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?
Best Answer
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.