This is a bit of an elementary question, but
suppose $\pi: \mathbb{S}^3\to \mathbb{S}^2$ is the Hopf fibration, are there reasonably computable obstructions to when a map $f:M\to \mathbb{S}^2$ can be lifted to a map $\tilde{f}:M\to \mathbb{S}^3$?
If it matters everything can be taken in the smooth category. Also, I am most interested in the case that $M$ is an open subset of a oriented closed surface if this simplifies things at all.
My understanding is that if $M$ is a disk this there is no obstruction,
I apologize if this is trivial but it is not by area of expertise...