Finite homotopy models for spherical Čech complexes
For $n \ge 1$ give $S^n \subset \mathbb{R}^{n+1}$ the geodesic metric $d(x,y) = \arccos\langle x, y\rangle$, and for $r > 0$ let $C_n(r)$ be the intrinsic Čech complex of $S^n$ at scale $r$: the nerve of the open balls $B(x;r) = \{z \in S^n : d(x,z) < r\}$, $x \in S^n$. Its vertex set is $S^n$, and a finite set $\sigma \subset S^n$ is a simplex when $\bigcap_{x \in \sigma} B(x;r) \ne \varnothing$. The geometric realization $|C_n(r)|$ carries the CW topology.
For $0 < r \le \pi/2$ the balls form a good cover, and the nerve lemma gives $|C_n(r)| \simeq S^n$. Adamaszek and Adams showed that, as $r$ increases, $C_1(r)$ takes the homotopy types $S^1, S^3, S^5, \ldots$. Adams, Jauhari and Mallick proved that for every $n \ge 1$ the homotopy type of $C_n(r)$ changes infinitely many times as $r$ ranges over $(0, \pi)$. With closed balls in place of open ones, the assertion below already fails for the circle.
Problem. Is $|C_n(r)|$ homotopy equivalent to a finite CW complex for every $n \ge 2$ and every $r > 0$? The finite complex may depend on $n$ and $r$; no bound on its number of cells or its dimension is required.
Source: Adams, Jauhari and Mallick (2026), Conjecture 8.4, for all n ≥ 1 (arXiv 2025). Stated by AIM — Open Applied Problems for n ≥ 2, the case n = 1 being known.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, Čech complexes, homotopy types, 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},
}
@misc{aim-spherical-cech-finite-homotopy-models,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite homotopy models for spherical Čech complexes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/539-spherical-cech-finite-homotopy-models.md},
note = {Entry 539 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-00901, "Finite homotopy models for spherical Čech complexes", clopen.solutions.