# Coarse geometry

Ulrich Bunke Fakultät für Mathematik, Universität Regensburg, 93040
Regensburg<EMAIL_ADDRESS>
###### Abstract

In this chapter we introduce coarse and bornological coarse spaces. We explain
the concept of a coarse homology theory and discuss the examples of coarse
ordinary homology and coarse $K$-homology in some detail.

###### Contents

  1. 1 Introduction
  2. 2 Coarse spaces
  3. 3 Bornological coarse spaces
  4. 4 Coarse equivalence
  5. 5 Coarse homology theories
  6. 6 Equivariant coarse ordinary homology
  7. 7 Equivariant coarse topological $K$-homology
  8. 8 Summary

##  1 Introduction

Coarse geometry was invented by J. Roe [35], [37]. The original motivation
came from index theory of Dirac type operators on complete Riemannian
manifolds [36]. But coarse geometry is also a framework to study geometric
properties of groups and assembly maps for algebraic $K$-theory [1], [2]. Via
the cone construction [27], [33] it subsumes the controlled topology approach
[19], [41] to algebraic $K$-theory. Coarse geometry also has been used to
study topological insulators in mathematical physics [22], [31].

  * •

In this survey we explain the category of $G$-bornological coarse spaces [13]
as a basic framework to study large scale invariants of metric spaces with
$G$-action.

  * •

We introduce the concept of a coarse equivalence and the idea of a coarse
invariant. As examples we discuss the sets of coarse components and the Higson
corona.

  * •

We introduce the concept of an equivariant coarse homology theory.

  * •

We provide a complete description of equivariant coarse ordinary homology and
equivariant coarse $K$-theory.

Let us point out that though coarse homology theories constitute a central
aspect of coarse geometry this survey only touches a very small portion of the
field of coarse geometry. Furthermore, the list of given references is by far
not complete and just provides entry points for further reading.

##  2 Coarse spaces

In metric geometry, the mutual relation between points of a set is encoded by
a distance function $d:X\times X\to[0,\infty]$, where the generalised real
number $d(x,y)$ is interpreted as the distance from $x$ to $y$. The pair
$(X,d)$ is called a metric space. Note that in contrast to the usual
conventions we allow infinite distances in order to model spaces with more
than one coarse component later on.

The space $\mathbb{R}^{n}$ with the euclidean distance $d_{eu}$ is an example
of a metric space. Every subset of a metric space becomes a metric space with
the restricted distance function.

A distance function often encodes much more structure about the geometry of
$X$ as one is interested in or can determine in practise.

######  Example 2.1.

Assume that $G$ is a group. If we choose a subset $S$ of $G$, then we can
define a distance $d_{S}(g,h)$ as the minimal number of elements of $S\cup
S^{-1}$ needed to express $gh^{-1}$. This could be infinite if $S$ does not
generate $G$. But even if $S$ and $T$ are finite generating sets of $G$, then
the distance functions $d_{T}$ and $d_{S}$ are different in general. But one
is only interested in geometric properties of $G$ that do not depend on that
choice. $\Box$

Let $(X,d)$ be a metric space. If one wants to concentrate on the small scale
structure of $X$ locally, then one usually only considers the topology
determined by the distance function which is generated by the open balls

$B(x,r):=\\{y\in X\mid d(x,y)<r\\}$ (1)

for all $r$ in $(0,\infty)$ and $x$ in $X$. If one wants to be able to compare
the local scales at different points of $X$, then it is natural to work with
the uniform structure on $X$ generated by the entourages

$U_{r}:=\\{(x,y)\in X\times X\mid d(x,y)<r\\}$ (2)

for all $r$ in $(0,\infty)$. Similarly, in coarse geometry one is interested
in the large scale structure of the metric space only.

A mathematical way to introduce structures is to describe a category whose
objects represent the structures of interest, and whose morphisms are
structure preserving maps. In the cases discussed above we arrive at the
categories $\mathbf{Top}$ of topological spaces and continuous maps and
$\mathbf{Unif}$ of uniform spaces and uniform maps. In the following we
describe the category $\mathbf{Coarse}$ of coarse spaces which is designed to
encode the large scale structure of metric spaces. Note that not every
topological or uniform space comes from a metric space. Similarly, coarse
spaces represented by metric spaces only exhaust a small portion of the
category of coarse spaces.

######  Definition 2.2.

A coarse structure $\mathcal{C}$ on a set $X$ is a collection of subsets of
$X\times X$ whose elements are called coarse entourages. One requires the
following axioms:

  1. 1.

The diagonal $\mathrm{diag}(X):=\\{(x,x)\mid x\in X\\}$ belongs to
$\mathcal{C}$.

  2. 2.

The set $\mathcal{C}$ is closed under forming finite unions and taking
subsets.

  3. 3.

The set $\mathcal{C}$ is closed under the operations flip

$U\mapsto U^{-1}:=\\{(y,x)\mid(x,y)\in U\\}$

and composition

$(U,V)\mapsto U\circ V:=\\{(x,y)\in X\times X\mid(\exists z\in X\mid(x,z)\in
U\>\&\>(z,y)\in V)\\}\ .$

A coarse space is a pair $(X,\mathcal{C})$ of a set with a coarse structure
$\mathcal{C}$. If $(X^{\prime},\mathcal{C}^{\prime})$ is a second coarse space
and $f:X\to X^{\prime}$ is a map of underlying sets, then $f$ is called
controlled if $f\times f$ sends coarse entourages of $X$ to coarse entourages
of $X^{\prime}$. We thus obtain the category $\mathbf{Coarse}$ of coarse
spaces and controlled maps.

######  Example 2.3.

A distance function $d$ on $X$ gives rise to the metric coarse structure
$\mathcal{C}_{d}$ defined as the smallest coarse structure containing the
metric entourages (2).

Assume that $d^{\prime}$ is a second distance function on $X$. If for every
$s$ in $(0,\infty)$ there exists an $r$ in $(0,\infty)$ such that $d(x,y)<s$
implies $d^{\prime}(x,y)<r$, then we have
$\mathcal{C}_{d}\subseteq\mathcal{C}_{d^{\prime}}$ and the identity of $X$ is
a morphism $(X,\mathcal{C}_{d})\to(X,\mathcal{C}_{d^{\prime}})$ of coarse
spaces. If in the condition above we can interchange the roles of $d$ and
$d^{\prime}$, then $(X,\mathcal{C}_{d})=(X,\mathcal{C}_{d^{\prime}})$. $\Box$

######  Example 2.4.

If $S$ and $T$ are two finite generating sets of a group $G$, then we have the
equality of coarse structures $\mathcal{C}_{d_{S}}=\mathcal{C}_{d_{T}}$ on
$G$. Thus by considering the coarse structure on $G$ determined by any choice
of such a generating set and studying geometric properties of $G$ which only
depend on the coarse structure we get rid of the dependence on the choice of
the generating set. $\Box$

Any subset $Y$ of a coarse space $(X,\mathcal{C})$ has an induced coarse
structure given by the coarse entourages of $X$ that are contained in $Y\times
Y$.

