Equivariant coarse homotopy theory and coarse algebraic K -homology
Ulrich Bunke, Alexander Engel, Daniel Kasprowski, Christoph Winges
EEquivariant coarse homotopy theoryand coarse algebraic K -homology Ulrich Bunke ∗ Alexander Engel † Daniel Kasprowski ‡ Christoph Winges § September 25, 2018
Abstract
We study equivariant coarse homology theories through an axiomatic framework.To this end we introduce the category of equivariant bornological coarse spacesand construct the universal equivariant coarse homology theory with values in thecategory of equivariant coarse motivic spectra.As examples of equivariant coarse homology theories we discuss equivariant coarseordinary homology and equivariant coarse algebraic K -homology.Moreover, we discuss the cone functor, its relation with equivariant homologytheories in equivariant topology, and assembly and forget-control maps. This is apreparation for applications in subsequent papers aiming at split-injectivity resultsfor the Farrell–Jones assembly map. Contents
1. Introduction 3 ∗ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] † Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, [email protected] ‡ Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Mathematisches Institut, Endenicher Allee 60,53115 Bonn, [email protected] § Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Mathematisches Institut, Endenicher Allee 60,53115 Bonn, [email protected] a r X i v : . [ m a t h . K T ] S e p . General constructions 7
2. Equivariant bornological coarse spaces 7
BornCoarse . . . . . . . . . . . . . . . . . . . . . 11
3. Equivariant coarse homology theories 164. Equivariant coarse motivic spectra 19
5. Continuity 26
6. Change of groups 35
II. Examples 42
7. Equivariant coarse ordinary homology 42 can,min ⊗ S min,max . . . . . . . . . . . . 447.3. Additional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.4. Change of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8. Equivariant coarse algebraic K -homology 50 K -theory functor . . . . . . . . . . . . . . . . . . . . . . . . 508.2. X -controlled A -objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.3. Coarse algebraic K -homology . . . . . . . . . . . . . . . . . . . . . . . . . 558.4. Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608.5. Change of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.6. Variations on the definition . . . . . . . . . . . . . . . . . . . . . . . . . . 632 II. Cones and assembly maps 66
9. Cones 66
References 108
1. Introduction
In this paper we study equivariant coarse homology theories. We start with the equivariantgeneralization of the coarse homotopy theory developed by Bunke–Engel [BE16]. To thisend we introduce the category of equivariant bornological coarse spaces and construct theuniversal equivariant coarse homology theory with values in the category of equivariantcoarse motivic spectra.As examples of equivariant coarse homology theories we discuss equivariant ordinary coarsehomology and equivariant coarse algebraic K -homology of an additive category.An important application of equivariant coarse homotopy theory is in the study of assemblymaps which appear in isomorphism conjectures of Farrell–Jones or Baum–Connes type.The main tools for the transition between equivariant homology theories and equivariantcoarse homology theories are the cone functor and the process of coarsification. In thispaper we give a detailed account of the cone functor and the construction of equivarianthomology theories from equivariant coarse homology theories. Then we introduce thecoarsification functor and the forget-control map, and discuss its relation with the assemblymaps.The third part of the present paper provides the technical background for subsequentpapers: 3. In [BEKW17] we show that a certain large scale geometric condition called finitedecomposition complexity implies that a motivic version of the forget-control map isan equivalence.2. In [BEKW] we study the descent principle. It states that under certain conditionsthe fact that the forget-control map becomes an equivalence after restriction of theaction to all finite subgroups implies that it is split injective for the original (ingeneral infinite) group.We combine this with the results of [BEKW17] and the technical results of the presentpaper to deduce split injectivity results for the original Farrell–Jones assembly map.3. In [BE17] we study more formal aspects of the process of coarsification of homologytheories and provide a general account for coarse assembly maps.A bornological coarse space is a set equipped with a coarse and a bornological structuresuch that these structures are compatible with each other. The category of bornologicalcoarse spaces BornCoarse was introduced in [BE16] as a general framework for coarsegeometry and coarse topology. Interesting invariants of bornological coarse spaces up tocoarse equivalence are coarse homology theories. In [BE16] the examples of coarse ordinaryhomology and various coarse versions of topological K -homology were discussed. In orderto study general properties of coarse homology theories the category of motivic coarsespectra Sp X was constructed as the target of the universal coarse homology theoryYo s : BornCoarse → Sp X . One of the motivations to consider equivariant coarse algebraic topology is that it appearsas a building block of proofs that certain assembly maps are equivalences (Farrell–Jonesconjecture [BL11], [BFJR04] and Baum–Connes conjecture [Yu95]).In Section 2.1 we define for every group Γ the category Γ
BornCoarse of Γ-bornologicalcoarse spaces. We provide various constructions of Γ-bornological coarse spaces fromΓ-sets, metric spaces with isometric Γ-action, or Γ-simplicial complexes. We further showthat the category of Γ-bornological coarse spaces admits coproducts and fiber products. Italso has an interesting symmetric monoidal structure ⊗ .In Section 3 we introduce the notion of an equivariant coarse homology theory, and theterminology necessary to state its defining properties:1. coarse invariance,2. coarse excision,3. vanishing on flasques, and4. u -continuity. 4trongness (introduced in Section 4.4) is an additional property which an equivariantcoarse homology might have. It is important in order to interpret the forget-control mapas a transformation between equivariant coarse homology theories.In the present paper the most important additional property is continuity. We introducethis notion in Section 5. Continuity is crucial if one wants to relate the forget-control mapwith the assembly map for the family of finite subgroups.Following the line of thought of [BE16], in Section 4 we construct the universal equivariantcoarse homology theory Yo s : Γ BornCoarse → Γ Sp X with values in the category of equivariant coarse motivic spectra. Similarly, in Section 5we construct a universal continuous equivariant coarse homology theoryYo sc : Γ BornCoarse → Γ Sp X c . As examples of equivariant coarse homology theories, in Part II we introduce equivariantcoarse ordinary homology H X Γ and equivariant coarse algebraic K -homology K A X Γ ofan additive category A . Our main results are the verification that the definitions indeedsatisfy the four defining properties of an equivariant coarse homology theory. We also showthat these examples have the additional properties of being strong, strongly additive andcontinuous. We calculate the evaluations of these equivariant homology theories on simpleΓ-bornological coarse spaces. These calculations are important if one wants to understandwhich equivariant homology theories they induce after pull-back with the cone functor.In most of the present paper we consider the theory for a fixed group Γ. But in Section 6we consider a homomorphism of groups H → Γ and provide various transitions from H -bornological coarse spaces to Γ-bornological coarse spaces and back. The most importantexamples are restriction and induction. Our examples of equivariant coarse homologytheories are defined for all groups, so in particular for H and Γ. The group-changeconstruction on the level of bornological coarse spaces are accomplished by naturaltransformations relating the evaluations of the H - and Γ-equivariant versions of theequivariant homology theories.In Section 9 we introduce the category of Γ-uniform bornological coarse spaces Γ UBC and the cone functor O : Γ UBC → Γ BornCoarse . The construction of the cone is motivated by Bartels–Farrell–Jones–Reich [BFJR04] andMitchener [Mit01, Mit10], and its main ingredient in the construction is the hybrid coarsestructure first introduced by Wright [Wri02] and studied in detail in [BE16]. Our maintechnical results are homotopy invariance and excisiveness of the cone. While the cone O ( X ) depends on the coarse structure on X , its germ at ∞ , denoted by O ∞ ( X ), isessentially independent of the coarse structure. So O ∞ is very close to an equivarianthomology theory. But it is still defined on Γ UBC and does not satisfy a wedge axiom.5n Section 10 we first review some general features of equivariant homotopy theory andthen derive an equivariant homology theory O ∞ hlg : Γ Top → Γ Sp X from the functor O ∞ . We then introduce the classifying space E F Γ for a family F ofsubgroups of Γ and define the motivic assembly map as the map α E F Γ : O ∞ hlg ( E F Γ) → O ∞ hlg ( ∗ )induced by the morphism E F Γ → ∗ . We discuss some conditions on a Γ-bornologicalcoarse space Q implying that the twisted version α E F ⊗ Yo s ( Q ) of the assembly mapbecomes an equivalence.In Section 11 we introduce the universal coarsification functor F ∞ : Γ BornCoarse → Γ Sp X and the forget-control map β : F ∞ → Σ F , a natural transformation of functors from Γ BornCoarse to Γ Sp X . The coarse geometryapproach to the isomorphism conjectures provides conditions on a Γ-bornological coarsespace X implying that the forget-control map β X : F ∞ ( X ) → Σ F ( X ) is an equivalence, orbecomes an equivalence after application of a suitable equivariant coarse homology theory.Since one is also interested in the assembly maps of equivariant homotopy theory like α E F Γ we provide a comparison between this assembly map and the forget-control map. Acknowledgements
U. Bunke and A. Engel were supported by the SFB 1085 “Higher In-variants” funded by the Deutsche Forschungsgemeinschaft DFG. A. Engel was furthermoresupported by the Research Fellowship EN 1163/1-1 “Mapping Analysis to Homology”,also funded by the DFG. D. Kasprowski and C. Winges acknowledge support by the MaxPlanck Society.Parts of the present work were obtained during the Junior Hausdorff Trimester Program“Topology” at the Hausdorff Research Institute for Mathematics (HIM) in Bonn.We would also like to thank Mark Ullmann for helpful discussions.6 art I.General constructions
2. Equivariant bornological coarse spaces
In this section we introduce the equivariant version of the category of bornological coarsespaces introduced in [BE16]. We assume familiarity with [BE16, Section 2].Let Γ be a group. If Γ acts on a bornological coarse space X by automorphisms, then itacts on the set of coarse entourages C of X . We let C Γ denote the partially ordered subsetof C of entourages of X which are fixed set-wise. Definition 2.1.
A Γ -bornological coarse space is a bornological coarse space X togetherwith an action of Γ by automorphisms such that C Γ is cofinal in C .A morphism between Γ-bornological coarse spaces is a morphism of bornological coarsespaces which is in addition Γ-equivariant. (cid:7) We let Γ
BornCoarse denote the category of Γ-bornological coarse spaces and morphisms.By considering a bornological coarse space as a Γ-bornological coarse space with the trivialaction we get a fully faithful functor C : BornCoarse → Γ BornCoarse . (2.1) Example 2.2.
Let X be a set with an action of Γ and A ⊆ P ( X × X ) Γ be a family ofΓ-invariant subsets. Then we can form the coarse structure C := C(cid:104) A (cid:105) generated by A . Inthis case C Γ is cofinal in C . Hence a Γ-coarse structure can be generated by a family ofΓ-invariant entourages. (cid:7) Remark 2.3.
Let
Coarse denote the category of coarse spaces (i.e., sets with coarsestructures) and controlled morphisms. We can consider Γ equipped with the minimalcoarse structure
C(cid:104) diag Γ (cid:105) as a group object in Coarse .Let X be a set with an action of a group Γ and C be a coarse structure on X . Thefollowing conditions are equivalent:1. C Γ is cofinal in C .2. For every entourage U in C the set (cid:83) γ ∈ Γ ( γ × γ )( U ) also belongs to C .3. The action is a morphism Γ × X → X in Coarse .The proof is straightforward.We denote the category of Γ-coarse spaces consisting of coarse spaces with a Γ-actionsatisfying the above conditions and equivariant and controlled maps by Γ
Coarse . (cid:7) xample 2.4. We consider Γ as a Γ-set with the left action. We furthermore let B min be the minimal bornology on Γ consisting of the finite subsets. Finally, we let the coarsestructure C can on Γ be generated by the Γ-invariant sets Γ( B × B ) for all B in B min .Then (Γ , C can , B min ) is a Γ-bornological coarse space called the canonical Γ-bornologicalcoarse space associated to Γ. We will denote it by Γ can,min . (cid:7) Example 2.5.
Let X be a set with an action of Γ. It gives rise to the Γ-bornologicalcoarse space X min,min with the minimal structures B min consisting of the finite subsets of X and C min := C(cid:104) diag X (cid:105) . We will use the notation X min,min for this Γ-bornological coarsespace.For example, the identity of the underlying set of Γ is a morphism Γ min,min → Γ can,min ofΓ-bornological coarse spaces. (cid:7) Example 2.6.
Let X again be a set with an action of Γ. Then we can equip X with themaximal bornological and coarse structures. In this way we get a Γ-bornological coarsespace X max,max .Our notation convention is such that the first subcript indicates the coarse structure, whilethe second subscript reflects the bornological structure.For example, we also have a Γ-bornological coarse space X min,max and morphisms ofΓ-bornological coarse spaces X min,max → X max,max and X min,max → X min,min given by theidentity of the underlying set of X .Note that in general X max,min does not make sense since the minimal bornology is notcompatible with the maximal coarse structure. (cid:7) Example 2.7.
Let (
X, d ) be a metric space with an isometric Γ-action. For r in (0 , ∞ )we consider the invariant entourages U r := { ( x, y ) ∈ X × X | d ( x, y ) ≤ r } . The coarse structure associated to the metric is defined by these entourages, i.e., given by C d := C(cid:104){ U r | r ∈ (0 , ∞ ) }(cid:105) . Furthermore, the bornology associated to the metric is generated by the metrically boundedsubsets, i.e., given by B d := B(cid:104){ B ( x, r ) | x ∈ X , r ∈ (0 , ∞ ) }(cid:105) , where B ( x, r ) denotes the metric ball of radius r centered at x .The associated bornological coarse space X d := ( X, C d , B d ) is a Γ-bornological coarse space.The identity of the underlying set of X is a morphism X min,max → X d of Γ-bornologicalcoarse spaces. (cid:7) Remark 2.8.
Let Γ be a countable group equipped with a proper left invariant metric d .Then the Γ-bornological coarse spaces Γ d and Γ can,min are equal. (cid:7) xample 2.9. Let X be a Γ-bornological coarse space with the coarse structure C andthe bornology B . For every invariant entourage U in C we consider the coarse structure C U := C(cid:104){ U }(cid:105) . The coarse structure C U is compatible with B . We let X U := ( X, C U , B )denote the resulting Γ-bornological coarse space. The identity of the underlying set is amorphism of Γ-bornological coarse spaces X U → X . If U (cid:48) is a second invariant entouragesuch that U ⊆ U (cid:48) , then we also have a morphism X U → X U (cid:48) . This construction isimportant for the formulation of the u -continuity condition in Definition 3.10. (cid:7) Example 2.10.
Let X be a Γ-bornological coarse space with coarse structure C andbornology B , and let Z be a Γ-invariant subset of X . Then we define the induced coarsestructure and bornology on Z as follows:1. C Z := { ( Z × Z ) ∩ U | U ∈ C} B Z := { Z ∩ B | B ∈ B} .Then Z X := ( Z, C Z , B Z ) is a Γ-bornological coarse space. The inclusion Z X → X is amorphism of Γ-bornological coarse spaces. (cid:7) Example 2.11.
If Γ acts on the underlying set of a bornological coarse space ( X, C , B ),then we can define a Γ-bornological coarse space Γ X := ( X, Γ C , Γ B ), whereΓ C := C (cid:42)(cid:40) (cid:91) γ ∈ Γ ( γ × γ )( U ) | U ∈ C (cid:41)(cid:43) and Γ B := B(cid:104){ U [ γB ] | U ∈ Γ C , γ ∈ Γ and B ∈ B}(cid:105) . (2.2)In general we must enlarge the bornology B to Γ B as described above in order to keep itcompatible with the new coarse structure.If X was a Γ-bornological coarse spaces, then Γ X = X . (cid:7) Definition 2.12.
Let Γ denote a group and let X be a set with an action of Γ. If B is abornology on X , then we let B Γ denote the bornology on X which is generated by the setsΓ B for all B in B .Let ( X, C , B ) be a Γ-bornological coarse space. Then the new bornology B Γ is compatiblewith the original coarse structure. The Γ -completion of ( X, C , B ) is defined to be theΓ-bornological coarse space ( X, C , B Γ ).Let ( X, C , B ) be a Γ-bornological coarse space and Y be a subset of X . The subset Y iscalled Γ -bounded if it belongs to B Γ . (cid:7) Example 2.13.
Let ( X, B ) be a bornological space with an action of Γ by proper maps.We say that Γ acts properly if for every B in B the set { γ ∈ Γ | γB ∩ B (cid:54) = ∅} is finite.We define the coarse structure C B on X to be generated by the Γ-invariant entourages U B :=Γ( B × B ) for all B in B . If Γ acts properly, then the bornological structure is compatiblewith this coarse structure and we get a Γ-bornological coarse space ( X, C B , B ). (cid:7) xample 2.14. Let X be a Γ-complete Γ-bornological coarse space with coarse structure C and bornology B . We equip the quotient set ¯ X := Γ \ X with the maximal bornology ¯ B such that the projection q : X → ¯ X is proper, i.e.,¯ B = B(cid:104){ ¯ B ⊆ ¯ X | q − ( ¯ B ) ∈ B}(cid:105) . We furthermore equip ¯ X with the minimal coarse structure ¯ C such that q is controlled, i.e.,¯ C := C(cid:104){ ( q × q )( U ) | U ∈ C}(cid:105) . Then ¯ C and ¯ B are compatible and we obtain a bornological coarse space ( ¯ X, ¯ C , ¯ B ).This construction produces a functor Q : Γ BornCoarse → BornCoarse , Q ( X, C , B ) := ( ¯ X, ¯ C , ¯ B ) . It is easy to check that the morphism of bornological coarse spaces q : ( X, C , B ) → ( ¯ X, ¯ C , ¯ B )can be interpreted as the unit of an adjunction Q : Γ BornCoarse (cid:28)
BornCoarse : C , where C is the inclusion (2.1). (cid:7) Lemma 2.15.
The category Γ BornCoarse admits arbitrary coproducts and cartesianproducts.Proof.
The coproduct of a family of Γ-bornological coarse spaces is represented by thecoproduct of the underlying bornological coarse spaces with the induced Γ-action.Similarly, the cartesian product of a family of Γ-bornological coarse spaces is representedby the cartesian product of the family of the underlying bornological coarse spaces withthe induced Γ-action.For more information about limits and colimits in Γ
BornCoarse we refer to Section 2.2.
Example 2.16.
We consider a family ( X i ) i ∈ I of Γ-bornological coarse spaces. We definethe free union (cid:70) free i ∈ I X i as follows:1. The underlying Γ-set of the free union is the disjoint union of Γ-sets (cid:70) i ∈ I X i .2. The coarse structure of the free union is generated by entourages (cid:70) i ∈ I U i for allfamilies ( U i ) i ∈ I , where U i is an entourage of X i for every i in I .3. The bornology is generated by the set { B | B ∈ B ( X i ) , i ∈ I } of subsets of (cid:70) i ∈ I X i .If I is finite, then the free union is the coproduct of the family. In general, we have amorphism of Γ-bornological coarse spaces (cid:97) i ∈ I X i → free (cid:71) i ∈ I X i induced by the identity of the underlying sets. (cid:7) xample 2.17. Let X and X (cid:48) be two Γ-bornological coarse spaces. Then we can formthe Γ-bornological coarse space X (cid:48) ⊗ X whose coarse structure is the one of the cartesianproduct and the bornology is generated by the products B (cid:48) × B for bounded subsets B (cid:48) of X (cid:48) and B of X . This construction defines a symmetric monoidal structure − ⊗ − : Γ BornCoarse × Γ BornCoarse → Γ BornCoarse with tensor unit given by the one-point space.Let Y be a Γ-set. We can form the space Y min,min ⊗ X . For a second Γ-set Y (cid:48) we have acanonical isomorphism( Y (cid:48) × Y ) min,min ⊗ X ∼ = Y (cid:48) min,min ⊗ ( Y min,min ⊗ X ) . (cid:7) Γ BornCoarse
In this section we show that the category Γ
BornCoarse admits all limits of non-emptydiagrams and various colimits. We furthermore discuss some special cases.Let Γ be a group. In the following arguments we let ι : Γ BornCoarse → Γ Set (2.3)be the forgetful functor. For a Γ-bornological coarse space X we let B X and C X denote itsbornology and coarse structure. Proposition 2.18.
1. The category Γ Coarse admits all small limits.2. The category Γ BornCoarse admits all limits of diagrams indexed by non-emptysmall categories.Proof.
We give the proof for Γ
BornCoarse . The statement for Γ
Coarse can be obtainedby ignoring all comments pertaining to bornologies and allowing in addition the index set I below to be empty.Note that the non-emptiness assumption on I only enters into the part of the proofconcerning bornologies. See also Remark 2.19.Let I be a non-empty small category and X : I → Γ BornCoarse be a functor. We will show that lim I X exists.The category Γ Set is complete. In a first step we form the Γ-set (cid:101) Y := lim I ιX . i in I we have a map of Γ-sets e i : (cid:101) Y → ιX ( i ).On (cid:101) Y we define the bornology B Y := B(cid:104){ e − i ( B ) | i ∈ I and B ∈ B X ( i ) }(cid:105) . Since B X ( i ) is Γ-invariant for every i in I , we see that B Y is Γ-invariant.We can view (cid:101) Y as a subset of (cid:81) i ∈ I ιX ( i ). On (cid:101) Y we define the coarse structure C Y := C (cid:10)(cid:8)(cid:0) (cid:89) i ∈ I U i (cid:1) ∩ ( (cid:101) Y × (cid:101) Y ) | ( U i ) i ∈ I ∈ (cid:89) i ∈ I C X ( i ) (cid:9)(cid:11) . Using that C X ( i ) has a cofinal subset of invariant entourages for every i in I , we see that thecoarse structure C Y is generated by invariant entourages and is hence a Γ-coarse structure.Finally, the relation (cid:0) (cid:89) i ∈ I U i (cid:1) [ e − j ( B )] = e − j ( U j [ B ])for j in I and B a subset of X j shows that the compatibility of B X ( j ) with C X ( j ) for all j in I implies that B Y and C Y are compatible.We therefore have defined an object Y := ( (cid:101) Y , C Y , B Y ) in Γ BornCoarse . We now showthat e j : Y → X ( j ) is a morphism of Γ-bornological coarse spaces for all j in I . For everyfamily ( U i ) i ∈ I we have e j (( (cid:81) i ∈ I U i ) ∩ ( (cid:101) Y × (cid:101) Y )) ⊆ U j . This implies that e j is controlled.Furthermore, B Y is defined such that e j is proper for every j in I .The morphisms e i : Y → X ( i ) give a transformation e : Y → X in Fun ( I, Γ BornCoarse ),where Y is the constant functor with value Y . We now show that ( Y, e ) has the universalproperty of the limit .We consider a pair (
Z, f ) with Z in Γ BornCoarse and with f : Z → X a morphism in Fun ( I, Γ BornCoarse ). Because the underlying Γ-set of Y is the limit of the diagram ιX there is a unique map h : ιZ → ιY of Γ-sets such that ιf = ιe ◦ h .It suffices to show that h is a morphism in Γ BornCoarse . Let U be an entourage of Z .Then f i ( U ) is an entourage of X ( i ) for every i in I . We have h ( U ) ⊆ (cid:0) (cid:89) i ∈ I ( f i × f i )( U ) (cid:1) ∩ ( (cid:101) Y × (cid:101) Y ) , i.e., h ( U ) is contained in one of the generating entourages of C Y . Hence h is a controlledmap.In order to show that h is proper, we consider a generating bounded subset e − i ( B ) for i in I and B in B X ( i ) . Then h − ( e − i ( B )) = f − i ( B ) is bounded in Z . Since I is non-empty,the set of subsets e − i ( B ) for all i in I and B in B X ( i ) cover Y . Therefore, every elementof B Y is contained in a finite union of such subsets. We conclude that h is proper.12 emark 2.19. Note that the last piece of the argument goes wrong if the index categoryof the diagram is empty. Then Y = ∗ , and we need the subset {∗} (which is not of theform e − i ( B )) to generate the bornology. The empty limit does not exist since the categoryΓ BornCoarse does not have a final object. (cid:7)
We now turn to colimits. Let I be a small category and X : I → Γ BornCoarse be a functor. The category Γ
Set is cocomplete. We form the Γ-set (cid:101) Y := colim I ιX , where ι is the forgetful functor from (2.3). For every i in I we have a map e ( i ) : ιX ( i ) → (cid:101) Y of Γ-sets. Definition 2.20.
We say that the diagram X is colim-admissible if we have e − i (( e j ( U ) ◦ · · · ◦ e j k ( U k ))[ { y } ]) ∈ B X ( i ) for every y in (cid:101) Y , every i in I , every r in N , every family ( j , . . . , j r ) of objects of I , andevery family of entourages U k in C X ( j k ) for k in { , . . . , r } . (cid:7) Proposition 2.21.
1. The category Γ Coarse admits all small colimits.2. The category Γ BornCoarse admits colimits for all colim-admissible diagrams.Proof.
We give the proof for Γ
BornCoarse . The statement for Γ
Coarse can be obtainedby ignoring all comments pertaining to bornologies. Note that the assumption of colim-admissibility only enters to see that the bornology and the coarse structure on the colimitare compatible.Assume that X : I → Γ BornCoarse is a colim-admissible diagram. We show that colim I X exists.In a first step we form the Γ-set (cid:101) Y := colim I ιX . On (cid:101) Y we define the coarse structure C Y := C(cid:104){ ( e i × e i )( U ) | i ∈ I and U ∈ C X ( i ) }(cid:105) . Using the fact that C X ( i ) has a cofinal subset of invariant entourages we see that the coarsestructure C Y is generated by invariant entourages and is hence a Γ-coarse structure.13e define the bornology B Y to be the subset of P ( (cid:101) Y ) consisting of the sets B satisfying e − i (( e j ( U ) ◦ · · · ◦ e j k ( U k ))[ B ]) ∈ B X ( i ) for every i in I , every r in N , every family ( j , . . . , j r ) of objects of I , and every familyof entourages U k in C X ( j k ) for k in { , . . . , r } . Since the diagram is colim-admissible, allone-point sets belong to B Y . Furthermore B Y is obviously closed under forming finiteunions and subsets. Consequently, B Y is a bornology on Y . Since B X ( i ) and C X ( i ) areΓ-invariant for every i in I we see that B Y is Γ-invariant. We finally observe that C Y and B Y are compatible by construction.We define now the object Y := ( (cid:101) Y , C Y , B Y ) of Γ BornCoarse . By construction the maps e i : X ( i ) → Y are morphisms in Γ BornCoarse for all i in I . The family of morphisms( e i ) i ∈ I provides a morphism e : X → Y in Fun ( I, Γ BornCoarse ), where Y is the constantfunctor with value Y .We now show that ( Y, e ) has the universal property of a colimit.Consider a pair (
Z, f ) with Z in Γ BornCoarse and f : X → Z . Since the underlyingΓ-set of Y is the colimit of the diagram ιX there is a unique map h : ιY → ιZ of Γ-setssuch that ιf = h ◦ ιe .It suffices to show that h is a morphism.Let i in be I and U be in C X ( i ) . Then h ( e i ( U )) = f i ( U i ) is an entourage of Z . This impliesthat h is controlled.Let now B be a bounded subset of Z . Since C Z and B Z are compatible and f j is controlled, f j ( U )[ B ] is bounded for every j in I and U in C X j . We now have e − i ( e j ( U )[ h − ( B )]) ⊆ e − i ( h − ( h ( e j ( U ))[ B ])) = f − i ( f j ( U )[ B ]) . Since f i is proper, we conclude that e − i ( e j ( U )[ h − ( B )]) is bounded. Since i, j in I and U in C X ( j ) were arbitrary this shows that h − ( B ) is bounded in Y . We conclude that h is aproper map. Example 2.22.
If ( Y , Z ) is an equivariant complementary pair (see Definition 3.7) on X ,then we have a push-out colim Y ∩ Z (cid:47) (cid:47) (cid:15) (cid:15) Z (cid:15) (cid:15) colim Y (cid:47) (cid:47) X We first note that colim Y is admissible since it is a filtered colimit of inclusions. It isstraightforward to check that the diagram is colim-admissible and that the bornology onthe space X is the one of the colimit. The only non-trivial fact to check is that the coarsestructure on X given by the colimit is not too small. Let U be an entourage of X . Assume Here we use the general relation U [ f − ( B )] ⊆ f − ( f ( U )[ B ]) for a map f : X → Y , entourage U of X and subset B of Y . i is in I such that Y i ∪ Z = X . Since the family is big, there exists j in I such that U [ Y i ] ⊆ Y j . Let e : Y j → X and f : Z → X be the inclusions. Then U ⊆ e ( U ∩ ( Y j × Y j )) ∪ f ( U ∩ ( Z × Z )) . (cid:7) Example 2.23.
By Proposition 2.18 the category Γ
BornCoarse admits fiber products.Here we give an explicit description. We consider a diagram X a (cid:15) (cid:15) Y b (cid:47) (cid:47) Z of Γ-bornological coarse spaces. We then form the cartesian product X × Y in the categoryΓ BornCoarse . We define the Γ-bornological coarse space X × Z Y to be the subset of X × Y of pairs ( x, y ) with a ( x ) = b ( y ) with the induced bornological and coarse structure.It is straightforward to check that the square X × Z Y (cid:15) (cid:15) (cid:47) (cid:47) X a (cid:15) (cid:15) Y b (cid:47) (cid:47) Z is cartesian in Γ BornCoarse . (cid:7) Example 2.24.
Assume that (
Y, Z ) is a coarsely excisive pair (see Definition 4.13) on X .Then we have a push-out Y ∩ Z (cid:47) (cid:47) (cid:15) (cid:15) Z (cid:15) (cid:15) Y (cid:47) (cid:47) X It is straightforward to check that the diagram is colim-admissible and that the bornologyon the space X is the one of the colimit. The only non-trivial fact to check is that thecoarse structure on X given by the colimit is not too small. Let U be an entourage of X .Then there is an entourage W of X such that U ⊆ W and U [ Z ] ∩ U [ Y ] ⊆ W [ Z ∩ Y ]. Let e : Y → X and f : Z → X be the inclusions. Then U ⊆ e ( W ∩ ( Y × Y )) ∪ f ( W ∩ ( Z × Z )) . (cid:7) Example 2.25.
Let H be a group acting on a Γ-bornological coarse space X such thatthe set of H -invariant entourages of X is cofinal in all entourages.We let B H ( X ) denote H -completion of X obtained from X by replacing the bornology B of X by the bornology B H ( B ) generated by the subsets HB for all B in B . Then thecoequalizer ( H min,min ⊗ X ) max −B ⇒ B H ( X ) π → X/H for the H -action exists. Here the index − max −B indicates that we replaced the bornologyby the maximal bornology. The two arrows are given by ( h, x ) (cid:55)→ x and ( h, x ) (cid:55)→ hx . One15hecks easily that they are both morphisms of Γ-bornological coarse spaces: they are bothproper since their domain has the maximal bornology, and if U is a H -invariant entourageof X then both maps send diag( H ) × U to U and hence the maps are controlled in viewof our assumption on the coarse structure of X .Finally, we check that the coequalizer diagram is colim-admissible. It suffices to checkthat for every H -invariant entourage U of X and point Hx in X/H the set U [ π − ( Hx )] isbounded in B H ( X ). This is the case since U [ x ] belongs to B and so U [ π − ( Hx )] = HU [ x ]belongs to B H ( B ). (cid:7) Example 2.26.
Let M : Γ BornCoarse → Γ Set be the functor which sends a Γ-bornological coarse space to its underlying Γ-set. In viewof Proposition 2.18, it preserves all limits over non-empty small index categories. It is infact the right-adjoint of an adjunction( − ) min,max : Γ Set (cid:28) Γ BornCoarse : M , where ( − ) min,max sends a Γ-set S to the Γ-bornological coarse space obtained by equipping S with the minimal coarse structure and maximal bornology. Indeed, for a Γ-bornologicalcoarse space X we have a natural identificationHom Γ BornCoarse ( S min,max , X ) ∼ = Hom Γ Set ( S, M ( X )) . (cid:7)
3. Equivariant coarse homology theories
The following notions are the obvious generalizations from the non-equivariant situationconsidered in [BE16].Two morphisms f, f (cid:48) : X → X (cid:48) between Γ-bornological coarse spaces are close to eachother if the subset { ( f ( x ) , f (cid:48) ( x )) | x ∈ X } of X (cid:48) × X (cid:48) is an entourage, i.e. if they are closeas morphisms between the underlying bornological coarse spaces. Definition 3.1.
A morphism between Γ-bornological coarse spaces is an equivalence if itadmits an inverse morphism up to closeness. (cid:7)
Example 3.2.
We consider a Γ-bornological coarse space X , a Γ-invariant subset A of X ,and a Γ-invariant entourage U of X . Then we can form the U -thickening U [ A ] which isagain Γ-invariant. We now assume that U contains the diagonal. Then we have a naturalinclusion i : A → U [ A ]. This inclusion is in general not an equivalence of Γ-bornologicalcoarse spaces.For example, let the group Z act on C by ( n, z ) (cid:55)→ e πiθn z , where θ is an irrational realnumber. Then the subset C \ { } of C is Z -invariant. Every non-trivial thickening ofthis subset contains the point 0. This point is fixed by the action, but C \ { } does notcontain any fixed point which could serve as the image of 0 under a potential inverse ofthe inclusion C \ { } → C . (cid:7) X be a Γ-bornological coarse space and A be a Γ-invariant subset of X . Definition 3.3.
The subset A is called nice if for every invariant entourage U of X containing the diagonal the inclusion A → U [ A ] is an equivalence. (cid:7) Example 3.4.
Let X be a Γ-bornological coarse space and let Y be a bornological coarsespace considered as a Γ-bornological coarse space with the trivial Γ-action.For every subset A of Y the subset A × X of the Γ-bornological coarse space Y × X (or of Y ⊗ X ) is nice. (cid:7) A filtered family of subsets of a set X is a family ( Y i ) i ∈ I of subsets indexed by a filteredpartially ordered set I such that the map I → P ( X ) given by i (cid:55)→ Y i is order-preserving.Let X be a Γ-bornological coarse space. Recall from [BE16] that a big family on X is afiltered family of subsets ( Y i ) i ∈ I of X such that for every entourage U of X and i in I there exists j in I such that U [ Y i ] ⊆ Y j . Definition 3.5. An equivariant big family on X is a big family consisting of Γ-invariantsubsets. (cid:7) Example 3.6.
Let X be a Γ-bornological coarse space and A be a Γ-invariant subsetof X . Then the family { A } := ( U [ A ]) U ∈C Γ is an equivariant big family. (cid:7) Let X be a Γ-bornological coarse space. Recall from [BE16] that a complementary pair( Z, Y ) on X is a pair of a subset Z of X and a big family Y = ( Y i ) i ∈ I on X such thatthere exists i in I with Z ∪ Y i = X . Definition 3.7. An equivariant complementary pair on X is a complementary pair ( Z, Y )such that Z is a Γ-invariant subset and Y is an equivariant big family. (cid:7) Definition 3.8.
A Γ-bornological coarse space X is flasque if it admits a morphism f : X → X such that1. f is close to id X .2. For every entourage U of X the subset (cid:83) n ∈ N ( f n × f n )( U ) is an entourage of X .3. For every bounded subset B of X there exists an integer n such that Γ B ∩ f n ( X ) = ∅ .We say that flasqueness of X is implemented by f . (cid:7) Remark 3.9.
In Condition 3 above one could require the weaker condition B ∩ f n ( X ) = ∅ instead of Γ B ∩ f n ( X ) = ∅ . Then much of the theory would go through, but we losethe possibility of descending the group change functors “ H -completion”, “quotient” and“induction” to the motivic level. (cid:7) C be a cocomplete stable ∞ -category. We consider a functor E : Γ BornCoarse → C . If Y = ( Y i ) i ∈ I is a filtered family of Γ-invariant subsets of X , then we set E ( Y ) := colim i ∈ I E ( Y i ) . (3.1)In this formula we consider the subsets Y i as Γ-bornological coarse spaces with thestructures induced from X .The set { , } max,max is a Γ-bornological coarse space with the trivial Γ-action. Definition 3.10.
A Γ -equivariant C -valued coarse homology theory is a functor E : Γ BornCoarse → C with the following properties:1. (Coarse invariance) For all X ∈ Γ BornCoarse the projection { , } max,max ⊗ X → X is sent by E to an equivalence.2. (Excision) E ( ∅ ) (cid:39) Z, Y ) on aΓ-bornological coarse space X the square E ( Z ∩ Y ) (cid:47) (cid:47) (cid:15) (cid:15) E ( Z ) (cid:15) (cid:15) E ( Y ) (cid:47) (cid:47) E ( X )is a push-out.3. (Flasqueness) If a Γ-bornological coarse space X is flasque, then E ( X ) (cid:39) X the natural mapcolim U ∈C Γ E ( X U ) (cid:39) −→ E ( X )is an equivalence (see Example 2.9 for notation).If the group Γ is clear from the context, then we will often just speak of an equivariantcoarse homology theory. (cid:7) Remark 3.11.
Condition 1 in the above definition is equivalent to the condition that E sends equivalences (Definition 3.1) of Γ-bornological coarse spaces to equivalences in C :The projection { , } max,max ⊗ X → X is an equivalence of Γ-bornological coarse spaces.If E preserves equivalences, then it sends this morphism to an equivalence.Vice versa, if E satisfies Condition 1, then it sends pairs of close maps to equivalent maps.If f is an equivalence with inverse g up to closeness, then E ( f ◦ g ) and E ( g ◦ f ) areequivalent to E (id) and therefore themselves equivalences. This implies by functorialitythat E ( f ) is an equivalence. (cid:7) ∞ -category C have all small products. Let ( X i ) i ∈ I be a familyof Γ-bornological coarse spaces. If E is a C -valued equivariant coarse homology theory,then by excision for every index i in I we have a projection E ( (cid:70) free i ∈ I X i ) → E ( X i ). Thecollection of these projections induces a morphism E (cid:0) free (cid:71) i ∈ I X i (cid:1) → (cid:89) i ∈ I E ( X i ) (3.2) Definition 3.12. E is called strongly additive if (3.2) is an equivalence for every family( X i ) i ∈ I of Γ-bornological coarse spaces. (cid:7) Let E be an equivariant coarse homology theory and let S be a Γ-set. Lemma 3.13. If E is strongly additive, then the twist E ( − ⊗ S max,max ) is strongly additive.Proof. This follows from the fact that for every family ( X i ) i ∈ I of Γ-bornological coarsespaces we have an isomorphism (cid:0) free (cid:97) i ∈ I X i (cid:1) ⊗ S max,max ∼ = free (cid:97) i ∈ I ( X i ⊗ S max,max )of Γ-bornological coarse spaces.
4. Equivariant coarse motivic spectra
In this section we define the stable ∞ -category of coarse motives Γ Sp X . This is completelyanalogous to [BE16, Sec. 3 & 4]. The category Γ Sp X is designed such that equivariant C -valued coarse homology theories (Definition 3.10) are the same as colimit-preservingfunctors Γ Sp X → C . The precise formulation is Corollary 4.10.Let Spc be the ∞ -category of spaces, i.e., the universal presentable ∞ -category generatedby ∗ . We start with the category PSh (Γ BornCoarse ) :=
Fun (Γ BornCoarse op , Spc )of
Spc -valued presheaves on Γ
BornCoarse . Letyo : Γ
BornCoarse → PSh (Γ BornCoarse ) (4.1)be the Yoneda embedding. Precomposition with it induces an equivalence
Fun lim ( PSh (Γ BornCoarse ) op , Spc ) (cid:39) PSh (Γ BornCoarse ) , Fun lim denotes limit-preserving functors [Lur09, Thm. 5.1.5.6] (see also [BE16,Rem. 3.9]). We use this equivalence in order to evaluate presheaves on other presheaves.For a filtered family Y = ( Y i ) i ∈ I of invariant subsets on some Γ-bornological coarse space X we write yo( Y ) := colim i ∈ I yo( Y i ) ∈ PSh (Γ BornCoarse ) . (4.2)For E in PSh (Γ BornCoarse ) we set E ( Y ) := E (yo( Y )) . Remark 4.1.
For a Γ-bornological coarse space X we have the equivalence E ( X ) (cid:39) E (yo( X )) . Furthermore we have E ( Y ) (cid:39) lim i ∈ I E ( Y i ) (4.3)for a filtered family Y = ( Y i ) i ∈ I of invariant subsets on X . (cid:7) Let E be an object of PSh (Γ BornCoarse ). Definition 4.2.
We say that E satisfies descent if1. E ( ∅ ) (cid:39) ∗ , and2. for every equivariant complementary pair ( Z, Y ) on a Γ-bornological coarse space X the square E ( X ) (cid:47) (cid:47) (cid:15) (cid:15) E ( Z ) (cid:15) (cid:15) E ( Y ) (cid:47) (cid:47) E ( Z ∩ Y ) (4.4)is cartesian.Presheaves which satisfy descent are called sheaves . (cid:7) Remark 4.3.
One can show that there is a subcanonical Grothendieck topology τ χ onΓ BornCoarse such that the τ χ -sheaves are exactly the presheaves which satisfy descentfor equivariant complementary pairs. Since we will not be using this fact in this paper wewill omit the arguments. (cid:7) We let Sh (Γ BornCoarse ) denote the full subcategory of
PSh (Γ BornCoarse ) of sheaves.We can characterize sheaves as presheaves which are local with respect to the morphismsyo( Y ) (cid:116) yo( Z ∩Y ) yo( Z ) → yo( X ) (4.5)yo( ∅ ) → ∗ . emark 4.4. In order to fix set-theoretic issues we assume that all Γ-bornological coarsespaces and the index sets I for the big families belong to some Grothendieck universe ofsmall sets. The class of local objects is then generated by a small set of morphisms. Thecategory Sh (Γ BornCoarse ) then belongs to a bigger universe. (cid:7)
We have a sheafification adjunction
PSh (Γ BornCoarse ) (cid:28) Sh (Γ BornCoarse ) : inclusion .
Remark 4.5.
For a Γ-bornological coarse space X the presheaf yo( X ) is a compact objectof PSh (Γ BornCoarse ). If Y is a big family, then yo( Y ) is an in general infinite colimit ofcompact objects and hence not compact anymore. Consequently, the morphisms (4.5) arenot morphisms between compact objects. The localization Sh (Γ BornCoarse ) is thereforea presentable ∞ -category, but it is not compactly generated. (cid:7) Let E be an object of Sh (Γ BornCoarse ). Definition 4.6. E is coarsely invariant if it is local with respect to the morphismsyo( { , } max,max ⊗ X ) → yo( X )induced by the projection for every Γ-bornological coarse space X .2. E vanishes on flasques if it is local with respect to the morphismsyo( ∅ ) → yo( X )for every flasque Γ-bornological coarse space X .3. E is u -continuous if it is local for the morphismcolim U ∈C Γ yo( X U ) → yo( X )for every Γ-bornological coarse space X (where C Γ denotes the invariant entouragesof the space X ).The above notions are just the equivariant analogues of the corresponding notions from[BE16, Sec. 3]. (cid:7) Definition 4.7.
We define the ∞ -category of Γ -equivariant motivic coarse spaces Γ Spc X as the full localizing subcategory of Sh (Γ BornCoarse ) of coarsely invariant, u -continuoussheaves which vanish on flasques. (cid:7) The locality condition is generated by a small set of morphisms. Therefore we have alocalization adjunction L : PSh (Γ BornCoarse ) (cid:28) Γ Spc X : inclusion . (4.6)We define Yo := L ◦ yo : Γ
BornCoarse → Γ Spc X . The ∞ -category Γ Spc X is a presentable ∞ -category.21 efinition 4.8. We define the category of equivariant motivic coarse spectra as thestabilization Γ Sp X := Γ Spc X ∗ [Σ − ]in the realm of presentable ∞ -categories. (cid:7) Then Γ Sp X is a stable presentable ∞ -category which fits into an adjunctionΣ mot + : Γ Spc X (cid:28) Γ Sp X : Ω mot . We further define the Yoneda functorYo s := Σ mot + ◦ Yo : Γ
BornCoarse → Γ Sp X . Definition 4.9.
We call Yo s : Γ BornCoarse → Γ Sp X the universal equivariant coarsehomology theory. (cid:7) For a Γ-bornological coarse space X we consider the object Yo s ( X ) of Γ Sp X as the motiveof X .Let C be a cocomplete stable ∞ -category. Let Γ CoarseHomologyTheories C denote thefull subcategory of Fun (Γ BornCoarse , C ) of functors which are Γ-equivariant C -valuedcoarse homology theories in the sense of Definition 3.10. By Fun colim (Γ Sp X , C ) we denotethe full subcategory of Fun (Γ Sp X , C ) of colimit preserving functors.The construction of Γ Sp X has the following consequence (see [BE16, Cor. 4.6]): Corollary 4.10.
The functor Yo s is a Γ Sp X -valued equivariant coarse homology theory.Furthermore, precomposition with Yo s induces an equivalence of ∞ -categories Fun colim (Γ Sp X , C ) → Γ CoarseHomologyTheories C . If Y = ( Y i ) i ∈ I is an equivariant big family on a Γ-bornological coarse space X , then wedefine the equivariant motivic coarse spectrumYo s ( Y ) := Σ mot + ◦ L ◦ yo( Y ) . (4.7)Note that we have Σ mot + ◦ L ◦ yo( Y ) (cid:39) colim i ∈ I Yo s ( Y i ).We will use the notationYo s ( X, Y ) := Cofib(Yo s ( Y ) → Yo s ( X )) . (4.8)By construction we have the following properties: Corollary 4.11. . We have a fiber sequence Yo s ( Y ) → Yo s ( X ) → Yo s ( X, Y ) → ΣYo s ( Y )
2. For an equivariant complementary pair ( Z, Y ) on X the natural morphism Yo s ( Z, Z ∩ Y ) → Yo s ( X, Y ) is an equivalence.3. If X → X (cid:48) is an equivalence of Γ -bornological coarse spaces, then the inducedmorphism Yo s ( X ) → Yo s ( X (cid:48) ) is an equivalence in Γ Sp X .4. If X is a flasque Γ -bornological coarse space, then Yo s ( X ) (cid:39) .5. For every Γ -bornological coarse space X with coarse structure C the natural map colim U ∈C Γ Yo s ( X U ) (cid:39) −→ Yo s ( X ) is an equivalence. Let X be a Γ-bornological coarse space and A be a Γ-invariant subset of X . Recall that { A } denotes the equivariant big family generated by A (Example 3.6). Corollary 4.12. If A is nice, then we the natural map Yo s ( A ) → Yo s ( { A } ) is an equiva-lence.Proof. Since A is nice, for every invariant entourage U of X the inclusion A → U [ A ] is anequivalence. The assertion now follows since Yo s preserves equivalences.Let X be a Γ-bornological coarse space and Y, Z be invariant subsets such that Y ∪ Z = X . Definition 4.13.
We say that (
Y, Z ) is a coarsely excisive pair , if:1. For every entourage U of X there exists an entourage W of X such that U [ Y ] ∩ U [ Z ] ⊆ W [ Y ∩ Z ] .
2. There exists a cofinal set of invariant entourages V of X such that V [ Y ] ∩ Z is nice.Note that Condition 2 is a new aspect of the equivariant theory. (cid:7) Let X be a Γ-bornological coarse space and Y, Z be invariant subsets such that Y ∪ Z = X . Corollary 4.14. If ( Y, Z ) is a coarsely excisive pair, then we have a cocartesian square Yo s ( Y ∩ Z ) (cid:47) (cid:47) (cid:15) (cid:15) Yo s ( Z ) (cid:15) (cid:15) Yo s ( Y ) (cid:47) (cid:47) Yo s ( X )23 roof. The proof of [BE16, Lem. 3.38] goes through literally. In the proof we need theequivalence Yo s ( V [ Y ] ∩ Z ) (cid:39) Yo s ( { V [ Y ] ∩ Z } )for sufficiently large invariant entourages V of X . This is ensured by Condition 2 in theDefinition 4.13 of coarse excisiveness.Let X be a Γ-bornological coarse space. Given two bornological and Γ-invariant maps p = ( p − , p + ) with p − : X → ( −∞ ,
0] and p + : X → [0 , ∞ ) we can form the coarse cylinder I p X as in the non-equivariant case [BE16, Sec. 4.3]. With its natural Γ-action it is aΓ-bornological coarse space. The projection I p X → X is a morphism. We will call it anequivariant cylinder in order to stress that the datum p was Γ-invariant.Let X be a Γ-bornological coarse space and I p X be an equivariant coarse cylinder. Corollary 4.15.
The projection I p X → X induces an equivalence Yo s ( I p X ) → Yo s ( X ) .Proof. We observe that the proof of [BE16, Prop. 4.16] goes through. At all places in theargument where Corollary 4.12 is used the corresponding subset is nice, see Example 3.4.We say that two morphisms f + , f − : X → X (cid:48) between Γ-bornological coarse spaces are homotopic if there exists a cylinder I p X such that p ± are Γ-invariant, bornological and inaddition controlled, and if there exists a morphism h : I p X → X (cid:48) such that f ± = h ◦ i ± .This leads to an extension of the notion of coarse invariance. Corollary 4.16. If f + and f − are homotopic, then Yo s ( f + ) and Yo s ( f − ) are equivalent. Recall that the category Γ
BornCoarse has a symmetric monoidal structure − ⊗− : Γ
BornCoarse × Γ BornCoarse → Γ BornCoarse (4.9)with tensor unit ∗ . Lemma 4.17. Γ Sp X has an induced closed symmetric monoidal structure ⊗ such that thefunctor Yo s : Γ BornCoarse → Γ Sp X is symmetric monoidal. The functor ⊗ commuteswith colimits in each variable separately.Proof. We get an induced symmetric monoidal structure on
PSh (Γ BornCoarse ) by theDay convolution product. The unit is given by yo( ∗ ), the Yoneda embedding is a strongsymmetric monoidal functor, and PSh (Γ BornCoarse ) is closed symmetric monoidal.For a Γ-bornological coarse space Q the functor − ⊗ Q : Γ BornCoarse → Γ BornCoarse
PSh (Γ BornCoarse ) restricts to one on Sh (Γ BornCoarse ) andthe sheafification adjunction is a symmetric monoidal adjunction.The functor − ⊗ Q furthermore respects closeness of morphisms and therefore coarseequivalences, and it respects flasqueness and u -continuity. So we get an induced symmetricmonoidal structure on Γ Spc X and Yo : Γ BornCoarse → Γ Spc X is symmetric monoidal.Since Γ Spc X is presentable, we can equip its stabilization Γ Sp X with a unique symmet-ric monoidal structure ⊗ such that stabilization Σ mot + : Γ Spc
X → Γ Sp X is symmetricmonoidal [GGN15, Thm. 5.1].It follows from the construction that ⊗ commutes with colimits in each variable separately. In this section we discuss an additional property (strongness) which an equivariant coarsehomology theory can have. Another condition (continuity) will be discussed in Section 5.By definition, a flasque Γ-bornological coarse space X admits a morphism f : X → X satisfying the conditions listed in Definition 3.8. The first condition is the condition that f is close to the identity. This fact is usually used in order to deduce that Yo s ( f ) (cid:39) id Yo s ( X ) .In the following we will use this weaker condition in order to define a more general notionof flasqueness. Definition 4.18. X is called weakly flasque if it admits a morphism f : X → X satisfying1. Yo s ( f ) (cid:39) id Yo s ( X ) .2. For every entourage U of X the subset (cid:83) n ∈ N ( f n × f n )( U ) is again an entourageof X .3. For every bounded subset B of X there exists an integer n such that Γ B ∩ f n ( X ) = ∅ .We say that f implements weak flasqueness of X . (cid:7) Let C be a cocomplete stable ∞ -category and consider a C -valued equivariant coarsehomology theory E . Definition 4.19. E is called strong if E ( X ) (cid:39) X . (cid:7) Let us incorporate now the condition of strongness on the motivic level.
Definition 4.20.
We define the version of equivariant motivic spectra Γ Sp X wfl as thelocalization of the category Γ Sp X at the set of morphisms 0 → Yo s ( X ) for all weaklyflasque Γ-bornological coarse spaces X . (cid:7) s wfl : Γ BornCoarse → Γ Sp X wfl . We consider now the ∞ -category of strong Γ-equivariant coarse homology theories. Theconstruction of Γ Sp X wfl has the following immediate consequence: Corollary 4.21.
The functor Yo s wfl is a Γ Sp X wfl -valued equivariant coarse homologytheory. Furthermore, precomposition with Yo s wfl induces an equivalence of ∞ -categories Fun colim (Γ Sp X wfl , C ) → strong Γ CoarseHomologyTheories C .
5. Continuity
The purpose of this section is to introduce the notion of continuity for equivariant coarsehomology theories. This property will be crucially needed in Section 11.2. We will firstintroduce the notion of trapping exhaustions in Section 5.1. Section 5.2 contains the actualdefinition of continuous equivariant coarse homology theories, and Section 5.3 incorporatescontinuity motivically. In the last Section 5.4 we will show how one can force continuityfor an equivariant coarse homology theory.
In this section we will introduce the notion of a trapping exhaustion of a Γ-bornologicalspace and discuss some examples and basic properties of this notion. We will also introducethe stronger notion of a co-Γ-bounded exhaustion.Let X be a bornological space and let F be a subset of X . Definition 5.1.
The subset F is called locally finite if B ∩ F is finite for every boundedsubset B of X . (cid:7) Example 5.2.
Every finite subset of X is locally finite. (cid:7) Example 5.3. If X has the minimal bornology on X , i.e., a subset is bounded if and onlyif it is finite, then every subset of X is locally finite. (cid:7) Example 5.4. If X has the maximal bornology on X , i.e., every subset of X is bounded,then the locally finite subsets of X are exactly the finite subsets. (cid:7) Let f : X → X (cid:48) be a proper map between bornological spaces. Let F be a subset of X . Lemma 5.5. If F is locally finite, then f ( F ) is locally finite.Proof. We use the relation f ( F ) ∩ B ⊆ f ( F ∩ f − ( B )).26e consider in the following a Γ-bornological space X and a filtered family of invariantsubsets Y = ( Y i ) i ∈ I . Definition 5.6.
The family Y is called a trapping exhaustion if for every locally finite,invariant subset F of X there exists i in I such that F ⊆ Y i . (cid:7) Example 5.7.
The family consisting of all locally finite, invariant subsets is a trappingexhaustion.It might happen that a Γ-bornological coarse space does note admit any non-emptyinvariant locally finite subset. Consider e.g. Γ with the maximal bornology. In this casethe empty family is a trapping exhaustion. (cid:7)
In the following we will introduce a particular kind of trapping exhaustions which we callco-Γ-bounded exhaustions.We consider a Γ-bornological space X and a filtered family of invariant subsets Y = ( Y i ) i ∈ I .We use the notation and terminology introduced in Definition 2.12. Definition 5.8.
The family Y is called a co- Γ -bounded exhaustion if1. Y is an exhaustion, i.e., (cid:83) i ∈ I Y i = X , and2. Y is co-Γ-bounded, i.e., there exists i in I such that X \ Y i is Γ-bounded. (cid:7) We consider Γ-bornological spaces X and Z and a filtered family Y := ( Y i ) i ∈ I of invariantsubsets of X . In the following we denote by Z ⊗ X the Γ-bornological space whosebornology is generated by products A × B for all bounded subsets A of Z and B of X . Lemma 5.9. If Z is bounded and Y is a co- Γ -bounded (resp., trapping) exhaustion of X , ( Z × Y i ) i ∈ I is a co- Γ -bounded (resp., trapping) exhaustion of Z ⊗ X .Proof. The co-Γ-bounded case is straightforward.For the trapping case assume that F is an invariant, locally finite subset of Z ⊗ X . Sincethe projection p : Z ⊗ X → X is proper, by Lemma 5.5 the subset p ( F ) of X is locallyfinite. Hence there exists i in I such that p ( F ) ⊆ Y i , and therefore F ⊆ Z × Y i . Lemma 5.10. If Y is a co- Γ -bounded exhaustion of a Γ -bornological space X then it is atrapping exhaustion.Proof. Let F be an invariant, locally finite subset of X . Since Y is a co-Γ-boundedexhaustion of X there exists an index i in I and a bounded subset B of X such thatΓ B ∪ Y i = X . Since the union of Y is X and F ∩ B is finite there exists an index j in I such that i ≤ j and F ∩ B ⊆ Y j . Since Y j is invariant, then also F ⊆ Y j .27 xample 5.11. Let Z be a Γ-bounded Γ-bornological space and let Z = ( Z i ) i ∈ I be anexhaustion by not necessarily Γ-invariant subsets. For every i in I we consider the subset D i := Γ( Z i × { } )of Z × Γ. We consider the Γ-bornological space Z ⊗ Γ, where Γ has any Γ-invariantbornology. The family D := ( D i ) i ∈ I is a co-Γ-bounded (and hence trapping) exhaustion of Z ⊗ Γ. (cid:7) We consider [0 , ∞ ) as a Γ-bornological space with the trivial action and the bornologygenerated by the subsets [0 , n ] for all integers n . In the following we will construct aninteresting trapping exhaustion of the Γ-bornological space[0 , ∞ ) ⊗ Z ⊗ Γwhich will play an important role in Section 11.2.Note that in general ([0 , ∞ ) × D i ) i ∈ I is not trapping.We consider the set of functions I N with its partial order induced from I . Then thepartially ordered set I N is filtered. For a function κ in I N we define the set Y κ := (cid:91) n ∈ N [ n − , n ] × D κ ( n ) . Lemma 5.12. If Z is Γ -bounded, then Y := ( Y κ ) κ ∈ I N is a trapping exhaustion of the space [0 , ∞ ) ⊗ Z ⊗ Γ . Note that the exhaustion Y is not co-Γ-bounded. Proof.
The members of Y are Γ-invariant subsets. For every integer n the family given by([ n − , n ] × D i ) i ∈ I is a co-Γ-bounded exhaustion of [ n − , n ] ⊗ Z ⊗ Γ, since [ n − , n ] isbounded and ( D i ) i ∈ I is a co-Γ-bounded exhaustion of Z ⊗ Γ. So it is trapping.Let F be a Γ-invariant locally finite subset of [0 , ∞ ) × Z × Γ. Then F ∩ [ n − , n ] × Z × Γis also locally finite and Γ-invariant. For every integer n we can choose κ ( n ) in I such that( F ∩ [ n − , n ] × Z × Γ) ⊆ [ n − , n ] × D κ ( n ) . This describes a function κ in I N such thatby construction F ⊆ Y κ .Let f : X (cid:48) → X be a proper map between Γ-bornological spaces. Lemma 5.13. If Y is a co- Γ -bounded (resp., trapping) exhaustion of X , then f − Y is aco- Γ -bounded (resp., trapping) exhaustion of X (cid:48) .Proof. The co-Γ-bounded case is a direct consequence of properness and the fact thatforming preimages commutes with forming complements.For the trapping case one uses in addition Lemma 5.5.28 .2. Continuous equivariant coarse homology theories
In this section we will introduce an additional continuity condition on an equivariantcoarse homology theory. We then verify that a continuous equivariant coarse homologytheory preserves coproducts.
Remark 5.14.
In Section 7.3 we show continuity of equivariant coarse ordinary homologytheory and in Proposition 8.17 continuity of equivariant coarse algebraic K -homology. InExample 8.31 we describe a version of the coarse algebraic K -homology theory which isnot continuous. (cid:7) We can extend the notions of locally finite subsets and trapping / co-Γ-bounded exhaustionsto Γ-bornological coarse spaces by just considering the underlying Γ-bornological spaces.Let C be a stable cocomplete ∞ -category and E : Γ BornCoarse → C an equivariant coarse homology theory. We use the convention (3.1) for the evaluation E ( Y ) on a filtered family Y of invariant subsets of a Γ-bornological coarse space. We havea natural morphism E ( Y ) → E ( X ) . (5.1) Definition 5.15. E is called continuous if for every trapping exhaustion Y of someΓ-bornological coarse space X the morphism (5.1) is an equivalence. (cid:7) Remark 5.16.
A continuous equivariant coarse homology theory E is determined by itsvalues on locally finite, invariant spaces. More precisely, let F ( X ) be the filtered partiallyordered set of locally finite, invariant subsets of X . In view of Example 5.7 we have atrapping exhaustion Y := ( F ) F ∈F ( X ) of X . So we get E ( X ) (cid:39) E ( Y ) = colim F ∈F ( X ) E ( F )showing the claim.If F ( X ) is empty, i.e., X does not admit non-empty locally finite Γ-invariant subsets (seeExample 5.7), and E is continuous, then E ( X ) (cid:39) (cid:7) By excision an equivariant coarse homology theory preserves coproducts of finite familiesof Γ-bornological coarse spaces. By the following lemma, for a continuous equivariantcoarse homology theory, we can drop the word finite . Lemma 5.17.
A continuous equivariant coarse homology theory preserves coproducts.Proof.
Let ( X i ) i ∈ I be a family of Γ-bornological coarse spaces. We must show that thenatural map (cid:77) i ∈ I E ( X i ) → E (cid:0) (cid:97) i ∈ I X i (cid:1)
29s an equivalence. All invariant subsets of (cid:96) i ∈ I X i have the induced bornological coarsestructures.Consider the following diagram, where the horizontal maps are equivalences by continuityand where F ( − ) is the trapping exhaustion consisting of all locally finite, invariant subsets. (cid:76) i ∈ I colim F i ∈F ( X i ) E ( F i ) ! (cid:15) (cid:15) (cid:39) (cid:47) (cid:47) (cid:76) i ∈ I E ( X i ) (cid:15) (cid:15) colim F ∈F ( (cid:96) i ∈ I X i ) E ( F ) (cid:39) (cid:47) (cid:47) E ( (cid:96) i ∈ I X i )It remains to show that the map marked with ! is an equivalence. The bornology of thecoproduct is described in [BE16, Lemma 2.24]. A subset of the coproduct is bounded ifand only if its intersection with X i for every i in I is bounded. This implies that for everyinvariant, locally finite subset F of (cid:96) i ∈ I X i there exists a minimal finite subset J ( F ) of I such that F ⊆ (cid:96) i ∈ J ( F ) X i . We write F i := F ∩ X i . Then F = (cid:91) i ∈ J ( F ) F i is a finite, coarsely disjoint decomposition. Hence E ( F ) (cid:39) (cid:77) i ∈ J ( F ) E ( F i ) (cid:39) (cid:77) i ∈ I E ( F i ) . That the map marked with ! is an equivalence now follows from the equivalence (cid:77) i ∈ I colim F i ∈F ( X i ) E ( F i ) (cid:39) colim ( F i ) i ∈F ( (cid:96) i ∈ I X i ) (cid:77) i ∈ I E ( F i ) . In this section we will explain how to incorporate continuity on the motivic level and wewill discuss basic properties of this procedure. Recall the Yoneda embedding (4.1) and thenotation (4.2). We have a natural morphismyo( Y ) → yo( X ) . (5.2)Let E be an object of PSh (Γ BornCoarse ). Definition 5.18.
We call E continuous if it is local with respect to the morphisms (5.2)for all trapping exhaustions Y of Γ-bornological coarse spaces X . (cid:7) emark 5.19. Let E be an object of PSh (Γ BornCoarse ) and recall (4.3). The collectionof restriction morphisms E ( X ) → E ( Y i ) for all i in I induce a natural morphism E ( X ) → E ( Y ) . (5.3)Then E is continuous if and only if the morphism (5.3) is an equivalence for every trappingexhaustion Y of a Γ-bornological coarse space X . (cid:7) We now incorporate continuity on the motivic level by adding this relation to the list inDefinition 4.7.
Definition 5.20.
We define the ∞ -category of continuous Γ-equivariant motivic coarsespaces Γ Spc X c to be the full localizing subcategory of Sh (Γ BornCoarse ) of coarselyinvariant, continuous and u -continuous sheaves which vanish on flasques. (cid:7) The locality condition is generated by a small set of morphisms. Therefore we have alocalizing adjunction L c : PSh (Γ BornCoarse ) (cid:28) Γ Spc X c : inclusion . (5.4)We define Yo c := L c ◦ yo : Γ BornCoarse → Γ Spc X c . We furthermore have a localizing adjunction C : Γ Spc X (cid:28) Γ Spc X c : inclusion and the relations L c (cid:39) C ◦ L , Yo c (cid:39) C ◦ Yo . where L is as in (4.6).The ∞ -category Γ Spc X is a presentable ∞ -category. Definition 5.21.
We define the category of continuous equivariant motivic coarse spectraas the stabilization Γ Sp X c := Γ Spc X c, ∗ [Σ − ]in the realm of presentable ∞ -categories. (cid:7) Then Γ Sp X c is a stable presentable ∞ -category which fits into an adjunctionΣ motc, + : Γ Spc X c (cid:28) Γ Sp X c : Ω motc . We further define the following stable continuous version of the Yoneda functor:Yo sc := Σ motc, + ◦ Yo c : Γ BornCoarse → Γ Sp X c . (5.5)31e obtain the functor C s in the following commuting square from the universal propertyof the stabilization: Γ Spc X Σ mot + (cid:47) (cid:47) C (cid:15) (cid:15) Γ Sp X C s (cid:15) (cid:15) Γ Spc X c Σ motc, + (cid:47) (cid:47) Γ Sp X c (5.6)We furthermore have the relation Yo sc (cid:39) C s ◦ Yo s . (5.7)Let C be a cocomplete stable ∞ -category. We let Cont Γ CoarseHomologyTheories C denote the full subcategory of Fun (Γ BornCoarse , C ) of functors which are continuousΓ-equivariant coarse homology theories in the sense of Definition 3.10.The construction of Γ Sp X c has the following immediate consequence: Corollary 5.22.
Precomposition with Yo sc induces an equivalence of ∞ -catgeories Fun colim (Γ Sp X c , C ) → Cont Γ CoarseHomologyTheories C . The Yoneda functor Yo sc has all the properties listed in Section 4.2. In addition it satisfies: Corollary 5.23.
We have Yo sc ( Y ) (cid:39) Yo sc ( X ) for every trapping exhaustion Y of a Γ -bornological coarse space X . So in particular, Yo sc ( X ) is determined by the collection of invariant, locally finite subsetsof X : Yo sc ( X ) (cid:39) colim F ∈F ( X ) Yo sc ( F ) . Note that Yo sc is a Γ Sp X c -valued continuous Γ-equivariant coarse homology theory. HenceLemma 5.17 implies: Corollary 5.24.
The functor Yo sc : Γ BornCoarse → Γ Spc X c preserves coproducts. If Y = ( Y i ) i ∈ I is a co-Γ-bounded (or trapping, respectively) exhaustion of a Γ-bornologicalspace X and Z is a second Γ-bornological coarse space, then Y × Z := ( Y i × Z ) i ∈ I is notnecessarily trapping in X ⊗ Z . As a consequence the symmetric monoidal structure ⊗ does not descend to continuous motivic coarse spectra. But if Z is bounded, then Y × Z is again a trapping exhaustion of X ⊗ Z by Lemma 5.9. We can conclude: Corollary 5.25. If Z is a bounded Γ -bornological coarse space, then the functor − ⊗ Z : Γ BornCoarse → Γ BornCoarse descends to a functor − ⊗ mot Z : Γ Sp X c → Γ Sp X c such that Yo sc ( X ⊗ Z ) (cid:39) Yo sc ( X ) ⊗ mot Z . (5.8)32 .4. Forcing continuity
To every C -valued equivariant coarse homology theory E we can naturally associate acontinuous version E cont . This is actually the best approximation of E by some continuous C -valued equivariant coarse homology theory. We will show that there is an adjunction( − ) cont : Γ CoarseHomologyTheories C (cid:28) Cont Γ CoarseHomologyTheories C : inclusion . Let us first construct the functor ( − ) cont . For simplicity of the presentation we will onlydescribe it on objects. The honest construction of the functor is similar and just involvesmore complicated diagrams.Denote by Γ BornCoarse mb the full subcategory of Γ BornCoarse spanned by the Γ-bornological coarse spaces which carry the minimal bornology. Given a functor E : Γ BornCoarse → C with target a stable cocomplete ∞ -category, we define the functor E cont by left Kanextension: Γ BornCoarse mb E (cid:47) (cid:47) (cid:15) (cid:15) C Γ BornCoarse E cont (cid:55) (cid:55) Since the image of every morphism originating in a Γ-bornological coarse space withminimal bornology is locally finite, the point-wise formula for the left Kan-extensionimplies that the canonical map colim F ∈F ( X ) E ( F X ) ∼ −→ E cont ( X ) (5.9)is an equivalence. Lemma 5.26. If E is an equivariant coarse homology theory, then E cont is a continuousequivariant coarse homology theory.Proof. We start with showing that E cont is coarsely invariant. Let X be a Γ-bornologicalspace. Since the subsets { , } × F of { , } × X for all locally finite invariant subsets F of X are cofinal in all locally finite invariant subsets of { , } max,max ⊗ X we get thesecond equivalence in the chain E cont ( { , } max,max ⊗ X ) (cid:39) colim F (cid:48) ∈F ( { , }⊗ X ) E ( F (cid:48){ , } max,max ⊗ X ) (cid:39) colim F ∈F ( X ) E ( { , } max,max ⊗ F X ) (cid:39) colim F ∈F ( X ) E ( F X ) (cid:39) E cont ( X ) . F X (the induced structures on the subset F ). The thirdequivalence in the above chain of equivalences follows from the coarse invariance of E .Next we show that E cont satisfies excision. If ( Z, Y ) is an invariant complementary pair onthe Γ-bornological coarse space X , then ( F ∩ Z, F ∩ Y ) is an invariant complementarypair on F X . Hence E cont ( Z ∩ Y ) (cid:47) (cid:47) (cid:15) (cid:15) E cont ( Y ) (cid:15) (cid:15) E cont ( Z ) (cid:47) (cid:47) E cont ( X )is the colimit of the push-out diagrams over F in F ( X ): E ( F ∩ Z ∩ Y ) (cid:47) (cid:47) (cid:15) (cid:15) E ( F ∩ Y ) (cid:15) (cid:15) E ( F ∩ Z ) (cid:47) (cid:47) E ( F X )(all subsets of F are equipped with the bornological coarse structures induced from F X )and hence itself a push-out diagram.We now show that E cont vanishes on flasques. Assume that X is a flasque Γ-bornologicalcoarse space and that flasqueness is implemented by the morphism f : X → X . If F is an invariant, locally finite subset of X , then (cid:101) F := (cid:83) n ∈ N f n ( F ) is again an invariant,locally finite subset of X . Furthermore, (cid:101) F X is flasque with flasqueness implemented bythe restriction f | (cid:101) F . The inclusion F → (cid:101) F belongs to the structure maps for the colimitover F ( X ). Since E ( (cid:101) F ) (cid:39) E cont ( X ) (cid:39) u -continuity bycolim U ∈C E cont ( X U ) (cid:39) colim U ∈C colim F ∈F ( X ) E ( F X U ) (cid:39) colim U ∈C colim F ∈F ( X ) E ( F ( F × F ) ∩ U ) (cid:39) colim F ∈F ( X ) colim U ∈C E ( F ( F × F ) ∩ U ) (cid:39) colim F ∈F ( X ) E ( F X ) (cid:39) E cont ( X ) . This finishes the proof that E cont is an equivariant coarse homology theory.We now argue that E cont is continuous. Let X be a Γ-bornological coarse space and Y = ( Y i ) i ∈ I be a trapping exhaustion of X . For every invariant, locally finite subset F of X exists an index i in I such that F ⊆ Y i . Furthermore, for every i in I we have aninclusion F ( Y i ) ⊆ F ( X ). This implies that E cont ( Y ) (cid:39) colim i ∈ I colim F ∈F ( Y i ) E ( F Y i ) (cid:39) colim F ∈F ( X ) E ( F X ) (cid:39) E cont ( X ) . This finishes the proof of Lemma 5.26. 34 roposition 5.27.
There exists an adjunction ( − ) cont : Γ CoarseHomologyTheories C (cid:28) Cont Γ CoarseHomologyTheories C : inclusion . Proof.
The collection of maps F → X for all locally finite invariant subsets F of X inducesa transformation of functors η : ( − ) cont → idon Γ CoarseHomologyTheories C . By Remark 5.16, if E is continuous, then the trans-formation η E : E cont → E is an equivalence. Moreover the transformations η E cont and( η E ) cont (i.e., the functor ( − ) cont applied to η E ) are equivalences. By [Lur09, Prop. 5.2.7.4]we get the desired adjunction.Let the cocomplete stable ∞ -category C admit all small products. Let E be a C -valuedequivariant coarse homology theory. Lemma 5.28. If E is strongly additive, then so is E cont .Proof. Let ( X i ) i ∈ I be a family of Γ-bornological coarse spaces and set X := (cid:70) free i ∈ I X i (Example 2.16). For a subset F of X and i in I we write F i := F ∩ X i . Then F is locallyfinite in X if and only if F i is locally finite in X i for every i in I . Furthermore, we havean isomorphism of Γ-bornological coarse spaces F X ∼ = (cid:70) free i ∈ I F i,X i . This impliescolim F ∈F ( X ) (cid:89) i ∈ I E ( F i,X i ) (cid:39) (cid:89) i ∈ I colim F i ∈F ( X i ) E ( F i,X i ) . We must show that the right vertical map in the diagramcolim F ∈F ( X ) E ( F X ) (cid:39) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) E cont ( X ) (cid:15) (cid:15) colim F ∈F ( X ) (cid:81) i ∈ I E ( F i,X i ) (cid:39) (cid:15) (cid:15) (cid:81) i ∈ I colim F i ∈F ( X i ) E ( F i,X i ) (cid:39) (cid:47) (cid:47) (cid:81) i ∈ I E cont ( X i )is an equivalence. Indeed, the horizontal maps are equivalences by the definition of E cont and the upper vertical map is an equivalence since E is strongly additive.
6. Change of groups
In this section we describe various change of groups constructions. They induce adjunctionsbetween the corresponding categories of equivariant motivic coarse spectra which are verysimilar to the base change functors in motivic homotopy theory.35ssume that we are given an equivariant coarse homology theory defined for every group(as it is the case for all our examples). The compatibility of the equivariant coarse homologytheory with the change of groups functors is expressed by natural transformations. Thesetransformations are additional data and will be discussed for every example of equivariantcoarse homology theory separately.What we describe here is the beginning of a story which should finally capture all group-change functors in a sort of spectral Mackey functor formalism (see, e.g., Barwick [Bar17]and Barwick–Glasman–Shah [BGS15]). The obvious task here is to capture the relationsbetween these functors on the motivic level (like iterated restrictions or inductions and theMackey relation, Lemma 6.8) together with all their higher coherences in a proper way.All change of groups transformations are associated to a homomorphism of groups ι : H → Γ . Every Γ-bornological coarse space gives rise to an H -bornological coarse space, where theaction of H is induced from the action of Γ via ι . In this way we get a restriction functorRes Γ H : Γ BornCoarse → H BornCoarse . If the homomorphism ι is not clear from the context, then we add it to the notation andwrite Res Γ H ( ι ).The functor Res Γ H induces a pull-back functor Res Γ H,pre for presheaves, which preservesall limits and colimits. Since
PSh (Γ BornCoarse ) is a presentable ∞ -category, by Lurie[Lur09, Cor. 5.5.2.9] the functor Res Γ H,pre is the right-adjoint of an adjunctionRes Γ ,preH : PSh (Γ BornCoarse ) (cid:28) PSh ( H BornCoarse ) : Res Γ H,pre . The functor Res Γ H sends equivariant complementary pairs on a Γ-bornological coarsespace X to equivariant complementary pairs on the H -bornological coarse space Res Γ H ( X ).Consequently, the restriction functor Res Γ H,pre preserves sheaves. In the following, wedecorate the Yoneda functors by the relevant group. Using Res Γ ,preH ◦ yo Γ (cid:39) yo H ◦ Res Γ H ,we see that Res Γ ,preH sends the generators of the localization listed in Definition 4.6 for thegroup Γ to corresponding generators for H . We conclude that Res Γ H,pre preserves coarselyinvariant sheaves, sheaves which vanish on flasque spaces, and u -continuous sheaves. Hencewe get an adjunction Res Γ , ∗ H : Γ Spc X (cid:28) H Spc X : Res Γ H, ∗ . Here Res Γ H, ∗ is given by the restriction of Res Γ H,pre to coarse motivic spaces, and its leftadjoint satisfies Res Γ , ∗ H (cid:39) L H ◦ Res Γ ,preH , where L H is the localization as in (4.6) (we have added a subscript H in order to indicatethe relevant group). 36ence by passing to the stabilizations we get an adjunctionRes Γ ,MotH : Γ Sp X (cid:28) H Sp X : Res Γ H,Mot , where Res Γ H,Mot is defined by the extension of the functor Res Γ H,pre to stable objects. Theobvious equivalence Res Γ ,preH ◦ yo Γ (cid:39) yo H ◦ Res Γ H implies the equivalence Res Γ ,MotH ◦ Yo s Γ (cid:39) Yo sH ◦ Res Γ H . where we have decorated the Yoneda functors by the relevant group. We consider a Γ-bornological coarse space X . Let C and B denote the coarse structureand the bornology of X . We define a new compatible bornology B H on X generated bythe ι ( H )-completions ι ( H ) B of the bounded subsets B of X . We observe that C and B H are compatible and that B H is N Γ ( ι ( H ))-invariant (as a subset of P ( X )), where N Γ ( ι ( H ))denotes the normalizer of the subgroup ι ( H ) in Γ. Definition 6.1.
The H -completion of X is the N Γ ( ι ( H ))-bornological coarse space definedby B H ( X ) := ( X, C , B H ). (cid:7) In this way we define a functor B H : Γ BornCoarse → N Γ ( ι ( H )) BornCoarse . The pull-back along B H induces an adjunction B preH : PSh (Γ BornCoarse ) (cid:28) PSh ( N Γ ( ι ( H )) BornCoarse ) : B H,pre . It is easy to see that B H,pre preserves sheaves, u -continuity and coarse invariance, since itsleft-adjoint adjoint B preH preserves the corresponding generating morphisms. See Section 6.1for a similar argument.If X is a flasque Γ-bornological coarse space with flasqueness implemented by f : X → X ,then B H ( X ) is flasque with flasqueness implemented by the same map. Here it is importantto define flasqueness with Condition 3 in Definition 3.8 and not the weaker one discussedin Remark 3.9. Consequently, B H,pre preserves presheaves which vanish on flasques.Similarly as in the case of the restriction, we get an adjunction B MotH : Γ Sp X (cid:28) N Γ ( ι ( H )) Sp X : B H,Mot and an equivalence B MotH ◦ Yo s Γ (cid:39) Yo sN Γ ( ι ( H )) ◦ B H . b H : B H ( X ) → Res Γ N Γ ( ι ( H )) ( X ). Thetransformation b H induces a natural transformation of functors b H : B H → Res Γ N Γ ( ι ( H )) : Γ BornCoarse → N Γ ( ι ( H )) BornCoarse . By functoriality, we get a transformation b H : E ◦ B H → E ◦ Res Γ N Γ ( ι ( H )) for any equivariant coarse homology theory E . Remark 6.2. If H is a finite group, then B H ∼ = Res Γ N Γ ( ι ( H )) . (cid:7) We consider a Γ-bornological coarse space X . We form the quotient set H \ X which carriesan action of the group W Γ ( H ) := N Γ ( ι ( H )) /ι ( H ) , where N Γ ( ι ( H )) denotes the normalizer of the subgroup ι ( H ) in Γ.Let π : X → H \ X denote the projection. We equip H \ X with the maximal bornologysuch that the projection π : B H ( X ) → H \ X is proper. Furthermore, we equip H \ X withthe minimal coarse structure such that π : X → H \ X is controlled. In this way we definea functor Q H : Γ BornCoarse → W Γ ( H ) BornCoarse . The projection maps define a natural transformation π H : B H → Res W Γ ( H ) N Γ ( ι ( H )) ( Q H ) of N Γ ( ι ( H )) BornCoarse -valued functors.For X in Γ BornCoarse one can interpret Res W Γ ( H ) N Γ ( ι ( H )) Q H ( X ) as a coequalizer. For abornological coarse space Z let Z max −B denote the bornological coarse space obtained byreplacing its bornology by the maximal bornology. The two maps H × X → X in thelemma below are given by the projection ( h, x ) (cid:55)→ x and the action ( h, x ) (cid:55)→ hx . Thediagram and the colimit are considered in N Γ ( ι ( H )) BornCoarse , where the action of σ in N Γ ( ι ( H )) on H × X is given by σ ( h, x ) := ( σhσ − , σx ). Lemma 6.3.
We have an isomorphism
Res W Γ ( H ) N Γ ( ι ( H )) Q H ( X ) ∼ = colim (cid:0) ( H min,max ⊗ X ) max −B ⇒ B H ( X ) (cid:1) . Proof.
One checks that the two morphisms in the coequalizer diagram are controlled. Sincewe replaced the bornology on the domain by the maximal one they are obviously proper.The coequalizer diagram is colim-admissible, see Example 2.25. One checks that thedescription of the structures on Q H ( X ) given above coincides with the explicit descriptionof the structures of the colimit given in the proof of Proposition 2.21.38he functor Q H induces a pull-back in presheaves Q H,pre . This functor preserves all limitsand colimits. Again by Lurie [Lur09, Cor. 5.5.2.9] it fits into an adjunction Q preH : PSh (Γ BornCoarse ) (cid:28) PSh ( W Γ ( H ) BornCoarse ) : Q H,pre . Note that the functor Q H induces a bijection between equivariant complementary pairson a Γ-bornological coarse space X and on the W Γ ( H )-bornological coarse space Q H ( X ).Consequently, Q H,pre preserves sheaves.It is furthermore clear that Q H,pre preserves coarsely invariant sheaves and u -continuoussheaves. If the Γ-bornological coarse space X is flasque with flasqueness implementedby f : X → X , then the induced map ¯ f : Q H ( X ) → Q H ( X ) implements flasqueness of Q H ( X ). It follows that Q H,pre preserves coarse motivic spaces.Hence Q H,pre restricts to equivariant coarse motivic spaces. Similar as before we get anadjunction Q MotH : Γ Sp X (cid:28) W Γ ( H ) Sp X : Q H,Mot . We have the relation Yo sW Γ ( H ) ◦ Q H (cid:39) Q MotH ◦ Yo s Γ . We consider two groups Γ and Γ (cid:48) . For a Γ-bornological coarse space X and a Γ (cid:48) -bornologicalcoarse space X (cid:48) we can form the product X ⊗ X (cid:48) which is a (Γ × Γ (cid:48) ) bornological coarsespace. We fix the Γ-bornological coarse space X and consider the functor P X := X ⊗ − : Γ (cid:48) BornCoarse → (Γ × Γ (cid:48) ) BornCoarse . As in the preceding cases one can check that the restriction P X,pre : PSh ((Γ × Γ (cid:48) ) BornCoarse ) → PSh (Γ (cid:48) BornCoarse )along P X preserves coarse motivic spaces and induces an adjunction P MotX : Γ (cid:48) Sp X (cid:28) (Γ × Γ (cid:48) ) Sp X : P X,Mot such that P MotX ◦ Yo s Γ (cid:48) (cid:39) Yo s (Γ × Γ (cid:48) ) ◦ P X . We now come back to our original situation and consider the homomorphism ι : H → Γ ofgroups. We define a (Γ × H )-bornological coarse space ˆΓ as follows:1. The underlying bornological coarse space of ˆΓ is Γ min,min .39. The group (Γ × H ) acts on the set ˆΓ by( γ, h ) γ (cid:48) := γγ (cid:48) ι ( h ) − . We define the functorˆ P Γ := Res Γ × H × H Γ × H ◦ P ˆΓ : H BornCoarse → (Γ × H ) BornCoarse , where the restriction is along the homomorphismid Γ × diag H : Γ × H → Γ × H × H .
This functor extends to motivesˆ P Mot Γ := Res Γ × H × H,Mot Γ × H ◦ P Mot ˆΓ . By construction we have an adjunctionˆ P Mot Γ : H Sp X (cid:28) (Γ × H ) Sp X : ˆ P Γ ,Mot and the relation Yo s Γ × H ◦ ˆ P Γ (cid:39) ˆ P Mot Γ ◦ Yo sH . We consider the canonical embedding κ : H → Γ × H into the second factor. Note thatthen we have Γ × H = N Γ × H ( H ) and hence W Γ × H ( H ) ∼ = Γ. We thus have the quotientfunctor Q H ( κ ) : (Γ × H ) BornCoarse → Γ BornCoarse (it is useful to add the embedding κ as an argument since there is also the other obvioushomomorphism ( ι, id) : H → Γ × H ). We define the induction functor as the compositionInd Γ H : H BornCoarse → Γ BornCoarse , Ind Γ H := Q H ( κ ) ◦ ˆ P Γ . The induction functor extends to motivesInd Γ ,MotH := Q MotH ( κ ) ◦ ˆ P Mot Γ : H Sp X → Γ Sp X such that Ind Γ ,MotH ◦ Yo sH (cid:39) Yo s Γ ◦ Ind Γ H . Remark 6.4.
The underlying Γ-set of Ind Γ H ( X ) is Γ × H X with the left-action of Γ onthe left factor. Here Γ × H X stands for the quotient set H \ (Γ × X ) with respect to theaction given by h ( γ, x ) = ( γι ( h − ) , hx ) . The bornology on Γ × H X is generated by the images of the subsets { γ } × B for allbounded subsets of X , and the coarse structure is generated by the images of diag Γ × U for all entourages U of X . (cid:7) K := ker( ι : H → Γ).
Lemma 6.5.
We have
Ind Γ H ∼ = Ind Γ H ◦ B K .Proof. Straightforward.
Remark 6.6.
Using Lemma 6.3 one can easily check that we have an isomorphismInd Γ H ( X ) ∼ = colim (( H min,max ⊗ Γ min,min ⊗ X ) max −B ⇒ B H (Γ min,min ⊗ X ))in Γ BornCoarse . The two arrows are given by ( h, γ, x ) (cid:55)→ ( γ, x ) and ( h, γ, x ) (cid:55)→ ( γh − , hx ). The group Γ acts on the factor Γ min,min by left multiplication. We usedthe more complicated description of the induction as a composition of various previouslydefined functors in order to deduce that induction descends to motives. (cid:7) Proposition 6.7. If K is finite and the image of ι has finite index in Γ , then we have anadjunction Ind Γ H : H BornCoarse (cid:28) Γ BornCoarse : Res Γ H . Proof.
As we observed earlier, the underlying set of Ind Γ H ( X ) is given by Γ × H X . Recallthat the functions X → Γ × H X, x (cid:55)→ [ e, x ]and Γ × H Res Γ H ( Y ) → Y, [ γ, y ] (cid:55)→ γy define the unit and counit, respectively, of an adjunction H Set (cid:29) Γ Set . We must checkthat these functions define morphisms of equivariant bornological coarse spaces.Since K is finite and normal in H , we have B K X ∼ = X for every H -bornological coarsespace X . We claim that the function X → Γ × X , x (cid:55)→ ( e, x )defines a natural morphism B K X → B H (Γ min,min ⊗ X ). This function is obviouslycontrolled. To see that it is proper, we note that the generating bounded subsets of B H (Γ min,min ⊗ X ) are of the form ι ( H )( { γ } × B ). The intersection of such a subset with { e } × X is equal to { e } × KB , hence bounded because K is finite. Then the unit is givenby the composition X ∼ = B K ( X ) → Res Γ H ( B H (Γ min,min ⊗ X )) π −→ Res Γ H ( Q H (Γ min,min ⊗ X )) = Res Γ H (Ind Γ H ( X )) . Consider now the composition F : Γ × Res Γ H Y → Γ × H Res Γ H Y → Y of the projection map with the function which will provide the desired counit. It sufficesto show that this composition is a morphism of bornological coarse spaces B H (Γ min,min ⊗ We use the word “function” in order to denote maps between underlying sets Γ H ( Y )) → Y . If U is an entourage of Y , then F (diag Γ × U ) = (cid:83) γ ∈ Γ ( γ × γ )( f × f )( U )is an entourage of Y , see Remark 2.3. If B is a bounded subset of Y , then F − ( B ) = (cid:83) γ ∈ Γ { γ − } × γB is bounded since (cid:83) h ∈ H { γι ( h ) − } × ι ( h ) B is bounded for every γ in Γand the image of ι has finite index in Γ. This shows that F , and hence the counit, is amorphism.We now consider homomorphisms H → Γ and H (cid:48) → Γ, and set K := ker( H → Γ) and K (cid:48) := ker( H (cid:48) → Γ). Note that W H ( K ) ∼ = H/K =: ¯ H and W H (cid:48) ( K (cid:48) ) ∼ = H (cid:48) /K (cid:48) =: ¯ H (cid:48) . Lemma 6.8.
Assume that H (cid:48) \ Γ /H is finite.For an H -bornological coarse space X we have the relation Res Γ H (cid:48) ◦ Ind Γ H ( X ) ∼ = (cid:97) [ γ ] ∈ ¯ H (cid:48) \ Γ / ¯ H Res ¯ H (cid:48) H (cid:48) ◦ Ind ¯ H (cid:48) ¯ H ∩ γ − ¯ H (cid:48) γ ◦ Res ¯ H ¯ H ∩ γ − ¯ H (cid:48) γ ( c γ ) ◦ Ind ¯ HH ( X ) . where c γ : ¯ H ∩ γ − ¯ H (cid:48) γ → ¯ H is given by ¯ h (cid:55)→ γ − ¯ hγ .Proof. One just makes all definitions explicit.The induction on the motivic level is given byInd Γ ,MotH := Q MotH ◦ ˆ P Mot Γ . This functor fits into the adjunctionInd Γ ,MotH : H Sp X (cid:28) Γ Sp X : Ind Γ H,Mot and is compatible with the Yoneda functor:Ind Γ ,MotH ◦ Yo sH (cid:39) Yo s Γ ◦ Ind Γ H . Since the Yoneda functor preserves finite coproducts (since it is excisive), the Lemma 6.8implies:
Corollary 6.9. If H (cid:48) \ Γ /H is finite, then Res Γ ,MotH (cid:48) ◦ Ind Γ ,MotH (cid:39) (cid:77) [ γ ] ∈ ¯ H (cid:48) \ G/ ¯ H Res ¯ H (cid:48) ,MotH (cid:48) ◦ Ind ¯ H (cid:48) ,Mot ¯ H ∩ γ − ¯ H (cid:48) γ ◦ Res ¯ H,Mot ¯ H ∩ γ − ¯ H (cid:48) γ ( c γ ) ◦ Q MotK . Part II.Examples
7. Equivariant coarse ordinary homology
In this section we introduce equivariant coarse ordinary homology theory. Its constructionis completely analogous to the non-equivariant case [BE16, Sec. 6.3].42et X be a Γ-bornological coarse space and let n be a natural number. An n -chain on X is a function c : X n +1 → Z . Its support is defined by supp( c ) := { x ∈ X n +1 | c ( x ) (cid:54) = 0 } .We typically think of n -chains as infinite linear combinations of points in X n +1 .Let U be a coarse entourage of X and let B be a bounded subset. A point ( x , . . . , x n ) in X n +1 is U -controlled if ( x i , x j ) ∈ U for every 0 ≤ i, j ≤ n . The point ( x , . . . , x n ) in X n +1 meets B if there exists 0 ≤ i ≤ n such that x i lies in B .An n -chain c is U -controlled if every point of its support is U -controlled. The n -chain c is controlled if it is U -controlled for some coarse entourage U of X . Furthermore, c is locallyfinite if for every bounded subset B of X the set of points in supp( c ) which meet B isfinite. We let C X n ( X ) denote the abelian group of controlled locally finite n -chains.The boundary operator ∂ : C X n ( X ) → C X n − ( X ) (for all n ≥
1) is defined to be ∂ := (cid:80) ni =0 ( − i ∂ i , where ∂ i is the linear extension of the operator X n +1 → X n omitting the i ’th entry. One checks that ∂ is well-defined and a differential of a chain complex. By C X ( X ) we denote the chain complex of locally finite and controlled chains on X . Definition 7.1.
For every natural number n we let C X Γ n ( X ) denote the subgroup of C X n ( X ) of locally finite and controlled n -chains which are in addition Γ-invariant. (cid:7) For every natural number n the boundary operator ∂ : C X n +1 ( X ) → C X n ( X ) restricts toa boundary operator ∂ : C X Γ n +1 ( X ) → C X Γ n ( X ) between the subgroups of Γ-invariants.Hence we have defined a subcomplex C X Γ ( X ) of C X ( X ). If f : X → X (cid:48) is a morphismbetween Γ-bornological coarse spaces, then the induced map C X ( f ) : C X ( X ) → C X ( X (cid:48) )of chain complexes preserves the subcomplexes of Γ-invariants. Therefore, we obtain afunctor C X Γ : Γ BornCoarse → Ch , where Ch denotes the category of chain complexes.In order to go from chain complexes to spectra we use the Eilenberg–MacLane correspon-dence E M : Ch → Sp . (7.1)One way to define this functor is as the composition E M : Ch → Ch [ W − ] → Sp , where the first functor is the localization of the category of chain complexes at the quasi-isomorphisms, and the second functor is the mapping spectrum functor map ( Z [0] , . . . ) ofthe stable ∞ -category Ch [ W − ], where Z [0] is the chain complex with Z placed in degreezero. Definition 7.2.
We define the functor H X Γ : BornCoarse → Sp by H X Γ := E M ◦ C X Γ . (cid:7) Theorem 7.3. H X Γ is an equivariant coarse homology theory.Proof. We observe that the arguments given in the proof of the [BE16, Thm. 6.15] extendword-by-word to the equivariant case. 43 .2. Calculations for spaces of the form Γ can,min ⊗ S min,max In this section we will do some computations of equivariant coarse ordinary homologygroups. In particular, we will relate it to ordinary group homology.
Example 7.4.
If the Γ-bornological coarse space X has the trivial Γ-action, then we havean isomorphism H X Γ ( X ) ∼ = H X ( X ). (cid:7) In order to provide more examples we consider the group homology functor H (Γ , − ) : Mod ( Z [Γ]) → Sp which can be defined as the composition Mod ( Z [Γ]) → Ch Z [Γ] → Ch Z [Γ] [ W − ] (cid:39) ←− Ch free Z [Γ] [ W − ] Z ⊗ Z [Γ] − −−−−→ Ch Z [ W − ] EM −−→ Sp . The first functor sends a Z [Γ]-module to a chain complex of Z [Γ]-modules concentrated indegree 0. The second functor is the localization at quasi-isomorphisms. The equivalencein the third step is induced by the inclusion of the full subcategory of chain complexes offree Z [Γ]-modules. It is essentially surjective by the existence of free resolutions. Finally,the functor Z ⊗ Z [Γ] − is well-defined since it preserves quasi-isomorphisms between chaincomplexes of free Z [Γ]-modules.Since the above definition involves the inverse of an equivalence it does not directly providean explicit formula. But for calculations it is useful to choose an explicit model for H (Γ , − ).The standard choice is as follows. We consider the chain complex of Z [Γ]-modules C (Γ)given by · · · → Z [Γ n +1 ] → Z [Γ n ] → · · · → Z [Γ] . The differential C (Γ) n +1 → C (Γ) n is defined as the linear extension of the map( γ , . . . , γ n +1 ) → n +1 (cid:88) i =0 ( − i ( γ , . . . , (cid:98) γ i , . . . , γ n +1 ) , where (cid:98) γ i indicates that this component gets omitted. The group Γ acts diagonally on theproducts Γ n and this induces the Z [Γ]-module structure on C (Γ). We now consider thefunctor C (Γ , − ) : Mod ( Z [Γ]) → Ch Z , V (cid:55)→ Z ⊗ Z [Γ] ( C (Γ) ⊗ Z V ) , which sends a Z [Γ]-module V to its standard complex C (Γ , V ). Here Γ acts diagonally on C (Γ) ⊗ Z V . Then we have an equivalence of functors H (Γ , − ) (cid:39) E M ◦ C (Γ , − ) . If S is a Γ-set, then we form the Γ-bornological coarse space S min,max given by the Γ-set S with the maximal bornological and the minimal coarse structure. In this way we get afunctor ( − ) min,max : Γ Set → Γ BornCoarse .
44e have furthermore a functorΓ
Set → Mod ( Z [Γ]) , S (cid:55)→ Z [ S ] . We now have two functors Γ
Set → Sp given by S (cid:55)→ H X Γ (Γ can,min ⊗ S min,max ) and S (cid:55)→ H (Γ , Z [ S ]) . Proposition 7.5 (cf. [Eng18, Prop. 3.8]) . There is a natural equivalence H X Γ (Γ can,min ⊗ S min,max ) (cid:39) H (Γ , Z [ S ]) . Proof.
We claim that there is a natural isomorphism between C X Γ (Γ can,min ⊗ S min,max )and the standard complex C (Γ , Z [ S ]). To do so, we identify Z [Γ n +1 ] ⊗ Z Z [ S ] ∼ = Z [Γ n +1 × S ],where Γ n +1 × S carries the diagonal Γ-action. Then we define the homomorphism φ n : C n (Γ , Z [ S ]) ∼ = Z ⊗ Z [Γ] Z [Γ n +1 × S ] → C X Γ (Γ can,min ⊗ S min,max ) (7.2)as the linear extension of1 ⊗ ( γ , γ , . . . , γ n , s ) (cid:55)→ (cid:88) γ ∈ Γ (( γγ , γs ) , . . . , ( γγ n , γs )) . (7.3)Note that all summands are different points on (Γ × S ) n +1 so that the infinite summakes sense, and it is Γ-invariant by construction. Every point (( γγ , γs ) , . . . , ( γγ n , γs )) iscontrolled by the entourage Γ { ( γ i , γ j ) | ≤ i, j ≤ n } × diag S of the Γ-bornological coarsespace Γ can,min ⊗ S min,max . To show that this chain is also locally finite, it suffices to checkthat there are only finitely many points in the support of the chain (7.3) which meetbounded sets of the form B × S , where B is some finite subset of Γ. This is clear since Γacts freely on Γ n +1 . This finishes the argument for the assertion that (7.2) is well-defined.It is straightforward to check that the collection { φ n } n is a chain map.We now argue that the map (7.2) is an isomorphism. To this end we define an inverse ψ : C X n (Γ can,min ⊗ S min,max ) → Z ⊗ Z Z [Γ n +1 × S ] ∼ = C n (Γ , Z [ S ]) . Let c = (cid:88) x ∈ (Γ × S ) n +1 n x x be an invariant, controlled and locally finite n -chain on Γ can,min ⊗ S min,max . We now define ψ ( c ) := (cid:88) ( γ ,...,γ n ,s ) ∈ Γ n × S n ((1 ,s ) , ( γ ,s ) ..., ( γ n ,s )) ⊗ (1 , γ , . . . , γ n , s ) . Assume that c is U -controlled. Then only summands with { γ , . . . , γ n } ⊆ U [ { } ] contributeto the sum. Since U [ { } ] is bounded and c is locally finite we see that the number ofnon-trivial summands is finite. This implies that ψ ( c ) is well-defined.45t is straightforward to check that φ and ψ are inverse to each other: To see that ψ ◦ φ = id,use that 1 ⊗ ( γ , γ , . . . , γ n , s ) = 1 ⊗ (1 , γ − γ , . . . , γ − γ n , γ − s ) . The equality φ ◦ ψ = id follows from the Γ-invariance of an n -chain c = (cid:80) x ∈ (Γ × S ) n +1 n x x together with the observation that n (( γ ,s ) ,..., ( γ n ,s n )) = 0 unless s = · · · = s n . The latterfact is due to S carrying the minimal coarse structure.One easily checks that φ is natural for maps between Γ-sets. Remark 7.6.
Assume that S is a transitive Γ-set. If we fix a point s in S and let Γ s denote the stabilizer subgroup of s , then we have an isomorphism of Z [Γ]-modules Z [ S ] ∼ = Ind ΓΓ s Z . The induction isomorphism in group homology now gives the chain of equivalences H (Γ , Z [ S ]) (cid:39) H (Γ , Ind ΓΓ s Z ) (cid:39) H (Γ s , Z ) . Since this identification involves the choice of the base point s it is not reasonable to stateany naturality (for morphisms of Γ-sets) of this equivalence. (cid:7) The following definition and proposition depend on definitions and results of the followingsections. They are not needed later and can safely be skipped on first reading.We can apply Proposition 7.5 in order to study the equivariant homology theory H Z Γ on Γ-topological spaces induced by the equivariant coarse homology H X Γ and the twistYo s (Γ can,min ); see Definition 10.29 and Equation (10.17) for the notation. Definition 7.7.
We define the equivariant ordinary homology theory by H Z Γ := H X ΓYo s (Γ can,min ) O ∞ hlg : Γ Top → Sp . (cid:7) By the results from Section 10.1 the associated equivariant homology theory is determinedon Γ-CW complexes by the restriction of the functor H Z Γ to transitive Γ-sets. Thefollowing result describes this functor explicitly.We consider the following two functors Orb (Γ) → Sp given by S (cid:55)→ H Z Γ ( S ) , S (cid:55)→ H (Γ , Z [ S ]) . Proposition 7.8.
For every transitive Γ -set S we have a natural equivalence H Z Γ ( S ) (cid:39) Σ H (Γ , Z [ S ]) of spectra. roof. By definition we have a natural equivalence H Z Γ ( S ) (cid:39) H X Γ ( O ∞ ( M ( U ( S ))) ⊗ Yo s (Γ can,min )) (cid:39) H X Γ ( O ∞ ( S disc,max,max ) ⊗ Yo s (Γ can,min )) . By Proposition 9.35, O ∞ ( S disc,max,max ) (cid:39) ΣYo s ( S min,max ). This gives the equivalence H Z Γ ( S ) (cid:39) Σ H X Γ ( S min,max ⊗ Γ can,min ) . By Proposition 7.5 we have a natural equivalence H X Γ ( S min,max ⊗ Γ can,min ) (cid:39) H (Γ , Z [ S ])which finishes this proof. Recall Definition 5.15 of continuity of an equivariant coarse homology theory.
Lemma 7.9.
The equivariant coarse homology theory H X Γ is continuous.Proof. Let X be a Γ-bornological coarse space and Y := ( Y i ) i ∈ I be a trapping exhaustion.If c is a chain in C X Γ n ( X ), then supp( c ) is a Γ-invariant subset of X n +1 which meetsevery bounded subset of X in a finite set. For i in { , . . . , n } let p i : X n +1 → X be theprojection. Then we consider the Γ-invariant subset F := n (cid:91) i =0 p i (supp( c )) . Note that c belongs to the image of the map C X Γ n ( F X ) → C X Γ n ( X ) induced by theinclusion of F into X .Observe that F is locally finite. Hence there exists an index i in I such that F ⊆ Y i . Weconclude that C X Γ n ( X ) ∼ = colim i ∈ I C X Γ n ( Y i ) . The argument above implies that we have an isomorphism of chain complexes C X Γ ( X ) ∼ = colim i ∈ I C X Γ ( Y i ) . Since the Eilenberg–MacLane correspondence (7.1) preserves filtered colimits we concludethat H X Γ ( X ) (cid:39) colim i ∈ I H X Γ ( Y i )which was to be shown.Recall Definition 4.19 of strongness of an equivariant coarse homology theory. Lemma 7.10.
The equivariant coarse homology theory H X Γ is strong. roof. We can essentially repeat the proof of [BE16, Prop. 6.18].Let f : X → X implement weak flasqueness of a Γ-bornological coarse space X . Then wecan define the chain map S := ∞ (cid:88) n =0 C X ( f n ) : C X ( X ) → C X ( X ) . We refer to [BE16, Prop. 6.18] for the verification that this map is well-defined. We thenhave the identity of endomorphisms of C X Γ ( X )id C X ( X ) + C X ( f ) ◦ S = S .
Applying the Eilenberg–MacLane correspondence
E M this givesid H X Γ ( X ) + H X Γ ( f ) ◦ E M ( S ) = E M ( S ) . Since we already know that H X Γ is a coarse homology theory we have the equivalence H X Γ ( f ) (cid:39) id H X Γ ( X ) . Hence we get id H X Γ ( X ) + E M ( S ) (cid:39) E M ( S ), and this implies thatwe must have H X Γ ( X ) (cid:39) Lemma 7.11.
The equivariant coarse homology theory H X Γ is strongly additive.Proof. Let ( X i ) i ∈ I be a family of Γ-bornological coarse spaces. By inspection of thedefinitions, C X Γ (cid:16) free (cid:71) i ∈ I X i (cid:17) ∼ = (cid:89) i ∈ I C X Γ ( X i ) . We then use that the Eilenberg–MacLane correspondence (7.1) preserves products.
In this section we provide natural transformations which relate the equivariant coarsehomology theory with the change of group functors considered in Section 6.Let ι : H → Γ be a homomorphism of groups. Let X be a Γ-bornological coarse space.Since every Γ-invariant chain is H -invariant we have an inclusion C X Γ ( X ) (cid:44) → C X H (Res Γ H X ) . This gives a natural transformation between equivariant coarse homology theoriesres Γ H : H X Γ → H X H ◦ Res Γ H . (7.4)48e let the subset X (cid:48) of X be a set of representatives of H -orbits in X . Then we have arestriction map r : C X n ( X ) → Z X (cid:48) × X n , c (cid:55)→ c | X (cid:48) × X n . We want to define a homomorphism C X Γ n ( X ) → C X W Γ ( H ) n ( Q H ( X )) (7.5)by linear extension of the projection map X (cid:48) × X n → ( H \ X ) n +1 composed with therestriction r . We will argue now that this is well-defined, i.e., that the sums appearing inthis extension are finite. For x in X we denote by [ x ] its orbit in H \ X . We consider apoint ([ x ] , . . . , [ x n ]) in ( H \ X ) n +1 . Then [ x ] ∩ X (cid:48) consists of a unique point x .Let c be in C X n ( X ) and assume that c is U -controlled for some entourage U of X . Thenwe have supp( c ) ∩ { x } × X n ⊆ ( U [ x ]) n +1 . Since U [ x ] is bounded the number of points of supp( c ) which meet U [ x ] is finite.One checks that the homomorphism (7.5) does not depend on the choice of the set ofrepresentatives X (cid:48) and has values in W Γ ( H )-invariant chains. Furthermore, it is compatiblewith the differential and takes values in controlled and locally finite chains.We therefore get a natural transformation q H ( ι ) : H X Γ → H X W Γ ( H ) ◦ Q H ( ι ) . (7.6)Recall from Section 6.5 that P ˆΓ ( X ) = Γ min,min ⊗ X with the action of Γ × H given by( γ, h )( γ (cid:48) , x ) = ( γγ (cid:48) ι ( h ) − , hx ). We define a morphism of chain complexes C X Hn → C X Γ × Hn ( P ˆΓ ( X ))by linear extension of the map( x , . . . , x n ) (cid:55)→ (cid:88) γ (cid:48) ∈ Γ (( γ (cid:48) , x ) , ( γ (cid:48) , x ) , . . . , ( γ (cid:48) , x n )) . In this way we get a transformationˆ p Γ : H X H → H X Γ × H ◦ ˆ P Γ . (7.7)We finally get a natural transformationind Γ H (cid:39) q H ( κ ) ◦ ˆ p Γ : H X H → H X Γ ◦ Ind Γ H , (7.8)where κ : H → Γ × H is the inclusion of the second factor. Proposition 7.12. If ι : H → Γ is injective, then the transformation (7.8) is an equiva-lence of H -equivariant coarse homology theories.Proof. By Remark 6.4 the map X → Γ × H X = Ind Γ H ( X ) given by x (cid:55)→ [1 , x ] is anembedding of an H -invariant coarse component. Restriction along this map gives a mapof chain complexes C X Γ (Ind Γ H ( X )) → C X H ( X ) which induces the inverse to (7.8).49 . Equivariant coarse algebraic K -homology In this section, we define for every additive category A with Γ-action its Γ-equivariantcoarse algebraic K -homology K A X Γ : Γ BornCoarse → Sp . The construction associates to a Γ-bornological coarse space X an additive category ofequivariant X -controlled A -objects V Γ A ( X ), and defines K A X Γ to be the (non-connective)algebraic K -theory spectrum of this category. K -theory functor We describe the properties of the K -theory functor that we will use subsequently. See[BFJR04, Sec. 2.1] for similar statements. Let Add denote the 1-category of small additivecategories and exact functors. In the following, all additive categories will be small so thatwe can omit this adjective safely.The K -theory functor is a functor K : Add → Sp which has the following properties (we will recall the occurring notions further below):1. (Normalization) It sends (a skeleton of) the additive category of finitely generatedfree modules over a ring R to the non-connective K -theory (see e.g. [Sch04]) of thatring.2. (Invariance) It sends isomorphic exact functors to equivalent maps.3. (Colimits) If A = colim i A i is a filtered colimit of additive subcategories, then thenatural map colim i K ( A i ) (cid:39) −→ K ( A ) is an equivalence.4. (Additivity) If Φ , Ψ : A → A (cid:48) are exact functors between additive categories, thenwe have an equivalence K (Φ) + K (Ψ) (cid:39) K (Φ ⊕ Ψ) of morphisms between K -theoryspectra.5. (Exactness) If A is a Karoubi filtration of C , then we have a fiber sequence K ( A ) → K ( C ) → K ( C / A ) ∂ −→ K ( A ) .
6. (Products) If we have a family ( A i ) i ∈ I of additive categories, then the natural map K ( (cid:81) i ∈ I A i ) → (cid:81) i ∈ I K ( A i ) is an equivalence.7. (Flasqueness) It sends flasque additive categories to zero.50et us recall the definition of some notions appearing above.A category is additive if admits a zero object and biproducts such that the operationHom( A, B ) × Hom(
A, B ) → Hom(
A, B ) which sends f, g to f + g : A ∆ −→ A ⊕ A f ⊕ g −−→ B ⊕ B id + id −−−→ B defines an abelian group structure on morphism sets. In applications, it is useful to considerthe equivalent characterization of additive categories as categories which are enriched overabelian groups, have a zero object and admit finite coproducts Lurie [Lur14, Sec. 1.1.2].A morphism between additive categories is a functor between the underlying categorieswhich preserves the zero object and finite coproducts. Equivalently, one can require thatthe functor is compatible with the enrichement in abelian groups, see Mac Lane [ML98,Prop. VIII.2.4].Given morphisms Φ , Ψ : A → A (cid:48) between additive categories, we define a new morphismΦ ⊕ Ψ : A → A (cid:48) by choosing for every object A of A an object of A (cid:48) representing the sumΦ( A ) ⊕ Ψ( A ). Since the sum of two functors is unique up to unique isomorphism, there isan essentially unique map K (Φ ⊕ Ψ) by virtue of Property (2).
Definition 8.1.
An additive category A is called flasque if there exists a functor Σ : A → A such that id A ⊕ Σ ∼ = Σ. (cid:7) Note that Property (4) implies Property (7): Assume that A is flasque and that Σ : A → A is a functor satisfying id A ⊕ Σ ∼ = Σ. By Property 4 we then have an equivalence K (Σ) + id K ( A ) (cid:39) K (Σ), which implies that K ( A ) (cid:39) A ⊆ C be a full additive subcategory. For C, D in C , let Hom C ( C, A , D ) denotethe set of all morphisms in Hom C ( C, D ) which factor through some object in A . ThenHom C ( C, A , D ) is a subgroup of Hom C ( C, D ). We let C / A be the category with the sameobjects as C and whose morphisms are given byHom C / A ( C, D ) := Hom C ( C, D ) / Hom C ( C, A , D ) . Note that C / A is again an additive category. It has the universal property that exactfunctors on C / A correspond bijectively to exact functors on C which vanish on A .Let C be an additive category. Definition 8.2.
The inclusion A ⊆ C of a full additive subcategory is a Karoubi filtration if every diagram A f −→ C g −→ B in C , where A, B are objects of A , admits an extension to a commutative diagram A f (cid:47) (cid:47) (cid:15) (cid:15) C g (cid:47) (cid:47) ∼ = (cid:15) (cid:15) BD (cid:47) (cid:47) inc (cid:47) (cid:47) D ⊕ D ⊥ pr (cid:47) (cid:47) (cid:47) (cid:47) D (cid:79) (cid:79) for some object D of A . (cid:7)
51n [Kas15, Lem. 5.6] it is shown that Definition 8.2 is equivalent to the standard definitionof a Karoubi filtration as considered in [CP97].An algebraic K -theory functor as described here can be furnished by restricting the K -theory functor constructed by Schlichting [Sch06] to additive categories. For Property (1),see [Sch06, Thms. 5 & 8]. Property (2) is so elementary that it is rarely stated explicitly,but can be easily read off from the construction in [Sch06]. Property (3) is a combinationof [Qui73, Eq. (9)] and [Sch06, Cor. 5]. Property (4) follows from Property (5) (cf. also[Wal85, Prop. 1.3.2]); the latter is proved in [Sch04, Thm. 2.10]. Property (6) is shown in[KW17, Thm 1.2]; for connective K -theory, this is originally due to Carlsson [Car95]. X -controlled A -objects Let X be a bornological coarse space with the bornology B and the coarse structure C ,and let A be an additive category with a (strict) Γ-action. The poset B is ordered by thesubset inclusion and will be regarded as a category. For a functor A : B → A we define γA : B → A to be the functor sending a bounded set B to γ ( A ( γ − ( B ))). Definition 8.3. An equivariant X -controlled A -object is a pair ( A, ρ ) consisting of afunctor A : B → A and a family ρ = ( ρ ( γ )) γ ∈ Γ of natural isomorphisms ρ ( γ ) : A → γA satisfying the following conditions:1. A ( ∅ ) ∼ = 0.2. For all B, B (cid:48) in B , the commutative square A ( B ∩ B (cid:48) ) (cid:47) (cid:47) (cid:15) (cid:15) A ( B ) (cid:15) (cid:15) A ( B (cid:48) ) (cid:47) (cid:47) A ( B ∪ B (cid:48) )is a push-out.3. For all B in B , there exists a finite subset F of B such that the inclusion F ⊆ B induces an isomorphism A ( F ) ∼ = −→ A ( B ).4. For all pairs of elements γ, γ (cid:48) of Γ we have ρ ( γγ (cid:48) ) = γρ ( γ (cid:48) ) ◦ ρ ( γ ), where γρ ( γ (cid:48) ) isthe natural transformation from γA to γγ (cid:48) A induced from ρ ( γ (cid:48) ). (cid:7) Let (
A, ρ ) be an equivariant X -controlled A -object. Lemma 8.4.
1. a) The canonical morphism (cid:77) x ∈ F A ( { x } ) (cid:80) x ∈ F A ( { x }⊆ F ) −−−−−−−−−→ A ( F ) is an isomorphism for every finite subset F of X . ) For two finite subsets F, F (cid:48) of X with F ⊆ F (cid:48) we have a commuting square (cid:76) x ∈ F A ( { x } ) (cid:15) (cid:15) (cid:80) x ∈ F A ( { x }⊆ F ) (cid:47) (cid:47) A ( F ) (cid:15) (cid:15) (cid:76) x ∈ F (cid:48) A ( { x } ) (cid:80) x ∈ F (cid:48) A ( { x }⊆ F (cid:48) ) (cid:47) (cid:47) A ( F (cid:48) ) .
2. a) For a bounded subset B of X there exists a unique minimal finite subset F B of B such that A ( F B ) → A ( B ) is an isomorphism.b) If B and F B are as in 2a, then for any subset B (cid:48) of X with F B ⊆ B (cid:48) ⊆ B , themorphisms A ( F B ) → A ( B (cid:48) ) and A ( B (cid:48) ) → A ( B ) are isomorphisms.Proof. Item 1 is a direct consequence of Definition 8.3(1) and (2).Suppose now that
F, F (cid:48) are two finite subsets of B as in Definition 8.3(3). Then we obtaina push-out square A ( F ∩ F (cid:48) ) (cid:47) (cid:47) (cid:15) (cid:15) A ( F ) ∼ = (cid:15) (cid:15) A ( F (cid:48) ) ∼ = (cid:47) (cid:47) A ( B )in which all arrows are inclusions of direct summands. It follows that A ( F ∩ F (cid:48) ) → A ( B )is also an isomorphism. This implies the existence of F B .For a bounded subset B (cid:48) with F B ⊆ B (cid:48) ⊆ B , inspection of a similar push-out squareimplies that A ( F B ) → A ( B (cid:48) ) is an isomorphism, and the claim follows.Let ( A, ρ ) be an equivariant X -controlled A -object. Definition 8.5.
The function σ which sends a bounded subset B of X to the finite subset F B from Lemma 8.4 is called the support function of ( A, ρ ). (cid:7) The support function is an order preserving, equivariant function from B to the set offinite subsets of X with the property that σ ( σ ( B )) = σ ( B ) for every bounded subset B .Let ( A, ρ ) , ( A (cid:48) , ρ (cid:48) ) be equivariant X -controlled A -objects and let U be an invariant en-tourage of X . Definition 8.6. An equivariant U -controlled morphism f : ( A, ρ ) → ( A (cid:48) , ρ (cid:48) ) is a naturaltransformation f : A ( − ) → A (cid:48) ( U [ − ]) , such that ρ (cid:48) ( γ ) ◦ f = ( γf ) ◦ ρ ( γ ) for all elements γ of Γ. (cid:7)
53e let Mor U (( A, ρ ) , ( A (cid:48) , ρ (cid:48) )) denote the set of equivariant U -controlled morphisms. Fur-thermore, we define the set of controlled morphisms from A to A (cid:48) byHom V Γ A ( X ) (( A, ρ ) , ( A (cid:48) , ρ (cid:48) )) := colim U ∈C Γ Mor U (( A, ρ ) , ( A (cid:48) , ρ (cid:48) )) . We denote the resulting category of equivariant X -controlled A -objects and equivariantcontrolled morphisms by V Γ A ( X ).We observe that the composition of a U -controlled and a U (cid:48) -controlled morphism is a U ◦ U (cid:48) -controlled morphism. We conclude that composition in V Γ A ( X ) is well-defined. Lemma 8.7.
The category V Γ A ( X ) is additive.Proof. Let (
A, ρ ) , ( A (cid:48) , ρ (cid:48) ) be equivariant X -controlled A -objects. Denote by A ⊕ A (cid:48) theirdirect sum in Fun ( B , A ). Note that ( A ⊕ A (cid:48) , ρ ⊕ ρ (cid:48) ) is an X -controlled A -object becausefinite unions of bounded sets are bounded. Since finite unions of coarse entourage arecoarse entourages and there are canonical isomorphisms of Γ-setsNat( A ⊕ A (cid:48) , C ◦ U [ − ]) ∼ = Nat( A, C ◦ U [ − ]) × Nat( A (cid:48) , C ◦ U [ − ])and Nat( C, ( A ⊕ A (cid:48) ) ◦ U [ − ]) ∼ = Nat( C, A ◦ U [ − ]) × Nat(
C, A (cid:48) ◦ U [ − ])(we use the symbol Nat to denote the morphism sets in Fun ( B , A )) for all U in C Γ andequivariant X -controlled A -objects C , it follows that ( A ⊕ A (cid:48) , ρ ⊕ ρ (cid:48) ) is also a direct sumin V Γ A ( X ).Similarly, addition of morphisms in Fun ( B , A ) induces the addition operation of morphismsin V Γ A ( X ).Next we discuss the functoriality of V Γ A ( X ) in the variables X and A . For a Γ-equivariantexact functor Φ : A → A (cid:48) of additive categories, there exists an induced exact functor V ΓΦ ( X ) : V Γ A ( X ) → V Γ A (cid:48) ( X ) which sends an object ( A, ρ ) to (Φ ◦ A, Φ( ρ )). Therefore, wehave a functor V Γ − ( X ) : Fun ( B Γ , Add ) → Add . Let φ : ( X, B , C ) → ( X (cid:48) , B (cid:48) , C (cid:48) ) be a morphism of Γ-bornological coarse spaces, and let( A, ρ ) be an equivariant X -controlled A -object. Since φ is proper, we can define a functor φ ∗ A : B (cid:48) → A by φ ∗ A ( B ) := A ( φ − ( B )) , and we define φ ∗ ρ ( γ )( B ) = ρ ( γ )( φ − ( B )) . All properties of Definition 8.3 except (3) are immediate. To see that (3) also holds, wenote that σ ( φ − ( B )) ⊆ φ − ( φ ( σ ( φ − ( B ))) ⊆ φ − ( B ) and apply Lemma 8.4 to see that φ ∗ A ( φ ( σ ( φ − ( B )))) → φ ∗ A ( B ) is an isomorphism.54et f : ( A, ρ ) → ( A (cid:48) , ρ (cid:48) ) be an equivariant U -controlled morphism. Then there exists some V in C (cid:48) , Γ such that ( φ × φ )( U ) ⊆ V . Then U [ φ − ( B )] ⊆ φ − ( V [ B ]) for all bounded subsets B of X , so we obtain an induced V -controlled morphism φ ∗ f = { f φ − ( B ) : φ ∗ A ( B ) → φ ∗ A ( V [ B ]) } B ∈B (cid:48) . This defines a functor φ ∗ : V Γ A ( X ) → V Γ A ( X (cid:48) ) . We thus have constructed a functor V Γ A : Γ BornCoarse → Add . K -homology Let Γ be a group and A be an additive category with a Γ-action. Definition 8.8.
We define the coarse algebraic K -homology K A X Γ associated to A as K A X Γ := K ◦ V Γ A : Γ BornCoarse → Sp . (cid:7) This section discusses the homological properties of K A X Γ . Our first goal is to prove thefollowing theorem. Theorem 8.9.
The functor K A X Γ is an equivariant coarse homology theory. We divide the proof of Theorem 8.9 into a sequence of lemmas.
Lemma 8.10.
The functor K A X Γ is u -continuous.Proof. Let X be a Γ-bornological coarse space, and let U be an invariant entourage of X .The natural map X U → X induces a functor Φ U : V Γ A ( X U ) → V Γ A ( X ). Since the definitionof equivariant X -controlled A -objects is independent of the coarse structure, Φ U is theidentity on objects. Additionally, since inclusions of direct summands are monomorphisms,Φ U is faithful.This allows us to view V Γ A ( X U ) as a subcategory of V Γ A ( X ), and we have V Γ A ( X ) = (cid:91) U ∈C Γ V Γ A ( X U )since every morphism in V Γ A ( X ) is U -controlled for some U in C Γ .Since the algebraic K -theory functor is compatible with filtered colimits (Property (3)),the claim of the lemma follows.Let φ, ψ : X → X (cid:48) be morphisms of Γ-bornological coarse spaces.55 emma 8.11. If φ and ψ are close, then φ ∗ and ψ ∗ are isomorphic.Proof. Let U (cid:48) be a symmetric entourage of X (cid:48) containing the diagonal such that ( φ ( x ) , ψ ( x ))lies in U (cid:48) for all x in X . Note that this implies φ − ( B (cid:48) ) ⊆ ψ − ( U (cid:48) [ B (cid:48) ]) and ψ − ( B (cid:48) ) ⊆ φ − ( U (cid:48) [ B (cid:48) ]) for all bounded subsets B (cid:48) of X (cid:48) .Let ( A, ρ ) be an equivariant X -controlled A -object. The maps A ( φ − ( B (cid:48) )) → A ( ψ − ( U (cid:48) [ B (cid:48) ]))define a natural morphism f : φ ∗ A → ψ ∗ A , and similarly we have a natural morphism g : ψ ∗ A → φ ∗ A . Since the composition g ◦ f is given by the natural transformation { A ( φ − ( B (cid:48) ) ⊆ φ − (( U (cid:48) ) [ B (cid:48) ])) : φ ∗ A → φ ∗ A ◦ ( U (cid:48) ) [ − ] } B (cid:48) ∈B (cid:48) , we have g ◦ f = id φ ∗ A . Similarly, f ◦ g = id ψ ∗ A . It follows that φ ∗ ∼ = ψ ∗ . Corollary 8.12.
The functor K A X Γ is coarsely invariant.Proof. This is a direct consequence of Lemma 8.11 together with the Property (2) of thealgebraic K -theory functor. Lemma 8.13.
The functor K A X Γ vanishes on flasque Γ -bornological coarse spaces.Proof. Let X be a Γ-bornological coarse space with flasqueness implemented by φ : X → X .We claim that the functor Σ := (cid:77) n ∈ N ( φ n ) ∗ : V Γ A ( X ) → V Γ A ( X )is well-defined (up to canonical isomorphism).For every bounded subset B of X , there exists some n in N such that ( φ n ) − ( B ) = ∅ ,so the direct sum (cid:76) n ∈ N ( φ n ) ∗ A exists for every equivariant X -controlled object ( A, ρ ).Let f : ( A, ρ ) → ( A (cid:48) , ρ (cid:48) ) be a U -controlled morphism. Then (cid:76) n ∈ N ( φ n ) ∗ f is V -controlled,where V := (cid:83) n ∈ N ( φ × φ ) n ( U ) is again a coarse entourage of X by assumption on φ . So Σis an exact functor.Since φ is close to id X , we conclude from Lemma 8.11 that φ ∗ ◦ Σ and Σ are isomorphic.Hence, id V Γ A ( X ) ⊕ Σ ∼ = id V Γ A ( X ) ⊕ ( φ ∗ ◦ Σ) ∼ = Σ , and we deduce the lemma from Property (7) of the algebraic K -theory functor.Let X be a Γ-bornological coarse space and Z a Γ-invariant subset of X . For an equivariant X -controlled object ( A, ρ ), we denote by ( A | Z , ρ | Z ) the restriction to Z , that is A | Z is therestriction of A to B ∩ Z and ρ | Z the appropriate restriction of ρ .56et X be a Γ-bornological coarse space and let Y = ( Y i ) i ∈ I be an equivariant big family in X . For each i in I , the canonical exact functor V Γ A ( Y i ) → V Γ A ( X ) is injective on objectsand fully faithful, so we can regard V Γ A ( Y i ) as a full subcategory of V Γ A ( X ). Define V Γ A ( Y ) := (cid:91) i ∈ I V Γ A ( Y i ) , considered as a full subcategory of V Γ A ( X ). Lemma 8.14.
The inclusion V Γ A ( Y ) → V Γ A ( X ) is a Karoubi filtration.Proof. Let (
A, ρ ) , ( A (cid:48) , ρ (cid:48) ) be objects in V Γ A ( Y ), ( C, ρ C ) be an object in V Γ A ( X ), and let f : A → C and g : C → A (cid:48) be morphisms. Choose i in I such that both A and A (cid:48) areobjects in V Γ A ( Y i ), and pick an invariant and symmetric entourage U which contains thediagonal such that f and g are U -controlled. Let j in I be such that U [ Y i ] ⊆ Y j . Since f isa natural transformation A → C ◦ U [ − ] and g | C | X \ Yj = 0, the following diagram commutes: A f (cid:47) (cid:47) (cid:15) (cid:15) C ∼ = (cid:15) (cid:15) g (cid:47) (cid:47) A (cid:48) C | Y j inc (cid:47) (cid:47) C | Y j ⊕ C | X \ Y j pr (cid:47) (cid:47) C | Y j (cid:79) (cid:79) Hence, the inclusion V Γ A ( Y ) → V Γ A ( X ) is a Karoubi filtration. Proposition 8.15.
The functor K A X Γ is excisive.Proof. The category of ∅ -controlled A -objects is the zero category, which has trivial K -theory by Property (1).Let X be a Γ-bornological coarse space, and let ( Z, Y ) be an equivariant complementarypair on X . Both inclusions V Γ A ( Z ∩ Y ) → V Γ A ( Z ) and V Γ A ( Y ) → V Γ A ( X ) are Karoubifiltrations by Lemma 8.14. Therefore, we obtain by Property (5) of the algebraic K -theoryfunctor a map of fiber sequences K ( V Γ A ( Z ∩ Y )) (cid:47) (cid:47) (cid:15) (cid:15) K ( V Γ A ( Z )) (cid:47) (cid:47) (cid:15) (cid:15) K ( V Γ A ( Z ) / V Γ A ( Z ∩ Y )) (cid:15) (cid:15) ∂ (cid:47) (cid:47) K ( V Γ A ( Z ∩ Y )) (cid:15) (cid:15) K ( V Γ A ( Y )) (cid:47) (cid:47) K ( V Γ A ( X )) (cid:47) (cid:47) K ( V Γ A ( X ) / V Γ A ( Y )) ∂ (cid:47) (cid:47) K ( V Γ A ( Y ))Consider the induced exact functor Φ : V Γ A ( Z ) / V Γ A ( Z ∩ Y ) → V Γ A ( X ) / V Γ A ( Y ).Let ( A, ρ ) in V Γ A ( X ) and consider the natural morphisms f : ( A | Z , ρ | Z ) → ( A, ρ ) and p : ( A, ρ ) → ( A | Z , ρ | Z ). Clearly, pf = id ( A | Z ,ρ | Z ) . Pick i in I such that X \ Z ⊆ Y i . Thenid ( A,ρ ) − f p factors through ( A | Y i , ρ | Y i ), so f and p define mutually inverse isomorphismsin V Γ A ( X ) / V Γ A ( Y ). We conclude that Φ ◦ Ψ ∼ = id V Γ A ( X ) / V Γ A ( Y ) , so Φ is an equivalence ofcategories. 57t follows from Property (2) of the algebraic K -theory functor that K ( V Γ A ( Z ∩ Y )) (cid:47) (cid:47) (cid:15) (cid:15) K ( V Γ A ( Z )) (cid:15) (cid:15) K ( V Γ A ( Y )) (cid:47) (cid:47) K ( V Γ A ( X ))is a push-out. By Property (3) of algebraic K -theory we have K ( V Γ A ( Y )) (cid:39) K A X Γ ( Y ).This proves excision. Remark 8.16.
