# Principal fibre bundle

A $G$- 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 $u ^ {*} ( \pi _ {G} ) \rightarrow \pi _ {G}$ 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). 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

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

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)) 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).