universal covering space UniversalCoveringSpace 2002-02-02 03:50:12 2003-06-18 17:05:20 Definition % 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 Let $X$ be a topological space. A {\em universal covering space} is a covering space $\tilde{X}$ of $X$ which is connected and simply connected. If $X$ is based, with basepoint $x$, then a {\em based cover} of $X$ is cover of $X$ which is also a based space with a basepoint $x'$ such that the covering is a map of based spaces. Note that any cover can be made into a based cover by choosing a basepoint from the pre-images of $x$. The universal covering space has the following universal property: If $\pi:(\tilde X,x_0)\to(X,x)$ is a based universal cover, then for any connected based cover $\pi':(X',x')\to (X,x)$, there is a unique covering map $\pi'':(\tilde X,x_0)\to(X',x')$ such that $\pi=\pi'\circ\pi''$. Clearly, if a universal covering exists, it is unique up to unique isomorphism. But not every topological space has a universal cover. In fact $X$ has a universal cover if and only if it is semi-locally simply connected (for example, if it is a locally finite CW-complex or a manifold).