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

TL;DR

This paper proves that the isomorphism between embedded contact homology and Seiberg-Witten Floer cohomology preserves their absolute gradings, linking the dimensions of moduli spaces to the ECH index.

Contribution

It demonstrates that Taubes' isomorphism maintains the absolute gradings by homotopy classes of plane fields, connecting moduli space dimensions to the ECH index.

Findings

Taubes' isomorphism preserves absolute gradings.

The expected dimension of Seiberg-Witten moduli spaces relates to the ECH index.

The result links symplectic cobordism properties to homology gradings.

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.