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

John M. Lee (Author)

Book Description

ISBN-10: 0387950265 | ISBN-13: 978-0387950266 | Publication Date: May 25, 2000 | Edition: 1

Manifolds play an important role in topology, geometry, complex analysis, algebra, and classical mechanics. Learning manifolds differs from most other introductory mathematics in that the subject matter is often completely unfamiliar. This introduction guides readers by explaining the roles manifolds play in diverse branches of mathematics and physics. The book begins with the basics of general topology and gently moves to manifolds, the fundamental group, and covering spaces.

Editorial Reviews

Review

"This book is an introduction to manifolds on the beginning graduate level. It provides a readable text allowing every mathematics student to get a good knowledge of manifolds in the same way that most students come to know real numbers, Euclidean spaces, groups, etc. It starts by showing the role manifolds play in nearly every major branch of mathematics.

The book has 13 chapters and can be divided into five major sections. The first section, Chapters 2 through 4, is a brief and sufficient introduction to the ideas of general topology: topological spaces, their subspaces, products and quotients, connectedness and compactness.

The second section, Chapters 5 and 6, explores in detail the main examples that motivate the rest of the theory: simplicial complexes, 1- and 2-manifolds. It introduces simplicial complexes in both ways---first concretely, in Euclidean space, and then abstractly, as collections of finite vertex sets. Then it gives classification theorems for 1-manifolds and compact surfaces, essentially following the treatment in W. Massey's \ref[ Algebraic topology: an introduction, Reprint of the 1967 edition, Springer, New York, 1977; MR0448331 (56 \#6638)].

The third section (the core of the book), Chapters 7--10, gives a complete treatment of the fundamental group, including a brief introduction to group theory (free products, free groups, presentations of groups, free abelian groups), as well as the statement and proof of the Seifert-Van Kampen theorem.

The fourth major section consists of Chapters 11 and 12, on covering spaces, including proofs that every manifold has a universal covering and that the universal covering space covers every other covering space, as well as quotients by free proper actions of discrete groups.

The last Chapter 13 covers homology theory, including homotopy invariance and the Mayer-Vietoris theorem.

The book gives an ample opportunity to the reader to learn the subject by working out a large number of examples, exercises and problems. The latter are collected at the end of each chapter."  (B.N. Apanasov, Mathematical Reviews) 

From the Author

There is a second edition of this book available as of January 2011: amzn.com/1441979395. Check it out!

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Product Details

• Paperback: 400 pages
• Publisher: Springer; 1 edition (May 25, 2000)
• Language: English
• ISBN-10: 0387950265
• ISBN-13: 978-0387950266
• Product Dimensions: 9.2 x 6.1 x 0.8 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 5.0 out of 5 stars  See all reviews (10 customer reviews)
• Amazon Best Sellers Rank: #1,022,751 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

44 of 44 people found the following review helpful:

5.0 out of 5 stars Review of a non-mathematician, May 7, 2002

By A Customer

This review is from: Introduction to Topological Manifolds (Graduate Texts in Mathematics) (Paperback)

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.

22 of 22 people found the following review helpful:

5.0 out of 5 stars A very readable text, April 26, 2002

By 

Carey Allen (San Francisco Bay Area) - See all my reviews

This review is from: Introduction to Topological Manifolds (Graduate Texts in Mathematics) (Paperback)

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.

14 of 16 people found the following review helpful:

5.0 out of 5 stars The easiest introduction, May 2, 2006

By 

David Charlton - See all my reviews
(REAL NAME)   

This review is from: Introduction to Topological Manifolds (Graduate Texts in Mathematics) (Paperback)

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.