Principal U(1)-bundle

In mathematics, especially differential geometry, principal ${\displaystyle \operatorname {U} (1)}$-bundles (or principal ${\displaystyle \operatorname {SO} (2)}$-bundles) are special principal bundles with the first unitary group ${\displaystyle \operatorname {U} (1)}$ (isomorphic to the second special orthogonal group ${\displaystyle \operatorname {SO} (2)}$) as structure group. Topologically, it has the structure of the one-dimensional sphere, hence principal ${\displaystyle \operatorname {U} (1)}$-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 ${\displaystyle \operatorname {U} (1)}$-bundles are used in many areas of mathematics, for example for the formulation of the Seiberg–Witten equations or monopole Floer homology. Since ${\displaystyle \operatorname {U} (1)}$ is the gauge group of the electromagnetic interaction, principal ${\displaystyle \operatorname {U} (1)}$-bundles are also of interest in theoretical physics. Concretely, the ${\displaystyle \operatorname {U} (1)}$-Yang–Mills equations are exactly Maxwell's equations. In particular, principal ${\displaystyle \operatorname {U} (1)}$-bundles over the two-dimensional sphere ${\displaystyle S^{2}}$, 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.

Principal ${\displaystyle \operatorname {U} (1)}$-bundles are generalizations of canonical projections ${\displaystyle B\times \operatorname {U} (1)\twoheadrightarrow B}$ for topological spaces ${\displaystyle B}$, so that the source is not globally a product but only locally. More concretely, a continuos map ${\displaystyle p\colon E\twoheadrightarrow B}$with a continuous right group action ${\displaystyle E\times \operatorname {U} (1)\rightarrow E}$, which preserves all preimages of points, hence ${\displaystyle p(eg)=p(e)}$ for all ${\displaystyle e\in E}$ and ${\displaystyle e\in E}$, and also acts free and transitive on all preimages of points, which makes all of them homeomorphic to ${\displaystyle \operatorname {U} (1)}$, is a principal ${\displaystyle \operatorname {U} (1)}$-bundle.^[1][2]

Since principal bundles are in particular fiber bundles with the group action missing, their nomenclature can be transfered. ${\displaystyle E}$ is also called the total space and ${\displaystyle B}$ is also called the base space. Preimages of points are then the fibers. Since ${\displaystyle \operatorname {U} (1)}$ is a Lie group, hence in particular a smooth manifold, the base space ${\displaystyle B}$ is often chosen to be a smooth manifold as well since this automatically makes the total space ${\displaystyle E}$ into a smooth manifold as well.

Principal ${\displaystyle \operatorname {U} (1)}$-bundles can be fully classified using the classifying space ${\displaystyle \operatorname {BU} (1)}$ of the first unitary group ${\displaystyle \operatorname {U} (1)}$, which is exactly the infinite complex projective space ${\displaystyle \mathbb {C} P^{\infty }}$. For a topological space ${\displaystyle B}$, let ${\displaystyle \operatorname {Prin} _{\operatorname {U} (1)}(B)}$ denote the set of equivalence classes of principal ${\displaystyle \operatorname {U} (1)}$-bundles over it, then there is a bijection with homotopy classes:^[3]

${\displaystyle \operatorname {Prin} _{\operatorname {U} (1)}(B)\cong [B,\operatorname {BU} (1)]\cong [B,\mathbb {C} P^{\infty }].}$

${\displaystyle \mathbb {C} P^{\infty }}$ is a CW complex with its ${\displaystyle n}$-skeleton being ${\displaystyle \mathbb {C} P^{k}}$ for the largest natural number ${\displaystyle k\in \mathbb {N} }$ with ${\displaystyle 2k\leq n}$. For a ${\displaystyle n}$-dimensional CW complex ${\displaystyle B}$, the cellular approximation theorem^[4] states that every continuous map ${\displaystyle B\rightarrow \mathbb {C} P^{\infty }}$ is homotopic to a cellular map factoring over the canonical inclusion ${\displaystyle \mathbb {C} P^{k}\hookrightarrow \mathbb {C} P^{\infty }}$. As a result, the induced map ${\displaystyle [B,\mathbb {C} P^{k}]\hookrightarrow [B,\mathbb {C} P^{\infty }]}$ is surjective, but not necessarily injective as higher cells of ${\displaystyle \mathbb {C} P^{\infty }}$ allow additional homotopies. In particular if ${\displaystyle B}$ is a CW complex of three or less dimensions, then ${\displaystyle k=1}$ and with ${\displaystyle \mathbb {C} P^{1}\cong S^{2}}$, there is a connection to cohomotopy sets with a surjective map:

${\displaystyle \pi ^{2}(B)\rightarrow \operatorname {Prin} _{\operatorname {U} (1)}(B).}$

${\displaystyle \mathbb {C} P^{\infty }}$ is also the Eilenberg–MacLane space ${\displaystyle K(\mathbb {Z} ,2)}$,^[5] which represents singular cohomology,^[6] compare to Brown's representability theorem:

${\displaystyle \operatorname {Prin} _{\operatorname {U} (1)}(B)\cong H^{2}(B,\mathbb {Z} ).}$

(The composition ${\displaystyle \pi ^{2}(B)\rightarrow H^{2}(B,\mathbb {Z} )}$ is the Hurewicz map.) A corresponding isomorphism is given by the first Chern class. Although characteristic classes are defined for vector bundles, it is possible to also define them for certain principal bundles.

