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A [[G-fibration| $  G $-
fibration]]  $  \pi _ {G} :  X \rightarrow B $
such that the group  $  G $
acts freely and perfectly on the space  $  X $. 
The significance of principal fibre bundles lies in the fact that they make it possible to construct associated fibre bundles with fibre  $  F $
if a representation of  $  G $
in the group of homeomorphisms  $  F $
is given. Differentiable principal fibre bundles with Lie groups play an important role in the theory of connections and holonomy groups. For instance, let  $  H $
be a topological group with  $  G $
as a closed subgroup and let  $  H/G $
be the homogeneous space of left cosets of  $  H $
with respect to  $  G $; 
the fibre bundle  $  \pi _ {G} :  H \rightarrow H/G $
will then be principal. Further, let  $  X _ {G} $
be a Milnor construction, i.e. the join of an infinite number of copies of  $  G $, 
each point of which has the form

$$ 
\langle  g, t \rangle  =  \langle  g _ {0} t _ {0} , g _ {1} t _ {1} ,\dots \rangle ,
$$

where  $  g _ {i} \in G $, 
$  t _ {i} \in [ 0, 1] $, 
and where only finitely many  $  t _ {i} $
are non-zero. The action of  $  G $
on  $  X _ {G} $
defined by the formula  $  h \langle  g, t\rangle = \langle  hg, t\rangle $
is free, and the fibre bundle  $  \omega _ {G} :  X _ {G} \rightarrow X _ {G} $
$   \mathop{\rm mod}  G $
is a numerable principal fibre bundle.

Each fibre of a principal fibre bundle is homeomorphic to  $  G $.

A morphism of principal fibre bundles is a morphism of the fibre bundles  $  f:  \pi _ {G} \rightarrow \pi _ {G  ^  \prime   } $
for which the mapping of the fibres  $  f {\pi _ {G} }  ^ {-} 1 ( b) $
induces a homomorphism of groups:

$$ 
\theta _ {b}  = \ 
\xi _ {b} ^ {\prime - 1 } f \pi _ {G}  ^ {-} 1 ( b) \xi _ {b} : \ 
G  \rightarrow  G  ^  \prime  ,
$$

where  $  \xi _ {b} ( g) = gx $, 
$  \pi _ {G} ( x) = b $. 
In particular, a morphism is called equivariant if  $  \theta _ {b} = \theta $
is independent of  $  b $, 
so that  $  gf ( x) = \theta ( g) f ( x) $
for any  $  x \in X $, 
$  g \in G $. 
If  $  G = G  ^  \prime  $
and  $  \theta =  \mathop{\rm id} $, 
an equivariant morphism is called a  $  G $-
morphism. Any  $  ( G, B) $-
morphism (i.e. a  $  G $-
morphism over  $  B $) 
is called a  $  G $-
isomorphism.

For any mapping  $  u:  B  ^  \prime  \rightarrow B $
and principal fibre bundle  $  \pi _ {G} :  X \rightarrow B $
the [[Induced fibre bundle|induced fibre bundle]]  $  u  ^ {*} ( \pi _ {G} ) \rightarrow \pi _ {G} $
is principal with the same group  $  G $; 
moreover, the mapping  $  U:  u  ^ {*} ( \pi _ {G} ) \rightarrow \pi _ {G} $
is a  $  G $-
morphism which unambiguously determines the action of  $  G $
on the space  $  u  ^ {*} ( x) $. 
For instance, if the principal fibre bundle  $  \pi _ {G} $
is trivial, it is isomorphic to the principal fibre bundle  $  \phi  ^ {*} ( \eta ) $, 
where  $  \eta $
is the  $  G $-
bundle over a single point and  $  \phi $
is the constant mapping. The converse is also true, and for this reason principal fibre bundles with a section are trivial. For each numerable principal fibre bundle  $  \pi _ {G} :  X \rightarrow B $
there exists a mapping  $  f:  B \rightarrow X _ {G} $
$   \mathop{\rm mod}  G $
such that  $  f ^ { * } ( \omega _ {G} ) $
is  $  G $-
isomorphic to  $  \pi _ {G} $, 
and for the principal fibre bundles  $  f _ {0} ^ { * } ( \omega _ {G} ) $
and  $  f _ {1} ^ { * } ( \omega _ {G} ) $
to be isomorphic, it is necessary and sufficient that  $  f _ {0} $
and  $  f _ {1} $
be homotopic (cf. [[Homotopy|Homotopy]]). This is the principal theorem on the homotopy classification of principal fibre bundles, which expresses the universality of the principal fibre bundle  $  \omega _ {G} $(
obtained by Milnor's construction), with respect to the classifying mapping  $  f $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.L. Bishop,   R.J. Crittenden,   "Geometry of manifolds" , Acad. Press  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Nomizu,   "Lie groups and differential geometry" , Math. Soc. Japan  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Fibre spaces and their applications'' , Moscow  (1958)  (In Russian; translated from English)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Steenrod,   "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>

====Comments====
Let  $  \pi _ {G} :  X \rightarrow B $
be a principal fibre bundle. It is called numerable if there is a sequence  $  ( u _ {n} ) _ {n \geq  0 }  $
of continuous mappings  $  B \rightarrow [ 0, 1] $
such that the open sets  $  U _ {n} = u _ {n}  ^ {-} 1 (( 0, 1 ] ) $
form an open covering (cf. [[Covering (of a set)|Covering (of a set)]]) of  $  B $
and  $  X $
is trivializable over each  $  U _ {n} $(
i.e. the restricted bundles  $  \pi _ {G} :  X \rightarrow U _ {n} $
are trivial, cf. [[Fibre space|Fibre space]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dieudonné,   "A history of algebraic and differential topology 1900–1960" , Birkhäuser  (1989)</TD></TR></table>