$f\colon\mathcal{X}\to\mathcal{Y}$ be a map.

  * •

We call $f$ a _$\sigma$ -map_ if for every $m\in\mathbb{N}$ there exists
$n\in\mathbb{N}$ such that $f(X_{m})\subset Y_{n}$. Note that such an $f$ is
continuous in the final topologies if and only if all the restricted maps
$f|_{X_{m}}\colon X_{m}\to Y_{n}$ are continuous.

  * •

We call a continuous $\sigma$-map $f$ _proper_ if the preimages of all
$\sigma$-compact subspaces of $\mathcal{Y}$ under $f$ are $\sigma$-compact
subspaces of $\mathcal{X}$, or equivalently, if the restricted maps
$f|_{X_{m}}\colon X_{m}\to Y_{n}$ are proper continuous maps.

Using these notions we can introduce the categories of spaces on which we will
consider (co-)homology theories.

######  Definition 5.4.

  * •

Let $\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$ be the category whose
objects are pairs $(\mathcal{X},\mathcal{A})$ of $\sigma$-compact
$\Gamma$-spaces with $\mathcal{A}\subset\mathcal{X}$ a closed
subspace555Recall that subspaces are always $\Gamma$-invariant by Definition
5.2. and whose morphisms between $(\mathcal{X},\mathcal{A})$ and
$(\mathcal{Y},\mathcal{B})$ are continuous $\Gamma$-equivariant $\sigma$-maps
$\mathcal{X}\to\mathcal{Y}$ which restrict to maps
$\mathcal{A}\to\mathcal{B}$.

  * •

Let $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ be the category whose
objects are pairs $(\mathcal{X},\mathcal{A})$ of $\sigma$-locally compact
$\Gamma$-spaces with $\mathcal{A}\subset\mathcal{X}$ a closed subspace and
whose morphisms between $(\mathcal{X},\mathcal{A})$ and
$(\mathcal{Y},\mathcal{B})$ are proper continuous $\Gamma$-equivariant
$\sigma$-maps $\mathcal{X}\to\mathcal{Y}$ which restrict to maps
$\mathcal{A}\to\mathcal{B}$.

  * •

The category $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ has objects the
$\sigma$-locally compact $\Gamma$-spaces and its morphisms between
$\mathcal{X}$ and $\mathcal{Y}$ are proper continuous $\Gamma$-equivariant
$\sigma$-maps $\mathcal{U}\to\mathcal{Y}$ where
$\mathcal{U}\subset\mathcal{X}$ is an open subspace. The composition of two
such morphisms $\mathcal{X}\supset\mathcal{U}\xrightarrow{f}\mathcal{Y}$ and
$\mathcal{Y}\supset\mathcal{V}\xrightarrow{g}\mathcal{Z}$ is defined to be the
morphism $\mathcal{X}\supset f^{-1}(\mathcal{V})\xrightarrow{g\circ
f|_{f^{-1}(\mathcal{V})}}\mathcal{Z}$.

  * •

The category $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}$ is a mixture of
the last two: The objects $(\mathcal{X},\mathcal{A})$ are the same as in
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ and the morphisms are of the
form $\mathcal{X}\supset\mathcal{U}\to\mathcal{Y}$ as in
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ but with the additional
property that they map $\mathcal{A}\cap\mathcal{U}$ into $\mathcal{B}$. If
this is the case, then the restriction
$\mathcal{A}\cap\mathcal{U}\to\mathcal{B}$ is also morphisms in
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$.

  * •

The categories $\mathsf{CH}_{\Gamma}^{2}$, $\mathsf{LCH}_{\Gamma}^{2}$,
$\mathsf{LCH}_{\Gamma}^{+,2}$ and $\mathsf{LCH}_{\Gamma}^{+}$ are the
restrictions of the above categories to pairs of compact Hausdorff spaces and
locally compact Hausdorff spaces, respectively. Equivalently, one can define
them by removing all $\sigma$’s in the above four parts of the definition.
Everything that will be said in the remainder of this section about the above
four categories is also true in complete analogy for these
non-$\sigma$-counterparts and therefore we shall not mention the corresponding
statements explicitly.

  * •

If the index $\Gamma$ is omitted from the notation, this means we do not
consider group actions, i. e. $\Gamma=1$.

The definition of the morphisms in the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ may appear a bit strange at
first sight, but note that they are exactly in one-to-one correspondence with
the basepoint-preserving continuous $\Gamma$-equivariant $\sigma$-maps between
the _one-point $\sigma$-compactifications_

$\mathcal{X}^{+}\coloneqq\bigcup_{n\in\mathbb{N}}X_{n}^{+}=\bigcup_{n\in\mathbb{N}}X_{n}\cup\\{\infty\\}\,,$

i. e. the $\sigma$-compact spaces defined by the corresponding sequence of
one-point compactifications. In other words, the functor
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}\to\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$
which maps a space $\mathcal{X}$ to the pair of spaces
$(\mathcal{X}^{+},\\{\infty\\})$ and a morphism
$\mathcal{X}\supset\mathcal{U}\xrightarrow{f}\mathcal{Y}$ to the continuous
$\sigma$-map

$f^{+}\colon(\mathcal{X}^{+},\infty)\to(\mathcal{Y}^{+},\infty)\quad
x\mapsto\begin{cases}f(x)&x\in\mathcal{U}\\\
\infty&x\in\mathcal{X}^{+}\setminus\mathcal{U}\end{cases}$

is a full and faithful functor.

Conversely, a right inverse to the above functor is the composition of the
full and faithful inclusion of categories
$\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}\to\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$
with the faithful inclusion of categories
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}\to\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}$
and the functor
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}\to\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$
which maps pairs of spaces $(\mathcal{X},\mathcal{A})$ to their difference
$\mathcal{X}\setminus\mathcal{A}$ and morphisms
$(\mathcal{X},\mathcal{A})\to(\mathcal{Y},\mathcal{B})$ which are given by the
data $\mathcal{X}\supset\mathcal{U}\xrightarrow{f}\mathcal{Y}$ to the morphism
$\mathcal{X}\setminus\mathcal{A}\supset
f^{-1}(\mathcal{Y}\setminus\mathcal{B})\xrightarrow{f|_{f^{-1}(\mathcal{Y}\setminus\mathcal{B})}}\mathcal{Y}\setminus\mathcal{B}$.

######  Remark 5.5.

The most convenient way to construct a morphism $\mathcal{U}\to\mathcal{V}$ in
the category $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ is by providing a
morphism $(\mathcal{X},\mathcal{A})\to(\mathcal{Y},\mathcal{B})$ with
$\mathcal{X}\setminus\mathcal{A}=\mathcal{U}$ and
$\mathcal{Y}\setminus\mathcal{B}=\mathcal{V}$ in the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ and applying the above
construction, i. e. applying the last two of those three functors.

Now that we have explained our categories of spaces, we can go on to cartesian
products and homotopies.

######  Definition 5.6.

Let $\mathcal{X}$ be a $\sigma$-locally compact $\Gamma$-space and
$\mathcal{Y}$ a $\sigma$-locally compact $\Gamma^{\prime}$-spaces given by the
filtrations $(X_{m})_{m\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$,
respectively. Their _cartesian product_ $\mathcal{X}\times\mathcal{Y}$ is the
$\sigma$-locally compact $\Gamma\times\Gamma^{\prime}$-space defined by the
sequence of locally compact Hausdorff $\Gamma\times\Gamma^{\prime}$-spaces
$(X_{n}\times Y_{n})_{n\in\mathbb{N}}$.

Note that the union $\bigcup_{n\in\mathbb{N}}X_{n}\times Y_{n}$ equipped with
the final topology is homeomorphic to the cartesian product of the spaces
$\bigcup_{n\in\mathbb{N}}X_{n}$ and $\bigcup_{n\in\mathbb{N}}Y_{n}$ and hence
the interpretation of the expression $\mathcal{X}\times\mathcal{Y}$ as a
topological space is unambiguous.

The following two Lemmas and the corollary are obvious. The first Lemma says
that the cross product is functorial in both variables, the second treats
inclusions of one of the factors as slices and projections onto one of the
factors, and the corollary explains adequate notions of homotopy.

######  Lemma 5.7.

The assignment

$\big{(}(\mathcal{X},\mathcal{A}),\,(\mathcal{Y},\mathcal{B})\big{)}\mapsto(\mathcal{X}\times\mathcal{Y},\mathcal{A}\times\mathcal{Y}\cup\mathcal{X}\times\mathcal{B})$

gives rise to functors

$\displaystyle\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$
$\displaystyle\times\boldsymbol{\sigma}\mathsf{CH}_{\Gamma^{\prime}}^{2}$
$\displaystyle\to$
$\displaystyle\boldsymbol{\sigma}\mathsf{CH}_{\Gamma\times\Gamma^{\prime}}^{2}$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$
$\displaystyle\times\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma^{\prime}}^{2}$
$\displaystyle\to$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma\times\Gamma^{\prime}}^{2}$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}$
$\displaystyle\times\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma^{\prime}}^{+,2}$
$\displaystyle\to$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma\times\Gamma^{\prime}}^{+,2}$

and the assignment
$(\mathcal{X},\mathcal{Y})\mapsto\mathcal{X}\times\mathcal{Y}$ gives rise to a
functor

$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$
$\displaystyle\times\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma^{\prime}}^{+}$
$\displaystyle\to$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma\times\Gamma^{\prime}}^{+}\,.$

In each case, the morphisms are mapped in the obvious way. They are compatible
with each other in the sense that they commute with the three functors
mentioned earlier.∎

######  Lemma 5.8.

Fixing $(\mathcal{Y},\mathcal{B})$ (or only $\mathcal{Y}$) in the above lemma
gives rise to functors

$\displaystyle-\times(\mathcal{Y},\mathcal{B})\colon$
$\displaystyle\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$
$\displaystyle\to\boldsymbol{\sigma}\mathsf{CH}_{\Gamma\times\Gamma^{\prime}}^{2}$
$\displaystyle-\times(\mathcal{Y},\mathcal{B})\colon$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$
$\displaystyle\to\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma\times\Gamma^{\prime}}^{2}$
$\displaystyle-\times(\mathcal{Y},\mathcal{B})\colon$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}$
$\displaystyle\to\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma\times\Gamma^{\prime}}^{+,2}$
$\displaystyle-\times\mathcal{Y}\colon$
$\displaystyle\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$
$\displaystyle\to\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma\times\Gamma^{\prime}}^{+}$

and for each $y\in\mathcal{Y}$ the maps
$\mathcal{X}\mapsto\mathcal{X}\times\\{y\\}\subset\mathcal{X}\times\mathcal{Y}$
are proper continuous $\sigma$-maps and constitute a natural transformation
from the respective identity functors to each of the above functors.

Conversely, if $\mathcal{Y}$ is $\sigma$-compact and $\mathcal{B}=\emptyset$,
then the projection maps $\mathcal{X}\times\mathcal{Y}\to\mathcal{X}$ are
proper continuous $\sigma$-maps and constitute a natural transformation from
$-\times\mathcal{Y}$ to the identity functors in each of these categories and
this natural transformation is left inverse to the abovementioned ones.

######  Corollary 5.9.

For $\mathcal{Y}=I\coloneqq[0,1]$ the unit interval with trivial
$\Gamma^{\prime}=1$-action and $\mathcal{B}=\emptyset$, the above natural
tranformations give rise to an adequate notion of homotopy in each of the four
categories $\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$,
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$,
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}$ and
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$.

###  5.2 Generalized (co-)homology theories and the Rips complex

From all of the categories of spaces that we have introduced in the last
section,
$\mathsf{CH}_{\Gamma}^{2},\mathsf{LCH}_{\Gamma}^{2},\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$
and $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ are almost admissible
categories in the sense of Eilenberg and Steenrod [ES52], except for the
slight difference that the objects are not just topological spaces but contain
a little bit of additional information in the form of the group action and
possibly the filtration by closed subspaces.

######  Definition 5.10.

By a (co-)homology theory (or _$\Gamma$ -equivariant (co-)homology theory_) on
$\mathsf{CH}_{\Gamma}^{2},\mathsf{LCH}_{\Gamma}^{2},\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$
or $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ we understand one in the
sense of Eilenberg and Steenrod which satisfies the homotopy, exactness and
excision axioms, but not necessarily the dimension or the additivity axioms.

In the following, we will always be interested in such a homology theory
$\mathrm{E}^{\Gamma}_{*}$ or such a cohomology theory
$\mathrm{E}_{\Gamma}^{*}$ on the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$. The purpose of this section is
to show that applying them to the Rips complex yields coarse (co-)homology
theories on $\mathsf{CGBG}^{2}_{\Gamma}$ which satisfy the strong homotopy
axiom. From now on, $\Gamma$ is a discrete group.

We first construct the Rips complex for locally finite countably generated
proper isocoarse $\Gamma$-spaces. If $E$ is a $\Gamma$-invariant entourage,
then we define $P_{E}(X)$ to be the geometric realization of the simplicial
complex with vertex set $X$ and with $x_{0},\dots,x_{n}\in X$ spanning an
$n$-simplex iff $(x_{i},x_{j})\in E$ for all $0\leq i,j\leq n$. The space
$P_{E}(X)$ is locally compact, because $X$ is locally finite, and it is a
proper $\Gamma$-space, because $E$ is $\Gamma$-invariant and the action of
$\Gamma$ on $X$ is proper.

Let $S\coloneqq\\{E_{n}\mid n\in\mathbb{N}\\}$ be a countable generating set
of the coarse structure. We may assume ${\Delta_{X}}=E_{0}\subset
E_{1}\subset\dots$ and that each entourage $E$ is contained in one of the
$E_{n}$, because otherwise we simply replace $E_{n}$ by the finite union of
all the entourages that can be obtained from $E_{1},\dots,E_{n}$ by up to $n$
operations of the form 3.-5. of Definition 2.1. Furthermore, we may assume
that all $E_{n}$ are $\Gamma$-equivariant, because the $\Gamma$-action is
isocoarse.

######  Definition 5.11.

The _Rips complex_ of $X$ with respect to the generating set $S$ is the
$\sigma$-locally compact $\Gamma$-space
$\mathcal{P}(X,S)\coloneqq\bigcup_{m\in\mathbb{N}}P_{m}(X)$ where we have
abbreviated $P_{m}(X)\coloneqq P_{E_{m}}(X)$.

Note that as a topological space, the Rips complex is simply the full
simplicial complex of $X$, but as a $\sigma$-space it depends on the the
choice of the generating set, because the sequence of the $P_{m}(X)$ is an
essential part of the data. Nevertheless, the following Lemma justifies to
drop the generating set from the notation and simply write $\mathcal{P}(X)$.

