Whitehead theorem

A weak homotopy equivalence between connected CW complexes is a homotopy equivalence. The analogous statement holds for Kan complexes.

Table of contents
- Rational Whitehead theorem Whitehead theorem
- Homological Whitehead theorem Whitehead theorem

Rational Whitehead theorem ( $H_{} (f; Q) invertible \Rightarrow π_{} (f) \otimes Q invertible$ )

For maps between simply connected spaces, a rational homology equivalence is a rational homotopy equivalence. It follows from the relative Hurewicz theorem modulo torsion, or from rationalization and the homological Whitehead theorem.

Homological Whitehead theorem ( $H_{} (f; Z) invertible \Rightarrow π_{} (f) invertible$ )

An integral homology equivalence between simply connected spaces is a weak homotopy equivalence. A first nonzero relative homotopy group would contradict the Relative Hurewicz theorem. Between CW complexes the ordinary Whitehead theorem then gives a homotopy equivalence.

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