Allen Hatcher of Cornell appears to be undertaking the quixotic goal of writing accessible “introductory” textbooks for the entirety of Algebraic Topology. The first volume is already the definitive introductory work in the subject, covering the Fundamental Group, Homology, Cohomology, and Homotopy and is available online.

The style is geometric, so those whose nascent views on Algebraic Topology are functorial may be better served by Rotman’s book, but it is hard to recommend highly enough since Hatcher is an excellent expositor, and the book is clearly written for students rather than a vanity piece aimed at colleagues, a major pet peeve of mine.

I’ll second that recommendation. I liked the book so much that I even bought the dead trees version.

Hatcher also has a diagram that communicates better than any words how complicated the homotopy groups of spheres are.

I too really like the Algebraic Topology book! Unfortunately, I didn’t have the chance to take any courses from Hatcher when I was a student at CU, but one of the figures in the book is essentially a picture I showed to my then undergraduate advisor, Marshall Cohen, who said AH was writing a book and that he’d pass it on. It’s gratifying to see that AH must have liked the picture too. Walt, can you guess which picture I am talking about? (NOT the way cool diagram you mentioned, of course!)