nLab principal SO(5)-bundle

Idea

Principal SO(5)-bundles are special principal bundles with the fifth special orthogonal group SO(5) as structure group (gauge group).

Characteristic classes

Proposition

A principal SO(5)-bundle $P$ fulfills:

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

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

(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(5)-bundle $P$ fulfills:

$$w_5(P) \equiv e(P) \mod 2.$$

(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)

Proposition

A principal SO(5)-bundle $P$ fulfills:

$$e(P) =W_5(P) =\beta w_4(P).$$

with the fifth integral Stiefel-Whitney class and the Bockstein homomorphism $\beta\colon H^4(M,\mathbb{Z}_2)\rightarrow H^5(M,\mathbb{Z})$ of the short exact sequence $0\rightarrow\mathbb{Z}\hookrightarrow\mathbb{Z}\twoheadrightarrow\mathbb{Z}_2\rightarrow 0$. (In general, a principal $SO(2n+1)$-bundle $P$ fulfills $e(P)=W_{2n+1}(P)=\beta w_{2n}(P)$.)

(Milnor & Stasheff 74, Prob. 15-D, Hatcher 17, Ch. 3, Ex. 3)

Liftings

Proposition

A principal SO(5)-bundle $f\colon X\rightarrow B SO(5)$ lifts to a principal Sp(2)-bundle $\widehat{f}\colon X\rightarrow B Sp(2)$ 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(5)-bundle $SO(6)\twoheadrightarrow S^5$. Such principal bundles are classified by:
  $$\pi_5B SO(5) \cong\pi_4 SO(5) \cong\mathbb{Z}_2^2.$$

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

• 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]