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''fibre bundle with a structure group''
 
''fibre bundle with a structure group''
  
 
A generalization of the concept of the direct product of two topological spaces.
 
A generalization of the concept of the direct product of two topological spaces.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g0430102.png" /> be a topological group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g0430103.png" /> an effective right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g0430105.png" />-space, i.e. a topological space with a given right action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g0430106.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g0430107.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g0430108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g0430109.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301011.png" /> be the subset of those pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301012.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301013.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301014.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301015.png" /> be the orbit space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301016.png" /> be the mapping sending each point to its orbit. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301017.png" /> is continuous, then the tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301018.png" /> is called a principal fibre bundle with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301019.png" />.
+
Let $  G $
 +
be a topological group and $  X $
 +
an effective right $  G $-
 +
space, i.e. a topological space with a given right action of $  G $
 +
such that $  x g = x $
 +
for some $  x \in X $,  
 +
g \in G $,  
 +
implies $  g= 1 $.  
 +
Let $  X  ^ {*} \subset  X \times X $
 +
be the subset of those pairs $  ( x , x  ^  \prime  ) $
 +
for which $  x  ^  \prime  = x g $
 +
for some g \in G $,  
 +
let $  B = X / G $
 +
be the orbit space, and let $  p : X \rightarrow B $
 +
be the mapping sending each point to its orbit. If the mapping $  X  ^ {*} \rightarrow G : ( x , x g ) \mapsto g $
 +
is continuous, then the tuple $  \xi = ( X , p , B ) $
 +
is called a principal fibre bundle with structure group $  G $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301020.png" /> be a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301021.png" />-space. The topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301022.png" /> admits a right action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301023.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301025.png" />. The composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301026.png" /> induces a mapping: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301027.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301028.png" /> is the orbit space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301029.png" /> under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301030.png" />). The quadruple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301031.png" /> is called a fibre bundle with structure group associated with the principal fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301032.png" />, and the quadruple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301033.png" /> is a fibre bundle with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301037.png" />, base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301038.png" /> and structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301039.png" />. Thus, a principal fibre bundle with a given structure group is a part of the structure of any fibre bundle with (that) structure group, and it uniquely determines the fibre bundle for any left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301040.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301041.png" />.
+
Let $  F $
 +
be a left $  G $-
 +
space. The topological space $  X \times F $
 +
admits a right action of $  G $
 +
by  $  ( x, f  ) g = ( x g , g ^ {-} 1 f  ) $,
 +
$  f \in F $.  
 +
The composition $  X \times F \rightarrow ^ { \mathop{\rm pr} _ {X} } X \rightarrow  ^ {p} B $
 +
induces a mapping: $  X _ {F} = ( X \times F ) / G \rightarrow ^ {p _ {F} } B $(
 +
where $  X _ {F} $
 +
is the orbit space of $  X \times F $
 +
under the action of $  G $).  
 +
The quadruple $  ( X _ {F} , p _ {F} , B , F  ) $
 +
is called a fibre bundle with structure group associated with the principal fibre bundle $  \xi $,  
 +
and the quadruple $  ( X _ {F} , p _ {F} , F, \xi ) $
 +
is a fibre bundle with fibre $  F $,  
 +
base $  B $
 +
and structure group $  G $.  
 +
Thus, a principal fibre bundle with a given structure group is a part of the structure of any fibre bundle with (that) structure group, and it uniquely determines the fibre bundle for any left $  G $-
 +
space $  F $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301043.png" /> are two principal fibre bundles with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301044.png" />, then a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301045.png" /> is a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301046.png" />-spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301047.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301048.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301049.png" />. A principal fibre bundle with structure group is called trivial is it is isomorphic to a fibre bundle of the following type:
+
If $  \xi = ( X , p , B ) $,
 +
$  \xi  ^  \prime  = ( X ^ { \prime } , p  ^  \prime  , B  ^  \prime  ) $
 +
are two principal fibre bundles with structure group $  G $,  
 +
then a morphism $  \xi \rightarrow \xi  ^  \prime  $
 +
is a mapping of $  G $-
 +
spaces $  h : X \rightarrow X ^ { \prime } $.  
 +
$  h $
 +
induces a mapping $  f : B \rightarrow B  ^  \prime  $.  
 +
A principal fibre bundle with structure group is called trivial is it is isomorphic to a fibre bundle of the following type:
  
