# Coarse cohomology of the complement

Arka Banerjee  Department of Mathematics and Statistics
Auburn University
Auburn, AL 36849<EMAIL_ADDRESS>and  Boris Okun  Department of
Mathematical Sciences
University of Wisconsin–Milwaukee
Milwaukee, WI 53201<EMAIL_ADDRESS>
###### Abstract.

In this paper we define the coarse (co)homology of the complement of a
subspace in a metric space, generalizing the coarse (co)homology of Roe. We
give a model space which encodes coarse geometric structure of the complement.
We also introduce a new approach to coarse Poincaré duality spaces. We prove a
version of coarse Alexander duality for these spaces and give a homological
criterion for a space to be a coarse PD($n$) space. Our approach is inspired
by the work of Kapovich and Kleiner, but is somewhat different, and we
believe, simpler.

##  1\. Introduction

The goal of this paper is to recast the coarse Alexander duality theorems of
Kapovich and Kleiner [kk05] into the framework of the coarse homology and
cohomology theories of Roe [r93, r03]. Recall that for $A$, a subcomplex of
$S^{n}$, the classical Alexander duality is given by the isomorphisms
$\tilde{\operatorname{H}}^{*}(S^{n}-A)\cong\tilde{\operatorname{H}}_{n-1-*}(A)$
and
$\tilde{\operatorname{H}}_{*}(S^{n}-A)\cong\tilde{\operatorname{H}}^{n-1-*}(A)$.
The results of [kk05] are similar in spirit in the context of subsets of
coarse Poincaré duality spaces, but their actual statements are quite
technical. Our main result is the following theorem (Theorem 6.2):

######  Theorem 1.1.

If $X$ is a coarse PD($n$) space, then for any
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$

$\displaystyle\operatorname{H}{\mathrm{X}}^{k}(X-A)$
$\displaystyle\cong\operatorname{H}{\mathrm{X}}_{n-k}(A),$
$\displaystyle\operatorname{H}{\mathrm{X}}_{k}(X-A)$
$\displaystyle\cong\operatorname{H}{\mathrm{X}}^{n-k}(A).$

Our main innovation is in the definition of the left-hand sides of these
formulas, and in fact in the introduction of a notion of the coarse
complement.

One might think naively that, since in the coarse geometry one is allowed to
replace any subset with its metric neighborhood, the coarse complement $X-A$
should be the complement of a (perhaps bigger and bigger) metric neighborhood,
$X-N_{r}(A)$. If $A$ is bounded, this is indeed the right picture, as all
these complements are coarsely equivalent to each other (and in fact to $X$).
However, for unbounded subsets it is not a right notion. For example, if $X$
is a Euclidean plane and $A$ is a line in $X$, then $X-N_{r}(A)$ are still all
coarsely equivalent to $X$, contrary to the idea that a line coarsely
separates a plane, and therefore the coarse topology should change.

Instead, we consider complements of _expanding_ neighborhoods $X-N_{f}(A)$,
where $N_{f}(A)=\bigcup_{x\overset{}{\in}A}B(x,f(x))$ for proper functions
$f:A\to[0,\infty)$. These complements form a directed system of metric spaces
under inclusion (in the direction of slower growth of $f$), and we define
coarse (co)chain complexes of the complement $X-A$ as the appropriate limits
of the usual coarse complexes of these complements.

Our second observation, inspired by [msw11], is that almost all relevant
information is captured by the notion of _coarse containment_. We write
$A\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$ to
mean that $A\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{r}(B)$ for some
$r$. In some sense coarse geometry is the study of the poset of the
equivalence classes of subsets of $X$ induced by this relation. From this
point of view the coarse complement
$X\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ is the collection of
classes of subsets which are coarsely disjoint from $A$, i.e. have bounded
intersection with any metric neighborhood of $A$. This leads us to a very
explicit description of the limiting complexes in terms of support, which is
very similar to Roe’s definition of the coarse cochains. This description is
key to our proof of Alexander duality.

Moreover, this description allows us to identify the coarse type of
$X\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ with the coarse type of
the metric space $(X/\bar{A},d_{A})$ whose metric comes from the pseudometric
$d_{A}(x,y)=\min\\{d(x,A)+d(y,A),d(x,y)\\}$ on $X$. This identification can be
helpful in computations of $\operatorname{H}{\mathrm{X}}^{*}(X-A)$.

Our definition of a coarse PD($n$) space is a slight generalization of the
definition of Kapovich and Kleiner, in terms of controlled chain homotopy
equivalence between coarse chain and cochain complexes. We do not require
bounded geometry or simplicial structure. The classical Poincaré duality maps
on manifolds are given by the cap product with the fundamental class in one
direction and the slant product with the orientation class in the other. We
define a coarse version of these classes (called _orientation pair_) and prove
a similar statement for coarse PD($n$) spaces (Theorem 8.2):

######  Theorem 1.2.

$X$ is a coarse PD($n$) space if and only if it has an orientation pair.

As a corollary, we obtain that uniformly acyclic topological manifolds are
coarse PD($n$) spaces, a similar statement is claimed in [kk05] with only a
sketch of the proof.

######  Remark.

This theorem, as well as the Alexander duality, indicate that the term “coarse
PD($n$) space” is perhaps a misnomer, a more appropriate name would be “coarse
homology manifold”.

### Overview

This paper is organized as follows. In Section 2, we introduce coarse language
and state basic results about coarse containment. We also describe several
variations of Alexander–Spanier cochains and the corresponding cohomology
theories.

Section 3 contains our main definition of the coarse (co)homology of a
complement. The main result there is Lemma 3.2, which gives an equivalent
description of the cochain complex. In Section 4, we discuss a more geometric
version of our approach. In Section 5 we discuss the notion of coarse
separation and show that elements in
$\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}_{2})$ corresponds to
different ways in which $A$ coarsely separates $X$. In Section 6 we introduce
the notion of a coarse Poincaré duality space, and prove the main result of
this paper, the coarse Alexander Duality, Theorem 6.2. A standard argument
then implies a coarse version of the Jordan separation theorem. We study
coarse versions of the standard (co)homological products in Section 7 and use
them in Section 8 to derive a homological criterion for a space to be a coarse
PD($n$) space, and to prove that uniformly acyclic $n$-manifolds are coarse
PD($n$) spaces. Finally, in Section 9 we show that (under a mild set-theoretic
assumption) integral coarse chains and coarse cochains are dual of each other
in the sense of taking $\operatorname{Hom}(\cdot,\mathbb{Z})$.

### Acknowledgements

We thank Kevin Schreve for many helpful conversations and valuable comments on
the early draft of the paper. The second author would like to thank the Max
Planck Institute for Mathematics for its support and excellent working
conditions.

##  2\. Preliminaries

### Coarse containment

We adopt the notation from [msw11]. Let $(X,d)$ be a metric space. For
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$ 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 $B$, $A\mathrel{\overset{\scriptscriptstyle R}{\subset}}B$, if
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{R}(B)$. $A$ is _coarsely
contained_ in $B$,
$A\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$, if
$A\mathrel{\overset{\scriptscriptstyle R}{\subset}}B$ for some $R$. Two
subsets are _coarsely equal_ ,
$A\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}B$, if
$A\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$ and
$B\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$. All
bounded subsets are obviously coarsely equal, and we declare the empty set to
be coarsely equal to bounded subsets. Formally, this requires special casing
some of the metric arguments in the paper. A way to avoid doing this is to add
one extra point to $X$ and all its subsets and extend the metric. We will
simply ignore this annoyance.

We will use the following observation about the intersection of metric
neighborhoods in later sections:

######  Lemma 2.1.

Let $A$, $B$ and $C$ be subsets of metric space $X$. If
$C\mathrel{\overset{\scriptscriptstyle R}{\subset}}A$ and
$C\mathrel{\overset{\scriptscriptstyle r}{\subset}}B$, then

$C\mathrel{\overset{\scriptscriptstyle R}{\subset}}A\cap N_{R+r}(B).$

###### Proof.

If $x\overset{}{\in}C$, then there exist $y\overset{}{\in}A$ such that
$d(x,y)<R$. Therefore $y\overset{}{\in}A\cap N_{R+r}(B)$, and hence
$x\overset{}{\in}N_{R}(A\cap N_{R+r}(B))$ by the triangle inequality. ∎

### Coarse intersection

Recall from [msw11] the notion of _coarse intersection_ :
$A\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}B\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}C$
if for all sufficiently large $R$, $N_{R}(A)\cap
N_{R}(B)\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}C$. 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.
$A\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}B\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C$
means that for any $R$, $N_{R}(A)\cap
N_{R}(B)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C$.
It is immediate from Lemma 2.1 that this condition is equivalent to the
condition that for any $R$, $A\cap
N_{R}(B)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C$.

The following Lemma shows that coarse containment and intersection behave as
expected.

######  Lemma 2.2.

Let $A$, $B$, $C$ and $D$ be subsets of metric space $X$. If
$A\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$,
$A\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C$, and
$B\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}D$,
then $A\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}D$.

###### Proof.

By hypothesis, $A\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{R}(B)$ and
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{R}(C)$ for some $R$. Also
we have $N_{R}(B)\cap
N_{R}(C)\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{R^{\prime}}(D)$ for
some $R^{\prime}$. It follows that
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{R^{\prime}}(D)$. ∎

A subset $C$ is _coarsely disjoint_ from $A$ if
$A\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}C\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}*$,
in other words if $A\cap N_{R}(C)$ is bounded for all $R$. Note that coarse
disjointness is independent of the ambient space, it depends only on the
metric on $A\cup C$.

### Coarse difference

We will write
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ to mean
that $C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$
and
$A\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}C\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}*$.
One should think of the coarse difference
$B\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ as a collection of
subsets, it rarely has a well-defined coarse type in $X$. Nevertheless, the
notion of coarse containment between differences is well-defined. We will
write $(D\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}C)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}(B\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A)$ to mean
that
$Y\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}D\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}C$ implies
$Y\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$. Unwinding
the definitions, this really means that any subset of any metric neighborhood
of $D$ which has bounded intersection with $C$ is contained in some metric
neighborhood of $B$ and has bounded intersection with $A$.

######  Lemma 2.3.

$B\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}D\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$
if and only if $(D\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}(D\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}B)$.

###### Proof.

The forward direction is obvious. To prove the converse, suppose
$B\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}D\not\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$,
then the intersection $D$ and some metric neighborhood of $B$ contains a
sequence of points going away from $A$. This sequence is in
$D\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ but not in
$D\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}B$. ∎

### Coarse maps

Suppose $f:X\to Y$ is a map between metric spaces. $f$ is _proper_ if
$f^{-1}(B)$ is bounded in $X$ whenever $B$ is bounded in $Y$. $f$ is _coarse_
if it is proper and has an upper control: there exists a function
$\rho:[0,\infty)\to[0,\infty)$, so that

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

$f$ is a _coarse embedding_ if there exists two proper functions
$\rho_{-},\rho_{+}:[0,\infty)\to[0,\infty)$ such that

$\rho_{-}(d(x,y))\leq d(f(x),f(y))\leq\rho_{+}(d(x,y)).$

$f$ is a _coarse equivalence_ if it is a coarse embedding and
$Y\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}f(X)$.

Two maps $f,g:X\to Y$ are _close_ if $d(f(x),g(x))$ is uniformly bounded. It
is easy to see that a coarse map $f:X\to Y$ is a coarse equivalence if and
only if it has a coarse inverse: i.e. if there exists a coarse map $g:Y\to X$,
such that the compositions $fg$ and $gf$ are close to the identities.

The following lemma is immediate from the definitions.

######  Lemma 2.4.

Maps with an upper control preserve the coarse containment, and the preimages
of coarsely disjoint sets under a coarse map are coarsely disjoint.

A map of pairs $f:(X,A)\to(Y,B)$ is _relatively proper_ if for any
$D\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$,
$f^{-1}(D)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$,
and it is _relatively coarse_ if it is relatively proper and has an upper
control.

Relatively coarse maps induce coarse maps on the coarse complements, in the
following sense.

######  Lemma 2.5.

Let $f:(X,A)\to(Y,B)$ be a relatively coarse map and
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$. Then
$f(C)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}Y\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}B$ and
$f|_{C}$ is a coarse map.

###### Proof.

To prove the first claim, we need to show that for any
$D\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$,
$f(C)\cap D$ is bounded. Since $f$ is relatively proper,
$f^{-1}(D)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$.
Therefore,

$C\cap
f^{-1}(D)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}A\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}*,$

and thus $C\cap f^{-1}(D)$ is bounded. Since $f(C)\cap D=f(C\cap f^{-1}(D))$
and $f$ has an upper control $f(C)\cap D$ is bounded.

To prove $f|_{C}$ is a coarse map, we only need to show $f|_{C}$ is a proper
map because we already have an upper control by definition. Let
$D\mathrel{\overset{\scriptscriptstyle}{\subset}}Y$ be bounded. Then
$D\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$ and,
since $f$ is relatively proper,
$f^{-1}(D)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$.
Thus

$f|_{C}^{-1}(D)=f^{-1}(D)\cap
C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}C\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}*.$

Hence $f|_{C}^{-1}(D)$ is bounded, and $f|_{C}$ is proper. ∎

### Alexander–Spanier complexes

We will refer to points in $X^{n+1}$ as _$n$ -simplices_. Let $G$ be an
abelian group (we will be mostly concerned with $G=\mathbb{Z}$ or
$\mathbb{Z}/2$). We will think of functions $X^{n+1}\to G$ as $n$-cochains or
$n$-chains on $X$, depending on the context, and in the latter case will use
additive notation $c=\sum_{\sigma\overset{}{\in}X^{n+1}}c_{\sigma}\sigma$.

