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

$$ \begin{align*} G^{(n)} = \{(g_1, \dots, g_n) \in G^n \colon s(g_i) = r(g_{i+1}) \}. \end{align*} $$

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

$$ \begin{align*} C_i^j = \Gamma_c(G^{(i)}, s^* F^j), \end{align*} $$

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

$$ \begin{align*} C_{i,j}^k = \Gamma_c(G^{(i)}, s^* F_j^k), \end{align*} $$

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

$$ \begin{align*} \mathbb{H}_\bullet(G, F_\bullet) \cong \mathbb{H}_\bullet(H, f^* F_\bullet) \end{align*} $$

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

$$ \begin{align*} B_n(\phi, F)_x = \Gamma_c((x/\phi)^{(n)}, s^* \pi_x^* F). \end{align*} $$

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]

(1) $$ \begin{align} (L_n \phi_! F_{\bullet})_{x} = H_n( x / \phi , \pi_x^* F_\bullet). \end{align} $$

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.

4.1. Examples

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

(4) $$ \begin{align} H_{-1}(R^s(X,\phi))\cong \mathbb{Z},\quad H_{0}(R^s(X,\phi))\cong \mathbb{Z}^2,\quad H_{1}(R^s(X,\phi))\cong \mathbb{Z}. \end{align} $$

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.

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

$$ \begin{align*} \Gamma =\langle a,b \mid b^{-1}ab=a^{-1}\rangle \end{align*} $$

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

$$ \begin{align*} H_0(G, \underline{\mathbb Z}) \cong \mathbb{Z}[1/9] = \mathbb{Z}[1/3] \end{align*} $$

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,

$$ \begin{align*} H_k(\Gamma, \mathbb{Z}) \to H_k(g^i(\Gamma), \mathbb{Z}), \end{align*} $$

see [Reference BrownBro94, §III.9].

At degree $k = 1$ , let us write

$$ \begin{align*} H_1(\Gamma, \mathbb{Z}) = \Gamma^{\mathrm{ab}} \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, H_1(g(\Gamma), \mathbb{Z}) = g(\Gamma)^{\mathrm{ab}} \cong 3 \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, \end{align*} $$

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

$$ \begin{align*} H_1(\Gamma, \mathbb{Z}) \to H_1(g(\Gamma), \mathbb{Z}),\quad [a] \mapsto [g(a)],\quad [b] \mapsto 3 [g(b)], \end{align*} $$

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

$$ \begin{align*} H_1(G, \underline{\mathbb Z}) \cong \mathbb{Z}[1/3] \oplus \mathbb{Z}/2\mathbb{Z}. \end{align*} $$

For $k \ge 2$ , we have $H_k(G, \underline {\mathbb Z}) = 0$ . One way to see this is to use the isomorphism

$$ \begin{align*} H_k(G, \underline{\mathbb Z}) \cong H^{2-k}(Y, \underline{\mathbb Z} \times_{\mathbb{Z}/2\mathbb{Z}} \underline{o}) \end{align*} $$

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

$$ \begin{align*} H^0(Y, \underline{\mathbb Z} \times_{\mathbb{Z}/2\mathbb{Z}} \underline{o}) = \Gamma(Y, \underline{\mathbb Z} \times_{\mathbb{Z}/2\mathbb{Z}} \underline{o}) = 0 \end{align*} $$

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].)

6.1. 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

$$ \begin{align*} H_p\bigg( R_c \ltimes \prod_{v \in P^s(c)} K_v \bigg) \cong H_{p + d}\bigg( R_c \ltimes \prod_{v \in P^s_f(c)} K_v \bigg). \end{align*} $$

Proof. As in the proof of Theorem 4.2, consider the groupoid homomorphism

$$ \begin{align*} \pi\colon R_c \ltimes \prod_{v \in P^s(c)} K_v \to R_c \ltimes \prod_{v \in P^s_f(c)} K_v \end{align*} $$

