# Coarse cohomology of configuration space and coarse embedding

Arka Banerjee  Department of Mathematics and Statistics
Auburn University
Auburn, AL 36849<EMAIL_ADDRESS>
###### Abstract.

We introduce a notion of equivariant coarse cohomology of the complement of a
subspace in a metric space. We use this cohomology to define a notion of
coarse cohomology of the configuration space of a metric space and develop
tools to compute this cohomology under various conditions. As an application
of this theory, we show that certain classes in the coarse cohomology of
configuration space obstruct coarse embedding between two metric spaces.

##  1\. Introduction

Van Kampen [VK33] developed an obstruction theory for embeddings of
$n$-dimensional simplicial complexes into $\mathbb{R}^{2n}$. A modern approach
to his theory uses (co)homology of the configuration space. In this article,
we develop an analogous obstruction theory for coarse embedding by introducing
a notion of coarse cohomology of configuration space.

Let us first briefly describe the classical van Kampen obstruction for a
topological space $X$. Let $\delta(X)$ be the diagonal set $\\{(x,x)\mid
x\overset{}{\in}X\\}\overset{}{\subset}X\times X$. Consider the following
deleted product

$\tilde{X}:=X\times X-\delta(X)=\\{(x,y)\overset{}{\in}X\times X\mid x\neq
y\\}$

$\tilde{X}$ has a natural free action by $\mathbb{Z}_{2}$ by switching the
coordinates. Consider the corresponding $\mathbb{Z}_{2}$ covering map
$q:\tilde{X}\rightarrow\tilde{X}/\mathbb{Z}_{2}$. The space
$\tilde{X}/\mathbb{Z}_{2}$ which we denote by $\operatorname{Conf}(X)$, is the
unordered configuration space of two points in $X$. There exists a classifying
map from the $\mathbb{Z}_{2}$-bundle
$q:\tilde{X}\rightarrow\tilde{X}/\mathbb{Z}_{2}$ to the universal
$\mathbb{Z}_{2}$-bundle $S^{\infty}\rightarrow\mathbb{R}P^{\infty}$ as
follows.

${\tilde{X}}$${S^{\infty}}$${\operatorname{Conf}(X)}$${\mathbb{R}P^{\infty}}$$\scriptstyle{\tilde{\phi}}$$\scriptstyle{q}$$\scriptstyle{\phi}$

If there is an embedding $g:X\hookrightarrow\mathbb{R}^{m}$, then we can
choose $\tilde{\phi}$ so that it factors through $S^{m-1}$. More precisely, we
can choose $\tilde{\phi}$ to be the following map $\tilde{X}\rightarrow
S^{m-1}\overset{}{\subset}S^{\infty}$

$(x,y)\mapsto\frac{g(x)-g(y)}{|g(x)-g(y)|}$

In this case, $\phi$ maps $\operatorname{Conf}(X)$ to
$\mathbb{R}P^{m-1}\overset{}{\subset}\mathbb{R}P^{\infty}$.

So the induced map
$\phi^{*}:\operatorname{H}^{m}(\mathbb{R}P^{\infty})\rightarrow\operatorname{H}^{m}(\operatorname{Conf}(X))$
is trivial as it factors through $\operatorname{H}^{m}(\mathbb{R}P^{m-1})$. In
particular, if
$\eta^{m}\overset{}{\in}\operatorname{H}^{m}(\mathbb{R}P^{\infty};\mathbb{Z}_{2})$
denotes the nonzero class, then $\phi^{*}(\eta^{m})$ would be trivial. In
other words, the cohomology class $\phi^{*}(\eta^{m})$ gives an obstruction
for the embedding of $X$ into $\mathbb{R}^{m}$. We will call the class
$\phi^{*}(\eta^{m})$ the van Kampen obstruction class of degree $m$ and denote
it by $vk^{m}(X)$.

Let us now turn our attention to the coarse world. A map $f:X\rightarrow Y$
between two metric spaces is said to be a _coarse embedding_ if there exist
two proper non-decreasing maps
$\rho_{-},\rho_{+}:[0,\infty)\rightarrow[0,\infty)$ such that

$\rho_{-}(d(x,y))\leq d(f(x),f(y))\leq\rho_{+}(d(x,y))\quad\text{for all}\quad
x,y\overset{}{\in}X$

Roe [r93] defined the notion of coarse cohomology of a metric space which can
be thought of as a coarse analog of Alexander–Spanier cohomology in the
topological setting. The main motivation of this paper is to define coarse
$vk^{*}$ that obstructs coarse embedding between metric spaces.

The study of obstruction to the coarse embedding of finitely generated groups
into proper contractible $n$-manifold was initiated by Bestvina, Kapovich, and
Kleiner [BKK]. For several interesting classes of metric spaces (for example
CAT(0) space, Gromov-hyperbolic space) one can attach a boundary at infinity
to the space. Suppose $\partial_{\infty}X$ denotes such boundary of a space
$X$. A popular theme in the study of large scale geometry is to find
topological properties of $\partial_{\infty}X$ that provide information about
the coarse geometry of $X$. A special case of the work of
Bestvina–Kapovich–Kleiner [BKK] roughly says the following: if
$\partial_{\infty}X$ does not embed into $\mathbb{R}^{n}$ due to the van
Kampen embedding obstruction, then $X$ does not coarsely embed into
$\mathbb{R}^{n+1}$ with the Euclidean metric. One of the motivations behind
the present work was to understand their obstruction in the language of Roe’s
coarse cohomology.

To this end, recall that the van Kampen obstruction class $vk^{*}(X)$ lives in
the cohomology of the configuration space of $X$, which is the same as the
$\mathbb{Z}_{2}$-equivariant cohomology of $X\times X-\delta(X)$ where the
coefficients are $\mathbb{Z}_{2}$ with the trivial $\mathbb{Z}_{2}$ action.
This suggests that a coarse version of van Kampen obstruction class should
live in some coarse version of $\mathbb{Z}_{2}$-equivariant cohomology of
complement of $\delta(X)$ in $X\times X$. This motivates us to define a notion
of equivariant coarse cohomology of the complement of subspace in a metric
space.

Building on Roe’s theory, authors in [BB20] defined a notion of coarse
cohomology of complement of a subspace in a metric space. In this paper, we
extend the work in [BB20] to the equivariant setting. We then define the
coarse cohomology of the configuration space of $X$ simply to be the
$\mathbb{Z}_{2}$-equivariant coarse cohomology of the complement of
$\delta(X)$ in $X\times X$, where the action of $\mathbb{Z}_{2}$ on $X\times
X$ is by switching coordinate.

Once we have a proper notion of the coarse cohomology of the configuration
space, we can search for a coarse $vk^{*}$ in that cohomology that obstructs
coarse embeddings. Indeed, when $X$ is a separable metric space, we find a
class in the $n^{th}$ degree of the coarse cohomology of the configuration
space of $X$, which we denote by $cvk^{n}(X)$, that obstructs coarse embedding
of $X$ into $\mathbb{R}^{n-1}$. In general, the class $cvk^{n}(X)$ can be used
to get obstruction to coarse embedding into any other metric space. We define
the coarse obstruction dimension $\operatorname{cobdim}(X)$ of a separable
metric space $X$ to be the largest $n$ such that $cvk^{n}(X)$ is nonzero. One
of our main theorem is the following:

######  Theorem 1.1.

If $X$ admits a coarse embedding into $Y$, then
$\operatorname{cobdim}(X)\leq\operatorname{cobdim}(Y)$.

So one way to determine when a given space does not admit coarse embedding
into another space is to compare their $\operatorname{cobdim}$. However, the
equivariant coarse cohomology of the complement where $cvk^{*}$ lives, is hard
to compute in general. A big part of this paper is devoted to the computation
of equivariant coarse cohomology of the complement for certain spaces which
may be of independent interest. We use these computations to estimate
$\operatorname{cobdim}$ of certain spaces and obtain coarse embedding
obstruction between certain classes of spaces. Below we highlight some of our
results in this direction.

  * •

We show that $\operatorname{cobdim}(\mathbb{R}^{n})=n$. Hence Theorem 1.1
implies that $X$ does not admit coarse embedding into $\mathbb{R}^{n-1}$ if
$\operatorname{cobdim}(X)\geq n$.

  * •

Suppose $X=K\times[0,\infty)/K\times\\{0\\}$ is the open cone on a finite
simplicial complex $K$. A metric $d$ on $X$ is called expanding if for any two
disjoint simplices $\sigma,\tau\overset{}{\in}K$ and $S\geq 0$, there exists
$r\geq 0$ such that $d(\sigma\times[r,\infty),\tau\times[r,\infty))\geq S$. If
there is a class
$c\overset{}{\in}\operatorname{H}_{n}(\operatorname{Conf}(X))$ such that
$vk^{n}(X)(c)\neq 0$, then we show that $\operatorname{cobdim}(X)\geq n+1$
whenever $X$ is equipped with a proper, expanding metric (Example 7.2). Hence
such $X$ does not admit coarse embedding into $\mathbb{R}^{n}$ by Theorem 1.1
and the previous example. This was initially proved by
Bestvina–Kapovich–Kleiner [BKK].

  * •

If $X$ is a proper, uniformly acyclic $n$-manifold with uniformly locally
acyclic boundary, then $\operatorname{cobdim}(X)\leq n$ (Theorem 9.12).
Examples of such spaces include universal cover of compact aspherical
$n$-manifolds. Hence Theorem 1.1 implies that, if
$\operatorname{cobdim}(X)\geq n$, then $X$ does not admit coarse embedding
into the universal cover of any compact aspherical $(n-1)$-manifold.

  * •

If $\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=0$ for $*\leq n-1$, then
$\operatorname{cobdim}(X)\geq n$ (Theorem 10.1). This implies, in particular,
that any proper, uniformly contractible $n$-manifold has
$\operatorname{cobdim}\geq n$. Hence it follows from the last example that any
proper, uniformly contractible $n$-manifold does not admit coarse embedding
into the universal cover of any compact aspherical manifold (see Corollary
10.2 for a more general version). This recovers a result of Yoon [Yoon].

  * •

If a finitely generated group $G$ acts properly on $X$ by isometries, then
there exists a coarse embedding of $G$ (equipped with a word metric coming
from a finite generating set) into $X$ by mapping $G$ into an orbit of the
action. That means, from the coarse point of view, any space $X$ with a proper
$G$-action has to be at least as large as $G$. More precisely, using Theorem
1.1 we show that if $G$ acts properly, cocompactly on a contractible manifold
$M$ (possibly with boundary), then $\dim(M)\geq\operatorname{cobdim}(G)$
(Corollary 9.15).

######  Overview.

In section 2 we describe several variations of Alexander–Spanier cochains and
the corresponding cohomology theories. We also introduce coarse language and
define coarse (co)homology. In section 3, we define the equivariant coarse
cohomology of the complement and give some examples using Theorem 3.5 that
relate, for certain cases, equivariant coarse cohomology to the
Alexander–Spanier cohomology of the quotient. In section 4 we prove Theorem
3.5. In section 5, we define the coarse cohomology of the configuration space.
In section 6, we give a class in the coarse cohomology of the configuration
space that obstructs coarse embedding maps. Then we define the coarse
obstruction dimension of a space and prove Theorem 6.3 which is a slightly
stronger version of Theorem 1.1. In section 7, we give a relation between
classical van Kampen obstruction and coarse van Kampen obstruction. We use
this relation to compute coarse van Kampen obstruction for certain Euclidean
Cones on simplicial complexes. In section 8, we produce a coarse version of
the Gysin sequence for the $\mathbb{Z}_{2}$-bundle to compute the coarse
cohomology of configuration space. In section 9 and 10, we use the coarse
Gysin sequence to estimate $\operatorname{cobdim}$ of certain spaces.

######  Acknowledgment.

Part of this paper was written when the author was a graduate student in the
University of Wisconsin–Milwaukee. The author would like to thank his advisor
Boris Okun for his insights and advice during the preparation of this paper.
The author would also like to thank Kevin Schreve for helpful conversations.

##  2\. Preliminaries

### Alexander–Spanier complexes

We will refer to points in $X^{n+1}$ as _$n$ -simplices_. Let $R$ be a module
over some ring with unity. We will think of functions $X^{n+1}\to R$ as
$n$-cochains or $n$-chains on $X$, depending on the context, and in the latter
case will use additive notation
$c=\sum_{\sigma\overset{}{\in}X^{*+1}}c_{\sigma}\sigma$.

The basic complex is the complex of finitely supported chains

$\operatorname{C}_{*}(X;R):=\\{c=\sum_{i=0}^{n}c_{i}\sigma_{i}\mid
c_{i}\overset{}{\in}R\text{ and }\sigma_{i}\overset{}{\in}X^{*+1}\\}$

equipped with the usual boundary map, defined on the basis by

$\partial(x_{0},\cdots,x_{n}):=\sum_{i=0}^{n}(-1)^{i}(x_{0},\dots,\hat{x}_{i},\dots,x_{n}).$

Note that the boundary map $\partial$ is well-defined on a larger complex of
_locally finite_ chains $\operatorname{C}_{*}^{lf}(X;R)$ which consists of
chains $c$ satisfying the following property: for any bounded
$B\overset{}{\subset}X$ only finitely many simplices in $c$ have vertices in
$B$.

The algebraic dual of $\operatorname{C}_{*}(X)$ is the complex of all
Alexander–Spanier cochains

$\operatorname{C}^{*}(X;R)=\\{\phi:X^{*+1}\to R\\}$

with the coboundary operator

$d(\phi)(x_{0},\dots,x_{n})=\sum_{i=0}^{n}(-1)^{i}\phi(x_{0},\dots,\hat{x}_{i},\dots,x_{n}).$

These are acyclic complexes. All the cochain complexes that we will consider
in this paper would be subcomplexes of the complex $(\operatorname{C}^{*},d)$.

For a cochain $\phi\overset{}{\in}\operatorname{C}^{*}(X)$, let $||\phi||$ be
the intersection of the diagonal $\Delta=\\{(x,x,\ldots,x)\mid
x\overset{}{\in}X\\}\overset{}{\subset}X^{*+1}$ and the closure of the support
of the function $\phi:X^{*+1}\rightarrow M$. Let
$\operatorname{C}^{*}_{0}(X;R)$ be the complex of locally zero cochains:

$\operatorname{C}^{*}_{0}(X;R)=\\{\phi\overset{}{\in}\operatorname{C}^{*}(X;R)\mid\lVert\phi\rVert=\emptyset\\}.$

Note that the restriction of $d$ gives a well defined map
$\operatorname{C}^{*}_{0}(X)\rightarrow\operatorname{C}^{*+1}_{0}(X)$.
Consequently, $d$ induces a well defined map
$\operatorname{C}^{*}_{as}(X;M)\rightarrow\operatorname{C}^{*+1}_{as}(X;R)$,
where

$\operatorname{C}^{*}_{as}(X;R)=\operatorname{C}^{*}(X;R)/\operatorname{C}^{*}_{0}(X;R).$

Alexander–Spanier cohomology, denoted by $\operatorname{H}^{*}(X;M)$, is the
cohomology of the complex $(\operatorname{C}^{*}_{as}(X;M),d)$.

