Visibility complexes and the Baues problem for triangulations in the plane
type: Item
instance of: scholarly article
title: Visibility complexes and the Baues problem for triangulations in the plane
zbMATH Open document ID: 0927.52021
DOI: 10.1007/PL00009377
author: Paul Edelman
author: Victor Reiner
published in: Discrete \& Computational Geometry
publication date: 1999-12-06T00:00:00Z
review text: Let \(S\) be finite set of points in \(d\)-dimensional space. The polytopal subdivisions of \(S\) form a partial order whose maximal chains yield a simplicial complex. The generalized Baues conjecture by \textit{L. J. Billera, M. M. Kapranov} and \textit{B. Sturmfels} [Proc. Am. Math. Soc. 122, No.~2, 549-555 (1994; Zbl 0812.52007)] states that this simplicial complex has the homotopy type of a sphere. The conjecture was proven for \(d=1\), but disproven in general by \textit{J. Rambau} and \textit{G. M. Ziegler} [Discrete Comput. Geom. 16, No.~3, 215-237 (1996; Zbl 0865.52008)], who gave counterexamples for \(d=5\). In this paper, the authors consider the most natural and important case of polytopal subdivisions, i.e., triangulations of a set of points in general position in \(d=2\). They prove that for this case, the generalized Baues conjecture does hold. The proof is quite intricate and uses deletion-contraction techniques, as well as a newly-defined visibility complex for a simplicial complex.
Mathematics Subject Classification ID: 52B99
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 52C99
zbMATH DE Number: 1215507
zbMATH Keywords: planar subdivison
zbMATH Keywords: homotopy type
zbMATH Keywords: spheres
zbMATH Keywords: generalized Baues problem
zbMATH Keywords: complex of a triangulation
zbMATH Keywords: visibility complex
reviewed by: Sándor P. Fekete
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1215507