Projections of polytopes and the generalized Baues conjecture
type: Item
instance of: scholarly article
title: Projections of polytopes and the generalized Baues conjecture
zbMATH Open document ID: 0865.52008
DOI: 10.1007/BF02711510
published in: Discrete \& Computational Geometry
publication date: 1997-06-30T00:00:00Z
review text: With a projection \(\pi:P \to\pi (P)\) of polytopes a partially ordered set of all ``locally coherent strings'' is associated: Roughly speaking it consists of the families of proper faces of \(P\) that project to valid subdivisions of \(\pi(P)\), partially ordered by inclusion. Inspired by work of Baues, Billera stated the following generalized Baues conjecture: Conjecture: For every projection \(\pi:P \to\pi(P)\) of a \(d\)-polytope \(P\subset \mathbb{R}^d\) to a \(d'\)-polytope \(\pi (P) \subset \mathbb{R}^{d'}\), the poset \(\omega (P,\pi)\) of all locally \(\pi\)-coherent strings is homotopy equivalent to the \((d-d'-1)\)-sphere. Known results where the generalized Baues conjecture holds are: \(\bullet\) \(d'\leq 1\); \(\bullet\) \(P\) is a simplex and \(\dim (\pi(P)) \leq 2\); \(\bullet\) \(P\) is a hypercube and \(\dim (\pi(P)) \leq 2\) or \(\dim (P)- \dim (\pi(P)) \leq 3\); The authors establish: Theorem: The generalized Baues conjecture holds for \(d-d'\leq 2\) and is false in general for \(d'\geq 2\) and \(d-d'\geq 3\). As a counterexample they produce a nondegenerate projection of a 5-dimensional, simplicial, 2-neighborly polytope \(P\) with 10 vertices and 42 facets to a hexagon \(\pi(P) \subset \mathbb{R}^2\).
reviewed by: Charles Leytem
Mathematics Subject Classification ID: 52B05
zbMATH DE Number: 949785
zbMATH Keywords: projection of a polytope
zbMATH Keywords: locally coherent string
zbMATH Keywords: fiber polytope
zbMATH Keywords: Baues conjecture
Wikidata QID: Q123139072
author: Jörg Rambau
author: Günter M. Ziegler
describes a project that uses: PORTA
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 949785