###### Proof.

There exists a natural homeomorphism
$\displaystyle\bigcup_{{\mathfrak{q}}\in{\mathfrak{P}}_{2}}s_{{\mathfrak{q}}}^{-1}(0)\to
X$. In the following argument we identify these two spaces, and in particular
consider $\displaystyle X$ as a subspace of
$\displaystyle|{\mathcal{K}}_{2}|$. First we need the following fact, which is
essentially the same as Lemma 7.66 in [10]:

> ($\displaystyle\star$): For any open set $\displaystyle W$ of
> $\displaystyle|{\mathcal{K}}_{2}|$ containing a compact subset
> $\displaystyle K$ of $\displaystyle X$, there exists a strongly smooth
> function $\displaystyle g:|{\mathcal{K}}_{2}|\to[0,1]$ that has a compact
> support contained in $\displaystyle W$ and $\displaystyle 1$ on a
> neighborhood of $\displaystyle K$.

Applying the claim ($\displaystyle\star$) to $\displaystyle
K:=\varphi_{{\mathfrak{p}}}(({\mathcal{K}}^{\prime}_{1})_{{\mathfrak{p}}})$
and $\displaystyle
W:=B_{\delta}(\varphi_{{\mathfrak{p}}}(({\mathcal{K}}^{\prime}_{1})_{{\mathfrak{p}}}))$,
we obtain $\displaystyle f_{{\mathfrak{p}}}:|{\mathcal{K}}_{2}|\to[0,1]$.
Applying the claim ($\displaystyle\star$) to $\displaystyle K:=X$ and
$\displaystyle
W:=\bigg{\\{}x\in|{\mathcal{K}}_{2}|\,\bigg{|}\sum_{{\mathfrak{q}}\in{\mathfrak{P}}_{1}}f_{{\mathfrak{q}}}(x)>1/2\bigg{\\}}$,
we obtain $\displaystyle g:|{\mathcal{K}}_{2}|\to[0,1]$. Finally, for each
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{1}$ we define
$\displaystyle(\chi^{+}_{1})_{{\mathfrak{p}}}:|{\mathcal{K}}_{2}|\to[0,1]$ by

$\displaystyle(\chi^{+}_{1})_{{\mathfrak{p}}}(x):=\begin{cases}g(x)f_{{\mathfrak{p}}}(x)\biggl{(}\sum_{{\mathfrak{q}}\in{\mathfrak{P}}_{1}}f_{{\mathfrak{q}}}(x)\biggr{)}^{-1}&(x\in
W)\\\ 0&(x\notin W).\end{cases}$

∎

For each $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{1}$, we define
$\displaystyle(\chi_{1})_{{\mathfrak{p}}}:|{\mathcal{K}}_{1}|\to[0,1]$ by

$\displaystyle(\chi_{1})_{{\mathfrak{p}}}|_{({\mathcal{K}}_{1})_{{\mathfrak{q}}}}:=(\chi^{+}_{1})_{{\mathfrak{i}}({\mathfrak{p}})}\circ\varphi_{{\mathfrak{q}}}|_{({\mathcal{K}}_{1})_{{\mathfrak{q}}}}.$

Then
$\displaystyle\chi_{1}=((\chi_{1})_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}_{1}}$
is a partition of unity of
$\displaystyle(X,\widetriangle{{\mathcal{U}}}_{1},{\mathcal{K}}^{\prime}_{1},\delta)$.
We also take a partition of unitiy of
$\displaystyle(X,\widetriangle{{\mathcal{U}}}_{2},{\mathcal{K}}^{\prime}_{2},\delta)$,
which we denote by
$\displaystyle\chi_{2}=((\chi_{2})_{{\mathfrak{q}}})_{{\mathfrak{q}}\in{\mathfrak{P}}_{2}}$.

Now we can complete the proof as follows:

$\displaystyle\displaystyle(\widetriangle{f}_{1})_{*}(X,\widetriangle{{\mathcal{U}}}_{1},\widetriangle{\omega}_{1},\widetriangle{{\mathfrak{S}}}^{\varepsilon}_{1})$
$\displaystyle\displaystyle\quad=\sum_{{\mathfrak{p}}\in{\mathfrak{P}}_{1}}((f_{1})_{{\mathfrak{p}}})_{*}(({\mathcal{U}}_{1})_{{\mathfrak{p}}},(\chi_{1})_{{\mathfrak{p}}}(\omega_{1})_{{\mathfrak{p}}},({\mathfrak{S}}^{\varepsilon}_{1})_{{\mathfrak{p}}})$
$\displaystyle\displaystyle\quad=\sum_{{\mathfrak{p}}\in{\mathfrak{P}}_{1}}((f_{2})_{{\mathfrak{i}}({\mathfrak{p}})})_{*}(({\mathcal{U}}_{2})_{{\mathfrak{i}}({\mathfrak{p}})},(\chi^{+}_{1})_{{\mathfrak{i}}({\mathfrak{p}})}(\omega_{2})_{{\mathfrak{i}}({\mathfrak{p}})},({\mathfrak{S}}^{\varepsilon}_{2})_{{\mathfrak{i}}({\mathfrak{p}})})$
$\displaystyle\displaystyle\quad=\sum_{{\mathfrak{q}}\in{\mathfrak{P}}_{2}}((f_{2})_{{\mathfrak{q}}})_{*}(({\mathcal{U}}_{2})_{{\mathfrak{q}}},(\chi_{2})_{{\mathfrak{q}}}(\omega_{2})_{{\mathfrak{q}}},({\mathfrak{S}}^{\varepsilon}_{2})_{{\mathfrak{q}}})$
$\displaystyle\displaystyle\quad=(\widetriangle{f}_{2})_{*}(X,\widetriangle{{\mathcal{U}}}_{2},\widetriangle{\omega}_{2},\widetriangle{{\mathfrak{S}}}^{\varepsilon}_{2}).$

The first and fourth equality follows from the definition. The second equality
holds since
$\displaystyle(\omega_{1})_{{\mathfrak{p}}}=(\varphi_{{\mathfrak{p}}})^{*}((\omega_{2})_{{\mathfrak{i}}({\mathfrak{p}})})\,(\forall{\mathfrak{p}}\in{\mathfrak{P}}_{1})$
and

$\displaystyle\text{\rm
supp}\,(\chi^{+}_{1})_{{\mathfrak{p}}}\cap\Pi(({\mathfrak{S}}^{\varepsilon}_{2})^{-1}_{{\mathfrak{i}}({\mathfrak{p}})}(0))\subset
B_{\delta}(\varphi_{{\mathfrak{p}}}(({\mathcal{K}}^{\prime}_{1})_{{\mathfrak{p}}}))\cap\Pi(({\mathfrak{S}}^{\varepsilon}_{2})^{-1}_{{\mathfrak{i}}({\mathfrak{p}})}(0))\subset\varphi_{{\mathfrak{p}}}(({\mathcal{K}}_{1})_{{\mathfrak{p}}})$

when $\displaystyle\delta$ and $\displaystyle\varepsilon$ are sufficiently
small (this can be proved by arguments similar to the proof of Lemma 8.10).
Proof of the third equality is similar to the proof of Lemma 8.13. This
completes the proof of Lemma 8.14. ∎

Let us state and prove Stokes’ formula.

######  Proposition 8.16.

For sufficiently small $\displaystyle\varepsilon>0$, there holds

$\displaystyle\partial(\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},\widetriangle{\omega},\widetriangle{{\mathfrak{S}}}^{\varepsilon}))=(-1)^{|\omega|+1}\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},d\widetriangle{\omega},\widetriangle{{\mathfrak{S}}}^{\varepsilon}).$

###### Proof.

Take $\displaystyle{\mathcal{K}}^{\prime}<{\mathcal{K}}$,
$\displaystyle\delta>0$ and let
$\displaystyle(\chi_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}}$ be a
partition of unity of
$\displaystyle(X,\widetriangle{{\mathcal{U}}},{\mathcal{K}}^{\prime},\delta)$.
When $\displaystyle\varepsilon>0$ is sufficiently small,

$\displaystyle\displaystyle\partial(\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},\widetriangle{\omega},\widetriangle{{\mathfrak{S}}}^{\varepsilon}))-(-1)^{|\omega|+1}\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},d\widetriangle{\omega},\widetriangle{{\mathfrak{S}}}^{\varepsilon})$
$\displaystyle\displaystyle\quad=(-1)^{|\omega|+1}\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},d\chi_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})$
$\displaystyle\displaystyle\quad=(-1)^{|\omega|+1}\sum_{{\mathfrak{p}},{\mathfrak{q}}\in{\mathfrak{P}}}(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},\chi_{{\mathfrak{q}}}d\chi_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})$
$\displaystyle\displaystyle\quad=(-1)^{|\omega|+1}\sum_{{\mathfrak{p}},{\mathfrak{q}}\in{\mathfrak{P}}}(f_{{\mathfrak{q}}})_{*}({\mathcal{U}}_{{\mathfrak{q}}},\chi_{{\mathfrak{q}}}d\chi_{{\mathfrak{p}}}\omega_{{\mathfrak{q}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{q}}})$
$\displaystyle\displaystyle\quad=(-1)^{|\omega|+1}\sum_{{\mathfrak{q}}\in{\mathfrak{P}}}(f_{{\mathfrak{q}}})_{*}({\mathcal{U}}_{{\mathfrak{q}}},\chi_{{\mathfrak{q}}}(\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}d\chi_{{\mathfrak{p}}})\omega_{{\mathfrak{q}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{q}}})=0.$

