Coarse sheaf cohomology

TL;DR

This paper develops a sheaf cohomology theory for metric spaces using a Grothendieck topology, revealing geometric features like the number of ends and establishing coarse homotopy invariance, with applications to Riemannian manifolds.

Contribution

It introduces a resolution of the constant sheaf via cochains and demonstrates coarse homotopy invariance of the cohomology, enabling computations on Riemannian manifolds and relating to the Higson corona.

Findings

Cohomology in degree 0 detects the number of ends of the space.

Higher cohomology groups vanish if the asymptotic dimension is finite.

Finite abelian groups are effective coefficients for finitely generated groups.

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.