For any collection of entourages on $X$ we can consider the smallest coarse
structure containing the collection. A coarse structure represented by a
metric admits a countable set of generators, namely the family of metric
entourages $(U_{n})_{n\in\mathbb{N}}$. In the case of a path metric space even
the one-member family $(U_{1})$ suffices to generate the coarse structure.

######  Example 2.5.

If $((X_{i},\mathcal{C}_{i}))_{i\in I}$ is a family of coarse spaces, then the
free union $\bigsqcup^{\mathrm{free}}_{i\in I}(X_{i},\mathcal{C}_{i})$ is the
set $X:=\bigsqcup_{i\in I}X_{i}$ with the coarse structure generated by the
entourages $\bigcup_{i\in I}U_{i}$ for all $(U_{i})_{i\in I}$ in $\prod_{i\in
I}\mathcal{C}_{i}$. If $I$ is infinite, then except for degenerate cases the
coarse structure on the free union is not countably generated and therefore
does not come from a metric. $\Box$

Note that the category of coarse spaces $\mathbf{Coarse}$ still captures the
full information about the underlying sets, i.e. there is a forgetful functor
$\mathbf{Coarse}\to\mathbf{Set}$. This functor has left- and right adjoints
which send a set $S$ to the coarse space $S_{min}$ with coarse structure
generated by $\mathrm{diag}(S)$, or $S_{max}$ with the maximal coarse
structure given by the power set of $S\times S$. Honest large scale geometry
starts with the introduction of the notion of a coarse equivalence which will
be explained in Section 4.

##  3 Bornological coarse spaces

For most applications of large scale geometry the coarse structure needs to be
complemented by a notion of local finiteness. To this end one introduces the
notion of a bornology.

######  Definition 3.1.

A bornology $\mathcal{B}$ on a set $X$ is a collection of subsets of $X$
called the bounded subsets. One requires the following axioms:

  1. 1.

Every finite subset belongs to $\mathcal{B}$.

  2. 2.

$\mathcal{B}$ is closed under forming finite unions and taking subsets.

A bornological space is a pair $(X,\mathcal{B})$ of a set $X$ with a bornology
$\mathcal{B}$. If $(X^{\prime},\mathcal{B}^{\prime})$ is a second bornological
space, then a map $X\to X^{\prime}$ is called proper if preimages of bounded
sets are bounded. It is called bornological if images of bounded sets are
bounded.

Any set $X$ has a minimal bornology consisting of all finite subsets and a
maximal bornology where all subsets are bounded.

A distance function on $X$ determines the metric bornology $\mathcal{B}_{d}$
defined as the smallest bornology containing the metric balls $B(x,r)$ from
(1) for all $x$ in $X$ and $r$ in $(0,\infty)$. Any subset $Y$ of a
bornological space has an induced bornology consisting of the bounded subsets
of $X$ which are contained in $Y$.

We consider the bornological space $(X,\mathcal{B})$.

######  Definition 3.2.

A subset $Y$ of $X$ is called locally finite if the induced bornology on $Y$
is the minimal one.

The collection of locally finite subsets of $X$ again forms a bornology
$\mathcal{B}^{\perp}$. If
$f:(X,\mathcal{B})\to(X^{\prime},\mathcal{B}^{\prime})$ is proper, then
$f:(X,\mathcal{B}^{\perp})\to(X^{\prime},\mathcal{B}^{{}^{\prime},\perp})$ is
bornological.

A coarse structure $\mathcal{C}$ and a bornology $\mathcal{B}$ on the same set
$X$ are said to be compatible if for every bounded subset $B$ of $X$ and
coarse entourage $U$ of $X$ the thickening

$U[B]:=\\{y\in X\mid(\exists x\in B\mid(y,x)\in U)\\}$ (3)

is again bounded.

######  Definition 3.3 ([12, Def. 2.7]).

A bornological coarse space is a triple $(X,\mathcal{C},\mathcal{B})$ of a set
$X$ with a coarse structure $\mathcal{C}$ and a compatible bornology
$\mathcal{B}$. A morphism between bornological coarse spaces
$f:(X,\mathcal{C},\mathcal{B})\to(X^{\prime},\mathcal{C}^{\prime},\mathcal{B}^{\prime})$
is map of sets $f:X\to X^{\prime}$ which is controlled and proper.

In this way we get the category $\mathbf{BornCoarse}$ of bornological coarse
spaces.

The metric coarse structure and bornology associated to a distance function
$d$ on $X$ are compatible. We will denote the associated bornological coarse
space by $X_{d}$.

######  Example 3.4.

For a set $X$ we have the bornological coarse spaces $X_{min,min}$,
$X_{min,max}$ and $X_{max,max}$, where the first subscript indicates the
coarse structure and the second the bornology. If $X$ is infinite, then the
maximal coarse structure and the minimal bornology on $X$ are not compatible.
$\Box$

######  Example 3.5.

The free union $\bigsqcup_{i\in
I}^{\mathrm{free}}(X_{i},\mathcal{C}_{i},\mathcal{B}_{i})$ of a family of
bornological coarse spaces $((X_{i},\mathcal{C}_{i},\mathcal{B}_{i}))_{i\in
I}$ is the free union of the underlying coarse spaces with the bornology
generated by $\bigcup_{i\in I}\mathcal{B}_{i}$. For example, $X_{min,min}$ is
the free union of the family of one-point spaces $(\\{x\\})_{x\in X}$, while
$X_{min,max}$ is the coproduct. $\Box$

The category $\mathbf{BornCoarse}$ has a symmetric monoidal structure
$\otimes$ defined such that
$(X,\mathcal{C},\mathcal{B})\otimes(X^{\prime},\mathcal{C}^{\prime},\mathcal{B}^{\prime})$
is given by $(X\times
X^{\prime},\mathcal{C}^{\prime\prime},\mathcal{B}^{\prime\prime})$, where
$\mathcal{C}^{\prime\prime}$ is generated by all entourages $U\times
U^{\prime}$ with $U$ in $\mathcal{C}$ and $U^{\prime}$ in
$\mathcal{C}^{\prime}$, and $\mathcal{B}^{\prime\prime}$ is generated by
$B\times B^{\prime}$ with $B$ in $\mathcal{B}$ and $B^{\prime}$ in
$\mathcal{B}^{\prime}$. Note the forgetful functor
$\mathbf{BornCoarse}\to\mathbf{Coarse}$ is symmetric monoidal if we equip the
target with the cartesian structure. On the other hand, $\otimes$ differs from
the cartesian structure on $\mathbf{BornCoarse}$.

