The Hairy Ball Theorem states that $S^2$ has no nonvanishing tangent vector fields. But if we did have such a field then we could normalise each vector so that it lay on the unit circle of the tangent plane at that point. These unit circles form a bundle over $S^2$ with fibre $S^1$. The Hairy Ball Theorem is therefore equivalent to saying that this bundle has no global section.
Since this is a nontrivial $S^2$-bundle with fibre $S^1$ I thought that it might be the Hopf Fibration. Is it? If not, what is the total space?