1 ® x + y xl ® x. . L l l

degx^ + deg x^ = deg x

The element x.

x

and deg x^ > 0, deg

is called "primitive" if

The reduced coproduct of

x,

denoted

> 0.

A x = x ® l + 1 ® Ax

is

MOD 2 COHOMOLOGY OF A FINITE H-SPACE

209

Ax = Ax - x ® 1 - 1 ® x = ^ x! ® x. . L i i The primitives are a module denoted by PH (X).

If

IH (X)

PH (X;k)

or

is the augmentation ideal, then the

module of indecomposables is denoted

QH (X)

and is

QH*(X) = IH*(X)/IH*(X)2 . For technical reasons it will be convenient henceforth to assume that

X

is a finite simply connected H-space.

Then, using no more than the above concepts about Hopf algebras, it is possible to prove the following structure theorems. Hopf shows that * * nl nr H (X;Q) = H (S x ... x s r ;Q) = A(x ,x ,...,x ) n l n2 nr where

deg x

n.

are odd.

i

In particular this tells us no even sphere can be a Lie group.

Borel has similar theorems.

For

p

odd

f. * n J H (X;Z ) = 8 A(x.) % ILp [y.j/y1 : v p / \ LJj J where

deg x^

are odd and

deg y

are even.

For p = 2 f. H*(X;Z2 ) = ® A(x.) ® Z2 [yj]/y5



Here there is no restriction on the degrees of the and

x^

y . . One might begin to wonder if all possible

exterior and polynomial algebras are actually realizable as

210

LIN

the cohomology of finite H-spaces. There is a process called

Some results are known.

p-localization which does not

always yield a finite H-space, but the

mod p

and rational

cohomology are nevertheless finite dimensional Hopf algebras.

Using this process, Adams [2] shows that an odd

dimensional sphere localized at an odd prime is an H-space. This seems to indicate that the prime 2 may be more restrictive.

In fact, Adams shows in his paper on Hopf

Invariant One [1] that the only spheres that are H-spaces at the prime two are

1 3 S , S

and

7 S .

A much more sophisticated approach is necessary to prove such a result.

Recall that the

mod p

cohomology of

any space is a module over the Steenrod algebra. H-space

Given an

X, although one cannot always build its

classifying space, one can build a piece of the classifying space known as the projective plane,

^2^'

coefficients, the cohomology of

is related to

'Prfi

With field X,

by

the following exact triangle [6]: H*(P2X) —

-- » IH*(X)

IH*(X) ® IH*(X) where

i

is a map of degree minus one, and

degree two and earlier. exactness

If

A x

A

is a map of

is the reduced coproduct described is a primitive element of

x = i(y).

H (X)

then by

A computation shows that under these

MOD 2 COHOMOLOGY OF A FINITE H-SPACE circumstances,

2

A(x ® x) = y .

coefficients are

Z ,

Further if the field

then the triangle is an exact

triangle of maps over the Steenrod algebra. to the case y

is

p

211

equals two, then

y

2

= Sq

If we restrict 1 y

if degree of

n+1. In the case when

X

is an odd sphere

Sn ,

if

Sn

were an H-space wou^ exist and H ^ P ^ S 11;^) = 3 ^[y l / y . This comes from the exact triangle. Adams [1] shows that for

n ^ 1,3,7 So"1 -

where

a,.

1

“ lb i

belong to the Steenrod algebra and the

b^

either belong to the Steenrod algebra or are secondary cohomology operations.

Hence if

there is no space the other hand,

is not

1, 3, or 7

H*(Pc>Sn ;2y = S

1

3 and

S

are Lie groups and

the Cayley numbers of norm one. are H-spaces are

n

1

3 S , S

and

On 7 S

is

So the only spheres that 7 S .

From this analysis one

sees the potentially powerful restrictions that are placed on finite H-spaces by the structure of the Steenrod algebra.

Whereas in the study of Lie groups,

mathematicians appealed to the existence of a Lie algebra to analyze the cohomology, in finite H-space theory we replace this analysis with the use of tricks involving the Steenrod algebra.

212

LIN From the point of view of finite dimensional Hopf

algebras, odd spheres produce the simplest Hopf algebras, exterior primitive on one generator.

As we move away from

odd spheres, one might consider the next most elementary case of several generators, all primitive.

Recall there is

a natural vector space map PH*(X;k) ----- >QH*(X;k). We say that

H (X;k)

is an epimorphism.

is primitively generated if this map This reduces to being able to choose

the generators in a Borel decomposition (if to be primitive.

k = Z^)

all

Several people have studied finite

H-spaces with primitively generated cohomology.

The

following is a sample of the known facts: (1)

Milnor and Moore [17]:

H*(X;Zp)

generated if and only if

is primitively

H^(X;Z^)

is associative,

commutative and has no p^*1 powers. (2) Kane, Lin [12,14]: if and only if

H*(X;Z^)

H^fX;^)

is primitively generated

is associative and

commutative. (3) Samelson and Leray [17]:

If

odd degree generators and then

H (X;Z^)

(4) Browder [5]: x H (X;Zp)

H (X;Z ) * P

is exterior on

is associative

is primitively generated. H (X;Z)

has no p-torsion if and only if

is exterior on generators of odd degree.

(5) Browder [4]: then

H*(X;Z ) P

x H (X;Z)

If

H (X;Zp )

has no p

2

is primitively generated

torsion.

213

MOD 2 COHOMOLOGY OF A FINITE H-SPACE (6)

Zabrodsky [24]:

Let

p

be an odd prime and let

be a homotopy associative H-space. is primitively generated then

Then if

H (X;2^)

X

H (X;2^)

is exterior

on generators of odd degree. (7)

Harper [8]:

Let

finite H-space generated and (8)

X

be an odd prime.

with

H (X;Z)

Hubbuck [9]: X

p

Let

X

H (X;Z^)

There exists a

primitively

has p-torsion. be homotopy commutative.

Then

has the homotopy type of a torus.

By looking at results 6 and 7 it is already evident that the primitively generated case is quite subtle also at odd primes.

For the rest of the talk I will focus on a

result of Thomas for finite H-spaces with primitively generated

mod 2

cohomology.

The main reason for looking

at the prime two is that there seem to be restrictions on the degrees of the odd generators at the prime two.

Adams’

result about the p-local spheres indicates that at odd primes the cohomology algebra can have odd generators in any degrees. Returning to Thomas’ work.

In the sixties, Thomas

observed that the following theorem is true (9)

Thomas [20]: and

= 1

If

H*(X;2^)

m°d 2

is primitively generated

then

PHn (X;Z2 ) = Sq^PHn_,e(X;Z2 ) and

Sq£PHn (X;Z2 ) = 0.

214

LIN

A weaker form of this theorem is PH and (X;Z2 ) = 0 or

r > 0

and

k > 0.

This theorem tells us that the cohomology is very tightly woven by the action of the Steenrod algebra.

For

example, a primitive in degree 65 is connected to a 3-dimensional primitive by

Sq

32

Sq

16 8 4 2 Sq Sq Sq . Thomas

proves in other papers that the first nonvanishing cohomology group occurs in degrees 1, 3, 7 or 15

mod 2 and in

the absence of two torsion, it occurs in degrees 1, 3, or 7.

This, of course, depends on the primitively generated

Hopf algebra structure of the

mod 2

cohomology.

The next logical step in studying finite H-spaces is to try to extend results of Thomas to finite H-spaces with nonprimitive generators in their

mod 2

cohomology.

First

of all it is useful to note that there are examples of finite H-spaces that are not primitively generated and for which Thomas’ formulas do not hold. exceptional group

Eg.

One example is the

Its cohomology looks like:

Z9[x-,xc-,x ]

If this Hopf algebra were primitively generated, results of Thomas would imply

2 x^ = Sq x^.

But since there

MOD 2 COHOMOLOGY OF A FINITE H-SPACE is no 7-dimensional generator, we conclude cannot be primitively generated.

215

H (EgjZ^)

It is interesting to note

that every other possible Steenrod algebra connection predicted by Thomas’ theorem in fact exists in

H (E^Z^):

X5 = Sc12x3 X9 = S* \ X 17 X23 X27

= Sq8xg = o 8x 15 = Sq = Sq4x23

X29 “

o 2 q x27

The nonprimitive generator is -

2

Ax1cr = x„ 15 3

®

x^-

We have

2

9

+ xr 8 xr . 5 5

It can be shown that it is impossible to choose another representative for Based on theknown theoremsimilar

x^,_

to make it primitive.

examples one is

to Thomas’theorem

led to conjecture that a for primitively

generated finite H-spaces should exist for all finite H-spaces. First let us examine Thomas’ theorem and try to prove it in a slightly different way.

Thomas’ original proof

relies on the cohomology of the projective plane.

Adams

used the fact that for a sphere which has a single primitive generator, the cohomology of the projective plane is a polynomial algebra on a single generator truncated at height three.

Thomas proves if there are several primitive

216

LIN

generators, then the cohomology of the projective plane contains a polynomial algebra over the Steenrod algebra on several generators all truncated at height three.

His

analysis then reduces to studying the action of the Steenrod algebra on a truncated polynomial algebra. The problem with this approach for H-spaces with nonprimitively generated cohomology is that the cohomology of the projective plane no longer contains a polynomial algebra.

Efforts to work with algebras over the Steenrod

algebra that are not polynomial have not led to many fruitful results.

Fortunately there is another approach.

The first step follows from work of Browder and Kane [5,11].

Essentially they show that in the

homology, any primitive squared is zero. statement could be phrased as follows. element of

Hn (Z;2y

CffPfXjZ^).

Then there is no element

with the projection of having

x ® x

mod 2

The dual Let

x

with nonzero projection

Az

in

z

in

be an x

in

H^fXiZ^)

QH^XjZ^) ® Q H ^ X ; ^ )

as a nonzero summand.

In the case that

x

happens to be primitive this reduces to the fact that if 1 o i(y) = x and y lies in H (P^XiZ^) then y is



nonzero.

So regardless of the fact that

be primitively generated, if nonzero.

i(y) = x

H (X;Z^) then

y

2

may not is

217

MOD 2 COHOMOLOGY OF A FINITE H-SPACE The second step follows from the exact triangle. BE

be a space with

H^^fBE;^)

with

is a

H

v

in

2n

QBE = E. u^ = 0

Let

and

u

be an element of

cr*(u)

nonzero.

_ ^ ^ Av = a u ® a u.

(E ; Z with

great deal of flexibility in constructing time

E

E.

Then there There is a Most of the

will be a stage of a Postnikov system.

simplest form

E

Let

In its

will be a two stage system.

Note that any finite H-space elements such as

v

to try to map

to

X

X

does not have

in its cohomology. E.

If

E

The third step is

is a two stage system, we

have the following diagram:

13 1x'— !-- » K — ^— » Kj where

f (i ) = x, p ( i ) = c r u . v n' ^ K nJ

only if

gf

is null homotopic.

H-deviation by

D£: X x X

f(xy)f(y) *f(x)

Then

f

exists if and

If we denote the

E, that is

D£(x,y) =

then we have Af*(v) = x ® x + D^(v).

By step 1 if we project into must have

x ® x

QH*(X) ® QH*(X), D^(v)

as a nonzero summand.

is reduced to an analysis of

Hence our problem

D^fv).

The simplest case is the one described by Thomas’ theorem.

In this case all the variables described by the

218 map

LIN f: X -» K

are primitive.

Hence

follows in this simple case that

£

D~

is an H-map.

factors through

In this case the computations become quite simple.

It QK^.

We

illustrate this case briefly. Let’s suppose that we are in the process of proving 2r+2r+^k-l 2r 2r+^k-l PH * K (X,Z2 ) = Sq ?EZ K (X;Z2 ) 2r 2r+2r+1k-l Sq PiT * K (x;Z2 ) = 0 Assumethe case

r = 0

proved in Browder [4].) when

r = 1.

for r > 0,

has

and

k > 0.

been shown. (This

is

So it remains to prove the case

The cohomology is finite dimensional so we

may also assume that for

k' > k,

PH4 k + 1 (X;Z2 ) = Sq2PH4 k _ 1 (X;Z2 )

and

Sq2PH4 k + 1 (X;Z2 ) = 0. Let for some

x

be in

4k+1

PH

(X;Z2 ).

To prove that

2

x = Sq w

w:

Note that the Adem relations imply 0 4k+2 e 2e 4k 0 le 4kc 1 Sq = Sq Sq +Sq Sq Sq 1 1 Sq Sq = 0,

2 2 1 2 1 Sq Sq =Sq Sq^Sq .

We have Sq

4k 2 x4k+1 = Sq xgk_1

4k 1 1 Sq Sq x4k+1 = Sq xgk+1 2

1

1

Sq Sq Xgk_^ = Sq xgk+^

by induction by the case by the case

r = C r = 0

219

MOD 2 COHOMOLOGY OF A FINITE H-SPACE There is a diagram Sq1

Notice that all the variables may be chosen to be primitive because

H (X;^)

is primitively generated.

Let K = K(Z2 ,4k+l,8k-l,8k+l,8k+l) K x = K(Z2>8k+2,8k+l,8k+2) g: K

describe the relations in the diagram. = S q ^ S q 1! 4k+l Sq 18k+l 0 4k. 0 2. s (l8k+l} = Sq 4k+l " Sq 18k-1 ^ o 20 1. 8 ( W8k+2 o ) = Sq Sq l8k+l Sq X8k+1 ' 8k+27

g

is obviously an infinite loop map.

fibre of with

u

2

Let

Bg: BK -» BK^. Then there is a =0

and

** a u

nonzero.

u € H

be the 4k+9

This is because

2 = u . Hence by step 2, there is a = a u 0 o u.

BE

v € H

8k+2

(E)

(BE) Sq

4k+2 u

with

One checks that

.

2 1. 1. . j (v) = Sq i8k+1 + Sq i8k + Sq i ' _ All four steps described above are satisfied. have a commutative diagram

Hence we

— Av

220

LIN OK-

15

-

„E

V

'

X - t —

1P *

K - S — »Kx

and Af*(v) = x ® x + D^(v) = x ® x + D*j*(v) € x ® x + im Sq 4k+1

Finally,

PH

(X;Z^) D im Sq

Sq1PHeVen(X;Z2 ). 4k-1 PH (XjZg).

2 2

= jD

2 + im Sq .

because x = Sq2w

for some

w €

This completes the induction and proves 4k+1

Sq Sq

=0

It follows that

PH Now

1

1

since

12

9

4 k -1

(X;Z2 ) = Sq PH

1

= Sq Sq Sq

(X;Z2 ) .

together with the case

r = 0

proves Sq2PH4k+1(X;Z2 ) = 0 . It should now be evident to the reader how to proceed to prove PH

2r+2r+^k-l 2r 2r+^k-l Z K (X;Z2 ) = Sq PHZ K (X;^) and 2r 2r+2r+1k-l SqZ PH Z K (X;Z2 ) = 0 .

We rely on the results for

r* < r

and downward induction.

The key factorizations are c 2r+2r+1k Q 2rQ 2r+1k Sq = Sq Sq

T 1 _ 2* 2 q ai i=0

MOD 2 COHOMOLOGY OF A FINITE H-SPACE

s / s /

=

Y

s/e.

221

.

j=0 The advantage to this approach is that the information about

x

is now contained in the H-deviation

D£.

D^(v)

is simply the value of a higher order operation on elements of

H*(X) 0 H*(X).

Papers have been written on Cartan

formula for higher order operations [21].

Hence, use of

these papers can provide information about the action of the Steenrod algebra even in the nonprimitively generated case.

Very roughly speaking, if the variables involved in

the operation have nonzero reduced coproducts (that is, they are not primitive) and D^(v)

E

is a 2-stage system, then

has terms of the form secondary operation tensor

primary plus primary tensor secondary on the elements of the reduced coproduct. For example in the case of

H (EgjZ^)

we have a two

stage Postnikov system defined by the diagram

with

-

Axiir= 15

2

x„ 0 3

D^(v)

2

x„ +xcr0 Xr- and 9 5 5 has a summand of the form

which is secondary on

2

x^

® x9

tensor primary on

x^.

222

LIN 2

One can show that

® x9

or

x9 = * 1 , 1 = Sq4sq2x3 • Thus, the method of Cartan formulae for secondary operations allows one to carry the computations one step further.

We are now close to the frontiers of present day

research on the subject.

For most of the work done

presently, it is assumed that the finite H-spaces is associative.

mod 2

homology of the

This is certainly true for

all known examples, and it is suspected that all finite H-spaces admit an H-structure whose

mod 2

homology forms

an associative ring. The example of problems one faces. induction that QH0v e n ^ ^ }

QH

Eg

gives us a preview of the kind of

Suppose we try to prove by downward

4k+1

9

4 k -1

(X;2y = Sq QH

_

).

(XjZ^),

assuming

As before there is a

commutative diagram

except now the variables are not necessarily all primitive. The Cartan formulae computes the form

D^(v)

and contains a term of

^ i(x 4 k_2 ^ ® X4k+1 * ^ence> we

X4k+1 = *1 l^X4k-2^ + decomP°sables*

conclude

223

MOD 2 COHOMOLOGY OF A FINITE H-SPACE If we try to proceed further downward the inductive assumption is weaker; we can only assume that for X4k'+1 = ^1 l^X4k'-2^ + decomposables.

k' > k,

From this there is

the following diagram:

The diagram no longer commutes, but it commutes modulo decomposables.

and

^

are secondary operations and

their indeterminacy prevents us from knowing that the relations are precise.

The universal example

E

will be a

3-stage Postnikov system with element v in H ^ +^(E) $$ with Av = a u ® o u. The problem is there will be no

_

lift to

^

E

because the relations are true modulo

decomposables. This suggests that we look at There is a lifting of an H-map.

fiX

to

^ o v

in

gk+1 H (DE).

QE, which can be shown to be

We have a commutative diagram

n2K1 In. I J

224

LIN Williams and Zabrodsky [23,25] have shown that

o v

represents an obstruction to homotopy commutativity known as the c^-obstruetion.

Given a map

h: Z -» W,

which is a

map defined between homotopy commutative H-spaces W

c^(h): Z x Z

QW

is defined by the loop

and

c^(h)(z^,z^):

h(z2 zl)

h(Zlz2 )

h (z 1 )h(z2 )

h(z2 ,

translated to the basepoint. c^(h)

Z

If

W

becomes a cohomology class. , *x 2

, * >

is a

K(7T,n),

then

Zabrodsky shows

^ , *.2

c2^° v) = (a ) u ® (a ) u* Computing

c^ff o v) c^(f

yields

x ^ crv) = cr x ® cr x + c^(f) [p

Once againa simple argument shows a summand of in

H (fiX).

H (OX)

and

c2 ^ )

must be

(a v )• Hence information is obtained

H (X).

Kane [10].

The Eilenberg-Moore spectral sequence E^

term is isomorphic as coalgebras to

Since

E^ = T o r ^ ^ j {TL^TL^ ) , one easily

deduces that the primitives of

H (OX)

suspensions or transpotence elements. to conclude

a x ® o x

There is a bridge connecting the primitives of

collapses and the H*(QX)

that

. v)•

a x = a y

where

y

are either This fact allows one

is either related to

MOD 2 COHOMOLOGY OF A FINITE H-SPACE ^

or

y

2k

is in the image of

Sq

225

. Hence our analysis

allows us to continue the inductive argument.

For complete

details, see Lin [13]. The main objective of this argument is to obtain information about the action of the Steenrod algebra in the nonprimitively generated case. unsolved.

Several problems remain

I list a few here.

Problem 1 What restrictions are there on the homology algebra of a finite H-space at each prime, assuming the algebra is associative? Problem 2 Is the first nonvanishing homotopy group in degrees 1 , 3 or 7? Problem 3 If

X

is a finite H-space, is the

cohomology isomorphic to the

mod 2

mod 2 cohomology

of a Lie group product a bunch of seven spheres? Problem 4 Is

QH2 +2

k_ 1 (X;Z2 ) = Sq2 kQH2 ^k+ 1 ^_ 1 (X;Z2 )?

Problem 5 Can the following Hopf algebras be the

mod 2

cohomology of H-spaces?

4

(1 ) (n)

® A(xn ,x13)

Z2 [x1 5 ]/x1 5 ®

a

(x 2 3 ,x 27> x 29).

Problem 6 Given a finite H-space H-structure on

X

X

is there always an

such that the homology forms

an associative ring? As one can see, there are several open problems still to be solved concerning the

mod 2

cohomology of a finite

226

LIN

H-space.

The use of the secondary operation still seems to

be a powerful tool in attacking such questions.

REFERENCES [1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math., 72 (1960), 20-104. [2] J. F. Adams, The sphere considered as an H-space mod p, Quart. J. of Math., Oxford Ser., 12 (1961), 52-60. [3] A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math., 57 (1953), 115-207. [4]

W. Browder, Higher torsion in H-spaces, Trans. AMS, 108 (1963), 353-375.

[5]

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

[6 ] W. Browder and E. Thomas, On the projective plane of an H-space, 111. J. of Math., 7 (1963), 492-502. [7]

G. Cooke, J. Harper and A. Zabrodsky, Torsion free mod p H-spaces of low rank, Topology, 18 (1979).

[8 ] J. Harper, H-spaces with torsion, Memoirs of AMS, 22, No. 223 (1979). [9]

J. Hubbuck, On homotopy commutative H-spaces, Topology, 8 (1969), 119-126.

[10] R. Kane, On loop space without p-torsion, Pacific J. of Math., 60 (1975), 189-201. [11] R. Kane, The module of indecomposables for finite H-spaces II, TAMS, 249 (1979), 425-433. [12] R. Kane, Primitivity and finite H-spaces, Quart. J. of Math., 26 (1975), 309-313.3.3.

MOD 2 COHOMOLOGY OF A FINITE H-SPACE

227

[13] J. Lin, Higher order operations in the mod 2 cohomology of a finite H-space, Amer. J. of Math., 105 (1983), 855-938. [14] J. Lin, Torsion in H-spaces I, II, Ann. of Math., 103 (1976), 456-87; 107 (1978), 41-88. [15] J. Lin, Two torsion and the loop space conjecture, Ann. of Math., 115 (1982), 35-91. [16] J. Milnor, Construction of universal bundles, II, Ann. of Math., 63 (1956), 430-436. [17]

J. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math., 81 (1965), 211-264.

[18] J. Stasheff, Homotopy associativity of H-spaces, I, II, Trans. AMS, 108 (1963), 275-292; 108 (1963), 293-312. [19] E. Thomas, On the mod 2 cohomology of certain H-spaces, Comment. Math. Helv., 37 (1962), 132-140. [20] E. Thomas, Steenrod squares and H-spaces I, II, Ann. of Math., 77 (1963), 306-317; 81 (1965), 473-495. [21]

E. Thomas, Whitney Cartan product Zeit., 118 (1970), 115-138.

formulae, Math.

[22] F. Williams, Higher homotopy commutativity, Trans. AMS, 139 (1969), 191-206. [23] F. Williams, Primitive obstructions in the cohomology of loop spaces, to appear. [24] A. Zabrodsky, Implications in the cohomology of H-spaces, 111. J. of Math., 14 (1970), 363-375. [25] A. Zabrodsky, Cohomology operations and homotopy commutative H-spaces, in The Steenrod Algebra and Its Applications, Springer Verlag, 168 (1970), 308-317.

James P. Lin Department of Mathematics University of California, San Diego La Jolla, California 92093

X

MAPS BETW EEN C L A S S IF Y IN G SPACES A . Z a b ro d s k y

IN TRO D U C TIO N I n t h i s p a p e r we p r e s e n t a n o t h e r a p p l i c a t i o n o f H . M i l l e r ’ s p r o o f o f th e S u l l i v a n C o n je c t u r e ( [ M i 1 l e r ] ) ^ ) . O u r m a in th e o r e m s a r e th e f o l l o w i n g .

THEOREM 1 : or

G

Let

be either a locally finite

a compact connect Lie group.

( disc r e t e )

Let

H

be

( disc r e t e )

either a finite

g r o u p , a compact Lie group or a group of the

homotopy type of a finite complex with torsion free integral c o h omology . connective complex

is

Let

k

be

K - theory .

Then■

null homotopic if and only if

Remarks'-

either periodic or A map

f : BG -» BH

k ( f) = 0.

( a ) T h ro u g h o u t t h is p a p e r a lo c a lly f i n i t e g ro u p

m eans a c o u n ta b le g r o u p w h ic h i s a u n io n o f a n in c r e a s in g sequence o f f i n i t e g ro u p s . (b ) v e r s io n s w h e re

T h e o re m 1 a n d i t s c o r o l l a r i e s h a v e P

i s a n y s e t o f p r im e s .

T h e s e s h o u ld b e

q u i t e c le a r fr o m th e p r o o f s o f th e th e o r e m s .

228

mod P

MAPS BETWEEN CLASSIFYING SPACES

229

A s a c o n s e q u e n c e o f th e o r e m 1 (a n d i t s p r o o f ) o n e h a s th e f o l l o w i n g :

THEOREM

2'

G

Let

torsion free theorem 1.

be a compact connected Lie group with

f : BG

Then

BH

H

be a group as in

is null homotopic if and

H * (f,Z ) = 0.

only if

COROLLARY 1 :

Let

or c o mplex )

n - p la n e

G

be a group as in bundle on

BG

theorem 1.

Is t r lu la l

A

(real

if and only

s ta b ly t r l u l a l .

if it is

COROLLARY n - p la n e

h o m o l o g y .Let

integral

2'

G

Let

BG

bundle on

t r lu la l on

the

2n

be as in

Is t r lu la l

skeleton of

f : BS 3 -» BS3

Example:

theorem 2.

A complex

if and only if it is

BG.

i s n u l l hom o t o p ic i f a n d o n ly i f

i4 3 f |s C BS - *.

A t th e e n d o f t h i s p a p e r we d is c u s s som e p r o p e r t ie s o f f u n c t i o n s p a c e s w h ic h seem t o i n d ic a t e th e d i f f i c u l t i e s g e n e r a liz in g th e o r e m 1 .

Example

1:

Let

p 3

W = m a p ^ (B Z /p Z ,B S )

We g iv e th e f o l l o w i n g e x a m p le s :

b e a n od d p r im e a n d l e t b e th e s p a c e o f p o in t e d m aps

in

230

ZABRODSKY

BZ /p Z

3

BS .

T h e n a l l p a th c o m p o n e n ts

of

th e p a th c o m p o n e n t o f th e c o n s t a n t m ap) w e a k h o m o to p y ty p e o f a p o i n t .

W (e x c e p t fo r

do n o t h a v e th e

M o r e o v e r , th e y a l l h a v e

i n f i n i t e l y m any n o n v a n is h in g h o m o to p y g r o u p s .

Example 2-

Let

W = m a p ^ (B Z /2 n Z ,B S 3 ) n > 1 .

If

W'

is a

p a th c o m p o n e n t o f

W w h o s e e le m e n ts do

n o t in d u c e th e z e r o

map o n th e

c o h o m o lo g y th e n

is n o t

mod 2

W'

c o n tr a c tib le .

T h e p a p e r i s o r g a n iz e d a s f o l l o w s :

I n s e c t io n 1 ( " t h e

i n p u t ” ) we la y th e f o u n d a t io n f o r th e p r o o f o f o u r m a in th e o r e m s b y in t r o d u c in g som e n o t a t io n s a n d s t a t i n g th e o re m s o f C a r I s s o n - M i1 l e r , M i l l e r a n d F r e i d l a n d e r - M i s l i n . d e e p th e o r e m s a r e th e m a in t o o l s i n o u r p r o o f s . th e o r e m s a r e p r o v e d i n S e c t io n 2 .

These

T h e m a in

I n S e c t io n 3 we p r o v e

som e lem m as u s e d i n th e c o u r s e o f p r o o f o f th e m a in th e o r e m s .

S e c t io n 4 c o n t a in s e x a m p le s 1 a n d 2 .

T h is p a p e r r e p r e s e n t s a s l i g h t e x t e n s io n o f a t a l k d e liv e r e d i n th e J o h n M o o re a lg e b r a ic to p o lo g y a n d a lg e b r ia c K - t h e o r y c o n fe r e n c e h e ld i n P r in c e t o n U n i v e r s i t y i n O c to b e r 1 9 8 3 .

M o s t o f th e w o rk le a d in g t o t h i s p a p e r

w as d o n e w h ile th e a u t h o r w as v i s i t i n g R o c h e s te r i n th e F a l l o f 1 9 8 3 .

th e U n i v e r s i t y o f

We w o u ld l i k e t o e x p r e s s

o u r d e e p g r a t i t u d e t o th e s e i n s t i t u t i o n s .

M any th a n k s a r e

231

MAPS BETWEEN CLASSIFYING SPACES

d u e t o J o h n H a r p e r a n d J o e N e is e n d o r f e r w ho w e re w i l l i n g

to

l i s t e n , c o r r e c t , a d v is e a n d s u p p o r t, a n d t o H a y n e s M i l l e r w h o s e r e c e n t w o r k seem s t o b e a n u n e x h a u s t ib le s o u r c e o f n ew id e a s le a d in g t o a b e t t e r u n d e r s ta n d in g o f v a r io u s p r o b le m s i n h o m o to p y t h e o r y .

§1.

TEE INPUT

We s h a l l u s e th e f o l l o w i n g n o t a t i o n s : m a p ( X ,Y ) :

th e s p a c e o f ( u n p o in t e d ) m aps

m a p ^ ( X ,Y ) : th e s p a c e o f p o in t e d m aps

X -» Y .

X

Y.

C q ( X , Y ) : th e p a t h c o m p o n e n t o f th e c o n s t a n t map i n m a p (X ,Y ). Cq ( X , Y ) :

th e p a t h c o m p o n e n t o f th e c o n s t a n t map i n

m apx ( X , Y ) . V q (L ,X ): Vq ( L ,X )

H*(BG,Z/pZ).

But

PH

(QKQ ,Z/pZ)

i s g e n e r a te d o v e r

d im e n s io n a l e le m e n ts i n

ker

and

ft

s4(p )

H * ( B G ,Z /p Z )

n o ( n o n z e r o ) o d d d im e n s io n a l e le m e n ts i n H°m^(p)(M,H*(BZ/pZ,Z/pZ) = 0.

b y od d

By 1.2

c o n t a in s

ke r p,

hence

[BG.X] = *

and

f ~

2 .2 .

The case

G =

a f i n i t e p -g ro u p ,

a s h o rt e x a c t sequence

1-» C q

s h a l l u s e in d u c t io n o n

n - lo g ^ [G |

2 .1 ).

s a t is f y

Let

fib r a tio n k *(f o 1 .3

t

BH

BG^ — - ■ » BG im p lie s

Bo)

f o Bcr ~ *

(th e case

k * (f) = 0.

- > B Z /p Z .

)*

is a f i n i t e

c o m p le x ) .

: map (B Z /p Z ,B H ) — ^ *

th e re e x is ts x

k (B t)

f^ : B Z /p Z

We n= 1

is

O ne h a s a

B y in d u c t io n

a n d f e v j^ B G . B H ) .

By

and

A p p ly in g 1 .5 o n e c o n c lu d e s :

V^a ( B G ,B H ) . U

BH

One h a s

1.

* s C0 (B G 0 , B H )(Q C 0 (BG 0 , B H ) ~ m ap^ (B G Q , QBH)

OBH ~ H (B

f : BG

H = (F T F ).

t Z /p Z ----- » G ------» cy

w ith

i s i n j e c t i v e , c o n s e q u e n tly

In p a r tic u la r ,

f ~ f^ o B t . k

x

(£q ) =

0

By 3 .3

a n d b y 2 .1

f 0 ~ *'

2 .3 .

The case

G = (F ),H = (F T F ).

F o llo w in g H o p k in s a n d

M i l l e r ( s e e [ M i l l e r ] ^ s e c t io n 9 ) f o r e v e r y p r im e

p

th e re

MAPS BETWEEN CLASSIFYING SPACES

e x i s t s a s i m p l i c i a l c o m p le x BG

235

W a n d a s i m p l i c i a l map

BS

3

Let

and

tt.(e

7k) :

is surjective.

is a generator of

* map.

7

It follows h

could be

244

ZABRODSKY 4

lifted to f.

Adjointing

that

3

h : S

-» map(BZ/pZ,BS )

h

and

h|* x BZ/pZ = f.

H^(h,Z/pZ)w^ = u ® 1 + 1 ® f*w 4 « where 4 4 u € H (S ,Z/pZ)

and

contradiction as v

are generators.

P^w^ = 2 w^+^ ^

If

tt

mapped into

(map^(BZ/pZ,BS^) ,f) = 0

BS

3

so

This means that w^ € H^(BS^,Z/pZ) ,

But this leads to a

while

P^v = 2 v^+^ ^

does not satisfy

*

~ 4 h : S x BZ/pZ

one gets a map

h|S^ x * = h

with

u ® 1 + 1 ® ^*w4 =

(unless for

f* w4 = 0 ) *

n > r-1

then one

can solve the lifting problem: map (BZ/pZ,BS3 )

(BS3 )(r) where

j

is the

r- 1

BS3

— i-- >

connective fibering.

By adjointing and readjointing a different way one will get a lifting £ BZ/pZ —

-- » map((BS3 )^rl ( B S 3 ))j 0*

Now, one has a fibration

nfBS3 ) ) ^ ^

Tr.[fi(BS3 )j-r_ 2 )] = 0

i = r_l-

4.1

all maps

f°r

3 fi[(BS

BS

3

-* (BS3 ) ^

- “U BS3 ,

By [Zabrodsky] theorem are phantom maps and

7t . map^(n[ (BS3 ) ( r _ x } ] , BS3 ) a TT Ext[H t _ 1 (Q(BS3 ) ^r_1-j*Q) • 7rt_j(BS3 )/torsion] = Ext(H3 +j(Q(BS3 ) r _1 ,Q), ^ ( B S 3 ))

which is zero for

j > 0.

Hence,

Cq((Q(BS3 )r

,BS3 )

MAPS BETWEEN CLASSIFYING SPACES

and as

mapx (BS3 ,BS3 ) — j

V^((BS3 ) ^ , BS3 ). (J

mapx fBS3 ^r),BS3 )j. C V;*((BS3 ) ^ ,BS3 ) (J map^fBS3 ) , B S 3 )



245

In particular, mapx fBS^BS3 ),i

and consequently

map(BS3 ,BS3 ) — 5^

map(BS3 ) ^ , B S 3 ) .

This observation implies that the lifting f : BZ/pZ -» map((BS3 )^r^BS3 ) 3 3 could be further lifted to a map BZ/pZ -» map (BS ,BS )

and

a lifting map ((BZ/pZ,BS3 )

BS

3

/

i3

----- » BS

contradicting the first part of the example.

Example 2. 1:

A similar fact holds for

There is no

map^(BZ/2nZ,BS3 ), n >

map f :S4 x K(Z/2nZ,l) -» BS3

so that

H^(f,Z/2Z)w4 = u ® 1 + 1» O nx)2 , x eH 1 (K,Z/2Z). one uses the secondary operation excess[(Sq2 Sq^)Sq2 + Sq^Sq^"] > 4. nonlinear on 4 dim classes in satisfies in

,dx )

H^(fiX;k) ~ H^(AH(X),d^ ) . By contrast, a

commutative differential graded algebra (henceforth c.d.g.a.) is assumed to have a differential of degree + 1 . For us the source of c.d.g.a.’s will be the Sullivan minimal models [26] in rational homotopy theory, and we adopt the conventions of [14].

We define the Poincare

252

ANICK

series

Pq (z)

a d.g.a. (resp. c.d.g.a.)

G

to be the

Hilbert series of its homology (resp. cohomology) algebra. Artinian local rings in commutative algebra can also be studied using these methods. for a local ring field

k £ R/m.

Ext^(k,k)

R

with maximal ideal

m

(R,m,k)

and residue

The tie-in comes primarily because

is an associative graded Hopf algebra.

