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
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.
Readership
Undergraduate and graduate students interested in differential
topology
