Sucharita Mallick | Searcharxiv

arXiv2026

Homotopy connectivity of Čech complexes of spheres

Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $π$. The intrinsic Čech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of Čech complexes of spheres at each scale between $0$ and $π$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Čech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the Čech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,π)$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of Čech complexes of finite subsets of $S^n$ in terms of their packings.

math.AT↗

arXiv2025

Anti-Vietoris--Rips metric thickenings and Borsuk graphs

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 $α> \frac{2π}{3}$.

math.AT↗