This book contains accounts of talks held at a symposium in honor of John C. Moore in October 1983 at Princeton Universi
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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 BROWNGITLER 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 HSPACE
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 KTHEORY OF SPACES, CONCORDANCE AND STABLE HOMOTOPY THEORY
XVII THE MAP BSG → A(*) → QS^0
XVIII VECTOR BUNDLES, PROJECTIVE MODULES AND THE KTHEORY OF SPHERES
XIX LIMITS OF INFINITESMAL GROUP COHOMOLOGY
XX ALGEBRAIC KTHEORY OF GROUP SCHEME ACTIONS
Annals of M athematics Studies Number 113
ALGEBRAIC TOPOLOGY AND ALGEBRAIC KTHEORY
E D I T E D BY
WILLIAM BROWDER
Proceedings o f a conference O ctober 2428, 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 acidfree 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
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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 exstudents, 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 Wuyi 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 HSPACE 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 KTHEORY 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 KTHEORY OF SPHERES by Richard G. Swan
432
LIMITS OF INFINITESMAL GROUP COHOMOLOGY by Eric M. Friedlander and Brian J. Parshall
523
ALGEBRAIC KTHEORY OF GROUP SCHEME ACTIONS by R. W. Thomason
539
PREFACE From October 24 to October 28, 1983, a conference entitled 'Algebraic Topology and Algebraic Ktheory’ 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 Ktheory, 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 Ktheory at Princeton University during the 198384 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 KTheory
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 2primary 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
pprimary 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 pprimary 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. coH
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
qsphere
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 2nfold loop space of the localized sphere or else equivalently first take the
2n+l
connective cover of the
localized sphere and then take the 2nfold loop space of the connective cover.
The order of the identity of the
group so obtained is
pn . This implies the stated result
on
pprimary 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 Ctheory [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 EileribergMacLane 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
mcell to the (ml)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 mcell 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 rth 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 jfold 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 visavis coproducts, the lemma follows.^
LEMMA 1.6
If
k:
QX
Z
is a map, then
homotopy inverse if and only if
2k:
2QX
k
has a left 2Z
has a left
homotopy inverse.
Proof: We cofibration.
may
suppose without loss thatk
Then so also is 2k.
isa
15
EXPONENTS IN HOMOTOPY THEORY Let
tr*.
2Z
C = 2Z/2QX
has a left homotopy inverse n/
orvTT 2Z  > 2QX ^ C
evaluation map adjoint
Z
be the natural map. g, then the composite
is a homotopy equivalence.
2QX
QX
X
If
extends to a map
k
2Z » 2Z
Hence the
2Z
X
is a left homotopy inverse for
whose k.
The opposite implication being clear, the lemma follows.
f§
PROPOSITION 1.7. connected QY
mod p
If r
X
and
f
are bouquets of simply p
Moore spaces with
is a map which induces a
then,
Y
mod p
r
^ 2,
and
f'
QX »
homology monomorphism,
has a left homotopy inuerse.
The proposition above follows at once from the preceding lemmas.
PROPOSITION 1.8. nnr r\ QP(n,p )
There is a commutative diagram ™2n+l f r, »QE {p }
,r2n+l, r. , » V {p }
i=
i
i
nnr r\ OP(n,p )
v or»2 n+l, r. > QP (p )
^ n + l r T, > T {p }
^ *
r^2n+l f r, »OS {p }
=
rto2n+lr r. > OS {p }
such that the rows are split fibration sequences, and the inclusion map of the right hand column is equivalent with the natural map
V^n+^{pr} *T^n+*{pr}.
16
COHEN, MOORE, AND NEISENDORFER
Proof:
There is a commutative diagram with rows and
columns fibration sequences y2 n+i{pr} ,
T^n+l f r, , T {P }
nr T\* P(n,p )
i2n+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 nfold
the nfold
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 nth 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
nth
A
power
space, and
map, then fork
a field
is the nth 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 nth 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 mth 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 rth
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)AxP ^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
rki 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
rth Bockstein differential
v = 2n,
specified by
and the
i* 0 v = u.
with degree
u = 2nl,
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 (21) 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 3fold loop
space of the 3connected 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
^gpl^^
ir2 p  l ^ ^ >
(with image generated by
^pl
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 pth JamesHopf
h^: fiP^(p) » fl2(P^(p)^)
® ... 0 v, ptimes.
Since
satisfies
Pm (p) ^ En (p)
has 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), 121168.
[2]
The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. 110 (1979), 549565.
[3]
B. Gray, On the sphere of origin of infinite families in the homotopy groups of spheres, Topology 8 (1969), 219232.
[4]
I. M. James, On the suspension sequence, Ann. of Math. 65 (1957), 74107.
[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), 752780.
[8]
J. A. Neisendorfer, 3primary exponents, Math. Proc. Camb. Phil. Soc. 90 (1981), 6383.
[9]
J. A. Neisendorfer, Properties of certain Hspaces, Quart. J. Math. Oxford (2), 34 (1983), 201209.
[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), 407412. [12] JP. Serre, Groupes d ’homotopie et classes de groupes abeliens, Ann. of Math. 54 (1951), 425505. 2 [13] H. Toda, On the double suspension E , J. Inst. Polytech. Osaka City Univ. Ser. A. 7 (1956), 103145.
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 1tonMilnor 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 1tonMilnor 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 Hspace 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 semisplitting 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 rL
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
prth power [7] and the
C(n) has a null homotopic pth 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 coHspace
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 tth
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 semisplitting 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"11
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 coH 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
JP(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 Hspace 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»kl, 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 coHspace.
has additive exponent
j*
««
p . H
has the
But, if
50
NEISENDORFER Restricting the map
C(n) x Ur+^
T^n+^{pr}
main diagram (c.f. section 1 ) gives a map T^n+l{pr
of the
t• *
By Corollary 2.5, there are homotopy
equivalences
aT+ \ : Z(n,p
r+i
)
217r+^
where
Z(n,p
r+1
)
is
p
a bouquet of
mod p
PROPOSITION 4.4. into
Moore spaces.
Hence, 3.1 yields:
(2^t)(2ar+^)
The composition
factors
Ij: 2Z(n,pr+1) » 2Z(n,pr) V 22 Z(n,pr) » 22 T2 n+ 1 {pr} .
PROPOSITION 4.5.
There is a bouquet
X
spaces and a map
f : X » 2^T^n+*{pr}
such that adding
and
— i
gives a homotopy equivalence
of
mod pr
Moore f
r
g: X V 2Z(n,p ) V
JZrj, r. ^2 ^ 20 + 1 , r, 2 Z(n,p } » 2 T {p }.
Proof: By the proof of [4, Lemma 1.4], it is sufficient to show that 8.3,
i
the map
induces a i
mod p
induces a
With differential equal to
homology monomorphism.
mod p P
fr)
,
By
homology monomorphism. the following lemma
proves 4.5.
LEMMA 4.6.
Let
complexes with
C — — » C' — ^— » C' ' ji
monic.
isomorphic to the injection then
j
is an injection.
If C
bemaps of chain
d(C) = 0 and C
11
i
2C where
is d2x = x
51
THE EXPONENT OF A MOORE SPACE It is trivial.
Proof'
We can desuspend 4.5 in the following weak sense.
COROLLARY 4.7.
Y
There exists a bouquet 2Y = X
Moore spaces such that
mod pF
of
and a homotopy equivalence
h: Y V Z(n,pr) V 2Z(n,pr) » 2T2 n+ 1 {pr} such that
2h
For a bouquet of
Proof:
mod p
Hurewicz map is surjective.
oo
Z(n,p
Z(n,pr) =
V
VP
r (p )
)
with 9
Let
2T
n^ < n ^+ ^* n. {p1*} > P X(pr)
1
be the composition of
projection on these summands of p^
h
—1
with the
r r Z(n,p ) V 2Z(n,p ).
Let
be the composition r^ir r+ 1 ^ P (p )
rjf r+1 ^ ar+l w Z(n,p )  »
COROLLARY 4.8. of the map (b) homo topic.
(a)
If
i < j,
2l t +1
The composition
pr : P ^(pr+^)
r
T *+ l
(p
1=1
n.+l i
mod p
n
V P i=l
n. P 1 (pr).
homology map as
Moore spaces, the
i
) = 00
Then
t
J*
f
r*+ 1
Write
mod p
induces the same
X^P^
*
^ 2 n+l r r^ {p
ts the cofibre
P *(pr+*)
then the composition
XjP^
null
52
NEISENDORFER If
Proof: Pm (pr+*)
m < n, to
n
> 4,
and
Pn (pr),
then
f
and
g
are maps from
is homotopic to g
only if they induce the same
§5.
f
mod pr
if and
homology maps.
f§
CONSTRUCTING THE SEMISPLITTING 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 n1 , 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 i1
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, ll
a'
is the map which is on the second and
f§
IDENTIFYING A FIBRE Let
0: T^n+^{pr}
section 5.
II
be the map constructed in
To prove 1.1, we must show that the composition
0 c:
^as homotopy theoretic fibre Let
ni
1 [a'])
on the first summand and
third. §6 .
Here,
F
— * ffr+l
11^.
be the homotopy theoretic fibre of Ur
0t.
If
is the canonical fibration sequence,
56
NEISENDORFER
then 5.2(iii) implies that
f
factors through
F.
Hence,
we get the vertical maps of the horizontal fibration sequences below.
MI
Qrf
JL
17
MI
r+1
n .1 — 2 r+1
1 0c
JimiUnB Q27 r+1 r
We need to know that
IT
r+1
is a homotopy equivalence.
p
By the five lemma, it is sufficient to check that
is a
a
homotopy equivalence. The
mod p
product of the
homology of mod p
027^
is the infinite tensor
homologies of
The fibration sequence
OS
.
is an
commutative.
k > 1.
0 ^ S ^ +* + Q S ^ + ^{pt} + QS"^+ *
is clearly totally nonhomologous to zero 027?+l r t. S {p }
k 2pn X t {p },
T T rry Hspace [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 £pJ2 .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
221
"
.
