Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
aa r X i v : . [ m a t h . K T ] J un HOMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDSAND SMALE SPACES VALERIO PROIETTI AND MAKOTO YAMASHITA
Abstract.
Given an ample groupoid, we construct a spectral sequence with groupoid homologywith integer coefficients on the second sheet, converging to the K -groups of the groupoid C ∗ -algebrawhen the groupoid has torsion-free stabilizers and satisfies the strong Baum–Connes conjecture. Theconstruction is based on the triangulated category approach to the Baum–Connes conjecture byMeyer and Nest. For the unstable equivalence relation of a Smale space with totally disconnectedstable sets, this spectral sequence shows Putnam’s homology groups on the second sheet. Contents
Introduction 11. Preliminaries 32. Pullback and resolution groupoids 133. Approximation in the equivariant KK-category 174. Homology and K -theory 215. Examples 25Appendix A. Structure of groupoid equivariant KK-theory 29References 32 Introduction
In this paper, we look at the K -theory of ample Hausdorff groupoids, that is, étale groupoidson totally disconnected spaces, and its relation to groupoid homology. Such groupoids are closelyrelated to dynamical systems on Cantor sets, such as (sub)shifts of finite type (also called topologicalMarkov shifts) in symbolic dynamics. While this remains a fundamental example, the second halfof the last century saw a rapid development of the theory which resulted in several generalizationsinvolving various geometric, combinatorial, and functional analytic structures.One prominent example is the framework of Smale spaces introduced by Ruelle [Rue04], whodesigned them to model the basic sets of Axiom A diffeomorphisms [Sma67]. This turned out to be aparticularly nice class of hyperbolic dynamical systems, where Markov partitions provide a symbolicapproximation of the dynamics. Examples of Smale spaces include hyperbolic toral automorphisms,and more generally Anosov diffeomorphisms, see [Bow08] and references therein.Ample groupoids arise from Smale spaces with totally disconnected stable sets. This is especiallyuseful in the study of dynamical systems whose topological dimension is not zero, but whose dynamicsis completely captured by restricting to a totally disconnected transversal. Such spaces includegeneralized solenoids [Tho10b, Wil74] and substitution tiling spaces [AP98, Theorem 3.3], and canbe characterized as certain inverse limits [Wie14].Beyond the theory of dynamical systems, these groupoids also play an important role in the theoryof operator algebras, where they provide an invaluable source of examples of C ∗ -algebras. Theseare obtained by considering the (reduced) groupoid C ∗ -algebras [Ren80], generalizing the crossedproduct algebras for group actions. The resulting C ∗ -algebras capture interesting aspects of thehomoclinic and heteroclinic structure of expansive dynamics [Mat19, Put96, Tho10a], extending thecorrespondence between topological Markov shifts and the Cuntz–Krieger algebras. Date : June 14, 2020.2010
Mathematics Subject Classification.
Key words and phrases. groupoid, C ∗ -algebra, K -theory, homology, Baum–Connes conjecture, Smale space. K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 2 Another important class of Cantor systems comes from minimal homeomorphisms of the Cantorset. This study was initiated by Giordano, Putnam and Skau [GPS95], in which they classifiedminimal homeomorphisms up to orbit equivalence. Actions of Z k on the Cantor set, which are higherrank analogues, also naturally appear from tiling spaces. More generally, essentially free amplegroupoids appear in the study of actions of N k by local homeomorphisms on zero-dimensional spaces,where they are known as Deaconu–Renault groupoids [Dea95, ER07]. This is a convenient frameworkto understand higher-rank graph C ∗ -algebras. The étale groupoids, and related invariants such astopological full groups, of such systems proved to be a rich source of interesting examples in thestructure theory of discrete groups and operator algebras, see for example [JM13, Mat13, Phi05].The K -groups of groupoid C ∗ -algebra and groupoid cohomology with integer coefficients are knownto have close parallels, for example in various cohomological invariants of tiling spaces. In fact,groupoid homology [CM00] has even closer properties to K -groups, and the comparison of theseinvariants (for topologically free, minimal, and ample Hausdorff groupoids) was recently popularizedby Matui [Mat12]. While his conjectural isomorphism in its original form (“HK conjecture”) hascounterexamples [Sca19], in situations where one expects low homological dimension we do have anisomorphism, see for example [FKPS19, Ort18].Our main result gives a correspondence between groupoid homology and K -groups for torsion-freeample groupoids satisfying the strong Baum–Connes conjecture [Tu99a], as follows. Theorem A (Theorem 4.5) . Let G be an ample groupoid with torsion-free stabilizers satisfying thestrong Baum–Connes conjecture. Then there is a convergent spectral sequence E pq = E pq = H p ( G, K q ( C )) ⇒ K p + q ( C ∗ r ( G )) , Similarly to discrete groups, amenable groupoids satisfy the (strong) Baum–Connes conjecture,which cover most of our concrete examples in this paper.Note that, for groupoids with low homological dimension, this spectral sequence degenerates fordegree reasons. Moreover the top-degree group in groupoid homology tends to be torsion-free, ex-plaining the positive cases where the HK conjecture holds.Turning to Smale spaces, there is another homology theory proposed by Putnam [Put14] We showthat one of the variants, H s ∗ , fits into this scheme for the groupoid of the unstable equivalence relationon the underlying space, as follows. Theorem B (Theorem 4.9) . Let ( Y, ψ ) be an irreducible Smale space with totally disconnected stablesets, and R u ( Y, ψ ) be the groupoid of the unstable equivalence relation. Then there is a convergentspectral sequence E pq = E pq = H sp ( Y, ψ ) ⊗ K q ( C ) ⇒ K p + q ( C ∗ ( R u ( Y, ψ ))) . This result gives a partial answer to a question raised by Putnam [Put14, Section 8.4.1]. Animmediate consequence is that the K -groups of C ∗ ( R u ( Y, ψ )) are of finite rank.Although we give an independent proof of Theorem B, it can also be obtained from the combinationof Theorem A and the result below.
Theorem C (Theorem 4.12) . For any étale groupoid G that is Morita equivalent to R u ( Y, ψ ) , wehave an isomorphism H sp ( Y, ψ ) ≃ H p ( G, Z ) . In order to prove the result above, we turn the definition of Putnam’s homology into a resolutionof modules which computes groupoid homology. As a corollary we obtain a Künneth formula for H s ∗ ,generalizing a result in [DKW16].Our proofs of Theorem A and B above are based on the triangulated category approach to theBaum–Connes conjecture by Meyer and Nest [Mey08, MN06, MN10]. Building on their theory of pro-jective resolutions and complementary categories from homological ideals, we show that an explicitprojective resolution can be obtained from adjoint functors and associated simplicial objects. Apply-ing this to the restriction functor KK G → KK X and induction functor KK X → KK G for X = G (0) gives the standard bar complex computing the groupoid homology. Then, the spectral sequence inTheorem A appears as a particular case of the “ABC spectral sequence” of [Mey08].This paper is organized as follows. In Section 1 we lay out the basic notation and definitions forall the background objects of the paper. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 3 In Section 2, we discuss the multiple pullback of groupoid homomorphisms, generalizing a con-struction in [CM00], which provides the spatial implementation of the groupoid bar complex in thecase of the inclusion map G (0) → G regarded as a groupoid homomorphism. For Smale spaces, welook at an s -bijective map f : (Σ , σ ) → ( Y, ψ ) from a shift of finite type, which underlies Putnam’shomology through multiple fiber products for f . A key technical result is a transversality result inProposition 2.9, which allows us to relate the multiple fiber products of f to the multiple groupoidpullbacks.In Section 3, we look at a simplicial object arising from adjoint functors and relate it to thecategorical approach to the Baum–Connes conjecture. In a triangulated category, homological idealswith enough projectives, and a pair of complementary subcategories, appear from adjunction offunctors [Mey08]. Our observation is that the canonical comonad construction from homologicalalgebra gives a concrete model of projective resolution. We then use this to show that, when G is an étale groupoid satisfying the strong Baum–Connes conjecture, any G -C ∗ -algebra A which isKK X -nuclear as a C ( X )-algebra belongs to the triangulated subcategory of KK G generated by theimage of the induction functor KK X → KK G for X = G (0) .We then combine these results in Section 4 to obtain our main results mentioned above. Now,let us summarize the ingredients which go into the correspondence between groupoid homology and K -theory. By the adjunction of the functors Ind GX : KK X → KK G and Res GX : KK G → KK X , for any G -C ∗ -algebra A we have an exact triangle in KK G , P → A → N → Σ P, with Res GX N ≃ P being orthogonal to all such N . If G has torsion-free stabilizers and satisfiesthe strong Baum–Connes conjecture and A is KK X -nuclear, we actually have P ≃ A in KK G . Inaddition, for any homological functor F , we have a spectral sequence from the Moore complex of thesimplicial object ( F ( L n +1 A )) ∞ n =0 with L = Ind GX Res GX , converging to F ( P ).For an ample groupoid G , the functor F = K ∗ ( G ⋉ − ), and A = C ( X ), this complex is isomorphicto the bar complex computing the groupoid homology of G . For the groupoid of the unstableequivalence relation on a Smale space ( Y, ψ ) with totally disconnected stable sets, we follow the samescheme, but replace X by the subgroupoid coming from an s -bijective factor map from a shift of finitetype. Then the resulting complex is isomorphic to the one defining Putnam’s homology H s ∗ ( Y, ψ ).Finally, in Section 5 we discuss some examples. For the groupoids of (substitution) tilings, ourconstruction is an analogue of the one for tiling space cohomology by Savinien and Bellissard [SB09],via a Poincaré duality type isomorphism between groupoid homology and cohomology. We alsocompare our construction with the counterexample to the HK conjecture from [Sca19].
Acknowledgements.
We are indebted to R. Nest for proposing the topic of this paper as a re-search project, and for numerous stimulating conversations. We are also grateful to R. Meyer forvaluable advice concerning equivariant K -theory and for his careful reading of our draft. Thanksalso to M. Dadarlat, R. Deeley, M. Goffeng, and I. F. Putnam for stimulating conversations andencouragement at various stages, which led to numerous improvements.This research was partly supported through V.P.’s “Oberwolfach Leibniz Fellowship” by the Math-ematisches Forschungsinstitut Oberwolfach in 2020. In addition, V.P. was supported by the Scienceand Technology Commission of Shanghai Municipality (STCSM), grant no. 13dz2260400. M.Y. ac-knowledges support by Grant for Basic Science Research Projects from The Sumitomo Foundationat early stage of collaboration. 1.
Preliminaries
In this section we recall the most important objects and notions at the basis of this paper. Wewill deal with C ∗ -algebras endowed with a groupoid action, and will consider these as objects of theequivariant Kasparov category. In addition, we will introduce a special class of topological dynamicalsystems, called Smale spaces, which will be a key example to which we apply our results.1.1. Groupoids and Morita equivalence.
Let G be a groupoid with base space X = G (0) . Welet s, r : G → X denote respectively the source and range maps. In addition, we let G x = s − ( x ), G x = r − ( x ), and for a subset A ⊂ X , we write G A = S x ∈ A G x , G A = S x ∈ A G x , and G | A = G A ∩ C A . OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 4 Definition 1.1.
A topological groupoid G is étale if s and r are local homeomorphisms, and ample if it is étale and G (0) is totally disconnected.If G is étale and g ∈ G , then by definition, for small enough neighborhoods U of s ( g ) there is aneighborhood U ′ of g such that s ( U ′ ) = U , and the restriction of s and r to U ′ are homeomorphismsonto the images. When this is the case, we write g ( U ) = r ( U ′ ) and use g as a label for the map U → g ( U ) induced by the identification of U ∼ U ′ ∼ g ( U ).Throughout the paper we assume that a topological groupoid is second countable, locally compactHausdorff, and admits a (left) Haar system, so that its (reduced) groupoid C ∗ -algebra makes sense.In particular, G and X are σ -compact and paracompact. Recall that the condition on Haar systemis automatic for étale groupoids, as we can take the counting measure on G x .A locally compact groupoid is amenable if there is a net of probability measures on G x for x ∈ G (0) which is approximately invariant, see [ADR00]. In this case, the full and reduced C ∗ -norms are equal,and the completion of the compactly supported functions in the regular representation is ∗ -isomorphicto the full groupoid C ∗ -algebra.The notion of Morita equivalence of groupoids in the sense of [MRW87] plays an important role inthis paper. We review it here below for convenience. First, recall a topological groupoid G is proper if the map ( r × s ) : G → X × X is proper. Furthermore, if Z is a locally compact, Hausdorff G -space,we say that G acts properly on Z if the transformation groupoid G ⋉ Z is proper. The map Z → G (0) underlying the G -action is called the anchor map . Definition 1.2.
The groupoids G and H are Morita equivalent if there is a locally compact Hausdorffspace Z such that • Z is a free and proper left G -space with anchor map ρ : Z → G (0) ; • Z is a free and proper right H -space with anchor map σ : Z → H (0) ; • the actions of G and H on Z commute; • ρ : Z → G (0) induces a homeomorphism Z/H → G (0) ; • σ : Z → H (0) induces a homeomorphism G \ Z → H (0) .This can be conveniently packaged by a bibundle over G and H : that is, a topological space Z with G and H acting continuously from both sides with surjective and open anchor maps, such thatthat the maps G × G (0) Z → Z × H (0) Z, ( g, z ) ( gz, z ) , Z × H (0) H → Z × G (0) Z, ( z, h ) ( z, zh )are homeomorphisms.An important class of Morita equivalences comes from generalized transversals [PS99]. For atopological space X and x ∈ X , let us denote the family of open neighborhoods of x by O ( x ). Definition 1.3.
Let G be a topological groupoid. A generalized transversal for G is given by atopological space T and an injective continuous map f : T → G (0) such that: • f ( T ) meets every orbit of G ; and • the condition Ar for neighborhoods of x ∈ G and f − ( rx ), i.e., ∀ x ∈ G f ( T ) , U ∈ O ( x ) , V ∈ O ( f − ( rx )) ∃ U ∈ O ( x ) , V ∈ O ( f − ( rx )) : U ⊂ U , V ⊂ V , ∀ y ∈ U ∃ ! z ∈ U, s ( y ) = s ( z ) , r ( z ) ∈ f ( V ) . If G is a second countable locally compact Hausdorff groupoid, there is a (finer) topology on thesubgroupoid H = G | f ( T ) such that H is étale and Morita equivalent to G [PS99, Theorem 3.6]. Theequivalence is implemented by the principal bibundle G f ( T ) with a natural finer topology from thatof G and T .1.2. Groupoid equivariant C ∗ -algebras. Let us fix our conventions for G -C ∗ -algebras. Definition 1.4. A C ( X ) -algebra is a C ∗ -algebra A endowed with a nondegenerate ∗ -homomorphismfrom C ( X ) to the center of the multiplier algebra M ( A ).Thus, if a ∈ A , we have a = f b = bf for some f ∈ C ( X ) and b ∈ A , and the second equalityholds for all f and b . For an open set U ⊂ X , we put A U = AC ( U ). For a locally closed subset Y ⊂ X , that is, if Y = U r V for some open sets U, V ⊂ X , we put A Y = A U /A U ∩ V , and we put A x = A { x } = A/AC ( X r { x } ) for x ∈ X . OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 5 A C ( X )-algebra is C ( X ) -nuclear if it is a continuous field of C ∗ -algebras over X such that everyfiber A x is nuclear. There is another way to define this in terms of completely positive maps factoringthrough M n ( C ( X )), see [Bau98]. Definition 1.5.
