A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics) 1st Edition by J. P. May (PDF)

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.

User’s Reviews

Editorial Reviews: About the Author J. P. May is professor of mathematics at the University of Chicago; he is the author or coauthor of many papers and books, including Simplicial Objects in Algebraic Topology and A Concise Course in Algebraic Topology, both in the Chicago Lectures in Mathematics series.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book itself is really promising – the material is not available in any other textbook, to my knowledge. I really like it as a supplement to my topology course; category theory organizes a lot of the theorems of topology so that’s a lot easier to remember just because everything fits together.However, the typos and in the book make it much less pleasant to use than it should be. For instance, in one of the commutative diagrams, ι is supposed to be a “morphism of diagrams” aka a natural transformation. Yet there is a map drawn that shows ι taking two objects of the underlying category to one another?The author also uses the phrase “pullback along f and g” without ever giving an explanation of what that means. Worse, the maps don’t even have the same target so even when I found somewhere that explained the phrase, it was wrong!Language from category theory is used throughout without explanation, but the real problem is that there’s no glossary or glossary of symbols.Also I bought this on amazon and the blurb was screwed up – elongated or illegible letters in some places. Basically I use it because there’s nothing else. But a copy editor could fix everything.

⭐Billed as notes from a required first-year graduate course honed from decades of teaching, its clear that this is a wash-out class, designed to separate the very best from the rest. Concise, clear, well-written, but definitely challenging. The text requires some real thought but is ultimately understandable.The homework problems, however, can be overwhelming if you don’t have a very strong background. One required some knowledge of loop spaces that is not at all touched on in the main text. Another required knowledge of the representation theory of free groups, something I could guess at but didn’t know (and, again, is not at all touched on in the main text: you’d have to run to the library to educate yourself on the topic just to solve the homework problem. But I guess that’s what school is about, isn’t it?). Another reviewer recommends having previously gone through Bott & Tu, but this alone is not really enough. You’d have to have a pretty strong grounding in group theory, and perhaps a passing acquaintance with cat theory basics, such as pushouts, cones, limits, before this book becomes a pleasant read. I have NOT read Hatcher’s ‘algebraic topology’, but have the sneaking suspicion its a pre-requisite as well.In short: this is a U-of-Chicago-style school-book: to get the most out of it, you’d have to supplement with readings from other texts, and have the ability to ask others when you get hung up on some or another point. Its perhaps a bit much for casual self-study at home, as I’m treating it.

⭐I have always believed that the “goodness” of a mathematical textbook is inversely proportional to its length. J. P. May’s book “A Concise Course in Algebraic Topology” is a superb demonstration of this. While the book is indeed extremely terse, it forces the reader to thoroughly internalize the concepts before moving on. Also, it presents results in their full generality, making it a helpful reference work.

⭐clear, excellent book

⭐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.

⭐clean and brand new copy

⭐Category theory is never my taste of mathematics. This book is by far the most abstract nonsense book I’ve read. If a mathematician has no ideas then he practises axiomatic approach—-Felix Klein (I’m not sure of this quote).That said, this book has really good coverage. I’m not sure this is a blessing or curse. I prefer a more pictorial and concrete approach with the same coverage.

⭐Amazing book. Extremely dense but probably the only treatment of algebraic geometry using the language of category theory.

⭐The book covers most of the areas of Algebraic Topology, but in a intuitive and not fully rigorous manner, in my opinion. Recommended.

⭐doesn’t have more explanation.

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