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).
Jeremy Rickard
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