The embedded contact homology index revisited

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

The absolute gradings on embedded contact homology and Seiberg –
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 andExpand
The Weinstein conjecture for stable Hamiltonian structures
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 connectedExpand
Embedded contact knot homology and a surgery formula
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
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 subsetExpand
The absolute gradings on embedded contact homology and Seiberg-Witten Floer cohomology
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 contactExpand
The embedded contact homology of toric contact manifolds
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'' contactExpand
Automatic transversality in contact homology I: regularity
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 failExpand
Embedded contact homology and its applications
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 areExpand
Embedded contact homology and Seiberg-Witten Floer cohomology I
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’sExpand
Automatic transversality in contact homology II: filtrations and computations
  • J. Nelson
  • 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, aExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 32 REFERENCES
The Weinstein conjecture for stable Hamiltonian structures
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 connectedExpand
Embedded contact homology and Seiberg-Witten Floer cohomology I
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’sExpand
Gluing pseudoholomorphic curves along branched covered cylinders II
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 assumeExpand
An index inequality for embedded pseudoholomorphic curves in symplectizations
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 pseudoholomorphicExpand
Counting pseudo-holomorphic submanifolds in dimension 4
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. RoughlyExpand
J-Holomorphic Curves and Symplectic Topology
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 wasExpand
Cohomology operations from S¹-cobordisms in Floer homology
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 theExpand
Rounding corners of polygons and the embedded contact homology of T 3
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 theExpand
Handlebody construction of Stein surfaces
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 toExpand
The periodic Floer homology of a Dehn twist.
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
...
1
2
3
4
...