Relatively endotrivial complexes

Let G be a finite group and k a field of characteristic p > 0. We use the notion of projectivity relative to a kG-module to define multiple notions of relatively endotrivial chain complexes, extending the definition of an endotrivial chain complex. These definitions are motivated by Lassueur's analogous construction of relatively endotrivial kG-modules. Similar to the study of endotrivial modules, we obtain equivalent characterizations of relative endotriviality and corresponding h-mark homomorphisms which determine the isomorphism class of a relatively endotrivial complex up to a torsion element. Additionally, we study the homomorphism induced by restriction of relatively endotrivial complexes to subgroups containing Sylow p-subgroups S of G. One result of this investigation is that the group Eₖ (G) of endotrivial complexes is isomorphic to a direct product of its torsion subgroup Hom (G, k^) and a fusion-stable subgroup of Eₖ (S).