Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$.


**Is it true that the family of all such subsets $\{E^H\}$ is finite when $H$ runs over all compact subsgroups of $G$?**



On fixed point sets of actions of compact Lie groups
The quotient $M/G$ carries a stratification by orbit type (see e.g. [this MO question](https://mathoverflow.net/questions/191988/local-structure-of-the-quotient-of-a-lie-group-action) for references). More precisely, for any closed subgroup $F\subseteq G$ the stratum $(M/G)\_{(F)}$ is the set of all orbits which are isomorphic to $G/F$. The set $E^H$ is the union of all strata such that $H$ is conjugate to a subgroup of $F$. Since $M$ is compact, the stratification is finite. So also only finitely many subsets of the form $E^H$ are possible.


