# fibration

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.

 Title fibration Canonical name Fibration Date of creation 2013-03-22 15:37:57 Last modified on 2013-03-22 15:37:57 Owner whm22 (2009) Last modified by whm22 (2009) Numerical id 5 Author whm22 (2009) Entry type Definition Classification msc 55R65 Related topic fibremap Related topic FibreBundle Related topic LocallyTrivialBundle Related topic LongExactSequenceLocallyTrivialBundle Related topic homotopyliftingproperty Related topic cofibration Defines fibration