A Class of Homeomorphisms on Homogeneous Spaces of a Group Action

TL;DR

This paper introduces a new class of homeomorphisms on compact homogeneous spaces of group actions, revealing insights into their structure and implications for convex spaces in topological vector spaces.

Contribution

It develops a novel class of homeomorphisms that aids in understanding decomposition problems and characterizes convex homogeneous spaces as trivial in certain topological vector spaces.

Findings

Homogeneous spaces with these homeomorphisms are connected to decomposition solutions.

Convex homogeneous spaces in locally convex topological vector spaces are necessarily singletons.

The class relates group actions on the space of continuous functions to the original space.

Abstract

We develop a class of homeomorphisms on a compact homogeneous space of a transitive group action and show how the class sheds new light on a decomposition problem. We further use this class to show that every such homogeneous space in a locally convex topological vector space which is also convex must necessarily be trivial, ie. a singleton set. Additionally, this class of homeomorphisms allows us to relate the induced group action on the space of continuous functions to the action on the homogeneous space.