Subject: new Hopf listings
From: Mark Hovey
Date: 01 Oct 2005 10:20:39 -0400
There are 8 new papers this time, from Arkowitz-Lupton, DavisDaniel(2),
DavisD-Sun, Felix-Lupton, Henn, Kreck-Lueck, and
Lupton-Phillips-Schochet-SmithSB.
Mark Hovey
New papers appearing on hopf between 9/5/05 and 10/1/05
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Lupton/ArkLupActions
Homotopy Actions, Cyclic Maps and their Duals
Martin Arkowitz and Gregory Lupton
MSC 2000 55Q05, 55M30, 55P30
M.Arkowitz@Dartmouth.edu
G.Lupton@csuohio.edu
Abstract:
An action of A on X is a map F: AxX to X such that F|_X = id: X to X.
The restriction F|_A: A to X of an action is called a cyclic map. Special
cases of these notions include group actions and the Gottlieb groups
of a space, each of which has been studied extensively. We prove
some general results about actions and their Eckmann-Hilton duals.
For instance, we classify the actions on an H-space that are
compatible with the H-structure. As a corollary, we prove that if
any two actions F and F' of A on X have cyclic maps f and
f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F')
give the same action of Omega(A) on Omega(X). We introduce a new
notion of the category of a map g and prove that g is cocyclic
if and only if the category is less than or equal to 1. From this
we conclude that if g is cocyclic, then the Berstein-Ganea
category of g is <= 1. We also briefly discuss the
relationship between a map being cyclic and its cocategory being
<= 1.
Note:
Appeared in Homology, Homotopy and Applications, vol. 7(1) (2005),
169-184.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/galois
Title: Rognes's theory of Galois extensions and the continuous action
of G_n on E_n
Author: Daniel G. Davis
E-mail: dgdavis@math.purdue.edu
Address: Purdue University
Abstract: Let us take for granted that $L_{K(n)}S0 \rightarrow E_n$ is
some kind of a G_n-Galois extension. Of course, this is in the setting of
continuous G_n-spectra. How much structure does this continuous G-Galois
extension have? How much structure does one want to build into this
notion to obtain useful conclusions? If the author's conjecture that
"E_n/I, for a cofinal collection of I's, is a discrete G_n-symmetric ring
spectrum" is true, what additional structure does this give the
continuous G_n-Galois extension? Is it useful or merely beautiful? This
paper is an exploration of how to answer these questions. This inactive
manuscript arose as a letter to John Rognes, whom he thanks for a helpful
conversation in Rosendal. This paper was written before John's preprints
(the initial version and the final one) on Galois extensions were
available. The author thanks Paul Goerss for his encouragement.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/subhg
Title: Attempting to construct homotopy orbits for profinite groups
Author: Daniel G. Davis
E-mail: dgdavis@math.purdue.edu
Address: Purdue University
Abstract: This note gives a heuristic argument for how one might
like to define X_{hG}, for G profinite; it represents a first step
in attempting to do this. The argument is not shown to work, and
though the heuristic seems plausible, the author does not know
how to complete the critical Definition 4.2. Also, the proof of
Theorem 5.2 is incomplete.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Sun/DavisSun
A number-theoretic approach to homotopy exponents of SU(n)
Donald M. Davis and Zhi-Wei Sun
dmd1@lehigh.edu
zwsun@nju.edu.cn
AMS Classifications: 55Q52, 57T20, 11A07, 11B65, 11S05
Abstract:
We use methods of combinatorial number theory to prove that,
for all n and p, some homotopy group pi_i(SU(n)) contains an
element of order p^{n-1+ord_p([n/p]!)}, where ord_p(m)
denotes the exponent of p in m.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/Felix-Lupton/FelLupEval
Title: Evaluation Maps in Rational Homotopy
Authors: Yves Felix and Gregory Lupton
e-mail: felix@math.ucl.ac.be and G.Lupton@csuohio.edu
AMS MSC2000: 55P62, 55Q05
arXiv: math.AT/0509632
Abstract: Let E be an H-space acting on a based space X.
Then we refer to ev: E -> X, the map obtained by acting on
the base point of X, as a ``generalized evaluation map."
