1. Introduction
In this paper, we prove two structural results for the homology groups of topological dynamical systems, continuing our work [Reference Proietti and YamashitaPY22a, Reference Proietti and YamashitaPY23]. One is a Künneth type formula for the product of two systems, while the other reduces the homological computation to the (compactly supported) cohomology of the underlying space under certain conditions.
Previously, we have focused on totally disconnected systems, which are represented by ample groupoids, that is, étale groupoids whose unit space is totally disconnected. In this paper, we do not work under this restriction, but we deal with (more general) dynamical systems of finite topological dimension. A motivating class of examples is that of Smale spaces [Reference RuelleRue04], which capture hyperbolicity on compact metric spaces, such as the dynamics of Anosov diffeomorphisms and more generally those on the basic sets of Axiom A diffeomorphisms. This has led to interesting intersection between the theory of dynamical systems and operator algebras.
We mainly work within the homology theory introduced by Crainic and Moerdijk [Reference Crainic and MoerdijkCM00]. It defines homology groups with coefficient in equivariant sheaves to any étale groupoid, based on sheaves, derived formalism and simplicial methods.
In another direction, given a locally compact étale groupoid (and more generally a locally compact groupoid with a continuous Haar system), there is a convolution product on the space of compactly supported continuous functions on the groupoid, which can be completed to a C $^*$ -algebra [Reference RenaultRen80]. The K-groups of this C $^*$ -algebra can be regarded as homological invariants of the original groupoid, which has better connection to index theory, classification of C $^*$ -algebras and other topics involving operator K-theory.
Comparison between these two theories has been a major driving force behind our previous works, and it plays an important role in this note too.
Let us summarize the main conceptual results in this note.
Theorem A. (Theorem 3.1)
Let G and H be étale groupoids, and ${\mathcal S}$ and ${\mathcal T}$ be G- and H-equivariant sheaves. Then there is a split short exact sequence
Theorem B. (Theorem 4.2)
Let $\tilde G$ be a locally compact principal groupoid such that $\tilde G^x$ is homeomorphic to $\mathbb {R}^n$ for some fixed n. Suppose there exists a generalized transversal T in $\tilde G^{(0)}$ , so that $G = \tilde G|_T$ is an étale groupoid. We then have an isomorphism
where, on the right-hand side, cohomology is computed with coefficients in the orbit-wise orientation sheaf over the unit space of $\tilde G$ .
In operator K-theory, Theorem A has a direct analogue which is the Künneth formula of K-groups of tensor product C $^*$ -algebras due to Schochet [Reference SchochetScho82]. Theorem B can also be interpreted as an analogue of Connes’s Thom isomorphism for K-groups [Reference ConnesCon81] (for a groupoid of the form $\mathbb {R}^n\ltimes X$ , this result reduces operator K-groups of the associated $C^*$ -crossed product to the topological K-groups of X, up to a degree shift determined by n).
However, the proofs are completely different from the K-theoretic ones. While the proof of Theorem A is a standard Eilenberg–Zilber type manipulation of multicomplexes, for Theorem B, we make an essential use of the sheaf theoretic idea and the formalism of derived functors.
Theorem B is particularly useful in computing the homology of groupoids coming from dynamical systems on manifolds and related structures. Moreover, the idea behind this theorem is also useful to study systems of number theoretic origin.
We look at a class of Smale system $(Y^{(c)},\phi )$ that appears in the theory of algebraic actions due to Schmidt [Reference SchmidtSchm95]. Given an algebraic number c, the space $Y^{(c)}$ is given as the Pontryagin dual of the additive group of a certain subring of $K=\mathbb {Q}(c)$ , and $\phi $ is the natural map induced by the multiplication by c (see §6 for details on this construction).
Theorem C. (Theorem 6.3)
Denote by G the (un)stable étale groupoid associated to $(Y^{(c)},\phi )$ . Then the groupoid homology of G can be presented as the inductive limit of exterior power of a certain subgroup $\Gamma $ of the additive group of K,
with respect to the connecting map $\theta _k \colon \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma \to \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma $ . The shift is given by a natural number d which depends on the infinite places of K. The map $\theta _k$ is the unique extension of $N\bigwedge ^k m_{c^{-1}}$ , with a natural number N and $m_{c^{-1}}$ denoting multiplication by $c^{-1}$ .
An similar computation can be carried out for the K-groups of the groupoid C $^*$ -algebra $C^*_rG$ .
Theorem D. (Theorem 6.8)
Denote by G the (un)stable étale groupoid associated to $(Y^{(c)},\phi )$ . With notation as in Theorem C, there is an isomorphism
Consequently, the K-group is isomorphic to the direct sum of groupoid homology with the same degree parity.
Corollary. (Corollary 6.9)
In the setting of Theorems C and D, we have
This confirms, for the class of groupoids from Theorem C, the ‘HK conjecture’ formulated by Matui [Reference MatuiMat16, Reference Matui, Carlsen, Larsen, Neshveyev and SkauMat17] in the setting of ample groupoids, which asks if there is an isomorphism between the K-groups and periodicized homology as above.
Theorem C is interesting for a few reasons. First, the computations given here generalize previous work by Burke and Putnam [Reference Burke and PutnamBP17]. Second, the groupoids appearing in this context are not necessarily ample, providing us with examples where the HK conjecture holds beyond its original assumptions (the groupoid associated to an irrational flow on the torus provides another example, see Example 4.5 for details). Third, a key intermediate result in this setting (Proposition 6.2) showcases a ‘variant’ of Theorem B above, more akin to the Thom isomorphism in cohomology than to a proper duality.
The paper is organized as follows. In §2, we recall a few basic notions and briefly summarize our previous work to set the conventions and background for this note.
In §3, we prove the Künneth formula (Theorem A) for the homology of the product of étale groupoids. In §4, we show the Poincaré duality-type result (Theorem B), and in the last two sections, we present some concrete computations for notable examples of topological dynamical systems arising from expanding maps on compact manifolds and analogues of solenoids in algebraic number fields.
2. Preliminaries
We fix conventions in use throughout the paper. We only briefly recall definitions and generally follow the treatment in [Reference Proietti and YamashitaPY22a, Reference Proietti and YamashitaPY23].
2.1. Topological groupoids and homology groups
We mainly work with second countable, locally compact, Hausdorff groupoids. Given such a groupoid G, we denote its base space by $G^{(0)}$ , with structure maps $s, r \colon G \to G^{(0)}$ , and the nth nerve space (for $n\geq 1$ ) given by
We say that G is étale if s and r are local homeomorphisms, and ample if it is étale and its base space is totally disconnected.
We consider the homology of étale groupoids as defined by Crainic and Moerdijk [Reference Crainic and MoerdijkCM00]. A G-sheaf is a sheaf F on $G^{(0)}$ endowed with a continuous action of G, modelled by a map of sheaves $s^* F \to r^* F$ on G. A G-sheaf F is said to be (c-)soft when the underlying sheaf on $G^{(0)}$ has that property, that is, for any (compact) closed subset $S \subset G^{(0)}$ and any section $x \in \Gamma (S, F)$ , there is an extension $\tilde x \in \Gamma (X, F)$ .
Our standing assumption is that the cohomological dimension of any open set of $G^{(0)}$ is bounded by some fixed integer N: to be precise, $H_c^k(U, F) = 0$ for any open subset $U \subset G^{(0)}$ and any $k> N$ .
This is guaranteed when $G^{(0)}$ is a subspace of a metrizable space of (Lebesgue) topological dimension N, which covers all the concrete examples we consider. To see this, observe that the topological dimension of any compact subset $A \subset G^{(0)}$ is bounded by N [Reference EngelkingEng78, §3.1], then the Čech cohomology $\check {H}^\bullet (A; F)$ will vanish in degree above N. By paracompactness, $\check {H}^\bullet (A; F)$ agrees with the sheaf cohomology $H^\bullet (A; F) = R^\bullet \Gamma _A(F)$ for the right derived functor of the functor $\Gamma _A(F) = \Gamma (A, F)$ , then we can combine [Reference GodementGod73, Remarque II.4.14.1 and Théoréme II.4.15.1] to get the claim.
Let F be a G-sheaf. Then there is a resolution of F as above by c-soft G-sheaves, and the homology with coefficient F, denoted $H_\bullet (G, F)$ , is defined as the homology of the total complex of the double complex $(C_i^j)_{0 \le i,j}$ with terms
which has homological degree $i - j$ . More generally, when $F_\bullet $ is a homological complex of G-sheaves bounded from below, take a resolution of each $F_j$ by c-soft G-sheaves $F_j^k$ as above. Then the hyperhomology with coefficient $F_\bullet $ , denoted by $\mathbb {H}_\bullet (G, F_\bullet )$ , is the homology of triple complex with terms
which has homological degree $i + j - k$ .
