Rigidity of Higson coronas

TL;DR

This paper demonstrates that under certain set-theoretic assumptions, the topological structure of Higson coronas uniquely determines the coarse equivalence class of the underlying metric spaces.

Contribution

It establishes a rigidity result linking Higson corona homeomorphisms to coarse equivalences under set-theoretic hypotheses like OCA and MA_{ℵ_1}.

Findings

Higson coronas are rigid under OCA and MA_{ℵ_1}.

Homeomorphic Higson coronas imply coarse equivalence of spaces.

Rigidity does not follow from ZFC alone.

Abstract

We show that under mild set theoretic hypotheses we have rigidity for algebras of continuous functions over Higson coronas, topological spaces arising in coarse geometry. In particular, we show that under $\mathsf{OCA}$ and $\mathsf {MA}_{\aleph_1}$, if two uniformly locally finite metric spaces $X$ and $Y$ have homeomorphic Higson coronas $\nu X$ and $\nu Y$, then $X$ and $Y$ are coarsely equivalent, a statement which provably does not follow from $\mathsf{ZFC}$ alone.