Coarse geometry and asymptotic dimension

Dissertation

zur Erlangung des Doktorgrades

der Mathematisch-Naturwissenschaftlichen Fakultäten der Georg-August-Universität zu Göttingen

vorgelegt von Bernd Grave

aus Damme

Göttingen 2005

D7 Referent: Prof. Dr. Thomas Schick Korreferent: Prof. Dr. Ulrich Stuhler Tag der mündlichen Prüfung: Montag, 23. Januar 2006

Abstract

We prove that two spaces, whose coarse structures are induced by metrisable compactifications, are coarsely equivalent if and only if their (Higson) coronas are homeomorphic.

We introduce translation C^{∗}-algebras for coarse spaces which admit a countable, uniformly bounded cover using projection-valued measures. This was already done in [Roe03]. Here we give a more complete exposition on the subject including a result about the independence of the translation C^{∗}- algebra from the involved projection-valued measure. Moreover, we eliminate some mistakes, e.g. we have to be careful about null sets when defining the support of an operator.

We introduce some characterisations of asymptotic dimension in the general setting of coarse spaces and prove some basic properties such as monotony, a formula for the asymptotic dimension of finite unions and estimates for the asymptotic dimension of the product of two coarse spaces.

We define coarse cell complexes and prove the obvious conjecture about their asymptotic dimension.

We prove asdim(X, E) = dim(K\X) + 1 if the coarse structure E is induced by a metrisable compactification K of X. As an application, we obtain coarse spaces X and Y such that asdim(X × Y) < asdim(X) + asdim(Y).

For CAT(κ)-spaces with κ < 0 having nicely n-covered spheres we prove that the asymptotic dimension is at most n. We apply this result to prove that asdim(X) = dim(X) for any complete, simply connected Riemannian manifold X with bounded, strictly negative sectional curvature.

Acknowledgements

First of all I wish to thank my advisor, Thomas Schick, for his support, many enlightening discussions and most especially his patience when I was struggling.

The Graduiertenkolleg Gruppen und Geometrie at Göttingen University provided the financial support and mathematical education which benefited me greatly in this endeavor. And I profited as well from visiting the research group in Noncommutative Geometry and Operator Algebras at Vanderbilt University.

Special thanks to Paul Mitchener for inspiring discussions about coarse geometry, his many helpful suggestions and his very special English lessons, in addition to those taught by Marjory Frauts.

I am also grateful to Elias Kappos, Behnam Norouzizadeh and Moritz Wiethaup for their careful read of this dissertation and their detailed feedback.

Last, but not least, I wish to thank all my friends both in the Göttingen math department and without for their kindness and the part they play in making Göttingen such an attractive place in which to work and live.

Contents

1 Some basics in coarse geometry 3 1.1 Finitely generated groups from the metric viewpoint . . . . . 3 1.2 The coarse category . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Some constructions . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Coarse structures induced by metrisable compactifications . . 13

2 Translation C^{∗}-algebras via projection-valued measures 19 2.1 Geometric Hilbert spaces and the calculus of supports …. 19 2.2 Translation C^{∗}-algebras . . . . . . . . . . . . . . . . . . . . . 23 2.3 Induced maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 General theory of asymptotic dimension 33 3.1 Asymptotic dimension of pseudometric spaces . . . . . . . . . 33 3.2 Asymptotic dimension of coarse spaces . . . . . . . . . . . . . 35 3.3 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Asymptotic dimension of coarse cell complexes 47

5 Asymptotic dimension and the covering dimension of the Higson corona 57

6 Asymptotic dimension of CAT(κ)-spaces for κ < 0 63 6.1 Spaces with nicely covered spheres . . . . . . . . . . . . . . . 63 6.2 Proof of Theorem 6.5 . . . . . . . . . . . . . . . . . . . . . . . 64 6.3 Triangulations of bounded distortion . . . . . . . . . . . . . . 67 6.4 Complete, simply connected manifolds with bounded, strictly

negative sectional curvature . . . . . . . . . . . . . . . . . . . 70

Bibliography 73

Index 77

Curriculum vitae 79

Chapter 1

Some basics in coarse geometry

1.1 Finitely generated groups from the metric viewpoint

In this section we give an example of a setting where coarse geometry naturally arises. We define the word metric on each finitely generated group. This metric depends on the chosen generating set, but - as we will see soon - the asymptotic behavior does not. So we can associate a coarse structure to each finitely generated group.

Let G be a group with a finite generating set A.

Definition 1.1 (word metric) The distance d_{(G,A)}(g_{1}, g_{2}) of g_{1}, g_{2} ∈ G in the word metric associated to the pair (G, A) is the length of the shortest word in A representing g_{1}^{−1}g_{2}.

Alternatively we could have used the Cayley graph of (G, A) to define this metric. Compare [BH99] for more details.

Lemma 1.2 If B is another finite generating set of G, there is λ > 0 such that d_{(G,A)} ≤ λ · d_{(G,B)}.

Proof. For each b ∈ B choose a word in A of minimal length representing b. Define λ to be the maximum of the length of these words. Given any word in B representing g ∈ G, we replace each letter by the chosen word in A and count letters. □

Definition 1.3 Let X, Y be pseudometric^{1} spaces and f: X → Y a (not necessarily continuous) map.

1Omitting the condition d(x_{1}, x_{2}) > 0 if x_{1} ≠ x_{2} in the definition of a metric space, we get the definition of a pseudometric space. In a pseudometric space being of distance zero

• f is called coarsely proper if the inverse image of any bounded set is bounded.

• f is called coarsely uniform if for every r > 0 there is s(r) > 0 such that d(f(x_{1}), f(x_{2})) ≤ s(r) for all x_{1}, x_{2} ∈ X with d(x_{1}, x_{2}) ≤ r.

• f is called a coarse map if it is coarsely proper and coarsely uniform.

• Let S be a set. Two maps f, g: S → X are called close if there is D > 0 such that d(f(s), g(s)) < D for all s ∈ S.

• f is called a coarse equivalence if it is a coarse map and if there exists another coarse map g: Y → X such that g ∘ f is close to id_{X} and f ∘ g is close to id_{Y}.

• X and Y are called coarsely equivalent if there exists a coarse equivalence from X to Y.

Compare our coarse maps with asymptotically Lipschitz maps in [Dra00].

Consider a finitely generated group. The metric spaces corresponding to different finite generating sets are obviously coarsely equivalent. Thus, every functor from the category of metric spaces and arbitrary maps which is invariant under coarse equivalence gives an invariant of finitely generated groups. The asymptotic dimension will be an example of such an invariant.

1.2 The coarse category

In this section an axiomatic description of the structure needed to do coarse geometry will be given. Compare [Roe96], [Mit01] and [Roe03].

Definition 1.4 (coarse structure) Let X be a set. A collection E of subsets of X × X is called a coarse structure, and the elements of E will be called entourages, if the following axioms are fulfilled:

(a) A subset of an entourage is an entourage.

(b) A finite union of entourages is an entourage.

(c) The diagonal Δ_{X} := {(x, x) | x ∈ X} is an entourage.

(d) The inverse E^{−1} of an entourage E is an entourage.

E^{−1} := {(y, x) ∈ X × X | (x, y) ∈ E}

is an equivalence relation. Equivalent points are contained in exactly the same open sets.

The map to the quotient can be used to carry over many properties of metric spaces to pseudometric spaces. From the asymptotic point of view pseudometric spaces should be as good as metric spaces.

(e) The composition E_{1}E_{2} of entourages E_{1} and E_{2} is an entourage.

E_{1}E_{2} := {(x, z) ∈ X × X | ∃_{y∈X}(x, y) ∈ E_{1} and (y, z) ∈ E_{2}}

The pair (X, E) is called a coarse space. Sometimes we will say that E ⊆ X × X is controlled if E is an entourage. A coarse space is called connected if every point of X × X is contained in an entourage.

Remark 1.5 For E_{1}, E_{2}, F_{1}, F_{2} ⊆ X × X we have

(E_{1} ∪ E_{2})(F_{1} ∪ F_{2}) = E_{1}F_{1} ∪ E_{1}F_{2} ∪ E_{2}F_{1} ∪ E_{2}F_{2} .

Definition 1.6 (bounded sets) Let (X, E) be a coarse space, A ⊆ X and E ∈ E. We define

E[A] := {x ∈ X | (x, a) ∈ E for some a ∈ A} .

For a point x ∈ X we will write E(x) instead of E[{x}]. Sets of the form E(x) with x ∈ X and E ∈ E are called bounded.

Proposition 1.7 (properties of bounded sets) Let (X, E) be a coarse space.

1. Subsets of bounded sets are bounded.

2. If B ⊆ X is bounded, then B × B ∈ E.

3. If B ⊆ X is bounded and E ∈ E, then E[B] is bounded.

4. Let B_{1}, B_{2} ⊆ X be bounded sets. The following are equivalent.

• B_{1} ∪ B_{2} is bounded.

• B_{1} × B_{2} ∈ E

• There exists an entourage E ∈ E such that E ∩ B_{1}×B_{2} ≠ ∅.

If (X, E) is a connected coarse space, then any finite union of bounded sets is bounded.

Proof. Observe that for entourages E_{1}, E_{2} ∈ E and A ⊆ X we have E_{1}E_{2}[A] = E_{1}[E_{2}[A]]. Therefore E[B] is bounded if E ∈ E and B ⊆ X is bounded. Let B_{1}, B_{2} ⊆ X be bounded sets and let E ∈ E such that E ∩ B_{1}×B_{2} ≠ ∅. Take (b_{1}, b_{2}) ∈ E ∩ B_{1} × B_{2} and observe that b_{1}, b_{2} ∈ (E ∪ Δ_{X})(b_{2}) and B_{1} ∪ B_{2} ⊆ (B_{1}×B_{1} ∪ B_{2}×B_{2})(E ∪ Δ_{X})(b_{2}). Hence B_{1} ∪ B_{2} is bounded. □

Definition 1.8 Let X be a set and M a collection of subsets of X × X. Since any intersection of coarse structures on X is itself a coarse structure, we can make the following definition. By CS(M) we denote the smallest coarse structure containing M, i.e. the intersection of all coarse structures containing M. We call CS(M) the coarse structure generated by M. In the same way, define the connected coarse structure generated by M and denote it by _{cn}CS(M).

Remark 1.9 There is an explicit description of the coarse structure CS(M) generated by M. Set M_{0} := M∪{Δ_{X}} and for n ∈ ℕ define M_{n} inductively as follows.

M_{n+1} := {M_{1} | M_{2} ∈ M_{n}, M_{1} ⊆ M_{2}} ∪ {M^{−1} | M ∈ M_{n}}
∪ {M_{1} ∪ M_{2} | M_{1}, M_{2} ∈ M_{n}} ∪ {M_{1}M_{2} | M_{1}, M_{2} ∈ M_{n}}

Obviously, M_{∞} := ⋃_{n∈ℕ} M_{n} is contained in CS(M) and since M_{∞} is indeed a coarse structure, we get CS(M) = M_{∞}.

Remark 1.10 Let (X, E) be a coarse structure and define E_{cn} := _{cn}CS(E). For E ∈ E_{cn} there are E′ ∈ E, k ∈ ℕ and sets A_{1}, …, A_{k}, B_{1}, …, B_{k} ⊆ X which are bounded with respect to E such that

E = E′ ∪ A_{1}×B_{1} ∪ ⋯ ∪ A_{k}×B_{k} .

Any set which is bounded with respect to E_{cn} is a finite disjoint union of sets which are bounded with respect to E.

There are notions of compatibility of coarse structure and topology.

Definition 1.11 Suppose we are given a topological space X. A coarse structure E on X is said to be compatible with the topology if (1) there is a neighborhood of the diagonal Δ_{X} which is an entourage and (2) the closure of any bounded set is compact.

In [Roe03] and [HR00], the term proper is being used for compatibility of coarse structure and topology. Compatibility of coarse structure and topology in the sense of [Mit01], implies compatibility as defined above.

Example 1.12 (bounded coarse structure) Let (X, d) be a pseudometric space. Set Δ_{r} := {(x, y) ∈ X ×X | d(x, y) < r} and define

E_{d} := CS({Δ_{r} | r > 0}) = {E ⊆ X × X | E ⊆ Δ_{r} for some r > 0}.

It is easy to verify that (X, E_{d}) is a connected coarse space compatible with the topology.

Let G be a finitely generated group. Lemma 1.2 tells us that word metrics arising from different finite generating systems induce the same bounded coarse structure E_{G}. Thus, in a natural manner, every finitely generated group G is a coarse space.

Example 1.13 (continuously controlled coarse structure) Let X be a Hausdorff space and \overline{X} a compactification of X, i.e. X is a dense and open subset of the compact set \overline{X}. The collection

E\overline{X} := \{E ⊆ X × X | \overline{E} ⊆ X × X ∪ Δ\overline{X}\}

of all subsets E ⊆ X×X, whose closure meets the boundary (\overline{X}×\overline{X})\(X×X) only in the diagonal, is a connected coarse structure on X. If \overline{X} is metrisable, the coarse structure E\overline{X} is compatible with the topology. The proof is not hard. Compare [Roe03].

Definition 1.14 (close maps) Let (X, E) be a coarse space and S a set. The maps f: S → X and g: S → X are called close if {(f(s), g(s)) | s ∈ S)} is an entourage.

Compare with Definition 1.3.

Definition 1.15 Let X, Y be coarse spaces and f: X → Y a map.

• We call f coarsely proper if the inverse image of bounded sets is bounded.

• We call f coarsely uniform if the image of each entourage under the map f × f: X × X → Y × Y is an entourage.

• We call f a coarse map if it is coarsely proper and coarsely uniform.

• We call f a coarse embedding if f is coarsely uniform and the inverse image of an entourage under f × f is an entourage.^{2}

Note that a coarse embedding is a coarse map.

We will denote the category of coarse spaces and coarse maps by C. This is the category used in coarse algebraic topology. Compare [Roe96], [Roe03] and [Mit01]. For some constructions it turns out to be more convenient to work in the category of coarse spaces and coarsely uniform maps. We will denote this category by D. When refering only to connected coarse structures, we will write C_{cn} and D_{cn} respectively. Observe that there are some forgetful functors between these categories.

2 In [Roe03], coarse embeddings are also called rough maps.

A difference between the categories C and D is that in D products and direct limits exist, while in C they do not. We will deal with products, direct limits, etc. in Section 1.3.

Definition 1.16 Let X, Y be coarse spaces and f: X → Y a map.

• f is called a D-equivalence if f is coarsly uniform and there exists a coarsely uniform map g: Y → X such that g ∘ f is close to id_{X} and f ∘ g is close to id_{Y}.

• f is called a C-equivalence if f is coarse and there exists a coarse map g: Y → X such that g ∘ f is close to id_{X} and f ∘ g is close to id_{Y}.

Proposition 1.17 Any C-equivalence is also a D-equivalence. Conversely, any D-equivalence f is a coarse embedding and hence a C-equivalence.

Proof. Obviously, any C-equivalence is a D-equivalence. Let E and E′ be the coarse structures for X and Y respectively. We prove that a D-equivalence f: X → Y is a coarse embedding. Let g: Y → X be a coarsely uniform map as in Definition 1.16. Take M′ ∈ E′ and set M := g × g(M′). Define E := {(x, g ∘ f(x)) | x ∈ X} ∈ E. It is sufficient to prove (f × f)^{−1}(M′) ⊆ EME^{−1}. Let (x_{1}, x_{2}) ∈ (f × f)^{−1}(M′). With y_{i} := f(x_{i}) we have (y_{1}, y_{2}) ∈ M′. Set x̃_{i} := g(y_{i}) = g ∘ f(x_{i}). Using (x_{i}, x̃_{i}) ∈ E and (x̃_{1}, x̃_{2}) ∈ M, we get (x_{1}, x_{2}) ∈ EME^{−1}. □

Definition 1.18 (coarse equivalence) We call f: X → Y a coarse equivalence if f is a C-equivalence. A map g: Y → X as in Definition 1.16 is called a coarse inverse of f. We say that X and Y are coarsely equivalent if there exists a coarse equivalence from X to Y.

Proposition 1.19 Let f: X → Y be a map and M a collection of subsets of X × X. If E is a coarse structure on Y and f(M) ∈ E for all M ∈ M, then f: (X, CS(M)) → (Y, E) is coarsely uniform.

Proof. The proposition follows easily from Remark 1.9 and the fact that f ×f(E_{1} E_{2}) ⊆ f ×f(E_{1}) f ×f(E_{2}) for all E_{1}, E_{2} ⊆ X × X. □

Proposition 1.20 Let (X, E_{X}) and (Y, E_{Y}) be coarse spaces and suppose that f_{1} : X → Y and f_{2} : X → Y are close maps. If f_{1} is coarsely proper, coarsely uniform or coarse, the same is true for f_{2}. If f_{1} is a coarse embedding or a coarse equivalence, then so is f_{2}.

Proof. Define M := {(f_{1}(x), f_{2}(x)) | x ∈ X} and note that M ∈ E_{Y}. Observe that f_{2} × f_{2}(E) ⊆ M^{−1}(f_{1} × f_{1}(E))M for E ∈ E_{X}. This implies the statement about coarsely uniform maps.

For a bounded set B ⊆ Y we notice that f_{2}^{−1}(B) is contained in the bounded set f_{1}^{−1}(M[B]). Hence, the claim about coarsely proper maps follows. For E ∈ E_{Y} we have (f_{2} × f_{2})^{−1}(E) ⊆ (f_{1} × f_{1})^{−1}(MEM^{−1}). This implies the claim about coarse embeddings. Suppose that f_{1} is a coarse equivalence and g: Y → X a coarse inverse of f_{1}. Observe that f_{1} ∘g and f_{2} ∘g are close maps. Since g is coarsely uniform, we obtain that g ∘ f_{1} and g ∘ f_{2} are close maps. Since “being close” has the same properties as an equivalence relation, it follows that g is also a coarse inverse of f_{2}. □

1.3 Some constructions in the coarse category

Pull-back of a coarse structure

Let(X, E) be a coarse space, A an arbitrary set and f: A → X any map.

Definition 1.21 We call f^{∗}(E) := CS ({(f × f)^{−1}(E) | E ∈ E}) the pullback of E by f.^{3}

Clearly, f^{∗}(E) is a coarse structure on A such that f is a coarse embedding. If (X, E) is connected, the same is true for (A, f^{∗}(E)). Suppose f is a continuous and proper map between topological spaces. If E is compatible with the topology, the same is true for f^{∗}(E).

Definition 1.22 (restriction of a coarse structure to a subset) If A ⊆ X and f is the inclusion map, we call f^{∗}(E) the restriction of E to A and write E|_{A}.

Unions of coarse spaces

Let I be a set, (X_{i}, E_{i}) a coarse space for every i ∈ I and X = ⋃_{i∈I} X_{i}. Note that the sets X_{i} for i ∈ I are not supposed to be disjoint.

Definition 1.23 We call ⋁_{i∈I} E_{i} := CS (⋃_{i∈I} E_{i}) the minimal coarse structure on the union of (X_{i}, E_{i}).

Remark 1.24 If we are given a coarse structure E on X such that for all i ∈ I its restriction to X_{i} is E_{i}, then ⋁_{i∈I} E_{i} ⊆ E.

3 Since the collection {(f × f)^{−1}(E) | E ∈ E} is not necessarily closed under taking subsets, we need to take the coarse structure generated by this collection. Note that this is the only reason for putting CS here.

Direct limits of coarse spaces

For information about category theory compare [ML98]. Let I be a small category and let F: I → D or F: I → D_{cn} be a functor. In particular, we are given a coarse space (X_{i}, E_{i}) for each i ∈ Obj(I) and a coarsely uniform map F (h): (X_{i}, E_{i}) → (X_{j}, E_{j}) for each h ∈ Mor(i, j).

Proposition 1.25 (existence of direct limits in D and D_{cn}) The categories D and D_{cn} are co-complete, i.e. F has a colimit. More precisely,

\underrightarrow{lim} (X_{i}, E_{i}) = (\underrightarrow{lim}X_{i} , E\underrightarrow{lim})

where \underrightarrow{lim}X_{i} is the corresponding colimit in the category of sets and E\underrightarrow{lim} is the intersection of all (connected) coarse structures such that the maps f_{i}: X_{i} → \underrightarrow{lim}X_{i} are coarsely uniform for all i ∈ Obj(I).

Proof. Observe that E\underrightarrow{lim} = CS (⋃_{i∈Obj(I)} f_{i}(E_{i})). With Proposition 1.19 the claim follows easily from the definitions. □

The analogue of Proposition 1.25 in C and C_{cn} is not true. See Example 1.26. However, coproducts do exist in the categories C and C_{cn}.

Example 1.26 For each i ∈ ℕ take X_{i} := ℝ_{+} with the usual bounded coarse strucure and define f_{ji}: X_{i} → X_{j}, x ↦ max{ 0, x − j + i } for i ≤ j. Then \underrightarrow{lim}X_{i} is just a point.

In some cases we can give a more explicit description for the coarse structure of the direct limit.

Proposition 1.27 Let I be a directed set^{4}. If for all i ∈ I the map f_{i}: X_{i} → \underrightarrow{lim}X_{i} is injective, then

E\underrightarrow{lim} = \{D ∪ f_{j} × f_{j}(L) | j ∈ I, L ∈ E_{j}, D ⊆ Δ_{\underrightarrow{lim}X_{i}}\} .

Proof. For j ∈ I, L ∈ E_{j} and D ⊆ Δ_{\underrightarrow{lim}X_{i}} the set D ∪ f_{j} × f_{j}(L) is in E\underrightarrow{lim}. It remains to prove that E_{c} := \{D ∪f_{j} ×f_{j}(L) | j ∈ I, L ∈ E_{j}, D ⊆ Δ_{\underrightarrow{lim}X_{i}}\} is a coarse structure. In order to conclude that E_{c} is closed under composition of elements, we need the maps f_{i} to be injective. □

Proposition 1.28 Suppose that the identity morphisms are the only morphisms in the category I. In this case the colimit is just the coproduct and we denote the coarse structure of the direct limit by E∐. We get E∐ = {D ∪E_{i_{1}} ∪⋯∪E_{i_{k}} | k ∈ ℕ, i_{1}, …, i_{k} ∈ Obj(I), E_{i_{j}} ∈ E_{i_{j}}, D ⊆ Δ_{∐ X_{i}}}.

4 Compare [Spa95] for the definition of a directed set and the definition of direct limits in this special case.

Infinite disjoint unions of coarse structures

Example 1.29 ℝ and ℤ (both equipped with their usual bounded coarse strucure) are coarsely equivalent, but the corresponding countable coproducts ∐_{ℕ} ℝ and ∐_{ℕ} ℤ are not.

Proof. Any map ∐_{ℕ} ℝ → ∐_{ℕ} ℤ → ∐_{ℕ} ℝ will be different from the identity on any copy of ℝ. Thus, no such map can be close to id_{∐_{ℕ} ℝ}, since any entourage of the coproduct involves only off-diagonal elements of finitely many copies of ℝ. □

Because of the above example one might look for another more geometric coarse structure on the coproduct.

Definition 1.30 (disjoint union of coarse spaces) Let I be a set and assume for any i ∈ I we are given a coarse space (X_{i}, E_{i}). Set X := ⨆_{i∈I} X_{i}. We define

E⨆ := {A ⊆ ⨆_{i∈I} X_{i}^{2} | A ∩ X_{i}^{2} ∈ E_{i}} .

The underlying sets of coproduct and disjoint union are the same. We just take different coarse structures. For any finite set I, coproduct and disjoint union coincide.

Proposition 1.31 If for every i ∈ I the coarse spaces (X_{i}, E_{i}) and (Y_{i}, F_{i}) are coarsely equivalent, then ⨆_{i∈I}(X_{i}, E_{i}) and ⨆_{i∈I}(Y_{i}, F_{i}) are also coarsely equivalent.

Proof. If a coarse equivalence f_{i}: (X_{i}, E_{i}) → (Y_{i}, F_{i}) is given for all i ∈ I, then ⨆_{i∈I} f_{i}: ⨆_{i∈I}(X_{i}, E_{i}) → ⨆_{i∈I}(Y_{i}, F_{i}) is also a coarse equivalence. □

Products of coarse spaces

Definition 1.32 For i ∈ {1, …, k} let (X_{i}, E_{i}) be a coarse space. By p_{i}: X_{1} × ⋯ × X_{k} → X_{i} we denote the projection to the i-th factor. The product coarse structure is defined as follows.

