--- title: Homotopy description: Homotopy studies the continuous deformation of topological spaces and maps between them. Learners will understand how to determine if two spaces are topologically equivalent and compute homotopy groups to analyze higher-dimensional geometric structures. category: mathematics subcategory: algebraic-topology difficulty: beginner, intermediate, advanced url: /subject/homotopy --- # Homotopy Homotopy studies the continuous deformation of topological spaces and maps between them. Learners will understand how to determine if two spaces are topologically equivalent and compute homotopy groups to analyze higher-dimensional geometric structures. ## Available Resources 2 Books • 3 Courses • 4 Websites ## Websites ### 1. nLab: Homotopy Theory The nLab entry on homotopy theory, a research-level wiki page surveying the field from a higher-category viewpoint: homotopy categories, model categories, infinity-groupoids, and stable and abstract homotopy theory. It helps readers connect classical topology to modern structural frameworks and references. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/homotopy+theory **Tags:** homotopy-theory, algebraic-topology, model-categories, higher-category-theory, infinity-groupoids ### 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 ## Courses ### 1. Algebraic Topology Explore algebraic topology with N J Wildberger's homotopy course! Learn fundamental concepts and build a strong foundation in this fascinating area. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLIljB45xT85D7wczwyUQdwDe2duZ7wPTf **Tags:** algebraic-topology, homotopy, fundamental-group, university-lectures ### 2. Algebraic Topology Learn homotopy theory with Ryan Budney's Algebraic Topology course! Explore fundamental concepts and build a strong foundation. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLevbjS885lNS9Nk8ve13FE5tMdZejsTrE **Tags:** algebraic-topology, homotopy-theory, homology, university-lectures ### 3. 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 ## Books ### 1. Algebraic Topology **Author:** Allen Hatcher Graduate textbook covering the fundamental group and covering spaces, homology, cohomology with cup products and Poincaré duality, and homotopy theory. Readers learn to compute invariants of spaces and use them to distinguish spaces up to homotopy equivalence. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521541867?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, algebraic-topology ### 2. A Concise Course in Algebraic Topology **Author:** J. Peter May Fast-paced graduate text from the Chicago Lectures in Mathematics, treating the fundamental group, covering spaces, cofibrations and fibrations, homotopy groups, CW complexes, axiomatic homology and cohomology, and characteristic classes. Readers gain the homotopy-theoretic viewpoint used in modern research, best after a first course. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/9394852727?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, algebraic-topology --- *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