<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/g/g043/g043010/g04301050.png" /></td> </tr></table>
+
$$
 +
( B \times G ,  \mathop{\rm pr} _ {B} , B ) ,\ \
 +
( b , g ) g  ^  \prime  = ( b , g g  ^  \prime  ) ,\ \
 +
b \in B ,\  g , g ^  \prime  \in G .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301051.png" /> be a principal fibre bundle and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301052.png" /> be a continuous mapping of an arbitrary topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301053.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301054.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301055.png" /> be the subset of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301056.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301057.png" />. The projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301058.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301059.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301060.png" /> has the natural structure of a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301061.png" />-space, and the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301062.png" /> is a principal fibre bundle; it is induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301063.png" /> and is called an induced fibre bundle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301064.png" /> is the inclusion mapping of a subspace, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301065.png" /> is called the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301066.png" /> over the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301067.png" />.
+
Let $  ( X , p , B ) $
 +
be a principal fibre bundle and let $  f : B  ^  \prime  \rightarrow B $
 +
be a continuous mapping of an arbitrary topological space $  B  ^  \prime  $
 +
into $  B $.  
 +
Let $  X  ^  \prime  \subset  B  ^  \prime  \times X $
 +
be the subset of pairs $  ( b , x) $
 +
for which $  f ( b) = p ( x) $.  
 +
The projection $  \mathop{\rm pr} _ {B  ^  \prime  } : B  ^  \prime  \times X \rightarrow B  ^  \prime  $
 +
induces a mapping $  p  ^  \prime  : X ^ { \prime } \rightarrow B  ^  \prime  $.  
 +
The space $  X ^ { \prime } $
 +
has the natural structure of a right $  G $-
 +
space, and the triple $  ( X ^ { \prime } , p , B ) $
 +
is a principal fibre bundle; it is induced by $  f $
 +
and is called an induced fibre bundle. If $  f : B  ^  \prime  \rightarrow B $
 +
is the inclusion mapping of a subspace, then $  ( X ^ { \prime } , p  ^  \prime  , B  ^  \prime  ) $
 +
is called the restriction of $  ( X , p , B ) $
 +
over the subspace $  B  ^  \prime  $.
  
A principal fibre bundle with structure group is called locally trivial if its restriction to some neighbourhood of any point of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301068.png" /> is trivial. For a wide class of cases, the requirement of local triviality is unnecessary (e.g. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301069.png" /> is a compact Lie group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301070.png" /> a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301071.png" />-manifold). Hence, the term  "fibre bundle"  with structure group is often used in the sense of a locally trivial fibre bundle (or fibration).
+
A principal fibre bundle with structure group is called locally trivial if its restriction to some neighbourhood of any point of the base $  B $
 +
is trivial. For a wide class of cases, the requirement of local triviality is unnecessary (e.g. if $  G $
 +
is a compact Lie group and $  X $
 +
a smooth $  G $-
 +
manifold). Hence, the term  "fibre bundle"  with structure group is often used in the sense of a locally trivial fibre bundle (or fibration).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301073.png" /> be a pair of fibre bundles with the same structure group and the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301074.png" />-space as fibre. Given a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301075.png" /> of principal fibre bundles, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301076.png" /> induces a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301077.png" />, and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301078.png" /> is called a morphism of fibre bundles with structure group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301079.png" />.
+
Let $  ( X _ {F} , p _ {F} , F , \xi ) $,
 +
$  ( X _ {F} ^ { \prime } , p _ {F}  ^  \prime  , F , \xi  ^  \prime  ) $
 +
be a pair of fibre bundles with the same structure group and the same $  G $-
 +
space as fibre. Given a morphism $  h : \xi \rightarrow \xi  ^  \prime  $
 +
of principal fibre bundles, the mapping $  h \times  \mathop{\rm id} : X \times F \rightarrow X ^ { \prime } \times F $
 +
induces a continuous mapping $  \phi : X _ {F} \rightarrow X _ {F} ^ { \prime } $,  
 +
and the pair $  ( h , \phi ) $
 +
is called a morphism of fibre bundles with structure group, $  ( X _ {F} , p _ {F} , F , \xi ) \rightarrow ( X _ {F} ^ { \prime } , p _ {F}  ^  \prime  , F , \xi  ^  \prime  ) $.
  
