W. W. Comfort - Continuous cross sections on abelian groups equipped with the Bohr topology

All groups here are Abelian. 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 such that (with the natural homomorphism).

The symbol denotes G with its Bohr topology, i.e., the topology induced by .

A topological group H is an absolute ccs-group [resp., an absolute retract ] if His a ccs-subgroup [resp., is a retract] in every group of the form containing H as a (necessarily closed) subgroup. One then writes [resp., ].

Theorem 1. Every ccs-subgroup H of a group of the form is a retract of (and is homeomorphic to ); hence .

Theorem 2.  [resp., ] iff is a ccs-subgroup of its divisible hull [resp., is a retract of ].

Theorem 3. (a)  Every cyclic group is in .
(b)  The classes and are closed under finite products.

Theorem 4. Not every Abelian group is in .

Question [van Douwen, 1990]. Is every Abelian group in ?

^* Presented in Kingston by this co-author.