The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y -- U0 tie U1, these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Y relative to certain totally real subspaces associated to U0 and U1.

Abstract. We use the Ozsváth-Szabó theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the Ozsváth-Szabó Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of perfect knots in S3 for which ĈF r has a particularly simple form. For these knots, formal properties of the Ozsváth-Szabó theory enable us to make a complete calculation of the Floer homology. It turns out that most small knots are perfect. 1.

Abstract. In [22], we introduced absolute gradings on the three-manifold invariants developed in [21] and [20]. Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the “complexity bounds ” derived in [20], restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF + for a variety of threemanifolds. Moreover, we show how the structure of HF + constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given threemanifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson’s diagonalizability theorem and the Thom conjecture for CP 2. 1.

Abstract. We define a Floer-homology invariant for knots in an oriented threemanifold, closely related to the Heegaard Floer homologies for three-manifolds defined in [18]. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. Then, we establish a relationship with HF + for surgeries along the knot. Applications include calculation of HF + of threemanifolds obtained by surgeries on some special knots in S 3, and also calculation of HF + for certain simple three-manifolds which fiber over the circle. 1.

Abstract. Given a contact structure on a closed, oriented three-manifold Y, we describe an invariant which takes values in the three-manifold’s Floer homology ̂ HF (in the sense of [10]). This invariant vanishes for overtwisted contact structures and is non-zero for Stein fillable ones. The construction uses of Giroux’s interpretation of contact structures in terms of open book decompositions (see [4]), and the knot Floer homologies introduced in [14]. 1.

Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in [8] and [12]. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces. 1.

Abstract. In [23] we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y (c.f. [21]). In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (compare [24]). Applications include new restrictions on the Alexander polynomial of alternating knots. 1.

Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the four-ball genera of torus knots. As another illustration, we use calculate the invariant for several ten-crossing knots. 1.

Abstract. In an earlier paper, we used the absolute grading on Heegaard Floer homology HF + to give restrictions on knots in S 3 which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that all the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information in turn can be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p, q) which arise as integral surgeries on knots in S 3 with |p | ≤ 1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge’s knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts. 1.

Abstract. We calculate HF + for three-manifolds obtained by plumbings of spheres specified by certain graphs. Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres. These calculations can be used to determine also the Floer homology of other three-manifolds, including the product of a circle with a genus two surface. 1.