nLab principal SO(6)-bundle

Idea

Principal SO(6)-bundles are special principal bundles with the sixth special orthogonal group $SO(6)$ as structure group/gauge group. Applications include the frame bundle of an orientable 6-manifold.

Principal SO(6)-bundles in particular are induced by principal SU(4)-bundles using the canonical projection $SU(4)\cong Spin(6)\hookrightarrow SO(6)$. Principal SO(6)-bundles also induce principal SO(2)-bundles, principal SO(3)-bundles and principal SO(6)-bundles using the canonical inclusions $SO(2)\hookrightarrow SO(3)\hookrightarrow SO(4)\hookrightarrow SO(6)$.

Principal SO(6)-bundles also arise from any principal $G$-bundle with a six-dimensional Lie group $G$ using its adjoint representation $Ad\colon G\rightarrow SL_6(\mathbb{R})$, which induces a map $\mathcal{B}Ad\colon\mathcal{B}G\rightarrow B SO(6)$.

Characteristic classes

Proposition

A principal SO(6)-bundle $P$ fulfills:

$$w_2^2(P) \equiv p_1(P) \mod 2;$$

$$w_4^2(P) \equiv p_2(P) \mod 2;$$

$$w_6^2(P) \equiv p_3(P) \mod 2.$$

(In general, a principal $SO(n)$-bundle $P$ fulfills $w_{2k}^2(P)\equiv p_k(P) \mod 2$ for $2k\leq n$.)

(Milnor & Stasheff 74, Prob. 15-A, Gompf & Stipsicz 99, Ex. 1.4.21 d, Hatcher 17, Prop. 3.15 a)

Proposition

A principal SO(6)-bundle $P$ fulfills:

$$p_3(P) =e^2(P).$$

(In general, a principal $SO(2n)$-bundle $P$ fulfills $p_n(P)=e^2(P)$.)

(Milnor & Stasheff 74, Crl. 15.8, Hatcher 17, Prop. 3.15 b)

The two previous propositions together imply $w_6^2(P)\equiv e^2(P) \mod 2$ and one even has:

Proposition

A principal SO(6)-bundle $P$ fulfills:

$$w_6(P) \equiv e(P) \mod 2.$$

(In general, a principal $SO(n)$-bundle $P$ fulfills $w_n(P)\equiv e(P) \mod 2$.)

(Milnor & Stasheff 74, Prop. 9.5, Hatcher 17, Prop. 3.13 c)

Liftings

Proposition

A principal SO(6)-bundle $f\colon X\rightarrow B SO(6)$ lifts to a principal SU(4)-bundle $\widehat{f}\colon X\rightarrow B SU(4)$ if and only if its second Stiefel-Whitney class vanishes, hence the composition $w_2\circ f\colon X\rightarrow K(\mathbb{Z}_2,2)$ is nullhomotopic.

Examples

• One has $S^n\cong SO(n+1)/SO(n)$, hence there is a principal SO(6)-bundle $SO(7)\twoheadrightarrow S^6$. Such principal bundles are classified by:
  $$\pi_6B SO(6) \cong\pi_5 SO(6) \cong\mathbb{Z}.$$

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

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)

• 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]

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