The Hairy Ball Problem is PPAD-Complete.

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of computing an approximate zero is PPAD-complete. We also give a FIXP-hardness result for the general exact computation problem. In order to show that this problem lies in PPAD, we provide new results on multiple-source variants of END-OF-LINE, the canonical PPAD-complete problem. In particular, finding an approximate zero of a Hairy Ball vector field on an even-dimensional sphere reduces to a 2-source END-OF-LINE problem. If the domain is changed to be the torus of genus $g \geq 2$ instead (where the Hairy Ball Theorem also holds), then the problem reduces to a $2(g-1)$-source END-OF-LINE problem. These multiple-source END-OF-LINE results are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the IMBALANCE problem defined by Beame et al. in 1998.