The Wild World of 4-ManifoldsWhat a wonderful book! I strongly recommend this book to anyone, especially graduate students, interested in getting a sense of 4-manifolds. —MAA 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 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 small enough so that there is no room to undo the wildness. For example, only manifolds of dimension four can exhibit infinitely many distinct smooth structures. Indeed, their topology remains the least understood today. To put things in context, the book starts with a survey of higher dimensions and of topological 4-manifolds. In the second part, the main invariant of a 4-manifold—the intersection form—and its interaction with the topology of the manifold are investigated. In the third part, as an important source of examples, complex surfaces are reviewed. In the final fourth part of the book, gauge theory is presented; this differential-geometric method has brought to light how unwieldy smooth 4-manifolds truly are, and while bringing new insights, has raised more questions than answers. The structure of the book is modular, organized into a main track of about two hundred pages, augmented by extensive notes at the end of each chapter, where many extra details, proofs and developments are presented. To help the reader, the text is peppered with over 250 illustrations and has an extensive index. |
Contents
| 1 | |
| 5 | |
Background Scenery | 23 |
Smooth 4Manifolds and Intersection Forms | 105 |
A Survey of Complex Surfaces | 271 |
Gauge Theory on 4Manifolds | 325 |
Epilogue | 557 |
List of Figures and Tables | 559 |
| 567 | |
| 587 | |
Errata | 611 |
Back Cover | 615 |
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Common terms and phrases
2-handle 3-sphere 4-manifolds adjunction inequality admits algebraic almost-complex structure anti-self-dual attaching bound boundary Casson handles characteristic element Chern class CIP2 classification cobordism cobordism group cocycle cohomology complex surfaces conjecture connected sum CP¹ curvature defined denote diagram diffeomorphic differential dimension disk Donaldson elliptic surfaces embedded end-notes of chapter equivalent example exotic fiber fibration figure follows formula framing gauge genus geometry gluing h-cobordism H²(M holomorphic homeomorphic homology class homotopy ID² intersection form isomorphism J-holomorphic curves Kähler Kirby knot Lemma line bundle Math moduli space neighborhood obstruction obtain oriented Poincaré proof proved result Riemannian metric Rokhlin's theorem rrrrr rrrrr Seiberg-Witten equations Seiberg-Witten invariants Seiberg-Witten theory self-dual self-intersection simply-connected singular smooth manifold smooth structures sphere spin structure spinor split symplectic manifolds tangent bundle topological manifolds torus trivial unimodular vanishing