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

Abstract

Let $Y$ be a closed connected contact $3$–manifold. In [Geom. Topol. 14 (2010) 2497–2581], 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 $2$–plane fields. We show that Taubes’ isomorphism preserves these gradings, which implies that the absolute grading on ECH is a topological invariant. 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.