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)

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)) 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).

References

[a1]	J. Dieudonné, "A history of algebraic and differential topology 1900–1960", Birkhäuser (1989)