On the number of countably compact group topologies on a free Abelian group

We show under MA(σ-centered) the existence of at least (2ω)+ non-homeomorphic topological group topologies on the free Abelian group of size 2ω which make it countably compact and separable. In particular, under GCH the maximum possible number of such topologies is attained. As a corollary, we show the existence of a semigroup which possesses (2ω)+ non-homeomorphic semigroup topologies which make it a counterexample for Wallaceʹs Problem.