Differential Topology: First Steps (Dover Books on Mathematics) by Andrew H. Wallace (PDF)

Description

Keeping mathematical prerequisites to a minimum, this undergraduate-level text stimulates students’ intuitive understanding of topology while avoiding the more difficult subtleties and technicalities. Its focus is the method of spherical modifications and the study of critical points of functions on manifolds.No previous knowledge of topology is necessary for this text, which offers introductory material regarding open and closed sets and continuous maps in the first chapter. Succeeding chapters discuss the notions of differentiable manifolds and maps and explore one of the central topics of differential topology, the theory of critical points of functions on a differentiable manifold. Additional topics include an investigation of level manifolds corresponding to a given function and the concept of spherical modifications. The text concludes with applications of previously discussed material to the classification problem of surfaces and guidance, along with suggestions for further reading and study.

User’s Reviews

Editorial Reviews: About the Author Andrew Wallace is Professor Emeritus in the Department of Mathematics at the University of Pennsylvania.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐As someone who came to this book with some exposure to analysis, multivariate calculus, and topology, I found it really difficult to learn from this book. I knew absolutely nothing about differential topology when starting, and cannot say I know much more now. The chapters on manifolds and submanifolds were easy enough to understand, but starting with the chapters tangent spaces and critical points, and critical and noncritical levels I just became lost. The big picture overviews given at the beginning of sections were somewhat helpful, but all in all I think I came away with very little from reading this book, considering the amount of effort I put into reading it.

