Assembly and norm maps via genuine equivariant homotopy theory

Abstract

We give a new, conceptual proof and sharp generalization of a Theorem by Cary Malkiwiech [Mal17] about how the assembly map of the algebraic K-theory of a group ring (spectrum) with respect to a finite group G admits a dual coassembly map, such that the composition of assembly and coassembly is the well-studied norm map of K(R). Using the equivariant perspective on assembly of [DL98] and the precise un- derstanding of the 1-category of genuine G-spectra that the theory of spectral G-Mackey functors of [Bar17] a↵ords, we show the above theorem by contem- plating various universal properties, and that it holds for any additive functor Catperf ! Sp instead of K-theory.