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Coarse homology theories

In this paper we develop an axiomatic approach to coarse homology theories. We prove a uniqueness result concerning coarse homology theories on the category of `coarse CW-complexes'. This uniqueness result is used to prove a version of the coarse Baum-Connes conjecture for such spaces.


Introduction
Coarse geometry is the study of large scale structures on geometric objects. A large-scale analogue of cohomology, called coarse cohomology, was introduced by J. Roe in [10] in order to perform index theory on non-compact manifolds. The dual theory, coarse homology, has proved useful in formulating a coarse version of the Baum-Connes conjecture; see [8] and [14]. 1 This coarse Baum-Connes conjecture has similar consequences to the original Baum-Connes conjecture formulated in [2]. In particular, the coarse Baum-Connes conjecture implies the Novikov conjecture concerning the oriented homotopy-invariance of higher signatures on a manifold. The book [11] contains an overview of coarse geometric techniques applied to such problems.
In this paper we present an axiomatic approach to coarse (generalised) homology theories; the axioms used are coarse analogues of the classical Eilenberg-Steenrod axioms (see [4]).
A coarse (n − 1)-sphere is defined to be Euclidean space R n equipped with a suitable coarse structure compatible with the topology. This terminology comes from thinking of a coarse (n − 1)-sphere as an ordinary (n − 1)-sphere 'at infinity'. There is a similar notion of a coarse n-cell. Using these basic building blocks we can formulate the notion of a coarse CW -complex.
The main result of this article is the fact that a coarse homology theory is uniquely determined on the category of finite coarse CW -complexes by its values on the one-point coarse space and on coarse 0-cells. The proof of this result is analogous to the corresponding uniqueness result for generalised homology theories in classical algebraic topology.
As an application, we use this result together with the main results from [7] to study the coarse Baum-Connes conjecture. In particular, using coarse structures arising from 'continuous control at infinity' (see [1]) we obtain a result on the Novikov conjecture that appears to be new.
The remainder of the paper comprises: Section 2: The coarse category; section 3: Coarse homology theories; section 4: Relative coarse homology; section 5: Coarse CW -complexes; section 6: The coarse assembly map.
• The inverse of an entourage M : is contained in an entourage.
• The composition of entourages M 1 and M 2 : is contained in an entourage.
The coarse space X is called unital if the diagonal, D = {(x, x) | x ∈ X}, is contained in an entourage.
Two coarse structures {M α ⊆ X × X | α ∈ A} and {N β ⊆ X × X | β ∈ B} on a set X are said to be equivalent if every entourage M α is contained in some entourage N β , and every entourage N β ′ is contained in some entourage M α ′ . We do not distinguish between equivalent coarse structures.
Definition 2.2 Let X be a coarse space, and consider maps f : S → X and g : S → X for some set S . Then the maps f and g are called close or coarsely equivalent if the set: {(f (s), g(s)) | s ∈ S} is contained in an entourage.
There is a notion of a subset of a coarse space being of finite size. Definition 2.3 Let X be a coarse space, and consider a subset A ⊆ X and an entourage M ⊆ X × X . Then we define: Observe that a subset of a bounded set is bounded, and a finite union of bounded sets is bounded. If B ⊆ X is a bounded set, and M is an entourage, then the set M [B] is bounded.
In this article we explore invariants that depend only on the coarse structure of a given space. However, it can be useful to know when coarse structures are compatible with other structures that may be present. The following definition also comes from [7].
Definition 2.4 Let X be a coarse space. Then X is called a coarse topological space if it is equipped with a Hausdorff topology such that every entourage is open, and the closure of every bounded set is compact.
In fact it is easy to demonstrate that any coarse topological space is locally compact, and the bounded sets are precisely those which are precompact. If X is a coarse space equipped with some topology, we say that the topology and coarse structure are compatible when X is a coarse topological space.
The main examples of coarse structures we will use are the bounded coarse structure on a metric space, and the continuously controlled coarse structure arising from a compactification of a topological space.
Definition 2.5 If X is a proper metric space, the bounded coarse structure is the unital coarse structure formed by defining the entourages to be neighbourhoods of the diagonal:  [3].
