The first Rips homotopy transition for higher-dimensional spheres
For $n \ge 3$ give the unit sphere $S^n \subset \mathbb{R}^{n+1}$ the geodesic metric $d(x,y) = \arccos\langle x, y\rangle$, and let $K_n(r)$ be the ordinary geometric realization of its Vietoris–Rips complex at scale $r$, whose simplices are the finite subsets of diameter strictly less than $r$. Let $$r_n = \arccos\Bigl(-\frac{1}{n+1}\Bigr),$$ the distance between any two vertices of a regular $(n+1)$-simplex inscribed in $S^n$. The orientation-preserving symmetries of such a simplex form a copy of the alternating group $A_{n+2}$ in $\operatorname{SO}(n+1)$; let $Q_n = \operatorname{SO}(n+1)/A_{n+2}$, and write $X * Y$ for the join.
Lim, Mémoli and Okutan proved that $K_n(r) \simeq S^n$ for $0 < r \le r_n$. Adamaszek, Adams and Frick proved that the Vietoris–Rips metric thickening of $S^n$ at scale $r_n$, with diameter at most $r_n$, is homotopy equivalent to $S^n * Q_n$. For $n = 1$ and $n = 2$, $K_n(r)$ is homotopy equivalent to $S^n * Q_n$ for all $r$ in an interval to the right of $r_n$ (Adamaszek and Adams; Lim, Mémoli and Okutan).
Problem. Is it true that for every integer $n \ge 3$ there exists $\varepsilon_n \in (0, \pi - r_n)$ such that $K_n(r) \simeq S^n * Q_n$ whenever $r_n < r < r_n + \varepsilon_n$?
Source: Adamaszek, Adams and Frick (2018), Conjecture 5.7, for every n and together with the corresponding statements for metric thickenings; restated for the complexes by Lim, Mémoli and Okutan (2024) and by Adams, Bush and Virk (2025), Conjecture 7.2. Stated for n ≥ 3 by AIM — Open Applied Problems.
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{adamaszek2018metric,
author = {Adamaszek, Micha{\l} and Adams, Henry and Frick, Florian},
title = {Metric reconstruction via optimal transport},
journal = {SIAM J. Appl. Algebra Geom.},
volume = {2},
number = {4},
year = {2018},
pages = {597--619},
doi = {10.1137/17M1148025},
eprint = {1706.04876},
archivePrefix = {arXiv},
}
@article{lim2024vietoris,
author = {Lim, Sunhyuk and M{\'e}moli, Facundo and Okutan, Osman Berat},
title = {Vietoris--{R}ips persistent homology, injective metric spaces, and the filling radius},
journal = {Algebr. Geom. Topol.},
volume = {24},
number = {2},
year = {2024},
pages = {1019--1100},
doi = {10.2140/agt.2024.24.1019},
eprint = {2001.07588},
archivePrefix = {arXiv},
}
@article{adams2025connectivity,
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},
}
@misc{aim-spherical-rips-first-transition,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The first Rips homotopy transition for higher-dimensional spheres}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/512-spherical-rips-first-transition.md},
note = {Entry 512 at commit 24edaad, 2026-10-15},
}
Resolution claims
For every \(n\geq3\), we identify the homotopy type of the strict Vietoris--Rips complex of the unit round sphere on an interval immediately above the regular-simplex threshold \(r_n=\arccos(-1/(n+1))\). It is the join of \(S^n\) with \(\SO(n+1)/A_{n+2}\), where \(A_{n+2}\) acts by the rotational symmetries of the simplex. This proves the first-transition interval conjecture for higher-dimensional spheres. A recent critical-diameter theorem supplies a …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00835, "The first Rips homotopy transition for higher-dimensional spheres", clopen.solutions.