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])

$$ \begin{align*} E^2_{p q} = H_p\bigg( R_c \ltimes \prod_{v \in P^s_f(c)} K_v, L_q \pi_! \underline{\mathbb Z} \bigg) \Rightarrow H_{p+q}\bigg( R_c \ltimes \prod_{v \in P^s(c)} K_v, \underline{\mathbb Z} \bigg) \end{align*} $$

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

$$ \begin{align*} X^{(n)} = \prod_{v \in P^s_f(c)} K_v / \pi_v^{n m_v} O_v. \end{align*} $$

We have a projective system of the discrete $R_c$ -spaces $X^{(n)}$ with proper connecting maps and

(6) $$ \begin{align} R_c \ltimes \prod_{v \in P^s_f(c)} K_v = \varprojlim_n R_c \ltimes X^{(n)}. \end{align} $$

Now, $X^{(n)}$ is a transitive $R_c$ -set with stabilizer

$$ \begin{align*} \Gamma_n = \{a \in R_c \mid \text{for all } v \in P^s_f(c) \colon |a|_v \le |c|_v^{n} \}. \end{align*} $$

We thus have $H_\bullet (R_c \ltimes X^{(n)}) \cong H_\bullet (\Gamma _n)$ and

$$ \begin{align*} H_k\bigg( R_c \ltimes \prod_{v \in P^s_f(c)} K_v \bigg) \cong \varinjlim_n H_k(R_c \ltimes X^{(n)}) \end{align*} $$

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 $c^{-1}$ .

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:

$$ \begin{align*} H_k\bigg( R_c \ltimes \prod K_v,\underline{\mathbb Z} \bigg)\cong H_k\bigg(R_c,C\bigg( \prod K_v ,\mathbb{Z}\bigg)\bigg)\cong H_k\bigg(B R_c,C\bigg( \prod K_v ,\mathbb{Z}\bigg)\bigg), \end{align*} $$

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

$$ \begin{align*} |\{ x \in X \mid \phi^n(x) = x \}| = \sum_k (-1)^k \operatorname{\mathrm{Tr}}_{H^s_k(X, \phi) \otimes \mathbb{Q}}((\phi^{-n})_*) \end{align*} $$

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

$$ \begin{align*} (c^n- 1)^{-1} \mathcal O_K / (\mathcal O_K \cap (c^n - 1)^{-1} \mathcal O_K ). \end{align*} $$

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

$$ \begin{align*} \tilde f(a_j) = a_{j - 1} = N (\mathop{\textstyle\bigwedge}\nolimits^k m_{c^{-1}})(a)_j, \end{align*} $$

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

$$ \begin{align*} [D] \in \operatorname{\mathrm{KK}}^{\mathbb{R}^d}(C_0(\mathbb{R}^d) \otimes \mathbb{C}_d, \mathbb{C}) = \operatorname{\mathrm{KK}}^{\mathbb{R}^d}_d(C_0(\mathbb{R}^d), \mathbb{C}). \end{align*} $$

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

$$ \begin{align*} [D_c] \in \operatorname{\mathrm{KK}}^{R_c}_d(C_0(\mathbb{R}^d),\mathbb{C}). \end{align*} $$

Proposition 6.7. The class $[D_c]$ induces an isomorphism

$$ \begin{align*} K_{\bullet} \bigg( C_0\bigg(\prod_{v \in P^s(c)} K_v\bigg)\rtimes R_c\bigg)\cong K_{\bullet+d} \bigg( C_0\bigg(\prod_{v \in P^s_f(c)}K_v\bigg) \rtimes R_c \bigg). \end{align*} $$

Next let us present an analogue of Theorem 6.3. From the inverse limit presentation of equation (6), we obtain an inductive limit structure

(7) $$ \begin{align} C_0\bigg(\prod_{v \in P^s_f(c)} K_v\bigg) \rtimes R_c\cong \varinjlim_n C_0(X^{(n)})\rtimes R_c. \end{align} $$

