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. Readership Graduate students and research mathematicians interested in lowdimensional 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."  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 Table of Contents 4manifolds  Introduction
 Surfaces in 4manifolds
 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 4manifolds
 Stein surfaces
Appendices  Solutions
 Notation, important figures
 Bibliography
 Index
