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A 15-Vertex Triangulation of the Quaternionic Projective Plane

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Abstract

In 1992, Brehm and Kühnel constructed an 8-dimensional simplicial complex \(M^8_{15}\) with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold “like a projective plane” in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to \({\mathbb {H}}P^2\). This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of \(M^8_{15}\). As a result, we obtain that it is indeed a minimal triangulation of \({\mathbb {H}}P^2\).

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Acknowledgements

The author would like to thank his advisor Alexander A. Gaifullin for suggesting this interesting problem, for invaluable discussions, constant attention to this work and patience.

Funding

Funding was provided by the Russian Science Foundation (Grant No. 14-50-00005).

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Authors and Affiliations

  1. Steklov Mathematical Institute of RAS, Moscow, Russian Federation

    Denis Gorodkov

  2. Skolkovo Institute of Science and Technology, Moscow, Russian Federation

    Denis Gorodkov

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  1. Denis Gorodkov
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Correspondence to Denis Gorodkov.

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This work has been supported in part by the Moebius Contest Foundation for Young Scientists and by the Russian Science Foundation (Project 14-50-00005).

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Gorodkov, D. A 15-Vertex Triangulation of the Quaternionic Projective Plane. Discrete Comput Geom 62, 348–373 (2019). https://doi.org/10.1007/s00454-018-00055-w

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