Coarse homology theories and finite decomposition complexity
Ulrich Bunke, Alexander Engel, Daniel Kasprowski, Christoph Winges
CCoarse homology theories and finitedecomposition complexity
Ulrich Bunke ∗ Alexander Engel † Daniel Kasprowski ‡ Christoph Winges § December 20, 2017
Abstract
Using the language of coarse homology theories, we provide an axiomatic accountof vanishing results for the fibres of forget-control maps associated to spaces withequivariant finite decomposition complexity.
Contents G -FDC for bornological coarse spaces and the main theorem . . . . . . . . 113.3 Structure of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ∗ 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 ] D ec .4 Vanishing pairs and strong decomposability . . . . . . . . . . . . . . . . . 143.5 Properties of the Rips complex . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Vanishing is closed under decomposition . . . . . . . . . . . . . . . . . . . 243.7 Proof of Theorem 3.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 References 32
For a group G we consider the category G BornCoarse of G -bornological coarse spaces[BEKW17, Sec. 2]. In [BEKW17, Sec. 3 & 4] we introduced the notion of an equivariantcoarse homology theory and constructed the universal equivariant coarse homology theoryYo s : G BornCoarse → G Sp X whose target is the presentable stable ∞ -category of equivariant coarse motivic spectra.To every G -bornological coarse space X one can functorially associate the motivic forget-control map β X : F ∞ ( X ) → Σ F ( X ) (1.1)which is a morphism in G Sp X (see [BEKW17, Def. 11.10] and (1.3) below). If E is a C -valued equivariant coarse homology theory, then it factors in an essentially unique wayover G Sp X and we get the induced forget-control map E ( β X ) : E ( F ∞ ( X )) → Σ E ( F ( X ))for E . The goal of the present paper is to show the following theorem: Theorem 1.1.
Assume:1. E is weakly additive (see Definition 2.11).2. E admits weak transfers (see Definition 2.4).3. C is compactly generated.4. X has G -finite decomposition complexity ( G -FDC) (see Definition 3.14).5. G acts discontinuously on X (see Definition 3.15).Then the forget-control map for E is an equivalence E ( β X ) : E ( F ∞ ( X )) → Σ E ( F ( X )) .
2n the case that X is the group G with the canonical bornological coarse structure and theaction by left multiplication (we often denote this object by G can,min ), the forget-controlmap is closely related to the assembly map which appears in Farrell–Jones type conjectures,see [BEKW17, Sec. 11.2] for details. Theorem 1.1 will be used in [BEKWa] to obtainsplit-injectivity results for the assembly map.Theorem 1.1 applies to equivariant coarse algebraic K -homology K A X G with coefficientsin an additive category A with a strict G -action, see Example 3.20. In this case we willreprove Novikov-type results for several classes of groups including a large class of lineargroups in [BEKWa].In the following we give a more detailed description of the result. In [BEKW17, Sec. 9]we introduced the category of G -uniform bornological coarse space G UBC . This categoryis related with G BornCoarse by a forgetful functor F U : G UBC → G BornCoarse (which forgets the uniform structure) and a cone functor O : G UBC → G BornCoarse . These functors are connected by a natural transformation F U → O whose motivic cofibre O ∞ := Cofib(Yo s ◦F → Yo s ◦O ) : G UBC → G Sp X is called the germs-at- ∞ -functor. By construction we have the cone fibre sequenceYo s ◦F U → Yo s ◦O → O ∞ ∂ −→ Σ Yo s ◦F U (1.2)of functors from G UBC to G Sp X .The Rips complex construction provides a functor in the other direction from G BornCoarse to G UBC . We let G BornCoarse C be the category of pairs ( X, U ) of a G -bornologicalspace X and an invariant entourage U of X . A morphism ( X, U ) → ( X (cid:48) , U (cid:48) ) in G BornCoarse C is a morphism f : X → X (cid:48) in G BornCoarse such that ( f × f )( U ) ⊆ U (cid:48) .Formally, the category G BornCoarse C is the Grothendieck construction of the functor G BornCoarse → Cat which sends a G -bornological coarse space ( X, C , B ) to the poset C G of invariant entourages. In [BEKW17, Sec. 11.1] or Definition 3.17 we introduce theRips complex functor P : G BornCoarse C → G UBC , ( X, U ) (cid:55)→ P U ( X ) . We form the left Kan-extension of the precomposition of the cone sequence (1.2) with P along the forgetful functor G BornCoarse C → G BornCoarse , ( X, U ) (cid:55)→ X .
The result is a fibre sequence F → F → F ∞ β −→ Σ F (1.3)3f functors from G BornCoarse to G Sp X . If we apply this fibre sequence to a bornologicalcoarse space X , then we get a fibre sequence in G Sp X whose third morphism is the motivicforget-control map (1.1).Let X be a G -bornological coarse space and E be an equivariant coarse homology theory.In view of the fibre sequence (1.3), Theorem 1.1 is an immediate consequence of thefollowing theorem: Theorem 1.2 (Theorem 3.19) . Assume:1. E is weakly additive.2. E admits weak transfers.3. C is compactly generated.4. X has G -FDC.5. G acts discontinuously on X .Then E ( F ( X )) (cid:39) . Note that the pointwise formula for the left Kan-extension gives E ( F ( X )) (cid:39) colim S ∈C G E ( O ( P U ( X )))whose right-hand side explicitly appears in Theorem 3.19. In order to deduce Theorem 1.2from Theorem 3.19 furthermore note that the condition that C is compactly generatedimplies that phantom objects (Definition 2.9) are trivial.The notion of finite decomposition complexity was introduced by Guentner, Tessera andYu [GTY13] as a generalization of finite asymptotic dimension. It was used in [GTY12]to prove instances of the stable Borel conjecture. Subsequently, Ramras, Tessera and Yu[RTY14] employed this notion to prove instances of the K -theoretic Novikov conjecture.Kasprowski [Kas14] introduced G -FDC as an equivariant version of finite decompositioncomplexity and proved that if X has G -FDC, then the forget-control map induces anequivalence K A X G ( β X ) in equivariant coarse algebraic K -homology K A X G associatedto an additive category A .The proofs in [RTY14] and [Kas14] use properties of the coarse algebraic K -homologyfunctor K A X which go beyond the four general properties of an equivariant coarsehomology listed in Definition 2.3. The main new contribution of the present paper is theobservation that in addition to the axioms of a coarse homology theory we just need thefollowing two additional properties:1. existence of weak transfers (see Definition 2.4).2. weak additivity (see Definition 2.11).We show that equivariant coarse ordinary homology H X G and equivariant coarse algebraic K -homology K A X G associated to an additive category A with G -action both admit weaktransfers and are weakly additive. Therefore, Theorem 1.1 applies to these examples andour main result directly implies and generalizes the results in [GTY12] and [Kas14].4 cknowledgements U.B. and A.E. were supported by the SFB 1085
Higher Invariants funded by the DeutscheForschungsgemeinschaft DFG, and A.E. was further supported by the Research FellowshipEN 1163/1-1
Mapping Analysis to Homology of the DFG. D.K. and C.W. acknowledgesupport by the Max Planck Society. Parts of the work presented here were obtained duringthe Junior Hausdorff Trimester Program
Topology at the Hausdorff Research Institute forMathematics (HIM) in Bonn. We also thank Mark Ullmann for helpful discussions.
We consider a group G . In [BEKW17, Sec. 2] we introduced the category G BornCoarse of G -bornological coarse spaces and equivariant proper and controlled maps. The notionof the free union of a family ( X i ) i ∈ I of G -bornological coarse spaces plays an importantrole in the present paper. Definition 2.1.
We define the free union (cid:96) free i ∈ I X i in G BornCoarse as follows:1. The underlying G -set of (cid:96) free i ∈ I X i is the disjoint union (cid:96) i ∈ I X i .2. A set B ⊆ (cid:96) free i ∈ I X i is bounded if and only if:a) The set |{ i ∈ I | B ∩ X i (cid:54) = ∅}| is finite.b) The set B ∩ X i is bounded (as a subset of X i ) for all i in I .3. The coarse structure of (cid:96) free i ∈ I X i is generated by the entourages (cid:70) i ∈ I U i for all families( U i ) i ∈ I , where U i is an entourage of X i . (cid:7) Remark 2.2.
