# ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS

@article{Ekholm2004ORIENTATIONSIL,
title={ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS},
author={Tobias Ekholm and John B. Etnyre and Michael G. Sullivan},
journal={International Journal of Mathematics},
year={2004},
volume={16},
pages={453-532}
}
• Published 30 August 2004
• Mathematics
• International Journal of Mathematics
We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n + 1)-space from ℤ2 to ℤ. We demonstrate how the ℤ-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian…
93 Citations

## Figures from this paper

### Legendrian Contact Homology

• Mathematics
A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × R where P is an exact symplectic manifold is established. The class of such contact

### Morse flow trees and Legendrian contact homology in 1-jet spaces

Let L ⊂ J 1 (M) be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M. A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid

### The orientability problem in open Gromov–Witten theory

We give an explicit formula for the holonomy of the orientation bundle of a family of real Cauchy‐Riemann operators. A special case of this formula resolves the orientability question for spaces of

### Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold

• Mathematics
• 2013
We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact

### Legendrian contact homology in $P \times \mathbb{R}$

• Mathematics
• 2007
A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P x R, where P is an exact symplectic manifold, is established. The class of such contact

### Legendrian Contact Homology in P X R

• Mathematics
• 2005
A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such

### Exact Lagrangian immersions with a single double point

• Mathematics
• 2011
Let k > 2. We show that if a closed orientable 2k-manifold K, with Euler characteristic not equal to -2, admits an exact Lagrangian immersion into complex Euclidean 2k-space with one transverse

### Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology

We relate the version of rational symplectic field theory for exact Lagrangian cobordisms introduced in [6] to linearized Legendrian contact homology. More precisely, if L ⊂ Xis an exact Lagrangian

### An exact sequence for Legendrian links

A result of Bourgeois, Ekholm and Eliashberg [4] describes the linearized contact homology of the boundary of a symplectic cobordism obtained by Legendrian surgery in terms of the cyclic homology of

## References

SHOWING 1-10 OF 22 REFERENCES

### Legendrian Submanifolds in $R^{2n+1}$ and Contact Homology

• Mathematics
• 2002
Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex

### The contact homology of Legendrian submanifolds in R2n+1

• Mathematics
• 2005
We define the contact homology for Legendrian submanifolds in standard contact (2n + 1)-space using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex n-space. This

### Invariants of Legendrian Knots and Coherent Orientations

• Mathematics
• 2001
We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R and a relative version of Eliashberg and Hofer’s contact homology. We use

### Differential algebras of Legendrian links

The problem of classification of Legendrian knots (links) up to isotopy in the class of Legendrian embeddings (Legendrian isotopy) naturally leads to the following two subproblems. The first of them

### Differential algebra of Legendrian links

Let the space R = {(q, p, u)} be equipped with the standard contact form α = du − pdq. A link L ⊂ R3 is called Legendrian if the restriction of α to L vanishes. Two Legendrian links are said to be

### Coherent orientations in symplectic field theory

• Mathematics
• 2001
Abstract.We study the coherent orientations of the moduli spaces of holomorphic curves in Symplectic Field Theory, generalizing a construction due to Floer and Hofer. In particular we examine their

### Non-isotopic Legendrian submanifolds in R2n+1

• Mathematics
• 2005
In the standard contact (2n + 1)-space when n > 1, we construct infinite families of pairwise non-Legendrian isotopic, Legendrian n-spheres, n-tori and surfaces which are indistinguishable using

### Introduction to Symplectic Field Theory

• Mathematics
• 2000
We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds

### Twisting spun knots

1. Introduction. In [5] Mazur constructed a homotopy 4-sphere which looked like one of the strongest candidates for a counterexample to the 4-dimensional Poincaré Conjecture. In this paper we show