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