$\DeclareMathOperator{\Fix}{\operatorname{Fix}}$I am looking for any reference regarding the following problem:

Problem: Consider a smooth almost-free action of $S^1$ on a smooth sphere $S^n$. Then for all finite subgroups $\mathbb{Z}_m\subseteq S^1$, the fixed point set $\Fix(\mathbb{Z}_m)\subseteq S^n$ is either empty or a rational homology sphere.

This is well-known to be true when the action is linear, or by Smith theory when $m=p^k$ is the power of a prime $p$.

A stronger statement would be that for every action of $\mathbb{Z}_m$ on a sphere $S^n$, the fixed point set $\Fix(\mathbb{Z}_m)$ is a rational homology sphere. What I know is that when $m$ is not the power of a prime, then $\Fix(\mathbb{Z}_m)$ is not necessarily a smooth sphere, but I have not found any indication about its possible (rational) homotopy types.

What boggles me the most is that, I had imagined this problem to either well known to be true, or well known to be false, or extremely interesting – however, I have not found any evidence for any of these.


1 Answer 1


Back in the early 1950's Ed Floyd and Pierre Conner found that composite order cyclic groups could act rather differently that cyclic p-groups. In their tradition, the publication [J. M. Kister, Examples of periodic maps on Euclidean spaces without fixed points. Bull. Amer. Math. Soc. 67 (1961), 471–474] shows that the cyclic group of order 6 can act on a big enough Euclidean space with no fixed points. One point compactifying, one gets an action on a sphere with one fixed point.

There are quite a few survey papers on this subject. Here is one that I found quickly, that has lots of references: Reinhard Schultz, Bull. A.M.S., volume 11, October 1984. See also lots of other papers by him, and also Bob Oliver.

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    $\begingroup$ Thank you for the suggestions. I do not think that the action you mention can extend to an almost free circle action, though: even if a circle extension was possible, I would imagine that the point at infinity would be a fixed point for the whole circle. Also, I was aware of the survey you mention (very nice one), but again I see no mention of the problem I am interested in. Finally, I wonder if there are examples of cyclic group actions whose fixed point set has positive dimension but it is not a rational sphere. $\endgroup$ Nov 1, 2020 at 15:18

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