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

TL;DR

This paper extends the understanding of countably compact powers of Abelian groups, showing under certain set-theoretic assumptions that groups can have all smaller powers countably compact but not the larger one, including in ZFC.

Contribution

It proves the existence of topological groups with specific countable compactness properties for their powers, generalizing previous results and providing new constructions under set-theoretic assumptions.

Findings

Existence of groups with countably compact powers up to a certain size but not beyond.

Construction of such groups in ZFC for various cardinalities.

Extension of van Douwen-like theorems to small powers of countably compact groups.

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.