text:
\section{Introduction}

Symplectic field theory (SFT), proposed about 15 years ago by Eliashberg-Givental-Hofer \cite{EGH}, is a package which gives invariants of contact manifolds and symplectic manifolds with boundary as well as gluing formulas for Gromov-Witten invariants on closed symplectic manifolds. The transversality theory in the somewhere injective case and the Fredholm theory were carried out by Dragnev~\cite{Dr} and the SFT compactness was worked out by \cite{BEHWZ}, both very early on in the development of the theory. However, the rigorous foundations of the full theory are still under development, although parts of the theory are gradually becoming more rigorous, thanks to the efforts of many authors, primarily by Hofer-Wysocki-Zehnder~\cite{HWZ3}.  Partial work has been done by Hutchings-Nelson~\cite{HN} and Bao-Honda~\cite{BH} for cylindrical contact homology in dimension three and Bourgeois-Oancea~\cite{BO} on $S^1$-equivariant symplectic homology.  Also the Kuranishi perturbation theory of Fukaya-Ono~\cite{FO} and Fukaya-Oh-Ohta-Ono \cite{FO3} (which is closest in spirit to this work) and the approaches of Liu-Tian~\cite{LT} and \cite{Ru} may be used to define SFT, although this has not been done yet.

The goal of this paper is to make a further contribution to this effort and define the full contact homology differential graded algebra (dga) for any closed contact manifold in any dimension and show that its isomorphism class is an invariant of the contact manifold. More precisely, given a closed cooriented contact $(2n+1)$-dimensional manifold $(M,\xi)$ and some auxiliary data $\mathcal{D}$ which includes a nondegenerate contact form $\alpha$ for $\xi$, we define a direct limit dga $\mathfrak{A}(\mathcal{D})$ whose homology is called the {\em full contact homology algebra} $HC(\mathcal{D})$. We then prove that:

\begin{thm} \label{thm: invariance}
If $\mathcal{D}_1$ and $\mathcal{D}_2$ are two choices of auxiliary data for $(M,\xi)$, then $HC(\mathcal{D}_1)$ and $HC(\mathcal{D}_2)$ are isomorphic. Hence the isomorphism class of the algebra $HC(\mathcal{D})$ is an invariant of $(M,\xi)$.
\end{thm}

We denote the isomorphism class of $HC(\mathcal D)$ by $HC(\xi)$.  We also prove the following:

\begin{thm} \label{thm: cobordisms}
Let $W$ be a compact exact symplectic cobordism which restricts to $(M_+,\xi_+)$ at the positive boundary and to $(M_-,\xi_-)$ at the negative boundary.
Then there is an algebra homomorphism
$$\Phi_*:HC(\xi_+)\to HC(\xi_-).$$
In the case when $\xi_+=\xi_-$ and $W$ is the trivial cobordism, we have $\Phi_*=\operatorname{id}$.
\end{thm}

The construction of the direct limit dga $\mathfrak{A}(\mathcal{D})$ and the proofs of Theorems~\ref{thm: invariance} and \ref{thm: cobordisms} use a particularly simple version of the Kuranishi multivalued perturbation theory of \cite{FO,FO3}, which we call {\em semi-global Kuranishi structures}.  The abstract perturbations that we use are supported near the ends of finite energy $J$-holomorphic curves and are described using the asymptotic eigenfunctions of the asymptotic operators.  For example, if a moduli space
$$\mathcal{M}^{\operatorname{ind}=k}_J(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$$
of Fredholm index $\operatorname{ind}=k$ $J$-holomorphic maps $\dot F\to \widehat{W}$ which map to a completed exact symplectic cobordism $\widehat{W}$ and are asymptotic to collections $\boldsymbol\gamma_+$ and $\boldsymbol\gamma_-$ of Reeb orbits at the positive and negative ends is already compact (i.e., we do not need multiple-level curves in its symplectic field theory (SFT) compactification\footnote{Throughout this paper we will freely use the SFT compactness theorem of \cite{BEHWZ} without explicit mention.}), then {\em we only need one Kuranishi chart, which we call a semi-global Kuranishi chart}.

\begin{rmk}
We emphasize that our simplification {\em crucially uses the existence of at least one end limiting to a Reeb orbit} and can be used to define SFT in the case when $\widehat{W}$ is an exact symplectic cobordism, although we only deal with contact homology in this paper.  Our simplified methods are not readily applicable to the case when there is a {\em closed} multiply-covered  $J$-holomorphic curve. However, we expect the combination of our methods and those of \cite{FO} and \cite{FO3} to yield a rigorous definition of SFT.
\end{rmk}

Finally, during the preparation of this paper, Pardon~\cite{Pa} posted a paper proving the existence and invariance of contact homology in arbitrary dimensions.

