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...al genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology. 1. Introduction The present paper is a continuation of =-=[26]-=-, where we defined topological invariants for closed orientated, three-manifolds Y , equipped with a Spin c structure s ∈ Spin c (Y ). These invariants are a collection of Floer homology groups HF − (...

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...homology groups and Heegaard Floer homology, which is established in Section 4. Next, we appeal to the conjectured relationship between Heegaard Floer homology and Seiberg-Witten Floer homology (c.f. =-=[20]-=-). And finally, non-triviality of the corresponding Seiberg-Witten objects is established by work of Kronheimer-Mrowka [13], which in turn rests on work of Gabai [8] and EliashbergThurston [5]. 1.2. G...

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...d L-space surgeries. The appropriate generalization of the notion of lens spaces, for our purposes, is given in the following definition. Note that ̂ HF(Y ) is the three-manifold invariant defined in =-=[17]-=-. Definition 1.1. A closed three-manifold Y is called an L-space if H1(Y ; Q) = 0, and ̂HF(Y ) is a free Abelian group whose rank coincides with the number of elements in H1(Y ; Z), which we write as ...

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... AND ZOLTÁN SZABÓ 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 deco...

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...Given a link L ⊂ S 3 , we can form its branched double cover, a new three-manifold which we denote by Σ(L). In this paper, we study the Heegaard Floer homology of this three-manifold ̂ HF(Σ(L)) (c.f. =-=[17]-=-). The starting point for these investigations is a skein exact sequence which this link invariant L ↦→ ̂ HF(Σ(L)) satisfies. Specifically, fix a projection of L, and let L0 and L1 denote the two reso...

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...on g ∗ (K), using the knot filtration from [17], see also [22] and [21]. Specifically, letting ĈF(S 3 ) denote the chain complex whose homology calculates the Heegaard Floer homology ̂ HF(S 3 ) (c.f. =-=[15]-=-, see also Section 2 for a quick explanation), recall that a knot K in S 3 induces a filtration on the chain complex ĈF(S3 ). Letting F(K, m) ⊂ ĈF(S3 ) be the subcomplex generated by intersection poin...

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...section we will give a brief overview of the results in Heegaard Floer theory we will need in the following, with no pretension of completeness. The details can be found in Ozsváth and Szabó’s papers =-=[16, 15, 20, 14, 19, 13]-=-. 2.1 Heegaard Floer homology Let Y be a closed, connected, oriented 3–manifold. For any Spinc –structure t on Y Ozsváth and Szabó [16] defined an Abelian group HF +(Y,t) which is an isomorphism invar...