######  Lemma 5.12.

For two different choices of generating sets $S,S^{\prime}$ as above, the
resulting Rips complexes are canonically isomorphic in the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}$ of $\sigma$-locally compact
$\Gamma$-spaces and proper continuous $\Gamma$-equivariant maps via the
identity map
$\operatorname{id}\colon\mathcal{P}(X,S)\to\mathcal{P}(X,S^{\prime})$.

###### Proof.

If $S^{\prime}$ is given by the sequence ${\Delta_{X}}=E^{\prime}_{0}\subset
E^{\prime}_{1}\subset\dots$, then for every $m\in\mathbb{N}$ there is
$n\in\mathbb{N}$ such that $E_{m}\subset E^{\prime}_{n}$ and hence
$P_{E_{m}}(X)\subset P^{\prime}_{E_{n}}(X)$. Therefore the identity is a
$\sigma$-map. ∎

Given in addition a $\Gamma$-invariant subspace $A\subset X$, its Rips complex
$\mathcal{P}(A)$ with respect to the restricted generating set
$\\{E\cap(A\times A)\mid E\in S^{\prime}\\}$ is a closed subspace of
$\mathcal{P}(X)$. If $f\colon(X,A)\to(Y,B)$ is a $\Gamma$-equivariant coarse
map between two pairs of locally finite countably generated coarse
$\Gamma$-spaces, then affine linear extension defines a proper continuous
$\Gamma$-invariant $\sigma$-map
$\mathcal{P}(f)\colon(\mathcal{P}(X),\mathcal{P}(A))\to(\mathcal{P}(Y),\mathcal{P}(B))$,
i. e. a morphism in $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$, the subset
$P_{E}(X)$ being mapped to $P_{F}(Y)$ if $(f\times f)(E)\subset F$. It is
readily verified that close maps $X\to Y$ induce homotopic maps
$\mathcal{P}(X)\to\mathcal{P}(Y)$, where the homotopy is affine linear, but we
even have the following much stronger result.

######  Lemma 5.13.

Let $f,g\colon(X,A)\to(Y,B)$ be $\Gamma$-equivariant coarse maps between two
pairs of locally finite countably generated proper isocoarse $\Gamma$-spaces.
If $f,g$ are $\Gamma$-equivariantly coarsely homotopic, then their induced
maps between the Rips complexes
$\mathcal{P}(f),\mathcal{P}(g)\colon(\mathcal{P}(X),\mathcal{P}(A))\to(\mathcal{P}(Y),\mathcal{P}(B))$
are $\Gamma$-equivariantly homotopic.

###### Proof.

We recall the choices that we have already made at the beginning of the proof
of Lemma 4.4. Let
$H\colon(X\times[a,b],\mathcal{E}_{\mathcal{U}})\to(Y,\mathcal{E}_{Y})$ be a
$\Gamma$-equivariant generalized coarse homotopy between $f$ and $g$ which
maps $A\times[a,b]$ into $B$. For each point $x\in X$ we choose
$a=s_{x,0}<s_{x,1}<\dots<s_{x,k_{x}}=b$ such that $(s,s_{x,j})\in U_{x}$ and
$(s,s_{x,j+1})\in U_{x}$ for all $s\in[s_{x,j},s_{x,j+1}]$,
$j=0,\dots,k_{x}-1$ and we can assume that $s_{x,j}=s_{x\gamma,j}$ for all
$\gamma\in\Gamma$, $x\in X$ and $j=0,\dots,k_{x}-1=k_{x\gamma}-1$. Now, if
$x_{0},\dots,x_{n}$ span an $n$-simplex in $P_{E}(X)$ for some entourage
$E\in\mathcal{E}_{X}$ and if
$s\in\bigcap_{i=0}^{n}[s_{x_{i},j_{i}},s_{x_{i},j_{i}+1}]$, then the points

$(x_{0},s_{x_{0},j_{0}}),\dots,(x_{n},s_{x_{n},j_{n}}),(x_{0},s_{x_{0},j_{0}+1}),\dots,(x_{n},s_{x_{n},j_{n}+1})$

span an $2n$-simplex in $P_{(E\mathbin{\bar{\times}}{\Delta_{[a,b]}})\circ
E_{\mathcal{U}}}(X)$. Due to this property the map
$\tilde{H}\colon\mathcal{P}(X)\times[a,b]\to\mathcal{P}(Y)$ we are now about
to define will obviously be a $\sigma$-map.

We start the construction of $\tilde{H}$ by demanding that $\tilde{H}$ shall
agree with $H$ on the subset $\\{(x,s_{x,j})\mid x\in X\wedge
j=0,\dots,k_{x}\\}$. Then we extend it affine linearly to a map on
$X\times[a,b]$, that is

$\tilde{H}(x,(1-r)s_{x,j}+rs_{x,j+1})\coloneqq(1-r)\cdot H(x,s_{x,j})+r\cdot
H(x,s_{x,j+1})$

for all $j=0,\dots,k_{x}-1$ and $r\in[0,1]$. And finally, we extend it again
affine linearly to all of $\mathcal{P}(X)\times[a,b]$ by

$\tilde{H}\left(\sum_{x\in X}\lambda_{x}x,s\right)\coloneqq\sum_{x\in
X}\lambda_{x}\tilde{H}(x,s)\,.$

Continuity of $\tilde{H}$ is obvious and topological properness of $\tilde{H}$
follows directly from the coarse geometric properness of $H$. It is also a
$\sigma$-map, because it clearly maps $P_{E}(X)$ into $P_{(H\times
H)((E\mathbin{\bar{\times}}{\Delta_{[a,b]}})\circ E_{\mathcal{U}})}(Y)$, and
it obviously maps $\mathcal{P}(A)$ into $\mathcal{P}(B)$. Finally,
$\Gamma$-equivariance follows from the $\Gamma$-invariant choice of the
$s_{x,j}$. ∎

We are now going to apply these constructions to discretizations of pairs
$(X,A)$ of countably generated proper isocoarse $\Gamma$-spaces of
bornologically bounded geometry, i. e. objects in
$\mathsf{CGBG}^{2}_{\Gamma}$. According to Lemma 2.24 there are
$\Gamma$-invariant discretizations $\iota_{X}\colon X^{\prime}\subset X$ and
$A^{\prime}\subset A$ such that there are $\Gamma$-equivariant coarse
equivalences $\pi_{X}\colon X\to X^{\prime}$ and $\pi_{A}\colon A\to
A^{\prime}$ which are the identities on $X^{\prime}$ and $A^{\prime}$,
respectively. By performing the obvious modifications of $X^{\prime}$ and
$\pi_{X}$, if necessary, we may assume that $A^{\prime}\subset X^{\prime}$ and
$\pi_{A}=\pi_{X}|_{A}$. Then the inclusion
$\iota_{X}\colon(X^{\prime},A^{\prime})\to(X,A)$ is a $\Gamma$-equivariant
coarse equivalence with $\pi_{X}$ being a $\Gamma$-equivariant coarse inverse
up to closeness.

######  Definition 5.14.

Let $\mathrm{E}_{*}^{\Gamma}$ be a generalized $\Gamma$-equivariant homology
theory or $\mathrm{E}^{*}_{\Gamma}$ a generalized $\Gamma$-equivariant
cohomology theory. Then its _coarsification_ $\mathrm{EX}_{*}^{\Gamma}$ or
$\mathrm{EX}^{*}_{\Gamma}$ is defined by the groups

$\mathrm{EX}_{*}^{\Gamma}(X,A)\coloneqq\mathrm{E}_{*}^{\Gamma}(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime}))\qquad\text{or}\qquad\mathrm{EX}^{*}_{\Gamma}(X,A)\coloneqq\mathrm{E}^{*}_{\Gamma}(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime}))\,,$

respectively, where $(X^{\prime},A^{\prime})$ denotes some fixed
discretization of $(X,A)$ of the type described above.

######  Theorem 5.15.

Up to canonical isomorphism, the coarsifications
$\mathrm{EX}_{*}^{\Gamma}(X,A)$ and $\mathrm{EX}^{*}_{\Gamma}(X,A)$ are
independent of the choice of discretization. Furthermore, they are the groups
of a $\Gamma$-equivariant (co-)homology theory that satisfies the strong
homotopy axiom.

###### Proof.

If there is another pair of spaces with discretization
$(Y^{\prime},B^{\prime})\xleftrightharpoons[\iota_{Y}]{\pi_{Y}}(Y,B)$ of the
same type and if $f\colon(X,A)\to(Y,B)$ is a $\Gamma$-equivariant coarse map,
then the $\Gamma$-equivariant coarse map $f^{\prime}\coloneqq\pi_{Y}\circ
f\circ\iota_{X}\colon(X^{\prime},A^{\prime})\to(Y^{\prime},B^{\prime})$ is up
to closeness independent of the choice of $\pi_{Y}$ and consequently the
induced map on the Rips complexes
$\mathcal{P}(f)\colon(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime}))\to(\mathcal{P}(Y^{\prime}),\mathcal{P}(B^{\prime}))$
is up to homotopy in the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ independent of the choice of
$\pi_{Y}$ by Lemma 5.13. Thus, we obtain an induced group homomorphism
$f_{*}\colon\mathrm{EX}_{*}^{\Gamma}(X,A)\to\mathrm{EX}_{*}^{\Gamma}(Y,B)$ and
$f^{*}\colon\mathrm{EX}^{*}_{\Gamma}(Y,B)\to\mathrm{EX}^{*}_{\Gamma}(X,A)$
which is independent of the choice of $\pi_{Y}$.

Also, if $f_{1},f_{2}$ are generalized coarsely homotopic, then so are
$f_{1}^{\prime},f_{2}^{\prime}$ and therefore
$\mathcal{P}(f_{1}),\mathcal{P}(f_{2})$ are homotopic in the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$, so $(f_{1})_{*}=(f_{2})_{*}$
and $(f_{1})^{*}=(f_{2})^{*}$.

If there is a third pair of spaces with discretization
$(Z^{\prime},C^{\prime})\xleftrightharpoons[\iota_{Z}]{\pi_{Z}}(Z,C)$ and
$g\colon(Y,B)\to(Z,C)$ is a $\Gamma$-equivariant coarse map, then
$g^{\prime}\circ f^{\prime}$ is close to $(g\circ f)^{\prime}$ and hence
$\mathcal{P}(g^{\prime})\circ\mathcal{P}(f^{\prime})$ is homotopic to
$\mathcal{P}(g^{\prime}\circ f^{\prime})$, so $g_{*}\circ f_{*}=(g\circ
f)_{*}$ and $f^{*}\circ g^{*}=(g\circ f)^{*}$.

Applying all of this to the identity map between the same pair of spaces
$(X,A)$ but equipped with different discretizations $(X^{\prime},A^{\prime})$,
we see that the groups $\mathrm{EX}_{*}^{\Gamma}(X,A)$ and
$\mathrm{EX}^{*}_{\Gamma}(X,A)$ are independent of the choice of
discretization up to canonical isomorphism.

Thus, so far we have shown that the groups form a covariant or a contravariant
functor from $\mathsf{CGBG}_{\Gamma}^{2}$ to the category of abelian groups
which satisfies the strong homotopy axiom. The exactness axiom for
$\mathrm{EX}_{*}^{\Gamma}$, $\mathrm{EX}^{*}_{\Gamma}$ follows immediately
from the exactness axiom for $\mathrm{E}_{*}^{\Gamma}$,
$\mathrm{E}^{*}_{\Gamma}$.

Finally, but most complicatedly, it remains to show the excision axiom. Let
$(X\setminus C,A\setminus C)\subset(X,A)$ be an excision (cf. Definition 3.2).
We can choose discretizations $C^{\prime}\subset C$, $A^{\prime}\subset A$ and
$X^{\prime}\subset X$ as above such that $C^{\prime}\subset A^{\prime}\subset
X^{\prime}$ and then $(X^{\prime}\setminus C^{\prime},A^{\prime}\setminus
C^{\prime})\subset(X^{\prime},A^{\prime})$ is an excision, too. Let
$\alpha\colon\mathcal{P}(X)\to[0,1]$ be the continuous $\Gamma$-invariant
function which is $1$ on $A^{\prime}$, $0$ on $X^{\prime}\setminus A^{\prime}$
and then extended over all simplices affine linearily. In the same way we
construct $\beta\colon\mathcal{P}(X)\to[0,1]$ being $1$ on $C^{\prime}$ and
$0$ on $X^{\prime}\setminus C^{\prime}$. We define the closed subspace
$\mathcal{A}\coloneqq\alpha^{-1}[\frac{1}{3},1]$ and the open subspace
$\mathcal{C}\coloneqq\beta^{-1}(\frac{2}{3},1]$ of $\mathcal{P}(X^{\prime})$.
Note that the closure of $\mathcal{C}$ is contained in the interior of
$\mathcal{A}$, because $\beta\leq\alpha$, and therefore the inclusions
$(\mathcal{P}(X^{\prime})\setminus\mathcal{C},\mathcal{A}\setminus\mathcal{C})\to(\mathcal{P}(X^{\prime}),\mathcal{A})$
are topological excisions and thus induce isomorphisms on (co-)homology.

The proof can now be finished by showing that the subspaces
$\mathcal{P}(A^{\prime})\subset\mathcal{A}$, $\mathcal{P}(X^{\prime}\setminus
C^{\prime})\subset\mathcal{P}(X^{\prime})\setminus\mathcal{C}$ and
$\mathcal{P}(A^{\prime}\setminus
C^{\prime})\subset\mathcal{A}\setminus\mathcal{C}$ are deformation retracts,
because then homotopy invariance and the long exact sequences will show that
the inclusions
$(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime}))\to(\mathcal{P}(X^{\prime}),\mathcal{A})$
and $(\mathcal{P}(X^{\prime}\setminus
C^{\prime}),\mathcal{P}(A^{\prime}\setminus
C^{\prime}))\to(\mathcal{P}(X^{\prime})\setminus\mathcal{C},\mathcal{A}\setminus\mathcal{C})$
induce isomorphisms on (co-)homology.

