Anti-Vietoris--Rips metric thickenings and Borsuk graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866915242868473856 |
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| author | Adams, Henry Elchesen, Alex Mallick, Sucharita Moy, Michael |
| author_facet | Adams, Henry Elchesen, Alex Mallick, Sucharita Moy, Michael |
| contents | 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 $α> \frac{2π}{3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_08862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anti-Vietoris--Rips metric thickenings and Borsuk graphs Adams, Henry Elchesen, Alex Mallick, Sucharita Moy, Michael Algebraic Topology Metric Geometry 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 $α> \frac{2π}{3}$. |
| title | Anti-Vietoris--Rips metric thickenings and Borsuk graphs |
| topic | Algebraic Topology Metric Geometry |
| url | https://arxiv.org/abs/2503.08862 |