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

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 $n \geq 4$.