#### Volume 10, issue 2 (2006)

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A cylindrical reformulation of Heegaard Floer homology

### Robert Lipshitz

Geometry & Topology 10 (2006) 955–1096
 arXiv: math.SG/0502404
##### Abstract

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold $\Sigma ×\left[0,1\right]×R$, 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 $\stackrel{̄}{\partial }$–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.

##### Keywords
Heegaard Floer homology, symplectic field theory, holomorphic curves, three–manifold invariants
##### Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57R58, 57M27