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''induced 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  $  \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" />

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====
<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>