2. Notation and preliminaries

We start by introducing notational conventions for étale groupoids: see, for example, [Reference Sims and Perera58] for background on étale groupoids and their $C^*$-algebras. For a groupoid $\mathcal{G}$, we write $\mathcal{G}^{(0)}$ for the base space (also called the unit space) and $r,s:\mathcal{G}\to \mathcal{G}^{(0)}$ for the range and source maps. An ordered pair $(g, h) \in\mathcal{G}\times \mathcal{G}$ is composable if $s(g) = r(h)$ and its composition is denoted by $gh$. The inverse of $g \in \mathcal{G}$ is denoted $g^{-1}$. All groupoids considered in the present paper will be topological groupoids: we always assume the topologies are locally compact and Hausdorff. A bisection is a subset $B$ of $\mathcal{G}$ such that $r|_B:B\to r(B)$ and $s|_B:B\to s(B)$ are homeomorphisms. We will also always assume that $\mathcal{G}$ is étale, meaning that there is a basis for the topology consisting of open bisections. Under these assumptions, $\mathcal{G}^{(0)}$ is a closed and open subset of $\mathcal{G}$, and there is a canonical Haar system given by counting measures.

To a groupoid $\mathcal{G}$ satisfying the assumptions above, one can associate its reduced groupoid $C^*$-algebra using the method in [Reference Renault54]; see also [Reference Sims and Perera58] for an exposition focused on the (much simpler) special case of étale groupoids. The resulting $C^*$-algebra is denoted by $C^*_r(\mathcal{G})$.

A groupoid $\mathcal{G}$ is ample if its base space is totally disconnected (e.g. the Cantor set). It is principal if for each $x \in \mathcal{G}^{(0)}$, $s^{-1}(x)\cap r^{-1}(x)=\{x\}$. A subset $F$ of $\mathcal{G}^{(0)}$ is invariant if $s(r^{-1}(F))=F$; in this case, we write $\mathcal{G}|_F$ for $r^{-1}(F)$, which is a subgroupoid of $\mathcal{G}$ with the induced operations. Note that $\mathcal{G}|_F$ need not be locally compact or étale when equipped with the subspace topology; it does, however, have these properties if $F$ is open.

A $\mathcal{G}$-space is a locally compact Hausdorff topological space $X$ equipped with a continuous open map $\rho:X\to G^{(0)}$ called the anchor map and a continuous action map

\begin{equation*} \mathcal{G}_s\!\times_\rho X\to X \end{equation*}

denoted $(g,x)\mapsto gx$ that satisfies $(gh)x=g(hx)$ where defined; here $\mathcal{G}_s\!\times_\rho X$ denotes the fibred product $\{(g,x)\in \mathcal{G}\times X\mid s(g)=\rho(x)\}$ equipped with the topology it inherits from $\mathcal{G}\times X$.

Many of the groupoids considered in this paper are constructed from group actions. Let $X$ be a locally compact Hausdorff space equipped with an action of a discrete group $G$ ( $G$ will almost always be ${\mathbb Z}$ in this paper) by homeomorphisms. The associated transformation groupoid is denoted $G \lt imes X$. It is defined to be $G \times X $ as a topological space, with associated algebraic operations determined by:

1. (i) the base space is $\{e\}\times X\subseteq \mathcal{G}$ where $e$ is the identity element of ${G}$ (usually, we just identify the base space with $X$);

2. (ii) the range and source of $(\gamma,x)$ are $\gamma x$ and $x$, respectively;

3. (iii) the composition $(\gamma, \alpha x) \cdot (\alpha, y)$ is equal to $(\gamma \alpha, y)$;

4. (iv) the inverse of $(\gamma, x)$ is $(\gamma^{-1}, \gamma x)$.

We now recall the Crainic–Moerdijk homology of a groupoid $\mathcal{G}$ following [Reference Crainic and Moerdijk9, Section 3]. We will need to work in terms of sheaves: see, for example, [Reference Bredon3] or [Reference Kashiwara and Schapira31] for general background on sheaves and sheaf cohomology. We recall that a sheaf $\mathcal{S}$ over a space $X$ can be specified by either:

1. (i) for each open set, one specifies an abelian group of sections $\Gamma(U;\mathcal{S})$ satisfying appropriate conditions as the open sets vary (see, for example, [Reference Bredon3, pages 6-7] or [Reference Kashiwara and Schapira31, Definition 2.2.1]);

2. (ii) an étale space, which is a (possibly non-Hausdorff) topological space $\mathcal{S}$ equipped with a local homeomorphism $\pi:\mathcal{S}\to X$ satisfying appropriate conditions (see, for example, [Reference Bredon3, Definition 3.2]).

In particular, for us a sheaf will always mean a sheaf of abelian groups. For a sheaf $\mathcal{S}$ on a locally compact Hausdorff space $X$ and an open subset $U$ of $X$, we write $\Gamma_c(U;\mathcal{S})$ for the compactly supported sections of $\mathcal{S}$ over $U$: see, for example, [Reference Bredon3, pages 21-22] or [Reference Kashiwara and Schapira31, Line (2.5.2) on page 103].

The most important example of a sheaf for us will be the sheaf $\mathcal{Z}$ whose sections $\Gamma(U;\mathcal{Z})$ are locally constant functions from $U$ to ${\mathbb Z}$. Another useful example to bear in mind is the sheaf $\mathcal{R}$ whose sections $\Gamma(U;\mathcal{R})$ of continuous real-valued functions on $X$.

We recall some standard definitions.

For example, the sheaf $\mathcal{R}$ discussed above is $c$-soft. Indeed, given $K\subseteq U$ and $s\in \Gamma(U;\mathcal{R})$, choose open sets $V,W$ with $K\subseteq V\subseteq \overline{V}\subseteq W$ and $\overline{W}$ a compact subset of $\mathit{U}$. Use Urysohn’s lemma to choose a continuous function $g:X\to [0,1]$ that is equal to one on $V$, and zero outside $W$ (we use here that $\overline{W}$ is normal, even though $X$ might not be). Then define $t$ to be equal to $gs$ on $U$, and zero outside $U$. On the other hand, the sheaf $\mathcal{Z}$ is $c$-soft if $X$ is totally disconnected, but usually not for more general spaces.

We now specialize to equivariant sheaves on groupoids.

For example, the sheaves $\mathcal{Z}$ and $\mathcal{R}$ defined above are always $\mathcal{G}$-sheaves on $\mathcal{G}^{(0)}$. Note also that if $\mathcal{G}=G \lt imes X$ is a transformation groupoid associated to an action of a group $G$ on a locally compact space $X$, then a $\mathcal{G}$-sheaf on $\mathcal{G}^{(0)}=X$ is the same thing as a $G$-sheaf (also called an equivariant sheaf) on $X$ in the sense of [Reference Grothendieck25, Section 5].

Following [Reference Crainic and Moerdijk9, 1.7], for each $n\in \mathbb{N}$ write $\mathcal{G}^{(n)}$ for the collection of ‘composable strings’

\begin{equation*} x_0 \stackrel{g_1}{\leftarrow} x_1 \stackrel{g_2}{\leftarrow} \cdots \stackrel{g_n}{\leftarrow} x_n \end{equation*}

where $x_i\in \mathcal{G}^{(0)}$ and each $g_i\in \mathcal{G}$ satisfies $r(g_i)=x_{i-1}$ and $s(g_i)=x_i$. Note that $\mathcal{G}^{(0)}$ has the usual meaning, and $\mathcal{G}^{(1)}=\mathcal{G}$. We equip $\mathcal{G}^{(n)}$ with the topology it inherits as a subspace of $\mathcal{G}^n$ for $n\geq 1$, and the usual topology if $n=0$. For each $n$, define

(1)\begin{equation} \tau_n:\mathcal{G}^{(n)}\to \mathcal{G}^{(0)},\quad (x_0 \stackrel{g_1}{\leftarrow} \cdots \stackrel{g_n}{\leftarrow} x_n)\mapsto x_0 \end{equation}

and for a $\mathcal{G}$-sheaf $\mathcal{S}$ on $\mathcal{G}^{(0)}$, define

(2)\begin{equation} \mathcal{S}_n:=\tau_n^*\mathcal{S} \end{equation}

to be the pullback sheaf (also called the inverse image sheaf: see, for example, [Reference Bredon3, page 12] or [Reference Kashiwara and Schapira31, Definition 2.3.1]).

For the next definition, we work in terms of double complexes: for definiteness, we adopt the conventions on double complexes and associated total complexes in [Reference Weibel60, pages 7-9].

Definition 2.3. (Crainic and Moerdijk, [Reference Crainic and Moerdijk9, 3.4])

Let $\mathcal{A}$ be a $\mathcal{G}$-sheaf on $\mathcal{G}^{(0)}$. Following Crainic and Moerdijk [Reference Crainic and Moerdijk9, 3.3] (or use Lemma A.3) there exists a resolution

(3)\begin{equation} 0\to \mathcal{A}\to \mathcal{S}^0\to \mathcal{S}^1\to \cdots \end{equation}

of $\mathcal{A}$ by $c$-soft $\mathcal{G}$-sheaves; in other words, the sequence in line (3) is exact in the category of $\mathcal{G}$-sheaves, and each $\mathcal{S}^i$ is $c$-soft.

The groupoid homology of $\mathcal{G}$ with coefficients in $\mathcal{A}$, denoted $H^*(\mathcal{G};\mathcal{A})$, is the homology of the (direct sum total complex associated to the) double complex

(4)

where the vertical arrows are functorially induced from the resolution in line (3), and the horizontal arrows are as in [Reference Crainic and Moerdijk9, 1.7]. (We will not need the general definition, but discuss some special cases in Remark 2.6 below.)

Let us give some remarks on the definition.

Remark 2.6. We will not need to describe the horizontal arrows in line (4) in full generality, but let us mention a special case for relatively simple sheaves. Assume for simplicity that $\mathcal{S}$ is a sheaf of ${\mathbb{C}} $-valued functions (maybe satisfying some additional properties) on $\mathcal{G}^{(0)}$, whence each $\mathcal{S}^i$ also has sections given by ${\mathbb{C}} $-valued functions. For $n\geq 1$ and $i\in \{0,...,n\}$, following [Reference Crainic and Moerdijk9, 1.7], we define $d_i:\mathcal{G}^{(n)}\to \mathcal{G}^{(n-1)}$ by

\begin{equation*} d_i(x_0 \stackrel{g_1}{\leftarrow} \cdots \stackrel{g_n}{\leftarrow} x_n):=\left\{\begin{array}{ll} x_1 \stackrel{g_2}{\leftarrow} \cdots \stackrel{g_n}{\leftarrow} x_n~, & i=0 \\ x_0 \stackrel{g_1}{\leftarrow} \cdots x_{i-1} \stackrel{g_ig_{i+1}}{\longleftarrow} x_{i+1}\cdots \stackrel{g_n}{\leftarrow} x_n~, & 1\leq i \lt n \\ x_0 \stackrel{g_1}{\leftarrow} \cdots \stackrel{g_{n-1}}{\leftarrow} x_{n-1}~, & i=n\end{array}\right.. \end{equation*}

For $t\in \Gamma_c(\mathcal{G}^{(n)};\mathcal{S}_n^i)$ we define

\begin{equation*} (\delta_i t)(\underline{h}):=\sum_{\underline{g}\in d_i^{-1}(\underline{h})} t(\underline{g}) \end{equation*}

(this makes sense as $d_i$ is a local homeomorphism and $t$ is compactly supported), and $\delta:=\sum_{i=0}^n (-1)^i\delta_i$. Note in particular that for $t\in \Gamma_c(\mathcal{G}^{(1)};\mathcal{S}_1^i)$ and $x\in \mathcal{G}^{(0)}$ we have

(5)\begin{equation} (\delta t)(x)=\sum_{g\in s^{-1}(x)}t(g) - \sum_{g\in r^{-1}(x)}t(g). \end{equation}

Going back to generalities, we make the following notational conventions.

