Algebraic Topology

Below is a list of reviews of books on algebraic topology.

A Concise Course in Algebraic Topology
May, P. Free Download
Prerequisites: Basic category theory and abstract algebra

It is generally agreed that this short text is best used as a second pass at the subject, or in conjunction with something like Allen Hatcher’s Algebraic Topology (also in this list). The book is written with an eye toward modern Algebraic Topology, which emphasizes the use of model categorical ideas (without the baggage of the abstract definition thereof) and more categorically abstract argumentation in general. The book avoids visual intuition and geometric arguments where possible. There is a thematically similar sequel More Concise Algebraic Topology by the same author and Kate Ponto.

Algebraic Topology
Hatcher, A. Free Download
Prerequisites: abstract algebra (groups, rings, and modules primarily)

The most famous introduction to the subject and probably the most commonly assigned in an undergraduate (or graduate) course. Written with the goal of preparing students for any area of modern topology, including low dimensional and geometric topology (however, it does not cover differential topology). Depends heavily on visual arguments at times.

Algebraic Topology
Dieck, T. Free Download
Prerequisites: Technically speaking introductory algebra and basic knowledge of category theory would suffice, but it is probably best for a second pass after already knowing some algebraic topology.

Thorough introduction to homotopy and homology, plus some additional topics, from a very formal and highly categorical perspective. Not the best for developing visual intuition but it ties together various topics in algebraic topology well and provides a very holistic view of the subject at an introductory or slightly above introductory level. Fairly terse.