When two étale groupoids G and H are Morita equivalent, there are natural correspondences between the G-sheaves and H-sheaves inducing an isomorphism of groupoid homology. In particular, if $f \colon H \to G$ is a Morita equivalence homomorphism, we have
for any complex $F_\bullet $ of G-sheaves as above. Note that the existence of f is equivalent to G and H being equivalent in the sense studied in [Reference Muhly, Renault and WilliamsMuh87] (see [Reference Farsi, Kumjian, Pask and SimsFKPS19, Proposition 3.10]). For this reason, we will often say that G and H are ‘equivalent’ without any other specification.
2.2. Derived functor formalism
We briefly recall the derived functor formalism of groupoid homology from [Reference Crainic and MoerdijkCM00, §4]. Let G and $G'$ be étale groupoids, and $\phi \colon G \to G'$ be a continuous groupoid homomorphism. Then, for each $x \in G^{\prime (0)}$ , the comma groupoid $x / \phi $ is defined as the groupoid whose objects are the pairs $(y, g')$ , where $y \in G^{(0)}$ and $g' \in G_{x}^{\prime \phi (y)}$ , and an arrow from $(y_1, g^{\prime }_1)$ to $(y_2, g^{\prime }_2)$ is given by $g \in G_{y_1}^{y_2}$ such that $\phi (g) g^{\prime }_1 = g^{\prime }_2$ . This is an étale groupoid that comes with a homomorphism $\pi _x \colon x / \phi \to G$ .
When F is a G-sheaf, we consider a simplicial system of $G'$ -sheaves, denoted by $B_\bullet (\phi , F)$ , which at the level of stalks is given by
Assume that the groupoids have good homological properties to define groupoid homology. Then the above construction leads to the left derived functor $\mathcal L \phi _!$ from a category of homological complexes of G-sheaves bounded from below to a similar category of complexes of $G'$ -sheaves.
To be more concrete, let $F_\bullet $ be such a complex of G-sheaves. Then $\mathcal L \phi _! F_\bullet $ is represented by the total complex of the triple complex of $G'$ -sheaves with terms $B_i(\phi , F_j^k)$ with homological degree $i + j - k$ , where $F_j^\bullet $ is a bounded resolution of $F_j$ by c-soft G-sheaves. This is well defined up to quasi-isomorphism of $G'$ -sheaves. The nth derived functor, denoted by $L_n \phi _! F_\bullet $ , is the $G'$ -sheaf given as the nth homology of $\mathcal L \phi _! F_\bullet $ . By construction, the fibre of this sheaf is given by [Reference Crainic and MoerdijkCM00, Proposition 4.3]
When $G'$ is the trivial groupoid and $\phi $ is the unique homomorphism $G \to G'$ , this recovers the definition of $H_n(G, F_\bullet )$ .
In addition to the pullback functor (the inverse image functor) for sheaves, we will also make use of the direct image functor, simply defined as $g_*F(U)=F(g^{-1}(U))$ [Reference GodementGod73]. In the setting of equivariant sheaves, $g_*$ can also be defined, and it is still right adjoint to the pullback functor, see [Reference Crainic and MoerdijkCM00, §2.3] for details. It is worth noting that if g is proper, then $g_*$ coincides with the functor $g_!$ , sometimes called direct image functor with compact supports.
2.3. Smale spaces
Many examples of groupoids in this note appear as (reduction of) the stable and unstable groupoid of a Smale space. A Smale space is a certain kind of hyperbolic dynamics modelled on a compact metric space X with a self-homeomorphism $\phi $ . See [Reference PutnamPut14] for precise definition and conventions. In particular, there are two distinguished equivalence relations on X, defined as follows:
-
• two points x and y are stably equivalent (denoted $x\sim _s y$ ) if
$$ \begin{align*} \lim_{n\to \infty} d(\phi^{n}(x),\phi^{n}(y)) = 0; \end{align*} $$ -
• similarly, x and y are unstably equivalent (denoted $x\sim _u y$ ) if
$$ \begin{align*} \lim_{n\to \infty} d(\phi^{-n}(x),\phi^{-n}(y)) = 0. \end{align*} $$
The equivalence classes of the stable (respectively unstable) equivalence relation are called the stable sets (respectively unstable sets).
The graph of the stable (respectively unstable) equivalence relation has a structure of locally compact groupoid with a Haar system [Reference PutnamPut96] that we denote by $R^s(X, \phi )$ (respectively $R^u(X, \phi )$ ). Following the construction detailed in [Reference Putnam and SpielbergPS99], we obtain an étale groupoid by restricting $R^u(X,\phi )$ to an appropriate subspace contained in a finite union of stable sets.
3. Künneth formula
Suppose G and H are étale groupoids such that groupoid homology is definable, and ${\mathcal S}$ and ${\mathcal T}$ are equivariant sheaves of abelian groups over $G^{(0)}$ and $H^{(0)}$ , respectively. Furthermore, denote by p and q the canonical projections from $G \times H$ to G and H, respectively. We define the sheaf ${\mathcal S} \boxtimes {\mathcal T}$ as $p^*{\mathcal S} \otimes q^*{\mathcal T}$ . Note this is a $G\times K$ -equivariant sheaf over $G^{(0)} \times K^{(0)}$ .
Theorem 3.1. Under the above setting, there is a split short exact sequence
Proof. Let us take bicomplexes $A=A_{a,i}$ and $B=B_{b,j}$ computing $H_\bullet (G, {\mathcal S})$ and $H_\bullet (H, {\mathcal T})$ , respectively. These are obtained from c-soft cohomological complex of sheaves $\tilde {{\mathcal S}}^\bullet $ and $\tilde {\mathcal T}^\bullet $ , each quasi-isomorphic to ${\mathcal S}$ and ${\mathcal T}$ concentrated at degree $0$ . To obtain a homological complex, we invert the degree, so that $A_{a, i} = \Gamma _c(G^{(a)}, \tilde {\mathcal S}^{-i})$ for example.
Up to the identifications
the total complex of the triple complex $(A_{k,i} \otimes B_{k,j})$ computes $H_k(G \times H, {\mathcal S}\boxtimes {\mathcal T})$ . The claim follows by a standard argument if we can show that this is quasi-isomorphic to the total complex of the quadruple complex $A \otimes B = (A_{a, i} \otimes B_{b,j})_{a,b,i,j}$ .
Now, observe that $A\otimes B$ can be regarded as a bisimplicial object in the category of complexes, by totalizing in the i- and j-directions. By an Eilenberg–Zilber type theorem [Reference Goerss and JardineGJ09, Theorem IV.2.4], for fixed q, the total complex of the bisimplicial group $C_q(a,b) = \bigoplus _{q = i + j} A_{a, i} \otimes B_{b,j}$ is chain homotopic to the Moore complex of the simplicial group $C^{\prime }_q(k) = \bigoplus _{q = i + j} A_{k, i} \otimes B_{k,j}$ .
Now, take double complexes $C_{k,q} = \bigoplus _{k = a + b} C_q(a,b)$ and $C^{\prime }_{k,q} = C^{\prime }_q(k)$ . Since the degree k is concentrated in $k \ge 0$ while the degree q is in $q \le 0$ , the spectral sequences E and $E'$ associated with filtration by q-degree on $\operatorname {\mathrm {Tot}} C$ and $\operatorname {\mathrm {Tot}} C'$ are regular, in the sense that for any n, there is $s(n)$ such that we have $E^r_{p, q} = 0$ for $p + q = n$ , $p < s(n)$ . Then the spectral sequences converge to the total homologies, while we have the isomorphisms $E^r_{p,q} \cong E'^{r}_{p,q}$ for $r \ge 1$ by the above remark. We thus obtain the assertion.
As usual, the morphisms in the short exact sequence above are natural in any conceivable sense; however, the splitting is not.
Remark 3.2. Matui’s result [Reference MatuiMat16, Theorem 2.4] is a special case of the above result in the situation where the groupoids are totally disconnected and the coefficients are locally constant sheaves $\underline {\mathbb Z}$ . (Note that his convention of homology $H_n(G)$ differs from $H_n(G, \underline {\mathbb Z})$ unless G is totally disconnected.)
Remark 3.3. In [Reference Proietti and YamashitaPY23, Theorem 5.1], we have identified Putnam’s homology groups for Smale spaces [Reference PutnamPut14, Reference ProiettiPro20] with the étale groupoid homology groups considered in this paper, where the groupoid is the unstable equivalence relation associated to a non-wandering Smale space with totally disconnected stable sets (this implies the associated groupoid is ample). In a companion paper [Reference Proietti and YamashitaPY22b] to the present one, we remove the hypothesis on the stable sets and prove the identification of homology groups for a general non-wandering Smale space. Combining this result with Theorem 3.1 above, we obtain a general Künneth formula for the product of two Smale spaces and their homology groups as defined by Putnam, generalizing [Reference Proietti and YamashitaPY23, Theorem 5.2] and [Reference Deeley, Killough and WhittakerDKW16, Theorem 6.5].
4. Poincaré duality
Suppose we have a locally compact principal groupoid $\tilde G$ such that the map $r \colon \tilde G \to \tilde {G}^{(0)}$ is a fibre bundle whose fibres are (homeomorphic to) $\mathbb {R}^n$ . We assume that there is a generalized transversal T in $\tilde G^{(0)}$ , so that $G = \tilde G|_T$ is an étale groupoid. In keeping with our previous instalment [Reference Proietti and YamashitaPY23, Definition 2.3], the notion of generalized transversal we use here is that introduced in [Reference Putnam and SpielbergPS99]. We also assume that groupoid homology for G is definable.
Example 4.1. A motivating example is the unstable groupoid of a Smale space whose unstable classes are homeomorphic to $\mathbb {R}^n$ . In this case, we get the fibre bundle structure and (generalized) transversal from the bracket maps, cf. [Reference PutnamPut00].
Under our assumption, the structure map $s \colon \tilde G \to \tilde G^{(0)}$ is a model of the universal principal $\tilde G$ -bundle $E \tilde G \to B \tilde G$ . Then the Baum–Connes conjecture suggests a close relation between $H_\bullet (G, \underline {\mathbb Z})$ and the compactly supported cohomology of the space $\tilde G^{(0)}$ . Let us make this precise in the framework of groupoid homology.
Let us consider the orbit-wise orientation sheaf $\underline {o}$ on $\tilde G^{(0)}$ . Formally, its stalk at x is given by
The étale space structure on the total space $\bigcup _{x \in G^{(0)}} \underline {o}_x$ is given by the local choice of orientations on the fibres $\tilde {G}^x$ . Namely, whenever $U \subset \tilde {G}^{(0)}$ is an open set such that $\tilde {G}^U$ is homeomorphic to $\mathbb {R}^n \times U$ as a space over U, this homeomorphism, together with the standard orientation on $\mathbb {R}^n$ , defines points $\sigma _x \in \underline {o}_x$ . We take the subsets of the form ${\{ \sigma _x : x \in U \}}$ for such U as a base of the topology on the total space. Note that $\underline {o}$ admits a global section (equivalently, it is trivializable) if and only if there is a global orientation on the orbits of $\tilde G$ .
By construction, this sheaf has a natural action of $\mathbb {Z}/2\mathbb {Z}$ . Moreover, when F is a sheaf of commutative groups on $\tilde {G}^{(0)}$ , we obtain another sheaf $F \times _{\mathbb {Z}/2\mathbb {Z}} \underline {o}$ of commutative groups, whose fibres are $F_x \times _{\mathbb {Z}/2\mathbb {Z}} \underline {o}_x$ , endowed with the group structure $(a, \sigma ) + (b, \sigma ) = (a + b, \sigma )$ .
Theorem 4.2. Under the above setting, we have an isomorphism
Proof. Consider the G-space $E = \tilde G^T$ . Then we have a morphism of groupoid
induced by the range map $\tilde G^T \to T$ . We want to apply the constructions in [Reference Crainic and MoerdijkCM00, §4] to this setting. We have an equivalence between $G \ltimes E$ and $\tilde {G}^{(0)}$ , induced by the source map $s \colon E \to \tilde G^{(0)}$ . Let us consider the $(G \ltimes E)$ -sheaf $F = s^*(\underline {\mathbb Z} \times _{\mathbb {Z}/2\mathbb {Z}} \underline {o})$ . This relates to the compactly supported cohomology in the claim by the Morita invariance of groupoid homology [Reference Crainic and MoerdijkCM00, Corollary 4.6] and the fact that groupoid homology of a space as a trivial groupoid is just the compactly supported sheaf cohomology up degree inversion, which gives
We will compute the left-hand side through its left derived functors $L_k \phi _!$ and get $H_k(G, \underline {\mathbb Z})$ . Recall that the stalks of $L_k \phi _! F$ can be computed as groupoid homology of the groupoid $x/ \phi $ using equation (1). By our assumption on $\tilde G$ , the object space of $x / \phi $ can be identified with the disjoint union of $\tilde G^y$ for $y \in G_x$ . Given objects $g \in \tilde G^y$ and $g' \in \tilde G^z$ in $x / \phi $ , there is an arrow from g to $g'$ if and only if $g = g" g'$ for the unique $g" \in \tilde G^y_z$ . In particular, $x / \phi $ is Morita equivalent to the space $\tilde G^x \cong \mathbb {R}^n$ .
If we restrict the pullback sheaf $\pi _x^* F$ to $\tilde G^x$ , we get $s^*(\underline {\mathbb Z} \times _{\mathbb {Z}/2\mathbb {Z}} \underline {o})$ . Choosing a homeomorphism between $\tilde {G}^x$ and $\mathbb {R}^n$ , this is isomorphic to $\underline {\mathbb Z}$ . We then have
where we used the standard orientation on $\mathbb {R}^n$ to get the second isomorphism. Thus, the G-sheaf $L_k \phi _! F$ on $G^{(0)} = T$ has the stalks isomorphic to $\mathbb {Z}$ when $k = -n$ , and we have $L_k \phi _! F = 0$ otherwise. The next step is to check that $L_k \phi _! F$ is isomorphic to $\underline {\mathbb Z}$ as a G-sheaf.
Since we already know the fibres to be isomorphic to $\mathbb {Z}$ , it is enough to check that these isomorphisms can be chosen in a consistent way. The isomorphism in equation (3) was defined on a choice of orientation on $\tilde {G}^x$ , or equivalently, a choice of element in the two-point set $\underline {o}_x$ , but we claim that the overall isomorphism is independent of the choice.
Suppose we choose the other element of $\underline {o}_x$ . On one hand, the isomorphism ${\pi _x^* F \simeq \underline {\mathbb Z}}$ is replaced by the negative of the original one. On the other, the isomorphism ${H_c^n(\mathbb {R}^n, \underline {\mathbb Z}) \simeq \underline {\mathbb Z}}$ is also affected in the same way if we use an orientation on $\mathbb {R}^n$ different from the first one. Overall, these two changes cancel with each other and give what we wanted.
The G-invariance is also straightforward from this presentation. When $g \in \tilde G^x_y$ , the induced map $\tilde {G}^y \to \tilde {G}^x$ gives an isomorphism $H_k(y/\phi , \pi _y^* \underline {\mathbb Z}) \to H_k(x/\phi , \pi _x^* \underline {\mathbb Z})$ that is identified with the identity map on $\mathbb {Z}$ up to equation (3).
Now, the Leray-type spectral sequence from [Reference Crainic and MoerdijkCM00, Theorem 4.4]
is degenerate at the $E^2$ -sheet for degree reasons. Thus, we get that $H_p(G, \underline {\mathbb Z})$ is isomorphic to $H_{p-n}(G \ltimes E, F)$ . Combined with equation (2), we obtain the assertion.
Remark 4.3. (Cf. [Reference Gelfand and ManinGM03, §III.5])
We used the Leray-type spectral sequence in the above proof, but more conceptually, $\mathcal L \phi _! F$ is quasi-isomorphic to the degree shift of $\underline {\mathbb Z}$ . To be more precise, there is a zig-zag of quasi-isomorphisms of chain complexes of G-sheaves between $\mathcal L \phi _! F$ and $\underline {\mathbb Z}[n]$ , where $\underline {\mathbb Z}[n]$ is the chain complex that has a copy of $\underline {\mathbb Z}$ at degree $-n$ , and $0$ elsewhere.
In general, suppose that $\mathcal A$ is an abelian category and consider the category $\mathcal D^-(\mathcal A)$ of homological complexes with terms in $\mathcal A$ that are bounded from below. We are interested in the case where $\mathcal A$ is the category of G-sheaves, and hence $\mathcal D^-(\mathcal A)$ is where the left derived functors $\mathcal L_\bullet \phi _!$ are defined. When an object $X_\bullet $ in $\mathcal D^-(\mathcal A)$ has homology concentrated in degree k, we can take the truncation $\tau (X)_\bullet $ defined by
Then the inclusion $\tau (X)_\bullet \to X_\bullet $ and the projection $\tau (X)_\bullet \to H_k(X_\bullet )[-k]$ are quasi- isomorphisms of complexes.
Remark 4.4. Let $(X, \phi )$ be a non-wandering Smale space whose unstable sets are contractible. In [Reference PutnamPut14, Question 8.3.2], Putnam conjectured that the stable homology $H^s_\bullet (X, \phi )$ is isomorphic to $H^{\bullet }(X)$ up to a degree shift. In view of Remark 3.3, Theorem 4.2 answers this conjecture in the affirmative if the unstable sets are homeomorphic to $\mathbb {R}^d$ and there is a consistent choice of orientation on these spaces. As pointed out in [Reference Amini, Putnam and Saeidi GholikandiAPSG17], the original conjecture is false without such orientability, and Theorem 4.2 provides a necessary modification for non-orientable cases.
