Coarse sheaf cohomology

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.


Introduction
The sheaf-theoretic approach to coarse metric spaces has been applied in many different contexts [BE17,RS13,Sch99].Sheaf-theoretic methods play an important role in our paper.We also present three other computational tools.Cochain complexes assigned to a filtration of Vietoris-Rips complexes have not just been used in the coarse setting [Hau95].Many well-known coarse (co-)homology theories are coarse homotopy invariant [HR94,Mit01,Wul20].The cohomology of the Higson corona is of course as a composition of functors a coarse invariant which has been studied before [Kee94].Even in combination with other computational methods [HRY93] coarse invariants are hard to compute for the spaces one is most interested in which include Riemannian manifolds and finitely generated groups.
Coarse sheaf cohomology has been designed by the author in her thesis.Aside from an agenda to present new computational methods which may be suitable for a large number of spaces there are two immediate results: Theorem 1.If M is a non-positively curved closed Riemannian n-manifold and A a finite abelian group then Ȟq ct (M, A) = H q sing (S n−1 ; A) the left side denotes coarse sheaf cohomology with values in the constant sheaf A and the right side denotes singular cohomology with values in the group A.
This result can be immediately applied to define a coarse version of mapping degree associated to a coarse map between manifolds.
There are many interesting cohomology theories on coarse metric spaces.The most prominent examples are Roe's coarse cohomology [Roe93,Roe03,Hai10] and controlled operator K-theory [Roe96,HR00,Yu95,Yu00].If two metric spaces X, Y have the same coarse type then specifying a coarse equivalence X → Y is a proof.If on the other hand X, Y do not have the same coarse type then a coarse invariant which does not have the same values on X and Y gives a proof.In general a well designed cohomology theory delivers a rich source of invariants which are easy to compute.To this date cohomology of finitely generated free abelian groups has been calculted for Roe's coarse cohomology and also controlled operator K-theory.There is still a gap in knowledge about cohomology of other finitely generated groups.Riemannian manifolds on the other hand do not show interesting cohomology groups since every Riemannian n-manifold with nonpositive sectional curvature is coarsely homotopic to R n and most coarse cohomology theories are coarse homotopy invariant [Roe93].
Our coarse cohomology theory Ȟq ct (•; •) is a sheaf cohomology theory on a Grothendieck topology X ct assigned to a metric space X [Har20].If A is an abelian group then for the constant sheaf A on X we obtain in dimension 0 a copy of A for every end of X or an infinite direct sum of copies of A if X does not have finitely many ends [Har20].
In this paper we design a cochain complex (CY q b (X; A)) q assigned to a metric space X and abelian group A. The functor U ⊆ X → CY q b (U, A) forms a flabby sheaf on X ct .The sequence of sheaves 0 computes coarse sheaf cohomology of X with values in A X .
Theorem A. If X is a metric space then there is a flabby resolution CY q b (•; A) of the constant sheaf A X on X ct .We can compute sheaf cohomology with values in the constant sheaf using cochain complexes: Ȟq ct (X; A X ) = HY q b (X, A).For q ≥ 1 there is a comparison map HY q b (X; A) → HX q+1 (X; A) with Roe coarse cohomology.This map is neither injective nor surjective though.The main difference is that our cochains are defined as maps that need to be "blocky" while coarse cochains do not have this restriction.Thus general statements on cohomology are easier to prove for Roe coarse cohomology.While we hope that combinatorical computations are easier realized using blocky cochains.
There are several notions of homotopy on the coarse category which are all equivalent in some way.The homotopy theory we are going to employ uses the asymptotic product as coarse substitute for a product and the first quadrant in R 2 equipped with the Manhattan metric as a coarse substitute for an interval [Har19b].In effect this homotopy theory and the other coarse homotopy theories are only of use if one wants to compute cohomology of R n and maybe Riemannian manifolds.Nonetheless we prove that coarse sheaf cohomology is a coarse homotopy invariant using the resolution via cochains.A coarse map α : X → Y between metric spaces induces a cochain map α * : CY q b (Y ; A) → CY q b (X; A) which in turn induces a homomorphism α * : HY q b (Y ; A) → HY q b (X; A).Conversely the inverse image functor maps the constant sheaf A Y on Y ct to the constant sheaf A X on X ct .Thus there is an induced homomorphism α * in cohomology.One may wonder if both homomorphisms α * , α * coincide.And indeed they do.
To a proper metric space X we can assign a compact Hausdorff topological space ν(X), the Higson corona of X.This version of boundary reflects sheaf cohomology in the following way: There is a functor • ν which maps a sheaf F on X ct to a sheaf F ν on ν(X).Conversely the functor • maps a sheaf G on ν(X) to a sheaf Ĝ on X ct , Together they provide an equivalence of categories between "reflective" sheaves on X ct and sheaves on ν(X).In particular the constant sheaf A X on X ct is reflective and mapped to the constant sheaf A ν(X) on ν(X).We can compute cohomology with constant coefficients either way:

Theorem C. If X is a proper metric space and A an abelian group then
the qth cohomology of X with values in the constant sheaf A on X ct is isomorphic to the qth sheaf cohomology of the Higson corona ν(X) of X with values in the constant sheaf A ν(X) .Moreover if asdim(X) ≤ n then Ȟq ct (X, A X ) = 0 for q > n.
This paper provides enough computational methods to compute metric cohomology of finitely generated groups.Vanishing of Ȟ1 ct (Z, A) = 0 for finite A can be computed directly using cochains.Then our result on the Higson corona implies that Z is acyclic for finite coefficients.The same method can be employed to show that trees are acyclic for finite coefficients.Thus we computed metric cohomology of the free group F n with n < ∞ generators.Computing cohomology of the free abelian groups Z n with n < ∞ is more challenging.A coarse homotopy equivalence Z n−1 × Z ≥0 → Z ≥0 provides a Leray cover of Z n which has the same combinatorical information as the nerve of a Leray cover of the topological space S n−1 .Thus cohomology with finite coefficients can be derived.

Theorem D. If A is a finite abelian group then
There is a more general notion of coarse space which includes the class of coarse metric spaces.Most of our concepts work in more generality.We restrict our attention to metric spaces only since a wider audience (than coarse geometers) is interested in this class of coarse spaces only.The coarse sheaf cohomology theory is defined on coarse spaces with connected coarse structure.The resolution via cochains also works for this class of spaces.The homotopy theory is only defined for metric spaces and the results on the Higson corona work for proper metric spaces and coarse structures generated by a compactification of a paracompact, locally compact Hausdorff space.
This article is organized in 10 chapters.Some can be read independently but there are also a few dependencies as depicted in the following diagram.
The final chapter uses every aspect so far discussed.Sheaf-theoretic methods, the resolution via cochains, coarse homotopy and the Higson corona are employed in the computation of metric cohomology of Z n .

Coarse cohomology by Roe and the Higson corona
This chapter introduces terminology and concepts which are well known to coarse geometers.

COARSE COHOMOLOGY BY ROE AND THE HIGSON CORONA Elisa Hartmann
The set of entourages forms the coarse structure of X.If R ≥ 0 then the set This paper presents a resolution of the constant sheaf which consists of cochains which closely resemble coarse cochains of Roe's coarse cohomology.For this purpose we give a quick introduction to coarse cohomology by Roe which was invented by Roe in [Roe93,Roe03].
If X is a metric space then the set of q-simplices of the R-Vietoris-Rips complex of X is defined as R is bounded.Definition 3. If X is a metric space and A an abelian group then the coarse cochains CX q (X; A) is the set of functions X q+1 → A with cocontrolled support.It is a group by pointwise addition.The coboundary map ∂ q : CX q (X; A) → CX q+1 (X; A) is defined by This makes (CX q (X; A), ∂ q ) a cochain complex.Its homology is called coarse cohomology by Roe and denoted by HX * (X; A).
If α : X → Y is a coarse map then it induces a cochain map Two coarse maps which are close induce the same map in cohomology.If A = R and X = R n then A section in this paper transfers sheaves on a proper metric space to sheaves on its Higson corona.For this purpose we give a definition of the Higson corona which is equivalent to the usual one [Har19c].
Let X be a metric space.Two subsets A, B ⊆ X are called close (or not coarsely disjoint) if there exists an unbounded sequence (a i , b i ) i ⊆ A × B and some R ≥ 0 such that d(a i , b i ) ≤ R for every i.We write A B in this case.
Let R > 0 be a real number.A metric space X is called R-discrete if d(x, y) ≥ R for every x = y.If X is a metric space an R-discrete for some R > 0 subspace S ⊆ X is called a Delone set if the inclusion S → X is coarsely surjective.Every metric space contains a Delone set.If it is in addition proper then the finite sets of the Delone set are exactly the bounded sets.Definition 4. Let X be a proper metric space and S ⊆ X a Delone subset.Denote by Ŝ the set of nonprincipal ultrafilters on S.
Then define a relation on subsets of Ŝ: The relation on subsets of Ŝ determines a Kuratowski closure operator π = {F ∈ Ŝ : {F } π}.

Now define a relation λ on Ŝ:
Now the Higson corona ν(X) of X is defined ν(X) = Ŝ/λ as the quotient by λ.
If A ⊆ X is a subset of a metric space then cl(A) = Ā ∩ ν(X) where the closure is taken in the Higson compactification.We call (cl(A) c ) A⊆X the basic open sets and (cl(A)) A⊆X the basic closed sets.There are two observations: If A, B ⊆ X are two subsets then