The first equality follows from Stokes’ formula for K-charts: Lemma 8.2 (iii).
The second and fifth equality holds since
$\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}\chi_{{\mathfrak{p}}}=1$
on $\displaystyle\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon})^{-1}(0))$
when $\displaystyle\varepsilon$ is sufficiently small. The third equality
holds by the argument similar to the proof of Lemma 8.13. The fourth equality
is obvious. ∎

###  8.3. K-space

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a compact, oriented K-space
with a strongly smooth map
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to{\mathcal{L}}_{k+1}$,
a differential form $\displaystyle\widehat{\omega}$ on
$\displaystyle(X,\widehat{{\mathcal{U}}})$ and a CF-perturbation
$\displaystyle\widehat{{\mathfrak{S}}}=(\widehat{{\mathfrak{S}}}^{\varepsilon})_{\varepsilon\in(0,1]}$
on $\displaystyle(X,\widehat{{\mathcal{U}}})$ which is transversal to
$\displaystyle 0$ and $\displaystyle\text{\rm
ev}\,_{0}\circ\widehat{f}:(X,\widehat{{\mathcal{U}}})\to L$ is strongly
submersive with respect to $\displaystyle\widehat{{\mathfrak{S}}}$. Under
these assumptions we define a de Rham chain

$\displaystyle\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}}^{\varepsilon})\in
C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1})$

for sufficiently small $\displaystyle\varepsilon>0$, and check Stokes’ formula
and fiber product formula. We only consider K-spaces without boundaries, since
generalization to admissible K-spaces with boundaries (and corners) will be
straightforward.

By Lemma 9.10 in [10], there exist

  * •

$\displaystyle\widetriangle{{\mathcal{U}}}$: GCS of $\displaystyle X$,

  * •

$\displaystyle\widetriangle{\omega}$: differential form on
$\displaystyle\widetriangle{{\mathcal{U}}}$,

  * •

$\displaystyle\widetriangle{f}$: strongly smooth map from
$\displaystyle(X,\widetriangle{{\mathcal{U}}})$ to
$\displaystyle{\mathcal{L}}_{k+1}$,

  * •

$\displaystyle{\mathcal{K}}$: support system of
$\displaystyle\widetriangle{{\mathcal{U}}}$,

  * •

$\displaystyle\widetriangle{{\mathfrak{S}}}$: CF-perturbation of
$\displaystyle(\widetriangle{{\mathcal{U}}},{\mathcal{K}})$ which is
transversal to $\displaystyle 0$, and $\displaystyle\text{\rm
ev}\,_{0}\circ\widetriangle{f}:(X,\widetriangle{{\mathcal{U}}})\to L$ is
strongly submersive with respect to
$\displaystyle\widetriangle{{\mathfrak{S}}}$,

  * •

strict KG-embedding
$\displaystyle\widehat{\Phi}:\widehat{{\mathcal{U}}}_{0}\to\widetriangle{{\mathcal{U}}}$,
where $\displaystyle\widehat{{\mathcal{U}}_{0}}$ is an open substructure of
$\displaystyle\widehat{{\mathcal{U}}}$,

satisfying the following compatibilities:

  * •

$\displaystyle\widehat{\omega}|_{\widehat{{\mathcal{U}}_{0}}}=\widehat{\Phi}^{*}(\widetriangle{\omega})$,

  * •

$\displaystyle\widehat{f}|_{\widehat{{\mathcal{U}}_{0}}}=\widetriangle{f}\circ\widehat{\Phi}$,

  * •

$\displaystyle\widehat{{\mathfrak{S}}}|_{\widehat{{\mathcal{U}}_{0}}}=\widehat{\Phi}^{*}(\widetriangle{{\mathfrak{S}}})$.

Then we define

(65)
$\widehat{f}_{*}(X,\widehat{{\mathcal{U}}},\widehat{\omega},\widehat{{\mathfrak{S}}^{\varepsilon}}):=\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},\widetriangle{\omega},\widetriangle{{\mathfrak{S}}^{\varepsilon}})$

for sufficiently small $\displaystyle\varepsilon>0$. Well-definedness (i.e.
the RHS of (65) does not depend on choices of
$\displaystyle\widetriangle{{\mathcal{U}}}$,
$\displaystyle\widetriangle{\omega}$, etc. in the sense of
$\displaystyle\spadesuit$) follows from invariance by GG-embedding (Lemma
8.14) and arguments in [10] Section 9.2 (proof of Proposition 9.16
$\displaystyle\implies$ Theorem 9.14). Stokes’ formula in this setting
(Theorem 7.5) follows from Stokes’ formula for GCS (Theorem 8.16).

Finally we give a sketch of the proof of the fiber product formula (Theorem
7.6), imitating the proof of Proposition 10.23 in [10]. Namely, we prove

(66)
$(\widehat{f}_{12})_{*}(X_{12},\widehat{{\mathcal{U}}}_{12},\widehat{\omega}_{12},\widehat{{\mathfrak{S}}}_{12}^{\varepsilon})=(\widehat{f}_{1})_{*}(X_{1},\widehat{{\mathcal{U}}}_{1},\widehat{\omega}_{1},\widehat{{\mathfrak{S}}}_{1}^{\varepsilon})\circ_{j}(\widehat{f}_{2})_{*}(X_{2},\widehat{{\mathcal{U}}}_{2},\widehat{\omega}_{2},\widehat{{\mathfrak{S}}}_{2}^{\varepsilon})$

for sufficiently small $\displaystyle\varepsilon>0$, where
$\displaystyle(X_{12},\widehat{{\mathcal{U}}}_{12})$ denotes the fiber product
of $\displaystyle(X_{1},\widehat{{\mathcal{U}}}_{1})$ and
$\displaystyle(X_{2},\widehat{{\mathcal{U}}}_{2})$. For each $\displaystyle
i\in\\{1,2\\}$, there exist a GCS
$\displaystyle\widetriangle{{\mathcal{U}}}_{i}$ on $\displaystyle X_{i}$ and a
KG-embedding
$\displaystyle\widehat{{\mathcal{U}}}_{i}\to\widetriangle{{\mathcal{U}}}_{i}$,
namely a strict KG-embedding
$\displaystyle(\widehat{{\mathcal{U}}}_{i})_{0}\to\widetriangle{{\mathcal{U}}}_{i}$,
where $\displaystyle(\widehat{{\mathcal{U}}}_{i})_{0}$ is an open substructure
of $\displaystyle\widehat{{\mathcal{U}}}_{i}$. Let
$\displaystyle(\widehat{{\mathcal{U}}}_{12})_{0}$ denote the fiber product of
$\displaystyle(\widehat{{\mathcal{U}}}_{1})_{0}$ and
$\displaystyle(\widehat{{\mathcal{U}}}_{2})_{0}$. One may assume that, for
each $\displaystyle i\in\\{1,2\\}$ there exist
$\displaystyle\widetriangle{\omega}_{i}$, $\displaystyle\widetriangle{f}_{i}$,
$\displaystyle\widetriangle{{\mathfrak{S}}}_{i}$ satisfying compatibilities.

