Classification of the topological holonomy groups in $SO(3)$

TL;DR

This paper classifies all topological holonomy groups in SO(3), detailing finite, infinite, and dense subgroups, thus providing a comprehensive understanding of their structure and types.

Contribution

It provides a complete classification of topological holonomy groups in SO(3), including finite, abelian, and non-abelian infinite groups, expanding previous partial results.

Findings

Finite groups classified by Klein

Infinite abelian groups generated by one or two elements

Non-abelian infinite groups with orthogonal order-2 generators

Abstract

In this paper, we obtain classification of the topological holonomy groups in $SO(3)$. Such a group is given by one of the following: a finite group (such groups are classified by Klein); a commutative infinite group which is generated by one or two elements, and dense in a subgroup of $SO(3)$ isomorphic to $SO(2)$; a non-commutative infinite group generated by two elements of order $2$, $\infty$ such that these rotation axes are orthogonal; a non-commutative infinite group which is dense in $SO(3)$.