Families of Legendrian submanifolds via generating families
type: Item
instance of: scholarly article
title: Families of Legendrian submanifolds via generating families
zbMATH Open document ID: 1358.57029
DOI: 10.4171/QT/82
author: Joshua M. Sabloff
author: Michael G. Sullivan
published in: Quantum Topology
publication date: 2017-02-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.0528
review text: Let \(J^1M\) be the \((2n+1)\)-dimensional \(1\)-jet space of an \(n\)-dimensional smooth manifold \(M\) that is closed or diffeomorphic to \(\mathbb R^n\) outside of a compact set. The \(1\)-jet space is equipped with the standard contact structure \(\xi\). If \(\Lambda\subset J^1M\) is an \(n\)-dimensional compact Legendrian submanifold, then the topology of the space \(\mathcal{L}\) of all such submanifolds is formed by taking the quotient of the function space of Legendrian embeddings by orientation-preserving self-diffeomorphisms of the domain. Since the \(1\)-jet of a function \(g:M\to\mathbb R\) is a Legendrian submanifold of \(J^1M\), a function \(f:M\times\mathbb R^N\to\mathbb R\) is a generating family if \(0\) is a regular value of the function \(\partial_\eta f:M\times\mathbb R^N\to\mathbb R^N\). If \(\Sigma_f=\{(x,\eta)\in M\times\mathbb R^N:\partial_\eta f(x,\eta)=0\}\) is the fiber critical set, then a generating family \(f\) defines a Legendrian submanifold \(\Lambda_f=\{(x,\partial_\eta f(x,\eta),f(x,\eta)):(x,\eta)\in\Sigma_f\}\) as the \(1\)-jet of \(f\) along \(\Sigma_f\). If \(\delta:M\times\mathbb R^N\times\mathbb R^N\to\mathbb R\),\ \((x,\eta,\widetilde\eta)\mapsto f(x,\widetilde\eta)-f(x,\eta)\) is the difference function on the fiber product of the domain of \(f\) with itself, then, since the critical points of \(\delta\) with positive critical values correspond to the Reeb chords of \(\Lambda_f\), the generating family homology with coefficients in \(\mathbb Z/2\) is defined as \(\text{GH}_k(f)=H_{N+1+k}(\delta^\omega,\delta^\varepsilon;\mathbb Z/2)\), where \(\omega\) is a number larger than any critical value of \(\delta\) and where there are no critical values of \(\delta\) in \((0,\varepsilon)\). In this paper, the authors investigate families of Legendrian submanifolds in \(1\)-jet spaces by applying the theory of families of generating family homologies. They show that if the Legendrian \(\Lambda\subset J^1M\) has a generating family …
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 6684921
zbMATH Keywords: \(1\)-jet space
zbMATH Keywords: Legendrian submanifold
zbMATH Keywords: generating family
zbMATH Keywords: generating family homology
MaRDI profile type: Publication
arXiv ID: 1311.0528
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OpenAlex ID: W2963598498
description: scientific article