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Difference between revisions of "Locally trivial fibre bundle"

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such that for any point of the base  $  b \in B $
 
such that for any point of the base  $  b \in B $
 
there is a neighbourhood  $  U \ni b $
 
there is a neighbourhood  $  U \ni b $
and a homeomorphism  $  \phi _ {U} :  U \times F \rightarrow \pi  ^ {-} 1 ( U) $
+
and a homeomorphism  $  \phi _ {U} :  U \times F \rightarrow \pi  ^ {-1} ( U) $
 
such that  $  \pi \phi _ {U} ( u, f  ) = u $,  
 
such that  $  \pi \phi _ {U} ( u, f  ) = u $,  
 
where  $  u \in U $,  
 
where  $  u \in U $,  
 
$  f \in F $.  
 
$  f \in F $.  
The mapping  $  h _ {U} = \phi _ {U}  ^ {-} 1 $
+
The mapping  $  h _ {U} = \phi _ {U}  ^ {-1} $
 
is called a chart of the locally trivial bundle. The totality of charts  $  \{ h _ {U} \} $
 
is called a chart of the locally trivial bundle. The totality of charts  $  \{ h _ {U} \} $
 
associated with a covering of the base  $  \{ U \} $
 
associated with a covering of the base  $  \{ U \} $
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$$  
 
$$  
 
h _ {U} ( g x )  =  g h _ {U} ( x) ,\ \  
 
h _ {U} ( g x )  =  g h _ {U} ( x) ,\ \  
x \in \pi  ^ {-} 1 ( U) ,
+
x \in \pi  ^ {-1} ( U) ,
 
$$
 
$$
  
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For any locally trivial fibre bundle  $  \pi :  X \rightarrow B $
 
For any locally trivial fibre bundle  $  \pi :  X \rightarrow B $
 
and continuous mapping  $  f :  B _ {1} \rightarrow B $
 
and continuous mapping  $  f :  B _ {1} \rightarrow B $
the [[Induced fibre bundle|induced fibre bundle]] is locally trivial.
+
the [[induced fibre bundle]] is locally trivial.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</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">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</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">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>

Latest revision as of 20:28, 16 January 2024


A fibre bundle (cf. Fibre space) $ \pi : X \rightarrow B $ with fibre $ F $ such that for any point of the base $ b \in B $ there is a neighbourhood $ U \ni b $ and a homeomorphism $ \phi _ {U} : U \times F \rightarrow \pi ^ {-1} ( U) $ such that $ \pi \phi _ {U} ( u, f ) = u $, where $ u \in U $, $ f \in F $. The mapping $ h _ {U} = \phi _ {U} ^ {-1} $ is called a chart of the locally trivial bundle. The totality of charts $ \{ h _ {U} \} $ associated with a covering of the base $ \{ U \} $ forms the atlas of the locally trivial bundle. For example, a principal fibre bundle with a locally compact space and a Lie group $ G $ is a locally trivial fibre bundle, and any chart $ h _ {U} $ satisfies the relation

$$ h _ {U} ( g x ) = g h _ {U} ( x) ,\ \ x \in \pi ^ {-1} ( U) , $$

where $ G $ acts on $ G \times U $ according to the formula $ g ( g ^ \prime , u ) = ( g g ^ \prime , u ) $. For any locally trivial fibre bundle $ \pi : X \rightarrow B $ and continuous mapping $ f : B _ {1} \rightarrow B $ the induced fibre bundle is locally trivial.

References

[1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[2] N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)
[3] S.-T. Hu, "Homotopy theory" , Acad. Press (1959)
[4] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
How to Cite This Entry:
Locally trivial fibre bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_trivial_fibre_bundle&oldid=55149
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article