Floer homology and sutured manifolds

Low-dimensional topology has been revolutionized in the past few years by Ozsvath and Szabo when they introduced a package of gauge-theoretic invariants, called collectively Heegaard-Floer homology, for the study of three- and four-manifolds and knots [23, 24, 25, 26]. These invariants gained popularity because they were simpler to compute than their predecessors. The hat version of Heegaard-Floer homology can even be computed algorithmically (though not efficiently) by an algorithm of Sarkar and Wang [27].; Ozsvath and Szabo proved in [22] that knot Floer homology, denoted by HFK&d14; , detects the genus of a knot, and more generally, link Floer homology detects the Thurston norm of a link complement, see [19]. Ni [16] showed that HFk&d14; also detects bred knots. Both of these results make use of Gabai's sutured manifold theory [2] and some deep results from contact topology.; Gabai introduced sutured manifolds to study taut foliations on three-manifolds and using them he managed to prove several long standing conjectures in low-dimensional topology. Sutured manifolds are compact oriented three-manifolds with some set of dividing curves on the boundary.; In this work we introduce a gauge-theoretic invariant of sutured manifolds which we call sutured Floer homology, in short: SFH. This can be thought of as a generalization of the hat version of Heegaard Floer homology of both three-manifolds and knots. In fact, we can generalize the Sarkar-Wang algorithm to compute SFH.; Gabai introduced an operation called sutured manifold decomposition in which we cut a sutured manifold along a properly embedded surface and then reconnect the dividing curves along the boundary of the decomposing surface. Gabai proved that a sutured manifold carries a taut foliation (these manifolds are called taut) if and only if after a sequence of such decompositions we end up with a set of balls with a single suture on each. SFH of the latter is Z .; One of the main results of this work establishes that in most cases (including the ones considered by Gabai) the SFH of the result of a sutured manifold decomposition is a direct summand of the SFH of the original. Thus if a sutured manifold is taut then its SFH has a Z direct summand. This gives a simpler proof of the fact that HFK&d14; detects the genus of a knot. Moreover, we give a simpler and more transparent proof of Ni's theorem. We show that if a sutured manifold is not a product then after finitely many horizontal and product annulus decompositions we can find two decompositions for which the Z direct summands are distinct. Thus SFH of a non- product sutured manifold has a Z 2 direct summand and, consequently, knot Floer homology detects fibred knots.; This approach can be extended to give the following stronger result. Let K be a null-homologous genus g knot in a rational homology 3-sphere Y. Suppose that the coefficient ag of the Alexander polynomial DeltaK( t) of K is non-zero and rkHFK&d14; Y,K,g<4. Let N(K) denote a tubular neighborhood of K. Then Y N( K) has a depth ≤ 2 taut foliation transverse to ∂ N(K). Note that K is fibred if and only if the knot complement has a depth zero taut foliation, so this is indeed an extension of Yi's result.; The following is a direct corollary of the decomposition of formula. Let K be a knot in S3 of genus g and let n > 0. We show that if rk HFK&d14; (K, g) < 2n +1, in particular if K is an alternating knot such that the leading coefficient ag of its Alexander polynomial satisfies |ag| < 2 n+1, then K has at most n pairwise disjoint non-isotopic genus g Seifert surfaces. For n = 1 this implies that K has a unique minimal genus Seifert surface up to isotopy.; The results in this thesis have been accepted for publication [9, 10, 11].