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lifting of maps (Definition)

Let $ p\colon\thinspace E\to B$ and $ f\colon\thinspace X\to B$ be (continuous) maps. Then a lifting of $ f$ to $ E$ is a (continuous) map $ \tilde f\colon\thinspace X\to E$ such that $ p\circ \tilde f=f$. The terminology is justified by the following commutative diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ &{E}\ar[d]^{p}\ {X}\ar@{-->}[ur]^{\tilde f}\ar[r]_{f} & {B} } } \end{xy}$
which expresses this definition. $ \tilde f$ is also said to lift $ f$ or to be over $ f$.

This notion is especially useful if $ p\colon\thinspace E\to B$ is a fiber bundle. In particular lifting of paths is instrumental in the investigation of covering spaces.

This terminology is used in more general contexts: $ X$, $ E$ and $ B$ could be objects (and $ p$, $ f$ and $ \tilde f$ be morphisms) in any category.



"lifting of maps" is owned by Dr_Absentius.
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Also defines:  lifting, lift
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Cross-references: category, morphisms, objects, covering spaces, paths, fiber bundle, commutative diagram, maps, continuous
There are 17 references to this entry.

This is version 3 of lifting of maps, born on 2003-02-02, modified 2004-01-24.
Object id is 3961, canonical name is LiftingOfMaps.
Accessed 5880 times total.

Classification:
AMS MSC55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces)

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