Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics)

Rating:
Summary: Masterful
Review: This introduction to singular homology combines a strong historical sense with an easy mastery of modern methods. The massive contributions of Poincare and Brouwer are credited, and their geometrical motivations are clear. At the same time the book neither minimizes nor apologizes for modern algebraic machinery, but treats categories and acyclic models and more as natural means to simplify the subject. The book goes through Poincare duality and a good account of the Lefschetz fixed point theorems. It is at once very visual and algebraically slick. The only problem with this approach is that the author seems a bit uncomfortable descending into the nuts and bolts of the longer proofs of two key results (the acyclic model theorem, and the duality theorem). He handles the details unevenly and makes some actual mis-statements. Here the reader needs the experience and confidence to make some corections.