# The embedded contact homology index revisited

@article{Hutchings2008TheEC, title={The embedded contact homology index revisited}, author={Michael Hutchings}, journal={arXiv: Symplectic Geometry}, year={2008} }

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in the symplectization of Y whose relative homology classes have ECH index 1. A known index inequality… Expand

#### Figures from this paper

#### 61 Citations

The absolute gradings on embedded contact homology and Seiberg –

- 2013

Let Y be a closed connected contact 3–manifold. In [14], Taubes defines an isomorphism between the embedded contact homology (ECH) of Y and its Seiberg–Witten Floer cohomology. Both the ECH of Y and… Expand

The Weinstein conjecture for stable Hamiltonian structures

- Mathematics
- 2009

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected… Expand

Embedded contact knot homology and a surgery formula

- Mathematics
- 2018

Embedded contact knot homology (ECK) is a variation on Embedded contact homology (ECH), defined with respect to an open book decomposition compatible with a contact structure on some 3-manifold, M.… Expand

Quantitative embedded contact homology

- Mathematics
- 2010

Define a "Liouville domain" to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset… Expand

The absolute gradings on embedded contact homology and Seiberg-Witten Floer cohomology

- Mathematics
- 2012

Let Y be a closed connected contact 3-manifold. In the series of papers "Embedded contact homology and Seiberg-Witten Floer cohomology I-V", Taubes defines an isomorphism between the embedded contact… Expand

The embedded contact homology of toric contact manifolds

- Mathematics
- 2013

Embedded contact homology (ECH) is an invariant of a contact three-manifold. In Part I of this thesis, we provide a combinatorial description of the ECH chain complex of certain ``toric'' contact… Expand

Automatic transversality in contact homology I: regularity

- Mathematics
- 2014

This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail… Expand

Embedded contact homology and its applications

- Mathematics
- 2010

Embedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Taubes has shown that ECH is isomorphic to a version of Seiberg-Witten Floer homology (and both are… Expand

Embedded contact homology and Seiberg-Witten Floer cohomology I

- Mathematics
- 2008

This is the third of five papers whose purpose is to prove that the embedded contact homology of a compact, oriented 3–dimensional manifold with contact 1–form is isomorphic to the manifold’s… Expand

Automatic transversality in contact homology II: filtrations and computations

- Mathematics
- Proceedings of the London Mathematical Society
- 2019

This paper is the sequel to the previous paper [Ne15] which showed that sufficient regularity exists to define cylindrical contact homology in dimension 3 for dynamically separated contact forms, a… Expand

#### References

SHOWING 1-10 OF 32 REFERENCES

The Weinstein conjecture for stable Hamiltonian structures

- Mathematics
- 2009

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected… Expand

Embedded contact homology and Seiberg-Witten Floer cohomology I

- Mathematics
- 2008

This is the third of five papers whose purpose is to prove that the embedded contact homology of a compact, oriented 3–dimensional manifold with contact 1–form is isomorphic to the manifold’s… Expand

Gluing pseudoholomorphic curves along branched covered cylinders II

- Mathematics
- 2007

This paper and its prequel (“Part I”) prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U+ and U− in the symplectization of a contact 3-manifold. We assume… Expand

An index inequality for embedded pseudoholomorphic curves in symplectizations

- Mathematics
- 2001

Abstract.Let Σ be a surface with a symplectic form, let φ be a symplectomorphism of Σ, and let Y be the mapping torus of φ. We show that the dimensions of moduli spaces of embedded pseudoholomorphic… Expand

Counting pseudo-holomorphic submanifolds in dimension 4

- Mathematics
- 1996

The purpose of this article is to describe a certain invariant (called the Gromov invariant) for compact symplectic 4-manifolds which assigns an integer to each dimension 2-cohomology class. Roughly… Expand

J-Holomorphic Curves and Symplectic Topology

- Mathematics
- 2004

The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was… Expand

Cohomology operations from S¹-cobordisms in Floer homology

- Mathematics
- 1995

In this work, Floer homology is considered as a relative Morse theory for the symplectic action functional on the loop space of a symplectic manifold (M, to). It is assumed that M is closed and the… Expand

Rounding corners of polygons and the embedded contact homology of T 3

- Mathematics
- 2006

The embedded contact homology (ECH) of a 3‐manifold with a contact form is a variant of Eliashberg‐Givental‐Hofer’s symplectic field theory, which counts certain embedded J ‐holomorphic curves in the… Expand

Handlebody construction of Stein surfaces

- Mathematics
- 1998

The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained-they correspond to… Expand

The periodic Floer homology of a Dehn twist.

- Mathematics
- 2005

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits,… Expand