Hello everybody,
This is the first installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Feedback is welcomed. Instructions for subscribing or
unsubscribing to this list, as well as getting the papers on the list,
are at the end of this message. Please tell me if I am using an
out-of-date e-mail address for you.
Papers uploaded to Hopf between Dec. 20, 1994 and Jan. 1, 1995:
1. Hopkins-Ravenel-Wilson/moravaktheory
Morava Hopf algebras and spaces K(n) equivalent to finite Postnikov
systems,
by Michael J. Hopkins, Douglas C. Ravenel, and W. Stephen Wilson.
We have three somewhat independent sets of results.
Our first results are a mixed blessing.
We show that Morava $K$-theories don't see $k$-invariants for homotopy
commutative $H$-spaces which are finite Postnikov systems, i.e.
for those with only a finite number of homotopy groups.
Since $k$-invariants are what holds the space together, this
suggests that Morava $K$-theories will not be of much use
around such spaces.
On the other hand, this gives us the Morava $K$-theory of a wide
class of spaces which is bound to be useful.
In particular, this work allows the recent work in \cite{RWY}
to be applied to compute the Brown-Peterson cohomology
of all such spaces.
Their Brown-Peterson cohomology
turns out to be all in even degrees (as is their Morava $K$-theory)
and flat as a $BP^{*}$ module for the category of finitely presented
$BP^{*}(BP)$ modules.
Thus these examples have extremely nice
Brown-Peterson cohomology which is as
good as a Hopf algebra.
Our second set of results produces a large family of spaces which
behave as if they were finite Postnikov systems from the point
of view of Morava $K$-theory even though they are not.
This allows us to apply the above results to an even wider class
of spaces than finite Postnikov systems.
These examples come from spaces in omega spectra with certain
properties.
There are many well known examples with these properties.
In particular, we compute the $K(n)$ homology of all the spaces
in the $\Omega$-spectra for $P(q)$ and $k(q)$ where $q > n$.
In order to prove our results on finite Postnikov systems we
need our third set of results; a beginning of an analysis of
bicommutative Hopf algebras over $K(n)_*$.
2. JPMay/completions
Completions in algebra and topology
by J.P.C. Greenlees and J.P. May
Abstract
We discuss algebraic completions at ideals and localizations away from
ideals in a commutative ring, and we use the framework of ``Modern
foundations for stable homotopy theory'' to show how this algebra can be
mimicked topologically for ideals in the coefficient ring of an E
infinity ring spectrum. The algebraic fact that completion is not exact
forces us to work with the derived functors of completion, and we
explain how topological completions of spectra mimic an algebraic
description of these derived functors in terms of ``local homology
groups''. These constructs are relevant to cohomology theories. The dual
constructs relevant to homology theories involve Grothendieck's ``local
cohomology groups''. There are concomitant notions of ``\v{C}ech
homology and cohomology groups'', which fit into algebraic fibre
sequences that we mimic by fibre sequences of spectra. These lead to a
new theory of localizations of spectra away from ideals. When
specialized to MU-modules, these localizations shed light on the
chromatic filtration and the chromatic convergence theorem.
Contents:
Algebraic definitions: local and \v{C}ech cohomology and homology
Connections with derived functors; calculational tools
Topological analogues of the algebraic definitions
Completion at ideals and Bousfield localization
Localization away from ideals and Bousfield localization
The specialization to ideals in $MU_*$
This paper is to appear in ``The handbook of Algebraic Topology'',
edited by Ioan James.
3. JPMay/derived_categories
Derived categories in algebra and topology
by J.P. May
Abstract
An analogy between the derived category of modules over a
commutative ring and the stable homotopy category of spectra is elaborated
to a much closer analogy between the derived category of E infinity modules
over an E infinity algebra and the derived category of E infinity module
spectra
over an E infinity ring spectrum. In both the algebraic and topological
contexts, these new derived categories allow one to study ``modules up to
homotopy'' over ``commutative algebras up to homotopy'' in much the same way
that one studies ordinary modules in classical homological algebra. There are
many applications in algebraic topology, algebraic K-theory, and algebraic
geometry. This expository note explains the ideas and gives a brief summary
of the relevant definitions in both contexts.
This paper will appear in the proceedings of the Eleventh International
Conference on Topology, Trieste, 1993.
4. JPMay/equivariant_theory
Equivariant stable homotopy theory
by J.P.C. Greenlees and J.P. May
Abstract
After sketching the basic concepts of space level equivariant homotopy theory,
we introduce the basic ideas and constructions of spectrum level equivariant
homotopy theory, combining earlier work of Lewis and May with the
framework of ``Modern
foundations for stable homotopy theory''. We then illustrate ideas by
explaining the fundamental
localization and completion theorems that relate equivariant to
nonequivariant homology and
cohomology. A key idea is that ``completion theorems'' in cohomology are
sometimes consequences
of results that deserve to be called ``localization theorems'' in
homology. For example, for
finite groups G, the Atiyah-Segal completion theorem that computes the K
cohomology of BG is a
consequence of a localization theorem that computes the K homology of
BG. We describe a recent
result that gives the same kind of localization and completion theorems
for the spectrum MU(G)
that represents a stabilized version of equivariant complex cobordism
and for all module spectra
over MU(G). For example, this applies to equivariant versions of
Brown-Peterson and Morava
homology and cohomology theories. We also discuss equivariant
cohomotopy, a theory for which
the cohomological completion theorem is true, by Carlsson's proof of the
Segal conjecture,
but the homological localization theorem is false.
Contents:
Equivariant homotopy
The equivariant stable homotopy category
Homology and cohomology theories and fixed point spectra
Change of groups and duality theory
Mackey functors, $K(M,n)'s$ and $RO(G)$-graded cohomology
Philosophy of localization and completion theorems
How to prove localization and completion theorems
Examples of localization and completion theorems
This paper is to appear in ``The handbook of Algebraic Topology'', edited by Ioan James.
5. JPMay/modern_foundations
Modern foundations for stable homotopy theory
by A.D. Elmendorf, I. Kriz, and J.P. May}
Abstract
We describe the foundations of stable homotopy theory to be established in our
monograph ``Rings, algebras, and modules in stable homotopy theory'', in
preparation, which will have Michael Mandell as a fourth author.
Contents:
Spectra and the stable homotopy category
Smash products and twisted half-smash products
The category of S-modules and its derived category
The smash product of S-modules
A infinity and E infinity ring spectra and their modules
The smash product of R-modules and function R-modules
Tor and Ext in topology and algebra
Universal coefficient and Kunneth spectral sequences
Algebraic constructions in the derived category of R-modules
Algebra structures on localizations and on quotients by ideals
The specialization to MU-modules and algebras
The paper is already slightly obsolete, in that the definitive treatment
will be based
on a modified category of S-modules with a smash product that is not
only commutative and
associative but also unital.
This paper is to appear in ``The handbook of Algebraic Topology'',
edited by Ioan James.
6. JPMay/operads_motives
Operads, algebras, modules, and motives
by Igor Kriz and J.P. May
Abstract
With motivation from algebraic topology, algebraic geometry, and string
theory, we study various topics in differential homological algebra.
The work is divided into five largely independent Parts:
Definitions and examples of operads and their actions
Partial algebraic structures and conversion theorems
Derived categories from a topological point of view
Rational derived categories and mixed Tate motives
Derived categories of modules over $E_{\infty}$ algebras
In differential algebra, operads are systems of parameter chain complexes
for multiplication on various types of differential graded algebras
``up to homotopy", for example commutative algebras, n-Lie algebras,
n-braid algebras, etc. Our primary focus is the development of the
concomitant theory of modules up to homotopy and the study of both classical
derived categories of modules over DGA's and derived categories of modules
up to homotopy over DGA's up to homotopy. Examples of such derived
categories provide the appropriate setting for one approach to mixed Tate
motives in algebraic geometry, both rational and integral.
This monograph will appear in Asterisque.
--------------------------
Hello everybody,
This is the second installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Messages:
1. If you got this message from me (hovey---math.mit.edu) then you are
currently subscribed. I am sure that not every one who wants to be on
this list is on it, so I encourage subscribers to forward this to others
who are not on it and are interested. Current graduate students in
particular should know about it.
2. Last time the papers in the JPMay directory were not actually ready
to be downloaded. They are now. I will try to prevent this kind of
mistake from happening in the future.
3. The WWW web interface for the archive has improved. Open the URL
http://hopf.math.purdue.edu/pub/hopf.html using Lynx, Mosaic, or
Netscape.
Mark Hovey
Papers uploaded to Hopf between Jan. 2 and Jan. 18, 1995:
1. /pub/Nakano-Palmieri/nakano-palmieri.dvi
D.K. Nakano and J.H. Palmieri,
Support varieties for the Steenrod algebra
In this paper we study the cohomological varieties associated to the
finite-dimensional sub-Hopf algebras of the Steenrod algebra. A
stratification theorem like the Quillen/Avrunin-Scott stratification
theorem for finite groups is proven. With this stratification one can
then invoke results from restricted Lie algebra cohomology to study
these cohomological varieties. As a result, we get a description of
the cohomology of these Hopf algebras, modulo nilpotence; we also
prove a conjecture of Margolis about $P^{s}_{t}$-homology of a tensor
product of modules.
2. /pub/Palmieri/palmieri-quasi.dvi
A note on the cohomology of finite dimensional cocommutative Hopf algebras
John H. Palmieri
In the context of finite dimensional cocommutative Hopf algebras, we
prove versions of various group cohomology results: the Quillen-Venkov
theorem on detecting nilpotence in group cohomology, Chouinard's
theorem on determining whether a $kG$-module is projective by
restricting to elementary abelian $p$-subgroups of $G$, and Quillen's
theorem which identifies the cohomology of $G$, ``modulo nilpotent
elements.'' This last result is only proved for graded connected Hopf
algebras.
-----------------
Hello everybody,
This is the third installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
New Message:
Those with a World-Wide Web browser will notice some changes at
http://hopf.math.purdue.edu/pub/hopf.html . In particular, there is now
a link to a home page for the Purdue math department, which list
seminars and colloquia happening there. There are also some conference
announcements on the hopf.html page, including the Great Lakes K-theory
conference in March, the Midwest topology seminar in February, the
Lehigh conference in June, and the SUNY Geometry festival in
April. There is also a link to the K-theory calendar of events,
maintained by Dan Grayson at the University of Illinois at Chicago
Circle.
