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From Calculus to Cohomology

From Calculus to Cohomology
De Rham Cohomology and Characteristic Classes

  • Date Published: March 1997
  • availability: Available
  • format: Paperback
  • isbn: 9780521589567


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About the Authors
  • 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, Poincaré 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.

    • Only decent treatment of subject at this level
    • Thoroughly class tested text with loads of exercises
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    Reviews & endorsements

    '… a self-contained exposition.' L'Enseignement Mathématique

    'This is a very fine book. It treats de Rham cohomology in an intellectually rigourous yet accessible manner which makes it ideal for a beginning graduate student. Moreover, it gets beyond the minimal agenda that many authors have set … A welcome addition.' Mathematika

    ' … a very polished completely self-contained introduction to the theory of differential forms … the book is very well-written … I recommend the book as an excellent first reading about curvature, cohomology and algebraic topology to anyone interested in these themes from students to active researchers, and especially to those who deliver lectures concerning the mentioned fields.' Acta. Sci. Math.

    'The book is written in a precise and clear language, it combines well topics from differential geometry, differential topology and global analysis.' European Mathematical Society

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    Product details

    • Date Published: March 1997
    • format: Paperback
    • isbn: 9780521589567
    • length: 296 pages
    • dimensions: 238 x 192 x 16 mm
    • weight: 0.51kg
    • availability: Available
  • Table of Contents

    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 Poincaré-Hopf theorem
    13. Poincaré 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.

  • Authors

    Ib H. Madsen, Aarhus Universitet, Denmark

    Jxrgen Tornehave, Aarhus Universitet, Denmark

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