Let A and B be C ( X )-algebras which admit faithful C ( X )-equivariant nondegen-erate representations on Hilbert C ∗ - C ( X )-modules E and E ′ . Then their C ∗ -algebraic relative tensorproduct A ⊗ C ( X ) B is defined as the closure of the image of A ⊗ alg C ( X ) B in the adjointable operators L ( E ⊗ C ( X ) E ′ ).Although we do not need it, the above definition can be extended to arbitrary C ( X )-algebras[Kas88, Definition 1.6]. Remark . If A or B is C ( X )-nuclear, we have A ⊗ C ( X ) B ≃ ( A ⊗ max B ) ∆( X ) ≃ ( A ⊗ min B ) ∆( X ) , where ∆( X ) = { ( x, x ) | x ∈ X } ⊂ X × X , see [Bla96, Section 3.2].If f : Y → X is a continuous map, C ( Y ) is a C ( X )-algebra. It is a continuous field (hence C ( X )-nuclear) if and only if f is open [BK04]. This induces a functor f ∗ A = C ( Y ) ⊗ C ( X ) A from the category of C ( X )-algebras to that of C ( Y )-algebras. For Y = G and f = s , we write s ∗ A = C ( G ) ⊗ s C ( X ) A , and similarly for f = r . Definition 1.7.
Let G be a topological groupoid as above, with G (0) = X . A continuous action of G on a C ( X )-algebra A is given by an isomorphism of C ( G )-algebras α : C ( G ) ⊗ s C ( X ) A → C ( G ) ⊗ r C ( X ) A such that the induced homomorphisms α g : A s ( g ) → A r ( g ) for g ∈ G satisfy α gh = α g α h . In this case,we say that A is a G -C ∗ - algebra .For an étale groupoid G , the above amounts to giving α g as isomorphisms A U → A g ( U ) for smallenough neighborhoods U of s ( g ), compatible with the natural actions of C ( U ) ≃ C ( g ( U )) andmultiplicative in g .In [LG99], Le Gall constructed the equivariant KK-category of separable and trivially graded G -C ∗ -algebras with morphism sets KK G ( A, B ), generalizing Kasparov’s construction for transformationgroupoids. This will be our main framework to work in.
Remark . Le Gall uses a different convention for A ⊗ C ( X ) B , namely ( A ⊗ max B ) ∆( X ) . Thisis different from ours in general, however these definitions agree in all the relevant cases, such as B = C ( Y ) for a locally compact space Y endowed with an open map Y → X , see Remark 1.6.The algebraic balanced tensor product C c ( G ) ⊗ C ( X ) A admits an A -valued inner product inducedby the measures on the sets G x from the Haar system, and we denote its closure as a right Hilbert A -module as E GA = L ( G ; A ). The reduced crossed product G ⋉ α A is the C ∗ -algebra generated by C c ( G ) ⊗ s C ( X ) A represented on E GA , see [KS04, MW08] for the details. In this paper we always takereduced crossed products, although they will be isomorphic to the full ones in most of our concreteexamples as we mostly consider amenable groupoids.1.3. Equivariant sheaves over ample groupoids.
The nerve ( G ( n ) ) ∞ n =0 of G form a simplicialspace, with the face maps are given by d ni : G ( n ) → G ( n − , ( g , ..., g n ) ( g , ..., g n ) if i = 0( g , ..., g i g i +1 , ..., g n ) if 1 ≤ i ≤ n − g , ..., g n − ) if i = n ,with d = r and d = s as maps G → X , while the degeneracy maps are given by insertion of units.These structure maps are étale maps.Suppose further that G be an ample groupoid, and A be a commutative group. For a topologicalspace Y , we denote the group of compactly supported continuous functions from Y to A by C c ( Y, A ). OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 6 The groupoid homology of G with coefficients in A , denoted H ∗ ( G, A ), is the homology of the chaincomplex ( C c ( G ( n ) , A )) ∞ n =0 with differential ∂ n = n X i =0 ( − i ( d ni ) ∗ : C c ( G ( n ) , A ) → C c ( G ( n − , A ) , ( d ni ) ∗ ( f )( x ) = X d ni ( y )= x f ( y ) . (This is well defined as d ni is étale.)This is a special case of groupoid homology with coefficients in equivariant sheaves [CM00]. Letus quickly review this more general setting. When G is a topological groupoid with base space X , a G -equivariant sheaf (of commutative groups) over X is a sheaf (of commutative groups) F over X ,together with a morphism s ∗ F → r ∗ F of sheaves over G , with analogous multiplicativity conditionsto the case of G -C ∗ -algebras.In fact, when G is ample, such G -sheaves are represented by unitary C c ( G, Z ) -modules [Ste14].Here, we consider the convolution product on C c ( G, Z ), and a module M over C c ( G, Z ) is said tobe unitary if it has the factorization property C c ( G, Z ) M = M . The correspondence is given byΓ c ( U, F ) = C c ( U, Z ) M for open sets U ⊂ X if F is the sheaf corresponding to such a module M .A sheaf F on a topological space Y is called soft (resp. c-soft ) if, for any closed (resp. compact)subspace A and s ∈ Γ( A, F ), there is a global section s ′ ∈ Γ( Y, F ) such that s ′ | A = s . Proposition 1.9.
Let Y be a totally disconnected, second countable, locally compact Hausdorff space.Then any sheaf of commutative groups on Y is soft. This seems to be folklore, but can be directly deduced from [God73, Sections 3.3 ad 3.4] combinedwith the following observation. By assumption, Y is the union of its isolated points with eithera Cantor set X , or a disjoint union of countable copies of X . In particular, any accumulationpoint has an open neighborhood U homeomorphic to X , and any closed subset of U has a base ofneighborhoods consisting of compact open subsets of Y .Back to equivariant sheaves over (second countable) ample groupoids, with G , F , and M asabove, the homology of G with coefficient in F , denoted H ∗ ( G, F ), is the homology of the chaincomplex ( C c ( G ( n ) , Z ) ⊗ C c ( X, Z ) M ) ∞ n =0 with differentials coming from the simplicial structure as above.Concretely, the differential is given by ∂ n : C c ( G ( n ) , Z ) ⊗ C c ( X, Z ) M → C c ( G ( n − , Z ) ⊗ C c ( X, Z ) M∂ n ( f ⊗ m ) = n − X i =0 ( − i ( d ni ) ∗ f ⊗ m + ( − n α n ( f ⊗ m ) , where α n is the concatenation of the last leg of C c ( G ( n ) , Z ) with M induced by the module structuremap C c ( G, Z ) ⊗ M → M . By Proposition 1.9, this definition agrees with the one given in [CM00] asthere is no need to take c -soft resolutions of equivariant sheaves.More generally, if F • is a chain complex of G -sheaves modeled by a chain complex of unitary C c ( G, Z )-modules M • , we define H ∗ ( G, F • ), the hyperhomology with coefficient F • , as the homologyof the double complex ( C c ( G ( p ) , Z ) ⊗ C c ( X, Z ) M q ) p,q .As usual, a chain map of complexes of G -sheaves f : F • → F ′• is a quasi-isomorphism if it inducesquasi-isomorphisms on the stalks. When F • and F ′• are bounded from below, such maps induce anisomorphism of the hyperhomology [CM00, Lemma 3.2].1.4. Triangulated categorical structures.
The framework of triangulated categories is ideal forextending basic constructions from homotopy theory to categories of C ∗ -algebras. Much work in thisdirection has been carried out by Meyer and Nest in [Mey08, MN06, MN10].We follow their convention which we quickly recall here. The fundamental axiom requires thatthere is an autoequivalence Σ, and any morphism f : A → B should be part of an exact triangle: A → B → C → Σ A. An additive functor F between triangulated categories is said to be exact when it intertwines suspen-sions and preserves exact triangles. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 7 We say that T has countable direct sums if, given a sequence of objects ( A n ) ∞ n =1 in T , there is anobject L ∞ n =1 A n such that T ∞ M n =1 A n , B ! ≃ ∞ Y n =1 T ( A n , B )naturally in the A n and B . An exact functor F is compatible with direct sums if it commutes withcountable direct sums (see [Mey08, Proposition 3.14]).As before let G be a second countable, locally compact, Hausdorff groupoid with a (left) Haarsystem. Note that triangulated categories involving KK-theory have no more than countable directsums, because separability assumptions are needed for certain analytical results in the background. Proposition 1.10 ([Pro18a, Section A.3]) . The equivariant Kasparov category KK G is triangulated. See Section A.1 for some details. Here, the suspension functor Σ is given by Σ A = C ( R , A ). Notethat Bott periodicity implies Σ ≃ id, so that Σ is also a model of Σ − . The exact triangles aredefined as the triangles isomorphic to mapping cone triangles for equivariant ∗ -homomorphisms.We also note that functors such as A G ⋉ A and A D ⊗ A preserve mapping cones, hence definetriangulated functors into appropriated (equivariant) KK -categories. These are also compatible withcountable direct sums.We call a subcategory thick when it is closed under direct summands. Definition 1.11.
We call two thick triangulated subcategories L , N of T complementary if T ( P, N ) =0 for all P ∈ L , N ∈ N , and for any A ∈ T , there is an exact triangle P A → A → N A → Σ P A where P A ∈ L and N A ∈ N .Let us list some of the basic properties of a pair of complementary subcategories (see [MN06,Proposition 2.9]). • We have N ∈ N if and only if T ( P, N ) = 0 for all P ∈ L . Analogously, we have P ∈ L if andonly if T ( P, N ) = 0 for all N ∈ N . • The exact triangle as above, with P A ∈ L and N A ∈ N , is uniquely determined up toisomorphism and depends functorially on A . In particular, its entries define functors P : T → L , A P A N : T → N , A N. The functors P and N are respectively left adjoint to the embedding functor P → T andright adjoint to the embedding functor
N → T . • The localizations T / N and T / L exist and the compositions L −→ T −→ T / N , N −→ T −→ T / L are equivalences of triangulated categories.Most concrete examples come from homological ideals with enough projectives , as we quickly recallhere. Let T and S be triangulated categories with countable direct sums, and F : T → S be an exactfunctor compatible with direct sums. The system of morphisms I ( A, B ) = ker( F : T ( A, B ) → S ( F A, F B ))is an example of homological ideal compatible with countable direct sums.
Remark . We do not lose generality by assuming that S is a stable abelian category, and that F is a stable functor, see [MN10, Remark 19]. More concretely, we can always replace the targettriangulated category S by the category of representable contravariant functors S →
Ab, which arecokernels of the natural transforms S (- , A ) → S (- , B ) induced by some f : A → B .An object P ∈ T is called I -projective if I ( P, A ) = 0 for all objects A ∈ T . An object N ∈ T iscalled I -contractible if id N belongs to I ( N, N ). Let P I , N I ⊆ T be the full subcategories of projectiveand contractible objects, respectively. We say that I has enough projectives if for any A ∈ T , thereis an I -projective object P and a morphism P → A such that, in the associated exact triangle P → A → C → Σ P, OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 8 the morphism A → C belongs to I . With I = ker F as above, the latter condition is equivalent to F P → F A being a split surjection for all A .We denote by h P I i the localizing subcategory generated by the projective objects, i.e., the smallesttriangulated subcategory that is closed under countable direct sums and contains P I . In particular, h P I i is closed under isomorphisms, suspensions, and if A → B → C → Σ A is an exact triangle in T where any two of the objects A, B, C are in h P I i , so is the third. Note that N I is localizing, and any localizing subcategory is thick. Theorem 1.13 ([Mey08, Theorem 3.16]) . Let T be a triangulated category with countable direct sums,and let I be a homological ideal with enough projective objects. Suppose that I is compatible withcountable direct sums. Then the pair of localizing subcategories ( hP I i , N I ) in T is complementary.Remark . Note that if ( L , N ) is a complementary pair, then ker P has enough projectives and wehave L = P ker P , N = N ker P . Thus the above construction is universal, although I is not uniquelydetermined from ( hP I i , N I ). Definition 1.15.
Let F : T → S be an exact functor compatible with countable direct sums. Givenan object A ∈ T and a chain complex · · · P n · · · P A, δ n +1 δ n δ δ (1)we say that (1) is an (even) I -projective resolution of A if each P n is I -projective and the chaincomplex F ( P • ) F ( A ) 0 F ( δ ) is split exact.There is also an intrinsic formulation of I -exactness for chain complexes corresponding to thesecond condition above, and the above definition does not depend on the choice of F with I = ker F .Moreover, if I has enough projectives, any A has an I -projective resolution. In particular, two I -projective resolutions of A are chain homotopy equivalent, and we obtain functor T →
Ho( T ). Definition 1.16. An odd I -projective resolution is an I -projective resolution where the boundarymaps of positive index have degree one, i.e., the morphism δ n : P n → P n − gets replaced, for n ≥ δ n : P n → Σ P n − .Evidently, if ( P n , δ n ) is an odd projective resolution, then ( P ′ n , δ ′ n ) is an even resolution, where P ′ n = Σ − n P n , δ ′ n = Σ − n δ n , and δ ′ = δ .Let K : T → C be a covariant homological functor into a stable abelian category. We put K n ( A ) = K (Σ − n A ). Let us recall a few extra constructions on K motivated by homological algebra. Definition 1.17.
Let ( L , N ) be a complementary pair, with P : T → L . The localization of K withrespect to N is defined by L N K = K ◦ P .The defining morphisms P ( A ) → A induce a natural transformation L N K ⇒ K . Definition 1.18.
Let I be a homological ideal with countable direct sums and enough projectives.The p -th derived functor of K with respect to I is defined as L I p K ( A ) = H p ( K ( P • )) , where P • is any I -projective resolution of A .This is well-defined because projective resolutions are unique up to chain homotopy. Note thatunless K is compatible with I -exact sequences, one cannot expect L I K ≃ K . When ( L , N ) isa complementary pair, we can think of the localization L N K as the derived functor L ker P K for P : T → L up to the embedding of Remark 1.12.Building on the idea of Christensen [Chr98] to understand the Adams spectral sequence, Meyerconstructed the following spectral sequence.
OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 9 Theorem 1.19 ([Mey08, Theorems 4.3 and 5.1]) . Let I be a homological ideal with countable directsums and enough projectives, and let K : T → Ab be a homological functor. Then there is a convergentspectral sequence E rpq ⇒ L N I K p + q ( A ) , with the E -sheet E pq = L I p K q ( A ) . The E r -differentials d r : E rpq → E rp − r,q + r − come from a choice of phantom tower for A and theassociated exact couple , but their precise form will not be important for us.1.5. The Baum–Connes conjecture for groupoids.
