nLab principal O(2)-bundle

Idea

Principal O(2)-bundles are special principal bundles with the second orthogonal group O(2) as structure group(gauge group).

Characteristic classes

Proposition

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

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

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

(Hatcher 17, Prop. 3.15 a)

Examples

• One has $S^n\cong O(n+1)/O(n)$, hence there is a principal O(2)-bundle $O(3)\twoheadrightarrow S^2$. Such principal bundles are classified by:
  $$\pi_2B O(2) \cong\pi_1 O(2) \cong\mathbb{Z}.$$

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

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