### Coarse inclusion

We adopt the notation from [msw11]. Let $(X,d)$ be a metric space. For
$A\overset{}{\subset}X$ denote $N_{r}(A)=\\{x\overset{}{\in}X\mid
d(x,A)<r\\}$. We will call such neighborhoods _metric neighborhoods_ of $A$.
We will say that $A$ is _$r$ -contained_ in $B$, $A\overset{r}{\subset}B$, if
$A\overset{}{\subset}N_{r}(B)$. $A$ is _coarsely contained_ in $B$,
$A\overset{\mathrm{c}}{\subset}B$, if $A\overset{r}{\subset}B$ for some $r$.
Two subsets are _coarsely equal_ , $A\mathbin{\overset{\mathrm{c}}{=}}B$ if
$A\overset{\mathrm{c}}{\subset}B$ and $B\overset{\mathrm{c}}{\subset}A$.

### Coarse intersection

Now we recall from [msw11] the notion of _coarse intersection_ :
$A\mathbin{\overset{\mathrm{c}}{\cap}}B\mathbin{\overset{\mathrm{c}}{=}}C$ if
for all sufficiently large $r$, $N_{r}(A)\cap
N_{r}(B)\mathbin{\overset{\mathrm{c}}{=}}C$. The coarse intersection is not
always well-defined, it may happen that the coarse type of $N_{r}(A)\cap
N_{r}(B)$ does not stabilize as $r$ goes to infinity. However the notion
“coarse intersection is coarsely contained in” is well-defined.
$A\mathbin{\overset{\mathrm{c}}{\cap}}B\overset{\mathrm{c}}{\subset}C$ means
that for any $r$, $N_{r}(A)\cap N_{r}(B)\overset{\mathrm{c}}{\subset}C$. It is
not hard to see that this condition is equivalent to the condition that for
any $r$, $A\cap N_{r}(B)\overset{\mathrm{c}}{\subset}C$.

### Coarse (co)homology

In what follows, we will need to measure distances between simplices of
different dimensions. A convenient way to do this is to stabilize simplices by
repeating the last coordinate, as follows. Denote by $X^{\infty}$ the subset
of the product of countably many copies of $X$, consisting of eventually
constant sequences. Equip $X^{\infty}$ with the $\sup$ metric. Let
$i:X^{n+1}\to X^{\infty}$ denote the map
$(x_{0},\dots,x_{n})\mapsto(x_{0},\dots,x_{n},x_{n},x_{n},\dots)$. For a
function $\phi:X^{n+1}\to R$ define its stabilized support

$\lvert\phi\rvert=\\{i(\sigma)\mid\sigma\overset{}{\in}X^{n+1}\text{ and
}\phi(\sigma)\neq 0\\}\overset{}{\subset}X^{\infty}.$

Let $\Delta=i(X)$ denote the diagonal of $X^{\infty}$. Define the support of
$\phi$ at scale $r$ to be $\lvert\phi\rvert_{r}=\lvert\phi\rvert\cap
N_{r}(\Delta)$.

We now define coarse (co)homology theories, following Roe [r93] and Hair [h10]
using our language. The coarse cochain complex is

$\operatorname{C}{\mathrm{X}}^{*}(X;R):=\\{\phi\overset{}{\in}\operatorname{C}^{*}(X;R)\mid\lvert\phi\rvert_{r}\text{
is bounded for all }r\\}.$

An equivalent way to define coarse cochains is to require the support to be
coarsely disjoint from the diagonal:

$\operatorname{C}{\mathrm{X}}^{*}(X;R)=\\{\phi\overset{}{\in}\operatorname{C}^{*}(X;R)\mid\lvert\phi\rvert\mathbin{\overset{\mathrm{c}}{\cap}}\Delta\mathbin{\overset{\mathrm{c}}{=}}*\\}.$

The coboundary operator $d$ preserves this property, and the coarse cohomology
$\operatorname{H}{\mathrm{X}}^{*}(X;R)$ is defined to be the cohomology of the
complex $(\operatorname{C}{\mathrm{X}}^{*}(X;R).d)$. The coarse homology
$\operatorname{H}{\mathrm{X}}_{*}(X;G)$ is the homology of the following
subcomplex of $\operatorname{C}_{*}^{lf}(X;G)$

$\operatorname{C}{\mathrm{X}}_{*}(X;R):=\\{c\overset{}{\in}\operatorname{C}_{*}^{lf}(X;R)\mid\lvert
c\rvert\overset{\mathrm{c}}{\subset}\Delta\\}$

equipped with the restriction of the boundary operator $\partial$. Note that
in the presence of the support condition $\lvert
c\rvert\overset{\mathrm{c}}{\subset}\Delta$ local finiteness is equivalent to
$\lvert c\rvert$ having finite intersections with bounded subsets of
$X^{\infty}$.

######  Example 2.1.

Roe [r93] showed that the coarse cohomology is isomorphic to the compactly
supported Alexander–Spanier cohomology if the space is uniformly contractible.
In particular, this applies to the universal cover of finite aspherical
complexes. In this case, the coarse homology is isomorphic to the locally
finite homology [r93]*Chapter 2. For example,

$\operatorname{H}{\mathrm{X}}_{*}(\mathbb{R}^{n};R)=\operatorname{H}{\mathrm{X}}^{*}(\mathbb{R}^{n};R)=\begin{cases}R&*=n,\\\
0&\text{otherwise}.\end{cases}$

######  Remark 2.2.

We will need coarse homology only in Section 9 for a brief discussion of
coarse PD($n$) spaces. For the rest of the paper, we will work with coarse
cohomology.

##  3\. Equivariant coarse cohomology of the complement

We start by briefly reviewing the notion of coarse cohomology of the
complement from [BB20]. Roughly the idea is that the coarse complement of a
subset $A$ in $X$ is determined by the collection of subsets $\mathcal{S}$ of
$X$ which are coarsely disjoint from $A$:

$\mathcal{S}:=\\{B\overset{}{\subset}X\mid N_{r}(B)\cap N_{r}(A)\text{ is
bounded for any $r\geq 0$}\\}$

Coarse cohomology of the complement of $A$, denoted by
$\operatorname{H}{\mathrm{X}}^{*}(X-A)$, is defined to be the cohomology of
the following complex with the usual coboundary operator $d$ mentioned in the
previous section.

$\operatorname{C}{\mathrm{X}}^{n}(X-A;R)=\\{\phi\overset{}{\in}\operatorname{C}^{n}(X;R)\mid\phi|_{B}\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(B;R)\text{
for all }B\overset{}{\in}\mathcal{S}\\}$

Recall that $i:X\to X^{\infty}$ denote the map $x\mapsto(x,\dots,x,\dots)$ and
$\Delta=i(X)$. For a subset $A\overset{}{\subset}X$, we denote the set
$i(A)\overset{}{\subset}\Delta$ by $\Delta_{A}$. For our purpose, we will work
with the following equivalent description of
$\operatorname{C}{\mathrm{X}}^{*}(X-A)$ (see [BB20], Lemma 3.2), which is
closer to the spirit of Roe’s original definition of coarse cochains discussed
in the last section.

$\operatorname{C}{\mathrm{X}}^{n}(X-A;R)=\\{\phi\overset{}{\in}\operatorname{C}^{n}(X;R)\mid\lvert\phi\rvert\mathbin{\overset{\mathrm{c}}{\cap}}\Delta\overset{\mathrm{c}}{\subset}\Delta_{A}\\}$

######  Example 3.1.

When $X\mathbin{\overset{\mathrm{c}}{=}}A$, we can see that
$\operatorname{C}{\mathrm{X}}^{*}(X-A;R)$ coincides with
$\operatorname{C}^{*}(X;R)$ which is an acyclic complex. So in this case
$\operatorname{H}{\mathrm{X}}^{*}(X-A;R)$ is trivial in all degrees except at
degree $0$ where it is isomorphic to $R$.

######  Example 3.2.

If $A$ is bounded, then $\operatorname{C}{\mathrm{X}}^{*}(X-A;R)$ coincides
with $\operatorname{C}{\mathrm{X}}^{*}(X;R)$. Hence,
$\operatorname{H}{\mathrm{X}}^{*}(X-A,R)=\operatorname{H}{\mathrm{X}}^{*}(X;R)$
whenever $A$ is bounded.

######  Example 3.3.

As explained in [BB20]*Theorem 5.9, the elements of
$\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}_{2})$ correspond to the
‘coarse complementary components’ of $A$ inside $X$. More precisely, if $X$ is
a geodesic space and $A\overset{}{\subset}X$ and
$k=\dim_{\mathbb{Z}/2}\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$ is
finite, then there exists $r$ such that for any $R\geq r$, $X-N_{R}(A)$ has
exactly $k+1$ deep path components, where deep path components are those path
components which are not coarsely contained in $A$.

### Equivariant coarse cohomology of the complement

We now define an equivariant version of the coarse cohomology of the
complement. Let $G$ be a group acting on $X$ by isometries and $R$ be a
$G$-module. Suppose $G$ acts on $X^{n+1}$ by the diagonal action,
$g(x_{0},\dots,x_{n}):=(gx_{0},\dots,gx_{n})$. $G$-equivariant coarse
cohomology of the complement of $A$ in $X$, denoted by
$\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R)$, is defined to be the
cohomology of the following cochain complex

$\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R):=\\{\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(X-A;R)\mid\text{$\phi$
is $G$-equivariant}\\}$

with the usual coboundary operator $d$.

In particular, if $G$ is acting trivially on $X$ and $R$, then
$\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R)$ coincides with
$\operatorname{C}{\mathrm{X}}^{*}(X-A;R)$. In general
$\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{*}$ is hard to compute. We can
relate $\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{*}$ to a more computable
cohomology under certain acyclicity conditions which we define below.

######  Definition 3.4.

$X$ is called locally acyclic away from $A$ with coefficients in $R$ if there
exists an $r$ such that for any $x\overset{}{\in}X-N_{r}(A)$ and any open
neighborhood $U$ of $x$, there exists another open neighborhood of
$V\overset{}{\subset}U$ of $x$ such that the inclusion $V\hookrightarrow U$
induces trivial map in the singular homology with coefficients in $R$.

$X$ is uniformly acyclic with coefficients in $R$ if there exists a non-
decreasing function $\rho:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$
such that for any $B\overset{}{\subset}X$ the inclusion map $B\hookrightarrow
N_{\rho(\operatorname{\text{diam}}(B))}(B)$ induces trivial map in the
singular homology with coefficients in $R$.

$X$ is called uniformly acyclic away from $A\overset{}{\subset}X$ with
coefficients in $R$ if there exist two non-decreasing functions
$\rho,\mu:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ such that for any
$B\overset{}{\subset}X$ the inclusion map $B\hookrightarrow
N_{\rho(\operatorname{\text{diam}}(B))}(B)$ induces trivial map in the
singular homology with coefficients in $R$ if $d(B,A)\geq\mu(r)$.

For the rest of the paper $G\curvearrowright X$ will mean that $G$ is acting
on $X$ by isometries and $G\curvearrowright(X,A)$ will mean $G\curvearrowright
X$ and $GA=A$. We let

$\operatorname{Fix_{\mathrm{G}}}(X):=\\{x\overset{}{\in}X\mid gx=x\text{ for
all }g\overset{}{\in}G\\}.$

We now state a theorem that relates the equivariant coarse cohomology of the
complement and Alexander–Spanier cohomology for certain spaces.

######  Theorem 3.5.

Suppose $G\curvearrowright(X,A)$ and
$\operatorname{Fix_{\mathrm{G}}}(X)\neq\emptyset$. Let $R$ be an abelian group
with trivial $G$-action. Suppose $X$ is uniformly acyclic away from
$A\overset{}{\subset}X$ with coefficients in $R$ and locally acyclic away from
$A$ with coefficients in $R$.

  1. (1)

If $X\overset{\mathrm{c}}{\subset}A$, then

$\displaystyle\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R)=\begin{cases}R&\text{if
$*=0$},\\\ 0&\text{otherwise}\end{cases}$

  2. (2)

If $X$ is not coarsely contained in $A$, then

$\displaystyle\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R)=\begin{cases}0&\text{if
$*=0$},\\\
\varinjlim\tilde{\operatorname{H}}^{*-1}((X-N_{r}(A))/G;R)&\text{otherwise}\end{cases}$

where ${\tilde{\operatorname{H}}^{*}}(-)$ is the reduced Alexander–Spanier
cohomology.

We will defer the proof of Theorem 3.5 to the next section. We will finish
this section with an example.

######  Example 3.6.

Consider the action of $\mathbb{Z}_{2}$ on $\mathbb{R}^{n}$ by antipodal map.
Let $M$ be a codimension $k$ vector subspace. Then for any
$\mathbb{Z}_{2}$-module $R$ with trivial action and $i\geq 1$, we have

$\displaystyle\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{i}(\mathbb{R}^{n}-M;R)$
$\displaystyle=\varinjlim\tilde{\operatorname{H}}^{i-1}((\mathbb{R}^{n}-N_{r}(M))/\mathbb{Z}_{2};R)\quad[\text{by
Theorem }\ref{c:equicoa=coaquo}]$
$\displaystyle=\tilde{\operatorname{H}}^{i-1}((\mathbb{R}^{n}-M)/\mathbb{Z}_{2};R)$
$\displaystyle=\tilde{\operatorname{H}}^{i-1}(\mathbb{R}P^{k-1};R)$

The second equality holds because $\mathbb{R}^{n}-N_{r}(M)$ is
$\mathbb{Z}_{2}$-equivariantly homotopic to $\mathbb{R}^{n}-M$ and the last
equality follows because $\mathbb{R}^{n}-M$ is $\mathbb{Z}_{2}$-equivariantly
homotopic to $S^{k-1}$ with the antipodal $\mathbb{Z}_{2}$-action.

##  4\. Computation of equivariant coarse cohomology

In this section, our main goal is to prove Theorem 3.5. The proof is similar
to the proof of a non-equivariant version of Theorem 3.5 proved in
[B24]*Corollary 3.5. The key is to relate the coarse cohomology of complement
to the boundedly supported cohomology of the complement which we introduce
next.