Definition 2.7. Define $H_*(\mathcal{G}):=H_*(\mathcal{G};\mathcal{Z})$. Write $H_{**}(\mathcal{G})$ for the ${\mathbb Z}/2$-graded homology theory with even and odd groups defined by

(6)\begin{equation} H_{ev}(\mathcal{G}):=\bigoplus_{m\in {\mathbb Z}} H_{2m}(\mathcal{G}) \quad \text{and}\quad H_{od}(\mathcal{G}):= \bigoplus_{m\in {\mathbb Z}} H_{2m+1}(\mathcal{G}). \end{equation}

We will also need to discuss compactly supported sheaf cohomology of a locally compact Hausdorff space $X$: see, for example, [Reference Bredon3] or [Reference Kashiwara and Schapira31] for background. The sheaf cohomology of $X$ can be defined by taking a resolution

\begin{equation*} 0\to \mathcal{Z}\to \mathcal{S}^0\to \mathcal{S}^1\to \cdots \end{equation*}

of the sheaf $\mathcal{Z}$ of locally constant ${\mathbb Z}$-valued functions on $X$ by $c$-soft $\mathcal{G}$-sheaves, and defining $H_c^*(X)$ to be the homology of the associated complex

\begin{equation*} \Gamma_c(X;\mathcal{S}^0)\to\Gamma_c(X;\mathcal{S}^1)\to \cdots \end{equation*}

of compactly supported sections. We will also use the notation $H^{**}_c(X)=H_c^{ev}(X)\oplus H^{od}_c(X)$ for the corresponding ${\mathbb Z}/2$ graded theory and its even and odd parts. If $X$ is compact, we will generally drop the subscript ‘ $_c$’ and just write $H^*(X)$. For ‘reasonable’ spaces $X$, sheaf cohomology agrees with any other standard definition of cohomology. For more exotic spaces, sheaf cohomology generally behaves better than other standard theories for our purposes: the interested reader might compare the sheaf cohomology of the Cantor space $X$ (for which $H^0(X)$ identifies with the group of continuous ${\mathbb Z}$-valued functions $C(X,{\mathbb Z})$) with its singular cohomology (for which $H^0(X)$ identifies with the group of all functions from $X$ to ${\mathbb Z}$).

We conclude this section with two basic examples of groupoid homology to help orient the reader. In both cases, the claimed identifications are true essentially by definition.

Combining these Examples 2.8 and 2.9, it is useful for intuition to think of groupoid homology as being built from ‘cohomology of spaces in negative degrees, and cohomology of groups in positive degrees’.

Let us finally mention some notational conventions on abstract (co)chain complexes in an abelian category (typically, the category of modules over a ring, or of $\mathcal{G}$-sheaves for some groupoid). We will typically write an abstract chain complex as $C_\bullet$, with notation like ‘ $C_{\bullet+1}$’ meaning the chain complex whose $i^\text{th}$ object is $C_{i+1}$, and similarly for ‘ $C_{-\bullet}$’ with $i^\text{th}$ object $C_{-i}$ and so on. We will also write $C^\bullet$ for cochain complexes. Our (co)chain complexes will always be indexed by ${\mathbb Z}$, but we will occasionally write something like ‘ $(C_q)_{q\geq 0}$’ for a chain complex: this should be taken to mean a chain complex indexed by ${\mathbb Z}$, where $C_q=0$ for $q \lt 0$.

A map $f:C_\bullet\to D_\bullet$ between chain complexes is a sequence of maps $f_q:C_q\to D_q$ that is compatible with the boundary maps. Such a map $f$ induces a map on homology, and $f$ is a quasi-isomorphism if the induced map on homology is an isomorphism.

3. A trace pairing and Elliott invariant for groupoids

Our main goal in this section is to define a pairing between the space of invariant probability measures on the base space of a groupoid and its zeroth homology group. From there, we define an ‘Elliott invariant’ for groupoids modelled on the usual Elliott invariant for $C^*$-algebras.

To motivate this, let us first recall a convenient form of the Elliott invariant for $C^*$-algebras. This is purely motivation, and will not be used in the rest of the paper.

Definition 3.1. Let $A$ be a unital $C^*$-algebra. Then its (weak) Elliot invariant is the quadruple

\begin{equation*} (K_*(A),[1]_K,T(A),\rho_K) \end{equation*}

where $K_*(A)=K_0(A)\oplus K_1(A)$ is the ${\mathbb Z}/2$-graded $K$-group of $A$, $T(A)$ is its tracial state space equipped with the weak- $*$ topology and affine structure it inherits from the dual $A^*$, $\rho_K:T(A)\to \text{Hom}(K_0(A),\mathbb{R})$ is the pairing between traces and $K$-theory, and $[1]_K\in K_0(A)$ is the class of the unit.

This is essentially (it is not actually the same, as those authors use the space $\text{Aff}(T(A))$ of continuous affine real-valued functions on $T(A)$ in place of $T(A)$; either of the versions from Definition 3.1 above or [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6, Definition 2.3] is recoverable from the other, however) the same thing as the invariant denoted $KT_u$ in [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6, Definition 2.3]. It is not the classical Elliott invariant, which also takes the order on $K$-theory into account. However, it agrees with the classical version for the so-called ‘classifiable’ $C^*$-algebras as defined below: see, for example, [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6, Discussion around Definitions 2.2 and 2.3].

The precise meanings of the list of adjectives appearing above are not important for this paper: readers who are not experts in $C^*$-algebra classification theory can just read ‘classifiable’ as ‘nice and well-studied’. If one is only looking at classifiable $C^*$-algebras then it is reasonable to drop the adjective ‘weak’ in ‘(weak) Elliott invariant’, which is why we have included it in parentheses; we will typically drop it.

The name ‘classifiable’ comes from the fact that these are exactly those $C^*$-algebras that are classified by their (weak) Elliott invariant, i.e. the $C^*$-algebras are isomorphic if and only if their Elliott invariants are isomorphic. This is the culmination of the 40-year programme outlined in [Reference Elliott19]. It is difficult to summarize the huge amount of work that went into this, but let us say that the programme was completed through a combination of [Reference Kirchberg32, Reference Phillips42] (in the purely infinite case) and [Reference Elliott, Gong, Lin and Niu20, Reference Tikuisis, White and Winter59] (in the stably finite case). The preprint [Reference Carrión, Gabe, Schafhauser, Tikuisis and White6] contains a historical summary, as well as a different approach in the stably finite case.

We now turn back to groupoids, and our main goal of pairing invariant measures with the zeroth homology group. Throughout the rest of this section $\mathcal{G}$ is a locally compact, Hausdorff, étale groupoid.

The following definition is based on [Reference Renault55, Definition 2.3.8 and Exercise 2.3.9].

Definition 3.3. For a Borel measure $\mu$ on $\mathcal{G}^{(0)}$, define a Borel measure $r^*\mu$ on $\mathcal{G}$ by setting

\begin{equation*} \int_{\mathcal{G}} f d(r^*\mu):=\int_{\mathcal{G}^{(0)}} \sum_{g\in r^{-1}(x)}f(g)d\mu(x) \end{equation*}

for all $f\in C_c(\mathcal{G})$. Define $s^*\mu$ analogously. A Borel measure $\mu$ on $\mathcal{G}^{(0)}$ is invariant if $r^*\mu=s^*\mu$.

Using that there is a basis for the topology consisting of open bisections, one checks that $\mu$ is invariant if and only if for any open bisection $B$, $\mu(s(B))=\mu(r(B))$.

Let now $\mathcal{B}$ denote the $\mathcal{G}$-sheaf on $\mathcal{G}^{(0)}$ whose sections over an open set $U\subseteq \mathcal{G}^{(0)}$ consist of all Borel functions $f:U\to {\mathbb{C}} $ whose restriction to every compact subset of $U$ is bounded. (It might seem more natural to define sections to be bounded Borel functions on $U$, but this does not define a sheaf.)

Definition 3.5. A resolution

(7)\begin{equation} 0\to \mathcal{Z}\to \mathcal{S}^0\to \mathcal{S}^1 \to \cdots \end{equation}

of the sheaf $\mathcal{Z}$ of locally constant ${\mathbb Z}$-valued functions on $\mathcal{G}^{(0)}$ by $c$-soft $\mathcal{G}$-sheaves is said to be Borel if $\mathcal{S}^0$ is a subsheaf of $\mathcal{B}$ that contains $\mathcal{Z}$, and the map $\mathcal{Z}\to \mathcal{S}^0$ appearing in the resolution is the inclusion.

Given a Borel resolution of $\mathcal{Z}$ as above, let

\begin{equation*} C_0(\mathcal{G}):=\bigoplus_{p=0}^\infty \Gamma_c(\mathcal{G}^{(p)};\mathcal{S}_p^p) \end{equation*}

denote the zeroth chain group arising from the double complex as in line (4) above. We define a pairing

(8)\begin{equation} T(\mathcal{G})\times C_0(\mathcal{G})\to {\mathbb{C}} ,\quad (\mu,(a_p)_{p=0}^\infty )\mapsto \int_{\mathcal{G}^{(0)}} a_0d\mu. \end{equation}

Proposition 3.6. Let $\mathcal{G}$ be a locally compact, Hausdorff, étale groupoid with compact base space. Then the pairing defined above descends to a well-defined pairing

\begin{equation*} T(\mathcal{G})\times H_0(\mathcal{G})\to {\mathbb{C}} . \end{equation*}

(This includes the statement that Borel resolutions always exist, and that the pairing does not depend on the choice of Borel resolution).

Proof. We first show that given a Borel resolution as in line (7), the map in line (8) descends to $H_0(\mathcal{G})$. Indeed, it suffices for this to show that if $t\in \Gamma_c(\mathcal{G}^{(1)};\mathcal{S}_1^0)$ is a section and $\delta:\Gamma_c(\mathcal{G}^{(1)};\mathcal{S}_1^0)\to \Gamma_c(\mathcal{G}^{(0)};\mathcal{S}_0^0)$ is the relevant horizontal boundary map from line (4), then for any $\mu \in T(\mathcal{G})$ we have $\mu(\delta t)=0$. From the formula for $\delta$ in line (5), we have

\begin{equation*} (\delta t)(x)=\sum_{g\in s^{-1}(x)}t(g) - \sum_{h\in r^{-1}(x)}t(h). \end{equation*}

Hence, with notation as in Definition 3.3

\begin{align*} \mu(\delta t) & =\int_{\mathcal{G}^{(0)}} (\delta t)(x)d\mu(x) \\ & =\int_{\mathcal{G}^{(0)}}\sum_{r\in s^{-1}(x)}t(g)d\mu(x) - \int_{\mathcal{G}^{(0)}}\sum_{h\in r^{-1}(x)}t(h)d\mu(x) \\ & =\int_{\mathcal{G}} t d(s^*\mu)-\int_{\mathcal{G}}td(r^*\mu). \end{align*}

As $\mu$ is invariant, $r^*\mu=s^*\mu$, so this is zero.

We now show that Borel resolutions exist, and in fact construct a canonical Borel resolution. Let $0\to \mathcal{Z}\to \mathcal{B}$ be the canonical inclusion. Lemma A.3 gives an exact sequence of $\mathcal{G}$-sheaves

(9)\begin{equation} 0\to \mathcal{Z}\to \mathcal{B}\to \mathcal{I}^1\to \mathcal{I}^2\to \cdots \end{equation}

where each $\mathcal{I}^i$ is injective as a $\mathcal{G}$-sheaf, and also $c$-soft. Note also that $\mathcal{B}$ is $c$-soft, whence the resolution in line (9) is a Borel resolution as claimed.

Finally, consider any Borel resolution as in line (7). Then we have a commutative diagram

where the first vertical arrow $\mathcal{S}^0\to \mathcal{B}$ is the inclusion from the definition of ‘Borel resolution’ and the dashed vertical arrows are filled in using injectivity of each $\mathcal{I}^i$ as a $\mathcal{G}$-sheaf. (This is a standard argument from homological algebra: compare, for example, [Reference Weibel60, Comparison Theorems 2.2.6 and 2.3.7].) As both sequences are exact, these arrows constitute a quasi-isomorphism of complexes, so induce an isomorphism between the corresponding groupoid homologies by Proposition A.5. This quasi-isomorphism is compatible with the pairings with $T(\mathcal{G})$ (from the formula for the pairing, and as $\mathcal{S}^0\to \mathcal{B}$ is just inclusion), and so the independence of the pairing from the choice of Borel resolution follows.

Definition 3.7. Let $\mathcal{G}$ be a locally compact, Hausdorff, étale groupoid with compact base space. The groupoid Elliott invariant consists of the quadruple

\begin{equation*} (H_{**}(\mathcal{G}),[1]_H,T(\mathcal{G}),\rho_H) \end{equation*}

where: $H_{**}(\mathcal{G})=H_{ev}(\mathcal{G})\oplus H_{od}(\mathcal{G})$ is the ${\mathbb Z}/2$-graded Crainic–Moerdijk homology as in line (6); $T(\mathcal{G})$ is the space of invariant probability measures as in Definition 3.4; $\rho_H:T(\mathcal{G})\to \text{Hom}(H_{**}(\mathcal{G}),\mathbb{R})$ is the pairing between $T(\mathcal{G})$ and $H_{**}(\mathcal{G})$ that agrees with the pairing from Proposition 3.6 on $H_0(\mathcal{G})$ and is zero on the other summands; and $[1]_H\in H_{even}(\mathcal{G})$ is the class of the constant function with value one in $H_0(\mathcal{G})$.