= b
2£pxl
’" ' W  i
.
2£p12
,
i > 1
= ?*hzepz = Pc2 « = °
P^b i 0 = fb ... ] , x 2 fip  2 I 25 p il_2J
Notice that 5.2(i) implies that a2£l
k ^ = P n“l*
^°r
hold when a 2 ^_i•
QU
57
k > 1.
replaces
i >
fl(0 i)j) 1, then the
commutative diagram 22+1
p
22+1
'
r 1

22+1
22+1
yields the map of fibration sequences no2 fi+l, rx OS {p }
02 S25+1
_ nc2fi+1 » OS Op
no2^+ 1{p} r i OS
02 S2* +1
Hence, it is impossible if Thus, §7.
and
a^
r 1
2 2+1
 ,
» OS
r > 1.
are isomorphisms.
f§
APPENDIX ON CUBICAL DIAGRAMS OF FIBRATION SEQUENCES In order to avoid drawing high dimensional diagrams,
we introduce some convenient language. A partially ordered set can be regarded as a category in which the objects are the elements of the set. and
b
are elements with
regarded as a map from
a
If
a
a > b, then this relation is to
b.
60
NEISENDORFER If
C
and
D
are two partially ordered sets, then so
is the cartesian product and only if
c > c'
C
and
x D
where
(c,d) > (c',d')
if
d > d'.
We are especially interested in the partially ordered [£] = {0 ,1 ,...,21,2}
sets powers
and their nth cartesian
[£]n = [£] x ... x [£].
DEFINITION 7.1.
Anndimensional 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
kdimensional section of of
F
(x. d. . Jnk
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 ndimensional 1.
cube with side length of
F
We now define an extension
to an ndimensional cube with side length
Given any point
(a^.... a^)
in
[2]n ,
F
2.
let
d. ,...,d. J1 Jnk
denote the coordinates which are either 0 or
1.
s(a^
F
Denote by
a^)
the kdimensional 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 Jnk
= d. Jnk 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 kdimensional 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 (mk)dimensional section of
which passes through w
fsiH.
iH » isiH.
over the noninitial vertices of
m
is
be the kdimensional 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 ndimensional 1
cube of side length
is equivalent to a totally fibred
cube.
Proof: Let If (a^
F a )
be an ndimensional 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 41 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 lefthand 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 nl
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, SpringerVerlag, 1976.
[2]
F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, Torsiox in homotopy groups, Ann. of Math. 109 (1979), 121168. 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), 549565.
[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, 3primary exponents, Math. Proc. Caml Phil. Soc. 90 (1981), 6383.
[7]
J. A. Neisendorfer, Properties of certain Hspaces, Quarl J. Math, Oxford (2), 34 (1983), 201209.
[8 ] P. S. Selick, A reformulation of the Arf invariant one me p problem and applications to atomic spaces, Pacific Joui Math, 108 (1983), 431450.
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 3connected 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
ns
Map (Ps (2r),Sn ) ns I v J
I s r n I H lMap^(P (2 ),S ) .
be an admissible sequence,
that is,
denote the generator of & = H
ns
U SSn
and
x
J.1 ns+1
denote the
generator of Hn_s+1 [topw (PS(2r),Sn )] = Hn_s+1(nS_1Sn ) . Q t (x ) = i^QT(x ), Iv nsy ns' denote an element of
with
i. < s1. k ”
Map^(PS (2r),Sn )j If
8+2 ^ n>
Let
Let
Q Tx I ns+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 ns+ly Iv ns+1 '
Similarly,
if
we choose i
j
f
t
s
i, < s3. 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 DyerLashof 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 .... s2)(xns+ l)' we denote
by
Sq^ s  2 (xns+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*«s2 + others
r = 1
if
"others” = 0
if
and
n £ 3(4) with
a = 1 , and
S*Qs2 {xns+ l> = ‘S  A  s + l *
(“ O
77
^
r > 1
f°r ®*«
n.
If
n = 3(4),
then
Sq^Q^(x 0 ) « l nz
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 (xnS) = Q l K  s ^ (ii)
SqiQ ) ** s o(x z nsj.1 +1
tf
t 1 2 ' ^
cari ke computed by assuming that
gA  2 < xns+l> = Qs2( 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
(r1 )connected space
An
H^fY.Z/pZ) = Z/pZ
isomorphism, then
and given any selfmap
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
Hatomic at
p
Emaps
(r1)connected Hspace
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
1connected, then
Y
QY is Hatomic 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 Hatomic 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 Hatomic at
n / 6 , 8 , 9, 16, or 17.
if
THEOREM 3.3. Hatomic 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 selfHmap, 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 selfmap of
which is an Hmap. for
j < 2n4, 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
2n2 X
and
4n3
for
n > 2.
is isomorphic to that of
Map^h*(2},S^n
as a coalgebra over the Steenrod algebra, then isomorphism.
mod2
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 nl 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
= Qs2 (yns+l)
g ^ ^ V s * ^
Here notice that in case ^s2^ns+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 ) = s2 v ns+ 1 J
g*P*Qs_2 (xns+1) = p X  2 {yns+ 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 < i1 _ < ... < i,k “ = Qt(yn2 ) + < ? ' V yn3> 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 (xns+i) = Qs2 ^yn  s + P
suspension, It follows that a 1 + Q fy ) + others. s 2 sw ns'
Furthermore the degree of
is greater than that of
Qs2^ns+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)
^(xn2 ^ + e^2 ^Xn3 ^ *
(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 * ^ Xn2 > = Qt(yn2 ) + “ a ^ V
(* )
4 y ns
Thus
it follows that
= ^o^l" ^ Xn2^
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(xn2 * Xn3] =
’ SqiQl(xn3} = Q0(xn3)'
Pi V Xn2> = Xn2 * Xn3 + V
xn3>
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(xn3} and
and only if
j.
2 V n 3 '
n  2 J
Because
SqiQl(xn2} = °'
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 — Hspace
Q^Sn, where
2rth 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 2kl^ = ^l(x 4 kl^’ Apply
and
2

to obtain
h^O = S q ^ f x ^ ^ ) .
Q l{y4k1} + “ lQ3 (y4k2t
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 ns+1 s 2 ns+1 sns Sq^
2*
Sq^
=0
a = 1.
as before.
We will
of the righthand 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)): ns+s 2 Sq2 Q syns
^s2 * ^ns
0i
n 2 0i
Q s2 i yns
r2 t«+l + ... + 2 t 1  2 1 Q s2 i yns
= 0 Thus the term 2 *Sqw Q
and
qX .. s 2 ns+1
s = 3.
.
aQ y sns
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>
xn2’
(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>^
Xn3^
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>
^2Xn3’
(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 n4
Is an Hmap.
for
f^
is an
Then
= Q.x A + others i n4
fJ3.x q = Q.x 0 + others * l n3 l n3
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 n4
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 < 2n4.
where others Is zero In (111) If
Proof *
Map,w jp4 (2r),Sn],
be a selfmap 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
^*^3Xn4 = f*Q2Xn4 =
is an isomorphism in this
86
CAMPBELL, COHEN, PETERSON AND SELICK
LEMMA5.4.
Let
£
be a selfmap of
n > 5,
which is an Hmap.
on
for
Hj
j < 2n4,
2
If
induces an isomorphism
then
= Q 1^2Xn3’
2
Proof: ^ ^ 2 x n _ 3 = ^ 2 Xn3 + scluares
12

Sq = V n  4 ,
Sq Sq Q 0x ~ = Q nx ., ** **2 n —o u n^±
Since
P = 1,
Sq2QOx n3 =
2
Hence
a
S% Q 3Xn4 = 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 Hmap, 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 ^ n3 n4
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
n1 / 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 n3 0 n4
dimension, Let
is preserved.*S Preserve 2t+*. Consider
a selfmap ofMap^jp^ (2),Snj. Sq
Q2Xn3
and thus
Q
By 2.1, x 2  2 n2>l
suspension we see that
is
^2Xn3
is preserved. Let
n = 6(8),
we must show then
n > 6,
Q2Xn3
r = 1.
As
*S Preservec*
CLx A € Ker f also. 1 n4 *
^
However,
in the above case, 2Xn3 € ^er Q 1x . 1 n4
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*41,
n
t > 5,
/ 2^+1. r = 1.
As
we need to show ^2Xn4 ^ ^er **'
€ Ker f^.
Let
M
in the case of
n =
assume ^2Xn4
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*(y2n5} Sqig^(y2n5> = Q2Xn4 30
with
(2n7)connected
Thus there exists
which is a monomorphism on homology
f,
93
311(1
^ ( y 2 n5) = Q3Xn4 + Q lXn3’ Let > n3 Sn . Then
7r: Map^p4 (2),Sn
Since
g ' = gS
0 = irg' : S2n 6 » 03 Sn .
(^g)v,(y0 K ) = Q x Q , we see that ^ zno l n o
invariant one [9].
Clearly
p2 n5 (2 ) — 2
= 0
2 n _0
27Tg ^ *,
^ p2n5^2 ^
0
has Arf
so we have a diagram trg
Q 
s2ns 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
3fold 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 ]*
mod2
*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 hjh^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
^2Xn3
*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*41, such that
to get
g 1 : p2 n3 {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 selfmap 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^(QS2n1) ® 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 n2 *
want to sh°w
{(QaQ 2x 2 n~2 ^ that rank
f*
^ a+^ 
PH^fW^) < 1
J ^ 0}*
a ^ix2n2
*S Preservec*
k,£.
Since
We now consider
Qf.
(nf)*Q lQ2x2n3 = Q lQ2x2n3’
,
this
2a_ 1
Sq^
= ^ i x2 n2^2
Thus all primitives are preserved if
preserved.
