Extending Topological Abelian Groups by the Unit Circle

Abstract

A twisted sum in the category of topological Abelian groups is a short exact sequence $0\to Y\to X\to Z\to 0$ where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to $0\to Y\to Y\times Z\to Z\to 0$. We study the class $S_{\text{T}\text{G}}\left(𝕋\right)$ of topological groups G for which every twisted sum $0\to 𝕋\to X\to G\to 0$ splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with ${ℒ}_{\infty }$ 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 $S_{\text{T}\text{G}}\left(𝕋\right)$, we use the connection between extensions of the form $0\to 𝕋\to X\to G\to 0$ 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.