The triangulations of the 3-sphere with up to 8 vertices
TL;DR: The different combinatorial types of triangulations of the 3-sphere with up to 8 vertices are determined and it is shown that one cannot always preassign the shape of a facet of a 4-polytope.
About: This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-01-01 and is currently open access. It has received 65 citations till now. The article focuses on the topics: Facet (geometry).
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09 Apr 2002
TL;DR: Polytopes topology and combinatorics of simplicial complexes Commutative and homological algebra of Cubical complexes Cubical Complexes Toric and quasitoric manifolds Moment-angle Complexes Cohomology of moment-angle complexes as discussed by the authors.
Abstract: Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.
547 citations
Book•
16 Dec 1996
TL;DR: In this article, the universality theorem and the applications of university polytopes are discussed, as well as alternative construction techniques for three-dimensional polytope construction and problems.
Abstract: The objects and the tools.- The universality theorem.- Applications of university.- Three-dimensional polytopes.- Alternative construction techniques.- Problems.
229 citations
Additional excerpts
...More recently it was proved for the 4dimensional case that cs(8, 4) = 37 (Grünbaum, Sreedharan [32]), ss(8, 4) = 39 (Barnette [9]), ss(9, 4) = 1296 (Altshuler, Steinberg [3]), cs(9, 4) = 1142 (Altshuler, Bokowski, Steinberg [5]), and c(8, 4) = 1294, s(8, 4) = 1336 (Altshuler, Steinberg [2, 4])....
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15 Jul 2015
TL;DR: Toric topology emerged in the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra, and continues to attract experts from different fields.
Abstract: Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area
194 citations
TL;DR: This report summarizes what is known about the d-step conjecture and its relatives and includes the first example of a polytope that is not vertex-decomposable, showing that a certain natural approach to the conjecture will not work.
Abstract: The d-step conjecture arose from an attempt to understand the computational complexity of edge-following algorithms for linear programming, such as the simplex algorithm. It can be stated in terms of diameters of graphs of convex polytopes, in terms of the existence of nonrevisiting paths in such graphs, in terms of an exchange process for simplicial bases of a vector space, and in terms of matrix pivot operations. First formulated by W. M. Hirsch in 1957, the conjecture remains unsettled, though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures. If the conjecture is false, as we believe to be the case, then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture. This report summarizes what is known about the d-step conjecture and its relatives. A considerable amount of new material is included, but it does not seem to come close to settling the conjecture. Of special interest is the first example of a polytope that is not vertex-decomposable, showing that a certain natural approach to the conjecture will not work. Also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture.
147 citations
TL;DR: A construction of Billera and Lee is extended to obtain a large family of triangulated spheres and it is proved that logs(n) =20.69424n(1+o(1)).
Abstract: Lets(d, n) be the number of triangulations withn labeled vertices ofSd?1, the (d?1)-dimensional sphere. We extend a construction of Billera and Lee to obtain a large family of triangulated spheres. Our construction shows that logs(d, n)?C1(d)n[(d?1)/2], while the known upper bound is logs(d, n)≤C2(d)n[d/2] logn.
Letc(d, n) be the number of combinatorial types of simpliciald-polytopes withn labeled vertices. (Clearly,c(d, n)≤s(d, n).) Goodman and Pollack have recently proved the upper bound: logc(d, n)≤d(d+1)n logn. Combining this upper bound forc(d, n) with our lower bounds fors(d, n), we obtain, for everyd?5, that limn??(c(d, n)/s(d, n))=0. The cased=4 is left open. (Steinitz's fundamental theorem asserts thats(3,n)=c(3,n), for everyn.) We also prove that, for everyb?4, limd??(c(d, d+b)/s(d, d+b))=0. (Mani proved thats(d, d+3)=c(d, d+3), for everyd.)
Lets(n) be the number of triangulated spheres withn labeled vertices. We prove that logs(n)=20.69424n(1+o(1)). The same asymptotic formula describes the number of triangulated manifolds withn labeled vertices.
106 citations
References
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TL;DR: An enumeration of all the different combinatorial types of 4-dimensional simplicial convex polytopes with 8 vertices is given in this paper, which corrects an earlier enumeration attempt by M. Bruckner, and leads to a simple example of a diagram which is not a Schlegel diagram.
146 citations
TL;DR: All n -dimensional spheres are realizable as boundary complexes of polytopes as well as simplicial n -sphere with few vertices if their number does not exceed n + 3.
78 citations
36 citations