Differential Topology: First Steps | 
| Author: Andrew H. Wallace
Publisher: Dover Publications
Category: Book
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Avg. Customer Rating:  3 reviews
Sales Rank: 63253
Media: Paperback
Number Of Items: 1
Pages: 144
Shipping Weight (lbs): 0.4
Dimensions (in): 8.3 x 5.3 x 0.3
ISBN: 0486453170
Dewey Decimal Number: 514.72
EAN: 9780486453170
ASIN: 0486453170
Publication Date: October 27, 2006
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Product 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. 1968 edition.
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| Customer Reviews:
 Your First Time November 30, 2007
5 out of 6 found this review helpful
Wallace's is an ideal book for the budding mathematician with some interest in topology and familiarity with basic real analysis (e.g., Bartle). It takes the reader gently from first steps all the way through the complete classification of compact, smooth surfaces, with minimal fuss and bother in surprisingly few pages. (In other words, complete classification of these spaces is not as hard as one may have been led to believe elsewhere.) It is a truly wonderful book that a senior math major or beginning grad student can work their way through over a Christmas break, as I did and I hope they carry away the same fond memory that I do 35 years later.
a delight June 5, 2007
13 out of 18 found this review helpful
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.
 A quickie on differential topology September 23, 2001
28 out of 29 found this review helpful
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.....
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