Stone-Čech compactifications and homeomorphisms of products of the long line

Abstract

In this note we shall prove that the Stone-Čech 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-Čech 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 $(\bz/2\bz)^n\ltimes S_n$).

8 pages 2 figures