Metric compactifications and coarse structures

TL;DR

This paper establishes a categorical equivalence between totally bounded, locally compact metric spaces with $C_0$ coarse structures and compact metrizable spaces via Higson coronas, linking coarse geometry to topology.

Contribution

It proves that coarse equivalence corresponds to homeomorphic Higson coronas and that coarse structures induced by compactifications depend only on the topology of the remainder.

Findings

Coarse equivalence is characterized by homeomorphic Higson coronas.

Higson corona functor provides an equivalence of categories between $ extbf{TB}$ and $ extbf{K}.

Coarse structures from compactifications depend solely on the topology of the remainder.

Abstract

Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the $C_0$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories $\mathbf{TB}\to\mathbf{K}$, where $\mathbf{K}$ is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space $X$ induced by some metrizable compactification $\tilde{X}$ is determined only by the topology of the remainder $\tilde{X}\setminus X$.