Knowlpedia

Hopf fibration as a principal U(1)-bundle

The classic circle bundle with total space the 3-sphere and base the 2-sphere.

Let S3={(z1,z2)∈C2:∣z1∣2+∣z2∣2=1}S^3=\{(z_1,z_2)\in\mathbb C^2:|z_1|^2+|z_2|^2=1\} and let U(1)⊂C×U(1)\subset\mathbb C^\times act on S3S^3 by scalar multiplication:

(z1,z2)⋅u:=(z1u,z2u),u∈U(1).(z_1,z_2)\cdot u := (z_1u, z_2u),\qquad u\in U(1).

The Hopf fibration is the quotient map

π:S3⟶S3/U(1)≅CP1≅S2.\pi:S^3\longrightarrow S^3/U(1)\cong \mathbb{CP}^1 \cong S^2.

With this action, π\pi is a with U(1)U(1) (a ), i.e. a principal circle bundle over S2S^2.

Concretely, one can take π(z1,z2)=[z1:z2]∈CP1\pi(z_1,z_2)=[z_1:z_2]\in\mathbb{CP}^1, and the fiber over a point is exactly the U(1)U(1)-orbit of any representative.

Nontriviality

This bundle is nontrivial; in particular it has no global (compare ).

Examples
  1. Local trivializations from the standard affine charts. On the open set U2={[z1:z2]∈CP1:z2≠0}U_2=\{[z_1:z_2]\in\mathbb{CP}^1: z_2\neq 0\}, write w=z1/z2∈Cw=z_1/z_2\in\mathbb C. A point in S3S^3 over ww can be uniquely written as
    11+∣w∣2(w,1)⋅u,u∈U(1),\frac{1}{\sqrt{1+|w|^2}}(w,1)\cdot u,\qquad u\in U(1),
    giving a smooth identification π−1(U2)≅U2×U(1)\pi^{-1}(U_2)\cong U_2\times U(1). A similar trivialization holds on U1={z1≠0}U_1=\{z_1\neq 0\}.
  1. Clutching function. Using the two chart trivializations, the transition function on the overlap is the map S1→U(1)S^1\to U(1) of degree 11 (it winds once). This winding is the basic topological reason the bundle is not isomorphic to S2×U(1)S^2\times U(1).
  1. Natural connection. The Hopf bundle carries a canonical principal connection whose curvature is a multiple of the area form on the base; this is the geometric starting point for the .