universal covering space

Let X be a topological spaceMathworldPlanetmath. A universal covering space is a covering space X~ of X which is connected and simply connected.

If X is based, with basepoint x, then a 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 propertyMathworldPlanetmath: If π:(X~,x0)(X,x) is a based universal cover, then for any connected based cover π:(X,x)(X,x), there is a unique covering map π′′:(X~,x0)(X,x) such that π=ππ′′.

Clearly, if a universal covering exists, it is unique up to unique isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. 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 finitePlanetmathPlanetmath CW-complexMathworldPlanetmath or a manifold).

Title universal covering space
Canonical name UniversalCoveringSpace
Date of creation 2013-03-22 12:15:34
Last modified on 2013-03-22 12:15:34
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 7
Author bwebste (988)
Entry type Definition
Classification msc 54-00
Synonym universal cover
Related topic OmegaSpectrum