Introduction to Differential Topology
By T. Bröcker, K. Jänich
* Publisher: Cambridge University Press
* Number Of Pages: 172
* Publication Date: 1982-10-29
* ISBN-10 / ASIN: 0521284708
* ISBN-13 / EAN: 9780521284707
Product Description:
This book is intended as an elementary introduction to differential manifolds. The authors concentrate on the intuitive geometric aspects and explain not only the basic properties but also teach how to do the basic geometrical constructions. An integral part of the work are the many diagrams which illustrate the proofs. The text is liberally supplied with exercises and will be welcomed by students with some basic knowledge of analysis and topology.
Summary: Excellent graduate-level introduction, slightly marred by poor editing & translation
Rating: 5
Broecker & Jaenich's "Introduction to Differential Topology" is the best book for (reasonably proficient) first-year graduate students to acquire the basic tools for studying the topological aspects of smooth manifolds. Originally written in German in 1973 (as Einführung in die Differentialtopologie) and then translated into English in 1982, it has a high reputation among mathematicians, being praised by, e.g., Milnor and Brieskorn. In fact, Barden & Thomas were motivated to write their own book, An Introduction to Differential Manifolds, in part by the fact that this book was out of print; but now Cambridge has reprinted it.
Although an introduction, it would probably be considered too difficult by most undergraduates, as it moves rather quickly and assumes knowledge of basic topology and analysis. In contrast to, say, Guillemin & Pollack's Differential Topology, terms are always defined precisely (e.g., manifolds are 2nd countable and not assumed to be subsets of R^n) and there is relatively little motivating discussion, but rather it immediately launches into the subject. While it thoroughly covers the basics of differential topology - immersions and submanifolds, tangent and vector bundles, partitions of unity, transversality, isotopies, tubular neighborhoods, flows, Whitney's and Sard's theorems - there is no treatment of more advanced topics, such as Morse theory, surgery, or handlebodies (as in Hirsch's Differential Topology or Kosinski's Differential Manifolds), and there is only a brief mention of (co)bordism. Moreover, Riemann metrics are barely used and other diffeo-geometric/analytic aspects of smooth manifolds - differential forms, integration, Lie groups, de Rham cohomology, the Frobenius theorem - are not even hinted at, so this is definitely not fungible with Lee's Introduction to Smooth Manifolds or even Barden & Thomas.
Where the book really distinguishes itself is its conciseness, efficiency, and rigorousness. Despite keeping verbosity to a minimum (in contrast to, say, Lee), some very clear and complete explanations of key concepts are presented, such as the comparison of 3 different definitions of tangent spaces (the "algebraist's, physicist's, and geometer's" definitions), which helps to sort out any confusion the reader may have acquired from other sources. Usually, more general versions of theorems are given, yet with short proofs, such as that of Whitney's embedding theorem, Sard's theorem, the existence of collars, and the transversality theorems, and theorems are expressed in precise modern language, such as by the use of germs in the rank and inverse function theorems. Following Lang's Differential and Riemannian Manifolds (but more accessibly), dynamical systems and sprays are introduced and used to construct isotopies of embeddings and tubular neighborhoods. There's a refreshing lack of handwaving, with, e.g., connected sums and manifolds with corners being handled properly in the differential case; in fact, at the beginning of a chapter they state, "The differential topologist sometimes 'pushes' a submanifold aside, 'dents' it somewhere, 'bends' or 'deforms' it, and the handwaving which accompanies such operations all the more undermines the confidence of the observer. He believes the assertions are plausible but that they have not been proven. We propose to make such 'bending' precise by means of isotopies and embeddings...," and then they follow through on that promise. These reasons, combined with the book's wealth of useful technical lemmas and observations and many figures, all packed into only 150 pages, make it one of my 3 favorites (along with Kosinski and Milnor's Topology from the Differentiable Viewpoint) on the subject.
Every chapter includes 10-30 exercises, which are good practice for applying the theorems, with hints for the more difficult ones (which aren't that hard anyway). None of these exercises are used in the text.
There are a few faults with the book. First of all, as noted above, it would have been better to include more material, as neither more advanced topics in differential topology nor any of the analysis is covered, necessitating that this be supplemented with another text regardless of the emphasis of the course. Then there were a few errors/omissions (e.g., on p. 71 they fail to acknowledge that a theorem about locally compact spaces that they cite only holds if the target space is Hausdorff), and near the end of the book they start skipping steps in some proofs and are not as careful as in earlier chapters; the most egregious example of this is on p. 148, where they assume that a spray with certain special properties exists without demonstrating it. Also, there are a few sentences where the meaning is a bit hard to decipher, perhaps due to a poor translation (which is odd since the book was translated by the mathematician C. B. Thomas), and this is compounded by a more serious problem, namely, the copyediting was atrocious. I don't recall when I last saw this many meaning-altering misplaced commas or adverbs used as conjunctions; other editor's errors include a theorem number being used twice and different terminology alternately being used for the same thing. But being a former copyeditor, I am probably disturbed by this more than most people.
Overall this book, combined with Hirsch for the Morse theory and surgery, would constitute the ideal 1st-year graduate course in differential topology (for topology students). It also covers the core preparatory material for Kosinski as well. However, students with no prior exposure to the subject would probably be better served by looking at Guillemin & Pollack or Lee first.
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