Circle bundle

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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-manifolds[edit]

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[edit]

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 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.

Examples[edit]

  • 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. 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 .

Classification[edit]

The isomorphism classes of circle bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps . There is an extension of groups, , where . Circle bundles classified by maps into are known as principal -bundles, and are classified by an element of the second integral cohomology group of M, since . This isomorphism is realized by the Euler class. 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.

References[edit]