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# Homology and $K$-theory of dynamical systems
III. Beyond stably disconnected Smale spaces

Valerio Proietti  Graduate School of Mathematical Sciences, The University of
Tokyo, Japan Department of Mathematics, University of Oslo, P.O. box 1053,
Blindern, 0316 Oslo, Norway<EMAIL_ADDRESS>and  Makoto Yamashita
Department of Mathematics, University of Oslo, P.O. box 1053, Blindern, 0316
Oslo, Norway<EMAIL_ADDRESS>
(Date: v3: October 17, 2023; v2: October 28, 2022; v1: July 7, 2022.)

###### Abstract.

We study homological invariants of étale groupoids arising from Smale spaces,
continuing on our previous work, but going beyond the stably disconnected case
by incorporating resolutions in the space direction. We show that the homology
groups defined by Putnam are isomorphic to the Crainic–Moerdijk groupoid
homology with integer coefficients. We also show that the $K$-groups of
C∗-algebras of stable and unstable equivalence relations have finite rank. For
unstably disconnected Smale spaces, we provide a cohomological spectral
sequence whose second page is the (stable) homology groups, and converges to
the $K$-groups of the unstable C∗-algebra.

###### Key words and phrases:

groupoid homology, topological dynamics, Smale spaces, spectral sequences,
$K$-theory

###### 2020 Mathematics Subject Classification:

37B02; 55T25, 46L80

###### Contents

  1. 1 Preliminaries
  2. 2 Symmetric simplicial spaces
  3. 3 Comparison of homologies
  4. 4 Smale space with totally disconnected unstable sets
  5. 5 Finiteness for general Smale spaces

## Introduction

In this paper we study homological and $K$-theoretical invariants of a class
of hyperbolic dynamical systems known as _Smale spaces_ introduced by D.
Ruelle [ruelle:thermo], continuing our work
[valmak:groupoid][valmak:groupoidtwo]. Previously we have focused on stably
disconnected systems, which are represented by _ample groupoids_ , i.e.,
totally disconnected étale topological groupoids. Now we lift this restriction
and work general non-wandering Smale spaces.

Smale spaces capture hyperbolicity phenomena in a topological setting, akin to
the dynamics of Anosov diffeomorphisms, and more generally basic sets of Axiom
A diffeomorphisms, in the differentiable (smooth) setting. Other examples of
Smale spaces include shifts of finite type, generalized solenoids, systems
arising from self-similar group actions [MR2162164][MR2526786], substitution
tiling systems [MR1631708], etc.

Aside from the purely dynamical interest in these systems, there is a fruitful
interplay with the theory of operator algebras, via the construction of
groupoid C∗-algebras. Given a second countable locally compact groupoid $G$
with continuous Haar system, the convolution product on the space of compactly
supported continuous functions gives a complex algebra with involution that
can be completed to a C∗-algebra [MR584266]. This can be further generalized
to incorporate continuous actions of $G$ on C∗-algebras, and can be analyzed
by homological invariants such as the operator $K$-groups.

For Smale spaces, the C∗-algebras constructed from the stable and unstable
equivalence relation capture interesting aspects of the homoclinic and
heteroclinic structure of expansive dynamics
[put:algSmale][thomsen:smale][mats:ruellemarkov]. The C∗-algebras attached to
Smale spaces provide a unified framework that contain (AF-algebras inside) the
Cuntz–Krieger algebras, the Bunce–Deddens algebras, irrational rotation
algebras, among others. They are also known to have good structural properties
one would expect in the theory of classification of C∗-algebras, see
[deeley:strung][dgy:rrzero][kp:clas][phil:clas] and references therein.

Another intriguing homological invariant for dynamical system is the homology
of étale groupoids with finite cohomological dimension due to M. Crainic and
I. Moerdijk, which is based on sheaves, derived formalism, and simplicial
methods [cramo:hom]. For _ample_ groupoids, which correspond to dynamical
systems on totally disconnected spaces, a connection between this homology and
operator $K$-theory of the associated C∗-algebra was recently popularized by
H. Matui [MR2876963][MR3552533], and gained a lot of attention in the
C∗-algebra community. More precisely, Matui has conjectured that groupoid
homology (suitably periodicized) and $K$-groups of the groupoid C∗-algebra are
isomorphic. Although the original form of this conjecture is shown to be
incorrect by some counterexamples [scarparo:hk][deeley:hk], nonetheless it is
still confirmed in many interesting cases by concrete computations
[simsfarsi:hk][bdgw:matui].

In our previous works [valmak:groupoid][valmak:groupoidtwo] we provided a more
systematic picture behind this correspondence, constructing a homological
spectral sequence whose $E^{2}$-sheet is given by the homology groups of ample
groupoids $G$, and abuts to the $K$-groups of certain $G$-C∗-algebras
[valmak:groupoid], based on the Meyer–Nest theory of triangulated categorical
structure on equivariant $\mathit{KK}$-theory [meyernest:tri][meyer:tri]. When
the groupoid has _torsion-free_ stabilizers and satisfies (a stronger form of)
the Baum–Connes conjecture as studied by J.-L. Tu [MR1703305], the abutment is
indeed the $K$-groups of the groupoid C∗-algebra of $G$.

Turning back to Smale spaces, I. F. Putnam developed a theory of homology
based on an intricate resolution of Smale spaces by symbolic dynamics (_shifts
of finite type_), see [put:HoSmale][dkw:func, val:smale]. This theory can be
seen as an attempt to classify Axiom A systems by means of purely
combinatorial data, an insight that goes back to S. Smale himself, in parallel
with the theory of Morse–Smale systems.

More precisely, the main ingredient in his homology is a pair of Smale spaces
$(Y,\psi)$ and $(Z,\zeta)$ that map onto $(X,\phi)$ such that: $(Y,\psi)$ has
totally disconnected unstable sets, and the factor map induces a homeomorphism
of stable sets (an _$s$ -bijective map_); $(Z,\zeta)$ has totally disconnected
stable sets, and the factor map induces a homeomorphism of unstable sets (a
_$u$ -bijective map_). Then the iterated fiber products of these spaces give a
simplicial-cosimplicial system of shifts of finite type, leading to this
homology.

He then posed a number of interesting questions, and in particular predicted a
spectral sequence relating his homology to the operator $K$-groups of the
groupoid [put:HoSmale]*Question 8.4.1.

In [valmak:groupoidtwo], for the non-wandering Smale spaces _with totally
disconnected stable sets_ , we showed that Putnam’s stable homology coincides
with the Crainic–Moerdijk homology for the unstable equivalence relation
groupoid, reduced to a transversal, with integer coefficients. Combined with
the above spectral sequence from triangulated categorical structure of
equivariant $\mathit{KK}$-theory, it gave an answer to Putnam’s question in
this class of Smale spaces.

Motivated by this, in this paper we further generalize the identification of
the homology groups non-wandering Smale spaces, as follows.

######  Theorem A (Theorem 3.5).

Let $(X,\phi)$ be a non-wandering Smale space, and $T$ be a transversal for
the unstable equivalence relation groupoid $R^{u}(X,\phi)$. Then Putnam’s
stable homology $H^{s}_{\bullet}(X,\phi)$ is isomorphic to the groupoid
homology of the étale groupoid $G=R^{u}(X,\phi)|_{T}$ with coefficients in the
constant sheaf of integers:

$H^{s}_{k}(X,\phi)\cong H_{k}(G,\underline{\mathbb{Z}}).$

One crucial ingredient is a cosimplicial system of $G$-sheaves resolving the
constant sheaf $\underline{\mathbb{Z}}$, which naturally arise cosimplicial
system of Smale spaces over $(X,\phi)$ coming from iterated fiber product of
the $(Z,\zeta)$ above. We combine sheaf theoretic methods for the associated
complex together with techniques behind our previous result to obtain the
above theorem.

The above result, in combination results from [valmak:kpd], leads to a number
of concrete structural results on homology of Smale spaces. On one hand, we
obtain a Künneth type formula for the homology of Smale spaces (Theorem 3.6),
generalizing results in [valmak:groupoidtwo] and [dkw:dyn] to arbitrary non-
wandering Smale spaces. On the other, we give a solution to a question by
Putnam [put:HoSmale]*Question 8.3.2 on a relation between his homology of
$(X,\phi)$ and the cohomology of the underlying space $X$ (see Theorem 3.7 for
the precise statement).

We next turn to the problem of comparison between homology and $K$-theory. As
we remarked above, there is a homological spectral sequence whose
$E^{2}$-sheet is the Putnam homology and abuts to the $K$-groups of
$C^{*}R^{u}(X,\phi)$ for Smale spaces with totally disconnected stable sets.
Here we give the dual analogue of this result, for Smale spaces whose
_unstable_ sets are totally disconnected.

######  Theorem B (Corollary 4.2).

Let $(X,\phi)$ be a non-wandering Smale space with totally disconnected
unstable sets. There is a cohomological spectral sequence abutting to
$K_{\bullet}(C^{*}R^{u}(X,\phi))$, with $E_{2}$-sheet given by
$H^{s}_{\bullet}(X,\phi)$.

Again this gives an answer to [put:HoSmale]*Question 8.4.1 in this class of
Smale spaces. While the claim is quite analogous to our previous result, the
proof turns out to be somewhat different: for the case of stable sets being
totally disconnected, one starts with a factor map from a shift of finite type
that maps onto $X$ by an $s$-bijective map. This leads to an open subgroupoid
of the étale groupoid $G$, where the theory of projective resolutions in
triangulated categories [meyernest:tri][meyer:tri] is applicable through
restriction and induction functors between the equivariant
$\mathit{KK}$-categories.

For the case of unstable sets being totally disconnected, one starts with a
factor map from a shift of finite type by a $u$-bijective map. This leads to a
totally disconnected space $T_{0}$ on which the étale groupoid $G$ acts.
Moreover, we obtain a simplicial system $T_{\bullet}$ of totally disconnected
$G$-spaces, with a compatible action of the symmetric group, by taking the
iterated pullback construction of the structure map $T_{0}\to T$. Morally
speaking this represents a kind of injective resolution, which however does
not behave as nicely as projective resolutions in the framework of
$\mathit{KK}$-categories.

Here we instead adapt ideas behind G. Segal’s work [MR232393] on simplicial
spaces and spectral sequences. Using a variation of geometric realization
suitable in the setting of locally compact spaces, we obtain a spectral
sequence analogous to the Atiyah–Hirzebruch spectral sequence but without CW-
complexes around. The abutment to the desired $K$-groups, which corresponds to
the Baum–Connes conjecture in the case of projective resolutions, comes from
the sheaf theoretic comparison of cohomological invariants for the nerves of
the associated transformation groupoid on the $G$-simplicial spaces
$T_{\bullet}$.

Finally, in the setting of general non-wandering Smale spaces, while we cannot
directly relate the homology groups to the $K$-theory groups, the techniques
underlying Theorem B still allow us to prove the following finiteness result
for the $K$-groups of the stable and unstable equivalence relation groupoids
for Smale spaces.

######  Theorem C (Theorem 5.1).

Let $(X,\phi)$ be a non-wandering Smale space. Then the $K$-theory groups

$K_{\bullet}(C^{*}R^{s}(X,\phi)),\qquad K_{\bullet}(C^{*}R^{u}(X,\phi))$

of the associated groupoid C∗-algebras are of finite rank.

