Coarse assembly maps

Abstract

A coarse assembly map relates the coarsification of a generalized homology theory with a coarse version of that homology theory. In the present paper we provide a motivic approach to coarse assembly maps. To every coarse homology theory $E$ we naturally associate a homology theory $E\mathcal{O}^{\infty}$ and construct an assembly map $$\mu_{E} :\mathrm{Coarsification}(E\mathcal{O}^{\infty})\to E\ .$$ For sufficiently nice spaces $X$ we relate the value $E\mathcal{O}^{\infty}(X)$ with the locally finite homology of $X$ with coefficients in $E(*)$. In the example of coarse $K$-homology we discuss the relation of our motivic constructions with the classical constructions using $C^{*}$-algebra techniques.