Coarse sheaf cohomology, a survey
This chapter gives a survey on coarse sheaf cohomology or coarse cohomology with twisted coefficients as we call it [Har20].There are several facts on sheaf cohomology on topological spaces which hold in more generality for sheaf cohomology defined on a Grothendieck topology.Since the literature does not provide every aspect we are going to prove these facts by hand.
Let U ⊆ X be a subset of a metric space.A finite family of subsets U 1 , . . ., U n ⊆ U forms a coarse cover of U if for every entourage E ⊆ U × U the set is bounded.This is equivalent to saying that the set is a cocontrolled subset of X2 .To a metric space X we associate a Grothendieck topology X ct in the following way.The underlying category Cat(X ct ) is the poset of subsets of X. Subsets (U i ) i form a covering of U ⊆ X if they coarsely cover U .
A contravariant functor F on subsets of X is a sheaf on X ct if for every coarse cover U 1 , . . ., U n ⊆ U of a subset of X the following diagram is an equalizer If F is a sheaf on X ct then the right derived functor of the global sections functor is called coarse sheaf cohomology, written Ȟ * ct (X; F ).If 0 → F → G → H → 0 is a short exact sequence of sheaves on X ct then there is a long exact sequence in cohomology If A is an abelian group the sheafification of the constant presheaf A on X ct is called the constant sheaf A X on X.In this paper we are interested in the computation of Ȟq ct (X; A X ) in higher dimension.The zeroth cohomology group is related to the number of ends e(X) of the metric space X: Here im ϕ is the sheafification of the image presheaf U → im(ϕ(U )).
cohomology of F can be computed by taking homology of the cocomplex is an exact sequence with F i acyclic for every i.
The inclusion ε i → F i and the corestriction of This sequence gives rise to a long exact sequence Lemma 6.If X is a metric space every injective sheaf on X ct is flabby.
Proof.Let I be an injective sheaf on X ct and let V ⊆ U be an inclusion of subsets.We define a presheaf Z U,X on X ct by Denote by Z # U,X the sheafification.In a similar way we define Z V,X and is a subsheaf in a canonical way.Thus we have an exact sequence Since I is an injective object the sequence Hom Sh (Z # V,X , I) = Hom P Sh (Z V,X , I) = I(V ) which proves the claim.Lemma 7. If X is a metric space then flabby sheaves on X ct are acyclic.
If F is a flabby sheaf then it can be embedded in an injective sheaf I.The quotient of this inclusion is denoted G. Then we have an exact sequence with F flabby, I flabby by Lemma 6 and G is flabby by a standard argument.General theory on flabby sheaves also implies that the sequence is exact.Then the long exact sequence in cohomology to the short exact sequence 1, the exactness of 2 and Ȟq ct (X, for q ≥ 2. Since G satisfies the requirements for this Lemma we obtain the result for q ≥ 2 using inductively G and the isomorphism 3.

Standard resolution
This chapter proves Theorem A.
Let A be an abelian group.
A differential on C q (U, A) is defined by Definition 8.If X is a metric space, A an abelian group and q ≥ 0 then coarse cohomology HY q b (X, A) is defined to be the qth homology of the coarse cochain complex (CY q b (X, A), d q ) q≥0 .
Subsets U 1 , . . ., U n of a subset U ⊆ X of a metric space form a coarse disjoint union of U if they coarsely cover U and every two elements are disjoint.Lemma 9.If U ⊆ X is a subset of a metric space and A an abelian group then Let ϕ ∈ ker d 0 be a cocycle.Then x ∼ y if d 0 ϕ(x, y) = 0 defines an equivalence relation on X with equivalence classes (ϕ −1 (k)) k∈A .The ϕ −1 (k) form a coarse disjoint union since d 0 ϕ has cocontrolled support.We can assume all ϕ −1 (k) are not bounded otherwise we substract a cochain with bounded (cocontrolled) support.Thus ϕ is an element of A(U ).
If we are given a coarse disjoint union U 1 , . . ., U n of U and (a i ) Ui ∈ A(U ) then we can assume the U i are disjoint and not bounded.Then We show the identity axiom.Let ϕ ∈ CY q b (U, A) be a section with ϕ R | Ui = 0 for every i.By Lemma 11 the set as a finite sum of functions with cocontrolled support has cocontrolled support.We show the gluing axiom.Suppose As can easily be seen the cochain ϕ restricts to ϕ i for every i.
Here the empty intersection denotes U .Then

Now the projection to the ith factor of
Then φ ∈ C A1,...,An,U c (V, A) restricts to ϕ on U .
Lemma 13.The homology of C q (X, A) is concentrated in degree zero.
We note an exact sequence of cochain complexes: and for every subset U ⊆ X and ψ ∈ ker dX b q+1 there exists a coarse cover Then there exist a coarse cover (U i ) i and elements Since this set is bounded (x, y) must be contained in a bounded set too.
Proof.Suppose ψ ∈ C A1,...,An (U, A) and fix a i ∈ A i for every i.Then A i is bounded if ψ(a i , . . ., a i ) = 0. We add i to a list C. Likewise A i × A j is cocontrolled if there exists a map f : {0, . . ., q} → {i, j} with ψ(a f (0) , . . ., a f (q) ) = 0. We add the set {i, j} to the list C. We proceed likewise with A i × • • • × A j of up to q + 1 factors.We then define . By Lemma 15 we can conclude that U is a coarse cover.
Theorem 17.If X is a metric space and A an abelian group then the CY q b (X, A) are a flabby resolution of the constant sheaf A on X ct .We can compute Ȟq (X, A) = HY q b (X, A).
Proof.We prove that 0 By Lemma 10 the CY q b (X, A) are sheaves.They are flabby by Lemma 12.By Lemma 9 the sequence is exact at 0. If we combine Lemma 14 and Lemma 16 then we see that it is exact for q ≥ 1.