This was conjectured in [MR3692021], and has implications on the structure of
the _Ruelle algebras_ , i.e., the crossed products of $C^{*}R^{s}(X,\phi)$ and
$C^{*}R^{u}(X,\phi)$ by the natural integer actions induced by $\phi$. It also
shows that C∗-algebras of Smale spaces do _not_ exhaust the class of (simple)
classifiable, real rank zero C∗-algebras, which is a natural question in the
context of the classification program of nuclear C∗-algebras, see
[deeley:strung][dgy:rrzero] for further details.

The paper is organized as follows. In Section 1 we recall the basic notions
and fix our conventions used throughout the paper. We also briefly summarize
our previous work to set the background for this work.

Section 2 is the technical core of the paper, where we use methods from
simplicial homotopy theory and manipulate sheaf resolutions to achieve
approximation results in cohomology and $K$-theory. These results are useful
in view of the (co)simplicial structure arising from factor maps of shifts of
finite type.

In the next three sections we consider applications to Smale spaces. The main
results here are the comparison theorem between Putnam’s homology and groupoid
homology (Theorem A) in Section 3, the spectral sequence for _unstably_
disconnected Smale spaces relating homology and $K$-groups (Theorem B) in
Section 4, and finally the finiteness result on the $K$-groups of Smale spaces
(Theorem C) in Section 5.

### Acknowledgments

V.P.: this research was supported by: Foreign Young Talents’ grant (National
Natural Science Foundation of China), CREST Grant Number JPMJCR19T2 (Japan),
Marie Skłodowska-Curie Individual Fellowship (project number 101063362).

M.Y.: this research was funded, in part, by The Research Council of Norway
[project 300837].

##  1\. Preliminaries

We fix conventions in use throughout the paper. We only briefly recall
definitions and mostly refer to and generally follow the treatment in
[valmak:groupoid][valmak:groupoidtwo].

###  1.1. Topological groupoids

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 $n$-th nerve space (for
$n\geq 1$) given by

$G^{(n)}=\\{(g_{1},\dots,g_{n})\in G^{n}\colon s(g_{i})=r(g_{i+1})\\}.$

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.

When we work with operator algebras, we further assume that $G$ admits a
_continuous Haar system_ [MR584266]. A choice of continuous left Haar system
is denoted by $(\lambda^{x})_{x\in G^{(0)}}$, and the associated C∗-algebras
are denoted by $C^{*}(G,\lambda)$ and $C^{*}_{r}(G,\lambda)$, for the
universal and reduced model, respectively. Most concrete groupoids we consider
will be amenable so that this distinction disappears. When the choice of
$\lambda$ is already understood, we also write $C^{*}G$ and $C^{*}_{r}G$. When
$G$ is étale, we take $\lambda$ to be the counting measure.

Following [cramo:hom], we say that a (topological) groupoid homomorphism
$f\colon H\to G$ is a _Morita equivalence_ if

$\\{(x,g)\mid x\in H^{(0)},g\in G,f(x)=r(g)\\}\to G^{(0)},\quad(x,g)\mapsto
s(g)$

is an étale surjection, and

${H}$${G}$${H^{(0)}\times H^{(0)}}$${G^{(0)}\times
G^{(0)}}$$\scriptstyle{f}$$\scriptstyle{(r,s)}$$\scriptstyle{(r,s)}$$\scriptstyle{(f,f)}$

is a pullback diagram. Two groupoids $G$, $H$ are _Morita equivalent_ if there
is another groupoid $K$ and Morita equivalence homomorphisms $K\to G$, $K\to
H$. This is equivalent to the existence of a _bibundle_ $H\curvearrowright
X\curvearrowleft G$, see [valmak:groupoidtwo] and references therein.

In this paper, when $G$ is a groupoid which is not étale, we assume the
existence of an abstract transversal $T\subseteq G^{(0)}$ such that $G|_{T}$
can be given an étale topology, and there is a Morita equivalence $G\sim
G|_{T}$ (see [put:spiel] for more details on this). This assumption is
satisfied in most examples arising from geometric or dynamical situations.

Given groupoid homomorphisms $f_{i}\colon H_{i}\to G$, we consider _multiple
groupoid pullbacks_ $H_{1}\times_{G}\dots\times_{G}H_{n}$ as we did in
[valmak:groupoid]. This is a groupoid whose base is given by

$\\{(x_{1},g_{1},x_{2},\dots,g_{n-1},x_{n})\mid x_{i}\in(H_{i})^{(0)},g_{i}\in
G^{f_{i}(x_{i})}_{f_{i+1}(x_{i+1})}\\},$

and an arrow from $(x_{1},g_{1},\dots,x_{n})$ to
$(x^{\prime}_{1},g^{\prime}_{1},\dots,x^{\prime}_{n})$ is given by a tuple
$(h_{1},\dots,h_{n})$ for $h_{i}\in(H_{i})^{x^{\prime}_{i}}_{x_{i}}$ such that
the diagram

${f_{1}(x^{\prime}_{1})}$${f_{2}(x^{\prime}_{2})}$${\dots}$${f_{n}(x^{\prime}_{n})}$${f_{1}(x_{1})}$${f_{2}(x_{2})}$${\dots}$${f_{n}(x_{n})}$$\scriptstyle{g^{\prime}_{1}}$$\scriptstyle{g^{\prime}_{n-1}}$$\scriptstyle{f_{1}(h_{1})}$$\scriptstyle{g_{1}}$$\scriptstyle{f_{2}(h_{2})}$$\scriptstyle{g_{n-1}}$$\scriptstyle{f_{n}(h_{n})}$

is commutative.

######  Proposition 1.1.

Let $G$ be a topological groupoid, and $n$ be a nonnegative integer. Then the
fibered product groupoid $G^{\times_{G}(n+1)}$, with respect to the copies of
identity homomorphisms $G\to G$, is Morita equivalent to $G$.

###### Proof.

We give explicit Morita equivalence homomorphisms between $G$ and
$G^{\times_{G}(n+1)}$ in both ways (formally, we only need to know one). In
one direction, $f\colon G^{\times_{G}(n+1)}\to G$ is given by

$(G^{\times_{G}(n+1)})^{(0)}\to G^{(0)},\quad(x_{0},g_{1},\dots,x_{n})\mapsto
x_{n}$

at the level of base, and

$G^{\times_{G}(n+1)}\to G,\quad(g_{0},\dots,g_{n})\mapsto g_{n}$

at the level of arrows. In the other direction, $f^{\prime}\colon
G^{\times_{G}(n+1)}\to G$ is given by

$G^{(0)}\to(G^{\times_{G}(n+1)})^{(0)},\quad
x\mapsto(x,\mathrm{id}_{x},\dots,x)$

at the level of base, and

$G\to G^{\times_{G}(n+1)},\quad(g,g,\dots,g)$

at the level of arrows. ∎

###  1.2. Homology of étale groupoids

We consider the homology of étale groupoids as defined by Crainic and Moerdijk
[cramo:hom]. A _$G$ -sheaf_ is a sheaf $F$ on $G^{(0)}$ equipped with a
continuous action of $G$, concretely given by a map of sheaves $s^{*}F\to
r^{*}F$ on $G$. A $G$-sheaf $F$ is called _(c-)soft_ when the underlying sheaf
on $G^{(0)}$ has that property, i.e, for any (compact) closed subset $S\subset
G^{(0)}$ and any section $x\in\Gamma(S,F)$, there is an extension
$\tilde{x}\in\Gamma(X,F)$.

We are going to make the standing assumption that the cohomological dimension
of any open sets 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 why this is true, observe that when the topological
dimension of any compact subset $A\subset G^{(0)}$ is bounded by $N$
[MR0482697]*Section 3.1, then the Čech cohomology $\check{H}^{\bullet}(A;F)$
will vanish in degree above $N$. Then, 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 $\Gamma_{A}(F)=\Gamma(A,F)$, and we can combine
[MR0345092]*Remarque II.4.14.1 and Théorème 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 $(C_{i}^{j})_{0\leq i,j}$, whose terms are given by

$C_{i}^{j}=\Gamma_{c}(G^{(i)},s^{*}F^{j}),$

which has homological degree $i-j$. In more generality, when $F_{\bullet}$ is
a homological complex of $G$-sheaves bounded from below, consider 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

$C_{i,j}^{k}=\Gamma_{c}(G^{(i)},s^{*}F_{j}^{k}),$

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

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

for any complex $F_{\bullet}$ of $G$-sheaves as above.

Recall that a groupoid is called _elementary_ if it is the disjoint union of
groupoids $K_{i}$, each of which is isomorphic to the product of a second
countable locally compact space and a transitive principal groupoid on a
finite set. An ample groupoid is an _AF groupoid_ when it is an increasing
union of elementary groupoids [MR584266]*Definition III.1.1.

######  Proposition 1.2 (cf. [MR2876963]*Theorem 4.11).

Let $G$ be an AF groupoid, and $F$ be a $G$-sheaf. Then $H_{p}(G,F)=0$ for
$p>0$.

###### Proof.

The proof is a close analogue of that for [MR2876963]*Theorem 4.11. For each
$i$, the groupoid $K_{i}$ is Morita equivalent to the space $K_{i}\backslash
K_{i}^{(0)}$ regarded as a trivial groupoid. The latter space is second
countable and totally disconnected, hence has trivial groupoid homology (see
[valmak:groupoid]*Proposition 1.8). Consider a class of $H_{p}(G,F)$,
represented by a cycle $f\in\Gamma_{c}(G^{(p)},s^{*}F)$. Since $f$ has compact
support, there is an index $i$ with $f\in\Gamma_{c}(K_{i}^{(p)},s^{*}F)$. Then
we have $H_{p}(K_{i},F)=0$ by the Morita invariance of groupoid homology, thus
$f$ must be a boundary. ∎

###  1.3. Derived functor formalism

Let us briefly recall the formalism of derived functors in groupoid homology
from [cramo:hom]*Section 4. Let $G$ and $G^{\prime}$ be étale groupoids, and
$\phi\colon G\to G^{\prime}$ 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 $(y,g^{\prime})$, where $y\in G^{(0)}$
and $g^{\prime}\in G_{x}^{\prime\phi(y)}$, 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 equipped with a homomorphism $\pi_{x}\colon x/\phi\to G$.

When $F$ is a $G$-sheaf, we can consider a simplicial system of
$G^{\prime}$-sheaves, denoted by $B_{\bullet}(\phi,F)$, which at the level of
stalks is given by

$B_{n}(\phi,F)_{x}=\Gamma_{c}((x/\phi)^{(n)},s^{*}\pi_{x}^{*}F).$

The _left derived functor_ $\mathcal{L}\phi_{!}F_{\bullet}$ for a homological
complex of $G$-sheaves bounded from below $F_{\bullet}$ is represented by the
total complex of the triple complex of $G^{\prime}$-sheaves with terms
$B_{i}(\phi,F_{j}^{k})$ with homological degree $i+j-k$, where
$F_{j}^{\bullet}$ is a resolution of $F_{j}$ by c-soft $G$-sheaves. Note that
this is well defined up to a quasi-isomorphism of $G^{\prime}$-sheaves. The
_$n$ -th derived functor_, denoted by $L_{n}\phi_{!}F_{\bullet}$, is the
$G^{\prime}$-sheaf given as the $n$-th homology of
$\mathcal{L}\phi_{!}F_{\bullet}$. The fiber of this sheaf is given by
[cramo:hom]*Proposition 4.3 (by construction):

$(L_{n}\phi_{!}F_{\bullet})_{x}=H_{n}(x/\phi,\pi_{x}^{*}F_{\bullet}).$ (1)

If $G^{\prime}$ is the trivial groupoid and $\phi$ is the unique homomorphism
$G\to G^{\prime}$, then we recover the definition of $H_{n}(G,F_{\bullet})$.

