nLab principal U(2)-bundle

Idea

Principal U(2)-bundles (also principal Spin$^\mathrm{c}$(3)-bundles or principal Spin$^\mathrm{h}$(2)-bundles) are special principal bundles with the second unitary group $U(2)$ (exceptionally isomorphic to the third spinᶜ group $Spin^\mathrm{c}(3)$ and second spinʰ group $Spin^\mathrm{h}(2)$) as structure group/gauge group.

Principal U(2)-bundles in particular induce principal SO(3)-bundles using the canonical projection $U(2)\cong Spin^\mathrm{c}(3)\twoheadrightarrow SO(3)$ and are induced by principal SU(2)-bundles using the canonical inclusion $SU(2)\hookrightarrow U(2)$.

Characteristic classes

Proposition

A principal U(2)-bundle $P$ fulfills:

$$e(P) \equiv c_2(P).$$

(In general, a principal $U(n)$-bundle $P$ fulfills $e(P)=c_n(P)$.)

(Hatcher 17, Prop. 3.13 c)

Classification over 4-manifolds by characteristic classes

Proposition

Let $X$ be a $4$-manifold. Two principal U(2)-bundles $P,Q\twoheadrightarrow X$ are isomorphic if and only if their first and second Chern class are equal:

$$c_1(P) =c_1(Q) \in H^2(X,\mathbb{Z})$$

$$c_2(P) =c_2(Q) \in H^4(X,\mathbb{Z}).$$

(Gompf & Stipsicz 99, Thrm. 1.4.20)

Lemma

Let $X$ be a $4$-manifold. For every $(c_1,c_2)\in H^2(X,\mathbb{Z})\times H^4(X,\mathbb{Z})$, there exists a principal U(2)-bundle $P\twoheadrightarrow X$ with $c_1(P)=c_1$ and $c_2(P)=c_2$.

(Gompf & Stipsicz 99, Thrm. 1.4.20)

Adjoint bundle

Let $P$ be a principal $U(2)$-bundle, then its adjoint bundle splits as $Ad(P)\cong E\oplus\underline{\mathbb{R}}$ with a real vector bundle $E$ of rank $3$, which comes from $U(2)\cong(SU(2)\times U(1))/\mathbb{Z}_2$. Their characteristic classes are related by:

$$p_1Ad(P) =(c_1^2-4c_2)(P);$$

$$w_2Ad(P) =c_1(P)mod 2.$$

(Donaldson & Kronheimer 91, Eq. (2.1.39))

Associated vector bundle

For principal $U(2)$-bundles $P\twoheadrightarrow X$, there is an associated complex plane bundle $E=P\times_{U(2)}\mathbb{C}^2\twoheadrightarrow X$ using the balanced product. If $Q$ is the induced principal SU(3)-bundle (using the canonical inclusion $U(2)\hookrightarrow SU(3),U\mapsto diag(U,det(U)^{-1})$), then its adjoint bundle is given by:

$$Ad(Q) \cong Ad(P)\oplus(det(E)\otimes E)_\mathbb{R}.$$

If $P$ reduces to a principal SU(2)-bundle, this reduces to:

$$Ad(Q) \cong Ad(P)\oplus E_\mathbb{R}\oplus\underline{\mathbb{R}}.$$

(Both relations hold in general for the maps $SU(n)\hookrightarrow U(n)\rightarrow SU(n+1)$.)

(Donaldson & Kronheimer 91, p. 205)

Liftings

Proposition

A principal U(2)-bundle $f\colon X\rightarrow B U(2)$ lifts to a principal SU(2)-bundle $\widehat{f}\colon X\rightarrow B SU(2)\cong\mathbb{H}P^\infty$ if and only if its first Chern class vanishes, hence the composition $c_1\circ f\colon X\rightarrow K(\mathbb{Z},2)$ is nullhomotopic.

(Gompf & Stipsicz 99, Thrm. 1.4.20)

Proof

The fiber bundle $SU(2)\hookrightarrow U(2)\xrightarrow{det}U(1)$ induces a fiber bundle $\mathbb{H} P^\infty\cong B SU(2)\hookrightarrow B U(2)\xrightarrow{\mathcal{B}det}B U(1)\cong\mathbb{C}P^\infty$, hence a lift exists if and only if the composition $det\circ f\colon X\rightarrow B U(1)$ is nullhomotopic. Since the first Chern class $c_1\colon B U(1)\rightarrow K(\mathbb{Z},2)$ can be chosen as the identity (since $B U(1)$ is homeomorphic and $K(\mathbb{Z},2)$ is weakly homotopy equivalent to the infinite complex projective space $\mathbb{C}P^\infty$), postcomposing it to this map doesn’t change its homotopy class. Since the first Chern class is also preserved by the determinant, it then vanishes from the composition.

Examples

• One has $S^{2n+1}\cong U(n+1)/U(n)$, hence there is a principal U(2)-bundle $U(3)\twoheadrightarrow S^5$. Such principal bundles are classified by:
  $$\pi_5B U(2) \cong\pi_4 U(2) \cong\mathbb{Z}_{2}.$$

  $U(3)\twoheadrightarrow S^5$ is the unique non-trivial principal bundle, which can be detected by the fourth homotopy group:

  $$\pi_4 U(3) \cong 1;$$
  $$\pi_4\left(S^5\times U(2)\right) \cong\pi_4(S^5)\times\pi_4 U(2) \cong\mathbb{Z}_2.$$

Related concepts

Particular principal bundles:

• double cover (principal O(1)-bundle)
• principal O(2)-bundle
• principal U(1)-bundle (principal SO(2)-bundle)
• principal U(2)-bundle (principal $Spin^\mathrm{c}(3)$-bundle, principal $Spin^\mathrm{h}(2)$-bundle)
• principal U(3)-bundle
• principal SO(3)-bundle
• principal SO(4)-bundle ($Spin^\mathrm{h}(3)$-bundle)
• principal SO(5)-bundle
• principal SO(6)-bundle
• principal SO(7)-bundle
• principal SO(8)-bundle
• principal SU(2)-bundle (principal Sp(1)-bundle, principal Spin(3)-bundle)
• principal SU(3)-bundle
• principal SU(4)-bundle (principal Spin(6)-bundle)
• principal Sp(2)-bundle (principal Spin(5)-bundle)

References

• Friedrich Hirzebruch, Heinz Hopf, Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten (1958)

• Robert Gompf and András Stipsicz, 4-Manifolds and Kirby Calculus (1999), Graduate Studies

  in Mathematics, Volume 20 [ISBN: 978-0-8218-0994-5, doi:10.1090/gsm/020]

• Simon Donaldson, Peter Kronheimer: The Geometry of Four-Manifolds (1990, revised 1997), Oxford University Press and Claredon Press, [oup:52942, doi:10.1093/oso/9780198535539.001.0001, ISBN:978-0198502692, ISSN:0964-9174]

• Allen Hatcher: Vector bundles and K-Theory, book draft (2017) [webpage, pdf]