Functoriality, graded ring structure and Mayer-Vietoris
This chapter presents a few immediate applications of Theorem A.
Lemma 18.If two coarse maps α, β : X → Y are close then they induce the same map in cohomology.
Proof.The chain homotopy for coarse cohomology presented in [Roe03, Proposition 5.12] can be costumized for our setting.Define a map h : Since α, β are close, cocontrolled support of ϕ implies cocontrolled support of hϕ.Thus h is well-defined.
Throughout this section X denotes a metric space and R a commutative ring.We define a map on cochains Lemma 19.This product ∨ is well-defined.
Proof.We show φ ∨ ψ has cocontrolled support if one of φ, ψ does.Suppose φ has cocontrolled support.Then supp φ ∩ ∆ p R is bounded for every R ≥ 0. This implies the 0th factor is bounded.This proves the claim.
The formula is easy to check.From that we deduce that the ∨-product of two cocycles is a cocycle and the product of a cocycle with a coboundary is a coboundary.Thus ∨ gives rise to a cup-product ∪ on HY * b (X, R).Associativity and the distributive law can be checked on cochain level.This makes (HY * b (X; R), +, ∪) a graded ring.
Theorem 20. (Mayer-Vietoris) If U 1 , U 2 is a coarse cover of a metric space X then there is a long exact sequence in cohomology Here α is defined by ϕ → (ϕ| U1 , ϕ| U2 ) and β by (ϕ The result is obtained by taking the long exact sequence in cohomology of the exact sequence of cochain complexes. We denote by HX q b (X; A) the qth homology group of the cochain complex CX q b (X; A) given a metric space X and an abelian group A.
Proposition 21.Let X be a metric space and A an abelian group.Then Proof.Suppose q = 0.An element of ker d 0 is a constant function.If X is bounded every constant function represents an element of HX 0 b (X, A) = A. If X is not bounded then every constant function with bounded (cocontrolled) support must be zero.
If q > 0 we use the short exact sequence of cochain complexes 0 → CX q b (X, A) → C q (X, A) → CY q b (X, A) → 0. which splits since every element of CY q b (X, A) is represented by a blocky map ϕ : X q+1 → A. Now we produce the long exact sequence in cohomology.The first few terms are Thus HX 1 b (X, A) = 0.If X is not bounded then the first few terms of the long exact sequence in cohomology read 0 In the middle term we mod out the constant functions.
For q ≥ 1 the long exact in cohomology reads Thus we proved the claimed results.

Computations
Theorem A can also be applied to compute cohomology groups in a combinatorical manner.

Lemma 22.
If A is a finite abelian group then Ȟ1 ct (Z ≥0 , A) = 0. Proof.If ϕ ∈ ker dY b 1 then d 1 ϕ has cocontrolled support.This means for every n ∈ N the set supp(d 1 ϕ) ∩ ∆ 1 n is bounded.This amounts to saying that the function This function is blocky since φ assumes only finitely many values.Here we use that A is finite.Moreover we have This shows that ϕ − d 0 φ has cocontrolled support.Thus ϕ is a coboundary in CY 1 b (Z ≥0 , A).
Theorem 23.If T is a tree and A is a finite abelian group then Proof.Designate an element t 0 ∈ T as the root of the tree and define Let ϕ ∈ CY 1 b (S, A) be a cocycle.Then for every s ∈ S there is a unique 1-path a 0 = t 0 , . . ., a n = s joining t 0 to s. Define Let n ≥ 0 be a number.If s, t ∈ S and d(s, t) = n then there exist paths a 0 , . . ., a k , c 0 , . . ., c m joining t 0 to t and a 0 , . . ., a k , b 0 , . . ., b l joining t 0 to s with n is a would-be cocycle in ∆ 1 n+2 thus has bounded support.Thus we showed ϕ is a coboundary.Since the inclusion S ⊆ T is coarsely surjective we can conclude Ȟ1 ct (T, A) = Ȟ1 ct (S, A) = 0.