We will want to compare the groupoid Elliott invariant above to the Elliott invariant of the reduced groupoid $C^*$-algebra. The next lemma looks at the spaces of invariant measures and traces; it is well-known. For the statement, let $E:C^*_r(\mathcal{G})\to C_0(\mathcal{G}^{(0)})$ denote the canonical conditional expectation as in [Reference Sims and Perera58, Proposition 4.2.6].

Lemma 3.8. Let $\mathcal{G}$ be a locally compact, Hausdorff, étale groupoid with compact base space. For $\mu \in T(\mathcal{G})$, define a map

\begin{equation*} \tau_\mu:C^*_r(\mathcal{G})\to {\mathbb{C}} \end{equation*}

as the composition of the canonical conditional expectation $E:C^*_r(\mathcal{G})\to C(\mathcal{G}^{(0)})$ and integration against $\mu$. Then the map

\begin{equation*} T(\mathcal{G})\to T(C^*_r(\mathcal{G})),\quad \mu \mapsto \tau_\mu \end{equation*}

is well-defined, injective, affine, and continuous. Moreover, if $\mathcal{G}$ is principal then $\tau$ is an affine homeomorphism.

We will typically elide the difference between an invariant measure on ${\mathcal{G}}^{(0)}$ and the trace it defines on $C_r^\ast(\mathcal G)$, and write $\tau$ for both.

Definition 3.9. Let $\mathcal{G}$ be a groupoid satisfying the conditions of Definition 3.7. Then $\mathcal{G}$ is HK-good if the Elliott invariants

\begin{equation*} (K_*(C^*_r(\mathcal{G})),[1]_K,T(C^*_r(\mathcal{G})),\rho_K) \quad \text{and}\quad (H_{**}(\mathcal{G}),[1]_H,T(\mathcal{G}),\rho_H) \end{equation*}

are isomorphic in the following sense:

1. (i) there are algebraic isomorphisms $\varphi_0:K_0(C^*_r(\mathcal{G}))\to H_{ev}(\mathcal{G})$ and $\varphi_1:K_1(C^*_r(\mathcal{G}))\to H_{od}(\mathcal{G})$;

2. (ii) $\varphi_0([1_K])=[1_H]$;

3. (iii) the map $\tau:T(\mathcal{G})\to T(C^*_r(\mathcal{G}))$ of Lemma 3.8 is an affine homeomorphism such that the diagram

   

   commutes.

We make the following conjecture.

The conjecture is known to hold for many classifiable $C^*$-algebras that arise as the $C^*$-algebra of an ample groupoid. For example, [Reference Matui36, Theorem 4.10] essentially establishes it for AF groupoids, and [Reference Matui36, Theorem 4.14] essentially establishes it for $C^*$-algebras associated to shifts of finite type. Another class of examples partly generalizing those above comes from [Reference Bönicke, Dell’Aiera, Gabe and Willett1, Corollary 4.20]: this shows that if $\mathcal{G}$ is an ample, second countable, principal étale groupoid with compact base space and dynamic asymptotic dimension at most one, then it is an HK-good model for its groupoid $C^*$-algebra. As an application of the result of the previous sentence, Reardon [Reference Reardon53] has shown that the groupoids constructed by Putnam [Reference Putnam50] have dynamic asymptotic dimension one and hence are HK-good.

It also seems interesting to study HK-good groupoids that give rise to non-simple $C^*$-algebras. Again, examples are known in the ample case, such as some of the coarse groupoids studied in [Reference Bönicke, Dell’Aiera, Gabe and Willett1, Example 5.8].

The main goal of the rest of this paper is to give some examples of HK-good groupoids that go beyond the ample case.

4. An example: irrational rotation groupoids

In this section, we give our first non-trivial computation of a pairing between groupoid homology and invariant measures: the groupoids ${\mathbb Z} \lt imes S^1$ coming from an irrational rotation action on $S^1$, and the (unique) invariant measure coming from Lebesgue measure. It will follow from general machinery we develop below (see Example 7.1) that these examples are HK-good, but we thought it was worth doing one explicit computation from first principles (the current version was suggested by the referee: it replaces an earlier version with a more ad-hoc construction) to give intuition for the pairing, and also because of the importance of irrational rotation groupoids in $C^*$-algebra theory and noncommutative geometry. (The computation can be done even more simply using group hyperhomology as in Appendix B: we leave this to the interested reader).

Let $S^1$ be the circle, which we identify with the interval $[0,1]$ modulo the gluing $0\sim 1$, or equivalently, with $\mathbb{R}/{\mathbb Z}$. Let $\theta\in [0,1)$ be a fixed irrational number, and let $\alpha:S^1\to S^1$, $x\mapsto x+\theta$ be the action defined by the addition of $\theta$ modulo ${\mathbb Z}$. Let $\mathcal{G}:={\mathbb Z} \lt imes S^1$ denote the corresponding transformation groupoid, which we call an irrational rotation groupoid. As $\theta$ is irrational, there is a unique invariant probability measure on $S^1$ coming from the Lebesgue measure on $[0,1]$, which we denote $\tau$.

In order to compute the pairing of $H_0(\mathcal{G})$ with $\tau$, it will be useful to have a geometrically natural resolution of the sheaf $\mathcal{Z}$ of locally constant ${\mathbb Z}$-valued functions on $S^1$. Let $\mathcal{R}$ be the sheaf of continuous $\mathbb{R}$-valued functions on $S^1$, and let $\mathcal{S}^1$ denote the sheaf of continuous $S^1$-valued functions on $S^1$. Then the quotient map $q:\mathbb{R}\to S^1$ induces a morphism of sheaves fitting into a short exact sequence

(10)\begin{equation} 0\to \mathcal{Z}\to\mathcal{R}\xrightarrow{q}\mathcal{S}^1\to 0, \end{equation}

where the first arrow is the canonical inclusion. Moreover, it is not difficult to check that the sheaves $\mathcal{R}$ and $\mathcal{S}^1$ are $c$-soft (for $\mathcal{R}$ this would be true for any reasonable base space, but for $\mathcal{S}^1$, it is important that the base space is relatively simple: in particular, the fact that it is one-dimensional in any standard sense is enough) $\mathcal{G}$-sheaves.

The groupoid homology of $\mathcal{G}={\mathbb Z} \lt imes S^1$ is then by definition the homology of the double complex

as in line (4) on page (8).

Now, by definition of the homology of a double complex (and noting the conventions on row positions), a $0$-cycle representing a class in $H_0(\mathcal{G})$ consists of a pair

(11)\begin{equation} (a,b) \in \Gamma_c(\mathcal{G}^{(0)};\mathcal{R}_0)\oplus \Gamma_c(\mathcal{G}^{(1)};\mathcal{S}^1_1) \end{equation}

such that $\partial(a)=-\delta(b)$. We also use the resolution in line (10) to compute the cohomology groups of $S^1$, and then define maps from $H^*(S^1)$ into $H_0(\mathcal{G})$ as follows.

First, an element of $H^0(S^1)$ is represented by a constant function $a:S^1\to {\mathbb Z}$. We define a map

(12)\begin{equation} \varphi_0:H^0(S^1)\to H_0(\mathcal{G}),\quad a\mapsto (a,0); \end{equation}

it is straightforward to see that this is well-defined. On the other hand, for a cycle $b\in \Gamma_c(S^1;\mathcal{S}^1)$ representing a class in $H^1(S^1)$, we define an element of $\Gamma_c(\mathcal{G}^{(1)};\mathcal{S}^1_1)$ by

\begin{equation*} b_1(m,x):=\left\{\begin{array}{ll} b(x) & m=1 \\ 0 & \text{otherwise}\end{array}\right. \end{equation*}

Note that if $\alpha:\Gamma_c(S^1;\mathcal{S}^1)\to \Gamma_c(S^1;\mathcal{S}^1)$ is the map induced by the action, then

\begin{equation*} \delta(b_1)=b-\alpha(b) \end{equation*}

(compare line (5) on page 9 for the definition of $\delta$, and recall that we are treating $S^1$ as $\mathbb{R}/{\mathbb Z}$, so we write the operation on $S^1$ additively). As the map $b-\alpha(b)$ represents the trivial class in $H^1(S^1)$ (the action is homotopic to the identity, so induces the identity map on cohomology) there must be $a_b\in \Gamma_c(S^1;\mathcal{R})$ such that

\begin{equation*} b-\alpha(b)=-\partial(a_b). \end{equation*}

Adjusting $a_b$ by an integer constant, we assume for definiteness that $\lfloor a_b(0)\rfloor =0$; $a_b$ is then uniquely determined by $\delta(b)$. We may therefore define a map

(13)\begin{equation} \varphi_1:H^1(S^1)\to H_0(\mathcal{G}),\quad b\mapsto (a_b,b_1); \end{equation}

one checks directly this is a well-defined homomorphism.

Proposition 4.1. Let $\mathcal{G}={\mathbb Z} \lt imes S^1$ be an irrational rotation groupoid with associated angle $\theta$, and $\tau$ the unique invariant probability measure coming from Lebesgue measure.

Then the maps from lines (12) and (13) induce an isomorphism

(14)\begin{equation} \varphi_0\oplus \varphi_1:H^0(S^1)\oplus H^1(S^1)\to H_0(\mathcal{G}), \end{equation}

so in particular $H_0(\mathcal{G})\cong {\mathbb Z}\oplus {\mathbb Z}$. Let moreover $a$ be the generator of $H^0(S^1)$ given by the constant function with value $1$, and let $b$ be the generator of $H^1(S^1)$ given by the identity function $b(x)=x$. Then we have the formulas

\begin{equation*} \tau(\varphi_0(a))=1 \quad \text{and}\quad \tau(\varphi_1(b))=-\theta. \end{equation*}

The formulas above are consistent with the known $K$-theory of $C^*_r(\mathcal{G})$, and its pairing with the trace: see, for example, [Reference Pimsner and Voiculescu44] for the original reference, [Reference Davidson11, Theorem VI.5.2] for a textbook exposition, and [Reference Cuntz10, 2.5] and [Reference Pimsner and Araki43, Proposition 6] for elegant alternative arguments.

Proof of Proposition 4.1

Let $(a,b)\in \Gamma_c(\mathcal{G}^{(0)};\mathcal{R}_0)\oplus \Gamma_c(\mathcal{G}^{(1)};\mathcal{S}^1_1)$ be a cycle for $H_0(\mathcal{G})$. We first claim that we can adjust $(a,b)$ by boundaries of the form $\delta(c)$ with $c\in \Gamma_c(\mathcal{G}^{(2)};\mathcal{S}^1_2)$ so that $b$ is supported on $\{1\}\times S^1$.

Indeed, we may regard elements of $\Gamma_c(\mathcal{G}^{(1)};\mathcal{S}^1_1)$ as compactly supported, continuous functions from ${\mathbb Z}\times S^1$ to $S^1$. Let $m\in {\mathbb Z}$ be maximal with the property that the restriction of $b$ to $\{m\}\times S^1$ is non-zero. We may regard elements of $\Gamma_c(\mathcal{G}^{(2)};\mathcal{S}^1_2)$ as continuous functions from $ {\mathbb Z}\times {\mathbb Z}\times S^1$ to $S^1$ with compact support. If $m\geq 1$, define $c:{\mathbb Z}\times {\mathbb Z}\times S^1\to {\mathbb Z}$ by stipulating $c$ is supported on $\{(-1,m)\}\times S^1$, and satisfies $c(-1,m,x)=b(m,x)$. Then one checks that subtracting the boundary of $c$ from $b$ ‘cancels off’ the part of $b$ supported on $\{m\}\times S^1$, and does not otherwise change $b$ outside of $(\{-1\}\times S^1)\cup (\{m-1\}\times S^1)$. Continuing in this way, we may keep adjusting by boundaries so that $b$ is supported on $\{(n,x)\in {\mathbb Z}\times S^1\mid n\leq 0\}$. Applying a similar procedure to the negative part of the support, we may assume that $b$ is supported on $(\{0\}\times S^1)\cup (\{1\}\times S^1)$. Finally, subtracting off $\delta(c)$ where $c$ is supported on $\{(0,0)\}\times S^1$ and satisfies $c(0,0,x)=b(0,x)$, we may assume that $b$ is supported on $\{1\}\times S^1$ as claimed.

We next claim that further adjusting $(a,b)$ by classes of the form $\partial(e)$ with $e\in \Gamma_c(\mathcal{G}^{(1)};\mathcal{R}_1)$, we can assume that $b(1,x)=nx$ for some $n\in {\mathbb Z}$. Indeed, if $n$ is the winding number of $b$, then $b-nx$ has winding number zero, and thus lifts to a continuous map $e:S^1\to \mathbb{R}$. As $\partial$ is just induced by the quotient map $\mathbb{R}\to \mathbb{R}/{\mathbb Z}$, we have $\partial(e)=b-nx$, establishing the claim.