Poincare series of of its

Our notation is

Ext

R

algebra,

E =

The

is defined to be the Hilbert series Pr(z) = ^(z).

R

is equichar­

acteristic if and only if there is a right inverse the projection

p:R

R/m ~ k.

e

to

In this case the dual of

the bar resolution [19] gives a free associative graded algebra having m

G = T(fn )

and a differential

H (G,dL) ~ Ext0 (k,k). ** K K

= Hom^(wi,k)

and

on a vector space

T(V)

Here

m

d^

of degree

+1

denotes the dual

is the (graded) tensor algebra

V.

3 When

m

=0,

Roos [22] demonstrated that

related by a very simple formula to the subalgebra of

^ Ext^(k,k)

Pp(z)

H^fz), where

generated by

G

*s is

1

Ext^(k,k).

G

has the additional property of being finitely presented, with generators in degree one and relations in degree two; such an algebra will be called a one-two algebra. Topologists studying finite

CW

complexes which arise as

cofibers of maps between suspensions also find that loop space homology contains a certain finitely presented graded algebra [18].

A graded k-algebra

G

is degree-one­

253

GENERIC ALGEBRAS AND CW COMPLEXES generated (henceforth referred toasd.o.g.) generated as a k--algebra by

if

G

is

. The d.o.g. algebras

constitute a reasonably natural subcollection to consider, especially for abstractly defined algebras with no particular reason to assign degrees differently.

These

considerations justify special attention being devoted to finitely presented algebras, d.o.g. algebras, one-two algebras, and to the Hopf algebras in each of these subclasses.

§3.

A COMPLETELY SOLVED EXAMPLE By way of further motivation and to indicate what we

hope to do in general, we present here one completely solved case illustrating the "’generic” concept.

The

example comes from topology and ties in with the rationality/irrationality question. W

In [4] a CW

is built, out of two 2-cells and

P^(z)

complex

two 6-cells, for which

is not rational (homology is computed over

undertake to evaluate

for all

CW

Q ) . We

complexes

X

consisting of two 2-cells and two 6-cells, hoping that one series will emerge as the ’’most typical” one. Let

tt4 (0(S2

S2 )) ® Q

v

H4 (Q(S2 v S2 );Q), h

denoting the Hurewicz homomorphism.

is a free d.o.g. Hopf algebra and

H^(Q(S

a \*a2

2

are Prim itive

elements in degree four.

We may in fact identify

7r^(fi(S2 v S2 )) ® Q

the primitives of

2

Hx (fi(S

symbols

2

v S );Q) x

and

with via h,

y

as names for

F = H^(Q(S2 v S2 );Q). [a,b] = a b - ( - l ) ^

F.

makes sense to re-usethe the generators of

Furthermore, the bracket

^ba

the Whitehead product in P l’P 2 ,p3

so it

2

v S );Q)

in

p

2

corresponds up to sign

2

v S ) [23], so we allow

also to denote the corresponding primitives in

to

255

GENERIC ALGEBRAS AND CW COMPLEXES We are ready to do some calculations. the Poincare series of

Pnx(z) 1 = (1 +

where

G

QX

z

By [2] or [18]

is given by

)Hg ( z )

1

is the Hopf algebra

-

z ( l

-

2z

+

2z4 )

,

G = Q /

(6)

and

3 ^

c^ .p ..

We need to compute

H^,(z)

for various

j=l values of

{c.

.

Calculations of this sort are notoriously difficult. Fortunately, however, the parameters in this problem are few enough and occur in low enough dimension that every H^(z) Q

may be found.

First,

replace

is an algebraic closure of

Hilbert series

H^fz).

Q;

G

by

G

where

this will not affect the

Assuming this has been done,

consider the effects of a linear change of basis, (y]

(u v ] (£)



s -t -u -v e

on the generators of

.

a.

A = sv ~ tu * 0 • is replaced by

3 x\ i

) c ! .p ., L ij*j j=i

' ch

where "V

' ci l '



ci2 - Ci3 Let

2 st

= A

= det

-

°12 C13

c22 °23 = det

C11 C 12

C21 C22

sv+tu vt

2 uv

ci2 .

AQ = det

• ci3 ■ ’ c13 C11 ' ■ c23 C21 -

256

ANICK

= ( c ^ •c12>ci3 ) and c 2 = ^c21,C22’C23^ are _3 linearly independent in Q if and only if one or more of

Note that

the

A.’s J

is non-zero.

(A^.A^.A^)

In this event the vector

is perpendicular to both

pairs of vectors

and

c^.

The

ke classified into four sets

as follows (we omit the proofs): (A) that

A ^ 0

aj

A^

s,t,u,v

may be chosen so is parallel to

Under this change of basis,

and

elements

then

while (2st, sv + tu, 2uv)

(Aj.A^.A^). so

2

If

cj^ = c22 =

are linear combinations of the basis

p^

and

p^

only.

linearly independent, span

Since

and

are

(a^.a^) = span (p^.P^)

and

G = Q < x , y > / < a ^ ~ Q / = Q /. The last of these had its Hilbert series computed in [4]. It is an irrational series,

"

g

M

-

[,

with radius of convergence (B) s,t,u,v

If

A^ = A 1 A^ V s

C11 = C21 =

p^

and

A1 ^ 0

but

p^

(A^.A^.A^). so a {

only.

J H - z

and

[[x,y],y],

or

A ^ 0

A~ 5* 0, then

while

2

2

(s ,su,u )

are ^ near combinations of

Since they are linearly independent, G = Q / ~

Q / = Q /• x

J

Under this change of basis,

an/

W = t *

g3 •

- 1 + / r

with radius of convergence

p =

(C) If A. = A0 = A0 = v J 1 2 3 is the quotient of

0

Q

but some

c . . / 0, then ij

G

Its Hilbert series is always

= ---- 1----- 4 1 - 2z + z

Mz)

(D)

£ .618.

by the ideal generated by a

single degree four primitive.

for which

g

.

(7C)

p £ .544. If

G = Q

c..=0

for

i J

i = 1,2,

and

j = 1,2,3,

then

and "g = - n s

with a radius of convergence



t™ )

p = 1/2.

By (6) there is a corresponding classification for Pq

x

(x ) »

X € a^

belongs to a certain affine

(or the empty set if

an > dimfF^))

Furthermore, the polynomials which define

express constraints of the form all entries

0

or

c^.),"

” 0 = det

so these polynomials take their

coefficients from the prime subfield of

LEMMA 4.1.

Let

G

€ ^

(a matrix with

k.

denote the graded algebra

associated with the point

N c = (c^ .) € k , as above.

Then

00

for any formal power series

A(z) =

^

anzT1,

{c € k^ |

n=l H

(z) >> A(z)}

and

GC algebraic varieties in

(c € kN |H (z) > A(z)} GC k^.

are affine

262

ANICK

Proof: Setting

V = {c £ k^ | H

(z) >> A(z)},

we have

V

GC 00

=

D V, n=l-1 n

where V

n

= {c € kN I dim(Gc) 1 1 v nJ

By the Hilbert basis theorem, J and

V

is itself

To handle as before.

V =

> a }. ~ nJ

s D V 1 n n=l

s < 00

for some

an affine variety. W=

{c €

|H

Let

(z)> A(z)}, Gc

and for

n > 2

define

V n

set

n-l N W = V U U {c € k I dim(G?) > a.}. n n . . 1 1 viJ i=l

Each

W n

is an

algebraic affine variety defined by homogeneous polynomials 00

in the

(c. .) and 1J

W =

fl W . Like n=l n

V,

W

is itself an

affine variety. An interesting application comes next. generated connected graded k-algebra and only if

Gn = 0

for

n >

some

G

is

n^.

A finitely Artinian

if

If one knows

G ’s

degree vector in advance, one can tell at some finite stage whether or not

LEMMA 4.2.

G

is Artinian.

Fix a field

there exist integers that

If

G

Gluen any degree vector

n^ = n^(d)

G € n

and let

nr. = 0

if and only if

= {c €

G^

| dim(G^) > 1}

and m0

= n

U V ; i=0 n+1

these are affine varieties in

W q D W^ D

This descending chain of affine varieties must

W^ 2-*•• stabilize. and set

N k . We have

Deduce that

n. = n~ 1 U



where

c € W

if

c € W

,

no

+m „ . U

n

then

W

n0

If

= W

c

,. = ...

G € , d

for some G

n0

for some

n~

0

is Artinian, then

n, so

c£ W

is Artinian.

nQ

.

G ~

Conversely,

Lemma 4.2 follows.

This also shows that an algebra being non-Artinian imposes a closed polynomial condition on its defining coefficients. Lemmas 4.1 and 4.2 also apply if instead of finitely presented algebras we consider finitely presented graded Lie algebras with prescribed degree vectors for their presentations.

The condition

Ln = 0

for

n >

is usually described by calling the Lie algebra

some n^ L

nilpotent. For one-two algebras, note that there can be at most g

2

linearly independent relations in degree two if

the number of generators, (g;l

2

l\g ;2,...,2)

so

for

d =

actually includes all Hilbert

g

is

264

ANICK

series of one-two algebras with

g

generators.

Because of

this we sometimes classify all one-two algebras by the single parameter whether

g.

It would be interesting to know

n^ = n^(d)

is linear in

g

Clas Lofwall [oral communication] that

nq (^) > g~l

for these algebras.

has shown by examples

for one-two Lie algebras.

The promised well-ordering of

^

is a consequence of

Lemma 4.1.

THEOREM 4.3.

is well-ordered (downwards) by

the set

Proof:

For any fixed degree vector d

Suppose not, and suppose

and field >.

c^,c^,c^,...€ k

infinite sequence of points for which the series H

c

(z)

satisfied

k,

A ri*(z) < A /0 x(z) < ....

N

were an A^^(z) =

Let

vn = {c e kN I H (g C)(z ) > A (n)(z)}. By lemma 4.1 these are affine varieties and by Noetherianness V^+ ^.

But

c^ €

D V3 3 ...

D

stabilizes, say

=

- ^2+1’ a contradiction.

We shall construct in Section 7 examples in which

Sf

does contain infinite descending chains. We now hope to repeat this success with other kinds of graded objects.

Consider free associative d.g.a.*s

= (k < Xj,...,x

> ,6 ).

As long as

G

(0,6)

is finitely

generated as a (free) graded algebra, we can associate to

265

GENERIC ALGEBRAS AND CW COMPLEXES G

a ’’degree vector” or ’’generator degree multi-set”

d = (g;m^.m^,...»mg) occur. 1

"

o£ degrees in which the generators

Without loss of generality we may assume

< m. ) v

degree -1, each

. . Because

i-l 'm.-l

satisfies the

8

l

6 (xy) = 6 (x)y + (-1 )

derivation formula

Ix I 'x6 (y),

8

g is completely specified by the

N=

^

s^coefficients,

1=1

and we write

c = (c. v ij'

Not every however.

8=8

Q will 2

We require

guaranteed if each

8

be a valid differential,

=0,

2

8 (x^)

and fortunately this is

is zero.

’^(^(x^)) = 0 ”

in

turn expresses a homogeneous polynomial condition on the (c_).

So the collection of equivalence classes of free

d.g.a.’s with the given degree vector affine variety

W

lying in

algebras themselves are all k < x, 1

x >, g

d

is indexed by an

N k . Note that the free isomorphic to

and only the differential J

F = 6=5

c

A

changes as

c

LEMMA 4.4.

Let

moves within

W C kN

be

W.

as above, let Fc = (F,6 C )

denote the free finitely generated augmented d.g.a. indexed

266

ANICK

by the point

c = (c^.) e W,

and let

A(z) =

^

^e

n=l any formal power series.

Then

{c € W | P

(z) >> A(z)} (F°)

A

A

{c £ W | P

and

(z) > A(z}}

W

are subvarieties of

in

(Fc) kN

.

Proof:

Let

Sc(z) = P

(z)

(F°) and write

S°(z) =

^

s^z*,

i=0 so

Let

s* = dim(ker(6 C :Fn -» Fn_ 1 )/im(6 C :Fn+1 -> Fn )) .

tC =dim(im(6C :FC-+ FC .)), n v v n n- 1 "

sC = (f n v n

tC) - t° . , where nJ n+1

so that f = dim(F )is a fixed n v nJ

integer depending only upon the degree vector If

d.

V = {c € W I sC > a }, then n 1 1 n “ n' V = {c e W I n 1

=

(tc + t ® ) < f v n n+ 1 ' “ n

u

({c

-a } nJ

e V | t*< i} n {C e

w | t °+1 < j})

1 +j= V an

is an affine variety for each

n.

The remainder of the

proof is exactly as in the proof of lemma 4.1. By

themethod of theorem 4.3,

an immediatecorollary

is that

the set of Poincare series

of freed.g.a.’s

a given degree vector well-ordered.

d

having

for their generators is

This is interesting enough by itself, but a

more important consequence is the following.

267

GENERIC ALGEBRAS AND CW COMPLEXES THEOREM 4.5.

Fix

collection of all

n,

and

CW

and let

.

ty^ ^

(By convention, the

base point is not counted as a cell here.)

Proof: D

m,n

Let

= {sequences d = (g;cL ,dOJ...,d ) I g < m 1 2 gy 1

2 < d. < d~ < ... < d , 4 x x(w y wy) = -x(w y) =

2 3 w y wy

4, x 4 2 5 -w (xy) = -w y - w , 2 3 w y wy =

is yet another obstruction, with

4 2 5 -w y - w

in

P.

Continuing this process, which is

called resolving overlap ambiguities, allows one to find all obstructions [9]. obstruction set is

For the algebra

P,

the complete

V = {xy,xw} U (w^ynwy}n>i*

^be

language of [5], the set of 2-chains is ={xw^ynwy)n^

and

there are no

m > 3.[5,formula (16)]

uHp(z) f

m-chains for

gives

ro x 2x 8 x_ +z 9_ l = 1 1- (2z+z ) + x (z^ +z +z +z

10

_ L .

+ . . . i) -

(z9+z10+zn + ...) + (0) = 1 - 2z + z"^ + z9 = P^(z) as desired.

PROPOSITION 5.5. monomials in

a = {a^,a^,...,a^}

be any set of

F = k, and suppose

strongly free. and

Let

T(z) >> 0,

a

is

S(z) >> 0

Then there exist polynomials

with integer coefficients and without

constant terms, such that f

(1 - S(z)) (1 - T(z)) »

1 -

2 i=l

Equivalently, 5.1(B) Implies 5.1(A).

lx J

z

£

+

2 j=l

282

ANICK

Proof:

For simplicity write

H

f K I ) z

(z) for

and

L,

(i)

i=l Ha (z)

v J

for

Z

K-!



j=l

By [3, Theorem 3.1]

a

is combinatorially free, i.e.,

no

a. = ua.v 1 J

for monomials u

and v

no

ua. = cfcjV

for monomials u

and v

|u| < laj I•

If

equals an

occur in any other

a.

and

i £

j,

and

having

x ,

then x^

does not

and

1 - H (z) + H (z) = 1 - H f . (z) + H f ■>(z). Because J aK J w-{xi)v J a-{aj}v J of this, the

S(z)

k/ (o = {xn ,...,x }, 1 Z

and

T(z)

found for

will also work for

where

so we can assume without loss of

generality that

a^. = x^,

never occurs.

letting the length of a monomial x ^ ’s

k/,

Equivalently,

be the number of

a

which are factors (including multiplicities), we may

assume that every

a^.

has length

^(«j) > 2.

Def ine the length excess of a set

monomials to be

LE(j3) =

by induction on

a ’s

J

P = {p^,...,p^}

(£(£L)-2).

of

Our proof proceeds

j=l length excess, and we have already

reduced to the case where a. J

a =

equals

x

S .

x^

t .

for some indices

{xg |1 < j < r} Sj

LE(a) >0.

and

t

If s. J

LE(a) = 0, and

= {x^ |l < j < r). 3

t.. J

Let

The

each

GENERIC ALGEBRAS AND CW COMPLEXES combinatorial freeness of are disjoint.

Setting

283

guarantees that

a

S(z) =

^

z

1

*

= H^(z)

a

and

r

and

x.Ccr T M

=

> x

«

1

we have

.€t

i

0 ,

as where

285

GENERIC ALGEBRAS AND CW COMPLEXES (0 = {x. , . . . ,x }

1

o = {u. , . . . ,u }

and

g

1 1

are disjoint graded sets having r >

lu iI z

= S(z),

v 2

and

u .€cr

t = (v,. , . . . ,v. } 1 1

|x_. | = m . , lv iI

z

= T(z).

Choose a total

v .€ t

i

i

(o U

ordering on h,i,j

and

sJ

and

u.

i

a U

> u.

J

nr =

r

for which

whenever

x^

> ru >v.

|u. I < lu.l.

1 i1

1 J1

denotes an abstract graded set having

K l

= t..

j

j

Writing

H^(z)

§ /

for

m. z *

^ T (z )

^or

i=l

Y y

z

t'

the inequality (13) quickly becomes

j=l Hw (z) + S(z)T(z) » Because of this

H^(z) + S(z) + T ( z ) .

there exists an injection of graded sets

• 7 U ci U t --- > (o U (ctxt) ,

all unions being d i s j o i n t . k

Define two sets of elements in

a = {fx(nf.) I^e-Tr} U P = where

by

U (v1-fx(vi ) I v ^ t } ,

|nfi€-Y} U {u.-uh g 1 " zp 0

but

1



2 P^(z) = 1 - 2z + z

then either

2

satisifes

G €

or

a

the radius of

2

z^n

for all

n > 0.

dim(kXg>)n = gn * Since 1 . — < g

p < z^,

Also,

dim(Gn )
2.

d > 3. 1 jjj < d < 2 ,

2

2 S(z) = z + z + ... + z

Choose

^ and write

CO T^ S (zj

^

=

b^z*,

the

(bi)

being positive integers

i=0 obeying the recursion b.

= b. i + b. 0 + ... + b.

l

As long as

l-l

r
0 i some so

for

0 < i < n

a. 0.

Multiplying

|p^(z) * | = P^(z) * + 9(z)

( z ) —1

+ q (z ) ) » Ip ,^ 2)- 1 ! .

Hq (z) + (HG (z)prf(z)q(z)) » H g (z ) + (asz s

+

gives

...) »

Ip ^Cz ) *| ,

Ipd (z)_ 1 | .

(26)

The right-hand side of

(26) is a polynomial ofdegree

< s - 1,

so the terms

(a zS+...) s

dominate

|p (z)

cannot help

coefficient-wise. H~(z)

Cl.

Ur

£L(z)

to

must do this

on its own, i.e., (23) holds. Because of (23), if any H^(z) = |p^(z) * |, then this is the generic series.

G € ^ G

can be found having

is generic and

Ip ^(z) * I

This happens for one-two algebras

when there are enough relations.

2 Example 6.2.

Let

d = (g;1,...,l\r;2,...,2).

then there exists

G € ^

having

If

r > g— ,

H^(z) = |p^(z) * |.

-1 2 2 2 3 Proof. P^(z) = 1 + gz + (g - r)z + g(g - 2r)z + ..., 2 so

r >

implies

_2 2 Ip ^(z ) | = 1 + gz + (g - r)z . p -2 + 1

suffices to find

r' =

— ] linearly independent

It

GENERIC ALGEBRAS AND CW COMPLEXES quadratic relations

297

{a.,...,a ,} C k 1 1 r J 1 g

G = k/=x.y.y„ = 0 J iJj 2 i J £ :rr£

and and

y.x.x„ = y.y.y„ = x.x.y„ = 0. i j £ J iJ y € i y£ When

g

x s+l....X2s+1 r 1 = 2s

2

is odd, say aS

+ 2s + 1

V

g = 2s+l,

y l..... V

re-label

The set °f

relations is

2 a = {x.y. 1sx.x.-y. .,y.,x.y -y.y^.y^x .-y y .,y~ Ilyxa2 ,y a2 \xa^x,xa^y ,ya^x.,ya^y-,a^x. ,a^yx.,a^f }.

(32)

300

ANICK

We will prove that

dim(Jd+2) < 9

by exhibiting a linear

dependence among these ten elements of

Since

d im ( Gd + 2 ) = d lm ( F d+2 ^ " d im f J d+ 2 ^ 311(1 Hp(z) = ( l - z ) 2 = l + 2 z +

... + ( d +

+ (d + 2)zd+1 + (d + 3)zd+2 + ...

l)zd ,

(33)

this will prove the desired inequality, dim(Gd+2) > (d+3) - (9) = d - 6. When range

= xy -eyx,

note that for any

i

in the

0 < i < d, f i d-i. if i w d-i > x(y x )y = e (y x)(x y) ( iw d-iw i d-i> = (& )(e )(y xyx ) f d + K f i d-i. = (& )y(y x )x

in

F,

d+1 (xa^y) - (e Jfya^x) = 0

so

what the coefficients

nr

in

in (31) are.

F no matter

This is a linear

dependence among the elements (32).

2

When

= xy - yx + y , we view

operators on the ring variable

t.

R

xy-yx + y

Specifically, with

faithfully on

=0 R

as an algebra of

of formal Laurent series in one

viewed as multiplication by

2

F

is valid.

t For

x

viewed as

--and

y

the relation char(k)= 0 , F

acts

and we easily find that

x(y1xJ)y[tm ] = (m-i-j-1)(m-1)(m-2)...(m-j)tm 1 J 2 y(yixJ')x[ tm ] = y(y1x J)y[tm ] =

m(m-l)(m-2) ... (m-j ) (m-1)(m-2)...(m-j)tm 1 J 2 ,

GENERIC ALGEBRAS AND CW COMPLEXES w[tm ]

denoting the action of

particular, F.

w € F

on

301

tm € R.

In

x(y1xJ)y - y(y1xJ)x + (i+j+l)y(y1xJ)y = 0

Summing for

i+j = d

in

yields

xa2y = ya2x - (d+l)ya2y , the desired linear dependency.

When

to a transcendental extension of contain the formal elements

k

(34)

char(k) ^ 0, and allowing

(ta |a € k}

passing R

to

permits the same

argument to work. The generic algebras for the degree vector of example 6.3 are actually Artinian, at least when field.

k

is an infinite

Although the coefficients in the generic series do

not decline as rapidly as the coefficients of |(1 - 2z +

+ z^) ^|,

they do eventually reach zero.

One might suspect, whenever the right-hand side of (23) is a polynomial as opposed to a non-terminating infinite series, that the generic algebras would be Artinian. Experimentally this holds, i.e., for all d.o.g. degree vectors

d

having

|p^(z)

^

^

^or which the

generic series has been computed, the generic algebras are indeed Artinian.

Combining this fact with the open

question as to whether or not 5.1(E) implies 5.1(D) when the generators have degree one, we may formulate

Question 6.k.

Let

generic algebras in or be Artinian?

d = (g;l ^

l\r;t^,...,t^).

Must the

either have global dimension two

302

ANICK It is conceivable that the answer to question 6.4 is

"yes" for some fields and "no" for others.

We shall say

more about the role of the ground field in the next section. Question 6.4 is open even for one-two algebras.

Lemma

5.10 and example 6.2 answer it affirmatively except in the

2

2

undetermined range

< r
> 0, we have that g(A(z)) >> S(B(z)). g (A(z)) > «(B(z)).

A(z) >> B(z)

Likewise



(36)

£(A(z) + B(z)) = £(A(z)) >> 1

for

implies

A(z) > B(z)

if and only if

303

GENERIC ALGEBRAS AND CW COMPLEXES

In view of (5), the Hilbert series of the enveloping algebra of a Lie algebra must always equal some

H^(z) >> 0*

By lemma 6.1 we deduce, when

universal enveloping algebra of h l (Z )>

r^ip

LEMMA

£(H^(z))

(z)_ 1 D

a

A consequence is

are Lie relations, and

r

enveloping algebra of

is the

L, that

6.5. If G = k/, 1 g 1 r

a. 1

G

for

L,

G

where

is the universal

then

Hl (z) > u -1( |Pe£(z)_ 1 1) I , d, =(g; |x1 1....|xg |\r; |a1 |----

where

laj).

The weakest Lie algebra analog to question 6.4 would be whether L

when

when

H^(z) = 8 *(p^(z)

8 ^(p^(z) *} >>0,

|p^(z)

^

for generic Lie algebras and whether

L

is nilpotent

The first P^rt of this fails,

as example 5.4 shows.

The second part also fails, with the

non-nilpotent generic Lie algebra of Section 3 being a counterexample.

§7.

WHAT K im OF SET IS

a^

dimfG^) > a^ for some

n

for some

n.

H^(z) >

Conversely,

obviously implies

H^,(z) ^ A(z).

00

Deduce that

Z .= d

U {c € k^ldim(GC) > a + 1}, .1 1 v nJ ~ n J n=l

which is a

countable union of affine varieties by lemma 4.1. Thus

Y , is a countable intersection of open sets d

but, as we shall see, it need not itself be open.

305

GENERIC ALGEBRAS AND CW COMPLEXES

For several of the examples of Sections 3, 5, and 6 we made use of a change of field, replacing G

by

G 0^ K.

k

by

K D k

and

We formalize this in the next lemma, for

which we need no further proof.

LEMMA 7.2.

If

k C K

is any field extension and

any degree vector, then an*

G

K €

d

is

G €

for any

C»f). - d

d

It is conceivable that a collection of fields sharing the same characteristic could all yield different sets of Hilbert series for the same degree vector. limit on such variation, however.

There is a

Define a field to be

large if it contains an algebraic closure of an infinite transcendental extension of its prime subfield.

THEOREM 7.3.

Let

degree vector.

(a)

characteristic as if

k C K, -

k be

then

If k, d

a large K

field and let

Is any field having the same

then

9

d

v J

Part^cu^ar > _ d

in the order topology on power series. € SP

for

i =1,2,3,...

and

J

(a)

such that

is closed

d

Equivalently, if lim S,..(z) =S(z), X-*x>

V

'

G € *& such thatHn (z) = S(z). d Lir

thenthere exists

Proof.

d be any

Given H~(z) € Q

G ^ G . Let

Kq

Ct

,

choose

c = (c. .) € 1J

be the prime subfield of

K

306

ANICK

and let

be the smallest subfield of

Kn

and all the

of

Kq ,

0

Since

ij

K.

which contains

is a finite extension

1

is isomorphic with a subfield

G ’ be the so that

of

k.

Let

K^-algebra having the same presentation as

G = G'

identifying and

c. .'s.

K

K

K1 i

and

and

H^,(z) = H^(z). Also,

k 1 , we have

G" = G' 0,

i

H^t,(z) = H^,(z),

so

G,

k €

H^(z) = H^ft(z) €

a

, sis

desired. (b)

Let

S^j(z)

converges and let

be a sequence in

S(z)

be its limit.

only finitely many of the Consequently, for every

S ^ ^ ’s

n,

^

which

By theorem 4.3,

can lie below

some

S^^(z)

S(z).

satisfy

S(z) < S^j(z) < S(z) + z11. By part (a), we can replace k if., a

by any uncountable extension of so we shall assume that

k

k

without altering

is uncountable and

algebraically closed. Let

Vr = {c € kN |H c (z) ^ S(z) + z11} , V = G

{c e kN [H (z) > S(z)}, and GC Clearly

C

C ..., and

W = {c € kN |H (z) > S(z)}. GC 00 V = U is a countable n=l

union of affine subvarieties of equals

W, since some

no

V

n

S,.x(z) € W - V , so some (i)v ’ n

irreducible component of dim(W') > dim(W' fl V )

W.Furthermore,

W'

of

W

for each n.

uncountable algebraically closed

has But over an

field, a variety cannot

equal a countable union of varieties of lower dimension.

GENERIC ALGEBRAS AND CW COMPLEXES

307

00

We deduce that W' properly contains hence

W ^ V,

U (W' flV ) = W' fl V, n=l n

which means that

{c € kN |H fz) = S(z)} = W - V GC

is non-empty.

An obvious corollary of theorem 7.3(b) is that generic algebras always exist for every degree vector over large fields.

A similar result holds for generic connected

graded Lie algebras.

In one other situation we easily

obtain an existence result.

LEMMA 7.4.

Let

k

be any infinite field, Let A(z)

degree vector, and let c/>00 a

jhen

fT (z) >> A(z) (j

Furthermore, if any

G € ^

algebra exists in

d

be a

be the generic series for for every

G € B(z)

Sf .

is impossible,

as desired.

We present next a few examples showing that be infinite and that the sets

Y . and d

even for one-two Hopf algebras.

d

iP

can

can be ’’bad” ,

The examples depend upon

the construction in the following theorem.

THEOREM 7.6.

Fix m

k < x^,...,xm >

>1

and

n > 1.

F=

Let

let U

be any free d.o.g. k-algebra,

n-dimenstonal h-module with basis {u.,...,u }, 1 1 nJ

be an

and let

denote (possibly singular) matrices in H°mk(U,U).

Let

F

act on the right on

(u)*(x

.. . x

)=M

J1 Fix any Uq

in

u^ € U and let F,

J

... M Jg

via (u).

J1

be the monomial annihilator of

i.e.,

J = span{x. ... x €F J1 q J

Jg

U

| (u )*(x. ... x ) =0}. U J1 q

is a graded right ideal of

F

with Hilbert series

Hj(z). Then there is a one-two Hopf algebra

G,

presented

GENERIC ALGEBRAS AND CW COMPLEXES via (2m + n + 1)

2

(m

generators and

311

+ mn + m + 1)

relations, whose Hilbert series is given by ^ = 1 - (2m+n+l)z + (m^+mn+m+l)z^ - (l-mz)Hj(z)z^.

Hq ( z )

(36) Furthermore, G in which case

G

Proof. of

F

has global dimension three unless G

J = 0,

has global dimension two and is generic.

will be a quotient of semi-tensor product [25]

with another free algebra

E = k. m

Semi-tensor products of free

algebras are analyzed in [2, section 5], where they are called "generalized products". productE O F , E

define

To specify the semi-tensor

it to be the quotient of the free

11 F bythe m(m+ n+ 1) relations [w,x.]

(37A)

[v.,x.]

(37B)

[u.,x.]-[Mj(ui),vj] Equivalently, define the right action of

F

.

on

(37C) E

by

w*x. = 0 J v.*x. = 0 i J u .*x . = [M .(u.) ,v .] . i J L i' Take as

G

the quotient of

E O F

by the two-sided ideal

which

[uQ.w] generates.

(37D)

312

ANICK By [2, proposition 5.5],

product of

F

{[[...[[M

E

...M q

• ■• • Vj

[m]

equals the semi-tensor

with the quotient algebra

is the two-sided ideal of

where

G

where

I

generated by all

(u ),v ].v ], J1 Jq q-1

].w]|(jr

denotes the set

E/I,

(38)

. . . . J q ) € [m]q > ,

{l,2,...,m}.

Taking

u 1 > u~ > ... > u > v i > ... > v >w, 1 2 n 1 m the high term of (38) when

...M^ (u^) / 0

is

u V ...V W, * q J1 where

un

is the high term of

(39)

M. ...M.

e

Jq

(u~). Since the

V

0

set of all monomials of the form (39) is strongly free, E/I

has global dimension two.

Its Hilbert series is

H£/ i (z ) * = 1 - (m+n+l)z + ^ a a

being the set of all sequences

which

M. ...M. (u~) / 0. Jq J1 °

(j^....»j^) € [m]^

for

Furthermore,

J zq + Hj(z ) = Hp(z) = (! - mz) 1,

Hjyj(z) ^ = 1 - (m+n+l)z + z^(l-mz) ^ - Hj(z)z^. Because

G

dim. 2)

and

is the semi-tensor product of F

g l . dim. (G) < 3

E/I

(40)

(with gl.

(with gl. dim. 1), we have and by (40),

T HG( z ) '1 = T He /

i

, ^ 1 TT , >“1

(z )

Hp ( z )

2

\tT

, ^1

= (1 - m z J H g /j f z )

2

2

l-(2m+n+l)z+(m +mn+m+l)z -Hj(z)(l-mz)z ,

GENERIC ALGEBRAS AND CW COMPLEXES as promised.

2

(m +mn+m+l)

Since

G

has

(2m+n+l)

313

generators and

minimal quadratic relations, gl. dim.

(G) = 2

if and only if H^fz)

2

-1

= 1 - (2m+n+l)z + (m +mn+m+l)z

and this in turn holds if and only if

2

J = 0.

A slightly more general form of theorem 7.6 can be

Remark.

obtained by replacing

u^

by a submodule

Uq

of

U.

The

relation (37D) would be expanded to the set of relations {[u\w]}

as

u'

runs through a basis for

Uq , and the

total number of quadratic relations would be upped from

2

(m +mn+m+l) Hj(z)

to

in (36)

m(m+n+l)+dim(UQ). Lastly, the expression would be replaced by J

z |y|(dim(U0 )-dim(U0*y)) ,

being the set of all monomials on

Example 7.7.

Let

k

be any infinite field, let

(7; 1, ...,1\11 ;2, ...,2) , and

let

3!f

all Hilbert series of Hopf algebras in infinite, and the generic series of

(Xj,...,x }.

d =

be the collection of *0 . Then

P^(z) ^

is

is a limit point

in the order topology induced on power series by 1.

If instead

{t^, t^ ,t^, ...} ,

oneof them, say

Let

b

(For far stronger results

concerning (43), see [10] or even [27].)

have order

k

t^ ^ t^

t^

and let

t\

This concludes example 7.7.

By [22], example 7.7 may be translated directly into the language of

CW complexes and local rings.

No further

proof is needed for the next two examples.

Example 7.8.

Suppose

k

is an infinite field.

of Poincare series of commutative rings m

3

=0,

2 dim{m/m ) = 7,

and

limit point of

2k.

(R,/»,k) having

2 dim(m ) = 11

Furthermore, the generic series

The set

is infinite. 2 -1

(1 - 7z + llz )

is a

2k

317

GENERIC ALGEBRAS AND CW COMPLEXES Example 7.9.

Let

8.

A = k[x^, x^, x^]

with

FREE (Z/2)3 - ACTIONS ON FINITE COMPLEXES Proof.

We first note that if

quotients of

A,

then

K

denotes the field of

H^(M,K) = 0,

H^(M) ® K,

and

Proposition

1.9, we find that

since

M is totally finite.

H^(M,K) =

Applying

dim^ M

We consider the possibilities

is even.

d = 2, 4, 6.

be any composition series of minimal length for Proposition

1.5,

4.

obtain it(^) = (k^, ..., k^), where

Thus we

^ = (0 C

C

339

Let M;

& by

C ... C M ), where

q >

^ k^ =d. i

Lemma 1 now shows that occur, since on the case

^

the cases d = 2

ischosen

d = 6.

minimal.

The case

and

d = 4

cannot

Thus, we concentrate

q > 4

can again be

eliminated, using Lemma 1, part (b).

For

easily checked that the only values of

q = 4,

i^(^)

it is

not

eliminated by Lemma 1, part (b) are (a)

= (1,2,2,1)

(b)

lc(SP) = (2,1,2,1)

(c)

ic(y>) = (1,2,1,2)

We consider first the case (a). be taken by choosing a basis for

M

composition series, to have the form

The matrix of

d

compatible with the

can

340

CARLSSON f

0

f14

f15

f16

f24

f25

f26

f34 0

f35 0

f36

0

0

0

0

0

0

0

0

0

0

12 0

f13 0

0

0

0

0

0

0 0

f46 f56 0

The rank of the matrices

anc*

(f^j

must

each be one, since if it is zero, the matrix is identically zero, and we may shorten the composition series, contradicting the minimality of rf f r 24 25 f f *34 35

the matrix

Therefore, the rank of

must be at most one.

But it is

also at least one; otherwise, it would be identically zero, and we could shorten the composition series. ^24^35 + ^34^25 =

anC* us*ng t*ie fact that A is a f f 24 25 gu hu where u f f .gv hv. 34 35

U.F.D., we see that v

are relatively prime homogeneous polynomials, and

and

h

are homogeneous polynomials, not both zero.

