A cylindrical reformulation of Heegaard Floer homology

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold $Σ \times [0, 1] \times R$ , where $Σ$ is the Heegaard surface, instead of ${Sym}^{g} (Σ)$ . 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 $\bar{\partial}$ –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.

Keywords

Heegaard Floer homology, symplectic field theory, holomorphic curves, three–manifold invariants

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 57R58, 57M27

Correction: 9 January 2014

References

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Publication

Received: 14 May 2005

Revised: 9 October 2005

Accepted: 3 January 2006

Published: 9 August 2006

Proposed: Peter Ozsváth

Seconded: John Morgan, Ronald Fintushel

Authors

Robert Lipshitz
Department of Mathematics Stanford University Stanford, CA 94305-2125 USA

Volume 10, issue 2 (2006)

Robert Lipshitz

Abstract