The basic complex is the complex of finitely supported integral chains

$\operatorname{C}_{*}(X):=\\{c=\sum_{i=1}^{k}c_{i}\sigma_{i}\mid
c_{i}\overset{}{\in}\mathbb{Z},\,\sigma_{i}\overset{}{\in}X^{*+1}\\}$

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

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

Tensoring it with $G$ gives the complex of finitely supported chains with
coefficients in $G$.

$\operatorname{C}_{*}(X;G):=\\{c=\sum_{i=1}^{k}c_{i}\sigma_{i}\mid
c_{i}\overset{}{\in}G,\,\sigma_{i}\overset{}{\in}X^{*+1}\\}$

Taking $\operatorname{Hom}(\operatorname{C}_{*}(X),G)$ gives the complex of
all Alexander–Spanier cochains

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

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.

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

### 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 $\phi:X^{n+1}\to\mathbb{Z}$ define its stabilized support

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

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

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

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

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

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

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

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

######  Example.

Roe [r93] showed that for uniformly contractible proper metric spaces the
coarse cohomology is isomorphic to the compactly supported Alexander–Spanier
cohomology. 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 [r96]*Chapter 2. For example,

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

### Change of coefficients

A short exact sequence of abelian groups $0\to G^{\prime}\to G\to
G^{\prime\prime}\to 0$ induces short exact sequences of coarse complexes, and
hence long exact sequences

$\dots\to\operatorname{H}{\mathrm{X}}_{*}(X;G^{\prime})\to\operatorname{H}{\mathrm{X}}_{*}(X;G)\to\operatorname{H}{\mathrm{X}}_{*}(X;G^{\prime\prime})\to\operatorname{H}{\mathrm{X}}_{*-1}(X;G^{\prime})\to\dots$

and

$\dots\to\operatorname{H}{\mathrm{X}}^{*}(X;G^{\prime})\to\operatorname{H}{\mathrm{X}}^{*}(X;G)\to\operatorname{H}{\mathrm{X}}^{*}(X;G^{\prime\prime})\to\operatorname{H}{\mathrm{X}}^{*+1}(X;G^{\prime})\to\dots$

It follows that taking coarse complexes commutes with finite direct sums. For
a finitely generated abelian group $G$ this leads to an identification of the
coarse complexes with coefficients in $G$ with the tensor product with $G$ of
the integral coarse complexes, as follows. Pick a finitely generated free
abelian resolution of $G$:

$0\to F_{1}\to F_{0}\to G\to 0.$

Tensoring it with $\operatorname{C}{\mathrm{X}}_{*}(X;\mathbb{Z})$ and
comparing to the induced short exact sequence of the coarse chain complexes
gives a commutative diagram:

${0}$${\operatorname{C}{\mathrm{X}}_{*}(X;\mathbb{Z})\otimes
F_{1}}$${\operatorname{C}{\mathrm{X}}_{*}(X;\mathbb{Z})\otimes
F_{0}}$${\operatorname{C}{\mathrm{X}}_{*}(X;\mathbb{Z})\otimes
G}$${0}$${0}$${\operatorname{C}{\mathrm{X}}_{*}(X;F_{1})}$${\operatorname{C}{\mathrm{X}}_{*}(X;F_{0})}$${\operatorname{C}{\mathrm{X}}_{*}(X;G)}$${0}$

and similarly for cochains.

######  Remark.

We are not aware of general universal coefficient formulas in the coarse
setting. The difficulty here is that neither chains nor cochains are free
abelian groups. However, for proper uniformly contractible spaces, since the
coarse theories are isomorphic to the compactly supported cohomology and the
locally finite homology [r93], we have [m78b]*Cor. 4.18 and 4.25:

$\displaystyle
0\to\operatorname{Ext}(\operatorname{H}{\mathrm{X}}^{*+1}(X;\mathbb{Z}),G)\to$
$\displaystyle\operatorname{H}{\mathrm{X}}_{*}(X;G)\to\operatorname{Hom}(\operatorname{H}{\mathrm{X}}^{*}(X;\mathbb{Z}),G)\to
0,$ $\displaystyle 0\to\operatorname{H}{\mathrm{X}}^{*}(X;\mathbb{Z})\otimes
G\to$
$\displaystyle\operatorname{H}{\mathrm{X}}^{*}(X;G)\to\operatorname{Tor}(\operatorname{H}{\mathrm{X}}^{*+1}(X;\mathbb{Z}),G)\to
0.$

One might expect these formulas to hold in greater generality.

### Functoriality

Coarse maps induce maps on the coarse (co)homology. The following proposition
goes back to Roe, cf. [r93]*2.13.

######  Proposition 2.6.

A coarse map $f:X\to Y$ induces chain maps
$f_{*}:\operatorname{C}{\mathrm{X}}_{*}(X)\to\operatorname{C}{\mathrm{X}}_{*}(Y)$
and
$f^{*}:\operatorname{C}{\mathrm{X}}^{*}(Y)\to\operatorname{C}{\mathrm{X}}^{*}(X)$
by the usual formulas.

$f_{*}((x_{0},\dots,x_{n})):=(f(x_{0}),\dots,f(x_{n})),$ $f^{*}\phi:=\phi\circ
f_{*}.$

Close coarse maps induce chain homotopic maps.

###### Proof.

Note that $f$ induces a coarse map $f:X^{\infty}\to Y^{\infty}$, and that
proper maps preserve local finiteness, so the claims follow from Lemma 2.4.

For the second claim, take two close coarse maps $f,g:X\rightarrow Y$. Then
one can easily check that the map

$D_{n}(x_{0},x_{1},\ldots,x_{n}):=\sum_{i=0}^{n}(f(x_{0}),f(x_{1}),\dots,f(x_{i}),g(x_{i}),\dots,g(x_{n}))$

extends to a map
$\operatorname{C}{\mathrm{X}}_{n}(X)\to\operatorname{C}{\mathrm{X}}_{n+1}(Y)$
and gives a chain homotopy between $f_{*}$ and $g_{*}$. Similarly, the dual
$D_{n}^{*}$ maps $\operatorname{C}{\mathrm{X}}^{n+1}(Y)$ to
$\operatorname{C}{\mathrm{X}}^{n}(X)$ and gives a chain homotopy between
$f^{*}$ and $g^{*}$. ∎

In particular, coarsely equivalent spaces have isomorphic coarse (co)homology.

##  3\. Coarse (co)homology of complements

We now introduce coarse complements into the picture. Since the results in
this section apply to any choice of coefficients and to simplify notation, we
fix a coefficient group $G$ and omit it from the notation.

Let $A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$. A straightforward
approach to define the coarse cohomology of the complement $X-A$ is to look at
cochains on $X$ which restrict to coarse cochains on complements of expanding
neighborhoods of $A$, as follows. Given a proper function
$f:A\to\mathbb{R}_{\geq 0}$, the corresponding expanding neighborhood of $A$
is obtained by taking the union of open balls of radius $f(x)$ centered at $x$
as $x$ varies throughout $A$:

$N_{f}(A)=\bigcup_{x\overset{}{\in}A}B(x,f(x)).$

Note that $f$ can take $0$ as its value and so $N_{f}(A)$ might not be an
actual neighborhood of $A$.

The complements of expanding neighborhoods $X-N_{f}(A)$ form a directed system
under the inclusion, and we are interested in the limit in the direction of
slower growth of $f$. In other words, we want the complement to be as large as
possible.

Then the coarse chains and cochains on the complements of expanding
neighborhoods $X-N_{f}(A)$ form a direct and an inverse system, respectively.
We define the coarse chains and cochains of the complement to be the
(co)limits of these systems. Note that the system of complements $X-N_{f}(A)$
is cofinal in the system of sets coarsely disjoint from $A$, since for any
such set
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$, the map
$f:A\to\mathbb{R}_{\geq 0}$ given by $x\mapsto d(x,C)/2$ is proper and
$C\mathrel{\overset{\scriptscriptstyle}{\subset}}X-N_{f}(A)$. Thus, we have

$\displaystyle\operatorname{C}{\mathrm{X}}_{n}(X-A)$
$\displaystyle=\\{c=\sum_{\sigma\overset{}{\in}X^{n+1}}a_{\sigma}\sigma\mid\exists
C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A\quad
c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(C)\\},$
$\displaystyle\operatorname{C}{\mathrm{X}}^{n}(X-A)$
$\displaystyle=\\{\phi\overset{}{\in}\operatorname{C}^{n}(X)\mid\forall
C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A\quad\phi|_{C}\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(C)\\}.$

The usual (co)boundary operators make $\operatorname{C}{\mathrm{X}}_{*}(X-A)$
and $\operatorname{C}{\mathrm{X}}^{*}(X-A)$ into complexes, and we call their
(co)homology _the coarse (co)homology of the complement_
$\operatorname{H}{\mathrm{X}}_{*}(X-A)$ and
$\operatorname{H}{\mathrm{X}}^{*}(X-A)$. Note that $X-A$ here is a part of
notation, not a set-theoretic complement.

######  Remark.

If $A$ is a bounded subset of $X$, then every subset of $X$ is coarsely
disjoint from $A$. It follows that,
$\operatorname{C}{\mathrm{X}}^{*}(X-A)=\operatorname{C}{\mathrm{X}}^{*}(X)$
and
$\operatorname{C}{\mathrm{X}}_{*}(X-A)=\operatorname{C}{\mathrm{X}}_{*}(X)$.

### Relatively coarse maps

We now state an analog of Proposition 2.6 in the coarse complement setting.

######  Proposition 3.1.

A relatively coarse map $f:(X,A)\to(Y,B)$ between pairs induces chain maps
$f_{*}:\operatorname{C}{\mathrm{X}}_{*}(X-A)\to\operatorname{C}{\mathrm{X}}_{*}(Y-B)$
and
$f^{*}:\operatorname{C}{\mathrm{X}}^{*}(Y-B)\to\operatorname{C}{\mathrm{X}}^{*}(X-A)$.

###### Proof.

Suppose $c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(X-A)$. We want to
show that $f_{*}(c)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(Y-B)$.
Since $c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(X-A)$, there exists a
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ such that
$c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(C)$. By Lemma 2.5, $f|_{C}$
is a coarse map and hence Proposition 2.6 implies that
$f_{*}(c)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(f(C))$. Moreover by
Lemma 2.5,
$f(C)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}Y\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}B$. It follows
that $f_{*}(c)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(Y-B)$.

Next we prove that
$f^{*}(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(X-A)$ if
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(Y-B)$. Let
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$. It is
enough to show that
$f^{*}\phi|_{C}\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(C)$. By Lemma
2.5,
$f(C)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}Y\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}B$. Since
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(Y-B)$, we have
$\phi|_{f(C)}\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(f(C))$. Moreover,
$f|_{C}$ is a coarse map by Lemma 2.5. Hence we have
$f^{*}(\phi)|_{C}=({f|_{C}})^{*}(\phi|_{C})\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(C)$
by Proposition 2.6. and the claim follows. ∎

### Another description

Recall that $i:X\to X^{\infty}$ denote the map $x\mapsto(x,\dots,x,\dots)$ and
$\Delta=i(X)$. For a subset
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$, we denote the set
$i(A)\mathrel{\overset{\scriptscriptstyle}{\subset}}\Delta$ by $\Delta_{A}$.
The next two lemmas provide more explicit descriptions of
$\operatorname{C}{\mathrm{X}}_{n}(X-A)$ and
$\operatorname{C}{\mathrm{X}}^{n}(X-A)$.

######  Lemma 3.2.

$\displaystyle\operatorname{C}{\mathrm{X}}_{n}(X-A)$
$\displaystyle=\\{c\overset{}{\in}\operatorname{C}_{n}^{lf}(X)\mid\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\Delta_{A}\\},$
$\displaystyle\operatorname{C}{\mathrm{X}}^{n}(X-A)$
$\displaystyle=\\{\phi\overset{}{\in}\operatorname{C}^{n}(X)\mid\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}\\}.$

###### Proof.

The condition $c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(C)$ for some
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ implies
that $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{C}$.
Therefore $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\Delta_{A}$.
Vice versa, if $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\Delta_{A}$,
set $C=\bigcup_{i=0}^{n}p_{i}(\lvert c\rvert)$. Then
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ and $\lvert
c\rvert\mathrel{\overset{\scriptscriptstyle}{\subset}}C^{n+1}$, so
$c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(C)$. This proves the first
equation.