Bounded coarse structures are more general that they might at first appear because of the following result, which is proved in [12].
Proposition 2.7 Let X be a coarse space. Suppose that X is countably generated in the sense that there is a sequence of entourages (M n ) such that every entourage M is contained in a finite composition of the form M 1 M 2 · · · M n .
Then the coarse structure on X is equivalent to the bounded coarse structure arising from some metric.
A coarse map is a structure-preserving map between coarse spaces.
Definition 2.8 Let X and Y be coarse spaces. Then a map f : X → Y is said to be coarse if: • The mapping f × f : X × X → Y × Y takes entourages to subsets of entourages The main definitions of coarse geometry are motivated by looking at coarse maps between metric spaces equipped with the bounded coarse structure; see [11].
We can form the category of all coarse spaces and coarse maps. We call this category the coarse category. We call a coarse map f : X → Y a coarse equivalence if there is a coarse map g : Y → X such that the composites g • f and f • g are close to the identities 1 X and 1 Y respectively.
Let X and Y be coarse spaces, equipped with collections of entourages E(X) and E(Y ) respectively. Then we define the product of X and Y to be the Cartesian product X ×Y equipped with the coarse structure defined by forming finite compositions and unions of entourages in the set: Unfortunately, the above product is not a product in the category-theoretic sense since the projections π X : X × Y → X and π Y : X × Y → Y are not in general coarse maps. 4 Definition 2.9 A generalised ray is the topological space [0, ∞) equipped with some unital coarse structure compatible with the topology.
We reserve the notation R + for the space [0, ∞) equipped with the bounded coarse structure defined by the metric.
The following definition is a generalisation of the notion of a Lipschitz homotopy given in [5] and used in coarse geometry in [14].
Definition 2.10 Let X and Y be coarse spaces. Then a coarse homotopy is a map F : X × R → Y for some generalised ray R such that: • The function F (−, ∞) defined by the formula: is a coarse map.
The coarse maps F (−, ∞) and F (−, 0) are said to be linked by a coarse homotopy. Two coarse functions f : X → Y and g : X → Y are said to be coarsely homotopic if they are linked by a chain of coarse homotopies. A coarse map f : X → Y is called a coarse homotopy-equivalence if there is a coarse map g : Y → X such that the composites g • f and f • g are coarsely homotopic to the identities 1 X and 1 Y respectively.
Observe that any two coarsely equivalent maps are coarsely homotopic. The following result gives us an example of spaces that are coarsely homotopic but not coarsely equivalent.
Proposition 2.11 Let R n denote Euclidean space equipped with the bounded coarse structure. Then the inclusion i : R + ֒→ R n × R + defined by the formula i(s) = (0, s) is a coarse homotopy-equivalence.
Proof Define a coarse map p : R n × R + → R + by writing p(x, s) = x + s. The composition p • i is equal to the identity 1 R + . We can define a coarse homotopy linking the functions 1 R n ×R + and i • p by the formula: Actually, the above construction of a homotopy equivalence can be generalised to some situations involving continuous control. We will come back to this point in section 5.
We have seen that there is no good notion of products in the coarse category. Fortunately, the situation is rather better when we look at coproducts and some more general colimits.
Definition 2.12 Let {X λ | λ ∈ Λ} be a collection of coarse spaces. Let E λ denote the collection of entourages of the space X λ , and let B λ denote the collection of bounded subsets. Then the disjoint union, λ∈Λ X λ , is equipped with a coarse structure defined by forming the set of entourages: It is straightforward to verify that the disjoint union of a collection of coarse spaces defines a coproduct in the coarse category.
Definition 2.13 Let X be a coarse space equipped with an equivalence relation ∼ and quotient map π : X → X/ ∼. Then the quotient X/ ∼ is equipped with a coarse structure formed by defining the set of entourages to be the collection: In general the quotient map π : X → X/ ∼ is not a coarse map, since the inverse image of a bounded subset need not be bounded. However, the map π is a coarse map when each equivalence class of the relation ∼ is finite.

Coarse homology theories
A coarse homology theory is a functor on the category of coarse spaces that enables us to distinguish different coarse structures. It is the analogue in the world of coarse geometry of a generalised homology theory in the world of topology.