In coarse geometry group actions play an important role. If $G$ is a group,
then we can consider the symmetric monoidal category
$\mathbf{Fun}(BG,\mathbf{BornCoarse})$ of bornological coarse spaces with a
$G$-action by automorphisms. It contains the full symmetric monoidal
subcategory $G\mathbf{BornCoarse}$ of $G$-bornological coarse spaces
$(X,\mathcal{C},\mathcal{B})$ characterized by the condition that the subset
of $G$-invariant entourages $\mathcal{C}^{G}$ is cofinal in $\mathcal{C}$ with
respect to the inclusion relation, i.e., every element of $\mathcal{C}$ is
contained in a $G$-invariant one. In this case we say that $\mathcal{C}$ is a
$G$-coarse structure.

######  Example 3.6.

If $G$ acts isometrically on a metric space $(X,d)$, then the associated
bornological coarse space $X_{d}$ belongs to $G\mathbf{BornCoarse}$. $\Box$

######  Example 3.7.

If we consider the action of $\mathbb{Z}$ on the metric space
$(\mathbb{R},d_{eu})$ given by $(n,x)\mapsto 2^{n}x$, then
$\mathbb{R}_{d_{eu}}$ with this action belongs to
$\mathbf{Fun}(BG,\mathbf{BornCoarse})$, but not to $G\mathbf{BornCoarse}$.
$\Box$

######  Example 3.8.

The group $G$ itself has a canonical $G$-coarse structure $\mathcal{C}_{can}$
generated by the family of invariant entourages $(\\{(gh,gk)\mid g\in
G\\})_{(h,k)\in G\times G}$. Together with the minimal bornology we get an
object $G_{can,min}$ in $G\mathbf{BornCoarse}$. If $G$ admits a finite
generating set $S$, then $G_{can,min}=G_{d_{S}}$.  $\Box$

If $(X,\mathcal{C})$ is a coarse space, then there is a minimal compatible
bornology on $X$. The classical literature only considers bornological coarse
spaces whose bornology is the minimal one compatible with the coarse
structure. But following examples show that it is useful to decouple the
choice of the bornology from the coarse structure.

######  Example 3.9.

The orbit category $G\mathrm{Orb}$ of a group $G$ is the category of
transitive $G$-sets and equivariant maps. We have a functor

$i:G\mathrm{Orb}\to G\mathbf{BornCoarse},\quad S\mapsto S_{min,max}\ .$

If $S$ is infinite, then the maximal bornology is different from the minimal
one compatible with the coarse structure. This functor is the starting point
for the application of coarse geometry to the study of assembly maps (11).
$\Box$

######  Example 3.10.

Let $(M,g)$ be a complete Riemannian manifold with an action of a discrete
group $G$ by isometries. The Riemannian metric $g$ allows to measure the
length of curves and induces a Riemannian distance function $d_{g}$, and one
usually considers the $G$-bornological coarse space
$M_{d_{g}}=(M,\mathcal{C}_{d_{g}},\mathcal{B}_{d_{g}})$. But in index theory
of Dirac operators it will be useful to work with a larger bornology
$\mathcal{B}_{s_{g}}$. Let $s_{g}:M\to\mathbb{R}$ be the scalar curvature
function. Then the bornology $\mathcal{B}_{s_{g}}$ consists of the subsets $B$
of $M$ with the property that there exists a compact subset $K$ of $M$ such
that $\inf_{B\setminus K}s_{g}>0$. One checks that $\mathcal{B}_{s_{g}}$ is
compatible with the metric coarse structure and we get a bornological coarse
space $M_{d_{g},s_{g}}:=(M,\mathcal{C}_{d_{g}},\mathcal{B}_{s_{g}})$. The
identity of $M$ is a morphism $M_{d_{g},s_{g}}\to M_{d_{g}}$ in
$G\mathbf{BornCoarse}$. We will explain in Section 7 that this example allows
to capture the fact that the coarse index of an invariant $Spin$ Dirac
operator is supported away from the set where the scalar curvature is
positive. $\Box$

##  4 Coarse equivalence

Two morphisms $f_{0},f_{1}:X\to Y$ between $G$-bornological coarse spaces are
said to be close to each other if $(f_{0}\times f_{1})(\mathrm{diag}(X))$ is a
coarse entourage of $Y$. Equivalently we can require that the map
$\\{0,1\\}_{max,max}\otimes X\to Y$ given by $(i,x)\mapsto f_{i}(x)$ is a
morphism in $G\mathbf{BornCoarse}$. Closeness is an equivalence relation on
morphisms which is compatible with composition.

######  Definition 4.1.

A map $f:X\to Y$ in $G\mathbf{BornCoarse}$ is a coarse equivalence if it is
invertible up to closeness.

In detail this means that there exists a map $h:Y\to X$ such that $h\circ f$
is close to $\mathrm{id}_{X}$ and $f\circ h$ is close to $\mathrm{id}_{Y}$.

A subset $Y$ of a bornological coarse space $X$ is called dense if there
exists a coarse entourage $U$ of $X$ such that $U[Y]=X$, see (3). In this case
the inclusion $Y\to X$ is a coarse equivalence, where we equip $Y$ with the
induced bornological coarse structures. In the equivariant case this is not
always true.

