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

@article{Ekholm2007LegendrianCH,
title={Legendrian contact homology in \$P \times \mathbb\{R\}\$},
author={Tobias Ekholm and Tobias Ekholm and John B. Etnyre and John B. Etnyre and Michael C. Sullivan},
journal={Transactions of the American Mathematical Society},
year={2007},
volume={359},
pages={3301-3335}
}
• Published 1 July 2007
• Mathematics
• Transactions of the American Mathematical Society
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 manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds ofR" and, more generally, invariants of self transverse immersions into R n up to restricted regular homotopies. When n = 3, this application is the…

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## References

SHOWING 1-10 OF 16 REFERENCES

### 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

### 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

### Framed knot contact homology

We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is

### 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

### 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

### Pseudo holomorphic curves in symplectic manifolds

Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called

### Knot and braid invariants from contact homology II

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to

### 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

### Holomorphic curves in symplectic geometry

• Mathematics
• 1994
Introduction: Applications of pseudo-holomorphic curves to symplectic topology.- 1 Examples of problems and results in symplectic topology.- 2 Pseudo-holomorphic curves in almost complex manifolds.-