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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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
Mathematical Subject Classification 2010
Primary: 53D40
References
Publication
Received: 15 September 2012
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/