A locally trivial fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301080.png" /> admits the following characterization, which gives rise to another (also generally accepted) definition of a fibre bundle with structure group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301081.png" /> be an open covering of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301082.png" /> such that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301083.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301084.png" /> is trivial for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301085.png" />. The choice of trivializations and their equality on the intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301086.png" /> leads to continuous functions (called transfer functions) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301087.png" />. On the intersection of three neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301088.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301089.png" />, while the choice of other trivializations over every neighbourhood leads to new functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301090.png" />. In this way, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301091.png" /> form a one-dimensional Aleksandrov–Čech cocycle with coefficients in the sheaf of germs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043010/g04301092.png" />-valued functions (the coefficients are non-Abelian), and a locally trivial fibre bundle determines this cocycle up to a coboundary.
+
A locally trivial fibre bundle $  \eta = ( X _ {F} , p _ {F} , F , \xi ) $
 +
admits the following characterization, which gives rise to another (also generally accepted) definition of a fibre bundle with structure group. Let $  U = \{ u _  \alpha  \} $
 +
be an open covering of the base $  B $
 +
such that the restriction of $  \eta $
 +
to $  u _  \alpha  $
 +
is trivial for all $  \alpha $.  
 +
The choice of trivializations and their equality on the intersections $  u _  \alpha  \cap u _  \beta  $
 +
leads to continuous functions (called transfer functions) $  g _ {\alpha \beta }  : u _  \alpha  \cap u _  \beta  \rightarrow G $.  
 +
On the intersection of three neighbourhoods $  u _  \alpha  \cap u _  \beta  \cap u _  \gamma  $
 +
one has $  g _ {\alpha \beta }  \circ g _ {\beta \gamma }  \circ g _ {\gamma \alpha }  = 1 \in G $,  
 +
while the choice of other trivializations over every neighbourhood leads to new functions $  g _ {\alpha \beta }  ^  \prime  = h _  \alpha  g _ {\alpha \beta }  h _  \beta  ^ {-} 1 $.  
 +
In this way, the functions $  \{ g _ {\alpha \beta }  \} $
 +
form a one-dimensional Aleksandrov–Čech cocycle with coefficients in the sheaf of germs of $  G $-
 +
valued functions (the coefficients are non-Abelian), and a locally trivial fibre bundle determines this cocycle up to a coboundary.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , 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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , 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></table>

Latest revision as of 19:40, 5 June 2020


fibre bundle with a structure group

A generalization of the concept of the direct product of two topological spaces.

Let $ G $ be a topological group and $ X $ an effective right $ G $- space, i.e. a topological space with a given right action of $ G $ such that $ x g = x $ for some $ x \in X $, $ g \in G $, implies $ g= 1 $. Let $ X ^ {*} \subset X \times X $ be the subset of those pairs $ ( x , x ^ \prime ) $ for which $ x ^ \prime = x g $ for some $ g \in G $, let $ B = X / G $ be the orbit space, and let $ p : X \rightarrow B $ be the mapping sending each point to its orbit. If the mapping $ X ^ {*} \rightarrow G : ( x , x g ) \mapsto g $ is continuous, then the tuple $ \xi = ( X , p , B ) $ is called a principal fibre bundle with structure group $ G $.

Let $ F $ be a left $ G $- space. The topological space $ X \times F $ admits a right action of $ G $ by $ ( x, f ) g = ( x g , g ^ {-} 1 f ) $, $ f \in F $. The composition $ X \times F \rightarrow ^ { \mathop{\rm pr} _ {X} } X \rightarrow ^ {p} B $ induces a mapping: $ X _ {F} = ( X \times F ) / G \rightarrow ^ {p _ {F} } B $( where $ X _ {F} $ is the orbit space of $ X \times F $ under the action of $ G $). The quadruple $ ( X _ {F} , p _ {F} , B , F ) $ is called a fibre bundle with structure group associated with the principal fibre bundle $ \xi $, and the quadruple $ ( X _ {F} , p _ {F} , F, \xi ) $ is a fibre bundle with fibre $ F $, base $ B $ and structure group $ G $. Thus, a principal fibre bundle with a given structure group is a part of the structure of any fibre bundle with (that) structure group, and it uniquely determines the fibre bundle for any left $ G $- space $ F $.

