The minimum vertex number of a triangulation of real projective 5-space
A triangulation of $\mathbb{RP}^5$ is a finite simplicial complex whose geometric realization is homeomorphic to real projective $5$-space; no symmetry, polytopal-realizability or combinatorial-manifold condition is imposed. Let $\mu_5$ be the minimum number of vertices of a triangulation of $\mathbb{RP}^5$. By a theorem of Arnoux and Marin, every triangulation of $\mathbb{RP}^d$ with $d \ge 3$ has at least $(d+1)(d+2)/2 + 1$ vertices, so $\mu_5 \ge 22$. Lutz found a $24$-vertex triangulation of $\mathbb{RP}^5$ with the bistellar-flip program BISTELLAR. Guyer, Steinerberger and Yang gave another one, the antipodal quotient of the boundary of a centrally symmetric simplicial $6$-polytope with $48$ vertices. The minimum vertex numbers of triangulations of $\mathbb{RP}^2$, $\mathbb{RP}^3$ and $\mathbb{RP}^4$ are $6$, $11$ and $16$.
Problem. Is $\mu_5 = 24$?
Source: Guyer, Steinerberger and Yang (2026), Conjecture 3.1, after Lutz (2005), Question 13, who found a 24-vertex triangulation and asked whether one with 22 vertices exists.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- AT, GT
- Keywords
- AIM Open Applied Problems, real projective spaces, triangulated manifolds, vertex-minimal triangulations
References
@article{guyer2026,
author = {Guyer, Dan and Steinerberger, Stefan and Yang, Yirong},
title = {An efficient triangulation of $\mathbb{R}P^5$},
journal = {Discrete Comput. Geom.},
year = {2026},
doi = {10.1007/s00454-026-00882-2},
eprint = {2603.07808},
archivePrefix = {arXiv},
}
@misc{lutz2005,
author = {Lutz, Frank H.},
title = {Triangulated manifolds with few vertices: combinatorial manifolds},
year = {2005},
eprint = {math/0506372},
archivePrefix = {arXiv},
}
@article{arnoux1991,
author = {Arnoux, Pierre and Marin, Alexis},
title = {The {K}{\"u}hnel triangulation of the complex projective plane from the view point of complex crystallography, {P}art {II}},
journal = {Mem. Fac. Sci. Kyushu Univ. Ser. A},
volume = {45},
number = {2},
year = {1991},
pages = {167--244},
doi = {10.2206/kyushumfs.45.167},
}
@misc{aim-rp5-minimum-vertices,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The minimum vertex number of a triangulation of real projective 5-space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/595-rp5-minimum-vertices.md},
note = {Entry 595 at commit 24edaad, 2026-10-15},
}
Resolution claims
We give a computer-assisted proof that every finite simplicial triangulation of real projective five-space has at least twenty-four vertices. Together with the construction of Guyer, Steinerberger, and Yang, this determines the minimum. A cup-product argument produces an induced subcomplex on sixteen or seventeen vertices. Transfer to the spherical double cover then imposes a covering condition on its lifted faces. In …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.25171v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00846, "The minimum vertex number of a triangulation of real projective 5-space", clopen.solutions.