To prove the second, note that the condition $\phi|_{C}$ is coarse for all
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ means that
$\lvert\phi\rvert\cap C^{n+1}$ is coarsely disjoint from $\Delta_{C}$. Since
we can replace $C$ with its metric neighborhood, this is equivalent to
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}C^{n+1}\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{C}\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}*$,
which we can rewrite as
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{C}\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}*$.
The condition
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ is
equivalent to
$\Delta_{C}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\Delta_{A}$,
so we have $(\Delta\mathbin{\mathrel{\mathop{-}\limits^{\vbox to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\Delta_{A})\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}(\Delta\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\lvert\phi\rvert)$. By Lemma
2.3 this is equivalent to
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$,
as claimed. ∎

We now consider a slight generalization of the above (co)chain complexes. Let
$A,B\mathrel{\overset{\scriptscriptstyle}{\subset}}X$. Define:

$\displaystyle\operatorname{C}{\mathrm{X}}_{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
$\displaystyle=\\{c\overset{}{\in}\operatorname{C}_{*}^{lf}(X)\mid\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{B}\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern
0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\Delta_{A}\\},$
$\displaystyle\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
$\displaystyle=\\{\phi\overset{}{\in}\operatorname{C}^{*}(X)\mid\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{B}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}\\}.$

We claim that the homology of these complexes does not depend on the ambient
space $X$, it depends only on the metric on $B\cup A$:

######  Lemma 3.3.

The maps induced by inclusion

$\displaystyle
i_{*}:\operatorname{C}{\mathrm{X}}_{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}B\cup
A)\to\operatorname{C}{\mathrm{X}}_{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X),$
$\displaystyle
i^{*}:\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)\to\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}B\cup
A)$

are chain homotopy equivalences.

###### Proof.

Choose an approximate projection $\pi:X\to B\cup A$ with the property that
$\pi$ restricts to the identity on $B\cup A$ and for all $x\overset{}{\in}X$,
$d(x,\pi(x))<d(x,B\cup A)+1$. This map induces a map on $n$-simplices
$\pi:X^{n+1}\to(B\cup A)^{n+1}$, and we claim that it extends linearly to a
well-defined map
$\pi_{n}:\operatorname{C}{\mathrm{X}}_{n}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)\to\operatorname{C}{\mathrm{X}}_{n}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}B\cup
A)$.

Indeed, if
$c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$,
then there exists $R$ such that $\lvert
c\rvert\mathrel{\overset{\scriptscriptstyle R}{\subset}}\Delta_{B}$. Since
$\pi|_{N_{R}(\Delta_{B\cup A})}$ is $(R+1)$-close to the identity, it follows
that $\pi_{n}(c)$ is a well-defined chain in
$\operatorname{C}{\mathrm{X}}_{n}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}B\cup
A)$.

Next define

$D_{n}(x_{0},\dots,x_{n})=\sum_{i=0}^{n}(-1)^{i}(x_{0},\dots,x_{i},\pi(x_{i}),\dots,\pi(x_{n})).$

A similar argument shows that $D_{n}$ extends linearly to a well-defined map
$D_{n}:\operatorname{C}{\mathrm{X}}_{n}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)\to\operatorname{C}{\mathrm{X}}_{n+1}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.
Finally, we note that $\pi_{*}i_{*}=id$ and
$i_{*}\pi_{*}-id=D_{*}\partial+\partial D_{*}$.

The approximate projection $\pi$ induces a map
$\pi^{*}:\operatorname{C}^{*}(B\cup A)\to\operatorname{C}^{*}(X)$,
$\pi^{*}(\phi)=\phi\pi$. Thus we have $\lvert\pi^{*}(\phi)\rvert\cap
N_{R}(\Delta_{A})\mathrel{\overset{\scriptscriptstyle}{\subset}}(\pi|_{N_{R}(\Delta_{A})})^{-1}(\lvert\phi\rvert)$
and $\lvert\pi^{*}(\phi)\rvert\cap
N_{R}(\Delta_{B})\mathrel{\overset{\scriptscriptstyle}{\subset}}(\pi|_{N_{R}(\Delta_{B})})^{-1}(\lvert\phi\rvert)$.
Since $\pi|_{N_{R}(\Delta_{B\cup A})}$ is close to identity, it follows that
$\pi^{*}$ restricts to a map
$\pi^{*}:\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}B\cup
A)\to\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.
Similarly $D^{*}$ defined by $D^{*}(\phi)=\phi D_{*}$ gives a chain homotopy
between $\pi^{*}i^{*}$ and $id$. ∎

It follows that we can write unambiguously
$\operatorname{H}{\mathrm{X}}_{*}(B-A)$ and
$\operatorname{H}{\mathrm{X}}^{*}(B-A)$. Note that when $B=X$ or $A=\emptyset$
this agrees with the previous notation. Also, these definitions make sense
even if $B$ is empty, and in this case the complexes become the
Alexander–Spanier complexes of finitely supported chains and all cochains on
$X$ and hence

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

##  4\. Coarse complement as a space

As we already mentioned, $X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A$ rarely has
a coarse type of a subspace of $X$. However, it is still possible to compute
the coarse cohomology of the complement as the coarse cohomology of a single
space, but this space is not a subspace of $X$, but rather a quotient space.
We continue to suppress the coefficients $G$ from the notation.

We start by defining a pseudometric $d_{A}$ on $X$ by

$d_{A}(x,y)=\min(d(x,A)+d(y,A),d(x,y)).$

######  Lemma 4.1.

  1. (1)

$N_{r}^{d}(A^{n+1})=N_{r}^{d_{A}}(A^{n+1})$.

  2. (2)

$N_{r}^{d}(\Delta)\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{r}^{d_{A}}(\Delta)$.

  3. (3)

$N_{r}^{d_{A}}(\Delta)\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{2r}^{d}(\Delta\cup
A^{n+1})$.

###### Proof.

(1) and (2) follow from the facts that $d_{A}(x,A)=d(x,A)$ and $d_{A}(x,y)\leq
d(x,y)$.

To prove (3), let $\sigma\overset{}{\in}N_{r}^{d_{A}}(\Delta)$. So we have
$d_{A}(\sigma,x)<r$, for some $x\overset{}{\in}\Delta$. If $d(x,A^{n+1})\leq
r$, then $d(\sigma,A^{n+1})=d_{A}(\sigma,A^{n+1})\leq
d_{A}(\sigma,x)+d_{A}(x,A^{n+1})<2r$ and
$\sigma\overset{}{\in}N_{2r}^{d}(A^{n+1})$. If $d(x,A^{n+1})>r$ then, since
$d_{A}(x,\sigma)<r$, it follows from the definition of $d_{A}$ that
$d_{A}(x,\sigma)=d(x,\sigma)$ and $\sigma\overset{}{\in}N_{2r}^{d}(\Delta)$. ∎

Much of the coarse metric theory works the same for pseudometric spaces. In
particular, we have the usual definition of coarse (co)chains.

$\displaystyle\operatorname{C}{\mathrm{X}}_{n}(X,d_{A})$
$\displaystyle=\\{c\overset{}{\in}\operatorname{C}_{n}^{lf}(X,d_{A})\mid\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\subset}}}\Delta\\},$
$\displaystyle\operatorname{C}{\mathrm{X}}^{n}(X,d_{A})$
$\displaystyle=\\{\phi\overset{}{\in}\operatorname{C}^{n}(X)\mid\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\cap}}\Delta\mathbin{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{=}}*\\}.$

######  Proposition 4.2.

$\displaystyle\operatorname{C}{\mathrm{X}}_{n}(X,d_{A})$
$\displaystyle=\operatorname{C}{\mathrm{X}}_{n}(X-A),$
$\displaystyle\operatorname{C}{\mathrm{X}}^{n}(X,d_{A})$
$\displaystyle=\operatorname{C}{\mathrm{X}}^{n}(X-A).$

###### Proof.

Let $c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(X,d_{A})$. Since
$c\overset{}{\in}\operatorname{C}_{n}^{lf}(X,d_{A})$ and any bounded set in
$(X^{n+1},d)$ is bounded in $(X^{n+1},d_{A})$, we have
$c\overset{}{\in}\operatorname{C}_{n}^{lf}(X)$. Moreover, since $A^{n+1}$ is
bounded in $d_{A}$ metric, Lemma 4.1(1) implies that $\lvert c\rvert\cap
N^{d}_{r}(A^{n+1})$ is finite for any $r\geq 0$. So $\lvert c\rvert$ is
coarsely disjoint from $A^{n+1}$.

On the other hand, the condition $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\subset}}}\Delta$
and Lemma 4.1(3) give $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}\Delta\cup
A^{n+1}$. Hence, $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}\Delta\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c},d$}\vss}}}}\Delta_{A}$,
which together with local finiteness implies
$c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(X-A)$.

Conversely, let $c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(X-A)$. Then,
for any $r$, $\lvert c\rvert\cap N^{d}_{r}(\Delta_{A})$ is bounded, and hence
finite, since $c$ is locally finite.

Then, since $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}\Delta$,
$\lvert c\rvert\cap N^{d}_{r}(A^{n+1})$ is also finite, and therefore, by
Lemma 4.1(1), we have that $\lvert c\rvert\cap N_{r}^{A^{n+1}}(\Delta_{A})$ is
finite as well. Also, the condition $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}\Delta$
implies by Lemma 4.1(2) that $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\subset}}}\Delta$.
Hence $c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(X,d_{A})$. This proves
the first assertion.

Let $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X,d_{A})$. Then
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\cap}}\Delta\mathbin{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{=}}*$.
Since $A^{n+1}$ is bounded in $(X^{n+1},d_{A})$, this is equivalent to
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\subset}}}A^{n+1}$.
By Lemma 4.1(1) and (2), we have
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}A^{n+1}$.
It follows that
$(\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d}{\cap}}\Delta)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}(A^{n+1}\mathbin{\overset{\scriptscriptstyle\mathrm{c},d}{\cap}}\Delta)=\Delta_{A}$.
Hence, $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X-A)$.

Conversely, if $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X-A)$,
then
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}\Delta_{A}$,
and it follows that
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d}{\cap}}(\Delta\cup
A^{n+1})\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d}{\subset}}}A^{n+1}$.
By Lemma 4.1(1) and (3) this implies
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c},d_{A}}{\subset}}}A^{n+1}$.
Hence $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X,d_{A})$. This
proves the second assertion. ∎

Since $d_{A}(x,y)=0$ if and only if
$(x,y)\overset{}{\in}\overline{A}\times\overline{A}$, the pseudometric $d_{A}$
becomes a metric on the quotient space $X/\overline{A}$. One easily checks
that the quotient map $q:(X,d_{A})\to(X/\overline{A},d_{A})$ is a coarse
equivalence, and therefore it induces an isomorphism on the coarse
(co)homology. Using the previous proposition we obtain the following.

######  Proposition 4.3.

The quotient map $q:X\to X/\overline{A}$ induces isomorphisms
$\operatorname{H}{\mathrm{X}}_{*}(X/\overline{A})=\operatorname{H}{\mathrm{X}}_{*}(X-A)$
and
$\operatorname{H}{\mathrm{X}}^{*}(X/\overline{A})=\operatorname{H}{\mathrm{X}}^{*}(X-A)$.

######  Remark.

Another coarse model for $(X\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}A)$ can be
obtained by the “coning off” construction, popular in the study of relatively
hyperbolic groups. Here instead of taking a quotient by $A$ one attaches a
cone $cA$ on $A$, and declares $d(c,a)=1$ for all $a\overset{}{\in}A$.

##  5\. Separations and $\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$

We now consider the coarse cohomology with $\mathbb{Z}/2$ coefficients
$\operatorname{H}{\mathrm{X}}^{*}(X-A;\mathbb{Z}/2)$. It is the cohomology of
the cochain complex

$\operatorname{C}{\mathrm{X}}^{n}(X-A;\mathbb{Z}/2)=\\{\phi\overset{}{\in}\operatorname{C}^{n}(X;\mathbb{Z}/2)\mid\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}\\}.$

We will see that $\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$ measures
the number of coarse components of the complement of $A$. Making this precise
requires some preparation.

There is a simple interpretation of $1$-cocycles with
$\mathbb{Z}/2$-coefficients in terms of subsets of $X$. The power set $2^{X}$
is a boolean algebra, with multiplication given by the intersection and
addition by the symmetric difference. The complementation map $C\to X-C$ is a
free involution on $2^{X}$. Let $\mathcal{S}$ be the quotient space, we will
think of its elements as unordered pairs $\\{C,X-C\\}$, and refer to them as
_separations_ of $X$. Alternatively, since $X-C=X\mathbin{\triangle}C$, one
can think of $\mathcal{S}$ as a quotient of abelian groups
$2^{X}/\\{\emptyset,X\\}$. As an abelian group $2^{X}$ is isomorphic to
$\operatorname{C}^{0}(X;\mathbb{Z}/2)$, the isomorphism is given by taking
supports and the map $C\to 1_{C}$. Thus we get a commutative diagram, where
$F$ is the induced isomorphism.

${0}$${\\{\emptyset,X\\}}$${2^{X}}$${\mathcal{S}}$${0}$${0}$${\mathbb{Z}/2}$${\operatorname{C}^{0}(X;\mathbb{Z}/2)}$${Z^{1}(X;\mathbb{Z}/2)}$${0}$$\scriptstyle{1_{(\cdot)}}$$\scriptstyle{F}$$\scriptstyle{d}$$\scriptstyle{\lvert\cdot\rvert}$

More explicitly, $F(\\{C,X-C\\})=d(1_{C})$, and its inverse is given by the
following lemma.

