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PreprintCC BY 4.0arXiv

The minimum number of vertices of a triangulation of ℝℙ5

  • Xinrui Chang
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Abstract

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 the sixteen-vertex case, a direct capacity bound excludes twenty-two vertices. For twenty-three vertices, a cochain extension argument reduces every relevant completion to one of 115 critical signed complexes; integer weights and exact covering certificates exclude all of them. The argument requires no assumption that the triangulation is combinatorial.

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Cite

BibTeXRIS
BibTeX
@misc{chang2026minimum,
  title = {{The minimum number of vertices of a triangulation of $\mathbbRP^5$}},
  author = {Xinrui Chang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25171},
  eprint = {2610.25171},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25171},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The minimum number of vertices of a triangulation of ℝℙ^5
AU  - Xinrui Chang
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25171
UR  - https://arxiv.org/abs/2610.25171
AB  - 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 the sixteen-vertex case, a direct capacity bound excludes twenty-two vertices. For twenty-three vertices, a cochain extension argument reduces every relevant completion to one of 115 critical signed complexes; integer weights and exact covering certificates exclude all of them. The argument requires no assumption that the triangulation is combinatorial.
ER  -