$W^{\prime}\cap\Gamma^{N+M}$ and $\gamma:=x_{c}(x)$. We show that $x\in
W^{\prime}$. This is a consequence of the fact that
$W^{\prime}\cap[x_{c}\leq\gamma]$ is $\sigma$-compact with respect to $x_{hh}$
and $x_{v}$. ∎

######  Lemma 5.7.

Let

$W^{\prime}_{1}:=(W^{\prime}\cap\Gamma^{N+M})\bigcup[x_{h}=\infty].$

There exists a $K$-definable deformation retraction

$H^{\prime\mathrm{trop}}_{\Gamma f}\colon[0,\infty]\times W^{\prime}_{1}\to
W^{\prime}_{1}$

which satisfies the following properties.

  1. (1)

The deformation $H^{\prime\mathrm{trop}}_{\Gamma f}$ leaves the functions
$\xi_{i}$ invariant, is $G$-equivariant and preserves the fibres of the
morphism $f^{\prime}$.

  2. (2)

Let $W^{\prime}_{0}$ denote the image of the deformation retraction
$H^{\prime\mathrm{trop}}_{\Gamma f}$. There exists a $K$-definable open subset
$W^{\prime}_{o}$ of $W^{\prime}$ that contains
$W^{\prime}_{0}\smallsetminus[x_{h}=\infty]$ and $m\in\mathbb{N}$ and
$c\in\Gamma(K)$ such that $x_{i}\leq(m+1)x_{h}+c$ for every $1\leq i\leq N$.

  3. (3)

For every $x\in W$, if $W^{\prime}_{x}$ is pure of dimension $n$ then for
every $x$, $W^{\prime}_{0x}$ is pure of dimension $n$.

###### Proof.

We adapt the proof of [7, 11.5.1] to our setting. Before doing so, we give a
rough outline of this proof. The specific details can be found in loc.cit.
This is the case when $\Upsilon$ is a point and $M=0$. In this situation, we
choose a $G$-invariant $K$-definable cell decomposition $\mathcal{D}$ of
$\Gamma^{N}$ that respects the definable set $W^{\prime}\cap\Gamma^{N}$ as
well as all sets of the form $[x_{a}=x_{b}]$ or $[x_{a}=0]$ for all
coordinates $x_{a},x_{b}$. We define the restriction of the deformation
retraction $H^{\prime\mathrm{trop}}_{\Gamma f}$ to $\Gamma^{N}$ by specifying
its behaviour on each cell of $\mathcal{D}$. Let $\mathcal{D}_{0}$ denote the
sub collection of cells $C\in\mathcal{D}$ such that every coordinate $x_{i}$
is _$h$ -bounded on $C$_. By this we mean that there exists $m\in\mathbb{N}$,
$c\in\Gamma(K)$ and $x_{i}(z)\leq mx_{h}(z)+c$ for every $z\in C$.
Furthermore, when every coordinate is $h$-bounded on $C$, we shall say that
the cell itself is $h$-bounded.

The homotopy $H^{\prime\mathrm{trop}}_{\Gamma f}$ will fix every point in
every element of $\mathcal{D}_{0}$. Let $C\notin\mathcal{D}_{0}$. We construct
an element $e_{C}\in\mathbb{Q}_{+}^{N}$ such that if a coordinate $x_{i}$ is
$h$-bounded on $C$ then $x_{i}(e_{C})=0$. For $x\in C$ and $t\in[0,\infty)$,
let $H^{\prime\mathrm{trop}}_{\Gamma f}(t,x):=x-te_{C}$. We verify that the
definable function $H_{\Gamma}$ is continuous on $[0,\infty]\times C$ by
induction on the dimension of the cell $C$. An important observation which
makes this possible is the following. Since $x_{i}(te_{C})\geq 0$ for
$t\in[0,\infty)$ and any coordinate $x_{i}$, we must have that $x-te_{C}\notin
C$ for some $t$. Let $\tau(x)$ be the smallest such $t$. Observe that
$H^{\prime\mathrm{trop}}_{\Gamma f}(\tau(x),x)$ must lie in a lower
dimensional cell. It follows that the path $t\mapsto
H^{\prime\mathrm{trop}}_{\Gamma f}(t,x)$ begins at $x$ when $t=0$ and
traverses finitely many cells in decreasing dimensions till it finally ends up
in $\mathcal{D}_{0}$.

The proof in the relative case follows the same steps as in the sketch above.
We highlight only those points which require more than just the obvious
adaptation.

Step 1. _Preliminaries._

As in loc.cit., let $\mathbf{A}$ denote the convex subgroup of
$\Gamma(\mathbb{U})$ which is generated by $\Gamma(K)$ and define
$B:=\Gamma(\mathbb{U})/\mathbf{A}$. Given a definable subset
$X\subset\Gamma^{t}$ for some $t\in\mathbb{N}$, we define $\beta X$ to be the
image of $X$ in $B^{t}$. We choose a cell decomposition $\mathcal{D}$ of
$\Gamma^{N+M}$ which respects the definable set $W^{\prime}\cap\Gamma^{N+M}$,
all sets of the form $[x_{a}=x_{b}]$ and $[x_{a}=0]$ for all coordinates
$x_{a},x_{b}$ and is such that its push forward $f^{\prime}(\mathcal{D})$ is a
cell decomposition of $\Gamma^{M}$ which respects $W\cap\Gamma^{M}$. By
definition, for every $C^{\prime}\in\mathcal{D}$, its image in $\Gamma^{M}$
for the projection $f^{\prime}$ is a cell $C$. By [12, Proposition 3.3.5], if
$a\in C$ then the fibre $C^{\prime}_{a}$ is a cell in
$\Gamma^{N}\times\\{a\\}$. We can suppose that the decomposition $\mathcal{D}$
is such that for every $C\in\mathcal{D}$, $C$ is $h$-bounded iff for every
$a\in f^{\prime}(C)$, $C_{a}$ is $h$-bounded which in turn will be equivalent
to saying that there exists $a\in f^{\prime}(C)$ such that $C_{a}$ is
$h$-bounded. Indeed, we modify the existing decomposition $\mathcal{D}$ so
that it has the above property as follows. Let $C^{\prime}\in\mathcal{D}$ and
$C:=f^{\prime}(C^{\prime})$. The set $C_{h-bdd}$ of those points $x\in C$ such
that $C^{\prime}_{x}$ is $h$-bounded is definable. This is a consequence of
the cell decomposition. We can now refine the decomposition $\mathcal{D}$ so
that it respects the preimages of the definable sets $C_{h-bdd}$ and
$C\smallsetminus C_{h-bdd}$ for every $C$.

For a cell $C^{\prime}\in\mathcal{D}$, we now define the point
$e_{C^{\prime}}$. Let $\mathfrak{M}$ denote the last $M$ coordinates of
$\Gamma^{N+M}$ and $\mathfrak{N}$ denote the first $N$ coordinates. Let

$\beta^{\prime}C^{\prime}:=\beta
C^{\prime}\bigcap[x_{h}=0]\bigcap[x_{i}=0]_{i\in\mathfrak{M}}$

and let $e_{C^{\prime}}$ denote the barycentre of
$\beta^{\prime}C^{\prime}\cap[\sum_{i}x_{i}=1]$. By construction, if
$x_{i}\in\mathfrak{M}$ or if $x_{i}$ is $h$-bounded on $C^{\prime}$ then
$x_{i}(e_{C^{\prime}})=0$. It follows that if $C^{\prime}\in\mathcal{D}_{0}$
then $e_{C^{\prime}}=0$ Also, if $x_{i}$ is not $h$-bounded on $C^{\prime}$
then $x_{i}(e_{C^{\prime}})>0$.

For $x\in C^{\prime}$, we define $H^{\prime\mathrm{trop}}_{\Gamma
f}(x):=x-te_{C^{\prime}}$. Clearly, $H^{\prime\mathrm{trop}}_{\Gamma f}$ is
well defined. Furthermore, if $C=f^{\prime}(C^{\prime})$ then by construction,
we see that for every $x\in C$, the homotopy $H^{\prime\mathrm{trop}}_{\Gamma
f}$ restricts to a well defined homotopy along the fibre over $x$.

Step 2. _Continuity and end of the proof._

The continuity of $H^{\prime\mathrm{trop}}_{\Gamma f}$ and assertion (2) of
the Lemma can be shown by following the arguments in [7, Lemma 11.5.1] with
little change. Note that the homotopy $H^{\prime\mathrm{trop}}_{\Gamma f}$
preserves the closures of the cells. By Lemma 5.6,
$H^{\prime\mathrm{trop}}_{\Gamma f}$ preserves $W^{\prime}\cap\Gamma^{N+M}$.
Assertion (3) can be verified using the argument in loc.cit., and considering
the hyperplane $L\colon\sum_{i\in\mathfrak{N}}x_{i}=Mx_{h}+K$ where $M=N(m+1)$
and $K=Nc$.

∎

It remains to treat the case when for some $x\in W$,
$W^{\prime}_{x}\cap(\Gamma^{N}\times\\{x\\})$ is empty. We attempt a relative
version of [7, Lemma 11.5.2].

Let $\eta$ denote the generic point of $U$ and $V^{\prime}_{1\eta}$ be the
generic fibre of the morphism $\phi^{\prime}_{U}$. Recall that we assumed that
every generic point of every irreducible component of $V^{\prime}_{1U}$ is
contained in the fibre over $\eta$. Let
$V^{\prime}_{10\eta},\ldots,V^{\prime}_{1r\eta}$ denote the irreducible
components of $V^{\prime}_{1\eta}$ and for every $j$, we set $V^{\prime}_{1j}$
to be the Zariski closure of $V^{\prime}_{1j\eta}$ in $V^{\prime}_{1U}$. By
construction, we have that the $V^{\prime}_{1j\eta}$ are equidimensional and
there exists a Zariski open subset $U_{0}\subseteq U$ such that the maps
$V^{\prime}_{1j}\to U_{0}$ are flat. By [4, Corollary 9.6], we get that the
fibres of $V^{\prime}_{1j}$ are pure of dimension
$\mathrm{dim}(V^{\prime}_{1j\eta})$. Recall by construction that
$D^{\prime}_{1U}$ contains the intersections $V^{\prime}_{1j}\cap
V^{\prime}_{1j^{\prime}}$ for every $j,j^{\prime}$ such that $j\neq
j^{\prime}$. For every $j$, let $W^{\prime}_{j}$ be the image of
$\Upsilon^{\prime}_{bcf}\cap[\widehat{V^{\prime}_{1j}}\smallsetminus\bigcup_{j^{\prime}\neq
j}\widehat{V^{\prime}_{1j^{\prime}}}]$ for the map $\alpha^{\prime}$. Observe
that $W^{\prime}_{j}$ is open and

$W^{\prime}=\bigsqcup_{j}W^{\prime}_{j}\bigsqcup([x_{h}=\infty]\cap
W^{\prime}).$

For every $x\in W$, $W^{\prime}_{jx}$ is pure of dimension
$\mathrm{dim}(V^{\prime}_{1jx})$.

Let $x_{1},\ldots,x_{N}$ denote the first $N$ coordinates of
$\Gamma_{\infty}^{N+M}$. Recall that the preimage of the locus
$[x_{i}=\infty]$ via the function $x_{i}\circ\alpha^{\prime}$ is of the form
$\widehat{Z_{i}}$ where $Z_{i}$ is a Zariski closed subset of
$V^{\prime}_{1U}$.

Let $j\in\\{1,\ldots,r\\}$. Let $\mathfrak{P}_{j}\subset\\{1,\ldots,N\\}$ be
the subset of indices such that for $p\in\mathfrak{P}_{j}$, $Z_{p}\cap
V^{\prime}_{1j\eta}$ is of dimension strictly less than
$\mathrm{dim}(V^{\prime}_{1j\eta})$. We shrink $U_{0}$ further to get that for
every such $p\in\\{1,\ldots,N\\}$, $Z_{p}\cap V^{\prime}_{1j}$ is flat over
$U_{0}$ or empty. It follows that for every $x\in U_{0}$ and
$p\in\mathfrak{P}_{j}$, $Z_{p}\cap V^{\prime}_{1jx}$ is of dimension strictly
less than $\mathrm{dim}(V^{\prime}_{1j\eta})$.

