Dynamic asymptotic dimension and Matui's HK conjecture

We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side‐effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K‐theory of the C∗$C^*$ ‐algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK‐conjecture.


Introduction
Dynamic asymptotic dimension is a notion of dimension for actions of discrete groups on locally compact spaces, and more generally, for locally compactétale groupoids introduced by the last named author with Guentner and Yu in [24]. It is inspired by Gromov's theory of asymptotic dimension [22,Section 1.3]. At the same time it is strongly connected to other existing dimension theories for dynamical systems, for example the conditions introduced by Bartels, Lück and Reich [3] or Kerr's tower dimension [28].
The original article [24] focused on the fine structure of C˚-algebras associated withétale groupoids of finite dynamic asymptotic dimension, while later work by the same set of authors in [23] presented some consequences to K-theory and topology.
In the present work we aim to explore the implications of dynamic asymptotic dimension for groupoid homology and its relation to the K-theory of groupoid C˚-algebras. A homology theory forétale groupoids was introduced by Crainic and Moerdijk in [11]. More recently, groupoid homology attracted a considerable amount of interest from the topological dynamics and operator algebras communities following the work of Matui [34]. The main contribution of this article is the following: Theorem A. Let G be a locally compact, Hausdorff,étale, principal, σ-compact, ample groupoid with dynamic asymptotic dimension at most d. Then H n pGq " 0 for n ą d and H d pGq is torsion-free.
Our proof of Theorem A goes via a description of groupoid homology in terms of semi-simplicial spaces equipped with a G-action. As a byproduct this leads to a description of these homology groups in terms of the classical Tor groups of homological algebra, quite analogous to the well-known case of the homology of a discrete group. While this may be known to experts, it seems worthwhile recording it as we are not aware of its appearance in the literature.
Theorem B. Let G be a locally compact, Hausdorff,étale, ample groupoid with σcompact base space. There is a canonical isomorphism H˚pGq -Tor Note that having finite dynamic asymptotic dimension forces all the isotropy groups to be locally finite. Consequently, it is natural to restrict our attention to the class of principal groupoids to avoid the trouble caused by torsion in the isotropy groups.
The second theme of this work is an attempt to make the HK-conjecture more explicit. It is well-known and easy to see that there is a canonical homomorphism µ 0 : H 0 pGq Ñ K 0 pCr pGqq.
In general, not much seems to be known about this map, and there are only partial results on the existence of maps in higher dimensions. To formulate our progress in this direction, for an ample groupoid G we denote by rrGss its topological full group. Moreover, Matui constructs in [34] an index map I : rrGss Ñ H 1 pGq.
If moreover G is principal, second countable, and has dynamic asymptotic dimension at most 2, then µ 0 and µ 1 induce the injection H 0 pGq Ñ K 0 pCr pGqq and isomorphism H 1 pGq Ñ K 1 pCr pGqq from Corollary C.
The construction of µ 1 is straightforward if the index map I is surjective, and under further structural assumptions on G, Matui was already able to prove this. Our construction of the map µ 1 is completely general: we in fact give two independent constructions, an elementary one based on ideas of Putnam [45], and a more sophisticated one based on the work of Proietti and Yamashita [44].
It was shown in [24] that the dynamic asymptotic dimension yields an upper bound for the nuclear dimension of reduced groupoid C˚-algebras. In particular, if G is a second countable, principal, minimal ample groupoid with finite dynamic asymptotic dimension, then Cr pGq is classifiable. 1 Our result allows us to completely determine the classifying invariant (usually called the Elliott invariant and denoted Ellp¨q) in the 1-dimensional case: Corollary E. Let G be a locally compact, Hausdorff,étale, σ-compact, principal, ample groupoid with compact base space and with dynamic asymptotic dimension at most 1. Then EllpCr pGqq " pH 0 pGq, H 0 pGq`, r1 G p0q s, H 1 pGq, M pGq, pq.
In light of these results one is tempted to formulate a stronger version of the HK-conjecture in low dimensions, by asking that for ample principal groupoids with H n pGq " 0 for n ě 2 the canonical maps µ i are isomorphisms. To see that this cannot be the case, we construct a counterexample using groupoids with topological property (T) introduced in [14]. The example is based on the construction of counterexamples to the Baum-Connes conjecture by Higson, Lafforgue and Skandalis [26] and work of Alekseev and Finn-Sell [1].
Theorem F. There exists a locally compact, Hausdorff,étale, second countable, principal, ample groupoid G with H n pGq " 0 for all n ě 2 such that µ 0 : H 0 pGq Ñ K 0 pCr pGqq is not surjective. 1 This is due to Kirchberg and Phillips in the purely infinite case [30], [40], and due to many hands in the finite case, including Elliott, Gong, Lin, and Niu [20], [21], [15], and Tikuisis, White, and Winter [52] (see [9] for an alternative proof of classification in the finite case).
As an application of our results, we study a geometric class of examples. Given a metric space X with bounded geometry, Skandalis, Tu and Yu construct an ample groupoid GpXq which encodes many coarse geometric properties of the underlying space X [49]. The following result adds to this list of connections. It might be known to experts but does not seem to appear in the literature so far (except for degree zero, which has been treated in [2]).
Theorem G. Let X be a bounded geometry metric space and GpXq be the associated coarse groupoid. Then there is a canonical isomorphism H˚pGpXqq -H uf pXq between the groupoid homology of the coarse groupoid and the uniformly finite homology of X in the sense of Block and Weinberger [5].
As the dynamic asymptotic dimension of the coarse groupoid equals the asymptotic dimension of the underlying metric space (in the sense of Gromov), a combination of Theorems A (adapted to the non second countable case) and D yields the following purely geometric corollary.
Corollary H. Let X be a bounded geometry metric space.
(1) If asdimpXq ď 2 and H uf 2 pXq is free, or finitely generated, then K 0 pCů pXqq -H uf 0 pXq ' H uf 2 pXq, K 1 pCů pXqq -H uf 1 pXq. (2) If asdimpXq ď 3, H uf 3 pXq is free, or finitely generated, and X is nonamenable, then K 0 pCů pXqq -H uf 2 pXq, K 1 pCů pXqq -H uf 1 pXq ' H uf 3 pXq. The first part of this corollary applies for example if X is the fundamental group of a closed, orientable surface, and the second applies for example if X is the fundamental group of a closed, orientable, hyperbolic 3-manifold. The reader might compare this to [16,Theorem B, Theorem F, and Corollary H]: combining these implies related results after taking a "completed tensor product with C" (in an appropriate sense) for certain spaces.
Outline of the paper. In Section 2 we give a new picture of Crainic-Moerdijk homology by defining a G-equivariant homology theory for an appropriate notion of semi-simplicial G-spaces. This is done in section 2.2, by describing the homology groups as the left derived functor of the coinvariants in the sense of classical homological algebra. The necessary background is given in section 2.1. It turns out that the groups H˚pGq will be naturally isomorphic to the equivariant homology of a semi-simplicial G-space EG˚. We deduce Theorem B from this material.
The central piece of the vanishing result in Theorem A is tackled in Section 3. There we define a colouring of G, which will induce an appropriate cover of G. The nerve of this cover in the sense of section 3.3 is a semi-simplicial G-space, and that defines the homology of the colouring. The central idea is to define an anti-Čech sequence of G as a sequence of colourings with induced covers that are bigger and bigger, in the spirit of anti-Čech covers in coarse geometry, introduced by Roe (see [47, chapter 5]). We show that the inductive limit of an anti-Čech cover is well defined and that it converges to the Crainic-Moerdijk homology groups for principal σ-compact ample groupoids in theorem 3.29.
Section 4 is dedicated to the two constructions of the map µ 1 : H 1 pGq Ñ K 1 pCr pGqq, from Theorem D, and a proof that these give the same result. We also deduce Corollary C (in a stronger form) and Corollary E.
The composition of the global section functor and the coinvariant functor is denoted by Zr?s G . Lemma 2.5. If X is a free and proper space in Top G then there is an isomorphism of abelian groups ZrXs G -ZrGzXs.
Proof. Denote by rxs P GzX the class of x P X, and define a ZrGs-linear map ε : ZrXs Ñ ZrGzXs byε pf qprxsq " ÿ gPG ppxq f pg¨xq @x P X.
This obviously factors through ZrXs G , giving a map which we still denote bỹ ε : ZrXs G Ñ ZrGzXs.
Let us build an inverse toε. Since the action of G on X is proper, the quotient is locally compact Hausdorff, and hence a partition of unity argument shows that the family of functions f P ZrGzXs such that there exists a compact open subset V Ď X with supppf q " qpV q and such that q |V : V Ñ qpV q is a homeomorphism, generates ZrGzXs as an abelian group. Now given such a function f , we define f V :" f˝q |V P ZrXs and our first goal is to show that the class of f V in ZrXs G does not depend on the choice of V . So let V 1 Ď X be another compact open set with supppf q " qpV 1 q and such that q |V 1 : V 1 Ñ qpV 1 q is a homeomorphism. Then qpV q " qpV 1 q and hence every element x P V can be written as x " gy for some y P V 1 , g P G. Using that G isétale and compactness of V we can decompose V " Ť i V i and V 1 " i . Hence doing another partition of unity argument, we may assume that the sets V and V 1 themselves are related in this way, i.e. there exists a bisection S Ď G which induces a homeomorphism α S : V Ñ V 1 by α S pxq " hx where h P S is the unique element in S X G ppxq . But then This relation can be rewritten as f V " χ S´1¨fV 1 , and hence we have rf V s " rf V 1 s in ZrXs G as desired. Let ψ : ZrGzXs Ñ ZrXs G be the map given by ψpf q " rf V s. If V Ď X is open such that q |V : V Ñ qpV q is a homeomorphism, then for a given x P X there is at most one g P G ppxq such that gx P V since the action is free. It follows that for any function f P ZrGzXs supported in such a V we havẽ εrf V sprxsq " ÿ gPG ppxq f pq V pgxqq " f prxsq and henceε˝ψ " id.
Conversely, if f P ZrXs is a function supported in a set of the form S¨V for a bisection S Ď G and V Ď X is open such that q |V is a homeomorphism onto its image, then one easily checks thatεrf s is supported in qpV q. Hence rψpεrf sqs " rεrf s˝q |V s. Using freeness again, one checks that the latter class equals rχ S¨f s " rεpχ S´1 q¨f s " rf s. Since functions f as above generate ZrXs we are done.

2.2.
Semi-simplicial G-spaces and homology. For n a nonnegative integer, denote by rns the interval t0, ..., nu. Recall (see for example [54,Chapter 8]) the definition of the semi-simplicial category ∆ s : its objects are the nonnegative integers, and Hom ∆s pm, nq consists of the (injective) increasing maps f : rms Ñ rns. A semi-simplicial object in a category C is a contravariant functor from ∆ s to C. The collection of all semi-simplicial objects in a category is itself a category, with morphisms given by natural transformations.
Let ε n i : rn´1s Ñ rns be the only increasing map whose image misses i. We will omit the superscript n if it does not cause confusion. Any increasing map f : rms Ñ rns has a unique factorization f " ε i1 ε i2 ...ε i k with 0 ď i k ď ... ď i 1 ď n (see Lemma 8.1.2 in [54]). Thus, any semi-simplicial object is the data, for all n, of an object C n of C, together with arrows ε n i : C n Ñ C n´1 in C, called face maps, satisfying the (semi-)simplicial identities ε n´1 j ε n i " ε n´1 i ε n j´1 if i ă j. Similarly, if C˚and D˚are semi-simplicial objects in C, a morphism between them is a collection of morphisms f n : C n Ñ D n in C that are compatible with the face maps.
A semi-simplicial object in the category of locally compact Hausdorff topological spaces will be called a semi-simplicial topological space. Definition 2.6. A semi-simplicial G-space is a semi-simplicial object in the category Top G .
As an example, define EG˚to be the semi-simplicial G-space EG n " G rˆr G rˆr ... rˆr G (n+1 times) with ‚ anchor map p : EG n Ñ G 0 given by the common range of the tuple, ppγ 0 , ..., γ n q " rpγ 0 q " ... " rpγ n q for γ P EG n , ‚ left action given by left multiplication by G on all factors, ‚ if f : rms Ñ rns, then EGpf q : G n Ñ G m is defined by pγ 0 , ..., γ n q Þ Ñ pγ f p0q , ..., γ f pmq q (note that the face maps are given by B n`1 i : pγ 0 , ..., γ n q Þ Ñ pγ 0 , ...,γ i , ..., γ n q where the hat means that the entry is omitted). One checks that the moment and the face maps are G-equivariant andétale.
On the other hand, define the classifying space of G, denoted by BG˚, to be the semi-simplicial topological space defined by BG n " tpg 1 , ..., g n q P G n | rpg i q " spg i´1 q for all iu with face maps ǫ n i : pg 1 , ..., g n q Þ Ñ $ & % pg 2 , ..., g n q if i " 0, pg 1 , ..., g i´1 , g i g i`1 , ..., g n q if 1 ď i ă n, pg 1 , ..., g n´1 q if i " n.
The G-action on EG is free and proper and hence GzEG˚is a semi-simplicial topological space. The maps pg 1 , ..., g n q Þ Ñ rrpg 1 q, g 1 , g 1 g 2 , . . . , g 1¨¨¨gn s and rγ 0 , . . . , γ n s Þ Ñ pγ´1 0 γ 1 , . . . , γ´1 n´1 γ n q define maps of semi-simplicial topological spaces between BGå nd GzEG˚that are mutually inverse. A semi-simplicial G-space pX˚, tǫi u i q naturally induces a chain complex of ZrGsmodules pZrX˚s,B˚q by composing by the Zr?s functor, where the boundary maps areB n " n ÿ i"0 p´1q i pB n i qå nd thus a chain complex of abelian groups pZrX˚s G ,B˚q by composing by the coinvariant functor.
Definition 2.7. Let pX˚, tBi u i q be a semi-simplicial G-space. We define the equivariant homology group H G pXq to be the homology of the chain complex of abelian groups pZrX˚s G ,B˚q.
The relationship with the homology of Matui and Crainic-Moerdijk is now a consequence of Lemma 2.5. Namely, Matui introduced a chain complex of abelian groups to compute Crainic-Moerdijk homology groups in the special case of an ample groupoid. This chain complex is none other than pZrBG˚s,ǫ˚q. As EG˚is a free and proper semi-simplicial G-space with BG˚-GzEG˚, this homology is isomorphic to H G pEGq -H˚pBGq. In other words, we have proved the following.
Proposition 2.8. Let G be an ample groupoid. The complex introduced by Matui to compute H˚pGq identifies canonically with the homology of the complex pZrEG˚s G ,B˚q.

