A sharp Betti-number bound for 4-manifold triangulation complexity
Let $M$ be a closed simply connected PL 4-manifold, and let $\beta_2(M) = \dim_{\mathbb{Q}} H_2(M; \mathbb{Q})$. Triangulations are generalised: finitely many 4-simplices (pentachora) whose tetrahedral faces are identified in pairs by affine maps, so that two pentachora may share several faces and two faces of one pentachoron may be identified, but no face is identified with itself by a non-identity map; the result must be PL homeomorphic to $M$. The complexity $c(M)$ is the least number of pentachora in such a triangulation. Spreer and Tobin constructed triangulations with $2\beta_2(M) + 2$ pentachora of every connected sum of PL standard copies of $\mathbb{CP}^2$ and $S^2 \times S^2$, in any orientation. They proved $c(M) \ge (6\beta_2(M) - 4)/5$, and that a conjectured bound on the number of vertices of a triangulation in terms of its number of pentachora would give $c(M) \ge 2\beta_2(M)$. The smallest known triangulation of the K3 surface, due to Burke, has $54$ pentachora (as reported by Spreer and Tobin in 2025), where $2\beta_2 + 2 = 46$.
Problem. Prove or disprove that $$c(M) \ge 2\beta_2(M) + 2$$ for every closed simply connected PL 4-manifold $M$.
Source: Spreer and Tobin (2024) and, independently, Burke (2024); see Spreer and Tobin (2025), Conjecture 1.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- GT — Geometric Topology
- Secondary
- CO
- Keywords
- 4-manifolds, AIM Open Applied Problems, generalised triangulations, triangulation complexity
References
@misc{spreer2024,
author = {Spreer, Jonathan and Tobin, Lucy},
title = {Vertex bounds in triangulated $d$-manifolds and an application to 4-manifold complexity},
year = {2024},
note = {First version titled ``Face numbers of triangulations of manifolds''},
eprint = {2401.11152},
archivePrefix = {arXiv},
}
@inproceedings{burke2024,
author = {Burke, Rhuaidi Antonio},
title = {Practical software for triangulating and simplifying 4-manifolds},
booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
volume = {293},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
year = {2024},
pages = {29:1--29:23},
doi = {10.4230/LIPIcs.SoCG.2024.29},
eprint = {2402.15087},
archivePrefix = {arXiv},
}
@article{spreer2025,
author = {Spreer, Jonathan and Tobin, Lucy},
title = {Small triangulations of simply connected 4-manifolds},
journal = {J. Appl. Comput. Topol.},
volume = {9},
number = {4},
year = {2025},
pages = {Paper No. 24},
doi = {10.1007/s41468-025-00221-z},
eprint = {2501.18933},
archivePrefix = {arXiv},
}
@misc{aim-four-manifold-triangulation-complexity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A sharp Betti-number bound for 4-manifold triangulation complexity}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/579-four-manifold-triangulation-complexity.md},
note = {Entry 579 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-00867, "A sharp Betti-number bound for 4-manifold triangulation complexity", clopen.solutions.