Let ( X i ) i ∈ I be a family of G -bornological coarse spaces. For every j in I we have a canonical morphism X j → (cid:96) free i ∈ I X i . These morphisms induce a morphism (cid:97) i ∈ I X i → free (cid:97) i ∈ I X i , where (cid:96) i X i denotes the coproduct of the family ( X i ) i ∈ I in the category G BornCoarse .The coproduct can be realized such that this morphism is the identity on the underlyingsets. In general, it is not an isomorphism since the coarse structure on the free union islarger than the coarse structure on the categorical coproduct in G BornCoarse , while thebornology of the coproduct is larger than that of the free union. (cid:7)
Let C be a cocomplete stable ∞ -category and E : G BornCoarse → C be a functor. The following definition is taken from [BEKW17, Def. 3.10].5 efinition 2.3. The functor E is an equivariant C -valued coarse homology theory if itsatisfies the following properties:1. E is excisive for equivariant complementary pairs.2. E is coarsely invariant.3. E vanishes on flasque G -bornological coarse spaces.4. E is u -continuous. (cid:7) We consider a family ( X i ) i ∈ I of G -bornological coarse spaces and a point j in I . We set I j := I \ { j } . Then ( X j , (cid:96) free i ∈ I j X i ) is a coarsely excisive decomposition (see [BEKW17,Def. 4.12]) of (cid:96) free i ∈ I X i . Since E satisfies excision for coarsely excisive decompositions[BEKW17, Def. 4.13] we can define a projection p j : E ( free (cid:97) i ∈ I X i ) (cid:39) E ( X j ) ⊕ E ( free (cid:97) i ∈ I j X i ) → E ( X j ) . (2.1)Let I be a set and E : G BornCoarse → C be an equivariant coarse homology theory.Then we define a functor E I : G BornCoarse → C , X (cid:55)→ E ( free (cid:97) i ∈ I X ) . For every j in I the projection (2.1) then provides a natural transformation of functors p j : E I → E .
Definition 2.4. E has weak transfers for I if there exists a natural transformationtr I : E → E I such that p j ◦ tr I (cid:39) id E (2.2)for every j in I . (cid:7) Example 2.5.
Recall from [BEKW17, Def. 8.8], that equivariant coarse algebraic K -homology K A X G was constructed by assigning to a G -bornological coarse space X anadditive category V G A ( X ) of X -controlled objects and taking K -theory of this category.Let ( (cid:92) K A X G ) I := K ( (cid:89) I V G A ( X )) . Sending an object M of V G A ( (cid:96) free I ( X )) to the sequence ( M | X i ) i ∈ I , where X i is the i th copyof X in the free union, yields a natural transformation τ : ( K A X G ) I → ( (cid:92) K A X G ) I . τ is a natural equivalence. We thendefine tr I : K A X G K (∆) −−−→ ( (cid:92) K A X G ) I τ − −−→ ( K A X G ) I , where ∆ is the diagonal. Property (2.2) for the weak transfer follows from [BEKW17,Rem. 8.16]. (cid:7) Remark 2.6.
In [BEKWb] we introduce the notion of a coarse homology theory withtransfers. More precisely, we extend the category G BornCoarse to an ∞ -category G BornCoarse
Qtr whose additional morphisms encode a more general kind of transfers. In[BEKWb] we show that every coarse homology theory with transfers in particular hasweak transfers and we construct extensions of equivariant coarse ordinary homology H X G and equivariant coarse algebraic K -homology K A X G to coarse homology theories withtransfers. (cid:7) Let E : G BornCoarse tr → C be an equivariant C -valued coarse homology theory. Thefollowing Definition is taken from [BEKW17, Def. 3.12]. Definition 2.7. E is called strongly additive if for every family ( X i ) i ∈ I of G -bornologicalcoarse spaces the morphism ( p j ) j ∈ I : E ( free (cid:97) i ∈ I X i ) → (cid:89) j ∈ I E ( X j ) (2.3)is an equivalence. (cid:7) Many examples of equivariant coarse homology theories are strongly additive. The followinghas been shown in [BEKW17].
Theorem 2.8.
1. Equivariant coarse ordinary homology H X G is strongly additive [BEKW17, Lem. 7.11].2. Equivariant coarse algebraic K -homology K A X G with coefficients in an additivecategory A with a strict G -action is strongly additive [BEKW17, Prop. 8.19]. The arguments of the present paper use a weaker form of additivity we will now introduce.Let C be a stable ∞ -category. In the following an object K of C is called compact if it is ω -compact, i.e., the functor Map C ( K, − ) preserves filtered colimits. Definition 2.9.
1. An object C in C is called a phantom object if Map( K, C ) (cid:39) ∗ forevery compact object K of C . 7. A morphism f : C → D in C is called a phantom monomorphism if and only if forevery compact object K and morphism φ : K → C the condition f ◦ φ (cid:39) φ (cid:39) (cid:7) Example 2.10.
In general, if the fibre of a morphism in C is a phantom object, then themorphism is a phantom monomorphism. The converse is not true.If C is compactly generated, then every phantom object in C is equivalent to the zeroobject. This applies e.g. to the stable ∞ -categories Ch ∞ of chain complexes and of spectra Sp . (cid:7) Let E : G BornCoarse → C be an equivariant C -valued coarse homology theory. Definition 2.11. E is called weakly additive if the morphism( p i ) i ∈ I : E ( free (cid:97) i ∈ I X i ) → (cid:89) i ∈ I E ( X i )is a phantom monomorphism. (cid:7) Since every equivalence is a phantom monomorphism it is obvious that a strongly additiveequivariant coarse homology theory is weakly additive. Hence we have the followingcorollary of Theorem 2.8.
Corollary 2.12.
1. Equivariant coarse ordinary homology H X G is weakly additive [BEKW17, Lem. 7.11].2. Equivariant coarse algebraic K -homology K A X G with coefficients in an additivecategory A with a strict G -action is weakly additive [BEKW17, Prop. 8.19]. In this section we introduce the notion of G -equivariant finite decomposition complexity( G -FDC) for G -bornological coarse spaces. Finite decomposition complexity for metricspaces was introduced by Guentner, Tessera and Yu [GTY12] and the equivariant versionfor metric spaces was defined in [Kas14], see also [Kas16]. The main theorem of the sectionis Theorem 3.19 which can be interpreted as a version of [RTY14, Prop. 6.6] for generalweakly additive coarse homology theories with weak transfers.8 .1 Finite decomposition complexity Let G be a group. Recall that a G -coarse space is a coarse space ( X, C ) such that thesubset C G of invariant entourages is cofinal in C .Let T be an invariant entourage of a G -coarse space X . Furthermore, let U and V be G -invariant subsets of X . Definition 3.1.
The subsets
U, V are said to be T -disjoint if T [ U ] ∩ V = ∅ and U ∩ T [ V ] = ∅ . (cid:7) Let X be a G -set. Definition 3.2. An equivariant family of subsets of X is a family of subsets ( U i ) i ∈ I indexed by a G -set I such that U g ( i ) = g ( U i ) for every i in I and g in G . (cid:7) Let Y be a G -set, X be a G -coarse space with coarse structure C , and f : Y → X be anequivariant map of sets. Definition 3.3.
The induced G -coarse structure f − C on Y is the maximal G -coarsestructure on Y such that the map f is controlled. (cid:7) Let X be a G -set and X := ( X i ) i ∈ I be a partition of X into G -invariant subsets. Then weform the G -invariant entourage U ( X ) := (cid:71) i ∈ I X i × X i . If C is a G -coarse structure on X , then we define a new G -coarse structure C ( X ) := C(cid:104){ V ∩ U ( X ) | V ∈ C}(cid:105) . Let X be a G -coarse space with coarse structure C and U := ( U i ) i ∈ I be an equivariantfamily of subsets of X . Then we have a natural map of G -sets f : (cid:96) i ∈ I U i → X . Definition 3.4.
We define the G -coarse space (cid:96) sub i ∈ I U i to be the G -set (cid:96) i ∈ I U i with the G -coarse structure ( f − C )( U ). (cid:7) Note that the G -coarse structure ( f − C )( U ) is generated by the entouragessub U ( V ) := V ∩ U ( U )for all entourages V of the induced coarse structure f − C . Remark 3.5.
Note that the set of entourages { sub U ( V ) | V ∈ C G } of (cid:96) sub i ∈ I U i is cofinal inthe coarse structure of (cid:96) sub i ∈ I U i . (cid:7) emark 3.6. The coarse space (cid:96) sub i ∈ I U i is a coarsely disjoint union of the subsets U i whichall have the coarse structure induced from X via the inclusion U i → X . But the union isin general not free.There is a coarse map (cid:96) sub i ∈ I U i → X , but the coarse structure on the domain is in generalsmaller than the one induced from X . (cid:7) Let X be a G -bornological coarse space and ( U i ) i ∈ I an equivariant family of subsets of X . Definition 3.7.
We say that the family ( U i ) i ∈ I is nice if for every invariant entourage S of X containing the diagonal the natural morphism sub (cid:97) i ∈ I U i → sub (cid:97) i ∈ I S [ U i ]is an equivalence of G -bornological coarse spaces. (cid:7) Remark 3.8.
In the non-equivariant case (i.e., if G is the trivial group) every family ofsubsets of a coarse space is nice. In contrast, for non-trivial groups the inclusion of aninvariant subset of G -coarse space into its thickening need not be a coarse equivalence. (cid:7) Let X be a G -coarse space and let F be a class of G -coarse spaces. Definition 3.9.
