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Rigidity of Higson coronas

Alessandro Vignati

TL;DR

The paper addresses rigidity of Higson coronas in coarse geometry by translating the question into the study of unital $^*$-homomorphisms between Higson corona algebras $C_ν(X)\to C_ν(Y)$. It introduces the coarse weak Extension Principle $cwEP$ to control when such maps must be trivial, and proves that $cwEP$ holds under Todorčević's Open Colouring Axiom $\mathsf{OCA}$ and Martin's Axiom at $\aleph_1$ $ (\mathsf{MA}_{\aleph_1})$, via lifting theorems. A local-to-global strategy shows that locally trivial homomorphisms are globally trivial, enabling a rigidity conclusion: if Higson coronas $νX$ and $νY$ are homeomorphic, then $X$ and $Y$ are coarsely equivalent under $\mathsf{OCA}$ and $\mathsf{MA}_{\aleph_1}$. The work also clarifies the structure of embeddings between Higson coronas and provides a framework for understanding liftings in quotient structures of Roe-type algebras, contrasting results under CH. Overall, it demonstrates that certain set-theoretic axioms enforce rigidity that is not provable in 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 $νX$ and $νY$, then $X$ and $Y$ are coarsely equivalent, a statement which provably does not follow from $\mathsf{ZFC}$ alone.

Rigidity of Higson coronas

TL;DR

The paper addresses rigidity of Higson coronas in coarse geometry by translating the question into the study of unital -homomorphisms between Higson corona algebras . It introduces the coarse weak Extension Principle to control when such maps must be trivial, and proves that holds under Todorčević's Open Colouring Axiom and Martin's Axiom at , via lifting theorems. A local-to-global strategy shows that locally trivial homomorphisms are globally trivial, enabling a rigidity conclusion: if Higson coronas and are homeomorphic, then and are coarsely equivalent under and . The work also clarifies the structure of embeddings between Higson coronas and provides a framework for understanding liftings in quotient structures of Roe-type algebras, contrasting results under CH. Overall, it demonstrates that certain set-theoretic axioms enforce rigidity that is not provable in 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 and , if two uniformly locally finite metric spaces and have homeomorphic Higson coronas and , then and are coarsely equivalent, a statement which provably does not follow from alone.

Paper Structure

This paper contains 6 sections, 18 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Set theoretic preliminaries
  3. Trivial $^*$-homomorphisms and the coarse weak Extension Principle
  4. Local triviality
  5. From local triviality to triviality
  6. Nice liftings: Local existence

Key Result

Proposition 1.2

Let $X$ and $Y$ be u.l.f. spaces. A coarse proper map $\varphi\colon Y\to X$ gives a continuous map $\nu\varphi\colon\nu Y\to\nu X$. If $\varphi,\psi\colon Y\to X$ are close, then $\nu\varphi=\nu\psi$. Moreover, if $\varphi$ is a coarse embedding, $\nu\varphi$ is injective and if $\varphi$ is a coar

Theorems & Definitions (61)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 51 more