Relation between the Farrell-Jones conjectures in algebraic and Hermitian \(K\)-theory (Q2372793)

Let \(A\) be a ring and let \(G\) be a discrete group. For each integer \(n\), \textit{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 \textit{F. T. Farrell} and \textit{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 \textit{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.