Now let us define moduli spaces $\displaystyle{\mathcal{M}}_{k+1}(\beta)$,
$\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta)$ and
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta)$. For every
$\displaystyle(T,B)\in{\mathcal{G}}(k+1,\beta)$, one can define

(23) $\text{\rm ev}\,_{\text{\rm int}\,}:\prod_{v\in C_{0,\text{\rm
int}\,}(T)}{\mathring{{\mathcal{M}}}}_{k_{v}+1}(B(v))\to\prod_{e\in
C_{1,\text{\rm int}\,}(T)}L^{2}$

in the same manner as the definition of the map (21.2) in [10]. We also
consider

(24) $\text{\rm ev}\,_{\text{\rm ext}\,}:\prod_{v\in C_{0,\text{\rm
int}\,}(T)}{\mathring{{\mathcal{M}}}}_{k_{v}+1}(B(v))\to\prod_{e\in
C_{1,\text{\rm ext}\,}(T)}L\cong L^{k+1}$

where the isomorphism on the right is defined by labeling exterior vertices by
$\displaystyle\\{0,\ldots,k\\}$ in positive cyclic order so that the root is
labeled by $\displaystyle 0$.

We also consider the diagonal map:

(25) $\Delta:\prod_{e\in C_{1,\text{\rm int}\,}(T)}L\to\prod_{e\in
C_{1,\text{\rm int}\,}(T)}L^{2};\quad(x_{e})_{e}\mapsto(x_{e},x_{e})_{e}.$

Then we define $\displaystyle{\mathcal{M}}_{k+1}(\beta)$ by taking fiber
products of (23) and (25). Namely

$\displaystyle{\mathcal{M}}_{k+1}(\beta):=\bigsqcup_{(T,B)\in{\mathcal{G}}(k+1,\beta)}\bigg{(}\prod_{e\in
C_{1,\text{\rm int}\,}(T)}L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm
ev}\,_{\text{\rm int}\,}}\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)}{\mathring{{\mathcal{M}}}}_{k_{v}+1}(B(v))\bigg{)}.$

We define $\displaystyle\text{\rm ev}\,^{{\mathcal{M}}}=(\text{\rm
ev}\,^{{\mathcal{M}}}_{0},\ldots,\text{\rm
ev}\,^{{\mathcal{M}}}_{k}):{\mathcal{M}}_{k+1}(\beta)\to L^{k+1}$ by
restricting $\displaystyle\text{\rm ev}\,_{\text{\rm ext}}\,$.

The definition of $\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta)$ is similar.
For any $\displaystyle(T,B)\in{\mathcal{G}}(k+1,\beta)$ and $\displaystyle
v_{0}\in C_{0,\text{\rm int}\,}(T)$, we define

(26) $\text{\rm ev}\,_{\text{\rm int}\,}:\prod_{v\in C_{0,\text{\rm
int}\,}(T)\setminus\\{v_{0}\\}}{\mathring{{\mathcal{M}}}}_{k_{v}+1}(B(v))\times{\mathring{{\mathcal{N}}}}^{0}_{k_{v_{0}}+1}(B(v_{0}))\to\prod_{e\in
C_{1,\text{\rm int}\,}(T)}L^{2}$

and

(27) $\text{\rm ev}\,_{\text{\rm ext}\,}:\prod_{v\in C_{0,\text{\rm
int}\,}(T)\setminus\\{v_{0}\\}}{\mathring{{\mathcal{M}}}}_{k_{v}+1}(B(v))\times{\mathring{{\mathcal{N}}}}^{0}_{k_{v_{0}}+1}(B(v_{0}))\to
L^{k+1}$

in manners similar to (23) and (24). Then we define:

$\displaystyle\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta)$
$\displaystyle\displaystyle:=\bigsqcup_{\begin{subarray}{c}(T,B)\in{\mathcal{G}}(k+1,\beta)\\\
v_{0}\in C_{0,\text{\rm int}\,}(T)\end{subarray}}$
$\displaystyle\displaystyle\bigg{(}\prod_{e\in C_{1,\text{\rm
int}\,}(T)}L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)\setminus\\{v_{0}\\}}{\mathring{{\mathcal{M}}}}_{k_{v}+1}(B(v))\times{\mathring{{\mathcal{N}}}}^{0}_{k_{v_{0}}+1}(B(v_{0}))\bigg{)}.$

We define $\displaystyle\text{\rm ev}\,^{{\mathcal{N}}^{0}}=(\text{\rm
ev}\,^{{\mathcal{N}}^{0}}_{0},\ldots,\text{\rm
ev}\,^{{\mathcal{N}}^{0}}_{k}):{\mathcal{N}}^{0}_{k+1}(\beta)\to L^{k+1}$ by
restring $\displaystyle\text{\rm ev}\,_{\text{\rm ext}}\,$ defined in (27).
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta)$ and $\displaystyle\text{\rm
ev}\,^{{\mathcal{N}}^{\geq 0}}:{\mathcal{N}}^{\geq 0}_{k+1}(\beta)\to L^{k+1}$
are defined in a similar way.

Now we have defined sets $\displaystyle{\mathcal{M}}_{k+1}(\beta)$,
$\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta)$, and
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta)$. These sets have natural
topologies: the topology on $\displaystyle{\mathcal{M}}_{k+1}(\beta)$ is
described in Section 7.1.4 [8] (actually our situation is much simpler, since
we have neither interior marked points nor sphere bubbles). The topologies on
$\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta)$ and
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta)$ are defined in similar
ways, and details are omitted.

####  7.2.3. K-structures on compactified moduli spaces

In this subsubsection, we consider a system of K-structures on compactified
moduli spaces defined in the previous subsubsection. More accurately, we
consider admissible K-structures (see Section 25 in [10]), and define a
certain inductive system of such admissible K-spaces (this is a variant of
“inductive system of $\displaystyle A_{\infty}$ correspondence” in Definition
21.17 in [10]). Our goal in this subsubsection is to spell out an accurate
statement in Theorem 7.20.

Recall that we took $\displaystyle\varepsilon>0$ so that every nonconstant
holomorphic disk with boundary on $\displaystyle L$ has area at least
$\displaystyle 2\varepsilon$. We take $\displaystyle U\in{\mathbb{Z}}_{>0}$ so
that $\displaystyle\varepsilon(U-1)\geq 2\|H\|$.

######  Theorem 7.20.

For every $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$, $\displaystyle
m\in{\mathbb{Z}}_{\geq 0}$ and $\displaystyle P\in\\{\\{m\\},[m,m+1]\\}$,
there exist the following data:

  1. (i):

(Moduli spaces) Compact, oriented, admissible K-spaces

$\displaystyle\displaystyle{\mathcal{M}}_{k+1}(\beta:P)\qquad(\beta\in
H_{2}({\mathbb{C}}^{n},L),\,\omega_{n}(\beta)<\varepsilon(m+1-k)),$
$\displaystyle\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta:P)\qquad(\beta\in
H_{2}({\mathbb{C}}^{n},L),\,\omega_{n}(\beta)<\varepsilon(m-1-k)),$
$\displaystyle\displaystyle{\mathcal{N}}^{\geq
0}_{k+1}(\beta:P)\qquad(\beta\in
H_{2}({\mathbb{C}}^{n},L),\,\omega_{n}(\beta)<\varepsilon(m-k-U)),$

whose underlying topological spaces are

$\displaystyle P\times{\mathcal{M}}_{k+1}(\beta),\quad
P\times{\mathcal{N}}^{0}_{k+1}(\beta),\quad P\times{\mathcal{N}}^{\geq
0}_{k+1}(\beta),$

respectively. Dimensions of these K-spaces are given by

$\displaystyle\displaystyle\text{\rm
dim}\,{\mathcal{M}}_{k+1}(\beta:P)=\mu(\beta)+n+k-2+\text{\rm dim}\,P,$
$\displaystyle\displaystyle\text{\rm
dim}\,{\mathcal{N}}^{0}_{k+1}(\beta:P)=\mu(\beta)+n+k+\text{\rm dim}\,P,$
$\displaystyle\displaystyle\text{\rm dim}\,{\mathcal{N}}^{\geq
0}_{k+1}(\beta:P)=\mu(\beta)+n+k+1+\text{\rm dim}\,P.$

  2. (ii):

(Evaluation maps) Corner stratified strongly smooth maps (from K-spaces to
manifolds with corners, see Definition 26.6 (1) in [10])

$\displaystyle\displaystyle\text{\rm ev}\,^{{\mathcal{M}},P}$
$\displaystyle\displaystyle:{\mathcal{M}}_{k+1}(\beta:P)\to P\times L^{k+1},$
$\displaystyle\displaystyle\text{\rm ev}\,^{{\mathcal{N}}^{0},P}$
$\displaystyle\displaystyle:{\mathcal{N}}^{0}_{k+1}(\beta:P)\to P\times
L^{k+1},$ $\displaystyle\displaystyle\text{\rm ev}\,^{{\mathcal{N}}^{\geq
0},P}$ $\displaystyle\displaystyle:{\mathcal{N}}^{\geq 0}_{k+1}(\beta:P)\to
P\times L^{k+1},$

such that their underlying set-theoretic maps are:

$\displaystyle\text{\rm id}_{P}\times\text{\rm
ev}\,^{{\mathcal{M}}},\quad\text{\rm id}_{P}\times\text{\rm
ev}\,^{{\mathcal{N}}^{0}},\quad\text{\rm id}_{P}\times\text{\rm
ev}\,^{{\mathcal{N}}^{\geq 0}}.$

We require that the following maps to $\displaystyle P\times L$

$\displaystyle(\text{\rm id}_{P}\times\text{\rm pr}_{0})\circ\text{\rm
ev}\,^{{\mathcal{M}},P},\quad(\text{\rm id}_{P}\times\text{\rm
pr}_{0})\circ\text{\rm ev}\,^{{\mathcal{N}}^{0},P},\quad(\text{\rm
id}_{P}\times\text{\rm pr}_{0})\circ\text{\rm ev}\,^{{\mathcal{N}}^{\geq
0},P}$

are corner stratified weak submersions (see Definition 26.6 (2) in [10]). Here
$\displaystyle\text{\rm pr}_{0}:L^{k+1}\to L$ is defined by
$\displaystyle(p_{0},\ldots,p_{k})\mapsto p_{0}$.

  3. (iii):

(Energy zero part) An isomorphism of admissible K-structures

$\displaystyle{\mathcal{M}}_{k+1}(0:P)\cong P\times L\times D^{k-2}$

for every $\displaystyle k\geq 2$, so that $\displaystyle\text{\rm
ev}\,^{{\mathcal{M}},P}:{\mathcal{M}}_{k+1}(0:P)\to P\times L^{k+1}$ coincides
with $\displaystyle\text{\rm pr}_{P}\times(\text{\rm pr}_{L})^{k+1}$. Here
$\displaystyle D^{k-2}$ in the RHS is identified with the Stasheff cell; see
[7] Section 10. The coordinate near boundary is $\displaystyle 1/\log T$,
where $\displaystyle T$ denotes the length of neck region; see Remark 25.45 in
[10].

  4. (iv):

(Compatibility at boundaries) The following isomorphisms of admissible
K-spaces preserving orientations ($\displaystyle\partial$ in the LHS denotes
normalized boundaries, and the fiber product $\,{}_{\text{\rm
ev}\,_{i}}\\!\times_{\text{\rm ev}\,_{0}}$ is abbreviated as
$\,{}_{i}\\!\times_{0}$):

(28)
$\partial{\mathcal{M}}_{k+1}(\beta:m)\cong\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
1\leq i\leq k_{1}\\\
\beta_{1}+\beta_{2}=\beta\end{subarray}}(-1)^{\varepsilon_{0}}{\mathcal{M}}_{k_{1}+1}(\beta_{1}:m)\,_{i}\\!\times_{0}{\mathcal{M}}_{k_{2}+1}(\beta_{2}:m),$
(29) $\displaystyle\displaystyle\partial{\mathcal{N}}^{0}_{k+1}(\beta:m)\cong$
$\displaystyle\displaystyle\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
1\leq i\leq k_{1}\\\
\beta_{1}+\beta_{2}=\beta\end{subarray}}\biggl{(}(-1)^{\varepsilon_{1}}{\mathcal{N}}^{0}_{k_{1}+1}(\beta_{1}:m)\,_{i}\\!\times_{0}{\mathcal{M}}_{k_{2}+1}(\beta_{2}:m)$
$\displaystyle\displaystyle\qquad\qquad\sqcup(-1)^{\varepsilon_{2}}{\mathcal{M}}_{k_{1}+1}(\beta_{1}:m)\,_{i}\\!\times_{0}{\mathcal{N}}^{0}_{k_{2}+1}(\beta_{2}:m)\biggr{)},$
(30) $\displaystyle\displaystyle\partial{\mathcal{N}}^{\geq
0}_{k+1}(\beta:m)\cong{\mathcal{N}}^{0}_{k+1}(\beta:m)\sqcup$
$\displaystyle\displaystyle\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
1\leq i\leq k_{1}\\\
\beta_{1}+\beta_{2}=\beta\end{subarray}}\biggl{(}(-1)^{\varepsilon_{3}}{\mathcal{N}}^{\geq
0}_{k_{1}+1}(\beta_{1}:m)\,_{i}\\!\times_{0}{\mathcal{M}}_{k_{2}+1}(\beta_{2}:m)$
$\displaystyle\displaystyle\qquad\qquad\sqcup(-1)^{\varepsilon_{4}}{\mathcal{M}}_{k_{1}+1}(\beta_{1}:m)\,_{i}\\!\times_{0}{\mathcal{N}}^{\geq
0}_{k_{2}+1}(\beta_{2}:m)\biggr{)},$ (31)
$\displaystyle\displaystyle\partial{\mathcal{M}}_{k+1}(\beta:[m,m+1])\cong(-1)^{\varepsilon_{5}}{\mathcal{M}}_{k+1}(\beta:m)\sqcup(-1)^{\varepsilon_{6}}{\mathcal{M}}_{k+1}(\beta:m+1)\,\sqcup$
$\displaystyle\displaystyle\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
1\leq i\leq k_{1}\\\
\beta_{1}+\beta_{2}=\beta\end{subarray}}(-1)^{\varepsilon_{7}}{\mathcal{M}}_{k_{1}+1}(\beta_{1}:[m,m+1])\,_{i}\\!\times_{0}{\mathcal{M}}_{k_{2}+1}(\beta_{2}:[m,m+1]),$
(32)
$\displaystyle\displaystyle\partial{\mathcal{N}}^{0}_{k+1}(\beta:[m,m+1])\cong(-1)^{\varepsilon_{8}}{\mathcal{N}}^{0}_{k+1}(\beta:m)\sqcup(-1)^{\varepsilon_{9}}{\mathcal{N}}^{0}_{k+1}(\beta:m+1)\,\sqcup$
$\displaystyle\displaystyle\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
1\leq i\leq k_{1}\\\
\beta_{1}+\beta_{2}=\beta\end{subarray}}\biggl{(}(-1)^{\varepsilon_{10}}{\mathcal{N}}^{0}_{k_{1}+1}(\beta_{1}:[m,m+1])\,_{i}\\!\times_{0}{\mathcal{M}}_{k_{2}+1}(\beta_{2}:[m,m+1])$
$\displaystyle\displaystyle\qquad\qquad\sqcup(-1)^{\varepsilon_{11}}{\mathcal{M}}_{k_{1}+1}(\beta_{1}:[m,m+1])\,_{i}\\!\times_{0}{\mathcal{N}}^{0}_{k_{2}+1}(\beta_{2}:[m,m+1])\biggr{)},$
(33) $\displaystyle\displaystyle\partial{\mathcal{N}}^{\geq
0}_{k+1}(\beta:[m,m+1])\cong(-1)^{\varepsilon_{12}}{\mathcal{N}}^{\geq
0}_{k+1}(\beta:m)\sqcup(-1)^{\varepsilon_{13}}{\mathcal{N}}^{\geq
0}_{k+1}(\beta:m+1)$
$\displaystyle\displaystyle\sqcup(-1)^{\varepsilon_{14}}{\mathcal{N}}^{0}_{k+1}(\beta:[m,m+1])\sqcup$
$\displaystyle\displaystyle\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
1\leq i\leq k_{1}\\\
\beta_{1}+\beta_{2}=\beta\end{subarray}}\biggl{(}(-1)^{\varepsilon_{15}}{\mathcal{N}}^{\geq
0}_{k_{1}+1}(\beta_{1}:[m,m+1])\,_{i}\\!\times_{0}{\mathcal{M}}_{k_{2}+1}(\beta_{2}:[m,m+1])$
$\displaystyle\displaystyle\qquad\qquad\sqcup(-1)^{\varepsilon_{16}}{\mathcal{M}}_{k_{1}+1}(\beta_{1}:[m,m+1])\,_{i}\\!\times_{0}{\mathcal{N}}^{\geq
0}_{k_{2}+1}(\beta_{2}:[m,m+1])\biggr{)}.$

The signs $\displaystyle\varepsilon_{0},\ldots,\varepsilon_{16}$ are given as
follows:

$\displaystyle\displaystyle\varepsilon_{0}$
$\displaystyle\displaystyle=(k_{1}-i)(k_{2}-1)+n+k_{1},$
$\displaystyle\displaystyle\varepsilon_{1}$
$\displaystyle\displaystyle=\varepsilon_{0}+k+k_{1},\,\varepsilon_{2}=\varepsilon_{0}+k+k_{2},$
$\displaystyle\displaystyle\varepsilon_{3}$
$\displaystyle\displaystyle=\varepsilon_{1}+1,\,\varepsilon_{4}=\varepsilon_{2}+k_{1}+1,$
$\displaystyle\displaystyle\varepsilon_{5}$
$\displaystyle\displaystyle=1,\,\varepsilon_{6}=0,\,\varepsilon_{7}=\varepsilon_{0}+1,$
$\displaystyle\displaystyle\varepsilon_{8}$
$\displaystyle\displaystyle=1,\,\varepsilon_{9}=0,\,\varepsilon_{10}=\varepsilon_{1}+1,\,\varepsilon_{11}=\varepsilon_{2}+1,$
$\displaystyle\displaystyle\varepsilon_{12}$
$\displaystyle\displaystyle=1,\,\varepsilon_{13}=0,\,\varepsilon_{14}=1,\,\varepsilon_{15}=\varepsilon_{3}+1,\,\varepsilon_{16}=\varepsilon_{4}+1.$

  5. (v):

(Compatibility at corners I) First we introduce the following notations:

     * •

For any admissible K-space $\displaystyle X$ and $\displaystyle
l\in{\mathbb{Z}}_{\geq 1}$, let $\displaystyle\widehat{S}_{l}X$ denote the
normalized codimension $\displaystyle l$ corner of $\displaystyle X$; see [10]
Definition 24.17.

     * •

For every nonnegative integers $\displaystyle d$ and $\displaystyle m$, we
denote

$\displaystyle\widehat{S}_{d}\\{m\\}:=\begin{cases}\\{m\\}&(d=0)\\\
\emptyset&(d\geq
1),\end{cases}\qquad\widehat{S}_{d}[m,m+1]:=\begin{cases}[m,m+1]&(d=0)\\\
\\{m,m+1\\}&(d=1)\\\ \emptyset&(d\geq 2).\end{cases}$

Then, there are the following isomorphisms of admissible K-spaces (34), (35),
and (36): here we do not consider orientations of moduli spaces.

(34)
$\displaystyle\displaystyle\widehat{S}_{l}{\mathcal{M}}_{k+1}(\beta:P)\cong$
$\displaystyle\displaystyle\qquad\bigsqcup_{\begin{subarray}{c}(T,B)\in{\mathcal{G}}(k+1,\beta)\\\
\\#C_{1,\text{\rm int}\,}(T)+d=l\end{subarray}}\bigg{(}\prod_{e\in
C_{1,\text{\rm int}\,}(T)}\widehat{S}_{d}P\times
L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)}{\mathcal{M}}_{k_{v}+1}(B(v):\widehat{S}_{d}P)\bigg{)}$

where the fiber product in the RHS is taken over $\displaystyle\prod_{e\in
C_{1,\text{\rm int}\,}(T)}(\widehat{S}_{d}P\times L)^{2}$. Notice that the
fiber product makes sense, since

$\displaystyle(\text{\rm id}_{P}\times\text{\rm pr}_{0})\circ\text{\rm
ev}\,^{{\mathcal{M}},P}:{\mathcal{M}}_{k+1}(\beta:P)\to P\times L$

is a corner stratified weak submersion, as we assumed in (ii).

