--- title: Homotopy Groups description: Homotopy groups classify topological spaces by studying the equivalence classes of continuous maps from spheres. Learners will understand how to compute fundamental and higher homotopy groups, analyzing the connectivity and topological holes of spaces. category: mathematics subcategory: differential-topology difficulty: beginner, intermediate, advanced url: /subject/homotopy-groups --- # Homotopy Groups Homotopy groups classify topological spaces by studying the equivalence classes of continuous maps from spheres. Learners will understand how to compute fundamental and higher homotopy groups, analyzing the connectivity and topological holes of spaces. ## Available Resources 3 Books • 2 Courses • 4 Websites • 1 Papers ## Websites ### 1. 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. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/homotopy+group **Tags:** homotopy groups, homotopy theory, algebraic topology, higher category theory, reference ### 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. 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 ### 4. 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. Algebraic Topology (EMS Textbooks in Mathematics) **Author:** Tammo tom Dieck 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/3037190485?tag=edmonddante07-20 **Tags:** algebraic topology, homotopy groups, homotopy groups of spheres, homology, characteristic classes, bordism ### 2. A Concise Course in Algebraic Topology **Author:** J. P. May 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0226511839?tag=edmonddante07-20 **Tags:** algebraic topology, homotopy theory, homotopy groups, fibrations, cw complexes, spectral sequences ### 3. Algebraic Topology: An Introduction (GTM 56) **Author:** William S. Massey Graduate text built around the fundamental group rather than homology, covering surface classification, free products, and then covering spaces: the lifting criterion, the universal cover, the classification of coverings, and deck transformation groups. Unlike general algebraic topology texts that treat coverings as a chapter en route to homology, this book spends its whole length on the fundamental group and covering spaces, making it the deepest single-topic treatment available. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387902716?tag=edmonddante07-20 **Tags:** algebraic-topology, fundamental-group, covering-spaces, deck-transformations, surfaces ## Courses ### 1. Algebraic Topology II (MIT 18.906) **Author:** Haynes Miller 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. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-906-algebraic-topology-ii-spring-2020/ **Tags:** homotopy-theory, fibrations, spectral-sequences, characteristic-classes, steenrod-operations ### 2. Topology and Groups **Author:** Jonny Evans Flipped-classroom UCL course with short pre-recorded videos and matching written notes, building from the fundamental group through van Kampen to a full section on covering spaces, covering transformations, and the Galois theory of coverings. One of the few free resources giving covering spaces its own multi-hour block plus a dedicated section on the Galois correspondence and covering transformations, in paired video-plus-notes form. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://jde27.uk/tg/ **Tags:** algebraic-topology, fundamental-group, van-kampen-theorem, covering-spaces, galois-correspondence ## Papers ### 1. Introduction to Higher Homotopy Groups and Obstruction Theory **Author:** Michael Hutchings 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. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf **Tags:** homotopy groups, hurewicz theorem, fibrations, obstruction theory, eilenberg-maclane spaces, lecture notes --- *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/homotopy-groups