\vskip.1in\noindent
{\em Outline of the paper.}  We will be using the language of orbifolds and multisections, following Adem-Leida-Ruan~\cite{ALR} and Fukaya-Ono~\cite{FO}; this is reviewed in  Section~\ref{section: orbifolds and multisections}. After some preparation in Sections~\ref{section: almost complex structures and moduli spaces} and \ref{section: Fredholm theory}, we construct the semi-global Kuranishi chart in Section~\ref{section: semi-global Kuranishi charts}, with some modifications in Section~\ref{subsection: trimming}.   The semi-global Kuranishi structure is constructed in Section~\ref{section: construction of semi-global}, using the gluing results which are stated in Section~\ref{section: gluing}; they are average specimens of the ``gluing theorem'' type and are proved in Section~\ref{section: details of gluing}. Once the semi-global Kuranishi structure is constructed, the definition of contact homology, the definition of chain maps, and the invariance of contact homology (Theorems~\ref{thm: invariance} and \ref{thm: cobordisms}) follow the usual lines of argument and are carried out in Section~\ref{section: contact homology}.

We note that the discussion until the end of Section~\ref{section: gluing} is valid for curves of any genus; starting from Section~\ref{section: construction of semi-global} we specialize to contact homology.

\vskip.1in\noindent
{\em Acknowledgements.} We thank Francis Bonahon, Tobias Ekholm, Michael Hutchings, Eleny Ionel, Gang Liu, and Kaoru Ono for very helpful discussions.  We also thank Tobias Ekholm and John Pardon for pointing out some errors.

\section{Orbifolds and multisections} \label{section: orbifolds and multisections}

In this section we review the basics of (effective) orbifolds, orbibundles, and multisections from \cite{ALR} and \cite{FO}. The definitions in this section, while elementary, are a bit cumbersome to state.  Also, at the end of the day, all we do is replace the words ``manifold", ``vector bundle'', and ``section'' by ``orbifold'', ``orbibundle'', and ``multisection'', and treat them in exactly the same way for our purposes.  For the above reasons the reader may want to skip this section and return to it as the relevant concepts gradually start appearing in the rest of the paper.

\subsection{Orbifolds and orbibundles}\label{subsection: orbifolds and orbibundles}

\begin{defn} [Orbifold charts] \label{defn: orbifold charts}
Given a topological space $X$, a triple $(V,\Gamma, \phi)$ is an {\em orbifold chart of $X$} if
\begin{enumerate}
\item $V$ is a connected open subset of $\mathbb R^n$;
\item $\Gamma$ is a finite group that acts smoothly and effectively on $V$; and
\item $\phi:V\to X$ is a $\Gamma$-invariant continuous map such that the quotient map $V/\Gamma \to U\subseteq X$ is homeomorphism, where  $U$ is some open subset of $X$.
\end{enumerate}
\end{defn}

\begin{defn} [Embedding of orbifold charts]
If $(V_i,\Gamma_i,\phi_i)$ and $(V_j,\Gamma_j,\phi_j)$ are orbifold charts of $X$, we say that $(V_i,\Gamma_i,\phi_i)$ {\em embeds into $(V_j,\Gamma_j,\phi_j)$} if there exists a smooth embedding $\psi_{ji}:V_i\hookrightarrow V_j$ such that $\phi_j\circ \psi_{ji}=\phi_i$.
\end{defn}

\begin{rmk}
If $(V_i,\Gamma_i,\phi_i)$ embeds into $(V_j,\Gamma_j,\phi_j)$, then there exists a monomorphism $\theta_{ji}: \Gamma_i\to \Gamma_j$ such that $\psi_{ji}$ is $\theta_{ji}$-equivariant. See \cite{ALR} for details.
\end{rmk}