2. If you have a Mac, you may be interested in /pub/macweb1.00A3.sea.hqx,
which a BinHexed Stuffit archive of MacWeb, a WWW-browser similar to
Mosaic and Netscape.
Old message:
If you got this message from me (hovey---math.mit.edu) then you are
currently subscribed. I am sure that not every one who wants to be on
this list is on it, so I encourage subscribers to forward this to others
who are not on it and are interested. Current graduate students in
particular should know about it.
Mark Hovey
Papers uploaded to Hopf between Jan. 18 and Feb. 2, 1995:
1. /pub/Ravenel-Wilson/pnhopfring.dvi
\title{The Hopf ring for $P(n)$}
\author{Douglas C. Ravenel
\thanks{Partially supported by the National Science Foundation}
\\University of Rochester\\Rochester, New York 14627\\
{\small drav---troi.cc.rochester.edu}
\and
W. Stephen Wilson
\\Johns Hopkins University\\Baltimore, Maryland 21218\\
{\small wsw---math.jhu.edu}}
\maketitle
\begin{abstract}
We show that $E_*(\pn{n}{*})$, the $E$-homology of the
$\Omega$-spectrum for $P(n)$, is an $E_*$ free
Hopf ring for $E$ a complex oriented theory with $I_n$ sent to $0$.
This covers the cases $P(q)_*(\pn{n}{*})$ and
$K(q)_*(\pn{n}{*})$, $q \geq n$.
The generators of the Hopf ring are those necessary for the stable
result.
The motivation for this paper is to show that $P(n)$ satisfies
all of the conditions for the machinery of unstable cohomology
operations set up in Boardman-Johnson-Wilson.
This can then be used
to produce splittings analogous to those
for $BP$.
\end{abstract}
2. /pub/Jardine/README
(Note from Hovey: Jardine has set up his own home page. I will not be
announcing which papers he has there unless he uploads them to Hopf. If
you are using a WWW-browser, go to /pub/Jardine and you will see a link
to his home page.)
If you've got a web browser like mosaic or lynx,
go to Jardine's subdirectory on the UWO Math. Dept. WWW server, which
is right here.
Alternatively, Jardine's preprints are available by anonymous ftp at
jardine.math.uwo.ca in the subdirectory /pub/papers/jardine.
------------------
This is the fourth installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
It seems to me that everyone will have their own home page on the WWW
within five years or so. However, not everyone has WWW access at the
moment, so please continue putting papers on the Hopf archive even if
you do set up your own home page. It would be helpful as well, if you
develop your own home page, to put abstracts as well as dvi files there
and to make sure to date the papers by time of last modification.
Clarence is willing to put a simple html document on the archive with a
pointer to your home page, like the one Jardine put on hopf
(/pub/Jardine/jardine.html).
Don Davis has his own home page now:
http://www.lehigh.edu/dmd1/public/www-data/dmd1.html
Mark Hovey
Papers uploaded to Hopf between Feb. 3 and Feb. 16, 1995:
1. /pub/Bendersky-DDavis-Mahowald/spn.dvi
v1-periodic homotopy groups of Sp(n)
Martin Bendersky Donald M. Davis Mark Mahowald
In this paper we calculate the 2-primary v1-periodic homotopy groups of
the symplectic groups Sp(n). The proof utilizes new methods of
calculating the unstable Novikov spectral sequence. One corollary is
that some homotopy group of Sp(n) contains an element of order
2^{2n-1} .
2. /pub/Bousfield-DDavis/bous.dvi
The unstable Adams spectral sequence of $SO$ and $U$, and a splitting of
unstable Ext groups
by A.K. Bousfield and Donald M. Davis
We construct algebraic spectral sequences which are conjectured to agree
with the unstable Adams spectral sequences for the infinite unitary and
special
orthogonal groups $U$ and $SO$. A closely related conjecture is that the
unstable Ext groups of $H^*(\Sigma CP^\infty)$ and $H^*(RP^\infty)$ split as
direct sums of the unstable Ext groups for their subquotients consisting
of classes whose degrees have a fixed number of 1's in their binary expansions.
3. /pub/DDavis/harp.dvi
Equivalences of some v1-telescopes
DONALD M. DAVIS
Abstract.Certain naturally occurring spaces have isomorphic v1-periodic
homotopy groups. To each is associated a mapping telescope whose ordi-
nary homotopy groups equal the v1-periodic homotopy groups of the space.
It is proved that the mapping telescopes of the spaces are
homotopy equivalent.
Lehigh University, Bethlehem, Pennsylvania 18015
4. /pub/DDavis/survey.dvi
Computing v1 -periodic homotopy groups of spheres and some compact Lie groups
Donald M. Davis
Contents
1.Introduction
2.Definition of v1-periodic homotopy groups
3.The isomorphism v11ss (S2n+1) ss v11sss2n1 (Bqn)
4.J-homology
5.The v1-periodic homotopy groups of spectra
6.The v1-periodic UNSS for spheres
7.v1-periodic homotopy groups of SU(n)
8.v1-periodic homotopy groups of some Lie groups
References
HANDBOOK OF ALGEBRAIC TOPOLOGY
Edited by I.M. James
1995 Elsevier Science B.V. All rights reserved
5. /pub/DDavis-Yang/huaj.dvi
Tractable formulas for $v_{1}$-periodic homotopy groups of $SU(n)$ when
$n \leq p^{2}-p+1$.
by Donald M. Davis and Huajian Yang
Let $p$ be a fixed odd prime. In \cite{Davis}, it was proved that for
$\epsilon=0$ and 1,
$v_1^{-1}\pi_{2k-\epsilon}(SU(n))$ has order $p^{e(k,n)}$,
where $e(k,n)=\min\{\nu_p(j!S(k,j)):n\le j\le k\}$, with $S(k,j)$ the
Stirling number of the second kind. In this paper, we give a more tractable
formula for $e(k,n)$ when $n\le p^2-p+1$ by calculating the unstable
Novikov spectral sequence. We also determine the abelian group structure
when $\epsilon=1$; it was known to be cyclic when $\epsilon=0$.
6. /pub/Hovey-Sadofsky/tate-bousfield-class.dvi
Tate Cohomology Lowers Chromatic Bousfield Classes
By Mark Hovey and Hal Sadofsky
Let $G$ be a finite group. We use the results of \cite{greenlees-sadofsky}
to show that the Tate homology of $E(n)$-local spectra with respect to
$G$ produces $E(n-1)$ local spectra. We also show that the Bousfield
class of the Tate homology of $L_{n}X$ (for $X$ finite) is the same as
that of $L_{n-1}X$.
To be precise, recall that Tate homology is a functor from $G$-spectra
to $G$-spectra. To produce a functor $P_{G}$ from spectra to spectra, we
look at a spectrum as a naive $G$-spectrum on which $G$ acts trivially,
apply Tate homology, and take $G$-fixed points. This composite is the
functor we shall actually study, and we'll prove that $\langle
P_{G}(L_{n}X)\rangle = \langle L_{n-1}X \rangle$ when $X$ is finite.
When $G=\Sigma_{p}$, the symmetric group on $p$ letters, this is related
to a conjecture of Hopkins and Mahowald (usually framed in terms of
Mahowald's functor $RP_{-\infty}(-)).$
------------------------
This is the fifth installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between Feb. 17 and Mar. 9, 1995:
1. /pub/Aguade-Broto-Notbohm/cookemod2.abstract
A mod two analogue of a conjecture of Cooke
by
J. Aguad\'e, C. Broto and D. Notbohm
Abstract: We study spaces whose mod 2 cohomology
has the form: Poly(x)\otimes Exterior(Sq^1x).
We prove:
Theorem: There is a space X with this cohomology
if and only if x has degree 2, 4 or 8.
The 'only if' part can be considered as the mod 2
version of a conjecture of Cooke. Its proof is
similar to the proof for old primes (contained
in Cooke.dvi) but one should be slightly more
careful in small degrees. The most interesting
goal of this paper is probably the construction of
what we think to be remarkable space X with
H*(X;F_2) = F_2[x_8] \otimes E(Sq^1x_8)
2. /pub/Boardman/stabop.abstract
DVI FILE: stabop.dvi
TITLE: Stable operations in generalized cohomology
AUTHOR: J. Michael Boardman
TO APPEAR: Handbook of Algebraic Topology, ed. I.M.James,
Elsevier (Amsterdam, 1995).
We describe the structure of the stable operations on E-cohomology
following Adams, in a manner that generalizes to unstable operations. The
appropriate context is the language of comonads and coalgebras over a
comonad. The necessary category theory is developed in detail. Five examples
are presented: ordinary mod p cohomology, unitary cobordism MU, Brown-
Peterson cohomology BP, complex K-theory KU, and Morava K-theory K(n).
3. /pub/Boardman-Johnson-Wilson/bjw.abs
DVI FILE: bjw.dvi
TITLE: Unstable operations in generalized cohomology
AUTHORS: J. Michael Boardman, David Copeland Johnson, W. Stephen Wilson
TO APPEAR: Handbook of Algebraic Topology, ed. I.M.James,
Elsevier (Amsterdam, 1995).
We describe the structure of the unstable operations on E-cohomology
in terms of comonads, in the style of the companion paper on stable
operations. There are two variants, depending on whether we consider only
the additive operations, or all unstable operations. For practical use, we
unpack the comonad information and express it in terms of Hopf rings. Five
examples are discussed: ordinary mod p cohomology, unitary cobordism MU,
Brown-Peterson BP-cohomology, complex K-theory KU, and Morava K-theory K(n).
We give two applications to BP-cohomology. The first shows that the
presence of unstable operations imposes dimensional restrictions on the
Landweber filtration of the BP-cohomology of a finite complex. The second
constructs idempotent operations in degree k that recover the known
unstable splittings of BP-cohomology.
4. /pub/Shipley/convergence.new
(This is a significantly revised version of a paper already on the
archive. I reproduce here an abstract followed by a brief description of
the changes--Mark.)