Constructing a retraction $\mathcal{A}\to\mathcal{P}(A^{\prime})$ is
straightforward: A point $x\in\mathcal{A}$ has a decomposition $x=\lambda
a+(1-\lambda)b$ with $a\in\mathcal{P}(A^{\prime})$ and
$b\in\mathcal{P}(X^{\prime}\setminus A^{\prime})$ in which
$\lambda\geq\frac{1}{3}$. In particular, $\lambda\not=0$ and therefore $a$ is
uniquely determined. Due to this uniqueness, mapping $x\mapsto a$ gives us a
continuous $\Gamma$-equivariant retraction which is homotopic to the identity
via the obvious affine linear homotopy. Furthermore, both the retraction and
the homotopy respect the filtration, because if $x$ lies in a simplex, then
$a$ and the whole path between $x$ and $a$ lie in the same simplex. Exactly
the same procedure provides us with a deformation retraction
$\mathcal{P}(X^{\prime})\setminus\mathcal{C}\to\mathcal{P}(X^{\prime}\setminus
C^{\prime})$.

The tricky part is to construct the deformation retraction
$\mathcal{A}\setminus\mathcal{C}\to\mathcal{P}(A^{\prime}\setminus
C^{\prime})$ and here we will need the excisiveness condition. Let
${\Delta_{X^{\prime}}}=E_{0}\subset E_{1}\subset\dots$ be a sequence of
symmetric $\Gamma$-invariant entourages of $X^{\prime}$ such that every
entourage is contained in some $E_{n}$, and set $E_{-1}=\emptyset$. We choose
a point $x_{O}\in O$ in each $\Gamma$-orbit $O$ of $X^{\prime}$,
$O=x_{O}\Gamma$, and let $\Gamma_{x_{O}}$ denote the stabilizer of $x_{O}$.
Now, there is a unique $n_{O}\in\mathbb{N}$ such that
$x_{O}\in\operatorname{Pen}_{E_{n}}(A^{\prime}\setminus
C^{\prime})\setminus\operatorname{Pen}_{E_{n-1}}(A^{\prime}\setminus
C^{\prime})$ and therefore in particular
$O=x_{O}\Gamma\subset\operatorname{Pen}_{E_{n}}(A^{\prime}\setminus
C^{\prime})$. Choose $y_{O}\in A^{\prime}\setminus C^{\prime}$ such that
$(x_{O},y_{O})\in E_{n}$. Note that for all
$\gamma\in\Gamma,\gamma^{\prime}\in\Gamma_{x_{O}}$ we have
$(x_{O}\gamma,y_{O}\gamma^{\prime}\gamma)\in E_{n}$ by $\Gamma$-invariance of
$E_{n}$. Also, for $O\subset X^{\prime}\setminus C^{\prime},x_{O}\in
X^{\prime}\setminus C^{\prime}$, we have $y_{O}=x_{O}$.

We can now define a map
$r\colon\mathcal{P}(X^{\prime})\to\mathcal{P}(A^{\prime}\setminus C^{\prime})$
as the affine linear extension of the $\Gamma$-equivariant extension of the
mapping
$x_{O}\mapsto\sum_{\gamma^{\prime}\in\Gamma_{x_{O}}}y_{O}\gamma^{\prime}$,
that is

$r\colon\sum_{O\subset
X^{\prime},\gamma\in\Gamma}\lambda_{O,\gamma}x_{O}\gamma\mapsto\sum_{O\subset
X^{\prime},\gamma\in\Gamma,\gamma^{\prime}\in\Gamma_{x_{O}}}\lambda_{O,\gamma}y_{O}\gamma^{\prime}\gamma\,.$

This map is clearly well defined, $\Gamma$-equivariant, continuous and
restricts to the identity on $\mathcal{P}(A^{\prime}\setminus C^{\prime})$.

It is not a $\sigma$-map itself, but its restriction to
$\mathcal{A}\setminus\mathcal{C}$ is: If
$x\in\mathcal{A}\setminus\mathcal{C}$, then each simplex of
$\mathcal{P}(X^{\prime})$ which contains $x$ must have at least one vertex in
$A^{\prime}$ (because $\alpha(x)\geq\frac{1}{3}$) and one vertex in
$X^{\prime}\setminus C$ (because $\beta(x)\leq\frac{2}{3}$). Thus, if this
particular simplex lies in $P_{E}(X^{\prime})$, then all of its vertices lie
in
$\operatorname{Pen}_{E}(A^{\prime})\cup\operatorname{Pen}_{E}(X^{\prime}\setminus
C^{\prime})\subset\operatorname{Pen}_{E_{n}}(A^{\prime}\setminus C^{\prime})$,
where $n\in\mathbb{N}$ only depends on $E$. This shows
$(\mathcal{A}\setminus\mathcal{C})\cap P_{E}(X^{\prime})\subset
P_{E}(\operatorname{Pen}_{E_{n}}(A^{\prime}\setminus C^{\prime}))$ and $r$
clearly maps this subset into $P_{E_{n}\circ E\circ E_{n}}(A^{\prime}\setminus
C^{\prime})$.

The same applies to the homotopy between $r$ and the identity on
$\mathcal{A}\setminus\mathcal{C}$ given by affine linear interpolation.
Therefore, $\mathcal{P}(A^{\prime}\setminus
C^{\prime})\subset\mathcal{A}\setminus\mathcal{C}$ is a deformation retract. ∎

###  5.3 Single space (co-)homology theories and transgression maps

An important class of coarse (co-)homology theories are the coarsifications of
so-called single-space (co-)homology theories. We will show in this section
that they always satisfy the flasqueness axiom and admit the so-called
transgression maps.

######  Definition 5.16.

A (co-)homology theory on $\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$ or
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ is called a _single-
space(co-)homology theory_ (or _$\Gamma$ -equivariant single space
(co-)homology theory_, if we want to emphasize the group action), if it
factors through $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ up to natural
isomorphism. This property is called the _strong excision axiom_.
Equivalently, these (co-)homology theories can be axiomatized using solely the
absolute (co-)homology groups666Therefore the name. as follows.

  * •

A _$\Gamma$ -equivariant single-space homology theory
$\mathrm{E}^{\Gamma}_{*}$ for $\sigma$-locally compact spaces_ is a collection
of covariant homotopy functors $\mathrm{E}^{\Gamma}_{n}$ indexed by
$n\in\mathbb{Z}$ from the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ to the category of abelian
groups together with _boundary maps_
$\partial_{n}\colon\mathrm{E}^{\Gamma}_{n}(\mathcal{X}\setminus\mathcal{A})\to\mathrm{E}^{\Gamma}_{n-1}(\mathcal{A})$
such that the long sequences

$\dots\to\mathrm{E}^{\Gamma}_{n}(\mathcal{A})\to\mathrm{E}^{\Gamma}_{n}(\mathcal{X})\to\mathrm{E}^{\Gamma}_{n}(\mathcal{X}\setminus\mathcal{A})\xrightarrow{\partial_{n}}\mathrm{E}^{\Gamma}_{n-1}(\mathcal{A})\to\dots$

are exact and natural in
$(\mathcal{X},\mathcal{A})\in\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}$.

  * •

Dualy, a _$\Gamma$ -equivariant single-space cohomology theory
$\mathrm{E}^{\Gamma}_{*}$ for $\sigma$-locally compact spaces_ is a collection
of contravariant homotopy functors $\mathrm{E}_{\Gamma}^{n}$ indexed by
$n\in\mathbb{Z}$ from the category
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ to the category of abelian
groups together with _coboundary maps_
$\delta^{n}\colon\mathrm{E}_{\Gamma}^{n}(\mathcal{A})\to\mathrm{E}_{\Gamma}^{n+1}(\mathcal{X}\setminus\mathcal{A})$
such that the long sequences

$\dots\to\mathrm{E}_{\Gamma}^{n}(\mathcal{X}\setminus\mathcal{A})\to\mathrm{E}_{\Gamma}^{n}(\mathcal{X})\to\mathrm{E}_{\Gamma}^{n}(\mathcal{A})\xrightarrow{\delta_{n}}\mathrm{E}_{\Gamma}^{n+1}(\mathcal{X}\setminus\mathcal{A})\to\dots$

are exact and natural in
$(\mathcal{X},\mathcal{A})\in\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+,2}$.

In the same way, $\Gamma$-equivariant single-space (co-)homology theories can
be introduced on $\mathsf{CH}_{\Gamma}^{2}$, $\mathsf{LCH}_{\Gamma}^{2}$ and
$\mathsf{LCH}_{\Gamma}^{+}$.

This definition has already been well-established for a long time on the
categories $\mathsf{CH}^{2}$, $\mathsf{LCH}^{2}$, $\mathsf{LCH}^{+}$. In
[EW17, Definition 4.13] we gave $\sigma$-versions for
$\boldsymbol{\sigma}\mathsf{CH}^{2},\boldsymbol{\sigma}\mathsf{LCH}^{+}$ under
the name “generalized Steenrod (co-)homology theories”, where “Steenrod” was
meant to express that the strong excision axiom holds. However, we drop this
naming in the present paper to avoid confusion, because other authors use it
as a qualifier for the cluster axiom, which says that homology turns disjoint
unions of spaces into products of homology groups and co-homology turns
disjoint unions into direct sums (cf. [WY20, Definition B.2.2]). We will not
use the cluster axiom here, but it should be noted that it might be very
useful for the calculation of (co-)homology of the Rips complexes by cellular
methods.

For the remainder of this section, $\mathrm{E}^{\Gamma}_{*}$ will always
denote a $\Gamma$-equivariant single-space homology theory and
$\mathrm{E}_{\Gamma}^{*}$ will always denote a generalized
$\Gamma$-equivariant single-space cohomology theory for $\sigma$-locally
compact spaces and their coarsifications will be denoted by
$\mathrm{EX}^{\Gamma}_{*}$ and $\mathrm{EX}_{\Gamma}^{*}$, respectively, as
before.

######  Lemma 5.17.

Coarsifications of single-space (co-)homology theories satisfy the flasqueness
axiom.

###### Proof.

Let $X$ be a flasque witnessed by $\phi\colon X\to X$ and let $\iota\colon
X^{\prime}\subset X$ be a discretization as in Lemma 2.24 with
$\Gamma$-equivariant coarse equivalence $\pi\colon X\to X^{\prime}$. Then we
define the map
$(\mathcal{P}(X^{\prime}))^{+}\times[0,\infty]\to(\mathcal{P}(X^{\prime}))^{+}$
as being equal to $\mathcal{P}(\pi\circ\phi^{n}\circ\iota_{X})$ on
$\mathcal{P}(X^{\prime})\times\\{n\\}$, interpolating affine linearily
inbetween and mapping everything else (i. e.
$\\{\infty\\}\times[0,\infty]\cup\mathcal{P}(X^{\prime})\cup\\{\infty\\}$) to
$\infty$. From the assumptions on $\phi$, it is straightforward to see that
this is a $\Gamma$-equivariant proper continous $\sigma$-map. Furthermore, it
is readily checked that it descends to a homotopy in
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ which shows that
$\mathcal{P}(X^{\prime})$ is homotopy equivalent to the empty set. The long
exact sequence shows that (co-)homology vanishes on the empty set, and
therefore its coarsification vanishes on $X$. ∎

Given a coarse space $X$, we let $\mathrm{B}_{h}(X)$ denote the commutative
$\mathrm{C}^{*}\\!$-algebra of all bounded functions $X\to\mathbb{C}$ of
vanishing variation and
$\mathrm{B}_{0}(X)\coloneqq\mathrm{B}_{0}(X;\mathbb{C})$ the ideal of all
functions vanishing at infinity. The _Higson corona_ $\partial_{h}X$ is by
definition the maximal ideal space of the quotient $\mathrm{C}^{*}\\!$-algebra
$\mathrm{B}_{h}(X)/\mathrm{B}_{0}(X)$, i. e. the compact Hausdorff space with
$\mathrm{C}(\partial_{h}X)\cong\mathrm{B}_{h}(X)/\mathrm{B}_{0}(X)$. We call a
compact Hausdorff space $\partial X$ a _Higson dominated corona_ for the
coarse space $X$ if there is a continuous surjective map
$\partial_{h}X\twoheadrightarrow\partial X$.

If $X$ is even a topological coarse space, then $\mathrm{B}_{h}(X)$ contains
the sub-$\mathrm{C}^{*}\\!$-algebra of continuous functions
$\mathrm{C}_{h}(X)=\mathrm{B}_{h}(X)\cap\mathrm{C}(X)$ which contains
$\mathrm{C}_{0}(X)$. Its maximal ideal space is a compactification
$\overline{X}^{h}$ of $X$, called the _Higson compactification_. We have
$\mathrm{C}(\partial_{h}X)\cong\mathrm{B}_{h}(X)/\mathrm{B}_{0}(X)\cong\mathrm{C}_{h}(X)/\mathrm{C}_{0}(X)\cong\mathrm{C}(\overline{X}^{h}\setminus
X)$, so the stable Higson corona is the boundary of this compactification. A
compactification $\overline{X}$ of $X$ with boundary $\partial X$ is called
_Higson dominated_ , if there is a continuous surjection
$\overline{X}^{h}\twoheadrightarrow\overline{X}$ which restricts to the
identity on $X$, or equivalently, if the closure of each entourage $E\subset
X\times X$ intersects $\overline{X}\times\partial X\cup\partial
X\times\overline{X}$ only within $\partial X\times\partial X$. In this case
$\partial X\coloneqq\overline{X}\setminus X$ is a Higson dominated corona.

If $X$ comes equipped with a coarse $\Gamma$-action, then $\partial_{h}X$ and
$\overline{X}^{h}$ are $\Gamma$-spaces. In this case we will always assume
that Higson dominated coronas and compactifications $\partial X,\overline{X}$
are also $\Gamma$-spaces and that the surjections are $\Gamma$-equivariant.

Now assume that $X$ is a countably generated proper isocoarse $\Gamma$-space
of bornologically bounded geometry and $\partial X$ a Higson dominated corona.
Let $X^{\prime}\subset X$ be a $\Gamma$-equivariant discretization as in Lemma
2.24. Then for each entourage $E$ of $X^{\prime}$ there is a Higson dominated
compactification of $P_{E}(X^{\prime})$ with the same corona $\partial X$,
which can be constructed as follows: Note that the corona $\partial X$
corresponds to a unital sub-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra of
$\mathrm{C}(\partial_{h}X)\cong\mathrm{B}_{h}(X)/\mathrm{B}_{0}(X)\cong\mathrm{B}_{h}(X^{\prime})/\mathrm{B}_{0}(X^{\prime})\cong\mathrm{B}_{h}(P_{E}(X^{\prime}))/\mathrm{B}_{0}(P_{E}(X^{\prime}))\cong\mathrm{C}_{h}(P_{E}(X^{\prime}))/\mathrm{C}_{0}(P_{E}(X^{\prime}))$,
that is, it is of the form $C/\mathrm{C}_{0}(P_{E}(X^{\prime}))$ for a unital
sub-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra
$C\subset\mathrm{C}_{h}(P_{E}(X^{\prime}))$ and the maximal ideal space of $C$
is a Higson-dominated compactification $\overline{P_{E}(X^{\prime})}$ of
$P_{E}(X^{\prime})$ with boundary $\partial X$. Applying it to the sequence of
Rips complexes $P_{E_{0}}(X^{\prime})\subset
P_{E_{1}}(X^{\prime})\subset\dots$ we obtain a sequence of compact Hausdorff
$\Gamma$-spaces
$\overline{P_{E_{0}}(X^{\prime})}\subset\overline{P_{E_{1}}(X^{\prime})}\subset\dots$
which are the identity on $\partial X$. Thus, this sequence is a
$\sigma$-compactification $\overline{\mathcal{P}(X^{\prime})}$ of
$\mathcal{P}(X^{\prime})$ with boundary $\partial X$.

