Assume a finite group $G$ acts smoothly on a connected manifold $M$. Is it true that all connected components of the fixed point set $M^G$ have the same dimension? If not in general, then maybe at least for $M$ being spheres? I am interested in some other result than the classical Smith theory of $p$-groups - in this case for $\mathbb{Z}_p$-homology spheres, the fixed point sets are $\mathbb{Z}_p$-homology spheres as well and thus connected.
2026-03-25 21:01:18.1774472478
Dimensions of connected components of fixed point sets
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