Customer Reviews

Differential Topology: An Introduction (Dover Books on Mathematics)

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Gauld's "Differential Topology" is primarily a more advanced version of Wallace's Differential Topology: First Steps. The topics covered are almost identical, including an introduction to topology and the classification of smooth surfaces via surgery, and a few of the pictures and some of the terminology ("disconnecting surgery," "twisting surgery") are the same, too. But overall, Gauld is written at a higher level (even though it is also an introduction to the subject, for undergrads) and is much more rigorous. Also as with Wallace, the presentation focuses on topology, with no coverage of such analytic and geometric concepts such as Riemann metrics, differential forms, integration, Lie groups, etc., so this is not fungible with Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds, or Barden & Thomas's An Introduction to Differential Manifolds, in addition to being more elementary than these books. I probably would've rated this 5 stars if not for the unusual presentation of topological spaces and the sloppiness in the proof of the aforementioned classification.

The book begins with a 3-chapter introduction to point-set topology. Like Wallace, the amount of material covered is inadequate as a substitute for a course in general topology - only basic definitions (openness, neighborhoods, continuity) and Hausdorffness, connectedness, and compactness are covered. There is a decent amount of motivating discussion and examples, but this is done in by a serious flaw: The author uses his pet approach to topology via "nearness." Nearness is essentially the property of a point being an accumulation point (or limit point or point of closure - the terminology varies among authors) of a set. If one starts with a nearness relation, defined on the collection of points and subsets of a set, then one can define the concepts of neighborhood, openness, and continuity in terms of this relation. This is an idea that was pioneered in the '70s, first for the definition of continuity (which is particularly nice if one uses nearness) and then, by the author, for the definition of topology, where it is not as useful. This approach never caught on, so a beginning student who learns topology for the first time using this book may have problems when reading just about any other book on the subject; conversely, if that student has already learned topology elsewhere, this book will at first seem confusing. This aspect is probably the worst feature of this book.

After the brief introduction to topology, manifolds are defined and some useful analytic tools in R^n, such as the inverse function theorem, the Morse lemma, and bump functions, are treated. Another deficiency here is that manifolds are defined with no mention of the 2nd countability (or paracompactness) of the manifold. This isn't necessary for the purposes of this book since partitions of unity (and their applications) are never discussed and most of the theorems concern compact manifolds only, but the full definition should've been given for completeness (cf. Wallace, who does give the proper definition even though he, too, has no need for the 2nd countability). Morse theory is introduced fairly early in the book and then developed over the course of several chapters - the presentation is not as unified or complete as in Milnor's books, but is certainly superior to that of Wallace.

The book includes the standard proof of the easier Whitney embedding theorem (oddly, divided between chapters 7 and 15), as well as all 3 definitions of tangent spaces (similar to Broecker & Jaenich's Introduction to Differential Topology, but less concise) and some coverage of vector fields, albeit without precisely defining the tangent bundle (or mentioning vector bundles at all).

What really sets this book apart is the emphasis on differential structures and orientability via charts and bases of them. In other words, throughout the book, showing that a manifold (or map) is smooth or orientable, and comparing different smooth structure or orientations, is done by working with explicit choices of charts. It is this focus that is utterly lacking in Wallace. As an example, it is shown how to create infinitely many "different" differential structures on the real line by using different charts (but the author fails to note that these are all diffeomorphic). The many pictures of coordinate patches on manifolds and computations involving Jacobians really show the reader how one can deduce these properties in practice.

The treatment of surgery, from multiple perspectives, is also outstanding. First it is defined using embedded spheres and balls, and then the notion of differentiable gluing is used to improve the construction, to show that it is smooth (with no handwaving about smoothing corners). Next, the trace of the surgery is defined and that is related to the neighborhood of a critical point, using several different explicit representations of that neighborhood, with many excellent diagrams showing the different possibilities. Applying surgical techniques to classify surfaces starts well, even first using it to classify 1-manifolds, but the exposition starts to break down, with some results concerning moving critical points being proved in an informal manner, and a very confusing explanation accompanying an equally confusing diagram on page 200. Most disappointing is that Gauld substitutes homeomorphic for diffeomorphic in his ultimate theorem, but at least he (unlike Wallace) acknowledges this deficiency and points out that it can be overcome. After being so careful for most of the book, Chapter 14 should've been written at the same level of rigor.

The 2 appendices fill in some missing details in the proofs (Appendix A) and point toward extensions of the material (Appendix B). However, even in Appendix A, most of the proofs are just sketched. Moreover, and this is another irritating feature of the presentation, throughout the text a proof will sometimes just be given as one word: "Omitted." This means that the proof, or its sketch, is in Appendix A, but the author never tells the reader this. And then there are a number of places where a fact is used (such as that the genus of a surface is finite) but not justified, making it appear as if the author is making an unsupported assumption when in fact the claim is proved in the appendix; references to Appendix A really should've been added to the text. Appendix B touches upon other topics, such as other separation properties besides Hausdorffness, the classification of nonorientable surfaces (this should've been treated in the main body of the text, in more detail), Sard's theorem (not proved here), the smooth Brouwer fixed point theorem without homotopy theory, Hausdorff dimension (really out of place here), and the barest mention of dynamical systems. Most of this appendix is too little to be of any use.

This is a reprint of a camera-ready typescript (i.e., it looks like it was typed on an old typewriter), so it is hard on the eyes. Despite this, it is relatively free from typos, as well as other errors, until near the end of the book (e.g., in the last paragraph on p. 201 the variable S should have a tilde twice, on p. 202 the variable epsilon should've been defined before it was used, on p. 153 it should read "surgery of type (m,n)" not "to type (m,n)," on p. 112 the definition of U should include the words "range of h" not "domain of h" and the word "open" should've been inserted before "neighborhood" a few lines earlier).

There are a good amount of straightforward exercises following each chapter, with the results of some being cited in the text, so this could serve well as a textbook (it was designed for an experimental course), notwithstanding the issues that I've raised concerning some of its definitions. Don't be put off by the silly limericks that open the book, either. It's a shame - with a little more work and a little less idiosyncrasy, this could've been the best book on the differential topology for undergraduates, but even as is it makes a nice companion to Guillemin & Pollack's Differential Topology in that their overlap is not that large, so the 2 books combined do a good job of covering the subject.

Nicely written, in understandable language, this book should stand amongst the references of its kind.