### Boundedly supported cohomology of the complement

Recall that $i:X\to X^{\infty}$ denotes the map $x\mapsto(x,\ldots,x,\ldots)$
and $\Delta=i(X)$. For a subset $A\overset{}{\subset}X$, we denote the set
$i(A)\overset{}{\subset}\Delta$ by $\Delta_{A}$. Boundedly supported
cohomology of the complement, denoted by
$\operatorname{H}{\mathrm{B}}^{*}(X-A)$, is the cohomology of the following
cochain complex with $d$ being the coboundary operator.

$\operatorname{C}{\mathrm{B}}^{*}(X-A;R):=\\{\phi\overset{}{\in}\operatorname{C}^{*}(X;R)\mid\lVert\phi\rVert\overset{\mathrm{c}}{\subset}\Delta_{A}\\}$

Suppose $R$ is a $G$-module. We define the equivariant boundedly supported
cohomology of the complement with coefficients in $R$ to be the cohomology of
the following cochain complex.

$\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R):=\\{\phi\overset{}{\in}\operatorname{C}{\mathrm{B}}^{*}(X-A;R)\mid\phi\text{
is $G$-equivariant}\\}$

We denote the cohomology of the above complex by
$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)$.

Our next goal is to relate $\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}$ to
the equivariant Alexander–Spanier cohomology. We start by briefly recalling
the notion of equivariant Alexander–Spanier cohomology.

### Equivariant Alexander-Spanier cohomology

Honkasalo defined a notion of equivariant Alexander–Spanier cohomology in
[honkasalo]. Let $R$ be a $G$-module. The equivariant Alexander–Spanier
cochain complex is defined as follows

$\operatorname{C}^{*}_{\mathrm{G}}(X;R):=\\{\phi\overset{}{\in}\operatorname{C}^{*}_{as}(X;R)\mid\phi\text{\,
is $G$-equivariant}\\}$

with the usual coboundary operator $d$. We will denote the cohomology of this
complex by $\operatorname{H}^{*}_{\mathrm{G}}(X;R)$.

The following theorem of Honkasalo relates the $G$-equivariant
Alexander–Spanier cohomology with the Alexander–Spanier cohomology of the
quotient by the $G$-action.

######  Theorem 4.1 (cf. Corollary 6.8 [honkasalo]).

Let $R$ be an abelian group with trivial $G$ action. There is a natural
isomorphism
$\operatorname{H}^{*}_{\mathrm{G}}(X;R)\cong{\operatorname{H}^{*}}(X/G;R)$,
where the right-hand side is the ordinary Alexander–Spanier cohomology of
$X/G$.

######  Remark 4.2.

To define $\operatorname{C}^{*}_{\mathrm{G}}$ in general, one needs a
contravariant coefficient system—a contravariant functor from the category of
$G$-spaces $G/H$ $(H\leq G)$ and $G$-maps between them to the category of
$R$-modules. If $R$ is a $G$-module, it defines a contravariant coefficient
system $G/H\mapsto R$, each $G$-map $G/H\rightarrow G/K$ inducing identity
$R\rightarrow R$. With this coefficient system, the equivariant
Alexander–Spanier cochain complex takes the form of
$\operatorname{C}^{*}_{\mathrm{G}}(X;R)$ discussed above.

### Relation between $\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}$ and
$\operatorname{H}^{*}_{\mathrm{G}}$

To relate the equivariant boundedly supported cohomology with the equivariant
Alexander–Spanier cohomology, we will use the cohomology of the following
complex

$\mathscr{C}_{G}^{*}(X;R):=\\{\phi\overset{}{\in}\operatorname{C}^{*}(X;R)\mid\phi\text{
is $G$ equivariant}\\}$

with the usual coboundary operator $d$. Let $\mathcal{H}_{G}^{*}(X;R)$ denote
the cohomology of the above complex. We start with the following lemma.

######  Lemma 4.3.

Suppose $G$ acts on $X$ by isometries such that
$\operatorname{Fix_{\mathrm{G}}}(X)\neq\emptyset$. Then
$\mathcal{H}_{G}(X;R)=0$ for $*\geq 1$.

###### Proof.

Choose $a\overset{}{\in}\operatorname{Fix_{\mathrm{G}}}(X)$. Now consider the
following $G$-equivariant cone operator

$D:(x_{0},x_{1},\ldots,x_{n})\mapsto(a,x_{0},\ldots,x_{n}).$

For any simplex of dimension $\geq 1$, we observe that
$D\partial(\sigma)-\partial D(\sigma)=\sigma$. By considering the dual, we
have that for any cochain $\phi\overset{}{\in}\mathscr{C}_{G}^{*}(X;R)$ in
degree $\geq 1$,

$dD^{*}(\phi)-D^{*}d(\phi)=\phi.$

In particular, $dD^{*}(\phi)=\phi$ when $\phi$ is a cocycle. Since $D$ is
$G$-equivariant, we have
$D^{*}(\phi)\overset{}{\in}\mathscr{C}_{G}^{*+1}(X;R)$. Hence,
$\mathcal{H}^{*}_{G}(X)=0$ for $*\geq 1$. ∎

We now state our main proposition that relates
$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}$ and
$\tilde{\operatorname{H}}^{*}_{\mathrm{G}}$ under the assumption that
$\operatorname{Fix_{\mathrm{G}}}(X)\neq\emptyset$.

######  Proposition 4.4.

Suppose $G\curvearrowright(X,A)$ and
$\operatorname{Fix_{\mathrm{G}}}(X)\neq\emptyset$.

  1. (1)

If $X\overset{\mathrm{c}}{\subset}A$, then

$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)=\begin{cases}R&\text{if
$*=0$},\\\ 0&\text{otherwise.}\end{cases}$

  2. (2)

If $X$ is not coarsely contained in $A$, then

$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)=\begin{cases}0&\text{if
$*=0$,}\\\
\varinjlim\tilde{\operatorname{H}}^{*-1}_{G}(X-N_{r}(A);R)&\text{otherwise.}\end{cases}$

###### Proof.

(1) If $X\overset{\mathrm{c}}{\subset}A$, then
$\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)=\mathscr{C}_{G}^{*}(X;R)$
where $\mathscr{C}_{G}^{*}$ is as in the last lemma. In particular, we get
that

$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{0}(X-A;R)=\\{\text{constant
functions on $X$}\\}\cong R.$

For $*\geq 1$, we know that $\mathcal{H}_{G}^{*}(X;R)=0$ by Lemma 4.3. Hence,
$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)=0$ for $*\geq 1$.

(2) Elements in $\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{0}(X-A;R)$ are
constant functions on $X$ with support contained in $N_{r}(A)$ for some $r$.
This means, if $X$ is not coarsely contained in $A$ then
$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{0}(X-A;R)=0$.

To calculate $\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)$ for $*\geq
1$, consider the following short exact sequence of $G$-equivariant cochain
complexes,

$\displaystyle
0\rightarrow\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)\xrightarrow{j}\mathscr{C}_{G}^{*}(X;R)\xrightarrow{i}\varinjlim\operatorname{C}^{*}_{\mathrm{G}}(X-N_{r}(A);R)\rightarrow
0$

Indeed

$\displaystyle\ker(i)$
$\displaystyle=\\{\phi\overset{}{\in}\mathscr{C}_{G}^{*}(X;R)\mid\phi\overset{}{\in}\operatorname{C}^{*}_{0}(X-N_{r}(A);R)\text{
for some r}\\}$
$\displaystyle=\\{\phi\overset{}{\in}\mathscr{C}_{G}^{*}(X;R)\mid|\phi|\text{
is bounded}\\}=\operatorname{im}(j).$

The short exact sequence induces a long exact sequence of corresponding
reduced cohomologies. The reduced cohomology of the middle cochain complex
$\mathscr{C}_{G}^{*}(X;R)$ is trivial in all degrees above 0. Hence, the long
exact sequence implies that

$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)\cong\varinjlim\tilde{\operatorname{H}}^{*-1}_{\mathrm{G}}(X-N_{r}(A);R)\quad\text{
for }*\geq 1$

∎

Combining Proposition 4.4 and Theorem 4.1 we get the following.

######  Corollary 4.5.

Suppose $A\overset{}{\subset}X$ such that $X$ is not coarsely contained in
$A$, $G\curvearrowright(X,A)$ and
$\operatorname{Fix_{\mathrm{G}}}(X)\neq\emptyset$. Then

$\displaystyle\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)=\begin{cases}0&\text{if
$*=0$},\\\
\varinjlim\tilde{\operatorname{H}}^{*-1}((X-N_{r}(A))/G;R)&\text{otherwise}\end{cases}$

### Relation between $\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{*}$ and
$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}$

Let us come back to the coarse world now. The motivation for defining
$\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)$ is the following
theorem.

######  Theorem 4.6.

Suppose $G\curvearrowright(X,A)$ and $R$ is a $G$-module. Suppose $X$ is
uniformly acyclic away from $A$, and is locally acyclic away from $A$ with
coefficients in $R$. Then the inclusion
$\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R)\hookrightarrow\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R)$
induces an isomorphism on the cohomology:

$\operatorname{H}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R)\cong\operatorname{H}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A;R).$

The proof of the above theorem follows a sketch that Roe described for a proof
of a similar statement in [r03] that relates coarse cohomology with the
compactly supported cohomology. The main idea of the proof is a standard
“connect the dots” construction which enables us to attach a singular chain to
each simplex without going too far from the simplex when the space is
uniformly acyclic. We will state this more precisely in our next lemma.

For the rest of the section, we suppress the coefficient $R$ from the notation
unless it is important. Let $\operatorname{C}_{*}^{\mathrm{s}}(X)$ be the
singular chain complex on $X$. Let
$V:\operatorname{C}_{*}^{\mathrm{s}}(X)\to\operatorname{C}_{*}(X)$ be the
forgetful map, which maps a singular simplex to its vertices.

######  Lemma 4.7.

Suppose $X$ is uniformly acyclic away from $A$ and locally acyclic away from
$A$ and $G\curvearrowright(X,A)$. Then there exist two non-decreasing
sequences of control functions
$\rho_{n},\mu_{n}:[0,\infty)\rightarrow[0,\infty)$ and a $G$-equivariant chain
map $M:\operatorname{C}_{*}^{F}(X)\to\operatorname{C}_{*}^{\mathrm{s}}(X)$
where

$\operatorname{C}_{n}^{F}(X)=\langle\sigma^{n}\mid
d(\sigma^{n},A)\geq\mu_{n}(\operatorname{\text{diam}}\sigma^{n})\rangle\overset{}{\subset}\operatorname{C}_{n}(X)$

such that

  1. (1)

$\lvert
M(\sigma^{n})\rvert\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$.

  2. (2)

There exists $r$ such that for every $x\overset{}{\notin}N_{r}(A)$, and a
neighborhood $U$ of $x$, there is a neighborhood $W\overset{}{\subset}U$ of
$x$ such that $\lvert M(\sigma^{n})\rvert\overset{}{\subset}U$ whenever
$\sigma^{n}\overset{}{\in}{W}^{n+1}$ and
$\sigma^{n}\overset{}{\in}\operatorname{C}_{n}^{F}(X)$.

###### Proof.

We will define $\mu_{i},\rho_{i}$, and the chain map
$M:\operatorname{C}_{i}^{F}\rightarrow\operatorname{C}_{i}^{\mathrm{s}}$ by
induction on $i$.

For $i=0$, define $\mu_{0}$ and $\rho_{0}$ to be the constant functions with
image $\\{0\\}$. That means,
$\operatorname{C}_{0}^{F}(X)=\operatorname{C}_{0}(X)$ and we define
$M:\operatorname{C}_{0}^{F}(X)\rightarrow\operatorname{C}_{0}^{\mathrm{s}}(X)$
to be the identity map.

Since $X$ is uniformly acyclic away from $A$, there exists
$\rho,\mu:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ such that the map
$i_{*}:\operatorname{H}_{*}(B)\rightarrow\operatorname{H}_{*}(N_{\rho(\operatorname{\text{diam}}(B))}(B))$
is trivial if $d(B,A)\geq\mu(\operatorname{\text{diam}}(B))$. By the induction
hypothesis, suppose $\mu_{i},\rho_{i}$, and
$M:\operatorname{C}_{i}^{F}\rightarrow\operatorname{C}_{i}^{\mathrm{s}}$ are
already defined with the desired properties for all $i\leq n$. For $i=n+1$,
define

$\mu_{n+1}(x)=\max\\{\mu(2\rho_{n}(x))+2\rho_{n}(x),\mu_{n}(x)\\}.$

By construction, $\mu_{n+1}\geq\mu_{n}$. Next, we define $M$ on
$\operatorname{C}_{n+1}^{F}(X)$. Take
$\sigma\overset{}{\in}\operatorname{C}_{n+1}^{F}(X)$ with
$\operatorname{\text{diam}}(\sigma)=r$. By the induction hypothesis,
$M(\partial\sigma)$ is a singular cycle with diameter atmost $2\rho_{n}(r)$
due to property (2). Since
$\sigma\overset{}{\in}\operatorname{C}_{n+1}^{F}(X)$, we have
$d(\sigma,A)\geq\mu_{n+1}(x)$. It follows that

$\displaystyle d(|M(\partial\sigma)|,A)$ $\displaystyle\geq
d(\sigma,A)-\operatorname{\text{diam}}(|M(\partial\sigma)|)$
$\displaystyle\geq\mu_{n+1}(r)-2\rho_{n}(r)$
$\displaystyle\geq\mu(2\rho_{n}(r))+2\rho_{n}(r)-2\rho_{n}(r)$
$\displaystyle=\mu(2\rho_{n}(r)).$

So $M(\partial\sigma)$ is a cycle of diameter at most $2\rho_{n}(r)$ and
$d(|M(\partial\sigma)|,A)\geq\mu(2\rho_{n}(r))$. Since $X$ is
$(\mu,\rho)$-uniformly acyclic away from $A$, it follows that
$M(\partial\sigma)$ is a boundary of some singular chain of diameter at most
$\rho(2\mu_{n}(r))$. Let

$k(\sigma):=\inf\\{\operatorname{\text{diam}}(c)\mid\partial
c=M(\partial\sigma)\\}.$

We define $M(\sigma)$ to be a singular chain that bounds $M(\partial\sigma)$
and has diameter at most $2k(\sigma)$. By construction, $\partial
M(\sigma)=M(\partial\sigma)$. To make $M$ $G$-equivariant, we can first define
$M$ on a set of simplices from $\operatorname{C}_{n+1}^{F}(M)$ that contains
one element from each orbit of simplices under the action of $G$ and then
extend the map $G$-equivariantly. Finally we define
$\rho_{n+1}(r):=2\rho(2\mu_{n}(r))$. Now we show that $\rho_{n+1},\mu_{n+1}$,
and $M$ have the desired properties.

