Transitive actions of locally compact groups on locally contractible spaces

TL;DR

This paper proves that certain quotient spaces of locally compact groups are manifolds and that the acting group is a Lie group if the action is faithful, under conditions of local contractibility and connectedness.

Contribution

It establishes that locally contractible, connected quotients of locally compact groups are manifolds and characterizes the acting group as a Lie group when the action is faithful.

Findings

Quotients of locally compact groups by closed subgroups are manifolds if locally contractible and connected.

Faithful actions imply the group is a Lie group.

Provides conditions under which group actions produce manifold structures.

Abstract

Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.