9=0 , = 0.

we have Since

f 3v = 0 ,

g so

polynomial

f12f24 + f13f34 = 0 and

h

p,

since u,

Then there is a point vanish at

a,

since

311(1

u

v

v

Since

f^u +

are relatively prime.

are both non-constant.

a £ (0,0,0) k

g

^or some homogeneous

and

and

and

f12f25 + f13f35

are not both zero, we have

(^12*^13^ =

Suppose first that

v

Thus,

in

k

3

so that

is algebraically closed.

u

and Let

FREE (Z/2)3 - ACTIONS ON FINITE COMPLEXES m

a

be the maximal ideal of

A

associated to

We obtain a differential k-module

341

a; A/

m

= k.

M ® A/ , whose * m

differential has the form 0

X 14

x 15

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

X 16 X26

CD

0

CD

0 0

x56 0

But this matrix has rank less than or equal to two, so we have contradicted Proposition 1.8, using Proposition 1.9. If

u

is constant, say

cd ch _gv hv.

We have

s

g'

f 46 +

and

h'

h ’ f 56

so that

O'

and

h'

f f *24*25 f f *34*35 J

then we have

(g,h) = q(g',h'),

;.c .d. (g,h) , and both

u = c,

where

q =

are relatively prime.

are non-constant, we find that since ^46*^56^ = r( k ’S ) *

h ’(a) = g'(a)

0,

but

Selecting

a / (0,0,0),

that the matrix representing the differential in has the form 0

If

0

X 12 0

X 13 0

X 14 0

X 15 0

0

0

0

0

0

0

0

0

0

0

x36 0

0

0

0

0

0

0

0

0

0

0

0

0

X 16 X26

a € we find

M 0 A/m

342

CARLSSON

Again, the rank of this matrix is less than or equal to two, yielding the same contradiction as in the previous case.

Of course, the case

v = c

is handled identically.

We are now left with the case where either constant, and either and

g'

g'

or

h'

u

is constant.

or

v

is

Suppose

u

are constant; all other cases are equivalent to

this by a basis change. cc' c h ' c 'v h'v

Now,

r

Let

g' = c ’. Then

f f 24 25 f f 34 35

must be non-constant, otherwise the

matrix defining the differential contains constant entries, hence our DG-module is not minimal, which would reduce us to the rank which

r

closed. then over

4

case.

Let

a

be any point in

vanishes; it exists, since Let

m a

k

k

3

at

is algebraically

be the maximal ideal associated to

A/m , the matrix a

rf f r 24 25 f f 34 35

a;

vanishes, and the

differential has the form 0 0

X 12 0

X 13 0

X 14 0

X 15 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

X 16 X26 X36 X46 X56 0

Consequently, its rank is less than or equal to 2, again contradicting Proposition 1.8 using Proposition 1.9. settles case (a).

This

FREE (Z/2)3 - ACTIONS ON FINITE COMPLEXES We turn to case (b).

343

In this case, the matrix of

d

has the form 0

0

0

0

0

f 13

f14

f15

f16

f23 0

f24

f25

0

0

0

0

f34 0

f35 0

f26 f 36

0

0

0

0

0

0

0

0

0

0

f~. o4

and

f^

f56 0

f = foo^o^ f f10f0 = f f10f0r = 0 6, f23f35 13 34. = 13 35r 23 f34 34 = = ^o^ocr 23 35 “ f13f34

First, we note that so that either

f46

are identically identically zero, zero, or or $23 are f23

and and

We consider the first first case,, case, which

f a r are. e . oh

corresponds to the matrix 0

0

0

0

0

0

0

0

0

0

0

0 0

f14

f 15

f16

f24

f25

f26

0

f34 0

f35 0

0

0

0

0

0

0

0

0

f 36 f46 f56 0

But now we find that the length is at most contradicts Proposition 1.5.

If

f0 f ^. and O i

f^ f35

identically zero, the matrix has the form 0 u

00 u

tff13

tf14

tf15

r\ 0

0r\

rf

r

r

0

13

f14

f15

0

*23 0

f24 0

f25 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

f t 16 f16 rf 26 f 36

f46 f56 0

„ 2, are

which

344

CARLSSON

Again, the length is at most 1.5.

2,

contradicting Proposition

This settles case (b), and case (c) proceeds

identically.

REFERENCES [1]

CarIsson, G . , "On the rank of Abelian groups acting n k freely on (S ) ," Inventiones Mathematicae, 69,

393-400 (1982). [2]

Carlsson, G . , "On the homology of finite free (Z/^)11 complexes," Inventiones Mathematicae, 74, pp. 139-147

(1983).

Gunnar Carlsson Department of Mathematics University of California, San Diego La Jolla, CA 92093

XIV EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA J . P . May

In a conversation with me in the early 1970’s, John emphasized his view that the existing constructions of the stable category ’’passed to homotopy much too quickly".

His

point was that, ideally, there ought to be a construction which results by passage to homotopy from a category of spectra and maps which enjoys the same kinds of closure properties under both limits and colimits and under both function objects and smash products as does the category of spaces. Gaunce Lewis and I have since developed full details of just such a construction.

More interestingly, our

construction of the nonequivariant stable category readily generalizes to a construction of a good stable category of G-spectra for a compact Lie group

G,

stability meaning

that one can desuspend by arbitrary representations of

G.

Since there are a great many new phenomena encountered in the equivariant setting, our full dress treatment [13] is quite lengthy.

It breaks into two halves.

345

The first, by

346

MAY

Lewis and myself, is aimed at equivariant applications. The second, by Lewis, Steinberger, and myself, is aimed at the exploitation of equivariant techniques for the construction of useful nonequivariant spectra.

The purpose

of this note is to give a brief summary of some of the main features of our work, with emphasis on the second half. To give focus to the discussion, we state three theorems about the existence of nonequivariant spectra. For a based space group

,

Y

and a subgroup

the extended power

D^Y

7r of the symmetric is defined to be the

half-smash product Etr tx Y ^

= Ett

TT

where j

th

Ett

x

Y^/ E i r * {*},

7T

TT

is a free contractible TT-space and

smash power of

THEOREM A.

Y ^

Y.

There is an extended power functor

D^E

00

spectra

E

such that

is the

D^2 Y

on

00

is isomorphic to

2 D^Y

for

Y.

based spaces

00

Here

2

is the suspension functor from spaces to

spectra. The functor

was first constructed (although not

fully understood) in 1976, and some of its early

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA applications were announced in [14].

347

A great deal more has

been done since, particularly by Bruner and McClure, and [4]

gives details.

Cohen [7], Jones and Wegmann [10,11],

and Kuhn [12] have also exploited this functor, and Bruner’s work in [4] played a central technical role in the original proof of Ravenel’s nilpotency conjecture by Devinatz, Hopkins, and Smith [8,9]. The statement of Theorem A is incomplete.

A full

statement would explain that the spectrum level functor enjoys all of

the

good homological and homotopical

properties of

the

space level functor, and then

some.We

give a slightly more complete (but somewhat vague) statement of our second theorem. of

G

as real G-inner product spaces.

THEOREM B. spectra f : BG

Think of representations

-W

BG

Let -V

-» BG

G

be a compact Lie group.

for representations

-V

V

V C W

for inclusions

of

There exist G

and maps

which satisfy the

following properties. (1)

* —V Under suitable orientability hypotheses, H (BG ) a free H

(BG)-module on one generator

c^ of degree

-dim V,

and f* : H*(BG ^)

is the

morphism of

H (BG)-moduies specified by

>C(W-V)t , where complement

H*(BG ^)

W-V

\(W-V) of

V

f (Ly ) =

is the Euler class of the in

W.

is

348 (2)

MAY For a split G-spectrum

with underlying

nonequivariant spectrum ^ +^ E G ),

to

k*{BG~W )

k*(BG A

where G.

representation of

k,

}

is isomorphic

is the adjoint

Moreover, the diagram

----------

IR

-V

»

k^(BG“V )

IR

k® (2“ ^W+A ^EG+ )— ---^-sk^ (2~(W+A) £G+a SW_V )='k^(2_ ^V+A^EG+ ) commutes, where

Here and EG

EG+

S

V

e'- S

0

W-V S

is the canonical inclusion.

denotes the 1-point compactification of V

denotes the union of a free contractible G-space

and a disjoint basepoint.

Other terms will be

explained in due course. When

G

is finite, the spectra

BG

-V

play a basic

role in Carlsson’s proof of the Segal conjecture [6], and he gives an ad hoc construction adequate for his purposes. Property (2) for general theories

G k^

(as opposed to just

stable homotopy) is needed for the generalization of Carlsson’s work given in [5,15] and requires the more conceptual construction to be described here. In fact, we have two apparently very different constructions of regarding -V

BG

-V

. The most intuitive is obtained by

as a virtual bundle over

negative of the representation bundle such, it is classified by a map

BG, namely the EG x^V

BG.

As

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA -V: B G

» BO x {-n } C BO x Z,

349

n = dim V.

As was first understood by Boardman [3], one can associate a Thom spectrum define

BG

_y

M(f)

to be

to any map M(-V).

f : Y -» BO x Z,

and we

A systematic study of such

Thom spectra of maps defined on infinite complexes is given in [13, IX and X].

This definition makes the isomorphism

of part (1) quite clear, the orientability hypothesis being the existence of a Thom class [13, X§5], but sheds little light on part (2).

For that, a different, but equivalent,

construction is appropriate. Our second construction of of

D E

BG

-V

and our construction

are both special cases of the ’’twisted half-smash

product” in equivariant stable homotopy theory. unbased free G-space

X

and a based G-space

X kgY = X xgY/X

THEOREM C.

Let

G

xq

X k^E

00

such that G-spaces

Here

There is a on G-spectra

E

00

X tx^.2 Y

is isomorphic to

2 (X KqY)

for based

Y.

00

2 Y

is the suspension G-spectrum of

Theorem A is obtained by taking

G = ir and

replacing

E

spectrum.

Theorem B is obtained by setting

BG_V = EG

define

{ * } .

be a compact Lie group.

twisted half-smash product functor

Y,

For an

Y.

X = Ett

and

by the j-fold smash power of a nonequivariant

k „S_V, G

where

S_V

is the (-V)-sphere

350

MAY

G-spectrum.

For

V C W,

S ^

is equivalent to

^a S

and we set f = 1 KG (eAl): BG

-W

0 -W

= EG k (S aS

)

£ G kg (SW

W-V Va S

-W

W)^BG

-V v .

We shall return to these special cases after saying just enough about the details to be able to explain the construction of

X £> D'V

We say that

G-prepeetrum if each adjoint map D

351

D

of

such that is an inclusion

~ W-V a: DV -» Q DW

is a G-spectrum

is an if eacha

We have categories GSM 3 GQst 3 G M

of G-prespectra, inclusion G-prespectra, and G-spectra indexed on

d.

It is obvious that limits:

G9M

has arbitrary colimits and

we simply perform such constructions spacewise.

It is very easy to see that

GWd

is closed under limits.

If we take the pullback of a diagram of spectra or take an (infinite) product of spectra, the result is still a spectrum.

However,

GM

is not closed under colimits.

Pushouts and wedges of spectra give prespectra which are generally not spectra (or even inclusion prespectra). To remedy this defect, "spectrification” functor the forgetful functor two steps.

we observe that there is a L: G$d

Gtfd

&'• Q$d -» G&d.

We first go from

G5M

to

left adjoint to

This is obtained in GQ.d

by a fairly

uni 1luminating (transfinite) iteration of an image prespectrum functor or simply by categorical nonsense, quoting the Freyd adjoint functor theorem. GQd

to

G&Pd

We then go from

by an obvious union construction, setting

is

352

MAY (LD)(V) =

for an inclusion prespectrum colimits in the category L

f2W"VDW D.

Now, to construct we simply apply the functor

GiPd,

to the prespectrum level colimits. Similarly, to construct the smash product

G-spectrum where

E

and a G-space

(Da Y)(V) = DV aY

G-spectra F(Y,EV), cylinders that a map

F(Y,E)

Y,

we set

for a G-prespectrum

are defined directly by

D.

Function

F(Y,E)(V) = We now have

and thus a notion of homotopy.

f : E -» E'

of a

E aY = L(£Ea Y),

and the usual adjunctions hold. E a I+

E aY

We say

of G-spectra is a weak equivalence

if the H-fixed point map

f^: (EV)^

(E'V)^

is a

nonequivariant weak equivalence for each (closed) subgroup H

of

G

and each

V € s4.

The stable category

hGM

is

obtained from the homotopy category of G-spectra by adjoining formal inverses to the weak equivalences, so as to force the weak equivalences to become isomorphisms. The book [13] begins with a preamble comparing the resulting construction of the nonequivariant stable category with the earlier constructions of Boardman [2] and Adams [1]. 00

We obtain the suspension functor G-spectra by setting

00

2

from G-spaces to

V

y ,

2 Y = L{2 Y ), where

the obvious inclusion prespectrum with More generally, for an indexing space

V Z,

(2 Y)

th

space

denotes V 2 Y.

we define a

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA Z 00 shifted suspension functor A 2 Y G-spectra by setting

from G-spaces to

A^2 Y = L{2^ ^Y},

denotes the inclusion prespectrum with if

V D Z

and a point otherwise.

left adjoint to

the

space

spectrum E.

EZ

denoted

to a S

-Z

use the indexing spaces

and

(2^ ^Y}

V**1 space

n € Z,

classes of G-maps

n

2^ ^Y

Z 00 A 2

The functor

is

Z**1space functor, which assigns The

spectrum A^2 S®

z 2 S . Since oo

Z-sphere spectrum is just

H C G

where

and called the (-Z)-sphere spectrum.

for all integers

353

IRn

is The

00 = IR , we may

to obtain sphere spectra

(assuming we let

U

G

IRn € d,

^(E)

the

as we may).

Sn For

he the set of homotopy

(G/H)+ASn -» E.

A key technical theorem

(which is trivial in the nonequivariant case) asserts that f • E -» E ’ H

)

is a weak equivalence if and only if is an isomorphism for all

H

and

f^:

n.

Given this much, it is entirely straightforward to develop a good theory of

G-CW

level spheres

as the domains of attaching maps

of cells

(G/H)+ASn

(G/H)^ACSn , where

CE

spectra, using spectrum

is the cone

E a I.

In

particular, it is easy to prove the stable cellular approximation and G-Whitehead theorems. that a weak equivalence between G-homotopy equivalence.

G-CW

The latter asserts spectra is a

These results imply that the

stable category is equivalent to the homotopy category of G-CW

spectra and cellular maps.

354

MAY The functor

2 oo "A 2 "

and a "shift desuspension" functor functor

A

2

A

2

on G-spectra.

and

Q^E

permutations of loop coordinates. the functors

A

2

and

2 Q

differ by

a

It is easy to check that

(on hGiPd)

are naturally

equivalent, and it follows adjointly that naturally equivalent.

A^.

have the same component

G-spaces, but their structural homeomorphisms

2 2

The

has an inverse shift suspension functor

The G-spectra

and

00 2

above is the composite of

A^,

and

2

2

This implies that the functors

are adjoint equivalences,allowing us to

as a desuspension functor

2

-Z

.

are Q

2 2

view

This argument is

independent of the Freudenthal suspension theorem. illustrates a thematic scheme of proof in [13]:

It

use simple

verifications with right adjoints to prove results about left adjoints.

The remarkable efficacy of this scheme

prevents the lack of good point set level control of the functor

L

from being a hindrance to proofs.

What we have said so far makes sense for any indexing set

sd,

and we shall exploit this

s4 below.

It is also easy to check that isomorphic

G-universes where

freedom to use varying

G£fU

U

give rise to equivalent categories

is defined using the canonical indexing set

consisting of all indexing spaces in GSPU

to

G W

GSfU,

U.

for nonisomorphic universes

central to the theory.

The comparison of U

For a G-linear isometry

and

U*

i: U

is U ’,

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA there is a functor E'(iV).

i*: GTU' -> GTU

355

given by (i*E')(V) =

That is, we ignore those indexing spaces not in

the image of

i.

the prespectrum i ^(V');

This functor has a left adjoint V'-iV (i^D)(V') = 2 DV

level,

on the spectrum level,

A key case

i^.

On

where V =

i^E = Li^E.

is the inclusion

i: U

G

U.

We can only

define orbit spectra and fixed point spectra directly when working in a trivial universe, such (D/G)(V) = DV/G L(£E/G)

G as U .Here we

on the prespectrum level and

on the spectrum level and set

set

E/G =

E^(V) = (EV)^.

quickest way to construct a spectrum equivalent to for

E C G^U

product

is to pass to orbits over

X+Ai E.

G

The

X k^E

from the smash

The trouble with this definition is that,

while correct, it is useless for proving theorems, the problem being that the functor but difficult to study. G-CW

i

is trivial to define

For example, it fails to preserve

spectra. To proceed further, we exploit the topology of the

function

G-space

^(U,U')

where

and

are topologized as the unions of their

U

U'

of linear isometries

U

U',

finite dimensional subspaces. For example, the definition of smash products runs as follows.

For G-prespectra

of all indexing spaces in

D U,

and

D'

indexed on the set

we have an evident

"external" smash product indexed on the set of indexing spaces of the form by

V ® V'

in

U ® U;

D aD'

is specified

356

MAY (DaD')(V © V') = DV aD ’V ’,

with the obvious structural maps. E',

For G-spectra

E

and

we thus have the external smash product E aE '

=

L ( £ E a£ E ') .

To ”internalize” the smash product, we choose a G-linear isometry

f: U © U

product of

E

U

and

E'

the fact that ^(11^,11)

and define the internal smash to be

f^(EAE').

We then exploit

is G-contractible for all

j

to

prove that, after passage to the stable category, the resulting smash product is independent of the choice of

f

and is unital, associative, and commutative up to coherent natural isomorphism.

It is also quite simple to give an

explicit concrete definition of function G-spectra F(E,E'),

such as dual G-spectra

now redefine since

2 ^E 2ZnZE

equivalent to

to be

D(E) = F(E,S^).

E aS

and

2 ^E = EaS ^aS^

We can

is equivalent to are both

E.

Turning to twisted half smash products, we assume henceforward that the universe ensures that

00

^(U,IR )

U

is complete.

This

is G-free and contractible, so that

its orbit space is a classifying space for principal G-bundles.

For an unbased free

G-CW

complex

X,

there

is thus a G-map X -- » ^(U.IR00), and

x

is unique up to G-homotopy.

We think of

\

twisting function which intertwines the topology of

as a X

and

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA the topology of the indexing spaces of G-spectra. simplicity, we assume that

X

is finite.

{rIRn ii}

n. X(X)(A^) C IR 1

for all

in

i > 0.

IR

For

By compactness,

this allows us to choose an indexing sequence and an indexing sequence

357

{A^}

in

U

such that

We then have G-bundle

inclusions

specified by

x(x,a) = (x,\(x)(a))

a G A ^ . Let

T^

for

x G X

and

be the Thom complex of the complementary

bundle (taken as the 1-point compactification of its total space).

By elementary inspection of bundles, there are

canonical G-homeomorphisms A. (1)

S

n.

1a T 1

= X aS 1

and n .,-i-n. A ..-A. T.a S 1 = T. ,a S 1 1.

(2)

V '

1

1+1

For a G-prespectrum X k^D

prespectrum

(3) The (4)

D

indexed on

indexed on

n. {IR }

{A^},

define a

by setting

(X KGD)(Rni) = DA.a g T.. i^

structural map

a.:

is defined by

1

(D A ^ T .)

D A . ^ , ^

cx^(dA (x,b)As) = cr(dAa)a (x,c),

d G DA^, (x,b) G T^

with

x G X

and

where

n. b G IR 1 - x(x)(A^),

58



MAY

IR

n i+l~n i

,

and (x,b)As

le homeomorphism (2),

corresponds to

so that

b + s = c +

(x,c)Aa

under

x(x)(a)

in

R ^ - x f x ) (A. ) )®IRni+1 ni =(IRni+1-X (x) (A.+ 1 ) )©(X (x) (A.+1- A . ) ) . Dr

a G-spectrum E, define XkgE=L(\ k^E).

'ter passage to the stable category,

the resulting

x

Dectrum is independent of the choice of

t*GE. To

and is denoted

G-CW

best handle infinite

complexes

X,

one

’ves a more invariant reformulation of the definition

x xqE

Dove which allows one to construct Dlimits over the restrictions of

x

by passage to

to finite

ibcomplexes.

To check that x kq2COY is isomorphic to 200(X^qY) Dra based G-space Y, observe that the homeomorphism(1) id the definition (3) give rise to a homeomorphism ( X txG { 2 A i Y } ) ( [ R n i ) = S ^ f X

k gY )

ider which the structural map of (4) corresponds to le obvious identification. That is, wehave an 3omorphism A. n. a

x *g {2 M

= {2 "(X txQY)}

.

' prespectra indexed on

n {IR }.

An

easy formal argument

Dased on use of right adjoints) shows that

L(x xgD) =L(x k^LD)

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA

for G-prespectra

D.

Applied to

oo

A. D = {2 Y } , this gives A

oo

i

x «G2 Y = L(x !XG£2 Y) = L(x «G {2 This proves Theorem C. is equivalent to earlier.

X

+

a

\jr

i E,

X t*GE

Q

i: IR = U

C U,

as claimed

One manifestation of this principle is

that a free G-spectrum to

E

indexed on

f°r a free G-spectrum

and that — 00 hGJIR .

oo

The principle is that ’’free G-spectra live in the

trivial universe” .

— hGSPU

co

Y}} = 2 (X Kq Y) .

We should explain why

^

359

E'

U

is isomorphic in

E'

indexed on

00 IR

is uniquely determined up to isomorphism in

Now

x xQE

orbits over

G

^ X+A^,i E

and

result by passage to

from free G-spectra

\ x E

and

X+Ai E

indexed on

00 IR , as it turns out that

X LT( v X + Ai E)

are both equivalent to the untwisted half-smash

product

X+a E

indexed on

U.

i„(x x E)

and

For the first, using a more

general definition of twisted half-smash products which allows non-trivial target universes, we find that i^(x ix E) = (i.ox)

k

E,

iox: X ->^(U,U).

the G-contractibility of half-smash product

^(U,U)

(i°x) x E

that the twisted

is equivalent to the

untwisted half-smash product

X+a E.

i^(X+Ai E)

X+a i^i E,

is isomorphic to

evaluation map

i^i E

E

We then see from

For the second, and the natural

induces an isomorphism on the

(nonequivariant) homotopy groups

tr

.

Just as on the space

level, it follows that this map becomes a weak G-equivalence when smashed with the free G-space

X+ .

360

MAY CO

To prove Theorem A, we take acting by permutations.

For a spectrum

the j-fold external smash power indexed on space

Y,

U, and we set (2 Y)^'^

and

7T-spectra indexed on

E^^

E

ir C 2^.

indexed on

00 IR ,

is a 7r~spectrum

D E = Ett tx E^J^. tr 7r 2 (Y^^)

U,

-j

U = (IR ) , with

For a based

are isomorphic

and the relation

00 00 D 2 Y = 2 D Y 7r ir

follows. As said before, we take Theorem B.

BG

-V

= EG x S

-V

to prove

We sketch how (1) and (2) of that theorem

follow from this definition.

There is a general twisted

diagonal map 6: X kg (EaE) -- » (X kgE)a(X f^E'). With

X = EG,

E =

and

E’ = S

it specializes to

give a coaction —V

oo

—V

6: BG v -- > 2 BG+aBG \ and this coaction induces the x

—V

H (BG EG

).

When

G

H (BG)-module structure on

is finite, the skeletal filtration of

gives rise to a spectral sequence converging from

H*(G;H*(S~V ))

to

trivially on

H (S

*

H*(BG~V ). -V

),

Provided that

for example if

G

G

acts

is a p-group and

we take cohomology with mod p coefficients, the spectral sequence collapses to an identification of the free

* -V H (BG ) with

H (BG)-module generated by the fundamental class

ty € H n (S ^ ) ,

n = dim V.

The diagram

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA BG W = EG ix S W VJT f=lixG( e A l )



-- »

2°°BG a BG W +

|

|

—V W—V —W 8 BG V = EG o

in (1) of Theorem B follows.

and For

general compact Lie groups, we can use the same argument to identify

f

after using the alternative Thom spectrum

construction of

BG

-V

to calculate

* -V H (BG ).

In (2) of Theorem B, we start with a G-spectrum indexed on

U.

00 G i : IR = U

We have the inclusion

the underlying nonequivariant spectrum with G-action ignored. ^ G i’ (i kG ) map

k

is just

U,

and

i kG

We have an inclusion

and we say that

* G (i k^)

f: k

k

kG

such that

kG

is split if there is a

cf ^ 1: k -» k.

This holds

for such theories as cohomotopy, K-theory, and cobordism. When it holds, and not in general otherwise, we have isomorphisms (*)

kG (ixE) = k*(E/G)

for a free G-spectrum Here

i

and E

kg(2_AixE) = kx (E/G)

indexed on

CO IR

[13, 11.8.4].

is innocuous since free G-spectra live in the

trivial universe.

The presence of the adjoint

362

MAY

representation G

is finite.

from

EG

k

S

A

is essential, but of course

Now

-V

BG

-V

A = 0 if

is obtained by passage to orbits -V ®)

and, as already explained,

is equivalent to

EG+a S

-V

. Thus

(*)

gives the

isomorphism of (2) of Theorem B, and the diagram there is given by the naturality in applied to the map which

f

E

of the isomorphism (*)

1 tx (eAl): EG

the Thom spectrum of

-V: BG

BO x Z. Y

a: KOq (Y) -- » KOg (EG induced by the projection isomorphism in (*). classifying spaces

EG tx S

-V

KO^

x

,

from

G.

is equivalent to

More generally,

and the natural map Y) = KO(EG x Y)

EG x Y -» Y

and the first

There are canonical Grassmannian BO^fU)

and

K0,

these spaces [13, X§2]

00 x : EG -» ^(U,IR )

and

00 B0(IR ) ^ BO x Z

which

and the precise specification of leads to an evaluation map

00 e: 5>(U,IR )

G-map

-W

-V EG tx^S

consider an arbitrary G-space

Let

S

was obtained by passage to orbits over

We must still explain why

represent

k

xg B0g

00 (U) -- » B0(IR ).

be a G-map.

Y -» BO^fU), a(f)

It turns out that, for a

is represented by the

composite EG *gY --

^(U.IR0) xgB0g (U) ----— * B0(ffi°°) .

There is a Thom G-spectrum

M(f),

and one sees by

inspection of definitions [13, X.7.2] is isomorphic to

x x^M(f).

that

M(eo(x x^f))

We apply this fact with

Y

a

EQUIVARIANT CONSTRUCTIONS OF NONEQUIVARIANT SPECTRA point and

f = -V,

where the virtual representation

is viewed as an element of map from a point into M(f) = S

-V

RO(G) = KO^fpt) Cr

BO^fU). Cr

in this situation.

e°(x xq?): B G - . B O x Z we conclude that

and thus as a

Since -V

is indeed equivalent to

Moreover, as one would expect, the Thom diagonal 00

-V

Not surprisingly,

is the map we called

M(-V)

363

before, EG x^S

—V

M(-V) -*

+

2 BG aM(-V) corresponds under this equivalence to the coaction map

6

described above.

REFERENCES [1]

J. F. Adams. Stable homotopy and generalized homology. The University of Chicago Press, 1974.

[2]

J. M. Boardman. Stable homotopy theory. Mimeographed notes. University of Warwick, 1965, and Johns Hopkins University, 1969.

[3]

J. M. Boardman. Stable homotopy theory, chapter V duality and Thom spectra. Mimeographed notes. 1966.

[4]

R. Bruner, J. P. May, J. E. McClure, and M. Steinberger. ring spectra. Springer Lecture Notes in Mathematics.

Vol. 1176, 1986.

[5]

J. Caruso, J. P. May, and S. B. Priddy. The Segal conjecture for elementary Abelian p-groups, II; p-adic completion in equivariant cohomology. Topology. To appear.

[6]

G. Carlsson. Equivariant stable homotopy and Segal’s Burnside ring conjecture. Annals of Math. 120 (1984), 189-224.

[7]

R. L. Cohen. Stable proofs of stable splittings. Math. Proc. Camb. Phil. Soc. 88 (1980), 149-152.

364

MAY

[8]

E. S. Devinatz. A nilpotence theorem in stable homotopy theory. Ph.D. Thesis, Massachusetts Institute of Technology. 1985.

[9]

E. S. Devinatz, M. J. Hopkins, and J. H. Smith. preparation.

In

[10] J. D. S. Jones. Root invariants and cup-r products in stable homotopy theory. Preprint. 1983. [11] J. D. S. Jones and S. A. Wegman. Limits of stable homotopy and cohomotopy groups. Math. Proc. Camb. Phil. Soc. 94 (1983), 473-482. [12] N. J. Kuhn. Extended powers of spectra and a generalized Kahn-Priddy theory. Topology 23 (1985), 473-480. [13] L. G. Lewis, Jr., J. P. May, and M. Steinberger (with contributions by J. E. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics. Vol. 1213, 1986. [14] J. P. May.

H

ring spectra and their applications.

Proc. Symp. in Pure Mathematics, Vol, 32, Amer. Math. Soc. 1978. [15] J. P. May. The completion conjecture in equivariant cohomology. Springer Lecture Notes in Mathematics Vol. 1051, 1984, 620-637.

J. P. May University of Chicago Chicago, IL

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

Let

M

n

be a compact,

pseudo-isotopy of

M

whose restriction to

oo C manifold.

Recall that a

is a diffeomorphism F:Mn x I 1

x I U M x {0}

Mn x I

is the identity.

In his fundamental paper [1] J. Cerf described a ’’function theoretic” technique for studying the space of pseudoisotopies,

C(Mn )„

He in particular showed that there is

an isomorphism of homotopy groups 7rq (C(Mn )) = irq+1(F(Mn ), E(Mn )) where

F(M)

f:Mn x I boundary

is the space of all real valued functions I

whose restriction to some neighborhood of the

d(Mn xI)

is the projection, and

E(Mn ) C F(Mn )

is the subspace of those maps that have no critical points. It then became apparent that an understanding of the homotopy type of the spaces of functions on

M x I

whose

critical points are of a particular type is crucial to this

^The author was partially supported by NSF grant MCS82-03806 and by an A. P. Sloan Foundation fellowship.

365

366

COHEN

approach.

Recently, considerable information about these

spaces has been obtained by K. Igusa, [4,5]. In this paper we will use Igusa’s results and some stable homotopy theory to prove that the space of stable, generalized Morse functions on a manifold

Mn

has the

homotopy type of a product of smaller, simpler spaces. These spaces will

be shown to depend only

on the stable

homotopy type of

Mn and we will be able

to describe them

explicitly in both homotopy theoretic and in geometric settings.

In order to state these results more precisely

we first recall some definitions. Let

B.:IRm

IR be the function given by i+1

VX1

Xm>=Xl-

1

m

X5+ I V

j=2 If C

00

N

k=i+2

is a compact, smooth, m-manifold and

map, an element

critical point of neighborhood such that

U

y €N f

of

e(0) = y

f:N

IR

is a

is called a birth-death

of index i

if there is a

0 € IRm and an embedding

e:U

N

and

f(e(x}) = f(y) + B.(x) for all

x € U.

Now let tt:M

x I -> I

Mn

be a compact smooth n-manifold and

the projection.

As in [4] we let

the space of all smooth functions with

tr

in a neighborhood of

f'M x I

I

H(MxI)

be

that agree

d(MxI), and so that all

367

GENERALIZED MORSE FUNCTIONS degenerate points of

f

are birth-death.

Such functions

will be called generalized Morse functions. As described in [2] there is a directed system H ( M x I ) ---- » H(MxI2 ) ---- » H(MxI3 ) ---- » ... Roughly, the '’suspension map”

2

cr:H(MxI) -» H(MxI )

can be

described by the formula of(x,t) = f(x) + t^ where

x € M x I,

be modified near

and

t € I.

d(MxI)

(Note: This formula has to

in the usual manner so that the

boundary conditions are met.

See [2].)

We let

H(M)

denote the direct limit of this system. One of Igusa’s results in [4] is the following.

THEOREM.

If

n > 5

cr:H(Mn xI)

then the map

H(Mn xI^)

is

n-connected.

MAIN THEOREM.

If

n > 5

then there is a homotopy

equivalence H(Mn ) * where the spaces 1.

Y. k

2.

Let So

rr k>0

y

have the following properties'-

is 2k-connected. T7^(Mn )

be the unoriented bordism groups of

Mn .

?7x (M) =* p^(MOaM+ ) = Z/2[bi :i*2r-l] ® H^(M;Z/2).

Then, through dimension of graded groups

4k+l

there is an isomorphism

368

COHEN v v2k+l riun m v2k+2 irxYk 2; 2 tjx (M) © 2

Roughly speaking, the splitting of

.

H(Mn )

in this

theorem is obtained by using Igusa’s result in [4] that H(Mn ) 00

has the homotopy type of the infinite loop space

00

Q 2 (B0a M+ ),

and then appealing to a stable splitting

theorem of V. Snaith [6]. significance of the spaces

We will also study the geometric Y^.

In particular we will

describe a relationship between these spaces and certain subspaces of

H(M)

defined by allowing only those

generalized Morse functions whose critical points all have certain specified indices. This paper is organized as follows.

In section 1 we

describe some background material, recall some results from [1,4], and in particular describe the significance of in studying pseudo-isotopies. description of

H(M)

H(M)

In section 2 we use Igusa’s

in terms of the space of sections of

a certain jet bundle, and use his techniques from [5] to show that if we restrict to certain subbundles, we obtain spaces homotopy equivalent to

CO °0 0 2 (B0(k)AM+ ).

In section

3 we recall Snaith’s splitting theorem, define the spaces Y^, and prove the main theorem.

We then end the paper by

relating our results to the 2-index conjecture of Hatcher

369

GENERALIZED MORSE FUNCTIONS The author is grateful to D. Burghelea and W. C.

Hsiang for helpful conversations concerning this material, and to K. Igusa for helpful correspondence.

The author is

also grateful to the mathematics departments at the University of Chicago and at Princeton University for their hospitality during visits when some of this work was carried out.

§1. BACKGROUND MATERIAL Let

C(M),

F(M), E(M), and

H(MxI)

spaces defined in the introduction.

be the function

Consider the map

a:C(M) -» E(M) defined by that

0

p(G) = ttoG:MxI

is a fibration.

isotopy space

Mxl -> I.

In[l] Cerf proved

Indeed the fiber = p ^(tt)

^(M) = {GCC(M) *ttG=7r}.

homeomorphic to the path space

Notice that

space

F(M)

agree with

contractible.

Thus

is

Thus

C(M) is

E(M). Notice furthermore that the

consisting of all functions ir near

^(M)

{a:I -» Diff(M,dM) :a(0) = 1}

and so it is canonically contractible. homotopy equivalent to

is the

6(MxI),

f:MxI-»I

that

is convex and hence

tt^(E(M)) ^ tt^+^(F(M), E(M)).

In sum

we have the following.

THEOREM 1.1.

There is an isomorphism of homotopy groups,

irq+1 (F(M), E(M)) * *C(H).

370

COHEN Even the first of these homotopy groups

7r^(F(M),

E(M)) = tTq C(M) has important geometric significance. understand this, observe that the isotopy group on

Diff(M,dM)

To

^(M)

acts

by the function

p:^(M) x Diff(M,3M) -* Diff(M.dM) defined by letting given by

p(G,f)

be the diffeomorphism of

M

p(G,f)(x) = G(f(x),l) € Mx{l} C Mxl.

Two diffeomorphisms are said to be isotopic if they lie in the same orbit under this action.

Notice that this

is equivalent to the two diffeomorphisms lying in the same path component of

Diff(M.dM).