E_{1} ∗ ⋯ ∗ E_{k} := {E ⊆ (X_{1} × ⋯ × X_{k})^{2} | p_{i} × p_{i}(E) ∈ E_{i} for i ∈ {1, …, k}}

If (X, E) is a coarse space, we will sometimes write E^{∗k} for the product coarse structure on X^{k}.

It is easy to prove that E_{1}∗⋯∗E_{k} actually is a coarse structure. The product coarse structure E_{1} ∗ ⋯ ∗ E_{k} is connected if and only if the coarse structures E_{i} are connected. Moreover, we have the following formulas:

Δ_{X_{1}×···×X_{k}} = Δ_{X_{1}} × ⋯ × Δ_{X_{k}}
(E_{1} × ⋯ × E_{k})(E′_{1} × ⋯ × E′_{k}) = E_{1}E′_{1} × ⋯ × E_{k}E′_{k}

One remark on our notation: If E ⊆ X × X, we should not confuse the composition E^{k} = E ⋯ E and the product E^{×k} = E × ⋯ × E.

Remark 1.33 (compatibility with other products)

• Let (X, d_{X}) and (Y, d_{Y}) be metric spaces. Then the formula

d ((x_{1}, y_{1}), (x_{2}, y_{2})) := d_{X}(x_{1}, x_{2}) + d_{Y} (y_{1}, y_{2})

defines a metric on X × Y. It is easy to see that E_{d} = E_{d_{X}} ∗ E_{d_{Y}}.

• Let X and Y be Hausdorff spaces with compactifications \overline{X} and \overline{Y} respectively. Then \overline{X} × \overline{Y} is a compactification of X × Y. By E\overline{X}, E\overline{Y} and E_{\overline{X}×\overline{Y}} we denote the coarse structures on X, Y and X × Y induced by \overline{X}, \overline{Y} and \overline{X} × \overline{Y} respectively. The definitions easily imply E_{\overline{X}×\overline{Y}} = E\overline{X} ∗ E\overline{Y}.

Remark 1.34 (compatibility with coarse equivalence) Let (X, E), (X′, E′) and (Y, F) be coarse spaces and assume (X, E) and (X′, E′) to be coarsely equivalent. Then the spaces (X × Y, E ∗ F) and (X′ × Y, E′ ∗ F) are coarsely equivalent.

Quotients of coarse spaces

Definition 1.35 Let (X, E) be a coarse space. Let ∼ be an equivalence relation on X and π: X → X/∼ the corresponding quotient map. We define a coarse structure on X/∼ by

E_{∼} := CS ({π × π(E) | E ∈ E}) .

If E is connected, E_{∼} is connected. The map π is coarsely uniform by definition of the coarse structure on the quotient. If there is an unbounded equivalence class, then π is not coarsely proper.

Attaching coarse spaces

Let (X, E_{X}), (Y, E_{Y}) be coarse spaces, A ⊆ X and f: A → Y coarsely uniform. Attaching X to Y using f, we get X ∪_{f} Y := X ⊔ Y / ∼ and a quotient map π: X ⊔ Y → X ∪_{f} Y. Here ∼ is the equivalence relation generated by a ∼ f(a) for all a ∈ A. The map π is coarsely uniform, but not necessarily coarsely proper. We denote the coarse structure on the quotient by E_{X∪_{f} Y}.

Lemma 1.36 E_{Y} = E_{X∪_{f} Y} |_{Y} Proof. We certainly have E_{Y} ⊆ E_{X∪_{f} Y} |_{Y} and need to prove the converse. Let L ∈ E_{X∪_{f} Y} |_{Y}. There are L′_{X,1}, …, L′_{X,k} ∈ E_{X} and L_{Y,1}, …, L_{Y,k} ∈ E_{Y}, such that

L ⊆ π × π(L′_{X,1} ∪ L_{Y,1}) ⋯ π × π(L′_{X,k} ∪ L_{Y,k}).

Define L_{X,i} ⋯ L_{X,i+j} := f × f(L′_{X,i} ⋯ L′_{X,i+j}) ∈ E_{Y}. This is a partial replacement of the notation of composition which we will apply only in this proof. We conclude that

L ⊆ ⋃_{(σ_{1},...,σ_{k})∈{X,Y }^{n}} L_{σ_{1},1} ⋯ L_{σ_{k},k} ∈ E_{Y} .

□

Since π|_{X\A} is coarsely uniform, we have E_{X}|_{X\A} ⊆ E_{X∪_{f} Y} |_{X\A}. But in general these coarse structures do not coincide as the following example shows.

Example 1.37 Let Y be a single point, X′ any coarse space such that X′ × X′ is not an entourage, X = X′ × ℝ_{+} and A = X′ × {0}. For r > 0 the set {((x, t), (x̃, t̃)) | t, t̃ ∈ (0, r] , x, x̃ ∈ X′} is contained in E_{X∪_{f} Y} |_{X\A} but not in E_{X}|_{X\A}.

Remark 1.38 If the attaching map f is a coarse embedding, we have E_{X}|_{X\A} = E_{X∪_{f} Y} |_{X\A}.

Proof. The map π|_{X} is a coarse equivalence and π|_{X\A} is a canonical isomorphism. □

1.4 Coarse structures induced by metrisable compactifications

We will briefly recall the definitions of the Higson compactification and the Higson corona. For more detailed explanations see [Roe93], [HR00] or [Roe03]. Let X be a Hausdorff space with a coarse structure E such that the closure of every bounded set is compact. A bounded continuous function f on X is called a Higson function with respect to a coarse structure E if for all entourages E ∈ E the function

df: X × X → ℂ, (x_{1}, x_{2}) ↦ f(x_{1}) − f(x_{2})

restricted to E tends to zero at infinity. The Higson functions form a unital C^{∗}-algebra C_{h}(X). It follows from the Gelfand-Naimark theorem that C_{h}(X) is the algebra of bounded, continuous functions on a compactification hX of X. We call hX the Higson compactification of (X, E). The Higson corona νX is defined to be hX\X. For metrisable compactifications the relation between compactifications and coarse structures is especially simple:

Proposition 1.39 Let X be a Hausdorff space and K a metrisable compactification. The Higson compactification of the continuously controlled coarse structure on X induced by K is homeomorphic to K.

Proof. This is Proposition 2.48 of [Roe03]. □

Let X be a non-compact Hausdorff space and hX a metrisable compactification of X. By E_{hX} denote the continuously controlled coarse structure induced by hX. Choose a metric d on hX and define

X_{i} := \{x ∈ X | d(x, νX) ≥ \frac{1}{i}\}

for i ∈ ℕ (X_{0} := ∅). We have defined a sequence of compact subsets of X such that X_{0} ⊆ X_{1} ⊆ X_{2} ⊆ ⋯ ⊆ X and ⋃_{i∈ℕ} X_{i} = X. Moreover {hX\X_{i}}_{i∈ℕ} is a basis of neighborhoods for νX := hX\X.

Lemma 1.40 For ρ > 0 define Δ_{ρ} := {(x, y) ∈ X^{2} | d(x, y) < ρ}. We assert that E ∈ E_{hX} if and only if there is a sequence {ρ_{i}}_{i∈ℕ} of non-negative real numbers converging to zero such that E\X_{i}^{2} ⊆ Δ_{ρ_{i}} for all i ∈ ℕ.

The proof of this lemma is straightforward. □

Our intention is to construct a coarse structure E on νX × ℕ such that (X, E_{hX}) is coarsely equivalent to (νX × ℕ, E). We may assume that X_{1} ≠ ∅. Choose a map

f: νX × ℕ → X  (1.1)

such that f(\overline{x}, n) ∈ X_{n} has minimal distance to \overline{x} among all points in X_{n}. Since X_{n} is compact, this definition makes sense. Next, we define a coarse structure on νX × ℕ by pulling back the coarse structure E_{hX}. Thus f becomes a coarse embedding.

E := f^{∗}(E_{hX})

By π_{ℕ}: νX ×ℕ → ℕ we denote the projection onto ℕ and by E_{ℕ} the continuously controlled coarse structure on ℕ induced by the one-point compactification. Observe that E_{ℕ} is the maximal coarse structure which is compatible with the topology.

Lemma 1.41 We assert that E ∈ E if and only if the following two conditions are satisfied.

(a) π_{ℕ} × π_{ℕ}(E) ∈ E_{ℕ}

(b) There is a sequence {δ_{k}}_{k∈ℕ} of non-negative real numbers converging to zero such that d(\overline{x}, \overline{y}) < δ_{k} for all ((\overline{x}, i), (\overline{y}, j)) ∈ E with max{i, j} > k.

Proof. Define

a_{i} := max\{d(X_{i}, \overline{x}) | \overline{x} ∈ νX\}

for i ∈ ℕ. The sequence {a_{i}}_{i∈ℕ} is monotone decreasing, since X_{i} ⊆ X_{i+1} for all i ∈ ℕ. We claim that this sequence converges to zero. Assume a_{i} > a > 0 for all i ∈ ℕ. Take the centers of a finite cover of νX with balls of radius \frac{a}{3}. For each of these points there is a point in X within distance less than \frac{a}{3}. These finitely many points in X form a compact set and hence are contained in some X_{k}. We get a_{k} ≤ \frac{2}{3}a, which contradicts our assumption. Hence lim_{i→∞}a_{i} = 0. Define

n_{i} := max \{n ∈ ℕ | a_{i} < \frac{1}{n}\}

for all i with a_{i} < 1. The equation lim_{i→∞}a_{i} = 0 implies lim_{i→∞}n_{i} = ∞.

For M ⊆ ℕ × ℕ we define a completion M^{□ ∠} of M in the following way. Set M^{□ ∠} := {(u, v) ∈ ℕ^{2} | (x, y) ∈ M and (x ≤ u ≤ v ≤ y or y ≤ v ≤ u ≤ x)}. Let E ⊆ (νX × ℕ)^{2} and define

b_{i} := min {n ∈ ℕ | (n, i) ∈ π_{ℕ}×π_{ℕ} (E ∪ E^{−1})^{□ ∠}} .

Suppose that E satisfies conditions (a) and (b). First we will prove that f × f(E) ∈ E_{hX}. Since E_{ℕ} is compatible with the topology, condition (a) implies lim_{i→∞}b_{i} = ∞. We define a sequence {ρ_{i}}_{i∈ℕ}.

ρ_{i} := a_{i} + a_{b_{i}} + δ_{i}

Certainly we have lim_{i→∞}ρ_{i} = 0. Let ((\overline{x}, n), (\overline{y}, m)) ∈ E and define x := f(\overline{x}, n) and y := f(\overline{y}, m). If x ∉ X_{i}, then d(x, \overline{x}) ≤ a_{i}, d(y, \overline{y}) ≤ a_{b_{i}} and d(\overline{x}, \overline{y}) ≤ δ_{i}. Hence f × f(E)\X_{i}^{2} ⊆ Δ_{ρ_{i}}. Now we can apply Lemma 1.40 to conclude f × f(E) ∈ E_{hX}. It follows E ⊆ (f × f)^{−1}(f × f(E)) ∈ E. Suppose E ∈ E. There is an entourage E′ ∈ E_{hX} with E ⊆ (f × f)^{−1}(E′). For all i ∈ ℕ there is m_{i} ∈ ℕ such that a_{m_{i}} < \frac{1}{i}. Since \frac{1}{m_{i}} ≤ a_{m_{i}}, we get m_{i} > i for all i ∈ ℕ. We may assume that the sequence {m_{i}}_{i∈ℕ} is monotone.

Since X_{i} is bounded, E′[X_{i}] is bounded, i.e. E′[X_{i}] is contained in some X_{j}. Note that the image of νX × {0, …, i} under f is contained in X_{i}. Observe that E[νX × {0, …, i}] ⊆ f^{−1} (E′[X_{i}]) ⊆ f^{−1} (X_{j}). It follows that π_{ℕ}(E[νX × {0, …, i}]) ⊆ π_{ℕ}(f^{−1}(X_{j})) ⊆ {0, …, m_{j} − 1}. Together with an analog fact for E^{−1} this implies that E satisfies condition (a). Choose a sequence {ρ_{i}}_{i∈ℕ} converging to zero and such that E′\X_{i}^{2} ⊆ Δ_{ρ_{i}} for all i ∈ ℕ. Let ((\overline{x}, i), (\overline{y}, j)) ∈ E with max\{i, j\} > k. Assume i ≥ k. It follows that j ≥ b_{k}. We already proved π_{ℕ}×π_{ℕ}(E) ∈ E_{ℕ}. Hence lim_{i→∞}b_{i} = ∞.

Define x := f(\overline{x}, i) and y := f(\overline{y}, j). We have d(x, \overline{x}) ≤ a_{i} ≤ a_{k} < \frac{1}{n_{k}}, i.e. x ∉ X_{n_{k}}. Therefore d(x, y) < ρ_{n_{k}}. Note that

δ_{k} := a_{k} + a_{b_{k}} + ρ_{n_{k}} ⟶_{k→∞} 0 .

Obviously E satisfies (b) with {δ_{k}}_{k∈ℕ} as defined above. □

We take a map

g: X → νX × ℕ  (1.2)

such that for x ∈ X_{i}\X_{i−1} the image g(x) = (\overline{x}, i) is chosen in such a way that \overline{x} minimizes the distance to x among all points in νX.

Lemma 1.42 The map g is coarsely uniform.

Proof. Let E ∈ E_{hX}. Since E_{hX} is compatible with the topology, g × g(E) satisfies condition (a) of Lemma 1.41. Let {ρ_{i}}_{i∈ℕ} be a sequence as in Lemma 1.40 for the entourage E and define b_{i} := min{n ∈ ℕ | E[X_{i}\X_{i−1}]∩ X_{n} ≠ ∅\}. This implies lim_{i→∞}b_{i} = ∞. Defining δ_{i} := \frac{1}{i−1} + \frac{1}{b_{i}−1} + ρ_{i−1} we see that g × g(E) satisfies condition (b) of Lemma 1.41. □

Theorem 1.43 Any map f chosen as in (1.1) is a coarse equivalence and any map g chosen as in (1.2) is a coarse inverse of f.

Proof. We will use some of the notation from the proof of Lemma 1.41. We first consider the map g ∘ f: νX × ℕ → νX × ℕ. Let (x, k) ∈ νX × ℕ. Observe that d(f(x, k), x) ≤ d(X_{k}, x) ≤ a_{k} < \frac{1}{n_{k}}. Hence f(x, k) ∈ X_{k}\X_{n_{k}}. Defining (x̃, k̃) := g ∘ f(x, k) we get n_{k} + 1 ≤ k̃ ≤ k. Moreover d(x, x̃) ≤ 2a_{k}. It follows that conditions (a) and (b) of Lemma 1.41 are satisfied for {((x, k), g ∘ f(x, k)) | x ∈ νX × ℕ}. Now we consider the map f ∘g: X → X. Let x ∈ X_{i}\X_{i−1} and (\overline{x}, i) := g(x). Then d(x, \overline{x}) < \frac{1}{i−1}. Thus d(f ∘ g(x), x) ≤ \frac{2}{i−1}, i.e. f ∘ g is close to id_{X}. □

Corollary 1.44 Let X and Y be Hausdorff spaces with metrisable compactifications hX and hY respectively. The coarse spaces (X, E_{hX}) and (Y, E_{hY}) are coarsely equivalent if and only if the coronas νX and νY are homeomorphic.

Proof. To see that the Higson coronas of coarsely equivalent spaces are homeomorphic, we refer to Corollary 2.42 of [Roe03]. □

To every compact Hausdorff space K we assign a coarse space ψ(K) as follows.

ψ(K) := (K × [0, 1) , E_{K×[0,1]})

Corollary 1.45 ψ induces a bijection Ψ between homeomorphism classes of metrisable compact spaces^{5} and coarse equivalence classes of spaces whose coarse structure is induced by a metrisable compactification. The inverse of Ψ is induced by the correspondence ν assigning to each coarse space its Higson corona.

We define a ”product” of two compact spaces A and B as follows.

A ⊠ B := A×B×[0, 1] ≅ A×{1}×B×[0, 1] ∪ A×[0, 1]×B×{1}
= A×[0, 1]×B×[0, 1] \ A×[0, 1)×B×[0, 1)

Here ≅ stands for homeomorphism.

Proposition 1.46 If A and B are metrisable compact spaces, then

Ψ(A ⊠ B) = Ψ(A) × Ψ(B).

Here × denotes the product of coarse spaces as in Definition 1.32.

Proof. Because of Remark 1.34, the right hand side of the equation makes sense. The Higson corona of ψ(A) × ψ(B) = A×[0, 1)×B×[0, 1) is homeomorphic to A ⊠ B. Thus, Corollary 1.45 implies Ψ(A ⊠ B) = Ψ(A) × Ψ(B).

□

Example 1.47 By E_{·} we denote the coarse structure on ℝ_{+} induced by the one-point compactification and by E_{vis} the coarse structure induced on ℝ^{n} by the visual corona S^{n−1}. The usual bounded coarse structure on ℝ^{n} is contained in E_{vis}. Proposition 1.46 implies that (ℝ_{+}, E_{·})^{n} and (ℝ_{+}^{n}, E_{vis}|_{ℝ_{+}^{n}}) are coarsely equivalent.

5 Since every metrisable compact set is homeomorphic to a subset of [0, 1]^{ℕ}, the homeomorphism classes of metrisable compact spaces form a set.

Translation C^{∗}-algebras via projection-valued measures

To any coarse space (X, E) which admits a countable, uniformly bounded cover, we will assign translation C^{∗}-algebras E^{∗}(X, E) and C^{∗}(X, E). This may be considered as an analog of assigning to a locally compact space X the algebra C_{0}(X) of complex valued, continuous function vanishing at infinity. In [HR00], a representation of the algebra C_{0}(X) is used to define translation C^{∗}-algebras for coarse spaces which are at the same time nice topological spaces such that topology and coarse structure are compatible. John Roe proposed in [Roe03] to use projection-valued measures in order to define translation C^{∗}-algebras for more general coarse spaces. In this chapter we give a more complete exposition on the subject and eliminate some mistakes. The idea is to generalize an ad-hoc approach which has been used for defining translation C^{∗}-algebras of discrete coarse spaces. We prove that the translation C^{∗}-algebras are well defined up to isomorphism of C^{∗}-algebras, we compare the different approaches of defining translation C^{∗}-algebras and we discuss the matter whether a coarse map induces maps between the translation C^{∗}-algebras of the corresponding coarse spaces.

2.1 Geometric Hilbert spaces and the calculus of supports

Let (X, E) be a coarse space.

Definition 2.1 A family U of subsets of X is called a localizing decomposition of (X, E), if (1) U is a decomposition, i.e. a cover of X consisting of pairwise disjoint sets, (2) U is countable and (3) U is uniformly bounded, i.e. Δ_{U} := ⋃_{U∈U} U × U ∈ E.

Remark 2.2 Let (X, E) be a coarse space and U a localizing decomposition. We say that x, y ∈ X are U-equivalent if they are contained in the same U ∈ U. We denote the quotient of X with respect to U-equivalence by X/_{U} . Using the quotient map π: X → X/_{U}, we define the coarse structure E/_{U} := { π×π(E) | E ∈ E} on X/_{U} . In this sence, a localizing decomposition U induces a countable “discretization” (X/_{U}, E/_{U}) which is coarsely equivalent to (X, E).

Let 𝔄 be a σ-algebra over X.^{1}

Definition 2.3 A localizing σ-algebra over (X, E) is a σ-algebra 𝔄 together with a topology on X such that for each entourage E ∈ E the closure \overline{E} is still an entourage. Moreover, there has to exist a localizing decomposition U ⊆ 𝔄 such that ⋃_{U∈U} Ů is dense in X. We will call U an 𝔄-localizing decomposition. For each x ∈ X we define 𝔄(x) ⊆ 𝔄 to be all the sets A ∈ 𝔄 whose interior Å contains x. Such an A ∈ 𝔄(x) will be called an 𝔄-neighborhood of x ∈ X.

Example 2.4 Given a coarse space (X, E) which admits a uniformly bounded cover V = {V_{n} | n ∈ ℕ}, there are several localizing σ-algebras:

(1) The σ-algebra σ(V) generated by V together with the discrete topology.

(2) The σ-algebra 𝔓(X) consisting of all subsets of X together with the discrete topology.

(3) If X is a topological space and the coarse structure E is compatible with the topology, then the Borel algebra 𝔅(X), i.e. the smallest σ-algebra containing all open sets, is localizing.

The σ-algebra 𝔄 = {∅, X} is not localizing if X is unbounded.

Proof. Defining U_{n} = V_{n}\(V_{0}∪⋯∪V_{n−1}) we get a localizing decomposition U = {U_{n} | n ∈ ℕ} which is contained in the σ-algebras σ(V) and 𝔓(X). In order to prove (3), note that from V we can construct a countable, uniformly bounded cover consisting of open sets as follows: If E is an open entourage containing Δ_{X}, take the cover {E[V_{n}] | n ∈ ℕ}. From this cover we get a localizing decomposition as before. □

Definition 2.5 Let 𝔄 be a σ-algebra over X and H a Hilbert space. A projection-valued measure is a map λ: 𝔄 → B(H) with the following properties:

1 A σ-algebra 𝔄 over a set X is a family of subsets of X such that (1) X ∈ 𝔄, (2)

X\A ∈ 𝔄 whenever A ∈ 𝔄 and (3) the union of countably many elements of 𝔄 is in 𝔄.

Compare any book on measure theory.

(1) For every A ∈ 𝔄 the operator λ(A) is a projection, i.e. λ(A)^{∗} = λ(A) = λ(A)^{2}.

(2) λ(∅) = 0 and λ(X) = id_{H}.

(3) If A_{n} ∈ 𝔄 for all n ∈ ℕ and A_{i} ∩ A_{j} = ∅ whenever i ≠ j, then

λ ( ⋃_{n∈ℕ}A_{n}) = ∑_{n∈ℕ}λ(A_{n}).

Remark 2.6 Note that the infinite sum in Definition 2.5 has to be considered as a limit with respect to strong convergence.^{2} The sequence {∑_{n=0}^{k} λ(A_{n})}_{k∈ℕ} does in general not converge in the operator norm.

Lemma 2.7 Let λ: 𝔄 → B(H) be a projection-valued measure over X. For A, B ∈ 𝔄 we have λ(A)λ(B) = λ(B)λ(A) = λ(A ∩ B).

Proof. First suppose A ∩ B = ∅. In order to prove λ(A)λ(B) = 0, observe that the images of λ(A) and λ(B) are orthogonal, because λ(A) + λ(B) = λ(A ∪ B) is a projection. The general case now follows easily.

λ(A)λ(B) = (λ(A\B) + λ(A ∩ B))(λ(B\A) + λ(B ∩ A))
= λ(A ∩ B)λ(B ∩ A) = λ(A ∩ B)

The first equality uses additivity of projection-valued measures. The last one requires that a projection P is idempotent, i.e. P^{2} = P. □

Example 2.8 Take the Hilbert space H = L^{2}(X, μ) where μ is a measure on some localizing σ-algebra 𝔄. A projection-valued measure λ: 𝔄 → B(H) is given by

λ(A)f = χ_{A} · f for A ∈ 𝔄 and f ∈ H.

Here χ_{A}: X → ℂ is the characteristic function of A, i.e. χ_{A}(x) = 1 if x ∈ A and χ(x) = 0 if x ∈ X\A.

Example 2.9 Let (X, E) be a coarse space and U = {U_{i} | i ∈ ℕ} a localizing decomposition of X. Define 𝔄 := σ(U) to be the σ-algebra generated by U. Assume we are given a Hilbert space H_{i} for each i ∈ ℕ such that H_{i} = {0} if U_{i} = ∅. In this situation we can define a projection-valued measure λ: 𝔄 → 𝔅 (⨁_{i∈ℕ} H_{i}) as follows: For A ∈ 𝔄 define

λ(A) = orthogonal projection onto ⨁_{\begin{matrix}i∈ℕ \\ U_{i}⊆A \end{matrix}} H_{i} .

2 A sequence {T_{i}}_{i∈ℕ} in B(H) converges strongly to T ∈ B(H), if for any v ∈ H the sequence {T_{i}(v)}_{i∈ℕ} converges to T (v) in H.

Definition 2.10 Let (X, E) be a coarse space. A geometric Hilbert space over (X, E) is a projection-valued measure λ: 𝔄 → B(H) where 𝔄 is a localizing σ-algebra over (X, E) and H a separable Hilbert space.

