Absolutely Graded Floer homologies and intersection forms for four-manifolds with boundary

In an earlier paper (math.SG/0110169), we introduced absolute gradings on the three-manifold invariants developed in math.SG/0101206 and math.SG/0105202. Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the ``complexity bounds'' (from math.SG/0101206), restrictions on knots whose surgeries give rise to lens spaces, and calculations of $\HFp$ for a variety of three-manifolds. Moreover, we show how the structure of $\HFp$ constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given three-manifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for $\CP{2}$.