Now observe that this action extends to an action of the pseudo-isotopy space, p:C(M) x Diff(M.5M) -> Diff(M.dM) def ined in the same manner. We then have the induced pseudo-isotopy equivalence relation defined on

Diff(M,9M).

Notice that two isotopic diffeomorphisms are pseudo­ isotopic, and the obstruction to the converse is given by the number of path components of the pseudo-isotopy orbits. In particular these obstructions are encoded in

tTq C(M)

ir1(F(M) ,E(M)). In [1] Cerf proved the following-

THEOREM 1.2.

If

n > 5

and

tt^(F(M) ,E(MJ) = tTqC(M) = 0, diffeomorphisms are isotopic.

- 0,

then

and ft&rice pseudo-iso topic

=

371

GENERALIZED MORSE FUNCTIONS

Soon thereafter Hatcher and Wagoner generalized Cerf’s results to the nonsimply connected case.

In particular

they proved a stronger version of the following.

THEOREM 1.3.

If

n > 7

then

functor of the homotopy groups

TT^CfM11) 7T^(Mn )

is an algebraic and

tt^(Mn ).

A major step toward understanding the higher homotopy type of

C(M)

was made by Igusa when he proved the

following.

THEOREM 1.4.

(theorem 10.1 of [4]).

For

n > 5

the

composition TTk (H(Mn xI), E(Mn )) -» irk (F(M), E(M)) £ i r ^CfM11) is a surjection if

k < n+1.

This epimorphism is split if

k < n.

Thus for

k < n+1

every element of

7r^,_^C(Mn )

can be

represented by a k-parameter family of smooth maps whose critical points are all either nondegenerate or birth-death. It is easy to see that the homomorphisms E(M)) -+ Trk lC(tt)

ir^(H(MxI),

are compatible with the ’’suspension” maps

obtained by crossing the manifold with the interval; that is, the following diagram commutes:

372

COHEN

7rk (H(MxI),E(M))

---- > Y l C(M)

a |

[a

TTk (H(MxI2 ),E(MxI)) Therefore if we let E(M) = lhm E(MxI^) q

COROLLARY 1.5.

>irk lC(MxI)

C(M) = lim C(MxI^), £

and

we obtain the following:

For

n > 5

k

and for each

we have a

split short exact sequence 0 ^7Tk (H(Mn )) ---- >7Tk (H(M), E(M))

»TTk_1C(M)

0

As stated in the introduction, our interest in this paper is to gain information about will use a description of

H(MxI)

tt^H(M)

. To do this we

obtained by Igusa that,

as it turns out, in fact implies 1.4.

First we adopt some

notation. Let

N

be a manifold and

f

N a fiber bundle

equipped with a section

o^N-af.

space of sections of

that agree with

neighborhood of

f

Let

in a

H(Mn xI), through a range of

dimensions, with a space of sections an appropriate bundle J^(Nm )

a^

denote the

c/N.

The idea is to identify

Let

F^.(N,dN)

£*.

f

(Mxl),d(MxI))

is defined as follows.

be the space of k-jets of maps

(See [3] for example.)

The source map

N

IR.

of

GENERALIZED MORSE FUNCTIONS

373

Jk (Nm ) -> Nm is a p

fiberbundlewith fiber

Pm = the space of polynomials

in m-variables with degree < k.

The structure group of

this bundle is the group of k-jets of diffeomorphisms of (CRm ,0)

which can be reduced to the orthogonal group

The associated principal

0(m)

0(m).

bundle is isomorphic to the

t(N).

principal tangent bundle,

Now restrict to the case

N = Mn xl

and

k = 3.

Let

3

C Pn+-^ be the sub space of degree 3 polynomials 3 p € Pn+ 2

such that one of the following holds:

a.

0 6 IRn+^ is a regular point of

p,

b.

0 6 IRn+^ is a nondegenerate critical point of p,

c.

0 6 [Rn+^

or

3 F^+^ C Pn+2

is a birth-death critical point of is an

p.

0(n+l)-invariant subspace and so

determines a subbundle p:f -> Mnxl of

J3(Mn xI)

with fiber

F

Now notice that any map section f

at

x.

3 j f:MxI -> f Let

where as above,

:H(MxI) j3 :H(Mx; Of [4].

f 6 H(MxI)

by letting

cr^:MxI

f

tt:Mx I

I

form the section space

r

3 j f(x)

be the 3-jet of

be the section

3 cr^ = j tt,

is the projection.

r^.(MxI,d(MxI))

r^(MxI,a(MxI)).

determines a

We then can

and a map

The following is theorem 9.1

374

COHEN

THEOREM 1.6.

For

r^(Mn xI,a(MxI))

n > 5, the map is

n+1

j^:H(Mn xI) -*

connected.

For the sake of completeness, we end this section by including Igusa’s proof that theorem 1.6 implies theorem 1.4.

So assume theorem 1.6.

Proof of l.h.

Abbreviate

r^fMxI, d(MxI))

by

and

consider the long exact sequence of homotopy groups ... -+ 7Tk (H(MxI), E(M)} ^7Tk (rc ,E(M)) ^7Tk (rc ,H(MxI)) By 1.6,

7Tk (r^.,H(MxI)) = 0

Trk (H(MxI), E(M)) k < n

for E(M))

and a surjection for

Now recall that F(M) is contractible.

k < n+1

..

and so therefore

is an isomorphism for

k < n+1.

irk_^C(M) ^ 7rk (F(M), E(M)), and that Thus to prove 1.4 it is sufficient

to show that the boundary map

E(M)) -+ ?rk_^(E(M)) i

a split epimorphism. To do this Igusa constructed a contractible space so that the map r . X

r X

3 j :E(M) ~+

naturally factors through

is defined as follows.

polynomials so that either

0

Let

3 X C P n+1

be those

is a regular point, or a

nondegenerate critical point of index zero.

X

is clearly

contractible via a linear homotopy to the polynomial

GENERALIZED MORSE FUNCTIONS

375

n+1 11q € X

defined by

h^fX) =

^

x^.

Since

X C P^+ ^

is an

i=l 0(n+1)-invariant subspace, it determines a subbundle X C [ C J^(Mn xI)

having fiber

X.

We therefore get a

natural factorization j3 :E(Mn ) -> r (Mxl, a(MxI)) X But since

x

->rr .

has contractible fiber, the space of sections

T = T (Mxl, d(MxI)) X X

is also contractible.

As observed

above, this proves theorem 1.4.

§2.

RELATION WITH THE STABLE HOMOTOPY OF

B0(k)

In [5] Igusa showed that the space of sections T^ rf (MxI,a(MxI)) admits an loop space

00 00 Q 2 (B0aM+ ).

n+l-connected map to the infinite In this section we use Igusa’s

techniques to show that there are subbundles whose space of sections n+l-connected maps to Mn xl

=

= ^C(k) 00 00 Q 2 (B0(k)AM+ ).

f(k)

'^(^*1)) The bundle

of

f

admit f(k) -»

is defined as follows. Let

3 Fn+^(k) C F^+^ C Pn+2

of third degree polynomials

^ e subspace consisting

3 p € Pn+^

such that one of the

following holds n+1

a.

0 € IR

b.

0 6 IRn+^

is a regular point of

p

is a nondegenerate critical point of

index < k, or c.

0 6 IR

T1+1

is a birth-death critical point of

index < k - 1.

376

COHEN Clearly

F

. n+1

^n+^(k) 3

and

P n+1

is an

0(n+l)invariant subspace of

and so it determines a subbundle of the

3-jet bundle T(k) ---- » Mn xl with fiber

Fn+^(k).

Thus

C(k) C J^(MxI)

consists of

those 3-jets whose critical points are either nondegenerate of index

< k or birth-death of index

Now since the projection the section C(k).

tt:Mx I

o^ = j^7r:Mn xI -+ J^(MxI)

< k - 1. I

is a submersion,

factors through each

We then can define the spaces of sections

=

(Mn xl,6(Mn xI)). The following is the main result of this section.

THEOREM 2.1.

There are n-connected maps

00 00f f n Q 2 (BO(k)AM+ )

that make the following diagrams homotopy

commute:

rC(k)

rC(k+l)

| fk

| fk+l

Q°°2C0(B0(k)AM+ )-- — --- > QC02C°(BO(k+l)AM+ ).

Here

i

is induced by the inclusion of the subbundle

C(k) C f(k+l),

and

j:B0(k) -+ B0(k+1).

j

is induced by the inclusion

377

GENERALIZED MORSE FUNCTIONS

The following is the first step in the proof of this theorem.

PROPOSITION 2.2. 2n+^B0(k)

There are 2n+l-connected maps k.

that are compatible over

This is, the

following diagrams homotopy commute Fn+i(k)

---- * Fn+1(k+ l)

Jgk

|gk+ l

2n+1B0(k)

Proof.

Let

F +i(k) and - C

En+^(k)

---- » 2n+1B0(k+l)

be the space of all polynomials

with no constant or linear term.

Dp(0) = 0. where

.

Thus

p(0) = 0

Notice that we can express E^+^(k)

3 m = dim(Pn+^)-n-2

subspace.

Now since

^n+^(^)

Dp(0) £ 0

we have that

where

C

p G

as

IRm

is a closed

contains all

3 p € Pn+^

with

Fn+1(k) = IRx(IRm+n+1 - C) . where the

IR factor corresponds to the constant term of

the polynomial. 3.1

A standard point set argument (see lemma

of [5] for example) then yields the following.

LEMMA 2.3.

F^+ ^(k)

Is homotopy equivalent to the join

Sn*(IRm-C) * Sn*En+1(k) * 2n+1En+1(k).

378

COHEN Thus prop. 2.2 will be a corollary of the following.

(Compare [5; 3.2]).

LEMMA 2.4.

a.

cr:En+^(k) -»

a (P)(x !

Xn+l'Xn+2> = P + xn+s*

ts n+1~

connected. b.

Proof.

lim E (k) a BO(k). m

We continue to follow the arguments of [5], adapted

to our situation. For

i < k

polynomials point of Gi

A. C E .(k) i n+lv J

such that

p of index

0

i.

in IRn+*.

(p ) = N(p),

subspace of

be the set of

is a nondegenerate critical

Let

= 0(n+l)/0(i) x 0(n+l - i)

i-planes

by

p

let

be the Grassmannian of

Define a map

where

X1+1 IR

N(p)

is the i-dimensional

spanned by the negative eigenvectors of

the second derivative,

2 D p(0). As observed in [5],

is

surjective, and is a homotopy equivalence. Similarly, for En+i(k) p

such that

of index

i.

i defined on

B. i

Then

extends to a map to be the negative

to be the map which

associates to a polynomial subspace



A. = A. U B . U B . i l l l-l .

eigenspace map, and on

* ii *

p €

^he i“dimensional

N(p) © K(p) C IRn+^ . Again, as observed in [5],

is a homotopy equivalence. Notice that this says that

En+^(k)

is, by

definition, the pushout of the diagram Aq \

Ax / \ B0

A2 / B1

\

A3

.. .

/ B2

\

Ak_1 / Bk-2

\

Aj^ / Bk-1

where all the maps in this diagram are the inclusions. Since they are in fact cofibrations we obtain the following.

(Compare [5; 3.4].)

380

COHEN

Claim 2.5.

*s homotopy equivalent to the homotopy

pushout of the diagram

A A 4i\ A vA A-i

Si+1,0

Si+1,1

G1 n+1,0 where the maps

i^

by the identity on

Remark.

Si+1,2

’'‘*

G1 n+1,1 and

Si+1,k

G1 n+1,k-1 are the quotient maps induced

0(n+l).

Recall that the homotopy pushout of a diagram X1

X2

Xk-1

f l \ / gl

Xk

fk \

Y *1

/ V

1

Y *k-l

is defined to be the union of the mapping cylinders k

11 i= l

(Y

xl) U

1

X. U

gi - l 1

(Y.xl)/

i

yxl € Y. xl c (Y^xIJU yxl € Y.xl C (Y

U(Yi+^xI)

this is

is identified with

xI)U X ± U(Y.xI).

Using this description of that the map

where

1

cr:En+^(k)

En+^(k)

En+^(k) is

is it immediate

n+l-connectedsince

the map of pushouts of thetype given in 2.5

induced by the inclusion of the orthogonal groups 0(n+l) c__+ 0(n+2). Now consider the map

E^+ ^(k) -+ B0(k)

defined by

homotopy equivalence of 2.5, using the maps of the Grassmannians

the

381

GENERALIZED MORSE FUNCTIONS G for

j < k.

equivalence

, lim G = BO(r) — — >BO(k) n+l,r m,r v J l v J m The fact that in the limit one gets a homotopy litn E (k) ^ BO(k) m

after observing that for

is a standard argument,

r < k, the composition

Gn+1,r ---- >B0(r)

* B0{k)

is the same as the composition Gn+i

=0(n+l)/0(r) x 0(n+l-r) ^ 0(n+l)xl /0(r)xO(n+l-r)xl^

«-K>(» 2 ) / 0 ( r ) * I 1,0(.>H-r) = Gm 2 ,p

°n+2 ,r+l

= 0(n+l)/0(r+l)x0(n+l-r) ---- > B0(r+1)

BO(k) .

We leave the details of this argument to the reader. We remark that an argument of this type was carried out in considerable detail in [5]. This completes the proof of lemma 2.4 and therefore of proposition 2.2. We now proceed with the proof of theorem 2.1. The idea is to replace

C(k)

by a trivial bundle

whose space of sections has the same n-dimensional homotopy type of

Fr(k)-

Since the space of sections of a trivial

bundle can be identified with the space of pointed maps from the base to the fiber, we will be in a more manageable situation. Now recall that

C(k)

is an

0(n+l)

bundle with

associated principal bundle isomorphic to the principal tangent bundle,

T(Mn xI).

Thus to ’’trivialize”

f(k)

will add on a normal bundle in the following manner.

we

382

COHEN Let

v

be the normal bundle to an embedding

Mn xl rG(k)(D(ii). S(«)) defined as follows. define a map

If

a:Mn xI -+f(k)

2U(cr):D(r) -+2Uf(k)

by

is a section, 2U(a)(x) = (x,q(x)).

By the pull-back property this defines a section 2Ua:D(u) -> G.

LEMMA 2.6. n+1-connected.

(Mn xl ,3(Mn xI) -»

(D(u) , S(u) )

is

383

GENERALIZED MORSE FUNCTIONS This follows from the Freudenthal suspension

Proof.

theorem, after observing that

Mn xl

the fiber of

is n+l-connected by 2.2.

C(k)>

Fn+^(k),

is n+l-dimensional and

(We note that the details of the analogous result with replacing

£(k)

f

were carried out in [5].)

As observed earlier,

G(k)

has fiber

Sn * ^ n + l ^ ) ’ an^ ^as structure group

2n+^Fn+^(k) ^

0(n+l) x 0(n+l).

As in [5] one can define an n+l-connected equivariant map h :£>n * Fn+i(k ) _>F2n+2^k ^' by

0(2n+2).

bundle of

where

F2n+2^k ^

is acted upon

Since the associated principal

0(2n+2)

G(k) is trivial, this would allow us to define a

bundle map G(k) —

D(i>) x F2n+2(k)

I D(i>) The map

I — =-»D(»)

h is constructed as follows.

be the standard inclusion.

Define

h^(x)(y) = the inner product

Let

j :Sn

IR^n+^

h. :Sn -+ F~ ~ 1 2n+2

.

by J

Define

by n+1 V p > ( x1

X 2n+ 2 ) = P K + 2 .....X 2n+2> +

I

Xf

'

i=l h:Sn * F

^(k) -+ ^2n+2^^

is defined ^y the formula

h(y»t»P) = th^(y) + (l-tjh^fp), that

h

is equivariant is clear.

trivialization

where

t € I.

The fact

Thus we get the induced

h:G(k) -+ D(u) x F ^ ^ f k )

as above.

The

384

COHEN

restriction of

h

to the fibers, namely

h,

is easily

seen by 2.2 to be 3n+2-connected and hence the induced map of sections ^ :rG ( k ) ^ U^’S ^ ^ is n+l-connected. maps

X -* Y

Here

MaP(D (u)'S (l)):F2n+2^k ^

Map(X,A;Y)

is the space of all

which restrict to a fixed map on

Now by 2.2,

Map(D(u),S(u); ^2n+2^*0^

dimensional homotopy type of which has the same Map(D(u)/S(i>)

has the n+1-

Map(D(u),S(u); 2^n+^B0(k}),

2n+2-dimensional homotopy type as

2^n+^B0(k)).

2n+2 S-dual of

A.

M^,

But since

D(r)/S(u)

is the

we have that this last space is 00 00 Jl 2 (B0(k)AM+ ).

homotopy equivalent to

Combining this

with lemma 2.6 yields our n-connected maps fk :rf(k) ^ n“2“ (B°(k ) A M+)The fact that they are compatible as from the constructions.

k

varies is clear

This completes the proof of

theorem 2.1. We end this section by stating a stable analogue of theorem 2.1. Let bundle

2mf(k)

be the m-fold fiberwise suspension of the

C(k) -+Mn xI.

So 2mC(k) = C(k)xIm/~

where

(x,t) ~ (x',t)

The fiber of the bundle suspension

2mFn+j(k).

if

p(x) = p(x') 2mf(k) -+ Mn xl

Now let

and

t € d(Im ).

is the m-fold

Smf(k) -+ Mn xlm ^

be the

GENERALIZED MORSE FUNCTIONS pullback of

2mf(k)

over

(Mn x l )

x l m.

385

As we did above we

can construct a suspension map ,(Mxi,a(Mxi)) Now recall the map *

r (Mxim+1,a(Mxin+1)). smc(k) hiS'V4 . -> fk n+1 n+m+1

defined above.

This yields a map of bundles h:SmC(k) -» C(k,m) where with

f(k,m) Mn x l m+^

is the bundle defined analogously to replacing

Mn x l .

T(k)

Finally, define the

suspension map am :rc(k)(MxI,3(MxI)

rc ( k m ) (MxIm+1,a(MxIm+1))

to be the composition

r_, ,

,(M xi,a(M xi))

— —

»r

(Mxim+1 , a ( M x i m+1) )

Smf(k)

The following is an immediate consequence of both theorem 2.1

and its proof.

COROLLARY 2.6.

Let

rk = lim

T+1

T+1

rj(MxIx \ 3 ( M x r

x))

where the directed system is defined by the suspension maps om

as above.

Then there is a homotopy equivalence rk ^ n°°2” (B0(k)AM").

§3.

PROOF OF THE MAIN THEOREM AND ITS RELATION TO THE 2-INDEX CONJECTURE We now proceed with the proof of the main theorem as

stated in the introduction.

The key ingredient is the

386

COHEN

following stable splitting theorem of V. Snaith [6].

THEOREM 3.1.

For every

k

there is a stable map (map of

CO CO BO -> 2 B0(2k)

suspension spectra)

so that the

composition 2°°B0(2k) — -- » 2°°B0 ---- > 2°B0(2k) J rk is stably homotopic to the identity.

This theorem implies there is a splitting of suspension spectra 2°°B0 ^

V 2C°B0(2k+2)/B0(2k), k>0

or equivalently of infinite loop spaces nW2“B0 ^

TT

fi‘V >(B0(2k+2)/B0(2k)).

k>0

We will use this splitting, together with corollary 2.6 to prove the main theorem. So let ^2k+2

Tk

be as in 2.6 and consider the map

^jc:^2k

Educed By the inclusion of the subbundles

f (2k,m) [ (2k+2,m) .

COROLLARY 3.2. so that

Proof.

There is a retraction map

o i^

By 2.6

Pjc'^2k+2

is homotopic to the identity of

i^

follows from 3.1.

^ k ’

is homotopic to the inclusion

00 00 00 00 n 2 (B0(2k)AM+ ) -> Q 2 (B0(2k+2)AM+).

^2k

The result now

387

GENERALIZED MORSE FUNCTIONS Finally, define retraction map

to be the homotopy fiber of the

° k :^2k+2

^2k*

^ ° ^ owinS is now

immediate.

COROLLARY 3.3.

Y^

is homotopy equivalent to

oo oo r» n 2 (B0(2k+2)/B0(2k) aM_j_) , and

T2k+2 ^

x Yfc.

Thus the homotopy theoretic interpretation of the group

7T Y, q k

is as the stable homotopy group

7r^(B0(2k+2)/B0(2k)AM^). Geometrically, we can think of this group as follows: V k

= 7rq^F2k+2 ’F2k^ =

Thus an element of

is represented by a q-parameter

family of sections of the 3-jet bundle, r v J (M xlTm+1\ f ^.„n M xlTm+1 -» ); so that for each

t £ Dq

3-jet of a function

f

and

,q t^ €^ r IT1

x € Mxlm+^, ft(x)

is the

x ;Mn xlm+* -» IR satisfying the

following properties: a.

All critical points of ^

f^ t ,x

nondegenerate of index < 2k+2

are either or birth-death of

index < 2k+l. b.

If

t € 5Dq = Sq *, x

then all critical points of

are either nondegenerate of index < 2k or

birth-death of index < 2k-1.

388

COHEN c.

In a neighborhood of with

Tr:(MxIm ) xl

d(M*I

agrees

I.

00

00

Now by 3.1

2 (BOaM ) ^

00 00

-r-r-

and hence

), f^ x

Q 2 (BOaM ) ^

V 2 (B0(2k+2)/B0(2k)AM ) k>0 00 00

JT ^ 2 (B0(2k+2)/B0(2k)aM ) k>0

“ TT V k>0

Combining this with Igusa’s result that for

n > 5

H(Mn ) ^

CO 00

rr = lim rrf, > ^ fi 2 (BOaM ), ic * *

the first part of the main

theorem follows. We now prove part b. of the main theorem; that in dimensions < 4k+l v

there is an isomorphism ^ v2k+l

m v2k+2

= 2 where of

=

2

tt^(M0 a M+ )

is the unoriented bordism group

M. Now as already established,

homotopy group

ir^\^

*s t^ie stable

7r®(B0(2k+2)/B0(2k)AM+ ). To compute this

group consider the cofibration sequence of spectra B0(2k+1)/B0(2k)

a

M+ ---- » B0(2k+2)/B0(2k)

a

> B0(2k+2)/B0(2k+l)

M+ a

M+ .

Recall that there is a 4k+2 connected map of spectra 9k+1 a2k+l *B0(2k+1)/B0(2k) = M0(2k+1) ----- >2 MO, similarly a

and

4k+4 connected map

a2k+2:B0(2k+2)700(2k+1) = M0(2k+2)

» 22k+2M0.

Moreover, since

MO

is a wedge of Eilenberg-MacLane

spectra of type

K(2£/2), and since the inclusion

B0(2k+1)/B0(2k) ---- > B0(2k+2)/B0(2k)

induces a surjection

389

GENERALIZED MORSE FUNCTIONS

in mod 2 cohomology, there is a map a2k+i: ^0(2k+2)/B0(2k) »2

2k+l

MO

that extends

Thus through dimension

a2k+l’ 4k+l

to homotopy.

we get a split short

exact sequence of stable homotopy groups 0 ---- » TT®(22k+1M0AM+ )

» ir®(B0(2k+2)/B0(2k)AM+ )

» ir®(22k+2M0AM+ ) ---- » 0. Part

b

of the main theorem now follows.

We end this paper by describing some implications of these results to the 2-index conjecture of A. Hatcher [2]. Let

k n H j (M xl)

of functions

be the subspace of

f:Mn xI -» IR

in

H(MxI)

critical points all have an index

i

H(MxI)

consisting

whose nondegenerate such that

j < i < k,

and whose birth-death critical points all have an index such that

j < i < k-1.

Now let

, H.(Mn )=

J

i

lim , -> H.(Mn xIm 1).

m

J

The 2-index conjecture can be stated as follows.

CONJECTURE 3.4.

The inclusion

Hq

is a homotopy

equivalence.

To relate this to the results in this paper, first recall Igusa’s

n+1 connected map (theorem 1.6).

j3 :H(MnxI) ---- > rf(MnxI,a(MnxI)). k n H q (M xl) C H(M x I)

Restricting to in

^^(MxI.dfMxI)).

a map

^ H

q

this map has its image

After passing to the limit, we get

CM) -* rfe.

390

COHEN

CONJECTURE 3.5.



if

Is a homotopy equivalence

n > 5.

Remark.

By letting

k

tend to infinity the conjecture

becomes true by observing that Igusa’s H(MxI) -> r^.(MxI,d(MxI))

n+l-connected maps

respect the limiting process.

Notice that if both conjectures 3.4 and 3.5 are true, we would have that the composition gk :i l (Mll) C

— 3“ * Fk -----*■ 0"2"(B0(k)AM^)

j would be a homotopy equivalence.

The splitting in our main

theorem could then be realized in this setting in the following manner. Consider the suspension map .TT2k ,„2nA a ’-2k-l( >

defined as follows. map

f:MxIm

IR.

Let

^ TT2k+2rijrn^ > -2k+l^ > 2k

f € ?2k-l^^

rePresente(* ^y a

We then represent

cr(f) €

CT(f):(MxIm ) x I x I

»R

by

the map

def ined by cr(f)(x,s,t) = f(x) - s^ - t^. It is easy to see that the following diagrams homotopy commute: s 2 U < * “ >



js2k

nC°200(BO(2k)AM+ )

O

' "

)

|S2k+2 -j

» 0 ° 2°°(B0(lk+2) a M +

) .

GENERALIZED MORSE FUNCTIONS

391

Thus if conjectures 3.4 and 3.5 were true, corollaries 3.2

and 3.3 would imply the existence of retractions v a S " " )



g -itf* )

and indeed the existence of homotopy equivalences -2k+l(

)“

-2k-

1^ ) * V

Thus the main theorem would yield a good bit of information about the homotopy groups

^ 7TxH ^ +^(Mn ) .

REFERENCES [1]

J. Cerf, La Stratification naturelle des espaces de fonctions differentiables relies et le theoreme de la pseudo-isotopie, Publ. Math. I. H. E. S. 36 (1970).

[2]

A. Hatcher and J. Wagoner, Pseudo-isotopies of compact manifolds, Asterisque 6, Soc. Math, de France (1973), Paris.

[3]

M. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer-Verlag, (1976) New York.

[4]

K. Igusa, Higher singularities of smooth functions are unnecessary, to appear, Annals of Math.

[5]

K. Igusa, On the homotopy type of the space of Morse functions, preprint, 1983.

[6]

V. Snaith, Algebraic cobordism and K-theory, Memoirs of A.M.S. #221 (1979).

Ralph L. Cohen Stanford University Stanford, California 94305

XVI ALGEBRIAC K-THEORY OF SPACES, CONCORDANCE, AND STABLE HOMOTOPY THEORY Friedhelm Waldhausen

It is known [7] that there is a splitting, up to homotopy, A(X) ^ AS (X) x WhDIFF(X) as well as another G

oo 00

A°(X) ^ Q S (X+ )

X

p(X) .

It will be shown here that the factor

p^(X)

is trivial.

Hence we have

oo oo

THEOREM.

r jT F F

A(X) ^ Q S (X+ ) x WIi

f(X) .

The method of proof is to establish a version of the Kahn-Priddy theorem for

p.(X) . As

]Lt(X)

is a homology

theory there results a kind of growth condition for the homotopy groups.

But

M-(X)

is connected, so the growth

condition boils down to zero growth and thus we can conclude that

M-(X)

is trivial.

To explain what is meant by a Kahn-Priddy theorem we have to know about transfer maps.

392

First, there is a

ALGEBRAIC K-THEORY OF SPACES

transfer for the algebraic K-theory of spaces: made from spaces over

X , so if

X -» X

393

A(X)

is a fibration

with fibre of finite type, pullback induces a map A(X) , cf. [8].

is

A(X)

Next, if the fibration is a finite

covering projection then the transfer can be considered in the framework of the ’manifold approach’ of [7], in particular everything in theorem 1 of that paper is compatible with the transfer. and

S A (X)

It follows that

00 00 Q S (X+ )

have transfers for finite covering projections

which are compatible to the transfer on

A(X) , and

compatible to each other, in the sense that it is possible to fill in the broken arrows so that the following diagram commutes, up to homotopy, 0 ° V “(X+ )

> AS (X)

» A(X)

i

i

I

n "s"(X + ) -------- » AS (X)

> A(X) .

It could be checked directly that these transfers agree with the usual ones (which are defined for all homology theories) but we will not need this fact. Let

denote the symmetric group,

classifying space, and A(X)

ES^

its

the universal bundle.

Let

denote the reduced part, the factor in the splitting

A(X) ^ A(*) x A(X) , i.e., The transfer gives a map

A(X) = fibre(A(X) -» A(*)) .

394

WALDHAUSEN A(B2n ) ---- > A(B2n )

Let

p

> A(E2n ) * A(*) .

be a prime and let the subscript

localization at

p .

(p)

denote the

Following the method of Segal [2] it

was shown in [6] that the Kahn-Priddy theorem is valid for the algebraic K-theory of spaces in the sense that for every

p

the map tt.A(B2 ), x j v p '(p )

is surjective for every

* 7r.A(^) f . j (p)

j >

Our main task here will

0 . be toshow that the

analogous

map 7T.AS (B2 ), . ---- » 7T.AS (*)r x J v p'(p) J v y(p) is also surjective.

This would follow at once if we knew

that the map

A (X)

g

A(X)

were transfer commuting.

However we do not know this,

so we must proceed

differently. In [6] there were constructed maps ( ’operations’) en : A(*) ---- » A(B2n ) . They have the property, among others, that the composite of 0n

with the transfer map A(B2n ) ---- ► A(E2n ) «

is homotopic to the polynomial map of associated with the polynomial

LEMMA.

The map

operation

0n

S A (X)

A(X)

A(*) A(^)

to itself

x(x-l)...(x-n+1) .

Is compatible with the

in the sense that it is possible to fill in

the broken arrow so that the diagram

395

ALGEBRAIC K-THEORY OF SPACES

AS (*) ---- > A(*) i

I

AS (B2n ) ---- > A(B2n ) commutes up to (weak) homotopy.

s A (*) -» A(*)

Also,

is a

map of ring spaces.

Since

S A (*)

A(*)

is a coretraction, up to

homotopy, we obtain from

the lemma

COROLLARY 1.

map A^(*) -» A^fBE^)

There is a

composite with the transfer

whose

S S A (82^) -> A (*)

is the

g

A (*)

polynomial map on

associated with the polynomial

x(x-l)...(x-n+1) .

The desired Kahn-Priddy

S A (X) now

theorem for

follows from corollary 1 by a formal argument.

The

argument may be found in the introduction to [6].

(The

argument involves an application of Nakayama’s lemma, so S A (*)

one has to know the homotopy groups of generated.

As

S A (*)

is a factor of

A(*)

are finitely this follows

from Dwyer’s theorem that the homotopy groups of finitely generated [1].)

COROLLARY 2.

Thus,

For every prime

p , the (transfer) map

tt.AS (B2

J

v

), . ---- » P'(P)

is surjective for every

A(*)

j > 0 .

tt.AS (*),

J

v

> (P)

are

396

WALDHAUSEN It was explained earlier that the map

S A (X)

is transfer commuting.

CO 00 Q S (X+ )

Hence we know that in the

partially defined map of short exact sequences

u-.oVVbE J

v

), .

» ir,AS (B2 ), >

P+'(p)

J

oo oo

P 7( p )

v

» ir .?I(B2 ), , J

S

7T.0 S f x j (p)



» 7T.A (*), x j v '(p)

the left arrow can be filled in. arrow can also be filled in.

p'(p)

>K j

, x '(p)

It follows that the right

From corollary 2 we therefore

conclude

COROLLARY 3.

For every prime

p

and for every

j > 0

there is a surjective map 7T.p(B2 ), x ---- >7r.p(*), x . j p '(p ) j v '(p)

p(X)

Proof of theorem.

suffices to show that every prime induction on

p

is a homology theory, so it p(*)

is contractible; or that for

the localization j

is*

that the homotopy groups

s^ow ky p j are

trivial. The induction beginning is provided by the fact that p(*}

is connected.

2-connected:

In fact,

is known to be

this follows from the double splitting

theorem together with the fact [5] that the map 7TjA(*)

is an isomorphism for

Suppose now that i < j-1 .

j > 0

CO 00 S

ttSI

j < 2 . and that

=^

By the spectral sequence of a generalized

ALGEBRAIC K-THEORY OF SPACES

397

homology theory we obtain that the reduced group 7Tjp(X)^^ is trivial for every

X . Taking

X =

we therefore

conclude from corollary 3 that there is a surjective map 0 = 7T.p(B2 )f . ---- \ • j p (p) j Hence

y(p)

= ^ * This completes the inductive step

and hence the proof.

It remains to prove the lemma.

To prove the lemma we need a framework where an S A (X)

explicit description of A(X)

are available.

and of the map

S A (X) -»

The ’manifold approach’ of [7]

provides such a description in terms of smooth manifolds. Namely, supposing that partitions of

X

is a manifold, one considers

Xx[0,l] , that is, triples

XxO C M

, Xxl C N , and where

M

N . These form a simplicial category

and

described [loc.cit.].

F

(M,F,N)

is the common frontier of

X x [0,e]

shown in [7] that

h^fX)

as

There is a simplicial subcategory

h$™(X) ; briefly, those partitions where from

where

by attaching of

k

M

is obtained

m-handles.

It is

A(X) , or rather a connected component

of it, is obtained by the Quillen + construction from the (homotopy) direct limit, with respect to h#^(XxJn )

where

J

n, m, k,

denotes an interval.

q

that

A (X)

is similarly obtained from the

of the

It is also shown m ti ^(XxJ ) ,

398

WALDHAUSEN

where

^(X)

denotes the simplicial set of objects of the

simplicial category

h9^(X) . The map

thus represented by the inclusion map

A^(X)

A(X)

is

$™(XxJn )

M^(XxJn ) . It has been discussed in [7] that union of the

h9l ’m (XxJn ) , the

h#^(XxJn ) , has a composition law given by

gluing (at least in the limit with respect to

n) .

composition law restricts to one on

&m (XxJn )

(in the

limit again).

S A (X)

It results that both

The

and A(X)

are

H-spaces (infinite loop spaces, in fact) and that the map S A (X)

A(X) is a map of H-spaces.

This takes care of the

addition. We next come to the multiplication or, what is the appropriate general notion, the exterior pairing A(X)aA(X') -»A(XxX') . We claim that it restricts to a pairing

A^(X)aA^(X') -»A^(XxX') . This is seen by the

same argument as before.

Namely we check that the pairing

is definable in terms of an explicit construction on the simplicial category of partitions.

It will therefore

restrict to the corresponding construction on the subspace given by the simplicial set of partitions. The exterior pairing is induced by the fibrewise smash product which to a pair of spaces, over respectively, associates a space over

X XxX'.

and

X ’,

We want to

represent that, up to homotopy, by a construction with

ALGEBRAIC K-THEORY OF SPACES

manifolds. ^(X)

and

Let

(M,..)

and

(IT,..)

$(X') , respectively.

subspace of

399

be partitions in

We form the space (a

Xx[0,l]xX*x[0,1] )

MxM' U Xx[0,e]xX’x[0,l] U Xx[0,l]xX'x[0,e1] . Then for sufficiently small

and

a

e'

this space has the

homotopy type of the fibrewise smash product M x V

X-

,

it is a manifold (with corners), and, up to some bending of corners, it defines a partitition in

$(XxX'x[0,1]) .

We

have thus obtained a map, well defined up to some choices (’contractible choices’) h*J(X)

x

m

£ (X ')

---------»

m

£ £ ( X

and restricting in the desired way.

xX

, x[ 0 . 1 ] )

This completes the

account of the multiplication. The case of the operations is a little more delicate, and the verification takes much longer.

We need a

modification of the ’manifold approach’ where the simplicial category of the partitions is replaced by another simplicial category. needed only in the case than the general case.

The modified construction is

X = *

which is somewhat easier

We restrict to that case.

