The wild world of 4-manifolds /

Table Of Contents:
Preview vii
Contents of the Notes xv
Introduction 1(4)
Front matter 5(20)

Part I. Background Scenery Contents of Part I 25(82)

Higher Dimensions and the h--Cobordism Theorem 27(42)

The statement of the theorem 28(4)

Handle decompositions 32(8)

Handle moves 40(3)

Outline of proof 43(2)

The Whitney trick 45(2)

Low and high handles; handle trading 47(7)

Notes 54(15)

Topological 4--Manifolds and h--Cobordisms 69(38)

Casson handles 70(10)

The topological h--cobordism theorem 80(3)

Homology 3--spheres bound fake 4--balls 83(6)

Smooth failure: the twisted cork 89(2)

Notes 91(16)

Part II. Smooth 4--Manifolds and Intersection Forms Contents of Part II 107(166)

Getting Acquainted with Intersection Forms 111(28)

Preparation: representing homology by surfaces 112(3)

Intersection forms 115(12)

Essential example: the K3 surface 127(7)

Notes 134(5)

Intersection Forms and Topology 139(98)

Whitehead's theorem and homotopy type 140(9)

Wall's theorems and h--cobordisms 149(11)

Intersection forms and characteristic classes 160(8)

Rokhlin's theorem and characteristic elements 168(5)

Notes 173(64)

Classifications and Counterclassifications 237(36)

Serre's algebraic classification of forms 238(1)

Freedman's topological classification 239(4)

Donaldson's smooth exclusions 243(7)

Byproducts: exotic R4's 250(10)

Notes 260(13)

Part III. A Survey of Complex Surfaces Contents of Part III 273(54)

Running through Complex Geometry 275(10)

Surfaces 275(2)

Curves on surfaces 277(1)

Line bundles 278(5)

Notes 283(2)

The Enriques--Kodaira Classification 285(16)

Blow-down till nef 286(6)

How nef: numerical dimension 292(2)

Alternative: Kodaira dimension 294(1)

The Kahler case 295(1)

Complex versus diffeomorphic 296(3)

Notes 299(2)

Elliptic Surfaces 301(26)

The rational elliptic surface 302(4)

Fiber-sums 306(4)

Logarithmic transformations 310(4)

Topological classification 314(3)

Notes 317(10)

Part IV. Gauge Theory on 4--Manifolds Contents of Part IV 327(230)

Prelude, and the Donaldson Invariants 331(44)

Prelude 332(1)

Bundles, connections, curvatures 333(17)

We are special: self-duality 350(3)

The Donaldson invariants 353(4)

Notes 357(18)

The Seiberg--Witten Invariants 375(106)

Almost-complex structures 376(6)

Spin structures and spinors 382(14)

Definition of the Seiberg--Witten invariants 396(8)

Main results and properties 404(5)

Invariants of symplectic manifolds 409(3)

Invariants of complex surfaces 412(3)

Notes 415(66)

The Minimum Genus of Embedded Surfaces 481(50)

Before gauge theory: Kervaire--Milnor 482(4)

Enter the hero: the adjunction inequality 486(5)

Digression: the happy case of 3--manifolds 491(5)

Notes 496(35)

Wildness Unleashed: The Fintushel--Stern Surgery 531(26)

Gluing results in Seiberg--Witten theory 532(7)

Review: the Alexander polynomial of a knot 539(2)

The knot surgery 541(4)

Applications 545(2)

Notes 547(10)
Epilogue 557(2)
List of Figures and Tables 559(8)
Bibliography 567(20)
Index 587