Geometry & Topology

Floer homology and surface decompositions

András Juhász

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Abstract

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

Source
Geom. Topol., Volume 12, Number 1 (2008), 299-350.

Dates
Received: 13 November 2006
Accepted: 24 November 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800021

Digital Object Identifier
doi:10.2140/gt.2008.12.299

Mathematical Reviews number (MathSciNet)
MR2390347

Zentralblatt MATH identifier
1167.57005

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

Keywords
sutured manifold Floer homology surface decomposition

Citation

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


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