nLab
universal principal bundle

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

For GG a topological group there is a notion of GG-principal bundles PXP \to X over any topological space XX. Under continuous maps f:XYf : X \to Y there is a notion of pullback of principal bundles f *:GBund(Y)GBund(X)f^* : G Bund(Y) \to G Bund(X).

A universal GG-principal bundle is a GG-principal bundle, which is usually written EGBGE G \to B G, such that for every CW-complex XX the map

[X,BG]GBund(X)/ [X, B G] \to G Bund(X)/_\sim

from homotopy classes of continuous functions XBGX \to B G given by [f]f *EG[f] \mapsto f^* E G, is an isomorphism.

In this case one calls BGB G a classifying space for GG-principal bundles.

The universal principal bundle is characterized, up to equivalence, by its total space EGE G being contractible.

More generally, we can ask for a universal bundle for numerable bundles, that is principal bundles which admit a trivialisation over a numerable open cover. Such a bundle exists, and classifies numerable bundles over all topological spaces, not just paracompact spaces or CW-complexes.

References

Among the earliest references that consider the notion of universal bundles is

A review is for instance in

  • Stephen Mitchell, Universal principal bundles and classifying spaces (pdf)

Revised on May 13, 2013 13:36:52 by Anonymous Coward (134.100.220.127)