This text fits any course with the word "Manifold" in the title. It is a graduate level book.
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This text fits any course with the word "Manifold" in the title. It is a graduate level book.
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23 of 23 people found the following review helpful
By Malcolm
Format:Hardcover
There are few books really suitable for undergraduates who wish to get a feel for differential topology, and among them Guillemin and Pollack is probably the best. Assuming only multivariate calculus, linear algebra, and some point-set topology (with a typical analysis class covering everything in the first and third categories), G&P presents an intuitive introduction to smooth manifolds with many pictures and simple examples while avoiding much of the formalism. It is most similar to Milnor's Topology from the Differentiable Viewpoint, upon which it was based, but it has additional material, most notably on differential forms and integration.
Books on differential topology (a.k.a. smooth manifolds or differential manifolds) tend to divide neatly into 2 types. Every book begins with basic definitions of smooth manifolds, tangent vectors and spaces, differentials/derivatives, immersions, embeddings, submersions, submanifolds, diffeomorphisms, and partitions of unity. Also the inverse function theorem is at least cited, if not proved (the proof is left to the reader here), as well as Sard's theorem and some sort of embedding theorem, usually Whitney's "easy" one. But beyond that the 2 types of books diverge, with one type treating vector bundles, the Frobenius theorem, differential forms, Stokes's theorem, and de Rham cohomology, and then possibly continuing on to differential geometry or Lie groups, such as in Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds, or Warner's Foundations of Differentiable Manifolds and Lie Groups, whereas the other type focuses on Morse theory, normal bundles, tubular neighborhoods, transversality, intersection theory, degree, the Hopf degree theorem, the Poincare-Hopf index theorem, and then possibly continues on to surgery, handlebodies, or cobordism, such as in Wallace's Differential Topology: First Steps, Milnor's TFDV, or Hirsch's Differential Topology. The first type of book is most suitable for the analytic aspects of the subject whereas the second for the topological, so comparing Lee to Hirsch is really an apples-to-oranges comparison. Mathematicians must know both, of course, but physicists, for example, usually use the first type more (although Dubrovin et al.'s second book, Modern Geometry. Part 2: The Geometry and Topology of Manifolds,is more of the second type). This book largely falls into the second category, but the final chapter covers differential forms, integration, Stokes's theorem, a little de Rham cohomology, and the Gauss-Bonnet theorem, making it somewhat of a hybrid, like Berger's Differential Geometry: Manifolds, Curves, and Surfaces (which focuses more on differential geometry) and Bredon's Topology and Geometry (which focuses more on algebraic topology).
The book is at its best when explaining concepts such as smoothness, transversality, stability, Whitney's theorem, intersection number, orientation, Lefschetz fixed point theorem, etc., pictorially, discussing the concept for a while before giving the definition or theorem. Many results that also can be proved using algebraic topology, such as the Brouwer fixed-point theorem, the Borsuk-Ulam theorem or the Jordan separation theorem, are proved, making the book much more interesting than those like Lee or Lang that just focus on machinery.
The main drawback of the book is its carelessness in definitions, particularly at the beginning. As some other reviewers have noted, manifolds are defined as being subsets of some Euclidean space, and diffeomorphisms are defined as being (a type of) maps between open sets in Euclidean spaces, which obviates an explanation of transition functions, but then makes awkward the many places later where references are made to "arbitrary" manifolds, which are never defined. But beyond the introduction, this embedding in some R^N is never used in the proofs, which use only local coordinate neighborhoods, so the results hold more generally (of course, every manifold does embed in some R^N, but one cannot use the proof of Whitney's theorem given here since manifolds were defined as subsets of Euclidean space to begin with).
The other notable feature of this book is its exercises, of which there are many, most being rather easy, but some being important theorems (the Whitney immersion theorem, smooth Urysohn theorem, tubular neighborhood theorem, etc.), with frequent hints provided. Later in the book, some proofs are rendered as extended problem sets, with the proof broken down into steps and each step treated as a separate exercise. This allows the reader to build up the ability to derive these results on his/her own, as well as forcing the reader to actually practice these techniques and thus truly learn the subject matter.