######  Lemma 5.1.

Given a cocycle $\phi\overset{}{\in}Z^{1}(X;\mathbb{Z}/2)$ consider the
following relation on $X$: $x\sim y$ if $\phi(x,y)=0$. This relation is an
equivalence relation with at most two equivalence classes, i.e. a separation.

###### Proof.

Since $\phi$ is a cocycle with $\mathbb{Z}_{2}$-coefficients, for any
$x,y,z\overset{}{\in}X$ we have

$\phi(x,y)+\phi(x,z)+\phi(y,z)=0.$

This equation immediately implies the transitivity of the relation. Taking
$z=y=x$ in the equation shows that $\phi(x,x)=0$ and hence the relation is
reflexive. Taking $z=y$ shows then that $\phi(x,y)=\phi(y,x)$ and hence the
relation is symmetric.

The equation also implies that if neither $y$ nor $z$ is related to $x$, then
$y$ and $z$ are related, and therefore there are at most two equivalence
classes. ∎

We now introduce coarse supports into the picture. Some of the following
definitions are inspired by [m18]. Let
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$, a subset
$C\mathrel{\overset{\scriptscriptstyle}{\subset}}X$ is a _coarse complementary
component_ of $A$ if
$C\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}(X-C)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$.
$C$ of $A$ is _shallow_ if
$C\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$,
otherwise it is _deep_. The _$r$ -boundary_ of $C$ is

$\partial_{r}C:=\\{x\overset{}{\in}X-C\mid d(x,C)\leq r\\}.$

The following description follows easily from Lemma 2.1.

######  Lemma 5.2.

$C$ is a coarse complementary component of $A$ if and only if
$\partial_{r}(C)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$
for all $r$.

We will call a collection $\\{C_{\alpha}\\}$ of coarse complementary
components _uniform_ if for any $r$ there exists $R$ such that
$\partial_{r}(C_{\alpha})\mathrel{\overset{\scriptscriptstyle R}{\subset}}A$
for all $C_{\alpha}$. Since $\partial_{r}$ of an arbitrary union is contained
in the union of $\partial_{r}$’s, Lemma 5.2 implies

######  Lemma 5.3.

If $\\{C_{\alpha}\\}$ is uniform, then its union $\bigcup C_{\alpha}$ is a
coarse complementary component of $A$.

Let $\mathcal{C}_{A}$ denote the collection of all coarse complementary
components of $A$. As before, each $C\overset{}{\in}\mathcal{C}_{A}$
determines a separation $\\{C,X-C\\}\overset{}{\in}\mathcal{S}$, we will refer
to it as a _separation of $X$ with respect to $A$_. Such a separation is
_deep_ if both $C$ and $X-C$ are deep, and _shallow_ otherwise. We will say
that _$A$ separates $X$_ if there exists a deep separation of $X$ with respect
to $A$. Let $\mathcal{S}_{A}$ denote the collection of all such separations,
and let $\mathcal{S}\mathcal{S}_{A}$ be the subcollection of shallow
separations.

By Lemma 5.3, $\mathcal{C}_{A}$ is closed under finite unions, and since it is
closed under the complementation, it follows that $\mathcal{C}_{A}$ is a
subalgebra of the boolean algebra $2^{X}$, and $\mathcal{S}\mathcal{S}_{A}$
and $\mathcal{S}_{A}$ are subgroups of $\mathcal{S}$. Let
$\mathcal{D}\mathcal{S}_{A}=\mathcal{S}_{A}/\mathcal{S}\mathcal{S}_{A}$, its
nonzero elements are equivalence classes of deep separations of $X$ with
respect to $A$ modulo shallow ones.

Note that
$1_{C}\overset{}{\in}\operatorname{C}{\mathrm{X}}^{0}(X-A;\mathbb{Z}/2)$
precisely when $C$ is shallow, so $F$ maps isomorphically
$\mathcal{S}\mathcal{S}_{A}$ onto $BX^{1}(X-A;\mathbb{Z}/2)$. Moreover,

$\lvert
d(1_{C})\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X}=(C\times(X-C)\cup(X-C)\times
X)\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta=\Delta_{C}\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X-C},$

so $F$ restricts to an isomorphism $F:\mathcal{S}_{A}\to
ZX^{1}(X-A;\mathbb{Z}/2)$.

Thus we have:

######  Proposition 5.4.

$F$ induces an isomorphism
$\mathcal{D}\mathcal{S}_{A}\to\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$.

We will need the following observation in Section 6.

######  Lemma 5.5.

Suppose $A$ coarsely separates $X$. Then any deep coarse complementary
component $C$ of $X$ with respect to $A$ contains a deep coarse complementary
component $C^{\prime}\mathrel{\overset{\scriptscriptstyle}{\subset}}C$ such
that the separation $\\{C^{\prime},X-C^{\prime}\\}$ is deep.

###### Proof.

If $X-C$ is deep we can take $C^{\prime}=C$, so assume that $X-C$ is shallow.
By hypothesis, we have a deep separation $\\{D,X-D\\}$. Set $C^{\prime}=C\cap
D$. Then both $C^{\prime}=D-(X-C)$ and its complement
$X-C^{\prime}=(X-D)\cup(X-C)$ are deep. ∎

The next lemma shows that pairwise disjoint collections of deep coarse
complementary components bound from below the dimension of
$\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$.

######  Lemma 5.6.

Suppose $\\{C_{\alpha}\\}$ is a pairwise disjoint collection of deep coarse
complementary components of $X$ with respect to $A$. Then any proper
subcollection of $\\{C_{\alpha}\\}$ maps to a linearly independent subset of
$\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$.

###### Proof.

Since the coefficients are $Z/2$, we need to show that any finite sum
$\sum_{i=1}^{n}C_{\alpha_{i}}$ of the elements of the subcollection maps
nontrivially. Disjointness implies that the sum is just the union, so it is
deep, and so is its complement, since it contains a deep element not in the
subcollection. Therefore the corresponding separation is not in
$\mathcal{S}\mathcal{S}_{A}$. ∎

We now specialize to the case of geodesic metrics.

######  Proposition 5.7.

If $X$ is a geodesic space and
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$, then for any $r$, the
collection of path components of $X-N_{r}(A)$ is a uniform collection of
coarse complementary components.

###### Proof.

Fix $r$, and let $C$ be a path component of $X-N_{r}(A)$. We will show that
for any $R$, $\partial_{R}(C)\mathrel{\overset{\scriptscriptstyle
R+r}{\subset}}A$.

Let $x\overset{}{\in}\partial_{R}(C)$. Moreover, assume that
$x\overset{}{\notin}N_{r}(A)$, otherwise there is nothing to prove. Then there
exists $y\overset{}{\in}C$ such that $d(x,y)\leq R$. Since
$x\overset{}{\notin}C$ and $y\overset{}{\in}C$, $x$ and $y$ are in different
path components of $X-N_{r}(A)$, thus a geodesic between $x$ and $y$ must dip
into $N_{r}(A)$: there exists $z$ on the geodesic with $d(z,A)<r$. Note that
$d(x,z)\leq d(x,y)\leq R$. Then by triangle inequality $d(x,A)\leq
d(x,z)+d(z,A)<R+r$, as claimed. ∎

######  Proposition 5.8.

Let $X$ be a geodesic space and
$A,C\mathrel{\overset{\scriptscriptstyle}{\subset}}X$. The following are
equivalent:

  1. (1)

$C$ is a coarse complementary component of $A$.

  2. (2)

There exist $r$ such that $C-N_{r}(A)$ is a union of path components of
$X-N_{r}(A)$.

  3. (3)

There exist $r$ such that for any $R\geq r$, $C-N_{R}(A)$ is a union of path
components of $X-N_{R}(A)$.

###### Proof.

$\eqref{i:any r}\Rightarrow\eqref{i:r exists}$ is clear.

$\eqref{i:r exists}\Rightarrow\eqref{i:c-set}$ follows from Proposition 5.7.

$\eqref{i:c-set}\Rightarrow\eqref{i:any r}$ Let $C$ be a coarse complementary
component of $A$ in $X$, then there exists $r$ such that
$\partial_{1}(C)\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{r-1}(A)$.
Let $R\geq r$. We claim that $C^{\prime}:=C-N_{R}(A)$ is a union of path
components of $X-N_{R}(A)$.

Suppose $x\overset{}{\in}C^{\prime}$ and $y\overset{}{\in}X-N_{R}(A)$ are
connected by a path in $X-N_{R}(A)$. We need to show that
$y\overset{}{\in}C^{\prime}$. Choose points $\\{x=x_{0},\dots,x_{n}=y\\}$ on
the path so that $d(x_{i},x_{i+1})<1$ for all $i$. Suppose
$x_{i}\overset{}{\in}C^{\prime}$, then $d(x_{i+1},A)\geq
d(x_{i},A)-d(x_{i},x_{i+1})>R-1\geq r-1$, so
$x_{i+1}\overset{}{\notin}N_{r-1}(A)$, and therefore
$x_{i+1}\overset{}{\notin}\partial_{1}(C)$. On the other hand,
$d(x_{i+1},C)\leq d(x_{i},x_{i+1})<1$. Combining these together we get
$x_{i+1}\overset{}{\in}C^{\prime}$. So by induction
$y\overset{}{\in}C^{\prime}$. ∎

######  Theorem 5.9.

Let $X$ be a geodesic space and
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$, and suppose
$k=\dim_{\mathbb{Z}/2}\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$ is
finite. Then, for any $r$ the union of shallow path components of $X-N_{r}(A)$
is shallow: they all are contained in $N_{R}(A)$ for some $R$. Moreover, there
exist $r$ such that for any $R\geq r$, $X-N_{R}(A)$ has exactly $k+1$ deep
path components.

###### Proof.

If, for some $r$, the union of shallow path components is not shallow, then
there is a sequence $\\{C_{i}\\}$ of shallow path components of $X-N_{r}(A)$
such that $C_{i}$ is not contained in $N_{i}(A)$ for all
$i\overset{}{\in}\mathbb{N}$. Then the union of any subsequence is a deep
complementary component, and since there are infinitely many subsequences
whose unions are pairwise disjoint,
$\dim_{\mathbb{Z}/2}\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$ is
infinite by Lemma 5.6. This proves the first claim.

To prove the second, we observe that again by Lemma 5.6 for any $R$, there
exists at most $k+1$ deep path components in $X-N_{R}(A)$. For each of $2^{k}$
elements of $\operatorname{H}{\mathrm{X}}^{1}(X-A;\mathbb{Z}/2)$ we pick a
separation which maps onto it, and take both of its coarse complementary
components. So we get $2^{k+1}$ distinct coarse complementary components. By
Proposition 5.8 there exits $r$ such that for any $R\geq r$ these coarse
complementary components with $N_{R}(A)$ removed are unions of path components
of $X-N_{R}(A)$. Next we remove all shallow path components to obtain
$2^{k+1}$ coarse complementary components, each of which is a union of deep
path components of $X-N_{R}(A)$. Since any union of shallow path components is
shallow, the removals do not affect the image in cohomology, and therefore all
these unions are still distinct. Hence, $X-N_{R}(A)$ has at least $k+1$ deep
path components. This finishes the proof. ∎

##  6\. Alexander Duality

From now on, we will be mostly interested in integral (co)homology, so the
omitted coefficients are $\mathbb{Z}$.

######  Definition.

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

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

The next lemma shows that the maps in the definition of a coarse PD($n$) space
restrict to maps between $\operatorname{C}{\mathrm{X}}^{*}(X-A)$ and
$\operatorname{C}{\mathrm{X}}_{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.

######  Lemma 6.1.

(1) $\displaystyle p(\operatorname{C}{\mathrm{X}}^{*}(X-A))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}_{n-*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
$\displaystyle
p(\operatorname{C}{\mathrm{X}}^{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}_{n-*}(X-A)$
(2) $\displaystyle F(\operatorname{C}{\mathrm{X}}^{*}(X-A))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}^{*-1}(X-A)$
$\displaystyle
F(\operatorname{C}{\mathrm{X}}^{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}^{*-1}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
(3) $\displaystyle
q(\operatorname{C}{\mathrm{X}}_{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}^{n-*}(X-A)$
$\displaystyle q(\operatorname{C}{\mathrm{X}}_{*}(X-A))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}^{n-*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
(4) $\displaystyle
G(\operatorname{C}{\mathrm{X}}_{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}_{*+1}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
$\displaystyle G(\operatorname{C}{\mathrm{X}}_{*}(X-A))$
$\displaystyle\mathrel{\overset{\scriptscriptstyle}{\subset}}\operatorname{C}{\mathrm{X}}_{*+1}(X-A)$

###### Proof.

The proof is a routine exercise in coarse set theory.

(1) By assumption on $p$, $\lvert
p(\phi)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\phi\rvert$.
Since $p$ maps into $\operatorname{C}{\mathrm{X}}_{n-*}(X)$, $\lvert
p(\phi)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta$.
If $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(X-A)$, then
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$.
Combining these using Lemma 2.2 we get $\lvert
p(\phi)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$
and thus
$p(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n-*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.

