# The Hairy Ball Problem is PPAD-Complete

## TL;DR

This paper proves that finding approximate zeros of the Hairy Ball problem is PPAD-complete and exact zeros are FIXP-hard, establishing the computational complexity of this topological problem.

## Contribution

It introduces the first proof of PPAD-completeness for the Hairy Ball problem and develops new tools involving multiple-source END-OF-LINE variants.

## Key findings

- Approximate zero computation is PPAD-complete.
- Exact zero computation is FIXP-hard.
- New variants of END-OF-LINE are introduced and analyzed.

## Abstract

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 (a) computing an approximate zero is PPAD-complete, and (b) computing an exact zero is FIXP-hard. We also consider the Hairy Ball Theorem on toroidal instead of spherical domains and show that the approximate problem remains PPAD-complete. On a conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely on the investigation of multiple-source variants of END-OF-LINE, the canonical PPAD-complete problem. Our results on these new END-OF-LINE variants 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.