Geometry & Topology
- Geom. Topol.
- Volume 12, Number 1 (2008), 299-350.
Floer homology and surface decompositions
Sutured Floer homology, denoted by , is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if is a sutured manifold decomposition then is a direct summand of . To prove the decomposition formula we give an algorithm that computes from a balanced diagram defining that generalizes the algorithm of Sarkar and Wang.
As a corollary we obtain that if is taut then . Other applications include simple proofs of a result of Ozsváth and Szabó that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry.
Moreover, using these methods we show that if is a genus knot in a rational homology –sphere whose Alexander polynomial has leading coefficient and if then admits a depth taut foliation transversal to .
Geom. Topol., Volume 12, Number 1 (2008), 299-350.
Received: 13 November 2006
Accepted: 24 November 2007
First available in Project Euclid: 20 December 2017
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Juhász, András. Floer homology and surface decompositions. Geom. Topol. 12 (2008), no. 1, 299--350. doi:10.2140/gt.2008.12.299. https://projecteuclid.org/euclid.gt/1513800021