Extending topological Abelian groups by the unit circle

Autores: Bello, H. J.; Chasco Ugarte, María Jesús; Domínguez, X.
ISSN: 1085-3375
Páginas: 590159
Fecha de publicación: 2013
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 x Z -> Z -> 0. We study the class S-TG(T) 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-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 -> 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.