2.3.
Projective resolutions and Tor. The aim of this section is to identify H˚pGq with one of the standard objects studied in homological algebra. These results will not be used in the rest of the paper; we include them as they can be derived without too much difficulty from our other methods, and seem interesting. Remark 2.9. If R is a non-unital ring, then free R-modules are not necessarily projective. However, for any idempotent e P R it is easily seen that Hom R pRe, M q -eM naturally for any R-module M . If N ։ M is an epimorphism then elements in eM lift to eN , so it follows that Re is projective. Consequently, if R is a ring with enough idempotents, i.e. it contains a family pe i q iPI of mutually orthogonal idempotents such that R -À iPI Re i -À iPI e i R, then free R-modules are projective. More generally, a (non-degenerate) R-module is projective exactly when it is a direct summand of a free R-module.
We will restrict to ample groupoids G with σ-compact base space G 0 . The main reason is that it implies that G 0 (resp. G) can be written as a disjoint union of compact open sets (resp. compact open bisections). 3 By the above remark it follows that free G-modules are projective, and similarly for ZrG 0 s.
To state the main result of this subsection, note that G 0 admits left and right actions of G (with the anchor map being the identity) defined by gx :" rpgq and xg :" spgq.
These actions make ZrG 0 s into both a left and a right G-module. Thus the Tor groups Tor ZrGs pZrG 0 s, ZrG 0 sq (see for example [54,Definition 2.6.4]) of homological algebra make sense. Theorem 2.10. Let G be an ample groupoid with σ-compact base space. There is a canonical isomorphism H˚pGq -Tor ZrGs pZrG 0 s, ZrG 0 sq.
The rest of this section will be spent proving this theorem, which will proceed by a sequence of lemmas. To give the idea of the proof, recall (see for example [ of ZrGs-modules, where each P n is projective. The group Tor ZrGs n pZrG 0 s, ZrG 0 sq is then by definition the n th homology group of the compleẍ¨¨i In general, this fails for locally compact, Hausdorff, totally disconnected spaces, such as Counterexample 65 from [51] (R equipped with a rational sequence topology). One can also show that if X is this particular example, then ZrXs is not projective as a ZrXs-module, so free ZrXs-modules are not projective.
of abelian groups. We will prove Theorem 2.10 by showing that each ZrGs-module ZrEG n s is projective, and that we have an exact sequencë¨¨B where the boundary maps are the alternating sums of the face maps. As for any left non-degenerate G-module M by Lemma 2.2, Proposition 2.8 completes the proof. We now embark on the details of the proof. Recall that G 0 is equipped with a left G-action defined by stipulating that the anchor map is the identity and defining the moment map by gx :" rpgq, and that ZrG 0 s is a left G-module with the induced structure. Define B : ZrEG 0 s Ñ ZrG 0 s to be the map induced on functions by thé etale map r : G Ñ G 0 .
Then one computes on the spatial level that B i h " hB i´1 for 1 ď i ď n, and B 0 h is the identity map. Hence by functoriality and this is the identity. Define also h : G 0 Ñ EG 0 by hpxq " x. Then the map rh : G 0 Ñ G 0 is the identity, and so Bh˚is the identity map on ZrG 0 s. To summarise, h˚is a chain homotopy (as in for example [54,Definition 1.4.4]) between the identity map and the zero map. This implies that the complex has trivial homology (see for example [54,Lemma 1.4.5]), or, equivalently, is exact.
For the next lemma, let us define a right action of ZrG 0 s on ZrGs via pf φqpgq :" f pgqφpspgqq @f P ZrGs, φ P ZrG 0 s and similarly a left action of ZrG 0 s on ZrEG n s via pφaqpg 0 , ..., g n q :" φprpg 0 qqapg 0 , ..., g n q.
Note that this right action of ZrG 0 s on ZrGs commutes with the canonical left action of ZrGs on itself: indeed, the action of ZrG 0 s is just the action by rightmultiplication of the submodule ZrG 0 s of ZrGs, so this commutativity statement is associativity of multiplication. For notational convenience, we define EG´1 :" G 0 . Lemma 2.12. With notation as above, for any n ě´1 there is a canonical isomorphism of ZrGs-modules.
We claim that this map is an isomorphism. For injectivity, say we have an element ř n i"1 f i ba i that goes to zero. Splitting up the sum further, we may assume that each f i is the characteristic function χ Bi of a compact open bisection B i such that B i XB j " ∅ for i ‰ j. For each i, let φ i be the characteristic function of spB i q. As χ Bi " χ Bi φ i , we have that χ Bi ba i " χ Bi bφ i a i , we may further assume that each a i is supported in tpg 0 , ..., g n q P EG n | rpg 0 q P spB i qu. Now, we are assuming that ř n i"1 pχ Bi , a i q " 0. As B i X B j " ∅ for i ‰ j, the functions pχ Bi , a i q have disjoint supports, and therefore we have that pχ Bi , a i q " 0 for each i. Assume for contradiction that a i ‰ 0, so there exists ph 0 , ..., h n q P EG n with aph 0 , ..., h n q ‰ 0; our assumptions force rph 0 q P spB i q. Let g P B i be such that spgq " rph 0 q. Then pg, gh 0 , ..., gh n q P EG n`1 and pχ Bi , aq evaluates to aph 0 , ..., h n q ‰ 0 at this point, giving the contradiction and completing the proof of injectivity.
For surjectivity, as any element of ZrEG n`1 s is a finite Z-linear combination of characteristic functions of subsets of the form B 0ˆ¨¨¨ˆBn`1 X EG n`1 with each B i a compact open bisection in G, it suffices to show that any such characteristic function is in B. Set f P ZrGs to be the characteristic function of B 0 , and a P ZrEG n s to be the characteristic function of B´1 0 B 1ˆ¨¨¨B´1 0 B n X EG n . We leave it to the reader to check that f b a maps to the function we want.
For the next lemma, we consider ZrGs as a left ZrG 0 s-module via the leftmultiplication action induced by the inclusion ZrG 0 s Ď ZrGs.
In many interesting cases ZrGs is actually free over ZrG 0 s: for example, this happens for transformation groupoids associated to actions of discrete groups. However, freeness does not seem to be true in general.

Proof.
For a compact open bisection U , let χ U P ZrGs denote the characteristic function of that bisection. As G 0 is σ-compact we may choose a covering G " Ů iPI U i of G by disjoint compact open bisections, and for each i, let ZrG 0 sχ Ui denote the ZrG 0 s-submodule of ZrGs generated by χ Ui . Then as ZrG 0 s-modules, we have ZrGs -à iPI ZrG 0 sχ Ui .
It thus suffices to prove that each ZrG 0 sχ Ui is projective. For this, one checks that ZrG 0 sχ Ui is isomorphic as a ZrG 0 s-module to ZrG 0 sχ rpUiq , so projective by Remark 2.9, and we are done.
Corollary 2.14. If G 0 is σ-compact, the G-module ZrEG n s is projective for n ě 0.
Proof. We proceed by induction on n. For n " 0, ZrEG 0 s identifies with ZrGs as a left ZrGs-module, so free, and thus projective by Remark 2.9. Now assume we have the result for ZrEG n s. By Remark 2.9, ZrEG n s is a direct summand in a free ZrGs-module, say À J ZrGs. Hence, as ZrG 0 s-modules, ZrEG n s is a direct summand in À J ZrGs which is projective by Lemma 2.13. Therefore, ZrEG n s is projective as a ZrG 0 s-module. Let then N be a ZrG 0 s-module such that ZrEG n s'N is isomorphic as a ZrG 0 s-module to À iPI ZrG 0 s for some index set I. It follows that as ZrGs-moduleś ZrGs.

Hence ZrGs b
ZrG 0 s ZrEG n s is isomorphic to a direct summand of a free ZrGs-module, so projective as a ZrGs-module. Using Lemma 2.12, we are done.
Proof of Theorem 2.10. Lemma 2.11 and Corollary 2.14 together imply thaẗ¨¨B is a resolution of ZrG 0 s by projective modules. The groups Tor ZrGs pZrG 0 s, ZrG 0 sq are therefore by definition the homology groups of the compleẍ¨¨i However, using Lemma 2.2, this is the same as the compleẍ¨¨B and we have already seen that the homology of this is the same as the homology H˚pGq.

Colourings and homology
The goal of this section is to use what we shall call a colouring of a groupoid G to produce an associated nerve space, and prove that the nerve is a semi-simplicial G-space. This lets us define the homology of a colouring in terms of the homology of the associated semi-simplicial G-space. Throughout most of the section we assume that G is an ample groupoid with compact base space G 0 : the most important exception is the main result-Theorem 3.36-at the end, where we drop the compact base space assumption.
3.1. Colourings and nerves. In this subsection we introduce colourings of a groupoid and the associated nerve spaces and homology. Elements of the set t0, ..., du are called the colours of the colouring, and the colour of G i is i.
We will associate a cover of G to a colouring. Let PpGq denote the collection of subsets of G. Definition 3.2. Let G 0 , ..., G d be a colouring of G. The cover associated to the colouring is Typically, we will just write U for an element of U, and treat U as a (non-empty) subset of G. In particular, when considering intersections U 0 X U 1 for U 0 , U 1 P U we just mean the intersection of the corresponding subsets of G (ignoring the colour!). The precise definition however calls for pairs as we want each element of U to have a well-defined colour in t0, ..., du. Note that an element U P U could be equal to pgG The sequence pN n q 8 n"0 is denoted N˚, and called the nerve of the colouring. As noted before, the intersection in the definition above is to be interpreted as the intersection of the corresponding subsets of G (ignoring the colour). In particular we allow distinct U i appearing in a tuple as above to have different colours.
Our next goal is to give N˚the structure of a semi-simplicial G-space.
Definition 3.4. For each n, the anchor map r : N n Ñ G 0 takes pU 0 , ..., U n q to the unique x P G 0 such that U 0 is a subset of 4 G x . Let G sˆr N n be the fibered product as in line (1) above, and define an action by G sˆr N n Ñ N n , pg, pU 0 , ..., U n qq Þ Ñ pgU 0 , ..., gU n q.
Direct checks show this is a well-defined groupoid action of G on the set N n . Our next goal is to introduce a topology on N n , and prove that the action is continuous.
and equip N 0 with the topology generated by these sets. For each n ě 0, equip N n`1 0 with the product topology, and give N n Ď N n`1 0 the subspace topology. Lemma 3.6. For each n, the topology on N n is locally compact, Hausdorff, and totally disconnected. Moreover, the action defined in Definition 3.4 above is continuous.
Proof. We first look at the case n " 0. Given a compact open bisection V Ď G with spV q Ď G 0 i one readily verifies that the mapping g Þ Ñ pgG spgq i , iq defines a homeomorphism V Ñ U V,i . In particular, the sets U V,i for such compact open bisections V are themselves compact open Hausdorff subsets of N 0 . Consequently, N 0 is locally compact and locally Hausdorff. To check that it is Hausdorff, it suffices to check that if pU j q j is a net in N 0 that converges to both U and V , then U " V . Equivalently, say we have a net pg j G spgj q ij for all j, and so that g j Ñ g and h j Ñ h. First note that 4 and therefore all the U j are subsets of since the net converges, i j will be eventually constant so we may as well assume that i j " i for all j; we need to check that gG . By symmetry, it suffices to check that gk is in hG for all j, we can also find l j P G sphj q i for all j so that g j k j " h j l j . Using that G i is compact, we may pass to a subnet and so assume that pl j q converges to some l P G i , which is necessarily in G sphq i . Hence gk " lim g j k j " lim h j l j " hl, and so gk is in hG sphq i as required.
The first paragraph of the proof implies that N 0 admits a basis of compact open subsets and hence it is totally disconnected.
Continuity of the action follows on observing that if pg j , h j G sphq i q is a convergent net in G sˆr N 0 , then pg j h j G spgj hq i q is a convergent net in N 0 by continuity of the multiplication in G.
We now look at the case of general n. The facts that N n is Hausdorff and totally disconnected, as well as the continuity of the G-action, all follow directly from the corresponding properties for N 0 .
We claim that N n is closed in N n`1 0 ; as closed subsets of locally compact spaces are locally compact, this will suffice to complete the proof. To check closedness, for each j P t0, ..., nu, say pg k j q kPK is a net such that g k j Ñ g j as k Ñ 8, and such that pg k 0 G spg k 0 q i , ..., g k n G spg k n q i q is in N n for all k. We need to show that As G i is compact, we may assume that each net ph k j q kPK converges to some h j in G i . Hence , which is thus non-empty. Hence We thus have shown that each of the spaces N n is in the category Top G of Definition 2.3. To show that N˚is a semi-simplicial G-space, it remains to build the face maps and show that they are equivariant andétale. Definition 3.7. For each n ě 1 and each j P t0, ..., nu, define the j th face map to be the function where the hat "p " means to omit the corresponding element. Proof. Let pU 0 , ..., U n q be a point in N n , and write U k " g k G is an open neighbourhood of pU 0 , ..., U n q in N n . We claim that B j restricts to a homeomorphism on W . Indeed, let W j be the image of W under B j . Then an inverse is defined by sending a point pV 0 , ..., V n´1 q in W j to the point pV 0 , ..., hG sphq ij , ..., V n´1 q in W , where hG sphq ij occurs in the j th entry, and where h is the unique point in B j so that rphq " ppV 0 , ..., V n´1 q.
The G-equivariance and the claimed relations between the face maps are straightforward.
Definition 3.9. Let C be a colouring of G. The homology of the colouring, denoted H˚pCq, is the homology H G pN˚q of the semi-simplicial G-space N˚as in Definition 2.7.
The homology groups H˚pCq depend strongly on the colouring. For example, C could just consist of a partition of G 0 by compact open subsets, in which case one can check that the groups H n pCq are zero for n ą 0. We will, however, eventually show that an appropriate limit of the homologies H˚pCq as the colourings vary recovers the Cranic-Moerdijk-Matui homology H˚pGq for principal and σ-compact G.

Homology vanishing. Our goal in this subsection is to show that if
C " tG 0 , ..., G d u is a colouring of G, then H n pCq " 0 for n ą d. We will actually establish something a little more precise than this, as it will be useful later. The computations in this section are inspired by classical results in sheaf cohomology: see for example [19,Section 3.8]. More specifically, the precise formulas we use are adapted from [50,Tag 01FG].
Throughout this subsection, we fix an ample groupoid G with compact unit space G 0 , a colouring C " tG 0 , ..., G d u as in Definition 3.1, and associated nerve space N˚as in Definition 3.3.
Lemma 3.10. Let U be the cover associated to the colouring C as in Definition 3.2. Then any two elements of U that are the same colour and intersect non-trivially are the same. In particular, if pU 0 , ..., U n q is a point of some N n , then any two elements of the same colour are actually the same. with gk g " hk h . It follows that h´1g " k h k´1 g , so h´1g is in G i , as G i is a subgroupoid. Hence whenever gk is in gG spgq i , we have that gk " hph´1gkq is also in hG  We leave it to the reader to check that c is continuous and invariant under the action of G.
For a point x " pi 0 , ..., i n q P t0, ..., du n`1 , let σ x P S n`1 be the unique permutation determined by the conditions below: (ii) σ x is order-preserving when restricted to each subset S of t0, ..., nu such that the elements ti j | j P Su all have the same colour (i.e. so that the set ti j | j P Su consists of a single element of t0, ..., du).
Definition 3.12. For each n, let c : N n Ñ t0, ..., du n`1 be the colour map of Definition 3.11 and define N ą n :" c´1`tpi 0 , ..., i n q | i 0 ă¨¨¨ă i n u˘. Similarly, define N ě n :" c´1`tpi 0 , ..., i n q | i 0 ď¨¨¨ď i n u˘. We note that each N ą and N ě is a semi-simplicial G-space with the restricted structures from N˚: indeed, each N ą n and N ě n is a closed, open and G-invariant subset of N n as c is continuous and G-invariant, and the face maps of Definition 3.8 clearly restrict to maps N ą n Ñ N ą n and N ě n Ñ N ě n . For each a P t0, ..., nu define now h a : N n Ñ N n`1 by the formula h a pU q :" pU σ cpU q a p0q , ..., U σ cpU q a pa´1q , U σ cpU q paq , U σ cpU q a paq , U σ cpU q a pa`1q , ..., U σ cpU q a pnq q (in words, we use σ cpUq a to rearrange the order of the components of U , but also insert U σ cpU q paq into the a th position).
It is not too difficult to see that each h a is an equivariantétale map, and so induces a pushforward map ph a q˚: ZrN n s Ñ ZrN n`1 s of ZrGs-modules. We define h : ZrN n s Ñ ZrN n`1 s by stipulating that for each x P t0, ..., du n`1 , its restriction to each subset Zrc´1pxqs equals n ÿ a"0 p´1q a signpσ x a qph a q˚.
On the other hand, define p : ZrN n s Ñ ZrN ě n s by stipulating that for each x P t0, ..., du n`1 , its restriction to each subset Zrc´1pxqs equals signpσ x qσ x . Finally, let i : ZrN ě s Ñ ZrN˚s be the canonical inclusion. Proof. We look at the restriction of Bh to Zrc´1pxqs for some x; it suffices to prove the given identity for such restrictions. For notational simplicity, let σ :" σ x . We have then for this restriction that (ii) Terms of the form B i`1 ph a q˚where i ě a and σ a piq ‰ σpaq.
(iii) Terms of the form B å ph a q˚, or of the form B i`1 ph a q˚where σ a piq " σpaq.
We look at each type in turn.
(i) We leave it to the reader to compute that as maps on the spatial level, B i h a " h a´1 B σapiq . Moreover, pB i h a q˚occurs in the sum defining Bh with sign p´1q i p´1q a signpσ a q, and ph a´1 B σapiq q˚occurs in the sum defining hB with the sign p´1q a´1 p´1q σapiq signpσ 1 a´1 q, where σ 1 a´1 is the permutation defined as the composition with f and g the unique order preserving bijections. One can compute that signpσ a q " p´1q σapiq´i signpσ 1 a´1 q, essentially as σ a can be built from the same transpostitions as used to construct σ 1 a (conjugated by f and g) together with a cycle of length |σ a piq´i|`1, which has sign p´1q σapiq´i . In conclusion, the term p´1q i p´1q a signpσ a qpB i h a qå ppearing in the sum defining Bh is matched by the term ppearing in the sum defining hB; as these precisely match other than having opposing signs, they cancel. (ii) We compute that as maps on the spatial level B i`1 h a " h a B σapiq . Moreover, pB i`1 h a q˚occurs in the sum defining Bh with sign p´1q i`1 p´1q a signpσ a q, and ph a B σapiq q˚occurs in the sum defining hB with the sign p´1q a p´1q σapiq signpσ 1 a q, where σ 1 a is the permutation defined as the composition t0, ..., n´1u f / / t0, ..., p i, ..., nu σa / / t0, ..., z σ a piq, ..., nu g t0, ..., n´1u with f and g the unique order preserving bijections. Much as in case (i), we have that signpσ a q " p´1q σapiq´i signpσ 1 a´1 q, and can conclude that the term ppearing in the sum defining Bh is matched by the term p´1q a p´1q σapiq signpσ 1 a qph a B σapiq q˚" p´1q a p´1q i signpσ a qh a B σapiq qå ppearing in the sum defining hB; as these precisely match other than having opposite signs, they cancel.
At this point, one can check that we have canceled all the terms appearing in the sum defining hB using terms from Bh of types (i) and (ii). It remains to consider terms of type (iii), and show that the sum of these equals identity´i˝p as claimed. For each a, let i a be the index such that σ a pi a q " σpaq. The totality of terms of type (iii) looks like The first term is the identity, and the last term is i˝p (note that σ n " σ). Hence it suffices to show that each term p´1q ia`1 p´1q a signpσ a qpB ia`1 h a q˚`p´1q a`1 p´1q a`1 signpσ a`1 qpB a`1 h a`1 qi n the sum in the middle is zero. Now, one computes on the spatial level that B a`1 h a`1 " B ia`1 h a (this uses Lemma 3.10 to conclude that two coordinates with the same colour are actually the same), and so it suffices to prove that p´1q ia`1 p´1q a signpσ a q`p´1q a`1 p´1q a`1 signpσ a`1 q " 0, or having simplified slightly, that p´1q ia`a`1 signpσ a q`signpσ a`1 q " 0. Indeed, one checks that σ a`1 differs from σ a by a cycle moving the element in the i th a position to the a th (and keeping all other elements in the same order), and such a cycle has sign p´1q a´ia . Hence signpσ a q " signpσ a`1 qp´1q ia´a and we are done.
Corollary 3.14. The natural inclusion i : ZrN ě s Ñ ZrN˚s is a chain homotopy equivalence.
Proof. Lemma 3.13 implies in particular that the map i˝p is a chain map; as i is an injective chain map, this implies that p : ZrN n s Ñ ZrN ě n s is a chain map too. Lemma 3.13 implies that i˝p is chain-homotopic to the identity, while p˝i just is the identity. Hence p provides an inverse to i on the level of chain homotopies.
Our next goal is to show that the natural inclusion j : ZrN ą s Ñ ZrN ě s is again a chain homotopy equivalence. For each a P t0, ..., n´1u, let us write D n a for the subset of t0, ..., du n`1 consisting of those tuples pi 0 , ..., i n q such that i 0 ă i 1 ă¨¨i a " i a`1 ď i a`2 ď¨¨¨ď i n . For x P D n a , define k x : Zrc´1pxqs Ñ ZrN n`1 s by k x : pU 0 , ..., U n q Þ Ñ pU 0 , U 1 , ..., U a , U a , U a`1 , U a`2 , ..., U n q. Now define k : ZrN ě n s Ñ ZrN ě n`1 s by stipulating that for each x P t0, ..., du n`1 the restriction of k to Zrc´1pxqs is given by p´1q a k x if x is in D n a for some a P t0, ..., n´1u, and by zero otherwise. On the other hand, let q : ZrN ě n s Ñ ZrN ą n s be the natural projection that acts as the identity on each Zrc´1pxqs with c´1pxq Ď N ą , and as 0 otherwise. Proof. It suffices to check the formula for each restriction to a submodule of the form Zrc´1pxqs.
Say first that x " pi 0 , ..., i n q satisfies i 0 ă¨¨¨ă i n . Then j˝q acts as the identity on Zrc´1pxqs, whence the right hand side is zero. Note that k restricts to zero on Zrc´1pxqs, whence Bk is zero. On the other hand, the image of Zrc´1pxqs is contained in a direct sum of subgroups of the form Zrc´1pyqs where y " pj 0 , ..., j n´1 q P t0, ..., du n satisfies j 0 ă¨¨¨ă j n´1 . Hence kB " 0 too, and we are done with this case. Say then that x " pi 0 , ..., i n q does not satisfy i 0 ă¨¨¨ă i n . Then x is in D n a for some a. We then compute using the assumption that x is in D n a that B i k x is the identity map for i P ta, a`1, a`2u, and therefore that Looking instead at kB, note first that as x is in D a , we have that B a " B a`1 when restricted to c´1pxq, and so Putting this discussion together with the formula in line (3) gives that Comparing this with the formula in line (2), it follows that kB`Bk restricts to the identity on this summand Zrc´1pxqs. On the other hand, the same is true for the right hand side 'identity´j˝q', so we are done.
Proof. Lemma 3.15 implies that the natural projection q is an inverse on the level of chain homotopies.
Theorem 3.17. Let N˚be a nerve complex built from a colouring C " pG 0 , ..., G d q.
Then the canonical inclusion N ą Ñ N˚induces an isomorphism H˚pCq -HpN ą q.
In particular, H n pCq " 0 for n ą d.
Proof. Corollaries 3.14 and 3.16 imply that the inclusion ZrN ą s Ñ ZrN˚s is a chain homotopy equivalence. Functoriality of taking coinvariants then implies that the naturally induced map ZrN ą s G Ñ ZrN˚s G is a chain homotopy equivalence, so in particular induces an isomorphism on homology. The remaining statement follows as N ą n is empty for n ą d.

Maps between nerves and G.
In this subsection, we build maps between our nerve spaces N 0 and G that induce maps between the higher nerve spaces N n and the spaces EG n we used to define groupoid homology. This will allow us to compare the homology of colourings of G to the homology of G.
Definition 3.18. Let G 0 , ..., G d be a colouring of G as in Definition 3.1, and let K be a subset of G. We say that the colouring G 0 , ..., G d is: For n ě 0, recall that EG n " tpg 0 , ..., g n q P G n`1 | rpg 0 q " ... " rpg n qu, equipped with the subspace topology that it inherits from G n`1 . For n ě 1 and any subset K of G, let EG K n denote the subspace of EG n consisting of those tuples pg 0 , ..., g n q such that g´1 i g j P K for all i, j. Let EG K 0 be just EG 0 " G, whatever K is. (i) Φ 0 pgq Ě gK for all g P G; (ii) for all n, the map As the set G x X K is compact and discrete, it is finite. Write g 1 , ..., g n for the elements of this set, which are all in G i by assumption that x is in V i . For each j P t1, ..., nu, let W j Ď G i be a compact open bisection containing g j . We may assume the W j are disjoint by shrinking them if necessary. Define W :" Ş n j"1 rpW j q, which is a compact open neighbourhood of x. We may write the compact open set r´1pW q X K as a finite disjoint union of compact open bisections of the form W 1 X r´1pW q, ..., W n X r´1pW q, B 1 , ..., B m . Note that no B j can intersect G x X K, whence none of the sets rpB j q can contain x. Define This is an open set containing x. Moreover, r´1pV q is contained in W 1 Y¨¨¨Y W n , and therefore in Note now that V 0 , ..., V d covers G 0 by the assumption that the underlying colouring is K-Lebesgue. As G 0 has a basis of compact open sets and is compact, there is a finite cover, say U of G 0 consisting of disjoint compact open sets, and such that each U P U is contained in some V i . Define E i to be the union of those U P U such that i is the smallest element of t0, ..., du with U contained in V i . Then the sets E 0 , ..., E d are a partition of G 0 by compact open subsets, and each E i is contained in V i . Now, for each g P G, let ipgq P t0, ..., du be the unique i such that spgq is in E i ; as the partition G 0 " Ů n i"0 E d is into clopen sets, the map i : G Ñ t0, ...du this defines is continuous. Define We claim this has the right properties. First, note that for g P G, G spgq X K Ď G ipgq by definition of ipgq and the cover V 0 , ..., V d . Hence We now show that Φ 0 isétale. Continuity of the restriction of Φ 0 to each set s´1pE i q follows from the definition of the topology of N 0 , and continuity of Φ 0 on all of G follows from this as the sets s´1pE 0 q, ..., s´1pE d q are a closed partition of G. Let now g P G, and let B be a clopen bisection containing g such that the map i : G Ñ t0, ..., du is constant on B (such exists as i is continuous). Let C " Φ 0 pBq. Then the map C Ñ B defined by sending U to the unique element of B X rpU q is a well-defined continuous inverse to the restriction Φ 0 | B , completing the proof that Φ 0 isétale.
Equivariance of Φ 0 follows as if spgq " rphq, then spghq " sphq and ipghq " iphq, and so this set is non-empty. Hence pΦ 0 pg 0 q, ..., Φ 0 pg n qq is a well-defined element of N n . Equivariance of Φ n and the fact that it isétale are straightforward from the corresponding properties for Φ 0 , so we are done.
Lemma 3.20. Assume that G is principal, and K is a compact open subset of G that contains G 0 . Let G 0 , ..., G d be a K-bounded colouring with associated nerve N˚. Then there exists an equivariantétale map Ψ 0 : N 0 Ñ G with the following properties: (i) Ψ 0 pU q P U for all U P N 0 ; (ii) for all n, the map Ψ n : N n Ñ EG KK´1 n , pU 0 , ..., U n q Þ Ñ pΨ 0 pU 0 q, ..., Ψ 0 pU n qq is a well-defined, equivariant local homeomorphism.
To prove this, we need an ancillary lemma, which is based on the following structural result from [18,Lemma 3.4].
Lemma 3.21. Let H be a compact, ample, principal groupoid. Then there are m P N and (i) disjoint clopen subgroupoids H 1 , ..., H m of H, (ii) clopen subsets X 1 , ..., X m of H 0 (equipped with the induced, i.e. trivial, groupoid structure), and (iii) finite pair groupoids P 1 , ..., P m , such that H identifies with the disjoint union H " Ů m i"k H k as a topological groupoid, and such that each H k is isomorphic as a topological groupoid to X kˆPk . (i) σ splits the quotient map π : H 0 Ñ H 0 {H (so in particular, π isétale); (ii) r˝τ " identity and and s˝τ " σ˝π.
Proof. Assume first that H " XˆP , where X is a compact trivial groupoid and P is the pair groupoid on some finite set t0, ..., nu. Then H 0 {H identifies homeomorphically with X (so in particular is Hausdorff) via the map Making this identification, we may define σpxq " px, p0, 0qq and τ px, pi, iqq " px, pi, 0qq. These maps have the right properties when H " XˆP .
In the general case, Lemma 3.21 gives a decomposition of H into groupoids of the form XˆP as above, and we may build σ and τ on each separately using the method above.
Proof of Lemma 3.20. Let π i , σ i and τ i be as in Corollary 3.22 for H " G i . Define We first check that this is well-defined. Indeed, if hG sphq i " gG spgq i , then rphq " rpgq and h´1g P G i . Hence π i pspgqq " π i psph´1gqq " π i prph´1gqq " π i psphqq, and so σ i pπ i pspgqqq and σ i pπ i psphqqq are the same. As τ i pxq has source σ i pπ i pxqq for all x P G p0q i , this implies that both τ i pspgqq and τ i psphqq have the same source. As moreover g and h have the same range, the elements gτ i pspgqq and hτ i psphqq of G have the same source and range and are therefore the same as G is principal. Having seen that Ψ 0 is well-defined, equivariance of Ψ 0 is straightforward. The fact that Ψ 0 pU q P U for all U P N 0 follows as if we write U " gG spgq i , then Ψ 0 pU q " gτ i pspgqq, To see that Ψ 0 isétale, let gG spgq i be an element of N 0 , and and let B be a clopen bisection of g in G such that the set thG Using that both s and τ i areétale, we have that spτ i pBqq is open, and therefore that Ψ 0 pBq is open. We claim that the map is a local inverse to Ψ 0 ; as it is continuous, this will suffice to complete the proof. Indeed, for any h P B, On the other hand, for h P Ψ 0 pBq, as h is in the image of Ψ 0 pBq, we have that sphq is in the image of σ i , and therefore that σ i pπ i psphqqq " sphq, and so τ i psphqq " sphq. Hence Ψ 0 pκphqq " hτ i psphqq " h and we are done with showing that Ψ 0 isétale.
To see that Ψ n is well-defined, we need to check that if pU 0 , ..., U n q is in N n , then pΨ 0 pU 0 q, ..., Ψ 0 pU n qq is in EG KK´1 n . Write g j :" Ψ 0 pU j q for notational simplicity, so g j is in U j by the properties of Ψ 0 . Let h be an element of U 0 X¨¨¨X U n . Then for all j, the fact that the colouring is K-bounded implies that g´1 j h is in K for each j. Hence for any i, j, g´1 i g j " g´1hh´1g j P KK´1, completing the proof that Ψ n is well-defined. The facts that Ψ n isétale and equivariant follow from the corresponding properties for Ψ 0 , so we are done.
3.4. Anti-Čech homology. In this subsection, we show that the Crainic-Moerdijk-Matui homology groups H˚pGq can be realised by a direct limit of homology groups of appropriate colourings.
The key definition is as follows. we have that ι pmq pU q Ě U , and moreover so that for any compact open subset K of G, there exists m K such that for all m ě m K and all U P N pm´1q 0 , we have that ι pmq pU q Ě U K.
Definition 3.24. Let A " pC m q 8 m"1 be an anti-Čech sequence for G, with associated sequence of morphisms ι pmq : N pm´1q Ñ N pmq . We define the homology of A, denoted H˚pAq, to be the corresponding direct limit of the sequence of maps pι pmq : H˚pC m´1 q Ñ H˚pC m qq 8 m"1 . Anti-Čech sequences always exist under the assumptions that G is principal and σ-compact. This follows from the next three lemmas.
Lemma 3.25. Assume that G is principal, and that K is a compact open subset of G that contains G 0 and that satisfies K " K´1. Let G 0 , ..., G d and H 0 , ..., H e be colourings of G with associated nerves N˚and M˚respectively. Assume moreover that the colouring G 0 , ..., G d is K-bounded, and that the colouring H 0 , ..., H e is K 3 -Lebesgue.
Then there exists an equivariantétale map ι 0 : N 0 Ñ M 0 with the following properties: (i) ι 0 pU q Ě U K for all U P N 0 ; (ii) for all n, the map Proof. Using Lemma 3.20, there is an equivariantétale map Ψ 0 : N 0 Ñ G such that Ψ 0 pU q P U for all U P N 0 , and such that for all n, the prescription Ψ n : N n Ñ EG K 3 n`1 , pU 0 , ..., U n q Þ Ñ pΨ 0 pU 0 q, ..., Ψ 0 pU n qq gives gives a well-defined equivariantétale map (Lemma 3.20 has KK´1 in place of K 3 , but note that our assumptions imply that K 3 contains KK´1). Using Lemma 3.19, there is an equivariant local homeomorphism Φ 0 : G Ñ M 0 such that gK 3 Ď Φ 0 pgq for all g P G, and so that the prescription Φ n : EG K 3 n Ñ N n , pg 0 , ..., g n q Þ Ñ pΦ 0 pg 0 q, ..., Φ 0 pg n qq is a well-defined equivariant local homeomorphism.
Define then ι 0 :" Φ 0˝Ψ0 , and note that ι n :" Φ n˝Ψn for all n. Each ι n is then a well-defined equivariantétale map. Moreover, fix U P N 0 and write U " gG spgq i . As Ψ 0 pU q P U , we may write Ψ 0 pU q " gh for some h P G spgq i . Then h´1 is in G i , and so in K as each G i is a subset of K. Hence for an arbitrary element gk of gG spgq i with k P G i Ď K´1, we have that gk " ghh´1k P Ψ 0 pU qK 2 . As gk was an arbitrary element of U , this gives that U Ď Ψ 0 pU qK 2 , and so U K Ď Ψ 0 pU qK 3 . Using the properties of Ψ 0 and Φ 0 , we thus get that Proof. Say the elements of r´1pxq X K are g 0 " x, g 1 , ..., g n . As G is principal, we have spg i q ‰ spg j q for all i ‰ j. Hence for each g i , we may choose a clopen bisection D i containing g i such that spD i q X spD j q " ∅ for i ‰ j, so that r| Di is a homeomorphism, and so that D 0 is contained in G 0 .
Set C 0 :" Ş n i"0 rpB i q, which is a clopen set containing x, and for each i, set C i :" B i X r´1pC 0 q, which is a clopen set containing g i . The set r´1pC 0 q X K is compact. We may thus write it as where each E j is a clopen bisection such that r| Ej is a homeomorphism, and so that each E j does not intersect r´1px 0 q. Set and set B i :" r´1pB 0 q X C i . These sets have the right properties. Proof. Fix x P X, and let B 0 , ..., B n be a collection of sets with the properties in Lemma 3.26. For each i, let ρ i : B 0 Ñ B i be the inverse of r| Bi . Let P " t0, ..., nu 2 be the pair groupoid on the set t0, ..., nu, and define It is not difficult to check that f is a homeomorphism onto its image, which is a compact open subgroupoid of G. Write G x for the image of f . Moreover, by construction we have that for every y P G 0 x , the set r´1pyq X K is contained in G x . The collection tG 0 x | x P G 0 u is an open cover of G 0 , and thus has a finite subcover. Let G 0 , ..., G d be the collection of compact open subgroupoids of G whose base spaces appear in this subcover. This collection has the right properties.
Corollary 3.28. For any σ-compact principal G with compact base space, an anti-Cech sequence exists.
Proof. As G is σ-compact, there is a sequence L 0 Ď L 1 Ď¨¨¨of compact open subsets of G such that each L n equals L´1 n and contains G 0 , and such that any compact subset of G is eventually contained in all of the L n . Set K 0 " L 0 . Lemma 3.27 implies that there is a K 3 0 -Lebesgue collection of compact open subgroupoids of G, say G 0 , ..., G d , with associated nerve space N p0q . As this collection is finite, there exists some compact open subset M 0 of G that contains all of G 0 , ..., G d , and with the properties there. Now let M 1 be a compact open subset of G such that M 1 " M´1 1 , and that contains all the groupoids from this new colouring. Set K 2 :" K 1 Y L 2 Y M 1 and use Lemma 3.27 to build a K 3 2 -Lebesgue covering, and Lemma 3.25 to build a map ι p2q from N p1q to the associated nerve N p2q with the properties in that lemma. Iterating this process builds an anti-Čech sequence as desired.
Our main goal in this subsection is to prove the following theorem.
Theorem 3.29. Let G be principal and σ-compact, with compact base space. Let A be an anti-Čech sequence for G. Then the homology groups H˚pAq and H˚pGq are isomorphic.
The proof will proceed by some lemmas. First, we need a definition. (i) D˚is a nerve N˚, and for all x P C 0 there exists g P G such that αpxq and βpxq are both subsets of gK; (ii) D˚is of for the form EG L , and for all x P C 0 , αpxq´1βpxq is in K. Proof. Let K be as in the definition of closeness for α and β, and assume also that K is so large that the colouring underlying N pmq is K-bounded. Let l ě m be large enough so that if ι : N pmq Ñ N plq is the composition of the morphisms in the definition of the anti-Čech sequence, then for all U P N pmq , we have that ιpU q Ě U K 2 (such an l exists by definition of an anti-Čech sequence). It will suffice to show that ι˝α and ι˝β induce the same map H˚pCq Ñ H˚pN plq q.
Let now x be a point in C n for some n. For each j P t0, ..., nu let π j : C n Ñ C 0 be the map corresponding under the semi-simplicial structure to the map t0u Ñ t0, ..., nu that sends 0 to j (see Section 2.2 for notation). Define x j :" π j pxq. We claim that the intersection is non-empty. Indeed, pαpx 0 q, ..., αpx n qq is a point of N plq n , whence there is some g α in the intersection Ş n j"0 αpx j q, and similarly for g β with β replacing α. As α and β are close with respect to K, we have that αpx 0 q and βpx 0 q are both subsets of gK for some g P G, whence there are k α and k β in K such that g α " gk α and g β " gk β . Hence g α " g β k β k´1 α , so in particular g α is in g β K 2 . Now, by choice of ι, ιpβpx j qq Ě βpx j qK 2 for all j, whence g α is in ιpβpx j qq for all j. Moreover, g α P αpx j q Ď ιpαpx j qq for all j, so g α is a point in the claimed intersection.
For each n and each i P t0, ..., nu, we define a map For the next lemma, let K 0 Ď K 1 Ď K 2 ... be a sequence of compact open subsets of G, all of which contain G 0 , and whose union is G. We then get a sequence pEG Km q 8 m"0 of spaces. Note moreover that the corresponding limit lim Ñ H˚pG Kn q canonically identifies with H˚pGq: indeed, this follows directly from the observation that for each n, EG n is the increasing union of the EG Km n , and the fact that taking homology groups commutes with direct limits. Proof. The proof is very similar to that of Lemma 3.31. We leave the details to the reader.
Proof of Theorem 3.29. Let A be the given anti-Čech sequence with associated nerves and morphisms ι pmq : N pm´1q Ñ N pmq . Let m 1 " 1. Then the colouring underlying N p1q is K-bounded for some compact open subset K of G, which we may assume contains G 0 , and that satisfies K " K´1. Set K 1 :" K 2 . Then Lemma 3.20 gives a morphism Ψ p1q : N pm1q Ñ G K1 . On the other hand, by definition of an anti-Čech sequence there is m 2 ą m 1 such that N pm2q is K 1 -Lebesgue, whence Lemma 3.19 gives a morphism Φ p1q : G K1 Ñ N pm2q . Continuing, the colouring underlying N pm2q is K-bounded for some compact open subset K of G that we may assume contains K 1 and satisfies K " K´1. Set K 2 :" K 2 , so Lemma 3.20 gives a morphism Φ p2q : N pm2q Ñ G K2 . Continuing in this way, we get sequences 1 " m 1 ă m 2 ă m 3 ă¨¨¨of natural numbers and K 1 Ď K 2 Ď¨¨¨of compact open subsets of G together with morphisms We may fill in horizontal arrows in the diagram: on the top row, these should be appropriate compositions of the morphisms ι pmq coming from the definition of an anti-Čech sequence, while on the bottom row they should be induced by the canonical inclusions EG K k n Ñ EG K k`1 n coming from the fact that K k Ď K k`1 for all k. We thus get a (non-commutative!) diagram Notice that the limit of the horizontal maps in the top row is H˚pAq. Moreover, the definition of an anti-Čech sequence and the construction of the sequence pK k q forces G " Ť K k , so the limit of the horizontal maps on the bottom row is H˚pGq. Now, consider the compositions (6) H˚pEG where the second arrow is the canonical one that exists by definition of the direct limit. Any two morphisms into any N pm k q are close using that all the colourings are bounded. It follows therefore from Lemma 3.31 that for any k, the diagram H˚pAq H˚pAq commutes; here the vertical maps are the ones in line (6), and the bottom horizontal line is induced by the canonical inclusion EG K k Ñ EG K k`1 . Taking the limit in k of the maps in line (6), we thus get a well-defined homomorphism Φ : H˚pGq Ñ H˚pAq.
Precisely analogously, using Lemma 3.32 in place of Lemma 3.31, we get a homomorphism Ψ : H˚pAq Ñ H˚pGq. We claim that Φ and Ψ are mutually inverse, which will complete the proof. Indeed, for any k, the triangles appearing in line (5) commute up to closeness, whence the claim follows directly from Lemmas 3.31 and 3.32, so we are done.
3.5. Dynamic asymptotic dimension. In this subsection, we show that the homology of an ample σ-compact principal groupoid vanishes above its dynamic asymptotic dimension, and also that the top-dimensional homology group is torsion free.
The following definition is [ Here is our key use of dynamic asymptotic dimension. For compactness, we first claim that G i has compact closure. Indeed, say k 1 ...k n is an element of G i , where each k j is in tg P K | spgq, rpgq P W i u. Then by definition of W i , for each j P t1, ..., nu there is an element h j of K such that h j´1 k j h´1 j has range and source in U i . Hence for each j P t1, ..., nu, h j´1 k j h´1 j is in tg P K 3 | rpgq, spgq P U i u. Let H i be the subgroupoid of G generated by tg P K 3 | rpgq, spgq P U i u, so H i has compact closure by choice of the cover U 0 , ..., U d . Moreover, We now have that G i is contained in KH i K. However, KH i K is relatively compact by Lemma 3.34, so we see that G i is also relatively compact. We next claim that there is N P N such that any element of G i can be written as a product of at most N elements of tg P K | spgq, rpgq P W i u. Indeed, from [24,Lemma 8.10] there exists N P N such that each range fibre G x i has at most N elements. Now, let g " k 1¨¨¨kn be an element of G, where each k j is in K. Consider the 'path' k 1 , k 1 k 2 , . . . , k 1¨¨¨kn . If this path contains any repetitions, we may shorten it by just omitting all the elements in between. Hence the length n of this path can be assumed to be at most the number of elements of G rpgq , which is N as claimed.
To complete the proof that G i is compact, it now suffices to show that it is closed. For this, let pg j q be a net of elements of G i . Using the previous claim, we may write g j " k j 1 ...k j N , where each k i j is in tg P K | rpgq, spgq P W i u (possibly identity elements). Note that the latter set is compact, whence up to passing to subnets, we may assume that each net pk j i q converges to some k i P tg P K | rpgq, spgq P W i u.
Finally, we check that the colouring is K-Lebesgue. Note that for any x P G 0 , x is contained in V i for some i, whence spr´1pxq X Kq is contained in W i by definition of W i . Hence G i contains r´1pxq X K, giving the K-Lebesgue condition.
Theorem 3.36. Let G be a σ-compact ample groupoid. If G is principal and has dynamic asymptotic dimension at most d, then H n pGq " 0 for n ą d, and H d pGq is torsion-free.
Proof. We first prove the result under the additional assumption that G 0 is compact. In that case Lemma 3.35 lets us build an anti-Čech sequence A for G consisting of colourings pC pnq q each of which only has d`1 colours. As H˚pAq " lim nÑ8 H˚pC pnq q and as Theorem 3.17 implies that H n pC pmq q " 0 for all m and all n ą d, we see that H n pAq " 0 for all n ą d. Theorem 3.29 now gives the vanishing result.
For the claim that H d pGq is torsion-free result, keep notation as above, and let us write N pnq for the nerve space associated to C pnq . Then we have where the first equality is just by definition of H d pAq, the second is by definition of H d pC pnq q, the third is Theorem 3.17, and the fourth follows as taking homology commutes with direct limits. Now, using Lemma 2.5, ZrpN Let us now assume that G 0 is only locally compact. In this case use σ-compactness to write G 0 as an increasing union G 0 " Ť mPN K m of compact open subsets. Let G m :" tg P G | spgq, rpgq P K m u denote the restriction of G to K m . Note that each G m is a σ-compact principal ample groupoid in its own right and it is straightforward to check from the definition (see [24,Definition 5.1]) that the dynamic asymptotic dimension of each G m is dominated by the dynamic asymptotic dimension of the ambient groupoid G. Then G " Ť mPN G m and the inclusion maps G m ãÑ G induce isomorphisms H n pGqlim mÑ8 H n pG m q for each n (see for example [17,Proposition 4.7]). Applying the compact unit space case from above for each G m and passing to the limit, we obtain both the vanishing and the torsion-freeness results for G.

