A minimal triangulation of the quaternionic projective plane (original) (raw)
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A 15-Vertex Triangulation of the Quaternionic Projective Plane
Discrete & Computational Geometry, 2019
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 HP 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 HP 2. 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). e-mail: denis.gorod at mi.ras.ru
Equilibrium triangulations of the complex projective plane
Geometriae Dedicata, 1992
Starting with the well-known 7-vertex triangulation of the ordinary torus, we construct a 10-vertex triangulation of CP 2 which fits the equilibrium decomposition of CP 2 in the simplest possible way. By suitable positioning of the vertices, the full automorphism group of order 42 is realized by a discrete group of isometries in the Fubini-Study metric. A slight subdivision leads to an elementary proof of the theorem of Kuiper-Massey which says that CP 2 modulo conjugation is PL homeomorphic to the standard 4-sphere. The branch locus of this identification is a 7-vertex triangulation RP72 of the real projective plane. We also determine all tight simplicial embeddings of Cp2o and RP 2.
On K�hnel's 9-vertex complex projective plane
Geometriae Dedicata, 1994
We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C p2. The original proof of existence, due to Ktthnel, as well as the original proof of uniqueness, due to Ktthnel and Lassmann, were based on extensive computer search. Recently Arnoux and Matin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the K0hnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.
From the icosahedron to natural triangulations
2016
We present two constructions in this paper : (a) A 10-vertex triangulation CP 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 (RP 2 6 , the decahedron), its automorphism group is A 4 ; (b) a 12-vertex 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 CP 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 CP 2 9 triangulates CP 2. It is also shown that CP 2 10 and (S 2 × S 2) 12 induce the standard piecewise linear structure on CP 2 and S 2 × S 2 respectively.
On Kühnel's 9-vertex complex projective plane
Geometriae Dedicata, 1994
We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of ℂP2. The original proof of existence, due to Kühnel, as well as the original proof of uniqueness, due to Kühnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kühnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.
From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2
Discrete & Computational Geometry, 2010
We present two constructions in this paper: (a) a 10-vertex triangulation mathbbCP210\mathbb{C}P^{2}_{10}mathbbCP210 of the complex projective plane ℂP 2 as a subcomplex of the join of the standard sphere ( S24S^{2}_{4}S24 ) and the standard real projective plane ( mathbbRP26\mathbb{R}P^{2}_{6}mathbbRP26 , the decahedron), its automorphism group is A 4; (b) a 12-vertex 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 two-fold branched cover; we construct the triangulation mathbbCP210\mathbb{C}P^{2}_{10}mathbbCP210 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 mathbbCP29\mathbb{C}P^{2}_{9}mathbbCP29 triangulates ℂP 2. It is also shown that mathbbCP210\mathbb{C}P^{2}_{10}mathbbCP210 and (S 2×S 2)12 induce the standard piecewise linear structure on ℂP 2 and S 2×S 2 respectively.
Combinatorial triangulations of homology spheres
Let M be an n-vertex combinatorial triangulation of a Z 2 -homology d-sphere. In this paper we prove that if n ≤ d + 8 then M must be a combinatorial sphere. Further, if n = d + 9 and M is not a combinatorial sphere then M can not admit any proper bistellar move. Existence of a 12-vertex triangulation of the lens space L(3, 1) shows that the first result is sharp in dimension three.
Complexity of triangulations of the projective space
Arxiv preprint math.GT/0205326
It is known that any two triangulations of a compact 3-manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3-dimensional projective space, in terms of the number of tetrahedra.
arXiv (Cornell University), 2010
We present two constructions in this paper : (a) A 10-vertex triangulation CP 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 (RP 2 6 , the decahedron), its automorphism group is A 4 ; (b) a 12-vertex 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 CP 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 CP 2 9 triangulates CP 2. It is also shown that CP 2 10 and (S 2 × S 2) 12 induce the standard piecewise linear structure on CP 2 and S 2 × S 2 respectively.
Minimal triangulations of sphere bundles over the circle
For integers d ≥ 2 and ε = 0 or 1, let S 1,d−1 (ε) denote the sphere product S 1 × S d−1 if ε = 0 and the twisted sphere product S 1 × − S d−1 if ε = 1. The main results of this paper are : (a) if d ≡ ε (mod 2) then S 1,d−1 (ε) has a unique minimal triangulation using 2d + 3 vertices, and (b) if d ≡ 1 − ε (mod 2) then S 1,d−1 (ε) has minimal triangulations (not unique) using 2d + 4 vertices. In this context, a minimal triangulation of a manifold is a triangulation using the least possible number of vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while S 1,d−1 (ε) has at most one (2d + 3)-vertex triangulation (one if d ≡ ε (mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic (2d + 4)-vertex triangulations of these d-manifolds grows exponentially with d for either choice of ε. The result in (a), as well as the minimality part in (b), is a consequence of the following result : (c) for d ≥ 3, there is a unique (2d+3)-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension d. This amazing simplicial complex was first constructed by Kühnel in 1986. Generalizing a 1987 result of Brehm and Kühnel, we prove that (d) any triangulation of a non-simply connected closed d-manifold requires at least 2d + 3 vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality. : 57Q15, 57R05.