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We present two constructions in this paper: (a) a 10-vertex triangulation $\mathbb{C}P^{2}_{10}$ of the complex projective plane ℂP2 as a subcomplex of the join of the standard sphere ($S^{2}_{4}$) and the standard real projective plane ($\mathbb{R}P^{2}_{6}$, the decahedron), its automorphism group is A4; (b) a 12-vertex triangulation (S2×S2)12 of S2×S2 with automorphism group 2S5, the Schur double cover of the symmetric group S5. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S2×S2. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ℂP2 has S2×S2 as a two-fold branched cover; we construct the triangulation $\mathbb{C}P^{2}_{10}$ of ℂP2 by presenting a simplicial realization of this covering map S2×S2→ℂP2. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S2×S2, different from the triangulation alluded to in (b). This gives a new proof that Kühnel’s $\mathbb{C}P^{2}_{9}$ triangulates ℂP2. It is also shown that $\mathbb{C}P^{2}_{10}$ and (S2×S2)12 induce the standard piecewise linear structure on ℂP2 and S2×S2 respectively.

From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2