Splittings and cross-sections in topological groups

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)Ext(G,H) is trivial if every extension of topological abelian groups of the form 1→H→X→G→11→H→X→G→1 splits. We prove that Ext(A(Y),K)Ext(A(Y),K) is trivial for any free abelian topological group A(Y)A(Y) over a zero-dimensional kωkω-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/KX/K is a zero-dimensional kωkω-space, then there exists a continuous cross section from X/KX/K to X  . In the second part of the article we prove that Ext(G,H)Ext(G,H) is trivial whenever G is a product of locally precompact abelian groups and H   has the form Tα×RβTα×Rβ for arbitrary cardinal numbers α and β  . An analogous result is true if G=∏i∈IGiG=∏i∈IGi where each GiGi is a dense subgroup of a maximally almost periodic, Čech-complete group for which both Ext(Gi,R)Ext(Gi,R) and Ext(Gi,T)Ext(Gi,T) are trivial.