Coarse Sheaf Cohomology

Abstract

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups.