On a Vertex-Minimal Triangulation of $\mathbb R \mathrm P ^4$

  • Sonia Balagopalan
Keywords: Combinatorial manifolds, Vertex-minimal, Minimal triangulation, Projective space, Witt design


We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}\mathrm{P}^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated $\mathbb{R}\mathrm{P}^4$ and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate $n$-dimensional real projective space. All three constructions deliver the same combinatorial manifold, which is also the same as the only known $16$-vertex triangulation of $\mathbb{R}\mathrm{P}^4$. We also give a short, simple construction of the $22$-point Witt design, which is closely related to the complex we construct.

Author Biography

Sonia Balagopalan, Department of Mathematics and Statistics, Maynooth University
Lecturer, Department of Mathematics and Statistics, Maynooth University
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