OP-00792 Solution claimed

A finite three-ample complex with nontrivial fundamental group

For a simplicial complex $X$ with vertex set $V(X)$ and $U \subseteq V(X)$, let $X[U]$ be the induced subcomplex, and let $\operatorname{Lk}_X(v)$, the link of a vertex $v$, consist of the simplices $\sigma$ not containing $v$ for which $\sigma \cup \{v\}$ is a simplex of $X$. A nonempty complex $X$ is $r$-ample if, for every $U \subseteq V(X)$ with $|U| \le r$ and every simplicial subcomplex $A \subseteq X[U]$, there is a vertex $v \in V(X) \setminus U$ with $$\operatorname{Lk}_X(v) \cap X[U] = A.$$ Here $A$ may be empty and need not be induced. A two-ample complex is connected, and Farber gives a two-ample complex on $13$ vertices that is not simply connected; every four-ample complex is simply connected. Barmak constructed an infinite three-ample complex that is not simply connected.

Problem. Does there exist a finite three-ample simplicial complex $X$ with $\pi_1(|X|) \ne \{1\}$? No restriction to flag complexes is imposed.

Source: Farber (2024), Example 2.5, who asks for a finite example after Barmak's infinite three-ample complex that is not simply connected (Barmak 2023, Theorem 1).

Imported from github.com/MColbrook/AIM.

Classification

Primary: AT — Algebraic Topology
Secondary: CO
Keywords: AIM Open Applied Problems, ample simplicial complexes, fundamental group

References

@article{farber2024,
  author        = {Farber, Michael},
  title         = {Large simplicial complexes: universality, randomness, and ampleness},
  journal       = {J. Appl. Comput. Topol.},
  volume        = {8},
  number        = {6},
  year          = {2024},
  pages         = {1551--1574},
  doi           = {10.1007/s41468-023-00134-9},
  eprint        = {2301.07404},
  archivePrefix = {arXiv},
}
@article{barmak2023,
  author        = {Barmak, Jonathan Ariel},
  title         = {Connectivity of ample, conic, and random simplicial complexes},
  journal       = {Int. Math. Res. Not. IMRN},
  volume        = {2023},
  number        = {8},
  year          = {2023},
  pages         = {6579--6597},
  doi           = {10.1093/imrn/rnac030},
  eprint        = {2103.03952},
  archivePrefix = {arXiv},
}
@misc{aim-finite-three-ample-nonsimply-connected,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A finite three-ample complex with nontrivial fundamental group}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/645-finite-three-ample-nonsimply-connected.md},
  note         = {Entry 645 at commit 24edaad, 2026-10-15},
}

Resolution claims

A finite three-ample complex with a nontrivial fundamental group v1 GitHub ↗

Kaiwen Jin

We construct a finite three-ample simplicial complex on $32{,}768$ vertices which retracts simplicially onto a four-cycle. Thus the prescribed extension of every subcomplex on at most three vertices is compatible with a nontrivial fundamental group in a finite complex. The construction fixes a four-cycle and chooses the remaining permitted faces probabilistically. A finite union bound proves that all extension requests …

Full resolution · not verified · fidelity not established · posted on GitHub 14 Oct 2026

Status history

When	Change	By	Reason
2026-10-18	Needs verification → Solution claimed	—	Recorded here: AIM @ 24edaad, paper/finite_three_ample.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00792, "A finite three-ample complex with nontrivial fundamental group", clopen.solutions.