(35)
$\displaystyle\displaystyle\widehat{S}_{l}{\mathcal{N}}^{0}_{k+1}(\beta:P)\cong\bigsqcup_{\begin{subarray}{c}(T,B)\in{\mathcal{G}}(k+1,\beta)\\\
\\#C_{1,\text{\rm int}\,}(T)+d=l\\\ v_{0}\in C_{0,\text{\rm
int}\,}(T)\end{subarray}}\bigg{(}\prod_{e\in C_{1,\text{\rm
int}\,}(T)}\widehat{S}_{d}P\times L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm
ev}\,_{\text{\rm int}}\,}$
$\displaystyle\displaystyle\qquad\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)\setminus\\{v_{0}\\}}{\mathcal{M}}_{k_{v}+1}(B(v):\widehat{S}_{d}P)\times{\mathcal{N}}_{k_{v_{0}}+1}^{0}(B(v_{0}):\widehat{S}_{d}P)\bigg{)}.$
(36) $\displaystyle\displaystyle\widehat{S}_{l}{\mathcal{N}}^{\geq
0}_{k+1}(\beta:P)\cong\bigsqcup_{\begin{subarray}{c}(T,B)\in{\mathcal{G}}(k+1,\beta)\\\
\\#C_{1,\text{\rm int}\,}(T)+d=l\\\ v_{0}\in C_{0,\text{\rm
int}\,}(T)\end{subarray}}\bigg{(}\prod_{e\in C_{1,\text{\rm
int}\,}(T)}\widehat{S}_{d}P\times L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm
ev}\,_{\text{\rm int}}\,}$
$\displaystyle\displaystyle\qquad\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)\setminus\\{v_{0}\\}}{\mathcal{M}}_{k_{v}+1}(B(v):\widehat{S}_{d}P)\times{\mathcal{N}}_{k_{v_{0}}+1}^{\geq
0}(B(v_{0}):\widehat{S}_{d}P)\bigg{)}$
$\displaystyle\displaystyle\qquad\sqcup\,\bigsqcup_{\begin{subarray}{c}(T,B)\in{\mathcal{G}}(k+1,\beta)\\\
\\#C_{1,\text{\rm int}\,}(T)+d=l-1\\\ v_{0}\in C_{0,\text{\rm
int}\,}(T)\end{subarray}}\bigg{(}\prod_{e\in C_{1,\text{\rm
int}\,}(T)}\widehat{S}_{d}P\times L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm
ev}\,_{\text{\rm int}}\,}$
$\displaystyle\displaystyle\qquad\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)\setminus\\{v_{0}\\}}{\mathcal{M}}_{k_{v}+1}(B(v):\widehat{S}_{d}P)\times{\mathcal{N}}_{k_{v_{0}}+1}^{0}(B(v_{0}):\widehat{S}_{d}P)\bigg{)}.$

  6. (vi):

(Compatibility at corners II) Let $\displaystyle X$ be either
$\displaystyle{\mathcal{M}}_{k+1}(\beta:P)$,
$\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta:P)$ or
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta:P)$. Then, for every
$\displaystyle l,l^{\prime}\in{\mathbb{Z}}_{\geq 1}$, the canonical covering
map
$\displaystyle\pi_{l^{\prime},l}:\widehat{S}_{l^{\prime}}(\widehat{S}_{l}X)\to\widehat{S}_{l+l^{\prime}}X$
(see Proposition 24.16 in [10]) coincides with the map defined from the fiber
product presentation in (v).

Construction of these K-structures are mostly the same as the construction of
K-structure on $\displaystyle{\mathcal{M}}_{k+1}(\beta)$, which is explained
in [8] Section 7.1. Evaluation maps in (ii) and isomorphisms in (iv), (v)
naturally follow from the construction of these K-structures. Admissibility of
this K-structure is a consequence of the exponential decay estimate in [9];
see Section 25.5 in [10]. We do not spell out a detailed proof of Theorem
7.20. Nevertheless, in the rest of this subsubsection, we explain:

  * •

Explicit description of Kuranishi charts of
$\displaystyle{\mathcal{M}}_{k+1}(\beta:P)$, which we use in Section 7.4.

  * •

Total order on the set of moduli spaces, which we need to carry out inductive
argument.

  * •

Orientations: how signs $\displaystyle\varepsilon_{0},\ldots,\varepsilon_{16}$
are computed.

Explicit description of Kuranishi charts of
$\displaystyle{\mathcal{M}}_{k+1}(\beta:P)$.

Let $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$ and $\displaystyle\beta\in
H_{2}({\mathbb{C}}^{n},L)$. Let $\displaystyle{\mathcal{MM}}_{k+1}(\beta)$
denote the set consists of tuples $\displaystyle(u,z_{0},z_{1},\ldots,z_{k})$
such that $\displaystyle u:(D,\partial D)\to({\mathbb{C}}^{n},L)$ is a
$\displaystyle C^{\infty}$-map such that $\displaystyle\bar{\partial}u=0$ on a
neighborhood of $\displaystyle\partial D$ and $\displaystyle[u]=\beta$, and
$\displaystyle z_{0},z_{1},\ldots,z_{k}$ are distinct points on
$\displaystyle\partial D$ aligned in the anti-clockwise order.

######  Remark 7.21.

To define K-structure on $\displaystyle{\mathcal{M}}_{k+1}(\beta)$ we consider
an obstruction bundle $\displaystyle E$ and a perturbed Cauchy-Riemann
equation $\displaystyle\bar{\partial}u\in E$. One can take $\displaystyle E$
so that every section of $\displaystyle E$ is supported on a compact set of
$\displaystyle\text{int}D$, thus these perturbed holomorphic maps (with
boundary marked points) are elements in
$\displaystyle{\mathcal{MM}}_{k+1}(\beta)$.

Now we can state the explicit description of Kuranishi chart in Lemma 7.22
below.

######  Lemma 7.22.

Let $\displaystyle p\in{\mathcal{M}}_{k+1}(\beta:P)$ and
$\displaystyle{\mathcal{U}}_{p}=(U_{p},{\mathcal{E}}_{p},s_{p},\psi_{p})$ be a
K-chart at $\displaystyle p$. Let $\displaystyle(T,B)$ be an element in
$\displaystyle{\mathcal{G}}(k+1:\beta)$ such that

$\displaystyle p\in P\times\bigg{(}\prod_{e\in C_{1,\text{\rm
int}\,}(T)}L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}\,}}\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)}{\mathring{{\mathcal{M}}}}_{k_{v}+1}(B(v))\bigg{)}.$

Then, $\displaystyle U_{p}$ can be (set theoretically) embedded into

$\displaystyle\bigsqcup_{(T^{\prime},B^{\prime})}P\times\bigg{(}\prod_{e\in
C_{1,\text{\rm int}\,}(T^{\prime})}L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm
ev}\,_{\text{\rm int}\,}}\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T^{\prime})}{\mathcal{MM}}_{k_{v}+1}(B^{\prime}(v))\bigg{)}$

where $\displaystyle(T^{\prime},B^{\prime})$ runs over all reductions (see
Definition 7.19) of $\displaystyle(T,B)$.

Total order on the set of moduli spaces. To achieve compatibility conditions
(iv), (v) and (vi) in Theorem 7.20, we need a total order on the set of moduli
spaces, so that the following property is satisfied: if a moduli space
$\displaystyle X$ is a part of the normalized boundary $\displaystyle\partial
Y$ of another moduli space $\displaystyle Y$, then $\displaystyle X<Y$. To
achieve this property we take a total order which is described as follows:

  * •

If the $\displaystyle P$-part of a moduli space $\displaystyle X$ is
$\displaystyle\\{m\\}$ and the $\displaystyle P$-part of a moduli space
$\displaystyle Y$ is $\displaystyle[m^{\prime},m^{\prime}+1]$, then
$\displaystyle X<Y$ ($\displaystyle m$ and $\displaystyle m^{\prime}$ are
arbitrary elements in $\displaystyle{\mathbb{Z}}_{\geq 0}$).

  * •

If the $\displaystyle P$-part of $\displaystyle X$ is $\displaystyle\\{m\\}$
and the $\displaystyle P$-part of $\displaystyle Y$ is
$\displaystyle\\{m+1\\}$, then $\displaystyle X<Y$.

  * •

If the $\displaystyle P$-part of $\displaystyle X$ is $\displaystyle[m,m+1]$
and the $\displaystyle P$-part of $\displaystyle Y$ is
$\displaystyle[m+1,m+2]$, then $\displaystyle X<Y$.

  * •

For each $\displaystyle m\in{\mathbb{Z}}_{\geq 0}$, we take a total order on
moduli spaces with $\displaystyle P$-parts equal to $\displaystyle\\{m\\}$, in
the following manner:

    * –

For any $\displaystyle k,k^{\prime},k^{\prime\prime}\in{\mathbb{Z}}_{\geq 0}$
and $\displaystyle\beta,\beta^{\prime},\beta^{\prime\prime}\in
H_{2}({\mathbb{C}}^{n},L)$,

$\displaystyle{\mathcal{M}}_{k+1}(\beta:m)<{\mathcal{N}}^{0}_{k^{\prime}+1}(\beta^{\prime}:m)<{\mathcal{N}}^{\geq
0}_{k^{\prime\prime}+1}(\beta^{\prime\prime}:m).$

    * –

If $\displaystyle k,k^{\prime}\in{\mathbb{Z}}_{\geq 0}$ and
$\displaystyle\beta,\beta^{\prime}\in H_{2}({\mathbb{C}}^{n},L)$ satisfy
$\displaystyle\omega_{n}(\beta)+\varepsilon(k-1)<\omega_{n}(\beta^{\prime})+\varepsilon(k^{\prime}-1)$,
then
$\displaystyle{\mathcal{M}}_{k+1}(\beta:m)<{\mathcal{M}}_{k^{\prime}+1}(\beta^{\prime}:m)$,
$\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta:m)<{\mathcal{N}}^{0}_{k^{\prime}+1}(\beta^{\prime}:m)$,
and $\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta:m)<{\mathcal{N}}^{\geq
0}_{k^{\prime}+1}(\beta^{\prime}:m)$.

    * –

For each $\displaystyle c\in{\mathbb{R}}$, we choose arbitrary total orders on
sets

$\displaystyle\displaystyle\\{{\mathcal{M}}_{k+1}(\beta:m)\mid\omega_{n}(\beta)+\varepsilon(k-1)=c\\},$
$\displaystyle\displaystyle\\{{\mathcal{N}}^{0}_{k+1}(\beta:m)\mid\omega_{n}(\beta)+\varepsilon(k-1)=c\\},$
$\displaystyle\displaystyle\\{{\mathcal{N}}^{\geq
0}_{k+1}(\beta:m)\mid\omega_{n}(\beta)+\varepsilon(k-1)=c\\}.$

  * •

For each $\displaystyle m\in{\mathbb{Z}}_{\geq 0}$, we take a total order on
moduli spaces with $\displaystyle P$-parts equal to $\displaystyle[m,m+1]$, in
a manner similar to above.

######  Remark 7.23.

For each moduli space $\displaystyle X$, there are only finitely many moduli
spaces which are smaller than $\displaystyle X$. Therefore, we can assign
$\displaystyle\tau(X)\in(1/2,1)$ for each moduli space $\displaystyle X$, so
that $\displaystyle Y<X\implies\tau(Y)>\tau(X)$. We use these numbers in
Section 7.6.

Orientations. We explain how signs
$\displaystyle\varepsilon_{0},\ldots,\varepsilon_{16}$ are computed.

$\displaystyle\varepsilon_{0}$: First we orient
$\displaystyle{\mathcal{M}}_{k+1}(\beta)$ following Section 8.3 in [8]. It is
sufficient to orient its interior
$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta)=\\{(u,z_{0},\ldots,z_{k})\\}/\text{\rm
Aut}\,(D)$. We consider the set $\displaystyle\\{(u,z_{0},\ldots,z_{k})\\}$ as
an open subset of
$\displaystyle{\mathring{{\mathcal{M}}}}(\beta)\times(\partial D)^{k+1}$,
where

$\displaystyle{\mathring{{\mathcal{M}}}}(\beta):=\\{u:(D,\partial
D)\to({\mathbb{C}}^{n},L)\mid\bar{\partial}u=0,\,[u]=\beta\\}$

is canonically oriented by Theorem 8.1.1 in [8] and $\displaystyle\partial
D=\\{e^{i\theta}\mid\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}\\}$ is oriented so
that $\displaystyle\partial_{\theta}$ is of positive direction (anti-clockwise
orientation). On the other hand, following Convention 8.3.1 in [8],
$\displaystyle\text{\rm Aut}\,(D)$ is oriented so that the diffeomorphism

$\displaystyle\text{\rm Aut}\,(D)\to(\partial
D)^{3};\,\rho\mapsto(\rho^{-1}(1),\rho^{-1}(e^{2\pi i/3}),\rho^{-1}(e^{4\pi
i/3}))$

is orientation preserving. Finally, the quotient
$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta)$ is oriented so that a
natural isomorphism (determined uniquely up to homotopy)

$\displaystyle T({\mathcal{M}}_{k+1}(\beta))\oplus T(\text{\rm Aut}\,(D))\cong
T({\mathcal{M}}(\beta))\oplus T(\partial D)^{\oplus k+1}$

preserves orientations (see page 692 in [8]).

Now let us compute $\displaystyle\varepsilon_{0}$. When $\displaystyle i=1$,
Proposition 8.3.3 in [8] shows
$\displaystyle\varepsilon_{0}=(k_{1}-1)(k_{2}-1)+n+k_{1}$. For arbitrary
$\displaystyle i\in\\{1,\ldots,k_{1}\\}$, we can compute
$\displaystyle\varepsilon_{0}$ as

$\displaystyle\varepsilon_{0}=(k_{1}-1)(k_{2}-1)+n+k_{1}+(i-1)+(i-1)k_{2}=(k_{1}-i)(k_{2}-1)+n+k_{1}$

where the term $\displaystyle(i-1)$ comes from exchanging input boundary
marked points $\displaystyle 1,\ldots,i-1$ and $\displaystyle i$, and the term
$\displaystyle(i-1)k_{2}$ comes from exchanging output boundary marked points
$\displaystyle 1,\ldots,i-1$ and $\displaystyle i,\ldots,i+k_{2}-1$.

$\displaystyle\varepsilon_{1}$ and $\displaystyle\varepsilon_{2}$: We orient
$\displaystyle{\mathring{{\mathcal{N}}}}^{0}_{k+1}(\beta)=\\{(u,z_{0}=1,z_{1},\ldots,z_{k})\\}$
as an open subset of
$\displaystyle{\mathring{{\mathcal{M}}}}(\beta)\times(\partial D)^{k}$. Then
$\displaystyle\varepsilon_{1}=\varepsilon_{0}+k+k_{1}$ and
$\displaystyle\varepsilon_{2}=\varepsilon_{0}+k+k_{2}$ follow from the next
lemma. Following [8] page 694, we say that an isomorphism of K-spaces is
$\displaystyle(-1)$-oriented (resp. $\displaystyle(+1)$-oriented) if the
isomorphism reverses (resp. preserves) orientations.

######  Lemma 7.24.

Let us orient $\displaystyle\text{\rm Aut}\,(D,1)$ so that the embedding

$\displaystyle\text{\rm Aut}\,(D,1)\to(\partial
D)^{2};\,\rho\mapsto(\rho^{-1}(e^{2\pi i/3}),\rho^{-1}(e^{4\pi i/3}))$

preserves orientations. Then, the isomorphism of K-spaces

$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta)\cong{\mathring{{\mathcal{N}}}}^{0}_{k+1}(\beta)/\text{\rm
Aut}\,(D,1)$

is $\displaystyle(-1)^{k}$-oriented.

###### Proof.

Recall that
$\displaystyle{\mathring{{\mathcal{M}}}}_{k+1}(\beta)=\\{(u,z_{0},\ldots,z_{k})\\}/\text{\rm
Aut}\,(D)$. The natural diffeomorphism $\displaystyle\text{\rm
Aut}\,(D)\cong\partial D\times\text{\rm Aut}\,(D,1)$ preserves orientations,
and the first factor $\displaystyle\partial D$ acts on the $\displaystyle
0$-th marked point $\displaystyle z_{0}$, thus the factor
$\displaystyle(-1)^{k}$ comes from exchanging $\displaystyle z_{0}$ and
$\displaystyle z_{1},\ldots,z_{k}$. ∎

$\displaystyle\varepsilon_{3}$ and $\displaystyle\varepsilon_{4}$:
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta)$ can be identified (on a
neighborhood of $\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta)$) with
$\displaystyle[0,1)\times{\mathcal{N}}^{0}_{k+1}(\beta)$. Following Convention
8.2.1 in [8], we orient $\displaystyle[0,1)$ so that
$\displaystyle\partial/\partial t$ is of negative direction, where
$\displaystyle t$ denotes the canonical coordinate on
$\displaystyle{\mathbb{R}}$. Then

$\displaystyle\partial([0,1)\times{\mathcal{N}}^{0})=\\{0\\}\times{\mathcal{N}}^{0}\sqcup(-1)\cdot[0,1)\times\partial{\mathcal{N}}^{0}$

and $\displaystyle\varepsilon_{3}=\varepsilon_{1}+1$ immediately follows.
$\displaystyle\varepsilon_{4}=\varepsilon_{2}+k_{1}+1$ follows from

$\displaystyle[0,1)\times({\mathcal{M}}_{k_{1}+1}(\beta_{1})\,_{i}\\!\times_{0}{\mathcal{N}}^{0}_{k_{2}+1}(\beta_{2}))=(-1)^{\delta}{\mathcal{M}}_{k_{1}+1}(\beta_{1})\,_{i}\\!\times_{0}([0,1)\times{\mathcal{N}}^{0}_{k_{2}+1}(\beta_{2}))$

where $\displaystyle\delta=\text{\rm
dim}\,{\mathcal{M}}_{k_{1}+1}(\beta_{1})-n\equiv k_{1}\,(\text{mod}\,2)$.

$\displaystyle\varepsilon_{5},\ldots,\varepsilon_{16}$:
$\displaystyle{\mathcal{M}}_{k+1}(\beta:[m,m+1])$ and
$\displaystyle[m,m+1]\times{\mathcal{M}}_{k+1}(\beta)$ are different K-spaces
with a common underlying topological space. One can take a common open
substructure (see Definition 3.20 in [10]) of these two K-spaces. Thus, to fix
orientations, one can identify these two K-spaces. The same remark applies to
$\displaystyle{\mathcal{N}}^{0}_{k+1}(\beta:[m,m+1])$ and
$\displaystyle{\mathcal{N}}^{\geq 0}_{k+1}(\beta:[m,m+1])$. Now let us orient
$\displaystyle[m,m+1]$ so that $\displaystyle\partial/\partial t$ is of
positive direction, where $\displaystyle t$ denotes the canonical coordinate
on $\displaystyle{\mathbb{R}}$. Then

$\displaystyle\partial([m,m+1])=(+1)\cdot\\{m+1\\}\sqcup(-1)\cdot\\{m\\}.$

Then $\displaystyle\varepsilon_{5},\ldots,\varepsilon_{16}$ can be determined
from $\displaystyle\varepsilon_{0},\ldots,\varepsilon_{4}$ using the following
formulas ($\displaystyle X$ and $\displaystyle Y$ denote arbitrary K-spaces):

$\displaystyle\displaystyle\partial([m,m+1]\times X)$
$\displaystyle\displaystyle=\\{m+1\\}\times X\sqcup(-1)\cdot\\{m\\}\times
X\sqcup(-1)\cdot[m,m+1]\times\partial X,$
$\displaystyle\displaystyle[m,m+1]\times(X\times_{L}Y)$
$\displaystyle\displaystyle=([m,m+1]\times X)\times_{[m,m+1]\times
L}([m,m+1]\times Y).$

###  7.3. Spaces of continuous paths and loops

Let $\displaystyle\Pi^{\text{\rm con}}\,$ denote the set of continuous Moore
paths on $\displaystyle L$, namely:

$\displaystyle\Pi^{\text{\rm con}\,}:=\\{(T,\gamma)\mid T\in{\mathbb{R}}_{\geq
0},\,\gamma\in C^{0}([0,T],L)\\}.$

For each $\displaystyle q\in L$, let $\displaystyle c_{q}$ denote the constant
path at $\displaystyle q$ with length $\displaystyle 0$. Namely,
$\displaystyle c_{q}=(0,\gamma_{q})$ where $\displaystyle\gamma_{q}$ denotes
the unique map from $\displaystyle\\{0\\}$ to $\displaystyle q$.

To define a metric $\displaystyle d_{\Pi}$ on $\displaystyle\Pi^{\text{\rm
con}}\,$, we fix (throughout this paper) an auxiliary Riemannian metric on
$\displaystyle L$, and $\displaystyle d_{L}$ denotes the associated metric on
$\displaystyle L$. For later use we fix a constant
$\displaystyle\rho_{L}\in{\mathbb{R}}_{>0}$ which is smaller than the
injectivity radius of $\displaystyle L$ such that for any $\displaystyle
r\in(0,\rho_{L}]$ and $\displaystyle x\in L$ the ball with center
$\displaystyle x$ and radius $\displaystyle r$ is geodesically convex.

