zbMATH — the first resource for mathematics

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
Full Text:
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.