Circle bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle formula_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.
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 formula_2 being cohomologous to zero, i.e. exact. 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 formula_6 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.
Examples.
Since formula_15 and the characteristic classes pull back non-trivially, we have that the line bundle associated to the sheaf formula_16 has Chern class formula_17.
Classification.
The isomorphism classes of principal formula_7-bundles over a manifold "M" are in one-to-one correspondence with the homotopy classes of maps formula_19, where formula_20 is called the classifying space for U(1). Note that formula_21 is the infinite-dimensional complex projective space, and that it is an example of the Eilenberg–Maclane space formula_22 Such bundles are classified by an element of the second integral cohomology group formula_23 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 formula_25, 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 formula_7 bundle if and only if the associated map formula_27 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 formula_28. This follows from the extension of groups, formula_29, where formula_30.
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 formula_31; circle bundles with an affine connection are classified by formula_32 while formula_33 classifies line bundle gerbes.