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In mathematics, a circle bundle is a fiber bundle where the fiber is the circle .

Oriented circle bundles are also known as principal U(1)-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-manifoldsEdit

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 electrodynamicsEdit

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, (equivalently, the affine connection) 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 (contrary to popular belief), as no quantization is involved or required in the construction of the fiber bundles or connections.

ExamplesEdit

  • The Hopf fibration is an example of a non-trivial circle bundle.
  • The unit normal bundle of a surface is another example of a circle bundle.
  • The unit normal bundle of a non-orientable surface is a circle bundle that is not a principal   bundle. Only orientable surfaces have principal unit tangent bundles.
  • Another method for constructing circle bundles is using a complex line bundle   and taking the associated sphere (circle in this case) bundle. Since this bundle has an orientation induced from   we have that it is a principal  -bundle.[1] Moreover, the characteristic classes from Chern-Weil theory of the  -bundle agree with the characteristic classes of  .
  • For example, consider the analytification   a complex plane curve
 

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

ClassificationEdit

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(1). 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 (essentially because a circle is homotopically equivalent to  , the complex plane with the origin removed; and so a complex line bundle with the zero section removed is homotopically equivalent to a circle 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 complexesEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ https://mathoverflow.net/q/144092. Missing or empty |title= (help)