From the Icosahedron to Natural Triangulations of ℂP ^{2} and S ^{2}×S ^{2}
 Bhaskar Bagchi,
 Basudeb Datta
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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.
 Akhmedov, A., Park, B.D.: Exotic smooth structures on S ^{2}×S ^{2}, 12 pp. (2010). arXiv:1005.3346v5
 Arnoux, P., Marin, A. (1991) The Kühnel triangulation of the complex projective plane from the viewpoint of complex crystallography (part II). Mem. Fac. Sci. Kyushu Univ. Ser. A 45: pp. 167244 CrossRef
 Bagchi, B., Datta, B. (1994) On Kühnel’s 9vertex complex projective plane. Geom. Dedic. 50: pp. 113 CrossRef
 Bagchi, B., Datta, B. (2001) A short proof of the uniqueness of Kühnel’s 9vertex complex projective plane. Adv. Geom. 1: pp. 157163 CrossRef
 Bagchi, B., Datta, B. (2005) Combinatorial triangulations of homology spheres. Discrete Math. 305: pp. 117 CrossRef
 Bagchi, B., Datta, B. (2008) Lower bound theorem for normal pseudomanifolds. Exp. Math. 26: pp. 327351
 Banchoff, T.F., Kühnel, W. (1992) Equilibrium triangulations of the complex projective plane. Geom. Dedic. 44: pp. 413433 CrossRef
 Bing, R.H. Some aspects of the topology of 3manifolds related to the Poincaré conjecture. In: Saaty, T.L. eds. (1964) Lectures on Modern Mathematics. Wiley, New York, pp. 93128
 Brehm, U.: Private communication (2010)
 Brehm, U., Kühnel, W. (1987) Combinatorial manifolds with few vertices. Topology 26: pp. 465473 CrossRef
 Kühnel, W., Banchoff, T.F. (1983) The 9vertex complex projective plane. Math. Intell. 5: pp. 1122 CrossRef
 Kühnel, W., Laßmann, G. (1983) The unique 3neighbourly 4manifold with few vertices. J. Comb. Theory Ser. A 35: pp. 173184 CrossRef
 Lawson, T. (1982) Splitting S 4 on ℝP 2 via the branched cover of ℂP 2 over S 4. Proc. Am. Math. Soc. 86: pp. 328330
 Lutz, F.H. (1999) Triangulated Manifolds with Few Vertices and VertexTransitive Group Actions. Shaker Verlag, Aachen
 Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain threemanifolds, 7 pp. (2003). arXiv:math/0307245v1
 Saveliev, N. (1999) Lectures on the Topology of 3Manifolds: An Introduction to the Casson Invariant. Walter de Gruyter, Berlin
 Sparla, E. (1998) An upper and a lower bound theorem for combinatorial 4manifolds. Discrete Comput. Geom. 19: pp. 575593 CrossRef
 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