MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition).

Does it make sense to speak of a universal F-G bundle ? (ie a bundle with fiber F and structure group G)

Specifically, during a course, we defined the universal G-bundle in the standard fashion (Milnor construction for existence etc.). But then it was said that $ EG \times_G F $ (the associated bundle) is a universal F-G bundle.

I then have the same question for characteristic classes: can they also be defined for non-principal bundles?

share|cite|improve this question
up vote 4 down vote accepted

If $B$ is a principal $G$-bundle, $B \times_G F$ is an $F-G$ bundle. Conversely, if $C $ is an $F -G$ bundle, $Hom (F,C)$ (structure-preserving Homs) is a principal bundle. So the two notions are equivalent.

share|cite|improve this answer
Hom(F,C), right? – Ryan Reich Mar 31 '13 at 15:41

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.