⭐As noted in other reviews, this book lacks many actual proofs. E.g. as one reviewer specifically remarked, the “proof” of the centrally important “Morse lemma”, Thm. 4.1 on page 57, is merely a reference to the results of the previous sequence of exercises. As we all know, when you leave out proofs, errors tend to creep in.E.g. although I am far from expert and could be wrong, it seems that the last of these exercises, the key one used, i.e. 4.12, on p. 57, is actually false, at least in the generality stated. I.e. it is claimed that any smooth function at all can be represented in some local coordinates, as a constant linear combination of squares of coordinates, near any critical point of any rank. Indeed the exercise specifically refers to diagonalizing the quadratic term of the function using entries which may be 1, -1, or 0, whereas if the rank were assumed maximal there would be no zeroes.However it seems such a diagonal quadratic form will have non isolated singularities whenever it is less than maximal rank. Thus this exercise, if true, would apparently imply that all degenerate critical points are non isolated, contradicting the example in exercise 4.10, p.56. Indeed it seems clear that when n=2, the qualitative behavior of the function x^3+y^3 near the origin cannot be replicated by any quadratic of form ax^2 +b y^2, with a and b constant. It is even more obvious for n=1, that x^3 cannot be represented as claimed, since ax^2 cannot change sign. The method of proof suggested for the exercise is the same as in all the other standard books I consulted, but the generality of the stated result is increased, apparently unjustifiably. The problem seems to be in the step where the reader is asked to arrange the coordinates so that f11(0) ≠ 0, since this cannot be done without some non vanishing hypothesis. Even then, presumably when one tries to proceed by induction, one will need actually non degeneracy. (Apparently in the degenerate case, when the rank is r < n, one can achieve a sum of ± the squares of exactly r of the n coordinates, but the other n-r coordinates will still appear with non constant coefficients.) Fortunately the Morse lemma only needs the case in which the exercise is correct, namely the maximal rank case. Since Professor Wallace was a famous expert in the topic, if this objection is correct, it may be a reminder of the risk of talking about math without giving full proofs. It may also remind that one can succeed in providing insight even while making minor mistakes. The 4 stars is for the genuine insights available here.I just notice that I wrote a more glowing review 14 years ago, before actually reading the book closely. This illustrates that the book does offer the casual reader valuable insight but denies the close reader fully convincing arguments. ⭐Good overview ⭐Wallace's "Differential Topology" is certainly the most elementary book on the subject that I've seen (and I've read dozens of such books). I wouldn't even say it is for "advanced undergraduates" - it could, and should, be read with only a background in multivariate calculus and basic linear algebra. It was intended to introduce the topological aspects of the subject without too much analytic or algebraic formalism, to build up a student's intuition. Technical details in this thin book are kept to a minimum and much of the presentation is done pictorially. Another notable feature is that it covers more advanced material, such as surgery, that most elementary books do not. However, due to a lack of rigor in some proofs as well as the limited range of topics covered, graduate students, and even senior undergrads who have studied topology, would be better served by higher-level introductory books, such as Guillemin & Pollack's ⭐, Milnor's ⭐or ⭐, or Broecker & Jaenich's ⭐, although none of these books cover surgery. Another possibility is to read Gauld's, ⭐, which is a more advanced version of this book, but that has some problems of its own (cf. my review of it). This is not a textbook, but rather is designed for self-study; ideally it should be read as preparation for one of the above books or concurrently.The presentation is so heavily weighted toward topology, there's no mention of analytical concepts such as differential forms, integration, metrics, vector bundles, or Lie groups, or even Sard's theorem or transversality, so don't expect this to be a substitute for Lee's ⭐, Lang's ⭐, or Barden & Thomas's ⭐. Instead Wallace introduces Morse theory and surgery (which he refers to by the alternative term "spherical modification"), and uses them to present a proof of the classification of 2-d surfaces that is different from the most common one based upon triangulations. He also includes standard results on embeddings, submanifolds, and immersions, as well as an introductory chapter on point-set topology for those who have no familiarity with the area. The amount of general topology covered is very small - only enough to define open sets, continuity, connectedness, and compactness - so if you haven't studied the subject already you'll need to learn it elsewhere, whereas if you have studied it, the first chapter could be skimmed or skipped.The hallmark of this book is its informality, since the purpose of the book was to develop new students' topological and geometric intuition, before they become acquainted with more abstract concepts such as algebraic topology. Many of the proofs are not that rigorous, as steps are skipped and details omitted, and a number of important results are only cited or sketched. Sometimes pictures are relied upon for key steps in a proof, as most of proofs proceed by using embeddings in Euclidean space. This informality is both the biggest advantage of the book, as it can be read and grasped relatively quickly by even very inexperienced students, without getting bogged down in technicalities, and also its biggest weakness, since it is important for budding mathematicians to become proficient in proving theorems rigorously. At some point all the handwaving gets to be a little too much for my tastes.One omission of this sort that I find particularly irritating is his almost complete failure to pay any attention to the differential structure of manifolds. Virtually never is a manifold actually shown to be smooth; even when a certain manipulation is asserted to produce homeomorphic manifolds, surgery is applied as if they were diffeomorphic. He practically uses the 2 words interchangeably, and doesn't even justify this by mentioning that all topological 2-manifolds can be smoothed. See Gauld for an example of how this detail should be handled properly.In addition, some of the proofs use roundabout or inelegant methods, even when they are not necessarily easier to understand. For example, the proof of the existence of an embedding into Euclidean space for any manifold, rather than using the standard trick of having 2 nested sets of open coverings (which the author uses elsewhere), instead uses the explicit form of the bump function (the only proof the I can ever recall doing this), necessitating a tedious calculation, and embeds in a space that is of much, much higher dimension than necessary. His proof that an injective immersion of a compact manifold is an embedding is similarly convoluted and inefficient. Also, the proofs of cancellation of certain surgeries and rearrangement of surgeries are much harder to follow than the standard ones using handles or Morse theory, although in this case those given here at least have the benefit of being more elementary.Oddly, for a book that has such a wealth of pictures and relies so much upon visualization, there are a couple of key points late in the book where a picture would have been a huge help, namely, on p. 106, when discussing rearrangements of surgeries, and especially on pp. 123-5, in the discussion of cancellation of surgeries. You will need to draw careful pictures to see what he is talking about.There are exercises frequently sprinkled throughout the text that are used in fill in important missing steps in proofs, so doing them is really essential to learning the material. In fact, parts of the book just consist of a series of exercises with the author's guidance, such as the classification of non-orientable surfaces or the proof of the Morse lemma. However, I feel that some of them are perhaps demanding too much from beginners at this stage, as they are normally proved explicitly in even the more advanced books on the subject.The are few mathematical typos, with only a couple being serious. On p. 124 it reads "phi and phi prime...respectively" when the author intends to say, "phi prime and phi in reverse...respectively." On p. 107 the number 0 is missing from the sentence "modifications of type on the 2-sphere" (they are of type 0). And most amusingly, on pp. 40-41, after constructing 2 open coverings, with the closure of one inside the other, the author explains that this is not an "unnecessary complication" but rather a "convenience," and then seemingly proceeds to not make use of this. However, he really is using it, but the reader cannot see that because in 3 places there is a prime on the variable U that should not be there.In short, for students with virtually no experience with differential topology, this is a great place to start, but it is only a small "first step."

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