# Circle bundle

(Redirected from Principal circle bundle)

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

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

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

Given a circle bundle P over M and its projection

${\displaystyle \pi :P\to M}$

one has the homomorphism

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

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

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

## Classification

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