•••
for SOme
^ i x2n2^
2j
Slnce
Sq? Q lQ3x2n3 = ( ^ ) ^ 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 n2 ^
f*(Qix2n2^
Sq^Sq^Qa+^X2n_2 = ^ ^ 2 X2n2’ 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 ZnZ
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 nontrivial 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 ipowers
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+il
(x^
^
and
such that
Sq^ ^(xa
=
i Qi 9 i+al 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 Hspace,
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 NonExistence of Elements of Hopf Invariant One," Ann. of Math., 72(1960), 20104.
[2]
H. E. A. Campbell, F. P. Peterson, and P. S. Selick, "SelfMaps of Loop Spaces, I," Trans. AMS, V293 (Jan. 1986), 139.
[3]
F. R. Cohen, "The Unstable Decomposition of Q^2^X its Applications," Math. Zeit., 2(1983), 553568.
[4]
F. R. Cohen, "TwoPrimary Analogues of Selick*s Theorem and the KahnPriddy Theorem for the 3Sphere," Topology, 23(1984), 401421.
[5]
F. R. Cohen and P. S. Selick, "Suspending Loop Spaces and the Fibre of the Hspace Squaring map," preprint.
[6 ]
F. R. Cohen and L. R. Taylor, "The Homology of Function Spaces,'* Contemporary Math., 19(1983), 3950.
[7]
J. C. Moore, "On a Theorem of Borsuk," Fund. Math. 43(1956), 195201.
and
100
CAMPBELL, COHEN, PETERSON AND SELICK
[8 ]
F. P. Peterson, ’’SelfMaps of Loop Spaces of Spheres,” Contemporary Math., 12(1983), 287288.
[9]
P. S. Selick, A Reformulation of the Arf Invariant One Mod p Problem and Applications to Atomic Spaces, Pac. J. Mathematics, 108(1983), 431450.
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 BrownGitler 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,
kfold 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 kstrings.
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 BrownGitler 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
kinvariants 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 spacelike
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 spacelike 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 spacelike cohomology is a strictly stronger condition than having unstable
Amodule 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 Amodule
structure (in fact its a trivial Amodule) but it does not have spacelike 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
Amodules,
denotes the canonical antiautomorphism
A. E
Let
be any spectrum with spacelike
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
Ts < 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 EilenbergMacLane 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 BrownGitler 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
Amodule = 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 spacelike
cohomology, the differentials in the Adams spectral
^
sequence for
Y) ,
, . i—is*ti,s+r,t+rl d :E * E r r are zero for
ts < 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^
^k1
LetE^’5(k) = E®,1:(Xk
define maps
E^ ’t(kl) » E^ ’t(k)
a*: E ^ ’t(k) » e®*t_k([k/2 ]) V
E * ’t ( [ k / 2 ] )
» E ^ + 1 ’
t+k(kl) .
i
 Y).
107
ADAMS SPECTRAL SEQUENCE OF Q2S3 Recall the
Xalgebra,
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:(k1) >E^,t:(k) JL
I
® u) =
is given by ®
u
ts < and
108
BROWN AND COHEN
(ill)
E ® ,t:[k/2] » E®=,t+k(kl)
is
y*(XI ® u) = XiXki ® u *^(kjJ = 0
dx
(iv)
ts < 2 k+l.
for
We now prove (2.1), that Is, ts = 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(kl)+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 spacelike 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^’ ,
ts
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 k2 > ^ 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 2fold loop space.
This in particular
implies that there exist "cup1 " 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 "cup1" operation formula
b^
the braid on around the
halftwists. pairings
q
define an ArakiKudo
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 ArakiKudo 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 il' 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 kl C~ _* ^2 k'
£*2k2 ^
F(IR^,2 k2 )/22^._2
This induces a map
2
F (ER ^ k ) / ^ ^ , which in turn induces a map of Thom spectra ^
T « 2 k2 > ^ 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 JamesHopf 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 BrownGitler spectra, all of which are easy consequences of Theorem 1.1. Define
to be
anc* ^et
^n
t*ie
nSpanierWhitehead dual spectrum to
Thus
^ ( G^)
05 Hn_q(B [n/2 ]'
LEMMA 3.5.
Recall that
Proof: > n}
has spacelike 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 nmanifold
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 nmanifold 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
spacelike cohomology. The following characterization of BrownGitler 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
Amodules, and
ADAMS SPECTRAL SEQUENCE OF fi2S3 (1 1 )
If Z
Is the
(2k+l)Sdual of
Y^,
117 then
Z
has spacelike 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 Sduality to the statement that J
H G * n
is projective in the category of unstable, right Amodules.
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
spacelike cohomology.)
A proof of this conjecture
was recently announced by Lannes and Zarati [13].
The following characterization of BrownGitler 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
2equivalent
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 spacelike 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
(43) 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 JJ i+J i+J where
e = 0 or 1
4.3 and 4.4 for Let
21.
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(kl) » C(k)

be the
Note
—
(X^ = A
map given by t (Av
for all admissible
LEMMA 4.5.
J) = A
if l = k1
=0
if i > k  1
(I,i).
C(k) C C(kl)
One easily proves:
is a subcomplex, and
induces an isomorphism of chain complexes
nr*
linear
121
ADAMS SPECTRAL SEQUENCE OF n2S3 C(k1)/C(k) £ C(k)
.
_
^
nr
Note 4.5 implies » C(kl)
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(kl)
be the inclusion.
LEMMA 4.6.
a*: C(k) >C(k),
C(kl) » C(k) H^(Xk_^)
lift the maps
P*: C(k) ^ ( 2 * ^ ) »
» H*(2k ^X)
and
C(kl)
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 = k1, 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.
satisfyingu/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  j1 ) > 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)
andu
Then
a*(X2I ® u) = I ^Jk~ 1 (Ak_I _1 A2 I)AJ ® uSqk . By 4.6 since
(k
l  1) + l = k1 < i^,
\llX2I = X l\ 1 mod Jk1 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), 283295.
[3]
E. H. Brown and F. P. Peterson, Relations among characteristic classes I, Topology 3 (1964), 3952.
[4]
E. H. Brown and F. P. Peterson, On immersions of nmanifolds, Adv. in Math. 24 (1977), 7477.
[5]
E. H. Brown and F. P. Peterson, On the stable 9
r+ 9
decomposition of Q ST 287298. [6 ]
, Trans. A.M.S. 243 (1978),
G. Carlsson, G. B. Segal’s Burnside ring conjecture for (Z/2)k , Topology 22 (1983) 83103.
[7]
F. R. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), 99110.
[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), 225228.
[10] R. L. Cohen, The geometry of Q^S^ and braid orientations, Invent. Math. 54 (1979), 5367. [11] R. L. Cohen, Representations of BrownGitler spectra, Proc. Top. Symp. at Siegen, 1979, Springer Lecture Notes #788 (1980) 399417. [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), 249256. [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), 577583.
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 __ » pt8 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
^b8 ,t 8
pt16 ^b16, t16 „t24 * b16 * b24
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 Ktheory 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 (P5) X D Let
bo
denote the spectrum for connective real
Ktheory 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 boresolutions.
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 Adamstype 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 BrownComenetz 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 filtrationk 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 b8 ;
factors as c
t rb
7
fpt ., irb8 j
as always, denotes the collapse map. t = °°,
Sdual of
i f
pt b 8
If
K .
and
is (b1 )connected and so the composite
is trivial.
c,
K
Since it has filtration 4,
pb1 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 2divisible.)
let
and
T P^ 0 o—o
B = L  t + 7 ,
Then the argument of the
preceding paragraph gives a filtration4 map
T f T P^  » ^B8 *
whose Sdual is a filtration4 map can be lifted to
P* ~ b 8
(P^ v b8 '
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 __ , pt8 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
(ts,s)coordinates (e.g., [15; pp. 9395], [8 ; p.41], [10; p. 149]).
For a sequence of filtration4 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
,t4irv . (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].)
ts =
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+rl
ts
satisfying
:E
denotes the exponent of 2.
charts, but by [15; 7.1]
is nonzero on u^fts+l) = r+1 , Such charts are
they do correspond to a
DAVIS AND MAHOWALD
132
direct limit of charts derived from resolutions of t8 i ^b8 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
= 4T1 (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 ^ 12A ;
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(4ne+A, 1 + maxfi^Ci+1), v^(±+2))).
where
§3.
PROOF OF THEOREM 1.2. The proof that
^(P^)
injective uses
boresolutions. 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
^b8k+8 ,t8k+8
k
„t8 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
Ndual
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+l0b+8kl 2 P_t+ 8 k_1
Hence f ^
... \ # K+1 •*• \
T2K+4(kK)+l _ I • Choose
lifts
, ^ , 0 k = nb2.
to
137
HOMOTOPY GROUPS OF SOME MAPPING TELESCOPES
Then „ (dimension of and hence
^n+l^b+SkK . , . . « 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
T4k2K+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 b8k
JPi m i b8k
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
P4n1
’ 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 oddprimary 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 „t8 n P, _ b 8n
, ^tSfnl) k/(p2  p  1 )
The existence of such exactly
k/2 (pl) (for
k = 2(pl)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(pS^); 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
Ktheory
be its 3connected 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 plocal 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 plocal 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 ptorsion,
homotopy of the cofiber
D of
j
iv
C
of
and that the is bounded above
and ptorsion. 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
ptorsion 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 AtiyahHirzebruch 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 nullhomotopic.
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 Ktheory selfequivalence of a finite ptorsion 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 Ktheory 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 Kmodule 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(pl).
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(pS^),
spectra
X
which are
nontrivial.
spectrum has
K local, and
K^~equivalence of a finite ptorsion 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 nonnilpotent map with
(e.g., Smith [16] when
Mahowald [8] when
P
ptorsion,
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 Ktheory (Theorem 10.12).
The
proof actually addresses the more natural question of the existence of appropriate selfmaps, 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), 189199.
