A cylindrical reformulation of Heegaard Floer homology

Abstract

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold $\Sigma \times \left[0,1\right]\times R$, where $\Sigma$ is the Heegaard surface, instead of ${Sym}^g\left(\Sigma \right)$. We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the $\bar{\partial }$–operator in Heegaard Floer homology, and shorten several proofs. After proving invariance, we show that our construction is equivalent to the original construction of Ozsváth–Szabó. We conclude with a discussion of elaborations of Heegaard Floer homology suggested by our construction, as well as a brief discussion of the relation with a program of C Taubes.