We produce new convergence conditions for the homology spectral sequence
of a cosimplicial space by requiring that each codegree of the
cosimplicial space has finite type mod $p$ homology. Specifically, we
find conditions which ensure strong convergence if and only if the total
space has $p$-good components. We also find exotic convergence
conditions for cosimplicial spaces not covered by the strong convergence
conditions. These results give new convergence conditions, for
example, for the Eilenberg-Moore spectral sequence and for mapping
spaces.
This new version contains
several generalizations of the old results. Specifically, the
requirement of a non-empty total space is no longer needed. Also, Corollary
10.3 is a new strong convergence result requiring p-complete codegrees
instead of p-nilpotent codegrees. Of course, there have been other minor
changes and corrections.
5. /pub/MWeiss/betticurv.abstract
Curvature and Finite Domination, by Michael Weiss.
Abstract. Gromov obtained an upper bound on the Betti numbers of
a closed Riemannian manifold in terms of a lower bound on the sectional
curvature. It is shown that Gromov's upper bound is an upper
bound on the minimum number of cells in CW-spaces dominating the manifold.
6. /pub/MWeiss/embed.abstract
Calculus of Embeddings, by Michael Weiss
Abstract. This is a study of spaces of smooth embeddings emb(M,N) in
the spirit of immersion theory, and in the spirit of "Calculus". It
leads to very efficient calculations of emb(M,N) when dim(M) is small
compared to dim(N). Immersion theory
appears as the "first derivative" of embedding theory, and the game is
to find the higher derivatives, i.e. the "Taylor Series". The Taylor
series converges when the codimension, dim(N)-dim(M), is at least 3.
This follows from a multiple disjunction lemma proved recently by
Goodwillie (not in his thesis).
It's an announcement - no proofs.
7. /pub/MWeiss/ortho.abstract
Orthogonal Calculus, by Michael Weiss
Abstract. Orthogonal calculus is a way to explore spaces equipped with
a filtration indexed by the finite dimensional linear subspaces V of
an infinite dimensional euclidean space. Example: BO, filtered by
subspaces BO(V), or BTOP, filtered by subspaces BTOP(V).
Those who like to split big spaces may be
interested, and the hardy ones who still like surgery theory may also be
interested, since many of the moduli spaces in surgery theory come with such
a filtration.
Orthogonal calculus is modelled on Goodwillie calculus:
Among the spaces equipped with a filtration of the type above, some
are "polynomial of degree n", and the game is to approximate
arbitrary ones by polynomial ones (Taylor approximation). First order
approximations in orthogonal calculus have been used heavily by Bruce
Williams and me in papers related to surgery. They look like
generalized total Stiefel-Whitney classes, and second order approximations
look like generalized total Pontryagin classes plus generalized total
Stiefel-Whitney classes.
------------
This is the sixth installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Note that there is a new directory, /pub/pictures, containing pictures
of topologists in gif and jpeg format. You need some kind of viewing
program to see these, like xv (Unix), lview.exe (Windows), or JPEGView
(Mac). With Netscape or Mosaic you can just click on them by first going
to the usual spot, http://hopf.math.purdue.edu/pub/hopf.html . The file
INDEX gives some information about who is in the pictures.
Also note that the papers by J. P. May and coauthors announced in the
first installment of this list are now in more appropriate
directories. That is, there are now directories like Kriz-May and
Greenlees-May, whereas before they were all in JPMay.
Mark Hovey
Papers uploaded to Hopf between Mar. 10 and May 3, 1995:
1. /pub/RBruner/newQ8.abstract
Real connective $K$-theory and the quaternion group
Dilip Bayen and Robert R. Bruner
Mathematics Department
Wayne State University
Detroit, Michigan, 48202
dbayen---math.wayne.edu
rrb---math.wayne.edu
April, 1995
Let ko be the real connective K theory spectrum. We compute ko_*BG and
ko^*BG for groups G whose Sylow 2-subgroup is quaternion of order 8.
Using this we compute the coefficients t(ko)^G_* of the G fixed points
of the Tate spectrum t(ko) for G = Sl_2(3) and G = Q_8. The results
provide a counterexample to the optimistic conjecture of Greenlees and
May [Generalized Tate Cohomology, Conj 13.4], by showing, in
particular, that t(ko)^G is not a wedge of Eilenberg-Maclane spectra,
as occurs for groups of prime order.
---------
This is the seventh installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between May 3 and May 13, 1995:
1. /pub/Blanc/Blanc_hspace.abstract
%
% Homotopy operations and the obstructions to being an H-space
% David Blanc
%
% November 14, 1994
%
The question of whether a given space X possesses such an
H-space structure has been studied from a variety of viewpoints.
Here we address this question from the aspect of homotopy operations.
This is done by reformulating the question in terms of the
realizability of a certain morphism of abelian \Pi-algebras, which
translates in turn (using the author's obstruction theory for
realization of such morphisms) into the requirement that a certain
sequence of higher homotopy operations, taking values in \pi_{\star} X,
vanishes coherently.
We illustrate the theory by a couple of examples: it can be used
to calculate the obstruction to CP^2 being an H-space rationally;
we also show that the "torsion Whitehead product" (which we define)
may be thought of as ``the first higher order obstruction'' to
being an H-space, and give another example.
2. /pub/Blanc/Blanc_loop.abstract
%
% Loop spaces and homotopy operations
% David Blanc
%
% April 27, 1995
%
We describe two obstruction theories for a given topological space X to be
a loop space, both defined in terms of higher homotopy operations:
First, we explain how an H-space structure on X can be used to
define the action of the primary homotopy operations on the shifted
homotopy groups \pi_{\star-1} X (which are isomorphic to \pi_{\star} Y
if X\simeq\Omega Y). \ This action will behave properly
with respect to composition of operations if X is homotopy-associative,
and will lift to a topological action of the monoid of all maps between
spheres if and only if X is a loop space. The obstructions
to having such a topological action may be stated in terms of the author's
obstruction theories for realizing Pi-algebras and their morphisms.
A more concrete approach, which does not require a given H-space
structure on X, yields the following:
Theorem A: If X is a CW complex such that all Whitehead products vanish in
\pi_{\star} X, then X is homotopy equivalent to a loop space if and only if
a certain collection of higher homotopy operations vanish coherently.
The higher homotopy operations in question depend only on maps
between wedges of spheres, and take value in homotopy groups of
spheres. They are constructed by means of a certain collection of
convex polyhedra which may be of independent interest.
3. /pub/Blanc/model.abstract
%
% New Model Categories from Old
% David Blanc
%
% (revision: January 25, 1995)
%
Model categories, first introduced by Quillen, have proved useful in
a number of areas - most notably in his treatment of rational
homotopy, and in defining homology and other derived functors in
non-abelian categories. From a homotopy theorist's point of view,
one interesting example of such non-abelian derived functors is the
E^2-term of the mod p unstable Adams spectral sequence of
Bousfield and Kan. They identify this E^2-term as a sort of Ext
in the category CA of unstable coalgebras over the mod p Steenrod algebra.
The original purpose of this note was to provide an element in
this identification which appears to be missing from the literature:
namely, an explicit model category structure for the category cCA
of cosimplicial coalgebras as above. What one would really like is a model
category for arbitrary categories of cosimplicial universal coalgebras,
analogous to Quillen's treatment of simplicial universal algebras,
which is based on Quillen's ``small object argument'', and
implicitly uses a procedure for transfering model category structures
by means of adjoint functors (in the direction of the left adjoint; the
procedure is made explicit in the paper).
Unfortunately, Quillen's procedure cannot be dualized, in the categorical
sense. The reason is essentially set-theoretic: more can be said about
maps into a sequential colimit of sets than about maps out of a
sequential limit (and thus, for example, colim is exact, for R-modules,
while lim is not).
Therefore, for our purposes we describe alternative (and less elegant)
conditions for using adjoint functors to create new model category
structures. The dual version then allows us to define model category
structures for certain categories of cosimplicial universal coalgebras -
including cCA.
========================
4. /pub/Blanc/towers.abstract
%
% Colimits for the Pro category of towers of simplicial sets
% David Blanc
%
% January 18, 1995
%
The Pro category of towers of spaces (and of other categories) has been
studied in several contexts, and used for a variety of applications in
homotopy theory, shape theory, geometric topology, and algebraic
geometry - as well as in the study of v_n-periodicity in unstable
homotopy theory.
One problem in the usual version of the Pro category of towers is
that, while finite limits and colimits exist, and may be
constructed in a straightforward (levelwise) manner, the same does not hold for
infinite colimits; and these were needed for the application to
v_n-periodicity.
The construction presented here embeds a suitable subcategory of the
Pro category Tow of towers of simplicial sets in a certain category
Net of strict Ind-towers, in which we have explicit constructions
for all colimits, as well as finite limits. This category Net
can thus be thought of as a cocompletion of the Pro category of towers
of spaces.
There are other cocomplete categories in which Tow may be embedded -
for example, the category of all pro-simplicial sets, or the full
category of all inductive systems of towers. One advantage of
the approach described here is that one obtains a smaller, and more mangeable,
cocompletion, in this special case, and the construction of the
colimits may be made quite explicitly. A side effect of our approach
is the elimination of certain ``phantom phenomena'' from the
Pro category of towers.
5./pub/Elmendorf-Kriz-Mandell-May/ekmm.abstract
Title: Rings, modules, and algebras in stable homotopy theory
Authors: A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May
address: Purdue University Calumet, Hammond IN 46323
email: aelmendo---math.purdue.edu
address: The University of Michigan, Ann Arbor, MI 48109-1003
email: ikriz---math.lsa.umich.edu
address: The University of Chicago, Chicago, IL 60637
email: mandell---math.uchicago.edu
address: The University of Chicago, Chicago, IL 60637
email: may---math.uchicago.edu
Let $S$ be the sphere spectrum. We construct an associative,
commutative, and unital smash product in a complete and cocomplete
category $\sM_S$ of ``$S$-modules'' whose derived category $\sD_S$ is
equivalent to the classical stable homotopy category. This allows a
simple and algebraically manageable definition of ``$S$-algebras'' and
``commutative $S$-algebras'' in terms of associative, or associative
and commutative, products $R\sma_S R \darrow R$. These notions are
essentially equivalent to the earlier notions of $A_\infty$ and
$E_\infty$ ring spectra, and the older notions feed naturally into the
new framework to provide plentiful examples. There is an equally
simple definition of $R$-modules in terms of maps $R\sma_S M\darrow M$.
