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''induced fibration''
 
''induced fibration''
  
The [[Fibration|fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507201.png" /> induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507202.png" /> and the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507204.png" /> is the subspace of the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507205.png" /> consisting of the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507206.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507207.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507208.png" /> is the mapping defined by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i0507209.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072010.png" /> from the induced fibre bundle into the original fibre bundle defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072011.png" /> is a bundle morphism covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072012.png" />. For each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072013.png" />, the restrictions
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The [[Fibration|fibration]] $  f ^ { * } ( \pi ) : X  ^  \prime  \rightarrow B  ^  \prime  $
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induced by the mapping $  f : B  ^  \prime  \rightarrow B $
 +
and the fibration $  \pi : X \rightarrow B $,  
 +
where $  X  ^  \prime  $
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is the subspace of the direct product $  B  ^  \prime  \times X $
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consisting of the pairs $  ( b  ^  \prime  , x ) $
 +
for which $  f ( b  ^  \prime  ) = \pi ( x) $,  
 +
and $  f ^ { * } ( x) $
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072014.png" /></td> </tr></table>
+
$$
 +
F _ {b  ^  \prime  } : \
 +
( f ^ { * } ( \pi ) )  ^ {-} 1 ( b  ^  \prime  )  \rightarrow  \pi  ^ {-} 1 ( f ( b  ^  \prime  ) )
 +
$$
  
are homeomorphisms. Furthermore, for any fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072015.png" /> and morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072016.png" /> covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072017.png" /> there exist precisely one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072018.png" />-morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072021.png" /> and such that the following diagram is commutative: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072022.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072023.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072024.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072025.png" />
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are homeomorphisms. Furthermore, for any fibration $  \eta : Y \rightarrow B  ^  \prime  $
 +
and morphism $  H : \eta \rightarrow \pi $
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covering $  f $
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there exist precisely one $  B  ^  \prime  $-
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morphism $  K : \eta \rightarrow f ^ { * } ( \pi ) $
 +
such that $  F K = H $,  
 +
$  f ^ { * } ( \pi ) K= \eta $
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and such that the following diagram is commutative: $  Y $
 +
$  H $
 +
$  k $
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$  \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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072026.png" /></td> </tr></table>
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$$
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 +
\begin{array}{ccl}
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f ^ { * } ( X)  &\rightarrow ^ { F }  & X  \\
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size - 3 {f ^ { * } ( \pi ) } \downarrow  &{}  &\downarrow size - 3 \pi  \\
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B  ^  \prime  &\rightarrow _ { f }  & B  \\
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\end{array}
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 +
$$
  
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072027.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072028.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072029.png" /> is a section of the induced fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072030.png" /> and satisfies the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072031.png" />. For example, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072032.png" /> induces the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072033.png" /> with space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072034.png" /> and base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072035.png" /> that is the square of the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072036.png" /> and has the canonical section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050720/i05072037.png" />.
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For any section of a fibration $  \pi $,  
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the mapping $  \sigma : B  ^  \prime  \rightarrow f ^ { * } ( x) $
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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 $
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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>

Latest 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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article