Assume furthermore that $A\subset X$ is a coarse $\Gamma$-invariant subspace.
Then the image of $\mathrm{C}(\partial X)$ under the restriction map
$\mathrm{C}(\partial_{h}X)\cong\mathrm{B}_{h}(X)/\mathrm{B}_{0}(X)\to\mathrm{B}_{h}(A)/\mathrm{B}_{0}(A)\cong\mathrm{C}(\partial_{h}A)$
corresponds to a Higson dominated corona $\partial A$ of $A$. By construction
we have a surjection $\mathrm{C}(\partial X)\to\mathrm{C}(\partial A)$, so
$\partial A$ can be considered a subspace of $\partial X$. If
$X^{\prime}\subset X$ and $A^{\prime}\subset A$ are two discretizations as in
Lemma 2.24 with $A^{\prime}\subset X^{\prime}$, then we obtain corresponding
compactifications $\overline{P_{E}(A^{\prime})}$ with boundary $\partial A$
for each entourage $E$ of $X$. The function algebras
$\mathrm{C}(\overline{P_{E}(A^{\prime})})$ are exactly the images of
$\mathrm{C}(\overline{P_{E}(X^{\prime})})$ under the restriction maps
$\mathrm{C}_{h}(P_{E}(X^{\prime}))\to\mathrm{C}_{h}(P_{E}(A^{\prime}))$, so we
can consider $\overline{P_{E}(A^{\prime})}$ as a subspace of
$\overline{P_{E}(X^{\prime})}$. Thus, the $\sigma$-compactification
$\overline{\mathcal{P}(A^{\prime})}$ is also a subspace of
$\overline{\mathcal{P}(X^{\prime})}$ and we note that
$\overline{\mathcal{P}(A^{\prime})}\cap\partial X=\partial A$.

In the proof of Theorem 5.15 we have seen that different choices of
discretizations $(X^{\prime\prime},A^{\prime\prime})$ lead to homotopy
equivalences
$(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime}))\simeq(\mathcal{P}(X^{\prime\prime}),\mathcal{P}(A^{\prime\prime}))$
in the category $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ which are
canonical up to homotopy. They clearly extend to homotopy equivalences
$(\overline{\mathcal{P}(X^{\prime})},\overline{\mathcal{P}(A^{\prime})})\simeq(\overline{\mathcal{P}(X^{\prime\prime})},\overline{\mathcal{P}(A^{\prime\prime})})$
in the category $\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}^{2}$ which are the
identity on the boundaries $(\partial X,\partial A)$ and are also canonical up
to homotopy. Therefore, the next definition is independent of the choice of
discretizations.

######  Definition 5.18.

Let $(X,A)$ be a pair of countably generated proper isocoarse $\Gamma$-spaces
of bornologically bounded geometry, $\partial X$ a Higson dominated corona and
$\partial A$ the corresponding corona of $A$. The _transgression maps_ are the
connecting homomorphisms

$\mathrm{EX}_{*+1}^{\Gamma}(X,A)\to\mathrm{E}_{*}^{\Gamma}(\partial X,\partial
A)\quad\text{and}\quad\mathrm{E}^{*}_{\Gamma}(\partial X,\partial
A)\to\mathrm{EX}^{*+1}_{\Gamma}(X,A)$

associated to the pair of $\sigma$-locally compact spaces
$(\overline{\mathcal{P}(X^{\prime})}\setminus\overline{\mathcal{P}(A^{\prime})},\partial
X\setminus\partial A)$.

Important for applications (see, e. g. Theorems 6.6 and 6.12 below) is the
question when transgression maps are isomorphisms. The long exact sequences
and homotopy invariance immediately imply the following result.

######  Lemma 5.19.

If $\overline{\mathcal{P}(A^{\prime})}$ is a deformation retract of
$\overline{\mathcal{P}(X^{\prime})}$ in the category
$\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}$, then the transgression maps are
isomorphisms. ∎

In the absolute case $A=\emptyset$ there is a similar result if we consider a
reduced version of the transgression maps. To define them, let
$\mathcal{C}\mathcal{K}\coloneqq\mathcal{K}\times[0,1]/\mathcal{K}\times\\{1\\}$
and
$\mathcal{O}\mathcal{K}\coloneqq\mathcal{K}\times(0,1]/\mathcal{K}\times\\{1\\}$
denote the closed and open cone over the $\sigma$-compact $\Gamma$-space
$\mathcal{K}$ and $\Sigma\mathcal{X}\coloneqq\mathcal{X}\times(0,1)$ be be the
suspension of a $\sigma$-locally compact $\Gamma$-space $\mathcal{K}$. Thanks
to $\sigma$-compactness of $\mathcal{K}$, the closed cone is again $\sigma$
compact and the open cone is $\sigma$-locally compact. The obvious deformation
retraction of $\mathcal{X}\times[0,1]$ to $\mathcal{X}\times\\{1\\}$ induces a
homotopy equivalence in $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ between
$\mathcal{X}\times[0,1)$ and the empty space and hence all single space
(co-)homology theories vanish on them.

######  Definition 5.20.

The _suspension isomorphisms_ are the connecting homomorphisms

$\mathrm{E}^{\Gamma}_{*+1}(\Sigma\mathcal{X})\cong\mathrm{E}^{\Gamma}_{*}(\mathcal{X})\quad\text{and}\quad\mathrm{E}_{\Gamma}^{*}(\mathcal{X})\cong\mathrm{E}_{\Gamma}^{*+1}(\Sigma\mathcal{X})$

in the long exact sequences associated to the pairs of spaces
$(\mathcal{X}\times[0,1),\mathcal{X}\times\\{0\\})$. The _reduced_
(co-)homology groups of non-empty $\sigma$-compact $\Gamma$-spaces are defined
as

$\widetilde{\mathrm{E}}^{\Gamma}_{*}(\mathcal{K})\coloneqq\mathrm{E}^{\Gamma}_{*+1}(\mathcal{O}\mathcal{K})\quad\text{and}\quad\widetilde{\mathrm{E}}_{\Gamma}^{*}(\mathcal{K})\coloneqq\mathrm{E}_{\Gamma}^{*+1}(\mathcal{O}\mathcal{K})\,.$

As $\mathcal{C}\mathcal{K}$ is contractible to the apex of the cone $*$, the
long exact sequences associated to the pair
$(\mathcal{C}\mathcal{K},\mathcal{K}\times\\{0\\})$ are

$\displaystyle\cdots\to\mathrm{E}^{\Gamma}_{*+1}(\\{*\\})\to\widetilde{\mathrm{E}}^{\Gamma}_{*}(\mathcal{K})\to\mathrm{E}^{\Gamma}_{*}(\mathcal{K})\to\mathrm{E}^{\Gamma}_{*}(\\{*\\})\to\cdots$
$\displaystyle\cdots\to\mathrm{E}_{\Gamma}^{*}(\\{*\\})\to\mathrm{E}_{\Gamma}^{*}(\mathcal{K})\to\widetilde{\mathrm{E}}_{\Gamma}^{*}(\mathcal{K})\to\mathrm{E}_{\Gamma}^{*+1}(\\{*\\})\to\cdots$
$\displaystyle\,.$ (6)

These sequences imply in particular that reduced (co-)homology vanishes on
contractible spaces.

If $\mathcal{K}$ contains a $\Gamma$-invariant point, then the map
$\mathcal{K}\to\\{*\\}$ has a right inverse in
$\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}$ and the exact sequences yield
canonical isomorphisms

$\widetilde{\mathrm{E}}^{\Gamma}_{*}(\mathcal{K})\cong\ker(\mathrm{E}^{\Gamma}_{*}(\mathcal{K})\to\mathrm{E}^{\Gamma}_{*}(\\{*\\}))\quad\text{and}\quad\widetilde{\mathrm{E}}_{\Gamma}^{*}(\mathcal{K})\cong\operatorname{coker}(\mathrm{E}_{\Gamma}^{*}(\\{*\\})\to\mathrm{E}_{\Gamma}^{*}(\mathcal{K})).$

These isomorphisms are better-known definitions of the reduced groups, but we
emphasize that in the equivariant case they only work under the additional
assumption of the existence of a $\Gamma$-invariant point.

Given a closed non-empty subspace $\mathcal{A}\subset\mathcal{K}$, we have
$\mathcal{O}\mathcal{K}\setminus\mathcal{O}\mathcal{A}=\Sigma(\mathcal{K}\setminus\mathcal{A})$.
Thus, the long exact sequences associated to the pairs
$(\mathcal{O}\mathcal{K},\mathcal{O}\mathcal{A})$ become

$\displaystyle\cdots\to\widetilde{\mathrm{E}}^{\Gamma}_{*}(\mathcal{A})\to\widetilde{\mathrm{E}}^{\Gamma}_{*}(\mathcal{K})\to\mathrm{E}^{\Gamma}_{*}(\mathcal{K},\mathcal{A})\to\widetilde{\mathrm{E}}^{\Gamma}_{*-1}(\mathcal{A})\to\cdots$
$\displaystyle\cdots\to\widetilde{\mathrm{E}}_{\Gamma}^{*-1}(\mathcal{A})\to\mathrm{E}_{\Gamma}^{*}(\mathcal{K},\mathcal{A})\to\widetilde{\mathrm{E}}_{\Gamma}^{*}(\mathcal{K})\to\widetilde{\mathrm{E}}_{\Gamma}^{*}(\mathcal{A})\to\cdots$
$\displaystyle\,.$

By naturality of the long exact sequences under the morphisms
$(\mathcal{O}\mathcal{K},\mathcal{O}\mathcal{A})\supset(\Sigma\mathcal{K},\Sigma\mathcal{A})$
and compatibilty of the suspension maps with the connecting homomorphisms up
to a sign (this follows again by adapting standard arguments from algebraic
topology to our situation) we see that the above long exact sequences are
mapped up to sign to the long exact sequences in unreduced (co-)homology for
the pair $(\mathcal{K},\mathcal{A})$.

Defining reduced transgression as the connecting homomorphisms in reduced
theory associated to the pair $(\overline{\mathcal{P}(X^{\prime})},\partial
X)$ now shows the following

######  Lemma 5.21.

If $\overline{\mathcal{P}(X^{\prime})}$ is contractible in the category
$\boldsymbol{\sigma}\mathsf{CH}_{\Gamma}$, then the _reduced transgression
maps_

$\mathrm{EX}_{*+1}^{\Gamma}(X)\to\widetilde{\mathrm{E}}_{*}^{\Gamma}(\partial
X)\quad\text{and}\quad\widetilde{\mathrm{E}}^{*}_{\Gamma}(\partial
X)\to\mathrm{EX}^{*+1}_{\Gamma}(X)$

are isomorphisms. ∎

As an example, the central Theorem 5.7 of [EW17] says that the hypothesis is
satisfied in the non-equivariant case for the combing compactification of a
proper metric space equipped with an expanding and coherent combing.

Note that the Lemma is, as it stands, probably not very useful in the case of
an infinite $\Gamma$: The contraction point has the infinite stabilizer
$\Gamma$ and must therefore lie in the corona $\partial X$, because the
$\Gamma$-action on $X$ itself is proper. But a corona which contains a point
with stabilizer $\Gamma$ clearly destroys a lot of the coarse geometric
information contained in the group action.

Not all hope is lost, however, because sometimes it might well be possible to
prove vanishing of reduced (co-)homology of
$\overline{\mathcal{P}(X^{\prime})}$ by other means. For example, in the proof
of [EM08, Theorem 14] a related result was used for which it was sufficient
that a compactification is only equivariantly contractible with respect to all
compact subgroups of the (in this case locally compact) group $\Gamma$.

Let us close this section by mentioning the following relation between reduced
and unreduced co-assembly. If $\Gamma$ is finite, $X$ a countably generated
isocoarse $\Gamma$-space of bornologically bounded goemetry and $K\subset X$
is bounded and $\Gamma$-invariant, then we define the countably generated
isocoarse $\Gamma$-space $X^{\to}\coloneqq X\cup_{K}K\times\mathbb{N}$ of
bornologically bounded geometry by equipping it with the obvious coarse
structure (cf. the end of Section 4.3). Then reduced transgression for
$X^{\to}$ is equivalent to relative transgression of the pair
$(X^{\to},K\times\mathbb{N})$ and to unreduced transgression for $X$.

