consider $U$-invariant sets for an open subgroup of $G$. To formalize this we
discuss restrictions and inductions.

Let $U$ be an open subgroup of $G$. Write $\operatorname{res}_{G}^{U}\colon
G\text{-}\matheurm{Sets}\to U\text{-}\matheurm{Sets}$ for the restriction
functor. It induces a restriction functor

$\displaystyle\mathcal{S}^{G}(\Omega)$ $\displaystyle\to$
$\displaystyle\mathcal{S}^{U}(\operatorname{res}_{G}^{U}\Omega),$
$\displaystyle(\Sigma,c)$ $\displaystyle\mapsto$
$\displaystyle(\operatorname{res}_{G}^{U}\Sigma,\operatorname{res}_{G}^{U}c),$

that we will also denote by $\operatorname{res}_{G}^{U}$. We write
$\operatorname{ind}_{U}^{G}\colon\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)\to\mathcal{B}_{G}(\Lambda)$
for the canonical inclusion575757More precisely, writing
$\operatorname{res}_{G}^{U}\mathcal{B}$ for the subcategory of $\mathcal{B}$
whose morphisms have support in $U$ we have
$\operatorname{ind}_{U}^{G}\colon\operatorname{res}_{G}^{U}\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)\to\mathcal{B}_{G}(\Lambda)$..
This inclusion identifies $\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)$
with the subcategory of $\mathcal{B}_{G}(\Lambda)$ consisting of all objects
and morphisms with $G$-support in $U$. Let us briefly write

$\begin{array}[]{rlclcl}\otimes^{G}\colon&\mathcal{S}^{G}(\Omega)&\times&\mathcal{B}_{G}(\Lambda)&\to&\operatorname{Idem}\mathcal{B}_{G}(\Omega\times\Lambda);\\\
\otimes^{U}\colon&\mathcal{S}^{U}(\operatorname{res}_{G}^{U}\Omega)&\times&\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)&\to&\operatorname{Idem}\mathcal{B}_{U}(\operatorname{res}_{G}^{U}(\Omega\times\Lambda)),\end{array}$

for the tensor products. Directly from the definition it follows that for
$\mathbf{V}\in\mathcal{S}^{G}(\Omega)$ and
$\mathbf{B}\in\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)$ we have

(11.12)
$\operatorname{ind}_{U}^{G}(\operatorname{res}_{G}^{U}\mathbf{V}\otimes^{U}\mathbf{B})=\mathbf{V}\otimes^{G}\operatorname{ind}_{U}^{G}\mathbf{B}.$

Similarly, for morphisms $\rho\colon\mathbf{V}\to\mathbf{V}^{\prime}$ in
$\mathcal{S}^{G}(\Omega)$ and $\varphi\colon\mathbf{B}\to\mathbf{B}^{\prime}$
in $\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)$ we have

(11.13)
$\operatorname{ind}_{U}^{G}(\operatorname{res}_{G}^{U}\rho\otimes^{U}\varphi)=\rho\otimes^{G}\operatorname{ind}_{U}^{G}\varphi.$

Because of these identities we will later often drop
$\operatorname{ind}_{U}^{G}$ and $\operatorname{res}_{G}^{U}$ from the
notation and simply write $\otimes=\otimes^{G}=\otimes^{U}$.

###  11.f. The category $\mathcal{B}_{G,U}(\Lambda)$

Let $U$ be an open subgroup of $G$. We write $\mathcal{B}_{G,U}(\Lambda)$ for
the full subcategory of $\mathcal{B}_{G}(\Lambda)$ on all objects with
$G$-support in $U$. The induction
$\operatorname{ind}_{U}^{G}\colon\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)\to\mathcal{B}_{G}(\Lambda)$
from Subsection 11.e factors through the inclusion
$\mathcal{B}_{G,U}(\Lambda)\subseteq\mathcal{B}_{G}(\Lambda)$. Moreover,
$\mathcal{B}_{G,U}(\Lambda)$ is the full subcategory on objects in the image
of $\operatorname{ind}_{U}^{G}$. Because of support cofinality (3.2d) for
$\mathcal{B}$ the inclusion
$\mathcal{B}_{G,U}(\Lambda)\subseteq\mathcal{B}_{G}(\Lambda)$ induces an
equivalence on idempotent completions. Thus we can often work with
$\mathcal{B}_{G,U}(\Lambda)$ in place of $\mathcal{B}_{G}(\Lambda)$.

###  11.g. Tensor product with subcomplexes of $\mathbf{S}_{*}(X)$

Assume we are given

  1. (11.14a)

a compact open subgroup $U$ of $G$ and a compact $U$-invariant subspace $D$ of
the $U$-fixed points $X^{U}$ with a locally ordered triangulation.

Then we obtain the simplicial chain complex
$\mathbf{C}_{*}(D)\in\operatorname{ch}\mathcal{S}^{U}(\operatorname{res}_{G}^{U}S(X))$,
see Subsection 10.d. The tensor product with $\mathbf{C}_{*}(D)$ then yields a
functor

$\mathbf{C}_{*}(D)\,\otimes^{U}\,\\--\;\colon\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)\,\to\,\operatorname{ch}\mathcal{B}_{U}(\operatorname{res}_{G}^{U}(S(X)\times\Lambda)).$

Here we do not need the idempotent completion because of Lemma 11.7. Write
$i\colon\mathbf{C}_{*}(D)\to\operatorname{res}_{G}^{U}\mathbf{S}_{*}(X)$ for
the inclusion in
$\operatorname{ch}\mathcal{S}^{U}(\operatorname{res}_{G}^{U}S(X))$. Assume we
are given in addition

  1. (11.14b)

a chain map
$r\colon\operatorname{res}_{G}^{U}\mathbf{S}_{*}(X)\to\mathbf{C}_{*}(D)$ in
$\operatorname{ch}\mathcal{S}^{U}(\operatorname{res}_{G}^{U}S(X))$ with
$r\circ i=\operatorname{id}_{\mathbf{C}_{*}(D)}$ and a chain homotopy $H\colon
i\circ
r\simeq\operatorname{id}_{\operatorname{res}_{G}^{U}\mathbf{S}_{*}(X)}$.

In
$\operatorname{ch}\mathcal{B}_{U}(\operatorname{res}_{G}^{U}(S(X)\times\Lambda))$
we now obtain for each
$\mathbf{B}\in\mathcal{B}_{U}(\operatorname{res}_{G}^{U}\Lambda)$

$\textstyle{\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i\otimes^{U}{\operatorname{id}_{\mathbf{B}}}}$$\textstyle{\operatorname{res}_{G}^{U}\mathbf{S}_{*}(X)\otimes^{U}\mathbf{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\otimes^{U}{\operatorname{id}_{\mathbf{B}}}}$

where
$(r\otimes^{U}{\operatorname{id}_{\mathbf{B}}})\circ(i\otimes^{U}{\operatorname{id}_{\mathbf{B}}})=\operatorname{id}_{\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B}}$
and
$H\otimes^{U}{\operatorname{id}_{\mathbf{B}}}\colon(i\otimes^{U}{\operatorname{id}_{\mathbf{B}}})\circ\,(r\,\otimes^{U}{\operatorname{id}_{\mathbf{B}}})\simeq\operatorname{id}_{\operatorname{res}_{G}^{U}\mathbf{S}_{*}(X)\otimes\mathbf{B}}$.
Applying $\operatorname{ind}_{U}^{G}$ and using (11.12) we obtain

$\textstyle{\operatorname{ind}_{U}^{G}\big{(}\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ind}_{U}^{G}(i\otimes^{U}{\operatorname{id}_{\mathbf{B}}})}$$\textstyle{\mathbf{S}_{*}(X)\otimes^{G}\operatorname{ind}_{U}^{G}\mathbf{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ind}_{U}^{G}(r\otimes^{U}{\operatorname{id}_{\mathbf{B}}})}$

in $\operatorname{ch}\operatorname{Idem}\mathcal{B}_{G}(S(X)\times\Lambda)$.
The main advantage of
$\operatorname{ind}_{U}^{G}(\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B})$ over
$\mathbf{S}_{*}(X)\otimes^{G}\operatorname{ind}_{U}^{G}\mathbf{B}$ is that it
is smaller and has better chances of satisfying control conditions. Its
disadvantage is that, as $D$ is only $U$-invariant and not $G$-invariant,
morphisms
$\varphi\colon\operatorname{ind}_{U}^{G}\mathbf{B}\to\operatorname{ind}_{U}^{G}\mathbf{B}^{\prime}$
in $\mathcal{B}_{G,U}(\Lambda)$ do not induce maps
$\operatorname{ind}_{U}^{G}(\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B})\to\operatorname{ind}_{U}^{G}(\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B}^{\prime})$.
But $\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi$ is defined and we
can use the composition

(11.15)
$\textstyle{\operatorname{ind}_{U}^{G}\big{(}\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ind}_{U}^{G}(i\otimes^{U}{\operatorname{id}_{\mathbf{B}}})}$$\textstyle{\mathbf{S}_{*}(X)\otimes^{G}\operatorname{ind}_{U}^{G}\mathbf{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes^{G}\varphi}$$\textstyle{\operatorname{ind}_{U}^{G}\big{(}\mathbf{C}_{*}(D)\otimes^{U}\mathbf{B}^{\prime}\big{)}}$$\textstyle{\mathbf{S}_{*}(X)\otimes^{G}\operatorname{ind}_{U}^{G}\mathbf{B}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ind}_{U}^{G}(r\otimes^{U}{\operatorname{id}_{\mathbf{B}}})}$

instead. While this is not strictly compatible with composition, the
homotopies
$\operatorname{ind}_{U}^{G}(H\otimes^{U}{\operatorname{id}_{\mathbf{B}}})$
guarantee that it is compatible with composition up to coherent homotopy. From
now on we will simply the notation as alluded to in Subsection 11.e and drop
$\operatorname{ind}_{U}^{G}$ and $\operatorname{res}_{G}^{U}$ from the
notation and simply write $\otimes=\otimes^{G}=\otimes^{U}$. Thus (11.15)
abbreviates to

$\textstyle{\mathbf{C}_{*}(D)\otimes\mathbf{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i\otimes{\operatorname{id}_{\mathbf{B}}}}$$\textstyle{\mathbf{S}_{*}(X)\otimes\mathbf{B}\phantom{{}^{\prime}.}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi}$$\textstyle{\mathbf{C}_{*}(D)\otimes\mathbf{B}^{\prime}}$$\textstyle{\mathbf{S}_{*}(X)\otimes\mathbf{B}^{\prime}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\otimes{\operatorname{id}_{\mathbf{B}}^{\prime}}}$

In summary, we can use the data chosen in (11.14a) and (11.14b) to define a
homotopy coherent functor, see Definition C.1,

(11.16)
$F=(F^{0},F^{1},\dots)\;\colon\mathcal{B}_{G,U}(\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(S(X)\times\Lambda)$

as follows. For $\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$ we set

$F^{0}(\mathbf{B}):=\mathbf{C}_{*}(D)\otimes\mathbf{B}.$

For a chain

$\mathbf{B}_{0}\xleftarrow{\varphi_{1}}\mathbf{B}_{1}\xleftarrow{\varphi_{2}}\dots\xleftarrow{\varphi_{n}}\mathbf{B}_{n}$

of composable morphisms in $\mathcal{B}_{G,U}(\Lambda)$ we define

$\displaystyle F^{n}(\varphi_{1},\dots,\varphi_{n}):=(r\otimes$
$\displaystyle\operatorname{id}_{\mathbf{B}_{0}})\circ(\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi_{1})\circ(H\otimes\operatorname{id}_{\mathbf{B}_{1}})\circ\dots$
$\displaystyle\dots\circ(H\otimes\operatorname{id}_{\mathbf{B}_{n-1}})\circ(\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi_{n})\circ(i\otimes\operatorname{id}_{\mathbf{B}_{n}}).$

It is not difficult to check that this defines a homotopy coherent functor,
compare Example C.3.

In Section 12 we will discuss the effect of (11.16) on
$\operatorname{supp}_{2}$. For now we record the following easy facts.

######  Lemma 11.17.

  1. (11.17a)

If $\mathbf{B}$ is finite, then all chain modules $(F^{0}(\mathbf{B}))_{n}$ of
$F^{0}(\mathbf{B})$ are finite as well;

  2. (11.17b)

We have
$\operatorname{supp}_{1}(F^{0}(\mathbf{B}))_{n}\subseteq\operatorname{simp}_{n}(D)\times\operatorname{supp}_{1}\mathbf{B}$
and
$\operatorname{supp}_{G}(F^{0}(\mathbf{B}))_{n}\subseteq\operatorname{supp}_{G}\mathbf{B}$;

  3. (11.17c)

The $n$-th boundary map $\partial^{F^{0}(\mathbf{B})}_{n}$ of
$F^{0}(\mathbf{B})$ satisfies
$\operatorname{supp}_{G}\partial^{F^{0}(\mathbf{B})}_{n}\subseteq\operatorname{supp}_{G}\mathbf{B}$.

###### Proof.

This follows from (11.9a), (11.9b) and (11.9d). ∎

###  11.h. Projection back to $\Lambda$

The projection $\operatorname{pr}\colon S(X)\times\Lambda\to\Lambda$ induces a
functor

$P\colon\operatorname{ch}\mathcal{B}_{G}(X\times\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(\Lambda).$

We are interested in the composition of $P$ with $F$ from (11.16)

$P\circ
F\colon\mathcal{B}_{G,U}(\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(\Lambda).$

Let

$I\colon\mathcal{B}_{G,U}(\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(\Lambda)$

be the inclusion585858Recall that $\mathcal{B}_{G,U}(\Lambda)$ is a
subcategory of $\mathcal{B}_{G}(\Lambda)$.. We construct a natural
transformation $\tau\colon P\circ F\to I$. Write $\star$ for the one-point
space. Let $p(\mathbf{C}_{*}(D))$ be the image of $\mathbf{C}_{*}(D)$ under
the functor $\mathcal{S}^{U}(X)\to\mathcal{S}^{U}(\star)$ induced by the
projection $X\to\star$. We can identify $(P\circ F)(\mathbf{A})\cong
p(\mathbf{C}_{*}(D))\otimes\mathbf{B}$, for
$\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$. Let
$\mathbbm{1}_{G}:=(\star,\operatorname{id}_{\star})\in\mathcal{S}^{G}(\star)$.
We have
$\mathbbm{1}_{U}:=\operatorname{res}_{G}^{U}\mathbbm{1}_{G}=(\star,\operatorname{id}_{\star})\in\mathcal{S}^{U}(\star)$.
We can identify $I(\mathbf{B})\cong\mathbbm{1}_{U}\otimes\mathbf{B}$. The
projection $D\to\star$ induces an augmentation $\epsilon\colon
p(\mathbf{C}_{*}(D))\to\mathbbm{1}_{U}$. Now for
$\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$ we define

(11.18)
$\tau_{\mathbf{B}}:=\epsilon\otimes\operatorname{id}_{\mathbf{B}}\colon
p(C_{*}(D))\otimes\mathbf{B}\to\mathbbm{1}_{U}\otimes\mathbf{B}.$

We will need the notion of a strict natural transformation between homotopy
coherent functors, see Definition C.4.

######  Lemma 11.19.

Under the canonical identifications $(P\circ F)(\mathbf{B})\cong
p(\mathbf{C}_{*}(D))\otimes\mathbf{B}$ and
$I(\mathbf{B})\cong\mathbbm{1}_{U}\otimes\mathbf{B}$ the maps (11.18) define a
strict natural transformation $\tau\colon P\circ F\to I$, see Example C.5.

###### Proof.

This is a straight forward exercise in the definitions. ∎

######  Lemma 11.20.

Suppose that $D$ is contractible. Let
$\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$. Then in
$\operatorname{ch}\operatorname{Idem}\mathcal{B}_{G,U}(\Lambda)$ there are a
chain map $f\colon\mathbbm{1}_{U}\otimes\mathbf{B}\to
p(\mathbf{C}_{*}(D))\otimes\mathbf{B}$, and chain homotopies $h\colon
f\circ\tau_{\mathbf{B}}\simeq\operatorname{id}_{p(\mathbf{C}_{*}(D))\otimes\mathbf{B}}$,
$k\colon\tau_{\mathbf{B}}\circ
f\simeq\operatorname{id}_{\mathbbm{1}}{}_{U}\otimes\mathbf{B}$ such that

$\displaystyle\operatorname{supp}_{2}f,\operatorname{supp}_{2}h,\operatorname{supp}_{2}h^{\prime}$
$\displaystyle\subseteq\operatorname{supp}_{2}\mathbf{B},$
$\displaystyle\operatorname{supp}_{G}f,\operatorname{supp}_{G}h,\operatorname{supp}_{G}h^{\prime}$
$\displaystyle\subseteq\operatorname{supp}_{G}\mathbf{B}\subseteq U.$

In particular, $\tau_{\mathbf{B}}$ is a homotopy equivalence.

###### Proof.

Under the assumptions on $D$, $p(\mathbf{C}_{*}(D))$ and $\mathbbm{1}_{U}$ are
homotopy equivalent in $\operatorname{ch}\mathcal{S}^{U}(\star)$ (because the
simplicial chain complex of $D$ is homotopy equivalent to the simplicial chain
complex of a point). Thus in $\operatorname{ch}\mathcal{S}^{U}(\star)$ there
are a chain map $f_{0}\colon\mathbbm{1}_{U}\to p(\mathbf{C}_{*}(D))$, and
chain homotopies $h_{0}\colon
f_{0}\circ\epsilon\simeq\operatorname{id}_{p(\mathbf{C}_{*}(D))}$,
$k_{0}\colon\epsilon\circ f_{0}\simeq\operatorname{id}_{\mathbbm{1}_{U}}$. Now
set $f:=f_{0}\otimes\operatorname{id}_{\mathbf{B}}$,
$h:=h_{0}\otimes\operatorname{id}_{\mathbf{B}}$, and
$k:=k_{0}\otimes\operatorname{id}_{\mathbf{B}}$. The claims about
$\operatorname{supp}_{2}$ and $\operatorname{supp}_{G}$ follow from (11.9c)
and (11.9d). ∎

##  12\. Support estimates for homotopy coherent functors

Let $X$ be a $G$-space equipped with a $G$-invariant metric $d_{X}$. We assume
that for $K\subseteq X$ compact the pointwise isotropy group $G_{K}$ is open
in $G$. Let $J$ and $\Lambda$ be further $G$-spaces. Let $\mathcal{B}$ be a
Hecke category with $G$-support.

We will refine the construction of the homotopy coherent functor from
Subsection 11.g and will be interested in its effect on supports of objects
and morphisms. Its construction and analysis will depend on a list of data.
This section is very formal and will be used later to check that a sequence of
homotopy coherent functors does descend to $\matheurm{D}^{0}_{G}(\\--)$ as
discussed in Subsection 9.a.

###  12.a. Set-up

Throughout this section we fix

  * -

numbers $L\in\mathbb{N}$, $\rho>0$;

  * -

a compact open subgroup $U\subseteq G$ and a compact subset $M\subseteq G$
with $U\subseteq M$;

  * -

a subspace $D\subseteq X$ with a locally ordered triangulation all whose
simplices are of diameter $<\rho$;

  * -

a sequence of further subspaces $D=D^{(0)}\subseteq
D^{(1)}\subseteq\dots\subseteq D^{(L)}$ with $M\cdot D^{(l-1)}\subseteq
D^{(l)}$ for $l=1,\dots,L$; moreover we require that $D^{(L)}$ is pointwise
fixed by $U$;

  * -

a retraction $r^{0}\colon X\to D$ for the inclusion $i^{0}\colon D\to X$,
i.e., $r^{0}\circ i^{0}=\operatorname{id}_{D}$;

  * -

a homotopy $H^{0}\colon i^{0}\circ r^{0}\simeq\operatorname{id}_{X}$; we will
assume that $H^{0}$ is non-expanding, i.e., for all $\tau\in[0,1]$,
$x,x^{\prime}\in X$ we require
$d_{X}(H^{0}(x,\tau),H^{0}(x^{\prime},\tau))\leq d_{X}(x,x^{\prime})$; we also
assume that $H^{0}$ preserves the $D^{(l)}$, i.e., we require
$H^{0}(D^{(l)}\times[0,1])\subseteq D^{(l)}$;

  * -

a map $f\colon X\to J$;

  * -

a subset $E\subseteq J\times J$.

We do _not_ assume that $f$ is $G$-equivariant.

