Discrete & Computational Geometry

, Volume 46, Issue 3, pp 542–560

From the Icosahedron to Natural Triangulations of ℂP2 and S2×S2


DOI: 10.1007/s00454-010-9281-0

Cite this article as:
Bagchi, B. & Datta, B. Discrete Comput Geom (2011) 46: 542. doi:10.1007/s00454-010-9281-0


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.


Triangulated manifoldsComplex projective planeProduct of 2-spheresIcosahedron

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Theoretical Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia