CCoarse assembly maps
Ulrich Bunke ∗ Alexander Engel † October 17, 2018
Abstract
A coarse assembly map relates the coarsification of a generalized homology theorywith a coarse version of that homology theory. In the present paper we provide amotivic approach to coarse assembly maps. To every coarse homology theory E wenaturally associate a homology theory E O ∞ and construct an assembly map µ E : Coarsification( E O ∞ ) → E .
For sufficiently nice spaces X we relate the value E O ∞ ( X ) with the locally finitehomology of X with coefficients in E ( ∗ ). In the example of coarse K -homology wediscuss the relation of our motivic constructions with the classical constructionsusing C ∗ -algebra techniques. Contents K -homology . . . . . . . . . . . . . . . . . . . . . . . . 8 P ∗ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] † Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] a r X i v : . [ m a t h . K T ] O c t From coarse to local homology theories via F
207 Coarsifying spaces 218 Cone functors 259 The coarse assembly map 3110 Isomorphism results 33
11 Extension from locally compact, separable spaces 4212 Calculation of E O ∞ K -homology 5415 O ∞ ( X ) is representable 5516 Comparison with the analytic assembly map 5617 Index theory 64References 67 In the present paper we propose a definition of a coarse assembly map for every strongcoarse homology theory and every bornological coarse space. We further study conditionswhich imply that the coarse assembly map is an equivalence.Classically, for a separable bornological coarse space X of bounded geometry, the analyticcoarse assembly map is a homomorphism of Z -graded groups µ an X : QK an ,lf ∗ ( X ) → K X ∗ ( X ) (1.1)from the coarsified analytic locally finite K -homology groups QK an ,lf ∗ ( X ) to the coarse K -homology groups of X ; see Roe [Roe96], Higson–Roe [HR95], Yu [Yu95b], Roe–Siegel[RS12] or Definition 16.10. The coarse Baum–Connes conjecture predicts conditions onthe space X which imply that the analytic coarse assembly map is an isomorphism; see,e.g., Higson–Roe [HR95], Yu [Yu95a], Skandalis–Tu–Yu [STY02] or Wright [Wri05].2n the present paper we are interested in a refinement of the coarse assembly map froma Z -graded group homomorphism to a morphism between K -theory spectra, see (1.6).Furthermore we ask for generalizations of the coarse assembly map to other coarse homologytheories and study its functorial properties, e.g., the compatibility with Mayer–Vietorissequences. In the following we describe the set-up in which we will construct the coarse assembly map.The basic category is the category
BornCoarse of bornological coarse spaces introducedin [BE16]. Let C be a cocomplete stable ∞ -category, e.g., the ∞ -category of spectra Sp .A C -valued coarse homology theory is a functor E : BornCoarse → C which is coarsely invariant, coarsely excisive, u -continuous, and vanishes on flasques. Werefer to [BE16] for a detailed description of these properties. In order to study properties ofcoarse homology theories in general we constructed in [BE16] a universal coarse homologytheory Yo s : BornCoarse → Sp X with values in the stable ∞ -category of motivic coarse spectra Sp X . A C -valued coarsehomology theory as above is then equivalently described as a colimit preserving functor E : Sp X → C . Locally finite homology theories are defined on the category
TopBorn of topologicalbornological spaces and proper continuous maps [BE16, Sec. 6.5]. A functor H lf : TopBorn → C is a locally finite homology theory if, in addition to the usual homological conditions ofexcision and homotopy invariance, it satisfies the local finiteness condition that the naturalmap H lf ( X ) → lim B Cofib( H lf ( X \ B ) → H lf ( X )) (1.2)is an equivalence for every bornological topological spaces X , where the limit runs overthe bounded subsets of X . Every homology theory H has a corresponding locally finiteversion H lf [BE16, Def. 6.48].A particular class of coarse homology theories are coarsifications QH lf of locally finitehomology theories H lf [BE16, Sec. 6.6]. In contrast to general coarse homology theories,coarsifications of locally finite homology theories seem to be much more tractable becausethey can be studied by well-established methods of homotopy theory. One could askwhether there are other pairs (besides coarse K -homology K X and analytic locally finite K -homology K an ,lf ) of a coarse homology theory E and a locally finite homology theory H lf which are related by a coarse assembly map QH lf → E .3ocally finite homology theories are characterized by a limit condition (1.2). It is thereforecomplicated to construct maps out of locally finite homology theories. The main noveltyof the present paper is to introduce the notion of a local homology theory, essentially byreplacing the condition of being locally finite by the weaker condition of vanishing onflasques, see Definition 3.10.In the following we explain this in greater detail. We introduce the category of uniformbornological coarse spaces UBC . A C -valued local homology theory is then a functor F : UBC → C which is homotopy invariant, excisive, u -continuous, and vanishes on flasques. We willactually construct a universal local homology theoryYo s B : UBC → Sp B with values in motivic uniform bornological coarse spectra (Corollary 4.16). Note that thenature of the local finiteness condition (1.2) makes it impossible to construct a universallocally finite homology theory in a similar manner. Similarly as in the case of coarsehomology theories, a C -valued local homology theory is equivalently described as a colimitpreserving functor F : Sp B → C . Any locally finite homology theory H lf gives rise to a local homology theory which in thenotation of the present paper appears as H lf ◦ F C , U / in Lemma 3.13.A uniform bornological coarse space has an underlying bornological coarse space. But if wesimply forget the uniform structure, then we completely lose the local topological structureof the space. A more interesting transition from uniform bornological coarse spaces tobornological coarse spaces keeping the local structure is given by the cone construction.Indeed, with the help of the cone one can encode the uniform structure into a suitablecoarse structure.The cone construction will be investigated in various versions in Section 8; the main versionis in Definition 8.1, [BE16, Ex. 5.16] and [BEKWb, Def. 9.24]. It provides a functor O : UBC → BornCoarse . The cone and the germs at infinity O ∞ : UBC → Sp X of the cone (see Definition 8.2, [BE16, Ex. 5.23] and [BEKWb, Sec. 9.5]) can be used topull-back coarse homology theories to functors defined on UBC . If E is a strong (i.e.,vanishes on weakly flasques) coarse homology theory, then its pull-backs E O ∞ and E O are local homology theories, see Lemma 9.5.The idea to use some version of cones in order to pull-back coarse homology theories hassome history. We refer to Higson–Pederson–Roe [HPR96, Prop. 12.1] (coarse K -homology),4itchener [Mit10, Thm. 4.9] (coarsely excisive theories), Bartels–Farrell–Jones–Reich[BFJR04, Sec. 5] (equivariant coarse algebraic K -homology), or Weiss [Wei02] (algebraic K -theory of additive categories and retractive spaces) as entry points to the literature.Given an entourage U of a bornological coarse space X we can form the Rips complex P U ( X ) at scale U , see Example 2.6. It is a simplicial complex which will be equippedwith the metric induced by the spherical metric on its simplices. The metric induces acoarse and a uniform structure on P U ( X ), and the family of subsets ( P U ( B )) B ∈B (where B denotes the bornology of X ) generates the bornology of the uniform bornological coarsespace P U ( X ). There is a canonical embedding of sets X (cid:44) → P U ( X ) which induces anequivalence of bornological coarse spaces X U → F U ( P U ( X )).On the one hand the family ( F U ( P U ( X ))) U ∈C (where C denotes the coarse structure of X )of underlying bornological coarse spaces of the Rips complexes approximates the space X .One the other hand, forming the colimit of the motivic uniform bornological coarse spectrarepresented by the Rips complexes we obtain a functor P : BornCoarse → Sp B called the universal coarsification, see Definition 5.4. In detail, P ( X ) (cid:39) colim U ∈C Yo s B ( P U ( X )) . By Proposition 5.2 the functor P is a Sp B -valued coarse homology theory and can thereforebe interpreted as a colimit preserving functor P : Sp X → Sp B . Pull-back along P associates to every C -valued local homology theory F a C -valued coarsehomology theory F P , see Definition 5.5.For a locally finite homology theory H lf the coarsification of the local homology theory H lf ◦ F C , U / induced from H lf coincides with the coarsification QH lf from [BE16, Sec. 6.6]which we have discussed earlier, i.e., we have an equivalence QH lf (cid:39) ( H lf ◦ F C , U / ) P . Let us state the main construction of the paper. Let E be a coarse homology theory. Definition 1.1 (Definition 9.6) . If E is strong, then the coarse assembly map is thenatural transformation between coarse homology theories µ E : E O ∞ P → Σ E , derived from the boundary map of the cone sequence. .2 Isomorphism results and computations In Section 10 we study various conditions on the coarse homology theory E and thebornological coarse space X which imply that the coarse assembly map µ X,E : E O ∞ P ( X ) → Σ E ( X )is an equivalence. Let us mention the following two results which are analogues of instancesof the coarse Baum–Connes conjecture.Let X be a bornological coarse space and E be a strong coarse homology theory. Theorem 1.2 (Theorem 10.3) . If X admits a cofinal set of entourages U such that X U has finite asymptotic dimension, then the coarse assembly map µ E,X is an equivalence.
Let K be a simplicial complex and K d be the corresponding uniform bornological coarsespace whose structures are induced from the path metric induced by the spherical metricon the simplices. Then F U ( K d ) denotes the underlying bornological coarse space of K d .Furthermore let E be a C -valued coarse homology theory. Theorem 1.3 (Corollary 10.21) . Assume:1. E is strong, countably additive, and admits transfers.2. C is presentable.3. K has bounded geometry.4. K d is equicontinuously contractible.5. K d admits a coarse scaling.Then the coarse assembly map µ E,F U ( K d ) is an equivalence. We refer to Section 10 for a detailed description of the assumptions.Let E : BornCoarse → C be a coarse homology theory. In general the local homologytheory E O ∞ seems to be quite complicated. But if E is additive, then on nice spaces itbehaves like a locally finite homology theory. Concretely, we have the following result.Let X be a uniform bornological coarse space. Proposition 1.4 (Proposition 12.17) . Assume:1. C is presentable.2. E is countably additive, see (12.3) .3. X is small (Definition 11.1).4. X is homotopy equivalent in UBC to a countable, locally finite, finite-dimensionalsimplicial complex. hen we have a natural equivalence (Σ E ( ∗ ) ∧ Σ ∞ + ) lf ( X ) (cid:39) E O ∞ ( X ) . (1.3)The left-hand side of (1.3) is the value on the underlying bornological topological space of X of the locally finite version of the homology represented by the spectrum Σ E ( ∗ ), seeDefinition 12.1.The next proposition is a consequence of Proposition 12.17 applied to Rips complexes. Itprovides, under appropriate conditions, a calculation of the domain of the coarse assemblymap.Let E be a coarse homology theory and X be a bornological coarse space. Proposition 1.5 (Proposition 13.2) . Assume:1. C is presentable.2. E is countably additive.3. X is separable and of bounded geometry.Then we have a natural equivalence ((Σ E ( ∗ ) ∧ Σ ∞ + ) lf ◦ F C , U / ) P ( X ) (cid:39) E O ∞ P ( X ) . Assume that E → E (cid:48) is a natural transformation between coarse homology theories suchthat E ( ∗ ) → E (cid:48) ( ∗ ) is an equivalence. Then we can use Proposition 1.5 in order to showfor a bornological coarse space X that E ( X ) → E (cid:48) ( X ) is an equivalence if the assemblymaps µ E,X and µ E (cid:48) ,X are equivalences, see Theorem 13.3. The precise statement is thefollowing: Theorem 1.6 (Theorem 13.3) . Assume:1. C is presentable.2. E and E (cid:48) are strong and countably additive.3. E ( ∗ ) → E (cid:48) ( ∗ ) is an equivalence.4. X is separable and of bounded geometry.5. The assembly maps µ E,X and µ E (cid:48) ,X are equivalences.Then E ( X ) → E (cid:48) ( X ) is an equivalence. It is tempting to apply Theorem 1.6 to the transformation µ E : E O ∞ P → Σ E in order toshow that µ E,X is an equivalence. But in view of Assumption 1.6.5 this would lead to acircular argument. 7 .3 The case of coarse K -homology In the remaining part of this introduction we will discuss the application of our theory to K -homology. Note that coarse K -homology K X is countably additive and therefore theProposition 1.5 can be applied. We can choose an identification of spectra K an,lf ( ∗ ) (cid:39) K X ( ∗ ) (1.4)since both are equivalent to KU . This choice induces an equivalence K an,lf ( K ) (cid:39) ( K X ( ∗ ) ∧ Σ ∞ + ) lf ( K ) (1.5)for every countable and finite-dimensional simplicial complex K , see [BE16, Prop. 6.73]and [BE16, Ex. 6.72]. As a consequence, if X is separable and of bounded geometry, thenwe get the following formula for the domain of the coarse assembly map K X O ∞ P ( X ) Prop. 13 . (cid:39) ( K X ( ∗ ) ∧ Σ ∞ + ) lf ( F C , U / ( P ( X ))) (1.5) (cid:39) K an,lf ( F C , U / ( P ( X ))) (cid:39) QK an,lf ( X ) . The coarse assembly map µ K X ,X can therefore be interpreted as a morphism of spectra µ top X : QK an ,lf ( X ) → K X ( X ) (1.6)which induces a map on homotopy groups as in (1.1). This solves one of the problemsstated at the beginning of this introduction.Note that the group of automorphisms of the spectrum KU is huge. Hence there are manychoices for the identification (1.4). The equivalence (1.5) and hence the assembly map µ topX depend non-trivially on this choice. Therefore, in order to fix a canonical identification ofthe assembly maps µ an X and µ top X , we must fix the identification (1.4) appropriately.An idea in this direction would be to observe that both sides of (1.4) are ring spectra in anatural way. One could then require that (1.4) is an equivalence of ring spectra. We willnot discuss this problem further in the present paper.But we can show that µ topX induces an equivalence if and only if µ anX is an isomorphism of Z -graded groups, see Corollary 16.12. Consequently, Theorem 1.2 and Theorem 1.3 implyinstances of the coarse Baum-Connes conjecture. But note that these cases were knownbefore by the work of Higson–Roe [HR95], Wright [Wri05] and Yu [Yu98].The construction of the coarse assembly map as a transformation between coarse homologytheories automatically implies its compatibility with the boundary maps in Mayer–Vietorissequences. We derive the corresponding statement for our version of the assembly map µ topX in Corollary 14.1. In contrast, for the analytic assembly map µ anX (1.1) (recall that µ anX is only defined as a transformation between Z -graded group-valued functors) thiscompatibility is a non-trivial issue as we explain in the following. For a proper metricspace X the coarse analytic assembly map is obtained from the analytic assembly map(which we will recall in Definition 16.7) A X : K an ,lf ∗ ( X ) → K X ∗ ( X ) (1.7)8y the process of coarsification (Remark 16.9). The construction of the map A X requiresthe choice of an ample Hilbert space on X and uses Paschke duality, see Higson–Roe [HR00](see also the discussion in [BE16, Sec. 7.10]). There is also the alternative construction inSkandalis–Tu–Yu [STY02, Def. 2.9]. At the moment, Paschke duality or the alternativeconstrucion from [STY02] are understood as maps between K -theory groups, but notas morphisms of spectra. This is the reason that (1.7) is only a map between K -theorygroups. The analytic coarse assembly map (1.1) is natural in X , but because of the choicesmade during the construction this naturality is already difficult to establish.Siegel [Sie12] has shown, by studying various explicit descriptions of the boundary map in K -theory associated to an exact sequence of C ∗ -algebras, that A X is compatible with theboundary map in the Mayer–Vietoris sequence associated to a decomposition of X . Forthis result he adopts a fixed choice of an ample Hilbert space on X which provides theample Hilbert spaces on the pieces of X .In order to show that µ an X is compatible with the boundary map in the Mayer–Vietorissequence we would need uniqueness results for the identification of different ample Hilbertspaces up to contractible choice. It seems that such results have not appeared in theliterature so far — the currently available uniqueness results are only up to homotopy.The fact that we can show the compatibility of µ topX with Mayer–Vietoris sequences doesnot imply the corresponding compatibility for µ an X . The problem is that, at the moment,we do not have a natural identification between µ topX and µ an X , see Remark 16.11.In Section 15 we will observe that O ∞ ( X ) is actually a representable motive. If X is acomplete Riemannian manifold, then this will be used in Section 17 in order to provideexamples of classes in K X O ∞∗ ( X ) represented by coarse indices of Dirac operators.We now switch from X to M in order to denote the uniform bornological coarse spaceassociated to a complete Riemannian manifold M . More concretely, given a generalizedDirac operator /D of degree n on M we will construct a class σ ( /D ) in K X O ∞ n +1 ( M ), seeDefinition 17.3. This class is an analog of the symbol class of /D . For a discussion of theprecise relation with the classical notion of a symbol of /D we refer to the end of Section 17and especially Problem 17.6. The symbol class can further be promoted to a locally finite K -homology class Q ˜ σ an ,lf ( /D ) in QK an ,lf ( X ). We the argue that the coarse assembly map µ topM sends the symbol class to the coarse index (17.4): µ topM ( Q ˜ σ an ,lf ( /D )) = Ind X ( /D ) . This paper is written as an addendum to [BE16] to which we refer for details on coarseand on locally finite homology theories and for more references to the literature.In the present paper coarse K -homology is considered as a motivating example. But wedo not want to put the analytic details to much into the foreground. The index theoreticfacts used in the present paper are special cases of results to appear in [BE17] and [Bun].We also refer to these papers for more references to the previous literature on coarse indextheory. For readers interested precisely in coarse index theory it might be unsatisfyingthat we do not provide answers to the questions raised in Problems 16.6, 16.8 and 17.6.We think that a satisfying solution would require another much more analytic paper.9 cknowledgements The authors were supported by the SFB 1085 “Higher Invariants”funded by the Deutsche Forschungsgemeinschaft DFG. The second named author was alsosupported by the Research Fellowship EN 1163/1-1 “Mapping Analysis to Homology” ofthe Deutsche Forschungsgemeinschaft DFG.The first named author profited from many critical remarks by Denis-Charles Cisinski,Markus Land and the participants of the course on “Coarse Geometry” held in Regensburgin the years 2016/17, where parts of this material were first presented.
In this section we introduce the category of uniform bornological coarse spaces.To start with we consider a set equipped with a bornology and a coarse structure. Thebornology and the coarse structure are called compatible ([BE16, Def. 2.5]) if the bornologyis invariant under thickening with respect to coarse entourages.A set equipped with compatible bornological and coarse structures is an object of thecategory
BornCoarse of bornological coarse spaces. A morphism between bornologicalcoarse spaces is a proper and controlled map. We refer to [BE16, Sec. 2] for a detailedstudy of the category
BornCoarse .We now consider a set with a coarse structure and a uniform structure. The structures arecalled compatible ([BE16, Def. 5.4]) if there exists an entourage with is both coarse anduniform.Let ( X, U ) be a uniform space. Definition 2.1.
The coarse structure associated to the uniform structure is defined by C ( U ) := (cid:92) V ∈U C(cid:104){ V }(cid:105) . Here
C(cid:104){ V }(cid:105) denotes the coarse structure generated by V ([BE16, Ex. 2.11]). Example 2.2.
The coarse structure C ( U ) is not necessarily compatible with the uniformstructure U . Let us construct an example of such a space. We let X := { } ∪ { /n | n ∈ N } and the uniform structure U of X is defined to be the one induced from the natural metricon X coming from the canonical inclusion X ⊆ R . This uniform structure is generated bythe uniform entourages U r for all r > U r := { ( x, y ) ∈ X | d ( x, y ) < r } . Now amoment of reflection reveals C ( U ) = C(cid:104){ diag X }(cid:105) , which is not compatible with U . Example 2.3.
The notion of a quasi-metric on a set is defined similary as the notion ofa metric where one in addition allows that points have infinite distance. For example, adisjoint union of metric spaces is naturally a quasi-metric space. The definition of a coarsestructure associated to a metric [BE16, Ex. 2.17] generalizes immediately to the case ofquasi-metric spaces. Similarly, a quasi-metric also induces a uniform structure.10e consider a quasi-metric space with the induced coarse and uniform structures C and U .They are compatible. If the space is in addition a path quasi-metric space, then we havethe equality C = C ( U ). In particular, in this case C ( U ) is compatible with U .A bornological coarse space with a uniform structure ( X, C , B , U ) such that U and C arecompatible is an object of the category of uniform bornological coarse spaces UBC . Amorphism between uniform bornological coarse spaces is a morphism between bornologicalcoarse spaces which is in addition uniformly continuous. We refer to [BEKWb, Sec. 9.1]for more details.
