# locally trivial bundle

A locally trivial bundle is a continuous map
$\pi :E\to B$ of topological spaces^{} such that the following conditions
hold.
First, each point $x\in B$ must have a neighborhood^{} $U$ such that
the inverse image $\stackrel{~}{U}={\pi}^{-1}(U)$ is homeomorphic^{}
to $U\times {\pi}^{-1}(x)$.
Second, for some homeomorphism $g:\stackrel{~}{U}\to U\times {\pi}^{-1}(x)$,
the diagram

$$\text{xymatrix}\stackrel{~}{U}\text{ar}{[r]}^{(}0.3)g\text{ar}[d]{}_{\pi}\mathrm{\&}U\times \pi {}^{-1}(x)\text{ar}[d]{}^{\mathrm{id}\times \{x\}}U\text{ar}[r]{}^{\mathrm{id}}\mathrm{\&}U$$ |

must be commutative (http://planetmath.org/CommutativeDiagram).

Locally trivial bundles are useful because of their covering homotopy property and because each locally trivial bundle has an associated long exact sequence (http://planetmath.org/LongExactSequenceLocallyTrivialBundle) and Serre spectral sequence. Every fibre bundle is an example of a locally trivial bundle.

Title | locally trivial bundle |
---|---|

Canonical name | LocallyTrivialBundle |

Date of creation | 2013-03-22 13:15:01 |

Last modified on | 2013-03-22 13:15:01 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 10 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 55R10 |

Related topic | Fibration^{} |

Related topic | Fibration2 |

Related topic | HomotopyLiftingProperty |