Inflations of ideal triangulations
type: Item
instance of: scholarly article
title: Inflations of ideal triangulations
zbMATH Open document ID: 1304.57034
author: William Jaco
author: J. Hyam Rubinstein
published in: Advances in Mathematics
publication date: 2014-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.6921
reviewed by: Masakazu Teragaito
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 6358435
zbMATH Keywords: inflation
zbMATH Keywords: ideal triangulation
MaRDI profile type: Publication
OpenAlex ID: W2006007499
arXiv ID: 1302.6921
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DOI: 10.1016/J.AIM.2014.09.001
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review text: As summarized in the introduction of the paper persuasively, triangulations play a central role in the study and understanding of \(3\)-manifolds. In [\textit{W. Jaco} and \textit{J. H. Rubinstein}, J. Differ. Geom. 65, No. 1, 61--168 (2003; Zbl 1068.57023)], the authors introduced a technique for crushing a triangulation of a \(3\)-manifold along a normal surface. Later, this was extended to manifolds with boundary, crushing a triangulation along the boundary and arriving at a related ideal triangulation of the interior of the \(3\)-manifold. In this paper, a dual operation for ideal triangulations, called an inflation, is introduced. More precisely, starting from an ideal triangulation of the interior of a compact \(3\)-manifold with boundary, an inflation yields a strongly related triangulation of the compact \(3\)-manifold itself. And crushing along its boundary gives back the original ideal triangulation.NEWLINENEWLINEIn the last part of the paper, two concrete examples of inflations are exhibited. One is that of the two-tetrahedron ideal triangulation of the figure-eight knot complement, and the other one is that of the one-tetrahedron ideal triangulation of the Gieseking manifold, which is a non-orientable \(3\)-manifold doubly covered by the ideal triangulation of the figure-eight knot complement. Applications of inflations will be given in other papers by the same authors.
description: scientific article; zbMATH DE number 6358435