The Wild World of 4-ManifoldsAmerican Mathematical Soc., 2005/05/10 - 609 ページ 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 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. |
目次
Notes for Ch 1 Higher Dimensions and the hCobordism Theorem | 54 |
Topological 4Manifolds and hCobordisms | 69 |
Notes for Ch 2 Topological 4Manifolds and hCobordisms | 91 |
Smooth 4Manifolds and Intersection Forms | 105 |
Getting Acquainted with Intersection Forms | 111 |
Notes for Ch 3 Getting Acquainted with Intersection Forms | 134 |
Intersection Forms and Topology | 139 |
Notes for Ch 4 Intersection Forms and Topology | 173 |
Notes for Ch 8 Elliptic Surfaces | 317 |
Gauge Theory on 4Manifolds | 325 |
Prelude and the Donaldson Invariants | 331 |
Notes for Ch 9 Prelude and the Donaldson Invariants | 357 |
The SeibergWitten Invariants | 375 |
Notes for Ch 10 The SeibergWitten Invariants | 415 |
The Minimum Genus of Embedded Surfaces | 481 |
Notes for Ch 11 The Minimum Genus of Embedded Surfaces | 496 |
Classifications and Counterclassifications | 237 |
Notes for Ch 5 Classifications and Counterclassifications | 260 |
A Survey of Complex Surfaces | 271 |
Notes for Ch 6 Running through Complex Geometry | 283 |
The EnriquesKodaira Classification | 285 |
Notes for Ch 7 The EnriquesKodaira Classification | 299 |
Elliptic Surfaces | 301 |
The FintushelStern Surgery | 531 |
The FintushelStern Surgery | 547 |
Epilogue | 557 |
| 567 | |
| 587 | |
Errata | 611 |
他の版 - すべて表示
多く使われている語句
2–handle 3–sphere 4–dimensional admits algebraic almost-complex structure basic classes blow-up boundary called Casson handles characteristic element Chern class cobordism cobordism group cocycle cohomology complex structure complex surface conjecture Conn(L connected sum curvature defined denote diagram diffeomorphic dimension disk Donaldson elliptic surfaces embedded end-notes of chapter equivalent example exotic R4 exotic R4’s fiber fibration figure follows formula framing Freedman’s gauge genus geometry gluing h–cobordism theorem handle decomposition holomorphic homeomorphic homology class homotopy induced inside intersection form isomorphism J–holomorphic curves Kähler knot Lemma line bundle moduli space neighborhood obstruction obtain oriented proof proved result Riemannian metric Rokhlin’s theorem rrrrr rrrrr rrrrr Seiberg–Witten invariants Seiberg–Witten theory self-dual self-intersection signature signM simply-connected singular smooth 4–manifolds smooth structures sphere spin structure spinC spinor split symplectic manifolds tangent bundle topological manifolds torus trivial unimodular vanishing vector bundle