Definition 3.1 A coarse homology theory consists of a collection of functors, {HX p } p∈Z , from the category of coarse spaces to the category of Abelian groups such that the following axioms hold: • Coarse homotopy-invariance: For any two coarsely homotopic maps f : X → Y and g : X → Y , the induced maps f * : HX p (X) → HX p (Y ) and g * : HX p (X) → HX p (Y ) are equal.
• Excision axiom: Consider a decomposition X = A ∪ B of a coarse space X . Suppose that for all entourages m ⊆ X × X we can find an entourage A ֒→ X , and l : B ֒→ X . Then we have a natural map d : HX p (X) → HX p−1 (A ∩ B) and a long exact sequence: The coarse homology theory {HX p } is said to satisfy the large scale axiom if the groups HX p (+) are all trivial, where + denotes the one-point topological space.
A decomposition, X = A∪B , of a coarse space X is said to be coarsely excisive if the coarse excision axiom applies, that is to say for all entourages m ⊆ X × X we can find an entourage The long exact sequence: is called the coarse Mayer-Vietoris sequence.
The process of coarsening, described in [8] and [11], is used to construct coarse homology theories on the category of proper metric spaces equipped with their bounded coarse structures. This process can be significantly generalised.
Definition 3.2 Let X be a coarse space. A good cover of X is a cover {B i | i ∈ I} such that each set B i is bounded, and each point x ∈ X lies in at most finitely many of the sets B i .
Let U and V be good covers of the space X . Then we say that the cover V is a coarsening of the cover U , and we write The map φ is called a coarsening map.
• Let x ∈ X and suppose that i < j . Then there is a set U ∈ U j such that It is proved in [10] that any proper metric space, with its bounded coarse structure, admits a coarsening family. The following result is a generalisation of this fact.
Proposition 3.4 Let X be a unital coarse topological space. Then X has a coarsening family.
Proof Let {M i | i ∈ I} be a generating set for the collection of entourages in the space X , in the sense that every entourage M is contained in some Define J to be the set of finite sequences of elements of the set I . The set J can be ordered; we say Consider a finite sequence j = (i 1 , . . . , i k ) ∈ J . Form the composite M j = M i 1 . . . M i k and union A j = A i 1 ∪ · · · ∪ A i k , and look at the collection of bounded sets: Since each of the generators is symmetric and contains the diagonal, the composite M j contains each of the generators that form it. By maximality of the M i 1 -sparse set A i 1 , the collection U j is a cover of the space X . Choose a point x 0 ∈ X . Then the set is a discrete subset of a compact set, and is therefore finite. Thus the cover U j is good in the sense of definition 3.2.
We now need to check that the family of covers (U j ) forms a coarsening family. Consider the family of entourages (M j ). By construction, every set U ∈ U j takes the form M j (x) for some point x ∈ X , and every entourage M is contained in some entourage of the form M j .
All that remains is to check that for every point x ∈ X and every pair of It suffices to look at the special case where we have sequences j = (i 1 , . . . , i k ) and In particular, note that any generalised ray admits a coarsening family.
Recall that the nerve, |U i |, of a cover U i is a simplicial set with one vertex for every set in the cover. Vertices represented by sets U 1 , . . . , U n ∈ U i are spanned by an n-simplex if and only if the intersection U 1 ∩ · · · ∩ U n is non-empty. If (U i , φ ij ) is a coarsening family, then each map φ ij : U i → U j induces a map of simplicial sets φ ij * : |U i | → |U j |. The map φ ij * is proper because the covers we consider are locally finite. 7 Definition 3.5 Let {H lf p } be a generalised locally finite homology theory on the category of simplicial sets. Let (U i , φ ij ) be a coarsening family on the coarse space X . Then we write: As the above terminology suggests, it will turn out that the assignment X → HX p (X) forms a coarse homology theory.
Proposition 3.6 Let X and Y be coarse spaces equipped with coarsening families (U i ) i∈I and (V j ) j∈J respectively. Let f : X → Y be a coarse map. Then there is a functorially induced homomorphism f * : Proof Since the map f is a coarse map, it takes entourages to entourages. The given conditions on coarsening families ensure that for each element i ∈ I , Hence we have an induced map of simplicial sets f * : |U i | → |V j |. This map is proper since the map f is coarse. By definition of the nerve of a cover, up to proper homotopy the map f * is independent of any choice of coarsening family.
