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, $\varphi$, in $Y$ and a point, $x$, lying above $\varphi (0)$, returns a lift of $\varphi$, 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