Relatively endotrivial complexes

Let $G$ be a finite group and $k$ be a field of characteristic $p>0$. In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category $K^b({}_{kG}\mathbf{t}\mathbf{r}\mathbf{i}\mathbf{v})$ of $p$-permutation $kG$-modules. Using the notion of projectivity relative to a $kG$-module, we expand on this study by defining notions of “relatively” endotrivial chain complexes, analogous to Lassueur’s construction of relatively endotrivial $kG$-modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow $p$-subgroups $S$ of $G$.