We say that X is decomposable over F if for every invariant entourage T of X there exist pairwise T -disjoint and nice equivariant families ( U Ti ) i ∈ I and ( V Tj ) j ∈ J ofsubsets of X such that1. X = U T ∪ V T with U T := (cid:83) i ∈ I U Ti and V T := (cid:83) j ∈ J U Tj .2. The G -coarse spaces (cid:96) sub i ∈ I U Ti and (cid:96) sub j ∈ J V Tj belong to F . (cid:7) Let F be a class of G -coarse spaces. Definition 3.10.
We say that the class F is closed under decomposition if every G -coarsespace that is decomposable over F is contained in F . (cid:7) Remark 3.11.
Recall that the G -coarse structure C of a G -coarse space X induces a G -invariant equivalence relation (cid:83) U ∈C U on X . The equivalence classes for this relationare called coarse components. The group G acts on the set of coarse components π ( X ) inthe natural way.Furthermore recall that a subset Y of a coarse space X is called T -bounded for an entourage T of X if Y × Y ⊆ T . (cid:7) Let X be a G -coarse space. Definition 3.12. X is called semi-bounded if there exists an invariant entourage T of X such that every coarse component of X is T -bounded. (cid:7)
10e denote the class of semi-bounded G -coarse spaces by B . Definition 3.13.
Let D be the smallest class of G -coarse spaces that contains B and isclosed under decomposition. (cid:7) Let X be a G -coarse space. Definition 3.14. X has G -equivariant finite decomposition complexity (for short G -FDC )if it is contained in D . (cid:7) Let X be a G -coarse space with coarse structure C . Then X has a minimal bornology B C which is compatible with the coarse structure. It is generated by the subsets S [ x ] for all x in X and S in C . Definition 3.15.
The G -action on X is said to be discontinuous if for every point x of X and every B in B C the intersection Gx ∩ B is finite. (cid:7) G -FDC for bornological coarse spaces and the main theorem Let X be a G -bornological coarse space. Definition 3.16.
We say X has G -FDC if its underlying G -coarse space has G -FDC. (cid:7) We refer to [BEKW17, Def. 9.9] for the definition of the category of G -uniform bornologicalcoarse spaces.Let X be a G -bornological coarse space, Y be a G -invariant subset, and S be an invariantentourage. Definition 3.17.
We define the G -uniform bornological coarse space P XS ( Y ) as follows:1. The underlying G -set of P XS ( Y ) is the G -set of probability measures on X whosesupport is finite S -bounded and contained in Y .2. We consider P XS ( X ) as a G -simplicial complex with the G -equivariant sphericalquasi-metric. This quasi-metric induces the G -uniform structure on the subset P XS ( Y ).3. The quasi-metric from 2 also induces the G -coarse structure on the subset P XS ( Y ).4. The bornology of P XS ( X ) is generated by the subsets P XS ( B ) for all bounded subsets B of X . It induces the bornology of P XS ( Y ).We call P XS ( Y ) the Rips complex . (cid:7) In order to abbreviate the notation we will simplify the notation and write P S ( X ) := P XS ( X ) . emark 3.18. Let X be a G -bornological coarse space. If Y is a G -invariant subset, thenwe will use the notation Y also for the G -bornological coarse space obtained by equippingthis subset with the coarse structure and the bornological structure induced from X . Ifwe want to underline that the structures come from X , then we also use the more precisenotation Y X for this G -bornological coarse space.If S is an invariant entourage of X and we set S Y := ( Y × Y ) ∩ S , (3.1)then we have a natural morphism of G -uniform bornological coarse spaces P S Y ( Y ) → P XS ( Y ) . It is an isomorphism of the underlying bornological spaces. But the quasi-metric on theleft-hand side might be larger than the quasi-metric on the right-hand side. (cid:7)
The following is the main theorem of the present section: Let X be a G -bornological coarsespace with coarse structure C and let E be a weakly additive coarse homology theory withweak transfers. Theorem 3.19. If X has G -FDC and a discontinuous G -action, then colim S ∈C G E ( O ( P S ( X ))) is a phantom object. Example 3.20.
In view of Example 2.5, Remark 2.6 and Theorem 2.8, Theorem 3.19applies to equivariant coarse ordinary homology H X G and to equivariant coarse algebraic K -homology K A X G with coefficients in an additive category A with a strict G -action. (cid:7) The remaining part of this section is devoted to the proof of Theorem 3.19. We fix onceand for all a C -valued weakly additive (see Definition 2.11) coarse homology theory E with weak transfers (see Definition 2.4). Let X be a G -bornological coarse space with G -coarse structure C . Definition 3.21.
We call
X E -vanishing if for every G -invariant subset Y of X colim S ∈C G E ( O ( P S Y ( Y )))is a phantom object of C . (cid:7) Let V E denote the class of E -vanishing G -bornological coarse spaces.12 emark 3.22. Let Y be a G -invariant subset of X and let C ( Y ) denote the coarsestructure of Y X . Then we have an equality of sets { S Y | S ∈ C G } = C G ( Y ) (see (3.1) fornotation) of entourages of Y . Hence we have an equivalence of objects of C colim S ∈C G E ( O ( P S Y ( Y ))) (cid:39) colim T ∈C G ( Y ) E ( O ( P T ( Y ))) . We could define a notion of a weakly E -vanishing G -bornological coarse space X by justrequiring that colim S ∈C G E ( O ( P S ( X )))is a phantom object. Then X is E -vanishing if and only if all its G -invariant subsets areweakly vanishing.The motivation to define the notion of E -vanishing G -bornological coarse spaces as aboveis that it better suits the induction arguments below. (cid:7) Theorem 3.19 follows from the next theorem. Since every equivariant subspace of a G -bornological coarse space with G -FDC has again G -FDC, both theorems are actuallyequivalent. Theorem 3.23.
The class of G -bornological coarse spaces with G -FDC and discontinuous G -action is contained in V E . To prove Theorem 3.23 it suffices to show that the class V E contains all semi-bounded G -bornological coarse spaces with discontinuous G -action (Proposition 3.55) and is closedunder decomposition. We will proceed as follows:In Definition 3.34 we introduce the notion of an E -vanishing sequence of G -bornologicalcoarse spaces. Furthermore, in Definition 3.41 we define the concept of decomposability ofsequences of G -bornological coarse spaces. We let VS E denote the class of E -vanishingsequences. The main steps of the proof are now as follows:1. If a sequence of G -bornological coarse spaces is decomposable over VS E , then thesequence belongs to VS E (Theorem 3.48).2. If a G -bornological coarse space X is decomposable over V E , then the constantsequence X is decomposable over VS E (Corollary 3.43).3. By 1 and 2, if X is decomposable over V E , then X belongs to VS E .4. If X is a G -bornological coarse space and X belongs to VS E , then X belongs to V E (Lemma 3.36).5. By 3 and 4 we can conclude that if X is decomposable over V E , then it belongs to V E . 13 .4 Vanishing pairs and strong decomposability Let [0 , ∞ ) d denote the positive ray in R considered as a G -bornological coarse space withthe trivial G -action and the bornological coarse structure induced from the standardmetric.Let X be a G -uniform bornological coarse space with coarse structure C . Recall that thecone O ( X ) is obtained from the G -bornological coarse space [0 , ∞ ) d ⊗ X by replacingthe coarse structure C ⊗ of this tensor product by the hybrid structure C h . In particularwe have C h ⊆ C ⊗ . It follows that the projection pr : [0 , ∞ ) × X → X is a morphism([0 , ∞ ) × X, C h ) → ( X, C ) in the category of G -coarse spaces.For t in [0 , ∞ ) and x in X we will denote the corresponding point in O ( X ) by ( t, x ).Let X be a G -uniform bornological coarse space whose coarse structure is induced by ametric d . We set U r := { ( x, y ) ∈ X × X | d ( x, y ) ≤ r } . Let R be an entourage of O ( X ). Definition 3.24.
We define the propagation l ( R ) in [0 , ∞ ) by l ( R ) := inf { r ∈ [0 , ∞ ] | (pr × pr)( R ) ⊆ U r } . (cid:7) We consider a family ( X i ) i ∈ I of G -uniform bornological coarse spaces such that the coarsestructure of X i is induced by an invariant metric. Here we consider the metric as specified,but we will not introduce special notation for it. Recall that every entourage R of the freeunion (cid:96) free i ∈ I O ( X i ) is given by (cid:70) i ∈ I R i for a uniquely determined family ( R i ) i ∈ I , where R i is an entourage of X i . We define the propagation of R to be the element r ( R ) := sup i ∈ I l ( R i )in [0 , ∞ ]. Definition 3.25.
For r in (0 , ∞ ) we define (cid:96) semi( r ) i ∈ I O ( X i ) to be the G -bornological coarsespace given as follows:1. The underlying G -bornological space is the one of (cid:96) free i ∈ I O ( X i ).2. The G -coarse structure is generated by the entourages R of (cid:96) free i ∈ I O ( X i ) with propa-gation satisfying r ( R ) ≤ r .We further set semi (cid:97) i ∈ I O ( X i ) := colim r ∈ (0 , ∞ ) semi( r ) (cid:97) i ∈ I O ( X i ) . (cid:7) Note that the structure maps of the colimit above are given by the identity of the underlyingset. 14 emark 3.26.