Now let us define a metric $\displaystyle d_{\Pi}$ on
$\displaystyle\Pi^{\text{\rm con}}\,$ by

$\displaystyle
d_{\Pi}((T,\gamma),(T^{\prime},\gamma^{\prime}))):=\max\big{\\{}|T-T^{\prime}|,\,\max_{0\leq
s\leq 1}d_{L}(\gamma(sT),\gamma^{\prime}(sT^{\prime}))\big{\\}}.$

We define $\displaystyle\text{\rm ev}\,_{0},\text{\rm ev}\,_{1}:\Pi^{\text{\rm
con}}\,\to L$ by

$\displaystyle\text{\rm ev}\,_{0}(T,\gamma):=\gamma(0),\qquad\text{\rm
ev}\,_{1}(T,\gamma):=\gamma(T).$

Then we can define a concatenation map

$\displaystyle\Pi^{\text{\rm con}}\,\,_{\text{\rm
ev}\,_{1}}\\!\times_{\text{\rm ev}\,_{0}}\Pi^{\text{\rm
con}}\,\to\Pi^{\text{\rm
con}}\,;\qquad(\Gamma_{0},\Gamma_{1})\mapsto\Gamma_{0}*\Gamma_{1}$

by
$\displaystyle(T,\gamma)*(T^{\prime},\gamma^{\prime}):=(T+T^{\prime},\gamma*\gamma^{\prime})$,
where $\displaystyle\gamma*\gamma^{\prime}:[0,T+T^{\prime}]\to L$ is defined
by

$\displaystyle(\gamma*\gamma^{\prime})(t):=\begin{cases}\gamma(t)&(0\leq t\leq
T),\\\ \gamma^{\prime}(t-T)&(T\leq t\leq T+T^{\prime}).\end{cases}$

######  Lemma 7.25.

The concatenation map defined as above is continuous with respect to
$\displaystyle d_{\Pi}$.

###### Proof.

We have to show that, if sequences $\displaystyle(T_{j},\gamma_{j})_{j\geq 1}$
and $\displaystyle(T^{\prime}_{j},\gamma^{\prime}_{j})_{j\geq 1}$ satisfy

$\displaystyle\lim_{j\to\infty}d_{\Pi}((T_{j},\gamma_{j}),(T,\gamma))=\lim_{j\to\infty}d_{\Pi}((T^{\prime}_{j},\gamma^{\prime}_{j}),(T^{\prime},\gamma^{\prime}))=0$

then

(37)
$\lim_{j\to\infty}d_{\Pi}((T_{j}+T^{\prime}_{j},\gamma_{j}*\gamma^{\prime}_{j}),(T+T^{\prime},\gamma*\gamma^{\prime}))=0.$

If (37) is not the case, there exists a sequence $\displaystyle(s_{j})_{j\geq
1}$ in $\displaystyle[0,1]$ and $\displaystyle\varepsilon>0$ such that

(38)
$d_{L}((\gamma_{j}*\gamma^{\prime}_{j})(s_{j}(T_{j}+T^{\prime}_{j})),(\gamma*\gamma^{\prime})(s_{j}(T+T^{\prime})))\geq\varepsilon$

for every $\displaystyle j$. Then $\displaystyle T_{j},T^{\prime}_{j}>0$ for
sufficiently large $\displaystyle j$, thus we may assume that $\displaystyle
T_{j},T^{\prime}_{j}>0$ holds for every $\displaystyle j\geq 1$.

By taking a subsequence, we may also assume that either $\displaystyle
s_{j}\leq T_{j}/(T_{j}+T^{\prime}_{j})$ or $\displaystyle s_{j}\geq
T_{j}/(T_{j}+T^{\prime}_{j})$ holds for all $\displaystyle j$. In the
following we only consider the case $\displaystyle s_{j}\leq
T_{j}/(T_{j}+T^{\prime}_{j})$, since the proof in the other case is parallel.
Now we obtain

$\displaystyle(\gamma_{j}*\gamma^{\prime}_{j})(s_{j}(T_{j}+T^{\prime}_{j}))=\gamma_{j}\biggl{(}s_{j}\cdot\frac{T_{j}+T^{\prime}_{j}}{T_{j}}\cdot
T_{j}\biggr{)}.$

We set $\displaystyle\tau_{j}:=s_{j}(T_{j}+T^{\prime}_{j})/T_{j}\in[0,1]$.
Then the LHS of (38) is bounded from above by

(39)
$d_{L}(\gamma_{j}(\tau_{j}T_{j}),\gamma(\tau_{j}T))+d_{L}(\gamma(\tau_{j}T),(\gamma*\gamma^{\prime})(s_{j}(T+T^{\prime}))).$

Then the first term of (39) goes to $\displaystyle 0$ as $\displaystyle
j\to\infty$, since

$\displaystyle d_{L}(\gamma_{j}(\tau_{j}T),\gamma(\tau_{j}T))\leq
d_{\Pi}((T_{j},\gamma_{j}),(T,\gamma)).$

Since $\displaystyle\gamma*\gamma^{\prime}$ is (uniformly) continuous, to show
that the second term of (39) goes to $\displaystyle 0$ as $\displaystyle
j\to\infty$, it is sufficient to check:

(40) $\lim_{j\to\infty}\tau_{j}T-s_{j}(T+T^{\prime})=0.$

Using the obvious identity

(41)
$\tau_{j}T-s_{j}(T+T^{\prime})=s_{j}(T_{j}+T^{\prime}_{j})(T/T_{j})-s_{j}(T+T^{\prime}),$

we can check (40) by considering the following three cases:

  * •

$\displaystyle T>0$: this case $\displaystyle\lim_{j\to\infty}T/T_{j}=1$, thus
the RHS of (41) goes to $\displaystyle 0$.

  * •

$\displaystyle T=0$ and $\displaystyle T^{\prime}>0$: this case the first term
is $\displaystyle 0$, and the second term goes to $\displaystyle 0$ because
$\displaystyle 0\leq s_{j}\leq T_{j}/(T_{j}+T^{\prime}_{j})$ implies
$\displaystyle\lim_{j\to\infty}s_{j}=0$.

  * •

$\displaystyle T,T^{\prime}=0$: this case both the first and second terms are
$\displaystyle 0$.

This completes the proof. ∎

Next we consider the space of continuous Moore loops with marked points. For
every $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$, let
$\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm con}}\,$ denote the set consists
of $\displaystyle(T,\gamma,t_{0},\ldots,t_{k})$ such that

  * •

$\displaystyle T\in{\mathbb{R}}_{\geq 0}$ and $\displaystyle\gamma\in
C^{0}([0,T],L)$ such that $\displaystyle\gamma(0)=\gamma(T)$.

  * •

$\displaystyle 0=t_{0}\leq t_{1}\leq\ldots\leq t_{k}\leq T$.

For each $\displaystyle a\in H_{1}(L:{\mathbb{Z}})$, let
$\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm con}}\,(a)$ denote the subspace
of $\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm con}}\,$ which consists of
$\displaystyle(T,\gamma,t_{0},\ldots,t_{k})$ such that
$\displaystyle[\gamma]=a$. The set
$\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm con}}\,$ can be identified with
the set

$\displaystyle\\{(\Gamma_{0},\ldots,\Gamma_{k})\in(\Pi^{\text{\rm
con}}\,)^{k+1}\mid\text{\rm ev}\,_{1}(\Gamma_{i})=\text{\rm
ev}\,_{0}(\Gamma_{i+1})\,(0\leq i\leq k-1),\,\text{\rm
ev}\,_{1}(\Gamma_{k})=\text{\rm ev}\,_{0}(\Gamma_{0})\\}.$

Using this identification, a metric $\displaystyle d_{{\mathcal{L}}_{k+1}}$ on
$\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm con}}\,$ is defined by

$\displaystyle
d_{{\mathcal{L}}_{k+1}}((\Gamma_{0},\ldots,\Gamma_{k}),(\Gamma^{\prime}_{0},\ldots,\Gamma^{\prime}_{k})):=\max_{0\leq
i\leq k}d_{\Pi}(\Gamma_{i},\Gamma^{\prime}_{i}).$

We consider the topology on $\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,$ induced from $\displaystyle d_{{\mathcal{L}}_{k+1}}$.

The evaluation map

$\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}:{\mathcal{L}}_{k+1}^{\text{\rm con}}\,\to
L\quad(j\in\\{0,\ldots,k\\})$

and the concatenation map

$\displaystyle\text{\rm con}\,_{j}:{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,\,{}_{\text{\rm ev}\,^{{\mathcal{L}}}_{j}}\\!\times_{\text{\rm
ev}\,^{{\mathcal{L}}}_{0}}{\mathcal{L}}_{k^{\prime}+1}^{\text{\rm
con}}\,\to{\mathcal{L}}_{k+k^{\prime}}^{\text{\rm
con}}\,\quad(j\in\\{1,\ldots,k\\})$

are defined in the same way as in the case of smooth loops (see Section 4.1).
The evaluation map is obviously continuous, and the concatenation map is
continuous by Lemma 7.25. The evaluation map $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}$ will be abbreviated as $\displaystyle\text{\rm
ev}\,_{j}$ when there is no risk of confusion.

###  7.4. Strongly continuous maps to
$\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm con}}\,$

The goal of this subsection is to define strongly continuous maps from moduli
spaces of (perturbed) pseudo-holomorphic disks (with boundary marked points)
to spaces of continuous loops (with marked points), so that natural
compatibility conditions (spelled out in Proposition 7.26) are satisfied.
Throughout this subsection, the space
$\displaystyle{\mathcal{L}}_{k+1}^{\text{\rm con}}\,$ is equipped with the
topology defined from the metric $\displaystyle d_{{\mathcal{L}}_{k+1}}$,
which is defined in Section 7.3.

To state Proposition 7.26 we have to introduce the following notations. For
any $\displaystyle\beta\in H_{2}({\mathbb{C}}^{n},L)$ and
$\displaystyle(T,B)\in{\mathcal{G}}(k+1:\beta)$, one can define a continuous
map

(42) $\text{\rm ev}\,_{\text{\rm int}\,}:\prod_{v\in C_{0,\text{\rm
int}\,}(T)}P\times{\mathcal{L}}_{k_{v}+1}^{\text{\rm con}}\,(\partial
B(v))\to\prod_{e\in C_{1,\text{\rm int}\,}(T)}(P\times L)^{2}$

in a way similar to (23). Also let us recall the diagonal map (25). Then one
can define a continuous map

(43) $\biggl{(}\prod_{e\in C_{1,\text{\rm int}\,}(T)}P\times
L\biggr{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}\,}}\biggl{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T)}P\times{\mathcal{L}}_{k_{v}+1}^{\text{\rm con}}\,(\partial
B(v))\biggr{)}\to P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,(\partial\beta)$

by taking concatenations of loops, where the fiber product in the LHS is taken
over $\displaystyle\prod_{e\in C_{1,\text{\rm int}\,}(T)}(P\times L)^{2}$.

######  Proposition 7.26.

For every $\displaystyle k\in{\mathbb{Z}}_{\geq 0}$, $\displaystyle
m\in{\mathbb{Z}}_{\geq 0}$, and $\displaystyle P\in\\{\\{m\\},[m,m+1]\\}$, one
can define strongly continuous maps

(44) $\text{\rm ev}\,^{{\mathcal{M}}}:{\mathcal{M}}_{k+1}(\beta:P)\to
P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,(\partial\beta)\qquad(\omega_{n}(\beta)<\varepsilon(m+1-k)),$ (45)
$\text{\rm ev}\,^{{\mathcal{N}}^{0}}:{\mathcal{N}}^{0}_{k+1}(\beta:P)\to
P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,(\partial\beta)\qquad(\omega_{n}(\beta)<\varepsilon(m-1-k)),$ (46)
$\text{\rm ev}\,^{{\mathcal{N}}^{\geq 0}}:{\mathcal{N}}^{\geq
0}_{k+1}(\beta:P)\to P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,(\partial\beta)\qquad(\omega_{n}(\beta)<\varepsilon(m-k-U)),$

so that the following diagrams commute for every
$\displaystyle(T,B)\in{\mathcal{G}}(k+1,\beta)$:

  * •

$\displaystyle\textstyle{\biggl{(}\prod_{e}P\times
L\biggr{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}\biggl{(}\prod_{v}{\mathcal{M}}_{k_{v}+1}(B(v):P)\biggr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathcal{M}}_{k+1}(\beta:P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{\biggl{(}\prod_{e}P\times
L\biggr{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}\biggl{(}\prod_{v}P\times{\mathcal{L}}_{k_{v}+1}^{\text{\rm
con}}\,(\partial
B(v))\biggr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{(\ref{170615_1})}$$\displaystyle\textstyle{P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,(\partial\beta)}$

where the first horizontal map is defined from (34) by setting $\displaystyle
d=0$, and vertical maps are defined by (44).

  * •

$\displaystyle\textstyle{\biggl{(}\prod_{e}P\times
L\biggr{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}(\star)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathcal{N}}^{0}_{k+1}(\beta:P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{\biggl{(}\prod_{e}P\times
L\biggr{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}\biggl{(}\prod_{v}P\times{\mathcal{L}}_{k_{v}+1}^{\text{\rm
con}}\,(\partial
B(v))\biggr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{(\ref{170615_1})}$$\displaystyle\textstyle{P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,(\partial\beta)}$

where $\displaystyle(\star):=\prod_{v\neq
v_{0}}{\mathcal{M}}_{k_{v}+1}(B(v):P)\times{\mathcal{N}}^{0}_{k_{v_{0}}+1}(B(v_{0}):P)$
and the first horizontal map is defined from (35) by setting $\displaystyle
d=0$, and vertical maps are defined by (45).

  * •

$\displaystyle\textstyle{\biggl{(}\prod_{e}P\times
L\biggr{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}(\star)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathcal{N}}^{\geq
0}_{k+1}(\beta:P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{\biggl{(}\prod_{e}P\times
L\biggr{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}}\,}\biggl{(}\prod_{v}P\times{\mathcal{L}}_{k_{v}+1}^{\text{\rm
con}}\,(\partial
B(v))\biggr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{(\ref{170615_1})}$$\displaystyle\textstyle{P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,(\partial\beta)}$

where $\displaystyle(\star)=\prod_{v\neq
v_{0}}{\mathcal{M}}_{k_{v}+1}(B(v):P)\times{\mathcal{N}}^{\geq
0}_{k_{v_{0}}+1}(B(v_{0}):P)$ and the first horizontal map is defined from
(36) by setting $\displaystyle d=0$, and vertical maps are defined by (46).

In the rest of this subsection, we explain the definition of the strongly
continous map (44). Definitions of (45) and (46) are similar. For each
$\displaystyle p\in{\mathcal{M}}_{k+1}(\beta:P)$, let
$\displaystyle{\mathcal{U}}_{p}=(U_{p},{\mathcal{E}}_{p},s_{p},\psi_{p})$ be a
K-chart at $\displaystyle p$. To define a strongly continuous map (44), it is
sufficient to define a continuous map $\displaystyle\text{\rm
ev}\,^{{\mathcal{M}}}_{p}:U_{p}\to P\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,$ for each $\displaystyle p$, so that compatibility conditions with
coordinate changes are satisfied.

By Lemma 7.22, every $\displaystyle x\in U_{p}$ is represented by an element
of

$\displaystyle P\times\bigg{(}\prod_{e\in C_{1,\text{\rm
int}\,}(T^{\prime})}L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}\,}}\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T^{\prime})}{\mathcal{MM}}_{k_{v}+1}(B^{\prime}(v))\bigg{)}$

where $\displaystyle(T^{\prime},B^{\prime})$ is a reduction of
$\displaystyle(T,B)$. We denote the representative by

$\displaystyle(\pi,(u^{v},z^{v}_{0},\ldots,z^{v}_{k_{v}})_{v}).$

Now we define

$\displaystyle\text{\rm ev}\,^{{\mathcal{M}}}_{p}(x):=(\pi,\text{\rm
ev}\,((u^{v},z^{v}_{0},\ldots,z^{v}_{k_{v}})_{v}))\in
P\times{\mathcal{L}}_{k+1}^{\text{\rm con}}\,(\partial\beta),$

where $\displaystyle\text{\rm
ev}\,((u^{v},z^{v}_{0},\ldots,z^{v}_{k_{v}})_{v})$ is the concatenation
(defined by (43)) of $\displaystyle\text{\rm
ev}\,(u^{v},z^{v}_{0},\ldots,z^{v}_{k_{v}})\in{\mathcal{L}}_{k_{v}+1}^{\text{\rm
con}}\,(\partial B^{\prime}(v))$ (which is defined below) for all
$\displaystyle v\in C_{0,\text{\rm int}\,}(T^{\prime})$.

To define $\displaystyle\text{\rm
ev}\,(u^{v},z^{v}_{0},\ldots,z^{v}_{k_{v}})$, we distinguish the case
$\displaystyle u^{v}$ is constant and the case $\displaystyle u^{v}$ is
nonconstant.

  * •

The case $\displaystyle u^{v}$ is a constant map to $\displaystyle q\in L$: we
define

$\displaystyle\text{\rm
ev}\,(u^{v},z^{v}_{0},\ldots,z^{v}_{k_{v}}):=(c_{q},\ldots,c_{q})$

where $\displaystyle c_{q}$ denotes the constant path at $\displaystyle q$
with length $\displaystyle 0$.

  * •

The case $\displaystyle u^{v}$ is nonconstant: we use the length
parametrization. For every $\displaystyle j\in\\{0,\ldots,k_{v}\\}$ we define
$\displaystyle\theta_{j}\in[0,2\pi)$ by $\displaystyle
z^{v}_{j}/z^{v}_{0}=e^{\sqrt{-1}\theta_{j}}$, thus $\displaystyle
0=\theta_{0}<\theta_{1}<\cdots<\theta_{k_{v}}<2\pi$. Then we define

$\displaystyle\displaystyle T^{u^{v}}_{j}$
$\displaystyle\displaystyle:=\int_{\theta_{j}}^{\theta_{j+1}}\bigg{\lvert}\frac{d}{d\theta}u^{v}(z^{v}_{0}\cdot
e^{\sqrt{-1}\theta})\bigg{\rvert}\,d\theta,$
$\displaystyle\displaystyle\lambda^{u^{v}}_{j}$
$\displaystyle\displaystyle:[\theta_{j},\theta_{j+1}]\to[0,T^{u^{v}}_{j}];\quad\Theta\mapsto\int_{\theta_{j}}^{\Theta}\bigg{\lvert}\frac{d}{d\theta}u^{v}(z^{v}_{0}\cdot
e^{\sqrt{-1}\theta})\bigg{\rvert}\,d\theta,$
$\displaystyle\displaystyle\gamma^{u^{v}}_{j}$
$\displaystyle\displaystyle:[0,T^{u^{v}}_{j}]\to L;\quad t\mapsto
u^{v}(e^{\sqrt{-1}(\lambda^{u^{v}}_{j})^{-1}(t)}).$

$\displaystyle\frac{d}{d\theta}u^{v}(e^{\sqrt{-1}\theta})$ vanishes at only
finitely many $\displaystyle\theta$ since $\displaystyle u^{v}$ is
nonconstant, thus $\displaystyle T^{u^{v}}_{j}$ is positive and
$\displaystyle\lambda^{u^{v}}_{j}$ is strictly increasing. Hence
$\displaystyle(\lambda^{u^{v}}_{j})^{-1}:[0,T^{u^{v}}_{j}]\to[\theta_{j},\theta_{j+1}]$
is a well-defined continuous map. Then we define

$\displaystyle\text{\rm
ev}\,(u^{v},z^{v}_{0},\ldots,z^{v}_{k_{v}}):=(T^{u^{v}}_{j},\gamma^{u^{v}}_{j})_{0\leq
j\leq k_{v}}.$

This completes the definition of $\displaystyle\text{\rm
ev}\,^{{\mathcal{M}}}_{p}(x)$.

######  Remark 7.27.

For later purpose (proof of Lemma 7.28) we define
$\displaystyle\Lambda^{u^{v}}_{j}:[0,1]\to[0,T^{u^{v}}_{j}]$ by
$\displaystyle\Lambda^{u^{v}}_{j}(s):=\lambda^{u^{v}}_{j}((1-s)\theta_{j}+s\theta_{j+1})$.

The above definition of $\displaystyle\text{\rm ev}\,^{{\mathcal{M}}}_{p}(x)$
does not depend on choices of representatives of $\displaystyle x$. In
particular, the family of maps $\displaystyle(\text{\rm
ev}\,^{{\mathcal{M}}}_{p})_{p}$ is compatible with coordinate changes of
K-charts. To show that this family defines a strongly continuous map, we have
to check that $\displaystyle\text{\rm ev}\,^{{\mathcal{M}}}_{p}$ is continuous
for each $\displaystyle p$.

######  Lemma 7.28.

$\displaystyle\text{\rm ev}\,^{{\mathcal{M}}}_{p}:U_{p}\to
P\times{\mathcal{L}}_{k+1}^{\text{\rm con}}\,(\partial\beta)$ is continuous
for every $\displaystyle p\in{\mathcal{M}}_{k+1}(\beta:P)$.