To prove (1), observe that $k(\sigma)\leq\rho(2\rho_{n}(r))$ where $\sigma$ is
any $n$-simplex. Hence by construction,
$\operatorname{\text{diam}}(|M(\sigma)|)\leq 2k(\sigma)\leq
2\rho(\rho_{n}(r))=\rho_{n+1}(r)$. Hence
$|M(\sigma)|\overset{}{\subset}N_{\rho_{n+1}(\operatorname{\text{diam}}(\sigma))}(\sigma)$
as claimed.

To prove (2), recall that $X$ is locally acyclic away from $A$ and hence there
exists $r$ such that $X-N_{r}(A)$ is locally acyclic. By construction,
$M(\sigma)$ is a chain of diameter $2k(\sigma)$ with boundary
$M(\partial\sigma)$ , where $k(\sigma)$ is the infimum of the diameter of
chains bounding $M(\partial\sigma)$. Since $X-N_{r}(A)$ is locally acyclic,
$k(\sigma)$ can be made as small as possible by taking
$\operatorname{\text{diam}}(\sigma)$ small enough given $d(\sigma,A)\geq r$.
This gives us (2). ∎

######  Lemma 4.8.

Assume that $X\overset{}{\subset}Y$ and $G\curvearrowright(Y,X)$. Let
$f:\operatorname{C}_{*}(X)\rightarrow\operatorname{C}_{*}(Y)$ be a
$G$-equivariant chain map such that for any $n$-simplex $\sigma^{n}$,
$|f(\sigma^{n})|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$
for some non-decreasing sequence of function
$\rho_{n}:[0,\infty)\rightarrow[0,\infty)$. Then $f$ and the inclusion map
$i:\operatorname{C}_{*}(X)\rightarrow\operatorname{C}_{*}(Y)$ are homotopic
via a $G$-equivariant chain homotopy $D$ such that
$|D(\sigma)|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$.

###### Proof.

We can define $D$ by induction on the dimension of $\sigma$. Define
$D_{0}(x):=(x,f(x))$. Note that $D_{0}$ is $G$-equivariant, $f(x)-x=\partial
D(x)$ and
$D_{0}(x)\overset{}{\subset}N_{\rho_{0}(\operatorname{\text{diam}}\\{x\\})}(x)$.
Suppose we have defined
$D_{m}:\operatorname{C}_{m}(X)\rightarrow\operatorname{C}_{m+1}(X)$ such that
$D_{m}$ is $G$-equivariant,
$D_{m}(\sigma)\overset{}{\subset}N_{\operatorname{\text{diam}}(\sigma)}(\sigma)$
and $\partial D_{m}(\sigma)+D_{m-1}\partial(\sigma)=i(\sigma)-f(\sigma)$ for
any $m\leq n$. To define $D_{n+1}(\sigma)$ for an $(n+1)$-simplex $\sigma$,
consider $c=i(\sigma)-f(\sigma)-D_{n}\partial(\sigma)$. By induction
hypothesis, $c$ is a cycle and
$|c|\overset{}{\subset}N_{\rho(\operatorname{\text{diam}}(\sigma))}(\sigma)$.
Take a vertex $b$ from the chain $c$, and consider the cone operator
$T_{b}:\operatorname{C}_{n+1}(X)\rightarrow\operatorname{C}_{n+2}(X)$,
$(x_{0},\ldots,x_{n+1})\mapsto(b,x_{0},\ldots,x_{n+1})$. Define
$D_{n+1}(\sigma):=T_{b}(c)$. By construction, $\partial
D_{n+1}(\sigma)=c=i(\sigma)-f(\sigma)-D_{n}\partial(\sigma)$ and
$|D_{n+1}(\sigma)|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma))}(\sigma)$.
To make $D_{n+1}$ equivariant, we first define it on elements from each orbit
class of $(n+1)$-simplices and then extend it $G$-equivariantly. ∎

For an open cover $\mathcal{U}$ of $X$, let
$\operatorname{C}_{*}^{\mathcal{U}}(X)$ be the chain complex generated by
singular simplices supported in some open set in $\mathcal{U}$. To prove
Theorem 4.6, we actually need to fill in simplices by singular chains in
$\operatorname{C}_{*}^{\mathcal{U}}(X)$ for some $\mathcal{U}$. More
precisely, we need the following.

######  Proposition 4.9.

Suppose $X$ is uniformly acyclic away from $A$ and locally acyclic away from
$A$ and $G\curvearrowright(X,A)$. Let $\mathcal{U}$ be a $G$-invariant open
cover of $X$, i.e for any $U\overset{}{\in}\mathcal{U}$
$gU\overset{}{\in}\mathcal{U}$ for all $g\overset{}{\in}G$. Then there exist
two non-decreasing sequences of control functions
$\rho_{n},\mu_{n}:[0,\infty)\rightarrow[0,\infty)$, a $G$-equivariant chain
map $S:\operatorname{C}_{*}^{F}(X)\to\operatorname{C}_{*}^{\mathcal{U}}(X)$
where

$\operatorname{C}_{n}^{F}(X)=\langle\sigma^{n}\mid
d(\sigma^{n},A)\geq\mu_{n}(\operatorname{\text{diam}}\sigma^{n})\rangle\overset{}{\subset}\operatorname{C}_{n}(X)$

and a $G$-equivariant chain homotopy
$H:\operatorname{C}_{*}^{F}(X)\to\operatorname{C}_{*+1}(X)$ between
$VS:\operatorname{C}_{*}^{F}(X)\rightarrow\operatorname{C}_{*}(X)$ and the
inclusion map so that

  1. (1)

$\lvert
S(\sigma^{n})\rvert\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$.

  2. (2)

$\lvert
H(\sigma^{n})\rvert\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$.

  3. (3)

There exists $r>0$ so that for every $x\overset{}{\notin}N_{r}(A)$, there is a
neighborhood $W$ of $x$ such that for all $\sigma^{n}\overset{}{\in}W^{n+1}$,
$H(\sigma^{n})\overset{}{\in}\operatorname{C}_{n+1}^{\mathcal{U}}(X)$.

###### Proof.

Consider the map
$M:\operatorname{C}_{*}^{F}(X)\rightarrow\operatorname{C}_{*}^{\mathrm{s}}(X)$
provided by the Lemma 4.7. Let $\mathcal{U}$ be a $G$-equivariant cover of
$X$. We choose a barycentric subdivision map
$P:\operatorname{C}_{*}^{\mathrm{s}}(X)\rightarrow\operatorname{C}_{*}^{\mathcal{U}}(X)$.
We can choose $P$ in a $G$-equivariant way by the same trick as before: in
each dimension, first define it on an element from each orbit class of
simplices and then extend $G$-equivariantly. Define $S:=PM$. $S$ satisfies (1)
for some $\rho_{n}$ because $M$ and $P$ do the same.

It follows that
$|VS|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$
and therefore we can apply Lemma 4.8 to the map to get the homotopy $H$
between $VS$ and the $id$ with property (2).

Finally to prove (3), note that the map $M$ satisfies the property (3) by
Lemma 4.7(2), and hence so does $S=PM$ and $VS$. By construction,
$|H(\sigma)|$ has same the set of vertices as $|VS(\sigma)|$ for any $\sigma$.
Therefore $H$ satisfies (3). ∎

######  Remark 4.10.

If $X$ is uniformly acyclic, then every simplex is fillable and so in this
case we can replace $\operatorname{C}_{*}^{F}(X)$ with
$\operatorname{C}_{*}(X)$ in Proposition 4.9. Also, note that the local
acyclicity away from $A$ was only used to ensure property (3). So, if the
space is just uniformly acyclic away from $A$, we still get $H$ and $S$
satisfying the properties (1) and (2).

###### Proof of Theorem 4.6.

By the long exact sequence, we need to show that
$\operatorname{H}^{*}(\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{*}(X-A)/\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A))=0$.
In other words, for
$\phi\overset{}{\in}\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{n}(X-A)$ with
$d\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{n+1}(X-A)$ we
need to find
$\psi\overset{}{\in}\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{n-1}(X-A)$ so
that
$\phi-d\psi\overset{}{\in}\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{n}(X-A)$.
Choose a regular neighborhood $U$ of $A$ such that
$||\phi||\overset{}{\subset}U$. Since $G\curvearrowright(X,A)$ by hypothesis,
we have $gU=U$ for all $g\overset{}{\in}G$. For each
$x\overset{}{\in}X-||\phi||$, choose a metric neighborhood $U_{x}$ with
diameter $\leq 1$ such that $U_{x}^{n+1}\cap\lvert\phi\rvert=\emptyset$. Since
$\phi$ is $G$-equivariant, we can choose the association $x\mapsto U_{x}$ so
that $U_{gx}=gU_{x}$ for all $g\overset{}{\in}G$. Let $\mathcal{U}$ denote the
collection of $U_{x}$ together with $U$. By construction, this is a
$G$-invariant cover.

Now we apply Proposition 4.9 to this setup which outputs a complex
$\operatorname{C}_{*}^{F}(X)$, a $G$-equivariant chain map
$S:\operatorname{C}_{*}^{F}(X)\rightarrow\operatorname{C}_{*}^{\mathcal{U}}(X)$
and a $G$-equivariant chain homotopy
$H:\operatorname{C}_{*}^{F}(X)\rightarrow\operatorname{C}_{*+1}(X)$ between
$VS$ and the inclusion map with certain control on their support.

We now use $H$ to define a linear map
$D:\operatorname{C}_{*}(X)\to\operatorname{C}_{*+1}(X)$ by setting

$D(\sigma^{n})=\begin{cases}H(\sigma^{n})&\text{ if
}\sigma^{n}\overset{}{\in}\operatorname{C}_{n}^{F}(X)\\\ 0&\text{
otherwise.}\end{cases}$

We define $\tau:\operatorname{C}_{*}(X)\rightarrow\operatorname{C}_{*+1}(X)$
as the follows

$\tau=id+\partial D+D\partial$

When $\sigma\overset{}{\in}\operatorname{C}_{*}^{F}(X)$, by construction
$D(\sigma)=H(\sigma)$ and therefore $\tau(\sigma)=VS(\sigma)$. Suppose
$\tau^{*}$ denote the dual of $\tau$. By applying $\tau^{*}$ on $\phi$, we get
the following

$\tau^{*}\phi=\phi+dD^{*}\phi+D^{*}d\phi.$

We claim that
$\tau^{*}\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{n}(X-A)$.
If $\operatorname{\text{diam}}(\sigma^{n})\leq k$ and
$d(\sigma^{n},A)>\mu_{n}(k)$ for some $k\geq 0$, then
$\sigma^{n}\overset{}{\in}\operatorname{C}_{*}^{F}(X)$ and hence
$\tau(\sigma^{n})=VS(\sigma^{n})$. Moreover, if $\sigma^{n}$ is outside of the
$\rho_{n}(k)$-neighborhood of $U^{n+1}$, then $|\tau(\sigma^{n})|$ does not
touch $U^{n+1}$ because
$|\tau(\sigma^{n})|=|VS\sigma^{n}|\overset{}{\subset}N_{\rho_{n}(k)}(\sigma^{n})$.
This implies $(\tau^{*}\phi)(\sigma^{n})=\phi(\tau(\sigma^{n}))=0$. In other
words, $\sigma^{n}\overset{}{\notin}|\tau^{*}\phi|$. It follows that,

$|\tau^{*}\phi|\cap
N_{k}(\Delta)\overset{}{\subset}N_{\mu_{n}(k)}(\Delta_{A})\cup
N_{\rho_{n}(k)}(U^{n+1}).$

Since $U\overset{\mathrm{c}}{\subset}A$, we have

$|\tau^{*}\phi|\cap N_{k}(\Delta)\overset{\mathrm{c}}{\subset}\Delta_{A}.$

This proves the claim.

Next, we claim that
$D^{*}(\phi)\overset{}{\in}\operatorname{C}{\mathrm{B}}_{\mathrm{G}}^{n-1}(X-A)$.
By Proposition 4.9(3), we can choose a set $N_{r}(A)$ containing $U$ such that
for any $x\overset{}{\notin}N_{r}(A)$, there is a neighborhood $W_{x}$ of $x$
so that $|H(\sigma^{n})|\overset{}{\subset}U_{x}^{n+2}$ for any
$\sigma^{n}\overset{}{\in}W_{x}^{n+1}$. Hence for any
$x\overset{}{\notin}N_{r}(A)\cup||\phi||$, we have
$|H(\sigma^{n})|\overset{}{\notin}|\phi|$ and hence
$|D(\sigma^{n})|\overset{}{\notin}|\phi|$ for all
$\sigma^{n}\overset{}{\in}W_{x}^{n+1}$. Therefore,
$||D^{*}(\phi)||\overset{}{\subset}||\phi||\cup N_{r}(A)$. The claim follows
since $\phi\overset{}{\in}\operatorname{C}{\mathrm{B}}^{*}(X)$.

Finally, we claim that
$D^{*}d(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{n}(X-A)$.
By construction of $D$, if $d(\sigma^{n},A)\geq\mu_{n}(k)$ then $\lvert
D\sigma^{n}\rvert\overset{}{\subset}N_{\rho_{n}(k)}(\sigma^{n})$ where $k$ is
the diameter of $\sigma^{n}$. Since
$d(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A)$,
the claim follows.

Now setting $\psi=-D^{*}(\phi)$ we get what we want. ∎

We can now prove Theorem 3.5.

###### Proof of Theorem 3.5.

We observe that
$\operatorname{C}{\mathrm{X}}_{\mathrm{G}}^{*}(X-A;R)=\mathscr{C}_{G}^{*}(X;R)$
when $X$ is coarsely contained in $A$. Hence Theorem 3.5(1) follows from Lemma
4.3.

Theorem 3.5(2) follows immediately from Proposition 4.4 and Theorem 4.6.

∎

##  5\. Coarse cohomology of the configuration space

Let $(X,d)$ be a metric space. Equip $X^{2}=X\times X$ with the sup metric.
Consider the $\mathbb{Z}_{2}$-action on $X^{2}$ that flips the coordinates.
Since the fixed point set for this action is the diagonal subspace
$\delta(X)=\\{(x,x)\mid x\overset{}{\in}X\\}\overset{}{\subset}X^{2}$, we have
$\mathbb{Z}_{2}\curvearrowright(X^{2},\delta(X))$. Let $R$ be a
$\mathbb{Z}_{2}$-module. We define _coarse cohomology of the configuration
space of $X$_ with coefficients in $R$ to be the cohomology of the complex
$\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X);R)$.

Theorem 3.5 immediately gives us the following.

