Some `homological' properties of the stable Higson corona
Coarse geometry studies the large-scale geometry of metric spaces. Prominent examples are groups with word-length metric. The \(K\)-theory of the Roe \(C^*\)-algebra of a coarse space is the most popular coarse invariant. It is related to a purely topological invariant via the coarse Baum-Connes assembly map. Another coarse invariant that is in some sense dual to the Roe \(C^*\)-algebra is the stable Higson corona introduced by \textit{H. Emerson} and \textit{R. Meyer} [J. Inst. Math. Jussieu 5, No. 2, 161--186 (2006; Zbl 1092.19005)]. It is constructed using functions from the coarse space to the \(C^*\)-algebra of compact operators that have vanishing variation at infinity. The \(K\)-theory of the stable Higson corona is computed by a coarse coassembly map. It has been shown by \textit{H. Emerson} and \textit{R. Meyer} [Topology 46, No. 2, 185--209 (2007; Zbl 1119.19005)] that the coarse coassembly map is an isomorphism for a group if the Dirac-dual Dirac method applies to it; the converse holds for torsion-free groups with finite classifying space. One set of tools to study the \(K\)-theory of the Roe \(C^*\)-algebra is coarse homotopy theory. In particular, there is a coarse Mayer-Vietoris sequence, and the \(K\)-theory of the Roe \(C^*\)-algebra is invariant under coarse homotopy. This suffices to compute the \(K\)-theory of the Roe \(C^*\)-algebra for many examples. This article establishes analogous results for the stable Higson corona, and uses this to show that the coarse coassembly map is an isomorphism for some examples. These examples include proper metric spaces that are geodesic and CAT(0) or quasi-geodesic and Gromov hyperbolic. In the case of groups, these examples were already treated in the last section in [\textit{H. Emerson} and \textit{R. Meyer}, \(K\)-theory and noncommutative geometry. Proceedings of the ICM 2006 satellite conference, Valladolid, Spain, August 31--September 6, 2006. Zürich: European Mathematical Society (EMS). EMS Series of Congress Reports, 71--89 (2008; Zbl 1160.19003)].
scientific article; zbMATH DE number 6155609
scientific article