Because we are particularly interested inspectral sequences which approximate the K -groups of groupoid C ∗ -algebras, the Baum–Connesconjecture naturally plays a fundamental role. The notion of pair of complementary subcategoriesintroduced earlier allows for a general formulation of this conjecture in terms of localization at thecontractible objects.However, as our main focus is on torsion-free amenable groupoids, we will not need the full ma-chinery for our applications, hence we limit ourselves to simply recalling the main positive resultconcerning the conjecture for groupoids with the Haagerup property.Suppose G is a second countable, locally compact, Hausdorff groupoid with second countable,Hausdorff unit space X . In the following, the crossed product is understood to be reduced . Definition 1.20. A G -algebra A is said to be proper if there is a locally compact Hausdorff proper G -space Z such that A is a G ⋉ Z -algebra.Evidently, a commutative G -C ∗ -algebra is proper if and only if its spectrum is a proper G -space. Remark . If G is locally compact, σ -compact, Hausdorff, then there is a locally compact, σ -compact, and Hausdorff model of E G , the classifying space for proper actions of G ; in our case G is second countable hence E G is too [Tu99b, Proposition 6.15]. In Definition 1.20 for a proper G -algebra we can always assume Z to be a model of E G . Indeed if φ : Z → E G is a G -equivariantcontinuous map, then φ ∗ : C (E G ) → M ( C ( Z )) = C b ( Z ) can be precomposed with the structuremap Φ : C ( Z ) → Z M ( A ), making A into an G ⋉ E G -algebra.We will need the following result proved by J.-L. Tu. Theorem 1.22 ([Tu99a]) . Suppose that G acts properly on a continuous field of affine Euclideanspaces. Then there exists a proper G -space Z with an open surjective structure morphism Z → X ,and a G ⋉ Z -C ∗ -algebra P which is a continuous field of nuclear C ∗ -algebras over Z , and such that P ≃ C ( X ) in KK G . As a consequence, for any other algebra A ∈ KK G , we have that A ⊗ C ( X ) P is a proper G -C ∗ -algebra and KK G -equivalent to A .In this paper, for a general groupoid G we say that it satisfies the strong Baum–Connes conjectureif the conclusions of the previous theorem hold. This definition implies the standard version of theconjecture. More precisely, if D : P → C ( X ) is the isomorphism from Theorem 1.22, the followingdiagram is commutative ([EM10, Theorem 6.12], see also [MN06]).lim −→ Y ⊆ EG KK G ( C ( Y ) , A ) K ∗ ( G ⋉ A ) K ∗ ( G ⋉ ( A ⊗ C ( X ) P )) µ GA ≃ j G ( D b ⊗ id A ) The functor j G above is the descent morphism of Kasparov [Kas88] which has been generalized tothis context in [LG99, Laf07].The groupoids arising from Smale spaces are amenable [PS99, Theorem 1.1]. In particular, theyact properly on a continuous field of affine Euclidean spaces [Tu99a, Lemma 3.5], and the theoremabove applies. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 10 Induction and restriction for groupoid KK -theory. Suppose G is a groupoid as in theprevious subsection.Let H ⊆ G be an open subgroupoid with the same base space X = G (0) = H (0) . Note that H has a continuous Haar system automatically by restriction. We have a natural restriction functorRes GH : KK G → KK H . It admits a left adjoint, which is an analogue of induction, as follows. Fulldetails will appear elsewhere in a joint work of the first named author with C. Bönicke.Let B be an H -C ∗ -algebra, with structure map ρ : C ( X ) → Z ( M ( B )). As before, take the C ( G )-algebra B ′ = C ( G ) ⊗ s C ( X ) B. This has a right action of H , by combination of the right translation on C ( G ) and the action on B twisted by the inverse map of H . We then setInd GH ( B ) = B ′ ⋊ H = ( C ( G ) ⊗ s C ( X ) B ) ⋊ diag H. This can be regarded as the crossed product of B ′ by the transformation groupoid G ⋊ H for theright translation action of H on G . Moreover, notice that G also acts on B ′ by left translation on C ( G ). This induces a continuous action of G on Ind GH ( B ).Let A be a G -C ∗ -algebra. Then the Haar system on G induces an A -valued inner product on C c ( G ) ⊗ C ( X ) A , and by completion we obtain a right Hilbert A -module E GA = L ( G ; A ). We thenhave the following, see Section A.2 for details. Proposition 1.23.
Under the above setting, E GA implements an equivariant strong Morita equivalencebetween A and Ind GG A . Let κ denote the inclusion homomorphismInd GH Res GH ( A ) = ( C ( G ) ⊗ s C ( X ) A ) ⋊ H → ( C ( G ) ⊗ s C ( X ) A ) ⋊ G = Ind GG A, induced by H ⊆ G because H is open, and let ι denote the mapInd HH B = ( C ( H ) ⊗ s C ( X ) B ) ⋊ H → ( C ( G ) ⊗ s C ( X ) B ) ⋊ H = Res GH Ind GH ( B )induced by the ideal inclusion C ( H ) ⊆ C ( G ). Theorem 1.24.
The functor
Ind GH induces a functor KK H → KK G , and there is a natural isomor-phism KK G (Ind GH B, A ) ≃ KK H ( B, Res GH A ) defining an adjunction ( ǫ, η ) : Ind GH ⊣ Res GH with counit and unit natural morphisms ǫ A = [ κ ] ⊗ Ind GG A [ E GA ] ∈ KK G (Ind GH Res GH A, A ) , η B = [ ¯ E HB ] ⊗ Ind HH B [ ι ] ∈ KK H ( B, Res GH Ind GH B ) . In fact, Theorem 4.5 only requires this for H = X in ample groupoids G , for which [Bön18] isenough. Example . If G is the transformation groupoid Γ ⋉ X and H = X , the previous theorem amountsto KK Γ ⋉ X ( C (Γ) ⊗ B, A ) ≃ KK X ( B, A )for any C ( X )-algebra B and G -algebra A , where the Γ-action on C (Γ) ⊗ B is given by translationon the factor C (Γ).1.7. Smale spaces.
A Smale space is given by a self-homeomorphism on a compact metric spacewhich admit contracting and expanding directions. The precise definition requires the definition of abracket map satisfying certain axioms [Put14, Rue04], as follows.
Definition 1.26. A Smale space ( X, φ ) is given by a compact metric space (
X, d ) and a homeomor-phism φ : X → X such that: • there exist constant 0 < ǫ X and a continuous map { ( x, y ) ∈ X × X | d ( x, y ) ≤ ǫ X } → X, ( x, y ) [ x, y ]satisfying the bracket axioms :[ x, x ] = x, [ x, [ y, z ]] = [ x, z ] , [[ x, y ] , z ] = [ x, z ] , φ ([ x, y ]) = [ φ ( x ) , φ ( y )] , for any x, y, z in X when both sides are defined. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 11 • there exists 0 < λ < contraction axioms :[ x, y ] = y ⇒ d ( φ ( x ) , φ ( y )) ≤ λd ( x, y ) , [ x, y ] = x ⇒ d ( φ − ( x ) , φ − ( y )) ≤ λd ( x, y ) , whenever the brackets are defined.Suppose x ∈ X and 0 < ǫ ≤ ǫ X . We define the local stable sets and the local unstable sets around x as X s ( x, ǫ ) = { y ∈ X | d ( x, y ) < ǫ, [ y, x ] = x } X u ( x, ǫ ) = { y ∈ X | d ( x, y ) < ǫ, [ x, y ] = x } . The bracket [ x, y ] can be characterized as the unique element of X s ( x, ǫ ) ∩ X u ( y, ǫ ) when 2 d ( x, y ) <ǫ < ǫ X . This means that, locally, we can choose coordinates so that[ · , · ] : X u ( x, ǫ ) × X s ( x, ǫ ) → X is a homeomorphism onto an open neighborhood of x ∈ X for 0 < ǫ < ǫ X / x ∈ X is called non-wandering if for all opens U ⊆ X containing x there exists N ∈ N with U ∩ φ N ( U ) = ∅ . Periodic points are dense among the non-wandering points [Put15, Theorem4.4.1]. We say that X is non-wandering if any point of X is non-wandering. We will set a blanketassumption that Smale spaces are non-wandering . This holds in virtually all interesting examples.It can be shown that any non-wandering Smale space (
X, φ ) can be partitioned in a finite number of φ -invariant clopen sets X , . . . , X n , in a unique way, such that ( X k , φ | X k ) is irreducible for k = 1 , . . . , n [Put00]. Irreducibility means that for every (ordered) pair U, V of nonempty open sets in X , thereexists N ∈ N such that U ∩ φ n ( V ) = ∅ , n ≥ N . Example . The standard definition of a shift of finite type is given in [LM95, Definition 2.1.1].However, an equivalent and more convenient definition is to start out with a finite directed graph G .A directed graph G = ( G , G , i, t ) consists of finite sets G and G , called vertices and edges, suchthat each edge e ∈ G is given by a directed arrow from i ( e ) ∈ G to t ( e ) ∈ G . Then a shift of finitetype (Σ G , σ ) is defined as the space of bi-infinite sequences of pathsΣ G = { e = ( e k ) k ∈ Z ∈ ( G ) Z | t ( e k ) = i ( e k +1 ) } , together with the left shift map σ ( e ) k = e k +1 . The metric is such that d ( e, f ) ≤ − n − if e, f coincide on the interval [ − n, n ]. In particular, d ( e, f ) = 2 − means that e, f share the central edge,i.e., e = f . Then we can define[ e, f ] = ( . . . , f − , f − , e , e , e , . . . ) . The pair (Σ G , σ ) is a Smale space with constant ǫ = 1 / unstable equivalence relation of Smalespaces. Given x, y ∈ X , we say they are • stably equivalent , denoted by x ∼ s y , iflim n →∞ d ( φ n ( x ) , φ n ( y )) = 0; • unstably equivalent , x ∼ u y , iflim n →∞ d ( φ − n ( x ) , φ − n ( y )) = 0 . We denote the graphs of these relations as R s ( X, φ ) = { ( x, y ) ∈ X × X | x ∼ s y } , (2) R u ( X, φ ) = { ( x, y ) ∈ X × X | y ∼ u y } , and treat them as groupoids, with source, range, and composition maps given by s ( x, y ) = y, r ( x, y ) = x, ( x, y ) ◦ ( w, z ) = ( x, z ) if y = w . OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 12 The orbit of x ∈ X under the stable (resp. unstable) equivalence relation is called the global stable (resp. unstable ) set , and is denoted by X s ( x ) (resp. X u ( x )). They satisfy the following identities: X s ( x ) = [ n ≥ φ − n ( X s ( φ n ( x ) , ǫ )) (3) X u ( x ) = [ n ≥ φ n ( X s ( φ − n ( x ) , ǫ )) (4)for any fixed ǫ < ǫ X .This leads to locally compact Hausdorff topologies on the above groupoids [Put96]: consider theinduced topology on G ns = { ( x, y ) | y ∈ φ − n ( X s ( φ n ( x ) , ǫ )) } , G nu = { ( x, y ) | y ∈ φ n ( X s ( φ − n ( x ) , ǫ )) } as subsets of X × X . Then, as R s ( X, φ ) is the union of the increasing sequence, it has the inductivelimit topology of these spaces. Since the inclusion G ns → G n +1 s is open, R s ( X, φ ) is a locally compactHausdorff groupoid. Of course, analogous considerations make R u ( X, φ ) a locally compact Hausdorffgroupoid.To get an étale groupoid, we can take a transversal T ⊂ X and restrict the base space to T , putting G | T = G TT . A canonical choice is to take T = X s ( x ), with the inductive limit topology from (3), whichis an example of generalized transversal. Slightly generalizing this, for a subset P ⊆ X , we write X s ( P ) meaning the union of all X s ( x )’s for x ∈ P , with the disjoint union topology. Analogously wedefine X u ( P ) = S x ∈ P X u ( x ). Let us put R s ( X, P ) = R s ( X, φ ) | X u ( P ) , R u ( X, P ) = R u ( X, φ ) | X s ( P ) . As we indicated after Definition 1.3, since we consider finer topologies on the sets X s ( x ), X u ( x )than the ones induced by the inclusion into X , we need to endow R s ( X, P ) , R u ( X, P ) with a differenttopology, following [PS99]. Concretely, this is achieved by taking the “holonomy groupoid” topologyfor the maps in (5) (see for example [Kil09, Theorem 2.17], see also [Tho10a] under the name “topologyof local conjugacies”). For each pair ( x, y ) ∈ G ns , consider maps X u ( y, δ ) → X u ( x, δ ) , z φ − n ([ φ n ( z ) , φ n ( x )]) , (5)defined for any δ > φ n ( X u ( x, δ )) ⊆ X u ( φ n ( x ) , ǫ ) , φ n ( X u ( y, δ )) ⊆ X u ( φ n ( y ) , ǫ ) . This way, R s ( X, P ) becomes an étale groupoid, which is Morita equivalent to R s ( X, φ ). Here, theequivalence is implemented by the set R s ( X, φ ) X u ( P ) , together with the topology generated by thesets of the form U ∩ s − V for open sets U ⊂ R s ( X, φ ) and V ⊂ X u ( P ). Analogous considerationshold for R u ( X, P ). Theorem 1.28 ([PS99, Theorem 1.1]) . These groupoids are amenable.
Maps of Smale spaces.
A continuous and surjective map f : ( X, φ ) → ( Y, ψ ) between Smalespaces is called a factor map if it intertwines the respective self-maps, i.e., f ◦ φ = ψ ◦ f. (6)Equation (6) is enough to guarantee that f preserves the local product structure. In particular,there is ǫ f > x , x ] and [ f ( x ) , f ( x )] are defined and f ([ x , x ]) = [ f ( x ) , f ( x )]for all x , x with d ( x , x ) < ǫ f . Proposition 1.29 ([Put15, Lemma 5.2.10]) . If y ∈ Y is a periodic point with f − ( y ) = { x , . . . , x N } ,given ǫ X > ǫ > , there exists δ > such that f − ( Y u ( y , δ )) ⊆ N [ i =1 X u ( x i , ǫ ) . Definition 1.30.
A factor map f : ( X, φ ) → ( Y, ψ ) is called s -resolving if it induces an injective mapfrom X s ( x ) to Y s ( f ( x )) for each x ∈ X . It is called s -bijective , if moreover these induced maps arebijective. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 13 Theorem 1.31 ([Put05, Corollary 3]) . Let ( X, φ ) be an irreducible Smale space such that X s ( x, ǫ ) is totally disconnected for every x ∈ X and < ǫ < ǫ X . Then there is an irreducible shift of finitetype (Σ , σ ) and an s -bijective factor map f : (Σ , σ ) → ( X, φ ) . Theorem 1.32 ([Put15, Theorem 5.2.4]) . Let f : X → Y is an s -resolving map between Smale spaces.There is a constant N ≥ such that for any y ∈ Y there exist x , . . . , x n in X , with n ≤ N , satisfying f − ( Y u ( y )) = n [ k =1 X u ( x k ) . For any y ∈ Y the cardinality of the fiber f − ( y ) is less than or equal to N . Let us list several additional facts about s -resolving maps, which can be found in [Put15]. First,if each point in Y is non-wandering, then f is s -bijective. Second, the induced maps X s ( x ) → Y s ( f ( x )) and X u ( x ) → Y u ( f ( x )) are both continuous and proper in the inductive limit topologyof the presentation in (3) and (4). If, moreover, f is s -bijective, the map X s ( x ) → Y s ( f ( x )) is ahomeomorphism. Assume that X and Y are irreducible, and P is an at most countable subset of X such that no two points of P are stably equivalent after applying f . Then f × f : R u ( X, P ) → R u ( Y, f ( P ))is a homeomorphism onto an open subgroupoid of R u ( Y, f ( P )).2. Pullback and resolution groupoids
In this section we consider the groupoids associated to resolutions of Smale spaces and proveseveral key Morita equivalences.2.1.
Multiple pullback of groupoids.
We start by defining the appropriate notion of fiberedproduct between groupoids which will be used in the following proofs.
Definition 2.1.
Let α : H → G be a homomorphism of groupoids, and n ≥
2. We define the n -thfibered product of H with respect to α as the groupoid H × G n defined as follows: • the object space is the set( H × G n ) (0) = { ( y , g , y , . . . , g n − , y n ) | y k ∈ H (0) , g k ∈ G α ( y k ) α ( y k +1 ) }• the arrows from ( y , g , y , . . . , g n − , y n ) to ( y ′ , g ′ , y ′ , . . . , g ′ n − , y ′ n ) are given by the n -tuples( h , . . . , h n ) ∈ H y ′ y × · · · × H y ′ n y n such that the squares in α ( y ′ ) α ( y ′ ) · · · α ( y ′ n ) α ( y ) α ( y ) · · · α ( y n ) g ′ g ′ g ′ n − α ( h ) g α ( h ) g g n − α ( h n ) are all commutative.(Of course, we can put H × G = H ). We say that an arrow in H × G n is represented by the tuple( h , g ′ , h , . . . , g ′ n − , h n ) in the above situation. This way we can think of H × G n as a subset of H × G × · · · G × H , and in the setting of topological groupoids this gives a compatible topology on H × G n (for example, local compactness passes to H × G n ). Remark . The above definition makes sense for n -tuples of different homomorphisms α k : H k → G ,so that we can define H × G · · · × G H n as a groupoid. The case of n = 2 appears in [CM00]. Definition 2.3.
In the setting of Definition 2.1, define a groupoid G × G H × G n as follows: • the object space is the set( G × G H × G n ) (0) = { ( g , y , g , y , . . . , g n − , y n ) | y k ∈ H (0) , g ∈ G α ( y ) , g k ∈ G α ( y k ) α ( y k +1 ) ( k ≥ } OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 14 • a morphisms from ( g , y , g , y , . . . , g n − , y n ) to ( g ′ , y ′ , g ′ , y ′ , . . . , g ′ n − , y ′ n ) is given by k ∈ G rg ′ rg and an n -tuple ( h , . . . , h n ) ∈ H y ′ y × · · · × H y ′ n y n such that the squares in rg ′ α ( y ′ ) α ( y ′ ) · · · α ( y ′ n ) rg α ( y ) α ( y ) · · · α ( y n ) g ′ g ′ g ′ g ′ n − k α ( h ) g g α ( h ) g g n − α ( h n ) are all commutative.Again we say that an arrow of G × G H × G n is represented by ( k, g ′ , h , . . . , h n ) in the abovesituation. As in the case of H × G n , this induces a compatible topology in the setting of topologicalgroupoids. Proposition 2.4.
Let α : H → G be a homomorphism of topological groupoids. Then H × G n and G × G H × G n are Morita equivalent as topological groupoids.Proof. Consider the space Z = { ( g , h , g , h , . . . , g n − , h n ) | ( g , . . . , g n − ) ∈ G ( n ) , α ( rh k ) = sg k − } . We define a left action of G × G H × G n as follows. The anchor map is Z → ( G × G H × G n ) (0) , ( g , h , . . . , h n ) ( g , rh , g , . . . , rh n ) , and an arrow of G × G H × G n with source ( g , rh , g , . . . , rh n ) acts by( k, g ′ , h ′ , . . . , h ′ n ) . ( g , h , . . . , h n ) = ( g ′ , h ′ h , g ′ , . . . , h ′ n h n ) . On the other hand, there is a right action of H × G n defined as follows. The anchor map is Z → ( H × G n ) (0) , ( g , h , . . . , h n ) ( sh , g ′ , . . . , sh n ) , ( g ′ k = α ( h k ) − g k α ( h k +1 )) . An arrow of H × G n with range ( sh , g ′ , . . . , sh n ) acts by( g , h , . . . , h n ) . ( h ′′ , g , h ′′ , . . . , h ′′ n ) = ( g , h h ′′ , g , . . . , h n h ′′ n ) . We claim that Z implements the Morita equivalence (compatibility with topology will be obviousfrom the concrete “coordinate transform” formulas).Comparing between Z × ( G × G H × Gn ) (0) Z and Z × ( H × Gn ) (0) H × G n amounts to comparison of pairs( h k , h ′ k ) with rh k = rh ′ k on the one hand, and the composable pairs ( h k , h ′′ k ) ∈ H (2) on the other.There is a bijective correspondence between the two sides, given by the coordinate transform h ′ k = h k h ′′ k . Comparing Z × ( H × Gn ) (0) Z with G × G H × G n × ( G × G H × Gn ) (0) Z amounts to comparing: • on the side of Z × ( H × Gn ) (0) Z : (( g , h ) , ( g ′ , h ′ )) with ( g , α ( h ) , ( g ′ , α ( h ′ )) ∈ G (2) and sh = sh ′ , and ( h k , h ′ k ) ∈ H (2) with sh k = sh ′ k for k ≥ • on the side of G × G H × G n × ( G × G H × Gn ) (0) Z : ( k, g ′′ ) ∈ G (2) , ( h , h ′′ ) ∈ H (2) with sh = sg ′′ ,and ( h k , h ′′ k ) ∈ H (2) for k ≥ h ′ k = h ′′− k , g = g ′′ α ( h ) − , and g ′ = kg ′′ α ( h ) − . (cid:3) A slight generalization is obtained by considering the groupoid H × G a × G G × G H × G b for a, b ≥ H × G ( a + b +1) in Definition 2.1, with the difference that h a +1 is not in H y ′ a +1 y a +1 , andinstead in G α ( y ′ a +1 ) α ( y a +1 ) . Proposition 2.5.
The groupoid H × G a × G G × G H × G b is Morita equivalent to H × G ( a + b ) .Proof. Recall the construction in the proof of Proposition 2.4 for the Morita equivalence between G × G H × G b and H × G b : we have the space Z = { ( g , h , g , . . . , h b ) | ( g , . . . , g b − ) ∈ G ( b ) , α ( rh k ) = rg k } , which is a bimodule between these groupoids. Based on this, put˜ Z = { ( h , g , h , . . . , g a , g a +1 , h a +1 , g i +2 , . . . , h a + b ) | ( g , . . . , g a + b ) ∈ G ( a + b ) ,α ( rh k ) = rg k ( k ≤ a ) , α ( rh k ) = sg k ( k > a ) } . OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 15 This has obvious “composition” actions of H × G a × G G × G H × G b from the left and H × G ( a + b ) fromthe right. By a similar argument as before, we can see that ˜ Z implements a Morita equivalence. (cid:3) Next let us show the compatibility of fiber products and generalized transversals.
Proposition 2.6.
Let α : H → G be a homomorphism of topological groupoids, and f : T → H (0) bea generalized transversal. Consider the space ˜ T = { ( t , g , t , . . . , t n ) | t k ∈ T, g k ∈ G f ( t k ) f ( t k +1 ) } with the induced topology from the natural embedding into T n × G n − . The map ˜ f : ˜ T → ( H × G n ) (0) , ( t , g , t , . . . , t n ) ( f ( t ) , g , f ( t ) , . . . , f ( t n )) is a generalized transversal for H × G n .Proof. Let us check the conditions in Definition 1.3. First, ˜ T meets all orbits of H × G n . Indeed, if wetake a point ( y , g , y , . . . , g n − , y n ) ∈ ( H × G n ) (0) , we can find t k ∈ T and h k ∈ H f ( t k ) y k for k = 1 , . . . , n .Then there are unique g ′ k such that ( h , . . . , h n ) represents an arrow from ( y , g , y , . . . , g n − , y n ) to( f ( t ) , g ′ , . . . , f ( t n )).Next, let us check the condition Ar. Thus, take an arrow x represented by ( h , g , h , . . . , g n − , h n )with range rx = ( f ( t ) , g , f ( t ) , . . . , g n − , f ( t n )), open neighborhood U of x , and another V of rx .We may assume that these neighborhoods are of the form U = ( U ′ × U ′′ × U ′ × · · · × U ′ n ) ∩ H × G n , ( U ′ k ∈ O ( h k ) , U ′′ k ∈ O ( g k )) V = ( V ′ × V ′′ × V ′ × · · · × V ′ n ) ∩ ˜ T , ( V ′ k ∈ O ( t k ) , V ′′ k ∈ O ( g k ) . Then, for each k we can find ˜ U k ∈ O ( h k ) with ˜ U k ⊂ U ′ k , ˜ V k ∈ O ( t k ) with ˜ V k ⊂ V ′ k realizing thecondition Ar. We claim that U = ( ˜ U × U ′′ × · · · × ˜ U ′ n ) ∩ H × G n , V = ( ˜ V × V ′′ × · · · × ˜ V n ) ∩ ˜ T do the job. Indeed, if y = (˜ h , ˜ g , · · · , ˜ h n ) ∈ U , another element z = (˜ h ′ , ˜ g ′ , · · · , ˜ h ′ n ) as the samesource as y if and only if s ˜ h k = s ˜ h ′ k and f (˜ h k ) − ˜ g k f (˜ h k +1 ) = f (˜ h ′ k ) − ˜ g ′ k f (˜ h ′ k +1 ) hold for all k .Moreover, rz ∈ ˜ T if and only if rh ′ k ∈ f ( T ) for all k . The elements ˜ g ′ k are determined by the ˜ h ′ k , andwe can find such ˜ h ′ k uniquely by condition Ar for U ′ k and V ′ k . (cid:3) Suppose f : T → H (0) is a generalized transversal for H such that αf : T → G (0) is also a transver-sal for G . Then α induces a homomorphism of étale groupoids from H ′ = H | f ( T ) to G ′ = G | αf ( T ) . Corollary 2.7.
In the setting above, H × G n is Morita equivalent to H ′× G ′ n .Proof. The construction of Proposition 2.6 gives a generalized transversal for ˜ f : ˜ T → ( H × G n ) (0) .The étale groupoid obtained by this is isomorphic to H ′× G ′ n . (cid:3) Transversality for Smale spaces.
Let (
Y, ψ ) be a non-wandering Smale space with totallydisconnected unstable sets, and f : (Σ , σ ) → ( Y, ψ ) be an s -resolving (hence s -bijective) factor mapfrom a shift of finite type.Let Σ n denote the fibered product of n + 1 copies of Σ with respect to f . Then σ n = σ × · · · × σ | Σ n defines a Smale space, which is again a shift of finite type. If a = ( a , . . . , a n ) and b = ( b , . . . , b n ) arepoints of Σ n , they are unstably (resp. stably) equivalent if and only if a k is unstably (resp. stably)equivalent to b k for all k . Theorem 2.8.
In the setting above, set G = R u ( Y, ψ ) , H = R u (Σ , σ ) , and α = f × f : H → G bethe induced groupoid homomorphism. Then H × G n +1 is Morita equivalent to R u (Σ n , σ n ) as a locallycompact groupoid. We will apply this to the s -bijective maps from Theorem 1.31. A key step is the following propo-sition, which is our first technical result. Proposition 2.9.
Let f : (Σ , σ ) → ( Y, ψ ) be an s -bijective factor map from a shift of finite type.Suppose a , . . . , a n in Σ are points such that f ( a ) ∼ u f ( a k ) for all k . Then there are points b , . . . , b n in Σ satisfying a k ∼ u b k , f ( b ) = f ( b k ) for k = 0 , . . . , n . OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 16 Lemma 2.10.
Let d be the standard metric of Σ , and a, b ∈ Σ be points such that d ( a, b ) < ǫ Σ . Thenwe have d ([ a, b ] , b ) = d ([ b, a ] , a ) .Proof. As we saw in Example 1.27, the brackets are given by [ a, b ] = ( . . . , b − , a , a , . . . ) and [ b, a ] =( . . . , a − , a , b , . . . ). Hence both distances are computed (in the same way) from the minimum n > a n = b n . (cid:3) Proof of Proposition 2.9.
A graphical illustration for the case n = 1 is provided in Figure 1, with thevertical direction representing the stable direction and the horizontal one representing the unstableone. Because the maps ψ n for n ∈ Z preserve the unstable equivalence relation, we can assume d ( f ( a ) , f ( a k )) < ǫ < ǫ Y and f ( a k ) = [ f ( a k ) , f ( a )] holds for all k . Let 0 < δ < ǫ Y / f : Σ s ( a k , δ ) → Y s ( f ( a k ) , ǫ ) are homeomorphisms onto their images.Choose a periodic point y ∈ Y close to the points f ( a k ), so that y = [ y , f ( a k )] and z k =[ f ( a k ) , y ] are well-defined. Note that y does not depend on k . We claim that there are points b k ∈ Σ such that a k ∼ u b k and f ( b k ) = y .Write f − ( y ) = { c , . . . , c m } , with m ≤ N as in Theorem 1.32. Replacing ψ and σ by anappropriate power, we may assume that each c i is fixed by σ .Since f is s -bijective, there is a unique point ¯ z k ∈ Y s ( a k , δ ) satisfying f (¯ z k ) = z k . As y is fixedby ψ and z k ∼ u y , we have the convergence ψ − n ( z k ) → y . Consider the sequence ( σ − n (¯ z k )) ∞ n =0 .Since Σ is compact, we can take a cluster point w , which should be among the c i ’s. Then, as the c i ’sare fixed by σ , our sequence can only cluster around one of them. We thus obtain σ − n (¯ z k ) → c i k forsome i k , and we get ¯ z k ∼ u c i k . Again using s -bijectivity, there is a unique b k ∈ Σ s ( c i k ) such that f ( b k ) = y . It remains to prove that b k ∼ u a k . By Proposition 1.29, there is δ such that f − ( Y u ( y , δ )) ⊆ m [ i =1 Σ u ( c i , ǫ ′ ) , where 2 ǫ ′ < min( ǫ f , ǫ Σ ). Take M > ψ − M ( z k ) ∈ Y u ( y , δ ), so that we have σ − M (¯ z k ) ∈ Σ u ( c i k , ǫ ′ ). Then take points u , . . . , u n from Σ s ( c i k ) such that d ( c i k , u ) , d ( u , u ) , . . . , d ( u n , σ − M ( b k )) < ǫ ′ . Then we can inductively define v = [ σ − M (¯ z k ) , u ] , v = [ v , u ] , . . . , v n +1 = [ v n , σ − M ( b k )]since d ( v i , u i ) remains equal to d ( σ − M (¯ z k ) , c i k ) < ǫ ′ by Lemma 2.10.Mapping down by f , we have the same relation as above for the points ψ − M ( z k ), f ( u i ), and f ( v i ).This shows, for example, ψ M ( f ( v )) = [ z k , ψ M ( f ( u ))], and by induction, we obtain ψ M ( f ( v n +1 )) =[ z k , y ] = f ( a k ). Again s -bijectivity implies σ M ( v n +1 ) = a k , and we obtain a k ∼ u b k . (cid:3) Remark . Although we presented a somewhat metric geometrical proof, it is possible to turn partof it into a more direct argument using a symbolic presentation of Σ; as the points c i are representedby periodic sequences, ¯ z k and b k will be represented by sequences which are periodic in one direction.Combined with the consistency condition for f , it is possible to show a k ∼ u b k from this. f ( a ) ψ M ( f ( v )) f ( v ) ψ M ( f ( v n )) ψ M ( f ( u )) f ( u ) ψ M ( f ( u n )) y f ( a ) y z z Y u ( f ( a ) , ǫ ) Y s ( f ( a ) , ǫ ) c i u u n σ − M ( b ) σ − M (¯ z ) v v n v n +1 b ¯ z a Figure 1.
The configuration of points in the proof of Proposition 2.9.
OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 17 Proof of Theorem 2.8.
We have an embedding of the groupoid R u (Σ n , σ n ) into H × G n +1 by the cor-respondence ( a , . . . , a n ) ( a , id y , a , . . . , id y , a n ) ( y = f ( a ) = · · · = f ( a n ))at the level of objects, and by(( a , . . . , a n ) , ( b , . . . , b n )) (( a , b ) , . . . , ( a n , b n ))at the level of arrows. Proposition 2.9 implies that Σ n ⊂ ( H × G n +1 ) (0) meets all orbits of H × G n +1 .Moreover, a, b ∈ Σ n are connected by an arrow in ( H × G n +1 ) (0) if and only if they are connectedin R u (Σ n , σ n ). Thus, R u (Σ n , σ n ) y ( H × G n +1 ) Σ n x H × G n +1 gives a Morita equivalence betweenthe two groupoids. It is a routine task to see that this is compatible with the topology on the twogroupoids. (cid:3) Combining Proposition 2.4, Corollary 2.7, and Theorem 2.8, we obtain the following.
Theorem 2.12.
