**Graduate Studies in Mathematics**

Volume: 20;
1999;
558 pp;
Hardcover

MSC: Primary 57;
Secondary 53

Print ISBN: 978-0-8218-0994-5

Product Code: GSM/20

List Price: $79.00

Individual Member Price: $63.20

**Electronic ISBN: 978-1-4704-1797-0
Product Code: GSM/20.E**

List Price: $79.00

Individual Member Price: $63.20

# 4-Manifolds and Kirby Calculus

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*Robert E. Gompf; András I. Stipsicz*

The past two decades have brought explosive growth in
4-manifold theory. Many books are currently appearing that approach
the topic from viewpoints such as gauge theory or algebraic
geometry. This volume, however, offers an exposition from a
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 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.

Applications are featured, and there are over 300 illustrations and
numerous exercises with solutions in the book.

#### Table of Contents

# Table of Contents

## 4-Manifolds and Kirby Calculus

- Cover Cover11 free
- Other titles in this series i2 free
- Title page v6 free
- Contents vii8 free
- Preface xi12 free
- Part I. 4-manifolds 118 free
- Introduction 320
- Surfaces in 4-manifolds 3754
- Complex surfaces 6784
- Part II. Kirby calculus 97114
- Handelbodies and Kirby diagrams 99116
- Kirby calculus 139156
- More examples 197214
- Part III. Applications 237254
- Branched covers and resolutions 239256
- Elliptic and Lefschetz fibrations 283300
- Cobordisms, ℎ-cobordisms and exotic ℝ^{4,}s 339356
- Symplectic 4-manifolds 385402
- Stein surfaces 419436
- Part IV. Appendices 453470
- Solutions 455472
- Notation, important figures 533550
- Bibliography 541558
- Index 553570 free
- Back Cover Back Cover1576

#### Readership

Graduate students and research mathematicians interested in low-dimensional topology.

#### Reviews

This book is important and valuable in that both gives a comprehensive and accessible picture of an area which has developed rapidly in the past 20 years and also provides readers with techniques to begin research in the field. The book is pedagogically very strong, with many examples and exercises. The material will not go out of date, and however the field may develop in the future, this will be an important reference for many years to come.

-- Bulletin of the London Mathematical Society

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

A complete record of the folklore related to handle calculus … All of the mathematical statements are given in absolutely precise language, and the notation and terminology used are well chosen … a very comprehensive book … Most low-dimensional topologists will want to have access to this as a reference book … any student … will be rewarded with a thorough understanding of this fascinating field.

-- Bulletin of the AMS

The book under review introduces the current state of 4-manifold topology; it is almost unique in that it does so from the point of view of differential topology. Part I of the book … would be priceless for algebraic geometers and gauge theorists who want to learn the topological aspects of the theory. Part II … is essentially independent of Part I and would make for an excellent graduate text on its own.

-- Mathematical Reviews

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

Provides a unique and comprehensive account of almost all that is known about the topology of 4-manifolds and the existing techniques for studying them … This book is important and valuable in that it gives a comprehensive and accessible picture of an area which has developed rapidly in the past 20 years and also provides the reader with techniques to begin research in the field. The book is pedagogically very strong, with many examples and exercises (including solutions to selected exercises). The material will not go out of date, and however the field may develop in the future, this will be an important reference for many years to come.

-- Bulletin of the London Mathematical Society