A Triangulation of ℂP 3 as Symmetric Cube of S 2

Abstract

The symmetric group $\operatorname{Sym}(d)$ acts on the Cartesian product (S 2)d by coordinate permutation, and the quotient space $(S^{2})^{d}/\operatorname{Sym}(d)$ is homeomorphic to the complex projective space ℂP d . We used the case d=2 of this fact to construct a 10-vertex triangulation of ℂP 2 earlier. In this paper, we have constructed a 124-vertex simplicial subdivision $(S^{2})^{3}_{124}$ of the 64-vertex standard cellulation $(S^{2}_{4})^{3}$ of (S 2)3, such that the $\operatorname{Sym}(3)$ -action on this cellulation naturally extends to an action on $(S^{2})^{3}_{124}$. Further, the $\operatorname{Sym}(3)$ -action on $(S^{2})^{3}_{124}$ is “good”, so that the quotient simplicial complex $(S^{2})^{3}_{124}/\operatorname{Sym}(3)$ is a 30-vertex triangulation $\mathbb{C}P^{3}_{30}$ of ℂP 3. In other words, we have constructed a simplicial realization $(S^{2})^{3}_{124} \to\mathbb{C} P^{3}_{30}$ of the branched covering (S 2)3→ℂP 3.