Aside from the pullback functor (also called inverse image functor) for
sheaves, we will also need the _direct image_ functor, defined as
$g_{*}F(U)=F(g^{-1}(U))$ [MR0345092]. The functor $g_{*}$ can be defined in
the setting of groupoid-equivariant sheaves, and it is still right adjoint to
the pullback functor (see [cramo:hom]*Section 2.3 for details). Let us note
that, if $g$ is proper, then $g_{*}$ coincides with the functor $g_{!}$,
called direct image functor with compact supports. In the familiar setting of
topological spaces, $g_{!}$ is left adjoint to $g^{*}$ whenever the map $g$ is
étale, however we emphasize that, in the case of groupoid actions considered
here, the corresponding statement requires additional hypotheses
[cramo:hom]*Remark 5.2.

###  1.4. Triangulated category from groupoid equivariant $\mathit{KK}$-theory

Let us briefly summarize the contents of [valmak:groupoid][valmak:groupoidtwo]
in order to provide the background for this paper. We consider equivariant
(bivariant) Kasparov category $\mathit{KK}^{G}$ for a second countable locally
compact Hausdorff groupoid $G$ (in our case $G$ can be further assumed to be
étale) [val:kthpgrp].

Let $H$ be an open subgroupoid of $G$ satisfying $H^{(0)}=G^{(0)}$. Then we
can define the induction functor
$\operatorname{Ind}_{H}^{G}\colon\mathit{KK}^{H}\to\mathit{KK}^{G}$ as

$\operatorname{Ind}_{H}^{G}B=(C_{0}(G)\otimes_{C_{0}(G^{(0)})}B)\rtimes H,$

where $\otimes_{C_{0}(G^{(0)})}$ denotes the C∗-algebraic balanced tensor
product over $C_{0}(G^{0})$. This functor is left adjoint to the natural
restriction functor
$\operatorname{Res}_{H}^{G}\colon\mathit{KK}^{G}\to\mathit{KK}^{H}$.

Let us set $L=\operatorname{Ind}_{H}^{G}\operatorname{Res}_{H}^{G}$ and
$P_{n}=L^{n+1}A$ for $n\geq 0$. The adjunction guarantees that $P_{\bullet}$
has the structure of an augmented simplicial object over $A$ ($L$ is a
comonad). Applying the theory of projective resolutions in triangulated
categories, as a special case of the “ABC spectral sequence” [meyer:tri] for
the functor $A\mapsto K_{\bullet}(G\ltimes A)$, we obtain a spectral sequence
whose $E^{2}$-sheet shows the homology groups of the chain complex
$K_{\bullet}(G\ltimes P_{\bullet})$. The convergence is to
$K_{\bullet}(G\ltimes P)$ for a naturally defined object $P$, which should be
viewed as an approximation of $A$.

In broad terms, the object $P$ is constructed as follows: since the objects
$P_{n}$ satisfy a certain projectivity condition, it is possible to iterate
the construction of exact triangles

$P_{n}\to N_{n}\to N_{n+1}\to P_{n}[1],$

where we use the convention $A=N_{0}$ and $(-)[1]$ denotes suspension. The
$N_{k}$’s form an inductive system whose homotopy colimit we denote $N$. Now
the object $P$ is characterized as fitting a unique exact triangle

$P\to A\to N\to P[1].$

The assumption the stabilizers of $G$ are torsion-free implies that
$K_{\bullet}(G\ltimes P)$ can be identified with the left-hand side of the
Baum–Connes conjecture with coefficients in $A$, see [val:kthpgrp]. The
theorem below is a summary of the situation described so far.

######  Theorem 1.3 ([valmak:groupoid]*Corollary 3.6).

Let $G$ be an ample groupoid with torsion-free stabilizers satisfying the
Baum–Connes conjecture with coefficients. Let $H$, $A$, and $P_{n}$ be as
above. There exists a homological spectral sequence

$E^{2}_{pq}=H_{p}(K_{q}(G\ltimes P_{\bullet}))\Rightarrow K_{p+q}(G\ltimes
A).$

######  Remark 1.4.

If we choose the subgroupoid $H$ to be the unit space $G^{(0)}\subset G$, the
$E^{1}$-sheet of above spectral sequence becomes
$E^{1}_{pq}=K_{q}(C_{0}(G^{(p)})\otimes_{C_{0}(G^{(0)})}A)$. If moreover we
assume that $G$ is ample, the $E^{2}$-sheet can be identified with
$H_{p}(G,K_{q}(A))$, the groupoid homology of $G$ with the coefficient
$G$-sheaf corresponding to the $C_{c}(G,\mathbb{Z})$-module $K_{q}(A)$
[valmak:groupoid]*Section 3.

###  1.5. Smale spaces

A _Smale space_ is a compact metric space $X$ with a self-homeomorphism $\phi$
satisfying a certain hyperbolicity condition [ruelle:thermo]. We denote the
bracket map on $X$ by $[x,y]$ for $d(x,y)<\epsilon_{X}$, so that $x$ is stably
equivalent to $[x,y]$ while $y$ is unstably equivalent to it. _We assume that
a Smale space is non-wandering_.

The most essential feature of Smale spaces is given by the definition of two
equivalence relations, named respectively _stable_ and _unstable_ , which
reads as follows:

  * •

given $x,y\in X$, we say they are _stably equivalent_ (denoted $x\sim_{s}y$)
if

$\lim_{n\to\infty}d(\phi^{n}(x),\phi^{n}(y))=0;$

  * •

given $x,y\in X$, we say they are _unstably equivalent_ (denoted $x\sim_{u}y$)
if

$\lim_{n\to\infty}d(\phi^{-n}(x),\phi^{-n}(y))=0.$

The graph of the unstable equivalence relation has a structure of locally
compact groupoid with a Haar system [put:algSmale], that we denote by
$R^{u}(X,\phi)$. Note that $R^{s}(X,\phi)=R^{u}(X,\phi^{-1})$ and
$R^{s}(X,\phi)=R^{s}(X,\phi^{k})$ for any $k\geq 1$. The orbits of
$R^{s}(X,\phi)$ are referred to as _stable sets_. Following the construction
detailed in [put:spiel], we obtain an étale groupoid by restricting
$R^{u}(X,\phi)$ to an appropriate subspace contained in a finite union of
stable sets.

Given a finite directed graph
$\mathcal{G}=(\mathcal{G}^{0},\mathcal{G}^{1},i,t\colon\mathcal{G}^{1}\to\mathcal{G}^{0})$,
the associated _shift of finite type_ is the Smale space with underlying
metric space

$\Sigma_{\mathcal{G}}=\\{e=(e_{k})_{k\in\mathbb{Z}}\in(\mathcal{G}^{1})^{\mathbb{Z}}\mid
t(e_{k})=i(e_{k+1})\\}$

and homeomorphism given by $\sigma(e)_{k}=e_{k+1}$.

A Smale space $(X,\phi)$ is isomorphic to a shift of finite type as above if
and only if $X$ is totally disconnected, and $\phi$ is furthermore irreducible
if and only if $\mathcal{G}$ is connected [put:HoSmale]*Theorem 2.2.8. When
$(\Sigma,\sigma)$ is a shift of finite type, the groupoid
$R^{u}(\Sigma,\sigma)$ is AF. Moreover, the $K_{0}$-group of
$C^{*}R^{u}(\Sigma,\sigma)$ is isomorphic to the _stable dimension group_
$D^{s}(\Sigma,\sigma)$ introduced by Krieger [krieger:inv], which can be
described using stable sets of $(\Sigma,\sigma)$ [put:HoSmale]*Chapter 3. The
$K_{1}$-group is trivial.

###  1.6. Homology of Smale spaces

Let $(X,\phi)$ be a non-wandering Smale space. An _$s/u$ -bijective pair_ over
$(X,\phi)$ is given by a Smale space $(Y,\psi)$ with totally disconnected
unstable sets and an $s$-bijective map $f\colon Y\to X$, and another Smale
space $(Z,\zeta)$ with totally disconnected stable sets and a $u$-bijective
map $g\colon Z\to X$. Given such data, we set

$\displaystyle Y_{n}$
$\displaystyle=\underbrace{Y\times_{X}\dots\times_{X}Y}_{(n+1)\times},$
$\displaystyle\psi_{n}$ $\displaystyle=(\psi\times\dots\times\psi)|_{Y_{n}}.$

Then $(Y_{n},\psi_{n})$ is again a Smale space with totally disconnected
unstable sets. We denote the restriction of $f\times\dots\times f$ to $Y_{n}$
by $f_{n}$. This map is again $s$-bijective [put:notes]*Theorem 5.2.15.
Similarly, we define a Smale space with totally disconnected stable sets,

$\displaystyle Z_{n}$
$\displaystyle=\underbrace{Z\times_{X}\dots\times_{X}Z}_{(n+1)\times},$
$\displaystyle\zeta_{n}$
$\displaystyle=(\zeta\times\dots\times\zeta)|_{Z_{n}},$

together with the $u$-bijective map $g_{n}\colon Z_{n}\to X$. Then we have
that

$\displaystyle\Sigma_{L,M}$ $\displaystyle=Y_{L}\times_{X}Z_{M},$
$\displaystyle\sigma_{L,M}$
$\displaystyle=(\psi_{L}\times\zeta_{M})|_{\Sigma_{L,M}}$

is a Smale space with totally disconnected underlying space. Thus it is a
shift of finite type.

By definition, the _stable homology_ of $(X,\phi)$, denoted by
$H^{s}_{\bullet}(X,\phi)$, is the homology of the double complex with terms
$D^{s}(\Sigma_{L,M},\sigma_{L,M})$ for $L,M\geq 0$ [put:HoSmale][val:smale].
This construction is independent of the chosen $s/u$-bijective pair and it is
functorial for $s$-bijective maps.

We will also work with a normalized double complex as originally used in
[put:HoSmale]: the group $D^{s}(\Sigma_{L,M},\sigma_{L,M})$ admits a natural
action of $S_{L+1}\times S_{M+1}$. We will use the notation
$D^{s}(\Sigma_{L,M},\sigma_{L,M})_{,A}$ to denote the subgroup of elements $x$
satisfying $sx=(-1)^{\left|s\right|}x$ for $s\in S_{M+1}$. Notice that when
the map $g\colon Z\to X$ is at most $N$-to-$1$, the group
$D^{s}(\Sigma_{L,M},\sigma_{L,M})_{,A}$ is trivial for $M>N$.

Now, consider a Smale space $(X,\phi)$ whose stable sets are totally
disconnected, and take an $s$-bijective map
$g\colon(\Sigma,\sigma)\to(X,\phi)$ from a shift of finite type. We then have
étale groupoids $G=R^{u}(X,\phi)|_{T}$ and $H=R^{u}(\Sigma,\sigma)|_{T_{0}}$
for appropriate transversals $T\subset X$ and $T_{0}\subset\Sigma$ and such
that $g(T_{0})=T$, such that $H$ embeds as an open subgroupoid of $G$ (see
[put:funct] for the details). Then the construction of Section 1.4 gives the
following.

