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fibration (Definition)

A fibration is a map satisfying the homotopy lifting property. This is easily seen to be equivalent to the following:

A map $ f:X \to Y$ is a fibration if and only if there is a continuous function which given a path, $ \phi$, in $ Y$ and a point, $ x$, lying above $ \phi(0)$, returns a lift of $ \phi$, starting at $ x$.

Let $ D^2$ denote the set of complex numbers with modulus less than or equal to 1. An example of a fibration is the map $ g: D^2 \to [-1,1]$ sending a complex number $ z$ to $ re(z)$.

Note that if we restrict $ g$ to the boundary of $ D^2$, we do not get a fibration. Although we may still lift any path to begin at a prescribed point, we cannot make this assignment continuously.

Another class of fibrations are found in fibre bundles.



"fibration" is owned by whm22.
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See Also: fibre map, fibre bundle, locally trivial bundle, long exact sequence (locally trivial bundle), homotopy lifting property, cofibration

Also defines:  fibration
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Cross-references: fibre bundles, class, boundary, modulus, complex numbers, lift, point, path, continuous function, equivalent, homotopy lifting property, map
There are 3 references to this entry.

This is version 2 of fibration, born on 2006-01-11, modified 2006-01-11.
Object id is 7560, canonical name is Fibration2.
Accessed 2487 times total.

Classification:
AMS MSC: 55R65 (Algebraic topology :: Fiber spaces and bundles :: Generalizations of fiber spaces and bundles)
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