Remark 2.4.
A map between metric spaces f : ( X, d ) → ( X (cid:48) , d (cid:48) ) is called uniformlycontinuous if for every δ in (0 , ∞ ) there exists an (cid:15) in (0 , ∞ ) such that for all pairs ofpoints x, y of X with d ( x, y ) ≤ (cid:15) we have d ( f ( x ) , f ( y )) ≤ δ . A uniformly continuousmap between metric spaces in this sense is uniformly continuous as a map between theassociated uniform spaces. Example 2.5.
Let X be a simplicial complex. Then X has a canonical spherical metricwhich induces a coarse structure C and a compatible uniform structure U .A choice of a set A of sub-complexes generates a bornology B := B(cid:104) A (cid:105) . It is compatiblewith the coarse structure if for every entourage U in C and every sub-complex K in A there exists another sub-complex K (cid:48) in A with U [ K ] ⊆ K (cid:48) . The triple ( X, C , B , U ) is auniform bornological coarse space.If X (cid:48) and A (cid:48) is similar data and f : X → X (cid:48) is a simplicial map such that for every Y (cid:48) in A (cid:48) we have f − ( Y (cid:48) ) ∈ A , then f is a morphism of uniform bornological coarse spaces. Example 2.6. If X is a bornological coarse space and U is an entourage of X , then weconsider the simplicial complex P U ( X ) of probability measures on X which have finite, U -bounded support. For a subset Y of X we let P U ( Y ) denote the sub-complex of P U ( X )of measures supported on Y . We let A be the set of sub-complexes P U ( B ) for all boundedsubsets B of X . The constructions explained in Example 2.5 turn P U ( X ) into a uniformbornological coarse space.Let f : X → X (cid:48) be a morphism between bornological coarse spaces and U (cid:48) be an entourageof X (cid:48) such that ( f × f )( U ) ⊆ U (cid:48) . Then the push-forward of measures provides a morphism P U ( X ) → P U (cid:48) ( X (cid:48) ) between uniform bornological coarse spaces in a functorial way. Example 2.7.
Let X be a uniform bornological coarse space. If Y is a subset of X , then Y has an induced uniform bornological coarse structure. If not said differently, we willalways consider subsets with the induced structures. The inclusion Y → X is then amorphism between uniform bornological coarse spaces. In this section we introduce the notion of a local homology theory.11et ( X, U ) be a uniform space and let A and B be subsets of X with A ∪ B = X . For anentourage U let P ( X × X ) ⊆ U denote the set of elements of P ( X × X ) (the power set of X × X ) which are contained in U . The following is taken from [BE16, Def. 5.18]: Definition 3.1.
The pair ( A, B ) is a uniformly excisive decomposition of X if there existsa uniform entourage U and a function κ : P ( X × X ) ⊆ U → P ( X × X ) such that:1. The restriction of κ to U ∩ P ( X × X ) ⊆ U is U -admissible.2. For every W in P ( X × X ) ⊆ U we have W [ A ] ∩ W [ B ] ⊆ κ ( W )( A ∩ B ) . Remark 3.2.
Note that in Definition 3.1 we consider P ( X × X ) ⊆ U and P ( X × X ) aspartially ordered sets with the order relation given by the opposite of the inclusion relation.By definition, a function between partially ordered sets is order preserving.Condition 3.1.1 means that for every V in U there exists W in U ∩ P ( X × X ) ⊆ U such that κ ( W ) ⊆ V .For a coarse space ( X, C ) the notion of a coarsely excisive decomposition [BE16, Def. 3.37]is defined similarly. We again consider two subsets A and B of X such that X = A ∪ B . Definition 3.3.
The pair ( A, B ) is a coarsely excisive decomposition of the space X iffor every coarse entourage V of X there exists a coarse entourage W of X such that wehave V [ A ] ∩ V [ B ] ⊆ W ( A ∩ B ) . Let ( X, U ) be a uniform space. Lemma 3.4. If C ( U ) is compatible with U , then any uniformly excisive decomposition ( A, B ) of X is coarsely excisive for the coarse structure C ( U ) .Proof. Let U and κ be as in the Definition 3.1. Since C ( U ) is compatible with U , we canassume that U is also a coarse entourage and κ ( W ) is a coarse entourage for every W in U ∩ P ( X × X ) ⊆ U .Let V be an entourage in C ( U ). Then there exists an integer n such that V ⊆ U n . Weclaim that V [ A ] ∩ V [ B ] ⊆ ( U n +1 ◦ κ ( U ))( A ∩ B ) . Let z be a point in V [ A ] ∩ V [ B ]. Then there exists integers r and s with r ≤ n and s ≤ n and a sequence of points ( x , . . . , x r + s ) in X such that x ∈ A , x r + s ∈ B , and ( x i , x i +1 ) ∈ U for all i = 0 , . . . , r + s −
1. There exists i in { , . . . , r + s − } such that x i ∈ A and x i +1 ∈ B . But then x i +1 ∈ U [ x i ], i.e., x i +1 ∈ U [ A ] ∩ U [ B ]. Hence there exists a point y in A ∩ B such that ( x i +1 , y ) ∈ κ ( U ). This now implies that z ∈ ( U n +1 ◦ κ ( U ))( A ∩ B ) asasserted. Example 3.5.
On a path quasi-metric space every closed decomposition is coarsely anduniformly excisive.Let C be a cocomplete stable ∞ -category and consider a functor E : UBC → C .12 efinition 3.6. We say that E satisfies excision if E ( ∅ ) (cid:39) and for every uniformbornological coarse space X and uniformly and coarsely excisive closed decomposition ( A, B ) of X the square E ( A ∩ B ) (cid:47) (cid:47) (cid:15) (cid:15) E ( A ) (cid:15) (cid:15) E ( B ) (cid:47) (cid:47) E ( X ) is cocartesian. We refer to Remark 3.14 for comments on the condition that A and B are closed subsetsof X in the above definition.Let X and Y be two uniform bornological coarse spaces. We define the tensor product X ⊗ Y such that the underlying bornological coarse space is the tensor product of thecorresponding bornological coarse spaces [BE16, Ex. 2.30] and the uniform structure isthe product uniform structure.The unit interval [0 ,
1] has a canonical uniform bornological coarse structure (the maximalcoarse and bornological structure, and the metric uniform structure). The product [0 , ⊗ X is now defined, and the projection [0 , ⊗ X → X is a morphism of uniform bornologicalcoarse spaces since [0 ,
1] is bounded.Let E : UBC → C be a functor. Definition 3.7.
We say that E is homotopy invariant if for every uniform bornologicalcoarse space X the morphism E ([0 , ⊗ X ) → E ( X ) induced by the projection is anequivalence. A homotopy between morphisms f , f : X → Y of uniform bornological coarse spaces is amorphism h : [0 , ⊗ X → Y which restricts to f i at the endpoints of the interval.A uniform bornological coarse space X is called flasque with flasqueness implemented bya morphism f : X → X if f implements flasqueness in the sense of bornological coarsespaces [BE16, Def. 3.21] and f is in addition uniformly homotopic to the identity.Let E : UBC → C be a functor. Definition 3.8.
We say that E vanishes on flasques if E ( X ) (cid:39) for every flasque uniformbornological coarse space X . Let X be a uniform bornological coarse space and U be an entourage which is both coarseand uniform. Then for every coarse entourage V such that U ⊆ V we can replace the coarsestructure by the coarse structure generated by V and obtain a uniform bornological coarsespace X V . Hence, for a uniform bornological coarse space X the uniform bornological coarsespace X V is well-defined for sufficiently large coarse entourages V . We have a canonicalmorphism X V → X given by the identity of the underlying sets. Hence the colimit andthe canonical morphism in the following definition have a well-defined interpretation.Let E : UBC → C be a functor. 13 efinition 3.9. We say that E is u -continuous if for every uniform bornological coarsespace X the canonical morphism colim V ∈C E ( X V ) → E ( X ) is an equivalence. Let C be a cocomplete stable ∞ -category and let E : UBC → C be a functor. Definition 3.10. E is called a C -valued local homology theory if1. E satisfies excision,2. E is homotopy invariant,3. E vanishes on flasques, and4. E is u -continuous. We have a forgetful functor F U : UBC → BornCoarse (3.1)which forgets the uniform structure.
Lemma 3.11. If E is a C -valued coarse homology theory, then E ◦ F U is a C -valued localhomology theory.Proof. It is clear that E ◦ F U is homotopy invariant, u -continuous, and vanishes on flasques.The functor F U sends uniformly and coarsely excisive closed decompositions to coarselyexcisive decompositions. By [BE16, Lem. 3.38] the composition E ◦ F U is excisive. Remark 3.12.
The reason that the proof of the Lemma 3.11 is not completely trivial isthat excision for coarse homology theories was not defined in terms of coarsely excisivedecompositions but with complementary pairs. Our main reason for doing this was that theintersection of a coarsely excisive decomposition with a subset need not be coarsely excisive,while intersection with subsets preserves complementary pairs. In fact, the Definition 3.6suffers from the same defect which causes some work at other points later.Let
TopBorn be the category of topological bornological spaces ([BE16, Sec. 6.5]).We have a forgetful functor F C , U / : UBC → TopBorn which forgets the coarse structure and a part of the uniform structure, i.e., only remembersthe bornology and the topology induced from the uniform structure.A locally finite homology theory in the sense of [BE16, Def. 6.60] will be called closed if itsatisfies excision for closed decompositions.An example of a spectrum-valued closed locally finite homology theory is the analytic K -homology K an ,lf constructed in [BE16, Def. 6.92].We assume that C is a complete and cocomplete stable ∞ -category. Let E : TopBorn → C be a locally finite homology theory. 14 emma 3.13. If E is closed, then E ◦ F C , U / is a C -valued local homology theory.Proof. Homotopy invariance of E implies homotopy invariance of E ◦ F C , U / . The functorsends coarsely and uniformly excisive closed decompositions to closed decompositions.Since we assume that E is closed the composition E ◦ F C , U / satisfies excison.The functor F C , U / sends a flasque uniform bornological coarse space to a topologicalbornological space which is flasque in the sense of [BE16, Def. 6.52]. Since E vanishes onflasque topological bornological spaces by [BE16, Lem. 6.54] we conclude that E ◦ F C , U / vanishes on flasques.Since F C , U / forgets the coarse structure, the composition E ◦ F C , U / is u -continuous. Remark 3.14.
Note that in Definition 3.6 we can replace the condition that A and B are closed in X by the condition that these subsets are open in X . Then using the functor F C , U / we can pull-back locally finite homology theories satisfying open excision.A typical example of a locally finite homology theory satisfying open excision is the locallyfinite version of stable homotopy Σ ∞ + ( − ) lf , see [BE16, Ex. 6.56].We choose to work with the condition closed since the main example for the present paperis analytic locally finite K -homology which satisfies closed excision. In this section we construct the universal local homology theory. The construction here iscompletely analogous to the construction of the universal coarse homology theory carriedout in [BE16, Sec. 3 & 4]. We keep the present section as short as possible and refer to[BE16] for more background and references to the ∞ -category literature. Remark 4.1.
For an ∞ -category D we use the standard notation PSh ( D ) := Fun ( D op , Spc )for the ∞ -category space-valued presheaves. If D is an ordinary category, then we considerit as an ∞ -category using the nerve.In order to perform the localizations below we must assume that D is small. The category UBC is not small. Therefore in order to make the theory below precise we must replace
UBC by a small full subcategory which contains all isomorphism classes of uniformbornological coarse spaces we are interested in. This category we choose must be closedunder constructions like forming products, taking subspaces, etc. Furthermore, it mustcontain the Rips complexes P U ( X ) for X belonging to a similarly choosen small and fullsubcategory of BornCoarse . From now on we will drop these set-theoretic issues andpretent that
UBC is small.Let E be in PSh ( UBC ). 15 efinition 4.2. E satisfies descent if for every uniform bornological coarse space X anduniformly and coarsely excisive closed decomposition ( A, B ) of X the square E ( X ) (cid:47) (cid:47) (cid:15) (cid:15) E ( A ) (cid:15) (cid:15) E ( B ) (cid:47) (cid:47) E ( A ∩ B ) is cartesian. Definition 4.3.
We let Sh ( UBC ) be the full subcategory of PSh ( UBC ) of presheaveswhich satisfy descent. Its objects will be called sheaves. Lemma 4.4.
We have a localization L : PSh ( UBC ) (cid:28) Sh ( UBC ) : inclusion .
Proof.
The condition of descent can be written as a locality condition for a set of maps of
PSh ( UBC ). This implies existence of the localization.
Remark 4.5.
We think that the sheafification adjunction is exact. But since exactnessdoes not play a role in the present paper we refrain from working out the details.We let Y : UBC → PSh ( UBC ) be the Yoneda embedding.
Lemma 4.6.
For every uniform bornological coarse space X the presheaf Y ( X ) satisfiesdescent.Proof. See [BE16, Lem. 3.12] for a similar argument.Let E be in Sh ( UBC ). Definition 4.7. E is homotopy invariant if for every uniform bornological coarse space X the morphism E ( X ) → E ([0 , ⊗ X ) induced by the projection is an equivalence. We let Sh h ( UBC ) denote the full subcategory of Sh ( UBC ) of homotopy invariant sheaves.
Lemma 4.8.
We have an adjunction H : Sh ( UBC ) (cid:28) Sh h ( UBC ) : inclusion .
Proof.
Homotopy invariance can be written as locality with respect to a set of maps in
UBC . This implies the existence of the localization.We call H the homotopification.Let E be in Sh h ( UBC ). Definition 4.9.
We say that E vanishes on flasques if for every flasque uniform bornolog-ical coarse space X we have E ( X ) (cid:39) ∅ .
16e let Sh h,fl ( UBC ) denote the full subcategory of Sh h ( UBC ) of homotopy invariantsheaves which vanish on flasques.
Lemma 4.10.
We have an adjunction
Fl : Sh h ( UBC ) (cid:28) Sh h,fl ( UBC ) : inclusion .
Proof.
The condition of vanishing on flasques can be written as a locality condition withrespect to a set of maps in
PSh ( UBC ). This implies existence of the localization.For a uniform bornological coarse space X let ˜ C denote the subset of coarse entourageswhich are also uniform. Note that this subset is cofinal in C .Let E be in Sh h,fl ( UBC ). Definition 4.11.
We say that E is u -continuous if for every uniform bornological coarsespace X the natural morphism E ( X ) → lim U ∈ ˜ C E ( X U ) is an equivalence. We let Sh h,fl,u ( UBC ) denote the full subcategory of Sh h,fl ( UBC ) of homotopy invariantsheaves which vanish on flasques and are u -continuous. Lemma 4.12.
We have an adjunction U : Sh h,fl ( UBC ) (cid:28) Sh h,fl,u ( UBC ) : inclusion .
Proof.
The condition of being u -continuous can be written as a locality condition withrespect to a set of maps in PSh ( UBC ). This implies existence of the localization.
Lemma 4.13.
The category Sh h,fl,u ( UBC ) is presentable.Proof. The category Sh h,fl,u ( UBC ) is a reflective localization of a presheaf category.
Definition 4.14.
We call Sh h,fl,u ( UBC ) the category of motivic uniform bornologicalcoarse spaces and use the notation Spc B . The following is analogous to [BE16, Sec. 4.1].
Definition 4.15.
We define the stable ∞ -category of motivic uniform bornological coarsespectra Sp B as the stabilization of Spc B in the world of presentable stable ∞ -categories. We have a canonical functor Σ ∞ + : Spc
B → Sp B . By construction the category Sp B is a presentable stable ∞ -category.We have a functor Yo B s := Σ ∞ + ◦ U ◦ Fl ◦ H ◦ L ◦ Y : UBC → Sp B . In view of Lemma 4.6 we could omit L in this composition. For a uniform bornologicalcoarse space X we call Yo B s ( X ) the motive of X .17y construction the functor Yo B s is a Sp B -valued local homology theory. It is in fact theuniversal local homology theory. Corollary 4.16. If C is a cocomplete stable ∞ -category, then precomposition with Yo B s induces an equivalence between the ∞ -categories of C -valued local homology theories andcolimit preserving functors Sp B → C . For a local homology theory E we will use the notation E also to denote the correspondingcolimit preserving functor Sp B → C . Remark 4.17.
The existence of non-trivial local homology theories (see Lemma 3.11 andLemma 3.13) shows that the category Sp B is non-trivial. P In this section we extend the construction given in Example 2.6 to a coarse homologytheory P called the universal coarsification.Let BornCoarse C be the category of pairs ( X, U ) of bornological coarse spaces X andcoarse entourages U of X . A morphisms f : ( X, U ) → ( X (cid:48) , U (cid:48) ) is a morphism f : X → X (cid:48) of bornological coarse spaces such that ( f × f )( U ) ⊆ U (cid:48) . By Example 2.6 we have afunctor P : BornCoarse C → UBC which sends (
X, U ) to the uniform bornological coarse space P ( X, U ) associated to thesimplical complex P U ( X ) and the family A of sub-complexes P U ( B ) for all bounded subsets B of X . We furthermore have a forgetful functor F C : BornCoarse C → BornCoarse (5.1)which sends the pair (
X, U ) to X . Definition 5.1.
We define P : BornCoarse → Sp B as the left Kan-extension BornCoarse C Yo B s ◦ P (cid:47) (cid:47) F C (cid:15) (cid:15) Sp B BornCoarse P (cid:53) (cid:53) Proposition 5.2.
The functor P is an Sp B -valued coarse homology theory.Proof. If X is a bornological coarse space with coarse structure C , then by the point-wiseformula for the left Kan extension P ( X ) (cid:39) colim U ∈C Yo B s ( P ( X, U )) . (5.2)18e have equivalencescolim V ∈C P ( X V ) (cid:39) colim V ∈C colim U ∈C(cid:104) V (cid:105) Yo B s ( P ( X, U )) (cid:39) colim U ∈C Yo B s ( P ( X, U )) (cid:39) P ( X ) , where for the second equivalence we use a cofinality consideration. Hence P is u -continuous.Consider two morphisms f , f : ( X, U ) → ( X (cid:48) , U (cid:48) ) in BornCoarse C . If f and f are U (cid:48) -close, then P ( f ) and P ( f ) are homotopic and Yo B s ( P ( f )) (cid:39) Yo B s ( P ( f )) by thehomotopy invariance of Yo B s . This implies that P is coarsely invariant.Let ( X, U ) be an object of
BornCoarse C such that U contains the diagonal of X . For asubset Y of X note that P U ( Y ) is a closed subset of P U ( X ). If ( Z, Y ) with Y = ( Y i ) i ∈ I isa complementary pair, then for every i, j in I such that Z ∪ Y i = X and U [ Y i ] ⊆ Y j thepair ( P U ( Z ) , P U ( Y j )) is a closed decomposition of the path quasi-metric space P U ( X ) andhence uniformly and coarsely excisive (see Example 3.5). For sufficiently large j in I andsince Yo B s is excisive we get a cocartesian squareYo B s ( P U ( Z ) ∩ P U ( Y j )) (cid:47) (cid:47) (cid:15) (cid:15) Yo B s ( P U ( Z )) (cid:15) (cid:15) Yo B s ( P U ( Y j )) (cid:47) (cid:47) Yo B s ( P U ( X ))We form the colimit over j in I and over U in the coarse structure of X . The lower rightcorner yields P ( X ). For the lower left corner we first take the U -colimit and then the j -colimit. Then we obtain the object P ( Y ). In the upper right corner we get P ( Z ). Forthe upper left corner we note that P U ( Z ) ∩ P U ( Y j ) = P U ( Z ∩ Y j ) and finally get P ( Z ∩ Y ).Since we have exhibited the square P ( Z ∩ Y ) (cid:47) (cid:47) (cid:15) (cid:15) P ( Z ) (cid:15) (cid:15) P ( Y ) (cid:47) (cid:47) P ( X )as a colimit of cocartesian squares it is cocartesian itself. We conclude that the functor P satisfies excision.Finally, assume that a bornological coarse space X is flasque with flasqueness implementedby f : X → X . Let U be an entourage of X such that id X and f are U -close to each other.Then V := (cid:83) n ∈ N ( f n × f n )( U ) is again an entourage of X which contains U . Now note that( f × f )( V ) ⊆ V . Therefore P ( f ) : P ( X, V ) → P ( X, V ) is defined. This map implementsflasqueness of P ( X, V ), hence Yo B s ( P ( X, V )) (cid:39)
0. In view of (5.2) by cofinality we seethat P vanishes on flasques. Remark 5.3.
Note that the Proposition 5.2 would also be true (with a slightly differentargument for excision) if we would have worked with open instead of closed decompositionsin the definition of excision. 19e therefore have defined a colimit preserving functor P : Sp X → Sp B . Definition 5.4.
The functor P is the universal coarsification functor. Let E be a local homology theory. Definition 5.5.
The coarse homology theory E P := E ◦ P will be called the coarsificationof the theory E . Example 5.6.
In Example 3.13 we have seen that for any closed locally finite homologytheory E : TopBorn → C we can define a local homology theory E ◦ F C , U / : UBC → C .The coarsification ( E ◦ F C , U / ) P is equivalent to the coarse homology theory QE , which isthe coarsification of E from [BE16, Defn. 6.74]. F In this section we refine the forgetful functor F U : UBC → BornCoarse from (3.1) to alocal homology theory. We define F := Yo s ◦ F U : UBC → Sp X . Lemma 6.1. F is a local homology theory.Proof.
The proof is straightforward and similar to the one of Lemma 3.11.We therefore get a colimit-preserving functor F : Sp B → Sp X (6.1)For a C -valued coarse homology theory E we write E F := E ◦ F : Sp B → C for the associated local homology theory (compare with Lemma 3.11 where the notation E ◦ F U was used). Proposition 6.2.
We have a canonical equivalence id (cid:39) −→ F ◦ P . (6.2) Proof.
We have a functor I : BornCoarse C → BornCoarse which is defined on objects by I ( X, U ) := X U . By u -continuity of Yo s the left Kanextension of Yo s ◦ I along F C (see (5.1) for the definition of F C ) is equivalent to Yo s . Let20 X, U ) be in
BornCoarse C . Dirac measures provide a canonical inclusion X → P U ( X ) ofsets. This map is an equivalence X U (cid:39) → F U ( P ( X, U )) (6.3)of bornological coarse spaces. Hence we get an equivalence of functors from
BornCoarse C to Sp X Yo s ◦ I (cid:39) → Yo s ◦ F U ◦ P (cid:39) F ◦ Yo B s ◦ P . (6.4)Since F is colimit-preserving the equivalence (6.4) induces an equivalence of left Kanextensions along F C : Yo s → F ◦ P : BornCoarse → Sp X . We finally interpret P as a colimit-preserving functor Sp X → Sp B to get the desiredequivalence. Corollary 6.3.
Every coarse homology theory E is equivalent to the coarsification of thelocal homology theory E F . Similarly, every morphism between coarse homology theories isinduced by coarsification from a morphism between the associated local homology theories. Under certain finiteness conditions on the uniform bornological coarse space X we canconstruct a morphism c X : Yo B s ( X ) → P ( F ( X ))called the comparison morphism. We will furthermore show that it is an equivalence forsimplicial complexes of bounded geometry which are uniformly contractible. Part of thematerial here is inspired by Roe [Roe96, Ch. 2, Part “Coarse algebraic topology”].Let X be a coarse space with a uniform structure. Definition 7.1.
We say that the uniform structure is numerable if there exists an entourage U which is both coarse and uniform, and an equicontinuous, uniformly point-wise locallyfinite partition of unity ( χ α ) α ∈ A such that supp ( χ α ) is U -bounded for all α in A . Remark 7.2.
Here uniform point-wise local finiteness meanssup x ∈ X |{ α ∈ A | χ α ( x ) (cid:54) = 0 }| < ∞ . Let us spell out the meaning of equicontinuous explicitly: For every positive real number (cid:15) there exists a uniform entourage V of X such that for all α in A and ( x, x (cid:48) ) in V we havethe inequality | χ α ( x ) − χ α ( x (cid:48) ) | ≤ (cid:15) .Let X be a simplicial complex with the coarse and uniform structures both induced fromthe spherical path metric. Lemma 7.3. If X is finite-dimensional, then X is numerable. roof. We consider the entourage U of width 2. We define the equicontinuous partitionof unity ( χ v ) v ∈ X (0) using the baricentric coordinates of the simplices, where X (0) is the setof vertices of X . If σ is a simplex in X and x is a point in σ , then χ v ( x ) (cid:54) = 0 exactly if v is a vertex of σ . Hence for every point x the number of vertices v of X with χ v ( x ) (cid:54) = 0 isbounded by dim( X ) + 1.The support of χ v is U -bounded for every vertex v of X .Let X be a numerable uniform bornological coarse space. By numerability of the uniformstructure we can choose an entourage U which is coarse and uniform such that there existsan equicontinuous, uniformly point-wise locally finite partition of unity ( χ α ) α ∈ A on X suchthat supp ( χ α ) is U -bounded for every α in A . We choose a family of points ( x α ) α ∈ A in X such that x α ∈ supp ( χ α ) for all α in A . We can then define a map X → P U ( F U ( X )) , x (cid:55)→ (cid:88) α ∈ A χ α ( x ) δ x α . (7.1)This map is uniform. Note that at this point we use the uniformity of the point-wise locallyfiniteness condition since we measure distances in the simplices of P U ( F U ( X )) in thespherical metric and not in the maximum metric with respect to baricentric coordinates,cf. [BE16, Ex. 5.37].The map defined in (7.1) can also be regarded as a morphism of uniform bornologicalcoarse spaces ˜ c : X U → P U ( F U ( X )). It induces a morphismYo B s ( X U ) → Yo B s ( P U ( F U ( X ))) → P ( F ( X ))for every sufficiently large entourage U of X , and by u -continuity of Yo B s , a morphism c X : Yo B s ( X ) → P ( F ( X )) . Definition 7.4.
For a numerable uniform bornological coarse space the transformation c X is called the comparison map. Remark 7.5.
We must assume that X is numerable in order to produce a uniform map X → P U ( F U ( X )) by (7.1).In the classical approach to the coarsification of locally finite homology theories (see, e.g.,Higson–Roe [HR95, Sec. 3]) one only needs a coarse and continuous map. In this case thesame formula works, and we only have to assume that the members of the partition ofunity have uniformly controlled support. The existence of such a partition of unity followsfrom the compatibility of the uniform and the coarse structure if we in addition assumethat the underlying topological space of X is paracompact.In our approach we must work with uniform maps since this is required by functoriality ofthe cone functor O which we employ below in order to construct the assembly map. Lemma 7.6.
Up to equivalence the comparison map does not depend on the choice of thepartition of unity. roof. We consider a second choice of partition of unity (without loss of generality for thesame entourage U ) and denote the associated morphism by ˜ c (cid:48) : X U → P U ( F U ( X )). Then s (cid:55)→ (1 − s ) c + s ˜ c (cid:48) is a homotopy between ˜ c and ˜ c (cid:48) . Moreover ˜ c and ˜ c (cid:48) are U -close to eachother.Let f : X → X (cid:48) be a morphism of uniform bornological coarse spaces which are assumedto be numerable. Lemma 7.7.
We have an equivalence c X (cid:48) ◦ Yo B s ( f ) (cid:39) ( P ◦ F )( f ) ◦ c X . Proof.
After choosing partitions of unity for X and X (cid:48) with bounds U and U (cid:48) such that( f × f )( U ) ⊆ U (cid:48) we have a square (not necessarily commuting) of morphisms of uniformbornological coarse spaces X U ˜ c X (cid:47) (cid:47) f (cid:15) (cid:15) P U ( F U ( X )) P ( F U ( f )) (cid:15) (cid:15) X (cid:48) U (cid:48) ˜ c X (cid:48) (cid:47) (cid:47) P U (cid:48) , ( F U ( X (cid:48) ))We now observe that the compositions P ( F U ( f )) ◦ ˜ c X and ˜ c X (cid:48) ◦ f are close and (linearly)homotopic to each other.Let Y be a uniform bornological coarse space . Definition 7.8.
We say that Y is coarsifying if it is numerable and the comparison map c Y is an equivalence. Let E be a local homology theory. If Y is coarsifying, then the comparison map inducesan equivalence E ( c X ) : E ( Y ) (cid:39) → E P ( F U ( Y )) . Let X be a numerable uniform bornological coarse space . Definition 7.9.
A morphism f : X → Y in UBC is called a coarsifying approximationif Y is coarsifying and ( P ◦ F )( f ) is an equivalence. Let E be a local homology theory. If X → Y is a coarsifying approximation, then byconstruction we have an equivalence E P ( F U ( X )) (cid:39) E ( Y ) . We refer to [BE16, Sec. 6.8] for more information.Let us discuss now an important class of examples of coarsifying spaces.Let K be a simplicial complex. We get a uniform bornological coarse space K d by equipping K with the bornology of bounded subsets and the metric coarse and uniform structures.Below B q +1 is the unit ball in R q +1 and S q its boundary.23 efinition 7.10. K has bounded geometry if the number of vertices in the stars ofits vertices is uniformly bounded.2. K is equicontinuously contractible, if for every q in N and for every equicontinuousfamily of maps { ϕ i : S q → K } i ∈ I there exists an equicontinuous family of maps { Φ i : B q +1 → K } i ∈ I with Φ i | ∂B q +1 = ϕ i . Let A be a subcomplex of K , X be a metric space and f : K d → X d be a morphism ofbornological coarse spaces such that f | A is uniformly continuous. Lemma 7.11. If K is finite-dimensional and X is equicontinuously contractible, then f is close to a morphism of uniform bornological coarse spaces which extends f | A and is inaddition uniformly continuous.Proof. The proof given in [BE16, Lem. 6.97] (which covers the non-uniform version of thislemma) also works word-for-word here.
Proposition 7.12. If K is a simplicial complex of bounded geometry which is equicontin-uously contractible, then K d is coarsifying.Proof. Note that K is finite-dimensional and hence K d is numerable by Lemma 7.3. Theverification that the comparison map for K d is an equivalence is the core of the argumentof [BE16, Prop. 6.105], which is itself taken from Nowak–Yu [NY12, Proof of Thm. 7.6.2].As in the beginning of the proof of [BE16, Prop. 6.105], by extensive use of Lemma 7.11,we construct maps K d f (cid:41) (cid:41) P U ( F U ( K d )) f (cid:43) (cid:43) g (cid:111) (cid:111) P U ( F U ( K d )) f (cid:43) (cid:43) g (cid:107) (cid:107) P U ( F U ( K d )) f (cid:39) (cid:39) g (cid:106) (cid:106) · · · with g n ◦ ( f n ◦ · · · ◦ f ◦ f ) being homotopic in UBC to id K d , and ( f n ◦ · · · ◦ f ◦ f ) ◦ g n − being homotopic to f n in UBC . We claim that the induced comparison mapYo s B ( K d ) → colim n ∈ N Yo s B ( P U n ( F U ( K d ))) (cid:39) P ( F ( K d ))is an equivalence.Let T be any object of Sp B . Then we must check that lim n ∈ N Map (Yo s B ( P U n ( F U ( K d ))) , T ) → Map (Yo s B ( K d ) , T )is an equivalence. We first check that lim n ∈ N π ∗ ( Map (Yo s B ( P U n ( F U ( K d ))) , T )) → π ∗ ( Map (Yo s B ( K d ) , T )) (7.2) This is a slight strengthening of the notion of uniform contractibility which is commonly used in thecoarse geometry literature.
24s an isomorphism.Let us first check that (7.2) is surjective: Let t ∈ π ∗ ( Map (Yo s B ( K d ) , T )). Then we considerthe family ( g ∗ n t ) n ∈ N . We observe that f ∗ n g ∗ n t = ( g ∗ n − f ∗ f ∗ · · · f ∗ n )( g ∗ n t ) = g ∗ n − t . (7.3)Therefore our family belongs to the limit. Furthermore f ∗ f ∗ g ∗ t = t , hence the family is apreimage of t .Let now ( t n ) be any preimage of t . Then we have t n − g ∗ n f ∗ f ∗ · · · f ∗ n t n = f ∗ n +1 t n +1 − g ∗ n f ∗ f ∗ · · · f ∗ n f ∗ n +1 t n +1 = f ∗ n +1 t n +1 − f ∗ n +1 t n +1 = 0 , which shows injectivity.We verify now the Mittag-Leffler condition. We claim Im( f ∗ n +1 ) = Im( f ∗ n +1 · · · f ∗ n + k ) for all k ,which follows from the next calculation. Let t n +1 be in π ∗ ( Map (Yo s B ( P U n +1 ( F U ( K d ))) , T )).Then using the identities show above we get f ∗ n +1 · · · f ∗ n + k g ∗ n + k f ∗ f ∗ · · · f ∗ n +1 t n +1 (7.3) = f ∗ n +1 · · · f ∗ n + k − g ∗ n + k − f ∗ f ∗ . . . f ∗ n +1 t n +1 ...= g ∗ n f ∗ f ∗ . . . f ∗ n +1 t n +1 = f ∗ n +1 t n +1 finishing this proof. Example 7.13.
The following is taken from [BE16, Ex. 6.96] and originally goes back toGromov [Gro93, Ex. 1.D ]:Let G be a finitely generated group admitting a model for its classifying space BG whichis a finite simplicial complex. Then the universal cover EG of BG is a simplicial complexof bounded geometry which is equicontinuously contractible, i.e., EG d is coarsifying bythe above Proposition 7.12.The group G quipped with a word-metric becomes a metric spaces and hence a uniformbornological coarse G d . The action of G on EG provides a morphism f : G d → EG d whichdepends on the choice of a base-point in EG . The morphism ( P ◦ F )( f ) is an equivalence.Therefore we have shown that f : G d → EG d is a coarsifying approximation. In this section we describe the cone functor O : UBC → BornCoarse and its germs atinfinity O ∞ : UBC → Sp X . These functors play a crucial role in the construction ofthe coarse assembly map. After the introduction of the cone functor, we compare it with25ariants which occur in the literature on coarse geometry and which are useful in certainarguments.In short, the cone of a uniform bornological coarse space X is the bornological coarsespace O ( X ) obtained from the bornological coarse space F U ([0 , ∞ ) ⊗ X ) by replacing thecoarse structure by the hybrid structure (cf. [BE16, Sec. 5.1]) associated to the family ofsubsets Y := ([0 , n ] × X ) n ∈ N and the uniform structure on [0 , ∞ ) ⊗ X .In the following we spell out the definition of the cone explicitly. Let T denote the uniformstructure of X . Recall that a function φ : [0 , ∞ ) → T is cofinal if for every entourage U in T there exists an element t in [0 , ∞ ) such that φ ( s ) ⊆ U for all s in [ t, ∞ ). Definition 8.1.
We let O ( X ) be the bornological coarse space defined as follows:1. The underlying set of O ( X ) is [0 , ∞ ) × X .2. The bornology of O ( X ) is generated by the subsets [0 , n ] × B for n in N and boundedsubsets B of X
3. The coarse structure of O ( X ) is generated by the entourages of the form V ∩ U ( κ,φ ) ,where V is a coarse entourage of [0 , ∞ ) ⊗ X and U ( κ,φ ) := { (( s, x ) , ( t, y )) ∈ ([0 , ∞ ) × X ) | | s − t | ≤ κ (max { s, t } ) & ( x, y ) ∈ φ (max { s, t } ) } . for all cofinal functions φ : [0 , ∞ ) → T and functions κ : [0 , ∞ ) → [0 , ∞ ) satisfying lim t →∞ κ ( t ) = 0 . If f : X → X (cid:48) is a morphism of uniform bornological coarse spaces, then the map id [0 , ∞ ) × f : [0 , ∞ ) × X → [0 , ∞ ) × X is a morphism of bornological coarse spaces O ( f ) : O ( X ) → O ( X (cid:48) ) . We thus have described the cone functor O : UBC → BornCoarse . Let X be a uniform bornological coarse space. Then Y ( X ) := ([0 , n ] × X ) n ∈ N is a bigfamily in O ( X ). The inclusion X → { } × X → [0 , n ] × X induces a coarse equivalenceand hence induces an equivalence Yo s ( F U ( X )) → Yo s (([0 , n ] × X ) O ( X ) ) for every n in N .The collection of these equivalences for all n in N induces an equivalenceYo s ( F U ( X )) (cid:39) Yo s ( Y ( X )) . The pair sequence of ( O ( X ) , Y ( X )) therefore gives rise to the cone sequence of motiviccoarse spectra F U ( X ) → Yo s ( O ( X )) → O ∞ ( X ) → ΣYo s ( F U ( X )) , (8.1)where, by definiton, O ∞ ( X ) := Cofib (cid:0) Yo s ( Y ( X )) → Yo s ( O ( X )) (cid:1) . This construction is functorial for X in UBC .26 efinition 8.2.
We call the resulting functor O ∞ : UBC → Sp X the germs at infinityof the cone. We refer to [BE16, Ex. 5.16] and [BEKWb, Sec. 9] for more details.In the proof of Proposition 10.12 below it is useful to use a modified version of the coneover a uniform bornological coarse space X which we will denote by ˜ O ( X ). Definition 8.3.
We let ˜ O ( X ) be the bornological coarse space defined as follows:1. The underlying set of ˜ O ( X ) is [0 , ∞ ) × X .2. The bornology of ˜ O ( X ) is generated by the subsets [0 , n ] × B for n in N and boundedsubsets B of X
3. The coarse structure of ˜ O ( X ) is generated by the entourages of the form V ∩ U φ ,where V is a coarse entourage of [0 , ∞ ) ⊗ X and U φ := { (( s, x ) , ( t, y )) ∈ ([0 , ∞ ) × X ) × ([0 , ∞ ) × X ) | ( x, y ) ∈ φ (max { s, t } ) } for all cofinal functions φ : [0 , ∞ ) → T . Note that the underlying bornological spaces of O ( X ), ˜ O ( X ) and [0 , ∞ ) ⊗ X coincide.The identity map of the underlying sets induces a morphism i : O ( X ) → ˜ O ( X ) . (8.2) Lemma 8.4.
The morphism (8.2) induces an equivalence Yo s ( i ) : Yo s ( O ( X )) → Yo s ( ˜ O ( X )) . Proof.
We define a map of sets q : [0 , ∞ ) × X → [0 , ∞ ) × X , q ( t, x ) := ( √ t, x ) . The map q induces a morphism of bornological coarse spaces j : ˜ O ( X ) → O ( X ). Notethat the compositions i ◦ j and j ◦ i are both given on the level of sets by the map q . Itsuffices to show that the morphisms on Yo s ( O ( X )) or Yo s ( ˜ O ( X )), respectively, inducedby q are equivalent to the respective identities.We first consider the case of the modified cone ˜ O ( X ). In this case we shall see that q is coarsely homotopic to the identity (see [BE16, Defn. 4.17]). In order to define thehomotopy we let the map p + : ˜ O ( X ) → [0 , ∞ ) be given by p + ( t, x ) := t + 1 − √ t + 1and set p := ( p + , p + is bornological and controlled. Then we define thecoarse homotopy I p ˜ O ( X ) → ˜ O ( X ) , ( u, t, x ) (cid:55)→ (cid:16)(cid:16) − up + ( t ) (cid:17) t + up + ( t ) √ t, x (cid:17) (8.3)27see [BE16, Defn. 4.14] for notation of coarse cylinders). One easily checks that this mapis proper and controlled. Since Yo s is invariant under coarse homotopies (in particularusing [BE16, Cor. 4.18]) we conclude thatYo s ( q ) : Yo s ( ˜ O ( X )) → Yo s ( ˜ O ( X ))is equivalent to the identity.The case of the cone O ( X ) is more involved. By Definition 8.1 the hybrid structure on O ( X ) is generated by entourages of the form V ∩ U ( κ,φ ) . We fix the pair ( κ, φ ) and V . Wecan now choose a differentiable function σ : [0 , ∞ ) → [0 , ∞ ) such that lim t →∞ σ ( t ) = ∞ and p + : O ( X ) V ∩ U ( κ,φ ) → [0 , ∞ ) given by p + ( t ) := σ ( t )( t + 1 − √ t + 1)is controlled. To this end we must make sure that (1 + t ) σ (cid:48) ( t ) and σκ are both uniformlybounded. Note that p + is also bornological. We then define the coarse homotopy I p O ( X ) V ∩ U ( κ,φ ) → O ( X )between the maps induced by id [0 , ∞ ) × X and q by the same formula as in (8.3) as above.Indeed one checks that this map is proper and controlled. Hence we have an equivalenceof morphisms Yo s ( q ) (cid:39) Yo s ( id ) : Yo s ( O ( X ) V ∩ U ( κ,φ ) ) → Yo s ( O ( X )) . We now perform the colimit of these equivalences over the poset of data ( V, ( κ, φ )). By u -continuity we get the desired equivalence ofYo s ( q ) : Yo s ( O ( X )) → Yo s ( O ( X ))with the identity.Note that in the definition of the modified cone ˜ O ( X ) we have not fixed the decay rate(encoded in the function φ in Definition 8.3.3) of the entourages in the X -direction as t and s tend to ∞ . Let us fix such a function φ which we assume to be monotoneous andsuch that φ (0) = X × X . Definition 8.5.
We let ˜ O φ ( X ) be the bornological coarse space defined as follows:1. The underlying set of ˜ O φ ( X ) is [0 , ∞ ) × X .2. The bornology of ˜ O φ ( X ) is generated by the subsets [0 , n ] × B for n in N and boundedsubsets B of X .3. The coarse structure of ˜ O φ ( X ) is generated by entourages of the form V ∩ U φ , where V is a coarse entourage of [0 , ∞ ) ⊗ X . xample 8.6. Let X be a metric space. Recall that its coarse structure is generated bythe collection of entourages W r := { ( x, y ) ∈ X × X | d ( x, y ) ≤ r } for all r >
0. If we set φ ( t ) = W /t , then ˜ O φ ( X ) is the open cone over X as considered at many places in thecoarse geometry literature and usually called the Euclidean cone over X .We have a canonical morphism k φ : ˜ O φ ( X ) → ˜ O ( X ) (8.4)given by the identity of the underlying sets. Lemma 8.7. If φ is monotoneous and satisfies φ (0) = X × X , then the map (8.4) inducesan equivalence Yo s ( k φ ) : Yo s ( ˜ O φ ( X )) → Yo s ( ˜ O ( X )) . Proof. If φ (cid:48) is a second monotoneous function as in 8.3.3 such that φ ( t ) ⊆ φ (cid:48) ( t ) for all t in[0 , ∞ ), then U φ ⊆ U φ (cid:48) . Therefore the identity of the underlying maps induces a morphism k φ (cid:48) φ : ˜ O φ ( X ) → ˜ O φ (cid:48) ( X ) . By u -continuity we have an equivalenceYo s ( ˜ O ( X )) (cid:39) colim φ (cid:48) ≥ φ Yo s ( ˜ O φ (cid:48) ( X )) . It therefore suffices to show thatYo s ( k φ (cid:48) φ ) : Yo s ( ˜ O φ ( X )) → Yo s ( ˜ O φ (cid:48) ( X ))is an equivalence for all pairs φ, φ (cid:48) such that φ ( t ) ⊆ φ (cid:48) ( t ) for all t in [0 , ∞ ).We will show now that there exists a controlled function σ : [0 , ∞ ) → [0 , ∞ ) such that φ (cid:48) ( t ) ⊆ φ ( σ ( t )) and lim t →∞ σ ( t ) = ∞ . To this end set δ : [0 , ∞ ) → [0 , ∞ ) , δ ( s ) := sup { t ∈ [0 , ∞ ) | φ (cid:48) ( s ) ⊆ φ ( t ) } . This function is monotoneously increasing and satisfies lim s →∞ δ ( s ) = ∞ . The idea isnow to define σ to be δ . But to ensure that σ is controlled, we have to modify this ideaslightly. We choose t ∈ [0 , ∞ ) such that δ ( t ) ≥
2. We can find σ ( t ) for all t in [ t , ∞ ) bysolving the equation t = (cid:90) σ ( t )0 h ( s ) ds , where h is a function with h ≥ t ≤ (cid:90) δ ( t )0 h ( s ) ds . More concretely, we can take h ( t ) := max (cid:8) , sup s ∈ [1 ,t ] δ − (2 s ) (cid:9) , δ − ( u ) := sup { r ∈ [0 , ∞ ) | δ ( r ) ≤ u } . Note that if t ∈ [ t , ∞ ), then theinterval [ δ ( t ) / , δ ( t )] in the domain of integration yields the estimate (cid:90) δ ( t )0 h ( s ) ds ≥ δ ( t )2 δ − (2 δ ( t ) / ≥ t . For t ∈ [0 , t ] we set σ ( t ) = 0. The Lipschitz constant of σ on [ t , ∞ ) is bounded by 1. Itfollows that σ is controlled.We consider the map of sets q : [0 , ∞ ) × X → [0 , ∞ ) × X , q ( t, x ) := ( σ ( t ) , x ) . By construction it induces a morphism j : ˜ O φ (cid:48) ( X ) → ˜ O φ ( X ) . We now note that the compositions j ◦ k φ (cid:48) φ : ˜ O φ ( X ) → ˜ O φ ( X ) , k φ (cid:48) φ ◦ j : ˜ O φ (cid:48) ( X ) → ˜ O φ (cid:48) ( X )are both induced by q .It suffices to show that these morphisms are both coarsely homotopic to the identity.We set p := ( σ + 1 ,
0) and observe that the map I p ˜ O φ ( X ) → ˜ O φ ( X ) , ( u, t, x ) (cid:55)→ (cid:16)(cid:16) − uσ ( t ) + 1 (cid:17) t + uσ ( t ) + 1 σ ( t ) , x (cid:17) is a suitable homotopy (i.e., proper and controlled) that does the job. The same constructionalso works in the case of φ (cid:48) . Remark 8.8.
The cone ˜ O ( X ) has a big family Y ( X ) := ([0 , n ] ⊗ X ) n ∈ N and we can definea modified version of the germs at infinity˜ O ∞ ( X ) := Cofib (cid:0) Yo s ( Y ( X )) → Yo s ( ˜ O ( X )) (cid:1) . Similarly we can define˜ O ∞ φ ( X ) := Cofib (cid:0) Yo s ( Y ( X )) → Yo s ( ˜ O φ ( X )) (cid:1) . The inclusion X → [0 , n ] × X is a coarse equivalence for every n in N and the structureinduced by ˜ O ( X ) or ˜ O φ ( X ), respectively. In the latter case this is granted by the conditionthat φ (0) = X × X . Therefore we get fibre sequencesYo s ( F U ( X )) → Yo s ( ˜ O ( X )) → ˜ O ∞ ( X ) → ΣYo s ( F U ( X )) , and Yo s ( F U ( X )) → Yo s ( ˜ O φ ( X )) → ˜ O ∞ φ ( X ) → ΣYo s ( F U ( X )) , O ∞ ( X ) (cid:39) ˜ O ∞ ( X ) (cid:39) ˜ O ∞ φ ( X ) . So we could have defined the germs at infinity of the cone using a modified version of thecone. But since the modified cones do not come from a hybrid structure construction wecan not apply the general theorems (Homotopy Theorem and Decomposition Theorem)for hybrid spaces shown in [BE16, Sec. 5.2 & 5.3] in order to deduce the properties of thisfunctor, see e.g. Lemma 9.1 below. For this reason we prefer to work with O ( X ) insteadof ˜ O ( X ) or ˜ O φ ( X ). Example 8.9.
Let X be a geodesic, locally compact hyperbolic metric space. One canconstruct a nice compactification of X by attaching the Gromov boundary ∂X . Note that ∂X is a compact metric space. Higson–Roe [HR95] showed that X is coarsely homotopyequivalent to the Euclidean cone ˜ O φ ( ∂X ) over its Gromov boundary ∂X . Together withthe results of the present section we therefore get the equivalenceYo s ( X ) (cid:39) Yo s ( O ( ∂X )) . (8.5)Fukaya–Oguni [FO17] generalized the result of Higson–Roe to all proper coarsely convexspaces (examples are hyperbolic spaces, CAT(0) spaces and systolic complexes). Especially,we have the equivalence (8.5) where ∂X is a suitable version of Gromov’s boundary. In this section we define the coarse assembly map.Taking the functoriality of cone sequence (8.1) into account we get a fibre sequence offunctors from
UBC to Sp X F → Yo s ◦ O → O ∞ ∂ −→ Σ F (9.1)which we call the cone sequence. Lemma 9.1.
The functors Yo s ◦ O , O ∞ : UBC → Sp X satisfy excision for uniformlyand coarsely excisive decompositions, and they are homotopy invariant.Proof. This is shown in [BEKWb, Sec. 9.4 & 9.5].
Remark 9.2.
Since we consider excision for decompositions which are uniform and coarseat the same time it is not necessary to assume that our uniform spaces are Hausdorff, see[BEKWb, Rem. 9.26]. 31y Lemma 6.1 functor F vanishes on flasque spaces, but we do not expect that O vanisheson flasque spaces. Assume that X is a flasque uniform bornological coarse space with theflasqueness witnessed by the self-map f . Then in general O ( f ) is not close to the identity,but it is equivalent to it [BE16, Cor. 5.31]. In fact, the map O ( f ) exhibits the cone O ( X )as a weakly flasque bornological coarse space in the sense of [BEKWb, Def. 4.17], see[BEKWb, Proof of Prop. 11.20]. Definition 9.3 ([BEKWb, Def. 4.18]) . A coarse homology theory is called strong if itvanishes on weakly flasque bornological coarse spaces.
Example 9.4.
Ordinary coarse homology H X , algebraic K -theory K AX of an additivecategory A and coarse K -homology K X are strong coarse homology theories.Let C be a cocomplete stable ∞ -category and E : BornCoarse → C be a coarse homologytheory. We set E O ∞ := E ◦ O ∞ , E O := E ◦ O . Lemma 9.5. If E is strong, then both E O ∞ , E O : UBC → C are local homology theories.Proof. For a uniform bornological coarse space X we have a natural fibre sequence E ( F U ( X )) → E O ( X ) → E O ∞ ( X ) E ( ∂ ) −−→ Σ E ( F U ( X )) . (9.2)By Lemma 9.1 both functors E O and E O ∞ are homotopy invariant and satisfy excision,and by Lemma 3.11 the functor E ◦ F U also has these properties.Lemma 3.11 shows that the functor E ◦ F U is u -continuous. The functor O ∞ is invariantunder coarsenings ([BEKWb, Prop. 9.31] or Definition 12.8) which implies the equivalence O ∞ ( X U ) (cid:39) → O ∞ ( X ) for sufficiently large entourages U of X . In particular, the functor E O ∞ is u -continuous. It follows from the fibre sequence (9.2) that E O is u -continuous.If X is flasque, then F U ( X ) is flasque and O ( X ) is weakly flasque. Since E ( F U ( X )) (cid:39) E O ( X ) (cid:39) E , we conclude that E O ∞ ( X ) (cid:39) E be a strong coarse homology theory. Definition 9.6.
The coarse assembly map is the natural transformation between coarsehomology theories µ E : E O ∞ P → Σ E defined as the composition of E ( ∂ ◦ P ) with the identification ( E F ) P (cid:39) E from (6.2) . emark 9.7. It follows from the above fibre sequence (9.1) that for a bornological coarsespace X the coarse assembly map µ E,X : E O ∞ P ( X ) → Σ E ( X ) (9.3)is an equivalence if and only if E O P ( X ) (cid:39)
0. Therefore we have identified E O P as thecoarse homology theory which detects the obstructions to µ E being an equivalence. Remark 9.8.
At the moment the local homology theory E O ∞ appearing in the domainof the coarse assembly map might appear mysterious. In Proposition 12.17 we calculatethe evaluation of this homology theory on countable, finite-dimensional, locally finitesimplicial complexes under the assumption that E is countably additive.In general, using the fact that O ∞ ( X ) is representable (see Section 15) one can express thevalue of E O ∞ on a uniform bornological coarse space by the value of E on an explicitlygiven bornological coarse space.
10 Isomorphism results
In this section we discuss conditions which imply that the coarse assembly map µ E,X (Equation 9.3) is an equivalence. We will discuss the cases of finite asymptotic dimension,finite decomposition complexity, and scaleable spaces. Our goal is to show that in manycases the reasons for the validity of the coarse Baum–Connes conjecture for X in fact implyin greater generality that the coarse assembly map µ E,X is an equivalence for suitablecoarse homology theories E .Note that the coarse assembly map µ E : E O ∞ P → Σ E is a morphism between coarsehomology theories. So it is clear from the outset that the property of µ E,X of being anequivalence only depends on the coarse motivic spectrum Yo s ( X ). Let ( X, C , B ) be a bornological coarse space. Recall that X is called discrete as a coarsespace, if C = C(cid:104){ diag X }(cid:105) .Let X be a bornological coarse space and E be a strong coarse homology theory. Proposition 10.1. If X is discrete as a coarse space, then the coarse assembly map µ E,X is an equivalence.Proof.
Assume that X is a discrete bornological coarse space. Then we have X ∼ = P diag ( X )as uniform bornological coarse spaces if we equip X with the discrete uniform structure. By[BEKWb, Prop. 9.35] the boundary map of the cone sequence (9.1) induces an equivalence O ∞ ( X ) (cid:39) → ΣYo s ( F U ( X )) . This implies the result immediately. 33 orollary 10.2. If E is strong, then the coarse assembly map µ X,E is an equivalence forall bornological coarse spaces X such that Yo s ( X ) belongs to the subcategory Sp X (cid:104) disc (cid:105) generated under colimits by discrete bornological coarse spaces.
Let X be a bornological coarse space with coarse structure C and E be a strong coarsehomology theory. Theorem 10.3.
Assume that there exists a cofinal set of entourages U in C such that X U has finite asymptotic dimension.Then the coarse assembly map µ E,X : E O ∞ P ( X ) → Σ E ( X ) is an equivalence.Proof. The assumptions on the space X imply by [BE16, Thm. 6.114] that the motiveYo s ( X ) belongs to Sp X (cid:104) disc (cid:105) . We now apply the above Corollary 10.2.Note that the condition of having finite asymptotic dimension only depends on the coarsestructure of X . Guentner–Tessera–Yu [GTY12] introduced a weaker condition than finite asymptoticdimension called finite decomposition complexity FDC. In [BEKWa] we investigated underwhich assumptions on E the condition that a bornological coarse space X has FDC impliesthat E O P ( X ) (cid:39) E we need weak additivity and transfers. Letus define and discuss these notions now.The notion of a coarse homology theory with transfers was introduced in [BEKWa]. In[BEKWa, Sec. 2.2] we introduced the category BornCoarse tr of bornological coarse spaceswith transfers. It has the same objects as BornCoarse , but its morphisms X → X (cid:48) arecompositions f ◦ tr X,I of a transfer morphisms tr X,I : X → I min,min ⊗ X for some well-ordered set I and a morphism f : I min,min ⊗ X → X (cid:48) of bornological coarse spaces. In particular we have a canonically given inclusion functor BornCoarse → BornCoarse tr .A C -valued coarse homology theory with transfers is a functor E tr : BornCoarse tr → C such that the restriction E : BornCoarse → C of E tr along the canonical inclusion is acoarse homology theory and for every i in I the composition E tr ( X ) E tr ( tr X,I ) −−−−−−→ E tr ( I min,min ⊗ X ) excision (cid:39) E tr ( X ) ⊕ E tr (( I \ { i } ) min,min ⊗ X ) (10.1)34s equivalent to id E tr ( X ) ⊕ E tr ( tr X,I \{ i } ) . We say that a coarse homology theory E admits transfers if it has an extension to a coarsecohomology theory with transfers.For the definition of the property of E being weakly additive we refer to [BEKWa]. Butnote that E is weakly additive if it is strongly additive. Recall that E is strongly additiveif it sends free unions to products, i.e, E (cid:0) free (cid:71) i ∈ I X i (cid:1) (cid:39) (cid:89) i ∈ I E ( X i )for every family ( X i ) i ∈ I of bornological coarse spaces, where the map is induced by thefamily of projections ( E ( (cid:70) freei ∈ I X i ) → E ( X i )) i ∈ I given by excision. Example 10.4.
Examples of coarse homology theories which admit transfers are ordinarycoarse homology H X , algebraic K -homology K AX with coefficients in an additive category A , and coarse K -homology K X . We refer to [BEKWa] for the first two cases and [BE] forthe last case. In these references we actually considered the equivariant case for a group G . For the present application just need the case of a trivial group G = { } .In [BEKWa] we show that the cohomology theories H X and K AX are strongly additive.At the moment we do not know whether coarse K -homology K X is strongly additive, seethe discussion in [BE16, Rem. 7.76].Let E be a C -valued coarse homology theory. Assumption 10.5.
Assume:1. C is compactly generated.2. E is strong.3. E is weakly additive.4. E admits transfers. Remark 10.6.
The category of spectra Sp is compactly generated.Let X be a bornological coarse space and E be a coarse homology theory. Theorem 10.7. If E satisfies the Assumption 10.5 and X U has FDC for a cofinal set ofentourages U of X , then the coarse assembly map µ E,X is an equivalence.Proof.
This follows from [BEKWa, Thm. 1.6] and Remark 9.7.
Remark 10.8.
Note that finite asymptotic dimension implies FDC. Hence in the casethat the coarse homology theory E satisfies Assumption 10.5 the above Theorem 10.7generalizes Theorem 10.3. 35 emark 10.9. In [BE16, Lem. 7.74] we show that coarse K -homology K X is stronglyadditive on the subcategory of BornCoarse of locally countable bornological coarse spaces.This should suffices to conclude the statement of Theorem 10.7 under the assumption that X is locally countable. But we have not checked this in detail. In the literature on the coarse Baum–Connes conjecture it is an important observationthat the existence of a suitable scaling implies that the analytic coarse assembly map incoarse K -homology is an isomorphism [HR95]. In the following we show analogous resultsfor general coarse homology theories.Let X be a uniform bornological coarse space and s : X → X be a morphism of uniformbornological coarse spaces. We assume that the uniform structure of X is induced by ametric. Definition 10.10.
The morphism s is a scaling if it satisfies the following conditions:1. s is -Lipschitz.2. For every coarse entourage W and uniform entourage V of X there exists k in N such that ( s k × s k )( W ) ⊆ V .3. For every coarse entourage U of X the union (cid:83) k ∈ N ( s k × s k )( U ) is also a coarseentourage of X . Example 10.11.
Assume that X is a proper metric space whose structures are inducedfrom the metric. If s : X → X is a map which is 1 / s is ascaling in the sense of Definition 10.10. Note that in order to be a scaling in the sense of[HR95, Def. 7.1] one must in addition assume that s is coarsely and properly homotopicto the identity. These conditions will be added in Definition 10.15 which characterizescoarse scalings.Using the existence of a scaling for X we want to deduce that E O ( X ) (cid:39) E . Similarly as in the proof of [HR95, Thm. 7.2] the argumentis based on an Eilenberg swindle. In order to make this work in our abstract setting weneed to assume that the homology theory admits transfers.Let X be a uniform bornological coarse space and s : X → X be a morphism. Furthermorelet E be a coarse homology theory. Proposition 10.12.
Assume:1. s : X → X is a scaling.2. E ( F U ( s )) (cid:39) id E ( F U ( X )) .3. E admits transfers.4. E O ∞ ( s ) (cid:39) id E O ∞ ( X ) . hen E O ( X ) (cid:39) . Before starting the proof of the above proposition let us first prove the following statement.Recall Definition 8.3 of the modified cone ˜ O ( X ). We define the map of setsΦ : N × [0 , ∞ ) × X → [0 , ∞ ) × X , Φ( n, t, x ) := ( n + t, s n ( x )) . Lemma 10.13.
The map Φ is a morphism of bornological coarse spaces Φ : N min,min ⊗ ˜ O ( X ) → ˜ O ( X ) . Proof.