Now, if $b(1,x)=nx$, we have by line (5) that

\begin{equation*} \delta(b)(x)=nx - n(x-\theta) = n\theta \end{equation*}

As $\delta(b)=-\partial(a)$, we necessarily have $a=m+n\theta$ (as a constant function $S^1 \to \mathbb{R}$) where $m\in {\mathbb Z}$ is the unique integer such that $\lfloor m+n\theta \rfloor=0$. At this point, we have that every class for $H_0(\mathcal{G})$ can be represented in the form $(m+n\theta,nx)$ for some uniquely determined $m,n\in {\mathbb Z}$. Using very similar (and simpler) arguments, one can represent any class for $H^0(S^1)$ by a uniquely determined constant function $S^1\to {\mathbb Z}$, and one can represent any class for $H^1(S^1)$ by a uniquely determined function of the form $x\mapsto nx$ for some $n\in {\mathbb Z}$. These observations complete the proof that the map in line (14) is an isomorphism.

We now look at the pairings. The formula $\tau(\varphi_0(a))=1$ is clear. To compute $\varphi_1(b)$, we note that $b-\alpha(b)$ is the function $x\mapsto \theta$. This is equal to $-\partial(-\theta)$ (where $-\theta$ now denotes the constant function with value $\theta$). We therefore have that $\varphi_1(b)=(-\theta,b_1)$. Hence, $\tau(\varphi_1(b))=\int_0^1-\theta=-\theta$ as claimed.

5. Homology of crossed products by the integers

In this section, we compute the groupoid homology of transformation groupoids ${\mathbb Z} \lt imes X$ associated to an action of the integers on a locally compact Hausdorff space. We will write $G$ for ${\mathbb Z}$: this is mainly to avoid having to use the notation ‘ ${\mathbb Z}[{\mathbb Z}]$’ for the integral group ring of the integers, so we can instead write ‘ ${\mathbb Z} G$’.

For a $G$-module $M$, write $\alpha:M\to M$ for the action of the usual generator $1\in G={\mathbb Z}$; in particular, we use this notation when $M={\mathbb Z} G$ is the integral group ring acting on itself. There is then a free resolution (topologically, it comes from considering the circle with one $0$-cell and $1$-cell as a CW complex model for the classifying space $BG$: compare, for example, [Reference Brown4, Section I.4])

(15)\begin{equation} 0\leftarrow {\mathbb Z} \leftarrow {\mathbb Z} G \stackrel{\delta}{\leftarrow} {\mathbb Z} G \leftarrow 0 \end{equation}

of the trivial ${\mathbb Z} G$-module ${\mathbb Z}$ where the map $\delta$ is given by

(16)\begin{equation} \delta:x\mapsto x-\alpha(x). \end{equation}

Note moreover that if $M$ is any $G$-module, then we may define $\delta:M\to M$ by the same formula as in line (16). If we do this, we have canonical identifications

(17)\begin{equation} M^G=\text{Kernel}(\delta) \quad \text{and}\quad M_G=\text{Cokernel}(\delta) \end{equation}

where $M^G$ and $M_G$ are, respectively, the invariant submodule, and coinvariant quotient of $M$ for the $G$-action (see, for example, [Reference Brown4, pages 27 and 34]).

Here is the homology computation we need (see [Reference Deaconu13, Theorem 3.9] and [Reference Ortega41, Lemma 1.3] for related results in the case of ample base space).

Proposition 5.1. Let $X$ be a locally compact Hausdorff space equipped with an action of ${\mathbb Z}$. Then for each $n\in {\mathbb Z}$ there is a canonical short exact sequence

\begin{equation*} 0\to H^{-n}_c(X)_{\mathbb Z}\to H_{n}({\mathbb Z} \lt imes X)\to H_c^{1-n}(X)^{\mathbb Z} \to 0 \end{equation*}

(this includes the case $n=1$, where the sequence degenerates to an isomorphism $H_{1}({\mathbb Z} \lt imes X)\cong H_c^{0}(X)^{\mathbb Z}$, and $n \gt 1$, where it degenerates to $H_n({\mathbb Z} \lt imes X)=0$).

One could maybe do the computation a little more quickly using the hyperhomology spectral sequence as in [Reference Weibel60, Section 5.7], but we give a more direct approach (which in any case really just amounts to unpacking the spectral sequence argument).

Proof of Proposition 5.1

Let

(18)\begin{equation} 0\to \mathcal{Z}\to \mathcal{S}^0\to \mathcal{S}^1\to \cdots \end{equation}

be a resolution of the sheaf $\mathcal{Z}$ of locally constant ${\mathbb Z}$-valued functions on $X$ by $c$-soft $\mathcal{G}$-sheaves. As in Corollary B.9, we may treat $H_{n}(G \lt imes X)$ as the group hyperhomology $\mathbb{H}_*(G;X)$ of Definitions B.1 and B.5. Doing this, $H_*(G \lt imes X)$ is then the homology of the double complex

formed as the tensor product of the complex of line (15) and the complex $\Gamma_c(X;\mathcal{S}^{-\bullet})$ of compactly supported sections. For notational simplicity, write $M_q$ for the ${\mathbb Z} G$-module $\Gamma_c(X;\mathcal{S}^{-q})$, so the above becomes

Write $T_\bullet$ for the total complex associated to this, and note that the first column defines a subcomplex of $T_\bullet$ with quotient the second column, and thus we get a short exact sequence of chain complexes

\begin{equation*} 0\to M_\bullet \to T_\bullet \to M_{\bullet-1} \to 0 \end{equation*}

(the degree shift on the second copy of $M_{\bullet}$ comes about due to the different ways the two columns appear as summands in the double complex: compare [Reference Weibel60, 1.2.6]). This in turn gives rise to a long exact sequence of homology groups

with connecting maps $H_i(M_\bullet) \to H_i(M_\bullet)$ given by $\delta$ (this can be computed, for example, from the proof of [Reference Weibel60, Theorem 1.3.1]). On the other hand, $H_*(M_\bullet)$ computes the compactly supported sheaf homology $H_c^{-*}(X)$, and therefore the long exact sequence above becomes

Splitting this up into short exact sequences along the kernels and cokernels of the maps $\delta$ and applying line (17) gives the result.

Remark 5.2. As it will be useful later, let us give an explicit description of cycles in $H_0({\mathbb Z} \lt imes X)$ based on the proof of Proposition 5.1. Write $\mathcal{G}$ for ${\mathbb Z} \lt imes X$.

Let $\mathcal{Z}$ be the sheaf of locally constant ${\mathbb Z}$-valued functions on $X$, and consider a resolution by equivariant $c$-soft $\mathcal{G}$-sheaves that starts

(19)\begin{equation} 0\to \mathcal{Z}\stackrel{\partial}{\to} \mathcal{R}\stackrel{\partial}{\to} \mathcal{S}_B^1\stackrel{\partial}{\to} \cdots \end{equation}

where $\mathcal{R}$ is the sheaf of continuous $\mathbb{R}$-valued functions on $X$, and $\mathcal{S}_B^1$ is the sheaf of Borel $S^1$-valued functions on $X$. The map from $\mathcal{Z}$ to $\mathcal{R}$ is the canonical inclusion, and the map $\mathcal{R}\to \mathcal{S}_B^1$ is induced by the exponential map $x\mapsto e^{2\pi i x}$; it is not too difficult to see that this is exact, and that $\mathcal{R}$ and $\mathcal{S}_B^1$ are $c$-soft $\mathcal{G}$-sheaves; moreover, the image of $\mathcal{R}$ in $\mathcal{S}^1_B$ is the subsheaf $\mathcal{S}^1$ consisting of continuous $S^1$-valued functions on $X$.

Now, a cycle for $H_0({\mathbb Z} \lt imes X)$ is a pair $(f,v)$ where $f:X\to \mathbb{R}$ is a continuous compactly supported function, $v:X\to S^1$ is a compactly supported (i.e. equal to $1$ outside a compact set) bounded Borel function on $X$, and these functions satisfy $\partial(v)=0$ and $\delta(v)=-\partial(f)$. The fact that $\partial(v)=0$ implies that $v$ is continuous (as it is locally in the image of the exponential map from $\mathcal{R}$). On the other hand, the formula $\delta(v)=-\partial(f)$ says explicitly that $\exp(2\pi if)=\alpha(v)v^*$.

6. The pairing and Chern character for actions of the integers

Let $\alpha$ denote a homeomorphism of a locally compact space $X$, and also the associated actions of ${\mathbb Z}$ on $X$ and on $C_0(X)$; we fix this notation throughout the remainder of this section. Our aim in this section is to construct a Chern character map

(20)\begin{equation} \text{ch}_\alpha:K_*(C_0(X)\rtimes_\alpha{\mathbb Z})\to H_{**}({\mathbb Z} \lt imes X;{\mathbb Q}) \end{equation}

that becomes an isomorphism on tensoring the left-hand side by ${\mathbb Q}$, is compatible with the pairings with invariant probability measures, and that moreover can be made sense of as an integral isomorphism in some cases: for example, this happens if $X$ has only low-dimensional homology as in Corollary 6.10 below.

We should note that the existence of a Chern character rational isomorphism follows from our earlier results in [Reference Deeley and Willett18]. However, compatibility with the pairings and the fact that it is sometimes compatible with an integral isomorphism is not so obvious for the Chern character in [Reference Deeley and Willett18] (which is essentially due to Raven [Reference Raven52]), so we give a different approach here.

We first recall the approach to the Pimsner–Voiculescu sequence based on mapping tori: the original reference for this is [Reference Connes8, pages 48-49], or see, for example, [Reference Blackadar2, Sections 10.3–10.4] for a textbook exposition; the original reference for the Pimsner–Voiculescu sequence is [Reference Pimsner and Voiculescu44].

Definition 6.1. The mapping torus of $\alpha$, denoted $M_\alpha$, is the maximal ideal space (we could also directly define $M_\alpha$ to be the quotient of $X\times [0,1]$ by the relation $(x,0)\sim (\alpha^{-1}(x),1)$, but it is more useful for us to have a description of the continuous functions on the space) of the ${C}^{*}$-algebra

\begin{equation*}C_0(M_\alpha):=\{f\in C_b({\mathbb{R}},C_0(X))\mid f(t+1)=\alpha(f(t))\;\text{for all}\ t\in{\mathbb{R}\}}.\end{equation*}

Note that there is a short exact sequence

(21)\begin{equation} 0\to C_0(SX)\to C_0(M_\alpha) \to C_0(X)\to 0 \end{equation}

where $SX:=(0,1)\times X$ is the usual $C^*$-algebraic suspension of $X$, the left hand map is inclusion as ${\mathbb Z}$-equivariant functions from $\mathbb{R}$ to $C_0(X)$ that vanish at the integers, and the right-hand map is evaluation at zero. This gives rise to a six-term exact sequence

(22)

Moreover, if we use the periodicity isomorphisms to make identifications $K^i(SX)\cong K^{i-1}(X)$, then the boundary maps in this sequence identify with $\text{id}-\alpha_*$ (see, for example, [Reference Blackadar2, Proposition 10.4.1]).

The following result records the key $C^*$-algebraic facts we will need; this seems likely to be known to experts, but we provide proofs where we could not find appropriate references. For $i\in \{0,1\}$ and an action $\beta$ of $\mathbb{R}$ on a $C^*$-algebra $B$, we let $\varphi_{\beta}^i:K_i(B)\to K_{i+1}(B\rtimes_{\beta} \mathbb{R})$ denote the Connes-Thom isomorphism as defined in [Reference Connes8, Section II].

Proposition 6.2. Let $X$ be a locally compact Hausdorff space, and let $\alpha:X\to X$ be a homeomorphism inducing an action of ${\mathbb Z}$ on $X$. Let $\beta$ denote the action of $\mathbb{R}$ on $C_0(X)\rtimes_\alpha{\mathbb Z}$ (we can equivalently define $\beta$ to be the composition of the dual action of $\widehat{{\mathbb Z}}=\mathbb{R}/{\mathbb Z}$ on $C_0(X)\rtimes_\alpha{\mathbb Z}$, and the quotient map $\mathbb{R}\to \mathbb{R}/{\mathbb Z}$) determined by

\begin{equation*} \beta_t(f u^n)=e^{2\pi i tn} f u^n, \end{equation*}

where $f\in C_0(X)$, $u\in M(C_0(X)\rtimes {\mathbb Z})$ is the canonical unitary multiplier, $n\in {\mathbb Z}$, and $t\in \mathbb{R}$. Then the following holds:

1. (i) There is a distinguished class of isomorphisms

   \begin{equation*} m_\alpha: (C_0(X)\rtimes_\alpha {\mathbb Z})\rtimes_{\beta}\mathbb{R}\to C_0(M_\alpha)\otimes \mathcal{K}(\ell^2({\mathbb Z})), \end{equation*}
   all of which induce the same map on $K$-theory.

