Subject:
Date: Mon, 21 Apr 2003 10:13:15 -0400 (EDT)
From: Elliott Pearl
To:
CC:
Dear Don,
Please consider posting this message to the algebraic topology
discussion list.
I am editing a cumulative status report on the book Open Problems in
Topology (1990). The book contains the problem list from the 1986 Arcata
Conference collected by J. F. Adams, W. Browder and G. E. Carlsson.
The book is out-of-print but it is freely available at the Elsevier
math portal MathematicsWeb.
The whole book is at
http://www.mathematicsweb.org/homepage/sac/opit/toc.htm
The section on algebraic topology is at
http://www.mathematicsweb.org/homepage/sac/opit/28/article.pdf
This report will be published in Topology and its Applications.
Please help me by sending me any information on solutions to the
problems from this section on algebraic topology, even if this
information is quite old.
For reference, I am appending the information that I have already
collected for this section. The citation keys are MR numbers.
I can provide the full LaTeX source upon request.
Thanks,
Elliott Pearl
Problem 754
(A. Adem) Let $G$ be a finite $p$-group.
If $H^n(G;\mathbb{Z})$ has an element of order $p^r$ for some value of
$n$, does the same follow for infinitely many $n$?
No.
J. Pakianathan \cite{MR2000m:20086} constructed a counterexample.
Problem 766.
J. Klippenstein and V. Snaith \cite{MR90c:55011} proved a conjecture of
Barratt-Jones-Mahowald concerning framed manifolds having Kervaire
invariant one.
Problems 810--813.
(Morimoto) Do there exist smooth, one fixed point actions of compact Lie
groups (possibly finite groups) on $S^3$, $D^4$, $S^5$, or $S^8$
(respectively)?
When $G$ is a compact Lie group, if a $G$-manifold has exactly one
$G$-fixed point then the action is said to be a \emph{one fixed point
action}.
M. Furuta \cite{furuta} proved that there are no smooth one fixed point
actions on $S^3$ of finite groups.
This was also proved by N. P. Buchdahl, S. Kwasik and R. Schultz
\cite[Theorem I.1]{MR92b:57047}.
In \cite[Theorem II.2]{MR92b:57047}, they proved that there are no
locally linear, one fixed point actions on homology $4$-dimensional
spheres of finite groups; in \cite[Theorem II.4]{MR92b:57047}, this was
proved for homology $5$-dimensional spheres of finite groups.
M. Morimoto \cite{MR88j:57039,MR92h:57055} proved that there exist smooth
one fixed point actions on $S^6$ of $A_5$.
A. Bak and M. Morimoto \cite[Theorem 7]{MR93e:57058} proved that there
exist smooth one fixed point actions on $S^7$ of $A_5$.
A. Bak and M. Morimoto \cite{MR95e:19006} proved that there are smooth
one fixed point actions on $S^8$ of $A_5$.
Problems 822--824.
These problems concern the question of which smooth manifolds can occur as
the fixed point sets of smooth actions of a given compact Lie group $G$ on
disks (resp., Euclidean spaces). In the case where $G$ is a finite group
not of prime power order, complete answers go back to B. Oliver
\cite{MR97g:57059}. Specifically, for a compact smooth manifold $F$
(resp., a smooth manifold $F$ without boundary), Oliver has described
necessary and sufficient conditions for $F$ to occur as the fixed point
set of a smooth action of $G$ on a disk (resp., Euclidean space). Oliver's
description of the necessary and sufficient conditions imply affirmative
answers to Problems 822, 823 and 824. In the case where $G$ is of
$p$-power order for a prime $p$, a compact smooth manifold $F$ occurs as
the fixed point set of a smooth action of $G$ on a disk if and only if $F$
is mod $p$-acyclic and stably complex. This follows from Smith theory and
the results of L. Jones \cite{MR45:4427}. A similar result holds in the
case where $G$ is a compact Lie group such that the identity connected
component $G_0$ of $G$ is abelian (i.e., $G_0$ is a torus) and $G/G_0$ is
a finite $p$-group for a prime $p$. Moreover, K. Pawa{\l}owski
\cite{MR90i:57032} proved that for such a group $G$, a smooth manifold $F$
without boundary occurs as the fixed point set of a smooth action of $G$
on some Euclidean space if and only if $F$ is mod $p$-acyclic and stably
complex. The article of K. Pawa{\l}owski \cite{MR1943326} gives an up to
date survey of results and answers to the question of which smooth
manifolds can occur as the fixed point sets of smooth actions of a given
compact Lie group $G$ on Euclidean spaces, disks, and spheres.