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On the domain of the assembly map in algebraic \(K\)-theory. (English) Zbl 1038.19001
Let \(R\) be an associative ring with unit and let \(\Gamma\) be a group. In the notation of J. F. Davis and W. Lück [K-Theory 15, 201–252 (1998; Zbl 0921.19003)], the isomorphism conjecture of F. T. Farrell and L. E. Jones [J. Am. Math. Soc. 6, 249–297 (1993; Zbl 0798.57018)] states that the assembly map \[ H_*^{Or\Gamma}(E\Gamma(\mathcal V\mathcal C);\mathbf K R^{-\infty})\to K_*(R\Gamma) \] is an isomorphism (for \(R=\mathbb Z\)) where \(\mathcal V\mathcal C\) is the family of virtually cyclic subgroups of \(\Gamma\).
The motivation for the paper is that the groups \(H_*^{Or\Gamma}(E\Gamma(\mathcal F);\mathbf K R^{-\infty})\) are better understood for other families \(\mathcal F\) of subgroups of \(\Gamma\), in particular for the family \(\mathcal F=\mathcal{FIN}\) of finite subgroups. The main result is that \[ H_*^{Or\Gamma}(E\Gamma(\mathcal{FIN});\mathbf K R^{-\infty})\to H_*^{Or\Gamma}(E\Gamma(\mathcal V\mathcal C);\mathbf K R^{-\infty}) \] is split injective, thus – assuming the isomorphism conjecture – providing a readily understandable split summand of \(K_*(R\Gamma)\). The results are obtained through controlled K-theory.
For virtually cyclic \(\Gamma\) (in which case \(H_*^{Or\Gamma}(E\Gamma(\mathcal V\mathcal C);\mathbf K R^{-\infty})\to K_*(R\Gamma)\) is an equivalence for trivial reasons), this shows that the assembly map \[ H_*^{Or\Gamma}(E\Gamma(\mathcal{FIN});\mathbf K R^{-\infty})\to K_*(R\Gamma) \] is split injective, a result also obtained by D. Rosenthal [math.AT/0309106, “Splitting with continuous control in algebraic \(K\)-theory” (2003)]. Under assumptions about vanishing of lower \(K\)-groups (which are true for \(R=\mathbb Z\)), the author obtains parallel results for \(L\)-theory.

19D50 Computations of higher \(K\)-theory of rings
19B28 \(K_1\) of group rings and orders
19A31 \(K_0\) of group rings and orders
19G24 \(L\)-theory of group rings
Full Text: DOI EMIS EuDML arXiv
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