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Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la Atiyah-Floer conjecture, which I have then forgotten. What is the origin of Heegard Floer?

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up vote 7 down vote accepted

I think the crude answer is that there is (or maybe just should be) an extended 4 dimensional TQFT that assigns the Fukaya category of a symmetric product to a surface, and the usual Heegard-Floer Lagrangian to a 3 manifold. So, the usual definition of Heegard-Floer is the gluing formula for a Heegard splitting, and invariance is no miracle at all.

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Yes, of course. Denis Auroux gave a talk about it www-math.mit.edu/~auroux/papers/slides-fuksymg.pdf and it does remove the miraculousness. But this was a posteriori, in light of Lipshitz-Ozsvath-Thurston. Surely this is not how Ozsváth-Szabo came up with this? –  Max M Nov 2 '09 at 20:58
I think it's a caricature of their thought processes. This TQFT is supposed to come from gauge theory which was known at the time. In particular, I believe that the symmetric power showed as a space of some solutions to equations. I would say the point of that work of Auroux is that you can explicitly reconstruct the higher levels of the TQFT from the Heegard Floer theory. –  Ben Webster Nov 2 '09 at 21:20
I think you are right. I was hoping to get some info on the gauge theory. From what I gather, the idea is that the symmetric product is the space of solutions of vortex equations - explained front.math.ucdavis.edu/0606.5063 This is U(1) gauge theory,and presumably "monopole" version of Atiyah-Floer is what produces the Heegard Floer, which Ozsvath-Szabo then went on to study directly. I wonder if anyone can flesh out some details (e.g. how to the Lagrangian tori arise). –  Max M Nov 3 '09 at 2:03
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From Szabo's delightfully understated response (pdf) to receiving the Veblen prize:

The joint work with Peter Ozsváth which is noted here grew out of our attempts to understand Seiberg-Witten moduli spaces over three-manifolds where the metric degenerates along a surface. This led to the construction of Heegaard Floer homology that involved both topological tools, such as Heegaard diagrams, and tools from symplectic geometry, such as holomorphic disks with Lagrangian boundary constraints. The time spent on investigating Heegaard Floer homology and its relationship with problems in low-dimensional topology was rather interesting.

Of course, if one believes that Heegaard Floer homology is somehow the limit of monopole Floer homology as one degenerates the metric in some way that depends on the Heegaard diagram, then the independence of Heegaard Floer homology from the Heegaard diagram would fall out from the metric-independence of monopole Floer homology. Unfortunately, I can't seem to find references that give any sort of precise picture of how Ozsvath and Szabo came to think that this should be the case (though it might have been a baby analogue of the picture in this paper (pdf) by Yi-Jen Lee, written a few years later).

It perhaps bears mentioning that Heegaard Floer homology wasn't the first invariant that Ozsvath and Szabo constructed based on thinking about the interaction of the Seiberg-Witten equations with a Heegaard diagram--these papers, which extract an invariant from the theta-divisor of the Heegaard surface, appear to have been based on thinking about what happens to the Seiberg-Witten equations when one has a neck Sx[-T,T] (S is the Heegaard surface) with the metric on S at t=-T itself having long cylinders over the compressing circles for one handlebody, while the metric on S at t=T has long cylinders over the compressing circles for the other handlebody.

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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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