Let
$\displaystyle\chi^{i}=(\chi^{i}_{{\mathfrak{p}}_{i}})_{{\mathfrak{p}}_{i}\in{\mathfrak{P}}_{i}}$
be a partition of unity of
$\displaystyle(X_{i},\widetriangle{{\mathcal{U}}}_{i})$. For each
$\displaystyle{\mathfrak{p}}_{i}\in{\mathfrak{P}}_{i}$,
$\displaystyle\chi^{i}_{{\mathfrak{p}}_{i}}$ induces a strongly smooth map
$\displaystyle\widehat{\chi}^{i}_{{\mathfrak{p}}_{i}}:(X_{i},\widehat{{\mathcal{U}}}_{i})\to[0,1]$,
and there holds
$\displaystyle\sum_{{\mathfrak{p}}_{i}\in{\mathfrak{P}}_{i}}\widehat{\chi}^{i}_{{\mathfrak{p}}_{i}}=1$.
Then (66) reduces to proving

(67)
$\displaystyle\displaystyle(\widehat{f}_{12})_{*}(X_{12},(\widehat{{\mathcal{U}}}_{12})_{0},\widehat{\chi}^{1}_{{\mathfrak{p}}_{1}}\widehat{\chi}^{2}_{{\mathfrak{p}}_{2}}\widehat{\omega}_{12},\widehat{{\mathfrak{S}}}^{\varepsilon}_{12})=$
$\displaystyle\displaystyle\qquad\qquad(\widehat{f}_{1})_{*}(X_{1},(\widehat{{\mathcal{U}}}_{1})_{0},\widehat{\chi}^{1}_{{\mathfrak{p}}_{1}}\widehat{\omega}_{1},\widehat{{\mathfrak{S}}}^{\varepsilon}_{1})\circ_{j}(\widehat{f}_{2})_{*}(X_{2},(\widehat{{\mathcal{U}}}_{2})_{0},\widehat{\chi}^{2}_{{\mathfrak{p}}_{2}}\widehat{\omega}_{2},\widehat{{\mathfrak{S}}}^{\varepsilon}_{2})$

for every $\displaystyle{\mathfrak{p}}_{1}\in{\mathfrak{P}}_{1}$ and
$\displaystyle{\mathfrak{p}}_{2}\in{\mathfrak{P}}_{2}$. Now there holds

$\displaystyle(\widehat{f}_{1})_{*}(X_{1},(\widehat{{\mathcal{U}}}_{1})_{0},\widehat{\chi}^{1}_{{\mathfrak{p}}_{1}}\widehat{\omega}_{1},\widehat{{\mathfrak{S}}}^{\varepsilon}_{1})=((f_{1})_{{\mathfrak{p}}_{1}})_{*}(({\mathcal{U}}_{1})_{{\mathfrak{p}}_{1}},(\chi_{1})_{{\mathfrak{p}}_{1}}(\omega_{1})_{{\mathfrak{p}}_{1}},({\mathfrak{S}}_{1})^{\varepsilon}_{{\mathfrak{p}}_{1}}),$

and similar formulas hold for
$\displaystyle(\widehat{f}_{12})_{*}(X_{12},(\widehat{{\mathcal{U}}}_{12})_{0},\widehat{\chi}^{1}_{{\mathfrak{p}}_{1}}\widehat{\chi}^{2}_{{\mathfrak{p}}_{2}}\widehat{\omega}_{12},\widehat{{\mathfrak{S}}}^{\varepsilon}_{12})$
and
$\displaystyle(\widehat{f}_{2})_{*}(X_{2},(\widehat{{\mathcal{U}}}_{2})_{0},\widehat{\chi}^{2}_{{\mathfrak{p}}_{2}}\widehat{\omega}_{2},\widehat{{\mathfrak{S}}}^{\varepsilon}_{2})$.
Then (67) follows from the fiber product formula for single K-charts (Lemma
8.3). This completes a sketch of the proof of Theorem 7.6.

##  9\. Proof of $\displaystyle C^{0}$-approximation lemma (Theorem 7.31)

Recall that $\displaystyle{\mathcal{L}}_{k+1}$ is a subspace of
$\displaystyle\Pi^{k+1}$, which consists of
$\displaystyle(\Gamma_{0},\ldots,\Gamma_{k})\in\Pi^{k+1}$ such that
$\displaystyle\text{\rm ev}\,_{1}(\Gamma_{i})=\text{\rm
ev}\,_{0}(\Gamma_{i+1})\,(0\leq i\leq k-1)$ and $\displaystyle\text{\rm
ev}\,_{1}(\Gamma_{k})=\text{\rm ev}\,_{0}(\Gamma_{0})$. Then, Theorem 7.31 is
reduced to Lemma 9.1 below. We first give a proof of Lemma 9.1 assuming Lemma
9.2 (which is stated in the proof), and proceed to a proof of Lemma 9.2.

######  Lemma 9.1.

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a compact K-space and
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to\Pi^{\text{\rm
con}}\,$ be a strongly continuous map such that $\displaystyle\text{\rm
ev}\,_{j}\circ\widehat{f}:(X,\widehat{{\mathcal{U}}})\to L$ is strongly smooth
for $\displaystyle j=0,1$.

Let $\displaystyle Z$ be a closed subset of $\displaystyle X$ and
$\displaystyle\widehat{g}:(Z,\widehat{{\mathcal{U}}}|_{Z})\to\Pi$ be a
strongly smooth map (the notion of smooth map from a K-space to
$\displaystyle\Pi$ is defined in the same way as Definition 7.1) such that
$\displaystyle\text{\rm ev}\,_{j}\circ\widehat{g}=\text{\rm
ev}\,_{j}\circ\widehat{f}|_{Z}$ for $\displaystyle j=0,1$ and
$\displaystyle\widehat{g}$ is $\displaystyle\varepsilon$-close to
$\displaystyle\widehat{f}|_{Z}$ (with respect to $\displaystyle d_{\Pi}$).

If $\displaystyle\varepsilon<\rho_{L}$, there exist an open substructure
$\displaystyle\widehat{{\mathcal{U}}_{0}}$ of
$\displaystyle\widehat{{\mathcal{U}}}$ and a strongly smooth map
$\displaystyle\widehat{g^{\prime}}:(X,\widehat{{\mathcal{U}}_{0}})\to\Pi$ such
that the following conditions hold:

  * •

$\displaystyle\widehat{g^{\prime}}$ is $\displaystyle\varepsilon$-close to
$\displaystyle\widehat{f}|_{\widehat{{\mathcal{U}}_{0}}}$.

  * •

$\displaystyle\text{\rm ev}\,_{j}\circ\widehat{g^{\prime}}=\text{\rm
ev}\,_{j}\circ\widehat{f}|_{\widehat{{\mathcal{U}}_{0}}}$ for $\displaystyle
j=0,1$.

  * •

$\displaystyle\widehat{g^{\prime}}=\widehat{g}$ on
$\displaystyle\widehat{{\mathcal{U}}_{0}}|_{Z}$.

###### Proof.

Step 1. There exist

  * •

a GCS $\displaystyle\widetriangle{{\mathcal{U}}_{Z}}$ of $\displaystyle Z$,

  * •

a KG-embedding from $\displaystyle\widehat{{\mathcal{U}}}|_{Z}$ to
$\displaystyle\widetriangle{{\mathcal{U}}}_{Z}$, namely an open substructure
$\displaystyle\widehat{{\mathcal{U}}_{Z,0}}$ of
$\displaystyle\widehat{{\mathcal{U}}}|_{Z}$ and a strict KG-embedding
$\displaystyle\Phi_{1}:\widehat{{\mathcal{U}}_{Z,0}}\to\widetriangle{{\mathcal{U}}_{Z}}$,

  * •

a strongly continuous map
$\displaystyle\widetriangle{f_{Z}}:(Z,\widetriangle{{\mathcal{U}}}_{Z})\to\Pi^{\text{\rm
con}}\,$,

  * •

a strongly smooth map
$\displaystyle\widetriangle{g}:(Z,\widetriangle{{\mathcal{U}}}_{Z})\to\Pi$,

such that

  * •

$\displaystyle\Phi_{1}^{*}\widetriangle{f}_{Z}=\widehat{f}|_{\widehat{{\mathcal{U}}_{Z,0}}}$
and
$\displaystyle\Phi_{1}^{*}\widetriangle{g}=\widehat{g}|_{\widehat{{\mathcal{U}}_{Z,0}}}$,

  * •

$\displaystyle\widetriangle{g}$ is $\displaystyle\varepsilon$-close to
$\displaystyle\widetriangle{f}_{Z}$,

  * •

$\displaystyle\text{\rm ev}\,_{j}\circ\widetriangle{g}=\text{\rm
ev}\,_{j}\circ\widetriangle{f}_{Z}$ for $\displaystyle j=0,1$.

The GCS $\displaystyle\widetriangle{{\mathcal{U}}_{Z}}$ exists by Theorem 3.30
in [10]. By inspecting its proof (Section 11.1 in [10]), one can take
$\displaystyle\widetriangle{{\mathcal{U}}_{Z}}$ so that each chart of
$\displaystyle\widetriangle{{\mathcal{U}}_{Z}}$ is an open subchart of a
certain K-chart of $\displaystyle\widehat{{\mathcal{U}}_{Z}}$, thus one can
define $\displaystyle\widetriangle{g}$ and $\displaystyle\widetriangle{f}_{Z}$
by pulling back $\displaystyle\widehat{f}$ and $\displaystyle\widehat{g}$.

Step 2. By Proposition 7.52 and Lemma 7.53 in [10], there exist

  * •

a GCS $\displaystyle\widetriangle{{\mathcal{U}}}$ of $\displaystyle X$.

