Homotopy type of torus-grid Rips complexes at half the diameter
For an integer $k \ge 3$ give $T_{2k} = \{0, \dots, 2k-1\}^2$ the periodic grid metric $$d((a,b),(c,e)) = \min\{|a-c|, 2k-|a-c|\} + \min\{|b-e|, 2k-|b-e|\},$$ whose diameter is $2k$, and let $K_{2k}(k)$ be its closed Vietoris–Rips complex at scale $k$, whose simplices are the subsets of diameter at most $k$. For $k = 2$ the complex is the Rips complex of the $4$-cube at scale $2$, and Adams, Adetowubo, Barriga-Acosta, Feng and Sterling note that $K_4(2) \simeq \bigvee_9 S^3$, the case $k = 2$ of the formula below.
Problem. Is $$K_{2k}(k) \simeq S^3 \vee \bigvee_{j=1}^{4k} S^{2k-1}$$ for every integer $k \ge 3$, the wedge taken at a common base point?
Source: Adams, Adetowubo, Barriga-Acosta, Feng and Sterling (2025), Question 8.6, for k ≥ 2; stated for k ≥ 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, homotopy types, torus grids, Vietoris–Rips complexes
References
@article{adams2025torus,
author = {Adams, Henry and Adetowubo, Adenike Yeside and Barriga-Acosta, Hector and Feng, Ziqin and Sterling, John},
title = {{V}ietoris--{R}ips complexes of torus grids},
journal = {Mediterr. J. Math.},
volume = {22},
number = {7},
year = {2025},
pages = {173},
doi = {10.1007/s00009-025-02945-9},
eprint = {2502.07134},
archivePrefix = {arXiv},
}
@misc{aim-torus-grid-half-diameter-homotopy,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Homotopy type of torus-grid Rips complexes at half the diameter}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/532-torus-grid-half-diameter-homotopy.md},
note = {Entry 532 at commit 24edaad, 2026-10-15},
}
Resolution claims
For every integer \(k\geq3\), we determine the homotopy type of the closed Vietoris--Rips complex of the \(2k\) by \(2k\) periodic Manhattan grid at radius \(k\). It is \(S^3\vee\bigvee^{4k}S^{2k-1}\), as conjectured by Adams, Adetowubo, Barriga-Acosta, Feng and Sterling. Adjacent pairs of rows and columns give the high-dimensional spheres. Removing one facet from each leaves a complex with the integral homology of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00839, "Homotopy type of torus-grid Rips complexes at half the diameter", clopen.solutions.