Loose Legendrians and the plastikstufe
type: Item
instance of: scholarly article
title: Loose Legendrians and the plastikstufe
zbMATH Open document ID: 1301.57020
author: Emmy Murphy
author: Klaus Niederkrüger
author: Olga Plamenevskaya
author: András I. Stipsicz
published in: Geometry \& Topology
publication date: 2013-07-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.3895
reviewed by: Guangcun Lu
Mathematics Subject Classification ID: 57R17
zbMATH DE Number: 6185366
zbMATH Keywords: contact manifolds
zbMATH Keywords: loose Legendrian knots
zbMATH Keywords: plastikstufe
zbMATH Keywords: overtwisted contact manifolds
MaRDI profile type: Publication
OpenAlex ID: W2057636558
arXiv ID: 1211.3895
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DOI: 10.2140/GT.2013.17.1791
review text: The notion of overtwisted contact structures on three-dimensional manifolds introduced by \textit{Y. Eliashberg} [Invent. Math. 98, No. 3, 623--637 (1989; Zbl 0684.57012)] played an important role in the study of contact topology. A contact structure \(\xi\) on a \(3\)-dimensional manifold \(M\) is called \textit{overtwisted} if it contains an \textit{overtwisted disc}, which is an embedded closed disc \(D\subset M\) whose boundary \(\partial D\) is not only Legendrian for \(\xi\) but also transverse to \(\xi\) along \(\partial D\). Such contact structures on closed \(3\)-manifolds exhibit very nice flexibility: their classification up to isotopy coincides with their homotopy classification as plane fields. In other words they satisfy an \(h\)-principle. \textit{K. Dymara} [Ann. Global Anal. Geom. 19, No. 3, 293--305 (2001; Zbl 0985.57009)] showed that any Legendrian knot in the complement of an overtwisted disk also satisfies an \(h\)-principle.NEWLINENEWLINEThe paper under review gives a higher-dimensional generalization for the last claim. Recall that as a generalization of the overtwisted disk to higher dimensions \textit{K. Niederkrüger} [Algebr. Geom. Topol. 6, 2473--2508 (2006; Zbl 1129.53056)] introduced the notion `plastikstufe'. Let \(B\) be a closed \((n-1)\)-manifold. A \textit{plastikstufe \({\mathcal P}_B\) with core \(B\)} in a \((2n+1)\)-dimensional contact manifold \((M, \xi)\) is a submanifold \({\mathcal P}_B=D^2\times B\hookrightarrow M\) such that \(\xi\cap T{\mathcal P}_B\) is a singular foliation that is tangent to the fibers \(\{z\}\times B\) for every \(z\in D^2\), and that restricts on every slice \(D^2\times\{b\}\) to the foliation of the overtwisted disk sketched. A plastikstufe \({\mathcal P}_B\) is called a \textit{small plastikstufe} if there is an embedded open ball in \((M, \xi)\) containing \({\mathcal P}_B\). A contact manifold that contains a plastikstufe is called \textit{PS-overtwisted}. A \textit{leaf ribbon} in a plastikstufe …
description: scientific article; zbMATH DE number 6185366