Subject: new Hopf listings
From: Mark Hovey
Date: 29 Jan 2000 21:35:27 -0500
This seems a good time to remind you that if you have submitted a paper
to Hopf and it does not appear on this list, it is NOT because Clarence
has rejected it. Hopf is not an automated archive, so sometimes it
takes a while for papers to be moved into the appropriate spot. On the
other hand, it is always possible that Clarence or I have made a mistake,
so it doesn't hurt to send e-mail reminding us.
5 new papers this time.
Mark Hovey
New papers uploaded to hopf between 1/25/00 and 1/29/00.
1.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi/homotopy-ext
On spaces of self homotopy equivalences of p-completed classifying
spaces of finite groups and homotopy group extensions
By Carles Broto
and Ran Levi
Fix a prime p. A mod-p homotopy group extension of a group $\pi$ by
a group G is a fibration with base space $B\pi^\wedge_p$ and fibre
$BG^\wedge_p$. In this paper we study homotopy group extensions for
finite groups. We observe that there is a strong analogy between homotopy
group extensions and ordinary group extensions. The study involves
investigating the space of self homotopy equivalences of a
p-completed classifying space. In particular we show that under
the appropriate assumption on $G$, the identity component of this
space is homotopy equivalent to $BZ(G)$, the classifying space of
the centre of $G$. We proceed by studying the group of components.
We show that this group maps into a group of natural equivalences
of a certain functor with kernel and cokernel, which are
computable in terms of the first and second derived functors of
the inverse limit for a certain diagram of abelian groups.
2.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/simp-models
Combinatorial models for iterated loop spaces
By: Fred Cohen
and Ran Levi
The objective of this paper is to provide free simplicial group models
for the functors $\Omega^n X$ and $\Omega^n\Sigma^{n+k}X$. The models
are based on classical constructions in simplicial homotopy
theory. Specifically, Milnor's functor F, Kan's loops group functor G
and the Moore loop space construction $\Omega$ are used to produce these
models. The models are given in terms of free groups with specific
generators and the formulas defining the simplicial operators are
given. The utility of these models is that in them certain maps can be
written explicitly in a relatively easy way. To illustrate this a null
homotopy of the commutator map on a double loop space is given. Similar
ideas are used to give a model for pointed mapping spaces out of a
Riemann surface.
3.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Cohen-Levi/stunted-proj
ON THE HOMOTOPY TYPE OF INFINITE STUNTED PROJECTIVE SPACES
By: Fred Cohen
and Ran Levi
Let $X_n$ denote the infinite stunted projective space ${\Bbb
R}P^\infty/{\Bbb R}P^{n-1}$. In this note we study the homotopy type
of this family of spaces. In particular we show that for $n=2 $ and 4,
the space $X_n$ splits after looping once and for $n=3$ after looping
four times and passing to connected covers. In each case the factors
are loop spaces on naturally occuring finite complexes. These result
generalise to higher values of $n$, but in those cases without a
splitting result. The splittings enable us to carry out a calculation
of low dimensional homotopy and loop space homology for these spaces,
which complements a computer calculation of Sergeraert and Smirnov. A
number of interesting related facts and questions is also discussed.
4.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McGibbon-Strom/numphant
Numerical invariants of phantom maps
C. A. McGibbon and Jeffrey Strom
Wayne State University and Dartmouth College
Two numerical homotopy invariants of phantom maps, the Gray index
G(f) and the essential category weight E(f), are studied. The
possible values of these invariants are determined. In certain
cases bounds on these values are given in terms of rational
homotopy data.
Examples are provided showing that the Gray index can take any
positive finite value. For certain cases it is shown that every
essential phantom f: X --> Y has finite Gray index. However it
is also shown that there exist spaces, e. g. CP^\infty, which are
the domains of essential phantoms with infinite index.
The same type of analysis is carried out on the essential category
weight of a phantom map. If the loop space of X is homotopy
equivalent to a finite complex, then every phantom f: X --> Y has
E(f) = \infty. However, in certain other cases it is shown that
E(f) is strictly less than the rational Lusternik-Schnirelmann
category of the domain. A homotopy classification of phantoms
f: K(Z, n)--> S^m is given along with the values of E(f).
The invariants G and E provide decreasing filtrations on the set
of homotopy classes of phantoms from X to Y. A third filtration on
this set is introduced for certain special targets. When the rational
cohomology of the domain X is finitely generated, this
filtration enables one to reduce the search for essential phantoms
(into finite type targets) to a finite list of spheres.
5.
http://hopf.math.purdue.edu/cgi-bin/generate?/pub/TanK-XuK/dickson
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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before you type
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the dvi file using binary, by first typing
binary
then
put
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