In mathematics, especially differential geometry, principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles (or principal Sp ( 1 ) {\displaystyle \operatorname {Sp} (1)} -bundles) are special principal bundles with the second special unitary group SU ( 2 ) {\displaystyle \operatorname {SU} (2)} (isomorphic to the first symplectic group Sp ( 1 ) {\displaystyle \operatorname {Sp} (1)} ) as structure group. Topologically, it has the structure of the three-dimensional sphere, hence principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles without their group action are in particular sphere bundles. These are basically topological spaces with a sphere 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 SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles are used in many areas of mathematics, for example for the Fields Medal winning proof of Donaldson's theorem or instanton Floer homology. Since SU ( 2 ) {\displaystyle \operatorname {SU} (2)} is the gauge group of the weak interaction, principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles are also of interest in theoretical physics. In particular, principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles over the four-dimensional sphere S 4 {\displaystyle S^{4}} , which include the quaternionic Hopf fibration, can be used to describe hypothetical magnetic monopoles in five dimensions, known as Wu–Yang monopoles, see also four-dimensional Yang–Mills theory.
Definition Principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles are generalizations of canonical projections B × SU ( 2 ) ↠ B {\displaystyle B\times \operatorname {SU} (2)\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 × SU ( 2 ) → E {\displaystyle E\times \operatorname {SU} (2)\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 ∈ SU ( 2 ) {\displaystyle g\in \operatorname {SU} (2)} , and also acts free and transitive on all preimages of points, which makes all of them homeomorphic to SU ( 2 ) {\displaystyle \operatorname {SU} (2)} , is a principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -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 SU ( 2 ) {\displaystyle \operatorname {SU} (2)} 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 SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles can be fully classified using the classifying space BSU ( 2 ) {\displaystyle \operatorname {BSU} (2)} of the second special unitary group SU ( 2 ) {\displaystyle \operatorname {SU} (2)} , which is exactly the infinite quaternionic projective space H P ∞ {\displaystyle \mathbb {H} P^{\infty }} . For a topological space B {\displaystyle B} , let Prin SU ( 2 ) ( B ) {\displaystyle \operatorname {Prin} _{\operatorname {SU} (2)}(B)} denote the set of equivalence classes of principal SU ( 2 ) {\displaystyle \operatorname {SU} (2)} -bundles over it, then there is a bijection with homotopy classes:
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