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Introduction to Topological Manifolds (Graduate Texts in Mathematics) [Hardcover]

John Lee
5.0 out of 5 stars  See all reviews (9 customer reviews)

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Book Description

December 30, 2010 1441979395 978-1441979391 2nd ed. 2011

This book is an introduction to manifolds at the beginning graduate level, and accessible to any student who has completed a solid undergraduate degree in mathematics. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields.

Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched.  The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness.

Frequently Bought Together

Introduction to Topological Manifolds (Graduate Texts in Mathematics) + Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218) + Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics)
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Editorial Reviews


From the reviews of the second edition:

“An excellent introduction to both point-set and algebraic topology at the early-graduate level, using manifolds as a primary source of examples and motivation. … The author has … fulfilled his objective of integrating a study of manifolds into an introductory course in general and algebraic topology. This text is well-organized and clearly written, with a good blend of motivational discussion and mathematical rigor. … Any student who has gone through this book should be well-prepared to pursue the study of differential geometry … .” (Mark Hunacek, The Mathematical Association of America, March, 2011)

“This book is designed for first year graduate students as an introduction to the topology of manifolds. … The book can be read with advantage by any graduate student with a good undergraduate background, and indeed by many upper class undergraduates. It can be used for self study or as a text book for a fine geometrically flavored introduction to manifolds. One which provides excellent motivation for studying the machinery needed for more advanced work.” (Jonathan Hodgson, Zentralblatt MATH, Vol. 1209, 2011)

From the Author

There is a second edition of this book available as of January 2011: Check it out! --This text refers to an out of print or unavailable edition of this title.

Product Details

  • Series: Graduate Texts in Mathematics (Book 202)
  • Hardcover: 433 pages
  • Publisher: Springer; 2nd ed. 2011 edition (December 30, 2010)
  • Language: English
  • ISBN-10: 1441979395
  • ISBN-13: 978-1441979391
  • Product Dimensions: 9.2 x 6.1 x 1 inches
  • Shipping Weight: 1.8 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (9 customer reviews)
  • Amazon Best Sellers Rank: #344,713 in Books (See Top 100 in Books)

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Customer Reviews

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Most Helpful Customer Reviews
51 of 52 people found the following review helpful
5.0 out of 5 stars Review of a non-mathematician May 7, 2002
By A Customer
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.
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26 of 26 people found the following review helpful
5.0 out of 5 stars A very readable text April 26, 2002
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.
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21 of 23 people found the following review helpful
5.0 out of 5 stars The easiest introduction May 2, 2006
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.
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8 of 8 people found the following review helpful
5.0 out of 5 stars Optimal Introduction to Topology June 6, 2006
By Pioneer
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.
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11 of 12 people found the following review helpful
5.0 out of 5 stars The printing is not up to the standard of the writing January 8, 2009
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.
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