######  Example 4.2.

Consider the $C_{2}$-bornological coarse space $\mathbb{R}_{d_{eu}}$ such that
the non-trivial element of $C_{2}$ acts by multiplication by $-1$. Then the
inclusion $\mathbb{R}_{d_{eu}}\setminus\\{0\\}\to\mathbb{R}_{d_{eu}}$ is dense
and a coarse equivalence of underlying bornological coarse spaces, but not a
coarse equivalence in $C_{2}\mathbf{BornCoarse}$ since there exists no
equivariant map of sets $\mathbb{R}\to\mathbb{R}\setminus\\{0\\}$ at all. In
view of this example one sometimes considers the notion of a weak coarse
equivalence (see e.g. [9, Def. 2.18]) between $G$-bornological coarse spaces
which is map which becomes a coarse equivalence after forgetting the
$G$-action. $\Box$

######  Example 4.3.

Let $(M,g)$ be a connected compact Riemannian manifold with fundamental group
$G$. Its universal covering $(\tilde{M},\tilde{g})$ is a complete Riemannian
manifold with an isometric $G$-action. It has an induced $G$-invariant
distance function $d_{\tilde{g}}$. If $\tilde{m}_{0}$ is any point in
$\tilde{M}$, then the map $G\to\tilde{M}$, $g\mapsto gm_{0}$, induces a coarse
equivalence $G_{can,min}\to\tilde{M}_{d_{\tilde{g}}}$ in
$G\mathbf{BornCoarse}$. $\Box$

As a general rule, all concepts of coarse geometry should be invariant under
coarse equivalences. In the following we provide two examples, the set of
coarse components and the Higson corona. Further examples of coarsely
invariant concepts are the notions of bounded geometry [28], property $A$
[44], asymptotic dimension [24], or finite decomposition complexity [25].

######  Example 4.4.

In the following we describe the functor

$\pi_{0}^{{\tiny c}}:G\mathbf{BornCoarse}\to G\mathbf{Set}$

which associates to every $G$-bornological coarse space its $G$-set of coarse
components. Let $X$ be a $G$-bornological coarse space.

######  Definition 4.5.

The $G$-set of coarse components $\pi_{0}^{{\tiny c}}(X)$ is the set of
equivalence classes on $X$ with respect to the equivalence relation
$R_{\mathcal{C}}:=\bigcup_{U\in\mathcal{C}}U$ and induced $G$-action, where
$\mathcal{C}$ denotes the coarse structure of $X$.

Note that $\pi_{0}^{{\tiny c}}(X)$ is independent of the bornology. A map of
$G$-bornological coarse spaces $f:X\to Y$ functorially induces a map of
$G$-sets

$\pi_{0}^{{\tiny c}}(f):\pi_{0}^{{\tiny c}}(X)\to\pi_{0}^{{\tiny c}}(Y)\ .$

If $f_{0}$ is close to $f_{1}$, then we have $\pi_{0}^{{\tiny
c}}(f_{0})=\pi_{0}^{{\tiny c}}(f_{1})$. So the functor $\pi_{0}^{{\tiny c}}$
is coarsely invariant. It even sends weak coarse equivalences to isomorphisms.

For a $G$-set $X$ we have a canonical isomorphism $X\cong\pi_{0}^{{\tiny
c}}(X_{min,min})$. The group $G$ with its canonical coarse structure is
coarsely connected, i.e., we have $\pi_{0}^{{\tiny c}}(G_{can,min})\cong*$. If
the $G$-coarse structure comes from an invariant distance function $d$ on $X$,
then $x$ and $y$ belong to the same coarse component of $X$ if and only if
$d(x,y)<\infty$. $\Box$

######  Example 4.6.

In this example we discuss the Higson corona functor

$\partial:G\mathbf{BornCoarse}\to G\mathbf{Top}_{\mathrm{cp}}$ (4)

which associates to a $G$-bornological coarse space $X$ a compact Hausdorff
space $\partial X$ with $G$-action. Let $\phi:X\to\mathbb{C}$ be a bounded
function. For any entourage $U$ and subset $Y$ of $X$ we define the
$U$-variation of $\phi$ on $Y$ by

$\mathrm{Var}_{U}(\phi)(Y):=\sup_{(y,y^{\prime})\in U\cap(Y\times
Y)}|\phi(y)-\phi(y^{\prime})|\ .$

We then consider the $G$-$C^{*}$-subalgebra

$C_{h}(X):=\\{\phi\in C_{b}(X)\mid(\forall
U\in\mathcal{C}\mid\lim_{B\in\mathcal{B}}\mathrm{Var}_{U}(\phi)(X\setminus
B)=0)\\}$

of bounded functions with vanishing variation at infinity. We furthermore
consider the $G$-$C^{*}$-subalgebra $C_{0}(X)$ generated by the functions with
bounded support. We finally define the unital $G$-$C^{*}$-algebra

$C(\partial X):=C_{h}(X)/C_{0}(X)$ (5)

######  Definition 4.7 ([28]).

The Higson corona $\partial X$ of $X$ is the compact Hausdorff space
corresponding to $C(\partial X)$ under Gelfand duality with the induced
$G$-action.

If $f:X\to Y$ is a morphism of $G$-bornological coarse spaces, then the
canonical homomorphism $f^{*}:C_{b}(Y)\to C_{b}(X)$ sends the subalgebras
$C_{h}(Y)$ and $C_{0}(Y)$ to $C_{h}(X)$ and $C_{0}(X)$, respectively and
therefore induces an equivariant homomorphism of quotients
$\bar{f}^{*}:C(\partial Y)\to C(\partial X)$ and hence a continuous map of
coronas $\partial f:\partial X\to\partial Y$. We thus get the functor (4).

If $f_{0},f_{1}:X\to Y$ are close to each other and $\phi$ is in $C_{h}(Y)$,
then $f_{0}^{*}\phi-f_{1}^{*}\phi\in C_{0}(X)$. This implies that $\partial
f_{0}=\partial f_{1}$. Therefore the corona functor $\partial$ is a coarsely
invariant.

Applying properties of compact Hausdorff spaces to the Higson corona we get a
variety of coarsely invariant concepts.

In Section 7 we will explain that the topological $K$-theory $K(\partial X)$
of the Higson corona pairs interestingly with the coarse $K$-homology
$K\mathcal{X}(X)$ of $X$. $\Box$

##  5 Coarse homology theories

In the following we describe an axiomatization of equivariant coarse homology
theories. We consider a functor $E:G\mathbf{BornCoarse}\to\mathbf{M}$ whose
target $\mathbf{M}$ is a cocomplete stable $\infty$-category, e.g. the
category of spectra $\mathbf{Sp}$ or the derived category of abelian groups
$D(\mathbb{Z})$.

######  Remark 5.1.

Following [32, Sec. 1] an $\infty$-category $\mathbf{M}$ is called stable, if
it is pointed and admits finite limits and colimits, and if push-out squares
in $\mathbf{M}$ are the same as pull-back squares. The $\infty$-category of
spectra $\mathbf{Sp}$ is the universal presentable stable $\infty$-category
generated by an object $S$ called the sphere spectrum. If $E$ is a spectrum,
then we define its $\mathbb{Z}$-graded homotopy groups by

$\pi_{n}E:=[\Sigma^{n}S,E]\ ,$

where $[-,-]$ denote the group of maps in the homotopy category of
$\mathbf{Sp}$. In particular, a morphism between spectra is an equivalence if
and only if it induces an isomorphism between the homotopy groups.

For any stable $\infty$-category $\mathbf{M}$ and objects $A,B$ we have a
mapping spectrum $\mathrm{map}_{\mathbf{M}}(A,B)$ in $\mathbf{Sp}$.

The stable $\infty$-category $D(\mathbb{Z})$ is the Dwyer-Kan localization of
the category of chain complexes of abelian groups at the quasi-isomorphisms.
It is the universal presentable stable $\mathbb{Z}$-linear $\infty$-category
generated by $\mathbb{Z}$ considered as a chain complex in the natural way. If
$A$ is any chain complex, then we have an isomorphism

$\pi_{n}\mathrm{map}_{D(\mathbb{Z})}(\mathbb{Z},A)\cong H_{n}(A)\ .$

Again, a morphism in $D(\mathbb{Z})$ is an equivalence if and only if it
induces an isomorphism of homology groups. $\Box$

Classically, coarse homology theories are defined as $\mathbb{Z}$-graded
abelian group valued functors. In this case the boundary operators for Mayer-
Vietoris sequences have to be constructed as an additional datum. Working with
functors with values in a stable $\infty$-category turns Mayer-Vietoris into a
property of the functor $E:G\mathbf{BornCoarse}\to\mathbf{M}$, see 5.2.2 below
for a precise formulation.

######  Definition 5.2 ([13, Def. 3.10]).

$E$ is an equivariant coarse homology theory if

  1. 1.

$E$ is coarsely invariant.

  2. 2.

$E$ is excisive.

  3. 3.

$E$ annihilates flasques.

  4. 4.

$E$ is $u$-continuous.

In the following we explain the meaning of these conditions.

Coarse invariance means that $E$ sends coarse equivalences to equivalences.
This could equivalently be phrased as the condition that the projection onto
$X$ induces an equivalence

$\mathrm{pr}:E(\\{0,1\\}_{min,max}\otimes
X)\stackrel{{\scriptstyle\simeq}}{{\to}}E(X)$

for every $G$-bornological coarse space $X$. Coarse invariance is the main
condition which ensures that associating $E(X)$ to $X$ is a coarsely invariant
concept. There are interesting examples of equivariant coarse homology
theories which even send weak coarse equivalences to equivalences, e.g. [9,
Thm. 8.7] and the examples from [6, Cor. 5.3.13].

Excisiveness is the condition which makes $E$ homological. It means that
$E(X)$ can be determined by glueing the values of $E$ on pieces of $X$. For a
precise formulation we introduce the notion of a complementary pair
$(Z,\mathcal{Y})$ in $X$. Here $Z$ is an $G$-invariant subset of $X$ and
$\mathcal{Y}$ is a family $(Y_{i})_{i\in I}$ of $G$-invariant subsets indexed
by a filtered poset $I$ which is big in the sense that for any $i$ in $I$ and
invariant coarse entourage $U$ of $X$ there exists $i^{\prime}$ in $I$ such
that $U[Y_{i}]\subseteq Y_{i^{\prime}}$, see (3). In addition we require that
there exists $i$ in $I$ with $Y_{i}\cup Z=X$. We define
$E(\mathcal{Y}):=\mathrm{colim}_{i\in I}E(Y_{i})$ and consider the big family
$Z\cap\mathcal{Y}:=(Z\cap Y_{i})_{i\in I}$ on $Z$. Then $E$ is called excisive
if

$\textstyle{E(Z\cap\mathcal{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(\mathcal{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(X)}$

is a push-out square in $\mathbf{M}$ for every $G$-bornological coarse space
$X$ and complementary pair $(Z,\mathcal{Y})$. If $(Z,Y)$ is an invariant
covering of $X$, then one considers the complementary pair $(Z,\\{Y\\})$ with
the big family $\\{Y\\}:=(U[Y])_{U\in\mathcal{C}^{G}}$ of all invariant
thickenings of $Y$. We work with the big family $\\{Y\\}$ instead of $Y$ since
the intersection $Z\cap\\{Y\\}$ is a coarsely invariant concept in contrast to
$Z\cap Y$. But in many examples the canonical map $E(Y)\to E(\\{Y\\})$ is an
equivalence since the inclusions $Y\to U[Y]$ are coarse equivalences.

A $G$-bornological coarse space $X$ is called flasque if it admits a selfmap
$f:X\to X$ such that $f$ is close to $\mathrm{id}_{X}$, $f$ is non-expanding
in the sense that for any coarse entourage $U$ of $X$ the union
$\bigcup_{n\in\mathbb{N}}(f^{n}\times f^{n})(U)$ is again a coarse entourage
of $X$, and $f$ shifts $X$ to infinity in the sense that for every bounded
subset $B$ of $X$ there exists an $n$ in $\mathbb{N}$ such that $B\cap
f^{n}(X)=\emptyset$. A typical flasque space is $[0,\infty)\otimes X$ with $f$
defined by $f(n,x):=(n+1,x)$, where $[0,\infty)$ has the bornological coarse
structure induced from the inclusion into $\mathbb{R}_{d_{eu}}$. We say that
$E$ annihilates flasques if $E(X)\simeq 0$ for every flasque bornological
coarse space. Annihilation of flasques reflects a version of local finiteness
of $E$.

If $X$ is a $G$-bornological space with coarse structure $\mathcal{C}$, then
for any $U$ in $\mathcal{C}^{G}$ we can consider the $G$-bornological coarse
space $X_{U}$ with the smaller $G$-coarse structure generated by $U$. The
identity of $X$ induces maps $X_{U}\to X$ and $X_{U}\to X_{U^{\prime}}$ for
all $U^{\prime}$ in $\mathcal{C}^{G}$ with $U\subseteq U^{\prime}$. We say
that $E$ is $u$-continuous if the canonical map is an equivalence

$\mathrm{colim}_{U\in\mathcal{C}^{G}}E(X_{U})\stackrel{{\scriptstyle\simeq}}{{\to}}E(X)$

for every $G$-bornological coarse space $X$.

If we fix any compact object $M$ of $\mathbf{M}$, then we can form a coarsely
invariant $\mathbb{Z}$-graded group-valued functor
$X\mapsto\pi_{*}\mathrm{map}_{\mathbf{M}}(M,E(X))$ which is $u$-continuous and
vanishes on flasques. The excisiveness of $E$ gives rise to long exact Mayer-
Vietoris sequences for complentary pairs.

