Resumen:
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¿-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¿-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¿×Rß for arbitrary cardinal numbers ¿ and ß. An analogous result is true if G=¿i¿IGi where each Gi is a dense subgroup of a maximally almost periodic, ¿ech-complete group for which both Ext(Gi,R) and Ext(Gi,T) are trivial.