If
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$,
then
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{A}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}*$.
So $\lvert
p(\phi)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{X}\mathbin{\mathrel{\mathop{-}\limits^{\vbox
to0.0pt{\kern 0.0pt\hbox{$\scriptscriptstyle\mathrm{c}$}\vss}}}}\Delta_{A}$
and thus $p(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n-*}(X-A)$.

(2) By assumption on $F$, $\lvert
F(\phi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\phi\rvert$.
If $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(X-A)$ then
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$.
Combining these we get $\lvert
F(\phi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$.
So $F(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*-1}(X-A)$.

If
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$,
then
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{A}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}*$.
It follows that $\lvert
F(\phi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{A}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}*$
and $F(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*-1}(X-A)$.

(3) Let
$c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$,
so $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$.
By assumption on $q$, $\lvert
q(c)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
c\rvert$. Thus $\lvert
q(c)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$.
Hence $q(c)\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n-*}(X-A)$.

Let $c\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(X-A)$ then $\lvert
c\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{A}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}*$
Hence by the property of $q$, $\lvert
q(c)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{A}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
c\rvert$. By Lemma 2.2, we have
$q(c)\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n-*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.

(4) Let
$c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$,
so $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$.
By assumption on $G$, $\lvert
G(c)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
c\rvert$. Thus $\lvert
G(c)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{A}$.
Hence
$G(c)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*+1}(A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.

Let $c\overset{}{\in}\operatorname{C}{\mathrm{X}}^{*}(X-A)$, so $\lvert
c\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{A}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}*$.
By assumption on $G$, $\lvert
G(c)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
c\rvert$. Combining these, we get $\lvert
G(c)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{A}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}*$.
∎

######  Theorem 6.2 (Coarse Alexander Duality).

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

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

###### Proof.

By Lemmas 6.1 and 3.3, $p\otimes G$ and $q\otimes G$ induce maps between
$\operatorname{H}{\mathrm{X}}^{*}(X-A;G)$ and
$\operatorname{H}{\mathrm{X}}_{n-*}(A;G)$, and
$\operatorname{H}{\mathrm{X}}^{*}(A;G)$ and
$\operatorname{H}{\mathrm{X}}_{n-*}(X-A;G)$, which are inverses of each other.
∎

One immediate corollary of the above theorem is the following computation of
coarse (co)homology of coarse PD($n$) spaces.

######  Corollary 6.3.

Suppose $X$ is a coarse PD($n$) space, and $G$ be a finitely generated abelian
group. Then:

  1. (1)

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

Furthermore, any cycle representing a non-zero element of
$\operatorname{H}{\mathrm{X}}^{n}(X;G)$ coarsely contains $\Delta_{X}$ in its
support.

  2. (2)

If $A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$ and $X$ is not coarsely
contained in $A$, then $\operatorname{H}{\mathrm{X}}^{n}(A;G)=0$.

###### Proof.

(1) Taking $A=\emptyset$ in Theorem 6.2, we have

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

A similar argument computes homology.

For the second part, consider the map
$q:\operatorname{C}{\mathrm{X}}_{n}(X)\to\operatorname{C}^{0}(X)$ from the
definition of a coarse PD($n$) space. If an $n$-cycle $c$ is nonzero in
$\operatorname{H}{\mathrm{X}}_{n}(X;G)$, then its dual $0$-cocycle
$\phi=q\otimes G(c)$ is a nonzero constant function $\phi:X\to G$, in
particular $\lvert\phi\rvert=\Delta$. By definitions
$\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
c\rvert$ and $\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta$,
thus $\lvert
c\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}\Delta$.

(2) Since $0$-cocycles on $X$ are constant functions,
$X\not\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$
implies that the only $0$-cocycle in $\operatorname{C}{\mathrm{X}}^{0}(X-A;G)$
is $0$. Hence,
$\operatorname{H}{\mathrm{X}}^{n}(A;G)=\operatorname{H}{\mathrm{X}}^{0}(X-A;G)=0$.
∎

Combining Proposition 5.4 and the coarse Alexander duality shows that deep
separations of a coarse PD($n$) space are determined by the
$(n-1)^{\text{st}}$ coarse homology of the separating subspace:

######  Corollary 6.4.

Let $X$ be a coarse PD($n$) space and
$A\mathrel{\overset{\scriptscriptstyle}{\subset}}X$. Then

$\mathcal{D}\mathcal{S}_{A}=\operatorname{H}{\mathrm{X}}_{n-1}(A;\mathbb{Z}/2).$

As an application, we prove the following coarse version of the Jordan
separation theorem, which combined with Theorem 5.9 in the geodesic case,
essentially recovers the coarse Jordan separation theorem of Kapovich and
Kleiner [kk05]*Corollary 7.8, see also [p07]*Proposition 1.

######  Corollary 6.5 (Coarse Jordan separation).

Let $f:Y\to X$ be a coarse embedding, where $X$ is a coarse PD($n$) space.
Then:

  1. (1)

If $Y$ is a coarse PD($n-1$) space, then modulo shallow separations, there is
a unique deep coarse separation of $X$ with respect to $f(Y)$.

Moreover, if $C$ is a deep coarse complementary component of $X$ with respect
to $f(Y)$, then
$f(Y)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C$.

  2. (2)

If $Y\mathrel{\overset{\scriptscriptstyle}{\subset}}Z$, where $Z$ is a coarse
PD($n-1$) space, and $Z$ is not coarsely contained in $Y$, then $f(Y)$ does
not coarsely separate $X$.

###### Proof.

Since $f$ is a coarse equivalence between $Y$ and $f(Y)$, the separation
claims follow immediately from Corollaries 6.3 and 6.4.

To prove the coarse containment claim, note that the claim for a subset of $C$
implies the claim for $C$, so by Lemma 5.5 it is enough to consider the case
where both $C$ and $X-C$ are deep.

In this case $[d(1_{C})]\neq 0$ in
$\operatorname{H}{\mathrm{X}}^{1}(X-f(Y);\mathbb{Z}/2)$ and hence its dual
$[p(d(1_{C}))]\neq 0$ in
$\operatorname{H}{\mathrm{X}}_{n-1}(f(Y)\mathrel{\overset{\scriptscriptstyle}{\subset}}X;\mathbb{Z}/2)$.
By construction of Lemma 3.3 the cycle $p(d(1_{C}))$ is homologous to a cycle
$c$ in $\operatorname{C}{\mathrm{X}}_{n-1}(f(Y);\mathbb{Z}/2)$ with $\lvert
c\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}\lvert
p(d(1_{C}))\rvert$. Since $\lvert
d(1_{C})\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{C}$,
arguing like in Lemma 6.1(1) shows that $\lvert
p(d(1_{C}))\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C$.
By Corollary 6.3 we have
$f(Y)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
c\rvert$ and we conclude
$f(Y)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}C$. ∎

##  7\. Products

Our next goal is to obtain a characterization of coarse PD($n$) spaces in
terms of existence of a certain pair of (co)homology classes (Theorem 8.2).
Our proof mostly follows the proof of a similar characterization of the usual
homology manifolds by McCrory [m77], we just need to keep careful track of
supports. In this section we gather the required information about various
products. Here we closely follow [s81a]*Chapter 6. Our main observation is
that certain slant and cap product maps between
$\operatorname{C}{\mathrm{X}}_{*}(X)$ and $\operatorname{C}^{*}(X)$ satisfy
the support conditions of the maps $p$ and $q$ in the definition of a coarse
PD($n$) space (Lemma 7.4).

To define product maps between (co)chain complexes, it is convenient to work
with $\operatorname{C}_{*}(X)\otimes\operatorname{C}_{*}(Y)$, rather than with
$\operatorname{C}_{*}(X\times Y)$. On the other hand, certain homotopies are
easier to describe in $\operatorname{C}_{*}(X\times Y)$. So we will need to
transfer these homotopies in $\operatorname{C}_{*}(X\times Y)$ to homotopies
in $\operatorname{C}_{*}(X)\otimes\operatorname{C}_{*}(Y)$. To have control
over the support of the homotopies we need to fix a particular choice of chain
homotopy equivalence between $\operatorname{C}_{*}(X\times Y)$ and
$\operatorname{C}_{*}(X)\otimes\operatorname{C}_{*}(Y)$.

To simplify notation, denote
$\operatorname{C}_{*}(X)\otimes\operatorname{C}_{*}(Y)$ by
$\operatorname{C}_{*}(X\otimes Y)$ and
$\operatorname{Hom}(\operatorname{C}_{*}(X)\otimes\operatorname{C}_{*}(Y),\mathbb{Z})$
by $\operatorname{C}^{*}(X\otimes Y)$.

For a simplex $\sigma=(x_{0},\dots,x_{i},\dots,x_{k})$ let
${}_{i}\sigma=(x_{0},\dots,x_{i})$ and $\sigma_{k-i}=(x_{i},\dots,x_{k})$
denote its front and back faces respectively. The Alexander–Whitney map is
given by:

$\displaystyle A:\operatorname{C}_{*}(X\times
Y)\to\operatorname{C}_{*}(X)\otimes\operatorname{C}_{*}(Y)$
$\displaystyle\rho^{k}\mapsto\sum_{i}{}_{i}p_{X}(\rho)\otimes
p_{Y}(\rho)_{k-i}.$

Here $p_{X}$ and $p_{Y}$ denote the projections to $X$ and $Y$, respectively:
if $\rho^{k}=(x_{0},y_{0},\dots,x_{k},y_{k})$,
$p_{X}(\rho):=(x_{0},\dots,x_{k})$ and $p_{Y}(\rho):=(y_{0},\dots,y_{k})$.

To get a map in the other direction, take two simplices
$\sigma=(x_{0},\dots,x_{k})$ and $\tau=(y_{0},\dots,y_{l})$. Arrange pairs
$(x_{i},y_{j})$ as the vertices of an $k\times l$ rectangular grid in
$\mathbb{R}^{2}$. To a path $\gamma$ in this grid starting at $(x_{0},y_{0})$
and ending at $(x_{k},y_{l})$, always moving either north or east, associate a
simplex $\rho_{\gamma}$ in $X\times Y$, obtained by reading the labels along
the path. Define the simplicial cross product

$\displaystyle S:\operatorname{C}_{*}(X\otimes Y)$
$\displaystyle\to\operatorname{C}_{*}(X\times Y)$
$\displaystyle\sigma^{k}\otimes\tau^{l}$
$\displaystyle\mapsto\sum_{\gamma}(-1)^{n(\gamma)}\,\rho_{\gamma},$

where $n(\gamma)$ is the number of squares in the grid lying below the path
$\gamma$. Define the support $\lvert\sigma\otimes\tau\rvert$ of
$\sigma\otimes\tau$ to be the element of $(X\times Y)^{(k+1)(l+1)}$ whose
$(i,j)$ coordinate is $(x_{i},y_{j})$. Let $\operatorname{C}_{*}^{lf}(X\otimes
Y)$ consists of all integral chains of the form $c=\sum
a_{\sigma,\tau}\sigma\otimes\tau$ with the following support condition: there
are only finitely many products of simplices that have a vertex in any given
bounded set in $X\times Y$. We will be interested in the following subcomplex
of $\operatorname{C}_{*}^{lf}(X\otimes Y)$:

$\operatorname{C}{\mathrm{X}}_{*}(X\otimes
Y):=\\{c\overset{}{\in}\operatorname{C}_{*}^{lf}(X\otimes Y)\mid\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{X\times
Y}\\}.$

######  Lemma 7.1.

The maps $A$ and $S$ are coarsely support preserving chain homotopy
equivalences between $\operatorname{C}{\mathrm{X}}_{*}(X\times Y)$ and
$\operatorname{C}{\mathrm{X}}_{*}(X\otimes Y)$.

###### Proof.

Note that $\lvert
S(\sigma\otimes\tau)\rvert\mathrel{\overset{\scriptscriptstyle}{\subset}}\lvert\sigma\times\tau\rvert$
and $\lvert A(\rho)\rvert\mathrel{\overset{\scriptscriptstyle}{\subset}}\lvert
p_{X}(\rho)\times p_{Y}(\rho)\rvert$. The acyclic models produce chain
homotopies $H:\operatorname{C}_{*}(X\otimes
Y)\to\operatorname{C}_{*+1}(X\otimes Y)$ and
$H^{\prime}:\operatorname{C}_{*}(X\times Y)\to\operatorname{C}_{*+1}(X\times
Y)$ between their compositions and identities, which satisfy the same support
conditions. (The point is that the acyclic models use the same vertices.)
Since $\lvert p_{X}(\rho)\times
p_{Y}(\rho)\rvert\mathrel{\overset{\scriptscriptstyle\operatorname{diam}\rho}{\subset}}\lvert\rho\rvert$,
all these maps extend to coarsely support preserving maps between coarse chain
complexes. ∎

######  Remark.

A similar statement, in a slightly different language, is proved by Hair in
[h10], where an explicit formulas for the homotopies are given.

We now consider the case $X=Y$. Let $\delta$ denote the diagonal map from $X$
to $X\times X$, i.e. $\delta(x)=(x,x)$ for all $x\overset{}{\in}X$.

