Circle bundle


In mathematics, a circle bundle is a fiber bundle where the fiber is the circle.
Oriented circle bundles are also known as principal U-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.

As 3-manifolds

Circle bundles over surfaces are an important example of 3-manifolds. A more general class of 3-manifolds is Seifert fiber spaces, which may be viewed as a kind of "singular" circle bundle, or as a circle bundle over a two-dimensional orbifold.

Relationship to electrodynamics

The Maxwell equations correspond to an electromagnetic field represented by a 2-form F, with being cohomologous to zero. In particular, there always exists a 1-form A, the electromagnetic four-potential, such that
Given a circle bundle P over M and its projection
one has the homomorphism
where is the pullback. Each homomorphism corresponds to a Dirac monopole; the integer cohomology groups correspond to the quantization of the electric charge. The Bohm-Aharonov effect can be understood as the holonomy of the connection on the associated line bundle describing the electron wave-function. In essence, the Bohm-Aharonov effect is not a quantum-mechanical effect, as no quantization is involved or required in the construction of the fiber bundles or connections.

Examples

Since and the characteristic classes pull back non-trivially, we have that the line bundle associated to the sheaf has Chern class.

Classification

The isomorphism classes of principal -bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps, where is called the classifying space for U. Note that is the infinite-dimensional complex projective space, and that it is an example of the Eilenberg–Maclane space Such bundles are classified by an element of the second integral cohomology group of M, since
This isomorphism is realized by the Euler class; equivalently, it is the first Chern class of a smooth complex line bundle
A circle bundle is a principal bundle if and only if the associated map is null-homotopic, which is true if and only if the bundle is fibrewise orientable. Thus, for the more general case, where the circle bundle over M might not be orientable, the isomorphism classes are in one-to-one correspondence with the homotopy classes of maps. This follows from the extension of groups,, where.

Deligne complexes

The above classification only applies to circle bundles in general; the corresponding classification for smooth circle bundles, or, say, the circle bundles with an affine connection requires a more complex cohomology theory. Results include that the smooth circle bundles are classified by the second Deligne cohomology ; circle bundles with an affine connection are classified by while classifies line bundle gerbes.