###### Proof.

Step 1. Let $\displaystyle(x_{l})_{l\geq 1}$ be a sequence in $\displaystyle
U_{p}$ which converges to $\displaystyle x_{\infty}\in U_{p}$. For simplicity,
we work under the following assumptions:

  * •

$\displaystyle k=0$.

  * •

Each $\displaystyle x_{l}$ is represented by a single holomorphic map; we
denote the representative by $\displaystyle(\pi_{l},u_{l},z_{l,0})$.

  * •

$\displaystyle x_{\infty}$ is represented by an element of

$\displaystyle P\times\bigg{(}\prod_{e\in C_{1,\text{\rm
int}\,}(T_{\infty})}L\bigg{)}\,_{\Delta}\\!\times_{\text{\rm ev}\,_{\text{\rm
int}\,}}\bigg{(}\prod_{v\in C_{0,\text{\rm
int}\,}(T_{\infty})}{\mathcal{MM}}_{k_{v}+1}(B_{\infty}(v))\bigg{)}$

for some $\displaystyle(T_{\infty},B_{\infty})\in{\mathcal{G}}(k+1,\beta)$;
see Lemma 7.22. We denote the representative by
$\displaystyle(\pi_{\infty},(u^{v}_{\infty},z^{v}_{\infty,0},\ldots,z^{v}_{\infty,k_{v}})_{v\in
C_{0,\text{\rm int}\,}(T_{\infty})})$.

Since $\displaystyle\lim_{l\to\infty}\pi_{l}=\pi_{\infty}$ is obvious, it is
sufficient to show that

(47) $\lim_{l\to\infty}d_{{\mathcal{L}}_{1}}(\text{\rm
ev}\,(u_{l},z_{l,0}),\text{\rm
ev}\,((u^{v}_{\infty},z^{v}_{\infty,0},\ldots,z^{v}_{\infty,k_{v}})_{v}))=0.$

Step 2. We first consider the case $\displaystyle\sharp C_{0,\text{\rm
int}\,}(T_{\infty})=1$, and abbreviate $\displaystyle u^{v}_{\infty}$ as
$\displaystyle u_{\infty}$. We may assume that
$\displaystyle\lim_{l\to\infty}u_{l}=u_{\infty}$ in the $\displaystyle
C^{\infty}$-topology, and $\displaystyle z_{l,0}=1$ for every $\displaystyle
l$ (including $\displaystyle l=\infty$). When $\displaystyle u_{\infty}$ is
constant, then $\displaystyle u_{l}|_{\partial D}$ converges to a constant map
in $\displaystyle C^{\infty}$ (in particular $\displaystyle C^{0}$) topology,
then (47) follows. When $\displaystyle u_{\infty}$ is nonconstant, then
$\displaystyle u_{l}$ is nonconstant for sufficiently large $\displaystyle l$.
Moreover $\displaystyle\lim_{l\to\infty}T^{u_{l}}_{0}=T^{u_{\infty}}_{0}$ and
$\displaystyle\lim_{l\to\infty}\Lambda^{u_{l}}_{0}=\Lambda^{u_{\infty}}_{0}$
in the $\displaystyle C^{0}$-topology. Then (47) follows from the next lemma,
the proof of which is completely elementary and omitted.

######  Lemma 7.29.

Let $\displaystyle(\tau_{l})_{l}$ be a sequence in $\displaystyle
C^{0}([0,1],[0,1])$ such that

  * •

$\displaystyle(\tau_{l})_{l}$ has a $\displaystyle C^{0}$-limit
$\displaystyle\tau_{\infty}:[0,1]\to[0,1]$.

  * •

For every $\displaystyle l$ (including $\displaystyle\infty$),
$\displaystyle\tau_{l}$ is strictly increasing and
$\displaystyle\tau_{l}(0)=0$, $\displaystyle\tau_{l}(1)=1$.

Then $\displaystyle(\tau_{l})^{-1}$ converges to
$\displaystyle(\tau_{\infty})^{-1}$ in the $\displaystyle C^{0}$-topology.

Step 3. Now we consider the case $\displaystyle\sharp C_{0,\text{\rm
int}\,}(T_{\infty})>1$. We fix $\displaystyle\varepsilon>0$, which can be
arbitrarily small. For sufficiently large every $\displaystyle l$, there
exists a decomposition (depends on $\displaystyle\varepsilon$ but we drop it
from the following notations)

$\displaystyle D=V_{l}\sqcup\bigsqcup_{v\in C_{0,\text{\rm
int}\,}(T_{\infty})}U^{v}_{l}$

such that the following conditions are satisfied (see [8] Section 7.1.4):

  * •

$\displaystyle V_{l}$ is a compact set with the number of connected components
is $\displaystyle\sharp C_{0,\text{\rm int}\,}(T_{\infty})$, and
$\displaystyle U^{v}_{l}$ is a connected open set for every $\displaystyle
v\in C_{0,\text{\rm int}\,}(T_{\infty})$.

  * •

$\displaystyle V_{l}\cap\partial D$ is a disjoint union of $\displaystyle
2\sharp C_{0,\text{\rm int}\,}(T_{\infty})-1$ closed intervals.

  * •

$\displaystyle\int_{V_{l}\cap\partial
D}\bigg{\lvert}\frac{d}{d\theta}u_{l}(e^{\sqrt{-1}\theta})\bigg{\rvert}<\varepsilon$.

  * •

For every $\displaystyle v\in C_{0,\text{\rm int}\,}(T_{\infty})$,
$\displaystyle\overline{U^{v}_{l}}\cap\partial D$ is a disjoint union of
$\displaystyle k_{v}+1$ closed intervals. We denote them as $\displaystyle
I^{v}_{l}(0),\ldots,I^{v}_{l}(k_{v})$ aligned in anti-clockwise order.

  * •

There exists a sequence $\displaystyle(\rho^{v}_{l})_{l}$ in
$\displaystyle\text{\rm Aut}\,(D)$ such that
$\displaystyle(\rho^{v}_{l})^{-1}(U^{v}_{l})$ is independent on $\displaystyle
l$ (denoted by $\displaystyle U^{v}$), and $\displaystyle
u_{l}\circ\rho^{v}_{l}$ converges to $\displaystyle u^{v}_{\infty}$ (with
respect to the $\displaystyle C^{\infty}$-topology) on
$\displaystyle\overline{U^{v}}$.

  * •

$\displaystyle\overline{U^{v}}\cap\partial D$ is a disjoint union of
$\displaystyle k_{v}+1$ closed intervals, which we denote by $\displaystyle
I^{v}_{\infty}(0),\ldots,I^{v}_{\infty}(k_{v})$ in anti-clockwise order.

$\displaystyle V_{l}$

For each $\displaystyle v\in C_{0,\text{\rm int}\,}(T_{\infty})$ and
$\displaystyle j\in\\{0,\ldots,k_{v}\\}$, our assumption $\displaystyle
u_{l}\circ\rho^{v}_{l}\to u^{v}_{\infty}$ implies that

$\displaystyle\lim_{l\to\infty}d_{\Pi}(u_{l}|_{I^{v}_{l}(j)},u^{v}_{\infty}|_{I^{v}_{\infty}(j)})=0.$

The proof is same as our argument in Step 2. On the other hand, for every
$\displaystyle l$, the total lengths of $\displaystyle
u_{l}|_{V_{l}\cap\partial D}$ is at most $\displaystyle\varepsilon$. Since
$\displaystyle\text{\rm ev}\,(u_{l},z_{l,0})$ is obtained as the concatenation
of these paths, it is not difficult to see that

$\displaystyle\limsup_{l\to\infty}\,d_{{\mathcal{L}}_{1}}(\text{\rm
ev}\,(u_{l},z_{l,0}),\text{\rm
ev}\,((u^{v}_{\infty},z^{v}_{\infty,0},\ldots,z^{v}_{\infty,k_{v}})_{v}))\leq
10\varepsilon.$

Since $\displaystyle\varepsilon$ can be arbitrarily small, this implies the
convergence (47). ∎

###  7.5. $\displaystyle C^{0}$-approximation lemma and CF-perturbation

We first introduce the notion of “$\displaystyle\varepsilon$-closeness” for
strongly continuous maps from a K-space to a metric space:

######  Definition 7.30.

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a K-space,
$\displaystyle(Y,d)$ be a metric space, and
$\displaystyle\widehat{f},\widehat{g}:(X,\widehat{{\mathcal{U}}})\to Y$ be
strongly continuous maps. For any $\displaystyle\varepsilon>0$, we say that
$\displaystyle\widehat{f}$ and $\displaystyle\widehat{g}$ are
$\displaystyle\varepsilon$-close, if $\displaystyle
d(f_{p}(x),g_{p}(x))<\varepsilon$ for every $\displaystyle p\in X$ and
$\displaystyle x\in U_{p}$.

Let us state a key technical result in this subsection, which we call
$\displaystyle C^{0}$-approximation lemma. The notion of “open substrcture” of
a given K-structure is defined in Definition 3.20 in [10]. The constant
$\displaystyle\rho_{L}$ in the statement was introduced in the second
paragraph of Section 7.3.

######  Theorem 7.31.

Let $\displaystyle(X,\widehat{{\mathcal{U}}})$ be a compact K-space and
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,$ be a strongly continuous map such that $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{f}:(X,\widehat{{\mathcal{U}}})\to L$ is
strongly smooth for every $\displaystyle j\in\\{0,\ldots,k\\}$. Let
$\displaystyle Z$ be a closed subset of $\displaystyle X$ and
$\displaystyle\widehat{g}:(Z,\widehat{{\mathcal{U}}}|_{Z})\to{\mathcal{L}}_{k+1}$
be a strongly smooth map such that:

  * •

$\displaystyle\text{\rm ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{g}=\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{f}|_{Z}$ for every $\displaystyle
j\in\\{0,\ldots,k\\}$.

  * •

$\displaystyle\widehat{g}$ is $\displaystyle\varepsilon$-close to
$\displaystyle\widehat{f}|_{Z}$ with respect to $\displaystyle
d_{{\mathcal{L}}_{k+1}}$.

If $\displaystyle\varepsilon<\rho_{L}$, there exists an open substructure
$\displaystyle\widehat{{\mathcal{U}}_{0}}$ of
$\displaystyle\widehat{{\mathcal{U}}}$ and a strongly smooth map
$\displaystyle\widehat{g^{\prime}}:(X,\widehat{{\mathcal{U}}_{0}})\to{\mathcal{L}}_{k+1}$
such that the following conditions hold:

  * •

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

  * •

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

  * •

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

The proof of Theorem 7.31 is carried out in Section 9. Combining Theorem 7.31
with results from [10], we obtain Theorem 7.32 below. In the statement of
Theorem 7.32 we use the following notions from [10] without repeating their
definitions:

  * •

Thickening of K-structures: see Section 5.2 in [10].

  * •

Collared versions of K-structures, strongly smooth maps and CF-perturbations:
see Section 17.5 in [10].

The assumptions in Theorem 7.32 on K-structures and CF-perturbations are
similar to Situations 17.43 and 17.57 in [10], respectively.

######  Theorem 7.32.

Suppose that we are given the following data:

  * •

$\displaystyle k\in{\mathbb{Z}}_{\geq 0}$, $\displaystyle\tau\in(0,1)$ and
$\displaystyle\varepsilon\in(0,\rho_{L})$.

  * •

A $\displaystyle\tau$-collared K-space
$\displaystyle(X,\widehat{{\mathcal{U}}})$.

  * •

A $\displaystyle\tau$-collared strongly continuous map
$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,$ such that $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{f}:(X,\widehat{{\mathcal{U}}})\to L$ is
admissible for every $\displaystyle j\in\\{0,\ldots,k\\}$, and
$\displaystyle\text{\rm ev}\,^{{\mathcal{L}}}_{0}\circ\widehat{f}$ is strata-
wise weakly submersive.

  * •

For every $\displaystyle l\in{\mathbb{Z}}_{\geq 1}$, a
$\displaystyle\tau$-collared K-structure
$\displaystyle\widehat{{\mathcal{U}}^{+}_{l}}$ on
$\displaystyle\widehat{S}_{l}(X)$ which is a thickening of
$\displaystyle\widehat{{\mathcal{U}}}|_{\widehat{S}_{l}(X)}$.

  * •

For every $\displaystyle l_{1},l_{2}\in{\mathbb{Z}}_{\geq 1}$,
$\displaystyle(l_{1}+l_{2})!/(l_{1})!(l_{2})!$-fold covering of
$\displaystyle\tau$-collared $\displaystyle K$-spaces

$\displaystyle\widehat{S}_{l_{1}}(\widehat{S}_{l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{2}}})\to(\widehat{S}_{l_{1}+l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{1}+l_{2}}})$

such that the following diagrams commute for every $\displaystyle
l_{1},l_{2},l_{3}\in{\mathbb{Z}}_{\geq 1}$:

$\displaystyle\textstyle{\widehat{S}_{l_{1}}(\widehat{S}_{l_{2}}(\widehat{S}_{l_{3}}(X),\widehat{{\mathcal{U}}^{+}_{l_{3}}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{\widehat{S}_{l_{1}+l_{2}}(\widehat{S}_{l_{3}}(X),\widehat{{\mathcal{U}}^{+}_{l_{3}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{\widehat{S}_{l_{1}}(\widehat{S}_{l_{2}+l_{3}}(X),\widehat{{\mathcal{U}}^{+}_{l_{2}+l_{3}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{(\widehat{S}_{l_{1}+l_{2}+l_{3}}(X),\widehat{{\mathcal{U}}^{+}_{l_{1}+l_{2}+l_{3}}}),}$
$\displaystyle\textstyle{\widehat{S}_{l_{1}}(\widehat{S}_{l_{2}}(X,\widehat{{\mathcal{U}}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{\widehat{S}_{l_{1}}(\widehat{S}_{l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{2}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{\widehat{S}_{l_{1}+l_{2}}(X,\widehat{{\mathcal{U}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{(\widehat{S}_{l_{1}+l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{1}+l_{2}}}).}$

  * •

A $\displaystyle\tau$-collared CF-perturbation
$\displaystyle\widehat{{\mathfrak{S}}^{+}_{l}}$ of
$\displaystyle(\widehat{S}_{l}(X),\widehat{{\mathcal{U}}^{+}_{l}})$ for every
$\displaystyle l\in{\mathbb{Z}}_{\geq 1}$, such that the pullback of
$\displaystyle\widehat{{\mathfrak{S}}^{+}_{l_{1}+l_{2}}}$ by
$\displaystyle\widehat{S}_{l_{1}}(\widehat{S}_{l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{2}}})\to(\widehat{S}_{l_{1}+l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{1}+l_{2}}})$
coincides with the restriction of
$\displaystyle\widehat{{\mathfrak{S}}^{+}_{l_{2}}}$ for every $\displaystyle
l_{1},l_{2}\in{\mathbb{Z}}_{\geq 1}$.

  * •

A $\displaystyle\tau$-collared admissible map
$\displaystyle\widehat{f^{+}_{l}}:(\widehat{S}_{l}(X),\widehat{{\mathcal{U}}^{+}_{l}})\to{\mathcal{L}}_{k+1}$
for every $\displaystyle l\in{\mathbb{Z}}_{\geq 1}$ such that:

    * –

The pullback of $\displaystyle\widehat{f^{+}_{l_{1}+l_{2}}}$ by
$\displaystyle\widehat{S}_{l_{1}}(\widehat{S}_{l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{2}}})\to(\widehat{S}_{l_{1}+l_{2}}(X),\widehat{{\mathcal{U}}^{+}_{l_{1}+l_{2}}})$
coincides with the restriction of $\displaystyle\widehat{f^{+}_{l_{2}}}$ for
every $\displaystyle l_{1},l_{2}\in{\mathbb{Z}}_{\geq 1}$.

    * –

For every $\displaystyle j\in\\{0,\ldots,k\\}$, $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{f_{l}^{+}}:(\widehat{S}_{l}(X),\widehat{{\mathcal{U}}^{+}_{l}})\to
L$ coincides with $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{f}|_{\widehat{S}_{l}(X)}:(\widehat{S}_{l}(X),\widehat{{\mathcal{U}}}|_{\widehat{S}_{l}(X)})\to
L$ via the KK-embedding
$\displaystyle\widehat{{\mathcal{U}}}|_{\widehat{S}_{l}(X)}\to\widehat{{\mathcal{U}}^{+}_{l}}$.

    * –

$\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{0}\circ\widehat{f_{l}^{+}}:(\widehat{S}_{l}(X),\widehat{{\mathcal{U}}^{+}_{l}})\to
L$ is strata-wise strongly submersive with respect to
$\displaystyle\widehat{{\mathfrak{S}}^{+}_{l}}$.

    * –

$\displaystyle\widehat{f^{+}_{l}}$ is $\displaystyle\varepsilon$-close to
$\displaystyle\widehat{f}|_{\widehat{S}_{l}(X)}$.

Then, for any $\displaystyle\tau^{\prime}\in(0,\tau)$, there exist the
following data:

  * •

A $\displaystyle\tau^{\prime}$-collared K-structure
$\displaystyle\widehat{{\mathcal{U}}^{+}}$ on $\displaystyle X$, which is a
thickening of $\displaystyle\widehat{{\mathcal{U}}}$.

  * •

An isomorphism of $\displaystyle\tau^{\prime}$-collared K-structures
$\displaystyle\widehat{{\mathcal{U}}^{+}}|_{\widehat{S}_{l}(X)}\cong\widehat{{\mathcal{U}}^{+}_{l}}$
for every $\displaystyle l\in{\mathbb{Z}}_{\geq 1}$.

  * •

A $\displaystyle\tau^{\prime}$-collared CF-perturbation
$\displaystyle\widehat{{\mathfrak{S}}^{+}}$ of
$\displaystyle(X,\widehat{{\mathcal{U}}^{+}})$ such that
$\displaystyle\widehat{{\mathfrak{S}}^{+}}|_{\widehat{S}_{l}(X)}$ coincides
with $\displaystyle\widehat{{\mathfrak{S}}^{+}_{l}}$ via the isomorphism of
K-spaces
$\displaystyle\widehat{{\mathcal{U}}^{+}}|_{\widehat{S}_{l}(X)}\cong\widehat{{\mathcal{U}}^{+}_{l}}$.

  * •

A $\displaystyle\tau^{\prime}$-collared admissible map
$\displaystyle\widehat{f^{+}}:(X,\widehat{{\mathcal{U}}^{+}})\to{\mathcal{L}}_{k+1}$
such that:

    * –

$\displaystyle\widehat{f^{+}}$ is $\displaystyle\varepsilon$-close to
$\displaystyle\widehat{f}$.

    * –

For every $\displaystyle j\in\\{0,\ldots,k\\}$, $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{f^{+}}$ coincides with
$\displaystyle\text{\rm ev}\,^{{\mathcal{L}}}_{j}\circ\widehat{f}$ with
respect to the KK-embedding
$\displaystyle\widehat{{\mathcal{U}}}\to\widehat{{\mathcal{U}}^{+}}$.

    * –

$\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{0}\circ\widehat{f^{+}}:(X,\widehat{{\mathcal{U}}^{+}})\to
L$ is strata-wise strongly submersive with respect to
$\displaystyle\widehat{{\mathfrak{S}}^{+}}$.

###### Proof.

The $\displaystyle K$-structure $\displaystyle\widehat{{\mathcal{U}}^{+}}$ and
the CF-perturbation $\displaystyle\widehat{{\mathfrak{S}}^{+}}$ are defined by
Propositions 17.62 and 17.65 in [10], respectively; here we apply Proposition
17.65 (2) to $\displaystyle\text{\rm
ev}\,^{{\mathcal{L}}}_{0}\circ\widehat{f}:(X,\widehat{{\mathcal{U}}})\to L$.
Moreover
$\displaystyle\widehat{f^{+}_{1}}:\widehat{S}_{1}(X)\to{\mathcal{L}}_{k+1}$
extends to the $\displaystyle\tau^{\prime}$-neighborhood of
$\displaystyle\partial X$ (denoted by $\displaystyle N(\tau^{\prime})$). Then
we can apply Theorem 7.31 to conclude the proof, taking
$\displaystyle{\mathcal{N}}(\tau^{\prime})$ as $\displaystyle Z$ and the
extension of $\displaystyle\widehat{f^{+}_{1}}$ as $\displaystyle\widehat{g}$.
∎

######  Remark 7.33.

Nextly, we should state and prove a version of Theorem 7.32 such that
$\displaystyle(X,\widehat{{\mathcal{U}}})$ is a $\displaystyle\tau$-collared
K-space, and

$\displaystyle\widehat{f}:(X,\widehat{{\mathcal{U}}})\to[a,b]^{\boxplus\tau}\times{\mathcal{L}}_{k+1}^{\text{\rm
con}}\,$

is a $\displaystyle\tau$-collared strongly continuous map, where
$\displaystyle a<b$ are real numbers and
$\displaystyle[a,b]^{\boxplus\tau}:=[a-\tau,b+\tau]$. However, the statement
is similar to Theorem 7.32 and involves further notations, thus here we choose
not to write it down in detail.

