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

In Ozsváth and Szabó (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in Ozsváth and Szabó (Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math. (2001), to appear). 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" derived in Ozsváth and Szabó (Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math. (2001), to appear), restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF⁺ for a variety of three-manifolds. Moreover, we show how the structure of HF⁺ 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 ℂℙ².