Injectivity of the Rips shadow map on fundamental groups
Let $n \ge 1$, let $X \subset \mathbb{R}^n$ be a nonempty finite set and let $r > 0$. With Euclidean distances, let $$K = \operatorname{VR}_{<}(X; r) = \{\sigma \subseteq X : \operatorname{diam}(\sigma) < r\}$$ be the Vietoris–Rips complex of $X$ at scale $r$, and let its shadow be the union $$S(X; r) = \bigcup_{\sigma \in K} \operatorname{conv}(\sigma) \subset \mathbb{R}^n.$$ The vertex inclusion extends affinely on each simplex to a continuous map $p : |K| \to S(X; r)$. Chambers, de Silva, Erickson and Ghrist proved that $p$ induces an isomorphism on fundamental groups for every finite $X \subset \mathbb{R}^2$, and that it may fail to be surjective on fundamental groups in every dimension $n \ge 4$. Adamaszek, Frick and Vakili proved that it is surjective on fundamental groups for every finite $X \subset \mathbb{R}^3$.
Problem. Is $$p_* : \pi_1(|K|, x_0) \longrightarrow \pi_1(S(X; r), x_0)$$ injective for every such $n$, $X$ and $r$ and every base vertex $x_0 \in X$? That is, can a nontrivial loop in the abstract Rips complex become null-homotopic after projection onto its shadow? The question concerns each finite sample at each fixed scale; no dense-sampling or vanishing-scale limit is involved.
Source: Adamaszek, Frick and Vakili (2017), Conjecture 7.2, in all dimensions; for points in R^3, a question of Chambers, de Silva, Erickson and Ghrist (2010).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, fundamental group, Vietoris–Rips complexes
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{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-rips-shadow-fundamental-group,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Injectivity of the Rips shadow map on fundamental groups}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/499-rips-shadow-fundamental-group.md},
note = {Entry 499 at commit 24edaad, 2026-10-15},
}
Resolution claims
The affine projection from a Euclidean Vietoris--Rips complex to its shadow need not be injective on fundamental groups. We give six rational points in three-dimensional space whose Rips complex retracts onto a four-cycle, while its shadow is the union of two convex quadrilaterals containing the origin and is therefore contractible. Scaling and isometric inclusion give the same failure in every …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/rips_shadow_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00832, "Injectivity of the Rips shadow map on fundamental groups", clopen.solutions.