# principal bundle

Let $E$ be a topological space^{} on which a topological group^{} $G$ acts continuously and freely. The map $\pi :E\to E/G=B$ is called a *principal bundle ^{}* (or

*principal $G$-bundle*) if the projection map $\pi :E\to B$ is a locally trivial bundle.

Any principal bundle with a section^{} $\sigma :B\to E$ is trivial, since the map $\varphi :B\times G\to E$ given by $\varphi (b,g)=g\cdot \sigma (b)$ is an isomorphism^{}. In particular, any $G$-bundle which is topologically trivial is also isomorphic to $B\times G$ as a $G$-space. Thus any local trivialization of $\pi :E\to B$ as a topological bundle is an equivariant trivialization.

Title | principal bundle |
---|---|

Canonical name | PrincipalBundle |

Date of creation | 2013-03-22 13:07:18 |

Last modified on | 2013-03-22 13:07:18 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 8 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 55R10 |

Defines | principal G-bundle |