In addition to f : (Σ , σ ) → ( Y, ψ ) as above, let f ′ : T → Σ be a generalized transversalfor the locally compact groupoid R u (Σ , σ ) such that f f ′ : T → Y defines a transversal for R u ( Y, ψ ) .Denote the corresponding étale groupoids by H = R u (Σ , σ ) | f ′ ( T ) , G = R u ( Y, ψ ) | ff ′ ( T ) . The groupoid G × G H × G n +1 with respect to the natural inclusion H → G is Morita equivalent to R u (Σ n , σ n ) as a topological groupoid. Approximation in the equivariant KK -category In this section we study a special situation in which Theorem 1.13 can be applied. It yields apair of complementary subcategories which is completely characterized by a pair of adjoint functors.In the setting of the equivariant Kasparov category, we obtain this pair from the induction andrestriction functors, and use it to translate the strong Baum–Connes conjecture to a statement aboutthe localizing subcategory generated by those objects in the image of the induction functor.3.1.
Simplicial approximation from adjoint functors.
One powerful way to check that a homo-logical ideal has enough projectives is to relate it to adjoint functors between triangulated categories.More precisely, let S and T be triangulated categories with countable direct sums, and E : S → T and F : T → S be exact functors compatible with countable direct sums, with natural isomorphisms S ( A, F B ) ≃ T ( EA, B ) ( A ∈ S , B ∈ T ) . (7)Then I = ker F has enough projectives and the I -projective objects are retracts of EA for some A ∈ S [MN10, Section 3.6].Our next goal is to give an explicit projective resolution in this setting. In fact, this situation isquite standard in homological algebra: such adjoint functors give a comonad L = EF on T , fromwhich we obtain a simplicial object ( L n +1 A ) ∞ n =0 giving a “resolution” of A [Wei94, Section 8.6]. Proposition 3.1.
In the above setting, any A ∈ T admits an I -projective resolution P • consistingof P n = L n +1 A . The pair of subcategories ( h E Si , I ) is complementary.Proof. Let us denote the structure morphisms of the adjunction as ǫ B ∈ T ( LB, B ) , η A ∈ S ( A, F EA ) , so that the isomorphism (7) is given by S ( A, F B ) → T ( EA, B ) T ( EA, B ) → S ( A, F B ) f ǫ B E ( f ) g F ( g ) η A . As already observed in [MN10], the objects of the form EA are I -projective. Indeed, if g ∈T ( EA, B ) is in the kernel of F , the corresponding morphism in S ( A, F B ) is zero by the abovepresentation.Next, let us recall the comonad structure on L . There are natural transformations L → id T and L → L defining a coalgebra structure on L . The counit is simply given by the morphisms ǫ B , whilethe comultiplication is given by δ B = E ( η F B ) ∈ T ( LB, L B ). The compatibility condition betweenthese reduces to consistency between ǫ and η . OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 18 Now we are ready to define a structure of simplicial object on ( P n ) ∞ n =0 as in the assertion. The face morphisms d ni : P n → P n − (0 ≤ i ≤ n ) are d ni = L i ( ǫ L n − i A ) : L n +1 A → L n A, while the degeneracy morphisms s ni : P n → P n +1 (0 ≤ i ≤ n ) are s ni = L i ( δ L n − i A ) : L n +1 A → L n +2 A, see [Wei94, Paragraph 8.6.4]. The associated Moore complex on ( P n ) ∞ n =0 is given by δ n = n X i =0 ( − i d ni : P n → P n − , (8)together with the augmentation morphism δ = ǫ : P = LA → A .Now, it remains to check the I -exactness of the augmented complex, or as in Definition 1.15, thesplit exactness of · · · → F L A → F LA → F A → A in a natural way. We claim that the the complex · · · → F L A → F LA → F A → S is contractible. Again this is a consequence of a standard argument: the contracting homotopyis given by h n = η F L n A : F L n A → F L n +1 A for n ≥
0, see [Wei94, Proposition 8.6.10]. The secondstatement follows from Theorem 1.13. (cid:3)
We will apply the previous proposition in the setting of K -theory, more precisely for T = KK G , S = KK H , E = Ind GH , F = Res GH .3.2. The Baum–Connes conjecture for torsion-free groupoids.
Hereafter it is assumed that G is étale and that it satisfies the conclusion of Theorem 1.22. We are going to use the notion of R KK( X )-nuclearity as defined by Bauval [Bau98, Definition 5.1] (see also [Ska88]). Here, we call itKK X -nuclearity.Our next goal is to prove the following result. Theorem 3.2.
Suppose that G is an étale groupoid with torsion-free stabilizers satisfying the conclu-sion of Theorem , and that H ⊆ G is an étale subgroupoid with the same base space X . If A isa G -C ∗ -algebra which is KK X -nuclear as a C ( X ) -algebra, it belongs to the triangulated subcategorygenerated by the image of Ind GH : KK H → KK G . Above, H is an open subgroupoid of G because H (0) = X and H is étale. The key step is to provethe special case when H = X . Proposition 3.3.
Under the assumptions of Theorem , A belongs to the triangulated categorygenerated by the objects Ind GX B for C ( X ) -algebras B . The following lemma clarifies the local picture of proper actions.
Lemma 3.4.
Let G be an étale groupoid with torsion-free stabilizers, and G y Z a proper actionon a locally compact Hausdorff space with the anchor map ρ : Z → X . Then each z ∈ Z has an openneighborhood U satisfying: • U has a compact closure in Z ; • the saturation GU can be identified as G × X U as a G -space.Proof. This is essentially contained in Proposition 2.42 of the extended version of [Tu04], but let usgive a proof. First, observe that any w ∈ Z has trivial stabilizer. Indeed, on the one hand it canbe identified with the inverse image of ( w, w ) for the action map φ : G ⋉ Z → Z × X Z , hence is acompact set by the properness of the action. On the other hand, it is a subgroup of the stabilizer of ρ ( w ), which is a torsion-free group, hence it must be trivial.Next, fix an open neighborhood V of z , and put C = ( G ⋉ Z ) r V , where V is identified withan open subset of G ⋉ Z by taking the identity morphisms of v ∈ V . Since Z is locally compactHausdorff, φ is closed (with compact fibers) and in particular φ ( C ) is closed in Z × X Z , and it doesnot contain ( z, z ) by the above observation. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 19 Take an open neighborhood U of z such that U × X U does not meet φ ( C ). Then the restriction ofthe action map to G × X U is a bijection onto GU . Indeed, if ( g, u ) and ( g ′ , u ′ ) had the same imagein GU , we would have ( u, u ′ ) ∈ U × X U ∩ φ ( G ⋉ Z ) ⊂ φ ( V ) , which implies u = u ′ and then g = g ′ .Finally, as G ⋉ Z is an étale groupoid, the action map G × X U → Z is an open map. Then weobtain that the bijective continuous map G × X U → GU is a homeomorphism. (cid:3) For the next proof we use the equivariant E -theory of C ( Y )-algebras [PT00]. That is, the equi-variant E -groups E Y ( A, B ) (denoted by R E( Y ; A, B ) in [PT00]) define a triangulated category withcountable direct sums and a triangulated functor KK Y → E Y compatible with countable direct sums. Lemma 3.5.
Let Y be a second countable locally compact space, and ( V k ) ∞ k =0 be a countable andlocally finite open covering of Y . If A is a KK Y -nuclear C ( Y ) -algebra, and if N is a C ( Y ) -algebrasuch that N V k is KK V k -equivalent to for all k , then we have KK Y ( A, N ) = 0 .Proof.
By assumption on A , we have KK Y ( A, N ) ≃ E Y ( A, N ) [PT00, Theorem 4.7]. In order toshow the latter group vanishes, it is enough to show E Y ( N, N ) = 0.Put N k = N V ∪···∪ V k . We first claim that E Y ( N k , N ) = 0 for all k . By induction, it is enough toprove this for k = 1. We have an extension of C ( Y )-algebras0 → N → N → N V ∪ V r V → . By assumption N is contractible in KK Y (hence in E Y ). We also have the contractibility of N V ∪ V r V , as it is a quotient of the contractible object N V . Now, the functor B E Y ( B, N )satisfies excision [PT00, Theorem 4.17], which gives an exact sequence of the form0 = E Y ( N V ∪ V r V , N ) → E Y ( N , N ) → E Y ( N , N ) = 0 , and we obtain E Y ( N , N ) = 0.The inclusion maps make ( N k ) ∞ k =0 an inductive system, and N is its inductive limit as a C ( Y )-algebra. This inductive system is admissible in the sense of [MN06, Section 2.4] (this condition isautomatic for inductive systems in E Y , but this example is already admissible in KK Y ). In particular,there is an exact triangle of the formΣ N → M k N k → M k N k → N. Since we already have E Y ( L k N k , N ) ≃ Q k E Y ( N k , N ) = 0, we obtain E Y ( N, N ) = 0. (cid:3)
Lemma 3.6.
Let
X, Y be locally compact spaces, and f : Y → X be a continuous map. Suppose A is a KK X -nuclear C ( X ) -algebra, and B is a C ( Y ) -nuclear C ( Y ) -algebra. Then A ⊗ C ( X ) B is KK Y -nuclear as a C ( Y ) -algebra.Proof. Let ( E , F ) be a C ( X )-equivariant Kasparov cycle from A to A representing id A and suchthat the left action A → M ( K ( E )) is strictly C ( X )-nuclear. Similarly, take an analogous one( E ′ , F ′ ) for B . Then their “cup product” ( E , F ) ⊗ C ( X ) ( E ′ , F ′ ) [Kas88, Proposition 2.21] representsid A ⊗ C X ) B .We claim that this cup product has the underlying Hilbert bimodule E ⊗ C ( X ) E ′ ≃ ( E ⊗ E ′ ) ∆( X ) = ( E ⊗ E ′ ) / ( E ⊗ E ′ ) C (∆( X ) ∁ ) . By definition, it has the underlying bimodule
E ⊗ A ( A ⊗ C ( X ) B ) ⊗ A ⊗ C X ) B ( E ′ ⊗ B ( A ⊗ C ( X ) B )) . By the assumption on B , we have the identification A ⊗ C ( X ) B ≃ ( A ⊗ B ) ∆( X ) , see Remark 1.6.From this we obtain isomorphisms like E ⊗ A ( A ⊗ C ( X ) B ) ≃ ( E ⊗ B ) ∆( X ) , and consequently, theabove bimodule is isomorphic to ( E ⊗ E ′ ) ∆( X ) .Thus, it is enough to show that the left action map A ⊗ C ( X ) B → L (( E ⊗ E ′ ) ∆( X ) ) is strictly C ( Y )-nuclear. Let T : A → L ( E ) be a completely positive C ( X )-linear map factoring through M m ( C ( X )) (approximating the left action of A on E ), and T ′ : B → L ( E ) be a similar one factoringthrough M n ( C ( Y )). Then T ⊗ T ′ induces a completely positive map ( A ⊗ B ) ∆( X ) → L (( E ⊗ E ′ ) ∆( X ) ) OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 20 factoring through M m ( C ( X )) ⊗ C ( X ) M n ( C ( Y )) ≃ M mn ( C ( Y )). This construction is compatiblewith approximation in the pointwise convergence for the strict topology of adjointable morphisms. (cid:3) Remark . If A is moreover C ( X )-nuclear, then A ⊗ C ( X ) B is C ( Y )-nuclear with fibers A f ( y ) ⊗ B y . Proof of Proposition 3.3.
Let A be a KK X -nuclear G -algebra. By Theorem 1.22, there is a paracom-pact proper G -space Z and a G ⋉ Z -C ∗ -algebra P such that A is KK G -equivalent to A ⊗ C ( X ) P . ByLemma 3.6, A ⊗ C ( X ) P is KK Z -nuclear. Thus, we may assume that A is a KK Z -nuclear G ⋉ Z -C ∗ -algebra.Let U ⊂ Z be an open set satisfying the conditions of Lemma 3.4, and put V = GU . Then the G -algebra A V = C ( V ) A is isomorphic to Ind GX A U . Indeed, the latter is C ( G ) ⊗ C ( X ) A U , and the G -equivariant isomorphism V ≃ GU induces A V ≃ C ( G ) ⊗ C ( X ) A U .Now, take countably many open sets ( U i ) i ∈ I satisfying the conditions of Lemma 3.4, such that thesets V i = GU i cover Z and ( V i /G ) i is a countable and locally finite open cover of Z/G (this is possibleby paracompactness). We want to say that the (unreduced) “Čech complex” of objects A V i ∩···∩ V ik give a resolution of A in KK G ⋉ Z . Then, combined with the “induction functor” KK G ⋉ Z → KK G (which is really given by the restriction of C ( Z )-algebras to C ( X )-algebras), we get that A is indeedin h Ind GX KK X i . Suppose U and U ′ are open sets of Z satisfying the conditions of Lemma 3.4, andput V = GU and V ′ = GU ′ . Then there is an open set W satisfying the conditions of Lemma 3.4with V ∩ V ′ = GW . Indeed, we put W = U ∩ V ′ . This implies that the G -algebras A V i ∩···∩ V ik areall of the form Ind GX B .Now, set ˜ Z = ` i V i , and regard it as a G ⋉ Z -space by the canonical equivariant map ˜ Z → Z .The functors Ind Z ˜ Z : KK G ⋉ ˜ Z → KK G ⋉ Z and Res Z ˜ Z : KK G ⋉ Z → KK G ⋉ ˜ Z make sense. Concretely, if B is a G -equivariant C ( Z )-algebra, we haveRes Z ˜ Z B = M i B V i endowed with an obvious action of G , while for a G -equivariant C ( ˜ Z )-algebra B , we set Ind Z ˜ Z B tobe the same C ∗ -algebra as B regarded as a C ( Z )-algebra. Then we have the standard adjunctionKK G ⋉ Z (Ind Z ˜ Z B, B ′ ) ≃ Y i KK G ⋉ V i ( B V i , B ′ V i ) ≃ KK G ⋉ ˜ Z ( B, Res Z ˜ Z B ′ ) . From this, we see that L = Ind Z ˜ Z Res Z ˜ Z satisfies L k A = M i ,...,i k A V i ∩···∩ V ik . By Proposition 3.1, we obtain an exact triangle P → A → N → Σ P in KK G ⋉ Z , such that P is in the triangulated subcategory generated by objects of the form Ind G ⋉ ZU i B ,and N ∈ ker Res G ⋉ ZG ⋉ ˜ Z . It remains to prove that N = 0 in KK G ⋉ Z . Then it is enough to prove thatthe morphism A → N is zero.Since the action of G on Z is free and proper, there is an equivalence of categories between KK G ⋉ Z and KK Z/G , and similar statements hold for the G -invariant open sets V i . Under this correspondence, A corresponds to a KK Z/G -nuclear algebra. Now, Lemma 3.5 implies that KK G ⋉ Z ( A, N ) = 0. (cid:3)
We are now ready to prove the main result of the section.
Proof of Theorem . Consider the functorsRes GH : KK G → KK H , Ind GH : KK H → KK G as in Section 1.6. By Proposition 3.1, we have an complementary pair ( hP I i , N I ) for I = ker Res GH ,with hP I i being generated by the image of Ind GH as a triangulated subcategory. Moreover, we have anatural isomorphism of functors Ind GX ≃ Ind GH Ind HX . Combined with Proposition 3.3, we obtain that A belongs to hP I i . (cid:3) The following is a direct consequence of Theorem 1.22 and Theorem 3.2.
OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 21 Corollary 3.8.
Let G , H , and A be as in Theorem . Let P H ( A ) ∈ h Ind GH KK H i be the algebraappearing in the exact triangle P H ( A ) → A → N → Σ P H ( A ) that we get by applying Proposition . Then we have P H ( A ) ≃ A ⊗ C ( X ) P ≃ A . Corollary 3.9.
