Math 2.0

Algebraic topology Homotopy groups Theorem

Whitehead theorem

Builds on CW complexHomotopy groupHomotopy equivalenceWeak homotopy equivalence

Statement

Let XX and YY be connected CW complexes, let x0∈Xx_0 \in X, and let f ⁣:X→Yf \colon X \to Y be a continuous map. Assume that f∗ ⁣:πn(X,x0)→πn(Y,f(x0))f_* \colon \pi_n(X, x_0) \to \pi_n(Y, f(x_0)) is an isomorphism of homotopy groups for every n≥1n \ge 1. Then ff is a homotopy equivalence.

Remarks

Replace YY by the mapping cylinder of ff, so that ff becomes the inclusion of a subcomplex. The relative homotopy groups of the pair then vanish, and one deforms YY onto XX cell by cell, using compression of maps of disks. In other words, a weak homotopy equivalence between CW complexes is a homotopy equivalence. The CW hypothesis is needed: the map from the Warsaw circle to a point induces isomorphisms on all homotopy groups, but it is not a homotopy equivalence. The isomorphisms must be induced by one map: ℝP2×S3\R P^2 \times S^3 and S2×ℝP3S^2 \times \R P^3 have isomorphic homotopy groups, but they are not homotopy equivalent, as their homology differs.

Going deeper

For a student

The theorem justifies working with CW complexes in homotopy theory, where weak and genuine homotopy equivalences coincide. Combined with the Hurewicz theorem, it gives a homology version. A map between simply connected CW complexes that induces isomorphisms on all integral homology groups is a homotopy equivalence. For example, a simply connected CW complex with the integral homology of a point is contractible.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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CW complex, concept, no evidence levelCW complexconceptHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptHomotopy group, concept, no evidence levelHomotopy groupconceptWeak homotopy equivalence, concept, no evidence levelWeak homotopy equival…conceptWhitehead theorem, statement, no evidence levelWhitehead theoremstatement
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  1. CW complex concept
  2. Homotopy equivalence concept
  3. Homotopy group concept
  4. Weak homotopy equivalence concept
  5. Whitehead theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Whitehead_theorem
About this page
Tag6P1E2FJPSD
URIm2://6P1E2FJPSD
Kindstatement, theorem
WritingW0 stub
MSC55P10, 55Q05
StatusThis copy of the site has no build data. Run m2 status 6P1E2FJPSD.
AboutHomotopy equivalence, Weak homotopy equivalence
UsesCW complex, Homotopy group, Homotopy equivalence, Weak homotopy equivalence
Same as
Changed2 November 2026

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