Morse flow trees and Legendrian contact homology in 1-jet spaces
type: Item
instance of: scholarly article
title: Morse flow trees and Legendrian contact homology in 1-jet spaces
zbMATH Open document ID: 1162.53064
author: Tobias Ekholm
published in: Geometry \& Topology
publication date: 2007-12-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0509386
review text: Let \(M\) be a Riemannian \(n\)-manifold and let \(J^1(M):=T^*M \times\mathbb{R}\) be its 1-jet space. \(J^1(M)\) is endowed with the standard contact structure \(\xi\) defined as the kernel of the standard 1-form on \(T^*M\times\mathbb{R}\). An \(n\)-dimensional submanifold \(L\) of \(J^1(M)\) is called ``Legendrian'' if it is everywhere tangent to \(\xi\). In the paper under review, the author studies Legendrian submanifolds \(L\subset J^1 (M)\) with a view toward their Legendrian contact homology. In symplectic field theory [cf.: \textit{Y. Eliashberg, A. Givental} and \textit{H. Hofer}, in: Geom. Funct. Analysis, Basel: Birkhäuser, 560--673 (2000; Zbl 0989.81114)], Legendrian contact homology is a useful framework for finding isotopy invariants of Legendrian submanifolds of contact manifolds by appropriately counting rigid (pseudo-)holomorphic disks, where the analytical foundations of this toolkit were established in a series of foregoing papers by \textit{T. Ekholm, J. Etnyre} and \textit{M. Sullivan}, Trans. Am. Math. Soc. 359, No. 7, 3301--3335 (2007; Zbl 1119.53051)]. In the present paper, these investigations are continued by constructing an explicit correspondence between certain rigid flow trees in \(M\) associated to the Legendrian submanifold \(L\subset J^1(M)\) and boundary punctured pseudo-holomorphic disks in the cotangent bundle \(T^*M\) of particular boundary and asymptotic behaviour with respect to the projection of \(L\). This result makes it possible to reduce the computation of Legendrian contact homology in \(J^1(M)\) to a finite-dimensional problem in Morse theory, which in turn has powerful applications to establishing new invariants of knots in 3-space [cf.: \textit{L. Ng}, Geom. Topol. 9, 247--297, 1603--1637 (2005; Zbl 1111.57011 and Zbl 1112.57001)]. As for the basic objects linked by the author's main theorem, a flow tree in \(M\) determined by the Legendrian submanifold \(L\subset J^1(M)\) is a continuous map of a combinatorial tree \…
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 5220937
zbMATH Keywords: symplectic and contact homology
zbMATH Keywords: Legrendrian submanifolds
zbMATH Keywords: jet spaces
zbMATH Keywords: Legendrian contact homology
zbMATH Keywords: Floer homology symplectic aspects
zbMATH Keywords: Morse theory
zbMATH Keywords: symplectic field theory
reviewed by: Werner Kleinert
MaRDI profile type: Publication
OpenAlex ID: W3104377673
arXiv ID: math/0509386
cites work: Differential algebra of Legendrian links
cites work: Non-isotopic Legendrian submanifolds in \(\mathbb R^{2n+1}\)
cites work: The contact homology of Legendrian submanifolds in \(\mathbb R^{2n+1}\)
cites work: ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS
cites work: Invariants in contact topology
cites work: Q2739348
cites work: Surgery on Lagrangian and Legendrian singularities
cites work: Invariants of Legendrian knots and coherent orientations
cites work: The unregularized gradient flow of the symplectic action
cites work: Witten's complex and infinite dimensional Morse theory
cites work: Q4850917
cites work: Zero-loop open strings in the cotangent bundle and Morse homotopy
cites work: Knot and braid invariants from contact homology. I
cites work: Knot and braid invariants from contact homology. II
cites work: Removal of boundary singularities of pseudo‐holomorphic curves with Lagrangian boundary conditions
cites work: On gradient dynamical systems
DOI: 10.2140/GT.2007.11.1083
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description: scientific article