Differential Topology (Practitioner Series)
By Morris W. Hirsch
* Publisher: Springer
* Number Of Pages: 222
* Publication Date: 1997-01
* ISBN-10 / ASIN: 3540901485
* ISBN-13 / EAN: 9783540901488
Product Description:
This text provides a thorough knowledge of the basic topological ideas necessary for studying differential manifolds. These topics include immersions and imbeddings, approach techniques, and the Morse classification of surfaces and their cobordism. The author keeps the mathematical prerequisites to a minimum; this and the emphasis on the geometric and intuitive aspects of the subject make the book a useful introduction for the student. There are numerous exercises on many different levels, ranging from practical applications of the theorems to significant further development of the theory.
Summary: Excellent intermediate-level text
Rating: 5
While I agree with almost everything that reviewers Paul Thurston and Dr. Carlson say about this book, I would rate it a little higher, since this book fills a niche that not too many other books occupy: It is more advanced than truly introductory treatments such as Guillemin & Pollack's Differential Topology or Milnor's Topology from the Differentiable Viewpoint but more basic than Kosinski's Differential Manifolds, and far more comprehensive than specialized books such as Milnor's Lectures on the h-Cobordism Theorem. Thus it is useful for its coverage of a wide range of topics in differential topology - embeddings, vector bundles, transversality, degree and intersection numbers, cobordism, Morse theory, isotopies - which is rigorous yet still somewhat elementary.
Hirsch writes clearly with precise definitions using modern terminology (in contrast to, say, G&P or Milnor's definition of manifold as a subset of Euclidean space). The proofs are usually compact but easy to follow, and he often explains what he is going to do ahead of time. A lot is compressed into relatively few pages - his proof of the equivalence of C^r, r>=1, and smooth structures in Chapter 2 versus Munkres' 60 page proof (in Elementary Differential Topology) being a good illustration of this - and results are often proved in much generality.
The second chapter in particular stands out, which covers function spaces and approximations and contains a general theorem that immediately yields the denseness of diffeomorphisms, embeddings, immersions, submersions, proper maps, etc., in the strong C^r topology. There is an excursion into jets, which is not found much in the literature, and is not really my cup of tea either, but it is not used much elsewhere in the book. Analytic approximations are also mentioned, although a key result in this area is only cited rather than proved (since this is not a book on complex analysis).
Beginning with Chapter 3, transversality is rightly emphasized as a central concept, while Chapter 6 introduces the Morse theory that dominates that later few chapters. A complete proof of the Morse-Sard theorem is given, although only in the smooth case, with a more general theorem only being stated (but still, this is more than any of the other texts I mentioned). Standard results such as the easy Whitney embedding, the existence of collars and tubular neighborhoods, the Brouwer fixed point theorem, the hairy ball theorem, the Hopf theorem, the Morse lemma, and the Morse inequalities are proved in addition to more advanced theorems, such as the classification of vector bundles or Thom's "fundamental theorem of cobordism," although some results, such as the computation of some cobordism groups, are only stated.
The chapter on vector bundles is actually the longest in the book, and certainly seems sufficient, although I partially agree with Mr. Thurston that it is odd that the idea of tangent vectors as derivations is not even mentioned. (Tangent vectors are essentially defined as elements of a vector bundle that transform in a certain way under coordinate changes.)
It is good to see gluings handled properly without hand-waving about smoothings, and the technical theorems on isotopies and the characterization of the disk from a Morse-theoretic perspective are a welcome addition, but the treatment of cobordism is too brief and handles and surgery are only touched upon. The classification of compact surfaces at the end is kind of overkill with all this machinery as well, but proceeding to handle decompositions or the h-cobordism theorem would make this book much longer. Consider this, instead, as preparation for those topics, which can be learned from, say, Kosinski.
Note that this book contains nothing on differential forms, integration, Riemannian geometry, or Lie groups, as it is intended for students of topology itself, rather than those who wish to apply it to study analysis or physics on manifolds. It is certainly not geared toward physicists.
With one major type of exception, there aren't that many typos or other errors, but that persistent problem is that frequently the wrong letter is used for a variable, such as f is written for g, or f for F, or U for V, or 1 for 2, etc. Sometimes words or subscripts are omitted, too, and in a few places a cross-reference is given to the wrong theorem number. The proof of Theorem 9.2.1 is a bit sloppy, though, with 3 mathematical typos, an unproven assumption, an unnecessary step, an unmentioned restriction, a couple of paragraphs in reversed order, and a reference to a theorem (8.1.9) that wasn't established in enough generality to be applied here, but no other proof is remotely like this.
The are sometimes historical remarks at the ends of sections that contain references to significant extensions. The chapters also begin with relevant quotes from mathematicians, including Whitehead's remark, '"Transversal" is a noun; the adjective is "transverse."' (Someone should tell Lang.) Most sections end with exercises, with many of them being rather challenging. In fact, this book probably has more exercises than any of the other works I cited. Many important results are contained within the exercises, too, although these are not cited elsewhere in the book.
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