######  Remark 12.1 (Some explanations for the list of data).

Later

  * -

$X$ will be the extended Bruhat-Tits building for the reductive $p$-adic group
$G$;

  * -

the $D^{(i)}$ will be an increasing sequence of balls around a common center;

  * -

$r^{0}$ will be the radial projection and $H^{0}$ will be the associated
radial homotopy;

  * -

$U$ will be the pointwise isotropy group of $D^{(L)}$;

  * -

$\Lambda$ will be $|\mathbf{M}|^{\wedge}$ for some
$\mathbf{M}\in\mathcal{R}^{0}$;

  * -

$J$ will be $|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}$;

  * -

$f\colon X\to J$ will come from Theorem D.1.

Below $U$, $B$, $r^{0}$, and $H^{0}$ will be used to construct a homotopy
coherent functor

$\mathcal{B}_{G,U}(\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(S(X)\times\Lambda)$

as in Subsection 11.g. Let $\operatorname{bary}\colon S(X)\to X$ be the map
that sends a singular simplex to its barycenter. We can then compose with
$(f\circ\operatorname{bary}\times\operatorname{id}_{\Lambda})_{*}$ and obtain
a homotopy coherent functor

$\mathcal{B}_{G,U}(\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(J\times\Lambda)$

and will be interested in estimates for the support of objects and morphisms
under this latter functor, see Proposition 12.8 below. For these estimates we
will bound the $G$-support of morphisms and objects by $M$ and we will treat
chains of at most $L$-composable morphisms. The upper bound for
$\operatorname{supp}_{2}$ will be in terms of $E$.

We will work under the following assumptions throughout this section.

######  Assumption 12.2.

  1. (12.2a)

$\Big{\\{}{\begin{pmatrix}{f(gx)}\\\ {gf(x)}\end{pmatrix}}\;\Big{|}\;x\in
D^{(L)},g\in M\Big{\\}}\subseteq E$;

  2. (12.2b)

$\Big{\\{}{\begin{pmatrix}{f(H^{0}(x,\tau))}\\\
{f(x)}\end{pmatrix}}\;\Big{|}\;x\in D^{(L)},\tau\in[0,1]\Big{\\}}\subseteq E$;

  3. (12.2c)

$\Big{\\{}{\begin{pmatrix}{f(x^{\prime})}\\\
{f(x)}\end{pmatrix}}\;\Big{|}\;x,x^{\prime}\in
D^{(L)},d_{X}(x,x^{\prime})\leq(L+1)\rho\Big{\\}}\subseteq E$.

###  12.b. From $S(X)$ to $J$

Set

$\displaystyle E_{X}$ $\displaystyle:=$
$\displaystyle\big{\\{}{\begin{pmatrix}{x^{\prime}}\\\
{x}\end{pmatrix}}\;\big{|}\;x,x^{\prime}\in
D^{(L)},d_{X}(x,x^{\prime})\leq(L+1)\rho\big{\\}}$
$\displaystyle\qquad\cup\;\big{\\{}{\begin{pmatrix}{H^{0}(x,\tau)}\\\
{x}\end{pmatrix}}\;\big{|}\;x\in D^{(L)},\tau\in[0,1]\big{\\}}\quad\subseteq
X\times X;$ $\displaystyle E_{S}$ $\displaystyle:=$
$\displaystyle(\operatorname{bary}\times\operatorname{bary})^{-1}(E_{X})\qquad\subseteq
S(X)\times S(X).$

We note that $E_{X}\subseteq E_{X}^{\circ 2}$ and $E_{S}\subseteq E_{S}^{\circ
2}$. We use again the convention for products from (11.8).

######  Lemma 12.3.

Let $E^{\prime}\subseteq\Lambda\times\Lambda$,
$\Phi\colon\mathbf{B}\to\mathbf{B}^{\prime}\in\mathcal{B}_{G}(S(X)\times\Lambda)$,
and $\nu\in\mathbb{N}$. Suppose $\operatorname{supp}_{1}\mathbf{B}\subseteq
S(D^{(L)})\times\Lambda$, $\operatorname{supp}_{2}\Phi\subseteq
E_{S}^{\circ\nu}\times E^{\prime}$, and $\operatorname{supp}_{G}\Phi\subseteq
M$. Then
$\operatorname{supp}_{2}(f\circ\operatorname{bary}\times\operatorname{id}_{\Lambda})_{*}\Phi\subseteq
E^{\circ(\nu+1)}\times E^{\prime}$.

###### Proof.

By (4.7), and since $\operatorname{bary}$ and $\operatorname{id}_{\Lambda}$
are $G$-equivariant,
$\operatorname{supp}_{2}(f\circ\operatorname{bary}\times\operatorname{id}_{\Lambda})_{*}\Phi$
is contained in

$(f\circ\operatorname{bary}\operatorname{id}_{\Lambda})^{\times
2}(E_{S}^{\circ\nu}\times
E^{\prime})\circ\Big{\\{}{\begin{pmatrix}{f(g\operatorname{bary}(\sigma)),g\lambda}\\\
{gf(\operatorname{bary}(\sigma)),g\lambda}\end{pmatrix}}\,\Big{|}\,(\sigma,\lambda)\in
S(D^{(L)})\times\Lambda,g\in M\Big{\\}}.$

We have

$(f\circ\operatorname{bary}\operatorname{id}_{\Lambda})^{\times
2}(E_{S}^{\circ\nu}\times
E^{\prime})\subseteq((f\circ\operatorname{bary})^{\times
2}(E_{S}^{\circ\nu}))\times E^{\prime}\\\
\subseteq((f\circ\operatorname{bary})^{\times 2}(E_{S}))^{\circ\nu}\times
E^{\prime}\subseteq E^{\circ\nu}\times E^{\prime}$

and, by (12.2a)

$\Big{\\{}{\begin{pmatrix}{f(g\operatorname{bary}(\sigma)),g\lambda}\\\
{gf(\operatorname{bary}(\sigma)),g\lambda}\end{pmatrix}}\,\Big{|}\,(\sigma,\lambda)\in
S(D^{(L)})\times\Lambda,g\in M\Big{\\}}\subseteq
E\times\Big{\\{}{\begin{pmatrix}{\lambda}\\\
{\lambda}\end{pmatrix}}\;\big{|}\;\lambda\in\Lambda\Big{\\}}.$

Thus

$\operatorname{supp}_{2}(f\circ\operatorname{bary}\times\operatorname{id}_{\Lambda})_{*}\Phi\subseteq\Big{(}E^{\circ\nu}\times
E^{\prime}\Big{)}\circ\Big{(}E\times\Big{\\{}{\begin{pmatrix}{\lambda}\\\
{\lambda}\end{pmatrix}}\;\big{|}\;\lambda\in\Lambda\Big{\\}}\Big{)}\subseteq
E^{\circ(\nu+1)}\times E^{\prime}.$

∎

###  12.c. Construction of $r$ and $H$

We write $\mathbf{C}_{*}(D)\in\operatorname{ch}\mathcal{S}^{U}(S(X))$ for the
simplicial chain complex of $D$ and
$\mathbf{S}_{*}(X)\in\operatorname{ch}\mathcal{S}^{G}(S(X))$ for the singular
chain complex of $X$. Let $i\colon\mathbf{C}_{*}(D)\to\mathbf{S}_{*}(X)$ be
the inclusion. For $k,l\leq L$ we define $S^{k,l}\subseteq S(X)$ as the
collection of all singular simplices in $X$ that are contained in $D^{(l)}$
and are of diameter $<k\rho$.

######  Lemma 12.4.

There is a chain map $r\colon\mathbf{S}_{*}(X)\to\mathbf{C}_{*}(D)$ and a
chain homotopy $H\colon\mathbf{S}_{*}(X)\to\mathbf{S}_{*}(X)$ in
$\operatorname{ch}\mathcal{S}^{U}(S(X))$ with $r\circ
i=\operatorname{id}_{\mathbf{C}_{*}(B)}$, $H\colon i\circ
r\simeq\operatorname{id}_{\mathbf{S}_{*}(X)}$ satisfying the following: if
${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}r\cup\operatorname{supp}_{2}H$
with $\sigma\in S^{k,l}$ and $k\leq L-1,l\leq L$, then

  1. (12.4a)

$\sigma^{\prime}\in S^{k+1,l}$;

  2. (12.4b)

${\begin{pmatrix}{\sigma^{\prime}}\\\ {\sigma}\end{pmatrix}}\in E_{S}^{\circ
2}$.

###### Proof.

We will use $\operatorname{im}\sigma$ and $(\operatorname{im}\sigma)^{\rho}$
as introduced in Subsection 10.d. We write
$i^{1}\colon\mathbf{C}_{*}(D)\to\mathbf{S}_{*}(D)$ for the inclusion. By Lemma
10.5 there is $r^{1}\colon\mathbf{S}_{*}(D)\to\mathbf{C}_{*}(D)$ with
$r^{1}\circ i^{1}=\operatorname{id}_{\mathbf{C}_{*}(D)}$ and a chain homotopy
$H^{1}\colon\mathbf{S}_{*}(D)\to\mathbf{S}_{*+1}(D)$ for $i^{1}\circ
r^{1}\simeq\operatorname{id}_{\mathbf{S}_{*}(D)}$. Moreover, Lemma 10.5 also
yields

  1. (12.4c)

if ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}(r^{1})\cup\in\operatorname{supp}_{2}(H^{1})$,
then
$\operatorname{im}\sigma^{\prime}\subseteq(\operatorname{im}\sigma)^{\rho}$.

Next we use that $H^{0}$ is not expanding and apply Lemma 10.4 to the homotopy
$H^{0}$ and obtain a chain homotopy
$\widetilde{H}^{0}\colon(i^{0})_{*}\circ(r^{0})_{*}\simeq\operatorname{id}_{\mathbf{S}_{*}(X)}$
such that

  1. (12.4d)

if ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}(\widetilde{H}^{0})$, then
$\operatorname{diam}\operatorname{im}(\sigma^{\prime})<\operatorname{diam}\operatorname{im}(\sigma)+\rho$
and
$\operatorname{im}\sigma^{\prime}\subseteq\operatorname{im}(H^{0}\circ(\sigma\times\operatorname{id}_{[0,1]}))$.

We have $i=(i^{0})_{*}\circ i^{1}\colon\mathbf{C}_{*}(D)\to\mathbf{S}_{*}(X)$
and set

$\displaystyle r$ $\displaystyle:=$ $\displaystyle
r^{1}\circ(r^{0})_{*}\colon\mathbf{S}_{*}(X)\to\mathbf{C}_{*}(D);$
$\displaystyle H$ $\displaystyle:=$ $\displaystyle(i^{0})_{*}\circ
H^{1}\circ(r^{0})_{*}+(\widetilde{H}^{0})_{*}\colon i\circ
r\simeq\operatorname{id}_{\mathbf{S}_{*}(X)}.$

Suppose ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}r$ with $\sigma\in S^{k,l}$.
As $r$ is a map to $\mathbf{C}_{*}(D)$, $\sigma^{\prime}$ is simplex in the
triangulation of $D$, in particular $\sigma^{\prime}\in S^{k+1,l}$. Now
$r_{\sigma}^{\sigma^{\prime}}\neq 0$ implies that there is $\tau$ with
$(r^{1})_{\tau}^{\sigma^{\prime}}\neq 0$ and
$((r^{0})_{*})_{\sigma}^{\tau}\neq 0$. The latter means
$\tau=r^{0}\circ\sigma=H^{0}(\sigma(\\--),1)$. In particular,
${\begin{pmatrix}{\tau}\\\ {\sigma}\end{pmatrix}}\in E_{S}$. As $H^{0}$ is
non-expanding,
$\operatorname{diam}\operatorname{im}\tau\leq\operatorname{diam}\operatorname{im}\sigma<k\rho$.
By (12.4c),
$\operatorname{im}\sigma^{\prime}\subseteq(\operatorname{im}\tau)^{\rho}$.
Thus
$d_{X}(\operatorname{bary}(\sigma^{\prime}),\operatorname{bary}(\tau))\leq\rho+\operatorname{diam}\operatorname{im}\tau<(k+1)\rho\leq
L\rho$ and ${\begin{pmatrix}{\sigma^{\prime}}\\\ {\tau}\end{pmatrix}}\in
E_{S}$. Therefore ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in E_{S}^{\circ 2}$.

Suppose ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}H$ with $\sigma\in S^{k,l}$.
Then ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}\widetilde{H}$ or
${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}((i^{0})_{*}\circ
H^{1}\circ(r^{0}))$. In the first case, (12.4d) implies
$\operatorname{diam}\sigma^{\prime}<\operatorname{diam}\sigma+\rho<(k+1)\rho$.
Moreover, $\operatorname{im}\sigma^{\prime}\subseteq
H_{0}(\operatorname{im}\sigma\times[0,1])$. As $H^{0}$ preserves $D^{(l)}$,
this implies $\sigma^{\prime}\in S^{k+1,l}$. Also
$\operatorname{bary}(\sigma^{\prime})=H_{0}(x,t)$ for some $t\in[0,1]$ and
$x\in\operatorname{im}\sigma$. Let $\tau:=H_{0}(\sigma(\\--),t)$. Then
${\begin{pmatrix}{\sigma}\\\ {\tau}\end{pmatrix}}\in E_{S}$. Also
$\operatorname{bary}(\sigma^{\prime})\in\operatorname{im}\tau$. As
$H_{0}(\\--,t)$ is non-expanding,
$d_{X}(\operatorname{bary}(\sigma^{\prime}),\operatorname{bary}(\tau))<\operatorname{diam}\operatorname{im}\tau<\operatorname{diam}\operatorname{im}\sigma<k\rho$.
Thus ${\begin{pmatrix}{\tau}\\\ {\sigma}\end{pmatrix}}\in E_{S}$. Altogether,
${\begin{pmatrix}{\sigma^{\prime}}\\\ {\sigma}\end{pmatrix}}\in E_{S}^{\circ
2}$.

In the second case there are $\tau$, $\tau^{\prime}$ with
${\begin{pmatrix}{\sigma^{\prime}}\\\
{\tau^{\prime}}\end{pmatrix}}\in\operatorname{supp}_{2}(i^{0})_{*}$,
${\begin{pmatrix}{\tau^{\prime}}\\\
{\tau}\end{pmatrix}}\in\operatorname{supp}_{2}H^{1}$, and
${\begin{pmatrix}{\tau}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}(r^{0})_{*}$. Now
${\begin{pmatrix}{\sigma^{\prime}}\\\
{\tau^{\prime}}\end{pmatrix}}\in\operatorname{supp}_{2}(i^{0})_{*}$ means
$\tau^{\prime}=\sigma^{\prime}$, while ${\begin{pmatrix}{\tau}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}(r^{0})_{*}$ means
$\tau=r^{0}\circ\sigma=H^{0}(\sigma(\\--),1)$. In particular,
${\begin{pmatrix}{\tau}\\\ {\sigma}\end{pmatrix}}\in E_{S}$. Also, as $H^{0}$
is non-expanding,
$\operatorname{diam}\operatorname{im}\tau\leq\operatorname{diam}\operatorname{im}\sigma<k\sigma$.
Using (12.4c), ${\begin{pmatrix}{\tau^{\prime}}\\\
{\tau}\end{pmatrix}}\in\operatorname{supp}_{2}H^{1}$ implies
$\operatorname{im}\tau^{\prime}\subseteq(\operatorname{im}\tau)^{\rho}$. In
particular, $d_{X}(\operatorname{bary}(\tau^{\prime}),x)<\rho$ for some
$x\in\operatorname{im}\tau$. Moreover,
$\operatorname{diam}\operatorname{im}\sigma^{\prime}=\operatorname{diam}\operatorname{im}\tau^{\prime}<\operatorname{diam}\operatorname{im}\tau+\rho<(k+1)\rho$.
As $H^{1}$ is a map to $\mathbf{S}_{*}(D)$, we have
$\operatorname{im}\sigma^{\prime}=\operatorname{im}\tau^{\prime}\subseteq D$.
In particular, $\sigma^{\prime}\in S^{k+1,l}$. Now

$d_{X}(\operatorname{bary}(\sigma^{\prime}),\operatorname{bary}(\tau))=d_{X}(\operatorname{bary}(\tau^{\prime}),\operatorname{bary}(\tau))\\\
\leq
d_{X}(\operatorname{bary}(\tau^{\prime}),x)+d_{X}(x,\operatorname{bary}(\tau))<\rho+\operatorname{diam}\operatorname{im}\tau<\rho+k\rho=(k+1)\rho$

implies ${\begin{pmatrix}{\sigma^{\prime}}\\\ {\tau}\end{pmatrix}}\in E_{S}$.
Thus ${\begin{pmatrix}{\sigma^{\prime}}\\\ {\tau}\end{pmatrix}}\in
E_{S}^{\circ 2}$. ∎

###  12.d. Estimates over $S(X)\times\Lambda$

We now fix $r$ and $H$ as in Lemma 12.4. We obtain a homotopy coherent functor
as in (11.16)

$F=(F^{0},F^{1},\dots,)\colon\mathcal{B}_{G,U}(\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(S(X)\times\Lambda).$

Recall that $F_{0}(\mathbf{B})=\mathbf{C}(D)_{*}\otimes\mathbf{B}$. Its $n$-th
chain module is $F_{0}(\mathbf{B})_{n}=\mathbf{C}(D)_{n}\otimes\mathbf{B}$.
Its $n$-th boundary map is
${\partial}_{n}^{F_{0}(\mathbf{B})}={\partial}^{\mathbf{C}_{*}(D)}_{n}\otimes\operatorname{id}_{\mathbf{B}}$.

######  Lemma 12.5.

Let $\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$. Then

  1. (12.5a)

$\operatorname{supp}_{G}F_{0}(\mathbf{B})_{n}\subseteq U$,
$\operatorname{supp}_{2}F_{0}(\mathbf{B})_{n}\subseteq
E_{S}\times\operatorname{supp}_{2}\mathbf{B}$;

  2. (12.5b)

$\operatorname{supp}_{G}{\partial}_{n}^{F_{0}(\mathbf{B})}\subseteq U$,
$\operatorname{supp}_{2}{\partial}_{n}^{F_{0}(\mathbf{B})}\subseteq
E_{S}\times\operatorname{supp}_{2}\mathbf{B}$.