###  7.6. Wrap-up of the proof

Now we can complete the proof of Theorem 6.1 assuming results presented in
Sections 7.1 and 7.5.

Let $\displaystyle X$ be one of moduli spaces considered in Theorem 7.20 (i).
In Section 7.4, we defined a strongly continuous map from $\displaystyle X$ to
the space of continuous loops (with marked points). This map naturally extends
to a $\displaystyle 1$-collared strongly continuous map from $\displaystyle
X^{\boxplus 1}$, see Lemma-Definition 17.35 and Lemma 17.37 (3) in [10].
Taking $\displaystyle\tau(X)\in(1/2,1)$ as in Remark 7.23 and successively
applying Theorem 7.32 and Remark 7.33, $\displaystyle\bar{X}:=X^{\boxplus
1/2}$ is equipped with an admissible CF-perturbation and an admissible
strongly smooth map to the space of smooth loops (with marked points). In
conclusion, we obtain the following data for every $\displaystyle
k\in{\mathbb{Z}}_{\geq 0}$, $\displaystyle m\in{\mathbb{Z}}_{\geq 0}$ and
$\displaystyle P\in\\{\\{m\\},[m,m+1]\\}$.

  1. (i):

Compact admissible K-spaces

$\displaystyle\displaystyle\bar{{\mathcal{M}}}_{k+1}(\beta:P)\qquad(\omega_{n}(\beta)<\varepsilon(m+1-k)),$
$\displaystyle\displaystyle\bar{{\mathcal{N}}}^{0}_{k+1}(\beta:P)\qquad(\omega_{n}(\beta)<\varepsilon(m-1-k)),$
$\displaystyle\displaystyle\bar{{\mathcal{N}}}^{\geq
0}_{k+1}(\beta:P)\qquad(\omega_{n}(\beta)<\varepsilon(m-k-U)),$

and admissible CF-perturbations on these K-spaces.

  2. (ii):

Admissible maps

(48) $\displaystyle\displaystyle\text{\rm
ev}\,^{\bar{{\mathcal{M}}}_{k+1}}:\bar{{\mathcal{M}}}_{k+1}(\beta:P)$
$\displaystyle\displaystyle\to P\times{\mathcal{L}}_{k+1}(\partial\beta)$ (49)
$\displaystyle\displaystyle\text{\rm
ev}\,^{\bar{{\mathcal{N}}}^{0}_{k+1}}:\bar{{\mathcal{N}}}^{0}_{k+1}(\beta:P)$
$\displaystyle\displaystyle\to P\times{\mathcal{L}}_{k+1}(\partial\beta)$ (50)
$\displaystyle\displaystyle\text{\rm ev}\,^{\bar{{\mathcal{N}}}^{\geq
0}_{k+1}}:\bar{{\mathcal{N}}}^{\geq 0}_{k+1}(\beta:P)$
$\displaystyle\displaystyle\to P\times{\mathcal{L}}_{k+1}(\partial\beta)$

such that compositions with $\displaystyle\text{\rm id}_{P}\times\text{\rm
ev}\,_{0}$ (which are admissible maps to $\displaystyle P\times L$) are
strata-wise strongly submersive with respect to the CF-perturbations in (i).

  3. (iii):

Isomorphisms of admissible K-spaces, which are obtained from isomorphisms
(28)–(33) by replacing each moduli space $\displaystyle X$ with
$\displaystyle\bar{X}$. For example, we obtain

(51)
$\partial\bar{{\mathcal{M}}}_{k+1}(\beta:m)\cong\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
1\leq i\leq k_{1}\\\
\beta_{1}+\beta_{2}=\beta\end{subarray}}(-1)^{\varepsilon_{0}}\bar{{\mathcal{M}}}_{k_{1}+1}(\beta_{1}:m)\,_{i}\\!\times_{0}\bar{{\mathcal{M}}}_{k_{2}+1}(\beta_{2}:m)$

from (28). We require that these isomorphisms are compatible with CF-
perturbations in (i) and evaluation maps (ii).

Applying results in Section 7.1, one can define

(52) $\displaystyle\displaystyle x_{m}(a,k)$
$\displaystyle\displaystyle:=\sum_{\omega_{n}(\bar{a})<\varepsilon(m+1-k)}(-1)^{n+1}\text{\rm
ev}\,_{*}(\bar{{\mathcal{M}}}_{k+1}(\bar{a},\\{m\\}))$ (53)
$\displaystyle\displaystyle\bar{x}_{m}(a,k)$
$\displaystyle\displaystyle:=\sum_{\omega_{n}(\bar{a})<\varepsilon(m+1-k)}(-1)^{n+1}\text{\rm
ev}\,_{*}(\bar{{\mathcal{M}}}_{k+1}(\bar{a},[m,m+1])),$ (54)
$\displaystyle\displaystyle y_{m}(a,k)$
$\displaystyle\displaystyle:=\sum_{\omega_{n}(\bar{a})<\varepsilon(m-U-k)}(-1)^{n+k+1}\text{\rm
ev}\,_{*}(\bar{{\mathcal{N}}}_{k+1}^{\geq 0}(\bar{a},\\{m\\})),$ (55)
$\displaystyle\displaystyle\bar{y}_{m}(a,k)$
$\displaystyle\displaystyle:=\sum_{\omega_{n}(\bar{a})<\varepsilon(m-U-k)}(-1)^{n+k+1}\text{\rm
ev}\,_{*}(\bar{{\mathcal{N}}}_{k+1}^{\geq 0}(\bar{a},[m,m+1])),$ (56)
$\displaystyle\displaystyle z_{m}(a,k)$
$\displaystyle\displaystyle:=\sum_{\omega_{n}(\bar{a})<\varepsilon(m-1-k)}(-1)^{n+k+1}\text{\rm
ev}\,_{*}(\bar{{\mathcal{N}}}_{k+1}^{0}(\bar{a},\\{m\\})),$ (57)
$\displaystyle\displaystyle\bar{z}_{m}(a,k)$
$\displaystyle\displaystyle:=\sum_{\omega_{n}(\bar{a})<\varepsilon(m-1-k)}(-1)^{n+k+1}\text{\rm
ev}\,_{*}(\bar{{\mathcal{N}}}_{k+1}^{0}(\bar{a},[m,m+1])).$

######  Remark 7.34.

In the above formulas we abbreviate differential form $\displaystyle 1$ (see
Remark 7.3) and CF-perturbations. We also abbreviate superscripts of ev for
simplicity.

######  Remark 7.35.

Here we explain one issue which can be overlooked by abbreviating CF-
perturbations in the above formulas. As is clear from Section 7.1, to define a
de Rham chain one has to fix a parameter of CF-perturbation $\displaystyle
e\in(0,1]$ (here we use a letter $\displaystyle e$, not to be confused with
$\displaystyle\varepsilon$ used in the above formulas). Hence we take a
strictly decreasing sequence of positive real numbers
$\displaystyle(e_{m})_{m\geq 1}$ satisfying
$\displaystyle\lim_{m\to\infty}e_{m}=0$, and define $\displaystyle x_{m}(a,k)$
by

$\displaystyle
x_{m}(a,k):=\sum_{\omega_{n}(\bar{a})<\varepsilon(m+1-k)}(-1)^{n+1}\text{\rm
ev}\,_{*}(\bar{{\mathcal{M}}}_{k+1}(\bar{a},\\{m\\}),\widehat{{\mathfrak{S}}}^{e_{m}}).$

$\displaystyle y_{m}(a,k)$ and $\displaystyle z_{m}(a,k)$ are defined by
similar formulas, using $\displaystyle e_{m}$. Then
$\displaystyle\bar{x}_{m}(a,k)$ is defined by interpolating $\displaystyle
e_{m}$ and $\displaystyle e_{m+1}$ on the moduli space
$\displaystyle\bar{{\mathcal{M}}}_{k+1}(\bar{a},[m,m+1])$.
$\displaystyle\bar{y}_{m}(a,k)$ and $\displaystyle\bar{z}_{m}(a,k)$ are
defined in a similar way.

Let us check that the requirements in Theorem 6.1 are satisfied.
$\displaystyle x_{m}=e_{-}(\bar{x}_{m})$, $\displaystyle
y_{m}=e_{-}(\bar{y}_{m})$, $\displaystyle z_{m}=e_{-}(\bar{z}_{m})$ follow
from the above definition and (21). Moreover

$\displaystyle(x_{m+1}-e_{+}(\bar{x}_{m}))(a,k)\neq
0\implies\omega_{n}(\bar{a})\geq\varepsilon(m+1-k)$

show $\displaystyle x_{m+1}-e_{+}(\bar{x}_{m})\in F^{m}$. By similar arguments
one can show $\displaystyle y_{m+1}-e_{+}(\bar{y}_{m})\in F^{m-U-1}$ and
$\displaystyle z_{m+1}-e_{+}(\bar{z}_{m})\in F^{m-2}$.

The isomorphism (51) shows

$\displaystyle\partial\bar{x}_{m}(a,k)=\sum_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\
a_{1}+a_{2}=a\\\ 1\leq i\leq
k_{1}\end{subarray}}(-1)^{(k_{1}-m)(k_{2}-1)+(k_{1}-1)}\bar{x}_{m}(a_{1},k_{1})\circ_{i}\bar{x}_{m}(a_{2},k_{2})$

for every $\displaystyle(a,k)$ such that
$\displaystyle\omega_{n}(\bar{a})<\varepsilon(m+1-k)$, thus
$\displaystyle\partial\bar{x}_{m}-\frac{1}{2}[\bar{x}_{m},\bar{x}_{m}]\in
F^{m}$. By similar arguments one can show
$\displaystyle\partial\bar{y}_{m}-[\bar{x}_{m},\bar{y}_{m}]-\bar{z}_{m}\in
F^{m-U-1}$ and $\displaystyle\partial\bar{z}_{m}-[\bar{x}_{m},\bar{z}_{m}]\in
F^{m-2}$.

Finally $\displaystyle x_{m}(\bar{a},k)\neq
0\implies{\mathcal{M}}_{k+1}(\bar{a})\neq\emptyset$, thus
$\displaystyle\omega_{n}(\bar{a})\geq 2\varepsilon$ or $\displaystyle a=0$,
$\displaystyle k\geq 2$. Moreover
$\displaystyle[x_{m}(0,2)]=(-1)^{n+1}[{\mathcal{M}}_{3}(0)]=(-1)^{n+1}[L]$.
Similarly, $\displaystyle z_{m}(\bar{a},k)\neq
0\implies{\mathcal{N}}^{0}_{k+1}(\bar{a})\neq\emptyset$, thus
$\displaystyle\omega_{n}(\bar{a})\geq 2\varepsilon$ or $\displaystyle a=0$.
Moreover
$\displaystyle[z_{m}(0,0)]=(-1)^{n+1}[{\mathcal{N}}^{0}_{1}(0)]=(-1)^{n+1}[L]$.
∎

##  8\. Strongly smooth map from a K-space with a CF-perturbation gives a de
Rham chain

The goal of this section is to explain proofs of results presented in Section
7.1. Namely, given a strongly smooth map from a K-space (equipped with a
differential form and a CF-perturbation) to $\displaystyle{\mathcal{L}}_{k+1}$
(the space of loops with $\displaystyle k+1$ marked points), we define a de
Rham chain on $\displaystyle{\mathcal{L}}_{k+1}$ and prove Stokes’ formula and
fiber product formula. Here we imitate arguments in [10] Sections 7 and 9 in
our setting. In Section 8.1 we consider smooth maps from single K-charts. In
Section 8.2 we consider strongly smooth maps from spaces with good coordinate
system (GCS), and in Section 8.3 strongly smooth maps from K-spaces. In
Sections 8.2 and 8.3, we only consider spaces without boundaries (and
corners), since generalizations to spaces with boundaries are straightforward.

Throughout this section, $\displaystyle X$ denotes a separable, metrizable
topolgoical space.

###  8.1. Single K-chart

In this subsection, given a smooth map from a K-chart (equipped with a CF-
perturbation and a differential form) to $\displaystyle{\mathcal{L}}_{k+1}$,
we define a de Rham chain on $\displaystyle{\mathcal{L}}_{k+1}$. We also prove
Stokes’s formula and fiber product formula. We first consider K-charts without
boundary, and then proceed to the case of K-charts with boundaries.

####  8.1.1. K-chart without boundary

Suppose we are given the following data:

  * •

$\displaystyle{\mathcal{U}}=(U,{\mathcal{E}},s,\psi)$ is a K-chart of
$\displaystyle X$.

  * •

$\displaystyle f:U\to{\mathcal{L}}_{k+1}$ is a smooth map in the sense of
Definition 4.1.

  * •

$\displaystyle\omega\in{\mathcal{A}}^{*}_{c}(U)$.

  * •

$\displaystyle{\mathfrak{S}}=({\mathfrak{S}}^{\varepsilon})_{0<\varepsilon\leq
1}$ is a CF-perturbation of $\displaystyle\text{\rm supp}\,\omega$ such that
$\displaystyle\text{\rm ev}\,_{0}\circ f:U\to L$ is strongly submersive with
respect to $\displaystyle{\mathfrak{S}}$ (see Definition 7.24 and Definition-
Lemma 7.25 in [10]).

We are going to define $\displaystyle
f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon})\in C^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1})$ for every $\displaystyle\varepsilon\in(0,1]$.
Let
$\displaystyle({\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}_{{\mathfrak{r}}})_{{\mathfrak{r}}\in{\mathfrak{R}}}$
be a representative of $\displaystyle{\mathfrak{S}}$ (see Definitions 7.15,
7.16 and 7.19 in [10]), such that

  * •

$\displaystyle{\mathfrak{V}}_{{\mathfrak{r}}}=(V_{{\mathfrak{r}}},E_{{\mathfrak{r}}},\phi_{{\mathfrak{r}}},\widehat{\phi}_{{\mathfrak{r}}})$
is a manifold chart (we do not consider group actions: see Remark 10.1) of
$\displaystyle(U,{\mathcal{E}})$ such that
$\displaystyle(\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}}))_{{\mathfrak{r}}\in{\mathfrak{R}}}$
covers $\displaystyle U$. Let $\displaystyle
s_{{\mathfrak{r}}}:V_{{\mathfrak{r}}}\to E_{{\mathfrak{r}}}$ denote the pull
back of $\displaystyle s$ by $\displaystyle\phi_{{\mathfrak{r}}}$.

  * •

$\displaystyle{\mathcal{S}}_{{\mathfrak{r}}}=(W_{{\mathfrak{r}}},\eta_{{\mathfrak{r}}},\\{{\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}}\\}_{\varepsilon})$
is a CF-perturbation of $\displaystyle{\mathcal{U}}$ on
$\displaystyle{\mathfrak{V}}_{{\mathfrak{r}}}$. Namely

    * –

$\displaystyle W_{{\mathfrak{r}}}$ is an open neighborhood of $\displaystyle
0$ of a finite-dimensional oriented vector space
$\displaystyle\widehat{W_{{\mathfrak{r}}}}$.

    * –

$\displaystyle{\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}}:V_{{\mathfrak{r}}}\times
W_{{\mathfrak{r}}}\to E_{{\mathfrak{r}}}$ is a $\displaystyle C^{\infty}$ map
transversal to $\displaystyle 0$ for every $\displaystyle\varepsilon\in(0,1]$.

    * –

$\displaystyle\lim_{\varepsilon\to
0}{\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}}(y,\xi)=s_{{\mathfrak{r}}}(y)$
in compact $\displaystyle C^{1}$ topology on $\displaystyle
V_{{\mathfrak{r}}}\times W_{{\mathfrak{r}}}$.

    * –

$\displaystyle\eta_{{\mathfrak{r}}}\in{\mathcal{A}}^{\text{\rm
dim}\,W_{{\mathfrak{r}}}}_{c}(W_{{\mathfrak{r}}})$ such that
$\displaystyle\int_{W_{{\mathfrak{r}}}}\eta_{{\mathfrak{r}}}=1$.

  * •

$\displaystyle\text{\rm ev}\,_{0}\circ
f\circ\phi_{{\mathfrak{r}}}\circ\text{\rm
pr}_{V_{\mathfrak{r}}}:({\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0)\to
L$ is a submersion for every $\displaystyle{\mathfrak{r}}\in{\mathfrak{R}}$
and $\displaystyle\varepsilon\in(0,1]$.

$\displaystyle V_{{\mathfrak{r}}}$ and $\displaystyle E_{{\mathfrak{r}}}$ are
oriented so that $\displaystyle\phi_{{\mathfrak{r}}}$ and
$\displaystyle\widehat{\phi}_{{\mathfrak{r}}}$ preserve orientations.
$\displaystyle({\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0)$ is
oriented so that the isomorphism

$\displaystyle E_{{\mathfrak{r}}}\oplus
T({\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0)\cong
TV_{{\mathfrak{r}}}\oplus TW_{{\mathfrak{r}}}$

preserves orientations, following Convention 8.2.1 in [8].

We take a partition of unity
$\displaystyle\\{\chi_{{\mathfrak{r}}}\\}_{{\mathfrak{r}}\in{\mathfrak{R}}}$
subordinate to
$\displaystyle(\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}}))_{{\mathfrak{r}}\in{\mathfrak{R}}}$,
i.e., $\displaystyle\chi_{\mathfrak{r}}\in C^{\infty}_{c}(U,[0,1])$ and
$\displaystyle\text{\rm
supp}\,\chi_{\mathfrak{r}}\subset\phi_{\tau}(V_{\mathfrak{r}})$ for every
$\displaystyle{\mathfrak{r}}\in{\mathfrak{R}}$, and
$\displaystyle\sum_{{\mathfrak{r}}\in{\mathfrak{R}}}\chi_{{\mathfrak{r}}}\equiv
1$ on $\displaystyle\text{\rm supp}\,\omega$. Then, for each
$\displaystyle\varepsilon\in(0,1]$, we define

(58)
$f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon}):=\sum_{{\mathfrak{r}}\in{\mathfrak{R}}}f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}}),$

where the RHS is defined as

(59)
$f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}}):=(-1)^{\dagger}(({\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0),f\circ\phi_{{\mathfrak{r}}}\circ\text{\rm
pr}_{V_{{\mathfrak{r}}}},(\phi_{{\mathfrak{r}}}\circ\text{\rm
pr}_{V_{{\mathfrak{r}}}})^{*}(\chi_{{\mathfrak{r}}}\omega)\wedge(\text{\rm
pr}_{W_{{\mathfrak{r}}}})^{*}(\eta_{{\mathfrak{r}}})),$ (60)
$\dagger:=\text{\rm dim}\,W_{{\mathfrak{r}}}\cdot(\text{\rm
rk}\,{\mathcal{E}}+|\omega|).$

$\displaystyle({\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0)$ is an
oriented $\displaystyle C^{\infty}$-manifold, and $\displaystyle\text{\rm
ev}\,_{0}\circ f\circ\phi_{{\mathfrak{r}}}\circ\text{\rm
pr}_{V_{{\mathfrak{r}}}}$ is a submersion, thus the RHS in (59) makes sense.

######  Remark 8.1.

Strictly speaking, one has to fix an embedding of
$\displaystyle({\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0)$ to
Euclidean space to define a de Rham chain. However it is easy to check that
the de Rham chain does not depend on choice of embedding.

A priori, the above definition may depend on choices of representative and
partition of unity. In Lemma 8.2 below we prove well-definedness and Stokes’
formula.

######  Lemma 8.2.

  1. (i):

For any $\displaystyle\omega_{1},\omega_{2}\in{\mathcal{A}}^{*}_{c}(U)$,
$\displaystyle a_{1},a_{2}\in{\mathbb{R}}$ and
$\displaystyle\varepsilon\in(0,1]$,

$\displaystyle
f_{*}({\mathcal{U}},a_{1}\omega_{1}+a_{2}\omega_{2},{\mathfrak{S}}^{\varepsilon})=a_{1}f_{*}({\mathcal{U}},\omega_{1},{\mathfrak{S}}^{\varepsilon})+a_{2}f_{*}({\mathcal{U}},\omega_{2},{\mathfrak{S}}^{\varepsilon}).$

  2. (ii):

The above definition does not depend on choices of representative of the CF-
perturbation $\displaystyle{\mathfrak{S}}$ and partition of unity.

  3. (iii):

$\displaystyle\partial
f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon})=(-1)^{|\omega|+1}f_{*}({\mathcal{U}},d\omega,{\mathfrak{S}}^{\varepsilon})$
for every $\displaystyle\varepsilon\in(0,1]$.

