A-theoretic Farrell-Jones conjecture

Let $G$ be a group and let $\underline{\underline{E}}G$ be a model for the classifying space for virtually cyclic subgroups. Let ${\mathbf A}$ be the covariant spectrum-valued functor obtained from Waldhausen's nonconnective $A$-theory, so that ${\mathbf A}(BG)$ is its value on the classifying space of $G$.

A-theoretic Farrell--Jones conjecture. The assembly map

$$H_n^G(\operatorname{pr};{\mathbf A}):H_n^G(\underline{\underline{E}}G;{\mathbf A})\to H_n^G(G/G;{\mathbf A})=\pi_n({\mathbf A}(BG))$$

is bijective for all $n\in\mathbb{Z}$. This is the $A$-theory analogue of the algebraic Farrell--Jones conjecture; the text explains its controlled-topological origin and relates it to Waldhausen's construction.