Remark 2.11 The projection-valued measure in Example 2.8 is a geometric Hilbert space if and only if L^{2}(X, μ) is separable. The projection-valued measure in Example 2.9 is a geometric Hilbert space if and only if the Hilbert space H_{i} is separable for all i ∈ ℕ.

Definition 2.12 Let λ: 𝔄 → B(H) be a geometric Hilbert space over (X, E). We define the support of u ∈ H and T ∈ B(H) as follows:

Supp(u) = {x ∈ X | λ(A)u ≠ 0 for all A ∈ 𝔄(x)}
Supp(T) = {(x_{1}, x_{2}) ∈ X ×X | λ(A_{1})T λ(A_{2}) ≠ 0 for all A_{i} ∈ 𝔄(x_{i})}

Remark 2.13 Assume λ(A) = 0 and x ∈ Å. Then x ∉ Supp(u) for all u ∈ H. Similarly (x, y), (y, x) ∉ Supp(T) for T ∈ B(H) and y ∈ X. In particular, consider the case where {x} ∈ 𝔄 and λ({x}) = 0 for all x ∈ X. If we consider the discrete topology on X or equivalently if we set 𝔄(x) := {A ∈ 𝔄 | x ∈ A}, then Supp(u) = ∅ for all u ∈ H and Supp(T) = ∅ for all T ∈ B(H). This demonstrates why we introduced 𝔄-neighborhoods and why we do not use the definition of support given in [Roe03].

In order to get a calculus of supports as in Proposition 2.18, we need λ to fulfill an additional condition.

Remark 2.14 Let λ: 𝔄 → B(H) be any geometric Hilbert space. For A, A_{1}, A_{2} ∈ 𝔄, u ∈ H and T ∈ B(H) the following is always true.

λ(A)u = 0 ⟹ Å∩ Supp(u) = ∅
λ(A_{1}) T λ(A_{2}) = 0 ⟹ Å_{1}×Å_{2} ∩ Supp(T) = ∅

Definition 2.15 We say that a geometric Hilbert space leads to computable supports if for A, A_{1}, A_{2} ∈ 𝔄, u ∈ H and T ∈ B(H) the following is true.

A ∩ Supp(u) = ∅ ⟹ λ(A)u = 0  (2.1)
A_{1}×A_{2} ∩ Supp(T) = ∅ ⟹ λ(A_{1}) T λ(A_{2}) = 0  (2.2)

Remark 2.16 In the situation of Example 2.9, supports are computable (provided that the Hilbert spaces H_{i} are separable).

Lemma 2.17 Let λ: 𝔄 → B(H) be a geometric Hilbert space over (X, E). Assume that X is σ-compact.^{3} Then the geometric Hilbert space λ leads to computable supports.

3 A topological space is σ-compact if every open cover of X has a countable subcover.

Proof. Assume A ∈ 𝔄 and A ∩ Supp(u) = ∅. For every x ∈ A there is B_{x} ∈ 𝔄(x) such that λ(B_{x})u = 0. Set C_{x} = B_{x} ∩ A. Using Lemma 2.7, we get

λ(C_{x})u = λ(C_{x}) \underbrace{λ(B_{x})u}_{=0} = 0 .

Take a countable subcover C′ ⊆ C := {C_{x} | x ∈ A} of A. Using σ-additivity of λ, we get λ(A)u = 0. The proof of the corresponding claim about the support of T is similar. □

The assumption of σ-compactness of X is necessary in Lemma 2.17. Compare Remark 2.13.

Proposition 2.18 Let λ: 𝔄 → B(H) be a geometric Hilbert space over (X, E) which leads to computable supports. Let u, v ∈ H and T, S ∈ B(H). Take Δ_{U} = ⋃_{U∈U} U × U ∈ E for some localizing decomposition U. We get the following calculus of supports:

Supp(u + v) ⊆ Supp(u) ∪ Supp(v)  (2.3)
Supp(S + T) ⊆ Supp(S) ∪ Supp(T)  (2.4)
Supp(T u) ⊆ Δ_{U} Supp(T)Δ_{U}[Supp(u)]  (2.5)
Supp(ST) ⊆ Δ_{U} Supp(S)Δ_{U} Supp(T)Δ_{U}  (2.6)
Supp(T^{∗}) = Supp(T)^{−1}  (2.7)

Proof. The first two inclusions (2.3) and (2.4) follow directly from the definitions. The same is true for (2.7). In order to prove (2.5), suppose x ∉ Δ_{U} Supp(T)Δ_{U} Supp(u) and choose U_{x} ∈ U such that x ∈ U_{x}. For every U ∈ U\{U_{x}} we have

U ∩ Supp(u) = ∅  or  U_{x} × U ∩ Supp(T) = ∅.

Hence λ(U_{x})T u = ∑_{U∈U} λ(U_{x})T λ(U)u = 0, i.e. x ∉ Supp(T u). The prove of (2.6) is similar. Suppose (x, y) ∉ Δ_{U} Supp(S)Δ_{U} Supp(T)Δ_{U} and choose U_{x}, U_{y} ∈ U containing x and y respectively. For every U ∈ U we have

U_{x} × U ∩ Supp(S) = ∅  or  U × U_{y} ∩ Supp(T) = ∅.

Therefore λ(U_{x})ST λ(U_{y}) = ∑_{U∈U} λ(U_{x})Sλ(U)T λ(U_{y}) = 0 and it follows that (x, y) ∉ Supp(ST). □

2.2 Translation C^{∗}-algebras

In the following let λ: 𝔄 → B(H) be a geometric Hilbert space over (X, E) with computable supports. From Proposition 2.18 we get the following corollary.

Corollary 2.19 The bounded operators of controlled support on H, i.e. the bounded operators on H whose support is an entourage, form a ∗-algebra.

Definition 2.20 We denote the closure of the ∗-algebra of operators with controlled support in the operator norm by E_{λ}^{∗}(X, E). We may call it the big translation C^{∗}-algebra.

Remark 2.21 The closure of the ∗-algebra of operators with controlled support with respect to strong convergence is the full algebra of bounded operators B(H).

Proof. Let {U_{i}}_{i∈ℕ} be an 𝔄-localizing decomposition of (X, E) and set V_{i} := U_{0}∪⋯∪U_{i}. Observe that for T ∈ B(H) the sequence {λ(V_{i})T λ(V_{i})}_{i∈ℕ} converges strongly to T. □

Definition 2.22 An operator T ∈ B(H) is pseudolocal if T λ(B) − λ(B)T is a compact operator for every bounded set B ∈ 𝔄.

Lemma 2.23 The pseudolocal operators with controlled support form a ∗-algebra.

Proof. The locally compact operators are closed under addition, scalar multiplication and adjoints, i.e. if S and T are pseudolocal operators and z ∈ ℂ, then S + T, z · T and T^{∗} are also pseudolocal. The composition of pseudolocal operators is pseudolocal, as the following computation shows: If S and T are pseudolocal operators and B ∈ A is a bounded set, then

ST λ(B) − λ(B)ST = (Sλ(B) − λ(B)S)T + compact operator

is compact. □

Definition 2.24 By D_{λ}^{∗}(X, E) we denote the closure of the ∗-algebra of pseudolocal operators with controlled support in the operator norm.

Definition 2.25 An operator T ∈ B(H) is locally compact if T λ(B) and λ(B)T are compact operators for every bounded set B ∈ 𝔄.

Lemma 2.26 The locally compact operators with controlled support form a ∗-algebra.

Proof. The locally compact operators are closed under addition, scalar multiplication and adjoints. The composition of locally compact operators is locally compact, since composing a bounded operator and a compact operator gives a compact operator. □

Definition 2.27 The small translation C^{∗}-algebra of (X, E) with respect to λ is the closure of the ∗-algebra of locally compact operators with controlled support in the operator norm. We denote it by C_{λ}^{∗}(X, E). This algebra is often called Roe-C^{∗}-algebra.

Remark 2.28 Observe that every locally compact operator is pseudolocal. Hence C_{λ}^{∗}(X, E) ⊆ D_{λ}^{∗}(X, E) ⊆ E_{λ}^{∗}(X, E).

Lemma 2.29 The small translation C^{∗}-algebra C_{λ}^{∗}(X, E) is an ideal^{4} in the big translation C^{∗}-algebra E_{λ}^{∗}(X, E). The small translation C^{∗}-algebra C_{λ}^{∗}(X, E) is also an ideal in D_{λ}^{∗}(X, E).

Proof. Let S ∈ C_{λ}^{∗}(X, E) and T ∈ E_{λ}^{∗}(X, E). For a bounded set B ∈ 𝔄 the operator λ(B)S is compact. Therefore λ(B)ST is compact. It remains to prove that ST λ(B_{2}) is compact for any bounded set B_{2} ∈ 𝔄. Let U be an 𝔄-localizing decomposition of (X, E) and observe that B_{1} := Δ_{U} Supp(T)[B_{2}] ∈ 𝔄 is bounded and (X\B_{1}) × B_{2} ∩ Supp(T) = ∅. Using (2.2), we see that

S T λ(B_{2}) = \underbrace{Sλ(B_{1})}_{compact} T λ(B_{2}) + S \underbrace{λ(X\B_{1}) T λ(B_{2})}_{=0}

is compact. □

Example 2.30 If (X, E) is a coarse space such that X is bounded, then E^{∗}(X, E) is the entire algebra of bounded operators on H and C^{∗}(X, E) is the algebra of compact operators. Moreover, D_{λ}^{∗}(X, E) = {T ∈ B(H) | [T, λ(A)] is compact for all A ∈ 𝔄}.

Definition 2.31 A geometric Hilbert space λ: 𝔄 → B(H) over (X, E) is called ample if there is an 𝔄-localizing decomposition U ⊆ 𝔄 such that λ(U) is not compact for U ∈ U. Note that for a projection being compact and having finite dimensional image is the same.

Example 2.32 Let (X, E) be a coarse space and x_{0} ∈ X. The geometric Hilbert space λ: 𝔄 → B(H) defined by

λ(A) = \{ \begin{matrix}0 & if x_{0} ∉ A \\ id & if x_{0} ∈ A \end{matrix}

is not ample if X is unbounded.

4In the context of C^{∗}-algebras, by an ideal we mean a closed, two-sided ideal which is closed unter taking adjoints.

Theorem 2.33 Let λ: 𝔄 → B(H_{1}) and μ: 𝔅 → B(H_{2}) be ample, geometric Hilbert spaces over (X, E) with computable supports. Then

C_{λ}^{∗}(X, E) ≅ C_{μ}^{∗}(X, E)  and  E_{λ}^{∗}(X, E) ≅ E_{μ}^{∗}(X, E).

The following two lemmas yield a proof of this theorem.

Lemma 2.34 Let λ: 𝔄 → B(H) be a geometric Hilbert space over (X, E) with computable supports and U an 𝔄-localizing decomposition. By λ′ we denote the restriction of λ to the σ-algebra σ(U) generated by U. Then λ′ together with the discrete topology on X is another geometric Hilbert space and we have C_{λ}^{∗}(X, E) = C_{λ′}^{∗}(X, E) and E_{λ}^{∗}(X, E) = E_{λ′}^{∗}(X, E).

Proof. Whenever in this proof we make reference to a topology we mean the topology coming with 𝔄. We first claim

Supp_{λ}(T) ⊆ \overline{Supp_{λ′}(T)} for T ∈ B(H).

This implies E_{λ′}^{∗}(X, E) ⊆ E_{λ}^{∗}(X, E). To prove the claim, set Y := ⋃_{U∈U} Ů. If x ∈ Y, then σ(U)(x) = {U ∈ U | x ∈ U} ⊆ 𝔄(x). Therefore we get Supp_{λ}(T) ∩ Y × Y ⊆ Supp_{λ′}(T) ∩ Y × Y. For x ∈ X set U_{x} := \{U ∈ U | x ∈ \overline{U}\} and observe that ⋃_{U∈U_{x}} U is an 𝔄neighborhood of x. Thus, for (x, y) ∈ Supp_{λ}(T) there is U ∈ U_{x} and V ∈ U_{y} such that λ(U)T λ(V) ≠ 0. Take sequences {x_{i}}_{i∈ℕ} ⊆ U and {y_{i}}_{i∈ℕ} ⊆ V converging to x and y respectively. Note that (x_{i}, y_{i}) ∈ Supp_{λ′}(T) for all i ∈ ℕ and hence (x, y) ∈ \overline{Supp_{λ′}(T)}. Set Δ_{U} = ⋃_{U∈U} U × U. We claim

Supp_{λ′}(T) ⊆ Δ_{U} Supp_{λ}(T)Δ_{U} for all T ∈ B(H)

This claim implies E_{λ}^{∗}(X, E) ⊆ E_{λ′}^{∗}(X, E). To prove the claim, suppose U, V ∈ U and U × V ∩ Supp_{λ}(T) = ∅. Since supports are computable, we get λ(U)T λ(V) = 0. Therefore U × V ∩ Supp_{λ′}(T) = ∅. So far we proved E_{λ}^{∗}(X, E) = E_{λ′}^{∗}(X, E). In order to prove C_{λ}^{∗}(X, E) ⊆ C_{λ′}^{∗}(X, E), just note that each bounded set in σ(U) is also a bounded set in 𝔄. Each bounded set B ∈ 𝔄 is contained in B̃ := Δ_{U}[B] which is a bounded set in σ(U). Let T be locally compact with respect to σ(U). Using Lemma 2.7, we see that λ(B)T = λ(B)λ(B̃)T is compact, since λ(B̃)T is compact. □

Lemma 2.35 Given localizing decompositions U and V of the coarse space (X, E) let λ: σ(U) → B(H_{1}) and μ: σ(V) → B(H_{2}) be ample, geometric Hilbert spaces over (X, E). Then there is an isometric isomorphism ρ: H_{1} → H_{2} such that ad_{ρ}: B(H_{1}) → B(H_{2}), T ↦ ρT ρ^{−1} restricts to isomorphisms

E_{λ}^{∗}(X, E) ⟶^{≅} E_{μ}^{∗}(X, E) and C_{λ}^{∗}(X, E) ⟶^{≅} C_{μ}^{∗}(X, E).

Proof. Defining W = {U ∩ V | U ∈ U, V ∈ V} \ {∅} we get a localizing decomposition refining U and V. Set W_{U} = {W ∈ W | W ⊆ U} for U ∈ U and W_{V} = {W ∈ W | W ⊆ V } for V ∈ V. We will write H_{U} for the image of λ(U) and H_{V} for the image of μ(V). For every U ∈ U we choose a decomposition H_{U} = ⨁_{W ∈W_{U}} H_{1,W} such that H_{1,W} is infinite dimensional for all W ∈ W. Similarly, we choose a decomposition H_{V} = ⨁_{W ∈W_{V}} H_{2,W} for each V ∈ V. Now we extend the geometric Hilbert spaces λ and μ to λ̃: σ(W) → B(H_{1}) and μ̃: σ(W) → B(H_{2}) respectively. We define λ̃(W) to be the projection onto H_{1,W}. Similarly μ̃(W) will be the projection onto H_{2,W}. Remember that λ, λ̃, μ and μ̃ have computable supports. We choose an isometric isomorphism ρ: H_{1} → H_{2} such that the image of ρ|_{H_{1,W}} is H_{2,W} for all W ∈ W. Then ad_{ρ} is an isomorphism of C^{∗}-algebras. Using ρλ̃ = μ̃ρ, we see that Suppλ̃(T) = Suppμ̃(ρT ρ^{−1}) for all T ∈ B(H_{1}). Hence ad_{ρ} restricts to an isomorphism E∗ ; λ̃(X, E) ⟶^{≅} E∗ ; μ̃(X, E). Moreover, we note that for B ∈ σ(W) and T ∈ B(H_{1}) the operator λ̃(B)T is compact if and only if μ̃(B)ad_{ρ}(T) is compact. Therefore ad_{ρ} also restricts to an isomorphism C∗ ; λ̃(X, E) ⟶^{≅} C∗ ; μ̃(X, E). Applying Lemma 2.34 now completes the proof. □

There are some choices involved in order to get the isometric isomorphism ρ: H_{1} → H_{2} in the proof of Lemma 2.35. Therefore, the isomorphisms E_{λ}^{∗}(X, E) ⟶^{≅} E_{μ}^{∗}(X, E) and C_{λ}^{∗}(X, E) ⟶^{≅} C_{μ}^{∗}(X, E) are not uniquely determined and without further specifications there is no natural choice.

We compare our definition of small translation C^{∗}-algebra with the corresponding definition in Section 6.3 of [HR00].

Proposition 2.36 Let X be a locally compact, separable and metrizable space with a coarse structure E which is compatible with the topology. Suppose that

ρ: C_{0}(X) → B(H)

is a non-degenerate^{5} and ample^{6} representation of the C^{∗}-algebra C_{0}(X) on the separable Hilbert space H. There is an ample geometric Hilbert space λ over (X, E) with computable supports and for any such λ

C_{ρ}^{∗}(X, E) ≅ C_{λ}^{∗}(X, E) .

Proof. According to Theorem 2.33 we are free to replace λ by any ample geometric Hilbert space with computable supports. We will construct λ from ρ.

5 A C^{∗}-representation ρ of a C^{∗}-algebra A on a Hilbert space H is non-degenerate if {ρ(a)(v) | a ∈ A, v ∈ H} is dense in H.

6 A C^{∗}-representation ρ is called ample, if the operator ρ(f) is not compact if f ≠ 0.

Using Borel functional calculus (compare Remark 1.5.7 of [HR00]), we see that ρ extends to a representation ρ̃ of the C^{∗}-algebra B(X) of bounded Borel functions on X. It is easy to check that

λ: 𝔅(X) → B(H), A ↦ ρ̃(χ_{A})

is a projection-valued measure. Since X is separable and its coarse structure is compatible with the topology, there exists a countable, uniformly bounded, open cover of X. We may even assume that each U ∈ U has non-empty interior. (Compare Claim 6.3.14 of [HR00].) Hence λ is a geometric Hilbert space. Let U ∈ U and f ∈ C_{0}(U), f ≠ 0. Since ρ is ample, λ(U)ρ(f) = ρ(f) is not compact. Hence λ(U) is not compact and λ is ample. Applying Lemma 2.17 we see that λ leads to computable supports. Now that we have a convenient geometric Hilbert space, we compare the support of a bounded operator T with respect to λ (as defined in Definition 2.12) and the support with respect to ρ (as defined in [HR00]). Observe that

Supp_{ρ}(T) = \{(x_{1}, x_{2}) ∈ X^{2} | ∀\begin{matrix}U_{1}∈𝔅(X)(x_{1}) \\ U_{2}∈𝔅(X)(x_{2}) \end{matrix}∃\begin{matrix}f_{1}∈C_{0}(Ů_{1}) \\ f_{2}∈C_{0}(Ů_{2}) \end{matrix}ρ(f_{1})T ρ(f_{2}) ≠ 0\} .

Let x_{1}, x_{2} ∈ X and U_{i} ∈ 𝔅(X)(x_{i}). Suppose there are f_{i} ∈ C_{0}(Ů_{i}) such that ρ(f_{1})T ρ(f_{2}) ≠ 0. Observe the following relations between ρ and λ.

ρ(f_{1})λ(U_{1}) = ρ(f_{1})ρ̃(χ_{U_{1}}) = ρ̃(f_{1} · χ_{U_{1}}) = ρ(f_{1})
λ(U_{2})ρ(f_{2}) = ρ̃(χ_{U_{2}})ρ(f_{2}) = ρ̃(χ_{U_{2}} · f_{2}) = ρ(f_{2})

It follows

ρ(f_{1})λ(U_{1})T λ(U_{2})ρ(f_{2}) = ρ(f_{1})T ρ(f_{2}) ≠ 0

and hence λ(U_{1})T λ(U_{2}) ≠ 0. This proves Supp_{ρ}(T) ⊆ Supp_{λ}(T). Choose a metric d which induces the given topology on X. For U ⊆ X and k ∈ ℕ define

f_{U,k}: X → ℝ, x ↦ max{ 0, 1 − k · d(x, U) } .

Let (x_{1}, x_{2}) ∈ Supp_{λ}(T) and U_{i} ∈ 𝔅(X)(x_{i}). Since (X, d) is a metric space, there is k_{i} ∈ ℕ and V_{i} ∈ 𝔅(X)(x_{i}) such that f_{i} := f_{V_{i},k_{i}} ∈ C_{0}(Ů_{i}). We have the following relations between ρ and λ.

λ(V_{1})ρ(f_{1}) = ρ̃(χ_{V_{1}})ρ(f_{1}) = ρ̃(χ_{V_{1}} · f_{1}) = λ(V_{1})
ρ(f_{2})λ(V_{2}) = ρ(f_{2})ρ̃(χ_{V_{2}}) = ρ̃(f_{2} · χ_{V_{2}}) = λ(V_{2})

These relations imply

λ(V_{1})ρ(f_{1})T ρ(f_{2})λ(V_{2}) = λ(V_{1})T λ(V_{2}) ≠ 0

and hence ρ(f_{1})T ρ(f_{2}) ≠ 0. It follows Supp_{λ}(T) ⊆ Supp_{ρ}(T). We assume that T ∈ C_{ρ}^{∗}(X, E). Note that f_{A} := f_{A,1} ∈ C_{0}(X) for a bounded set A ⊆ X. Since ρ(f_{A})T is compact, we conclude that λ(\overline{A})T = λ(\overline{A})ρ(f_{A})T is compact. In the same way we get compactness for T λ(\overline{A}). This proves T ∈ C_{λ}^{∗}(X, E). Now let T ∈ B(H) be locally compact with respect to λ. If f ∈ C_{0}(X), then M_{f,n} := \{x ∈ X | |f(x)| ≥ \frac{1}{n}\} is compact for all n ∈ ℕ and thus bounded. Continuity of the representation ρ̃ yields

\xymatrix{ρ(f)λ(M_{f,n})T \ar[r]_{n→∞}^{∥·∥} & ρ(f)T} .

Since all operators on the left hand side are compact, ρ(f)T is compact. In the same way we get compactness for T ρ(f). □

Corollary 2.37 Up to isomorphism, the C^{∗}-algebra C_{ρ}(X) does not depend on the choice of the representation ρ (as long as ρ is an ample and non-degenerate representation on a separable Hilbert space).

2.3 Induced maps

Let (X, E_{X}) and (Y, E_{Y}) be coarse spaces which admit localizing decompositions and let f: (X, E_{X}) → (Y, E_{Y}) be a coarse map. We would like to obtain induced maps E^{∗}(f) and C^{∗}(f) on the translation C^{∗}-algebras such that the following diagramm commutes.

\xymatrix{C^{∗}(X, E_{X}) \ar[r]^{C^{∗}(f)} \ar@{^{(}->}[d] & C^{∗}(Y, E_{Y}) \ar@{^{(}->}[d] \\ E^{∗}(X, E_{X}) \ar[r]^{E^{∗}(f)} & E^{∗}(Y, E_{Y})}

Question 2.38 Let f: (X, E_{X}) → (Y, E_{Y}) be a coarse map and 𝔄 a localizing σ-algebra with respect to E_{X}. Is there a way of defining an induced localizing σ-algebra f_{∗}(𝔄)?

An attempt to define an induced localizing σ-ring^{7} was made in Remark 4.26 of [Roe03]. Roe defines

f_{∗}(𝔄) := {S ⊆ f(X) ⊆ Y | f^{−1}(S) ∈ 𝔄} .

Now f_{∗}(𝔄) is indeed a σ-ring, but it does not have to be localizing as the following example tells us.

7 A σ-ring ℜ over a set X is a family of subsets of X such that (1) ∅ ∈ ℜ, (2) A\B ∈ ℜ whenever A, B ∈ ℜ and (3) the union of countably many elements of ℜ is in ℜ.

Example 2.39 Take (X, E_{X}) = (Y, E_{Y}) = (ℝ, E_{eucl}) and the σ-algebra 𝔄 generated by the set of half open intervals {[n, n+1[ | n ∈ ℤ}). For the coarse map f: X → Y

f(x) = \{ \begin{matrix}n & if x ∈ [n − \frac{1}{4}, n + \frac{1}{4}] for some n ∈ ℤ \\ 2 · x − n − \frac{1}{2} & if x ∈ [n + \frac{1}{4}, n + \frac{3}{4}] for some n ∈ ℤ \end{matrix}

we get f_{∗}(𝔄) = {∅, ℝ} which is not localizing (in the sense of Definition 4.20 in [Roe03]), since ℝ is unbounded. Roe’s definition of f_{∗}(𝔄) does give a “localizing σ-ring” if the coarse map f “respects an 𝔄-localizing decomposition”, i.e. if f has the following additional property: There is an 𝔄-localizing decomposition U of X such that U_{1}, U_{2} ∈ U and U_{1} ≠ U_{2} implies f(U_{1}) ∩ f(U_{2}) = ∅. This is the case for example if f is injective.