Infinite coefficients
The computations in Chapter 5 only work for finite coefficients.This chapter shows that the coefficient Z does not produce interesting cohomology groups.
If X is a proper metric space and A a metric space denote by C h (X, A) the abelian group of Higson functions ϕ : X → A modulo functions with bounded support.Namely a bounded function ϕ : X → A is called Higson if for every entourage E ⊆ X × X and every ε > 0 there exists a compact subset K ⊆ X such that (x, y) Lemma 25.If X is a proper R-discrete for some R > 0 metric space and A a metric space then U → C h (U, A) for U ⊆ X with the obvious restriction maps is a sheaf on X ct .
Proof.Let U 1 , . . ., U n be a coarse cover of a subspace U ⊆ X.
We prove the base identity axiom.Let ϕ ∈ C h (U, A) be an element.If ϕ| U1 , . . ., ϕ| Un have bounded support B 1 , . . ., B n then ϕ has support contained in the set which is a finite union of bounded sets and therefore itself bounded.Thus ϕ has bounded support.Now we prove the gluing axiom.Suppose there are functions is bounded.Now we check the Higson property.If E ⊆ U 2 is an entourage then where B ⊆ U is bounded.Let ε > 0 be a number.Then for each i there exists a bounded subset Thus ϕ satisfies the Higson property.
Denote by C f (X, A) the abelian group of Freudenthal functions ϕ : X → A modulo functions with bounded support.Namely a bounded function ϕ : X → A is called Freudenthal if for every entourage E ⊆ X × X there exists a compact subset K ⊆ X such that (x, y) ∈ E \ (K × K) implies ϕ(x) = ϕ(y)., A) be an element.We show ϕ is Freudenthal.Let E ⊆ U 2 be an entourage and choose ε = 1/2.Then there exists a bounded set

Lemma 26. If X is a proper metric space and A is a countable abelian group then the constant sheaf
The ϕ i are Higson and glue to a Higson function Φ(a) : Without loss of generality the U i are pairwise disjoint and cover U and likewise the V j are pairwise disjoint and cover U .Then Thus Φ is a homomorphism.We show Φ is well defined.Suppose c ∈ A(U ) is represented by 0 on B c where B is finite.Then Φ(c) has bounded support.Thus Φ is welldefined.Now we construct the inverse Ψ : ) is Freudenthal then in particular its image im ϕ = {a 1 , . . ., a n } is finite.Since ϕ is Higson the ϕ −1 (a 1 ), . . ., ϕ −1 (a n ) are a pairwise coarsely disjoint union of U .Then define Ψ(ϕ) to be the element represented by Proposition 27.If X is a proper metric space the following sequence of sheaves on X ct is exact: ) is induced by the inclusion Z → R.This map is well-defined since every Freudenthal function is Higson.It is injective.Thus exactness at C f (•, Z) is guaranteed.
The map C h (U, R) → C h (U, S 1 ) is induced by the quotient map R → R/Z.A Higson function is in the kernel of this map exactly when its image is contained in Z. Thus exactness at obtained by defining ϕ 2 (x) to be the representative of ϕ(x) in the interval [0.75, 1.5].Thus the right morphism in the diagram is surjective. holds.
We provide another proof using the Tietze extension theorem.If an element in C h (U, R) is represented by ϕ ∈ C h (U ) then it extends to φ on hU .By the Tietze extension theorem we can extend φ to a bounded function φ on hX.Then φ| X represents an element in C h (X, R) that restricts to ϕ.
Example 29.We show C h (•, R) is flabby on the specific example Z constructing a concrete global lift of a Higson function ϕ : U → R on a subspace of Z.If z ∈ U c then there are z − , z + ∈ U with z − the largest number in U with z − < z and z + the smallest number in U with z < z + .Define This function φ is Higson: If ε > 0 then there exists an Remark 30.By the long exact sequence in cohomology we obtain If X = Z we can define This function satisfies the Higson condition but is not bounded.Post-composition with the projection R → S 1 gives a Higson function φ : Z → S 1 which does not have a lift.Compare this result with [Kee94].
Remark 31.It would be great if we could find an algorithm that computes coarse cohomology with constant coefficients of a finitely presented group.This does not work not even in degree 0. If we could decide whether Ȟ0 ct (G; A) vanishes then we can decide whether G is finite.This is in general not decidable.

