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...s −1 times the Alexander grading on EN. Combining this fact with the isomorphisms ĈF s(H,s) ∼ = ĈF(H, N, zs,s) ∼ = ( ĈF(H, −N, zs,s)) ∗ ∼ = (ĈF s(−H,s)) ∗ provided by Lemma 4.2 and Proposition 7.3 of =-=[19]-=-, we see that ĈF s(H,s) is dual to ĈF s(−H,s) as a filtered complex. The result then follows from the fact that the dual of the reduction is the reduction of the dual, which will be obvious from the c...

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...rsion, 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. Introduction In =-=[23]-=-, we defined topological invariants for closed, oriented three-manifolds Y with b1(Y ) = 0. Starting with a Heegaard diagram for Y , with Riemann surface Σ and attaching circles {α1, ..., αg} and {β1,...

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...LOER HOMOLOGIES AND INTERSECTION FORMS FOR FOUR-MANIFOLDS WITH BOUNDARY PETER OZSVÁTH AND ZOLTÁN SZABÓ Abstract. In [19], we introduced absolute gradings on the three-manifold invariants developed in =-=[18]-=- and [21]. 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” de...

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... 18 Oct 2001 PETER OZSVÁTH AND ZOLTÁN SZABÓ 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...

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...the Floer homology of other three-manifolds, including the product of a circle with a genus two surface. 1. Introduction The main goal of this paper is to calculate the groups HF + (Y ) introduced in =-=[8]-=- for three-manifolds Y which are boundaries of certain spherical plumbings. This class of three-manifolds includes all Seifert fibered rational homology spheres. To state the result, we require some p...

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...ÁTH AND ZOLTÁN SZABÓ 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 gradi...

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...related developments, but we want to give at least a few indications. The idea of working relative to a divisor is very common in symplectic geometry; some papers whose viewpoint is close to ours are =-=[12, 16, 3, 17]-=-. There is also at least one entirely different approach to Fukaya categories, using Lagrangian fibrations and Morse theory [8, 15, 4]. Finally, the example of the two-torus has been studied extensive...

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...Z[H 1 (Y ; Z)] allows us to define the homology of Ip with local coefficients in the group ring Z[H 1 (Y ; Z)], or in any module over this ring. Comparison with the Ozsvath-Szabo Floer homologies. In =-=[17]-=- and [18], Ozsvath and Szabo have constructed several versions of Floer homology for three-manifolds, which they denoted by ̂ HF,HF + ,HF − ,HF. Their theory is supposed to give the same output as Sei...