# 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

 $\begin{CD}\cdots @>{}>{}>\pi_{n}(F)@>{i_{*}}>{}>\pi_{n}(E)@>{\pi_{*}}>{}>\pi_{% n}(B)@>{\partial_{*}}>{}>\pi_{n-1}(F)@>{}>{}>\cdots\end{CD}$

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 $[\varphi]\in\pi_{n}(B)$, then $\varphi$ 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 $\partial_{*}[\varphi]$. The covering homotopy property of a locally trivial bundle shows that this is well-defined.

Title long exact sequence (locally trivial bundle) LongExactSequencelocallyTrivialBundle 2013-03-22 13:14:58 2013-03-22 13:14:58 bwebste (988) bwebste (988) 6 bwebste (988) Definition msc 55Q05 Fibration Fibration2 HomotopyLiftingProperty