The embedded contact homology index revisited

TL;DR

This paper advances embedded contact homology by refining gradings, extending index inequalities to cobordisms, establishing new inequalities, and introducing a useful filtration for computational purposes.

Contribution

It introduces an absolute grading, extends index inequalities to cobordisms, and defines a new filtration, enhancing the computational and theoretical framework of ECH.

Findings

Refined the relative grading to an absolute grading using homotopy classes of 2-plane fields.

Extended the ECH index inequality to symplectic cobordisms and simplified the proof.

Established inequalities on the ECH index for unions and multiple covers of holomorphic curves.

Abstract

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.