We consider compact smooth submanifolds codimension

0

in euclidean space

neighborhood of the origin. category from such manifolds.

R

d

M

of

containing a

We manufacture a simplicial First, we define a

400

WALDHAUSEN

simplicial set

#(d)

where a k-simplex is a smooth family,

parametrized by the simplex

A ,

considered.

#(d)

Next we regard

of manifolds of the type as the simplicial set of

objects in a simplicial category

h#(d)

simplicial partially ordered set): from

M

to

M'

if and only if

(in fact, a

there is a morphism

M C M'

and if furthermore

the two inclusion maps, of boundaries, SM ---- » Cl(M'-M) M(d+l)

M-I--- > M x [-1.+1]

.

So, by using that map, we can form the stabilization with respect to dimension, lim M ( d )

.

We can obtain a homotopy equivalent simplicial category of

R^

where

h$'(d)

which D^(r)

by restricting to those submanifolds

satisfy

D^(l) C Int(M) , M C Int(D^(2))

denotes the disk of radius

the isomorphism of

M

C1(D^(2)-D^(1))

obtain an isomorphism of

hQ'(d)

r .

with

In view of

S^ *x[0,l]

we

with one of the

simplicial categories of [7] , h C ’(d)

--- » W ^ S 6-1)

M-I--- ► C1(M - Dd ( l ) ) , and hence a homotopy equivalence h^(Sd_1) ---- »h£2(d) . It restricts to other homotopy equivalences

^(S^

#(d) , h^m (S^ *) -* hCm (d) , and so on. The stabilization map

hfi(d) -» h#(d+l)

corresponds,

under the homotopy equivalence, to a map h$(Sd_1) ---- » h$(Sd ) . Up to homotopy, that map factors through have a homotopy equivalence

d h#(D ) , so we

402

WALDHAUSEN lim h#(d) J

^

lim h#(D^) . cf

Similarly we have homotopy equivalences lim Q(d) ^ lim $(D^) , and so on. As a result, therefore, theorem 1 of [7] may be restated to say, among other things, that the inclusion map n

, .

i ^m ,

lim #k (d) ---- > l m M k (d) «g

is an approximation to the map

A (*) -» A(*) .

As regards the limits with respect to

m

and

k ,

there is the happy technical point that the details don’t really matter. map

S A (X)

The reason is that, as we already know, the

A(X)

is a coretraction, up to homotopy; this

will allow us to restrict the necessary checking, below, to a checking on representatives only.

All we need to know

about those limits, therefore, is that they exist in some weak sense; say, as homotopy direct limits with respect to stabilization maps which exist only after geometric realization and are well defined up to (weak) homotopy, and compatible to each other.

Thus we may simply take the

stabilization maps of [7] and transport them to the present situation by means of the homotopy equivalences above. The simplicial set category

h#m (d) )

#m (d)

(resp. the simplicial

has an additional structure, namely it

is a partial monoid in the sense of [3] with respect to gluing.

(The monoid is only partial because the result of

403

ALGEBRAIC K-THEORY OF SPACES

the gluing should be a manifold again, and should be of the correct type.)

As a result, lim #m (d) c?

(resp.

lim h#m (d) ) cf

is the underlying space of a T-space in the sense of [4]. Let

Bp( lim (h)(2m (d) )

T-space.

denote the realization of that

Then the loop space

QBp( lim (h)#m (d) )

as a ’group completion* for the H-space

serves

lim (h)$m (d) , and

there is a map lim (h)em (d) ---- » 0Br( lim (h)«f(d) ) which, up to (weak) homotopy, is universal for H-maps of lim (h)#m (d)

into group-like H-spaces [4].

holim 0Br ( lim £2m (d) ) in c?

The map

» holim QBr ( lim h£2m (d) ) in ct

may be identified, by [7] and the homotopy equivalences J Q lim Q{d) ^ lim #(D ) , etc., above, to the map A (*) -»

A (*) • We claim now that to show the existence of the broken arrow in the diagram AS (*) ---- » A(*) i

i

AS (B2n ) ---- » A(B2n ) it will suffice to show that the arrow exists if the source AS (*) 2i holim 0Br ( lim £2™(d) ) in c? is replaced by just em (d)

.

404

WALDHAUSEN This is seen by the following series of reductions.

First, suppose the broken arrow can always be found if restricted to 0Br ( lim d2m (d) ) . a Then, for varying

m, these arrows are automatically

compatible to each other, up to homotopy:

this follows

g

from the fact that to homotopy.

A (^2^) -* A(B2n )

is a coretraction, up

As a result, the arrows can therefore be

assembled to a map of the homotopy direct limit, and the resulting diagram is weakly homotopy commutative (the two composite arrows are homotopic when restricted to any compactum). For the next reduction we have to invoke the universal property of ’group completion*, we must therefore keep track of H-space structures.

The maps

0n

can all be

assembled into a single map

0 : A (*) --------» JT A(B2n ) n which is an H-map with respect to the additive structure on A(*)

and a suitable H-space structure on the product [6].

That H-space structure is manufactured from the exterior pairings A(B2p )

a

A(B2q ) ---- » A(B2p xB2q )----- > A(B2p+q)

together with the additive structure. exist, compatibly, on

A

S

Now both of these

(the account above gives this

405

ALGEBRAIC K-THEORY OF SPACES

only in the compact case, but the general case follows by an exhaustion argument from this).

Hence we have a map of

H-spaces TT AS (B2n ) -------- » U n n

A(B2n ) ,

and both of these are in fact group-like H-spaces by an easy formal argument [6].

It results that in the diagram Cm( d ) )

» A (* )

i

1

TT AS (B2n ) n

» I T A(B2n ) n

all the solid arrows are H-maps.

Now suppose that the

broken arrow can be filled in if restricted to

lhm #m (d) .

Using the fact that the bottom arrow is a coretraction, up to homotopy, we obtain that the filled-in arrow is necessarily an H-map.

Hence, by the universal property for

H-maps into group-like H-spaces, we conclude that the arrow extends to

OB^flhn #m (d)) .

Finally, suppose that in the diagram li.m £2m(d ) -------- » A (* ) i

AS (B2n )

I

-------- »A (B 2n )

the broken arrow can always be found if restricted to $m (d) .

By the coretraction argument again, the arrows are

then compatible and assemble to a map of the homotopy direct limit with respect to

d , which is homotopy

equivalent to the actual direct limit.

406

WALDHAUSEN It remains to see that the desired factorization

exists if restricted to Recall

$m (d) .

from [6] that the map

0n :A(*)

A(B2n )

is

defined in terms of the functor from pointed spaces to free pointed

2^-spaces,

Y |----> 0n (Y) = Yn / (coordinate axes U fat diagonal) . In detail,

0n (Y) = Yn/fn (Y)

of the tuples

(y^»-.-»yn )

least one of the

y^

least two of the In [6] We want the

where

fn (Y)

is the subspace

having the property that at

is equal to the basepoint or that at are equal to each other.

the construction has beendone simplicially. topological version here. There are routine

ways to pass back and forth between the simplicial and the topological contexts [8]. technical point.

But there is a little extra

Namely if

Y

is allowed to be a

topological space of the pointed homotopy type of a

CW

complex, there is, unfortunately, no reason to suppose that the diagonal map definition of type.

Y ^Y

0n (Y)

2

is a cofibration.

So the above

would not give the correct homotopy

On the other hand, for the purposes of [6] one is

interested in

0n

only up to homotopy (up to weak homotopy

equivalence of functors, to be precise).

There is

therefore a variety of ways to correct the defect. example one can combine

0n

For

with some correction functor,

such as the geometric realization of the singular complex.

407

ALGEBRAIC K-THEORY OF SPACES

Another technical point is the remark that it is not really necessary to work with pointed spaces throughout. Specifically, we want the following modification here. a (weakly) contractible category of the

2^-space

Fix

W . Consider the

2^-spaces having the

2^-homotopy type, in

the strong sense, of a finite free 2^-CW complex relative to

W . Then there are functors between the pointed

situation and the W-situation given by product with by quotienting out

W , respectively.

W

and

These functors

induce homotopy equivalences of the respective subcategories of weak homotopy equivalences. As a result we may modify basepoint of 2^-space

0n (Y)

0n

by allowing the

to be blown up *into some contractible

W .

Specifically, therefore, we have the following representative, up to homotopy, of category d n (R )

h#m (d) . Let

on the simplicial

be defined as the subspace of

given by the union of the coordinate axes and of the

fat diagonal; that is, before.

W

0n

Then for

M M I

n d W = f (R ) in

h$m (d)

in the notation used

the map is given by

► Mn U fn (Rd ) .

We want to modify the construction a little further so that the result can be a manifold (with corners). N& (fn (R^))

denote the e-neighborhood of

Suppose, for the moment, we know that

Mn

Let

£n (R^) . is in general

408

WALDHAUSEN n d f (R ) . Under this assumption

position with respect to

we obtain that, for sufficiently small

a ,

Mn U N£ (fn (Rd )) is indeed a manifold, and is essentially independent of a . By modifying the construction some more, we can re-interpret its result as a partition in the sense of [7] (cf. above).

Namely let

sufficiently large.

5 < a

and suppose that

is

r

Then

( Mn U N£ (fn (Rd )) ) - Int( N6(fn (Rd )) ) defines a

2 -equivariant partition in

NT (fn (Rd )) - Int(N6(fn (Rd )) and hence

a partition in

) ~ SN6(fn (Rd ))

$(C) where

C

* [0,1]

is the orbit

space C

=

9Ng(fn (Rd )) / 2n .

Thus, if the assumption of general position could be generally justified, we would have obtained a factorization em ( d )

» hem ( d )

I

I

» a (*)

f(C) ---- » h^(C) i

i

AS (C)

»

\

A(C)

\

AS (B2n ) ---- » A(B2n ) where we use the sublemma below to provide the broken arrows in the middle; i.e., to show that

$(C)

and

h$(C)

409

ALGEBRAIC K-THEORY OF SPACES

(rather than just to

S

A (C)

and

$m (C)

and

)

A(C) , respectively.

relate naturally

To complete the proof

of the lemma it thus remains to establish that sublemma and to justify the appeal to general position, above. As to the latter, there is certainly no problem as far

n as the part of concerned:

f (R )

coming from the coordinate axes is

we just restrict

(h)#m (d)

to the homotopy

equivalent simplicial subset (resp. -category) of the containing the disk

D^(l)

contained in the interior of

M

in its interior (and being D^(2) ; the latter has the

effect of ensuring that one and the same

r , above, will

do) . Concerning the remaining part of

fn (R^) , the fat

diagonal, there is a potential problem only atsuch points (y1,...,y ) € Mn

where one or more of the

boundary points of

y

are

M .

We will take for granted that in fact there is no such problem at all in the following special case: where near all the

y^

concerned, the boundaryis actually

flat (i.e., there is a neighborhood of of which

M

the case

y^

in

R^

inside

looks like euclidean half-space, up to a rigid

motion). More precisely, what we take for granted in this case is the following:

that near such a point,

M

n

xi

d

U N&(f (R ))

is a manifold (with corners) and essentially independent of

410 e

WALDHAUSEN (i.e., varies with

e

in a locally trivial way); the

here is allowed to be a sufficiently small constant or, more generally, a function which is

e

> 0

C^-close to such a

constant. Our theme will now be that in the general case there is no problem either, and that we can convince ourselves of this by means of suitably chosen isomorphisms to compare with the special case. To this end we note that we can restrict

(h)#m (d)

to

the homotopy equivalent simplicial subset (resp. -category) of the manifolds which are actually smooth rather than smooth with general corners [7] as so far. M , and given any

n-tuple

(y^,....yn ) € Mn

a ’trivializing’ diffeomorphism of make the boundary

dM

Thus, given we can find

whose effect is to

flat near the points

'm

*

We can, and will, assume here that in first approximation the diffeomorphism is the identity at each of the points *^l’*‘*’^n '

^ i s imPlies that the induced

diffeomorphism of near the point

(R^)n

is C^-close to the identity map

(y^...-,yn ) • Thus we obtain, locally, the

desired comparison. To draw the desired conclusion globally, we must impose a condition of uniformity on the construction.

For

example it would suffice to know that the trivializing diffeomorphisms could be found out to a certain distance,

411

ALGEBRAIC K-THEORY OF SPACES

uniformly, and

C^-close to the identity, again uniformly.

But this is no problem.

For it suffices to treat a finite

number of (parametrized families of)

M ’s at a time.

This

is enough to give the factorization on one finite piece at a time (i.e., finite simplicial subcategory in the case of h$m (d) , resp. finite simplicial subset in the case of #m (d) ) and is therefore enough to give, eventually, a factorization up to weak homotopy. We are left to show now

SUBLEMMA. (1)

X

If

there is a natural map

h#(X)

(-l)m , the map

the sign (2)

is a manifold then A(X)

hem (X)

extending, up to A(X) ;

there is a factorization S^X) i

> h#(X) 1

AS (X) ---- » A(X) .

The first part is in effect in [7]:

the required map

may be given in terms of a composite map h$(X) where

h9^^(X).

»hShf(X). ---- » A(X)

denotes the simplicial category of weak

homotopy equivalences of retractive spaces over

X

of

homotopically finite type, and where the second map is from

[ 8],

412

WALDHAUSEN The content of the sublemma really is that this map

can be described without the auxiliary use of a homotopy theoretic device such as

hM^^(X).

. This amounts to

showing that the ’manifold approach’ [7] can be extended to cover the

W.

construction of [8] (or rather a technical

variant referred to as the of [8]).

ST.

construction in section 1.3

Given then that theorem 1 of [7] can be restated

in terms of that construction, the compatibility asserted in part (2) of the sublemma will be an automatic consequence. Let

X

be a manifold.

M C Xx[0,l]

We consider submanifolds

as in the definition of the partitions [7],

but now we consider sequences of such, M

o

C M . C ... C M . 1 n

These form a simplicial set

^^(X)

. We make it into a

simplicial category in two ways which we will denote by the prefixes

i

and

h , respectively.

will be a morphism from for all all

p

M ’ fl M P q

to

only if

is not bigger than

C M P

for

p < q . In

{Mp}

and if

{Mp}

In each case there

i$7n (X)

to

{Mp}

there is, by definition, a morphism from if and only if, in addition to the above

cond itions, we have M U M' = M* p o p for all

p .

inclusion

M

Note there is really no condition on the M' . Thus, for example, the category

413

ALGEBRAIC K-THEORY OF SPACES

i9^o (X)

is contractible.

In general, the role of these

morphisms is to provide a systematic way of ignoring the M q ’s

in those filtrations.

morphism in

iS^T^fX)

as an isomorphism of the nonexistent

quotient-manifolds In

h$Tn (X)

(By abuse one could think of a

(MVMT) .) there is a morphism from

to

{^}

if and only if, in addition to the above conditions, we have that the inclusion maps M

p

are homotopy equivalences.

U M' M' o p (We omit using a refined

condition here, as in the definition of the simplicial category of partitions; stably, in the limit, such a distinction would not be essential anyway.) We also need a simplicial subset

^^( X )

corresponding simplicial subcategories

i#^(X)

and and

hS^T^(X) . Again we omit using a refined condition, and we let

$T^(X)

denote the simplicial set of the sequences

M q C ... C Mn

where all the inclusions are homotopy

equivalences. For varying

n , these simplicial objects assemble to

bisimplicial objects.

We assert that, in the limit with

respect to dimension, the square iS^(X)

1

» iSf^T. (X)

I

h»^(X) ---- > h&J. (X) may be identified to the square on p. 149 of [7] ,

414

WALDHAUSEN Nr (lim »(XxJd )) a

Nr (llm if (XxJd )) a

or rather the square obtained from that by homotopy direct limit with respect to

m .

This is seen as follows.

First, one shows the square

is homotopy cartesian; the method is that of proposition 5.1 of [7], essentially.

Next, there is a natural

transformation from the latter square to the former, and it will suffice to show that the transformation is a homotopy equivalence on three of the four corners.

This is

trivially true in the case of the lower left corner (both terms are contractible by the initial object argument).

In

the case of the upper left corner one uses a degreewise argument, namely one shows that for every

n

there is a

homotopy equivalence, in the limit with respect to dimension, between product

(11m $(XxJ ))n

nv J

on the one hand and the n-fold

on the other.

And finally, in the

case of the lower right corner, one reduces, by proposition 5.4 of [7] and an analogue of that in the filtered case at hand, to the map lim Nr {h9im (X)) ---- > h3-.9^f(X) m which is a homotopy equivalence by [8].

415

ALGEBRAIC K-THEORY OF SPACES

Since

is contractible, the

the simplicial object

[n] I— >i3^n (X)

up to homotopy, to the suspension of

’1-skeleton’

of

may be identified, i3^(X) . By

adjointness there is therefore a map into the loop space, is^x)

.

»n|is^-.(x)|

In view of the above assertion (the comparison of diagrams) that loop space may be identified, in the limit with respect to dimension, to

S A (X) . But also

i » 1(X) * #(X) , at least in the limit with respect to dimension (by the initial object argument, essentially).

Thus we obtain the

required factorization #(X) ---- » h#(X) i

i

AS (X)

> A(X) .

This completes the argument.

Remarks.

The vanishing of

J^(X)

statement that a certain map homotopy equivalence [7].

is equivalent to the

Wh^°m^(X)

Wh^^(X)

is a

The statement may be regarded as

a stable version (stable with respect to dimension) of Igusa*s theorem that Higher singularities of smooth functions are unnecessary.

It is therefore not surprising

that, conversely, the vanishing of from Igusa*s theorem.

jll(X)

may be deduced

416

WALDHAUSEN There is still another proof of the vanishing of

p(X) , using quite different methods again.

Namely there

is a method, due to Goodwillie, to obtain information about DIFF Wh

of a highly connected map.

to obtain information about

A

There is another method

of a highly connected map.

The computations obtained by these two methods are, of course, similar looking.

But, and this is the point, they

are not quite identical:

the only way to avoid a

contradiction is to conclude that

jli(X)

must be trivial.

These computations also provide a generalization of the vanishing of

jll{X)

. The ultimate statement is that,

generally, stable K-theory may be identified to Hochschild homology provided that the latter is understood, 00 00 throughout, over the universal ground ring, fi S . of the ground ring statement that

u(*)

00 00 0 S

The case

itself here is precisely the

(and hence

jll(X)

in general) is

trivial.

REFERENCES [1]

W. G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. Ill (1980), 239-251.

[2]

G. Segal, Operations in stable homotopy theory, New Developments in Topology, London Math. Soc. Lecture Note Series 11, Cambridge University Press (1974).

[3]

Configuration spaces and iterated loop spaces, Invent, math. 21 (1973), 213-221.

ALGEBRAIC K-THEORY OF SPACES

417

_________ , Categories and cohomology theories, Topology 13 (1974), 293-312. F. Waldhausen, Algebraic K-theory of topological spaces. I, Proc. Symp. Pure Math., Vol. 32, A.M.S. (1978), 35-60. _________ _ Operations in the algebraic K-theory of spaces, Springer Lecture Notes in Math. 967 (1982), 390-409. _________ _ Algebraic K-theory of spaces, a manifold approach, Canadian Math. Soc., Conf. Proc., Vol. 2, Part 1, A.M.S. (1982), 141-184. _________ _ Algebraic K-theory of spaces, Springer Lecture Notes in Math. 1126 (1983), 318-419.

Friedhelm Waldhausen Fakultat flir Mathematik Universitat Bielefeld 4800 Bielefeld F . R . Germany

XVII THE MAP

BSG -» A(*) -> QS°

Marcel Bokstedt and Friedhelm Waldhausen

§1.

INTRODUCTION Let

A(X)

be the algebraic K-theory of the space

X.

This can be defined in various ways, see [9], [10], [11], [12].

Let

BG

be the space classifying 0-dimensional

virtual spherical fiberbundles. There are maps F : BG -» A(X), [10],[11], maps

1 : A(X) -* K(Z); A(*) -» QS°,

i : QS° -» A(*) .

splitting

i

In

up to homotopy

are constructed. In this paper, we construct a splitting map Tr : A(*) -» QS°, QS°.

and compute the composite

BG

A(*) ->

We apply this construction to show that

7r3 (WhDiff(*)) ~ TL/2.

A further application is [4].

There

it is used that the splitting given here agrees with the. splitting in [11]; this will be proved in [5]. Recal1 that A(*) ^ lim B Aut (v^Sn )+ n,k where

Aut

denotes the simplicial monoid of homotopy

equivalences, and

+

denotes the Quillen plus

418

THE MAP construction. f : BG

BSG -> A(*) -*•QS°

In this description of

-» A(*)

as the

419

A(*),

we can define

inclusion

BG = lim(B Aut Sn ) C 11m B Aut(vkSn )+ = A(*) n n,k and

1 : A(*)

A(*)

= lim B Aut (vkSn )+ -> lim B Aut (H (vkSn ))+ = K(Z). n,k n,k n Let

K(Z) as the linearization map

BSG C BG

fiberbundles.

classify the oriented spherical

The composite BSG -» BG —L

A(*) - L K(Z)

is the trivial map. In §3 we will show that the composite BG —

A(*)



> QS°

studied in §2.

equals a certain map

In §2 we show that if i > 3 , ^i-l ~ iri(BG)

S

~ Z/2. ®

Q tt.(QS

rf,

q tt^

=

_ is given by

0(x,y) = qx + y.

The splitting of A(*) iri(A(*)) £• -n\ (QS°) $ C.. 0(x,T]x) = 0,

the map

0 f +i^ © 7T.(QS ) ------ » tt.(A(*)) -»

tt.(BSG)

g

) = iri

then

the generator of

In particular, for i > 3

q

0 :

QS°,

iri(QS°) -

is given by multiplication with 7

rj : BG

so that

induces a splitting

If

x € ir®,

1 > 2,

then

f*(x ) + i^{rpc) € C i+^* We

show that for some choices of x, this element

want to

is

nontrivial. The composite [7].

QS°

» A(*)

There it is shown that

K(Z)

is studied in

420

BOKSTEDT AND WALDHAUSEN

^41+3(0®°) = ^41+3

ls lnjectlve on the image

1T4i+2^K ^ ^

of the J-homomorphism. Recall from [1] that there are classes i > 1,

so that

J-homomorphism.

_2

77 p0 . 4

ol+l

x = T)2 . Then

is in the image of the

Similarly,

of the J-homomorphism.

-3 rj €

Choose

S

is also in the image

x =

» x =

f*(x) + ix (rjx) 6 Cg1+3>

+ i^(hx)) = I ^ (

S kgj+j € ^81+1’

tjx)

or

and

/ 0.

We have proved

THEOREM 1.1. 7^(08°)

The kernel of the map

Tr^ : ^(Af*)) -*

contains a nontrivial element of order 2 if

n = 2,3(mod 8);

n > 3.

On the other hand, it is known that

C^ < Z/2 [6], so

we have

COROLLARY 1.2.

ir3A(*) = ir® © Z/2.

It is known [9], [11]

that

A(*)

splits as a product

A(*) ^ QS° x WhDlff(*) x fi. It will be proved in [13] that

THEOREM 1.3. (i)

tt3

WhDlff(*) = Z/2

p = 0.

We conclude

THE MAP BSG -> A(*) -* QS°

421

(li) There are nontrivial two-torsion classes In ir8i+2^WhDlff^ ^

§2.

lr8i+3^Wh° lff^ ^

SPHERICAL FIBER BUNDLES AND

:

1 “ lm

r;.

In this paragraph we study a certain map

17 : BG -» G.

We first give a homotopy theoretical definition of calculate the induced maps of homotopy groups.

17, and

Finally, we

17 agrees with a geometrically defined map,

show that

which will be used in §3.

3

Let

X = Q Y



j] '

be a threefold loopspace.

be the Hopf map.

Definition 2.1.

induced by

: BX = fi^Y

= X

is the map

17. 00 CO

Example 2.2.

X == Q S . We identify

with the ring

7S of stable homotopy groups of spheres. S (nx)x : if* = ^(X) with

Let

^(BX) =

S

The map

is given by product

1— 7 € 7Ts^ .

Example 2.3.

Let

X = ZxBG

be the classifying space of

based stable spherical fibrations; space [3].

Then

QX = QBG

X

is an infinite loop

can be identified with the

space of stable homotopy equivalences of spheres, i.e., i : QX

(Q°S°)±1. This equivalence is not an H-space

422

BOKSTEDT AND WALDHAUSEN

equivalence, when

oo co

OS

o

= QS

is given the H-space

structure derived from loop sum.

But fi3 (QS°)

03i : nSt

is an equivalence of threefold loopspaces, so that ^

n3os o)

^

o 4x )•

~

We conclude from the previous example, that for

^ZxbgL : V i = V 2 x—BG) ■ *iri(G) “ Ti s 77 € ir^

is induced by composition with i < 2

we do not get any information.

not equivalent to

2 o Q (QS )

multiplication by Let

X

17

i > 3.

Actually,

For

3 fi X

as a threefold loopspace.

S : v2

induced map

for

i >

S

^3

*S triv^a ^> whereas

is nontrivial.

be an infinite loopspace.

Composition of

loops defines an infinite loop map 00 00

p : 0 S There are structure maps ^

0

n

x X : E2

X. n

x

2

Xn

X,

and a

n

commutative diagram

11 B2m

(idXj A)

11 E2m

x X -----52--- »

m>0 “

m>0

(2-4)

x

2

lie

(im xid)

Xm

m

00 00

QS There are two maps where

f^

x X f^ : S* x X

B2^ xX,

is the trivial map, and

generator of

tt^BX^) = Z/2.

f^

^± = ^ [ x

represents the

is The

THE MAP BSG -> A(*) -» QS°

423

Composition with the square above defines maps g. : S* x X i

11

defines a map image under *2^1~*2^o of

g

B2

VA m>0

x X

m

g : p

a

X.

The difference

X -» X. This difference is the

of the difference 17 :

*S eclua^ to

g-,-g 1 o

(i^g^-i^g^) x id.

QS°,

But

so that the adjoint

17^ : X -» QX.

is the map

In particular, the map

^xBG 1 ^ x ^

G

can

described as the difference between the adjoints of the maps

gi

(i=0 ,l)

0 g. : S 1 x (ZxBG) -» E2„ x (ZxBG) 2 — ^ 1 £ 22 Let ZxBG.

f

ZxBG.

be the standard (virtual) spherical fiberbundle on Let

denote fiberwise smashproduct.

a

classifies certain virtual bundles on

Then

S^x(ZxBG).

These

bundles are the identifications of the bundle

f

a

I x (ZxBG), using certain bundle maps

a

f

as clutching function, T ^ X A y) = y

a

where

tq

= id,

t\

:f

g^

f

on f

and

x.

We reformulate this description as follows.

LEMMA 2.5.

f

Let

The automorphisms

be the standard bundle over t. • f a f -» f a f

Z x BG.

(i=o,l)

induce

maps t. : Z x BG -> G. 1

The difference homotopy.

ti~tQ

equals

17 ■ * Z x BG -» G

up to

af

424

BOKSTEDT AND WALDHAUSEN Finally, consider the following situation.

a finite dimensional space. fibration over

B,

Let

f

Let

B

be

be a spherical

classified by a map f : B -» TL x BG.

Let

f ' be a spherical fibration over

B,

and

u

a fiber

homotopy trivialization: u : f'

a

f -» s” X B.

B The map over

u

B,

can be interpreted as an S-duality parametrized see [2].

A 2N-dual u' : S

N

u'

x B

of this map is a map f

a

such that the following diagram

B commutes up to fiber homotopy V. I

J

a

B

V.

f

a

n A U A 1d f 0 N x B) ----- » (S

f

(S

fo N

x B) a (S B

B

idAU1

f where

rj ■»

x B)

I -

A f A f A f• B B B



-- »

(S2N X B)

v(a,b,c,d) = u(a,c)A u(d,b).

The transfer

B Tr : B t : SN x B

LEMMA 2.6.

-> QS°

-“L f

is defined as the adjoint of the map

a S' S' B

aS



SN x B -» S*.

B

The following diagram is homotopy commutative B — Tr

I

^o „ i QS°

2 x BG

THE MAP

BSG -» A(*) -* QS°

425

i : SG -» QS° Is the standard identification

where

1

with the component of

The map

Tr

a suspension of

t:

Proof.

tn

SG

of

QS°.

can also be defined as the adjoint of

., ^ ~2N „ idAu' idAt : S x B -------» (S

>., x B) a f a f

B B jdA(uoTwist) , (SN x B) A {SN x B) B Let

f

Af Af A f B B B

be the map B

B

B

permuting the second andthird factor.

By assumption,

the

following diagram commutes up to fiber homotopy (SN XB)A(SN XB) i ^ U s ^ A f A f B v B B

. ldAM £ g £L , (SN XB)A(SN XB) B |v

j uAid

Tw23 ------------— ------------- >

^ f A f A f A f

f A f A f A f

B B B We conclude that

B B B

B -» G C QS°

is the difference

t' - t' 1 o

between the maps t! • B ' 1

induced

bythe automorphisms Tl

:f

'

A

B = T w^*

§3.

G

=

£

A

B

f

A f •-* f ’

B

A f

B

A

f

B

identity. The lemmafollows

A f '

B from 2.5.

TRANSFER AND SPLITTING In this paragraph we will construct a splitting map

Tr : A(*) -» QS°.

This splitting map will be used to prove

426

BOKSTEDT AND WALDHAUSEN

theorem 1.1.

In a later paper it will show that this map

agrees with the splitting maps in [10] and [11], cf [5]. We recall some properties of the transfer map [2]. Let

B

be a finite dimensional space.

Let

F

be a fibration with section, and suppose that fiber homotopy equivalent to a finite complex. 00

transfer map

:B

t

00

00

00

00

be

00

Q S (E+ ) -> Q S (pt+ ) =

with the map

t

is

00

Tr^, : B -» Q S 00

the composite of

F

Then there is a

00

0 S (E+ ). Let

E -» B

00

Q S ,

induced by

E -» pt.

We will need the following properties of the transfer: Let

S^"

E

a

B

be the fiberwise double suspension of

E.

B 00

3.1.

Tr^ ^ Tr 0 E

S

00

: B -» Q S . aE

B Let

E^,E^

be two fibrations over

can consider the fiberwise wedge

B

as above.

E = E^ v E^ + B. 00

3.2.

Tr^ ^ Tr-p

b

h2

+ Tr„

Then we

:B

00

Q S

2

These properties will be proved at the end of this section. Recall that the algebraic K-theory of a point can be def ined as A(*) = lim B Aut(v^Sn )+ .

Let

n,k k 2n f : B -» B Aut(v S ) be a finite dimensional approx-

imation.

There is an induced fibration

k 2n (v S )

E

B.

To this fibration, there is an associated transfer map Tr_ : B -» Q°S°. Jti

Let

a : B Aut(vkS2n)

B Aut(vkS2n+2)

be

THE MAP

BSG -* A(*)

induced by double suspension.

QS°

427

Then the map

the fiberwise double suspension of and because of 3.1

->

erf

E : (v^S^n+^)

induces E'

B,

Tr„ ^ Tiv, . By a homotopy colimit r, hi

argument, these maps extend to a map \r r\ 00 00 Tr : lim B Aut(vKSn ) - » Q S . rk n The stabilization map B Aut(v^Sn ) -» B Aut(v^+ ^Sn ) induced by adding a factor in the wedge, induces by 3.2 a diagram, which is homotopy commutative on all finite subspaces

11 k>0

11

lim (B Aut(v^+ ^S^n )) lim (B Aut(v^S^n )) ---- » k>-l -> n n li.Tr, k

l l Tr 00 00

Q S The map loop.

*[1]

*r 1 1

00 00

»Q

k S

here denotes loop sum with the identity

Again, you can extend to a map, defined on finite

subcomplexes k 2n. oo oo Tr : Z x lim B Aut(v S ) -» fi S n,k And by the universal property of the plus construction, this finally extends to a map 00 00 Q S .

Tr : A(*) Recall from |"8] that Q S Z x lim B2^ k

= Z x lim BX*\ k k

Ir ri

Z x lim B Aut v S n,k

map inducing the equivalence, so

split surjection.

The map

00 00

-» Q S

actually is the

Tr : A(*)

00 00 0 S is a

428

BOKSTEDT AND WALDHAUSEN Now, theorem 1.1 follows from the description of

^ZxBG

aS a tr a n s ^e r in 2 .8 . It remains to prove 3.1 and 3.2.

the transfer

3.3

Recall from [2] that

has the following properties:

Given a fibration

p :E

B

as above, and a map

g : X -» B, we have a pullback diagram A/

E

S -> E

;i

I

X Then

3.4

00 00 ^ ft S (g+ ) o

Given fibrations

t~

g— > B

^ Tg o g.

p. : E . *i

i

i

B.

as above, we can form

the fiberwise smashproduct P1

a

PQ : E 1

B1 B1

E9

a

» B1 x B .

B 1 B2

The following diagram commutes up to homotopy

tb

x tb 1

9

oo oo

B 1 * B2 ---

oo oo

0 S (Et ) x 0 S (E9 2+ ) +

Q S (E 1 x E2 ) We can now prove 3.1.

If

F -» F

with trivial base, then

Tr : S F

a

oo oo

S

*

is a fibration

,0

T H E M AP

B S G -» A(*) -> QS

429

is g iven by the Euler characteristic

x(F).

understood in the pointed sense here;

thus a sphere has

Euler characteristic

+1

or

-1

This is to be

depending on the parity

of the dimension. From 3.3 it follows, product fibration,

that if

F

F x B 00

then

B

00

Tp xg : B -» Q S (BxF)+

isa is the

composite

Applying 3.4 to the diagonal map

E. = E;

E^ =

x B ^ B

and then 3.3 to

B -» B x B, the statement 3.1 follows.

In order to prove 3.2, note that if

f^ : S

N

are duality maps of exspaces in the sense of [2],

x B

E.

then the

fiberwise coproduct followed by fiberwise wedge

is also a duality map.

The 2N-dual of this map is the

wedge of the 2N-duals of

f^

and

f^

followed by the fold

map

The transfer map

^xB -

is the adjoint of the composite

Tr

( 8 " ,s " ) x B

which equals the sum

Tr„

E1

+ Trr

2

430

BOKSTEDT AND WALDHAUSEN REFERENCES

[1]

Adams, J. F., On the groups J(X) I,II,III and IV, Topology 2: 181-195(1963), 3: 137-171, 193-222(1965) and 5: 21-71(1966).

[2]

Becker, J. C. and Gottlieb, D. H . , Transfer maps for fibrations and duality, Composito Mathematica 2: 107-133(1976).

[3]

Boardman, J. M. and Vogt, R. M . , Homotopy invariant structures on topological spaces, Lecture Notes in Mathematics no 347, Springer 1974.

[4]

Bokstedt, M . , The rational homotopy type of Q WhPi;^ ( * ) , Lecture Notes in Mathematics no 1051, Springer 1984 :25-37.

[5]

Bokstedt, M . , Equivarant transfer and the splitting of A(*), to appear.

[6]

Kassel, C . , Stabilisation de la K-theorie algebrique des espace topologique, Ann. Scient. Ec. Norm. Sup. 4 Serie, t. 16:123-149(1983)

[7]

0

Quillen, D. , Letter to Milnor on ImfTr^O-^r^nK^Z), Lecture Notes on Mathematics no 551, Springer 1976: 182-189.

[8]

Segal, G . , Configuration-Spaces and Iterated Loop-spaces, Invent. Math. 21: 213-221(1973).

[9]

Waldhausen, F., Algebraic K-theory of topological spaces, I, Proc Symp. Pure Math, vol 32 Part I: 35-60, A.M.S. (1978).

[10] Waldhausen, F., Algebraic K-theory of topological spaces, II,Lecture Notes in Mathematics 763 (1979): 356-394. [11] Waldhausen, F., Algebraic K-theory of spaces, a manifold approach, Canad. Math. Soc., Conf. Proc., vol. 2, part 1: 141-184, A.M.S. (1982).

