Homology at the second transition of torus-grid Rips complexes
For an integer $n \ge 3$ give $T_n = \{0, \dots, n-1\}^2$ the periodic grid metric $$d((a,b),(c,e)) = \min\{|a-c|, n-|a-c|\} + \min\{|b-e|, n-|b-e|\},$$ and let $K_n(r)$ be its closed Vietoris–Rips complex at scale $r$, whose simplices are the subsets of $T_n$ of diameter at most $r$. Write $\beta_i(K) = \operatorname{rank} H_i(K;\mathbb{Z})$. Adams, Adetowubo, Barriga-Acosta, Feng and Sterling proved that $K_{3k}(k) \simeq \bigvee_{6k^2-1} S^2$ for $k \ge 2$ and $K_{3k-1}(k) \simeq \bigvee_{6k-3} S^2 \vee \bigvee_{6k-2} S^3$ for $k \ge 3$.
Problem. Is it true that, for every integer $k \ge 5$, $$\widetilde H_i(K_{3k-2}(k);\mathbb{Z}) = 0 \quad (i \notin \{3, 4\}), \qquad \beta_3(K_{3k-2}(k)) = 6k - 2 + (-1)^k, \qquad \beta_4(K_{3k-2}(k)) = 6k - 3 + (-1)^k?$$ No assertion of torsion-freeness in degrees $3$ and $4$ is included.
Source: Adams, Adetowubo, Barriga-Acosta, Feng and Sterling (2025), Question 8.4, which asks the vanishing outside degrees 3 and 4 for k ≥ 3 and the Betti numbers for k ≥ 5; stated for k ≥ 5 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, integral homology, 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-transition-homology,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Homology at the second transition of torus-grid Rips complexes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/531-torus-grid-transition-homology.md},
note = {Entry 531 at commit 24edaad, 2026-10-15},
}
Resolution claims
We compute the fourth integral Betti number of the closed radius-five Vietoris--Rips complex on the thirteen-by-thirteen torus grid with its periodic Manhattan metric. It is two, giving a counterexample to the integer-rank formula in AIM Problem 531, which predicts twenty-six for this grid. The proof separates a subcomplex with the homotopy type of the torus from the simplices that cannot …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/torus_grid_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00840, "Homology at the second transition of torus-grid Rips complexes", clopen.solutions.