In other words, the coarse structure of (cid:96) semi i ∈ I O ( X i ) is generated by thoseentourages of the free union which in addition have finite propagation. (cid:7) We have canonical morphisms semi( r ) (cid:97) i ∈ I O ( X i ) → semi (cid:97) i ∈ I O ( X i ) → free (cid:97) i ∈ I O ( X i ) . Example 3.27.
Let X be a G -bornological coarse space, Y be an invariant subset of X ,and S be an invariant entourage of X . By Definition 3.17 the Rips complex P XS ( Y ) is a G -uniform bornological coarse space with a specified invariant metric. This is our mainsource of examples. (cid:7) In the following we will use the abbreviated notation ( X n ) for sequences ( X n ) n ∈ N ofbornological coarse spaces indexed by N . If not said differently, by C n we denote the coarsestructure of X n .Let ( X n ) be a sequence of G -bornological coarse spaces and let ( Y n ) be a sequence of G -invariant subsets, i.e., we have Y n ⊆ X n . Let C Gn denote the partially ordered set of G -invariant entourages of X n and consider a family ( S n ) in (cid:81) n ∈ N C Gn . Lemma 3.28.
For d in N the inclusion of the underlying G -bornological spaces defines amorphism of G -bornological coarse spaces semi( d ) (cid:97) n ∈ N O ( P X n S n ( Y n )) → semi (cid:97) n ∈ N O ( P ( S dn ) Yn ( Y n )) . Furthermore, the canonical map colim ( S n ) ∈ (cid:81) n ∈ N C Gn E (cid:0) semi (cid:97) n ∈ N O ( P ( S n ) Yn ( Y n )) (cid:1) → colim ( S n ) ∈ (cid:81) n ∈ N C Gn E (cid:0) semi (cid:97) n ∈ N O ( P X n S n ( Y n )) (cid:1) is an equivalence.Proof. The second claim follows from the first by u -continuity of E .Assume that the sequence ( R n ) defines an entourage of the free union with propagationbounded by d . Fix a natural number n and consider ( t, y ) and ( t (cid:48) , y (cid:48) ) in O ( P X n S n ( Y n )). If(( t, y ) , ( t (cid:48) , y (cid:48) )) belongs to R n , then the distance between y and y (cid:48) in P X n S n ( Y n ) is boundedby d . There exists a family of points ( x i ) di =0 in X n such that x is a vertex of the simplexcontaining y and x d is a vertex of a simplex containing y (cid:48) , and for every i in { , . . . , d − } we have ( x i , x i +1 ) ∈ S n . Then x and x n belong to Y n , and the distance between x and x d in P ( S dn ) Yn ( Y n ) is bounded by 1. Consequently, the distance between y and y (cid:48) in P ( S dn ) Yn ( Y n )is bounded by 3. 15 emark 3.29. The proof shows that we actually have a morphism of G -bornologicalcoarse spaces semi( d ) (cid:97) n ∈ N O ( P X n S n ( Y n )) → semi(3) (cid:97) n ∈ N O ( P ( S dn ) Yn ( Y n )) . (cid:7) Let ( X n ) be a sequence of G -bornological coarse spaces and ( S n ) and ( S (cid:48) n ) be familiesin (cid:81) n ∈ N C Gn such that ( S n ) ≤ ( S (cid:48) n ). Then we have a commuting square of morphisms of G -bornological coarse spaces: semi (cid:96) n ∈ N O ( P S n ( X n )) (cid:47) (cid:47) (cid:15) (cid:15) free (cid:96) n ∈ N O ( P S n ( X n )) (cid:15) (cid:15) semi (cid:96) n ∈ N O ( P S (cid:48) n ( X n )) (cid:47) (cid:47) free (cid:96) n ∈ N O ( P S (cid:48) n ( X n ))In the following lemma we consider the colimit in the vertical direction. Lemma 3.30.
We have an equivalence colim ( S n ) ∈ (cid:81) n ∈ N C Gn E (cid:0) semi (cid:97) n ∈ N O ( P S n ( X n )) (cid:1) → colim ( S n ) ∈ (cid:81) n ∈ N C Gn E (cid:0) free (cid:97) n ∈ N O ( P S n ( X n )) (cid:1) . Proof.
We produce an inverse equivalence. Let R be an entourage of the free unionassociated to the family ( R n ). Then by the same argument as in the proof of Lemma 3.28the inclusion of the underlying sets induces a morphism of bornological coarse spaces.( free (cid:97) n ∈ N O ( P S n ( X n ))) R → semi(3) (cid:97) n ∈ N O ( P S l ( Rn ) n ( X n )) . By u -continuity of E applied to the colimit over the entourages R of the free union we getthe desired inverse.Let ( X n ) be a sequence of G -bornological coarse spaces and let ( Y n ) be a sequence of G -invariant subsets. Then we consider the object C ∞ (( Y n )) of C given by C ∞ (( Y n )) := colim N ∈ N colim ( S n ) ∈ (cid:81) n ∈ N C Gn E (cid:0) semi (cid:97) N ≤ n O ( P ( S n ) Yn ( Y n )) (cid:1) . (3.2) Remark 3.31.
Note that the connecting map for the outer colimit involves the projection E (cid:0) semi (cid:97) N ≤ n O ( P ( S n ) Yn ( Y n )) (cid:1) → E (cid:0) semi (cid:97) N +1 ≤ n O ( P ( S n ) Yn ( Y n )) (cid:1) E along the coarsely excisive pair (cid:0) O ( P ( S N ) YN ( Y N )) , semi (cid:97) N +1 ≤ n O ( P ( S n ) Yn ( Y n )) (cid:1) . (cid:7) Remark 3.32.
In view of the first part of Remark 3.22 we could rewrite the definition of C ∞ (( Y n )) as follows: C ∞ (( Y n )) := colim N ∈ N colim ( T n ) ∈ (cid:81) n ∈ N C Gn ( Y n ) E (cid:0) semi (cid:97) N ≤ n O ( P T n ( Y n )) (cid:1) . In particular, the object C ∞ (( Y n )) of C is an intrinsic invariant of the sequence of G -bornological coarse spaces ( Y n ). (cid:7) We furthermore define the free version C ∞ free (( Y n )) := colim N ∈ N colim ( T n ) ∈ (cid:81) n ∈ N C G ( Y n ) E (cid:0) free (cid:97) N ≤ n O ( P T n ( Y n )) (cid:1) . Then Lemma 3.30 has the following consequence:
Corollary 3.33.
The natural morphism C ∞ (( Y n )) → C ∞ free (( Y n )) is an equivalence. Let ( X n ) be a sequence of G -bornological coarse spaces. Definition 3.34.
The sequence ( X n ) is called E -vanishing if for every sequence ( Y n ) of G -invariant subsets the object C ∞ (( Y n )) is a phantom object. (cid:7) We let VS E denote the class of E -vanishing sequences.Let ( X n ) and ( Y n ) be sequences of G -bornological coarse spaces. Lemma 3.35.
