
#1
Jan813, 07:32 PM

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This question asks whether every circle bundle comes from a 2 plane bundle. Paracompact space please  preferably a manifold.
By circle bundle I mean the usual thing, a fiber bundle with fiber, a circle, that is locally a product bundle. The transition functions lie in some group of homeomorphisms of the circle. A similar question can be asked for a sphere bundle. 



#2
Jan913, 10:13 AM

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P: 9,428

a more general question, but only over the base space P^2, whether every bundle of quadrics comes from a bundle of ambient projective spaces defined by a vector bundle, is answered affirmatively by Beauville, in his famous paper on prym varieties and intermediate jacobians, p.321, prop. 2.1.
http://math.unice.fr/~beauvill/pubs/prym.pdf The argument there uses sheaves and the Picard variety of a quadric, but may apply to your question. The idea seems to be to get a vector space from sections of the relative cotangent bundle of the map. But this is presumably a question that would have arisen very early. Have you looked in Steenrod's book on Topology of Fiber bundles? 



#3
Jan913, 12:10 PM

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P: 1,716

Here are the two examples that prompted my question.  The fundamental group of a Riemann surface acts properly discontinuously on the upper half plane as a subgroup of PSL(2:R). This action preserves the real axis U{∞}, RP[itex]^{1}[/itex]. The quotient of HxR[itex]^{2}[/itex] by this action is a circle bundle. This bundle also has a 2 fold cover which is another circle bundle. One can show that both of these bundles can be extended to vector bundles. 


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