Let ( X, B , C ) be a Γ-bornological coarse space and let Y be a Γ-invariantsubspace of X with the property that U [ Y ] = Y for every U in C . Then ( Y, X \ Y ) is acoarsely excisive pair. Inspecting the proof of Proposition 8.15, we obtain the followingcommutative diagram:0 (cid:47) (cid:47) (cid:15) (cid:15) K ( V Γ A ( Y )) (cid:39) (cid:47) (cid:47) (cid:15) (cid:15) K ( V Γ A ( Y )) (cid:39) (cid:15) (cid:15) K ( V Γ A ( X \ Y )) (cid:47) (cid:47) K ( V Γ A ( X )) (cid:47) (cid:47) K ( V Γ A ( X ) / V Γ A ( X \ Y ))In addition, we observe that an inverse to the right vertical equivalence is induced by thefunctor Ψ which is given by Ψ( A, ρ ) = ( A | Y , ρ | Y ). Since this functor is already well-definedas a functor Ψ : V Γ A ( X ) → V Γ A ( Y ), we see that the projection map K ( V Γ A ( X )) (cid:39) K ( V Γ A ( X \ Y )) ⊕ K ( V Γ A ( Y )) → K ( V Γ A ( Y ))arising from excision coincides with K (Ψ). (cid:7) Theorem 8.9 follows now by combining Lemma 8.10, Corollary 8.12, Lemma 8.13 andProposition 8.15. In the remainder of this section, we establish some additional propertiesof the equivariant coarse homology theory K A X Γ . For the next two propositions recallthe notions of continuity (Definition 5.15) and strongness (Definition 4.19). Proposition 8.17.
The equivariant coarse homology theory K A X Γ is continuous.Proof. Let (
A, ρ ) be an equivariant X -controlled object. Set S := { x ∈ X | A ( { x } ) (cid:29) } .By definition, we have S ∩ B = σ ( B ), where σ is the support function of A . Hence, S is alocally finite subset of X , so ( A, ρ ) lies in the full subcategory V Γ A ( S ) of V Γ A ( X ). Thisshows that V Γ A ( X ) = (cid:83) S ⊆ X locally finite V Γ A ( S ). By Property (3) of the algebraic K -theoryfunctor, it follows that K A X Γ is continuous. Proposition 8.18.
The equivariant coarse homology theory K A X Γ is strong.Proof. Let X be a Γ-bornological coarse space with weak flasqueness implemented by φ : X → X . As in the proof of Lemma 8.13, the functor Σ : V Γ A ( X ) → V Γ A ( X ) given byΣ := (cid:77) n ∈ N ( φ n ) ∗
58s well-defined. By assumption, we have id K A X Γ ( X ) = K A X Γ ( φ ). Now apply Property (4)of algebraic K -theory to deduce thatid K A X Γ ( X ) + K (Σ) (cid:39) id K A X Γ ( X ) + K A X Γ ( φ ) ◦ K (Σ) (cid:39) K (id V Γ A ( X ) ⊕ φ ∗ ◦ Σ) (cid:39) K (Σ) , so id K A X Γ ( X ) (cid:39) Proposition 8.19.
The equivariant coarse homology theory K A X Γ is strongly additive.Proof. Let ( X i ) i ∈ I be a family of Γ-bornological coarse spaces. The functorsΦ j : V Γ A ( free (cid:71) i ∈ I X i ) → V Γ A ( X j )sending a (cid:70) free i ∈ I X i -controlled A -object ( A, ρ ) to ( A | X j , ρ | X j ) for j in I assemble to a functorΦ : V Γ A ( free (cid:71) i ∈ I X i ) → (cid:89) i ∈ I V Γ A ( X i ) . For j in I , let ι j : X j → (cid:70) free i ∈ I X i denote the inclusion. We claim that the functorΨ : (cid:89) i ∈ I V Γ A ( X i ) → V Γ A ( free (cid:71) i ∈ I X i )which sends a sequence ( A i , ρ i ) i to (cid:76) i ∈ I ( ι i ) ∗ ( A i , ρ i ) is well-defined (up to canonicalisomorphism). We only have to check that the direct sum exists. This follows from thefact that for every bounded subset B of (cid:70) free i ∈ I X i the subset { i ∈ I | B ∩ X i (cid:54) = ∅} is finite,and that B ∩ X i is bounded for all i in I .Clearly, Ψ ◦ Φ is isomorphic to the identity. The composition Ψ ◦ Φ is also isomorphic tothe identity since (
A, ρ ) ∼ = (cid:76) i ∈ I ( A | X i , ρ | X i ) for all objects ( A, ρ ).Using Properties (2) and (6), we conclude that K ( V Γ A ( free (cid:71) i ∈ I X i )) K (Φ) −−−→ K ( (cid:89) i ∈ I V Γ A ( X i )) (cid:39) −→ (cid:89) i ∈ I K ( V Γ A ( X i ))is an equivalence. Note that the j -th component of this equivalence is given by the map K (Φ j ). Now apply Remark 8.16 to see that K (Φ j ) agrees with the projection map comingfrom excision. 59 .4. Calculations (Γ /H ) min,min Let H be a subgroup of Γ. Let A be an additive category with trivial Γ-action. Lemma 8.20.
We have an equivalence K A X Γ ((Γ /H ) min,min ) (cid:39) K ( Fun ( BH, A )) . Proof.
In view of the Property 2 of the K -theory functor it suffices to construct anequivalence of additive categoriesΦ : V Γ A ((Γ /H ) min,min ) → Fun ( BH, A ) . This functor sends an object (
A, ρ ) of V Γ A ((Γ /H ) min,min ) to the functor sending γ to ρ ( γ )( { eH } ) : A ( { eH } ) → A ( { eH } ).Furthermore, the functor Φ sends a morphism f : ( A, ρ ) → ( A (cid:48) , ρ (cid:48) )in V Γ A ((Γ /H ) min,min ) to the transformation f ( { eH } ) : A ( { eH } ) → A (cid:48) ( { eH } ).In order to define an inverse functor we choose a section s : G/H → G of the projection G → G/H . Then Ψ :
Fun ( BH, A ) → V Γ A ((Γ /H ) min,min )sends a functor F : BH → A to the following object ( A, ρ ) of V Γ A ((Γ /H ) min,min ): wechoose A ( B ) = (cid:76) b ∈ B F ( ∗ ) and ρ is defined on an element γ of Γ such that ρ ( γ )( B ) isthe morphism (cid:76) b ∈ B F ( ∗ ) → (cid:76) b ∈ γ − ( B ) F ( ∗ ) sending the summand with index b = gH tothe summand with index γ − gH via F ( h ), where h is the element of H which is uniquelydetermined by the equation γ − s ( gH ) = s ( γ − gH ) h .It is an easy exercise to construct the isomorphisms from the compositions Ψ ◦ Φ andΦ ◦ Ψ to the respective identity functors. X min,max ⊗ Γ can,min We consider the group Γ as a Γ-bornological coarse space Γ can,min . In applications ofcoarse homotopy theory to proofs of the Farrell–Jones conjecture the coarse algebraic K -homology K A X Γ twisted by Yo s (Γ can,min ) plays an important role. Therefore it isrelevant to calculate the spectra K A X ΓYo s (Γ can,min ) ((Γ /H ) min,max ) (cid:39) K A X Γ ((Γ /H ) min,max ⊗ Γ can,min ) . More generally, we will replace Γ /H by any Γ-set X .60 efinition 8.21 ([BR07, Def. 2.1]) . Let A be an additive category with a Γ-action and let X be a Γ-set. We define a new additive category denoted A ∗ Γ X as follows. An object A in A ∗ Γ X is a family A = ( A x ) x ∈ X of objects in A where we require that { x ∈ X | A x (cid:54) = 0 } isa finite set. A morphism φ : A → B is a collection of morphisms φ = ( φ x,g ) ( x,g ) ∈ X × Γ , where φ x,g : A x → g ( B g − x ) is a morphism in A . We require that the set of pairs ( x, g ) in X × Γwith φ x,g (cid:54) = 0 is finite. Addition of morphisms is defined componentwise. Composition ofmorphisms is defined as the convolution product. (cid:7) Remark 8.22.
In [BR07, Def. 2.1] additive categories with right Γ-action are used. Forus it is more convenient to consider left Γ-actions. (cid:7)
Let H be a subgroup of Γ. Definition 8.23.
We will denote A ∗ Γ (Γ /H ) by A [ H ]. (cid:7) If A is the category of finitely generated, free R -modules for some ring R , then A [ H ] isequivalent to the category of finitely generated, free R [ H ]-modules.The following calculation closely follows Bartels–Farrell–Jones–Reich [BFJR04, Sec. 6.1and Proof of Prop. 6.2]. Proposition 8.24.
For every Γ -set X we have an equivalence V Γ A ( X min,max ⊗ Γ can,min ) (cid:39) A ∗ Γ X .
Proof.
The desired equivalence is given by an exact functorΦ : V Γ A ( X min,max ⊗ Γ can,min ) → A ∗ Γ X .
We define Φ as follows:1. For an object (
A, ρ ) in V Γ A ( X min,max ⊗ Γ can,min ), we define Φ( A, ρ ) x as A ( { ( x, } ).2. For a morphism f : ( A, ρ ) → ( A (cid:48) , ρ (cid:48) ) we define Φ( f ) x,g as the composition A ( { x, } ) f −→ A (cid:48) ( { x } × F ) p g −→ A (cid:48) ( { x, g } ) ρ (cid:48) ( g ) −−→ gA (cid:48) ( { g − x, } ) , where F is a finite subset of Γ containing g and p g is the projection arising from theidentification (cid:76) f ∈ F A ( { x, f } ) ∼ = −→ A ( { x } × F ).Note that Φ( f ) x,g is independent of the choice of F .We will first show that Φ is fully faithful. A morphism f : ( A, ρ ) → ( A (cid:48) , ρ (cid:48) ) is determined byits values on A ( { x, } ) by equivariance. Since X min,max has the minimal coarse structure,the family (Φ( f ) x,g ) ( x,g ) ∈ X × Γ determines f . Hence Φ is faithful.Since the Γ-action on X min,max ⊗ Γ can,min is free, for every finite subset F of Γ and everyfamily of morphisms A ( { x, } ) → A (cid:48) ( { x } × F ) indexed by points x in X there exists aunique equivariant extension to a morphism f : ( A, ρ ) → ( A, ρ (cid:48) ). Let φ : Φ( A, ρ ) → Φ( A (cid:48) , ρ (cid:48) )61e any morphism in A ∗ Γ X . Let F := { g ∈ Γ | ∃ x ∈ X : φ g,x (cid:54) = 0 } , then F is a finitesubset of Γ. The family of morphisms (cid:32) ( A ( { x, } ) (cid:76) g ∈ F ρ (cid:48) g − ◦ φ x,g −−−−−−−−→ (cid:77) g ∈ F A (cid:48) ( { x, g } ) ∼ = −→ A (cid:48) ( { x } × F ) (cid:33) x ∈ X extends to a morphism f : ( A, ρ ) → ( A (cid:48) , ρ (cid:48) ) with Φ( f ) = φ . This shows that Φ is fullyfaithful.We now show that Φ is essentially surjective. Every finitely supported family ( A x ) x ∈ X ofobjects of A extends essentially uniquely to an equivariant object ( A, ρ ) with A ( { x, } ) = A x for all x in X . This uses the choice of finite sums of the objects A x .Let X be a Γ-set. Corollary 8.25.
We have an equivalence K A X Γ ( X min,max ⊗ Γ can,min ) (cid:39) K ( A ∗ Γ X ) . (Γ /H ) min,min ⊗ Γ ? ,max . Let H be a subgroup of Γ and let A be an additive category with Γ-action. Lemma 8.26. If H is finite, then K A X Γ ((Γ /H ) min,min ⊗ Γ max,max ) (cid:39) K A X Γ ((Γ /H ) min,min ⊗ Γ can,max ) (cid:39) K ( A [ H ]) . Otherwise K A X Γ ((Γ /H ) min,min ⊗ Γ max,max ) (cid:39) K A X Γ ((Γ /H ) min,min ⊗ Γ can,max ) (cid:39) . Proof.
We argue similarly as in the proof of Proposition 8.24 for X = (Γ /H ) min,min . If H is finite, then the set F appearing in the proof is still finite, but for a different reason.If H is infinite, then there are no non-trivial Γ-invariant (Γ /H ) min,min ⊗ Γ ? ,max controlledmodules (this does not depend on the coarse structures). Let H be a subgroup of Γ. Theorem 8.27.
There is an equivalence of H -equivariant coarse homology theories ind Γ H : K A X H (cid:39) −→ K A X Γ ◦ Ind Γ H .
62n the proof we will construct a equivalence in the other direction. We state the theoremin this form since this is the more common direction.
Proof.
Let ( X, B , C ) be a bornological coarse space. Recall from the Remark 6.4 that thebornological coarse space Ind Γ H X is given by the set Γ × H X with bornology generated bythe subsets { g } × B for B in B and coarse structure generated by the entourages diag Γ × U for U in C .Note that H × H X (cid:39) X is an H -invariant coarse component of Ind Γ H X . Hence, restrictingan object ( A, ρ ) of V Γ A (Ind Γ H X ) to ( A | X , ρ X ) yields a functor V Γ A (Ind Γ H X ) → V H A ( X ) . Similarly to the proof of Lemma 8.20, one checks that this functor is an equivalence.Let H be a subgroup of Γ. Sending a Γ-equivariant X -controlled object ( A, ρ ) to theobject ( A, { ρ ( h ) } h ∈ H ) yields a natural transformationres Γ H : K A X Γ → K A X H ◦ Res Γ H . (8.1) Let again A be an additive category with a Γ-action and ( X, B , C ) be a Γ-bornologicalcoarse space. Intuitively, an equivariant X -controlled A -object ( A, ρ ) is some (infinite) sumof objects in A parametrized by points in X together with an action of Γ. One may wantto keep track of the “global” object associated to an X -controlled object explicitly. Thepurpose of this section is to give an alternative definition of V Γ A ( X ) which accomplishesprecisely this, and discuss a variation of this definition which leads to an example of anon-continuous coarse homology theory.Since an equivariant X -controlled A -object usually involves an infinite number of objectsin A , we need to enlarge our coefficient category appropriately. Therefore, let A → (cid:98) A be afully faithful and Γ-equivariant embedding of A into an additive category (cid:98) A with Γ-actionwhich admits sufficiently large direct sums. The sum completion of A is a canonical choicefor (cid:98) A .For an object A in (cid:98) A , let Π( A ) denote the set of idempotents on A whose image splits offas a direct sum and is isomorphic to an object in A . Definition 8.28. A global equivariant X -controlled A -object is a triple ( A, φ, ρ ) consistingof 1. an object A in (cid:98) A ;2. a function φ : B → Π( A );3. a morphism ρ ( γ ) : A → γA for every element γ in Γ;63uch that the following conditions are satisfied:1. The function φ satisfies the following relations:a) φ ( ∅ ) = 0;b) φ ( B ∪ B ) = φ ( B ) + φ ( B ) − φ ( B ∩ B );c) φ ( B ∩ B ) = φ ( B ) ◦ φ ( B ).2. For every bounded subset B of X , there exists some finite subset F of B such that φ ( B ) = φ ( F ).3. For all pairs of elements γ , γ (cid:48) in Γ, we have ρ ( γ (cid:48) γ ) = γρ ( γ (cid:48) ) ◦ ρ ( γ ).4. For all elements γ of Γ and bounded subsets B of X , we have the equality φ ( γ − B ) = ρ ( γ ) − ◦ γφ ( B ) ◦ ρ ( γ ) . (cid:7) In contrast to the data specifying an equivariant X -controlled object, this definition doesnot fix a chosen image for each of the idempotents φ ( B ). Definition 8.29.
A morphism f : ( A, φ, ρ ) → ( A (cid:48) , φ (cid:48) , ρ (cid:48) ) of global equivariant X -controlled A -objects is a morphism f : A → A (cid:48) in (cid:98) A satisfying the following conditions:1. f is equivariant in the sense that ( γf ) ◦ ρ ( γ ) = ρ (cid:48) ( γ ) ◦ f ;2. f is controlled in the sense that the set (cid:92) { U ⊆ X × X | ∀ B, B (cid:48) ∈ B : U [ B ] ∩ B (cid:48) = ∅ ⇒ φ (cid:48) ( B (cid:48) ) f φ ( B ) = 0 } is an entourage of X . (cid:7) Morphisms of global equivariant X -controlled A -objects can be composed. We denote theresulting category by V Γ A ⊆ (cid:98) A ( X ). Similar to the discussion in Section 8.2, one shows that V Γ A ⊆ (cid:98) A ( X ) is additive. If f : X → X (cid:48) is a morphism of Γ-bornological coarse spaces, thenwe define a functor f ∗ : V Γ A ⊆ (cid:98) A ( X ) → V Γ A ⊆ (cid:98) A ( X (cid:48) )which sends ( A, φ, ρ ) to (
A, f ∗ φ, ρ ), were f ∗ φ ( B (cid:48) ) := φ ( f − ( B (cid:48) )) for all B (cid:48) in B (cid:48) . Fur-thermore, the construction is functorial with respect to commutative squares of additivefunctors A (cid:47) (cid:47) Φ (cid:15) (cid:15) (cid:98) A (cid:98) Φ (cid:15) (cid:15) A (cid:48) (cid:47) (cid:47) (cid:98) A (cid:48) in which the horizontal arrows are fully faithful and Γ-equivariant embeddings.We have already indicated how objects in V Γ A ( X ) correspond to objects in V Γ A ⊆ (cid:98) A ( X )and vice versa, namely by summing up all values of an equivariant X -controlled objectand choosing images of idempotents, respectively. In fact, it is not difficult to show thefollowing. 64 roposition 8.30. There is a zig-zag of equivalence between the functors V Γ A and V Γ A ⊆ (cid:98) A from Γ BornCoarse to Add . Example 8.31.
We now provide a modification of the definition of V Γ A ⊆ (cid:98) A which leads toa non-continuous coarse homology theory.Let ( A, φ, ρ ) be a global equivariant X -controlled A -object. Condition (2) implies, inthe presence of Condition (1), that the object ( A, φ ) is essentially determined by itsrestriction to the poset of finite subsets of X , and that the support of ( A, φ ), i.e. the set { x ∈ X | φ ( { x } (cid:54) = 0 } , is a locally finite subset of X .Dropping Condition (2), we obtain another additive category V Γ A , Ψ ( X ) which is alsofunctorial in Γ-bornological coarse spaces. Taking non-connective algebraic K -theory ofthis category also gives rise to an equivariant coarse homology theory K A X ΓΨ since theproofs of Lemma 8.10, Lemma 8.11, Lemma 8.13 and Proposition 8.15 go through withoutchange. However, the proof of continuity (Proposition 8.17) does not apply to K A X ΓΨ since the condition ensuring that the support of an object is locally finite has been omitted.In fact, the following example shows that the coarse homology theory K A X Ψ is notcontinuous: Suppose that there exists an object A in A whose class in K is non-trivial.Consider the bornological coarse space N min,max . Choose an ultrafilter F on N and definethe function φ : P ( N ) → Π( A ) by φ ( B ) := (cid:40) id A B ∈ F , B / ∈ F . Then (
A, φ ) is a global N min,max -controlled A -object since F is an ultrafilter. Moreover,the morphism of bornological coarse spaces N min,max → ∗ induces a homomorphism π K A X Ψ ( N min,max ) → π K A X Ψ ( ∗ ) ∼ = K ( A )which maps the class [( A, φ )] to [ A ] (cid:54) = 0.On the other hand, locally finite subsets of N min,max are precisely the finite subsets of N .Since each finite subset F is a union of coarse components of N min,max , the pair of subsets( F, N \ F ) is a complementary pair. From excision, we obtain a direct sum decomposition K A X Ψ ( N min,max ) (cid:39) K A X Ψ ( F min,max ) ⊕ K A X Ψ (( N \ F ) min,max ) , in which the projection K A X Ψ ( N min,max ) → K A X Ψ ( F min,max ) is induced by the functorrestricting N min,max -controlled objects to F (cf. Remark 8.16). Since the restriction of( A, φ ) to any finite subset of N is the zero object, we conclude that the class [( A, φ )] doesnot lie in the image of the comparison mapcolim F ⊆ N finite π K A X Ψ ( F min,max ) → π K A X Ψ ( N min,max ) . In particular, K A X Ψ is not continuous. (cid:7) art III.Cones and assembly maps
9. Cones
In this section we will introduce the cone and the ‘cone at infinity’ functors, and discusstheir properties. The cone at infinity is later used in Section 10.3 to define the universalassembly map, and in Section 10.4 to transform coarse homology theories into topologicalhomology theories. In conjunction with the Rips complex construction the cone at infinitywill be used in Section 11.3 to construct the universal coarse assembly map.The first three Sections 9.1–9.3 are technical preparations. The cone functor is then definedand discussed in Section 9.4 and Section 9.5. Γ -uniform bornological coarse spaces In this section we introduce the category Γ
UBC of Γ-uniform bornological coarse spaces.Its objects are Γ-bornological coarse spaces with an additional Γ-uniform structure. Theadditional datum of a Γ-uniform structure is needed in order to define hybrid structures,see Section 9.2.We start with recalling some basics on uniform spaces. Let X be a set and T be a subsetof P ( X × X ), the power set of X × X . Definition 9.1.
The set T is a uniform structure if it is non-empty, closed under compo-sition, inversion, supersets, finite intersection, every element of T contains the diagonal of X , and if for every U in T there exists V in T such that V ◦ V ⊆ U . (cid:7) Remark 9.2.
Note that any subset S of P ( X × X ) with the property that for every U in S there exists V in X such that V ◦ V ⊆ U generates a uniform structure on X by takingthe closure of S under composition, inversion, supersets and finite intersection. (cid:7) The elements of T are called uniform entourages . We will consider T as a filtered partiallyordered set whose order relation is the opposite of the inclusion relation. Definition 9.3. A uniform space is a pair ( X, T ) of a set X together with a uniformstructure T . (cid:7) Let ( X, T ) and ( X (cid:48) , T (cid:48) ) be uniform spaces and f : X → X (cid:48) be a map of sets. Definition 9.4.
The map f is called a uniform map if for every uniform entourage U (cid:48) of X (cid:48) we have ( f × f ) − ( U (cid:48) ) ∈ T . (cid:7)
66f a group Γ acts on a uniform space ( X, T ), then it acts on the set of uniform entourages T .We let T Γ denote the subset of T of Γ-invariant uniform entourages. Definition 9.5.
A Γ -uniform space is a uniform space ( X, T ) with an action of Γ byautomorphisms such that T Γ is cofinal in T . (cid:7) A uniform structure T on a Γ-set X such that X is a Γ-uniform space will be called aΓ -uniform structure . Example 9.6.
The uniform structure T d of a metric space ( X, d ) is generated by theuniform entourages U r := { ( x, y ) ∈ X × X | d ( x, y ) ≤ r } (9.1)for all r in (0 , ∞ ).If Γ acts isometrically on a metric space ( X, d ), then the associated uniform space X u,d := ( X, T d ) is a Γ-uniform space.If the metric is implicitly clear, then we will also write X u instead of X u,d .A uniformly continuous map between metric spaces induces a uniform map between theassociated uniform spaces.The standard metric turns R into a uniform space R u . The action of Z on R by dilatations( n, x ) (cid:55)→ n x is an action on R u by automorphisms of uniform spaces, but R u is not a Z -uniform space. (cid:7) Let Γ U be the category of Γ-uniform spaces and uniform equivariant maps. Example 9.7.
Let K be a simplicial complex with a simplicial Γ-action. On K we considerthe path quasi-metric induced by the spherical metric on the simplices. This quasi-metricis preserved by Γ and the associated uniform structure is a Γ-uniform structure. Thereforea simplicial complex with a simplicial Γ-action gives rise to a Γ-uniform space K u .If K → K (cid:48) is an equivariant simplicial map, then it is a uniform map K u → K (cid:48) u betweenthe Γ-uniform spaces. (cid:7) We now consider the combination of uniform and bornological coarse structures. Weconsider a Γ-set X with a Γ-coarse structure C and a Γ-uniform structure T . Definition 9.8.
We say that C and T are compatible if C Γ ∩ T Γ (cid:54) = ∅ . (cid:7) In words, the coarse and the uniform structures are compatible if there exists an invariantentourage which is both a coarse entourage and a uniform entourage.
Definition 9.9.
We define the category Γ
UBC of Γ-uniform bornological coarse spacesas follows:1. The objects of Γ
UBC are Γ-bornological coarse spaces with an additional compatibleΓ-uniform structure. 67. The morphisms of Γ
UBC are morphisms of Γ-bornological coarse spaces which arein addition uniform. (cid:7)
Example 9.10.
Let (
X, d ) be a quasi-metric space with an action of Γ by isometries anda Γ-invariant bornology B . Assume that the metric and the bornology are compatible inthe sense that for every r in (0 , ∞ ) and B in B we have U r [ B ] ∈ B , where U r is as in (9.1).Then we get a Γ-uniform bornological coarse space X du with the following structures:1. The coarse structure is generated by the coarse entourages U r for all r in (0 , ∞ ).2. The uniform structure is generated by the uniform entourages U r for all r in (0 , ∞ ).3. The bornology is B .If ( X (cid:48) , d (cid:48) ) is a second quasi-metric space with isometric Γ-action and f : X → X (cid:48) is aproper (this refers to the bornologies), Γ-equivariant contraction, then f : X du → X (cid:48) du is amorphism of Γ-uniform bornological coarse spaces. (cid:7) Example 9.11.
Let K be a Γ-simplicial complex equipped with the spherical quasi-metric.Then we can equip K with the bornology of metrically bounded subsets and obtain aΓ-uniform bornological coarse space K du . Alternatively we can equip it with the maximalbornology and get the Γ-uniform bornological coarse space K du,max .A morphism of Γ-simplicial complexes f : K → K (cid:48) always induces the two morphisms f : K du,max → K (cid:48) du,max and K du,max → K (cid:48) du of Γ-uniform bornological coarse spaces. If f isproper (in the sense that preimages of simplices are finite complexes), then it also inducesa morphism f : K du → K (cid:48) du . (cid:7) Example 9.12.
For a Γ-uniform bornological coarse space X let F T ( X ) denote theunderlying Γ-bornological coarse space obtained by forgetting the datum of the uniformstructure. Let X and Y be Γ-uniform bornological coarse spaces. Then we define theΓ-uniform bornological coarse space X ⊗ Y such that F T ( X ⊗ Y ) = F T ( X ) ⊗ F T ( Y ) andthe Γ-uniform structure on X ⊗ Y is generated by the products U × V for all pairs ofuniform entourages U of X and V of Y . (cid:7) In this section we will define hybrid coarse structures, which will feature in the definitionof cones in Section 9.4.We consider a Γ-uniform bornological coarse space X with coarse structure C , bornology B ,and uniform structure T . Let furthermore a Γ-invariant big family Y = ( Y i ) i ∈ I be given. Definition 9.13.
The pair ( X, Y ) is called hybrid data . (cid:7) In this situation we can define the hybrid coarse structure C h as follows.Note that P ( X × X ) Γ is a filtered poset with the opposite of the inclusion relation. Weconsider a function φ : I → P ( X × X ) Γ , i.e., an order-preserving map.68 efinition 9.14. The function φ is called T Γ -admissible, if for every U in T Γ there exists i in I such that φ ( i ) ⊆ U . (cid:7) Given a T Γ -admissible function φ : I → P ( X × X ) Γ we define the entourage U φ := (cid:8) ( x, y ) ∈ X × X | (cid:0) ∀ i ∈ I | ( x, y ) ∈ Y i × Y i or ( x, y ) ∈ φ ( i ) (cid:1)(cid:9) . Note that U φ is Γ-invariant. Definition 9.15.
The hybrid coarse structure C h is the coarse structure generated by theentourages U ∩ U φ for all U in C Γ and T Γ -admissible functions φ : I → P ( X × X ) Γ . (cid:7) Definition 9.16.
The hybrid space X h is defined to be the Γ-bornological coarse spacewith underlying set X , the hybrid coarse structure C h and the bornological structure B . (cid:7) A morphism of hybrid data f : ( X, Y ) → ( X (cid:48) , Y (cid:48) ) is a morphism of Γ-uniform bornologicalcoarse spaces which is compatible with the big families Y = ( Y i ) i ∈ I and Y (cid:48) = ( Y (cid:48) i (cid:48) ) i (cid:48) ∈ I (cid:48) inthe sense that for every i in I there exists i (cid:48) in I (cid:48) such that f ( Y i ) ⊆ Y (cid:48) i (cid:48) . Lemma 9.17. If f is a morphism of hybrid data, then the underlying map of sets is amorphism f : X h → X (cid:48) h of Γ -bornological coarse spaces.Proof. [BE16, Lem. 5.15]. Remark 9.18.
One could set up a category of hybrid data and understand the constructionof the hybrid structure as a functor from hybrid data to Γ-bornological coarse spaces. (cid:7)
In this section we discuss the Decomposition Theorem and the Homotopy Theorem forhybrid coarse structures. These two theorems constitute important technical results whichare needed to prove crucial properties of the cone functor in Section 9.5.Let
A, B be Γ-invariant subsets of a Γ-uniform space Y with uniform structure T . For anentourage U of Y we set T Γ ⊆ U := { V ∈ T Γ | V ⊆ U } . Definition 9.19.
The pair (
A, B ) is an equivariant uniform decomposition if1. Y = A ∪ B , and2. there is an invariant uniform entourage U of Y and a function s : P ( Y × Y ) Γ ⊆ U → P ( Y × Y ) Γ such that for every W in P ( Y × Y ) Γ ⊆ U we have the inclusion W [ A ] ∩ W [ B ] ⊆ s ( W )[ A ∩ B ]and the restriction s |T Γ ⊆ U is T Γ -admissible. (cid:7) s |T Γ ⊆ U is T Γ -admissible if for every entourage V in T Γ there is an entourage W in T Γ ⊆ U such that s ( W ) ⊆ V . Example 9.20.
Assume that K is a Γ-simplicial complex, and A and B are Γ-invariantsubcomplexes such that K = A ∪ B . Then ( A, B ) is an equivariant uniform decompositionof K u . This follows from [BE16, Ex. 5.19]. (cid:7) Let Y be a Γ-uniform bornological coarse space with an invariant big family Y = ( Y i ) i ∈ I .We further assume that ( A, B ) is an equivariant uniform decomposition of Y . We let Y h denote the associated bornological coarse space with the hybrid structure.We write A h for the Γ-bornological coarse space obtained from the Γ-uniform bornologicalcoarse structure on A induced from Y by first restricting the hybrid data to A and thenforming the hybrid structure. By A Y h we denote the Γ-bornological coarse space obtainedby restricting the structures of Y h to the subset A . It was shown in [BE16, Lem. 5.17] that A h = A Y h . Definition 9.21.
A Γ-uniform space ( Y, T ) is called Hausdorff, if (cid:84) U ∈T U = diag Y . (cid:7) Theorem 9.22. (Decomposition Theorem) If I = N and Y is Hausdorff, then the followingsquare in Γ Sp X is cocartesian: Yo s (( A ∩ B ) h , A ∩ B ∩ Y ) (cid:47) (cid:47) (cid:15) (cid:15) Yo s ( A h , A ∩ Y ) (cid:15) (cid:15) Yo s ( B h , B ∩ Y ) (cid:47) (cid:47) Yo s ( Y h , Y ) (9.2) Proof.
The proof of [BE16, Thm. 5.20] goes through word-for-word. One just works withinvariant entourages or invariant uniform neighbourhoods everywhere.We consider a Γ-uniform bornological coarse space Y with an invariant big family Y =( Y n ) n ∈ N . We consider the unit interval [0 ,
1] with the trivial Γ-action as a Γ-uniformbornological coarse space [0 , du with the structures induced from the metric. On thetensor product [0 , du ⊗ Y (Example 9.12) we consider the big family ([0 , × Y n ) n ∈ N . Let B denote the bornology of Y . Theorem 9.23 (Homotopy Theorem) . Assume that for every B in B there exists n in N such that B ⊆ Y n . Then the projection induces an equivalence Yo s (([0 , du ⊗ Y ) h ) → Yo s ( Y h ) . Proof.
The proof of [BE16, Thm. 5.25] goes through in the present equivariant case.70 .4. The cone functor
In this section we define the cone functor O : Γ UBC → Γ BornCoarse (9.3)and prove that decompositions of the spaces which are simultaneously uniformly andcoarsely excisive lead to corresponding coarsely excisive decompositions of the cones.We consider the metric space [0 , ∞ ) with the metric induced from the inclusion into R and the trivial Γ-action. We get a Γ-uniform bornological coarse space [0 , ∞ ) du . For aΓ-uniform bornological coarse space Y we form the Γ-uniform bornological coarse space[0 , ∞ ) du ⊗ Y (Example 9.12). This Γ-uniform bornological coarse space has a canonical big family givenby Y ( Y ) := ([0 , n ] × Y ) n ∈ N . (9.4)A morphism f : Y → Y (cid:48) of Γ-uniform bornological coarse spaces induces a morphism ofhybrid data ([0 , ∞ ) du ⊗ Y, Y ( Y )) → ([0 , ∞ ) du ⊗ Y (cid:48) , Y ( Y (cid:48) ))given by the map id [0 , ∞ ) × f on the underlying sets. Definition 9.24.
