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A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

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Abstract

We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP2. The automorphism group of X is isomorphic to S 4 × S 3. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP2 admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of X and show that the automorphism group of X can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision \( \bar X \) of the triangulation X such that the classical moment mapping μ: ℂP2 → Δ2 is a simplicial mapping of the triangulation \( \bar X \) onto the barycentric subdivision of the triangle Δ2. We study the relationship of the triangulation X with complex crystallographic groups.

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Authors and Affiliations

  1. Moscow State University, Moscow, 119991, Russia

    A. A. Gaifullin

  2. Institute for Information Transmission Problems, Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, Moscow, 127994, Russia

    A. A. Gaifullin

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  1. A. A. Gaifullin
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Correspondence to A. A. Gaifullin.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 266, pp. 33–53.

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Gaifullin, A.A. A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices. Proc. Steklov Inst. Math. 266, 29–48 (2009). https://doi.org/10.1134/S008154380903002X

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  • DOI: https://doi.org/10.1134/S008154380903002X

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