Books on differential topology (a.k.a. smooth manifolds or differential manifolds) tend to divide neatly into 2 types. Every book begins with basic definitions of smooth manifolds, tangent vectors and spaces, differentials/derivatives, immersions, embeddings, submersions, submanifolds, diffeomorphisms, and partitions of unity. Also the inverse function theorem is at least cited, if not proved (the proof is left to the reader here), as well as Sard's theorem and some sort of embedding theorem, usually Whitney's "easy" one. But beyond that the 2 types of books diverge, with one type treating vector bundles, the Frobenius theorem, differential forms, Stokes's theorem, and de Rham cohomology, and then possibly continuing on to differential geometry or Lie groups, such as in Lee's Introduction to Smooth Manifolds, Lang's Differential and Riemannian Manifolds, or Warner's Foundations of Differentiable Manifolds and Lie Groups, whereas the other type focuses on Morse theory, normal bundles, tubular neighborhoods, transversality, intersection theory, degree, the Hopf degree theorem, the Poincare-Hopf index theorem, and then possibly continues on to surgery, handlebodies, or cobordism, such as in Wallace's Differential Topology: First Steps, Milnor's TFDV, or Hirsch's Differential Topology. The first type of book is most suitable for the analytic aspects of the subject whereas the second for the topological, so comparing Lee to Hirsch is really an apples-to-oranges comparison. Mathematicians must know both, of course, but physicists, for example, usually use the first type more (although Dubrovin et al.'s second book, Modern Geometry. Part 2: The Geometry and Topology of Manifolds,is more of the second type). This book largely falls into the second category, but the final chapter covers differential forms, integration, Stokes's theorem, a little de Rham cohomology, and the Gauss-Bonnet theorem, making it somewhat of a hybrid, like Berger's Differential Geometry: Manifolds, Curves, and Surfaces (which focuses more on differential geometry) and Bredon's Topology and Geometry (which focuses more on algebraic topology).
The book is at its best when explaining concepts such as smoothness, transversality, stability, Whitney's theorem, intersection number, orientation, Lefschetz fixed point theorem, etc., pictorially, discussing the concept for a while before giving the definition or theorem. Many results that also can be proved using algebraic topology, such as the Brouwer fixed-point theorem, the Borsuk-Ulam theorem or the Jordan separation theorem, are proved, making the book much more interesting than those like Lee or Lang that just focus on machinery.
The main drawback of the book is its carelessness in definitions, particularly at the beginning. As some other reviewers have noted, manifolds are defined as being subsets of some Euclidean space, and diffeomorphisms are defined as being (a type of) maps between open sets in Euclidean spaces, which obviates an explanation of transition functions, but then makes awkward the many places later where references are made to "arbitrary" manifolds, which are never defined. But beyond the introduction, this embedding in some R^N is never used in the proofs, which use only local coordinate neighborhoods, so the results hold more generally (of course, every manifold does embed in some R^N, but one cannot use the proof of Whitney's theorem given here since manifolds were defined as subsets of Euclidean space to begin with).
The other notable feature of this book is its exercises, of which there are many, most being rather easy, but some being important theorems (the Whitney immersion theorem, smooth Urysohn theorem, tubular neighborhood theorem, etc.), with frequent hints provided. Later in the book, some proofs are rendered as extended problem sets, with the proof broken down into steps and each step treated as a separate exercise. This allows the reader to build up the ability to derive these results on his/her own, as well as forcing the reader to actually practice these techniques and thus truly learn the subject matter.
19 of 21 people found the following review helpful
Format:Hardcover
First, I must comment about the reviewer below (who is obviously a greater mathematician than I) - I wouldn't recommend Bredon's book to anyone who wants to study differential topology. Man, I fought through a year of algebraic topology with that book, and I'm not sure I got a darn thing out of it! Being of a more analytic, geometric mindset, however, Guillemin and Pollack's book was right up my alley.
First, the authors make the wonderful assumption in the beginning that all manifolds live in R^n for some large enough n. This made study a great deal easier for me, as fighting through charts and atlases may not be the best place to start manifold theory (I don't mean to shortchange other important methods for working with differentiable manifolds, but rather I want to emphasize that many students might get lost in the machinery before learning anything of the theory). The book moves casually along (as the authors suggest, this book is nice for a smell-the-flowers two semester grad school class; we finished in Wisconsin in about a semester and a half before moving on to other pastures). The authors' reluctance to mention functors is also quite nice (I have asked many an algebraic topologist to describe these little guys, and the best answer I've heard is "A functor is an arrow"). A bit of analysis knowledge is nice, particularly in chapter four, and linear algebra (which seems to be a lost art, at least over here in the states) is absolutely critical.
For those of you out there who want to learn a little of this vast and incredibly interesting subject, I would highly recommend this book (even over Milnor's "Topology from the Differential Viewpoint", although the price of Milnor is much nicer). I must agree that this book is outrageously overpriced, but I ended up sucking it in for a month to spare the change for it. If Bredon is your cup of tea, so be it, but I think that most will find this book much more to their liking. One caveat, however: you MUST do some exercises. The authors leave important theorems entirely to exercises (some that come to mind are the "Stack of Records" theorem, the Jordan curve theorem, the Hopf degree theorem, the Cauchy integral formula, etc.).
