From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes

by Ib H. Madsen, Jxrgen Tornehave

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 10 chapters study cohomology of open sets in Euclidean space, treat smooth manifolds and their cohomology and end with integration on manifolds. The last 11 chapters cover Morse theory, index of vector fields, Poincare duality, vector bundles, connections and curvature, Chern and Euler classes, and Thom isomorphism, and the book ends with the general Gauss-Bonnet theorem.

The text includes well over 150 exercises, and gives the background necessary for 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 anyone who wishes to know about cohomology, curvature, and their applications.
--back cover

Genres: Mathematics, Calculus

296 pages, Paperback

First published March 1, 1997

Reviews

[James Swenson · Rating 3 out of 5]

This is a nice introduction to an area of mathematics that was somehow skipped over in my mathematical education (I completed the Ph.D.).

I'd prefer to see more examples and exercises. [Note that the exercises are collected in an appendix: someone glancing through might mistakenly believe that there aren't any.]

[Sam · Rating 1 out of 5]

Horrible text. As unreadably dense as Rudin without being nearly as streamlined or elegant. Does not deliver on the promise of being accessible to advanced undergraduates with only an understanding of "standard calculus and linear algebra" as claimed in the foreword. While you technically don't need to know multilinear algebra, differential forms, analysis, algebra, category theory, and topology before reading the book, in practice, you will be left hopelessly lost without such knowledge---frankly, a pathetic excuse for an "accessible" textbook. Reeks of everything that is horrid about mathematicians assuming that what they know is as easy for others to understand for the first time as it is for them to recall their thousandth time.

To be specific,
1. proofs skip several steps without explanation,
2. notation is incredibly confusing,
3. it's riddled with errors.

No second edition or list of corrections that I can find.