Differential Topology / Edition 1

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Overview

Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea—transversality—the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincare-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.

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Product Details

  • ISBN-13: 9780132126052
  • Publisher: Prentice Hall Professional Technical Reference
  • Publication date: 2/27/1974
  • Series: Mathematics Series
  • Edition number: 1
  • Pages: 324
  • Product dimensions: 6.10 (w) x 9.10 (h) x 0.60 (d)

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Sort by: Showing 1 Customer Reviews
  • Anonymous

    Posted June 7, 2002

    an elegantly beautiful and useful book

    This is a gorgeous book. The material, elementary differential topology of manifolds (intersection numbers, indices of vector fields, fixed points, curvature), is among the most beautiful in mathematics, and the tools used (inverse and implicit function theorems, measure zero, tangent bundles, normal bundles, smooth maps, regular values, transversality, differential forms, stokes theorem, homotopy, stability, morse functions), are among the most important a student can learn. The exposition is so clear that everything covered seems easy, and guided exercises are included in which the student gains valuable experience carrying out significant arguments himself. Any student who has learned calculus of several variables and wonders what he can do with it is invited to look here for a spectacularly lovely answer. To give credit where it is due, the 211 page book is so beautiful largely because it is an unabashed rewrite of one of the best written works in mathematics, the unparalleled 57 page 'Topology from the differentiable viewpoint' by the great John Milnor. Milnor's book is a better value, although a bit harder to read at 1/4 the pages, and has the touch of the master that the present book, good as it is, lacks. Milnor never pretends. For example he proves on page 8 the full fundamental theorem of algebra with the minimum necessary tools, whereas Guillemin Pollack delay it for 110 pages, giving the false impression that it requires the machinery they have built up in the meantime. Also their proofs of some results which are trivial consequences of previous results, such as the transversality theorem on pages 68-9 seem unneccessarily lengthy. Their treatment of differential forms also is lifted almost bodily from Spivak's little book, Calculus on manifolds. The nice discussion on the other hand of the key concept of stability is very insightful and does not seem to derive from Milnor or Spivak. To me, a lengthy preliminary development of abstract ideas such as manifolds, transversality, tubular neighborhoods, homotopy, Sard's theorem, no matter how well done, belies the claim in the introduction that the goal is to prove siginificant results with a minimum of machinery. This book is rather a beautiful development of important geometric concepts growing out of differential calculus, illustrated by meaningful and lovely results along the way. But one can prove most of those same results more easily than they do, using integration techniques they postpone to the last chapter. They also give the impression that simple winding number arguments using angle forms are somehow untituitive by labeling them 'de Rham cohomology', before going on to use them. One last quibble is that several theorems are misleadingly weak, by virtue of non standard definitions, meant to make proofs easier. [For experts, to define the Euler characteristic of X as the self intersection of the normal bundle of the product XxX, of course renders the Poincare Hopf theorem trivial, and to define the curvature of a hypersurface as the jacobian determinant of the Gauss map makes the Gauss Bonnet theorem almost as trivial. Their ¿Jordan Brouwer separation theorem¿ is much weaker than the original one since the curves treated, being smooth, have tubular neighborhoods.] It is all right to make these Bowdlerizations of the material but students should not be tricked into thinking mathematics is all so easy, when in fact the presentation of the topics has been carefully chosen and even modified to make it look so. Finally, some of the references covering unfortunate omissions to the text, are virtually unavailable, such as Wallace and Gramain. Except for these philosophical reservations, I thoroughly enjoyed this useful enhancement to Milnor's original work.

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