######  Theorem 1.5 ([valmak:groupoidtwo]*Theorem 3.9).

Let $(X,\phi)$ be a non-wandering Smale space with totally disconnected stable
sets. We then have a homological spectral sequence

$E^{2}_{pq}=E^{3}_{pq}=H^{s}_{p}(X,\phi)\otimes K_{q}(\mathbb{C})\Rightarrow
K_{p+q}(C^{*}R^{u}(X,\phi)).$

In fact, Putnam’s homology agrees with the groupoid homology:

######  Theorem 1.6 ([valmak:groupoidtwo]*Theorem 4.1).

With $(X,\phi)$ and $G$ as above, we have

$H^{s}_{p}(X,\phi)\cong H_{p}(G,\underline{\mathbb{Z}}).$

##  2\. Symmetric simplicial spaces

Let us start with some definitions and conventions. Let
$A_{\bullet}=(A_{n})_{n=0}^{\infty}$ be a _simplicial topological space_ ,
that is, a simplicial object in the category of topological spaces. Its _thin
geometric realization_ is defined to be

$\left|A_{\bullet}\right|=\left(\coprod_{n=0}^{\infty}\Delta^{n}\times
A_{n}\right)/\sim,$

where $\Delta^{n}$ is the standard $n$-simplex, and $\sim$ is the equivalence
relation induced by the structure maps (the degeneracy and face maps).

Next further assume that $A_{\bullet}$ is a _symmetric simplicial space_ , or
equivalently, an $S_{\bullet}$-space for the _crossed simplicial group_
$S_{\bullet}$ [MR998125], i.e., a simplicial space with compatible actions of
$S_{n+1}$ on $A_{n}$. Alternatively, we can think of $A_{\bullet}$ as a
contravariant functor on the category of (unordered) finite sets. Then we can
form a “naive” geometric realization $\left\rceil A_{\bullet}\right\lceil$, as
the quotient of $\left|A_{\bullet}\right|$ with respect to the orbit relations
for the diagonal action of $S_{n+1}$ on $\Delta^{n}\times A_{n}$, for all $n$.
Moreover, we denote the image of $\Delta^{n}\times A_{n}$ in $\left\rceil
A_{\bullet}\right\lceil$ by $\left\rceil A_{\bullet}\right\lceil^{(n)}$.

###  2.1. Adaptation of Segal’s work

In this section we leverage Segal’s work [MR232393] on simplicial spaces and
spectral sequences to obtain cohomological approximation results in the
setting of groupoids and dynamical systems.

Let $T$ be a locally compact space, and $g\colon T_{0}\to T$ be a proper
continuous surjective map from a locally compact space $T_{0}$. Then we obtain
an $S_{\bullet}$-space $T_{\bullet}$ by setting $T_{n}$ to be the $(n+1)$-fold
fiber product of $T_{0}$ over $T$. Let us also denote by $T_{n}^{f}$ the
subset of points $(x_{0},\dots,x_{n})$ such that $x_{i}\neq x_{j}$ for $i\neq
j$, i.e. the union of free $S_{n+1}$-orbits in $T_{n}$. This is an open subset
of $T_{n}$.

Moreover, suppose that $g$ is at most $N$-to-one, for some $N$. Then
$\left\rceil T_{\bullet}\right\lceil$ is a locally compact space endowed with
a proper surjective map $\tilde{g}\colon\left\rceil T_{\bullet}\right\lceil\to
T$. We also have $\left\rceil T_{\bullet}\right\lceil=\left\rceil
T_{\bullet}\right\lceil^{(N)}$.

######  Proposition 2.1.

Suppose that $g$ is as above. We have

$H^{n}(T,\underline{\mathbb{Z}})\cong H^{n}(\left\rceil
T_{\bullet}\right\lceil,\underline{\mathbb{Z}})$

induced by $\tilde{g}$.

###### Proof.

For a topological space $X$, let $D^{+}(X)$ be the derived category of cochain
complexes of sheaves of commutative groups which are bounded below up to
quasi-isomorphisms. We thus have the right derived functor
$\mathcal{R}\tilde{g}_{*}\colon D^{+}(\left\rceil T_{\bullet}\right\lceil)\to
D^{+}(T)$ of the direct image functor $\tilde{g}_{*}$. We also have the (term-
wise application of) inverse image functor $\tilde{g}^{*}\colon D^{+}(T)\to
D^{+}(\left\rceil T_{\bullet}\right\lceil)$.

Since $\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil}=\tilde{g}^{*}\underline{\mathbb{Z}}_{T}$, we have a
natural morphism
$\underline{\mathbb{Z}}_{T}\to\mathcal{R}\tilde{g}_{*}\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil}$ in $D^{+}(T)$ induced by the adjunction between
$\mathcal{R}\tilde{g}_{*}$ and $\tilde{g}^{*}$.

We first claim that $R^{n}\tilde{g}_{*}\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil}=0$ for $0<n$. By the properness of $\tilde{g}$, for
any $x\in T$ we have

$(R^{n}\tilde{g}_{*}\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil})_{x}=H^{n}(\tilde{g}^{-1}(x),\underline{\mathbb{Z}})$

from the proper base change theorem, see [sga4-vbis]*Corollaire 4.1.2. Here
$\tilde{g}^{-1}(x)$ is homeomorphic to $\Delta^{m}$ with
$m=\left|g^{-1}(x)\right|$, hence the $n$-th cohomology group on the right
hand side must vanish. This also shows
$\tilde{g}_{*}\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil}=\underline{\mathbb{Z}}_{T}$.

Now, we have the Leray spectral sequence

$E_{2}^{p,q}=H^{p}(T,R^{q}\tilde{g}_{*}\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil})\Rightarrow H^{p+q}(\left\rceil
T_{\bullet}\right\lceil,\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil}).$

By the above claim we have $E_{2}^{p,q}=0$ for $0<q$, and
$E_{2}^{p,0}=H^{p}(T,\underline{\mathbb{Z}}_{T})$. As the higher differentials
are $d_{r}^{p,q}\colon E_{r}^{p,q}\to E_{r}^{p+r,q-r+1}$, we have
$E_{2}^{p,0}=E_{\infty}^{p,0}$ and $E_{\infty}^{p,q}=0$ for $q>0$. This
implies

$H^{n}(\left\rceil T_{\bullet}\right\lceil,\underline{\mathbb{Z}}_{\left\rceil
T_{\bullet}\right\lceil})\cong E^{n,0}_{\infty}\cong
H^{n}(T,\underline{\mathbb{Z}}_{T}),$

as claimed. ∎

######  Proposition 2.2.

Suppose $g$ is as above. We then have $K^{\bullet}(\left\rceil
T_{\bullet}\right\lceil)\cong K^{\bullet}(T)$ induced by $\tilde{g}$.

###### Proof.

First, we may assume that $T$ is compact. Indeed, denoting the one-point
compactification by $T^{+}$, we have a proper surjective map $T_{0}^{+}\to
T^{+}$. Then we have $\left\rceil T_{\bullet}\right\lceil^{+}=\left\rceil
T_{\bullet}^{+}\right\lceil$, hence it is enough to prove the claim for
$T_{\bullet}^{+}$.

Consider the representable $K$-groups
$\operatorname{RK}^{n}(X)=[X,\mathrm{KU}_{n}]$, where $\mathrm{KU}_{n}$ is the
$K$-theory spectrum, i.e., $\mathrm{KU}_{2k}=BU\times\mathbb{Z}$ and
$\mathrm{KU}_{2k+1}=\Omega BU=U$ for the infinite unitary group $U=\varinjlim
U_{k}$. Then, this is a generalized cohomology as considered in [MR232393],
while satisfying $\operatorname{RK}^{n}(X)\cong K^{n}(X)$ for compact $X$. We
thus need to check $\operatorname{RK}^{n}(\left\rceil
T_{\bullet}\right\lceil)\cong\operatorname{RK}^{n}(T)$.

By [MR232393]*Proposition 5.2, there is a spectral sequence

$E_{2}^{pq}=H^{p}(X,F^{q})\Rightarrow\operatorname{RK}^{p+q}(X).$

where $F^{2k}=\underline{\mathbb{Z}}$ and $F^{2k+1}=0$. (This is a version of
the Atiyah–Hirzebruch spectral sequence that works without any CW-complex
structure.) By Proposition 2.1, we have the isomorphism of $E_{2}$-sheet for
$X=T$ and $X=\left\rceil T_{\bullet}\right\lceil$. ∎

Following the scheme in [MR232393], we define a spectral sequence converging
to $K_{\bullet}(G\ltimes C_{0}(\left\rceil T_{\bullet}\right\lceil))$. The
filtration $\left\rceil T_{\bullet}\right\lceil^{(n)}$ on $\left\rceil
T_{\bullet}\right\lceil$, combined with Proposition 2.2 induces a spectral
sequence.

$E_{1}^{pq}=K^{p+q}(\left\rceil
T_{\bullet}\right\lceil^{(p)}\setminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)})\Rightarrow K^{p+q}(T).$

When $V$ is a vector space (over $\mathbb{Q}$) with a linear representation of
$S_{n}$, let us denote its isotypic summand for the sign representation by
$V_{A}$. This is the image of the projection

$p_{a}=\frac{1}{n!}\sum_{s\in S_{n}}(-1)^{\left|s\right|}s$

acting on $V$.

######  Lemma 2.3.

Let $X$ be a locally compact Hausdorff space with an action of $S_{n}$, such
that the stabilizer of any point $x\in X$ contains an odd permutation. Then
$H^{\bullet}(X,\underline{\mathbb{Q}})_{A}=0$.

###### Proof.

Let $Y=X/S_{n}$, and consider the projection map $\pi\colon X\to Y$. In this
situation, we claim that $H^{\bullet}(X,\underline{\mathbb{Q}})\cong
H^{\bullet}(Y,\pi_{*}\underline{\mathbb{Q}})$. Indeed, $\pi$ is a proper map
with discrete fiber, hence the proper base change theorem applies and gives
$\mathcal{R}\pi_{*}\underline{\mathbb{Q}}\simeq\pi_{*}\underline{\mathbb{Q}}$.
Then the claim is a straightforward consequence of
$\mathcal{R}\pi^{Y}_{*}\mathcal{R}\pi_{*}\simeq\mathcal{R}\pi^{X}_{*}$ for the
canonical maps $\pi^{X}$ and $\pi^{Y}$ from $X$ and $Y$ to the singleton
space.

Acting by the projection $p_{a}$, we obtain

$H^{\bullet}(X,\underline{\mathbb{Q}})_{A}\cong
H^{\bullet}(Y,p_{a}\pi_{*}\underline{\mathbb{Q}})$

Now, at the level of stalks we have

$(p_{a}\pi_{*}\underline{\mathbb{Q}})_{y}=\Gamma(\pi^{-1}(y),\mathbb{Q})_{A}=0$

for all $y\in Y$, hence $H^{\bullet}(Y,p_{a}\pi_{*}\underline{\mathbb{Q}})=0$.
∎

######  Proposition 2.4.

We have

$(K^{p+q}(\left\rceil T_{\bullet}\right\lceil^{(p)}\setminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}))\otimes\mathbb{Q}\cong(K^{q}(T_{p})\otimes\mathbb{Q})_{A}$

with respect to the natural action of $S_{p+1}$ on $T_{p}$.

