Holomorphic Disks and Three-manifold Invariants: Properties and Applications

In [27], we introduced Floer homology theories HF(Y, s), HF(Y, s), HF(Y, t), ĤF (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. 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.