The Wallace problem and countably compact torsion-free Abelian groups in ZFC

Abstract

We prove in ZFC that every torsion-free Abelian group of cardinality $\mathfrak{c}$ admits a Hausdorff countably compact group topology without nontrivial convergent sequences. In particular, this applies to the free Abelian group ${\mathbb{Z}}^{(\mathfrak{c})}$, the Baer-Specker group ${\mathbb{Z}}^{\omega }$ and ${\mathbb{Q}}^{(\mathfrak{c})}$. For the topology constructed on ${\mathbb{Z}}^{(\mathfrak{c})}$, the coordinatewise nonnegative cone is countably compact in the subspace topology. Consequently, there exists in ZFC a commutative Tychonoff countably compact topological semigroup which has two-sided cancellation but is not a group, giving a negative answer to Wallace's question. Combined with earlier results, the main theorem also yields in ZFC a Tychonoff countably compact topological semigroup containing a copy of the bicyclic semigroup and a functionally Hausdorff countably compact paratopological group that is not a topological group.

For a Lean 4 formalization of these results, see https://github.com/vo-rodrigues/wallace-problem-zfc-paper