THE MAP

BSG -» A(*) -> QS°

431

[12] Waldhausen, F., Algebraic K-theory of spaces, Lecture Notes in Mathematics No. 1126, Springer (1985): 318-419. [13] Waldhausen, F . , Algebraic K-theory of spaces, stable homotopy and concordance theory, these proceedings.

Marcel Bokstedt Fakultat fur Mathematik Universitat Bielefeld D-4800 Bielefeld F.R. Germany

Friedhelm Waldhausen Fakultat fur Mathematik Universitat Bielefeld D-4800 Bielefeld F.R. Germany

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

One of many things I learned from John Moore was the existence of a large number of useful analogies and relations between algebra and topology.

The first part of

this paper will be a survey of one such relationship: that existing between vector bundles and projective modules. The main applications of this so far have been to the construction of non-trivial examples of projective modules, the non-triviality being proved by passing to the associated vector bundle and using topological methods.

I

have taken this opportunity to present this material to a topologically oriented audience in the hope that others may be inspired to continue and extend this work.

As evidence

that more can be done, I have included some new material in §6 .

In part II, I will discuss the problem of determining the algebraic K-theory of the coordinate rings of spheres. Most of this part is purely algebraic.

In it, I have

extended the original results of Claborn, Fossum, and

432

BUNDLES, MODULES AND K-THEORY

433

Murthy to the case of non-degenerate affine quadric hypersurfaces over any field of characteristic not 2.

The

main purpose is to prese;nt evidence for a conjecture about Kq

of such a hypersurface.

S. Shatz has informed me that

D. S. Rim was working on these questions shortly before his untimely death.

Unfortunately it was not possible to

include an account of his results here.

Hopefully, a

detailed account of his work will be prepared in the near future. Finally, part III contains expositions of some previously unpublished work of Murthy, Mohan Kumar, and Nori. I would like to thank the following people: (1)

Pavaman Murthy for many discussions of the

material in this paper, for showing me his results, and for permission to include some of his unpublished work in §15 and 16. (2)

Mohan Kumar for information about his work with

Nori and for permission to include it in §17. (3)

S. Shatz and C. Weibel for information about

Rim’s work. (4)

W. Haboush for bringing to my attention the

problem discussed in §6. (5)

E. Friedlander, P. May, J. McClure, H. Miller,

and R. Thomason for discussions of the material in §14.

434

SWAN (6)

The referee for many helpful suggestions and

corrections.

Remark

(added April 6, 1985).

After this paper was

written, I succeeded in proving the main conjecture (see the end of §7).

The method is that discussed in §13 and

gives an affirmative answer to the first problem in §13. The details appear in my paper "K-theory of quadric hypersurfaces,” Ann. of Math., 122(1985), 113-153.

PART I. §1.

VECTOR BUNDLES AND PROJECTIVE MODULES

THE MAIN THEOREM In [FAC] Serre showed that a vector bundle on an

affine variety is essentially the same as a projective module over the coordinate ring of the variety. theorem here is a topological version of this. vector bundle over a topological space of continuous sections of ring and

C(X)

E

over

X

x -» f(x)s(x).

one defines

fs

E

is a

the space

T(E)

X:

if

s € T(E)

to be the section

We may consider real, complex, or

quaternionic bundles with

C(X)

being the ring of

continuous functions with values in respectively.

If

is a module over the

of continuous functions on

f € C(X),

X,

The main

IR,

C,

or

IH

BUNDLES, MODULES AND K-THEORY

THEOREM 1.1.

If

X

435

is a connected compact Hausdorff

space, the functor

F

gives an equivalence of categories

between the category of vector bundles on

X

category of finitely generated projective

C(X)-modules.

and the

A proof of this theorem may be found in [SwV] where it is also shown how to weaken the hypotheses of compactness and connectedness.

§2.

THE TANGENT BUNDLE TO

Sn

In [SD], Serre showed that every finitely generated projective module k[xQ,...#x ] P © R

r

£ R

s

P

(with

over a polynomial ring k

R =

a field) is stably free, i.e.,

for some finite

r

and

s.

This reduced his

well-known problem about whether projective R-modules are free to the question of whether stably free modules are free.

This question makes sense over any ring

induction, if

R

submodule i.e., y^

Rx

P © R ~ Rm+*

R under an isomorphism where

x = (x~,...,x ) v 0

R.

for some

nr

Conversely, if m+ 2 row, we have R = Q © Rx

y~,...,y

implies

m_|_^

P © R ~ R

P £ Rm . isa

is a unimodular row,

In fact, the m+ ^ are the entries of the matrix of the projection R ~

P © R

) x.y. = 1

By

is commutative or noetherian, it reduces

to the question of whether The image of

R.

€ R.

(x^,...,xm ) where

is a unimodular

436

SWAN

Q = {z e Rm + 1 | Y ZjYi = 0 } write

and

P(Xq,...,xm ) = Rm+*/Rx

P = Rm+1/Rx ~ Q.

I will

and call it the projective

module defined by the unimodular row

x.

All stably free

R-modules will

be free if and onlyif all P(Xq ,....x^)

free.

Pfx^,...,xm )

Clearly

Rm+^

will be free if and only if

has a basis whose first element is

x.

This is

equivalent to the existence of an invertible matrix whose first row is assume that

x.

If

det A = 1

are

R

A

is commutative, we can even

by multiplying some row of

A

by a

unit. The first free was found

example of a stably free module which is not by Samuel [Sa] and, independently, by

Kaplansky and Milnor (unpublished). Sn = {x € IRn+^ | ^ x^ = 1}. to

Sn .

Since a vector

Let

Consider the n-sphere

T

be the tangent bundle

(z~,...,z ) v 0 nJ

tangent to

Sn

that

is the projective module

T(T)

if and only if

^

T

x € Sn

zix i =

C^(Sn ). Therefore by Theorem 1.1, if module is not free since

at

is

it is clear

P(x q »•••»xn ) nj*0,l,3,7,

is non-trivial.

over this

For even

n

this can be seen without using Theorem 1.1 since a unimodular element in field on

T(T) would give a never zero vector

Sn [Sa].

The ring

C(Sn )

is too big to be of much algebraic

interest but we can easily reduce to the ring of polynomial functions on The row

Sn

given by

(x~,...,x ) v U n

A^ = IR[Xq ,...»x^]/(^ x? - 1).

is still unimodular over this ring.

BUNDLES, MODULES AND K-THEORY

THEOREM 2.1. free ouer

n ^ 0,1,3,7

If

437

P(x~,...,x ) v 0 ir

then

is not

A . n

In fact it is clear that

C(Sn )

^(x0 ’*‘’,Xn^ ~ n

T(T).

We remark that if

P(x~,...,x ) v 0 nJ € 0^ Ar

n = 0.1,3,7

is free over

for all

n.

A . Also n

then P = €P IR

is free over

See [SwV].

Although the statement of Theorem 2.1 is purely algebraic, the proof requires the topological fact that the tangent bundle

T

is non-trivial.

For many years there

was no algebraic proof that this (or any other stably free module) is not free.

Finally Kong [K] succeeded in giving

a purely algebraic proof of Theorem 2.1 for even

n.

His

method was to take Chern’s formula for the Euler class of T

as an integral over

Sn

and, by using suitable

algebraic approximations, to show that non-zero.

this class is

His work was inspired by that of Ozeki [0] on

algebraic Chern classes. Theorem 2.1 has been adapted to give some other interesting examples of projective modules.

Even before

the proof of the Qui1len-Suslin theorem that all projective modules over

k[Xgf...,x ]

principal ideal domain),

are free (k

a field or

Eisenbud observed that projective

modules over a localization of such a ring need not be free.

Therefore the property that all projectives are free

is not preserved under localization.

438

SWAN

COROLLARY 2.2 (Eisenbud 1970). = IR[Xq, ...,x^].

n j* 0,1,3,7

R[l/f]

module over x~,...,x 0 n

If

f =

Let

+ ... +

6 R

then the projective

defined by the unimodular row

is not free.

In fact this module becomes that of Theorem 2.1 under the change of rings

k[l/f] -+ A^.

After the proof of the Quillen-Suslin theorem, the central problem in this area became the so-called Bass-Quillen conjecture that for regular modules over

R[x-,...,x ] L 1 nJ

have the form

P

R

all projective

are extended from

R,

i.e.,

R[x^,....x^]. The main positive result

in this direction is Lindel’s theorem [Lin] that the conjecture is true if

R

is finitely generated (as a ring)

over a field or is a localization of such a ring.

In [QP]

Quillen observes that it would be sufficient to show that if

R

is a regular local ring and

parameter

(f € M - M

2

where

f

ffl is the maximal ideal)

then all R[l/f]-projectives are free. that the hypothesis that really needed.

f

is a regular

The following shows

be a regular parameter is

I learned this result from Murthy who says

that he heard it from Samuel.

THEOREM 2.3.

R = IRIx^, ...,x U 0 n

Let

series ring in

n+1

f = x? + ... + 0

n



be a formal power

indeterminates. Let R.

If

n t 0.1,3,7

thenthe

439

BUNDLES, MODULES AND K-THEORY

projective x~,...,x 0 n

Proof. row

R[l/f]-module defined by the unimodular row Is not free.

Suppose

Xq

x^

A and

we get a matrix B X q .....x^ in

B

is a matrix over det A = 1. over

and det B =

R[l/f]

By clearing denominators

IREx q ,....x^I with first row fm .By truncating the power series

to polynomials we get a matrix

IR[x„, ...,x ] with first row L 0 nJ det C = fm + g

where

C

over

x~,...,x O n

g € (x^

and

Xn ^

^°r *ar^e

Consider the polynomials as functions on

V

2

S = {x| 2

On

C = e2m + 0(eN ).

so det

sufficiently small and x~,...,x O n

- e }

N > 2m, det C

defines a free module over

implies that

the tangent bundle

lRn+*

2

restrict to the sphere S, f = e2

with first

to

and

°f radius If

e

is never C(S) vJ

e.

is 0

so

but this

Sis trivial.

Weibel, in answer to a question of

B. Nashier, has

shown that Kong’s methods can be used to give a purely algebraic proof of Theorem 2.3. The following consequence of Theorem 2.3 has been noted in [BR]

COROLLARY 2.4 ([BR]). and let

Let

R = lR[xn ,...,x ], u n (x0

f = x2 + ... + x2 € R. 0 n

P(Xq,....x^)

over

R[l/f]

If

n ^ 0,1,3,7

is not free.

. xn-' then

440

SWAN In fact, the module becomes that of theorem 2.3 under A

the change of rings

R — * R = IRIx q ,...

Alternatively, the same proof can be applied. Another question of interest is whether the BassQuillen conjecture holds for Laurent polynomial rings, i.e., are all [SwL]

R[T,T ^]-projectives extended from

R?

In

I gave an example to show that the answer is

negative.

In [BR], Bhatwadekar and Rao gave a much simpler

example based on Theorem 2.1.

COROLLARY 2.5 ([BR]). P

Let

Ar

be as in Theorem 2.1. A^fT.T *]

be the projective module over

unimodular row

is not extended from

If

P

P/(T+1)P while

§3.

0J

A

n

0

for

1

were extended then

P^ = P/(T-1)P An

over

C

but

and P^

P_^ =

is free

S2n+1

The examples of §2 are all over C.

P

is isomorphic to the module of Theorem 2.1.

A COMPLEX "TANGENT BUNDLE" TO

over

Then

nJ

n ^ 0,1,3,7.

would be isomorphic over

P__^

defined by the

((1 - x~)T + 1 + x~, x 1,...,x ).

vv

Let

IR and become trivial

There is no direct way to modify them to work since the tangent bundle to

structure for

n / 2 or 6.

that the tangent bundle to

Sn

has no complex

However we can use the fact s^n+*

is the direct sum of a

complex bundle and a trivial real line bundle.

We embed

441

BUNDLES, MODULES AND K-THEORY

g2n+l

unit sphere in

z~z~ + ... + z z 0 0 n n

=1.

C11*^

defined by

Let

E = {(z,t) € s^n+* x e n \ ^ ziti =0}.

Then E is 2n+ ^ S with

clearly a complex vector bundle over projection

p(z,t) = z. It is easy to see that the tangent 2n+ 2 S is the direct sum of E and the trivial

bundle to

real line bundle whose fiber over

z

is

IRiz.

More

important is the fact (easily verified) that the associated principal bundle of U(n)

E

U(n+1) -»S^n+^.

fibration is the image R2 nU(n ) .

Since

is the standard fibration The characteristic map of this 2n+ ^

2, the projective module

is not free.

W

ziW i " P(zQ,...,zn )

over

B^

442

SWAN In this argument,

of



C

could be replaced by any subring

with no essential change.

In [Ra], Raynaud showed

that the analogue of Theorem 3.1 holds with any field

K.

C

replaced by

Her argument uses etale cohomology to show

that the Borel-Serre proof [BSL] of the non-triviality of the fibration

U(n)

characteristic.

U(n+1) -» S^n+*

works in any

This proof excludes a few pairs

(K,n).

Recently, Mohan Kumar and Nori have found a remarkable new approach which gives a purely algebraic proof of this result with no exceptions.

An exposition of this method is

given in §17.

2

In [SwT], Towber and I showed that for a unimodular row

a,

b,

c

P(a ,b,c)

is free

over any commutative ring.

Suslin [Su] gave a remarkable generalization of this result.

He showed that if

aA ,...,a 0 n

is a unimodular row

m0 ring then P(aQ

over any commutative mQ...m^ = 0 mod n ! .

mn »•••>an)is free if

An example given in [SwT] shows that

this is the best possible result-

THEOREM 3.2. iru. ..m O n

Let

£ 0 mod n!

B

n

be as in Theorem 3.1.

then

m m P(z~ ,...,z ) 0 n

To prove this consider the by

£(Zq

z ) = afz^O, ...,z|V)

If

is not free.

map f:S^n+^ where

a > 0

S^n+*

given

is chosen

443

BUNDLES, MODULES AND K-THEORY

so that

f(z)

lies on

S

2x1+1

. The module of interest

clearly corresponds to the vector bundle Its characteristic map is generates

f*(E)

on

S^n+^.

g^n+l -- » BU(n)

g^n+l

d^II^^BUfn) = d^Z/n!Z

which

where

d = deg f = mQ...m^. The method of Mohan Kumar and Nori shows that this result also holds over any field. results over

§4.



See §17.

Other related

may be found in [MS].

THE CANONICAL BUNDLE ON

RPn

In [For], Forster showed that a finitely generated projective module of rank Krull dimension

d

r

over a noetherian ring of

can be generated by

example pointed out by Chase best possible [SwV].

d+r

elements.

shows that this resultis the

As usual, we start with a topological

example, in this case the canonical line bundle RPn .

If

then

T(E)

E = f 0 0T ^ needs

elements generate

n+r

where

is 1+a a 1 £ 0, To

0

T(E),

we

f

on

is a trivial line bundle

generators.

0m = E © E' = 0r-1 © f © E' generates

In fact, if

m

get an epimorphism

0m -* E

with

If a

rk E* = m-r.

H^(RPn ,Z/2Z), the Stiefel-Whitney class of so

An

w(E') = (1+a) * = 1+a +

so

f

... + a11. Since

m-r = rkE' > n. make thisexample more algebraic, consider

a quotient of

Sn

by the antipodal

RPn

map. We can give A =

as

444

SWAN

Ar = IR[Xq

^ xi “ U

A = A % and

where is

be regarded map to

the subset of odd functions. as a ring of functions on f

an element of the line where

f

we get a map x -» f(x)x.

grading

is the subring of even functions

x € RPn . A section of

x € RPn f(x)x

a 2/22

Clearly

RPn . Let x € Sn

associates to eachpoint

IRx which we canwrite as

must be an odd function of

A^

T(E)

by sending

It is easy to check that

A°-module of rank

1

and that

A^ can

f

x.

Therefore

to the section

A*

is a projective

T(f) £ C(RPn ) ® _ A 1 . A

Details may be found in [SwV].

THEOREM 4.1. rank

r

A^ © (A^)r *

which requires

Of course

is a projective

n+r

dim A^ = n

pP-module of

generators.

here.

Recently, I was able to use this result to settle a question which at first sight has nothing to do with projective modules or topology.

Recall that a Dedekind

ring is a commutative integral domain in which every ideal is projective.

The rings of integers of number fields and

the coordinate rings of smooth affine algebraic curves are typical examples.

It is well known that every ideal of a

Dedekind ring needs at most 2 generators.

A Prlifer ring is

defined in the same way but only finitely generated ideals

445

BUNDLES, MODULES AND K-THEORY

are assumed to be projective.

An old question of Gilmer

asks whether all finitely generated ideals in a Prlifer ring have 2 generators.

A recent example due to Schiilting shows

that this is not the case.

In the other direction, the

best positive result is due to Heitmann [H]: finitely generated ideal has

1+dim R

every

generators.

By

modifying the example of Theorem 4.1 I was able to show that Heitmann*s result is the best possible.

THEOREM 4.2.

For every

of Krull dimension requiring

n+1

n

n > 1

there is a Prufer ring

with a finitely generated ideal

generators.

To prove this we start with the ring 4.1.

R

We can embed

A^

in a Prufer ring

A^ R

of Theorem

with the same

quotient field by a transfinite series of adjunctions of elements of the form and

y

2

2

0

x^x.,...,x~x

0 1

of rings

2

and

2

On

R^

2

xy/(x +y )

are in a previously constructed ring.

is the required example. x^,

2

x /(x +y )

where

x

This ring

R

The ideal is the one generated by

0

€ A . The ring

R

is the direct limit

each of which is the coordinate ring of a

real algebraic variety

X ,

the map

p^:

-» RPn

proper and birational.

Borel and Haefliger [BH]

being

have

shown that compact real algebraic varieties have fundamental classes in

mod 2

homology which are preserved

446

SWAN

by proper birational maps.

In particular, it follows that

p* : Hn (RPn ,Z/2Z) -> Hn (Xa ,Z/2Z) to see that the ideal

2

Xq ,Xq X ^ ,....Xq X^

p*(an ) needs

§5.

)

of

It is easy

generated by

is projective of rank one and corresponds ^ Pa (?)

to the line bundle Pa (w(f)

I

is injective.

on

has, in dimension

Now

^ —1 w(pa (f)) =

n, the non-zero value

so the argument of Theorem 4.1 applies and I n+1

generators.

The details may be found in [SwP].

L0NSTED’S THEOREM. Lbnsted [L0] proved a remarkable theorem showing that,

if one is only interested in isomorphism classes, for a finite complex

X,

the ring

replaced by a noetherian ring.

C(X)

in Theorem 1.1 can be

In [SwT] I showed that it

could, in fact, be replaced by a ring locally of finite type over

IR, i.e., a localization of a finitely generated

ER-algebra.

THEOREM 5.1.

Let

X R

there is a subring finite type over

be a finite simplicial complex. of

Then

which is locally of

IR and such that there is a

1-1

correspondence between isomorphism classes of finitely generated projective bundles on

X.

R-modules and those of real vector

The same is true for complex or

quaternionic bundles and the rings

€ ®^R

and

IH

447

BUNDLES, MODULES AND K-THEORY

The correspondence is given by taking the vector bundle associated to the projective module Theorem 1.1

A proof is given in [SwT].

C(X)

by

A simpler proof

has recently been found by M. Carral [C].

It is a very

nice application of patching techniques and also has other interesting applications. This theorem makes it possible to produce examples where some condition is imposed on the totality of R-projectives.

For example, if

a noetherian ring of dimension projective module of rank

m

m = 2 mod 4, m

one can find

with a non-trivial

but such that all projectives

of rank j* m are free [SwT]. In certain simple cases the ring

R

of Theorem 5.1

can be given very explicitly.

THEOREM 5.2 ([SwT]). tahe

R

If

X = Sn

+

f?1

+

(A )0

to be the localization *x ]/( / x * ” 1)

1

in Theorem 5.1, we can

...

+

fj? N

for all

y n'S

aru^

N >

1,

S

f.i

where the set of all € A . n

This gives the following purely algebraic but apparently useless description of the homotopy groups of the classical groups.

Let

^n (^)

denote the set of

isomorphism classes of finitely generated projective R-modules of rank

n.

448

SWAN

COROLLARY 5.3 ([SwT]).

If

"„-i 0(l!|" pk(R)'

R

is as in Theorem 5.2 then *

V " v>-

Vc

V>-

V i Sp(k) ■

Details may be found in [SwT].

§6.

ALGEBRAIC GROUPS If

G

is a Lie group, it has the rational homotopy

type of a product of odd spheres and it is clear that Q ® K^op(G) ~ HeVen(G,Q)

will usually be far from trivial.

However, in this case, the algebraic situation is quite different from the topological one.

In [SGA6],

Grothendieck proved the following theorem.

A detailed

proof was given by R. Marlin [Ma].

THEOREM 6.1.

Let

G

be a semisimple simply connected

affine algebraic group over an algebraically closed field. Let

A

be the coordinate ring of

G.

Then

K^(A) = Z.

In other words, all algebraic vector bundles on

G

are stably trivial. The situation at first sight seems similar to that encountered in connection with Serre’s problem where it was known since Serre’s earliest work on the problem that K^fkfx^,...,xn ]) = Z

and one hoped to prove that all

449

BUNDLES, MODULES AND K-THEORY

projective modules are? free. all projectives over

A

This suggests asking whether

in Theorem 6.1 are free.

The only case of Theorem 6.1 where the algebraic and topological results agree is the case

G = SL^.

In this

case Murthy has shown that the answer is affirmative.

THEOREM 6.2 (MURTHY). Let SL^

the coordinate ring of projective

A = k[x,y,z,w]/(xy - zw -1) over any field

k.

be

Then all

A-modules are free.

The proof will be given in §15. In general, however, the answer to our question is negative.

THEOREM 6.3. €.

Let

A

be the coordinate ring of

SL^

over

Then there is a non-free projective A-module of rank. 2.

The proof will follow the pattern established in §2, 3, and 4.

The module is given by an explicit algebraic

construction but the proof of non-triviality requires topological methods.

I have not yet succeeded in finding a

purely algebraic proof of Theorem 6.3.

Possibly the

methods of §17 might be applied. Let

x^,...,x^

and

y^,...,y^ € A

whose values at

g € G = SL^(C)

first row of

and the first column of

g

be the functions

are the entries of the g *

so that

450

SWAN

(x 11 xn2 Define

:A

4

A

2

by

x0 x.l

4j , g

3

yl

UgSUp) =

im(qp)H = 6 (2/122) / 0.

452

SWAN PART II.

§7.

K-THEORY OF

SPHERES

ALGEBRAIC S P H E R E S .

It is natural to ask whether the ring

R = (An )g of

theorem 5.2 can be replaced by = 1R[Xq ,...»xn ]/( ^ - 1) itself. we let

Pn (R)

In other words,

denote the set of isomorphism classes of

finitely generated projective R-modules of rank IR VB^(X)

let

n

and

denote the set of isomorphism classes of real

vector bundles of rank

Pk{V ■ * VB^(Sn )

if

n

on

X,

the question is whether

Pk(C®V < (Sn)- 311(1Pk(W ®V ■ *

are bijective.

This seems to be a difficult

question and the following is about all that is known.

THEOREM 7.1.

(a)

pk (An ) -» VB^(Sn )

P, (€ ® A ) -» VB^(Sn ) jk n rC (b)

^ ( ^ n)

all

k.

(c)

(Murthy)

n < 3

® &n )

VBk (Sn )

n = 0

or

1. 1 and

is bijective for

The referee has observed that the

also holds, i.e.,

for

n = 0

or

= 0 and

k.

(b)

IH ® A^

k

are bijective for

VB^(Sn ) is bijective for

and all

Remark.

and

1

and all

® &n ) k.

IH analogue of

VB^(Sn )

is bijective

The main point is that

is a principal ideal domain as one sees by

regarding it as the twisted group ring of € ® Aj = €[t,t_1] .

over

453

BUNDLES, MODULES AND K-THEORY

Proof.

(a)

= IT -.0(1) n-l

The case

k= 0

is trivial.

which is

0 for

n^l

andI/2Z

By [SwV, Th.5] (or by §9 below), - 0

so

n j* 1

for

We have for

is a UFD

while

VB^(Sn )

for

n ^ 1

P^(A^) = TL/27L by ID

Theorem 9.2 below.

n = 1.

By [SwV, p. 273],

P-^)

1

VB^fS1)

is

onto and hence is an isomorphism. Similarly, TL for for

n =

VB^(Sn ) = 2Tn_^U(l)

2.

n / 2.

0 for

By [SwV, Th.5] (or §9),

€% A

The case

W

and

is a UFD

n = 2 will follow from (c).

The usual stability theorems [B] [SwT]

P (A ) n v n'

n 7* 2

is

An >

w

give

Ko Plc(Cc(S2)) = VB®(S2)

is just Z

— — > Z

I

I

P ^ C ® a 2)

> z

and all its maps must be isomorphisms.

3

Finally, all vector bundles over Sare trivial since

of any Lie group is trivial.

CCX0 .. \ x =Xq +

But

€ ® A^ =

~ X) = C[x,y,z,w]/(xy - zw - 1) where ix^,

y=

Xq - ix^,

z = x^ + ix^, w = -x^ + ix^

and all projectives over this ring are trivial by Theorem

6. 2. Considerably more is known about the stable case of Yi

'V '

the above problem, i.e., whether isomorphism and similarly for

THEOREM 7.2 (Fossum [Fos]). (a)

K^(An )

KO^(Sn)

K^(An)

C ® A^ and

For all

([SwT])Kq(IH ® An)

(c)

K0(C 9 An ) -£■* KU°(Sn) .

~

~

O

n

KO (S ).

IH ® A^.

n,

KSp°(Sn)

is onto

THEOREM 7.3 (Claborn, Fossum, Murthy [CF]). K0 (An )

is an

Is onto

(b)

~

K (S )

If

n < 4,

BUNDLES, MODULES AND K-THEORY

455

The proof is given in §12. Fossum shows that these maps are onto by showing that the generators of

KO^(Sn )

and

KU^(Sn )

given by Atiyah,

Bott, and Shapiro [ABS] can be defined algebraically.

His

construction works over any field of characteristic not In the following, I willignore the case fields such as

IH.

quadratic form over

Letq(x^,. k

a i € k,

q

then

relations elements

1).

[ABS].If

C(q)

non-degenerate

Let C(q)

isgenerated

e.e. = -e.e. for ji

e. e. ...e. with i- i0 l 1 Z r C(q). Let

dimensional

be a

be the Clifford

q has the form

ij

k-base for

.

of non-commutative

and define

R(q) = k[xQ,...,xn ]/(q algebra of

.

q = ^

by eQ,...,en e. = a. . i i

j{(C(q))

be the category of finite

e^

by sending

-» R(q)

acting as

1 ® e^

where on

M

$(R)

be the category of

projective

Define a functor

to the kernel of

e = -^{1 - ^

R(q)

with the

i.e.,

I V e(r ® m) = ~{r ® m - 2 x ^r ® eim )• kernel is a direct summand of

The

i. < i0 < ... < i forma 1 2 r

finitely generated projective R-modules.

e:R(q)

with

2

i ^ jand ^J

C(q)-modules and let

0:J(C(q)) ->$(R(q))

Since

R(q) ® M

2 e = e,

this

and so is a

R(q)-module.

In order to compare this construction with that in [ABS], [Fos],

2.

[SwT] we must also

look at graded

C(q)-modules.

C(q) has a uniqueZ/2Z

C(q) =C^ ©

such that all

grading

ei have degree 1. The

456

SWAN

e. ...e. i1 1 r

with even

with odd

r

r

form a base for

from a base for

C * . Let

category of finite dimensional Recall that

q 1 1

q(xQ, . . . .x^) + y

2

f2 = 1,

has generators

where

If

M

where

f

is a

defines a

acts as

and

i

1

and

i

then

on

M € ^(C(-q))

M

that

e!e" = -e"e!

Replacing

j

e"

i

by

for -e”

and gives a module in

-1

on

and

for all

j

J

^(C(q)).

J x.ej:N^ -*

,

M*.

M = M % e.) i'

such

eVe! = -a. = e!eV. i i

i

changes

a^

l

l

to

-a^

Using this notation, the

as follows (using the notation of [SwT]). and let

M

^(C(q)) ~ ^(C(--q)).

construction used in [ABS], [Fos], [SwT]

M° © M 1 € ^(C(-q))

so

(by the action of v J

i ^ j

M

C(q 1 l)-module

can be described as

e!:M^ -» M * , e’.’:M* -* i * i i j

e.M* =

is a

and as

with

with

M € 4

Since R(q)

then

Recall that

R, the group of

is the free abelian group of formal

Z-linear combinations of height 1 prime ideals of K = Q(R)

the

is the quotient field of

R

then

1 -» U(R) -* K* -» D(R) -* C(R) -» 0.

R.

If

468

SWAN

If

S

is a multiplicative set in

subgroup of

D(R)

height 1 with

^

0

generated by the

P fl S ^ 0.

»

R

_

0

let

D(R,S)

be the

primeideals of

P

of

Applying the snake lemma to

---- >

K

i i

*

=

---- >

K

*

»0

i

0 ---- » D(R,S) ---- > D(R)

---- > D(Rg )

»0

gives [Sa] (1)

0

U(R)

If let

$

-> U(Rg ) D(R,S)

-» C(R) -> C(Rg ) -* 0.

is a collection of invertible ideals of

R^ = R[I

all I € ^ ] . Then spec

{P €

spec R| P 3

I

for all I 6 i’}.

this

locally with respect to

R

I € &

are principal and = Rg

where

products of generators of ment shows that

R^

ideals

is flat over

be the subgroup generated by the P 3

I for

some

I € $>.

Then,

U(R^)

D(R,^)

(2)

0 -> U(R)

(c)

Suppose now that

that

q = q' 1 hwith

q

=

It is enough to check but if S

I € $. R. P

R/(u) = R(q')[v] height 1.

Therefore, (1)

so

is generated by

(u)

C(R)

is principal.

is the set of

Let

D(R,^) C D(R)

ofheight 1 with

-» C(R)

C(R^) -> 0.

is isotropic of rank

gives

(u)

> 4

so

Then

C(R ) = 0. y uJ

is a domain so

islocalthe

exactly as above, we get

h hyperbolic.

*] so J

R

A similar argu­

R = R(q) = k[k^,...,xn ,u,v]/(q'(x) + uv-1) R = k[x1,...,x ,u,u u L 1 n

Rf

and

Since n > 2, is prime of

Z = D(R,(u))

(u)and hence

C(R)

0

C(R) = 0 since

469

BUNDLES. MODULES AND K-THEORY

In general, if q = q' 1 h

over

rk q > 4

k ' . Let

take

R' = k' ® R

G = Gal(k’/k)

Theorem 90 so D(R)

since

ffl = Pp

1

R'

with

then

H^G.k'*) = 0

of height 1

is a sum of distinct primes

This shows that If

C(R) = 0.

k(V-ab)

by Hilbert’s But

D(R')^ =

R, i.e., if

A=Rp,

A' = k' ® A then k'/k

G

[Bo]

P

in

the result follows. C(R) = 0.

is binary isotropic then In general, if

is

P^. Since these are

K* -» D(R) is onto so

q

D(R') -» 0.

It follows that the image of

permuted transitively by

and

and

K' = Q(R‘).

x

is a product of fields since

separable algebraic.

(a)

K'

k* -> K* -» D(R')^ -» 0.

A'/SKA' = k' ® A/ffl

D(R')

1 -» k ’

is unramified over

P

quadratic with

and

x

Then, using Lemma 9.1, we get If

k ’/k

q = ax

2

R(q)

2

+ by ,

=k[u,u*] let

k' =

as above and, using the above notation we get

1 -* U(R') -» K'* -> D(R' ) -» 0 so H*(G,U(R')) -* 0 U(R') -> Z -> 0

and

and

G

1-» U(R) -> K* -» D(R) -►

C(R) = H 1(G,U(R')). acts on

Now 1

-> k ’* -*

TL by cr(n) = -n

u = x + V-b/a y -> x - V-b/a y = a ^u ^ . Taking

since G

cohomology gives 0 = H 1(G,k,X) -» H 1(G,R'X ) p

x

X

x

^

6(1) = [a].

is a norm from

2 + (b/a)y2

represents 1 .

Now

X

H (G,k' ) = k be

TL/ITL -£-» H2 (G,k’*). anc*

^

is easilY computed to

Therefore C(R) = 7L/27L

if and only if

k ’. This means that the norm form represents

a over

k,

i.e., that

q

a

470

SWAN (b)

Suppose first that

k[u,v,w]/(uv + w (v,w-l).

Then

2

-1).

and

In

A,

Let

soP

where

ut(w-l) + (w^-1) = 0

» C(R)

and C(R) = TL

(u,w+l)P = (u) (v,w+l) £ P

so

0

generated by

P.

with

2

= (w+1)

2

2

+ cz

G = Gal(k*/k).

v = y/~b y - y/~c z.

U(R')=k'*

P ^

A = R[P*]

=

Using (2) we get

C(R) C(A)

q = ax + by

has the above form with and

Let

and

Note that

so

(u,w+l) ~ P * ~ (v,w-l).

In general, let kfV^T.V^T.v^)

P' =

so u t + w + l = 0

and (v,w+1)(u,w+1)

and

R = R(q) =

is invertible

A = k[u,t,w]/(ut + w + 1) = k[u,t].

Z

so

t =v/(w-l).

1 -> R* -> A* -» Z so

2

P= (u,w-l) and

PP' = (w-1)

= (w-1) *P' = (1,t) R[t],

q = uv + w

w = Vr~a x,

and set

Then

k' =

R' = k' % R

u = V b y + yf-c z,

The above argument shows that

1 -* k ’* -» K'* -» D(R') -» C(R')

-»0.

This

breaks up into 1 -» k '* -> K'* -> H -> 1 1 -» H -» D(R') and taking

G

C(R') -» 0

cohomology gives 1 -» k* -» K* -» HG -» H^G.k'*) = 0 1 -» HG -> D(R) -> C(R')G -» H 1(G,H)

and we deduce a €G

sends

0 -* C(R) -» C(R')^ -» H^(G,H). P

to an isomorphic ideal if andonly if it

changes the sign of an even number of i.e., if and only if it fixes C(R’)^ = TL torsion,

Now an element

if and only if

the result follows.

V

>/"a,

vHb,

y/-c

V-abc = V ds q. Therefore ds q €k.

Since

H^(G,H)is

471

BUNDLES, MODULES AND K-THEORY

Suppose

q

haLS rank 3

ax^ + by^ + cz^,

R

= R(q) , k' = kfV^a), and R'

Remark.

k' O R .

Then

u = y + b V

V ds q € k.

R ‘ =k' [u, v,w]/(uv+w -1) d/a z,

d = ds q = -abc. can take

and

v = b(y - b

d/a z)

In particular, if

a = 1

so

k' = k

and

with

q

Let =

w = V ”a x,

where

represents 1

R(q)

This also shows that for

V ds q € k,

q

will represent 1 if and only if it is

isotropic.

Here

(V~a x - 1, then and

Vra by

aP = aP

wherea = (V”a

cr(a)a = -ab.

rk q = 3

is generated by

+ V~d z).

Now

If

G = Gal(k'/k) = {1 ,cr}

by - Vr~dz )(V~8l x - 1) ^

0 -» H 1(G,H) -» H2 (G,k'*) = k*/Nk'*

C(R‘)G - ^ ( G . H ) ->k*/Nk'* trivial if and only if ax

2

doesn’t represent 1,

COROLLARY 9.3.

If

is

b = f

+ by

2

P

2

- aq

2

with

represents 1.

G then C(R) -»C(R')

rk q =

under

-ab = b mod N k 1*.

2,

KQ (R(q))

non-isotropic and represents 1.

In fact

and

P =

and one computes easily that the image of

if and only if

we

itself has the

required form.

