A Concise Course in Algebraic Topology
by J. P. May · University of Chicago
Author-hosted PDF of the University of Chicago Press text, 251 pages. Terse graduate treatment of homotopy theory: cofibrations, fibrations, homotopy groups, the Hurewicz and Whitehead theorems, CW approximation, Eilenberg-MacLane spaces, Postnikov towers, and spectral sequences.
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More resources on Homotopy Groups
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Homotopy Groups
Community-maintained nLab wiki entry on homotopy groups, giving the definition for pointed spaces and higher-categorical generalizations, key properties and examples, and links to related entries across homotopy theory. Best used as a precise reference after a first course in algebraic topology.
Hatcher's Algebraic Topology (free PDF)
Allen Hatcher's standard graduate textbook, freely downloadable from Cornell, covering fundamental groups, homology, cohomology and homotopy theory, with CW complexes introduced in Chapter 0 and used throughout. Readers learn to build spaces from cells and compute their algebraic invariants.
Algebraic Topology (EMS Textbooks in Mathematics)
The one text here treating homotopy groups of spheres as a subject in its own right, with the exercise density needed to actually compute rather than just read. EMS graduate textbook with roughly 500 exercises. Part one develops homotopy and homology from the ground up; later chapters treat homotopy groups of spheres, duality, characteristic classes, and bordism in substantially more depth than shorter texts.
Algebraic Topology II (MIT 18.906)
Detailed lecture notes (about 160 pages) and problem sets on homotopy theory: fibrations and cofibrations, homotopy groups, Whitehead and Hurewicz theorems, obstruction theory, classifying spaces, spectral sequences, characteristic classes and Steenrod operations. Assumes a first graduate course in algebraic topology.
Introduction to Higher Homotopy Groups and Obstruction Theory
The most precisely-scoped free resource for this topic: the table of contents runs higher homotopy groups, Hurewicz, basepoint dependence, fiber bundles, LES of a fibration, obstruction theory, Whitehead. Course notes from Berkeley's second-semester algebraic topology sequence covering higher homotopy groups, the Hurewicz isomorphism, basepoint dependence, fiber bundles, the long exact sequence of a fibration, obstruction theory, Eilenberg-MacLane spaces, Whitehead's theorem, and Euler classes.