We establish several fundamental results about the rational
homotopy behaviour of a generalized evaluation map, all of
which apply to the usual evaluation map Map(X, X;1) -> X.
With mild hypotheses on X, we show that a generalized
evaluation map ev factors, up to rational homotopy,
through a map Gamma_ev: S_ev -> X where S_ev is a
(relatively small) finite product of odd-dimensional
spheres and the map induced by Gamma_ev on rational homotopy
groups is injective. This result has strong consequences:
if the image in rational homotopy groups of ev
is trivial, then the generalized evaluation map is
null-homotopic after rationalization; unless X satisfies a
very strong splitting condition, any generalized evaluation
map induces the trivial homomorphism in rational cohomology;
the map Gamma_ev is rationally a homotopy monomorphism and
a generalized evaluation map may be written as a composition
of a homotopy epimorphism and this homotopy monomorphism.
We include illustrative examples and prove numerous
subsidiary results of interest.
6.
http://hopf.math.purdue.edu/cgi-bin/generate?/Henn/kn-res-ded
Title: On finite resolutions of K(n)-local spheres
Author: Hans-Werner Henn
e-mail address: henn@math.u-strasbg.fr
Abstract: For odd primes p we construct finite resolutions of the
trivial module Z_p for the n-th Morava stabilizer group
by (direct summands of) permutation modules with respect to
finite p-subgroups. Furthermore we discuss
the problem of realizing these resolutions
by finite resolutions of the K(n)-local sphere via
spectra which are (direct summands of)
wedges of homotopy fixed point spectra
for the action of these finite p-subgroups
on the Lubin-Tate spectrum E_n.
7.
http://hopf.math.purdue.edu/cgi-bin/generate?/Kreck-Lueck/kreck+lueck0905
Title of Paper: Topological rigidity for non-aspherical manifolds
Author(s): Matthias Kreck and Wolfgang Lueck
AMS Classification number: 57N99, 57R67.
xxx_archive: math.GT/0509238
kreck@mathi.uni-heidelberg.de
lueck@math.uni-muensetr.de
The Borel Conjecture predicts that closed aspherical manifolds are
topological rigid. We want to investigate when a non-aspherical oriented
connected closed manifold M is topological rigid in the following sense.
If f:
N ---> M is an orientation preserving homotopy equivalence with a closed
oriented manifold as target, then there is an orientation preserving
homeomorphism h: N ---> M such that h and f induce up to conjugation the
same
maps on the fundamental groups. We call such manifolds Borel manifolds. We
give
partial answers to this questions for S^k x S^d, for sphere bundles over
aspherical closed manifolds of dimension less or equal to 3 and for
3-manifolds
with torsionfree fundamental groups. We show that this rigidity is
inherited
under connected sums in dimensions greater or equal to 5. We also classify
manifolds of dimension 5 or 6 whose fundamental group is the one of a
surface and whose second homotopy group is trivial.
8.
http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-Phillips-Schochet-SmithSB/RationalTaylor
Title: Banach Algebras and Rational Homotopy Theory
Authors: Gregory Lupton, N.Christopher Phillips, Claude L.~Schochet
and Samuel B. Smith
AMS MSC (2000): 46J05, 46L85, 55P62, 54C35, 55P15, 55P45
arXiv number: math.AT/0509269
g.lupton@csuohio.edu
claude@math.wayne.edu
smith@sju.edu
Abstract: Let A be a unital commutative Banach algebra with maximal
ideal space Max(A). We determine the rational H-type of $GL_n (A)$,
the group of invertible $n \times n$ matrices with coefficients in A
in terms of the rational cohomology of Max(A). We also address an
old problem of J. L. Taylor. Let $Lc_n (A)$ denote the space of
``last columns'' of $GL_n (A).$ We construct a natural isomorphism
\[
{\check{H}}^s (Max(A); Q)
\cong \pi_{2 n - 1 - s} (Lc_n (A)) \otimes Q
\]
for $n > (1/2) s + 1$ which shows that the rational cohomology
groups of Max(A) are determined by a topological invariant
associated to A. As part of our analysis, we determine the rational
H-type of certain gauge groups F(X,G) for G a Lie group or, more
generally, a rational H-space.
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