A Short Course in Differential Topology

This book is offered as an entry in Cambridge University Press’s “program of undergraduate and beginning graduate-level textbooks for core courses, new courses, and interdisciplinary courses in pure and applied mathematics.” It qualifies under the first title: differential topology is something everyone should know — no mathematics program (especially at the graduate level) should be without a few offerings in this area. For such studies, the present book is excellent.

It scores on a number of counts: It does a solid job on the big topics that launch the subject, i.e., manifolds, the tangent space, the cotangent space — the usual suspects, as Claude Rains would have it (also of differential geometry). The author is to be commended for inserting, before he gets to the all-important material on bundles, a charter on such things as the notion of rank of a smooth map between manifolds, the inverse function theorem, regular values, and even the difference between an immersion and an imbedding: he is dotting some important i’s and crossing some important t’s. But then it’s on to the topics of bundles, specifically vector bundles, naturally culminating with a discussion of the tangent and cotangent bundles. In this context it is noteworthy that Dundas spends a good deal of space on constructions on bundles, e.g. the induced bundle, Whitney sums, and normal bundles.

After that, he does on to deal with integrability, with an explicit emphasis on flows. This is of course a very good thing to do, given, for example, the approach to Morse theory using dynamical systems. And integration on manifolds is clearly a titanic theme in its own right, ubiquitous in modern mathematics. I’m reminded of what is probably the first place I saw this material, namely, Spivak’s timeless little classic, Calculus on Manifolds.

The book starts with some marvelous and — at least to me — unexpected motivations, to wit, a discussion of how a robot’s arm operating in 3-space sweeps out surprising manifolds (like the torus), a discussion of the configuration space of a pair of electrons, and a discussion of state spaces and fibre bundles. Engineers and physicists should be happy, particularly in these days of ecumenism: witness the role quantum field theory plays in low-dimensional topology.

This Short Course in Differential Topology is first and foremost a textbook for mathematics students of the right level, and so is full of exercises. Dundas includes an appendix containing hints: so, do the exercises. Speaking of appendices, he also includes one on point set topology, and it’s quite complete. Still, the student should already know this material, and this appendix should serve only as a reference. Finally, I very much like the way the book is laid out: big margins just asking to be doodled all over, excellent delineation of things (numbered definitions, theorems, examples, notes, and exercises, and so on), and a good and engaging writing style — cf. p. 117, for example: Exercise 5.5.14 included the phrase, “There is an even groovier description of TSⁿ …” You gotta love it. It looks like a very good book.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.