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''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====
<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>