Sharp vertex bounds for generalized triangulations in even dimensions
Let $d \ge 4$ be even. A generalised triangulation $T$ of a closed connected PL $d$-manifold is a finite set of $d$-simplices whose $(d-1)$-faces are glued in pairs by affine maps; two faces of the same simplex may be glued, and two simplices may be glued along several faces. No face may be identified with itself by a non-identity map, and the quotient space is the manifold. Write $f_0(T)$ for the number of vertices after the identifications and $f_d(T)$ for the number of $d$-simplices.
For closed connected surfaces $f_0 = f_2/2 + \chi$, so $f_0 \le f_2/2 + 2$, with equality exactly for the $2$-sphere. Spreer and Tobin showed that for every even $d$ and every even $f_d$ some triangulation of the $d$-sphere has $f_0 = f_d/2 + d$. They proved $f_0 \le f_d/2 + d$ for every closed balanced triangulation (one whose vertices can be coloured with $d+1$ colours so that each simplex has one vertex of each colour and the gluings preserve colours), with the cases of equality determined, and noted that for simplicial complexes the inequality follows from the Lower Bound Theorem. They also constructed a closed connected triangulation of a $4$-dimensional pseudomanifold with $14$ simplices and $f_0 > f_4/2 + 4$. For $d = 4$ the inequality would imply $f_4 \ge 2\beta_2(M)$ for every generalised triangulation of a simply connected closed $4$-manifold $M$.
Problem. Prove or disprove that every such $T$ satisfies $$f_0(T) \le \frac{f_d(T)}{2} + d.$$
Source: Spreer and Tobin (arXiv 2024), Conjecture 5.1 of the 2026 revision.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- GT — Geometric Topology
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, face numbers, triangulated manifolds
References
@misc{spreer2024vertex,
author = {Spreer, Jonathan and Tobin, Lucy},
title = {Vertex Bounds in Triangulated $d$-Manifolds and an Application to $4$-Manifold Complexity},
year = {2024},
eprint = {2401.11152},
archivePrefix = {arXiv},
note = {Version 2, 1 July 2026; version 1 (2024) titled ``Face numbers of triangulations of manifolds''},
}
@article{spreer2025small,
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 = {24},
doi = {10.1007/s41468-025-00221-z},
eprint = {2501.18933},
archivePrefix = {arXiv},
}
@misc{aim-even-manifold-vertex-bound,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp vertex bounds for generalized triangulations in even dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/580-even-manifold-vertex-bound.md},
note = {Entry 580 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-00862, "Sharp vertex bounds for generalized triangulations in even dimensions", clopen.solutions.