The inverse image functor
In this section we fix a coarse map α : X → Y between metric spaces.
If G is a sheaf on Y ct then the inverse image (or pullback sheaf) α * G is the sheafification of the presheaf on X ct which assigns U ⊆ X with G(α(U )).
Conversely if F is a sheaf on X ct then the direct image α * F is the sheaf on Y ct which assigns Lemma 32.The functor α * is left adjoint to α * .The functor α * is left exact and the functor α * is exact.The functor α * maps injectives to injectives.
Proof.The functor is a morphism of Grothendieck topologies and therefore gives rise to functors (α −1 ) p , (α −1 ) p between categories of presheaves [Tam94, Chapter I,2.3].The functor (α −1 ) p maps a presheaf F on Then [Tam94, Chapter I,3.6] discusses functors (α −1 ) s and (α −1 ) s between categories of sheaves.We obtain the direct image functor α * = (α −1 ) s and the inverse image functor α * = (α −1 ) s .Then [Tam94, Proposition I,3.6.2]implies that α * is left adjoint to α * , the functor α * is left exact and if α * is actually exact then the functor α * maps injectives to injectives.It remains to show that α * is exact.By [Tam94, Proposition I,3.6.7] the functor α * is exact if α −1 preserves finite fibre products and final objects.Indeed the inverse image of an intersection is an intersection of inverse images and the inverse image of the whole space is the whole space.
Proposition 33.If we equip N with the topological coarse structure associated to the one-point compactification N + of N we obtain a coarse space called * .This space * is not metrizable but a final object for metric spaces.The constant sheaf on * is flabby.
the first factor is finite exactly when the projection of E ′ to the second factor is finite.
Let X be a metric space and x 0 ∈ X a basepoint.We define a map We show this map is coarsely uniform: If R ≥ 0 and F ⊆ ρ ×2 (∆ R ) a subset such that the projection to the first factor is finite then choose arbitrary (x, y) ∈ ∆ R with (ρ(x), ρ(y)) ∈ F .Since the first factor of F is finite there is some S ≥ 0 with ⌊d(x, x 0 )⌋ ≤ S. Then Thus the projection of F to the second factor is finite.This implies ρ is coarsely uniform.If B ⊆ * is bounded then there exists some S ≥ 0 such that b ∈ B implies b ≤ S − 1.Then ρ −1 (B) is contained in a ball of diameter S around x 0 .Thus ρ is coarsely proper.This way we showed ρ is a coarse map.Suppose ϕ : X → * is another coarse map.Let H ′ ⊆ (ρ × ϕ)(∆ 0 ) be a subset such that the projection of H ′ to the first factor is finite.We have that H ′ is of the form for some subset A ⊆ X.Then the projection of H ′ to the first factor is ρ(A).Since ρ is coarsely proper the set A ⊆ ρ −1 • ρ(A) is bounded.This implies ϕ(A) is bounded which is the projection of H ′ to the second factor.Since we only used that ρ, ϕ are coarse maps we can use the same argument with the factors reversed.Thus we showed that (ρ × ϕ)(∆ 0 ) is an entourage in * .This implies ρ, ϕ are close, they represent the same coarse map.This way we showed that * is a final object for metric spaces.
If A, B ⊆ * are infinite subspaces then there exists a bijection ϕ : We conclude that a coarse cover of a subset U ⊆ * contains an element which is cofinite in U .Let A be an abelian group.Then Proof.If Z is a metric space and ρ Z : Z → * the unique coarse map we prove A Z = ρ * Z A * .This proves the claim since The sheaf ρ * Z A * is the sheafification of the following presheaf Now this is just the constant sheaf on Z. Now we compute the unit of the adjunction α * , α * .We denote by α −1 the presheaf inverse image functor.Then the unit of the adjunction α −1 , α * at A Y is given by The unit of the adjunction sheafification # and inclusion ι of presheaves in sheaves at α −1 A Y is given by This makes sense since ϕ assigns a value a Vi to V i ⊆ α(U ) where (V i ) i is a coarse disjoint union of α(U ).Since α is a coarse map the α −1 (V i ) form a coarse disjoint union of U .Then (a Vi ) α −1 (Vi) represents ϕ on U which in cocycle notation is ϕ • α| U .Now we compose the units: and obtain the desired result.
Theorem 35.The map induced by the inverse image functor coincides with the canonical map Proof.We apply [Ive84, Scolium II.5.2].We checked the proof of this result also works for sheaves on a Grothendieck topology.We choose f = α, G = A Y and T q = CY q (•, A) a resolution on Y .Then α * G = A X has a resolution CY q (•, A) on X.The morphism of complexes is given by commute.