We therefore obtain a homomorphism: by taking locally finite homology groups followed by the direct limit.
Corollary 3.7 The group HX p (X) does not, up to isomorphism, depend on the choice of coarsening family.
Lemma 3.8 Let f : X → Y and g : X → Y be coarsely homotopic maps between coarse spaces that admit coarsening families. Then the induced maps f * : HX p (X) → HX p (Y ) and g * : Proof Without loss of generality suppose we have a coarse homotopy F : and (W k ) k∈K be coarsening families for the spaces X , R, and Y respectively. Then for all elements i ∈ I and j ∈ J we can find an element k ∈ K such that F (U i × V j ) ≤ W k . Hence we have an induced proper map of simplicial sets: Now, let us write the cover V j of the generalised ray R as a sequence of bounded sets (V n ) where sup V n+1 ≥ sup V n and V n+1 ∩ V n = ∅. Then we can define a continuous map [0, ∞) → |V j | by sending a natural number n ∈ N to the vertex V n and a point t ∈ (n, n + 1) to the appropriate point on the edge joining the vertices V n and V n+1 . Hence we have a proper continuous map such that for each point x ∈ |U i | the function t → F * (x, t) is eventually constant.
The induced maps f * : |U i | → |W k | and g * : |U i | → |W k | are thus properly homotopic; we obtain the appropriate proper homotopy by renormalising the map F * : The maps f * and g * therefore induce the same map at the level of locally finite homology. Taking direct limits, the maps f * : HX p (X) → HX p (Y ) and g * : HX p (X) → HX p (Y ) must be equal. Proof Let (U i ) i∈I be a coarsening family for the space X . Then the spaces A and B have coarsening families (U i | A ) and (U i | B ) defined by writing: respectively.
is coarsely excisive, so if we look at interiors then: By the existence of Mayer-Vietoris sequences in ordinary homology (see for example [13]) we have natural maps d : and exact sequences: Taking direct limits, we have an exact sequence: Lemma 3.8 and lemma 3.9 together imply the following result. The coarse homology theories considered above do not satisfy the large scale axiom. Now, let X be a coarse paracompact topological space equipped with an open coarsening family (U i ). Let {ϕ U | U ∈ U i } be a partition of unity subordinate the the open cover U i . Given a point x ∈ X there are only finitely many sets U ∈ U i such that x ∈ U . The sum U ∈U i ϕ U (x)U represents a point in the simplex spanning the vertices represented by these sets. We can therefore define a proper continuous map κ : X → |U i | by the formula: Considering locally finite homology and taking direct limits we obtain a coarsening map: c : H lf n (X) → HX n (X) It is proved in [8] that the coarsening map, c, is an isomorphism whenever X is a proper metric space 8 with good local properties. In particular, it is easy to see that the map c is an isomorphism if the space X is a single point.

Relative coarse homology
Homology theories are usually defined by looking at pairs of topological spaces. There is a corresponding notion in the coarse setting.  • Large scale axiom: Let + denote the one-point coarse space. Then the groups HX p (+, ∅) are all trivial.
Let {HX p } be a relative coarse homology theory. Then we write HX p (X) = HX p (X, ∅). The assignment X → HX p (X) is a coarse homology theory in the sense of definition 3.1 because of the following result. where α = (i * , −j * ) and β = k * + l * .
Proof We have a commutative diagram of exact sequences: The coarse geometry of the cone OX is closely related to the topology of the space X -see for example [8], [9], and [11] for instances of this phenomenon.
Definition 4.4 Let X be a subspace of the sphere S n−1 . Let r : (0, ∞) → (0, ∞) be any map such that r(t) → ∞ as t → ∞ and the inequality: holds for all points s, t ∈ R ≥0 . Then the radial contraction associated to r is the map ρ : OX → OX defined by the formula: The following result is proved in [8]. It is easy to see that any radial contraction is a coarse map, and is coarsely homotopic to the identity map.
