Coefficients for the Farrell-Jones conjecture
The Farrell-Jones conjecture asserts that the \(K\)- or \(L\)-theory of a group ring \(RG\) can be obtained from that of the virtually cyclic subgroups of \(G\); more precisely, \(K_*(RG)\) is isomorphic, via the assembly map, to the direct limit \({\mathrm{colim}}\;H^G_*(E_{H \in \mathcal{F}}G;\mathbf{K}_R)\) where \(\mathcal{F}\) is the family of virtually cyclic subgroups of \(G\), \(H^G_*(-;\mathbf{K}_R)\) is \(G\)-equivariant homology and \(E_{\mathcal{F}}G\) is the classifying space for \(\mathcal{F}\), and likewise for \(L\)-groups. This paper presents a generalization of the Farrell-Jones conjecture which allows coefficients in additive categories with \(G\)-action. There are many advantages obtained from this generalization. The conjecture passes to subgroups and more generally pulls-back under arbitrary group homomorphisms. The injectivity and surjectivity parts of the conjecture have these inheritence properties separately. Further, the correct generalizations of the conjecture are obtained for twisted group rings and crossed product rings. The construction of the corresponding additive categories is matter of interest in its own right and can be expected to have wider applications in representation theory.
Farrell-Jones conjecture, controlled algebra, additive category with group action
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