Stone-\v{C}ech compactifications and homeomorphisms of products of the long line

TL;DR

This paper characterizes the Stone-ech compactification of products of the long line and semi-closed half-long line, and explores the structure of their homeomorphism groups, revealing connections to symmetric groups.

Contribution

It provides explicit descriptions of the Stone-ech compactifications for these long line products and analyzes the torsion subgroups of their homeomorphism groups.

Findings

Stone-ech compactification of ech^n is ech^n

Homeomorphism groups have torsion subgroups isomorphic to symmetric groups

Descriptions extend to semi-closed half-long line products

Abstract

In this note we shall prove that the Stone-\v{C}ech compactification of $\mathcal{L}^n$ is the space $\bar{\mathcal{L}}^n$ where $\bar{\mathcal{L}}$ is the extended long line, namely, $\mathcal{L}$ together with its ends $\pm \Omega$. We give a similar description for the Stone-\v{C}ech compactification of the cartesian power of the semi-closed half-long line $\mathcal{L}_+$. As an application we show that any torsion subgroup of the group of all homeomorphisms of $\cj^n$ (resp. $\cl^n$) is isomorphic to a subgroup of the symmetric group $S_n$ (resp. the semidirect product $(\bz/2\bz)^n\ltimes S_n$).