Since for every $x\in W$, $W^{\prime}_{jx}$ is pure of dimension
$\mathrm{dim}(V^{\prime}_{1jx})$, we deduce that for every
$p\in\mathfrak{P}_{j}$, $x_{p}$ is not identically $\infty$ on
$W^{\prime}_{jx}$. Indeed, our choice of $p$ implies that
$(V^{\prime}_{1j}\smallsetminus(\bigcup_{j\neq
j^{\prime}}V^{\prime}_{1j^{\prime}}\bigcup Z_{p}))\bigcap V^{\prime}_{1U_{0}}$
is a non-empty Zariski open subset of $V^{\prime}_{1U}$. By construction, if
$A:=(V^{\prime}_{1j}\smallsetminus(\bigcup_{j\neq
j^{\prime}}V^{\prime}_{1j^{\prime}}\bigcup Z_{p}))\bigcap V^{\prime}_{1U_{0}}$
then $\phi^{\prime}_{U}(A)=U_{0}$ and hence
$\widehat{\phi^{\prime}_{U}}(\widehat{A})=\widehat{U_{0}}$. Note that if
$x^{\prime}\in\Upsilon_{b}$ is such that $x^{\prime}\mapsto x$ then the
homotopies $H^{\prime}_{bf}$ and $H^{\prime}_{curvesf}$ restrict to well
defined homotopies on $\widehat{\phi^{\prime}_{U}}^{-1}(x^{\prime})$. As
$H_{b}$ satisfies property (5) of Theorem 1.1, we see that
$x^{\prime}\in\widehat{U_{0}}$. Since $H^{\prime}_{bf}$ and
$H^{\prime}_{curvesf}$ are also Zariski generalizing, we must have that
$\Upsilon_{bcf}\cap(\widehat{\phi^{\prime}_{U}})^{-1}(x^{\prime})$ intersects
$V^{\prime}_{1j}\smallsetminus(\bigcup_{j\neq
j^{\prime}}V^{\prime}_{1j^{\prime}}\bigcup Z_{p})$ non-trivially. It follows
that $x_{p}$ is not identically $\infty$ on $W^{\prime}_{jx}$.

Let $n_{j}$ be the cardinality of the set $\mathfrak{P}_{j}$. Consider the
embedding $\alpha^{\prime}_{j}\colon
W^{\prime}_{j}\to\Gamma_{\infty}^{n_{j}+M}$ given by
$z\mapsto((x_{i}(z))_{i\in\mathfrak{P}_{j}},(x_{i}(z))_{i\in\mathfrak{M}})$.
Note that ${\alpha^{\prime}_{j}}$ is a homeomorphism onto its image. Let
$W^{\prime\circ}_{j}$ be the preimage of the open set
$\alpha^{\prime}_{j}(W^{\prime}_{j})\cap\Gamma^{n_{j}+M}$. Observe that
$W^{\prime\circ}_{j}$ is $z$-open and $z$-dense in $W^{\prime}_{j}$. Let
$H^{\prime\mathrm{trop}}_{\Gamma j}$ be the deformation retraction induced on
$W^{\prime\circ}_{j}$ from Lemma 5.7. Let
$W^{\circ}:=\bigcup_{j}W^{\prime\circ}_{j}$ and
$W^{\prime\prime}:=(W^{\circ}\smallsetminus[x_{v}=\infty])\cup[x_{h}=\infty]$.

Let

$H^{\prime\mathrm{trop}}_{\Gamma f}\colon[0,\infty]\times W^{\prime\prime}\to
W^{\prime\prime}$

denote the deformation retraction whose restriction to $W^{\prime\circ}_{j}$
for every $j$ is induced by $H^{\prime\mathrm{trop}}_{\Gamma j}$. Let
$W^{\prime}_{0}$ denote the image of $H^{\prime\mathrm{trop}}_{\Gamma f}$.
Observe that $H^{\prime\mathrm{trop}}_{\Gamma f}$ satisfies the following
properties.

  1. (1)

The deformation $H^{\prime\mathrm{trop}}_{\Gamma f}$ leaves the $\xi_{i}$
invariant, is $G$-equivariant and preserves the fibres of the morphism
$f^{\prime}$.

  2. (2)

There exists a $K$-definable open subset $W^{\prime}_{o}$ of $W^{\prime}$ that
contains $W^{\prime}_{0}\smallsetminus[x_{h}=\infty]$, $m\in\mathbb{N}$ and
$c\in\Gamma(K)$ such that on $W^{\prime}_{o}\cap W^{\prime}_{j}$,
$x_{i}\leq(m+1)x_{h}+c$ for every $i\in\mathfrak{P}_{j}$.

###  5.5. Completing the proof of Theorem 1.1

In order to complete the proof of Theorem 1.1, we choose inflation homotopies
$H^{\prime}_{inff}$ and $H^{\prime}_{inff-primary}$ using Lemma 4.8 such that
the image of the composition $H^{\prime}_{bf}\circ H^{\prime}_{curvesf}\circ
H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$ is contained in a subspace
$P$ of $\Upsilon^{\prime}_{bcf}$ such that $\alpha^{\prime}_{|P}$ is a
homeomorphism. We can then define a homotopy $H^{\prime}_{\Gamma f}$ on $P$
via the tropical homotopy $H^{\prime\mathrm{trop}}_{\Gamma f}$ so that the
composition

$H^{\prime}_{\Gamma f}\circ H^{\prime}_{bf}\circ H^{\prime}_{curvesf}\circ
H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$

is well defined and fixes its image. Here we abuse notation and write
$H^{\prime}_{\Gamma f}$ for the homotopy on $P$.

For every $i\in\mathfrak{N}$, let
$y_{i}:=\mathrm{min}\\{x_{i},(m+1)x_{h}+c\\}$. Observe that for every $i$,
$(y_{i}\circ\alpha^{\prime})^{-1}(\infty)$ is contained in
$\widehat{D^{\prime}_{1U}}$ and hence also in $\widehat{D^{\prime}_{11U}}$.
When there is no ambiguity, we simplify notation and write $y_{i}$ in place of
the composition $y_{i}\circ\alpha^{\prime}$.

We now construct the homotopy $H^{\prime}_{inff}$ using Lemma 4.8. We deduce
from (1) of §5.1 that for some copy of $\mathbb{A}^{m}\subset E$, the morphism
$g^{\prime}_{U}$ restricts to a map $V^{\prime}_{1U}\smallsetminus
D^{\prime}_{11U}\to\mathbb{A}^{m}\times U$. By our choice of
$D^{\prime}_{11U}$ in §5.1 and Lemma 4.5, this map satisfies requirement (2)
of Lemma 4.8. Let

$H^{\prime}_{inff}\colon[0,\infty]\times\widehat{V^{\prime}_{1U}}\to\widehat{V^{\prime}_{1U}}$

be the deformation retraction constructed in Lemma 4.8 with respect to the
morphisms $\phi^{\prime}_{U}$ and
${g^{\prime}_{U}}_{|V^{\prime}_{1U}\smallsetminus D^{\prime}_{11U}}$, the
divisor $D^{\prime}_{11U}$, the family
$\\{\xi_{i}\\}_{i}\cup\\{y_{j}\circ\alpha^{\prime}\\}_{j}$ and the group $G$.
Note that $H^{\prime}_{inff}$ is compatible with the morphism
$\phi^{\prime}_{U}\colon V^{\prime}_{1U}\to U$. Furthermore,
$H^{\prime}_{bf}\circ H^{\prime}_{curvesf}\circ H^{\prime}_{inff}$ is a well
defined homotopy which respects the levels of the functions $\xi_{i}$ and is
equivariant for the action of the group $G$. Note that at this stage we cannot
say that the image of $H^{\prime}_{bf}\circ H^{\prime}_{curvesf}\circ
H^{\prime}_{inff}$ will be contained in $W^{\prime\prime}$ \- the domain of
the homotopy $H^{\prime}_{\Gamma f}$.

Let $Z^{\prime}_{U}\subset V^{\prime}_{U}$ be as introduced in §5.1. Recall
that by definition, $\rho_{0}\colon V^{\prime}_{U}\to\Gamma_{\infty}$ was
defined to be the schematic distance to $D^{\prime}_{U}$ and
$x_{h}\circ\alpha^{\prime}=\rho_{0}\circ b_{U}$. By construction
$V^{\prime}_{U}\smallsetminus Z^{\prime}_{U}\subset
V^{\prime}_{1U}\smallsetminus Z^{\prime}_{1U}$. Let $v_{i}\colon
V^{\prime}_{U}\to\Gamma_{\infty}$ be the coordinates of the map
$\alpha_{0}\colon\widehat{V^{\prime}_{U}}\to\Gamma_{\infty}^{N_{1}}$ i.e.
$v_{i}$ is the composition of $\alpha_{0}$ and the $i$-th projection
$\Gamma_{\infty}^{N_{1}}\to\Gamma_{\infty}$. For every $i$, let
$w_{i}:=\mathrm{min}\\{v_{i},(m+1)\rho_{0}+c\\}$. Observe that
$\rho_{0}^{-1}(\infty)=D^{\prime}_{U}$. Let $H^{\prime}_{inff-primary}$ be the
homotopy as provided by Lemma 4.8 for the morphism $V^{\prime}_{U}\to U$, the
morphism $g_{U}\colon V^{\prime}_{U}\to\mathbb{P}^{m}\times U$, the divisor
$D^{\prime}_{U}$, the functions $\\{w_{j}\\}_{j}\cup\\{\xi_{i}\\}_{i}$ and the
group $G$.

Observe that the composition $H^{\prime}_{bf}\circ H^{\prime}_{curvesf}\circ
H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$ is well defined. Let
$\beta\colon\widehat{V^{\prime}_{U}}\to\Upsilon^{\prime}$ be the retraction
associated to this composition and $B:=\beta(\widehat{V^{\prime}_{U}})$.

######  Lemma 5.8.

The morphism $\alpha^{\prime}\colon\Upsilon^{\prime}_{bcf}\to W^{\prime}$
restricts to a homeomorphism from $\beta(\widehat{V^{\prime}_{U}})$ onto its
image.

###### Proof.

The morphism $\alpha^{\prime}$ is injective and continuous. It suffices to
show that the restriction $\alpha^{\prime}_{|B}$ is a closed map from $B$ to
$\alpha^{\prime}(B)$. Note that $\alpha^{\prime}(B)$ is an iso-definable
subset of $\Upsilon^{\prime}_{bcf}$. This is a consequence of the fact that
$\beta(V^{\prime}_{U})\subset\Upsilon^{\prime}_{bcf}$ is iso-definable and
$\Gamma$-internal and hence
$\beta(V^{\prime}_{U})=\beta(\widehat{V^{\prime}_{U}})$. Let $Z\subset B$ be
an iso-definable closed subset. We claim that $\alpha^{\prime}(Z)$ is closed
in $\alpha^{\prime}(B)$. Let $a$ belong to the closure of $\alpha^{\prime}(Z)$
in $\alpha^{\prime}(B)$. By [7, Proposition 4.2.13], there exists a definable
type $q$ that concentrates on $\alpha^{\prime}(Z)$ with limit point $a$. Let
$a^{\prime}\in B$ be the preimage of $a$ and $q^{\prime}$ be the preimage of
the definable type $q$. We show that $a^{\prime}$ is a limit point of
$q^{\prime}$. Recall the coordinate $x_{c}$ on $\Gamma_{\infty}^{M}$ which
corresponds to the schematic distance $d_{bord}\colon V\to\Gamma_{\infty}$ to
$V\smallsetminus U$. Let $\lambda\in\Gamma$ be such that $x_{c}\circ
f^{\prime}(a)<\lambda$ where $f^{\prime}$ is the projection
$\Gamma_{\infty}^{N+M}\to\Gamma_{\infty}^{M}$. Note that if
$U_{\lambda}:=\\{x\in V|d_{bord}(x)\leq\lambda\\}$ and
$V^{\prime}_{1U_{\lambda}}:=({\phi^{\prime}_{U}})^{-1}(U_{\lambda})$ then
$\widehat{V^{\prime}_{1U_{\lambda}}}$ is definably compact. It follows that
$\beta(\widehat{V^{\prime}_{1U_{\lambda}}})\subset B$ is definably compact and
hence $\alpha^{\prime}$ restricts to a homeomorphism from
$\beta(\widehat{V^{\prime}_{1U_{\lambda}}})$ onto its image. Since $H_{b}$
preserves $d_{bord}$, we deduce that
$B_{\lambda}:=B\cap\widehat{V^{\prime}_{1U_{\lambda}}}=\beta(\widehat{V^{\prime}_{1U_{\lambda}}})$.
By construction, we have that $f^{\prime}\circ\alpha^{\prime}=\alpha\circ f$
where $f=\widehat{\phi^{\prime}_{U}}_{|\Upsilon_{bcf}}$. Hence,
$\alpha^{\prime}$ restricts to a homeomorphism from $B_{\lambda}$ onto
$[x_{c}\leq\lambda]\cap\alpha^{\prime}(B)$. Note that $B_{\lambda}$ and
$[x_{c}\leq\lambda]\cap\alpha^{\prime}(B)$ are closed neighbourhoods of
$a^{\prime}$ and $a$ respectively. It follows that $a^{\prime}$ is a limit
point of $q^{\prime}$. Since $Z$ is closed, $a^{\prime}\in Z$. It follows that
$a\in\alpha^{\prime}(Z)$. This concludes the proof.

