Coarse cohomology with twisted coefficients

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.

Twisted Coefficients on coarse Spaces and their Corona

“…Proof. This has already been proved in [7,Theorem 72]. Now Theorem 34 serves another proof: A coarse map f : X → Y between coarse spaces gives rise to a morphism of Grothendieck topologies Y ct , X ct in the same way as ν ′ (f ) gives rise to a morphism of Grothendieck topologies of the topological spaces ν ′ (Y ), ν ′ (X).…”

“…Proof. This is already [7,Theorem 74]. Now Theorem 34 gives rise to an alternative proof: If A, B ⊆ X are two subsets of a metric space then ν ′ (A), ν ′ (B) can be realized as closed subsets of ν ′ (X) by Lemma 39.…”

“…Coarse cohomology with twisted coefficients has been introduced in [7]. Coarse covers determine a Grothendieck topology on a coarse space and a coarse map gives rise to a morphism of Grothendieck topologies.…”

“…Then a coarse map f : X → Y between metric spaces is mapped to a continuous map ν ′ (f ) : ν ′ (X) → ν ′ (Y ) between coronas. Sheaf cohomology on coarse spaces has been introduced in [7]. We show the functor ν ′ preserves and reflects sheaf cohomology.…”

Coarse compactifications of proper metric spaces

“…The Proposition 49 shows both the topological and the coarse version of Freudenthal compactifiaction agree on proper geodesic metric spaces. The space of ends gives information about the number of ends of a coarse metric space [Har17]. It is both metrizable and totally disconnected.…”