Some ‘homological’ properties of the stable Higson corona

Abstract

We establish certain ‘homological properties’ of the stable Higson corona used by Emerson and Meyer to study the Dirac-dual-Dirac approach to the Baum–Connes conjecture [5]. These are used to obtain explicit isomorphisms between the K-theory groups of stable Higson coronas, and the K-theory groups of certain geometrically defined boundaries. This is sufficient to give a simple proof of the strong Novikov conjecture for torsion-free hyperbolic groups and torsion-free groups acting properly and cocompactly on CAT(0) spaces, and also provides an input into an index theorem in single operator theory [15], [16].