Assume that for every natural number n the bornological coarse space X n is equivalent to Y n . Then the sequence ( X n ) is E -vanishing if and only if the sequence ( Y n ) is E -vanishing.Proof. By the symmetry of the assertion it suffices to show that if ( Y n ) is E -vanishing,then also ( X n ) is E -vanishing.By assumption, for every natural number n we can find morphisms of G -bornologicalcoarse spaces f n : X n → Y n and g n : Y n → X n and entourages U n of X n and V n of Y n suchthat g n ◦ f n is U n -close to id X n , and f n ◦ g n is V n -close to id Y n .Assume that S n in C Gn is given. If we choose S (cid:48) n in C G ( X n ) such that(( g n ◦ f n ) × ( g n ◦ f n ))( S n ) ∪ U n ⊆ S (cid:48) n , g n ◦ f n induces a map P S n ( X n ) → P S (cid:48) n ( X n ) which is homotopic to id X n by a unit-speed homotopy. We have a similar statement for the composition f n ◦ g n . By homotopyinvariance of the cone functor [BEKW17, Cor. 9.38] and a cofinality consideration thisimplies that f n induces an equivalencecolim S ∈C Gn E ( O ( P S ( X n ))) (cid:39) colim T ∈C G ( Y n ) E ( O ( P T ( Y n ))) (3.3)in C for every natural number n .We assume that ( Y n ) is E -vanishing. Then by Definition 3.34 the object C ∞ (( Y n )) is aphantom object. We show that C ∞ (( X n )) is a phantom object, too.Let K be a compact object in C and φ : K → C ∞ (( X n )) be a morphism. We must showthat φ is equivalent to zero.By compactness of K there is a factorization ˜ φ as in the following diagram: E ( free (cid:96) N ≤ n O ( P S n ( X n ))) (cid:15) (cid:15) !! (cid:40) (cid:40) (cid:47) (cid:47) E ( free (cid:96) N ≤ n O ( P T n ( Y n ))) (cid:40) (cid:40) K (cid:51) (cid:51) ! (cid:52) (cid:52) ˜ φ (cid:57) (cid:57) φ (cid:47) (cid:47) C ∞ (( X n )) (cid:81) N ≤ n E ( O ( P S n ( X n ))) (cid:47) (cid:47) (cid:81) N ≤ n E ( O ( P T n ( Y n )))If we choose the sequence of entourages ( T n ) sufficiently large (depending on the choice of( S n )), then the dashed arrows exist. Note that by Corollary 3.33 and our assumption weknow that C ∞ free (( Y n )) is a phantom object. Since K is compact, after increasing N furtherand choosing ( T n ) sufficiently large the dotted arrow becomes equivalent to zero. Now weuse (3.3) again in order to see that if we choose ( S n ) and ( T n ) sufficiently large, then thearrow marked by ! is equivalent to zero. But then ˜ φ is equivalent to zero since we assumethat E is weakly additive and hence the arrow marked by !! is a phantom monomorphism.This finally implies that φ is equivalent to zero.If X is a G -bornological coarse space, then we can consider the constant sequence X of G -bornological spaces indexed by N .Let X be a G -bornological coarse space with coarse structure C . Lemma 3.36. If X is an E -vanishing sequence, then X is E -vanishing.Proof. We must show that for every G -invariant subset Y of X the objectcolim S ∈C G E ( O ( P S Y ( Y ))) (3.4)is a phantom object. 18et K be a compact object of C and consider a morphism f : K → colim S ∈C G E ( O ( P S Y ( Y ))) . We must show that f is equivalent to zero.In the following we build step by step the following diagram: K (cid:27) (cid:27) ˜ f (cid:15) (cid:15) f (cid:47) (cid:47) f (cid:48) (cid:35) (cid:35) colim S ∈C G E ( O ( P S Y ( Y ))) tr N (cid:15) (cid:15) E ( O ( P R Y ( Y ))) (cid:52) (cid:52) tr N (cid:15) (cid:15) colim S ∈C G E ( free (cid:96) n ∈ N O ( P S Y ( Y ))) ! (cid:15) (cid:15) E ( free (cid:96) n ∈ N O ( P R Y ( Y ))) !!! (cid:15) (cid:15) (cid:53) (cid:53) colim ( S n ) ∈ (cid:81) n ∈ N C G E ( free (cid:96) n ∈ N O ( P S n ( Y ))) E ( free (cid:96) n ∈ N O ( P ( R n ) Y ( Y ))) (cid:53) (cid:53) E ( free (cid:96) n ≤ N O ( P ( R n ) Y ( Y ))) (cid:79) (cid:79) (cid:47) (cid:47) colim N ∈ N colim ( S n ) ∈ (cid:81) n ∈ N C G E ( free (cid:96) n ≤ N O ( P ( S n ) Y ( Y ))) !! (cid:79) (cid:79) First of all, by compactness of K there exists an invariant entourage R of X such that wehave the factorization ˜ f . The next square expresses the naturality of the transfer tr N (seeDefinition 2.4). The map marked by ! is the canonical map induced by the inclusion ofthe index set of the colimit in the domain into the index set of the colimit of its target.By Corollary 3.33 for every G -invariant subset Y of X we have an equivalence C ∞ ( Y ) (cid:39) C ∞ free ( Y ) . (3.5)By assumption, C ∞ ( Y ) and hence by (3.5) also C ∞ free ( Y ) is a phantom object. Sincethe latter is the cofibre of the lower right vertical map marked by !!, and since K iscompact, we get the diagonal dotted factorization of the composition ! ◦ tr N ◦ f . Usingagain compactness of K we can choose N in N and ( R n ) in (cid:81) N ≤ n C G sufficiently largesuch that the factorization f (cid:48) exists and the diagram commutes.We now consider the projection (see (2.1)) p : E ( free (cid:97) n ∈ N O ( P ( R n ) Y ( Y ))) → E ( O ( P ( R N ) Y ( Y )))19rising from excision and the inclusion of the summand with index N + 1. The compositionof f (cid:48) with the canonical map E ( free (cid:97) n ≤ N O ( P R Y ( Y ))) → E ( free (cid:97) n ∈ N O ( P R Y Y )))and p is equivalent to zero. Hence the composition p ◦ !!! ◦ tr N ◦ ˜ f is equivalent to zero. Onthe other hand, by property (2.2) of the transfer, the latter is equivalent to K ˜ f −→ E ( O ( P R Y ( Y ))) → E ( O ( P R N ( Y ))) . This implies that f is equivalent to zero.We now consider a sequence ( X n ) of G -bornological coarse spaces. Lemma 3.37.
If the G -bornological coarse space X n is E -vanishing for every n in N , thenthe sequence of G -bornological coarse spaces ( X n ) is E -vanishing.Proof. We consider a sequence ( Y n ) of invariant subspaces of ( X n ). We must show that C ∞ (( Y n )) is a phantom object.By Corollary 3.33 it suffices to show that C ∞ free (( Y n )) is a phantom object.Let K be a compact object of C and f : K → colim N ∈ N colim ( S n ) ∈ (cid:81) n ∈ N C G ( X n ) E ( free (cid:97) N ≤ n O ( P ( S n ) Yn ( Y n )))be some morphism. We must show that f is equivalent to zero.Since K is compact there exists a factorization K f (cid:47) (cid:47) ˜ f (cid:37) (cid:37) colim N ∈ N colim ( S n ) ∈ (cid:81) n ∈ N C G ( X n ) E ( free (cid:96) N ≤ n O ( P ( S n ) Yn ( Y n ))) E ( free (cid:96) N ≤ n O ( P ( R n ) Yn ( Y n ))) (cid:52) (cid:52) for some sufficiently large choices of N in N and ( R n ) in (cid:81) n ∈ N C G ( X n ).By our assumption on the G -bornological coarse spaces X k for each k in N with N ≤ k there exists R (cid:48) k in C G ( X k ) such that the composition of ˜ f with the projection E ( free (cid:97) N ≤ n O ( P ( R n ) Y ( Y n ))) → E ( O ( P ( R k ) Y ( Y k ))) → E ( O ( P ( R (cid:48) k ) Y ( Y k )))20s equivalent to zero, where the first morphism is the projection onto the k th summand.By weak additivity of E , it follows that the map K ˜ f −→ E ( free (cid:97) N ≤ n O ( P ( R n ) Y ( Y n ))) → E ( free (cid:97) N ≤ n O ( P ( R (cid:48) n ) Y ( Y n )))is also trivial. This implies that f is equivalent to zero.Let X be a G -bornological space with bornology B , Y be a G -set and f : Y → X be anequivariant map of sets. Definition 3.38.
The induced bornology f − B on Y is defined to be the minimal bornologyon Y such that the map f is proper. (cid:7) Let X be a G -bornological coarse space and ( U i ) i ∈ I be an equivariant family of subsets. Definition 3.39.
We define the G -bornological coarse space (cid:96) sub i ∈ I U i as follows:1. The underlying G -set is (cid:96) i ∈ I U i .2. The coarse structure is the one defined in Definition 3.4.3. The bornology is induced from X via the canonical map (cid:96) i ∈ I U i → X . (cid:7) Remark 3.40.
In the situation of Definition 3.39 we have a morphism of G -bornologicalcoarse spaces (cid:96) sub i ∈ I U i → X . The bornology on the domain of that map is induced from X , but the coarse structure is in general smaller than the induced coarse structure. (cid:7) We consider a class
F S of sequences of G -bornological coarse spaces. Let ( X n ) be asequence of G -bornological coarse spaces. Definition 3.41.
The sequence ( X n ) is decomposable over F S if for every sequence ( T n ) in (cid:81) n ∈ N C Gn and for all natural numbers n there exist pairwise T n -disjoint and nice equivariantfamilies U T n := ( U T n i ) i ∈ I n and V T n := ( V T n i ) j ∈ J n of subsets of X such that:1. X n = U T n ∪ V T n with U T n := (cid:83) i ∈ I n U T n i and V T n := (cid:83) j ∈ J n V T n j .2. The sequences of G -bornological coarse spaces (cid:32) sub (cid:97) i ∈ I n U T n i (cid:33) , (cid:32) sub (cid:97) j ∈ J n V T n j (cid:33) belong to F S . (cid:7) Let ( X n ) be a sequence of G -bornological coarse spaces and ( Y n ) be a sequence of sub-spaces. Lemma 3.42. If F S is closed under taking sequences of subspaces and the sequence ( X n ) is decomposable over F S , then ( Y n ) is decomposable over F S . roof. We use the notation appearing in Definition 3.41. For every natural number n wecan interpret invariant entourages T n of Y n as invariant entourages of X n . We set U T n Y,i := U T n i ∩ Y n , V T n Y,j := V T n j ∩ Y .
