The geometry of four-manifolds

Front cover
......Page 1
Title
......Page 5
Copyright
......Page 6
CONTENTS......Page 7
PREFACE......Page 9
1.1.1 Homology......Page 13
1.1.2 Some elementary examples......Page 15
1.1.3 Unimodular forms
......Page 17
1.1.4 The tangent bundle: characterislic classes and spin structures
......Page 18
1.1.5 Self-duality and special isomorphisms
......Page 19
1.1.6 Self-duality and Hodge theory......Page 21
1.2.1 Homotopy type
......Page 27
1.2.2 Manifolds with boundary......Page 28
1.2.3 Stable classification and cobordism......Page 29
1.2.4 h-cobordisms; embedded surfaces and the Whitney Lemma
......Page 33
1.3 Summary or results proved in this book
......Page 36
1.3.1 Realisation of forms
......Page 37
1.3.2 New invariants of smooth four-manifolds
......Page 38
1.3.3 Geometry: topological manifolds and homeomorphisms
......Page 39
Notes......Page 40
2.1.1 Bundles and connections
......Page 43
2.1.2 Curvature and differential operators
......Page 47
2.1.3 Anti-self-dual connections over four-manifolds
......Page 50
2.1.4 Bundle theory and characteristic classes
......Page 51
2.1.5 Holomorphic bundles
......Page 55
2.2.1 Flat connections......Page 60
2.2.2 Proof of the integrability theorem for holomorphic structures 
......Page 62
2.3.1 Gauge fixing......Page 65
2.3.2 Application of the implicit function theorem
......Page 68
2.3.3 Uhlenbeck's theorem
......Page 70
2.3.5 Estimating higher derivatives: proof of (2.3.8)
......Page 72
2.3.6 Method 0f continuity
......Page 75
2.3.7 Closedness: connections control gauge transformations
......Page 76
2.3.8 Openness-the implicit function theorem
......Page 77
2.3.9 Completion of proof......Page 79
2.3.10 Alternative approach
......Page 81
Notes......Page 84
3.1.1 Spinors and the Dirac equation
......Page 87
3.1.2 Spinors and complex structures
......Page 90
3.1.3 Connections and projections......Page 93
3.2.1 Definitions
......Page 95
3.2.2 The inversion theorem......Page 98
3.2.3 Double complexes and spectral sequences......Page 100
3.1.5 The inversion theorem for ASD connections
......Page 105
3.3.1 The correspondence......Page 108
3.3.2 Formal aspects......Page 110
3.3.3 Conformally invariant operators
......Page 112
3.3.4 The double complex......Page 116
3.3.5 Contribution from infinity......Page 120
3.3.6 Euclidean interpretation 
......Page 123
3.4.1 The basic instanton
......Page 127
3.4.2 Completion of the moduli space......Page 129
3.4.3 Coordinates on an open set in the moduli space
......Page 131
3.4.4 Interpretation of the completion
......Page 133
3.4.5 The case of the torus
......Page 135
Notes......Page 136
4.1.1 Example (i). One-instantons over S^4
......Page 138
4.1.3. Example (iii). ASD SU(2) connections over CP^2
......Page 139
4.1.5 Example (v). ASD SU(2) connections over S^2 x S^2
......Page 140
4.2.1 The orbit space
......Page 141
4.2.2 Reductible connections
......Page 143
4.2.3 The moduli space......Page 146
4.2.4 Fredholm theory
......Page 147
4.2.5 Local models for the moduli space
......Page 149
4.2.6 Discussion of examples
......Page 151
4.3.1 Review of standard theory......Page 153
4.3.2 The Fredholm case
......Page 157
4.3.3 Applications to moduli spaces
......Page 158
4.3.4 Unique continuation......Page 162
4.3.5 Proofs of transversality results......Page 164
4.3.6 Other perturbations
......Page 167
4.4 Compactification of moduli spaces
......Page 168
4.4.1 The compactification
......Page 169
4.4.2 Patching arguments......Page 170
4.4.1 Proof of the compactness theorem......Page 175
4.4.4 The removable singularities Theorem: regularity of L^2_1 solutions
......Page 178
4.4.5 Cutting off connections......Page 179
4.4.6 Completion of proof of removable Singularities theorem
......Page 181
Notes
......Page 182
TOPOLOGY AND CONNECTIONS......Page 184
5.1.1 Families of connections
......Page 185
5.1.2 Cohomology......Page 188
5.1.3 K-theory and the index of a family
......Page 194
5.1.4 Links of the reducible connections......Page 198
5.2 Three geometric constructions
......Page 199
5.2.1 Determinant line bundles
......Page 200
5.2.2 Codimension-two submanifolds......Page 203
5.2.3 Differential forms......Page 207
5.3.1 Concentrated connections: statement of the result......Page 210
5.3.2 Proof of a local version......Page 211
5.4.1 The orientation bundle Λ
......Page 215
5.4.2 Triviality 0f Λ 
