# zbMATH — the first resource for mathematics

Relation between the Farrell-Jones conjectures in algebraic and Hermitian $$K$$-theory. (Relation entre les conjectures de Farrell-Jones en $$K$$-théories algébrique et hermitienne.) (French. English summary) Zbl 1126.19005
Let $$A$$ be a ring and let $$G$$ be a discrete group. For each integer $$n$$, J.-L. Loday defined in [“$$K$$-théorie algébrique et représentations de groupes”, Ann. Sci. Éc. Norm. Supér. (4) 9, 309–377 (1976; Zbl 0362.18014)] an assembly map
$\lambda_n: h_n(BG,\mathcal K_A) \rightarrow K_n(AG)$ between the homology groups $$h_n(BG,\mathcal K_A) = \pi_n(BG_+\wedge \mathcal K_A)$$ of $$BG$$ with values in the algebraic $$K$$-theory spectrum $$\mathcal K_A$$ of $$A$$ and the $$K$$-theory groups of the group ring $$AG$$. In the case $$A = \mathbb Z$$ the morphisms $$\lambda_n$$ have been conjectured to be isomorphisms for all $$n$$ by F. T. Farrell and L. E. Jones [“Isomorphism conjectures in algebraic $$K$$-theorie”, J. Am. Math. Soc. 6, No. 2, 249–297 (1993; Zbl 0798.57018)].
In a similar manner, for a ring $$A$$ with involution $$\bar{ }$$ and $$1/2 \in A$$, one can define assembly maps $\alpha_n :h_n(BG,\mathcal L_A) \rightarrow \,_{\varepsilon}L_n(AG)$ by replacing the $$K$$-theory spectrum $$\mathcal K_A$$ of $$A$$ by the $$\epsilon$$-Hermitian $$K$$-theory spectrum $$\mathcal L_A$$ of $$A$$ and the $$K$$-groups $$K_n(AG)$$ by the corresponding $$\epsilon$$-Hermitian $$K$$-groups $$\,_{\epsilon}L_n(AG)$$. Here $$\epsilon$$ is a central element in $$A$$ satisfying $$\epsilon \bar{\epsilon} = 1.$$ Again Farrell and Jones conjectured that the morphisms $$\alpha_n$$ are isomorphisms in the case that $$A = \mathbb Z [\frac{1}{2}].$$
Using results of M. Karoubi’s [“Le théorème fondamental de la $$K$$-théorie hermitienne”, Ann. Math. (2) 112, 259–282 (1980; Zbl 0483.18008)], the author shows the following: If the Farrell-Jones Conjecture holds in algebraic $$K$$-theory, then the validity of the Farrell-Jones Conjecture in Hermitian $$K$$-theory is equivalent to the fact that for some integer $$n$$ the maps $$\alpha_n$$ and $$\alpha_{n-1}$$ are isomorphisms.
##### MSC:
 19G38 Hermitian $$K$$-theory, relations with $$K$$-theory of rings
Full Text:
##### References:
 [1] Bartels, A.; Reich, H., On the Farrell-Jones conjectures for higher algebraic $$K$$-theory, (2003) · Zbl 1073.19002 [2] Farrell, F. T.; Jones, L., Isomorphism conjectures in algebraic $$K$$-theory, J. Amer, Math. Soc., 6, 2, 249-297, (1993) · Zbl 0798.57018 [3] H. Bass, A. Heller; Swan, R. G., The Whitehead group of polynomial extension, Inst. hautes études sci., 22, 61-79, (1964) · Zbl 0248.18026 [4] Hsiang, W. C., Borel’s conjecture, Novikov’s conjecture and the $$K$$-theoritic analogue, (1989), World scientific book, Singapour · Zbl 0744.57018 [5] Karoubi, M., Le théorème fondamental de la $$K$$-théorie hermitienne, Annals of mathematics, 112, 259-282, (1980) · Zbl 0483.18008 [6] Loday, J.-L.$$, K$$-théorie algébrique et représentation de groupes, Ann. Sci. Ecole Normale Sup. Sér. 4, 9, 3, 309-377, (1976) · Zbl 0362.18014 [7] Whitehead, G. W., Generalised homology theories, Trans. A. M. S, 102, 227-283, (1962) · Zbl 0124.38302
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.