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

clean and brand new copy

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.

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

May gives a solid and (as the name suggests) concise overview of the basics of homotopy and homology theory. His presentation of the fundamental groupoid is lucid and he gives a short proof of the groupoid version of Van Kampen. Using only category theory, he then derives the group version of Van Kampen we all know and love. However, May does not develop enough of the theory of fundamental groupoids to obtain the fundamental group of the circle via a Van Kampen-type theorem. For reasons of space, this can be forgiven. But I would recommend chapters 6-9 of Ronald Brown’s “Topology and Groupoids” for an account which does not have this downfall. If you are not versed in category theory, May leaves some gaps for you to fill. Ultimately, May’s book is a solid reference with great exercises, but I would not recommend it as an introduction as it leaves quite a bit of work up to the reader and is not very geometric/visual.