A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences

Abstract

We show that if $\kappa \leq \omega$ and there exists a group topology without non-trivial convergent sequences on an Abelian group $H$ such that $H^n$ is countably compact for each $n<\kappa$ then there exists a topological group $G$ such that $G^n$ is countably compact for each $n <\kappa$ and $G^{\kappa}$ is not countably compact. If in addition $H$ is torsion, then the result above holds for $\kappa=\omega_1$. Combining with other results in the literature, we show that: $a)$ Assuming ${\mathfrak c}$ incomparable selective ultrafilters, for each $n \in \omega$, there exists a group topology on the free Abelian group $G$ such that $G^n$ is countably compact and $G^{n+1}$ is not countably compact. (It was already know for $\omega$). $b)$ If $\kappa \in \omega \cup \{\omega\} \cup \{\omega_1\}$, there exists in ZFC a topological group $G$ such that $G^\gamma$ is countably compact for each cardinal $\gamma <\kappa$ and $G^\kappa$ is not countably compact.