Hardcover ISBN:  9780821809945 
Product Code:  GSM/20 
List Price:  $84.00 
MAA Member Price:  $75.60 
AMS Member Price:  $67.20 
Electronic ISBN:  9781470417970 
Product Code:  GSM/20.E 
List Price:  $79.00 
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Book DetailsGraduate Studies in MathematicsVolume: 20; 1999; 558 ppMSC: Primary 57; Secondary 53;
The past two decades have brought explosive growth in 4manifold 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 secondyear graduate level and offers an overview of current research. Part II is devoted to an exposition of Kirby calculus, or handlebody theory on 4manifolds. It is both elementary and comprehensive. Part III offers in depth a broad range of topics from current 4manifold research. Topics include branched coverings and the geography of complex surfaces, elliptic and Lefschetz fibrations, \(h\)cobordisms, symplectic 4manifolds, and Stein surfaces.
Applications are featured, and there are over 300 illustrations and numerous exercises with solutions in the book.ReadershipGraduate students and research mathematicians interested in lowdimensional topology.

Table of Contents

Part 1. 4manifolds

Chapter 1. Introduction

Chapter 2. Surfaces in 4manifolds

Chapter 3. Complex surfaces

Part 2. Kirby calculus

Chapter 4. Handlebodies and Kirby diagrams

Chapter 5. Kirby calculus

Chapter 6. More examples

Part 3. Applications

Chapter 7. Branched covers and resolutions

Chapter 8. Elliptic and Lefschetz fibrations

Chapter 9. Cobordisms, $h$cobordisms and exotic $\mathbb {R}^{4,}$s

Chapter 10. Symplectic 4manifolds

Chapter 11. Stein surfaces

Part 4. Appendices

Chapter 12. Solutions

Chapter 13. Notation, important figures


Additional Material

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 The material will not go out of date, and however the field may develop in the future, this will be an important reference for many years to come.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 lowdimensional 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 4manifold 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 4manifold 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 4manifold 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 4manifolds 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


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The past two decades have brought explosive growth in 4manifold 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 secondyear graduate level and offers an overview of current research. Part II is devoted to an exposition of Kirby calculus, or handlebody theory on 4manifolds. It is both elementary and comprehensive. Part III offers in depth a broad range of topics from current 4manifold research. Topics include branched coverings and the geography of complex surfaces, elliptic and Lefschetz fibrations, \(h\)cobordisms, symplectic 4manifolds, and Stein surfaces.
Applications are featured, and there are over 300 illustrations and numerous exercises with solutions in the book.
Graduate students and research mathematicians interested in lowdimensional topology.

Part 1. 4manifolds

Chapter 1. Introduction

Chapter 2. Surfaces in 4manifolds

Chapter 3. Complex surfaces

Part 2. Kirby calculus

Chapter 4. Handlebodies and Kirby diagrams

Chapter 5. Kirby calculus

Chapter 6. More examples

Part 3. Applications

Chapter 7. Branched covers and resolutions

Chapter 8. Elliptic and Lefschetz fibrations

Chapter 9. Cobordisms, $h$cobordisms and exotic $\mathbb {R}^{4,}$s

Chapter 10. Symplectic 4manifolds

Chapter 11. Stein surfaces

Part 4. Appendices

Chapter 12. Solutions

Chapter 13. Notation, important figures

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 lowdimensional 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 4manifold 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 4manifold 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 4manifold 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 4manifolds 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