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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
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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
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