Sheaves on the Higson corona
This chapter proves Theorem C.
Lemma 36.Let X be a metric space.If U 1 , . . ., U n is a coarse cover of X then there exists a coarse cover V 1 , . . ., V n of X with V i U c i for every i = 1, . . ., n. Proof.By [Har19a, Lemma 15] there exists a cover W 1 , • • • , W n of X as a set such that W i U c i for every i.For every i there exists an in between set C i with W i C c i and C i U c i .Then for every i the sets A 1 i := W c i , A 2 i := C i are a coarse cover.Taking the intersection over those coarse covers provides a coarse cover and in the other case ε i = 1 for every i.Thus Now we join appropriate elements of B and obtain the desired coarse cover: Given a sheaf F on X ct we define a sheaf F ν on ν(X): Proposition 37.If X is a proper metric space and F a sheaf on X ct then F ν is a sheaf on ν(X).
. By [Har19d, Lemma 32] the subsets U 1 , . . ., U n , A c are a coarse cover of X.By Lemma 36 there exists a finite coarse cover V 1 , . . ., V n , B of X such that V i U c i for every i and B A. Then V 1 , . . ., V n are a coarse cover of A. We show the base identity axiom: Let φ, ψ ∈ F ν (cl(U c ) c ) be elements with for every i.Since φ Vi = ψ Vi for every i the identity axiom on coarse covers implies φ A = ψ A .Now we show the base gluablity axiom.Let φ i ∈ F ν (cl(U c i ) c ) be a section for every i such that φ i | cl((Ui∩Uj ) c ) c = φ j | cl((Ui∩Uj ) c ) c for every i, j.Then the (φ i ) Vi glue to a section φ A on A by the gluablity axiom on coarse covers.
If α : F → F ′ is a morphism of sheaves on X ct then we define for every basic open cl(U c ) c : This definition makes sense since . By gluing along basic open covers we obtain for every open π ⊆ ν(X) a map α ν (π) : Moreover id ν F = id F ν and (α • β) ν = α ν • β ν .Thus we have proved • ν is a functor.Namely if Sheaf(X ct ) denotes the category of sheaves on X ct and Sheaf(ν(X)) denotes the category of sheaves on ν(X) then is a functor between categories of sheaves.
Lemma 38.Let X be a proper metric space.
an open cover and U c A.
Given a subset A ⊆ X of a proper metric the relations U c A and Thus we can define a well defined restriction map Ĝ is an isomorphism.
Proposition 39.Let X be a proper metric space.If G is a sheaf on ν(X) then Ĝ is a reflective sheaf on X ct .
Proof.Let A 1 , . . ., A n be a coarse cover of A ⊆ X.
We prove the identity axiom.Let s ∈ Ĝ(A) be a section with s| Ai = 0 for every i.Then there exists By Lemma 38 there is some ) c an open cover and U c A. Thus the identity axiom for open covers implies s U c = 0.This proves s = 0 in Ĝ(A).Now we prove the gluablity axiom.Let s i ∈ Ĝ(A i ) be a section for every i with s i | Aj = s j | Ai for every i, j.Suppose s i are represented by (s i ) As in the first part of this proof there is some subset and U c A. By the gluablity axiom on open covers the (s i ) U c i glue to a section s It is easy to see that η : G → η G defines a natural transformation.This way we showed that η is a natural isomorphism between id Sheaf(ν(X)) and • ν • •.Now let F be a sheaf on X ct .Then for every A ⊆ X there is a map This map is well defined since [(s A ′ ) A ′ U c ] = 0 implies there is some U c A such that for every A ′ U c the section s A ′ = 0 vanishes.This in particular implies that s A = 0. Now we show ǫ The right side denotes sheaf cohomology on ν(X).
Proof.We first show if G is a flabby sheaf on ν(X) then Ĝ is a flabby sheaf on X ct .For every Thus ker β ⊇ im α.This way we proved ker β = im α, the sequence Ĝ1 → Ĝ2 → Ĝ3 is exact at Ĝ2 .
If F is a reflective sheaf on X ct then there exists a flabby resolution of sheaves on ν(X).Since • is an exact functor we obtain an exact resolution of flabby sheaves with an isomorphism F ν → F .The global section functor on the reduced sequences gives the same result.
Proposition 42.If A is an abelian group and X a proper metric space then A X ν = A ν(X) on ν(X) and Âν(X) = A X on X ct are isomorpic.In particular A X is a reflective sheaf.
Lemma 47.If I is a metric space, ϕ ∈ CX q b (I, A) a cochain and (h t ) t a family of coarse maps I → I with the properties 1. d(z 0 , z 1 ) ≥ d(h t (z 0 ), h t (z 1 )) for every z 0 , z 1 ∈ I, t; 2. d(z 0 , 0) = d(h t (z 0 ), 0) for every z 0 ∈ I, t and some 0 ∈ I; then for every R ≥ 0 there exists S ≥ 0 such that The proof of Theorem 51 can be illustrated by an example.Proposition 48 carries out the essential step of the proof for X = Z ≥0 .Proposition 48.If I 0 := Z ≥0 × Z ≥0 ⊆ Z 2 the projection π : I 0 → Z ≥0 (x, y) → x + y induces an isomorphism in cohomology π * inverse to the induced map ι * associated to the inclusion ι : x → (x, 0).
Proof.In the following proof z i is short for x i y i and z abbreviates x 0 y 0 , . . ., x q y q or x 0 y 0 , . . ., x q−1 y q−1 .
We show the map induces the same map p * in cohomology as the identity on I 0 .

