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. In this way, they present intelligent
treatments of important theorems, such as the Lefschetz fixed-point
theorem, the Poincare-Hopf index theorem, and Stokes theorem. 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.
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