Let be a topological group. A universal bundle for is a principal bundle such that for any principal bundle , with a CW-complex, there is a map , unique up to homotopy, such that the pullback bundle is equivalent to , that is such that there is a bundle map .
with , such that any bundle map of any bundle over extending factors uniquely through .
There is a useful criterion for universality: a bundle is universal if and only if all the homotopy groups of , its total space, are trivial. This allows us to construct the universal bundle any subgroup from that of a larger group. Assume and that is a universal bundle for . Then also acts freely on which is contractable so must be a universal bundle for .
In 1956, John Milnor gave a general construction of the universal bundle for any topological group (see Annals of Mathematics, Second Series, Volume 63 Issue 2 and Issue 3 for details). His construction uses the infinite join of the group with itself to define the total space of the universal bundle.
|Date of creation||2013-03-22 13:07:21|
|Last modified on||2013-03-22 13:07:21|
|Last modified by||mps (409)|