Subject: new Hopf listings
From: Mark Hovey
Date: 19 Apr 2007 10:10:09 -0400
6 new papers this month, from Benson, Chebolu-Christensen-Minac,
DavisD-Mahowald, Muro-Schwede-Strickland, Oliver-Ventura, and
Panin-Pimenov-Roendigs.
Mark Hovey
New papers appearing on hopf between 3/19/07 and 4/19/07
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Benson/loops
An algebraic model for chains on $\Omega BG\phat$
Dave Benson
Department of Mathematics, University of Aberdeen,
Aberdeen AB24 3UE
Abstract:
We provide an interpretation of the homology of the loop space on the
$p$-completion of the classifying space of a finite group in terms of
representation theory, and demonstrate how to compute it. We then give
the following reformulation. If $f$ is an idempotent in $kG$ such that
$f.kG$ is the projective cover of the trivial module $k$, and $e=1-f$,
then we exhibit isomorphisms for $n\ge 2$:
H_n(\Omega BG\phat;k) \cong \Tor_{n-1}^{e.kG.e}(kG.e,e.kG)
H^n(\Omega BG\phat;k) \cong \Ext^{n-1}_{e.kG.e}(e.kG,e.kG).
Further algebraic structure is examined, such as products and
coproducts, restriction and Steenrod operations.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/ghostnumber
TITLE: Ghosts in modular representation theory
AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac
Department of Mathematics
University of Western Ontario
London, ON N6A 5B7, Canada
AMS Subject classsification: Primary 20C20, 20J06; Secondary 55P42
ABSTRACT:
A \emph{ghost} over a finite group $G$ is a map between modular
representations of $G$ which is invisible in Tate cohomology. Motivated
by the failure of the \emph{generating hypothesis}---the statement that
ghosts between finite-dimensional $G$-representations factor through a
projective---we define the \emph{compact ghost number} of $kG$ to be the
smallest integer $l$ such that the composition of any $l$ ghosts between
finite-dimensional $G$-representations factors through a projective. In
this paper we study ghosts and the compact ghost numbers of
$p$-groups. We begin by showing that a weaker version of the generating
hypothesis, where the target of the ghost is fixed to be the trivial
representation $k$, holds for all $p$-groups. We do this by proving
that a map between finite-dimensional $G$-representations is a ghost if
and only if it is a \emph{dual ghost}. We then compute the compact
ghost numbers of all cyclic $p$-groups and all abelian $2$-groups with
$C_2$ as a summand. We obtain bounds on the compact ghost numbers for
abelian $p$-groups and for all $2$-groups which have a cyclic subgroup
of index $2$. Using these bounds we determine the finite abelian groups
which have compact ghost number at most $2$. %Finally, using universal
ghosts, we establish various sets of conditions which %guarantee the
existence of a non-trivial ghost out of a $G$-representation. Our
methods involve techniques from group theory, representation theory,
triangulated category theory, and constructions motivated from homotopy
theory.
COMMENTS: This version replaces an earlier one with file name ghost.tex.
This is
a substantial improvement with many new results and major reorganisation
of the
paper.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Mahowald/overlook
Nonimmersions of RP^n implied by tmf, revisited
Donald M. Davis and Mark Mahowald
In a 2002 paper, the authors and Bruner used the new spectrum
tmf to obtain some new nonimmersions of real projective spaces.
In this note, we complete/correct two oversights in that paper.
The first is to note that in that paper a general nonimmersion
result was stated which yielded new nonimmersions for RP^n with
n as small as 48, and yet it was stated there that the first new result
occurred when n=1536. Here we give a simple proof of those
overlooked results.
Secondly, we fill in a gap in the proof of the 2002 paper. There it was
claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We
realized recently that this is not clear. However, here we show that
it is true up multiplication by a unit in the appropriate ring, and so we
retrieve all the nonimmersion results claimed in the original paper.
Finally, we present a complete determination of
tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty)
in positive dimensions.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/Muro-Schwede-Strickland/tcwm17
Author(s): Fernando Muro, Stefan Schwede, Neil Strickland
Abstract: We exhibit examples of triangulated categories which are
neither the stable category of a Frobenius category nor a full
triangulated subcategory of the homotopy category of a stable model
category. Even more drastically, our examples do not admit any
non-trivial exact functors to or from these algebraic respectively
topological triangulated categories.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Ventura/ov2
Saturated fusion systems over $2$-groups
Bob Oliver & Joana Ventura
AMS classification: Primary 20D20. Secondary 20D45, 20D08
Abstract:
We develop methods for listing, for a given 2-group $S$, all
nonconstrained centerfree saturated fusion systems over $S$. These are
the saturated fusion systems which could, potentially, include minimal
examples of exotic fusion systems: fusion systems not arising from any
finite group. To test our methods, we carry out this program over four
concrete examples: two of order $27$ and two of order $2^{10}$. Our
long term goal is to make a wider, more systematic search for exotic
fusion
systems over 2-groups of small order.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Panin-Pimenov-Roendigs/BGL-post
Author: Ivan Panin
Author2: Konstantin Pimenov
Author3: Oliver Roendigs
Title: On Voevodsky's algebraic K-theory spectrum BGL
Under a certain normalization assumption we prove that the Voevodsky's
spectrum BGL which represents algebraic $K$-theory is unique over the
integers. Following an idea of Voevodsky, we equip the spectrum BGL
with the structure of a commutative ring spectrum in the motivic stable
homotopy category. Furthermore, we prove that under a certain
normalization assumption this ring structure is unique over the integers
We pull this structure back to get a distinguished monoidal structure on
BGL for an arbitrary Noetherian base scheme.
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