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locally trivial bundle (Definition)

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

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \widetilde{U}\ar[r]^(0.3)g\ar[d]... ...-1}(x)\ar[d]^{\mathrm{id}\times\{x\}}\ U\ar[r]^{\mathrm{id}} & U } } \end{xy}$
must be commutative.

Locally trivial bundles are useful because of their covering homotopy property and because each locally trivial bundle has an associated long exact sequence and Serre spectral sequence. Every fibre bundle is an example of a locally trivial bundle.



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"locally trivial bundle" is owned by mps. [ full author list (2) | owner history (1) ]
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See Also: fibre map, fibration, homotopy lifting property

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Cross-references: fibre bundle, spectral sequence, property, homotopy, covering, homeomorphism, homeomorphic, inverse image, neighborhood, point, topological spaces, continuous map
There are 2 references to this entry.

This is version 7 of locally trivial bundle, born on 2002-12-11, modified 2006-02-15.
Object id is 3727, canonical name is LocallyTrivialBundle.
Accessed 2442 times total.

Classification:
AMS MSC55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

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