A Heegaard-Floer TQFT for link cobordisms

We introduce a Heegaard-Floer homology functor from the category of oriented links in closed 3 3 -manifolds and oriented surface cobordisms in 4 4 -manifolds connecting them to the category of F [ v ] \mathbb {F}[\mathsf {v}] -modules and F [ v ] \mathbb {F}[\mathsf {v}] -homomorphisms between them, where F \mathbb {F} is the field with two elements. In comparison with the previously defined Topological Quantum Field Theories (TQFTs) for decorated links and link cobordisms, the construction of this paper has the advantage of being independent from the decoration. In fact, the main result of the paper is a partial invariance theorem for all versions of the Heegaard-Floer cobordism maps for decorated link cobordisms under the change of decoration. Some of the basic properties and applications of this functor are also explored. In particular, an invariant of the smooth isotopy classes of slice surfaces bounding a knot in S 3 S^3 is constructed as an element in a corresponding knot Floer homology group.