Extending Topological Abelian Groups by the Unit Circle

作者：Bello Hugo J; Jesus Chasco Maria; Dominguez Xabier^*

来源：Abstract and Applied Analysis, 2013, 2013: 590159.

DOI：10.1155/2013/590159

A twisted sum in the category of topological Abelian groups is a short exact sequence 0 -%26gt; Y -%26gt; X -%26gt; Z -%26gt; 0 where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to 0 -%26gt; Y -%26gt; Y x Z -%26gt; Z -%26gt; 0. We study the class S-TG(T) of topological groups G for which every twisted sum 0 -%26gt; T -%26gt; X -%26gt; G -%26gt; 0 splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with L-infinity 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-TG(T), we use the connection between extensions of the form 0 -%26gt; T -%26gt; X -%26gt; G -%26gt; 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.

• 出版日期2013