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, \gamma)--> (M', \gamma') is a sutured manifold decomposition then SFH(M',\gamma') is a direct summand of SFH(M, \gamma). To prove the decomposition formula we give an algorithm that computes SFH(M,\gamma) from a balanced diagram defining (M,\gamma) that generalizes the algorithm of Sarkar and Wang. As a corollary we obtain that if (M, \gamma) is taut then SFH(M,\gamma) is non-zero. Other applications include simple proofs of a result of Ozsvath and Szabo 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 a_g non-zero and if the rank of \hat{HFK}(Y,K,g) < 4 then the knot complement admits a depth < 2 taut foliation transversal to the boundary of N(K).Comment: 40 pages, 10 figures. Improved, expanded expositio

The sutured Floer homology polytope

“…I used SFH in [14] to give a more elegant and direct proof of the fact that knot Floer homology detects the genus of a knot. That proof only relies on Gabai's theory of sutured manifolds and the following two results.…”

Sutured Floer homology distinguishes between Seifert surfaces

“…The perpendicular two-plane field v ⊥ 0 is trivial on ∂M if and only if (M, γ) is strongly balanced [Ju08,Prop. 3.4].…”

“…Then Juhász showed that SF H(S 3 (R)) is a knot invariant [Ju08]; that is, the top term of knot Floer homology [OS04b,Ra03] is isomorphic to the sutured Floer homology of the complement: SF H(S 3 (R)) ∼ = HF K(K, genus(R)).…”

“…These small knots have either a unique minimal genus Seifert surface, or all of their minimal genus Seifert surfaces can be identified with Murasugi sums of bands. However, Juhász showed that the sutured Floer homology of the complement of a Murasugi sum is the tensor product of the sutured Floer homology of the complement of each summand [Ju08,Cor. 8.8].…”

“…The latter gives us a way of explicitly computing the sutured Floer homology groups SF H(X) and SF H(Y ) (see Propositions 2 and 3). Then the groups SF H(S 3 (R 1 )) and SF H(S 3 (R 2 )) are the tensor product SF H(X) ⊗ SF H(Y ) [Ju08,Prop. 8.6], where the Spin c grading can be derived from the appropriate Mayer-Vietoris maps on the level of the first homology groups [Ju10a, Prop.…”