Existence and uniqueness of group structures on covering spaces over groups

Katsuya Eda ¹   ; Vlasta Matijević ²

¹ Department of Mathematics Waseda University Tokyo 169-8555, Japan
² Department of Mathematics University of Split Teslina 12 21000 Split, Croatia

Fundamenta Mathematicae, Tome 238 (2017) no. 3, pp. 241-267

Résumé

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.

Détail

DOI : 10.4064/fm990-10-2016

Keywords: rightarrow covering map connected space topological group point identity examine there exists group operation which makes topological group identity homomorphism groups prove answer positive cases overlay map locally compact group locally compactly connected generalize previous results overlay maps compact groups covering maps locally path connected groups furthermore prove cases group structure unique