When $R$ is commutative, the category $\sM_R$ of $R$-modules also has
an associative, commutative, and unital smash product, and its derived
category $\sD_R$ has properties just like the stable homotopy
category.
Working in the derived category $\sD_R$, we construct spectral
sequences that specialize to give generalized universal coefficient
and K\"{u}nneth spectral sequences. Classical torsion products and Ext
groups are obtained by specializing our constructions to
Eilenberg-Mac~Lane spectra and passing to homotopy groups, and the
derived category of a discrete ring $R$ is equivalent to the derived
category of its associated Eilenberg-Mac~Lane $S$-algebra.
We also develop a homotopical theory of $R$-ring spectra in $\sD_R$,
analogous to the classical theory of ring spectra in the stable
homotopy category, and we use it to give new constructions as
$MU$-ring spectra of a host of fundamentally important spectra whose
earlier constructions were both more difficult and less precise.
Working in the module category $\sM_R$, we show that the category of
finite cell modules over an $S$-algebra $R$ gives rise to an
associated algebraic $K$-theory spectrum $KR$. Specialized to the
Eilenberg-Mac~Lane spectra of discrete rings, this recovers Quillen's
algebraic $K$-theory of rings. Specialized to suspension spectra
$\Sigma^{\infty}(\Omega X)_+$ of loop spaces, it recovers Waldhausen's
algebraic $K$-theory of spaces.
Replacing our ground ring $S$ by a commutative $S$-algebra $R$, we
define $R$-algebras and commutative $R$-algebras in terms of maps
$A\sma_R A\darrow A$, and we show that the categories of $R$-modules,
$R$-algebras, and commutative $R$-algebras are all topological model
categories. We use the model structures to study Bousfield
localizations of $R$-modules and $R$-algebras. In particular, we prove
that $KO$ and $KU$ are commutative $ko$ and $ku$-algebras and
therefore commutative $S$-algebras.
We define the topological Hochschild homology $R$-module $THH^R(A;M)$
of $A$ with coefficients in an $(A,A)$-bimodule $M$ and give spectral
sequences for the calculation of its homotopy and homology groups.
Again, classical Hochschild homology and cohomology groups are
obtained by specializing the constructions to Eilenberg-Mac~Lane
spectra and passing to homotopy groups.
6. /pub/Green-Leary/extra.abstract
Chern classes and extraspecial groups
David J. Green and Ian J. Leary
djg---math.uchicago.edu
leary---mpim-bonn.mpg.de
Abstract:
The mod-$p$ cohomology ring of the extraspecial $p$-group of exponent~$p$ is
studied for odd~$p$. We investigate the subquotient~$ch(G)$ generated by
Chern classes modulo the nilradical. The subring of~$ch(G)$ generated by
Chern classes of one-dimensional representations was studied by Tezuka and
Yagita. The subring generated by the Chern classes of the faithful irreducible
representations is a polynomial algebra. We study the interplay
between these two families of generators, and obtain some relations between
them.
----------
This is the eighth installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. This list is maintained by Mark Hovey
(hovey---math.mit.edu). Instructions at the end.
Perhaps I should mention that generally I find out about new papers on
hopf the day or day after Clarence makes them available. The erratic
schedule of this list is due to variability in when people put new
papers on the archive.
New business:
1. I have decided to distribute this list through Don Davis' topology
mailing list. The effect of this on you should be minimal, except that
you will receive the other submissions to Don's list as well as my
mailings. This means that you should subscribe and unsubscribe through
Don, not me. The instructions at the end are suitably revised to
reflect this. Let me know your reactions to this, if any.
2. I now have a home page on the world-wide web!!! It contains the back
issues of these mailings, and my own papers, as well as some various
computer and Emacs-related links. No graphics though. The URL is
http://www.mit.edu:8001/afs/athena.mit.edu/user/h/o/hovey/Public/homepage.html
Add it to your bookmarks so you don't have to type it more than once!
Mark Hovey
Papers uploaded to Hopf between May 13 and July 10, 1995:
1. /pub/DJGreen/m24.abstract
Author: David J Green
Title : The 3-local cohomology of the Mathieu group M_24
Status: To appear in Glasgow Math. J.
Date : Submitted 8th August 1994. Resubmitted 11th November 1994.
Abstract:
The localisation at the prime 3 of the integral cohomology ring of the
Mathieu group $M_{24}$ is calculated. The Chern classes of the Todd
representation in $GL_{11} (F_2)$ generate the even-degree part
of this ring. The mod-3 cohomology ring is also calculated.
[These results have been used by C. B. Thomas to prove that the elliptic
cohomology of the classifying space $BM_{24}$ is generated by Chern classes,
and is therefore concentrated in even dimensions.]
1991 Mathematics Subject Classification: 20J06 (primary), 20D08
2. /pub/DJGreen/p5.abstract
Author: David J Green
Title : Chern classes and extraspecial groups of order $p^5$
Date : 7th June 1995
A presentation is obtained for the Chern subring modulo nilradical of
both extraspecial $p$-groups of order $p^5$, for $p$ an odd prime.
Moreover, it is proved that, for every extraspecial $p$-group of
exponent $p$, the top Chern classes of the irreducible representations
do not generate the Chern subring modulo nilradical. Finally, a
related question about symplectic invariants is discussed, and solved
for $Sp_4 (F_p)$.
The main innovation in this work is to consider extraspecial groups
as central products, and to partition the maximal elementary abelian
subgroups of the central product into those which lift to abelian
subgroups of the corresponding direct product, and those which do not.
1991 Mathematics Subject Classification: 20J06
3. /pub/Henderson/Ext_Mon_HA.abstract
(I think this is an updated version of a paper that was already on the
archive-- Mark)
Hopf Algebra Extensions of Monogenic Hopf Algebras
Gregory D. Henderson
Pennsylvania State University
William M. Singer has described a cohomology theory of connected Hopf
algebras which classifies extensions of a cocommutative Hopf algebra by a
commutative Hopf algebra in much the same way as the cohomology of groups
classifies extensions of a group by an abelian group. We compute these
cohomology groups for monogenic Hopf algebras, construct an action of the
base ring on the cohomology groups in the case of trivial matched pairs,
and use these results to further study Singer's cohomology.
4. /pub/Thomason/thomason_SymMon_equals_Spectra.abstract
Symmetric monoidal categories model all connective spectra
R. W. Thomason
The classical infinite loopspace machines in fact
induce an equivalence of categories between a localization
of the category of symmetric monoidal categories and the
stable homotopy category of -1-connective spectra.
5. /pub/Welker-Ziegler-Zivaljevic/compare.abstract
Abstract : Comparison Lemmas and Applications for Diagrams of Spaces
V. Welker, G.M. Ziegler, R.Zivaljevic
We provide a ``toolkit'' of basic lemmas for the comparison of homotopy
types of (homotopy) limits of diagrams of spaces over finite partially
ordered sets, among them several new ones.
In the setting of this paper, we obtain simple inductive proofs that
provide explicit homotopy equivalences.
(In an appendix we provide the link to the general setting
of diagrams of spaces over an arbitrary small category.)
We show how this toolkit of old and new diagram lemmas can be used on
quite different fields of applications. In this paper we demonstrate
this with respect to
* the ``generalized homotopy-complementation
formula'' by Bj\"orner
* the topology of toric varieties
(which turn out to be homeomorphic to homotopy limits, and for which
the homotopy limit construction provides a suitable spectral sequence),
* in the study of homotopy types of arrangements of subspaces,
where we establish a new, general combinatorial formula for the
homotopy types of ``Grassmannian'' arrangements, and
* in the analysis of homotopy types of subgroup complexes.
6. /pub/Wolbert/current.abstract
Toward an algebraic classification of module spectra
by J. Wolbert
Department of Mathematics, University of Chicago,
Chicago, IL 60637, USA
Abstract:
The category of modules over an $S$-algebra (\Ai\ or \Ei\ ring spectrum)
has many of the good properties of the category of
spectra. When the homotopy groups
of the $S$-algebra in question form a sufficiently nice ring, it is
possible to see the deviation of the category of modules over an
$S$-algebra from the corresponding algebraic module category. In
particular, many algebraic modules are realized as homotopy groups
of topological modules over $S$-algebras.
Examples studied include real and complex $K$-theory, both connective
and periodic.
Further, Bousfield localization by a smashing spectrum is
shown to yield a category of modules over the localized sphere.
For periodic $K$-theory, these methods yield an algebraic criterion
to determine when a local spectrum is a module over the $K$-theory
$S$-algebra, real or complex.
-------
This is the ninth installment of a mailing list of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. This list is maintained by Mark Hovey
(hovey---math.mit.edu). Instructions at the end.
Papers uploaded to Hopf between July 11 and July 16, 1995:
1. /pub/Hovey-Palmieri-Strickland/axiomatic.abstract
Axiomatic Stable Homotopy Theory
Mark Hovey, John Palmieri and Neil Strickland
We define and investigate a class of categories with formal properties
similar to those of the homotopy category of spectra. This class
includes suitable versions of the derived category of modules over a
commutative ring, or of comodules over a commutative Hopf algebra, and
is closed under Bousfield localization. We study various notions of
smallness, questions about representability of (co)homology functors,
and various kinds of localization. We prove some theorems analogous
to those of Hopkins and Smith about detection of nilpotence and
classification of thick subcategories. We define the class of
Noetherian stable homotopy categories, and investigate their special
properties. Finally, we prove that a number of categories occurring
in nature (including those mentioned above) satisfy our axioms.
(Note from Mark: For a hyperlinked dvi
version of this file (for use with xhdvi) see my home page, whose URL is
in the instructions at the end).
2. /pub/Kashiwabara-Strickland-PTurner/dlk.abstract
The Morava K-Theory Hopf Ring for BP
Takuji Kashiwabara, Neil Strickland and Paul Turner
Let $K$ be a $p$-local complex-oriented homology theory. The
$K$-homology of the even spaces in the $\Omega$-spectrum for $BP$ form
a Hopf ring. In~\cite{rawi:hrc} Ravenel and Wilson chararacterise
this Hopf ring by a purely algebraic universal property, and also
prove that the $K$-homology of each component of each even space is
polynomial under the star product. The star-indecomposables in this
Hopf ring form an algebra under the circle product.