In the following we use the notation U T n Y := ( U T n Y,i ) i ∈ I . We observe by an inspection of thedefinitions that we have an isomorphism sub Y (cid:97) i ∈ I n U T n Y,i ∼ = sub X (cid:97) i ∈ I n U T n Y,i of G -bornological coarse spaces. Here the subscript Y or X at the sub-symbol indicatesthat the coarse structures are generated by the entourages sub U TnY (( S n ) Y ) (or sub U TnY ( S n ),respectively) for all entourages S n of X n , and that the bornologies are induced from thebornologies of Y n (or X n , respectively). We conclude that (cid:16)(cid:96) sub Y i ∈ I n U T n Y,i (cid:17) is a sequence ofsubspaces of the sequence (cid:16)(cid:96) sub X i ∈ I n U T n i (cid:17) and hence belongs to F S by assumption.A similar reasoning applies to the V -sequences.Recall that V E and VS E denote the classes of E -vanishing G -bornological coarse spaces(Definition 3.21) and of E -vanishing sequences of G -bornological coarse spaces (Defini-tion 3.34).Let X be a G -bornological coarse space with coarse structure C , and let X be thecorresponding constant sequence of G -coarse spaces. Corollary 3.43. If X is decomposable over V E , then the constant sequence X is decom-posable over VS E .Proof. Let ( T n ) in (cid:81) n ∈ N C G be given. Since X is decomposable over V E , for each naturalnumber n there exist pairwise T n -disjoint nice equivariant families ( U T n i ) i ∈ I n and ( V T n i ) i ∈ J n of subsets of X such that X = U T n ∪ V T n with U T n := (cid:91) i ∈ I n U T n i , V T n := (cid:91) j ∈ J n V T n j , and such that the G -bornological coarse spaces (cid:96) sub i ∈ I n U T n i and (cid:96) sub j ∈ J n V T n j are E -vanishing.By Lemma 3.37, both sequences of G -bornological coarse spaces ( (cid:96) sub i ∈ I n U T n i ) and ( (cid:96) sub j ∈ J n V T n j )are E -vanishing sequences. We conclude that the sequence X is decomposable over VS E . 22 .5 Properties of the Rips complex Remark 3.44.
As a preparation of what follows we recall the following conventions.If Y is a G -set with a G -invariant quasi-metric d , then for every r in R we define theinvariant entourage U r := { ( x, y ) ∈ Y × Y | d ( x, y ) ≤ r } . The G -coarse structure C d on Y induced by the metric is the coarse structure generatedby the enourages U r for all r in R .All this applies in particular to the quasi-metric G -space P XS ( Y ) for a G -bornologicalcoarse space X and invariant subset Y of X and invariant entourage S of X . (cid:7) Let X be a G -bornological coarse space, Y be an invariant subset of X , and S be aninvariant entourage of X such that diag( X ) ⊆ S . Note that this implies that S k ⊆ S k +1 for all integers k and that ( S n [ Y ]) n ∈ N is an increasing family of invariant subsets of X . Lemma 3.45.
For every integer k we have the inclusion U k − [ P XS ( Y )] ⊆ P XS ( S k [ Y ]) ⊆ U k +1 [ P XS ( Y )] . In particular, ( P XS ( S n [ Y ])) n ∈ N is a big family in the G -bornological coarse space P S ( X ) .Proof. In the following argument we identify the zero skeleton of P S ( X ) with X . This ispossible since by assumption S contains the diagonal.Let µ be a point in U k − [ P XS ( Y )]. Then it is contained in some simplex. We choose somevertex x of that simplex. Then x ∈ U k − [ P XS ( Y )]. The shortest path which connects x with a point in P XS ( Y ) is contained in the one-skeleton. Hence we can conclude that x ∈ S k − [ Y ]. But then all vertices of the simplex containing µ are contained in S k [ Y ].Hence µ ∈ P XS ( S k [ Y ]).Let now µ be a point in P XS ( S k [ Y ]). We again choose a vertex x of the simplex containing µ . Then the distance of x from P XS ( Y ) is bounded by k . Hence the distance of µ from P XS ( Y ) is bounded by k + 1, i.e., we have µ ∈ U k +1 [ P XS ( Y )].Let X be a G -bornological coarse space and V := ( V i ) i ∈ I be an equivariant pairwise disjointfamily of subsets. Let S and T be invariant entourages of X such that S contains thediagonal, and set S (cid:48) := S ∪ sub V ( T ) (Definition 3.4).Let n be a natural number. Lemma 3.46. If ( P XS ( V i )) i ∈ I is U n -disjoint in P XS ( X ) , then ( P XS (cid:48) ( V i )) i ∈ I is U n − -disjointin P XS (cid:48) ( X ) . roof. Since S (cid:48) contains the diagonal we can identify X with the zero skeleton of the Ripscomplex P XS (cid:48) ( X ). Since every point in the Rips complex has distance at most one to apoint in the 0-skeleton, it suffices to show that the family ( V i ) i ∈ I is U n -disjoint in P XS (cid:48) ( X ).We argue by contradiction. Let d denote the distance in P S (cid:48) ( X ). Assume that i and j belong to I such that i (cid:54) = j and d ( V i , V j ) < n . Then there exists a sequence ( x k ) k =0 ,...,n − in X with x ∈ V i , x n − ∈ V j , and ( x k , x k +1 ) ∈ S (cid:48) for all k in { , . . . , n − } .We let k in { , . . . , n − } be the minimal element such that x k ∈ V i and x k +1 / ∈ V i .Such an element k exists since V i ∩ V j = ∅ .We let k in { , . . . , n − } be the maximal element such that for all m in { k , . . . , k − } we have ( x m , x m +1 ) (cid:54)∈ (cid:83) i ∈ I ( T ∩ ( V i × V i )). We have k < k ≤ n − V i ) i ∈ I is pairwise disjoint. There exists l in I such that x k ∈ V l . Hence d ( V i , V l ) ≤ k − k ≤ n −
1. This contradicts the assumption that the family ( V i ) i ∈ I is U n -disjoint.Let X be a G -bornological coarse space, let U be an invariant subspace and let k be anatural number. We consider again invariant entourages S and T of X containing thediagonal and form the invariant entourage S (cid:48) := S ∪ ( T ∩ ( S k [ U ] × S k [ U ])) . Lemma 3.47.
We have ( S (cid:48) ) m [ U ] ⊆ S k + m [ U ] .Proof. We consider a point x in ( S (cid:48) ) m [ U ]. Then there exist a sequence ( x , . . . , x m ) in X with x ∈ U , x m = x and ( x i , x i +1 ) ∈ S (cid:48) for all i in { , . . . , m − } . Let l in { , . . . , m − } be maximal with x l ∈ S k [ U ]. By definition of S (cid:48) we have ( x i , x i +1 ) ∈ S for all i in { l + 1 , . . . , m − } . Thus x ∈ S m − l [ S k [ U ]] ⊆ S m + k [ U ]. Recall that VS E denotes the class of E -vanishing sequences (Definition 3.34). Theorem 3.48.
The class VS E is closed under decomposition. This subsection is devoted to the proof of this theorem.Suppose that the sequence ( X n ) is a sequence of G -bornological coarse spaces which isdecomposable over VS E (see Definition 3.41). In view of Definition 3.34 we have to showfor every sequence ( Y n ) of G -invariant subsets that C ∞ (( Y n )) defined in (3.2) is a phantomobject.It immediately follows from Definition 3.34 that the class VS E is closed under takingsequences of G -invariant subspaces. By Lemma 3.42 every sequence of G -invariant sub-spaces ( Y n ) is also decomposable over VS E . So we must see that the decomposability ofthe sequence ( Y n ) implies that it is E -vanishing.24t therefore suffices to show that C ∞ (( X n )) is a phantom. Note that C ∞ (( X n )) (cid:39) colim r ∈ (0 , ∞ ) colim N ∈ N colim ( S n ) ∈ (cid:81) n ∈ N C Gn E ( semi( r ) (cid:97) N ≤ n O ( P S n ( X n ))) . Let K be a compact object in C and f : K → colim r ∈ (0 , ∞ ) colim N ∈ N colim ( S n ) ∈ (cid:81) n ∈ N C Gn E ( semi( r ) (cid:97) N ≤ n O ( P S n ( X n )))be a morphism. By compactness of K it factorizes over a morphism˜ f : K → colim N ∈ N E ( semi( r ) (cid:97) N ≤ n O ( P S n ( X n )))for some real number r , and a sequence ( S n ) (we will assume that S n contains the diagonalof X n for every integer n ) in (cid:81) n ∈ N C Gn . It suffices to show that there exists a real number r (cid:48)(cid:48) with r ≤ r (cid:48)(cid:48) , and a sequence ( S (cid:48)(cid:48) n ) in (cid:81) n ∈ N C Gn with ( S n ) ≤ ( S (cid:48)(cid:48) n ) such that the induced(by the inclusion of Rips complexes) map˜ f (cid:48)(cid:48) : K → colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P S (cid:48)(cid:48) n ( X n ))) (3.6)is equivalent to zero.Let ( U n ) and ( V n ) be sequences of invariant subsets of the sequence ( X n ) such that forevery natural number n we have X n = U n ∪ V n . Using the second assertion of Lemma 3.45and excision for E we obtain a pushout:colim N,k ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P X n S n ( S kn [ U n ] ∩ S kn [ V n ]))) (cid:47) (cid:47) (cid:15) (cid:15) colim N,k ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P X n S n ( S kn [ U n ]))) (cid:15) (cid:15) colim N,k ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P X n S n ( S kn [ V n ]))) (cid:47) (cid:47) colim N ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P S n ( X n ))) (3.7) Remark 3.49.
