This is a panorama of the topology of simply connected smooth manifolds of dimension four. Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but too small to have room to undo them. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. The first part of the book puts things in context with a survey of higher dimensions and of topological 4-manifolds. The second part investigates the main invariant of a 4-manifold--the intersection form--and its interaction with the topology of the manifold. The third part reviews complex surfaces as an important source of examples. The fourth and final part of the book presents gauge theory. This differential-geometric method has brought to light the unwieldy nature of smooth 4-manifolds; and although the method brings new insights, it has raised more questions than answers. The structure of the book is modular and organized into a main track of approximately 200 pages, which is augmented with copious notes at the end of each chapter, presenting many extra details, proofs, and developments. To help the reader, the text is peppered with over 250 illustrations and has an extensive index. Readership Graduate students and research mathematicians interested in low-dimensional topology. Reviews "The book gives an excellent overview of 4-manifolds, with many figures and historical notes. Graduate students, nonexperts, and experts alike will enjoy browsing through it." *-- Robion C. Kirby, University of California, Berkeley* "What a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds." *-- MAA Reviews* "The author records many spectacular results in the subject ... (the author) gives the reader a taste of the techniques involved in the proofs, geometric topology, gauge theory and complex and symplectic structures. "The book has a large and up-to-date collection of references for the reader wishing to get a more detailed or rigorous knowledge of a specific topic. The exposition is user-friendly, with a large number of illustrations and examples." *-- Mathematical Reviews* |