We start constructing geometric Hilbert spaces λ_{X} and λ_{Y} such that f does induce maps between the corresponding translation C^{∗}-algebras.

Lemma 2.40 There are geometric Hilbert spaces λ_{X}: σ(U_{X}) → B(H_{X}) and λ_{Y} : σ(U_{Y}) → B(H_{Y}), a map F: U_{X} → U_{Y} and a partial isometry φ: H_{X} → H_{Y} such that

• U_{X} and U_{Y} are localizing decompositions of X and Y respectively,

• λ_{X} and λ_{Y} are ample and lead to computable supports,

• the map F: U_{X} → U_{Y} satisfies F (U) ⊆ f(U) for all U ∈ U_{X},

• the map F: U_{X} → U_{Y} is almost injective, i.e. F (U) = F (U′) implies U = U′ or F (U) = ∅,

• the partial isometry satisfies φ(λ_{X}(U)(H_{X})) ⊆ λ_{Y} (F (U))(H_{Y}) for all U ∈ U_{X}.

Proof. Take any localizing decomposition U_{X} of (X, E_{X}). We enumerate the elements of U_{X}, denote them by U_{0}, U_{1}, U_{2}, … and set W_{i} := f(U_{i}). For i ∈ ℕ define V_{i} = W_{i}\ ⋃_{j<i} W_{j} and note that V := {V_{0}, V_{1}, V_{2}, …} is a localizing decomposition of im(f). The empty set might be an element of V. For i ∈ ℕ define F (U_{i}) := V_{i}. Let Ũ_{Y} be a localizing decomposition of Y and define

U_{Y} := V ∪ {U\ im(f) | U ∈ Ũ_{Y} } .

Note that U_{Y} is itself a localizing decomposition of Y. Choose a separable Hilbert space H_{U} for each U ∈ U_{X} with U ≠ ∅ and similarly choose H_{V} for each V ∈ U_{Y} with V ≠ ∅. Define H_{∅} = {0} if ∅ ∈ U_{X} or ∅ ∈ U_{Y}. Define H_{X} = ⨁_{U∈U_{X}} H_{U} and H_{Y} = ⨁_{V ∈U_{Y}} H_{V}. Choose a partial isometry φ: H_{X} → H_{Y} such that φ (H_{U}) ⊆ H_{F (U)} for all U ∈ U_{X}. We consider the geometric Hilbert space λ_{X}: σ(U_{X}) → B (H_{X}) which is determined by defining λ_{X}(U) to be the projection onto H_{U} for any U ∈ U_{X}. Likewise we define λ_{Y} : σ(U_{Y}) → B (H_{Y}). Note that λ_{X} and λ_{Y} are ample if H_{U} and H_{V} are infinite dimensional for all U ∈ U_{X} and V ∈ U_{Y}. Moreover λ_{X} and λ_{Y} lead to computable supports. (Compare Remark 2.16.) □

Lemma 2.41 In the situation of Lemma 2.40, the partial isometry φ induces the map

ad_{φ}: B(H_{X}) → B(H_{Y}), T ↦ φT φ^{∗}

and ad_{φ} restricts to maps between the translation C^{∗}-algebras.

\xymatrix{E_{λ_{X}}^{∗} (X, E_{X}) \ar[r]^{ad_{φ}} & E_{λ_{Y}}^{∗} (Y, E_{Y})}  (2.8)
\xymatrix{C_{λ_{X}}^{∗} (X, E_{X}) \ar[r]^{ad_{φ}} & C_{λ_{Y}}^{∗} (Y, E_{Y})}  (2.9)

Proof. Let T ∈ B(H_{X}). We will prove

Supp_{λ_{Y}} (ad_{φ}(T)) ⊆ f × f (Supp_{λ_{X}} (T)) .

This implies ad_{φ} (E_{λ_{X}}^{∗} (X, E_{X})) ⊆ E_{λ_{Y}}^{∗} (Y, E_{Y}). Let U_{1}, U_{2} ∈ U_{X} such that U_{1} × U_{2} ⊈ Supp_{λ_{X}} (T). The definition of λ_{X} implies U_{1} × U_{2} ∩ Supp_{λ_{X}} (T) = ∅ and hence λ_{X}(U_{1})T λ_{X}(U_{2}) = 0, since supports are computable. Observe that

λ_{Y} (F (U_{1})) ad_{φ}(T) λ_{Y} (F (U_{2})) = λ_{Y} (F (U_{1})) φ T φ^{∗} λ_{Y} (F (U_{2}))

= λ_{Y} (F (U_{1})) φ ( ∑_{U∈U_{X}} λ_{X}(U)) T ( ∑_{U∈U_{X}} λ_{X}(U)) φ^{∗} λ_{Y} (F (U_{2}))
= λ_{Y} (F (U_{1})) φ \underbrace{λ_{X}(U_{1}) T λ_{X}(U_{2})}_{=0} φ^{∗} λ_{Y} (F (U_{2})) = 0 .

If (y_{1}, y_{2}) ∈ Supp_{λ_{Y}} (ad_{φ}(T)), there are U_{1}, U_{2} ∈ U_{X} such that y_{i} ∈ F (U_{i}). Hence λ_{Y} (F (U_{1}))ad_{φ}(T)λ_{Y} (F (U_{2})) ≠ 0. As we have just seen, this implies U_{1} × U_{2} ⊆ Supp_{λ_{X}} (T) and therefore

(y_{1}, y_{2}) ∈ F (U_{1}) × F (U_{2}) ⊆ f(U_{1}) × f(U_{2}) ⊆ f × f (Supp_{λ_{X}} (T)) .

It remains to prove ad_{φ} (C_{λ_{X}}^{∗} (X, E_{X})) ⊆ C_{λ_{Y}}^{∗} (Y, E_{Y}). For this purpose suppose T ∈ C_{λ_{X}}^{∗} (X, E_{X}). Let B ∈ σ(U_{Y}) be a bounded set. Then λ_{X} (Δ_{U_{X}} [f^{−1}(B)]) T is compact, since Δ_{U_{X}} [f^{−1}(B)] ∈ σ(U_{X}) is bounded and T is locally compact. Therefore

λ_{Y} (B)ad_{φ}(T) = λ_{Y} (B)φT φ^{∗} = λ_{Y} (B)φ \underbrace{λ_{X} (Δ_{U_{X}} [f^{−1}(B)]) T}_{compact} φ^{∗}

and ad_{φ}(T)λ_{Y} (B) are compact operators. This proves that ad_{φ}(T) is locally compact. □

We denote the maps (2.8) and (2.9) by E_{λ_{X},λ_{Y} ,φ}^{∗}(f) and C_{λ_{X},λ_{Y} ,φ}^{∗}(f) respectively. Note that these maps do not only depend on λ_{X} and λ_{Y}, but also on the partial isometry φ.

Proposition 2.42 Let (X, E_{X}) and (Y, E_{Y}) be coarse spaces which admit countable, uniformly bounded covers. If (X, E_{X}) and (Y, E_{Y}) are coarsely equivalent, then C^{∗}(X, E_{X}) ≅ C^{∗}(Y, E_{Y}) and E^{∗}(X, E_{X}) ≅ E^{∗}(Y, E_{Y}).

Proof. Let f: (X, E_{X}) → (Y, E_{Y}) be a coarse equivalence with coarse inverse g: (Y, E_{Y}) → (X, E_{X}). Take a localizing decomposition W of im(f ∘ g) ⊆ Y which does not contain the empty set. Define

E := Δ_{W} ∪ {(y, f ∘ g(y)) | y ∈ Y \ im(f ∘ g)}

and observe that U_{Y} := {E[W] | W ∈ W} is a localizing decomposition of the coarse space (Y, E_{Y}). Let U ∈ U_{Y}. Since there is W ∈ W with W ⊆ U, the inverse image of U under f is not empty. Define U_{X} := {f^{−1}(U) | U ∈ U_{Y} }. Choose an infinite dimensional, separable Hilbert space H_{U} for each U ∈ U_{Y} and set H := ⨁_{U∈U_{Y}} H_{U}. Consider the geometric Hilbert space λ_{Y} : σ(U_{Y}) → B(H) which is determined by setting λ(U) to be the projection onto H_{U}. Furthermore, consider the geometric Hilbert space λ_{X}: σ(U_{X}) → B(H) which is given by defining λ(f^{−1}(U)) to be the projection onto H_{U}. If we define F: U_{X} → U_{Y} by F (f^{−1}(U)) = U for all U ∈ U_{Y} and φ = id_{H}, the conclusions of Lemma 2.40 are satisfied. Moreover, the map ad_{φ} is just the identity on B(H). Hence, Lemma 2.41 yields E_{λ_{X}}^{∗} (X, E_{X}) ⊆ E_{λ_{Y}}^{∗} (Y, E_{Y}) and C_{λ_{X}}^{∗} (X, E_{X}) ⊆ C_{λ_{Y}}^{∗} (Y, E_{Y}). The reverse inclusion for the big translation C^{∗}-algebras follows from

Supp_{λ_{X}} (T) = (f × f)^{−1} (Supp_{λ_{Y}} (T)) for all T ∈ B(H)

and the fact that f is a coarse embedding. Observe that λ_{Y} (B)T = λ_{X}(f^{−1}(B))T for each bounded sets B ∈ σ(U_{Y}). We conclude that an operator T ∈ B(H) is locally compact with respect to Y if and only if T is locally compact with respect to X. □

General theory of asymptotic dimension

The following definitions are an asymptotic analog of the covering dimension of topological spaces. Asymptotic dimension was first introduced in [Gro93]. For dimension theory of topological spaces see [HW41], [Eng78] and [Fed88].

Finiteness of asymptotic dimension seems to play an important role for some isomorphism conjectures in K-theory. Guoliang Yu proved in [Yu98] that the Baum-Connes assembly map is injective for groups of finite asymptotic dimension admitting a finite classifying space. This implies the Novikov conjecture on the homotopy invariance of higher signatures. Arthur Bartels proved injectivity of the assembly map in algebraic K-theory for the same class of groups. Compare [Bar03].

3.1 Asymptotic dimension of pseudometric spaces

Definition 3.1 (asymptotic dimension of pseudometric spaces) Let (X, d) be a pseudometric space, M a collection of subsets of X and n ∈ ℕ.

• The multiplicity of M is defined to be the maximal number of sets with non-empty intersection and will be denoted by μ(M).

• The mesh of M is defined to be the supremum of the diameter of sets from M.

• The Lebesgue number of a cover U of X is the largest positive number L such that for every x ∈ X the open ball of radius L with center x is contained in some U ∈ U. We will write L(U) for the Lebesgue number of U.

• asdim(X, d) ≤ n if for all L > 0 there exist D > 0 and a cover U of X such that

(1) μ(U) ≤ n + 1,

(2) L(U) ≥ L and

(3) mesh(U) ≤ D, i.e. U is uniformly bounded.

• asdim(X, d) = n if asdim(X, d) ≤ n and asdim(X, d) ≰ n − 1.

• asdim(X, d) = ∞ if there is no n ∈ ℕ with asdim(X, d) ≤ n.

In Section 3.2 we will generalize this definition to coarse spaces. As in dimension theory of topological spaces, there are some alternative definitions. For details see [Gro93] and [Dra00]. We will give a second definition of asymptotic dimension, which has been generalized to coarse spaces in [Roe03]. We will prove that both definitions coincide, in the more general setting of coarse spaces.

Definition 3.2 Let (X, d) be a pseudometric space.

• Let L > 0. A family V of subsets of X is called L-disjoint, if the distance of two sets from V is always bigger than L.

• asdim(X, d) ≤ n if for all L > 0 there is a cover U of X such that

(1) the cover U consists of n + 1 families U_{1}, …, U_{n+1},

(2) each family U_{i} is L-disjoint and

(3) the cover U is uniformly bounded.

Remark 3.3 (monotony) Let (X, d) be a pseudometric space. If (A, d|_{A}) is a subspace of (X, d), then asdim(A, d|_{A}) ≤ asdim(X, d).

Example 3.4 Let T be a tree and d the natural metric on T. (All edges are supposed to be of length 1.) Then asdim(T, d) ≤ 1 and asdim(T, d) = 1 if and only if T is not bounded.

Proof. Choose x_{0} ∈ vert(T). Let L > 0 and define L′ to be the smallest natural number bigger than 2 · L. Consider the map

f: T → ℕ, x ↦ [\frac{d(x_{0}, x)}{L′}] .

We define A := \overline{f^{−1}(2 · ℕ)} and B := \overline{f^{−1}(2 · ℕ + 1)}, thus X = A ∪ B. There is an equivalence relation on A: Let a_{1}, a_{2} ∈ A. With γ_{i} we denote the geodesic path from x_{0} to a_{i}.

a_{1} ∼ a_{2} ⟺ f(a_{1}) = f(a_{2}) and γ_{1}(τ) = γ_{2}(τ)

where τ := L′ · (f(a_{1}) − \frac{1}{2}). If a_{1} ≁ a_{2}, then d(a_{1}, a_{2}) ≥ L′. Furthermore, we see diam([a]) ≤ 3 · L′. Here [a] denotes the equivalence class of a ∈ A. Of course we have the same equivalence relation on B. The open L-neighborhoods of the equivalence clases give an open cover of T with Lebesgue number L and multiplicity two. □

Proposition 3.5 Assume X ⊆ ℝ^{n} contains arbitrarily big balls, i.e. for each n ∈ ℕ there is a point x_{n} ∈ X such that the ball B_{ℝ^{n}}(x_{n}, n) with center x_{n} and radius n is contained in X. In this case asdim(X, d_{eucl.}) = n.

Proof. First we prove asdim(ℝ^{n}, d_{eucl.}) ≤ n. Then an application of Remark 3.3 leads to asdim(X, d_{eucl.}) ≤ n. By Q_{a}(x) := \{y ∈ ℝ^{n} | |y_{i} − x_{i}| < \frac{a}{2} for all i ∈ \{1, …, n\}\} we denote the specified n-cube around x ∈ ℝ^{n} with edges of length a > 0. We define v := (1, …, 1) ∈ ℤ^{n} and consider the following n + 1 families of disjoint, open cubes:

Q_{i} := \{Q_{a} (a · (z + \frac{i}{n + 1} · v)) | z ∈ ℤ^{n}\} for i ∈ \{0, …, n\}

The cover U_{a} := Q_{0} ∪ ⋯ ∪ Q_{n} is uniformly bounded and open and has multiplicity μ(U_{a}) = n + 1. We calculate the Lebesgue number of this cover. The boundaries of the covering sets decompose ℝ^{n} into cubes with edges of length \frac{a}{n+1}. Any n-cube is limited by 2n faces. Now take x ∈ ℝ^{n}. Choose the small closed cube x is belonging to. At most n of its limiting faces are at a distance not exceeding \frac{a}{2·(n+1)}. There is i ∈ \{0, …, n\} such that a cube of

Q_{i} contains all these faces. Therefore B_{ℝ^{n}} (x, \frac{a}{2·(n+1)}) is contained in this cube. If we want the cover U_{a} to have Lebesgue number L(U_{a}) ≥ L, we just choose a ≥ 2 · (n + 1) · L. Now suppose asdim(X, d_{eucl.}) =: k < n. There is a uniformly bounded, open cover U of X with multiplicity μ(X) ≤ k + 1 < n + 1. Let d be the mesh of the cover. Let ε > 0. We get an open cover of B_{ℝ^{n}}(0, 1) with sets of diameter at most ε and of multiplicity not exceeding k + 1 by translating a ball of diameter \frac{d}{ε} contained in X to the origin and multiplying with \frac{ε}{d}. Applying Theorem 1.6.12 of [Eng78] yields dim (B_{ℝ^{n}}(0, 1)) ≤ k < n, but this is wrong. Thus asdim(X, d_{eucl.}) ≥ n. □

Corollary 3.6 asdim(ℝ^{n}, d_{eucl.}) = n

3.2 Asymptotic dimension of coarse spaces

In [Roe03], John Roe generalized Definition 3.2. We are now going to generalize Definition 3.1.

Definition 3.7 (asymptotic dimension of coarse spaces) Let (X, E) be a coarse space.

• Let L ∈ E be an entourage and U a cover of X. We say that U has

appetite L if ∀_{x∈X} ∃_{U∈U} L(x) ⊆ U.

• We call a cover U uniformly bounded if Δ_{U} := ⋃_{U∈U} U × U is an entourage.

• Let n ∈ ℕ. We say asdim(X, E) ≤ n if for every^{1} entourage L ∈ E there exists a cover U of X such that

(1) the multiplicity μ(U) is at most n + 1,

(2) U has appetite L and

(3) U is uniformly bounded.

Remark 3.8 If (X, d) is a pseudometric space and E_{d} the corresponding bounded coarse structure, then asdim(X, d) as defined in Definition 3.1 and asdim(X, E_{d}) coincide.

Definition 3.9 Let G be a finitely generated group. Lemma 1.2 yields that different word metrics on G all induce the same coarse structure E_{G}. We define the asymptotic dimension of the finitely generated group G.

asdim(G) := asdim(G, E_{G})

Proposition 3.10 A cover U is uniformly bounded if and only if

∃_{D∈E} ∀_{U∈U} ∃_{x∈X} U ⊆ D(x) .  (3.1)

Proof. Let (X, E) be a coarse space and let U be a cover of X. Defining D := Δ_{U} we see that (3.1) follows from being uniformly bounded. Suppose conversely that we have an entourage D as in (3.1). Now we conclude that ⋃_{U∈U} U × U ⊆ DD^{−1} ∈ E, i.e. U is uniformly bounded. □

Example 3.11 Let X be a Hausdorff space. We call the collection

T := CS ({K ⊆ X × X | K compact })

the trivial coarse structure on X. Observe that this coarse structure is not compatible with the topology if X is not compact and points in X are not open. The asymptotic dimension of (X, T ) is zero.

1 An entourage L ∈ E is called symmetric if L = L^{−1}. We need to consider only symmetric entourages which contain the diagonal, because for any entourage L ∈ E we have L ⊆ L ∪ L^{−1} ∪ Δ_{X} ∈ E.

Proof. The projections π_{i}: X × X → X, (x_{1}, x_{2}) ↦ x_{i} are continuous. Let L ∈ E be a symmetric entourage and define K := π_{1}(\overline{L\Δ_{X}}). Then U := {K} ∪ {{x} | x ∉ K} is a uniformly bounded cover with appetite L and multiplicity zero. □

Remark 3.12 Note that the covers in Definition 3.7 do not have to be open. If we demand the covers to be open, we obtain asdim_{open}(X, T ) = ∞ for every connected, non-compact Hausdorff space X, because a cover of X cannot be open and uniformly bounded with respect to T at the same time.

Comparison with Roe’s definition

The following is a redraft of Roe’s definition of asymptotic dimension.

Definition 3.13 Let (X, E) be a coarse space. We say asdim_{Roe}(X, E) ≤ n if for every entourage L ∈ E there is a cover U of X such that

(1) U = U_{1} ∪ ⋯ ∪ U_{n+1},

(2) each of the families U_{1}, …, U_{n+1} is L-disjoint (i.e. whenever A, B ∈ U_{i} and A ≠ B, then A × B ∩ L = ∅) and

(3) U is uniformly bounded.

There is a small difference between Definition 3.13 and the definition given in [Roe03]. In Roe’s original definition the cover U is supposed to be countable. We will not make any assumptions on the cardinality of U.

Remark 3.14 Let (X, d) be a pseudometric space and E the corresponding bounded coarse structure. It is easy to see that asdim_{Roe}(X, E) and the asymptotic dimension of (X, d) as defined in Definition 3.2 coincide.

A third version of asymptotic dimension will appear in Theorem 3.16.

Definition 3.15 asdim_{fam}(X, E) ≤ n if for every entourage L ∈ E there is a cover U of X such that (1) U = U_{1} ∪ ⋯ ∪ U_{n+1} where each of the families U_{i} consists of disjoint sets, (2) U has appetite L and (3) U is uniformly bounded.

Theorem 3.16 Let (X, E) be a coarse space. Then

asdim(X, E) = asdim_{Roe}(X, E) = asdim_{fam}(X, E) .

Proof. We first prove asdim_{Roe} ≥ asdim_{fam}. Assume asdim_{Roe}(X, E) = n ∈ ℕ. Let L be a symmetric entourage which contains the diagonal. For L^{2} := LL ∈ E, there exists a cover U as in Definition 3.13. Since A × B ∩ L^{2} = ∅ is equivalent to L[A] ∩ L[B] = ∅, the cover U_{L} := {L[U] | U ∈ U} meets all conditions required in Definition 3.15. Note that ⋃_{U∈U} L[U] × L[U] ⊆ L (⋃_{U∈U} U × U) L^{−1} ∈ E. In a second step we have to prove asdim_{fam} ≥ asdim, but this is obvious, since condition (1) of Definition 3.15 implies condition (1) of Definition 3.7. It remains to prove asdim ≥ asdim_{Roe}. For this purpose we need to construct a uniformly bounded cover V consisting of L-disjoint families from a uniformly bounded cover U with appetite L^{n+1}. The idea is to take all intersections of n + 1 sets from U as one family, the intersections of exactly n sets from U as a second family, etc. However, we have to ensure these families to be L-disjoint. Assume that asdim(X, E) = n ∈ ℕ. Let L ∈ E be a symmetric entourage that contains the diagonal. Let U be a uniformly bounded cover of X with appetite L^{n+1} and multiplicity at most n + 1. For an entourage E and U ⊆ X we define Int_{E}(U) := {x ∈ X | E(x) ⊆ U}. Observe that E_{1} ⊆ E_{2} implies Int_{E_{2}}(U) ⊆ Int_{E_{1}}(U). Some more definitions are needed to get V.

U_{i} := {U_{1} ∩ ⋯ ∩ U_{i} | U_{1}, …, U_{i} ∈ U pairwise distinct }
S_{i} := ⋃_{U∈U_{i}} Int_{L^{n+2−i}}(U)
V_{i} := {Int_{L^{n+2−i}}(U)\S_{i+1} | U ∈ U_{i}}
V := V_{1} ∪ ⋯ ∪ V_{n+1}

Now V is a cover of X. Actually, V is a refinement of the cover U. Therefore V is uniformly bounded. It remains to prove that each of the families V_{1}, …, V_{n+1} is L-disjoint. Let A, B ∈ V_{i} such that A ≠ B. There are A_{1}, …, A_{i}, B_{1}, …, B_{i} ∈ U such that A = Int_{L^{n+2−i}}(A_{1} ∩ ⋯ ∩ A_{i})\S_{i+1} and B = Int_{L^{n+2−i}}(B_{1} ∩ ⋯ ∩ B_{i})\S_{i+1}. The sets A_{1}, …, A_{i} are supposed to be pairwise distinct as are the sets B_{1}, …, B_{i}. Let (a, b) ∈ A × B ∩ L and observe the following facts.

a, b ∉ S_{i+1}  (3.2)
a ∈ A ⊆ Int_{L^{n+1−i}}(A_{1} ∩ ⋯ ∩ A_{i})  (3.3)
b ∈ B ⊆ Int_{L^{n+1−i}}(B_{1} ∩ ⋯ ∩ B_{i})  (3.4)
a ∈ L[B] ⊆ L[Int_{L^{n+2−i}}(B_{1} ∩ ⋯ ∩ B_{i})]  (3.5)
b ∈ L[A] ⊆ L[Int_{L^{n+2−i}}(A_{1} ∩ ⋯ ∩ A_{i})]  (3.6)

Since L[Int_{L^{j}}(U)] = {x | ∃_{y∈X}L^{j}(y) ⊆ U, x ∈ L(y)} ⊆ {x | L^{j−1}(x) ⊆ U} = Int_{L^{j−1}}(U), we get the following conclusions from (3.5) and (3.6):

a ∈ Int_{L^{n+1−i}}(B_{1} ∩ ⋯ ∩ B_{i})
b ∈ Int_{L^{n+1−i}}(A_{1} ∩ ⋯ ∩ A_{i})

Finally a, b ∈ Int_{L^{n+2−(i+1)}}(A_{1} ∩ ⋯ ∩ A_{i} ∩ B_{1} ∩ ⋯ ∩ B_{i}). Since A ≠ B, we know that the set {A_{1}, …, A_{i}, B_{1}, …, B_{i}} contains at least i + 1 elements. Thus a, b ∈ S_{i+1}, but this is a contradiction to (3.2). □

Remark 3.17 For further reference we record the results of our constructions in the previous proof. Let (X, E) be a coarse space, L ∈ E and n ∈ ℕ.