###### Proof.

(i) is obvious as far as we fix a representative of
$\displaystyle{\mathfrak{S}}$ and a partition of unity. Thus to prove (ii), it
is sufficient to prove the following claim:

> Let
> $\displaystyle{\mathcal{S}}_{i}=(W_{i},\eta_{i},\\{{\mathfrak{s}}^{\varepsilon}_{i}\\}_{\varepsilon})\,(i=1,2)$
> be two equivalent CF-perturbations of $\displaystyle{\mathcal{U}}$ on a
> manifold chart $\displaystyle(V,E,\phi,\widehat{\phi})$ and
> $\displaystyle\omega\in{\mathcal{A}}^{*}_{c}(U)$ such that
> $\displaystyle\text{\rm supp}\,\omega\subset\phi(V)$. Then
>
>
> $\displaystyle\displaystyle(({\mathfrak{s}}^{\varepsilon}_{1})^{-1}(0),f\circ\phi\circ\text{\rm
> pr}_{V},(\phi\circ\text{\rm pr}_{V})^{*}\omega\wedge(\text{\rm
> pr}_{W_{1}})^{*}\eta_{1})$
> $\displaystyle\displaystyle=(-1)^{\dagger}(({\mathfrak{s}}^{\varepsilon}_{2})^{-1}(0),f\circ\phi\circ\text{\rm
> pr}_{V},(\phi\circ\text{\rm pr}_{V})^{*}\omega\wedge(\text{\rm
> pr}_{W_{2}})^{*}\eta_{2}),$ $\displaystyle\displaystyle\dagger:=(\text{\rm
> dim}\,W_{1}-\text{\rm dim}\,W_{2})(\text{\rm rk}\,{\mathcal{E}}+|\omega|)$

By definition of equivalence (see Definition 7.5 in [10]), we may assume that
there exists a linear projection
$\displaystyle\Pi:\widehat{W}_{2}\to\widehat{W}_{1}$ such that
$\displaystyle(\Pi)_{!}(\eta_{2})=\eta_{1}$ and
$\displaystyle{\mathfrak{s}}^{\varepsilon}_{2}=(\text{\rm
id}_{V}\times\Pi)^{*}({\mathfrak{s}}^{\varepsilon}_{1})$. Then, pushout of
$\displaystyle\text{\rm
id}_{V}\times\Pi:({\mathfrak{s}}^{\varepsilon}_{2})^{-1}(0)\to({\mathfrak{s}}^{\varepsilon}_{1})^{-1}(0)$
sends $\displaystyle(\phi\circ\text{\rm pr}_{V})^{*}\omega\wedge(\text{\rm
pr}_{W_{2}})^{*}\eta_{2}$ to $\displaystyle(-1)^{\dagger}(\phi\circ\text{\rm
pr}_{V})^{*}\omega\wedge(\text{\rm pr}_{W_{1}})^{*}\eta_{1}$, which completes
the proof.

To prove (iii), by (i) and (ii) we may assume that $\displaystyle\text{\rm
supp}\,\omega$ is sufficiently small and
$\displaystyle\chi_{{\mathfrak{r}}}\equiv 1$ on $\displaystyle\text{\rm
supp}\,\omega$ for some $\displaystyle{\mathfrak{r}}\in{\mathfrak{R}}$. In
this case (iii) is obvious except for signs, which can be checked by simple
computations. ∎

Now we can state the fiber product formula. Suppose, for each $\displaystyle
i\in\\{1,2\\}$, we have $\displaystyle X_{i}$,
$\displaystyle{\mathcal{U}}_{i}=(U_{i},{\mathcal{E}}_{i},s_{i},\psi_{i})$,
$\displaystyle f_{i}:U_{i}\to{\mathcal{L}}_{k_{i}+1}$,
$\displaystyle\omega_{i}$, $\displaystyle{\mathfrak{S}}_{i}$ as before. Then,
for every $\displaystyle\varepsilon\in(0,1]$ one can define

$\displaystyle(f_{i})_{*}({\mathcal{U}}_{i},\omega_{i},{\mathfrak{S}}_{i}^{\varepsilon})\in
C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k_{i}+1})$

for each $\displaystyle i\in\\{1,2\\}$. On the other hand, for each
$\displaystyle j\in\\{1,\ldots,k_{1}\\}$, one can take a fiber product of
K-charts

$\displaystyle{\mathcal{U}}_{12}={\mathcal{U}}_{1}\,{}_{\text{\rm
ev}\,_{j}\circ f_{1}}\\!\times_{\text{\rm ev}\,_{0}\circ
f_{2}}{\mathcal{U}}_{2}.$

One can also define a fiber product of CF-perturbations
$\displaystyle{\mathfrak{S}}_{1}\times{\mathfrak{S}}_{2}$ on
$\displaystyle{\mathcal{U}}_{12}$. Finally we define a differential form
$\displaystyle\omega_{12}$ on $\displaystyle{\mathcal{U}}_{12}$ by

$\displaystyle\omega_{12}:=(-1)^{(\text{\rm dim}\,U_{1}-\text{\rm
rk}\,{\mathcal{E}}_{1}-|\omega_{1}|-n)|\omega_{2}|}\cdot\omega_{1}\times\omega_{2},$

and a smooth map

$\displaystyle f_{12}:U_{1}\,{}_{\text{\rm ev}\,_{j}\circ
f_{1}}\\!\times_{\text{\rm ev}\,_{0}\circ
f_{2}}U_{2}\to{\mathcal{L}}_{k_{1}+k_{2}};\quad(x_{1},x_{2})\mapsto\text{\rm
con}\,_{j}(f_{1}(x_{1}),f_{2}(x_{2})).$

Then one can state the fiber product formula as follows. The proof is obvious
except for signs, which can be checked by direct computations.

######  Lemma 8.3.

In the situation described above, there holds

$\displaystyle(f_{12})_{*}({\mathcal{U}}_{12},\omega_{12},{\mathfrak{S}}^{\varepsilon}_{12})=(f_{1})_{*}({\mathcal{U}}_{1},\omega_{1},{\mathfrak{S}}^{\varepsilon}_{1})\circ_{j}(f_{2})_{*}({\mathcal{U}}_{2},\omega_{2},{\mathfrak{S}}^{\varepsilon}_{2})$

for every $\displaystyle\varepsilon\in(0,1]$.

####  8.1.2. K-chart with boundary

Suppose we are given the following data:

  * •

$\displaystyle{\mathcal{U}}=(U,{\mathcal{E}},s,\psi)$ is an admissible K-chart
on $\displaystyle X$, i.e. $\displaystyle U$ is an admissible manifold with
boundaries (and corners), the vector bundle $\displaystyle{\mathcal{E}}$ and
the section $\displaystyle s$ are also admissible.

  * •

$\displaystyle f:U\to{\mathcal{L}}_{k+1}$ is an admissible map in the sense of
Definition 7.7 (i).

  * •

$\displaystyle\omega\in{\mathcal{A}}^{*}_{c}(U)$ is admissible.

  * •

$\displaystyle{\mathfrak{S}}=({\mathfrak{S}}^{\varepsilon})_{0<\varepsilon\leq
1}$ is an admissible CF-perturbation of $\displaystyle\text{\rm supp}\,\omega$
such that $\displaystyle\text{\rm ev}\,_{0}\circ f:U\to L$ is strata-wise
strongly submersive with respect to $\displaystyle{\mathfrak{S}}$.

Under these assumptions, our goal is to define $\displaystyle
f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon})\in C^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1})$ for every $\displaystyle\varepsilon\in(0,1]$.

Let
$\displaystyle({\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}_{{\mathfrak{r}}})_{{\mathfrak{r}}\in{\mathfrak{R}}}$
be a representative of $\displaystyle{\mathfrak{S}}$, and
$\displaystyle(\chi_{{\mathfrak{r}}})_{{\mathfrak{r}}\in{\mathfrak{R}}}$ be a
partition of unity subordinate to
$\displaystyle(\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}}))_{{\mathfrak{r}}\in{\mathfrak{R}}}$
such that $\displaystyle\sum_{{\mathfrak{r}}}\chi_{{\mathfrak{r}}}\equiv 1$ on
$\displaystyle\text{\rm supp}\,\omega$. Then we define

(61)
$f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon}):=\sum_{{\mathfrak{r}}\in{\mathfrak{R}}}f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}})$

where each term in the RHS is defined below. The proof of well-definedness
(the RHS does not depend on choices of representatives of
$\displaystyle{\mathfrak{S}}$ and partition of unity) is straightforward and
omitted.

Let $\displaystyle D:=\text{\rm dim}\,U$. We may assume that $\displaystyle
V_{{\mathfrak{r}}}$ is an open neighborhood of
$\displaystyle(t_{1},\ldots,t_{D})\in({\mathbb{R}}_{\geq 0})^{D}$. We define

$\displaystyle{\mathcal{R}}:{\mathbb{R}}^{D}\to({\mathbb{R}}_{\geq
0})^{D};\,(t_{1},\ldots,t_{D})\mapsto(t^{\prime}_{1},\ldots,t^{\prime}_{D})$

by $\displaystyle t^{\prime}_{i}:=\begin{cases}t_{i}&(t_{i}\geq 0)\\\
0&(t_{i}<0)\\\ \end{cases}$ for every $\displaystyle 1\leq i\leq D$. We take
$\displaystyle\kappa\in C^{\infty}({\mathbb{R}},[0,1])$ such that
$\displaystyle\kappa\equiv 1$ on a neighborhood of
$\displaystyle{\mathbb{R}}_{\geq 0}$, and $\displaystyle\kappa\equiv 0$ on a
neighborhood of $\displaystyle{\mathbb{R}}_{\leq-1}$. Now let us define the
following data:

  * •

$\displaystyle\bar{V}_{{\mathfrak{r}}}:={\mathcal{R}}^{-1}(V_{{\mathfrak{r}}})$,
$\displaystyle\bar{E}_{{\mathfrak{r}}}:={\mathcal{R}}^{*}E_{{\mathfrak{r}}}$.

  * •

$\displaystyle\bar{{\mathfrak{s}}}^{\varepsilon}_{{\mathfrak{r}}}:=({\mathcal{R}}|_{\bar{V}_{{\mathfrak{r}}}}\times\text{\rm
id}_{W_{{\mathfrak{r}}}})^{*}({\mathfrak{s}}^{\varepsilon}_{{\mathfrak{r}}})$.

  * •

$\displaystyle\bar{f}_{{\mathfrak{r}}}:=f\circ\phi_{{\mathfrak{r}}}\circ{\mathcal{R}}|_{\bar{V}_{{\mathfrak{r}}}}$.

  * •

$\displaystyle\overline{\chi_{{\mathfrak{r}}}\omega}(t_{1},\ldots,t_{D}):=\kappa(t_{1})\cdots\kappa(t_{D})\cdot(\phi_{{\mathfrak{r}}}\circ{\mathcal{R}}|_{\bar{V}_{{\mathfrak{r}}}})^{*}(\chi_{{\mathfrak{r}}}\omega)$.

Finally, we define

(62)
$f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}}):=(-1)^{\dagger}((\bar{{\mathfrak{s}}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0),\bar{f}_{{\mathfrak{r}}}\circ\text{\rm
pr}_{\bar{V}_{{\mathfrak{r}}}},\text{\rm
pr}^{*}_{\bar{V}_{{\mathfrak{r}}}}(\overline{\chi_{{\mathfrak{r}}}\omega})\wedge\text{\rm
pr}^{*}_{W_{{\mathfrak{r}}}}(\eta_{{\mathfrak{r}}})),$

where the sign $\displaystyle\dagger$ is defined by (60). Note that
$\displaystyle\text{\rm ev}\,_{0}\circ\bar{f}_{{\mathfrak{r}}}\circ\text{\rm
pr}_{\bar{V}_{{\mathfrak{r}}}}:(\bar{{\mathfrak{s}}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0)\to
L$ is a submersion, thus the map
$\displaystyle\bar{f}_{{\mathfrak{r}}}\circ\text{\rm
pr}_{\bar{V}_{{\mathfrak{r}}}}:(\bar{{\mathfrak{s}}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0)\to
L$ is smooth.

######  Remark 8.4.

The RHS of (62) may depend on the choice of the cutoff function
$\displaystyle\kappa$. Here we fix such $\displaystyle\kappa$ and drop it from
our notation in the following arguments.

The fiber product formula holds in the obvious manner and omitted. Stokes’
formula is formulated as follows.

######  Proposition 8.5.

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

###### Proof.

This follows from

$\displaystyle\displaystyle
d(\overline{\chi_{{\mathfrak{r}}}\omega})(t_{1},\ldots,t_{D})$
$\displaystyle\displaystyle=\kappa(t_{1})\cdots\kappa(t_{D})(\phi_{{\mathfrak{r}}}\circ{\mathcal{R}}|_{\bar{V}_{{\mathfrak{r}}}})^{*}(d(\chi_{{\mathfrak{r}}}\omega))$
$\displaystyle\displaystyle+\sum_{i=1}^{D}\kappa(t_{1})\cdots
d\kappa(t_{i})\cdots\kappa(t_{D})(\phi_{{\mathfrak{r}}}\circ{\mathcal{R}}|_{\bar{V}_{{\mathfrak{r}}}})^{*}(\chi_{{\mathfrak{r}}}\omega)$

and $\displaystyle\partial U$ is oriented so that $\displaystyle
TU\cong{\mathbb{R}}_{\text{out}}\oplus T(\partial U)$ preserves orientations.
∎

####  8.1.3. K-chart with boundary over an interval

Suppose we are given the following data:

  * •

$\displaystyle{\mathcal{U}}=(U,{\mathcal{E}},s,\psi)$ is an admissible K-chart
on $\displaystyle X$.

  * •

$\displaystyle f:U\to[a,b]\times{\mathcal{L}}_{k+1}$ is an admissible map in
the sense of Definition 7.13 (i).

  * •

$\displaystyle\omega\in{\mathcal{A}}^{*}_{c}(U)$ is admissible.

  * •

$\displaystyle{\mathfrak{S}}=({\mathfrak{S}}^{\varepsilon})_{0<\varepsilon\leq
1}$ is an admissible CF-perturbation of $\displaystyle\text{\rm supp}\,\omega$
such that $\displaystyle\text{\rm ev}\,_{0}\circ f:U\to[a,b]\times L$ is
strata-wise strongly submersive with respect to $\displaystyle{\mathfrak{S}}$.

Under these assumptions, our goal is to define $\displaystyle
f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon})\in\bar{C}^{\text{\rm
dR}}_{*}({\mathcal{L}}_{k+1})$ for every $\displaystyle\varepsilon\in(0,1]$.
For simplicity of notations, in the following we assume that $\displaystyle
a=-1$, $\displaystyle b=1$.

Let
$\displaystyle({\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}_{{\mathfrak{r}}})_{{\mathfrak{r}}\in{\mathfrak{R}}}$
be a representative of $\displaystyle{\mathfrak{S}}$, and
$\displaystyle(\chi_{{\mathfrak{r}}})_{{\mathfrak{r}}\in{\mathfrak{R}}}$ be a
partition of unity subordinate to
$\displaystyle(\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}}))_{{\mathfrak{r}}\in{\mathfrak{R}}}$
such that $\displaystyle\sum_{{\mathfrak{r}}}\chi_{{\mathfrak{r}}}\equiv 1$ on
$\displaystyle\text{\rm supp}\,\omega$. Then we define

(63)
$f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon}):=\sum_{{\mathfrak{r}}\in{\mathfrak{R}}}f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}})$

where each term in the RHS is defined below. The proof of well-definedness is
omitted.

To define $\displaystyle
f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}})$,
it is sufficient to consider the following three cases:

  1. (i):

$\displaystyle f\circ\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}})$ is contained
in $\displaystyle(-1,1)\times{\mathcal{L}}_{k+1}$.

  2. (ii):

$\displaystyle f\circ\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}})$ is contained
in $\displaystyle[-1,1)\times{\mathcal{L}}_{k+1}$ and intersects
$\displaystyle\\{-1\\}\times{\mathcal{L}}_{k+1}$.

  3. (iii):

$\displaystyle f\circ\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}})$ is contained
in $\displaystyle(-1,1]\times{\mathcal{L}}_{k+1}$ and intersects
$\displaystyle\\{1\\}\times{\mathcal{L}}_{k+1}$.

The case (i) is similar to the case in the previous subsubsection and omitted.
In the following we only consider the case (ii), since the case (iii) is
completely parallel.

Let $\displaystyle D:=\text{\rm dim}\,U$. We may assume that $\displaystyle
V_{{\mathfrak{r}}}$ is an open neighborhood of
$\displaystyle(0,t_{2},\ldots,t_{D})$ in $\displaystyle({\mathbb{R}}_{\geq
0})^{D}$ and $\displaystyle
f_{\mathbb{R}}\circ\varphi_{{\mathfrak{r}}}(t_{1},\ldots,t_{D})=t_{1}-1$ .
Here $\displaystyle f_{\mathbb{R}}$ denotes $\displaystyle\text{\rm
pr}_{{\mathbb{R}}}\circ f$. Similarly, we set $\displaystyle
f_{{\mathcal{L}}}:=\text{\rm pr}_{{\mathcal{L}}_{k+1}}\circ f$.

We define $\displaystyle\bar{V}_{{\mathfrak{r}}}$,
$\displaystyle\bar{E}_{{\mathfrak{r}}}$ and
$\displaystyle\bar{{\mathfrak{s}}}^{\varepsilon}_{{\mathfrak{r}}}$ in the same
way as in the previous subsubsection. We also define
$\displaystyle\bar{f}_{{\mathfrak{r}}}$ and
$\displaystyle\overline{\chi_{{\mathfrak{r}}}\omega}$ as follows:

  * •

$\displaystyle\bar{f}_{{\mathfrak{r}}}:\bar{V}_{{\mathfrak{r}}}\to{\mathbb{R}}\times{\mathcal{L}}_{k+1}$
is defined by

$\displaystyle\text{\rm
pr}_{{\mathcal{L}}_{k+1}}\circ\bar{f}_{{\mathfrak{r}}}:=f_{{\mathcal{L}}}\circ\phi_{{\mathfrak{r}}}\circ{\mathcal{R}}|_{\bar{V}_{{\mathfrak{r}}}},\qquad\text{\rm
pr}_{{\mathbb{R}}}\circ\bar{f}_{{\mathfrak{r}}}(t_{1},\ldots,t_{D}):=t_{1}-1.$

  * •

$\displaystyle\overline{\chi_{{\mathfrak{r}}}\omega}(t_{1},\ldots,t_{D}):=\kappa(t_{2})\cdots\kappa(t_{D})\cdot(\phi_{{\mathfrak{r}}}\circ{\mathcal{R}}|_{\bar{V}_{{\mathfrak{r}}}})^{*}(\chi_{{\mathfrak{r}}}\omega)$.

Here $\displaystyle\kappa\in C^{\infty}({\mathbb{R}},[0,1])$ is taken in the
previous subsubsection. Then we define

$\displaystyle
f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}}):=(-1)^{\dagger}((\bar{{\mathfrak{s}}}^{\varepsilon}_{{\mathfrak{r}}})^{-1}(0),\bar{f}_{{\mathfrak{r}}}\circ\text{\rm
pr}_{\bar{V}_{{\mathfrak{r}}}},\tau_{+},\tau_{-},\text{\rm
pr}^{*}_{\bar{V}_{{\mathfrak{r}}}}(\overline{\chi_{{\mathfrak{r}}}\omega})\wedge\text{\rm
pr}^{*}_{W_{{\mathfrak{r}}}}(\eta_{{\mathfrak{r}}})),$

where the sign $\displaystyle\dagger$ is defined as before and
$\displaystyle\tau_{-}$, $\displaystyle\tau_{+}$ are defined as follows.
$\displaystyle\tau_{+}$ is defined in the obvious way, since $\displaystyle
f\circ\phi_{{\mathfrak{r}}}(V_{{\mathfrak{r}}})$ does not intersect
$\displaystyle\\{1\\}\times{\mathcal{L}}_{k+1}$. On the other hand,
$\displaystyle\tau_{-}$ is defined as a restriction of

$\displaystyle(\bar{V}_{{\mathfrak{r}}}\cap\\{t_{1}\leq 0\\})\times
W_{{\mathfrak{r}}}\to{\mathbb{R}}_{\leq-1}\times((V_{{\mathfrak{r}}}\cap\\{t_{1}=0\\})\times
W_{{\mathfrak{r}}});\,(t_{1},\ldots,t_{D},w)\mapsto(t_{1}-1,t_{2},\ldots,t_{D},w).$

This completes the definition of $\displaystyle
f_{*}({\mathcal{U}},\chi_{{\mathfrak{r}}}\omega,{\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}^{\varepsilon}_{{\mathfrak{r}}})$,
thus the definition of $\displaystyle
f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon})$.

The fiber product formula is stated and proved in the obvious way. Stokes’
formula

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

where $\displaystyle\partial_{h}{\mathcal{U}}$ is the restriction of
$\displaystyle{\mathcal{U}}$ to $\displaystyle\partial_{h}U$, and

$\displaystyle\displaystyle
e_{+}(f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon}))=(f_{{\mathcal{L}}}|_{U_{1}})_{*}({\mathcal{U}}|_{U_{1}},\omega|_{U_{1}},{\mathfrak{S}}^{\varepsilon}|_{U_{1}}),$
$\displaystyle\displaystyle
e_{-}(f_{*}({\mathcal{U}},\omega,{\mathfrak{S}}^{\varepsilon}))=(f_{{\mathcal{L}}}|_{U_{-1}})_{*}({\mathcal{U}}|_{U_{-1}},\omega|_{U_{-1}},{\mathfrak{S}}^{\varepsilon}|_{U_{-1}})$

can be checked directly and proofs are omitted.

