Minimum Topological Group Topologies

TL;DR

This paper investigates the existence of the minimal Hausdorff group topology on various classes of topological groups, showing that many homeomorphism groups lack such a minimal topology, while some oligomorphic groups do possess it.

Contribution

It characterizes when homeomorphism groups and oligomorphic groups have the minimum Hausdorff group topology, providing new insights into the structure of these groups.

Findings

Homeomorphism groups of certain spaces lack minimum Hausdorff group topology.

Homeomorphism groups of spaces with dense open one-manifolds have minimum group topology.

Some oligomorphic groups possess the minimum group topology.

Abstract

A Hausdorff topological group topology on a group $G$ is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on $G$. For every compact metrizable space $X$ containing an open $n$-cell, $n\ge2$, the homeomorphism group $H(X)$ has no minimum Hausdorff group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space $X$ containing a dense open one-manifold, $H(X)$ has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.