## Algebraic & Geometric Topology

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

Daniel Cristofaro-Gardiner

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 4 (2013), 2239-2260.

Dates
Received: 15 September 2012
Revised: 26 February 2013
Accepted: 3 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715637

Digital Object Identifier
doi:10.2140/agt.2013.13.2239

Mathematical Reviews number (MathSciNet)
MR3073915

Zentralblatt MATH identifier
1279.53080

Subjects
Primary: 53D40: Floer homology and cohomology, symplectic aspects

#### Citation

Cristofaro-Gardiner, Daniel. The absolute gradings on embedded contact homology and Seiberg–Witten Floer cohomology. Algebr. Geom. Topol. 13 (2013), no. 4, 2239--2260. doi:10.2140/agt.2013.13.2239. https://projecteuclid.org/euclid.agt/1513715637

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