Profinite equivariant stable homotopy theory

We establish several model structures on based and unbased G-spaces related to the collections C of subgroups H of G that are used to create weak equivalences and fibrations using fixed points: a map f is a weak equivalence if after taking fixed points f^H is a weak equivalence of topological spaces, for every subgroup H in G that is in the collection C. Similarly for fibrations.

There are three different, but related versions of the Elmendorf theorem, depending on the choice of collection C. The main advantage of the Elmendorf theorem in this context (in analogy to the classical case as well) is the construction of a universal space for a family as a cofibrant replacement using the Elmendorf theorem.<br><br>In the stable case, we investigate a good model of profinite G-spectra given by orthogonal G-spectra using the approach of Mandell and May and investigate how different model structures behave with respect to the change of groups functors (induced by subgroup inclusions or quotients by subgroups).