# long exact sequence (locally trivial bundle)

Let $\pi :E\to B$ is a locally trivial bundle, with fiber $F$. Then there is a long exact sequence of homotopy groups

$$\begin{array}{ccccccccccc}\hfill \mathrm{\cdots}\hfill & \hfill \to \hfill & \hfill {\pi}_{n}(F)\hfill & \hfill \stackrel{{i}_{*}}{\to}\hfill & \hfill {\pi}_{n}(E)\hfill & \hfill \stackrel{{\pi}_{*}}{\to}\hfill & \hfill {\pi}_{n}(B)\hfill & \hfill \stackrel{{\partial}_{*}}{\to}\hfill & \hfill {\pi}_{n-1}(F)\hfill & \hfill \to \hfill & \hfill \mathrm{\cdots}\hfill \end{array}$$ |

Here ${i}_{*}$ is induced by the inclusion $i:F\hookrightarrow E$ as the fiber over the basepoint of $B$, and ${\partial}_{*}$ is the following map: if $[\phi ]\in {\pi}_{n}(B)$, then $\phi $ lifts to a map of $({D}^{n},\partial {D}^{n})$ into $(E,F)$ (that is a map of the $n$-disk into $E$, taking its boundary to $F$), sending the basepoint on the boundary to the base point of $F\subset E$. Thus the map on $\partial {D}^{n}={S}^{n-1}$, the $n-1$-sphere, defines an element of ${\pi}_{n-1}(F)$. This is ${\partial}_{*}[\phi ]$. The covering homotopy property of a locally trivial bundle shows that this is well-defined.

Title | long exact sequence (locally trivial bundle) |
---|---|

Canonical name | LongExactSequencelocallyTrivialBundle |

Date of creation | 2013-03-22 13:14:58 |

Last modified on | 2013-03-22 13:14:58 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 6 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 55Q05 |

Related topic | Fibration^{} |

Related topic | Fibration2 |

Related topic | HomotopyLiftingProperty |