## Geometry & Topology

### A cylindrical reformulation of Heegaard Floer homology

Robert Lipshitz

#### 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.

#### Article information

Source
Geom. Topol., Volume 10, Number 2 (2006), 955-1096.

Dates
Revised: 9 October 2005
Accepted: 3 January 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799745

Digital Object Identifier
doi:10.2140/gt.2006.10.955

Mathematical Reviews number (MathSciNet)
MR2240908

Zentralblatt MATH identifier
1130.57035

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57R58: Floer homology 57M27: Invariants of knots and 3-manifolds

#### Citation

Lipshitz, Robert. A cylindrical reformulation of Heegaard Floer homology. Geom. Topol. 10 (2006), no. 2, 955--1096. doi:10.2140/gt.2006.10.955. https://projecteuclid.org/euclid.gt/1513799745

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