Handle attaching in symplectic homology and the chord conjecture
type: Item
instance of: scholarly article
title: Handle attaching in symplectic homology and the chord conjecture
zbMATH Open document ID: 1012.53066
author: K. Cieliebak
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2002-09-10T00:00:00Z
review text: In his 1986 paper [Russ. Math. Surv. 41, No. 6, 1-21 (1986; Zbl 0649.58010)], \textit{V. I. Arnold} conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord, i. e. an orbit of the Reeb vector field intersecting the knot at least twice. The author of this paper formulates the same question in the following more general situation: Let \(N^{2n-1}\) be a closed oriented manifold of dimension \(2n-1\). A contact form on \(N\) is a \(1\)-form \(\lambda\) such that \(\lambda\wedge (d\lambda)^{n-1}\) is a positive volume form. The kernel \(\xi = \text{ker }\lambda\) of a contact form is called a (cooriented) contact structure, and the pair \((N,\xi)\) a contact manifold. A submanifold \(L\subset (N,\xi)\) is called isotropic if it is tangent to \(\xi\). An isotropic submanifold of the maximal possible dimension \(n-1\) is called Legendrian. A contact form determines a unique vector field \(Y_\lambda\), called the Reeb vector field, by the conditions \(i_{Y_\lambda}dX = 0\) and \(\lambda(Y_\lambda) = 1\). A characteristic chord for a contact form \(\lambda\) and a Legendrian submanifold is an orbit of the Reeb vector field intersecting \(L\) at least twice. Then the author says that the (generalised) chord conjecture holds for a triple \((N,\lambda,L)\) if \(L\) possesses a characteristic chord for \(\lambda\), and that it holds for a triple \((N,\xi,L)\) if it holds for every triple \((N,\lambda,L)\) with \(\xi = \text{ker} \lambda\). The author observes that since Legendrian isotopies extend to ambient contact isotopies, if the chord conjecture holds for \((N,\xi,L)\) then it holds for \((N,\xi,\widetilde L)\) with \(\widetilde L\) isotopic to \(L\) (through Legendrian embeddings). He also notices that the existence of a chord for a triple \((N,\lambda,L)\) can be checked directly from the Reeb flow of \(\lambda\) but the question becomes nontrivial if we fix the contact structure and allow the contact form a…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 58C40
zbMATH DE Number: 1798901
zbMATH Keywords: contact form
zbMATH Keywords: Legendrian submanifold
zbMATH Keywords: Reeb flow
zbMATH Keywords: Legendrian knot
MaRDI profile type: Publication
full work available at URL: s100970100036
OpenAlex ID: W2027438934
Wikidata QID: Q123337356
DOI: 10.1007/S100970100036
description: scientific article; zbMATH DE number 1798901