Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

TL;DR

This paper establishes new bounds on Gromov-Hausdorff distances between metric spaces, especially spheres of different dimensions, by linking topological obstructions via Vietoris-Rips complexes and Borsuk-Ulam theorems.

Contribution

It introduces a novel approach connecting Gromov-Hausdorff distances with equivariant topology and Vietoris-Rips complexes, extending bounds on distances between spheres of various dimensions.

Findings

Bounded how discontinuous odd maps between spheres must be.

Improved lower bounds on Gromov-Hausdorff distances between spheres of different dimensions.

Provided new upper bounds for Gromov-Hausdorff distances between adjacent spheres.

Abstract

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be. Topology has few tools to obstruct the existence of discontinuous functions. However, an arbitrary function $f\colon X\to Y$ induces a continuous map between their Vietoris--Rips simplicial complexes, where the allowable choices of scale parameters depend on how much the function $f$ distorts distances. We can then use equivariant topology to obstruct the existence of certain continuous maps between Vietoris--Rips complexes. With these ideas we bound how discontinuous an odd map between spheres $S^k\to S^n$ with $k>n$ must be, generalizing a result by Dubins and Schwarz (1981), which is the case $k=n+1$. As an application, we recover or improve upon all of the lower bounds from Lim, M{\'e}moli, and Smith (2022) on the Gromov--Hausdorff distances between spheres of different dimensions. We also provide new upper bounds on the Gromov--Hausdorff distance between spheres of adjacent dimensions.