Let G , H , and A be as in Theorem . Then we have a convergent spectral sequence E pq = H p ( K q ( G ⋉ L • +1 A )) ⇒ K p + q ( G ⋉ A ) , (9) where L n A = (Ind GH Res GH ) n ( A ) .Proof. The reduced crossed product functorKK G → KK , A G ⋉ A is exact and compatible with direct sums, whileKK → Ab , B K ( B )is a homological functor. Thus, their composition K ( G ⋉ − ) : KK G → Abis a homological functor, cf. [MN10, Examples 13 and 15]. Now we can apply Theorem 1.19 to get aspectral sequence H p ( K q ( G ⋉ P • )) ⇒ K p + q ( G ⋉ P H ( A )) , where P • is a (ker Res GH )-projective resolution of A . The (ker Res GH )-projective resolution from Propo-sition 3.1 gives the left hand side of (9). Now the claim follows from Corollary 3.8. (cid:3) Remark . It would be an interesting question to cast the above constructions to groupoid equivari-ant E -theory [Pop04], since we mostly use formal properties of KK G . However, since our definitionof the functor Ind GH involves reduced crossed products, there seem to be some details to be checked.(Note that H need not be a proper subgroupoid.)4. Homology and K -theory In this section we relate the construction of the previous section to groupoid homology for amplegroupoids with torsion-free stabilizers. As for the Smale spaces with totally disconnected stable sets,a similar construction will allow us to relate to Putnam’s homology.Suppose G is a second countable locally compact Hausdorff étale groupoid, and H is an opensubgroupoid with the same base space. Let us analyze the chain complex in (8) more concretely. Let s n : G ( n ) → X be the map ( g , . . . , g n ) sg n . Lemma 4.1.
Let A be an H -C ∗ -algebra. The C ( G ( n ) ) -algebra s ∗ n A is endowed with a continuousaction of the groupoid G × G H × G n .Proof. We give a concrete proof for n = 1, as the general case can be done following the same idea.We use ( C ( G ) ⊗ min A ) ∆( X ) as a model of C ( G ) ⊗ C ( X ) A , and analogous models for other relativeC ∗ -algebra tensor products as well. Recall that the arrow set of G × G H can be identified with theset of triples ( g, g , h ) where ( g, g ) ∈ G (2) , h ∈ H , and s ( g ) = s ( h ). Then C ( G × G H ) ⊗ s C ( G ) ( C ( G ) ⊗ C ( X ) A ) = ( C ( G × G H × G ) ⊗ A ) Y , where Y is the space of tuples ( g, g , h, g , x ) with ( g, g , h ) as above and x = s ( g ). On the otherhand, C ( G (2) ) ⊗ s sC ( X ) C ( H ) ⊗ s C ( G ) A ≃ ( C ( G (2) × H ) ⊗ A ) Z , where Z is the space of quadruples ( g, g , h, x ) where the components are related as above. Viathe obvious homeomorphism between Y and Z , we have the identification of these algebras. Thestructure map α : C ( H ) ⊗ s C ( G ) A → C ( H ) ⊗ r C ( G ) A of the H -C ∗ -algebra induces an isomorphismonto C ( G (2) ) ⊗ s sC ( X ) C ( H ) ⊗ r C ( G ) A ≃ ( C ( G (2) × H ) ⊗ A ) Z ′ , where Z ′ is the space of quadruples ( g, g , h, y ) with ( g, g , h ) as above and y = r ( h ). Finally, wehave C ( G × G H ) ⊗ r C ( G ) ( C ( G ) ⊗ C ( X ) A ) = ( C ( G × G H × G ) ⊗ A ) Y ′ , OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 22 where Y ′ is the space of tuples ( g, g , h, g ′ , y ) where ( g, g , h ) is as above, g ′ = gg h − , and y = s ( g ′ ) = r ( h ). Again the obvious bijection between Y ′ and Z ′ induces an isomorphism of the last twoalgebras, and combining everything we obtain an isomorphism C ( G × G H ) ⊗ s C ( G ) s ∗ A → C ( G × G H ) ⊗ r C ( G ) s ∗ A which is the desired structure morphism of G × G H -C ∗ -algebra. (cid:3) Proposition 4.2.
In the setting above, the functor L = Ind GH Res GH : KK G → KK G satisfies G ⋉ L n A ≃ ( G × G H × G n ) ⋉ s ∗ n A. Proof.
We have L n A = H × G n ⋉ s ∗ n A by expanding the definitions. (cid:3) Using the Morita equivalence between G × G H × G n and H × G n , we can replace the formula abovewith H × G n ⋉ s ∗ n − A . This enables us to transport the simplicial structure on ( G ⋉ L n +1 A ) ∞ n =0 to( H × G ( n +1) ⋉ s ∗ n A ) ∞ n =0 . The proof is again straightforward from definitions. Proposition 4.3.
The induced simplicial structure on ( H × G ( n +1) ⋉ s ∗ n A ) ∞ n =0 has face maps d ni rep-resented by the composition of KK -morphisms C ∗ r ( H × G ( n +1) ) → C ∗ r ( H × G i × G G × G H × G ( n − i ) ) → C ∗ r ( H × G n ) , where the first morphism is induced by the inclusion H × G ( n +1) → H × G i × G G × G H × G ( n − i ) as anopen subgroupoid, and the second morphism is given by the Morita equivalence of Proposition . Induction from unit space and groupoid homology.
Let us consider the case H = H (0) = X = G (0) . Proposition 2.4 says that we can replace G × G H × G ( n +1) by the Morita equivalent groupoid H × G ( n +1) . Now, this is just G ( n ) as a locally compact space with trivial groupoid structure. Herewe obtain the complex of groupoid homology in Section 1.3. Proposition 4.4.
There is an isomorphism of chain complexes ( K ( G ⋉ L • +1 C ( X )) , δ • ) ≃ ( C c ( G ( • ) , Z ) , ∂ • ) , ( K ( G ⋉ L • +1 C ( X )) , δ • ) ≃ . Proof.
Since G ( n ) is totally disconnected, we have K ( C ( G ( n ) )) ≃ C c ( G ( n ) , Z ) , K ( C ( G ( n ) )) = 0 . We have a (semi-)simplicial structure on K ( C ( G ( n ) )) from Proposition 4.3. It is a routine calculationto compare this with the one above from the nerve structure. (cid:3) Thus, we obtain an isomorphism of homology groups H p ( K q ( G ⋉ L • +1 C ( X )) , δ • ) ≃ H p ( G, K q ( C )) . Theorem 4.5.
Let G be an ample groupoid with torsion-free stabilizers satisfying the strong Baum–Connes conjecture. Then there is a convergent spectral sequence E rpq ⇒ K p + q ( C ∗ r ( G )) , with E pq = E pq = H p ( G, K q ( C )) .Proof. We obtain the spectral sequence by Corollary 3.9, and Proposition 4.4 gives the descriptionof E -sheet. By degree reasons the E -differential is trivial, so we have E pq = E pq . (cid:3) Remark . More generally, if A is a G -C ∗ -algebra, K ∗ ( A ) becomes a unitary module over C c ( G, Z )and we obtain a G -sheaf. We then have a spectral sequence with E pq = H p ( G, K q ( A )) that convergesto K p + q ( G ⋉ A ) when A is KK X -nuclear. Remark . Looking at the bidegree of differentials at the E -sheet, we see that the above spectralsequence collapses at the E -sheet if H k ( G, Z ) vanishes for k ≥
3. If, in addition, H ( G, Z ) istorsion-free, one has K ( C ∗ r G ) ≃ H ( G, Z ) ⊕ H ( G, Z ) , K ( C ∗ r G ) ≃ H ( G, Z ) . This covers the transformation groupoids of minimal Z -actions on the Cantor space considered in[Mat12] and the Deaconu–Renault groupoids of rank 1 and 2 (in particular k -graph groupoids for k = 1 ,
2) in [FKPS19], and groupoids of 1-dimensional generalized solenoids [Yi20]. The Exel–Pardo
OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 23 groupoid model [EP17] for Katsura’s realization [Kat08] of Kirchberg algebras also belong to thisclass [Ort18]. For the groupoid of tiling spaces (see Section 5.2) one can do slightly better; if G is agroupoid associated with some tiling in R d , one has the vanishing of H k ( G, Z ) for k > d and H d ( G, Z )is torsion-free. Comparing the rank of H ∗ ( G, Z ) and K ∗ ( C ∗ G ), we see that the higher differentialsare always zero on H d ( G, Z ), and the spectral sequence collapses if d ≤ Putnam’s homology for Smale spaces.
Let (
Y, ψ ) be an irreducible Smale space with totallydisconnected stable sets. Then there is an irreducible shift of finite type (Σ , σ ) and an s -bijectivefactor map f : (Σ , σ ) → ( Y, ψ ). Recall that Σ n stands for the fiber product of ( n + 1)-copies of Σ over Y , with Σ = Σ.For any shift of finite type (Σ , σ ) the group K ( C ∗ ( R u (Σ , σ ))) can be described as Krieger’s dimension group D s (Σ) [Kri80]. This group is generated by the elements [ E ] for compact open sets E in the stable orbits in Σ. We can restrict to a collection of stable orbits which form a generalizedtransversal, and also assume that E is contained in a local stable orbit as well [Pro18b, Lemma 1.3].The simplicial structure on the groups D s (Σ • ) is induced by natural face maps δ sk : Σ n → Σ n − ,which delete the k -th entry of a point in Σ n . This yields a well defined map between the correspondingdimension groups, via the assignment [ E ] [ δ sk ( E )]. This way the groups D s (Σ • ) form a simplicialobject, and the associated homology H s ∗ ( Y, ψ ), called stable homology of (
Y, ψ ), does not depend on f [Put14, Section 5.5; Pro18b].For a suitable choice of P ⊆ Σ, we have an open inclusion of étale groupoid f × f : R u (Σ , P ) → R u ( Y, f ( P )). We set G = R u ( Y, f ( P )) and H = ( f × f )( R u (Σ , P )). Notice that G is ample and H is AF [Put15, Tho10a]. Proposition 4.8.
There is an isomorphism of chain complexes ( K ( G ⋉ L • +1 C ( X )) , δ • ) ≃ ( D s (Σ • ) , d s ( f ) • ) , ( K ( G ⋉ L • +1 C ( X )) , δ • ) ≃ correspondences between groupoids. Ingeneral, if G and H are topological groupoids, a correspondence from G to H is a topological space Z together with commuting proper actions G y Z x H , such that the anchor map Z → H (0) is open(surjective) and induces a homeomorphism G \ Z ≃ H (0) . Of course, one source of such correspondenceis Morita equivalence. Another example is provided is continuous homomorphisms f : G → H , whereone puts Z = { [ g, h ] | f ( sg ) = rh } with the relation [ g g , h ] = [ g , f ( g ) h ].If G and H are (second countable) locally compact Hausdorff groupoids with Haar systems, acorrespondence Z induces a right Hilbert C ∗ r ( H )-module C ∗ r ( Z ) C ∗ r ( H ) with a left action of C ∗ r ( G )[MSO99]. If the action of C ∗ r ( G ) is in K ( C ∗ r ( Z ) C ∗ r ( H ) ), we obtain a map K ∗ ( C ∗ r ( G )) → K ∗ ( C ∗ r ( H )).While finding a good characterization of this condition in terms of Z seems to be somewhat tricky,in concrete examples as below it is not too difficult.On the other hand, composition of such Hilbert modules are easy to describe. If H ′ is anothertopological groupoid with Haar system, and Z ′ is a correspondence from H to H ′ , we have theidentification C ∗ r ( Z ) C ∗ r ( H ) ⊗ C ∗ r ( H ) C ∗ r ( Z ′ ) C ∗ r ( H ′ ) ≃ C ∗ r ( Z × H Z ′ ) C ∗ r ( H ′ ) . Proof of Proposition 4.8.
By Proposition 4.2 and Theorem 2.12, the C ∗ -algebra G ⋉ L n +1 C ( X ) isstrongly Morita equivalent to C ∗ ( R u (Σ n , σ n )). From this we have the identification of the underlyingmodules, and it remains to compare the corresponding simplicial structures. Let us give a concretecomparison of the maps K ( C ∗ r ( H × G n +1 )) → D s (Σ n − ) corresponding to the 0-th face maps, as thegeneral case is completely parallel.Let us put ˜ G = R u ( Y, ψ ), ˜ H = R u (Σ , σ ), and take a (generalized) transversal T ′ for R u (Σ n , σ n ),and put K = R u (Σ n , σ n ) | T ′ , K ′ = R u (Σ n − , σ n − ) | δ ( T ′ ) so that we have D s (Σ n , σ n ) ≃ K ( C ∗ r ( K )) , D s (Σ n − , σ n − ) ≃ K ( C ∗ r ( K ′ )) . We denote the generalized transversal of ˜ H × ˜ G n induced by P , as in Proposition 2.6, by ˜ T n .The map δ induces a groupoid homomorphism K → K ′ , and hence a correspondence Z δ from K to K ′ . Composing this with the Morita equivalence bibundle ˜ T n +1 ( ˜ H × ˜ G n +1 ) T ′ , we obtain a corre-spondence ˜ T n +1 ( ˜ H × ˜ G n +1 ) T ′ × K Z δ (10)from H × G n +1 to K ′ representing the effect of δ on the K -groups. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 24 As for the 0-th face map d of H × G • +1 , let Z be the Morita equivalence bibundle between G × G H × G n and H × G n from Proposition 2.4. Since H × G n +1 is an open subgroupoid of G × G H × G n , Z becomes a correspondence from H × G n +1 to H × G n . Composing this with the Morita equivalence ˜ T n ( ˜ H × ˜ G n ) δ ( T ′ ) between H × G n and K ′ , we obtain the correspondence Z × H × Gn ˜ T n ( ˜ H × ˜ G n ) δ ( T ′ ) (11)from H × G n +1 to K ′ representing the effect of d .It remains to check that the above correspondences are isomorphic, hence giving isomorphic Hilbertmodules. Expanding the ingredients of (11), we obtain the space W = { ( g , h , g , h , . . . , g n − , h n ) | ( g , . . . , g n − ) ∈ G ( n ) , h k ∈ ˜ H sg k − , ( sh , . . . , sh n ) ∈ δ ( T ′ ) } . On the other hand, (10) gives W × K K ′ with W ′ = { ( h , g , h , . . . , g n , h n +1 ) | ( g , . . . , g n ) ∈ G ( n ) ,h k ∈ ˜ H rg k , h n +1 ∈ ˜ H sg n , ( sh , . . . , sh n +1 ) ∈ T ′ } . The operation − × K K ′ “kills” the component h , and we obtain the identification with W . (cid:3) Thus, we obtain isomorphisms of homology groups H p ( K q ( G ⋉ L • +1 C ( X )) , δ • ) ≃ H sp ( Y, ψ ) ⊗ K q ( C ) . Theorem 4.9.
Let ( Y, ψ ) be an irreducible Smale space with totally disconnected stable sets. Thenthere is a convergent spectral sequence E rpq ⇒ K p + q ( C ∗ ( R u ( Y, ψ ))) , with E pq = E pq = H sp ( Y, ψ ) ⊗ K q ( C ) .Proof. The proof is parallel to that of Theorem 4.5, but this time we use Corollary 3.9 and Proposi-tion 4.8. (cid:3)
Corollary 4.10.
The K -groups K i ( C ∗ ( R u ( Y, ψ ))) have finite rank.Proof.
By the above theorem, for i = 0 ,
1, the rank of K i ( C ∗ ( R u ( Y, ψ ))) is bounded by that of L k H si +2 k ( Y, ψ ). The latter is of finite rank by [Put14, Theorem 5.1.12]. (cid:3)
Remark . By the Pimsner–Voiculescu exact sequence, the same can be said for the unstableRuelle algebra
Z ⋉ ψ C ∗ ( R u ( Y, ψ )). If the stable relation R s ( Y, ψ ) also has finite rank K -groups, theRuelle algebras will have finitely generated K -groups by [KPW17].In fact, Putnam’s homology is isomorphic to groupoid homology in the above setting, which givesan alternative proof of the previous result. Theorem 4.12.