Let $T:X\times X\to X\times X$ be the involution map switching factors. It
induces the chain involutions $T_{*}$ on
$\operatorname{C}{\mathrm{X}}_{*}(X\times X)$ and
$\operatorname{C}{\mathrm{X}}_{*}(X\otimes X)$, the latter is given by
$\sigma^{k}\otimes\tau^{l}\mapsto(-1)^{kl}\tau^{l}\otimes\sigma^{k}$.

Define a chain homotopy $D_{*}$ between $T_{*}$ and $id$ in
$\operatorname{C}_{*}(X\times X)$ by

$D_{*}(x_{0},\dots,x_{m})=\sum(-1)^{k}(x_{0},\dots,x_{k},T(x_{k}),\dots,T(x_{m})).$

Since metric neighborhoods of $\delta(X)$ in $X\times X$ are invariant under
$T$, it follows that $D$ extends to coarsely support preserving maps between
coarse chain complexes
$\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\times
X)$.

By restricting the map $S$ of Lemma 7.1 to
$\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\times
X)$, we get that
$\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\times
X)$ is coarse chain homotopy equivalent to the following complex

$\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\otimes
X):=\\{c\overset{}{\in}\operatorname{C}_{*}^{lf}(X\otimes X)\mid\lvert
c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{\delta(X)}\\}.$

Note that $S$ commutes with $T_{*}$: $ST_{*}=T_{*}S$. So we can concatenate
the homotopies from Lemma 7.1 with $D$ to obtain a chain homotopy $D^{T}$
between $T_{*}$ and $id$ in
$\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\otimes
X)$:

$T_{*}\mathbin{\overset{\scriptscriptstyle{H_{*}T_{*}}}{\simeq}}AST_{*}=AT_{*}S\mathbin{\overset{\scriptscriptstyle{AD_{*}S}}{\simeq}}AS\mathbin{\overset{\scriptscriptstyle{H^{\prime}_{*}}}{\simeq}}id.$

In other words, $D^{T}=H^{\prime}+ADS-HT_{*}$ and $T_{*}-id=\partial
D^{T}+D^{T}\partial$.

We summarize these observations in the following lemma.

######  Lemma 7.2.

  1. (1)

The homotopy $D^{T}_{*}:\operatorname{C}_{*}(X\otimes
X)\to\operatorname{C}_{*}(X\otimes X)$ satisfies

$\lvert
D^{T}_{*}(c)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}\lvert
c\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}.$

  2. (2)

The homotopy $D_{T}^{*}:\operatorname{C}^{*}(X\otimes
X)\to\operatorname{C}^{*}(X\otimes X)$ satisfies

$\lvert
D_{T}^{*}(\phi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}.$

  3. (3)

The map
$T_{*}:\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\otimes
X)\to\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\otimes
X)$ is controlled chain homotopic to the identity.

The Alexander–Whitney diagonal approximation is the composition:

$\operatorname{C}_{*}(X)\xrightarrow{\delta_{*}}\operatorname{C}_{*}(X\times
X)\xrightarrow{A}\operatorname{C}_{*}(X\otimes X).$

Its tensor product with itself is used for the cap and cup products in
$X\otimes X$:

$\displaystyle\delta^{A}_{*}:\operatorname{C}_{*}(X\otimes X)$
$\displaystyle\to\operatorname{C}_{*}(X\otimes X\otimes X\otimes X)$
$\displaystyle\sigma^{k}\otimes\tau^{l}$
$\displaystyle\mapsto\sum_{i,j}(-1)^{j(k-i)}({}_{i}\sigma\otimes{}_{j}\tau)\otimes(\sigma_{k-i}\otimes\tau_{l-j}).$

We now recall chain versions of cup, cap, and slant products from [s81a].

### Cup product

Suppose $\phi,\psi\overset{}{\in}\operatorname{C}^{*}(X\otimes X)$. Then the
cross product $\phi\times\psi\overset{}{\in}\operatorname{C}^{*}(X\otimes
X\otimes X\otimes X)$ is defined by

$(\phi\times\psi)(\sigma\otimes\tau\otimes\sigma^{\prime}\otimes\tau^{\prime}):=\phi(\sigma\otimes\tau)\psi(\sigma^{\prime}\otimes\tau^{\prime}).$

Let $\delta^{*}_{A}$ be the dual of the map $\delta_{*}^{A}$. The cup product
$\phi\mathbin{\smile}\psi\overset{}{\in}\operatorname{C}^{*}(X\otimes X)$ is
defined to be

$\delta_{A}^{*}(\phi\times\psi).$

###  Cap product

Suppose $\psi\overset{}{\in}\operatorname{C}^{n}(X)$ and
$\sigma^{k}\overset{}{\in}\operatorname{C}_{k}(X)$ is a $k$-simplex. The cap
product is defined by

$\psi\mathbin{\frown}\sigma=\psi(\sigma_{n})(_{k-n}\sigma).$

Suppose, $\phi\overset{}{\in}\operatorname{C}^{n}(X\otimes X)$ and
$\sigma^{k}\otimes\tau^{l}\overset{}{\in}\operatorname{C}_{*}(X\otimes X)$.
Then the cap product is defined by

$\phi\mathbin{\frown}(\sigma^{k}\otimes\tau^{l})=\sum_{i+j=n}(-1)^{i(l-j)}(\phi(\sigma_{i}\otimes\tau_{j}))({}_{k-i}\sigma\otimes{}_{l-j}\tau).$

###  Slant product

Suppose $\phi\overset{}{\in}\operatorname{C}^{n}(X\otimes X)$ and
$\sigma^{k}\overset{}{\in}\operatorname{C}_{k}(X)$ is a $k$-simplex. The slant
product $\phi/\sigma\overset{}{\in}\operatorname{C}^{n-k}(X)$ is defined by

$(\phi/\sigma^{k})(\tau^{n-k})=\phi(\sigma\otimes\tau).$

We first observe the following.

######  Lemma 7.3.

Suppose $\phi\overset{}{\in}\operatorname{C}^{*}(X\otimes X\otimes X\otimes
X)$ and $\delta:X\times X\to X\times X\times X\times X$ is the diagonal map.
Then
$\lvert\delta_{A}^{*}(\phi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta^{-1}(\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X\times
X)})$.

###### Proof.

Suppose
$\lvert\sigma^{k}\otimes\tau^{l}\rvert\overset{}{\in}\lvert\delta_{A}^{*}(\phi)\rvert\cap
N_{r}(\Delta_{X\times X})$. It is enough to show that
$\lvert\sigma^{k}\otimes\tau^{l}\rvert\overset{}{\in}\delta^{-1}(N_{r}(\lvert\phi\rvert)\cap
N_{r}(\Delta_{\delta(X\times X)}))$.

The assumption
$\lvert\sigma^{k}\otimes\tau^{l}\rvert\overset{}{\in}\lvert\delta_{A}^{*}(\phi)\rvert$
implies that

$\lvert{}_{i}\sigma\otimes{}_{j}\tau\otimes\sigma_{k-i}\otimes\tau_{l-j}\rvert\overset{}{\in}\lvert\phi\rvert$

for some $i,j$.

The assumption
$\lvert\sigma^{k}\otimes\tau^{l}\rvert\overset{}{\in}N_{r}(\Delta_{X\times
X})$ implies that

$\lvert\sigma^{k}\otimes\tau^{l}\rvert\overset{}{\in}\delta^{-1}(N_{r}(\Delta_{\delta(X\times
X)})),$

and also that

$\delta(\lvert\sigma^{k}\otimes\tau^{l}\rvert)\mathrel{\overset{\scriptscriptstyle}{\subset}}N_{r}(\lvert{}_{i}\sigma\otimes{}_{j}\tau\otimes\sigma_{k-i}\otimes\tau_{l-j}\rvert),$

and hence

$\lvert\sigma^{k}\otimes\tau^{l}\rvert\overset{}{\in}\delta^{-1}(N_{r}(\lvert\phi\rvert)).$

Combining these we obtain

$\lvert\sigma^{k}\otimes\tau^{l}\rvert\overset{}{\in}\delta^{-1}(N_{r}(\lvert\phi\rvert))\cap\delta^{-1}(N_{r}(\Delta_{\delta(X\times
X)}))=\delta^{-1}(N_{r}(\lvert\phi\rvert)\cap N_{r}(\Delta_{\delta(X\times
X)})).$

This finishes the proof. ∎

To state our next lemma, we need the following cochain complex

$\operatorname{C}{\mathrm{X}}^{*}(X\otimes
X-\delta(X)):=\\{\phi\overset{}{\in}\operatorname{C}^{*}(X\otimes
X)\mid\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{\delta(X)}\\}.$

Note that this complex is chain homotopy equivalent to
$\operatorname{C}{\mathrm{X}}^{*}(X\times X-\delta(X))$ via coarsely support
preserving homotopies.

Let $p_{1*},p_{2*}:\operatorname{C}_{*}(X\otimes X)\to\operatorname{C}_{*}(X)$
be the projection maps: $p_{1*}(\sigma\otimes\tau)=\sigma$ and
$p_{2*}(\sigma\otimes\tau)=\tau$. They induce maps
$p^{*}_{1},p^{*}_{2}:\operatorname{C}^{*}(X)\to\operatorname{C}^{*}(X\otimes
X)$. For the rest of the paper, we set
$\varepsilon^{\prime}_{n}=(-1)^{n(n+1)/2}$ and $\varepsilon_{n}=(-1)^{n}$.

######  Lemma 7.4.

Let $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X\otimes
X-\delta(X))$ be a cocycle and let
$c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(X)$ be a cycle. Then:

  1. (1)

The slant product map,
$\varepsilon^{\prime}_{n-k-1}\phi/\cdot:\operatorname{C}{\mathrm{X}}_{k}(X)\to\operatorname{C}^{n-k}(X)$
is a chain map and
$\lvert\phi/a\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
a\rvert$ for all $a\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(X)$.

  2. (2)

The cap product map
$\varepsilon^{\prime}_{n-k}\cdot\mathbin{\frown}c:\operatorname{C}^{k}(X)\to\operatorname{C}{\mathrm{X}}_{n-k}(X)$
is a chain map and
$\lvert\psi\mathbin{\frown}c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\psi\rvert$
for all $\psi\overset{}{\in}\operatorname{C}^{k}(X)$.

  3. (3)

The chain maps
$\operatorname{C}{\mathrm{X}}_{*}(X)\to\operatorname{C}{\mathrm{X}}_{*}(\delta(X)\mathrel{\overset{\scriptscriptstyle}{\subset}}X\otimes
X)$ given by $a\mapsto\varepsilon_{nk}\phi\mathbin{\frown}(c\otimes a)$ and
$a\mapsto T^{*}\phi\mathbin{\frown}(a\otimes c)$ are controlled chain
homotopic: the chain homotopy $D$ satisfies $\lvert
D(a)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta(\lvert
a\rvert)$.

  4. (4)

The chain maps
$\operatorname{C}^{*}(X)\to\operatorname{C}{\mathrm{X}}^{*}(X\otimes
X-\delta(X))$ given by
$\psi\mapsto\varepsilon_{nk}\phi\mathbin{\smile}p_{2}^{*}(\psi)$ and
$\psi\mapsto p_{1}^{*}(\psi)\mathbin{\smile}\phi$ are controlled chain
homotopic: the chain homotopy $D$ satisfies $\lvert
D(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta(\lvert\psi\rvert)\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}$.

###### Proof.

(1) Since $a\overset{}{\in}\operatorname{C}{\mathrm{X}}_{*}(X)$ and
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X\otimes X-\delta(X))$,
it follows that $(\phi/a)(\sigma)$ is well-defined and
$\lvert\phi/a\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
a\rvert$. The choice of signs together with $d\phi=0$ shows that
$\varepsilon^{\prime}_{n-k-1}\phi/\cdot$ is indeed a chain map:

$d(\varepsilon^{\prime}_{n-k-1}\phi/a)=(-1)^{n-k}\varepsilon^{\prime}_{n-k-1}\phi/\partial
a=\varepsilon^{\prime}_{n-k}\phi/\partial a.$

(2) Since $c\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(X)$,
$\psi\mathbin{\frown}c$ is a well-defined element of
$\operatorname{C}{\mathrm{X}}_{n-k}(X)$ and
$\lvert\psi\mathbin{\frown}c\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\psi\rvert$.
Again the choice of signs together with $\partial c=0$ shows that
$\varepsilon^{\prime}_{n-k}\cdot\mathbin{\frown}c$ is a chain map:

$\partial(\varepsilon^{\prime}_{n-k}\psi\mathbin{\frown}c)=(-1)^{n-k}\varepsilon^{\prime}_{n-k}d\psi\mathbin{\frown}c=\varepsilon^{\prime}_{n-k-1}d\psi\mathbin{\frown}c.$

(3) It follows as before that both maps are well-defined chain maps. For
$a\overset{}{\in}\operatorname{C}{\mathrm{X}}_{k}(X)$ we have

$T^{*}\phi\mathbin{\frown}(a\otimes
c)=\varepsilon_{nk}T^{*}\phi\mathbin{\frown}T_{*}(c\otimes
a)=\varepsilon_{nk}T_{*}(T^{*2}\phi\mathbin{\frown}(c\otimes
a))=\varepsilon_{nk}T_{*}(\phi\mathbin{\frown}(c\otimes a)).$

By assumptions on $c$ and $a$ we have $\lvert c\rvert\times\lvert
a\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{X\times
X}$ and $\lvert c\rvert\times\lvert
a\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta(\lvert
a\rvert)$. Therefore, by assumptions on $\phi$,

$\lvert\phi\mathbin{\frown}(c\otimes
a)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}(\lvert
c\rvert\times\lvert
a\rvert)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}(\lvert
c\rvert\times\lvert
a\rvert)\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta(\lvert
a\rvert).$

The claim follows, since the homotopy between $T_{*}$ and $id$ coarsely
preserves supports by Lemma 7.2.

