Algebraic topology is graduate mathematics. The reading order matters more here than in almost any subject on this site, because the text most courses assign — Allen Hatcher's — opens assuming you have already done a semester of point-set topology and a semester of abstract algebra. Readers who start there cold do not find it hard; they find it unreadable, and then conclude the subject is beyond them when what is missing is two prerequisites.
Be realistic about the timescale as well. Point-set topology is a semester of work. The fundamental group and covering spaces is another. Homology and cohomology is a third, and it is the one where most self-studiers stall. Reaching homology from a standing start is a matter of months of steady work; being comfortable with cohomology, duality and spectral sequences is a matter of years. The assumed background across the whole list is real analysis or metric spaces, group theory to the level of quotients, free groups and presentations, later modules and exact sequences, and — for the last few books — smooth manifolds and multivariable calculus.
The point-set prerequisite
Topology by James Munkres is the standard undergraduate course. Part one is point-set topology done thoroughly; part two is a first look at the fundamental group and covering spaces, so the book straddles the boundary and is the single most efficient starting point. Basic topology by M. A. Armstrong is the alternative: shorter, more geometric, and it reaches the fundamental group faster. It is a better read and a worse reference. Pick one — doing both is a waste of a month.
The fundamental group and covering spaces
Algebraic Topology. An Introduction by William Massey is devoted almost entirely to the fundamental group, covering spaces and the Seifert–van Kampen theorem, at a pace that assumes you want the details. Introduction to Topological Manifolds by John Lee is the best transitional book on this list: it uses manifolds to motivate the algebraic machinery, and it explains why the constructions are built the way they are rather than just building them.
Two genuine alternatives sit here. William Fulton's Algebraic topology is an idiosyncratic first course organised around winding numbers, de Rham cohomology and covering spaces rather than the usual singular-homology spine; it is a different route through the same country, not a substitute. Joseph Rotman's An introduction to algebraic topology is the algebraist's version, heavier on homological algebra and lighter on geometry.
One overlap to know about: Massey's A basic course in algebraic topology is essentially his earlier introduction plus a homology half bolted on. If you buy the later book you do not need the earlier one.
Homology and cohomology
Elements of Algebraic Topology by Munkres covers simplicial and singular homology slowly and completely, and it is the gentlest serious treatment available. James Vick's Homology theory is a compact alternative for readers who want the same material in fewer pages.
Then Hatcher. Algebraic Topology is the standard graduate text, it is legally free on the author's website, and it covers the fundamental group, homology, cohomology and homotopy theory in about 550 pages. It is discursive, geometric and enormously exercise-rich. Most readers find it works far better as a course companion than as a solo read, because it explains by argument rather than by definition-theorem-proof and rewards someone who already half-knows where the argument is going.
Geometry and the deep end
Glen Bredon's Topology and Geometry runs differential topology and algebraic topology together and assumes more of the reader than anything above it. Differential forms in algebraic topology by Raoul Bott and Loring Tu approaches the subject through de Rham theory; it requires smooth manifolds and calculus on manifolds, and readers who have that background often name it the most enjoyable book in the field. John Milnor's Topology from the differentiable viewpoint is a slim classic on degree, Sard's theorem and transversality that can be read early — it needs only multivariable calculus and a free weekend. Finally, Lecture notes in algebraic topology by James Davis and Paul Kirk is second-course material: spectral sequences, obstruction theory, characteristic classes.
Work down the full path in order, and do the exercises — in this subject they are the content, not the review. If you want to see what sits next door, algebraic geometry is the usual neighbour.
Follow the full ordered path here: Algebraic Topology Books, in the Order a Graduate Course Would Assign Them.
