Cross sections and homeomorphism classes of Abelian groups equipped with the Bohr topology

A closed subgroup H of a topological group G is a ccs-subgroup if there is a continuous cross section from G/H to G—that is, a continuous function Γ such that π∘Γ=id|G/H (with π :G→G/H the natural homomorphism). mbol G# denotes an Abelian group G with its Bohr topology, i.e., the topology induced by Hom(G,T). logical group H is an absolute ccs-group(#) (respectively, an absolute retract(#)) if H is a ccs-subgroup (respectively, is a retract) in every group of the form G# containing H as a (necessarily closed) subgroup. One then writes H∈ACCS(#) (respectively, H∈AR(#)). m 1. Every ccs-subgroup H of a group of the form G# is a retract of G# (and G# is homeomorphic to (G/H)#×H#); hence ACCS(#)⊆AR(#). m 2. H#∈ACCS(#) (respectively, H#∈AR(#)) if H# is a ccs-subgroup of its divisible hull (div(H))# (respectively, H# is a retract of (div(H))# ). m 3. (a) cyclic group is in ACCS(#). asses ACCS(#) and AR(#) are closed under finite products. ery Abelian group is in ACCS(#). llowing corollary to Theorem 3 answers a question of Kunen: ary. The spaces (div(Z))# and ((div(Z)/Z)×Z)# are homeomorphic.