TQFT structures in Heegaard Floer homology

In the early 2000s, Ozsv\'{a}th and Szab\'{o} introduced a collection of invariants for 3--manifolds and 4--manifolds called Heegaard Floer homology. To a 3--manifold they constructed a group, and to a 4--manifold which cobounds two 3--manifolds, they constructed a homomorphism between the manifolds appearing on the ends. Their invariants satisfy many of the axioms of a TQFT as described by Atiyah, however their construction has some additional restrictions which prevent it from fitting into Atiyah's framework. There is a refinement of Heegaard Floer homology for 3--manifolds containing a knot, due to Ozsv\'{a}th and Szab\'{o}, and independently Rasmussen, and a further refinement for 3--manifolds containing links, due to Ozsv\'{a}th and Szab\'{o}. It's a natural question as to whether one can define functorial maps associated to link cobordisms.In this thesis, we describe a package of cobordism maps for Heegaard Floer homology and link Floer homology. The cobordism maps satisfy an appropriate analogy of the axiomatic description of a TQFT formulated by Atiyah. To a ribbon graph cobordism between two based 3--manifolds, we associate a map between the Heegaard Floer homologies of the ends. To a decorated link cobordism, we obtain maps on the link Floer homologies of the ends. The maps associated to decorated link cobordisms reduce to the maps for ribbon graphs, in a natural way. As applications, we describe several formulas for mapping class group actions on the Heegaard Floer and knot Floer groups. We prove a new bound on a concordance invariant $\Upsilon_K(t)$ from knot Floer homology, and also see how the link cobordism maps give straightforward proofs of other bounds on concordance invariants from knot Floer homology. We also explore the interaction of the maps with conjugation actions on Heegaard Floer homology and link Floer homology, giving connected sum formulas for involutive Heegaard Floer homology and involutive knot Floer homology