Namespaces
Variants
Actions

Difference between revisions of "Locally trivial fibre bundle"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A fibre bundle (cf. [[Fibre space|Fibre space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605301.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605302.png" /> such that for any point of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605303.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605304.png" /> and a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605305.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605308.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l0605309.png" /> is called a chart of the locally trivial bundle. The totality of charts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053010.png" /> associated with a covering of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053011.png" /> forms the atlas of the locally trivial bundle. For example, a [[Principal fibre bundle|principal fibre bundle]] with a locally compact space and a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053012.png" /> is a locally trivial fibre bundle, and any chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053013.png" /> satisfies the relation
+
<!--
 +
l0605301.png
 +
$#A+1 = 19 n = 0
 +
$#C+1 = 19 : ~/encyclopedia/old_files/data/L060/L.0600530 Locally trivial fibre bundle
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/l/l060/l060530/l06053014.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053015.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053016.png" /> according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053017.png" />. For any locally trivial fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053018.png" /> and continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060530/l06053019.png" /> the [[Induced fibre bundle|induced fibre bundle]] is locally trivial.
+
A fibre bundle (cf. [[Fibre space|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|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|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>

Revision as of 22:17, 5 June 2020


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=47699
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article