A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) [Hardcover]

J. P. May (Author)

Book Description

Series: Chicago Lectures in Mathematics | Publication Date: October 1, 1999

Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields.

J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.

Editorial Reviews

About the Author

J. P. May is professor of mathematics at the University of Chicago. He is author or coauthor of many books, including Simplicial Objects in Algebraic Topology and Equivalent Homotopy and Cohomology Theory.

Product Details

• Hardcover: 254 pages
• Publisher: University Of Chicago Press (October 1, 1999)
• Language: English
• ISBN-10: 0226511820
• ISBN-13: 978-0226511825
• Product Dimensions: 9 x 6 x 0.8 inches
• Shipping Weight: 1.2 pounds
• Average Customer Review: 4.8 out of 5 stars  See all reviews (9 customer reviews)
• Amazon Best Sellers Rank: #9,898,092 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

31 of 33 people found the following review helpful:

5.0 out of 5 stars A Unique and Necessary Book, May 15, 2002

By 

Michael Spertus (Chicago, IL USA) - See all my reviews
(REAL NAME)   

This review is from: A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) (Paperback)

Ones first exposure to algebraic topology should be a concrete and pictorial approach to gain a visual and combinatorial intuition for algebraic topology. It is really necessary to draw pictures of tori, see the holes, and then write down the chain complexes that compute them. Likewise, one should bang on the Serre Spectral Sequence with some concrete examples to learn the incredible computational powers of Algebraic Topology. There are many excellent and elementary introductions to Algebraic Topology of this type (I like Bott & Tu because of its quick introduction of spectral sequences and use of differential forms to bypass much homological algebra that is not instructive to the novice).

However, as Willard points out, mathematics is learned by successive approximation to the truth. As you becomes more mathematically sophisticated, you should relearn algebraic topology to understand it the way that working mathematicians do. Peter May's book is the only text that I know of that concisely presents the core concepts algebraic topology from a sophisticated abstract point of view. To make it even better, it is beautifully written and the pedagogy is excellent, as Peter May has been teaching and refining this course for decades. Every line has obviously been thought about carefully for correctness and clarity.

As an example, ones first exposure to singular homology should be concrete approach using singular chains, but this ultimately doesn't explain why many of the artificial-looking definitions of singular homology are the natural choices. In addition, this decidedly old-fashioned approach is hard to generalize to other combinatorial constructions.

Here is how the book does it: First, deduce the cellular homology of CW-complexes as an immediate consequence of the Eilenberg-Steenrod axioms. Considering how one can extend this to general topological spaces suggests that one approximate the space by a CW-complex. Realization of the total singular complex of the space as a CW-complex is a functorial CW-approximation of the space. As the total singular complex induces an equivalence of (weak) homotopy categories and homology is homotopy-invariant, it is natural to define the singular homology of the original space to be the homology of the total singular complex. Although sophisticated, this is a deeply instructive approach, because it shows that the natural combinatorial approximation to a space is its total singular complex in the category of simplicial sets, which lets you transport of combinatorial invariants such as homology of chain complexes. This approach is essential to modern homotopy theory.

9 of 9 people found the following review helpful:

4.0 out of 5 stars Lucid and elegant, but not for beginners, March 4, 2003

By A Customer

This review is from: A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) (Paperback)

This tiny textbook is well organized with an incredible amount of information. If you manage to read this, you will have much machinery of algebraic topology at hand. But, this book is not for you if you know practically nothing about the subject (hence four stars). I believe this work should be understood to have compiled "what topologists should know about algebraic topology" in a minimum number of pages.

10 of 11 people found the following review helpful:

5.0 out of 5 stars An important book for topologists., August 28, 2000

By 

Nicolas Yus Suarez (Santiago Chile) - See all my reviews

This review is from: A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) (Paperback)

This is an excellent book written by a very wellknown topologist and it deserves a place in every topologist's shelves. It is certainly not for anybody with a passing interest in the subject. As its title indicates, it is very concise and a reader has to be willing to spend a lot of time filling in details. It is not a user friendly book; it is a very good MATH book, where everything as said precisely and succintly and the user who works hard will learn a lot of deep mathematics and be well prepared to start the road to the frontier.

Another characteristic is that there it includes many topics that are not available in any of the usual introductory books.