Floer homology and surface decompositions

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 rk(^HFK(Y,K,g))<4 then Y\N(K) admits a depth ≤2 taut foliation transversal to ∂N(K). © 2008 Mathematical Sciences Publishers.