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

$$ \begin{align*} K_i\bigg(C_0\bigg(\prod_{v \in P^s_f(c)} K_v\bigg) \rtimes R_c\bigg)\cong \varinjlim \mathop{\textstyle\bigwedge}\nolimits^{[i]} \Gamma_0, \end{align*} $$

with connecting maps given by the unique extension of

$$ \begin{align*} N\bigwedge^{[i]}m_{c^{-1}}\colon \bigwedge^{[i]} \Gamma_1\to \bigwedge^{[i]}\Gamma_0 \end{align*} $$

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

$$ \begin{align*} K_i\bigg(C_0\bigg(\prod K_v\bigg) \rtimes R_c\bigg)\cong \varinjlim_n K_i(C^*(\Gamma_n)). \end{align*} $$

Next, let us identify $K_i(C^*(\Gamma _n))$ with $\mathop {\textstyle \bigwedge }\nolimits ^{[i]} \Gamma _n$ . Consider the subgroup

$$ \begin{align*} \Gamma_n^k = \{a\in\Gamma_n \mid \text{for all } v\in P^u_f(c) \colon |a|_v \le |c|_v^{k} \} < \Gamma_n. \end{align*} $$

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

$$ \begin{align*} K_i(C^*(\Gamma^k_n)) \cong \mathop{\textstyle\bigwedge}\nolimits^{[i]}\Gamma^k_n. \end{align*} $$

As both sides are compatible with colimit, we obtain

$$ \begin{align*} K_i(C^*(\Gamma_n)) \cong \mathop{\textstyle\bigwedge}\nolimits^{[i]} \Gamma_n. \end{align*} $$

It remains to identify the connecting map

(8) $$ \begin{align} K_i(C^*(\Gamma_n)) \to K_i(C^*(\Gamma_{n + 1})). \end{align} $$

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

$$ \begin{align*} K_\bullet(C^*_r(G)) \to K_\bullet(C(G/H)\rtimes_r G) \cong K_\bullet(C^*_r(H)) \end{align*} $$

can be computed as follows. We fix a system of representatives $(g_i)_{i = 1}^N$ of $G/H$ . Then we get an isomorphism

$$ \begin{align*} C(G/H) \rtimes_r G \cong M_N(\mathbb{C})\otimes C_r^*(H) \end{align*} $$

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.

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.

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

$$ \begin{align*} K_0\bigg(C_0\bigg(\prod_{v \in P^s_f(c)} K_v\bigg) \rtimes R_c\bigg)^+ \cong \mathbb{Z}^+[1/N]\oplus \bigoplus_{i\geq 1} \varinjlim \mathop{\textstyle\bigwedge}\nolimits^{2i} \Gamma_0. \end{align*} $$

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

$$ \begin{align*} A_\infty = C_0\bigg(\prod_{v \in P^s_f(c)} K_v\bigg) \rtimes R_c. \end{align*} $$

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

$$ \begin{align*} [E]-[\varepsilon^t] = [E^\prime]+[\varepsilon^{r + t - s}] -[\varepsilon^t] = [E^\prime\oplus \varepsilon^{r - s}], \end{align*} $$

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

$$ \begin{align*} X = Y^{(c)} = (\mathbb{R} \times \mathbb{Q}_p \times \mathbb{Q}_q) / R_c \end{align*} $$

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

$$ \begin{align*} \Gamma_n = \bigg\{ \frac{a q^n}{p^k}\, \bigg\vert\, a \in \mathbb{Z}, k \in \mathbb{N} \bigg\}, \end{align*} $$

and hence we have $N = q$ . The exterior algebra $\mathop {\textstyle \bigwedge }\nolimits ^\bullet \Gamma _0$ is

$$ \begin{align*} \mathop{\textstyle\bigwedge}\nolimits^0 \Gamma_0 = \mathbb{Z},\quad \mathop{\textstyle\bigwedge}\nolimits^1 \Gamma_0 = \mathbb{Z}\bigg[\frac1p\bigg],\quad \mathop{\textstyle\bigwedge}\nolimits^k \Gamma_0 = 0 \quad (k> 1). \end{align*} $$

