## Geometry & Topology

### Floer homology and surface decompositions

András Juhász

#### 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 $ag≠0$ and if then $Y∖N(K)$ admits a depth $≤2$ taut foliation transversal to $∂N(K)$.

#### Article information

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

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

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

#### 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