PlanetMath: long exact sequence (locally trivial bundle)

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long exact sequence (locally trivial bundle)

(Definition)

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

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

Here is induced by the inclusion $i:F\hookrightarrow E$ as the fiber over the basepoint of , 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 (that is a map of the -disk into , taking its boundary to ), 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 -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.