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4-Manifolds and Kirby Calculus
Robert E. Gompf, University of Texas, Austin, TX, and András I. Stipsicz, ELTE, TTK, Budapest, Hungary
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1999; 558 pp; hardcover
Volume: 20
ISBN-10: 0-8218-0994-6
ISBN-13: 978-0-8218-0994-5
List Price: US$75 Member Price: US$60
Order Code: GSM/20

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.

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

Reviews

"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."

"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

4-manifolds
• Introduction
• Surfaces in 4-manifolds
• Complex surfaces
Kirby calculus
• Handlebodies and Kirby diagrams
• Kirby calculus
• More examples
Applications
• Branched covers and resolutions
• Elliptic and Lefschetz fibrations
• Cobordisms, $$h$$-cobordisms and exotic $${\mathbb{R}}^{4,}$$s
• Symplectic 4-manifolds
• Stein surfaces
Appendices
• Solutions
• Notation, important figures
• Bibliography
• Index