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

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 $(\bz/2\bz)^n\ltimes S_n$).

💡 Research Summary

The paper investigates the Stone‑Čech compactifications of finite Cartesian powers of the long line ℒ and of the semi‑closed half‑long line ℒ₊, and applies these compactifications to the structure of their homeomorphism groups. The long line ℒ is defined as the ordered topological space ω₁ ×