(4) Suppose $\psi\overset{}{\in}\operatorname{C}^{*}(X)$. Then
$p_{i}^{*}(\psi)\overset{}{\in}\operatorname{C}^{*}(X\otimes X)$. We have

$\displaystyle
p_{1}^{*}(\psi)\mathbin{\smile}\phi=\delta_{A}^{*}(p_{1}^{*}(\psi)\times\phi)$
$\displaystyle=\delta_{A}^{*}T^{*}_{X\times X}(\varepsilon_{nk}\phi\times
p_{1}^{*}(\psi))$ $\displaystyle=\delta_{A}^{*}T^{*}_{X\times
X}(\varepsilon_{nk}\phi\times T^{*}p_{2}^{*}(\psi)).$

Since $\cdot\otimes 1$ and $\varepsilon_{nk}\phi\times\cdot$ are chain maps,
the concatenation of the homotopies of $T^{*}$ and $T^{*}_{X\times X}$ to
identities gives a chain homotopy between the maps in the claim:

$D(\psi):=\delta_{A}^{*}T^{*}_{X\times X}(\varepsilon_{k-1}\phi\times
D_{T}^{*}p_{2}^{*}(\psi))+\delta_{A}^{*}D_{T_{X\times
X}}^{*}(\varepsilon_{nk}\phi\times p_{2}^{*}(\psi)).$

For the support of the first term we have:

$\displaystyle\left\lvert\delta_{A}^{*}T^{*}_{X\times X}(\phi\times
D_{T}^{*}p_{2}^{*}(\psi))\right\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta^{-1}\bigl{(}\lvert
T^{*}_{X\times X}(\phi\times
D_{T}^{*}p_{2}^{*}(\psi))\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X\times
X)}\bigr{)}$ by Lemma 7.3
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta^{-1}\bigl{(}\lvert
D_{T}^{*}p_{2}^{*}(\psi)\times\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X\times
X)}\bigr{)}$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta^{-1}\bigl{(}\lvert
D_{T}^{*}p_{2}^{*}(\psi)\rvert\times\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X\times
X)}\bigr{)}$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
D_{T}^{*}p_{2}^{*}(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
D_{T}^{*}p_{2}^{*}(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}$
by assumption on $\phi$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
p_{2}^{*}(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}$
by Lemma 7.2(2)
$\displaystyle\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}\delta(\lvert\psi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X}).$

Similarly, for the support of the second term we have:

$\displaystyle\left\lvert\delta_{A}^{*}D_{T_{X\times X}}^{*}\bigl{(}\phi\times
p_{2}^{*}(\psi)\bigr{)}\right\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta^{-1}\bigl{(}\lvert
D_{T_{X\times X}}^{*}(\phi\times
p_{2}^{*}(\psi))\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X\times
X)}\bigr{)}$ by Lemma 7.3
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta^{-1}\bigl{(}\lvert\phi\times
p_{2}^{*}(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X\times
X)}\bigr{)}$ by Lemma 7.2(2)
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\delta^{-1}\bigl{(}\lvert\phi\rvert\times\lvert
p_{2}^{*}(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X\times
X)}\bigr{)}$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\phi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\lvert
p_{2}^{*}(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}$
$\displaystyle\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
p_{2}^{*}(\psi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{\delta(X)}$
by assumption on $\phi$
$\displaystyle\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}\delta(\lvert\psi\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X}).$

∎

The next three formulas tell how cup, cap, and slant products interact with
each other on the chain level.

######  Lemma 7.5.

[s81a]*6.1.4-6 Let $\sigma$ and $\tau$ be simplices in $X$ and let
$\phi\overset{}{\in}\operatorname{C}^{*}(X\otimes X)$, and
$\psi\overset{}{\in}\operatorname{C}^{*}(X)$. Then:

  1. (1)

$\psi\mathbin{\smile}(\phi/\tau)=(p_{1}^{*}(\psi)\mathbin{\smile}\phi]/\tau$.

  2. (2)

$\phi/(\psi\mathbin{\frown}\tau)=(\phi\mathbin{\smile}p_{2}^{*}(\psi))/\tau$.

  3. (3)

$(\phi/\sigma)\mathbin{\frown}\tau=p_{1*}(\phi\mathbin{\frown}(\tau\otimes\sigma))$.

##  8\. Algebraic Poincaré Duality

In this section we find a criterion for a space to be a coarse PD($n$) space
in terms of existence of a certain pair of (co)homology classes.

######  Definition.

A pair of classes $U\overset{}{\in}\operatorname{H}{\mathrm{X}}^{n}(X\otimes
X-\delta(X))$ and $[X]\overset{}{\in}\operatorname{H}{\mathrm{X}}_{n}(X)$ is
an _orientation pair_ if
$U/[X]=1\overset{}{\in}\operatorname{H}{\mathrm{X}}^{0}(X-X)$. We will call
$U$ a _diagonal class_ and $[X]$ a _fundamental class_.

######  Remark.

This condition is stronger than it looks. Since the only representative of
$1\overset{}{\in}\operatorname{H}{\mathrm{X}}^{0}(X-X)$ is the constant
function, the above equation is satisfied on the (co)chain level, independent
of the choice of representatives for $U$ and $[X]$. It also follows that the
support of any representative of $[X]$ is coarsely equal to $\Delta$.

######  Lemma 8.1.

Let $U\overset{}{\in}\operatorname{H}{\mathrm{X}}^{n}(X\otimes X-\delta(X))$
and $[X]\overset{}{\in}\operatorname{H}{\mathrm{X}}_{n}(X)$ be a pair of
classes and $p_{1*}:\operatorname{C}_{*}(X\otimes
X)\to\operatorname{C}_{*}(X)$ is the projection map:
$p_{1*}(\sigma\otimes\tau)=\sigma$. The following conditions are equivalent.

  1. (1)

$(U,[X])$ is an orientation pair.

  2. (2)

$[X]\neq 0$ and $p_{1*}(U\mathbin{\frown}([X]\otimes[X]))=[X]$.

  3. (3)

For any $x\overset{}{\in}X$, $U(x\otimes[X])=1$.

###### Proof.

(1)$\Leftrightarrow$(2). By Lemma 7.5(3),
$p_{1*}(U\mathbin{\frown}([X]\otimes[X]))=(U/[X])\mathbin{\frown}[X]$. For
nonzero $[X]$ this equals to $[X]$ if and only if $U/[X]=1$.

(1)$\Leftrightarrow$(3) follows immediately from the formula
$(U/[X])(x)=U(x\otimes[X])$. ∎

######  Theorem 8.2.

$X$ is a coarse $PD(n)$ space if and only if it has an orientation pair.

###### Proof.

Let $X$ be a coarse $PD(n)$ space and let $p$ and $q$ be the associated
homotopy equivalences. Set
$c=p(1)\overset{}{\in}\operatorname{C}{\mathrm{X}}_{n}(X)$ where
$1\overset{}{\in}\operatorname{C}{\mathrm{X}}^{0}(X-X)$, and define
$\phi\overset{}{\in}\operatorname{C}^{*}(X\otimes X)$ by the rule
$\phi(\sigma^{k}\otimes\tau^{n-k})=-\varepsilon^{\prime}_{k+1}q(\tau)(\sigma)$.
Note that, if $\sigma\otimes\tau\mathrel{\overset{\scriptscriptstyle
r}{\subset}}\Delta_{X\times X}$ and $q(\tau)(\sigma)\neq 0$, then it follows
from the property of $q$ that $\sigma$ and $\tau$ will be $r^{\prime}$ close
to each other where $r^{\prime}$ depends only on $r$. Hence we have
$\phi\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X\times
X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\Delta_{\delta(X)}$,
i.e. $\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X\otimes
X-\delta(X))$. Moreover, $\phi$ is a cocycle, since

$d\phi(\sigma^{k}\otimes\tau^{n-k+1})=\phi(\partial\sigma\otimes\tau+\varepsilon_{k}\sigma\otimes\partial\tau)=-\varepsilon^{\prime}_{k}q(\tau)(\partial\sigma)-\varepsilon_{k}\varepsilon^{\prime}_{k+1}q(\partial\tau)(\sigma)\\\
=-(\varepsilon^{\prime}_{k}+\varepsilon_{k}\varepsilon^{\prime}_{k+1})dq(\tau)(\sigma)=0.$

Then for $x\overset{}{\in}X$ we have $\phi(x\otimes
c)=-\varepsilon^{\prime}_{1}q(c)(x)=qp(1)(x)=1(x)=1$. Thus by Lemma 8.1(3)
$([\phi],[c])$ is an orientation pair.

Conversely, suppose $X$ has a fundamental class $[X]=[c]$ and an orientation
class $U=[\phi]$. We claim that the chain maps
$\varepsilon^{\prime}_{n-k-1}\phi/\cdot:\operatorname{C}{\mathrm{X}}_{k}(X)\to\operatorname{C}^{n-k}(X)$
and
$\varepsilon^{\prime}_{n}\varepsilon^{\prime}_{n-k}\cdot\mathbin{\frown}c:\operatorname{C}^{k}(X)\to\operatorname{C}{\mathrm{X}}_{n-k}(X)$
are controlled chain homotopy equivalences as in the definition of a coarse
$PD(n)$ space.

By Lemma 7.4(3)
$\varepsilon_{n}T^{*}U\mathbin{\frown}([X]\otimes[X])=U\mathbin{\frown}([X]\otimes[X])$,
thus by Lemma 8.1(2), $\varepsilon_{n}T^{*}U$ is also a diagonal class for
$[X]$ and $T^{*}U/[X]=\varepsilon_{n}$.

Note that
$\varepsilon^{\prime}_{n}\varepsilon^{\prime}_{n-k}\varepsilon^{\prime}_{n-k-1}=\varepsilon_{nk}\varepsilon_{n}$.
Let $a\overset{}{\in}\operatorname{C}{\mathrm{X}}_{k}(X)$. Then we have

$\displaystyle\varepsilon_{nk}\varepsilon_{n}(\phi/a)\mathbin{\frown}c$
$\displaystyle=\varepsilon_{n}p_{1*}(\varepsilon_{nk}\phi\mathbin{\frown}(c\otimes
a))$ by Lemma 7.5(3)
$\displaystyle\simeq\varepsilon_{n}p_{1*}(T^{*}\phi\mathbin{\frown}(a\otimes
c))$ by Lemma 7.4(3)
$\displaystyle=\varepsilon_{n}(T^{*}\phi/c)\mathbin{\frown}a$ by Lemma 7.5(3)
$\displaystyle=a.$

Note that the only homotopy we used here is the composition of $p_{1*}$ with
the homotopy from Lemma 7.4(3), so it has the property $\lvert
D(a)\rvert\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert
a\rvert$.

Similarly,
$\varepsilon^{\prime}_{n}\varepsilon^{\prime}_{k-1}\varepsilon^{\prime}_{n-k}=\varepsilon_{nk}$,
and for $\phi\overset{}{\in}\operatorname{C}^{k}(X)$ we have

$\displaystyle\varepsilon_{nk}\phi/(\psi\mathbin{\frown}c)$
$\displaystyle=\varepsilon_{nk}(\phi\mathbin{\smile}p_{2}^{*}(\psi))/c$ by
Lemma 7.5(2) $\displaystyle\simeq(p_{1}^{*}(\psi)\mathbin{\smile}\phi)/c$ by
Lemma 7.4(4) $\displaystyle=\psi\mathbin{\smile}(\phi/c)$ by Lemma 7.5(1)
$\displaystyle=\psi.$

Lemma 7.4(4) implies that the homotopy above satisfies $\lvert
D(\phi)\rvert\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}\Delta_{X}\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}\lvert\phi\rvert$.
∎

######  Corollary 8.3.

Any proper, uniformly acyclic $n$-manifold is a coarse PD($n$) space.

###### Proof.

Suppose $X$ is a proper and uniformly acyclic $n$-manifold. Since $X$ is an
acyclic manifold, it has an orientation class
$V\overset{}{\in}\operatorname{H}^{n}(X\times X,X\times X-\delta(X))$ such
that $V|_{x\times(X,X-x)}$ is a generator of
$\operatorname{H}^{n}(x\times(X,X-x))$ for all $x\overset{}{\in}X$. It also
has a fundamental class
$[X]\overset{}{\in}\operatorname{H}_{n}^{\mathrm{l}f}(X)$ which similarly
restricts to generators of $\operatorname{H}_{n}(X,X-x)$. Choosing $V$ and
$[X]$ in compatible way we obtain $V(x\times[X])=1$ for all
$x\overset{}{\in}X$. Let $\phi^{\prime}$ and $c^{\prime}$ be singular
(co)cycles representing these classes. We now construct their coarse versions.