The one-dimensional comparison map and the HK-conjecture
In this section our first goal is to construct a canonical comparison map for an arbitrary ample groupoid G, extending earlier constructions of Matui under additional restrictions on the structure of G (see [34,Corollary 7.15] and [35,Theorem 5.2]). In fact we will provide two different constructions. The first approach is based on relative homology and K-theory groups, and is quite explicit and elementary; this is carried out in Subsection 4.1. The second (suggested by a referee) has its origins in the triangulated category perspective on groupoid equivariant Kasparov theory as recently exploited by Proietti and Yamashita [44,Corollary 4.4]; it is carried out in Subsection 4.2. The two approaches turn out to be equivalent, but this requires some fairly lengthy computations; these are carried out in Subsection 4.3. 5 In fact, it is even free: it can be regarded as a commutative ring (with pointwise multiplication) that is generated by idempotents, and hence its underlying group is free abelian by [4, Theorem 1.1].
In the remaining Subsection 4.4, we apply these results and our vanishing results from Theorem 3.36 to deduce a general theorem on validity of the HK conjecture for low-dimensional groupoids. The result is actually more precise than this: it shows not only that the HK conjecture is true, but identifies the isomorphisms with the comparison maps we have constructed.
4.1. Comparison maps from relative K-theory. In this subsection, we give an elementary construction of a map µ 1 : H 1 pGq Ñ K 1 pCr pGqq that works for any ample groupoid. This is based on two ingredients: a suitable description of the K-theory of the mapping cone of the inclusion ι : C 0 pG 0 q ãÑ Cr pGq, and a relative version of groupoid homology with respect to an open subgroupoid.
Let us start with our description of the K 0 -group of a mapping cone, which is inspired by results of Putnam in [45]. Recently, Haslehurst has independently developed a similar description to ours [25]. Neither Putnam's nor Haslehurst's constructions are quite the same as ours, but the latter in particular has substantial overlap. We will thus omit some details below that can be filled in using techniques from Haslehurst's paper [25,Section 3]; full details can also be found in the first version of the current paper on the arXiv.
For a C˚-algebra (or Hilbert module) B, define IB :" Cpr0, 1s, Bq, and recall that the mapping cone of a˚-homomorphism φ : A Ñ B is (7) Cpφq :" tpa 0 , bq P A ' IB | φpa 0 q " bp0q, bp1q " 0u. The first coordinate projection e 0 : A ' IB Ñ A restricts to a surjection Cpφq Ñ A with kernel ΣB :" C 0 pp0, 1q, Bq. This induces a six-term exact sequence in K-theory Our first aim is to give a different picture of the relative K 0 -group. In what follows, when φ : A Ñ B is a˚-homomorphism, we abuse notation by letting φ also denote the induced homomorphism M n pAq Ñ M n pBq for n P N. equipped with the inductive limit topology. Let " h denote homotopy in V 8 pA, B; φq, 6 and let « be the equivalence relation on V 8 pA, B; φq defined as follows: pp, v, qq « pp 1 , v 1 , q 1 q exactly when there are projections r, r 1 P M n pAq such that (10) pp, v, qq ' pr, φprq, rq " h pp 1 , v 1 , q 1 q ' pr 1 , φpr 1 q, r 1 q.
In the special case where φ is an inclusion of a C˚-subalgebra, we simply write V 8 pA, Bq instead of V 8 pA, B; φq. Moreover, if pp, v, qq P V 8 pA, Bq then p " vvå nd q " v˚v. Hence we will simply denote the elements of V 8 pA, Bq by v.
We equip V 8 pA, B; φq{« with the block sum operation defined by it is straightforward to check that this makes V 8 pA, B; φq{ « into a well-defined monoid with identity r0, 0, 0s.
Our goal is to construct an isomorphism η : K˚pA, B; φq Ñ V 8 pA, B; φq{ « when φ is non-degenerate, and A has an approximate unit of projections; we will apply our construction to the canonical˚-homomorphism ι : C 0 pG 0 q Ñ Cr pGq for an ample groupoid G, so these assumptions are satisfied. For this, we need an ancillary construction.
With Cpφq as in line (7), we obtain a short exact sequence is an isomorphism. We let U n pAq denote the unitary group of M n pAq for a unital C˚-algebra A, and let V pAq denote the the semigroup of equivalence classes of projections in Ť nPN M n pAq, so that K 0 pAq is the Grothendieck group of V pAq whenever A is unital, or more generally, when A has an approximate unit consisting of projections.
Assume now that A is a C˚-algebra with an approximate unit consisting of projections, and let φ : A Ñ B be a non-degenerate˚-homomorphism; note then that B and so Dpφq also have approximate units consisting of projections, so in particular K 0 pDpφqq is the Grothendieck group of V pDpφqq. Let pp 0 , p, p 1 q P M n pDpφqq be a projection representing a K-theory class. Using for example [27, Corollary 4.1.8], there is a continuous path of unitaries pu t q tPr0,1s in M n pBq such that p t " ut p 0 u t " ut φpp 0 qu t , for t P r0, 1s and u 0 " 1. We define η : V pDpφqq Ñ V 8 pA, B; φq by the formula (14) ηpp 0 , p, p 1 q :" pp 0 , φpp 0 qu 1 , p 1 q.
Here then is our promised picture of relative K-theory. Due to similarities to Haslehurst's methods [25], we do not give a proof here: the interested reader can find a complete proof in the original arXiv version of this paper. Theorem 4.3. Let φ : A Ñ B be a non-degenerate˚-homomorphism, where A has an approximate unit consisting of projections.
In particular, V 8 pA, B; φq{« is an abelian group that is isomorphic to K 0 pA, B; φq.
Moreover, the six term exact sequence from line (9) identifies with an exact sequence where the map K 1 pBq Ñ V8pA,B;φq « is given by rus 1 Þ Ñ rp b 1 Mn , u, p b 1 Mn s for u P U n pφppqBφppqq (where p P A is a projection); and V8pA,B;φq « Ñ K 0 pAq is given by rp, v, qs Þ Ñ rps 0´r qs 0 .
Remark 4.4. We note the following basic properties of cycles pp 1 , v 1 , q 1 q and pp 2 , v 2 , q 2 q in V 8 pA, B; φq; these can be justified using the same sort of rotation homotopies that establish the analogous properties for the classical K 0 and K 1 groups.
(i) If p 1 p 2 " q 1 q 2 " 0, then in V 8 pA, B; φq{« we have that Let us now turn to the second ingredient in the construction of the comparison map µ 1 : H 1 pGq Ñ K 1 pCr pGqq: relative groupoid homology. To define it, we return to Matui's picture of groupoid homology as it allows for an elementary description. For an open subgroupoid H Ď G the canonical inclusion induces for each n ě 0 a short exact sequence of abelian groups (15) 0 Ñ ZrH pnq s Ñ ZrG pnq s Ñ ZrG pnq s{ZrH pnq s Ñ 0.
Let Z n rG, Hs denote the quotient group. One easily checks that ZrH pnq s is invariant under the boundary maps B n for ZrG pnq s, and hence we obtain induced maps B 1 n turning pZrG, Hs, B 1 n q into a chain complex such that the sequence (15) is an exact sequence of chain complexes. Definition 4.5. Let G be an ample groupoid. The relative homology of G with respect to an open subgroupoid H is defined as the homology of the chain complex pZrG, Hs, B 1 n q, i.e. H n pG, Hq :" kerpB 1 n q{impB 1 n`1 q. From the short exact sequence (15) we obtain a long exact sequence of homology groups¨¨¨Ñ Important for us is the special case where H " G 0 : there one easily checks that H 0 pG, G 0 q " 0, and H 1 pG, G 0 q -ZrGs{impB 2 q. We are now ready to put everything together and construct the map µ 1 : H 1 pGq Ñ K 1 pCr pGqq. Lemma 4.6. Let G be an ample groupoid. Then there exists a well-defined homomorphismρ : ZrGs Ñ V 8 pC 0 pG 0 q, Cr pGqq{« determined byρp1 W q " r1W s andρp´1 W q " r1 W s for every compact open bisection W Ď G. Moreover, impB 2 q Ď kerpρq and henceρ factors through a well-defined homomorphism ρ : H 1 pG, G 0 q -ZrGs{impB 2 q Ñ V 8 pCpG 0 q, Cr pGqq{« .
Proof. As G is an ample groupoid, every function f P ZrGs can be written (nonuniquely) as a linear combination f " ř λ i 1 Wi where λ 1 , . . . , λ n P Z and W 1 , . . . , W n are pairwise disjoint compact open bisections of G. We have to show that the resulting class of the partial isometry à With this in mind we can establish the lemma: Let f P ZrGs. We may assume that f ě 0. Suppose we have f " ř λ i 1 Ui " ř µ j 1 Vj for λ i , µ j P N and two families pU i q i and pV j q j of pairwise disjoint compact open bisections of G. Since G is ample we may choose a common refinement of these two families by compact open bisections pW k q k . Let η k be equal to λ Similarly, we obtain À k η k 1 W k « À i λ i 1 Ui , from which we conclude thatρ is indeed well-defined. It is a group homomorphism by construction.
For the second part let U and V be compact open bisections of G such that spU q " rpV q. Then we haveρpB 2 p1 pUˆV qXG p2q qq "ρp1 U´1UV`1V q " 0 since Proof. Let ι : C 0 pG 0 q Ñ Cr pGq denote the canonical inclusion, which is nondegenerate. As G is ample, C 0 pG 0 q contains an approximate identity of projections. By Theorem 4.3 and Lemma 4.6 we obtain a commutative diagram with exact rows (16) 0 where the top row is part of the long exact sequence for the pair pG, G 0 q. By exactness, there exists a unique homomorphism µ 1 : H 1 pGq Ñ K 1 pCr pGqq filling in the dashed arrow such that the diagram commutes.