∎

We define $H^{\prime}_{\Gamma f}$ as follows. Observe firstly that
$\alpha^{\prime}(B)\subset W^{\prime\prime}$. This is a consequence of the
inflation properties of $H^{\prime}_{inff}$ and $H^{\prime}_{inff-primary}$.
Firstly, note that the image of $H^{\prime}_{inff-primary}$ is contained in
$\widehat{V^{\prime}_{1U}}\smallsetminus\widehat{Z^{\prime}_{1U}}$ which in
turn implies that the image of $H_{inff}\circ H^{\prime}_{inff-primary}$ is
contained $\widehat{V^{\prime}_{1U}}\smallsetminus\widehat{Z^{\prime}_{1U}}$.
Furthermore, for any $u\in U$, if $Z\subset V^{\prime}_{u}$ is a closed sub
variety of dimension strictly smaller than $\mathrm{dim}(V^{\prime}_{u})$ then
the intersection of $\widehat{Z}$ with the image of $H^{\prime}_{inff}\circ
H^{\prime}_{inff-primary}$ will be contained in $\widehat{D^{\prime}_{1u}}$.
Let $x\in\beta(\widehat{V^{\prime}_{U}})$ and $t\in[0,\infty]$. We set
$H^{\prime}_{\Gamma
f}(t,x):={\alpha^{\prime}}^{-1}(H^{\prime\mathrm{trop}}(t,\alpha^{\prime}(x))$.
By Lemma 5.8, $H^{\prime}_{\Gamma f}$ is a well defined homotopy. We have thus
shown that $H^{\prime}_{\Gamma f}\circ H^{\prime}_{bf}\circ
H^{\prime}_{curvesf}\circ H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$ is
a well defined homotopy.

We now show that $H^{\prime}_{\Gamma f}\circ H^{\prime}_{bf}\circ
H^{\prime}_{curvesf}\circ H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$
fixes its image. From the construction, we see that it suffices to verify that
$H^{\prime}_{inff}$ and $H^{\prime}_{inff-primary}$ fix this image. Recall
that we used $W^{\prime}_{0}$ to denote the image of
$H^{\prime\mathrm{trop}}_{\Gamma f}$. Let $\Upsilon^{\prime}_{0}$ denote the
image of $H^{\prime}_{\Gamma f}$. Let $w\in\Upsilon_{b}\subset\widehat{U}$. We
check that $H^{\prime}_{inff}$ and $H^{\prime}_{inff-primary}$ both fix
$W^{\prime}_{0w}$. To do so we borrow the notation from the previous section.
Recall that we have a decomposition

$W^{\prime}=\bigsqcup_{j}W^{\prime}_{j}\bigsqcup([x_{h}=\infty]\cap
W^{\prime}).$

On $W^{\prime}_{j}\cap W^{\prime}_{0}$, for every $i\in\mathfrak{P}_{j}$, we
must have that $y_{i}=x_{i}$. Furthermore, if $i\notin\mathfrak{P}_{j}$ then
$x_{i}$ is identically $\infty$ on $V^{\prime}_{1j}$. Since
$H^{\prime}_{inff}$ respects the levels of the functions $y_{i}$ for all $i$,
by [7, Lemma 8.3.1(2)], it fixes $W^{\prime}_{j}\cap W^{\prime}_{0}$.

We now show that $H^{\prime}_{inff-primary}$ fixes the image of the
composition $H^{\prime}_{\Gamma f}\circ H^{\prime}_{bf}\circ
H^{\prime}_{curvesf}\circ H^{\prime}_{inff}\circ H^{\prime}_{inff-primary}$.
The argument is identical to the one made above but makes use of the fact that
the image of the composition must lie in the complement of $Z^{\prime}_{1U}$
and is hence controlled completely by the coordinates $v_{i}$.

One observes that each of the homotopies in the composition respects the
levels of the functions $\xi_{i}$ and the action of the group $G$.
Furthermore, the induction hypothesis that gives rise to $H^{\prime}_{bf}$ and
(3) of Lemma 5.7 guarantee that for every $z\in\Upsilon_{b}$, the fibre over
$z$ is pure of dimension $n$ where $n$ is the dimension of the generic fibre
for the map $V^{\prime}_{1U}\to U$. We now show how to ensure that the
composition is Zariski generalizing. The argument is similar to the one that
appears in [7, §11.6]. By Lemma 5.8, $\alpha^{\prime}$ restricts to a
homeomorphism from $\beta(\widehat{V^{\prime}_{U}})$ onto its image. Hence, by
[7, Corollary 10.4.6], it suffices to verify that
$H^{\prime\mathrm{trop}}_{\Gamma}$ is Zariski generalizing. This can be done
as in §11.6 of loc.cit. Lastly, our construction of $H_{b}$ ensures that it
satisfies properties (5) and (6) of Theorem 1.1. This concludes the proof.

∎

##  6\. When the base is a curve

Theorem 1.1 asserts the existence of compatible deformations generically over
the base. In the proof of this theorem, at several stages, we shrunk the base
so as to obtain that the family behaved in a tame manner. When the base is a
curve, we do not need to shrink the base constantly to obtain tame properties
of the family. This allows us to prove the following theorem.

######  Theorem 6.1.

Let $S$ be a smooth connected $K$-curve and $X$ be a quasi-projective
$K$-variety. Let $\phi\colon X\to S$ be a surjective morphism such that every
irreducible component of $X$ dominates $S$. Let $\\{\xi_{i}\colon
X\to\Gamma_{\infty}\\}$ be a finite collection of $K$-definable functions.
Recall that the functions $\xi_{i}$ extend to functions
$\xi_{i}\colon\widehat{X}\to\Gamma_{\infty}$. Let $G$ be a finite algebraic
group acting on $X$ such that the action of $G$ respects the fibres of the
morphism $\phi$. Let $s\in S(K)$. There exists a Zariski open subset $U\subset
S$ containing $s$ and compatible homotopies $(H,\Upsilon)$ of $\widehat{U}$
and $(H^{\prime},\Upsilon^{\prime})$ of $\widehat{X_{U}}$
333$X_{U}:=X\times_{S}U$ such that the following hold.

  1. (1)

The homotopy $H$ is in fact a deformation retraction.

  2. (2)

The images $\Upsilon\subset\widehat{U}$ and
$\Upsilon^{\prime}\subset\widehat{X_{U}}$ are $\Gamma$-internal.

  3. (3)

The homotopy $H^{\prime}$ respects the functions $\xi_{i}$ i.e.
$\xi_{i}(H^{\prime}(t,p))=\xi_{i}(p)$ for every $p\in\widehat{X}$ and $t\in
I$.

  4. (4)

The action of the group $G$ on $X$ extends to an action on $\widehat{X}$. The
deformation $H^{\prime}$ can be taken to be $G$-invariant i.e. for every $g\in
G^{\prime}$, $H^{\prime}(t,g(p))=g(H^{\prime}(t,p))$.

  5. (5)

The homotopy $H^{\prime}$ is Zariski generalizing i.e. if $W\subset X_{U}$ is
a Zariski open subset then $H^{\prime}$ restricts to a well defined homotopy
on $\widehat{W}$.

###  6.1. Initial reductions

######  Remark 6.2.

  1. (1)

_By arguments similar to those that appear in the proof of Lemma 4.2 and using
Lemma 4.1, one shows that if Theorem 6.1 is true for $G$-equivariant morphisms
444The action of $G$ on $S$ is assumed to be trivial. $\phi\colon X\to S$
between projective $K$-varieties whose fibres are pure and where $S$ is a
smooth projective connected $K$-curve then the theorem is true in general.
Henceforth, unless otherwise stated, we will assume $\phi\colon X\to S$ is a
flat morphism between projective $K$-varieties such that the fibres of $\phi$
are pure and $S$ is a smooth, connected curve. _

  2. (2)

_The functions $\xi_{i}\colon V^{\prime}\to\Gamma_{\infty}$ can be taken to be
$v+g$-continuous [7, §11.2]. It follows by Lemma 10.4.3 in loc.cit. that for
every $i$, $\xi_{i}^{-1}(\infty)$ is a sub-variety of $V^{\prime}$. For every
$i$, let $Z_{i}:=\xi_{i}^{-1}(\infty)$. Since $S$ is a curve, we can realize
$Z_{i}$ as the union of a horizontal divisor $Z_{ih}$ and a vertical divisor
$Z_{iv}$. Indeed, $Z_{ih}$ is the closure of those irreducible components of
$Z_{i}$ that dominate $S$ while $Z_{iv}$ is the union of those irreducible
components that map to a $K$-point in $S$._

###  6.2. The inflation homotopy

######  Remark 6.3.

_Let $s\in S(K)$ be as in the statement of Theorem 6.1. We apply the first
part of Lemma 4.3 to obtain a Zariski open affine neighbourhood $U$ of $s$
such that the morphism $\phi$ factors through a finite $G$-equivariant
morphism $f\colon X_{U}\to\mathbb{P}^{m}\times U$ where $G$ acts trivially on
$\mathbb{P}^{m}\times U$ and $m=\mathrm{dim}(X_{s})$. Let
$T\subset\mathbb{P}^{m}\times U$ be as given by Lemma 4.6. Let
$H:=\mathbb{P}^{m}\smallsetminus\mathbb{A}^{m}$. We enlarge $T$ so that it
contains $H\times U$. Let $D:=f^{-1}(T)$. We enlarge $T$ so that $D$ contains
the closed sub-varieties $Z_{ih}$ introduced above and remains $G$-invariant._

_We apply the second part of Lemma 4.3 to obtain a variety $X^{\prime}_{U}$
that fits into a commutative diagram_

${X^{\prime}_{U}}$${E\times U}$${\mathbb{P}^{m-1}\times
U}$${X_{U}}$${\mathbb{P}^{m}\times
U}$${U}$$\scriptstyle{f^{\prime}}$$\scriptstyle{b^{\prime}}$$\scriptstyle{b\times
id}$$\scriptstyle{p}$$\scriptstyle{\phi}$$\scriptstyle{f}$

_such that $b\colon E\to\mathbb{P}^{m}$ is the blow up at a $K$-point $z$, the
restriction of $(p\circ f^{\prime})$ to $b^{\prime-1}(D)$ is finite surjective
onto $\mathbb{P}^{m-1}\times U$ and the square in the diagram is cartesian. We
define $D^{\prime}\subset X^{\prime}_{U}$ to be the union of $b^{\prime-1}(D)$
and the preimage via $(b\times id)\circ f^{\prime}$ of
$\\{z\\}~\times\mathbb{P}^{m}$. Let $\phi^{\prime}:=\phi\circ b^{\prime}$.
Lastly, for every $i$, we set $\xi^{\prime}_{i}:=\xi_{i}\circ b^{\prime}$. _

###  6.3. Relative curves homotopy

Our goal in this section is to construct a homotopy on a family of curves
similar to the construction in §5.2. The difference between the results
presented here and those before is that we no longer have the freedom to
shrink the base $U$ arbitrarily. As a result, we proceed a little differently
and make use crucially of the fact that the base is of dimension $1$.

We adapt the construction in [7, §11.3] of the relative curve homotopy to the
fibration $p^{\prime}:=p\circ f^{\prime}:X^{\prime}_{U}\to F\times U$ where
$F=\mathbb{P}^{m-1}$. As in §5.2, there exists an affine open subset
$F_{0}\subset F$ such that $p^{-1}(F_{0}\times U)\subset E\times U$ is
isomorphic to $\mathbb{P}^{1}\times(F_{0}\times U)$. Let $W:=F_{0}\times U$,
$A:={p^{\prime}}^{-1}(W)\subset X^{\prime}_{U}$ and $B:=p^{-1}(W)\subset
E\times U$.

_The homotopy $H^{\prime}_{curvesf}$._

We fix three points - $0,1$ and $\infty$ on $\mathbb{P}^{1}$. Recall that
given a divisor $P$ on $B$, [7, §10.2] implies that there exists a well
defined definable map $\psi_{P}\colon[0,\infty]\times B\to\widehat{B/W}$ which
fixes $\widehat{P}$. These homotopies played a crucial role in §5.2.

######  Lemma 6.4.

There exists a divisor $P\subset B$ which satisfies the following properties.

  1. (1)

For any divisor $P^{\prime}\subset B$ that contains $P$, the definable map
$\psi_{P^{\prime}}\colon[0,\infty]\times B\to\widehat{B/W}$ lifts uniquely to
a definable map $h\colon[0,\infty]\times A\to\widehat{A/W}$ such that :

    1. (a)

For every $i$, the function $h$ preserves the levels of the functions
$\xi_{i}^{\prime}$.