• If we are given a uniformly bounded cover U of X such that U = U_{1} ∪ ⋯ ∪ U_{n+1} and each of the families U_{1}, …, U_{n+1} is L^{2}-disjoint, we get the uniformly bounded cover U_{L} := {L[U] | U ∈ U} consisting of n + 1 families of disjoint subsets of X and with appetite L.

• Given a uniformly bounded cover U of X with multiplicity n + 1 and appetite L^{n+1}, we constructed a uniformly bounded cover V such that V = V_{1} ∪ ⋯ ∪ V_{n+1} and each of the families V_{1}, …, V_{n+1} is L-disjoint.

Suppose X is a metric space and E its bounded coarse structure. Given a uniformly bounded cover with multiplicity m and Lebesgue number L, the previous construction yields a uniformly bounded cover which consists of m families of disjoint subsets of X and with Lebesgue number \frac{L}{2n+2}.

Asymptotic dimension via anti-Čech systems

The asymptotic dimension of proper metric spaces can be characterized using anti-Čech systems. Compare Theoreom 9.9 of [Roe03]. We will see that this characterisation does not work for all coarse spaces. However, it does work for coarse structures induced from metrizable compactifications. We recall some definitions from John Roe’s book.

Definition 3.18 Let (X, E) be a coarse space.

• A uniformly bounded cover U of X is called uniform if each bounded subset of X meets only finitely many elements of U.

• A collection of uniform covers is called an anti-Čech system if for every entourage L ∈ E it contains a cover having appetite L.

• We define a partial order on an anti-Čech system. U ≤ V if V has appetite Δ_{U}. This way, any anti-Čech system becomes a directed set.

Example 3.19 Consider the coproduct ∐_{n∈ℕ}(ℝ, E_{eucl.}). According to Proposition 3.30, this coarse space has asymptotic dimension one. On the other hand, for every uniformly bounded cover there exists a copy of ℝ where the cover consists of the sets {x} for x ∈ ℝ. Hence, there don’t exist uniform covers for this coarse space and the following proposition can not be true in general.^{2} Proposition 3.20 Let X be a Hausdorff space and let K be a metrisable compactification of X.^{3} Let E be the coarse structure induced by K. Then asdim(X, E) ≤ n if and only if there is an anti-Čech system consisting of covers with multiplicity at most n + 1.

2 Even if this coarse space does not admit an anti-Čech system, it does have a coarsening sequence in the sense of [Mit03], i.e. this space belongs to the category where coarse homology theory can be defined.

Remark 3.21 If we take just uniformly bounded covers in the definition of anti-Čech systems, Proposition 3.20 is true for all coarse spaces - just by definitions.

Proof of Proposition 3.20. Because of Remark 3.21, we only need to prove that we can choose all covers to be uniform. Let us call a cover V a nice refinement of a cover U if we have an injective map i: V → U such that V ⊆ i(V) for all V ∈ V. Let L be a symmetric entourage that contains the diagonal and let U be a uniformly bounded cover of X consisting of n+1 families each of them being L^{2}-disjoint. Note that every nice refinement of U has the same properties. Since K is metrisable, we can write X = ⋃_{i∈ℕ} K_{i} with K_{i} compact and K_{1} ⊆ K_{2} ⊆ ⋯ ⊆ X. We proceed inductively in order to define a nice refinement of U which finally will lead to a uniform cover of appetite L. Set U_{0} := U. Observe that L[K_{i}] can be covered by a finite subcover U_{fin,i} of U_{i−1}. Define

U_{i} := U_{fin,i} ∪ { U\L[K_{i}] | U ∈ U_{i−1}\U_{fin,i} } .

The cover U_{i} is a nice refinement of U_{i−1}. Finally, we get a nice refinement

U_{red} := {U | ∃_{k∈ℕ}∀_{i≥k}U ∈ U_{fin,i}}

of the original cover U. In order to see that U_{red} is indeed a cover of X, let x ∈ K_{i} ⊆ X. The union of all U ∈ U_{fin,i} containing x is bounded and therefore contained in K_{j} for some j ∈ ℕ. It follows that any U ∈ U_{fin,j} which contains x is an element of U_{red}. Observe that V_{L} := {L[U] | U ∈ U_{red}} is a uniformly bounded cover of X with appetite L and multiplicity at most n + 1. Moreover V_{L} is uniform. In order to see this, let B ⊆ X be bounded. This implies B ⊆ K_{i} for some i ∈ ℕ. Hence B ∩ U ≠ ∅ for most N_{i} different U ∈ U_{red} where N_{i} is the cardinality of U_{fin,i}. Thus, the covers V_{L} for symmetric entourages L which contain the diagonal form the desired anti-Čech system. □

3.3 Basic properties of asymptotic dimension

Theorem 3.22 If f: (X, E) → (Y, F) is a coarse embedding, then

asdim(X, E) ≤ asdim(Y, F).

3 This implies that X is locally compact and paracompact. Note that a compact set K is metrisable if and only if its topology has a countable base.

Proof. Suppose that n := asdim(Y, F) < ∞. Let E ∈ E be an entourage and set F := f × f(E). Note that E ⊆ (f × f)^{−1}(F). There is a uniformly bounded cover U of Y with appetite F and multiplicity at most n + 1. The inverse image of U is a uniformly bounded cover of X with appetite E and the same multiplicity as U. □

Corollary 3.23 (monotony of asymptotic dimension) Let (X, E) be a coarse space and A ⊆ X. Remember that the inclusion map is a coarse embedding. Hence asdim(A, E|_{A}) ≤ asdim(X, E).

Corollary 3.24 (coarse invariance of asymptotic dimension) Asymptotic dimension is a coarse invariant, i.e. given a coarse equivalence f: (X, E_{X}) → (Y, E_{Y}), we have asdim(X, E_{X}) = asdim(Y, E_{Y}).

Definition 3.25 Let (X, E) be a coarse space and A ⊆ X. We call A a substantial part of X if asdim(A, E|_{A}) = asdim(X, E).

For a coarsely uniform map (X, E) → (Y, F) which is also injective, there is no relation between the asymptotic dimensions of (X, E) and (Y, F).

Example 3.26 Let (X, E) be a coarse space. Observe that the power set P(X×X) is a coarse structure on X. The map id: (X, E) → (X, P(X×X)) is coarsely uniform, but asdim(X, E) ≥ 0 = asdim(X, P(X ×X)).

Example 3.27 Let n ∈ {2, 3, …}. By E_{·} we denote the coarse structure coming from the one-point compactification of ℝ^{n} and by E_{eucl.} the bounded coarse structure corresponding to the euclidean metric of ℝ^{n}. The map id: (ℝ^{n}, E_{eucl.}) → (ℝ^{n}, E_{·}) is coarse, but asdim(ℝ^{n}, E_{eucl.}) = n > 1 = asdim(ℝ^{n}, E_{·}).

Proof. Since E_{eucl.} ⊆ E_{·}, the map id is coarsely uniform. A set B is bounded with respect to E_{eucl.} if and only if B is precompact. The same is true for E_{·}. Thus id is coarsely proper. It remains to prove asdim(ℝ^{n}, E_{·}) = 1. Note that

| · | : (ℝ^{n}, E_{·}) → (ℝ_{+}, E_{·})
x ↦ |x|

is a coarse equivalence with inverse ı: (ℝ_{+}, E_{·}) → (ℝ^{n}, E_{·}) , r ↦ (r, 0, …, 0). In fact | · | ∘ ı = id_{ℝ^{n}} and ı ∘ | · | is close to id_{ℝ^{n}} with respect to E_{·}. This yields asdim(ℝ^{n}, E_{·}) = asdim(ℝ_{+}, E_{·}). It remains to calculate the asymptotic dimension of (ℝ_{+}, E_{·}). Since (ℝ_{+}, E_{·})is a ray, we can apply Corollary 4.11. As an alternative, we refer to Example 9.7 of [Roe03]. □

Proposition 3.28 Let (X, E) be a coarse space and E_{cn} = _{cn}CS(E) the connected coarse structure generated by E. Then asdim(X, E) = asdim(X, E_{cn}).

Proof. Suppose n := asdim(X, E_{cn}) < ∞. Let E ∈ E ⊆ E_{cn}. There is a cover U of X with appetite E and multiplicity at most n + 1 which is uniformly bounded with respect to E_{cn}. Each U ∈ U can be written as the disjoint union of finitely many sets U_{1}, …, U_{k} which are bounded with respect to E and such that the union of any two of the sets U_{1}, …, U_{k} is not bounded with respect to E. Define comp(U) := {U_{1}, …, U_{k}} and observe that U′ := ⋃_{U∈U} comp(U) is a cover of X with multiplicity at most n + 1. Furthermore, U′ has appetite E and is uniformly bounded with respect to E. Hence asdim(X, E) ≤ n. Set n := asdim(X, E). Let E ∈ E_{cn} be a symmetric entourage. This implies that E = E′ ∪ (A_{1} × A_{σ(1)}) ∪ ⋯ ∪ (A_{k} × A_{σ(k)}) with E′ ∈ E, k ∈ ℕ, A_{1}, …, A_{k} bounded subsets of X (not necessarily pairwise distinct) and σ a permutation of the set {1, …, k} with the additional property σ ∘ σ = id. Set M := Δ_{X} ∪ A_{1}^{2} ∪ ⋯ ∪ A_{k}^{2} and observe that E′′ := (E′ ∪ Δ_{X})M ∈ E. Let U′′ be a cover of X which is uniformly bounded with respect to E and which has appetite E′′ and multiplicity at most n + 1. Note that there are sets U_{1}, …, U_{k} ∈ U′′ such that E′[A_{i}] ∪ A_{i} ⊆ U_{i}. We define the cover U := U′′ ∪ {U_{1} ∪ ⋯ ∪ U_{k}}\{U_{1}, …, U_{k}} of X. Observe that U is uniformly bounded with respect to E_{cn} and has multiplicity at most n + 1. Moreover, U has appetite E. To see this, let x ∈ X. If x ∈ A_{i}, then E(x) = E′(x) ∪ A_{σ(i)} ⊆ U_{i} ∪ U_{σ(i)} ⊆ U_{1} ∪ ⋯ ∪ U_{k}. If x ∉ {A_{1} ∪ ⋯ ∪ A_{k}}, then E(x) = E′(x) ⊆ E′′(x). □

Proposition 3.29 (asymptotic dimension of finite unions) Let (X, E) be a coarse space and A, B ⊆ X with A ∪ B = X.

asdim(X, E) = max{asdim(A, E|_{A}), asdim(B, E|_{B})}

Proof. The proof of ≥ follows from monotony. To see ≤ , we generalize an argument of Bell and Dranishnikov (see [BD01]). Let n be the maximum of asdim(A, E|_{A}) and asdim(B, E|_{B}) and take a symmetric entourage L ∈ E which contains Δ_{X}. For U ⊆ P(X) and V ⊆ X we define

N_{L}(V, U) := V ∪ ⋃_{\begin{matrix}U∈U \\ L∩U×V ≠∅ \end{matrix}} U .

There is a uniformly bounded cover U = U_{1} ∪ ⋯ ∪ U_{n+1} of A consisting of L-disjoint families U_{i}. Moreover, there is a uniformly bounded cover V = V_{1} ∪ ⋯ ∪ V_{n+1} of B consisting of (LΔ_{U}LΔ_{U}L)-disjoint families V_{i}. For i ∈ {1, …, n + 1} set

W_{i} := {N_{L}(V, U_{i}) | V ∈ V_{i}} ∪ {U ∈ U_{i} | L ∩ U × V = ∅ for all V ∈ V_{i}}.

Observe that N_{L}(V, U_{i}) ⊆ Δ_{U}L[V]. Hence, we get a uniformly bounded cover W = W_{1} ∪ ⋯ ∪ W_{n+1} of X where W_{i} is L-disjoint for 1 ≤ i ≤ n + 1. This proves asdim(X, E) ≤ n. □

Proposition 3.30 (asymptotic dimension of coproducts) Let Λ be any set and (X_{λ}, E_{λ}) a coarse space for every λ ∈ Λ. Define X := ∐_{λ∈Λ}X_{λ}. If (X, E) and (X, E_{cn}) are the coproducts in the categories D

and D_{cn} respectively, then

asdim (X, E_{cn}) = asdim (X, E) = sup_{λ∈Λ} asdim(X_{λ}, E_{λ}) .

Proof. Set n := sup_{λ∈Λ} asdim(X_{λ}, E_{λ}). Monotony of asymptotic dimension

implies asdim(X, E_{cn}) ≥ n and asdim(X, E) ≥ n. We will now prove asdim(X, E) ≤ n. Take an entourage L ∈ E which contains Δ_{X}. Then there are λ_{1}, …, λ_{k} ∈ Λ and L_{λ_{i}} ∈ E_{λ_{i}} such that L = L_{λ_{1}}∪⋯∪L_{λ_{k}}∪Δ_{X}. For i ∈ {1, …, k} choose a uniformly bounded cover U_{λ_{i}} of X_{λ_{i}} with appetite L_{λ_{i}} and multiplicity at most asdim(X_{λ_{i}}, E_{λ_{i}}) + 1. For λ ∈ Λ\{λ_{1}, …, λ_{k}} set U_{λ} := {{x} | x ∈ X_{λ}}. The union U := ⋃_{λ∈Λ} U_{λ} is a uniformly bounded cover of X with appetite L whose multiplicity does not exceed n + 1. For the equality asdim(X, E_{cn}) = asdim(X, E) compare Proposition 3.28. □

Proposition 3.31 (asymptotic dimension of disjoint unions) Let Λ be any set and (X_{λ}, E_{λ}) a coarse space for every λ ∈ Λ.

asdim ( ⨆_{λ∈Λ}X_{λ}, E⨆) = sup_{λ∈Λ} asdim(X_{λ}, E_{λ})

Proof. The proof of ≥ follows from monotony of asymptotic dimension. Set X := ⨆_{λ∈Λ}X_{λ} and n := sup_{λ∈Λ} asdim(X_{λ}, E_{λ}). In order to prove the inequal-

ity asdim(X, E⨆) ≤ n, let L ∈ E⨆. Set L_{λ} := L ∩ X_{λ}^{2} ∈ E_{λ} and choose uniformly bounded covers U_{λ} of X_{λ} with appetite L_{λ} and multiplicity at most n + 1. Now U := ⋃_{λ∈Λ} U_{λ} is a uniformly bounded cover of X with appetite L and multiplicity at most n + 1. □

Proposition 3.32 (asdim of certain direct limits) Let j ∈ I. If I is a directed set and if for all k ≥ j the map f_{k}: X_{k} → \underrightarrow{lim}X_{i} is injective, then

asdim (\underrightarrow{lim}(X_{i}, E_{i})) = sup_{k≥j} asdim(X_{k}, E_{k}) .

Proof. With Proposition 1.27 in mind, the proof is not difficult. Observe that injectivity of f_{k} for all k ≥ j implies that f_{i} is a coarse embedding for i ≥ j. The proof of ≥ now is given by applying Theorem 3.22. We write (X, E) for the direct limit. Let L ∈ E be an entourage which contains Δ_{X}. Proposition 1.27 implies that L is the union of Δ_{X} and an entourage L_{i} ∈ E_{i} for some i ∈ I. We may assume i ≥ j. If U_{i} is a uniformly bounded cover of X_{i} with appetite L_{i} and multiplicity at most asdim(X_{i}, E_{i}) + 1, then U := U_{i} ∪ {{x} | x ∈ X\X_{i}} is a uniformly bounded cover of X with appetite L. Moreover U has the same multiplicity as U_{i}. □

Proposition 3.33 (asymptotic dimension of products) Let (X, E_{X}) and (Y, E_{Y}) be coarse spaces.

asdim(X × Y, E_{X} ∗ E_{Y}) ≤ asdim(X, E_{X}) + asdim(Y, E_{Y})
asdim(X, E_{X}) ≤ asdim(X × Y, E_{X} ∗ E_{Y}) if Y ≠ ∅

Proof. Compare [Roe03] for the special case of bounded coarse structures and Remark 5.8 for the case of continuously controlled coarse structures induced by metrisable compactifications. Set n := asdim(X) and m := asdim(Y). Let E ∈ E_{X} ∗ E_{Y}. There are symmetric entourages E_{X} ∈ E_{X} and E_{Y} ∈ E_{Y} containing the diagonals Δ_{X} and Δ_{Y} respectively such that E ⊆ E_{X} × E_{Y}. There is a uniformly bounded cover U of X with appetite E_{X}^{n+m+1} and multiplicity μ(U) ≤ n + 1. There is also a uniformly bounded cover V of Y with appetite E_{Y}^{n+m+1} and multiplicity μ(V) ≤ m + 1. We get a uniformly bounded cover U × V := {U × V | U ∈ U, V ∈ V} of X × Y with appetite E^{n+m+1} and multiplicity ≤ (n + 1) · (m + 1) = n · m + n + m + 1. Thus, we need to improve the multiplicity. We proceed similar as in the proof of Theorem 3.16 and begin with some definitions. Let k ∈ {2, …, n + m + 2}.

A_{k} := {U_{1} ∩ ⋯ ∩ U_{p} × V_{1} ∩ ⋯ ∩ V_{q} |
p + q = k, U_{i} ∈ U, V_{i} ∈ V pairwise distinct }
B_{k} := ⋃_{A∈A_{k}} Int_{E^{n+m+3−k}}(A) and B_{n+m+3} := ∅
W_{k} := {Int_{E^{n+m+3−k}}(U) \ B_{k+1} | U ∈ A_{k}}
W := W_{2} ∪ ⋯ ∪ W_{n+m+2}

Notice that W is a uniformly bounded cover of X × Y consisting of the n + m + 1 disjoint families W_{2}, …, W_{n+m+2}. It remains to prove that W_{k} is E-disjoint for k ∈ {2, …, n + m + 2}. For this purpose let M, N ∈ W_{k} with M ≠ N and suppose M×N ∩ E ≠ ∅. Choose ((x_{M}, y_{M}), (x_{N}, y_{N})) ∈ M ×N ∩ E. There are p_{M}, q_{M} ∈ ℕ with p_{M} + q_{M} = k and M_{1}, …, M_{p_{M}} ∈ U, M′_{1}, …, M′_{q_{M}} ∈ V such that

M = Int_{E^{n+m+3−k}}(\underbrace{M_{1} ∩ ⋯ ∩ M_{p_{M}} × M′_{1} ∩ ⋯ ∩ M′_{q_{M}}}_{=:MM′}) \ B_{k+1}.

Similarly, there are p_{N}, q_{N} ∈ ℕ with p_{N} + q_{N} = k and N_{1}, …, N_{p_{N}} ∈ U, N′_{1}, …, N′_{q_{N}} ∈ V such that

N = Int_{E^{n+m+3−k}}(\underbrace{N_{1} ∩ ⋯ ∩ N_{p_{N}} × N′_{1} ∩ ⋯ ∩ N′_{q_{N}}}_{=:NN′}) \ B_{k+1}.

Observe that the following relations hold.

(x_{M}, y_{M}) ∉ B_{k+1}
(x_{M}, y_{M}) ∈ M ⊆ Int_{E^{n+m+2−k}}(MM′)
(x_{M}, y_{M}) ∈ E[N] ⊆ E[Int_{E^{n+m+3−k}}(NN′)] ⊆ Int_{E^{n+m+2−k}}(NN′)

It follows that (x_{M}, y_{M}) ∈ Int_{E^{n+m+3−(k+1)}}(MM′ ∩ NN′). Since M ≠ N, the set {M_{1}, …, M_{p_{M}} , M′_{1}, …, M′_{q_{M}} , N_{1}, …, N_{p_{N}} , N′_{1}, …, N′_{q_{N}} } contains at least k + 1 different elements. Hence (x_{M}, y_{M}) ∈ B_{k+1}. But this is a contradiction to what we found before. □

The equality asdim(X × Y) = asdim(X) + asdim(Y) is not true in general. Compare Corollary 5.9.

There are no general restrictions on the asymptotic dimensions of quotients as the following three examples show.

Example 3.34 Let X = ℝ^{2} be the real plane and E the bounded coarse structure induced by the euclidean metric on ℝ^{2}. Define an equivalence relation by (x, y) ∼ (x′, y′) if and only if x = x′. The quotient X/ ∼ is clearly ℝ and E_{∼} is the bounded coarse strucure on ℝ again induced by the euclidean metric on ℝ. In this case asdim(X, E) > asdim(X/∼, E_{∼}).

Example 3.35 Let 𝔽_{2} = ⟨a, b |⟩ be the free group with two generators. We say that two words are equivalent if the number of letters a counted with exponents as signs is equal in both words and the same for the letter b. The quotient is the group ℤ^{2}. In this case asdim(𝔽_{2}) = 1 < 2 = asdim(ℤ^{2}).

Remark 3.36 Let (X, E) be a coarse space. Assume there is an equivalence relation ∼ on X. If the equivalence relation happens to be an entourage, i.e. {(x, y) | x ∼ y} ∈ E, then the projection onto the quotient is a coarse equivalence.

Let (X, E_{X}), (Y, E_{Y}) be coarse spaces, A ⊆ X and f: A → Y coarsely uniform. Attaching X to Y using f we get X ∪_{f} Y := X ⊔ Y / a∼f(a) and a quotient map π: X ⊔ Y → X ∪_{f} Y. We denote the coarse structure on the quotient by E_{X∪_{f} Y}.

Lemma 3.37 Whenever asdim(X, E_{X}) = asdim(X\A, E_{X∪_{f} Y} |_{X\A}), we can conclude that asdim(X ∪_{f} Y) = max{asdim(X), asdim(Y)}.

Proof. Note that X ∪_{f} Y = X\A ∪ Y. Thus asdim(X ∪_{f} Y, E_{X∪_{f} Y}) = max {asdim(X\A, E_{X∪_{f} Y} |_{X\A}), asdim(Y, E_{X∪_{f} Y} |_{Y})} by Proposition 3.29. In Lemma 1.36 we proved E_{X∪_{f} Y} |_{Y} = E_{Y}. Together with the assumption this implies the claim. □

Asymptotic dimension of coarse cell complexes

Coarse cell complexes

In order to define coarse cell complexes, we need to describe their building blocks. Compare the discussion in [Mit03] for a better understanding of the following definition of a ray.

Definition 4.1 Let E be a connected coarse structure on ℝ_{+} which is compatible with the standard topology on ℝ_{+}. We call R = (ℝ_{+}, E) a ray if the coarse structure E satisfies the following two conditions.

• If M and N are entourages, the same is true for

M + N := {(u + x, v + y) | (u, v) ∈ M, (x, y) ∈ N} .

• If M is an entourage, so is

M^{□ ∠} := {(u, v) | (x, y) ∈ M and (x ≤ u ≤ v ≤ y or y ≤ v ≤ u ≤ x)} .

Proposition 4.2 By E_{eucl.} we denote the usual bounded coarse structure on ℝ_{+}. The coarse space (ℝ_{+}, E_{eucl.}) is a ray and E_{eucl.} ⊆ E whenever (ℝ_{+}, E) is a ray.

For a proof consult Proposition 2.5 of [Mit03].

Proposition 4.3 By E_{·} we denote the continuously controlled coarse structure on ℝ_{+} induced by the one-point compactification. The coarse space (ℝ_{+}, E_{·}) is a ray and E ⊆ E_{·} whenever (ℝ_{+}, E) is a ray.

Proof. The inclusion E ⊆ E_{·} is true for every coarse structure which is compatible with the topology of ℝ_{+}. It remains to prove that (ℝ_{+}, E_{·}) is a ray. This is straightforward. □

Definition 4.4 Let R be a ray and n ∈ ℕ.

S^{n−1}R := (R ∐ R)^{n} × {0}
D^{n}R := (R ∐ R)^{n} × R

We call S^{n}R a coarse n-sphere and D^{n}R a coarse n-cell.

There are two different coarse structures on R ∐ R. There is the coproduct coarse structure E∐ which is not connected and there is the connected coarse structure _{cn}CS(E∐) generated by E∐. The arguments in this chapter work for both coarse structures, but only equipped with the connected coarse structure the spaces S^{n}R and D^{n}R deserve the names sphere and cell respectively.

