Compact Lie groups isolated up to conjugacy

TL;DR

This paper characterizes compact subgroups of Lie groups that are isolated up to conjugacy, based solely on their intrinsic structure, and explores the properties of induced maps between subgroup spaces.

Contribution

It provides a characterization of isolated conjugacy classes of compact subgroups in Lie groups independent of embedding details.

Findings

Characterization depends only on the subgroup's intrinsic structure

Continuous homomorphisms induce open maps between subgroup spaces

Identification of isolated subgroups up to conjugacy in Lie groups

Abstract

The set $\mathcal S(G)$ of compact subgroups of a Hausdorff topological group $G$ can be equipped with the Vietoris topology. A compact subgroup $K\in\mathcal S(G)$ is isolated up to conjugacy if there is a neighborhood $\mathcal U\subseteq\mathcal S(G)$ of $K$ such that every $L\in\mathcal U$ is conjugate to $K$. In this paper, we characterize compact subgroups of a Lie group that are isolated up to conjugacy. Our characterization depends only on the intrinsic structure of $K$, the ambient Lie group $G$ and the embedding of $K$ into $G$ are irrelevant. In addition, we prove that any continuous homomorphism from a compact group $G$ onto a compact Lie group $H$ induces a continuous open map from $\mathcal S(G)$ onto $\mathcal S(H)$.