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The Baum-Connes and the Farrell-Jones conjectures in \(K\)- and \(L\)-theory. (English) Zbl 1120.19001

Friedlander, Eric M. (ed.) et al., Handbook of \(K\)-theory. Vol. 1 and 2. Berlin: Springer (ISBN 3-540-23019-X/hbk). 703-842 (2005).
In the article the authors give an extensive account of the present state of research on the Baum-Connes conjecture about the topological \(K\)-theory of the reduced \(C^*\)-algebra and on the Farrell-Jones conjecture about the \(K\)- and \(L\)-theory of the group ring of a discrete group. Although these conjectures came up in different areas of mathematics, namely operator theory on the one hand [P. Baum, A. Connes and N. Higson, Contemp. Math. 167, 241–291 (1994; Zbl 0830.46061)] and higher-dimensional manifold theory on the other [F. T. Farrell and L. E. Jones, J. Am. Math. Soc. 6, 249–297 (1993; Zbl 0798.57018)], they have a common origin, which was first addressed in J. F. Davis and W. Lück [K-Theory 15, 201–252 (1998; Zbl 0921.19003)], the principle of equivariant assembly. The unified approach has already initiated additional interactions between the two fields and provides potential for transporting further techniques for proving one of the conjectures from the one field to the other. For that reason the authors have put some special emphasis on this common origin, while most other overview articles on the matter typically either concentrate on the Farrell-Jones conjecture or concentrate on the Baum-Connes conjecture individually.
The conjectures at hand all claim that a certain homomorphism, the according equivariant assembly map, is an isomorphism. The domain of the equivariant assembly map always is a generalized homology group of some classifying space, while the target is the gadget of interest, i.e., the \(K\)-theory of the reduced \(C^*\)-algebra in the case of the Baum-Connes conjecture and the \(K\)- and \(L\)-groups of the group ring in the case of the Farrell-Jones conjecture. Once one of the conjectures has been verified for a specific group, this allows to actually compute the corresponding gadget of interest via standard methods from algebraic topology. Besides this computational aspect the verification of the conjectures also gives qualitative information, as in each case the according assembly map has a geometric interpretation, namely in terms of index theory in the case of the Baum-Connes conjecture and in terms of surgery theory in the case of the Farrell-Jones conjecture. The geometric interpretations also lead to a vast amount of applications of the conjectures, like, e.g., the Bass conjecture, the Borel conjecture, the Kadison conjecture or the Novikov conjecture, just to name some of the most prominent examples.
The present article provides information on all relevant aspects of the subject. A big portion of it is devoted to the vast amount of applications, other parts deal with the present state of the conjectures, the various methods for proving the conjectures and the techniques for computing the relevant homology groups. The different aspects are discussed in such a way that the corresponding sections can be read almost independently from each other. Each section gives comprehensive, detailed, and rather complete information on the particular aspect, which altogether makes the article a valuable reference for both the novice and the expert in the field.
For the entire collection see [Zbl 1070.19002].

MSC:

19-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to \(K\)-theory
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
19A31 \(K_0\) of group rings and orders
19B28 \(K_1\) of group rings and orders
19D99 Higher algebraic \(K\)-theory
19G24 \(L\)-theory of group rings
19K99 \(K\)-theory and operator algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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