  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.

Symplectic Geometry;Geometric Topology