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Book
vii, 286 p. : ill. ; 26 cm.
  • 1. Introduction-- 2. The alternating algebra-- 3. De Rham cohomology-- 4. Chain complexes and their cohomology-- 5. The Mayer-Vietoris sequence-- 6. Homotopy-- 7. Applications of De Rham cohomology-- 8. Smooth manifolds-- 9. Differential forms on smooth manifolds-- 10. Integration on manifolds-- 11. Degree, linking numbers and index of vector fields-- 12. The Poincare-Hopf theorem-- 13. Poincare duality-- 14. The complex projective space CPn-- 15. Fiber bundles and vector bundles-- 16. Operations on vector bundles and their sections-- 17. Connections and curvature-- 18. Characteristic classes of complex vector bundles-- 19. The Euler class-- 20. Cohomology of projective and Grassmanian bundles-- 21. Thom isomorphism and the general Gauss-Bonnet formula.
  • (source: Nielsen Book Data)9780521580595 20160528
De Rham cohomology is the cohomology of differential forms. This book offers a self-contained exposition to this subject and to the theory of characteristic classes from the curvature point of view. It requires no prior knowledge of the concepts of algebraic topology or cohomology. The first ten chapters study cohomology of open sets in Euclidean space, treat smooth manifolds and their cohomology and end with integration on manifolds. The last eleven chapters include Morse theory, index of vector fields, Poincare duality, vector bundles, connections and curvature, and the book ends with the general Gauss-Bonnet theorem. The text includes well over 150 exercises, and gives the background to the modern developments in gauge theory and geometry in four dimensions, but it also serves as an introductory course in algebraic topology. It will be invaluable to anyone studying cohomology, curvature, and their applications.
(source: Nielsen Book Data)9780521580595 20160528
Science Library (Li and Ma)
MATH-215B-01
Book
x, 221 p. : ill. ; 25 cm.
  • Introduction.- Manifolds and Maps.- Function Spaces.- Transversality.- Vector Bundles and Tubular Neighborhoods.- Degrees, Intersection Numbers and the Euler Characteristic.- Morse Theory.- Corbodism.- Isotopy.- Surfaces.- Bibliography.- Appendix.- Index.
  • (source: Nielsen Book Data)9780387901480 20160528
This book gives the reader a thorough knowledge of the basic topological ideas necessary for studying differential manifolds. These topics include immersions and imbeddings, approach techniques, and the Morse classification of surfaces and their cobordism. The author keeps the mathematical prerequisites to a minimum; this and the emphasis on the geometric and intuitive aspects of the subject make the book an excellent and useful introduction for the student. There are numerous exercises on many different levels ranging from practical applications of the theorems to significant further development of the theory and including some open research problems.
(source: Nielsen Book Data)9780387901480 20160528
Science Library (Li and Ma)
MATH-215B-01