###### Proof.

We have
$\operatorname{supp}_{2}\operatorname{id}_{\mathbf{C}_{n}(D)}\subseteq\\{{\begin{pmatrix}{\sigma}\\\
{\sigma}\end{pmatrix}}\mid\sigma\in S(D)\\}\subset E_{S}$. Diameters of
simplices in the triangulation of $D$ are $<\rho$. If
$(\partial_{n}^{\mathbf{C}_{*}(D)})_{\sigma}^{\sigma^{\prime}}\neq 0$, then
$\operatorname{im}\sigma^{\prime}\subseteq\operatorname{im}\sigma$ and so
$d_{X}(\operatorname{bary}{\sigma^{\prime}},\operatorname{bary}{\sigma})<\rho$.
Therefore $\operatorname{supp}_{2}\partial_{n}^{\mathbf{C}_{*}(D)}\subseteq
E_{S}$. We have
$\operatorname{supp}_{G}\mathbf{B}=\operatorname{supp}_{G}\operatorname{id}_{\mathbf{B}}\subseteq
U$, as $\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$. Both (12.5a), and (12.5b)
follow now from Lemma 11.9. ∎

We will now use summands and corners as in Remarks 4.14 and 4.15 relative to
$S^{k,l}\subseteq S(X)$. For $\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$ we
abbreviate
$(\mathbf{S}_{n}(X)\otimes\mathbf{B})_{k,l}:=(\mathbf{S}_{n}(X)\otimes\mathbf{B})|_{S^{s,l}\times\Lambda}$
and write

$(\mathbf{S}_{n}(X)\otimes\mathbf{B})_{k,l}\xrightarrow{i_{k,l}}\mathbf{S}_{n}(X)\otimes\mathbf{B}\xrightarrow{r^{k,l}}(\mathbf{S}_{n}(X)\otimes\mathbf{B})_{k,l}$

for the corresponding inclusion and retraction. For
$\Phi\colon\mathbf{S}_{n}(X)\otimes\mathbf{B}\to\mathbf{S}_{n^{\prime}}(X)\otimes\mathbf{B}^{\prime}$
we set
$\Phi_{k,l}^{k^{\prime},l^{\prime}}:=r^{k^{\prime},l^{\prime}}\circ\Phi\circ
i_{k,l}$. We also set
$(i\otimes\operatorname{id}_{\mathbf{B}})^{k^{\prime},l^{\prime}}:=r^{k^{\prime},l^{\prime}}\circ(i\otimes\operatorname{id}_{\mathbf{B}})$,
and
$(r\otimes\operatorname{id}_{\mathbf{B}})_{k,l}:=(r\otimes\operatorname{id}_{\mathbf{B}})\circ
i^{k,l}$. This means

$\displaystyle(\Phi_{k,l}^{k^{\prime},l^{\prime}})_{\sigma,s}^{\sigma^{\prime},s^{\prime}}$
$\displaystyle=$
$\displaystyle\begin{cases}\Phi_{\sigma,s}^{\sigma^{\prime},s^{\prime}}\hskip
30.56937pt&\sigma\in S^{k,l},\sigma^{\prime}\in S^{k^{\prime},l^{\prime}};\\\
0&\text{else};\end{cases}$
$\displaystyle((i\otimes\operatorname{id}_{\mathbf{B}})^{k^{\prime},l^{\prime}})_{\sigma,s}^{\sigma^{\prime},s^{\prime}}$
$\displaystyle=$
$\displaystyle\begin{cases}(i\otimes\operatorname{id}_{\mathbf{B}})_{\sigma,s}^{\sigma^{\prime},s^{\prime}}&\sigma^{\prime}\in
S^{k^{\prime},l^{\prime}};\\\ 0&\text{else};\end{cases}$
$\displaystyle((r\otimes\operatorname{id}_{\mathbf{B}})_{k,l})_{\sigma,s}^{\sigma^{\prime},s^{\prime}}$
$\displaystyle=$
$\displaystyle\begin{cases}(r\otimes\operatorname{id}_{\mathbf{B}})_{\sigma,s}^{\sigma^{\prime},s^{\prime}}&\sigma\in
S^{k,l};\\\ 0&\text{else}.\end{cases}$

######  Lemma 12.6.

Let $k,l\leq L$,
$\mathbf{B},\mathbf{B}^{\prime}\in\mathcal{B}_{G,U}(\Lambda)$,
$\varphi\colon\mathbf{B}\to\mathbf{B}^{\prime}\in\mathcal{B}_{G,U}(\Lambda)$
with $\operatorname{supp}_{G}\varphi\subseteq M$. Set
$E^{\prime}:=\operatorname{supp}_{2}\varphi\cup\operatorname{supp}_{2}\mathbf{B}$.
Then

  1. (12.6a)

$i_{k,l+1}\circ(\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi)_{k,l}^{k,l+1}=(\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi)\circ
i_{k,l}$ (provided $l+1\leq L)$;

  2. (12.6b)

$\operatorname{supp}_{2}\big{(}(\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi)_{k,l}^{k,l+1}\big{)}\subseteq
E_{S}\times E^{\prime}$ (provided $l+1\leq L)$;

  3. (12.6c)

$i_{k+1,l}\circ(H\otimes\operatorname{id}_{\mathbf{B}})_{k,l}^{k+1,l}=(H\otimes\operatorname{id}_{\mathbf{B}})\circ
i_{k,l}$ (provided $k+1\leq L)$;

  4. (12.6d)

$\operatorname{supp}_{2}\big{(}(H\otimes\operatorname{id}_{\mathbf{B}})_{k,l}^{k+1,l}\big{)}\subseteq
E_{S}^{\circ 2}\times E^{\prime}$ (provided $k+1\leq L)$;

  5. (12.6e)

$i_{k,l}\circ(i\otimes\operatorname{id}_{\mathbf{B}})^{k,l}=i\otimes\operatorname{id}_{\mathbf{B}}$;

  6. (12.6f)

$\operatorname{supp}_{2}\big{(}(i\otimes\operatorname{id}_{\mathbf{B}})^{k,l}\big{)}\subseteq
E_{S}^{\circ 2}\times E^{\prime}$;

  7. (12.6g)

$(r\otimes\operatorname{id}_{\mathbf{B}})_{k,l}=(r\otimes\operatorname{id}_{\mathbf{B}})\circ
i_{k,l}$;

  8. (12.6h)

$\operatorname{supp}_{2}\big{(}(r\otimes\operatorname{id}_{\mathbf{B}})_{k,l}\big{)}\subseteq
E_{S}^{\circ 2}\times E^{\prime}$.

###### Proof.

From (11.9d) we obtain
$\operatorname{supp}_{G}\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi\subseteq\operatorname{supp}_{G}\varphi\subseteq
M$. Using $M\cdot D^{(l)}\subseteq D^{(l+1)}$ we obtain
$(\operatorname{supp}_{G}\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi)\cdot
S^{k,l}\subseteq M\cdot S^{k,l}\subseteq S^{k,l+1}$. If
${\begin{pmatrix}{\sigma^{\prime},\lambda^{\prime}}\\\
{\sigma,\lambda}\end{pmatrix}}\in\operatorname{supp}_{2}\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi$,
then, by (11.9c), ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}\operatorname{id}_{\mathbf{S}_{*}(X)}$
and so $\sigma=\sigma^{\prime}$. Now Lemma 4.17 gives (12.6a). Using (11.9c)
and (4.16) we obtain

$\operatorname{supp}_{2}(\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi)_{k,l}^{k,l+1}\subseteq\Big{\\{}{\begin{pmatrix}{\sigma,\lambda^{\prime}}\\\
{\sigma,\lambda}\end{pmatrix}}\Big{|}\sigma\in
S^{k,l+1},{\begin{pmatrix}{\lambda^{\prime}}\\\
{\lambda}\end{pmatrix}}\in\operatorname{supp}_{2}\varphi\Big{\\}}$

and this implies (12.6b).

By (11.9d) $\operatorname{supp}_{G}H\otimes\operatorname{id}_{\mathbf{B}}$,
$\operatorname{supp}_{G}r\otimes\operatorname{id}_{\mathbf{B}}$, and
$\operatorname{supp}_{G}i\otimes\operatorname{id}_{\mathbf{B}}$ are all
contained in $\operatorname{supp}_{G}\mathbf{B}\subseteq U$. As $U$ fixes
$D^{(l)}$, we have $U\cdot S^{k,l}=S^{k,l}$. If
${\begin{pmatrix}{\sigma^{\prime},\lambda^{\prime}}\\\
{\sigma,\lambda}\end{pmatrix}}\in\operatorname{supp}_{2}H\otimes\operatorname{id}_{\mathbf{B}}$,
then, by (11.9c), ${\begin{pmatrix}{\sigma^{\prime}}\\\
{\sigma}\end{pmatrix}}\in\operatorname{supp}_{2}H$. If, in addition,
$\sigma\in S^{k,l}$, then, by (12.4a), $\sigma^{\prime}\in S^{k+1,l}$ and by
(12.4b) ${\begin{pmatrix}{\sigma^{\prime}}\\\ {\sigma}\end{pmatrix}}\in
E_{S}^{\circ 2}$. Now (12.6c) follows from Lemma 4.17 and (12.6d) follows from
(11.9c) and (4.16). As (12.4a) and (12.4b) also apply to $r$, and hold
directly by definition for $i$, (12.6e), (12.6f), and (12.6h) follow from the
same argument. Finally, (12.6g) is the definition of
$(r\otimes\operatorname{id}_{\mathbf{B}})_{k,l}$. ∎

######  Proposition 12.7.

Let
$\mathbf{B}_{0}=(S_{0},\pi_{0},B_{0})\xleftarrow{\varphi_{1}}\dots\xleftarrow{\varphi_{l}}\mathbf{B}_{l}=(S_{l},\pi_{l},B_{l})$
be a chain of composable morphisms in $\mathcal{B}_{G,U}(\Lambda)$. Assume
$l\leq L$ and $\operatorname{supp}_{G}(\varphi_{j})\subseteq M$ for all $j$.
Let
$E^{\prime}:=\bigcup\operatorname{supp}_{2}\varphi_{j}\cup\bigcup_{j}\operatorname{supp}_{2}\mathbf{B}_{j}\subseteq\Lambda\times\Lambda$.
Then there are
$\Phi_{i}\colon\mathbf{B}^{\prime}_{i}\to\mathbf{B}^{\prime}_{i+1}\in\mathcal{B}_{G,U}(S(X)\times\Lambda)$,
$i=0,\dots,2l$ such that

$F^{l}(\varphi_{1},\dots,\varphi_{l})=\Phi_{2l}\circ\dots\circ\Phi_{0}$

and $\operatorname{supp}_{G}\Phi_{i}\subseteq M$,
$\operatorname{supp}_{2}\Phi\subseteq E_{S}^{\circ 2}\times E^{\prime}$,
$\operatorname{supp}_{1}\mathbf{B}^{\prime}_{i}\subseteq S(D)\times\Lambda$
for all $i$.

###### Proof.

Recall that by construction

$F^{l}(\varphi_{1},\dots,\varphi_{l})=\\\
\big{(}r\otimes\operatorname{id}_{\mathbf{B}_{0}}\big{)}\circ\big{(}\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi_{1}\big{)}\circ\big{(}H\otimes\operatorname{id}_{\mathbf{B}_{1}}\big{)}\circ\dots{\\\
}\dots\circ\big{(}H\otimes\operatorname{id}_{\mathbf{B}_{l-1}}\big{)}\circ\big{(}\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi_{l}\big{)}\circ\big{(}i\otimes\operatorname{id}_{\mathbf{B}_{l}}\big{)}.$

Using (12.6a),(12.6c),(12.6e), and (12.6g) we can rewrite this as

$F^{l}(\varphi_{1},\dots,\varphi_{l})=\\\
\big{(}r\otimes\operatorname{id}_{\mathbf{B}_{0}}\big{)}_{l,l}\circ\big{(}\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi_{1}\big{)}^{l,l}_{l,l-1}\circ\big{(}H\otimes\operatorname{id}_{\mathbf{B}_{1}}\big{)}^{l,l-1}_{l-1,l-1}\circ\dots{\\\
}\dots\circ\big{(}H\otimes\operatorname{id}_{\mathbf{B}_{l-1}}\big{)}_{1,1}^{2,1}\circ\big{(}\operatorname{id}_{\mathbf{S}_{*}(X)}\otimes\varphi_{l}\big{)}_{1,0}^{1,1}\circ\big{(}i\otimes\operatorname{id}_{\mathbf{B}_{l}}\big{)}^{1,0}.$

By (11.9d) (and since $U\subseteq M$) for each of these factors
$\operatorname{supp}_{G}$ is contained in $M$. By (12.6b),(12.6d),(12.6f), and
(12.6h), $\operatorname{supp}_{2}$ of each factor, is contained in
$E_{S}^{\circ 2}\times E^{\prime}$595959Here we use that $E_{S}\subseteq
E_{S}^{\circ 2}$ for our specific $E_{S}$.. Finally, the domains of the
factors are of the form $(\mathbf{S}_{n}(X)\otimes\mathbf{B}_{i})_{k,l}$ (or
$\mathbf{C}(D)\otimes\mathbf{B}_{0}$) and have therefore
$\operatorname{supp}_{1}$ contained in $S(D^{(L)})$. ∎

###  12.e. Estimates over $J\times\Lambda$

We now consider the homotopy coherent functor

$\widetilde{F}:=(f\circ\operatorname{bary}\times\operatorname{id}_{\Lambda})_{*}\circ
F\colon\mathcal{B}_{G,U}(\Lambda)\to\operatorname{ch}\mathcal{B}_{G}(J\times\Lambda).$

######  Proposition 12.8.

  1. (12.8a)

Let $\mathbf{B}\in\mathcal{B}_{G,U}(\Lambda)$ and
$E^{\prime}:=\operatorname{supp}_{2}\mathbf{B}$. Then

$\begin{array}[]{rlrl}\operatorname{supp}_{G}\widetilde{F}_{0}(\mathbf{B})_{n}&\subseteq
U,&\operatorname{supp}_{2}\widetilde{F}_{0}(\mathbf{B})_{n}&\subseteq E^{\circ
2}\times E^{\prime},\\\
\operatorname{supp}_{G}{\partial}_{n}^{\widetilde{F}_{0}(\mathbf{B})}&\subseteq
U,&\operatorname{supp}_{2}{\partial}_{n}^{\widetilde{F}_{0}(\mathbf{B})}&\subseteq
E^{\circ 2}\times E^{\prime};\end{array}$

  2. (12.8b)

Let
$\mathbf{B}_{0}=(S_{0},\pi_{0},B_{0})\xleftarrow{\varphi_{1}}\dots\xleftarrow{\varphi_{l}}\mathbf{B}_{l}=(S_{l},\pi_{l},B_{l})$
be a chain of composable morphisms in $\mathcal{B}_{G,U}(\Lambda)$. Assume
$l\leq L$ and $\operatorname{supp}_{G}(\varphi_{j})\subseteq M$ for all $j$.
Let
$E^{\prime}:=\bigcup\operatorname{supp}_{2}\varphi_{j}\cup\bigcup_{j}\operatorname{supp}_{2}\mathbf{B}_{j}$.
Then

$\begin{array}[]{rl}\operatorname{supp}_{2}\widetilde{F}^{l}(\varphi_{1},\dots,\varphi_{l})&\subseteq(M^{2l}(E^{\circ
3}\times E^{\prime}))^{\circ(2l+1)},\\\
\operatorname{supp}_{G}\widetilde{F}^{l}(\varphi_{1},\dots,\varphi_{l})&\subseteq
M^{2l+1}.\end{array}$

The point here is that the upper bounds are in terms of $E$, $E^{\prime}$, $M$
and $l$ and independent of $\mathbf{B}$ and the $\varphi_{i}$.

###### Proof.

Lemma 12.5 directly implies
$\operatorname{supp}_{G}\widetilde{F}_{0}(\mathbf{B})_{n}$,
$\operatorname{supp}_{G}{\partial}_{n}^{\widetilde{F}_{0}(\mathbf{B})}\subseteq
U$. Lemma 12.5 together with Lemma 12.3 implies
$\operatorname{supp}_{2}\widetilde{F}_{0}(\mathbf{B})_{n}\subseteq E^{\circ
2}\times E^{\prime}$,
$\operatorname{supp}_{2}{\partial}_{n}^{\widetilde{F}_{0}(\mathbf{B})}\subseteq
E^{\circ 2}\times E^{\prime}$606060As $U$ fixes $D$ pointwise we could here
use $E$ instead of $E^{\circ 2}$, but the precise form of the estimates is not
important.. Thus (12.8a) holds.

Applying $(f\circ\operatorname{bary}\times\operatorname{id}_{\Lambda})_{*}$ to
the factorization from Proposition 12.7 gives a factorization

$\widetilde{F}^{l}(\varphi_{1},\dots,\varphi_{l})=\widetilde{\Phi}_{2l}\circ\dots\circ\widetilde{\Phi}_{0}.$

The estimates for the $\Phi_{i}$ from Proposition 12.7 translate to the
$\tilde{\Phi}_{i}$. We get
$\operatorname{supp}_{G}\widetilde{\Phi}_{i}\subseteq M$ directly from
Proposition 12.7. Using in addition Lemma 12.3 we get
$\operatorname{supp}_{2}\widetilde{\Phi}_{i}\subseteq E^{\circ 3}\times
E^{\prime}$. As $\operatorname{supp}_{G}$ is submultiplicative we get
$\operatorname{supp}_{G}\widetilde{F}^{l}(\varphi_{1},\dots,\varphi_{l})\subseteq
M^{2l+1}$. Using the composition formula 4.3 for $\operatorname{supp}_{2}$ it
is not difficult to obtain an explicit bound for
$\operatorname{supp}_{2}\widetilde{F}^{l}(\varphi_{1},\dots,\varphi_{l})$, for
example $(M^{2l}(E^{\circ 3}\times E^{\prime}))^{\circ(2l+1)}$ works. Thus
(12.8b) holds. ∎

##  13\. Construction of the transfer

Let $G$ be a reductive $p$-adic group and $X$ be the associated extended
Bruhat-Tits building. For each ${\underline{t}}\in\mathbb{N}^{\times 2}$ we
will construct the data considered in Section 12, i.e.,

  * -

numbers $L_{\underline{t}}\in\mathbb{N}$, $\rho_{\underline{t}}>0$;

  * -

compact open subgroups $U_{\underline{t}}\subseteq G$ and a compact subset
$M_{\underline{t}}\subseteq G$ with $U_{\underline{t}}\subseteq M$;

  * -

a subspace $D_{\underline{t}}\subseteq X$ with a locally ordered triangulation
all whose simplices are of diameter $<\rho_{\underline{t}}$;

  * -

a sequence of further subspaces
$D_{\underline{t}}=D^{(0)}_{\underline{t}}\subseteq
D^{(1)}_{\underline{t}}\subseteq\dots\subseteq
D^{(L_{\underline{t}})}_{\underline{t}}$ with $M_{\underline{t}}\cdot
D^{l-1}_{\underline{t}}\subseteq D_{\underline{t}}^{l}$ for $l=1,\dots,L_{t}$,
and such that $D^{L}_{\underline{t}}$ is pointwise fixed by
$U_{\underline{t}}$;

  * -

a retraction $r^{0}_{\underline{t}}\colon X\to D_{\underline{t}}$ for the
inclusion $i^{0}_{\underline{t}}\colon D_{\underline{t}}\to X$, i.e.,
$r^{0}_{\underline{t}}\circ
i^{0}_{\underline{t}}=\operatorname{id}_{D_{\underline{t}}}$;

  * -

a homotopy $H^{0}_{\underline{t}}\colon i^{0}_{\underline{t}}\circ
r^{0}_{\underline{t}}\simeq\operatorname{id}_{X}$, that is not expanding,
i.e., for all $\tau\in[0,1]$, $x,x^{\prime}\in X$ we will have
$d_{X}(H^{0}_{\underline{t}}(x,\tau),H^{0}_{\underline{t}}(x^{\prime},\tau))\leq
d_{X}(x,x^{\prime})$;

  * -

a map $f_{\underline{t}}\colon
X\to|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}$;

  * -

a subset
$E_{\underline{t}}\subseteq|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}\times|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}$.

Let $\mathbf{M}\in\mathcal{R}^{0}$. As in Section 12 (with
$\Lambda=|\mathbf{M}|^{\wedge}$) we obtain then for every ${\underline{t}}$ a
homotopy coherent functor

$F_{\underline{t}}=(F_{\underline{t}}^{0},F_{\underline{t}}^{1},\dots)\colon\mathcal{B}_{G,U_{\underline{t}}}(|\mathbf{M}|^{\wedge})\to\operatorname{ch}\mathcal{B}_{G}(S(X)\times|\mathbf{M}|^{\wedge})$

and its composition

$\widetilde{F}_{\underline{t}}:=((f_{\underline{t}}\circ\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}\circ
F_{\underline{t}}\colon\mathcal{B}_{G,U_{\underline{t}}}(|\mathbf{M}|^{\wedge})\to\operatorname{ch}\mathcal{B}_{G}(|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}\times|\mathbf{M}|^{\wedge}).$

We will verify Assumption 12.2 for each ${\underline{t}}$. Thus the support
estimates from Proposition 12.8 will be available. These estimates will allow
us to combine the
$(f_{\underline{t}}\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}\circ
F_{\underline{t}}$ to obtain a homotopy coherent functor

$\widetilde{F}_{\mathbf{M}}\colon\matheurm{D}^{0}_{G}(\mathbf{M})\to\operatorname{ch}\matheurm{D}^{0}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G))$

and Lemmas 11.19 and 11.20 will imply that the $\tilde{F}_{\mathbf{M}}$ induce
the desired transfer for

$\mathbf{K}\matheurm{D}^{0}_{G}\big{(}\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times-\big{)}\to\mathbf{K}\matheurm{D}^{0}_{G}(-)$

in $\mathcal{R}^{0}\text{-}\matheurm{Spectra}$.

###  13.a. Retractions to balls in $X$

We write $d_{X}$ for the $\operatorname{CAT}(0)$-metric of $X$. We fix a base
point $x_{0}\in X$ and write $D_{R}$ for the closed ball in $X$ centered at
$x_{0}$. We write $\pi_{R}\colon X\to D_{R}$ for the radial projection. We
define the radial homotopy $H_{R}\colon i_{R}\circ
r_{R}\simeq\operatorname{id}_{X}$ with $H_{R}(x,\tau)=\pi_{\tau
d_{X}(x,x_{0})+(1-\tau)R}(x)$. The $\operatorname{CAT}(0)$-condition implies
that $H_{R}$ is contracting, i.e.,

$d_{X}\big{(}H^{X}_{R}(x,\tau),H_{R}^{X}(x^{\prime},\tau)\big{)}\leq
d_{X}(x,x^{\prime})$

for all $R\geq 0$, $\tau\in[0,1]$ and $x,x^{\prime}\in X$. Let $U:=G_{x_{0}}$
be the stabilizer of $x_{0}$ in $G$. As the action of $G$ on $X$ is smooth and
proper, this is a compact open subgroup of $G$.