First we show that Φ is proper. Let B be a bounded subset in X and u be in N and consider the bounded subset [0 , u ] × B in ˜ O ( X ). Then Φ − ([0 , u ] × B ) is contained in[0 , u ] × [0 , ∞ ) × X . The restriction of Φ to { n } × ˜ O ( X ) is proper for every n in N sincethe maps s n : X → X and t (cid:55)→ n + t : [0 , ∞ ) → [0 , ∞ ) are proper. Therefore we canconclude that Φ − ([0 , u ] × B ) is bounded.We now show that Φ is controlled. It is easy to check using 10.10.3 and the fact that t (cid:55)→ n + t is 1-Lipschitz that Φ is a morphism of bornological coarse spaces N min,min ⊗ F U ([0 , ∞ ) ⊗ X ) → F U ([0 , ∞ ) ⊗ X ) . Let ψ : [0 , ∞ ) → [0 , ∞ ) be a function such that lim t →∞ ψ ( t ) = 0. For simplicity we canassume that ψ is monotoneously decreasing. It determines a function φ : [0 , ∞ ) → T by φ ( t ) := U ψ ( t ) as used in Definition 8.3.3. Let W be a coarse entourage of X and V := U r × W be a coarse entourage of [0 , ∞ ) ⊗ X for r in (0 , ∞ ). Then we must show that(Φ × Φ)( diag ( N ) × V ∩ U φ ) ⊆ U φ (cid:48) for φ (cid:48) ( t ) = U ψ (cid:48) ( t ) with ψ (cid:48) having the same properties as ψ . This boils down to the assertionthat for all t in [0 , ∞ ) we have d ( s n ( x ) , s n ( y )) ≤ ψ (cid:48) ( t ) for all n ∈ N with t ≥ n and( x, y ) ∈ W with d ( x, y ) ≤ ψ ( t − n − r ) (here we use the monotonicity of ψ ). Here we set ψ ( t ) := ψ (0) for negative t .We define the monotonously decreasing function e : N → [0 , ∞ ] , e ( n ) := sup { d ( s n ( x ) , s n ( y )) | ( x, y ) ∈ W } . By 10.10.2 we have lim n →∞ e ( n ) = 0. We define ψ (cid:48) ( t ) := max { min { ψ ( t − n − r ) , e ( n ) } | n ∈ N & t ≥ n } . In view of 10.10.1 this function would do the job if lim t →∞ ψ (cid:48) ( t ) = 0. Let (cid:15) in (0 , ∞ ) begiven. Then we choose n in N so large that e ( n ) ≤ (cid:15) for all n in N with n ≥ n . Letfurthermore t in [0 , ∞ ) be so large that ψ ( t ) ≤ (cid:15) for all t in [ t , ∞ ). If t in [0 , ∞ ) satisfies t ≥ n + t + r , then ψ (cid:48) ( t ) ≤ (cid:15) . 37 roof of Proposition 10.12. Let E tr be an extension of E to a coarse homology theorywith transfers. An application of the relation (10.1) yields a decomposition E tr (Φ ◦ tr ˜ O ( X ) , N ) (cid:39) E tr (Φ (cid:48) ◦ tr ˜ O ( X ) , N ≥ ) + id E ( ˜ O ( X )) , (10.2)where Φ (cid:48) is the restriction of Φ to N ≥ min,min ⊗ ˜ O ( X ). We consider the following commutingdiagram in BornCoarse tr :˜ O ( X ) ˜ O ( s ) (cid:15) (cid:15) tr X, N ≥ (cid:47) (cid:47) N ≥ min,min ⊗ ˜ O ( X ) ( n (cid:55)→ n − ⊗ ˜ O ( s ) (cid:15) (cid:15) Φ (cid:48) (cid:47) (cid:47) ˜ O ( X )˜ O ( X ) tr X, N (cid:47) (cid:47) N min,min ⊗ ˜ O ( X ) Φ (cid:47) (cid:47) ˜ O ( X ) T (cid:79) (cid:79) where T : ˜ O ( X ) → ˜ O ( X ) is given by T ( t, x ) := ( t + 1 , x ). Note that the morphism T isclose to the identity.Since T is close to the identity the commutativity of the above diagram implies E tr (Φ (cid:48) ◦ tr ˜ O ( X ) , N ≥ ) (cid:39) E tr ( T ) ◦ E tr (Φ) ◦ E tr ( tr ˜ O ( X ) , N ) ◦ E tr ˜ O ( s ) (cid:39) E tr (Φ ◦ tr ˜ O ( X ) , N ) ◦ E ˜ O ( s ) , from which we get, using (10.2), E tr (Φ ◦ tr ˜ O ( X ) , N ) (cid:39) E tr (Φ ◦ tr ˜ O ( X ) , N ) ◦ E ˜ O ( s ) + id E ˜ O ( X ) . (10.3)We now consider the diagram (note that we are now using the cone instead of the modifiedcone as above) E ( F U ( X )) ι (cid:47) (cid:47) E ( F U ( s )) (cid:3) (cid:3) (cid:27) (cid:27) E O ( X ) δ (cid:115) (cid:115) (cid:47) (cid:47) E O ( s ) (cid:3) (cid:3) (cid:27) (cid:27) E O ∞ ( X ) (cid:47) (cid:47) E O ∞ ( s ) (cid:3) (cid:3) (cid:27) (cid:27) Σ E ( F U ( X )) Σ E ( F U ( s )) (cid:3) (cid:3) (cid:27) (cid:27) E ( F U ( X )) ι (cid:47) (cid:47) E O ( X ) (cid:47) (cid:47) E O ∞ ( X ) (cid:47) (cid:47) Σ E ( F U ( X ))whose horizontal sequences are two copies of the cone sequence and the non-labeled verticalmaps are induced by the identity. The diagram is a picture of two morphisms betweenfibre sequences (one is the identity) which we want to compare. The Condition 4 yields amorphism δ : E O ( X ) → E ( F U ( X )) such that E O ( s ) − id E O ( X ) (cid:39) ι ◦ δ . (10.4)Condition 2 then implies that ι ◦ δ ◦ ι (cid:39) . (10.5)In view of Lemma 8.4 we get the same relations if we replace the cone by the modifiedcone. Equivalence (10.3) now implies that (using in the second line (10.4) for the modifiedcone) id E ˜ O ( X ) (cid:39) E tr (Φ ◦ tr ˜ O ( X ) , N ) − E tr (Φ ◦ tr ˜ O ( X ) , N ) ◦ E ˜ O ( s ) (cid:39) E tr (Φ ◦ tr ˜ O ( X ) , N ) − E tr (Φ ◦ tr ˜ O ( X ) , N ) ◦ ( ι ◦ δ + id E ˜ O ( X ) ) (cid:39) − E tr (Φ ◦ tr ˜ O ( X ) , N ) ◦ ι ◦ δ
38f we compose this equivalence from the right with ι and use (10.5), then we get ι (cid:39) − E tr (Φ ◦ tr N , ˜ O ( X ) ) ◦ ι ◦ δ ◦ ι (cid:39) . Hence we get id E ˜ O ( X ) (cid:39) , which in view of Lemma 8.4 implies E O ( X ) (cid:39) F U ( s ) is coarsely homotopic to the identity map. In the literature this is a standardassumption on a scaling; see, e.g., Higson–Roe [HR95].Condition 10.12.4 is more problematic. If s is homotopic to the identity in the sense of UBC ,then 10.12.4 is satisfied by the homotopy invariance of the functor E O ∞ : UBC → C ,see Lemma 9.1. Unfortunately, in applications s is rarely homotopic to the identity inthe sense of UBC . The standard assumption made in, e.g., Higson–Roe [HR95] is that F C , U / ( s ) is homotopic to the identity map, i.e., that s is homotopic to the identity in thesense of TopBorn (i.e., after forgetting the coarse and the uniform structures, but thehomotopies are still required to be proper). If E is countably additive, then E O ∞ hasbetter homotopy invariance properties on nice spaces which we will use in the following tomake the standard assumption of Higson–Roe also work in our situation.Let X be a uniform bornological coarse space and let E be a C -valued coarse homologytheory. Lemma 10.14.
Assume:1. X is homotopy equivalent (in UBC ) to a countable, locally finite, finite-dimensionalsimplicial complex.2. E is countably additive.3. C is presentable.4. F C , U / ( s ) is homotopic to id F C , U / ( X ) .Then E O ∞ ( s ) (cid:39) id E O ∞ ( X ) .Proof. This is an immediate consequence of Corollary 12.18.In the following definition of a coarse scaling we introduce a class of scalings with additionalproperties ensuring that Proposition 10.12 is applicable.Let X be a uniform bornological coarse space whose uniform structure is induced by ametric and let s : X → X be a scaling. Definition 10.15.
The scaling s is a coarse scaling if it satisfies in addition:1. F U ( s ) is coarsely homotopic to the identity.2. F C , U / ( s ) is properly homotopic to the identity. emark 10.16. A scaling in the sense of [HR95, Def. 7.1] is a coarse scaling; see alsoExample 10.11.The following corollary is an analog of Higson-Roe [HR95, Thm. 7.2]. Assumption 10.17.4does not occur in [HR95] because the analogue of our E O ∞ is the functor X (cid:55)→ K ∗ ( D ∗ ( X ))in the notation of [HR95] which has good homotopy invariance properties replacing theapplication of our Lemma 10.14. Corollary 10.17.
Assume:1. E is countably additive and admits transfers.2. C is presentable.3. The uniform structure of X is induced by a metric.4. X is homotopy equivalent (in UBC ) to a countable, locally finite, finite-dimensionalsimplicial complex.5. X admits a coarse scaling (see Definition 10.15).Then E O ( X ) (cid:39) and the cone boundary E O ∞ ( X ) → Σ E ( X ) is an equivalence.Proof. This follows from Proposition 10.12. Lemma 10.14 verifies Assumption 10.12.4.
Example 10.18.
A typical example of a uniform bornological coarse space which admitsa coarse scaling is a Euclidean cone. Let Y be a subset of the unit sphere in a Hilbertspace and let X be the cone over Y with the metric induced from the Hilbert space. Weconsider X as a uniform bornological coarse space with all structures induced from themetric. Then the map s : X → X , s ( x ) := x/ Y has a finite-dimensional, locally finite triangulation with a uniform bound on the sizeof its simplices, then so does X . In this case Corollary 10.17 can be applied to X .Let X be a uniform bornological coarse space and E a C -valued coarse homology theory. Theorem 10.19.
Assume:1. E is strong, countably additive, and admits transfers.2. C is presentable.3. The uniform structure of X is induced by a metric.4. X is homotopy equivalent (in UBC ) to a countable, locally finite, finite-dimensionalsimplicial complex.5. X admits a coarse scaling (see Definition 10.15).6. X is coarsifying (Definition 7.8). hen E O P ( F U ( X )) (cid:39) and therefore the coarse assembly map µ E,F U ( X ) is an equivalence.Proof. Since X is coarsifying and E O is a local homology theory (Lemma 9.5) we have anequivalence E O P ( F U ( X )) (cid:39) E O ( X ). We now apply Corollary 10.17 in order to concludethat E O ( X ) (cid:39) Example 10.20.
Let Y and X be as in Example 10.18. In general we can not expect X tobe coarsifying even if Y is compact and the Hilbert space is finite-dimensional. Especially,we do not expect that the analogue of [HR95, Prop. 4.3] is true in our generality. By usingProposition 7.12 one can prove that X is coarsifying if Y is a finite simplicial complex.Hence one can apply Theorem 10.19 to Euclidean cones over finite complexes.Therefore we get the analogue of [HR95, Cor. 7.3] under the additional assumption of Y being a finite simplicial complex (instead of a finite-dimensional compact metric space).Every complete, simply-connected, non-positively curved Riemannian manifold is coarselyhomotopy equivalent to the Euclidean cone over a finite-dimensional sphere. Because afinite-dimensional sphere has a finite triangulation, Theorem 10.19 provides a generalizationof [HR95, Cor. 7.4].Because of the Assumptions 10.19.4 and 10.19.6 we are not able to apply Theorem 10.19 tocones over arbitrary compact metric spaces. In particular, we do not obtain the analogue of[HR95, Cor. 8.2] asserting the coarse Baum–Connes conjecture for all hyperbolic (proper)metric spaces.We do not know whether we should expect that the assmbly map µ E,F U ( X ) is an equivalencefor all hyperbolic (proper) metric spaces or Euclidean cones over finite-dimensional compactmetric spaces and arbitrary coarse homology theories E satisfying the Assumptions 10.19.1and 10.19.2.The next corollary specializes Theorem 10.19 by utilizing a convenient condition on thespace X to be coarsifying.Let K be a simplicial complex and K d be the associated uniform bornological coarse space. Corollary 10.21.
Assume:1. E is strong, countably additive, and admits transfers.2. C is presentable.3. K has bounded geometry.4. K d is equicontinuously contractible.5. K d admits a coarse scaling.Then the coarse assembly map µ E,F U ( K d ) is an equivalence.Proof. Combine Proposition 7.12 with Proposition 10.19.41 xample 10.22. If X is a tree or an affine Bruhat–Tits building of bounded geometry,then Corollary 10.21 applies to X . Hence we obtain the analogue of [HR95, Cor. 7.5]. Example 10.23.
Recall that the stable ∞ -category Sp of spectra is presentable. Examplesof Sp -valued coarse homology theories which are strong, countably additive and admittransfers are ordinary coarse homology H X , algebraic K -homology K AX with coefficientsin an additive category A , and coarse K -homology K X .Examples of spaces admitting coarse scalings and which are homotopy equivalent (in UBC )to uniformly contractible simplicial complexes of bounded geometry are simply-connectedcomplete Riemannian manifolds M with sectional curvatures satisfying − C ≤ sec ≤ C . The coarse scaling s is in this case given by, e.g., s ( x ) := exp(log( x ) / x in M , exp : T x M → M is the Riemannian exponentialmap and log : M → T x M is its inverse. Remark 10.24. If E is the coarse K -homology K X , then using the comparison betweenthe analytic and the topological assembly maps obtained in Section 16 below one can infact deduce the isomorphism statements of Higson–Roe [HR95, Sec. 7] (under additionalassumptions on X related to Conditions 10.19.4 and 10.19.6; see the discussion at the endof Example 10.20) formally as special cases of Proposition 10.19 or Corollary 10.21. Butnote that specialized to these cases our proof is not really different from the proof givenby Higson–Roe [HR95]. In fact, in our approach we have just separated the geometric andhomological arguments from the analysis which is hidden in the verification that K X is acoarse homology theory, the existence of transfers, and the comparison between the twoassembly maps.
11 Extension from locally compact, separable spaces
In this section we will describe an extension process from locally compact, separablespaces to uniform bornological coarse spaces. It will be used to construct analytic local K -homology. Definition 11.1.
A uniform bornological coarse space X is called small if the underlyingtopological space of X is separable and locally compact, and the bornology consists of therelatively compact subsets. We let
UBC small denote the full subcategory of
UBC consisting of small uniform bornolog-ical coarse spaces. Adapting Definition 3.10 we can talk about local homology theories E : UBC small → C . We call them restricted. Definition 11.2. E is called a closed restricted local homology theory, if E is a restrictedlocal homology theory which satisfies in addition excision for closed decompositions. Remark 11.3.
Note that the closed decompositions considered in Definition 11.2 neednot be uniformly or coarsely excisive. 42 subset Y of a uniform bornological coarse space X is called small if Y with the induceduniform bornological coarse structure is small. Note that a small subset is closed (becauseof the requirement on the bornology), and a closed subset of a small uniform bornologicalcoarse space is small.We let UBC loc be the category of pairs (
X, Y ), where X is a uniform bornological coarsespace and Y is a small subset of X . A morphism f : ( X, Y ) → ( X (cid:48) , Y (cid:48) ) is a morphism f : X → X (cid:48) in UBC such that f ( Y ) ⊆ Y (cid:48) . We have functors (cid:96) : UBC loc → UBC small , ( X, Y ) (cid:55)→ Y and F loc : UBC loc → UBC , ( X, Y ) (cid:55)→ X .
Let E : UBC small → C be a functor whose target is a cocomplete stable ∞ -category. Wedefine the functor Ex( E ) : UBC → C as the left Kan-extension UBC loc E ◦ (cid:96) (cid:47) (cid:47) F loc (cid:15) (cid:15) CUBC
Ex( E ) (cid:54) (cid:54) Let X be a uniform bornological coarse space. Lemma 11.4. If X is small, then Ex( E ( X )) (cid:39) E ( X ) .Proof. Let X be a uniform bornological coarse space. By the pointwise formula for theleft Kan-extension we have Ex( E )( X ) (cid:39) colim Y E ( Y ) , (11.1)where Y runs over all small subsets of X . If X itself is small, then X is the final elementof the index set of the colimit. Proposition 11.5. If E is a closed restricted local homology theory, then Ex( E ) is a localhomology theory.Proof. We will use the point-wise formula (11.1) for the left Kan-extension. The subspacesof the form [0 , ⊗ Y of [0 , ⊗ X for small subspaces Y are cofinal in all small subspacesof [0 , ⊗ X . Hence homotopy invariance of the restricted homology theory E implieshomotopy invariance of Ex( E ).If Y is a small subset on X , then a coarsely and uniformly excisive closed decomposition( A, B ) induces a closed decomposition ( Y ∩ A, Y ∩ B ). Note that we do not expect thatthe latter is coarsely or uniformly excisive.The analog of (11.1) for closed pairs expresses the excision square for Ex( E ) as a colimitover the small subsets Y of X of the corresponding squares for E . Since E satisfies closedexcision, the colimit square is a colimit of cocartesian squares and hence cocartesian.43ssume that the uniform bornological coarse space X is flasque with flasqueness imple-mented by f : X → X . Let Y be a small subset. Let h : [0 , ⊗ X → X be the homotopyfrom id to f . We define inductively ˜ Y := Y and ˜ Y n := ˜ Y n − ∪ h ([0 , ⊗ ˜ Y n − ). Note that˜ Y small. We show now inductively that ˜ Y n is small. Note that a morphism from a smalluniform bornological coarse space to a uniform bornological coarse space has the propertythat every point in the target has a neighbourhood (take a bounded one) whose preimageis relatively compact. Using the induction hypothesis that ˜ Y n − is small this implies that h ([0 , ⊗ ˜ Y n − ) is closed and locally compact. Moreover, it is separable.We set ˜ Y := (cid:83) n ≥ ˜ Y n . This union is locally finite. Hence ˜ Y is still locally compact andhas the induced bornology of relatively compact subsets. Furthermore it is separable. Themorphism f restricts to ˜ Y and is homotopic to id ˜ Y by restriction of the homotopy h . Weconclude that ˜ Y is flasque and Y ⊆ ˜ Y .Hence, if X is a flasque uniform bornological coarse space, then the index set of the colimitin (11.1) contains a cofinal subset of flasque small subsets. Since E vanishes on flasques,it follows that Ex( E ) vanishes on X .Finally, u -continuity of E implies u -continuity of Ex( E ).Our main example of a closed restricted local homology theory is the functor K an : UBC small → Sp defined by (see [BE16, Sec. 6.7]) K an ( X ) := KK ( C ( X ) , C ) . By [BE16, Lem. 6.89] the functor K an satisfies excision for closed decompositions and ishomotopy invariant. By [BE16, Prop. 6.91] it is locally finite and therefore in particularvanishes on flasques by [BE16, Lem. 6.54]. Finally, K an ( X ) does not depend on the coarsestructure of X . Hence K an is u -continuous. Definition 11.6.
We define the analytic local K -homology by K an , loc := Ex( K an ) . Remark 11.7.
The analytic local K -homology is the analogue of the functor L ( K an ) : TopBorn → Sp appearing in [BE16, Def. 6.92] of analytic locally finite K -homology. In particular, we donot expect that K an , loc is locally finite on all of UBC .44 E O ∞ The goal of this section is to provide a computation of E O ∞ ( X ) in terms of the value E ( ∗ ) of E at the one-point space, see Proposition 12.17. For this calculation we mustadopt some finiteness assumptions on X and require that E is countably additive.In this section we assume that C is a presentable stable ∞ -category.Let F : UBC small → C be a functor and let X be a small (Definition 11.1) uniformbornological coarse space. Definition 12.1.
We define the locally finite evaluation of F at X by F lf ( X ) := lim W Cofib( F ( X \ W ) → F ( X )) , (12.1) where W runs over all open subsets of X with compact closure. Similary as in [BE16, Rem. 6.49] one can turn the above definition into a construction ofa functor F lf : UBC small → C . Remark 12.2.
Here are the details. We consider the category
UBC small, B of pairs ( X, W ),where X is in UBC small and W is an open subset of X with compact closure. A morphism f : ( X, W ) → ( X (cid:48) , W (cid:48) ) is a morphism f : X → X (cid:48) in UBC small with f ( W ) ⊆ W (cid:48) . Wehave the functors p : UBC small, B → UBC small , p ( X, W ) := X and ˜ F : UBC small, B → C , ˜ F ( X, W ) := Cofib( F ( X \ W ) → F ( X )) . We then define the functor F lf as the right Kan extension of ˜ F along p : UBC small, B ˜ F (cid:47) (cid:47) p (cid:15) (cid:15) CUBC small F lf (cid:54) (cid:54) The formula (12.1) now follows from the pointwise formula for the evaluation of the rightKan extension.