    2. (b)

The function $h$ is $G$-invariant.

  2. (2)

Recall the divisor $D^{\prime}\subset X^{\prime}_{U}$ from Remark 6.3 which is
finite over $F\times U$. We have that $f^{\prime}(D^{\prime}\cap A)\subseteq
P$.

  3. (3)

Let $W^{\prime}\subset W$ be the open subspace over which the divisor $P$ is
finite. The restriction $W^{\prime}\to U$ is surjective.

###### Proof.

We use arguments as in the proof of Lemma 4.6 to show that there exists a
constructible set $P_{1}\subset B$ such that for every $w\in W$,
$P_{1w}:=P_{1}\cap B_{w}$ is a finite set of those points $c\in B_{w}$ such
that $\mathbf{card}\\{f^{\prime-1}(c)\\}<\mathrm{sup}_{b\in
B_{w}}\\{\mathbf{card}\\{f^{\prime-1}(b)\\}$. We can further enlarge $P_{1}$
so that for every $w\in W$, if $FB_{w}\subset B_{w}$ denotes the forward
branching points [7, Definition 7.4.2] of the morphism
$f^{\prime}_{w}:=f^{\prime}_{|A_{w}}$ then the convex hull of $P_{1w}$
contains $FB_{w}$. One sees from the proof of [7, Lemma 11.3.1] that $P_{1}$
can be chosen so that it continues to be constructible and finite over $W$
i.e. for every $w\in W$, $P_{1w}$ is finite.

Let $\overline{P_{1}}\subset B$ denote the Zariski closure of $P_{1}$ in $B$.
We claim that $\psi_{\overline{P_{1}}}$ lifts uniquely to a definable map
$h_{\overline{P_{1}}}\colon[0,\infty]\times A\to\widehat{A/W}$. It suffices to
show that if given $w\in W$, then the restriction
${\psi_{\overline{P_{1}}}}_{|[0,\infty]\times B_{w}}$ lifts to a definable map
$h_{\overline{P_{1}},w}\colon[0,\infty]~\times A_{w}\to\widehat{A_{w}}$. This
can be accomplished by the arguments in [7, Proposition 7.4.6] and Lemma 4.5.
Note that in [7, Proposition 7.4.6], one constructs a path in
$\widehat{A_{w}}$ (which is a lift of a path in $\widehat{B_{w}}$) starting
from a Zariski closed point which is distinct from a point of ramification
upto a forward branching point. However, the only reason to have the
hypothesis that the Zariski closed point be distinct from the ramification
locus is to use Lemma 7.3.1 in loc.cit. which we accomplish by Lemma 4.5. Let
$W_{1}\subset W$ be the open subspace over which $\overline{P_{1}}\to W$ is
finite. We claim that the projection $W_{1}\to U$ is surjective. If $W_{1}$
was not surjective then we deduce that $\overline{P_{1}}$ must contain a
subspace of the form $E\times\\{u\\}$ for some $u\in U$ using the notation
from Remark 6.3. Observe that
$\mathrm{dim}(\overline{P_{1}})=\mathrm{dim}(E)=m$. Hence we must have that
$E\times\\{u\\}$ is an irreducible component of $\overline{P_{1}}$ which
implies that $P_{1}\cap(E\times\\{u\\})$ is dense in $E\times\\{u\\}$. By
construction of $P_{1}$, this is not possible. We have thus verified the
claim.

We now enlarge $\overline{P_{1}}$ to a divisor $P$ so that the lift $h$ of
$\psi_{P}$ respects the levels of the functions $\xi^{\prime}_{i}$ . We
proceed as follows. We enlarge $\overline{P_{1}}$ so that it contains the
divisor $\\{\infty\\}\times W\subset B$. Observe that
$B_{0}:=B\smallsetminus(\\{\infty\\}\times W)=\mathbb{A}^{1}\times W$ is
affine. It follows that $A_{0}:=f^{\prime-1}(B_{0})$ is affine as well. By the
proof of [7, Lemma 10.2.3], we see that for every $i$, there exists a finite
family $\\{\epsilon_{ji}\colon B\to\Gamma_{\infty}\\}$ of definable functions
such that if a definable function $[0,\infty]\times B\to\widehat{B/W}$
preserves $\epsilon_{ji}$ for every $j$ then any lift $[0,\infty]\times
A\to\widehat{A/W}$ must preserve $\xi^{\prime}_{i}$.

Since $B_{0}$ is affine, for every $i,j$, the restriction
$(\epsilon_{ij})_{|B_{0}}$ factorizes through functions of the form
$\mathrm{val}(g)$ where $g$ is a regular function on $B_{0}$. Hence there
exists finitely many regular functions $g_{1},\ldots,g_{r}$ on $B_{0}$ such
that if $Z_{1},\ldots,Z_{r}$ denotes the zeroes of the functions and
$P_{2}=\bigcup_{t}Z_{t}$ then the homotopy $\psi_{P_{2}}$ respects the levels
of the functions $\epsilon_{ij}$ for every $i,j$. We modify $P_{2}$ slightly
as follows. For every $t$, let $Z_{ht}$ be the union of those components of
$Z_{t}$ which are generically finite over $W$. If $W_{2}\subset W$ denotes the
locus over which the Zariski closure $\overline{Z_{ht}}$ in $B$ is finite then
we claim that $W_{2}\to U$ is surjective. Indeed, the Zariski closure
$\overline{Z_{ht}}$ is such that it cannot contain a subset of the form
$E\times\\{u\\}$ for dimension reasons. Let
$P_{3}:=\bigcup_{t}\overline{Z_{ht}}$.

Let $P=\overline{P_{1}}\bigcup P_{3}\bigcup f^{\prime}(D^{\prime}\cap A)$. By
construction, $\psi_{P}$ lifts to a homotopy on $A$ and respects the levels of
the functions $\xi^{\prime}_{i}$. To conclude a proof of the lemma, we must
show that the lift $h$ is $G$-invariant. This follows from the uniqueness of
the lift $h$.

∎

Using Lemma 4.5, we deduce that our choice of $D^{\prime}\subset
X^{\prime}_{U}$ implies that we can apply Lemma 4.8 to the morphism
$X^{\prime}_{U}\to U$, the map $X^{\prime}_{U}\smallsetminus
D^{\prime}\to\mathbb{A}^{m}\times U$, the functions $\xi_{i}\colon
X^{\prime}_{U}\smallsetminus D^{\prime}\to\Gamma_{\infty}$ and the group $G$.

######  Lemma 6.5.

There exists a constructible set $C\subset X^{\prime}_{U}$ with the following
properties.

  1. (1)

The set $C$ maps surjectively onto $\mathbb{P}^{m-1}\times U$ via $p^{\prime}$

  2. (2)

Let $H^{\prime}_{inff}$ be the homotopy obtained by applying Lemma 4.8 to the
morphism $X^{\prime}_{U}\to U$, the map $X^{\prime}_{U}\smallsetminus
D^{\prime}\to\mathbb{A}^{m}\times U$, the functions $\xi_{i}\colon
X^{\prime}_{U}\smallsetminus D^{\prime}\to\Gamma_{\infty}$ and the group $G$.
The image $H^{\prime}_{inff}(e,\widehat{X^{\prime}_{U}})$ is contained in
$\widehat{C}$.

  3. (3)

Let $I_{2}:=[0,\infty]$. There exists a $v+$g-continuous homotopy
$h^{\prime}_{curvesf}\colon I_{2}~\times C\to\widehat{C/\mathbb{P}^{m}\times
U}$. The image
$\Upsilon^{\prime}_{2}:=h^{\prime}_{curvesf}(0,C)\subset\widehat{X^{\prime}_{U}/\mathbb{P}^{m-1}\times
U}$ is iso-definable and relatively $\Gamma$-internal over
$\mathbb{P}^{m-1}\times U$.

###### Proof.

We verify that $C:=A\cup D^{\prime}$ satisfies the assertions of the lemma by
identical constructions and arguments as in Lemma 5.1 using Lemma 6.4 in place
of [7, Lemma 11.3.2] ∎

###  6.4. Theorem 6.1 and consequences

###### Proof.

(Theorem 6.1) The proof is identical to that given in §5.3 of Theorem 1.1.
Note that we do not claim that the homotopy $H^{\prime}$ on $\widehat{X_{U}}$
fixes its image i.e. that it is a deformation retraction. Hence, we do not
require the relative tropical homotopy from §5.4.

∎

In the case that the morphism $\phi\colon X\to S$ in Remark 6.2 is of relative
dimension $1$ i.e. for every $s\in S(K)$, $X_{s}$ is a $K$-curve, we can
verify that there exists deformation retractions of $\widehat{X}$ and
$\widehat{S}$ which are compatible with $\widehat{\phi}$ and whose images are
$\Gamma$-internal.

The method of proof is to first show the result locally around an arbitrary
point. The following lemma then allows us to glue the various relative
homotopies to obtain a relative homotopy of the total family whose image is
relatively $\Gamma$-internal.

We employ the following notation in the lemma below. Given a projective
variety $V$, recall from [7, §3.10] the notion of a definable metric $m\colon
V\times V\to\Gamma_{\infty}$. Let $D\subset V$ be a closed sub-variety of $V$.
As in the proof of Lemma 10.3.2 in loc.cit., we define $\rho_{D}\colon
V\to\Gamma_{\infty}$ as follows. For $x\in V$, set
$\rho_{D}(x):=\mathrm{sup}\\{m(x,d)|~d\in D\\}$. When there is no ambiguity
about the divisor $D$ chosen, we simplify notation and write $\rho_{D}$ in
place of $\rho$.

######  Lemma 6.6.

Let $f\colon V^{\prime}\to V$ be a morphism of projective $K$-varieties. Let
$D\subset V$ be a closed sub-variety and $D^{\prime}:=f^{-1}(D)$. Let
$U:=V\smallsetminus D$ and $U^{\prime}:=V^{\prime}\smallsetminus D^{\prime}$.
Let $h\colon[0,\infty]\times U^{\prime}\to\widehat{U^{\prime}/U}$ be a
homotopy. Let $\epsilon\in[0,\infty)$ be $K$-definable. There exists a
$K$-definable homotopy $g_{h,\epsilon}\colon[0,\infty]\times
V^{\prime}\to\widehat{V^{\prime}/V}$ such that if $x\in V$ and
$\rho_{D}(x)\leq\epsilon$ then
$g_{h,\epsilon}(0,V^{\prime}_{x})=h(0,V^{\prime}_{x})$.

###### Proof.

We begin by verifying a certain technical condition that is necessary for the
proof. Let $m^{\prime}$ be a definable metric on $V^{\prime}$. Let $m$ be a
definable metric on $V$. Let $\gamma\colon U\to[0,\infty]$ be defined as
$\gamma(u):=\mathrm{sup}\\{\gamma^{\prime}(u^{\prime})|u^{\prime}\in
f^{-1}(u)\\}$ where $\gamma^{\prime}(u^{\prime})$ is the smallest element in
$[0,\infty]$ such that $h(\gamma^{\prime}(u^{\prime}),u^{\prime})$ belongs to
the ball

$B(u^{\prime},m^{\prime},\rho_{D}(f(u^{\prime}))):=\\{x\in{U^{\prime}}|m^{\prime}(u^{\prime},x)\geq\rho_{D}(f(u^{\prime}))\\}.$

We claim that $\gamma$ is locally bounded on $U$ in the sense of [7, §10.1].

Let $u^{\prime}\in U^{\prime}$ and $t_{0}\in\Gamma$. By continuity,
$h^{-1}(\widehat{B(u^{\prime},m^{\prime},\rho_{D}(f(u^{\prime}))})$ is a
$v+g$-closed neighbourhood of the point
$(\infty,u^{\prime})\in[0,\infty]\times U^{\prime}$. The $v+g$-topology of
$[0,\infty]\times U^{\prime}$ is the product of the $v+g$-topology on
$[0,\infty]$ and the $v+g$-topology on $U^{\prime}$. Hence we can assume that
the closed neighbourhood of $(\infty,u^{\prime})$ above contains an open
neighbourhood $O_{1}\times O_{2}$ where $O_{1}$ is of the form $(s,\infty]$
and $O_{2}$ is a $v$-open 555It is $g$-open as well but we will use only that
it is $v$-open. neighbourhood of $u^{\prime}$. We can further shrink $O_{2}$
so that for every $o\in O_{2}$, $\rho_{D}(f(o))=\rho_{D}(f(u^{\prime}))$. This
is a consequence of the fact that $\rho_{D}$ is a $v+g$-continuous function
and hence $\rho_{D}^{-1}(\rho_{D}(f(u^{\prime})))$ is a $v$-open neighbourhood
of $f(u^{\prime})$. We can hence replace $O_{2}$ with $O_{2}\cap
f^{-1}(\rho_{D}^{-1}(\rho_{D}(f(u^{\prime}))))$. It follows that for every
$o\in O_{2}$, $\gamma^{\prime}(o)\leq s$. We have thus shown that
$\gamma^{\prime}\colon U^{\prime}\to\Gamma$ is locally bounded. We now deduce
that as a consequence $\gamma$ is locally bounded. Indeed, let $u\in U$. There
exists $\beta\in\Gamma$ large enough so that $B(u,m,\beta)\subset{U}$. Since
$f$ is a projective morphism, we see that $f^{-1}(B(u,m,\epsilon))$ is bounded
and definable. By [7, Lemma 10.1.7], we get that
$\gamma^{\prime}_{|f^{-1}(B(u,m,\epsilon))}$ is bounded.

