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]

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?

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$^{1}$. The quotient of HxR$^{2}$ 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.

 

[blue_raver22]

Does every sphere bundle come from a 3 plane bundle?

I can provide a response to this question from a mathematical perspective.

Firstly, it is important to clarify that a circle bundle is a fiber bundle where the base space is a paracompact manifold and the fiber is the circle. The transition functions of a circle bundle are required to be homeomorphisms of the circle, which means they preserve the topological structure of the circle.

To answer the question, it is not necessarily true that every circle bundle comes from a 2 plane bundle. In general, a circle bundle can come from a higher-dimensional bundle, such as a 3 or 4 plane bundle. However, there are certain conditions that can guarantee that a circle bundle comes from a 2 plane bundle, such as the base space being a simply connected manifold.

On the other hand, every sphere bundle does come from a 3 plane bundle. This is known as the clutching construction, where a sphere bundle can be constructed by gluing together two copies of the 3-dimensional disk bundle over the base space.

In summary, while not every circle bundle necessarily comes from a 2 plane bundle, every sphere bundle does come from a 3 plane bundle. It is important to consider the specific conditions and properties of the base space when studying fiber bundles in order to understand their underlying structures and connections to higher-dimensional bundles.