The cone functor (9.3) is defined such that it sends the Γ-uniformbornological coarse space Y to the Γ-bornological coarse space O ( Y ) := ([0 , ∞ ) du ⊗ Y ) h and a morphism f : Y → Y (cid:48) of Γ-uniform bornological coarse spaces to the morphism O ( f ) : O ( Y ) → O ( Y (cid:48) )of Γ-bornological coarse spaces given by the map id [0 , ∞ ) × f of the underlying sets. (cid:7) Example 9.25.
If the Γ-uniform bornological coarse space Y is discrete as a uniform andas a coarse space, then O ( Y ) is flasque: Flasqueness of O ( Y ) can be implemented by themap ( t, y ) (cid:55)→ ( t + t +1 , y ).If the uniform structure of Y is discrete, but the coarse structure of Y is strictly largerthan the discrete one, then the above morphism f does not implement flasqueness of O ( Y ),because Condition 2 in Definition 3.8 is violated. (cid:7) Let X and Y be Γ-uniform bornological coarse spaces. Recall that F T is the functor fromΓ-uniform bornological coarse spaces to Γ-bornological coarse spaces which forgets theuniform structure. Lemma 9.26. If Y is discrete as a coarse space, then O ( X ) ⊗ F T ( Y ) ∼ = O ( X ⊗ Y ) . roof. Immediate from the definitions.Let Y be a Γ-uniform bornological coarse space with coarse and uniform structures C , T ,and let A, B be Γ-invariant subsets of Y . Lemma 9.27. If ( A, B ) is an equivariant uniform (Definition 9.19) and coarsely excisivedecomposition of Y , then ([0 , ∞ ) × A, [0 , ∞ ) × B ) is a coarsely excisive pair on O ( Y ) .Proof. We have ([0 , ∞ ) × A ) ∪ ([0 , ∞ ) × B ) = [0 , ∞ ) × Y .Let s and U be as in Definition 9.19, let φ : N → P ( Y × Y ) Γ be a T Γ -admissible functionand let κ : [0 , ∞ ) → [0 , ∞ ) be monotoneously decreasing such that lim u →∞ κ ( u ) = 0. Thepair ψ := ( φ, κ ) determines the invariant entourage U ψ := { (( a, x ) , ( b, y )) ∈ ([0 , ∞ ) × Y ) × | | a − b | ≤ κ (max { a, b } ) & ( x, y ) ∈ φ ( (cid:100) a (cid:101) ) ∩ φ ( (cid:100) b (cid:101) ) } . For W in C Γ and r in (0 , ∞ ) we consider the entourage W r := U r × W of [0 , ∞ ) d ⊗ Y . Theentourages of the form U ψ ∩ W r for all ψ as above, r in (0 , ∞ ) and W in C Γ are cofinal inthe hybrid coarse structure of O ( Y ).We now fix ψ , W and r as above. We must show that there exist r (cid:48) in (0 , ∞ ), W (cid:48) in C Γ ,and ψ (cid:48) such that( U ψ ∩ W r )[[0 , ∞ ) × A ] ∩ ( U ψ ∩ W r )[[0 , ∞ ) × B ] ⊆ ( U ψ (cid:48) ∩ W (cid:48) r (cid:48) )[[0 , ∞ ) × ( A ∩ B )] . (9.5)Using coarse excisiveness of ( A, B ) we can choose an invariant entourage W (cid:48) of X suchthat W [ A ] ∩ W [ B ] ⊆ W (cid:48) [ A ∩ B ]. We further set r (cid:48) := r . Then W r [[0 , ∞ ) × A ] ∩ W r [[0 , ∞ ) × B ] ⊆ W (cid:48) r [[0 , ∞ ) × ( A ∩ B )] . (9.6)By T Γ -admissibility of φ there is u in N such that φ ( u ) ⊆ U for all u in N with u ≥ u .We define φ (cid:48) : N → P ( Y × Y ) Γ , φ (cid:48) ( u ) := (cid:26) W (cid:48) u < u s ( φ ( u )) u ≥ u Then φ (cid:48) is T Γ -admissible. We further set ψ (cid:48) := ( φ (cid:48) , κ ). We claim that( U ψ ∩ W r )[[0 , ∞ ) × A ] ∩ ( U ψ ∩ W r )[[0 , ∞ ) × B ] ⊆ U ψ (cid:48) [[0 , ∞ ) × ( A ∩ B )] . (9.7)Consider a point ( u, z ) in ( U ψ ∩ W r )[[0 , ∞ ) × A ] ∩ ( U ψ ∩ W r )[[0 , ∞ ) × B ]. Then there exists( a, x ) in [0 , ∞ ) × A such that we have | a − u | ≤ κ (max { a, u } ) and ( z, x ) ∈ φ ( (cid:100) u (cid:101) ) ∩ φ ( (cid:100) a (cid:101) ) , and there exist ( b, y ) in [0 , ∞ ) × B such that | b − u | ≤ κ (max { b, u } ) and ( z, y ) ∈ φ ( (cid:100) u (cid:101) ) ∩ φ ( (cid:100) b (cid:101) ) .
72n particular, we have z ∈ ( φ ( (cid:100) u (cid:101) ) ∩ W )[ A ] ∩ ( φ ( (cid:100) u (cid:101) ) ∩ W )[ B ]. If u ≥ u , then we have z ∈ φ (cid:48) ( (cid:100) u (cid:101) )[ A ∩ B ] by the corresponding property of s (see Definition 9.19). Let w in A ∩ B be such that we have ( z, w ) ∈ φ (cid:48) ( (cid:100) u (cid:101) ). Then (( u, z ) , ( u, w )) ∈ U ψ (cid:48) [[0 , ∞ ) × ( A ∩ B )].If u < u , then again (( u, z ) , ( u, w )) ∈ U ψ (cid:48) [[0 , ∞ ) × ( A ∩ B )] because of the choice of W (cid:48) .The relations (9.6) and (9.7) together imply (9.5). Remark 9.28.
In the above proof, in contrast to the general Decomposition Theorem 9.22for hybrid structures, we do not use that Y is Hausdorff. (cid:7) In this section we will define the “cone at infinity” functor. It fits into the cone fibersequence. We discuss invariance under coarsenings and calculate it for discrete spaces.Furthermore, we show that the “cone at infinity” is excisive and homotopy invariant.If Y is a Γ-uniform bornological coarse space Y , then O ( Y ) has a canonical big family Y ( Y ) given by (9.4). Recall the notation (4.8). Definition 9.29.
We define the functor O ∞ : Γ UBC → Γ Sp X by O ∞ ( Y ) := Yo s ( O ( Y ) , Y ( Y )) . (cid:7) Recall that F T is the functor from Γ-uniform bornological coarse spaces to Γ-bornologicalcoarse spaces which forgets the uniform structure. For n in N let ([0 , n ] × Y ) O ( Y ) denotethe Γ-bornological coarse space given by the subset [0 , n ] × Y of O ( Y ) with the inducedstructures. The inclusion F T ( Y ) → ([0 , n ] × Y ) O ( Y ) , y (cid:55)→ (0 , y )is an equivalence of Γ-bornological coarse spaces for every integer n . Hence we have anequivalence Yo s ( Y ) (cid:39) Yo s ( Y ( Y ) O ( Y ) ) . Corollary 9.30.
For every Γ -uniform bornological coarse space Y we have a natural fibersequence Yo s ( F T ( Y )) → Yo s ( O ( Y )) → O ∞ ( Y ) ∂ −→ ΣYo s ( F T ( Y )) (9.8) in Γ Sp X .Proof. The fiber sequence is associated to the pair ( O ( Y ) , Y ( Y )), see Corollary 4.11.1.73et Y be a Γ-uniform bornological coarse space. Then we consider the Γ-bornologicalcoarse space O ( Y ) − obtained from the Γ-uniform bornological coarse space R ⊗ Y bytaking the hybrid coarse structure 9.15 associated to the big family (( −∞ , n ] × Y ) n ∈ N .Note that the subset [0 , ∞ ) × Y of O ( Y ) − with the induced structures is the cone O ( Y ).We then have maps of Γ-bornological coarse spaces F T ( Y ) i −→ O ( Y ) j −→ O ( Y ) − d −→ F T ( R du ⊗ Y ) . The first two maps i and j are the inclusions, and the last map d is given by the identityof the underlying sets. Proposition 9.31.
We have a commutative diagram in Γ Sp X Yo s ( O ( Y )) j (cid:47) (cid:47) Yo s ( O ( Y ) − ) (cid:39) (cid:15) (cid:15) d (cid:47) (cid:47) Yo s ( F T ( R du ⊗ Y )) (cid:39) (cid:15) (cid:15) Yo s ( O ( Y )) (cid:47) (cid:47) O ∞ ( Y ) ∂ (cid:47) (cid:47) ΣYo s ( F T ( Y )) (9.9) Remark 9.32.
This proposition identifies a segment of the cone sequence (9.8) with asequence represented by maps between Γ-bornological coarse spaces. It in particular showsthat the cone O ∞ ( Y ) is represented by the Γ-bornological coarse space O ( Y ) − . (cid:7) Proof of Proposition 9.31.
We consider the diagram of motivic coarse spectraYo s ( F T ( Y )) i (cid:47) (cid:47) (cid:15) (cid:15) Yo s ( O ( Y )) (cid:47) (cid:47) j (cid:15) (cid:15) Yo s ( F T ([0 , ∞ ) ⊗ Y )) (cid:15) (cid:15) Yo s ( F T (( −∞ , ⊗ Y )) (cid:47) (cid:47) Yo s ( O ( Y ) − ) d (cid:47) (cid:47) Yo s ( F T ( R du ⊗ Y )) (9.10)The left and right vertical and the lower left horizontal map are given by the canonicalinclusions. The upper right horizontal map is the identity map of the underlying sets.This diagram commutes since it is obtained by applying Yo s to a commuting diagram ofbornological coarse spaces.The left square in (9.10) is cocartesian since the pair (( −∞ , × Y, O ( Y )) in O ( Y ) − iscoarsely excisive. Furthermore, since (( −∞ , × Y, [0 , ∞ ) × Y ) is coarsely excisive in F T ( R du ⊗ Y ) the outer square is cocartesian. It follows that the right square is cocartesian.Since the upper right and the lower left corners in (9.10) are trivial by flasqueness of therays the diagram is equivalent to the compositionYo s ( F T ( Y )) i (cid:47) (cid:47) (cid:15) (cid:15) Yo s ( O ( Y )) (cid:47) (cid:47) j (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) Yo s ( O ( Y ) − ) d (cid:47) (cid:47) Yo s ( F T ( R du ⊗ Y ))of cocartesian squares. Note that O ∞ ( Y ) is defined as the cofiber of the left upperhorizontal map i . Hence the left square yields the middle vertical equivalence in (9.9).The outer square yields the equivalence Yo s ( F T ( R du ⊗ Y )) (cid:39) ΣYo s ( Y ). The right squarethen identifies d with the boundary map ∂ of the cone sequence.74ext we will observe that O ∞ ( Y ) is essentially independent of the coarse structure on Y .Let Y be a Γ-uniform bornological coarse space with coarse structure C , bornology B anduniform structure T . Let C (cid:48) be a Γ-coarse structure on Y such that C ⊆ C (cid:48) and C (cid:48) is stillcompatible with the bornology. We write Y (cid:48) for the Γ-uniform bornological coarse spaceobtained from Y by replacing the coarse structure C by C (cid:48) . Then the identity map of theunderlying sets is a morphism Y → Y (cid:48) of Γ-uniform bornological coarse spaces. We willcall such a morphism a coarsening .Let Y be a Γ-uniform bornological coarse space. Proposition 9.33. If Y → Y (cid:48) is a coarsening, then the induced map O ∞ ( Y ) → O ∞ ( Y (cid:48) ) is an equivalence.Proof. By definition of O ∞ ( Y ) we have O ∞ ( Y ) (cid:39) colim n ∈ N Yo s (([0 , ∞ ) du ⊗ Y ) h , [0 , n ] × Y ) , (9.11)where the subsets [0 , n ] × Y of ([0 , ∞ ) du ⊗ Y ) h have the induced bornological coarsestructure. By u -continuity of Yo s we have O ∞ ( Y ) (cid:39) colim n ∈ N colim U Yo s (([0 , ∞ ) du ⊗ Y ) U , ([0 , n ] × Y ) U ) , (9.12)where U runs over the Γ-invariant entourages of ([0 , ∞ ) du ⊗ Y ) h . Here for a subset X (cid:48) of X the notation X (cid:48) U denotes the set X (cid:48) with the structures induced from X U , i.e., X (cid:48) U is ashort-hand notation for X (cid:48) X U . For every integer n , there is a cofinal set of entourages U such that the pair ([0 , n ] × Y, [ n, ∞ ) × Y )is coarsely excisive on ([0 , ∞ ) × Y ) U . In fact, this is a coarsely excisive pair for anyentourage U that allows propagation from { n } × Y in the direction of the ray. Since theYoneda functor Yo s is excisive we get the equivalence O ∞ ( Y ) (cid:39) colim n ∈ N colim U Yo s (([ n, ∞ ) × Y ) U , ( { n } × Y ) U ) . (9.13)In general, for a Γ-bornological coarse space X with coarse structure C and an invariantsubset Z we have an equivalencecolim U ∈C Γ Yo s ( Z U ) (cid:39) colim U ∈C Γ Yo s ( Z ( Z × Z ) ∩ U ) (9.14)(here we must not omit the colimit). We insert this into (9.13) and get O ∞ ( Y ) (cid:39) colim n ∈ N colim U Yo s (([ n, ∞ ) × Y ) U n , ( { n } × Y ) U n ) , (9.15)75here we use the abbreviation U n := (([ n, ∞ ) × Y ) × ([ n, ∞ ) × Y )) ∩ U . We can nowinterchange the order of the colimits and get O ∞ ( Y ) (cid:39) colim U colim n ∈ N Yo s (([ n, ∞ ) × Y ) U n , ( { n } × Y ) U n ) . (9.16)We argue now that in this formula we can replace the colimit over the invariant entourages U of ([0 , ∞ ) du ⊗ Y ) h by the colimit over all invariant entourages U (cid:48) of ([0 , ∞ ) du ⊗ Y (cid:48) ) h .To this end we consider the generating entourages U ψ ∩ W (cid:48) of O ( Y (cid:48) ) (see the proof ofLemma 9.27 for notation). Since T and C are compatible, there exists an integer n sufficiently large such that φ ( n ) ∈ C . But then, since φ is monotoneous, we have φ ( x ) ∈ C for every x in [ n , ∞ ). We conclude that for every integer n with n ≥ n we have( U ψ ∩ W (cid:48) ) ∩ (cid:0) ([ n, ∞ ) × Y ) × ([ n, ∞ ) × Y ) (cid:1) ⊆ U ψ ∩ ( W (cid:48) ∩ φ ( n ))and W (cid:48) ∩ φ ( n ) ∈ C .This gives O ∞ ( Y ) (cid:39) colim U (cid:48) colim n ∈ N Yo s (([ n, ∞ ) × Y ) U (cid:48) n , ( { n } × Y ) U (cid:48) n ) , (9.17)where now U (cid:48) runs over the invariant entourages of ([0 , ∞ ) du ⊗ Y (cid:48) ) h . Going the argumentabove backwards with Y replaced by Y (cid:48) we end up with O ∞ ( Y ) (cid:39) colim n ∈ N Yo s (([0 , ∞ ) du ⊗ Y (cid:48) ) h , [0 , n ] × Y ) (cid:39) O ∞ ( Y (cid:48) ) (9.18)and this completes the proof.In the following proposition we use the invariance under coarsening in order to calculatethe value of the O ∞ -functor on Γ-uniform bornological coarse spaces whose underlyinguniform structure is discrete.For a Γ-uniform bornological coarse space X which is discrete as a uniform space let X disc denote the Γ-uniform bornological coarse space obtained by replacing the coarse structureby the discrete coarse structure. Remark 9.34. If X is not discrete as a uniform space, then the discrete coarse structureis not compatible with the uniform structure. (cid:7) Let X be a Γ-uniform bornological coarse space. Proposition 9.35. If X is discrete as a uniform space, then we have an equivalence O ∞ ( X ) (cid:39) ΣYo s ( F T ( X disc )) in Γ Sp X . roof. Since X disc → X is a coarsening, by Proposition 9.33 we have an equivalence O ∞ ( X disc ) (cid:39) −→ O ∞ ( X ) . By Example 9.25 we know that O ( X disc ) is flasque and hence Yo s ( O ( X disc )) (cid:39)
0. The fibersequence obtained in Corollary 9.30 yields an equivalence O ∞ ( X disc ) (cid:39) ΣYo s ( F T ( X disc ))as desired.Next we discuss excision and homotopy invariance for O ∞ .Let Y be a Γ-uniform bornological coarse space and A, B be Γ-invariant subsets of Y . Corollary 9.36. If ( A, B ) is an equivariant uniformly and coarsely excisive decomposition,then the following square in Γ Sp X is cocartesian: O ∞ ( A ∩ B ) (cid:47) (cid:47) (cid:15) (cid:15) O ∞ ( B ) (cid:15) (cid:15) O ∞ ( A ) (cid:47) (cid:47) O ∞ ( Y ) (9.19) Proof.
Since (
A, B ) is coarsely excisive the squareYo s ( F T ( A ) ∩ F T ( B )) (cid:47) (cid:47) (cid:15) (cid:15) Yo s ( F T ( B )) (cid:15) (cid:15) Yo s ( F T ( A )) (cid:47) (cid:47) Yo s ( F T ( Y ))is cocartesian. Furthermore, by Lemma 9.27 the square O ( A ∩ B ) (cid:47) (cid:47) (cid:15) (cid:15) O ( B ) (cid:15) (cid:15) O ( A ) (cid:47) (cid:47) O ( Y )is cocartesian. Now it just remains to use the cone sequence (9.8) in order to concludethat the square (9.19) is cocartesian. Remark 9.37.
If we assume that the underlying uniform space of Y is Hausdorff, thenwe could drop the assumption that ( A, B ) is coarsely excisive. In this case it will followfrom the Decomposition Theorem 9.22 applied to the equivariant uniform decomposition([0 , ∞ ) × A, [0 , ∞ ) × B ) of [0 , ∞ ) du ⊗ Y that (9.19) is cocartesian. (cid:7) Corollary 9.38.
The functor O ∞ : Γ UBC → Γ Sp X is homotopy invariant.Proof. Let Y be a Γ-uniform bornological coarse space. Let h : [0 , du ⊗ Y → Y be theprojection. By functoriality of the cone we get the morphism O ( h ) : O ([0 , du ⊗ Y ) → O ( Y ) .
77e now observe that O ([0 , du ⊗ Y ) ∼ = ([0 , ∞ ) du ⊗ [0 , du ⊗ Y ) h ∼ = ([0 , du ⊗ [0 , ∞ ) du ⊗ Y ) h . By the Homotopy Theorem 9.23 we get an equivalenceYo s ( O ([0 , du ⊗ Y )) (cid:39) Yo s (([0 , du ⊗ [0 , ∞ ) d ⊗ Y ) h ) (cid:39) Yo s (([0 , ∞ ) du ⊗ Y ) h ) (cid:39) Yo s ( O ( Y )) . Since the projections [0 , du ⊗ [0 , n ] du ⊗ Y → [0 , n ] du ⊗ Y induce equivalences of underlyingΓ-bornological coarse spaces we conclude that the projection h induces an equivalence O ∞ ([0 , du ⊗ Y ) → O ∞ ( Y )in Γ Sp X .
10. Topological assembly maps
The overall theme of this section is the interplay between equariant coarse homologytheories and equivariant homology theories.We start in Section 10.1 with the general discussion of equivariant homology theories, andthen in Section 10.2 we modify the “cone at infinity” functor O ∞ to get an equivariantΓ Sp X -valued homology theory O ∞ hlg : Γ Top → Γ Sp X . We introduce classifying spaces E F Γ for families F of subgroups and define the motivicassembly map α F : O ∞ hlg ( E F Γ) → O ∞ hlg ( ∗ )in Section 10.3. Using the cone sequence we define further versions α X,Q of the motivicassembly map with twist Q and discuss some instances where it is an equivalence. Finally,in Section 10.4 we use the functor O ∞ hlg in order to derive equivariant homology theoriesfrom coarse homology theories. In this section we will recall the notion of a (strong) equivariant homology theory onΓ-topological spaces.Our basic category of topological spaces is the convenient category
Top of compactlygenerated weakly Hausdorff spaces. A map between topological spaces is a weak equivalenceif it induces an isomorphism between the sets of connected components and isomorphisms ofhomotopy groups in all positive degrees and for all choices of base points. The ∞ -category78btained from (the nerve of) Top by inverting these weak equivalences is a model for thepresentable ∞ -category Spc of spaces. In particular, we have the localization functor κ : Top → Spc . (10.1)A Γ-topological space is a topological space with an action of the group Γ by automorphisms.We denote the category of Γ-topological spaces and equivariant continuous maps by Γ Top .A weak equivalence between Γ-topological spaces is a Γ-equivariant map which inducesweak equivalences on fixed-point spaces for all subgroups of Γ. We will model this homotopytheory by presheaves on the orbit category of Γ.The orbit category
Orb (Γ) of Γ is the category of transitive Γ-sets and equivariant maps.A Γ-set can naturally be considered as a discrete Γ-topological space. In this way we get afully faithful functor
Orb (Γ) → Γ Top . For a transitive Γ-set S and Γ-topological space X we consider the topological space Map Γ Top ( S, X ) of equivariant maps from S to X . Remark 10.1.
We consider a transitive Γ-set S . If we fix a base point s in S and denotethe stabilizer of s by Γ s , then we get an identification Map( S, X ) (cid:39) X Γ s , where X Γ s isthe subspace of Γ s -fixed points. (cid:7) We define a functor (cid:96) : Γ
Top → PSh ( Orb (Γ)) by (cid:96) ( X )( S ) := κ (Map Γ Top ( S, X )) for S ∈ Orb (Γ) . (10.2) Remark 10.2.
A map between topological spaces is a weak equivalence if and only ifits image under κ is an equivalence. Consequently, a map between Γ-topological spacesis a weak equivalence if and only if its image under (cid:96) is an equivalence. By Elmendorf’stheorem [May96, Thm. VI.6.3] (which boils down to the assertion that (cid:96) is essentiallysurjective) the functor (cid:96) induces an equivalenceΓ Top [ W − ] (cid:39) −→ PSh ( Orb (Γ)) , (10.3)where Γ Top [ W − ] denotes the ∞ -category obtained from Γ Top by inverting the weakequivalences. Occasionally we will use the fact that a weak equivalence in Γ
Top betweenΓ-CW-complexes is actually a homotopy equivalence in Γ
Top . (cid:7) Let C be a cocomplete ∞ -category. By the universal property of the presheaf category wehave an equivalence of ∞ -categories Fun colim ( PSh ( Orb (Γ)) , C ) (cid:39) Fun ( Orb (Γ) , C ) , (10.4)where the superscript colim stands for colimit-preserving. The localization functor (10.2)induces a faithful restriction functor Fun colim ( PSh ( Orb (Γ) , C ) → Fun (Γ Top , C ) . (10.5)From now on we assume that C is cocomplete and stable.Our preferred definition of the notion of an equivariant C -valued homology theory wouldbe the following.Let E : Γ Top → C be a functor. 79 efinition 10.3. E is called a strong equivariant C -valued homology theory if it is in theessential image of (10.5). (cid:7) Assume that we are given a functor E as above. If it sends weak equivalences to equiv-alences, then, using the equivalence (10.3), it extends essentially uniquely to a functor PSh ( Orb (Γ)) → C . The functor E is an equivariant C -valued homology theory if thisextension preserves colimits. In general it seems to be complicated to check these conditionsif E is given by some geometric construction. For this reason we add the adjective strong in order to distinguish this notion from the Definition 10.4 of an equivariant C -valuedhomology theory that we actually work with.Let E : Γ Top → C be a functor. We extend E to pairs ( X, A ) of Γ-topological spaces andsubspaces by setting E ( X, A ) := Cofib (cid:0) E ( ∗ ) → E ( X ∪ A Cone( A )) (cid:1) , where Cone( A ) denotes the cone over A and ∗ is the base point of the cone. Definition 10.4.
The functor E is called an equivariant C -valued homology theory if ithas the following properties:1. (Homotopy invariance) For every Γ-topological space X the projection induces anequivalence E ([0 , × X ) → E ( X ) .
2. (Excision) If (
X, A ) is a pair of Γ-topological spaces and U is an invariant open subsetof A such that U is contained in the interior of A , then the inclusion ( X \ U, A \ U ) → ( X, A ) induces an equivalence E ( X \ U, A \ U ) → E ( X, A ) .
3. (Wedge axiom) For every family ( X i ) i ∈ I of Γ-topological spaces the canonical map (cid:77) i ∈ I E ( X i ) (cid:39) −→ E (cid:16) (cid:97) i ∈ I X i (cid:17) is an equivalence. (cid:7) Remark 10.5.
In order to verify that E satisfies excision one must show that E sendsthe square Cone( A \ U ) (cid:47) (cid:47) (cid:15) (cid:15) ( X \ U ) ∪ A \ U Cone( A \ U ) (cid:15) (cid:15) Cone( A ) (cid:47) (cid:47) X ∪ A Cone( A )to a push-out square. For homotopy invariant functors E this is equivalent to the propertythat E sends the right vertical map in the square above to an equivalence.80or homotopy invariant functors E excision follows from the stronger condition of closedexcision, i.e., that for every decomposition ( A, B ) of X into closed invariant subsets thediagram 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 a push-out square. This can be seen as follows. Assume that E is homotopy invariantand satisfies closed excision. Let X , A , and U be as above. Then we have a closeddecomposition ( X \ U ∪ A \ U Cone( A \ U ) , Cone( U ))of X ∪ A Cone( A ) with intersection Cone( U \ U ). By closed excision we get the cocartesiansquare E (Cone( U \ U )) (cid:47) (cid:47) (cid:15) (cid:15) E ( X \ U ∪ A \ U Cone( A \ U )) (cid:15) (cid:15) E (Cone( U )) (cid:47) (cid:47) E ( X ∪ A Cone( A ))Since E is homotopy invariant it sends the left vertical map to an equivalence since conesare contractible. Consequently, the right vertical map is an equivalence, too. (cid:7) Remark 10.6.
Let E be an equivariant C -valued homology theory. In general we cannot expect that it factorizes over the localization (10.2).Using the equivalence (10.4) the restriction of E to the orbit category gives rise to a strongequivariant C -valued homology theory E % which comes with a natural transformation E % → E . Using the theory developed by Davis–L¨uck [DL98, Sec. 3] one can check that E % ( X ) (cid:39) −→ E ( X )for all Γ-CW-complexes X . (cid:7) In Section 9.5 we have seen that the “cone at infinity” functor O ∞ is a homotopy invariantand excisive functor from Γ UBC to Sp X . In this section we modify this functor in orderto get an equivariant homology theory O ∞ hlg : Γ Top → Γ Sp X .If X is a Γ-uniform space, then we can consider X as a Γ-uniform bornological coarsespace X max,max by equipping the uniform space X in addition with the maximal coarsestructure and the maximal bornology. In this way we get a functor M : Γ U → Γ UBC , X (cid:55)→ M ( X ) := X max,max . (10.6)81et Y be a Γ-set and Q be a Γ-bornological coarse space. The projection Y max,max → ∗ induces a morphism Yo s ( Y max,max ) ⊗ Yo s ( Q ) → Yo s ( Q ) (10.7)in Γ Sp X . Lemma 10.7.
If the underlying set of Q is a free Γ -set, then (10.7) is an equivalence.Proof. Since Q is a free Γ-set we can choose a Γ-equivariant map of sets κ : Q → Y . Themap ( κ, id) : Q → Y max,max ⊗ Q is then a morphism of Γ-bornological coarse spaces. It isan inverse to the projection up to equivalence. Hence the projection is an equivalence ofΓ-bornological coarse spaces and therefore Yo s ( Y max,max ⊗ Q ) → Yo s ( Q ) is an equivalence.We now use that Yo s is a symmetric monoidal functor (see Section 4.3) in order to rewritethe domain of the morphism as in (10.7).The functor O ∞ ◦ M : Γ U → Γ Sp X (10.8)behaves very much like an equivariant homology theory. Of course, it is not defined onΓ Top , but on Γ U . On the other hand it is homotopy invariant by Corollary 9.38, andsends equivariant uniform decompositions to push-outs by Corollary 9.36. The drawbackis that it in general only preserves finite coproducts.In order to improve these points we define a new functor O ∞ hlg : Γ Top → Γ Sp X by first restricting O ∞ ◦ M to the subcategory of Γ-compact Γ-metrizable spaces and thenleft-Kan extending the result to Γ Top . In the following we describe the details.Let X be a Γ-toplogical space. Recall that X is Γ -compact if there exists a compact subset K of X such that Γ K = X , and that X is Γ -metrizable if there exists a Γ-invariant metricon X which induces the topology of X .We denote by Γ Top cm the full sub-category of Γ Top spanned by all the Γ-compact andΓ-metrizable Γ-topological spaces. Associated to any X in Γ Top cm we define N ( X ) := { N ⊆ X × X | N contains a Γ-invariant neighborhood of the diagonal } . Furthermore we set U ( X ) := ( X, N ( X )). For a Γ-invariant metric on X which is compatiblewith the topology we let T d denote the associated metric uniform structure on X . Lemma 10.8.