For any $\delta$, let $U_{\delta}:=\\{u\in U|\rho_{D}(u)=\delta\\}$ and
$\gamma_{1}(\delta):=\mathrm{sup}\\{\gamma(u)|u\in U_{\delta}\\}$. Since
$U_{\delta}$ is bounded, by [7, Lemma 10.1.7] and the locally boundedness on
the function $\gamma$, $\gamma_{1}(\delta)\in\Gamma$. Observe that since
$\gamma_{1}$ is piece-wise affine, there exists $m\in\mathbb{N}$ and
$c_{0}\in\Gamma$ such that if $\delta\geq 0$ then $\gamma_{1}(\delta)\leq
m\delta+c_{0}$. Let $\gamma_{10}(\delta):=m\delta+c_{0}$. Let
$\epsilon^{\prime}\in\Gamma(K)$ be such that $\epsilon^{\prime}>\epsilon$. Let
$\gamma_{2}$ be a continuous function $\Gamma\to\Gamma$ which is defined as
follows. For every $x\leq\epsilon$, $\gamma_{2}(x)=0$. For
$x\in[\epsilon,\epsilon^{\prime}]$,

$\gamma_{2}(x):=(\gamma_{10}(\epsilon^{\prime})/(\epsilon^{\prime}-\epsilon))x-(\gamma_{10}(\epsilon^{\prime})/(\epsilon^{\prime}-\epsilon))\epsilon$

and for $x\geq\epsilon^{\prime}$, $\gamma_{2}(x):=\gamma_{10}(x)$. By
construction, $\gamma_{2}$ is continuous. Let $\gamma_{3}\colon
U^{\prime}\to\Gamma$ be continuous function
$x\mapsto\gamma_{2}(\rho_{D}(f(x)))$.

Let $g_{h,\epsilon}\colon[0,\infty]\times V^{\prime}\to\widehat{V^{\prime}/V}$
be defined as follows. We set

$(g_{h,\epsilon})_{|[0,\infty]\times U^{\prime}}:=h^{\prime}[\gamma_{3}]$

and for every $t\in[0,\infty],d\in D^{\prime}$,

$g_{h,\epsilon}(t,d):=d.$

We now show that the canonical extension $G_{h,\epsilon}$ is continuous. We
proceed as in [7, Lemma 10.3.2]. It suffices to verify that $G_{h,\epsilon}$
is continuous at a point $(t,d)$ with $t\in[0,\infty]$ and
$d\in\widehat{D^{\prime}}$. By definition, $G_{h,\epsilon}(t,d)=d$. Let
$\mathbf{O}\subset\widehat{V^{\prime}}$ be a neighbourhood of $d$. We must
show that there exists an open neighbourhood $W$ of $(t,d)$ in
$[0,\infty]~\times\widehat{V^{\prime}}$ such that $G_{h,\epsilon}$ maps the
simple points of $W$ to $\mathbf{O}$.

Let $M$ be a model of ACVF such that $\mathbf{O}$ is pro-definable over $M$
and $d$ is an $M$-definable type. Let $z$ be a realization of $d_{|M}$. Let
$\alpha\in[0,\infty)$ be the smallest value such that
$B(z,m^{\prime},\alpha)^{-}:=\\{y\in
V^{\prime}|m^{\prime}(y,z)>\epsilon\\}\subset\mathbf{O}$. Observe that
$\alpha\in\Gamma(M(z))$. Since $d$ is stably dominated, we get that
$\alpha\in\Gamma(M)$. Let $W_{0}$ be the set of all $x\in V^{\prime}$ such
that $B(x,m^{\prime},\alpha)^{-}$ is contained in $\mathbf{O}$. The set
$W_{0}$ is $v+g$-open and definable with parameters in $M$. Since $z\in
W_{0}$, we see that $d\in\widehat{W_{0}}$.

Suppose the open set $W$ does not exist. As the simple points are dense, there
exists sequences $t_{i}\mapsto t$, $v_{i}\mapsto d$ but
$G_{h,\epsilon}(t_{i},v_{i})\notin\mathbf{O}$. Since $v_{i}\mapsto d$ and
$\widehat{W_{0}}$ is an open neighbourhood of $d$, we get that there exists
$i_{0}$ such that if $i\geq i_{0}$ then $v_{i}\in W_{0}$. It follows that
$B(v_{i},m^{\prime},\alpha)^{-}\subset\mathbf{O}$. The assumption that
$v_{i}\mapsto d$ implies that $f(v_{i})\mapsto f(d)$ which in turn implies
that $\rho_{D}(f(v_{i}))\mapsto\infty$. Hence after increasing $i_{0}$
suitably, we get that if $i\geq i_{0}$ then
$G_{h,\epsilon}(t_{i},v_{i})\in\widehat{B(v_{i},m^{\prime},\alpha)^{-}}$. This
implies that $G_{h,\epsilon}(t_{i},v_{i})\in\mathbf{O}$ which gives a
contradiction. Clearly, from the construction,
$g_{h,\epsilon}(0,V^{\prime}_{x})=h(0,V^{\prime}_{x})$ for every $x\in
U_{\epsilon}$. ∎

####  6.4.1. Deformation retractions for curves

As stated above, our goal is to show that when $\phi\colon X\to S$ is of
relative dimension $1$, there exist deformation retractions of $\widehat{X}$
and $\widehat{S}$ which are compatible with the morphism $\widehat{\phi}$. We
require the following lemmas to ensure that the homotopies we construct do
indeed fix their image.

We introduce the following notation to simplify the statement and proof of
Lemma 6.7. Let $F$ be a valued field and $C$ be an $F$-curve. Let $f\colon
C\to\mathbb{P}^{1}_{F}$ be a finite morphism. Let $D\subset\mathbb{P}^{1}$ be
a divisor. We say that the pair $(f,D)$ is _homotopy liftable_ if the standard
homotopy with stopping divisor
$\psi_{D}\colon[0,\infty]\times\mathbb{P}^{1}\to\widehat{\mathbb{P}^{1}}$
lifts to a homotopy $h\colon[0,\infty]\times C\to\widehat{C}$ via the morphism
$\widehat{f}$. Given a $v+g$-continuous function
$\gamma\colon\mathbb{P}^{1}\to[0,\infty]$, recall the cut-off homotopy
$h[\gamma\circ f]$ from [7, Lemma 10.4.6].

######  Lemma 6.7.

Let $F$ be an algebraically closed valued field and let $C$ be an $F$-curve.
Let $f_{1}\colon C\to\mathbb{P}^{1}_{F}$ and $f_{2}\colon
C\to\mathbb{P}^{1}_{F}$ be finite morphisms. Let
$D_{1},D_{2},D_{11},D_{22}\subset\mathbb{P}^{1}$ be finite $F$-definable sets
with the following properties.

  1. (1)

The pairs $(f_{1},D_{1})$, $(f_{1},D_{11})$, $(f_{2},D_{2})$ and
$(f_{2},D_{22})$ are homotopy liftable.

  2. (2)

We have the inclusions $D_{11}\subseteq D_{1}$ and $D_{22}\subseteq D_{2}$.

  3. (3)

Let $D^{\prime}_{1}:=f_{1}^{-1}(D_{1})$,
$D^{\prime}_{11}:=f_{1}^{-1}(D_{11})$, $D^{\prime}_{2}=f_{2}^{-1}(D_{2})$ and
$D^{\prime}_{22}:=f_{2}^{-1}(D_{22})$. Assume there exists a finite
$F$-definable set $D^{\prime}\subseteq D^{\prime}_{1}\cap D^{\prime}_{2}$
containing $D^{\prime}_{11}$ and $D^{\prime}_{22}$ such that
$f_{1}(D^{\prime})\cap f_{2}(D^{\prime})$ contains $\\{0,\infty\\}$.

Let $h_{1}$ and $h_{11}$ ($h_{2}$ and $h_{22}$) lift the standard homotopies
$\psi_{D_{1}}$ and $\psi_{D_{11}}$ ($\psi_{D_{1}}$ and $\psi_{D_{11}}$)
respectively via $\widehat{f_{1}}$ ($\widehat{f_{2}}$). Let
$\gamma_{1},\gamma_{2}\colon\mathbb{P}^{1}\to[0,\infty]$ be $v+g$-continuous
functions. We then have that the image of the composition
$h_{2}[\gamma_{2}\circ f_{2}]\circ h_{1}[\gamma_{1}\circ f_{1}]$ is the
intersection of the images of the homotopies $h_{2}[\gamma_{2}\circ f_{2}]$
and $h_{1}[\gamma_{1}\circ f_{1}]$.

###### Proof.

We simplify notation and write $h_{1}[\gamma_{1}]$ in place of
$h_{1}[\gamma_{1}\circ f_{1}]$. Let $\Upsilon^{\prime}_{1}$,
$\Upsilon^{\prime}_{2}$, $\Upsilon^{\prime}_{11}$ and $\Upsilon^{\prime}_{22}$
denote the images of the homotopies $h_{1}$,$h_{2}$, $h_{11}$, and $h_{22}$
respectively. Let $\Upsilon^{\prime}$ denote the convex hull in $\widehat{C}$
of the union of the finite set $D^{\prime}$ and
$\Upsilon^{\prime}_{11}\cup\Upsilon^{\prime}_{22}$. Furthermore, let
$\Upsilon^{\prime}_{1}[\gamma_{1}]$ and $\Upsilon^{\prime}_{2}[\gamma_{2}]$
denote the images of the homotopies $h_{1}[\gamma_{1}]$ and
$h_{2}[\gamma_{2}]$. Since $h_{1}$ and $h_{2}$ are deformation retractions, we
see that it suffices to check that if $x\in\widehat{C}$ lies in the image of
the composition $h_{2}[\gamma_{2}]\circ h_{1}[\gamma_{1}]$ then $x$ belongs to
$\Upsilon^{\prime\prime}:=\Upsilon^{\prime}_{1}[\gamma_{1}]\cap\Upsilon^{\prime}_{2}[\gamma_{2}]$.

By construction of the standard homotopies and their lifts via finite
morphisms in [7, §7.5], we see that it suffices to show that if $x\in C$ then
$h_{2}[\gamma_{2}](0,h_{1}[\gamma_{1}](0,x))$ belongs to
$\Upsilon^{\prime\prime}$. We prove the following lemma.

######  Lemma 6.8.

Observe that $h_{11}(0,x)\in\Upsilon^{\prime}_{11}\subset\Upsilon^{\prime}$.
Let $t$ be the largest element in $[0,\infty]$ such that
$h_{11}(t,x)\in\Upsilon^{\prime}$ and $p_{11}:=h_{11}(t,x)$. Likewise, let
$t^{\prime}$ be the largest element in $[0,\infty]$ such that
$h_{22}(t^{\prime},x)\in\Upsilon^{\prime}$ and $p_{22}:=h_{22}(t^{\prime},x)$.
Then $p_{22}=p_{11}$.

###### Proof.

By construction, $\Upsilon^{\prime}$ is such that for every
$a,b\in\Upsilon^{\prime}$ there exists no path in $\widehat{C}$ from $a$ to
$b$ that does not intersect $\Upsilon^{\prime}$ outside of the points $a$ and
$b$. If $p_{11}\neq p_{22}$ then we deduce that there exists a path from
$p_{11}$ to $p_{22}$ which does not lie in $\Upsilon^{\prime}$ outside of
$\\{p_{11},p_{22}\\}$. This is not possible. Hence $p_{11}=p_{22}$. ∎

Let $p:=p_{11}=p_{22}$. Let $P_{1}$ be the path from $x$ to $p$ given by
$r\mapsto h_{11}(r,x)$ for $r\in[t,\infty]$. Let $P_{2}$ denote the path
$r\mapsto h_{22}(r,x)$ from $x$ to $p$ for $r\in[t^{\prime},\infty]$. Observe
that the morphism $f^{\prime}_{1}$ maps the paths $P^{\prime}_{1}$ and
$P^{\prime}_{2}$ to paths from $f_{1}(x)$ to $f_{1}(p)$. Since there is
exactly one injective path up to re-parametrization from $f_{1}(x)$ to
$f_{1}(p)$ in $\widehat{\mathbb{P}^{1}}$ and this path lifts uniquely to a
path in $\widehat{C}$, we deduce that the images of $P^{\prime}_{1}$ and
$P^{\prime}_{2}$ must coincide.

