From the Icosahedron to Natural Triangulations of CP(2) and S(2) x S(2)

Bagchi, Bhaskar and Datta, Basudeb (2011) From the Icosahedron to Natural Triangulations of CP(2) and S(2) x S(2). In: Discrete & Computational Geometry, 46 (3). pp. 542-560.

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Abstract

We present two constructions in this paper: (a) a 10-vertex triangulation CP(10)(2) of the complex projective plane CP(2) as a subcomplex of the join of the standard sphere (S(4)(2)) and the standard real projective plane (RP(6)(2), the decahedron), its automorphism group is A(4); (b) a 12-vertex triangulation (S(2) x S(2))(12) of S(2) x 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) x S(2). Both constructions have surprising and intimate relationships with the icosahedron. It is well known that CP(2) has S(2) x S(2) as a two-fold branched cover; we construct the triangulation CP(10)(2) of CP(2) by presenting a simplicial realization of this covering map S(2) x S(2) -> CP(2). The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S(2) x S(2), different from the triangulation alluded to in (b). This gives a new proof that Kuhnel's CP(9)(2) triangulates CP(2). It is also shown that CP(10)(2) and (S(2) x S(2))(12) induce the standard piecewise linear structure on CP(2) and S(2) x S(2) respectively.

Item Type:	Journal Article
Publication:	Discrete & Computational Geometry
Publisher:	Springer
Additional Information:	Copyright of this article belongs to Springer.
Keywords:	Triangulated manifolds;Complex projective plane;Product of 2-spheres;Icosahedron
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	16 Sep 2011 07:05
Last Modified:	16 Sep 2011 07:05
URI:	http://eprints.iisc.ac.in/id/eprint/40430

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