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An Efficient Triangulation of $\mathbb{R}P^5$

Dan Guyer, Stefan Steinerberger, Yirong Yang

Abstract

We present a $6$-dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a $24$-vertex triangulation of the $5$-dimensional real projective space. This $6$-polytope is highly symmetric with an automorphism group of order $192$, and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of $\mathbb{R}P^5$. Our method also produces two triangulations of $\mathbb{R}P^6$ on $45$ and $49$ vertices; both improve the previously best known construction in dimension $6$ that used $53$ vertices.

An Efficient Triangulation of $\mathbb{R}P^5$

Abstract

We present a -dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a -vertex triangulation of the -dimensional real projective space. This -polytope is highly symmetric with an automorphism group of order , and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of . Our method also produces two triangulations of on and vertices; both improve the previously best known construction in dimension that used vertices.
Paper Structure (10 sections, 3 theorems, 10 equations, 3 figures, 1 table)

This paper contains 10 sections, 3 theorems, 10 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and Results
  2. Introduction
  3. Main result
  4. Triangulations of $\mathbb{R}P^5$ and $\mathbb{R}P^6$
  5. Background
  6. A $24$-vertex triangulation of $\mathbb{R}P^5$
  7. Triangulations of $\mathbb{R}P^6$
  8. Remarks and Future Directions
  9. Construction of $\mathcal{P}_{6,48}$
  10. Future Directions.

Key Result

Proposition 1.1

If $d\geq 3$, then any triangulation of real projective space $\mathbb{R}P^d$ must have at least $(d+2)(d+1)/2 + 1$ vertices.

Figures (3)

  • Figure 1: Identifying antipodal vertices $\pm i$ for $i \in [6]$ of the icosahedron yields the unique minimal triangulation of $\mathbb{R}P^2$.
  • Figure 2: The union of induced subgraphs on $V_S$ for every $S \in \mathcal{S}$ and the neighborhood of $P_1$.
  • Figure 3: Polytopal realizations of different combinatorial types of links of the $2$-faces of $\partial \mathcal{P}_{6,48}$, ordered by the number of vertices.

Theorems & Definitions (4)

  • Proposition 1.1: Arnoux--Marin, 1991
  • Theorem : Main Result
  • Theorem 2.1
  • Conjecture 3.1