4.1. Examples
4.1.1. Substitution tilings
In the case of groupoids for substitution tilings, the isomorphism of Theorem 4.2 appears in [Reference Proietti and YamashitaPY22a, §5.2].
4.1.2. Anosov diffeomorphisms
Let $\phi $ be an Anosov diffeomorphism $\phi $ of a compact manifold X, with dimension of stable sets n. By the stable manifold theorem [Reference Hirsch, Pugh, Chern and SmaleHP70] (see also [Reference Brin and StuckBS02, §5.6]), the stable set of any point $x \in X$ is an immersed copy of $\mathbb {R}^n$ . Then the monodromy groupoid of stable foliation on X satisfies the assumption for $\tilde G$ .
When X agrees with the set of the non-wandering points of $\phi $ , we have a non-wandering Smale space $(X, \phi )$ , and the above monodromy groupoid is just $R^s(X, \phi )$ .
Example 4.5. As a concrete example, let us consider the Smale spaces associated to hyperbolic toral automorphisms $(X,\phi )=(\mathbb {R}^2/\mathbb {Z}^2,A)$ , where A is a $2$ -by- $2$ matrix with integer entries and determinant equal to $1$ (see [Reference PutnamPut14, §7.4]). This implies the $\mathbb {R}$ -linear endomorphism associated to A descends to a map of the $2$ -torus. Note that A is called ‘hyperbolic’ when its eigenvalues $\unicode{x3bb} _1,\unicode{x3bb} _2$ satisfy $\unicode{x3bb} _1<1$ , $\unicode{x3bb} _2>1$ .
The stable and unstable orbits of A coincide with the lines spanned by the eigenvectors associated to $\unicode{x3bb} _1$ and $\unicode{x3bb} _2$ . Denote by $R^s(X,\phi )$ the stable equivalence relation associated to $(X,\phi )$ , suitably reduced to a transversal so that $R^s(X,\phi )$ is étale (in this case, a transversal is given for example by the $\unicode{x3bb} _2$ -eigenline). Applying Theorem 4.2, we obtain
Even though the HK conjecture is only formulated for ample groupoids, it is worth pointing out that this homology calculation corresponds (after periodicization) to the K-groups of the stable C $^*$ -algebra associated to $(X,\phi )$ . Indeed, this algebra is the foliation C $^*$ -algebra of the Kronecker flow along the $\unicode{x3bb} _1$ -eigenline, and hence it is Morita equivalent to the rotation algebra with angle the slope of the eigenline. The K-groups of this algebra are well known to be $\mathbb {Z}^2$ in both even and odd degree.
The Morita equivalence arises from an equivalence of groupoids as explained in detail in [Reference Putnam and SpielbergPS99, Ch. 3]. Since the homology groups are Morita invariant [Reference Crainic and MoerdijkCM00, §4], the calculation in equation (4) is valid for the topological groupoid $\mathbb {Z}\ltimes S^1$ underlying the irrational rotation algebra with angle the slope of the $\unicode{x3bb} _1$ -eigenline.
Example 4.6. Let $M = L / \Gamma $ be an infra-nilmanifold, that is, a quotient of a nilpotent, connected and simply connected Lie group L by a torsion-free group $\Gamma $ of affine automorphisms of L such that $\Lambda = L \cap \Gamma $ is a finite index subgroup of $\Gamma $ . Moreover, let $\psi $ be a hyperbolic affine automorphism of M [Reference DekimpeDek11]. Then $R^u(M, \psi )$ and $R^s(M,\psi )$ satisfy the assumption of Theorem 4.2. Indeed, an unstable set of $(M,\psi )$ can be identified with the subspace of the Lie algebra $\mathfrak {l}$ of L spanned by eigenvectors of the ‘linear part’ of $\psi $ corresponding to its eigenvalues bigger than $1$ , while a stable set can be identified with the span of the other eigenvectors.
Remark 4.7. If we denote by S the $C^*$ -algebra of the stable equivalence relation, Takai [Reference Takai, Araki and EffrosTak86] has conjectured that $K_i(S)$ is isomorphic to $K^{i+n}(X)$ , which can be understood as an instance of the Baum–Connes conjecture for foliations (for general formulations of the Baum–Connes conjecture for groupoids, see for example [Reference Bönicke and ProiettiBP24, Reference TuTu99]). Theorem 4.2 gives an affirmative answer to the homological version of the Takai conjecture.
4.1.3. Self-similar action
Another class of examples comes from the theory of self-similar actions, which is also closely related to Example 4.6.
Let $\Gamma $ be a finitely generated group, and $\phi $ be an injective and surjective contracting virtual endomorphism of $\Gamma $ . Then $\Gamma $ is virtually nilpotent and admits a self-similar action $(\Gamma , X)$ where the alphabet set X is a system of representatives of $\Gamma / \operatorname {\mathrm {dom}} \phi $ . Its limit $\Gamma $ -space $\mathcal X_{\Gamma , X}$ can be identified with a nilpotent connected and simply connected Lie group L on which $\Gamma $ acts by affine transformations in a proper and cocompact way [Reference NekrashevychNek05, §6.1].
In general, when $(\Gamma , X)$ is a contracting, recurrent and regular self-similar action, we have the associated Smale space ${\mathcal S}_{\Gamma , X}$ (the limit solenoid of $(\Gamma , X)$ ), and its unstable sets can be identified with $\mathcal X_{\Gamma , X}$ , see [Reference NekrashevychNek09]. Thus, under the above assumption on $(\Gamma , \phi )$ , the unstable groupoid $G = R^u({\mathcal S}_{\Gamma , X})$ satisfies the assumption of Theorem 4.2.
In the next section, we look at examples from compact Riemannian manifolds that fall in this setting.
5. Expanding maps on compact manifolds
Let us combine the Poincaré duality and transfer maps in homology to obtain a more elaborate computation of groupoid homology.
Suppose that M is an n-dimensional connected compact Riemannian manifold and $g \colon M \to M$ is an expanding map. Then g admits a fixed point x, $\Gamma = \pi _1(M, x)$ is a torsion-free group of polynomial growth (hence virtually nilpotent) and $\mathbb {R}^n$ is a universal cover for M, see [Reference NekrashevychNek05, §6.1].
With the virtual endomorphism $\phi $ represented by $g^{-1}$ , we are in the setting of §4.1.3. Then the Smale space ${\mathcal S}_{\Gamma , X}$ is given by $Y = \varprojlim _g M$ and the associated self-homeomorphism $\phi $ of Y. Again the groupoid $\tilde G = R^u(Y, \phi )$ is equivalent to the étale groupoid $G = \Gamma \ltimes \Omega $ , where $\Omega = \varprojlim \Gamma / g^i(\Gamma )$ , and we have
Let us write this as a group cohomology with coefficient.
Since G is a transformation groupoid, we also have
with respect to the induced action of $\Gamma $ on $C(\Omega , \mathbb {Z})$ . As M is a model of the Eilenberg–MacLane space $K(\Gamma , 1)$ , $\Gamma $ is a Poincaré duality group [Reference BrownBro94, §VIII.10], and we have
where D is an infinite cyclic group with the ‘sign’ representation of $\Gamma $ .
Again the equivalence between $\Gamma \ltimes (\Gamma / g^i (\Gamma ))$ and $g^i(\Gamma )$ leads to a presentation of these (co)homology groups as inductive limits of the groups $H_k(g^i(\Gamma ), \mathbb {Z}) \cong H_k(\Gamma , \mathbb {Z})$ with connecting maps given by the transfer maps. This corresponds to the isomorphism
that follows from the Baum–Connes conjecture for coefficients for $\Gamma $ , see also [Reference Chaiser, Coates-Welsh, Deeley, Farhner, Giornozi, Huq, Lorenzo, Oyola-Cortes, Reardon and StockerCCWD+23, Reference DeeleyDee23] for the case of flat manifolds.
5.1. Klein bottle
Let us describe a concrete example in the class of §4.1.3 that arises from a non-orientable surface. Consider the group
and its action on $\mathbb {R}^2$ given by $a(x,y)=(x+1,y)$ , $b(x,y)=(-x,y+1)$ . Let K be the orbit space of this action, which is a model of the Klein bottle space. Since $\mathbb {R}^2$ is the universal cover of K, we have $\Gamma \cong \pi _1(K, [0])$ , and $\mathbb {R}^2$ is a model of $E \Gamma $ .
Let g be the uniform scaling on $\mathbb {R}^2$ given by $g(x, y) = (3 x, 3 y)$ . This induces to an expanding endomorphism of K that fixes $[0]$ , which we denote again by g. The associated endomorphism of $\Gamma $ can be written as $g_*(a) = a^3$ , $g_*(b) = b^3$ up to the above identification, and the virtual endomorphism $\phi $ represented by $g^{-1}\colon g(\Gamma ) \to \Gamma $ satisfies the assumptions of §4.1.3.
