Overlays and group actions

TL;DR

This paper redefines overlays as a form of covering maps with star refinement properties, providing new characterizations and applications to topological group structures and local isometries.

Contribution

It introduces novel characterizations of overlays using chain lifts and simplicial coverings, and applies these to topological group and metric space contexts.

Findings

Overlays can be characterized by chain lift properties.

Overlays induce simplicial coverings of nerves.

Surjective overlays between metrizable spaces are local isometries.

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 $x\in X$.