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

Discrete & Computational Geometry, Aug 2010

We present two constructions in this paper: (a) a 10-vertex triangulation $$\mathbb{C}P^{2}_{10}$$ of the complex projective plane ℂP 2 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 A 4; (b) a 12-vertex triangulation (S 2×S 2)12 of S 2×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×S 2. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ℂP 2 has S 2×S 2 as a two-fold branched cover; we construct the triangulation $$\mathbb{C}P^{2}_{10}$$ of ℂP 2 by presenting a simplicial realization of this covering map S 2×S 2→ℂP 2. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S 2×S 2, different from the triangulation alluded to in (b). This gives a new proof that Kühnel’s $$\mathbb{C}P^{2}_{9}$$ triangulates ℂP 2. It is also shown that $$\mathbb{C}P^{2}_{10}$$ and (S 2×S 2)12 induce the standard piecewise linear structure on ℂP 2 and S 2×S 2 respectively.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-010-9281-0.pdf

Bhaskar Bagchi, Basudeb Datta. From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2, Discrete & Computational Geometry, 2010, 542-560, DOI: 10.1007/s00454-010-9281-0