C(R’)

This is

f.q € k,

i.e.,

Therefore if

q

is

2 Z ---»Z.

= Z/2Z

Otherwise

K^fRfq)) = Pic R(q) = C(R(q))

if q

Let

q

since

R(q)

be a binary quadratic form which is

non-iso tropic and represents

1.

If

M

is

K^(R(q)) = 0.

is a Dedekind ring.

LEMMA 9.4

q =

is a finitely

472

SWAN R(q)

generated torsion module over K^(R(q))

Let

Proof.

R = R(q)

= dim^(R/P) mod 2 Dedeking ring, ideals of ICR.

R

2

q = x + ay . Then

is clear that d is R

=

0

2

withf,g € B

then

2

+ g (ay -1) has even degree.

. t f = ry + ...,

is a

B = k[y]. R/I as a B-module

+ gx))= (f^ - g^x^)

deg f Z deg g + 1

R

on principal ideals. Let

dim R/I = dim B/(f^ - g^x^). But

2

Since

d(P)

d(I) = dim^(R/I) for

B (B Bx where

has order ideal

f

P.

by

is the group of invertible fractional

and it

I = (f + gx)

d:D(R) -> Z/2Z

for prime ideals

D(R)

in

is even.

and define

I claim that

2

If

dim^, M

if and only if

[M] = 0

then

so

f^ - g^x^ =

This is clear if

and in the remaining case if

g = sy

t-1 ^ + ...

2 2 2t (r + as )y + ... andr

2 + as

. . 2 2 2 then f - g x 2 = q(r,s)

=

Z 0

since

q

is non-isotropic. It follows that d:C(R) -» Z/2Z

d defines a homomorphism

whenever

q

also represents 1 we have and

P = (x-a,y-|3) has

is non-isotropic. q(a,/3) = 1for some

d(P)

= 1

so

d

KQ (R) = C(R)

THEOREM 9.5. 0: ABS(q)

If

Z/22

q

KQ (R(q)) .

sends

M to

a,/3 € k

is onto.

Finally, by taking a composition series for that

Since q

M

it is clear

din^M.

isa binary quadratic form then

BUNDLES, MODULES AND K-THEORY

We need only check that

Proof.

0

is non-trivial when

is non-isotropic and represents 1. An explicit isomorphism

C(q)-module, Therefore

e

on

Write

C(q) £ ^ ( k ) so if

473

q = x

2

*s given by

M £ k

2

is a simple

R(q) 0 M £ R x R

0(M) = {(r,s) € R x R|(l-x)r = ybs,

I 3 (l-x,y),

§10.

1 C I

and

[I] = -[R/I]

since

2

+ by .

(l+x)s = yr} £ I = {s € R|(l+x)s € yR} = (l-x,y)

Kq (R),

q

(l-x,y)

is maximal.

since In

which is non-trivial by Lemma 9.4

R/I £ k.

K-THEORY OF R(q) If

A

K^fJfA))

is a noetherian ring I will write

where

^(A)

is the category of

G^(A)

for

finitely

generated A-modules.

This group is

[QK]

G notation less easily confused with

but I find the

(A) = K^($(A))

where

$(A)

is the category of finitely

generated projective A-modules. if

A

R(q)

is regular

Of course

G^(A) = K^,(A)

In particular this is true for

as observed in §9. If

A

is a k-algebra,

G i(A) = ckr[Gi(k) of

[QK].

often denoted by K^(A)

k

G^A)].

since it is just

commutative,

[A]

k If

a field, I will write i = 0, this is independent

Gq(A)/Z[A].

Note that, if

A

generates a subgroup isomorphic to

This is easily seen by considering

G^(A) G^(Ap) = TL

where

If

P

is a prime of height

0.

is TL.

A is reduced this

474

SWAN

argument shows that

Z[A]

Gq(A) ~ TL © Gq(A)

GQ (Ap)

so that

when A is reduced.

For

will at least haveG ^ A ) = G ^ k ) © G ^ A ) augmentation

e:A

i >0

if

A

we

has

a good

k with Tor- dim. k < co . A

The following is a well known classical argument [Mu, Prop. 6.], [JK].

LEMMA 10.1.

Let

f € k[x^,....x^],

k[x^ ,...,xn ]/(f )

... -* G^(k) -» G^(A) -» G^(B) -»

there is an exact sequence G._^(k) -» G^ ^(A) -» ...

Note that

Note that

Proof.

g(x,u,v) € B have u|h g

and

Also ug = 0

f ^ 0.

represents

torsion free, k[u]

B

...

G.(k)

0

while

B -» B^

B.

e(x^) = 0 ,

e(u)

B^ =

k[x^...... x^.u.v] we so

It follows that in

a good

is injective since if

then in

ug(x,u,v) = (f+uv)h(x,u,v) since

has

so G i(B) = G.(k) © G.(B).

B/(u) = A[v]

k[x^,....x^.u.u *].

A

G^A) - ^ G ^ B ) .

has a good augmentation

e(v) = -f(0___ ,0)

1,

. In particular, if

i = 0 then

B

A =

B = k[x^,....x^u, v]/(f+uv) . Then

and

augmentation or if

f ^ 0,

u|fh

and therefore

f+uv

divides

g

From this it is clear that

and therefore flat, over

so B

is

k[u]. The map

induces a map of localization sequences

i

G.(k[u]) —

i

G.(k[u,u_1]) —

i

i ~

... — * GjfB/u)-- * G 1(B)

Gj.jfk) ^

— * G.(Bu )



» G 1_1(BAi)—

»...

BUNDLES # MODULES AND K-THEORY

Now

O i(k[u]) = G^(k)

and the upper sequence reduces to

split short exact sequences G i_l(k ) ^ 0 [QK]

[QK].

475

Also

0 -» G^(k)

G^(k[u,u ^]) -»

G (B/u) = G.(A[v]) = G.(A)

by

and the lemma follows easily.

Remarks. f = 0

(1) The condition

then

f / 0

is really needed.

B = k[x^,....x^.u,v]/(uv)

G^(k[u,v]/(uv))

and we

can assume

so

If

G^(B) =

n = 0. The

localization sequence gives ... with

G.(B/u) -* G. (B) -» G.(Bu ) -» G._1(B/u) ^ ...

B/u = k[v]

G^fB/u)

and

B^ = k[u,u *].

G^fB) -» G^B^) = G i(k[v,v *])

monomorphism so we get

If

A

applies using can assume (3) by

k

and

is a split

in this case (see [RoK, p. 521]).

B

are regular the same argument

the localization sequence for K^[GQ]

and we

is any regular ring.

The map

J(A) -* M(B)

G^(k[v]) =

G^(B) ~ G^(k[v]) ©G i(k[u,u *]) ~

G^(k) © G^fk) © G i_^(k) (2)

The map

G.(A) -» G.(B) sending

M

to

in Lemma 10.1 A[v] ®^M

is induced

considered as a

B-module. (4)

A nice application of this type of argument is

given in [Br].

THEOREM 10.2.([JK]). (1)

If

For any regular ring k,

q = x ^ + ... + xnyn

KjfRfq)) = K.(k) © K i_ 1(k).

then

476

SWAN

(2)

If

R = k[xr ...»xn »y x--- ’V

( J x.y. - z(l-z))

2^ 7

K.(R) = K.(k) ffiK.(k).

then

Note that (2) appliesto

R(q)

2 q = x 1y 1 + ... + xnyn + z

if

1 and ^ € k.

To prove Theorem 10.2,

useLemma 10.1

to reduce to the

R(q) = k[x,y]/(xy-l) = k[x,x *] or R(q) =

case

k[z]/(z (1—z)) = k x k. In particular, for case (1) [CF].

and

Z

k

a field,

in case (2).

K^fRfq))

An explicit

generator is easily found using remark (3).

result as an R-module. where

P = (x1 v 1

k[y^,...,yn ] Thus

in

This was first proved in

Note that Theorem 7.2c follows.

k[z]/(z-l), tensor with

is 0

K^(R)

We start with

and regard the

is generated by

[R/P]

x ,z-l). n '

From now on I will concentrate on the case where more detailed information is available.

i = 0 It is not

clear to me at present how to formulate the conjecture of §7 for

if

i > 0.

Also, from now on,

k

will again

be a field of characteristic not 2.

COROLLARY 10.3. [RoB]. k

then (2)

(1)

KQ (R(q)) = Z ® H

If where

In all other cases

rk q

is odd and

2rH = 0 ,

V ds q €

r < | ' and

M(£q , . . . , e^ ) = R V P(& q , . . . , e^)

P (e n , . . . , £ ) = (V a x v 0 my v n n

x0 2m-l

V ds q

u , v - , . . . ,v .w l/ f^ u .v . + w^-1) m l m JVZ , i i J

- V- a2 i/a2 i - l x

where

We need only check that

= 7L i f and on ly i f

l +

is generated by any one o f w ith

2 .

K q (R ')

R' = k T y . ^ 1

v i = a2 i - l ( x2 i - l x

... + if

exten sion and tra n s fe r we g e t

n

is eq u iva len t to one o f

k ' = k (V -a 0/a1 , .. .,V-a /a 7) v 2 1 n n-lJ

L et

w = V a

of

2

is even and

if

G

considered in Theorem 10.2. I f q = a^x^ +

we

is odd.

be a g a lo is extension w ith group

such that over

the forms sl

k'

a.

1

is

± l.

M( l , . . . , l )

to

M(a^t...,am )

a(V a ) = a~yf~a , and nJ O n v

TL

i

*■

'i

ABSk ,(q) — —— » Pic R'(q) and it follows that

e = 2

2i

Z —

>Z

and the top map is an

isomorphism.

Remark.

This shows that if

is non-isotropic then

C(q) = A x A

division algebra while

LEMMA 11.2 (MURTHY). det: K0 (R(q))

rk q = 3,

A = M^(k)

If

where if

rk q = 3.

V ds q € k,

q

and

A

and

q

is a

is isotropic.

V ds q € k

then

Pic(R(q)).

A much stronger result, also due to Murthy, is proved in §16.

COROLLARY 11.3. is isotropic

If

rk q = 3

and if

V ds q € k

0:ABS(q) £ K^fRfq)).

then

In dealing with the remaining cases it will be convenient to make use of the non-regular rings S(q) = k[xx

,xn ]/(q(x1

Xn ))

cf. [CF].

Let

1 = (x^ ...,xn ) C S(q).

LEMMA 11.4. (b)

If

(a)

If

rk q = 1,

rk q = 2 then

GQ (S(q)) = Z/2Z.

or if

q

484

SWAN (1)

If q is isotropic, G^fSfq)) = Z. If q is not isotropic, G^fSfq)) = Z/2Z

(ii)

[S(q)/3R].

generated by (c)

If

rk q = 3

(i)

If q is isotropic,

(it)

(a)

(b)

generated by

q

S(q) = k[x]/(x )

is isotropic,

G^Sfq))

If

q = ax

k* = k(V-ab)

and

k'* x and sends

Then

0.

Sx= k'[x,x

Here

to

Suppose

[S/x]

dim^ S/(ax+Py) = 2.

The map

k'

2

is

0

in

Gq (S/x )

d

-- » Gq (S/x )

_______ x f = a + P V ds q 6 k ’

Therefore

+ by

G^fS^} = K^fS^) =

so we get

This is

and we

*] where

G^(S/x) = G^(k[y]/(y^)) = Z.

d(f) = [S/(ax+py)].

[S/9R]

and

y -* xV-a/b.

x € K^fS^)

^ G0 (S)

0

2

d ^ ( S ^ ) -- >

non-isotropic, the localization sequence gives G q (S) -» Gq (Sx )

or

the result is clear.

S(q) = k[u,v]/(uv)

use remark (1) following Lemma 10.1.

Gq (S/x)

is 0

[S(q)/1].

2

Since

If

G^(S(q)) = Z/2Z.

If q is non-iso tropic, Z/2Z

Proof.

then

with

a,P € k.

G^(S/x) Gq (S)

since and

clearly generates. (c)

Suppose write

If q

q

is non-isotropic.

q = ax

G l(Sz ) R(q')[z,z

is isotropic use Lemma 10.1 and (a). Up to a scalar factor we can

2 + by2 + z2 . The localization sequence gives

Go (S/z) " G0 (s) where

K^(R(q1)) = TL since

G0 (Sz ) ■* °-

q' = -ax^ - b y / q'

NoW

By 9.3,

does not represent 1,

SZ =

Gq (S^) = q

being

BUNDLES, MODULES AND K-THEORY

non-isotropic. G0 (S/z) q'

As usual,

so we get

KjtS )

is non-isotropic,

generated by

z € G^($z)

maps to

GQ (S/z)

S/(z) = S(q')

has

This follows from

Gq = 7L/27L

we get

S(q) =

G^(S(q)) = Z/2Z

K^(R(q')) = IR

and

K^C)

in

11.4(ii).

which he proves by

considering the localization sequence for

to

Since

[S/1].

IK[x,y,z]/(x^+y^+z^)

p = Spec C,

[S/z] €

GQ (S) -» 0.

The referee points out that when

Remark.

485

X-p = Spec R(q').

X = Proj S(q),

This sequence reduces

K 1(W) -» K 1(R(q‘))/K1(IR) -* 0

where

0

is

onto [MeS].

LEMMA 11.5.

rk q = 3,

If

non-isotropic, and generated by

Proof.

q

1

represents

[R(q)/(x,y,z-l)]

2

Let

V ds q € k ,

q’ = q - t

and

if

q

is

then

q = ax

2

K^(R(q)) = Z/2Z + by

2

2

+ z .

S = S(q’) =

k[x,y,z,t]/(ax^+by^+z^-t^). By 10.1, GQ (S) £ GQ (S(ax2+by2 )) is not isotropic.

which is

G^(S/t)

G^fS/t)

0

But

S/I —

[S/1] S/I

0.

0

if

so

By 11.4c(ii),

but this is SI

q

G^(St)

S t = R(q)[t,t_1]

Gq(S) -» G^fRfq))

is generated by

since we have

by 11.4 since

The localization sequence gives

-*G0 (S/t) -» Gq (S) -*G0 (St) -*0. we deduce

Z/2Z

0

in

G^(S)

1 = (x,y,z-t)

486

SWAN

so that

S/I = k[t].

Therefore

G^(R(q)). The generator G^fSfax^+by^))

Gq (S)

Gq (S ) =

[S(ax^+by^)/(x,y)]

maps to

[S/(x,y,z-t]

in

of

G^(S).

Now

S^.

= k[x,y,z, t, t 1]/(q') = (k[f ,r),f]/(q(f ,T},f)-l))[t,t-1] = R[t,t *]

and the generator is

R/(J,T),f-l))[t, t *]

corresponds to the given generator under

THEOREM 11.6.

If

rk q = 3

G q (S^) ~ G^(R).

0:ABS(q)

then

except possibly in the case where

KQ (R(q))

V ds q € k

Note that in the exceptional case ABS(q) = 0 conjecture is equivalent to

V ds q £ k.

If

q represents

1

Let

R

ABS(q) = Z/2Z 0

by

is R' = k'

R' module and regarding

module gives a commutative diagram

i

i

ABSk (q)

TL

-

i

---- > Kq (R)

TL

i TL/2TL

The upper map is an isomorphism by 11.3.

11.2

in this case.

is non-isotropic and

then

ABSk ,(q) - H - » K 0 (R')

sufficient

does

so the

k' = k(V ds q), R = R(q), and

The transfer map, taking an

it as an

q

11.5 itis enough to show that

non-trivial. ® R.

K(R(q)) = 0

By 11.3 we can assume that

8.10 so by

q

and

1.

not represent

Proof.

which

> Z/2Z. It is clearly

to show that the right vertical map is onto.

and the proof of 9.2b,

K^(R')

is generated by

[M]

By

487

BUNDLES,, MODULES AND K-THEORY

where

M = R'/P,

with

z

acting as

acting as and

P = (z-l,x - V-b/a y ) . Now

y.

1,

Therefore

X~l > R -> N -> 0 since

§12.

M

V-b/a y

and

has

[N] = 0

[M] = [M/N] but

xf,yf = xe € N.

which

is

y e = 1

(z-1)

P fl R is a non-maximal prime

N = Re C M so

and

is generated by

Ann^(e) = P fl R

(z-1) € P fl R

ideal.

acting as

As an R module,

f = V-b/a. But

since

x

M ~ k'[y]

by

0 -+ R

M/N = R/(x,y,z-l)

By 11.5 this generates

K^(R).

RANK k AND 5 Here I will only consider a few cases needed to finish

the proof of Theorem 7.3. Consider

R = R(q 1 1) =

o k[x^,...,xn>w]/(q(x^,....x^) + w - 1). Then and

Set

R = k[x^,....x^,t]/(q+t(t-2)) R^ = k[x^

x ,t,t *]/(q(x)+t(t-2)) =

k[y^,...,yn ,z,z *]/(q(y) +l-2z) z = t 1.

Therefore

k[y^ ,...

where

y

G^(k)

(1) where

*

and

* The localization sequence gives

lies in that of

G^(R)

Since the image

we get

G 1(Rt) -» G0 (S(q)) -» G0 (R(q 1 1)) -» 0 S(q) = k[x^,...,xn ]/(q) To compute

for

= x^

R t = (k[y1 .... yn ,z]/(q(y)+l-2z))z =

G^(R) nG^(Rt) -» G^(R/t) -> G^(R) -+ TL -> 0. of

t = w+1.

G^(R^)

A = k[yx.... yR ]

as in §11.

we use the localization sequence getting

488

SWAN

Gj(A) -* GjfAj^) -» Gq (A/( 1+q)) -» GQ (A) -» T ■* 0

k* ■* Gi n, any

linear section

of

i. A^

n

A..,

=0

is generated by linear subvarieties and

L':yn = x. = ... = x = 0 . 0 1 n

are unchanged if an even number of

exchanged in these equations.

L:x^ =

The classes

^ ’Yj

are

495

BUNDLES, MODULES AND K-THEORY

Case (b). (i) For y J0

A.,

i < n,

any linear subvariety, e.g.,

= ...= y = 0 , n

z=0,

x. - = . . . = x = 0 . l+l n

i > n,

any section by a linear

(ii) For

A,.,

subspace of the correct dimension. The proof uses the localization sequence Aj, (X) -+ A^X-Y) -» 0 be defined by where q' X--Y

and induction on

= 0.

is

q

with

Then X q

Y

and

is an affine space so

n.

We choose

is a cone over y^

A*(X-Y) = 0

for

A #(X(q))

is onto for

i > 0.

Y

to

X(q')

set equal to

the proof of Lemma 13.4 below one shows that A^+ ^(Y)

A,.(Y) -»

0.

i ^ 0.

Also As in

A^(X(q') -»

In this way we see that

is generated by the indicated elements.

Computing intersections then shows that there are no relations and also gives the relations mentioned in case (a)(iii).

Note that Theorem 13.1(1)

Theorems 13.2

Problem.

is immediate from

and 13.3.

Determine the Chow ring of

X(q).

As far as I know, this problem is also still open for Severi-Brauer varieties.

Some cases are treated in [MeS].

Some information can be obtained from Theorem 13.2 but the map

TL by taking the intersection multiplicity with a linear subspace of Pn+^ i.e., by

A t(X) -> A i(Pn+1)

LEMMA 13.4.

Let

1.

q = h^ 1 q' q'

hyperbolic and and let

of complementary dimension,

where

h is binary

is non-isotropic. Let

X = X(q) C Pn+^

deg A^(X) = TL for

be defined by

i < d-1

rk q = n + 2

q = 0.

deg A^ = 2Z

and

Then for

d < i < n.

Of course

For

Proof.

dimension

A^(X) = 0

i < d-1, i

i > n =

dim X.

contains a linear space of

which has degree 1.

deg A^(X) D 2Z i

X

for

For any

i,

since a linear space section of dimension

has degree 2.

Clearly deg A^(X)= 2TL.

Case d = 0. In this case we must show that all

i.

Let

Y C X

linear subspace of finite. where

Then p

deg Y

have dimension Pn+*

deg A^(X) = 2Z i. Let

of dimension

n+l-i

L

be a with

is the number of points of

is counted with multiplicity

for

|k(p)-k | .

Y fl L

L*Y If

BUNDLES, MODULES AND K-THEORY

deg Y

is odd, some

rational over over

k

|k(p):k|

is odd.

Since

k(p), q is isotropic over

because

|k(p):k|

497

p € X

k(p)

is odd [Lam].

is

and hence

But this implies

d > 0.

Case d > 0. Write

q = h C q' = q*(x) + uv.

Theorem 13.3 we let

Y = X fl {u=0}. Then

affine variety defined by affine space

An .

A^fA11) -» 0 i < n.

Now

vertex

X - Y

q'(x) + v = 0

Y C Pn

A ^ Y ) -> A^(X)

is defined by

(x.,...,x ,v) v 1 n J

p = (0,...,0,1)

over

Y

A^(Y) -» A^(X)

is onto for

q'(x)

so

is the

which is just the

The localization sequence

shows that

coordinates are

As in the proof of

=0

where the

is the cone with

Z = X(q’) C Pn * =

{(Xj,...,x ,0) € Pn } . The localization sequence gives A..(p)

A (Y) -* A^(Y-p)

shows that

0.

Since the rational point

deg Aq(X) = TL we can assume

A ^ Y ) « A.(Y-p). (x^,...,x )

Now

is an

Y - p L z

A*

by

i > 0

fffXj

p

getting

xn -v ) =

bundle and so, by a standard

property of the Chow ring, it induces an isomorphism JTX -A^(Z) £ A^+ ^(Y-p) IT *(D).

sending a cycle

One checks immediately that

D in

Z

to

deg IT*(D) = deg D

and the lemma follows by induction. Now, given of degree

2r

q,

we can find a galois extension

such that

q,

over

k'/k

k ’, has one of the

498

SWAN

forms considered in Theorem 13.3.

Let

and

X* -» X

X' = k' ® X = X^,(q).

have maps

Since

f:Ai(X) +— - A^fX’

with

compositions are multiplication by deg:A^(X’) -+ Z

preserves the

must be fixed by these

i,

G

for

£:A^(X) -* Z

+ x y nrm

over

a

b

and

k'.

represented by

yQ - xi = . . .

LEMMA 13.5.

=

n = dim X

so for

q = Xq Yq + with generators &

X q = ... = x^ = 0

a

a

fixes

crV ds q =

If

and

and

- V ds q

if

a

then

a

b

b.

and

2

q = a~x~ + ... + a ,-x M 0 0 n+1 n+1

U0V0 + •■• + UmVm

Xq - V-a^/a^ x ^ , etc.

with

Over k'

a a

switches some of the and

b

If

Remark. dimension choose

I Z V - a ^ ^ ^ = ^ a2k^ ^

n

represents

This means

a + b € Am (X')

X'

of since we can

X' fl {Uj = ... = um = 0} = {u^ = ... = um = 0,

U0V0 = 0}-

u i ’v i

q

is even, a plane section of

m = ^n

q

are fixed if and

only if an even number of pairs are switched. a fixes

write

U0 = a0 (x0 + V~a l/a0 X l)' V0 =

Then

up to constant factors and

that

Both

j* 2 . Since

and that

2

Let

Proof.

G = Gal(k'/k).

(X1) = Z x Z mv J

a € G

cf.[RoB]

interchanges

is finite we

= xn = ° .

V ds q € k ‘.

fixes

A

k

A^XJ/2-torsion = Z.

n = 2m

Then

over

action we see that A,.(X')

i^-n,

and

Suppose now that

G

X = X(q)

BUNDLES, MODULES AND K-THEORY

THEOREM 13.6.

X = X(q)

Let

h^ 1 q' with

h

(1)

subspace of (2)

i

D' -» Pic X' -» 0

_^(X),

X.

1 -» k'* -» K ’* -> P'

n = dim 0 and

0 ->P' -♦ D' -» Pic X' -» 0.

Using Hilbert’s Theorem 90 as 1 ^ k*

K* -> P ,G -> 0

and

0 -» Pic X -» Pic X'G . But The results on

in §9 we get

0 -» P ,G -> D'G -» Pic X ,G Pic X ’

Pic R(q)

so

is torsion free by 13.3.

in §9

using the localization sequence. Murthy.

This gives

can now be deduced

This approach is due to

501

BUNDLES, MODULES AND K-THEORY

A detailed investigation of the hypersurfaces would clearly be very worthwhile.

X(q)

It is not even clear to

me how to classify these hypersurfaces up to isomorphism or birational equivalence.

§14.

THE DESCENT SPECTRAL SEQUENCE

It is easy to see that Thoerem 7.2a does not extend to the higher

KL

since the algebraic

are too big.

However,

using Theorem 10.2 and Suslin’s recent computation of the K-theory of

C

[SuK] one sees easily that the groups with

finite coefficients are isomorphic KU 1(Sn ;Z/m).

KL(C ® A^Z/m) £

I have not checked that the explicit map of

Theorem 7.2 induces these isomorphisms but it seems quite likely that this is so.

It should not be too difficult to

verify this using work of Friedlander [Fr].

One would then

like to deduce the real case by using a comparison theorem and descent spectral sequences. Q Suppose A = : B C B i s a galois extension of rings (see e.g., [SwN]}.

An old question of Lichtenbaum [Lie]

asks whether there is a spectral sequence H^(G,K_ (B)) => K (A). Q P Q

This question has been discussed

by D. W. Anderson (unpublished) and, more recently, by R. Thomason (cf. [Th]).

In the topological case one would

like a corresponding spectral sequence Hp (G,KUq (X)) =» K0P+q(X)

where

G = Z/2Z

acting by complex

502

SWAN

conjugation.

If we had these spectral sequences and a map

between them we could hope to deduce K0-q(X;Z/m)

from

K^(A;Z/m) £

Kq (B;Z/m) ~ KU-q(X;Z/m).

Bloch (unpublished) has given a counterexample to the existence of the algebraic descent sequence even when is a field.

However, it may still be possible to make the

descent argument work.

In particular, for our purposes it

would be enough to have the sequences for with

B

mod 2

G = TL/2TL and

coefficients.

Thomason [Th]

has shown that the spectral sequence

can be derived if a certain homotopy limit problem has a positive solution.

I will outline briefly the relation

between the present formulation and Thomason’s work.

The

following descent theorem of Speiser is often referred to as ’’Hilbert’s Theorem 90” presumably because of its close relationship to that result.

LEMMA 14.1 (Speiser). rings with group such that

Let

M

be a galois extension of

be a B-module with

a(bm) = a(b)a(m). Then

Conversely, if

Proof.

G.

A C B

Let

N

is an

B ®^M ~ A-module, N

G

^

G

action

M.

Q (B ®^N) .

By a faithfully flat base extension (in fact A -* B)

we can assume that

B

is split,

B=

TT

o€G the verification is trivial.

A.

In this case

BUNDLES, MODULES AND K-THEORY

COROLLARY 14.2.

The category of

to the category of true for

503

A-modules is equivalent

B-modules with

G

action.

The same is

the categories of finitely generatedprojective

modules#(A)

For

and

$(B).

the second part we use the fact that

B is

finitely generated projective as an A-module. Let

G operate on

#>(B)

where

M*7 = M

with new

where

fa = f

on morphisms. Let

category of

G.

by o'-M

action by bMn = cr(b)m

Let

G

M*7

and

EG be the translation

Its objects are the elements of

Homfcr.T) = {t o *}. o

B

from the right

act on

EG

G

and

from the right by

erg.

LEMMA

14.3. CatG (EG,#(B)) £ $(A).

Proof.

The

left side is the category of G-equivariant

functors and natural maps. F(l) = M, then gives

M

M*7

If

F

is such a functor and

and

F

on

which defines a

G

action on

F(cr) = vP

Hom(l.a) = {o} M

as in

Lemma 14.1.

COROLLARY 14.4. Quillen’s

Q

CatG (EG, Q#(B)) ~ Q$(A)

construction.

where

Q

is

504

SWAN Taking nerves as in [Th] we deduce an isomorphism is

simplicial sets

Map^(EG,NQ^(B)) ~NQ$(A).

Thomason’s

homotopy limit problem asks if the same is true of the geometric realizations, i.e., if |Map^(EG,NQ^(BJ)| ^ Map^(|EG|,BQ$(B)).

If so, filtering

|EG|

yields the required spectral sequence.

by skeletons

We refer to [Th]

for more details and further results. The homotopy limit problem is closely related to the Segal conjecture [Th]. On the other hand, the topological analogue seems closely related to the Sullivan conjecture [Sul]. We approximate M

and

M

G

where

BU

and

G = Z/2Z

BO

by finite Grassmannians

acts by complex conjugation.

Note that a complex linear space stable under defined over

IR by

14.1

so

G M

G

is

is just the real

Grassmannian. that

The strong Sullivan conjecture [Sul] asserts Q Map^(EG,M) ^ M after completing at 2. Taking

mapping spaces gives EG

X G X Mapp (EG,M } ss (M )

by skeletons as in [Th] gives

and filtering

H^(G,II_^(M^)) =>

I T p _ _ q ( . Taking direct limits as

M

tends to

BU

gives the required spectral sequence X IT_p_q(B0 ).

Of course this is still only conjectural since

the Sullivan conjecture has as yet only been proved for trivial

G

actions [Mi].

505

BUNDLES, MODULES AND K-THEORY

PART III. §15.

WORK OF MUKTHY, MOHAN KUMAR AND NORI

PROOF OF THEOREM 6.2. The results of this section are due to Murthy.

proofs follow his fairly closely.

The

For convenience we

restate the theorem.

THEOREM 15.1.

(Murthy).

Let

k[w,x,y,z]/(wx + yz - 1).

k

be any field and let

A =

Then all projective A-modules

are free.

We recall first a theorem of Plumstead [P]. be a commutative ring

and let

Rs + Rt = R.

Let

R

Then

the

diagram R ---- > R

s

i

i

R. t

>R _ st

has the Milnor patching property, i.e., given finitely generated projectives with

a:(P^)t ~ (^2^s*

P^

over

Rg

t*ie Pu Hkack

and P^

over

R^

P = PfP^’^ ’0^ pi

I P2 ---- >s is finitely generated P

P^.

-1--- CPl>t

projective over

R and

Pg

P^,

This is easily seen by patching sheaves over

Spec R = Spec Rg U Spec R t

or, even more easily, by

506

SWAN

checking the asserted properties locally (see below). Conversely any finitely generated R-projective obtained by patching

Pg

and

(f - g)

over

R[t]

when

t -» 1.

to

Here M[t]

f when

over

t

R

are

M[t] £ N[t]

0 and to

g

means R[t]

THEOREM 15.2 (Plumstead [P]). a,/3:(P^)t £ (^2^s

f,g:M £ N

if there is an isomorphism

which reduces

can be

P .

We say that two isomorphisms homotopic

P

With the above notations, if anc* a ~ P

over

PfP^.P^a)

then

ss P ( P r P2 , p ) .

For the proof see [P]. To prove Theorem 15.1 we apply this with Here

A^ = k[y,z,x,x

and

A = Ax + Ay.

A^ = k[w,x,y,y *]

have all

projectives free so any finitely generated projective A-module

P

An , i.e., xy we can write

has the form

P = P(An ,An ,a) v x y y

a € GL (A ). n v xy'

Now

xy

It follows easily that

a.tz + ... + a trzr € GL (A [t]) 1 r n v xyL = cXq.

a:An £ xy

= k[x,x *,y,y *,z] L J J i* a = o' + a.z + ... + a z where a. € 0 1 r i

Mn (k[x,x ^,y,y *]).

A

where

so that

ct(t) =

a €

GLn (k[x,x_1,y,y_1]). Murthy's original argument now runs as follows. t? - Spec k[x,y].

Then

U =

+

a = a(l) ^ a(0) v J

Therefore by 15.2 we can assume that

Consider

so

-{0},

a

BUNDLES, MODULES AND K-THEORY

non-affine scheme, has the form

U = V U W

V = Spec k[x,x ^,y], W = Spec k[x,y,y *], V fl W = Spec k[x,x ^,y,y ^].

Use

vector bundles over

W getting a

U.

V

and

a

507

with and

to patch trivial bundle ^

over

The inclusion

k[x,y] C A

2 1*:Spec A -» U C A

and

^ f (^)

associated to

Therefore it is enough to show that

is trivial.

P.

Now

induces a map is clearly the bundle

extends to a coherent sheaf

%

which is associated to a finitely generated module since U.

M.

We can replace

M -» M

M

3

on

M

A

2

B = k[x,y]

by its double dual

M**

inducesan isomorphism over all points of

However,M**

is projective by Lemma 15.3

and all

projective B-modules are free. The following is a standard result.

LEMMA 15.3

If

R

is regular of dimension

finitely presented then

Proof.

We can find

and finitely F'

-* X

0.

dimension B

x

I B

I

---- > B y xy Bx + By 5* B.

which is not a patching diagram since = Bn

and let

M

Let

F

be the pullback of F x

F ---- > F y xy If we apply

A

pullback is

P.

^---F £ xy

- to this diagram weget a diagram whose

D

This yields a map

A

-» P

which

Jd

localizes to an M

x

£ F x

and

M

y

isomorphism overA^ £ F y

Therefore A^

A

N C M P

with

and

N = M x x

A.

The same argument shows that N**

A

is an

is free by 15.3.

PROOF OF LEMMA 11.2.

follow closely a letter of Murthy to Serre. is a field of characteristic not

non-degenerate quadratic form. and

N = y

Ax + Ay =

-» A

Here again all results are due to Murthy.

k

and

and hence is an isomorphism since

isomorphism and

A^. Also

localizes to isomorphisms over

A^

§16.

over

are finitely generated so we can

find a finitelygenerated M^,.

and

ds q = -abc.

Here

2

and q =

2 ax

I will

As in §11, q

is a + by

2

I

+ cz'

BUNDLES, MODULES AND K-THEORY

THEOREM 16.1 (MURTHY). Let V ds q € k

or if

F © Q

the form

q.

This conclusion

V ds q € k

F

free and

q

If

then every projective

with

In particular,

rkq =3.

Q

509

is isotropic

R(q)-module has

of rank 1.

de^K^fRfq))

Pic R(q)

fails however if

q

since then

SK^(R(q)) ^ 0

COROLLARY 16.2 (MURTHY).

If

for

such

represents 1

and

by 11.5.

k = Rand

r k q = 3,

the

following are equivalent. (1)

R(q) project lues have the form free © rank 1.

(2)

The tangent bundle

(3)

q

is isotropic or negative definite.

(1) => (2).

Proof.

to(q = 1} is trivial.

The tangent bundle is stably free since

it is represented by the unimodular row (2ax,2by,2cz). F © Q

stably free,

F

free.

(2)

Theorem 2.1. (3) =$(1) by Theorem

(3) by

free,

rk Q = 1

implies

Q

But

is

16.1. Towber [To] has shown that (2) holds over a global field of

F

if and only if it holds over all real completions

F. The proof of Theorem 16.1 is based on a version of

Seshadri’s Theorem. Euclidean) = En (R)

for

if

R

We say that

R

is

GE

(generalized

is a principal ideal domain and n - Assuming

R

SL^fR)

noetherian, this is

510

SWAN

equivalent to the statement that any row be reduced to the form

(d,0,...,0)

(a^f...,a )

can

by elementary

transf ormations.

THEOREM 16.3 (SESHADRI). noetherian.

A/p

such that A[p

Let

or all

projective

Let

be commutative and

be a set of invertible prime Ideals

$

GE

is

A

p € ^].

for all P

Let

A-module.

p € 3>. Let

be a finitely generated

B ®^P = free

If

B = A[# ^] =

© rank 1 then

P = free © rank 1.

For the proof see [B].

COROLLARY 16.4 (SESHADRI). projectives over

We let

R[t]

If

R Is a

Dedehind ring, all

are extended from

$ = (p[t]|p / 0}

R.

so B = K[t]

and use

Pic

R[t] = Pic R. We now turn to the proof of 16.1 for We can write

Case 1 .

q

isotropic.

