In this 2h-per-week lecture course we will cover the foundations of differential topology, which are often assumed to be known in more advanced classes in geometry, topology and related fields. Possible topics are: transversality, Sard's theorem, de Rham cohomology, vector bundles and their classification, characteristic classes, Morse theory. The precise choice will depend on background and needs of the attending students. Basic prerequisites are the concepts of multivariable calculus, including differential forms, vector fields and the implicit function theorem, as well as the definition of differentiable manifolds. In particular, the course is also suitable for advanced bachelor students.
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Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject.
It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra.
By relying on a unifying idea—transversality—the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem.
An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course. Activate Remote Access.
Readership Undergraduate and graduate students interested in differential topology. My Holdings.