Weak split extensions of topological Abelian groups

Abstract

In the category of topological Abelian groups, we consider the usual notion of an extension $E=(B\to X\to A)$ of $B$ by $A$, together with the notion of a weakly split extension, i.e., an extension for which the projection $X\to A$ admits a continuous section $A\to X$. Given a weakly split extension $E$, the topological Abelian group $X$ is homeomorphic to $B\times A$, although in general it is not algebraically isomorphic to $B\times A$. For two topological Abelian groups $A$ and $B$, we study the Abelian group $E_{\mathrm{TA}}^{\mathrm{ws}}(A,B)$ of weakly split extensions of $B$ by $A$, modulo extension isomorphisms. We show that $E_{\mathrm{TA}}^{\mathrm{ws}}(A,B)$ can be described as the group of all continuous sum structures defined on the product space $B\times A$ (up to topological isomorphism), with $B$ as a topological subgroup and $A$ as a topological quotient. We also provide an alternative description of $E_{\mathrm{TA}}^{\mathrm{ws}}(A,B)$ as a quotient $Z_c(A,B)/B_c(A,B)$, where $Z_c(A,B)$ consists of cocycles given by continuous maps $A\times A\to B$, and $B_c(A,B)$ denotes the corresponding coboundaries. Furthermore, we compare $E_{\mathrm{TA}}^{\mathrm{ws}}(A,B)$ with the group of standard extensions $E_A(A,B)$, where $A$ and $B$ denote the underlying Abelian groups, and relate these constructions by means of a six-term exact sequence. Although the Bohr topology of discrete Abelian groups has been investigated by many workers, there still remain many parts that are not well understood. Here, as an application of the methods developed in the paper, new examples of nontrivial $ws$-extensions for discrete Abelian groups equipped with the Bohr topology are provided and some related open questions are also proposed.