
On a Vertex-Minimal Triangulation of $\mathbb R \mathrm P ^4$
Author(s) -
Sonia Balagopalan
Publication year - 2017
Publication title -
the electronic journal of combinatorics/the journal of combinatorics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.703
H-Index - 52
eISSN - 1097-1440
pISSN - 1077-8926
DOI - 10.37236/5956
Subject(s) - combinatorics , mathematics , vertex (graph theory) , triangulation , cover (algebra) , graph , geometry , mechanical engineering , engineering
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.