Contents -- Preface -- 1. Introduction and Background -- 1.1 Interaction of Physics and Mathematics -- 1.2 Manifolds: Smoothness and Other Structures -- 1.3 The Basic Questions -- 1.4 Some Basic Topological Exotica -- 1.4.1 Whitehead continua -- 1.4.2 Weierstrafl functions -- 1.5 The Physics of Certain Mathematical Structures -- 1.6 The Physics of Exotic Smoothness -- 1.7 In Sum -- 2. Algebraic Tools for Topology -- 2.1 Introduction -- 2.2 Prerequisites -- 2.3 Concepts in Algebraic Topology -- 2.3.1 Homotopy groups -- 2.3.2 Singular homology -- 2.4 Interplay between Homotopy and Homology -- 2.5 Examples -- 2.6 Axiomatic Homology Theory -- 2.7 Conclusion -- 3. Smooth Manifolds, Geometry -- 3.1 Introduction -- 3.2 Smooth Manifolds -- 3.3 de Rham Cohomology -- 3.4 Geometry: A Physical/Historical Perspective -- 3.5 Geometry: Differential Forms -- 4. Bundles, Geometry, Gauge Theory -- 4.1 Introduction -- 4.2 Bundles -- 4.3 Geometry and Bundles -- 4.3.1 Connections -- 4.4 Gauge Theory: Some Physics -- 4.5 Physical Generalizations, Yang-Mills, etc. -- 4.6 Yang-Mills Gauge Theory: Some Mathematics -- 5. Gauge Theory and Moduli Space -- 5.1 Introduction -- 5.2 Classification of Vector and Principal Fiber Bundles -- Pullback bundle and homotopy theory of bundles -- Operations on vector bundles -- K-theory of vector bundles -- 5.3 Characteristic Classes -- The Weil homomorphism -- Chern-Weil theory -- Stiefel- Whitney classes -- 5.4 Introduction of Spin and Spinc Structures -- 5.5 More on Yang-Mills Theories -- Non-abelian Yang-Mills theory -- 5.6 The Concept of a Moduli Space -- 5.7 Donaldson Theory -- 5.8 From Donaldson to Seiberg-Witten Theory -- These are the famous Seiberg-Witten equations. -- 6. A Guide to the Classification of Manifolds -- 6.1 Preliminaries: From Morse Theory to Surgery -- 6.1.1 Morse theory and handle bodies.

Morse functions and topology -- 6.1.2 Cobordism and Morse theory -- 6.1.3 Handle bodies and surgery -- Handle and Handlebody -- Example: Handle attachment to D2 -- Handlebody decomposition -- Framing -- Surgery -- 6.2 Application of Surgery to Low-dimensional Manifolds -- 6.2.1 1- and 2-manifolds: algebraic topology -- 6.2.2 3-manifolds: surgery along knots and Thurston's Geometrization Program -- Framing and Integer Surgery along a Link -- Blow-up and Blow-down Surgery -- 6.3 Higher-dimensional Manifolds -- 6.3.1 The simply-connected h-cobordism theorem -- Algebraic versus geometric intersection numbers -- Whitney trick and generalized Poincar6 conjecture -- 6.3.2 The non-simply-connected s-cobordism theorem* -- 6.4 Topological 4-manifolds: Casson Handles* -- The skeleton of a Casson handle -- Self-plumbing and kinky handle -- The Casson handle -- 6.5 Smooth 4-manifolds: Kirby Calculus -- Kirby diagram -- Handle sliding -- Framing for 2-handles after the handle slide -- 6.6 Why is Dimension 4 so Special? -- Why does Whitney's trick fail in dimension -- Consequences for the handle calculus -- Akbulut corks -- 6.7 Constructing 4-manifolds from Intersection Forms -- 6.7.1 The intersection form -- Representations of the intersection form -- Some properties of the intersection form -- 6.7.2 Classification of quadratic forms and 4-manifolds -- Basic algebraic information about integral quadratic forms -- Classification of integral quadratic forms -- 6.7.3 Some simple manifold constructs -- Algebraic subsets of the Rn -- Manifolds from equivalence classes and group actions -- Gluing and sewing of spaces -- 6.8 Freedman's Classification -- 7. Early Exotic Manifolds -- 7.1 Introduction -- 7.2 Some Physical Background: Yang-Mills -- 7.3 Mathematical Background: Sphere Bundles -- 7.4 Milnor's Exotic Bundles -- 7.5 Coordinate Patch Presentation.

7.6 Geometrical Consequences -- 7.7 Eells-Kuiper Smoothness Invariant -- 7.8 Higher-dimensional Exotic Manifolds(Spheres) -- 7.9 Classification of Manifold Structures -- 8. The First Results in Dimension Four -- 8.1 The Smoothing of the Euclidean Space -- 8.2 Freedman's Work on the Topology of 4-manifolds -- 8.3 Applications of Donaldson Theory -- 8.4 The First Constructions of Exotic R4 -- 8.4.1 The first exotic R4 -- 8.5 The Infinite Proliferation of Exotic R4 -- 8.5.1 The existence of two classes -- The failure of the smooth h-cobordism theorem and ribbon R4 -- Akbulut corks and exotic R4's -- Structures on the set R of smoothings of R4 -- 8.6 Explicit Descriptions of Exotic R4's -- 8.7 Other Non-compact 4-manifolds -- 9. Seiberg-Witten Theory: The Modern Approach -- 9.1 The Construction of the Moduli Space -- 9.2 Seiberg-Witten Invariants -- Vanishing results: -- Non-vanishing results: -- 9.3 Gluing Formulas -- 9.4 Changing of Smooth Structures by Surgery along Knots and Links -- 9.5 The Failure of the Complete Smooth Classification -- 9.6 Beyond Seiberg-Witten: The Cohomotopy Approach -- 10. Physical Implications -- 10.1 The Principle of Relativity -- 10.2 Extension of Metrics -- 10.3 Exotic Cosmology -- 10.4 Global Anomaly Cancellation of Witten -- 11. From Differential Structures to Operator Algebras and Geometric Structures -- 11.1 Exotic Smooth Structures and General Relativity -- Connection change by a logarithmic transform -- Application to the theory of general relativity -- 11.2 Differential Structures: From Operator Algebras to Gee metric Structures on 3-manifolds -- 11.2.1 Differential structures and operator algebras -- 11.2.2 From Akbulut corks to operator algebras -- 11.2.3 Algebraic K-theory and exotic smooth structures -- 11.2.4 Geometric structures on 3-manifolds and exotic differential structures -- Bibliography -- Index.