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

(9) $$ \begin{align} H_0(R_c \ltimes \mathbb{Q}_q) &\cong \mathbb{Z}\bigg[\frac1q\bigg],& H_1(R_c \ltimes \mathbb{Q}_q) &\cong \mathbb{Z}\bigg[\frac1p\bigg],& H_k(R_c \ltimes \mathbb{Q}_q) &= 0 \quad (k \neq 0, 1). \end{align} $$

This gives a description of the groupoid homology $H_k(G, \underline {\mathbb Z})$ .

As for the stable equivalence relation, Proposition 6.2 gives

$$ \begin{align*} H_k(R_c \ltimes (\mathbb{R} \times \mathbb{Q}_p)) \cong H_{k + 1}(R_c \ltimes \mathbb{Q}_p). \end{align*} $$

This, together with equation (9) (switching the role of p and q) gives

$$ \begin{align*} H_{-1}(G', \underline{\mathbb Z}) \cong \mathbb{Z}\bigg[\frac1p\bigg],\quad H_0(G', \underline{\mathbb Z}) \cong \mathbb{Z}\bigg[\frac1q\bigg],\quad H_k(G', \underline{\mathbb Z}) = 0 \quad (k \neq 0, -1). \end{align*} $$

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

$$ \begin{align*} c = \frac{1 + \sqrt{-5}}{2}. \end{align*} $$

The relevant finite places of K correspond to the prime ideals

$$ \begin{align*} \mathfrak p_1 = (2, 1 + \sqrt{-5}),\quad \mathfrak p_2 = (3, 1 + \sqrt{-5}). \end{align*} $$

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

$$ \begin{align*} (x) = \mathfrak p_i^m \mathfrak a \end{align*} $$

for some ideal $\mathfrak a < \mathcal O_K$ such that $\mathfrak p_i \mathbin {\not |} \mathfrak a$ . Then,

$$ \begin{align*} (2) = \mathfrak p_1^2,\quad (1 + \sqrt{-5}) = \mathfrak p_1 \mathfrak p_2 \end{align*} $$

imply that we have

$$ \begin{align*} |c|_1 = |\pi_1|_1^{-1},\quad |c|_2 = |\pi_2|_2. \end{align*} $$

Writing $v_\infty $ for the unique infinite place, we have

$$ \begin{align*} P^s(c)= \{v_2\},\quad P^u(c)= \{v_\infty, v_1\}. \end{align*} $$

From this, we get

$$ \begin{align*} R_c = \bigg\{ \frac{x}{2^k (1 + \sqrt{-5})^k}\, \bigg\vert\, x \in \mathcal O_K, k \in \mathbb{N} \bigg\}. \end{align*} $$

(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

$$ \begin{align*} Y^{(c)} = (\mathbb{C} \times K_1 \times K_2) / R_c \end{align*} $$

and the diagonal action of c gives

$$ \begin{align*} R^u(X, \phi)|_{X^s(x_0)} \cong R_c \ltimes K_2,\quad R^s(X, \phi)|_{X^u(x_0)} \cong R_c \ltimes (\mathbb{C} \times K_1) \end{align*} $$

for $x_0 = (0, 0, 0)$ .

To compute its groupoid homology for unstable groupoid, the general constructions from above give

$$ \begin{align*} \Gamma_0 = \bigg\{ \frac{x}{2^k}\, \bigg\vert\, x \in \mathcal O_K, k \in \mathbb{N} \bigg\},\quad \Gamma_1 = \bigg\{ \frac{x}{2^k}\, \bigg\vert\, x \in \mathfrak p_2, k \in \mathbb{N} \bigg\}. \end{align*} $$

