OP-00814 Solution claimed

Odd Euler characteristic for flag-no-square four-manifolds

A finite simplicial complex is flag if every clique of its $1$-skeleton is a simplex, and flag-no-square if moreover its $1$-skeleton has no induced cycle of length four. The $4$-manifolds first known to admit flag-no-square triangulations are quotients of hyperbolic $4$-space. Kalmanovich, Nevo and Sorcar showed that a $4$-manifold with a flag-no-square triangulation need not be aspherical, and constructed, for every sufficiently large even integer $s$, a connected $4$-manifold of Euler characteristic $s$ with a flag-no-square triangulation.

Problem. Does there exist a closed connected topological $4$-manifold $M$ with odd Euler characteristic that admits a finite flag-no-square triangulation? Here closed means compact without boundary; orientability is not required.

Source: Kalmanovich, Nevo and Sorcar (2026), Problem 4.1. Stated for closed connected topological 4-manifolds, orientable or not, by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: GT — Geometric Topology
Secondary: CO
Keywords: AIM Open Applied Problems, Euler characteristic, flag-no-square triangulations, four-manifolds

References

@article{kalmanovich2026,
  author        = {Kalmanovich, Daniel and Nevo, Eran and Sorcar, Gangotryi},
  title         = {On Flag-No-Square 4-Manifolds},
  journal       = {Discrete Comput. Geom.},
  volume        = {76},
  number        = {1},
  year          = {2026},
  pages         = {298--312},
  doi           = {10.1007/s00454-025-00774-x},
  eprint        = {2310.13495},
  archivePrefix = {arXiv},
}
@misc{aim-odd-euler-flag-no-square,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Odd Euler characteristic for flag-no-square four-manifolds}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/628-odd-euler-flag-no-square.md},
  note         = {Entry 628 at commit 24edaad, 2026-10-15},
}

Resolution claims

Odd Euler characteristic for flag-no-square four-manifolds v1 GitHub ↗

Kaiwen Jin

We construct a closed connected PL four-manifold with odd Euler characteristic that admits a finite flag-no-square triangulation. Here flag means that every clique is a simplex, and no-square excludes induced four-cycles in the graph. The construction identifies antipodal points on the boundary of a deleted vertex star. In a regular hyperbolic tessellation, each graph edge outside that vertex changes the …

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/odd_euler_flag_no_square.pdf.

Citing this problem

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

OP-00814, "Odd Euler characteristic for flag-no-square four-manifolds", clopen.solutions.