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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.

MSC:
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
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References:
[1] A Bartels, T Farrell, L Jones, H Reich, On the isomorphism conjecture in algebraic \(K\)-theory, Topology 43 (2004) 157 · Zbl 1036.19003
[2] M Bökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic \(K\)-theory of spaces, Invent. Math. 111 (1993) 465 · Zbl 0804.55004
[3] G E Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics 34, Springer (1967) · Zbl 0162.27202
[4] G Carlsson, On the algebraic \(K\)-theory of infinite product categories, \(K\)-Theory 9 (1995) 305 · Zbl 0829.18005
[5] G Carlsson, E K Pedersen, Controlled algebra and the Novikov conjectures for \(K\)- and \(L\)-theory, Topology 34 (1995) 731 · Zbl 0838.55004
[6] J F Davis, W Lück, Spaces over a category and assembly maps in isomorphism conjectures in \(K\)- and \(L\)-theory, \(K\)-Theory 15 (1998) 201 · Zbl 0921.19003
[7] F T Farrell, L E Jones, Isomorphism conjectures in algebraic \(K\)-theory, J. Amer. Math. Soc. 6 (1993) 249 · Zbl 0798.57018
[8] F T Farrell, L E Jones, The lower algebraic \(K\)-theory of virtually infinite cyclic groups, \(K\)-Theory 9 (1995) 13 · Zbl 0829.19002
[9] F T Farrell, L E Jones, Rigidity for aspherical manifolds with \(\pi_1{\subset}\mathrm{GL}_m(\mathbbR)\), Asian J. Math. 2 (1998) 215 · Zbl 0912.57012
[10] W Lück, Chern characters for proper equivariant homology theories and applications to \(K\)- and \(L\)-theory, J. Reine Angew. Math. 543 (2002) 193 · Zbl 0987.55008
[11] E K Pedersen, C A Weibel, \(K\)-theory homology of spaces, Lecture Notes in Math. 1370, Springer (1989) 346 · Zbl 0674.55006
[12] D Rosenthal, Splitting with continuous control in algebraic \(K\)-theory · Zbl 1068.19008
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