##  6 Further examples

In the following subsections we take a closer look at coarsifications of some
specific directly constructed (co-)homology theories on $\sigma$-spaces and
partly also their transgression maps

###  6.1 Singular (co-)homology

Locally finite singular homology $\mathrm{HS}^{\mathrm{lf}}_{*}(-,-;M)$ and
compactly supported singular cohomology $\mathrm{HS}_{\mathrm{c}}^{*}(-,-;M)$
with coefficients in an abelian group $M$ are known to be (co-)homology
theories on $\mathsf{LCH}^{2}$. Let $\mathrm{CS}^{\mathrm{lf}}_{*}(-,-,M)$ and
$\mathrm{CS}_{\mathrm{c}}^{*}(-,-,M)$ denote their respective (co-)chain
complexes. If $\Gamma$ is in addition a discrete group which acts on $M$ from
the left, then we can perform the following modifications to the (co-)chain
complexes to obtain (co-)homology theories on
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$. First we pass to the
subcomplexes
$\mathrm{CS}^{\mathrm{lf},\Gamma}_{*}(-,-;M)\subset\mathrm{CS}^{\mathrm{lf}}_{*}(-,-;M)$
of $\Gamma$-equivariant chains and the quotient complex
$\mathrm{CS}_{\mathrm{c},\Gamma}^{*}(-,-;M)\coloneqq\Gamma\backslash\mathrm{CS}_{\mathrm{c}}^{*}(-,-;M)$
just as we did in cf. Sections 4.1 and 4.2 to obtain functors on
$\mathsf{LCH}_{\Gamma}^{2}$. And second, if
$\mathcal{X}=\bigcup_{n\in\mathbb{N}}X_{n}$ is a $\sigma$-locally compact
$\Gamma$-space with closed subspace
$\mathcal{A}=\bigcup_{n\in\mathbb{N}}A_{n}$, then we define

$\displaystyle\mathrm{CS}^{\mathrm{lf},\Gamma}_{*}(\mathcal{X},\mathcal{A};M)$
$\displaystyle\coloneqq\varinjlim_{n\in\mathbb{N}}\mathrm{CS}^{\mathrm{lf},\Gamma}_{*}(X_{n},A_{n};M)\,,$
$\displaystyle\mathrm{CS}_{\mathrm{c},\Gamma}^{*}(\mathcal{X},\mathcal{A};M)$
$\displaystyle\coloneqq\varprojlim_{n\in\mathbb{N}}\mathrm{CS}_{\mathrm{c},\Gamma}^{*}(X_{n},A_{n};M)\,,$

where all the maps in the directed systems are injective or surjective,
respectively. The usual proves go through to show that these (co-)chain
complexes define (co-)homology theories $\mathrm{HS}^{\mathrm{lf}}_{*}(-,-;M)$
and $\mathrm{HS}_{\mathrm{c}}^{*}(-,-;M)$ on
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{2}$ and for the sake of
computations we note that we have
$\mathrm{HS}^{\mathrm{lf},\Gamma}_{*}(\mathcal{X},\mathcal{A};M)\cong\varinjlim_{n\in\mathbb{N}}\mathrm{HS}^{\mathrm{lf},\Gamma}_{*}(X_{n},A_{n};M)$
and a Milnor-${\varprojlim}^{1}$-sequence

$0\to{\varprojlim_{n\in\mathbb{N}}}^{1}\mathrm{HS}_{\mathrm{c},\Gamma}^{*-1}(X_{n},A_{n};M)\to\mathrm{HS}_{\mathrm{c},\Gamma}^{*}(\mathcal{X},\mathcal{A};M)\to\varprojlim_{n\in\mathbb{N}}\mathrm{HS}_{\mathrm{c},\Gamma}^{*}(X_{n},A_{n};M)\to
0\,.$

Now, if $(X^{\prime},A^{\prime})$ is a pair of locally finite countably
generated proper isocoarse $\Gamma$-spaces, then its ordinary
$\Gamma$-equivariant coarse (co-)chain complex is nothing else but the
$\Gamma$-equivariant locally finite/compactly supported simplicial (co-)chain
complex of the pair of $\sigma$-simplicial complexes
$(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime}))$ (defined in the completely
analogous way). The usual proofs using barycentric subdivision can be adapted
to the present $\sigma$-situation with $\Gamma$-actions to show that the
singular and simplicial (co-)homology of
$(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime}))$ coincide. This shows
$\mathrm{HX}^{\Gamma}_{*}(X^{\prime},A^{\prime};M)\cong\mathrm{HS}^{\mathrm{lf},\Gamma}_{*}(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime});M)$
and
$\mathrm{HX}_{\Gamma}^{*}(X^{\prime},A^{\prime};M)\cong\mathrm{HS}_{\mathrm{c},\Gamma}^{*}(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime});M)$,
that is, the ordinary $\Gamma$-equivariant coarse (co-)homology is the
coarsification of $\Gamma$-equivariant locally finite/compactly supported
singular (co-)homology.

###  6.2 Alexander–Spanier (co-)homology

Unfortunately, singular (co-)homology does not satisfy strong excision, so
they do not provide transgression maps. Instead one has to use other
(co-)homology theories like Alexander–Spanier cohomology and its dual.

Alexander–Spanier cohomology is known to be a single-space cohomology theory
on $\mathsf{LCH}^{+,2}$ (see in particular [Spa66, Section 6.6] or [Mas78b,
§1.7] for a proof of the strong excision property). By taking inverse limits
of Alexander–Spanier cochain complexes one can easily generalize this
cohomology theory to a single-space cohomology theory on
$\boldsymbol{\sigma}\mathsf{LCH}^{+,2}$ whose coarsification is the (non-
equivariant) ordinary coarse cohomology. As a consequence we obtain a
transgression maps from the Alexander–Spanier cohomology of Higson-dominated
coronas to the coarse cohomology of a space, and one can indeed show that they
agree with Roe’s original transgression map from [Roe93, Section 5.3]. All of
this was done in [EW17, Section 4.4].

Care has to be taken if one tries to dualize Alexander–Spanier cohomology (or
Čech cohomology) in order to obtain a homology theory which satisfies the
strong excision axiom, because most attempts of dualization destroy the long
exact sequences. Massey pointed out how to do it correctly [Mas78a, Mas78b],
providing us with a single-space homology theory on $\mathsf{LCH}^{+,2}$,
which we call Alexander–Spanier homology. It can be generalized easily to
$\boldsymbol{\sigma}\mathsf{LCH}^{+,2}$ by taking direct limits. If
$(X^{\prime},A^{\prime})$ is a pair of uniformly locally finite countably
generated coarse spaces, then all the pairs of Rips complexes
$(P_{E}(X^{\prime}),P_{E}(A^{\prime}))$ are finite dimensional and the
Alexander–Spanier homology of these pairs are isomorphic to their locally
finite cellular homology by [Mas78b, §4.9(4)] and hence also to their locally
finite simplicial homology. Consequently, for spaces of coarsely bounded
geometry the coarsification of Alexander–Spanier homology is (non-equivariant)
ordinary coarse homology and we obtain transgression maps from ordinary coarse
homology to the Alexander–Spanier homology of Higson-dominated coronas.

The proofs of the basic properties of Alexander–Spanier (co-)homology are
somewhat involved, and therefore some non-trivial effort would probably have
to be invested to see if $\Gamma$-actions can be implemented into the
definitions in the naive ways seen in Sections 4.1, 4.1 and 6.2. We are not
going to pursue this question any further.

###  6.3 Coarse $\mathrm{K}$-theory and coassembly

Given a $\sigma$-locally compact $\Gamma$-space
$\mathcal{X}=\bigcup_{n\in\mathbb{N}}X_{n}$ and a
$\Gamma$-$\mathrm{C}^{*}\\!$-algebra $D$, the associated function $*$-algebra

$\mathrm{C}_{0}(\mathcal{X};D)\coloneqq\\{f\colon\mathcal{X}\to D\mid\forall
n\in\mathbb{N}\colon f|_{X_{n}}\in\mathrm{C}_{0}(X_{n};D)\\}$

equipped with the countable family of $\mathrm{C}^{*}\\!$-seminorms
$\|f\|_{n}\coloneqq\|f|_{X_{n}}\|$ and the pull-back action $\gamma\cdot
f\colon x\mapsto\gamma\cdot(f(x\gamma))$ is a so-called
$\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra, i. e. it is a complex
topological $*$-algebra whose topology is Hausdorff, generated by a countable
family of $\Gamma$-equivariant $\mathrm{C}^{*}\\!$-seminorms and is complete
with respect to this family of seminorms. Equivalently, it is the inverse
limit in the category of topological complex $*$-algebras of an inverse system
of $\Gamma$-$\mathrm{C}^{*}\\!$-algebras indexed over the natural numbers (cf.
[Phi88]).

Refering to a previous remark, the $\Gamma$-action on a
$\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra is isometric with respect to all
the seminorms. Instead one could also pose the weaker condition that the
$\Gamma$-action is only continuous with respect to the topology on the whole
algebra. The latter results in what we would call
$\Gamma$-$\sigma$-$\mathrm{C}^{*}\\!$-algebras, i. e. the $\sigma$ and
$\Gamma$ are interchanged to account for the different order in which they are
implemented into the definition.

Via pullback of functions, we obtain a contravariant functor
$\mathrm{C}_{0}(-;D)\colon\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}\to\boldsymbol{\sigma}\mathsf{C}^{*}_{\Gamma}$
into the category of $\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebras and
$\Gamma$-equivariant $*$-homomorphisms, the latter being automatically
continuous. In the case $D=\mathbb{C}$ this functor is actually an equivalence
between the categories
$(\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+})^{\mathrm{op}}$ and the
category of commutative $\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebras
$\mathsf{c}\boldsymbol{\sigma}\mathsf{C}^{*}_{\Gamma}$, i. e. a
Gelfand–Naimark type duality.

If $-\rtimes\Gamma$ is any crossed product functor on the category of
$\Gamma$-$\mathrm{C}^{*}\\!$-algebras, then we obtain a corresponding crossed
product functor from $\boldsymbol{\sigma}\mathsf{C}^{*}_{\Gamma}$ to
$\boldsymbol{\sigma}\mathsf{C}^{*}$ by applying it to an associated inverse
system of $\Gamma$-$\mathrm{C}^{*}\\!$-algebras and taking the inverse limit
afterwards. It is a direct consequence of [Phi88, Proposition 5.3(2)] that the
resulting crossed product functor is exact, if we started with an exact
crossed product functor. We can now extend $\mathrm{K}^{\mathrm{top}}_{*}$ to
$\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebras by turning (4) into a
definition, as was done in [EM07, EM08].

######  Definition 6.1.

For any exact crossed product functor and any
$\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra $C$ we define

$\mathrm{K}^{\mathrm{top}}_{*}(\Gamma,C)\coloneqq\mathrm{K}_{*}((C\mathbin{{\otimes}_{\mathrm{max}}}\mathsf{P})\mathbin{{\rtimes}_{\mathrm{max}}}\Gamma)\,.$

Furthermore, the _$\Gamma$ -equivariant $\mathrm{K}$-theory with coefficients
in a $\Gamma$-$\mathrm{C}^{*}\\!$-algebra $D$_ is defined as

$\mathrm{K}_{\Gamma}^{*}(-;D)\coloneqq\mathrm{K}^{\mathrm{top}}_{-*}(\Gamma,\mathrm{C}_{0}(-;D))\,,$

where we have used Phillips’ $\mathrm{K}$-theory for
$\sigma$-$\mathrm{C}^{*}\\!$-algebras [Phi89, Phi91] .

######  Lemma 6.2.

The functors $\mathrm{K}_{\Gamma}^{*}(-,D)$ and
$\mathrm{K}_{-*}(\mathrm{C}_{0}(-;D)\rtimes\Gamma)$ for any exact crossed
product functor $-\rtimes\Gamma$ are $\Gamma$-equivariant single-space
cohomology theories on $\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ and they
all agree on proper $\sigma$-locally compact $\Gamma$-spaces.

###### Proof.

The first statement follows directly from the long exact sequences and
homotopy invariance of Phillips’ $\mathrm{K}$-theory together with continuity
and exactness of the maximal tensor product and the crossed product functors
in consideration.

If $X$ is a proper locally compact $\Gamma$-space, then $\mathrm{C}_{0}(X;D)$
is a proper $\Gamma$-$\mathrm{C}^{*}\\!$-algebra and [Ech17, Remark 3.4.16]
implies that all crossed products $\mathrm{C}_{0}(X;D)\rtimes\Gamma$, even the
non-exact ones, coincide. Thus, the same is true for
$\mathrm{C}_{0}(\mathcal{X};D)\rtimes\Gamma$ if $\mathcal{X}$ is a proper
$\sigma$-locally compact $\Gamma$-space.

As observed in [EM07, Section 2.7], the Baum–Connes assembly maps yield
isomorphisms
$\mathrm{K}^{\mathrm{top}}_{-*}(\Gamma,C)=\mathrm{K}_{-*}((C\mathbin{{\otimes}_{\mathrm{max}}}\mathsf{P})\mathbin{{\rtimes}_{\mathrm{max}}}\Gamma)\cong\mathrm{K}_{-*}(C\mathbin{{\rtimes}_{\mathrm{max}}}\Gamma)$
for all proper $\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebras $C$. Applying
this to $C=\mathrm{C}_{0}(\mathcal{X};D)$ for proper $\sigma$-locally compact
$\Gamma$-spaces $\mathcal{X}$ finishes the proof. ∎

######  Definition 6.3.

By the second part of the preceding lemma, the coarsifications of
$\mathrm{K}_{\Gamma}^{*}(-,D)$ and
$\mathrm{K}_{-*}(\mathrm{C}_{0}(-;D)\rtimes\Gamma)$ for all exact crossed
product functors $-\rtimes\Gamma$ agree and we call it the _$\Gamma$
-equivariant coarse $\mathrm{K}$-theory_ $\mathrm{KX}_{\Gamma}^{*}(-,D)$.