  * •

a KG-embedding from $\displaystyle\widehat{{\mathcal{U}}}$ to
$\displaystyle\widetriangle{{\mathcal{U}}}$, namely and open substructure
$\displaystyle\widehat{{\mathcal{U}}_{0,+}}$ of
$\displaystyle\widehat{{\mathcal{U}}}$ and a strict KG-embedding
$\displaystyle\Phi_{2}:\widehat{{\mathcal{U}}_{0,+}}\to\widetriangle{{\mathcal{U}}}$,

  * •

an extension from $\displaystyle\widetriangle{{\mathcal{U}}_{Z}}$ to
$\displaystyle\widetriangle{{\mathcal{U}}}$ (see Definition 7.50 of [10]),
namely an open substructure $\displaystyle\widetriangle{{\mathcal{U}}_{Z,0}}$
of $\displaystyle\widetriangle{{\mathcal{U}}_{Z}}$ and a strict extension
$\displaystyle\Phi_{3}:\widetriangle{{\mathcal{U}}_{Z,0}}\to\widetriangle{{\mathcal{U}}}$,

  * •

a strongly continuous map
$\displaystyle\widetriangle{f}:(X,\widetriangle{{\mathcal{U}}})\to\Pi^{\text{\rm
con}}\,$ such that
$\displaystyle\Phi_{2}^{*}\widetriangle{f}=\widehat{f}|_{\widehat{{\mathcal{U}}_{0,+}}}$
and
$\displaystyle\Phi_{3}^{*}\widetriangle{f}=\widetriangle{f_{Z}}|_{\widetriangle{{\mathcal{U}}_{Z,0}}}$.

Now we can state a $\displaystyle C^{0}$-approximation result for GCS in Lemma
9.2 below, which we assume for the moment.

######  Lemma 9.2.

There exist

  * •

$\displaystyle\widetriangle{{\mathcal{U}}_{Z,00}}$: an open substructure of
$\displaystyle\widetriangle{{\mathcal{U}}_{Z,0}}$,

  * •

$\displaystyle\widetriangle{{\mathcal{U}}_{0}}$: an open substructure of
$\displaystyle\widetriangle{{\mathcal{U}}}$,

  * •

a strict extension from $\displaystyle\widetriangle{{\mathcal{U}}_{Z,00}}$ to
$\displaystyle\widetriangle{{\mathcal{U}}_{0}}$,

  * •

a strongly smooth map
$\displaystyle\widetriangle{g^{\prime}}:(X,\widetriangle{{\mathcal{U}}_{0}})\to\Pi$
such that

    * –

$\displaystyle\widetriangle{g^{\prime}}$ is $\displaystyle\varepsilon$-close
to $\displaystyle\widetriangle{f}|_{\widetriangle{{\mathcal{U}}_{0}}}$,

    * –

$\displaystyle\text{\rm ev}\,_{j}\circ\widetriangle{g^{\prime}}=\text{\rm
ev}\,_{j}\circ\widetriangle{f}\,(j=0,1)$ on
$\displaystyle\widetriangle{{\mathcal{U}}_{0}}$,

    * –

$\displaystyle\widetriangle{g^{\prime}}=\widetriangle{g}$ on
$\displaystyle\widetriangle{{\mathcal{U}}_{Z,00}}$.

Step 3. Take an open substructure $\displaystyle\widehat{{\mathcal{U}}_{0}}$
of $\displaystyle\widehat{{\mathcal{U}}}$ with a strict KG-embedding
$\displaystyle\widehat{{\mathcal{U}}_{0}}\to\widetriangle{{\mathcal{U}}_{0}}$
such that
$\displaystyle\widehat{{\mathcal{U}}_{0}}|_{Z}\to\widetriangle{{\mathcal{U}}_{0}}$
factors through a strict KG-embedding
$\displaystyle\widehat{{\mathcal{U}}_{0}}|_{Z}\to\widetriangle{{\mathcal{U}}_{Z,00}}$.
Finally, we can define
$\displaystyle\widehat{g^{\prime}}:(X,\widehat{{\mathcal{U}}_{0}})\to\Pi$ by
pulling back
$\displaystyle\widetriangle{g^{\prime}}:(X,\widetriangle{{\mathcal{U}}_{0}})\to\Pi$
by
$\displaystyle\widehat{{\mathcal{U}}_{0}}\to\widetriangle{{\mathcal{U}}_{0}}$.
∎

The rest of this section is devoted to the proof of Lemma 9.2. First we need
to prove Lemmas 9.3, 9.4 and 9.5.

######  Lemma 9.3.

For any nonempty finite set $\displaystyle I$, there exists a $\displaystyle
C^{\infty}$-map

$\displaystyle G:\\{(x_{i},t_{i})_{i\in I}\mid x_{i}\in
L,\,t_{i}\in[0,1],\,\max_{i,j\in I}d_{L}(x_{i},x_{j})<\rho_{L},\,\sum_{i\in
I}t_{i}=1\\}\to L$

such that the following properties hold:

  1. (i):

If $\displaystyle t_{i}=\begin{cases}1&(i=i_{0})\\\ 0&(i\neq
i_{0})\end{cases}$ for some $\displaystyle i_{0}\in I$, then $\displaystyle
G(x_{i},t_{i})_{i}=x_{i_{0}}$.

  2. (ii):

If there exists $\displaystyle y\in L$ and $\displaystyle r\in(0,\rho_{L}]$
such that $\displaystyle d_{L}(y,x_{i})<r\,(\forall i\in I)$, then
$\displaystyle d_{L}(y,G(x_{i},t_{i})_{i})<r$.

###### Proof.

We fix an arbitrary total order on $\displaystyle I$. For each $\displaystyle
t=(t_{i})_{i\in I}$, let $\displaystyle i_{0}:=\max\\{i\mid t_{i}\neq 0\\}$,
and for every $\displaystyle\theta\in[0,1]$ let

$\displaystyle(t^{\theta})_{i}:=\begin{cases}t_{i}(1-\theta
t_{i_{0}})/(1-t_{i_{0}})&(i<i_{0})\\\ \theta t_{i_{0}}&(i=i_{0})\\\
0&(i>i_{0}).\end{cases}$

Now we define $\displaystyle G$ so that it satisfies (i), and for any
$\displaystyle x=(x_{i})_{i}$, the map

$\displaystyle[0,1]\to L;\,\theta\mapsto G(x,t^{\theta})$

is the shortest geodesic connecting the end points. Now $\displaystyle G$
satisfies (ii) since any geodesic ball with radius $\displaystyle
r\in(0,\rho_{L}]$ is geodesically convex, by the definition of
$\displaystyle\rho_{L}$ (see Section 7.3). ∎

######  Lemma 9.4.

For any $\displaystyle(T,\gamma)\in\Pi^{\text{\rm con}}\,$ and
$\displaystyle\varepsilon>0$, there exists
$\displaystyle(T^{\prime},\gamma^{\prime})\in\Pi$ such that $\displaystyle
d_{\Pi}((T,\gamma),(T^{\prime},\gamma^{\prime}))<\varepsilon$.

###### Proof.

Take $\displaystyle T^{\prime}:=T+\varepsilon/2$, and
$\displaystyle\gamma^{\prime}:[0,T^{\prime}]\to L$ so that $\displaystyle
d_{L}(\gamma(sT),\gamma^{\prime}(sT^{\prime}))<\varepsilon/2$ for any
$\displaystyle s\in[0,1]$, which is possible since $\displaystyle
C^{\infty}([0,1],L)$ is dense in $\displaystyle C^{0}([0,1],L)$ with respect
to the $\displaystyle C^{0}$-topology. ∎

######  Lemma 9.5.

Let $\displaystyle U$ be a $\displaystyle C^{\infty}$-manifold, $\displaystyle
f:U\to\Pi^{\text{\rm con}}\,$ be a continuous map, such that
$\displaystyle\text{\rm ev}\,_{j}\circ f:U\to L$ is of $\displaystyle
C^{\infty}$ for $\displaystyle j=0,1$.

Let $\displaystyle V$ be a submanifold of $\displaystyle U$, $\displaystyle
g:V\to\Pi$ be a smooth map such that $\displaystyle\text{\rm ev}\,_{j}\circ
g=\text{\rm ev}\,_{j}\circ f|_{V}\,(j=0,1)$ and $\displaystyle g$ is
$\displaystyle\varepsilon$-close to $\displaystyle f|_{V}$.

Then for any $\displaystyle x\in V$, there exists an open neighborhood
$\displaystyle W$ of $\displaystyle x$ in $\displaystyle U$ and a smooth map
$\displaystyle g^{\prime}:W\to\Pi$ such that $\displaystyle g^{\prime}=g$ on
$\displaystyle W\cap V$, $\displaystyle\text{\rm ev}\,_{j}\circ
g^{\prime}=\text{\rm ev}\,_{j}\circ f|_{W}\,(j=0,1)$, and $\displaystyle
g^{\prime}$ is $\displaystyle\varepsilon$-close to $\displaystyle f|_{W}$.

