Anti-Vietoris--Rips metric thickenings and Borsuk graphs

Abstract

For $X$ a metric space and $r\ge 0$, the anti-Vietoris-Rips metric thickening $\mathrm{AV}{\mathrm{R}}^{\mathrm{m}}(X;r)$ is the space of all finitely supported probability measures on $X$ whose support has spread at least $r$, equipped with an optimal transport topology. We study the anti-Vietoris-Rips metric thickenings of spheres. We have a homeomorphism $\mathrm{AV}{\mathrm{R}}^{\mathrm{m}}(S^n;r)\cong S^n$ for $r>\pi$, a homotopy equivalence $\mathrm{AV}{\mathrm{R}}^{\mathrm{m}}(S^n;r)\simeq {\mathbb{RP}}^n$ for $\frac{2\pi }{3}<r\le \pi$, and contractibility $\mathrm{AV}{\mathrm{R}}^{\mathrm{m}}(S^n;r)\simeq \ast$ for $r=0$. For an $n$-dimensional compact Riemannian manifold $M$, we show that the covering dimension of $\mathrm{AV}{\mathrm{R}}^{\mathrm{m}}(M;r)$ is at most $(n+1)p-1$, where $p$ is the packing number of $M$ at scale $r$. Hence the $k$-dimensional Čech cohomology of $\mathrm{AV}{\mathrm{R}}^{\mathrm{m}}(M;r)$ vanishes in all dimensions $k\ge (n+1)p$. We prove more about the topology of $\mathrm{AV}{\mathrm{R}}^{\mathrm{m}}(S^n;\frac{2\pi }{3})$, which has vanishing cohomology in dimensions $2n+2$ and higher. We explore connections to chromatic numbers of Borsuk graphs, and in particular we prove that for $k>n$, no graph homomorphism $\mathrm{Bor}(S^k;r)\to \mathrm{Bor}(S^n;\alpha )$ exists when $\alpha >\frac{2\pi }{3}$.