Smooth classification of complex line bundles on the complex projective line
= Smooth classification of complex line bundles on the complex projective line
{title2=$L\cong\mathcal O_{\mathbb{CP}^1}(k)$}
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 $\mathbb C^*$; its <winding number> determines $k$, and a <partition of unity> removes the zero-winding factor.