# The contact homology of Legendrian submanifolds in R2n+1

@article{Ekholm2005TheCH,
title={The contact homology of Legendrian submanifolds in R2n+1},
author={Tobias Ekholm and John B. Etnyre and Michael G. Sullivan},
journal={Journal of Differential Geometry},
year={2005},
volume={71},
pages={177-305}
}
• Published 1 October 2005
• Mathematics
• Journal of Differential Geometry
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 homology provides new invariants of Legendrian isotopy which indicate that the theory of Legendrian isotopy is very rich. Indeed, in [4] the homology is used to detect infinite families of pairwise non-isotopic Legendrian submanifolds which are indistinguishable using previously known invariants. The…
136 Citations

## Figures from this paper

### 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

### Company ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN

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

### 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

### Bilinearised Legendrian contact homology and the augmentation category

• Mathematics
• 2012
In this paper we construct an $\mathcal{A}_\infty$-category associated to a Legendrian submanifold of jet spaces. Objects of the category are augmentations of the Chekanov algebra

### Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications

• Mathematics
• 2020
We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the

### Cellular Legendrian contact homology for surfaces, part III

• Mathematics, Biology
International Journal of Mathematics
• 2019
A by-hand Legendrian surfaces are constructed for which specific properties of their gradient flow trees hold, enabling the proof in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. Math].

### Maximal Page Crossing Numbers of Legendrian Surfaces in Closed Contact 5-Manifolds

• Mathematics
• 2021
We introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface L embedded in a closed contact 5-manifold (M, ξ) which admits an “admissable” open book (B, f) (supporting

### Legendrian contact homology for attaching links in higher dimensional subcritical Weinstein manifolds

Let $\Lambda$ be a link of Legendrian spheres in the boundary of a subcritical Weinstein manifold $X$. We show that the computation of the Legendrian contact homology of $\Lambda$ can be reduced to a

### Legendrian knots and constructible sheaves

• Mathematics
• 2017
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is

## References

SHOWING 1-10 OF 38 REFERENCES

### 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

### NON-ISOTOPIC LEGENDRIAN SUBMANIFOLDS IN R

• 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

### Zero-loop open strings in the cotangent bundle and Morse homotopy

• Mathematics
• 1997
0. Introduction. Many important works in symplectic geometry and topology are regarded as the symplectization or the quantization of the corresponding results in ordinary geometry and topology. One

### 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

### Reidemeister torsion in symplectic Floer theory and counting pseudo-holomorphic tori

The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance—the

### 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

### K-theoretic invariants for Floer homology

Abstract. This paper defines two K-theoretic invariants, Wh1 and Wh2, for individual and one-parameter families of Floer chain complexes. The chain complexes are generated by intersection points of

### Morse theory for periodic solutions of hamiltonian systems and the maslov index

• Mathematics
• 1992
In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the

### Symplectic fixed points and holomorphic spheres

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the