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Preface Introduction and Background Interaction of Physics and Mathematics Manifolds: Smoothness and Other Structures The Basic Questions Some Basic Topological Exotica Whitehead continua Weierstrafl functions The Physics of Certain Mathematical Structures The Physics of Exotic Smoothness In Sum Algebraic Tools for Topology Introduction Prerequisites Concepts in Algebraic Topology Homotopy groups Singular homology Interplay between Homotopy and Homology Examples Axiomatic Homology Theory Conclusion Smooth Manifolds, Geometry Introduction Smooth Manifolds de Rham Cohomology Geometry: A Physical/Historical Perspective Geometry: Differential Forms Bundles, Geometry, Gauge Theory Introduction Bundles Geometry and Bundles Connections Gauge Theory: Some Physics Physical Generalizations, Yang-Mills, etc Yang-Mills Gauge Theory: Some Mathematics Gauge Theory and Moduli Space Introduction Classification of Vector and Principal Fiber Bundles Pullback bundle and homotopy theory of bundles Operations on vector bundles K-theory of vector bundles Characteristic Classes The Weil homomorphism Chern-Weil theory Stiefel- Whitney classes Introduction of Spin and Spinc Structures More on Yang-Mills Theories Non-abelian Yang-Mills theory The Concept of a Moduli Space Donaldson Theory From Donaldson to Seiberg-Witten Theory These are the famous Seiberg-Witten equations A Guide to the Classification of Manifolds Preliminaries: From Morse Theory to Surgery Morse theory and handle bodies Morse functions and topology Cobordism and Morse theory Handle bodies and surgery Handle and Handlebody Example: Handle attachment to D Handlebody decomposition Framing Surgery Application of Surgery to Low-dimensional Manifolds

• and -manifolds: algebraic topology -manifolds: surgery along knots and Thurston’s Geometrization Program Framing and Integer Surgery along a Link Blow-up and Blow-down Surgery Higher-dimensional Manifolds The simply-connected h-cobordism theorem Algebraic versus geometric intersection numbers Whitney trick and generalized Poincar conjecture The non-simply-connected s-cobordism theorem* Topological -manifolds: Casson Handles* The skeleton of a Casson handle Self-plumbing and kinky handle The Casson handle Smooth -manifolds: Kirby Calculus Kirby diagram Handle sliding Framing for -handles after the handle slide Why is Dimension so Special? Why does Whitney’s trick fail in dimension Consequences for the handle calculus Akbulut corks Constructing -manifolds from Intersection Forms The intersection form Representations of the intersection form Some properties of the intersection form Classification of quadratic forms and -manifolds Basic algebraic information about integral quadratic forms Classification of integral quadratic forms Some simple manifold constructs Algebraic subsets of the Rn Manifolds from equivalence classes and group actions Gluing and sewing of spaces Freedman’s Classification Early Exotic Manifolds Introduction Some Physical Background: Yang-Mills Mathematical Background: Sphere Bundles Milnor’s Exotic Bundles Coordinate Patch Presentation Geometrical Consequences Eells-Kuiper Smoothness Invariant Higher-dimensional Exotic Manifolds(Spheres) Classification of Manifold Structures The First Results in Dimension Four The Smoothing of the Euclidean Space Freedman’s Work on the Topology of -manifolds Applications of Donaldson Theory The First Constructions of Exotic R The first exotic R The Infinite Proliferation of Exotic R The existence of two classes The failure of the smooth h-cobordism theorem and ribbon R Akbulut corks and exotic R’s Structures on the set R of smoothings of R Explicit Descriptions of Exotic R’s Other Non-compact -manifolds Seiberg-Witten Theory: The Modern Approach The Construction of the Moduli Space Seiberg-Witten Invariants Vanishing results: Non-vanishing results: Gluing Formulas Changing of Smooth Structures by Surgery along Knots and Links The Failure of the Complete Smooth Classification Beyond Seiberg-Witten: The Cohomotopy Approach Physical Implications The Principle of Relativity Extension of Metrics Exotic Cosmology Global Anomaly Cancellation of Witten From Differential Structures to Operator Algebras and Geometric Structures Exotic Smooth Structures and General Relativity Connection change by a logarithmic transform Application to the theory of general relativity Differential Structures: From Operator Algebras to Gee metric Structures on -manifolds Differential structures and operator algebras From Akbulut corks to operator algebras Algebraic K-theory and exotic smooth structures Geometric structures on -manifolds and exotic differential structures Bibliography Index