• 3-Manifolds with boundary
• A-infinity modules
• Heegaard Floer Homology

First introduced by Ozsvath and Szabo, in the last decade Heegaard Floer homology has been one of the central objects of study in low dimensional topology. In its simplest version (called the hat version) it assigns to each closed oriented 3-manifold Y an F_2-vector space. In the cylindrical reformulation due to Lipshitz, this is obtained as the homology of a chain complex constructed by counting moduli spaces of some special holomorphic curves in a space associated to Y . Such invariants have been proved to be extremely effective when addressing the topology of 3-manifolds and knots, the geometric structures they carry and their relations with the 4-dimensional world.

In this sense a very interesting problem is to extend such invariants also to 3-manifolds with boundary. A more practical motivation has a computational nature. Even if a breakthrough of Sarkar and Wang showed that the simplest version of Heegaard Floer homology can be computed in a purely combinatorial way, their algorithm is really inefficient and works ad hoc for each example.

In the present work we expose a construction in this direction due to Lipshitz, Ozsvath and Thurston called bordered Heegaard Floer homology.