Amazon.com: A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) (9780226511832): J. P. May: Books

Product Description

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.

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

Paperback: 254 pages
Publisher: University Of Chicago Press; 1 edition (September 1, 1999)
Language: English
ISBN-10: 0226511839
ISBN-13: 978-0226511832
Product Dimensions: 8.8 x 6 x 0.6 inches
Shipping Weight: 11.2 ounces (View shipping rates and policies)

Average Customer Review: 4.9 out of 5 stars See all reviews (7 customer reviews)

Amazon.com Sales Rank: #431,314 in Books (See Bestsellers in Books)

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#64 in	Books > Professional & Technical > Professional Science > Mathematics > Geometry & Topology > Algebraic Geometry
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Inside This Book (learn more)

First Sentence:
We introduce algebraic topology with a quick treatment of standard material about the fundamental groups of spaces, embedded in a geodesic proof of the Brouwer fixed point theorem and the fundamental theorem of algebra. Read the first page

Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
excisive triad, path lifting function, cup product form, cofiber sequences, wedge summand, based homotopy classes, suspension theorem, boundary collar, excision theorem, disjoint basepoint, modern algebraic topology, homotopy commutative, stable homotopy theory, simplicial spaces, cohomology operations, complex cobordism, homotopy groups, dimension axiom, homotopy category, suspension isomorphism, periodicity theorem, cellular chains, covering space theory, generalized cohomology theory, homotopy equivalence

Key Phrases - Capitalized Phrases (CAPs): (learn more)
Eilenberg-Mac Lane

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Citations (learn more)

This book cites 20 books:

Theory of Unitary Group Representation (Chicago Lectures in Mathematics) by George W. Mackey in Front Matter
Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon (Chicago Lectures in Mathematics) by Michael Christ in Front Matter
Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics) by Yakov B. Pesin in Front Matter
Fuchsian Groups (Chicago Lectures in Mathematics) by Svetlana Katok in Front Matter
Stable Homotopy and Generalised Homology (Chicago Lectures in Mathematics) by J. F. Adams in Front Matter

See all 20 books this book cites

8 books cite this book:

Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Chicago Lectures in Mathematics) by Phillip Griffiths in Front Matter
Algebraic Topology by Allen Hatcher in Back Matter
Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon (Chicago Lectures in Mathematics) by Michael Christ in Front Matter
Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon (Chicago Lectures in Mathematics) by Michael Christ in Front Matter
Algebraic Topology from a Homotopical Viewpoint by Marcelo Aguilar in Back Matter

See all 8 books citing this book

Books on Related Topics (learn more)

Algebraic Topology by Allen Hatcher

Discusses:

Simplicial Homotopy Theory by Paul Gregory Goerss

Discusses:

Lecture Notes in Algebraic Topology by James F. Davis

Discusses:

Algebraic K-Theory and Its Applications by Jonathan Rosenberg

Discusses:

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	77% buy the item featured on this page: A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) 4.9 out of 5 stars (7) $20.52
	12% buy Algebraic Topology 3.9 out of 5 stars (16) $36.89
	4% buy Topology (2nd Edition) 4.3 out of 5 stars (29) $107.73
	4% buy Algebra 4.0 out of 5 stars (23) $43.70

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Customer Reviews

7 Reviews

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4 star:		(1)
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Average Customer Review

4.9 out of 5 stars (7 customer reviews)

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Most Helpful Customer Reviews

25 of 27 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)

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.

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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 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.

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8 of 9 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 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.

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Most Recent Customer Reviews

5.0 out of 5 stars The opposite of Hatcher
This book is clear, and direct. It tells you want you want to know.

Published on November 6, 2007 by D. Spivak

5.0 out of 5 stars Excellent Modern Treatment of Algebraic Topology
One of the reasons that Algebraic Topology is difficult to learn is that often the more general constructions (which are algebraic) are difficult to motivate visually. Read more

Published on February 21, 2002 by Nicholas Cox-Steib

5.0 out of 5 stars [too much] for a book that will just sit on your bookself
this is not a bad book, but it isnt for real. the back of the book says: ...treatment is sophisticated, no prior knowledge of the subject is assumed.

i think not. Read more

Published on August 5, 2001

5.0 out of 5 stars Peter May Should get some sort of award for this book
This is a wonderful book. Through years of teaching the course he has refined to a perfection. It would take another 30 years to write such a book. Read more

Published on March 2, 2001 by jeanfarogh

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