Robert E. Gompf, University of Texas, Austin, TX.Andras I. Stipsicz, ELTE, TTK, Budapest, Hungary.

• 4-manifolds: IntroductionSurfaces in 4-manifoldsComplex surfacesKirby calculus: Handlebodies and Kirby diagramsKirby calculusMore examplesApplications: Branched covers and resolutionsElliptic and Lefschetz fibrationsCobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$sSymplectic 4-manifoldsStein surfacesAppendices: SolutionsNotation, important figuresBibliographyIndex

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 The material will not go out of date, and however the field may develop in the future, this will be an important reference for many years to come.”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

Robert E. Gompf, University of Texas, Austin, TX.Andras I. Stipsicz, ELTE, TTK, Budapest, Hungary.

• 4-manifolds: IntroductionSurfaces in 4-manifoldsComplex surfacesKirby calculus: Handlebodies and Kirby diagramsKirby calculusMore examplesApplications: Branched covers and resolutionsElliptic and Lefschetz fibrationsCobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$sSymplectic 4-manifoldsStein surfacesAppendices: SolutionsNotation, important figuresBibliographyIndex

“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

Robert E. Gompf, University of Texas, Austin, TX.Andras I. Stipsicz, ELTE, TTK, Budapest, Hungary.

• 4-manifolds: IntroductionSurfaces in 4-manifoldsComplex surfacesKirby calculus: Handlebodies and Kirby diagramsKirby calculusMore examplesApplications: Branched covers and resolutionsElliptic and Lefschetz fibrationsCobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$sSymplectic 4-manifoldsStein surfacesAppendices: SolutionsNotation, important figuresBibliographyIndex

“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