Holomorphic disks and three-manifolds invariants: Properties and applications

Abstract

In [27], we introduced Floer homology theories $HF^-(Y,\mathfrak{s})$, $HF^\infty(Y,\mathfrak{s})$, $HF^+(Y,\mathfrak{t})$, $\widehat{HF}(Y,\mathfrak{s}) $, and $HF_{\mbox{red}}(Y,\mathfrak{s})$ associated to closed, oriented three-manifolds $Y$ equipped with a ${\mathrm{Spin}}^c$ structures $\mathfrak{s} \in {\mathrm{Spin}}^c(Y)$. 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 $HF^{\pm}$ 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.