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

Abstract

We show that if $κ\leq ω$ 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 $n<κ$ then there exists a topological group $G$ such that $G^n$ is countably compact for each $n <κ$ and $G^κ$ is not countably compact. If in addition $H$ is torsion, then the result above holds for $κ=ω_1$. Combining with other results in the literature, we show that: $a)$ Assuming ${\mathfrak c}$ incomparable selective ultrafilters, for each $n \in ω$, 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 $ω$). $b)$ If $κ\in ω\cup \{ω\} \cup \{ω_1\}$, there exists in ZFC a topological group $G$ such that $G^γ$ is countably compact for each cardinal $γ<κ$ and $G^κ$ is not countably compact.