###  13.b. Choosing the data

We choose for $j\in\mathbb{N}$ numbers $\epsilon_{j}>0$, $\eta_{j}>0$,
$L_{j}\in\mathbb{N}$ and compact subsets $M_{j}\subseteq G$ such that

$\epsilon_{j}\to 0,\eta_{j}\to 0,L_{j}\to\infty\;\text{as}\;j\to\infty$

and that for any $K\subseteq G$ compact we have $K\subseteq M_{j}$ for all but
finitely many $j$. We also assume that each $M_{j}$ contains an open
subgroup616161To construct $M_{j}$ we can choose a metric on $G$ and take
$M_{j}$ to be the closed ball of radius $j$ around the unit in $G$. Since $G$
is a td-group, $M_{j}$ contains a compact open subgroup of $G$.. We set
$L^{+}_{j}:=L_{j}\cdot\max\\{d_{X}(x_{0},gx_{0})\mid g\in M_{j}\\}$.

Let $N$ be the first number appearing in Theorem D.1. Given
$t_{0}\in\mathbb{N}$, we obtain from Theorem D.1 applied to $M=M_{t_{0}}$ and
$\epsilon=\epsilon_{t_{0}}$ a number $\beta_{t_{0}}$ and
$\mathcal{V}_{t_{0}}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite. Given further
$t_{1}\in\mathbb{N}$, we obtain again from Theorem D.1 applied to
$\eta=\eta_{t_{1}}$ and $L=L^{+}_{t_{1}}$ numbers $R_{t_{0},t_{1}}>0$,
$\rho^{\prime}_{t_{0},t_{1}}>0$ and a map

$f_{t_{0},t_{1}}\colon X\to|\mathbf{J}^{N}_{\mathcal{V}_{t_{0}}}(G)|^{\wedge}$

such that

  1. (13.1a)

for $x\in D_{R_{t_{0},t_{1}}+L^{+}_{t_{1}}}$, $g\in M_{t_{0}}$ we have

$d_{\mathbf{J}\text{-}\mathrm{fol}}\big{(}f_{t_{0},t_{1}}(gx),gf_{t_{0},t_{1}}(x)\big{)}<(\beta_{t_{0}},\eta_{t_{1}},\epsilon_{t_{0}});$

  2. (13.1b)

for $x\in D_{R_{t_{0},t_{1}}+L^{+}_{t_{1}}}$, $R^{\prime}\geq R_{t_{0},t_{1}}$
we have

$d_{\mathbf{J}\text{-}\mathrm{fol}}\big{(}f_{t_{0},t_{1}}(x),f_{t_{0},t_{1}}(\pi_{R^{\prime}}(x))\big{)}<(\beta_{t_{0}},\eta_{t_{1}},\epsilon_{t_{0}});$

  3. (13.1c)

for all $x,x^{\prime}\in X$ with $d_{X}(x,x^{\prime})<\rho^{\prime}_{t_{1}}$
we have

$d_{\mathbf{J}\text{-}\mathrm{fol}}\big{(}f_{t_{0},t_{1}}(x),f_{t_{0},t_{1}}(x^{\prime})\big{)}<(\beta_{t_{0}},\eta_{t_{1}},\epsilon_{t_{0}}).$

Here and later we abbreviate
$d_{\mathbf{J}\text{-}\mathrm{fol}}=d_{\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\text{-}\mathrm{fol}}$.
For ${\underline{t}}=(t_{0},t_{1})$ we now define respectively choose

  * -

$L_{\underline{t}}:=L_{t_{1}}$,
$\rho_{\underline{t}}:=\rho^{\prime}_{\underline{t}}/(L_{t}+1)$,
$M_{\underline{t}}:=M_{t_{0}}$, $D_{\underline{t}}:=D_{R_{t}}$; as balls in a
building the $D_{\underline{t}}$ can be triangulated;

  * -

$D_{\underline{t}}^{(l)}:=D_{R_{t_{0},t_{1}}+\frac{l}{L_{\underline{t}}}L^{+}_{t_{1}}}$
for $l=0,\dots,L_{t}$;

  * -

$U_{\underline{t}}$ a compact open subgroup that fixes
$D_{\underline{t}}^{(L_{\underline{t}})}$ pointwise and is contained in
$M_{\underline{t}}$;

  * -

$i^{0}_{\underline{t}}\colon D_{\underline{t}}\to X$ the inclusion;

  * -

$r^{0}_{\underline{t}}:=\pi_{R_{{\underline{t}}}}\colon X\to
D_{\underline{t}}$, the radial projection;

  * -

$f_{\underline{t}}:=f_{t_{0},t_{1}}$;

  * -

$H^{0}_{\underline{t}}:=H_{R_{\underline{t}}}$ the radial homotopy
$i_{\underline{t}}^{0}\circ r_{\underline{t}}^{0}\simeq\operatorname{id}_{X}$;

  * -

$E_{\underline{t}}:=\big{\\{}{\begin{pmatrix}{z}\\\
{z^{\prime}}\end{pmatrix}}\;\big{|}\;d_{\mathbf{J}\text{-}\mathrm{fol}}(z,z^{\prime})<(\beta_{t_{0}},\eta_{t_{1}},\epsilon_{t_{0}})\big{\\}}$.

We note that (13.1a), (13.1b) and (13.1c) imply that Assumption 12.2 holds. We
can now apply the construction from Section 12 and obtain for
$\mathbf{M}\in\mathcal{R}^{0}$ the homotopy coherent functor

$((f_{\underline{t}}\circ\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}\circ
F_{\underline{t}}\colon\mathcal{B}_{G,U_{\underline{t}}}(|\mathbf{M}|^{\wedge})\to\operatorname{ch}\mathcal{B}_{G}(|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}\times|\mathbf{M}|^{\wedge}).$

###  13.c. The $\tilde{F}_{t}$ combine to a homotopy coherent functor on
$\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})$.

We will now use the $G$-control structure $\mathfrak{D}^{0}({\bf\Sigma})$ from
Definition 7.7 for ${\bf\Sigma}=\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)$.

######  Lemma 13.2.

Let $E:=\Big{\\{}{\begin{pmatrix}{z^{\prime},{\underline{t}}}\\\
{z,{\underline{t}}}\end{pmatrix}}\;\Big{|}\;{\underline{t}}\in\mathbb{N}^{\times
2},{\begin{pmatrix}{z^{\prime}}\\\ {z}\end{pmatrix}}\in
E_{\underline{t}}\Big{\\}}$. Then
$E\in\mathfrak{D}^{0}_{2}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G))$.

###### Proof.

We need to verify the foliated control condition from Definition 7.7. Let
$\epsilon>0$ be given. Choose $k_{0}$ such that $\epsilon_{t_{0}}<\epsilon$
for all $t_{0}\geq k_{0}$. Let $t_{0}\geq k_{0}$ be given. Set
$\beta:=\beta_{t_{0}}$. Let $\eta>0$ be given. Choose $k_{1}$ such that
$\eta_{t_{1}}<\eta$ for all $t_{1}\geq k_{1}$. Let $t_{1}\geq k_{1}$ be given.
Set ${\underline{t}}:=(t_{0},t_{1})$. Let
$z,z^{\prime}\in|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}$ be given
with ${\begin{pmatrix}{z^{\prime},{\underline{t}}}\\\
{z,{\underline{t}}}\end{pmatrix}}\in E$. By definition of $E$ we then have
${\begin{pmatrix}{z^{\prime}}\\\ {z}\end{pmatrix}}\in E_{{\underline{t}}}$. By
definition of $E_{{\underline{t}}}$ we then have
$d_{\mathbf{J}\text{-}\mathrm{fol}}(z,z^{\prime})<(\beta_{t_{0}},\eta_{t_{1}},\epsilon_{t_{0}})$.
Since $\beta=\beta_{t_{0}}$, $\eta_{t_{1}}<\eta$ and
$\epsilon_{t_{0}}<\epsilon$ this implies

$d_{\mathbf{J}\text{-}\mathrm{fol}}(z,z^{\prime})<(\beta,\eta,\epsilon),$

as required. ∎

Following Remark 7.17 we will view $\matheurm{D}^{0}_{G}(\mathbf{M})$ and
$\matheurm{D}^{0}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$
respectively as subcategories of
${\sideset{}{{}^{\prime}}{\textstyle\prod}}_{\mathbb{N}^{\times
2}}\mathcal{B}_{G}(|\mathbf{M}|^{\wedge})$ and
${\sideset{}{{}^{\prime}}{\textstyle\prod}}_{\mathbb{N}^{\times
2}}\mathcal{B}_{G}(|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M}|^{\wedge})$
respectively. Thus objects are sequences
$\mathbf{B}=(\mathbf{B}_{\underline{t}})_{{\underline{t}}\in\mathbb{N}^{\times}}$
of objects in $\mathcal{B}_{G}(|\mathbf{M}|^{\wedge})$ and
$\mathcal{B}_{G}(|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M}|^{\wedge})$
respectively satisfying the conditions spelled out in Remarks 7.17. Similarly
morphisms are equivalence classes of sequences
$(\varphi_{\underline{t}})_{{\underline{t}}\in\mathbb{N}^{\times 2}}$. We
recall that sequences
$(\varphi_{\underline{t}})_{{\underline{t}}\in\mathbb{N}^{\times 2}}$ and
$(\varphi^{\prime}_{\underline{t}})_{{\underline{t}}\in\mathbb{N}^{\times 2}}$
are equivalent, if there is $k_{0}$ such that for all $t_{0}\geq k_{0}$ there
is $k_{1}$ such that for all $t_{1}\geq k_{1}$ we have
$\varphi_{t_{0},t_{1}}=\varphi^{\prime}_{t_{0},t_{1}}$. In particular, we can
ignore all $\varphi_{t_{0},t_{1}}$ with $t_{0}$ or $t_{1}$ small. We define
$\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})$ as the full subcategory of
$\matheurm{D}^{0}_{G}(\mathbf{M})$ on all objects
$\mathbf{B}=(\mathbf{B}_{\underline{t}})_{{\underline{t}}\in\mathbb{N}^{\times
2}}$ with $\operatorname{supp}_{G}\mathbf{B}_{\underline{t}}\subseteq
U_{\underline{t}}$ for all ${\underline{t}}$. We can now define the homotopy
coherent functor

$F_{\mathbf{M}}=(F_{\mathbf{M}}^{0},F_{\mathbf{M}}^{1},\dots)\;\colon\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})\to\operatorname{ch}\matheurm{D}^{0}_{G}\big{(}\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M}\big{)}.$

Let
$\mathbf{B}=(\mathbf{B}_{\underline{t}})_{{\underline{t}}\in\mathbb{N}^{\times
2}}$ be an object of $\matheurm{D}^{0}_{G}(\mathbf{M})$. We define

$F_{\mathbf{M}}^{0}(\mathbf{B}):=\Big{(}((f_{\underline{t}}\circ\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}(F_{\underline{t}}^{0}(\mathbf{B}_{\underline{t}}))\Big{)}_{{\underline{t}}\in\mathbb{N}^{\times
2}}.$

Let
$\mathbf{B}_{0}\xleftarrow{\varphi_{1}}\dots\xleftarrow{\varphi_{l}}\mathbf{B}_{l}$
be a chain of composable morphisms in $\matheurm{D}^{0}_{G}(\mathbf{M})$.
Write
$\varphi_{i}=\big{(}(\varphi_{i})_{\underline{t}}\big{)}_{{\underline{t}}\in\mathbb{N}^{\times
2}}$. We define

$F_{\mathbf{M}}^{l}(\varphi_{1},\dots,\varphi_{l}):=\Big{(}((f_{\underline{t}}\circ\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}\big{(}F_{\underline{t}}^{l}\big{(}(\varphi_{1})_{\underline{t}},\dots,(\varphi_{l})_{\underline{t}}\big{)}\big{)}\Big{)}_{{\underline{t}}\in\mathbb{N}^{\times
2}}.$

We write $\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}$ for the $n$-chain
module of $F_{\mathbf{M}}^{0}(\mathbf{B})$ and
$\partial_{n}^{F^{0}_{\mathbf{M}}(\mathbf{B})}$ for the $n$-th boundary map.

######  Lemma 13.3.

  1. (a)

$\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}$ as above is an object in
$\matheurm{D}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$;

  2. (b)

$\partial_{n}^{F^{0}_{\mathbf{M}}(\mathbf{B})}$ as above is a morphism in
$\matheurm{D}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$;

  3. (c)

$F_{\mathbf{M}}^{l}(\varphi_{1},\dots,\varphi_{l})$ as above is a morphism in
$\operatorname{ch}\matheurm{D}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$.

###### Proof.

(a) We need to verify the four conditions (7.18a) to (7.18d).

For (7.18a) we need to check
$\operatorname{supp}_{1}\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}\in\matheurm{D}^{0}_{1}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$.
Let

$F:=\bigcup_{{\underline{t}}\in\mathbb{N}^{\times
2}}f_{\underline{t}}(\operatorname{bary}_{n}(D_{{\underline{t}}}))\times\\{{\underline{t}}\\}.$

We claim that
$F\in\mathfrak{D}_{1}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G))$. This
amounts to checking the three conditions in (7.7a) for $F$: Finiteness over
$\mathbb{N}^{\times 2}$ is clear as $D_{{\underline{t}}}$ is compact and has
only finitely many simplices. For the other two conditions we use that
$f_{\underline{t}}=f_{t_{0},t_{1}}$ is a map to
$|\mathbf{J}^{N}_{\mathcal{V}_{t_{0}}}(G)|^{\wedge}\subseteq|\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)|^{\wedge}$.
Thus $F$ has finite dimensional support. As $\mathcal{V}_{t_{0}}$ is finite
$|J^{N}_{\mathcal{V}_{t_{0}}}(G)|$ is a finite subcomplex of
$|J_{{\mathcal{C}\mathrm{vcy}}}(G)|$. It follows that $F$ has compact support
in $|J_{{\mathcal{C}\mathrm{vcy}}}(G)|$. So
$F\in\mathfrak{D}_{1}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G))$. Set
$F^{\prime}:=\operatorname{supp}_{1}\mathbf{B}\in\mathfrak{D}_{1}^{0}(\mathbf{M})$.
By (11.9b)

$\operatorname{supp}_{1}((f_{\underline{t}}\circ\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}(F_{{\underline{t}}}^{0}(\mathbf{B}_{\underline{t}}))=f_{\underline{t}}(\operatorname{bary}_{n}(D_{\underline{t}}))\times\operatorname{supp}_{1}\mathbf{B}_{t}.$

Thus

$\displaystyle\operatorname{supp}_{1}\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}$
$\displaystyle=$ $\displaystyle\bigcup_{{\underline{t}}\in\mathbb{N}^{\times
2}}\operatorname{supp}_{1}((f_{\underline{t}}\circ\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}(F_{{\underline{t}}}^{0}(\mathbf{B}_{\underline{t}})\times\\{{\underline{t}}\\})$
$\displaystyle=$
$\displaystyle\big{\\{}(z,\lambda,{\underline{t}})\mid(z,{\underline{t}})\in
F,(\lambda,{\underline{t}})\in F^{\prime}\big{\\}}.$

Thus
$\operatorname{supp}_{1}\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}\in\mathfrak{D}^{0}_{1}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$
by (7.19a).

For (7.18b) we need to check
$\operatorname{supp}_{2}\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}\in\mathfrak{D}^{0}_{2}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$.
By (12.8a)

$\operatorname{supp}_{2}(f_{\underline{t}}\times\operatorname{bary})\subseteq
E_{\underline{t}}^{\circ
2}\times\operatorname{supp}_{2}\mathbf{B}_{\underline{t}}.$

We now use
$E\in\mathfrak{D}_{2}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G))$ from
Lemma 13.2. Then

$\operatorname{supp}_{2}(F^{0}_{\mathbf{M}}(\mathbf{B}))_{n}\;\subseteq\;\Big{\\{}{\begin{pmatrix}{z^{\prime},\lambda^{\prime},{\underline{t}}}\\\
{z,\lambda,{\underline{t}}}\end{pmatrix}}\,\Big{|}\,{\begin{pmatrix}{z^{\prime},{\underline{t}}}\\\
{z,{\underline{t}}}\end{pmatrix}}\in E^{\circ
2},{\begin{pmatrix}{\lambda^{\prime},{\underline{t}}}\\\
{\lambda,{\underline{t}}}\end{pmatrix}}\in\operatorname{supp}_{2}\mathbf{B}\Big{\\}}$

which belongs to
$\mathfrak{D}_{2}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$
by (7.19b).

For (7.18c) we need to check
$\operatorname{supp}_{G}\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}\in\mathfrak{D}^{0}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$.
By (11.17b) we have
$\operatorname{supp}_{G}\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}\subseteq\operatorname{supp}_{G}\mathbf{B}$.
The latter is relatively compact as
$\mathbf{B}\in\matheurm{D}_{G}^{0}(\mathbf{M})$. Thus
$\operatorname{supp}_{G}\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}$ is
relatively compact as well.

Finally, the finiteness condition (7.18d) for
$\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}$ follows from (11.17a) as
the $\mathbf{B}_{{\underline{t}}}$ are finite.
(b) We need to verify (7.18e) and (7.18f) for
$\partial_{n}^{F^{0}_{\mathbf{M}}(\mathbf{B})}$. This can be done exactly in
the same way as for the corresponding properties (7.18b) and (7.18c) for
$\big{(}F^{0}_{\mathbf{M}}(\mathbf{B})\big{)}_{n}$ under (a). For (7.18f) this
uses (11.17c) in place of (11.17b).
(c) We need to verify (7.18e) and (7.18f) for
$F_{\mathbf{M}}^{l}(\varphi_{1},\dots,\varphi_{l})$.

Choose $k_{0}$ such that $\operatorname{supp}_{G}\varphi_{i}\subseteq
M_{k_{0}}$ for $i=1,\dots,l$. Since we are working in
$\matheurm{D}^{0}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$
we can ignore all ${\underline{t}}=(t_{0},t_{1})$ with $t_{0}\leq k_{0}$. Thus
for the rest of the argument we can and will always assume $t_{0}\geq k_{0}$.
From (12.8b) we obtain that

$\operatorname{supp}_{G}\big{(}((f_{\underline{t}}\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}(F_{\underline{t}}^{l})\big{(}(\varphi_{1})_{\underline{t}},\dots,(\varphi_{l})_{\underline{t}}\big{)}\big{)}\subseteq
M_{k_{0}}^{2l+1}$

for all ${\underline{t}}$. In particular, the union of these $G$-supports is
relatively compact in $G$. Thus
$\operatorname{supp}_{G}F_{\mathbf{M}}^{l}(\varphi_{1},\dots,\varphi_{l})\in\mathfrak{D}_{G}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$
as required by (7.18f).

