Overlays and group actions

Abstract

Overlays were introduced by R. H. Fox [6] as a subclass of covering maps. We offer a different view of overlays: it resembles the definition of paracompact spaces via star refinements of open covers. One introduces covering structures for covering maps and $p:X\to Y$ is an overlay if it has a covering structure that has a star refinement. We prove two characterizations of overlays: one using existence and uniqueness of lifts of discrete chains, the second as maps inducing simplicial coverings of nerves of certain covers. We use those characterizations to improve results of Eda-Matijevi\' c concerning topological group structures on domains of overlays whose range is a compact topological group. In case of surjective maps $p:X\to Y$ between connected metrizable spaces we characterize overlays as local isometries: $p:X\to Y$ is an overlay if and only if one can metrize $X$ and $Y$ in such a way that $p|B(x,1):B(x,1)\to B(p(x),1)$ is an isometry for each $x\in X$.