Then, K can be identified with the limit space $\mathcal J_{\Gamma , X}$ of the associated self-similar action. Thus, the associated limit solenoid ${\mathcal S}_{\Gamma , X}$ is $Y = \varprojlim _g K$ , with the induced self-homeomorphism again denoted by $\phi $ . Moreover, the action of $\Gamma $ on the nilpotent Lie group L, which appears as the limit $\Gamma $ -space as above, is conjugate to the above action of $\Gamma $ on $\mathbb {R}^2$ by the universality of $\mathbb {R}^2$ as the total space of the classifying space for principal $\Gamma $ -bundles.
Let us write the coset spaces for the image of powers of g as $\Omega _i= \Gamma / g_*^i(\Gamma )$ . We have natural projection maps $\Omega _{i+1}\to \Omega _i$ and their projective limit $\Omega = \varprojlim _i \Omega _i$ . Then Y can be identified with the homotopy quotient $\Omega \times _K \mathbb {R}^2$ . Thus, the groupoid $\tilde G = R^u(Y, \phi )$ can be identified with the transformation groupoid of $\mathbb {R}^2$ acting on Y by translation. Taking the image T of $\Omega \times \{(0, 0)\}$ as the transversal, the associated étale groupoid $G = \tilde G|_T$ is the transformation groupoid of the canonical action of $\Gamma $ on $\Omega $ .
Turning to the groupoid homology, at degree $k = 0$ , we have
by [Reference Proietti and YamashitaPY23, Theorem 4.1 and Proposition 6.3]. More generally, as remarked in [Reference Proietti and YamashitaPY23, §6.2], $H_k(G, \underline {\mathbb Z})$ is the direct limit of group homology groups $H_k(\Gamma , \mathbb {Z})$ , where the connecting map is induced by the equivalence between $\Gamma \ltimes \Omega _i$ and $g^i(\Gamma ) \cong \Gamma $ . Concretely, these maps are the transfer maps of group homology,
see [Reference BrownBro94, §III.9].
At degree $k = 1$ , let us write
where the images of a and $g(a)$ become the generator of the summand $\mathbb {Z}/2\mathbb {Z}$ , and those of b and $g(b)$ become generators of $\mathbb {Z}$ and $3 \mathbb {Z}$ , respectively. The transfer map is given by
see for example [Reference BrownBro94, Exercise III.9.2].
Thus, the inductive system $(H_1(g^i(\Gamma ), \mathbb {Z}))_i$ can be identified with the constant system $\mathbb {Z} \oplus \mathbb {Z}/2\mathbb {Z}$ whose connecting map is given by $(x, y) \mapsto (3 x, y)$ . In particular, we obtain
For $k \ge 2$ , we have $H_k(G, \underline {\mathbb Z}) = 0$ . One way to see this is to use the isomorphism
given by Theorem 4.2. This forces $H_k(G, \underline {\mathbb Z}) = 0$ for $k> 2$ by degree reasons, and at $k = 2$ , we have
because there is no global orientation on the orbits of $\tilde G$ . (Alternatively, one can also use the duality between group homology and cohomology for $\Gamma $ to directly check ${H_k(\Gamma , \mathbb {Z}) = 0}$ , see [Reference BrownBro94, §VIII.10].)
6. Number theoretic generalization of solenoid
We have seen an example in the previous section where our duality theorem has a fairly straightforward application. Here we consider a slightly more complicated system, obtained by generalizing the notion of a solenoid. Although Theorem 4.2 cannot be applied directly, the argument in our proof of Poincaré duality will be useful in this setting too, leading us to a statement which is somewhat analogous to the Thom isomorphism.
Let us consider the following class of Smale spaces from [Reference SchmidtSchm95, §7]. Let $c \in \bar {\mathbb {Q}} \subset {\mathbb {C}}$ be an algebraic number such that $|g c| \neq 1$ for any element g in the absolute Galois group $\mathrm {Gal}(\bar {\mathbb {Q}} / \mathbb {Q})$ . Consider the algebraic number field $K = \mathbb {Q}(c)$ generated by c. For each place v of K, let us fix an absolute value function $|a|_v$ representing v. We denote the set of finite and infinite places of K by $P^K_f$ and $P^K_\infty $ , respectively.
Now, set
(By our assumption, each $v \in P^K_\infty $ satisfies $|c|_v \neq 1$ .) Then the ring
is a cocompact subring of the direct product of local fields $\prod _{v \in P(c)} K_v$ , and the quotient
with respect to the translation action can be identified with the Pontryagin dual of the additive group of $R_c$ .
Remark 6.1. To the best of our knowledge, there are some gaps in the proof of this duality presented in [Reference SchmidtSchm95]. An alternative argument is sketched in [Reference Enstad, Jakobsen, Luef and OmlandEJLO22, Proposition 6.8] and we also provide a shorter proof in Appendix A.
The periodic points of the self-homeomorphism
form a dense subset of $Y^{(c)}$ [Reference SchmidtSchm95, §5], and the $K_v$ -direction is contracting (respectively expanding) for $\phi $ if and only if $|c|_v < 1$ (respectively $|c|_v> 1$ ). Thus, the system $(Y^{(c)}, \phi )$ is a non-wandering Smale space.
Consider the subsets
of $P(c)$ . Then the stable equivalence class of each point of $Y^{(c)}$ is identified with $\prod _{v \in P^s(c)} K_v$ . Thus, the unstable groupoid of $(Y^{(c)}, \phi )$ is equivalent to the transformation groupoid $R_c \ltimes \prod _{v \in P^s(c)} K_v$ . Similarly, the stable groupoid is equivalent to $R_c \ltimes \prod _{v \in P^u(c)} K_v$ .
6.1. Groupoid homology
Let us compute the homology of the above groupoids. In the following, when omitted, the coefficient sheaf for homology is $\underline {\mathbb Z}$ .
Proposition 6.2. Let $P^s_f(c) = P^s(c) \cap P^K_f$ and $d = \sum _{v \in P^s(c) \cap P^K_\infty } \dim _{\mathbb {R}} K_v$ . We then have
Proof. As in the proof of Theorem 4.2, consider the groupoid homomorphism
induced by the projection of base space to the factors labelled by $P^s_f(c)$ . The fibres of this homomorphism can be identified with $\mathbb {R}^d$ , and the action of $R_c$ preserves orientation. We thus have $L_{-d} \pi _! \underline {\mathbb Z} \cong \underline {\mathbb Z}$ and $L_q \pi _! \underline {\mathbb Z} = 0$ for $q \neq -d$ on $ \prod _{v \in P^s_f(c)} K_v$ . Then the Leray-type spectral sequence (see [Reference Crainic and MoerdijkCM00, Theorem 4.4])
is degenerate at the $E^2$ -sheet for degree reasons, and we obtain the claim.
For each $v \in P^s_f(c)$ , let $O_v \subset K_v$ denote the corresponding local ring and $\pi _v \in O_v$ be a generator of its maximal ideal. Moreover, take $m_v \in \mathbb {N}_{>0}$ such that $|c|_v = |\pi _v|_v^{m_v}$ and set
We have a projective system of the discrete $R_c$ -spaces $X^{(n)}$ with proper connecting maps and
Now, $X^{(n)}$ is a transitive $R_c$ -set with stabilizer
We thus have $H_\bullet (R_c \ltimes X^{(n)}) \cong H_\bullet (\Gamma _n)$ and
can be identified with the inductive limit of the homology groups $H_k(\Gamma _n)$ with respect to the transfer maps $\theta \colon H_k(\Gamma _n) \to H_k(\Gamma _{n + 1})$ associated with the finite index inclusion $\Gamma _{n + 1} < \Gamma _n$ (see [Reference BrownBro94, Ch. III, §9]).
The transfer map can be made more explicit as follows. Let us denote by $m_{c^{-1}}$ the isomorphism $\Gamma _{n + 1} \to \Gamma _n$ given by multiplication by $c^{-1}$ .
Theorem 6.3. Let $N = [\Gamma _0 \colon \Gamma _1]$ and $\theta '$ be the endomorphism of $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ uniquely extending $N \mathop {\textstyle \bigwedge }\nolimits ^k m_{c^{-1}} \colon \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _1 \to \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ . The groupoid homology $H_k( R_c \ltimes \prod _{v \in P^s_f(c)} K_v )$ is isomorphic to the inductive limit $\varinjlim \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ for the connecting maps $\theta '$ .
