Math 2.0

Algebraic topology Cell complexes and simplicial methods Theorem

CW approximation

Builds on CW complexWeak homotopy equivalenceHomotopy equivalenceWhitehead theoremTopological space

Statement

Let XX be a topological space.

  1. There are a CW complex ZZ and a weak homotopy equivalence f ⁣:Z→Xf \colon Z \to X.
  2. Let f′ ⁣:Z′→Xf' \colon Z' \to X be another weak homotopy equivalence from a CW complex Z′Z'. Then there is a homotopy equivalence h ⁣:Z→Z′h \colon Z \to Z' such that f′∘hf' \circ h is homotopic to ff.

The pair (Z,f)(Z, f) is called a CW approximation of XX.

Remarks

Part 1 is proved by building ZZ one dimension at a time. For each path component one starts with a point. At stage nn one attaches cells to make the map surjective on πn\pi_n and further cells to kill the kernel on πn\pi_n. Part 2 uses the fact that a weak homotopy equivalence f′ ⁣:Z′→Xf' \colon Z' \to X induces a bijection [Z,Z′]→[Z,X][Z, Z'] \to [Z, X] for every CW complex ZZ, together with the Whitehead theorem.

Example: the Warsaw circle WW has all homotopy groups trivial, so the inclusion of a point is a weak homotopy equivalence and a point is a CW approximation of WW. The space WW itself is not contractible, so a weak equivalence need not be a homotopy equivalence when the source or target is not a CW complex.

Going deeper

For a student

CW approximation lets one prove results about singular homology and homotopy groups for CW complexes only and then extend them to all spaces. It also shows that the homotopy category of CW complexes is equivalent to the category of all spaces with weak homotopy equivalences inverted. Functorial versions are given by the geometric realization of the singular simplicial set, ∣Sing⁡X∣→X\abs{\operatorname{Sing} X} \to X.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
CW complex, concept, no evidence levelCW complexconceptElement, definition, no evidence levelElementdefinitionHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptHomotopy group, concept, no evidence levelHomotopy groupconceptProposition, concept, no evidence levelPropositionconceptWeak homotopy equivalence, concept, no evidence levelWeak homotopy equival…conceptSet, definition, no evidence levelSetdefinitionWhitehead theorem, statement, no evidence levelWhitehead theoremstatementConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionTopological space, definition, no evidence levelTopological spacedefinitionCW approximation, statement, no evidence levelCW approximationstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. CW complex concept
  2. Element definition
  3. Homotopy equivalence concept
  4. Homotopy group concept
  5. Proposition concept
  6. Weak homotopy equivalence concept
  7. Set definition
  8. Whitehead theorem statement
  9. Conjunction definition
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Empty set object
  14. Family of sets definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Subset definition
  20. Union of a family of sets definition
  21. Topological space definition
  22. CW approximation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/CW_complex
About this page
TagPDFK69ZMST
URIm2://PDFK69ZMST
Kindstatement, theorem
WritingW0 stub
MSC55P10, 57Q05
StatusThis copy of the site has no build data. Run m2 status PDFK69ZMST.
AboutCW complex, Weak homotopy equivalence
UsesCW complex, Weak homotopy equivalence, Homotopy equivalence, Whitehead theorem, Topological space
Changed2 November 2026

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