Circle bundle

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In mathematics, a circle bundle is a fiber bundle where the fiber is the circle , or, more precisely, a principal U(1)-bundle. It is homotopically equivalent to a complex line bundle. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.

[edit] Relationship to electrodynamics

The Maxwell equations correspond to an electromagnetic field represented by a 2-form F, with π ^* F being cohomologous to zero. In particular, there always exists a 1-form A such that

π ^* F = dA

Given a circle bundle P over M and its projection

one has the homomorphism

where π ^* is the pullback. Each homomorphism corresponds to a Dirac monopole; the integer cohomology groups correspond to the quantization of the electric charge.

[edit] Examples

The Hopf fibrations are examples of non-trivial circle bundles.

[edit] Classification

The isomorphism classes of circle bundles over a manifold M are in one-to-one correspondence with the elements of the second integral cohomology group of M. This isomorphism is realized by the Euler class.

Equivalently, the isomorphism classes correspond to homotopy classes of maps to the infinite-dimensional complex projective space , which is the classifying space of U(1). See classifying space for U(n).

In homotopy theory terms, the circle and the complex plane without its origin are equivalent. Circle bundles are, by the associated bundle construction, equivalent to smooth complex line bundles because the transition functions of both can be made to live in C*. In this situation, the Euler class of the circle bundle or real two-plane bundle is the same as the first Chern class of the line bundle.

See also: Wang sequence.