Proof. First, since $\Gamma _n$ is a torsion-free commutative group, its integral homology $H_\bullet (\Gamma _n, \mathbb {Z})$ is naturally isomorphic to the exterior algebra $\mathop {\textstyle \bigwedge }\nolimits ^\bullet \Gamma _n$ generated by $\Gamma _n$ , see [Reference BrownBro94, Theorem V.6.4]. Second, the restriction of $\theta $ to $H_k(\Gamma _{n+1})$ is the multiplication map by N (this is easy to see by comparison with singular homology, see for example [Reference HatcherHat02, §3G]). Moreover, $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _n$ is torsion-free and $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _{n + 1}$ is its finite index subgroup. Thus, the transfer $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _n \to \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _{n + 1}$ is the unique extension of $N \mathrm {id}$ on $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _{n + 1}$ . Then the composition of transfer $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0 \to \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _1$ and the induced isomorphism $\mathop {\textstyle \bigwedge }\nolimits ^k m_{c^{-1}}$ gives $\theta '$ , and hence we obtain the claim.
Next let us consider the automorphism of groupoid homology induced by the homeomorphism $\phi $ . As the basepoint, we can take $(0)_v \in Y^{(c)}$ , which is fixed by $\phi $ . This choice of point is convenient because the action of $\phi $ preserves its stable equivalence class, which can be identified with the multiplication by c on $\prod _{v \in P^s(c)} K_v$ , and the groupoid automorphism of $R_c \ltimes \prod _{v \in P^s(c)} K_v$ again given by multiplication by c on all factors.
Note that the isomorphism of Proposition 6.2 is compatible with the automorphism induced by $\phi $ and the analogous one on $R_c \ltimes \prod _{v \in P^s_f(c)} K_v$ , up to multiplying by $-\mathrm {id}$ certain factors of $K_v$ when $K_v \cong \mathbb {R}$ . This is because multiplication by c on $K_v$ , for Archimedean v, is properly homotopic to the identity map $\mathrm {id}$ or possibly to $-\mathrm {id}$ when $K_v \cong \mathbb {R}$ .
Remark 6.4. In the above argument, we used the invariance of groupoid homology under proper homotopy, which can be obtained by considering the following chain of isomorphisms:
where we have group homology in the middle and the last group is the homology of the classifying space with coefficient of the local system induced by $\prod K_v$ . See also [Reference Kashiwara and SchapiraKS94, Proposition 2.7.5] for a proof of homotopy invariance in the setting of sheaves.
In [Reference PutnamPut14, Theorem 6.1.1], Putnam gave a remarkable analogue of the Lefschetz formula for non-wandering Smale spaces $(X, \phi )$ , which gives the number of periodic points as
using the transformation on his stable homology $H^s_k(X, \phi )$ induced by $\phi ^{-n}$ .
For the Smale spaces $(Y^{(c)}, \phi )$ , we can make both sides more explicit.
Proposition 6.5. For $n \ge 1$ , the fixed points for $\phi ^n$ on $Y^{(c)}$ are bijectively parametrized by
Proof. Since $Y^{(c)}$ is defined as the quotient space $\prod K_v/R_c$ , the points with period n have coordinates x (for all places v) such that $(c^n - 1) x$ belongs to $R_c$ . These points are parametrized by $(c^n - 1)^{-1} R_c) / (R_c \cap (c^n - 1)^{-1} R_c)$ . We can further simplify this quotient and arrive at the claim by writing $R_c$ as an increasing union of copies of $\mathcal O_K$ and noticing the compatibility of the quotient with the inductive structure.
Proposition 6.6. The automorphism of $H_k(R_c \ltimes \prod _{v \in P^s_f(c)} K_v) \otimes \mathbb {Q}$ induced by $\phi ^{-1}$ has trace equal to that of $N \mathop {\textstyle \bigwedge }\nolimits ^k m_{c^{-1}}$ acting on $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0 \otimes \mathbb {Q}$ .
Proof. Given $a \in \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ , let us write $a_j$ for the element of $\varinjlim \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ represented by a in the jth copy of $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ . Unpacking the correspondence in Theorem 6.3, the automorphism on $\varinjlim \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ induced by $\phi $ is the map $f(a_j) = a_{j + 1}$ for $a \in \mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ . Thus, $\phi ^{-1}$ induces the transform
and hence we obtain the claim.
6.2. Comparison with K-groups
The discussion above has a straightforward counterpart in K-theory for the corresponding groupoid $C^*$ -algebras. As we are working with the transformation groupoid $R_c \ltimes \prod _{v \in P^s(c)} K_v$ , the algebra of interest is the crossed product $C^*$ -algebra $C_0(\prod _{v \in P^s(c)} K_v)\rtimes R_c$ . (Of course, the stable case is analogous.)
Let us first see the analogue of Proposition 6.2 in K-theory.
Let $\mathbb {C}_d$ denote the (complex) Clifford algebra of the Euclidean space $\mathbb {R}^d$ , which is a $\mathbb {Z}_2$ -graded C $^*$ -algebra. Then there is a Dirac morphism, given by a (strictly) equivariant unbounded Fredholm module D between $C_0(\mathbb {R}^d) \otimes \mathbb {C}_d$ and $\mathbb {C}$ for the translation action of $\mathbb {R}^d$ on $C_0(\mathbb {R}^d)$ , and hence a class
This class $[D]$ is invertible by Connes’s Thom isomorphism theorem in $\operatorname {\mathrm {KK}}$ -theory [Reference Fack and SkandalisFS81]. This can be also interpreted as the strong Baum–Connes conjecture for $\mathbb {R}^d$ [Reference Higson and KasparovHK01] (see also [Reference Nishikawa and ProiettiNP20, Corollary 2.3]).
Now, with d as in Proposition 6.2, take the group embedding of $R_c$ into $\mathbb {R}^d$ corresponding to the completion at infinite places. Since D is given by a strictly $\mathbb {R}^d$ -equivariant Fredholm module, we obtain a class
Proposition 6.7. The class $[D_c]$ induces an isomorphism
Proof. Let us set $A= C_0(\prod _{v \in P^s(c)} K_v)$ and $B=C_0(\prod _{v \in P^s_f(c)} K_v$ ), so that we have ${A \cong B \otimes C_0(\mathbb {R}^d)}$ . Then $f=\mathrm {id}_B\otimes [D_c]$ defines an invertible class in $\operatorname {\mathrm {KK}}^{R_c}(A\otimes \mathbb {C}_d,B)$ .
Taking Kasparov’s descent (see [Reference KasparovKas88, pp. 172]), we see that the crossed product algebras $(A \otimes \mathbb {C}_d) \rtimes R_c = A \rtimes R_c \otimes \mathbb {C}_d$ and $B \rtimes R_c$ are KK-equivalent. Taking into account the degree-shift induced by the Clifford algebra, we obtain the claim.
Next let us present an analogue of Theorem 6.3. From the inverse limit presentation of equation (6), we obtain an inductive limit structure
For a $\mathbb {Z}$ -module A, let us define $\mathop {\textstyle \bigwedge }\nolimits ^{[i]}A$ as the direct sum of all exterior powers of A whose degree is congruent to i modulo $2$ . Analogously, we define $H_{[i]}(G,\underline {\mathbb Z})$ as the direct sum of homology groups with corresponding degree parity.
Theorem 6.8. There is an isomorphism
with connecting maps given by the unique extension of
as in Theorem 6.3.
Proof. On one hand, taking K-groups is compatible with taking inductive limits of $C^*$ -algebras. On the other, $X^{(n)}$ is a transitive $R_c$ -set with stabilizer equal to $\Gamma _n$ . Combining these and equation (7), we have
Next, let us identify $K_i(C^*(\Gamma _n))$ with $\mathop {\textstyle \bigwedge }\nolimits ^{[i]} \Gamma _n$ . Consider the subgroup
We then have $\bigcup _k \Gamma ^k_n = \Gamma _n$ . Each group $\Gamma ^k_n$ is finitely generated, because it is generated by a finite union of groups that are isomorphic to the ring of integers $\mathcal O_K$ , which is free abelian of rank equal to the degree of K. Thus $\Gamma ^k_n$ , being a torsion free and finitely generated commutative group, is also free abelian. From this, we obtain
As both sides are compatible with colimit, we obtain
It remains to identify the connecting map
Generally, suppose one has a finite index inclusion of discrete groups $H < G$ . Consider the G-equivariant map $\mathbb {C} \to C(G/H)$ and the induced map $C_r^*(G)\to C(G/H)\rtimes _r G$ . Then the map of the reduced group C $^*$ -algebras
can be computed as follows. We fix a system of representatives $(g_i)_{i = 1}^N$ of $G/H$ . Then we get an isomorphism
that sends $\delta _{g_i} \in C(G/H)$ to $e_{i,i} \in M_N$ and $\unicode{x3bb} _g \in C^*_r(G)$ to $\sum _i e_{\sigma _g(i),i}\otimes \delta _{h(i,g)}$ , where $\sigma _g(i)$ and $h(i,g)$ are characterized by the relation $g g_i = g_{\sigma _g(i)}h(i,g)$ .