If $ \xi = ( X , p , B ) $, $ \xi ^ \prime = ( X ^ { \prime } , p ^ \prime , B ^ \prime ) $ are two principal fibre bundles with structure group $ G $, then a morphism $ \xi \rightarrow \xi ^ \prime $ is a mapping of $ G $- spaces $ h : X \rightarrow X ^ { \prime } $. $ h $ induces a mapping $ f : B \rightarrow B ^ \prime $. A principal fibre bundle with structure group is called trivial is it is isomorphic to a fibre bundle of the following type:

$$ ( B \times G , \mathop{\rm pr} _ {B} , B ) ,\ \ ( b , g ) g ^ \prime = ( b , g g ^ \prime ) ,\ \ b \in B ,\ g , g ^ \prime \in G . $$

Let $ ( X , p , B ) $ be a principal fibre bundle and let $ f : B ^ \prime \rightarrow B $ be a continuous mapping of an arbitrary topological space $ B ^ \prime $ into $ B $. Let $ X ^ \prime \subset B ^ \prime \times X $ be the subset of pairs $ ( b , x) $ for which $ f ( b) = p ( x) $. The projection $ \mathop{\rm pr} _ {B ^ \prime } : B ^ \prime \times X \rightarrow B ^ \prime $ induces a mapping $ p ^ \prime : X ^ { \prime } \rightarrow B ^ \prime $. The space $ X ^ { \prime } $ has the natural structure of a right $ G $- space, and the triple $ ( X ^ { \prime } , p , B ) $ is a principal fibre bundle; it is induced by $ f $ and is called an induced fibre bundle. If $ f : B ^ \prime \rightarrow B $ is the inclusion mapping of a subspace, then $ ( X ^ { \prime } , p ^ \prime , B ^ \prime ) $ is called the restriction of $ ( X , p , B ) $ over the subspace $ B ^ \prime $.

A principal fibre bundle with structure group is called locally trivial if its restriction to some neighbourhood of any point of the base $ B $ is trivial. For a wide class of cases, the requirement of local triviality is unnecessary (e.g. if $ G $ is a compact Lie group and $ X $ a smooth $ G $- manifold). Hence, the term "fibre bundle" with structure group is often used in the sense of a locally trivial fibre bundle (or fibration).

Let $ ( X _ {F} , p _ {F} , F , \xi ) $, $ ( X _ {F} ^ { \prime } , p _ {F} ^ \prime , F , \xi ^ \prime ) $ be a pair of fibre bundles with the same structure group and the same $ G $- space as fibre. Given a morphism $ h : \xi \rightarrow \xi ^ \prime $ of principal fibre bundles, the mapping $ h \times \mathop{\rm id} : X \times F \rightarrow X ^ { \prime } \times F $ induces a continuous mapping $ \phi : X _ {F} \rightarrow X _ {F} ^ { \prime } $, and the pair $ ( h , \phi ) $ is called a morphism of fibre bundles with structure group, $ ( X _ {F} , p _ {F} , F , \xi ) \rightarrow ( X _ {F} ^ { \prime } , p _ {F} ^ \prime , F , \xi ^ \prime ) $.

A locally trivial fibre bundle $ \eta = ( X _ {F} , p _ {F} , F , \xi ) $ admits the following characterization, which gives rise to another (also generally accepted) definition of a fibre bundle with structure group. Let $ U = \{ u _ \alpha \} $ be an open covering of the base $ B $ such that the restriction of $ \eta $ to $ u _ \alpha $ is trivial for all $ \alpha $. The choice of trivializations and their equality on the intersections $ u _ \alpha \cap u _ \beta $ leads to continuous functions (called transfer functions) $ g _ {\alpha \beta } : u _ \alpha \cap u _ \beta \rightarrow G $. On the intersection of three neighbourhoods $ u _ \alpha \cap u _ \beta \cap u _ \gamma $ one has $ g _ {\alpha \beta } \circ g _ {\beta \gamma } \circ g _ {\gamma \alpha } = 1 \in G $, while the choice of other trivializations over every neighbourhood leads to new functions $ g _ {\alpha \beta } ^ \prime = h _ \alpha g _ {\alpha \beta } h _ \beta ^ {-} 1 $. In this way, the functions $ \{ g _ {\alpha \beta } \} $ form a one-dimensional Aleksandrov–Čech cocycle with coefficients in the sheaf of germs of $ G $- valued functions (the coefficients are non-Abelian), and a locally trivial fibre bundle determines this cocycle up to a coboundary.

References

[1] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[2] N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)
How to Cite This Entry:
G-fibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G-fibration&oldid=47030
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article