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Homotopy connectivity of Čech complexes of spheres

Henry Adams, Ekansh Jauhari, Sucharita Mallick

TL;DR

This work analyzes the intrinsic Čech complexes $\\check{C}(S^n;r)$ of the $n$-sphere with geodesic metric, deriving both lower and upper bounds on their homotopy connectivity across scales $r\in(0,\pi)$. The lower bounds come from Barmak’s conicity criterion applied to finite dense subsets, while the upper bounds are obtained via Lovász’s bound relating neighborhood complexes to chromatic numbers of Borsuk graphs. Together these bounds show that the homotopy type of $\\check{C}(S^n;r)$ changes infinitely often as $r$ traverses $(0,\pi)$, and the authors conjecture only countably many changes. The paper also links homological dimension of Čech complexes of finite subsets to packings and explores the circle case in detail as a benchmark, highlighting both the strengths and gaps of current methods and proposing several open directions for extending the theory to broader manifolds and scales.

Abstract

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.

Homotopy connectivity of Čech complexes of spheres

TL;DR

This work analyzes the intrinsic Čech complexes of the -sphere with geodesic metric, deriving both lower and upper bounds on their homotopy connectivity across scales . The lower bounds come from Barmak’s conicity criterion applied to finite dense subsets, while the upper bounds are obtained via Lovász’s bound relating neighborhood complexes to chromatic numbers of Borsuk graphs. Together these bounds show that the homotopy type of changes infinitely often as traverses , and the authors conjecture only countably many changes. The paper also links homological dimension of Čech complexes of finite subsets to packings and explores the circle case in detail as a benchmark, highlighting both the strengths and gaps of current methods and proposing several open directions for extending the theory to broader manifolds and scales.

Abstract

Let be the -sphere with the geodesic metric and of diameter . The intrinsic Čech complex of at scale is the nerve of all open balls of radius in . In this paper, we show how to control the homotopy connectivity of Čech complexes of spheres at each scale between and in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case , 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 . Our bounds imply the new result that for , the homotopy type of the Čech complex of at scale changes infinitely many times as varies over ; we conjecture only countably many times. Additionally, we lower bound the homological dimension of Čech complexes of finite subsets of in terms of their packings.

Paper Structure

This paper contains 14 sections, 11 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Metric spaces
  5. Topological spaces
  6. Simplicial complexes
  7. Graphs
  8. A lower bound on connectivity
  9. An upper bound on connectivity
  10. The case of the circle
  11. Comparing the lower bound
  12. Comparing the upper bound
  13. Homological dimension
  14. Conclusion

Key Result

Theorem 1.1

For $n\ge 1$ and $\delta\in(0,\pi)$, if $\mathop{\mathrm{conn}}\nolimits(\mathrm{\check{C}}(S^n;\pi-\delta))=k-1$, then

Figures (3)

  • Figure 1: Intervals where $\mathrm{\check{C}}(S^2;r)$ may have connectivity $k-1$, given by Theorem \ref{['thm:main']}. The blue endpoints are plotted using only approximate values of $\mathop{\mathrm{cov}}\nolimits_{S^2}(2k+2)$ or of $2\mathop{\mathrm{cov}}\nolimits_{S^2}(k+1)$; see tarnai1991covering.
  • Figure 2: The homotopy types of $\mathrm{\check{C}}(S^1;r)$ as $r$ varies AA-VRS1 are indicated by black bars. Theorem \ref{['thm:main']} gives intervals where $\mathrm{\check{C}}(S^1;r)$ may have connectivity $k-1$, which are indicated by colored bars. Left endpoints of orange bars (when $k-1$ is even) are tight.
  • Figure 3: We have $\chi(\mathrm{Bor}(S^2;\delta))=4$ for all $s_2<\delta<\pi$, as illustrated by the covering of $S^2$ by four sets each of diameter $s_2$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • proof
  • Remark 3.1
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Corollary 4.3
  • proof
  • ...and 16 more