COARSE HOMOTOPY INVARIANCE Elisa Hartmann
For t ∈ N 0 we define an auxilary map We obtain p(x, y) = h y (x, y).The (h t ) t satisfy the conditions of Lemma 46 and Lemma 47. Suppose q ≥ 2. Let ϕ ∈ CX q b (I 0 , A) be a cocyle.Then (h * t − id * I0 )ϕ ∈ im d q−1 since h t , id I0 are close.Thus there is some ψ Then define the map x 0 y 0 , . . ., x q−1 y q−1 .
This map is well-defined since for each fixed point in I q 0 , only finitely many terms in the above sum are defined.If R ≥ 0 then Lemma 47 implies that there exists some S ≥ 0 such that each summand of ψ| ∆ q−1 R has support contained in (∆ S [0]) q .This implies that supp( ψ| ∆ q−1 R ) ⊆ (∆ S [0]) q .Thus ψ has cocontrolled support.Now ψ may or may not be blocky.We have to go the extra step to produce a map with cocontrolled support which is also blocky.To obtain such a map we are going to add a coboundary.
Since I 0 is one-ended a cocycle ϕ ∈ CY 0 (I 0 , A) is represented by a constant a ∈ A function on I 0 except on a bounded set.Then p * ϕ is constant a ∈ A except on a bounded set.Thus p * is the same map as the identity on HY 0 (I 0 , A).Now denote I := R ≥0 × R ≥0 and equip this space with the Manhattan metric.Namely if (s 1 , t 1 ), (s 2 , t 2 ) ∈ I then d((s 1 , t 1 ), (s 2 , t If X is a metric space and x 0 ∈ X a point then the asymptotic product of X and I is defined to be X * I = {(x, i) ∈ X × I | d(x, x 0 ) = d(i, (0, 0))} The paper [Har19b] shows that X * I is the pullback of d(•, x 0 ) and d(•, (0, 0)).Moreover we can define a well-defined homotopy theory: If X is a metric space define maps ι 0 : X → X * I x → (x, (d(x, x 0 ), 0)) ι 1 : X → X * I x → (x, (0, d(x, x 0 ))) Thus ψ is blocky, namely if ϕ ∈ C q A1,...,An (X * I, A) then ψ ∈ C q−1 p −1 Ai∩Aj (X * I, A).This way we have proved that ψ is a cochain.
The proof of Lemma 50 shows that p * induces the identity on HY 0 b (X * I, A).Namely if ϕ ∈ ker dY b 0 then there are only boundedly many x ∈ X such that ϕ has mixed values on {x} * I. Thus ϕ is the same map as (x, (s, t)) → ϕ(x, (s + t, 0)) up to bounded error.
A metric space X is called coarsely contractible if the map d(x 0 , •) is a coarse homotopy equivalence.
Lemma 52.If X is a coarsely contractible metric space then Ȟq ct (X, A) = 0 for every q > 0 and finite abelian group A.
Proof.By definition the map d(x 0 , •) is a coarse homotopy equivalence.Therefore it induces an isomorphism in cohomology by Theorem 51.Since Z ≥0 is acyclic so is X by Theorem 44.

Cohomology of free abelian groups
This chapter proves Theorem D.
Lemma 53.If X is a CAT(0) space then X × R ≥0 is coarsely contractible.
Here ŝ = s s+t and t = t s+t .The map h joins ι • π to id X×Z ≥0 .It remains to show that h is coarse.If R ≥ 0 and d(((x 1 , i 1 ), (s 1 , t 1 )), ((x 2 , i 2 ), (s 2 , t 2 )) ≤ R then (5) The inequalities 4 and 5 add to an inequality d(x, x 0 ) ≤ S.This inequality and |i| ≤ S shows that h is coarsely proper.This way we have showed that h is a coarse map.
Lemma 53 in particular implies that R ≥0 × R n−1 is a coarsely contractible subspace of R n .In fact R i × R ≥0 × R n−1−i and R i × R <0 × R n−1−i are coarsely contractible subspaces of R n and so is every finite intersection of them.Lemma 54.If F is a sheaf on a metric space X and (U i ) i a Leray cover of X, namely a coarse cover such that every finite intersection The right side denotes Čech-cohomology of the cover (U i ) i .
Proof.For sheaves on a topological space there exist a number of proofs for this result.We mimic the proof of [Har77, Theorem III.4.5].Embed F in a flabby sheaf G an take the quotient F 1 .Then there is a short exact sequence of sheaves 0 → F → G → F 1 → 0. (6)

Theorem B .
If two coarse maps α, β : X → Y between metric spaces are coarsely homotopic then they induce the same mapα * , β * : Ȟq ct (Y ; A) → Ȟq ct (X; A)in cohomology with values in a constant sheaf A.
Now Ā is equivalent (as a compactification) to hA and thus also generated by C h (A).Since both C h (X)| A and C h (A) separate points from closed sets they are both contained in C h , the algebra of bounded functions on A which extend to hA [Men95, Proposition 2].The [Men95, Proposition 2] also states that C h ⊆ C * (A) is the smallest unital, closed C * -algebra with this property.Since both C h (X)| A and C h (A) are unital, closed the equality

Proof.
Since ν(X) is compact there only exists one proximity relation on ν(X) which induces the topology on ν(X).Thus the relation δ defined by π 1 δπ 2 if π 1 ∩ π 2 = ∅ and the relation induced by on the quotient coincide.Since both π := cl(U c 1 ) ∩ • • • ∩ cl(U c n ) and cl(A) are closed sets we obtain π ∩ cl(A) = π ∩ cl(A) = ∅.Thus π cl(A).Then there exist U c , B ⊆ X with π ⊆ cl(U c ), cl(A) ⊆ cl(B) and U c B. This in particular implies that U c A. Then cl
The map α is called coarse if α is both coarsely uniform and coarsely proper.Two maps α, β : X → Y between metric spaces are called close if the set α × β(∆ 0 ) is an entourage in Y .The coarse category consists of metric spaces as objects and coarse maps modulo close as morphisms.Isomorphisms in this category are called coarse equivalences.