Given a principal ${\displaystyle \operatorname {U} (1)}$-bundle ${\displaystyle E\twoheadrightarrow B}$, there is an associated vector bundle ${\displaystyle E\times _{\operatorname {U} (1)}\mathbb {C} \twoheadrightarrow B}$. Intuitively, the spheres at every point are filled over the canonical inclusions ${\displaystyle \operatorname {U} (1)\subset \mathbb {C} }$. Due to the single rank, the vetor bundle is only described by the first Chern class${\displaystyle c_{1}\colon \operatorname {Vect} _{\mathbb {C} }(B)\rightarrow H^{2}(B,\mathbb {Z} )}$, which is an isomorphism over CW complexes.^[7]

Principal bundles also have an adjoint vector bundle, which is trivial for principal ${\displaystyle \operatorname {U} (1)}$-bundles.

• By definition of complex projective space, the canonical projection ${\displaystyle S^{2n+1}\twoheadrightarrow \mathbb {C} P^{n}}$ is a principal ${\displaystyle \operatorname {U} (1)}$-bundle. With ${\displaystyle \mathbb {C} P^{1}\cong S^{2}}$, known as Riemann sphere, the complex Hopf fibration ${\displaystyle h_{\mathbb {C} }\colon S^{3}\twoheadrightarrow S^{2}}$ is a special case. For the general case, the classifying map is the canonical inclusion:

${\displaystyle \mathbb {C} P^{n}\hookrightarrow \mathbb {C} P^{\infty }\cong \operatorname {BU} (1).}$

• One has ${\displaystyle S^{2n+1}\cong \operatorname {U} (n+1)/\operatorname {U} (n)}$, which means that there is a principal ${\displaystyle \operatorname {U} (1)}$-bundle ${\displaystyle \operatorname {U} (2)\twoheadrightarrow S^{3}}$. Such bundles are classified by:^[8]

${\displaystyle \pi _{3}\operatorname {BU} (1)\cong \pi _{2}\operatorname {U} (1)\cong \pi _{2}S^{1}\cong 1.}$

Hence the bundle is trivial, which fits that ${\displaystyle \operatorname {SU} (2)\cong S^{3}}$ and ${\displaystyle \operatorname {U} (n)\cong \operatorname {SU} (n)\times \operatorname {U} (1)}$.

• One has ${\displaystyle S^{n}\cong \operatorname {SO} (n+1)/\operatorname {SO} (n)}$, which means that (using ${\displaystyle \operatorname {U} (1)\cong \operatorname {SO} (2)}$) there is a principal ${\displaystyle \operatorname {U} (1)}$-bundle ${\displaystyle \operatorname {SO} (3)\twoheadrightarrow S^{2}}$. Such bundles are classified by:^[8]

${\displaystyle \pi _{2}\operatorname {BU} (1)\cong \pi _{1}\operatorname {U} (1)\cong \pi _{1}S^{1}\cong \mathbb {Z} .}$

One has ${\displaystyle \operatorname {SO} (3)\cong \mathbb {R} P^{3}}$ and the composition of the canonical double cover ${\displaystyle S^{3}\twoheadrightarrow \mathbb {R} P^{3}}$ with the principal bundle ${\displaystyle \mathbb {R} P^{3}\twoheadrightarrow S^{2}}$ is exactly the complex Hopf fibration ${\displaystyle S^{3}\twoheadrightarrow S^{2}}$. Since the first Chern class of the complex Hopf fibration is ${\displaystyle -1}$, the first Chern class of the principal bundle ${\displaystyle \mathbb {R} P^{3}\twoheadrightarrow S^{2}}$ is ${\displaystyle -2}$.

• Principal SU(2)-bundle

• Freed, Daniel (1991). Instantons and 4-Manifolds. Cambridge University Press. ISBN 978-1-4613-9705-2.
• Hatcher, Allen (2001). Algebraic Topology. Cambridge University Press. ISBN 0-521-79160-X.
• Mitchell, Stephen (2011). "Notes on principal bundles and classifying spaces" (PDF).
• Hatcher, Allen (2017). Vector Bundles and K-Theory (PDF).