Weak-split extensions of topological Abelian groups

In the category of topological Abelian groups, we consider the usual notion of extension $E=(B\to X\to A)$ of $B$ by $A$ and the notion of weak-split extension (when $X\to A$ has a continuous split $A\to X$). Given a weak-split extension $E$, the topological Abelian group $X$ is homeomorphic to $B\times A$, but in general, $X$ need not be algebraic isomorphic to $B\times A$. In this paper, for two topological Abelian groups $A,B$, we study the Abelian group $E_{\mathbf{T}\mathbf{A}}^{ws}(A,B)$ of weak-split extensions of $B$ by $A$ modulo extension isomorphisms. We prove that $E_{\mathbf{T}\mathbf{A}}^{ws}(A,B)$ can be described as the Abelian group of all possible continuous sums given in the product topological space $B\times A$ (modulo topological isomorphism) having $B$ as a topological subgroup and $A$ as a topological quotient. We also give an alternative description of $E_{\mathbf{T}\mathbf{A}}^{ws}(A,B)$ as a quotient ${\mathcal{Z}}_c(A,B)/{\mathcal{B}}_c(A,B)$, where ${\mathcal{Z}}_c(A,B)$ are cocycles represented by certain continuous maps of the form $A\times A\to B$, and similarly for the coboundaries ${\mathcal{B}}_c(A,B)$. For two topological Abelian groups $A,B$, we compare the Abelian group of $ws$-extensions $E_{\mathbf{T}\mathbf{A}}^{ws}(A,B)$ with the Abelian group of standard extensions $E_{\mathbf{A}}(A,B)$ where now $A,B$ also denote the subjacent Abelian groups. We relate these different types of extensions using an exact sequence with six terms. 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.