  Let $G$ be a finite group, $p$ a prime, and $k$ a field of characteristic
$p$. We introduce the notion of an endotrivial chain complex of $p$-permutation
$kG$-modules, which are the invertible objects in the bounded homotopy category
of $p$-permutation $kG$-modules, and study the corresponding Picard group
$\mathcal{E}_k(G)$ of endotrivial complexes. Such complexes are shown to induce
splendid Rickard autoequivalences of $kG$. The elements of $\mathcal{E}_k(G)$
are determined uniquely by integral invariants arising from the Brauer
construction and a degree one character $G \to k^\times$. Using ideas from
Bouc's theory of biset functors, we provide a canonical decomposition of
$\mathcal{E}_k(G)$, and as an application, give complete descriptions of
$\mathcal{E}_k(G)$ for abelian groups and $p$-groups of normal $p$-rank 1.
Taking Lefschetz invariants of endotrivial complexes induces a group
homomorphism $\Lambda: \mathcal{E}_k(G) \to O(T(kG))$, where $O(T(kG))$ is the
orthogonal unit group of the trivial source ring. Using recent results of
Boltje and Carman, we give a Frobenius stability condition elements in the
image of $\Lambda$ must satisfy.
