Equivariant \(p\)-adic homotopy theory (Q1612287)

This paper extends the algebraic results of the \(p\)-adic homotopy theory of finite type nilpotent spaces [the author, Topology 40, No. 1, 43-94 (2001; Zbl 0974.55004)] to the case of the action of a finite group \(G\). The author shows that the cochain functor with coefficients in \(\overline F_p\) gives an equivalence between the \(p\)-adic equivariant homotopy category of finite type nilpotent \(G\)-spaces and a full subcategory of the homotopy category of diagrams of \(E_\infty\overline F_p\)-algebras indexed on the orbit category of \(G\). This is proved as a consequence of Elmendorf's theorem and Kan's work on diagrams in closed model categories plus the equivalence in the nonequivariant context. The author also extends to the case of nonconnected spaces the results of his earlier paper [loc. cit.] characterizing the subcategory of \(E_\infty\) algebras that are quasi-isomorphic to the cochain complexes of finite-type nilpotent or simply connected spaces.