fibration Fibration2 2006-01-11 16:05:52 2006-01-11 16:24:59 Definition fibration % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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.