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Full Description
This book offers 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 are 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 250 illustrations and has an extensive index.
Contents
Introduction Front matter Part I: Background scenery: Introduction Contents of part I Chapter 1: Higher dimensions and the $h$-cobordism theorem Chapter 2: Topological 4-manifolds and $h$-cobordisms Part II: Smooth 4-manifolds and intersection forms: Introduction Contents of part II Chapter 3: Getting acquainted with intersection forms Chapter 4: Intersection forms and topology Chapter 5: Classifications and counterclassifications Part III: A survey of complex surfaces: Introduction Contents of part III Chapter 6: Running through complex geometry Chapter 7: The Enriques-Kodaira classification Chapter 8: Elliptic surfaces Part IV: Gauge theory on 4-manifolds: Introduction Contents of part IV Chapter 9: Prelude, and the Donaldson invariants Chapter 10: The Seiberg-Witten invariants Chapter 11: The minimum genus of embedded surfaces Chapter 12: Wildness unleashed: The Fintushel-Stern surgery Epilogue List of figures and tables List of figures graphic Bibliography Index.
