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Differential Topology: First Steps

Andrew H. Wallace
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
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This is a Dover reprint of a book first published by W. A. Benjamin in 1968, part of a remarkable series of short mathematics books that also included Spivak's brilliant Calculus on Manifolds. It is good to see it back in print.

Wallace takes a very straightforward approach to structuring the book. His introduction sets the stage with its very first sentence: "What is differential topology about?" The book is Wallace's attempt to answer that question by showing his reader some specific results while keeping the pre-requisites at a minimum. The resulting approach is very geometric; in particular, Wallace assumes no knowledge of abstract algebra. (He does assume "some knowledge of the behavior of quadratic forms under linear transformations of the variables", something that one can no longer expect most undergraduates to have.) The book does not formally assume knowledge of general topology, but the brief summary in chapter 1 probably serves best as a refresher than as an introduction to the subject.

Chapters two through five introduce the basic theory of differentiable manifolds: the definition, submanifolds, tangent spaces, critical points. The meat of the book is chapter six, on spherical modifications. The book wraps up with a chapter on two-manifolds that applies all the ideas introduced before in a way that allows for a lot of pictures.

Wallace's final chapter, entitled "Second Steps", points towards further topics and gives some references. It does not, of course, contain any references to the literature after 1968, so readers will want to look elsewhere for an updated list of references.

In his review of the book in the American Mathematical Monthly (March, 1970), D. J. Sterling says that "This little gem should find its way into many libraries if for no other reason than the fact that it is an excellent example of really good mathematical exposition on the advanced undergraduate level." Well, sure, but one hopes it will also find itself in the hands of many undergraduates who would like to know a little about differential topology and what it is about.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He remembers this book from when he was an undergraduate, but he lost his original copy, and is therefore happy to have the reprint.


The table of contents is not available.