In this paper we take $K$ to be 2-periodic Morava $K$-theory, and
study the resulting ring $R$ of indecomposables. In
propositions~\ref{pr:pres} and~\ref{pr:iso} we give an algebraic
universal property which characterises $R$, and relate this to a
better-known description of the stable ring $K_*(BP)$. In
theorem~\ref{th:split} we nearly provide a splitting of $R$ as a
product of indecomposable factors, each of which is isomorphic modulo
nilpotents to $K_*(BP)$. In the case $n=1$, there are no nilpotents
and $R$ is the subring of an infinite product of copies of $K_*(BP)$
defined by a certain asymptotic condition; this is proved as
theorem~\ref{th:tauiso}. We give a very simple description of the
Dyer-Lashof operation on $R$ in these terms.
3. /pub/Strickland/fpfp.abstract
Functorial Philosophy for Formal Phenomena
Neil Strickland
The purpose of this paper is to introduce the ``schematic viewpoint''
in algebraic topology. This seems to be the most natural framework in
which to discuss the algebraic structures which arise from
complex-oriented cohomology theories. Many of the parts which are
original are joint work with Mike Hopkins and Matthew Ando.
We give a definition of (formal) schemes which is well adapted to the
particular technicalities which arise in the study of Morava K-theory
and completed E(n)-theory. We show how to interpret the generalised
(co)homology of $CP^\infty$, $Z\times BU$, $B\Sigma_{p^m}$, projective
bundles and Thom spaces of complex vector bundles, and various other
spaces, using the language of formal group theory.
4. /pub/Strickland/sigma4.abstract
Notes on K(B\Sigma_4) at q=4
Neil Strickland
In this document we describe the Morava $K$ theory (with $n=p=2$) of
$\Sigma_4$ and its subgroups in excruciating detail. We use Chern
classes and their transfers as generators, and describe the ring
structure and all transfer and restriction maps. Much of the
calculation was done using Mathematica.
5. /pub/Strickland/subgp.abstract
Finite Subgroups of Formal Groups
Neil Strickland
In this paper we discuss various moduli problems involving the
classification of finite subgroups or related structures on formal
groups of finite height. Analogous problems for elliptic curves have
of course been widely studied. The moduli spaces which we consider
turn out to be surprisingly well-behaved. They are all
Cohen-Macaulay, and most of them are smooth. The original motivation
for this work came from algebraic topology, in particular the study of
power operations in certain homology theories constructed by Morava.
I learnt most of what I know about these questions from Mike Hopkins,
and a great deal of the theory presented here was developed in
discussions with him. See section~\ref{se:at} for a brief discussion
of how moduli problems arise in algebraic topology.
6. /pub/Strickland/rme.abstract
Rational Morava E-theory and DS^0
Neil Strickland and Paul Turner
The extended-power spectrum $DS^0$ has two coproducts and two products,
which interact in an intricate way. Given an $H_\infty$ ring spectrum
$E$, the resulting algebraic structure on $E^*(DS^0)$ gives a framework
in which to encode information about power operations. (However, we
will not study power operations in this paper).
Fix a prime $p$ and an integer $n>0$. We shall take $E$ to be a
suitable completed and extended version of $E(n)$, which we shall call
Morava $E$-theory. It is represented by a spectrum which we shall
also call $E$. It is known (by unpublished work of Miller and
Hopkins) that $E$ is an $E_\infty$ ring spectrum (but we shall not use
this fact).
In the present work, we discuss the ring $L(\ds)$ obtained from
$E^0(DS^0)$ by making a certain algebraic extension and inverting $p$.
Let $\Lambda$ be the group $(Q_p/Z_p)^n$, and
$\Lambda^* = \Hom(\Lambda,Q/Z) = Z_p^n$
its dual. Write $\burn$ for the Burnside semiring of $\Lambda^*$, in
other words the semiring of isomorphism classes of finite sets with an
action of $\Lambda^*$. Write $F(\burn,L)$ for the set of functions
from $\burn$ to $L$. Our central result is to give an isomorphism of
$L(DS^0)$ with $F(\burn,L)$, and show that this respects all structure
in sight.
---------
This is the tenth installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Papers uploaded to Hopf between July 16 and July 28, 1995:
1. /pub/Monks/bases
TITLE: Change of basis, monomial relations, and $P_t^s$ bases for the
Steenrod algebra
AUTHOR: Ken Monks
Department of Mathematics
University of Scranton
Scranton, PA 18510
email: monks---uofs.edu
FILENAME: BASES.DVI
ABSTRACT:
The relationship between several common bases for the mod 2 Steenrod algebra
is explored and a new family of bases consisting of monomials in distinct
$P_t^s$'s is developed. A recursive change of basis formula is produced to
convert between the Milnor basis and each of the bases for which the change
of basis matrix in every grading is upper triangular. In particular, it is
shown that the basis of admissible monomials, the new $P_t^s$ bases, and two
bases due to D. Arnon, are all bases having this property, and the
corresponding change of basis formula is produced for each of them. Some
monomial relations for the mod 2 Steenrod algebra are then obtained by
exploring the change of basis transformations.
2. /pub/Monks/polymods
TITLE: Polynomial Modules Over the Steenrod Algebra
and Conjugation in the Milnor Basis
AUTHOR: Ken Monks
Department of Mathematics
University of Scranton
Scranton, PA 18510
email: monks---uofs.edu
FILENAME: polymods.dvi
ABSTRACT:
Let $P_s=\Bbb F_2\left[ x_1,\ldots ,x_s\right]$ be the mod 2 cohomology
of the $s $-fold product of $\Bbb R\text{P}^\infty $ with the usual
structure as a module over the Steenrod algebra. A monomial in $P_s$ is
said to be hit if it is in the image of the action $\overline{A} \otimes
P_s\rightarrow P_s$ where $ \overline{A}$ is the augmentation ideal of
$A$. We extend a result of Wood to determine a new family of hit
monomials in $P_s$. We then use similar methods to obtain a
generalization of antiautomorphism formulas of Davis and Gallant.
3. /pub/Monks/pstpaper
TITLE: The nilpotence height of $P_t^s$
AUTHOR: Ken Monks
Department of Mathematics
University of Scranton
Scranton, PA 18510
email: monks---uofs.edu
FILENAME: PSTHEIGHT.DVI
ABSTRACT:
The method of Walker and Wood is used to completely determine the
nilpotence height of the elements $\Pst$ in the Steenrod algebra at the
prime 2. In particular, it is shown that $(\Pst)^{2\lfloor s/t
\rfloor+2}=0$ for all $s\ge 0$, $t\ge 1$. In addition, several
interesting relations in $A$ are developed in order to carry out the
proof.
----------
This is the eleventh installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Papers uploaded to Hopf between July 29 and August 9, 1995:
1. /pub/Benson-Greenlees/Liegroupca.abstract
\title{Commutative algebra for cohomology rings of classifying spaces
of compact Lie groups.}
\author{D.~J.~Benson}
\address{Department of Mathematics, University of Georgia, Athens, GA 30602,
USA}
\email{djb---byrd.math.uga.edu}
\author{J.~P.~C.~Greenlees }
\address{School of Mathematics and Statistics, Hicks Building,
Sheffield S3 7RH. UK.}
\email{j.greenlees---sheffield.ac.uk}
\date{}
\begin{abstract}
We apply the techniques of highly structured ring and module
spectra to prove a duality theorem for the cohomology ring
of the classifying space of a compact Lie group. This generalizes results
of Benson--Carlson \cite{bc3,bc5} and Greenlees \cite{groupca} in the case
of finite groups. In particular, we prove a functional equation for the
Poincar\'e series in the oriented Cohen--Macaulay case. We make
essential use of the theorem of Elmendorf--May \cite{em} that
Borel cohomology is represented by a highly structured ring
spectrum.
\end{abstract}
2. /pub/Bisson-Joyal/cr_one.abstract
CRone.abstract
The Dyer-Lashof Algebra in Bordism
Terrence Bisson and Andr\'e Joyal
(June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada)
We present a theory of Dyer-Lashof operations in unoriented bordism
(the canonical splitting $N_*(X)\simeq N_*\otimes H_*(X)$, where
$N_*(\ )$ is unoriented bordism and $H_*(\ )$ is homology mod 2,
does not respect these operations). For any finite covering space we
define a ``polynomial functor'' from the category of topological
spaces to itself. If the covering space is a closed manifold
we obtain an operation defined on the bordism of any $E_\infty$-space.
A certain sequence of operations called squaring operations are
defined from two-fold coverings; they satisfy the Cartan formula and
also a generalization of the Adem relations that is formulated by
using Lubin's theory of isogenies of formal group laws. We call a
ring equipped with such a sequence of squaring operations a
$D$-{\it ring}, and observe that the bordism ring of any free
$E_\infty$-space is free as a $D$-ring. In particular, the bordism
ring of finite covering manifolds is the free $D$-ring on one generator.
In a second compte-rendu we discuss the (Nishida) relations between
the Landweber-Novikov and the Dyer-Lashof operations, and show how
to represent the Dyer-Lashof operations in terms of their actions on
the characteristic numbers of manifolds.
3. /pub/Bisson-Joyal/cr_two.abstract
CRtwo.abstract
Nishida Relations in Bordism and Homology
Terrence Bisson and Andr\'e Joyal
(June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada)
This is the second of a series of Compte Rendus. In the first [1]
we have presented a theory of Dyer-Lashof operations in
unoriented bordism. Here we shall discuss the (Nishida) relations
between Dyer-Lashof and Landweber-Novikov operations.
They are used to represent the algebra $N_*\Sigma$ of
covering manifolds in terms of their homology characteristic numbers.
The proofs are based on the properties of the covering
space operations and the notions of $D$-ring and $Q$-ring
introduced in [1].
4. /pub/Elmendorf-May/SGalgebras.abstract
Algebras over equivariant sphere spectra
by
A.D.Elmendorf and J.P.May
Abstract: We study algebras over the sphere spectrum S_G of
a compact Lie group G. In particular, we show how to construct
S_G algebras from S-algebras, where S is the nonequivariant
sphere spectrum. This gives a reservoir of equivariant
examples to which recently developed algebraic techniques in
stable homotopy theory can be applied. A special case will be
used in a companion paper of Benson and Greenlees to
study the ordinary cohomology of the classifying space BG.
