| Preface | p. vii |
| Introduction and Background | p. 1 |
| Interaction of Physics and Mathematics | p. 1 |
| Manifolds: Smoothness and Other Structures | p. 4 |
| The Basic Questions | p. 6 |
| Some Basic Topological Exotica | p. 9 |
| Whitehead continua | p. 9 |
| Weierstrab functions | p. 10 |
| The Physics of Certain Mathematical Structures | p. 10 |
| The Physics of Exotic Smoothness | p. 13 |
| In Sum | p. 13 |
| Algebraic Tools for Topology | p. 15 |
| Introduction | p. 15 |
| Prerequisites | p. 16 |
| Concepts in Algebraic Topology | p. 20 |
| Homotopy groups | p. 21 |
| Singular homology | p. 25 |
| Interplay between Homotopy and Homology | p. 31 |
| Examples | p. 32 |
| Axiomatic Homology Theory | p. 33 |
| Conclusion | p. 34 |
| Smooth Manifolds, Geometry | p. 35 |
| Introduction | p. 35 |
| Smooth Manifolds | p. 35 |
| de Rham Cohomology | p. 41 |
| Geometry: A Physical/Historical Perspective | p. 49 |
| Geometry: Differential Forms | p. 52 |
| Bundles, Geometry, Gauge Theory | p. 55 |
| Introduction | p. 55 |
| Bundles | p. 55 |
| Geometry and Bundles | p. 62 |
| Connections | p. 64 |
| Gauge Theory: Some Physics | p. 69 |
| Physical Generalizations, Yang-Mills, etc. | p. 82 |
| Yang-Mills Gauge Theory: Some Mathematics | p. 83 |
| Gauge Theory and Moduli Space | p. 85 |
| Introduction | p. 85 |
| Classification of Vector and Principal Fiber Bundles | p. 86 |
| Characteristic Classes | p. 101 |
| Introduction of Spin and Spin[subscript C] Structures | p. 114 |
| More on Yang-Mills Theories | p. 119 |
| The Concept of a Moduli Space | p. 124 |
| Donaldson Theory | p. 126 |
| From Donaldson to Seiberg-Witten Theory | p. 135 |
| A Guide to the Classification of Manifolds | p. 151 |
| Preliminaries: From Morse Theory to Surgery | p. 153 |
| Morse theory and handle bodies | p. 153 |
| Cobordism and Morse theory | p. 159 |
| Handle bodies and surgery | p. 161 |
| Application of Surgery to Low-dimensional Manifolds | p. 169 |
| 1- and 2-manifolds: algebraic topology | p. 169 |
| 3-manifolds: surgery along knots and Thurston's Geometrization Program | p. 173 |
| Higher-dimensional Manifolds | p. 177 |
| The simply-connected h-cobordism theorem | p. 177 |
| The non-simply-connected s-cobordism theorem | p. 180 |
| Topological 4-manifolds: Casson Handles | p. 182 |
| Smooth 4-manifolds: Kirby Calculus | p. 187 |
| Why is Dimension 4 so Special? | p. 190 |
| Constructing 4-manifolds from Intersection Forms | p. 193 |
| The intersection form | p. 193 |
| Classification of quadratic forms and 4-manifolds | p. 197 |
| Some simple manifold constructs | p. 200 |
| Freedman's Classification | p. 203 |
| Early Exotic Manifolds | p. 205 |
| Introduction | p. 205 |
| Some Physical Background: Yang-Mills | p. 206 |
| Mathematical Background: Sphere Bundles | p. 207 |
| Milnor's Exotic Bundles | p. 208 |
| Coordinate Patch Presentation | p. 211 |
| Geometrical Consequences | p. 213 |
| Eells-Kuiper Smoothness Invariant | p. 215 |
| Higher-dimensional Exotic Manifolds(Spheres) | p. 215 |
| Classification of Manifold Structures | p. 221 |
| The First Results in Dimension Four | p. 231 |
| The Smoothing of the Euclidean Space | p. 231 |
| Freedman's Work on the Topology of 4-manifolds | p. 234 |
| Applications of Donaldson Theory | p. 237 |
| The First Constructions of Exotic R[superscript 4] | p. 239 |
| The first exotic R[superscript 4] | p. 241 |
| The Infinite Proliferation of Exotic R[superscript 4] | p. 243 |
| The existence of two classes | p. 245 |
| Explicit Descriptions of Exotic R[superscript 4]'s | p. 248 |
| Other Non-compact 4-manifolds | p. 250 |
| Seiberg-Witten Theory; The Modern Approach | p. 253 |
| The Construction of the Moduli Space | p. 254 |
| Seiberg-Witten Invariants | p. 257 |
| Gluing Formulas | p. 259 |
| Changing of Smooth Structures by Surgery along Knots and Links | p. 260 |
| The Failure of the Complete Smooth Classification | p. 263 |
| Beyond Seiberg-Witten: The Cohomotopy Approach | p. 264 |
| Physical Implications | p. 267 |
| The Principle of Relativity | p. 267 |
| Extension of Metrics | p. 270 |
| Exotic Cosmology | p. 272 |
| Global Anomaly Cancellation of Witten | p. 275 |
| From Differential Structures to Operator Algebras and Geometric Structures | p. 281 |
| Exotic Smooth Structures and General Relativity | p. 281 |
| Differential Structures: From Operator Algebras to Geometric Structures on 3-manifolds | p. 291 |
| Differential structures and operator algebras | p. 292 |
| From Akbulut corks to operator algebras | p. 297 |
| Algebraic K-theory and exotic smooth structures | p. 301 |
| Geometric structures on 3-manifolds and exotic differential structures | p. 303 |
| Bibliography | p. 307 |
| Index | p. 317 |
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