###  8.2. Space with GCS

In this subsection, we assume that $\displaystyle X$ is compact. Let
$\displaystyle\widetriangle{{\mathcal{U}}}=(\\{{\mathcal{U}}_{{\mathfrak{p}}}\\}_{{\mathfrak{p}}\in{\mathfrak{P}}},\\{\Phi_{{\mathfrak{p}}{\mathfrak{q}}}\\}_{{\mathfrak{p}}\geq{\mathfrak{q}}})$
be a GCS (good coordinate system) without boundary on $\displaystyle X$ (see
Section 10). We start from the following definition.

######  Definition 8.6.

A strongly smooth map $\displaystyle\widetriangle{f}$ from
$\displaystyle(X,\widetriangle{{\mathcal{U}}})$ to
$\displaystyle{\mathcal{L}}_{k+1}$ is a family
$\displaystyle(f_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}}$ which
satisfies the following conditions:

  * •

For every $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$, $\displaystyle
f_{{\mathfrak{p}}}$ is a smooth map from $\displaystyle U_{{\mathfrak{p}}}$ to
$\displaystyle{\mathcal{L}}_{k+1}$.

  * •

For every $\displaystyle{\mathfrak{q}}\leq{\mathfrak{p}}$, there holds
$\displaystyle
f_{\mathfrak{p}}\circ\varphi_{{\mathfrak{p}}{\mathfrak{q}}}=f_{{\mathfrak{q}}}|_{U_{{\mathfrak{p}}{\mathfrak{q}}}}$.

Let $\displaystyle{\mathcal{K}}$ be a support system of
$\displaystyle\widetriangle{{\mathcal{U}}}$ and
$\displaystyle\widetriangle{{\mathfrak{S}}}$ be a CF-perturbation of
$\displaystyle(\widetriangle{{\mathcal{U}}},{\mathcal{K}})$ (see Definitions
5.6 and 7.47 in [10]). Here we recall the definition of support system (in the
case $\displaystyle Z=X$):

######  Definition 8.7.

A support system of $\displaystyle\widetriangle{{\mathcal{U}}}$ is
$\displaystyle{\mathcal{K}}=({\mathcal{K}}_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}}$
where $\displaystyle{\mathcal{K}}_{{\mathfrak{p}}}$ is a compact set of
$\displaystyle U_{{\mathfrak{p}}}$ for each
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$ which is a closure of an open
subset $\displaystyle{\mathring{{\mathcal{K}}}}_{{\mathfrak{p}}}$, and
$\displaystyle\bigcup_{{\mathfrak{p}}\in{\mathfrak{P}}}\psi_{{\mathfrak{p}}}({\mathring{{\mathcal{K}}}}_{{\mathfrak{p}}}\cap
s^{-1}_{{\mathfrak{p}}}(0))=X$.

We assume that $\displaystyle\widetriangle{{\mathfrak{S}}}$ 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
$\displaystyle\widetriangle{{\mathfrak{S}}}$. Also, let
$\displaystyle\widetriangle{\omega}=(\omega_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}}$
be a differential form on $\displaystyle(X,\widetriangle{{\mathcal{U}}})$.
Given these data, we are going to define

(64)
$\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},\widetriangle{\omega},\widetriangle{{\mathfrak{S}}}^{\varepsilon})\in
C^{\text{\rm dR}}_{*}({\mathcal{L}}_{k+1})$

for sufficiently small $\displaystyle\varepsilon>0$. Note that the support
system $\displaystyle{\mathcal{K}}$ is a part of the data to define (64),
though it is implicit in the above formula.

######  Remark 8.8.

In contrast to the case of single K-charts, where de Rham chain is defined for
all $\displaystyle\varepsilon\in(0,1]$, the de Rham chain (64) is defined only
when $\displaystyle\varepsilon>0$ is sufficiently small.

In Section 8.2.1 we state and prove some technical lemmas. In Section 8.2.2 we
define (64) and check its well-definedness, invariance under GG-embedding, and
Stokes’ formula. We only consider spaces with GCS without boundaries, since
generalization to spaces with boundaries (and corners) is straightforward. The
arguments of this subsection partly follow arguments in Sections 7.5–7.7 in
[10] with some modifications.

####  8.2.1. Technical lemmas

Recall that
$\displaystyle|{\mathcal{K}}|:=\bigg{(}\bigsqcup_{{\mathfrak{p}}\in{\mathfrak{P}}}{\mathcal{K}}_{{\mathfrak{p}}}\bigg{)}/\sim$
equipped with the topology from $\displaystyle|\widetriangle{{\mathcal{U}}}|$
is metrizable; see Definition 5.6 (3) in [10]. We fix a metric $\displaystyle
d$ on $\displaystyle|{\mathcal{K}}|$ which is compatible with this topology.

For any $\displaystyle S\subset|{\mathcal{K}}|$ and $\displaystyle\delta>0$ we
set

$\displaystyle B_{\delta}(S):=\\{x\in|{\mathcal{K}}|\mid
d(S,x)<\delta\\},\qquad\bar{B}_{\delta}(S):=\\{x\in|{\mathcal{K}}|\mid
d(S,x)\leq\delta\\}.$

Recall the notion of “support set” of CF-perturbations from Definition 7.72 in
[10]:

> For each $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$, let
> $\displaystyle\\{({\mathfrak{V}}_{{\mathfrak{r}}},{\mathcal{S}}_{{\mathfrak{r}}})\mid{\mathfrak{r}}\in{\mathfrak{R}}\\}$
> be a representative of the CF-perturbation
> $\displaystyle{\mathfrak{S}}_{{\mathfrak{p}}}$ of
> $\displaystyle{\mathcal{U}}_{{\mathfrak{p}}}$. Then for each
> $\displaystyle\varepsilon\in(0,1]$, we define
> $\displaystyle\Pi(({\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})^{-1}(0)))$
> to be the set consisting of all $\displaystyle x\in U_{{\mathfrak{p}}}$ such
> that there exists $\displaystyle{\mathfrak{r}}\in{\mathfrak{R}}$,
> $\displaystyle y\in V_{{\mathfrak{r}}}$, $\displaystyle\xi\in
> W_{{\mathfrak{r}}}$ such that
>
> $\displaystyle\phi_{{\mathfrak{r}}}(y)=x,\qquad
> s^{\varepsilon}_{{\mathfrak{r}}}(y,\xi)=0,\qquad\xi\in\text{\rm
> supp}\,\eta_{{\mathfrak{r}}}.$
>
> This definition is independent of the choice of representative. Then we
> define
>
>
> $\displaystyle\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon})^{-1}(0)):=\bigcup_{{\mathfrak{p}}\in{\mathfrak{P}}}({\mathcal{K}}_{{\mathfrak{p}}}\cap\Pi(({\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})^{-1}(0)))\subset|{\mathcal{K}}|$
>
> and call it the support set of
> $\displaystyle\widetriangle{{\mathfrak{S}}}^{\varepsilon}$.

######  Lemma 8.9.

For any neighborhood $\displaystyle U$ of
$\displaystyle\bigcup_{{\mathfrak{p}}\in{\mathfrak{P}}}({\mathcal{K}}_{{\mathfrak{p}}}\cap
s_{{\mathfrak{p}}}^{-1}(0))$ in $\displaystyle|{\mathcal{K}}|$, there exists
$\displaystyle\varepsilon_{0}>0$ such that $\displaystyle
0<\varepsilon<\varepsilon_{0}\implies\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon})^{-1}(0))\subset
U$.

###### Proof.

If this is not the case, there exists a sequence
$\displaystyle(\varepsilon_{m})_{m\geq 1}$ of positive real numbers converging
to $\displaystyle 0$, and a sequence $\displaystyle(x_{m})_{m\geq 1}$ such
that $\displaystyle
x_{m}\in\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon_{m}})^{-1}(0))\setminus
U$ for every $\displaystyle m\geq 1$. Up to subsequence
$\displaystyle(x_{m})_{m}$ has a limit in $\displaystyle|{\mathcal{K}}|$ which
we denote by $\displaystyle x$.

There exists $\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}$ such that
$\displaystyle x_{m}\in{\mathcal{K}}_{{\mathfrak{q}}}$ for infinitely many
$\displaystyle m$, thus we may assume that $\displaystyle
x_{m}\in{\mathcal{K}}_{{\mathfrak{q}}}$ for all $\displaystyle m$, then
$\displaystyle x\in{\mathcal{K}}_{{\mathfrak{q}}}$. It is sufficient to show
that $\displaystyle s_{{\mathfrak{q}}}(x)=0$, which implies $\displaystyle
x_{m}\in U$ for sufficiently large $\displaystyle m$, a contradiction.

Take a manifold chart $\displaystyle(V,E,\phi,\widehat{\phi})$ of
$\displaystyle(U_{{\mathfrak{q}}},{\mathcal{E}}_{{\mathfrak{q}}})$ such that
$\displaystyle x\in\phi(V)$ and $\displaystyle{\mathfrak{S}}_{{\mathfrak{q}}}$
is locally represented by
$\displaystyle(W,\eta,({\mathfrak{s}}^{\varepsilon})_{\varepsilon})$ where
$\displaystyle{\mathfrak{s}}^{\varepsilon}:V\times W\to E$ and
$\displaystyle\eta$ is a compactly supported form on $\displaystyle W$. Let
$\displaystyle s:V\to E$ denote the pullback of $\displaystyle
s_{{\mathfrak{q}}}$ by $\displaystyle\phi$. Then
$\displaystyle{\mathfrak{s}}^{\varepsilon}(y,\xi)\to s(y)\,(y\in V,\,\xi\in
W)$ as $\displaystyle\varepsilon\to 0$ in the compact $\displaystyle C^{1}$
topology on $\displaystyle V\times W$.

Take $\displaystyle y\in V$ so that $\displaystyle\phi(y)=x$ and
$\displaystyle y_{m}\in V$ for sufficiently large $\displaystyle m$ so that
$\displaystyle\phi(y_{m})=x_{m}$. Then $\displaystyle y_{m}\to y$ and there
exists $\displaystyle\xi_{m}\in\text{\rm supp}\,\eta$ such that
$\displaystyle{\mathfrak{s}}^{\varepsilon_{m}}(y_{m},\xi_{m})=0$. Since
$\displaystyle\text{\rm supp}\,\eta$ is compact, this shows that
$\displaystyle s(y)=0$, which implies $\displaystyle s_{{\mathfrak{q}}}(x)=0$.
∎

######  Lemma 8.10.

For any support system $\displaystyle{\mathcal{K^{\prime}}}<{\mathcal{K}}$
(see Definition 5.6 (2) in [10]), there exist $\displaystyle\delta>0$ and
$\displaystyle\varepsilon_{0}>0$ such that

$\displaystyle 0<\varepsilon<\varepsilon_{0}\implies
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon})^{-1}(0))\subset{\mathcal{K}}_{{\mathfrak{p}}}\,(\forall{\mathfrak{p}}\in{\mathfrak{P}}).$

###### Proof.

We take a support system $\displaystyle{\mathcal{K}}^{\prime\prime}$ such that
$\displaystyle{\mathcal{K}}^{\prime}<{\mathcal{K}}^{\prime\prime}<{\mathcal{K}}$.

Step 1. We first show that there exists $\displaystyle\delta>0$ such that

$\displaystyle\bar{B}_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap\bigcup_{{\mathfrak{q}}\in{\mathfrak{P}}}({\mathcal{K}}_{{\mathfrak{q}}}\cap
s_{{\mathfrak{q}}}^{-1}(0))\subset{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}\,(\forall{\mathfrak{p}}\in{\mathfrak{P}}).$

Suppose that this is not the case. Then there exist
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$ and a sequence
$\displaystyle(x_{m})_{m\geq 1}$ on
$\displaystyle\bigcup_{{\mathfrak{q}}\in{\mathfrak{P}}}({\mathcal{K}}_{{\mathfrak{q}}}\cap
s_{{\mathfrak{q}}}^{-1}(0))\setminus{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}$
such that $\displaystyle d(x_{m},{\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\to
0$ as $\displaystyle m\to\infty$.

Since $\displaystyle{\mathfrak{P}}$ is a finite set, there exists
$\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}$ such that $\displaystyle
x_{m}\in{\mathcal{K}}_{{\mathfrak{q}}}\cap s_{{\mathfrak{q}}}^{-1}(0)$ for
infinitely many $\displaystyle m$. We may assume that $\displaystyle
x_{m}\in{\mathcal{K}}_{{\mathfrak{q}}}\cap s_{{\mathfrak{q}}}^{-1}(0)$ for all
$\displaystyle m$, and there exists $\displaystyle
x\in{\mathcal{K}}_{{\mathfrak{q}}}\cap{\mathcal{K}}^{\prime}_{{\mathfrak{p}}}$
such that $\displaystyle d(x_{m},x)\to 0$. Then $\displaystyle
s_{{\mathfrak{q}}}(x)=0$, since $\displaystyle x_{m}\to x$ in
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}$ and $\displaystyle
s_{{\mathfrak{q}}}(x_{m})=0$ for all $\displaystyle m$.

Since
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}\cap{\mathcal{K}}_{{\mathfrak{p}}}\neq\emptyset$
in $\displaystyle|\widetriangle{{\mathcal{U}}}|$, at least one of the
following three cases holds:

  * •

$\displaystyle{\mathfrak{p}}\geq{\mathfrak{q}}$: This case
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}\setminus U_{{\mathfrak{p}}}$ is
compact, thus $\displaystyle d({\mathcal{K}}_{{\mathfrak{q}}}\setminus
U_{{\mathfrak{p}}},{\mathcal{K}}^{\prime}_{{\mathfrak{p}}})>0$. Since
$\displaystyle\lim_{m\to\infty}x_{m}=x$ and $\displaystyle
x\in{\mathcal{K}}^{\prime}_{{\mathfrak{p}}}$, we obtain $\displaystyle
x_{m}\in U_{{\mathfrak{p}}}$ for sufficiently large $\displaystyle m$. Since
$\displaystyle{\mathcal{K}}^{\prime}_{{\mathfrak{p}}}\subset{\mathring{{\mathcal{K}}}}^{\prime\prime}_{{\mathfrak{p}}}$
(this follows from
$\displaystyle{\mathcal{K}}^{\prime}<{\mathcal{K}}^{\prime\prime}$; see
Definition 5.6 (2) in [10]), we obtain $\displaystyle
x_{m}\in{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}$ for sufficiently large
$\displaystyle m$, contradicting our assumption.

  * •

$\displaystyle{\mathfrak{p}}\leq{\mathfrak{q}}$ and $\displaystyle\text{\rm
dim}\,U_{{\mathfrak{p}}}=\text{\rm dim}\,U_{{\mathfrak{q}}}$: This case
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}\setminus U_{{\mathfrak{p}}}$ is
compact since $\displaystyle\Phi_{{\mathfrak{q}}{\mathfrak{p}}}$ is an open
embedding, and we obtain a contradiction as in the previous case.

  * •

$\displaystyle{\mathfrak{p}}<{\mathfrak{q}}$ and $\displaystyle\text{\rm
dim}\,U_{{\mathfrak{p}}}<\text{\rm dim}\,U_{{\mathfrak{q}}}$: Since
$\displaystyle\lim_{m\to\infty}x_{m}=x\in{\mathcal{K}}^{\prime}_{{\mathfrak{p}}}\subset{\mathring{{\mathcal{K}}}}^{\prime\prime}_{{\mathfrak{p}}}$
and $\displaystyle x_{m}\notin{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}$
for every $\displaystyle m$, we obtain $\displaystyle
x_{m}\notin\varphi_{{\mathfrak{q}}{\mathfrak{p}}}(U_{{\mathfrak{q}}{\mathfrak{p}}})$
for sufficiently large $\displaystyle m$. On the other hand $\displaystyle
x=\lim_{m\to\infty}x_{m}$ satisfies $\displaystyle s_{{\mathfrak{q}}}(x)=0$
and $\displaystyle
D_{x}s_{{\mathfrak{q}}}:T_{x}U_{{\mathfrak{q}}}/T_{x}U_{{\mathfrak{p}}}\to({\mathcal{E}}_{{\mathfrak{q}}})_{x}/({\mathcal{E}}_{{\mathfrak{p}}})_{x}$
is an isomorphism, which implies that $\displaystyle
s_{{\mathfrak{q}}}(x_{m})\neq 0$ for sufficiently large every $\displaystyle
m$, contradicting our assumption.

Step 2. Taking $\displaystyle\delta$ as in Step 1, we prove that there exists
$\displaystyle\varepsilon_{0}>0$ such that

$\displaystyle 0<\varepsilon<\varepsilon_{0}\implies
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon})^{-1}(0))\subset{\mathcal{K}}_{{\mathfrak{p}}}\quad(\forall{\mathfrak{p}}\in{\mathfrak{P}}).$

If this is not the case, there exist a sequence
$\displaystyle(\varepsilon_{m})_{m\geq 1}$ of positive real numbers converging
to $\displaystyle 0$, $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$, and a
sequence $\displaystyle(x_{m})_{m\geq 1}$ such that

$\displaystyle
x_{m}\in(B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon_{m}})^{-1}(0)))\setminus{\mathcal{K}}_{{\mathfrak{p}}}$

for every $\displaystyle m$. Up to subsequence $\displaystyle(x_{m})_{m}$ has
a limit $\displaystyle x$ in $\displaystyle|{\mathcal{K}}|$. By Lemma 8.9,
$\displaystyle
x\in\bigcup_{{\mathfrak{q}}\in{\mathfrak{P}}}({\mathcal{K}}_{{\mathfrak{q}}}\cap
s_{{\mathfrak{q}}}^{-1}(0))$. Also $\displaystyle
x\in\bar{B}_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})$, thus by Step 1
we get $\displaystyle x\in{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}$.

There exists $\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}$ such that
$\displaystyle x_{m}\in{\mathcal{K}}_{{\mathfrak{q}}}$ for infinitely many
$\displaystyle m$, then we may assume that $\displaystyle
x_{m}\in{\mathcal{K}}_{{\mathfrak{q}}}$ for all $\displaystyle m$, in
particular $\displaystyle x\in{\mathcal{K}}_{{\mathfrak{q}}}$. Then
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}\cap{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}\neq\emptyset$,
thus at least one of the following three cases holds:

  * •

$\displaystyle{\mathfrak{p}}\geq{\mathfrak{q}}$: This case
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}\setminus U_{{\mathfrak{p}}}$ is
compact, thus $\displaystyle d({\mathcal{K}}_{{\mathfrak{q}}}\setminus
U_{{\mathfrak{p}}},{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}})>0$. Since
$\displaystyle\lim_{m\to\infty}x_{m}=x\in{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}$,
we obtain $\displaystyle x_{m}\in U_{{\mathfrak{p}}}$ for sufficiently large
$\displaystyle m$. By
$\displaystyle{\mathcal{K}}^{\prime\prime}_{{\mathfrak{p}}}\subset{\mathring{{\mathcal{K}}}}_{{\mathfrak{p}}}$,
we obtain $\displaystyle x_{m}\in{\mathcal{K}}_{{\mathfrak{p}}}$ for
sufficiently large $\displaystyle m$, contradicting our assumption.

  * •

$\displaystyle{\mathfrak{p}}\leq{\mathfrak{q}}$ and $\displaystyle\text{\rm
dim}\,U_{{\mathfrak{p}}}=\text{\rm dim}\,U_{{\mathfrak{q}}}$: Again
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}\setminus U_{{\mathfrak{p}}}$ is
compact, and we obtain a contradiction as in the previous case.

  * •

$\displaystyle{\mathfrak{p}}<{\mathfrak{q}}$ and $\displaystyle\text{\rm
dim}\,U_{{\mathfrak{p}}}<\text{\rm dim}\,U_{{\mathfrak{q}}}$: Since
$\displaystyle\lim_{m\to\infty}x_{m}=x\in{\mathring{{\mathcal{K}}_{{\mathfrak{p}}}}}$
and $\displaystyle x_{m}\notin{\mathcal{K}}_{{\mathfrak{p}}}$ for every
$\displaystyle m\geq 1$, we obtain $\displaystyle
x_{m}\notin\varphi_{{\mathfrak{q}}{\mathfrak{p}}}(U_{{\mathfrak{q}}{\mathfrak{p}}})$
for sufficiently large $\displaystyle m$. Since the CF-perturbation
$\displaystyle{\mathfrak{S}}^{\varepsilon_{m}}_{{\mathfrak{q}}}$ converges to
$\displaystyle s_{{\mathfrak{q}}}$ in compact $\displaystyle C^{1}$-topology,
$\displaystyle s_{{\mathfrak{q}}}(x)=0$ thus $\displaystyle
D_{x}s_{{\mathfrak{q}}}:T_{x}U_{{\mathfrak{q}}}/T_{x}U_{{\mathfrak{p}}}\to({\mathcal{E}}_{{\mathfrak{q}}})_{x}/({\mathcal{E}}_{{\mathfrak{p}}})_{x}$
is an isomorphism, we obtain $\displaystyle
x_{m}\notin\Pi(({\mathfrak{S}}_{{\mathfrak{q}}}^{\varepsilon_{m}})^{-1}(0))$
for sufficiently large $\displaystyle m$, contradicting our assumption.

