The classification of endotrivial complexes

Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We give a complete classification of endotrivial complexes, i.e. determine the Picard group $\mathcal{E}_k(G)$ of the tensor-triangulated category $K^b({}_{kG}\mathbf{triv})$, the bounded homotopy category of $p$-permutation modules, which Balmer and Gallauer recently considered. For $p$-groups, we identify $\mathcal{E}_k(-)$ with the rational $p$-biset functor $CF_b(-)$ of Borel-Smith functions and recover a short exact sequence of rational $p$-biset functors constructed by Bouc and Yal\c{c}in. As a consequence, we prove that every $p$-permutation autoequivalence of a $p$-group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yal\c{c}in, showing the kernel of the Bouc homomorphism for an arbitrary finite group $G$ is described by superclass functions $f: s_p(G) \to \mathbb{Z}$ satisfying the oriented Artin-Borel-Smith conditions.