Symposium on Algebraic Topology

117 Pages·1971·1.792 MB·English

by  D. W. Anderson (auth.), Peter J. Hilton (eds.)

Lecture Notes ni
Mathematics
A collection of informal reports and seminars
Edited by .A Dold, Heidelberg and .B Eckmann, Z0rich
249
m u i s o p m y S
no Algebraic Topology
Edited by Peter .J Hilton
Battelle Seattle Research Center, Seattle, WA/USA
galreV-regnirpS
Berlin. Heidelberg- New York 1791
AMS Subject Classifications (1970): 55Bxx, 55Dxx, 55Fxx, 55Jxx, 57Dxx
ISBN 3-540-05715-3 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-05715-3 Springer-Verlag New York • Heidelberg • Berlin
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© bSyp ringer-Verlag Berlin. Heidelberg .1791 Library of Congress Catalog Card Number .104581-97 Printed in Germany.
Offsetdmck: Julius Beltz, Hemsbach/Bergstr.
Dedicated to the memory of Tudor Ganea (1922-1971)
FOREWORD
During the academic year 1970-1971 the University of Washington instituted
a program of concentration in the area of algebraic topology in conjunction with the
Battelle Seattle Research Center. As part of that program the Center acted sa host
to a symposium which took place during the week of February 22-26, 1971. Several
topologlsts were invited from universities in the United States~ and there were
present, in addition to those invited, the regular members of the University of
Washington mathematics faculty, the mathematicians associated with Battelle, the
mathematicians visiting the University of Washington in conjunction with the year's
activities in topology, and several other topologists who were interested to attend.
Some of the talks given were of a very informal nature and, in those cases, the
speakers preferred not to provide a manuscript. On the other hand, in most cases,
the speaker did write up his talk subsequently so that this volume contains a fairly
complete record of the scientific program.
It is a pleasure to acknowledge the kindness of many people at the Battelle
Seattle Research Center who helped to make the occasion such a very pleasant and
productive one. In particular, I would llke to mention Mr. Louis .M Bonnefond, Miss
Kay Killingstad and Miss Penny Raines who made all of the necessary arrangements and
insured that the symposium ran with the smoothness which one has come to associate
with Battelle in Seattle. Further, I would like to express my own appreciation to
Mrs. Lorraine Pritchett for having helped so very much in the final preparation of
the manuscripts.
A further and more somber duty devolves upon me. The February symposium
was the last scientific meeting attended by my good friend and colleague, Tudor
Ganea, before his death. We topologists will all miss him very much indeed. At the
symposium he was not able to give a talk but he did distribute a preprint containing
VI
a list of unsolved proble~ in hie particular area of interest. I have therefore
included his catalog of problems in the proceedings o£ the syaposiua. I have also
dedicated this voltme to his memoxy, a gesture which, I mJ sure, ~rlll c~nd the
assent of all of the partic£psnts,
Battelle Seattle Research Center, August, 1971 Peter Hilton
Contents
D. W. Anderson:
Chain Functors and Homology Theories . . . . . . . . . . . . . . . . I
E. Dror:
A Generalization of the Whitehead Theorem . . . . . . . . . . . . . 13
T. Ganea:
Some Problems on Numerical Homotopy Invariants . . . . . . . . . . . 23
S. Gitler and J. Milgram:
Unstable Divisibility of the Chern Character . . . . . . . . . . . . 31
P. J. Hilton, G. Mislin, and J. Roitberg:
Sphere Bundles Over Spheres and Non-Cancellation Phenomena ..... 34
A. Liulevicius:
On the Algebra BP. (BP) . . . . . . . . . . . . . . . . . . . . . . 47
J. Milgram:
Surgery, BpL , BToP, and the PL Bordism Rings . . . . . . . . . . . . 54
G. Mislin:
The Genus of an H-Space . . . . . . . . . . . . . . . . . . . . . . 75
J. C. Moores
Bockstein Spectral Sequence, Modified Bockstein Spectral Sequences,
and Hopf Algebras Over Certain Frobenius Rings . . . . . . . . . . . 84
J. C. Moore and F. P. Peterson:
Nearly Frobenius Algebras and Their Module Categories ....... 94
D. L. Rector:
Loop Structures on the Homotopy Type of 3 S . . . . . . . . . . . . . 99
J. D. Stasheff:
Sphere Bundles Over Spheres as H-Spaces Mod p> 2 .......... 106
Addresses of Contributors . . . . . . . . . . . . . . . . . . . . . 111
,,,,,~1.AHC FUNCTORS ,AND HOMOLOGY THEORIES
D, W, Anderson
In his paper on homotopy everything H-spaces [7], .G Segal
showed that there was a relationship between O-spectra and certain
types of functors from the category of finite basepointed sets to the
category of topological spaces. These functors he called special
Y-spaces. We shall introduce the concept of a chain functor below,
which is essentially the same notion as a special Y-group, but our
treatment of this concept will be entirely different from Segal's.