If (X, A) is a pair of subspaces of the sphere S n−1 then (OX, OA) is a pair of coarse spaces so we have natural maps ∂ : H p (X, A) → H p−1 (A, ∅) and a long exact sequence where the maps i * and j * are induced by the inclusions i : (A, ∅) ֒→ (X, ∅) and j : (X, ∅) ֒→ (X, A) respectively.
Finally, suppose we have a pair (X, A) and a subset K ⊆ A such that K ⊆ A 0 . Since the space K is compact we can find a real number ε > 0 such that the neighbourhood N ε (K) is a subset of the space A. Hence for all R > 0 there exists S > 0 such that: Looking at cones, we see that the decomposition OX = OA ∪ O(X\K) is coarsely excisive, and the excision axiom follows. This completes the proof.

Coarse CW -complexes
The cone of the sphere S n−1 is the Euclidean space R n and the cone of the ball D n is the half-space R n × R ≥0 . By proposition 2.11 the coarse space R n × R + is coarsely homotopy-equivalent to the ray R + .
This prompts the following definition. If DX n is the space R n × [0, ∞) equipped with some coarse structure that makes it a coarse n-cell, then we refer to the space SX n−1 = {(x, 0) | x ∈ R n } as the boundary of the coarse cell DX n .
Definition 5.2 A coarse (n − 1)-sphere is a boundary of some coarse n-cell.
We have already seen that the space R n × R + , equipped with the bounded coarse structure, is a coarse n-cell. However, other coarse n-cells are possible if we look at continuous control.

Proposition 5.3
Let R be a generalised ray, with the coarse structure defined by looking at continuous control at infinity with respect to the one point compactification of the space [0, ∞). Then the coarse space (R R) n × R is a coarse n-cell.
Proof The topological space R n × [0, ∞) can be compactified by adding a 'hemisphere at infinity'. The coarse space (R R) n × R can be viewed as the space R n ×[0, ∞) with coarse structure defined by looking at continuous control with respect to this compactification.
As in proposition 2.11 we can define a coarse map p : R n ×R + → R + by writing p(x, s) = x + s. The composition p • i is equal to the identity 1 R + , and we can define a coarse homotopy linking the functions 1 R n ×R + and i • p by the formula: Definition 5.4 Let (X, A) be a pair of coarse spaces, and let f : A → Y be a continuous map. Then we define the space X ∪ f Y to be the quotient: Proposition 5.5 Let π X : X → X ∪ f Y and π Y : Y → X ∪ f Y be the canonical maps associated to the quotient X ∪ f Y . Then the images π X [X] and π Y [Y ] are coarsely equivalent to the original spaces X and Y respectively. Further, we have a coarsely excisive decomposition Proof The maps π X and π Y are injective, and the coarse structures of the images π X [X] and π Y [Y ] are defined to be those inherited from the spaces X and Y , respectively, under these maps. This establishes the first part of the proposition.
. Let m be an entourage for the space X ∪ f Y . We want to find an entourage M such that: By definition of the coarse structure on the space X ∪ f Y we can write: where m X and m Y are entourages for the spaces X and Y respectively, Let D X and D Y be the diagonals in the spaces X and Y respectively. Consider any point p ∈ A and let M be an entourage containing the image: and we are done.
Let DX n be a coarse n-cell, with boundary SX n−1 . Consider a coarse map f : SX n−1 → Y . Then the coarse space DX n ∪ f Y is called the coarse space obtained from Y by attaching a coarse n-cell through the map f .

Definition 5.6
A finite coarse CW -complex is a coarse space X together with a sequence X 0 ⊆ X 1 ⊆ · · · ⊆ X n = X of subspaces such that: • The space X 0 is a finite disjoint union of generalised rays.
• The space X k is coarsely equivalent to a space obtained by attaching a finite number of coarse k -cells to the space X k−1 .
Assuming that X n \X n−1 = ∅, the number n is called the dimension of the finite coarse CW -complex X .
The main purpose of this section is to prove that the axioms determine a coarse homology theory completely on the category of spaces coarsely homotopyequivalent to finite coarse CW -complexes once we know what the coarse homology of a generalised ray and a one-point set is.
