universal bundle

Let G be a topological groupMathworldPlanetmath. A universal bundle for G is a principal bundleMathworldPlanetmath p:EGBG such that for any principal bundle π:EB, with B a CW-complexMathworldPlanetmath, there is a map φ:BBG, unique up to homotopyMathworldPlanetmath, such that the pullback bundle φ*(p) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to π, that is such that there is a bundle mapMathworldPlanetmath φ.


with φ(B)=φ, such that any bundle map of any bundle over B extending φ factors uniquely through φ.

As is obvious from the universal propertyMathworldPlanetmath, the universal bundle for a group G is unique up to unique homotopy equivalenceMathworldPlanetmathPlanetmath.

The base space BG is often called a classifying space of G, since homotopy classes of maps to it from a given space classify G-bundles over that space.

There is a useful criterion for universality: a bundle is universalPlanetmathPlanetmath if and only if all the homotopy groups of EG, its total space, are trivial. This allows us to construct the universal bundle any subgroup from that of a larger group. Assume HG and that p:EGBG is a universal bundle for G. Then H also acts freely on EG which is contractable so pH:EH=EBBH=EB/H must be a universal bundle for H.

In 1956, John Milnor gave a general construction of the universal bundle for any topological group G (see Annals of Mathematics, Second Series, Volume 63 Issue 2 and Issue 3 for details). His construction uses the infinite join of the group G with itself to define the total space of the universal bundle.

Title universal bundle
Canonical name UniversalBundle
Date of creation 2013-03-22 13:07:21
Last modified on 2013-03-22 13:07:21
Owner mps (409)
Last modified by mps (409)
Numerical id 14
Author mps (409)
Entry type Definition
Classification msc 55R10
Classification msc 55R15
Defines classifying space