Actions of homeomorphism groups of manifolds admitting a nontrivial finite free action

TL;DR

This paper investigates how the identity component of homeomorphism groups of certain manifolds with free finite group actions influence actions on related manifolds, revealing structural constraints and generalizations of known constructions.

Contribution

It characterizes actions of $ ext{Homeo}_0(M)$ on manifolds with free finite actions, showing they are mostly standard or trivial, and extends the 'coning-off' concept for spheres.

Findings

If $M$ is not an $ extbf{F}_p$-homology sphere, then $N$ decomposes as $M imes K$ with trivial action on $K$.

For $M=S^n$, nontrivial actions generalize the 'coning-off' construction.

The action of $ ext{Homeo}_0(M)$ on $M$ is standard under the given conditions.

Abstract

In this paper, we study the action of $\text{Homeo}_0(M)$, the identity component of the group of homeomorphisms of an $n$-dimensional manifold $M$ with an $\mathbb{F}_p$-free action, on another manifold $N$ of dimension $n+k<2n$. We prove that if $M$ is not an $\mathbb{F}_p$-homology sphere, then $N\cong M\times K$ for a homology manifold $K$ such that the action of $\text{Homeo}_0(M)$ on $M$ is standard and on $K$ is trivial. In particular, for $M=S^n$ a sphere, any nontrivial action is a generalization of the "coning-off" construction.