Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
Henry Adams, Johnathan Bush, Clause, Nate, Florian Frick, Gómez, Mario, Harrison, Michael, R. Amzi Jeffs, Evgeniya Lagoda, Sunhyuk Lim, Facundo Mémoli, Michael Moy, Nikola Sadovek, Matt Superdock, Daniel Vargas, Qingsong Wang, Ling Hui Zhou · arXiv (Cornell University) · 2022
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{é}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.