In mathematics, a circle bundle is a fiber bundle where the fiber is the circle 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.
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.
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 = d A . Given a circle bundle P over M and its projection
π : P → M one has the homomorphism
π ∗ : H 2 ( M , Z ) → H 2 ( P , Z ) where π ∗ is the pullback. Each homomorphism corresponds to a Dirac monopole; the integer cohomology groups correspond to the quantization of the electric charge.
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 U ( 1 ) bundle. Orientable surfaces have principal unit tangent bundles.The isomorphism classes of circle bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps M → B O 2 . There is an extension of groups, S O 2 → O 2 → Z 2 , where S O 2 ≡ U ( 1 ) . Circle bundles classified by maps into B U ( 1 ) are known as principal U ( 1 ) -bundles, and are classified by an element of the second integral cohomology group H 2 ( M , Z ) of M, since [ M , B U ( 1 ) ] ≡ [ M , C P ∞ ] ≡ H 2 ( 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 → B Z 2 is null-homotopic, which is true if and only if the bundle is fibrewise orientable.
