WiSe 19/20: Seminar zur Diskreten Geometrie
Rainer Sinn
Zusätzl. Angaben / Voraussetzungen
Das Seminar findet zeitlich in Absprache mit den Teilnehmern statt.
Kommentar
Shellability is a fundamental property of polytopes and of some simplicial complexes that allows to construct these combinatorial objects by tidily gluing together their maximal faces.
Bruggesser and Mani (1970) proved that the boundary complex of every simplicial polytope is shellable. This fact was then used to prove the Euler-Poincare´ formula, which says that the alternating sum of the number f_i of i-dimensional faces of a convex polytope is zero. This is an extension of the well- known Euler formula for 3-polytopes: v-e+f=2, where v, e and f are the number of vertices, edges and maximal faces of the polytope.
From the f-vector, whose components are the numbers f_i, one can compute another vector, called h-vector, whose components count the number of certain facets in a shellable simplicial complex. The components of the h-vector of the boundary of a simplicial polytope are symmetric (they satisfy the Dehn-Sommerville equations).
Another crucial application of shellability is the Upper Bound Theorem, proved by McMullen in 1970, which states that the maximal number of k-faces for a d-polytope with n vertices is attained by the cyclic polytope.
In the setting of simplicial complexes, shellability is a strong property in the hierarchy of pure complexes: vertex-decomposable => shellable => constructible => Cohen-Macaulay, and each of these classes of simplicial complexes is interesting by its own.
Reisner (1976) gave a beautiful topological characterization of Cohen-Macaulay complexes in terms of the homology of the links of their faces. This was the starting point of the Stanley-Reisner theory that connects simplicial complexes to monomial ideals.
Shellability, unlike Cohen-Macaulayness, is independent of the field. However, there are interesting examples of complexes that are Cohen-Macaulay over every field, but are not shellable.
Moreover, even if shellability is a combinatorial property, it depends on the triangulation of the complex: there exist, for example, non-shellable triangulations of the 3-ball and of the 3-sphere.
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