# From the Icosahedron to Natural Triangulations of ℂ*P* ^{2} and *S* ^{2}×*S* ^{2}

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## Abstract

We present two constructions in this paper: (a) a 10-vertex triangulation \(\mathbb{C}P^{2}_{10}\) of the complex projective plane ℂ*P* ^{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 2*S* _{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 ℂ*P* ^{2} has *S* ^{2}×*S* ^{2} as a two-fold branched cover; we construct the triangulation \(\mathbb{C}P^{2}_{10}\) of ℂ*P* ^{2} by presenting a simplicial realization of this covering map *S* ^{2}×*S* ^{2}→ℂ*P* ^{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 ℂ*P* ^{2}. It is also shown that \(\mathbb{C}P^{2}_{10}\) and (*S* ^{2}×*S* ^{2})_{12} induce the standard piecewise linear structure on ℂ*P* ^{2} and *S* ^{2}×*S* ^{2} respectively.

## Keywords

Triangulated manifolds Complex projective plane Product of 2-spheres Icosahedron## References

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