# 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) |

**How to Cite This Entry:**

Locally trivial fibre bundle.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Locally_trivial_fibre_bundle&oldid=47699