### Abstract

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 ℂℙ^{2}.

Original language | English (US) |
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Pages (from-to) | 179-261 |

Number of pages | 83 |

Journal | Advances in Mathematics |

Volume | 173 |

Issue number | 2 |

DOIs | |

State | Published - Feb 10 2003 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Casson's invariant
- Floer homology
- Intersection forms
- Lens space surgeries