At the beginning of the paper six types of assembly maps \(\text{asmb}^G_n : {\mathcal H}^G_n(E_{\mathcal F}) \to {\mathcal H}^G_n(+)\) are given having as source and target certain \(G\)-homology groups related to \(K\)- and \(L\)-groups of group rings and group \(C^*\)-algebras, where \(G\) is a discrete group. There are conjectures about these maps, called the isomorphism conjectures, which asserts that all types of \(\text{asmb}^G_n\) are bijective for all \(n \in \mathbb{Z}\). The first three of these conjectures are known as the Farrell-Jones conjectures for algebraic and homotopy \(K\)-and \(L\)-theories, and also the next two ones are called the Bost conjecture and the Baum-Connes conjecture respectively. The authors prove that these conjectures are inherited under colimits over directed systems of groups (with not necessarily injective structure maps). The proof of this statement depends heavily on the construction of \(G\)-homology theories. Here the authors use a variation of the notion of an equivariant homology theory introduced by W. Lück [J. Reine Angew. Math. 543, 193–234 (2002; Zbl 0987.55008)]. For a precise description of the statement of the result above, as an illustrative sample one will observe the case of the assembly map listed above first.
Let \(R\) be a ring with \(G\)-action by ring automorphisms, let \(R\rtimes G\) be the twisted group ring and denote by \(K_n(R\rtimes G)\) its algebraic \(K\)-theory in the non-connective sense. Then one has a \(G\)-equivariant homology theory \(H^G_*(-; {\mathbf K}_R)\) with the property that \(H^G_n(G/H; {\mathbf K}_R)\cong K_n(R\rtimes H)\) for every subgroup \(H \subset G\) and \(n \in \mathbb{Z}\). Here for brevity one call \(\text{asmb}^G_n\) the composition of this isomorphism \(H^G_n(+; {\mathbf K}_R)\cong K_n(R\rtimes G)\) and the homomorphism \(H^G_n(E_{{\mathcal VC}_{\text{yc}}}(G); {\mathbf K}_R) \to H^G_n(+; {\mathbf K}_R)\) induced by the projection from \(E_{{\mathcal VC}_{\text{yc}}}(G)\) to the one-point space \(+\), where \(E_{{\mathcal VC}_{\text{yc}}}(G)\) denotes the classifying space associated to the family \({\mathcal VC}_{\text{yc}}\) of virtually cyclic subgroups of \(G\). Let \((G_i)_{i \in I}\) be a directed system of groups with \(G=\text{colim}_{i \in I}G_i\). Then the main result of this paper states, in the present case, that (i) if \(\text{asmb}^H_n : H^H_n(E_{{\mathcal VC}_{\text{yc}}}(H); {\mathbf K}_R) \to K_n(R\rtimes H)\) is bijective for all \(n \in \mathbb{Z}\), all \(i \in I\) and all subgroups \(H \subset G_i\), then for every subgroup \(K \subset G\) \(\text{asmb}^K_n : H^K_n(E_{{\mathcal VC}_{\text{yc}}}(K); {\mathbf K}_R) \to K_n(R\rtimes K)\) is bijective for all \(n \in \mathbb{Z}\); and that (ii) if suppose that all structure maps of \((G_i)_{i \in I}\) are injective and that \(\text{asmb}^{G_i}_n : H^{G_i}_n(E_{{\mathcal VC}_{\text{yc}}}(G_i); {\mathbf K}_R) \to K_n(R\rtimes G_i)\) is bijective for all \(n \in \mathbb{Z}\) and \(i \in I\), then \(\text{asmb}^G_n : H^G_n(E_{{\mathcal VC}_{\text{yc}}}(G); {\mathbf K}_R) \to K_n(R\rtimes G)\) is bijective for all \(n \in \mathbb{Z}\). It is also mentioned that the corresponding statement is true for all the other assembly maps except in one case. In fact the version (i) of the fifth assembly map does not hold, which is checked by producing a counterexample.
In addition to the above in Section 5 the authors consider the fibered version of these isomorphism conjectutes and prove a generalization of the result of F. T. Farrell and P. A. Linnell [Proc. Lond. Math. Soc., III. Ser. 87, No. 2, 309–336 (2003; Zbl 1031.19001)].
For the entire collection see [Zbl 1147.19001].