Set
$E^{\prime}_{\underline{t}}:=\bigcup_{j}\operatorname{supp}_{2}((\varphi_{j})_{\underline{t}})\cup\bigcup_{j}\operatorname{supp}_{2}((\mathbf{B}_{j})_{\underline{t}})$.
From (12.8b) we obtain

(13.4)
$\operatorname{supp}_{2}(((f_{\underline{t}}\circ\operatorname{bary})\times\operatorname{id}_{|\mathbf{M}|^{\wedge}})_{*}(F_{\underline{t}}^{l})\big{(}(\varphi_{1})_{\underline{t}},\dots,(\varphi_{l})_{\underline{t}}\big{)}\\\
\subseteq\Big{\\{}{\begin{pmatrix}{z^{\prime},\lambda^{\prime}}\\\
{z,\lambda}\end{pmatrix}}\Big{|}{\begin{pmatrix}{z^{\prime}}\\\
{z}\end{pmatrix}}\in\big{(}M_{k_{0}}^{2l}\cdot E_{\underline{t}}^{\circ
3}\big{)}^{\circ(2l+1)},{\begin{pmatrix}{\lambda^{\prime}}\\\
{\lambda}\end{pmatrix}}\in\big{(}M_{k_{0}}^{2l}\cdot
E^{\prime}_{\underline{t}}\big{)}^{\circ(2l+1)}\Big{\\}}.$

We use again
$E\in\mathfrak{D}_{2}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G))$ from
Lemma 13.2 and set

$\displaystyle\tilde{E}:=\big{(}M_{k_{0}}^{2l}\cdot E^{\circ
3}\big{)}^{\circ(2l+1)}\in\mathfrak{D}_{2}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)).$

We also have, as the $\varphi_{j}$ and the $\mathbf{B}_{j}$ are from
$\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})$,

$E^{\prime}:=\Big{\\{}{\begin{pmatrix}{\lambda^{\prime},{\underline{t}}}\\\
{\lambda,{\underline{t}}}\end{pmatrix}}\;\Big{|}\;{\begin{pmatrix}{\lambda^{\prime}}\\\
{\lambda}\end{pmatrix}}\in
E^{\prime}_{\underline{t}}\Big{\\}}\in\mathfrak{D}_{2}^{0}(\mathbf{M})$

and also $\tilde{E}^{\prime}:=(M_{k_{0}}^{2l}\cdot
E^{\prime})^{\circ(2l+1)}\in\mathfrak{D}_{2}^{0}(\mathbf{M})$. Now (13.4)
implies

$\operatorname{supp}_{2}F_{\mathbf{M}}^{l}(\varphi_{1},\dots,\varphi_{l})\subseteq\Big{\\{}{\begin{pmatrix}{z^{\prime},\lambda^{\prime},{\underline{t}}}\\\
{z,\lambda,{\underline{t}}}\end{pmatrix}}\;\Big{|}\;{\begin{pmatrix}{z^{\prime},{\underline{t}}}\\\
{z,{\underline{t}}}\end{pmatrix}}\in\tilde{E},{\begin{pmatrix}{\lambda^{\prime},{\underline{t}}}\\\
{\lambda,{\underline{t}}}\end{pmatrix}}\in\tilde{E}^{\prime}\Big{\\}}.$

The latter belongs to
$\mathfrak{D}_{2}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$
by (7.19b). Thus
$\operatorname{supp}_{2}F_{\mathbf{M}}^{l}(\varphi_{1},\dots,\varphi_{l})\in\mathfrak{D}_{2}^{0}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})$
as required by (7.18e). ∎

###### Conclusion of the proof of Theorem 6.7.

We have the following diagram.

$\textstyle{\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I}$$\scriptstyle{F_{\mathbf{M}}}$$\textstyle{\operatorname{ch}_{{\mathrm{fin}}}\matheurm{D}^{0}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{pr}_{\mathbf{M}})_{*}}$$\textstyle{\operatorname{ch}_{{\mathrm{fin}}}\matheurm{D}^{0}_{G}(\mathbf{M})}$

The functor $(\operatorname{pr}_{\mathbf{M}})_{*}$ is induced by the
projection
$\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M}\to\mathbf{M}$,
$F_{\mathbf{M}}$ is the homotopy coherent functor defined above (well-defined
by Lemma 13.3), and $I$ is the inclusion. We can then apply K-theory and
obtain

$\textstyle{\mathbf{K}\big{(}\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{K}(I)}$$\scriptstyle{\mathbf{K}(F_{\mathbf{M}})}$$\textstyle{\mathbf{K}\big{(}\matheurm{D}^{0}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{p}_{\mathbf{M}}=\mathbf{K}((\operatorname{pr}_{\mathbf{M}})_{*})}$$\textstyle{\mathbf{K}\big{(}\matheurm{D}^{0}_{G}(\mathbf{M})\big{)}.}$

Strictly speaking the homotopy coherent functor $F_{\mathbf{M}}$ induces a
zig-zag in K-theory, see Remark C.24. Since
$\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})\big{)}\to\matheurm{D}^{0}_{G}(\mathbf{M})$
induces an equivalence of idempotent completions, compare Subsection 11.f, the
map $\mathbf{K}(I)$ is a weak equivalence. It is not difficult to see from the
construction that the diagram is natural in $\mathbf{M}$, basically because
the tensor product functor is natural for induced functors in both entries.
There is a strict natural transformation
$\tau_{\mathbf{M}}\colon(\operatorname{pr}_{\mathbf{M}})_{*}\circ
F_{\mathbf{M}}\to I$ by weak equivalences. It is given by applying the
construction from Subsection 11.h for each $t\in\mathbb{N}^{\times 2}$, see
Lemma 11.19. On each object $\tau_{\mathbf{M}}$ evaluates to a chain homotopy
equivalence, see Lemma 11.20. It follows that
$(\operatorname{pr}_{\mathbf{M}})_{*}\circ F_{\mathbf{M}}$ and $I$ induce
homotopic maps in K-theory, see Remark C.26. Altogether,

$\textstyle{\mathbf{K}\big{(}\matheurm{D}^{0}_{G}(\mathbf{M})\big{)}}$$\textstyle{\mathbf{K}\big{(}\matheurm{D}^{0}_{G,\mathbf{U}}(\mathbf{M})\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{K}(I)}$$\scriptstyle{\sim}$$\scriptstyle{\mathbf{K}(F_{\mathbf{M}})}$$\textstyle{\mathbf{K}\big{(}\matheurm{D}^{0}_{G}(\mathbf{J}_{{\mathcal{C}\mathrm{vcy}}}(G)\times\mathbf{M})\big{)}}$

is the required section $\mathbf{tr}_{\mathbf{M}}$ for
$\mathbf{p}_{\mathbf{M}}$. ∎

This concludes the proof of the ${\mathcal{C}\mathrm{vcy}}$-Farrell–Jones
Conjecture for reductive $p$-adic groups, i.e., of Theorem 5.15.

##  14\. Reduction from ${\mathcal{C}\mathrm{vcy}}$ to
${\mathcal{C}\mathrm{om}}$

######  Theorem 14.1 (Reduction).

Let $G$ be a td-group and let $\mathcal{B}$ be Hecke category with $G$-support
satisfying (Reg) from Definition 3.11. Assume that the
${\mathcal{C}\mathrm{vcy}}$-assembly map (5.11) for $\mathcal{B}$

$\operatorname*{hocolim}_{P\in\matheurm{P}{\mathcal{C}\mathrm{vcy}}(G)}\mathbf{K}\big{(}\matheurm{C}_{G}(P)\big{)}\;\to\;\mathbf{K}\big{(}\matheurm{C}_{G}(\ast)\big{)}$

is an equivalence. Then the ${\mathcal{C}\mathrm{op}}$-assembly map (3.9) for
$\mathcal{B}$

$\operatorname*{hocolim}_{G/U\in\matheurm{Or}_{{\mathcal{C}\mathrm{op}}}(G)}\mathbf{K}\big{(}\mathcal{B}[G/U]\big{)}\to\mathbf{K}\big{(}\mathcal{B}[G/G]\big{)}\simeq\mathbf{K}\mathcal{B}$

is also an equivalence.

Modulo further results the proof of Theorem 14.1 will be given in Subsection
14.b.

###  14.a. Functoriality in the orbit category

Recall that the definition of $\matheurm{C}_{G}(P)$ used the (free) $G$-space
$|P|^{\wedge}$. The construction of this space is not functorial for
$P\in\matheurm{P}_{+}\matheurm{Or}(G)$, only for
$P\in\matheurm{P}_{+}{\mathcal{A}\mathrm{ll}}(G)$. For this reason,
$\matheurm{C}_{G}(P)$ is not functorial in $\matheurm{P}_{+}\matheurm{Or}(G)$.
But this is not a serious issue and we will now tweak the definitions to
remedy this.

Let $\mathcal{B}$ be a category with $G$-support and
$\mathfrak{E}=(\mathfrak{E}_{1},\mathfrak{E}_{2},\mathfrak{E}_{G})$ be a
$G$-control structure on $X$. Let $\mathcal{Y}$ be a collection of subsets of
$X$ as in Definition 4.19. We say that two maps $\pi,\rho\colon S\to X$ are
_$\mathfrak{E}_{2}$ -equivalent_ if

$\Big{\\{}{\begin{pmatrix}{\rho(s)}\\\ {\pi(s)}\end{pmatrix}}\;\Big{|}\;s\in
S\Big{\\}}\in\mathfrak{E}_{2}.$

In the definition of $\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$ we can
replace the maps $\pi\colon S\to X$ with their $\mathfrak{E}_{2}$-equivalence
classes and obtain the category
$\overline{\mathcal{B}}_{G}(\mathfrak{E},\mathcal{Y})$. In more detail,
objects of $\overline{\mathcal{B}}_{G}(\mathfrak{E},\mathcal{Y})$ are triples
$\mathbf{B}=(S,[\pi],B)$ such that $(S,\pi,B)$ is an object of
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$. The point here is that, if $\pi$
and $\rho$ are $\mathfrak{E}_{2}$-equivalent, then $(S,\pi,B)$ is an object of
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$, iff $(S,\rho,B)$ is an object of
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$. A morphism
$\mathbf{B}=(S,[\pi],B)\to\mathbf{B}^{\prime}=(S^{\prime},[\pi^{\prime}],B^{\prime})$
in $\overline{\mathcal{B}}_{G}(\mathfrak{E},\mathcal{Y})$ is a matrix
$\varphi=(\varphi^{s^{\prime}}_{s}\colon B(s)\to B^{\prime}(s^{\prime}))_{s\in
S,s^{\prime}\in S^{\prime}}$ of morphisms in $\mathcal{B}$ that also defines a
morphism $(S,\pi,B)\to(S^{\prime},\pi^{\prime},B^{\prime})$ in
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$. Again, the point here is that, if
$\pi$ and $\rho$ are $\mathfrak{E}_{2}$-equivalent, $\pi^{\prime}$ and
$\rho^{\prime}$ are $\mathfrak{E}_{2}$-equivalent, then $\varphi$ defines a
morphism in $(S,\pi,B)\to(S^{\prime},\pi^{\prime},B^{\prime})$ in
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$, iff $\varphi$ defines a morphism
in $(S,\rho,B)\to(S^{\prime},\rho^{\prime},B^{\prime})$ in
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$. The functor

(14.2)
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})\xrightarrow{\sim}\overline{\mathcal{B}}_{G}(\mathfrak{E},\mathcal{Y}),\quad(S,\pi,B)\mapsto(S,[\pi],B)$

is an equivalence. Essentially,
$\overline{\mathcal{B}}_{G}(\mathfrak{E},\mathcal{Y})$ is obtained from
$\mathcal{B}_{G}(\mathfrak{E},\mathcal{Y})$ by identifying $(S,\pi,B)$ and
$(S,\rho,B)$ along a canonical isomorphism, whenever $\pi$ and $\rho$ are
$\mathfrak{E}_{2}$-equivalent.

Recall $\matheurm{C}_{G}(P)=\mathcal{B}_{G}(\mathfrak{C}(P),\mathcal{Y}(P))$
from Definition 5.9. We now define
$\overline{\matheurm{C}}_{G}(P):=\overline{\mathcal{B}}_{G}(\mathfrak{C}(P),\mathcal{Y}(P))$.
From (14.2) we obtain a natural equivalence

(14.3) $\matheurm{C}_{G}(P)\xrightarrow{\sim}\overline{\matheurm{C}}_{G}(P).$

Let $f\colon P\to P^{\prime}$ be a map in $\matheurm{P}_{+}\matheurm{Or}G$. It
induces a map $|f|\colon|P|\to|P^{\prime}|$. There is always a lift of $|f|$
to $|P|^{\wedge}\to|P^{\prime}|^{\wedge}$, but it is typically not unique.

######  Lemma 14.4.

Let $\widehat{f}_{0},\widehat{f}_{1}\colon|P|^{\wedge}\to|P|^{\wedge}$ be
lifts of $|f|$. Then
$\widehat{f}_{0}\times\operatorname{id}_{\mathbb{N}},\widehat{f}_{1}\times\operatorname{id}_{\mathbb{N}}\colon|P|^{\wedge}\times\mathbb{N}\to|P|^{\wedge}\times\mathbb{N}$
are $\mathfrak{C}_{2}(P)$-equivalent.

###### Proof.

To simplify notation, we just treat the case $P=G/H$,
$P^{\prime}=G/H^{\prime}$. Then $f=|f|\colon G/H=|G/H|\to
G/H^{\prime}=|G/H^{\prime}|$. Any $G$-map $f\colon G/H\to G/H^{\prime}$ is of
the form $xH\mapsto xgH^{\prime}$ where $g^{-1}Hg\subseteq H^{\prime}$. Now
for the two lifts $\widehat{f}_{0},\widehat{f}_{1}\colon G=|G/H|^{\wedge}\to
G=|G/H^{\prime}|^{\wedge}$ there are $h^{\prime}_{0},h^{\prime}_{1}\in
H^{\prime}$ with $\widehat{f}_{0}(x)=xgh^{\prime}_{0}$,
$\widehat{f}_{1}(x)=xgh^{\prime}_{1}$ for all $x\in G$. With
$\beta:=d(h^{\prime}_{0},h^{\prime}_{1})$ then for all $x\in G$

$d_{H^{\prime}\text{-}\mathrm{fol}}(\widehat{f}_{0}(x),\widehat{f}_{1}(x))\leq(\beta,0).$

This implies that
$\widehat{f}_{0}\times\operatorname{id}_{\mathbb{N}},\widehat{f}_{1}\times\operatorname{id}_{\mathbb{N}}$
are $\mathfrak{C}_{2}(P)$-equivalent. ∎

Lemma 14.4 implies that the assignment

$P\mapsto\overline{\matheurm{C}}_{G}(P)$

is functorial in $\matheurm{P}_{+}\matheurm{Or}(G)$. This allows us to
consider the bottom row in the following diagram

(14.5)
$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}{\mathcal{C}\mathrm{op}}(G)}\mathbf{K}\big{(}\matheurm{C}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}{\mathcal{C}\mathrm{vcy}}(G)}\mathbf{K}\big{(}\matheurm{C}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbf{K}\big{(}\matheurm{C}_{G}(\ast)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}{\mathcal{C}\mathrm{op}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}{\mathcal{C}\mathrm{vcy}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(\ast)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{{\mathcal{C}\mathrm{op}}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{{\mathcal{C}\mathrm{vcy}}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(\ast)\big{)}.}$

We claim that the vertical maps in this diagram are all equivalences. For the
first three vertical maps this follows from (14.3). For the first two vertical
maps in the bottom row, this is an application of Lemma A.6.

###  14.b. The category $\matheurm{C}^{+}_{G}(P)$

We will need slightly bigger categories than $\matheurm{C}_{G}(P)$ and
$\overline{\matheurm{C}}_{G}(P)$, where we relax the condition on the supports
in $G$. This will be important in Subsection 14.h. For smooth $P$ this will
not change the K-theory, but we do not know this for general $P$.

Define the $G$-control structure
$\mathfrak{C}^{+}(P)=(\mathfrak{C}^{+}_{1}(P),\mathfrak{C}^{+}_{2}(P),\mathfrak{C}^{+}_{G}(P))$
by $\mathfrak{C}^{+}_{1}(P):=\mathfrak{C}_{1}(P)$,
$\mathfrak{C}^{+}_{2}(P):=\mathfrak{C}_{2}(P)$ and

$\mathfrak{C}^{+}_{G}(P):=\text{All subsets of $G$}.$

We define

$\matheurm{C}^{+}_{G}(P):=\mathcal{B}_{G}(\mathfrak{C}^{+}(P),\mathcal{Y}(P))\quad\text{and}\quad\overline{\matheurm{C}}^{+}_{G}(P):=\overline{\mathcal{B}}_{G}(\mathfrak{C}^{+}(P),\mathcal{Y}(P)).$

As $\mathfrak{C}_{G}(P)\subseteq\mathfrak{C}^{+}_{G}(P)$, there are inclusions
$\matheurm{C}_{G}(P)\to\matheurm{C}^{+}_{G}(P)$ and
$\overline{\matheurm{C}}_{G}(P)\to\overline{\matheurm{C}}^{+}_{G}(P)$.

######  Lemma 14.6.

Suppose that $P$ is smooth, i.e., that
$P\in\matheurm{P}_{+}{\mathcal{O}\mathrm{p}}(G)$. Then the inclusions
$\matheurm{C}_{G}(P)\to\matheurm{C}^{+}_{G}(P)$ and
$\overline{\matheurm{C}}_{G}(P)\to\overline{\matheurm{C}}^{+}_{G}(P)$ induce
equivalences in K-theory.

###### Proof.

Proposition 5.16 holds, with exactly the same proof, for
$\matheurm{C}^{+}_{G}(P)$ in place of $\matheurm{C}_{G}(P)$ as well626262This
works in the generality of categories with $G$-support in place of
$\mathcal{H}_{G}(\mathcal{A})$.. This implies the result for
$\matheurm{C}_{G}(P)\to\matheurm{C}^{+}_{G}(P)$. For
$\overline{\matheurm{C}}_{G}(P)\to\overline{\matheurm{C}}^{+}_{G}(P)$ it
follows now from (14.2). ∎

######  Theorem 14.7.

Suppose that $\mathcal{B}$ satisfies (Reg) from Definition 3.11. Then for all
$P\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{vcy}}(G)$ the canonical map

(14.8)
$\operatorname*{hocolim}_{(Q,f)\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)\downarrow
P}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+}_{G}(Q)\big{)}\;\xrightarrow{\sim}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+}_{G}(P)\big{)}$

is an equivalence.

The proof of Theorem 14.7 will be given later. We do not know whether or not
Theorem 14.7 holds also for $\overline{\matheurm{C}}_{G}(\\--)$ instead of
$\overline{\matheurm{C}}^{+}_{G}(\\--)$. We note that for Theorem 14.7 it is
important that we used
$\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)$; in contrast the
category $\matheurm{P}{\mathcal{C}\mathrm{om}}(G)\downarrow P$ is typically
empty.

###### Proof of Theorem 14.1 modulo Theorem 14.7.

Consider the following diagram.

$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{op}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{op}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\alpha}_{1}}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{2}}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\alpha}_{2}}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{vcy}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathit{FJ}}$$\scriptstyle{\sim}$$\textstyle{\displaystyle\operatorname*{hocolim}_{P\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{vcy}}(G)}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+}_{G}(P)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}_{G}(\ast)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+}_{G}(\ast)\big{)}}$

We will argue below that all maps labeled with an $\sim$ are equivalences. A
diagram chase shows then that all other maps in the diagram are equivalences
as well. In particular the left vertical composition is an equivalence.
Proposition 3.13 in combination with the equivalences from (14.5) implies that
this vertical composition is equivalent to (3.9).

The analog of the map labeled $\mathit{FJ}$ for $\matheurm{C}_{G}(P)$ instead
of $\overline{\matheurm{C}}_{G}(P)$ and
$\matheurm{P}{\mathcal{C}\mathrm{vcy}}(G)$ instead of
$\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{vcy}}(G)$ is an equivalence by
assumption of Theorem 14.1. The equivalences from (14.5) imply that the map
labeled $\mathit{FJ}$ is an equivalence as well.

Lemma 14.6 implies that the top and bottom horizontal maps are equivalences.
Any compact subgroup of a td-group is contained in a compact open subgroup,
see Lemma 2.3. This implies that for all
$P\in\matheurm{P}{\mathcal{C}\mathrm{om}}(G)$ the category
$P\downarrow\matheurm{P}{\mathcal{C}\mathrm{op}}(G)$ is non-empty. Thus Lemma
A.6 implies that $\alpha_{1}$ and $\widehat{\alpha}_{1}$ are equivalences.

The map $\widehat{\alpha}_{2}$ is an equivalence by Theorem 14.7 and the
transitivity Lemma A.2 for homotopy colimits. ∎

###  14.c. The category $\overline{\matheurm{C}}^{+,0}_{G}(P)$

To prove Theorem 14.7 we introduce a further category as an intermediate step.
In many ways this is similar to the passage from
$\matheurm{D}_{G}({\bf\Sigma};\mathcal{B})$ to
$\matheurm{D}^{0}_{G}({\bf\Sigma};\mathcal{B})$, see Definition 7.15.