We are also interested in the reduced $\mathrm{K}$-theory groups of
$\sigma$-compact $\Gamma$-spaces $\mathcal{K}$. In [EM08, Section 2.3],
Emerson and Meyer gave a definition thereof as the $\mathrm{K}$-theory of a
certain $\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra
$\mathrm{C}^{\mathrm{red}}(\mathcal{K};D)$. We recall their construction and
show that it agrees with the reduced groups obtained via Definition 5.20.
Given a non-empty compact Hausdorff space $K$, we let
$\mathrm{C}^{\mathrm{red}}(K;D)$ denote the
sub-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra of
$\mathrm{C}(K;{\mathcal{M}}^{s}(D))$ consisting of those functions for which
$f(x)-f(y)\in D\otimes{\mathfrak{K}}$ for all $x,y\in K$. If
$\mathcal{K}=\bigcup_{n\in\mathbb{N}}K_{n}$ is a $\sigma$-compact Hausdorff
space, then we define the $\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra

$\mathrm{C}^{\mathrm{red}}(\mathcal{K};D)\coloneqq\\{f\colon\mathcal{X}\to{\mathcal{M}}^{s}(D)\mid\forall
n\in\mathbb{N}\colon
f|_{X_{n}}\in\mathrm{C}^{\mathrm{red}}(K_{n};D)\\}=\varprojlim_{n}\mathrm{C}^{\mathrm{red}}(K_{n};D)\,.$

######  Lemma 6.4.

The reduced groups of the cohomology theories $\mathrm{K}^{*}(-;D)$,
$\mathrm{K}_{\Gamma}^{*}(-;D)$ and
$\mathrm{K}_{-*}(\mathrm{C}(-;D\otimes{\mathfrak{K}})\rtimes\Gamma)$ are
canonically naturally isomorphic to
$\mathrm{K}_{-*}(\mathrm{C}^{\mathrm{red}}(-;D))$,
$\mathrm{K}^{\mathrm{top}}_{-*}(\Gamma,\mathrm{C}^{\mathrm{red}}(-;D))$ and
$\mathrm{K}_{-*}(\mathrm{C}^{\mathrm{red}}(-;D\otimes{\mathfrak{K}})\rtimes\Gamma)$,
respectively, in such a way that the corresponding sequences (6) are
isomorphic to the long exact sequences induced by

$0\to\mathrm{C}(-;D)\otimes{\mathfrak{K}}\to\mathrm{C}^{\mathrm{red}}(-;D)\to{\mathcal{M}}^{s}(D)/D\otimes{\mathfrak{K}}\to
0\,.$

###### Proof.

We reuse the proof of Lemma 4.17, but this time with the
$\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebras
$I=\mathrm{C}(\mathcal{K};D)\otimes{\mathfrak{K}}$ and
$A=\mathrm{C}^{\mathrm{red}}(\mathcal{K};D)$. Note that here we have

$\operatorname{Cone}(\iota^{\prime}\colon
D\otimes{\mathfrak{K}}\to\mathrm{C}(\mathcal{K};D))\cong\mathrm{C}_{0}(\mathcal{O}\mathcal{K};D)\otimes{\mathfrak{K}}$

and that the $*$-homomorphism $\operatorname{Cone}(\iota^{\prime})\to
D\otimes{\mathfrak{K}}\cong\mathrm{C}(\\{*\\};D)\otimes{\mathfrak{K}}$ is
exactly the evaluation at the apex of the cone. The claim follows. ∎

The $\Gamma$-equivariant coarse $\mathrm{K}$-theory is the target of the so-
called coassembly map, which we recall next. Let $X$ be a coarsely connected
proper metric isometric $\Gamma$-space. Then for any discretization
$X^{\prime}$ as in Lemma 2.24, all the Rips complexes
$P_{n}(X^{\prime})=P_{E_{n}}(X^{\prime})$ carry an obvious $\Gamma$-isocoarse
coarse structure compatible with the topology such that all of the inclusions
$X\supset X^{\prime}\subset P_{m}(X^{\prime})\subset P_{n}(X^{\prime})$ for
$m\leq n$ are coarse equivalences. It follows that
$\mathfrak{c}^{\mathrm{red}}(\mathcal{P}(X^{\prime});D)\coloneqq\varprojlim_{n}\mathfrak{c}^{\mathrm{red}}(P_{n}(X^{\prime});D)$
is a $\Gamma$-$\mathrm{C}^{*}\\!$-algebra isomorphic to
$\mathfrak{c}^{\mathrm{red}}(X;D)$, because all of the maps in the inverse
system are isomorphisms. Furthermore we define the
$\sigma$-$\Gamma$-$\mathrm{C}^{*}\\!$-algebra
$\overline{\mathfrak{c}}^{\mathrm{red}}(\mathcal{P}(X^{\prime});D)\coloneqq\varprojlim_{n}\overline{\mathfrak{c}}^{\mathrm{red}}(P_{n}(X^{\prime});D)$.
By [EM06] they fit into a short exact sequence

$0\to\mathrm{C}_{0}(\mathcal{P}(X^{\prime});D\otimes{\mathfrak{K}})\to\overline{\mathfrak{c}}^{\mathrm{red}}(\mathcal{P}(X^{\prime});D)\to\mathfrak{c}^{\mathrm{red}}(\mathcal{P}(X^{\prime});D)\cong\mathfrak{c}^{\mathrm{red}}(X;D)\to
0\,.$

If $A\subset X$ is a closed subspace and a suitable pair of discretizations
$(X^{\prime},A^{\prime})$ has been chosen, then we obtain analogously the
short exact sequence

$0\to\mathrm{C}_{0}(\mathcal{P}(X^{\prime})\setminus\mathcal{P}(A^{\prime});D\otimes{\mathfrak{K}})\to\overline{\mathfrak{c}}(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime});D)\to\mathfrak{c}(X,A;D)\to
0$

where
$\overline{\mathfrak{c}}(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime});D)\coloneqq\ker(\overline{\mathfrak{c}}(\mathcal{P}(X^{\prime});D)\to\overline{\mathfrak{c}}(\mathcal{P}(A^{\prime});D))$.

######  Definition 6.5.

The connecting homomorphisms in $\mathrm{K}$-theory

$\begin{array}[]{rr}\mu^{*}\colon&{\begin{array}[]{rl}\mathrm{K}_{1-*}(\mathfrak{c}^{\mathrm{red}}(X;D))&\to\mathrm{KX}^{*}(X;D)\\\
\mathrm{K}_{1-*}(\mathfrak{c}(X,A;D))&\to\mathrm{KX}_{\phantom{\Gamma}}^{*}(X,A;D)\end{array}}\\\
\mu_{\mathrm{top}}^{*}\colon&{\begin{array}[]{rl}\mathrm{K}^{\mathrm{top}}_{1-*}(\Gamma;\mathfrak{c}^{\mathrm{red}}(X;D))&\to\mathrm{KX}_{\Gamma}^{*}(X;D)\\\
\mathrm{K}^{\mathrm{top}}_{1-*}(\Gamma;\mathfrak{c}(X,A;D))&\to\mathrm{KX}_{\Gamma}^{*}(X,A;D)\end{array}}\\\
\mu_{\rtimes}^{*}\colon&{\begin{array}[]{rl}\mathrm{K}_{1-*}(\mathfrak{c}^{\mathrm{red}}(X;D)\rtimes\Gamma)&\to\mathrm{KX}_{\Gamma}^{*}(X;D)\\\
\mathrm{K}_{1-*}(\mathfrak{c}(X,A;D)\rtimes\Gamma)&\to\mathrm{KX}_{\Gamma}^{*}(X,A;D)\end{array}}\end{array}$

associated to the above short exact sequences are all called _co-assembly
maps_.

The three co-assembly maps in the absolute case have originally been defined
in [EM06],[EM07] and [EWZ19], respectively.

It is usually $\mu_{\mathrm{top}}^{*}$ which is of interest, because it is
known to be an isomorphism in many important cases, whereas not much can be
said about $\mu_{\rtimes}^{*}$ [EM08, p. 72]. According to [EWZ19, Lemma 5.2],
the co-assembly map $\mu_{\mathrm{top}}^{*}$ factors through
$\mu_{\rtimes}^{*}$ in the absolute case, but the same also holds true for the
corresponding relative versions. Therefore, if one is only interested in
surjectivity (as was the case in [EWZ19]), then $\mu_{\rtimes}^{*}$ is good
enough.

######  Theorem 6.6.

The transgression maps associated to the cohomology theories
$\mathrm{K}^{*}(-,D)$, $\mathrm{K}_{\Gamma}^{*}(-,D)$ and
$\mathrm{K}_{-*}(\mathrm{C}_{0}(-;D\otimes{\mathfrak{K}})\rtimes\Gamma)$
factor through the assembly maps as follows:

$\begin{array}[]{rcl}\widetilde{\mathrm{K}}^{*-1}(\partial
X;D)\to&\mathrm{K}_{1-*}(\mathfrak{c}^{\mathrm{red}}(X;D))&\to\mathrm{KX}^{*}(X;D)\\\
\mathrm{K}^{*-1}(\partial X,\partial
A;D)\to&\mathrm{K}_{1-*}(\mathfrak{c}(X,A;D))&\to\mathrm{KX}^{*}(X,A;D)\\\
\widetilde{\mathrm{K}}_{\Gamma}^{*-1}(\partial
X;D)\to&\mathrm{K}^{\mathrm{top}}_{1-*}(\Gamma;\mathfrak{c}^{\mathrm{red}}(X;D))&\to\mathrm{KX}_{\Gamma}^{*}(X;D)\\\
\mathrm{K}_{\Gamma}^{*-1}(\partial X,\partial
A;D)\to&\mathrm{K}^{\mathrm{top}}_{1-*}(\Gamma;\mathfrak{c}(X,A;D))&\to\mathrm{KX}_{\Gamma}^{*}(X,A;D)\\\
\mathrm{K}_{1-*}(\mathrm{C}^{\mathrm{red}}(\partial
X;D)\rtimes\Gamma)\to&\mathrm{K}_{1-*}(\mathfrak{c}^{\mathrm{red}}(X;D)\rtimes\Gamma)&\to\mathrm{KX}_{\Gamma}^{*}(X;D)\\\
\mathrm{K}_{1-*}(\mathrm{C}_{0}(\partial X\setminus\partial
A;D\otimes{\mathfrak{K}})\rtimes\Gamma)\to&\mathrm{K}_{1-*}(\mathfrak{c}(X,A;D)\rtimes\Gamma)&\to\mathrm{KX}_{\Gamma}^{*}(X,A;D)\end{array}$

Thus, if a transgression map is an isomorphism (e. g. because
$\overline{\mathcal{P}(X^{\prime})}$ is contractible or
$\overline{\mathcal{P}(A^{\prime})}$ is a deformation retract of
$\overline{\mathcal{P}(X^{\prime})}$), then the corresponding co-assembly map
is surjective.

###### Proof.

The claims follow directly from the commutative diagrams with exact rows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}_{0}(\mathcal{P}(X^{\prime});D\otimes{\mathfrak{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}^{\mathrm{red}}(\overline{\mathcal{P}(X^{\prime})};D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}^{\mathrm{red}}(\partial
X;D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}_{0}(\mathcal{P}(X^{\prime});D\otimes{\mathfrak{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathfrak{c}}^{\mathrm{red}}(\mathcal{P}(X^{\prime});D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{c}^{\mathrm{red}}(X;D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}_{0}(\mathcal{P}(X^{\prime})\setminus\mathcal{P}(A^{\prime});D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}_{0}(\overline{\mathcal{P}(X^{\prime})}\setminus\overline{\mathcal{P}(A^{\prime})};D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}_{0}(\partial
X\setminus\partial
A;D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}_{0}(\mathcal{P}(X^{\prime})\setminus\mathcal{P}(A^{\prime});D\otimes{\mathfrak{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathfrak{c}}(\mathcal{P}(X^{\prime}),\mathcal{P}(A^{\prime});D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{c}(X,A;D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

which exist because functions on a Higson-dominated compactification restrict
to functions of vanishing variation. ∎

###  6.4 Coarse $\mathrm{K}$-homology and assembly

Using $\Gamma$-equivariant $\mathrm{E}$-theory777Of course,
$\mathrm{KK}$-theory could also be used. $\mathrm{E}$-theory was only selected
because of the author’s greater familiarity with this theory., we define the
$\Gamma$-equivariant $\mathrm{K}$-homology with coefficients in a separable
$\Gamma$-$\mathrm{C}^{*}\\!$-algebra $D$ as the functors

$\mathrm{K}^{\Gamma}_{*}(-;D)\coloneqq\mathrm{E}^{\Gamma}_{*}(\mathrm{C}_{0}(-);D)\,.$

Note that this definition requires the function algebras $\mathrm{C}_{0}(-)$
to be separable, so we only consider these functors on the full subcategory of
$\mathsf{LCH}_{\Gamma}^{+}$ of all _second countable_ locally compact
$\Gamma$-spaces. This is only a minor restriction to the theory of Section 5,
because we are mainly interested in two special cases: First, Rips complexes
of discretizations of countably generated proper isocoarse $\Gamma$-spaces of
bornologically bounded geometry are always second countable and, second, their
compactification with a Higson dominated corona is second countable, if the
corona is second countable, i. e. a compact metrizable space.

Due to the well-known properties of bivariant $\mathrm{K}$-theory, the
functors $\mathrm{K}^{\Gamma}_{*}(-;D)$ are $\Gamma$-equivariant single-space
homology theories on the above-mentioned category. Furthermore, as direct
limits preserve exact sequences, they can be extended to $\Gamma$-equivariant
single-space homology theories on the full subcategory of
$\boldsymbol{\sigma}\mathsf{LCH}_{\Gamma}^{+}$ consisting of all second
countable $\sigma$-locally compact $\Gamma$-spaces
$\mathcal{X}=\bigcup_{n\in\mathbb{N}}X_{n}$ by
$\mathrm{K}^{\Gamma}_{*}(\mathcal{X};D)\coloneqq\lim_{n\to\infty}\mathrm{K}^{\Gamma}_{*}(X_{n};D)$.

We want to compare the resulting transgression maps with the coarse assembly
maps
$\mu\colon\mathrm{KX}^{\Gamma}_{*}(X;D)\to\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X;D))$.
The definition of the latter is based upon other pictures of
$\mathrm{K}$-homology. We choose the one using Yu’s localization algebras
[Yu97] (see also [QR10] and [EWZ19]) for our discussion. A lot about
localization algebras has been proven in the comprehensive book [WY20], but
note that the “localized Roe algebras” considered there are slightly bigger
than Yu’s original localization algebras and hence a bit of care has to be
taken when applying their results.

Following up our Section 4.4 about Roe algebras, let $X$ be a coarsely
connected proper metric space equipped with a proper isometric
$\Gamma$-action. We assume that sufficiently large $\Gamma$-$X$-module
$(H,\rho,u)$ has been chosen and let $\mathfrak{H}\coloneqq H\otimes D$ be
equipped with the induced representations. Recall that we defined
$\mathrm{C}^{*}_{\Gamma}(X;D)$ as the norm closure in
${\mathfrak{B}}(\mathfrak{H})$ of the $*$-subalgebra of all
$\Gamma$-equivariant locally compact operators of finite propagation, and if
$A$ is a subspace of $X$, then $\mathrm{C}^{*}_{\Gamma}(A\subset X;D)$ denoted
the ideal generated by all operators which are supported in an
$R$-neighborhood of $A$ for some $R>0$ and
$\mathrm{C}^{*}_{\Gamma}(X,A;D)\coloneqq\mathrm{C}^{*}_{\Gamma}(X;D)/\mathrm{C}^{*}_{\Gamma}(A\subset
X;D)$.

######  Definition 6.7.

The _localization algebra_ $\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)$ is the
norm closure of the sub-$*$-algebra of
$\mathrm{C}_{b}([1,\infty),\mathrm{C}^{*}_{\Gamma}(X))$ of all bounded
uniformly continuous functions
$L\colon[1,\infty)\to\mathrm{C}^{*}_{\Gamma}(X)$ such that the propagation of
$L(t)$ is finite for all $t\geq 1$ and converges to zero as $t\to\infty$.
Furthermore, we denote by $\mathrm{C}^{*}_{\mathrm{L},\Gamma}(A\subset
X;D)\subset\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)$ the ideal generated by all
$L$ for which there is a function $\varepsilon\colon[1,\infty)\to(0,\infty)$
with $\varepsilon(t)\xrightarrow{t\to\infty}0$ such that $L(t)$ is supported
in an $\varepsilon(t)$-neighborhood of $A$ for all $t\geq 1$, and we define
$\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X,A;D)\coloneqq\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)/\mathrm{C}^{*}_{\mathrm{L},\Gamma}(A\subset
X;D)$.