5. /pub/Greenlees-May/gmMU.abstract
Localization and completion theorems for MU-module spectra
by
J.P.C.Greenlees and J.P.May
Abstract: Let G be a finite extension of a torus. Working with highly
structured ring and module spectra, let M be any module over MU; examples include all the standard homotopical MU-modules, such as the Brown-Peterson
and Morava K-theory spectra. We prove localization and completion theorems
for the computation of M_*(BG) and M^*(BG). The G-spectrum MU_G that
represents stablilized equivariant complex cobordism is an algebra over the
equivariant sphere spectrum S_G and there is an MU_G module M_G whose
underlying MU-module is M. This allows the use of topological analogues of constructions in commutative algebra. The computation of M_*(BG) and M^*(BG)
is expressed in terms of spectral sequences whose respective E_2 terms are
computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU_*^G and its module M_*^G.
The central feature of the proof is a new norm map in equivariant stable
homotopy theory, the construction of which involves the new concept of a
global I_*-functor with smash product.
--------
This is the twelfth installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Let me request that all submitters to the Hopf archive remember that your
abstracts should be in human readable form. The first of the following
abstracts had to be heavily edited by me to make it at all readable.
My apologies for any errors thereby introduced.
Mark Hovey
Papers uploaded to Hopf between August 9 and August 16, 1995:
1. /pub/Lindenstrauss/thh6.abstract
(This paper is a revised version of thh3.dvi previously on the archive-Mark)
Topological Hochschild Homology of Extensions of Z/pZ by Polynomials,
and of Z[x]/(xn) and Z[x]/(xn-1)
Ayelet Lindenstrauss
Princeton University
Princeton, NJ 08544
Introduction
The Dennis trace is a map from the algebraic K-theory of a ring into
Hochschild homology. The Hochschild homology groups are easier to
calculate,but they are relatively simple. The chance of finding
invariants which have interesting pullbacks to K-theory increases if we
succeed in factoring the Dennis trace map through intermediate groups,
and obtaining trace maps which land in groups with more structure.
The K-groups of a ring R are homotopy groups of BGL +(R). The
Hochschild homology groups are homotopy groups of a space CH(R),the
realization as a simplicial space of the standard (bar) complex used to
calculate Hochschild homology. The Dennis trace map is the map on
homotopy groups induced by a continuous map
BGL +(R) --> CH(R) ;
which can be factored through a space called THH(0)(R), which is the 0th
space in the topological Hochschild homology spectrum THH(R). The
existence of this intermediate step was conjectured by Goodwillie,and
shown by Bokstedt in [2]. Topological Hochschild homology is constructed
analogously to the bar complex CH(R), with Eilenberg-MacLane spectra
K(R;m) and smash products substituted for the ring R and tensor
products. The factoring of the Dennis trace (which is explained for
example in [4]) consists of a map
BGL +(R) --> THH (0)(R)
which is the 0-level part of a spectrum version of the Dennis trace, and
a map
THH (0)(R) --> CH(R)
induced by the component map . Bokstedt calculated the homotopy type of
this spectrum (which actually is an spectrum) for R = Z and R = Z/pZ.
Thus the Dennis trace map can be made to factor through
ss (THH (0)(R)) = ssS(THH (R)):
Bokstedt used his calculations of THH(Z) to obtain results concerning
the algebraic K-theory of Z.
Perhaps it is also possible to factor the Dennis trace map through
an object similar both to cyclic homology and to topological Hochschild
homology- a sort of `cyclic topological Hochschild
homology'. Bokstedt,Hsiang and Madsen (see [4]) succeeded in creating
such a theory in a different context which involves objects that can be
treated using the same FSP formalism as the functor X 7! R[X] which we
will use here, and indeed constructed spectra which were p-equivalent to
the K-theory spectrum for any given prime p.
The first part of this paper briefly describes the construction of
topological Hochschild homology, following [2], and the method and
results of Bokstedt's calculation of THH(Z) and THH(Z/pZ) in [3].
The second part of the paper calculates
HS(THH(R);Z/pZ)
for any ring R which is of the form Z/qZ[x]/(f(x)) (q prime,f(x) in
Z/qZ[x]),and for the rings Z[x]/(xn) and Z[x]/(xn-1). The calculation
uses the same kind of spectral sequence Bokstedt used in [2]. The result
of these calculations is a splitting
HS (THH(R); Z/pZ) = HH(R; Z/pZ) cross HS(THH(Z);Z/pZ)
of the ring of stable homology classes,where the multiplication is
induced by a shuffle-product.
The third part of this paper describes the explicit form of the
homotopy type of THH(R) for the rings Z/qZ[x]/(f(x)) (for any polynomial
f), Z[x]/(xn), and Z[x]=(xn-1).
2. /pub/JPMay/modnew.abstract
Equivariant and nonequivariant module spectra
by J.P. May
Abstract: Let $G$ be a compact Lie group, let $R_G$ be a commutative
algebra over the sphere $G$-spectrum $S_G$, and let $R$ be its underlying
nonequivariant algebra over the sphere spectrum $S$. When $R_G$ is split
as an algebra, as holds for example for $R_G=MU_G$, we show how to
``extend scalars'' to construct a split $R_G$-module $R_G\sma_R M$ from
an $R$-module $M$. This allows the wholesale construction of highly
structured equivariant module spectra from highly structured nonequivariant
module spectra. In particular, it applies to construct $MU_G$-modules from
$MU$-modules and therefore gives conceptual constructions of equivariant
Brown-Peterson and Morava $K$-theory spectra.
-----------
This is the thirteenth installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between August 16 and August 30, 1995:
1. /pub/GLewis/eqv_splt_sptr
SPLITTING THEOREMS FOR CERTAIN EQUIVARIANT SPECTRA
L. Gaunce Lewis, Jr.
Let $\Gamma $ be a compact Lie group, $\Pi $ be a normal subgroup
of $\Gamma $, $G=\Gamma / \Pi $, $X$ be a $G$-space and $Y$ be a
$\Gamma $-space. There are a number of results in the literature
giving a direct sum decomposition of the group of $\Gamma $-equivariant
stable homotopy classes of maps from the suspension spectrum of
$X$ to the suspension spectrum of $Y$. Here, these results are extended
to a decomposition of the group $[B,C]_\Gamma $ of equivariant stable
homotopy classes of maps from an arbitrary $G$-spectrum $B$ to any
$\Gamma $-spectrum $C$ carrying a cosplitting (a new type of structure
introduced here). Any naive $\Gamma $-spectrum, and any spectrum
derived from such by a change of universe functor, carries a cosplitting.
This decomposition of $[B,C]_\Gamma $ is a consequence of the fact
that, if $C$ is cosplit and $({\scr{F}}^\prime ,\scr{F})$ is any
pair of families of subgroups of $\Gamma $, then there is a splitting
of the cofibre sequence
$$(E{\scr{F}}_+ \wedge C)^\Pi \rightarrow
(E{\scr{F}}^\prime _+ \wedge C)^\Pi \rightarrow
(E({\scr{F}}^\prime ,\scr{F}) \wedge C)^\Pi $$
constructed from the universal spaces for the families. Both the
decomposition of the group $[B,C]_\Gamma $ and the splitting of the
cofibre sequence are proven here not just for complete $\Gamma
$-universes, but for arbitrary $\Gamma $-universes. Various technical
results about incomplete $\Gamma $-universes that should be of
independent interest are included in this paper. These include versions
of the Adams and Wirthm\"uller isomorphisms for incomplete
universes. Also included is a vanishing theorem for the fixed point
spectrum $(E({\scr{F}}^\prime ,\scr{F} ) \wedge C)^\Pi $ which gives
computational force to the intuition that what really matters about a
$\Gamma $-universe $U$ is which orbits $\Gamma /\Lambda $ embed as
$\Gamma $-spaces in $U$.
--------------
This is the fourteenth installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between August 30 and September 9, 1995:
1. /pub/Henn/kmod
COMMUTATIVE ALGEBRA OF UNSTABLE $K$ - MODULES,
LANNES' $T$ - FUNCTOR AND
EQUIVARIANT MOD - P COHOMOLOGY
by
Hans--Werner Henn
Let $p$ be a fixed prime and
let $K$ be an unstable algebra over the mod - $p$ Steenrod algebra $A$
such that $K$ is finitely generated as graded $\FF_p$ - algebra.
Let $K_{fg}-\Ua$ denote the abelian category of finitely generated
$K$ - modules with a compatible unstable $A$ - module structure.
We study various concepts of commutative algebra in this setting.
The r\^ole of the prime ideal spectrum of a commutative ring is here
taken by a category $\Rav (K)$ which, roughly speaking, consists of
the $A$ - invariant prime ideals of $K$ together with certain
``Galois information''; sheafs will correspond to functors on this
category, and the r\^ole of the sheaf associated to a module will
be taken by the components of Lannes' $T$ - functor. We discuss the
notions of support, of ${ \gl a}$ - torsion modules
(for an invariant ideal ${ \gl a}$ of $K$) and of localization away
from the Serre subcategory $\Ta ors ({ \gl a})$ of ${ \gl a}$ - torsion
modules in our setting. We show that the category $K_{fg}-\Ua$ has
enough injectives and use these injectives to study these localizations
and their derived functors; they are closely related to the derived
functors of the ${ \gl a}$ - torsion functor $F_{{ \gl a}}$.
Our results are formally analogous to Grothendieck's results in the
classical situation of modules over a noetherian commutative ring R [Gr].
Important for applications is the case $K=H^*BG$, the mod - $p$
cohomology of a classifying space of a compact Lie group
(or a suitable discrete group), and $M=H^*_GX$ where $X$ is a
(suitable) $G$ - $CW$ - complex. In these cases the category $\Rav (K)$
and the functor on $\Rav (K)$ associated to $H^*_GX$
can be described in terms of group theoretic and geometric data, and our
theory yields a far-reaching generalization of a result of Jackowski
and McClure [JM] resp. of Dwyer and Wilkerson [DW2].
As a concrete application of our theory we describe the size of the
kernel of the restriction map from the unknown mod - $2$ cohomology of the
$S$ - arithmetic group $GL(n,\Z[1/2])$ to the known cohomology of its
subgroup $D_n$ of diagonal matrices.
