Circle bundle

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

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 $\pi^{\!*}F$ 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

• 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.

Classification

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 $[M,BU(1)] \equiv [M,\mathbb CP^\infty] \equiv H^2(M)$. This isomorphism is realized by the Euler class. A circle bundle is a principal $U(1)$ bundle if and only if the associated map $M \to B\mathbb Z_2$ is null-homotopic, which is true if and only if the bundle is fibrewise orientable.

References

• Weisstein, Eric W., "Circle Bundle", MathWorld.
• Chern, Shiing-shen (1977), "Circle bundles", Lecture Notes in Mathematics, 597/1977, Springer Berlin/Heidelberg, pp. 114–131, ISBN 978-3-540-08345-0.