Definition 4.5 (coarse cell complex) We call (Y, E) a coarse cell complex if it is obtained by “inductively” attaching coarse cells to a disjoint union of coarse cells. More precisely, suppose that for all k ∈ ℕ we are given a coarse space (Y_{k}, E_{k}) and a set I_{k}. Furthermore, for all k ∈ ℕ and i ∈ I_{k} there is

• a number n_{k,i} ∈ ℕ,

• a ray R_{k,i} and

• a coarsely uniform map f_{k,i}: S^{n_{k,i}−1}R_{k,i} → Y_{k−1}.

Define f_{k} := ∐_{i∈I_{k}} f_{k,i} : ∐_{i∈I_{k}} S^{n_{k,i}−1}R_{k,i} → Y_{k−1} and observe that f_{k} is coarsely uniform. If

Y_{0} = ∐_{i∈I_{0}} D^{n_{0,i}}R_{0,i}  and  Y_{k} = (∐_{i∈I_{k}} D^{n_{k,i}}R_{k,i}) ∪_{f_{k}} Y_{k−1} ,

then \underrightarrow{lim}(Y_{k}, E_{k}) is a coarse cell complex. Define δ(Y) := sup{n_{k,i} | k ∈ ℕ, i ∈ I_{k}} and call δ(Y) the cell dimension of the coarse cell complex Y.

Asymptotic dimension of coarse cell complexes

Our goal is to prove the following theorem.

Theorem 4.6 Let Y be a coarse cell complex. Then asdim Y = δ(Y) + 1 where δ(Y) is the cell dimension of Y.

In order to get Theorem 4.6, we start with some lemmas.

Let R be a ray, X′ any coarse space and A′ ⊆ X′. Define X := D^{n−1}R ⊔ X′ and A := S^{n−2}R ⊔ A′ and denote the coarse structure of X by E_{X}. Notice that R^{n} ⊆ D^{n−1}R. We will consider the set

P_{n} := {(x_{1}, …, x_{n}) ∈ R^{n} | x_{n} > 0 and x_{i} ≤ x_{n} for i < n} ⊆ X\A.

Lemma 4.7 For any coarse space (Y, E_{Y}) and any coarsely uniform map f: (A, E_{X}|_{A}) → (Y, E_{Y}), we have E_{X}|_{P_{n}} = E_{X∪_{f} Y} |_{P_{n}}.

Proof. It follows directly from the definition of the quotient coarse structure that E_{X}|_{P_{n}} ⊆ E_{X∪_{f} Y} |_{P_{n}}. We need to prove the converse inclusion. Let E ∈ E_{X∪_{f} Y} |_{P_{n}}. There are symmetric entourages L_{i} ∈ E_{D^{n−1}R} containing the diagonal Δ_{D^{n−1}R} such that E ⊆ L_{1}^{A} ⋯ L_{k}^{A} where L_{i}^{A} := L_{i} ∪ A×A. If we set L := L_{1} ⋯ L_{k}, we get E ⊆ L_{1}^{A} ⋯ L_{k}^{A} ⊆ L ∪ L(A×A)L. Thus, it remains to prove L(A×A)L ∩ P_{n}^{2} ∈ E_{X}|_{P_{n}}. Since the projection p_{n}: P_{n} → R, (x_{1}, …, x_{n}) ↦ x_{n} is coarsely uniform, the image L̃ := p_{n} × p_{n}(L) of L is in E_{R} and compatibility with the topology implies L̃[0] ⊆ [0, r_{E}]. We conclude L(A×A)L ⊆ {(x_{1}, …, x_{n}) | x_{n} ≤ r_{E}}^{2} and hence L(A×A)L ∩ P_{n}^{2} ⊆ ([0, r_{E}]^{n})^{2} ∈ E_{X}|_{P_{n}}. □

Lemma 4.8 asdim(P_{n}, E_{X}|_{P_{n}}) ≥ n

Proof. Let U be a uniformly bounded cover of P_{n} with appetite Δ_{P_{n},1} := {(x, y) ∈ P_{n} × P_{n} | d_{eucl.}(x, y) ≤ 1}. Note that by Proposition 4.2 the set Δ_{P_{n},1} is an entourage. We will prove that the multiplicity of U is at least n+1. By p_{i}: P_{n} → R, (x_{1}, …, x_{n}) ↦ x_{i}, we denote the projection of P_{n} onto the i-th factor. Set E_{i} := p_{i}×p_{i}(Δ_{U}) ∈ E_{R} and Δ_{1} := {(x, y) ∈ R^{2} | |x−y| ≤ 1}. Choose r > 1 such that E_{n}Δ_{1}E_{n−1}E_{n−2} ⋯ E_{2}E_{1}[0] ⊆ [0, r). Define

a_{n} := (0, …, 0, 1) ∈ P_{n}  and  a_{j} := (\underbrace{0, …, 0}_{j}, r, …, r) ∈ P_{n}.

The points a_{0}, …, a_{n} are the vertices of the n-simplex

S := {λ_{0}a_{0} + ⋯ + λ_{n}a_{n} | λ_{0}, …, λ_{n} ≥ 0 and λ_{0} + ⋯ + λ_{n} = 1}.

By S_{j} we denote the (n−1)-face of S opposite to a_{j}. For x = (x_{1}, …, x_{n}) the following conclusions hold.

x ∈ S_{0} ⇒ x_{1} = 0
x ∈ S_{j} ⇒ x_{j} = x_{j+1}  (j = 1, …, n−2)
x ∈ S_{n−1} ⇒ 0 ≤ x_{n} − x_{n−1} ≤ 1
x ∈ S_{n} ⇒ x_{n} = r

We claim that for each U ∈ U there is a face S_{i} such that U ∩ S_{i} = ∅. Suppose there is U ∈ U with U ∩ S_{i} ≠ ∅ for all i ∈ {0, …, n}. Take an element x^{(0)} ∈ U ∩ S_{0}. Then x_{1}^{(0)} = 0 and hence p_{1}(U) ⊆ E_{1}[0]. Now take x^{(1)} ∈ U ∩ S_{1}. We have x_{2}^{(1)} = x_{1}^{(1)} and therefore p_{2}(U) ⊆ E_{2}E_{1}[0]. Inductively, we get p_{j}(U) ⊆ E_{j} ⋯ E_{1}[0] for j ∈ {1, …, n − 1}. Finally, we take x^{(n−1)} ∈ U ∩ S_{n−1}. This time we have x_{n}^{(n−1)} ∈ Δ_{1}(x_{n−1}^{(n−1)}) and hence p_{n}(U) ⊆ E_{n}Δ_{1}E_{n−1} ⋯ E_{1}[0] ⊆ [0, r). On the other hand, since U ∩ S_{n} ≠ ∅, we get r ∈ p_{n}(U). This is a contradiction. Of course U_{S} := {U ∈ U | U ∩ S ≠ ∅} is a cover of S. Take any map g: U_{S} → {a_{0}, … a_{n}}, such that U ∈ U_{S} does not intersect the (n − 1)- face opposite to g(U). Defining U_{i} := ⋃_{U∈g^{−1}(a_{i})} U we obtain the cover {U_{0}, … U_{n}} of S with Lebesgue number at least 1 and such that U_{i} contains a_{i} and U_{i} ∩ S_{i} = ∅. We intend to apply Theorem 4.9. Therefore, take a triangulation K of S such that the diameter of any simplex of K is at most 1. Choose a map h: vert(K) → {a_{0} …, a_{n}} such that for each v ∈ h^{−1}(a_{i}) the star of v, i.e. the union of all simplices containing v, is contained in U_{i}. Applying Theorem 4.9 we get an n-simplex of K which is contained in U_{0} ∩ ⋯ ∩ U_{n}. Hence U_{0} ∩ ⋯ ∩ U_{n} ≠ ∅. This yields μ(U) ≥ n + 1. □

Theorem 4.9 (Sperner’s Lemma) Let K be a triangulation of a closed n-simplex S. Suppose we are given a map h: vert(K) → vert(S) with the following property: If v ∈ vert(K) lies in any face of S, then h(v) lies in the same face. Under these assumptions, there is at least one n-simplex T of the triangulation K whose n + 1 vertices correspond under h to the n + 1 different vertices of S.

A proof of Sperner’s Lemma can be found in [Fed88]. Sperner’s Lemma is a combinatorial version of the fact that the sphere S^{n−1} is not a retract of the closed ball of radius 1 around the origin in ℝ^{n}.

Lemma 4.10 We can estimate the asymptotic dimension of coarse spheres and coarse cells as follows. Let R be a ray and n ∈ ℕ. Then

asdim D^{n−1}R ≤ n and asdim S^{n−1}R ≤ n.

Proof. Notice that Proposition 3.30 yields

asdim D^{n−1}R = asdim R^{n} = asdim S^{n−1}R.

For i ∈ {1, …, n} consider the projection p_{i}: R^{n} → R, (x_{1}, …, x_{n}) ↦ x_{i} and remember that Δ_{1} := {(x, y) ∈ R × R | |x − y| ≤ 1} ∈ E_{R}. Let E ∈ E_{R}^{∗n} be a symmetric entourage. Set E_{i} := p_{i} × p_{i}(E) ∪ Δ_{1} ∈ E_{R} and Ẽ := (E_{1} ∪ ⋯ ∪ E_{n})^{□ ∠} ∈ E_{R}. This implies E ⊆ Ẽ^{×n} ∈ E_{R}^{∗n}. We define some bounded sets out of which we will build a cover of R^{n}.

K_{0} := {0}
K_{i+1} := Ẽ[K_{i}]  for i ∈ ℕ
K_{i} := ∅  for i < 0
κ_{i} := sup K_{i}  for i ∈ ℕ

Observe that [0, κ_{i}) ⊆ K_{i}. Now we define a uniformly bounded cover U of R^{n} consisting of n + 1 families each of them being E-disjoint.

U_{i} := K_{i} \ K_{i−n}
U_{i} := {U_{i_{1}} × ⋯ × U_{i_{n}} | i_{1}, …, i_{n} ≡ i mod n + 1}
U := U_{0} ∪ ⋯ ∪ U_{n}

We first prove that U is indeed a cover of R^{n}. Let (x_{1}, …, x_{n}) ∈ R^{n}. Choose i_{1}, …, i_{n} ∈ ℕ such that (x_{1}, …, x_{n}) ∈ K_{i_{1}}\K_{i_{1}−1} × ⋯ × K_{i_{n}}\K_{i_{n}−1}. There is i ∈ {1, …, n+1} such that for all k ∈ {1, …, n} we have i_{k} ≢ i mod n+1. Define j_{k} := min{j ∈ ℕ | j ≥ i_{k} and j − n ≡ i mod n + 1}. It follows that j_{k} − n < i_{k} ≤ j_{k} and hence K_{i_{k}}\K_{i_{k}−1} ⊆ U_{j_{k}}. This implies (x_{1}, …, x_{n}) ∈ U_{j_{1}} × ⋯ × U_{j_{n}} ∈ U_{i−1}. Since Δ_{1} ⊆ Ẽ, the closure of Ẽ (with respect to the usual topology on ℝ^{2}) is contained in Ẽ^{3}. This implies (κ_{i}, κ_{i+1}) ∈ Ẽ^{3} for i ∈ ℕ and (x, κ_{i}) ∈ Ẽ^{3}

for i ∈ ℕ and x ∈ K_{i}\K_{i−1}. Hence Δ_{U} ⊆ \binom{}{E^{×n}}^{3n+6}, i.e. Δ_{U} is uniformly bounded. Observe that for i ∈ {0, …, n} the family U_{i} is E-disjoint. This is a direct consequence of the definitions. □

Corollary 4.11 If R is a ray, then asdim D^{n}R = asdim S^{n}R = n + 1.

Proof. Apply Lemma 4.8 and Lemma 4.10. □

Lemma 4.12 Let I be a set. For i ∈ I let R_{i} be a ray and n_{i} ∈ ℕ. Define X := ∐_{i∈I} D^{n_{i}}R_{i} and A := ∐_{i∈I} S^{n_{i}−1}R_{i}. Let (Y, E_{Y}) be any coarse space and E_{X} the coarse structure of X. Set n := sup{ n_{i} | i ∈ I}. If f: (A, E_{X}|_{A}) → (Y, E_{Y}) is coarsely uniform, then

asdim (X\A, E_{X∪_{f} Y} |_{X\A}) ≤ max {n + 1, asdim (im f, E_{Y} |_{im f})} .

Proof. Suppose n < ∞ and set m := max {n + 1, asdim (im f, E_{Y} |_{im f})}. For every entourage Ĕ ∈ E_{X∪_{f} Y} |_{X} there is E ∈ E_{X} and L ∈ E_{Y} |_{im f} such that Ĕ ⊆ E ∪EL̆E where L̆ := (f ×f)^{−1}(L). Note that far away from A the entourages E and Ĕ coincide, i.e. E\E[A]^{2} = Ĕ\E[A]^{2}. We may assume Δ_{A} ⊆ L̆ and Δ_{X} ⊆ E. Note that D^{n_{i}}R_{i} consists of at most 2^{n} copies of R_{i}^{n_{i}+1}. Choose one of these copies for each i ∈ I and denote their union by X′. Furthermore, set A′ := A ∩ X′. With Proposition 3.29 in mind it is enough to prove that asdim (X′\A′, E_{X∪_{f} Y} |_{X′\A′}) ≤ m. Similarly as in Lemma 4.10 we can choose E_{i} ∈ E_{R_{i}} for i ∈ I such that E|_{R_{i}^{n_{i}+1}} ⊆ E_{i}^{×(n_{i}+1)} and E_{i} = E_{i}^{□ ∠}. Since X is a coproduct, Proposition 1.28 implies that we can choose E_{i} in such a way that there is a finite set I_{0} ⊆ I such that E_{i} ⊆ Δ_{R_{i}} for all i ∈ I\I_{0}. Moreover, we may assume Δ_{1} := {(x, y) ∈ ℝ_{+}^{2} | |x − y| ≤ 1} ⊆ E_{i} for i ∈ I_{0}.

Define

K_{i,0} := {0}  for i ∈ I_{0},
K_{i,k} := ∅  for k < 0, i ∈ I_{0} and
K_{i,k+1} := E_{i}^{5}[K_{i,k}]  for k ∈ ℕ, i ∈ I_{0}.

Furthermore, we define a uniformly bounded cover U = U_{0} ∪ ⋯ ∪ U_{m} of X′ as follows: Let k ∈ ℕ, l ∈ {1, …, m + 1} and i ∈ I_{0}.

U_{i,k} := E_{i}^{2} [K_{i,k}\K_{i,k−m}]
U_{l} := {U_{i,k_{1}} × ⋯ × U_{i,k_{n_{i}+1}} | i ∈ I_{0}, k_{1}, …, k_{n_{i}+1} ≡ l mod m + 1}

So far we did not cover the parts of X′ corresponding to I\I_{0}. Define

U_{0} := U_{m+1} ∪ {{x} | x ∈ R_{i}^{n_{i}+1}, i ∈ I\I_{0}}.

The proof that U is a uniformly bounded cover of X′ is similar to an argument in the proof of Lemma 4.10. Observe that U has appetite (E|_{X′})^{2} and that each of the families U_{0}, …, U_{m} is E-disjoint. We want to construct a nice cover V of A′. For M ⊆ X × X define f_{∗}M := f × f(M ∩ A×A). Notice that f_{∗}E, f_{∗}Δ_{U} ∈ E_{Y}. Take a uniformly bounded cover Ṽ := Ṽ_{0} ∪ ⋯ ∪ Ṽ_{m} of im f with appetite f_{∗}E ∪ (f_{∗}E)L(f_{∗}E) and such that each family Ṽ_{k} is f_{∗}Δ_{U}-disjoint. By taking the inverse image under f|_{A′} we get a cover V = V_{0}∪⋯∪V_{m} of A′ with appetite (EL̆E)|_{A′} = (E∪EL̆E)|_{A′} and such that each of the families V_{0}, …, V_{m} is Δ_{U}-disjoint. In general V will not be uniformly bounded with respect to E_{X}|_{A′}. Now we construct a cover of E[A] ∩ X′\A′ starting from the cover V of A′. Let k ∈ {0, …, m}. For V ∈ V_{k} define

V^{♭} := ⋃_{i∈I}(V ∩ R_{i}^{n_{i}}) × E_{i}[0]  and  V^{♯} := (V^{♭} ∪ ⋃_{{U∈U_{k} \atop U∩V^{♭}≠∅}} U) \A .

Observe that V_{k}^{♯} := {V^{♯} | V ∈ V_{k}} is uniformly bounded with respect to E_{X∪_{f} Y} |_{X′\A′}. From U_{k} and V_{k}^{♯} we build the family

W_{k} := {U\A | U ∈ U_{k}, U ∩ V^{♭} = ∅ for all V ∈ V_{k}} ∪ V_{k}^{♯}.

Note that any U ∈ U_{k} does intersect V^{♭} for at most one V ∈ V_{k}. We obtain a cover W := W_{0} ∪ ⋯ ∪ W_{m} of X′\A′ which is uniformly bounded with respect to E_{X∪_{f} Y} |_{X′\A′} and has appetite Ĕ|_{X′\A′}. □

Proof of Theorem 4.6. Let Y_{k}, I_{k}, R_{k,i}, n_{k,i} and f_{k,i} be as in Definition 4.5. By induction we will prove

asdim Y_{k} = sup{n_{j,i} | j ≤ k, i ∈ I_{j}} + 1.  (4.1)

The case k = 0 follows from Proposition 3.30 and Corollary 4.11. Set X := ∐_{i∈I_{k}} D^{n_{k,i}}R_{k,i} and A := ∐_{i∈I_{k}} S^{n_{k,i}−1}R_{k,i}. By Corollary 4.11 the asymptotic dimension of X is s_{k} + 1 where s_{k} := sup{n_{k,i} | i ∈ I_{k}}. We need information on the asymptotic dimension of (X\A, E_{X∪_{f_{k}} Y_{k−1}}|_{X\A}). Observe that

X\A = ∐_{i∈I_{k}} D^{n_{k,i}}R_{k,i}\S^{n_{k,i}−1}R_{k,i} .

Using Lemma 4.7, Lemma 4.8 and monotony of asymptotic dimension, we get asdim (X\A, E_{X∪_{f_{k}} Y_{k−1}}|_{X\A}) ≥ s_{k} + 1. Applying Lemma 4.12 we get

asdim (X\A, E_{X∪_{f_{k}} Y_{k−1}}|_{X\A}) ≤ max{s_{k} + 1, asdim Y_{k−1}}. Lemma 1.36 assures us that there is no ambiguity about the coarse structure of Y_{k−1}. Notice that Y_{k} = X ∪_{f_{k}} Y_{k−1} = X\A ∪ Y_{k−1}. Hence, Proposition 3.29 completes the proof of (4.1). Since all the maps Y_{k} → Y are injective, the claim of Theorem 4.6 now follows from Proposition 3.32. □

We conclude this chapter by discussing some possible modifications in the definition of a coarse cell complex and the effect on the asymptotic dimension.

Remark 4.13 We might consider building a coarse cell out of different rays, e.g. R_{1}∐ R_{1} × ⋯ × R_{n}∐ R_{n} × R_{n+1}. Using Proposition 4.3, the arguments given in this chapter can be easily adjusted to prove asdim Y = δ(Y) + 1 for a coarse cell complex Y built with modified cells.

Remark 4.14 If we want to use the disjoint union ⨆ instead of the coproduct ∐ in Definition 4.5, we have to demand the maps f_{k} to be coarsely uniform, since this is no longer automatic. If for all k ∈ ℕ the map f_{k} is coarsely uniform, we can prove the formula of Theorem 4.6 in the same way as before. In the proof of Lemma 4.12 we do not have to distinguish two subsets of I. We may define I_{0} := I in this case. We do not need the fact that I_{0} is a finite set if we work with disjoint unions instead of coproducts.

Metric coarse cell complexes

If (X, d_{X}) and (Y, d_{Y}) are pseudometric spaces, A ⊆ X and f: A → Y is a (not necessarily continuous) map, then X ∪_{f} Y is defined as a pseudometric space. We will use 𝕊^{n} := ℝ^{n+1}×{0} and 𝔻^{n} := ℝ^{n}×ℝ_{+} with the restrictions of the euclidean metric as coarse n-sphere and coarse n-cell respectively.

Definition 4.15 Let I be any set and X_{i} and Y_{i} pseudometric spaces. A family of maps {f_{i}: X_{i} → Y_{i}}_{i∈I} is said to be coarsely uniform in a uniform way if there is a (not necessarily continuous) map s: ℝ_{+} → ℝ_{+} such that

∀_{r>0} ∀_{i∈I} ∀_{x,x̃∈X_{i}} d(x, x̃) ≤ r ⇒ d(f_{i}(x), f_{i}(x̃)) ≤ s(r).

Definition 4.16 Suppose that for all k ∈ ℕ there is a pseudometric space (Y_{k}, d_{k}) and a set I_{k}. Moreover, suppose that for all k ∈ ℕ and i ∈ I_{k} there is a number n_{k,i} ∈ ℕ and a map f_{k,i}: 𝕊^{n_{k,i}−1} → Y_{k−1} such that for each k ∈ ℕ the family {f_{k,i}}_{i∈I_{k}} is coarsely uniform in a uniform way. Define f_{k} := ⨆_{i∈I_{k}} f_{k,i} : ⨆_{i∈I_{k}} 𝕊^{n_{k,i}−1} → Y_{k−1} and observe that f_{k} is coarsely uniform. If

Y_{0} = ⨆_{i∈I_{0}} 𝔻^{n_{0,i}}  and  Y_{k} = ( ⨆_{i∈I_{k}} 𝔻^{n_{k,i}} ) ∪_{f_{k}} Y_{k−1} ,

then (Y, d) := \underrightarrow{lim}(Y_{k}, d_{k}) is a metric coarse cell complex.

There are two coarse structures on Y. We write E_{d} for the bounded coarse structure of Y induced by the pseudometric d. Similarly, we write E_{d_{k}} for the bounded coarse structure of Y_{k} induced by the pseudometric d_{k}. Denote the coarse structure of \underrightarrow{lim}(Y_{k}, E_{d_{k}}) by E. Using Proposition 1.27, we see that E ⊆ E_{d}. The following example yields that in general E and E_{d} do not coincide.

Example 4.17 Consider any metric coarse cell complex (Y, d) with I_{k} ≠ ∅ for all k ∈ ℕ. Define Δ_{1} := {(y_{1}, y_{2}) ∈ Y × Y | |y_{1} − y_{2}| ≤ 1} and observe that Δ_{1} ∈ E_{d}. Proposition 1.27 implies Δ_{1} ∉ E.

Remark 4.18 The arguments given in this chapter can be easily adjusted to prove

asdim Y_{k} = sup{n_{j,i} | j ≤ k, i ∈ I_{j}} + 1

for metric coarse cell complexes. Here we use the notations of Definition 4.16.

Proposition 4.19 asdim(Y, E) = δ(Y) + 1.

Proof. The claim follows from Remark 4.18 and Proposition 3.32. □

Proposition 4.20 If the metric coarse cell complex Y can be built in finitely many steps, i.e. if there is k_{0} ∈ ℕ such that I_{k} = ∅ for all k ≥ k_{0}, then asdim(Y, E_{d}) = δ(Y) + 1.

Proposition 4.20 follows directly from Remark 4.18.

The following example of a metric coarse cell complex yields that the formula of Theorem 4.6 is not true in general.

Example 4.21 We inductively define a metric coarse cell complex (Y, d) and start with Y_{0} := ℝ^{2} × ℝ_{+} = 𝔻^{2} ⊆ ℝ^{3}. For each k ∈ ℕ\{0} and i ∈ I := {0} × ℤ × ℕ ⊆ ℝ^{3} we attach a coarse 1-cell 𝔻^{1} to Y_{0} using the map

f̃_{k,i}: 𝕊^{0} → Y_{0}, (x, 0) ↦ (2^{k}x, 0, 0) + i .

Denote the metric space obtained after attaching a cell for all k ∈ {1, …, n} and i ∈ I as just explained by Y_{n}. Observe that the inclusion i_{k}: Y_{0} ↪ Y_{k} is not isometric. On the other hand, all the attaching maps f_{k,i} := i_{k−1} ∘ f̃_{k,i} are 2-Lipschitz.^{1} Hence Y := \underrightarrow{lim}Y_{k} is a metric coarse cell complex. The metric coarse cell complex Y is coarsely equivalent to the consecutively defined metric coarse cell complex Y′: We start with Y′_{0} := ℝ × ℝ_{+} = 𝔻^{1} and for each k ∈ ℕ\{0} and y ∈ ℤ × ℕ ⊂ Y′ we attach a coarse cell 𝔻^{1} using the constant attaching map f′_{k,y}: 𝕊^{0} → Y′_{0} whose image is just {y}. Finally, we obtain asdim(Y, E_{d}) = asdim(Y′) = 2 < 2 + 1 = δ(Y) + 1.

