Geometry & Topology

Floer homology and surface decompositions

András Juhász

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Sutured Floer homology, denoted by SFH, 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 (M,γ)(M,γ) is a sutured manifold decomposition then SFH(M,γ) is a direct summand of SFH(M,γ). To prove the decomposition formula we give an algorithm that computes SFH(M,γ) from a balanced diagram defining (M,γ) that generalizes the algorithm of Sarkar and Wang.

As a corollary we obtain that if (M,γ) is taut then SFH(M,γ)0. 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 K is a genus g knot in a rational homology 3–sphere Y whose Alexander polynomial has leading coefficient ag0 and if  rkHFK̂(Y,K,g)<4 then YN(K) admits a depth 2 taut foliation transversal to N(K).

Article information

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57R58: Floer homology

sutured manifold Floer homology surface decomposition


Juhász, András. Floer homology and surface decompositions. Geom. Topol. 12 (2008), no. 1, 299--350. doi:10.2140/gt.2008.12.299.

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