Journal ArticleDOI
A census of tight triangulations
Wolfgang Kühnel,Frank H. Lutz +1 more
TLDR
A triangulation of a manifold (or pseudomanifold) is called tight if any simplex-wise linear embedding into any Euclidean space is tight as mentioned in this paper.Abstract:
A triangulation of a manifold (or pseudomanifold) is called a tight triangulation if any simplexwise linear embedding into any Euclidean space is tight. Tightness of an embedding means that the inclusion of any sublevel selected by a linear functional is injective in homology and, therefore, topologically essential. Tightness is a generalization of convexity, and the tightness of a triangulation is a fairly restrictive property. We give a review on all known examples of tight triangulations and formulate a (computer-aided) enumeration theorem for the case of at most 15 vertices and the presence of a vertex-transitive automorphism group. Altogether, six new examples of tight triangulations are presented, a vertex-transitive triangulation of the simply connected homogeneous 5-manifold SU(3)/SO(3) with vertex-transitive action, two non-symmetric 12-vertex triangulations of S
3 × S
2, and two non-symmetric triangulations of S
3 × S
3 on 13 vertices.read more
Citations
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Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere
Anders Björner,Frank H. Lutz +1 more
TL;DR: A computer program based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices is presented and it is shown that if a d-manifold, with d ≥ 5, admits any triangulation on n vertices, it admits a noncombinatorial triangu lation on n + 12 vertices.
Posted Content
Triangulated Manifolds with Few Vertices: Combinatorial Manifolds
TL;DR: A survey on combinatorial properties of triangulated manifolds can be found in this paper, where the authors discuss various lower bounds on the number of vertices of simplicial manifold.
Journal ArticleDOI
Stacked polytopes and tight triangulations of manifolds
TL;DR: In this paper, it was shown that in any dimension d>=4, tight-neighborly triangulations as defined by Lutz, Sulanke and Swartz are tight.
Journal ArticleDOI
Tight combinatorial manifolds and graded Betti numbers
TL;DR: In this paper, it was shown that Kuhnel and Lutz's conjecture holds only for simplicial polytopes with vertices whose vertex links are simplicial polygons, and that this conjecture holds also for triangulations of manifolds and pseudomanifolds.
Journal ArticleDOI
On stellated spheres and a tightness criterion for combinatorial manifolds
Bhaskar Bagchi,Basudeb Datta +1 more
TL;DR: It is shown that every tight member of W"1(d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuhnel and Lutz asserting that tight homology manifolds should be strongly minimal.
References
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Book
Map Color Theorem
TL;DR: In this article, the four color problem is considered and the thread problem is formulated as follows: 1.1. Chromatic number, 2.2.3.4.5.6.
Journal ArticleDOI
Simply Connected Five-Manifolds
TL;DR: In this article, the authors complete the classification of simply connected 5-manifolds with the vanishing second Stiefel-Whitney class and show that any such isomorphism may be realized by a diffeomorphism of X with M.
Journal ArticleDOI
The lower bound conjecture for 3- and 4-manifolds
TL;DR: For simplicial polytopes in dimension 4 and 5, Theorem 6.1 has been proved in this article, which is a strong affirmative resolution of the lower bound conjecture in dimension 3 and 4.
Journal ArticleDOI
Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere
Anders Björner,Frank H. Lutz +1 more
TL;DR: A computer program based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices is presented and it is shown that if a d-manifold, with d ≥ 5, admits any triangulation on n vertices, it admits a noncombinatorial triangu lation on n + 12 vertices.