4.2.
Comparison maps from the ABC spectral sequence. In this subsection we recall the spectral sequence constructed by Proietti and Yamashita in [44, Corollary 4.4] (which is in turn based on the ABC spectral sequence of Meyer [37]), and show that this gives rise to canonical comparison maps µ k : H k pGq Ñ K k pCr pGqq for k P t0, 1u. This approach to the construction of comparison maps was suggested by the referee. Throughout this section, G is a second countable ample groupoid. We will need to work extensively with G-C˚-algebras and the G-equivariant Kasparov category KK G : see [31] for background. We will also need to treat KK G as a triangulated category as described in [44,Section A.4]. For background on the material we will need about triangulated categories and homological ideals and functors, see [38].
Let I be the homological ideal (see [ where for each p P N, P p :" L p`1 pAq and Moreover, one computes from the explicit description of Ind G G 0 in [7, Section 2.1] that L p`1 pAq identifies canonically with C 0 pG pp`1q q, where we recall that G ppq :" tpg 1 , ..., g p q P G p | spg i q " rpg i`1 q for all i P t1, ..., n´1uu and we write G p0q and G 0 interchangeably. Here and throughout, we equip G ppq with the left G-action induced by (19) g : ph 1 , h 2 , ..., h p q Þ Ñ pgh 1 , h 2 , ..., h p q and C 0 pG ppq q with the corresponding G-action. We will have need of the following definition and lemma multiple times. The corresponding completion is a Hilbert C 0 pXq-module that we will denote L 2 pY, pq.
Let now A be a G-C˚-algebra, and let r˚A be the pullback along r : G Ñ G 0 (see for example [29, 2.7(d)]). The reduced crossed product of A by G, denoted A¸r G, is defined to be a certain completion of C c pGq¨r˚A, where the latter is equipped with a natural˚-algebra structure: we refer the reader to [29,Section 3] for more details. For an element f P C c pGq¨r˚A and h P G, we write f phq for the corresponding element of pr˚Aq h " A rphq . In the special case A " C 0 pG ppq q with the left G-action from line (19), we see that f phq identifies with an element of C 0 pG rphq sˆr G pp´1q q. The next lemma is well-known and will be used by us several times below: it follows from direct checks based on the formulas we give that we leave to the reader. Lemma 4.9. For each p ě 2, let pr : G ppq Ñ G pp´1q be the map pg 1 , ..., g p q Þ Ñ pg 2 , ..., g p q, and let pr : G p1q Ñ G p0q " G 0 be the map g Þ Ñ spgq. For each p ě 1, f P C c pGq¨r˚C 0 pG ppq q, ξ P L 2 pG ppq , prq, and g P G define pκ p pf qξqpg 1 , ..., g p q :" ÿ hPG rpg 1 q f phqpg 1 , ..., g p qξph´1g 1 , g 2 , ..., g p q. 7 The function xξ, ηy is well-defined as the sum is finite; it is continuous as p isétale; and it is compactly supported as it is supported in ppsupppξq X supppηqq.