\begin{defn} [Orbifolds]  \label{def: orbifold}
Let $X$ be a Hausdorff, second countable topological space.  A {\em smooth orbifold structure on $X$} is a family $\mathcal{O}=\{(V_i,\Gamma_i, \phi_i)\}_{i\in\mathcal{I}}$ of orbifold charts such that
\begin{enumerate}
\item $\{ U_i \}_{i\in\mathcal{I}}$ forms an open cover of $X$; and
\item for any $x\in U_i \cap U_j \subseteq X$, there exist a neighborhood $U_k\subseteq U_i\cap U_j$ of $x$, a chart $(V_k,\Gamma_k,\phi_k)$ for $U_k$, and embeddings of orbifold charts $\psi_{ik}:(V_k,\Gamma_k,\phi_k) \hookrightarrow (V_i,\Gamma_i,\phi_i)$ and $\psi_{jk}:(V_k,\Gamma_k,\phi_k)\hookrightarrow (V_j,\Gamma_j,\phi_j)$.
\end{enumerate}
The pair $\mathbb X=(X, \mathcal{O})$ is a {\em smooth orbifold}.
\end{defn}

\begin{defn} [Smooth orbifold maps]
A {\em smooth orbifold map}
$$\lambda:(X,\mathcal{O})\to (X',\mathcal{O}')$$
is a continuous map $\lambda:X\to X'$ such that, for any $x\in X$, there exist orbifold charts $(V,\Gamma,\phi)$ around $x$ and $(V',\Gamma',\phi')$ around $\lambda(x)$, and a lift $\widehat \lambda : V\to V'$ of $\lambda|_{V/\Gamma}$ such that $\phi'\circ \widehat \lambda=\lambda \circ \phi$.
\end{defn}

\begin{defn} [Orbibundle charts]
Let $ \pi: \mathbb E\to \mathbb X$ be a smooth orbifold map. The triple $(E\to V, \Gamma, \widetilde \phi)$ is an {\em orbibundle chart of $\mathbb E \to \mathbb X$ of rank $n$}, if
\begin{enumerate}
\item $(V,\Gamma,\phi)$ and $(E,\Gamma,\widetilde\phi)$ are orbifold charts of $\mathbb X$ and $\mathbb{E}$;
\item $E=V\times \mathbb R^n$ and $\widehat \pi:E \to V$ is the projection onto the first factor;
\item $\Gamma$ acts diagonally on $E=V\times \mathbb{R}^n$, where the action on $\mathbb{R}^n$ is linear; and
\item $\pi \circ \widetilde \phi=\phi \circ \widehat \pi$ and $\widetilde \phi$ induces a homeomorphism $E/\Gamma \stackrel\sim\to  \pi^{-1}(U)$.
\end{enumerate}
We call $(E\to V,\Gamma)$ a {\em local orbibundle model of rank $n$}.
\end{defn}

\begin{defn} [Embedding of orbibundle charts]
If
$$(E_i\stackrel{\widehat{\pi}_i}\to V_i,\Gamma_i,\widetilde\phi_i) \quad \mbox{and} \quad (E_j\stackrel{\widehat{\pi}_j}\to V_j,\Gamma_j,\widetilde\phi_j)$$
are orbibundle charts of $\mathbb E \to \mathbb X$ of rank $n$, we say $(E_i\to V_i,\Gamma_i,\widetilde\phi_i)$ {\em embeds into $(E_j\to V_j,\Gamma_j,\widetilde\phi_j)$} if there exists an embedding of vector bundles $\widetilde \psi_{ji}:E_i\hookrightarrow E_j$ (i.e., an embedding which lies over an embedding $\psi_{ji}: (V_i,\Gamma_i,\phi_i)\hookrightarrow (V_j,\Gamma_j,\phi_j)$ of orbifold charts, takes vector space fibers to vector space fibers isomorphically, and satisfies $\widehat\pi_j\circ \widetilde \psi_{ji}= \psi_{ji}\circ \widehat{\pi}_i$) such that $\widetilde \phi_j \circ \widetilde \psi_{ji}=\widetilde \phi_i$.
\end{defn}