We identify the path $P_{1}$ with the closed interval $[t,\infty]$. Our
discussion above implies that the path $r\mapsto h^{\prime}_{22}(r,x)$ moves
along $P_{1}$ for $r\in[t^{\prime},\infty]$. Let $t^{\prime}_{1}$ be the
largest point on $[t,\infty]$ that belongs to
$\Upsilon^{\prime}_{1}[\gamma_{1}]$ and similarly, let $t^{\prime}_{2}$ be the
largest element that belongs to $\Upsilon^{\prime}_{2}[\gamma_{2}]$. Recall
that $h^{\prime}_{1}$ and $h^{\prime}_{2}$ are cut-offs of the homotopies
$h^{\prime}_{11}$ and $h^{\prime}_{22}$ respectively. Hence, we see that
$h^{\prime}_{1}[\gamma_{1}]$ defines an injective path from $\infty$ to
$t^{\prime}_{1}$ and fixes $t^{\prime}_{1}$. Likewise,
$h^{\prime}_{2}[\gamma_{2}]$ defines an injective path from $\infty$ to
$t^{\prime}_{2}$ and fixes $t^{\prime}_{2}$.

Suppose $t^{\prime}_{1}\leq t^{\prime}_{2}$. In this case,
$t^{\prime}_{1}\in\Upsilon^{\prime}_{2}[\gamma_{2}]$ and

$h^{\prime}_{2}[\gamma_{2}]\circ
h^{\prime}_{1}[\gamma_{1}](0,x)=t^{\prime}_{1}$

which belongs to the intersection
$\Upsilon^{\prime}_{1}[\gamma_{1}]\cap\Upsilon^{\prime}_{2}[\gamma_{2}]$. Now,
suppose $t^{\prime}_{1}>t^{\prime}_{2}$. This implies that
$t^{\prime}_{2}\in\Upsilon^{\prime}_{1}[\gamma_{1}]$ and we see that the image
of the composition is $t^{\prime}_{2}$ which lies in the intersection
$\Upsilon^{\prime}_{1}[\gamma_{1}]\cap\Upsilon^{\prime}_{2}[\gamma_{2}]$. This
completes the proof. ∎

######  Corollary 6.9.

Let $S$ be a smooth connected $K$-curve and $X$ be a quasi-projective
$K$-variety. Let $\phi\colon X\to S$ be a surjective morphism such that every
irreducible component of $X$ dominates $S$. We assume in addition that the
fibres of $\phi$ are of dimension $1$. Let $\\{\xi_{i}\colon
X\to\Gamma_{\infty}\\}$ be a finite collection of $K$-definable functions.
Recall that the functions $\xi_{i}$ extend to functions
$\xi_{i}\colon\widehat{X}\to\Gamma_{\infty}$. Let $G$ be a finite algebraic
group acting on $X$ such that the action of $G$ respects the fibres of the
morphism $\phi$. There exists compatible deformation retractions
$(H,\Upsilon)$ of $\widehat{S}$ and $(H^{\prime},\Upsilon^{\prime})$ of
$\widehat{X}$ such that

  1. (1)

The images $\Upsilon\subset\widehat{S}$ and
$\Upsilon^{\prime}\subset\widehat{X}$ are $\Gamma$-internal.

  2. (2)

The homotopy $H^{\prime}$ respects the functions $\xi_{i}$ i.e.
$\xi_{i}(H^{\prime}(t,p))=\xi_{i}(p)$ for every $p\in\widehat{X}$ and $t\in
I$.

  3. (3)

The action of the group $G$ on $X$ extends to an action on $\widehat{X}$. The
homotopy $H^{\prime}$ can be taken to be $G$-equivariant i.e. for every $g\in
G^{\prime}$ and $p\in\widehat{X}$, $H^{\prime}(t,g(p))=g(H^{\prime}(t,p))$.

  4. (4)

The homotopy $H^{\prime}$ is Zariski generalizing i.e. if $U\subset X$ is a
Zariski open subset then $H^{\prime}$ restricts to a well defined homotopy on
$\widehat{U}$.

###### Proof.

Similar to Remark 6.2, we can assume at the outset that $X$ and $S$ are
projective, $S$ is a smooth connected $K$-curve, and the fibres of the
morphism $\phi\colon X\to S$ are pure.

Let $s\in S(K)$. We show that there exists a Zariski open neighbourhood $U$ of
$s$ and a homotopy $H^{\prime}\colon[0,\infty]\times
X_{U}\to\widehat{X_{U}/U}$ whose image $H(0,X_{U})$ is relatively
$\Gamma$-internal over $U$. Note that this is slightly different from the
assertion in Theorem 6.1 since we are asking for a single homotopy and not a
chain of homotopies as would appear if we went through the construction in the
proof of loc.cit.

By the first part of Lemma 4.3, there exists a Zariski open set $U_{0}$ that
contains $s$ and is such that the morphism $\phi\colon X\to S$ factors through
a finite morphism $f_{0}\colon X_{U_{0}}\to\mathbb{P}^{1}\times U_{0}$. We
have the following commutative diagram.

${X_{U_{0}}}$${\mathbb{P}^{1}\times
U_{0}}$${U_{0}}$$\scriptstyle{f_{0}}$$\scriptstyle{\phi}$$\scriptstyle{p_{2}}$

We fix coordinates on $\mathbb{P}^{1}$. Let $B_{0}:=\mathbb{P}^{1}\times
U_{0}$. Observe that any closed subset of $B_{0}$ that is generically finite
over $U_{0}$ will be finite over $U_{0}$. We apply Lemma 6.4 to obtain a
divisor $P_{00}\subset B_{0}$ that satisfies the following properties

  1. (1)

$P_{00}$ is finite over $U_{0}$ and contains the closed subset
$\\{0,\infty\\}\times U_{0}$.

  2. (2)

For any divisor $T\subset B_{0}$ that contains $P_{00}$ and is finite over
$U_{0}$, the homotopy $\psi_{T}\colon[0,\infty]\times
B_{0}\to\widehat{B_{0}/U_{0}}$ (cf. [7, §10.2]) lifts uniquely to a homotopy
$h_{T}\colon[0,\infty]\times A\to\widehat{A/U}$. Furthermore, for every $j$,
$h_{T}$ preserves the levels of the function $\xi_{j}$ and is $G$-invariant.

The fact that $\psi_{T}$ is a homotopy follows from the fact that $T$ is
finite over $U_{0}$ which then implies that the lift $h_{T}$ is a homotopy by
[7, Lemma 10.1.1]. Observe that the image of $h_{T}$ is relatively
$\Gamma$-internal.

Let $D:=\\{x_{1},\ldots,x_{n}\\}$ be the complement of the set $U_{0}\subset
S$. For every $1\leq i\leq n$, there exists a Zariski open neighbourhood
$U_{i}$ of $x_{i}$ and a divisor $P_{ii}\subset\mathbb{P}^{1}\times{U_{i}}$
that satisfies the analogous properties as fulfilled by $P_{00}$ which are
listed above.

Let $W:=\bigcap_{0\leq i\leq n}U_{i}$. For every $i$, let
$P^{\prime}_{ii}:=f_{i}^{-1}(P_{ii})$. Let $P^{\prime}$ denote the Zariski
closure in $X$ of the closed subset $\bigcup_{i}P^{\prime}_{ii}$. For every
$i$, define $P_{i}:=f_{i}(P^{\prime}\cap X_{U_{i}})$ and
$P^{\prime}_{i}:=f_{i}^{-1}(f_{i}(P^{\prime}\cap X_{U_{i}}))$. Observe that
$P_{i}$ also satisfies the analogous versions of the points (1) - (2) as
fulfilled by $P_{00}$ above. Hence, we see that for every $i$, the homotopies
$\psi_{P_{i}}$ lift to homotopies $h_{i}$ which respect the definable function
$\xi_{j}$ and the action of the group $G$.

Let $D_{i}:=S\smallsetminus U_{i}$. Let $m$ be a metric on $S$. Recall the
function $\rho_{D}\colon S\to[0,\infty]$ defined as
$x\mapsto\mathrm{sup}\\{m(x,d)|d\in D\\}$. Let $\epsilon\in[0,\infty](K)$ and
$\delta\in[0,\infty](K)$ be such that the following is satisfied. For every
$i$, $B(x_{i},m,\epsilon)\subset\\{x\in S|\rho_{D_{i}}(x)\leq\delta\\}$ where
$B(x_{i},m,\epsilon):=\\{y\in S|~m(x_{i},y)\geq\epsilon\\}$.

For every $i$, we extend the homotopy $h_{i}$ to the whole of $X$ such that
the image of the composition of the extended $h_{i}$ will be relatively
$\Gamma$-internal over $S$. We proceed as follows.

Note that

$\\{x\in S|\rho_{D_{0}}(x)\geq\epsilon\\}=\bigcup_{x\in
D_{0}}~B(x,m,\epsilon).$

By Lemma 6.6, we can extend $h_{0}$ to a homotopy
$h_{f0}\colon[0,\infty]~\times X\to\widehat{X/S}$ such that for every $x\in S$
with $\rho_{D_{0}}(x)\leq\epsilon$,
$h_{f0}(0,X^{\prime}_{x})=h_{0}(0,X^{\prime}_{x})$. This implies in particular
that the image of $h$ is relatively $\Gamma$-internal over $\\{x\in
S|\rho_{D_{0}}(x)\leq\epsilon\\}$. Likewise, for every $i\geq 1$, we apply
Lemma 6.6 to extend $h_{i}$ to a homotopy $h_{fi}\colon[0,\infty]~\times
X\to\widehat{X/S}$ such that the image of $h_{fi}$ is relatively
$\Gamma$-internal over $\\{x\in S|\rho_{D_{i}}(x)\leq\delta\\}$. Hence the
image of $h_{fi}$ is relatively $\Gamma$-internal over $B(x_{i},m,\epsilon)$
since $B(x_{i},m,\epsilon)\subset\\{x\in S|\rho_{D_{i}}(x)\leq\delta\\}$.

Let $h:=h_{fn}\circ\ldots h_{f0}\colon[0,\infty]\cup\ldots\cup[0,\infty]\times
X\to\widehat{X/S}$ be the composition of the homotopies described above.
Observe from the construction that the image of $h$ is relatively
$\Gamma$-internal over $S$. Let $\Upsilon\subset\widehat{X/S}$ denote the
image of the homotopy $h$. By Lemma 6.7, the homotopy $h$ fixes its image.

By [7, Theorem 6.4.4], there exists a pseudo-Galois cover $\alpha\colon
S^{\prime}\to S$ and a finite collection of $K$-definable functions
$\mu_{j}\colon S^{\prime}\to\Gamma_{\infty}$ such that, for $I$ a generalised
interval, a homotopy $\beta\colon I\times S\to\widehat{S}$ which lifts to a
homotopy $a^{\prime}\colon I\times S^{\prime}\to\widehat{S^{\prime}}$ that
preserves the levels of the functions $\mu_{j}$ also induces a homotopy
$a\colon I\times\widehat{\Upsilon}\to\widehat{\Upsilon}$ that is $G$-invariant
and respects the levels of the functions $\xi_{i}$. The homotopy $a$ is
compatible with the homotopy $\beta$ for the morphism $\widehat{\phi}$. We
apply [7, Remark 11.1.3 (2)] to complete the proof while noting that since
each of the deformations in the composition are Zariski generalizing, the
composition is Zariski generalizing as well. ∎

##  7\. Locally trivial morphisms

In this section, we show that compatible homotopies can be constructed for
locally trivially morphisms. Unfortunately, our method doesn’t guarantee that
the homotopy on the source fixes its image and hence it might not be a
deformation retraction. It is possible that in this situation an additional
hypothesis is required. We demonstrate this explicitly for
$\mathbb{P}^{1}$-bundles which satisfy an additional finiteness requirement.
The method presented below adapts the proof of the relative curve homotopy in
[7].

###  7.1. Locally trivial morphisms

Let $m$ be a definable metric on a $K$-variety $S$. Suppose $D\subset S$ is a
Zariski closed subset. Recall the $v+g$-continuous function, $\rho_{D}\colon
S\to\Gamma_{\infty}$ given by $x\mapsto\mathrm{sup}\\{m(x,d)|d\in D\\}$.

