Stable equivariant abelianization, its properties, and applications

Let G be a finite group. For a based G-space X and a Mackey functor M, a topological Mackey functor X ⊗ ˜ M is constructed, which will be called the stable equivariant abelianization of X with coefficients in M. When X is a based G-CW complex, X ⊗ ˜ M is shown to be an infinite loop space in the sense of G -spaces. This gives a version of the RO ( G ) -graded equivariant Dold–Thom theorem. Applying a variant of Elmendorfʹs construction, we get a model for the Eilenberg–Mac Lane spectrum HM. The proof uses a structure theorem for Mackey functors and our previous results.