This extends to an injective representation
κ p : C 0 pG ppq q¸r G Ñ LpL 2 pG ppq , prqq whose image is exactly KpL 2 pG ppq , prqq.
In particular, the Kasparov class rκ p , L 2 pG ppq , prq, 0s P KKpC 0 pG ppq q¸r G, C 0 pG pp´1q qq is an isomorphism, induced by a canonical Morita equivalence C 0 pG ppq q¸r G M " C 0 pG pp´1q q.
We will need to work with a phantom tower in the sense of [37,Definition 3.1] associated to the I-projective resolution in line (17) above. This is an augmentation of that resolution to a diagram of the form (20) Aι satisfying certain conditions: we refer the reader to [37] for more details. Note that as in [37] we are using circled arrows for morphisms of degree one and plain arrows for morphisms of degree zero, but the circles are not in the same place as those of [37,Definition 3.1]. This is because (following [44]), we are working with even I-projective resolutions, whereas in [37, Definition 3.1], Meyer works with odd resolutions, where the maps δ p in line (17) have degree one (compare [44,Definition 2.15]). Analogously to the explanation below [44,Definition 2.15], it is straightforward to switch between these two degree conventions for phantom towers: if Σ : KK G Ñ KK G is the suspension automorphism, and we are given a phantom tower associated to an odd I-projective resolution with objects N p and P p as in [37,Definition 3.1], replacing these by Σ´pN p and Σ´pP p yields a phantom tower associated to the corresponding even resolution as in diagram (20) above. Now, following [37, Lemma 3.2], the I-projective resolution in line (17) can be augmented in an essentially unique way to a phantom tower as in line (20). Let J : KK G Ñ Ab Z{2 be the functor from KK G to the category of Z{2-graded abelian groups defined by J :" K˚˝j G , where j G : KK G Ñ KK is descent (see for example [7, Section 1.3]), and K˚: KK Ñ Ab Z{2 is K-theory; note that on objects, JpAq :" K˚pA¸r Gq. This is a stable homological functor (see [38,Definitions 2.12 and 2.14]), so we may use it to build the ABC spectral sequence of [37,Sections 4 and 5] based on the phantom tower from line (20). If G is amenable and has torsionfree isotropy groups, [44,Corollary 4.4] shows that the ABC spectral sequence for A " C 0 pG 0 q is a convergent spectral sequence (21) E r p,q ñ K p`q pCr pGqq. We call this the Proietti-Yamashita (or PY) spectral sequence.
As we will need to do explicit computations, let us give more details about how the general construction of the ABC spectral sequence specialises to our case, following [37,Sections 4 and 5]. For each p, q P N as in [37, page 189], define 8 E 1 p,q :" K q pP p¸r Gq -" K 0 pC 0 pG ppq qq q even 0 q odd 8 Note that the conventions of [37, Sections 4 and 5] would use "K p`q " where we have "Kq" in the definitions of E 1 pq and of D 1 pq ; the difference comes from our choice of using even projective resolutions -which necessitates introducing p-fold desuspensions -as explained above where the isomorphism uses the canonical Morita equivalence P p¸r G " C 0 pG pp`1q q¸r G M " C 0 pG ppq q from Lemma 4.9. Define also D 1 p,q :" K q pN p`1¸r Gq. Continuing following [37, page 189], there is an exact couple (see for example [54,Section 5.9] for background) defined by j pq :" J q pφ p q, and k pq : E 1 p,q Ñ D 1 p´1,q defined by k pq :" J q pψ p q. As in [44,Corollary 4.4], one computes that the second page E 2 has entries given by H p pGq q even 0 q odd .
With notation as in [37, page 172], [37,Theorem 5.1] implies that the filtration on K p`q pCr pGqq that corresponds to the convergence in line (21)  0 should be interpreted as the identity map on A). Now, taking p " 0, q " 0, and r " 2 we get a canonical map (25) µ 0 : H 0 pGq " E 2 0,0 Ñ J 0 : I 1 pC 0 pG 0 qq J 0 : I 0 pC 0 pG 0 qq , which is induced exactly by Jpδ 0 q. Note however that by definition J 0 : I 0 pC 0 pG 0 qq " 0, and by definition of J, J 0 : I 1 pC 0 pG 0 qq is a subgroup of K 0 pCr pGqq. Hence we may identify µ 0 with the map µ 0 : H 0 pGq Ñ K 0 pCr pGqq induced by δ 0 P KK G pC 0 pGq, C 0 pG 0 qq. Let us record this more explicitly in a lemma.
Lemma 4.10. Let δ 0 P KK G pC 0 pGq, C 0 pG 0 qq " KK G pLpC 0 pG 0 qq, C 0 pG 0 qq be the co-unit of adjunction. Let rX G s P KKpC 0 pG 0 q, C 0 pGq¸rGq be the KK class arising from the canonical Morita equivalence C 0 pG 0 q M " C 0 pGq¸r G of Lemma 4.9. Then there is a canonical homomorphism µ 0 : H 0 pGq Ñ K 0 pCr pGqq 9 As mentioned by Meyer on [37, page 193], this map is functorially determined by A, and does not depend on any of the choices of projective resolution or phantom tower involved. defined by taking the composition J 0 pδ 0 q˝K 0 prX G sq : K 0 pC 0 pG 0 qq Ñ K 0 pCr pGqq, then identifying K 0 pC 0 pG 0 qq " ZrG 0 s, and noting that this map descends to the quotient H 0 pGq of ZrG 0 s.
On the other hand, taking p " 1, q " 0, and r " 2 in lines (23) and (24), we get a canonical map (26) µ 1 : H 1 pGq " E 2 1,0 Ñ J 1 : I 2 pC 0 pG 0 qq J 1 : I 1 pC 0 pG 0 qq induced by Jpψ 1 q and taking image in J 1 pι 1 0 q`J 1 pAq˘Ď J 0 pN 1 q, which is isomorphic to the right hand side in line (26) above. Note that by definition (compare line (22)), the group J 1 : I 1 pC 0 pG 0 qq is the kernel of the map J 1 pι 1 0 q : J 1 pAq Ñ J 0 pN 1 q. However, thanks to exactness of the first triangle (27) Aι (20) and the fact that J is homological, we have an exact sequence J 1 pP 0 q Jpψ0q / / J 1 pAq The group J 1 pP 0 q is by definition equal to K 1 pC 0 pGq¸r Gq, which is isomorphic to K 1 pC 0 pG 0 qq by the Morita equivalence from Lemma 4.9, so is zero as G is ample. Hence by exactness, J 1 pι 1 0 q is injective, and so J 1 : I 1 pC 0 pG 0 qq " 0. Putting this information into line (26), together with the fact that J 1 : I 2 pC 0 pG 0 qq is naturally a subgroup of K 1 pCr pGqq, we get a canonical map µ 1 : H 1 pGq Ñ K 1 pCr pGqq induced by ψ 1 P KK G pC 0 pG p2q q, N 1 q. Let us again record this more explicitly as a lemma.
Lemma 4.11. Let ψ 1 P KK G pC 0 pG p2q q, N 1 q " KK G pP 1 , N 1 q be the morphism from the following part of a phantom tower for A " C 0 pG 0 q Let J 1 pι 1 0 q| ImpJ1pι 1 0 qq be the corestriction of Jpι 1 0 q to an isomorphism J 1 pAq Ñ ImagepJ 1 pι 1 0 qq Ď J 0 pN 1 q. Let rX G p2q s P KKpC 0 pGq, C 0 pG p2q q¸r Gq be the KK-isomorphism arising from the canonical Morita equivalence C 0 pGq M " C 0 pG p2q q¸r G of Lemma 4.9. Then there is a canonical homomorphism µ 1 : H 1 pGq Ñ K 1 pCr pGqq defined by taking the composition 10 pJ 1 pι 1 0 q| ImpJ1pι 1 0 qq˘´1˝J 0 pψ 1 q˝K 0 prX G p2q sq : KerpJ 0 pδ 1 qq Ñ K 1 pCr pGqq, and noting that this restriction passes to the quotient by the image of J 0 pδ 2 q : K 0 pC 0 pG p2q q¸r Gq Ñ K 0 pC 0 pGq¸r Gq 11 . Remark 4.12. In the context of the HK conjecture, it is natural to ask if there are "higher" maps µ k : H k pGq Ñ K k pCr pGqq arising from the Proietti-Yamashita spectral sequence in a canonical way. This does not seem clear, even for k " 2: here lines (23) and (24) would give a map For k " 0 and k " 1, the denominator on the right hand side turned out to be the zero group; for k " 2, this is no longer clear, so the natural map arising from the spectral sequence could a priori take its image in a proper quotient of K 0 pCr pGqq, not in K 0 pCr pGqq itself. For k " 3, the situation is similar: one has and this map a priori takes values in a quotient of K 1 pCr pGqq (note that we need to increase to r " 3 so the condition "0 ď p ď r" needed to apply line (23) is satisfied; as the differentials on the E 2 page are all zero for degree reasons, we still have an identification H 3 pGq " E 3 3,0 however). For k " 4, the situation seems worse: one has 4,0 could in principle be a proper subquotient of H 4 pGq, so a priori one only gets a map from a subquotient of H 4 pGq to a subquotient of K 0 pCr pGqq; the situation is similar to this for all k ě 4.
The recent principal counterexamples to the HK conjecture of Deeley [13] suggest that the a priori obstructions to the existence of the higher comparison maps discussed above really do pertain; however, we did not yet attempt the relevant computations. Much of what follows is essentially routine "book-keeping" computations; however, as some of it is of quite an involved nature, we thought it was worthwhile to record the details.
The special cases L 2 pG, rq and L 2 pG, sq of Definition 4.8 will be particularly important for us: we introduce the shorthand K r :" KpL 2 pG, rqq and K s :" KpL 2 pG, sqq for the compact operators on these modules. These C˚-algebras are equipped with the canonical G-actions coming from the left action of G on itself for L 2 pG, rq, and the right action of G on itself for L 2 pG, sq. We also write M r : C 0 pGq Ñ K r and M s : C 0 pGq Ñ K s ; the former is equivariant, while the latter is only equivariant if we consider C 0 pGq as a G-C˚-algebra via the right action 10 Note that the restriction to the kernel KerpJ 0 pδ 1 qq of Jpδ 1 q is needed to ensure that the image of J 0 pψ 1 q˝K˚prX G p2q sq is contained in the image of J 1 pι 1 0 q; we mention this fact explicitly as it is slightly buried in the spectral sequence machinery. 11 This last holds as commutativity of the rightmost triangle in diagram (28) and exactness of the second triangle from the right imply that Jpψ 1 q˝Jpδ 2 q " Jpψ 1 q˝Jpφ 2 q˝Jpψ 2 q " 0˝Jpψ 2 q " 0.
of G on itself (however, we will never do this: C 0 pGq always either has the left G-action in what follows, or the trivial action if we have passed through descent). With notation as in Lemma 4.10, µ 0 is the map induced on H 0 pGq by where δ 0 P KK G pC 0 pGq, C 0 pG 0 qq equals the counit of adjunction ǫ C0pG 0 q for the adjoint pair pE, F q discussed at the start of Subsection 4.2. Using the description of this counit in [7, Theorem 2.3], one computes that (30) δ 0 " rM r , L 2 pG, rq, 0s.
For the statement of the next lemma, let ι : C 0 pG 0 q Ñ Cr pGq denote the canonical inclusion. Recall also (compare [7, Section 1.3]) that the descent of a Hilbert G-A-module E is defined to be E¸r G :" E b A A¸r G. To have concrete formulas to work with, let E denote the upper-semicontinuous field of Hilbert modules over G 0 associated with E. Then E¸r G can alternatively be constructed as the completion of the vector space of compactly supported continuous sections Γ c pG; r˚Eq with respect to the A¸r G-valued inner product for ξ, ξ 1 P Γ c pG; r˚Eq and g P G. Moreover, if π : B Ñ LpEq is a G-equivariant representation of B, then for f P Γ c pG; r˚Bq, ξ P Γ c pG; r˚Eq, and g P G, the formula pf¨ξqpgq :" ÿ hPG rpgq π rphq pf phqqW h`ξ ph´1gqd efines a representation π¸r G : B¸r G Ñ LpE¸r Gq. The next lemma follows from direct (if somewhat lengthy) computations that we leave to the reader. Lemma 4.13. For ξ P L 2 pG, sq, f P C c pGq, g P G, and h P G rpgq define pωpξ b f qpgqqphq :" ξphqf ph´1gq.
Then ω extends to a unitary isomorphism of Hilbert Cr pGq-modules ω : L 2 pG, sq b ι Cr pGq -L 2 pG, rq¸r G.
Moreover, for κ 1 as in Lemma 4.9, ω satisfies ωpκ 1 paq b 1 Cr pGq qω˚" pM r¸r Gqpaq for all a P C 0 pGq¸r G. In particular, there is a˚-homomorphism β : K s Ñ K r¸r G defined by β : K s Ñ K r¸r G, k Þ Ñ ωpk b 1 Cr pGq qω˚.
Finally, ω˚pK r¸r Gqω " KpL 2 pG, sq b ι Cr pGqq, and so the element rL 2 pG, sq b ι Cr pGq, ad ω˚˝Mr¸G , 0s P KKpK r¸r G, Cr pGqq is an isomorphism in KK, induced by a Morita equivalence bimodule.
We are now ready to show that the description of µ 0 arising from the spectral sequence agrees with the classical map H 0 pGq Ñ K 0 pCr pGqq introduced by Matui.
Proposition 4.14. The map µ 0 : H 0 pGq Ñ K 0 pCr pGqq of Lemma 4.10 is the map induced by the canonical inclusion ι : C 0 pG 0 q Ñ Cr pGq on the level of K 0 -groups.
Proof. Lemma 4.13 implies that j G pδ 0 q is represented by the class rκ 1 b 1 Cr pGq , L 2 pG, sq b ι Cr pGq, 0s P KKpC 0 pGq¸r G, Cr pGqq, (where we have used the isomorphism K s -C 0 pGq¸r G, which is a special case of Lemma 4.9). Lemma 4.9 implies that rX G s is represented by the opposite of the Morita equivalence Kasparov cycle pκ 1 , L 2 pG, sq, 0q. Hence rX G s b j G pδ 0 q is represented by the tensor product of these, which one computes directly is the cycle pι, Cr pGq, 0q in KKpC 0 pG 0 q, Cr pGqq. The image of this under the K-theory functor is indeed the map induced by ι, so we are done.
We now move on to µ 1 . Let us start by more explicitly describing the exact triangle (31) C 0 pG 0 q˝ι (27) above. Note that the class rid, L 2 pG, rq, 0s P KK G pK r , C 0 pG 0 qq is an isomorphism (as it arises from an equivariant Morita equivalence) and from the formula in line (30) for δ 0 we clearly have rM r , K r , 0s b rid, L 2 pG, rq, 0s " δ 0 , so up to replacing C 0 pG 0 q by K r using the isomorphism rid, L 2 pG, rq, 0s P KK G pK r , C 0 pG 0 qq, we may replace the exact triangle in line (31) above with (32) K rι where M r P KK G pC 0 pGq, K r q is the class corresponding to the˚-homomorphism M r : C 0 pGq Ñ K r , and we have abused notation slightly by keeping the same label for the top horizontal map. The remaining parts of the diagram can be described explicitly in terms of the mapping cone of M r ; we recall this next. Now, by definition of the exact triangles in KK G (see [44,Appendix A.4]), in diagram (32) we may take N 1 " CpM r q, and ι 1 0 and φ 1 are then the KK-classes given by the left and right maps respectively in the canonical short exact sequence Hence we may replace the first part of the phantom tower in line (28) above with the diagram According to the proof of [37, Lemma 3.2], the morphism ψ 1 appearing above is unique, subject to the condition that the right hand triangle commutes (and the various other conditions defining a phantom tower, which are satisfied by the diagram above). Our next task towards computing µ 1 is to give an explicit description of ψ 1 ; for this, it will be helpful to see why M r˝δ1 " 0 (note that this is indeed the case, by definition of an I-projective resolution).
We first find an explicit representation for δ 1 P KK G pC 0 pG p2q q, C 0 pGqq. Let pr 1 : G sˆr G Ñ G be the projection onto the first factor, and define F 0 :" L 2 pG sˆr G, pr 1 q.
Similarly, let pr 2 : G rˆr G Ñ G be the projection on the second factor and define F 1 :" L 2 pG rˆr G, pr 2 q.
Equip F 0 with the G-action defined by the left translation action of G on the first factor, and F 1 with the G-action defined by the diagonal left action of G; note that both F 0 and F 1 are then G-C 0 pGq Hilbert modules, where the action on C 0 pGq is (as usual) induced from the left action of G on itself. Define a representation π 0 : C 0 pG p2q q Ñ LpF 0 q by pointwise multiplication, and a representation π 1 : C 0 pG p2q q Ñ LpF 1 q by pπ 1 pf qξqpg, hq :" f pg, g´1hqξpg, hq; both of these representations are equivariant and take values in the compact operators on the corresponding modules. Using the explicit description of the induction functor from [7, Section 2.1] and of the co-unit of adjunction from [7, Theorem 2.3], one computes that the elements Lpǫ C0pG 0 q q and ǫ LpC0pG 0 qq of KK G pC 0 pG p2q q, C 0 pGqq satisfy ǫ LpC0pG 0 qq " rπ 0 , F 0 , 0s and Lpǫ C0pG 0 q q " rπ 1 , F 1 , 0s; from the formula in line (18), we thus have that (33) δ 1 " rπ 0 , F 0 , 0s´rπ 1 , F 1 , 0s.
On the other hand, from the fact that line (17) is a projective resolution, we have that δ 1 b δ 0 " 0; as we are identifying δ 0 with rM r , K r , 0s P KK G pC 0 pGq, K r q, we therefore see that in KK G pC 0 pG p2q q, K r q. In order to construct ψ 1 and also its image under the descent morphism, we need to make this identity explicit; this is our next task. We may identify L 2 pG, sq b ι Cr pGq with the completion of C c pG sˆr Gq for the inner product valued in the dense subset C c pGq of Cr pGq defined by xξ, ηy s pgq :" ÿ hPG rpgq ÿ kPG sphq ξpk, h´1qηpk, h´1gq.
Given these descriptions, direct checks show that the map G rˆr G Ñ G sˆr G, pg, hq Þ Ñ pg, g´1hq induces a unitary Cr pGq-module isomorphism w : L 2 pG, sq b ι Cr pGq Ñ L 2 pG, rq b ι Cr pGq.
Lemma 4.15. There is a unitary, equivariant, isomorphism Moreover if L 2 pG p2q , prq and κ 2 are as in Lemma 4.9, then there is a commutative diagram where all the maps are unitary isomorphisms. The maps Θ 0 and Θ 1 satisfy respectively. The map w : L 2 pG, sq b ι Cr pGq Ñ L 2 pG, rq b ι Cr pGq is that defined above, and satisfies Proof. We begin by defining a unitary isomorphism of Hilbert C 0 pG 0 q-modules such that U pπ 0 b 1qU˚" π 1 b 1 as follows: a straightforward computation gives identifications (34) F 0 b Mr L 2 pG, rq -L 2 pG p2q , r˝pr 1 q, F 1 b Mr L 2 pG, rq -L 2 pG rˆr G, r˝pr 2 q.
It is then easy to see that the map G rˆr G Ñ G p2q , pg, hq Þ Ñ pg, g´1hq gives rise to the desired equivalence. Now recall, that K r splits as a tensor product L 2 pG, rq b C0pG 0 q L 2 pG, rq op . After making this identification we can define u as We now move on to the commutative diagram. Notice that pF 0 b Mr K r q¸r G b pL 2 pG, rq¸r Gq -pF 0 b Mr L 2 pG, rqq¸r G, and similarly pF 1 b Mr K r q¸r G b pL 2 pG, rq¸r Gq -pF 1 b Mr L 2 pG, rqq¸r G. Applying these identifications to the top row of the diagram in the lemma and using identifications similar to the ones in line (34) in the bottom row (and slightly abusing notation by still denoting the maps Θ 0 and Θ 1 ) shows that it will be enough to exhibit a commutative diagram of the form where W is induced by the map G p2qˆs˝p r,r G Ñ G p2qˆr˝p r,r G, ph 1 , h 2 , gq Þ Ñ ph 1 , h 2 , h´1 2 gq. Similar to the discussion just prior to the present lemma, the modules involved in the diagram all have a canonical dense subspace of compactly supported functions defined on a suitable fibred product of G p2q or G rˆr G with G. Hence it will be enough to describe a "dual" commutative diagram of homeomorphisms The map inducing U¸r G is given by ph 1 , h 2 , gq " ph 1 , h´1 1 h 2 , gq and the map ω inducing W is given by If we set θ 0 ph 1 , h 2 , gq :" ph 1 , h 2 , h 1 h 2 gq and θ 1 ph 1 , h 2 , gq :" ph 1 , h´1 1 h 2 , h 1 gq, then the diagram commutes. It follows that the induced diagram of Hilbert modules commutes. Moreover, direct checks show that Θ 0 intertwines the representations π 0¸r G and κ 2 b1 L 2 pG,sqbιCr pGq and Θ 1 intertwines π 1¸r G and κ 2 b1 L 2 pG,rqbιCr pGq .
Using this lemma, we may finally compute an explicit formula for both ψ 1 and its descent j G pψ 1 q. Define ith the IK r -valued inner product given by adding the componentwise inner products pointwise for each t P r0, 1s. Define F to be the collection of all triples pξ 0 , η, ξ 1 q in the direct sum F 0 ' F K ' F 1 such that 12 pξ 0 b 1 Kr , 0q " ηp0q and ηp1q " p0, ξ 1 b 1 Kr q.
Let DpM r q be as in line (11), and define a DpM r q-valued inner product on F by xpξ p0q 0 , η p0q , ξ irect checks then show that F is a well-defined Hilbert G-DpM r q-module. Now, with u as in Lemma 4.15 consider the unitary which commutes with the direct sum action of G. This is a self-adjoint unitary, so by the spectral theorem we can write V " p´q 13 , where p and q are orthogonal projections that commute with the G-action and satisfy p`q " 1. For t P r0, 1s, define V t " p`e iπt q, so tV t u tPr0,1s is a path of G-invariant unitaries connecting V 0 " 1 and V 1 " V . For f P C 0 pG p2q q, η P F K , and t P r0, 1s, define This defines a G-equivariant representation π K : C 0 pG p2q q Ñ F K . Moreover, it is compatible with the representations π 0 and π 1 of C 0 pG p2q q on F 0 and F 1 respectively, in the sense that for f P C 0 pG p2q q and pξ 0 , η, ξ 1 q P F , the formula π F pf qpξ 0 , η, ξ 1 q :" pπ 0 pf qξ 0 , π K pf qη, π 1 pf qξ 1 q 12 Note that even though Kr is typically non-unital, ξ i b 1 Kr still makes sense as an element of F i b Mr Kr for i P t0, 1u -for example, one can show that if pu j q jPJ is an approximate unit for Kr, then pξ i b u j q jPJ is Cauchy in F i b Mr Kr, and then define ξ i b 1 Kr to be its limit. 13 There are also concrete formulas: defines a Hilbert G-DpM q-module representation π F : C 0 pG p2q q Ñ LpF q. This representation takes values in the compact operators on F , and thus we get a welldefined Kasparov element r ψ :" rπ F , F, 0s P KK G pC 0 pG p2q q, DpM r qq.
The short exact sequence of line (12) and split exactness of KK G -theory then gives a canonical isomorphism (35) KK G pC 0 pG p2q q, DpM r qq -KK G pC 0 pG p2q q, C 0 pGqq ' KK G pC 0 pG p2q q, CpM r qq.
Write ψ for the image of r ψ in KK G pC 0 pG p2q q, CpM r qq under the canonical quotient map arising from the direct sum decomposition above.
Lemma 4.16. The element ψ fits into the canonical phantom tower in KK G as the map labeled ψ 1 As we already noted, the proof of [37,Lemma 3.2] shows that ψ 1 always exists, and is uniquely determined by the property ψ 1 b re 0 s " δ 1 . We thus need to show that ψ b re 0 s " δ 1 .
We have canonical quotient maps f j : DpM r q Ñ C 0 pGq defined by f j : pa 0 , b, a 1 q Þ Ñ a j for j P t0, 1u. Let s : C 0 pGq Ñ DpM r q, a Þ Ñ pa, cpM r paqq, aq be the canonical splitting of the short exact sequence from line (12). Clearly f 0˝s " f 1˝s , whence the map b prf 0 s´rf 1 sq : KK G pC 0 pG p2q q, DpM qq Ñ KK G pC 0 pG p2q q, C 0 pGqq vanishes on s˚pKK G pC 0 pG p2q q, C 0 pGqqq. Let i : CpM r q Ñ DpM r q, pa 0 , bq Þ Ñ pa 0 , b, 0q be the canonical inclusion. Then according to the isomorphism in line (35) we have r ψ " i˚pψq ' s˚pαq for some α P KK G pC 0 pG p2q q, C 0 pGqq, so the abovediscussed vanishing of¨b prf 0 s´rf 1 sq on the image of s˚gives r ψ b prf 0 s´rf 1 sq " i˚pψq b prf 0 s´rf 1 sq " ψ b prf 0˝i s´rf 1˝i sq.
On the other hand, we clearly have f 0˝i " e 0 and f 1˝i " 0, so the above implies that r ψ b prf 0 s´rf 1 sq " ψ b re 0 s It thus suffices to show that r ψ b prf 0 s´rf 1 sq " δ 1 . For this, we note that r ψrf j s " rF j , π j , 0s for j P t0, 1u so we need to show that this is exactly the formula in line (33), so we are done.
Our next goal is to compute the image of the diagram in line (36) under descent. Unfortunately, this necessitates more notation. Let ι : C 0 pG 0 q Ñ Cr pGq denote the canonical inclusion, and let Cpιq and Dpιq be the corresponding C˚-algebras from lines (7) and (11). Define X D to consist of all triples pξ 0 , η, ξ 1 q in L 2 pG, sq ' I`L 2 pG, sq b ι Cr pGq˘' L 2 pG, sq such that ξ i b 1 Cr pGq " ηpiq for i P t0, 1u. Direct checks based on Lemma 4.13 shows that this is canonically a Morita equivalence DpM r q¸r G-Dpιq-bimodule. Similarly, if X C consists of all pairs in L 2 pG, sq ' I`L 2 pG, sq b ι Cr pGq˘such that ξ 0 b1 Cr pGq " ηp0q and ηp1q " 0, we see that X C is a Morita equivalence CpM r q¸rG-Cpιq bimodule. Moreover, if X G is the Morita equivalence C 0 pGq¸r G-C 0 pG 0 q bimodule from Lemma 4.9, then the following diagram (built from the general short exact sequence of line (12)) is easily seen to commute in KK For ease of notation let E s and E r denote the Hilbert Cr pGq-modules L 2 pG, sqb ι Cr pGq and L 2 pG, rq b ι Cr pGq, respectively. Let E be the collection of triples pξ 0 , η, ξ 1 q in such that pξ 0 b 1 Cr pGq , 0q " ηp0q and p0, ξ 1 b 1 Cr pGq q " ηp1q; this is a Hilbert Dpιq-module in the natural way. It is moreover equipped with a left C 0 pGq-action defined as follows. Let w : E s Ñ E r be the unitary isomorphism from Lemma 4.15, and define v :"ˆ0 wẘ 0˙P LpE s ' E r q.
As in the discussion defining r ψ, we may write v " p´q for complementary projections p and q, and define v t :" p`e iπt q. Then the formula (39) π E :" M s ' v t pM s b 1 Cr pGq , 0qvt ' M r defines the desired C 0 pGq-action on E. Define r Ψ :" rπ E , E, 0s P KKpC 0 pGq, Dpιqq, and define Ψ to be the image of r Ψ in KKpC 0 pGq, Cpιqq under the canonical quotient map arising from the direct sum decomposition KKpC 0 pGq, Dpιqq -KKpC 0 pGq, Cpιqq ' KKpC 0 pGq, C 0 pG 0 qq that in turn arises from the split short exact sequence in line (12).
The next lemma is the last main ingredient needed to compute µ 1 . To state it, let ι : ΣCr pGq Ñ Cpιq and e 0 : Cpιq Ñ C 0 pG 0 q be the canonical maps associated to the mapping cone. Let also rss and rrs respectively denote the elements rM s , L 2 pG, sq, 0s and rM r , L 2 pG, rq, 0s of KKpC 0 pGq, C 0 pG 0 qq.
Proof. That the left triangle in line (36) has image equal to the left triangle in line (40) under descent (and modulo the given Morita equivalences) follows from the commutative diagram of short exact sequences in KK We next claim that j G pψq b CpMrq¸r G rX C s " rX G p2q s b C0pGq Ψ. Thanks to the commutative diagram in line (37) above, it suffices to show that This will moreover show that the bottom horizontal arrow in line (40) is correctly labeled, as it is clear that if f 0 , f 1 : Dpιq Ñ C 0 pG 0 q are the canonical -homomorphisms, then r Ψ b prf 0 s´rf 1 sq " rss´rrs; thus to complete the proof it suffices to establish the identity in line (42).
The Kasparov product j G p r ψq b DpMrq¸rG rX D s is represented by the triplè Our first goal is to identify this triple with the triplè where we emphasise that we are using the representationπ E " pM s , IpM s b 1, M r b 1q, M r q (as opposed to π E defined in line (39)).
In fact we will deal with the ambient modules of E and F respectively, which allows us to treat each component separately to improve readability. We first deal with the first and third components. For these we have isomorphisms (43) pF 0¸r Gq b κ1 L 2 pG, sq Ñ L 2 pG p2q , prq b Ms L 2 pG, sq identifying the first and third components of F¸r G b X D and L 2 pG p2q , prq bπ E E, respectively. These isomorphisms are produced in a similar fashion so will only explain the procedure for (44): We apply the isomorphism ω from Lemma 4.13 to the transformation groupoid G rˆr G of the left action of G on itself (in place of G) to obtain an isomorphism where S : G rˆr G Ñ G denotes the source map of G rˆr G given by Spg, hq " g´1h.
Using this we get a chain of identifications A tedious but routine calculation shows that this isomorphism intertwines the actions π 1¸G b 1 and κ 2 b 1. It remains to identify the middle components of F¸rGbX D and L 2 pG p2q , prqbπ E E, respectively. For every t P r0, 1s we have the following chain of isomorphisms, where we use Lemma 4.13 in line 2 and the isomorphisms Θ 0 and Θ 1 from Lemma 4.15 in line 4.
The commutative diagram in Lemma 4.15 together with the construction of the families of unitaries pV t q t and pv t q t imply that pΘ 0 ' Θ 1 qppV t¸r G b 1 L 2 pG,rq¸G q " p1 L 2 pG p2q ,prq b v t qpΘ 0 ' Θ 1 q and hence the chain of isomorphisms above intertwines the representations pV t pπ 0 b 1, 0qVt q¸r G b 1 and κ 2 b v t p1, 0qvt of C 0 pG p2q q¸r G. This completes the identification of the Kasparov triples.
Finally, we apply a standard trick in KK-theory to replace the representing module by a non-degenerate one, i.e. we pass to the modulè The latter module however is easily seen to be isomorphic to in such a way that the isomorphism intertwines the representations κ 2 bp1, v t p1, 0qvt , 1q and κ 2 b 1 E .
Finally, we are ready to give our concrete formula for the comparison map µ 1 . Proof. According to the description in Theorem 4.7, it will suffice to show that for any compact open bisection V in G, if η is as in Theorem 4.3 we have that ηpr1 V s b Ψq " r1V s in V 8 pC 0 pG 0 q, Cr pGqq{ «. Now, if we write λ V : C Ñ Lpπ E p1 V qEq for the unital scalar representation, then Recalling (see lines (38) and (39)) that π E " M s ' v t pM s b 1 Cr pGq , 0qvt ' M r , we compute that M s p1 V q¨L 2 pG, sq -1 spV q C 0 pG 0 q as a right C 0 pG 0 q-module, and that M r p1 V q¨L 2 pG, rq -1 rpV q C 0 pG 0 q as a right C 0 pG 0 q-module. On the other hand v t pM s p1 V q b 1 Cr pGq , 0qvt¨pL 2 pG, sq b ι Cr pGq ' L 2 pG, rq b ι Cr pGqq is isomorphic to p t pCr pGq ' Cr pGqq as a right Cr pGq-module, where It follows that r1 V s b r Ψ P KKpC, Dpιqq is represented by the class of the projection On the other hand, note that p t " u t p 0 ut , where so pu t q tPr0,1s defines a continuous path of unitaries in M 2 pCr pGqq (or in the unitization of this C˚-algebra if it is not unital). It follows from the definition of η given in line (14) that Computing,ˆ1 Ψq " r1V s.

