Definition. Hopf bundle [5A3M]

2026-02-10 · Lîm Tsú-thuàn

Identify the unit odd dimensional sphere S2n−1S^{2n-1} with the submanifold

{(z1,…,zn)∈Cn∣∣z0∣2+⋯+∣zn∣2=1}\{ (z_1, \dots, z_n) \in \mathbb{C}^n \mid |z_0|^2 + \cdots + |z_n|^2 = 1 \}

an S1S^1-action on S2n−1S^{2n-1} given by

eiθ⋅(z1,…,zn)=(eiθz1,…,eiθzn)e^{i\theta} \cdot (z_1, \dots, z_n) = (e^{i\theta}z_1, \dots, e^{i\theta}z_n)

The complex projective space CPn−1\mathbb{C}P^{n-1} is isomorphic to S2n−1/U(1)S^{2n-1} / U(1), the group U(1)U(1) corresponds to S1S^1. This quotient map is a principal S1S^1 bundle called Hopf bundle.

  1. [eiθz]∼[z][e^{i\theta}z] \sim [z] by definition of CPn−1\mathbb{C}P^{n-1}
  2. Use Ui:={[z]∈CPn−1∣zi≠0}U_i := \{ [z] \in \mathbb{C}P^{n-1} \mid z_i \ne 0 \} (because ⋃i=0n−1Ui=CPn−1\bigcup_{i=0}^{n-1} U_i = \mathbb{C}P^{n-1}, these describe all points in the projective space). The map πi:π−1(Ui)→S1\pi_i : \pi^{-1}(U_i) \to S^1 defined by z↦zi∣zi∣z \mapsto \frac{z_i}{|z_i|} then πi(eiθ⋅z)=eiθz∣eiθz∣=eiθz∣z∣=eiθ⋅z∣z∣=eiθ⋅πi(z)\begin{aligned} \pi_i(e^{i\theta} \cdot z) &= \frac{e^{i\theta} z}{|e^{i\theta} z|} \\ &= \frac{e^{i\theta} z}{|z|} \\ &= e^{i\theta} \cdot \frac{z}{|z|} \\ &= e^{i\theta} \cdot \pi_i(z) \end{aligned}