The embedded contact homology index revisited

By Michael Hutchings. 2008

94 citations recorded by the source corpus.

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 implies that such curves are (mostly) embedded and satisfy some additional constraints. In this paper we prove four new results about the ECH index. First, we refine the relative grading on ECH to an absolute grading, which associates to each union of Reeb orbits a homotopy class of oriented 2-plane fields on Y. Second, we extend the ECH index inequality to symplectic cobordisms between three-manifolds with Hamiltonian structures, and simplify the proof. Third, we establish general inequalities on the ECH index of unions and multiple covers of holomorphic curves in cobordisms. Finally, we define a new relative filtration on ECH, or any other kind of contact homology of a contact 3-manifold, which is similar to the ECH index and related to the Euler characteristic of holomorphic curves. This does not give new topological invariants except possibly in special situations, but it is a useful computational tool.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6a18604100b806f59e61651bab68e7bb502a34ee

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Michael Hutchings
2. Clifford H. Taubes
3. Daniel Cristofaro-Gardiner
4. P. Ghiggini
5. Andrew Cotton-Clay
6. V. Colin
7. Richard Siefring
8. K. Honda

In Geometry and Topology

1. Michael Hutchings
2. Daniel Cristofaro-Gardiner
3. P. Ghiggini
4. Clifford H. Taubes
5. K. Honda
6. Richard Siefring
7. V. Colin
8. Andrew Cotton-Clay

In Mathematical Physics

1. Michael Hutchings
2. Daniel Cristofaro-Gardiner
3. Clifford H. Taubes
4. K. Honda
5. P. Ghiggini
6. Richard Siefring
7. Andrew Cotton-Clay
8. V. Colin

Citations in the corpus

• Quantitative embedded contact homology (2010, methodology): “The detailed definition of the grading will not be needed here and can be found in [5, 6].”
• Intersection theory of punctured pseudoholomorphic curves (2009, methodology): “3 have been proved by Hutchings in [14] and [15].”
• Cylindrical contact homology for dynamically convex contact forms in three dimensions (2014, unknown): “…u as above, we define its Fredholm index by Here τ is a trivialization of ξ over γ + and γ − , and c τ ( u ) denotes the relative first Chern class c 1 ( u ∗ ξ, τ ), see [17, § 2.5] or [18, § 3.2]; the relative first Chern class vanishes when the trivialization τ extends to a trivialization of u ∗ ξ .”
• Embedded contact homology and Seiberg–Witten Floer cohomology II (2008, background): “Hutchings [3; 2] explains how Cech can be given a relative Z=pZ grading such that ı decreases by 1 this degree.”
• The asymptotics of ECH capacities (2012, background): “The definition of the ECH index is explained in [2]; all we need to know here is that the ECH index defines a relative Z/d-grading on the chain complex, where d denotes the divisibility of c1(ξ) + 2PD(Γ) in H (2)(Y ;Z) mod torsion.”
• Beyond ECH capacities (2014, background): “In particular, if is not both elliptic and L–negative, then (33) is proved in [8].”
• Sutures and contact homology I (2010, background): “1Both Seiberg–Witten Floer homology and ECH have absolute gradings by homotopy classes of oriented 2–plane fields on M (see [32; 25]), and it is natural to conjecture that Taubes’s isomorphism between them respects these gradings.”
• Proof of the Arnold chord conjecture in three dimensions, II (2010, background): “The definition of the ECH index is not needed in this paper and may be found in [5].”
• Symplectic Floer homology of area-preserving surface diffeomorphisms (2008, background): “The grading (x) of a fixed point x is given by (−1) (x) = sign (det(1− dφ)) (for an absolute grading over Z by homotopy classes of 2-plane fields, see [H3]).”
• Lecture Notes on Embedded Contact Homology (2013, background): “In fact, both Seiberg-Witten Floer cohomology and ECH have absolute gradings by homotopy classes of oriented two-plane fields [34, 22], and Taubes’s isomorphism (1.”