\begin{defn} [Orbibundles] \label{def: orbibundle}
Let $\pi: \mathbb E\to \mathbb X$ be a smooth orbifold map. A {\em smooth rank $n$ orbibundle structure on $\mathbb E\to \mathbb X$} consists of a family of rank $n$ orbibundle charts $\{(E_i\to V_i,\Gamma_i,\widetilde\phi_i)\}_{i\in\mathcal{I}}$ that satisfy
\begin{enumerate}
\item $\mathbb X=(X, \{(V_i,\Gamma_i, \phi_i)\}_{i\in\mathcal{I}})$; and
\item for any $x\in U_i \cap U_j$ where $U_i=V_i/\Gamma_i$ and $U_j=V_j/\Gamma_j$, there exist a neighborhood $U_k\subseteq U_i\cap U_j$ of $x$, an orbibundle chart $(E_k\to V_k,\Gamma_k,\widetilde\phi_k)$ such that $U_k=V_k/\Gamma_k$, and embeddings of orbibundle charts
    $$\widetilde \psi_{ik}:(E_k\to V_k,\Gamma_k,\widetilde\phi_k)\hookrightarrow (E_i\to V_i,\Gamma_i,\widetilde\phi_i)$$
    and
    $$\widetilde \psi_{jk}:(E_k\to V_k,\Gamma_k,\widetilde\phi_k)\hookrightarrow (E_j\to V_j,\Gamma_j,\widetilde\phi_j)$$
    that lie over and are compatible with embeddings of orbifold charts
    $$\psi_{ik}:(V_k,\Gamma_k,\phi_k) \hookrightarrow (V_i,\Gamma_i,\phi_i)\quad \mbox{and} \quad \psi_{jk}:(V_k,\Gamma_k,\phi_k)\hookrightarrow (V_j,\Gamma_j,\phi_j).$$
\end{enumerate}
We refer to $\mathbb{X}$ as the {\em base} of the orbibundle and $\mathbb{E}$ as the {\em total space} of the orbibundle.
\end{defn}

\subsection{Multisections} \label{subsection: multisections}

Given a topological space $W$, we denote its $n$-fold symmetric product by
$$\operatorname{Sym}^n(W):=W^n/S_n.$$
Here $S_n$ is the symmetric group of $n$ elements which acts on $W^n$ by
$$\sigma(x_1,\dots,x_n)=(x_{\sigma(1)},\dots,x_{\sigma(n)}),$$
where $\sigma\in S_n$ and $(x_1,\dots,x_n)\in W^n$.
We denote the equivalence class of $(x_1,...,x_n)$ in $\operatorname{Symp}^n(W)$ by $[x_1,...,x_n]$.
If a group $\Gamma$ acts on $W$, then $\Gamma$ also acts on $\operatorname{Sym}^n(W)$ diagonally by
$$g[x_1,\dots,x_n]=[g(x_1),\dots,g(x_n)],$$
where $g\in \Gamma$ and $[x_1,\dots,x_n]\in \operatorname{Sym}^n(W)$.
There is an inclusion
$$\iota^{n}:\operatorname{Sym}^{m}(W)\to \operatorname{Sym}^{mn}(W),$$
\[
[w_{1},\dots,w_{m}]\mapsto [\underbrace{w_{1},\dots,w_{1}}_{n\text{ copies}},\dots,\underbrace{w_{m},\dots,w_{m}}_{n\text{ copies}}].
\]

Let $(E\to V, \Gamma)$ be a local rank $n$ orbibundle model,
in particular, $E=\mathbb R^n \times V$.
Let $\operatorname{Sym}^m (E)=\operatorname{Sym}^m(\mathbb R^n) \times V$.
Then $\operatorname{Sym}^m(E)\to V$ is a fiber bundle with an equivariant $\Gamma$-action.

\begin{defn}[Multisections of a local model]
A {\em degree $m$ multisection (or $m$-multisection) of the local orbibundle model $(E\to V,\Gamma)$}
is a $\Gamma$-equivariant section of $\operatorname{Sym}^m(E)\to V$.
\end{defn}

\begin{defn}
An $m$-multisection $\mathbf s$ of $(E\to V,\Gamma)$ is {\em liftable} if there exists a section $\widetilde {\mathbf s}=(s_1,s_2,\dots,s_m):V\to E^m$ which is not necessarily $\Gamma$-equivariant, such that $\mathbf s=p \circ \widetilde{\mathbf s}$, where $E^m=\mathbb R^{mn}\times V $ and $p:E^m\to \operatorname{Sym}^m(E)$ is the quotient map. We call each such $s_i$ a {\em branch of $\mathbf s$}.
\end{defn}