Assume that X is in Γ Top cm .1. N ( X ) is a Γ -uniform structure.2. If d is a Γ -invariant metric compatible with the topology, then N ( X ) = T d .3. The assignment X (cid:55)→ U ( X ) defines a functor U : Γ Top cm → Γ U . roof. By assumption we can choose a Γ-invariant metric d which is compatible with thetopology. We claim that for every Γ-invariant neighborhood N of the diagonal there existssome T in T d such that T ⊆ N . The case N = X × X is trivial and we assume that N isa proper subset. We define a function b : X → (0 , ∞ ) , x (cid:55)→ sup { (cid:15) ∈ (0 , ∞ ) | B (cid:15) ( x ) × B (cid:15) ( x ) ⊆ N } , where B (cid:15) ( x ) denotes the open ball of radius (cid:15) around x (with respect to d ). Since d iscompatible with the topology the argument of the sup is non-empty for every x in X and the value b ( x ) is indeed positive. Furthermore, the supremum is attained. Moreover,the supremun can not be infinite since we assume that N is a proper subset. Finally, weobserve that b is Γ-equivariant with respect to the trivial Γ-action on (0 , ∞ ).We now show that the function b is 1-Lipschitz. Let x, y be two points in X and let δ betheir distance. If both b ( x ) and b ( y ) are less than δ , then so is their distance. Therefore,we can assume that b ( x ) ≥ b ( y ) and b ( x ) ≥ δ . By the triangle inequality we have B b ( x ) − δ ( y ) × B b ( x ) − δ ( y ) ⊆ B b ( x ) ( x ) × B b ( x ) ( x ) ⊆ N This implies b ( x ) ≥ b ( y ) ≥ b ( x ) − δ . In particular, | b ( x ) − b ( y ) | ≤ δ and b is indeed1-Lipschitz and thus continuous.By assumption we can choose a compact subset K of X such that Γ K = X . Since b iscontinuous, the restriction b | K : K → (0 , ∞ ) attains a positive minimal value, which wedenote by (cid:15) . By Γ-equivariance of b , we then have b ( x ) ≥ (cid:15) for all x in X . We concludethat the metric uniform entourage { ( x, y ) ∈ X × X | d ( x, y ) < (cid:15) } is contained in N .Note that every metric uniform entourage is a neighbourhood of the diagonal since d iscompatible with the topology. Since both N ( X ) and T d are closed under taking supersets,we have shown that N ( X ) = T d . This shows the first two assertions of the Lemma.We now show the third assertion. Let f : X → X (cid:48) be an equivariant continuous mapbetween two Γ-compact and Γ-metrizable Γ-topological spaces. We must show that f : ( X, N ( X )) → ( X (cid:48) , N ( X (cid:48) )) is uniformly continuous. Let V (cid:48) belong to N ( X (cid:48) ). Then V (cid:48) contains a Γ-invariant neighbourhood of the diagonal U (cid:48) . Since f is equivariantand continuous, ( f − × f − )( U ) is a Γ-invariant neighbourhood of the diagonal of X contained in ( f − × f − )( V (cid:48) ). Consequently, ( f − × f − )( V (cid:48) ) ∈ N ( X ). This implies that f : ( X, N ( X )) → ( X (cid:48) , N ( X (cid:48) )) is uniformly continuous.Let X be a Γ-topological space, and let A and B be closed Γ-invariant subsets of X suchthat A ∪ B = X . Lemma 10.9. If X is Γ -compact and Γ -metrizable, then ( A, B ) is an equivariant uniformdecomposition of U ( X ) (Definition 9.19).Proof. We choose a Γ-invariant metric d on X which is compatible with the topology. ByLemma 10.8, ( X, T d ) = U ( X ). 83or a subset Z of Y and e in (0 , ∞ ) we consider the e -thickening U e [ Z ] := { y ∈ Y | d ( y, Z ) ≤ e } (note that U e is defined in (9.1) with a ≤ -relation, too) of Z . If Z is invariant, then U e [ Z ]is again invariant.If A ∩ B = ∅ , then by Γ-compactness of Y the subsets A and B are uniformly separatedand ( A, B ) is an equivariant uniform decomposition.Assume now that A ∩ B (cid:54) = ∅ . It suffices to define a monotoneous function s : (0 , ∞ ) → (0 , ∞ )such that:1. lim e → s ( e ) = 0.2. For all e in (0 , ∞ ) we have U e [ A ] ∩ U e [ B ] ⊆ U s ( e ) [ A ∩ B ] . We define a function s : (0 , ∞ ) → [0 , ∞ ] by s ( e ) := inf { e (cid:48) ∈ (0 , ∞ ) | U e [ A ] ∩ U e [ B ] ⊆ U e (cid:48) [ A ∩ B ] } . By construction, this function is monotoneous. Since A ∩ B (cid:54) = 0, and since by Γ-compactnessof Y there exists R in (0 , ∞ ) such that U R [ A ∩ B ] = Y , the function s is finite. Condition 2follows from this observation.We claim that A ∩ B = (cid:92) e> U e [ A ] ∩ U e [ B ] . (10.9)It is clear that A ∩ B ⊆ (cid:92) e> U e [ A ] ∩ U e [ B ] . On the other hand, assume that y ∈ Y \ ( A ∩ B ). Without loss of generality (interchangethe roles of A and B , if necessary) we can assume that y ∈ Y \ A . Since A is closed thereexists e in (0 , ∞ ) such that y (cid:54)∈ U e [ A ]. Hence y ∈ Y \ (cid:84) e> U e [ A ] ∩ U e [ B ]. This shows theopposite inclusion (cid:92) e> U e [ A ] ∩ U e [ B ] ⊆ A ∩ B. We now show Condition 1. Assume the contrary. Then there exists (cid:15) in (0 , ∞ ) suchthat s ( e ) ≥ (cid:15) > e in (0 , ∞ ). For every integer n there exists a point y n in U /n [ A ] ∩ U /n [ B ] such that y n (cid:54)∈ U (cid:15) [ A ∩ B ]. We can assume (after replacing y n by γ n y n forsuitable elements γ n of Γ) by Γ-compactness that y n → y for n → ∞ . Then y (cid:54)∈ U (cid:15)/ [ A ∩ B ]but y ∈ A ∩ B by (10.9). This is a contradiction, and so we have verified Condition 1.84 efinition 10.10. We define the functor O ∞ hlg : Γ Top → Γ Sp X as the left Kan extension Γ Top cm O ∞ ◦M◦U (cid:47) (cid:47) (cid:15) (cid:15) Γ Sp X Γ Top O ∞ hlg (cid:54) (cid:54) of O ∞ ◦ M ◦ U along the fully faithful functor Γ Top cm (cid:44) → Γ Top . (cid:7) Proposition 10.11. O ∞ hlg is an equivariant Γ Sp X -valued homology theory.Proof. We first note that the objectwise formula for the left Kan-extension gives O ∞ hlg ( X ) (cid:39) colim ( Y → X ) ∈ Γ Top cm /X O ∞ ( M ( U ( Y ))) . (10.10)Let Y be a Γ-compact and Γ-metrizable Γ-topological space and ( φ, f ) : Y → [0 , × X be an equivariant map. Then we have the factorization( φ, f ) : Y ( φ, id Y ) −−−−→ [0 , × Y (id [0 , ,f ) −−−−−→ [0 , × X .
Note that [0 , × Y is again a Γ-compact and Γ-metrizable Γ-topological space. This showsthat the category of maps of the form (id [0 , , f ) : [0 , × Y → [0 , × X for morphisms ofΓ-topological spaces f : Y → X with Y in Γ Top cm is cofinal in Γ Top cm / ([0 , × X ). ByLemma 10.8, we have an isomorphism of Γ-uniform spaces U ([0 , × Y ) ∼ = [0 , u ⊗ U ( Y ) . By the homotopy invariance of O ∞ (Corollary 9.38) the projection [0 , u ⊗ U ( Y ) → U ( Y )induces an equivalence O ∞ ( M ( U ([0 , × Y ))) (cid:39) O ∞ ( M ( U ( Y ))). Hence we conclude that O ∞ hlg ([0 , × X ) (cid:39) O ∞ hlg ( X ), i.e., that O ∞ hlg is homotopy invariant.Assume now that ( X i ) i ∈ I is a filtered family of Γ-topological spaces and set X := colim i ∈ I X i .Then every morphism Y → X for a Y in Γ Top cm factorizes as Y → X i → X for some i in I since Y is Γ-compact. It follows that O ∞ hlg ( X ) (cid:39) colim ( Y → X ) ∈ Γ Top cm /X O ∞ ( M ( U ( Y ))) (cid:39) colim i ∈ I colim ( Y → X i ) ∈ Γ Top cm /X i O ∞ ( M ( U ( Y ))) (cid:39) colim i ∈ I O ∞ hlg ( X i ) . This in particular implies the wedge axiom for O ∞ hlg .85n order to show that O ∞ hlg satisfies excision, by Remark 10.5 it suffices to show the strongerresult that for every decomposition ( A, B ) of a Γ-topological space X into two closedinvariant subsets we have a push-out square O ∞ hlg ( A ∩ B ) (cid:47) (cid:47) (cid:15) (cid:15) O ∞ hlg ( A ) (cid:15) (cid:15) O ∞ hlg ( B ) (cid:47) (cid:47) O ∞ hlg ( X ) . (10.11)Let r : Y → X be an object of Γ Top cm /X . Then ( r − ( A ) , r − ( B )) is a closed decompositionof Y into Γ-compact and Γ-metrizable subspaces. Note that the objects of the form r − ( A ) → A of Γ Top cm /A for all r : Y → X are cofinal in Γ Top cm /A .We conclude that (10.11) is a colimit of commuting squares O ∞ ( M ( U ( r − ( A ) ∩ r − ( B )))) (cid:47) (cid:47) (cid:15) (cid:15) O ∞ ( M ( U ( r − ( A )))) (cid:15) (cid:15) O ∞ ( M ( U ( r − ( B )))) (cid:47) (cid:47) O ∞ ( M ( U ( Y ))) (10.12)Since Y is a Γ-compact and Γ-metrizable Γ-topological space, by Lemma 10.9 the decom-position ( r − ( A ) , r − ( B )) is an equivariant uniform decomposition of U ( Y ). Therefore thesquare (10.12) is a push-out square by Corollary 9.36. Being a colimit of push-out squaresthe square (10.11) is therefore also a push-out square. Remark 10.12.
Instead of O ∞ ◦ M we could consider any functor A : Γ U → C for some cocomplete stable ∞ -category C , which is excisive for closed decompositions andhomotopy invariant. The construction above produces an equivariant C -valued homologytheory A hlg : Γ Top → C as a left Kan-extensionΓ Top cm A ◦U (cid:47) (cid:47) (cid:15) (cid:15) C Γ Top A hlg (cid:55) (cid:55) The proof that this is indeed an equivariant C -valued homology theory is word-for-wordthe same as the proof of Proposition 10.11.Note that the proof also shows that for a filtered family of Γ-topological spaces ( X i ) i ∈ I the following natural morphism is an equivalencecolim i ∈ I A hlg ( X i ) (cid:39) −→ A hlg (colim i ∈ I X i ) . (10.13)We will use this equivalence later in this paper. (cid:7)
86 Γ-uniform space X has an underlying Γ-topological space which we will denote by τ ( X ).The topology of τ ( X ) is generated by the sets V [ x ] for all points x and uniform entourages V of X . We actually get a functor τ : Γ U → Γ Top . (10.14) Remark 10.13.
Let Y be a Γ-compact and Γ-metrizable Γ-topological space, and let X be a Γ-uniform space. Then every continuous map Y → τ ( X ) induces a uniform map U ( Y ) → X . Therefore, we obtain a natural morphism A hlg ( τ ( X )) → A ( X ) . (10.15)Since the inclusion functor Γ Top cm (cid:44) → Γ Top is fully faithful, the morphism (10.15) is anequivalence for X := U ( Y ) if Y is a Γ-compact and Γ-metrizable Γ-topological space. (cid:7) In this section we will define the classifying space E F Γ for a family F of subgroups of Γ,and we will define the corresponding universal assembly map α F : O ∞ hlg ( E F Γ) → O ∞ hlg ( ∗ ) . We will also introduce the motivic assembly map α X,Q : O ∞ hlg ( X ) ⊗ Yo s ( Q ) → ΣYo s ( Q )for a Γ-topological space X twisted by a Γ-bornological coarse space Q . Definition 10.14. A family of subgroups F of Γ is a non-empty subset of the set ofsubgroups of Γ which is closed under conjugation and taking subgroups. (cid:7) Example 10.15.
Examples of families of subgroups are:1. { } – the family containing only the trivial subgroup2. Fin – the family of all finite subgroups3.
Vcyc – the family of all virtually cyclic subgroups4.
All – the family of all subgroupsIn Section 11 we will mostly work with the family
Fin since one can model the correspondingclassifying space (see Definition 10.17) by the Rips complex. (cid:7)
For a family of subgroups F we let Orb F (Γ) ⊆ Orb (Γ) be the full subcategory of transitiveΓ-sets whose stabilizers belong to F . 87 xample 10.16. Note that
Orb
All (Γ) =
Orb (Γ).There is furthermore an equivalence B Γ → Orb { } (Γ) op which sends the unique object ∗ of B Γ to the Γ-set Γ (with the left action). On morphisms this equivalence sends the morphism γ in Hom B Γ ( ∗ , ∗ ) = Γ to the automorphism of the Γ-set Γ given by right-multiplicationwith γ . (cid:7) For two families of subgroups F , F (cid:48) satisfying F ⊆ F (cid:48) we have an inclusion
Orb F (Γ) (cid:44) → Orb F (cid:48) (Γ)of orbit categories. On presheaves this inclusion of orbit categories defines a restrictionfunctor which is the right-adjoint of an adjunctionInd F (cid:48) F : PSh ( Orb F (Γ)) (cid:28) PSh ( Orb F (cid:48) (Γ)) : Res F (cid:48) F . (10.16)We let ∗ F denote the final object in Orb F (Γ). Definition 10.17.
The object E F Γ := Ind
All F ( ∗ F )of PSh ( Orb (Γ)) is called the classifying space of Γ for the family F . (cid:7) Remark 10.18.
Assume that X is a Γ-CW-complex with an equivalence (cid:96) ( X ) (cid:39) E F Γ.Then the following Lemma 10.19 shows that for every subgroup H of Γ the fixed point set X H is contractible or empty depending on whether H belongs to F or not. This propertyis the usual characterization of a classifying space of Γ for the family F . We say that X isa model for E F Γ. A model for E F Γ is unique up to contractible choice. (cid:7)
Let y : Orb (Γ) → PSh ( Orb (Γ)) denote the Yoneda embedding.
Lemma 10.19.
For T ∈ Orb (Γ) we have
Map( y ( T ) , E F Γ) (cid:39) (cid:40) ∅ if T (cid:54)∈ Orb F (Γ) , ∗ if T ∈ Orb F (Γ) . Proof.
For a presheaf E on Orb F (Γ) and object T in Orb (Γ) we have (we interpret theHom-sets as discrete spaces)Ind
All F ( E )( T ) (cid:39) colim ( S → T ) ∈ Orb F (Γ) op /T E ( S ) . Consequently, we haveMap( y ( T ) , E F Γ) (cid:39) colim ( S → T ) ∈ Orb F (Γ) op /T ∗ F ( S ) . Note that ∗ F ( S ) (cid:39) ∗ and Orb F (Γ) op /T is empty if T is not in Orb F (Γ) and otherwisehas the identity on T as final object. 88 emark 10.20. We must consider models for E F Γ since O ∞ hlg or the equivariant homologytheory A hlg constructed in Remark 10.12 are not expected to be strong equivarianthomology theories (Definition 10.3) and therefore can not be applied to the object E F Γ ofthe ∞ -category PSh ( Orb (Γ)). But on the other hand, since these functors are homotopyinvariant, the evaluations O ∞ hlg ( E F Γ cw ) or A hlg ( E F Γ cw ) on a model E F Γ cw for E F Γ arewell-defined up to equivalence.From now on E F Γ cw will denote some choice of a model. (cid:7) The projection E F Γ cw → ∗ is a morphism in Γ Top .If F : Γ Top → C is an equivariant C -valued homology theory, then the Farrell–Jones andBaum–Connes type question is for which family F this projection induces an equivalence F ( E F Γ cw ) → F ( ∗ ) . The coarse geometric approach to the question uses that this projection induces a morphismin Γ Sp X α E F Γ cw : O ∞ hlg ( E F Γ cw ) → O ∞ hlg ( ∗ ) . Definition 10.21.
The morphism α E F Γ cw is called the universal assembly map for thefamily of subgroups F . (cid:7) One could ask if there is an interesting family F (i.e., not the family of all subgroups of Γ)for which the universal assembly map is an equivalence.Most of the study of the assembly map is based on an identification of this map with aforget-control map, or equivalently, a boundary operator of a cone sequence. We developthis point of view below.The following diagram is one of the starting points of [BLR08]. We consider a Γ-topologicalspace X and the projection X → ∗ . It induces the vertical maps in the diagram in Γ Sp X whose horizontal parts are segments of the cone sequence Corollary 9.30:colim ( Y → X ) ∈ Γ Top cm /X Yo s ( Y max,max ) (cid:47) (cid:47) (cid:15) (cid:15) colim ( Y → X ) ∈ Γ Top cm /X Yo s ( O ( M ( U ( Y )))) (cid:15) (cid:15) (cid:47) (cid:47) O ∞ hlg ( X ) α X (cid:15) (cid:15) Yo s ( ∗ ) (cid:47) (cid:47) Yo s ( O ( ∗ )) (cid:47) (cid:47) O ∞ hlg ( ∗ ) . Note that Yo s ( O ( ∗ )) (cid:39) O ( ∗ ) is flasque. Furthermore, if we tensor-multiply(with respect to − ⊗ − ) the diagram with Yo s ( Q ) for a Γ-bornological coarse space Q whose underlying Γ-set is free, then by Lemma 10.7 the left vertical map becomes anequivalence. Definition 10.22.
We define the obstruction motive to be the object M ( X ) := colim ( Y → X ) ∈ Γ Top cm /X Yo s ( O ( M ( U ( Y ))))of the category Γ Sp X . (cid:7) Q be a Γ-bornological coarse space. Corollary 10.23.
If the underlying Γ -set of Q is free, then we have a fiber sequence M ( X ) ⊗ Yo s ( Q ) → O ∞ hlg ( X ) ⊗ Yo s ( Q ) α X,Q −−−→
ΣYo s ( Q ) → Σ( M ( X ) ⊗ Yo s ( Q )) . Definition 10.24.
The morphism α X,Q := α X ⊗ id Yo s ( Q ) is called the motivic assemblymap with twist Q . (cid:7) Let Q be a Γ-bornological coarse space. Corollary 10.25.
Let the underlying Γ -set of Q be free. Then the motivic assembly mapwith Q -twist α X,Q is an equivalence if and only if M ( X ) ⊗ Yo s ( Q ) (cid:39) . A typical example for Q is Γ can,min , and variants like Γ can,max , Γ max,max , Γ min,min , etc.In general we expect the assembly map α E F Γ ,Q to become an equivalence only after applyingsuitable equivariant coarse homology theories. But the following is an example where theassembly map is an equivalence already motivically.Let F be a family of subgroups of Γ and consider a Γ-bornological coarse space Q . Lemma 10.26.
Assume that:1. E F Γ cw is Γ -compact and Γ -metrizable.2. Q is discrete as a coarse space.3. Q has stabilizers in F .4. Q is Γ -finite.Then we have M ( E F Γ cw ) ⊗ Yo s ( Q ) (cid:39) . Remark 10.27.
Note that if the family F is VCyc , i.e., the family of all virtually cyclicsubgroups, then the condition of E F Γ cw being Γ-compact is very restrictive: by a conjectureof Juan-Pineda–Leary [JPL06] this implies that Γ is virtually cyclic itself.This conjecture is proven for hyperbolic groups (Juan-Pineda–Leary [JPL06]), elementaryamenable groups (Kochloukova–Martinez-Perez–Nucinkis [KMPN09] and Groves–Wilson[GW13]) and for one-relator groups, acylindrically hyperbolic groups, 3-manifold groups,CAT(0) cube groups, linear groups (von Puttkamer–Wu [vPW16, vPW17]). (cid:7) Proof.
Since E F Γ cw is Γ-compact and Γ-metrizable the colimit in the Definition 10.22 of M ( E F Γ) cw stabilizes. We consider Q as a Γ-uniform bornological coarse space Q disc withthe discrete uniform structure. Using Lemma 9.26, we get the equivalence M ( E F Γ cw ) ⊗ Yo s ( Q ) (cid:39) Yo s ( O ( M ( U ( E F Γ cw ))) ⊗ Q disc ) . E F Γ cw has the following universal property: for every Γ-space X with stabilizersin F which is homotopy equivalent to a CW -complex the space Map Γ ( X, E F Γ cw ) iscontractible. It follows from the universal property of E F Γ cw that there is a uniquehomotopy class of maps ι : Q → E F Γ cw . Furthermore, the maps ι ◦ pr Q , pr E F Γ cw : E F Γ cw × Q → E F Γ cw are homotopic. Since Q is Γ-finite and E F Γ cw is Γ-compact Γ-metrizablethese maps and homotopies are automatically uniform maps when we consider E F Γ cw as auniform space by applying U . Hence id M ( U ( E F Γ cw )) ⊗ Q disc is homotopic to the composition M ( U ( E F Γ cw )) ⊗ Q disc pr Q −−→ Q disc ( ι, id Q ) −−−→ M ( U ( E F Γ cw )) ⊗ Q disc of morphisms between Γ-uniform bornological coarse spaces. Using that O is homotopyinvariant we conclude that id M ( E F Γ cw ) ⊗ Yo s ( Q ) factorizes over Yo s ( O ( Q disc )). We now useExample 9.25 in order to conclude that Yo s ( O ( Q disc )) (cid:39)
0. This finishes this proof.For the following we just combine Lemma 10.26 and Corollary 10.25.
Corollary 10.28. If E F Γ cw is Γ -compact and Γ -metrizable, then the motivic assemblymap α E F Γ cw , Γ min, ? is an equivalence for ? ∈ { min , max } . Let E be an equivariant C -valued coarse homology theory, i.e., a colimit preserving functor E : Γ Sp X → C . By Proposition 10.11 we can use the cone O ∞ hlg in order to pull-back E to an equivariant homology theory: Definition 10.29.
Let us define E O ∞ hlg := E ◦ O ∞ hlg : Γ Top → C , which is an equivariant C -valued homology theory. (cid:7) Recall that for an equivariant coarse motivic spectrum L the functor − ⊗ L : Γ Sp X → Γ Sp X preserves colimits. If E is a C -valued equivariant coarse homology theory, then we candefine a new equivariant coarse homology theory E L ( − ) := E ( − ⊗ L ) : Γ Sp X → C . (10.17)We will refer to L as a twist .The Farrell–Jones/Baum–Connes type question for E Yo s ( Q ) O ∞ hlg is now the question forwhich family F of subgroups the morphism E Yo s ( Q ) ( α E F Γ ) is an equivalence of spectra.The next corollary is a consequence of Corollary 10.25.Let Q be a Γ-bornological coarse space. 91 orollary 10.30. Assume that the underlying Γ -set of Q is free. Then the assembly map E Yo s ( Q ) ( α E F Γ ) is an equivalence if and only if E Yo s ( Q ) ( M ( E F Γ)) (cid:39) . Given the coarse homology theory E and a twist L it is of particular interest to charac-terize the homology theory E L O ∞ hlg . As explained in Section 10.1, the restriction of anequivariant homology theory from Γ Top to the full subcategory Γ CW of Γ- CW complexesis determined by its restriction to the orbit category Orb (Γ). So we must understand theevaluations E L O ∞ hlg ( S ) for all transitive Γ-sets S . The main tool is Proposition 9.35. Itgives E L O ∞ hlg ( S ) (cid:39) Σ E L (Yo s ( S min,max )) . We have calculated these evaluations for equivariant coarse ordinary homology and forequivariant coarse algebraic K -homology explicitly:1. see Section 7.2 for H X ΓYo s (Γ can,min ) , and2. see Section 8.4 for K A X ΓYo s (Γ ? , ? ) .
11. Forget-control and assembly maps
Let X be a coarse space and U an entourage of X .Let µ : P ( X ) → [0 ,
1] be a probability measure on the measurable space ( X, P ( X )). Definition 11.1.
The measure µ is called finite U -bounded if there is a finite U -boundedsubset F of X with µ ( F ) = 1. We then define the support supp( µ ) to be the smallestsubset of X with measure one.We let P U ( X ) denote the topological space of finite U -bounded probability measureson X equipped with the topology induced by the evaluation against finitely supportedfunctions. (cid:7) Every point x in X gives rise to a Dirac measure δ x at x . If U contains the diagonal of X , then we get a map X → P U ( X ), x (cid:55)→ δ x , of sets. A probability measure µ which isfinite U -bounded can be written as a finite convex combination of Dirac measures. Moreconcretely, µ = (cid:88) x ∈ supp( µ ) µ ( { x } ) δ x . The set P U ( X ) has a natural structure of a simplicial complex. In other words, P U ( X ) isthe simplicial complex with vertex set X such that a subset { x , . . . , x m } spans a simplexif and only if ( x i , x j ) ∈ U for all i, j in { , . . . , m } . Let P U ( X ) u denote the uniform space92ith the uniform structure induced by the canonical path quasi-metric on the simplicialcomplex, see Example 9.7. Note that the topology induced by this uniform structure isthe topology on P U ( X ) induced by the evaluation against finitely supported functions, i.e.,we have τ ( P U ( X ) u ) = P U ( X ) , where τ is the functor associating the underlying topological space to a uniform space,cf. (10.14). Eilenberg–Steenrod [ES52, p. 75] called this the metric topology. In generalthis topology differs from the weak topology on the simplicial complex in the sense ofEilenberg–Steenrod [ES52, p. 75].If X (cid:48) is a second coarse space with entourage U (cid:48) and f : X → X (cid:48) is a map such that( f × f )( U ) ⊆ U (cid:48) , then we get a map of simplicial complexes f ∗ : P U ( X ) → P U (cid:48) ( X (cid:48) ) , µ (cid:55)→ f ∗ µ , where the measure f ∗ µ is the push-forward of the measure µ .If X is a Γ-coarse space and the entourage U is Γ-invariant, then Γ acts on the simplicialcomplex P U ( X ) such that the latter becomes a Γ-simplicial complex, so in particular itbecomes a Γ-topological space. Furthermore we obtain a Γ-uniform space P U ( X ) u . If X (cid:48) is a second Γ-coarse space, U (cid:48) is a Γ-invariant entourage, and f : X → X (cid:48) is equivariantwith ( f × f )( U ) ⊆ U (cid:48) , then f ∗ : P U ( X ) → P U (cid:48) ( X (cid:48) ) is equivariant.Let X be a Γ-coarse space. Definition 11.2.
The Γ-topological spaceRips( X ) := colim U ∈C Γ P U ( X )is called the Rips complex of X . Note that the colimit is taken in Γ Top . (cid:7) Remark 11.3.
Note that in general the simplicial complex P U ( X ) is not locally finite. Inthis case its topology does not exhibit it as a CW-complex (this happens if and only if thesimplicial complex is locally finite).But by a result of Dowker [Dow52, Thm. 1 on P. 575] P U ( X ) has the homotopy type of aCW-complex (see also Milnor [Mil59, Thm. 2] for a short proof of this). Concretely, theunderlying identity map of the set induces a homotopy equivalence CW( P U ( X )) → P U ( X ),where CW( P U ( X )) denotes P U ( X ) retopologized as a CW-complex. (cid:7) Let Γ can be the group Γ considered as a Γ-coarse space with its canonical coarse structure.Recall the localization (10.2).
Lemma 11.4.
We have an equivalence (cid:96) (Rips(Γ can )) (cid:39) E Fin Γ .Proof. By definition, (cid:96) (Rips(Γ)) is the presheaf on
Orb (Γ) given by S (cid:55)→ κ (Map Γ Top ( S, Rips(Γ can ))) , κ is as in (10.1). We now observe that the stabilizers of the points of Rips(Γ can ) arefinite subgroups of Γ. Consequently, the presheaf belongs to the subcategory of presheavessupported on the orbits with finite stabilizers. Since, by definition, E Fin
Γ corresponds to thefinal object in this subcategory there exists a morphism (cid:96) (Rips(Γ can )) → E Fin
Γ. In orderto show that it is an equivalence it suffices to show that the spaces Map Γ Top (Γ /H, Rips( X ))for all finite subgroups H of Γ are equivalent to ∗ , i.e., that these spaces are connectedand that their homotopy groups are trivial.We now study these homotopy groups. Since the spheres S n are compact for all n in N and the structure mapsMap Γ Top (Γ /H, P U (Γ can )) → Map Γ Top (Γ /H, P U (cid:48) (Γ can ))for U ⊆ U (cid:48) are inclusions of CW-complexes, we have π ∗ (Map Γ Top (Γ /H, Rips(Γ can ))) ∼ = colim U ∈C Γ π ∗ (Map Γ Top (Γ /H, P U (Γ can ))) . We have a homeomorphismMap Γ Top (Γ /H, P U (Γ can )) ∼ = P U (Γ can ) H . We fix an integer n and consider a map f : S n → P U (Γ can ) H . The image f ( S n ) is a compact subset of P U (Γ can ) H . In the case n = 0, since Γ can iscoarsely connected, we can increase the entourage U such that the image f ( S ) belongs toa connected component of P U (Γ can ). Let now n be arbitrary. We can now assume that f ( S n ) belongs to a connected component of P U (Γ can ). It is hence bounded in diameter bysome integer N . We conclude that under the map P U (Γ can ) → P U N (Γ can ) the image f ( S n )is mapped to a subset of a single simplex of P U N (Γ can ). The intersection of the H -fixedpoints with this simplex is a convex subset and hence itself contractible. We conclude that f is homotopic to a constant map. Definition 11.5. If X is a Γ-bornological coarse space and U an invariant entourage of X ,then we equip the Γ-simplicial complex P U ( X ) with the bornology generated by the subsets P U ( B ) for all bounded subsets B of X . Equipped with this bornology and the coarsestructure induced by the metric we obtain a Γ-bornological coarse space which we denoteby P U ( X ) bd . We furthermore write P U ( X ) bdu for the corresponding Γ-uniform bornologicalcoarse space. Note in contrast that the notation P U ( X ) d (or P U ( X ) du , respectively) wouldmean the Γ-bornological coarse space (or Γ-uniform bornological coarse space, respectively)whose bornological and coarse (and uniform, respectively) structures are induced from themetric, see Example 9.11. (cid:7) Let X be a Γ-bornological coarse space and U be an invariant entourage of X . The Diracmeasures provide a morphism of Γ-bornological coarse spaces δ : X U → P U ( X ) bd . (11.1)94 emma 11.6. Assume:1. Γ is torsion-free.2. The underlying Γ -set of X is free.Then the morphism (11.1) is an equivalence of Γ -bornological coarse spaces.Proof. We first observe that Γ acts freely on P U ( X ). Indeed, for γ in Γ and µ in P U ( X )satisfying γµ = µ the subgroup of Γ generated by γ has a finite orbit contained in supp( µ ).Since Γ is torsion-free and X is a free Γ-set this can only happen if γ = 1.To define an inverse morphism g : P U ( X ) bd → X U we first choose representatives for theorbits P U ( X ) / Γ. Then we define g ( µ ) for every chosen representative µ to be a point insupp( µ ), and extend equivariantly. Then g ◦ δ = id X U and δ ◦ g is close to id P U ( X ) .The following corollary follows immediately from the above lemma. Corollary 11.7. If Γ is a finitely generated torsion-free group, then Γ can,min → P U (Γ) bd is an equivalence for every invariant generating entourage U of Γ . Assume that Q is a Γ-bornological coarse space. Let X be a Γ-bornological coarse spaceand U be an invariant entourage of X . Lemma 11.8.
If the underlying Γ -set of Q is free, then δ × id Q : X U ⊗ Q → P U ( X ) bd ⊗ Q is an equivalence of Γ -bornological coarse spaces. Note that we do not assume that X is a free Γ-set. Proof.
First we note that Γ acts freely on the set P U ( X ) × Q . We choose representativesfor the orbits ( P U ( X ) × Q ) / Γ. Then we choose for every representative ( µ, q ) ∈ P U ( X ) × Q a point x ∈ supp( µ ) and set g ( µ, q ) := ( x, q ). Then we extend this to an equivariant map g : P U ( X ) × Q → X × Q . This map of sets is a morphism g : P U ( X ) bd ⊗ Q → X U ⊗ Q .
By construction g ◦ ( δ × id Q ) = id X × Q and g ◦ ( δ × id Q ) is close to the identity.For the following recall the cone functor O from Section 9.4 and the “cone at infinity”functor O ∞ from Section 9.5.Let X be a Γ-bornological coarse space. 95 efinition 11.9. We define the following equivariant coarse motivic spectra: F ( X ) := colim U ∈C Γ Yo s ( O ( P U ( X ) bdu )) ,F ∞ ( X ) := colim U ∈C Γ O ∞ ( P U ( X ) bdu ) ,F ( X ) := colim U ∈C Γ Yo s ( P U ( X ) bd ) . (cid:7) Using standard Kan-extension techniques one can refine the above description to functors
F, F ∞ , F : Γ BornCoarse → Γ Sp X , see Remark 11.11 for details. The fiber sequence from Corollary 9.30 provides a naturalfiber sequence of functors F ( X ) → F ( X ) → F ∞ ( X ) β X −→ Σ F ( X ) . (11.2) Definition 11.10.
We call β X the forget-control map . (cid:7) Remark 11.11.
We let Γ
BornCoarse C denote the category of pairs ( X, U ), where X isa Γ-bornological coarse space and U is an invariant entourage of X containing the diagonal.A morphism ( X, U ) → ( X (cid:48) , U (cid:48) ) is a morphism f : X → X (cid:48) in Γ BornCoarse such that( f × f )( U ) ⊆ U (cid:48) . We have a forgetful functorΓ BornCoarse C → Γ BornCoarse , ( X, U ) (cid:55)→ X . (11.3)Let (cid:101) E : Γ BornCoarse C → C be a functor to some cocomplete target C and let E be the left Kan extension of (cid:101) E along(11.3). The evaluation of E on a Γ-bornological coarse space X is then given as follows: Lemma 11.12.
We have an equivalence E ( X ) (cid:39) colim U ∈C Γ ( X ) (cid:101) E ( X, U ) . Proof.
By the pointwise formula for the left Kan extension we have an equivalence E ( X ) (cid:39) colim (( X (cid:48) ,U (cid:48) ) ,f : X (cid:48) → X ) ∈ Γ BornCoarse C /X (cid:101) E ( X (cid:48) , U (cid:48) ) . If (( X (cid:48) , U (cid:48) ) , f : X (cid:48) → X ) belongs to Γ BornCoarse C /X , then we have a morphism( X (cid:48) , U (cid:48) ) → ( X, f ( U (cid:48) ) ∪ diag( X ))in Γ BornCoarse C /X . This easily implies that the full subcategory of objects of the form(( X, U ) , id X ) of Γ BornCoarse C /X with U in C Γ ( X ) is cofinal in Γ BornCoarse C /X . (cid:7)
96e have a functor P : Γ BornCoarse C → Γ UBC , ( X, U ) (cid:55)→ P U ( X ) bdu . We construct the fibre sequence (11.2) by applying the left Kan extension to the fibresequence of functors Γ
BornCoarse C → Γ Sp X Yo s ◦ F T ◦ P → Yo s ◦ O ◦ P → O ∞ ◦ P → ΣYo s ◦ F T ◦ P obtained by precomposing the sequence from Corollary 9.30 with P . (cid:7) Let X be a Γ-bornological space. In the following two corollaries we identify the Γ-coarsemotivic spectrum F ( X ). Corollary 11.13. If Γ is torsion-free and the underlying Γ -set of X is free, then F ( X ) (cid:39) Yo s ( X ) . Proof.
