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Covering group theory for topological groups
Valerii N. Berestovskii and Conrad Plaut
We develop a covering group theory for a large category of "coverable" topological groups, which includes all metrizable, connected, locally connected groups, and even many totally disconnected groups. Our covering group theory produces a categorial notion of fundamental group. In contrast to the traditional theory, our fundamental group is a naturally a topological group, and is the inverse limit of discrete groups. Our work also reveals 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. The paper includes a number of examples and open problems.