Grid diagrams and Heegaard Floer invariants

Abstract

We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in $\mathbb{Z}/2\mathbb{Z}$). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod $2$ Ozsváth-Szabó mixed invariants of closed four-manifolds, also in terms of grid diagrams.