From the Icosahedron to Natural Triangulations of ℂP ^{2} and S ^{2}×S ^{2}
 Bhaskar Bagchi,
 Basudeb Datta
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We present two constructions in this paper: (a) a 10vertex 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 12vertex 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 twofold 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.
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 Title
 From the Icosahedron to Natural Triangulations of ℂP ^{2} and S ^{2}×S ^{2}
 Journal

Discrete & Computational Geometry
Volume 46, Issue 3 , pp 542560
 Cover Date
 20111001
 DOI
 10.1007/s0045401092810
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Triangulated manifolds
 Complex projective plane
 Product of 2spheres
 Icosahedron
 Industry Sectors
 Authors

 Bhaskar Bagchi ^{(1)}
 Basudeb Datta ^{(2)}
 Author Affiliations

 1. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, 560 059, India
 2. Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India