The classical examples of coarse homology theories are usually defined as
group valued functors on certain subcategories of $G\mathbf{BornCoarse}$. But
most of them can be extended to all of $G\mathbf{BornCoarse}$ and admit a
model in the sense defined above. We refer to [13] for coarse equivariant
ordinary homology (see also Section 6), to [13], [5] for coarse algebraic $K$
with coefficients in an additive category, to [17] and [6] for coarse
algebraic $K$-theory of spaces and coarse algebraic $K$ with coefficients in a
left-exact $\infty$-category, to [9] for equivariant coarse topological
$K$-theory (see also Section 7), and to [18] for equivariant coarse cyclic and
Hochschild homology. In the non-equivariant case every locally finite homology
theory admits a coarsification [37, Sec. 5.5], [12, Sec. 7], [11].

One can derive the notion of an equivariant coarse cohomology by dualizing the
axioms in 5.2, see [10]. For alternative sets of axioms, mainly for group-
valued functors, see e.g. [33], [43].

######  Example 5.3.

In order to demonstrate the usage of the axioms we do some calculations. Let
$E$ be an equivariant coarse homology theory. We first show that for any $X$
in $G\mathbf{BornCoarse}$ we have an equivalence

$E(\mathbb{R}_{d_{eu}}^{n}\otimes X)\simeq\Sigma^{n}E(X)\ .$ (6)

This can be shown by induction on $n$. The basic step uses the complementary
pair $((-\infty,0]\times X,([-n,\infty)\times X)_{n\in\mathbb{N}})$ on
$\mathbb{R}_{d_{eu}}\otimes X$. By excision we have a push-out square

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
55.16234pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{colim}_{n\in\mathbb{N}}E([-n,0]\otimes
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
79.16234pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.56667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
79.16234pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{E((-\infty,0]\times
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
116.50949pt\raise-29.56667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{colim}_{n\in\mathbb{N}}E([-n,\infty)\times
X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.30789pt\raise-40.06667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
89.30789pt\raise-40.06667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E(\mathbb{R}_{d_{eu}}\otimes
X)}$}}}}}}}\ignorespaces}}}}\ignorespaces\ .$

By coarse invariance the upper left corner is equivalent to $E(X)$, and the
lower left and upper right corners are zero since $E$ vanishes on flasques.
Hence this square is equivalent to

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
15.40277pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
61.10437pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.62222pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
61.10437pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.60437pt\raise-29.56667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
39.40277pt\raise-40.06667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.40277pt\raise-40.06667pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E(\mathbb{R}_{d_{eu}}\otimes
X)}$}}}}}}}\ignorespaces}}}}\ignorespaces\ .$

$\Box$

######  Example 5.4.

Let $G$ be trivial. We consider the subspace $Sq:=\\{n^{2}\mid
n\in\mathbb{N}\\}$ of $\mathbb{R}_{d_{eu}}$ of square integers in
$\mathbb{R}$. For every entourage $U$ we have a decomposition of $Sq_{U}$ into
a bounded connected subset and an infinite discrete subset. Let us assume that
$E$ is additive in the sense that $E(X_{min,min})\cong\prod_{X}E(*)$ (see [12,
Def. 6.4]) for any set $X$. All examples mentioned above have this additional
property. We then have

$E(Sq_{U})\simeq E(*)\oplus\prod_{i=n}^{\infty}E(*)$

for a suitable $n$. For $n$ in $\mathbb{N}$ we consider the map

$E(*)\oplus\prod_{i=n}^{\infty}E(*)\to E(*)\oplus\prod_{i=n+1}^{\infty}E(*)\
,\quad(a,(x_{i})_{i\in\\{n,\dots,\infty\\}})\mapsto(a+x_{n},(x_{i})_{i\in\\{n+1,\dots,\infty\\}})\
.$

Then using $u$-continuity of $E$ we get

$E(Sq)\simeq\mathrm{colim}_{n\in\mathbb{N}}\left(E(*)\oplus\prod_{i=n}^{\infty}E(*)\right)\simeq
E(*)\oplus\frac{\prod_{\mathbb{N}}E(*)}{\bigoplus_{\mathbb{N}}E(*)}\ .$

$\Box$

##  6 Equivariant coarse ordinary homology

A version of ordinary coarse (co)homology was first defined by Roe [35]. For
further constructions see [3], [39], [26]. Following [13, Sec. 7], in this
section we sketch the construction of the equivariant coarse homology functor

$H\mathcal{X}^{G}:G\mathbf{BornCoarse}\to D(\mathbb{Z})$

in the sense of 5.2. Let $X$ be a $G$-bornological coarse space. Let $U$ be a
coarse entourage of $X$ and $B$ be a bounded subset. A point
$(x_{0},\dots,x_{i})$ in $X^{n+1}$ is called $U$-controlled if
$(x_{i},x_{i^{\prime}})\in U$ for all $i,i^{\prime}$ in $\\{0,\dots,n\\}$. The
point meets $B$ if $x_{i}\in B$ for some $i$ in $\\{0,\dots,n\\}$.

We consider the group $C\mathcal{X}_{n,U}^{G}(X)$ of all $G$-invariant
$U$-controlled and locally finite functions $\phi:X^{n+1}\to\mathbb{Z}$. The
latter two properties require that the support of $\phi$ consists of
$U$-controlled points and that every bounded subset $B$ meets the support of
$\phi$ in a finite subset. We define the differential
$\partial:C\mathcal{X}_{n+1,U}^{G}(X)\to C\mathcal{X}_{n,U}^{G}(X)$ by the
usual formula

$\partial\phi(x_{0},\dots,x_{n}):=\sum_{i=0}^{n+1}(-1)^{i}\sum_{x\in
X}\phi(x_{0},\dots,x_{i-1},x,x_{i},\dots,x_{n})\ .$

The conditions on $\phi$ ensure that the sum has only finitely many non-zero
terms. For every invariant entourage $U$ we get a chain complex
$C\mathcal{X}^{G}_{U}(X)$ which we consider as an object of $D(\mathbb{Z})$.
If $X\to X^{\prime}$ is a proper map such that $(f\times f)(U)\subseteq
U^{\prime}$, then we get a map of chain complexes

$f_{*}:C\mathcal{X}^{G}_{U}(X)\to C\mathcal{X}^{G}_{U^{\prime}}(X^{\prime})\
,\quad(f_{*}\phi)(x_{0}^{\prime},\dots,x^{\prime}_{n}):=\sum\phi(x_{0},\dots,x_{n})$

where the sum is taken over the fibre of the map $f^{n+1}:X^{n+1}\to
X^{\prime,n+1}$. It again has finitely many non-zero terms. We then define

$H\mathcal{X}^{G}(X):=\mathrm{colim}_{U\in\mathcal{C}^{G}}C\mathcal{X}^{G}_{U}(X)$

in $D(\mathbb{Z})$. The construction is functorial in $X$.