Remark 12.3. If F is induced from a functor F (cid:48) : TopBorn → C by F = F (cid:48) ◦ F C , U / ,then we have an equivalence F lf (cid:39) F (cid:48) ,lf ◦ F C , U / , where F (cid:48) ,lf is exactly the locally finite evaluation as defined in [BE16, Def. 6.48].We have a natural morphism F ( X ) → F lf ( X ). Lemma 12.4. If F is homotopy invariant, then so is F lf . roof. The proof if the same as the one of [BE16, Lem. 6.67]. One must observe that thesubsets of the form [0 , × W of [0 , ⊗ X are cofinal in the open subsets with compactclosure. Lemma 12.5. If F satisfies closed or open excision, then so does F lf Proof.
The argument is the same as for [BE16, Lem. 6.68].
Remark 12.6. If F satisfies excision in the sense of Definition 3.6, then it is not clearwhat kind of excison properties F lf has. The problem is that the intersection with X \ W does not necessarily preserve coarsely or uniformly excisive pairs.Let X be a small uniform bornological coarse space. Definition 12.7.
A coarsening X (cid:48) of X is a small uniform bornological coarse spaceobtained from X by replacing the coarse structure by a larger one which is still compatiblewith the bornology. Note that the identity of the underlying sets is a morphism X → X (cid:48) of uniform bornologicalcoarse spaces.Let F : UBC small → C be a functor. Definition 12.8.
We say that F is invariant under coarsening if for every small uniformbornological coarse space and coarsening X → X (cid:48) the induced morphism F ( X ) → F ( X (cid:48) ) is an equivalence. Example 12.9.
The functor O ∞ : UBC small → Sp X is invariant under coarsening, see[BEKWb, Prop. 9.31].The functor K an : UBC small → Sp is invariant under coarsening, since K an ( X ) does notdepend on the coarse structure of X at all.Note that countable, locally finite simplicial complexes naturally provide small uniformbornological coarse spaces.Let X be a countable, locally finite simplicial complex with a decomposition ( A, B ) intosub-complexes.
Lemma 12.10. If F satisfies excision in the sense of Definition 3.6 and is invariantunder coarsening, then we have a push-out square F lf ( A ∩ B ) (cid:47) (cid:47) (cid:15) (cid:15) F lf ( A ) (cid:15) (cid:15) F lf ( B ) (cid:47) (cid:47) F lf ( X ) (12.2)46 roof. We use that the cofibre of a map of cocartesian squares is a cocartesian square.In the limit (12.1) we can restrict W to run only over the interiors of finite sub-complexes.Then X \ W is again a simplicial complex and ( A \ W, B \ W ) is a decomposition of itinto closed sub-complexes.Note that in the terms F ( X \ W ) in (12.1) we must equip the set X \ W with the uniformbornological coarse structures induced from X , and not with the structures coming from apath-metric on X \ W . Although the uniform and bornological structures on X \ W arealso induced from the path-metric of X \ W , this will be in general not true for the coarsestructure. But since F is invariant under coarsening, we can, without changing the valueof F on the spaces X \ W , equip these spaces with the coarse structures associated to theintrinsic path-metrics.Using Example 3.5 we now see that excisiveness of F in the sense of Definition 3.6 can beapplied to the decompositions ( A \ W, B \ W ) of the complexes X \ W occuring in thelimit (12.1). We therefore have expressed the square (12.2) as a limit of cofibres of mapsof cocartesian squares, i.e., as a limit of cocartesian squares. Since C is stable, cartesianand cocartesian squares in C are the same. Hence (12.2) itself is a cocartesian square.Let F : UBC small → C be a functor and assume that F is excisive (in any of the sensesdiscussed above). If X is a small uniform bornological coarse space with the discreteuniform and coarse structures and x is a point in X , then we have a natural projectionmorphism F ( X ) → F ( { x } ), see [BE16, Ex. 4.11]. Definition 12.11. F is called additive if for every small uniform bornological coarse space X with the discrete uniform and coarse structures and the minimal bornology the naturalmorphism F ( X ) → (cid:89) x ∈ X F ( { x } ) induced by the projections is an equivalence. Let us underline that in Definition 12.11 we really mean the product and not the sum.Let F : UBC small → C be a functor. Lemma 12.12.
Assume:1. F satisfies excision in the sense of Definition 3.6.2. F is homotopy invariant.3. F invariant under coarsening.4. F is additive.Then for every countable, locally finite, finite-dimensional simplicial complex X the naturalmorphism F ( X ) → F lf ( X ) is an equivalence. roof. We argue by a finite induction over the dimension. The assertion is true for zerodimensional complexes since they are discrete and F , F lf are additive (see [BE16, Lem. 6.63]for the latter). Assume now that the assertion is true for complexes of dimension n − X is n -dimensional, then we can decompose X into a closed tubular neighbourhood Y of thickness 1 / n − Z of n -simplices of size 2 / / /
3. See the picture on Page 48.Decomposition X = Y ∪ Z used in the proof of Lemma 12.12This closed decomposition is coarsely and uniformly excisive. Hence we can apply excisionfor F in the sense of Definition 3.6. For F lf we use Lemma 12.10.We use homotopy invariance in order to replace the evaluation on Y by the evaluation onthe n − X n − itself. Furthermore, we can contract the n -simplices of size 2 / Z to the set C of their centers. Finally, we contract Y ∩ Z to the set W of the boundariesof these simplices of size 2 / F lf is also invariant under coarsening)in order to replace the induced coarse structures by the coarse structures induced bythe intrinsic path-quasi-metric on the n − X n − and on W and the discrete48oarse structure on the set C of centers of n -simplices. Then we can apply the inductionassumption to X n − , W (which is also n − C . Example 12.13.
Analytic local K -homology K an satisfies the assumptions of the aboveLemma 12.12. Excision for closed decompositions and homotopy invariance is shown in[BE16, Lem. 6.89] and additivity in [BE16, Lem. 6.90]. It is invariant under coarseningssince K an ( X ) does not depend on the coarse structure of X .Recall that a coarse homology theory E is countably additive if we have an equivalence E ( N min,min ) (cid:39) (cid:89) N E ( ∗ ) (12.3)induced by the canonical projections, where N min,min denotes the set N equipped with theminimal coarse and bornological structures.Let E be a coarse homology theory. Proposition 12.14. If E is countably additive and X is a countable, locally finite, finite-dimensional simplicial complex, then the natural morphism E O ∞ ( X ) → ( E O ∞ ) lf ( X ) is an equivalence.Proof. We will check that the assumptions of Lemma 12.12 are satisfied. By Lemma 9.1the functor O ∞| UBC small satisfies excision in the sense of Definition 3.6 and is homotopyinvariant. Therefore E O ∞ has these properties. Furthermore, by Example 12.9 the functor E O ∞ is invariant under coarsening.Let X be a uniform bornological coarse space which is discrete both as a uniform and as acoarse space. Then O ∞ ( X ) (cid:39) ΣYo s ( F U ( X ))by [BEKWb, Prop. 9.33]. Using that E is countably additive at the marked equivalence inthe following chain of equivalences, we have for a small uniform bornological coarse space X with the discrete uniform and coarse structures and the minimal bornology (note thatthe space X is countable under these assumptions) E O ∞ ( X ) (cid:39) E (Σ F U ( X )) (cid:39) Σ E ( F U ( X )) ! (cid:39) Σ (cid:0) (cid:89) x ∈ X E ( { x } ) (cid:1) (cid:39) (cid:89) x ∈ X Σ E ( { x } ) (cid:39) (cid:89) x ∈ X E O ∞ ( { x } )showing that E O ∞ is additive, and therefore finishing this proof.49ollowing Weiss–Williams [WW95], for a homotopy invariant functor F : UBC small → C we can construct a best approximation of F by a homology theory. It is given by the Kanextension procedure described in the proof of [BE16, Prop. 6.73] which produces a functorand a natural transformation F % : UBC small → C , F % → F .
The objectwise formula for F % is F % ( X ) := colim (∆ n → X ) F (∆ n ) , where the colimit runs over the category of simplices of X .Since C is a presentable stable ∞ -category, it is tensored over Sp . Since F is homotopyinvariant, the projection ∆ n → ∗ induces an equivalence F (∆ n ) → F ( ∗ ) (cid:39) Σ ∞ + ( ∗ ) ∧ F ( ∗ ).Using the equivalence colim (∆ n → X ) Σ ∞ + ( ∗ ) (cid:39) Σ ∞ + ( X ) and the fact that ∧ commutes withcolimits we therefore have an equivalence F % ( X ) (cid:39) Σ ∞ + ( X ) ∧ F ( ∗ ) . (12.4)This implies that the functor F % is homotopy invariant and satisfies open excision. Henceits locally finite evaluation ( F % ) lf : UBC small → C satisfies open excision, is homotopyinvariant and is countably additive [BE16, Lem. 6.63]. Alternatively one can use the factthat Remark 12.3 applies to F % by (12.4). Note that in the argument of Σ ∞ + we droppedthe obvious forgetful functor from UBC small to topological spaces.
Remark 12.15. If F is induced from a functor F (cid:48) : TopBorn → C by F = F (cid:48) ◦ F C , U / ,then F % (cid:39) F (cid:48) , % ◦ F C , U / , where F (cid:48) , % is precisely the functor defined in the proof of [BE16, Prop. 6.73]. Lemma 12.16. If F satisfies the assumptions stated in Lemma 12.12, and X is a countable,locally finite, finite-dimensional simplicial complex, then the natural morphism ( F % ) lf ( X ) → F lf ( X ) is an equivalence.Proof. The argument is the same as for Lemma 12.12. We just observe that ( F % ) lf is alsoexcisive for decompositions of simplicial complexes into closed sub-complexes, and that itis also invariant under coarsening.Let E : BornCoarse → C be a coarse homology theory and X be a uniform bornologicalcoarse space. Proposition 12.17.
Assume:1. C is presentable. . E is countably additive, see (12.3) .3. X is homotopy equivalent in UBC to a countable, locally finite, finite-dimensionalsimplicial complex.Then we have a natural equivalence (Σ E ( ∗ ) ∧ Σ ∞ + ) lf ( X ) (cid:39) E O ∞ ( X ) . Proof.
We first observe that Σ E ( ∗ ) (cid:39) E O ∞ ( ∗ ). Then we just combine Proposition 12.14and Lemma 12.16 with F := E O ∞ .Note that the functor X (cid:55)→ (Σ E ( ∗ ) ∧ Σ ∞ + ) lf ( X ) is naturally defined on TopBorn and islocally finite, homotopy invariant (in the sense of
TopBorn , i.e., for proper homotopieswhich are not necessarily uniform), and satisfies open excision. Therefore the functor X (cid:55)→ E O ∞ ( X ) for spaces X in UBC (which are homotopy equivalent in the sense of
UBC to a countable, locally finite, finite-dimensional simplicial complexes), also has thesestronger homological properties. In particular:Let E : BornCoarse → C be a coarse homology theory. Furthermore, let X, X (cid:48) be in
UBC and f, g : X → X (cid:48) be morphisms in UBC . Corollary 12.18.
Assume:1. C is presentable.2. E is countably additive.3. X and X (cid:48) are homotopy equivalent in UBC to countable, locally finite and finite-dimensional simplicial complexes.4. F C , U / ( f ) and F C , U / ( g ) are properly homotopic (there is a homotopy [0 , × X → X (cid:48) which is continuous and proper after forgetting the coarse and uniform structures).Then E O ∞ ( f ) is equivalent to E O ∞ ( g ) .
13 Comparison of coarse homology theories
In ordinary homotopy theory a transformation between spectrum-valued homology theorieswhich induces an equivalence on a point is an equivalence at least on all CW -complexes.In the present section we consider an analogous statement for coarse homology theories.Let C be a presentable stable ∞ -category. Assume that we have a transformation E → E (cid:48) of C -valued coarse homology theories which induces an equivalence E ( ∗ ) → E (cid:48) ( ∗ ). Inthis section we provide sufficient conditions on a space X and on the theories E and E (cid:48) which imply that E ( X ) → E (cid:48) ( X ) is an equivalence. The main result is formulated inCorollary 13.4. 51et X be a bornological coarse space. The following definitions are from [BE16, Def. 6.100],[BE16, Def. 6.102] and [BE16, Def. 7.32]. Definition 13.1. X has strongly bounded geometry if it has the minimal bornologycompatible with the coarse structure and for every coarse entourage U of X thenumber of points in U -bounded subsets of X is uniformly bounded.2. X has bounded geometry if it is equivalent to a bornological coarse space with stronglybounded geometry.3. X is called separable it admits a coarse entourage U and a countable family of points ( x i ) i ∈ I that (cid:83) i ∈ I U [ x i ] = X . Let X be a bornological coarse space and E be a C -valued coarse homology theory. Proposition 13.2.
Assume:1. C is presentable.2. E is countably additive.3. X is separable and of bounded geometry.Then we have a natural equivalence ((Σ E ( ∗ ) ∧ Σ ∞ + ) lf ◦ F C , U / ) P ( X ) (cid:39) E O ∞ P ( X ) . Proof.
Since both sides of the equivalence are coarsely invariant we can assume that X is a countable bornological coarse space of strongly bounded geometry. Then for everyentourage U of X the complex P U ( X ) is a countable, locally finite, finite-dimensionalsimplicial complex. Hence by Proposition 12.17 we get an equivalence(Σ E ( ∗ ) ∧ Σ ∞ + ) lf ( F C , U / ( P U ( X ))) (cid:39) E O ∞ ( P U ( X )) . Forming the colimit over the entourages U of X and using (5.2) we get the equivalence((Σ E ( ∗ ) ∧ Σ ∞ + ) lf ◦ F C , U / ) P ( X ) (cid:39) E O ∞ P ( X )as claimed.Let E → E (cid:48) be a transformation between C -valued coarse homology theories and let X be a bornological coarse space. Theorem 13.3.
Assume:1. C is presentable.2. E and E (cid:48) are strong and countably additive.3. E ( ∗ ) → E (cid:48) ( ∗ ) is an equivalence.4. X is separable and of bounded geometry. . The assembly maps µ E,X and µ E (cid:48) ,X are equivalences (Definition 9.6).Then E ( X ) → E (cid:48) ( X ) is an equivalence.Proof. We have a commuting diagram((Σ E ( ∗ ) ∧ Σ ∞ + ) lf ◦ F C , U / ) P ( X ) Prop. 13 . (cid:39) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) E O ∞ P ( X ) (cid:39) µ E,X (cid:47) (cid:47) (cid:15) (cid:15) Σ E ( X ) (cid:15) (cid:15) ((Σ E (cid:48) ( ∗ ) ∧ Σ ∞ + ) lf ◦ F C , U / ) P ( X ) Prop. 13 . (cid:39) (cid:47) (cid:47) E (cid:48) O ∞ P ( X ) (cid:39) µ E (cid:48) ,X (cid:47) (cid:47) Σ E (cid:48) ( X )The left vertical morphism is an equivalence by Condition 3. We conclude that the rightvertical morphism is an equivalence, too.We can use Theorems 10.3, 10.7 and 10.19 in order to check Condition 5 in the statementof Theorem 13.3.Let E → E (cid:48) be a transformation between C -valued coarse homology theories and X be abornological coarse space. Corollary 13.4.
Assume:1. C is presentable.2. E and E (cid:48) are strong and countably additive.3. E ( ∗ ) → E (cid:48) ( ∗ ) is an equivalence.4. X is separable and of bounded geometry.Furthermore assume one of the following three conditions:1. There is a cofinal set of coarse entourages U of X such that X U has finite asymptoticdimension.2. a) C is compactly generated.b) E and E (cid:48) are weakly additive and admit transfers.c) There is a cofinal set of entourages U of X such that X U has finite decompositioncomplexity.3. a) E and E (cid:48) are strong, countably additive, and admit transfers.b) There exists a uniform bornological coarse space Y with Yo s ( X ) (cid:39) Yo s ( F U ( Y )) and the following holds true:i. The uniform structure of Y is induced by a metric.ii. Y is homotopy equivalent (in UBC ) to a countable, locally finite, finite-dimensional simplicial complex.iii. Y admits a coarse scaling (see Definition 10.15). v. Y is coarsifying (Definition 7.8).Then E ( X ) → E (cid:48) ( X ) is an equivalence. Remark 13.5.
The case of finite asymptotic dimension in the above corollary has beenshown in [BE16, Thm. 6.116] in the slightly more general form that the coarse homologytheories are not assumed to be strong.
14 Coarse assembly map for coarse K -homology Let us apply the theory developed so far to coarse K -homology to interpret the coarseassembly map as a morphism from the coarsification of the locally finite K -homology tothe coarse K -homology, see (14.5).Note that the spectra KK ( C , C ) and K X ( ∗ ) are both equivalent to the complex K -theoryspectrum KU . We can fix once and for all an identification of spectra KK ( C , C ) (cid:39) K X ( ∗ ) . (14.1)Since KU has many self-equivalences this choice is not unique.Since ( K an ,lf ◦ F C , U / ) : UBC small → Sp is a locally finite homology theory, we have by [BE16, Prop. 6.73] the first equivalence inthe following chain of equivalences of functors( K an ,lf ◦ F C , U / ) (cid:39) ( K an ,lf ( ∗ ) ∧ Σ ∞ + ) lf (cid:39) ( K ( C , C ) ∧ Σ ∞ + ) lf (cid:39) ( K X ( ∗ ) ∧ Σ ∞ + ) lf (14.2)on UBC small . The second equivalence uses the definition of the analytic locally finite K -homology K an ,lf ( ∗ ) (cid:39) KK ( C ( ∗ ) , C ) (cid:39) KK ( C , C ), and the third equivalence involvesthe choice (14.1).We recall that K X is additive [BE16, Prop. 7.77], so in particular it is countably additive.If X is homotopy equivalent to a countable, locally finite, finite-dimensional simplicialcomplex, then using Proposition 12.17 and (14.2) we get an equivalence( K an ,lf ◦ F C , U / )( X ) (cid:39) Σ − K X O ∞ ( X ) . (14.3)If X is a separable bornological coarse space of bounded geometry, then we have a naturalequivalence (see Proposition 13.2)( K an ,lf ◦ F C , U / ) P ( X ) (cid:39) Σ − K X O ∞ P ( X ) . (14.4)We can now interpret the coarse assembly map µ K X ,X as a morphism µ topX : ( K an ,lf ◦ F C , U / ) P ( X ) → K X ( X ) (14.5)54rom the coarsification of the locally finite K -homology to the coarse K -homology of X .Observe that this construction produces a morphism of spectra. It is natural in X . It doesnot involve Paschke duality or similar results from functional analysis. On the other handwe must assume that X has bounded geometry and is separable.In the following we spell out explicitly the statement about the compatibility of the coarseassembly map with the coarse Mayer-Vietoris sequences. We use the notation introducedin [BE16, Sec. 6.6] for the coarsification( K an ,lf ◦ F C , U / ) P ∗ ( X ) = QK an ,lf ∗ ( X ) (14.6)of the locally finite analytic K -homology. Corollary 14.1.
Let X be a separable bornological coarse space of bounded geometry andlet ( A, B ) be a coarsely excisive decomposition of X . Then the following square commutes: QK an ,lf ∗ +1 ( X ) ∂ K an ,lfMV (cid:47) (cid:47) µ topX (cid:15) (cid:15) QK an ,lf ∗ ( A ∩ B ) µ topA ∩ B (cid:15) (cid:15) K X ∗ +1 ( X ) ∂ K X MV (cid:47) (cid:47) K X ∗ ( A ∩ B ) O ∞ ( X ) is representable In this section we show that the motivic coarse spectrum O ∞ ( X ) and the boundary mapof the cone sequence are representable. We will use these facts in Section 17 in order toprovide examples of local homology classes for the theory K X O ∞ .Let X be a uniform bornological coarse space. Then we consider the bornological coarsespace O ( X ) − obtained from the uniform bornological coarse space R ⊗ X by taking thehybrid coarse structure [BE16, Def. 5.10] associated to the big family (( −∞ , n ] × X ) n ∈ N .Note that the subset [0 , ∞ ) × X of O ( X ) − with the induced structures is the cone O ( X ).We then have maps of bornological coarse spaces F U ( X ) i −→ O ( X ) j −→ O ( X ) − d −→ F U ( R ⊗ X ) . The first two maps i and j are the inclusions, and the last map d is given by the identityof the underlying sets.The following proposition identifies a segment of the cone sequence (9.1) with a sequencerepresented by maps between bornological coarse spaces. It in particular shows that thecone O ∞ ( X ) is represented by the bornological coarse space O ( X ) − . Proposition 15.1.
