AMS Chelsea Publishing: Differential Topology
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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 Poincaré-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.
