A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics Series)

Rating:
Summary: An Outstanding Achievement.
Review: As a matter of fact all the materials written by J. P. May are precise, concise and useful. He is not of kind of those people who write 1000 pages and reach at obvious matters. This book is really a good introduction to the modern aspects of algebraic topology. It has less than 250 pages. I liked the treatment very much and appreciate it for teaching me a lot of mathematics. I dare to say that if someone else wants to write a book including all materials treated in this book, then the book would consist of at least 1000 pages. There is more to this book than just classical algebraic topology.

Rating:
Summary: Excellent Modern Treatment of Algebraic Topology
Review: 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. In fact, I have often found that attempts at visuallizing lead to confusion. J. Peter May avoids confusing illistrations in this book. Constructions are motivated by the results they consort. Most importantly May employes a thoroughly modern point of view. For example: the language of cofibrations/fibrations is used throughout, the handy idea of fundamental groupoid is introduced early in the treatment of the fundamental groups, there are a couple of chapters dedecated to homological algebra intersperced, both homology and cohomology are developed from the axiomatic point of view. May concludes the text with introductions to several more advanced topics such as cobordism, K-theory, and characteristic classes. The list of books that May offers in the suggestions for further reading section at the end is fairily comprehensive.

Rating:
Summary: A Unique and Necessary Book
Review: 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.

Rating:
Summary: [too much] for a book that will just sit on your bookself
Review: 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. I bought it recently and with the aid of some background books been reading through it; I havent found it especially difficult. The converage is vast. Prof. May deserves some sort of award for this book.I would encourage him to write a HANDBOOK out of this. After all, as far as I know there isnt a sort of First Handbook for this subject. BUY IT, YOU WONT REGRET IT.

Rating:
Summary: An important book for topologists.
Review: 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.

Rating:
Summary: Lucid and elegant, but not for beginners
Review: 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.