Note that in the previous example the inclusion Y_{k} ↪ Y is not a coarse embedding.

Remark 4.22 Let Y be a metric coarse cell complex such that all attaching maps f_{k,i} are contracting. In this case (Y_{k}, d_{k}) is a subspace of (Y, d), i.e. the inclusion map Y_{k} ↪ Y is isometric. Hence asdim(Y, E_{d}) ≥ δ(Y) + 1.

Proof. Compare Proposition 3.5. □

Question 4.23 Does asdim Y ≤ δ(Y) + 1 hold for any metric coarse cell complex Y? Does the inequality hold if all attaching maps are contracting?

1 Let X and Y be pseudometric spaces. A map f: X → Y is called λ-Lipschitz if d(f(x), f(x′)) ≤ λ · d(x, x′) for all x, x′ ∈ X. The map f is called contracting if it is 1-Lipschitz.

Asymptotic dimension and the covering dimension of the Higson corona

We will need dimension theory of topological spaces in this section, in particular some knowledge about the covering dimension dim. Compare [HW41],

[Eng78] and [Fed88]. We refer to the beginning of Section 1.4 for the definitions of the Higson compactification and the Higson corona.

For a proper metric space X, we can recover the bounded coarse structure from its Higson compactification hX, since the continuously controlled coarse structure with respect to hX and the bounded coarse structure coincide. (This is Proposition 2.47 of [Roe03].) Dranishnikov, Keesling and Uspenskij proved in [DKU98] and [Dra00] the following theorem.

Theorem 5.1 Let (X, d) be a metric space of finite asymptotic dimension and νX its Higson corona. Then asdim(X, d) = dim(νX).

The following examples show that Theorem 5.1 cannot be generalized to arbitrary coarse spaces.

Example 5.2 Let X be a non-compact, normal Hausdorff space which can be written as a union of countably many compact sets. The Stone-Čech compactification βX induces on X the trivial coarse structure T , and the Higson compactification of (X, T ) is homeomorphic to βX. We get the following results about dimensions: On the one hand we know that asdim(X, T ) = 0. On the other hand dim(βX\X) = 0 if and only if βX\X is totally disconnected. If X has the property that for every compact subset K ⊆ X the space X\K is connected, then βX\X is also connected. Hence, for such X we have dim(βX\X) ≠ 0.

Proof. Obviously T is contained in the coarse structure induced by any compactification. We write E_{βX} for the coarse structure on X induced by βX. It remains to prove that E_{βX} ⊆ T . There are compact sets K_{0} ⊂ K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ X such that X = ⋃_{i∈ℕ} K_{i}. Let E ∈ E_{βX}. Assume that for all i ∈ ℕ there is (x_{i}, y_{i}) ∈ E\(Δ_{X} ∪ K_{i}×K_{i}). This implies x_{i} ≠ y_{i}. We may assume that {x_{i} | i ∈ ℕ}∩{y_{i} | i ∈ ℕ} = ∅, since we can choose subsequences with this property. Observe that {x_{i} | i ∈ ℕ} ⊆ X and {y_{i} | i ∈ ℕ} ⊆ X are closed sets. Let (x, y) ∈ (βX\X)^{2} be an element in the closure of the sequence {(x_{i}, y_{i})}_{i∈ℕ}. Since the closure of E in βX × βX is contained in X ×X ∪ Δ_{βX}, we conclude x = y. Using Urysohn’s Lemma, we get a continuous function f: X → [0, 1] with f(x_{i}) = 1 and f(y_{i}) = 0 for all i ∈ ℕ. We may as well think of f as a continuous function with domain βX. Since inverse images of closed sets are closed, we get f(x) = 1 and f(y) = 0. This is a contradiction to x = y. Hence E ⊆ Δ_{X} ∪ K ×K for a compact set K ⊆ X, i.e. E ∈ T . Every bounded, continuous function on X is a Higson function with respect to T . Hence, the Higson compactification of (X, T ) is βX. Compare Example 3.11 and [Fed88], Chapter 1, Section 3.2, Theorem 5 for the statements on dimensions. Assume that βX\X is the disjoint union of two open and non-empty sets A and B. Then the characteristic function χ_{A} is a continuous projection in C(βX\X) ≅ C_{b}(X)/C_{0}(X).^{1} Choose a function f ∈ C_{b}(X) whose image under π: C_{b}(X) → C_{b}(X)/C_{0}(X) is χ_{A}. Observe that g := \frac{1}{2}(f + f^{∗}) is a self-adjoint preimage of χ_{A} with respect to π. In particular g is real-valued. Consider g as an element of C(βX). There is a compact set K ⊆ X such that g(x) ≠ \frac{1}{2} for all x ∈ X\K. Hence

X\K = \{x ∈ X\K | g(x) < \frac{1}{2}\} ∪ \{x ∈ X\K | g(x) > \frac{1}{2}\}

is not connected. □

Example 5.3 Let X be a proper metric space and E_{·} the continuously controlled coarse structure induced by the one-point compactification of X. In this case νX is just a single point and asdim(X, E_{·}) = 1 = dim(νX) + 1.

Proof. Compare Example 9.7 of [Roe03] for the calculation of asymptotic dimension. If X = ℝ^{n}, we can apply Example 3.27. □

1 Let M be a locally compact Hausdorff space. We denote the space of continuous functions on M with values in ℂ by C(M). The space of continuous, bounded functions on M with values in ℂ is called C_{b}(M) and by C_{0}(M) ⊆ C_{b}(M) we denote the subspace of those functions vanishing at infinity.

Example 5.4 Consider ℝ^{n} with the visual corona S^{n−1} and denote by E_{vis} the coarse structure coming from the corresponding compactification. Then asdim(ℝ^{n}, E_{vis}) = n = dim(S^{n−1}) + 1.

Proof. Example 1.47, the calculation of the asymptotic dimension of coarse cells and Proposition 3.29 yield the desired formula for the asymptotic dimension. □

Let hX be a metrisable compactification of X and denote the corresponding continuously controlled coarse structure by E_{hX}. The last two examples suggest a relation between the asymptotic dimension of (X, E_{hX}) and the covering dimension of the corona νX := hX\X. The goal of this chapter is to prove the following theorem.

Theorem 5.5

asdim(X, E_{hX}) = dim(νX) + 1

The proof of Theorem 5.5 is given by the next two lemmas.

Lemma 5.6

dim(νX) + 1 ≤ asdim(X, E_{hX})

Proof. Consider the map f: νX×ℝ_{+} → νX×ℕ, (x, t) ↦ (x, [t]) and denote by E′ the pull-back of the coarse structure E described in Lemma 1.41. Observe that f is a coarse equivalence. The coarse structure E′ has a description which is very similar to the description of E in Lemma 1.41. Theorem 1.43 now implies that (X, E_{hX}) and (νX × ℝ_{+}, E′) are coarsely equivalent. Let asdim(X, E_{hX}) = n. We have a uniformly bounded cover of νX × ℝ_{+} with appetite

E := \{((x, t), (y, s)) ∈ (νX × ℝ_{+})^{2} | d(x, y) < \frac{1}{max\{t, s\}}, |t − s| < 1\}

and multiplicity ≤ n+1. Since E((x, t)) is open for all (x, t) ∈ νX ×ℝ_{+}, this cover has an open refinement which also has appetite E and multiplicity at most n + 1. Denote this cover by U. Let ε > 0. From U we will construct an open cover U_{ε} of νX × [0, 1] with multiplicity ≤ n + 1 and mesh(U_{ε}) ≤ ε. This implies dim(νX × [0, 1]) ≤ n. We start constructing U_{ε}. Observe that L := π_{ℝ_{+}} × π_{ℝ_{+}}(Δ_{U}) ⊆ ℝ_{+}^{2} is symmetric and contains Δ_{1} := {(t, s) ∈ ℝ_{+}^{2} | |t−s| < 1}. Choose t_{ε} > 0 such that d(x, y) ≤ \frac{ε}{2} for all ((x, t), (y, s)) ∈ Δ_{U} with max\{s, t\} ≥ t_{ε}. Consider the continuous map f_{ε}: ℝ_{+} → ℝ_{+} with f_{ε}(0) = t_{ε} and

f_{ε} (\frac{n · ε}{4}) = sup L^{n}(t_{ε}) for n = \{1, 2, …\}

which is affine on the interval [\frac{n·ε}{4}, \frac{(n+1)·ε}{4}] for each n ∈ ℕ. Observe that f_{ε} is strictly monotone increasing. Define U_{ε} to be the pull-back of U under the continuous map id_{νX} ×f_{ε}. It is clear that U_{ε} has the same multiplicity as the cover U. Take V ∈ U and set U := (id_{νX} ×f_{ε})^{−1}(V). It remains to check that the diameter of U is at most ε. Let (x, t), (y, s) ∈ U. The definition of t_{ε} implies d(x, y) ≤ \frac{ε}{2}. Suppose s ≤ t. Notice that f_{ε}(t) ∈ L(f_{ε}(s)). This implies |t − s| ≤ \frac{ε}{2}. Together, we get d((x, t), (y, s)) ≤ ε. This completes the construction of the cover U_{ε}. Using some results about the covering dimension of products (compare [Fed88], Chapter 2, Section 6.3, Theorem 18 and Chapter 5, Section 5.3, Corollary 5), we get dim(νX) + 1 = dim(νX × [0, 1]) ≤ n. □

Lemma 5.7

asdim(X, E_{hX}) ≤ dim(νX) + 1

Proof. Assume dim(νX) + 1 = n. We will prove asdim(νX × ℕ, E) ≤ n where E is the coarse structure described in Lemma 1.41. Let E ∈ E be a symmetric entourage containing the diagonal Δ_{νX×ℕ} and {δ_{k}}_{k∈ℕ} a sequence of non-negative real numbers converging to zero such that d(x, x′) < δ_{k} for ((x, m), (x′, m′)) ∈ E with max{m, m′} ≥ k. We may assume this sequence to be monotone. For each k ∈ ℕ_{+} there is a finite cover of νX with open balls of radius \frac{1}{2k} and multiplicity at most n. Ostrand’s theorem^{2} provides a finite open cover V_{k} of νX with mesh(V_{k}) ≤ \frac{1}{k} consisting of n disjoint families V_{k,1}, …, V_{k,n}. For V ∈ V_{k,j} define j_{V} := j. Since νX is compact, V_{k} has positive Lebesgue number L(V_{k}) > 0. We define inductively a sequence {l_{i}}_{i∈ℕ}. Set l_{0} := 1 and for i > 0 take l_{i} ∈ ℕ such that \frac{1}{l_{i}} ≤ L(V_{l_{i−1}}) and l_{i} > l_{i−1}. We choose maps ϕ_{i}: V_{l_{i}} → V_{l_{i−1}} such that W ⊆ ϕ_{i}(W) for all W ∈ V_{l_{i}}. For V ∈ V_{l_{i−1}} we define V̆ := ⋃_{W ∈ϕ_{i}^{−1}(V)} W.

Observe that V̆ ⊆ V. Consider the projection π_{ℕ}: νX ×ℕ → ℕ and define E_{ℕ} := π_{ℕ}×π_{ℕ}(E)∪Δ_{1} where Δ_{1} := {(i, j) ∈ ℕ^{2} | |i − j| ≤ 1}. Observe that E_{ℕ} is an entourage of the continuously controlled coarse structure induced by the one-point compactification of ℕ. Set K_{0} := {0} and K_{k} := E_{ℕ}[K_{k−1}] and notice that K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ ℕ and ⋃_{k∈ℕ} K_{k} = ℕ. We define a sequence {k_{i}}_{i∈ℕ}. Set k_{−2} := 0. For all i ∈ ℤ with i ≥ −1 choose k_{i} ∈ ℕ such that k_{i} > k_{i−1} + 2n and δ_{m} < \frac{1}{l_{i+1}} for all m ∉ K_{k_{i}−2}. We need some last preparations. For i ∈ ℕ\{0} and j ∈ {1, …, n} we set A_{i,j} := K_{k_{i}}\K_{k_{i−1}+2j−2} and B_{i,j} := K_{k_{i}+2j}\K_{k_{i}}.

2 We use the following version of Ostrand’s theorem: A normal space X satisfies dim X ≤ n if and only if for every locally finite open cover U there exists an open cover V = V_{1} ∪ ⋯ ∪ V_{n+1} where each of the families V_{i} consists of disjoint sets and for each V ∈ V there exists U ∈ U such that V ⊆ U. Compare [Eng78].

Finally, we define a cover U of νX × ℕ.

U_{0} := ⋃_{V ∈V_{l_{0}}} V × K_{k_{0}+2j_{V}}

U_{i} := {V × A_{i,j_{ϕ_{i}(V )}} ∪ V̆ × B_{i,j_{V}} | V ∈ V_{l_{i}}}
U := {U_{0}} ∪ ⋃_{i∈ℕ\{0}}U_{i}

To check that U is uniformly bounded, let ((x, m_{x}), (y, m_{y})) ∈ Δ_{U}. Suppose that m_{x}, m_{y} ∉ K_{k_{−1}}. There are natural numbers k_{x} ,k_{y}, i_{x} and i_{y} such that the following is true.

m_{x} ∈ K_{k_{x}}\K_{k_{x}−1} ⊆ K_{k_{ix}} \K_{k_{ix−1}}
m_{y} ∈ K_{k_{y}}\K_{k_{y}−1} ⊆ K_{k_{iy}} \K_{k_{iy−1}}

We have |i_{x} −i_{y}| ≤ 1. Hence π_{ℕ}×π_{ℕ}(Δ_{U}) ∈ E_{ℕ} where E_{ℕ} is the continuously controlled coarse structure on ℕ induced by the one-point compactification. Without loss of generality, assume that m_{x} ≤ m_{y}. Set d_{m} := 1 if m ∈ K_{k_{1}}. Set d_{m} := \frac{1}{l_{i−2}} if m ∈ K_{k_{i}}\K_{k_{i−1}} and i > 1. Observe that lim_{m→∞} d_{m} = 0 and d(x, y) ≤ \frac{1}{l_{iy−2}} = d_{m_{y}}. Now we prove that U has multiplicity at most n+1. Let (x, m) ∈ νX ×ℕ. If m ∈ K_{k_{0}}, then (x, m) ∈ U_{0} and (x, m) is not contained in any other covering set. Suppose that m ∈ K_{k_{i}}\K_{k_{i−1}} with i ∈ ℕ\{0}. If (x, m) ∈ U for some U ∈ U, then U ∈ U_{i} ∪U_{i−1}. Observe that x is contained in at most one set of each of the collections V_{l_{i},1}, …, V_{l_{i},n}, V_{l_{i−1},1}, …, V_{l_{i−1},n}. Moreover, m ∉ A_{l,j} if l ≠ i, and m ∉ B_{l,j} if l ≠ i−1. Hence (x, m) is contained in at most 2n sets of the cover U. There is l ∈ ℕ\{0} such that m ∈ K_{k_{i−1}+2l}\K_{k_{i−1}+2l−2}. If l ≥ n, then m ∉ B_{i−1,j} for j ∈ {1, …, n − 1}. If 1 < l < n, then m ∉ B_{i−1,j} for j ∈ {1, …, l − 1} and m ∉ A_{i,j} for j ∈ {l + 1, …, n}. If l = 1, then m ∉ A_{i,j} for j ∈ {2, …, n}. Hence, there are at most n + 1 sets of the cover U containing (x, m). Finally we prove that U has appetite E. Let (x, m) ∈ νX × ℕ. Set K_{k} := ∅ for k < 0. There is a unique k ∈ ℕ such that m ∈ K_{k}\K_{k−1}. It follows that π_{ℕ}(E((x, m))) ⊆ K_{k+1}\K_{k−2}. There is a unique i ∈ {−1} ∪ ℕ with k_{i−1} ≤ k < k_{i}. We conclude that π_{ℕ}(E((x, m))) ⊆ K_{k_{i}}\K_{k_{i−1}−2}. If i ≤ 0, then E((x, m)) ⊆ νX × K_{k_{0}} ⊆ U_{0}. Suppose i > 0. Observe that π_{νX}(E((x, m))) is contained in the ball B_{δ_{m}}(x) of radius δ_{m} around x. There is W ∈ V_{l_{i}} such that B_{δ_{m}}(x) ⊆ W. Set V := ϕ_{i}(W) and observe that W ⊆ V ∈ V_{l_{i−1}}. It remains to prove that W × K_{k+1}\K_{k−2} is contained in some U ∈ U. If k ≥ k_{i−1} + 2j_{V}, then

W × K_{k+1}\K_{k−2} ⊆ W × K_{k_{i}}\K_{k_{i−1}+2j_{V} −2}
= W × A_{i,j_{V}} ⊆ W × A_{i,j_{V}} ∪ W̆ × B_{i,j_{W}} ∈ U_{i} .

If k < k_{i−1} + 2j_{V} and i > 1, then

W × K_{k+1}\K_{k−2} ⊆ W × K_{k_{i−1}+2j_{V}} \K_{k_{i−1}−2}
⊆ V × A_{i−1,j_{ϕ_{i−1}(V )}} ∪ V̆ × B_{i−1,j_{V}} ∈ U_{i−1} .

If k < k_{i−1} + 2j_{V} and i = 1, then

W × K_{k+1}\K_{k−2} ⊆ V × K_{k_{0}+2j_{V}} ⊆ U_{0} ∈ U .

This completes the proof that the cover U has appetite E. □

Remark 5.8 For any coarse spaces X and Y whose coarse structures are induced by metrisable compactifications, we get another proof of

asdim(X × Y) ≤ asdim(X) + asdim(Y).

Proof. By Corollary 1.45 the coarse space X is coarsely equivalent to ψ(νX) and the same for Y. This and Proposition 1.46 imply that X × Y is coarsely equivalent to ψ(νX ⊠ νY). Hence, the Higson corona of X × Y is homeomorphic to νX ⊠ νY. Using Theorem 5.5 and the product theorem of dimension theory (i.e. [Fed88], Chapter 2, Section 6.3, Theorem 18), we get asdim(X ×Y) = dim(νX ⊠ νY) + 1 = dim(νX ×νY ×[0, 1]) + 1 ≤ dim(νX) + dim(νY) + 2 = asdim(X) + asdim(Y). □

Corollary 5.9 There are coarse spaces X and Y such that

asdim(X × Y) < asdim(X) + asdim(Y).

Proof. There are metric compacta K and L such that dim(K × L) < dim(K) + dim(L). Compare the section about Pontryagin’s compacta, i.e Capter 5, Section 4.2 of [Fed88]. The same reasoning as in the proof of Remark 5.8 translates this to asymptotic dimensions. □

Remark 5.10 In [BL] the authors construct metric spaces X and Y such that asdim(X × Y) < asdim(X) + asdim(Y).

Asymptotic dimension of CAT(κ)-spaces for κ < 0

Let κ < 0. We write M_{κ}^{n} for the n-dimensional hyperbolic space with constant sectional curvature κ. We need some basic facts about CAT(κ)- spaces and some knowledge in hyperbolic geometry. A good introduction on these subjects can be found in [BH99]. We want to calculate the asymptotic dimension of CAT(κ)-spaces. As the following example shows, we should not expect that the asymptotic dimension and the covering dimension of a CAT(κ)-space always coincide.

Example 6.1 Consider an n-simplex σ in M_{κ}^{n}. Take a k-dimensional face f of σ. There is an isometric embedding i: M_{κ}^{k} ↪ M_{κ}^{n} such that f ⊆ im i. Now σ ∪ im i is a CAT(0)-space with covering dimension n and asymptotic dimension k.

Hence, for every pair of natural numbers (k, n) with k ≤ n, there is a CAT(κ)-space with asymptotic dimension k and covering dimension n.

6.1 Spaces with nicely covered spheres

Let (X, d) be a metric space and x_{0} ∈ X a basepoint. For r ≥ 0 we define S_{r}(x_{0}) = {x ∈ X | d(x, x_{0}) = r} and D_{r}(x_{0}) = {x ∈ X | d(x, x_{0}) < r}. We consider S_{r}(x_{0}) ⊆ X and D_{r}(x_{0}) ⊆ X with the metric obtained by restricting the metric d of X.

Definition 6.2 We say that X has nicely n-colored spheres (with respect to the basepoint x_{0} ∈ X) if there is ρ > 0 and a cover U_{k} of S_{k·ρ}(x_{0}) for all k ∈ ℕ such that

• there is a lower bound λ > 0 on the Lebesgue numbers of U_{k} for k ∈ ℕ,

• there is an upper bound on the mesh of the covers U_{k} and

• U_{k} = U_{k,1} ∪ ⋯ ∪ U_{k,n} with multiplicity μ(U_{k,i}) ≤ 1 for i ∈ {1, …, n}.

Definition 6.3 We say that X has nicely n-covered spheres (with respect to the basepoint x_{0} ∈ X) if there is ρ > 0 and a cover U_{k} of S_{k·ρ}(x_{0}) for all k ∈ ℕ such that

• there is a lower bound λ > 0 on the Lebesgue numbers of U_{k} for k ∈ ℕ,

• there is an upper bound on the mesh of the covers U_{k} and

• U_{k} has multiplicity at most n.

Lemma 6.4 A metric space X has nicely n-colored spheres if and only if it has nicely n-covered spheres.

Proof. Compare Remark 3.17. □

Theorem 6.5 Let κ < 0. For a CAT(κ)-space X with nicely n-covered spheres, we have asdim(X) ≤ n.

Remark 6.6 In some cases the upper bound on the asymptotic dimension of X given by Theorem 6.5 is lower than the covering dimension of X. Compare Example 6.1 and observe that in this case the upper bound provided by Theorem 6.5 is sharp.

6.2 Proof of Theorem 6.5

Because of Lemma 6.4 we may assume that X has nicely n-colored spheres. We will prove Theorem 6.5 by constructing uniformly bounded covers M_{ρ,N} of X with multiplicity at most n+1. In a second step we prove that for every L > 0 we can adjust the parameters ρ and N so that M_{ρ,N} has Lebesgue number at least L.

Step 1: Constructing covers

Let x_{0} be a basepoint of X and ρ > 0 such that for k ∈ ℕ the sphere S_{k} := S_{k·ρ}(x_{0}) is covered by U_{k} = U_{k,1} ∪ ⋯ ∪ U_{k,n} with Lebesgue number L(U_{k}) ≥ λ, mesh(U_{k}) ≤ D and μ(U_{k,i}) = 1 for all i ∈ {1, …, n}. Observe that U_{0} = {{x_{0}}}. For k ∈ ℕ we define D_{k} := D_{k·ρ}(x_{0}). Since X is a CAT(κ)-space, the space X is uniquely geodesic. We denote the geodesic segment from x to y by [x, y]. Two parameters will appear in the following construction: the distance ρ between spheres and the magnitude of shift N ∈ ℕ towards the basepoint. Note that whenever we have a space with nicely n-colored spheres of distance ρ, we can choose any integral multiple of ρ as a new distance of spheres, i.e. we may choose ρ as large as we want.

For k ∈ ℕ we define a map θ_{k}: X\D_{k} → S_{k} by assigning to x ∈ X\D_{k} the intersecting point of the geodesic segment [x_{0}, x] with S_{k}.

Lemma 6.7 The map θ_{k} is contracting for any k ∈ ℕ.

Proof. For x ∈ X set r(x) = d(x_{0}, x). Let x, y ∈ X\D_{k}. Without loss of generality assume r(x) ≤ r(y). Take z ∈ [x_{0}, y] with r(z) = r(x). Consider the triangle with vertices x_{0}, x and y. Take a comparison triangle in M_{κ}^{2} and denote its vertices by \overline{x_{0}}, \overline{x} and \overline{y}. Then

d(θ_{k}(x), θ_{k}(y)) ≤ d (\overline{θ_{k}(x)}, \overline{θ_{k}(y)}) ≤ d (\overline{x}, \overline{z}) ≤ d (\overline{x}, \overline{y}) = d(x, y) .

