Some ‘homological’ properties of the stable Higson corona

In (5), (4) Emerson and Meyer dene and study the stable Higson corona, a 'large-scale' geometric invariant of a metric space, and its rela- tionship with the Dirac-dual-Dirac approach to the Baum-Connes conjec- ture. Motivated by close analogies between the stable Higson corona and Roe algebra, we establish certain 'homological properties' of the former object with respect to an appropriate coarse category. These give more elementary proofs of some of its properties, as well as an input into an index theorem in single operator theory (14), (15). In particular, we obtain explicit isomorphisms (and inverses) between the K-theory of the stable Higson corona and that of simpler objects in some cases; this is in partic- ular sucient to reprove (split) injectivity of the Baum-Connes assembly map for certain groups.