Algebraic Topology
N J Wildberger
Learn homology with N J Wildberger's Algebraic Topology course. Explore fundamental concepts with this engaging video playlist!
More resources on Homology
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.
Topology Without Tears
Free textbook by Sidney A. Morris introducing point-set topology from open sets and the euclidean topology through continuity, metric spaces, compactness, connectedness, and Tychonoff's theorem. Readers learn to write rigorous topology proofs, with later chapters extending to topological groups.
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.
Elements of Algebraic Topology
Graduate textbook that builds algebraic topology from simplicial complexes upward, developing simplicial homology and cohomology, simplicial approximation, CW complexes, duality in manifolds, and the universal coefficient and Künneth theorems with full proofs and exercises.
Homology Theory: An Introduction to Algebraic Topology
The shortest rigorous route to duality theorems: where a reference text runs 450 pages, Vick reaches Poincaré and Alexander duality in under 250, making it the practical read-cover-to-cover option. A graduate text confined to homology and cohomology. It covers singular theory, the Eilenberg-Steenrod axioms, cup and cap products, Poincaré and Alexander duality, plus a chapter on covering spaces and the first homology group.
Algebraic Topology — Cambridge Part II Lecture Notes
Roughly eighty pages of Cambridge undergraduate notes whose longest chapter develops simplicial homology from chain complexes: homological algebra, explicit calculations, the Mayer-Vietoris sequence, homotopy invariance, homology of spheres and surfaces, Euler and Lefschetz numbers.