Smooth classification of complex line bundles on the complex projective line ($L\cong {\mathcal{O}}_{{\mathbb{CP}}^1}(k)$)

Every smooth complex line bundle on ${\mathbb{CP}}^1$ is isomorphic to exactly one twisting bundle $\mathcal{O}(k)$. Trivializing on the two standard affine charts leaves a transition function on ${\mathbb{C}}^{\ast }$; its winding number determines $k$, and a partition of unity removes the zero-winding factor.