At this this point it is important to work with the semi-free union (see Defi-nition 3.25). Indeed, in general the family of invariant subsets (cid:0)(cid:96) N ≤ n O ( P X n S n ( S kn [ V n ])) (cid:1) k ∈ N is not a big family in the G -bornological coarse space (cid:96) free N ≤ n O ( P S n ( X n )) (cid:7) By the decomposability assumption on ( X n ) for every natural number n we can choosenice equivariant S nn -disjoint families ( U n,i ) i ∈ I n and ( V n,j ) j ∈ J n of subspaces of X n such that X n = U n ∪ V n with U n := (cid:91) i ∈ I n U n,i , V n := (cid:91) j ∈ J n V n,j (cid:32) sub (cid:97) i ∈ I n U n,i (cid:33) , (cid:32) sub (cid:97) j ∈ J n V n,j (cid:33) belong to VS E . This choice will be fixed for the rest of the subsection.Since above we have chosen S nn -disjoint families (the n th power is important), by the firstassertion of Lemma 3.45 for every k in N there exists an N ( k ) in N such that for everyinteger n with N ( k ) ≤ n the families( S kn [ U n,i ]) i ∈ I n , ( S kn [ V n,j ]) j ∈ J n (3.8)are S n -disjoint, and the families( P X n S n ( S kn [ U n,i ])) i ∈ I n , ( P X n S n ( S kn [ V n,j ])) j ∈ J n (3.9)are U r -disjoint. Note that N ( k ) also depends on r , but we will not indicate this in thenotation.For a G -uniform bornological coarse space A , whose uniform structure is induced by aninvariant metric and a real number r , we let O r ( A ) denote the G -uniform bornologicalcoarse space obtained from O ( A ) by replacing the coarse structure by the coarse structuregenerated by all entourages of propagation (Definition 3.24) bounded by r .We abbreviate Y kn,i,j := S kn [ U n,i ] ∩ S kn [ V n,j ]and consider the family Y kn := ( Y kn,i,j ) i ∈ I n ,j ∈ J n . Below, we use the abbreviation sub k ( S n ) := sub Y kn ( S n ) , (3.10)see Definition 3.4. Lemma 3.50.
For all natural numbers n and k with N ( k ) ≤ n we have an isomorphismof G -bornological coarse spaces O r ( P X n S n ( S kn [ U n ] ∩ S kn [ V n ])) ∼ = O r ( P sub k ( S n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j )) . (3.11) Proof.
First of all the underlying G -simplicial complexes P X n S n ( S kn [ U n ] ∩ S kn [ V n ]) and P sub k ( S n ) ( (cid:96) sub i ∈ I n ,j ∈ J n Y kn,i,j ) are isomorphic since the families (3.8) are S n -disjoint. The dis-tance of the different components on the right-hand side in the metric of P X n S n ( X n ) is biggerthan r since the families (3.9) are U r -disjoint. Hence the entourages on P X n S n ( S kn [ U n ] ∩ S kn [ V n ])26nd on P sub k ( S n ) ( (cid:96) sub i ∈ I n ,j ∈ J n Y kn,i,j ) of propagation less or equal to r (measured in the re-spective quasi-metric) are equal (under the natural identification). The bornologies ofboth spaces coincide by construction.Since we consider the cone O r we can conclude that the natural map (3.11) induces anisomorphism of G -bornological coarse spaces.Using Lemma 3.50 we can rewrite the pushout square (3.7) in the form:colim N,k ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P sub k ( S n ) ( sub (cid:96) i ∈ I n ,j ∈ J n Y kn,i,j ))) (cid:47) (cid:47) (cid:15) (cid:15) colim N,k ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P X n S n ( S kn [ U n ]))) (cid:15) (cid:15) colim N,k ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P X n S n ( S kn [ V n ]))) (cid:47) (cid:47) colim N ∈ N E ( semi( r ) (cid:96) N ≤ n O ( P S n ( X n )))Since we take the colimit over N the fact that the isomorphism (3.11) exists only forsuffiently large n (and the condition of being sufficiently large depends on k ) does notcause any problem.Let ∂ : colim N ∈ N E ( semi( r ) (cid:97) N ≤ n O ( P S n ( X n ))) → colim N,k ∈ N Σ E ( semi( r ) (cid:97) N ≤ n O ( P sub k ( S n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j )))denote the boundary map of the Mayer–Vietoris sequence. Using compactness of K , themorphism ∂ ◦ ˜ f factorizes over a morphism g : K → colim N ∈ N E ( semi( r ) (cid:97) N ≤ n O ( P sub k ( S n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j )))for some k in N which will be fixed from now on. Lemma 3.51.
There exist a real number r (cid:48) such that r (cid:48) ≥ r and a sequence ( T n ) in (cid:81) n ∈ N C G such that with ( S (cid:48) n ) := ( S n ∪ sub k ( T n )) the map K → colim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P sub k ( S (cid:48) n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j ))) induced by g is equivalent to zero.Proof. As in Lemma 3.50 for all integers n with N ( k ) ≤ n we also have an isomorphism O r ( P (cid:96) sub i ∈ In S kn [ U n,i ]sub k, (cid:48) ( S n ) ( (cid:97) i ∈ I n S kn [ U n,i ] ∩ S kn [ V n ])) ∼ = O r ( P sub k ( S n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j )) , (3.12)27here we use the abbreviation sub k, (cid:48) ( S n ) := sub U kn ( S n ) (3.13)with U kn := ( S kn [ U n,i ]) i ∈ I n , and we use the equality S kn [ V n ] = (cid:91) j ∈ J n S kn [ V n,j ]of subsets of X .The sequence ( (cid:96) sub i ∈ I n S kn [ U n,i ] ∩ S kn [ V n ])) N ( k ) ≤ n of G -bornological coarse spaces is a sequenceof subspaces of the sequence ( (cid:96) sub i ∈ I n S kn [ U n,i ]) N ( k ) ≤ n . Since we assume that the family( U n,i ) i ∈ I n is nice the bornological coarse space (cid:96) sub i ∈ I n S kn [ U n,i ] is equivalent to (cid:96) sub i ∈ I n U n,i .By assumption the sequence ( (cid:96) sub i ∈ I n [ U n,i ]) is E -vanishing. By Lemma 3.35 we can nowconclude that the sequence ( (cid:96) sub i ∈ I n S kn [ U n,i ]) is E -vanishing. This implies that the sequence( (cid:96) sub i ∈ I n S kn [ U n,i ] ∩ S kn [ V n ]) is E -vanishing. Hence using Remark 3.5, there exist a real number r (cid:48) such that r ≤ r (cid:48) and a sequence ( T n ) in (cid:81) n ∈ N C Gn with ( S n ) ≤ ( T n ) such that K g −→ colim N ∈ N E ( semi( r ) (cid:97) N ≤ n O ( P sub k ( S n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j ))) → colim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P sub k ( T n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j )))is equivalent to zero. We now note that sub k ( T n ) = sub k ( S (cid:48) n ).It follows from Lemma 3.51 that K ∂ ◦ ˜ f −−→ colim N,k ∈ N Σ E ( semi( r ) (cid:97) N ≤ n O ( P sub k ( S n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y kn,i,j ))) → colim N,k (cid:48) ∈ N Σ E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P sub k (cid:48) ( S (cid:48) n ) ( sub (cid:97) i ∈ I n ,j ∈ J n Y k (cid:48) n,i,j )))is equivalent to zero. Consequently, using the naturality of the Mayer–Vietoris sequencesand the compactness of K (in order to be able to choose k (cid:48) ), the composition K ˜ f (cid:48) −→ colim N ∈ N E ( semi( r ) (cid:97) N ≤ n O ( P S n ( X n ))) → colim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P S (cid:48) n ( X n )))(where r (cid:48) and ( S (cid:48) n ) are as in Lemma 3.51) lifts to a morphism( ˆ f U , ˆ f V ) : K → colim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48) n ( S k (cid:48) n [ U n ]))) ⊕ colim N E ( semi( r (cid:48) ) (cid:97) N ≤ n ( P X n S (cid:48) n ( S k (cid:48) n [ V n ])))for some integer k (cid:48) such that k (cid:48) ≥ k . 28he notation sub k (cid:48) , (cid:48) ( T n ) is as in (3.13), while sub k ( T n ) is as in (3.10) for the specific choiceof the integer k made just before the statement of Lemma 3.51.Let ( T (cid:48)(cid:48) n ) be a sequence in (cid:81) n ∈ N C Gn such that ( T n ) ≤ ( T (cid:48)(cid:48) n ). We define the sequence( S (cid:48)(cid:48) n ) := ( S (cid:48) n ∪ sub k (cid:48) , (cid:48) ( T (cid:48)(cid:48) n )) . (3.14)By Lemma 3.47 and Lemma 3.46 there exists an integer N such that N ( k ) ≤ N andfor all integers n with N ≤ n the family ( S k (cid:48) n [ U n,i ]) i ∈ I n is ˜ S (cid:48)(cid:48) n -disjoint and the family( P S (cid:48)(cid:48) n ( S k (cid:48) n [ U n,i ])) i ∈ I n is r (cid:48) -disjoint.The following is similar to Lemma 3.50. Lemma 3.52.
