The existence of initially $\omega_1$-compact group topologies on free Abelian groups is independent of ZFC

summary:It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size ${\frak C}$ admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its $\omega$-th power countably compact. In particular, a free Abelian group does not admit a Hausdorff $p$-compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff initially $\omega_1$-compact group topology. We also show that the existence of such a group topology is independent of ${\frak C} = \aleph_2$