We assign the weight ${1\over m}$ to each branch $s_i$ of ${\bf s}$, if ${\bf s}$ is a degree $m$ multisection.

\begin{defn}
A liftable $m$-multisection $\mathbf s$ of $(E\to V,\Gamma)$ and a liftable $n$-multisection $\mathbf t$ of $(E\to V,\Gamma)$ are {\em transverse} (denoted by $\mathbf s \pitchfork \mathbf t$) if $s_i$ is transverse to $t_j$ for all $1\leq i \leq m$ and $1\leq j \leq n$.
\end{defn}

In the above definition, for each $i\in\{1,\dots,m\}$ and $j\in\{1,\dots,n\}$, the space $s_i^{-1}(t_j)$ is a submanifold of $V$ (it may or may not invariant under $\Gamma$),
and we assign the weight ${1\over mn}$ to it.
In this way, $\mathbf s^{-1}(\mathbf t):=\cup_{i,j} s_i^{-1}(s_j)$ is an immersed weighted branched submanifold of $V$, and it is invariant under action of $\Gamma$.

An $m$-multisection of $(E\to V,\Gamma)$ can be mapped to an $\ell m$-multisection of $(E\to V,\Gamma)$ by postcomposing with $\iota^\ell$, which is interpreted as acting on each fiber separately.

\begin{defn}
An $\ell$-multisection $\mathbf s$ and an $m$-multisection $\mathbf s'$ of $(E\to V, \Gamma)$ are said to be {\em equivalent} if $\iota^m \circ \mathbf s=\iota^\ell \circ \mathbf s'$.
\end{defn}

If ${\bf s}$ and ${\bf s}'$ are equivalent and ${\bf s}\pitchfork {\bf t}$, then $\mathbf s^{-1}(\mathbf t)$ and $\mathbf s^{'-1}(\mathbf t)$ represent the same weighted branched immersed submanifold of $V$.

\begin{defn}[Multisections of an orbibundle] \label{defn: multisections}
A {\em multisection $\mathbf s$ of an orbibundle $\mathbb E \to \mathbb X$} is a family $\{(E_i\to V_i,\Gamma_i,\widetilde\phi_i,\mathbf s_i)\}_{i\in\mathcal{I}}$ such that
\begin{enumerate}
\item $\{(E_i\to V_i,\Gamma_i,\widetilde\phi_i)\}_{i\in\mathcal{I}}$ is a family of orbibundle charts of $\mathbb E \to \mathbb X$;
\item $\mathbf s_i$ is an $m_i$-multisection of $(E_i\to V_i,\Gamma_i)$;
\item for any embedding
$$\widetilde\psi_{ji}:(E_i\to V_i,\Gamma_i,\widetilde\phi_i)\hookrightarrow (E_j\to V_j,\Gamma_j,\widetilde\phi_j)$$
of orbibundle charts that lies over the embedding
$$\psi_{ji}:(V_i,\Gamma_i,\phi_i) \hookrightarrow (V_j,\Gamma_j,\phi_j)$$
of base orbifold charts, the $m_j$-multisection $\mathbf s'_{i}:=({\widetilde\psi_{ji}}^{-1})^{m_j} \circ \mathbf s_j\circ \psi_{ji}$ and the $m_i$-multisection $\mathbf s_i$ of $(E_i\to V_i,\Gamma_i)$ are equivalent.
\end{enumerate}
The multisection ${\bf s}$ is {\em liftable} if each ${\bf s_i}$ is liftable.
\end{defn}

In Definition~\ref{defn: multisections},
$$\widetilde\psi_{ji}^{m_j}:\operatorname{Sym}^{m_j}(E_i)\to \operatorname{Sym}^{m_j}(E_j),$$
$$({\widetilde\psi_{ji}}^{-1})^{m_j}:\operatorname{Sym}^{m_j}(E_j)|_{{\widetilde\psi_{ji}}^{m_j}(\operatorname{Sym}^{m_j}(E_i))}\to \operatorname{Sym}^{m_j}(E_i)$$
are the obvious maps induced by
$$\widetilde\psi_{ji}:E_i\to E_j \quad \mbox{and} \quad \widetilde \psi_{ji}^{-1}:E_j|_{\widetilde \psi_{ji}(E_i)}\to E_i.$$

\begin{defn}[Boundedness of degree]
A multisection $\mathbf s$ of an orbibundle $\mathbb E \to \mathbb X$ {\em has bounded degree} if, for any fixed $(E_j\to V_j, \Gamma_j, \widetilde \phi_j)$, the set of $m_i$ for which there exists an embedding $(E_i\to V_i, \Gamma_i, \widetilde \phi_i)\hookrightarrow (E_j\to V_j, \Gamma_j, \widetilde \phi_j)$ is bounded.
\end{defn}