An obvious variant of [WY20, Lemma 6.3.6], which can be proven with the same
methods, says that the inclusion $A\subset X$ induces via the theory of
covering isometries an isomorphism
$\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(A;D))\cong\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(A\subset
X;D))$. Therefore we obtain a long exact sequence

$\displaystyle\dots$
$\displaystyle\to\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(A;D))\to\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D))\to\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X,A;D))$
$\displaystyle\to\mathrm{K}_{*-1}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(A;D))\to\dots\,.$
(7)

The relation between localization algebras and our definition of
$\mathrm{K}$-homology is the following. As ${\mathfrak{K}}(\mathfrak{H})\cong
D\otimes{\mathfrak{K}}$, it is straightforward to verify (cf. [QR10, Corollary
4.2]) that

$\displaystyle\delta\colon\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}_{0}(X)$
$\displaystyle\to\mathrm{C}_{b}([1,\infty),D\otimes{\mathfrak{K}})/\mathrm{C}_{0}([1,\infty),D\otimes{\mathfrak{K}})$
$\displaystyle L\otimes f$ $\displaystyle\mapsto[t\mapsto
L(t)\circ(\rho(f)\otimes\operatorname{id}_{D})]$

defines an $\Gamma$-equivariant asymptotic morphism. It vanishes on the ideal
$\mathrm{C}^{*}_{\mathrm{L},\Gamma}(A\subset
X;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}_{0}(X\setminus A)$ and hence
we also obtain an induced $\Gamma$-equivariant asymptotic morphism

$\delta\colon\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X,A;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}_{0}(X\setminus
A)\to\mathrm{C}_{b}([1,\infty),D\otimes{\mathfrak{K}})/\mathrm{C}_{0}([1,\infty),D\otimes{\mathfrak{K}})\,.$

Both of them define canonical $\mathrm{E}^{\Gamma}$-theory elements in
$\mathrm{E}^{\Gamma}_{0}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}_{0}(X),D)$
and
$\mathrm{E}^{\Gamma}_{0}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X,A;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}_{0}(X\setminus
A),D)$, respectively, which we shall also denote by the letter
$\delta$.888Here we define the groups $\mathrm{E}^{\Gamma}_{*}(A,B)$ for non-
separable $A$ as the inverse limit of the directed system over all
$\mathrm{E}^{\Gamma}_{*}(A^{\prime},B)$ with $A^{\prime}\subset A$ separable.
We obtain the composed group homomorphism

$\displaystyle\Delta\colon\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D))$
$\displaystyle\cong\mathrm{E}_{*}(\mathbb{C},\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D))$
$\displaystyle\xrightarrow{-\otimes\operatorname{id}_{\mathrm{C}_{0}(X)}}\mathrm{E}^{\Gamma}_{*}(\mathrm{C}_{0}(X),\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}_{0}(X))$
$\displaystyle\xrightarrow{\delta\circ-}\mathrm{E}^{\Gamma}_{*}(\mathrm{C}_{0}(X),D)=\mathrm{K}^{\Gamma}_{*}(X;D)$

and similarily
$\Delta\colon\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X,A;D))\to\mathrm{K}^{\Gamma}_{*}(X,A;D)$
in the relative case. By exploiting the homological properties of domains and
targets of the maps $\Delta$ (in particular that the domains are functorial
under uniformly continuous coarse $\Gamma$-maps via the theory of covering
isometries, that the maps $\Delta$ are natural under such maps and that the
connecting homomorphisms are in both cases constructed via mapping cones) we
see that the maps $\Delta$ map (7) to the long exact sequence of
$\mathrm{K}^{\Gamma}_{*}(-,D)$. It appears to be a wide spread believe that
the following assumption is true for general $\Gamma$ and $D$, but the author
does not know whether anyone has ever checked all the details.

######  Assumption 6.8.

We assume that $\Gamma$ and $D$ have been chosen for which the maps $\Delta$
are isomorphisms in the absolute case for every coarsely connected proper
metric space $X$ equipped with a proper isometric $\Gamma$-action and hence by
a five lemma argument also in the relative case.

The assumption is known to hold in the non-equivariant case ($\Gamma=1$)
without coefficients ($D=\mathbb{C}$). One proof thereof is sketched in [WY20,
Construction 6.7.5 and Exercises B.3.2,B.3.3], albeit for a different version
of the localization algebra. Note, however, that this proof relies solely on
homological properties and a comparison on one-point spaces. Therefore, it
goes through for original version of the localizaton algebras, too, and even
with arbitrary coefficient $\mathrm{C}^{*}\\!$-algebras $D$.

Another proof for the case $\Gamma=1$ and $D=\mathbb{C}$ can be found in
[QR10, Proposition 4.3] and relies on Paschke duality. Now, the isomorphism
$\omega_{4}$ in [Roe02] is an equivariant version of Paschke duality without
coefficients and an non-equivariant version with coefficients was mentioned in
the discussion preceding Definition 4.6 in [HPR97]. Both sources do not give
any details, but it seems reasonable to assume that their constructions can be
unified to a Paschke duality in the equivariant case with coefficients, from
which 6.8 can be derived in full generality. A related result is the
description of non-equivariant $\mathrm{KK}$-groups using a variant of the
localization algebra in [DWW18].

In order to keep this example section concise, we refrain from validating 6.8
in full generality and just assume that we only consider the non-empty class
of cases, in which the statement is true. However, as localization algebras
and $\mathrm{E}$-theory are defined in a very similiar manner, it would be
interesting to know if the assumption can be proven in full generality
directly without taking the detour via Paschke duality.

The reason why we do not just choose the isomorphism 6.8 as definition of
$\Gamma$-equivariant $\mathrm{K}$-homology with coefficients is that we prefer
to have a definition that works also for non-proper group actions, but the
proofs of the homological properties of
$\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(-;\mathbb{C}))$ in [WY20],
in particular the existence of the all important covering isometries
underlying functoriality, rely heavily on properness. Note also that the
equivariant Paschke duality in [Roe02] works only for proper $\Gamma$-actions,
so the two definitions might truely be different in the non-proper case.
Unfortunately, this makes the proof of Theorem 6.12 below much more
complicated than the argument using Paschke duality which was presented in
[HR00, Remark 12.3.8] for the non-equivariant case without coefficients.

######  Definition 6.9.

Let $(X,A)$ be a pair of coarsely connected proper metric spaces equipped with
a proper isometric $\Gamma$-action. Under 6.8, the _(uncoarsified) coarse
assembly maps_

$\begin{array}[]{rrcccl}\mu\colon&\mathrm{K}^{\Gamma}_{*}(X;D)&\cong&\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D))&\xrightarrow{(\mathop{\textrm{ev}_{1}})_{*}}&\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X;D))\\\
\mu\colon&\mathrm{K}^{\Gamma}_{*}(X,A;D)&\cong&\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X,A;D))&\xrightarrow{(\mathop{\textrm{ev}_{1}})_{*}}&\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X,A;D))\end{array}$

are induced by the evaluation at one.

The coarsified coarse assembly map will be obtained by taking the direct limit
over all the uncoarsified assembly maps of the Rips complexes
$(P_{n}(X^{\prime}),P_{n}(A^{\prime}))$ of a discretization
$(X^{\prime},A^{\prime})\subset(X,A)$. As we have chosen a description of the
assembly maps which requires proper metrics, we can’t content ourselves with
the canonical topological coarse structures on the Rips complexes, but we have
to metrize them. Most authors equip the Rips complexes $P_{n}(X^{\prime})$
with the spherical metric from [Roe91b], but they have the drawback that the
inclusions $X^{\prime}\subset P_{n}(X^{\prime})$ are in general not coarse
equivalences if the original space $X$ was not a path-metric space. Therefore,
we use the $\Gamma$-invariant metrics from the following lemma, such that all
of the inclusions $X\supset X^{\prime}\subset P_{m}(X^{\prime})\subset
P_{n}(X^{\prime})$ for $m\leq n$ are even isometric coarse equivalences.

######  Lemma 6.10.

We can assign to each discrete proper metric isometric $\Gamma$-space
$X^{\prime}$ a $\Gamma$-invariant metric $d_{\mathcal{P}(X^{\prime})}$ on
$\mathcal{P}(X^{\prime})$ such that

  * •

its restriction to $P_{0}(X^{\prime})=X^{\prime}$ is the original metric
$d_{X^{\prime}}$;

  * •

its restriction to each $P_{n}(X^{\prime})$ is a $\Gamma$-invariant _proper_
metric inducing the already existing topology on the complex;

  * •

for all $m\leq n$, $P_{m}(X^{\prime})$ is coarsely dense in
$P_{n}(X^{\prime})$;

  * •

if $f\colon X^{\prime}\to Y^{\prime}$ is a coarse $\Gamma$-map between
discrete proper metric isometric $\Gamma$-spaces and $X^{\prime}$ is
$\delta$-separated for some $\delta>0$, i. e. the distance between each two
distinct points of $X^{\prime}$ is at least $\delta$, then all of the
restrictions

$\mathcal{P}(f)|_{P_{n}(X^{\prime})}\colon
P_{n}(X^{\prime})\to\mathcal{P}(Y^{\prime})$

are low-scale Lipschitz coarse maps.

The last property is very useful when working with the localization algebras,
because the direct description of the functoriality of their
$\mathrm{K}$-theory using families of covering isometries (in the concrete
version of [EWZ19, Definition 4.29 & Proposition 4.30]) is only available for
uniformly continuous coarse maps. Note that $\delta$-separatedness is never a
problem, because all discretizations either already have this property by
construction (cf. the proof of Lemma 2.6 applied to the entourage
$E_{\delta}$) or can be thinned out such that they satisfy this property.

###### Proof.

For $R>0$, the $R$-spherical metric on a $k$-simplex is the path metric
obtained by identifying it with the subset

$S^{k}_{+}(R)\coloneqq\\{(x_{0},\dots,x_{k})\mid x_{0},\dots,x_{k}\geq 0\wedge
x_{0}^{2}+\dots+x_{k}^{2}=R^{2}\\}\subset\mathbb{R}^{k+1}\,.$

On each $P_{n}(X^{\prime})$ with $n\geq 1$ we first define the auxiliary
metric $d^{1}_{P_{n}(X^{\prime})}$ as the largest metric such that its
restriction to each simplex is at most the $2n$-spherical metric. This metric
is clearly a proper path metric and it induces the given topology, but it may
take the value $\infty$ and is not yet the restriction of the metric
$d_{\mathcal{P}(X^{\prime})}$ that we are looking for. Also, note that
$X^{\prime}$ is $\pi n$-dense in $P_{n}(X^{\prime})$ with respect to
$d^{1}_{P_{n}(X^{\prime})}$. On $P_{0}(X^{\prime})=X^{\prime}$ we simply
define $d^{1}_{P_{0}(X^{\prime})}=d_{X^{\prime}}$ instead.

Thanks to the choice $R=2n$, similar arguments to those in the proof of
[Roe91b, Proposition 3.4(iv)] (note also [Roe91a]) show that the inclusion
$X^{\prime}\subset P_{n}(X^{\prime})$ is distance non-decreasing with respect
to $d_{X^{\prime}}$ and $d^{1}_{P_{n}(X^{\prime})}$ for each $n\in\mathbb{N}$.
Hence there is a largest metric $d^{2}_{P_{n}(X^{\prime})}$ on
$P_{n}(X^{\prime})$ which is less or equal to $d^{1}_{P_{n}(X^{\prime})}$ and
restricts to $d_{X^{\prime}}$ on $X^{\prime}$. These new metrics have the
property that the inclusions $P_{m}(X^{\prime})\subset P_{n}(X^{\prime})$ are
also distance non-decreasing with respect to $d^{2}_{P_{m}(X^{\prime})}$ and
$d^{2}_{P_{n}(X^{\prime})}$ for all $m\leq n$.

Using all of these properties, it is now readily verified that there is a
largest metric on $d_{\mathcal{P}(X^{\prime})}$ on $\mathcal{P}(X^{\prime})$
such that all of the restrictions to $P_{n}(X^{\prime})$ are less or equal to
$d^{2}_{P_{m}(X^{\prime})}$ and this metric clearly satisfies the first three
properties on the list and is $\Gamma$-invariant.

Now, if $f\colon X^{\prime}\to Y^{\prime}$ and $\delta>0$ are as in the forth
item on the list, then for each $n\in\mathbb{N}$ there is an
$S_{n}\in\mathbb{N}$ such that $d_{X^{\prime}}(x,y)\leq n$ implies
$d_{Y^{\prime}}(f(x),f(y))\leq S_{n}$. Note that $\mathcal{P}(f)$ restricts to
a $\frac{S_{n}}{n}$-Lipschitz map from
$(P_{n}(X^{\prime}),d^{1}_{P_{n}(X^{\prime})})$ to
$(P_{S_{n}}(Y^{\prime}),d^{1}_{P_{S_{n}}(Y^{\prime})})$ for every $n\geq 1$
and, therefore, if $x,y\in P_{n}(X^{\prime})$ with
$d_{\mathcal{P}(X^{\prime})}(x,y)<\frac{\delta}{2}$, then
$d_{\mathcal{P}(Y^{\prime})}(f(x),f(y))\leq L\cdot
d_{\mathcal{P}(X^{\prime})}(x,y)$ with
$L\coloneqq\max(S_{1},\frac{S_{2}}{2},\dots,\frac{S_{n}}{n})$, and that is
what we mean when we say that the restriction is low-scale $L$-Lipschitz. ∎

######  Definition 6.11.

Taking direct limits over the uncoarsified assembly maps of the Rips
complexes, we obtain the _(coarsified) coarse assembly maps_ ,

$\begin{array}[]{rrcl}\mu\colon&\mathrm{KX}^{\Gamma}_{*}(X;D)&=&\varinjlim_{n\in\mathbb{N}}\mathrm{K}^{\Gamma}_{*}(P_{n}(X^{\prime});D)\\\
&&\to&\varinjlim_{n\in\mathbb{N}}\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(P_{n}(X^{\prime});D))\cong\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X;D))\\\
\mu\colon&\mathrm{KX}^{\Gamma}_{*}(X,A;D)&=&\varinjlim_{n\in\mathbb{N}}\mathrm{K}^{\Gamma}_{*}(P_{n}(X^{\prime}),P_{n}(A^{\prime});D)\\\
&&\to&\varinjlim_{n\in\mathbb{N}}\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(P_{n}(X^{\prime}),P_{n}(A^{\prime});D))\cong\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X,A;D))\,.\end{array}$

######  Theorem 6.12.

Given a Higson dominated corona $\partial X$ of a coarsely connected proper
metric space $X$ equipped with a proper isometric $\Gamma$-action, then under
6.8 the transgression maps factor through the assembly maps as follows:

$\begin{array}[]{rcl}\mathrm{KX}^{\Gamma}_{*}(X;D)\xrightarrow{\mu}&\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X;D))&\xrightarrow{\nu}\tilde{\mathrm{K}}^{\Gamma}_{*-1}(\partial
X;D)\\\
\mathrm{KX}^{\Gamma}_{*}(X,A;D)\xrightarrow{\mu}&\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X,A;D))&\xrightarrow{\nu}\mathrm{K}^{\Gamma}_{*-1}(\partial
X,\partial A;D)\end{array}$

In particular, if the transgression maps are isomorphisms, then the assembly
maps are injective.