If we use $G=\Gamma _n$ and $H=\Gamma _{n+1}$ , by commutativity, the image of $\unicode{x3bb} _g \in C^*_r(\Gamma _{n+1})$ under the map is a diagonal matrix embedding of g. Thus, equation (8) is an extension of the multiplication by N on $K_\bullet (C^*(\Gamma _{n+1}))$ . Taking into account the isomorphisms ${\mathop {\textstyle \bigwedge }\nolimits ^{[i]} \Gamma _n \cong \mathop {\textstyle \bigwedge }\nolimits ^{[i]} \Gamma _0}$ given by composition of $m_{c^{-1}}$ , we obtain the claim.
Corollary 6.9. The HK conjecture holds for the groupoid $R_c \ltimes \prod _{v \in P^s_f(c)} K_v$ .
Remark 6.10. While the original HK-conjecture [Reference MatuiMat16, Conjecture 2.6] was formulated for ample groupoids, it has an obvious generalization to the setting of étale groupoids. Unstable and stable groupoids of Smale spaces, reduced to transversal subspaces, give rich example of such groupoids. While there are counterexamples to the conjecture (in the ample case) [Reference DeeleyDee23, Reference ScarparoSca20], Corollary 6.9 holds for the stable and unstable groupoids of the Smale space $(Y^{(c)}, \phi )$ . In this case, Propositions 6.2 and 6.7 can be viewed as a reduction step to the ample case.
Remark 6.11. We note that there exist maps $\mu _i\colon H_i(G)\to K_i(C^*_rG)$ for $i = 0, 1$ in the case of ample groupoids, and they have been studied in [Reference Bönicke, Dell’Aiera, Gabe and WillettBDGW23, Reference MatuiMat12]. For higher degrees, one can still construct analogous maps by looking at kernels of higher differentials from the associated spectral sequence from [Reference Proietti and YamashitaPY22a].
Before looking at some examples, let us identify the positive cone of the $K_0$ -group appearing in Theorem 6.8. We are going to need the following cancellation theorem.
Lemma 6.12. [Reference HusemollerHus94, Theorem 9.1.2]
Let X be a CW-complex of dimension n and set $s=\lceil n/2\rceil $ . Every complex vector bundle E over X of rank $r \ge s$ splits as $E\cong E^\prime \oplus \varepsilon ^{r-s}$ , where $E'$ is a vector bundle over X and $\varepsilon ^k$ is the trivial bundle of rank k.
Let us write $\mathbb {Z}^+[1/N]$ for the subset of positive numbers in $\mathbb {Z}[1/N]$ (as a subset of $\mathbb {R}$ ).
Proposition 6.13. Suppose that $N> 1$ . Under the identification in Theorem 6.8, we have
Proof. Let us write $A_n = C_0(X^{(n)}) \rtimes R_c$ and $\phi ^{n+1}_n$ be the connecting map of the inductive system in equation (7). We further write
Let $x \in K_0(A_\infty )$ be a positive element. By the density of $\bigcup _n A_n$ in $A_\infty $ , we find a positive element $x_n^\prime \in K_0(A_n)$ . Using the Morita equivalence between $R_c\ltimes X^{(n)}$ and $\Gamma _n$ , we can further replace $x_n^\prime $ with $x_n\in K_0(C^*\Gamma _n)$ , which is still positive as the Morita equivalence preserves positivity. Again by density of $\bigcup _k C^* \Gamma _n^k$ in $C^* \Gamma _n$ for the subgroups $\Gamma _n^k$ from the proof of Theorem 6.8, we get a projection $p \in M_d(C^* \Gamma _n^k)$ whose class maps to x.
Now, under the isomorphism $K_0(C^*\Gamma ^k_{n})\cong \mathop {\textstyle \bigwedge }\nolimits ^{[0]}\Gamma ^k_{n}$ , the canonical tracial state $\tau _n$ of the group algebra $C^* \Gamma ^k_n$ picks up the constant term in $\mathop {\textstyle \bigwedge }\nolimits ^0 \Gamma ^k_n \cong \mathbb {Z}$ , which must be a positive integer for x. When we increase n, the action of $\phi ^{n+1}_n$ on the constant term is a multiplication by N. We thus obtain the degree $0$ part of x to be in $\mathbb {Z}^+[1/N]$ as claimed.
For the converse implication, let us take an element $x \in K_0(A_\infty )$ whose degree $0$ term belongs to $\mathbb {Z}^+[1/N]$ . By reasoning as above, we find its representative $x^k_n\in K_0(C^*\Gamma ^k_n)$ , whose degree $0$ part is positive in $\mathop {\textstyle \bigwedge }\nolimits ^0 \Gamma ^k_n \cong \mathbb {Z}$ .
Recall that $\Gamma ^k_n$ is free abelian. Moreover, its rank d is independent of n and (big enough) k, as we have $d = \dim _{\mathbb {Q}} \Gamma _n \otimes \mathbb {Q}$ . We thus have $C^*\Gamma ^k_n\cong C(\mathbb T^d)$ for all k and n. The degree $0$ part $\tau _n(x^k_n)$ of $x^k_n$ agrees with the rank of the corresponding vector bundle on $\mathbb T^d$ in the latter picture.
Now, take $m> 0$ big enough such that $r = \tau _{n+m}(\phi _{n,\ast }^{n+m}(x_n^k)) = N^{m} \tau _n(x)$ is larger than $s = \lceil d/2\rceil $ . Then the corresponding element in $K_0(C^* \Gamma _{n + m}^{\ell }) \cong K^0(\mathbb T^d)$ is represented by $[E] - [\varepsilon ^t]$ , where E is a vector bundle of rank $r + t$ . By Lemma 6.12, we have
which represents a positive element. This completes the proof.
6.3. Degree $1$ case
The case of $K(c) = \mathbb {Q}$ , that is, $c \in \mathbb {Q}$ , is considered by Burke and Putnam [Reference Burke and PutnamBP17], where they computed the stable and unstable Putnam homology groups. To simplify the presentation, let us consider the case of two prime factors as follows.
Let $p < q$ be two prime numbers and put $M = p q$ , $c = {q}/{p}$ , $R_c = \mathbb {Z}[1/M]$ . Our Smale space is given by the compact space
and the self-homeomorphism $\phi ([x, y, z]) = [ c x, c y, c z ]$ . Consequently, the étale groupoid $G = R^u(X, \phi )|_{X^s(x_0)}$ is the transformation groupoid $R_c \ltimes \mathbb {Q}_q$ , while ${G' = R^s(X, \phi )|_{X^u(x_0)}}$ is the transformation groupoid $R_c \ltimes (\mathbb {R} \times \mathbb {Q}_p)$ .
Now the subgroup $\Gamma _n < R_c$ is given by
and hence we have $N = q$ . The exterior algebra $\mathop {\textstyle \bigwedge }\nolimits ^\bullet \Gamma _0$ is
The map $N \mathop {\textstyle \bigwedge }\nolimits ^k m_{c^{-1}}$ is a multiplication by q for $k = 0$ and a multiplication by p for $k = 1$ . We thus obtain
This gives a description of the groupoid homology $H_k(G, \underline {\mathbb Z})$ .
As for the stable equivalence relation, Proposition 6.2 gives
This, together with equation (9) (switching the role of p and q) gives
Remark 6.14. In view of Remark 3.3, the computation of Putnam’s stable homology for $(Y^{(c)}, \phi )$ with $c \in \mathbb {Q}$ carried out in [Reference Burke and PutnamBP17] already gives equation (9). The method presented here is arguably more direct and does not require a deep understanding of the dynamics of the system; however, the method in [Reference Burke and PutnamBP17] is intimately tied to Markov partitions and allows to understand the system in terms of symbolic coding.
6.4. Degree $2$ , imaginary case
Next consider the case of quadratic extensions. To illustrate the situation associated with prime ideals that are not singly generated, we look at the case $K = \mathbb {Q}(\sqrt {-5})$ , so that $\mathcal O_K = \mathbb {Z}[\sqrt {-5}]$ is not a principal ideal domain.
To achieve this, let us take
The relevant finite places of K correspond to the prime ideals
For $i = 1, 2$ , we write $v_i$ for the corresponding place, $|a|_i$ for the absolute value, $K_i$ for the completed local field, $O_i < K_i$ for the local ring and $\pi _i \in O_i$ for the uniformizer. Thus, $x \in \mathcal O_K$ has absolute value $|x|_i = |\pi _i|_i^m$ if and only if the principal ideal $(x) < \mathcal O_K$ decomposes as
for some ideal $\mathfrak a < \mathcal O_K$ such that $\mathfrak p_i \mathbin {\not |} \mathfrak a$ . Then,
imply that we have
Writing $v_\infty $ for the unique infinite place, we have
From this, we get
(We cannot have $3$ in the denominator as the prime ideal $\mathfrak p_3 = (3, 1 - \sqrt {-5})$ satisfies $(3) = \mathfrak p_2 \mathfrak p_3$ , and hence $\frac 13$ would have absolute value bigger than $1$ for the corresponding absolute value.) Thus, the Smale space $(Y^{(c)}, \phi )$ is given by
and the diagonal action of c gives
for $x_0 = (0, 0, 0)$ .