[2]
J. F. Adams, A periodicity theorem in homological algebra, Proc. Camb. Phil. Soc. 62(1966), 365377.
[3]
J. F. Adams, On the groups J(X). (1966), 2171.
[4]
IV, Topology 5
J. F. Adams, Operations of the n ^ kind of Ktheory, 00
and what we don’t know about RP , London Math. Soc. Lecture Notes Series 11(1974), 19. [5]
D. W. Anderson and D. M. Davis, A vanishing theorem in homological algebra, Comm. Math. Helv. 48(1973), 318327.
[6]
A. K. Bousfield, The localization of spectra with respect to homology, Topology 18(1979), 257281.
[7]
M. C. Crabb and K. Knapp, Adams periodicity in stable homotopy, Topology 24(1985) 475486.
[8]
D. M. Davis and M. E. Mahowald, v^ and r^periodicity in stable homotopy theory, Am. J. Math. 103(1981), 615659.
[9]
D. M. Davis, Odd primary boresolutions and Ktheory localizations, 111. Jour. Math 30(1986) 79100.
MAPPING TELESCOPES AND K^LOCALIZATION
167
[10] K. Y. Lam, K0equivalences and existence of nonsingular bilinear maps, Pac. J. Math. 82(1979), 145153. [11] M. E. Mahowald, The image of J in the EHP sequence, Ann. of Math. 116(1982), 65112. [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), 287312. [13] H. R. Miller and C. W. Wilkerson, Vanishing lines for modules over the Steenrod algebra, J. Pure and Appl. Alg. 22(1981), 293307. [14] D. C. Ravenel, Localization with respect to certain periodic homology theories, Am. J. Math. 106(1984), 351414. [15] L. Schwartz, Ktheorie des corps finis et homotopie stable du classifiant d ’un groupe de Lie, J. Pure and Appl. Alg. 34(1984), 291300. [16] L. Smith, On realizing complex cobordism modules, Am. J. Math. 92(1970), 793856.
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 AdamsNovikov
spectral sequence (ANSS) converging to the pcomponent 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 padic numbers.
It is known to contain the pSylow
subgroup of the group of units of the ring of integers of any nth 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, n1
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 1.191.24 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
EilenbergMacLane spectrum) are known not to ([5], 2.2 and 3.1).
THEOREM 2. n > 0
(Mitchell [4]).
For each prime
there is a finite plocal spectrum
v_1,BP fX ) = 0 n1 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 ptorsion finite complex is Bousfield equivalent to some
and that these classes
generate the ‘pcomponent* 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
BPmodule 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
AtiyahHirzebruch 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), 257281.
GEOMETRIC REALIZATION OF THE CHROMATIC RESOLUTION
179
[2]
A. K. Bousfield, The Boolean algebra of spectra, Comment. Math. Helv. 54(1979), 368377.
[3]
H. R. Miller, D. C. Ravenel, and W. S. Wilson, Periodic Phenomena in the AdamsNovikov spectral sequence, Ann. of Math. 106(1977), 469516.
[4]
S. A. Mitchell, Finite complexes with A(n)free cohomology, Topology 24(1985), 227248.
[5]
D. C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106(1984), 351414.
[6]
L. Smith, On realizing complex bordism modules, Amer. J. Math. 92(1970), 793856.
[7]
H. Toda, On realizing exterior parts of the Steenrod algebra, Topology 10(1971), 5365.
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
Adiagram 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 homotopytheoretical 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],
kfold loop spaces [15],
kfold 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 ntype 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
Adiagrams 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
Urestricted
Adiagrams
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
Adiagrams is equivalent to the homotopy
theory of (strictly commutative) Adiagrams 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 requivalences. A functor
f:
A
B
between two simplicial
categories will be called a weak requivalence 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 requivalence and a weak requivalence 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 nonidentity 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
Adiagrams 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
Urestricted
Adiagrams (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 requivalence
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
requivalence.
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
requivalence. 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^Adiagram (1.3(iii)), then
theorem 2.1 implies that every infinitely homotopy commutative Adiagram of simplicial sets is weakly equivalent to a diagram induced from an actual Adiagram 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
LCOFINAL AND RCOFINAL FUNCTORS. Lcofinal (resp.
the nerve of
f :A
B
be
Rcofinal) 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 €
Bdiagram
(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^
Bdiagram
strong deformation retract of the
B S^
is that
hom(fA,)
Bdiagram
is a
f^hom(A,).
In fact the arguments of [4, 6.4] yield
3.3 € A,
PROPOSITION.
For every diagram hom(fA,)
the Bdiagrams
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
32(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) SpringerVerlago.
[2]
W. G. Dwyer and D. M. Kan, Simplicial localizations of categories, J. Pure and Appl. Alg. 17, (1980) 267284.
[3]
W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Topology 19, (1980), 427440.
[4]
W. G. Dwyer and D. M. Kan, A classification theorem for diagrams of simplicial sets, Topology 23, (1984) 139155.
[5] W. G. Dwyer and D. M. Kan, Equivariant homotopy classification, J. Pure and Appl. Alg. 35 (1985), 269285. [6] W. G. Dwyer and D. M. Kan, Singular functors and realization functions, Proc. Kon. Ned. Akad. van Wetensch. A87 = Ind. Math. 46 (1984), 147153. [7]
D. G. Quillen, Homotopical algebra, Lecture Notes in Math, 43 (1967), SpringerVerlag.
[8]
G. Segal, Categories and cohomology theories, Topology 13 (1974), 293312.
[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), 217252. [11] C. Wilkerson, Classification of spaces of the same ntype for all n, Proc. AMS 60 (1976), 279286. [12] A. K. Bousfield and E. M. Friedlander, Homotopy theory of Tspaces, spectra and bisimplicial sets, Lecture Notes in Math. 658 (1977), SpringerVerlag, 80130. [13] E. Dror, W. G. Dwyer and D. M. Kan, Self homotopy equivalences of Postnikov conjugates, Proc. AMS 74 (1979), 183186.
EQUIVALENCES BETWEEN HOMOTOPY THEORIES OF DIAGRAMS [14] M. J. Hopkins, Formulations of cocategory and the iterated suspension, Asterique 113114 (1984), 212226. [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 HSPACE James P. Lin^
In the first part of my talk I will develop some preliminary notions about finite Hspaces.
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 Hspaces 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 Hspaces 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 Hspaces. To begin, an Hspace 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 HSPACE
207
Hspaces occur naturally in many contexts.
The loops
on any space
B
multiplication.
(denoted
QB)
is an Hspace with the loop
This includes Eilenberg MacLane spaces,
and all strictly associative Hspaces by results of Milnor and Stasheff [16,18].
Another interesting class of
Hspaces 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 Hspace.
A further characteristic of Lie groups that
distinguish them from other Hspaces 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 Hspace, an Hspace which has the homotopy type of a finite complex.
Hence all Lie groups
are finite Hspaces, but very few other loop spaces are finite Hspaces. 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 Hspace 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 Hspaces. For » X x X
k
a field, one observes that the diagonal map
and the Hstructural 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 HSPACE
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 Hspace.
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 Hspaces. There is a process called
Some results are known.
plocalization which does not
always yield a finite Hspace, 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 Hspace. 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 Hspaces 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. Hspace
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 HSPACE 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 Hspace 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 Hspaces 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 Hspaces 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 Hspace 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
Hspaces 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 ptorsion 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 HSPACE (6)
Zabrodsky [24]:
Let
p
be an odd prime and let
be a homotopy associative Hspace. 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 Hspace 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 ptorsion. 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 Hspaces 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 plocal 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 3dimensional 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 Hspaces is to try to extend results of Thomas to finite Hspaces with nonprimitive generators in their
mod 2
cohomology.
First
of all it is useful to note that there are examples of finite Hspaces 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 HSPACE is no 7dimensional 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 Hspaces should exist for all finite Hspaces. 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 Hspaces 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 HSPACE 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 Hspace 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.
Hdeviation 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 Hmap.
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+^kl 2r 2r+^kl PH * K (X,Z2 ) = Sq ?EZ K (X;Z2 ) 2r 2r+2r+1kl 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 HSPACE 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,8kl,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 18k1 ^ 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 ). 4k1 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+^kl 2r 2r+^kl Z K (X;Z2 ) = Sq PHZ K (X;^) and 2r 2r+2r+1kl 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 HSPACE
s / s /
=
Y
s/e.
221
.
j=0 The advantage to this approach is that the information about
x
is now contained in the Hdeviation
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 2stage 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 Hspaces is associative.
mod 2
homology of the
This is certainly true for
all known examples, and it is suspected that all finite Hspaces admit an Hstructure 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^X4k2^ + decomP°sables*
conclude
223
MOD 2 COHOMOLOGY OF A FINITE HSPACE 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
3stage 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 Hmap.
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 Hspaces 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 EilenbergMoore 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 HSPACE ^
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 Hspace 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 Hspace, 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 Hspaces?
4
(1 ) (n)
® A(xn ,x13)
Z2 [x1 5 ]/x1 5 ®
a
(x 2 3 ,x 27> x 29).
Problem 6 Given a finite Hspace Hstructure 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
Hspace.
The use of the secondary operation still seems to
be a powerful tool in attacking such questions.
REFERENCES [1] J. F. Adams, On the nonexistence of elements of Hopf invariant one, Ann. of Math., 72 (1960), 20104. [2] J. F. Adams, The sphere considered as an Hspace mod p, Quart. J. of Math., Oxford Ser., 12 (1961), 5260. [3] A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math., 57 (1953), 115207. [4]
W. Browder, Higher torsion in Hspaces, Trans. AMS, 108 (1963), 353375.
[5]
W. Browder, Torsion in Hspaces, Ann. of Math., 74 (1961), 2451.
[6 ] W. Browder and E. Thomas, On the projective plane of an Hspace, 111. J. of Math., 7 (1963), 492502. [7]
G. Cooke, J. Harper and A. Zabrodsky, Torsion free mod p Hspaces of low rank, Topology, 18 (1979).
[8 ] J. Harper, Hspaces with torsion, Memoirs of AMS, 22, No. 223 (1979). [9]
J. Hubbuck, On homotopy commutative Hspaces, Topology, 8 (1969), 119126.
[10] R. Kane, On loop space without ptorsion, Pacific J. of Math., 60 (1975), 189201. [11] R. Kane, The module of indecomposables for finite Hspaces II, TAMS, 249 (1979), 425433. [12] R. Kane, Primitivity and finite Hspaces, Quart. J. of Math., 26 (1975), 309313.3.3.