For $P\in\matheurm{P}_{+}{\mathcal{A}\mathrm{ll}}(G)$ we define the
$G$-control structure

$\mathfrak{C}^{+,0}(P)=(\mathfrak{C}^{+,0}_{1}(P),\mathfrak{C}^{+,0}_{2}(P),\mathfrak{C}^{+,0}_{G}(P))$

as follows. Set
$\mathfrak{C}^{+,0}_{1}(P):=\mathfrak{C}^{+}_{1}(P)=\mathfrak{C}_{1}(P)$,
$\mathfrak{C}^{+,0}_{G}(P):=\widehat{\mathfrak{C}}_{G}(P)$. We define
$\mathfrak{C}^{+,0}_{2}(P)$ to consist of all
$E\in\mathfrak{C}^{+}_{2}(P)=\mathfrak{C}_{2}(P)$ satisfying

$\qquad{\begin{pmatrix}{\lambda^{\prime},t^{\prime}}\\\
{\lambda,t}\end{pmatrix}}\in E\;\implies\;{t}^{\prime}={t}.$

Set

$\matheurm{C}^{+,0}_{G}(P):=\mathcal{B}_{G}(\mathfrak{C}^{+,0}(P),\mathcal{Y}(P))\quad\text{and}\quad\overline{\matheurm{C}}^{+,0}_{G}(P):=\overline{\mathcal{B}}_{G}(\mathfrak{C}^{+,0}(P),\mathcal{Y}(P)).$

######  Proposition 14.9.

There are homotopy pushout diagrams of functors
$\matheurm{P}_{+}{\mathcal{A}\mathrm{ll}}(G)\to\matheurm{Spectra}$

$\textstyle{\mathbf{K}\big{(}\matheurm{C}^{+,0}_{G}(\\--)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\matheurm{C}^{+,0}_{G}(\\--)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(\\--)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(\\--)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\matheurm{C}^{+,0}_{G}(\\--)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\matheurm{C}^{+}_{G}(\\--)\big{)}}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(\\--)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+}_{G}(\\--)\big{)}}$

###### Proof.

For the diagram on the left this is analog to the proof of Proposition 8.9,
more precisely to the construction of either of the pushout squares from Lemma
8.11. For the diagram on the right it follows now from (14.2). ∎

Proposition 14.9 implies that it suffices to prove Theorem 14.7 for
$\overline{\matheurm{C}}^{+,0}_{G}(\\--)$ in place of
$\overline{\matheurm{C}}^{+}_{G}(\\--)$.

###  14.d. Structure of the remainder of the proof

For the proof of Theorem 14.7 for $\overline{\matheurm{C}}^{+,0}_{G}(\\--)$ we
will introduce further variations of $\matheurm{C}_{G}(\\--)$ and argue along
the following diagram

(14.10)
$\textstyle{\displaystyle\operatorname*{hocolim}_{(Q,f)\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)\downarrow
P}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(Q)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(P)\big{)}}$$\textstyle{\displaystyle\operatorname*{hocolim}_{\underline{\Gamma}}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(M)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{(1)}$$\scriptstyle{\sim}$$\scriptstyle{(3)}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(M)[\Gamma]\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{(2)}$$\scriptstyle{\sim}$$\scriptstyle{(4)}$$\textstyle{\displaystyle\operatorname*{hocolim}_{\underline{\Gamma}}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0,\sharp}_{G}(M)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0,\sharp}_{G}(M)[\Gamma]\big{)}}$$\textstyle{\displaystyle\operatorname*{hocolim}_{\underline{\Gamma}}\mathbf{K}\big{(}\matheurm{C}^{+,0,\sharp}_{G}(M)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{(4)}$$\scriptstyle{\sim}$$\scriptstyle{(7)}$$\textstyle{\mathbf{K}\big{(}\matheurm{C}^{+,0,\sharp}_{G}(M)[\Gamma]\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\sim}$$\scriptstyle{(6)}$

Details will be worked out in the remainder of this section.

###  14.e. The group $\Gamma$

Here we discuss the equivalences (1) and (2) from (14.10).

Fix $P=(G/V_{1},\dots,G/V_{n})$ with $V_{i}\in{\mathcal{C}\mathrm{vcy}}$. Let
$K_{i}\subseteq V_{i}$ be the maximal compact open subgroup of $V_{i}$. Set
$M:=(G/K_{1},\dots,G/K_{n})$. The quotients $\Gamma_{i}:=V_{i}/K_{i}$ are
either infinite cyclic or trivial. Let
$\Gamma:=\Gamma_{1}\times\cdots\times\Gamma_{n}$. Then $\Gamma$ is a finitely
generated free abelian group of rank at most $n$. There are canonical maps
$h_{i}\colon\Gamma_{i}\to\operatorname{end}_{\matheurm{Or}(G)}(G/K_{i})$,
sending $cK_{i}\in C_{i}$ to $G/K_{i}\to G/K_{i},gK_{i}\mapsto gcK_{i}$. These
combine to an action of $\Gamma$ on $M$ by morphisms in
$\matheurm{P}\matheurm{Or}(G)$. This induces an action of $\Gamma$ on
$\overline{\matheurm{C}}^{+,0}_{G}(M)$ and we can form
$\overline{\matheurm{C}}^{+,0}_{G}(M)[\Gamma]$. The projection $\pi\colon M\to
P$ is $\Gamma$-equivariant for the trivial action of $\Gamma$ on $P$. Thus
$\pi_{*}\colon\overline{\matheurm{C}}^{+,0}_{G}(M)\to\overline{\matheurm{C}}^{+,0}_{G}(P)$
is $\Gamma$-equivariant for the trivial action on
$\overline{\matheurm{C}}^{+,0}_{G}(P)$ and induces a functor

$R\colon\overline{\matheurm{C}}^{+,0}_{G}(M)[\Gamma]\;\to\;\overline{\matheurm{C}}^{+,0}_{G}(P).$

The functor $R$ induces (2) in (14.10). We write $\underline{\Gamma}$ for the
category with exactly one object $\ast_{\Gamma}$ whose endomorphisms are given
by $\Gamma$. The action of $\Gamma$ on $M$ determines a functor
$h\colon\underline{\Gamma}\to\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)\downarrow
P$ that sends $\ast_{\Gamma}$ to $\pi\colon M\to P$. In turn $h$ induces a map

$\operatorname*{hocolim}_{\underline{\Gamma}}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(M)\big{)}\;\to\;\operatorname*{hocolim}_{(Q,f)\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)\downarrow
P}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(Q)\big{)};$

this is (1) in (14.10). The inclusion
$\overline{\matheurm{C}}^{+,0}_{G}(M)\to\overline{\matheurm{C}}^{+,0}_{G}(M)[\Gamma]$
induces the assembly map from the second row of (14.10)

(14.11)
$\operatorname*{hocolim}_{\underline{\Gamma}}\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(M)\big{)}\to\mathbf{K}\big{(}\overline{\matheurm{C}}^{+,0}_{G}(M)[\Gamma]\big{)}.$

######  Lemma 14.12.

  1. (a)

The functor $R$ induces an equivalence in K-theory, i.e., (2) in (14.10) is an
equivalence;

  2. (b)

the functor $h$ induces an equivalence, i.e., (1) in (14.10) is an
equivalence.

###### Proof.

(a) We will first show that for
$\mathbf{B}\in\overline{\matheurm{C}}^{+,0}_{G}(P)$, there is
$(\widetilde{\mathbf{B}},p)\in\operatorname{Idem}\overline{\matheurm{C}}^{+,0}_{G}(M)$
such that $\mathbf{B}\cong R(\widetilde{\mathbf{B}},p)$ in
$\operatorname{Idem}\overline{\matheurm{C}}^{+,0}_{G}(P)$. Write
$\mathbf{B}=(S,[\pi],B)$. Let $U_{0}\subseteq U_{1}\subset\dots$ be a sequence
of compact open subgroups of $G$ with $\operatorname{diam}U_{i}\to 0$ as
$i\to\infty$. Write $\pi(s):=(g_{1}(s),\dots,g_{n}(s),t(s))\in
G^{n}\times\mathbb{N}=|P|^{\wedge}\times\mathbb{N}$. Set

$K(s):=\operatorname{supp}B(s)\cap
g_{1}(s)U_{t(s)}(g_{1}(s))^{-1}\cap\dots\cap g_{n}(s)U_{t(s)}(g_{n}(s))^{-1}.$

Using (3.2d) we set $\widetilde{\mathbf{B}}:=(s,\Pi,B|_{K(s)})$. By design
$\widetilde{\mathbf{B}}\in\overline{\matheurm{C}}^{+,0}_{G}(P)$. Define
$i\colon\mathbf{B}\to\widetilde{\mathbf{B}}$ and
$r\colon\widetilde{\mathbf{B}}\to\mathbf{B}$ with

$i_{s}^{s^{\prime}}=\begin{cases}i_{B(s),K(s)}&s=s^{\prime}\\\
0&\text{else}\end{cases}\quad\text{and}\quad
r_{s}^{s^{\prime}}=\begin{cases}r_{B(s),K(s)}&s=s^{\prime}\\\
0&\text{else}\end{cases}.$

Again by (3.2d) we have $r\circ i=\operatorname{id}_{\mathbf{B}}$. Thus
$p:=i\circ r$ is an idempotent on $\widetilde{\mathbf{B}}$ and
$\mathbf{B}\cong R(\widetilde{\mathbf{B}},p)$ as promised. It is an exercise
in the definitions to check that $R$ is bijective on morphism sets.
Altogether, $R$ induces an equivalence on idempotent completions and thus in
K-theory.
(b) This follows from cofinality Lemma A.1 for homotopy colimits; we need to
verify that $h$ is right cofinal. Fix $f\colon Q\to P$ from
$\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)\downarrow P$. We need
to show that $(Q,f)\downarrow h$ is contractible. Objects in $(Q,f)\downarrow
h$ are morphisms $Q\xrightarrow{g}M$ in
$\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)$ such that $f=\pi\circ
g$. Morphisms in $(Q,f)\downarrow h$ are commutative diagrams

$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{=}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\gamma}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{P}$

We check that $(Q,f)\downarrow h$ is equivalent to a point. Recall
$P=(G/V_{1},\dots,G/V_{n})$ and $M=(G/K_{1},\dots,G/K_{n})$, where $K_{i}$ is
the maximal compact open subgroup of $V_{i}$. Write
$Q=(G/L_{1},\dots,G/L_{m})$ and $f=(u,\varphi)$ with
$u\colon\\{1,\dots,n\\}\to\\{1,\dots,m\\}$ and $\varphi(i)\colon G/L_{u(i)}\to
G/V_{i}$. Then there are $x_{1},\dots,x_{n}\in G$ with
$\varphi(i)yL_{u(i)}=yx_{i}V_{i}$ for all $y\in G$, and
$x_{i}^{-1}L_{u(i)}x_{i}\subseteq V_{i}$. As $x_{i}^{-1}L_{u(i)}x_{i}$ is
compact and $K_{i}$ is the unique maximal compact subgroup, we have
$x_{i}^{-1}L_{u(i)}x_{i}\subseteq K_{i}$. Let $g:=(u,\psi)\colon Q\to M$ with
$\psi(i)\colon G/L_{u(i)}\to G/K_{i}$ given by $\psi(i)yL_{u(i)}=yx_{i}K_{i}$
for all $y\in G$. Then $f=\pi\circ g$. Thus $(Q,f)\downarrow h$ is non-empty.
If $g^{\prime}=(u^{\prime},\psi^{\prime})\colon Q\to M$ is another map with
$f=\pi\circ g^{\prime}$, then $u=u^{\prime}$ and there are $v_{i}\in V_{i}$
with $\psi^{\prime}(i)yL_{u(i)}=yx_{i}v_{i}K_{i}$. Now the $v_{i}$ define an
isomorphism $\gamma\colon M\to M$ with $\gamma\circ g=g^{\prime}$ and this
$\gamma$ is the only morphism with this property. Thus any two objects in
$(Q,f)\downarrow h$ are uniquely isomorphic. ∎

###  14.f. The category $\overline{\matheurm{C}}^{+,0,\sharp}(M)$

Here we discuss the equivalences (3), (4), (5) and (6) from (14.10).

Let $M=(G/K_{1}\times\cdots\times G/K_{n})$ be as before. We give a slight
variation of the categories $\matheurm{C}^{+,0}_{G}(M)$ and
$\overline{\matheurm{C}}^{+,0}_{G}(M)$. The fact that $|M|^{\wedge}\to|M|$ has
compact fibers (since $M\in P\matheurm{Or}_{\mathcal{C}\mathrm{om}}$) will
allow us to base the definition on $|M|$ instead of $|M|^{\wedge}$. See also
Remark 5.4.

Let $\mu_{i}$ be a Haar measure on $K_{i}$. We can integrate the left-
invariant metric $d_{G}$ on $G$ to a left-invariant metric $d_{G,i}$ on $G$
that is in addition right $K$-invariant,
$d_{G,i}(g,g^{\prime}):=\int_{K_{i}}d_{G}(gk,g^{\prime}k)d\mu_{i}(k)$ and
obtain a left-invariant metric

$d_{G/K_{i}}(gK_{i},g^{\prime}K_{i}):=\min_{k\in K}d_{G,i}(gk,g^{\prime})$

on $G/K_{i}$. We obtain a left-invariant metric $d_{|M|}$ on $|M|$ with

$d_{|M|}\big{(}(g_{1}K_{1},\dots,g_{n}K_{n}),(g^{\prime}_{1}K_{1},\dots,g^{\prime}_{n}K_{n})\big{)}:=\max_{i}d_{G/K_{i},}(g_{i}K_{i},g^{\prime}_{i}K_{i}).$

We define the $G$-control structure
$\mathfrak{C}^{+,0,\sharp}(M)=\big{(}\mathfrak{C}^{+,0,\sharp}_{1}(M),\mathfrak{C}^{+,0,\sharp}_{2}(M),\mathfrak{C}^{+,0,\sharp}_{G}(M)\big{)}$
on $|M|\times\mathbb{N}$ as follows:

  * $\bullet$

$\mathfrak{C}^{+,0,\sharp}_{1}(M)$ consists of all subsets $F$ of
$|M|\times\mathbb{N}$ for which $F\cap|M|\times\\{t\\}$ is finite for all
$t\in\mathbb{N}$;

  * $\bullet$

$\mathfrak{C}^{+,0,\sharp}_{2}(M)$ consists of all subsets $E$ of
$\big{(}|M|\times\mathbb{N}\big{)}^{\times 2}$ satisfying the following two
conditions

    * –

_$0$ -control over $\mathbb{N}$_: if
${\begin{pmatrix}{\lambda^{\prime},t^{\prime}}\\\ {\lambda,t}\end{pmatrix}}\in
E$ then $t=t^{\prime}$;

    * –

_metric control over $|M|$_: for any $\eta>0$ there is $t_{0}$ such that for
all $t\geq t_{0}$ and all $\lambda,\lambda^{\prime}$ we have

${\begin{pmatrix}{\lambda^{\prime},t}\\\ {\lambda,t}\end{pmatrix}}\in
E\quad\implies\quad d_{|M|}(\lambda,\lambda^{\prime})<\eta;$

  * $\bullet$

$\mathfrak{C}^{+,0,\sharp}_{G}(M)$ consists of all subsets of $G$.

Let $\mathcal{Y}^{\sharp}(M)$ be the collection of all subsets $Y$ of
$|M|\times\mathbb{N}$, for which there is $d$ with
$Y\subseteq|M|\times\mathbb{N}_{\leq d}$. We define

$\displaystyle\matheurm{C}^{+,0,\sharp}(M)$ $\displaystyle:=$
$\displaystyle\mathcal{B}_{G}(\mathfrak{C}^{+,0,\sharp}(M),\mathcal{Y}^{\sharp}(M));$
$\displaystyle\overline{\matheurm{C}}^{+,0,\sharp}(M)$ $\displaystyle:=$
$\displaystyle\overline{\mathcal{B}}_{G}(\mathfrak{C}^{+,0,\sharp}(M),\mathcal{Y}^{\sharp}(M)).$

Let $p\colon|M|^{\wedge}\to|M|$ be the canonical projection. As the $K_{i}$
are compact and therefore of finite diameter, it is not difficult to check
that $\mathfrak{C}^{+,0,\sharp}_{2}(M)$ is exactly the image of
$\mathfrak{C}^{+,0}_{2}(M)$ under
$(p\times\operatorname{id}_{\mathbb{N}})^{\times
2}\colon(|M|^{\wedge}\times\mathbb{N})^{\times
2}\to(|M|\times\mathbb{N})^{\times 2}$. This in turn implies that
$p\times\operatorname{id}$ induces an equivalence
$\matheurm{C}^{+,0}_{G}(M)\xrightarrow{\sim}\matheurm{C}^{+,0,\sharp}_{G}(M)$.
Applying (14.2) we obtain the equivalence
$\matheurm{C}^{+,0}_{G}(M)\xrightarrow{\sim}\matheurm{C}^{+,0,\sharp}_{G}(M)$
and this yields the equivalences (3) and (4) from (14.10). Also from (14.2) we
obtain an equivalence
$\matheurm{C}^{+,0,\sharp}_{G}(M)\xrightarrow{\sim}\overline{\matheurm{C}}^{+,0,\sharp}_{G}(M)$
and this induces the equivalences (5) and (6) from (14.10).

###  14.g. The limit category

We briefly digress to recall a result from [4]. Consider a nested sequence of
categories

$\mathcal{C}=\mathcal{C}_{0}\supseteq\mathcal{C}_{1}\supseteq\mathcal{C}_{2}\supseteq\cdots.$

Associated to this is the _sequence category_ $\mathcal{S}(\mathcal{C}_{*})$.
This is the subcategory of $\prod_{m\geq 0}\mathcal{C}$, whose objects are
sequences $(C_{m})_{m\geq 0}$ such that for any $l$ there is $m_{0}$ with
$C_{m}\in\mathcal{C}_{l}$ for all $m\geq m_{0}$. Morphisms are sequences
$(\varphi_{m})_{m\geq 0}$ such that for any $l$ there is $m_{0}$ with
$\varphi_{m}\in\mathcal{C}_{l}$ for all $m\geq m_{0}$. The sum
$\bigoplus_{m\geq 0}\mathcal{C}$ is a subcategory of
$\mathcal{S}(\mathcal{C}_{*})$ and we call the quotient category

$\mathcal{L}(\mathcal{C}_{*}):=\mathcal{S}(\mathcal{C}_{*})\bigg{/}\bigoplus_{m\geq
0}\mathcal{C}_{0}$

the _limit category_. We reviewed $l$-uniform regular coherence and exactness
in Subsection 3.b. Assume now that

  1. (14.13a)

for any $d\in\mathbb{N}$ there is $m_{0}$ such that the inclusion
$\mathcal{C}_{m+1}[\mathbb{Z}^{d}]\to\mathcal{C}_{m}[\mathbb{Z}^{d}]$ is exact
for all $m\geq m_{0}$;

  2. (14.13b)

for any $d\in\mathbb{N}$ there are $m_{0},l\in\mathbb{N}$ such that
$\mathcal{C}_{m}[\mathbb{Z}^{d}]$ is $l$-regular coherent for all $m\geq
m_{0}$.

Here we use the trivial action to form $\mathcal{C}_{m}[\mathbb{Z}^{d}]$. Then
[4, Thm. 14.1] asserts the following. Consider an action of a finitely
generated free abelian group $\Gamma$ on $\mathcal{C}$ with the following
property: for $\gamma\in\Gamma$, $j\in\mathbb{N}$ there is $i\in\mathbb{N}$
such that $\gamma(\mathcal{C}_{i})\subseteq\mathcal{C}_{j}$. Then the assembly
map

(14.14)
$\operatorname*{hocolim}_{\underline{\Gamma}}\mathbf{K}\big{(}\mathcal{L}(\mathcal{C}_{*})\big{)}\xrightarrow{\sim}\mathbf{K}\big{(}\mathcal{L}(\mathcal{C}_{*})[\Gamma]\big{)}$

is an equivalence. (In [4] the result is formulated using an equivariant
homology theory instead of a homotopy colimit, but the two formulations are
equivalent, see [22].)

###  14.h. Analysis of $\matheurm{C}^{+,0,\sharp}_{G}(M)$

Finally, we discuss the equivalence (7) from (14.10).