###### Proof.

The last condition “$\displaystyle g^{\prime}$ is
$\displaystyle\varepsilon$-close to $\displaystyle f|_{W}$” can be achieved by
taking $\displaystyle W$ sufficiently small, since $\displaystyle f$ is
continuous and $\displaystyle\varepsilon$-closeness is an open condition. Thus
it is sufficient to define $\displaystyle W$ and a smooth map $\displaystyle
g^{\prime}:W\to\Pi$ such that $\displaystyle g^{\prime}=g$ on $\displaystyle
W\cap V$ and $\displaystyle\text{\rm ev}\,_{j}\circ g^{\prime}=\text{\rm
ev}\,_{j}\circ f|_{W}\,(j=0,1)$.

Let $\displaystyle W$ be a sufficiently small neighborhood of $\displaystyle
x$ , so that there exists a $\displaystyle C^{\infty}$-map $\displaystyle
r:W\to W\cap V$ satisfying $\displaystyle r|_{W\cap V}=\text{\rm id}_{W\cap
V}$, and

$\displaystyle d_{L}(\text{\rm ev}\,_{j}(f(y)),\text{\rm ev}\,_{j}(g\circ
r(y)))<r_{\text{inj}}(L)$

for every $\displaystyle y\in W$ and $\displaystyle j\in\\{0,1\\}$, where
$\displaystyle r_{\text{inj}}(L)$ denotes the injectivity radius of
$\displaystyle L$. We define $\displaystyle\xi_{j}(y)\in T_{\text{\rm
ev}\,_{j}(g\circ r(y))}L$ by $\displaystyle\exp(\xi_{j}(y))=\text{\rm
ev}\,_{j}(f(y))$.

We set $\displaystyle g(z):=(T(z),\gamma(z))$ for any $\displaystyle z\in
W\cap V$. For every $\displaystyle y\in W$ and $\displaystyle j\in\\{0,1\\}$,
we define $\displaystyle\xi^{y}_{j}(\theta)\in
T_{\gamma(r(y))(\theta)}L\,(0\leq\theta\leq T(r(y)))$ so that

$\displaystyle\xi^{y}_{0}(0)=\xi_{0}(y),\quad\nabla_{\theta}\xi^{y}_{0}\equiv
0,\quad\xi^{y}_{1}(T(r(y))=\xi_{1}(y),\quad\nabla_{\theta}\xi^{y}_{1}\equiv
0.$

Taking a $\displaystyle C^{\infty}$-function $\displaystyle\chi:[0,1]\to[0,1]$
such that $\displaystyle\chi\equiv 0$ near $\displaystyle 0$ and
$\displaystyle\chi\equiv 1$ near $\displaystyle 1$, we define
$\displaystyle\xi^{\prime}(y)\in C^{\infty}(\gamma(r(y))^{*}TL)$ by

$\displaystyle\xi^{\prime}(y)(\theta):=\chi(\theta/T(r(y)))\cdot\xi^{y}_{1}(\theta)+(1-\chi(\theta/T(r(y))))\cdot\xi^{y}_{0}(\theta)\qquad(0\leq\theta\leq
T(r(y))).$

Finally, we define $\displaystyle\gamma^{\prime}(y):[0,T(r(y))]\to L$ by

$\displaystyle\gamma^{\prime}(y)(\theta):=\exp(\gamma(r(y))(\theta),\xi^{\prime}(y)(\theta))$

and $\displaystyle g^{\prime}(y):=(T(r(y)),\gamma^{\prime}(y))$. It is easy to
check that this map $\displaystyle g^{\prime}$ is smooth and satisfies
required conditions. ∎

Now let us start the proof of Lemma 9.2. Let $\displaystyle{\mathfrak{P}}$
denote the index set of $\displaystyle\widetriangle{{\mathcal{U}}}$, and
$\displaystyle{\mathfrak{P}}_{Z}\subset{\mathfrak{P}}$ denote the index set of
$\displaystyle\widetriangle{{\mathcal{U}}_{Z}}$. Let
$\displaystyle\\{{\mathcal{U}}_{{\mathfrak{p}}}\\}_{{\mathfrak{p}}\in{\mathfrak{P}}}$
be K-charts of $\displaystyle\widetriangle{{\mathcal{U}}}$, and we denote
$\displaystyle{\mathcal{U}}_{{\mathfrak{p}}}=(U_{{\mathfrak{p}}},\ldots)$ for
each $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$. Let
$\displaystyle\\{{\mathcal{U}}^{Z}_{{\mathfrak{p}}}\\}_{{\mathfrak{p}}\in{\mathfrak{P}}_{Z}}$
be K-charts of $\displaystyle\widetriangle{{\mathcal{U}}_{Z,0}}$ and we denote
$\displaystyle{\mathcal{U}}^{Z}_{{\mathfrak{p}}}=(U^{Z}_{{\mathfrak{p}}},\ldots)$
for each $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{Z}$. We also take
support systems
$\displaystyle\\{{\mathcal{K}}_{{\mathfrak{p}}}\\}_{{\mathfrak{p}}\in{\mathfrak{P}}}$
of $\displaystyle\widetriangle{{\mathcal{U}}}$ and
$\displaystyle\\{{\mathcal{K}}^{Z}_{{\mathfrak{p}}}\\}_{{\mathfrak{p}}\in{\mathfrak{P}}_{Z}}$
of $\displaystyle\widetriangle{{\mathcal{U}}_{Z,0}}$, such that
$\displaystyle(\varphi_{3})_{{\mathfrak{p}}}({\mathcal{K}}^{Z}_{{\mathfrak{p}}})\subset{\mathring{{\mathcal{K}}}}_{{\mathfrak{p}}}$
for every $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{Z}$, where
$\displaystyle\Phi_{3}=(\varphi_{3},\widehat{\varphi}_{3})$ is a strict
extension from $\displaystyle\widetriangle{{\mathcal{U}}_{Z,0}}$ to
$\displaystyle\widetriangle{{\mathcal{U}}}$.

A subset $\displaystyle{\mathfrak{F}}\subset{\mathfrak{P}}$ is called a filter
if $\displaystyle{\mathfrak{p}},{\mathfrak{q}}\in{\mathfrak{P}}$,
$\displaystyle{\mathfrak{p}}\geq{\mathfrak{q}}$,
$\displaystyle{\mathfrak{p}}\in{\mathfrak{F}}$ imply
$\displaystyle{\mathfrak{q}}\in{\mathfrak{F}}$. In particular the empty set is
a filter (see Section 12.3 in [10]). Now Lemma 9.2 is proved by applying Lemma
9.6 below for $\displaystyle{\mathfrak{F}}={\mathfrak{P}}$, and setting

$\displaystyle({\mathcal{U}}_{0})_{{\mathfrak{p}}}:={\mathcal{U}}_{{\mathfrak{p}}}|_{V_{{\mathfrak{p}}}}\,(\forall{\mathfrak{p}}\in{\mathfrak{P}}),\quad({\mathcal{U}}_{Z,00})_{{\mathfrak{p}}}:=({\mathcal{U}}_{Z,0})_{{\mathfrak{p}}}|_{W_{{\mathfrak{p}}}}\,(\forall{\mathfrak{p}}\in{\mathfrak{P}}_{Z}).$

######  Lemma 9.6.

For any filter $\displaystyle{\mathfrak{F}}$ of $\displaystyle{\mathfrak{P}}$,
there exist
$\displaystyle(V_{{\mathfrak{p}}},g^{\prime}_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{F}}}$
and $\displaystyle(W_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}_{Z}}$
such that the following conditions are satisfied:

  * •

$\displaystyle V_{{\mathfrak{p}}}$ is an open neighborhood of
$\displaystyle{\mathcal{K}}_{{\mathfrak{p}}}$ in $\displaystyle
U_{{\mathfrak{p}}}$ for every $\displaystyle{\mathfrak{p}}\in{\mathfrak{F}}$.

  * •

$\displaystyle g^{\prime}_{{\mathfrak{p}}}:V_{{\mathfrak{p}}}\to\Pi$ is a
smooth map. Moreover, $\displaystyle g^{\prime}_{{\mathfrak{p}}}$ is
$\displaystyle\varepsilon$-close to $\displaystyle
f_{{\mathfrak{p}}}|_{V_{{\mathfrak{p}}}}$ and $\displaystyle\text{\rm
ev}\,_{j}\circ g^{\prime}_{{\mathfrak{p}}}=\text{\rm ev}\,_{j}\circ
f_{{\mathfrak{p}}}|_{V_{{\mathfrak{p}}}}$ for $\displaystyle j=0,1$.

