Monotonicity of connectivity in spherical Rips filtrations
For an integer $n \ge 2$ equip the unit sphere $S^n \subset \mathbb{R}^{n+1}$ with the geodesic metric $d(x, y) = \arccos \langle x, y \rangle$. For $r > 0$ let $K_n(r)$ be the geometric realization, with the CW topology, of the Vietoris–Rips complex of $S^n$ at scale $r$: its vertices are all points of $S^n$ and its simplices are the finite subsets of diameter strictly less than $r$. For a path-connected space $Y$ let $$\operatorname{conn}(Y) = \sup\{k \in \mathbb{Z}_{\ge 0} : \pi_i(Y) = 0 \text{ for every } 1 \le i \le k\},$$ with value $+\infty$ if all these groups vanish; the spaces $K_n(r)$ are path-connected.
Lim, Mémoli and Okutan proved that $K_n(r) \simeq S^n$ for $0 < r \le \arccos(-1/(n+1))$, and $K_n(r)$ is contractible for $r \ge \pi$. Adams, Bush and Virk proved that if $0 < \delta < \pi$ and $\operatorname{conn}(K_n(\pi - \delta)) = k - 1$, then $$\operatorname{cov}_{S^n}(2k + 2) \le \delta < 2 \operatorname{cov}_{\mathbb{RP}^n}(k),$$ where $\operatorname{cov}_X(m)$ is the least radius of $m$ closed balls covering $X$ and $\mathbb{RP}^n$ carries the quotient metric, of diameter $\pi/2$; hence the homotopy type of $K_n(r)$ changes infinitely often as $r$ increases to $\pi$. For the circle, Adamaszek and Adams proved that $K_1(r) \simeq S^{2k+1}$ for $\frac{2\pi k}{2k+1} < r \le \frac{2\pi(k+1)}{2k+3}$, $k \ge 0$, so that the connectivity of $K_1(r)$ never decreases. Hausmann asked whether the connectivity of the Vietoris–Rips complexes of a Riemannian manifold grows with the scale until they become contractible; Virk constructed a compact Riemannian manifold for which it drops at some pair of scales.
Problem. Is $$\operatorname{conn}(K_n(r)) \le \operatorname{conn}(K_n(s))$$ for every integer $n \ge 2$ and all $0 < r < s$? Only the connectivities are compared; the problem does not assert that the inclusion $K_n(r) \hookrightarrow K_n(s)$ is a homotopy equivalence.
Source: Adams, Bush and Virk (2025), Conjecture 7.1, for every n (arXiv 2024), as the case of spheres of Hausmann's question (1995, Problem 3.12). Stated for n ≥ 2 by AIM — Open Applied Problems; the circle case follows from Adamaszek and Adams (2017).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, homotopy connectivity, spheres, Vietoris–Rips complexes
References
@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},
}
@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 Univ. Press},
address = {Princeton, NJ},
year = {1995},
pages = {175--188},
doi = {10.1515/9781400882588-013},
}
@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{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},
}
@article{virk2022counter,
author = {Virk, {\v{Z}}iga},
title = {A counter-example to {H}ausmann's conjecture},
journal = {Found. Comput. Math.},
volume = {22},
number = {2},
year = {2022},
pages = {469--475},
doi = {10.1007/s10208-021-09510-2},
}
@misc{aim-spherical-rips-connectivity-monotonicity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Monotonicity of connectivity in spherical Rips filtrations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/511-spherical-rips-connectivity-monotonicity.md},
note = {Entry 511 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-00902, "Monotonicity of connectivity in spherical Rips filtrations", clopen.solutions.