2

q = uv + az .

V~sl € k.

In this case, assume p = (u,z-l).

Then

p*(u,z+l) = (u),

A/p p

is

a = = k[v]

1. Let

A = R(q) and let

is GE. Since

invertiblewith p *

= (l,(z+l)u *}

511

BUNDLES, MODULES AND K-THEORY

and

B = A[p *] = A[(l+z)u ^].

then

v = -t(z-l),

z = ut-1

If we let so

t = (z+l)u ^

B = k[t,u]

must be algebraically independent since B transcendence degree 2 free by

16.4,

Case 2.

V~a € k.

k'

invertible with

with

2

p*(v,az -1) =

p * = (l.v(az^-l)

A[p *] = A[u *] = k[u,u *,z]

Remark.

In case 1, K^fRfq)) = Pic R(q)

In Case

A/p = k'[v]

2

(az -1)

= TLgenerated

abc =

-1.

isotropic.

If a.b.c, € Therefore we

case choose

= (ax + yz)(x - a

-1

*2

-1 €

k

canassumethat

p = (ax + yz, cz

2

- 1).

2

yz) mod cz -1

2

—1 p = (l,t)

where

B = A[p *] = A[t]

so

p

that

is

\Tc € k. ax

In thi

2

so

2

2

soq

Note that

A/p = k[x,y,z]/(ax+yz,cz -1) = k(V~c)[y]. p*(ax-yz,cz -1) = (cz -1)

is

Here we can assume

*2 k then

Now

is invertible with

2 -1 t = (ax-yz)(cz -1)

by

See 9.2b.

Finally we consider the case where q V~ds q € k.

p i

and 16.3 applies.

2 all projectives are free.

non-isotropic and

so

= (1,-u *).

Therefore

p.

has

16.3

2 p = (u,az -1)

= k(V~a). Here

t.u

Since B-projectives are

16.1follows from

Here we consider where

over k.

and

and

(not a polynomial ring in

t).

+ by

2

512

SWAN Since

B = k[y,z,t] = k[u,z,t]

2 -1 u = y + ctz = ac(by+xz)(cz -1) . One checks easily

where that

2

ax = yz + t(cz -1),

u2 - ct2 = -ac. , 2 q' = av

where

sur jection

2 + bu

R[z]

B

Let with

R = k[u,t]/(u2-ct2+ac) = R(q') v =a

-1

t.

Then we have a

which must be an isomorphism since

both rings are domains of dimension 2. B-projectives have the form implies the same for

Remark.

By 16.4 all

free © rank 1

and 16.3

A.

This argument shows that Spec R(q)

is

birationally equivalent to a ruled surface over Spec R(q').

Something like 16.1

could have been antici­

pated from this and work of Murthy on ruled surfaces [Mu].

§17.

THE THEOREM OF MOHAN KUMAR AND NORI As indicated in §3, the theorem to be proved is the

following.

THEOREM 17.1 (MOHAN KUMAR-NORI) . Let

R

be any (non-zero)

commutative ring. Let A = R[x^,...,x ,y^,...,yn ]/(^x^yj-l). Let

m^,...,m

be non-negative integers with

mod (n-l) !. Then the unimodular row

m^m^.•

£

ml mn (x^ ,...,xr )

defines a non-free 'projective A-module.

By letting

k = R/M

with

M

maximal we see that it

is sufficient to consider the case where

R = k

is a

0

513

BUNDLES, MODULES AND K-THEORY

field.

The proof makes use of the Chow ring

A^S p e c B)

(see §13).

defined as the g;roup

A*(B) =

In the affine case this can be W^(B)

of [CF]

simple algebraic definition.

which has a very

However the proof makes

essential use of Theorem 13.2 for which, as yet, no simple algebraic proof is known.

If P C B i s a prime ideal of

height

isan irreducible subvariety of

i

then

codimension [Spec B/P] set

Spec B/P

i

in Spec B

of

A^B).

[B/P] = 0

in

module such that filtration B/P.

and so represents an element

I will write

A 1(B)

if ht P > i .

Ann(M)

has height

0 = M. C M. C 0 1

and set

[B/P]for this. If

M

We

is a

i, choose any

...C M = M n

with

[M] = ^ [B/P^.] € A*(B).

M ./M. 1 = j j-1

The existence of

such a filtration follows easily by noetherian induction and the fact that

Ass(M) / 0

if

M/0.

common refinement we see easily that In fact a filtration of quotient

B/P

B/Pand the rest

-» M' -> M M ” -» 0

implies

By choosing a

[M]

is well defined.

as above will have one

B/Q

with

Q > P.

Clearly

[M] = [M'] + [M"].

All

0

this

is, of course, trivial if one uses the definition in [CF]. The

proof of 17.1 makes use of the ring

k[xx ..... xn ,y i ... yn ,z]/( ^

LEMMA 17.2.

If

- z(l-z)).

n > 0, A°(B) = Z

An (B) = TL generated by

[B/I],

B=

generated by

I = (x^

[B],

Xn ,Z^

aru^

514

SWAN fo r

A 1(B) = 0

[B]

generated by

and

for

is defined like

A^.

n = 0

A. -.(B'), J t An (B)

i / 0.

for

Consider the

A.(B/x ) -»A.(B) -»A.(B ) -» 0. J n j j xn

j / 2n. J r B

A°(B) = TL x TL

A X(B) = 0

B = k[x-,...,x ,z,y-,...,y - ,x x L 1 n Jn-1 n n

A.(B ) = 0 jv x^y

for

n = 0.

If

This is clear for

localization sequence Since

n.

[B/I],

and

(cf. 13.3)

Proof.

or

1 * 0

for

Also n-1.

where

B'

Since the lemma is trivial n-1.

Since

A*(B) = 0 for

is cyclic generated by

we see that

B/x = B T y ] n LJn J

we can assume it for it follows that

H J

A^fB'fy^]) £

i / n or0

B/I = B ’/ I ’fy^]

and

where

I ' = (x.,...,x 1 ,z). Now by Theorem 10.2, v 1 n-1 J Kq(B) =

Z

©

Z

and therefore

Theorem 13.2 shows that

so

An (B) =

Z.

this proof works even if

Since

rk An (B) = 1

[B/xn ] = 0

in

n = 1.

i = 0;

COROLLARY 17.3.

F1KQ (B) = KQ (B)

fo r

F iKQ (B) = KQ (B)

fo r

F*KQ (B) = 0

and

A^(B),

1 < i < n;

for

i > n;

V>:A1(B) £ grXK0 (X) (see 13.2).

This follows from the fact that

z ) c B N

for

N

for

large since

515

BUNDLES, MODULES AND K-THEORY

•£

if

0,

iur W

M ,-

( 1- z ) ^

and

.M

= (

2

17 . 4 .

J

z^

M and

m1 ,m0 , . . . m

1 2

/J n

=

so

N

z^ € J

V

and a s im ila r

a re rev ersed .

n

indices m^ . - . - . m^ .

The left side is generated by

therefore isomorphic to

, ml

mod (x^ ...................)

,1 - B /J , m. +l , mn i . . . ,m l , m0 , . . . , m 1 2 n 2 n

and s i m i l a r l y f o r ea c h of the

Proof.

0

.M _ n

x i Y |)

i s a u n i t mod

argum ent a p p l i e s i f

LEMMA

f \

z ( 1- z )

A

M >>

B/K

x^ 1

and is

where

K = {t € B|tx^l e J }. 1 m n+1, m ~ ,...,m 1 2 n

Clearly

K 3 J, 1, m ~ 2 j •^ and write

. For the reverse inclusion let t € K m n m. m.,+ 1 m0 . , m ,, N tx1 1 = a ix 1 l + a0x0 2 + . . . + a x n+bz. 1 11 2 2 n n f \ m. m0 m . N (t-a.x.jx^ 1 = a0x0 2 + . . . + a x n + bz maps to v ll'l 2 2 nn

rrii

Then 0

in

C = B/(x0m2 v 2

( ^ x iy i-z(l-z), is a unit in

C

x mn,z^) = k[x1 ,...,x ,y1 n J L 1 n^l

x^m2,...,xnmn,z^). we can let

Since

u^ = y^(l-z) *

y ,z]/ ^n J

1-z and get

un -z]/( ^ x 1u 1-z,x2m2 ......... xnmn,zN ) =

C = k[xx

,f f X

. k[x, ,...,x ,u......u n]/(( / x.u.) , x0m02, ...,x mn) . L 1 n 1 n J vv L l 2 n J

1r

Let

D = k[x0 ,...,x ,u0 n z

ri = x0u~ + ' 22 wx^m l = 0

... + in

C

u ]/ (x0m2, ...,x mn) n z n

x u . Then then

But if

in

77

m0 2

D[Xj,u^]

since

m . Therefore n

R

N C = D[x1 ,u. ]/(x1u 1+T7) . n n L 1 1J v 1 1 "

in D[x^,u^]

wx^m l = v(x^u^+tj)^.

= 0

and If

we have

N >> 0, (x^u^+77)^ = x^m l ^s where

R

depends only on

w x ^ l = vsx1m l+ ^ 1 1

but

x1 1

is

516

SWAN

regular in hence in

D[x^,u^]

C.

so

w = x^sv

Applying this to

and

shows

t =

so

1,m ~ ,...,m 2 n

COROLLARY 17.5. m ^m2 * ..m

Proof.

B/J nn 1

has a filtration with

m n

in

By Lemma 17.4 we have a filtration with B/J^.m^,...,mn -

argument on each quotient using

LEMMA 17.6.

n

then

m^

Repeat this

m^, etc.

If there is an epimorphism

projective of rank

[B/J] =

Thus

An (B).

quotients each isomorphic to

Proof.

B/I.

quotients all isomorphic to

m^m^...m [B/I]

have

D[x^,u^]

w = t-a^x^

j (xr r tmo a.x.+svx- mod 2,...#x mn,zN>) 1 1 1 v 2 n J t € J

in

m^m^.•.m

Q

J

with

Q

= 0 mod (n-l)!.

By a standard property of Chern classes [GC] we cn (Q) = [B/J]

in

An (B). This just says that the

top Chern class of a bundle is the intersection of the zero section with a section meeting it properly. [Q] - [Bn ] € Kq (B). KQ (B) = F^qCB)

Then

by 17.3, we have

in

f =

c j f ) = c J Q ) = [B/J],

yp(g) = cn (Q) = [B/J]. Therefore m^m^...mn [B/I]

Let

An (B). But

f e grnK0 (B)

Since and

W>(£) = [B/J] = An (B) = Z

generated by

BUNDLES, MODULES AND K-THEORY

[ B /I ]

and

W '(f) = ( n - l ) ! f

m^m^. . .m

for

(x™! v 1

in

x mn) n J

N >> 0. B. . 1-z

) = ) a.x.m i J' L i i

(1) v j

0

M

Therefore

Let

f:B? 1-z

have kernel

B1?l - z -»J1 1-z

0

d iv id e s

17 . 3 .

N N z (1-z) €

As we observed above,

(x^l.-.-.x mn) v 1 n J

n

so ( n - 1 ) !

J, 1—z

M.

Since J

by J z

=B , z

localizes to

(2) 0 -» Mz -* Bj(1_z) -» Jz(z_1} -» 0 but

J,., > = B > z(l-z) z(l-z)

soM

is the stably free module

z

P(x^m l,...,xnmn)

defined by the unimodular row

(x^l v 1

over

x mn) n 1

free over mapg :A y\,z (1)

B .. z(l-z)

If

A, the same will be true over Bz (i_z )

with

B

> z(l-z)

is

via the

g(x i) = x i-g(y1) =

^(1-z) Therefore

Mz

will be free and we can patch

with (3) v J

0 -» Bn _1 -* Bn -» B

z

z

z

-» 0 .

Explicitly, we choose an isomorphism take

P(x1m l,...,xmn) v 1 n J

Mz ~ (Bz

J f, ^ ~ B . induced by J C B, z(l-z) z(l-z) J

fact that (2)and (3) split to extend

and use the

to an isomorphism

//

N N

------

1

i -e -

M z

I

(i)z * (3 )i_z-

)i_z »

//

(B11-1) , — — » (B n ) 1 v z'l-: v z yl-z By p a tc h in g we g e t

( J l - z > z -------- > 0

// (V l-z

-------- * ° -

518

SWAN 0-»X-»Q->J->0

where

Q

is projective of rank

n,

and 17.1

follows from

17.2. Further applications of this type of argument may be found in [MK1] and [MK2].

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[B]

H. Bass, Algebraic K-Theory, Benjamin, New York 1968.

[BH]

A. Borel et A. Haefliger, La classe d ’homologie fondamentale d ’un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513.

[BR]

S. M. Bhatwadekar and R. A. Rao, On a question of Quillen, Trans. Amer. Math. Soc. 279 (1983), 801-810.

[BSL]

A. Borel et J-P. Serre, Groupes de Lie et puissances reduits de Steenrod, Amer. J. Math. 75 (1953), 409-448.

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N. Bourbaki, Commutative Algebra, Addison Wesley, Reading, MA 1972.

[Br]

J. P. Brennan, An algebraic periodicity theorem for spheres, Proc. Amer. Math. Soc. 90 (1984), 215-218.

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M. Carral, Modules projectifs sur les anneaux de fonctions, J. Algebra 87 (1984), 202-212.

[CF]

L. Claborn and R. Fossum, Generalizations of the notion of class group, 111. J. Math. 12 (1968), 228-253.

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[FAC]

J.-P. Serre, Faisceaux algebriques coherents, Ann. of Math. 61 (1955), 197-278.

[For]

0. Forster, Uber die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring, Math. Z. 84 (1964), 80-87.

[Fos]

R. Fossum, Vector bundles over spheres are algebraic, Invt. Math. 8 (1969), 222-225.

[Fr]

E. Friedlander, Etale K-theory II, Ann. Sci. Ec. Norm. Sup. 15 (1 982 ), 231-256.

[GC]

A. Grothendieck, La theorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137-154.

[GR]

A. V. Geramita and L. Roberts, Algebraic vector bundles on projective space, Invent. Math. 10 (1970), 298-304.

[GQ]

D. Grayson, Higher algebraic K-theory II (after D. Quillen), pp. 217-240 in Algebraic K-Theory, Lect. Notes in Math. 551 Springer-Verlag, Berlin 1976.

[H]

R. C. Heitmann, Generating ideals in Priifer domains, Pacific J. Math. 62 (1976), 117-126.

[Hi]

H. Hironaka, Smoothing of algebraic cycles of small dimension, Amer. J. Math. 90 (1968), 1-54.

[HP]

W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry II, Cambridge 1952.

[Ja]

N. Jacobson, Basic Algebra II, W. H. Freeman, San Francisco 1980.

[JC]

J. P. Jouanolou, Comparaison des K-theories algebriques et topologiques de quelques varietes algebriques, Strasbourg 1971 and C. R. Acad. Sci. Paris 272 (1971), 1373-1375.

[JK]

J. P. Jouanolou, Quelques calculs en K-theorie des schemas, in Algebraic K-Theory I, Lect. Notes in Math, 341,, Springer-Verlag, Berlin 1973.

[JRR]

J. P. Jouanolou, Riemann-Roch sans denominateurs, Invent. Math. 11 (1970), 15-26.

[K]

M. Kong, Euler classes of inner product modules, J. Algebra 49 (1977), 276-303.

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[Lam]

T. Y. Lam, Algebraic Theory of Quadratic Forms, Benjamin, Reading, MA 1973.

[Lie]

S. Lichtenbaum, in S. M. Gersten, Problems about higher K-functors, pp. 43-56 in Algebraic K-theory I, Lect. Notes in Math, 341, Springer-Verlag, Berlin 1973.

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H. Lindel, On a question of Bass, Quillen, and Suslin concerning projective modules over polynomial rings, Invent. Math. 65 (1981), 319-323.

[Lo]

K. Lonsted, Vector bundles over finite CW-complexes are algebraic, Proc. Amer. Math. Soc. 38 (1973), 27-31.

[Ma]

R. Marlin, Anneaux de Grothendieck des varietes de drapeaux, Bull. Soc. Math. France 104 (1976), 337-348.

[MeS]

A. S. Merkur’ev and A.A. Suslin, K-cohomology of the Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk. SSSR ser. mat. 46 (1982), 1011-1061 (=Math. USSR Izv. 21 (1983).

[Mi]

H. Miller, The Sullivan conjecture, Bull. Amer. Math. Soc. 9 (1983), 75-78.

[MK1]

N. Mohan Kumar, Some theorems on generation of ideals in affine algebras, to appear.

[MK2]

N. Mohan Kumar, Stably free modules, to appear.

[Mu]

M. Pavaman Murthy, Vector bundles over affine surfaces birationally equivalent to ruled surfaces, Ann. of Math. 89 (1969), 242-253.

[MS]

M. Pavaman Murthy and R. G. Swan, Vector bundles over affine surfaces, Invent. Math. 36 (1976), 125-165.

[0]

H. Ozeki, Chern classes of projective modules, Nagoya Math. J. 23 (1963), 121-152.

[P]

B. Plumstead, The conjectures of Eisenbud and Evans, Amer. J. Math. 105 (1983), 1417-1433.

[QK]

D. Quillen, Higher algebraic K-theory I, pp. 85-147 in Algebraic K-Theory I, Lect. Notes in Math. 341, Springer-Verlag, Berlin 1973.

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521

[QP]

D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167-171.

[Ka]

M. Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1-26.

[RoB]

L. G. Roberts, Base change for

Kq of algebraic

varieties, pp. 122-134 in Algebraic K-Theory II, Lect. Notes in Math. 342, Springer-Verlag, Berlin 1973. [RoC]

L. G. Roberts,

of a curve of genus zero, Trans.

Amer. Math. Soc. 188 (1974), 319-326. [RoK]

L. Roberts, K-theory of some reducible affine varieties, J. Alg. 35 (1975), 516-527.

[RoR]

L. G. Roberts, Comparison of algebraic and topological K-theory, in Algebraic K-Theory II, Lect. Notes in Math, 342, Springer-Verlag, Berlin 1973.

[Sa]

P. Samuel, Sur les anneaux factoriels, Bull. Soc. Math. France 89 (1961), 155-173.

[SD]

J-P. Serre, Modules projectifs et espaces fibres a fibre vectorielle, Sem. Dubreil-Pisot 11, Paris 1957/58.

[SGA6] A. Grothendieck, P. Berthelot, L. Illusie, et a l ., Theorie des intersections et theoreme de Riemann-Roch (SGA6), Lect. Notes in Math. 225, Springer-Verlag, Berlin 1971. [Su]

A. Suslin, Stably free modules, Mat. Sb. 102 (1977), 537-550.

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D. Sullivan, Geometric Topology, Part I: Localization, periodicity and Galois symmetry, MIT, Cambridge, MA 1970.

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R. G. Swan and J. Towber, A class of projective modules which are nearly free, J. Algebra 36 (1975), 427-434.

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J. Towber, Tangent bundles to affine quadric surfaces over local and global fields, Comm. Algebra 7 (1979), 525-532.

Richard G. Swan University of Chicago Chicago, IL

XIX LIMITS OF INFINITESIMAL GROUP COHOMOLOGY Eric M. Friedlander and Brian J. Parshall^

Let

G

be a connected, reductive algebraic group over

an algebraically closed field We assume that providing r > 0,

G

G

k

of characteristic

is defined and split over

with a Frobenius morphism

the scheme-theoretic kernel

G

o'-G

r

of

p > 0.

F , P G.

For any

i» o' G

G

is

an "infinitesimal group scheme" with coordinate ring k[G ].

A rational G-module is by definition a comodule

for the coordinate ring

k[G]

of

G.

V

Because the category

of rational modules contains enough injectives, the rational cohomology groups usual way. H (G^.V)

H (G,V)

can be defined in the

Similar remarks apply to

G^,

and in this case

is canonically isomorphic to the cohomology

H*(k[Gr]**,V)

of the dual algebra

kfG^]**.

(For readers

familiar with the foundational paper of Eilenberg-Moore [7], these cohomology groups identify with the cotor groups Cotork[G

^Both authors’ research supported by the National Science Foundation.

523

524

FRIEDLANDER AND PARSHALL

In this paper we investigate the system r = 1,2,...}

for rational G-modules

V.

{H (G^.V),

In Theorem 2.1,

we prove (subject to a restriction on the size of

p) that

the restriction map in cohomology H*(G,V) -» lim H*(G ,V) «r is an isomorphism for any finite dimensional rational G-module

V.

This result is a consequence of our basic

stability theorem (Theorem 1.3) asserting that the (inductive) system

(Hn (Gr ,k), r = 1,2,...}

stable value for each

n > 0.

achieves a

Sections 1 and 2 also

contain various consequences of these theorems, thereby completing results obtained in [2]. The reader seeking motivation for the study of rational cohomology should consider the following theorem of Cline, Parshall, Scott, and van der Kallen [5]: as above, n > 0 and

V

for

a finite dimensional rational G-module, and

a cohomological degree, there exist integers

R > 0

G

such that for any

d > D

and

r > R

D > 0

the

restriction map Hn ( G , V ^ ) -> Hn (G(Fpd) . V ^ ) = Hn (G(Fpd),V) is an isomorphism.

In this isomorphism,

H (G(Fpd),V)

denotes the Eilenberg-MacLane cohomology of the finite Chevalley group ^ rr 1 H (G,Vv J)

G(F^d)

of

F^d-rational points of

denotes the rational cohomology of

G

G with

and

LIMITS OF INFINITESIMAL GROUP COHOMOLOGY (r ) Vv J

coefficients in the G-module

obtained from

composing the given representation with

p

o .

value is called the ’’generic cohomology” of coefficients in of

{H (G^.k),

V.

525 V

by

This common G

with

As discussed below, the stable value

r = 1,2,...}

appears to play some

universal role in the computation of the generic cohomology. In a future paper,

2

we plan to complement the

qualitative results of this paper with numerous explicit computations of cohomology of discrete groups.

§1.

THE STABLE COHOMOLOGY RING We adopt the following hypotheses and conventions for

the duration of this paper.

G

denotes a simply connected,

reductive algebmic group over an algebraically closed field G

k

of characteristic

p > 0.

(The hypothesis that

be simply connected is merely one of convenience— cf.

[5;2.7]).

We assume

G

with Frobenius morphism torus

T

is defined and split over cr:G

and a Borel subgroup

denote the root system of lattice of

G.

T

in

0 IR,

T

We fix a maximal split

B

containing in

A+ C A

weights determined by the choice of

2

F^

G,

A 3 $

T.

We let

the weight

the set of dominant B.

As usual,

B

Cohomology of infinitesimal and discrete groups, Math. Ann. 273 (1986), 353-374.

526

FRIEDLANDER AND PARSHALL

determines a setof simple roots ordering on the root

A.

Let

a e $,

each simple

{a..}

and a partial

aV denote the coroot associated to

and for r > 1 A* = (A e A+ | < pr ,

ct..},

where

< , > is the usual pairing.

For a dominant weight

A,

we denote by

-A|

the

rational G-module obtained by inducing the one-dimensional rational B-module defined by -A|

-A

from

B

to

G.

Then

is an indecomposable rational G-module of highest

weight

= -w q (A) , where

A

Weyl group of showing

G.

-A|

is the long word in the

wq

This is easily checked directly by

has an irreducible socle (which has high

^ weight

G

A ).

It is well-known that the dimension of

is given by the classical Weyl dimension formula.

-A|

The

properties of these induced modules which we require are discussed in more detail in [2;§§2,5] and [5;§3]. We shall use below the so-called Steinberg modules St^

defined for

r>0.

If

p

denotes the weight defined

as one-half the sum of the positive roots of = -(p

r

- i)pi

G

$,

then

St^

is an irreducible, self-dual G-module whose

restriction to

G

is an injective

G -module (cf. [4;§6],

[ii])-

LEMMA 1.1. injection

For any -A|

G

V = (pr - 1)p - A-

r > 1

and any

^ G St^ ® -p |

A e A*,

there is an

of rational G-modnles, where

LIMITS OF INFINITESIMAL GROUP COHOMOLOGY The natural B-module homomorphism

Proof'-X -» -(p to

527

G

r

^ iQ - l)p ® —jlx I

determines by induction from

the asserted injection.

An increasing filtration G--module

V

B

§§

{Fg , s > 0}

of a rational

is said to be a good filtration if each Q is G-isomorphic to some -X| . Implicit

section

in what follows is our use of recent work of Donkin [6] sharpening earlier work of Wang [13]:

the tensor product

of two rational G-modules admitting good filtrations also admits a good filtration provided that G does not a simple factor of type

or

Eg

A basic property of modules

V

filtration is that

E^

Hn (G,V) = 0

Because of the identification follows that if in addition vector in

G

V, then no section

for

when

contain

p = 2.

having a good n > 0

[5;3.4].

1 1 H (GfW) = Ext^(k,W),

it

has no non-zero fixed Fg/Fs_i of

its good

filtration is isomorphic to the trivial module k.

We

make

use of this result in the proof of the following:

LEMMA 1.2. type

Erf

Assume that or

Eg

when

non-zero dominant weight

G

does not contain a factor of

p = 2.

For any

n > 0

X,

Hn (Gr ,-A|G ) = 0

for

r »

0.

and any

528

FRIEDLANDER AND PARSHALL

Proof' For

Choose

r > r'

r'

sufficiently large so that

A a A*, .

consider the Lyndon-HochschiId-Serre

spectral

sequence [2;4.5] E ® >1: = H S(Gr/Gr ,,Ht(Gr .,Str ,®-fxX |G ))^eS+t(Gr ,Str .®-^|G ) where (pr

-X|

G

^ G St^, ® -p |

- l)p*

Because

G^ ,-injectives,

St^,

is as in Lemma 1.1 with

\i £

and hence

are

we conclude that

St^, ®

H (G^St^, 0 -p | ) =

H^fG^/G^,,(St^, ® -p*|^)^r'). Because dim(Str#)> Q dim(-p | ) by an easy Weyl dimension formula calculation, we have that Thus,

(St^. ® -p*|^)^r' = Hom^

(Str ,,-p^|^) = 0. r'

H*(Gr>Str , ® -p*|^) = 0. By remarks above,

^ G St^, ® -p |

* G Q Q = (St^, ® -p | )/(—X| )

and hence

admit good filtrations in which

no section is isomorphic to

k.

We may thus use a

dimension shifting argument to reduce the proof of the lemma to the case X s* 0,

n = 0.

0 G H (G,-A| ) = 0

Because

this follows from [2;6.3.1].

In [9] the rational G-algebra

(

H (G^,k)

determined subject to a condition on to

x

p > h,

the Coxeter number of

xf11

H (G^,k) = A v

, where

A

X

for

G,

p

was

which was improved in [1].

Namely,

is the coordinate ring of the

variety of nilpotent elements in the

affine space

corresponding to the Lie algebra of

G.

Wenow proceed to

investigate the inductive system of rational G-algebras

LIMITS OF INFINITESIMAL GROUP COHOMOLOGY {H (Gr>k),

r = 1,2,...}

quotient maps

define

whose maps are determined by the

G^ -> G^/G^ =

using the fact that

Gr

529

(cf. [7;3.6]).

acts trivially on

Namely,

H (G^.k),

to be the rational G-algebra satisfying H*(G ,k) = H * ^ v r J r

so that

= A . Then we can consider the rational

G--algebra H* = lim H*. r r The following theorem justifies our referring to as the stable cohomology ring of

THEOREM 1.3.

Assume that

p > h.

H

G.

Then for any

n > 0,

we

have an isomorphism Hn ^ Hn r

Proof'

For

for

r »

0.

r > 1, we consider the Lyndon-Hochschild-Serre

spectral sequence [2;4.5] E®'t = H s(Gr/G1,Ht(G1>k)) =* H s+t(Gr ,k). It suffices to prove that for

t>0,

s + t < n,

H S(Gr/G^,Ht(G^,k)) = 0 r>>n.

(which improves the bound on A = H^

p

admits a good filtration.

character computation of

A

By [1;3.7, 4.4, 4.6] required in [9;2.4]), Moreover, the formal

given in [9;2.5] guarantees

that the trivial module does not appear as a section of a

530

FRIEDLANDER AND PARSHALL

good filtration of

A^

for

t > 0.

(Alternatively, this

follows from the identification [9;2.6] of

A

coordinate ring of the nulleone

G,

together with

A

plus the remarks

the fact that

G

Because we may identify

H^(Gr/Gr H^(Gr k))

Q

with

H*(G

j.A*), the required

H S(Gr/G^.H^G^,k)) follows from Lemma 1.2.

(a)

Remarks (l.k)' (one takes

of

has a dense orbit in

preceding (1.2).)

vanishing of

A

with the

By [3; 1.4] applied to each

in that proposition to be

conclude that

H

|

G^ 0.

A

careful inspection of the proof of (1.2) would enable one to give an explicit bound that

-» Hn (b)

For

restriction on

R > 0

depending on

n

such

is surjective for all r > R. G p

not necessarily split over

the

imposed in Theorem 1.3 implies that the

Frobenius morphism

G

a ’G

is a graph automorphism and for the split form on

G.

has the form a'

H i(G,H^ ® V) induced by the inclusion guarantees for every that

E ^ ’^ -» ’E * ’^

j i J

r

J > 0

s

.

Since Theorem 1.3 R > 0

such

is an isomorphism for all (i,j)

with

provided that

the exitence of

s > r £ R, the corollary now follows

easily (cf. [5; p. 152]).

§2.



H

PROJECTIVE SYSTEMS OF COHOMOLOGY GROUPS The relationship between the rational cohomology of

and that of its infinitesimal subgroups studied by J. Sullivan [12]. that the natural map

G^

was first

Among other things, he proved

Hn (G,V) -» lim Hn (Gr>V)

induced by

r the restriction maps is injective when finite dimensional rational G-module. established for all

n

shown there to hold for imposed on

G

in [2;§7] n < 2.

n = 1

for

V

a

Injectivity was

and surjectivity was Subject to a restriction

p, we answer below a question raised in

[2;p.ll3] by proving surjectivity for all

n.

534

FRIEDLANDER AND PARSHALL

THEOREM 2.1.^

G

Let

and

p

V

be as in (1.3) and let

be a finite dimensional rational G-module.

Then the

restriction maps induce an isomorphism H*(G,V) ^ lim H*(G ,V). «r Proof: By [2;6.3.1],

H°(G,V) = lim H°(G ,V). «-

non-zero dominant weight G-acyclicity of

-A|

G

A,

Lemma 1.2

We next show that for each

exists an

r > s

V = k.

and the

[5;3.4] imply the theorem for

G V = -A| .

Hn (Gs ,k)

For a

n > 0,

s > 0

such that the restriction map

there

Hn (Gr ,k)

is the zero map, thereby proving the theorem for Namely, using Theorem for

t > R

1.3 choose

and let r = R + s.

R > 0

suchthat

Because the

composite Hn (Gr/Gg ,k) ->Hn (Gr ,k) ->Hn (Gs ,k) is the zero map and because r

th ^ ^ twist

TTn(r) TTnfr) J -» Hrv J

Hn (Gr/Gg ,k)

Hn (Gr>k)

P . ,. of the isomorphism

conclude that the restriction map

TTn

is the TTn

Hn (Gr>k) -» Hn (Gg ,k)

, we is

also the zero map. Because 0,

Hn (Gr ,V)

is finite dimensional for any

r > 0, and finite dimensional rational G-module

n >

V, the

3 Since circulating this paper in preprint form, W. van der Kallenhas discovered an elegant proof of Theorem 2.1 with no restriction on p. Namely, van der Kallen argues that the finite dimensionality of relevant cohomology vector spaces guarantees the existence of an inverse limit "Hochschild-Serre" spectral sequence converging to H (G,M) 0 ^ with E^-term collapsing to li.m H (G/Gg ,H (Gg ,M)).

535

LIMITS OF INFINITESIMAL GROUP COHOMOLOGY exactness of

lim ( ) 0,

V

W

admitting a good

H^G.W) = lim Hn (G ,W) = 0 lim Hn_1(G ,W/V) -» lim Hn (G ,V) -» 0

r > 0.

H^G.V)

-> 0

Hn (G,V) -» lim Hn (G ,V) k) & 0

we have

H^(G,k) = 0

2

2

so that the restriction map

H (G,k)

isomorphism for any

Nevertheless, we obtain the

r > 0.

H (G^.k)

[5;3.5], is not an

stability criterion below which is based on the fact, proved in [2 ;7.5], that

H 1 (G,V) -» H ^ G . V )

dimensional and all sufficiently large and

G

has a simple factor of type

here that

H°(G,V) = 0.)

r.

for (When

V

finite p = 2

one must assume

536

FRIEDLANDER AND PARSHALL

PROPOSITION 2.2.

Let

G

defined and split over

be a reductive algebraic group F , P

has a simple factor of type

and E^

assumethat or

Eg.

Let

V

0 < i < n-2,

while if

factor of type n-1.

p = 2

H^G.V)

assume that

is an isomorphism for all

Proof:

= 0 for 0

j < n

provided

n = 0

r >> 0.

by [2;6.3.1], while if

follows from [2;7.5] as mentioned above. induction on

n > 2.

filtration.

W

k

W

fPfG^.V) = Hn ^(Gr ,W/V)

W.

Lemma

for

r >>

Hn (G,W) = Hn_1(G,W/V) by [5;3.4], so that

H 1(G,W/V) = 0 G

Since

to assume that the trivial module does not

1.2 then implies that

and

occurs

times as a section in the good filtration of

appear as a section in the good filtration of

2

as a

by hypothesis, we may by [9;2.1c] replace

W/H^(G,W)

Also,

V

having a good

(cf. also the remarks preceding Lemma 1.2).

H^(G,V) = 0

it

We argue by

By [5;3.4], the trivial module

dim(H^(G,W))

j = n.

n = 1

Using [9;3.4], we embed

G-submodule of a rational G-module

0.

< i
Y X

act on are

Y

so that

f

and the

G-equivariant. Then

£

induces a homotopy equivalence (4.1)

Proof:

f*: G(G,X)

The idea is to immerse

» G(G,Y)

Y

as an open in a

projective space bundle with a codimension one projective space bundle as complement, and then appeal to 3.1 and 2.7.

ALGEBRAIC K-THEORY OF GROUP SCHEME ACTIONS Locally the pair

(V,Y)

r V = © G , Y = V a

pair

a' V -+ V

is isomorphic to the trivial

under translation.

is a local automorphism of

a local automorphism of action of

V

on

determined by

a

Y

V

If

and

compatible with

Y, then

551

a € GL , r j3: Y -> Y

a

is

under the

P(v) = P(v+0) = a(v) + P(0)

up to a translation by

is

P(0) - 0 € V.

Thus the local automorphism group of the trivialized pair (V,Y)

is the affine group

(4.2) v J

Aff

r

= GL

r

k

r © G

a

This group is a subgroup of (4.3) Affr =

' €Lr I///I .00...0 1 1 J -

The global pair

f//// //}

\////\//] _„

[ 000 / / J - [ / / / | / / J

(V,Y)



r+1

is then determined by local

trivializations and a cocycle of transition functions with values in

Aff^.

The G-action is given under local

trivializations as a set of homomorphisms compatible with the cocyle. taking values in with a G-action.

G -+ Aff^,

Regarding the cocycle as

,one obtains a vector bundle As the cocyle reduces to

Aff^,

fff

there is

a natural short exact sequence of G-vector bundles (4.4)

0 -- ------------------->0 The vector bundle

1/

is assembled by the same cocycle

as the vector bundle space sections of

V.

The subsheaf Y,

z

V

hence

H

This is the dual sheaf of -1

(1) of

and the action of

action of

V,

on Y.

U

W on

is the sheaf of so

y f = 8 .

is the sheaf of sections of z “^(1)

corresponds to the

552

THOMASON Consider the dual sequence of G-vector bundles to

(4.4) V

(4.5)

0