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

TL;DR

The paper characterizes local countable compactness of topological groups via their continuous actions on universal bundles and joins, extending classical equivalences with exponentiability.

Contribution

It establishes new characterizations of local countable compactness through actions on joins, cones, and topological properties, generalizing known equivalences with exponentiability.

Findings

Topological groups act continuously on their iterated joins and unions.

Characterizations of local countable compactness via actions on joins and cones.

Extension of the equivalence between local compactness and exponentiability.

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.