4.4.
The HK-conjecture in low dimensions. In this subsection, we apply the previous result and Theorem 3.36 to deduce consequences for the HK conjecture. As discussed in Subsection 4.2, a recent result of Proietti and Yamashita in [44] established the existence of a convergent spectral sequence (45) E 2 p,q " H p pG, K q pAqq ñ K p`q pA¸r Gq for any G-algebra A, provided that G is a second countable ample groupoid with torsion free isotropy, which satisfies the strong Baum-Connes conjecture.
Combining this with our results from previous sections we obtain the following application to the HK-conjecture. The reader should compare this to [44,Remark 4.5], which establishes a similar result on the HK conjecture under a vanishing hypothesis on H k pGq for k ě 3; the main difference between Theorem 4.19 below and [44,Remark 4.5] is that the former gives a concrete criterion when vanishing holds.
Theorem 4.19. Let G be a second countable principal ample groupoid with dynamic asymptotic dimension at most two. Then there is a short exact sequence 0 Ñ H 0 pGq µ0 Ñ K 0 pCr pGqq Ñ H 2 pGq Ñ 0, and µ 1 : H 1 pGq Ñ K 1 pCr pGqq is an isomorphism. If moreover H 2 pGq is free (e.g. if it is finitely generated), then the HK-conjecture holds for G, i.e.
Proof. First of all we can apply the spectral sequence (45) since G is principal and any groupoid with finite dynamic asymptotic dimension is in particular amenable (this follows from the proof of [23,Theorem A.9]), and hence satisfies the strong Baum-Connes conjecture by the main result of [53]. Since H n pGq " 0 for all n ě 3 by Theorem 3.36 the spectral sequence (45) collapses on the second page and we conclude that K 1 pCr pGqq -H 1 pGq and that K 0 pCr pGqq fits into a short exact sequence 0 Ñ H 0 pGq Ñ K 0 pCr pGqq Ñ H 2 pGq Ñ 0.
If moreover H 2 pGq is free abelian (which holds if it is finitely generated, as it is torsion-free by Theorem 3.36), then the sequence above splits and we get a direct sum decomposition K 0 pCr pGqq -H 0 pGq ' H 2 pGq.
With a view towards the classification program for simple nuclear C˚-algebras, we obtain the following consequence. Proof. We have canonical isomorphisms µ i : H i pGq -K i pCr pGqq. The isomorphism µ 0 clearly extends to an isomorphism of ordered groups respecting the position of the unit. Since G is principal, there is an affine homeomorphism between the set M pGq of G-invariant probability measures on G 0 and the tracial state space T pCr pGqq of the reduced groupoid C˚-algebra (see, for instance, [32, Section 4.1]). Finally, from the definition of µ 0 it is clear that the pairings are compatible.

Examples and applications
In this final section we discuss several applications of our results for specific classes of groupoids and exhibit some interesting examples. 5.1. Free actions on totally disconnected spaces. In [10, Theorem 1.3], Conley et. al. show that for a large class of countable groups Γ, any free action on a second countable, locally compact, zero-dimensional space has dynamic asymptotic dimension at most the asymptotic dimension of Γ, and that the latter is finite. If X is compact, for example a Cantor set, then the dynamic asymptotic dimension will therefore be exactly equal to the asymptotic dimension of Γ by [24, Theorem 6.5]. The class described by the authors of [10] is technical and we refer there for details; suffice to say that it includes many interesting examples such as all polycylic groups, all virtually nilpotent groups, the lamplighter group pZ{2Zq ≀ Z, and the Baumslag-Solitar group BSp1, 2q.
Hence for such actions, Theorem 3.36 implies that H n pΓ, ZrXsq " 0 for all n ą asdimpΓq.