∎

######  Lemma 8.11.

For sufficiently small $\displaystyle\delta>0$ there holds

$\displaystyle{\mathfrak{q}}<{\mathfrak{p}}\implies
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap{\mathcal{K}}_{{\mathfrak{q}}}\subset
U_{{\mathfrak{p}}{\mathfrak{q}}}.$

###### Proof.

Suppose that this is not the case. Then there exist
$\displaystyle{\mathfrak{p}}$, $\displaystyle{\mathfrak{q}}$ such that
$\displaystyle{\mathfrak{q}}<{\mathfrak{p}}$ and a sequence
$\displaystyle(\delta_{m})_{m}$ converging to $\displaystyle 0$ and a sequence
$\displaystyle(x_{m})_{m}$ such that $\displaystyle
x_{m}\in(B_{\delta_{m}}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap{\mathcal{K}}_{{\mathfrak{q}}})\setminus
U_{{\mathfrak{p}}{\mathfrak{q}}}$ for every $\displaystyle m$. Then the
sequence $\displaystyle(x_{m})_{m}$ has a limit point $\displaystyle
x\in{\mathcal{K}}_{{\mathfrak{q}}}\setminus U_{{\mathfrak{p}}{\mathfrak{q}}}$,
then $\displaystyle d(x,{\mathcal{K}}^{\prime}_{{\mathfrak{p}}})>0$,
contradicting $\displaystyle x_{m}\in
B_{\delta_{m}}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\,(\forall m)$. ∎

####  8.2.2. Definition and well-definedness

We start from data $\displaystyle X$,
$\displaystyle\widetriangle{{\mathcal{U}}}$,
$\displaystyle\widetriangle{\omega}$, $\displaystyle{\mathcal{K}}$,
$\displaystyle\widetriangle{{\mathfrak{S}}}$ and
$\displaystyle\widetriangle{f}$. We take the following choices (our definition
of partition of unity is slightly different from that of [10], Definition
7.64).

  * •

A support system $\displaystyle{\mathcal{K}}^{\prime}<{\mathcal{K}}$.

  * •

A positive real number $\displaystyle\delta$ such that

    * –

There exists $\displaystyle\varepsilon>0$ such that

$\displaystyle 0<\varepsilon^{\prime}<\varepsilon\implies
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon^{\prime}})^{-1}(0))\subset{\mathcal{K}}_{{\mathfrak{p}}}\quad(\forall{\mathfrak{p}}\in{\mathfrak{P}}).$

    * –

$\displaystyle
2\delta<d({\mathcal{K}}^{\prime}_{{\mathfrak{p}}},{\mathcal{K}}^{\prime}_{{\mathfrak{q}}})$
for any $\displaystyle{\mathfrak{p}},{\mathfrak{q}}\in{\mathfrak{P}}$ such
that $\displaystyle U_{{\mathfrak{p}}}\cap U_{{\mathfrak{q}}}=\emptyset$.

    * –

$\displaystyle{\mathfrak{q}}<{\mathfrak{p}}\implies
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap{\mathcal{K}}_{{\mathfrak{q}}}\subset
U_{{\mathfrak{p}}{\mathfrak{q}}}$.

    * –

$\displaystyle{\mathcal{K}}^{\prime}_{{\mathfrak{p}}}(\delta):=\\{x\in{\mathcal{K}}_{{\mathfrak{p}}}\mid
d({\mathcal{K}}^{\prime}_{{\mathfrak{p}}},x)\leq\delta\\}\subset{\mathring{{\mathcal{K}}_{{\mathfrak{p}}}}}$
for every $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$.

  * •

Partition of unity
$\displaystyle\chi=(\chi_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}}$
of
$\displaystyle(X,\widetriangle{{\mathcal{U}}},{\mathcal{K}}^{\prime},\delta)$.
Namely, the following conditions are satisfied:

    * –

$\displaystyle\chi_{{\mathfrak{p}}}:|{\mathcal{K}}|\to[0,1]$ is a strongly
smooth function for every $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$.
Namely, for every $\displaystyle{\mathfrak{q}}\in{\mathfrak{P}}$,
$\displaystyle\chi_{{\mathfrak{p}}}|_{{\mathcal{K}}_{{\mathfrak{q}}}}$ can be
extended to a $\displaystyle C^{\infty}$-function defined on a neighborhood of
$\displaystyle{\mathcal{K}}_{{\mathfrak{q}}}$ in $\displaystyle
U_{{\mathfrak{q}}}$.

    * –

$\displaystyle\text{\rm supp}\,\chi_{{\mathfrak{p}}}\subset
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})$ for every
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$.

    * –

$\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}\chi_{{\mathfrak{p}}}=1$
on a neighborhood of
$\displaystyle\bigcup_{{\mathfrak{p}}\in{\mathfrak{P}}}s_{{\mathfrak{p}}}^{-1}(0)\subset|{\mathcal{K}}|$.

Then we define

$\displaystyle\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},\widetriangle{\omega},\widetriangle{{\mathfrak{S}}}^{\varepsilon}):=\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},\chi_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}}).$

$\displaystyle(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},\chi_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})$
in the RHS makes sense, since for every
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$ there holds

$\displaystyle\text{\rm supp}\,\chi_{{\mathfrak{p}}}\cap
U_{{\mathfrak{p}}}\subset
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}})\cap
U_{{\mathfrak{p}}}\subset{\mathcal{K}}^{\prime}_{{\mathfrak{p}}}(\delta)\subset{\mathring{{\mathcal{K}}}}_{{\mathfrak{p}}}$

and
$\displaystyle{\mathfrak{S}}_{{\mathfrak{p}}}\in{\mathcal{S}}({\mathcal{K}}_{{\mathfrak{p}}})$.
We need to recall the following definition (Definition 7.79 in [10]):

######  Definition 8.12.

Let $\displaystyle{\mathcal{A}}$ and $\displaystyle{\mathcal{X}}$ be sets, and
$\displaystyle(F_{a})_{a\in{\mathcal{A}}}$ be a family of maps such that
$\displaystyle F_{a}:(0,\varepsilon_{a})\to{\mathcal{X}}$ for each
$\displaystyle a\in{\mathcal{A}}$. We say that $\displaystyle F_{a}$ is
independent on choices of $\displaystyle a$ in the sense of
$\displaystyle\spadesuit$ if the following holds:

> $\displaystyle\spadesuit$: For any $\displaystyle
> a_{1},a_{2}\in{\mathcal{A}}$, there exists $\displaystyle
> 0<\varepsilon_{0}<\min\\{\varepsilon_{a_{1}},\varepsilon_{a_{2}}\\}$ such
> that $\displaystyle F_{a_{1}}(\varepsilon)=F_{a_{2}}(\varepsilon)$ for every
> $\displaystyle\varepsilon\in(0,\varepsilon_{0})$.

######  Lemma 8.13.

The above definition of
$\displaystyle\widetriangle{f}_{*}(X,\widetriangle{{\mathcal{U}}},\widetriangle{\omega},\widetriangle{{\mathfrak{S}}}^{\varepsilon})$
is independent on choices of $\displaystyle{\mathcal{K}}^{\prime}$,
$\displaystyle\delta$, $\displaystyle\chi$ in the sense of
$\displaystyle\spadesuit$.

###### Proof.

Let us take two choices
$\displaystyle(({\mathcal{K}}^{\prime})^{i},\delta^{i},\chi^{i})_{i=1,2}$. We
are going to prove

$\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},\chi^{1}_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})=\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},\chi^{2}_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})$

when $\displaystyle\varepsilon$ is sufficiently small.

By our assumption,
$\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}\chi^{1}_{{\mathfrak{p}}}=\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}\chi^{2}_{{\mathfrak{p}}}=1$
on a neighborhood of
$\displaystyle\bigcup_{{\mathfrak{p}}\in{\mathfrak{P}}}s^{-1}_{{\mathfrak{p}}}(0)$.
Thus, Lemma 8.9 implies that, for sufficiently small
$\displaystyle\varepsilon>0$ and every
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}$, there holds
$\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}\chi^{1}_{{\mathfrak{p}}}=\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}\chi^{2}_{{\mathfrak{p}}}=1$
on
$\displaystyle{\mathcal{K}}_{{\mathfrak{p}}}\cap\Pi(({\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})^{-1}(0))$.
Then we obtain

$\displaystyle\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},\chi^{1}_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})$
$\displaystyle\displaystyle=\sum_{{\mathfrak{p}}_{1},{\mathfrak{p}}_{2}}(f_{{\mathfrak{p}}_{1}})_{*}({\mathcal{U}}_{{\mathfrak{p}}_{1}},\chi^{1}_{{\mathfrak{p}}_{1}}\chi^{2}_{{\mathfrak{p}}_{2}}\omega_{{\mathfrak{p}}_{1}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}_{1}})$
$\displaystyle\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}}(f_{{\mathfrak{p}}})_{*}({\mathcal{U}}_{{\mathfrak{p}}},\chi^{2}_{{\mathfrak{p}}}\omega_{{\mathfrak{p}}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}})$
$\displaystyle\displaystyle=\sum_{{\mathfrak{p}}_{1},{\mathfrak{p}}_{2}}(f_{{\mathfrak{p}}_{2}})_{*}({\mathcal{U}}_{{\mathfrak{p}}_{2}},\chi^{1}_{{\mathfrak{p}}_{1}}\chi^{2}_{{\mathfrak{p}}_{2}}\omega_{{\mathfrak{p}}_{2}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}_{2}}).$

Then it is sufficient to show

$\displaystyle(f_{{\mathfrak{p}}_{1}})_{*}({\mathcal{U}}_{{\mathfrak{p}}_{1}},\chi^{1}_{{\mathfrak{p}}_{1}}\chi^{2}_{{\mathfrak{p}}_{2}}\omega_{{\mathfrak{p}}_{1}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}_{1}})=(f_{{\mathfrak{p}}_{2}})_{*}({\mathcal{U}}_{{\mathfrak{p}}_{2}},\chi^{1}_{{\mathfrak{p}}_{1}}\chi^{2}_{{\mathfrak{p}}_{2}}\omega_{{\mathfrak{p}}_{2}},{\mathfrak{S}}^{\varepsilon}_{{\mathfrak{p}}_{2}})$

for any $\displaystyle{\mathfrak{p}}_{1},{\mathfrak{p}}_{2}\in{\mathfrak{P}}$.
We may assume either $\displaystyle{\mathfrak{p}}_{1}\leq{\mathfrak{p}}_{2}$
or $\displaystyle{\mathfrak{p}}_{1}\geq{\mathfrak{p}}_{2}$, since otherwise
$\displaystyle U_{{\mathfrak{p}}_{1}}\cap U_{{\mathfrak{p}}_{2}}=\emptyset$,
thus
$\displaystyle\chi^{1}_{{\mathfrak{p}}_{1}}\chi^{2}_{{\mathfrak{p}}_{2}}\equiv
0$ by $\displaystyle
2\delta<d({\mathcal{K}}^{\prime}_{{\mathfrak{p}}_{1}},{\mathcal{K}}^{\prime}_{{\mathfrak{p}}_{2}})$.
Also this equality is clear when
$\displaystyle{\mathfrak{p}}_{1}={\mathfrak{p}}_{2}$. Thus we may assume that
$\displaystyle{\mathfrak{p}}_{1}<{\mathfrak{p}}_{2}$.

In the argument below, we abbreviate $\displaystyle{\mathfrak{p}}_{i}$ by
$\displaystyle i$. For example
$\displaystyle{\mathcal{U}}_{{\mathfrak{p}}_{i}}$ is abbreviated by
$\displaystyle{\mathcal{U}}_{i}$. Let us introduce the following notations:

  * •

$\displaystyle{\mathcal{U}}_{21}$ denotes
$\displaystyle{\mathcal{U}}_{1}|_{U_{21}}=(\varphi_{21})^{*}{\mathcal{U}}_{2}$.

  * •

$\displaystyle f_{21}$ denotes $\displaystyle
f_{1}|_{U_{21}}=f_{2}\circ\varphi_{21}$.

  * •

$\displaystyle\omega_{21}$ denotes
$\displaystyle\omega_{1}|_{U_{21}}=(\varphi_{21})^{*}\omega_{2}$.

  * •

$\displaystyle{\mathfrak{S}}_{21}$ denotes
$\displaystyle{\mathfrak{S}}_{1}|_{U_{21}}=(\varphi_{21})^{*}{\mathfrak{S}}_{2}$.

Then there holds

$\displaystyle(f_{1})_{*}({\mathcal{U}}_{1},\chi^{1}_{1}\chi^{2}_{2}\omega_{1},{\mathfrak{S}}^{\varepsilon}_{1})=(f_{21})_{*}({\mathcal{U}}_{21},\chi^{1}_{1}\chi^{2}_{2}\omega_{21},{\mathfrak{S}}^{\varepsilon}_{21})$

since $\displaystyle\text{\rm supp}\,\chi^{2}_{{\mathfrak{p}}_{2}}\subset
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}_{2}})$ and $\displaystyle
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}_{2}})\cap{\mathcal{K}}_{{\mathfrak{p}}_{1}}\subset
U_{{\mathfrak{p}}_{2}{\mathfrak{p}}_{1}}$. On the other hand

$\displaystyle(f_{2})_{*}({\mathcal{U}}_{2},\chi^{1}_{1}\chi^{2}_{2}\omega_{2},{\mathfrak{S}}^{\varepsilon}_{2})=(f_{21})_{*}({\mathcal{U}}_{21},\chi^{1}_{1}\chi^{2}_{2}\omega_{21},{\mathfrak{S}}^{\varepsilon}_{21})$

when $\displaystyle\varepsilon$ is sufficiently small, since
$\displaystyle\text{\rm supp}\,\chi^{1}_{{\mathfrak{p}}_{1}}\subset
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}_{1}})$ and $\displaystyle
B_{\delta}({\mathcal{K}}^{\prime}_{{\mathfrak{p}}_{1}})\cap\Pi((\widetriangle{{\mathfrak{S}}}^{\varepsilon})^{-1}(0))\subset{\mathcal{K}}_{{\mathfrak{p}}_{1}}$
by Lemma 8.10. ∎

Next we prove the invariance by GG-embedding (for the definition of GG-
embedding, see Definition 3.24 in [10]). Lemma 8.14 below is an analogue of
Proposition 9.16 in [10].

######  Lemma 8.14.

Let us consider
$\displaystyle(\widetriangle{{\mathcal{U}}}_{i},{\mathcal{K}}_{i},\widetriangle{{\mathfrak{S}}}_{i},\widetriangle{f}_{i},\widetriangle{\omega}_{i})_{i=1,2}$
and $\displaystyle\widetriangle{\Phi}$ such that

  * •

For each $\displaystyle i\in\\{1,2\\}$, the tuple
$\displaystyle(\widetriangle{{\mathcal{U}}}_{i},{\mathcal{K}}_{i},\widetriangle{{\mathfrak{S}}}_{i},\widetriangle{f}_{i},\widetriangle{\omega}_{i})$
satisfies the conditions to define
$\displaystyle(\widetriangle{f}_{i})_{*}(X,\widetriangle{{\mathcal{U}}}_{i},\widetriangle{\omega}_{i},\widetriangle{{\mathfrak{S}}}^{\varepsilon}_{i})$
for sufficiently small $\displaystyle\varepsilon>0$.

  * •

$\displaystyle\widetriangle{\Phi}:\widetriangle{{\mathcal{U}}_{1}}\to\widetriangle{{\mathcal{U}}_{2}}$
is a GG-embedding. Namely,
$\displaystyle\widetriangle{\Phi}=({\mathfrak{i}},(\Phi_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}_{1}})$
where $\displaystyle{\mathfrak{i}}:{\mathfrak{P}}_{1}\to{\mathfrak{P}}_{2}$ is
an order preserving map, and
$\displaystyle\Phi_{{\mathfrak{p}}}=(\varphi_{{\mathfrak{p}}},\widehat{\varphi}_{{\mathfrak{p}}}):({\mathcal{U}}_{1})_{{\mathfrak{p}}}\to({\mathcal{U}}_{2})_{{\mathfrak{i}}({\mathfrak{p}})}$
is an embedding of K-charts for each
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{1}$, such that compatibilities
in Definition 3.24 in [10] are satisfied.

  * •

$\displaystyle{\mathcal{K}}_{1}$, $\displaystyle{\mathcal{K}}_{2}$ and
$\displaystyle\widetriangle{{\mathfrak{S}}}_{1}$,
$\displaystyle\widetriangle{{\mathfrak{S}}}_{2}$ are compatible with
$\displaystyle\widetriangle{\Phi}$ (see Definition 9.3 (3), (4) in [10]).
Moreover
$\displaystyle\widetriangle{\omega}_{1}=\widetriangle{\Phi}^{*}\widetriangle{\omega}_{2}$,
$\displaystyle\widetriangle{f}_{1}=\widetriangle{f}_{2}\circ\widetriangle{\Phi}$.

Then, for sufficiently small $\displaystyle\varepsilon>0$, there holds

$\displaystyle(\widetriangle{f}_{1})_{*}(X,\widetriangle{{\mathcal{U}}}_{1},\widetriangle{\omega}_{1},\widetriangle{{\mathfrak{S}}}^{\varepsilon}_{1})=(\widetriangle{f}_{2})_{*}(X,\widetriangle{{\mathcal{U}}}_{2},\widetriangle{\omega}_{2},\widetriangle{{\mathfrak{S}}}^{\varepsilon}_{2}).$

###### Proof.

Let us take a support system $\displaystyle{\mathcal{K}}^{\prime}_{i}$ of
$\displaystyle\widetriangle{{\mathcal{U}}_{i}}$ for $\displaystyle i=1,2$ such
that:

  * •

$\displaystyle{\mathcal{K}}^{\prime}_{i}<{\mathcal{K}}_{i}$ for $\displaystyle
i=1,2$.

  * •

$\displaystyle{\mathcal{K}}^{\prime}_{1}$ and
$\displaystyle{\mathcal{K}}^{\prime}_{2}$ are compatible with
$\displaystyle\widetriangle{\Phi}$. Namely,
$\displaystyle\varphi_{{\mathfrak{p}}}(({\mathcal{K}}^{\prime}_{1})_{{\mathfrak{p}}})\subset{\mathring{({\mathcal{K}}^{\prime}_{2})}}_{{\mathfrak{i}}({\mathfrak{p}})}$
for every $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{1}$.

First we need Lemma 8.15 below, whose proof is almost the same as the proof of
Proposition 7.67 in [10]. In the statement, we take a metric on
$\displaystyle|{\mathcal{K}}_{2}|$ and its pullback to
$\displaystyle|{\mathcal{K}}_{1}|$ via a natural embedding map
$\displaystyle|{\mathcal{K}}_{1}|\to|{\mathcal{K}}_{2}|$. This embedding map
exists since $\displaystyle{\mathcal{K}}_{1}$ and
$\displaystyle{\mathcal{K}}_{2}$ are compatible with
$\displaystyle\widetriangle{\Phi}$.

######  Lemma 8.15.

For any $\displaystyle\delta>0$, there exists
$\displaystyle\chi^{+}_{1}=((\chi^{+}_{1})_{{\mathfrak{p}}})_{{\mathfrak{p}}\in{\mathfrak{P}}_{1}}$
which satisfies the following conditions:

  * •

$\displaystyle(\chi^{+}_{1})_{{\mathfrak{p}}}$ is a strongly smooth map from
$\displaystyle|{\mathcal{K}}_{2}|$ to $\displaystyle[0,1]$ for every
$\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{1}$.

  * •

$\displaystyle\text{\rm supp}\,(\chi^{+}_{1})_{{\mathfrak{p}}}\subset
B_{\delta}(\varphi_{{\mathfrak{p}}}(({\mathcal{K}}^{\prime}_{1})_{{\mathfrak{p}}}))$
for every $\displaystyle{\mathfrak{p}}\in{\mathfrak{P}}_{1}$. Here
$\displaystyle B_{\delta}$ denotes the $\displaystyle\delta$-neighborhood in
$\displaystyle|{\mathcal{K}}_{2}|$.

  * •

$\displaystyle\sum_{{\mathfrak{p}}\in{\mathfrak{P}}_{1}}(\chi^{+}_{1})_{{\mathfrak{p}}}=1$
on a neighborhood of
$\displaystyle\bigcup_{{\mathfrak{q}}\in{\mathfrak{P}}_{2}}s_{{\mathfrak{q}}}^{-1}(0)\subset|{\mathcal{K}}_{2}|$.