#### Volume 13, issue 4 (2013)

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

### Daniel Cristofaro-Gardiner

Algebraic & Geometric Topology 13 (2013) 2239–2260
##### 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.

##### Keywords
embedded contact homology, Seiberg–Witten theory, absolute gradings
Primary: 53D40
##### Publication
Revised: 26 February 2013
Accepted: 3 March 2013
Published: 13 June 2013
##### Authors
 Daniel Cristofaro-Gardiner Mathematics Department University of California, Berkeley 970 Evans Hall Berkeley, CA 94720 USA http://www.math.berkeley.edu/~gardiner/