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Introduction to Topological Manifolds (Graduate Texts in Mathematics) 1st Edition

5.0 out of 5 stars 8 customer reviews
ISBN-13: 978-0387950266
ISBN-10: 0387950265
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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) 

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Product Details

  • Series: Graduate Texts in Mathematics (Book 202)
  • Paperback: 385 pages
  • Publisher: Springer; 1 edition (May 23, 2008)
  • Language: English
  • ISBN-10: 0387950265
  • ISBN-13: 978-0387950266
  • Product Dimensions: 6.1 x 0.9 x 9.2 inches
  • Shipping Weight: 1.2 pounds
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #2,356,304 in Books (See Top 100 in Books)

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Format: 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.
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Format: 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.
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Format: Paperback
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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Format: Paperback
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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