######  Proposition 7.1.

Let $S$ be a projective $K$-variety. Let $m$ be a definable metric on $S$.
Suppose that for every $x\in S(K)$, we are given a Zariski open neighbourhood
$U_{x}\subset S$ of $x$ defined over $K$. Let $D_{x}:=S\smallsetminus D_{x}$.
We have that there exists finitely many points
$\\{x_{1},\ldots,x_{n}\\}\subset S(K)$ and
$\\{\epsilon_{1},\ldots,\epsilon_{n}\\}~\subset\Gamma(K)$ such that

$S=\bigcup_{1\leq i\leq n}\\{x\in
U_{x_{i}}|\rho_{D_{x_{i}}}(x)\leq\epsilon_{i}\\}.$

###### Proof.

Let $\langle a_{i}\rangle$ be a sequence in $\Gamma(K)$. We define a sequence
$\langle x_{i}\rangle_{i}\subset S(K)$ as follows. Let $x_{1}\in S(K)$. Given
$x_{i}$, let $x_{i+1}$ be a point in $F_{i}:=S\smallsetminus(\bigcup_{1\leq
j\leq i}U_{x_{j}})$ if $F_{i}$ is not empty. Observe that the sequence must
terminate. Let us assume that it terminates at $n\in\mathbb{N}$ i.e.
$F_{n}=\emptyset$.

We choose an $n-1$-tuple $(\delta_{1},\ldots,\delta_{n-1})\in\prod_{1\leq
i\leq n-1}[a_{i},\infty)(K)$ which satisfies the following properties.
Firstly, $\delta_{n-1}$ is such that $\\{y\in
S|\rho_{F_{n-1}}(y)\geq\delta_{n-1}\\}\cap D_{x_{n}}=\emptyset$. This is
possible because $F_{n-1}$ is disjoint from $D_{x_{n}}$ and $\rho_{F_{n-1}}$
is $v+g$-continuous and hence bounded in $\Gamma$ on $D_{x_{n}}$.

Suppose $1\leq i\leq n-1$. Having chosen $\delta_{i}$, we choose
$\delta_{i-1}$ such that the following is true.

  1. (1)

$\delta_{i-1}\in[a_{i-1},\infty)$ and $\delta_{i-1}>\delta_{i}$.

  2. (2)

If

$F_{i-1,\delta_{i}}:=\\{x\in F_{i-1}|\rho_{F_{i}}(x)\leq\delta_{i}\\}$

and

$D_{x_{i},\delta_{i}}:=\\{x\in D_{x_{i}}|\rho_{F_{i}}(x)\leq\delta_{i}\\}$

then $\\{y\in S|\rho_{F_{i-1}}(y)\geq\delta_{i-1}\\}\cap
D_{x_{i},\delta_{i}}=\emptyset$.

The existence of $\delta_{i-1}$ can be deduced as follows. Observe that
$F_{i}=D_{x_{i}}\cap F_{i-1}$. Hence, we see that $D_{x_{i},\delta_{i}}$ is
disjoint from $F_{i-1}$. We now use the fact that $\rho_{F_{i-1}}$ is
$v+g$-continuous and hence bounded on $D_{x_{i},\delta_{i}}$ .

We choose an $n-1$-tuple
$(\epsilon_{1},\ldots,\epsilon_{n-1})\in[0,\infty)(K)$ such that the following
holds. For $1\leq i\leq n-1$, if

$A_{i}:=S\smallsetminus(\bigcup_{1\leq j\leq i}\\{x\in
U_{x_{j}}|\rho_{D_{x_{j}}}(x)\leq\epsilon_{j}\\})$

then $A_{i}\subseteq\\{x\in S|~\rho_{F_{i}}(x)\geq\delta_{i}\\}$. We construct
$(\epsilon_{1},\ldots,\epsilon_{n-1})$ as follows. We set
$\epsilon_{1}=\delta_{1}$. Suppose, we have chosen
$(\epsilon_{1},\ldots,\epsilon_{i})$ with $i\leq{n-2}$ appropriately. Let
$\epsilon_{i+1}\in\Gamma(K)$ be such that $\epsilon_{i+1}>\delta_{i}$. We
claim that $A_{i+1}\subseteq\\{x\in S|~\rho_{F_{i+1}}(x)\geq\delta_{i+1}\\}$.

Let $z\in A_{i+1}$ and suppose that $z\notin\\{x\in
S|\rho_{F_{i+1}}(x)\geq\delta_{i+1}\\}$ i.e. $\rho_{F_{i+1}}(z)<\delta_{i+1}$.
Since $z\in A_{i+1}$, we have that $z\in A_{i}$ and hence $z\in\\{x\in
S|~\rho_{F_{i}}(x)\geq\delta_{i}\\}$. Let $y\in F_{i}$ be such that
$\rho_{F_{i}}(z)=m(z,y)$. We must have that $y\in F_{i,\delta_{i+1}}$. Indeed,
suppose that for some $y^{\prime}\in F_{i+1}$,
$m(y^{\prime},y)\geq\delta_{i+1}$. It follows that

$\displaystyle m(y^{\prime},z)$
$\displaystyle\geq\mathrm{inf}\\{m(y^{\prime},y),m(y,z)\\}$
$\displaystyle\geq\mathrm{inf}\\{\delta_{i+1},\delta_{i}\\}$
$\displaystyle\geq\delta_{i}$ $\displaystyle\mbox{ (by our choice of
}\delta_{i}).$

This contradicts our initial assumption that $\rho_{F_{i+1}}(z)<\delta_{i+1}$.

By assumption, $\rho_{D_{x_{i+1}}}(z)>\epsilon_{i+1}$. Using arguments as
above and the fact that $\epsilon_{i+1}>\delta_{i}>\delta_{i+1}$, we get that
if $x\in D_{x_{i+1}}$ is such that $\rho_{D_{x_{i+1}}}(z)=m(x,z)$ then $x\in
D_{x_{i+1},\delta_{i+1}}$.

Observe that

$\displaystyle m(x,y)$ $\displaystyle\geq\mathrm{inf}\\{m(x,z),m(y,z)\\}$
$\displaystyle\geq\mathrm{inf}\\{\epsilon_{i+1},\delta_{i}\\}$
$\displaystyle\geq\delta_{i}.$

However this is not possible by our choice of $\delta_{i}$.

Finally, we choose $\epsilon_{n}\in\Gamma(K)$ such that the following holds.
By construction, $F_{n}:=D_{x_{n}}\cap F_{n-1}$ is empty. Recall that we chose
$\delta_{n-1}$ such that $\\{y\in S|\rho_{F_{n-1}}(y)\geq\delta_{n-1}\\}\cap
D_{x_{n}}=\emptyset$. It follows that $\rho_{D_{x_{n}}}$ is bounded in
$\Gamma$ when restricted to $\\{y\in S|\rho_{F_{n-1}}(y)\geq\delta_{n-1}\\}$.
Let $\epsilon_{n}$ be such that $\\{a\in
S|\rho_{D_{x_{n}}}(a)\geq\epsilon_{n}\\}\cap\\{y\in
S|\rho_{F_{n-1}}(y)\geq\delta_{n-1}\\}=\infty$. One checks easily from the
construction that $(\epsilon_{1},\ldots,\epsilon_{n})$ satisfies the required
property.

∎

Proposition 7.1 used in conjunction with Lemma 6.6 allows us to construct
compatible deformations for certain locally trivial morphisms.

######  Corollary 7.2.

Let $V$ be a projective $K$-variety and $\psi\colon[0,\infty]\times
V\to\widehat{V}$ be a homotopy whose image is $\Gamma$-internal. Let
$\phi\colon X\to S$ be a morphism of projective varieties such that for every
$s\in S(K)$, there exists a Zariski open neighbourhood $U_{s}$ of $s$ and a
$U_{s}$-isomorphism $f_{U_{s}}\colon X_{U_{s}}\to U_{s}\times V$. Let
$\\{\xi_{i}\colon S\to\Gamma_{\infty}\\}$ be finitely many $K$-definable
functions. We then have that there exists a homotopy $H^{\prime}\colon
I\times\widehat{X}\to\widehat{X}$ and a deformation retraction $H\colon
I\times\widehat{S}\to\widehat{S}$ which are compatible for the morphism
$\widehat{\phi}$. Furthermore, the deformation $H$ respects the levels of the
functions $\xi_{i}$ for every $i$.

###### Proof.

Recall from [7, Remark 9.8.4] that if $V_{1}$ and $V_{2}$ are varieties over a
valued field and $\psi\colon I\times{V_{1}}\to\widehat{V_{1}}$ is
$v+g$-continuous then the map $(t,u,v)\mapsto\psi(t,u)\otimes v$ defines a
$v+g$-continuous map $\psi^{\prime}\colon I\times(V_{1}\times
V_{2})\to\widehat{(V_{1}\times V_{2})/V_{2}}$. It follows from this and our
assumptions on $\phi$ and $V$ that for every $s\in S(K)$ there exists a
Zariski open neighbourhood $U_{s}\subseteq S$ of $s$ and a homotopy
$h^{\prime}_{U_{s}}\colon[0,\infty]~\times
X_{U_{s}}\to\widehat{X_{U_{s}}/U_{s}}$ such that the image is relatively
$\Gamma$-internal.

Let $m$ be a definable metric on $V$. We apply Proposition 7.1 to the given
data and obtain finite sets $\\{s_{1},\ldots,s_{m}\\}\subset S(K)$ and
$\\{\epsilon_{1},\ldots,\epsilon_{m}\\}\subset\Gamma(K)$ such that
$S=\bigcup_{1\leq i\leq m}\\{x\in S|\rho_{D_{s_{i}}}(x)\leq\epsilon_{i}\\}$
where $D_{s_{i}}:=S\smallsetminus U_{s_{i}}$. For every $1\leq i\leq m$, we
simplify notation and write $U_{i}:=U_{s_{i}},D_{i}:=D_{s_{i}}$ and
$h^{\prime}_{U_{s_{i}}}:=h^{\prime}_{i}$.

By Lemma 6.6, we can extend the homotopy $h^{\prime}_{i}:[0,\infty]\times
X_{i}\to\widehat{X_{i}/U_{i}}$ to a homotopy
$h^{\prime}_{fi}\colon[0,\infty]\times X\to\widehat{X/S}$ such that for every
$s$ with $\rho_{D_{i}}(s)\leq\epsilon_{i}$,
$h^{\prime}_{fi}(0,X_{is})=h^{\prime}_{i}(0,X_{is})$. In particular the image
of $h^{\prime}_{fi}$ is relatively $\Gamma$-internal over $\\{s\in
S|~\rho_{D_{i}}(s)\leq\epsilon_{i}\\}$. Let $h^{\prime}_{f}:=\Sigma_{1\leq
i\leq m}h^{\prime}_{fi}\colon[0,\infty]~\cup~\ldots[0,\infty]\times
X\to\widehat{X/S}$ be the composition of the homotopies $h^{\prime}_{fi}$. Let
$I_{1}$ be the glueing of the $m$-copies of the intervals $[0,\infty]$. By
construction, $h^{\prime}_{f}\colon I_{1}\times X\to\widehat{X/S}$ and its
image $\Upsilon^{\prime}$ is relatively $\Gamma$-internal.

By [7, Theorem 6.4.4] there exists a finite pseudo-Galois cover $f\colon
S^{\prime}\to S$ and a a finite number of $K$-definable functions
$\\{\mu_{j}\colon S\to\Gamma_{\infty}\\}_{j}$ such that a homotopy $H\colon
I_{2}\times S\to\widehat{S}$ that lifts to a homotopy $I_{2}\times
S^{\prime}\to\widehat{S^{\prime}}$ and respects the level of the functions
$\\{\mu_{j}\\}_{j}$ must also give a homotopy $H^{\prime}_{1}\colon
I_{2}\times\Upsilon\to\widehat{\Upsilon}$ that is compatible with the map
$\Upsilon\to S$. By [7, Theorem 11.1.1], there exists a deformation $H$ that
satisfies these properties, has a $\Gamma$-internal image and also respects
the definable functions $\xi_{i}$. By construction, the composition
$H^{\prime}:=H^{\prime}_{1}\circ H^{\prime}_{f}\colon I_{2}+I_{1}\times
X\to\widehat{X}$ is a homotopy compatible with $H$ and has $\Gamma$-internal
image. ∎

######  Corollary 7.3.

Let $C$ be a projective $K$-curve and $S$ be a projective $K$-variety. Let
$\phi\colon X\to S$ be a projective morphism such that for every $s\in S$,
there exists a Zariski open neighbourhood $U$ of $s$ and a $U$-isomorphism
$f_{U}\colon X_{U}\to C\times U$. Let $\\{\xi_{i}\colon
S\to\Gamma_{\infty}\\}$ be a finite family of $K$-definable functions.

