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 handle body 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

Part 1. 4-Manifolds

Chapter 1. Introduction

Chapter 2. Surfaces in 4-manifolds

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 R4’s

Chapter 10. Symplectic 4-manifolds

Chapter 11. Stein surfaces

Part 4. Appendices

Chapter 12. Solutions

Chapter 13. Notation, important figures

About the Authors

Robert E. Gompf is an American mathematician best known for his work in 4-dimensional topology and symplectic topology. He is particularly famous for developing powerful techniques for constructing and studying smooth 4-manifolds, including the introduction of what are now called Gompf nuclei and major contributions to the theory of symplectic 4-manifolds. He is a professor at the University of Texas at Austin and is also the coauthor, with András I. Stipsicz, of the influential textbook 4-Manifolds and Kirby Calculus.

András I. Stipsicz is a Hungarian mathematician whose research focuses primarily on 4-dimensional topology, low-dimensional topology, and contact and symplectic geometry. He has made important contributions to the study of 4-manifolds, Stein surfaces, Lefschetz fibrations, and contact structures. Stipsicz is a professor at the Alfréd Rényi Institute of Mathematics in Budapest and has collaborated extensively with Gompf. Their book 4-Manifolds and Kirby Calculus is considered an important reference for researchers and advanced students studying the topology of 4-dimensional manifolds.