######  Definition 6.1.

The functor $H\mathcal{X}^{G}:G\mathbf{BornCoarse}\to D(\mathbb{Z})$ is called
the equivariant coarse ordinary homology theory.

It satisfies the axioms from 5.2.

######  Example 6.2.

In the case of the trivial group we omit $G$ from the notation. By an explicit
calculation we have

$H\mathcal{X}_{k}(*)=\left\\{\begin{array}[]{cc}\mathbb{Z}&k=0\\\
0&\mbox{else}\end{array}\right.\ .$

By specializing (6) we get

$H\mathcal{X}_{k}(\mathbb{R}^{n}_{d_{eu}})\cong\left\\{\begin{array}[]{cc}\mathbb{Z}&k=n\\\
0&\mbox{else}\end{array}\right.$

$\Box$

######  Example 6.3.

For non-trivial groups equivariant coarse homology is related with group
homology via

$H\mathcal{X}^{G}(G_{can,min}\otimes S_{min,max})\simeq H(G,\mathbb{Z}[S])\ ,$

where $S$ is a $G$-set, $\mathbb{Z}[S]$ in $G\mathbf{Mod}(\mathbb{Z})$ is the
associated $G$-module, and $H(G,-)$ is the group homology functor
$G\mathbf{Mod}(\mathbb{Z})\to D(\mathbb{Z})$ [21, Prop. 3.8], [13, Prop. 7.5].
$\Box$

##  7 Equivariant coarse topological $K$-homology

Group-valued coarse topological $K$-homology for proper metric spaces is
usually defined in terms of the $K$-theory of Roe algebras [37], [28], [42].
In this section, following [9] we sketch the construction of a spectrum-valued
equivariant coarse $K$-homology theory in the sense of 5.2 using Roe
$C^{*}$-categories.

Let $X$ be a $G$-bornological coarse space. An $X$-controlled Hilbert space is
a triple $(H,\rho,p)$ of a Hilbert space $H$, a unitary representation
$\rho=(\rho_{g})_{g\in G}:G\to U(H)$, and a mutually orthogonal family of
finite dimensional orthogonal projections $p:=(p_{x})_{x\in X}$ on $H$ such
that $\sum_{x\in X}p_{x}=\mathrm{id}_{H}$ strongly, the set $\\{x\in X\mid
p_{x}\not=0\\}$ is locally finite (see 3.2), and such that
$\rho_{g}p_{x}\rho_{g}^{-1}=p_{gx}$ for all $g$ in $G$ and $x$ in $X$.

A controlled morphism between $X$-controlled Hilbert spaces
$A:(H,\rho,p)\to(H^{\prime},\rho^{\prime},p^{\prime})$ is a bounded operator
$A:H\to H^{\prime}$ such that $\rho^{\prime}_{g}A=A\rho_{g}$ for all $g$ in
$G$, and such that the set $\\{(x,y)\in X\times X\mid
p^{\prime}_{x}Ap_{y}\not=0\\}$ is a coarse entourage of $X$. We let
$C^{*}((H,\rho,p),(H^{\prime},\rho^{\prime},p^{\prime}))$ denote the closure
in $B(H,H^{\prime})$ of the set of controlled morphisms from $(H,\rho,p)$ to
$(H^{\prime},\rho^{\prime},p^{\prime})$.

We obtain a $C^{*}$-category $\mathbf{V}^{G}(X)$ [23], [7] whose objects are
the $X$-controlled Hilbert spaces, and whose morphism spaces are the spaces
$C^{*}((H,\rho,p),(H^{\prime},\rho^{\prime},p^{\prime}))$ with the obvious
involution given by taking the adjoint operator.

If $X$ itself is locally finite and $\dim(p_{x})=1$ for all $x$ in $X$, then
the endomorphism $C^{*}$-algebra

$C^{*}(H,\rho,p):=\mathrm{End}_{\mathbf{V}^{G}(X)}((H,\rho,p))$

is called the equivariant uniform Roe algebra associated to $X$.

If $f:X\to X^{\prime}$ is a morphism in $G\mathbf{BornCoarse}$, then we get a
functor

$\mathbf{V}^{G}(f):\mathbf{V}^{G}(X)\to\mathbf{V}^{G}(X^{\prime})$

which sends $(H,\rho,p)$ to $(H,\rho,f_{*}p)$ with
$(f_{*}p)_{x^{\prime}}:=\sum_{x\in f^{-1}(x^{\prime})}p_{x}$ and the morphism
$A$ in $C^{*}((H,\rho,p),(H^{\prime},\rho^{\prime},p^{\prime}))$ to the same
operator $A$, now considered as an element in
$C^{*}((H,\rho,f_{*}p),(H^{\prime},\rho^{\prime},f_{*}p^{\prime}))$.

We get a functor

$\mathbf{V}^{G}:G\mathbf{BornCoarse}\to C^{*}\mathbf{Cat}\ .$

We now use the spectrum-valued $K$-theory functor for $C^{*}$-categories

$K:C^{*}\mathbf{Cat}\to\mathbf{Sp}\ ,$ (7)

see [29], [7] for details.

######  Definition 7.1.

We define the equivariant coarse topological $K$-homology functor by

$K\mathcal{X}^{G}:=K\circ\mathbf{V}^{G}:G\mathbf{BornCoarse}\to\mathbf{Sp}\ .$
(8)

By [9, Thm. 7.3] this functor is an equivariant coarse homology theory in the
sense of 5.2. On bornological coarse spaces associated to proper metric spaces
with isometric group actions the corresponding group valued functor coincides
with the classical construction of equivariant coarse $K$-homology [37], see
[8, Th. 6.1].