We then have that there exists a homotopy $H^{\prime}\colon
I\times\widehat{X}\to\widehat{X}$ and a deformation $H\colon
I\times\widehat{S}\to\widehat{S}$ which are compatible for the morphism
$\widehat{\phi}$. Furthermore, the deformation $H$ respects the levels of the
functions $\xi_{i}$ for every $i$.

###### Proof.

Let $C$ be as in the statement of the corollary. [7, Theorem 7.5.1] implies
the existence of a deformation $\psi\colon[0,\infty]\times C\to\widehat{C}$
whose image is $\Gamma$-internal. Applying Corollary 7.2 proves the result. ∎

####  7.1.1. Compatible deformations for $\mathbb{P}^{1}$-bundles

Note that Corollary 7.3 doesn’t guarantee that the existence of compatible
deformation retractions. It is possible that this is true only after an
additional hypothesis. We present such an instance of adding a hypothesis to
guarantee the existence of compatible deformations in the case of
$\mathbb{P}^{1}$-bundles.

Let $S$ be a quasi-projective $K$-variety.

######  Definition 7.4.

_A morphism $\phi\colon X\to S$ is a_ $\mathbb{P}^{1}$-bundle _if there exists
a finite Zariski open covering $\\{U_{i}\\}_{1\leq i\leq m}$ of $S$ such that
for every $i$, there exists an isomorphism $g_{i}\colon
X_{i}:=\phi^{-1}(U_{i})\to U_{i}\times\mathbb{P}^{1}_{k}$ over $U_{i}$. We
encode the data of the $\mathbb{P}^{1}$-bundle using the tuple $(\phi\colon
X\to S,\\{U_{i}\\}_{i},\\{g_{i}\\}_{i})$._

######  Remark 7.5.

_For $i,j$, the transition maps_

$\displaystyle g_{ij}:=[g_{j}\circ g_{i}^{-1}]_{U_{i}\cap
U_{j}}\colon(U_{i}\cap U_{j})\times\mathbb{P}^{1}_{k}\to(U_{i}\cap
U_{j})\times\mathbb{P}^{1}_{k}$

_are isomorphisms over $U_{i}\cap U_{j}$ i.e. for every $u\in U_{i}\cap
U_{j}$,
$g_{ij}^{u}:={g_{ij}}_{|\phi^{-1}(u)}\colon\mathbb{P}^{1}_{K}\to\mathbb{P}^{1}_{K}$
is a well defined automorphism. Let $X_{ij}:=X_{i}\cap X_{j}$. _

The projective bundles we consider will satisfy the following finiteness
hypothesis. For the remainder of this section, we fix a system of coordinates
on $\mathbb{P}^{1}_{K}$.

($\mathbf{F}$) _There exists a divisor $D\subset X$ such that_

  1. (1)

_The morphism_ $\phi$ restricts to a finite map from $D$ onto $S$.

  2. (2)

_For every_ $1\leq i\leq m$, $g_{i}(D\cap X_{i})$ contains
$U_{i}\times\\{0,\infty\\}$.

######  Remark 7.6.

_Let $(\phi\colon X\to S,\\{U_{i}\\}_{i},\\{g_{i}\\}_{i}\\}$ be a rank-$1$
projective bundle. Let $X_{i}=\phi^{-1}(U_{i})$. To construct a homotopy
$H^{\prime}\colon I\times X\to\widehat{X/S}$ it suffices to construct a family
of homotopies $H^{\prime}_{i}\colon I\times Y_{i}\to\widehat{Y_{i}/U_{i}}$
which satisfy the obvious glueing conditions. Precisely, if $u\in U_{i}\cap
U_{j}$ then $H^{\prime}_{i}$ restricts to a homotopy on the fibre
$\widehat{Y^{\prime}_{iu}}=\widehat{\mathbb{P}^{1}_{K}}$ such that for every
$t\in I$ and $x\in\widehat{Y^{\prime}_{iu}}$,
$H^{\prime}_{i}(t,x)=\widehat{g_{ji}}(H^{\prime}_{j}(t,\widehat{g_{ij}}(x)))$._

######  Theorem 7.7.

Let $\\{\phi\colon X\to S,\\{U_{i}\\}_{i},\\{g_{i}\\}_{i}\\}$ be a
$\mathbb{P}^{1}$-bundle which satisfies the hypothesis $(\mathbf{F})$. There
exists a pair of deformation retractions $(H^{\prime}\colon
I\times\widehat{X}\to\widehat{X},\Upsilon^{\prime})$ and $(H\colon
I\times\widehat{S}\to\widehat{S},\Upsilon)$ which are compatible for the
morphism $\widehat{\phi}$ and whose images
$\Upsilon^{\prime}\subset\widehat{X},\Upsilon\subset\widehat{S}$ are
$\Gamma$-internal subsets.

###### Proof.

For every $i$, let $Y_{i}:=U_{i}\times\mathbb{P}^{1}_{K}$. By assumption,
there exists a closed subset $D\subset X$ such that $\phi_{|D}\colon D\to S$
is finite and for every $i\in~\\{1,\ldots,n\\}$, $g_{i}(D\cap X_{i})\subset
Y_{i}$ contains $U_{i}\times\\{0,\infty\\}$. Let $E_{i}:=g_{i}(D\cap X_{i})$.
Let $p_{i}\colon Y_{i}\to U_{i}$ be the projection map. We have that
${p_{i}}_{|E_{i}}\colon E_{i}\to U_{i}$ is finite. Recall from the paragraph
above [7, Lemma 9.5.3], the construction of the homotopy $\psi_{E_{i}}$. The
homotopy $\psi_{E_{i}}$ is such that for every $u\in U_{i}$, $\psi_{E_{i}}$
restricts to the homotopy on the fibre $Y_{iu}=\mathbb{P}^{1}_{K}$ defined by
the standard homotopy with cut-off determined by $E_{i}\cap Y_{iu}$. By [7,
Lemma 10.2.1], the homotopy $\psi_{E_{i}}\colon[0,\infty]\times
Y_{i}\to\widehat{Y_{i}/U_{i}}$ is well defined and continuous.

By pulling back via the isomorphism $g_{i}$, we have for every $i$, a homotopy

$\psi^{\prime}_{i}\colon[0,\infty]\times X_{i}\to\widehat{X_{i}/U_{i}}$

whose image is relatively $\Gamma$-internal. We claim that these homotopies
glue to give a homotopy on $\widehat{X/S}$. We verify the claim as follows.

Let $i,j\in\\{1,\ldots,n\\}$. Let $u\in U_{i}\cap U_{j}$. The homotopies
$\psi^{\prime}_{i}$ and $\psi^{\prime}_{j}$ restrict to define homotopies
$\psi^{\prime}_{i,u},\psi^{\prime}_{j,u}\colon[0,\infty]\times
X_{u}\to\widehat{X_{u}}$ where $X_{u}:=\phi^{-1}(u)$. We show that these
homotopies coincide. Since, $i$ and $j$ were chosen arbitrarily, this will
imply that the homotopies $\psi^{\prime}_{i}$ glue together. Since
$\psi^{\prime}_{i}$ and $\psi^{\prime}_{j}$ are definable maps, it suffices to
consider the case when $u$ is defined over $K$.

The isomorphisms $g_{i}$ and $g_{j}$ imply a definable automorphism
$g_{iju}:=g_{j,u}\circ
g_{i,u}^{-1}\colon\mathbb{P}_{K(u)}^{1}\to\mathbb{P}_{K(u)}^{1}$ where
$g_{i,u}:={g_{i}}_{|X_{u}}$. Let
$\psi_{E_{i},u},\psi_{E_{j},u}\colon[0,\infty]\times\mathbb{P}^{1}\to\widehat{\mathbb{P}^{1}}$
be the homotopies induced by $\psi_{E_{i}}$ and $\psi_{E_{j}}$. As in Remark
7.6, it suffices to verify that for every $x\in\mathbb{P}_{K(u)}^{1}$ and
$t\in[0,\infty]$,
$\widehat{g_{iju}}(\psi_{E_{i},u}(t,x))=\psi_{E_{j},u}(t,g_{iju}(x))$. Since,
$\widehat{\mathbb{P}^{1}}$ is definable, we reduce to showing the above
equality when $x\in{\mathbb{P}^{1}_{K(u)}}(K^{max})$ and
$t\in[0,\infty](K^{max})=[0,\infty](\mathbb{R})$. In this situation, we make
use of the fact that $\widehat{\mathbb{P}^{1}_{K(u)}}(K^{max})$ is
homeomorphic to the Berkovich space $B_{K^{max}}(\mathbb{P}^{1})$.

If $x\in E_{i}$ then for every $t$,
$\widehat{g_{iju}}(\psi^{\prime}_{E_{i},u}(t,x))=\psi^{\prime}_{E_{j},u}(t,g_{iju}(x))$.
Let $\Upsilon_{i}\subset\widehat{\mathbb{P}^{1}}$ denote the convex hull of
$E_{iu}$ and likewise, $\Upsilon_{j}\subset\widehat{\mathbb{P}^{1}}$ denote
the convex hull of $E_{ju}$. Note that
$\widehat{g_{iju}}^{-1}(\Upsilon_{j})=\Upsilon_{i}$. Let
$x\in\mathbb{P}^{1}(K^{max})\smallsetminus\Upsilon_{i}$ and $O$ be the
connected component of
$\widehat{\mathbb{P}^{1}}(K^{max})\smallsetminus\Upsilon_{i}$ that contains
$x$. As $\widehat{g_{iju}}$ is a homeomorphism and
$\widehat{g_{iju}}(\Upsilon_{i})=\Upsilon_{j}$, there exists a connected
component $O^{\prime}$ of $\widehat{\mathbb{P}^{1}}\smallsetminus\Upsilon_{j}$
such that $\widehat{g_{iju}}(O)=O^{\prime}$. Note that these connected
components are Berkovich open balls. Since the morphism $g_{iju}$ is
algebraic, the restriction ${g_{iju}}_{|O}$ at the level of algebras must be
of the form $x\mapsto ax+b$ where $a\in K^{*}$ and $b\in K$. It can then be
checked by explicit calculation that
$\widehat{g_{iju}}(\psi^{\prime}_{E_{i},u}(t,x))=\psi^{\prime}_{E_{j},u}(t,g_{iju}(x))$.

We have thus shown that there exists a definable homotopy
$\psi\colon[0,\infty]\times X\to\widehat{X/S}$ whose image is relatively
$\Gamma$-internal. Let $\Upsilon$ denote the image of $\psi$. Let
$\overline{S}$ be a projective $K$-variety that contains $S$ as an open dense
subvariety. Let $d_{bord}\colon\overline{S}\to\Gamma_{\infty}$ be the
schematic distance to $S_{bord}:=\overline{S}\smallsetminus S$. Observe that
$\Upsilon$ is $\sigma$-compact with respect to the function
$d_{bord}\circ\widehat{\phi}$. By [7, Theorem 6.4.4], there exists a pseudo-
Galois cover $f\colon S^{\prime}\to S$ and a morphism
$\kappa\colon\Upsilon^{\prime}:=\Upsilon\times_{S}S^{\prime}\to
S^{\prime}\times\Gamma_{\infty}^{M}$ for some $M\in\mathbb{N}$ such that the
restriction
$\widehat{\kappa}_{|\widehat{\Upsilon^{\prime}}}\colon\widehat{\Upsilon^{\prime}}\to\widehat{S^{\prime}}\times\Gamma^{M}_{\infty}$
is a homeomorphism onto its image. By loc.cit., we see that there exists a
finite number of $K$-definable functions $\\{\mu_{j}\colon
S\to\Gamma_{\infty}\\}_{j}$ such that a homotopy $H\colon I\times
S\to\widehat{S}$ that lifts to a homotopy $I\times
S^{\prime}\to\widehat{S^{\prime}}$ and respects the level of the functions
$\\{\mu_{j}\\}_{j}$ must also give a homotopy $H^{\prime}_{1}\colon
I\times\Upsilon\to\widehat{\Upsilon}$ that is compatible with the map
$\Upsilon\to S$. By [7, Theorem 11.1.1], the homotopy $H$ can be chosen so
that it satisfies these properties and in addition has a $\Gamma$-internal
image. By construction, the composition $H^{\prime}_{1}\circ\psi\colon
I+[0,\infty]\times X\to\widehat{X}$ is compatible with $H$ and has
$\Gamma$-internal image. Clearly, $H^{\prime}_{1}\circ\psi$ is a deformation
retraction. ∎

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