The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure. Given a Heegaard splitting of Y = U 0 ∪ Σ U 1 , these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U 0 and U 1 .

We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants, including an Euler characteristic calculation, and a description of the behavior under connected sums. Then, we establish a relationship with HF þ for surgeries along the knot. Applications include calculation of HF þ of three-manifolds obtained by surgeries on some special knots in S 3 ; and also calculation of HF þ for certain simple three-manifolds which fiber over the circle. r

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 .

In [27], we introduced Floer homology theories HF, HF (Y, s),and HF red (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ Spin c (Y ). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with Seiberg-Witten theory. The properties include a relationship between the Euler characteristics of HF ± and Turaev's torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology.

Let L ⊂ S 3 be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S 3 , branched along L. When L is an alternating link, HF of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E 2 term is a suitable variant of Khovanov's homology for the link L, converging to the Heegaard Floer homology of Σ(L).

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.All of these properties (and various refinements) are stated precisely in Section 3. The invariants are constructed in Section 4, and the properties are verified in Sections 4-6.1.2. Absolute gradings. The principles used in construction of the cobordism invariant also allow us to define an absolute Q-lift of the relative Z-gradings of the Floer

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(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, tau gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper17.abs.htm

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 the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information can in turn 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.