--- title: Homology description: Homology is a mathematical procedure that associates a sequence of abelian groups or modules with a topological space. Learners will understand how to compute these groups to detect and classify multi-dimensional holes in spaces. category: mathematics subcategory: algebraic-topology difficulty: beginner, intermediate, advanced url: /subject/homology --- # Homology Homology is a mathematical procedure that associates a sequence of abelian groups or modules with a topological space. Learners will understand how to compute these groups to detect and classify multi-dimensional holes in spaces. ## Available Resources 3 Books • 1 Courses • 5 Websites • 1 Papers ## Courses ### 1. Algebraic Topology Learn homology with N J Wildberger's Algebraic Topology course. Explore fundamental concepts with this engaging video playlist! **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLIljB45xT85D7wczwyUQdwDe2duZ7wPTf **Tags:** algebraic-topology, homology, simplicial-complexes, euler-characteristic, topology ## Websites ### 1. Topology Without Tears **Author:** Sidney A. Morris 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. **Difficulty:** Intermediate | **Price:** Free **Link:** https://www.topologywithouttears.net/ **Tags:** point-set-topology, general-topology, metric-spaces, compactness, proof-writing ### 2. Hatcher's Algebraic Topology (free PDF) **Author:** Allen Hatcher 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. **Difficulty:** Intermediate | **Price:** Free **Link:** https://pi.math.cornell.edu/~hatcher/AT/ATpage.html **Tags:** algebraic-topology, cw-complexes, homology, homotopy-theory, fundamental-group ### 3. Algebraic Topology — Cambridge Part II Lecture Notes **Author:** Oscar Randal-Williams 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.dpmms.cam.ac.uk/~or257/teaching/notes/at.pdf **Tags:** algebraic-topology, simplicial-homology, fundamental-group, mayer-vietoris, lecture-notes ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ### 5. nLab Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ncatlab.org **Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki ## Books ### 1. Elements of Algebraic Topology **Author:** James R. Munkres 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201627280?tag=edmonddante07-20 **Tags:** algebraic-topology, simplicial-homology, cohomology, cw-complexes, simplicial-approximation ### 2. Homology Theory: An Introduction to Algebraic Topology **Author:** James W. Vick 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387941266?tag=edmonddante07-20 **Tags:** algebraic-topology, singular-homology, cohomology, poincare-duality, eilenberg-steenrod-axioms ### 3. Elements of Algebraic Topology **Author:** James R. Munkres Textbook on homology theory built from simplicial complexes, then singular homology, cohomology, duality on manifolds, and the Künneth and universal coefficient theorems. Readers learn to compute homology groups concretely and follow the homological algebra underlying them. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201045869?tag=edmonddante07-20 **Tags:** algebraic-topology, homology, cohomology, simplicial-complexes, homological-algebra ## Papers ### 1. Part II Algebraic Topology (Cambridge lecture notes) **Author:** Dexter Chua (lectures by Henry Wilton) A compact, exam-scoped treatment reaching the classification theorem in about fifteen pages, so a learner can see the whole covering-space-to-subgroup dictionary in one sitting before committing to a full textbook. Cambridge Part II notes taken by Dexter Chua from Henry Wilton's Michaelmas 2015 lectures. Chapter three covers covering spaces, path and homotopy lifting, the fundamental group of the circle, universal covers, and the Galois correspondence. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://dec41.user.srcf.net/notes/II_M/algebraic_topology.pdf **Tags:** algebraic-topology, covering-spaces, universal-cover, galois-correspondence, homotopy-lifting --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/homology