Existence and uniqueness of group structures on covering spaces over groups
Volume 238 / 2017
Fundamenta Mathematicae 238 (2017), 241-267
MSC: Primary 22D05, 57M10; Secondary 14E20.
DOI: 10.4064/fm990-10-2016
Published online: 23 March 2017
Abstract
Let $f:X\rightarrow Y$ be a covering map from a connected space $X$ onto a topological group $Y$ and let $x_{0}\in X$ be a point such that $f(x_{0})$ is the identity of $Y.$ We examine if there exists a group operation on $X$ which makes $X$ a topological group with identity $x_{0}$ and $f$ a homomorphism of groups. We prove that the answer is positive in two cases: if $f$ is an overlay map over a locally compact group $Y$, and if $Y$ is locally compactly connected. In this way we generalize previous results for overlay maps over compact groups and covering maps over locally path-connected groups. Furthermore, we prove that in both cases the group structure on $X$ is unique.