Lemma 5.7 Let DX n be a coarse n-cell with boundary SX n−1 . Suppose that the cell DX n is coarsely homotopy-equivalent to a generalised ray R. Form the space X = DX n ∪ f Y for some coarse map f : SX n → Y . Then we have a Mayer-Vietoris sequence equipped with the coarse structure inherited from the generalised ray R and let C be the space {(x, t) ∈ DX n | t ≥ 1}. Then the space C is coarsely equivalent to the space DX n , and therefore coarsely homotopy-equivalent to the generalised ray R.
Then the space B is coarsely equivalent to the space Y . The space X can be written as the union X = B ∪ C , and the intersection B ∩ C is coarsely equivalent to the coarse sphere SX n−1 . Hence by proposition 5.5 we have a Mayer-Vietoris sequence / / HX p (SX n−1 ) / / HX p (Y ) ⊕ HX p (R) / / HX p (X) / / HX p−1 (SX n−1 ) / / as required.
The proof of our main result is now virtually identical to the proof of the corresponding result in classical algebraic topology. 9 assembly map is an isomorphism for all finite coarse CW -complexes. This result is an easy consequence of the machinery developed in the previous section together with some results presented in [7].
We begin by recalling some relevant definitions.
Definition 6.1 Let X be a locally compact Hausdorff topological space, and let A be a C * -algebra. Then an (X, A)-module is a Hilbert A-module E equipped with a morphism ϕ : C 0 (X) → L(E) of C * -algebras. The module E is called adequate if ϕ[C 0 (X)]E = E , and the operator ϕ(f ) is compact only when the function f is the zero function.
Here L(E) denotes the C * -algebra of all operators that admit adjoints on the Hilbert A-module E . We usually omit explicit mention of the morphism ϕ when talking about (X, A)-modules. Note that when we talk about an operator between Hilbert A-modules being compact, we mean compact in the sense of operators between Hilbert modules, rather than compact as a bounded linear operator between Banach spaces.
Definition 6.2 Let E be an (X, A)-module, and consider an operator T ∈ L(E). Then we define the support of T , supp(T ), to be the set of all pairs (x, y) ∈ X × X such that given functions f ∈ C 0 (X) and g ∈ C 0 (X), the equality f T g = 0 implies that either f (x) = 0 or g(y) = 0. Definition 6.3 Let X be a coarse topological space, and let E be an (X, A)modules. Consider an operator T ∈ L(E).
• The operator T is said to be locally compact if the operators T f and f T are both compact for all functions f ∈ C 0 (X).
• The operator T is said to be pseudolocal if the commutator f T − T f is compact for all functions f ∈ C 0 (X).
• The operator T is said to be controlled if the support, supp(T ), is an entourage.
Definition 6.4 Let X be a coarse topological space, and let E be an adequate (X, A)-module. Then we define the C * -algebra C * A (X) to be the C * -algebra generated by all controlled and locally compact operators on the module E . We define the C * -algebra D * A (X) to be the C * -algebra generated by all controlled and pseudolocal operators on the module E .
As our terminology suggests, the (X, A)-module E which we are using in the above definition does not really matter to us. The K -theory groups K n C * A (X) and K n D * A (X) do not depend on the precise choice of adequate (X, A)-module; see [7] for a proof of this fact. By proposition 5.5 of [7] the assignments X → K n C * A (X) are coarsely-invariant functors on the category of coarse topological spaces. The C * -algebra C * A (X) is an ideal in the C * -algebra D * A (X). We can thus form the quotient algebra D * A (X)/C * A (X) and so we have an exact sequence: is naturally isomorphic to the KK -theory group KK −n (C 0 (X), A). The functor X → KK −n (C 0 (X), A) is a locally finite homology theory; it can be considered to be locally finite K -homology with coefficients in the C * -algebra A. Proposition 6.5 Let X be a coarse paracompact topological space. Let KX n (X; A) denote the coarse homology theory defined by coarsening the functor X → KK −n (C 0 (X), A). 10 Then there is a map α ∞ : KX n (X; A) → K n C * A (X) such that we have a commutative diagram: Proof Let (U i ) be an open coarsening family on the space X . Recall from the comments at the end of section 3 that we can define a proper continuous map κ : X → |U i | by the formula: The coarsening map c is defined to be the direct limit of the induced maps κ i * : KK −n (C 0 (X), A) → KK −n (C 0 (|U i |), A). The map α ∞ can be defined to be the direct limit of the maps: 11 10 See section 3. 11 The space X is coarsely equivalent to each nerve |U i |, so the groups K n C * A (X) and K n C * A (|U i |) are isomorphic The map α ∞ : KX n (X; A) → K n C * A (X) is called the coarse assembly map. The coarse Baum-Connes conjecture asserts that this map is an isomorphism for all metric spaces of bounded geometry. 12 The machinery we have developed in this article together with some results from [7] tells us that the coarse assembly map is an isomorphism for all finite coarse CW -complexes.
