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

Abstract

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 homology (ECH) of Y and its Seiberg-Witten Floer cohomology. Both the ECH of Y and the Seiberg-Witten Floer cohomology of Y admit absolute gradings by homotopy classes of oriented two-plane fields. We show that Taubes' isomorphism preserves these gradings. To do this, we prove another result relating the expected dimension of any component of the Seiberg-Witten moduli space over a completed connected symplectic cobordism to the ECH index of a corresponding homology class.