MOD 2 COHOMOLOGY OF A FINITE HSPACE
227
[13] J. Lin, Higher order operations in the mod 2 cohomology of a finite Hspace, Amer. J. of Math., 105 (1983), 855938. [14] J. Lin, Torsion in Hspaces I, II, Ann. of Math., 103 (1976), 45687; 107 (1978), 4188. [15] J. Lin, Two torsion and the loop space conjecture, Ann. of Math., 115 (1982), 3591. [16] J. Milnor, Construction of universal bundles, II, Ann. of Math., 63 (1956), 430436. [17]
J. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math., 81 (1965), 211264.
[18] J. Stasheff, Homotopy associativity of Hspaces, I, II, Trans. AMS, 108 (1963), 275292; 108 (1963), 293312. [19] E. Thomas, On the mod 2 cohomology of certain Hspaces, Comment. Math. Helv., 37 (1962), 132140. [20] E. Thomas, Steenrod squares and Hspaces I, II, Ann. of Math., 77 (1963), 306317; 81 (1965), 473495. [21]
E. Thomas, Whitney Cartan product Zeit., 118 (1970), 115138.
formulae, Math.
[22] F. Williams, Higher homotopy commutativity, Trans. AMS, 139 (1969), 191206. [23] F. Williams, Primitive obstructions in the cohomology of loop spaces, to appear. [24] A. Zabrodsky, Implications in the cohomology of Hspaces, 111. J. of Math., 14 (1970), 363375. [25] A. Zabrodsky, Cohomology operations and homotopy commutative Hspaces, in The Steenrod Algebra and Its Applications, Springer Verlag, 168 (1970), 308317.
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
hS^ x * = h
with
u ® 1 + 1 ® ^*w4 =
(unless for
f* w4 = 0 ) *
n > r1
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 )jr_ 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_1j*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 tiein 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 onetwo 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 kalgebra
G
is degreeone
253
GENERIC ALGEBRAS AND CW COMPLEXES generated (henceforth referred toasd.o.g.) generated as a kalgebra 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, onetwo 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 2cells and
P^(z)
complex
two 6cells, for which
is not rational (homology is computed over
undertake to evaluate
for all
CW
Q ) . We
complexes
X
consisting of two 2cells and two 6cells, 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 reusethe 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 nonzero.
(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=l1 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
nl 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 kalgebra 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
G°
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 nonArtinian 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 onetwo 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 onetwo algebras with
g
generators.
Because of
this we sometimes classify all onetwo 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 onetwo Lie algebras.
The promised wellordering of
^
is a consequence of
Lemma 4.1.
THEOREM 4.3.
is wellordered (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 multiset”
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
il '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 wellordered.
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 2chains is ={xw^ynwy)n^
and
there are no
m > 3.[5,formula (16)]
uHp(z) f
mchains 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^eTr} U P = where
by
U (v1fx(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
ll
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 righthand side of
(26) is a polynomial ofdegree
< s  1,
so the terms
(a zS+...) s
dominate
p (z)
cannot help
coefficientwise. 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 onetwo 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
relabel
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 di. if i w di > x(y x )y = e (y x)(x y) ( iw diw i di> = (& )(e )(y xyx ) f d + K f i di. = (& )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
xyyx + 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 ] = (mij1)(m1)(m2)...(mj)tm 1 J 2 y(yixJ')x[ tm ] = y(y1x J)y[tm ] =
m(ml)(m2) ... (mj ) (m1)(m2)...(mj)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 righthand side of (23) is a polynomial as opposed to a nonterminating 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 onetwo 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
nonnilpotent 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 nonempty.
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 onetwo 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. kalgebra,
ndimenstonal hmodule 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 onetwo 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^  (lmz)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 semitensor product [25]
with another free algebra
E = k. m
Semitensor products of free
algebras are analyzed in [2, section 5], where they are called "generalized products". productE O F , E
define
To specify the semitensor
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 twosided ideal
which
[uQ.w] generates.
(37D)
312
ANICK By [2, proposition 5.5],
product of
F
{[[...[[M
E
...M q
• ■• • Vj
[m]
equals the semitensor
with the quotient algebra
is the twosided ideal of
where
G
where
I
generated by all
(u ),v ].v ], J1 Jq q1
].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^(lmz) ^  Hj(z)z^. Because
G
dim. 2)
and
is the semitensor 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)(lmz)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 nonconstant.
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 kmodule
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 nonconstant, 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 nonconstant, otherwise the
matrix defining the differential contains constant entries, hence our DGmodule 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,
393400 (1982). [2]
Carlsson, G . , "On the homology of finite free (Z/^)11 complexes," Inventiones Mathematicae, 74, pp. 139147
(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 Gspectra 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
halfsmash product Etr tx Y ^
= Ett
TT
where j
th
Ett
x
Y^/ E i r * {*},
7T
TT
is a free contractible TTspace 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 Ginner 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(WV)t , where complement
H*(BG ^)
WV
\(WV) of
V
f (Ly ) =
is the Euler class of the in
W.
is
348 (2)
MAY For a split Gspectrum
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
WV S
is the canonical inclusion.
denotes the 1point compactification of V
denotes the union of a free contractible Gspace
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 halfsmash
product” in equivariant stable homotopy theory. unbased free Gspace
X
and a based Gspace
X kgY = X xgY/X
THEOREM C.
Let
G
xq
X k^E
00
such that Gspaces
Here
There is a on Gspectra
E
00
X tx^.2 Y
is isomorphic to
2 (X KqY)
for based
Y.
00
2 Y
is the suspension Gspectrum 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 halfsmash product functor
Y,
For an
Y.
X = Ett
and
by the jfold smash power of a nonequivariant
k „S_V, G
where
S_V
is the (V)sphere
350
MAY
Gspectrum.
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
WV 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
Gprepeetrum if each adjoint map D
351
D
of
such that is an inclusion
~ WV a: DV » Q DW
is a Gspectrum
is an if eacha
We have categories GSM 3 GQst 3 G M
of Gprespectra, inclusion Gprespectra, and Gspectra 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
Gspectrum where
E
and a Gspace
(Da Y)(V) = DV aY
Gspectra F(Y,EV), cylinders that a map
F(Y,E)
Y,
we set
for a Gprespectrum
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 Gspectra is a weak equivalence
if the Hfixed 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 Gspectra 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 Gspectra by setting
00
2
from Gspaces 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 Gspectra by setting
from Gspaces 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 Gmaps
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
Zsphere 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
GCW
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 GWhitehead theorems. that a weak equivalence between Ghomotopy equivalence.
GCW
The latter asserts spectra is a
These results imply that the
stable category is equivalent to the homotopy category of GCW
spectra and cellular maps.
354
MAY The functor
2 oo "A 2 "
and a "shift desuspension" functor functor
A
2
A
2
on Gspectra.
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
Gspaces, but their structural homeomorphisms
2 2
The
has an inverse shift suspension functor
The Gspectra
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
Guniverses 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 Glinear 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. GCW
i
is trivial to define
For example, it fails to preserve
spectra. To proceed further, we exploit the topology of the
function
Gspace
^(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 Gprespectra
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 Gspectra
E
and
we thus have the external smash product E aE '
=
L ( £ E a£ E ') .
To ”internalize” the smash product, we choose a Glinear 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 Gcontractible 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 Gspectra F(E,E'),
such as dual Gspectra
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 Gfree and contractible, so that
its orbit space is a classifying space for principal Gbundles.
For an unbased free
GCW
complex
X,
there
is thus a Gmap X  » ^(U.IR00), and
x
is unique up to Ghomotopy.
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 Gspectra. 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 Gbundle
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 1point compactification of its total space).
By elementary inspection of bundles, there are
canonical Ghomeomorphisms A. (1)
S
n.
1a T 1
= X aS 1
and n .,in. A ..A. T.a S 1 = T. ,a S 1 1.
(2)
V '
1
1+1
For a Gprespectrum 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+1X (x) (A.+ 1 ) )©(X (x) (A.+1 A . ) ) . Dr
a Gspectrum 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
GCW
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 Gspace 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 Gprespectra
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 Gspectrum to
E
indexed on
f°r a free Gspectrum
and that — 00 hGJIR .
oo
The principle is that ’’free Gspectra 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 Gspectra
\ 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 halfsmash
product
X+a E
indexed on
U.
i„(x x E)
and
For the first, using a more
general definition of twisted halfsmash products which allows nontrivial target universes, we find that i^(x ix E) = (i.ox)
k
E,
iox: X >^(U,U).
the Gcontractibility of halfsmash product
^(U,U)
(i°x) x E
that the twisted
is equivalent to the
untwisted halfsmash 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 Gequivalence when smashed with the free Gspace
X+ .
360
MAY CO
To prove Theorem A, we take acting by permutations.
For a spectrum
the jfold external smash power indexed on space
Y,
U, and we set (2 Y)^'^
and
7Tspectra 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 pgroup 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 Gspectrum indexed on
U.
00 G i : IR = U
We have the inclusion
the underlying nonequivariant spectrum with Gaction 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, Ktheory, and cobordism. When it holds, and not in general otherwise, we have isomorphisms (*)
kG (ixE) = k*(E/G)
for a free Gspectrum Here
i
and E
kg(2_AixE) = kx (E/G)
indexed on
CO IR
[13, 11.8.4].
is innocuous since free Gspectra 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 )
Gmap
W
V EG tx^S
consider an arbitrary Gspace
Let
S
was obtained by passage to orbits over
We must still explain why
represent
k
xg B0g
00 (U)  » B0(IR ).
be a Gmap.
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 Gspectrum
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 pgroups, II; padic completion in equivariant cohomology. Topology. To appear.
