The Hopf line bundle over S² by clutching

On the plus disk use points $[z:1]$, $\mathrm{\mid }z\mathrm{\mid }\le 1$, and the tautological frame $s_{+}(z)=(z,1)$. On the minus disk use $[1:w]$, $\mathrm{\mid }w\mathrm{\mid }\le 1$, with $w=z^{-1}$ on the equator, and frame $s_{-}(w)=(1,w)$. These vectors span the represented lines by [F2].

For $\mathrm{\mid }z\mathrm{\mid }=1$, one has $s_{+}(z)=(z,1)=z(1,z^{-1})=z{s}_{-}(z^{-1})$. Thus a physical vector with plus coefficient $v$ has minus coefficient $zv$. By [F1], the clutching function is exactly $g(z)=z$, not its inverse.

Interchanging the plus and minus charts reverses the coordinate change, so [F1] gives $g^{-1}(z)=z^{-1}$. Dualizing a line bundle inverts its scalar transition functions, hence this second clutching is ${\gamma }^{\ast }$. This completes the sign calculation without a choice principle.