Circle bundle

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In mathematics, a circle bundle is a fiber bundle where the fiber is the circle \scriptstyle \mathbf{S}^1.

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 \pi^{\!*}F being cohomologous to zero. In particular, there always exists a 1-form A such that

 \pi^{\!*}F = dA.

Given a circle bundle P over M and its projection

\pi:P\to M

one has the homomorphism

\pi^*:H^2(M,\mathbb{Z}) \to H^2(P,\mathbb{Z})

where \pi^{\!*} 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 unit tangent bundle of a surface is another example of a circle bundle.
  • The unit tangent bundle of a non-orientable surface is a circle bundle that is not a principal U(1) bundle. Orientable surfaces have principal unit tangent bundles.

Classification[edit]

The isomorphism classes of circle bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps M \to BO_2. There is an extension of groups, SO_2 \to O_2 \to \mathbb Z_2, where SO_2 \equiv U(1). Circle bundles classified by maps into BU(1) are known as principal U(1)-bundles, and are classified by an element element of the second integral cohomology group \scriptstyle H^2(M,\mathbb{Z}) of M, since [M,BU(1)] \equiv [M,\mathbb CP^\infty] \equiv H^1(M). This isomorphism is realized by the Euler class. A circle bundle is a principal U(1) bundle if and only if the associated map M \to B\mathbb Z_2 is null-homotopic, which is true if and only if the bundle is fibrewise orientable.

See also: Wang sequence.

References[edit]