Holomorphic disks and three-manifold invariants: Properties and applications

In [27], we introduced Floer homology theories HF^-(Y, s)t HF^∞(Y,s), HF⁺(Y,t), HF(Y,s),and. HF _red(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.