Homeomorphisms between finite powers of topological spaces (Q1082635)

The main result of the paper is the following. For every positive integer r and every infinite cardinal number \(\kappa\) there is a compact connected homogeneous topological space X (in fact a topological group) of weight \(\kappa\) such that \(X^ n\approx X^ m\) iff \(n\equiv m\) (mod r). This answers a question of \textit{V. Trnková} [Proc. 5th Prague Topol. Symp. 1981, 631-641 (1983; Zbl 0501.54026)]. The idea of the proof is to take an Abelian group G of size \(\kappa\) with the desired property, endow it with the discrete topology and let X be the character group of G.