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Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or group cohomology.)

Assume there are only finitely many fixed points.

Can there be more than one?

Remarks: (1) If X is a manifold, there can not be more than one, by the Lefschetz-Hopf fixed point theorem. But to generalize the proof to the general case, one would need to define the "index" of an isolated fixed point and I have no idea how to accomplish that, if one does not work with manifolds only. (2) The statement is true for graphs. (3) The statement is easily seen to be wrong for general contractible spaces, e.g. take a free action on $S^{\infty}$.

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  • $\begingroup$ I think the number of fixed points must be congruent to $1 \bmod p$ (subdivide and homotope until the action is cellular, then compute the Euler characteristic from cellular homology), but I don't know how to rule out the possibility that there are, say, $p + 1$ of them. $\endgroup$
    – Qiaochu Yuan
    Commented Jun 13, 2015 at 7:45
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    $\begingroup$ I also don't know what argument you have in mind for the manifold case. Say $p = 2$. Why can't there be three fixed points with indices $1, 1, -1$? $\endgroup$
    – Qiaochu Yuan
    Commented Jun 13, 2015 at 7:46
  • $\begingroup$ What do you mean "this seems to hold?" Your question is "Can there be more than one?", so I'm a bit confused. $\endgroup$
    – Francesco Polizzi
    Commented Jun 13, 2015 at 9:23
  • $\begingroup$ Sorry for the confusing grammar - I changed that now. $\endgroup$
    – Jens Reinhold
    Commented Jun 13, 2015 at 15:57
  • $\begingroup$ @Qiaochu: I think that for any map of finite order, all local fixed points have index +1. $\endgroup$
    – Jens Reinhold
    Commented Jun 13, 2015 at 16:01

1 Answer 1

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If a finite $p$-group $P$ acts on a finite-dimensional $CW$-complex $X$ which is acyclic mod $p$, then the fixed point set $X^P$ is also acyclic mod $p$. This is a special case of "Smith theory" (see Theorem II of "Fixed-Point Theorems for Periodic Transformations", P. A. Smith, American Journal of Mathematics Vol. 63, No. 1 (Jan., 1941), pp. 1-8 for an early version of the theory where $P$ is cyclic).

For a cellular action of $P=\mathbb{Z}/p$ on a contractible finite $CW$ complex with only finitely many fixed points, the reduced mod $p$ cellular chain complex would be a bounded acyclic chain complex of $\mathbb{F}_pP$-modules, with all components free modules in non-zero degree, and the degree zero component the direct sum of a free module and a trivial module of dimension one less than the number of fixed points. But this is impossible by representation theory if that trivial module is non-zero, since a trivial $\mathbb{F}_pP$-module doesn't have finite projective dimension.

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