Coarse Cohomology with twisted Coefficients

TL;DR

This paper introduces a sheaf cohomology theory for coarse spaces using a Grothendieck topology based on coarse covers, establishing functoriality, Mayer-Vietoris sequences, and relating cohomology to the number of ends.

Contribution

It develops a new sheaf cohomology framework for coarse spaces, connecting coarse geometry with algebraic topology tools and functorial properties.

Findings

Sheaf cohomology is functorial with respect to coarse maps.

Established a coarse Mayer-Vietoris sequence.

Cohomology with constant coefficients relates to the number of ends.

Abstract

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.