# Whitehead theorem

###### Theorem 1 (J.H.C. Whitehead)

If $f:X\to Y$ is a weak homotopy equivalence and $X$ and $Y$
are path-connected and of the homotopy type^{} of CW complexes, then $f$ is a strong homotopy equivalence.

###### Remark 1

It is essential to the theorem that isomorphisms^{} between ${\pi}_{k}(X)$ and ${\pi}_{k}(Y)$ for all $k$ are induced by a map $f:X\to Y;$ if an isomorphism exists which is not induced by a map, it need not be the case that the spaces are homotopy equivalent.

For example, let $X=\mathbb{R}{P}^{m}\times {S}^{n}$ and $Y=\mathbb{R}{P}^{n}\times {S}^{m}.$
Then the two spaces have isomorphic homotopy groups^{} because they both have a universal covering space homeomorphic^{} to ${S}^{m}\times {S}^{n},$ and it is a double covering in both cases. However, for $$ $X$ and $Y$ are not homotopy equivalent, as can be seen, for example, by using homology^{}:

${H}_{m}(X;\mathbb{Z}/2\mathbb{Z})$ | $\cong $ | $\mathbb{Z}/2\mathbb{Z},\text{but}$ | ||

${H}_{m}(Y;\mathbb{Z}/2\mathbb{Z})$ | $\cong $ | $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}.$ |

(Here, $\mathbb{R}{P}^{n}$ is $n$-dimensional real projective space, and ${S}^{n}$ is the $n$-sphere.)

Title | Whitehead theorem^{} |
---|---|

Canonical name | WhiteheadTheorem |

Date of creation | 2013-03-22 13:25:48 |

Last modified on | 2013-03-22 13:25:48 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 10 |

Author | antonio (1116) |

Entry type | Theorem |

Classification | msc 55P10 |

Classification | msc 55P15 |

Classification | msc 55Q05 |

Related topic | ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces |

Related topic | WeakHomotopyEquivalence |

Related topic | ApproximationTheoremAppliedToWhitneyCrMNSpaces |