On the Higson-roe Corona

Higson-Roe compacti cations rst arose in connection with C -algebra approaches to index theory on noncompact manifolds. Vanishing and/or equivariant splitting results for the cohomology of these compacti cations imply the integral Novikov Conjecture for fundamental groups of nite aspherical CW complexes. We survey known results on these compacti cations and prove some new results { most notably that the n cohomology of the Higson-Roe compacti cation of hyperbolic spaceH consists entirely of 2-torsion for n even and that the rational cohomology of the Higson-Roe compacti cation of R is nontrivial in all dimensions 1 k n. x1. The Higson-Roe Compactification Higson's compacti cation X rst appeared in [H] in connection with a K-theoretic analysis of Roe's index theorem for noncompact Riemannian manifolds. Higson de ned X to be to be the maximal ideal space of the commutativeC -algebra of smooth functions whose gradient vanishes at in nity. In [R1], Roe modi ed Higson's de nition to make sense for more general spaces. Here is Roe's de nition: De nition. If M is a space and : M ! C is a continuous function, de ne Vr( ) : M ! R by Vr( ) = supfj (y) (x)j : y 2 Br(x)g Then Ch(M) is the space of all bounded continuous functions : M ! C so that for each r > 0, Vr( )! 0 at in nity. Lemma 5.3 of [R1] proves that Ch(M) is a C -algebra, so it makes sense to de ne the Higson-Roe compacti cation, M of M to be the maximal ideal space of Ch(M). 1991 Mathematics Subject Classi cation. Primary 54D35, 54F45, 57M10, 57S30, 46L80, 53C20.