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Topologie differentielle --- Classes caracteristiques --- Classes et nombres caracteristiques --- Topologie differentielle --- Classes caracteristiques --- Classes et nombres caracteristiques

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Category theory. Homological algebra --- Characteristic classes --- Homology theory --- Loop spaces --- Classes caractéristiques --- Homologie --- Espaces de lacets --- Characteristic classes. --- Homology theory. --- Loop spaces. --- Classes caractéristiques

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This text is an introduction to the theory of differentiable manifolds and fiber bundles. The only requisites are a solid background in calculus and linear algebra, together with some basic point-set topology. The first chapter provides a comprehensive overview of differentiable manifolds. The following two chapters are devoted to fiber bundles and homotopy theory of fibrations. Vector bundles have been emphasized, although principal bundles are also discussed in detail. The last three chapters study bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres. Chapter 5, with its focus on the tangent bundle, also serves as a basic introduction to Riemannian geometry in the large. This book can be used for a one-semester course on manifolds or bundles, or a two-semester course in differential geometry. Gerard Walschap is Professor of Mathematics at the University of Oklahoma where he developed this book for a series of graduate courses he has taught over the past few years.

Geometry, Differential --- Fiber bundles (Mathematics) --- Differentiable manifolds --- Characteristic classes --- Géométrie différentielle --- Faisceaux fibrés (Mathématiques) --- Variétés différentiables --- Classes caractéristiques --- Differential geometry --- Geometry, Differential. --- Géométrie différentielle --- Faisceaux fibrés (Mathématiques) --- Variétés différentiables --- Classes caractéristiques

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Differential topology --- 515.1 --- Topology --- 515.1 Topology --- Algebraic topology --- Homology theory --- Topologie algébrique --- Topologie différentielle --- Congresses --- Congrès --- Topologie différentielle --- Foliations (Mathematics) --- Feuilletages (mathématiques) --- Topologie différentielle. --- Feuilletages (mathématiques) --- Lie, Groupes de --- Topologie differentielle --- Classes caracteristiques --- Classes et nombres caracteristiques

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Ch. 1. Introduction -- Ch. 2. The Alternating Algebra -- Ch. 3. de Rham Cohomology -- Ch. 4. Chain Complexes and their Cohomology -- Ch. 5. The Mayer-Vietoris Sequence -- Ch. 6. Homotopy -- Ch. 7. Applications of de Rham Cohomology -- Ch. 8. Smooth Manifolds -- Ch. 9. Differential Forms on Smooth Manifolds -- Ch. 10. Integration on Manifolds -- Ch. 11. Degree, Linking Numbers and Index of Vector Fields -- Ch. 12. The Poincare-Hopf Theorem -- Ch. 13. Poincare Duality -- Ch. 14. The Complex Projective Space CP n -- Ch. 15. Fiber Bundles and Vector Bundles -- Ch. 16. Operations on Vector Bundles and their Sections -- Ch. 17. Connections and Curvature -- Ch. 18. Characteristic Classes of Complex Vector Bundles -- Ch. 19. The Euler Class -- Ch. 20. Cohomology of Projective and Grassmannian Bundles -- Ch. 21. Thom Isomorphism and the General Gauss-Bonnet Formula -- App. A. Smooth Partition of Unity -- App. B. Invariant Polynomials -- App. C. Proof of Lemmas 12.12 and 12.13 -- App. D. Exercises

Characteristic classes --- Classes caractéristiques --- Formes différentielles --- 514.74 --- Algebraic and analytic methods in geometry --- Characteristic classes. --- 514.74 Algebraic and analytic methods in geometry --- Classes caractéristiques --- Formes différentielles --- Differential forms --- Homology theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Forms, Differential --- Continuous groups --- Geometry, Differential --- Classes, Characteristic --- Differential topology --- Differential forms. --- Homology theory. --- Homologie --- Variétés différentiables --- Topologie differentielle --- Analyse sur une variété --- Classes et nombres caracteristiques