A 15-vertex triangulation of the quaternionic projective plane
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arXiv
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| Format: | Preprint |
| Published: |
2016
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| _version_ | 1866910359114219520 |
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| author | Gorodkov, Denis |
| author_facet | Gorodkov, Denis |
| contents | In 1992, Brehm and Kühnel constructed a 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 $\mathbb{H}P^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1603_05541 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | A 15-vertex triangulation of the quaternionic projective plane Gorodkov, Denis Algebraic Topology Combinatorics Geometric Topology In 1992, Brehm and Kühnel constructed a 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 $\mathbb{H}P^2$. |
| title | A 15-vertex triangulation of the quaternionic projective plane |
| topic | Algebraic Topology Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/1603.05541 |