I met a question asking me to classify the $2$-dimensional vector bundles of the sphere $S^2$.
I did not know how to classify the vector bundles in general. The only example I know was the line bundles of $S^1$: the cylinder and Moebius band. I guess this might be a result from the covering spaces, regarding how to glue the fibers on $\mathbb{R}$, but may not be true. Could anybody provide some inspiration using a concrete example?