Covering group theory for topological groups

Abstract

We develop a covering group theory for a large category of “coverable” topological groups, with a generalized notion of “cover”. Coverable groups include, for example, all metrizable, connected, locally connected groups, and even many totally disconnected groups. Our covering group theory produces a categorial notion of fundamental group, which, in contrast to traditional theory, is naturally a (prodiscrete) topological group. Central to our work is a link between the fundamental group and global extension properties of local group homomorphisms. We provide methods for computing the fundamental group of inverse limits and dense subgroups or completions of coverable groups. Our theory includes as special cases the traditional theory of Poincaré, as well as alternative theories due to Chevalley, Tits, and Hoffmann–Morris. We include a number of examples and open problems.