Hecke functors and the equivariant Dold-Thom theorem
Let G be a finite group and S be a finite G-set. \({\mathbb{Z}}[S]\) denotes the free abelian group generated by S with the obvious G-action. Let k be a commutative ring with unit. A Hecke functor over k is an additive contravariant functor \(H: L^ G\to k\)-Mod, where \(L^ G\) is the category whose objects are all finite G-sets and \(L^ G(S,T)=Hom_{{\mathbb{Z}}[G]}({\mathbb{Z}}[S],{\mathbb{Z}}[T])\), and k-Mod is the category of k-modules. The author shows that the equivariant homotopy category \(Ho(M_{k[G]})\) of topological k[G]-modules and continuous k[G]-homomorphisms is determined in terms of Hecke functors. Moreover he shows that if the ring k is a PID such that 1/\(| G| \in k\), then the Bredon homotopy groups give an equivalence between the category \(Ho(M_{k[G]})\) and the category of the graded Hecke functors. This extends the Dold-Thom theorem [\textit{A. Dold} and \textit{R. Thom}, Ann. Math., II. Ser. 67, 239-281 (1958; Zbl 0091.371)] to the equivariant case.
equivariant homotopy category of topological modules
scientific article; zbMATH DE number 3881505
scientific article