Discrete & Computational Geometry

, Volume 46, Issue 3, pp 542-560

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

  • Bhaskar BagchiAffiliated withTheoretical Statistics and Mathematics Unit, Indian Statistical Institute
  • , Basudeb DattaAffiliated withDepartment of Mathematics, Indian Institute of Science Email author 

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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.


Triangulated manifolds Complex projective plane Product of 2-spheres Icosahedron