{\em In this paper, we assume that our multisections have bounded degree and are liftable.}

Let $\mathbf s$ and $\mathbf t$ be two multisections of $\mathbb E\to \mathbb X$ such that $\mathbf s \pitchfork \mathbf t$.
On each orbibundle chart $(E_i\to V_i, \Gamma_i, \widetilde \phi_i)$ of $\mathbb E\to \mathbb X$,
we know $\mathbf s_i^{-1}(t_i)$ is a weighted branched immersed submanifold of $V_i$ that is invariant under the action of $\Gamma_i$,
and we define $\mathbf s^{-1}(\mathbf t)$ to be the weighted branched immersed suborbifold of $\mathbb X$ obtained by patching together $\{\mathbf s_i^{-1}(\mathbf t_i)\}_{i\in \mathcal I}$.

\begin{defn}
A sequence of multisections $\{\mathbf s^n\}_{n\in \mathbb{N}}$ of $\mathbb E\to \mathbb X$ {\em converges to a multisection $\mathbf s$ of $\mathbb E \to \mathbb X$ in the $C^k$-topology} (where $k$ may be $\infty$), if for each fixed orbibundle chart $(E_i\to V_i,\Gamma_i,\widetilde\phi_i)$ of $\mathbb E\to \mathbb X$,
there exist $\ell_i$-multisections $\{\mathbf s'^n_i\}_{n\in \mathbb{N}}$ and $\mathbf s'_i$ of $(E_i\to V_i,\Gamma_i,\widetilde\phi_i)$ such that
$\mathbf s'^n_i$ is equivalent to $\mathbf s^n_i$, $\mathbf s'_i$ is equivalent to $\mathbf s_i$, and $\mathbf s'^n_i$ converges  to $\mathbf s'^n_i$ in the $C^k$-topology.
\end{defn}

The following is proved in \cite[Theorem 3.11]{FO}.

\begin{lemma}[Multisection perturbation lemma] \label{lemma: multisection perturbation lemma}
Let $\mathbf s,\mathbf t$ be two multisections of an orbibundle $\mathbb E \to \mathbb X=(X,\mathcal{O})$. If the base $X$ is compact, then there exists a sequence $\{\mathbf s^n\}_{n\in \mathbb{N}}$ of multisections of $\mathbb E \to \mathbb X$ such that $\{\mathbf s^n\}_{n\in\mathbb{N}}$ converges to $\mathbf s$ in the $C^\infty$-topology and $\mathbf s^n$ is transverse to  $\mathbf t$.
\end{lemma}

\section{Almost complex structures and moduli spaces} \label{section: almost complex structures and moduli spaces}

Let $(M^{2n+1},\xi)$ be a contact manifold and $\alpha$ a nondegenerate contact form. Denote the Reeb vector field of $\alpha$ by $\mathbf R_\alpha$ and the set of closed orbits $\gamma$'s of $\mathbf R_\alpha$ with action $\mathcal A(\gamma)\leq L$ by $\mathcal P_\alpha^L$.

{\em The discussion until the end of Section~\ref{section: gluing} is valid for curves of any genus.}

\subsection{Almost complex structures} \label{subsection: almost complex structures}

We will review some notions from \cite{BH}.

