On Generalized Covering Groups of Topological Groups

TL;DR

This paper extends the classical concept of covering homomorphisms in topological groups by characterizing generalized coverings through open epimorphisms with prodiscrete kernels, and shows these coverings are fibrations.

Contribution

It generalizes the classical covering group theory to include prodiscrete kernels and establishes that all such generalized coverings are fibrations.

Findings

Generalized coverings correspond to open epimorphisms with prodiscrete kernels.

Every generalized covering of a connected locally path connected topological group is a fibration.

The intersection property of covering subgroups is established for generalized coverings.

Abstract

It is well-known that a homomorphism p between topological groups K, G is a covering homomorphism if and only if p is an open epimorphism with discrete kernel. In this paper we generalize this fact, in precisely, we show that for a connected locally path connected topological group G, a continuous map p is a generalized covering if and only if K is a topological group and p is an open epimorphism with prodiscrete (i.e, product of discrete groups) kernel. To do this we first show that if G is a topological group and H is any generalized covering subgroup of fundamental group of G, then H is as intersection of all covering subgroups, which contain H. Finally, we show that every generalized covering of a connected locally path connected topological group is a fibration.