We have a commutative diagram in Sp X Yo s ( O ( X )) j (cid:47) (cid:47) Yo s ( O ( X ) − ) ι (cid:39) (cid:15) (cid:15) d (cid:47) (cid:47) Yo s ( F U ( R ⊗ X )) s (cid:39) (cid:15) (cid:15) Yo s ( O ( X )) (cid:47) (cid:47) O ∞ ( X ) ∂ (cid:47) (cid:47) ΣYo s ( F U ( X )) (15.1)55 roof. We consider the diagram of motivic coarse spectraYo s ( F U ( X )) i (cid:47) (cid:47) (cid:15) (cid:15) Yo s ( O ( X )) (cid:47) (cid:47) j (cid:15) (cid:15) Yo s ( F U ([0 , ∞ ) ⊗ X )) (cid:15) (cid:15) Yo s ( F U (( −∞ , ⊗ X )) (cid:47) (cid:47) Yo s ( O ( X ) − ) d (cid:47) (cid:47) Yo s ( F U ( R ⊗ X )) (15.2)The left and right vertical and the lower left horizontal map are given by the canonicalinclusions. The upper right horizontal map is induced from the identity of the underlyingsets. This diagram commutes since it is obtained by applying Yo s to a commuting diagramof bornological coarse spaces.The left square in (15.2) is cocartesian since the pair (( −∞ , × X, O ( X )) in O ( X ) − iscoarsely excisive. Furthermore, because (( −∞ , × X, [0 , ∞ ) × X ) is coarsely excisive in F U ( R ⊗ X ) the outer square is cocartesian. It follows that the right square is cocartesian.Since the upper right and the lower left corners in (15.2) are trivial by flasqueness of thespaces the diagram is equivalent toYo s ( F U ( X )) i (cid:47) (cid:47) (cid:15) (cid:15) Yo s ( O ( X )) (cid:47) (cid:47) j (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) Yo s ( O ( X ) − ) d (cid:47) (cid:47) Yo s ( F U ( R ⊗ X ))Note that O ∞ ( X ) is defined as the cofibre of the upper left horizontal map i . Hence theleft square yields the middle vertical equivalence ι in (15.1).The outer square yields the equivalence s : Yo s ( F U ( R ⊗ X )) (cid:39) → ΣYo s ( F U ( X )) . Then the right square identifies d with the boundary map ∂ of the cone sequence.
16 Comparison with the analytic assembly map
The classical instance of the coarse assembly map is the coarse analytic assembly mapfor coarse K -homology which is constructed using C ∗ -algebra theory. The main goalof the present section is the comparison of the coarse analytic assembly map defined inDefinition 16.10 and the coarse assembly map defined in Definiton 9.6. Our main result isa non-canonical identification of these two maps in Corollary 16.12.We start with Proposition 16.4 comparing the analytic assembly map (Definition 16.7) andthe cone boundary. To this end we must identify the domains of these maps appropriately.Let X be a proper metric space. We consider X with the bornology of metrically boundedsubsets. We will also use the notation X for the uniform bornological coarse space withthe coarse and uniform structures induced from the metric.56n addition to the metric coarse structure on X we will need other coarse structures C on X which are compatible with the bornology, but not necessarily associated to any othermetric on X . We will assume that such a coarse structure C contains open entourages.For the moment we write X C for the corresponding bornological coarse space. In ourapplication C will be a hybrid coarse structure.An ample continuously X -controlled Hilbert space is a pair ( H, ρ ), where H is a separableHilbert space and ρ is a non-degenerate representation of the C ∗ -algebra C ( X ) on H such that no non-trivial element of C ( X ) acts by a compact operator. Note that thisdefinition does not involve the coarse structure and also applies to X C in place of X .Using ρ we can talk about local compactness, propagation and pseudolocality of operators T in B ( H ).1. T is locally compact if ρ ( f ) T and T ρ ( f ) are compact operators for all f in C ( X ).2. T has propagation controlled by the entourage U if ρ ( f ) T ρ ( g ) = 0 for all functions f and g in C ( X ) with U [ supp ( g )] ∩ supp ( f ) = ∅ .3. T is pseudolocal if [ T, ρ ( f )] is a compact operator for every function f in C ( X ).A continuously X -controlled Hilbert space is called very ample, if it is unitarily equivalentto a direct sum of countably many copies of some ample continuously X -controlled Hilbertspace. Remark 16.1. An X C -controlled Hilbert space in the sense of [BE16, Def. 7.1] is a pair( H, φ ), where H is a Hilbert space and φ is a unital representation of the algebra of allbounded C -valued functions on X C such that φ ( χ B ) H is separable for every bounded subset B of X . We write H ( Y ) := φ ( χ Y ) H for the image of the projection φ ( χ Y ) associated toa subset Y of X . By definition, the X C -controlled Hilbert space ( H, φ ) is determined onpoints [BE16, Def. 7.3] if the natural inclusions induce an isomorphism H ∼ = (cid:76) x ∈ X H ( { x } ).The X C -controlled Hilbert space is called ample [BE16, Def. 7.12] if it is determined onpoints and there exists an entourage U of X C such that H ( U [ x ]) is infinite-dimensionalfor every point x in X . In contrast to the continuously controlled case this definition ofampleness depends on the coarse structure.We now explain a construction which associates to an ample continuously X -controlledHilbert space ( H, ρ ) an ample X C -controlled Hilbert space ( H, φ ).We first observe that the representation ρ of C ( X ) on H naturally extends to a represen-tation of the algebra L ∞ ( X ) of bounded Borel-measurable functions on X . We can choosean open entourage U of X C and a partition of X into U -bounded subsets with non-emptyinterior ( B α ) i ∈ I . For every i in I we let H i := ρ ( χ B i ) and P i : H → H i be the orthogonalprojection. Note that H i is ∞ -dimensional. For every i in I we choose a point b i in B i .Then we set φ := (cid:80) i ∈ I δ b i P i . We get the X C -controlled Hilbert space ( H, φ ).If U is also an entourage of X , then ( H, φ ) is an ample X -controlled Hilbert space.57et X be a proper metric space and ( H, φ ) be a continuously X -controlled Hilbert space.Let us fix an open entourage U of X . If ( H, ρ ) is very ample, then every other continuously X -controlled Hilbert space ( H (cid:48) , ρ (cid:48) ) with ρ (cid:48) non-degenerate admits a pseudolocal isometricinclusion into ( H, ρ ) of U -controlled propagation [HR00, Lem. 12.4.6]. In the reference thecoarse structures are assumed to come from a metric, but this is not necessary. Since weassume that the entourage U is open and X (as a proper metric space) is locally compactwe can find a locally finite open covering of X by U -bounded open subsets. Now the proofof [HR00, Lem. 12.4.6] carries over verbatim.We have an exact sequence of C ∗ -algebras0 → C ∗ ( X C , H, ρ ) → D ∗ ( X C , H, ρ ) → Q ∗ ( X C , H, ρ ) → , (16.1)where the entries have the following description.1. D ∗ ( X C , H, ρ ) is the sub- C ∗ -algebra of B ( H ) generated by the pseudolocal operatorsof U -controlled propagation for some U in C .2. C ∗ ( X C , H, ρ ) is the Roe algebra, which is generated by locally compact operators of U -controlled propagation for some U in C .3. Q ∗ ( X C , H, ρ ) is defined as the quotient. Remark 16.2.
Let X C be a bornological coarse space and let ( H, φ ) be an X C -controlledHilbert space. By [BE16, Def. 7.29] we have a Roe algebra C ∗ ( X C , H, φ ). If X is a propermetric space and ( H, φ ) is obtained from an ample continuously X -controlled Hilbertspace ( H, ρ ) by the construction described in Remark 16.1, then we have the equality C ∗ ( X C , H, φ ) = C ∗ ( X C , H, ρ ) (16.2)as subalgebras of B ( H ). Indeed, the local compactness conditions and the propagationconditions defined in the continuously controlled and controlled contexts are equivalent.Note that the algebras D ∗ ( X C , H, ρ ) and Q ∗ ( X C , H, ρ ) can not be defined in the controlledcontext. Their definition requires continuous control.Let X be a proper metric space and we choose a very ample continuously X -controlledHilbert space ( H, ρ ). Then for every integer n we have an isomorphism of groups K an ,lfn ( X ) def ∼ = KK n ( C ( X ) , C ) Paschke ∼ = K n +1 ( Q ∗ ( X, H, ρ )) , (16.3)given by the Paschke duality isomorphism, see e.g. Paschke [Pas81], Higson [Hig95] or alsoHigson–Roe [HR95, Prop. 5.2]. Furthermore, by [BE16, Thm. 7.64] we have a canonicalisomorphism of groups K n ( C ∗ ( X C , H, ρ )) ∼ = → K X n ( X C ) . (16.4)By using appropriate product metrics we consider R × X and its subspace [0 , ∞ ) × X asproper metric spaces. We note that the hybrid coarse structure on R × X contains openentourages. Hence the definitions above apply to O ( X ) − and O ( X ).Let i : X → [0 , ∞ ) × X and j : [0 , ∞ ) × X → R × X denote the inclusions.58 ata . We now make the following choices.1. We choose a very ample continuously X -controlled Hilbert space ( H, ρ ).2. We choose a very ample continuously [0 , ∞ ) ⊗ X -controlled Hilbert space ( H + , ρ + ).3. We choose a very ample continuously R ⊗ X -controlled Hilbert space ( ˜ H, ˜ ρ ).4. We choose a pseudolocal unitary embedding u : H → H + which is U -controlled as amorphism i ∗ ( H, ρ ) → ( H + , ρ + ), where U is an open coarse entourage of the hybridstructure of O ( X ).5. We choose a pseudolocal unitary embedding v : H + → ˜ H which is ˜ U -controlled as amorphism j ∗ ( H + , ρ + ) → ( ˜ H, ˜ ρ ), where ˜ U is an open coarse entourage of the hybridstructure of O ( X ) − .The embedding u induces an embedding u ∗ of algebras by A (cid:55)→ uAu ∗ , and similarly for v . Proposition 16.4.
We assume that X is isomorphic in UBC to a countable, locally finiteand finite-dimensional simplicial complex. The choices made above then determine naturallyan equivalence of fibre sequences of spectra K ( C ∗ ( X, H, ρ )) (cid:47) (cid:47) K ( D ∗ ( X, H, ρ )) (cid:47) (cid:47) K ( Q ∗ ( X, H, ρ )) ∂ C ∗ (cid:47) (cid:47) Σ K ( C ∗ ( X, H, ρ )) K X ( X ) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) K X ( O ( X )) (cid:39) (cid:79) (cid:79) (cid:47) (cid:47) K X ( O ∞ ( X )) b (cid:39) (cid:79) (cid:79) ∂ (cid:47) (cid:47) Σ K X ( X ) (cid:39) (cid:79) (cid:79) (16.5) Proof.
We have the following commuting diagram of spectra:Σ K ( C ∗ ( X, H, ρ )) u ∗ (cid:47) (cid:47) Σ K ( C ∗ ( O ( X ) , H + , ρ + )) v ∗ (cid:47) (cid:47) Σ K ( C ∗ ( O ( X ) − , ˜ H, ˜ ρ )) d (cid:47) (cid:47) Σ K ( C ∗ ( R ⊗ X, ˜ H, ˜ ρ )) K ( Q ∗ ( X, H, ρ )) u ∗ (cid:47) (cid:47) ∂ C ∗ (cid:79) (cid:79) K ( Q ∗ ( O ( X ) , H + , ρ + )) v ∗ (cid:47) (cid:47) (cid:79) (cid:79) K ( Q ∗ ( O ( X ) − , ˜ H, ˜ ρ )) δ (cid:79) (cid:79) d (cid:47) (cid:47) K ( Q ∗ ( R ⊗ X, ˜ H, ˜ ρ )) (cid:79) (cid:79) K ( D ∗ ( X, H, ρ )) u ∗ (cid:47) (cid:47) (cid:79) (cid:79) K ( D ∗ ( O ( X ) , H + , ρ + )) v ∗ (cid:47) (cid:47) (cid:79) (cid:79) K ( D ∗ ( O ( X ) − , ˜ H, ˜ ρ )) (cid:79) (cid:79) d (cid:47) (cid:47) K ( D ∗ ( R ⊗ X, ˜ H, ˜ ρ )) (cid:79) (cid:79) K ( C ∗ ( X, H, ρ )) (cid:79) (cid:79) u ∗ (cid:47) (cid:47) K ( C ∗ ( O ( X ) , H + , ρ + )) (cid:79) (cid:79) v ∗ (cid:47) (cid:47) K ( C ∗ ( O ( X ) − , ˜ H, ˜ ρ )) (cid:79) (cid:79) d (cid:47) (cid:47) K ( C ∗ ( R ⊗ X, ˜ H, ˜ ρ )) (cid:79) (cid:79) (16.6)The vertical sequences are fibre sequences of K -theory spectra associated to versions ofthe short exact sequence (16.1) of C ∗ -algebras. The horizontal maps d in the third columnare induced by the identity of ˜ H . By the comparison theorem [BE16, Thm. 7.70] and59he equality (16.2) (for the transition from the controlled to the continuously controlledsituation) we get the vertical equivalences in the following diagram K ( C ∗ ( X, H, ρ )) u ∗ (cid:47) (cid:47) K ( C ∗ ( O ( X ) , H + , ρ + )) v ∗ (cid:47) (cid:47) K ( C ∗ ( O ( X ) − , ˜ H, ˜ ρ )) d (cid:47) (cid:47) K ( C ∗ ( R ⊗ X, ˜ H, ˜ ρ )) K X ( X ) (cid:39) (cid:79) (cid:79) i ∗ (cid:47) (cid:47) K X ( O ( X )) (cid:39) (cid:79) (cid:79) j ∗ (cid:47) (cid:47) K X ( O ( X ) − ) (cid:39) (cid:79) (cid:79) ∂ (cid:48) (cid:47) (cid:47) Σ K X ( X ) (cid:39) (cid:79) (cid:79) where ∂ (cid:48) := ∂ ◦ ι with ι as in (15.1) and we use Proposition 15.1 for the commutativity ofthe right square. In particular, the lowest and the highest row in (16.6) are equivalent tosegments of a fibre sequence of spectra.We now use the following facts:1. The algebra Q ∗ ( X C , H, ρ ) does not depend on the coarse structure of X C , Higson–Roe[HR95, Lem. 6.2]. In the reference it is assumed that the coarse structure comesfrom a metric. But the argument only uses that C has open entourages U and thatwe can find locally finite, open and U -bounded coverings of X .Applied to [0 , ∞ ) ⊗ X and R ⊗ X we conclude that the canonical maps Q ∗ ( O ( X ) , H + , ρ + ) → Q ∗ ([0 , ∞ ) ⊗ X, H + , ρ + ) and Q ∗ ( O ( X ) − , ˜ H, ˜ ρ ) → Q ∗ ( R ⊗ X, ˜ H, ˜ ρ ) (16.7)are isomorphisms of C ∗ -algebras.2. By Paschke duality we have an isomorphism of groups K ∗ ( Q ∗ ([0 , ∞ ) ⊗ X, H + , ρ + )) ∼ = K an ,lf ∗ ([0 , ∞ ) × X ) . Since [0 , ∞ ) × X is flasque we have the marked isomorphism in the chain K ∗ ( Q ∗ ( O ( X ) , H + , ρ + )) ∼ = K ∗ ( Q ∗ ([0 , ∞ ) ⊗ X, H + , ρ + )) ∼ = K an ,lf ∗ ([0 , ∞ ) ⊗ X ) ! ∼ = 0 .
3. A sequence of spectra of the form · · · → → A ∼ → B → → . . . is clearly a segment of a fibre sequence. Hence the above Points 1 and 2 togetherimply that the second row in (16.6) is a fibre sequence. It then follows that the thirdrow in (16.6) is a fibre sequence, too.4. We want to show that δ in (16.6) is an equivalence. We have a map K an ,lf ∗ ( X ) ∼ = K an ,lf ∗ +1 ( R ⊗ X ) ∼ = K ∗ +1 ( Q ∗ ( O ( X ) − , ˜ H, ˜ ρ )) δ −→ K ∗ ( C ∗ ( O ( X ) − , ˜ H, ˜ ρ )) ∼ = K X ∗ ( O ( X ) − ) . (16.8)The first isomorphism is the suspension isomorphism for locally finite homologytheories and the second isomorphism is given by Paschke duality together with the60somorphism (16.7). And finally, the last isomorphism comes from the comparisontheorem [BE16, Thm. 7.70].If assume that X is isomorphic in the category UBC to a countable, locally finiteand finite-dimensional simplicial complex, then we have equivalences K X ( O ( X ) − ) Prop. 15 . (cid:39) K X O ∞ ( X ) Prop. 12 . (cid:39) ( K X O ∞ ) lf ( X ) . The transformation (16.8) is natural with respect to restrictions to subspaces of X .By the result of Siegel [Sie12] this transformation is also compatible with the boundaymaps of the Mayer–Vietoris sequences associated to open coverings. Since the target(by Proposition 12.14) and the domain of it both behave like locally finite homologytheories and (16.8) induces an isomorphism on bounded contractible subsets, (16.8)is an isomorphism.The fact that δ is an equivalence implies that K ( D ∗ ( O ( X ) − , ˜ H, ˜ ρ )) (cid:39) Remark 16.5.
An alternative option to show that K ( D ∗ ( O ( X ) − , ˜ H, ˜ ρ )) (cid:39) K ( D ∗ ( X, H, ρ )) (cid:39) K ( D ∗ ( O ( X ) , H + , ρ + ) directly using the invariance ofthe functor K ∗ ◦ D ∗ under coarse homotopies [HR95, Lem. 7.8].Putting all these facts together we get the diagram of vertical and horizontal fibre sequencesΣ K ( C ∗ ( X, H, ρ )) (cid:47) (cid:47) Σ K ( C ∗ ( O ( X ) , H + , ρ + )) (cid:47) (cid:47) Σ K ( C ∗ ( O ( X ) − , ˜ H, ˜ ρ )) (cid:47) (cid:47) Σ K ( C ∗ ( R ⊗ X, ˜ H, ˜ ρ )) K ( Q ∗ ( X, H, ρ )) (cid:47) (cid:47) ∂ C ∗ (cid:79) (cid:79) (cid:47) (cid:47) (cid:79) (cid:79) K ( Q ∗ ( O ( X ) − , ˜ H, ˜ ρ )) (cid:39) (cid:79) (cid:79) (cid:39) (cid:47) (cid:47) K ( Q ∗ ( R ⊗ X, ˜ H, ˜ ρ )) (cid:79) (cid:79) K ( D ∗ ( X, H, ρ )) (cid:39) (cid:47) (cid:47) (cid:79) (cid:79) K ( D ∗ ( O ( X ) , H + , ρ + )) (cid:47) (cid:47) (cid:79) (cid:79) (cid:79) (cid:79) (cid:47) (cid:47) K ( D ∗ ( R ⊗ X, ˜ H, ˜ ρ )) (cid:79) (cid:79) K ( C ∗ ( X, H, ρ )) (cid:79) (cid:79) u ∗ (cid:47) (cid:47) K ( C ∗ ( O ( X ) , H + , ρ + )) (cid:39) (cid:79) (cid:79) v ∗ (cid:47) (cid:47) K ( C ∗ ( O ( X ) − , ˜ H, ˜ ρ )) (cid:79) (cid:79) d (cid:47) (cid:47) K ( C ∗ ( R ⊗ X, ˜ H, ˜ ρ )) (cid:79) (cid:79) It provides the asserted morphism of fibre sequences.The vertical morphisms and the fillers of the squares in (16.5) may depend non-triviallyon the choice of the Data 16.3. In particular we ask:
Problem 16.6.
Does the map b in (16.5) depend non-trivially on the choice of Data 16.3made in addition to 16.3.1? Assume that we have just chosen a very ample continuously X -controlled Hilbert space( H, ρ ), i.e., the Datum 16.3.1.
Definition 16.7.
