Induced fibre bundle

induced fibration

The fibration $ f ^ { * } ( \pi ) : X ^ \prime \rightarrow B ^ \prime $ induced by the mapping $ f : B ^ \prime \rightarrow B $ and the fibration $ \pi : X \rightarrow B $, where $ X ^ \prime $ is the subspace of the direct product $ B ^ \prime \times X $ consisting of the pairs $ ( b ^ \prime , x ) $ for which $ f ( b ^ \prime ) = \pi ( x) $, and $ f ^ { * } ( x) $ is the mapping defined by the projection $ ( b ^ \prime , x ) \rightarrow b ^ \prime $. The mapping $ F : f ^ { * } ( X) \rightarrow X $ from the induced fibre bundle into the original fibre bundle defined by the formula $ F ( b ^ \prime , x ) = x $ is a bundle morphism covering $ f $. For each point $ b ^ \prime \in B $, the restrictions

$$ F _ {b ^ \prime } : \ ( f ^ { * } ( \pi ) ) ^ {- 1} ( b ^ \prime ) \rightarrow \pi ^ {- 1} ( f ( b ^ \prime ) ) $$

are homeomorphisms. Furthermore, for any fibration $ \eta : Y \rightarrow B ^ \prime $ and morphism $ H : \eta \rightarrow \pi $ covering $ f $ there exist precisely one $ B ^ \prime $-morphism $ K : \eta \rightarrow f ^ { * } ( \pi ) $ such that $ F K = H $, $ f ^ { * } ( \pi ) K= \eta $ and such that the following diagram is commutative: $ Y $ $ H $ $ k $ $ \eta $

Figure: i050720a

$$ \begin{array}{ccl} f ^ { * } ( X) &\rightarrow ^ { F } & X \\ size - 3 {f ^ { * } ( \pi ) } \downarrow &{} &\downarrow size - 3 \pi \\ B ^ \prime &\rightarrow _ { f } & B \\ \end{array} $$

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 $ \pi $, the mapping $ \sigma : B ^ \prime \rightarrow f ^ { * } ( x) $ defined by the formula $ \sigma ( b ^ \prime ) = ( b ^ \prime , s f ( b ^ \prime ) ) $ is a section of the induced fibration $ f ^ { * } ( \pi ) $ and satisfies the relation $ F \sigma = s f $. For example, the mapping $ \pi : X \rightarrow B $ induces the fibration $ \pi ^ {2} $ with space $ \pi ^ {*} ( x) $ and base $ X $ that is the square of the fibration $ \pi $ and has the canonical section $ s ( x) = ( x , x ) $.

References

[1]	C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)
[2]	N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)
[3]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)