For all integers n with N ≤ n we have an isomorphism of G -bornologicalcoarse spaces O r (cid:48) ( P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ U n ])) ∼ = O r (cid:48) ( P sub k (cid:48) , (cid:48) ( S (cid:48)(cid:48) n ) ( sub (cid:97) i ∈ I S k (cid:48) n [ U n,i ])) . Proof.
First of all the underlying G -simplicial complexes P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ U n ]) , P sub k (cid:48) , (cid:48) ( S (cid:48)(cid:48) n ) ( sub (cid:97) i ∈ I S k (cid:48) n [ U n,i ])are isomorphic since the family ( S k (cid:48) n [ U n,i ]) i ∈ I n is S (cid:48)(cid:48) n -disjoint. The distance of the differentcomponents on the right-hand side in the metric of P X n S (cid:48) n ( X n ) is bigger than r (cid:48) . Since weconsider the cone O r (cid:48) we now can conclude that the G -coarse structures on both sidescoincide. The bornologies coincide by construction.We get an equivalencecolim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ U n ]))) (cid:39) colim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P sub k (cid:48) , (cid:48) ( S (cid:48)(cid:48) n ) ( sub (cid:97) i ∈ I S k (cid:48) n [ U n,i ]))) (3.15) Lemma 3.53.
There exist a real number r (cid:48)(cid:48) such that r (cid:48)(cid:48) ≥ r (cid:48) and a sequence ( S (cid:48)(cid:48) n ) in (cid:81) n ∈ N C Gn with ( S (cid:48)(cid:48) n ) ≥ ( S (cid:48) n ) such that the map K ˆ f U −→ colim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48) n ( S k (cid:48) n [ U n ]))) → colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ U n ]))) is equivalent to zero. roof. In view of the equivalence (3.15) we must show that for an appropriate choice ofthe sequence ( T (cid:48)(cid:48) n ) (which determines the sequence ( S (cid:48)(cid:48) n ) by (3.14)) the morphism K → colim N ∈ N E ( semi( r (cid:48) ) (cid:97) N ≤ n O ( P sub k (cid:48) , (cid:48) ( S (cid:48) n ) ( sub (cid:97) i ∈ I S k (cid:48) n [ U n,i ]))) → colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P sub k (cid:48) , (cid:48) ( S (cid:48)(cid:48) n ) ( sub (cid:97) i ∈ I S k (cid:48) n [ U n,i ])))is equivalent to zero. As in the proof of Lemma 3.51 we argue that the sequence of G -bornological coarse spaces ( (cid:96) sub i ∈ I S k (cid:48) n [ U n,i ]) is E -vanishing. Using Remark 3.5 andcompactness of K we find the desired real number r (cid:48)(cid:48) and sequence ( S (cid:48)(cid:48) n ).Using Lemma 3.53, we find a real number r (cid:48)(cid:48) with r (cid:48)(cid:48) ≥ r (cid:48) and a sequence ( S (cid:48)(cid:48) n ) in (cid:81) n ∈ N C n with ( S (cid:48)(cid:48) n ) ≥ ( S (cid:48) n ) such that the map K → colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ U n ]))) ⊕ colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ V n ])))induced by ( ˆ f U , ˆ f V ) and the natural inclusions is equivalent to zero.Composition of the induced map with the morphismcolim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ U n ]))) ⊕ colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P X n S (cid:48)(cid:48) n ( S k (cid:48) n [ V n ]))) → colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P S (cid:48)(cid:48) n ( X n )))from the Mayer–Vietoris sequence gives the map (3.6)˜ f (cid:48) : K ˜ f −→ colim N ∈ N E ( semi( r ) (cid:97) N ≤ n O ( P S n ( X n ))) → colim N ∈ N E ( semi( r (cid:48)(cid:48) ) (cid:97) N ≤ n O ( P S (cid:48)(cid:48) n ( X n )))which is then also equivalent to zero.This finishes the proof of Theorem 3.48. As already observed in Section 3.3 the results shown so far imply:
Corollary 3.54.
The class V E of E -vanishing G -bornological coarse spaces is closed underdecomposition. Recall Definition 3.12 of the notion of a semi-bounded G -coarse space. Furthermore recallDefinition 3.15 of a discontinuous G -action on a G -bornological space.30 roposition 3.55. The class of G -bornological coarse spaces whose underlying G -coarsespace is semi-bounded and has a discontinuous G -action is contained in the class of E -vanishing spaces V E .Proof. Let X be a G -bornological coarse spaces whose underlying space is semi-boundedand has a discontinuous G -action. Then every invariant subset with the induced G -bornological coarse structure has the same properties. So it suffices to show that E ( O ( P S ( X ))) (cid:39) S of X .By assumption we can choose an invariant entourage T of X such that every coarsecomponent of X is bounded by T . Then for every invariant entourage S of X with T ⊆ S we have an isomorphism P S ( X ) ∼ = (cid:71) i ∈ π ( X ) ∆ X i of quasi-metric spaces, where X i is the component labelled by i and ∆ X i is the set offinitely supported probability measures on X i .Let π be a choice of a subset of π ( X ) of representatives of the orbit set π ( X ) /G . Forevery i in π we let G i be the stabilizer of i . This group acts simplicially on X i . Then weobtain a G -equivariant isomorphism of G -quasi-metric spaces P S ( X ) ∼ = (cid:97) i ∈ π G × G i ∆ X i . Let i be in π and x i be a point in X i . The set Gx i ∩ X i is finite since the G -actionon X is discontinuous and X i belongs to the minimal bornology compatible with thecoarse structure of X . Hence, Gx i ∩ X i is the set of vertices of a simplex ∆ i in ∆ X i . Thebarycenter b i of ∆ i is fixed under the action of G i .There is a unique G -equivariant map b : π ( X ) → P S ( X ) such that b ( x i ) = b i for all i in π . We consider π ( X ) as a discrete quasi-metric space. The map b is a homotopyequivalence. Indeed, an inverse is given by the map p : P S ( X ) → π ( X ) which sendsevery point µ in P S ( X ) to the coarse component represented by any choice of a vertexof a simplex containing µ . Then p ◦ b = id π ( X ) and b ◦ p is homotopic to id P S ( X ) by aunit-speed homotopy. The cone functor sends this homotopy to a coarse homotopy.We equip π ( X ) with the bornology induced from the bornology of X via the map b . Itfollows from the compatibility of the coarse and the bornological structure on X that theprojection p is proper.We conclude that E ( O ( P S ( X ))) (cid:39) E ( O ( π ( X ))) . Since π ( X ) is a discrete G -bornological coarse space the cone O ( π ( X )) is a flasque G -bornological coarse space. Consequently, 0 (cid:39) E ( O ( π ( X ))). This implies0 (cid:39) E ( O ( P S ( X ))) . roof of Theorem 3.19. The theorem is an immediate consequence of Corollary 3.54 andProposition 3.55: The class of G -coarse spaces with G -FDC is by definition the smallestclass of G -coarse spaces which is closed under decomposition and contains all semi-boundedspaces. Moreover, any decomposition of G -bornological space with discontinuous G -actionyields families of G -bornological coarse spaces with discontinuous G -action. References [BEKWa] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. On the descent principleand injectivity of the forget-control map. In preparation.[BEKWb] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Transfers in coarsehomology. In preparation.[BEKW17] U. Bunke, A. Engel, D. Kasprowski, and Ch. Winges. Equivariant coarsehomotopy theory and coarse algebraic K -homology. arXiv:1710.04935v1, 2017.[GTY12] E. Guentner, R. Tessera, and G. Yu. A notion of geometric complexity andits application to topological rigidity. Invent. Math. , 189(2):315–357, 2012.[GTY13] E. Guentner, R. Tessera, and G. Yu. Discrete groups with finite decompositioncomplexity.
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