Splittings and cross-sections in topological groups

Abstract

This paper deals with the splitting of extensions of topological abelian groups. Given topological abelian groups G and H, we say that Ext(G,H) is trivial if every extension of topological abelian groups of the form 1→H→X→G→1 splits. We prove that Ext(A(Y),K) is trivial for any free abelian topological group A(Y) over a zero-dimensional kω-space Y and every compact abelian group K. Moreover we show that if K is a compact subgroup of a topological abelian group X such that the quotient group X/K is a zero-dimensional kω-space, then there exists a continuous cross section from X/K to X. In the second part of the article we prove that Ext(G,H) is trivial whenever G is a product of locally precompact abelian groups and H has the form Tα×Rβ for arbitrary cardinal numbers α and β. An analogous result is true if G=∏i∈IGi where each Gi is a dense subgroup of a maximally almost periodic, Čech-complete group for which both Ext(Gi,R) and Ext(Gi,T) are trivial.