######  Proposition 5.1.

If $X$ is unbounded, uniformly acyclic, and locally acyclic with coefficients
in $R$ where $R$ is an abelian group with trivial $\mathbb{Z}_{2}$ action.
Then

$\displaystyle\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X);R)=\begin{cases}0&\text{if
$*=0$},\\\
\varinjlim\tilde{\operatorname{H}}^{*-1}((X^{2}-N_{r}(\delta(X)))/\mathbb{Z}_{2};R)&\text{otherwise}.\end{cases}$

.

######  Example 5.2.

$\mathbb{R}^{n}$ satisfies the hypothesis of Proposition 5.1. Moreover, for
any $r$, there is a $\mathbb{Z}_{2}$-equivariant deformation retraction of
$(\mathbb{R}^{n})^{2}-\delta(\mathbb{R}^{n})$ to
$(\mathbb{R}^{n})^{2}-N_{r}(\delta(\mathbb{R}^{n}))$. Therefore, applying
Proposition 5.1 we obtain the following where $\mathbb{Z}_{2}$ is assumed to
be the $\mathbb{Z}_{2}$-module with trivial action.

$\displaystyle\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\mathbb{R}^{n})^{2}-\delta(\mathbb{R}^{n});\mathbb{Z}_{2})$
$\displaystyle=\begin{cases}0&\text{if $*=0$}\\\
\tilde{\operatorname{H}}^{*-1}(((\mathbb{R}^{n})^{2}-\delta(\mathbb{R}^{n}))/\mathbb{Z}_{2};\mathbb{Z}_{2})&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}0&\text{if $*=0$}\\\
\tilde{\operatorname{H}}^{*-1}(\mathbb{R}P^{n-1};\mathbb{Z}_{2})&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}\mathbb{Z}_{2}&\text{if $2\leq*\leq n$}\\\
0&\text{otherwise}\end{cases}$

######  Example 5.3.

Recall that $\ell^{\infty}$ is the space of bounded sequences with the sup-
norm metric. $(\ell^{\infty})^{2}-\delta(\ell^{\infty})$ deformation retracts
$\mathbb{Z}_{2}$-equivariantly to
$(\ell^{\infty})^{2}-N_{r}(\delta(\ell^{\infty}))$. Since
$(\ell^{\infty})^{2}-\delta(\ell^{\infty})$ is acyclic,
$((\ell^{\infty})^{2}-\delta(\ell^{\infty}))/\mathbb{Z}_{2}$ is a classifying
space for $\mathbb{Z}_{2}$ and hence is homotopy equivalent to
$\mathbb{R}P^{\infty}$. Hence arguing as before we obtain the following.

$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty});\mathbb{Z}_{2})=\begin{cases}\mathbb{Z}_{2}&\text{for}\quad*\geq
2\\\ 0&\text{otherwise}\end{cases}$

In the topological setting, an injective continuous map induces a map between
the cohomology of the corresponding configuration spaces. In the coarse
setting, the role of injective continuous maps is played by _coarse expanding
maps_ which we define next.

######  Definition 5.4.

A map $f:(X,A)\rightarrow(Y,B)$ between pairs is said to be a relatively
proper map if $f^{-1}(N_{r}(B))\overset{\mathrm{c}}{\subset}A$ for any $r$.

A map $f:(X,A)\rightarrow(Y,B)$ between pairs is said to be a relatively
coarse map if $f$ is relatively proper and $f$ has an upper control i.e there
exists a non-decreasing function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ such
that $d(f(x),f(y))\leq\mu(d(x,y))$ for all $x,y\overset{}{\in}X$.

A map $f:X\rightarrow Y$ is a coarse expanding map if the induced map
$f:(X^{2},\delta(X))\rightarrow(Y^{2},\delta(Y))$ is a relatively coarse map.

Recall that a map $f:X\rightarrow Y$ between two metric spaces is said to be a
_coarse embedding_ if there exist two proper non decreasing maps
$\rho_{-},\rho_{+}:[0,\infty)\rightarrow[0,\infty)$ such that

$\rho_{-}(d(x,y))\leq d(f(x),f(y))\leq\rho_{+}(d(x,y))\quad\text{for all}\quad
x,y\overset{}{\in}X.$

One can see that any coarse embedding map is a coarse expanding map. We now
recall the following from [BB20].

######  Lemma 5.5.

For any module $R$, a relatively coarse map $f:(X,A)\to(Y,B)$ between pairs
induces chain map
$f^{*}:\operatorname{C}{\mathrm{X}}^{*}(Y-B;R)\to\operatorname{C}{\mathrm{X}}^{*}(X-A;R)$
by the canonical formula

$(f^{*}\phi)(x_{0},x_{1},\ldots,x_{n})=\phi(f(x_{0}),f(x_{1}),\ldots,f(x_{n})).$

The following lemma follows immediately from Lemma 5.5.

######  Lemma 5.6.

If $f:X\rightarrow Y$ is a coarse expanding map and $R$ is a module, then the
map $(x,y)\mapsto(f(x),f(y))$ induces a map
$f^{*}:\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(Y^{2}-\delta(Y);R)\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X);R)$.

##  6\. Coarse van Kampen obstruction

In this section, we find an obstruction to the existence of coarse expanding
maps between two metric spaces. Our key observation is the next proposition
which says that when the target space $Y$ in Lemma 5.6 is $\ell^{\infty}$, the
induced map $f^{*}$ depends only on $X$, not on $f$.

######  Proposition 6.1.

Any two coarse expanding maps from $X$ to $\ell^{\infty}$ induce the same map
from
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty});R)$
to $\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X);R)$.

###### Proof.

For convenience, we will suppress the coefficient $R$ from the notation in the
proof.

Suppose $f,g:X^{2}\rightarrow(\ell^{\infty})^{2}$ are two maps induced by two
coarse expanding maps from $X$ to $\ell^{\infty}$. For convenience, we will
denote the induced map between $\operatorname{C}_{*}(X^{2})$ and
$\operatorname{C}_{*}((\ell^{\infty})^{2})$ by $f$ and $g$ as well. Let
$f^{*}$ and $g^{*}$ be the corresponding map from
$\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
to $\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$. We
will show that there is a chain homotopy
$\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))\rightarrow\operatorname{C}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$
between $f^{*}$ and $g^{*}$.

Since $(\ell^{\infty})^{2}$ is uniformly contractible, the proof of
Proposition 4.9 (see remark 4.10) gives us a $\mathbb{Z}_{2}$-equivariant
chain map
$S:\operatorname{C}_{*}((\ell^{\infty})^{2})\rightarrow\operatorname{C}_{*}^{\mathrm{s}}((\ell^{\infty})^{2})$
such that
$|S(\sigma^{n})|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma)$
for some sequence of functions
$\rho_{n}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$. Moreover, the composition
$VS$ is chain homotopic to the identity map by a $\mathbb{Z}_{2}$-equivariant
chain homotopy $H$ such that
$|H(\sigma^{n})|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$.

We now construct a chain homotopy
$D_{*}:\operatorname{C}_{*}(X^{2})\rightarrow\operatorname{C}_{*+1}^{\mathrm{s}}((\ell^{\infty})^{2})$
between the two maps $S\circ f,S\circ
g:\operatorname{C}_{*}(X^{2})\rightarrow\operatorname{C}_{*}^{\mathrm{s}}((\ell^{\infty})^{2})$
with certain properties: in particular we want the homotopy to avoid
$\delta(\ell^{\infty})$.

For convenience let us fix some notations. For a singular chain
$c\overset{}{\in}\operatorname{C}_{*}^{\mathrm{s}}(X)$, let

$\operatorname{\text{size}}(c):=\sup_{\tau\overset{}{\in}|c|}\\{\operatorname{\text{diam}}(\tau)\\}.$

For a simplex $\sigma\overset{}{\in}(X^{2})^{*}$, let

$r_{\sigma}:=\min\\{d(|S(f(\sigma))|,\delta(\ell^{\infty})),d(|S(g(\sigma))|,\delta(\ell^{\infty}))\\}$

and

$R_{\sigma}:=\max\\{d(|S(f(\sigma))|,\delta(\ell^{\infty})),d(|S(g(\sigma))|,\delta(\ell^{\infty}))\\}.$

Let $C_{\sigma}$ be the following annulus around $\delta(\ell^{\infty})$:

$\\{z\mid r_{\sigma}\leq d(z,\delta(\ell^{\infty}))\leq R_{\sigma}\\}.$

We note that $C_{\sigma}$ is contractible because it is homotopy equivalent to
$S^{\infty}$. We will now define
$D_{i}:\operatorname{C}_{i}(X^{2})\rightarrow\operatorname{C}_{i+1}^{\mathrm{s}}((\ell^{\infty})^{2})$
with the following properties.

  1. (1)

$S\circ f-S\circ g=\partial D_{i}+D_{i-1}\partial$

  2. (2)

$\lvert D_{i}(\sigma)\rvert\overset{}{\subset}C_{\sigma}$.

  3. (3)

$\operatorname{\text{size}}(D_{i}(\sigma))\leq\operatorname{\text{size}}(S(f(\sigma))-S(g(\sigma))-D_{i-1}\partial(\sigma))$

  4. (4)

$D_{i}$ is $\mathbb{Z}_{2}$-equivariant.

We first define
$D_{0}:\operatorname{C}_{0}(X^{2})\rightarrow\operatorname{C}_{1}^{\mathrm{s}}((\ell^{\infty})^{2})$.
For a $0$-simplex $\sigma\overset{}{\in}Y$, join $f(\sigma)$ and $g(\sigma)$
by a path $\gamma$ in $C_{\sigma}$. Such a path exists because the annulus
$C_{\sigma}$ is contractible. Next, we subdivide $\gamma$ so that each subarc
is of diameter $\leq 1$. We define $D_{0}(\sigma)$ to be this subdivided path.
By construction, $D_{0}$ satisfies the first three desired properties. To make
it $G$-equivariant, we first define $D_{0}$ on a simplex from each $G$-orbit
of simplices and then move it $\mathbb{Z}_{2}$-equivariantly. It is
straightforward to see that $D_{0}$ still satisfies the first property because
$\partial$ commutes with the $G$-action. It satisfies the second property
because $gC_{\sigma}=C_{g\sigma}$ for any $g\overset{}{\in}\mathbb{Z}_{2}$.
$D_{0}$ satisfies the third property because isometric action preserves size
of chains.

Inductively, let us assume that
$D_{i}:\operatorname{C}_{i}(X^{2})\rightarrow\operatorname{C}_{i+1}^{\mathrm{s}}((\ell^{\infty})^{2})$
has been defined for all $i\leq n$ with the desired properties. To define
$D_{n+1}(\sigma)$, let $K=S(f(\sigma))-S(g(\sigma))-D_{n}(\partial(\sigma))$.
By induction hypothesis (1), $\partial K=D_{n-1}(\partial^{2}(\sigma))=0$ and
hence $K$ is a cycle. By (2), $\lvert K\rvert\overset{}{\subset}C_{\sigma}$.
Since $C_{\sigma}$ is acyclic, we can get a singular chain $c$ which is
supported in $C_{\sigma}$ such that $\partial c=K$. After applying the
appropriate subdivision, we can make $c$ to satisfy
$\operatorname{\text{size}}(c)\leq\operatorname{\text{size}}(K)$ without
changing its boundary. Define $D_{n+1}(\sigma)$ to be that $c$. By
construction, $D_{n+1}$ satisfies conditions (1), (2), and (3). To make
$D_{n+1}$ satisfy (4), we can use the same trick as before: first define it on
a simplex from each $\mathbb{Z}_{2}$-orbit of $(n+1)$-simplices and then
extend equivariantly. For the similar reason as $D_{0}$, this $D_{n+1}$ has
all the desired four properties.

Recall that
$V:\operatorname{C}_{*}^{\mathrm{s}}\rightarrow\operatorname{C}_{*}$ denotes
the map that sends a singular simplex to its vertices. Since
$V\partial=\partial V$, applying $V$ on both sides of (1), we have the
following

$VS(f-g)=\partial VD_{i}+VD_{i-1}\partial$

Recall that $VS$ is homotopic to the identity map with a
$\mathbb{Z}_{2}$-equivariant chain homotopy $H$ such that
$|H(\sigma^{n})|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$.

$VS=id+\partial H_{i}+H_{i-1}\partial.$

Combining the above two, we get

$f-g=\partial(VD_{i}+H_{i}(f-g))+(VD_{i-1}+H_{i-1}(f-g))\partial$

Dualizing the above we get $f^{*}$ and $g^{*}$ are chain homotopic via the
chain homotopy $(VD)^{*}+(f-g)^{*}H^{*}$.

Since
$|H(\sigma^{n})|\overset{}{\subset}N_{\rho_{n}(\operatorname{\text{diam}}(\sigma^{n}))}(\sigma^{n})$
and $H$ is $\mathbb{Z}_{2}$-equivariant, it follows that the dual $H^{*}$ maps
$\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
to
$\operatorname{C}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$.
Therefore $(f-g)^{*}H^{*}$ maps
$\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
to $\operatorname{C}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$. It
is now enough to prove that $(VD)^{*}$ maps
$\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
to $\operatorname{C}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$. Let
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
and $\sigma\overset{}{\in}|(VD)^{*}(\phi)|\cap N_{r}(\Delta)$ for some $r\geq
0$. Since $\sigma\overset{}{\in}N_{r}(\Delta)$, and $f$ and $g$ are coarse
expanding maps, we have $|S(f(\sigma))|\overset{}{\subset}N_{s}(\Delta)$ and
$|S(g(\sigma))|\overset{}{\subset}N_{s}(\Delta)$ for some $s$ that depends
only on $r$. Property (3) then implies that
$|VD(\sigma)|\overset{}{\subset}N_{s}(\Delta)$ for some $s$ that depends only
on $r$. Since
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
and $\phi(VD(\sigma))\neq 0$, it then follows that
$|VD(\sigma))|\overset{}{\subset}N_{t}(\Delta_{\delta(\ell^{\infty})})$ where
$t$ depends only on $s$ and hence depends only on $r$. It now follows from
property (2) that $\sigma\overset{}{\in}N_{p}(\Delta_{\delta(X))}$ for some
$p$ that depends only on $t$ and hence only on $r$. Hence, we proved that for
each $r\geq 0$, there exists $p\geq 0$ such that $|(VD)^{*}(\phi)|\cap
N_{r}(\Delta)\overset{}{\subset}N_{p}(\Delta_{\delta(X)}$. Finally,
$(VD)^{*}(\phi)$ is $\mathbb{Z}_{2}$-equivariant by property (4). Hence,
$(VD)^{*}(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$.
This finishes the proof.