2. (ii) the isomorphism

   \begin{equation*} \psi_\alpha^i:=(m_\alpha)_*\circ \varphi_\beta^i:K_i(C_0(X)\rtimes_\alpha {\mathbb Z})\to K^{i+1}(M_\alpha) \end{equation*}
   fits into a commutative diagram
   

   where the left-hand vertical map is Bott periodicity, and the horizontal maps are induced from the canonical inclusions $C_0(SX)\to C_0(M_\alpha)$ (see line (22)) and $C_0(X)\to C_0(X)\rtimes_\alpha {\mathbb Z}$.

3. (iii) The six-term exact sequence in line (22), the commutative diagram from part (ii), and the Pimsner–Voiculescu exact sequence as constructed in [Reference Blackadar2, Sections 10.3–4] induce a commutative diagram of short exact sequences

   

   with all vertical maps isomorphisms.

Proof. For part (i), let $\pi_X:C_0(X)\to \mathcal{B}(H_X)$ be a faithful nondegenerate representation. Let

\begin{equation*} \pi: C_0(X)\rtimes_\alpha {\mathbb Z}\rtimes_\beta \mathbb{R}\to \mathcal{B}(H_X\otimes \ell^2({\mathbb Z})\otimes L^2(\mathbb{R})) \end{equation*}

be the regular representation, i.e. if we identify $H_X\otimes \ell^2({\mathbb Z})\otimes L^2(\mathbb{R})=L^2({\mathbb Z}\times \mathbb{R},H_X)$, then $\pi$ is induced by the covariant triple $(\widetilde{\pi_X},u,v)$ where

\begin{equation*} ((\widetilde{\pi_X}(f))\xi)(n,t):=\pi_X(\alpha_{-n}(f))\xi(n,t) \end{equation*}

and

\begin{equation*} (u_m\xi)(n,t):=e^{2\pi i mt} \xi(n-m,t),\quad (v_s\xi)(n,t):=\xi(n,t-s). \end{equation*}

Let $F:L^2(\mathbb{R}) \to L^2(\mathbb{R})$ denote the unitary isomorphism induced by the Fourier transform, and let $\lambda$ denote the translation action of ${\mathbb Z}$ on $\mathbb{R}$. Then one computes that $1\otimes 1 \otimes F$ conjugates (the integrated form of) $(\widetilde{\pi_X},u,v)$ to the regular representation of $C_0(\mathbb{R}\times X)\rtimes_{\lambda\times \alpha}{\mathbb Z}$ on $H_X\otimes \ell^2({\mathbb Z})\otimes L^2(\mathbb{R})=\ell^2({\mathbb Z},L^2(\mathbb{R})\otimes H_X)$, and thus conjugates the image of $C_0(X)\rtimes_\alpha {\mathbb Z}\rtimes_\beta \mathbb{R}$ to $C_0(\mathbb{R}\times X)\rtimes_{\lambda\otimes \alpha}{\mathbb Z}$.

Let $\sigma$ denote the automorphism of $\mathcal{K}(\ell^2({\mathbb Z}))$ induced by conjugation by the bilateral shift. Then more direct computations show that $C_0(\mathbb{R}\times X)\rtimes_{\lambda\otimes \alpha}{\mathbb Z}$ canonically identifies with

\begin{equation*} C_b(\mathbb{R}\times X,\mathcal{K}(\ell^2({\mathbb Z})))^{\mathbb Z}=\{f\in C_b(\mathbb{R}\times X,\mathcal{K}(\ell^2({\mathbb Z})))\mid f(t+1,\alpha(x))=\sigma(f(t,x))\}. \end{equation*}

Choose now any path $(\sigma_t)_{t\in [0,1]}$ of automorphisms of $\mathcal{K}(\ell^2({\mathbb Z}))$ connecting $\sigma$ to the identity that is induced by a norm-continuous part of unitaries connecting the bilateral shift to the identity. Use this path to ‘untwist’ $C_0(\mathbb{R}\times X)\rtimes_{\lambda\otimes \alpha}{\mathbb Z}$, showing that it is isomorphic to

\begin{align*} \{f\in C_b(\mathbb{R}\times X,\mathcal{K}(\ell^2({\mathbb Z})))\mid f(t+1,\alpha(x)) =f(t,x)\} & = C_0((\mathbb{R}\times X)/{\mathbb Z},\mathcal{K}(\ell^2({\mathbb Z}))) \\ & = C_0(M_\alpha)\otimes \mathcal{K}(\ell^2({\mathbb Z})). \end{align*}

This isomorphism of $C_0(\mathbb{R}\times X)\rtimes_{\lambda\otimes \alpha}{\mathbb Z}$ with $C_0(M_\alpha)\otimes \mathcal{K}(\ell^2({\mathbb Z}))$ depends on the choice of the path $(\sigma_t)$, but any two such paths are homotopic, so the map induced on $K$-theory is well-defined.

For part (ii), note that the canonical inclusion $C_0(X)\to C_0(X)\rtimes_\alpha {\mathbb Z}$ is $\mathbb{R}$-equivariant for the trivial $\mathbb{R}$ action $\text{tv}$ on $C_0(X)$, and the action $\beta$ on $C_0(X)\rtimes_\alpha {\mathbb Z}$. Hence, by naturality of the Connes-Thom isomorphism, we get a commutative diagram

Up to applying a Fourier transform, $K_1(C_0(X)\rtimes_{\text{tr}}\mathbb{R})$ identifies with $C_0(\mathbb{R}\times X)$ and the Connes-Thom map $\varphi^0_{\text{tv}}$ identifies with the Bott periodicity isomorphism, so using also the identifications established in the proof of part (i) we get a commutative diagram

where the top-left horizontal map is induced by inclusion (and is an isomorphism) and the left-hand vertical map is Bott periodicity. Up to composing with the isomorphism $m_\alpha$, the outer rectangle is the diagram in the statement.

Part (iii) follows as the Pimsner–Voiculescu exact sequence in [Reference Blackadar2, Sections 10.3–4] is exactly defined by forcing the given diagram to commute.

The next result we need is a ‘dual version’ of a result of Connes: compare [Reference Connes8, Section II, Theorem 3]. It seems to be well-known, but we could not find the precise statement we wanted in the literature so sketch a proof.

Proposition 6.3. Let $X$ be a compact Hausdorff space equipped an action $\alpha$ of ${\mathbb Z}$ by homeomorphisms, and let $\tau:C(X)\to {\mathbb{C}} $ be an invariant probability measure.

Any class in $K^1(M_\alpha)$ can be represented by a continuous function $u:[0,1]\to \mathcal{U}(M_n(C(X)))$ that is continuously differentiable on $(0,1)$, and with the property that $u(1)=\alpha(u(0))$. Having chosen such a representative for each class, the formula

(23)\begin{equation} \Lambda_\tau:K^1(M^\alpha)\to \mathbb{R},\quad [u]\mapsto \frac{1}{2\pi i} \int_0^1 \tau (u'(t)u(t)^*) dt, \end{equation}

(where ‘ $u'$’ denotes the derivative of $u$ in the $t$-variable) gives a well-defined homomorphism that does not depend on the choice of representatives.

Moreover, with notation as in Proposition 6.2, the diagram

(24)

commutes.

Proof. For a function $u:[0,1]\to \mathcal{U}(M_n(C(X)))$, the real number $\Lambda_\tau[u]$ is the de la Harpe–Skandalis determinant of the path as in [Reference de la Harpe and Skandalis12, Section 1] (the notation there would be $\widetilde{\Delta}_\tau(u)$). The fact that $\Lambda_\tau$ is a well-defined group homomorphism (including that any $K_1$-class can be represented by a function that is differentiable in the $t$-variable) then follows directly from [Reference de la Harpe and Skandalis12, Lemme 1].

We need some more notation. Let $B$ be a general unital $C^*$-algebra equipped with an action $\beta$ of $\mathbb{R}$, and let $\tau$ denote an $\mathbb{R}$-invariant tracial state on $B$. Let $\widehat{\tau}$ denote the ‘dual’ (unbounded) trace on $B\rtimes_\beta \mathbb{R}$ determined on $C_c(\mathbb{R},B)$ by $\widehat{\tau}(f)=\tau(f(0))$. Let $\delta_\beta$ denote the (densely defined) derivation on $B\rtimes_\beta \mathbb{R}$ determined as on [Reference Connes8, page 32] by the formula ${\displaystyle \delta_\beta(a)=\lim_{t\to 0} t^{-1}(\beta_t(a)-a)}$. As in [Reference Connes8, Section II, Theorem 3] there is then a well-defined homomorphism

\begin{equation*} \Theta_{\tau}:K_1(B\rtimes_\beta \mathbb{R})\to \mathbb{R},\quad [u] \mapsto \frac{1}{2\pi i}\widehat{\tau}(\delta(u)u^*) \end{equation*}

(this includes the statement that any class in $K_1(B\rtimes_\beta \mathbb{R})$ can be represented by a unitary such that the right-hand side makes sense).

Now, let $p\in M_n(C(X)\rtimes_\alpha {\mathbb Z})$ represent a class $[p]$ in $K_0(C(X)\rtimes_\alpha {\mathbb Z})$. According to [Reference Connes8, Section II, Proposition 4], there is an exterior equivalent action $\beta'$ of $\mathbb{R}$ on $C(X)\rtimes_\alpha {\mathbb Z}$ that fixes $p$. If $(u_t)$ is the unitary one-cocycle implementing the exterior equivalence, then there is an isomorphism

\begin{equation*} \theta:C(X)\rtimes_\alpha {\mathbb Z}\rtimes_\beta \mathbb{R}\to C(X)\rtimes_\alpha {\mathbb Z}\rtimes_{\beta'} \mathbb{R} \end{equation*}

determined on $C_c(\mathbb{R},C(X)\rtimes_\alpha {\mathbb Z})$ by $(\theta b)(t):=u_tb(t)$. Consider the diagram below

We have that $\theta_*\circ \varphi^0_{\beta'}=\varphi_\beta^0$ (see [Reference Connes8, Section 1, Proposition 3]), so it suffices to show that this diagram commutes on the class $[p]$. We do this one square at a time.

For the leftmost square, we can represent $\varphi_\beta^0[p]$ as the Fourier transform of the function $u(t)=e^{2\pi i \chi(t)p}$ where $\chi:\mathbb{R}\to [0,1]$ is a smooth nondecreasing function such that $\chi(t)=0$ for $t\leq 0$ and $\chi(t)=1$ for $t\geq 1$ (compare [Reference Connes8, pages 37-38]). As the Fourier transform interchanges $\delta$ and differentiation, and as it interchanges evaluation at $0$ with integration, we thus have that

\begin{equation*} \Theta_{\tau}[u]=\frac{1}{2\pi i} \int_{\mathbb{R}} \tau(2\pi i\chi'(t)p uu^{*})dt=\tau(p)\int_{\mathbb{R}}\chi'(t)dt=\tau(p). \end{equation*}

Commutativity of the central square is again a direct computation using that $\widehat{\tau}$ is a trace and $\theta$ is induced by a unitary multiplier. Commutativity of the rightmost square is another direct computation, based on the description of $m_\alpha$ in terms of the Fourier transform, and the fact that the Fourier transform interchanges evaluation at $0$ with integration, and that it interchanges the derivation $\delta$ with differentiation; the integral ends up being only over $[0,1]$ rather than $\mathbb{R}$ as we are implicitly also taking the standard trace on $\mathcal{K}(\ell^2({\mathbb Z}))$, which has the effect of ‘adding up’ the ${\mathbb Z}$ translates of this.

We now turn to homology. Let $X$ be a locally compact Hausdorff space equipped with an action $\alpha$ of ${\mathbb Z}$. Recall from Remark 5.2 that cycles for $H_0({\mathbb Z} \lt imes X)$ can be thought of as pairs $(f,v)$ where $f:X\to \mathbb{R}$ is continuous and compactly supported, and $v:X\to S^1$ is continuous and compactly supported (i.e. equal to one outside a compact set), and satisfies $\alpha(v)v^*=e^{2\pi i f}$. Moreover, cycles for $H^1_c(M_\alpha)$ (respectively, $H^1_c(SX)$) can be thought of as continuous and compactly supported functions $u:M_\alpha\to S^1$ (respectively, $u:SX\to S^1$), and cycles for $H^0_c(X)$ can be thought of as locally constant and compactly supported functions $f:X\to {\mathbb Z}$.

