The only examples I found of nonprincipal circle bundle are nonorientable, like the Klein bottle that is an S^1 bundle over S^1 which is not principal and nontrivial. That makes me ask the question.

Is it true that every orientable circle bundle is principal?

  • 5
    This is essentially true because $Diff^+(S^1)\simeq U(1)$, so $BDiff^+(S^1)\simeq BU(1)$ from the fibration $Diff^+(S^1)/U(1)\to EDiff^+(S^1)/U(1) \to EDiff^+(S^1)/Diff^+(S^1)$ with contractible fibers. – Ian Agol Oct 6 '13 at 17:03
up vote 15 down vote accepted

Yes,it is true.You can find this result from "Geometry of differential forms" by Morita (Page 241)

PROPOSITION 6.15. Every oriented $S^1$ bundle admits the structure of principal $S^1$ bundle.

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