To compute its groupoid homology for unstable groupoid, the general constructions from above give
Thus, $\Gamma _0 / \Gamma _1 \cong \mathcal O_K / \mathfrak p_2$ and we have $N = [\Gamma _0 : \Gamma _1] = 3$ . As for the exterior algebra, we have
Similarly, we have
as $\mathfrak p_2$ is free of rank $2$ as a $\mathbb {Z}$ -module, with basis $3$ and $1 + \sqrt {-5}$ .
Next let us identify the extensions of
to $\mathop {\textstyle \bigwedge }\nolimits ^k \Gamma _0$ and the inductive limit with respect to this map. When $k = 0$ , by $N = 3$ , it is the multiplication by $3$ on $\mathbb {Z}$ , and hence the limit is $\mathbb {Z}[\frac 13]$ . When $k = 1$ , it is the multiplication by $6 / (1 + \sqrt {-5}) = 1 - \sqrt {-5}$ , and hence the limit is
When $k = 2$ , this map is
However, in $\mathop {\textstyle \bigwedge }\nolimits ^2 \Gamma _0$ , we have
Hence, the above map extends to multiplication by $2$ on $\mathop {\textstyle \bigwedge }\nolimits ^2 \Gamma _0$ . Thus, the limit is given by
Summarizing, the étale groupoid $G = R^u(X, \phi )|_{X^s(x_0)}$ has the integral homology groups
on which the induced action of $\phi ^{-1}$ is respectively by multiplication by $3$ , $1 - \sqrt {-5}$ and $2$ .
The homology of $G' = R^s(X, \phi )|_{X^u(x_0)}$ can be computed in a similar way as above, combined with Proposition 6.2, and we get
on which the induced action of $\phi $ is respectively by multiplication by $2$ , $1 + \sqrt {-5}$ and $3$ , and $H_k(G^\prime ) = 0$ for $k \neq 0, -1, -2$ .
6.5. Degree $2$ , real case
Let us next consider a case with non-trivial unit. We look at a unit in a real quadratic field.
Concretely, let us take
and hence $K = \mathbb {Q}(\sqrt {5})$ . In this case, we find that c is invertible in $\mathcal O_K = \mathbb {Z}[c]$ , with $-c^{-1}$ being the Galois conjugate of c (the non-trivial automorphism of K is given by $a+b\sqrt {5}\mapsto a-b\sqrt {5}$ ).
Then the relevant places are the Archimedean places $v_\infty $ and $v_\infty '$ , for which the corresponding absolute values are the usual one and its twist by the automorphism of K that sends c to $-c^{-1}$ . Thus, we have $R_c = \mathcal O_K$ , which is isomorphic to $\mathbb {Z}^2$ as a commutative group. Then $Y^{(c)}$ , being its Pontryagin dual, can be identified with $\mathbb T^2$ .
As a Smale space, the corresponding homeomorphism $\phi $ is induced by the matrix presentation of multiplication by c on $\mathcal O_K \cong \mathbb {Z}^2$ , that is,
This way, we obtain the hyperbolic toral automorphism as in Example 4.5.
In this case, Theorem 4.2 for $\tilde G = R^u(X, \phi )$ and $G = R^u(X, \phi )|_{X^s(x_0)}$ gives
with $\phi ^{-1}$ acting by $\pm 1$ for $k = \pm 1$ and by $-c^{-1}$ on $\mathcal O_K$ for $k = 0$ . We also have an analogous presentation of homology for $G = R^s(X, \phi )|_{X^u(x_0)}$ .
Remark 6.15. In general, the conjugacy classes of hyperbolic matrices in $\mathrm {SL}_n(\mathbb {Z})$ with distinct eigenvalues bijectively correspond to the ideal classes in the integer rings of certain totally real fields, see [Reference Latimer and MacDuffeeLM33, Reference TausskyTau49, Reference WallaceWal84].
Acknowledgments
V.P. is supported by Foreign Young Talents’ grant (National Natural Science Foundation of China), CREST Grant Number JPMJCR19T2 (Japan Science and Technology Agency of the Ministry of Education, Culture, Sports, Science and Technology), and Marie Skłodowska-Curie Individual Fellowship (project number 101063362). M.Y. is funded, in part, by The Research Council of Norway [project 300837]. Part of the work for this project was carried out during M.Y.’s stay at the Research Institute for Mathematical Sciences (RIMS), Kyoto University, Japan. He thanks N. Ozawa and others at RIMS for their hospitality. The authors would like to thank C. Bruce and T. Omland for comments on duality and S-integers.
A Appendix. Duality for S-integers
Let K be a number field. We denote a place of K by v, and the associated absolute value by $|a|_v$ for $a \in K$ . The associated completed local field is denoted by $K_v$ , while (for a finite place v) its maximal compact subring is denoted by $O_v$ . Then the adele ring of K is given by the restricted product
of the $K_v$ relative to $O_v$ .
Let S be a finite set of places of K which contains all the infinite places. We denote by $R_S$ the ring of S-integers, that is,
and by $\mathbb A_{K,S}$ the ring of S-adeles, that is,
In the setting of §6, we take $S = P(c)$ so that $R_c = R_S$ .
When G is a locally compact commutative group, we denote its Pontryagin dual by $\hat {G}$ . Our goal is to establish the following.
Theorem A.1. The dual $\hat R_S$ of the additive group of $R_S$ is isomorphic to
Let us recall some standard facts (see for example [Reference WeilWei74, §II.5]).
Proposition A.2. For any place v, there exists a self-duality pairing on $K_v$ such that the complement of $O_v$ is identified with $O_v$ itself.
Corollary A.3. The Pontryagin dual of $O_v$ is isomorphic to $K_v / O_v$ .
It is worth emphasizing that the isomorphism above is not canonical: it depends on a specific choice of pairing for $K_v$ . We will consider this choice fixed for the rest of the appendix. Having Proposition A.2, it is easy to prove the standard self-duality $\mathbb A_K\cong \hat {\mathbb A}_K$ of the adele ring. Then, considering the extension $K\to \mathbb A_K\to \mathbb A_K/K$ yields the following well-known result:
Proposition A.4. The dual group $\hat K$ is isomorphic to $\mathbb A_K / K$ .
Proposition A.5. (Strong approximation, cf. [Reference CasselsCas86, Theorem 10.4.1])
Let V be a set of places, and suppose that there is some place $v_0$ such that $v_0 \not \in V$ . Then the diagonal inclusion of K into the restricted product $\prod _{v \in V} (K_v, O_v)$ is dense.
The next proposition is the key to our proof.
Proposition A.6. There is an isomorphism
Proof. Let us first check that this is well-defined. For any given $a \in K$ , we have $|a|_v \le 1$ , hence $[a]_{K_v / O_v} = 0$ , except for finitely many places. This means that $([a]_{K_v / O_v})_{v \not \in S}$ indeed defines an element of $\bigoplus _{v \not \in S} K_v / O_v$ .
We then claim that this is a surjective homomorphism. Take an element $([x_v]_{K_v / O_v})_v \in \bigoplus _{v \not \in S} K_v / O_v$ , so that $x_v \in O_v$ except for finitely many v. In particular, $(x_v)_v$ is an element in the restricted product $\prod _{v \not \in S} (K_v, O_v)$ . By Proposition A.5, the open neighborhood ${K + \prod _v O_v}$ of K in $\prod _{v \not \in S} (K_v, O_v)$ contains $(x_v)_v$ . This implies that $([x_v]_{K_v / O_v})_v$ is in the image of K.
Finally, we check that the above homomorphism is injective. Take an element $a \in K$ that goes to $0 \in \bigoplus _{v \not \in S} K_v / O_v$ . This means that $|a|_v \le 1$ for $v \not \in S$ , which is the defining condition for $R_S$ , hence $[a]_{K / R_S} = 0$ .
Proposition A.7. The inclusion $\mathbb A_{K, S} \to \mathbb A_K$ induces an isomorphism ${\mathbb A_{K,S} / R_S \cong \mathbb A_K / K}$ .
Proof. Since $R_S = K \cap \mathbb A_{K, S}$ , the induced map $\mathbb A_{K,S} / R_S \to \mathbb A_K / K$ is injective. Let us check that it is also surjective, or equivalently, $\mathbb A_K = K + \mathbb A_{K, S}$ . Observe that $\mathbb A_K / \mathbb A_{K, S}$ is isomorphic to $\bigoplus _{v \not \in S} K_v / O_v$ . Then Proposition A.6 implies the claim.
Corollary A.8. There is an exact sequence
Proof of Theorem A.1
Taking the dual groups of the terms in the exact sequence (A.1), it is enough to check that the dual of $\prod _{v \not \in S} O_v$ is isomorphic to $K / R_S$ . By Corollary A.3, this dual is isomorphic to $\bigoplus _{v \not \in S} K_v / O_v$ . Again Proposition A.6 implies the claim.