  * •

$\displaystyle(g^{\prime}_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{F}}}$
is compatible with coordinate changes. Specifically, for any
$\displaystyle{\mathfrak{p}},{\mathfrak{p}}^{\prime}\in{\mathfrak{F}}$ such
that $\displaystyle{\mathfrak{p}}\geq{\mathfrak{p}}^{\prime}$, there holds
$\displaystyle
g_{{\mathfrak{p}}^{\prime}}=g_{{\mathfrak{p}}}\circ\varphi_{{\mathfrak{p}}{\mathfrak{p}}^{\prime}}$
on $\displaystyle
V_{{\mathfrak{p}}^{\prime}}\cap\varphi_{{\mathfrak{p}}{\mathfrak{p}}^{\prime}}^{-1}(V_{{\mathfrak{p}}})$.

  * •

$\displaystyle W_{{\mathfrak{p}}}$ is an open neighborhood of
$\displaystyle{\mathcal{K}}^{Z}_{{\mathfrak{p}}}$ in $\displaystyle
U^{Z}_{{\mathfrak{p}}}$ for every
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{Z}$.

  * •

$\displaystyle(g^{\prime}_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{F}}}$
is compatible with $\displaystyle\widetriangle{g}$. Specifically:

    * –

For any $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{Z}\cap{\mathfrak{F}}$,
there holds
$\displaystyle(\varphi_{3})_{{\mathfrak{p}}}(W_{{\mathfrak{p}}})\subset
V_{{\mathfrak{p}}}$ and $\displaystyle
g^{\prime}_{{\mathfrak{p}}}=g_{{\mathfrak{p}}}\circ(\varphi_{3})_{{\mathfrak{p}}}$
on $\displaystyle W_{{\mathfrak{p}}}$.

    * –

For any
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}_{Z}\setminus{\mathfrak{F}}$ and
$\displaystyle{\mathfrak{p}}\in{\mathfrak{F}}$ satisfying
$\displaystyle{\mathfrak{p}}<{\mathfrak{q}}$, there holds $\displaystyle
g_{{\mathfrak{q}}}\circ\varphi_{{\mathfrak{q}}{\mathfrak{p}}}=g^{\prime}_{{\mathfrak{p}}}$
on $\displaystyle
V_{{\mathfrak{p}}}\cap(\varphi_{{\mathfrak{q}}{\mathfrak{p}}})^{-1}(W_{{\mathfrak{q}}})$.

###### Proof.

The proof is by induction on the cardinality of $\displaystyle{\mathfrak{F}}$.
There is nothing to prove when $\displaystyle{\mathfrak{F}}=\emptyset$. To
discuss the induction step, let $\displaystyle{\mathfrak{F}}$ be a filter,
$\displaystyle{\mathfrak{p}}_{0}$ be its maximal element (namely
$\displaystyle{\mathfrak{p}}\in{\mathfrak{F}},{\mathfrak{p}}\geq{\mathfrak{p}}_{0}\implies{\mathfrak{p}}={\mathfrak{p}}_{0}$),
and suppose that there exist
$\displaystyle(V_{{\mathfrak{p}}},g^{\prime}_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\}}$
and $\displaystyle(W_{{\mathfrak{q}}})_{{\mathfrak{q}}\in{\mathfrak{P}}_{Z}}$
satisfying the conditions in the lemma.

For each $\displaystyle x\in{\mathcal{K}}_{{\mathfrak{p}}_{0}}$, we take an
open neighborhood $\displaystyle W_{x}$ of $\displaystyle x$ in $\displaystyle
U_{{\mathfrak{p}}_{0}}$ and a smooth map $\displaystyle
g^{\prime}_{x}:W_{x}\to\Pi$ in the way described below. We consider three
cases (here we consider $\displaystyle x$ as a point in
$\displaystyle|\widetriangle{{\mathcal{U}}}|=\bigsqcup_{{\mathfrak{p}}\in{\mathfrak{P}}}U_{{\mathfrak{p}}}/\sim$,
and $\displaystyle{\mathcal{K}}^{Z}_{{\mathfrak{p}}}$,
$\displaystyle{\mathcal{K}}_{{\mathfrak{p}}}$ as subspaces of
$\displaystyle|\widetriangle{{\mathcal{U}}}|$).

  * (i):

There exists
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}_{Z}\setminus({\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\})$
such that $\displaystyle x\in{\mathcal{K}}^{Z}_{{\mathfrak{q}}}$.

  * (ii):

$\displaystyle x\notin{\mathcal{K}}^{Z}_{{\mathfrak{q}}}$ for any
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}_{Z}\setminus({\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\})$,
but there exists
$\displaystyle{\mathfrak{q}}\in{\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\}$
such that $\displaystyle x\in{\mathcal{K}}_{{\mathfrak{q}}}$.

  * (iii):

$\displaystyle x\notin{\mathcal{K}}^{Z}_{{\mathfrak{q}}}$ for any
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}_{Z}\setminus({\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\})$,
and $\displaystyle x\notin{\mathcal{K}}_{{\mathfrak{q}}}$ for any
$\displaystyle{\mathfrak{q}}\in{\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\}$.

In case (i), take maximal
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}_{Z}\setminus({\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\})$
such that $\displaystyle x\in{\mathcal{K}}^{Z}_{{\mathfrak{q}}}$. This
condition implies $\displaystyle{\mathfrak{q}}>{\mathfrak{p}}_{0}$ since
$\displaystyle U_{{\mathfrak{q}}}\cap U_{{\mathfrak{p}}_{0}}\neq\emptyset$ and
$\displaystyle{\mathfrak{q}}\in{\mathfrak{F}}$. Then take $\displaystyle
W_{x}$ such that:

  * •

$\displaystyle\overline{W_{x}}\cap{\mathcal{K}}^{Z}_{{\mathfrak{q}}^{\prime}}=\emptyset$
for any
$\displaystyle{\mathfrak{q}}^{\prime}\in{\mathfrak{P}}_{Z}\setminus({\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\})$
which does not satisfy
$\displaystyle{\mathfrak{q}}^{\prime}\leq{\mathfrak{q}}$ (note this condition
implies that
$\displaystyle{\mathcal{K}}^{Z}_{{\mathfrak{q}}}\cap{\mathcal{K}}^{Z}_{{\mathfrak{q}}^{\prime}}=\emptyset$),

  * •

$\displaystyle
W_{x}\subset(\varphi_{{\mathfrak{q}}{\mathfrak{p}}_{0}})^{-1}(W_{{\mathfrak{q}}})$.

Then we define $\displaystyle g^{\prime}_{x}:W_{x}\to\Pi$ by $\displaystyle
g^{\prime}_{x}:=g_{{\mathfrak{q}}}\circ\varphi_{{\mathfrak{q}}{\mathfrak{p}}_{0}}|_{W_{x}}$.

In case (ii), take maximal
$\displaystyle{\mathfrak{q}}\in{\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\}$
such that $\displaystyle x\in{\mathcal{K}}_{{\mathfrak{q}}}$. This condition
implies $\displaystyle{\mathfrak{q}}<{\mathfrak{p}}_{0}$. Then take
$\displaystyle W_{x}$ such that:

  * •

$\displaystyle\overline{W_{x}}\cap{\mathcal{K}}_{{\mathfrak{q}}^{\prime}}=\emptyset$
for any
$\displaystyle{\mathfrak{q}}^{\prime}\in{\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\}$
which does not satisfy
$\displaystyle{\mathfrak{q}}^{\prime}\leq{\mathfrak{q}}$ (note this condition
implies that
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}\cap{\mathcal{K}}_{{\mathfrak{q}}^{\prime}}=\emptyset$),

  * •

$\displaystyle\overline{W_{x}}\cap{\mathcal{K}}^{Z}_{{\mathfrak{q}}^{\prime}}=\emptyset$
for any
$\displaystyle{\mathfrak{q}}^{\prime}\in{\mathfrak{P}}_{Z}\setminus({\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\})$,

  * •

$\displaystyle(\varphi_{{\mathfrak{p}}_{0}{\mathfrak{q}}})^{-1}(W_{x})\subset
V_{{\mathfrak{q}}}$.

When $\displaystyle W_{x}$ is sufficiently small, Lemma 9.5 shows that there
exists a smooth map $\displaystyle g^{\prime}_{x}:W_{x}\to\Pi$ such that

  * •

$\displaystyle g^{\prime}_{x}$ is $\displaystyle\varepsilon$-close to
$\displaystyle f_{{\mathfrak{p}}_{0}}|_{W_{x}}$.

  * •

$\displaystyle\text{\rm ev}\,_{j}\circ g^{\prime}_{x}=\text{\rm ev}\,_{j}\circ
f_{{\mathfrak{p}}_{0}}|_{W_{x}}$ for $\displaystyle j=0,1$.

  * •

$\displaystyle
g^{\prime}_{x}\circ\varphi_{{\mathfrak{p}}_{0}{\mathfrak{q}}}=g^{\prime}_{{\mathfrak{q}}}$
on $\displaystyle(\varphi_{{\mathfrak{p}}_{0}{\mathfrak{q}}})^{-1}(W_{x})$.