We have H s ∗ ( Y, ψ ) ≃ H ∗ ( G, Z ) .Proof. Let us consider G ( n +1) as an H × G ( n +1) -space by the anchor map( g , . . . , g n ) ( g , . . . , g n ) ∈ G ( n ) = ( H × G ( n +1) ) (0) and the action map ( h , g , h , . . . , h n )( g , . . . , g n ) = ( g h − , g ′ , . . . , g ′ n )in the notation of Definition 2.1. Then H ( H × G ( n +1) , C c ( G ( n +1) , Z )) is a unitary C c ( X, Z )-moduleby the action from the left, and the associated sheaf F n on X is a G -sheaf by the left translationaction of G . At x ∈ X , the stalk can be presented as( F n ) x = H ( H × G ( n +1) , C c (( G ( n +1) ) x , Z )) = C c (( G ( n +1) ) x , Z ) H × G ( n +1) . (12)Indeed, the sheaf corresponding to the C c ( X, Z )-module C c ( G ( n +1) , Z ) has the stalk C c (( G ( n +1) ) x , Z )at x , and taking coinvariants by H × G ( n +1) commutes with taking stalks.We then have H ( G, F n ) ≃ H ( G × G H × G ( n +1) , Z ) ≃ H ( H × G ( n +1) , Z ) . The simplicial structure on ( G × G H × G ( n +1) ) n leads to the complex of G -sheaves · · · → F → F → F , (13) OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 25 and H s ∗ ( Y, ψ ) is the homology of the complex obtained by applying the functor H ( G, − ) to (13).We first claim that the augmented complex · · · → F → F → F → Z (14)is exact. It is enough to check the exactness at the level of stalks. In terms of the presentation (12),we have the chain complex · · · → C c (( G (2) ) x , Z ) H × G → C c ( G x , Z ) H → Z with differential d (( g , . . . , g n +1 )) = ( g g , g , . . . , g n +1 ) − ( g , g g , . . . , g n +1 ) + · · · + ( − n − ( g , . . . , g n g n +1 ) + ( g , . . . , g n ) , with ( g , . . . , g n ) ∈ ( G ( n ) ) x denoting the image of δ ( g ,...,g n ) ∈ C c (( G ( n ) ) x , Z ) in the coinvariant space,and the augmentation d ( g ) = 1 at n = 0. This has a contracting homotopy given by Z → C c ( G x , Z ) H , a → a (id x ) and C c (( G ( n ) ) x , Z ) H × Gn → C c (( G ( n +1) ) x , Z ) H × Gn +1 , ( g , . . . , g n ) → (id x , g , . . . , g n ) , hence (14) is indeed exact.We next claim that H k ( G, F n ) = 0 for k >
0. Let H n +1 be a subgroupoid of G which is Moritaequivalent to H × G ( n +1) (this exists by choosing a good transversal for (Σ n , σ n )). Then the mod-ule H ( H × G ( n +1) , C c ( G ( n +1) , Z )) representing F n is isomorphic to H ( H n +1 , C c ( G, Z )). Thus, it isenough to check the claim when n = 0.Let us write M = H ( H, C c ( G, Z )), and consider the double complex of modules C c ( G ( p +1) × X H ( q ) , Z ) for p, q ≥
0, with differentials coming both from the simplicial structures on ( G ( p ) ) ∞ p =0 and( H ( q ) ) ∞ q =0 , cf. [CM00, Theorem 4.4]. For fixed p , this is a resolution of C c ( G ( p ) , Z ) ⊗ C c ( X, Z ) M , hencethe double complex computes H ∗ ( G, F ). For fixed q , this is a resolution of H ( G, C c ( G × X H ( q ) , Z )) ≃ C c ( H ( q ) , Z ), and this double complex also computes H ∗ ( H, Z ). Since H is Morita equivalent to anAF groupoid, H k ( H, Z ) = 0 by [Mat12, Theorem 4.11]. We thus obtain H k ( G, F n ) = 0.Finally, consider the hyperhomology H ∗ ( G, F ∗ ). On the one hand, by the above vanishing of H k ( G, F n ), this is isomorphic to the homology of the complex ( H ( G, F n )) n , i.e., H s ∗ ( Y, ψ ). On theother hand, since ( F n ) n is quasi-isomorphic to Z concentrated in degree 0, we also have H ∗ ( G, F ∗ ) ≃ H ∗ ( G, Z ). (cid:3) We then have the following Künneth formula from the corresponding result for groupoid homology[Mat16, Theorem 2.4].
Corollary 4.13.
Let ( Y , ψ ) and ( Y , ψ ) be Smale spaces with totally disconnected stable sets. Thenwe have a split exact sequence → M a + b = k H sa ( Y , ψ ) ⊗ H sb ( Y , ψ ) → H sk ( Y × Y , ψ × ψ ) → M a + b = k − Tor( H sa ( Y , ψ ) , H sb ( Y , ψ )) → . Remark . As usual, the splitting is not canonical. This generalizes [DKW16, Theorem 6.5], inwhich one of the factors is assumed to be a shift of finite type. Indeed, if ( Y , ψ ) is a shift of finitetype, the first direct sum reduces to D s ( Y , ψ ) ⊗ H sk ( Y , ψ ), while the second direct sum of torsiongroups vanishes as the dimension group D s ( Y , ψ ), being torsion-free, is flat.5. Examples
Solenoid.
One class of motivating example is that of one-dimensional solenoids [vD30, Wil67].Let us first explain the easiest example, the m ∞ -solenoid. Consider the space Y = { ( z , z , . . . ) | z k ∈ S , z k = z mk +1 } , which is the projective limit of S S S · · · z m ← [ z z m ← [ z z m ← [ z . (15) OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 26 A compatible metric is given by d (( z k ) k , ( z ′ k ) k ) = X k m − k d ( z k , z ′ k ) , where d is any metric on S compatible with its topology; for example, one may take the arc-lengthmetric d ( e is , e it ) = | s − t | when | s − t | ≤ π .There is a natural “shift” self-homeomorphism φ : Y → Y, ( z , z , . . . ) ( z m , z m = z , z m = z , . . . ) , with inverse given by φ − (( z , z , . . . )) = ( z , z , . . . ). Then ( Y, φ ) is a Smale space.Denote by π the canonical projection Y → S on the first factor. As each step of (15) is an m -to-1map, π − ( z ) can be identified with the Cantor set Σ = Q ∞ n =1 { , , . . . , m − } for any z ∈ S . Thisallows us to write local stable and unstable sets around z = ( z k ) k , as Y s ( z, ǫ ) = π − ( z ) ∼ = Σ , Y u ( z, ǫ ) = { ( e itm − k z k ) ∞ k =0 | | t | < δ ǫ } (16)for small enough ǫ >
0, with δ ǫ > ǫ . Note that π defines a fiber bundle with fiber Σ,and Y u ( z, ǫ ) × Σ → Y corresponding to the bracket map gives local trivializations.Now, the groupoid R u ( Y, φ ) is the transformation groupoid
R ⋉ α Y for the flow α t ( z , z , . . . ) = ( e it z , e itm − z , . . . , e itm − k z k , . . . ) ( t ∈ R ) . Restricted to the transversal π − (1), we obtain the “odometer” transformation groupoid Z ⋉ β Σ,where Σ is identified with lim ←− k Z m k , and the generator 1 ∈ Z acts by the +1 map on Z m k .There is a well-known factor map from the two-sided full shift on m letters onto ( Y, φ ). Namely,writing Σ ′ = { , , . . . , m − } Z = { ( a n ) ∞ n = −∞ | ≤ a n < m } , we have a continuous map f : Σ ′ → Y by f (( a n ) n ) = ( z k ) ∞ k =0 , z k = exp πi ∞ X j =0 m − j − a − k + j . Then we have f σ = φf for σ : Σ ′ → Σ ′ defined by σ (( a n ) n ) = ( a n − ) n .This allows us to compute all relevant invariants separately. As for the K -groups, by Connes’sThom isomorphism, K ( C ∗ R u ( Y, φ )) ≃ K ( Y ) ≃ Z (cid:20) m (cid:21) , K ( C ∗ R u ( Y, φ )) ≃ K ( Y ) ≃ Z . As for groupoid homology, we have H ∗ ( Z ⋉ β Σ , Z ) ≃ H ∗ ( Z , C (Σ , Z ))where right hand side is the groupoid homology of Z with coefficient C (Σ , Z ) endowed with the Z -module structure induced by β . This leads to H ( Z ⋉ β Σ , Z ) ≃ C (Σ , Z ) β ≃ Z (cid:20) m (cid:21) , H ( Z ⋉ β Σ , Z ) ≃ C (Σ , Z ) β ≃ Z , with coinvariants and invariants of β , while H n ( Z ⋉ β Σ , Z ) = 0 for n >
1. The computation for H s ∗ ( Y, φ ) will be more involved, but one finds [Put14, Section 7.3] that H s ( Y, φ ) ≃ D s (Σ ′ , σ ) ≃ Z (cid:20) m (cid:21) , H s ( Y, φ ) ≃ Z , and H sn ( Y, φ ) = 0 for n >
1. Thus the spectral sequences of Theorems 4.5 and 4.9 collapse at the E -sheet, and there is no extension problem. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 27 Substitution tiling.
We follow the convention of [KP00], and consider substitution tilings offinite local complexity. Thus, we are given a set P of prototiles in R d and a substitution rule ω for P .The associated hull Ω admits a self-homeomorphism induced by ω , again denoted by ω , giving aSmale space (Ω , ω ). Under reasonable assumptions on ω , the translation action τ of R d on Ω is freeand minimal. Then, analogously to the case of solenoids, the groupoid of the unstable equivalencerelation is the transformation groupoid R d ⋉ τ Ω. Moreover, by [SW03], there is a transversal X ⊂ Ωthat is homeomorphic to a Cantor set, such that ( R d ⋉ τ Ω) | X is the transformation groupoid Z d ⋉ α X for some action α : Z d y X , see also [KP03, Section 5].Let us quickly explain how a spectral sequence of more classical nature arises in this setting. ByConnes’s Thom isomorphism, the right hand side is K n ( C ∗ R u (Ω , ω )) ≃ K n + d (Ω) . Now, Ω can be identified with a projective limit of a self-map of branched d -dimensional manifold ob-tained by gluing (collared) prototiles [AP98]. This leads to the Atiyah–Hirzebruch spectral sequence E p,q = ˇ H p (Ω , K q ( C )) ⇒ K p + q (Ω) , (17)that is, E p,q is the p -th Čech cohomology of Ω with constant sheaf Z when q is even, and E p,q = 0otherwise (for dimension reasons we also have E p,q = 0 if p > d ). Since Ω is a compact Hausdorffspace, this is also equal to the sheaf cohomology as derived functor. Since the action τ is free and R d is contractible, Ω is a model of the classifying space BG and the universal principal bundle EG for the groupoid G = R u (Ω , ω ) = R d ⋉ τ Ω (up to nonequivariant homotopy). In particular, we caninterpret the sheaf cohomology on Ω as groupoid cohomology of G , see [Moe98, Tu06].Let us relate our construction to this. Using the transversal X , we have H ∗ ( G | X , Z ) ≃ H ∗ ( Z d , C ( X, Z )) , H ∗ ( G | X , Z ) ≃ H ∗ ( Z d , C ( X, Z )) , where we consider C ( X, Z ) as a module over Z d by the action induced by α . Moreover we have H k ( Z d , M ) ≃ H d − k ( Z d , M ) for any Z d -module M , see for example [Bro94, Section VIII.10]. Thisshows that H k ( G, Z ) ≃ H k ( G | X , Z ) ≃ H d − k ( G | X , Z ) ≃ H d − k ( G, Z )for the étale groupoid G | X , and the spectral sequence of Theorem 4.5 is comparable to (17).Let us also remark that these observations imply H sk (Ω , ω ) ≃ ˇ H d − k (Ω) , giving a positive answer to [Put14, Question 8.3.2] in the case of tiling spaces. Remark . A spectral sequence of the form (17) is given in [SB09], as an analogue of the Serrespectral sequence for the Anderson–Putnam fibration structure Ω → Γ k over the k -collared prototilespace. Our construction should be rather regarded as a Serre spectral sequence for the fibrationΩ → ( S ) d from [SW03], and it would be an interesting question to compare these.5.3. Semidirect product by torsion-free groups.
Let Γ be a torsion-free group satisfying thestrong Baum–Connes conjecture. Let G be an ample groupoid with torsion-free stabilizers satisfyingthe strong Baum–Connes conjecture, and suppose that Γ acts on G . Proposition 5.2.
The groupoid Γ ⋉ G satisfies the assumption of Theorem .Proof. Let us first check that the stabilizers are torsion-free, or equivalently, that there are no elementsof finite order in the stabilizers. Suppose that ( γ, g ) ∈ Γ ⋉ G is in the stabilizer of sg , if γ = e then itis of infinite order by assumption on Γ, and if γ = e then g is in the stabilizer of sg , which is againof infinite order by assumption on G .Next, let us check the assumption on proper actions. Take a proper G -algebra P G equivalent to C ( X ) in KK G , and a proper Γ-algebra P Γ equivalent to C in KK Γ . Then the Γ ⋉ G -algebra P Γ ⊗ P G is equivalent to C ( X ) in KK Γ ⋉ G . (cid:3) Let (
Y, ψ ) be a Smale space with totally disconnected stable sets. Then the groupoid Z⋉ ψ R u ( Y, ψ )behind the unstable Ruelle algebra R u fits into the above setting. Indeed, as a generalized transversalof R u ( Y, ψ ) take X = Y s ( P ) for some set P of periodic points of ψ . Then X is stable under ψ , and Z ⋉ ψ ( R u ( Y, ψ ) | X ) is Morita equivalent to Z ⋉ ψ R u ( Y, ψ ). OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 28 Remark . Recall that Y is of the form lim ←− Y for a constant projective system of some compactmetric space Y and a suitable self-map g : Y → Y [Wie14], analogous to the standard presentationof the m ∞ -solenoid above. Suppose further that g is open and the groupoid C ∗ -algebra of stablerelation R s ( Y, ψ ) has finite rank K -groups. Then, combining [KPW17] and [DGMW18], we see that K ∗ ( R u ) fits in an exact sequence K ∗ +1 ( C ( Y )) K ∗ +1 ( C ( Y )) K ∗ ( R u ) K ∗ ( C ( Y )) K ∗ ( C ( Y )) , − [ E g ] 1 − [ E g ] where E g is the C ( Y )-bimodule associated with g . It would be an interesting problem to comparethe two ways to compute K ∗ ( R u ).The setting of this section can also be applied to the study of Deaconu–Renault groupoids (see[FKPS19, Section 6] for proofs). Let X be a locally compact Hausdorff space, and σ be an action of N k on X by surjective local homeomorphisms. The associated Deaconu–Renault groupoid G = G ( X, σ )is defined by G = { ( x, p − q, y ) ∈ X × Z k × X : σ p ( x ) = σ q ( y ) } . There is a natural cocycle c : G ( X, σ ) → Z k given by c ( x, n, y ) := n , and the resulting skew-product groupoid G × c Z k is free and AF. By considering the automorphisms α p : (( x, m, y ) , n ) (( x, m, y ) , n + p ), we obtain a semidirect product groupoid ˜ G = G × c Z k ⋊ α Z k , which is homologicallysimilar to G . Moreover, H ∗ ( G ) is the group homology of Z k with coefficients in H ( G × c Z k ). Onthe K -theory side, Takai–Takesaki duality implies that C ∗ G is stably isomorphic to C ∗ ˜ G . Hence, forthe purpose of comparing homology and K -theory, we can use ˜ G in place of G .5.4. A non-example.
