On the "three-space" problem for locally quasi-convex topological groups

Abstract

We study, in the context of abelian topological groups, the “three-space” problem for the property of being locally quasi-convex, after a paper of M. Bruguera. Our main contributions are: establishing a 3-lemma suitable to work with topological groups (which allows to translate the basic elements of homological algebra to the category of topological groups) and obtaining the analogue, for topological groups, of Dierolf's result in topological vector spaces:¶Theorem. Given two abelian locally quasi-convex groups H and G there exists a non-locally-quasi-convex extension of H and G if and only if there exists a non-locally-quasi-convex extension of S (the circle group) and G.