Subject:
new Hopf listings
From:
Mark Hovey
Date:
05 Feb 2005 06:30:00 -0500
To:
dmd1@lehigh.edu
6 new papers this month, by Chacholski-Pitsch-Scherer, Ching,
DavisDaniel, Dugger, Flores-Scherer, and May-Sigurdsson.
Mark Hovey
New papers appearing on hopf between 1/10/05 and 2/5/05
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chacholski-Pitsch-Scherer/hopullbacks
Title: Homotopy pull-back squares up to localization
Authors: Wojciech Chacholski, Wolfgang Pitsch, Jerome Scherer
AMS classification numbers: Primary 55P60, 55R70; Secondary 55U35, 18G55
ArXiv submission number: math.AT/0501250
email: wojtek@math.kth.se, pitsch@mat.uab.es, jscherer@mat.uab.es
Abstract: We characterize the class of homotopy pull-back squares by
means of elementary closure properties. The so called Puppe theorem
which identifies the homotopy fiber of certain maps constructed as
homotopy colimits is a straightforward consequence. Likewise we
characterize the class of squares which are homotopy pull-backs ``up to
Bousfield localization". This yields a generalization of Puppe's theorem
which allows to identify the homotopy type of the localized homotopy
fiber. When the localization functor is homological localization this is
one of the key ingredients in the group completion theorem.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Ching/operad_bar
Bar constructions for topological operads and the Goodwillie derivatives
of the identity
Michael Ching
mcching@math.mit.edu
Massachusetts Institute of Technology
Includes 19 PS figures with filenames *.pstex
Abstract:
We describe a cooperad structure on the simplicial bar construction on a
reduced operad of based spaces or spectra and, dually, an operad
structure on the cobar construction on a cooperad. We show that if the
homology of the original operad (respectively, cooperad) is Koszul, then
the homology of the bar (respectively, cobar) construction is the Koszul
dual. We use our results to construct an operad structure on the
partition poset models for the Goodwillie derivatives of the identity
functor on based spaces and show that this induces the `Lie' operad
structure on the homology groups of those derivatives. We also extend
the bar construction to modules over operads (and, dually, to comodules
over ooperads) and show that a based space naturally gives rise to a
right module over the operad formed by the derivatives of the identity.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/enhfps2
Title: Homotopy fixed points for L_K(n)(E_n ^ X) using the continuous action
Author: Daniel Davis
E-mail: dgdavis@math.purdue.edu
Address: Purdue University, Department of Mathematics, 150 N. University
Street, West Lafayette, IN 47907-2067
Abstract: Let G be a closed subgroup of G_n, the extended Morava stabilizer
group. Let E_n be the Lubin-Tate spectrum, let X be an arbitrary spectrum
with trivial G-action, and define E^(X) to be L_K(n)(E_n ^ X). We prove
that E^(X) is a continuous G-spectrum with a G-homotopy fixed point spectrum,
defined with respect to the continuous action. Also, we construct a descent
spectral sequence whose abutment is the homotopy groups of the G-homotopy
fixed point spectrum of E^(X). We show that the homotopy fixed points of
E^(X) come from the K(n)-localization of the homotopy fixed points of the
spectrum (F_n ^ X).
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/spenrich
Spectral enrichments of model categories
Daniel Dugger
ddugger@math.uoregon.edu
Abstract:
We prove that every stable, combinatorial model category has a natural
enrichment by symmetric spectra (really a natural equivalence class of
enrichments). This in some sense generalizes the simplicial
enrichment of model categories provided by the Dwyer-Kan hammock
localization. As a particular application, we associate to every
object in a stable, combinatorial model category a certain "homotopy
endomorphism ring spectrum". The homotopy type of this ring spectrum
is preserved by Quillen equivalences, and so serves as an invariant of
model categories.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Flores-Scherer/cwandfusion
Title: Cellularization of classifying spaces and fusion properties of
finite groups
Authors: Ramon J. Flores, Jerome Scherer
AMS classification numbers: Primary 55P60, 20D200; Secondary 55R37, 55Q05
ArXiv submission number: math.AT/0501442
email: ramonj@mat.uab.es, jscherer@mat.uab.es
Abstract: One way to understand the mod p homotopy theory
of classifying spaces of finite groups is to compute their
B\Z/p-cellularization. In the easiest cases this is a
classifying space of a finite group (always a finite p-group).
If not, we show that it has infinitely many non-trivial
homotopy groups. Moreover they are either p-torsion free
or else infinitely many of them contain p-torsion. By means
of techniques related to fusion systems we exhibit concrete
examples where p-torsion appears.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/May-Sigurdsson/MSMaster
Parametrized homotopy theory
J. P. May and J. Sigurdsson
may@math.uchicago.edu, jsigurds@nd.edu
University of Chicago, University of Notre Dame
Primary 19D99, 55N20, 55P42; Secondary 19L99, 55N22, 55T25
Abstract: We provide rigorous modern foundations for parametrized
(equivariant, stable) homotopy theory in this four part monograph.
In Part I, we give preliminaries on the necessary point-set topology,
on base change and other relevant functors, and on generalizations
of various standard results to the context of proper actions of
non-compact Lie groups.
In Part II, we give a leisurely development of the homotopy theory
of ex-spaces that emphasizes several issues of independent
interest. It includes much new material on the general theory
of topologically enriched model categories. The essential
point is to resolve problems in the homotopy theory of ex-spaces
that have no nonparametrized counterparts. In contrast to
previously encountered situations, model theoretic techniques
are intrinsically insufficient for this purpose. Instead, a
rather intricate blend of model theory and classical homotopy
theory is required.
In Part III, we develop the homotopy theory of parametrized spectra.
We work equivariantly and with highly structured smash products and
function spectra. The treatment is based on equivariant orthogonal
spectra, which are simpler for the purpose than alternative kinds
of spectra. Again, there are many difficulties that have no
nonparametrized counterparts and cannot be dealt with model
theoretically.
In Part IV, we give a fiberwise duality theorem that allows fiberwise
recognition of dualizable and invertible parametrized spectra. This
allows application of the formal theory of duality in symmetric
monoidal categories to the construction and analysis of transfer maps.
A construction of fiberwise bundles of spectra, which are like bundles
of tangents along fibers but with spectra replacing spaces as fibers,
plays a central role. Using it, we obtain a simple conceptual proof
of a generalized Wirthmuller isomorphism theorem that calculates the
right adjoint to base change along an equivariant bundle with manifold
fibers in terms of a shift of the left adjoint. Due to the generality
of our bundle theoretic context, the Adams isomorphism theorem
relating orbit and fixed point spectra is a direct consequence.
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