# 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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article