[6]
G. Carlsson. Equivariant stable homotopy and Segal’s Burnside ring conjecture. Annals of Math. 120 (1984), 189224.
[7]
R. L. Cohen. Stable proofs of stable splittings. Math. Proc. Camb. Phil. Soc. 88 (1980), 149152.
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 cupr 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), 473482. [12] N. J. Kuhn. Extended powers of spectra and a generalized KahnPriddy theory. Topology 23 (1985), 473480. [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, 620637.
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,
pseudoisotopy 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 MCS8203806 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, mmanifold 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 birthdeath
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 nmanifold 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 birthdeath.
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
nconnected.
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 2kconnected. T7^(Mn )
be the unoriented bordism groups of
Mn .
?7x (M) =* p^(MOaM+ ) = Z/2[bi :i*2rl] ® 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 pseudoisotopies. 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 2index 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 pseudoisotopy space, p:C(M) x Diff(M.5M) > Diff(M.dM) def ined in the same manner. We then have the induced pseudoisotopy 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 pseudoisotopy 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 pseudoiso 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 kparameter family of smooth maps whose critical points are all either nondegenerate or birthdeath. 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^Naf.
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 kjets 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 mvariables with degree < k.
The structure group of
this bundle is the group of kjets 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 birthdeath 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 3jet 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+lconnected map to the infinite In this section we use Igusa’s
techniques to show that there are subbundles whose space of sections n+lconnected 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 birthdeath 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
3jet bundle T(k)  » Mn xl with fiber
Fn+^(k).
Thus
C(k) C J^(MxI)
consists of
those 3jets whose critical points are either nondegenerate of index
< k or birthdeath 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 nconnected 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+lconnected 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+^)n2
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*(IRmC) * 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)
iplanes
by
p
let
be the Grassmannian of
Define a map
where
X1+1 IR
N(p)
is the idimensional
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 ll .
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 / Bk2
\
Aj^ / Bk1
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 Ai
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,k1 are the quotient maps induced
0(n+l).
Recall that the homotopy pushout of a diagram X1
X2
Xk1
f l \ / gl
Xk
fk \
Y *1
/ V
1
Y *kl
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+lconnectedsince
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+lr) ^ 0(n+l)xl /0(r)xO(n+lr)xl^
«K>(» 2 ) / 0 ( r ) * I 1,0(.>Hr) = Gm 2 ,p
°n+2 ,r+l
= 0(n+l)/0(r+l)x0(n+lr)  > 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 ndimensional 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 pullback property this defines a section 2Ua:D(u) > G.
LEMMA 2.6. n+1connected.
(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+lconnected by 2.2.
C(k)>
Fn+^(k),
is n+ldimensional 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+lconnected 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) + (ltjh^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+2connected and hence the induced map of sections ^ :rG ( k ) ^ U^’S ^ ^ is n+lconnected. 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+2dimensional homotopy type as
2^n+^B0(k)).
2n+2 Sdual 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 nconnected 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 mfold 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 mfold
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 2INDEX 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 qparameter
family of sections of the 3jet bundle, r v J (M xlTm+1\ f ^.„n M xlTm+1 » ); so that for each
t £ Dq
3jet 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 birthdeath of
index < 2k+l. b.
If
t € 5Dq = Sq *, x
then all critical points of
are either nondegenerate of index < 2k or
birthdeath of index < 2k1.
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
rr
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 EilenbergMacLane
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 2index 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 birthdeath critical points all have an index such that
j < i < k1.
Now let
, H.(Mn )=
J
i
lim , > H.(Mn xIm 1).
m
J
The 2index 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+lconnected 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 ’2kl( >
defined as follows. map
f:MxIm
IR.
Let
^ TT2k+2rijrn^ > 2k+l^ > 2k
f € ?2kl^^
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 pseudoisotopie, Publ. Math. I. H. E. S. 36 (1970).
[2]
A. Hatcher and J. Wagoner, Pseudoisotopies of compact manifolds, Asterisque 6, Soc. Math, de France (1973), Paris.
[3]
M. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, SpringerVerlag, (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 Ktheory, Memoirs of A.M.S. #221 (1979).
Ralph L. Cohen Stanford University Stanford, California 94305
XVI ALGEBRIAC KTHEORY 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 KahnPriddy 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 KahnPriddy theorem we have to know about transfer maps.
392
First, there is a
ALGEBRAIC KTHEORY OF SPACES
transfer for the algebraic Ktheory 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 KahnPriddy theorem is valid for the algebraic Ktheory 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(xl)...(xn+1) .
Is compatible with the
in the sense that it is possible to fill in
the broken arrow so that the diagram
395
ALGEBRAIC KTHEORY 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(xl)...(xn+1) .
The desired KahnPriddy
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.
2connected:
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 < j1 .
j > 0
CO 00 S
ttSI
j < 2 . and that
=^
By the spectral sequence of a generalized
ALGEBRAIC KTHEORY 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
mhandles.
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
Hspaces (infinite loop spaces, in fact) and that the map S A (X)
A(X) is a map of Hspaces.
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 KTHEORY 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
$(XxX'x[0,1]) .
We
have thus obtained a map, well defined up to some choices (’contractible choices’) h*J(X)
x
m
£ (X ')
»
m
£ £ ( X
and restricting in the desired way.
xX
, x[ 0 . 1 ] )
This completes the
account of the multiplication. The case of the operations is a little more delicate, and the verification takes much longer.
We need a
modification of the ’manifold approach’ where the simplicial category of the partitions is replaced by another simplicial category. needed only in the case than the general case.
The modified construction is
X = *
which is somewhat easier
We restrict to that case.
We consider compact smooth submanifolds codimension
0
in euclidean space
neighborhood of the origin. category from such manifolds.
R
d
M
of
containing a
We manufacture a simplicial First, we define a
400
WALDHAUSEN
simplicial set
#(d)
where a ksimplex is a smooth family,
parametrized by the simplex
A ,
considered.
#(d)
Next we regard
of manifolds of the type as the simplicial set of
objects in a simplicial category
h#(d)
simplicial partially ordered set): from
M
to
M'
if and only if
(in fact, a
there is a morphism
M C M'
and if furthermore
the two inclusion maps, of boundaries, SM  » Cl(M'M) M(d+l)
MI > M x [1.+1]
.
So, by using that map, we can form the stabilization with respect to dimension, lim M ( d )
.
We can obtain a homotopy equivalent simplicial category of
R^
where
h$'(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 61)
MI ► 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 $(D^) , and so on. As a result, therefore, theorem 1 of [7] may be restated to say, among other things, that the inclusion map n
, .
i ^m ,
lim #k (d)  > l m M k (d) «g
is an approximation to the map
A (*) » A(*) .
As regards the limits with respect to
m
and
k ,
there is the happy technical point that the details don’t really matter. map
S A (X)
The reason is that, as we already know, the
A(X)
is a coretraction, up to homotopy; this
will allow us to restrict the necessary checking, below, to a checking on representatives only.
All we need to know
about those limits, therefore, is that they exist in some weak sense; say, as homotopy direct limits with respect to stabilization maps which exist only after geometric realization and are well defined up to (weak) homotopy, and compatible to each other.
Thus we may simply take the
stabilization maps of [7] and transport them to the present situation by means of the homotopy equivalences above. The simplicial set category
h#m (d) )
#m (d)
(resp. the simplicial
has an additional structure, namely it
is a partial monoid in the sense of [3] with respect to gluing.
(The monoid is only partial because the result of
403
ALGEBRAIC KTHEORY OF SPACES
the gluing should be a manifold again, and should be of the correct type.)
As a result, lim #m (d) c?
(resp.
lim h#m (d) ) cf
is the underlying space of a Tspace in the sense of [4]. Let
Bp( lim (h)(2m (d) )
Tspace.
denote the realization of that
Then the loop space
QBp( lim (h)#m (d) )
as a ’group completion* for the Hspace
serves
lim (h)$m (d) , and
there is a map lim (h)em (d)  » 0Br( lim (h)«f(d) ) which, up to (weak) homotopy, is universal for Hmaps of lim (h)#m (d)
into grouplike Hspaces [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 Hspace structures.
The maps
0n
can all be
assembled into a single map
0 : A (*) » JT A(B2n ) n which is an Hmap with respect to the additive structure on A(*)
and a suitable Hspace structure on the product [6].
That Hspace 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 KTHEORY OF SPACES
only in the compact case, but the general case follows by an exhaustion argument from this).
Hence we have a map of
Hspaces TT AS (B2n )  » U n n
A(B2n ) ,
and both of these are in fact grouplike Hspaces 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 Hmaps.
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 filledin arrow is necessarily an Hmap.
Hence, by the universal property for
Hmaps into grouplike Hspaces, 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 KTHEORY 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 Wsituation 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 eneighborhood 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 reinterpret 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 KTHEORY 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 halfspace, 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
ntuple
(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 KTHEORY 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$7n (X)
to
{Mp}
there is, by definition, a morphism from if and only if, in addition to the above
cond itions, we have M U M' = M* p o p for all
p .
inclusion
M
Note there is really no condition on the M' . Thus, for example, the category
413
ALGEBRAIC KTHEORY OF SPACES
i9^o (X)
is contractible.
In general, the role of these
morphisms is to provide a systematic way of ignoring the M q ’s
in those filtrations.
morphism in
iS^T^fX)
as an isomorphism of the nonexistent
quotientmanifolds In
h$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 nfold
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 KTHEORY OF SPACES
Since
is contractible, the
the simplicial object
[n] I— >i3^n (X)
up to homotopy, to the suspension of
’1skeleton’
of
may be identified, i3^(X) . By
adjointness there is therefore a map into the loop space, is^x)
.
»nis^.(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 Ktheory 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), 239251.
[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), 213221.
ALGEBRAIC KTHEORY OF SPACES
417
_________ , Categories and cohomology theories, Topology 13 (1974), 293312. F. Waldhausen, Algebraic Ktheory of topological spaces. I, Proc. Symp. Pure Math., Vol. 32, A.M.S. (1978), 3560. _________ _ Operations in the algebraic Ktheory of spaces, Springer Lecture Notes in Math. 967 (1982), 390409. _________ _ Algebraic Ktheory of spaces, a manifold approach, Canadian Math. Soc., Conf. Proc., Vol. 2, Part 1, A.M.S. (1982), 141184. _________ _ Algebraic Ktheory of spaces, Springer Lecture Notes in Math. 1126 (1983), 318419.
