Planar Rips complexes as wedges of spheres
Let $X \subset \mathbb{R}^2$ be a nonempty finite set, with Euclidean distances, and let $r > 0$. Let $$K = \operatorname{VR}_{<}(X; r) = \{\sigma \subseteq X : \operatorname{diam}(\sigma) < r\}$$ be the Vietoris–Rips complex of $X$ at scale $r$; for finite $X$ the non-strict version gives the same complexes after a small change of scale. Although $X$ is planar, $K$ can have any dimension. Chambers, de Silva, Erickson and Ghrist proved that the projection of $|K|$ onto its shadow in $\mathbb{R}^2$ induces an isomorphism on fundamental groups, so that the fundamental group of $|K|$ is free. Adamaszek, Frick and Vakili proved that a normal pseudomanifold of dimension $d \ge 2$ that is an induced subcomplex of $K$ is the boundary of the $(d+1)$-dimensional cross-polytope. Sipani and Kasilingam proved that the pseudomanifolds of dimension at least two that arise as planar Rips complexes are exactly the cross-polytopal spheres, and they note that every finite two-dimensional planar Rips complex is homotopy equivalent to a wedge of circles and $2$-spheres.
Problem. Assume that $K$ is connected. Must $|K|$ be homotopy equivalent to a finite wedge of spheres, $$|K| \simeq \bigvee_{j=1}^{m} S^{d_j}, \qquad d_j \ge 1,$$ where $m = 0$ stands for a point? Equivalently, is each connected component of every finite planar Rips complex contractible or homotopy equivalent to such a wedge?
Source: Adamaszek, Frick and Vakili (2017), Problem 7.3, as a question that had been around for some time; Conjecture 1.2 of Sipani and Kasilingam (2026). Stated for connected complexes by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- CO, MG
- Keywords
- AIM Open Applied Problems, homotopy types, Vietoris–Rips complexes, wedges of spheres
References
@article{adamaszek2017,
author = {Adamaszek, Micha{\l} and Frick, Florian and Vakili, Adrien},
title = {On homotopy types of {E}uclidean {R}ips complexes},
journal = {Discrete Comput. Geom.},
volume = {58},
number = {3},
year = {2017},
pages = {526--542},
doi = {10.1007/s00454-017-9916-5},
eprint = {1602.04131},
archivePrefix = {arXiv},
}
@article{sipani2026,
author = {Sipani, Vinay and Kasilingam, Ramesh},
title = {On the classification of planar-{R}ips complexes and their corresponding unit disk graphs},
journal = {Homology Homotopy Appl.},
volume = {28},
number = {3},
year = {2026},
pages = {1--30},
doi = {10.4310/HHA.2026.v28.n3.a1},
eprint = {2406.01082},
archivePrefix = {arXiv},
}
@article{chambers2010,
author = {Chambers, Erin W. and de Silva, Vin and Erickson, Jeff and Ghrist, Robert},
title = {Vietoris--{R}ips complexes of planar point sets},
journal = {Discrete Comput. Geom.},
volume = {44},
number = {1},
year = {2010},
pages = {75--90},
doi = {10.1007/s00454-009-9209-8},
eprint = {0712.0395},
archivePrefix = {arXiv},
}
@misc{aim-planar-rips-wedges-of-spheres,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Planar Rips complexes as wedges of spheres}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/498-planar-rips-wedges-of-spheres.md},
note = {Entry 498 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-00887, "Planar Rips complexes as wedges of spheres", clopen.solutions.