Fibred Farrell–Jones isomorphism conjecture

Let $G$ be a discrete group and let $\mathcal F$ be a family of subgroups of $G$. For a group homomorphism $\varphi:K{\longrightarrow} G$, define the induced family

$$\varphi^*\mathcal F=\{H\leq K\mid \varphi(H)\in\mathcal F\}.$$

Let $E(\varphi^*\mathcal F)$ be a model for the universal $K$-space for this family, and let $H^K_n(-;\mathbb K)$ be the relevant equivariant homology theory. Fibred Farrell–Jones isomorphism conjecture. The pair $(G,\mathcal F)$ satisfies the Fibred Isomorphism Conjecture if, for every group homomorphism $\varphi:K{\longrightarrow} G$, the assembly map

$$H^K_n(E(\varphi^*\mathcal F);\mathbb K)\longrightarrow H^K_n(\mathit{pt};\mathbb K)$$

is an isomorphism for all $n\in{\mathbb Z}$. This fibred formulation is designed to apply in more general situations than the ordinary Farrell–Jones conjecture; the paper subsequently uses it for surface braid groups.