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 the associated computational problem of (a) computing approximate zero is PPAD-complete, and (b) exact FIXP-hard. also consider toroidal instead spherical domains show remains PPAD-complete. On conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely investigation multiple-source variants END-OF-LINE, canonical PPAD-complete problem. Our these new END-OF-LINE independent interest provide tools for showing membership in PPAD. In particular, we use them to first full proof PPAD-completeness IMBALANCE defined by Beame et al. 1998.