Extensions of topological Abelian groups and three-space problems

TL;DR

This paper investigates the class of topological abelian groups where all twisted sums with the circle group split, exploring their properties, examples, and connections to three-space problems and quasi-characters.

Contribution

It characterizes the class of groups with splitting twisted sums, including locally precompact and certain limit groups, and studies their stability under various operations.

Findings

Contains locally precompact groups

Includes sequential direct limits of locally compact groups

Closed under open subgroups, quotients, and coproducts

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 $0\to Y\to Y \times Z \to Z\to 0$. \par We study the class $\ST$ 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 locally precompact groups, sequential direct limits of locally compact groups and topological groups with $\mathcal{L}_\infty$ topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups and coproducts. As a technique to find further examples of groups in $\ST$ we use the relation of this class with the existence of quasi-characters on $G$ and with 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.