###### Proof.

By construction, $\left\rceil
T_{\bullet}\right\lceil^{(p)}\setminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}$ is the quotient of
$\mathring{\Delta}^{p}\times T_{p}^{f}$ by the diagonal action of $S_{p+1}$.
By [MR2820377]*Proposition 5.5, we have

$(K^{p+q}(\left\rceil T_{\bullet}\right\lceil^{(p)}\setminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}))\otimes\mathbb{Q}\cong(K^{p+q}(\mathring{\Delta}^{p}\times
T_{p}^{f})^{S_{p+1}})\otimes\mathbb{Q}.$

On the right hand side, the action of $S_{p+1}$ on
$K^{p+q}(\mathring{\Delta}^{p}\times T_{p}^{f})$ can be identified with the
twist of the natural one on $K^{q}(T_{p}^{f})$ by the sign representation.
Moreover, rationalization and taking $S_{p+1}$-invariants commute, hence we
have $(K^{q}(T_{p}^{f})\otimes\mathbb{Q})_{A}$.

It remains to show
$(K^{q}(T_{p}^{f})\otimes\mathbb{Q})_{A}\cong(K^{q}(T_{p})\otimes\mathbb{Q})_{A}$.
By the $6$-term exact sequence involving
$K^{\bullet}(T_{p}^{f})\otimes\mathbb{Q}$,
$K^{\bullet}(T_{p})\otimes\mathbb{Q}$, and $K^{\bullet}(T_{p}\setminus
T_{p}^{f})\otimes\mathbb{Q}$, which is $S_{p+1}$-equivariant, the claim is
equivalent to $(K^{\bullet}(T_{p}\setminus
T_{p}^{f})\otimes\mathbb{Q})_{A}=0$.

Let $X$ be the one point compactification of $T_{p}\setminus T_{p}^{f}$. Then
Lemma 2.3 implies that

$(K^{i}(X)\otimes\mathbb{Q})_{A}\cong\biggl{(}\bigoplus_{k}H^{i+2k}(X,\underline{\mathbb{Q}})\biggr{)}_{A}$

is trivial for $i=0,1$, hence we obtain the claim. ∎

######  Corollary 2.5.

We have a convergent cohomological spectral sequence

$E_{1}^{pq}=(K^{q}(T_{p})\otimes\mathbb{Q})_{A}\Rightarrow
K^{p+q}(T)\otimes\mathbb{Q},$

with the $E_{1}$-differential $E_{1}^{pq}\to E_{1}^{(p+1)q}$ induced by the
simplicial structure of $T_{\bullet}$.

###  2.2. Equivariant $K$-theory spectral sequence

We now consider the case where $G$ is an étale groupoid with torsion-free
stabilizers satisfying the strong Baum–Connes [val:kthpgrp] conjecture, with
base $T=G^{(0)}$. Suppose that $T_{0}$ is a $G$-space, mapping onto $T$ by a
surjective map which is at most $N$-to-one. Then $\left\rceil
T_{\bullet}\right\lceil$ is also a locally compact $G$-space, and we have an
equivariant $*$-homomorphism $C_{0}(T)\to C_{0}(\left\rceil
T_{\bullet}\right\lceil)$.

######  Proposition 2.6.

Under the above setting, the inclusion $C^{*}_{r}G\to G\ltimes
C_{0}(\left\rceil T_{\bullet}\right\lceil)$ induces an isomorphism
$K_{\bullet}(G\ltimes C_{0}(\left\rceil T_{\bullet}\right\lceil))\cong
K_{\bullet}(C^{*}_{r}G)$.

###### Proof.

By Theorem 1.3 and Remark 1.4, we have a spectral sequence

$E^{1}_{pq}=K^{q}(G^{(p)}\times_{T}\left\rceil T_{\bullet}\right\lceil)\cong
K_{q}(G^{(p)}\ltimes C_{0}(\left\rceil T_{\bullet}\right\lceil))\Rightarrow
K_{p+q}(G\ltimes C_{0}(\left\rceil T_{\bullet}\right\lceil)),$

and a similar one for $C^{*}_{r}G$. Now, we observe that
$G^{(p)}\times_{T}T_{0}$ maps onto $G^{(p)}$ by a finite-to-one map, and

$G^{(p)}\times_{T}\left\rceil T_{\bullet}\right\lceil\cong\left\rceil
G^{(p)}\times_{T}T_{\bullet}\right\lceil.$

Combining this with Proposition 2.2, we have

$K^{q}(G^{(p)}\times_{T}\left\rceil T_{\bullet}\right\lceil)\cong
K^{q}(G^{(p)}).$

Thus, the inclusion $C^{*}_{r}G\to G\ltimes C_{0}(\left\rceil
T_{\bullet}\right\lceil)$ induces an isomorphism of spectral sequences at the
$E^{1}$-sheet. ∎

This time we obtain the convergent spectral sequence

$E_{1}^{pq}=K_{p+q}(G\ltimes C_{0}(\left\rceil
T_{\bullet}\right\lceil^{(p)}\smallsetminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}))\Rightarrow K_{p+q}(C^{*}_{r}G).$ (2)

###  2.3. Soft resolution from simplicial totally disconnected space

Now, suppose that $G$ is an étale groupoid, and $T_{0}$ is a totally
disconnected $G$-space with a proper structure map $g\colon T_{0}\to T$ that
is at most $N$-to-one. Our goal here is to show that, given these assumptions,
groupoid homology for $G$ is definable. We denote the $(n+1)$-fold fiber
product of $T_{0}$ over $T$ by $T_{n}$, and by $g_{n}$ the induced map
$T_{n}\to T$.

######  Proposition 2.7.

Let $F$ be a sheaf over $T_{n}$. Then $(g_{n})_{*}F$ is a (c-)soft sheaf on
$T$.

###### Proof.

To check the c-softness of $(g_{n})_{*}F$, we want to show the following
property (see [MR0345092]*Théorème II.3.4.1): for any point $x\in T$, there is
a neighborhood $U$ of $x$ such that, for any closed set $S$ of $X$ contained
in $U$ and $s\in\Gamma(S,(g_{n})_{*}F)$, there is an extension
$\tilde{s}\in\Gamma(U,(g_{n})_{*}F)$ of $s$. We take $U$ to be a relatively
compact open neighborhood of $x$.

By the second countability and total disconnectedness of $T_{n}$, $F$ is soft.
Then, if $S$ is a closed set as above (hence compact), any section of $F$ on
the compact set $g_{n}^{-1}(S)$ extends to a section on $g_{n}^{-1}(U)$. We
thus need to show that the sections of $(g_{n})_{*}F$ on $S$ can be identified
with those of $F$ on $g_{n}^{-1}(S)$.

We have ([MR0345092]*II.3.3.1)

$\displaystyle\Gamma(S,(g_{n})_{*}F)$ $\displaystyle=\varinjlim_{V\supset
S}\Gamma(g_{n}^{-1}(V),F),$ $\displaystyle\Gamma(g_{n}^{-1}(S),F)$
$\displaystyle=\varinjlim_{V^{\prime}\supset
g_{n}^{-1}(S)}\Gamma(V^{\prime},F),$

where $V$ runs over open subsets of $U$ containing $S$, while $V^{\prime}$
runs over those containing $g_{n}^{-1}(S)$. We thus have the equality of these
spaces if the sets $g_{n}^{-1}(V)$ form a fundamental system of neighborhoods
of $g_{n}^{-1}(S)$.

Let $V^{\prime}$ be an open subset of $g_{n}^{-1}(U)$ containing
$g_{n}^{-1}(S)$. Then $W=g_{n}^{-1}(\bar{U})\smallsetminus V^{\prime}$ is a
compact subset of $T_{n}$ by the properness of $g_{n}$. Then
$V=U\smallsetminus g_{n}(W)$ is an open neighborhood of $S$ such that
$g_{n}^{-1}(V)\subset V^{\prime}$. ∎

We also have the following correspondence for compactly supported sections.

######  Proposition 2.8.

Let $F$ be a sheaf of commutative groups on $T_{n}$. Then
$\Gamma_{c}(S,(g_{n})_{*}F)=\Gamma_{c}(g_{n}^{-1}(S),F)$ for any open set
$S\subset T$.

###### Proof.

This follows from the properness of $g_{n}$. ∎

Now, let $F$ be a $G$-sheaf, and put $F^{n}=(g_{n})_{*}(g_{n}^{*}F)$. This is
a soft sheaf on $T$ by Proposition 2.7. By Proposition 2.8, we get

$\Gamma_{c}((G\ltimes
T_{n})^{(k)},s^{*}g_{n}^{*}F)\cong\Gamma_{c}(G^{(k)},s^{*}F^{n}).$ (3)

Moreover, pullback along the structure maps of $(T_{n})_{n}$ as a simplicial
space induces a structure of cosimplicial object on $(F^{n})_{n=0}^{\infty}$
in the category of sheaves over $T$. Namely, let $d^{n}_{i}\colon Z_{n}\to
Z_{n-1}$ and $s^{n}_{i}\colon Z_{n}\to Z_{n+1}$ be the standard face and
degeneracy maps. Then for any open set $U\subset T$,

$\delta^{n}_{i}|_{U}\colon\Gamma(g_{n-1}^{-1}(U),g_{n-1}^{*}F)=\Gamma(U,F^{n-1})\to\Gamma(g_{n}^{-1}(U),g_{n}^{*}F)=\Gamma(U,F^{n}),\quad\sigma\mapsto\sigma\circ
d^{n}_{i},$

defines a morphism $\delta^{n}_{i}\colon F^{n-1}\to F^{n}$. Similarly we get
$\sigma^{n}_{i}\colon F^{n+1}\to F^{n}$, and we obtain a cosimplicial
structure on $(F^{n})_{n=0}^{\infty}$.

######  Proposition 2.9.

The cochain complex of $(F^{n})_{n=0}^{\infty}$ is a resolution of $F$.

###### Proof.

We need to check that, for each $t\in T$, the sequence of stalks

$0\to F_{t}\to F^{0}_{t}\to F^{1}_{t}\to\dots$

is exact. Put $Z_{t}=g_{0}^{-1}(t)$, which is a finite set by the properness
of $g_{0}=g$. Then $F^{n}_{t}=C(Z_{t}^{n+1},F_{t})$, and the above complex is
given by the canonical simplicial structure on $(Z_{t}^{n+1})_{n=0}^{\infty}$.
This is well-known to be exact. ∎

To have a soft resolution of bounded length, let us introduce the
“alternating” model. The natural action of $S_{n+1}$ on $T_{n}$ induces an
action on $F^{n}$. We define a subsheaf $F^{n,a}\subset F^{n}$ as follows:

$\Gamma(U,F^{n,a})=\\{x\in\Gamma(U,F^{n})\mid\operatorname{supp}x\subset
T_{n}^{f},\forall s\in S_{n+1}\colon sx=(-1)^{\left|s\right|}x\\}.$

######  Proposition 2.10.

The sheaf $F^{n,a}$ is again c-soft.

###### Proof.

As in the proof of Proposition 2.7, let $U$ be a relatively compact open set
of $T$, and $S$ be a compact subset of $U$. Given a section
$x\in\Gamma(S,F^{n,a})$, we need to find an extension
$\tilde{x}\in\Gamma(V,F^{n,a})$ of $x$ to some open neighborhood $V$ of $S$.