From our point of view, the category of fir~Ite basepointed sets will
arise naturally.
Chain functors seem to be a very convenient way to describe
homology theories and their associated spectra. Because spectra can
be constructed very explicitly from chain functors, we get several
new results. For example, we obtain constructions of the spectra for
the various connective K-theorles (including Im J theory) which
lead to strictly associative multiplications on these spectra, as
well as infinitely homotopy commutative multiplications.
As a second example of a result of this construction, we obtain
an interesting spectral sequence, which I call the n-th order Ellen-
berg-Moore spectral sequence. We shall define a functor Torn**(A)
for a commutative augmented Graded algebra A over a field K. If
n = ,l Torn**(A) = Tor.A(K,K). If X = O~Y, where Y is an in-
finite loop space, the n-th order Eilenberg-Moore spectral sequence
graded group associated to a filtration of H.(Y;K). Here H.(X;K)
is the ordinary homology of X with coefficients in ,K given the
usual Pontrjagin product structure. If ~ is abelian group, and
if the group ring K[~] is considered to be concentrated in degree ,0
we obtain the relation H.(K(N,n);K) = Tor~,0(K[N]). (The condition
above that Y be an infinite loop space is not actually necessary.)
The theory of chain functors is made more useful by the theory of
permutative categories. Permutative categories arise in nature, and
give rise to all of the "geometrically" defined homology theories
except for the bordism theories. Also, every theory defined by a
permutatlve category has associated to it equlvariant cohomology theo-
ries, in much the manner of equivariant K-theory. This will be dis-
cussed in a subsequent paper [1].
The theory of chain functors is adequate to describe all homology
theories and all homology operations. However, not all homology opera-
tioD~ naturally present themselves as natural transformations of chain
functors, but only as homotopy natural transformations. Quillen's map
BGL(~)+-~BGL(C) ]5[ is one term in such a homotopy natural trans-
formation which is of some importance. This will be discussed in
.)a[
.1 Chain functors
A chain functor ~ is a zero preserving covariant functor from
the category of finite basepolnted sets to the category of slmplicial
groups which satisfies the following relation.
(1.1) For any two basepointed sets ,X ,Y the natural map
(XvY) +- ~(X) x ~(Y) is a homotopy equlv~lence (of simpllclal sets).
The assumption that ~ takes values in the category of simpll-
clal groups is made for technical convenience. We could, with some
slight increase in effort, replace "group" by "monold" or even "set".
In the first instance, we obtain what we call semichain functors. In
the second, we obtain Segal's "special r-spaces".
The assumption that ~ takes values on finite sets is also un-
necessary. We could replace it by the assumption that ~ was com-
patible, up to homotopy, with direct limits. Indeed, one can use
direct limits to extend a chain functor from the category of finite
sets to all sets. Notice that (i.I) will again hold in this context
if we extend in this way.
If ~ is a chain functor, and if X is a simplicial set, we can
define a bisimplicial set ~(X) by ~(X)~,n) = ~(X~))~). The set
of chains .~ )X( of X defined by ~ is the diagonal simpllcial
set of ~(x).