###### Proof.

Let $\overline{X}$ be the compactification of $X$ corresponding to the corona
$\partial X$. The map $\nu$ is defined in the first case as follows. Functions
in $\mathrm{C}_{0}(\overline{X})$ have vanishing variation and hence commute
with operators in $\mathrm{C}^{*}_{\Gamma}(X;D)$ up to compact operators as in
the the proof of [Roe93, Proposition]. Hence there is a $\Gamma$-equivariant
$*$-homomorphism

$\mathrm{C}^{*}_{\Gamma}(X;D)\otimes\mathrm{C}(\overline{X})\to{\mathcal{M}}^{s}(D)/D\otimes{\mathfrak{K}}\,,\quad
T\otimes f\mapsto[Tf]$

which vanishes on $\mathrm{C}^{*}_{\Gamma}(X;D)\otimes\mathrm{C}_{0}(X)$ and
hence factors through
$\mathrm{C}^{*}_{\Gamma}(X;D)\otimes\mathrm{C}_{0}(\partial X)$. Identifying
$\mathrm{C}_{0}(\mathcal{O}(\partial X))$ with
$\operatorname{Cone}(\mathbb{C}\to\mathrm{C}(\partial X))$ and applying the
$*$-homomorphism above slice-wise, we obtain a $\Gamma$-equivariant
$*$-homomorphism

$\delta^{\prime}\colon\mathrm{C}^{*}_{\Gamma}(X;D)\otimes\mathrm{C}_{0}(\mathcal{O}(\partial
X))\to\operatorname{Cone}\big{(}{\mathcal{M}}^{s}(D)\to{\mathcal{M}}^{s}(D)/D\otimes{\mathfrak{K}}\big{)}\,.$

The inclusion of $D\otimes{\mathfrak{K}}$ as a canonical ideal in the mapping
cone to the right induces an isomorphism on $\mathrm{E}$-theory and hence we
can define the map $\nu$ as the composition

$\displaystyle\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X;D))$
$\displaystyle\cong\mathrm{E}_{*}(\mathbb{C},\mathrm{C}^{*}_{\Gamma}(X;D))$
$\displaystyle\xrightarrow{-\otimes\operatorname{id}_{\mathrm{C}_{0}(\mathcal{O}(\partial
X))}}\mathrm{E}^{\Gamma}_{*}(\mathrm{C}_{0}(\mathcal{O}(\partial
X)),\mathrm{C}^{*}_{\Gamma}(X;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}_{0}(\mathcal{O}(\partial
X)))$
$\displaystyle\xrightarrow{\delta^{\prime}\circ-}\mathrm{E}^{\Gamma}_{*}\left(\mathrm{C}_{0}(\mathcal{O}(\partial
X)),\operatorname{Cone}\big{(}{\mathcal{M}}^{s}(D)\to{\mathcal{M}}^{s}(D)/D\otimes{\mathfrak{K}}\big{)}\right)$
$\displaystyle\cong\mathrm{E}^{\Gamma}_{*-1}(\mathrm{C}_{0}(\mathcal{O}(\partial
X)),D\otimes{\mathfrak{K}})=\tilde{\mathrm{K}}^{\Gamma}_{*}(X;D)\,.$

The definition of $\nu$ is functorial under $\Gamma$-equivariant coarse maps
and thus it identifies with the corresponding maps $\nu_{n}$ associated to the
compactification $\overline{X}_{n}$ of $X_{n}\coloneqq P_{n}(X^{\prime})$ by
$\partial X$. We have to show that the diagram

$\textstyle{\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n};D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathop{\textrm{ev}_{1}})_{*}}$$\scriptstyle{\cong}$$\scriptstyle{\Delta}$$\textstyle{\mathrm{K}_{*}(\mathrm{C}^{*}_{\Gamma}(X_{n};D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{n}}$$\textstyle{\mathrm{K}^{\Gamma}_{*}(X_{n};D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\tilde{\mathrm{K}}^{\Gamma}_{*-1}(\partial
X;D)}$

commutes and then the claim follows by taking the direct limit for
$n\to\infty$. This will be done by considering for each
$x\in\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n};D))$ the
following diagram in the $\mathrm{E}^{\Gamma}$-category, in which all solid
arrows come from $*$-homomorphisms and all dashed arrows from asymptotic
morphisms or $\mathrm{E}^{\Gamma}$-theory elements.

$\textstyle{{\begin{array}[]{l}\mathrm{C}_{0}(0,1)\\\
\otimes\mathrm{C}_{0}(X_{n})\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{x\otimes\operatorname{id}}$$\textstyle{\operatorname{Cone}{\begin{pmatrix}\mathrm{C}_{0}(\mathcal{O}(\overline{X}_{n}))\\\
\downarrow\\\ \mathrm{C}_{0}(\mathcal{O}(\partial
X))\end{pmatrix}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\otimes\operatorname{id}}$$\textstyle{{\begin{array}[]{l}\mathrm{C}_{0}(0,1)\\\
\otimes\mathrm{C}_{0}(\mathcal{O}(\partial
X))\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\otimes\operatorname{id}}$$\textstyle{{\begin{array}[]{l}\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n};D)\\\
\otimes\mathrm{C}_{0}(0,1)\\\
\otimes\mathrm{C}_{0}(X_{n})\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\delta\otimes\operatorname{id}}$$\textstyle{\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n};D)\otimes\operatorname{Cone}{\begin{pmatrix}\mathrm{C}_{0}(\mathcal{O}(\overline{X}_{n}))\\\
\downarrow\\\ \mathrm{C}_{0}(\mathcal{O}(\partial
X))\end{pmatrix}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1}}$$\textstyle{{\begin{array}[]{l}\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n};D)\\\
\otimes\mathrm{C}_{0}(0,1)\\\ \otimes\mathrm{C}_{0}(\mathcal{O}(\partial
X))\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{2}}$$\textstyle{{\begin{array}[]{l}\mathrm{C}_{0}(0,1)\\\
\otimes
D\otimes{\mathfrak{K}}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Cone}{\begin{pmatrix}\mathrm{C}_{0}(0,1]\otimes{\mathcal{M}}^{s}(D)\\\
\downarrow\\\
\operatorname{Cone}({\mathcal{M}}^{s}(D)\to\frac{{\mathcal{M}}^{s}(D)}{D\otimes{\mathfrak{K}}})\end{pmatrix}}}$$\textstyle{{\begin{array}[]{l}\mathrm{C}_{0}(0,1)\otimes\\\
\operatorname{Cone}{\begin{pmatrix}{\mathcal{M}}^{s}(D)\\\ \downarrow\\\
\frac{{\mathcal{M}}^{s}(D)}{D\otimes{\mathfrak{K}}}\end{pmatrix}}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\begin{array}[]{l}\mathrm{C}_{0}(0,1)\\\
\otimes
D\otimes{\mathfrak{K}}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Cone}{\begin{pmatrix}\mathrm{C}_{0}(0,1]\otimes{\mathfrak{K}}\otimes
D\\\ \downarrow\\\
D\otimes{\mathfrak{K}}\end{pmatrix}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\begin{array}[]{l}\mathrm{C}_{0}(0,1)\\\
\otimes
D\otimes{\mathfrak{K}}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

The rows of the diagram are all of the form
$J\to\operatorname{Cone}(A\twoheadrightarrow
A/J)\leftarrow\mathrm{C}_{0}(0,1)\otimes A/J$ for different
$\mathrm{C}^{*}\\!$-algebras $A$ with ideals $J$. Also, the arrows going
upwards are also induced by $*$-homomorphsism of the form
$J\to\operatorname{Cone}(A\twoheadrightarrow A/J)$. All the arrows marked with
“$\simeq$” are isomorphisms in $\mathrm{E}^{\Gamma}$-theory, because they are
inclusions of ideals with contractible quotient $\mathrm{C}^{*}\\!$-algebras.
Furthermore, the mapping cone in the middle of the bottom row is itself
isomorphic to $\mathrm{C}_{0}(0,1)\otimes D\otimes{\mathfrak{K}}$ and the maps
leading into it then come from the inclusion of $(0,1)$ into itself as the
left or right half, up to orientation. Hence they represent the same element
in $\mathrm{E}^{\Gamma}$-theory up to a sign.

Using [WY20, Lemma 6.1.2], which clearly also holds true for operators on
Hilbert modules, and the fact that functions on $\overline{X}_{n}$ restrict to
functions of vanishing variation on $X_{n}$, we see that

$\displaystyle\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)\mathbin{{\otimes}_{\mathrm{max}}}\mathrm{C}(\overline{X}_{n})$
$\displaystyle\to\mathrm{C}_{b}([1,\infty),{\mathcal{M}}^{s}(D))/\mathrm{C}_{0}([1,\infty),{\mathcal{M}}^{s}(D))$
$\displaystyle L\otimes f$ $\displaystyle\mapsto[t\mapsto
L(t)\circ(\rho(f)\otimes\operatorname{id}_{D})]$

is an equivariant asymptotic morphism. It clearly restricts to $\delta$ on the
ideal $\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)\otimes\mathrm{C}_{0}(X_{n})$
and also passes to an equivariant asymptotic morphism between the quotients

$\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X;D)\otimes\mathrm{C}(\partial
X)\to\frac{\mathrm{C}_{b}([1,\infty),{\mathcal{M}}^{s}(D)/D\otimes{\mathfrak{K}})}{\mathrm{C}_{0}([1,\infty),{\mathcal{M}}^{s}(D)/D\otimes{\mathfrak{K}})}\,.$

Applying these constructions slicewise yields the asymptotic morphisms
$\delta_{1}$ and $\delta_{2}$. Note furthermore that $\delta_{2}$ is
$1$-homotopic in the obvious way to
$\delta^{\prime}\circ\mathop{\textrm{ev}_{1}}\otimes\operatorname{id}$ and
hence these two maps define the same $\mathrm{E}^{\Gamma}$-theory class. It is
clear that the diagram commutes.

The left column now is the image $y$ of $x$ in
$\mathrm{K}^{\Gamma}_{*+1}((0,1)\times
X_{n};D)\cong\mathrm{K}^{\Gamma}_{*}(X_{n};D)$ and the right column is
$\nu_{n}\circ(\mathop{\textrm{ev}_{1}})_{*}(x)\in\mathrm{K}^{\Gamma}_{*}(\mathcal{O}(\partial
X);D)\cong\tilde{\mathrm{K}}^{\Gamma}_{*-1}(\partial X;D)$. The middle column
represents an element in
$\mathrm{K}^{\Gamma}_{*+1}(\mathcal{O}(\overline{X}_{n})\times\\{1\\}\cup\mathcal{O}(\partial
X)\times(0,1];D)$ which is mapped to both $y$ and
$\nu_{n}\circ(\mathop{\textrm{ev}_{1}})_{*}(x)$ in the obvious way. By
definition of the connecting homomorphisms $\partial$ associated to the pair
$(\mathcal{O}(\overline{X}_{n}),\mathcal{O}(\partial X))$ via the mapping cone
construction, this shows exactly that
$\partial(y)=\nu_{n}\circ(\mathop{\textrm{ev}_{1}})_{*}(x)$.

The proof of the relative case works completely analogously but a lot simpler
by showing that for each
$x\in\mathrm{K}_{*}(\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n},A_{n};D))$ a
diagram

$\textstyle{\mathrm{C}_{0}(X_{n}\setminus
A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{x\otimes\operatorname{id}}$$\textstyle{\operatorname{Cone}{\begin{pmatrix}\mathrm{C}_{0}(\overline{X}_{n}\setminus\overline{A_{n}})\\\
\downarrow\\\ \mathrm{C}_{0}(\partial X\setminus\partial
A))\end{pmatrix}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\otimes\operatorname{id}}$$\textstyle{{\begin{array}[]{l}\otimes\mathrm{C}_{0}(0,1)\\\
\otimes\mathrm{C}_{0}(\partial X\setminus\partial
A)\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\otimes\operatorname{id}}$$\textstyle{{\begin{array}[]{l}\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n},A_{n};D)\\\
\otimes\mathrm{C}_{0}(X_{n}\setminus
A_{n})\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\delta}$$\textstyle{{\begin{array}[]{l}\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n},A_{n};D)\\\
\otimes\operatorname{Cone}{\begin{pmatrix}\mathrm{C}_{0}(\overline{X}_{n}\setminus\overline{A_{n}})\\\
\downarrow\\\ \mathrm{C}_{0}(\partial X\setminus\partial
A))\end{pmatrix}}\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\begin{array}[]{l}\mathrm{C}^{*}_{\mathrm{L},\Gamma}(X_{n},A_{n};D)\\\
\otimes\mathrm{C}_{0}(0,1)\\\ \otimes\mathrm{C}_{0}(\partial
X\setminus\partial
A)\end{array}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\otimes{\mathfrak{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Cone}({\mathcal{M}}^{s}(D)\to\frac{{\mathcal{M}}^{s}(D)}{D\otimes{\mathfrak{K}}})}$$\textstyle{\mathrm{C}_{0}(0,1)\otimes\frac{{\mathcal{M}}^{s}(D)}{D\otimes{\mathfrak{K}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

commutes in the $\mathrm{E}^{\Gamma}$-category, where $\Delta(x)$ is the left
column and $\nu((\mathop{\textrm{ev}_{1}})_{*}(x))$ is the composition of the
right column with the bottom row. ∎

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