Simplicial complex

About: Simplicial complex is a research topic. Over the lifetime, 3416 publications have been published within this topic receiving 61839 citations. The topic is also known as: simplicial complexes.

Combinatorics and commutative algebra

Abstract: This text offers an overview of two of the main topics in the connections between commutative algebra and combinatorics. The first concerns the solutions of linear equations in non-negative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the upper bound conjecture for spheres. An introductory chapter giving background information in algebra, combinatorics and toplogy aims to broaden access to this material for non-specialists. This edition contains a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Also included is information on subcomplexes and subdivisions of simplicial complexes, and an application to spline theory.

Simplicial homotopy theory

Abstract: Simplicial sets.- Model Categories.- Classical results and constructions.- Bisimplicial sets.- Simplicial groups.- The homotopy theory of towers.- Reedy model categories.- Cosimplicial spaces: applications.- Simplicial functors and homotopy coherence.- Localization.

Simplicial objects in algebraic topology

Abstract: Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces. J. Peter May gives a lucid account of the basic homotopy theory of simplicial sets (discrete analogs of topological spaces) which have played a central role in algebraic topology ever since their introduction in the late 1940s. "Simplicial Objects in Algebraic Topology presents much of the elementary material of algebraic topology from the semi-simplicial viewpoint. It should prove very valuable to anyone wishing to learn semi-simplicial topology. [May] has included detailed proofs, and he has succeeded very well in the task of organizing a large body of previously scattered material."--Mathematical Review

Geometry of the complex of curves I: Hyperbolicity

Abstract: The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface. We also show that the action of pseudo-Anosov mapping classes on the complex is hyperbolic, with a uniform bound on translation distance.