TAILIEUCHUNG - Đề tài " Holomorphic disks and threemanifold invariants: Properties and applications "

In [27], we introduced Floer homology theories HF − (Y, s), HF ∞ (Y, s), HF (Y, s),and HFred (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spinc structures s ∈ Spinc (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. | Annals of Mathematics Holomorphic disks and three-manifold invariants Properties and applications By Peter Ozsv ath and Zolt an Szab o Annals of Mathematics 159 2004 1159 1245 Holomorphic disks and three-manifold invariants Properties and applications By Peter OzsvATH and Zoltán Szabo Abstract In 27 we introduced Floer homology theories HF- Y s lll iY s HF Y t HF Y s and HFred Y s associated to closed oriented three-manifolds Y equipped with a Spinc structures s G Spinc 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. Introduction The present paper is a continuation of 27 where we defined topological invariants for closed oriented three-manifolds Y equipped with a Spinc structure s G Spinc Y . These invariants are a collection of Floer homology groups HF- Y s HF Y s HF Y s and HF Y s . Our goal here is to study these invariants calculate them in several examples establish their fundamental properties and give some applications. We begin in Section 2 with some of the properties of the groups including their behaviour under orientation reversal of Y and conjugation of its Spinc structures. Moreover we show that for any three-manifold Y there are at most finitely many Spinc structures s G Spinc Y with the property that HF Y s is PSO was supported by NSF grant number DMS-9971950 and a Sloan Research Fellowship. ZSz was supported by a Sloan Research Fellowship and a Packard Fellowship. throughout this introduction and indeed through most of this paper we will suppress the orientation system o used in the definition. This is justified in part by the fact that our .

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