∎

######  Coarse van Kampen obstruction class.

Let $X$ be a separable metric space and $g:X\rightarrow\ell^{\infty}$ be a
coarse expanding map. Such a map exists because any separable metric space
admits isometric embedding into $\ell^{\infty}$ by the work of Fréchet
([Frechet]). We consider the induced map
$g^{*}:\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty});\mathbb{Z}_{2})\rightarrow\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X);\mathbb{Z}_{2})$.
Recall from Example 5.3 that
$\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty});\mathbb{Z}_{2})=\mathbb{Z}_{2}$
if $n\geq 2$. Let $e^{n}$ be the nontrivial element in
$\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
for $n\geq 2$. Proposition 6.1 implies that $g^{*}(e^{n})$ depends only on the
space $X$, not on $g$. We call the class $g^{*}(e^{n})$ to be the _$n^{th}$
degree coarse van Kampen obstruction class_ of $X$ and denote it by
$cvk^{n}(X)$ where $n\geq 2$.

######  Assumption.

From now on all our metric spaces will be separable. In the non-equivariant
setting, our coefficient ring will be $\mathbb{Z}_{2}$ and in the equivariant
setting, the coefficient ring will be $\mathbb{Z}_{2}$ with the trivial
$\mathbb{Z}_{2}$-action unless stated otherwise. In those cases, we will omit
the coefficient from the notation.

######  Definition 6.2 (Coarse obstruction dimension).

The Coarse obstruction dimension of a space $X$, denoted by
$\operatorname{cobdim}(X)$, is 0 if $X$ is bounded and otherwise, it is the
largest $n$ such that $cvk^{n}(Y)\neq 0$.

Now we prove the main theorem of this section.

######  Theorem 6.3.

If there exists a coarse expanding map $X\rightarrow Y$, then
$\operatorname{cobdim}(X)\leq\operatorname{cobdim}(Y)$.

###### Proof.

If $\operatorname{cobdim}(X)=0$, then there is nothing to prove. If
$\operatorname{cobdim}(X)=1$, then $X$ is unbounded by definition. This means
$Y$ is also unbounded and hence $\operatorname{cobdim}(X)\geq 1$. Suppose
$\operatorname{cobdim}(X)=n\geq 2$. Let $g:Y\rightarrow\mathbb{R}^{\infty}$ be
a coarse expanding map. Consider the following composition.

$X\times X\xrightarrow{f}Y\times
Y\xrightarrow{g}\ell^{\infty}\times\ell^{\infty}$

By Proposition 5.6, the above maps induce the following maps between coarse
cohomology of the configuration spaces

$\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))\xrightarrow{g^{*}}\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(Y^{2}-\delta(Y))\xrightarrow{f^{*}}\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$

Let
$e^{n}\overset{}{\in}\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))$
be the generator. Then $cvk^{n}(Y)=g^{*}(e^{n})$ and
$cvk^{n}(X)=f^{*}g^{*}(e^{n}))=f^{*}(cvk^{n}(Y))$. By assumption
$cvk^{n}(X)\neq 0$ and hence $cvk^{n}(Y)\neq 0$. So, we get
$\operatorname{cobdim}(Y)\geq n$. ∎

One immediate corollary of the above theorem is the following.

######  Corollary 6.4.

If $X$ and $Y$ are coarsely equivalent, then
$\operatorname{cobdim}(X)=\operatorname{cobdim}(Y)$.

In Example 3.6, we saw that
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(\operatorname{Conf}(\mathbb{R}^{n}))=0$
for all $*>n$. Hence $\operatorname{cobdim}(\mathbb{R}^{n})\leq n$. Using
Theorem 6.3, we get the following.

######  Corollary 6.5.

If $\operatorname{cobdim}(X)\geq n$, then $X$ does not admit a coarse
expanding map into $\mathbb{R}^{n-1}$.

##  7\. Relation to the classical van Kampen obstruction

Let us recall the classical van Kampen obstruction class. Let $X$ be any
topological space. Any continuous embedding
$f:X\hookrightarrow\mathbb{R}^{\infty}$ induces a map

$f:\operatorname{H}^{*}((\mathbb{R}^{\infty})^{2}-\delta(\mathbb{R}^{\infty})/{\mathbb{Z}_{2}})\rightarrow\operatorname{H}^{*}((X^{2}-\delta(X))/\mathbb{Z}_{2}).$

This map depends only on $X$ because the quotient map

$q:(\mathbb{R}^{\infty})^{2}-\delta(\mathbb{R}^{\infty})\rightarrow((\mathbb{R}^{\infty})^{2}-\delta(\mathbb{R}^{\infty}))/{\mathbb{Z}_{2}}$

is a universal $\mathbb{Z}_{2}$-bundle. Let
$\eta^{*}\overset{}{\in}\operatorname{H}^{*}((\mathbb{R}^{\infty})^{2}-\delta(\mathbb{R}^{\infty})/{\mathbb{Z}_{2}})$
be the generator. The cohomology class $f(\eta^{*})$ is called the van Kampen
obstruction class in degree $*$ and will be denoted by $vk^{*}(X)$. Note that
$(\ell^{\infty})^{2}-\delta(\ell^{\infty})\rightarrow((\ell^{\infty})^{2}-\delta(\ell^{\infty}))/{\mathbb{Z}_{2}}$
can also be considered as a universal $\mathbb{Z}_{2}$-bundle because
$(\ell^{\infty})^{2}-\delta(\ell^{\infty})$ is contractible. Hence we can use
$\ell^{\infty}$ instead of $\mathbb{R}^{\infty}$ to define $vk^{*}(X)$. We use
this viewpoint in the next proposition.

######  Proposition 7.1.

Suppose
$i:\tilde{\operatorname{H}}^{*}((X^{2}-\delta(X))/\mathbb{Z}_{2})\rightarrow\varinjlim\tilde{\operatorname{H}}^{*}((X^{2}-N_{r}(\delta(X)))/\mathbb{Z}_{2})$
is the map induced by inclusions $X^{2}-N_{r}(\delta(X))\hookrightarrow
X^{2}-\delta(X)$ for each $r>0$. Then given $n\geq 2$, $cvk^{n}(X)$ is
nontrivial if $i(vk^{n-1}(X))$ is nontrivial.

###### Proof.

Let $f:X\rightarrow\ell^{\infty}$ be an isometry. Fix $n\geq 2$. Suppose
$i(vk^{n-1}(X))\neq 0$. We consider the following commutative diagram where
all the horizontal maps are canonically induced by $f$. We want to show that
the top horizontal map is nontrivial.

${\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))}$${\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))}$${\varinjlim\tilde{\operatorname{H}}^{n-1}(((\ell^{\infty})^{2}-N_{r}(\delta(\ell^{\infty})))/\mathbb{Z}_{2})}$${\varinjlim\tilde{\operatorname{H}}^{n-1}((X^{2}-N_{r}(\delta(X)))/\mathbb{Z}_{2})}$${\tilde{\operatorname{H}}^{n-1}(((\ell^{\infty})^{2}-\delta(\ell^{\infty}))/\mathbb{Z}_{2})}$${\tilde{\operatorname{H}}^{n-1}((X^{2}-\delta(X))/\mathbb{Z}_{2})}$$\scriptstyle{f}$$\scriptstyle{\cong}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{i}$

The top left isomorphism map is due to the Proposition 5.1.

The image of the generator of
$\tilde{\operatorname{H}}^{n-1}(((\ell^{\infty})^{2}-\delta(\ell^{\infty}))/\mathbb{Z}_{2})$
under the bottom horizontal map is $vk^{n-1}(X)$ if $n\geq 2$. Since
$i(vk^{n-1}(X))\neq 0$, the commutativity of the above diagram implies that
the middle horizontal map is nontrivial. Again using commutativity of the
diagram, we conclude that the top horizontal map is nontrivial. ∎

######  Example 7.2 (Bestvina-Kapovich-Kleiner obstruction).

Suppose $X=K\times[0,\infty)/K\times\\{0\\}$ is the open cone on a finite
simplicial complex $K$. A metric $d$ on $X$ is called expanding if for any two
disjoint simplices $\sigma,\tau\overset{}{\in}K$ and $S\geq 0$, there exists
$r\geq 0$ such that $d(\sigma\times[r,\infty),\tau\times[r,\infty))\geq S$.
One of the result in Bestvina, Kapovich and Kleiner [BKK] can be stated as
follows: If there is class
$c\overset{}{\in}\operatorname{H}_{n}(\operatorname{Conf}(X))$ such that
$vk^{n}(X)(c)\neq 0$ then $X$ with a proper, expanding metric cannot be
coarsely embedded inside $\mathbb{R}^{n}$. One can see that $cvk^{n+1}(X)\neq
0$ for such $X$ as follows: since the metric on $X$ is expanding any
representative of
$c\overset{}{\in}\operatorname{H}_{*}(\operatorname{Conf}(X))$ can be
homotoped to a cycle that lives in $(X^{2}-N_{r}(\delta(X)))/\mathbb{Z}_{2}$
for any $r$. Considering the dual, this implies that the restriction map
$\tilde{\operatorname{H}}^{*}((X^{2}-\delta(X))/\mathbb{Z}_{2})\rightarrow\varinjlim\tilde{\operatorname{H}}^{*}((X^{2}-N_{r}(\delta(X)))/\mathbb{Z}_{2})$
sends $vk^{n}(X)$ to a nontrivial element. Hence, $i(vk^{n}(X))$ is nontrivial
where $i$ is as in Proposition 7.1. Hence Proposition 7.1 implies
$cvk^{n+1}(X)\neq 0$. Therefore, $\operatorname{cobdim}(X)\geq n+1$ and hence
Corollary 6.5 recovers Bestvina–Kapovich–Kleiner’s result.

##  8\. A Coarse Gysin sequence

Recall that coarse van Kampen obstruction class lives in
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$. In this
section, we relate $\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}$ to
$\operatorname{H}{\mathrm{X}}^{*}$ and apply that to compute
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$ for
certain $X$.

Suppose $\mathbb{Z}_{2}$ is acting on some metric space $X$ by isometries and
$A$ is the subset of $X$ that is fixed by the action. We consider the
following exact sequence

$0\rightarrow\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\xrightarrow{i}\operatorname{C}{\mathrm{X}}^{*}(X-A)\xrightarrow{p}\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\xrightarrow{r}\operatorname{C}^{*}(A)\rightarrow
0$

where $i$ is the inclusion map and
$p(\phi):\sigma\mapsto\phi(\sigma)+\phi(g\sigma)$ where $g$ is the generator
of $\mathbb{Z}_{2}$ and $r$ is the restriction map. The image of $p$ consists
of those cochains in $\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)$
that send any simplex supported on $A$ to zero. It is easy to see that
collection of such cochains gives a subcomplex of
$\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)$. We denote this
complex by $\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)$, and the
corresponding cohomology by
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)$. Hence, the above
four-term short exact sequence splits into the following two short exact
sequences.

$\displaystyle
0\rightarrow\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\xrightarrow{i}\operatorname{C}{\mathrm{X}}^{*}(X-A)\xrightarrow{p}\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)\rightarrow
0$ $\displaystyle
0\rightarrow\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)\xrightarrow{i}\operatorname{C}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\xrightarrow{r}\operatorname{C}^{*}(A)\rightarrow
0$

These two short exact sequences give us two long exact sequences in the
cohomology which we record as our next lemma.

######  Lemma 8.1.

Let $X$ be a metric space and $\mathbb{Z}_{2}\curvearrowright X$ such that $A$
is the fixed point set of the action. Then we have the following two long
exact sequences

(8.1)
$\displaystyle\cdots\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\rightarrow\operatorname{H}{\mathrm{X}}^{*}(X-A)\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)\rightarrow\operatorname{H}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X-A)\rightarrow\cdots$
(8.2)
$\displaystyle\cdots\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\rightarrow\operatorname{H}^{*}(\operatorname{C}^{*}(A))\rightarrow\operatorname{H}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X-A,A)\rightarrow\cdots$

######  Lemma 8.2.

Suppose $A$ is a unbounded subset of $X$ and $X$ is not coarsely contained in
$A$. If $\mathbb{Z}_{2}\curvearrowright(X,A)$ where $A$ is the fixed point set
of the action, then

$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)=\begin{cases}0&\text{if
$*=0$}\\\
\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\oplus\mathbb{Z}_{2}&\text{if
$*=1$}\\\ \operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)&\text{if
$*\geq 2$}\end{cases}$

###### Proof.

Since $X$ is not coarsely contained in $A$, we have
$\operatorname{H}{\mathrm{X}}^{0}_{\mathbb{Z}_{2}}(X-A)$ and
$\operatorname{H}{\mathrm{X}}^{0}_{\mathbb{Z}_{2}}(X-A,A)=0$. When $A$ is
unbounded, the homology of the complex $\operatorname{C}^{*}(A)$ vanishes
everywhere except in the degree $0$ where it is $\mathbb{Z}_{2}$. The second
long exact sequence in the Lemma 8.1 implies

(8.3)
$0\rightarrow\mathbb{Z}_{2}\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X-A,A)\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X-A)\rightarrow
0\rightarrow\\\ \cdots
0\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A,A)\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\rightarrow
0\cdots$

Where $*\geq 2$.

Since the coefficient ring is $\mathbb{Z}_{2}$, the first five terms give us a
split short exact sequence. Hence
$\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X-A,A)=\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X-A)\oplus\mathbb{Z}_{2}$.
Also, we can see from the long exact sequence that
$\operatorname{H}{\mathrm{X}}^{*}(X-A,A)=\operatorname{H}{\mathrm{X}}^{*}(X-A)$
for all $*\geq 2$.

∎

Hence under the hypothesis of Lemma 8.2, we can rewrite the first long exact
sequence of Lemma 8.1 as follows.

######  Lemma 8.3 (Coarse Gysin sequence).

Suppose $\mathbb{Z}_{2}\curvearrowright X$ and $A\overset{}{\subset}X$ is the
fixed point set of the action. Moreover, assume that $A$ is unbounded and $X$
is not coarsely contained in $A$. Then there is a long exact sequence of the
following form:

$0\rightarrow\operatorname{H}{\mathrm{X}}^{0}_{\mathbb{Z}_{2}}(X-A)\rightarrow\operatorname{H}{\mathrm{X}}^{0}(X-A)\rightarrow
0\\\
\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X-A)\rightarrow\operatorname{H}{\mathrm{X}}^{1}(X-A)\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X-A)\oplus\mathbb{Z}_{2}\rightarrow\cdots\\\
\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\rightarrow\operatorname{H}{\mathrm{X}}^{*}(X-A)\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X-A)\rightarrow\cdots$

where $*\geq 2$.

We are now ready to state our main result of this section.

