# Circle bundle

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

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, the electromagnetic four-potential, (equivalently, the affine connection) 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. 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

• 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. Only orientable surfaces have principal unit tangent bundles.
• Another method for constructing circle bundles is using a complex line bundle $L\to X$  and taking the associated sphere (circle in this case) bundle. Since this bundle has an orientation induced from $L$  we have that it is a principal $U(1)$ -bundle. Moreover, the characteristic classes from Chern-Weil theory of the $U(1)$ -bundle agree with the characteristic classes of $L$ .
• For example, consider the analytification $X$  a complex plane curve
${\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{x^{n}+y^{n}+z^{n}}}\right)$

Since $H^{2}(X)=\mathbb {Z} =H^{2}(\mathbb {CP} ^{2})$  and the characteristic classes pull back non-trivially, we have that the line bundle associated to the sheaf ${\mathcal {O}}_{X}(a)={\mathcal {O}}_{\mathbb {P} ^{2}}(a)\otimes {\mathcal {O}}_{X}$  has Chern class $c_{1}=a\in H^{2}(X)$ .

## Classification

The isomorphism classes of principal $U(1)$ -bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps $M\to BU(1)$ , where $BU(1)$  is called the classifying space for U(1). Note that $BU(1)=CP^{\infty }$  is the infinite-dimensional complex projective space, and that it is an example of the Eilenberg–Maclane space $K(\mathbb {Z} ,2).$  Such bundles are classified by an element of the second integral cohomology group $H^{2}(M,\mathbb {Z} )$  of M, since

$[M,BU(1)]\equiv [M,\mathbb {C} P^{\infty }]\equiv H^{2}(M)$ .

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 $\mathbb {C} ^{*}$ , 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 $U(1)$  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. 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 $M\to BO_{2}$ . This follows from the extension of groups, $SO_{2}\to O_{2}\to \mathbb {Z} _{2}$ , where $SO_{2}\equiv U(1)$ .

### 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 $H_{D}^{2}(M,\mathbb {Z} )$ ; circle bundles with an affine connection are classified by $H_{D}^{2}(M,\mathbb {Z} (2))$  while $H_{D}^{3}(M,\mathbb {Z} )$  classifies line bundle gerbes.