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 Ktheory of the space
X.
This can be defined in various ways, see [9], [10], [11], [12].
Let
BG
be the space classifying 0dimensional
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 , ^il ~ 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 Jhomomorphism. Recall from [1] that there are classes i > 1,
so that
Jhomomorphism.
_2
77 p0 . 4
ol+l
x = T)2 . Then
is in the image of the
Similarly,
of the Jhomomorphism.
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 twotorsion 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 Hspace
422
BOKSTEDT AND WALDHAUSEN
equivalence, when
oo co
OS
o
= QS
is given the Hspace
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
(24)
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 Sduality parametrized see [2].
A 2Ndual 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^ ^ Trp
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 Ktheory 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 2Ndual of this map is the
wedge of the 2Nduals 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: 181195(1963), 3: 137171, 193222(1965) and 5: 2171(1966).
[2]
Becker, J. C. and Gottlieb, D. H . , Transfer maps for fibrations and duality, Composito Mathematica 2: 107133(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 :2537.
[5]
Bokstedt, M . , Equivarant transfer and the splitting of A(*), to appear.
[6]
Kassel, C . , Stabilisation de la Ktheorie algebrique des espace topologique, Ann. Scient. Ec. Norm. Sup. 4 Serie, t. 16:123149(1983)
[7]
0
Quillen, D. , Letter to Milnor on ImfTr^O^r^nK^Z), Lecture Notes on Mathematics no 551, Springer 1976: 182189.
[8]
Segal, G . , ConfigurationSpaces and Iterated Loopspaces, Invent. Math. 21: 213221(1973).
[9]
Waldhausen, F., Algebraic Ktheory of topological spaces, I, Proc Symp. Pure Math, vol 32 Part I: 3560, A.M.S. (1978).
[10] Waldhausen, F., Algebraic Ktheory of topological spaces, II,Lecture Notes in Mathematics 763 (1979): 356394. [11] Waldhausen, F., Algebraic Ktheory of spaces, a manifold approach, Canad. Math. Soc., Conf. Proc., vol. 2, part 1: 141184, A.M.S. (1982).
THE MAP
BSG » A(*) > QS°
431
[12] Waldhausen, F., Algebraic Ktheory of spaces, Lecture Notes in Mathematics No. 1126, Springer (1985): 318419. [13] Waldhausen, F . , Algebraic Ktheory of spaces, stable homotopy and concordance theory, these proceedings.
Marcel Bokstedt Fakultat fur Mathematik Universitat Bielefeld D4800 Bielefeld F.R. Germany
Friedhelm Waldhausen Fakultat fur Mathematik Universitat Bielefeld D4800 Bielefeld F.R. Germany
XVIII VECTOR BUNDLES, PROJECTIVE MODULES AND THE KTHEORY 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 nontrivial examples of projective modules, the nontriviality 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 Ktheory 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 KTHEORY
433
Murthy to the case of nondegenerate 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 "Ktheory of quadric hypersurfaces,” Ann. of Math., 122(1985), 113153.
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 KTHEORY
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
wellknown problem about whether projective Rmodules 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
Rmodules 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 nsphere
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 nontrivial.
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 KTHEORY
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 nontrivial.
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 nonzero.
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 Qui1lenSuslin 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 QuillenSuslin theorem, the central problem in this area became the socalled BassQuillen 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 KTHEORY
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/(T1)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 KTHEORY
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 BorelSerre proof [BSL] of the nontriviality 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, ...,zV)
If
is not free.
map f:S^n+^ where
a > 0
S^n+*
given
is chosen
443
BUNDLES, MODULES AND KTHEORY
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' = 0r1 © f © E' generates
In fact, if
m
get an epimorphism
0m * E
with
If a
rk E* = mr.
H^(RPn ,Z/2Z), the StiefelWhitney class of so
An
w(E') = (1+a) * = 1+a +
so
f
... + a11. Since
mr = 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
pPmodule 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 KTHEORY
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 nonzero 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
ERalgebra.
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
11
correspondence between isomorphism classes of finitely generated projective bundles on
X.
Rmodules and those of real vector
The same is true for complex or
quaternionic bundles and the rings
€ ®^R
and
IH
447
BUNDLES, MODULES AND KTHEORY
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 Rprojectives.
For example, if
a noetherian ring of dimension projective module of rank
m
m = 2 mod 4, m
one can find
with a nontrivial
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 Rmodules 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 KTHEORY
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
Amodules 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 nonfree projective Amodule 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 nontriviality 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.
KTHEORY 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 Rmodules 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 KTHEORY
Proof.
(a)
= IT .0(1) nl
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 KTHEORY
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)
nondegenerate
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
kbase for
.
of noncommutative
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
$(R)
be the category of
projective
Define a functor
to the kernel of
e = ^{1  ^
R(q)
with the
i.e.,
I V e(r ® m) = ~{r ® m  2 x ^r ® eim )• kernel is a direct summand of
The
i. < i0 < ... < i forma 1 2 r
finitely generated projective Rmodules.
e:R(q)
with
2
i ^ jand ^J
C(q)modules and let
0:J(C(q)) >$(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
Zlinear 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) + uv1) 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 KTHEORY
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(Vab)
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 + Vb/a y > x  Vb/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,wl).
Then
2
1).
and
In
A,
Let
soP
where
ut(wl) + (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,wl).
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/(wl).
1 > R* > A* » Z so
2
P= (u,wl) and
PP' = (w1)
= (w1) *P' = (1,t) R[t],
q = uv + w
w = Vr~a x,
and set
Then
k' =
R' = k' % R
u = V b y + yfc 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
Vabc = V ds q. Therefore ds q €k.
Since
H^(G,H)is
471
BUNDLES, MODULES AND KTHEORY
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))
nonisotropic 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
noniso 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 Bmodule
+ 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
t1 ^ + ...
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 nonisotropic. It follows that d:C(R) » Z/2Z
d defines a homomorphism
whenever
q
also represents 1 we have and
P = (xa,y3) has
is nonisotropic. 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 KTHEORY
We need only check that
Proof.
0
is nontrivial when
is nonisotropic 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(lx)r = ybs,
I 3 (lx,y),
§10.
1 C I
and
[I] = [R/I]
since
2
+ by .
(l+x)s = yr} £ I = {s € R(l+x)s € yR} = (lx,y)
Kq (R),
q
(lx,y)
is maximal.
since In
which is nontrivial by Lemma 9.4
R/I £ k.
KTHEORY 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 Amodules.
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 Amodules. 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 kalgebra,
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 uh 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,
ufh
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 KTHEORY
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
Bmodule. (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(lz))
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]/(xyl) = 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 Rmodule. where
P = (x1 v 1
k[y^,...,yn ] Thus
in
This was first proved in
Note that Theorem 7.2c follows.
k[z]/(zl), tensor with
is 0
K^(R)
We start with
and regard the
is generated by
[R/P]
x ,zl). 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 2ml
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 , .. .,Va /a 7) v 2 1 n nlJ
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 nonisotropic 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 nonregular 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(Vab)
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 * xVa/b.
x € K^fS^)
^ G0 (S)
0
2
d ^ ( S ^ )  >
nonisotropic, 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 noniso 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 nonisotropic.
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 KTHEORY
nonisotropic. G0 (S/z) q'
As usual,
so we get
KjtS )
is nonisotropic,
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
Xp = 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
nonisotropic, and generated by
Proof.
q
1
represents
[R(q)/(x,y,zl)]
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,zt)
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,zt]
in
of
G^(S).
Now
S^.
= k[x,y,z, t, t 1]/(q') = (k[f ,r),f]/(q(f ,T},f)l))[t,t1] = R[t,t *]
and the generator is
R/(J,T),fl))[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 nonisotropic 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
nontrivial. ® 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 KTHEORY
where
M = R'/P,
with
z
acting as
acting as and
P = (zl,x  Vb/a y ) . Now
y.
1,
Therefore
X~l > R > N > 0 since
§12.
M
Vb/a y
and
has
[N] = 0
[M] = [M/N] but
xf,yf = xe € N.
which
is
y e = 1
(z1)
P fl R is a nonmaximal prime
N = Re C M so
and
is generated by
Ann^(e) = P fl R
(z1) € P fl R
ideal.
acting as
As an R module,
f = Vb/a. But
since
x
M ~ k'[y]
by
0 + R
M/N = R/(x,y,zl)
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(t2)) R^ = k[x^
x ,t,t *]/(q(x)+t(t2)) =
k[y^,...,yn ,z,z *]/(q(y) +l2z) 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)+l2z))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 KTHEORY
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^XY) » 0 be defined by where q' XY
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*(XY) = 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 SeveriBrauer 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 nonisotropic. Let
X = X(q) C Pn+^
deg A^(X) = TL for
be defined by
i < d1
rk q = n + 2
q = 0.
deg A^ = 2Z
and
Then for
d < i < n.
Of course
For
Proof.
dimension
A^(X) = 0
i < d1, 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+li
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 KTHEORY
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^(Yp)
shows that
0.
Since the rational point
deg Aq(X) = TL we can assume
A ^ Y ) « A.(Yp). (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^+ ^(Yp) 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  Va^/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/2torsion = Z.
n = 2m
Then
over
action we see that A,.(X')
i^n,
and
Suppose now that
G
X = X(q)
BUNDLES, MODULES AND KTHEORY
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 KTHEORY
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 Ktheory 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 K0q(X;Z/m)
from
K^(A;Z/m) £
Kq (B;Z/m) ~ KUq(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 Bmodule with
a(bm) = a(b)a(m). Then
Conversely, if
Proof.
G.
A C B
Let
N
is an
B ®^M ~ Amodule, 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 KTHEORY
COROLLARY 14.2.