######  Proposition 8.4.

Suppose $X$ is unbounded and $\mathbb{Z}_{2}$ is acting on $X^{2}$ by
permuting the coordinates. Then there is a long exact sequence of the
following form:

(8.4)
$0\rightarrow\operatorname{H}{\mathrm{X}}^{0}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{0}(X^{2}-\delta(X))\rightarrow
0\\\
\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\oplus\mathbb{Z}_{2}\rightarrow\cdots\\\
\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\cdots$

where $*\geq 2$.

###### Proof.

Since $X$ is unbounded $\delta(X)$ is unbounded and $X^{2}$ is not coarsely
contained in $\delta(X)$. The proposition then follows if we replace $X$ by
$X^{2}$ and $A$ by $\delta(X)$ in Lemma 8.3. ∎

######  Remark 8.5.

If $X$ is unbounded and $\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))=0$,
then it follows from Proposition 8.4 that
$\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$. As a
consequence, we can write the beginning part of the coarse Gysin sequence
(8.4) as follows

(8.5) $\displaystyle
0\rightarrow\mathbb{Z}_{2}\rightarrow\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{2}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\cdots$

This observation will be useful for us in our next theorem where we compute
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$ for some
spaces where $\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))$ is
concentrated in some degree.

######  Theorem 8.6.

Suppose $X$ is a metric space such that for some $n\geq 1$

$\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=\begin{cases}\mathbb{Z}_{2}&*=n,\\\
0&\text{otherwise}.\end{cases}$

Moreover, suppose that
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for
some $*\geq n+1$. Then

$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\left\\{\begin{array}[]{ll}\mathbb{Z}_{2}&\quad\text{if
$n\geq 2$ and }2\leq*\leq n,\\\ 0&\quad\text{otherwise}.\end{array}\right.$

###### Proof.

Observe that $X$ is unbounded because $\operatorname{H}{\mathrm{X}}^{0}(X)=0$.
Elements of $\operatorname{H}{\mathrm{X}}^{0}(X^{2}-\delta(X))$ and
$\operatorname{H}{\mathrm{X}}^{0}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$ are
constant functions on $X^{2}$ with support contained in a neighborhood of
$\delta(X)$. There are no such elements because
$\operatorname{H}{\mathrm{X}}^{0}(X^{2}-\delta(X))=0$ by hypothesis. Hence
$\operatorname{H}{\mathrm{X}}^{0}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$.

Next, we will show that
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ if
$*\geq n+1$. Consider the following part of the coarse Gysin sequence where
$*\geq 2$.

(8.6)
$\displaystyle\rightarrow\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{*+1}(X^{2}-\delta(X))\rightarrow$

Since $\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=0$ for $*\geq n+1$,
the middle map is an isomorphism for $*\geq n+1$. Hence,

$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\operatorname{H}{\mathrm{X}}^{n+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\quad\text{
if }*\geq n+1.$

That means, if
$\operatorname{H}{\mathrm{X}}^{n+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\neq 0$
then $\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\neq
0$ for all $*\geq n+1$. But
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for
some $*\geq n+1$. Therefore
$\operatorname{H}{\mathrm{X}}^{n+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ and
consequently,

$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0\text{
for all }*\geq n+1$

We divide the rest of the calculations into three cases.

Case 1 ($n=1$): We only have to show that
$\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$. By the
hypothesis, $\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))=\mathbb{Z}_{2}$
and $\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$.
Hence, we get the following from the coarse Gysin sequence

$\displaystyle
0\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\mathbb{Z}_{2}\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\oplus\mathbb{Z}_{2}\rightarrow
0$

Since the third map is surjective, the map is injective and consequently, the
second map is trivial. Hence,
$\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$.

Case 2 ($n=2$): Since $\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))=0$,
we observed in 8.5 that
$\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$.
Furthermore, we have
$\operatorname{H}{\mathrm{X}}^{2}(X^{2}-\delta(X))=\mathbb{Z}_{2}$ by
hypothesis and we already showed
$\operatorname{H}{\mathrm{X}}^{3}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$. So a
part of the sequence (8.5) takes the following form.

$0\rightarrow\mathbb{Z}_{2}\rightarrow\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\mathbb{Z}_{2}\rightarrow\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow
0$

So there is an injective map and a surjective map from $\mathbb{Z}_{2}$ into
$\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$. It
follows that
$\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\mathbb{Z}_{2}$.

Case 3 ($n>2$): In this case,
$\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))=0$. The remark 8.5 gives us
$\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$.

Since $\operatorname{H}{\mathrm{X}}^{2}(X^{2}-\delta(X))=0$, the beginning
part of the sequence (8.5) takes the following form.

$0\rightarrow\mathbb{Z}_{2}\rightarrow\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow
0\rightarrow\cdots$

Hence
$\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\mathbb{Z}_{2}$.

Since $\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=0$ for $*\leq n-1$,
it follows from $(\ref{mid})$ that

$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\operatorname{H}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\quad\text{
when }\quad 2\leq*\leq n-2.$

That implies

$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\mathbb{Z}_{2}\text{
when }2\leq*\leq n-1.$

Finally to compute
$\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$, consider
the following part of the coarse Gysin sequence.

$\dots\rightarrow\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{n}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\operatorname{H}{\mathrm{X}}^{n+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\rightarrow\cdots$

Since $\operatorname{H}{\mathrm{X}}^{n}(X^{2}-\delta(X))\neq 0$ by assumption,
it follows from the above sequence that
$\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\neq 0$. We
already know that the fourth term is trivial in the above sequence. That means
the third map is surjective. Since
$\operatorname{H}{\mathrm{X}}^{n}(X^{2}-\delta(X))=\mathbb{Z}_{2}$, we can
conclude that
$\operatorname{H}{\mathrm{X}}^{n}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\mathbb{Z}_{2}$.
∎

######  Remark 8.7.

It follows from the first part of the proof of the above theorem that if
$\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=0$ for all $*\geq n+1$ and
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for
some $*\geq n+1$, then
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for all
$*\geq n+1$. In particular, this implies that $cvk^{*}(X)$ is trivial for
$*\geq n+1$ and hence $\operatorname{cobdim}(X)\leq n$ for such $X$.

##  9\. Upper bound of cobdim

In this section, our goal is to estimate $\operatorname{cobdim}$ for proper,
uniformly acyclic $n$-manifolds. The key tool that we are going to use here is
a coarse Alexander duality theorem that holds for such spaces. In fact, such
duality holds for a more general class of metric spaces called coarse PD($n$)
spaces, first introduced by Kapovich and Kleiner in [KK] where they proved a
coarse Alexander duality theorem for these spaces. A different treatment of
coarse PD($n$) space and coarse Alexander duality using coarse cohomology is
given in [BB20]. Let us now recall the definition of a coarse PD($n$) space
from [BB20].

######  Definition 9.1.

A metric space $X$ is a _coarse PD( $n$) space_, if there exist chain maps
$p:\operatorname{C}^{*}(X;\mathbb{Z})\to\operatorname{C}{\mathrm{X}}_{n-*}(X;\mathbb{Z})$
and
$q:\operatorname{C}{\mathrm{X}}_{n-*}(X;\mathbb{Z})\to\operatorname{C}^{*}(X;\mathbb{Z})$,
so that $pq$ and $qp$ are chain homotopic to identities via chain homotopies
$G:\operatorname{C}{\mathrm{X}}_{*}(X;\mathbb{Z})\to\operatorname{C}{\mathrm{X}}_{*+1}(X;\mathbb{Z})$
and
$F:\operatorname{C}^{*}(X;\mathbb{Z})\to\operatorname{C}^{*-1}(X;\mathbb{Z})$
which are controlled:

$\displaystyle\forall\phi\overset{}{\in}\operatorname{C}^{*}(X;\mathbb{Z})$
$\displaystyle\qquad\lvert
p(\phi)\rvert\overset{\mathrm{c}}{\subset}\lvert\phi\rvert,$
$\displaystyle\forall\phi\overset{}{\in}\operatorname{C}^{*}(X;\mathbb{Z})$
$\displaystyle\quad\lvert
F(\phi)\rvert\mathbin{\overset{\mathrm{c}}{\cap}}\Delta\overset{\mathrm{c}}{\subset}\lvert\phi\rvert,$
$\displaystyle\forall
c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(X;\mathbb{Z})$
$\displaystyle\quad\lvert
q(c)\rvert\mathbin{\overset{\mathrm{c}}{\cap}}\Delta\overset{\mathrm{c}}{\subset}\lvert
c\rvert,$ $\displaystyle\forall
c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(X;\mathbb{Z})$
$\displaystyle\qquad\lvert G(c)\rvert\overset{\mathrm{c}}{\subset}\lvert
c\rvert.$

######  Example 9.2.

Any proper, uniformly acyclic $n$-manifold is a coarse PD($n$) space
[BB20]*Corollary 8.3. In particular, the universal cover of a closed,
aspherical $n$-manifold is a coarse PD($n$) space.

We now recall the coarse Alexander duality theorem from [BB20].

######  Theorem 9.3 (Coarse Alexander duality [BB20]).

If $X$ is a coarse PD($n$) space, then for any $A\overset{}{\subset}X$ and
finitely generated abelian group $G$

$\operatorname{H}{\mathrm{X}}^{*}(X-A;G)\cong\operatorname{H}{\mathrm{X}}_{n-*}(A;G).$

By Theorem 9.3, if $X$ is a coarse PD($n$) space, then
$\operatorname{H}{\mathrm{X}}^{*}(X-X;G)=\operatorname{H}{\mathrm{X}}_{n-*}(X;G)$.
Observe that
$\operatorname{C}{\mathrm{X}}^{*}(X-X;G)=\operatorname{C}^{*}(X;G)$ and
$\operatorname{C}^{*}(X;G)$ is acyclic for $*\geq 1$. Hence, for any coarse
PD($n$) space $X$ we obtain

$\operatorname{H}{\mathrm{X}}_{*}(X;\mathbb{Z}_{2})=\operatorname{H}{\mathrm{X}}^{n-*}(X-X;\mathbb{Z}_{2})=\begin{cases}\mathbb{Z}_{2}&*=n,\\\
0&\text{otherwise}.\end{cases}$

For the rest of the paper, the omitted coefficient will mean $\mathbb{Z}_{2}$.

######  Lemma 9.4.

If $X$ is a coarse PD($n$) space, then

$\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=\begin{cases}\mathbb{Z}_{2}&*=n,\\\
0&\text{otherwise}.\end{cases}$

###### Proof.

If $X$ is a coarse PD($n$) space, then $X^{2}$ is a coarse PD($2n$) space.
Applying Theorem 9.3, we get

$\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=\operatorname{H}{\mathrm{X}}_{2n-*}(\delta(X))=\begin{cases}\mathbb{Z}_{2}&*=n,\\\
0&\text{otherwise}.\end{cases}$

∎

######  Lemma 9.5.

If $X$ is a proper, uniformly acyclic $n$-manifold where $n\geq 1$, then
$\operatorname{H}{\mathrm{X}}^{i}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for
$i\geq 2n+2$.

###### Proof.

If $X$ is bounded, then $X^{2}\mathbin{\overset{\mathrm{c}}{=}}\delta(X)$ and
the claim follows from Theorem 3.5(1). So, we assume that $X$ is unbounded.
This implies that $X^{2}$ is not coarsely contained in $\delta(X)$. By
hypothesis, $X^{2}$ is uniformly acyclic and locally acyclic. Hence we can
apply Corollary 3.5 to the pair $(X^{2},\delta(X))$ to conclude that
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=\varinjlim\tilde{\operatorname{H}}^{*-1}((X^{2}-N_{r}(\delta(X))/\mathbb{Z}_{2})$.
Since $(X^{2}-N_{r}(\delta(X)))/\mathbb{Z}_{2}$ is a $2n$-manifold,
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for all
$*\geq 2n+2$. ∎

######  Theorem 9.6.

If $X$ is a proper, uniformly acyclic $n$-manifold, then
$\operatorname{cobdim}(X)\leq n$.

###### Proof.

For $n=0$, the claim is trivial. For $n\geq 1$, Lemma 9.4 and Lemma 9.5
implies that $X$ satisfies the hypothesis of Theorem 8.6. Hence by Theorem
8.6, we have
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for
$*\geq n+1$. This implies $\operatorname{cobdim}(X)\leq n$. ∎

######  Remark 9.7.

Note that to prove Theorem 9.6, we did not need the full strength of the
Theorem 8.6. We needed to show
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))=0$ for
$*\geq n+1$ which only requires vanishing of
$\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))$ for $*\geq n+1$ and
vanishing of
$\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$ for some
$*\geq n+1$ (see remark 8.7).

Our next goal is to improve Theorem 9.6 to include manifolds with boundaries.
For that, we need to impose a condition on the metric of the boundary. This is
the purpose of the following definition which is inspired by the uniformly
locally $k$-connected space defined in [DFW].

######  Definition 9.8.

A metric space $(X,d)$ is uniformly locally acyclic, if for every
$\epsilon>0$, there is a $\delta>0$ such that any ball of radius $\delta$ is
acyclic inside a ball of radius $\epsilon$.

######  Lemma 9.9.

Let $(X,d)$ be a uniformly locally acyclic metric space. Then the space
$X\times[1,\infty)$ has a metric that makes the space uniformly acyclic away
from $X\times\\{1\\}$ and the map $x\mapsto(x,1)$ is an isometric embedding of
$X$ into $X\times[1,\infty)$.

###### Proof.

The construction of the metric follows the one appearing in Lemma 2.2 of
[DFW]. Choose a continuous strictly increasing function
$\phi:[1,\infty)\rightarrow[1,\infty)$ with $\phi(1)=1$. Let $d$ be the
original metric on $X$ and define a function $\rho^{\prime}$ by

  1. (1)

$\rho^{\prime}((x,t),(x^{\prime},t))=\phi(t)d(x,x^{\prime})$.

  2. (2)

$\rho^{\prime}((x,t),(x,t^{\prime}))=|t-t^{\prime}|$.