......Page 216
5.4.3 SO(3) and other structure groups......Page 217
Notes......Page 218
6.1.1 The stability condition......Page 220
6.1.2 Analogy with the Fredholm alternative
......Page 223
6.1.3 Tile Weitzenböck formula and some corollaries
......Page 224
6.1.4 Generalizations......Page 227
6.2.1 The gradient flow equation
......Page 229
6.2.2 Outline of proof: closure of G^c  orbits......Page 230
6.2.3 Calculations with the gradient flow equation......Page 232
6.2.4 Weak convergence of connections......Page 235
6.2.5 The limit of the complex gauge transformations
......Page 237
6.2.6 Completion of existence proof 
......Page 240
6.2.7 Semi-stable bundles and compactification of moduli spaces
......Page 242
6.3.1 Short-time solutions
......Page 245
6.3.2 Long-time existence......Page 247
6.4.1 Versal deformations
......Page 249
6.4.2 Comparison of deformation theories......Page 252
6.4.3 Examples......Page 253
6.5.1 Symplectic geometry and moment maps......Page 256
6.5.2 Kähler manifolds
......Page 258
6.5.3 Connections over Kähler manifolds
......Page 262
6.5.4 The curvature of the determinant line bundle
......Page 264
6.5.5 Quillen's calculation......Page 267
Notes
......Page 272
7.1.1 Pseudo-differential operators......Page 275
7.1.2 Alternative proof
......Page 279
7.1.3 Excision for families
......Page 283
7.1.4 Line bundles over the compactified moduli space
......Page 285
7.1.5 Application of the Weitzenböck formula 
......Page 287
7.1.6 Orientations of moduli spaces
......Page 293
7.2 Gluing anti-self-dual connections
......Page 295
7.2.1 Preliminaries......Page 296
7.2.2 Constructing solutions......Page 298
7.2.3 L^p Theory
......Page 303
7.2.4 The gauge fixing problem 
......Page 307
7.2.5 Application of the index formula......Page 310
7.2.6 Dislinguishing the solutions
......Page 314
7.2.7 Conclusions......Page 315
7.2.8 Multiple connected sums......Page 317
7.3.1 The main result......Page 320
7.3.2 The linearized problem......Page 321
7.3.3 The non-linear case......Page 324
7.3.4 Completion of proof......Page 326
Notes......Page 327
8.1.1 The quadratic form E_8 + E_8
......Page 329
8.1.2 Other definite forms......Page 331
8.1.3 Restriction and compactification
......Page 333
8.2.1 Scaling......Page 334
8.2.2 Summary of results
......Page 336
8.3.1 Concentrated conneclions and definite forms
......Page 338
8.3.2 Proof of Theorem (1.3.2)
......Page 339
8.3.3 Comments......Page 344
8.3.4 The homology class of the link
......Page 345
8.3.5 Cohomology classes and the spin condition......Page 348
Notes......Page 352
INVARIANTS OF SMOOTH FOUR-MANIFOLDS......Page 353
9.1.1 Definition
......Page 354
9.1.2 Independence of metric......Page 355
9.1.3 Calculation for a K3 surface
......Page 357
9.2.1 SU(2) bundles, the stable range......Page 361
9.2.2 SO(3) bundles......Page 365
9.2.3 Extensions and variations of the definition......Page 367
9.3.1 Vanishing by automorphisms
......Page 375
9.3.2 Shrinking the neck
......Page 377
9.3.3 The second mechanism......Page 380
9.3.4 The general case......Page 383
Notes......Page 386
THE DIFFERENTIAL TOPOLOGY OF ALGEBRAIC SURFACES
......Page 387
10.1.1 Statement of results
......Page 388
10.1.2 The main idea......Page 390
10.1.3 Gieseker's projective embedding......Page 391
10.1.4 Technical facts about moduli spaces......Page 395
10.1.5 Restriction to curves
......Page 397
10.1.6 The detailed argument......Page 398
10.2.1 Extensions
......Page 400
10.2.2 Rank-two bundles over surfaces and configurations of points......Page 403
10.2.3 Moduli problems......Page 407
10.3 Moduli spaces of bundles over a double plane
......Page 412
10.3.1 General properties......Page 413
10.3.2 Conics and configurations 0f points
......Page 414
10.3.3 Two remarks......Page 415
10.3.4 Description of moduli spaces......Page 416
10.3.5 Bundles with C_1 = 0
......Page 418
10.3.6 Deformations and multiplicities......Page 420
10.4.1 Statement of results
......Page 422
10.4.2 Compactification
......Page 424
10.4.3 The universal bundle and slant product
......Page 425
Notes
......Page 429
I Equations in Banach spaces......Page 431
II Sobolev spaces
......Page 432
III Elliptic operators......Page 433
IV. Sobolev inequalities and non-linear problems
......Page 435
V. Further L^p theory; integral operators
......Page 437
REFERENCES......Page 439
INDEX......Page 449
Back cover
......Page 454