Induced fibre bundle
The fibration induced by the mapping and the fibration , where is the subspace of the direct product consisting of the pairs for which , and is the mapping defined by the projection . The mapping from the induced fibre bundle into the original fibre bundle defined by the formula is a bundle morphism covering . For each point , the restrictions
are homeomorphisms. Furthermore, for any fibration and morphism covering there exist precisely one -morphism such that , and such that the following diagram is commutative:
Fibre bundles induced from isomorphic fibrations are isomorphic, a fibre bundle induced by a constant mapping is isomorphic to the trivial fibre bundle.
For any section of a fibration , the mapping defined by the formula is a section of the induced fibration and satisfies the relation . For example, the mapping induces the fibration with space and base that is the square of the fibration and has the canonical section .
|||C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)|
|||N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)|
|||D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)|
Induced fibre bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Induced_fibre_bundle&oldid=12456