A Question about circle bundles

lavinia

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.

mathwonk

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?

lavinia

Mathwonk I know zero Algebraic Geometry but will look at the paper. Maybe it is time to learn something.

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.

mathwonk

well it seems like a wonderful question. steenrod discusses bundles with a given group. maybe that prejudices the result.