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.