# fibration

A fibration^{} is a map satisfying the homotopy lifting property. This is easily seen to be equivalent^{} to the following:

A map $f:X\to Y$ is a fibration if and only if there is a continuous function^{} which given a path, $\varphi $, in $Y$ and a point, $x$, lying above $\varphi (0)$, returns a lift of $\varphi $, starting at $x$.

Let ${D}^{2}$ denote the set of complex numbers with modulus less than or equal to 1. An example of a fibration is the map $g:{D}^{2}\to [-1,1]$ sending a complex number $z$ to $re(z)$.

Note that if we restrict $g$ to the boundary of ${D}^{2}$, we do not get a fibration. Although we may still lift any path to begin at a prescribed point, we cannot make this assignment continuously.

Another class of fibrations are found in fibre bundles.

Title | fibration |

Canonical name | Fibration |

Date of creation | 2013-03-22 15:37:57 |

Last modified on | 2013-03-22 15:37:57 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 5 |

Author | whm22 (2009) |

Entry type | Definition |

Classification | msc 55R65 |

Related topic | fibremap |

Related topic | FibreBundle |

Related topic | LocallyTrivialBundle |

Related topic | LongExactSequenceLocallyTrivialBundle |

Related topic | homotopyliftingproperty |

Related topic | cofibration |

Defines | fibration |