## 4-Manifolds and Kirby CalculusSince the early 1980s, there has been an explosive growth in 4-manifold theory, particularly due to the influx of interest and ideas from gauge theory and algebraic geometry. This book offers an exposition of the subject from the topological point of view. It bridges the gap to other disciplines and presents classical but important topological techniques that have not previously appeared in the literature. Part I of the text presents the basics of the theory at the second-year graduate level and offers an overview of current research. Part II is devoted to an exposition of Kirby calculus, or handlebody theory on 4-manifolds. It is both elementary and comprehensive. Part III offers in-depth treatments of a broad range of topics from current 4-manifold research. Topics include branched coverings and the geography of complex surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. The authors present many important applications. The text is supplemented with over 300 illustrations and numerous exercises, with solutions given in the book. I greatly recommend this wonderful book to any researcher in 4-manifold topology for the novel ideas, techniques, constructions, and computations on the topic, presented in a very fascinating way. I think really that every student, mathematician, and researcher interested in 4-manifold topology, should own a copy of this beautiful book. --Zentralblatt MATH This book gives an excellent introduction into the theory of 4-manifolds and can be strongly recommended to beginners in this field ... carefully and clearly written; the authors have evidently paid great attention to the presentation of the material ... contains many really pretty and interesting examples and a great number of exercises; the final chapter is then devoted to solutions of some of these ... this type of presentation makes the subject more attractive and its study easier. --European Mathematical Society Newsletter |

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### Contents

2 | |

Handlebodies and Kirby diagrams | 99 |

Kirby calculus | 139 |

More examples | 197 |

Branched covers and resolutions | 239 |

18 | 265 |

Elliptic and Lefschetz fibrations | 283 |

37 | 326 |

Stein surfaces | 419 |

Solutions | 455 |

47 | 500 |

67 | 516 |

77 | 527 |

Notation important figures | 533 |

541 | |

545 | |

### Other editions - View all

4-Manifolds and Kirby Calculus Robert E. Gompf,András I. Stipsicz,András Stipsicz No preview available - 1999 |

### Common terms and phrases

0-framed 2-handle 4-manifold admits almost-complex structure attaching circle blow-up blowing boundary cancel canonical Casson handle compact complex surface component connected sum construction contact structures Corollary corresponding critical points D2-bundle defined denote determined diffeomorphic diffeomorphism type disjoint dotted circle double branched cover elliptic fibration elliptic surfaces embedded embeds example exotic R4 fiber sum Figure framing coefficient genus given h-cobordism handle decomposition handle slide handlebody hence Hint holomorphic homeomorphic homology class homotopy intersection form invariants isomorphic isotopy Kirby diagram knot Lefschetz fibration Lefschetz pencil Legendrian Lemma line bundle manifold meridian minimal monodromy neighborhood Note obtain oriented pair Proposition Prove Section Seiberg-Witten Seifert surface self-intersection shows simply connected singular fibers smooth 4-manifold smooth structures Solution of Exercise spin structure spinc Stein surface submanifold surgery symplectic 4-manifold topological torus trivial twist unique unknot vanishing cycle