Monotonicity of connectivity in spherical Čech filtrations
For an integer $n \ge 2$ equip the unit sphere $S^n \subset \mathbb{R}^{n+1}$ with the geodesic metric $d(x, y) = \arccos \langle x, y \rangle$, and for $r > 0$ let $$C_n(r) = \Bigl\{\sigma \subset S^n : \sigma \text{ finite and } \bigcap_{x \in \sigma} \{z \in S^n : d(x, z) < r\} \ne \varnothing\Bigr\}$$ be the intrinsic open Čech complex, the nerve of the open geodesic balls of radius $r$, intersected inside the sphere. Write $|C_n(r)|$ for its geometric realization with the CW topology. For a path-connected space $Y$ let $$\operatorname{conn}(Y) = \sup\{k \in \mathbb{Z}_{\ge 0} : \pi_i(Y) = 0 \text{ for all } 1 \le i \le k\},$$ allowing the value $+\infty$; the spaces $|C_n(r)|$ are path-connected.
For $0 < r \le \pi/2$ the balls form a good cover, so $|C_n(r)| \simeq S^n$ by the nerve lemma, and for $n \ge 2$ the complexes are simply connected at every scale (for $\pi/2 < r < \pi$ by a theorem of Virk). Adams, Jauhari and Mallick proved that if $0 < \delta < \pi$ and $\operatorname{conn}(|C_n(\pi - \delta)|) = k - 1$, then $$\operatorname{cov}_{S^n}(2k + 2) \le \delta \le 2 \operatorname{cov}_{S^n}(k + 1),$$ where $\operatorname{cov}_{S^n}(m)$ is the least radius of $m$ closed balls covering $S^n$; hence the homotopy type of $|C_n(r)|$ changes infinitely many times as $r$ varies over $(0, \pi)$. For the circle, Adamaszek and Adams showed that $|C_1(r)| \simeq S^{2k+1}$ for $\frac{\pi k}{k+1} < r \le \frac{\pi(k+1)}{k+2}$, $k \ge 0$, and that it is contractible for $r \ge \pi$, so that its connectivity never decreases.
Problem. Is $$\operatorname{conn}(|C_n(r)|) \le \operatorname{conn}(|C_n(s)|)$$ for every $n \ge 2$ and all $0 < r < s < \pi$?
Source: Adams, Jauhari and Mallick (2026), Conjecture 8.1, for every n ≥ 1 (arXiv 2025). Stated for n ≥ 2 by AIM — Open Applied Problems; the circle case follows from Adamaszek and Adams (2017).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, Čech complexes, homotopy connectivity, spheres
References
@article{adams2026cech,
author = {Adams, Henry and Jauhari, Ekansh and Mallick, Sucharita},
title = {Homotopy connectivity of {\v{C}}ech complexes of spheres},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {3},
year = {2026},
pages = {1742--1762},
doi = {10.1007/s00454-026-00847-5},
eprint = {2502.00122},
archivePrefix = {arXiv},
}
@article{adamaszek2017circle,
author = {Adamaszek, Micha{\l} and Adams, Henry},
title = {The {V}ietoris--{R}ips complexes of a circle},
journal = {Pacific J. Math.},
volume = {290},
number = {1},
year = {2017},
pages = {1--40},
doi = {10.2140/pjm.2017.290.1},
eprint = {1503.03669},
archivePrefix = {arXiv},
}
@article{virk2020intrinsic,
author = {Virk, {\v{Z}}iga},
title = {1-dimensional intrinsic persistence of geodesic spaces},
journal = {J. Topol. Anal.},
volume = {12},
number = {1},
year = {2020},
pages = {169--207},
doi = {10.1142/S1793525319500444},
}
@misc{aim-spherical-cech-connectivity-monotonicity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Monotonicity of connectivity in spherical Čech filtrations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/538-spherical-cech-connectivity-monotonicity.md},
note = {Entry 538 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00900, "Monotonicity of connectivity in spherical Čech filtrations", clopen.solutions.