2. /pub/Moller/normalizer
J.M. Moller: Normalizers of maximal tori
Normalizers and p-normalizers of maximal tori in p-compact groups can be
characterized by the Euler characteristic of the associated
homogeneous spaces. Applied to centralizers of elementary abelian
p-groups these criteria show that
the normalizer of a maximal torus of the
centralizer is given by the centralizer of a preferred homomorphism
to the normalizer of the maximal torus; i.e. that ``normalizer'' commutes with
``centralizer''.
3. /pub/Moller/survey
J.M. Moller: Homotopy Lie groups
This is a survey of Dwyer and Wilkerson's p-compact groups intended for a
general mathematical audience.
----------
This is the fifteenth installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
My home page appears to be broken. I'll try to fix it, but in any case
I have been slow about updating it recently. Sorry about
that. In case you are looking for more recent back issues of this list,
Don Davis' home page should have them. The URL for his discussion list
is http://www.lehigh.edu/dmd1/public/www-data/algtop.html .
Mark Hovey
Papers uploaded to Hopf between September 9 and September 23, 1995:
1. /pub/Ando/PowerOpsEll.dvi
Power operations in elliptic cohomology and representations of loop groups
Matthew Ando
The first part of this paper describes power operations in elliptic
cohomology in terms of isogenies of the underlying elliptic curve.
The second part discusses a relationship between equivariant elliptic
cohomology and representations of loop groups. The third part
investigates the representation theoretic considerations which give
rise to the power operations discussed in the first part.
2. /pub/Lindenstrauss/ram2.ps
(Note: At present this file is only available in .ps form--Mark)
Abstract of The Topological Hochschild Homology of the Gaussian Integers
by Ayelet Lindenstrauss:
Topological Hochschild homology is calculated explicitly for the rings
${\bf Z}[\sqrt2]$, ${\bf Z}[\sqrt{-2}]$, and ${\bf Z}[i]$.
The 2-torsion of the topological Hochschild homology is calculated
for the ring of integers in any quadratic extension of the rationals.
3. /pub/Rudyak/ts.dvi
THE SPECTRA k AND kO ARE NOT THOM SPECTRA
Yu. B. Rudyak
Abstract. Here is proved that neither k nor kO are Thom spectra.
This was conjectured by Mahowald in 1979.
Mathematisches Institut Universitat Heidelberg,
Im Neuenheimer Feld 288, D-69120 Heidelberg 1, Germany.
----------
This is the sixteenth installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between September 24 and October 24, 1995:
This time we have updated versions of two papers that were already on
the archive: /pub/Ando/PowerOpsEll and /pub/Lindenstrauss/ram2 . The
paper by Ando is on power operations, elliptic cohomology, and
representations of loop groups, and the paper by Lindenstrauss is about
calculating the topological Hochschild homology of the Gaussian
integers. I don't know how extensive the revisions are.
We also have one new paper:
1. /pub/Henderson/Spec_Seq_Ext_HA.abstract
SPECTRAL SEQUENCES FOR THE CLASSIFICATION
OF EXTENSIONS OF HOPF ALGEBRAS
Gregory D. Henderson
October 15, 1995
We construct spectral sequences which provide a way to compute the
cohomology theory that classifies extensions of graded connected Hopf algebras
over a commutative ring as described by William M. Singer. Specifically, for
(A,B) an abelian matched pair of graded connected R-Hopf algebras, we construct
a pair of spectral sequences relating H^*(B,A) to Ext_B(R,Cotor_A(R,R)). To
illustrate these spectral sequences, we examine the special case of B a
monogenic graded connected Hopf algebra and also analyze an extension of
Hopf algebras given by James P. Lin.
-----------
This is the seventeenth installment of abstracts of algebraic
topology papers recently uploaded to Clarence Wilkerson's
archive. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between October 24 and October 31, 1995:
We have one new paper:
1. /pub/Dwyer-Mitchell/curves
On the $K$-theory spectrum of a smooth curve over a finite field
by
W. G. Dwyer and S. A. Mitchell
We study the algebraic $K$-theory spectrum associated to a smooth
curve (either complete or affine) over a finite field, and determine
(for instance)
- the topological $K$-theory groups of the spectrum
- the Bousfield localization of the spectrum with respect
to topological $K$-theory
- the topological $K$-theory groups of the zero space in
the associated $\Omega$-spectrum
This determination is done in terms of classical algebraic invariants
of the curve. We also prove that the above Bousfield localization of
the spectrum is a retract (in positive dimensions) of the spectrum
itself. Actually, to obtain the results it is necessary to
$\ell$-complete the spectrum at an odd prime $\ell$ which is different
from the characteristic of the finite field; similarly, "topological
$K$-theory" as used above means $\ell$-completed topological
$K$-cohomology.
---------
We have one new paper at Clarence Wilkerson's
archive. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between October 31 and November 7, 1995:
1. /pub/Ravenel-Wilson-Yagita/bpcohfrommork
Brown-Peterson cohomology from Morava K-theory
Douglas C. Ravenel W. Stephen Wilson and Nobuaki Yagita
We give some structure to the Brown-Peterson cohomology
(or its $p$-completion) of a wide class of spaces.
The class of spaces are those with Morava K-theory even dimensional.
We can say that the Brown-Peterson cohomology is even dimensional
(concentrated in even degrees) and is flat as a $BP^*$-module
for the category of finitely presented $BP^*(BP)$-modules.
At first glance this would seem to be a very restricted class
of spaces, but the world abounds with naturally occurring examples:
Eilenberg-MacLane spaces, loops of finite Postnikov systems,
classifying spaces
of all finite groups whose Morava K-theory is known (including the
symmetric groups), $QS^{2n}$, $BO(n)$, $MO(n)$, $BO$, $\ImJ$, etc.
We finish with an explicit algebraic construction of the Brown-Peterson
cohomology of a product of Eilenberg-Maclane spaces.
(Note from Mark: Actually Igor Kriz has recently given a finite group
whose Morava K-theory is not concentrated in even degrees.)
---------
We have three new papers at Clarence Wilkerson's
archive. Instructions at the end.
My home page has moved, and also has been somewhat updated. The new URL
is
http://www-math.mit.edu/~hovey/
Mark Hovey
Papers uploaded to Hopf between November 7 and November 25, 1995:
1./pub/Dwyer/decompositions
Homology decompositions for classifying spaces of finite groups
by W. G. Dwyer
We look at ways of expressing the classifying space of a finite group
G, at least up to mod p homology, as a homotopy colimit of classifying
spaces of subgroups of G. What results is a general theory which
includes as special cases the decompositions of Jackowski-McClure and
of Jackowski-McClure-Oliver.
2. /pub/Slack/tfodd
Infinite loop spaces with odd torsion free homology
by Michael Slack
Abstract: It is shown that an infinite loop space with no odd torsion in its
integral homology also has no odd torsion in its homotopy. Combined with known
results of Steve Wilson, this gives a complete classification; all such spaces
are products of the Wilson spaces, which are the building blocks of the spaces
in the omega spectrum for BP.
Comments, questions, and corrections are all welcomed. For other
papers by Slack (published, preprints, and in progress), you can visit his
homepage at http://www.wmich.edu/math-stat/faculty/slack/.
3. /pub/JWu/Simplicial-group-1
On Combinatorial Descriptions of Homotopy
Groups of K(ss; 1)
Jie Wu
November 13, 1995
Abstract
We will give a combinatorial description of homotopy groups of
K(ss;1). Thus, by Kan-Thurston Theorem, a combinatorial descrip-
tion of the homotopy groups of a simply connected suspension space
is given.In particular, all of the homotopy groups of the 3-sphere are
combinatorially given.
4. /pub/JWu/Wu.copy
ON FIBREWISE SIMPLICIAL MONOIDS AND
MILNOR-CARLSSON'S CONSTRUCTIONS
JIE WU
In [?], G. Carlsson introduced a simplicial group construction which
gives a generalization of Milnor's F(K) construction [?]. Roughly
speaking, if we construct a simplicial group which is a free product of
a simplicial group G over a pointed simplicial set X, then we get a
simplicial group construction for (BG ^ X), where BG is the classifying
space of G. In this article, we give a categorial view of this construction.
Let C be a category. A fibrewise simplicial object over C, roughly
speaking, is a diagram over C with indices in a simplicial set. This is
an abstract view of fibrewise topology [?] or sheaf theory. If the
category C has coproducts, then the abstract F-construction is defined
to be certain coadjoint functor from the category of fibrewise
simplicial objects over C to the category of simplicial objects over C.
Suppose that there is a functor T from C to the category of pointed
simplicial sets such that T preserves coproducts up to homotopy. Then
there is an induced functor T from the category of fibrewise simplicial
objects over C to the category of pointed bisimplicial sets. Theorem ??
shows that T is homotopy equivalent to T ffiF. Let C be a category
of monoids. Notice that the bar-construction B preserves coproduct up to
homotopy [?]. A corollary of this abstract theorem is the Carlsson
theorem.
An application of Carlsson's construciton to homotopy theory is to
give a representation of the homotopy groups of simply connected
suspension spaces to certain combinatorial groups as centers
[?]. Applications of Carlsson's construction to minimal simplicial
groups are given in [?]. In this paper, we pay more attention to the
geometry of the Carlsson construction. The word length filtration is
considered. The resulting cofibres are certain smash product pinched out
certain reduced diagonal elements (Proposition ??). Our construction in
the monoid case is a generalization of the James construction [?]. We
construct certain natural map Hn : (Y ^X ) ! n(Y n^ (X(n)=4n )), which
is similar to the James-Hopf map,for any path connected CW-complex Y and
any pointed CW-complex X, where X(n)is the n-th fold self smash product
of X and 4n = f(x1 ^: :^:xn) 2 X(n)j xi= xi+1for some ig (Theorem ??). A
direct application of these natural maps is to give a decomposition
of H(F P 1^ X) for F =R, C or H. Let F P21= F P 1=F P 1.
Research at MSRI is supported in part by NSF grant DMS-9022140.