Lemma 6.4. Let $X$ be a locally compact Hausdorff space equipped with an action $\alpha$ of ${\mathbb Z}$. There is an isomorphism $H_0({\mathbb Z} \lt imes X)\to H^1_c(M_\alpha)$ explicitly realized on the level of cycles by sending $(f,v)$ to the function $u:M_\alpha\to S^1$ defined by $u(t,x)=e^{2\pi i tf(x)}v(x)$. This isomorphism fits moreover into a commutative diagram

where the bottom row is as in Proposition 5.1. The same holds if we use rational coefficients.

Analogously to Lemma 5.3 we may define a map $\Lambda_\tau^H:H^1_c(M_\alpha)\to \mathbb{R}$ using the same formula as for $\Lambda_\tau:K^1(M_\alpha)\to \mathbb{R}$ (see line (23) above—this involves first approximating a cycle $u:M_\alpha\to S^1$ by a cycle that is continuously differentiable in the $t$ variable). We then have another compatibility lemma about pairings.

Lemma 6.5. With notation as in Lemma 6.4, the diagram

commutes. The same holds with rational coefficients.

Proof. Let $(f,v)$ be a cycle for $H_0({\mathbb Z} \lt imes X)$, so $\tau[f,v]=\int_X fd\tau$ by definition. On the other hand, the isomorphism $H_0({\mathbb Z} \lt imes X)\to H^1_c(M_\alpha)$ from Lemma 6.4 sends $(f,v)$ to the function $u:M_\alpha\to S^1$ defined by $u(t,x)=e^{2\pi i tf(x)}v(x)$. Computing,

\begin{equation*} \Lambda_\tau[u]=\frac{1}{2\pi i } \int_0^1 \tau(u'(t)u(t)^*)dt = \frac{1}{2\pi i}\int_0^1 \tau(2\pi i fu(t)u(t)^*) dt=\tau(f), \end{equation*}

so we are done.

We will also need to compare the mapping cone sequence in cohomology to the sequence in groupoid homology. The following result is in some sense an (easier) analogue of ${\mathbb Z}$-equivariant Bott periodicity for groupoid homology. It is also essentially a special case of [Reference Proietti and Yamashita46, Theorem B], but we provide a direct proof in the case we need for the sake of keeping the paper self-contained.

Proposition 6.6. Let $X$ be a locally compact Hausdorff space equipped with an action $\alpha$ of ${\mathbb Z}$. Let $\mathbb{R}$ be equipped with the usual translation of ${\mathbb Z}$, and equip $\mathbb{R}\times X$ with the diagonal action. Then there is a canonical isomorphism

\begin{equation*} H_i({\mathbb Z} \lt imes X)\cong H_{i-1}({\mathbb Z} \lt imes (\mathbb{R}\times X)) \end{equation*}

for each $i\in {\mathbb Z}$.

We need an ancillary lemma, which is standard algebraic topology: we provide a proof as we could not find an appropriate reference. For the statement, recall that a sheaf $\mathcal{S}$ on a locally compact space $X$ is $c$-fine if the homomorphism sheaf $\mathcal{H}om(\mathcal{S},\mathcal{S})$ is $c$-soft (see, for example, [Reference Bredon3, Section II.9]).

Proof. We define a resolution $0\to \mathcal{Z}\to \mathcal{S}^0\to \mathcal{S}^1\to 0$ of the sheaf $\mathcal{Z}$ of locally constant ${\mathbb Z}$-valued functions on $\mathbb{R}$: $\mathcal{S}^0$ is the sheaf whose sections on an open set $U\subseteq \mathbb{R}$ are the functions $s:U\to {\mathbb Z}$ such that for all $x\in U$, ${\displaystyle \lim_{y\to x^-}s(y)}$ exists, ${\displaystyle\lim_{y\to x^+}s(y)}$ exists and equals $s(x)$, and $s$ has a discrete set of discontinuities in $U$; $\mathcal{S}^1$ is the sheaf whose sections on an open set $U$ are functions $s:U\to {\mathbb Z}$ with discrete support; the map $\mathcal{Z}\to \mathcal{S}^0$ is the canonical inclusion; and the map $\mathcal{S}^0\to \mathcal{S}^1$ is defined over an open set $U$ by sending a section $s$ to the section $t$ defined by ${\displaystyle t(x):=s(x)-\lim_{y\to x^-}s(y)}$. One checks directly that each $\mathcal{S}^i$ is a $c$-fine ${\mathbb Z}$-equivariant sheaf, and that $\mathcal{S}^1$ identifies with $\mathcal{S}^0/\mathcal{Z}$.

Define now a map $\int:\Gamma_c(\mathbb{R};\mathcal{S}^1)\to {\mathbb Z}$ by sending a section $s$ to $\sum_{x\in \mathbb{R}} s(x)$ (this makes sense as $s$ is finitely supported). The map $\int$ extends to a map of complexes $\int:\Gamma_c(\mathbb{R};\mathcal{S}^\bullet)\to \mathcal{C}^\bullet$ where $\mathcal{C}^\bullet$ is ${\mathbb Z}$ in degree one, and zero in all other degrees. As the homology groups of the complex $\Gamma_c(\mathbb{R};\mathcal{S}^\bullet)$ (i.e. the cohomology $H_c^i(\mathbb{R}))$ are ${\mathbb Z}$ in degree one, and zero in all other degrees, this map is a quasi-isomorphism.

Let now $\mathcal{T}^\bullet$ be any resolution of the sheaf $\mathcal{Z}$ of locally constant ${\mathbb Z}$-valued functions on $X$ by $c$-fine ${\mathbb Z}$-sheaves, and define $(\mathcal{S}\otimes \mathcal{T})^i$ to be the sheaf $\mathcal{S}^0\otimes \mathcal{T}^i\oplus \mathcal{S}^1\otimes \mathcal{T}^{i-1}$. As each $\mathcal{S}^i$ is torsion free, the tensor product sheaf $(\mathcal{S}\otimes \mathcal{T})^\bullet$ is a resolution of the sheaf of locally constant ${\mathbb Z}$-valued functions on $\mathbb{R}\times X$. Moreover, it consists of ${\mathbb Z}$-equivariant sheaves, and each constituent sheaf in the resolution is $c$-fine by [Reference Bredon3, Corollary II.9.18].

Now, as $\mathcal{S}^\bullet$ and $\mathcal{T}^\bullet$ are $c$-fine, [Reference Bredon3, Proposition II.5.1] identifies $\Gamma_c(\mathbb{R}\times X;(\mathcal{S}\otimes \mathcal{T})^\bullet)$ with the tensor product complex $\Gamma_c(\mathbb{R};\mathcal{S}^\bullet)\otimes \Gamma_c(X;\mathcal{T}^\bullet)$. As $\int$ induces a quasi-isomorphism $\Gamma_c(\mathbb{R};\mathcal{S}^\bullet)\to \mathcal{C}^\bullet$, $\int\otimes \text{id}$ induces a quasi-isomorphism

(25)\begin{equation} \smallint\otimes \text{id}:\Gamma_c(\mathbb{R}\times X;(\mathcal{S}\otimes \mathcal{T})^\bullet)\to \Gamma_c(X;\mathcal{T}^{\bullet-1}) \end{equation}

by the Künneth formula (the purely algebraic version for chain complexes—see, for example, [Reference Weibel60, Theorem 3.6.3]). As the left-hand side in line (25) computes $H_c^{*}(\mathbb{R}\times X)$ and the right-hand side computes $H_c^{*-1}(X)$, we are done.

Lemma 6.8. Let $X$ be a locally compact Hausdorff space equipped with an action $\alpha$ of ${\mathbb Z}$. For each $n$, the short exact sequences of Proposition 5.1 form the bottom row in a commutative diagram of short exact sequences

where the left-hand vertical map is induced by integration along the fibre $\mathbb{R}$, the middle vertical map is induced again by the isomorphism from Lemma 5.6 plus the Morita equivalence ${\mathbb Z} \lt imes (\mathbb{R}\times X)\sim M_\alpha$, and all the vertical maps are isomorphisms. The same holds if we use rational rather than integer coefficients.

Proof. Equip $\mathbb{R}$ with the translation action of ${\mathbb Z}$, and let $\int:H^n_c(\mathbb{R}\times X)\to H^{n-1}_c( X)$ be the integration along the fibre isomorphism if Lemma 6.7. Moreover, Lemma 5.6 implies that external product with this generator also induces isomorphisms on groupoid homology $H_{n-1}({\mathbb Z} \lt imes (\mathbb{R}\times X))\cong H_n({\mathbb Z} \lt imes X)$. We thus get a commutative diagram

Morita invariance (see [Reference Crainic and Moerdijk9, Corollary 4.6]) gives that the middle entry of the top row is isomorphic to $H^{n+1}_c(M_\alpha)$ and the five lemma implies that the central vertical map is an isomorphism as claimed. (This reference requires a finite-dimensionality assumption on $X$, and there are some technical difficulties with derived categories to remove it. A simpler argument in the infinite-dimensional case can be deduced using hyperhomology $\mathbb{H}_*({\mathbb Z};\mathbb{R}\times X)$ and a Cartan–Leray spectral sequence argument as in [Reference Brown4, page 173, particularly line (7.8)].) The same argument works with rational coefficients (or use the universal coefficient theorem).

Theorem 6.9. Let $X$ be a locally compact Hausdorff space, and let $\alpha:X\to X$ be a homeomorphism inducing an action, also denoted $\alpha$, of ${\mathbb Z}$ on $X$.

There is a natural Chern character

\begin{equation*} \text{ch}_\alpha:K_*(C_0(X)\rtimes_\alpha {\mathbb Z})\to H_{**}({\mathbb Z} \lt imes X;{\mathbb Q}). \end{equation*}

that becomes an isomorphism on tensoring the left-hand side by ${\mathbb Q}$, and that fits into a commutative diagram of short exact sequences

(26)

with the left and right vertical maps induced by the classical Chern character. Moreover, one can replace $\text{ch}_\alpha$ by an integral isomorphism if there is an integral Chern isomorphism for $M_\alpha$ in the sense of Definition C.1.

Finally, if $X$ is compact, then for any invariant probability measure $\tau$ on $X$ the diagram

(27)

commutes. (This can be usefully compared to material in [Reference Guo, Proietti and Wang26, Section 3].)

Proof. The Chern character $\text{ch}_\alpha$ is defined by composing the maps

\begin{equation*} K_*(C(X)\rtimes_\alpha{\mathbb Z})\to K^{*+1}(M_\alpha) \stackrel{\text{ch}}{\to} H^{**+1}(M_\alpha;{\mathbb Q})\to H_{**}({\mathbb Z} \lt imes X;{\mathbb Q}) \end{equation*}

where the first arrow is from Proposition 22, the second is the classical Chern character, and the third is from Lemma 6.4 on the summand $H^1(M_\alpha)$, and from Lemma 6.8 on the summands in all other degrees. It is a rational isomorphism as the central map is a rational isomorphism, and as the left and right maps are isomorphisms.

For commutativity of the diagram in line (26), we consider the large diagram

where the top two rows are as in Proposition 6.2, part (iii) (which shows that part of the diagram commutes), the maps between the middle two rows are the classical Chern characters (so that part of the diagram commutes by naturality of the Chern character), and the maps between the bottom two rows are as in Lemma 6.4 in degree one and as in Lemma 6.8 for all other degrees (and so that part of the diagram commutes by those lemmas).

To check compatibility of the pairings, consider the diagram

where the top composition defines the Chern character, and the last two vertical arrows are defined by first projecting onto $H^1(M_\alpha;{\mathbb Q})$ and $H^0({\mathbb Z} \lt imes X;{\mathbb Q})$ and then taking the respective pairings. The first square commutes by Proposition 6.3, and the last square by Lemma 6.5, so it remains to check commutativity of the central square.

For this, recall that for a compact space $Y$, the component of the Chern character $K^1(Y) \to H^1(Y;{\mathbb Q})$ with values in $H^1_c$ is defined by taking a unitary $u\in M_n(C(Y))$ to the continuous function $\text{det}(u):Y\to S^1$ (this is even defined integrally, but we do not need it). It thus suffices to check that if $u\in M_n(C(M_\alpha))$ is a unitary that is continuously differentiable in the $t$ variable (and equal to one outside a compact set), then

\begin{equation*} \tau(\text{det}(u)'\text{det}(u^*))=\tau(u'u^*) \end{equation*}

(where the prime denotes the derivative in the $t$-direction). This follows as if we locally write $u=e^{2\pi i x}$ with $x$ self-adjoint, then $\text{det}(u)=e^{2\pi i \text{tr}(x)}$ and so $\text{det}(u)'=\text{tr}(x')\text{det}(u)$, while on the other hand $u'=x'u$. The result follows from these local computations.

