# locally trivial bundle

A locally trivial bundle is a continuous map $\pi:E\to B$ of topological spaces such that the following conditions hold. First, each point $x\in B$ must have a neighborhood $U$ such that the inverse image $\widetilde{U}=\pi^{-1}(U)$ is homeomorphic to $U\times\pi^{-1}(x)$. Second, for some homeomorphism $g:\widetilde{U}\to U\times\pi^{-1}(x)$, the diagram

 $\xymatrix{\widetilde{U}\ar[r]^{(}0.3)g\ar[d]_{\pi}&U\times\pi^{-1}(x)\ar[d]^{% \mathrm{id}\times\{x\}}\\ U\ar[r]^{\mathrm{id}}&U}$

must be commutative (http://planetmath.org/CommutativeDiagram).

Locally trivial bundles are useful because of their covering homotopy property and because each locally trivial bundle has an associated long exact sequence (http://planetmath.org/LongExactSequenceLocallyTrivialBundle) and Serre spectral sequence. Every fibre bundle is an example of a locally trivial bundle.

Title locally trivial bundle LocallyTrivialBundle 2013-03-22 13:15:01 2013-03-22 13:15:01 mps (409) mps (409) 10 mps (409) Definition msc 55R10 Fibration Fibration2 HomotopyLiftingProperty