######  Example 7.2.

By an explicit calculation we get

$K\mathcal{X}(*)\simeq KU\ ,$

where $KU$ denotes the complex $K$-theory spectrum. By specializing (6) we get

$K\mathcal{X}(\mathbb{R}^{n}_{d_{eu}})\simeq\Sigma^{n}KU\ .$ (9)

$\Box$

######  Example 7.3.

The functor

$K^{G}:G\mathrm{Orb}\to\mathbf{Sp}\ ,\quad S\mapsto
K\mathcal{X}^{G}(S_{can,min}\otimes G_{can,min})$ (10)

is the functor constructed in [20]. Its values are given by $K^{G}(G/H)\simeq
K(C^{*}_{r}(H))$ [9, Prop. 9.2.3], where $C_{r}^{*}(H)$ is the reduced group
$C^{*}$-algebra of the subgroup $H$ of $G$. The functor $K^{G}$ features the
left-hand side of the Baum-Connes/Davis-Lück assembly map

$\mathrm{colim}_{G_{{\mathcal{F}in}}\mathrm{Orb}}K^{G}\to K^{G}(*)\ ,$ (11)

(see [30], [15] for the comparison of the two versions of assembly maps),
where $G_{{\mathcal{F}in}}\mathrm{Orb}$ is the full subcategory of
$G\mathrm{Orb}$ of transitive $G$-sets with finite stabilizers. Coarse
geometry can be used to show that this assembly map is split injective. For
example it is known by [40] that the assembly map (11) is split injective if
$G_{can,min}$ admits a coarse embedding into a Hilbert space. In [14] a
completely different argument (axiomatizing [25]) applies coarse geometry to
show split-injectivity of (11) and its versions for functors on the orbit
category derived from other coarse homology theories as in (10) under finite
decomposition complexity assumptions on $G$. $\Box$

######  Example 7.4.

Combining 4.3 and 7.3 for $S=*$ we get an equivalence

$K\mathcal{X}^{G}(\tilde{M}_{d_{\tilde{g}}})\simeq K(C^{*}_{r}(G))\ .$

$\Box$

######  Example 7.5.

The equivariant coarse $K$-homology naturally captures the index of
equivariant Dirac type operators on complete Riemannian manifolds $(M,g)$ with
a proper action of $G$ by isometries. Assume that $M$ admits an equivariant
spin structure and let $D/\hskip 2.27626pt$ denote the associated Dirac
operator. As observed in [38], [8] it then has a well-defined index class

$\mathrm{index}\mathcal{X}(D/\hskip
2.27626pt)\>\>\mbox{in}\>\>K\mathcal{X}^{G}_{-\dim(M)}(M_{d_{g},s_{g}})\ .$

The appearance of the bornology $\mathcal{B}_{s_{g}}$ (see 3.10) reflects the
fact that the index class is essentially supported away from the subset of $M$
where the scalar curvature is positive.

If $G$ is trivial and $s_{g}$ admits a uniform positive lower bound outside of
a compact subset, then $D/\hskip 2.27626pt$ is Fredholm, and its Fredholm
index $\mathrm{index}(D/\hskip 2.27626pt)$ in $K_{-\dim(M)}(\mathbb{C})$ can
be expressed using the functoriality of the coarse $K$-homology as
$\mathrm{index}(D/\hskip 2.27626pt)=p_{*}\mathrm{index}\mathcal{X}(D/\hskip
2.27626pt)$ in $K\mathcal{X}_{-\dim(M)}(*)\cong\pi_{-\dim(M)}KU$, where
$p:M_{d_{g},s_{g}}\to*$ is the projection. $\Box$

######  Example 7.6.

Recall the Higson corona defined in 4.7. For simplicity we consider the case
of the trivial group.

######  Proposition 7.7.

There exists a binatural pairing

$K(\partial X)\otimes K\mathcal{X}(X)\to\Sigma KU\ .$

In order to construct this pairing we let
$\mathbf{Hilb}(\mathbb{C})/\mathbf{Hilb}_{c}(\mathbb{C})$ be the Calkin
$C^{*}$-category whose objects are Hilbert spaces, and whose morphisms are the
quotients of bounded by compact operators. Following [34] there is a functor

$C(\partial
X)\otimes\mathbf{V}(X)\to\mathbf{Hilb}(\mathbb{C})/\mathbf{Hilb}_{c}(\mathbb{C})$
(12)

(the domain is a tensor product of $C^{*}$-categories [16, Sec. 7]) which
sends the object $(H,p)$ to $H$ and the morphism $[\phi]\otimes
A:(H,p)\to(H^{\prime},p^{\prime})$ to $[Am(\phi)]:H\to H^{\prime}$ in
$\mathbf{Hilb}(\mathbb{C})/\mathbf{Hilb}_{c}(\mathbb{C})$. Here $\phi$ in
$C_{h}(X)$ denotes a representative of the class $[\phi]$ in the quotient (5),
$m(\phi)=\sum_{x\in X}\phi(x)p_{x}$ is strongly convergent in $B(H)$, and
$[Am(\phi)]$ denotes the class of $Am(\phi):H\to H^{\prime}$ in the Calkin
category. Using $K(\partial X):=K(C(\partial X))$, the symmetric monoidal
structure of the $K$-theory functor (7) and (8) for the first map, and (12)
for the second, the pairing is given by the composition

$K(\partial X)\otimes K\mathcal{X}(X)\to K(C(\partial
X)\otimes\mathbf{V}(X))\to
K(\mathbf{Hilb}(\mathbb{C})/\mathbf{Hilb}_{c}(\mathbb{C}))\simeq\Sigma KU\ .$

The pairing of coarse index classes of Dirac type operators on complete
Riemannian manifolds with $K$-theory classes from the Higson corona can be
expressed in terms of Callias-type operators [4]. In order to exhibit a non-
trivial example we consider the natural map
$p:\partial\mathbb{R}^{n}_{d_{eu}}\to S^{n-1}$. If we choose a generator
$o_{S^{n-1}}$ of the reduced $K$-group
$\tilde{K}^{n-1}(S^{n-1})\cong\mathbb{Z}$, then the pairing

$p^{*}o_{S^{n-1}}\otimes-:K\mathcal{X}_{n}(\mathbb{R}^{n}_{d_{eu}})\to\pi_{1}\Sigma
KU\cong\mathbb{Z}$

is an isomorphism reflecting the equivalence (9). $\Box$

##  8 Summary

In this survey we introduced the category of $G$-bornological coarse spaces as
a basic framework for coarse geometry. We explained the concepts of a coarse
equivalence and of an equivariant coarse homology theory. We gave complete
constructions of equivariant coarse ordinary homology and equivariant coarse
$K$-homology and indicated some basic calculations.

### Keywords:

coarse space, bornological coarse space, coarse equivalence, coarse homology
theory, Higson corona, coarse ordinary homology, controlled Hilbert space, Roe
algebra, Roe category, coarse $K$-homology, canonical coarse structure on a
group, coarse index class, assembly map

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