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 k ω -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 k ω -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 T α × R β for arbitrary cardinal numbers α and β . An analogous result is true if G = ∏ i ∈ I G i where each G i is a dense subgroup of a maximally almost periodic, Cech-complete group for which both Ext ( G i , R ) and Ext ( G i , T ) are trivial.