Unshellable triangulations of spheres
type: Item
instance of: scholarly article
title: Unshellable triangulations of spheres
zbMATH Open document ID: 0746.57007
DOI: 10.1016/S0195-6698(13)80103-5
published in: European Journal of Combinatorics
publication date: 1992-06-27T00:00:00Z
review text: To shell a triangulated \(n\)-disk or \(n\)-sphere means to remove its \(n\)-simplices one by one so that after each step (except the last) a triangulated \(n\)-disk remains. It is easy to see that for \(n\leq 2\) every triangulation of the \(n\)-disk or the \(n\)-sphere can be shelled, and it has been generally known that for \(n\geq 3\) unshellable triangulations exist. But whereas explicit constructions of unshellable triangulations of disks have been available in the literature [\textit{M.-E. Rudin}, Bull. Am. Math. Soc. 64, 90--91 (1958; Zbl 0082.37602); \textit{R. H. Bing}, Lect. on Modern Math. 2, 93--128 (1964; Zbl 0126.39104); \textit{R. E. Goodrick}, Proc. Camb. Philos. Soc. 64, 31--36 (1968; Zbl 0179.52602)], those of spheres have apparently not been. In the present article the author proves that for each \(n\geq 3\) there exists a combinatorial triangulation of the \(n\)-sphere such that after removing any \(n\)-simplex from it we are left with a noncollapsible (and a fortiori unshellable) triangulation of the \(n\)-disk.
reviewed by: Joze Vrabec
Mathematics Subject Classification ID: 57Q15
zbMATH DE Number: 29893
zbMATH Keywords: simplicial shelling
zbMATH Keywords: noncollapsible triangulation of the \(n\)-disk
zbMATH Keywords: combinatorial triangulation of the \(n\)-sphere
MaRDI profile type: Publication
cites work: Q5331776
cites work: Q5570655
cites work: Q5584509
cites work: Piecewise Linear Critical Levels and Collapsing
cites work: Triangulations of the 3-Ball With Knotted Spanning 1-Simplexes and Collapsible r-th Derived Subdivisions
cites work: A non-shellable 3-sphere
full work available at URL: s0195-6698(13)80103-5
OpenAlex ID: W2057249490
Recommended article: Triangulations and quadrangulations of the sphere
Recommended article: Q4883301
Recommended article: Triangulations of the product of spheres with few vertices
Recommended article: Combinatorial triangulations of homology spheres
Recommended article: On nontriangulable polyhedra
Recommended article: Irreducible triangulations of surfaces
Recommended article: Q3981366
Recommended article: Simplicial shellable spheres via combinatorial blowups
Recommended article: Ambiguous incidences of unbounded polyhedra
Recommended article: Triangulations of the sphere, bitrades and abelian groups
author: W. B. Raymond Lickorish
description: scientific article; zbMATH DE number 29893