We have equivalences F ( X ) = colim U ∈C Γ Yo s ( P U ( X ) bd ) Lemma . (cid:39) colim U ∈C Γ Yo s ( X U ) Corollary . . (cid:39) Yo s ( X ) , which proves the claim.Let Q and X be Γ-bornological coarse spaces. Corollary 11.14.
If the underlying Γ -set of Q is free, then F ( X ) ⊗ Yo s ( Q ) (cid:39) Yo s ( X ) ⊗ Yo s ( Q ) . Proof.
Here we use Lemma 11.8, that Yo s is symmetric monoidal, and that the functor − ⊗ Yo s ( Q ) : Γ Sp X → Γ Sp X preserves colimits. Therefore we can write down an analogous sequence of equivalences asin the above proof of Corollary 11.13. In this section we will compare the assembly map for the family of finite subgroups withthe forget-control map.For every two Γ-bornological coarse spaces X and L we have the forget-control morphism(Definition 11.10) β X,L : colim U ∈C Γ O ∞ ( P U ( X ) bdu ) ⊗ Yo s ( L ) → colim U ∈C Γ ΣYo s ( P U ( X ) bd ⊗ L ) .
97e have furthermore the assembly map (see Definition 10.24) α Rips( X ) ,L : O ∞ hlg (Rips( X )) ⊗ Yo s ( L ) → O ∞ hlg ( ∗ ) ⊗ Yo s ( L ) (cid:39) ΣYo s ( L )induced by the morphism Rips( X ) → ∗ of Γ-topological spaces.Recall [BE16, Def. 2.28] that a coarse space ( X, C ) is called coarsely connected if for anytwo points x, y in X there exists an entourage U in C such that ( x, y ) ∈ U . Definition 11.15.
A Γ-bornological coarse space X is eventually coarsely connected ifthere exists a coarse entourage U such that X U is coarsely connected.Note that an eventually coarsely connected space is in particular coarsely connected. (cid:7) While one is interested in α Rips( X ) , Γ can,min , using descent methods (like in [BEKW]) wewill only be able to derive split-injectivity of β X, Γ max,max in several cases. The followingtheorem allows us to compare both maps. This comparison does not hold directly butonly after forcing continuity.Let X be a Γ-bornological coarse space. Recall [BE16, Def. 6.100] that X has stronglybounded geometry if it is equipped with the minimal compatible bornology and if forevery entourage U of X there exists a uniform finite upper bound on the cardinalities of U -bounded subsets of X . Furthermore recall the functor C s from (5.6). Theorem 11.16.
Assume:1. X has strongly bounded geometry.2. X is Γ -finite.3. The action of Γ on X is proper (Example 2.13).4. X is eventually coarsely connected.Then the morphisms C s ( α Rips( X ) , Γ can,min ) and C s ( β X, Γ max,max ) are equivalent. Remark 11.17.
Note that X being Γ-finite and the action of Γ on X being proper impliesthat X has the minimal bornology.Furthermore, if X is Γ-finite and U is a Γ-invariant entourage of X , then the assumptionthat every U -bounded subset is finite, already implies a uniform upper bound on thecardinality of U -bounded subsets. Hence Assumption 1 in Theorem 11.16 above could beequivalently replaced by the seemingly weaker assumption that every U -bounded subset isfinite. (cid:7) The rest of this section is devoted to the proof of Theorem 11.16.We consider a Γ-simplicial complex K . Recall the notation introduced in Example 9.11:1. K u denotes the Γ-uniform space associated to K .98. K u,max,max denotes the Γ-uniform bornological coarse space which has the uniformstructure of K u , but the maximal coarse and bornological structures.3. K u,d,max denotes the Γ-uniform bornological coarse space with the uniform structureof K u , the metric coarse structure and the maximal bornological structure.4. K d,max denotes the Γ-bornological coarse space underlying K u,d,max .We denote by β max,maxX,L : colim U ∈C Γ O ∞ ( P U ( X ) u,max,max ) ⊗ Yo s ( L ) → colim U ∈C Γ ΣYo s ( P U ( X ) max,max ⊗ L )and β d,maxX,L : colim U ∈C Γ O ∞ ( P U ( X ) u,d,max ) ⊗ Yo s ( L ) → colim U ∈C Γ ΣYo s ( P U ( X ) d,max ⊗ L )the forget-control maps.The proof of Theorem 11.16 consists of a sequence of lemmas.Let X and L be Γ-bornological coarse spaces. Lemma 11.18.
Assume:1. X has strongly bounded geometry.2. X is Γ -finite.3. The underlying set of L is a free Γ -set.Then α Rips( X ) ,L and β max,maxX,L are equivalent.Proof. The assumptions on X imply that P U ( X ) is Γ-compact and locally finite, henceΓ-metrizable, for every invariant coarse entourage U of X . Therefore, by (10.15), we havean equivalence O ∞ hlg ( P U ( X )) (cid:39) O ∞ ( P U ( X ) u,max,max ) . Using Definition 11.2 of the Rips complex as a filtered colimit of Γ-topological spaces andthe relation (10.13) we get O ∞ hlg (Rips( X )) (cid:39) colim U ∈C Γ O ∞ hlg ( P U ( X )) (cid:39) colim U ∈C Γ O ∞ ( P U ( X ) u,max,max ) . Hence we get the following commutative diagram:colim U ∈C Γ O ∞ ( P U ( X ) u,max,max ) ⊗ Yo s ( L ) β max,maxX,L (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) colim U ∈C Γ ΣYo s ( P U ( X ) max,max ⊗ L ) (cid:39) (cid:15) (cid:15) O ∞ hlg (Rips( X )) ⊗ Yo s ( L ) α Rips( X ) ,L (cid:15) (cid:15) O ∞ hlg ( ∗ ) ⊗ Yo s ( L ) (cid:39) (cid:47) (cid:47) ΣYo s ( L )99he vertical arrow on the right is an equivalence by Lemma 10.7.Recall the functor C s from (5.6). Let X be a Γ-bornological coarse space. Lemma 11.19.
Assume:1. X has strongly bounded geometry.2. X is eventually coarsely connected.Then C s ( β max,maxX, Γ can,min ) and C s ( β d,maxX, Γ can,min ) are equivalent.Proof. The morphism P U ( X ) u,d,max → P U ( X ) u,max,max of Γ-uniform bornological coarsespaces induces a diagram C s ( O ∞ ( P U ( X ) u,d,max ) ⊗ Yo s (Γ can,min )) β d,maxX, Γ can,min (cid:47) (cid:47) (cid:15) (cid:15) C s (ΣYo s ( P U ( X ) d,max ⊗ Γ can,min )) (cid:15) (cid:15) C s ( O ∞ ( P U ( X ) u,max,max ) ⊗ Yo s (Γ can,min )) β max,maxX, Γ can,min (cid:47) (cid:47) C s (ΣYo s ( P U ( X ) max,max ⊗ Γ can,min ))Since P U ( X ) u,d,max → P U ( X ) u,max,max is a coarsening, the left vertical arrow is an equiva-lence by Proposition 9.33. To show that the right vertical arrow is an equivalence for largeentourages U we will need continuity.For an invariant entourage U of X we denote the set of finite subcomplexes of P U ( X )by F ( P U ( X )). It is a filtered partially ordered set with respect to the inclusion relation.For every finite subcomplex F we define a Γ-invariant subcomplex D F := Γ( F × { } ) of P U ( X ) × Γ. We consider the family of Γ-invariant subsets D := ( D F ) F ∈F ( P U ( X )) of P U ( X ) × Γ. By Example 5.11 the family D is a co-Γ-bounded exhaustion of both spaces P U ( X ) max,max ⊗ Γ can,min and P U ( X ) d,max ⊗ Γ can,min . By continuity, it suffices to show thatthe bornological coarse structures on D induced from P U ( X ) max,max ⊗ Γ can,min and from P U ( X ) d,max ⊗ Γ can,min , respectively, agree.Since the bornologies of P U ( X ) max,max ⊗ Γ can,min and P U ( X ) d,max ⊗ Γ can,min agree, we onlyhave to care about the coarse structures. Let U be large enough, such that P U ( X ) isconnected.The coarse structure on D F induced by P U ( X ) d,max ⊗ Γ can,min is generated by the entourages( D F × D F ) ∩ ( U r × V B ) , where U r is a metric entourage of P U ( X ) of size r in (0 , ∞ ) and V B := Γ( B × B ) for afinite subset B is one of the generating entourages of the canonical structure of Γ.The coarse structure on D F induced by P U ( X ) max,max ⊗ Γ can,min is generated by theentourages ( D F × D F ) ∩ (( P U ( X ) × P U ( X )) × V B )100or finite subsets B of Γ. It is clear that the coarse structure of the latter is larger thanthe one of the first, and it remains to show to other inclusion.We have( D F × D F ) ∩ (( P U ( X ) × P U ( X )) × V B ) ∼ = (cid:91) ( γ,γ (cid:48) ) ∈ V B ( γF × γ (cid:48) F ) × { ( γ, γ (cid:48) ) } . Since F is a finite subcomplex and P U ( X ) is connected, there exists an r in (0 , ∞ ) suchthat BF × BF ⊆ U r . This implies that( γF × γ (cid:48) F ) × { ( γ, γ (cid:48) ) } ⊆ U r × V B for all pairs ( γ, γ (cid:48) ) in V B . We conclude that( D F × D F ) ∩ (( P U ( X ) × P U ( X )) × V B ) ⊆ ( D F × D F ) ∩ ( U r × V B ) . This finishes the proof.Let X be a Γ-bornological coarse space. Lemma 11.20.
Assume:1. X is Γ -finite.2. X has strongly bounded geometry.3. the Γ -action on X is proper.Then the maps C s ( β d,maxX, Γ can,min ) and C s ( β X, Γ can,max ) are equivalent.Proof. Recall from Definition 11.5 that P U ( X ) bd denotes the Γ-bornological coarse spacewhose coarse structure is induced from the metric and whose bornology is generated bythe subsets P U ( B ) for all bounded subsets B of X . Recall furthermore that we write P U ( X ) bdu for the corresponding Γ-uniform bornological coarse space.Recall from (5.7) that Yo sc (cid:39) C s ◦ Yo s . It suffices to produce diagramsYo sc ([0 , k ] ⊗ P U ( X ) bd ⊗ Γ can,max ) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) C s ( O ( P U ( X ) bdu ) ⊗ Yo s (Γ can,max )) (cid:39) (cid:15) (cid:15) Yo sc ([0 , k ] ⊗ P U ( X ) d,max ⊗ Γ can,min ) (cid:47) (cid:47) C s ( O ( P U ( X ) u,d,max ) ⊗ Yo s (Γ can,min ))for every natural number k which are compatible with increasing k and U .To produce the diagram we will use continuity with the exhaustion Y = ( Y κ ) κ ∈F ( P U ( X )) N from Lemma 5.12, where F ( P U ( X )) denotes the set of all finite subcomplexes of P U ( X ).Recall that for κ in F ( P U ( X )) N we set Y κ := (cid:91) n ∈ N [ n − , n ] × D κ ( n ) . (11.4)101ince P U ( X ) bd and P U ( X ) d,max are Γ-bounded the exhaustion is trapping by Lemma 5.12for both spaces O ( P U ( X ) u,d,max ) ⊗ Yo s (Γ can,min ) and O ( P U ( X ) bdu ) ⊗ Yo s (Γ can,max ) . Note that the hybrid coarse structure does not play a role here since trapping exhaustionsare a bornological concept.Since the definition of the exhaustion is independent of k and compatible with increasing U , it remains for us to show that the bornological coarse structures on Y κ induced from O ( P U ( X ) bdu ) ⊗ Yo s (Γ can,max ) and O ( P U ( X ) u,d,max ) ⊗ Yo s (Γ can,min ), respectively, agree inorder to obtain (by continuity) the equivalences in the above diagram. Since the coarsestructures of O ( P U ( X ) bdu ) ⊗ Yo s (Γ can,max ) and O ( P U ( X ) u,d,max ) ⊗ Yo s (Γ can,min ) agree, weonly have to consider the bornologies.Every bounded subset of O ( P U ( X ) u,d,max ) ⊗ Yo s (Γ can,min ) or O ( P U ( X ) bdu ) ⊗ Yo s (Γ can,max )is contained in [0 , n ] × P U ( X ) × Γ for some n . It therefore suffices to see that the inducedbornologies on ([0 , n ] × P U ( X ) × Γ) ∩ Y κ coincide. We can now further finitely decompose([0 , n ] × P U ( X ) × Γ) ∩ Y κ ⊆ n +1 (cid:91) i =1 [ i − , i ] × D κ ( i ) . It suffices to show that the induced bornologies on [ i − , i ] × D κ ( i ) coincide. For this wehave to show that for every F ∈ F ( P U ( X )) the bornologies on D F induced from P U ( X ) bd and P U ( X ) d,max , respectively, agree.Since X is Γ-finite and the Γ-action on X is proper, X carries the minimal bornology.Consequently, every bounded subset of P U ( X ) bd is contained in a finite subcomplex. Hence,the bornology on D F induced by P U ( X ) bd is generated by the sets D F ∩ ( F (cid:48) × Γ) for all F (cid:48) in F ( P U ( X )). This set is equal to (cid:0) (cid:91) γ ∈ Γ γF × { γ } (cid:1) ∩ ( F (cid:48) × Γ) = (cid:91) { γ ∈ Γ | γF ∩ F (cid:48) (cid:54) = ∅} ( γF ∩ F (cid:48) ) × { γ } . Note that the index set of the union on the right hand side is finite since the Γ-action isproper.The bornology induced by P U ( X ) d,max is generated by the sets D F ∩ ( P U ( X ) × B ) for allfinite subsets B of Γ. This set can be written in the form (cid:91) γ ∈ B γF × { γ } . The families of subsets (cid:16) (cid:91) { γ ∈ Γ | γF ∩ F (cid:48) (cid:54) = ∅} ( γF ∩ F (cid:48) ) × { γ } (cid:17) F (cid:48) ∈F ( P U ( X )) and (cid:16) (cid:91) γ ∈ B γF × { γ } (cid:17) B ⊆ Γ , | B | < ∞ generate the same bornologies. This finishes the proof of the lemma.102et X be a Γ-bornological coarse space. Lemma 11.21.
Assume:1. X has strongly bounded geometry.2. X is Γ -finite.Then the morphisms C s ( β X, Γ can,max ) and C s ( β X, Γ max,max ) are equivalent.Proof. The morphism of Γ-bornological coarse spaces Γ can,max → Γ max,max induces thecommutative diagram C s ( O ∞ ( P U ( X ) bdu ) ⊗ Yo s (Γ can,max )) (cid:47) (cid:47) (cid:15) (cid:15) ΣYo sc ( P U ( X ) bd ⊗ Γ can,max ) (cid:15) (cid:15) C s ( O ∞ ( P U ( X ) bdu ) ⊗ Yo s (Γ max,max )) (cid:47) (cid:47) ΣYo sc ( P U ( X ) bd ⊗ Γ max,max ) (11.5)The functor O ∞ from Γ UBC to Sp X is homotopy invariant and excisive for equivariantuniform decompositions. Since X is Γ-finite and has strongly bounded geometry, for everyinvariant entourage U the complex P U ( X ) is a Γ-finite simplicial complex. Using excisionand homotopy invariance we conclude that the left vertical map in the above diagram isan equivalence if C s ( O ∞ ( S ) ⊗ Yo s (Γ can,max )) → C s ( O ∞ ( S ) ⊗ Yo s (Γ max,max ))is an equivalence for every Γ-uniform bornological coarse space S which is a transitiveΓ-set that has the minimal bornology and the discrete uniform structure. Note that inthis case O ∞ ( S ) (cid:39) O ∞ ( S disc,min,min ) (cid:39) ΣYo s ( S min,min ) , since S disc,min,min → S is a coarsening and O ( S disc,min,min ) is flasque.If we can show that for every Γ-bounded Γ-bornological coarse space X the mapYo sc ( X ⊗ Γ can,max ) → Yo sc ( X ⊗ Γ max,max )is an equivalence, then we can conclude that the right vertical map in (11.5) is anequivalence, and by the above argument the left vertical map is an equivalence, too. Thelemma then follows by taking the colimit over all invariant entourages U of X .By continuity, it suffices to show that if F is a locally finite, invariant subset of X × Γ ? ,max (the coarse structure on Γ does not matter since local finiteness is a bornological concept),then the bornological coarse structures on F induced by X × Γ can,max and X × Γ max,max ,respectively, agree. Since the bornologies are the same, we only have to care about thecoarse structures.Every entourage of F Γ can,max is an entourage of F Γ max,max . So it remains to show the otherinclusion. We choose a bounded subset A of X such that Γ A = X . Let U be an invariantentourage of X containing the diagonal. Then U [ A ] is bounded. Furthermore, we have103 ⊆ Γ( A × U [ A ]). The set W (cid:48) := F ∩ ( U [ A ] × Γ) is finite since F is locally finite and U [ A ] × Γ is bounded. We let W denote the projection of W (cid:48) to Γ. Then we have( U × Γ × Γ) ∩ ( F × F ) ⊆ ( U × Γ( W × W )) ∩ ( F × F ) . Now note that Γ( W × W ) is an entourage of Γ can,max . This shows that every entourage of F Γ max,max is an entourage of F Γ can,max .Theorem 11.16 follows from combining Lemma 11.18 (with L = Γ can,min ), Lemma 11.19,Lemma 11.20 and Lemma 11.21. Recall Definition 11.9 of the three functors F , F , and F ∞ . In this section we analyzethe homological properties of the functor F ∞ . It turns out that this functor is almost acoarse homology theory. The only problematic axiom is vanishing on flasques. In order toimprove on this point recall the definition of Γ Sp X wfl from Definition 4.20 and considerthe composition F ∞ wfl : Γ BornCoarse F ∞ −−→ Γ Sp X → Γ Sp X wfl . In a similar manner, we derive functors F and F wfl from F and F , respectively. Forevery Γ-bornological coarse space X we have a fiber sequence in Γ Sp X wfl F ( X ) → F wfl ( X ) → F ∞ wfl ( X ) β X, wfl −−−→ Σ F ( X ) . (11.6)The morphism β X, wfl is a version of the forget-control morphism from Definition 11.10. Proposition 11.22.
The functor F ∞ wfl is an equivariant Γ Sp X wfl -valued coarse homologytheory.Proof. We verify the axioms.1. (Coarse invariance) We consider a Γ-bornological coarse space X . For i in { , } let ι i : X → { , } max,max ⊗ X denote the corresponding inclusions. It suffices to show that F ∞ wfl ( ι ) and F ∞ wfl ( ι ) areequivalent. For every invariant entourage U of X we consider the invariant entourage (cid:101) U := { , } × U of { , } max,max ⊗ X . Then the map[0 , du ⊗ P U ( X ) bdu → P (cid:101) U ( { , } max,max ⊗ X ) bdu given by ( t, µ ) (cid:55)→ (1 − t ) ι , ∗ µ + tι , ∗ µ is a homotopy between the morphisms of Γ-uniform bornological coarse spaces P U ( X ) bdu → P (cid:101) U ( { , } max,max ⊗ X ) bdu ι and ι .By the homotopy invariance of the functor O ∞ (Corollary 9.38) we conclude thatthe morphisms O ∞ ( P U ( X ) bdu ) → O ∞ ( P (cid:101) U ( { , } max,max ⊗ X ) bdu )induced by ι and ι are equivalent. Since the entourages of the form (cid:101) U for all U in C Γ are cofinal in the entourages of { , } ⊗ X , we get the equivalence of F ∞ wfl ( ι ) and F ∞ wfl ( ι ) as desired.2. (Excision) Let X be a Γ-bornological coarse space and Z an invariant subset. For aninvariant entourage U of X the subset P U ( Z ) of P U ( X ) bdu is invariant and closed.Let ( Y , Z ) be an equivariant complementary pair on X with Y = ( Y i ) i ∈ I . Let i in I be such that Y i ∪ Z = X . Let i in I be such that U [ Y i ] ⊆ Y i . Then for every i in I with i ≥ max { i , i } we have P U ( Y i ) ∪ P U ( Z ) = P U ( X ). The pair of invariantsubsets ( P U ( Y i ) , P U ( Z )) is then an equivariant uniform decomposition of P U ( X ) bdu .By Corollary 9.36 and Remark 9.37 the functor O ∞ sends equivariant uniformdecompositions to push-outs. We conclude that for i in I with i ≥ max { i , i } wehave a push-out O ∞ ( P U ( Z ∩ Y i ) bdu ) (cid:47) (cid:47) (cid:15) (cid:15) O ∞ ( P U ( Y i ) bdu ) (cid:15) (cid:15) O ∞ ( P U ( Z ) bdu ) (cid:47) (cid:47) O ∞ ( P U ( X ) bdu )Since colimits of push-out squares are push-out squares, we now take the colimitsover the invariant entourages U in C Γ and over i in I to get the push-out square F ∞ ( Z ∩ Y ) (cid:47) (cid:47) (cid:15) (cid:15) F ∞ ( Y ) (cid:15) (cid:15) F ∞ ( Z ) (cid:47) (cid:47) F ∞ ( X )We get the desired push-out F ∞ wfl ( Z ∩ Y ) (cid:47) (cid:47) (cid:15) (cid:15) F ∞ wfl ( Y ) (cid:15) (cid:15) F ∞ wfl ( Z ) (cid:47) (cid:47) F ∞ wfl ( X )3. (Flasqueness) We assume that X is flasque with the flasqueness implemented by theequivariant map f : X → X . For an invariant entourage U of X with the property(id , f )(diag X ) ⊆ U we form the entourage of X (cid:101) U := (cid:91) n ∈ N ( f n × f n )( U ) . f × f )( (cid:101) U ) ⊆ (cid:101) U . Therefore we have a morphism P (cid:101) U ( f ) : P (cid:101) U ( X ) bdu → P (cid:101) U ( X ) bdu . Like every simplicial map it is distance decreasing. Moreover, for every µ ∈ P (cid:101) U ( X )we have d ( µ, P (cid:101) U ( f )( µ )) ≤ . Finally, if B is a bounded subset of X and n is an integer such that (cid:101) U [ B ] ∩ f n ( X ) = ∅ ,then P (cid:101) U ( f n )( P (cid:101) U ( X )) ∩ P (cid:101) U ( B ) = ∅ . We conclude that P (cid:101) U ( f ) implements flasqueness of the bornological coarse space P (cid:101) U ( X ) bd . The set of invariant entourages of the form (cid:101) U as above is cofinal in allinvariant entourages of X . Therefore, we get F ( X ) (cid:39) F ( X ) (cid:39) O ( P (cid:101) U ( f )) implements weak flasqueness of O ( P (cid:101) U ( X ) bdu ). In thefollowing we verify the conditions stated in Definition 4.18.Since f is U -close to id X , as in 1 we can conclude that the map P (cid:101) U ( f ) bdu is uniformlyhomotopic to id P (cid:101) U ( X ) bdu . By the homotopy invariance of O we conclude thatYo s ( O ( P (cid:101) U ( f ))) (cid:39) id Yo s ( O ( P (cid:101) U ( f ) bdu )) as required in Definition 4.18.1.In order to save notation we define the map Q : P ([0 , ∞ ) × P (cid:101) U ( X ) × [0 , ∞ ) × P (cid:101) U ( X )) → P ([0 , ∞ ) × P (cid:101) U ( X ) × [0 , ∞ ) × P (cid:101) U ( X ))by Q ( V ) := (cid:91) n ∈ N (cid:0) ([0 , ∞ ) × P (cid:101) U ( f )) n × ([0 , ∞ ) × P (cid:101) U ( f )) n (cid:1) ( V ) . Let now V be an entourage of O ( P (cid:101) U ( X ) bdu ). We must show that Q ( V ) is again anentourage of O ( P (cid:101) U ( X ) bdu ). After enlarging V we can assume that it is of the form V = U ψ ∩ W r as in the proof of Lemma 9.27, where the function φ (which is the firstcomponent of ψ ) is such that φ ( i ) is a uniform entourage of the form U r ( i ) for every i in N , see (9.1). Since P (cid:101) U ( f ) is distance decreasing we see that Q ( W r ) ⊆ W r . Since P (cid:101) U ( f ) preserves the first coordinate of the cone and is distance decreasing we alsosee that Q ( U ψ ) ⊆ U ψ . Hence we actually get Q ( V ) ⊆ V .Finally, for every bounded subset A of O ( P (cid:101) U ( X ) bdu ) there exists r in (0 , ∞ ) and abounded subset B of X such that A ⊆ [0 , r ] × P (cid:101) U ( B ). We can choose an integer n such that f n ( X ) ∩ B = ∅ . Then O ( P (cid:101) U ( f )) n ( O ( P (cid:101) U ( X ) bdu )) ∩ A = ∅ .We conclude that Yo s wfl ( O ( P (cid:101) U ( X ) bdu )) (cid:39) . U and again using thecofinality of the resulting family of entourages (cid:101) U we get F wfl ( X ) (cid:39) F ∞ wfl ( X ) (cid:39) .
4. ( u -continuity) This is just a cofinality check:colim U ∈C Γ F ∞ wfl ( X U ) (cid:39) colim U ∈C Γ colim V ∈C(cid:104) U (cid:105) Γ O ∞ wfl ( P V ( X ) bdu ) (cid:39) colim V ∈C Γ O ∞ wfl ( P V ( X ) bdu ) (cid:39) F ∞ wfl ( X ) . This finishes the proof of Proposition 11.22.
Remark 11.23.
Let X be flasque. In the above proof we have shown that F wfl ( X ) (cid:39) F ( X ) (cid:39) (cid:7) Let E be a strong Γ-equivariant C -valued coarse homology theory. Then we have anessentially unique factorization E wfl : Γ Sp X wfl → C . The composition E wfl ◦ F ∞ wfl : Γ BornCoarse → C is then a Γ-equivariant C -valued coarse homology theory. We have an equivalence E ◦ F ∞ (cid:39) E wfl ◦ F ∞ wfl . Corollary 11.24. If E is a strong equivariant C -valued coarse homology theory, then E ◦ F ∞ : Γ BornCoarse → C is a Γ -equivariant coarse homology theory. Let E be an equivariant coarse homology theory and Q be a Γ-bornological coarse space. Lemma 11.25. If E is strong, then E Q is also strong.Proof. If X is a weakly flasque Γ-bornological coarse space with weak flasqueness imple-mented by f : X → X , then f ⊗ id Q implements weak flasqueness of X ⊗ Q . This impliesthe lemma.If the underlying Γ-set of Q is free, then by Corollary 11.14 we have E Q ( F ( X )) (cid:39) E ( F ( X ) ⊗ Yo s ( Q )) (cid:39) E ( X ⊗ Q ) (cid:39) E Q ( X ) . In particular, the functor E Q ◦ F : Γ BornCoarse → C is an equivariant coarse homology theory.Let Q be a Γ-bornological coarse space and E be an equivariant coarse homology theory.107 orollary 11.26. If E is strong and the underlying Γ -set of Q is free, then the forget-control map β : E Q ◦ F ∞ → Σ E Q ◦ F is a transformation between equivariant coarse homology theories. This aspect of the theory (in the case of a trivial group Γ) is further studied in [BE17].
References [Bar17] Clark Barwick. Spectral Mackey functors and equivariant algebraic K -theory(I). Adv. Math. , 304:646–727, 2017.[BE16] U. Bunke and A. Engel. Homotopy theory with bornological coarse spaces.arXiv:1607.03657v3, 2016.[BE17] U. Bunke and A. Engel. Coarse assembly maps. arXiv:1706.02164v2, 2017.[BEKW] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Injectivity results forcoarse homology theories. In preparation.[BEKW17] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Coarse homologytheories and finite decomposition complexity. arXiv:1712.06932, 2017.[BFJR04] A. Bartels, T. Farrell, L. Jones, and H. Reich. On the isomorphism conjecturein algebraic K -theory. Topology , 43(1):157–213, 2004.[BGS15] C. Barwick, S. Glasman, and J. Shah. Spectral Mackey functors and equivariantalgebraic K -theory (II). arXiv:1505.03098, 2015.[BL11] A. Bartels and W. L¨uck. The Farrell-Hsiang method revisited. Math. Ann. ,354:209–226, 2011. arXiv:1101.0466.[BLR08] A. Bartels, W. L¨uck, and H. Reich. The K -theoretic Farrell–Jones conjecturefor hyperbolic groups. Invent. math. , 172:29–70, 2008. arXiv:math/0701434.[BR07] A. Bartels and H. Reich. Coefficients for the Farrell–Jones Conjecture.
Adv. Math. , 209:337–362, 2007.[Car95] G. Carlsson. On the algebraic K -theory of infinite product categories. K -Theory , 9(4):305–322, 1995.[CP97] M. C´ardenas and E. K. Pedersen. On the Karoubi filtration of a category. K -Theory , 12(2):165–191, 1997.[DL98] J. F. Davis and W. L¨uck. Spaces over a Category and Assembly Maps inIsomorphism Conjectures in K - and L -Theory. K -Theory , 15:201–252, 1998.[Dow52] C. H. Dowker. Topology of Metric Complexes. Amer. J. Math. , 74(3):555–577,1952. 108Eng18] Alexander Engel. Wrong way maps in uniformly finite homology and homologyof groups.
J. Homotopy Relat. Struct. , 13(2):423–441, 2018.[ES52] S. Eilenberg and N. Steenrod.
Foundations of Algebraic Topology , volume 15of
Princeton Mathematical Series . Princeton University Press, 1952.[GGN15] D. Gepner, M. Groth, and Th. Nikolaus. Universality of multiplicative infiniteloop space machines.
Algebr. Geom. Topol. , 15(6):3107–3153, 2015.[GW13] J. R. J. Groves and J. S. Wilson. Soluble groups with a finiteness conditionarising from Bredon cohomology.
Bull. London Math. Soc. , 45(1):89–92, 2013.[JPL06] D. Juan-Pineda and I. J. Leary. On classifying spaces for the family ofvirtually cyclic subgroups. In
Recent Developments in Algebraic Topology, AConference to Celebrate Sam Gitler’s 70h Birthday, Dec. 2003 , volume 407 of
Contemporary Mathematics , 2006.[Kas15] D. Kasprowski. On the K -theory of groups with finite decomposition com-plexity. Proc. London Math. Soc. , 110(3):565–592, 2015.[KMPN09] D.H. Kochloukova, C. Martinez-Perez, and B.E.A. Nucinkis. Cohomologi-cal finiteness conditions in Bredon cohomology.
Bull. London Math. Soc. ,43(1):124–136, 2009.[KW17] D. Kasprowski and Ch. Winges. Shortening binary complexes and commuta-tivity of K -theory with infinite products. arXiv:1705.09116, 2017.[Lur09] J. Lurie. Higher topos theory , volume 170 of
Annals of Mathematics Studies
Equivariant homotopy and cohomology theory , volume 91 of
CBMSRegional Conference Series in Mathematics . Published for the ConferenceBoard of the Mathematical Sciences, Washington, DC; by the AmericanMathematical Society, Providence, RI, 1996.[Mil59] J. Milnor. On Spaces Having the Homotopy Type of a CW-Complex.
Trans-actions of the American Mathematical Society , 90(2):272–280, 1959.[Mit01] P. D. Mitchener. Coarse homology theories.
Algebr. Geom. Topol. , 1:271–297,2001.[Mit10] P. D. Mitchener. The general notion of descent in coarse geometry.
Algebr.Geom. Topol. , 10:2419–2450, 2010.[ML98] S. Mac Lane.
Categories for the Working Mathematician . Springer, secondedition, 1998.[Qui73] D. Quillen. Higher algebraic K -theory. I. pages 85–147. Lecture Notes inMath., Vol. 341, 1973. 109Sch04] M. Schlichting. Delooping the K -theory of exact categories. Topology ,43(5):1089–1103, 2004.[Sch06] M. Schlichting. Negative K -theory of derived categories. Math. Z. , 253(1):97–134, 2006.[vPW16] T. von Puttkamer and X. Wu. On the finiteness of the classifying space forthe family of virtually cyclic subgroups. arXiv:math/1607.03790, 2016.[vPW17] T. von Puttkamer and X. Wu. Linear Groups, Conjugacy Growth, andClassifying Spaces for Families of Subgroups. To appear in Int. Math. Res.Notices, arXiv:math/1704.05304v2, 2017.[Wal85] F. Waldhausen. Algebraic K -theory of spaces. In Algebraic and geometrictopology (New Brunswick, N.J., 1983) , volume 1126 of
Lecture Notes in Math. ,pages 318–419. Springer, Berlin, 1985.[Wri02] N. J. Wright. C coarse geometry . PhD thesis, Pennsylvania State University,2002.[Yu95] G. Yu. Baum–Connes Conjecture and Coarse Geometry. K -Theory-Theory