# weak homotopy equivalence

A continuous map $f:X\to Y$ between path-connected based
topological spaces^{} is said to be a weak homotopy equivalence if for each $k\ge 1$ it induces an isomorphism^{} ${f}_{*}:{\pi}_{k}(X)\to {\pi}_{k}(Y)$ between the
$k$th homotopy groups^{}. $X$ and $Y$ are then said to be weakly
homotopy equivalent.

###### Remark 1.

It is not enough for ${\pi}_{k}\mathit{}\mathrm{(}X\mathrm{)}$ to be isomorphic to ${\pi}_{k}\mathit{}\mathrm{(}Y\mathrm{)}$ for all $k\mathrm{.}$ The definition requires these isomorphisms to be induced by a space-level map $f\mathrm{.}$

###### Remark 2.

More generally, two spaces $X$ and $Y$ are defined to be weakly homotopy equivalent if there is a sequence of spaces and maps

$$X\to {X}_{1}\leftarrow {X}_{2}\to {X}_{3}\leftarrow \mathrm{\cdots}\to {X}_{n}\leftarrow Y$$ |

in which each map is a weak homotopy equivalence.

Title | weak homotopy equivalence |

Canonical name | WeakHomotopyEquivalence |

Date of creation | 2013-03-22 13:25:45 |

Last modified on | 2013-03-22 13:25:45 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 9 |

Author | antonio (1116) |

Entry type | Definition |

Classification | msc 55P10 |

Synonym | weak equivalence |

Related topic | HomotopyEquivalence |

Related topic | WeakHomotopyAdditionLemma |

Related topic | ApproximationTheoremForAnArbitrarySpace |

Related topic | OmegaSpectrum |

Related topic | WhiteheadTheorem |

Defines | weakly homotopy equivalent |

Defines | weakly equivalent |