$\mathbb{M}^*$, $\mathbb{N}^*$, and $\mathbb{H}^*$

TL;DR

Under the assumption of the Continuum Hypothesis, the paper proves that every autohomeomorphism of the Čech-Stone remainder of the natural numbers lifts to the product space, and also establishes the existence of an order-reversing autohomeomorphism of the Čech-Stone remainder of the half-line.

Contribution

The paper demonstrates that CH implies all autohomeomorphisms of N* lift to M*, and constructs an order-reversing autohomeomorphism of H* under CH, extending previous results.

Findings

Every autohomeomorphism of N* lifts to M* under CH.

Existence of an order-reversing autohomeomorphism of H* under CH.

Complemented previous results on lifting properties and automorphisms.

Abstract

Let $\mathbb{M} = \mathbb N \times [0,1]$. The natural projection $\pi: \mathbb{M} \rightarrow \mathbb N$, which sends $(n,x)$ to $n$, induces a projection mapping $\pi^*: \mathbb{M}^* \rightarrow \mathbb N^*$, where $\mathbb{M}^*$ and $\mathbb N^*$ denote the \v{C}ech-Stone remainders of $\mathbb{M}$ and $\mathbb N$, respectively. We show that $\mathsf{CH}$ implies every autohomeomorphism of $\mathbb N^*$ lifts through the natural projection to an autohomeomorphism of $\mathbb{M}^*$. That is, for every homeomorphism $h: \mathbb N^* \rightarrow \mathbb N^*$ there is a homeomorphism $H: \mathbb{M}^* \rightarrow \mathbb{M}^*$ such that $\pi^* \circ H = h \circ \pi^*$. This complements a recent result of the second author, who showed that this lifting property is not a consequence of $\mathsf{ZFC}$. Combining this lifting theorem with a recent result of the first author, we also prove that $\mathsf{CH}$ implies there is an order-reversing autohomeomorphism of~$\mathbb H^*$, the \v{C}ech-Stone remainder of the half line $\mathbb H = [0,\infty)$.