Finite groups of diffeomorphisms are topologically determined by a vector field

Abstract

In a previous work it is shown that every finite group $G$ of diffeomorphisms of a connected smooth manifold $M$ of dimension $\geq 2$ equals, up to quotient by the flow, the centralizer of the group of smooth automorphisms of a $G$-invariant complete vector field $X$ (shortly $X$ describes $G$). Here the foregoing result is extended to show that every finite group of diffeomorphisms of $M$ is described, within the group of all homeomorphisms of $M$, by a vector field. As a consequence, it is proved that a finite group of homeomorphisms of a compact connected topological $4$-manifold, whose action is free, is described by a continuous flow.