The category of
to the category of true for
503
Amodules is equivalent
Bmodules 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 Amodule. 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)) £ $(A).
Proof.
The
left side is the category of Gequivariant
functors and natural maps. F(l) = M, then gives
M
M*7
If
F
is such a functor and
and
F
on
which defines a
G
action on
F(cr) = vP
Hom(l.a) = {o} M
as in
Lemma 14.1.
COROLLARY 14.4. Quillen’s
Q
CatG (EG, Q#(B)) ~ Q$(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 KTHEORY
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 Amodules
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 Rprojective 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 Amodule
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 KTHEORY
nonaffine 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 Bmodules 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
nondegenerate 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 KTHEORY
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
Amodule.
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,zl).
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 KTHEORY
and
B = A[p *] = A[(l+z)u ^].
then
v = t(zl),
z = ut1
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*(axyz,cz 1) = (cz 1)
is
Here we can assume
*2 k then
Now
is invertible with
2 1 t = (axyz)(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.
nonisotropic and
so
= (1,u *).
Therefore
p.
has
16.3
2 p = (u,az 1)
= k(V~a). Here
t.u
Since Bprojectives 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]/(u2ct2+ac) = R(q') v =a
1
t.
Then we have a
which must be an isomorphism since
both rings are domains of dimension 2. Bprojectives 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 KUMARNORI) . Let
R
be any (nonzero)
commutative ring. Let A = R[x^,...,x ,y^,...,yn ]/(^x^yjl). Let
m^,...,m
be nonnegative integers with
mod (nl) !. Then the unimodular row
m^m^.•
£
ml mn (x^ ,...,xr )
defines a nonfree 'projective Amodule.
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 KTHEORY
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 j1
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(lz)).
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 Jn1 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 n1.
where
B'
Since the lemma is trivial n1.
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 n1 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 KTHEORY
•£
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 € Btx^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 (ta.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 iz(lz), 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^(lz) *
y ,z]/ ^n J
1z and get
un z]/( ^ x 1u 1z,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 = ta^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 (nl)!.
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 KTHEORY
[ B /I ]
and
W '(f) = ( n  l ) ! f
m^m^. . .m
for
(x™! v 1
in
x mn) n J
N >> 0. B. . 1z
) = ) a.x.m i J' L i i
(1) v j
0
M
Therefore
Let
f:B? 1z
have kernel
B1?l  z »J1 1z
0
d iv id e s
17 . 3 .
N N z (1z) €
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(lz) z(lz)
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(lz)
If
A, the same will be true over Bz (i_z )
with
B
> z(lz)
is
via the
g(x i) = x ig(y1) =
^(1z) 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(lz) z(lz) 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 »
//
(B111) , — — » (B n ) 1 v z'l: v z ylz By p a tc h in g we g e t
( J l  z > z  > 0
// (V lz
 * ° 
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 KTheory, 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), 461513.
[BR]
S. M. Bhatwadekar and R. A. Rao, On a question of Quillen, Trans. Amer. Math. Soc. 279 (1983), 801810.
[BSL]
A. Borel et JP. Serre, Groupes de Lie et puissances reduits de Steenrod, Amer. J. Math. 75 (1953), 409448.
[BSRR] A. Borel et JP. Serre, La theoreme de RiemannRoch (d’apres A. Grothendieck), Bull. Soc. Math, France 86 (1958), 97136. [Bo]
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), 215218.
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M. Carral, Modules projectifs sur les anneaux de fonctions, J. Algebra 87 (1984), 202212.
[CF]
L. Claborn and R. Fossum, Generalizations of the notion of class group, 111. J. Math. 12 (1968), 228253.
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[FAC]
J.P. Serre, Faisceaux algebriques coherents, Ann. of Math. 61 (1955), 197278.
[For]
0. Forster, Uber die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring, Math. Z. 84 (1964), 8087.
[Fos]
R. Fossum, Vector bundles over spheres are algebraic, Invt. Math. 8 (1969), 222225.
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E. Friedlander, Etale Ktheory II, Ann. Sci. Ec. Norm. Sup. 15 (1 982 ), 231256.
[GC]
A. Grothendieck, La theorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137154.
[GR]
A. V. Geramita and L. Roberts, Algebraic vector bundles on projective space, Invent. Math. 10 (1970), 298304.
[GQ]
D. Grayson, Higher algebraic Ktheory II (after D. Quillen), pp. 217240 in Algebraic KTheory, Lect. Notes in Math. 551 SpringerVerlag, Berlin 1976.
[H]
R. C. Heitmann, Generating ideals in Priifer domains, Pacific J. Math. 62 (1976), 117126.
[Hi]
H. Hironaka, Smoothing of algebraic cycles of small dimension, Amer. J. Math. 90 (1968), 154.
[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 Ktheories algebriques et topologiques de quelques varietes algebriques, Strasbourg 1971 and C. R. Acad. Sci. Paris 272 (1971), 13731375.
[JK]
J. P. Jouanolou, Quelques calculs en Ktheorie des schemas, in Algebraic KTheory I, Lect. Notes in Math, 341,, SpringerVerlag, Berlin 1973.
[JRR]
J. P. Jouanolou, RiemannRoch sans denominateurs, Invent. Math. 11 (1970), 1526.
[K]
M. Kong, Euler classes of inner product modules, J. Algebra 49 (1977), 276303.
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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 Kfunctors, pp. 4356 in Algebraic Ktheory I, Lect. Notes in Math, 341, SpringerVerlag, Berlin 1973.
[Lin]
H. Lindel, On a question of Bass, Quillen, and Suslin concerning projective modules over polynomial rings, Invent. Math. 65 (1981), 319323.
[Lo]
K. Lonsted, Vector bundles over finite CWcomplexes are algebraic, Proc. Amer. Math. Soc. 38 (1973), 2731.
[Ma]
R. Marlin, Anneaux de Grothendieck des varietes de drapeaux, Bull. Soc. Math. France 104 (1976), 337348.
[MeS]
A. S. Merkur’ev and A.A. Suslin, Kcohomology of the SeveriBrauer varieties and the norm residue homomorphism, Izv. Akad. Nauk. SSSR ser. mat. 46 (1982), 10111061 (=Math. USSR Izv. 21 (1983).
[Mi]
H. Miller, The Sullivan conjecture, Bull. Amer. Math. Soc. 9 (1983), 7578.
[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), 242253.
[MS]
M. Pavaman Murthy and R. G. Swan, Vector bundles over affine surfaces, Invent. Math. 36 (1976), 125165.
[0]
H. Ozeki, Chern classes of projective modules, Nagoya Math. J. 23 (1963), 121152.
[P]
B. Plumstead, The conjectures of Eisenbud and Evans, Amer. J. Math. 105 (1983), 14171433.
[QK]
D. Quillen, Higher algebraic Ktheory I, pp. 85147 in Algebraic KTheory I, Lect. Notes in Math. 341, SpringerVerlag, Berlin 1973.
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[QP]
D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167171.
[Ka]
M. Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 126.
[RoB]
L. G. Roberts, Base change for
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varieties, pp. 122134 in Algebraic KTheory II, Lect. Notes in Math. 342, SpringerVerlag, Berlin 1973. [RoC]
L. G. Roberts,
of a curve of genus zero, Trans.
Amer. Math. Soc. 188 (1974), 319326. [RoK]
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L. G. Roberts, Comparison of algebraic and topological Ktheory, in Algebraic KTheory II, Lect. Notes in Math, 342, SpringerVerlag, Berlin 1973.
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P. Samuel, Sur les anneaux factoriels, Bull. Soc. Math. France 89 (1961), 155173.
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JP. Serre, Modules projectifs et espaces fibres a fibre vectorielle, Sem. DubreilPisot 11, Paris 1957/58.
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[SwL]
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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 schemetheoretic 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 Gmodule 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 EilenbergMoore [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 Gmodules
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 Gmodule
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 Gmodule, 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 EilenbergMacLane cohomology of the finite Chevalley group ^ rr 1 H (G,Vv J)
G(F^d)
of
F^drational points of
denotes the rational cohomology of
G
G with
and
LIMITS OF INFINITESIMAL GROUP COHOMOLOGY (r ) Vv J
coefficients in the Gmodule
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), 353374.
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 Gmodule obtained by inducing the onedimensional rational Bmodule defined by A
A
from
B
to
G.
Then
is an indecomposable rational Gmodule 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 wellknown 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 socalled Steinberg modules St^
defined for
r>0.
If
p
denotes the weight defined
as onehalf the sum of the positive roots of = (p
r
 i)pi
G
$,
then
St^
is an irreducible, selfdual Gmodule 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 Gmodnles, where
LIMITS OF INFINITESIMAL GROUP COHOMOLOGY The natural Bmodule homomorphism
Proof'X » (p to
527
G
r
^ iQ  l)p ® —jlx I
determines by induction from
the asserted injection.
An increasing filtration Gmodule
V
B
§§
{Fg , s > 0}
of a rational
is said to be a good filtration if each Q is Gisomorphic 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 Gmodules 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 nonzero 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
nonzero dominant weight
G
does not contain a factor of
p = 2.
For any
n > 0
X,
Hn (Gr ,AG ) = 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 LyndonHochschiIdSerre
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 Galgebra
(
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 Galgebras
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 Galgebra satisfying H*(G ,k) = H * ^ v r J r
so that
= A . Then we can consider the rational
Galgebra 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 LyndonHochschildSerre
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 Gmodule. 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 Gmodule.
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). «
nonzero dominant weight Gacyclicity 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 Gmodule
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 "HochschildSerre" 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 < n2,
while if
factor of type n1.
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
Gsubmodule of a rational Gmodule
0.
< i
Y X
act on are
Y
so that
f
and the
Gequivariant. 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 KTHEORY 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 Gaction is given under local
trivializations as a set of homomorphisms compatible with the cocyle. taking values in with a Gaction.
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 Gvector 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 Gvector bundles to
(4.4) V
(4.5)
0