Scarparo has found a counterexample to the HK conjecture [Sca19]. In hisexample G is the transformation groupoid of an action α of the infinite dihedral group Γ = Z ⋉ Z onthe Cantor set X . Thus, it is amenable and in particular satisfies the strong Baum–Connes conjecture.However, α is not free, and the simplicial approximation P ( C ( X )) arising from restriction to the unitspace is indeed not KK G -equivalent to C ( X ). Let us explain the ingredients in more detail.Let ( n i ) ∞ i =0 be a strictly increasing sequence of integers such that, for i ≥ n i +1 /n i ∈ N for all i .We take the model X = lim ←− Z n i . Then Z acts by the odometer action, i.e., 1 ∈ Z acts by the +1 mapon each factor Z n i . There is a consistent action of Z , where the nontrivial element g = [1] ∈ Z actsby multiplication by −
1, giving rise to an action α of Γ on X . Note that α is topologically free butnot free, nor does it have torsion-free stabilizers.Put G = Γ ⋉ α X , and M = (cid:26) mn i : m ∈ Z , i ≥ (cid:27) . The C ∗ -algebra C ∗ G = Γ ⋉ α C ( X ) is an AF algebra, with K ( C ∗ G ) ≃ ( M ⊕ Z if n i +1 /n i is even for infinitely many iM ⊕ Z otherwise , see [BEK93]. On the other hand, the groupoid homology is H ( G, Z ) ≃ M,H k ( G, Z ) ≃ ,H k − ( G, Z ) ≃ ( Z if n i +1 /n i is even for infinitely many i Z otherwisefor k >
1, see [Sca19]. This shows that groupoid homology cannot form a spectral sequence convergingto K ∗ ( C ∗ G ), much less being isomorphic to it.Fortunately, there is a somewhat concrete description of P ( C ( X )) in this case. Consider theantipodal action of Z on S n , that is, g acts by the restriction of the multiplication by − R n +1 .Then the contractible space S ∞ = lim −→ S n is a model of the universal bundle E Z . We want to makesense of an analogue of Poincaré dual for this.Let Y n = C ( T ∗ S n ) denote the function algebra of the total space of the cotangent bundle of S n ,and Y ′ n denote the Z -graded C ∗ -algebra of continuous sections of the C ∗ -algebra bundle Cl C ( T ∗ S n ) OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 29 over S n with complex Clifford algebras Cl C ( T ∗ x S n ) as fibers. These admit naturally induced actionsof Z , and Y n is KK Z -equivalent to Y ′ n [Kas16, Theorem 2.7].Let us recall the (equivariant) Poincaré duality between C ( S n ) and Y ′ n [Kas88, Section 4]. Thenatural Clifford module structure on the differential forms of S n , together with D ′ n = d + d ∗ , givean unbounded model of a K -homology class [ D ′ n ] ∈ K Z ( Y ′ n ). Composed with the product map m : Y ′ n ⊗ C ( S n ) → Y ′ n , we obtain the class [ D n ] = m ⊗ Y ′ n [ D ′ n ] ∈ K Z ( Y ′ n ⊗ C ( S n )). The dual class[Θ n ] ∈ K Z ( C ( S n ) ⊗ Y ′ n ) is defined as a certain class localized around the diagonal.Let j : S n → S n +1 be the embedding at the equator (which is a Z -equivariant continuous map),and let j ′ : Y ′ n → Y ′ n +1 be the KK Z -morphism dual to the restriction map j ∗ : C ( S n +1 ) → C ( S n ).Thus, we have j ′ = [Θ n +1 ] ⊗ C ( S n +1 ) ⊗ Y n +1 (id Y ′ n ⊗ j ∗ ⊗ id Y ′ n ) ⊗ Y n ⊗ C ( S n ) [ D n ] , see [Kas88, Theorem 4.10]. Lemma 5.4.
We have j ′ ⊗ Y ′ n +1 [ D ′ n +1 ] = [ D ′ n ] in K Z ( Y ′ n ) .Proof. As a KK Z -morphism, [ D ′ n ] is the dual of the embedding η n : C → C ( S n ), hence the claimreduces to η n +1 = jη n . (cid:3) Take the homotopy colimit Y ′∞ = lim −→ Y ′ n in KK Z (to be precise, we are working in the enlargedcategory of Z -graded C ∗ -algebras). By the above lemma, the morphisms [ D ′ n ] induce a morphism[ D ′∞ ] ∈ KK Z ( Y ′∞ , C ). Transporting this by the KK Z -equivalence, we obtain Y ∞ = lim −→ Y n and[ D ∞ ] ∈ KK Z ( Y ∞ , C ). Lemma 5.5.
The image of [ D ∞ ] in KK( Y ∞ , C ) is a KK -equivalence.Proof. In the nonequivariant KK -category, Y n is equivalent to C or C ⊕ Σ C depending on the parityof n , and there is a distinguished summand which is equivalent to C (at the even degree) spannedby the K -theoretic fundamental class of T ∗ S n . Moreover, the morphism corresponding to [ D ′ n ] is aprojection onto this summand.The KK-morphisms corresponding to j ′ preserve the fundamental class while killing the otherdirect summand. Thus, the limit is equivalent to C , spanned by the image of the fundamental classes,and [ D ∞ ] gives the equivalence. (cid:3) Since Z acts freely on T ∗ S n , each Y n is orthogonal to the kernel of restriction functor KK Z → KK.The discussion so far can be readily adjusted to the groupoid G , as follows. Here, Y n ⊗ C ( X ) is a G -C ∗ -algebra for which Y n only sees the action of Z . Proposition 5.6.
The G -C ∗ -algebra Y n ⊗ C ( X ) belongs to the triangulated subcategory generated bythe image of Ind GX : KK X → KK G .Proof. First, G ⋉ ( T ∗ S n × X ) is a free groupoid. Indeed, it is the transformation groupoid of theaction Γ y T ∗ S n × X , but any element γ ∈ Γ that has a fixed point in X is either conjugate to ( g, g, g is the nontrivial element of Z and we identify Γ with Z × Z as a set.) By thefreeness of Z y T ∗ S n , these elements cannot have fixed points in T ∗ S n × X .We thus obtain that Y n ⊗ C ( X ) belongs to the triangulated subcategory generated by the image ofInd G ⋉ ( T ∗ S n × X ) T ∗ S n × X , see the proof of Proposition 3.3. Using the triangulated functor KK G ⋉ ( T ∗ S n × X ) → KK G given by restricting the scalars of C ( T ∗ S n × X )-algebras to C ( X ), we obtain the assertion. (cid:3) Corollary 5.7.
We have P I C ( X ) ≃ Y ∞ ⊗ C ( X ) for I = ker Res GX , with the corresponding KK G -morphism Y ∞ ⊗ C ( X ) → C ( X ) given by [ D ∞ ] ⊗ id C ( X ) . Consequently, the spectral sequence of groupoid homology converges to the K -groups of G ⋉ ( Y ∞ ⊗ C ( X )). Appendix A. Structure of groupoid equivariant KK -theory Let G be a locally compact Hausdorff groupoid with continuous Haar system, with X = G (0) . Proposition A.1.
With respect to the structure morphism r ∗ : C ( X ) → C b ( G ) = M ( C ( G )) , the C ( X ) -algebra C ( G ) is C ( X ) -nuclear. OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 30 Proof.
The range map G → X is open because there exists a continuous Haar system, see [Ren80,Proposition 2.4]. This implies that C ( G ) is a continuous field of C ∗ -algebras over X . Since the fibersare commutative, we obtain the C ( X )-nuclearity. (cid:3) In particular, for any C ( X )-algebra A , the C ( G )-algebra r ∗ A can be modeled by C ( G ) ⊗ C ( X ) A ≃ ( C ( G ) ⊗ max A ) ∆( X ) ≃ ( C ( G ) ⊗ min A ) ∆( X ) . Of course, the same holds for s ∗ : C ( X ) → C b ( G ). Proposition A.2.
Let f : A → B be an equivariant homomorphism of G -C ∗ -algebras. Then I = ker f is a G -C ∗ -algebra.Proof. Since I is an ideal of A , it inherits a structure of C ( X )-algebra. We need to show that thereis an isomorphism of C ( G )-algebras s ∗ I = C ( G ) ⊗ s C ( X ) I → r ∗ I = C ( G ) ⊗ r C ( X ) I defining a continuous action of G . By the nuclearity of C ( G ) as a C ∗ -algebra,0 → C ( G ) ⊗ I → C ( G ) ⊗ A → C ( G ) ⊗ B → s ∗ I is the kernel of s ∗ A → s ∗ B induced by f . By the C ( X )-nuclearity of C ( G ), we can write s ∗ I = ( C ( G ) ⊗ I ) ∆( X ) , etc. Then we have a commutative diagram0 0 00 I ′ A ′ B ′ C ( G ) ⊗ I C ( G ) ⊗ A C ( G ) ⊗ B s ∗ I s ∗ A s ∗ B
00 0 0with I ′ = C (( G × X ) r ( G × X X ))( C ( G ) ⊗ I ), etc., and we know the exactness of the verticalsequences and top and middle horizontal sequences. Then the bottom sequence is also exact, whichestablishes the claim.Then looking at the action map s ∗ A → r ∗ A, we see that s ∗ I is mapped onto r ∗ I = ker( r ∗ A → r ∗ B ) bijectively. (cid:3) A.1.
Triangulated structure.
Let f : A → B be an equivariant homomorphism of G -C ∗ -algebras.As usual, its mapping cone is given byCon( f ) = { ( a, b ∗ ) ∈ A ⊕ C ((0 , , B ) | f ( a ) = b (1) } , which inherits a structure of G -C ∗ -algebra from A and B .An exact triangle in KK G is a diagram of the form A → B → C → Σ A such that there exists a homomorphism f : A ′ → B ′ of G -C ∗ -algebras and a commutative diagram A B C Σ A Σ B ′ Con( f ) A ′ B ′ , OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 31 in KK G , where vertical arrows are equivalences and the rightmost downward arrow is equal to theleftmost downward arrow up to applying Σ and Bott periodicity Σ B ′ ≃ B ′ in KK G .Thus, we are really defining a triangulated category structure on the opposite category of KK G .Generally the opposite category of a triangulated category is again triangulated with “the same”exact triangles with suspension and desuspension exchanged, but for KK G we have Σ ≃ id and wecan ignore that issue.The crucial step is to check the axiom (TR1), in particular that any KK G -morphism is representedby a G -equivariant ∗ -homomorphism up to KK G -equivalence, due to Oyono-Oyono [Laf07, LemmaA.3.2]. Having established that, the rest is quite standard; one can follow [MN06, Appendix A] tocheck that the triangles of the form Σ B → Con( f ) → A → B satisfy the axioms (TR2), (TR3), and (TR4) for the opposite category of KK G .Finally, suppose that an equivariant ∗ -homomorphism f : A → B is surjective with a C ( X )-linearcompletely positive section B → A . Then the G -C ∗ -algebra I = ker f is isomorphic to Con( f ) inKK G , by the embedding homomorphism I → Con( f ) , a ( a, . It follows that there is an exact triangle of the form
I A B Σ I. f A.2.
Induction functor for subgroupoids.
Suppose that G acts freely and properly the fromright on a second countable, locally compact, Hausdorff space Y . Then the transformation groupoid Y ⋊ G is Morita equivalent to the quotient space Y /G as a groupoid. This induces the strong Moritaequivalence between G ⋉ C ( Y ) ≃ C ∗ ( Y ⋊ G ) and C ( Y /G ). In particular, for the case Y = G andaction given by right translation, we get the isomorphism between G ⋉ C ( G ) and K ( L r ( G )), where L r ( G ) is the right Hilbert C ( X )-module completion of C c ( G ) with C ( X )-module structure from r ∗ : C ( X ) → C b ( G ) and inner product from the Haar system. Proof of Proposition . As in the assertion, let A be a G -C ∗ -algebra. We have two actions of G : on the one hand, it acts on s ∗ A by the combination of right translation on G and the originalaction on A , while on the other hand it acts on r ∗ A by the right translation on G and trivially on A .Then, the structure morphism α : s ∗ A → r ∗ A of the action intertwines these two actions. Morally s ∗ A can be thought of as a space of sections f ( g ) ∈ A sg for g ∈ G , with the action of G given by f g ′ ( g ) = g ′− f ( gg ′ ) for ( g, g ′ ) ∈ G (2) , while r ∗ A as a space of sections f ( g ) ∈ A rg with G acting by f g ′ ( g ) = f ( gg ′ ) for ( g, g ′ ) ∈ G (2) . We have ( αf )( g ) = gf ( g ) for the sections of the first kind, andthese formulas give ( αf g ′ )( g ) = gf ( gg ′ ) = ( αf ) g ′ ( g ).Now, Ind GG Res GG ( A ) is the crossed product of s ∗ A by G , while K ( L r ( G )) ⊗ C ( X ) A is the crossedproduct of r ∗ A by G . Consequently we get an isomorphism between these algebras. The extra actionof G on Ind GG Res GG ( A ) comes from the action of G on s ∗ A given by the combination of the lefttranslation on G and the trivial action on A . Under the above isomorphism, this corresponds to theaction on r ∗ A given by the combination of left translation on G and the original action on A . Thus,it corresponds to the diagonal action of G on K ( L r ( G )) ⊗ C ( X ) A . (cid:3) More generally, the same argument gives an isomorphism φ : Ind GH Res GH A ≃ ( C ( G ) ⋊ H ) ⊗ r C ( X ) A, where G acts diagonally on the algebra on the right.The functor B Ind GH B = ( C ( G ) ⊗ C ( X ) B ) ⋊ H from H -C ∗ -algebras to G -C ∗ -algebras preservessplit extensions by equivariant completely positive maps, and is compatible with homotopy andstabilization (tensor product with K ( ℓ )). While this should be enough to have an extension to afunctor KK H → KK G by the universality properties of these categories, let us give a more concretedescription at the level of Kasparov cycles.Consider an H -equivariant right Hilbert module E over B . By using an approximate unit in B ,we can equip E with a compatible C ( X )-action. We can form the Hilbert module C ( G ) ⊗ E over G ( G ) ⊗ B , and restrict on the diagonal to get s ∗ E = ( C ( G ) ⊗ E ) ∆( X ) over s ∗ B ≃ ( C ( G ) ⊗ B ) ∆( X ) .This still has an action of H , analogous to the right action of H on s ∗ B . OMOLOGY AND K -THEORY OF TORSION-FREE AMPLE GROUPOIDS AND SMALE SPACES 32 Assume moreover ( π, E, T ) is an equivariant Kasparov module between H -C ∗ -algebras. So E isa graded right Hilbert module over B , T is an odd adjointable (or self adjoint) endomorphism, and π : C → L ( E ) is a ∗ -representation, with commutation relations as in [LG99]. Then s ∗ E as a rightHilbert module over s ∗ B , with a left module structure over s ∗ C . Moreover we can extend T to s ∗ T on s ∗ E as the restriction of 1 C ( G ) ⊗ T , with the right commutation properties (they hold beforerestriction to ∆( X )). Finally, we take the crossed product by the right action of H ,Ind GH ( π, E, T ) = j H ( s ∗ π, s ∗ E, s ∗ T ) . This way, we obtain a map Ind GH : KK H ( C, B ) → KK G (Ind GH C, Ind GH B ), realizing the extension ofInd GH to KK H . References [ADR00] C. Anantharaman-Delaroche and J. Renault. (2000).
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DOI:10.1017/s0004972719000522 . ↑ Research Center for Operator Algebras, East China Normal University, 3663 ZhongShan North Road,Putuo District, Shanghai 200062, China
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