Fibrations and coset spaces for locally compact groups

TL;DR

The paper addresses when the natural quotient map $q:G/K\to G/L$ between coset spaces is a fibration. It proves that if $L$ is a locally compact pro-Lie group, then $q$ is a Serre fibration, and if $G$ is paracompact, a Hurewicz fibration, generalizing and unifying earlier results by Skljarenko, Madison and Mostert. The approach combines Palais' Slice Theorem with Antonyan's results, and employs a Zorn-lemma lifting argument over intermediate subgroups to force a reduction to $K$. This work clarifies the homotopy-theoretic structure of homogeneous spaces for a broad class of locally compact groups and provides a solid foundation for further inquiries into fibrations of coset spaces.

Abstract

Let $G$ be a topological group and let $K,L\subseteq G$ be closed subgroups, with $K\subseteq L$. We prove that if $L$ is a locally compact pro-Lie group, then the map $q:G/K\to G/L$ is a fibration. As an application of this, we obtain two older results by Skljarenko, Madison and Mostert.

Theorems & Definitions (20)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Definition 3.4
• Lemma 3.5
• proof
• Definition 3.6