Fix again $M\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)$ and
write $M=(G/K_{1},\dots,G/K_{n})$ with $K_{r}\in{\mathcal{C}\mathrm{om}}$.
Using Lemma 2.3 we fix a choice of compact open subgroups
$(U_{r,i})_{r=1,\dots,n,i\in\mathbb{N}_{\geq 1}}$ of $G$ such that for each
$r=1,\dots,n$ we have

$U_{r,1}\supseteq U_{r,2}\supseteq\dots\;\;\supseteq\bigcap_{i}U_{r,i}=K_{r}.$

Set $U_{r,0}:=G$636363This ensure that $\mathcal{C}_{0}$ defined later carries
a $\Gamma$-action. and $Q_{i}:=(G/U_{1,i},\dots,G/U_{n,i})$. We write
$p^{i}\colon|M|\to|Q_{i}|$ and, for $j\geq i$,
$p_{j}^{i}\colon|Q_{j}|\to|Q_{i}|$ for the canonical projections. We define
the $G$-control structure
$\mathfrak{E}^{i}=(\mathfrak{E}^{i}_{1},\mathfrak{E}^{i}_{2},\mathfrak{E}^{i}_{G})$
on $|M|$, where $\mathfrak{E}^{i}_{1}$ is the collection of all finite subsets
of $|M|$, $\mathfrak{E}^{i}_{2}$ consists of all $E\subseteq|M|\times|M|$
satisfying

(14.15) ${\begin{pmatrix}{\lambda^{\prime}}\\\ {\lambda}\end{pmatrix}}\in
E\;\implies\;p^{i}(\lambda)=p^{i}(\lambda^{\prime}),$

and $\mathfrak{E}^{i}_{G}$ is the collection of all subsets of $G$. We
abbreviate $\mathcal{C}_{i}:=\mathcal{B}_{G}(\mathfrak{E}^{i})$ and obtain a
nested sequence of categories
$\mathcal{C}_{0}\supseteq\mathcal{C}_{1}\supseteq\mathcal{C}_{2}\supseteq\dots$.
The metric control condition for $E\in\widetilde{\mathfrak{C}}^{0}_{2}(M)$ is
equivalent to

  * -

for any $i$ there is $t_{0}$ such that for all $t\geq t_{0}$ and all
$\lambda,\lambda^{\prime}$ we have

${\begin{pmatrix}{\lambda^{\prime},t}\\\ {\lambda,t}\end{pmatrix}}\in
E\implies p^{i}(\lambda)=p^{i}(\lambda^{\prime}).$

From this observation it is easy to deduce that there is an equivalence
$\matheurm{C}^{+,0,\sharp}_{G}(M)\xrightarrow{\sim}\mathcal{L}(\mathcal{C}_{*})$
that sends an object $\mathbf{B}=(S,\pi,B)$ to the sequence
$(S_{i},\pi_{i},B_{i})_{i\in\mathbb{N}}$, where
$S_{i}=\pi^{-1}(|M|\times\\{i\\})$, $B_{i}=B|_{S_{i}}$ and $\pi_{i}$ is the
composition

$S_{i}\xrightarrow{\pi|_{S_{i}}}|M|\times\\{i\\}\xrightarrow{\equiv}|M|.$

We point out that for the equivalence
$\matheurm{C}^{+,0,\sharp}_{G}(M)\xrightarrow{\sim}\mathcal{L}(\mathcal{C}_{*})$
it is important that we work with
$\mathfrak{C}^{+}_{G}(P)=\mathfrak{C}^{+,0,\sharp}_{G}(M)=\text{All subsets of
$G$}$; if one were to use only compact subsets of $G$ instead, then the limit
category would be strictly bigger.

Let now $\Gamma=\Gamma_{1}\times\cdots\times\Gamma_{n}$ be as in Subsection
14.e. The group $\Gamma_{r}$ does not necessarily normalize $U_{r,i}$, but for
each $\gamma\in\Gamma_{r}$ we still have
$\bigcap_{i}(U_{r,i})^{\gamma}=(K_{r})^{\gamma}=K_{r}$. Using the compactness
of the $U_{r,i}$ for $i>0$, it is easy to check that for fixed $\gamma$ and
$j$ there is $i$ with $(U_{r,i})^{\gamma}\subseteq U_{r,j}$. This implies that
the action of $\Gamma$ on $M$ induces an action acts on the nested sequence
$C_{*}$ as in Subsection 14.g. Moreover
$\matheurm{C}^{+,0,\sharp}_{G}(M)\to\mathcal{L}(\mathcal{C}_{*})$ is
$\Gamma$-equivariant and an equivalence. It induces therefore an equivalence
$\matheurm{C}^{+,0,\sharp}_{G}(M)[\Gamma]\to\mathcal{L}(\mathcal{C}_{*})[\Gamma]$.
It follows that (7) from (14.10) is an equivalence, if and only if (14.14) is
an equivalence for our present choice of $\mathcal{C}_{*}$.

######  Lemma 14.16.

The nested sequence $\mathcal{C}_{*}$ defined above satisfies (14.13a). If
$\mathcal{B}$ satisfies (Reg) from Definition 3.11, then $\mathcal{C}_{*}$
also satisfies (14.13b).

###### Proof.

We start by examining $\mathcal{C}_{i}=\mathcal{B}_{G}(\mathfrak{E}^{i})$ for
$i>0$. For $\lambda\in|Q_{i}|$ let $G_{\lambda}$ be the isotropy group of
$\lambda$. This is a finite intersection of conjugates of the $U_{i,r}$ and
thus compact open in $G$. Let $\mathbf{B}=(S,\pi,B)\in\mathcal{C}_{i}$ and
$s\in S$. The control condition (14.15) implies
$\operatorname{supp}B(s)\subseteq G_{p_{i}(\pi(s))}$, in other words
$B(s)\in\mathcal{B}|_{G_{p_{i}(\pi(s))}}$. For $\lambda\in|Q_{i}|$ we obtain,
still using (14.15), a functor

$\displaystyle F_{i,\lambda}\colon\quad\mathcal{C}_{i}$ $\displaystyle\to$
$\displaystyle(\mathcal{B}|_{G_{\lambda}})_{\oplus}$ $\displaystyle\mathbf{B}$
$\displaystyle\mapsto$ $\displaystyle\bigoplus_{p_{i}(\pi(s))=\lambda}B(s).$

By definition of $\mathfrak{E}^{i}_{1}$ the set $S$ is finite and so the above
sum is finite and $F_{i,\lambda}(\mathbf{B})=0$ for all but finitely many
$\lambda$. The $F_{i,\lambda}$ combine to an equivalence

(14.17)
$\begin{array}[]{ccc}\mathcal{C}_{i}&\xrightarrow{\sim}&\bigoplus_{\lambda\in|Q_{i}|}(\mathcal{B}|_{G_{\lambda}})_{\oplus}\\\\[3.0pt]
\mathbf{B}&\mapsto&(F_{i,\lambda}(\mathbf{B}))_{\lambda\in|Q_{i}|}.\end{array}$

Since the property $l$-uniformly regular passes to direct sums over arbitrary
index sets and is invariant under equivalence of additive categories and the
passage going from an additive category $\mathcal{A}$ to
$\mathcal{A}[\mathbb{Z}^{d}]$ is compatible with infinite direct sums over
arbitrary index sets, (Reg) from Definition 3.11 implies (14.13b).

For $j>i$ and $\lambda\in|Q_{j}|$ we have $G_{\lambda}\subseteq
G_{p_{j}^{i}(\lambda)}$ as $p_{j}^{i}$ is $G$-equivariant. The inclusion
$\mathcal{C}_{j}\subseteq\mathcal{C}_{i}$ corresponds under (14.17) to the
functor

$\bigoplus_{\lambda\in|Q_{j}|}(\mathcal{B}|_{G_{\lambda}})_{\oplus}\to\bigoplus_{\kappa\in|Q_{i}|}(\mathcal{B}|_{G_{\kappa}})_{\oplus}$

that sends the $\lambda$-summand into the $\kappa$-summand by the functor
induced from the inclusion $G_{\lambda}\subseteq G_{\kappa}$ where
$\kappa=p_{j}^{i}(\lambda)$. By [7, Lem. 7.51] each of these is exact. This
implies (14.13a). ∎

Lemma 14.16 allows us to apply [4, Thm. 14.1] as reviewed in Subsection 14.g
to the nested sequence constructed above. So (14.14) and therefore also (7)
appearing in (14.10) is an equivalence.

###### Formal conclusion of the proof of Theorem 14.7 and Theorem 14.1..

Altogether we have shown that the top horizontal map in (14.10) is an
equivalence. As noted before, because of the pushout from Proposition 14.9,
this also implies that (14.8) is an equivalence as claimed in Theorem 14.7. We
have already explained in Subsection 14.b that Theorem 14.1 follows from
Theorem 14.7. ∎

######  Remark 14.18.

At least in the case
$M\in\matheurm{P}\matheurm{Or}_{\mathcal{C}\mathrm{om}}(G)$ we can give an
explanation of the value of $\mathbf{K}\big{(}\matheurm{C}_{G}(M)\big{)}$.
Write $M=(G/K_{1},\dots,G/K_{n})$ with $K_{r}\in{\mathcal{C}\mathrm{om}}$ and
fix a choice of compact open subgroups
$(U_{r,i})_{r=1,\dots,n,i\in\mathbb{N}_{\geq 1}}$ of $G$ such that for each
$r=1,\dots,n$ we have $U_{r,1}\supseteq
U_{r,2}\supseteq\dots\;\;\supseteq\bigcap_{i}U_{r,i}=K_{r}$. Put
$Q_{i}=(G/K_{1,i},\ldots,G/K_{n,i})$ for $i\in\mathbb{N}_{\geq 1}$.

Then there is zigzag of weak homotopy equivalences from
$\mathbf{K}\big{(}\matheurm{C}_{G}(M)\big{)}$ to the homotopy inverse limit
$\operatorname{holim}_{i\to\infty}\mathbf{K}\big{(}\matheurm{C}_{G}(M_{i})\big{)}$.
In particular there is for every $n\in\mathbb{Z}$ a short exact sequence

$0\to\operatorname{invlim}^{1}_{i\to\infty}\pi_{n+1}\bigl{(}\mathbf{K}(\matheurm{C}_{G}(Q_{i}))\big{)}\to\pi_{n}\bigl{(}\mathbf{K}(\matheurm{C}_{G}(M))\big{)}\\\
\to\operatorname{invlim}_{i\to\infty}\pi_{n}\bigl{(}\mathbf{K}(\matheurm{C}_{G}(Q_{i}))\big{)}\to
0.$

Since we do not need this result in this paper, we omit its proof. It is very
unlikely that the corresponding statement holds, if we drop the assumption
that each $K_{i}$ is compact.

##  Appendix A Homotopy colimits

###  A.i. Cofinality and transitivity

We recall two basic facts about homotopy colimits [25, §9].

Let $F\colon\mathcal{A}\to\mathcal{B}$ be functor. Let $B\in\mathcal{B}$. We
write $B\downarrow F$ for the following category. Objects are pairs
$(A,\beta)$ with $A\in\mathcal{A}$ and $\beta\colon B\to F(A)$ in
$\mathcal{B}$. Morphisms $(A,\beta)\to(A^{\prime},\beta^{\prime})$ are
morphisms $\alpha\colon A\to A^{\prime}$ in $\mathcal{A}$ with
$F(\alpha)\circ\beta=\beta^{\prime}$. We write $F\downarrow B$ for the
following category. Objects are pairs $(A,\beta)$ with $A\in\mathcal{A}$ and
$\beta\colon F(A)\to B$ in $\mathcal{B}$. Morphisms
$(A,\beta)\to(A^{\prime},\beta^{\prime})$ are morphisms $\alpha\colon A\to
A^{\prime}$ in $\mathcal{A}$ with $\beta\circ F(\alpha)=\beta^{\prime}$. If
$F$ is the inclusion of the subcategory $\mathcal{A}$, then we write
$B\downarrow\mathcal{A}$ and $\mathcal{A}\downarrow B$ instead of $B\downarrow
F$ and $F\downarrow B$.

A functor $F\colon\mathcal{A}\to\mathcal{B}$ between small categories is said
to be _right cofinal_ , if the nerve of $B\downarrow F$ is contractible (and
in particular non-empty) for every $B\in\mathcal{B}$.

######  Lemma A.1 (Cofinality).

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a right cofinal functor between
small categories. Then for any
$\mathbf{D}\colon\mathcal{B}\to\matheurm{Spectra}$ the canonical map

$\operatorname*{hocolim}_{\mathcal{A}}F^{*}\mathbf{D}\;\xrightarrow{\sim}\;\operatorname*{hocolim}_{\mathcal{B}}\mathbf{D}$

is an equivalence.

###### Proof.

This is [25, 9.4]. ∎

######  Lemma A.2 (Transitivity).

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a functor between small categories.
Consider functors $\mathbf{D}\colon\mathcal{A}\to\matheurm{Spectra}$,
$\mathbf{E}\colon\mathcal{B}\to\matheurm{Spectra}$ and a natural
transformation $\tau\colon\mathbf{D}\to F^{*}\mathbf{E}$. For
$B\in\mathcal{B}$ write $v_{B}\colon F\downarrow B\to\mathcal{A}$ for the
forgetful functor. Suppose that for any $B\in\mathcal{B}$ the canonical map

$\operatorname*{hocolim}_{F\downarrow
B}v_{B}^{*}\mathbf{D}\;\xrightarrow{\sim}\;\mathbf{E}(B)$

is an equivalence. Then the canonical map

$\operatorname*{hocolim}_{\mathcal{A}}\mathbf{D}\;\xrightarrow{\sim}\;\operatorname*{hocolim}_{\mathcal{B}}\mathbf{E}$

is an equivalence as well.

###### Proof.

The assumption implies that $\mathbf{E}$ is the homotopy push down
$F_{*}\mathbf{D}$ of $\mathbf{D}$ along $F$. The assertion follows from [25,
9.4]. ∎

###  A.ii. Homotopy colimits and categories with product

Let $\mathcal{P}$ be a small category with all non-empty finite products. Let
$\mathcal{P}_{+}$ be the category obtained from $\mathcal{P}$ by adding a
terminal object $\ast$. For example for $\mathcal{P}=\matheurm{P}\mathcal{A}$,
we have $\mathcal{P}_{+}=\matheurm{P}_{+}\mathcal{A}$. Let
$\mathbf{D}\colon\mathcal{P}_{+}\to\matheurm{Spectra}$ be a functor. The
unique maps $P\to\ast$ induce the canonical map

$\operatorname*{hocolim}_{P\in\mathcal{P}}\mathbf{D}(P)\;\to\;\mathbf{D}(\ast).$

######  Lemma A.3.

Fix $Q\in\mathcal{P}$. Then the canonical map

$\operatorname*{hocolim}_{P\in\mathcal{P}}\mathbf{D}(P\times
Q)\;\to\;\mathbf{D}(Q)$

is an equivalence.

###### Proof.

Consider

$\mathcal{P}\;\xrightarrow{f_{Q}}\;\big{(}\mathcal{P}\downarrow
Q\big{)}\;\xrightarrow{v_{Q}}\;\mathcal{P}\;\xrightarrow{\mathbf{D}}\;\matheurm{Spectra}$

where $f_{Q}$ sends $P$ to the canonical projection $P\times Q\to Q$ and
$v_{Q}$ is the forgetful functor. It is easy to verify that $f_{Q}$ is right
cofinal. Obviously $(Q,\operatorname{id}_{Q})$ is terminal in
$\mathcal{P}\downarrow Q$. Thus, using Lemma A.1,

$\operatorname*{hocolim}_{P\in\mathcal{P}}\mathbf{D}(P\times
Q)=\operatorname*{hocolim}_{\mathcal{P}}f_{Q}^{*}v_{Q}^{*}\mathbf{D}\xrightarrow{\sim}\operatorname*{hocolim}_{\mathcal{P}\downarrow
Q}v_{Q}^{*}\mathbf{D}\xrightarrow{\sim}v_{Q}^{*}\mathbf{D}(Q,\operatorname{id}_{Q})=\mathbf{D}(Q).$

∎

######  Proposition A.4.

Let $\mathcal{A}$ be a small category. Let $\mathcal{E}$ be the smallest
collection of functors
$\mathbf{E}\colon\mathcal{P}_{+}\mathcal{A}\to\matheurm{Spectra}$ that is
closed under hocolimits, retracts and contains all functors of the form
$P\mapsto\mathbf{D}(P\times Q)$ with $Q\in\mathcal{P}$ and
$\mathbf{D}\colon\mathcal{P}\to\matheurm{Spectra}$. Then for any
$\mathbf{E}\in\mathcal{E}$ the canonical map

$\operatorname*{hocolim}_{P\in\mathcal{P}}\mathbf{E}(P)\;\xrightarrow{\sim}\;\mathbf{E}(\ast).$

is an equivalence.

###### Proof.

Homotopy colimits and retracts of equivalences are again equivalences. The
result follows thus from Lemma A.3. ∎

######  Lemma A.5.

The nerv of $\mathcal{P}$ is contractible (provided that $\mathcal{P}$ is non-
empty).

###### Proof.

Let $Q$ be a fixed object of $\mathcal{P}$. Let
$c_{Q},x_{Q}\colon\mathcal{P}\to\mathcal{P}$ be the functors $c_{Q}(P)=Q$,
$c_{Q}(\varphi)=\operatorname{id}_{Q}$, $x_{Q}(P)=P\times Q$,
$x_{Q}(\varphi)=(\varphi\times\operatorname{id}_{Q})$. There are evident
natural transformations $\operatorname{id}_{\mathcal{P}}\leftarrow x_{Q}\to
c_{Q}$. On the nerv $\operatorname{id}_{\mathcal{P}}$ induces the identity,
$c_{Q}$ induces a constant map, and the two maps are homotopic. ∎

######  Lemma A.6.

Let $\mathcal{Q}$ be a further category with all non-empty finite products.
Let $F\colon\mathcal{Q}\to\mathcal{P}$ be a product preserving functor. Assume
that $P\downarrow F$ is non-empty for all $P\in\mathcal{P}$. Then $F$ is right
cofinal. In particular, for any
$\mathbf{D}\colon\mathcal{P}\to\matheurm{Spectra}$ the canonical map

$\operatorname*{hocolim}_{\mathcal{Q}}F^{*}\mathbf{D}\;\xrightarrow{\sim}\;\operatorname*{hocolim}_{\mathcal{P}}\mathbf{D}$

is an equivalence.

###### Proof.

Using that $\mathcal{Q}$ has all non-empty finite products and that $F$
preserves finite products, it is not difficult to check that $P\downarrow F$
also has all non-empty finite products. Lemma A.5 implies that $F$ is right
cofinal. The statement about homotopy colimits follows from Lemma A.1. ∎

##  Appendix B K-theory of $\operatorname{dg}$-categories

In order to obtain induced maps in K-theory for homotopy coherent functors we
use K-theory for $\operatorname{dg}$-categories. We briefly review its
construction. We write $\matheurm{Cat}_{\operatorname{dg}}$ for the category
of small $\operatorname{dg}$-categories [36]. For a
$\operatorname{dg}$-category $\mathcal{C}$ and objects
$C,C^{\prime}\in\mathcal{C}$, we write $\mathcal{C}(C,C^{\prime})$ for the
chain complex of morphisms from $C$ to $C^{\prime}$. For a
$\operatorname{dg}$-category $\mathcal{C}$ the category $H_{0}(\mathcal{C})$
has the same objects as $\mathcal{C}$ and for $C,C^{\prime}\in\mathcal{C}$ the
set of morphisms $C\to C^{\prime}$ in $H_{0}(\mathcal{C})$ is given by
$H_{0}(\mathcal{C}(C,C^{\prime}))$. A functor
$F\colon\mathcal{C}\to\mathcal{D}$ of $\operatorname{dg}$-categories is a
_quasi-equivalence_ if

$F_{*}\colon\mathcal{C}(C,C^{\prime})\to\mathcal{D}(F(C),F(C^{\prime}))$

is a quasi-isomorphism for all $C,C^{\prime}$, and it induces an equivalence
$H_{0}(\mathcal{C})\to H_{0}(\mathcal{C})$.

######  Theorem B.1 ([19, 36, 48]).

There exists a functor

$\mathbf{K}\colon\matheurm{Cat}_{\operatorname{dg}}\to\matheurm{Spectra}$

with the following properties:

  1. (a)

Restricted to additive categories $\mathbf{K}$ is weakly equivalent to the
usual (non-connective) K-theory functor;

  2. (b)

For an additive category $\mathcal{A}$ the inclusion of $\mathcal{A}$ into the
category $\operatorname{ch}_{{\mathrm{fin}}}\mathcal{A}$ of finite chain
complexes induces a weak equivalence
$\mathbf{K}(\mathcal{A})\to\mathbf{K}(\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A}))$;

  3. (c)

If $F\colon\mathcal{C}\to\mathcal{C}^{\prime}$ in
$\matheurm{Cat}_{\operatorname{dg}}$ is a quasi-equivalence, then
$\mathbf{K}(F)$ is a weak equivalence.

###### Sketch of proof.

K-theory of a $\operatorname{dg}$-category $\mathcal{C}$ can be defined as the
Waldhausen K-theory of the category of perfect $\mathcal{C}$-modules [36, Sec.
5.2] and [19].

