Algebraic Topology

Rating:
Summary: The Last Text on Introductory Algebraic Topology
Review: No serious introductory text on basic algebraic topology has ever achieved this level of clarity, readability and depth. Its richness in examples (in both the main text and the problems) exposes a beginner to the underlying mechanisms of geometry in algebraic topology; its choice and arrangement of topics strike a perfect balance between accesibility and substantiveness; its lively and motivating exposition makes a student reluctant to attend the often boring topology classes. For a novice, this should be the first reading on the subject before (s)he is ruined by the many existing daunting texts; for a veteran, this can be very nourishing, especially if (s)he is already ruined by those either unreadable or shallow 'introduction's.

Rating:
Summary: You would not regret if you buy this.
Review: There are many really good textbooks on algebraic topology and each has its own merit: Bredon for his effort in explaining everything that can be dealt without using spectral equences, Fomenko & Novikov for their effort in unifying differential geometry and algebraic/differential topology.
Hatcher's book is intended as one of the series that cover every aspect of the subject. Separate books on vector bundles and K-theory, and spectral sequences respectively, are to appear sometime in the future. Thus this one covers ordinary homology/cohomology and homotopy theory only. His writing style is helpful and user-friendly, not demanding extensive "mathematical maturity".
I am not sure if this is "the" textbook on algebraic topology, but I bet this is among the best ones. You would not regret if you buy this, even when an electronic version is available online (for free) from the author's home page.

Rating:
Summary: You would not regret if you buy this.
Review: There are many really good textbooks on algebraic topology and each has its own merit: Bredon for his effort in explaining everything that can be dealt without using spectral equences, Fomenko & Novikov for their effort in unifying differential geometry and algebraic/differential topology.
Hatcher's book is intended as one of the series that cover every aspect of the subject. Separate books on vector bundles and K-theory, and spectral sequences respectively, are to appear sometime in the future. Thus this one covers ordinary homology/cohomology and homotopy theory only. His writing style is helpful and user-friendly, not demanding extensive "mathematical maturity".
I am not sure if this is "the" textbook on algebraic topology, but I bet this is among the best ones. You would not regret if you buy this, even when an electronic version is available online (for free) from the author's home page.

Rating:
Summary: Very good book, but don't buy it!
Review: This book is avaliable free to download from Allen Hatcher's webpage. You will also find other books he has written.

http://www.math.cornell.edu/~hatcher/

Rating:
Summary: It's worth your money!
Review: This book is not just for topologists! If you're like me, then you've spent countless nights sans Hatcher's book trying to figure out the fundamental group of a beer can. Look no further, the answers are here!

Be sure to check out the vivid detail Hatcher brings to the Van Kampen theorem. I've not actually read that part myself, as I do not trust german mathematics.

Rating:
Summary: It's worth your money!
Review: This book is not just for topologists! If you're like me, then you've spent countless nights sans Hatcher's book trying to figure out the fundamental group of a beer can. Look no further, the answers are here!

Be sure to check out the vivid detail Hatcher brings to the Van Kampen theorem. I've not actually read that part myself, as I do not trust german mathematics.