In mathematics, especially differential geometry, principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles (or principal SO ( 2 ) {\displaystyle \operatorname {SO} (2)} -bundles) are special principal bundles with the first unitary group U ( 1 ) {\displaystyle \operatorname {U} (1)} (isomorphic to the second special orthogonal group SO ( 2 ) {\displaystyle \operatorname {SO} (2)} ) as structure group. Topologically, it has the structure of the one-dimensional sphere, hence principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles without their group action are in particular circle bundles. These are basically topological spaces with a circle glued to every point, so that all of them are connected with each other, but globally aren't necessarily a product and can instead be twisted like a Möbius strip. Principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles are used in many areas of mathematics, for example for the formulation of the Seiberg–Witten equations or monopole Floer homology. Since U ( 1 ) {\displaystyle \operatorname {U} (1)} is the gauge group of the electromagnetic interaction, principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles are also of interest in theoretical physics. Concretely, the U ( 1 ) {\displaystyle \operatorname {U} (1)} -Yang–Mills equations are exactly Maxwell's equations. In particular, principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles over the two-dimensional sphere S 2 {\displaystyle S^{2}} , which include the complex Hopf fibration, can be used to describe hypothetical magnetic monopoles in three dimensions, known as Dirac monopoles, see also two-dimensional Yang–Mills theory.
Definition Principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles are generalizations of canonical projections B × U ( 1 ) ↠ B {\displaystyle B\times \operatorname {U} (1)\twoheadrightarrow B} for topological spaces B {\displaystyle B} , so that the source is not globally a product but only locally. More concretely, a continuous map p : E ↠ B {\displaystyle p\colon E\twoheadrightarrow B} with a continuous right group action E × U ( 1 ) → E {\displaystyle E\times \operatorname {U} (1)\rightarrow E} , which preserves all preimages of points, hence p ( e g ) = p ( e ) {\displaystyle p(eg)=p(e)} for all e ∈ E {\displaystyle e\in E} and g ∈ G {\displaystyle g\in G} , and also acts free and transitive on all preimages of points, which makes all of them homeomorphic to U ( 1 ) {\displaystyle \operatorname {U} (1)} , is a principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundle. Since principal bundles are in particular fiber bundles with the group action missing, their nomenclature can be transferred. E {\displaystyle E} is also called the total space and B {\displaystyle B} is also called the base space. Preimages of points are then the fibers. Since U ( 1 ) {\displaystyle \operatorname {U} (1)} is a Lie group, hence in particular a smooth manifold, the base space B {\displaystyle B} is often chosen to be a smooth manifold as well since this automatically makes the total space E {\displaystyle E} into a smooth manifold as well.
Classification Principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles can be fully classified using the classifying space BU ( 1 ) {\displaystyle \operatorname {BU} (1)} of the first unitary group U ( 1 ) {\displaystyle \operatorname {U} (1)} , which is exactly the infinite complex projective space C P ∞ {\displaystyle \mathbb {C} P^{\infty }\!} . For a topological space B {\displaystyle B} , let Prin U ( 1 ) ( B ) {\displaystyle \operatorname {Prin} _{\operatorname {U} (1)}(B)} denote the set of equivalence classes of principal U ( 1 ) {\displaystyle \operatorname {U} (1)} -bundles over it, then there is a bijection with homotopy classes:
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