Proof. For part (i), we will show that there is an integral Chern isomorphism for $M_\alpha$ in the sense of definition C.1. We first note that the short exact sequences

(28)\begin{equation} 0\to H^i_c(SX)_{\mathbb Z}\to H^i(M_\alpha)\to H^i(X)^{\mathbb Z}\to 0 \end{equation}

from Proposition 5.1 imply that $H^i(M_\alpha)=0$ for $i \gt 4$, so we only need to work in degrees four or less. Moreover, this short exact sequence gives an isomorphism $H^4_c(SX)_{\mathbb Z}\to H^4(M_\alpha)$; this implies in particular that the map $H^4_c(SX)\to H^4(M_\alpha)$ is surjective, and as cup products are zero on a suspension (see Lemma C.5) we are in the situation of Lemma C.6 where the map $c$ of Definition C.3 is a homomorphism for $M_\alpha$.

Hence, we have a commutative diagram

of such Chern class maps. The left and right vertical maps are induced by the isomorphism $c:K^*(X)\cong H^{**}_c(X)$; by naturality of $c$, this is also an isomorphism of ${\mathbb Z}$-modules whence it induce isomorphisms on invariants and coinvariants. The result follows from the five lemma.

For part (ii), if $K^i(X)$ is free, then so are $K^i(X)^{\mathbb Z}$ (as subgroups of free groups are free) and $H^i(X)^{\mathbb Z}$ (as the assumed integral Chern isomorphism must also induce an isomorphism on fixed point subgroups by naturality). Hence, the Pimnser–Voiculescu type sequences all split (unnaturally), and the result follows by choosing the splittings appropriately (this is possible as everything in sight is free: we leave the algebra to the reader).

7. Examples of crossed products by the integers

In this section, we discuss a number of crossed product groupoids and their associated $C^*$-algebras. Each is obtained by a free action of the integers on a compact metric space. First, as promised in Section 3, we give a result covering the irrational rotation algebras.

Our next example provides a negative result: we give an example of a minimal action on a manifold so that the transformation groupoid is not HK-good.

Example 7.2. Let $M=S^3 \times \mathbb{R}P^4$. Then, by [Reference Fathi and Herman21, Theorem 1], there exists $\varphi : M \rightarrow M$ a minimal diffeomorphism that is homotopic to the identity and uniquely ergodic. The Pimsner–Voiculescu sequence and the fact that $\varphi$ is homotopic to the identity imply that

\begin{equation*} K_0( C(M) \rtimes {\mathbb Z}) \cong K_1(C(M) \rtimes {\mathbb Z}) \cong K^0(M) \oplus K^1(M). \end{equation*}

Since

\begin{equation*} K^0(S^3\times \mathbb{R}P^4) \cong K^1(S^3\times \mathbb{R}P^4) \cong K^0(\mathbb{R}P^4)\oplus K^1(\mathbb{R}P^4)\cong {\mathbb Z}\oplus ({\mathbb Z}/4) \end{equation*}

(see, for example, [Reference Karoubi30, IV.6.47] for the $K$-theory of real projective space), we get

\begin{equation*} K_0( C(M) \rtimes {\mathbb Z}) \cong K_1(C(M) \rtimes {\mathbb Z}) \cong {\mathbb Z}^2 \oplus ({\mathbb Z}/4)^2. \end{equation*}

On the other hand, the homology of the transformation groupoid can be computed using Proposition 5.1 and the well-known cohomology groups of real projective space (see, for example, [Reference Hatcher27, Theorem 3.12]). One gets

\begin{equation*} H_{ev}({\mathbb Z} \lt imes M) \cong H_{od}({\mathbb Z} \lt imes M) \cong {\mathbb Z}^2 \oplus ({\mathbb Z}/2)^4. \end{equation*}

Hence, ${\mathbb Z} \lt imes M$ is not HK-good.

Returning to positive results, there are other (and in fact many) transformation groupoids associated to integer actions that are HK-good. A number of the constructions build on work in [Reference Deeley, Putnam and Strung15, Reference Deeley, Putnam and Strung16]. In particular, we will use a specific dynamical system constructed in [Reference Deeley, Putnam and Strung15]: we summarize the key points in the next theorem.

Example 7.4. Let $G$ be a finitely generated abelian group and take $Y$ be a connected finite CW-complex with dimension at most three, with

\begin{equation*} H^0(Y)\cong {\mathbb Z}\ \hbox{and}\ H^2(Y) \cong G, \end{equation*}

and other cohomology groups trivial. Using Example C.8, the K-theory of $Y$ is

\begin{equation*} K^0(Y) \cong {\mathbb Z} \oplus G\ \hbox{and}\ K^1(Y)\cong \{0 \}. \end{equation*}

Let $X=Z\times Y\times Q$ where $Z$ as in Theorem 7.3 and $Q$ the Hilbert cube. In particular,

\begin{equation*} K^*(X) \cong K^*(Y)\ \hbox{and}\ H^*(X) \cong H^*(Y). \end{equation*}

Applying [Reference Fathi and Herman21, Theorem 1] there exists a minimal, uniquely ergodic homeomorphism $\varphi: X \rightarrow X$ that is homotopic to the identity. The Pimsner–Voiculescu exact sequence and the fact that $\varphi$ is homotopic to the identity imply that

\begin{equation*} K_0(C(X)\rtimes {\mathbb Z}) \cong K_1(C(X)\rtimes {\mathbb Z}) \cong {\mathbb Z} \oplus G. \end{equation*}

The groupoid homology can be computed similarly from Proposition 5.1, giving

\begin{equation*} H_{ev}({\mathbb Z} \lt imes X)\cong H_{od}({\mathbb Z} \lt imes X) \cong {\mathbb Z} \oplus G. \end{equation*}

These groupoids are HK-good by Corollary 6.10, part (i).

It is worth noting that if we take $G={\mathbb Z} \oplus ({\mathbb Z}/4)$, then the $C^*$-algebra $C(X) \rtimes {\mathbb Z}$ is isomorphic to the crossed product $C^*$-algebra in Example 7.2: thus, we see that this $C^*$-algebra admits both an HK-good and a non-HK-good model.

The next class of examples we consider is crossed products where the underlying space has finitely generated $K$-theory. The starting point is a slight generalization of [Reference Deeley, Putnam and Strung17, Theorem 4.5].

Theorem 7.5. Suppose $d\in \mathbb{N} \setminus \{0\}$ and $F_0$, $F_1$ are finite abelian groups. Then there exists a connected finite CW-complex $Y$ with dimension at most three and $\varphi: X\rightarrow X$ a minimal uniquely ergodic homeomorphism such that

\begin{equation*} K_0(C(X)\rtimes {\mathbb Z}) \cong {\mathbb Z}^d \oplus F_0\ \hbox{and}\ K_1(C(X)\rtimes {\mathbb Z})\cong {\mathbb Z}^d\oplus F_1 \end{equation*}

where $X=Z\times Y\times Q$ (with $Z$ as in Theorem 7.3 and [Reference Deeley, Putnam and Strung15] and $Q$ the Hilbert cube).

The next result is immediate from Corollary 6.10, part (i).

Next, a refinement of the previous two results is discussed. We state it because of the importance of projectionless unital $C^*$-algebras and as in this case the Elliott invariant of the relevant $C^*$-algebra can be completely determined (rather than just the $K$-theory as in Theorem 7.5).

Example 7.8. In this example, we give a self-contained development of the construction used in Theorem 7.5 in the special case when $d=1$, $F_0={\mathbb Z}/3$, and $F_1={\mathbb Z}/2$. We hope this example is useful to readers less familiar with [Reference Deeley, Putnam and Strung17].

Take $S^1\vee S^1\vee S^1$ with base point the wedge point and the finite order self-homomorphism given by cyclically permuting the copies of $S^1$. Form the reduced mapping cone, $C_f$ of the map $f: S^1\vee S^1\vee S^1 \rightarrow S^1$ defined by mapping each circle in $S^1\vee S^1\vee S^1$ to $S^1$ identically. The reduced mapping cone construction is equivariant, so we get a finite order self-homomorphism, $\beta_1: C_f \rightarrow C_f$. Using the long exact sequence in cohomology,

\begin{equation*} H^0(C_f)\cong {\mathbb Z}\quad \hbox{and}\quad H^2(C_f)\cong {\mathbb Z} \oplus {\mathbb Z} \end{equation*}

where the part of the long exact sequence that is relevant for later use is

\begin{equation*} 0 \rightarrow {\mathbb Z} \rightarrow {\mathbb Z}^3 \rightarrow H^2(C_f) \rightarrow 0 \end{equation*}

with map ${\mathbb Z} \rightarrow {\mathbb Z}^3$ given by $n \mapsto (n,n,n)$.

Let $e_1$, $e_2$, and $e_3$ denote, respectively, the images of $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ in $H^2(C_f)$. Equivariance implies that $\beta^*(e_1) = e_2$ and that $\beta^*(e_2)=e_3=-e_1-e_2$ where the second equality follows from the specific nature of the map (i.e., $n \mapsto (n,n,n)$). Therefore, we have that the map $\beta^*: {\mathbb Z}^2 \cong H^2(C_f)\rightarrow H^2(C_f)\cong {\mathbb Z}^2$ is given by the matrix

\begin{equation*} A=\left( \begin{array}{cc} 0 & -1 \\ 1 & -1 \end{array}\right). \end{equation*}

Let $Y$ be $S^3 \vee C_f$ with base point the wedge point. Define $\beta : S^3\vee C_f \rightarrow S^3\vee C_f$ via a reflection in a plane containing the base point on the $S^3$ factor and $\beta_1$ on $C_f$. By construction,

\begin{equation*} H^0(Y) \cong {\mathbb Z},~H^2(Y) \cong {\mathbb Z}^2,~H^3(Y)\cong {\mathbb Z}, \end{equation*}

and all other cohomology groups are trivial. Moreover, $\beta^*$ is given by the identity on $H^0(Y)$, the negation of the identity on $H^3(Y)$, and the matrix $A$ on $H^2(Y)$.

A similar argument (or one can use the Chern character) implies that for $K$-theory, we have

\begin{equation*} K^0(Y) \cong {\mathbb Z} \oplus {\mathbb Z}^2 \quad \hbox{and}\quad K^1(Y)\cong {\mathbb Z} \end{equation*}

and $\beta^*: K^0(Y) \rightarrow K^0(Y)$ is given by $id \oplus A$ and $\beta^*: K^1(Y) \rightarrow K^1(Y)$ is given by the negation of the identity.

Let $X=Z\times Y\times Q$ where $Z$ as in Theorem 7.3 (also see [Reference Deeley, Putnam and Strung15]) and $Q$ the Hilbert cube. By [Reference Deeley, Putnam and Strung16, Theorem 5.11], there exists a minimal uniquely ergodic homeomorphism $\varphi: X \rightarrow X$ such that $\varphi^*$ is given by $\beta^*$ on both $K$-theory and cohomology (here we are identifying $K^*(Y)$ and $K^*(X)$ and likewise for cohomology). Hence, the map $id - \varphi^*$ is given by the direct sum of the zero map on ${\mathbb Z}$ with the matrix

\begin{equation*} B=\left( \begin{array}{cc} 1 & 1 \\ -1 & 2 \end{array}\right) \end{equation*}

in even degree and multiplication by $2$ in odd degree. The Smith normal form of $B$ is

\begin{equation*} \left( \begin{array}{cc} 1 & 0 \\ 0 & 3 \end{array}\right). \end{equation*}

Using the Pimsner–Voiculescu exact sequence, we get that

\begin{equation*} K_0(C(X)\rtimes {\mathbb Z}) \cong {\mathbb Z} \oplus {\mathbb Z}/3 \quad \hbox{and}\quad K_1(C(X)\rtimes {\mathbb Z}) \cong {\mathbb Z} \oplus {\mathbb Z}/2. \end{equation*}

A similar computation in groupoid homology using Proposition 5.1 gives that

\begin{equation*} H_{ev}({\mathbb Z} \lt imes X) \cong {\mathbb Z} \oplus {\mathbb Z}/3 \quad \hbox{and}\quad H_{od}({\mathbb Z} \lt imes X) \cong {\mathbb Z} \oplus {\mathbb Z}/2. \end{equation*}

That the pairings are compatible can be seen directly, or follows from Corollary 6.10, part (i). Moreover, the crossed product $C(X) \rtimes {\mathbb Z}$ has a unique tracial state and fits within classification because the minimal system is uniquely ergodic.