By Proposition 2.7, we can find an open neighborhood $V_{0}$ of $S$ and a
section $\tilde{x}_{0}\in\Gamma(V_{0},F^{n})$ restricting to $x$. Let
$\bar{x}_{0}\in\Gamma(g_{n}^{-1}(V_{0}),g_{n}^{*}F)$ be the section
corresponding to $\tilde{x}_{0}$. It is enough to find a $S_{n+1}$-invariant
open neighborhood $V^{\prime}$ of $g_{n}^{-1}(S)$ in $g_{n}^{-1}(V_{0})$ such
that the restriction $\bar{x}_{0}|_{V^{\prime}}$ satisfies
$s(\bar{x}_{0}|_{V^{\prime}})=(-1)^{\left|s\right|}\bar{x}_{0}|_{V^{\prime}}$
for $s\in S_{n+1}$. Indeed, the argument of the proof of Proposition 2.7 gives
a neighborhood $V$ of $S$ satisfying $g_{n}^{-1}(V)\subset V^{\prime}$, and
the restriction of $\tilde{x}_{0}$ to $V$ is a section of $F^{n,a}$.

For each $u\in g_{n}^{-1}(S)$, find an open neighborhood $V_{u}$ such that
$s^{*}\tilde{x}_{0}=(-1)^{\left|s\right|}\tilde{x}_{0}$ holds on $V_{u}$ for
any $s\in S_{n+1}$. Moreover, if $u$ is not in $T_{n}^{f}$, we ask
$\tilde{x}_{0}$ to be trivial on $V_{u}$. This is possible since the stalks
$s^{*}\tilde{x}_{0}(u)$ are the inductive limits of $s^{*}\tilde{x}_{0}$
around open neighborhoods of $u$. By compactness there are finitely many
$u_{1},\dots,u_{k}$ such that $V^{\prime}=\bigcup_{i}V_{u_{i}}$ covers
$g_{n}^{-1}(S)$. ∎

A standard argument shows that $(F^{n,a})_{n}$ is a subcomplex of
$(F^{n})_{n}$. By assumption about the support and $T_{n}^{f}$, this is a
bounded complex.

######  Proposition 2.11.

The complex $(F^{n,a})_{n}$ is still a resolution of $F$.

###### Proof.

Again we need to check the exactness of the augmented complex of stalks,

$0\to F_{t}\to F^{0,a}_{t}\to F^{1,a}_{t}\to\dots\to F^{N,a}_{t}\to 0.$

The stalk of $F^{n,a}$ at $t$ is given by

$F^{n,a}_{t}=\\{f\in C(Z_{t}^{n+1},F_{t})\mid\operatorname{supp}f\subset
Z_{t}^{n+1}\smallsetminus D_{n+1},\forall s\in S_{n+1}\colon
sf=(-1)^{\left|s\right|}f\\}$

for $Z_{t}=g_{0}^{-1}(t)$, where $D_{n+1}$ is the degenerate part (the union
of non-free orbits). A usual normalization argument for simplicial complexes
(for example [eilstee:algtop]*Theorem VI.6.9), together with the fact that
$(Z_{t}^{n+1})_{n=0}^{\infty}$ is a contractible simplicial set, shows that
the above augmented complex is indeed exact. ∎

This allows us to define $H_{\bullet}(G,F)$, as follows.

######  Proposition 2.12.

Let $F$ be a $G$-sheaf. Then the groupoid homology $H_{\bullet}(G,F)$ is well-
defined as the homology of the double complex with components

$\Gamma_{c}(G^{(k)},s^{*}F^{n,a}),\qquad n,k\geq 0$

with total degree $k-n$.

##  3\. Comparison of homologies

Now we are ready to apply the above machinery to Smale spaces. Let us start
with some preliminaries. We fix an $s/u$-bijective pair $(Y,\psi,f,Z,\zeta,g)$
over $(X,\phi)$ as before. Consider a subspace $T\subseteq X$ and suppose it
is a transversal for the unstable equivalence relation. By Proposition 2.12,
the étale groupoid $G=R^{u}(X,\phi)|_{T}$ indeed satisfies the assumption
about $G$-sheaves to define homology.

######  Proposition 3.1.

We can choose $T$ so that $T_{n}=g_{n}^{-1}(T)$ is a transversal for the
unstable equivalence relation on $Z_{n}$ for all $n$.

###### Proof.

We choose a periodic point $x_{0}\in X$, and $T$ to be a local stable set
around $x_{0}$. Then $T_{n}$ is a union of local stable sets around the points
of $g_{n}^{-1}(x_{0})$ [put:notes]*Lemma 5.2.10. Now, let $z\in Z_{n}$. Since
$T$ is a transversal, we can find $x_{1}\in T$ such that
$g_{n}(z)\sim_{u}x_{1}$. As $g_{n}$ is $u$-bijective, we can find
$z_{1}\sim_{u}z$ such that $g_{n}(z_{1})=x_{1}$. Our choice of $T$ works if
the system is mixing. If it is not, we use Smale’s decomposition theorem
[put:funct]*Theorem 2.1 and the following discussion and apply the argument
above to each mixing factor. ∎

In the following we will always use $T$ satisfying the claim of Proposition
3.1. Put $G=R^{u}(X,\phi)|_{T}$. Denote by $\underline{\mathbb{Z}}$ by the
locally constant $G$-equivariant sheaf of integers over $X$.

######  Proposition 3.2.

The transversal $T_{n}\subset Z_{n}$ has an action of $G$ with anchor map
$g_{n}\colon T_{n}\to T$. We have $G\ltimes T_{n}\cong
R^{u}(Z_{n},\zeta_{n})|_{T_{n}}$.

###### Proof.

Let $z\in T_{n}$, $x=g_{n}(z)$, and $(x^{\prime},x)\in G$. Since $g_{n}$ is
$u$-bijective, there is a unique point $z^{\prime}\in Z_{n}$ such that
$z^{\prime}\sim_{u}z$ and $g(z^{\prime})=x^{\prime}$. We define the action of
$(x^{\prime},x)$ on $z$ to be this $z^{\prime}$. Conversely, if
$z,z^{\prime}\in T_{n}$ are unstably equivalent, $z^{\prime}$ is the image of
the action of $(g(z^{\prime}),g(z))$ on $z$. This gives the identification of
groupoids. ∎

Recall that $g_{n}$ is proper on stable sets [put:notes]*Theorem 5.2.5. In
particular, $g_{n}$ is proper as a map $T_{n}\to T$.

Let us establish a generalization of Theorem 1.6.

######  Lemma 3.3.

For each $M$, the natural map $\Sigma_{0,M}\to Z_{M}$ is $s$-bijective.
Moreover, we have

$\Sigma_{L,M}=\underbrace{\Sigma_{0,M}\times_{Z_{M}}\dots\times_{Z_{M}}\Sigma_{0,M}}_{(L+1)\times}.$

###### Proof.

Let $z=(z_{0},\dots,z_{M})$ and
$z^{\prime}=(z^{\prime}_{0},\dots,z^{\prime}_{M})$ be points of $Z_{M}$ which
are stably equivalent. This condition is equivalent to component-wise
equivalences

$z_{0}\sim_{s}z^{\prime}_{0},\quad\dots,\quad z_{M}\sim_{s}z^{\prime}_{M}.$

Suppose that we are given a point in the fiber $(\Sigma_{0,M})_{z}$, that is,
a tuple of the form

$(y_{0},z_{0},\dots,z_{M})\quad(f(y_{0})=g(z_{0})=\dots=g(z_{M})).$

Write $x=f(y_{0})=g(z_{i})$ and $x^{\prime}=g(z^{\prime}_{i})$. Then there is
a unique point $y^{\prime}_{0}$ in the stable equivalence class of $y$ which
maps to $x^{\prime}$. Then
$(y^{\prime}_{0},z^{\prime}_{0},\dots,z^{\prime}_{M})$ is the unique point of
$\Sigma_{L,M}$ stably equivalent to $(y_{0},z_{0},\dots,z_{M})$ and maps to
$z^{\prime}$. The relation between $\Sigma_{L,M}$ and the fibered product of
$\Sigma_{0,M}$ over $Z_{M}$ is obvious. ∎

Now, take a transversal $T\subset X$ as in Proposition 3.1, and consider the
étale groupoid $G=R^{u}(X,\phi)|_{T}$. On the one hand,
$T_{i}=g_{i}^{-1}(T)\subset Z_{i}$ is a totally disconnected $G$-space. On the
other, the factor map $Y_{j}\to X$ defines an open subgroupoid $H_{j}<G$ for
each $j$. Similarly, $\Sigma_{j,i}\to Z_{i}$ defines an open subgroupoid
$H_{j,i}<G\ltimes T_{i}$.

######  Proposition 3.4.

The groupoid $H_{j}$ is Morita equivalent to $H_{0}^{\times_{G}(j+1)}$.
Moreover, $H_{j,i}$ is the transformation groupoid $H_{j}\ltimes T_{i}$, which
is also Morita equivalent to the multiple groupoid pullback of $(j+1)$ copies
of $H_{0,i}$ over $G\ltimes T_{i}$.

###### Proof.

The main ingredient of the proof is the following transversality
[valmak:groupoidtwo]*Proposition 2.9: given $y_{0},\dots y_{j}$ such that
$f(y_{i})$ are mutually unstably equivalent in $X$, we can find
$y^{\prime}_{0},\dots,y^{\prime}_{j}$ such that $y^{\prime}_{i}$ is unstably
equivalent to $y_{i}$ in $Y$, and additionally
$f(y^{\prime}_{i})=f(y^{\prime}_{0})$, for each $i$. The first statement is
proved in [valmak:groupoidtwo]*Theorem 2.8. The rest can be proved in similar
ways. ∎

######  Theorem 3.5.

We have $H_{k}(G,\underline{\mathbb{Z}})\cong H^{s}_{k}(X,\phi)$ for all
$k\in\mathbb{Z}$.

###### Proof.

Consider the inclusion map $f_{j}\colon H^{\times_{G}(j+1)}\to
G^{\times_{G}(j+1)}$. We obtain a bicomplex of $G^{\times_{G}(j+1)}$-sheaves
$F_{j\bullet}^{\bullet}=\mathcal{L}(f_{j})_{!}\underline{\mathbb{Z}}$
[cramo:hom]*Section 4.2. This is defined through a choice of c-soft resolution
$S^{\bullet}$ of $\underline{\mathbb{Z}}$ as $H^{\times_{G}(j+1)}$-sheaves,
then by taking the bicomplex $B_{\bullet}(f_{j},S^{\bullet})$, where
$B_{n}(f_{j},F)$ is defined as $(\beta_{n})_{!}\alpha_{n}^{*}F$ for

${G^{\prime(0)}}$${K^{(n)}\times_{G^{\prime(0)}}G^{\prime}}$${K^{(0)}}$$\scriptstyle{\alpha_{n}=r\pi_{1}}$$\scriptstyle{\beta_{n}=s\pi_{2}}$

with $K=H^{\times_{G}(j+1)}$, $G^{\prime}=G^{\times_{G}(j+1)}$, and $\pi_{i}$
denoting the projections to $K^{(n)}$ and $G^{\prime}$.

