Extending Topological Abelian Groups by the Unit Circle

Abstract

A twisted sum in the category of topological Abelian groups is a short exact sequence 0→Y→X→Z→0 where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to 0→Y→Y×Z→Z→0. We study the class STG𝕋 of topological groups G for which every twisted sum 0→𝕋→X→G→0 splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with ℒ∞ topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses of STG𝕋, we use the connection between extensions of the form 0→𝕋→X→G→0 and quasi-characters on G, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of 𝒦-space, which were interpreted for topological groups by Cabello.