Coarse Cohomology with twisted Coefficients

Abstract

This paper studies sheaf cohomology on coarse spaces.

A totally bounded uniformity on coarse metric spaces

“…The [10, Proposition 93] guarantees that for choice A there exists at least one endpoint if the space X is coarsely proper coarsely geodesic. The proof of [10,Proposition 93] is similar to the one of Königs Lemma in graph theory.…”

“…The starting point of this research was an observation in the studies of [10]: coarse cohomology with twisted coefficients looked like singular cohomology on some kind of boundary. We tried to find a functor from the coarse category to the category of topological spaces that would reflect that observation.…”

“…Then (im f ) c ⊆ Y contains a countable subset (s i ) i that is coarsely disjoint to im f . Then by[10, Proposition 93] there is a coarse ray ρ : Z + → Y , an unbounded subsequence (s i k ) k and an entourage E ⊆ Y 2 such that(s i k ) k ⊆ E[ρ(Z + )] Now ρ represents a point r ∈ O(Y ) and O(f ) is surjective. Thus there is some p ∈ O(X) such that O(f )(p) = r.Suppose p is represented by ϕ : Z + → X then f • ϕ represents r and has image in im f .…”

Coarse Homotopy on metric Spaces and their Corona

“…If X is a metric space we associate to X a Grothendieck topology determined by coarse covers. Sheaf cohomology on coarse covers is coined coarse cohomology with twisted coefficients in [18]. Now coarse covers on X determine the finite open covers on ν ′ (X).…”

Coarse cohomology theories

“…Note that there are also other approaches to a general framework for coarse cohomology theories. Let us mention exemplary the work of Schmidt [Sch99] and Hartmann [Har17].…”