By the Morita equivalence of Proposition 1.1, we can regard
$F_{j\bullet}^{\bullet}$ as a complex of $G$-sheaves. Suppose that $S^{i}$ is
given by a bounded c-soft resolution of $\underline{\mathbb{Z}}$ as a
$G^{\times_{G}(j+1)}$-sheaf. Then the stalk of $F_{jn}^{i}$ at $x\in G^{(0)}$
is given by

$(F_{jn}^{i})_{x}=F_{jn}(S^{i})_{x}=\Gamma_{c}((H^{\times_{G}(j+1)})^{(n)}\times_{G^{(j)}}(G^{\times_{G}(j+1)})_{x},S^{i}_{x}),$
(4)

where we identified the base of $G^{\times_{G}(j+1)}$ with $G^{(j)}$, and $x$
with $(x,\mathrm{id}_{x},\dots,x)$ to make sense of
$(G^{\times_{G}(j+1)})_{x}$.

Recall that we have a bounded c-soft resolution $F^{\bullet,a}$ of the
$G$-sheaf $\underline{\mathbb{Z}}$ from the simplicial totally disconnected
$G$-space $T_{\bullet}$. We will use the corresponding
$G^{\times_{G}(j+1)}$-sheaves $S^{\bullet}$ in the above scheme. Moreover, we
get a homological differential in the $j$-direction from the simplicial
structure, hence we obtain a triple complex $F_{\bullet\bullet}^{\bullet}$ of
c-soft $G$-sheaves. We compute the groupoid hyperhomology
$\mathbb{H}_{\bullet}(G,F_{\bullet\bullet}^{\bullet})$ in two ways to obtain
the claim of the theorem.

To relate this to the groupoid homology
$H_{\bullet}(G,\underline{\mathbb{Z}})$, it is enough to see that the complex
$F_{\bullet}^{\bullet}$ is quasi-isomorphic to $\underline{\mathbb{Z}}$ (at
degree $0$), or equivalently, to the complex $F^{\bullet,a}$.

Given a $G$-sheaf $S$ and any $n$, we claim that the complex of $G$-sheaves
$F_{\bullet n}(S)$, defined analogously to $F_{jn}^{i}$ above but with $S$
instead of $S^{i}$, form a homological resolution of $S$. This is a stalk-wise
computation, and we just need to check that the complex

$\cdots\to F_{1n}(S)_{x}\to F_{0n}(S)_{x}\to S_{x}\to 0$

is exact. From (4), we see that $F_{j}(S)_{x}$ is the space of finitely
supported maps from $A_{x}^{j+1}$ to $S_{x}$, where

$A_{x}=\\{(h,g)\mid h\in H^{(n)},g\in G_{x},s(h)=r(g)\\}.$

Then we can construct a contracting homotopy $h\colon F_{jn}(S)_{x}\to
F_{j+1,n}(S)_{x}$ for the above complex, by fixing $a\in A_{x}$ and setting

$h(c)(a_{0},\dots,a_{j+1})=\delta_{a,a_{0}}c(a_{1},\dots,a_{j+1}).$

Next, if we treat $F_{\bullet\bullet}(S)$ as a bicomplex, this is still a
resolution of $S$. Indeed the above already shows that this is quasi-
isomorphic to a complex with terms $S$ at each nonnegative degree. Taking into
account the differential in the $n$-direction, we see that it is

${\dots}$${S}$${S}$${S}$${0.}$$\scriptstyle{0}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{0}$

Using the above argument for $S=F^{i,a}$, we get that
$F^{\bullet}_{\bullet\bullet}=(F^{i}_{jn})_{i,j,n}$ is a resolution of
$F^{\bullet,a}$, hence we get the isomorphism
$\mathbb{H}_{\bullet}(G,F_{\bullet\bullet}^{\bullet})\cong
H_{\bullet}(G,\underline{\mathbb{Z}})$.

It remains to show that our hyperhomology also computes
$H^{s}_{\bullet}(X,\phi)$. By definition
$\mathbb{H}(G,F^{\bullet}_{\bullet\bullet})$ is the homology of the quadruple
complex with terms

$E_{jnk}^{i}=\Gamma_{c}(G^{(k)},s^{*}F_{jn}^{i})$

with total degree $j+n+k-i$. The claim follows if we can show that the
homology in the $(n,k)$-direction gives $D(\Sigma_{j,i})_{,A}$ concentrated at
degree $0$ when we fix $j$ and $i$.

By the Morita equivalence between $H_{j}$ and $H^{\times_{G}(j+1)}$, this
homology can be interpreted as
$\mathbb{H}_{\bullet}(G,\mathcal{L}(f_{j})_{!}F^{i,a}$ if we regard $F^{i,a}$
as an $H_{j}$-sheaf. Then by
$\mathbb{H}_{\bullet}(G,\mathcal{L}(f_{j})_{!}F^{i,a})\cong
H_{\bullet}(H_{j},F^{i,a})$, our claim reduces to the computation of
$H_{\bullet}(H_{j},F^{i,a})$.

Let $C_{n}$ denote the subgroup of
$\Gamma_{c}(H_{j,i}^{(k)},\underline{\mathbb{Z}})$ consisting of the sections
$x$ supported on the free orbits and such that $sx=(-1)^{\left|s\right|}x$ for
$s\in S_{i+1}$. Then $H_{\bullet}(H_{j},F^{i,a})$ is the homology of
$C_{\bullet}$ by the c-softness of $F^{i,a}$. Hence we want to show

$\displaystyle H_{0}(C_{\bullet})$ $\displaystyle\cong
D^{s}(\Sigma_{j,i})_{,A},$ $\displaystyle H_{k}(C_{\bullet})$
$\displaystyle=0\quad(k>0).$

Let $H^{\prime}_{j,i}$ be the quotient groupoid $H_{j,i}/S_{i+1}$. On the one
hand, $H^{\prime}_{j,i}$ is AF because $H_{j,i}$, which is Morita equivalent
to $R^{u}(\Sigma_{j,i},\sigma_{j,i})$, is AF (notice that the quotient by the
action of $S_{i+1}$ does not create any stabilizers). On the other,
$H_{\bullet}(H_{j},F^{i,a})$ can be regarded as the groupoid homology of
$H^{\prime}_{j,i}$ with the induced action of $F^{i,a}$. Then we can conclude
that the higher homology groups vanish by virtue of Proposition 1.2.

Finally, the $0$-th homology is the group of $H_{j}$-coinvariants of
$C_{0}=\Gamma_{c}(T,F^{i,a})$. By the concrete description of
$D^{s}(\Sigma_{j,i})_{,A}$ given by [put:HoSmale]*Theorems 3.3.3 and 4.2.8, we
have $H_{0}(C_{\bullet})\cong D^{s}(\Sigma_{j,i})_{,A}$. More precisely, any
element of the alternating subgroup of $D^{s}(\Sigma_{j,i})$ (with respect to
the action of $S_{i+1}$) is represented by a linear combination of compact
open subsets of a stable set with alternating coefficients. Such compact open
sets must belong to the free part $T_{i}^{f}\subset T_{i}$. ∎

Combining Theorem 3.5 with the main results in [valmak:kpd] we obtain the
following results. The first is a Künneth formula for the product of two Smale
spaces, a result that generalizes [valmak:groupoidtwo]*Theorem 5.2 and
[dkw:dyn]*Theorem 6.5.

######  Theorem 3.6.

Let $(Y_{1},\psi_{1})$ and $(Y_{2},\psi_{2})$ be non-wandering Smale spaces.
Then we have a split exact sequence

$0\to{\bigoplus_{a+b=k}}H^{s}_{a}(Y_{1},\psi_{1})\otimes
H^{s}_{b}(Y_{2},\psi_{2})\to H^{s}_{k}(Y_{1}\times
Y_{2},\psi_{1}\times\psi_{2})\\\
\quad\to{\bigoplus_{a+b=k-1}}\operatorname{Tor}(H^{s}_{a}(Y_{1},\psi_{1}),H^{s}_{b}(Y_{2},\psi_{2}))\to
0.$

###### Proof.

This is obtained by combining Theorem 3.5 with a general Künneth type theorem
for groupoid homology [valmak:kpd]*Theorem A. ∎

As usual, the morphisms in the short exact sequence of Theorem 3.6 above are
natural in any reasonable sense, however the splitting is not.

The second result can be interpreted as a Poincaré duality-type theorem in the
setting of Smale spaces. It reduces the computation of the homology groups to
the standard (compactly supported) cohomology groups for the topological space
underlying the dynamical system.

Suppose that $(X,\phi)$ is a Smale space such that the unstable sets
$X^{u}(x)$ are homeomorphic to $\mathbb{R}^{n}$, for some fixed $n$. Then we
can consider a local system $\underline{o}$ for the group
$\mathbb{Z}/2\mathbb{Z}$ corresponding to the leafwise orientation on unstable
sets. Formally speaking, its stalk at $x\in X$ is given by

$\underline{o}_{x}=({\textstyle\bigwedge}^{n}T_{x}X^{u}(x,\epsilon)\setminus\\{0\\})/\mathbb{R}_{>0}$

for a small enough $\epsilon>0$.

######  Theorem 3.7.

Let $(X,\phi)$ be as above. Then we have an isomorphism

$H^{s}_{k}(X,\phi)\cong
H^{n-k}(X,\underline{\mathbb{Z}}\times_{\mathbb{Z}/2\mathbb{Z}}\underline{o}).$

###### Proof.

This is obtained by combining Theorem 3.5 with [valmak:kpd]*Theorem B. ∎

######  Remark 3.8.

In [put:HoSmale]*Question 8.3.2 Putnam asked if his homology groups of
$(X,\phi)$ are isomorphic, up to a degree shift, to the usual cohomology of
the underlying space $X$ whenever the (un)stable sets are contractible. The
above result gives a precise form of this conjectural isomorphism.

######  Example 3.9.

Let $(X,\phi)$ be a compact Riemannian manifold with an Anosov diffeomorphism.
If the non-wandering set of this system is $X$ itself, we obtain a Smale space
which satisfies the assumption of the above theorem. In particular, consider a
hyperbolic matrix $A\in\mathrm{SL}_{m}(\mathbb{Z})$. Then multiplication by
$A$ on $\mathbb{Z}^{m}$ induces an Anosov diffeomorphism $\phi_{A}$ on its
Pontryagin dual $\mathbb{T}^{m}$. By Theorem 3.7, we see that
$H^{s}_{k}(\mathbb{T}^{m},\phi_{A})$ is a free commutative group

$H^{s}_{k}(\mathbb{T}^{m},\phi_{A})\cong\mathbb{Z}^{r},\quad
r=\binom{m}{n-k},$

where $n$ is the number of eigenvalues $\lambda$ of $A$ (counted with
multiplicity) satisfying $\left|\lambda\right|>1$.

##  4\. Smale space with totally disconnected unstable sets

Now, let $(X,\phi)$ be a Smale space with totally disconnected unstable sets,
and let us fix a $u$-bijective map $f\colon(\Sigma,\sigma)\to(X,\phi)$ from a
shift of finite type. Let us take the constructions from Section 2.3, with
$(Z,\zeta)=(\Sigma,\sigma)$. In particular, we have an étale groupoid
$G=R^{u}(X,\phi)|_{T}$ for a transversal $T$ as in Proposition 3.1, and the
sheaves $(F^{n,a})_{n}$ giving a resolution of $\underline{\mathbb{Z}}_{T}$.