We then define $\rho:(X\times[1,\infty))^{2}\rightarrow[0,\infty)$ to be

$\rho((x,t),(x^{\prime},t^{\prime}))=\inf\sum_{i=1}^{l}\rho^{\prime}((x_{i},t_{i}),(x_{i-1},t_{i-1}))$

where the sum is over all chains

$(x,t)=(x_{0},t_{0}),(x_{1},t_{1}),\ldots,(x_{l},t_{l})=(x^{\prime},t^{\prime})$

and each segment is either horizontal or vertical. It pays to move towards 1
before moving in the $X$-direction. Also $\phi(1)=1$ implies that
$X\times\\{1\\}$ with the subspace metric is isometric to $X$ via the map
$(x,1)\mapsto x$. Now we will describe a $\phi$, so that the corresponding
metric $\rho$ makes $X\times[1,\infty)$ uniformly acyclic away from
$X\times\\{1\\}$. Since $X$ is uniformly locally acyclic, we have an infinite
positive decreasing sequence $\\{r_{i}\\}$ with $r_{1}=1$ such that for every
$x\overset{}{\in}X$, the inclusions
$\dots\overset{}{\subset}B_{r_{i}}^{d}(x)\overset{}{\subset}B_{r_{i-1}}^{d}(x)\overset{}{\subset}$
are nullhomotopic maps. Set $\phi(t)=\frac{1}{r_{t}}$ for
$t\overset{}{\in}\mathbb{N}$. For nonintegral values of $t$, we set

$\phi(t)=\phi([t])+(t-[t])\phi([t]+1)$

Suppose, $N_{i}=\frac{\phi(i)}{\phi(i-1)}$. Now we consider the ball
$B^{\rho}_{k}(x,i)\overset{}{\subset}X\times[1,\infty)$. Note that
$B^{\rho}_{k}(x,i)\overset{}{\subset}B^{d}_{\frac{k}{N_{i-k}}}(x)\times[i-k,i+k]$
and that $B^{\rho}_{k}(x,i)$ contracts in itself to
$B^{\rho}_{k}(x,i)\cap(X\times[i-k,i])\overset{}{\subset}B^{d}_{\frac{k}{N_{i-k}}}(x)\times[i-k,i]$.
Also,
$B^{d}_{\frac{k}{N_{i-k-1}}}(x)\times\\{i-k-1\\}\overset{}{\subset}B_{k+2}^{\rho}(x,i)$.
So, $B^{\rho}_{k}(x,i)$ can be contracted inside $B_{k+2}^{\rho}(x,i)$ by
pushing it down to $(i-k-1)$-level and contracting it there. ∎

The following gluing lemma in a slightly different form can be found in [BH].

######  Lemma 9.10 ([BH], Lemma I.5.24).

Let $X_{1}$ and $X_{2}$ be two proper metric spaces. Let
$A_{i}\overset{}{\subset}X_{i}$ be the closed subsets and $f:A_{1}\rightarrow
A_{2}$ be an isometry. Let $Y$ be the space obtained by gluing $(X_{i},A_{i})$
along $A_{i}$ via the map $f$. Define $d:Y\times Y\rightarrow\mathbb{R}$ as
follows

$d(x,y)=\begin{cases}d_{i}(x,y)&\text{if $x,y\overset{}{\in}X_{i}$}\\\
\inf_{a\overset{}{\in}A_{1}}\\{d_{1}(x,a)+d_{2}(f(a),y)\\}&\text{if
$x\overset{}{\in}X_{1},y\overset{}{\in}X_{2}$}.\end{cases}$

Then,

  1. (1)

$d$ is a proper metric on $Y$.

  2. (2)

The canonical inclusions $X_{i}\hookrightarrow Y$ are isometric embedding.

######  Proposition 9.11.

Any uniformly acyclic, proper $n$-manifold with uniformly locally acyclic
boundary admits an isometric embedding into a uniformly acyclic, proper
$n$-manifold.

###### Proof.

Since $\partial M$ is uniformly locally acyclic, Lemma 9.9 allows us to equip
$\partial M\times[1,\infty)$ with a metric so that it is uniformly acyclic
away from $\partial M\times\\{1\\}$. Let
$\rho,\mu:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be two functions such that
any ball $B(x,r)$ in $\partial M\times[1,\infty)$ is acyclic inside
$B(x,\rho(r))$ whenever $d(x,\partial M\times\\{1\\})\geq\mu(r)$. We glue
$\partial M\times[1,\infty)$ to $M$ along $\partial M$ by the attaching map
$(x,1)\mapsto x$. Let $Y$ be the resulting space. By Lemma 9.10, there is a
proper metric on $Y$ such that the canonical maps $M\hookrightarrow Y$ and
$\partial M\times[1,\infty)\hookrightarrow Y$ are isometric embedding.

We claim that $Y$ is uniformly acyclic. Since $M$ is uniformly acyclic, there
exists a function $\tau:[0,\infty)\rightarrow[0,\infty)$ such that, for any
$r\geq 0$, any ball of radius $r$ in $M$ is acyclic inside a concentric ball
of radius $\tau(r)$. Take a ball $B_{r}(x)$ of radius $r$ in $Y$. If
$x\overset{}{\in}M$ and $d(x,\partial M)\geq r$, then $B_{r}(x)$ is contained
inside $M$ because points in $\partial M\times[1,\infty)$ are at least as far
from $x$ as points in $\partial M$ by the construction of the metric on $Y$.
Hence, $B_{r}(x)$ is acyclic inside $B_{\tau(r)}(x)$. If
$x\overset{}{\in}\partial M\times[1,\infty)$ and $d(x,\partial
M)\geq\mu(r)+r$, then $B(x,r)\overset{}{\subset}\partial M\times[1,\infty)$
and hence is acyclic in $B(x,\rho(r))$. If $d(x,\partial M)\leq\mu(r)+r$, we
can deformation retract $B_{r}(x)\cap(\partial M\times[1,\infty))$ inside
$\partial M\times[1,\infty)$ by sliding it along $[1,\infty)$ until it lands
on $\partial M\times\\{1\\}$. Note that this deformation takes place inside
$\partial M\times[1,\mu(r)+2r]$, and the diameter shrinks as one approaches
$\partial M\times\\{1\\}$. Hence the deformation takes place in a set of
diameter atmost $\mu(r)+2r+2r$. Also, the deformed ball is now contained in
$M$ and has diameter at most $2r$ and hence it is acyclic inside a set of
diameter at most $\tau(2r)$ by uniform acyclicity of $M$. Hence, we conclude
that $B_{r}(x)$ is acyclic inside a set of diameter at most
$\mu(r)+4r+\tau(2r)$. Hence $Y$ is uniformly acyclic. ∎

As a consequence of the above proposition, we get the following.

######  Theorem 9.12.

If $X$ is a proper, uniformly acyclic $n$-manifold with uniformly locally
acyclic boundary, then $\operatorname{cobdim}(X)\leq n$.

###### Proof.

By Proposition 9.11, we have a uniformly acyclic proper $n$-manifold $Y$ such
that $X$ embeds isometrically in $Y$. By Theorem 6.3, it follows that
$\operatorname{cobdim}(X)\leq\operatorname{cobdim}(Y)$. By Corollary 9.6, we
know $\operatorname{cobdim}(Y)\leq n$ and hence $\operatorname{cobdim}(X)\leq
n$. ∎

Now we can state the following improvement of Corollary 6.5. The proof is
immediate from Theorem 6.3 and Theorem 9.12.

######  Corollary 9.13.

If $\operatorname{cobdim}(X)\geq n$, then $X$ cannot be coarsely embedded into
a proper, uniformly acyclic $(n-1)$-manifold with a uniformly locally acyclic
boundary.

######  Definition 9.14 (Cocompact action dimension).

The cocompact action dimension $\operatorname{cadim}(G)$ of a group $G$ is the
least dimension of a contractible manifold (possibly with boundary) that
admits a proper cocompact $G$-action.

######  Corollary 9.15.

$\operatorname{cadim}(G)\geq\operatorname{cobdim}(G)$.

###### Proof.

Suppose $\operatorname{cadim}(G)=n$. Then $G$ admits a proper, cocompact
action on an acyclic $n$-manifold $M$. Choose a point $x_{0}\overset{}{\in}M$.
By Milnor–Schwarz Lemma, the map $g\mapsto g.x_{0}$ gives a coarse equivalence
$f:G\rightarrow M$. Since $M$ is acyclic and it admits a cocompact action, $M$
is uniformly acyclic. Similarly, since $\partial M$ is locally acyclic and it
admits a cocompact action, it is uniformly locally acyclic. Since $M$ is
proper, by Corollary 9.13, we get $\operatorname{cobdim}(G)\leq
n=\operatorname{cadim}(G)$.

∎

######  Remark 9.16.

The action dimension, denoted by $\operatorname{actdim}(G)$, of a group $G$ is
the least dimension of a contractible manifold that admits a proper
$G$-action. It is easy to see that
$\operatorname{cadim}(G)\geq\operatorname{actdim}(G)$, however, we do not know
of any groups where $\operatorname{cadim}>\operatorname{actdim}$. In any case,
we believe that $\operatorname{cobdim}(G)$ gives a lower bound to
$\operatorname{actdim}(G)$. A naive approach might be as follows: Suppose, $G$
admits a proper action on a contractible $n$-manifold $M$. Then the map
$f:G\rightarrow M$, sending $G$ to one of its orbit gives a coarse embedding.
The image of $f$ is uniformly contractible in $M$: for any $r$ there exists
$s$ such that any ball of radius $r$ in $M$ centered at a point in $f(G)$ is
uniformly contractible inside a concentric ball of radius $s$. This should
imply that there is a bounded function $p:f(G)\rightarrow(0,\infty)$ such that
the space $X=\cup_{x\overset{}{\in}f(G)}N_{p(x)}(x)$ is uniformly
contractible. Since $G$ acts on $X$ cocompactly, $X$ has uniformly locally
acyclic boundary. Theorem 9.12 then implies that $\operatorname{cobdim}(X)\leq
n$. Since $G$ coarsely embed into $X$ by the map $f$, we get
$\operatorname{cobdim}(G)\leq\operatorname{cobdim}(X)\leq n$, proving the
desired claim. However, the author does not know how to prove that there
exists such $X$.

##  10\. Lower bound of cobdim

######  Theorem 10.1.

If $\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))=0$ for $*\leq n-1$, then
$\operatorname{cobdim}(X)\geq n$.

###### Proof.

For $n=0$, the claim is trivial. If $n=1$, the assumption says
$\operatorname{H}{\mathrm{X}}^{0}(X^{2}-\delta(X))=0$. This means $X$ is
unbounded, otherwise, non zero constant functions from $X^{2}$ to the
$\mathbb{Z}_{2}$ give nontrivial elements in
$\operatorname{H}{\mathrm{X}}^{0}(X^{2}-\delta(X))$. Hence,
$\operatorname{cobdim}(X)\geq 1$ in this case.

Suppose $n\geq 2$ and $f:X\rightarrow\ell^{\infty}$ is an isometry. We first
show that
$f^{*}:\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(\ell^{\infty})^{2}-\delta(\ell^{\infty}))\xrightarrow{f^{*}}\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$
is a nontrivial map. To see that, consider the following part of the maps
between the concerned coarse Gysin sequences. Our goal is to show the second
vertical map is nontrivial.

${\operatorname{H}{\mathrm{X}}^{1}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))}$${\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))\oplus\mathbb{Z}_{2}}$${\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))}$${{}\dots}$${\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))}$${\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))\oplus\mathbb{Z}_{2}}$${\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))}$${{}\ldots}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{j}$

By commutativity of the diagram, Our claim follows if we can show that $j$ is
injective and the first vertical map is nontrivial. Since
$\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))=0$, it follows from (8.5)
that $j$ is injective. Next, we show that the first vertical map in the above
commutative diagram is non trivial. It is equivalent to showing that the
following map is nontrivial.

$f^{*}:\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}),\delta(\ell^{\infty}))\rightarrow\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}((X^{2}-\delta(X),\delta(X))$

We can show that using the following diagram.

${0}$${\operatorname{H}^{0}(\operatorname{C}^{*}(\delta(\ell^{\infty}))}$${\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}),\delta(\ell^{\infty}))}$${\operatorname{H}{\mathrm{X}}^{1}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))}$${0}$${0}$${\operatorname{H}^{0}(\operatorname{C}^{*}(\delta(X)))}$${\operatorname{H}{\mathrm{X}}^{1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X),\delta(X))}$${\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))}$${0}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{j}$

The long exact sequence is due to the second long exact sequence of Lemma 8.1
combined with the fact that
$\operatorname{H}{\mathrm{X}}^{1}((\ell^{\infty})^{2}-\delta(\ell^{\infty})),\operatorname{H}^{1}(\operatorname{C}^{*}(\delta(\ell^{\infty}))),\operatorname{H}{\mathrm{X}}^{1}(X^{2}-\delta(X))$
and $\operatorname{H}^{1}(\operatorname{C}^{*}(\delta(X)))$ all are trivial.
The second vertical map is an isomorphism as both the domain and the range are
the constant functions defined on the respective spaces. The fourth vertical
map is an isomorphism because both the domain and the range are trivial. So,
applying the Five lemma, we conclude that the middle vertical map is an
isomorphism, as desired.

Hence we can conclude that the map
$f^{*}:\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(\ell^{\infty})^{2}-\delta(\ell^{\infty}))\rightarrow\operatorname{H}{\mathrm{X}}^{2}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))$
is nontrivial.

Let us now consider the maps between the following parts of the coarse Gysin
sequences where $*\geq 2$.

${\operatorname{H}{\mathrm{X}}^{*}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))}$${\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(\ell^{\infty})^{2}-\delta(\ell^{\infty}))}$${\operatorname{H}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}((\ell^{\infty})^{2}-\delta(\ell^{\infty}))}$${\operatorname{H}{\mathrm{X}}^{*}(X^{2}-\delta(X))}$${\operatorname{H}{\mathrm{X}}^{*}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))}$${\operatorname{H}{\mathrm{X}}^{*+1}_{\mathbb{Z}_{2}}(X^{2}-\delta(X))}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{*}}$

Note that the first terms of both sequences above are trivial for $*\leq n-1$.
That implies that the third horizontal map in the above diagram is injective.
Hence by commutativity of the diagram, if the second vertical map is
nontrivial then so is the third vertical map whenever $*\leq n-1$. We saw
previously that, when $*=2$ the second vertical map is injective. It now
follows by induction that the third vertical map is injective when $*\leq
n-1$. In particular, when $*=n-1$, injectivity of the third vertical map means
$cvk^{n}(X)\neq 0$. Hence, $\operatorname{cobdim}(X)\geq n$.

∎

######  Corollary 10.2.

If $X$ is a coarse $PD(n)$ space, then $X$ cannot be coarsely embedded into a
proper, uniformly acyclic $(n-1)$-manifold with a uniformly locally acyclic
boundary.

###### Proof.

If $X$ is a coarse $PD(n)$ space, then it satisfies the hypotheses of Theorem
10.1. Hence, $\operatorname{cobdim}(X)\geq n$. The claim now follows from
Corollary 9.13. ∎

######  Example 10.3.

If $X$ is a proper, uniformly acyclic $n$-manifold, then it satisfies all the
hypotheses of the above corollary. Hence $\operatorname{cobdim}(X)\geq n$.
Theorem 9.12 implies that $\operatorname{cobdim}(X)\leq n$. Hence
$\operatorname{cobdim}(X)=n$ whenever $X$ is a proper, uniformly acyclic
$n$-manifold.

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