Introduction to Topological Manifolds (Graduate Texts in Mathematics) | 
 enlarge | Author: John M. Lee
Publisher: Springer
Category: Book
List Price: $49.95
Buy New: $37.94
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Rating:  8 reviews
Sales Rank: 238075
Media: Paperback
Edition: 1
Pages: 400
Number Of Items: 1
Shipping Weight (lbs): 1.3
Dimensions (in): 9 x 6 x 1
ISBN: 0387950265
Dewey Decimal Number: 514.3
EAN: 9780387950266
Publication Date: May 25, 2000
Availability: Usually ships in 1-2 business days
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Product Description
This book is an introduction to manifolds at the beginning graduate level. 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. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word "manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields. In his beautifully-conceived Introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout. John M. Lee is currently Professor of Mathematics at the University of Washington in Seattle. In addition to pursuing research in differential geometry and partial differential equations, he has been teaching undergraduate and graduate courses on manifolds at U.W. and Harvard University for more than fifteen years.
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| Customer Reviews: Read 3 more reviews...
Review of a non-mathematician May 7, 2002
34 out of 34 found this review helpful
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.
A very readable text April 26, 2002
Carey Allen (San Francisco Bay Area)
18 out of 18 found this review helpful
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.
Optimal Introduction to Topology June 7, 2006
Pioneer
2 out of 2 found this review helpful
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.
Nice introduction September 2, 2005
Saurabh Mahapatra (Boston, MA)
I like John's style in writing this book. When I started studying manifolds, I had little understanding of some of the deep concepts of topology. This book was an excellent introduction.
What is so cool about this book are the exercises, and they are all connected with each other. It makes a sumptuous meal.
The other book on smooth manifolds is definitely on its way to being a classic among beginners.
A welcome text to my collection April 15, 2007
anonymous
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.
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