Gromov-Hausdorff Distances Between Spheres: Asymptotics and Vietoris-Rips Coindex Bounds

Abstract

For $1\le m<n$, let $c_{m,n}$ be the infimum of the scales $r$ at which the Vietoris-Rips simplicial complex of the unit round sphere ${\mathbb{S}}^m$, equipped with its geodesic metric, admits a continuous odd map from ${\mathbb{S}}^n$. Previous work established the lower bound $\frac{1}{2}{c}_{m,n}\le d_{\mathrm{GH}}({\mathbb{S}}^m,{\mathbb{S}}^n)$ through a quantitative Borsuk-Ulam theorem and raised the question of whether it is always sharp. We show that this bound can underestimate the distance by a factor arbitrarily close to two, the largest possible factor. Combining projective-code upper bounds for $c_{m,n}$ with lower bounds for $d_{\mathrm{GH}}$ obtained from obstructions to homomorphisms between Borsuk graphs, we obtain explicit finite-dimensional bounds. Along sequences of pairs $1\le m<n$, we have $c_{m,n}\to \frac{\pi }{2}$ whenever $\log n/m\to 0$, while $d_{\mathrm{GH}}({\mathbb{S}}^m,{\mathbb{S}}^n)\to \frac{\pi }{2}$ whenever $m/n\to 0$. When both conditions hold (for example when $n=m^2$) the ratio of the distance to $\frac{1}{2}{c}_{m,n}$ tends to two. For each fixed $m\ge 3$, the inequality is strict for all sufficiently large $n$.