The map A X : K an ,lfn ( X ) (16.3) ∼ = K n +1 ( Q ∗ ( X, H, ρ )) ∂ C ∗ −−→ K n ( C ∗ ( X, H, ρ )) (16.4) ∼ = K X n ( X ) is called the analytic assembly map.
61f we now choose the full Data 16.3, then we get the following commuting diagram: K an ,lfn ( X ) ∼ =(16.3) (cid:47) (cid:47) A X (cid:39) (cid:39) ∼ = comp (cid:15) (cid:15) K n +1 ( Q ∗ ( X, H, ρ )) ˜ ∂ C ∗ (cid:47) (cid:47) K X n ( X ) K an ,lfn ( X ) ∼ =(14.3) (cid:47) (cid:47) K X n +1 ( O ∞ ( X )) ∂ (cid:47) (cid:47) ∼ = b (cid:79) (cid:79) K X n ( X ) (16.9)where we are using the notation ˜ ∂ C ∗ := (16.4) ◦ ∂ C ∗ . We note that the isomorphism comp is determined by the condition that the left square commutes. It involves the choice ofa spectrum equivalence (14.1) via the equivalence (14.3) and also the Data 16.3 via theisomorphism b . Problem 16.8.
Can one canonically choose the spectrum equivalence (14.1) and theData 16.3 such that the isomorphism comp becomes the identity?
Remark 16.9.
In this remark we explain the process of coarsification involved in thetransition from the analytic assembly map A X to the analytic coarse assembly map µ an X .We assume that X is a separable bornological coarse space of bounded geometry. If U isan entourage of X , then P ( X, U ) is a countable, finite-dimensional, locally finite simplicialcomplex and hence a proper metric space. Consequently, we can apply the above theory tothe metric space P ( X, U ). After choosing a very ample continuously P ( X, U )-controlledHilbert space ( H U , ρ U ) we get the analytic assembly map A P ( X,U ) : K an ,lf ∗ ( P ( X, U )) → K X ∗ ( P ( X, U ))by Definition 16.7.We have an Equivalence (6.3) of bornological coarse spaces (recall that in the presentsection we drop the forgetful functor F U from the notation) X U → P ( X, U ) (16.10)For every two coarse entourages U , U (cid:48) of X with U ⊆ U (cid:48) we choose an isometry i U,U (cid:48) : H U → H U (cid:48) which induces a pseudolocal morphism f U,U (cid:48) , ∗ ( H U , ρ U ) → ( H U (cid:48) , ρ U (cid:48) ), where f U,U (cid:48) : P ( X, U ) → P ( X, U (cid:48) ) is the natural embedding. We then have a commuting diagram K an ,lf ∗ ( P ( X, U )) f U,U (cid:48) , ∗ (cid:15) (cid:15) ∼ = (cid:47) (cid:47) A P ( X,U ) (cid:42) (cid:42) K ∗ +1 ( Q ∗ ( P ( X, U ) , H U , ρ U )) i U,U (cid:48) , ∗ (cid:15) (cid:15) ∂ C ∗ (cid:47) (cid:47) K X ∗ ( P ( X, U )) (cid:15) (cid:15) K X ∗ ( X U ) (16.10) ∼ = (cid:111) (cid:111) K an ,lf ∗ ( P ( X, U (cid:48) )) A P ( X,U (cid:48) ) (cid:52) (cid:52) ∼ = (cid:47) (cid:47) K ∗ +1 ( Q ∗ ( P ( X, U (cid:48) ) , H U (cid:48) , ρ U (cid:48) )) ∂ C ∗ (cid:47) (cid:47) K X ∗ ( P ( X, U (cid:48) )) K X ∗ ( X U (cid:48) ) (16.10) ∼ = (cid:111) (cid:111)
62e now form the colimit of the horizontal maps over the entourages U of X . In view of u -continuity of K X ∗ we get the homomorphism µ an X : QK an ,lf ∗ ( X ) → K X ∗ ( X ) (16.11)finishing the construction of the coarse analytic assembly map. Definition 16.10.
The homorphism µ an X is called the coarse analytic assembly map. Remark 16.11.
In this remark we compare the coarse analytic assembly map (16.11)with the assembly map (14.5). We assume that X is a separable bornological coarse spaceof bounded geometry. We assume that the coarse structure admits a countable cofinalmonotoneously increasing family.We must choose the Data 16.3 for P ( X, U ) in place of X compatibly with the inclusions f U,U (cid:48) : P ( X, U ) → P ( X, U (cid:48) ). In order to simplify matters and to avoid a discussion ofrelations in the index poset for the colimit over the coarse entourages of X , we reduce thisconstruction to a cofinal monotoneously increasing family ( U n ) n ∈ N of entourages. We thenhave diagrams K an ,lfn ( P ( X, U )) ∼ =(16.3) (cid:47) (cid:47) A X (cid:42) (cid:42) ∼ = comp (cid:15) (cid:15) K n +1 ( Q ∗ ( P ( X, U ) , H U , ρ U )) ˜ ∂ C ∗ (cid:47) (cid:47) K X n ( P ( X, U )) K X n ( X ) (16.10) ∼ = (cid:111) (cid:111) K an ,lfn ( P ( X, U )) ∼ =(14.3) (cid:47) (cid:47) K X n +1 ( O ∞ ( P ( X, U ))) ∂ (cid:47) (cid:47) ∼ = b (cid:79) (cid:79) K X n ( P ( X, U )) K X n ( X ) ∼ =(16.10) (cid:111) (cid:111) (16.12)for all coarse entourages U of X and connecting maps between such diagrams for inclusions U → U (cid:48) . If we form the colimit over the coarse entourages of X , then the colimits of theouter squares yield the diagram QK an ,lfn ( X ) µ anX (cid:47) (cid:47) ∼ = Q comp (cid:15) (cid:15) K X n ( X ) QK an ,lfn ( X ) µ topX (cid:47) (cid:47) K X n ( X ) (16.13)where we use (14.5) and (14.6) for the identification of the lower horizontal map called µ topX , and Remark 16.9 for the upper horizontal map. The isomorphism Q comp possiblydepends on the choices of the ample Hilbert space data, the various embeddings, and aspectrum equivalence (14.1).The upshot of the above discussion is the following statement: Corollary 16.12.
There is an equivalence between the coarse analytic assembly map µ anX and coarse assembly map µ topX . In particular, if one of these maps is an isomorphism thenso is the other. The equivalence is canonical up to an automorphism of QK an ,lfn ( X ). At the moment weare not able to make the comparison more canonical.63 Let M be a complete Riemannian manifold and /D be a generalized Dirac operator on M of degree n . The Dirac operator acts on sections of a Z / Z -graded Dirac bundle E → M with a right action by the Clifford algebra Cl n . In this situation one can define the coarseindex class Ind ( /D ) ∈ K X n ( M )of the Dirac operator, see Higson–Roe [HR00] or Zeidler [Zei16]. In the original constructionof the coarse index class by Roe [Roe96] the degree was incorporated differently. Moreover,the index class arises as a K -theory class of the Roe algebra associated to the Dirac bundle.It requires some argument in order to interpret the index class as a coarse K -homologyclass in the theory K X as indicated above, see [BE16, Sec. 7.9]. A detailed constructionof the index class, even in the equivariant case and with support conditions, will be givenin [BE17, Def. 9.6].The goal of this section is to construct a K -homology class σ ( /D ) in K X n +1 ( O ∞ ( M )) suchthat µ K X ,M ( σ ( /D )) = Ind ( /D )in K X n ( M ). The class σ ( /D ) is the analogue of the symbol class of /D (this motivates thenotation).Let ( M, g ) be a complete Riemannian manifold and /D be a generalized Dirac operator on M of degree n . We will need the following two operations with Dirac operators.1. ([Bun, Ex. 4.3]) Assume that g (cid:48) is a second complete Riemannian metric on M .In [Bun] we explain a construction which starts from /D and produces a canonicalDirac operator /D (cid:48) associated to the metric g (cid:48) . The idea is to write /D locally as atwisted spin -Dirac operator. If we change the metric, then we change the spin -Diracoperator correspondingly and keep the twisting fixed.2. ([Bun, Ex. 4.4]) There is a natural way to extend the Dirac operator /D of degree n to a Dirac operator ˜ /D of degree n + 1 on the Riemannian product ˜ M := R × M . Wedenote by ˜ E (cid:48) → ˜ M the pull-back of the bundle E → M with the induced metric andconnection, and then form the graded bundle ˜ E := ˜ E (cid:48) ⊗ Cl . Under the identification Cl n +1 ∼ = Cl n ⊗ Cl it has a right action of the Clifford algebra Cl n +1 , where Cl n acts on ˜ E (cid:48) and Cl acts on the Cl -factor of ˜ E by right-multiplication. The Cliffordaction T M ⊗ E → E extends to a Clifford action T ˜ M ⊗ ˜ E → ˜ E such that ∂ t actsby left-multiplication by the generator of Cl on the Cl -factor of ˜ E , where t is thecoordinate of the R -factor in ˜ M = R × M .Let f : R → R be a function Assumption 17.1.
We assume that f is smooth, positive, and that f ( t ) = 1 for t in ( −∞ , and lim t →∞ f ( t ) = ∞ .
64e form the new complete Riemannian metric g f := dt + f ( t ) · pr ∗ g on R × M , where pr : R × M → M is the projection. We denote the resulting Riemannianmanifold by ˜ M f . Then we let ˜ /D f denote the Dirac operator associated to the metric g f obtained from ˜ /D by the Construction 2 mentioned above. We then have a class Ind ( ˜ /D f ) in K X n +1 ( ˜ M f ) . The Riemannian manifold ˜ M f can be considered as a bornological coarse space with thestructures induced by the metric. We now observe that the identity map of underlyingsets induces a morphism of bornological coarse spaces p f : ˜ M f → O ( M ) − . Proposition 17.2.
The class p f, ∗ ( Ind ( ˜ /D f )) in K X n +1 ( O ( M ) − ) is independent of thechoice of f as long as f satisfies Assumption 17.1.Proof. This is shown in [Bun, Prop. 4.11].The motivic equivalence Yo s ( O ( M ) − ) (cid:39) O ∞ ( M ) obtained in Proposition 15.1 inducesan equivalence of coarse K -homology spectra K X ( O ( M ) − ) (cid:39) K X O ∞ ( M ). Hence we canconsider p f, ∗ ( Ind ( ˜ /D f )) as a class in K X n +1 ( O ∞ ( M )). Definition 17.3.
We define the class σ ( /D ) in K X O ∞ n +1 ( M ) to be the class p f, ∗ ( Ind ( ˜ /D f )) for some choice of function f satisfying Assumption 17.1. Recall the boundary map ∂ : K X O ∞∗ +1 ( M ) → K X ∗ ( M ) of the cone sequence from (9.1). Proposition 17.4.
We have the equality ∂ ( σ ( /D )) = Ind ( /D ) (17.1) in K X ∗ ( M ) .Proof. This is shown in detail in [Bun, Lem. 4.14]. But for completeness we recall theidea. We have a commuting diagram of bornological coarse spaces˜ M f i (cid:47) (cid:47) p (cid:15) (cid:15) ˜ M O ( M ) − d (cid:47) (cid:47) R ⊗ M i ∗ ( Ind ( ˜ /D f )is independent of f as long as f ≥ d ∗ ( p ∗ ( Ind ( ˜ /D f ))) = Ind ( ˜ /D ) in K X n +1 ( R × M ). We now use the square K X ( O ( M ) − ) d (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) K X ( R ⊗ M ) s (cid:39) (cid:15) (cid:15) K X O ∞ ( M ) ∂ (cid:47) (cid:47) Σ K X ( M )obtained from Proposition 15.1 and the compatibility of the coarse index class with thesuspension equivalence s expressed by the equality s ∗ ( Ind ( ˜ /D )) = Ind ( /D ) (for the lattersee Zeidler [Zei16, Thm. 5.5] or [BE17, Thm. 11.1]).Let us discuss now the problem of identifying the class σ ( /D ) defined in Definition 17.3with the classical symbol class σ an ,lf ( /D ) whose definition will be recalled below.We consider a function χ : R → [ − ,
1] which is smooth, odd, and satisfies χ ( t ) > t in (0 , ∞ ) and lim t →±∞ χ ( t ) = ±
1. Then we get a ( C ( M ) , Cl n )-Kasparov module( H, ρ, χ ( /D )), see Higson–Roe [HR00, Sec. 10.6]. Definition 17.5.
The classical symbol class σ an ,lf ( /D ) of /D is defined to be the class ofthe ( C ( M ) , Cl n ) -Kasparov module ( H, ρ, χ ( /D )) in K an ,lfn ( M ) def = KK ( C ( M ) , Cl n ) . It follows from the details of the construction of the Paschke duality isomorphism that A M ( σ an ,lf ( /D )) = Ind ( /D ) . (17.2)If the Riemannian manifold M has bounded geometry, then M is isomorphic (in thecategory UBC small ) to a countable, locally finite, finite-dimensional simplicial complex.Indeed, there exists a triangulation of M as a simplicial complex K of bounded geometrysuch that M is bi-Lipschitz equivalent to K , Attie [Att04, Thm. 2.1]. Here K is equippedwith the spherical metric.In this case, after choosing all the Data 16.3 and a spectrum equivalence (14.1) we canconsider the Diagram (16.9) (with X replaced by M ).The class σ ( /D ) corresponds under the isomorphism induced by the Equivalence (14.3) toa class ˜ σ an ,lf ( /D ) in K an ,lfn ( M ). We note that the class ˜ σ an ,lf ( /D ) depends on the choice ofthe spectrum equivalence (14.1). Problem 17.6.
Do we have the equality comp ( σ an ,lf ( /D )) = ˜ σ an ,lf ( /D ) ? (17.3)66t is clear that the dependence on the choice of the spectrum equivalence (14.1) on bothsides of the Equation (17.3) cancels out because of the way we define the map comp . Theanswer to the Question (17.3) can only be positive if the answer to the question formulatedin Problem 16.6 is no .We can map the classes ˜ σ an ,lf ( /D ) and σ an ,lf ( /D ) into the group QK an ,lfn ( M ) by combiningthe procedures explained in Remark 16.11 with the map (7.1) to get classes that we denoteby Q ˜ σ an ,lf ( /D ) and Qσ an ,lf ( /D ). From (17.1) we get that ˜ σ an ,lf ( /D ) is mapped to Ind ( /D ) bythe boundary map of the cone sequence, and by (17.2) we know that σ an ,lf ( /D ) is mappedto Ind ( /D ) by A M . We conclude that we have µ anM (cid:0) Qσ an ,lf ( /D ) (cid:1) = Ind ( /D ) and µ topM (cid:0) Q ˜ σ an ,lf ( /D ) (cid:1) = Ind ( /D ) . (17.4)From this together with the Diagram (16.13) we get the following result: Lemma 17.7.
If the coarse assembly maps µ anM and µ topM are injective, then we have Q comp (cid:0) Qσ an ,lf ( /D ) (cid:1) = Q ˜ σ an ,lf ( /D ) . (17.5)Note that by Theorem 10.7 the coarse assembly map µ topM is an isomorphism if M has finitedecomposition complexity. Yu [Yu00] proved that µ anM is an isomorphism is M has boundedgeometry and is coarsely embeddable into a Hilbert space. More generally, Kasparov–Yu[KY06] proved injectivity of µ anM if M has bounded geometry and is coarsely embeddableinto a uniformly convex Banach space and Chen–Wang–Yu [CWY15] proved injectivityof µ anM if M has bounded geometry and is coarsely embeddable into a Banach space withProperty (H).The Equality (17.5) should be independent of results on the coarse Novikov conjecture(i.e., independent of injectivity of the coarse assembly map). So let us phrase this as aseparate question, which is a weakening of the Question (17.3): Problem 17.8.
Do we always have the equality Q comp ( Qσ an ,lf ( /D )) = Q ˜ σ an ,lf ( /D ) ? (17.6) References [Att04] O. Attie. A Surgery Theory for Manifolds of Bounded Geometry, 2004. Onlineavailable at http://arxiv.org/abs/math/0312017/ .[BE] U. Bunke and A. Engel. Equivariant coarse K -homology. In preparation.[BE16] U. Bunke and A. Engel. Homotopy theory with bornological coarse spaces.arXiv:1607.03657v3, 2016.[BE17] U. Bunke and A. Engel. The coarse index class with support. arXiv:1706.06959,2017. 67BEKWa] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Coarse motives withtransfer and finite decomposition complexity. In preparation.[BEKWb] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Equivariant coarsehomotopy theory and coarse algebraic K -homology. In preparation.[BFJR04] A. Bartels, T. Farrell, L. Jones, and H. Reich. On the isomorphism conjecturein algebraic K -theory. Topology , 43(1):157–213, 2004.[Bun] U. Bunke. Abstract boundary value problems. In preparation.[CWY15] X. Chen, Q. Wang, and G. Yu. The coarse Novikov conjecture and Banachspaces with Property (H).
J. Funct. Anal. , 268:2754–2786, 2015.[FO17] T. Fukaya and Sh.-I. Oguni. A coarse Cartan–Hadamard theorem with applica-tions to the coarse Baum–Connes conjecture. arXiv:1705.05588, 2017.[Gro93] M. Gromov. Asymptotic Invariants of Infinite Groups. In G. A. Niblo and M. A.Roller, editors,
Geometric Group Theory , volume 182 of
London MathematicalSociety Lecture Note Series , 1993.[GTY12] E. Guentner, R. Tessera, and G. Yu. A notion of geometric complexity and itsapplication to topological rigidity.
Invent. math. , 189(2):315–357, 2012.[Hig95] N. Higson. C ∗ -Algebra Extension Theory and Duality. J. Funct. Anal. , 129:349–363, 1995.[HPR96] N. Higson, E. K. Pedersen, and J. Roe. C ∗ -algebras and controlled topology. K -Theory , 11:209–239, 1996.[HR95] N. Higson and J. Roe. On the coarse Baum–Connes conjecture. In S. C. Ferry,A. Ranicki, and J. Rosenberg, editors, Novikov conjectures, index theorems andrigidity, Vol. 2 , London Mathematical Society Lecture Notes 227. CambridgeUniversity Press, 1995.[HR00] N. Higson and J. Roe.
Analytic K -Homology . Oxford University Press, 2000.[KY06] G. Kasparov and G. Yu. The coarse geometric Novikov conjecture and uniformconvexity. Adv. Math. , 206:1–56, 2006.[Mit10] P. D. Mitchener. The general notion of descent in coarse geometry.
Alg. Geom.Topol. , 10:2419–2450, 2010.[NY12] P. W. Nowak and G. Yu.
Large Scale Geometry . European MathematicalSociety, 2012.[Pas81] W.L. Paschke. K -Theory for Commutants in the Calkin Algebra. Pacific J.Math. , 95(2):427–434, 1981.[Roe96] J. Roe.
Index Theory, Coarse Geometry, and Topology of Manifolds , volume 90of
CBMS Regional Conference Series in Mathematics . American MathematicalSociety, 1996. 68RS12] J. Roe and P. Siegel. Sheaf theory and Paschke duality. arXiv:math/1210.6420,2012.[Sie12] P. Siegel. The Mayer–Vietoris Sequence for the Analytic Structure Group.arXiv:math/1212.0241, 2012.[STY02] G. Skandalis, J.L. Tu, and G. Yu. The coarse Baum–Connes conjecture andgroupoids.
Topology , 41:807–834, 2002.[Wei02] M. Weiss. Excision and restriction in controlled K -theory. Forum Math. ,14(1):85–119, 2002.[Wri05] N. J. Wright. The coarse Baum-Connes conjecture via C coarse geometry. J. Funct. Anal. , 220(2):265–303, 2005.[WW95] M. Weiss and B. Williams. Pro-excisive functors. In S. C. Ferry, A. Ranicki, andJ. Rosenberg, editors,
Novikov conjectures, index theorems and rigidity, Vol. 2 K -Theory , 9:223–231,1995.[Yu95b] G. Yu. Coarse Baum–Connes Conjecture. K -Theory , 9:199–221, 1995.[Yu98] G. Yu. The Novikov conjecture for groups with finite asymptotic dimension. Ann. Math. , 147:325–355, 1998.[Yu00] G. Yu. The coarse Baum–Connes conjecture for spaces which admit a uniformembedding into Hilbert space.
Invent. math. , 139(1):201–240, 2000.[Zei16] R. Zeidler. Positive scalar curvature and product formulas for secondary indexinvariants.