Extending Topological Abelian Groups by the Unit Circle

Author:

Bello Hugo J.1ORCID,Chasco María Jesús1,Domínguez Xabier2ORCID

Affiliation:

1. Departamento de Física y Matemática Aplicada, University of Navarra, 31080 Pamplona, Spain

2. Departamento de Métodos Matemáticos y de Representación, University of A Coruña, 15071 A Coruña, Spain

Abstract

A twisted sum in the category of topological Abelian groups is a short exact sequence0YXZ0where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to0YY×ZZ0. We study the classSTG𝕋of topological groupsGfor which every twisted sum0𝕋XG0splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups withtopologies. 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 ofSTG𝕋, we use the connection between extensions of the form0𝕋XG0and quasi-characters onG, 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 of𝒦-space, which were interpreted for topological groups by Cabello.

Funder

Spanish MICINN

Publisher

Hindawi Limited

Subject

Applied Mathematics,Analysis

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. On local quasi-convexity as a three-space property in topological abelian groups;Journal of Mathematical Analysis and Applications;2021-07

2. The Ext group in the categories of topological abelian groups and topological vector spaces;Topology and its Applications;2017-04

3. Splittings and cross-sections in topological groups;Journal of Mathematical Analysis and Applications;2016-03

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