In case (iii), take $\displaystyle W_{x}$ so that
$\displaystyle\overline{W_{x}}\cap{\mathcal{K}}_{{\mathfrak{q}}}=\emptyset$
for every
$\displaystyle{\mathfrak{q}}\in{\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\}$,
and
$\displaystyle\overline{W_{x}}\cap{\mathcal{K}}^{Z}_{{\mathfrak{q}}}=\emptyset$
for every
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}_{Z}\setminus({\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\})$.
When $\displaystyle W_{x}$ is sufficiently small, Lemmas 9.4 and 9.5 (applied
to $\displaystyle V=\\{x\\}$) show that there exists a smooth map
$\displaystyle g^{\prime}_{x}:W_{x}\to\Pi$ such that

  * •

$\displaystyle g^{\prime}_{x}$ is $\displaystyle\varepsilon$-close to
$\displaystyle f_{{\mathfrak{p}}_{0}}|_{W_{x}}$.

  * •

$\displaystyle\text{\rm ev}\,_{j}\circ g^{\prime}_{x}=\text{\rm ev}\,_{j}\circ
f_{{\mathfrak{p}}}|_{W_{x}}$ for $\displaystyle j=0,1$.

Since $\displaystyle{\mathcal{K}}_{{\mathfrak{p}}_{0}}$ is compact, one can
take finitely many points $\displaystyle\\{x_{i}\\}_{i\in I}$ such that
$\displaystyle\\{W_{x_{i}}\\}_{i\in I}$ covers
$\displaystyle{\mathcal{K}}_{{\mathfrak{p}}_{0}}$. For each $\displaystyle
i\in I$, we take $\displaystyle g^{\prime}_{x_{i}}:W_{x_{i}}\to\Pi$ and denote
it as $\displaystyle g^{\prime}_{x_{i}}=(\gamma_{i},T_{i})$. Let us take a
$\displaystyle C^{\infty}$-function
$\displaystyle\chi_{i}:U_{{\mathfrak{p}}_{0}}\to{\mathbb{R}}_{\geq 0}$ for
each $\displaystyle i\in I$, such that $\displaystyle\text{\rm
supp}\,\chi_{i}\subset W_{x_{i}}$ and $\displaystyle\sum_{i\in I}\chi_{i}=1$
on a neighborhood of $\displaystyle{\mathcal{K}}_{{\mathfrak{p}}_{0}}$, which
we denote by $\displaystyle V_{{\mathfrak{p}}_{0}}$. Then we define
$\displaystyle g^{\prime}_{{\mathfrak{p}}_{0}}:V_{{\mathfrak{p}}_{0}}\to\Pi$
by $\displaystyle g^{\prime}_{{\mathfrak{p}}_{0}}:=(T,\gamma)$, such that
$\displaystyle T:=\sum_{i\in I}\chi_{i}T_{i}$, and for every $\displaystyle
y\in V_{{\mathfrak{p}}_{0}}$

$\displaystyle\gamma(y):[0,T(y)]\to L;\quad\theta\mapsto
G(\gamma_{i}(y)(T_{i}(y)\theta/T(y)),\chi_{i}(y))_{i\in I}$

where $\displaystyle G$ is defined in Lemma 9.3. If $\displaystyle i\in I$
satisfies $\displaystyle y\in V_{x_{i}}$, then $\displaystyle
g^{\prime}_{x_{i}}(y)$ is $\displaystyle\varepsilon$-close to $\displaystyle
f_{{\mathfrak{p}}_{0}}(y)$, and $\displaystyle\varepsilon<\rho_{L}$, thus
$\displaystyle g^{\prime}_{{\mathfrak{p}}_{0}}(y)$ is
$\displaystyle\varepsilon$-close to $\displaystyle f_{{\mathfrak{p}}_{0}}(y)$.

Now we can finish the proof by replacing $\displaystyle V_{{\mathfrak{p}}}$
with a smaller neighborhood of $\displaystyle{\mathcal{K}}_{{\mathfrak{p}}}$
for each
$\displaystyle{\mathfrak{p}}\in{\mathfrak{F}}\setminus\\{{\mathfrak{p}}_{0}\\}$,
and $\displaystyle W_{{\mathfrak{q}}}$ with a smaller neighborhood of
$\displaystyle{\mathcal{K}}^{Z}_{{\mathfrak{q}}}$ for each
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}_{Z}$. ∎

##  10\. Some basic notions in the theory of Kuranishi structures

Here we recall some basic notions in the theory of Kuranishi structures
(abbreviated as K-structures), mainly to fix notations used throughout this
paper. When we use notions which are not recalled here, we directly refer to
[10]. Throughout this section, $\displaystyle X$ denotes a separable,
metrizable topological space.

Kuranishi chart (K-chart)

A K-chart of $\displaystyle X$ is a tuple
$\displaystyle{\mathcal{U}}=(U,{\mathcal{E}},s,\psi)$ such that:

  * •

$\displaystyle U$ is a $\displaystyle C^{\infty}$-manifold,

  * •

$\displaystyle{\mathcal{E}}$ is a $\displaystyle C^{\infty}$-vector bundle on
$\displaystyle U$,

  * •

$\displaystyle s$ is a $\displaystyle C^{\infty}$-section of
$\displaystyle{\mathcal{E}}$,

  * •

$\displaystyle\psi:s^{-1}(0)\to X$ is a homeomorphism onto an open set of
$\displaystyle X$.

$\displaystyle\text{\rm dim}\,{\mathcal{U}}:=\text{\rm dim}\,U-\text{\rm
rk}\,{\mathcal{E}}$ is called the dimension of $\displaystyle{\mathcal{U}}$.
An orientation of $\displaystyle{\mathcal{U}}$ is a pair of orientations of
$\displaystyle U$ and $\displaystyle{\mathcal{E}}$. A K-chart at
$\displaystyle p\in X$ is a K-chart
$\displaystyle{\mathcal{U}}=(U,{\mathcal{E}},s,\psi)$ such that $\displaystyle
p\in\text{\rm Im}\,\psi$. We denote $\displaystyle o_{p}:=\psi^{-1}(p)\in
s^{-1}(0)$.

######  Remark 10.1.

In the standard definition (see Definition 3.1 in [10]), one assumes that
$\displaystyle U$ is an orbifold and $\displaystyle{\mathcal{E}}$ is an
orbibundle. However, in the present paper we are working with pseudo-
holomorphic disks without sphere bubbles, thus we do not need to take
quotients by finite group actions on moduli spaces, hence it is sufficient to
work with vector bundles on manifolds.

Embedding of K-charts

Let
$\displaystyle{\mathcal{U}}_{i}=(U_{i},{\mathcal{E}}_{i},s_{i},\psi_{i})\,(i=1,2)$
be K-charts of $\displaystyle X$. An embedding of K-charts
$\displaystyle\Phi:{\mathcal{U}}_{1}\to{\mathcal{U}}_{2}$ is a pair
$\displaystyle\Phi=(\varphi,\widehat{\varphi})$ such that:

  * •

$\displaystyle\varphi:U_{1}\to U_{2}$ is an embedding of $\displaystyle
C^{\infty}$-manifolds,

  * •

$\displaystyle\widehat{\varphi}:{\mathcal{E}}_{1}\to{\mathcal{E}}_{2}$ is an
embedding of $\displaystyle C^{\infty}$-vector bundles over
$\displaystyle\varphi$,

  * •

$\displaystyle\widehat{\varphi}\circ s_{1}=s_{2}\circ\varphi$,

  * •

$\displaystyle\psi_{2}\circ\varphi=\psi_{1}$ on $\displaystyle s_{1}^{-1}(0)$,

  * •

for any $\displaystyle x\in s_{1}^{-1}(0)$, the covariant derivative

(68)
$D_{\varphi(x)}s_{2}:\frac{T_{\varphi(x)}U_{2}}{(D_{x}\varphi)(T_{x}U_{1})}\to\frac{({\mathcal{E}}_{2})_{\varphi(x)}}{\widehat{\varphi}(({\mathcal{E}}_{1})_{x})}$

is an isomorphism.

When K-charts $\displaystyle{\mathcal{U}}_{1}$ and
$\displaystyle{\mathcal{U}}_{2}$ are oriented, we say that
$\displaystyle\Phi=(\varphi,\widehat{\varphi})$ is orientation preserving, if
an isomorphism

$\displaystyle\det TU_{2}\otimes(\det
TU_{1})^{\vee}\cong\det{\mathcal{E}}_{2}\otimes(\det{\mathcal{E}}_{1})^{\vee}$

induced by (68) preserves orientations.

