A cylindrical reformulation of Heegaard Floer homology

Abstract

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold Σ× [0, 1] × R, where Σ is the Heegaard surface, instead of Symg(Σ). 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 ∂̄-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.