Discontinuous actions on cones, joins, and $n$-universal bundles

Abstract

We prove that locally countably-compact Hausdorff topological groups $\mathbb{G}$ act continuously on their iterated joins $E_n\mathbb{G}:=\mathbb{G}^{*(n+1)}$ (the total spaces of the Milnor-model $n$-universal $\mathbb{G}$-bundles) as well as the colimit-topologized unions $E\mathbb{G}=\varinjlim_n E_n\mathbb{G}$, and the converse holds under the assumption that $\mathbb{G}$ is first-countable. In the latter case other mutually equivalent conditions provide characterizations of local countable compactness: the fact that $\mathbb{G}$ acts continuously on its first self-join $E_1\mathbb{G}$, or on its cone $\mathcal{C}\mathbb{G}$, or the coincidence of the product and quotient topologies on $\mathbb{G}\times \mathcal{C}X$ for all spaces $X$ or, equivalently, for the discrete countably-infinite $X:=\aleph_0$. These can all be regarded as weakened versions of $\mathbb{G}$'s exponentiability, all to the effect that $\mathbb{G}\times -$ preserves certain colimit shapes in the category of topological spaces; the results thus extend the equivalence (under the separation assumption) between local compactness and exponentiability.