Exotic smoothness and physics : differential topology and spacetime models / Torsten Asselmeyer-Maluga, Carl H. Brans.

Format:

Book

Author/Creator:

Asselmeyer-Maluga, Torsten.

Contributor:

Brans, Carl Henry, 1935-

Anne and Joseph Trachtman Memorial Book Fund.

Language:

English

Subjects (All):

Differential topology.

Space and time--Mathematical models.

Space and time.

Mathematical physics.

Physical Description:

xiv, 322 pages : illustrations ; 24 cm

Place of Publication:

Hackensack, NJ ; London : World Scientific, [2007]

Summary:

The discovery of on-standard smoothness structures on topologically trivial manifolds promises new paths of research in space-time models of theoretical physics, particularly in general relativity. Asselmeyer-Maluga (physics, the Fraunhofer Institute) and Brans (theoretical physics, Loyola U.) introduce some of the developments in the mathematics of differential topology, in particular the non-standard ("exotic") smoothness on topologically simple spaces. They introduce the basics, such as the interaction of physics and mathematics, and the characteristics of basic topological exotica, then explain the algebraic tools for topology, the geometry of smooth folds, the relations between bundles and geometry with gauge theory, the relations between gauge theory and moduli space, the classification of manifolds, early exotic manifolds, the first results in dimension four, the modern approach to the Seiberg-Witten theory, physical implications, and the move from differential structures to operator algebras and geometric structures. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)

Contents:

1.1 Interaction of Physics and Mathematics 1

1.2 Manifolds: Smoothness and Other Structures 4

1.4 Some Basic Topological Exotica 9

1.4.1 Whitehead continua 9

1.4.2 Weierstrab functions 10

1.5 The Physics of Certain Mathematical Structures 10

1.6 The Physics of Exotic Smoothness 13

2 Algebraic Tools for Topology 15

2.3 Concepts in Algebraic Topology 20

2.3.1 Homotopy groups 21

2.3.2 Singular homology 25

2.4 Interplay between Homotopy and Homology 31

2.6 Axiomatic Homology Theory 33

3 Smooth Manifolds, Geometry 35

3.2 Smooth Manifolds 35

3.3 de Rham Cohomology 41

3.4 Geometry: A Physical/Historical Perspective 49

3.5 Geometry: Differential Forms 52

4 Bundles, Geometry, Gauge Theory 55

4.2 Bundles 55

4.3 Geometry and Bundles 62

4.3.1 Connections 64

4.4 Gauge Theory: Some Physics 69

4.5 Physical Generalizations, Yang-Mills, etc. 82

4.6 Yang-Mills Gauge Theory: Some Mathematics 83

5 Gauge Theory and Moduli Space 85

5.2 Classification of Vector and Principal Fiber Bundles 86

5.3 Characteristic Classes 101

5.4 Introduction of Spin and Spin[subscript C] Structures 114

5.5 More on Yang-Mills Theories 119

5.6 The Concept of a Moduli Space 124

5.7 Donaldson Theory 126

5.8 From Donaldson to Seiberg-Witten Theory 135

6 A Guide to the Classification of Manifolds 151

6.1 Preliminaries: From Morse Theory to Surgery 153

6.1.1 Morse theory and handle bodies 153

6.1.2 Cobordism and Morse theory 159

6.1.3 Handle bodies and surgery 161

6.2 Application of Surgery to Low-dimensional Manifolds 169

6.2.1 1- and 2-manifolds: algebraic topology 169

6.2.2 3-manifolds: surgery along knots and Thurston's Geometrization Program 173

6.3 Higher-dimensional Manifolds 177

6.3.1 The simply-connected h-cobordism theorem 177

6.3.2 The non-simply-connected s-cobordism theorem 180

6.4 Topological 4-manifolds: Casson Handles 182

6.5 Smooth 4-manifolds: Kirby Calculus 187

6.6 Why is Dimension 4 so Special? 190

6.7 Constructing 4-manifolds from Intersection Forms 193

6.7.1 The intersection form 193

6.7.2 Classification of quadratic forms and 4-manifolds 197

6.7.3 Some simple manifold constructs 200

6.8 Freedman's Classification 203

7 Early Exotic Manifolds 205

7.2 Some Physical Background: Yang-Mills 206

7.3 Mathematical Background: Sphere Bundles 207

7.4 Milnor's Exotic Bundles 208

7.5 Coordinate Patch Presentation 211

7.6 Geometrical Consequences 213

7.7 Eells-Kuiper Smoothness Invariant 215

7.8 Higher-dimensional Exotic Manifolds(Spheres) 215

7.9 Classification of Manifold Structures 221

8 The First Results in Dimension Four 231

8.1 The Smoothing of the Euclidean Space 231

8.2 Freedman's Work on the Topology of 4-manifolds 234

8.3 Applications of Donaldson Theory 237

8.4 The First Constructions of Exotic R[superscript 4] 239

8.4.1 The first exotic R[superscript 4] 241

8.5 The Infinite Proliferation of Exotic R[superscript 4] 243

8.5.1 The existence of two classes 245

8.6 Explicit Descriptions of Exotic R[superscript 4]'s 248

8.7 Other Non-compact 4-manifolds 250

9 Seiberg-Witten Theory; The Modern Approach 253

9.1 The Construction of the Moduli Space 254

9.2 Seiberg-Witten Invariants 257

9.3 Gluing Formulas 259

9.4 Changing of Smooth Structures by Surgery along Knots and Links 260

9.5 The Failure of the Complete Smooth Classification 263

9.6 Beyond Seiberg-Witten: The Cohomotopy Approach 264

10 Physical Implications 267

10.1 The Principle of Relativity 267

10.2 Extension of Metrics 270

10.3 Exotic Cosmology 272

10.4 Global Anomaly Cancellation of Witten 275

11 From Differential Structures to Operator Algebras and Geometric Structures 281

11.1 Exotic Smooth Structures and General Relativity 281

11.2 Differential Structures: From Operator Algebras to Geometric Structures on 3-manifolds 291

11.2.1 Differential structures and operator algebras 292

11.2.2 From Akbulut corks to operator algebras 297

11.2.3 Algebraic K-theory and exotic smooth structures 301

11.2.4 Geometric structures on 3-manifolds and exotic differential structures 303.

Notes:

Includes bibliographical references (pages 307-322) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

981024195X :

9789810241957

OCLC:

56449715