long exact sequence (locally trivial bundle)

Let π:EB is a locally trivial bundle, with fiber F. Then there is a long exact sequence of homotopy groups


Here i* is induced by the inclusion i:FE as the fiber over the basepoint of B, and * is the following map: if [φ]πn(B), then φ lifts to a map of (Dn,Dn) 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 FE. Thus the map on Dn=Sn-1, the n-1-sphere, defines an element of πn-1(F). This is *[φ]. 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 FibrationMathworldPlanetmath
Related topic Fibration2
Related topic HomotopyLiftingProperty