Volume 12, issue 1 (2008)

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Floer homology and surface decompositions

András Juhász

Geometry & Topology 12 (2008) 299–350

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 Y N(K) admits a depth 2 taut foliation transversal to N(K).

sutured manifold, Floer homology, surface decomposition
Mathematical Subject Classification 2000
Primary: 57M27, 57R58
Received: 13 November 2006
Accepted: 24 November 2007
Published: 12 March 2008
Proposed: Peter Ozsváth
Seconded: Ron Fintushel, Tom Mrowka
András Juhász
Department of Mathematics
Princeton University
Princeton NJ 08544