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I'm far away from being an expert, but I think the Heegaard Floer homology was invented by Peter Ozsváth and Zoltán Szabó, so I would recommend the following link to you: click me

If this Introduction is not enough, you should perhaps read "the original work" (in fact the Heegard Floer homology was developed in a long series of papers): P. S. Ozsváth and Z. Szabó. Holomorphic disks and topological invariants for closed three-manifolds. To appear in Annals of Math., math.SG/0101206.

EDIT: Perhaps the Introduction of the book Floer homology, gauge theory, and low-dimensional topology is useful if you are interested in the motivation of Heegard Floer homology.

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I'm far away from being an expert, but I think the Heegaard Floer homology was invented by Peter Ozsváth and Zoltán Szabó, so I would recommend the following link to you: click me

If this Introduction is not enough, you should perhaps read "the original work" (in fact the Heegard Floer homology was developed in a long series of papers): P. S. Ozsváth and Z. Szabó. Holomorphic disks and topological invariants for closed three-manifolds. To appear in Annals of Math., math.SG/0101206.