Difference between revisions of "Induced fibre bundle"
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''induced fibration'' | ''induced fibration'' | ||
| − | The [[Fibration|fibration]] | + | The [[Fibration|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 | + | 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 $ | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050720a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050720a.gif" /> | ||
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Figure: i050720a | 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. | 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 | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)</TD></TR></table> | ||
Revision as of 22:12, 5 June 2020
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) |
How to Cite This Entry:
Induced fibre bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Induced_fibre_bundle&oldid=12456
Induced fibre bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Induced_fibre_bundle&oldid=12456
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article