From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$

TL;DR

This paper introduces minimal triangulations of complex projective plane and S^2×S^2 with specific automorphism groups, revealing their relationships with the icosahedron and providing new proofs of known structures.

Contribution

It constructs explicit minimal triangulations of CP^2 and S^2×S^2, demonstrating their connections to the icosahedron and offering new proofs of existing triangulations.

Findings

Triangulation of CP^2 with 10 vertices and automorphism group A_4.

Triangulation of S^2×S^2 with 12 vertices and automorphism group 2S_5.

New simplicial realization of the branched cover from S^2×S^2 to CP^2.

Abstract

We present two constructions in this paper: (a) A 10-vertex triangulation $\CC P^{2}_{10}$ of the complex projective plane $\CC P^{2}$ as a subcomplex of the join of the standard sphere ($S^{2}_4$) and the standard real projective plane ($\RR P^{2}_{6}$, the decahedron), its automorphism group is $A_4$; (b) a 12-vertex triangulation $(S^{2} \times S^{2})_{12}$ of $S^{2} \times S^{2}$ with automorphism group $2S_5$, the Schur double cover of the symmetric group $S_5$. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of $S^{2} \times S^{2}$. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that $\CC P^{2}$ has $S^{2} \times S^{2}$ as a two-fold branched cover; we construct the triangulation $\CC P^{2}_{10}$ of $\CC P^{2}$ by presenting a simplicial realization of this covering map $S^{2} \times S^{2} \to \CC P^{2}$. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of $S^{2} \times S^{2}$, different from the triangulation alluded to in (b). This gives a new proof that Kuehnel's $\CC P^{2}_{9}$ triangulates $\CC P^{2}$. It is also shown that $\CC P^{2}_{10}$ and $(S^{2} \times S^{2})_{12}$ induce the standard piecewise linear structure on $\CC P^{2}$ and $S^{2} \times S^{2}$ respectively.