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Simplicial cell decompositions of $\mathbb{CP}^{\hspace{.3mm}n}$

Basudeb Datta, Jonathan Spreer

TL;DR

The authors address explicit triangulations of the complex projective space $\mathbb{CP}^{n}$ by realizing it as the quotient of the $n$-fold product $ (\mathbb{S}^{2})^{n}$ by the symmetric group $\mathrm{Sym}(n)$. They construct a regular simplicial cell decomposition $X^{n}$ of $(\mathbb{S}^{2})^{n}$ using a $S^2_3$ crystallization and a staircase monotone-path subdivision, and then form the good-action quotient $X^{n}/\mathrm{Sym}(n)$ to obtain a simplicial cell decomposition of $\mathbb{CP}^{n}$; its first derived subdivision yields a triangulation of $\mathbb{CP}^{n}$ for all $n\ge 2$, including new explicit triangulations for $n\ge 4$. They provide explicit $f$-vectors for the quotients $T_{n}=X^{n}/\mathrm{Sym}(n)$, plus isomorphism signatures and graph-encodings, and release computational code to construct these decompositions in Regina (DS2024). This work delivers a concrete, implementable framework for triangulating complex projective spaces and demonstrates the utility of good group actions and derived subdivisions in quotient constructions.

Abstract

According to a well-known result in geometric topology, we have \linebreak $\left (\mathbb{S}^2 \right)^{n}\!\!/\operatorname{Sym}(n) = \mathbb{CP}^{n}$, where $\operatorname{Sym}(n)$ acts on $\left (\mathbb{S}^2 \right)^{n}$ by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $\mathbb{CP}^{n}$ for each $n \geq 2$. In more detail, we start with the standard two triangle crystallisation $S^2_3$ of the $2$-sphere $\mathbb{S}^2$, in its $n$-fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the $\operatorname{Sym}(n)$ quotient of this subdivision yields a simplicial cell decomposition of $\mathbb{CP}^n$. The first derived subdivision of this cell complex is a simplicial triangulation of $\mathbb{CP}^n$. To the best of our knowledge, this is the first explicit description of triangulations of $\mathbb{CP}^n$ for $n \geq 4$.

Simplicial cell decompositions of $\mathbb{CP}^{\hspace{.3mm}n}$

TL;DR

The authors address explicit triangulations of the complex projective space by realizing it as the quotient of the -fold product by the symmetric group . They construct a regular simplicial cell decomposition of using a crystallization and a staircase monotone-path subdivision, and then form the good-action quotient to obtain a simplicial cell decomposition of ; its first derived subdivision yields a triangulation of for all , including new explicit triangulations for . They provide explicit -vectors for the quotients , plus isomorphism signatures and graph-encodings, and release computational code to construct these decompositions in Regina (DS2024). This work delivers a concrete, implementable framework for triangulating complex projective spaces and demonstrates the utility of good group actions and derived subdivisions in quotient constructions.

Abstract

According to a well-known result in geometric topology, we have \linebreak , where acts on by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of for each . In more detail, we start with the standard two triangle crystallisation of the -sphere , in its -fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the quotient of this subdivision yields a simplicial cell decomposition of . The first derived subdivision of this cell complex is a simplicial triangulation of . To the best of our knowledge, this is the first explicit description of triangulations of for .
Paper Structure (14 sections, 11 theorems, 7 equations, 1 figure)

This paper contains 14 sections, 11 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Suitability for a simplicial cell complex to allow quotients
  4. Simplicial cell decompositions of $(\mathbb{S}^2)^{n}$
  5. Simplicial cell decompositions of $\mathbb{CP}^{\space n}$
  6. Implementation
  7. Code to produce $X^n$ and $X^n/\operatorname{Sym}(n)$
  8. Helper functions to produce our simplicial cell decompositions
  9. Code to produce sphere products
  10. Code to produce $\mathbb{CP}^n$
  11. Explicit presentation of our simplicial cell complexes
  12. $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{CP}^2$
  13. $\mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{CP}^3$
  14. $\mathbb{CP}^4$

Key Result

Proposition 3.2

Let $G$ be a group of automorphisms of a simplicial complex $K$. If the action of $G$ on $K$ is good, then $|K/G| \cong |K|/G$.

Figures (1)

  • Figure 1: The dual graph of the subdivision of $\Delta^2 \times \Delta^2$ into $6$ simplices of dimension $4$. Notice that this subdivision is a graph encoding.

Theorems & Definitions (29)

  • Definition 3.1
  • Proposition 3.2: Corollary 2.7 in BD12
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • ...and 19 more