To see this fact, first note that by an argument similar to that given to prove theorem 11.2 of [7] the functors K n C * A are coarsely homotopy-invariant, and by corollary 9.5 of [7] these functors satisfy the coarse excision axiom. Hence the sequence of functors {K n C * A } is a coarse homology theory. We now need some computations. Lemma 6.6 Let + denote the one-point space. Then the K -theory groups K n D * A (+) are all trivial.
Proof The C * -algebra D * A (+) consists of all operators on some infinite-dimensional Hilbert A-module. An Eilenberg swindle argument tells us that the K -theory groups of such a C * -algebra are all trivial.
Lemma 6.7 Let R be a generalised ray. Then the groups K n C * A (R) are all trivial.
Proof The generalised ray R is flasque in the sense described in section 10 of [7]. Hence, by proposition 10.1 of [7], the K -theory groups K n C * A (R) are all trivial.
Lemma 6.8 Let R be a generalised ray. Then the groups KX n (R; A) are all trivial.
Proof We can find a coarsening family (U i ) for the space R such that each nerve |U i | is properly homotopic to the ray [0, ∞). An Eilenberg swindle argument tells us that the groups KK −n (C 0 [0, ∞), A) are all trivial. Taking direct limits, the groups KX n (R; A) must also be trivial.
The above three lemmas tell us that the map α : KK −n (C(X); A) → K n C * A (X) is an isomorphism whenever the space X is a single point or a generalised ray. In particular, the coarse assembly map α ∞ is an isomorphism for such spaces. Theorem 6.9 The coarse assembly map α ∞ : KX n (X; A) → K n C * A (X) is an isomorphism whenever the space X is coarsely homotopy-equivalent to a finite coarse CW -complex.
Proof The result is immediate from the previous three lemmas and theorem 5.10.
A well-known descent argument (see [11]) enables us to deduce some results about the Novikov conjecture. Corollary 6.10 Let Γ be a finitely presented group such that the corresponding metric space |Γ| 13 , equipped with the bounded coarse structure, is coarsely homotopy-equivalent to a finite coarse CW -complex. Then the group Γ satisfies the Novikov conjecture.
The above result is not new. For example, any metric space coarsely homotopyequivalent to a finite coarse CW -complex is certainly uniformly embeddable into a Hilbert space, so the above corollary is included in the main result of [15]. However, another result is possible if we look at continuously controlled coarse structures rather than bounded coarse structures. See for example [7] and the final chapter of [11] for the descent arguments that are necessary in this case. Definition 6.11 Let X be a locally compact proper metric space, equipped with a compactification X . Let X b denote the space X equipped with the bounded coarse structure, and let X cc denote the space X equipped with the continuously controlled coarse structure arising from the compactification X . Then we call the compactification X a coarse compactification if the identity map 1 : X b → X cc is a coarse map. Corollary 6.12 Let Γ be a discrete group such that: • The classifying space BΓ can be represented as a finite complex.
• The corresponding universal space EΓ admits a coarse compactification, EΓ, such that the space EΓ cc is coarsely homotopy-equivalent to a finite coarse CW -complex, and the given Γ-action on the space EΓ extends continuously to the compactification EΓ.
Then the group Γ satisfies the Novikov conjecture.
Actually, the descent argument is sufficiently general to prove that the Novikov conjecture holds for any subgroup of a group of the kind described above.