Differential Topology: First Steps (Dover Books on Mathematics) [Paperback]

Andrew H. Wallace (Author)

Book Description

ISBN-10: 0486453170 | ISBN-13: 978-0486453170 | Publication Date: October 27, 2006

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. 1968 edition.

Product Details

• Paperback: 144 pages
• Publisher: Dover Publications (October 27, 2006)
• Language: English
• ISBN-10: 0486453170
• ISBN-13: 978-0486453170
• Product Dimensions: 8.5 x 5.5 x 0.3 inches
• Shipping Weight: 5.6 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #1,031,005 in Books (See Top 100 in Books)

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25 of 26 people found the following review helpful:

4.0 out of 5 stars A quickie on differential topology, September 22, 2001

By 

Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: Differential Topology; First Steps (Mathematics Monograph Series) (Paperback)

In this book, the author has given a quick taste of a very important subject, both in mathematics and in applications. Differential topology has found a niche in economics, physics, financial engineering, computer graphics, and computational biology, and it will no doubt find many more in years to come. It is also an area of mathematics that is still advancing, and there are many unsolved problems that can lead to interesting research programs. The author reviews elementary topology in the first chapter and then immediately introduces differentiable manifolds in the next. The presentation is very clear, and the author does not hesitate to use pictures to motivate and illustrate the main points. All of the discussion in these two chapters can be read easily by someone with a background in undergraduate calculus and some linear algebra. Special subsets of differentiable manifolds, the submanifolds, are considered in chapter 3, with the important embedding theorem proved. The theory of critical points follows in the next chapter. Although Morse theory is not mentioned, the author does define nondegenerate critical points, and shows, via a collection of exercises, the well-known result that a differentiable function in a neighborhood of such a point can be written as a quadratic form. A stronger embedding theorem is proven, namely one that allows an embedding of a compact manifold in such a way that the critical points are all nondegenerate. This discussion is generalized in the next chapter to critical and noncritical levels. The author motivates well the study of the neighborhood of a critical level by first discussing the properties of critical levels in the torus. The changing of the topology as one sweeps through the critical levels in this chapter is viewed as the process of spherical modification in the next one. The author does define what is meant by spherical modification, but does not use the usual terminology to discuss it, such as "cobordism" etc. he does however discuss the process of isotopy, and illustrates general position by means of intersections of curves. He illustrates these results in chapter 7 in the classification of two-dimensional manifolds. The usual proof is done in terms of simplicial complexes, but here the author proves it for differentiable 2-manifolds using critical point theory. The author ends the book by discussing how the subject could be pursued if the tools of algebraic topology were brought in. He discusses the killing of homotopy groups and motivates the theorem that an orientable compact 3-dimensional manifold can be obtained from a 3-sphere by cutting out a finite number of disjoint solid tori and filling the holes again with solid tori, with suitable identification of boundaries. He does not however prove when such constructions lead to the same 3-manifold, for this would lead to a resolution of the three-dimensional Poincare conjecture.....

10 of 10 people found the following review helpful:

4.0 out of 5 stars pictorial approach is great for total beginners, but lacks rigor, January 19, 2009

By 

Malcolm (Tokyo) - See all my reviews

This review is from: Differential Topology: First Steps (Dover Books on Mathematics) (Paperback)

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 Differential Topology, Milnor's Topology from the Differentiable Viewpoint or Collected Papers of John Milnor. Volume III: Differential Topology, or Broecker & Jaenich's Introduction to Differential Topology, although none of these books cover surgery. Another possibility is to read Gauld's, Differential Topology: An Introduction (Dover Books on Mathematics), 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 Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds (Graduate Texts in Mathematics), or Barden & Thomas's An Introduction to Differential Manifolds. 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."

25 of 32 people found the following review helpful:

5.0 out of 5 stars a delight, June 5, 2007

By 

mathwonk - See all my reviews

This review is from: Differential Topology: First Steps (Dover Books on Mathematics) (Paperback)

deep mathematics made crystal clear and even elementary (to the senior college math major).

there are very few professional research mathematicians who write for beginners as does andrew wallace. i recommend all his books, although i have only read three of them, this one which classifies surfaces via morse theory, his intro to alg top via fundamental groups, and his other intro to alg top via covering spaces, classification of surfaces by triangulation, and fundamental groups

for those who do not know, morse theory is a beautiful and simple geometric theory that extends the second derivative test from calculus of two variables. think back at the picture of a surface in three space, the graph of a function of two variables, and recall the concept of a "level curve", or curve in the domain where the function is constant.

These level curves arise from passing a horizontal plane through the graph surface and projecting the intersection curve down to the x,y plane. In the case of a paraboloid, or bowl, graph of z = X^2 + Y^2, the curves look like circles or ellipses getting wider as you slice higher and higher. Thus the level curves down in the x,y plane form concentric closed curves. It is especially interesting that at the center, the level set is not a curve at all, but a single point, the minimum point of the graph.

If we consider a saddle surface, graph of Z = X^2 - Y^2, the slice by the horizontal plane through the origin is two lines, and all others, above and below, are hyperbolas. Thus again one can see from the geometry of the level curves, the geometry of the original graph surface. Here the second derivative test says there is no extremum.

We also know that for an infinite "trough" Z = X^2, in X,Y,Z space, the test fails, as any small perturbation can change the nature of the critical point at the origin. Morse theory says that, just as the second derivative test describes the shape of the graph at points where the second derivatives form an invertible matrix, so also the geometry of a surface can be reconstructed from the level curves of a single function defined on the surface, and having only such non degenerate critical points.

I.e. if at all critical points, the second derivative is non degenerate, then the geometry of the surface is entirely determined by knowing the index of the second derivative matrix at those critical points. E.g. a sphere is characterized by supporting a smooth function with exactly two critical points, one max and one min.

In between two successive critical points, the geometry of the surface does not change, and it looks like a "cylinder" i.e. a product of an interval with a single level curve. A torus, or surface of a doughnut, is characterized by having a function with one max, one min, and two saddle points. this is really making the solution theory of differential equations come alive and visible.