Theorem 0.1. Let F = R,C or H and let X be a pointed space. Suppose that H
is a multiplicative homology theory such that (1) both H (F P 1) and H
(F P21) are free H (pt)-modules;and (2) the inclusion of the bottom cell
Sd !F P 1 induces a monomorphism in the homology. Then there is a
product filtration fFrH (F P 1^ X)gr0 of H(F P 1 ^X) such that F0 =H
(pt) and
Fr=Fr1 = (d1)rH (X (r)=4r);
where d = dimR F and is the suspension. Furthermore, this filtration is
natural for X.
-----------
We have four new papers at Clarence Wilkerson's
archive. This is a good time to remind submitters of papers: Abstracts
must be readable by humans!! I do some editing to improve readability,
but I can only do so much, and this time I didn't even feel like doing
that. Instructions at the end.
Mark Hovey
Papers uploaded to Hopf between November 25 and December 8, 1995:
1./pub/Chacholski/B-M
A Generalization of The Triad Theorem of Blakers-Massey.
Wojciech Chacholski
Let F-->A-->X and H-->A-->E be fibration sequences.
Take the homotopy push-out B=hocolim(X<--A-->E), and then
take the homotopy pull-back Y=holim(X-->B<--E).
There is a natural map q:A-->Y. The main statement of the paper
is that if F and H are connected, then the homotopy fiber of
the suspenson of q is built from the smash of F and H using
homotopy push-outs, wedges and telescopes.
2. /pub/Chacholski/barc
CLOSED CLASSES
WOJCIECH CHACHOLSKI
1. Introduction
A non empty class C of connected spaces is said to be a closed class if
it is closed under weak equivalences and pointed homotopy colimits. Aclosed
class can be characterized as a non empty class of connected spaces which is
closed under weak equivalences and is closed under certain simple operations:
arbitrary wedges, homotopy push-outs and homotopy sequential colimits. The
notion of a closed class was introduced by E. Dror Farjoun [6].
Two important constructions give rise to examples ofclosed classes. The
first one is the Bousfield-Dror periodization functor PA [2]. The class ofthose
spaces X, such that PA X is weakly contractible, forms a closed class. By looki*
*ng
just at the properties of this class wecan prove, for example, that PA X is
weakly equivalent toPA X (see [2], [4]). The second construction is E. Dror
Farjoun's colocalization functor C WA. The class of those spaces X, for which
there exists a space Y, such that X is weakly equivalent toC WAY , forms a
closed class. This class is denoted by C(A) and is called the class of A-cellul*
*ar
spaces. By looking just at the properties ofthe class C (A) we can prove, for
example, that CWAX is weakly equivalent to CWA X (see [4], [6]).
We say that a closed class C is closed under extensions by fibrations, if
for every fibration sequence (Z! E ! B), such that Z andB belong to C, E
belongs to C. A closed class C is closed under extensions by fibrations if and
only if for every diagram F : I! C ,such that the classifying space BI belongs
to C, the unpointed homotopy colimit hocolimIF belongs to C.
The purpose of this paper is to understand to what extent a closedclass
is closed under extensions by fibrations and under taking unpointed homotopy
colimits. We start with proving a theorem that, in particular, implies:
fflLet F : I ! Spaces? be a pointed diagram, such that the classifying
space BI belongs to C. If for every i 2 I, F (i)b elongs to C , thenso
does the unpointed homotopy colimit hocolimIF.
fflLet (Z ! E ! B) be a fibration sequence with a section. If Z and B
belong to C, then so does E.
fflLet F : I ! C andG : I ! C be diagrams and : F ! G be a natural
transformation. If hocolimIF belongs to C, then so does hocolimIG.
2 WOJCIECH CHACHOLSKI
Surprisingly these and many other results are the consequences of just one
statement, see theorem 5.1.
We continue with investigating the properties of a base space B (respec-
tively of the classifying space BI),which will guarantee that a closed class C *
*is
closed under extensions by fibrations with base B (respectively C is closed un-
der taking the unpointed homotopy colimit of diagrams F : I ! C ). We study
the following class:
D(C ) =fB I jifF : I ! C isa diagram, then hocolimIF 2 Cg
The main result of this paper is:
Theorem. The class D(C) is a closed class and it is closedunder extensions
by fibrations.
3. /pub/Chacholski/thesis
On The Functors CW_{A} and P_{A}.
Wojciech Chacholski
I am looking at the relation between Bousfield's localization
functor P_{A} and Dror Frajoun's colocalization functor CW_{A}.
I am studying the question: to what extent the following sequence
is exact:
Spaces --P_{A}--> Spaces --CW_{A}--> Spaces --P_{A}--> Spaces
The image of P_{A} is equal to the kernel of CW_{A}.
The correlation between the kernel of P_{A} and the image of CW_{A}
is more complicated. I proved that The kernel of P_{A} is the
closure of the image of CW_{A} under taking extensions by fibrations.
In the paper I am giving algorithms to construct the functor
CW_{A} out of P_{A} and vice versa. I am using these algorithms
to show that S^{n} is in the image of CW_{\Omega S^{n+1}} if and
only if n=1,3,7 and that for every n S^{n} is in the kernel of
P_{\Omega S^{n+1}}.
4. /pub/JWu/Min_Simpl_Set
\title{On products in the minimal simplicial sets}
\author{Jie Wu}
It is well known that every fibrant simplicial set~$X$ is homotopy
equivalent to a minimal simplicial set~$Y$. If~$X$ is a loop space,
then there is an induced multiplication on~$Y$ given by the composite
$$Y\times Y\to X\times X\to X\to Y.$$
In this case, $Y$ is a minimal simplicial set together with a
multiplication. It is an old story in topology to look for a product
in the minimal simplicial sets. J.F.~Adams showed that the two-stage
Postnikov system~$X$ with $\pi_n(X)\ne0$ and $\pi_{n+1}(X)\ne0$ is
homotopy equivalent to a minimal simplicial group~\cite{A}. J.~Milnor
gave a counterexample that $\Omega(S^{n+1}\la n+1,n+2,n+3\ra)$ is not
homotopy equivalent to a minimal simplicial group, where $S^{n+1}\la
n+1,n+2,n+3\ra$ is the $3$-stage Postnikov system by taking the first
three homotopy groups of $S^{n+1}$~\cite{Wu1}. G.~Whitehead gave some
non-associative minimal simplicial abelian groupoids.
In this paper, we study minimal simplicial $H$-sets, \ie, minimal
simplicial sets with multiplications. We always assume that a
simplicial $H$-set $X$ has a strict unit element~$e$. A simplicial
$H$-set $X$ is called \emph{strong homotopy associative} if
\begin{enumerate}
\item $\mu\circ(\mu\times1)\simeq\mu\circ(1\times\mu)\colon X\times
X\times X\to X\quad\text{rel.\ }e\times X\times X$;
\item $\mu\circ(\mu\times1)\simeq\mu\circ(1\times\mu)\colon X\times
X\times X\to X\quad\text{rel.\ }X\times e\times X$; and
\item $\mu\circ(\mu\times1)\simeq\mu\circ(1\times\mu)\colon X\times
X\times X\to X\quad\text{rel.\ }X\times X\times e$,
\end{enumerate}
where $\mu\colon X\times X\to X$ is the multiplication. It was pointed
out by J.~Stasheff that if a multiplication $\mu\colon X\times X\to X$
is homotopy associative, then there exists a multiplication
$\mu'\colon X\times X\to X$ which is strong homotopy
associative~\cite{Sta}. Notice that the homotopy groups $\pi_*(X)$ can
be identified with the cycles in~$X$ if~$X$ is minimal, where $x\in
X_n$ is a cycle if $d_jx=*$ for all~$j$. A simplicial $H$-set is said
to be \emph{right (left) group-like} if~$X$ has a strict right (left)
inverse map. Our main theorem is as follows.
\begin{thm}
Let $X$ be a connected strong homotopy associative minimal simplicial
$H$-set. Then:
\begin{enumerate}
\item The associativity
$$(ab)c=a(bc)$$
holds if one of $a,b,c$ is in $\pi_*(X)$.
\item For $a\in X_n$, there exists a unique left inverse~$b$ in~$X_n$
such that $ba=e$ and there exists a unique right inverse~$c$ in~$X_n$
such that $ac=e$.
\item The commutativity
$$ab=ba$$
holds if $a\in\pi_*(X)$.
\item $X$ is generated by $\pi_*(X)$ as a simplicial $H$-set.
\item The fibration
$$F_n(X)\to P_n(X)\to P_{n+1}(X)$$
is a central extension and it is also a principal $F_n(X)\cong
K(\pi_n(X),n)$ bundle, where $\{P_n(X)\}_{n\ge0}$ is the
Moore-Postnikov system of~$X$.
\item Let $H\colon\pi_*(X)\to\bar H_*(X;Z)$ be the Hurewicz map. Then
there exists a (graded) subset~$S$ of~$\pi_*(X)$ such that
\begin{enumerate}
\item $H(x)\ne0$ for $x\in S$;
\item $H(x_1)\ne H(x_2)$ for $x_1\ne x_2$ in~$S$; and
\item $X$ is generated by~$S$ as a right (or left) group-like
simplicial $H$-set.
\end{enumerate}
\end{enumerate}
\end{thm}
Assertions~(1) to~(4) give a general description of the relations
between the homotopy groups and the total space. Assertion~(5)
shows that the Postnikov system of a connected homotopy associative
minimal simplicial $H$-set is very nice. Assertion~(6) gives a
relation between the total space and its homology. This supports the
Moore conjecture in some sense although it is still very unclear if
the exponents of the homotopy groups are related to the homology
groups. We should point out that the minimal subcomplex of a loop
space is non-associative in general. This paper is our starting work
to understand the product structures in the ``minimal models'' for the
loop spaces. The understanding of these product structures may help us
to study some homotopy problems such as the Freyd conjecture, the
Moore conjecture, and the Kavarre invariants problem.
As an example, the complete answer when a two-stage Postnikov
system~$X$ with $\pi_n(X)=\Z/2$ and $\pi_{n+1}(X)=\Z/2$ is homotopy
equivalent to a minimal simplicial group is given as follows.
\begin{thm}
Let $n,i>0$ and let $X$ be a two-stage Postnikov system with
$\pi_n(X)=\Z/2$ and $\pi_{n+i}(X)=\Z/2$. Then~$X$ is homotopy
equivalent to a minimal simplicial group if and only if the Postnikov
invariant of~$X$ is trivial or~$Sq^{i+1}$.
\end{thm}
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