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

$$ \begin{align*} \mathop{\textstyle\bigwedge}\nolimits^2 \Gamma_0 = \bigg\{ \frac{1 \wedge \sqrt{-5}}{2^j} k\, \bigg\vert\, j \in \mathbb{N}, k \in \mathbb{Z} \bigg\},\quad \mathop{\textstyle\bigwedge}\nolimits^k \Gamma_0 = 0 \quad (k> 2). \end{align*} $$

Similarly, we have

$$ \begin{align*} \mathop{\textstyle\bigwedge}\nolimits^2 \Gamma_1 = \bigg\{ \frac{3 \wedge (1 + \sqrt{-5})}{2^j} k\, \bigg\vert\, j \in \mathbb{N}, k \in \mathbb{Z} \bigg\} \end{align*} $$

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

$$ \begin{align*} N \mathop{\textstyle\bigwedge}\nolimits^k m_{c^{-1}} \colon \mathop{\textstyle\bigwedge}\nolimits^k \Gamma_1 \to \mathop{\textstyle\bigwedge}\nolimits^k \Gamma_0 \end{align*} $$

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

$$ \begin{align*} \varinjlim \Gamma_0 = \mathbb{Z}\bigg[\sqrt{-5}, \frac12, \frac1{1 - \sqrt{-5}}\bigg]. \end{align*} $$

When $k = 2$ , this map is

$$ \begin{align*} \mathop{\textstyle\bigwedge}\nolimits^2 \Gamma_1 \to \mathop{\textstyle\bigwedge}\nolimits^2 \Gamma_0, \quad \frac{3 \wedge (1 + \sqrt{-5})}{2^j} k \mapsto 3 \frac{(1 - \sqrt{-5}) \wedge 2}{2^j} k = 6 \frac{1 \wedge \sqrt{-5}}{2^j} k. \end{align*} $$

However, in $\mathop {\textstyle \bigwedge }\nolimits ^2 \Gamma _0$ , we have

$$ \begin{align*} \frac{3 \wedge (1 + \sqrt{-5})}{2^j} k = 3 \frac{1 \wedge \sqrt{-5}}{2^j} k. \end{align*} $$

Hence, the above map extends to multiplication by $2$ on $\mathop {\textstyle \bigwedge }\nolimits ^2 \Gamma _0$ . Thus, the limit is given by

$$ \begin{align*} \varinjlim \mathop{\textstyle\bigwedge}\nolimits^2 \Gamma_0 \cong \mathbb{Z}[ \tfrac12]. \end{align*} $$

Summarizing, the étale groupoid $G = R^u(X, \phi )|_{X^s(x_0)}$ has the integral homology groups

$$ \begin{align*} H_0(G) \cong \mathbb{Z}\bigg[\frac13\bigg],\quad &H_1(G) \cong \mathbb{Z}\bigg[\sqrt{-5}, \frac12, \frac1{1 - \sqrt{-5}}\bigg],\quad H_2(G) \cong \mathbb{Z}\bigg[\frac12\bigg],\\ & H_k(G) = 0 \quad (k \neq 0, 1, 2), \end{align*} $$

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

$$ \begin{align*} H_{-2}(G^\prime) \cong \mathbb{Z}\bigg[\frac12\bigg],\quad H_{-1}(G^\prime ) \cong \mathbb{Z}\bigg[\sqrt{-5}, \frac12, \frac1{1 + \sqrt{-5}}\bigg],\quad H_0(G^\prime) \cong \mathbb{Z}\bigg[\frac16\bigg], \end{align*} $$

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

$$ \begin{align*} c = \frac{1 + \sqrt{5}}2, \end{align*} $$

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,

$$ \begin{align*} \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}. \end{align*} $$

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

$$ \begin{align*} H_k(G) \cong \begin{cases} \mathbb{Z} & (k = -1, 1),\\ \mathcal O_K \cong \mathbb{Z}^2 & (k = 0),\\ 0 & (\text{otherwise}), \end{cases} \end{align*} $$

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 $G = R^s(X, \phi )|_{X^u(x_0)}$ .