On the orbit space of a compact linear Lie group with commutative connected component

TL;DR

This paper investigates conditions under which the topological quotient of a compact linear Lie group with a torus connected component forms a manifold, extending previous work on finite linear groups to more general cases.

Contribution

It characterizes when the quotient of such Lie groups is a topological or smooth manifold, focusing on cases where the connected component is a positive-dimensional torus.

Findings

Identifies conditions for the quotient to be a manifold

Extends previous finite group results to connected Lie groups

Provides criteria for smooth and topological manifold structures

Abstract

This paper is devoted to the study of topological quotients of compact linear Lie groups. More precisely, it investigates the question of when such a quotient is a topological or a smooth manifold. The topological quotient of a finite linear group was studied by Mikhailova in 1984. Here the connected component of the original Lie group G is assumed to be a torus of positive dimension.