By subdividing $c^{\prime}$ as necessary we can obtain another representative
$c$, whose singular simplices have uniformly bounded diameter. Since $X$ is
proper and $c$ is locally finite, any bounded set intersects $\lvert c\rvert$
in finitely many simplices, and therefore $c$ is a coarse cycle.

Using uniform acyclicity of $X\times X$ we can inductively fill coarse
simplices by singular chains in a controlled manner. Indeed, 0-simplices are
naturally identified, and if the faces of a coarse simplex are already filled
by singular chains, then their signed sum is a cycle, which by uniform
acyclicity bounds a singular chain contained in a controlled neighborhood of
the simplex.

This gives a chain map $B:\operatorname{C}_{*}(X\times
X)\to\operatorname{C}_{*}^{\mathrm{s}}(X\times X)$ with the property $\lvert
B(\sigma^{k})\rvert\mathrel{\overset{\scriptscriptstyle\rho_{k}(\operatorname{diam}(\sigma^{k}))}{\subset}}\lvert\sigma^{k}\rvert$,
where $\rho_{k}$ is a suitable sequence of control functions.

Set $\phi(\sigma^{n})=\phi^{\prime}(B(\sigma^{n}))$. Since $\phi^{\prime}$
vanishes on singular simplices in $X\times X-\delta(X)$, $\phi(\sigma)=0$ if
$\rho_{n}(\operatorname{diam}(\sigma))<d(\sigma,\delta(X))$. Thus
$\phi\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X\times X-\delta(X))$.

Let $U:=S^{*}(\phi)\overset{}{\in}\operatorname{C}{\mathrm{X}}^{n}(X\otimes
X-\delta(X))$ where $S:\operatorname{C}_{*}(X\otimes
X)\to\operatorname{C}_{*}(X\times X)$ is the simplicial cross product map
defined in Section 7. By construction, $U(x\otimes c)=\phi(S(x\otimes
c))=\phi(x\times c)=1$ for all $x\overset{}{\in}X$. Therefore, by Lemma 8.1
$(U,[c])$ is an orientation pair, and the claim follows. ∎

######  Remark.

Essentially the same argument shows that proper, uniformly acyclic ENR
homology $n$-manifolds are also coarse PD($n$) spaces.

##  9\. Algebraic duality

In this section we show that chains and cochains with $\mathbb{Z}$
coefficients are dual to each other in the sense of taking
$\operatorname{Hom}(\cdot,\mathbb{Z})$. This requires some algebraic
preliminaries, most of which can be found in [em02].

Let $R$ be a commutative ring with $1$. Consider a countable direct product
$R^{\mathbb{N}}$. Let $e_{i}$ denote the image in $R^{\mathbb{N}}$ of $1$ in
the $i^{\text{th}}$ factor. Given an $R$-module $H$ and homomorphism
$\phi:R^{\mathbb{N}}\to H$, the support of $\phi$ is
$\lvert\phi\rvert=\\{i\overset{}{\in}\mathbb{N}\mid\phi(e_{i})\neq 0\\}$. $H$
is _slender_ if for every homomorphism $\phi:R^{\mathbb{N}}\to H$,
$\lvert\phi\rvert$ is finite. A ring $R$ is slender if it is a slender module
over itself.

The following is a classical result, going back to Specker. For the sake of
completeness, we provide a proof.

######  Lemma 9.1 (cf. Corollary III.2.4 in [em02]).

$\mathbb{Z}$ is slender.

###### Proof.

Let $\phi:\mathbb{Z}^{\mathbb{N}}\to\mathbb{Z}$ be a homomorphism. We will
produce an element $v\overset{}{\in}\mathbb{Z}^{\mathbb{N}}$ of the form $\sum
2^{n_{k}}e_{k}$ with fast growing powers of $2$ such that $\phi(e_{k+1})=0$
for $k\geq\lvert\phi(v)\rvert$, and hence $\phi$ has finite support. (Here and
below $\lvert\phi(\cdot)\rvert$ denotes the absolute value of $\phi(\cdot)$.)

For each $k$, let $v=h_{k}+t_{k}$ be the splitting of $v$ into the head of
length $k-1$ and the infinite tail. At each step $k$ we already have $h_{k}$
(starting with $h_{1}=0$), and we choose $n_{k}>n_{k-1}$ so that
$2^{n_{k}}>k+\lvert\phi(h_{k})\rvert$.

Now let $k\geq\lvert\phi(v)\rvert$ be arbitrary. Since
$\phi(v)=\phi(h_{k})+\phi(t_{k})$, we have

$\lvert\phi(t_{k})\rvert\leq\lvert\phi(v)\rvert+\lvert\phi(h_{k})\rvert\leq
k+\lvert\phi(h_{k})\rvert<2^{n_{k}}.$

But $\phi(t_{k})$ is a multiple of $2^{n_{k}}$, since $t_{k}$ is. So
$\phi(t_{k})=0$ and therefore
$2^{n_{k+1}}\phi(e_{k+1})=\phi(t_{k+1})-\phi(t_{k})=0$, proving the claim. ∎

######  Theorem 9.2 (cf. Corollary III.3.4 in [em02]).

If $H$ is slender and $I$ is not an $\omega$-measurable cardinal, then for any
$\phi:R^{I}\to H$, if $\phi_{|\bigoplus_{I}R}=0$, then $\phi=0$.

We recall that a cardinal is _$\omega$ -measurable_ if it supports a non-
principal ultrafilter which is closed under countable intersections. The
existence of $\omega$-measurable cardinals cannot be proved in ZFC, and it is
unknown if the existence is consistent with ZFC. Moreover, if they exist, the
first $\omega$-measurable cardinal $\kappa$ must be strongly inaccessible,
i.e. if $\lambda<\kappa$, then $2^{\lambda}<\kappa$ and $\kappa$ cannot be
obtained as $\lambda$-union of $\lambda$ cardinals.

Let $\mathcal{S}$ be a family of subsets of $I$ which is closed under taking
subsets and finite unions. These conditions ensure that the set of
$\mathcal{S}$-supported functions $M_{\mathcal{S}}=\\{a:I\to R\mid\lvert
a\rvert\overset{}{\in}\mathcal{S}\\}$ is a submodule of $R^{I}$, so we will
call such $\mathcal{S}$ a _supportive_ family. The dual of $\mathcal{S}$,
given by
$\mathcal{S}^{*}=\\{T\mathrel{\overset{\scriptscriptstyle}{\subset}}I\mid\forall
S\overset{}{\in}\mathcal{S}\quad\\#(S\cap T)<\infty\\}$ is also supportive.
Note that there is a natural bilinear pairing
$\langle\,,\rangle:M_{\mathcal{S}}\times M_{\mathcal{S}^{*}}\to R$ given by
$\langle a,b\rangle=\sum_{i\overset{}{\in}I}a(i)b(i)$.

######  Lemma 9.3.

Suppose $R$ is slender, the cardinality $I$ is less than the first
$\omega$-measurable cardinal, and $\mathcal{S}$ is a supportive family which
covers $I$. Then the evaluation map
$e:M_{\mathcal{S}^{*}}\to\operatorname{Hom}(M_{\mathcal{S}},R)$ given by
$e(b)(a)=\langle a,b\rangle$ is an isomorphism.

###### Proof.

The injectivity of $e$ follows easily from the hypothesis that $\mathcal{S}$
covers $I$.

To prove surjectivity, let
$f\overset{}{\in}\operatorname{Hom}(M_{\mathcal{S}},R)$. We claim that $\lvert
f\rvert\overset{}{\in}\mathcal{S}^{*}$. Indeed, if the intersection $\lvert
f\rvert\cap S$ is infinite for some $S\overset{}{\in}\mathcal{S}$, then it
contains an infinite countable subset $C$, and thus
$R^{C}\mathrel{\overset{\scriptscriptstyle}{\subset}}M_{\mathcal{S}}$ and
$\lvert f_{|R^{C}}\rvert=C$, contradicting slenderness of $R$.

We also claim that $\lvert f\rvert=\emptyset$ implies $f=0$. If $f(a)\neq 0$
for some $a\overset{}{\in}M_{\mathcal{S}}$, then since $M_{\mathcal{S}}$
contains $R^{\lvert a\rvert}$ we get a contradiction to Theorem 9.2 for
$f|_{R^{\lvert a\rvert}}$.

The first claim shows that the characteristic function $1_{\lvert
f\rvert}\overset{}{\in}M_{\mathcal{S}^{*}}$, and the second claim implies that
$f-e(1_{\lvert f\rvert})=0$. ∎

Let $X$ be a metric space with two subspaces $A$ and $B$. The families

$\displaystyle\mathcal{H}$
$\displaystyle=\\{S\mathrel{\overset{\scriptscriptstyle}{\subset}}X\mid
S\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B\text{
and }\forall r\quad\\#(S\cap N_{r}(A))<\infty\\},$ $\displaystyle\mathcal{C}$
$\displaystyle=\\{T\mathrel{\overset{\scriptscriptstyle}{\subset}}X\mid
T\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}B\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A\\}$

are supportive and cover $X$.

######  Lemma 9.4.

The families $\mathcal{H}$ and $\mathcal{C}$ are dual to each other:
$\mathcal{H}^{*}=\mathcal{C}$ and $\mathcal{C}^{*}=\mathcal{H}$.

###### Proof.

If $Y$ belongs to both families, then
$Y\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$, by
the first condition of $\mathcal{H}$. Then
$Y\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{=}}Y\mathbin{\overset{\scriptscriptstyle\mathrm{c}}{\cap}}B$,
so $Y\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}A$, by
definition of $\mathcal{C}$. So $Y$ has to be finite by the second condition
of $\mathcal{H}$. This implies that if $S\overset{}{\in}\mathcal{H}$ and
$T\overset{}{\in}\mathcal{C}$, then $S\cap T$ is finite, which means
$\mathcal{H}^{*}\supset\mathcal{C}$ and $\mathcal{C}^{*}\supset\mathcal{H}$.

Now let $Y\overset{}{\in}\mathcal{C}^{*}$. Since $\mathcal{C}$ contains all
subsets which are coarsely disjoint from $B$, $Y$ cannot contain an infinite
subset which is coarsely disjoint from $B$. This implies that
$Y\mathrel{\mathrel{\overset{\scriptscriptstyle\mathrm{c}}{\subset}}}B$. Since
$\mathcal{C}$ also contains all metric neighborhoods of $A$, $Y$ must also
satisfy the second condition of $\mathcal{H}$, so
$Y\overset{}{\in}\mathcal{H}$. Thus
$\mathcal{C}^{*}\mathrel{\overset{\scriptscriptstyle}{\subset}}\mathcal{H}$.

Finally, if $Y\overset{}{\notin}\mathcal{C}$, then $Y$ contains an infinite
locally finite subset which is coarsely disjoint form $A$ and coarsely
contained in $B$. This subset is in $\mathcal{H}$, so
$Y\overset{}{\notin}\mathcal{H}^{*}$. Thus
$\mathcal{H}^{*}\mathrel{\overset{\scriptscriptstyle}{\subset}}\mathcal{C}$. ∎

From now on we will always assume that the cardinality of our space $X$ is
less than the first $\omega$-measurable cardinal.

######  Lemma 9.5.

The evaluation map $e$ of cochains on chains induces isomorphisms of (co)chain
complexes:

$\displaystyle\operatorname{C}{\mathrm{X}}_{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
$\displaystyle=\operatorname{Hom}(\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X),\mathbb{Z}),$
$\displaystyle\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$
$\displaystyle=\operatorname{Hom}(\operatorname{C}{\mathrm{X}}_{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X),\mathbb{Z}).$

###### Proof.

Suppose $R=\mathbb{Z}$, $X=X^{n+1}$, and $B=\Delta_{B}$ and $A=\Delta_{A}$.
Then the corresponding $\mathcal{H}$ is the collection of supports of chains
in
$\operatorname{C}{\mathrm{X}}_{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.
It follows that
$M_{\mathcal{H}}=\operatorname{C}{\mathrm{X}}_{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X)$.
For a similar reason,
$M_{\mathcal{C}}=\operatorname{C}{\mathrm{X}}^{*}(B-A\mathrel{\overset{\scriptscriptstyle}{\subset}}X))$.
So, we need to show that
$M_{\mathcal{H}}=\operatorname{Hom}(M_{\mathcal{C}},\mathbb{Z})$ and
$M_{\mathcal{C}}=\operatorname{Hom}(M_{\mathcal{H}},\mathbb{Z})$.

Lemma 9.3 gives
$M_{\mathcal{C}^{*}}=\operatorname{Hom}(M_{\mathcal{C}},\mathbb{Z})$ and Lemma
9.4 gives $\mathcal{C}^{*}=\mathcal{H}$. Combining these, we get
$M_{\mathcal{H}}=\operatorname{Hom}(M_{\mathcal{C}},\mathbb{Z})$. Similarly,
we can show that
$M_{\mathcal{C}}=\operatorname{Hom}(M_{\mathcal{H}},\mathbb{Z})$. Finally, it
is clear that the evaluation map commutes with the differentials. ∎

## References