nLab universal principal bundle







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.


See at classifying space.



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)
  • Dale Husemoeller, Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) gBooks pdf

For more see the references at classifying space.

For equivariant bundles

Discussion of universal equivariant principal bundles:

Last revised on June 21, 2023 at 10:28:01. See the history of this page for a list of all contributions to it.