Customer Reviews

Introduction to Topological Manifolds (Graduate Texts in Mathematics)

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Being a physicist I've always been fascinated with the use of manifolds and differential geometry in mechanics, field theory, etc ... Most differential geometry books I've encountered only devote about 1 chapter to manifolds and smooth manifolds at that. However this text takes its time to teach the reader what the author states he thinks is the minimum amount of general knowledge about topological manifolds (no discussion of smooth/analytic manifolds is included). The author takes his time developing everything from scratch, not even assuming any experience with (point set) topology, so this book is particularly suited for those who shy away from the subject just because they're not mathematicians and don't know topology. The only prerequisites are advanced calculus and linear algebra, nothing too fancy. The writing itself is very clear and while rigorous this book does not get lost in the boring lemma-theorem-proof vicious cycle so many other math books fall flat at. Throughout the book are scattered exercises for the reader to do (about 10-20 each chapter) and there are problems at the end of each chapter (no solutions/hints included). All-in-all I feel this text has offered me a much greater understanding of manifolds and the general theory dealing with them. Highly recommended.

An excellent text for a beginning graduate level class. This is NOT a comprehensive text covering the material in exhaustive detail, but it is an excellent overview of surfaces, simplicial complexes, homotopy, homology, and the briefest peek at cohomology. The sequence is efficient, and the author does a good job of motivating the discussions, rather than simply dumping an abstraction into your lap. As always, one should be familiar with point-set and groups before jumping in. If you are looking for a text at an undergraduate level, see Armstrong's Basic Topology or Kinsey's Topology of Surfaces.

This is an extremely well-written book that covers approximately the same material as Munkres' topology textbook, but with slightly less depth on point-set topology and a much greater emphasis (obviously) on manifolds. Every math book claims to be appropriate for "graduate students or advanced undergraduates," but in this case the claim is completely justified. This book has hardly any prerequisites, and should be readable by almost any serious math major. At the same time, it covers most of the "standard" theorems in a first graduate topology course, and is a great starting point before jumping into a more advanced book on manifolds.

My only complaint about the book is something that is very taste-dependent: in many parts, I thought the exposition was almost too simple. This will be very good for some readers: Lee presents his proofs in great detail, often using long chains of very simple reasoning, so by the time he's done, you can't help but be pursuaded that the theorem is true, and at the same time it is often surprising that he has proven a complicated or subtle theorem using such elementary steps. The down side I found is that it sometimes makes it harder to get the big picture of a complicated proof. Though I dislike the overly-terse proof style that is the norm in most textbooks, I found myself wishing that some of Lee's several-page elementary arguments would have been compressed into one page of slightly higher-level reasoning.

As I said, this is a matter of taste, and certainly doesn't lower my rating of the book. Some will like it, some won't, but even when I didn't like it I still found the book very easy to follow. I still prefer Munkres over Lee for reference purposes, but if your goal is to learn this material quickly and with minimal discomfort, this I'd choose Lee over Munkres hands down.

I began learning topology beyond real analysis with this book, and I found it to be a well-balanced text. This book covers every fundamental subject one needs to know without delving too much into a particular area of topology. The book begins with general topology and becomes increasingly algebraic as one progresses. Manifolds are emphasized throughout with ample examples and exercises. The presentation is very lucid and rigorous without being too pedantic.

There are more comprehensive books on topology, but this book is more apt for an introduction. I think that when one first learns about a mathematical subject, motivation is important. As a text goes deeper and deeper into the technicalities of a particular topic, the newcomer appreciates the concepts less and less and wonders where it is all leading to. This book affords just the right amount of material without causing one to reach the edge of boredom and lose sight of the bigger picture. In addition, a lot of motivation for learning the material is provided by the interspersed discussions on manifolds which are the most important topological spaces. The book prepares one for the entire field of topology in a concise manner.

Basic knowledge of metric spaces and group theory is recommended. If you are learning topology for the first time, you should definitely consider this book.

By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written. But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.

If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.

The only other books I have seen that deserve the title of both a reference and a textbook are by Lang. That being said, the exposition is at the other end of the spectrum with the goal being to teach the reader and not develop topology from the foundations in excruciating detail. The strength of the book (the same applies to the other titles by the same author) is that there is no compromise in rigor and the user-friendliness of the book relies upon the motivating discussions spread throughout the text. The author seems to have a long teaching experience and so he wisely advises the reader to work through the exercises. There is a pitfall with Lee's books: because the explanations are so lucid and intuitively satisfactory you might fool yourself that you know the material but this is not the case until you solve most of the problems.

I used Lee's 'Introduction to Smooth Manifolds' & 'Introduction to Curvature' for a few months, and I felt like it would be a good idea to complete the collection and acquire some more knowledge about topological manifolds using this book.

Overall, I think that Lee's 'Introduction to Topological Book' is an excellent book, as it is one of the few books that give both a profound intuition to the geometry of the subject, supplemented by rigorous proofs.
In my opinion, the book has only two disadvantages:
1) Not enough concrete examples.
2) It should has covered a bit more material, such as CW complexes.

All together, HIGHLY RECOMMENDED! The first chapter, giving some motivation to study manifolds is awesome!

I used this book in a graduate level topology class.
I had one semester of undergraduate topology experience prior to using this book.

The exposition in this book is excellent. Lee is very clear.

I have read various chapters of Munkres and prefer Lee. Munkres may cover more topics, but Lee, at least to me, was much more clear and gave me a better idea of what was going on and why.




I don't mean to bash Munkres! It is probably my best (or second best- to Gamelin and Greene...?) topology reference.

This book is a clear way to learn manifolds. In my opinion Lee is a great author with a clear knowledge of diferencial geometry. It worth to make this trip.

I picked this book mainly because a friend recommended this whole series to me. While I cannot say this book would make a great introduction to point set topology (I think Munkres is still the best for that), it has all that one would want to get going with manifold theory. What I liked most about this text is probably the rigor. This text will motivate the topics and give rigorous proof to many theorems. There are also many good examples to illustrate his point. I'd recommend this book, and the follow-up text "Introduction to Smooth Manifolds" to anyone interested in graduate level mathematics. Since the two texts will likely cost you less than $100, they'll make a nice addition to your math library.