TITLE: Is there a version of Seiberg-Witten-Floer or Heegard-Floer homology for 3-manifolds with boundary?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.