Is there a version of Seiberg-Witten-Floer or Heegard-Floer homology for 3-manifolds with boundary? 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?
 A: 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.)
A: 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:


*

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

*Bordered Floer homology determines sutured Floer homology for any choice of the sutures.
A: 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.
