Finite homotopy models for spherical Rips complexes
For an integer $n \ge 2$ equip the unit sphere $S^n \subset \mathbb{R}^{n+1}$ with the geodesic metric $d(x, y) = \arccos \langle x, y \rangle$, of diameter $\pi$. For $r > 0$ let $$\operatorname{VR}_{<}(S^n; r) = \{\sigma \subset S^n : \sigma \text{ finite}, \ \operatorname{diam}(\sigma) < r\}$$ be the Vietoris–Rips complex, and $|\operatorname{VR}_{<}(S^n; r)|$ its geometric realization with the usual CW topology. By Hausmann's theorem it is homotopy equivalent to $S^n$ for $r$ small enough, and it is contractible for $r \ge \pi$. For the circle, Adamaszek and Adams proved that $\operatorname{VR}_{<}(S^1; r) \simeq S^{2k+1}$ for $\frac{2\pi k}{2k+1} < r \le \frac{2\pi(k+1)}{2k+3}$, $k \ge 0$; with $\operatorname{diam}(\sigma) \le r$ in place of $< r$, the complex at $r = \frac{2\pi k}{2k+1}$, $k \ge 1$, is homotopy equivalent to an uncountable wedge of $2k$-spheres. Adams, Bush and Virk bounded the homotopy connectivity of $\operatorname{VR}_{<}(S^n; r)$ by covering radii of $S^n$ and of $\mathbb{RP}^n$, and deduced that its homotopy type changes infinitely often as $r$ increases to $\pi$.
Problem. Is $|\operatorname{VR}_{<}(S^n; r)|$ homotopy equivalent to a finite CW complex for every $n \ge 2$ and every $r > 0$? The finite complex may depend on $n$ and $r$.
Source: Adams, Bush and Virk (2025), Conjecture 7.3, for all n and r ≥ 0 (arXiv 2024). Stated by AIM — Open Applied Problems for n ≥ 2 and r > 0.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, homotopy types, spheres, Vietoris–Rips complexes
References
@article{adams2025,
author = {Adams, Henry and Bush, Johnathan and Virk, {\v{Z}}iga},
title = {The connectivity of {V}ietoris--{R}ips complexes of spheres},
journal = {J. Appl. Comput. Topol.},
volume = {9},
number = {2},
year = {2025},
pages = {15},
doi = {10.1007/s41468-025-00214-y},
eprint = {2407.15818},
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},
}
@incollection{hausmann1995,
author = {Hausmann, Jean-Claude},
title = {On the {V}ietoris--{R}ips complexes and a cohomology theory for metric spaces},
booktitle = {Prospects in Topology ({P}rinceton, {NJ}, 1994)},
series = {Ann. of Math. Stud.},
volume = {138},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {1995},
pages = {175--188},
doi = {10.1515/9781400882588-013},
}
@misc{aim-spherical-rips-finite-homotopy-type,
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 Rips complexes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/510-spherical-rips-finite-homotopy-type.md},
note = {Entry 510 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-00903, "Finite homotopy models for spherical Rips complexes", clopen.solutions.