The homology groups H.(X;,) of X defined by * are given by
~.(X;,) = ~.(..(X)). To verify that this defines a homology theory,
we make a few observations. If A is a subsimplicial set of X, for
each n20, X~) =A~)v (X/A~)), so that .(A~))-~Ker(.(X~))
~- ~((X/A)~)) is, by (i.i), a homotopy equivalence. Quillen's
spectral sequence [4] for the homotopy groups of a blsimplicial group
shows immediately that .9 )A( ~- Ker(~. )X( *- .~ (X/A)) is a homotopy
equivalence. Since (X/A) ~) is a retract of X~) for all ,n
~(X~)) ~- ~_((X/A)~)) is surJective for all ,n so that ~.(A) ~-
~X) *- )A/X(.~ is homotopy equivalent to a fibration. Thus the
functor satisfies all of the necessary axioms for a homology theory
except for the homotopy axiom.
To verify the homotopy axiom, we must make a general construc-
tion. If ,X Y are two basepointed sets, X A ~(Y) is the one point
union of as many copies of ~(Y) as there are non-basepoint elements
in X. Each element of X determines an inclusion Y-* X ^ Y, and
if we apply ~ to this inclusion, we obtain ~(y) .- (XAY). ~ If we
take the one point union of these maps, we obtain X A ~(Y) ~- ~(XAY).
Similarly, if X, Y are simplicial sets with basepoints, we obtain
X A )Y(.~ ~- .~. (XAY). If we take X to be the one point compactiflca-
tion of the 1-simplex, we can easily see that f0" fl: Y'+ Z homo-
topic implies that ~.(fo) , ~.(fl): ~.(Y) ~- ~.(Z) are homotopic.
Thus H.(-;~) satisfies the axioms for a homology theory.
Notice that there is a spectrum Spec(~) = [~.(sO),~.(SI),...)
obtained from the maps 1 ^ S ~.(S n) *- ~°(SIAs n) = ~.(S n+l) (here we
choose simplicial representatives for the n S so that
S n+l 1 = ^ S sn+l). There is a natural transformation of functors
X ^ .~ (S n) *- ~.(xAsn). Since ni(~. (xAsn)) = ~(xAsn;~) = Si_n(X~),
we obtain a natural transformation of functors ~(X;Spec(~)) -+
~(X;~). This transformation induces an isomorphism on the groups of
S ,0 and so is an isomorphism of homology theories.
As a technical point, notice that the maps X ^ ~.(Y) -+~.(XAY)
extend to X ® ~.(Y) ÷- ~.(XAY). In this manner we can keep all of our
constructions group valued.
There is an obvious notion of a natural transformation of chain
functors, and every such natural transformation defines a homology
operation. Since our groups are nonabelian, even the homology opera-
tion multiplication by 2, which on the chain level is essentially
the squaring map, will not be represented by a natural transformation
of chain functors unless the groups ~.(X) are free slmpllclal groups
for all X. Fortunately, there is a "freeln~'coD~truction ~+~F,
together with a natural transformation ~F_+ ~ such that for all X,
~F.(x) is free, and ~F.(x) +- ~.(X) is a homotopy equivalence. For
details see [I].
We can use an idea of Segal's [7] to show that every slmplicial
group spectrum • = {Gn] defines a chain functor C(~). This is done
by letting C(~)(n +) = lim F((si)n, Gi), where F denotes the func-
tion complex of basepoint preserving maps, + n is the set with n
non-basepoint elements, and the map Z((Si) n) +- (si+l) n is the pro-
duct of the suspensions of the projections Z((si) n) -+ZS i = S i+l
onto the factors. Notice that C(~) (n +) has the same homotopy type
as limF(Siv...vSI, Gi ,) so that C(~) satisfies (i.i). Clearly,
)~I(C is a chain functor, and C(~) (i +) naturally has the homotopy
type of llm OiGi . One can easily show that C(~).(S )n has the