We know by the Serre Swan theorem that smooth vector fields over a smooth manifold form a projective module over the ring of smooth functions. We also know that the hairy ball theorem implies that the module of vector spaces on the sphere $\mathfrak X(S^2)$ is not a free module. Hence, it must be a "real" projective module, not a free module. So, there must exist some non-trivial module $F$ over $C^\infty(S^2)$ such that $\mathfrak X(S^2) \oplus F \simeq \oplus_i C^\infty(S^2)$.
- Do we know what $F$ is, explicitly? Do we know what the right hand side of the equality $\mathfrak X(S^2) \oplus F \simeq \oplus_i C^\infty(S^2)$ should be explicitly? (what is the rank of the free module?)
- Can we compute $F$ "in general" for a given (nice) smooth manifold $M$?