Coordinate changes

Let
$\displaystyle{\mathcal{U}}_{i}=(U_{i},{\mathcal{E}}_{i},s_{i},\psi_{i})\,(i=1,2)$
be K-charts of $\displaystyle X$. A coordinate change in weak sense (resp.
strong sense) from $\displaystyle{\mathcal{U}}_{1}$ to
$\displaystyle{\mathcal{U}}_{2}$ is a triple
$\displaystyle\Phi_{21}=(U_{21},\varphi_{21},\widehat{\varphi}_{21})$
satisfying (i) and (ii) (resp. (i), (ii) and (iii)):

  * (i):

$\displaystyle U_{21}$ is an open subset of $\displaystyle U_{1}$,

  * (ii):

$\displaystyle(\varphi_{21},\widehat{\varphi}_{21})$ is an embedding of
K-charts $\displaystyle{\mathcal{U}}_{1}|_{U_{21}}\to{\mathcal{U}}_{2}$.

  * (iii):

$\displaystyle\psi_{1}(s_{1}^{-1}(0)\cap U_{21})=\text{\rm
Im}\,\psi_{1}\cap\text{\rm Im}\,\psi_{2}$.

Kuranishi structure (K-structure)

A K-structure $\displaystyle\widehat{{\mathcal{U}}}$ of $\displaystyle X$ (of
dimension $\displaystyle d$) consists of

  * •

a K-chart (of dimension $\displaystyle d$)
$\displaystyle{\mathcal{U}}_{p}=(U_{p},{\mathcal{E}}_{p},s_{p},\psi_{p})$ at
$\displaystyle p$ for every $\displaystyle p\in X$,

  * •

a coordinate change in weak sense
$\displaystyle\Phi_{pq}=(U_{pq},\varphi_{pq},\widehat{\varphi}_{pq}):{\mathcal{U}}_{q}\to{\mathcal{U}}_{p}$
for every $\displaystyle p\in X$ and $\displaystyle q\in\text{\rm
Im}\,(\psi_{p})$,

such that

  * •

$\displaystyle o_{q}\in U_{pq}$ for every $\displaystyle q\in\text{\rm
Im}\,\psi_{p}$,

  * •

for every $\displaystyle p\in X$, $\displaystyle q\in\text{\rm Im}\,\psi_{p}$
and $\displaystyle r\in\psi_{q}(s_{q}^{-1}(0)\cap U_{pq})$, there holds
$\displaystyle\Phi_{pr}|_{U_{pqr}}=\Phi_{pq}\circ\Phi_{qr}|_{U_{pqr}}$ where
$\displaystyle U_{pqr}:=\varphi^{-1}_{qr}(U_{pq})\cap U_{pr}$.

The pair $\displaystyle(X,\widehat{{\mathcal{U}}})$ is called a space with
Kuranishi structure (abbreviated as a K-space) of dimension $\displaystyle d$.
We say $\displaystyle\widehat{{\mathcal{U}}}$ is oriented if each K-chart
$\displaystyle{\mathcal{U}}_{p}$ is oriented and $\displaystyle\varphi_{pq}$
preserves orientations for every $\displaystyle p\in X$ and $\displaystyle
q\in\text{\rm Im}\,\psi_{p}$.

######  Remark 10.2.

In Definition 3.11 [10] the notion of a relative K-space
$\displaystyle(X,Z;\widehat{{\mathcal{U}}})$ is introduced, however in this
paper we only need the absolute case ($\displaystyle X=Z$).

Strongly continuous/smooth maps from K-spaces

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a K-space and $\displaystyle
Y$ be a topological space. A strongly continuous map
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to Y$ assigns a
continuous map $\displaystyle f_{p}:U_{p}\to Y$ for every $\displaystyle p\in
X$ such that $\displaystyle f_{p}\circ\varphi_{pq}=f_{q}$ on $\displaystyle
U_{pq}$ for every $\displaystyle p\in X$ and $\displaystyle q\in\text{\rm
Im}\,\psi_{p}$. When $\displaystyle Y$ has a structure of a $\displaystyle
C^{\infty}$-manifold, $\displaystyle\widehat{f}$ is called strongly smooth if
$\displaystyle f_{p}:U_{p}\to Y$ is of $\displaystyle C^{\infty}$ for every
$\displaystyle p\in X$. Moreover, $\displaystyle\widehat{f}$ is called weakly
submersive, if $\displaystyle f_{p}$ is a submersion for every $\displaystyle
p\in X$.

Good coordinate system (GCS)

Finally we recall the definition of a good coordinate system (GCS). A GCS
$\displaystyle\widetriangle{{\mathcal{U}}}$ of $\displaystyle X$ consists of

$\displaystyle(({\mathfrak{P}},\leq),\\{{\mathcal{U}}_{{\mathfrak{p}}}\\}_{{\mathfrak{p}}\in{\mathfrak{P}}},\\{\Phi_{{\mathfrak{p}}{\mathfrak{q}}}\\}_{{\mathfrak{q}}\leq{\mathfrak{p}}})$

such that:

  * •

$\displaystyle({\mathfrak{P}},\leq)$ is a finite partially ordered set.

  * •

$\displaystyle{\mathcal{U}}_{{\mathfrak{p}}}=(U_{{\mathfrak{p}}},{\mathcal{E}}_{{\mathfrak{p}}},s_{{\mathfrak{p}}},\psi_{{\mathfrak{p}}})$
is a K-chart for each $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$, and
$\displaystyle\bigcup_{{\mathfrak{p}}\in{\mathfrak{P}}}\text{\rm
Im}\,\psi_{{\mathfrak{p}}}=X$.

  * •

$\displaystyle\Phi_{{\mathfrak{p}}{\mathfrak{q}}}$ is a coordinate change
$\displaystyle{\mathcal{U}}_{{\mathfrak{q}}}\to{\mathcal{U}}_{{\mathfrak{p}}}$
in strong sense.

  * •

If $\displaystyle{\mathfrak{r}}\leq{\mathfrak{q}}\leq{\mathfrak{p}}$, there
holds
$\displaystyle\Phi_{{\mathfrak{p}}{\mathfrak{r}}}|_{U_{{\mathfrak{p}}{\mathfrak{q}}{\mathfrak{r}}}}=\Phi_{{\mathfrak{p}}{\mathfrak{q}}}\circ\Phi_{{\mathfrak{q}}{\mathfrak{r}}}|_{U_{{\mathfrak{p}}{\mathfrak{q}}{\mathfrak{r}}}}$,
where $\displaystyle
U_{{\mathfrak{p}}{\mathfrak{q}}{\mathfrak{r}}}:=\varphi^{-1}_{{\mathfrak{q}}{\mathfrak{r}}}(U_{{\mathfrak{p}}{\mathfrak{q}}})\cap
U_{{\mathfrak{p}}{\mathfrak{r}}}$.

  * •

If $\displaystyle\text{\rm Im}\,\psi_{{\mathfrak{p}}}\cap\text{\rm
Im}\,\psi_{{\mathfrak{q}}}\neq\emptyset$, then either
$\displaystyle{\mathfrak{p}}\leq{\mathfrak{q}}$ or
$\displaystyle{\mathfrak{q}}\leq{\mathfrak{p}}$ holds.

  * •

Let us define a relation $\displaystyle\sim$ on
$\displaystyle\bigsqcup_{{\mathfrak{p}}\in{\mathfrak{P}}}U_{{\mathfrak{p}}}$
as follows: $\displaystyle x\sim y$ if and only if one of the following holds:

    * –

$\displaystyle{\mathfrak{p}}={\mathfrak{q}}$ and $\displaystyle x=y$.

    * –

$\displaystyle{\mathfrak{p}}\leq{\mathfrak{q}}$ and $\displaystyle
y=\varphi_{{\mathfrak{q}}{\mathfrak{p}}}(x)$.

    * –

$\displaystyle{\mathfrak{q}}\leq{\mathfrak{p}}$ and $\displaystyle
x=\varphi_{{\mathfrak{p}}{\mathfrak{q}}}(y)$.

Then the relation $\displaystyle\sim$ is an equivalence relation, and the
quotient
$\displaystyle\bigg{(}\bigsqcup_{{\mathfrak{p}}\in{\mathfrak{P}}}U_{{\mathfrak{p}}}\bigg{)}/\sim$,
equipped with the quotient topology, is Hausdorff.

The definitions of strongly continuous/smooth maps naturally extend to spaces
with GCS. Finally, there exists a natural notion of embeddings from a
K-structure to a GCS (KG-embedding; see Definition 3.29 in [10]). For any
K-structure $\displaystyle\widehat{{\mathcal{U}}}$ on a compact space
$\displaystyle X$, there exist a GCS
$\displaystyle\widetriangle{{\mathcal{U}}}$ and a KG-embedding
$\displaystyle\widehat{{\mathcal{U}}}\to\widetriangle{{\mathcal{U}}}$; see
Theorem 3.30 in [10].

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