Let $M$ be a topological manifold with a continuous, free, and proper action of a topological group $G$ on $M$. Is $M/G$ a topological manifold? Is $M \to M/G$ a locally trivial principal bundle?

If no, are there mild assumptions on $G$ in order to ensure $M/G$ is a topological manifold and/or $M \to M/G$ is a locally trivial principal bundle? By "mild" I mean that $G$ is necessarily not a Lie group. I am aware that "upgrading" $G$ to a Lie group allows us to "downgrade" $M$ to a completely regular Hausdorff space to ensure that $M \to M/G$ is locally trivial.

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    $\begingroup$ Did you check what the results towards the Hilbert-Smith conjecture imply in your setting? Specifically, can the group of p-adic integers act freely? $\endgroup$ Nov 6, 2023 at 16:35
  • $\begingroup$ Possibly this is relevant for you: mathoverflow.net/questions/116408/… $\endgroup$
    – nicrot000
    Nov 12, 2023 at 19:29


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