We review the construction of Schlichting [48, Sec. 6.4]646464Strictly
speaking Schlichting only considers $\operatorname{dga}$s, not
$\operatorname{dg}$-categories, but his construction generalizes in a straight
forward manner.. Let $\mathcal{C}$ be a $\operatorname{dg}$-category. A
$\mathcal{C}$-module is a $\operatorname{dg}$-functor
$M\colon\mathcal{C}^{{\mathrm{op}}}\to\operatorname{ch}(\mathbb{Z}\text{-}\matheurm{Mod})$.
A map of $\mathcal{C}$-modules $M\to M^{\prime}$ is a natural transformation
by chain maps. A $\mathcal{C}$-module is free on $v\in M(C)_{k}$ if $f\mapsto
f^{*}(v)$ defines an isomorphism $\mathcal{C}(-,C)[k]\to M(-)$ of
$\mathcal{C}$-modules. A finite cell $\mathcal{C}$-module is a
$\mathcal{C}$-module $M$ together with a finite filtration $M_{0}\subseteq
M_{1}\subseteq\dots\subseteq M_{n}=M$ such that the quotients $M_{i}/M_{i-1}$
are free on a finite number of generators. We write
$\mathcal{C}$-$\matheurm{cell}$ for the category of finite cell
$\mathcal{C}$-modules. If $\mathcal{A}$ is an additive category, then the
category of finite cell $\mathcal{A}$-modules can be identified with the
$\operatorname{dg}$-category $\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A})$
of finite chain complexes over $\mathcal{A}$. For
$\mathcal{C}=\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A})$ the Yoneda
embedding of $\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A})$ into the
category of $\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A})$-modules
identifies $\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A})$ with
$\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A})$-$\matheurm{cell}$. A
sequence

$M\xrightarrow{i}M^{\prime}\xrightarrow{p}M^{\prime\prime}$

in $\mathcal{C}$-$\matheurm{cell}$ is said to be _exact_ , if $M(C)\to
M^{\prime}(C)\to M^{\prime\prime}(C)$ is exact in
$\operatorname{ch}(\mathbb{Z})$ for each $C\in\mathcal{C}$. With this notion
of exact sequences $\mathcal{C}$-$\matheurm{cell}$ is an exact category.

The exact category $\mathcal{C}$-$\matheurm{cell}$ has enough injectives and
projectives.656565$I$ is injective (projective) in an exact category
$\mathcal{E}$ if for any exact sequence $A\xrightarrow{i}B\xrightarrow{p}C$
the map $\mathcal{E}(B,I)\xrightarrow{i^{*}}\mathcal{E}(A,I)$ (the map
$\mathcal{E}(P,B)\xrightarrow{p_{*}}\mathcal{E}(P,C)$) is onto [35, Sec. 5].
The classes of projectives and injectives in this exact category coincide and
are given by the contractible cell modules. A map
$\mathcal{C}$-$\matheurm{cell}$ factors over an injective (projective), if and
only if it chain nullhomotopic. (All this can be proven using cones of
$\mathcal{C}$-modules exactly as in [48, Sec.6.4].)

A _Frobenius category_ is an exact category $\mathcal{E}$ with enough
projectives and injectives such that the classes of injectives and projectives
coincide. The associated stable category is the quotient of $\mathcal{E}$ by
the subcategory of projectives (injectives), i.e., morphisms are identified if
they factor over a projective (injective). A map of Frobenius categories is an
exact functor that preserves projectives (injectives). The category of cell
modules $\mathcal{C}$-$\matheurm{cell}$ is a Frobenius category and the
associated stable category is the homotopy category of
$\mathcal{C}$-$\matheurm{cell}$.

Schlichting’s [48, Sec. 12.1] non-connective K-theory functor

$\mathbf{K}^{\text{Frob}}\colon\text{Frobenius
categories}\to\matheurm{Spectra}$

has the following property: if a map $\mathcal{E}\to\mathcal{E}^{\prime}$ of
Frobenius categories induces an equivalence of stable categories, then
$\mathbf{K}^{\text{Frob}}(\mathcal{E})\to\mathbf{K}^{\text{Frob}}(\mathcal{E})$
is an equivalence [48, Cor. 1].

For a $\operatorname{dg}$-category $\mathcal{C}$ one then defines

$\mathbf{K}(\mathcal{C}):=\mathbf{K}^{\text{Frob}}(\mathcal{C}\text{-}\matheurm{cell}).$

For an additive category $\mathcal{A}$ we have
$\mathcal{A}$-$\matheurm{cell}\simeq\operatorname{ch}_{{\mathrm{fin}}}(\mathcal{A})$,
see above. Now (a) follows from [48, Thm. 5]. For (b) we can use that for an
additive category $\mathcal{A}$ the inclusion
$\mathcal{A}\to\operatorname{ch}_{fin}(\mathcal{A})$ induces an equivalence on
cell modules and therefore in K-theory. For (c) we use that for a quasi-
equivalence $\mathcal{C}\to\mathcal{C}^{\prime}$ the induced functor on the
stable categories associated to the categories of cell modules (i.e., on the
homotopy categories of cell modules) is an equivalence. ∎

We record the following consequence of the quasi-isomorphism invariance of
K-theory.

######  Corollary B.2.

Let $F,F^{\prime}\colon\mathcal{C}\to\mathcal{D}$ be
$\operatorname{dg}$-functors and let $\tau\colon F\to F^{\prime}$ be a natural
transformation of $\operatorname{dg}$-functors. Assume that $H_{0}(\tau)\colon
H_{0}(F)\to H_{0}(F^{\prime})$ is an isomorphism between the functors
$H_{0}(F),H_{0}(F^{\prime})\colon H_{0}(\mathcal{C})\to H_{0}(\mathcal{D})$.
Then $\mathbf{K}(F)$ and $\mathbf{K}(F^{\prime})$ are homotopic.

###### Proof.

This follows from Theorem B.1 (c) using the path $\operatorname{dg}$-category
$P(\mathcal{D})$ associated to $\mathcal{D}$ (see [24, 2.9] and [50, Def.
4.1]) by a standard argument.

Objects of $P(\mathcal{D})$ are diagrams $D\xrightarrow{f}D^{\prime}$ in
$\mathcal{D}$ with $f$ closed of degree $0$ whose homology class is an
isomorphism in $H_{0}(\mathcal{D})$. Given objects
$D_{0}\xrightarrow{f_{0}}D^{\prime}_{0}$,
$D_{1}\xrightarrow{f_{1}}D^{\prime}_{1}$, the complex
$P(\mathcal{D})(D_{0}\xrightarrow{f_{0}}D^{\prime}_{0},D_{1}\xrightarrow{f_{1}}D^{\prime}_{1})$
is the homotopy fiber of

$\displaystyle\mathcal{D}(D_{0},D^{\prime}_{0})\oplus\mathcal{D}(D_{1},D^{\prime}_{1})$
$\displaystyle\to$ $\displaystyle\mathcal{D}(D_{0},D^{\prime}_{1})$
$\displaystyle(\varphi,\varphi^{\prime})$ $\displaystyle\mapsto$
$\displaystyle f_{1}\circ\varphi-\varphi^{\prime}\circ f_{0}.$

Composition is defined by the formula

$(\varphi_{1},\varphi^{\prime}_{1},s_{1})\circ(\varphi_{0},\varphi^{\prime}_{0},s_{0}):=(\varphi_{1}\circ\varphi_{0},\varphi^{\prime}_{1}\circ\varphi^{\prime}_{0},\varphi^{\prime}_{1}\circ
s_{0}+s_{1}\circ\varphi_{0}).$

There are $\operatorname{dg}$-functors $p,p^{\prime}\colon
P(\mathcal{D})\to\mathcal{D}$ with $p(D\xrightarrow{f}D^{\prime})=D$,
$p^{\prime}(D\xrightarrow{f}D^{\prime})=D^{\prime}$. We also have
$i\colon\mathcal{D}\to P(\mathcal{D})$ with
$i(D)=(D\xrightarrow{\operatorname{id}_{D}}D)$ and $\Psi\colon\mathcal{C}\to
P(\mathcal{D})$ with $\Psi(C)=(F(C)\xrightarrow{\tau_{C}}F^{\prime}(C))$. Then
$p\circ i=p^{\prime}\circ i=\operatorname{id}_{\mathcal{D}}$. Moreover, $i$ is
a quasi-isomorphism [50, Lem. 4.3]. Using Theorem B.1 (c) this implies
$\mathbf{K}(p)\simeq\mathbf{K}(p^{\prime})$. We also have $F=p\circ\Psi$ and
$F^{\prime}=p^{\prime}\circ\Psi$. Thus
$\mathbf{K}(F)\simeq\mathbf{K}(F^{\prime})$. ∎

##  Appendix C Homotopy coherent functors

Our construction of the transfer depends on induced maps in K-theory for
homotopy coherent functors. Here we give a self contained construction of such
induced maps that is tailored to our specific needs. It is very pedestrian, by
no means a complete theory, and makes no direct contact with the elegant
language of stable $\infty$-categories. We include this for completeness.

######  Definition C.1.

Let $\mathcal{A}$ and $\mathcal{B}$ be additive categories. A _homotopy
coherent functor_
$F=(F^{0},F^{1},\dots)\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$
consists of the following data.

  1. (C.1a)

For any object $A\in\mathcal{A}$ an object
$F^{0}(A)\in\operatorname{ch}\mathcal{B}$;

  2. (C.1b)

For any chain

$A_{0}\xleftarrow{\varphi_{1}}A_{1}\xleftarrow{\varphi_{2}}\dots\xleftarrow{\varphi_{n}}A_{n}$

of morphisms in $\mathcal{A}$ a map
$F^{n}(\varphi_{1},\dots,\varphi_{n})\colon F^{0}(A_{n})\to F^{0}(A_{0})$ of
degree $n-1$.

$F$ is required to satisfy

$dF^{n}(\varphi_{1},\dots,\varphi_{n})=\sum_{j=1}^{n-1}(-1)^{j}\Big{(}F^{n-1}(\varphi_{1},\dots,\varphi_{j}\circ\varphi_{j+1},\dots\varphi_{n})\\\
-F^{j}(\varphi_{1},\dots,\varphi_{j})\circ
F^{n-j}(\varphi_{j+1},\dots,\varphi_{n})\Big{)}.$

Here $d$ is the differential in
$\hom_{\operatorname{ch}\mathcal{B}}(F(A_{n}),F(A_{0}))$666666We use
$d(\psi)=d_{B^{\prime}}\circ\psi-(-1)^{|\psi|}\psi\circ d_{B}$ for $\psi\colon
B\to B^{\prime}$ in $\operatorname{ch}\mathcal{B}$ of degree $|\psi|$.. The
$F^{n}$ are also be required to be multi-linear in the $\varphi_{i}$.

######  Example C.2.

Every functor $f\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$ can be
viewed as a homotopy coherent functor $F$ with $F^{0}(A)=f(A)$,
$F^{1}(\varphi)=f(\varphi)$, $F^{n}\equiv 0$ for $n\geq 2$.

######  Example C.3 (Deformation of a functor to a homotopy coherent functor).

Let $f\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$ be a
$\mathbb{Z}$-linear functor. Suppose we are given for each $A\in\mathcal{A}$ a
diagram in $\operatorname{ch}\mathcal{B}$

$B_{A}\xrightarrow{i_{A}}f(A)\xrightarrow{r_{A}}B_{A}$

with $r_{A}\circ i_{A}=\operatorname{id}_{B_{A}}$ and a chain homotopy
$H_{A}\colon i_{A}\circ r_{A}\simeq\operatorname{id}_{f(A)}$. Then we obtain a
homotopy coherent functor $F=(F^{0},F^{1},\dots)$ with $F(A)=B_{A}$ for
$A\in\mathcal{A}$ and

$F^{n}(\varphi_{1},\dots,\varphi_{n})=\\\ r_{A_{0}}\circ f(\varphi_{1})\circ
H_{A_{1}}\circ f(\varphi_{2})\circ\dots\circ f(\varphi_{n-1})\circ
H_{A_{n-1}}\circ f(\varphi_{n})\circ i_{A_{n}}\hskip 12.91663pt$

for

$A_{0}\xleftarrow{\varphi_{1}}A_{1}\xleftarrow{\varphi_{2}}\dots\xleftarrow{\varphi_{n}}A_{n}$

in $\mathcal{A}$.

######  Definition C.4.

Let $F,G\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$ be homotopy coherent
functors. A strict natural transformation $\tau\colon F\to G$ consists of
chain maps $\tau_{A}\colon F^{0}(A)\to G^{0}(A)$ for all $A\in\mathcal{A}$
such that for any chain

$A_{0}\xleftarrow{\varphi_{1}}A_{1}\xleftarrow{\varphi_{2}}\dots\xleftarrow{\varphi_{n}}A_{n}$

of morphisms in $\mathcal{A}$ the diagram

$\textstyle{F(A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{A_{n}}}$$\scriptstyle{F^{n}(\varphi_{1},\dots,\varphi_{n})}$$\textstyle{G(A_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G^{n}(\varphi_{1},\dots,\varphi_{n})}$$\textstyle{F(A_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{A_{0}}}$$\textstyle{G(A_{0})}$

commutes.

######  Example C.5.

Let $f\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$,

$B_{A}\xrightarrow{i_{A}}f(A)\xrightarrow{r_{A}}B_{A}$

and $H_{A}$ be as in Example C.3. Let
$F\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$ be the homotopy coherent
functor associated to this data.

Let $g\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$ a further functor,
which we also view as a homotopy coherent functor, see Example C.2.

Let $\tau\colon f\to g$ be a natural transformation. Suppose we are given
chain maps $p_{A}\colon B_{A}\to g(A)$ such that $\tau_{A}=p_{A}\circ r_{A}$,
$p_{A}=\tau_{A}\circ i_{A}$, $0=\tau_{A}\circ H_{A}$. Then $p_{A}$ determines
a strict natural transformation $F\to g$.

######  Definition C.6.

We write $\matheurm{Int}$ for the following category. The objects of
$\matheurm{Int}$ are linearly ordered sets of the form
$[0,n]:=\\{0<1<\cdots<n\\}$ with $n\in\mathbb{N}_{>0}$. Maps $[0,n]\to[0,m]$
are strictly order preserving maps
$\sigma\colon\\{0<\cdots<n\\}\to\\{0<\cdots<m\\}$ with $\sigma(0)=0$,
$\sigma(n)=\sigma(m)$. In particular $\sigma$ is injective and $n\leq m$.

######  Remark C.7.

There is an identification
$\Delta_{\text{inj}}^{\mathrm{op}}\cong\matheurm{Int}$. Here $\Delta$ is the
usual simplicial category of finite ordered sets of the form
$\\{0<\cdots<n\\}$ and $\Delta_{\text{inj}}$ is the subcategory obtained by
restricting to injective maps.

######  Remark C.8.

Often we will identify $[0,n]\in\matheurm{Int}$ with the category associated
to $[0,n]$ as a poset. Note that a composable chain of morphisms

$A_{0}\xleftarrow{\varphi_{1}}A_{1}\xleftarrow{\varphi_{2}}\dots\xleftarrow{\varphi_{n}}A_{n}$

in $\mathcal{A}$ is the same thing as a functor
$[0,n]^{{\mathrm{op}}}\to\mathcal{A}$.

Moreover, any finite linearly order set $I$ with at least two elements is
canonically isomorphic to some $[0,n]$. In particular, we can evaluate any
homotopy coherent functor $F\colon\mathcal{A}\to\operatorname{ch}\mathcal{B}$
on any functor $I^{{\mathrm{op}}}\to\mathcal{A}$.

######  Definition C.9.

For $[0,n]\in\operatorname{ob}\matheurm{Int}$ we define the chain complex
$C_{*}([0,n])$ as follows. As an abelian group $C_{*}([0,n])$ is generated by
pairs $(I,J)$ with $\\{0<n\\}\subseteq J\subseteq I\subseteq[0,n]$. The degree
of $(I,J)$ is the cardinality $|I\setminus J|$ of $I\setminus J$. The boundary
$d$ is determined as follows. Write $I\setminus J=\\{i_{1}<\cdots<i_{l}\\}$.
Then

$(I,J)\mapsto\sum_{j=0}^{l}(-1)^{j}\Big{(}(I\setminus\\{i_{j}\\},J)-(I,J\cup\\{i_{j}\\})\Big{)}.$

For $\sigma\colon[0,n]\to[0,m]$ we define $\sigma_{*}\colon C_{*}([0,n])\to
C_{*}([0,m])$ by

$\sigma_{*}(I,J):=\begin{cases}(\sigma(I),\sigma(J))&|I\setminus
J|=|\sigma(I)\setminus\sigma(J)|\\\ 0&\text{else}\end{cases}.$

Altogether we obtain the functor676767In the language of [22, end of Section
3] this is the covariant $\mathbb{Z}\matheurm{Int}$-chain complex
$C_{*}(E^{\operatorname{bar}}\matheurm{Int})$ for
$E^{\operatorname{bar}}\matheurm{Int}$ the bar-model for the covariant
classifying $\matheurm{Int}$-$CW$-complex of the category $\matheurm{Int}$.

$C_{*}\colon\matheurm{Int}\to\operatorname{ch}(\mathbb{Z}\text{-}\matheurm{Mod}).$

######  Remark C.10.

The chain complex $C_{*}([0,n])$ can also be described as the cellular chain
complex of the $(n\,\text{-})1$-dimensional cube.

######  Remark C.11.

In the category of functors
$\matheurm{Int}\to\mathbb{Z}\text{-}\matheurm{Mod}$ the complex $C_{*}(-)$ is
degreewise free; we have

$C_{k}(-)\cong\bigoplus_{i=1}^{\infty}\bigoplus_{\tiny\begin{array}[]{c}\\{0<i\\}\subseteq
J\subseteq[0,i],\\\ |[0,i]\setminus
J|=k\end{array}}\mathbb{Z}\matheurm{Int}(J,-).$

######  Remark C.12.

There is a concatenation map

(C.13) $C_{*}([0,n])\otimes C_{*}([0,m])\to C_{*}([0,n+m])$

defined by $(I,J)\otimes(I^{\prime},J^{\prime})\mapsto(I\cup
n+I^{\prime},J\cup n+J^{\prime})$.

######  Lemma C.14.

The augmentation map $\epsilon\colon C_{*}([0,n])\to\mathbb{Z}$ with
$\epsilon(I,I)=1$ is a homology isomorphism.

###### Proof.

We proceed by induction on $n$. For $n=1$ the augmentation is an isomorphism.
For $n>1$ we define $\tilde{C}_{*}([0,n])$ as the sub-complex of
$C_{*}([0,n])$ spanned by all $(I,J)$ with $1\notin I$. Clearly
$\tilde{C}_{*}([0,n])\cong C_{*}([0,{n-1}])$. The map defined by

$(I,J)\mapsto\begin{cases}(I,J\cup\\{1\\})&1\in I\setminus J\\\
0&\text{else}\end{cases}$

induces a contraction on the quotient $C_{*}([0,n])/\tilde{C}_{*}([0,n])$.
Thus the inclusion $\tilde{C}_{*}([0,n])\to C_{*}([0,{n}])$ induces an
isomorphism in homology. ∎

In the following we abbreviate
$\mathcal{A}(A,A^{\prime}):=\operatorname{mor}_{\mathcal{A}}(A,A^{\prime})$.

######  Definition C.15.

Let $\mathcal{A}$ be a small $\mathbb{Z}$-linear category. For
$[0,n]\in\matheurm{Int}$ and $A,A^{\prime}\in\operatorname{ob}\mathcal{A}$ we
define

$\displaystyle\mathcal{A}_{[0,n]}(A,A^{\prime}):=\bigoplus_{A_{1},\dots,A_{n-1}\in\mathcal{A}}\mathcal{A}(A_{1},A^{\prime})\otimes\mathcal{A}$
$\displaystyle(A_{2},A_{1})\otimes\dots\otimes\mathcal{A}(A,A_{n-1}).$

There is an evident concatenation map

(C.16)
$\mathcal{A}_{[0,n]}(A^{\prime},A^{\prime\prime})\otimes\mathcal{A}_{[0,m]}(A,A^{\prime})\to\mathcal{A}_{[0,n+m]}(A,A^{\prime\prime}).$

Using composition in $\mathcal{A}$, this construction is functorial in
$\matheurm{Int}$ and we obtain for fixed $A,A^{\prime}\in\mathcal{A}$ a
contravariant functor

$\mathcal{A}_{\\--}(A,A^{\prime})\colon\matheurm{Int}\to\mathbb{Z}\text{-}\matheurm{Mod};\quad[0,n]\mapsto\mathcal{A}_{[0,n]}(A,A^{\prime}).$

######  Definition C.17.

Let $\mathcal{A}$ be a small $\mathbb{Z}$-linear category. The
$\operatorname{dg}$-category $\mathfrak{C}_{\mathcal{A}}$ is defined as