Recall also that the Smale spaces $(\Sigma_{n},\sigma_{n})$ admit presentation
by certain graphs [put:HoSmale]. One starts from a graph $\mathcal{G}$
presenting $(\Sigma,\sigma)$ with vertex set $\mathcal{G}^{0}$ and edge set
$\mathcal{G}^{1}$. Then one defines graphs $\mathcal{G}_{n}$ representing
$(\Sigma_{n},\sigma_{n})$, whose vertex sets are subsets of
$(\mathcal{G}^{0})^{n+1}$, and the edge sets are subsets of
$(\mathcal{G}^{1})^{n+1}$. There are natural actions of $S_{n+1}$ on
$(\mathcal{G}^{i})^{n+1}$ inducing the one on $\Sigma_{n}$.

######  Proposition 4.1.

We have

$K_{i}(G\ltimes C_{0}(\left\rceil
T_{\bullet}\right\lceil^{(p)}\smallsetminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}))\cong H_{0}(G,F^{p,a})$

when $i\equiv p\bmod 2$ and $K_{i}(G\ltimes C_{0}(\left\rceil
T_{\bullet}\right\lceil^{(p)}\smallsetminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}))=0$ otherwise.

###### Proof.

Put $A=T_{p}/S_{p+1}$ and $B=\left\rceil
T_{\bullet}\right\lceil^{(p)}\smallsetminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}$. As $A$ is a totally disconnected $G$-space,
$G\ltimes A$ is an ample groupoid. Moreover the $G$-C∗-algebra $C_{0}(B)$ is
naturally a $G\ltimes A$-C∗-algebra. By Theorem 1.3 and Remark 1.4, we have a
spectral sequence

$E^{2}_{pq}=H_{p}(G\ltimes A,K_{q}(C_{0}(B)))\Rightarrow K_{p+q}(G\ltimes
C_{0}(B)).$

Moreover, as in the proof of Theorem 3.5, $G\ltimes A$ is an AF groupoid. By
Proposition 1.2 we have $E^{2}_{pq}=0$ for $p>0$ for the above spectral
sequence, hence it collapses at the $E^{2}$-sheet. We are left with
establishing the identification of $H_{0}(G\ltimes A,K_{q}(C_{0}(B)))$ with
either $H_{0}(G,F^{p,a})$ or $0$ depending on the parity of $q$.

By Morita invariance of groupoid homology, we may replace the transversal
$T^{p}\subset\Sigma_{p}$ by another $S_{p+1}$-invariant transversal. For each
$z\in\mathcal{G}_{p}^{0}\subset(\mathcal{G}^{0})^{p+1}$, let $A_{z}$ be the
space of one-sided infinite paths representing a local stable set in
$\Sigma_{p}$. As a new transversal, take one that can be identified with the
union of the $A_{z}$, and call it $T^{\prime}$. Then the free part $T^{\prime
f}$ is the subset of paths that pass through a vertex
$w\in\mathcal{G}_{p}^{0}$ which is in a free orbit of the $S_{p+1}$ action.

Now, let $B^{\prime}$ be the quotient of $\mathring{\Delta}^{p}\times
T^{\prime f}$ which corresponds to $B$ under the Morita equivalence. It is
enough to check that the induced homomorphism

$j\colon K_{q}(C_{0}(B^{\prime}))\to K_{q-p}(C_{0}(T^{\prime f})),$

is injective and identifies $K_{q-p}(C_{0}(T_{p}^{f}))$ with
$\Gamma_{c}(T_{p}^{f},\underline{\mathbb{Z}})$ or $0$ depending on
$(q-p)\bmod\leavevmode\nobreak\ 2$.

By the above presentation of $T^{\prime f}$, the $S_{p+1}$-C∗-algebra
$C_{0}(T^{\prime f})$ is the union of $S_{p+1}$-invariant finite-dimensional
subalgebras of the form $C(K_{n})$, where $K_{n}$ is a free $S_{p+1}$-set.
(Concretely, $K_{n}$ is the set of paths of length $n$ in $\mathcal{G}_{p}$
passing through a free orbit.) Let $B_{n}$ be the quotient of
$\mathring{\Delta}^{p}\times K_{n}$. Then $C_{0}(B)$ is again the union of
$C(B_{n})$ which implies $K_{q}(C_{0}(B))=\varinjlim K_{q}(C(B_{n}))$. On
$K_{q}(C_{0}(B_{n}))$, the analogous claim about $j$ is obvious. ∎

We now have a dual analogue of the homological spectral sequence in Theorem
1.5.

######  Corollary 4.2.

Let $(X,\phi)$ be as above. There is a cohomological spectral sequence
abutting to $K_{\bullet}(C^{*}R^{u}(X,\phi))$, with $E_{2}$-sheet given by
$H^{s}_{\bullet}(X,\phi)$. More precisely:

$E_{2}^{pq}=H^{s}_{p}(X,\phi)\otimes K_{q}(\mathbb{C})\Rightarrow
K_{p+q}(C^{*}R^{u}(X,\phi)).$

###### Proof.

The $E_{2}$-sheet of the spectral sequence (2) is given by the cohomology of
$H_{0}(G,F^{p,a})$ for even $q$, and is trivial for odd $q$.

We first claim that, up to inverting the degree, this cohomology is equal to
$H_{\bullet}(G,\underline{\mathbb{Z}})$. Indeed, as $(F^{p,a})_{p}$ is a
resolution of $\underline{\mathbb{Z}}$, the groupoid homology is computed from
the double complex with terms $\Gamma_{c}(G^{(q)},F^{p,a})$. When we fix $p$,
the resulting homological complex has homology groups

$H_{q}(G,F^{p,a})\cong H_{q}(G\ltimes A,F^{\prime})$

for $A=T_{p}/S_{p+1}$ as before, and the $G\ltimes A$-sheaf $F^{\prime}$
corresponding to $F^{p,a}$. Again by Proposition 1.2, this homology is trivial
for $q>0$. Then the homology of the total complex is the same as the
cohomology of the cochain complex $H_{0}(G,F^{\bullet,a})$.

Thus, we have

$\displaystyle E_{2}^{pq}$ $\displaystyle\cong
H_{-p}(G,\underline{\mathbb{Z}})\quad(\text{$q$ even}),$ $\displaystyle
E_{2}^{pq}$ $\displaystyle\cong 0\quad(\text{$q$ odd}),$

which implies the claim. ∎

This corollary, combined with the dual result from [valmak:groupoidtwo]
(summarized in Theorem 1.5 above), provides a partial solution a conjecture in
[put:HoSmale]*Question 8.4.1, asking about the possibility of computing
$K$-theory groups of C∗-algebras attached to Smale spaces from the
corresponding homology groups. We can consider the conjecture settled in the
case where either the stable or unstable foliations are totally disconnected.

##  5\. Finiteness for general Smale spaces

In general, we cannot directly relate the $K$-groups of $C^{*}R^{u}(X,\phi)$
to the groupoid homology of $R^{u}(X,\phi)|_{T}$. However, the considerations
from Section 2 still lead to the following finiteness result.

######  Theorem 5.1.

Let $(X,\phi)$ be a Smale space. Then $K_{\bullet}(C^{*}R^{u}(X,\phi))$ is of
finite rank.

###### Proof.

Let us fix a $u$-bijective map $g\colon(Z,\zeta)\to(X,\phi)$ from another
Smale space with totally disconnected stable sets, with $g$ at most $N$-to-
one. Take a transversal $T\subset X$ as in Proposition 3.1, and consider the
étale groupoid $G=R^{u}(X,\phi)|_{T}$ and the totally disconnected $G$-space
$T_{0}=g^{-1}(T)$. Let us take the constructions from Sections 2.2 and 2.3.
Then the rationalization of the cohomological spectral sequence (2) becomes

$E_{1}^{pq}=K_{p+q}(G\ltimes C_{0}(\left\rceil
T_{\bullet}\right\lceil^{(p)}\setminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}))\otimes\mathbb{Q}\Rightarrow
K_{p+q}(C^{*}_{r}G)\otimes\mathbb{Q}.$ (5)

Let us fix $p$. Again by Theorem 1.3, there is a homological spectral sequence

$E^{1}_{p^{\prime}q^{\prime}}=K_{q^{\prime}}(C_{0}(G^{(p^{\prime})}\ltimes(\left\rceil
T_{\bullet}\right\lceil^{(p)}\setminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)})))\Rightarrow
K_{p^{\prime}+q^{\prime}}(G\ltimes C_{0}(\left\rceil
T_{\bullet}\right\lceil^{(p)}\setminus\left\rceil
T_{\bullet}\right\lceil^{(p-1)}))$

compatible with the actions of $S_{p+1}$. By Proposition 2.4, rationalization
at the $E^{1}$-sheet agrees with

$(K_{q^{\prime}+p}(C_{0}(G^{(p^{\prime})}\ltimes
T_{p}))\otimes\mathbb{Q})_{A},$

which converges to $(K_{q}(C^{*}(G\ltimes T_{p}))\otimes\mathbb{Q})_{A}$. This
implies that the spectral sequence (5) satisfies

$E_{1}^{pq}\cong(K_{q}(C^{*}(G\ltimes T_{p}))\otimes\mathbb{Q})_{A}.$

The right hand side can be considered as a submodule of
$K_{q}(C^{*}(R^{u}(Z_{p},\zeta_{p})))\otimes\mathbb{Q}$. On one hand, we know
that $K_{q}(C^{*}(R^{u}(Z_{p},\zeta_{p})))$ is of finite rank by
[valmak:groupoidtwo]*Corollary 3.10. On the other, we have $E_{1}^{pq}=0$ for
$p>N$, hence the assertion. ∎

Recall that taking the inverse homeomorphism we get the stable equivalence
relation:

$R^{s}(X,\phi)=R^{u}(X,\phi^{-1}).$

In particular, $K_{\bullet}(C^{*}R^{s}(X,\phi))$ is also of finite rank by the
above theorem.

######  Remark 5.2.

In [MR3692021], Kaminker, Putnam, and Whittaker considered $K$-theoretic
duality for the _Ruelle algebras_

$\displaystyle\mathcal{R}_{s}(X,\phi)$
$\displaystyle=C^{*}(R^{s}(X,\phi))\rtimes_{\phi}\mathbb{Z},$
$\displaystyle\mathcal{R}_{u}(X,\phi)$
$\displaystyle=C^{*}(R^{u}(X,\phi))\rtimes_{\phi}\mathbb{Z}.$

They showed that these algebras are odd Spanier–Whitehead dual to each other,
and that this duality implies
$\mathcal{R}_{s}(X,\phi)\cong\mathcal{R}_{u}(X,\phi)$ under the assumption
that $K_{\bullet}((C^{*}R^{s}(X,\phi))$ and $K_{\bullet}((C^{*}R^{u}(X,\phi))$
are of finite rank. They have also conjectured that this assumption is
unnecessary, and our result above implies that this is indeed the case.

######  Remark 5.3.

In view of the classification program of C∗-algebras and the results in
[deeley:strung][dgy:rrzero], it is natural to ask whether the range of
$K$-theory on the class of (simple) classifiable, real rank zero C∗-algebras
is exhausted by the (un)stable algebras of Smale spaces up to restricting to a
transversal and passing to hereditary subalgebras.

However, the above finiteness implies that this is not possible. Indeed, there
are algebras as above whose $K$-groups are not of finite rank. For example,
the simple AF algebra associated to the following graph

$\cdots$,

or $A^{\otimes\infty}$ for a classifiable algebra $A$ whose $K$-groups have
rank more than $1$ will do.

## References