̂HF(Y, s),and HFred(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. 1.

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.

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the Symplectic Thom Conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative self-intersection in four-manifolds. 1.

Abstract. Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a non-trivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a non-vanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations. 1.

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In this paper, we derive new adjunction inequalities for embedded surfaces with non-negative self-intersection number in four-manifolds. These formulas are proved by using relations between Seiberg-Witten invariants which are induced from embedded surfaces. To prove these relations, we develop the relevant parts of a Floer theory for four-manifolds which bound circle-bundles over Riemann surfaces. 1.

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg–Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally elliptic (including the cyclic quotient and “polygonal”) singularities, and Brieskorn–Hamm complete intersections. Some of the verifications are based on a result which describes (in terms of the plumbing graph) the Reidemeister–Turaev sign refined torsion (or, equivalently, the Seiberg– Witten invariant) of a rational homology 3–manifold M, provided that M is given by a negative definite plumbing. These results extend previous work of Artin, Laufer and S S-T Yau, respectively of Fintushel–Stern and Neumann–Wahl.

Our main purpose in this article is to prove that the moduli space of solutions to the PU(2) monopole equations is a smooth manifold of the expected dimension for simple, generic parameters such as (and including) the Riemannian metric on the given four-manifold: see Theorem 1.3. In [16] we proved transversality using an

Abstract. The aim of this article is to introduce and study certain topological invariants for oriented, rational homology three-spheres Y. These groups are relatively Z-graded Abelian groups associated to Spin C structures over Y. Given a Heegaard splitting of Y = U0 ∪Σ U1, 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 U0 and U1. 1.

The main result of this paper describes a formula for the Seiberg-Witten invariant of a 4-manifold X which admits a nontrivial free S 1-action. A free circle action on X is classified by its orbit space, a 3-manifold M, and its Euler class χ ∈ H 2 (M; Z). If χ = 0, then