Circle 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 ${\displaystyle \pi ^{\!*}F}$ being cohomologous to zero. In particular, there always exists a 1-form A, the electromagnetic four-potential, (equivalently, the affine connection) such that

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

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. 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.

• 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. Only 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 ${\displaystyle c_{1}=a\in H^{2}(X)}$.

The isomorphism classes of principal ${\displaystyle U(1)}$-bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps ${\displaystyle M\to BU(1)}$, where ${\displaystyle BU(1)}$ is called the classifying space for U(1). Note that ${\displaystyle BU(1)=CP^{\infty }}$ is the infinite-dimensional complex projective space, and that it is an example of the Eilenberg–Maclane space ${\displaystyle K(\mathbb {Z} ,2).}$ Such bundles are classified by an element of the second integral cohomology group ${\displaystyle H^{2}(M,\mathbb {Z} )}$ 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; equivalently, it is the first Chern class of a smooth complex line bundle (essentially because a circle is homotopically equivalent to ${\displaystyle \mathbb {C} ^{*}}$, 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 ${\displaystyle U(1)}$ 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. 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 ${\displaystyle M\to BO_{2}}$. This follows from the extension of groups, ${\displaystyle SO_{2}\to O_{2}\to \mathbb {Z} _{2}}$, where ${\displaystyle SO_{2}\equiv U(1)}$.

• Wang sequence

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