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Recently, the Seiberg-Witten-Floer homology created by Kronheimer and Mrowka has important applications in Taubes' proofs of Weinstein conjecture and Arnold Chord Conjecture. Also, Cagatay Kutluhan, Yi-Jen Lee, and Clifford Taubes have a series of papers on the arxiv proving the equivalence of Heegard-Floer and Seiberg-Witten-Floer homologies. However, these are only defined for closed 3-manifolds.

My question is if there exists any version of HF or SWFH defined for 3-manifolds with boundary?

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Thanks to Andy and Dylan, I'll follow the references to learn more about the subjects. –  Cheuk Hwang May 27 '11 at 20:38
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3 Answers 3

Lipshitz, Ozsváth, and Thurston have a version of Heegaard Floer homology for 3-manifolds with boundary, which they call bordered Floer homology. To a surface F (with some additional data), it associates a dg-algebra $A(F)$, and to a 3-manifold with boundary, it associates a dg-module CFD and an $A_{\infty}$ module CFA. If you decompose a closed 3-manifold M along a surface F, the Heegaard Floer chain group of M is quasi-isomorphic to the $A_{\infty}$ tensor product of CFA of the left and CFD of the right over $A(F)$.

Some references are: http://arxiv.org/abs/0810.0687 (the original paper on the topic);

http://arxiv.org/abs/1003.0598 (discussing bimodules)

http://arxiv.org/abs/0810.0695 (an easier introduction)

http://www.math.columbia.edu/~lipshitz/CambridgeSlides.pdf (slides from a talk by Robert Lipshitz)

and many others can be found on the arXiv by searching for bordered Floer homology.

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It's worth noting that the algebra $\mathcal{A}(F)$ is not arbitrary, but, as Auroux showed, is Morita equivalent to a version of the Fukaya category of $F$. See [front.math.ucdavis.edu/1001.4323]. –  Dylan Thurston May 27 '11 at 1:36
    
Also, we currently only have a bordered version of a relatively weak version of Heegaard Floer homology ($\widehat{HF}$). In particular, that version is not enough to recover the closed 4-manifold invariants, although it still gives interesting information about 3-manifolds. –  Dylan Thurston May 27 '11 at 1:43
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Andy Manion has already plugged our answer for Heegaard Floer homology. On the Seiberg-Witten side, not as much is known, but Tim Nguyen's thesis starts to attack the problem. However, there isn't as yet a complete answer on the Seiberg-Witten side.

(Of course the two theories for closed 3-manifolds are now known to be isomorphic, but this doesn't actually help as much as you'd like. In particular, the isomorphism is not known to be natural, and not known to be compatible with the 4-manifold invariants.)

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Apparently, John Baldwin and Jonathan Bloom are giving talks about their joint work (still in preparation) about a bordered monopole Floer theory. (To be honest, I haven't been to any of their talks on the subject) –  Marco Golla Apr 22 '13 at 8:59
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A different kind of answer is provided by Juhász's sutured Floer homology, usually denoted with $SFH$. In brief, $SFH$ is an invariant of a pair $(M,\Gamma)$, where $M$ is a manifold with nonempty boundary $\Gamma$ is a collection of curves (called sutures) $\partial M$ satisfying certain conditions; the complex and its differential are very similar to the ones used to compute $\widehat{HF}$.

A good reference for the construction of $SFH$ is Juhász's paper Holomorphic discs and sutured manifolds; there are some applications to contact topology, for example Honda, Kazez and Matic's papers The contact invariant in sutured Floer homology and Contact structures, sutured Floer homology and TQFT.

Let me just make a couple of remarks:

  1. Bordered Floer homology depends on some choice on the boundary, too (namely, a parametrisation $F \simeq \partial M$).

  2. Bordered Floer homology determines sutured Floer homology for any choice of the sutures.

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It's also true that sutured Floer homology, plus some natural maps between them, determines much of bordered Floer homology, due to work of Zarev: see front.math.ucdavis.edu/1010.3496 . –  Dylan Thurston Jun 1 '11 at 23:22
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