The recent revolution in differential topology related to the discovery of non-standard ("exotic") smoothness structures on topologically trivial manifolds such as R4 suggests many exciting opportunities for applications of potentially deep importance for the spacetime models of theoretical physics, especially general relativity. This rich panoply of new differentiable structures lies in the previously unexplored region between topology and geometry. Just as physical geometry was thought to be trivial before Einstein, physicists have continued to work under the tacit - but now shown to be incorrect - assumption that differentiability is uniquely determined by topology for simple four-manifolds. Since diffeomorphisms are the mathematical models for physical coordinate transformations, Einstein's relativity principle requires that these models be physically inequivalent. This book provides an introductory survey of some of the relevant mathematics and presents preliminary results and suggestions for further applications to spacetime models.
Author(s): Torsten Asselmeyer-Maluga; Carl H. Brans
Year: 2007
Language: English
Pages: 339
Contents......Page 12
Preface......Page 8
1.1 Interaction of Physics and Mathematics......Page 16
1.2 Manifolds: Smoothness and Other Structures......Page 19
1.3 The Basic Questions......Page 21
1.4.1 Whitehead continua......Page 24
1.5 The Physics of Certain Mathematical Structures......Page 25
1.7 In Sum......Page 28
2.1 Introduction......Page 30
2.2 Prerequisites......Page 31
2.3 Concepts in Algebraic Topology......Page 35
2.3.1 Homotopy groups......Page 36
2.3.2 Singular homology......Page 40
2.4 Interplay between Homotopy and Homology......Page 46
2.5 Examples......Page 47
2.6 Axiomatic Homology Theory......Page 48
2.7 Conclusion......Page 49
3.2 Smooth Manifolds......Page 50
3.3 de Rham Cohomology......Page 56
3.4 Geometry: A Physical/Historical Perspective......Page 64
3.5 Geometry: Differential Forms......Page 67
4.2 Bundles......Page 70
4.3 Geometry and Bundles......Page 77
4.3.1 Connections......Page 79
4.4 Gauge Theory: Some Physics......Page 84
4.5 Physical Generalizations, Yang-Mills, etc.......Page 97
4.6 Yang-Mills Gauge Theory: Some Mathematics......Page 98
5.1 Introduction......Page 100
5.2 Classification of Vector and Principal Fiber Bundles......Page 101
Pullback bundle and homotopy theory of bundles......Page 103
K-theory of vector bundles......Page 107
5.3 Characteristic Classes......Page 116
The Weil homomorphism......Page 122
Chern-Weil theory......Page 124
Stiefel- Whitney classes......Page 128
5.4 Introduction of Spin and Spinc Structures......Page 129
5.5 More on Yang-Mills Theories......Page 134
Non-abelian Yang-Mills theory......Page 135
5.6 The Concept of a Moduli Space......Page 139
5.7 Donaldson Theory......Page 141
5.8 From Donaldson to Seiberg-Witten Theory......Page 150
These are the famous Seiberg-Witten equations.......Page 164
6. A Guide to the Classification of Manifolds......Page 166
6.1.1 Morse theory and handle bodies......Page 168
Morse functions and topology......Page 170
6.1.2 Cobordism and Morse theory......Page 174
Handle and Handlebody......Page 176
Example: Handle attachment to D2......Page 177
Handlebody decomposition......Page 178
Framing......Page 179
Surgery......Page 180
6.2.1 1- and 2-manifolds: algebraic topology......Page 184
6.2.2 3-manifolds: surgery along knots and Thurston’s Geometrization Program......Page 188
Blow-up and Blow-down Surgery......Page 190
6.3.1 The simply-connected h-cobordism theorem......Page 192
Whitney trick and generalized Poincar6 conjecture......Page 194
6.3.2 The non-simply-connected s-cobordism theorem*......Page 195
6.4 Topological 4-manifolds: Casson Handles*......Page 197
The skeleton of a Casson handle......Page 198
Self-plumbing and kinky handle......Page 199
The Casson handle......Page 200
Handle sliding......Page 202
Framing for 2-handles after the handle slide......Page 203
Why does Whitney’s trick fail in dimension......Page 205
Consequences for the handle calculus......Page 206
Akbulut corks......Page 207
6.7.1 The intersection form......Page 208
Representations of the intersection form......Page 210
Some properties of the intersection form......Page 211
Basic algebraic information about integral quadratic forms......Page 212
Classification of integral quadratic forms......Page 213
Manifolds from equivalence classes and group actions......Page 215
Gluing and sewing of spaces......Page 217
6.8 Freedman’s Classification......Page 218
7.1 Introduction......Page 220
7.2 Some Physical Background: Yang-Mills......Page 221
7.3 Mathematical Background: Sphere Bundles......Page 222
7.4 Milnor’s Exotic Bundles......Page 223
7.5 Coordinate Patch Presentation......Page 226
7.6 Geometrical Consequences......Page 228
7.8 Higher-dimensional Exotic Manifolds(Spheres)......Page 230
7.9 Classification of Manifold Structures......Page 236
8.1 The Smoothing of the Euclidean Space......Page 246
8.2 Freedman’s Work on the Topology of 4-manifolds......Page 249
8.3 Applications of Donaldson Theory......Page 252
8.4 The First Constructions of Exotic R4......Page 254
8.4.1 The first exotic R4......Page 256
8.5 The Infinite Proliferation of Exotic R4......Page 258
The failure of the smooth h-cobordism theorem and ribbon R4......Page 260
Structures on the set R of smoothings of R4......Page 261
8.6 Explicit Descriptions of Exotic R4’s......Page 263
8.7 Other Non-compact 4-manifolds......Page 265
9. Seiberg-Witten Theory: The Modern Approach......Page 268
9.1 The Construction of the Moduli Space......Page 269
9.2 Seiberg-Witten Invariants......Page 272
Vanishing results:......Page 273
9.3 Gluing Formulas......Page 274
9.4 Changing of Smooth Structures by Surgery along Knots and Links......Page 275
9.5 The Failure of the Complete Smooth Classification......Page 278
9.6 Beyond Seiberg-Witten: The Cohomotopy Approach......Page 279
10.1 The Principle of Relativity......Page 282
10.2 Extension of Metrics......Page 285
10.3 Exotic Cosmology......Page 287
10.4 Global Anomaly Cancellation of Witten......Page 290
11.1 Exotic Smooth Structures and General Relativity......Page 296
Connection change by a logarithmic transform......Page 301
Application to the theory of general relativity......Page 304
11.2 Differential Structures: From Operator Algebras to Gee metric Structures on 3-manifolds......Page 306
11.2.1 Differential structures and operator algebras......Page 307
11.2.2 From Akbulut corks to operator algebras......Page 312
11.2.3 Algebraic K-theory and exotic smooth structures......Page 316
11.2.4 Geometric structures on 3-manifolds and exotic differential structures......Page 318
Bibliography......Page 322
Index......Page 332
