Locally trivial fibre bundle

A fibre bundle (cf. Fibre space) $ \pi : X \rightarrow B $ with fibre $ F $ such that for any point of the base $ b \in B $ there is a neighbourhood $ U \ni b $ and a homeomorphism $ \phi _ {U} : U \times F \rightarrow \pi ^ {-} 1 ( U) $ such that $ \pi \phi _ {U} ( u, f ) = u $, where $ u \in U $, $ f \in F $. The mapping $ h _ {U} = \phi _ {U} ^ {-} 1 $ is called a chart of the locally trivial bundle. The totality of charts $ \{ h _ {U} \} $ associated with a covering of the base $ \{ U \} $ forms the atlas of the locally trivial bundle. For example, a principal fibre bundle with a locally compact space and a Lie group $ G $ is a locally trivial fibre bundle, and any chart $ h _ {U} $ satisfies the relation

$$ h _ {U} ( g x ) = g h _ {U} ( x) ,\ \ x \in \pi ^ {-} 1 ( U) , $$

where $ G $ acts on $ G \times U $ according to the formula $ g ( g ^ \prime , u ) = ( g g ^ \prime , u ) $. For any locally trivial fibre bundle $ \pi : X \rightarrow B $ and continuous mapping $ f : B _ {1} \rightarrow B $ the induced fibre bundle is locally trivial.

References

[1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[2]	N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)
[3]	S.-T. Hu, "Homotopy theory" , Acad. Press (1959)
[4]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)