5.2.
Smale spaces with totally disconnected stable sets. A Smale space consists of a self-homeomorphism ϕ : X Ñ X of a compact metric space X, such that the space can be locally decomposed into the product of a coordinate whose points get closer together as ϕ is iteratively applied, and a coordinate whose points get farther apart under the map ϕ. We refer to [46] for basic definitions and details. Given a Smale space pX, ϕq one can define two equivalence relations on X as follows: x " s y if and only if lim nÑ8 dpϕ n pxq, ϕ n pyqq " 0, and x " u y if and only if lim nÑ8 dpϕ´npxq, ϕ´npyqq " 0.
Let X s pxq and X u pxq denote the stable and unstable equivalence classes of a point x P X respectively. Upon choosing a finite set P of ϕ-periodic points one can constructétale principal groupoids G u pX, P q and G s pX, P q with unit space X s pP q " Ť xPP X s pxq and X u pP q " Ť xPP X u pP q, respectively. In particular, if these unit spaces are totally disconnected, the groupoids are ample. Note further, that for an irreducible Smale space pX, ϕq, the groupoids G u pX, P q and G s pX, P q only depend on P up to equivalence. In particular, the choice of P is irrelevant when computing their homology. Deeley and Strung prove in [12] that for an irreducible Smale space one has the estimate dadpG u pX, P qq ď dim X.
Combining this with our Theorem 3.36 and Corollary 4.19, and also [43, Theorem 4.1] we can compute the K-theory of the resulting C˚-algebras from Putnam's homology for Smale spaces.
Corollary 5.1. Let pX, ϕq be an irreducible Smale space with totally disconnected stable sets. Then H s n pX, ϕq " 0 for all n ą dimpXq; and if dimpXq ď 2 and H s 2 pX, ϕq is free abelian (e.g. when it is finitely generated), then K 0 pC˚pG u pX, P qqq -H s 0 pX, ϕq ' H s 2 pX, ϕq and K 1 pC˚pG u pX, P qqq -H s 1 pX, ϕq.
This result includes most of the previously known examples (see e.g. [56]) that were based on separate computations of the K-theory and homology, and hence provides a more conceptual explanation. 5.3. Bounded geometry metric spaces. A metric space X has bounded geometry if for each r ą 0 there is a uniform bound on the cardinalities of all r-balls in X; important examples come from groups with word metric, or from suitable discretisations of Riemannian manifolds.
Skandalis, Tu, and Yu [49] construct an ample groupoid GpXq which captures the coarse geometry of X. In particular, the reduced groupoid C˚-algebra Cr pGpXqq can be canonically identified with the uniform Roe algebra Ců pXq. Let us briefly recall the construction. Let βX denote the Stone-Čech compactification of X, i.e. the maximal ideal space of ℓ 8 pXq. For each r ą 0, let E r be the closure of tpx, yq P XˆX | dpx, yq ď ru inside βXˆβX, 14 which is a compact open set. Then the coarse groupoid GpXq of X has as underlying set Ť 8 r"0 E r . The operations are the restriction of the pair groupoid operations from βXˆβX, and the topology is the weak topology coming from the union Ť 8 r"0 E r , i.e. a subset U of G is open if and only if U X E r is open for each r. Then GpXq is a principal, ample, σ-compact groupoid with compact base space homeomorphic to βX, see [47,Theorem 10.20].
Our first goal is to identify the groupoid homology H˚pGpXqq with the uniformly finite homology of X introduced by Block and Weinberger in [5,Section 2]. We begin by recalling the relevant definitions. Let C n pXq denote the collection of all bounded functions c : X n`1 Ñ Z such that there exists r ą 0 such that if cpx 0 , ..., x n q ‰ 0, then the diameter of the set tx 0 , ..., x n u is at most r. For each i P t0, ..., nu, let B i : X n`1 Ñ X n be defined by B i px 0 , ..., x n q :" px 0 , ..., p x i , ..., x n q. Define B i : C n pXq Ñ C n´1 pXq by pB i cq " ÿ B i y"x cpxq and define B : C n pXq Ñ C n´1 pXq by B :" ř n i"0 p´1q i B i . Then we have B˝B " 0, so we get a chain complex. The uniformly finite homology of X, denoted H uf pXq, is by definition the associated homology of this complex.
Having introduced all the main actors we can now prove the following theorem.
Theorem 5.2. Let GpXq be the coarse groupoid associated to a bounded geometry metric space X. Then there is a canonical isomorphism H˚pGpXqq -H uf pXq.
Proof. For brevity let us denote the coarse groupoid by G throughout the proof. Now, for a P ZrEG n s, define a P ZrEG n s by apg 0 , ..., g n q :" " ř hPGx aph, hg 1 , ..., hg n q g 0 " x P G 0 0 g 0 R G 0 .
One can check (we leave this to the reader) that the equivalence classes ras and ras in ZrEG n s G of a and a respectively are the same, and moreover that a is the unique element of ras that is supported on tpg 0 , ..., g n q P EG n | g 0 P G 0 u. We define maps α : ZrEG n s G Ñ C n pXq and β : C n pXq Ñ ZrEG n s G as follows. First, pαrasqpx 0 , ..., x n q :" appx 0 , x 0 q, ..., px 0 , x n qq. This makes sense using that G contains the pair groupoid XˆX. We note that αras is bounded as a is. Moreover, the fact that a has compact support implies that it is supported in a set of the form E 0ˆEr1ˆ¨¨¨ˆErn X EG n for compact 14 The authors of [49] take the closure in βpXˆXq instead of in βXˆβX, but by [ open sets E ri as in the definition of G. It follows that αras is supported on the set of tuples with diameter at most 2 maxtr 1 , ..., r n u and is thus a well-defined element of C n pXq.
To define β, let first ppx, x 0 q, px, x 1 q, ..., px, x n qq P EG n where each pair px, x i q is in the pair groupoid. For c P C n pXq, define pβcqppx, x 0 q, px, x 1 q, ..., px, x n qq :" Due to the support condition on elements of C n pXq, there exists r ą 0 such that c is supported in the set tpx 0 , ..., x n q P X n`1 | dpx i , x j q ď r for all i, ju. One can check using the bounded geometry condition that this implies that the closure of the support S of βc in EG n X pXˆXq n`1 canonically identifies with the Stone-Cech compactification of S, and thus that βc extends uniquely to a function on the compact open set S as it is bounded, and so a function on EG n by setting it to be zero outside S. We also denote βc the corresponding class in ZrEG n s G . Now, having built the maps α and β, note that both define maps of complexes as the face maps in both cases are given by omitting the i th element in a tuple. To see that they are mutually inverse isomorphisms, one computes directly that αpβpcqq " c, and that βpαrasq " ras; we leave this to the reader. The result follows.
Having identified the homology groups of the coarse groupoid with a more classical object, we would now like to apply our main results and draw some consequences for the computation of the K-theory groups of uniform Roe algebras Ců pXq which can be canonically identified with Cr pGpXqq. Since the spectral sequence (45) is only available in the case of second countable groupoids we need to do some additional work. To this end it is useful to consider a slightly different construction of the coarse groupoid.
Following [49, Section 2.2], let Γ X denote the collection of all subsets A Ď XX such that the first coordinate map r : XˆX Ñ X and second coordinate maps s : XˆX Ñ X are both injective when restricted to A, and such that sup px,yqPA dpx, yq ă 8. As in [49, Section 3.1], every A P Γ X defines a bijection t A : spAq Ñ rpAq with the property that sup xPspAq dpx, t A pxqq ă 8. Every such bijection extends to a homeomorphism φ A : spAq Ñ rpAq between the respective closures in βX. As in [49, Definition 3.1], we write G pXq for the collection tφ A | A P Γ X u, which is a pseudogroup, i.e. closed under compositions and inverses. As in [49,Section 3.2], the coarse groupoid GpXq of X can be realized as the groupoid of germs associated to this pseudogroup (see [ Define G A to be the spectrum of the C˚subalgebra of C 0 pGpXqq generated by tχ A | φ A P Au, and let X A be the spectrum of the C˚-subalgebra of CpβXq " ℓ 8 pXq generated tχ spAq | φ A P Au. The following comes from [49, Lemma 3.3] and its proof. Lemma 5.3. Let A be an admissible sub-pseudogroup of G pXq. Then the groupoid operations naturally factor through the canonical quotient maps GpXq Ñ G A and GpXq p0q Ñ X A , making G A anétale, locally compact, Hausdorff groupoid with base space X A , which is moreover second countable if A is countable.
Moreover, the quotient map p : βX " GpXq p0q Ñ X A gives rise to an action of G A on βX, and there is a canonical isomorphism of topological groupoids GpXq -βX¸G A .
Proof. The only part not explicitly in [49,Lemma 3.3] or its proof is the second countability statement. This follows as if A is countable, then the C˚-subalgebra of C 0 pGpXqq generated by tχ A | φ A P Au is separable.
Lemma 5.4. Let X be a bounded geometry metric space. Then there exists a countable admissible sub-pseudogroup A of G pXq such that G A is principal.
Proof. First, choose a countable admissible sub-pseudogroup A 1 of G pXq as follows. For each n P N, a greedy algorithm based on bounded geometry (compare the discussion in [49, Section 2.2, part (a)]) gives a finite decomposition of tpx, yq P XˆX | dpx, yq ď nu such that tpx, yq P XˆX | dpx, yq ď nu " mn ğ i"1 A pnq i and so that each A pnq i is in Γ X . Let A 1 be the sub-pseudogroup of G pXq generated by all the A pnq i . It is countable (as generated by a countable set) and it is admissible by construction. Given φ P A 1 we can apply [42,Proposition 2.7] to decompose its domain dompφq " A 0,φ \ A 1,φ \ A 2,φ \ A 3,φ into disjoint clopen sets, where A 0,φ is the set of fixed points of φ and φpA i,φ q X A i,φ " H for i " 1, 2, 3. Let A be the sub-pseudogroup generated by A 1 and tid A i,φ | φ P A 1 , 0 ď i ď 4u. Then A is still countable and admissible. We claim that G A is principal. So let rφ, ωs P G A such that its source and range are are equal, i.e. φpωq " ω. We may assume that φ P A 1 as there is nothing to show if φ was already the identity function on some clopen set. We then have φ| A 0,φ " id A 0,φ and hence rφ, ωs " rid, ωs as desired.
Let us now assume that X has asymptotic dimension at most d, or equivalently (see [24,Theorem 6.4]) that GpXq has dynamic asymptotic dimension at most d. For each n P N, let E n :" tpx, yq P XˆX | dpx, yq ď nu, where the closure is taken in βXˆβX. Then E n is a compact open subset of GpXq. Hence using the assumption that the dynamic asymptotic dimension of GpXq is at most d, there exists a decomposition βX " U X X (this follows as clopen sets in βX are in one-one correspondence with arbitrary subsets of X in this way). Let B denote the sub-pseudogroup of G pXq generated by A as in Lemma 5.4, and by tid V pnq i | n P N, i P t0, ..., duu. Then we have the following result.
Lemma 5.5. Let X be a bounded geometry metric space with asymptotic dimension at most d. Then there is a second countable,étale, locally compact, Hausdorff principal groupoid G with dynamic asymptotic dimension at most d, and such that G acts on βX giving rise to a canonical isomorphism βX¸G -GpXq.
Proof. We claim that G " G B works. Note that B is countable (as generated by a countable set) and admissible (as it contains A , which is admissible). Hence most of the statement follows from Lemma 5.3. As we are only adding further identity functions in the passage from A to B, we also retain principality by the same proof as in Lemma 5.4. We only need to show that the dynamic asymptotic dimension of G B is at most d.
For each n P N, let us write rE n s for the image of E n Ď GpXq under the quotient map GpXq Ñ G B . Then rE n s is compact and open: indeed, with notation as in the construction of A its characteristic function is equal to mn ÿ Corollary 5.7. Let X be a bounded geometry metric space with asymptotic dimension d. Then H uf n pXq " 0 for all n ą d and H uf d pXq is torsion-free. Moreover, (1) if asdimpXq ď 2 and H uf 2 pXq is free, or finitely generated, then K 0 pCů pXqq -H uf 0 pXq ' H uf 2 pXq, and K 1 pCů pXqq -H uf 1 pXq, (2) if asdimpXq ď 3, X is non-amenable, and H uf 3 pXq is free, or finitely generated, then K 0 pCů pXqq -H uf 2 pXq, and K 1 pCů pXqq -H uf 1 pXq ' H uf 3 pXq.
Proof. The first case follows from Theorem 3.36 and Corollary 4.19 as in our earlier examples. For p2q note that if asdimpXq ď 3 then the only possibly non-zero differentials on the E 3 -page are the maps d 3 3,2l : H uf 3 pXq Ñ H uf 0 pXq for l ě 0. The sequence converges on the E 4 -page and hence there are short exact sequences 0 Ñ cokerpd 3 3,0 q Ñ K 0 pCů pXqq Ñ H uf 2 pXq Ñ 0, and 0 Ñ H uf 1 pXq Ñ K 1 pCů pXqq Ñ kerpd 3 3,0 q Ñ 0 For a non-amenable space X the group H uf 0 pXq vanishes by [5,Theorem 3.1]. Since H uf 3 pXq is free the result follows.
Examples 5.8. Let Γ be a countable group equipped with a left invariant bounded geometry metric 15 . Let Γ act on the group ℓ 8 pΓ, Zq of bounded Z-valued functions on Γ via the action induced by the left translation action of Γ on itself. Then it is well-known that H uf pΓq identifies with the group homology H˚pΓ, ℓ 8 pΓ, Zqq of Γ with coefficients in ℓ 8 pΓ, Zq: see for example [8, last paragraph on page 1515] (this discusses the case of H uf with real coefficients, but the same argument works for integer coefficients). Now, assume that Γ is a d-dimensional Poincaré duality group, for example if Γ is the fundamental group of a closed d-manifold with contractible universal cover. Then H uf d pΓq -H d pΓ, ℓ 8 pΓ, Zqq -H 0 pΓ, ℓ 8 pΓ, Zqqℓ 8 pΓ, Zq Γ -Z, where the first isomorphism is the general fact noted above, the second is Poincaré duality, the third is the definition of the zeroth cohomology group, and the fourth is straightforward. In particular, H uf d pΓq is free. This discussion applies in particular if Γ is the fundamental group of a closed orientable surface. In this case Γ is quasi-isometric to either the hyperbolic plane or to the Euclidean plane, whence the asymptotic dimension of Γ is two, and we may apply the first part of Corollary 5.7 to conclude that K 0 pCů pΓqq -H uf 0 pΓq ' Z, and K 1 pCů pΓqq -H uf 1 pΓq.
If moreover the underlying surface has genus at least two, then Γ is non-amenable, so H uf 0 pΓq vanishes, and K 0 pCů pΓqq -Z. The discussion also applies if Γ is the fundamental group of a closed, orientable, hyperbolic 3-manifold. In this case Γ is non-amenable, and Γ is quasi-isometric to hyperbolic 3-space, so of asymptotic dimension three. We may thus apply the second part of Corollary 5.7 to conclude that K 0 pCů pΓqq -H uf 2 pΓq, and K 1 pCů pΓqq -H uf 1 pΓq ' Z. 15 Such a metric exists and is essentially unique by [55, Proposition 2.3.3], for example.

Examples with topological property (T).
The HK-conjecture asserts that for a principal ample groupoid we have abstract isomorphisms À jPN H 2j`i pGq -K i pCr pGqq. If G has homological dimension 1 one might be tempted to strengthen this conjecture and ask for the canonical maps µ 0 and µ 1 to be isomorphisms. Here, we show that this strong version of the conjecture fails.
In order to exhibit the examples we need some preliminary facts about topological property (T). Topological property (T) for groupoids was introduced in [14, Definition 3.6], and we refer the reader there for the definition. Let R G " tπ x | x P G 0 u denote the family of regular representations of G, i.e. π x : C c pGq Ñ Bpℓ 2 pG x qq, π x pf qδ g " ÿ hPG rpgq f phqδ hg .
Then we have the following generalization of [14,Proposition 4.19].
Lemma 5.9. Let G be an ample groupoid with compact unit space acting on a compact space X. If G has property (T) with respect to R G then G˙X has property (T) with respect to R G˙X .
Proof. Let p : X Ñ G 0 denote the anchor map of the action. Let pK, cq be a Kazhdan pair for R G . Now let L :" tpg, xq P G˙X | g P Ku " pG˙Xq X pKˆXq, which is compact. We claim that L is a Kazhdan set for R G˙X . Indeed consider the regular representation π G˙X x : C c pG˙Xq Ñ Bpℓ 2 pG ppxq qq associated with an arbitrary point x P X and let ξ P ℓ 2 pG ppxq q be a unit vector. Since pK, cq is Kazhdan for R G , there exists a function f P C c pGq with support in K such that f I ď 1 and π G ppxq pf qξ´π G ppxq pΨpf qqξ ě c, where Ψ : C c pGq Ñ CpG 0 q is given by Ψpf qpxq " ř gPG x f pgq. Since K is compact we can cover it with finitely many compact open bisections V 1 , . . . , V n and using a partition of unity argument, we can write f " ř f i where supppf i q Ď V i . Then there must be some 1 ď i ď n such that π G ppxq pf i qξ´π G ppxq pΨpf i qqξ ě c n .
Using that f i is supported in a bisection one directly verifies that π G ppxq pf i q " π G ppxq pΨpf i qqπ G ppxq p1 Vi q and Ψpf i q " Ψpf i q1 rpViq and combining this with the previous observation, we conclude that there exists an i such that π G ppxq p1 Vi qξ´π G ppxq p1 rpViq qξ ě c n .
Now let V i˙X denote the compact open set pV iˆX q X G˙X and let f 1 :" 1 Vi˙X . Then f 1 is clearly supported in L with f 1 I ď 1 and since π G˙X x pf 1 q " π G ppxq p1 Vi q and π G˙X x pΨpf 1 qq " π G ppxq p1 rpViq q we conclude that π G˙X x pf 1 qξ´π G˙X x pΨpf 1 qqξ ě c n .
Hence pL, c n ) is a Kazhdan pair for R G˙X . Now let Γ be a residually finite group and L " pN i q i a sequence of finite index normal subgroups. Let N 8 denote the trivial subgroup of Γ and let π i : Γ Ñ Γ{N i be the quotient map. We denote by G L the associated HLS groupoid, i.e. the group bundle Ů iPNYt8u Γ{N i equipped with the topology generated by the singleton sets tpi, γqu for i P N and γ P Γ, and the tails tpi, π i pγqq | i ą N u for each fixed γ P Γ and N P N. It is well-known that this groupoid is Hausdorff if and only if for each γ P Γzteu the set ti P N | γ P N i u is finite. This is in particular the case if the sequence is nested and has trivial intersection.
Following a construction of Alekseev and Finn-Sell in [1] we associate a principal groupoid to this data as follows. Let X :" Ů iPNYt8u Γ{N i . Then X carries a canonical action of the HLS groupoid G L given by left multiplication. For γN i P Γ{N i let ShpγN i q " ď jěi π´1 i,j pγN i q be the shadow of γN i in X. Now let p X be the spectrum of the smallest G L -invariant C˚-subalgebra B Ď ℓ 8 pXq containing Since B is G L -invariant, p X also carries an action of G L and we can form the transformation groupoid G :" G L˙p X. As explained just after Remark 2.2 in [1], this groupoid is principal. Moreover, X Ď p X is a dense open G L -invariant subset with complement p XzX -p Γ L " lim Ð Ý Γ{N i . Hence we obtain isomorphisms The following result relies on property pτ q as defined in [33, Definition 4.3.1].
Proposition 5.10. Suppose Γ is a finitely generated, residually finite group and L " pN i q i is a sequence of finite index normal subgroups with property pτ q. Then the following hold: (1) G has topological property pT q with respect to the family of regular representations in the sense of [14]. (2) The sequence K 0 pCr pG| X qq Ñ K 0 pCr pGqq Ñ K 0 pCr pG| x XzX qq is not exact in the middle.
Proof. Since the regular representations of G extend to Cr pGq by definition of the reduced groupoid C˚-algebra, the result follows from [14,Proposition 4.15] and Lemma 5.9. Part (2) follows from (1) and [14,Proposition 7.14].
We can now provide some concrete principal ample groupoids where µ 0 is not surjective.
Let Γ " F 2 and choose a nested sequence pN i q i of finite index normal subgroups in F 2 with property pτ q such that the associated HLS groupoid is Hausdorff.
Example 5.11. To have a concrete example of such a sequence in mind consider the nested family pL i q i of finite index normal subgroups L i :" kerpSL 2 pZq Ñ SL 2 pZ{5 i qq of SL 2 pZq. Embed F 2 in SL 2 pZq as a finite index normal subgroup and let N i :" L i X F 2 . Then pN i q i is a nested family of finite index normal subgroups of F 2 with trivial intersection. Since F 2 has finite index in SL 2 pZq and SL 2 pZq has property pτ q with respect to the family pL i q i we conclude that F 2 has pτ q with respect to the family pN i q i .
Let us first compute the homology of the associated groupoid G. Consider the long exact sequence in homologÿ¨¨Ñ H n pG| X q Ñ H n pGq Ñ H n pG| x XzX q Ñ H n´1 pG| X q Ñ¨¨¨Ñ H 0 pG| x XzX q corresponding to the decomposition p X " X \ p XzX. Since G| X is a disjoint union of principal and proper groupoids, we have H n pG| X q " à iPNYt8u H n pΓ{N i˙Γ {N i q " 0 for all n ě 1, and From the long exact sequence we conclude that for all n ě 2 the restriction to the boundary induces isomorphisms H n pGq -H n pF 2˙x F 2L q.
It is a well-known fact that H n pF 2˙x F 2L q -H n pF 2 , Cp x F 2L , Zqq and the homology of the free group F 2 is well-known to be trivial for all n ě 2. Hence H n pGq " 0 for all n ě 2. Now by construction H 0 pF 2˙x F 2L q " Cp x F 2L , Zq{xf´γ.f | γ P F 2 y and from the Pimsner-Voiculescu exact sequence for actions of free groups from [41, Theorem 3.5] we obtain K 0 pCp We clearly have impβq " xf´γ.f | γ P F 2 y so that after the identification above, µ 0 is the identity. Now consider the commutative diagram The top row is exact in the middle, as it is part of the long exact sequence in homology corresponding to the open invariant subset X Ď p X. The bottom row however is not exact in the middle by Proposition 5.10. The map on the right hand side is an isomorphism by our reasoning above.
We claim that the map µ G 0 is not surjective. Suppose for contradiction that it was. Let x P K 0 pCr pGqq be an element which maps to zero in K 0 pCr pG| x XzX q but is not in the image of i 0 . Since µ G 0 is surjective, we can find an element y P H 0 pGq such that µ G 0 pyq " x. But then by commutativity of the right hand square we have µ 0 pppyqq " 0 and since µ 0 is injective we conclude that ppyq is zero and hence y " ipzq for some z P H 0 pG| X q. Moreover, by commutativity of the left square we have x " µ G 0 pyq " i 0 pµ 0 pzqq, which contradicts our assumption that x R impi 0 q. Remark 5.12. At this point, it seems there are three known reasons for failure of the HK conjecture. The first, due to Scarparo [48] is the presence of torsion in isotropy groups. The second, due to Deeley [13] is due to torsion phenomena in K-theory; however, Deeley's results do not contradict the "rational" HK conjecture one gets after tensoring with Q, analogously to the classical fact that the Chern character is a rational isomorphism between K-theory and cohomology. The third is exotic analytic phenomena connected to the failure of the Baum-Connes conjecture as discussed above (this is admittedly not exactly a failure of the HK conjecture, but it seems to us as evidence that the HK conjecture should sometimes fail when the Baum-Connes conjecture fails). Based on these counterexamples, the following "folk conjecture" (arrived at independently by several people) seems reasonable: if