Anti-Vietoris--Rips metric thickenings and Borsuk graphs

TL;DR

This work introduces and analyzes AVR^m(X; r), the anti-Vietoris--Rips metric thickening, focusing on infinite spaces such as spheres. It establishes that AVR^m(S^n; r) is homotopy equivalent to RP^n for 2π/3 < r ≤ π and is S^n when r > π, with contractibility at r = 0; it also proves a dimension bound on AVR^m(M; r) yielding vanishing cohomology above (n+1)p. A fiber-bundle viewpoint for AVR^m(S^n; π) and a total anti-VR construction are developed to facilitate topological obstructions, including a no-go theorem for graph homomorphisms Bor(S^k; r) → Bor(S^n; α) when k > n and α > 2π/3. The results link topology of AVR^m spaces to chromatic properties of Borsuk graphs and outline broad open questions about Morse-type thresholds and chromatic behavior across scales.

Abstract

For $X$ a metric space and $r\ge 0$, the anti-Vietoris-Rips metric thickening $\mathrm{AVR^m}(X;r)$ 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 $\mathrm{AVR^m}(S^n;r) \cong S^n$ for $r > π$, a homotopy equivalence $\mathrm{AVR^m}(S^n;r) \simeq \mathbb{RP}^{n}$ for $\frac{2π}{3} < r \le π$, and contractibility $\mathrm{AVR^m}(S^n;r) \simeq *$ for $r=0$. For an $n$-dimensional compact Riemannian manifold $M$, we show that the covering dimension of $\mathrm{AVR^m}(M;r)$ 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 $\mathrm{AVR^m}(M;r)$ vanishes in all dimensions $k\geq (n+1)p$. We prove more about the topology of $\mathrm{AVR^m}(S^n;\frac{2π}{3})$, 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 $\mathrm{Bor}(S^k;r) \to \mathrm{Bor}(S^n;α)$ exists when $α> \frac{2π}{3}$.

Figures (7)

• Figure 1: $\mathrm{AVR}^m(S^1;\pi)$ (left) is homeomorphic to the Möbius band (right).
• Figure 2: (Left) The map $\rho\colon \mathrm{AVR}^m(S^n;r) \to \mathrm{AVR}^m(S^n;\pi)$ in the proof of Theorem \ref{['thm:avrmSn-homotopy-type']}, with $n=1$. There are the two cases for how $\rho$ is defined. In one case, $\mu = \lambda_0\delta_{x_0}+(1-\lambda_0)\delta_{x'_0}$ and $\rho(\mu)=\lambda_1\delta_{x_1}+(1-\lambda_1)\delta_{-x_1}$. In the other case, $\rho(\mu')=\delta_{v}$. (Right) A depiction of the flashlight homotopy $H$ between $id_{\mathrm{AVR}^m(S^n;r)}$ and $\rho$. Here $H(\mu,t)=\lambda_t \delta_{x_{t}}+(1-\lambda_t)\delta_{x'_{t}}$.
• Figure 3: For $M$ a compact $n$-dimensional manifold with $p=\mathrm{pack}_M(r)$, the covering dimension of $\mathrm{AVR}^m(M;r)$ is at most $(n+1)p-1=(p-1)+np$. Above is a manifold $M$ of dimension $n=2$ with $r>0$ such that $p=\mathrm{pack}_M(r)=5$. In blue is a single $(p-1)$-dimensional simplex in $\mathrm{AVR}^m(M;r)$. The gray open neighborhoods show that each of the $p$ vertices of this simplex have $n$ degrees of freedom. Hence we expect $\mathrm{AVR}^m(M;r)$ to have dimension $(p-1)+np=4+2\cdot 5=14$.
• Figure 4: A pictorial representation of an element $v$ of $Z \times I$ and its image under the map $\varphi \colon Z \times I \to \mathrm{AVR}^m(S^n;\tfrac{2 \pi}{3})$. Note that $v=(V,y,x,r)$ and $v'=(V,y,x,0)$ live in the space $Z\times I$ and $\varphi(v)$ lives in a different space $\mathrm{AVR}^m(S^n;\frac{2\pi}{3})$.
• Figure 5: Plot of $s_n$ and $r_n$ vs. $n$ for $n=1$ to $100$, where $r_n$ is the diameter of the vertex set of an inscribed regular $(n+1)$-simplex in $S^n$, and where $s_n$ is the diameter of a radially projected $n$-face.

Theorems & Definitions (50)

• Lemma 3.1
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• proof
• Definition 4.4
• Lemma 4.5
• proof
• Corollary 4.6
• proof