Subject: new Hopf listings
From: Mark Hovey
Date: 09 Apr 2000 07:00:34 -0400
6 new papers this time.
Mark Hovey
New papers uploaded to hopf between 3/4/00 and 4/9/00.
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bendersky-DavisD/comp1
Compositions in the v1-periodic homotopy groups of spheres
Martin Bendersky and Donald M. Davis
mbenders@shiva.hunter.cuny.edu, dmd1@lehigh.edu
21 pages, completed March 7, 2000, submitted to Forum Mathematicum
Abstract
Let p_i in pi_{n+8i-1}(S^n) denote an element which suspends to
a generator of the image of the stable 2-primary J-homomorphism.
We determine the image of the composite p_j o p_k in v1-periodic
homotopy v_1^{-1} pi_{n+8i+8j-2}(S^n). The method is to use Adams
operations to compute the 1-line of an unstable homotopy spectral
sequence constructed by Bendersky and Thompson. Odd-primary
analogues are also obtained.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model
Spectra and symmetric spectra in general model categories
by Mark Hovey
Wesleyan University
hovey@member.ams.org
April, 2000
This is a revised version. The basic idea is to automate the passage
from unstable to stable homotopy theory, so that it applies in
particular to the A^1 category of Voevodsky. So if we start with a model
category C and a left Quillen endofunctor G of C, we want to make a new
model category, the stabilization of C, where G becomes a Quillen
equivalence. The simplest way to do this is with ordinary spectra.
Thanks to Hirschhorn's localization technology, we can construct the
stable model structure on ordinary spectra with almost no hypotheses on
C and G. A new feature of this revision is that we show that, under
strong smallness hypotheses on G and C, the stable equivalences coincide
with the appropriate generalization of stable homotopy isomorphisms. In
particular, this holds for the A^1 category.
If C has a tensor product, and G is given by tensoring with a cofibrant
object K, then we also can construct symmetric spectra. The
localization techniques apply here as well, so we get a stable model
structure of symmetric spectra without having to assume anything like the
Freudenthal suspension theorem. In particular, this is a new
construction of the stable model structure on simplicial symmetric
spectra. Symmetric spectra form a monoidal model category, unlike
ordinary spectra, but we are unable to prove that the monoid axiom holds
in general.
Also new to this revision is a much more careful comparison between
symmetric spectra and ordinary spectra when both are defined. Symmetric
spectra and ordinary spectra are not always Quillen equivalent; we need
the cyclic permutation map on K tensor K tensor K to be homotopic to the
identity. Under some additional technical hypotheses (which again are
satisfied in the A^1 category), we construct a zigzag of Quillen
equivalences between symmetric spectra and ordinary spectra.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Oliver-Segev/2dim
Fixed point free actions on $Z$-acyclic 2-complexes
by Bob Oliver and Yoav Segev
E-mail: bob@math.univ-paris13.fr, yoavs@math.bgu.ac.il
We show that a finite group has an "essential" fixed point free action on
an acyclic 2-complex if and only if it is one of the simple groups in the
following list:
- $PSL_2(2^k)$ for $k\ge2$,
- $PSL_2(q)$ for $q\equiv3,5$ (mod 8) and $q\ge5$,
- $Sz(2^k)$ for odd $k\ge3$.
More precisely, for any finite group $G$, and any 2-dimensional acyclic
$G$-CW complex $X$ without fixed points, there is a normal subgroup $H$ in
$G$ such that $G/H$ is in the above list, and such that the $G$-action on
$X$ looks "essentially" like the $G/H$-action which we construct.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rezk/rezk-simpl-alg-proper
Title: Every homotopy theory of simplicial algebras
admits a proper model
Author: Charles Rezk
rezk@math.nwu.edu
Abstract:
We show that any closed model category of simplicial algebras over an
algebraic theory is Quillen equivalent to a proper closed model
category. By ``simplicial algebra'' we mean any category of algebras
over a simplicial algebraic theory, which is allowed to be
multi-sorted. The results have applications to the construction of
localization model category structures.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Scheerer-Stanley-Tanre/Qcat
Fibrewise construction applied to Lusternik-Schnirelmann category
by Hans Scheerer, Donald Stanley and Daniel Tanr\'e
scheerer@math.fu-berlin.de
Don.Stanley@agat.univ-lille1.fr
Daniel.Tanre@agat.univ-lille1.fr
Abstract: In this paper a variant of Lusternik-Schnirelmann category is
presented which is denoted by Qcat(X). It is obtained by applying a
base-point free version of Q = Omega-infinity Sigma-infinity fibrewise
to the Ganea fibrations. We prove cat(X) >= Qcat(X) >= scat(X),
where scat(X) denotes Y. Rudyak's strict category weight. However,
Qcat(X) approximates cat(X) better, because e.g. in the case of a
rational space Qcat(X)=cat(X) and scat(X) equals the Toomer invariant.
We show that Qcat(X x Y) <= Qcat(X)+Qcat(Y). The invariant Qcat is
designed to measure the failure of the formula cat(X x S^r)=cat(X)+1. In
fact for 2-cell complexes Qcat(X)< cat(X) if and only if cat(X x S^r) <=
cat(X) for some r >= 1.
We note that the paper is written in the more general context of a
functor L from the category of spaces to itself satisfying certain
conditions; L= Q, Omega^n Sigma^n, Sp^infinity or L_f are just particular
cases.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dicknew
(This is a revised version)
Dickson Invariants hit by the Steenrod Squares
BY K. F. Tan and Kai Xu
Abstract: Let $D_3$ be the Dickson invariant ring of $F_2[X_1,X_2,X_3]$
by GL(3,F_2)$. In this paper, we prove each element in $D_3$ is hit by
the Steenrod square in $F_2[X_1,X_2,X_3]$. Our method provides a clue in
attacking the question in the general case. (This paper contains some
tedious computations which will be dropped in the simplified version
that will be written later.)
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