some examples of universal bundles

The universal bundle for a topological groupMathworldPlanetmath G is usually written as π:EGBG. Any principal G-bundle for which the total space is contractibleMathworldPlanetmath is universalPlanetmathPlanetmathPlanetmath; this will help us to find universal bundles without worrying about Milnor’s construction of EG involving infinite joins.

  • G=2: E2=S and B2=P.

  • G=n: En=S and Bn=S/n. Here n acts on S (considered as a subset of a separablePlanetmathPlanetmath complex Hilbert space) via multiplicationPlanetmathPlanetmath with an n-th root of unityMathworldPlanetmath.

  • G=n: En=n and Bn=Tn.

  • More generally, if G is any discrete group then one can take BG to be any Eilenberg-Mac Lane spaceMathworldPlanetmath K(G,1) and EG to be its universal cover. Indeed EG is simply connected, and it follows from the lifting theorem that πn(EG)=0 for n0. This example includes the previous three and many more.

  • G=S1: ES1=S and BS1=P.

  • G=SU(2): ESU(2)=S and BSU(2)=P.

  • G=O(n), the n-th orthogonal groupMathworldPlanetmath: EO(n)=V(,n), the manifold of frames of n orthonormal vectors in , and BO(n)=G(,n), the Grassmanian of n-planes in . The projection map is taking the subspace spanned by a frame of vectors.

Title some examples of universal bundles
Canonical name SomeExamplesOfUniversalBundles
Date of creation 2013-03-22 13:12:05
Last modified on 2013-03-22 13:12:05
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 11
Author bwebste (988)
Entry type Example
Classification msc 55R15
Classification msc 55R10
Synonym universal family of spaces
Related topic CategoryOfQuantumAutomata
Defines Hilbert bundle