Abstract: Previously, the Heegaard Floer invariants of 3-manifolds and links have only been defined up to isomorphism. This suffices for many applications, but to be able to talk about diffeomorphism or cobordism maps on such groups one needs a functorial construction. We will provide the missing ingredient for functoriality, and define the action of the based mapping class group. Our approach relies on the bifurcation analysis of 2-parameter families of gradient vector fields. This is joint work with Dylan Thurston and Peter Ozsváth.