The American Mathematical Society has reissued Differential Topology as a volume in its AMS Chelsea Book Series. When it was first published thirty-six years ago, the book had a reputation for its somewhat unorthodox approach. It is amusing to think that a book continuously in print over that period of time can still carry a faint aura of unorthodoxy.
The authors aim to provide a mostly elementary approach to differential topology for advanced undergraduates or beginning graduate students. Part of their motivation was to emphasize the intuitive content of the subject. When they wrote the book, many of the big theorems (for example, Borsuk-Ulam, Lefschetz fixed- point, and Jordan-Brouwer) were customarily proved using the powerful machinery of algebraic topology. In that context, these theorems fall out almost as byproducts of exercising the machine. Here the authors make the big theorems the main subject of the book and largely abandon algebraic topology. Differential topology has a strong geometric flavor, and the authors choose to emphasize that via intersection theory and transversality.
One unusual thing the authors do is to define k-dimensional manifolds as subsets of a big ambient Euclidean space Rn locally diffeomorphic to Rk and dispense with the business of charts and atlases. This seems to annoy some purists. (Indeed, it can obscure the distinction between intrinsic properties and properties of an embedding.). Yet it seems like a reasonable pedagogical approach and is in line with an emphasis on developing geometric intuition. The reader may then be surprised by seeing Whitney’s embedding theorem proved a few dozen pages later. While that might seem curious in this setting, it does establish that no k-dimensional manifold is so pathological that it can’t be embedded in a big enough Euclidean space.
Yet another unusual aspect of the book is that many big theorems appear only in the exercises. That makes the book a bit of a do-it-yourself project, but the reader is guided into the proof and usually hints and suggestions are provided. The authors expect readers to have had a year of analysis, a semester of linear algebra, and familiarity with the basic topological concepts in Euclidean space.
The book is divided into four chapters. The first and fourth (Manifolds and Smooth Maps and Integration on Manifolds) have more or less standard material. The second and third (Transversality and Intersection and Oriented Intersection Theory) are far less standard, and they embody the heart of the authors’ intuitive and geometric approach to the subject. The reader is first introduced to intersection theory mod 2, and that’s enough to prove the Jordan-Brouwer and Borsuk-Ulam theorems. Then oriented intersection theory provides the additional tools to prove the Lefschetz fixed-point, Poincaré-Hopf index and Hopf degree theorems.
It is apparent (and the authors acknowledge) that this book owes its inspiration to Milnor’s Topology from the Differentiable Viewpoint. The current book doesn’t have the charm of Milnor’s, but it remains a strong introduction to the subject. It would serve well for self-study or for classroom use.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.