Here \overline{θ_{k}(x)} and \overline{θ_{k}(y)} are the points on the chosen comparison triangle corresponding to θ_{k}(x) and θ_{k}(y) respectively. □

For k ∈ ℕ we choose a map Θ_{k}: U_{k+n} → U_{k}. We will see in Corollary 6.10 that for ρ large enough we may assume that Θ_{k} has the following property:

θ_{k}(U) ⊆ Θ_{k}(U) for k ∈ ℕ and U ∈ U_{k+n}  (6.1)

Let N ∈ ℕ, k ∈ n · ℕ and i ∈ {1, …, n}. For U ∈ U_{k,i} we define as follows:

A(U) := \{ \begin{matrix}D_{N+n} & if k = 0 \\ θ_{k}^{−1}(U) ∩ D_{k+N+n}\D_{k+N+i−1} & if k > 0 \end{matrix}
B(U) := θ_{k}^{−1}(U) ∩ D_{k+N+i}\D_{k+N}
U^{#} := A(U) ∪ ⋃_{V ∈Θ_{k}^{−1}(U)} B(V)

We get the following cover of X.

M_{ρ,N} := {U^{#} | k ∈ n · ℕ, U ∈ U_{k}}

Besides ρ and N this cover does depend on the choice of the maps Θ_{k}.

Lemma 6.8 The multiplicity of M_{ρ,N} is at most n + 1. Moreover, if the maps Θ_{k} have property (6.1), then M_{ρ,N} is uniformly bounded.

Proof. Let x ∈ X. Choose k ∈ n · ℕ and j ∈ {1, …, n} such that x ∈ D_{k+N+j}\D_{k+N+j−1}. Observe that for i ∈ {1, …, n} the point θ_{k}(x) belongs to at most one set from U_{k,i}. Hence x ∈ A(U) for at most j sets U ∈ U_{k}. Similarly x ∈ B(U) for at most n + 1 − j sets U ∈ U_{k}. Assume that the maps Θ_{k} have property (6.1). Let U ∈ U_{k} Observe that for x, y ∈ U^{♯} we have

d(x, y) ≤ d(x, U) + diam(U) + d(U, y) ≤ (N + 2n) · ρ + D + (N + 2n) · ρ .

Hence M_{ρ,N} is uniformly bounded. □

Step 2: Adjusting parameters

In order to conclude the proof of Theorem 6.5, it remains to prove that for every L > 0 we can choose the parameters ρ and N such that M_{ρ,N} has Lebesgue number at least L.

Lemma 6.9 Let δ > 0 and B ⊆ S_{ρ·k+a}(x_{0}) with diam(B) < a. The map θ_{k}|_{B}: B → S_{k} is δ-Lipschitz if a > \frac{2}{\sqrt{−κ}} · max \{1, log \frac{2}{δ}\}.

Proof. Set r = ρ · k + \frac{a}{2}. Let x, y ∈ B. Consider the geodesic triangle with vertices x_{0}, x and y. Denote the vertices of a comparison triangle in M_{κ}^{2} by \overline{x_{0}}, \overline{x} and \overline{y} respectively. We write \overline{θ_{k}(x)} and \overline{θ_{k}(y)} for the points on the chosen comparison triangle corresponding to θ_{k}(x) and θ_{k}(y) respectively. By \overline{x}_{m} we denote the point on [\overline{x_{0}}, \overline{x}] with distance \frac{a}{2} to \overline{x}. Similarly, we define a point \overline{y}_{m} on the segment [\overline{x_{0}}, \overline{y}]. Let α be the angle at \overline{x_{0}} between [\overline{x_{0}}, \overline{x}] and [\overline{x_{0}}, \overline{y}]. By γ_{r} we denote the circular arc parametrized by arc length from \overline{x}_{m} to \overline{y}_{m} in S_{r}(\overline{x_{0}}) ⊂ M_{κ}^{2} and by γ_{ρ·k} the corresponding arc from \overline{θ_{k}(x)} to \overline{θ_{k}(y)} in S_{ρ·k}(\overline{x_{0}}). Observe

d(θ_{k}(x), θ_{k}(y)) ≤^{(1)} d (\overline{θ_{k}(x)}, \overline{θ_{k}(y)}) ≤^{(2)} length of γ_{ρ·k}
=^{(3)} \frac{α}{\sqrt{−κ}} · sinh (\sqrt{−κ} · ρ · k)
< \frac{α}{2 · \sqrt{−κ}} exp (\sqrt{−κ} · ρ · k)
= \frac{α}{2 · \sqrt{−κ}} exp (\sqrt{−κ} · r) · exp (−\sqrt{−κ} · \frac{a}{2})
<^{(4)} \frac{α}{2 · \sqrt{−κ}} exp (\sqrt{−κ} · r) · \frac{δ}{2}
<^{(5)} \frac{α}{2 · \sqrt{−κ}}( exp (\sqrt{−κ} · r) − exp (−\sqrt{−κ} · r) ) · δ
= \frac{α}{\sqrt{−κ}} · sinh (\sqrt{−κ} · r) · δ =^{(3)} δ · length of γ_{r}
≤^{(6)} δ · d (\overline{x}, \overline{y}) = δ · d(x, y) .

Since X is a CAT(κ)-space, we can conclude (1). The fact that γ_{ρ·k} is a path from \overline{θ(x)} to \overline{θ(y)} implies inequality (2). For (3) compare Proposition 6.17 in Part I of [BH99] about the Riemannian metric of M_{κ}^{2} in polar coordinates. The assumption a > \frac{2}{\sqrt{−κ}} · log \frac{2}{δ} implies exp (−\sqrt{−κ} · \frac{a}{2}) < \frac{δ}{2} and hence (4). The assumption a > \frac{2}{\sqrt{−κ}} yields \sqrt{−κ} · r > 1. This implies exp(\sqrt{−κ} · r) < 2 · (exp (\sqrt{−κ} · r) − exp (−\sqrt{−κ} · r)) which proves inequality (5). Since d(\overline{x}, \overline{y}) ≤ a, the geodesic segment [\overline{x}, \overline{y}] lies in M_{κ}^{2}\D_{r}(\overline{x_{0}}). For a moment consider M_{κ}^{2}\D_{r}(\overline{x_{0}}) and S_{r}(\overline{x_{0}}) as metric spaces with their length metric. ^{1} Note that the radial projection M_{κ}^{2}\D_{r}(\overline{x_{0}}) → S_{r}(\overline{x_{0}}) with center \overline{x_{0}} is a contracting map which maps the geodesic segment [\overline{x}, \overline{y}] onto γ_{r}. This implies (6). □

Remember that λ is a lower bound on the Lebesgue number and D an upper bound on the mesh of the covers U_{i} for i ∈ ℕ.

Corollary 6.10 If ρ · n > D and ρ · n > \frac{2}{\sqrt{−κ}} · max \{1, log \frac{2·D}{λ} \}, then we can choose Θ_{k} such that θ_{k}(U) ⊆ Θ_{k}(U) for all k ∈ ℕ and U ∈ U_{k+n}.

Proof. Set δ = \frac{λ}{D}. Lemma 6.9 implies that θ_{k} maps any U ∈ U_{k+n} into a ball B ⊆ S_{k} of radius λ. Since U_{k} has Lebesgue number λ, we may choose Θ_{k}(U) such that B ⊆ Θ_{k}(U). □

Corollary 6.11 Let L > 0. Choose ρ > 2 · L and such that Corollary 6.10 applies. Choose N ∈ ℕ such that (N − 1) · ρ > 2L and (N − 1) · ρ > \frac{2}{\sqrt{−κ}} ·max \{1, log \frac{2·L}{λ}\}. Under these assumptions M_{ρ,N} has Lebesgue number at least L.

Proof. Let x ∈ X and let j ∈ ℕ be the smallest natural number such that the ball D_{L}(x) of radius L around x is contained in D_{j+N}. We claim that D_{L}(x) ⊆ X\D_{j+N−2}. Assume D_{L}(x) ∩ D_{j+N−2} ≠ ∅. This implies D_{L}(x) ⊆ {x ∈ X | d(x, D_{j+N−2}) < 2L} ⊆ D_{j+N−1}. Here the last inclusion follows from ρ > 2L. But this is a contradiction to the definition of j. Take k ∈ n · ℕ and l ∈ {1, …, n} such that j = k + l. Since θ_{k+N−1} is contracting, the image of D_{L}(x) is contained in a ball B ⊆ S_{k+N−1} of radius L. Set δ = \frac{λ}{L}. Lemma 6.9 implies that θ_{k}(B) is contained in a ball B′ ⊆ S_{k} of radius λ. Let U ∈ U_{k,i} be a covering set containing B′. Then D_{L}(x) ⊆ θ_{k}^{−1}(U). Hence D_{L}(x) ⊆ A(U) if i < l ≤ n. On the other hand, we get D_{L}(x) ⊆ B(U) if 1 < l ≤ i. Suppose l = 1. Set V := Θ_{k−n}(U). Note that Corollary 6.10 yields θ_{k}^{−1}(U) ∩ D_{k+N}\D_{k+N−1} ⊆ A(V). Hence D_{L}(x) ⊆ V^{#}. □

This completes the proof of Theorem 6.5.

6.3 Triangulations of bounded distortion

Simplicial complexes

A (combinatorial) simplicial complex C consists of a set vert(C) and a collection Σ(C) of finite, non-empty subsets of vert(C) such that Σ(C) is closed under taking subsets, i.e. σ ∈ Σ(C) and τ ⊆ σ imply τ ∈ Σ(C). The collection Σ(C) must contain {v} for every v ∈ vert(C). An element v ∈ vert(C) is called a vertex of C and σ ∈ Σ(C) is called a simplex of C. The dimension of σ is defined to be the cardinality of σ minus one. The dimension of C is defined to be the supremum of the dimensions of its simplices, i.e. dim(C) = sup{dim(σ) | σ ∈ Σ(C)}. By C^{(k)} we denote the k-skeleton of C, i.e. the simplicial complex obtained from C by removing all simplices of dimension greater than k. A simplicial complex is said to be locally finite if each vertex is contained in only finitely many simplices. By δ_{v}: vert(C) → ℂ we denote the function which maps v to 1 and all the other vertices to 0. We identify vert(C) as follows with a subset of ℓ^{2}(vert(C)):

1Compare [BH99] for details on length metrics.

vert(C) ↪ ℓ^{2}(vert(C)) , v ↦ \frac{1}{\sqrt{2}} · δ_{v}

Observe that the distance of any two vertices in ℓ^{2}(vert(C)) is one. We write |σ| for the convex hull of σ in ℓ^{2}(vert(C)) and call |σ| a geometric simplex. The euclidean realization |C| of C is defined to be the union of the geometric simplices. There are two natural metrics on |C|. The affine metric is just the restriction of the standard metric on ℓ^{2}(vert(C)) to |C|. The geodesic metric is the length metric associated to the affine metric. We will consider |C| with the geodesic metric. Note that restricted to a single simplex both metrics coincide. We write Star(v) for the open star of v ∈ vert(C), i.e. the interior of the union of all geometric simplices |σ| ⊆ |C| with v ∈ σ. The stability of a finite dimensional simplicial complex C is defined to be the biggest natural number k such that there exist two different k-simplices σ, τ ∈ Σ(C) such that σ ∩ τ is a (k − 1)-simplex of C.

Lemma 6.12 Let C be a simplicial complex of dimension n and stability k. The cover V = {Star(v) | v ∈ vert(C)} of |C| is uniformly bounded and has multiplicity n + 1. Moreover V has Lebesgue number

λ_{k} = \frac{1}{\sqrt{2 · k · (k + 1)}} .

Proof. The mesh of the cover V is at most 2, i.e. V is uniformly bounded. The multiplicity of V is n+1, since a simplex of C has at most n+1 vertices. The Lebesgue number of V is exactly the distance of the center of a k-simplex to a face of dimension k − 1. Hence, an elementary computation yields the claim. □

Spaces with triangulations of bounded distortion

Definition 6.13 A triangulation of bounded distortion of a metric space X is a tuple (C, ψ, l_{1}, l_{2}) where C is a simplicial complex, l_{1}, l_{2} > 0 and ψ: |C| → X is a homeomorphism such that for every σ ∈ Σ(C) the map ψ|_{|σ|} is (l_{1}, l_{2})-bi-Lipschitz, i.e.

l_{1} · d(x, y) ≤ d(ψ(x), ψ(y)) ≤ l_{2} · d(x, y)

for all x, y ∈ |σ|. We say that the triangulation is of dimension dim(C).

Note that according to our definition the simplicial complex of a triangulation of bounded distortion does not have to be locally finite.

Proposition 6.14 Let X be a metric space and (C, ψ, l_{1}, l_{2}) a triangulation of bounded distortion such that C is of dimension n and stability k. Under these assumptions U = {ψ(Star(v)) | v ∈ vert(C)} is a cover of X with the following properties:

• The cover U is uniformly bounded. More precisely, the diameter of any U ∈ U is at most 2 · l_{2}.

• When restricted to a single connected component, the cover U has Lebesgue number L(U) ≥ l_{1} ·λ_{k} where λ_{k} is defined as in Lemma 6.12.

• The cover U has multiplicity n + 1.

For the rest of this section let X be a CAT(κ)-space which admits a triangulation of bounded distortion (C, ψ, l_{1}, l_{2}) of dimension n. Choose a basepoint x_{0} ∈ X. As above we set S_{r} := {x ∈ X | d(x, x_{0}) = r}.

Remark 6.15 By restricting the cover U defined in Proposition 6.14 to the spheres S_{r} with r > 0 we see that X has nicely (n + 1)-covered spheres. Hence, Theorem 6.5 yields asdim(X) ≤ n + 1.

The original goal is to prove asdim(X) ≤ n, but this seems to be much more difficult - at least in full generality. However, if there is a radial structure for X as defined below, we do obtain asdim(X) ≤ n.

Definition 6.16 Let ρ > 0. For k ∈ ℕ define C_{k} to be the smallest simplicial complex containing as simplices all σ ∈ Σ(C) such that ψ(|σ|) has non-empty intersection with S_{k·ρ}. Let D > 0. Assume that for all k ∈ ℕ there is an equivalence relation on vert(C_{k}) such that

• in each n-simplex of C_{k} there are two equivalent vertices and

• the number D is an upper bound for the diameter of the image under ψ of any equivalence class.

A family of equivalence relations as described above will be called a radial structure for X.

One might try to get a radial structure by defining the endpoints of “radial” edges to be equivalent.

Proposition 6.17 If X has a radial structure, then asdim(X) ≤ n.

Proof. Consider the collection U_{k} = {ψ(Star(v)) | v ∈ vert(C_{k})}. We say U_{1}, U_{2} ∈ U_{k} are equivalent if they contain equivalent vertices. For U ∈ U_{k} define Ũ to be the union of all V ∈ U_{k} which are equivalent to U. By restricting Ũ_{k} := {Ũ | U ∈ U_{k}} to S_{k·ρ} we get a uniformly bounded cover of S_{k·ρ} with multiplicity at most n and Lebesgue number at least L(U). Applying Theorem 6.5 completes this proof. □

Question 6.18 Is there a CAT(κ)-space with a triangulation of bounded distortion which does not admit a radial structure?

6.4 Complete, simply connected manifolds with bounded, strictly negative sectional curvature

Theorem 6.20 has been conjectured by Mikhael Gromov. Compare Example 1.E′_{1} of [Gro93].

Definition 6.19 Let X be a Riemannian manifold. For x ∈ X and a twodimensional subspace p ⊆ T_{x}X, we denote the sectional curvature of the plane p by K(p) ∈ ℝ. We say that the sectional curvature of X is bounded between κ′ and κ or simply that X has bounded sectional curvature if κ′ ≤ K(p) ≤ κ for all x ∈ X and all two-dimensional subspaces p ⊆ T_{x}X. We say that X has strictly negative sectional curvature if there is κ < 0 such that K(p) ≤ κ for all x ∈ X and all two-dimensional subspaces p ⊆ T_{x}X.

Theorem 6.20 For a complete, simply connected Riemannian manifold X with bounded, strictly negative sectional curvature we have

asdim(X) = dim(X).

We will prove this theorem by applying Theorem 6.5. A different approach (using the boundary at infinity and a new invariant called capacity dimension) has been taken in [Buy] and [BL] to prove similar results. These results include asdim(X) ≤ dim(X) if in addition to our assumptions X is cobounded, i.e. if there is a bounded subset B ⊂ X such that the orbit of B under the isometry group of X covers X. Compare Theorem 1.1 of [Buy] and Corollary 1.2 and Proposition 6.2 of [BL]. Before proceeding with the proof of Theorem 6.20, we will look at the differential geometry of the spheres in X.

Lemma 6.21 Let X be a complete, simply connected Riemannian manifold of dimension n with bounded, strictly negative sectional curvature. Then there are κ_{1}, κ_{2} ∈ ℝ and ρ > 0 such that for all x_{0} ∈ X and r ≥ 1

(a) the sectional curvature of the sphere S_{r}(x_{0}) = {x ∈ X | d(x, x_{0}) = r} is bounded between κ_{1} and κ_{2} and

(b) the injectivity radius of S_{r}(x_{0}) is at least ρ.

Proof. Let x_{0} ∈ X and r ≥ 1. First we calculate bounds for the sectional curvature of S_{r}(x_{0}). Suppose the sectional curvature of X is bounded between −κ′ and −κ. Define f: X → ℝ, x ↦ d(x_{0}, x). Note that for each x ∈ X the second fundamental tensor S_{grad f}: T_{x}S_{r}(x_{0}) → T_{x}S_{r}(x_{0}) is just the restriction of the Hessian^{2} Hess(f) to the directions tangential to the spheres. Let x ∈ X with f(x) = r ≥ 1. Denote the eigenvalues of Hess(f) at x by τ_{1}(x), ⋯, τ_{n−1}(x). Using Riccati comparison arguments, we get

\sqrt{κ} ≤ \sqrt{κ} · coth (\sqrt{κ} · r) ≤^{(♭)} τ_{i}(x) ≤^{(♯)} \sqrt{κ′} · coth (\sqrt{κ′} · r) ≤ \sqrt{κ′} · coth (\sqrt{κ′}) .

For (♭) compare 1.7.3 of [Kar87]. For (♯) compare 1.7.1 of [Kar87] or Sections 1.4 and 1.6 of [Mey89]. Hence, the norm of the second fundamental form is bounded in terms of κ′ and κ. Using the Gauß equation (see [Car92] or [Jos02]) in order to relate the curvature of X and the spheres, we get universal bounds

κ_{1} := κ − κ′ − κ′ · coth^{2} (\sqrt{κ′}) and κ_{2} := −2 · κ + κ′ · coth^{2} (\sqrt{κ′})

for the sectional curvature of S_{r}(x_{0}). Applying Corollary 4 of [AL] yields that the injectivity radius of S_{r}(x_{0}) is at least ρ := \frac{π}{\sqrt{κ′}·coth(\sqrt{κ′})}. □

Proof of Theorem 6.20. First note that X is a CAT(κ)-space. Compare Theorem II.1A.6 and the Cartan-Hadamard Theorem (II.4.1) of [BH99]. We need completeness of X to apply the Cartan-Hadamard Theorem. Lemma 6.21 yields that M := ⋃_{n∈ℕ} S_{n}(x_{0}) ⊂ X is a Riemannian manifold of bounded geometry, i.e. M has bounded sectional curvature and positive injectivity radius. Hence, we may apply the following theorem.

Theorem 6.22 If M is an n-dimensional Riemannian manifold of bounded geometry, then M admits a triangulation of bounded distortion and the dimension of this triangulation is n.

2 If f: X → ℝ is a differentiable function on a Riemannian manifold X, ∇ the Levi-Civita connection and Y a tangent vector field on X, then Hess(f)(Y) = ∇_{Y} grad f.

Theorem 6.22 is an immediate consequence of Theorem 1.14 in [Att94]. Moreover, Theorem 6.22 and Proposition 6.14 imply that X has nicely covered spheres. Applying Theorem 6.5 now yields asdim(X) ≤ dim(X). Suppose asdim(X) =: k < n := dim(X). This implies that there is a uniformly bounded, open cover U of X with multiplicity at most k + 1. Set D := mesh(U). Choose a basepoint x_{0} ∈ X. Let ε > 0. We consider the following diffeomorphism f_{ε}: X → X. Define f_{ε}(x) to be the point on the geodesic segment from x_{0} to x with distance \frac{ε}{D} ·d(x_{0}, x) to x_{0}. Observe that f_{ε}(x) is \frac{ε}{D} -Lipschitz, since the corresponding map on ℝ^{n} is \frac{ε}{D} -Lipschitz and X is a CAT(0)-space. Now the image of the cover U under f_{ε} gives an open cover of X with sets of diameter at most ε and with multiplicity at most k + 1. Applying Theorem 1.6.12 of [Eng78] yields dim(X) ≤ k < n. This is a contradiction. Hence asdim(X) ≥ dim(X). □

Remark 6.23 We actually proved asdim(X) ≥ dim(X) for any complete, simply connected Riemannian manifold with strictly negative sectional curvature.

Question 6.24 Is there a complete, simply connected Riemannian manifold with strictly negative, but unbounded sectional curvature and asdim(X) > dim(X)?

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Index

affine metric, 68 ample, 25 anti-Čech system, 39 appetite, 36 asdim, 34, 36, 37 attaching coarse spaces, 12

bounded, 5 bounded coarse structure, 6 bounded distortion, 68 bounded sectional curvature, 70

C, 7 C-equivalence, 8 C_{cn}, 7 cell dimension, 48 close maps, 4, 7 cnCS, 6 coarse cell complex, 48 coarse embedding, 7, 40 coarse invariance, 41 coarse inverse, 8 coarse map, 4, 7 coarse n-cell, 48 coarse n-sphere, 48 coarse space, 5 coarse structure, 4 coarsely equivalent, 4, 8 coarsely proper, 4, 7 coarsely uniform, 4, 7 compactification, 7 compatibility of coarse structure and

topology, 6 composition of entourages, 5 computable supports, 22 connected coarse structure, 5 continuously controlled, 7 contracting, 55, 65 controlled, 5 coproduct, 10, 43 CS, 6

D, 7

D-equivalence, 8

D_{cn}, 7

diagonal of a set, 4 direct limit, 10, 43 directed set, 10 discretization, 20 L-disjoint, 34, 37 disjoint union, 11, 43

E(...), 5 E[...], 5 E^{∗}(…), 24 E_{G}, 7 entourage, 4

generated coarse structure, 6 geodesic metric, 68 geometric Hilbert space, 22

Higson compactification, 14 Higson corona, 14 Higson function, 13

Lebesgue number, 33

Lipschitz, 55

localizing σ-algebra, 20 localizing decomposition, 19

mesh, 33

metric coarse cell complex, 54 monotony, 34, 41 multiplicity, 33

one-point compactification, 41

product coarse structure, 11, 44 projection-valued measure, 20 proper, 6 pull-back, 9

quotient coarse structure, 12 quotione coarse structure, 45

radial structure, 69 ray, 47 restriction, 9 Roe-C^{∗}-algebra, 25

sectional curvature, 70 Sperner’s Lemma, 50 stability, 68 star, 68 Stone-Čech compactification, 57 strictly negative sectional curvature,

70 substantial part, 41 support, 22 symmetric entourage, 36

translation C^{∗}-algebra, 24 tree, 34 triangulation of bounded distortion,

68 trivial coarse structure, 36, 57

uniform cover, 39 uniformly bounded, 34, 36 union of coarse spaces, 9, 42

word metric, 3

Lebenslauf

Persönliche Daten

Name  Bernd Grave
Geburtsdatum  6. Oktober 1975
Geburtsort  Damme, Niedersachsen
Staatsangehörigkeit deutsch

Schulausbildung

1982 – 1988 Grundschule und Orientierungsstufe in Holdorf
1988 – 1995 Gymnasiums in Damme
1995  Abitur

Zivildienst

1995 – 1996 Zivildienst in einem Altenpflegeheim in Holdorf

universitäre Ausbildung

1996 – 1999 Studium der Mathematik und der Physik an der

Westfälischen-Wilhelms-Universität Münster

1998  Vordiplom in Mathematik und in Physik
1999 – 2000 Studium der Mathematik an der Universidad

Complutense de Madrid

2000 – 2001 Studium der Mathematik an der Universität Münster
2002  Diplom in Mathematik
seit 2002  Promotionsstudent im Graduiertenkolleg Gruppen und

Geometrie an der Georg-August-Universität Göttingen

2004  Forschungsaufenthalt an der Vanderbilt University

in Nashville, Tennessee

Göttingen, den 19. Dezember 2005 Bernd Grave