8. Matui’s long exact sequence for an open inclusion

We now start working towards showing that the examples constructed by the first author, Putnam, and Strung in [Reference Deeley, Putnam and Strung17] are HK-good; this will take the rest of the main body of the paper. The first step, carried out in this section, will be to study a long-exact sequence for an open inclusion of groupoids developed by Matui in [Reference Matui37]. Matui works primarily with ample groupoids, i.e. those with zero-dimensional base spaces. Here we generalize Matui’s work to other base spaces.

Throughout this section, $\mathcal{G}$ denotes a locally compact, Hausdorff, étale groupoid. Let also $\mathcal{G}'$ be an open subgroupoid of $\mathcal{G}$ with the same base space. Then $\mathcal{G}'$ is étale in its own right, and for each $n$, $(\mathcal{G}')^{(n)}$ is an open subspace of $\mathcal{G}^{(n)}$. Given a $c$-soft $\mathcal{G}$-sheaf $\mathcal{S}$ on $\mathcal{G}^{(n)}$, we write $\mathcal{S}_{(\mathcal{G}')^{(n)}}$ for the sheaf on $\mathcal{G}^{(n)}$ defined by setting all stalks outside $(\mathcal{G}')^{(n)}$ to be zero (compare [Reference Bredon3, page 11], or note also that this is the same thing as $i_*i^*\mathcal{S}$, where $i:(\mathcal{G}')^{(n)}\to \mathcal{G}^{(n)}$ is the inclusion, and we use the direct and inverse image of a sheaf as in [Reference Bredon3, Section I.3]). We write moreover $F^{(n)}:=\mathcal{G}^{(n)}\setminus (\mathcal{G}')^{(n)}$ for the closed ‘difference’ space, and write $\mathcal{S}_{F^{(n)}}:=\mathcal{S}/\mathcal{S}_{(\mathcal{G}')^{(n)}}$ for the corresponding quotient sheaf. Then for each $n$, using $c$-softness we have a short exact sequence

(29)\begin{equation} 0\to \Gamma_c(\mathcal{G}^{(n)};\mathcal{S}_{(G')^{(n)}}) \to \Gamma_c(\mathcal{G}^{(n)};\mathcal{S}) \to \Gamma_c(\mathcal{G}^{(n)};\mathcal{S}_{F^{(n)}}) \to 0. \end{equation}

(this follows, for example, from [Reference Bredon3, Theorem II.9.9] and on noting that $\mathcal{S}_{(\mathcal{G}')^{(n)}}$ is $c$-soft, either by applying [Reference Bredon3, Corollary II.9.13], or arguing directly).

Let now $\mathcal{A}$ be a $\mathcal{G}$-sheaf on $\mathcal{G}^{(0)}$, and consider a resolution

(30)\begin{equation} 0\to \mathcal{A}\to \mathcal{S}^0\to \mathcal{S}^1\to \cdots \end{equation}

of $\mathcal{A}$ by $c$-soft $\mathcal{G}$-sheaves as in line (3) above. Then the short exact sequences in line (29) give rise to a short exact sequence of double complexes as in line (4), and therefore to a short exact sequence of total complexes which in the $n^\text{th}$ entry looks like

(31)\begin{equation} 0\!\to\! \bigoplus_{p-q=n}\Gamma_c(\mathcal{G}^{(p)};(\mathcal{S}_p^q)_{(G')^{(p)}})\! \to \!\bigoplus_{p-q=n}\Gamma_c(\mathcal{G}^{(p)};\mathcal{S}_p^q)\! \to \!\bigoplus_{p-q=n}\Gamma_c(\mathcal{G}^{(p)};(\mathcal{S}_p^q)_{F^{(p)}})\! \to\! 0. \end{equation}

Recall moreover that for any sheaf $\mathcal{S}$ on ${\mathcal{G}}^{(n)}$ there is a canonical identification

\begin{equation*} \Gamma_c((\mathcal{G}')^{(n)};\mathcal{S}|_{(\mathcal{G}')^{(n)}})=\Gamma_c(\mathcal{G}^{(n)};\mathcal{S}_{(\mathcal{G}')^{(n)}}) \end{equation*}

(compare [Reference Bredon3, Proposition I.6.6]). It follows that regarding the resolution of line (30) as a resolution of $\mathcal{A}$ considered as a $\mathcal{G}'$ sheaf, we have that the left hand complex appearing in line (31) computes the homology $H_*(\mathcal{G}';\mathcal{A})$.

Definition 8.1. With notation as above, let $\mathcal{A}$ be a $\mathcal{G}$-sheaf on $\mathcal{G}^{(0)}$, which we also consider as a $\mathcal{G}'$ sheaf on $(\mathcal{G}')^{(0)}=\mathcal{G}^{(0)}$. We define $H_n(\mathcal{G}/\mathcal{G}';\mathcal{A})$ to be the homology of the double complex

\begin{equation*} \Big(\Gamma_c(\mathcal{G}^{(p)};(\mathcal{S}_p^{-q})_{F^{(p)}})\Big)_{q=,...-1,0;~p=0,1,2,...} \end{equation*}

Summarizing the discussion above and using that short exact sequences of complexes induce long-exact sequences in homology, we deduce the following result.

Proposition 8.2. With notation as in Definition 8.1, there is a long exact sequence of homology groups

\begin{equation*} \cdots H_{n-1}(\mathcal{G}';\mathcal{A}) \leftarrow H_n(\mathcal{G}/\mathcal{G}';\mathcal{A})\leftarrow H_n(\mathcal{G};\mathcal{A})\leftarrow H_n(\mathcal{G}';\mathcal{A})\leftarrow H_{n+1}(\mathcal{G}/\mathcal{G}';\mathcal{A}) \leftarrow \cdots \end{equation*}

The long exact sequence from Proposition 8.2 is not immediately useful without a way to compute the groups $H_n(\mathcal{G}/\mathcal{G}';\mathcal{A})$. In the remainder of this section, we will adapt an ‘excision’ type result of Matui [Reference Matui37, Section 3] to our context; this will allow us to compute $H_n(\mathcal{G}/\mathcal{G}';\mathcal{A})$ for the orbit breaking groupoids of interest to us. Matui’s result (along with our result) are the homology version of Putnam’s $K$-theory result, [Reference Putnam51].

Keeping in the situation of a pair $(\mathcal{G},\mathcal{G}')$ of an étale groupoid and open subgroupoid with the same base space (and following Putnam [Reference Putnam51, Section 6] and Matui [Reference Matui37, Section 3]), we define $\Delta:= \mathcal{G}\setminus \mathcal{G}'$.

In [Reference Putnam51, Theorems 6.7 and 6.8], Putnam shows that we have identifications of sets $\mathcal{H}=\mathcal{G}|_{r(\Delta)}$ and $\mathcal{H}'=\mathcal{G}'|_{r(\Delta)}$ (but not of topological spaces); we use these identifications to equip $\mathcal{H}$ and $\mathcal{H}'$ with groupoid structures. The next theorem essentially summarizes [Reference Putnam51, Theorems 6.7 and 6.8] mentioned above, and also results of Matui [Reference Matui37, Theorem 3.7 and Proposition 3.9].

Theorem 8.5 With conventions as above, both $\mathcal{H}$ and $\mathcal{H}'$ are locally compact, Hausdorff, étale groupoids, with $\mathcal{H}'$ clopen in $\mathcal{H}$. Moreover, for any $n\in \mathbb{N}\cup\{0\}$, the natural inclusion

\begin{equation*} \mathcal{H}^{(n)}\setminus (\mathcal{H}')^{(n)}\to \mathcal{G}^{(n)}\setminus (\mathcal{G}')^{(n)} \end{equation*}

is a homeomorphism.

Proposition 8.6. With conventions as above, let $\mathcal{A}$ be a $\mathcal{G}$-sheaf on $\mathcal{G}^{(0)}$, let $\iota:\mathcal{H}^{(0)}\to \mathcal{G}^{(0)}$ denote the canonical continuous map (we say ‘continuous map’ rather than ‘inclusion’ as while this injective continuous map is set-theoretically an inclusion, it is typically not a homeomorphism onto its image), and write $\iota^*\mathcal{A}$ for the pullback $\mathcal{H}$-sheaf. Then the natural map of pairs $(\mathcal{H},\mathcal{H}')\to (\mathcal{G},\mathcal{G}')$ induces an isomorphism

\begin{equation*} H_*(\mathcal{H}/\mathcal{H}';\iota^*\mathcal{A})\cong H_*(\mathcal{G}/\mathcal{G}';\mathcal{A}) . \end{equation*}

Proof. By definition, $H_*(\mathcal{G}/\mathcal{G}';\mathcal{A})$ is the homology of the double complex

\begin{equation*} \Big(\Gamma_c(\mathcal{G}^{(p)};(\mathcal{S}_p^{-q})_{F^{(p)}})\Big)_{q=,...-1,0;~p=0,1,2,...} \end{equation*}

as in the statement of Proposition 8.2, where $\mathcal{S}^\bullet$ is a resolution of $\mathcal{A}$ by $c$-soft sheaves. Using the natural identification

\begin{equation*} \Gamma_c(\mathcal{G}^{(p)};(\mathcal{S}_p^{-q})_{F^{(p)}})=\Gamma_c(F^{(p)};(\mathcal{S}_p^{-q})|_{F^{(p)}}) \end{equation*}

of [Reference Bredon3, Proposition II.6.6] (compare also [Reference Bredon3, Section II.10]), $H_*(\mathcal{G}/\mathcal{G}';\mathcal{A})$ is the same as the homology of the double complex

(32)\begin{equation} \Big(\Gamma_c(F^{(p)};(\mathcal{S}_p^{-q})|_{F^{(p)}})\Big)_{q=,...-1,0;~p=0,1,2,...} . \end{equation}

On the other hand, note that as pullbacks are exact functors (this is straightforward from the definition: compare [Reference Kashiwara and Schapira31, line (2.3.3)]), and as pullbacks of $c$-soft sheaves by $\iota$ are $c$-soft (this follows as $\iota$ is injective and takes compact sets to compact sets) we have that $\iota^*\mathcal{S}^\bullet$ is a $c$-soft resolution of $\iota^*\mathcal{A}$. Hence, $H_*(\mathcal{H}/\mathcal{H}';\iota^*\mathcal{A})$ is the homology of the double complex

(33)\begin{equation} \Big(\Gamma_c(\mathcal{H}^{(p)}\setminus (\mathcal{H}')^{(p)};(\iota^*\mathcal{S}_p^{-q})|_{\mathcal{H}^{(p)}\setminus (\mathcal{H}')^{(p)}})\Big)_{q=,...-1,0;~p=0,1,2,...}. \end{equation}

However, the last statement of Theorem 8.5 implies that for each $p$, the (set-theoretic) inclusion $\mathcal{H}^{(p)}\to \mathcal{G}^{(p)}$ induces an identification of topological spaces $F^{(p)}=\mathcal{H}^{(p)}\setminus (\mathcal{H}')^{(p)}$. Hence, the double complexes in lines (32) and (33) are the same.

9. Orbit breaking subgroupoids

In this section, we specialize the results of Section 7 to so-called orbit-breaking groupoids associated to actions of the integers. We finish with a general result comparing homology of orbit breaking groupoids and $K$-theory of the associated $C^*$-algebras. Throughout, we will fix a locally compact Hausdorff space $X$ and a homeomorphism $\varphi : X \to X$ inducing a free action of ${\mathbb Z}$ on $X$.

From the dynamical system $(X, \varphi)$, one can construct the transformation groupoid ${\mathbb Z} \lt imes X$ as we discussed in Section 1 above. To match the existing literature better, we instead work with the equivalence relation

\begin{equation*} \mathcal{R}_{\varphi} := \{(\varphi^{n}(x),x ) \mid x \in X, n \in \mathbb{Z} \}, \end{equation*}

which is a groupoid with the operations restricted from the pair groupoid $X\times X$. Since the dynamical system is free, the map

\begin{equation*} {\mathbb Z} \lt imes X \to \mathcal{R}_{\varphi}, \quad (n,x) \mapsto (\varphi^{n}(x),x) \end{equation*}

is a bijection. We equip $\mathcal{R}_{\varphi}$ with the topology such that this bijection is a homeomorphism; this makes $\mathcal{R}_\varphi$ into a topological groupoid, isomorphic to ${\mathbb Z} \lt imes X$.

Let now $Y$ be a closed non-empty subset of $X$, and assume for simplicity that $Y$ has finite covering dimension. We say that $Y$ meets every orbit at most once if $\varphi^n(Y) \cap Y = \varnothing$ for each $n \neq 0$. Following Putnam (compare [Reference Putnam48], [Reference Putnam49, Example 2.6], and [Reference Putnam51]) we define $\mathcal{R}_Y \subseteq \mathcal{R}_{\varphi}$ to be the subequivalence relation of $\mathcal{R}_\varphi$ obtained from splitting every orbit that passes through $Y$ into two equivalence classes: precisely, define