--- title: CW Complexes description: CW complexes are topological spaces constructed by inductively gluing cells of increasing dimensions. Learners will understand how to build these spaces, compute their homology groups, and use them to simplify homotopy calculations. category: mathematics subcategory: algebraic-topology difficulty: beginner, intermediate, advanced url: /subject/cw-complexes --- # CW Complexes CW complexes are topological spaces constructed by inductively gluing cells of increasing dimensions. Learners will understand how to build these spaces, compute their homology groups, and use them to simplify homotopy calculations. ## Where to start Start with Hatcher's Algebraic Topology, a free PDF textbook with a detailed chapter on building CW complexes and computing with them. If you only use one resource, stay with Hatcher, the standard text on the subject. For the point-set details Hatcher leaves to an appendix, such as the weak topology and products, continue with Cellular Structures in Topology by Rudolf Fritsch and Renzo A. Piccinini, a paid book. ## Available Resources 2 Books • 4 Websites • 1 Papers ## Websites ### 1. nLab: CW-complex Entry on the collaborative nLab wiki giving the definition of CW complexes, relative CW complexes and cell attachments, with properties, examples and their role in model-category and homotopy theory. Useful for readers who already know basic algebraic topology and want precise definitions. **Difficulty:** Advanced | **Price:** Free **Link:** https://ncatlab.org/nlab/show/CW-complex **Tags:** cw-complexes, algebraic-topology, homotopy-theory, cell-complexes ### 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. 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 ### 2. Cellular Structures in Topology **Author:** Rudolf Fritsch, Renzo A. Piccinini A book-length treatment of CW-complexes as topological spaces: closure-finiteness, the weak topology, products, subdivision, and the relationship to simplicial complexes, plus Milnor's theory of spaces of CW type and CW structures in fibration theory. Fills the exact gap Hatcher leaves: Hatcher builds CW complexes and computes with them but pushes closure-finiteness, weak-topology pathologies, products of CW complexes and the CW-versus-simplicial comparison into a short appendix. This book proves those results carefully over a full volume. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521063876?tag=edmonddante07-20 **Tags:** cw-complexes, cellular-structures, simplicial-complexes, point-set-topology, algebraic-topology ## Papers ### 1. On Spaces Having the Homotopy Type of a CW-Complex **Author:** John Milnor Nine-page 1959 Transactions paper defining the class of spaces homotopy equivalent to a CW complex and proving it is closed under function-space, product and covering constructions; the standard reference behind claims that a given space has CW homotopy type. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.ams.org/journals/tran/1959-090-02/S0002-9947-1959-0100267-4/S0002-9947-1959-0100267-4.pdf **Tags:** cw-complexes, homotopy-type, homotopy-theory, function-spaces --- *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/cw-complexes