First, the authors make the wonderful assumption in the beginning that all manifolds live in R^n for some large enough n. This made study a great deal easier for me, as fighting through charts and atlases may not be the best place to start manifold theory (I don't mean to shortchange other important methods for working with differentiable manifolds, but rather I want to emphasize that many students might get lost in the machinery before learning anything of the theory). The book moves casually along (as the authors suggest, this book is nice for a smell-the-flowers two semester grad school class; we finished in Wisconsin in about a semester and a half before moving on to other pastures). The authors' reluctance to mention functors is also quite nice (I have asked many an algebraic topologist to describe these little guys, and the best answer I've heard is "A functor is an arrow"). A bit of analysis knowledge is nice, particularly in chapter four, and linear algebra (which seems to be a lost art, at least over here in the states) is absolutely critical.
For those of you out there who want to learn a little of this vast and incredibly interesting subject, I would highly recommend this book (even over Milnor's "Topology from the Differential Viewpoint", although the price of Milnor is much nicer). I must agree that this book is outrageously overpriced, but I ended up sucking it in for a month to spare the change for it. If Bredon is your cup of tea, so be it, but I think that most will find this book much more to their liking. One caveat, however: you MUST do some exercises. The authors leave important theorems entirely to exercises (some that come to mind are the "Stack of Records" theorem, the Jordan curve theorem, the Hopf degree theorem, the Cauchy integral formula, etc.).
27 of 32 people found the following review helpful
By A Customer
Format:Hardcover
I had to study this for my degree. It was one of those books that one person bought and was passed around mainly due to it's outrageous cost. It has a lack of rigour that is not made up by being more intuitive or giving the reader insight into why differential topology is such a great subject.
Transversality is rightly given prominence, but you don't really walk away with a good feel for it's importance or power. Degrees, linking numbers etc I got for 10 GBP with Milnor's Topology from a Differential Point of View.
As an introduction to differential topology - with a little point set and alot of algebraic throw in - Bredon's Geometry and Topology sets the gold standard, with Darling's Differential Forms and Connections doing a good job on the differential geometry front and Milnor's book above providing bedtime reading beforehand. You can buy all three together for around the same price as this book.
Most Recent Customer Reviews
1.0 out of 5 stars
DON'T BUY THIS EDITION: Very poor printing
I loved to study this book several years ago as an undegraduate. Now I have to teach those subjects, so I decided to buy a copy for myself. Read more
Published 21 months ago by Jairo S. Bochi
5.0 out of 5 stars
ams chelsea edition
the AMS Chelsea edition appears to be a digital facsimile of the original with pixillated letters. the typeface is visibly deteriorated - a cleaner image comes from an ordinary... Read more
Published on October 29, 2010 by Lucius Schoenbaum
5.0 out of 5 stars
great old school (60's, 70's) math book
Back in the day there must have been a movement towards thin sleek books. Of course, there's tradeoffs. Read more
Published on March 27, 2010 by Mitchell Chan
5.0 out of 5 stars
A spectacular book
I agree with the reviewer who is not a "higher mathematician". Neither am I; in fact, I repeatedly found that both Milnor and Hirsch became remarkably clearer after reading the... Read more
Published on December 22, 2009 by V. Nagar
2.0 out of 5 stars
Poor beginning, good middle, ends as one long exercise.
I took differential topology as an undergraduate. We used this text. Neither I, nor any of the other students, had had any prior introduction to topological manifolds prior to... Read more
Published on August 13, 2009 by Mayer A. Landau
5.0 out of 5 stars
Good book
The book is very good, it is concise but I really like how the authors describe conceptual issues, it is easy to see the motivation behind most of the material we have covered so... Read more
Published on February 7, 2009 by Dan Barkley
5.0 out of 5 stars
Great, but it can't be your only book on the subject
As someone who normally likes to see things developed in more generality I was inclined to dislike this book, but it quickly won me over. Read more
Published on July 30, 2008 by Justin Hilburn
4.0 out of 5 stars
As a reference
This text is really great, once you know the material. It's a bit hard to learn from, but it has so much knowledge in it for people to look up. Read more
Published on December 7, 2007 by Tyler Lee Kelly
3.0 out of 5 stars
A casual introduction
We're using this text right now for my differential topology class. Over all, I find it rather hard to learn from... Read more
Published on February 10, 2007 by From Detroit
4.0 out of 5 stars
A good start....
Differential topology has influenced many areas of mathematics, and also has many applications in physics, engineering, comptuer graphics, network engineering, and economics. Read more
Published on September 29, 2001 by Dr. Lee D. Carlson
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