\begin{defn} \label{Lsimple form}
A  contact form $\alpha$ is called {\em $L$-simple}\footnote{In \cite{BH} there is a notion called ``$L$-supersimple", the difference between ``$L$-simple" and ``$L$-supersimple" is that ``$L$-simple" allows the existence of elliptic orbits of action $\leq L$.}
if each simple orbit $\gamma\in \mathcal P_\alpha^L$ of $\mathbf R_\alpha$ is nondegenerate and around $\gamma$ there exists an embedding
$$\Phi_\gamma:\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} \times D_\delta \to M,$$
for some small $\delta>0,$ where
$$D_\delta=\Big\{(x_1,\dots,x_n,y_1,\dots,y_n)\in \mathbb{R}^{2n} ~|~\sum_i x_i^2+y_i^2\leq \delta^2 \Big\},$$ such that
$$\gamma=\Phi_\gamma(\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} \times\{0\})$$ and
$$\alpha=(c(\gamma)+Q)dt+\sum_{i=1}^n (x_idy_i-y_idx_i).$$
Here $c(\gamma)>0$ is a constant, $t$ is the coordinate for $\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z}$, and $Q$ is a quadratic polynomial in $(x_1,\dots,x_n,y_1,\dots,y_n)$.
\end{defn}

If we denote the Hamiltonian vector field of $H:=c(\gamma)+Q$ on $D_\delta$ with respect to the symplectic form $\sum_i dx_i\wedge dy_i$ by $X_H$,
then the vector field $\partial_t+X_H\in \ker d\alpha$. Hence there exists a positive function $g$ on $\Phi_\gamma(\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} \times D_\delta)$ such that
\begin{itemize}
\item[(g)] $g \mathbf R_\alpha=\partial_t+X_H$.
\end{itemize}
Since $Q$ is quadratic, $X_H$ depends linearly on $(x_1,\dots,x_n,y_1,\dots,y_n)$, i.e.,
$$X_H=X(x_1,\dots,x_n,y_1,\dots,y_n)^T,$$
for some $X\in M_{2n\times 2n}(\mathbb{R})$.

\begin{defn}
Given a contact form $\alpha$, an almost complex structure $J$ on $\mathbb{R} \times M$ is {\em $\alpha$-tame} if
\begin{enumerate}
\item $J$ is independent of $s$, where $s$ is the $\mathbb{R}$-coordinate;
\item $J(\partial_s)=g\mathbf R_\alpha$ for some positive function $g$ on $M$; and
\item there exists a $2n$-plane field $\xi'$ on $M$ such that $J\xi'=\xi'$ and $d\alpha(v,Jv)>0$ for all nonzero $v\in \xi'$.
\end{enumerate}
\end{defn}

Given a Reeb orbit $\gamma$ of $\mathbf R_\alpha$, we write $\gamma^s$ for the simple orbit underlying $\gamma$ and $m(\gamma)$ for the multiplicity of $\gamma$ over $\gamma^s$. Choose a point $x_{\gamma^s}$ (a {\em marker}) on each $\gamma^s$.

\begin{defn}[$L$-simple for a symplectization] \label{defn: L-supersimple 1} Let $L>0$ and $\alpha$ be an $L$-simple contact form.

(1) An almost complex structure $J$ on $\mathbb{R}\times M$ is {\em $L$-simple for $\alpha$} if $J$ is $\alpha$-tame and inside a sufficiently small neighborhood $\Phi_\gamma(\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} \times D_\delta)$ (as given in Definition \ref{Lsimple form}) of each simple Reeb orbit $\gamma \in \mathcal P_\alpha^L$ we have
\begin{enumerate}
\item[(i)] $\xi'=\mathbb{R}\langle\frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1},\cdots, \frac{\partial}{\partial y_n}\rangle$;
\item[(ii)] $J: \frac{\partial}{\partial x_i} \mapsto \frac{\partial}{\partial y_i}$ for $1\leq i \leq n$; and
\item[(iii)] the function $g$ satisfying $J(\partial_s)=g\mathbf R_\alpha$ is given by (g) above.
\end{enumerate}
A pair $(\alpha,J)$ is said to be {\em $L$-simple pair}, if $\alpha$ is $L$-simple and $J$ is $L$-simple for $\alpha$.

(2) If $\gamma$ is simple, then the above coordinates
$$(t,x_1,\dots,x_n,y_1,\dots,y_n),$$
subject to the normalization
\begin{itemize}
\item[(N)] $\Phi_\gamma(0,0)=x_{\gamma^s}$.
\end{itemize}
are called {\em simple coordinates for $\gamma$}.

(3) If $\gamma$ is not simple, then, by abuse of notation, the coordinates
$$(t,x_1,\dots,x_n,y_1,\dots,y_n)$$
on $\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} \times D$ are called {\em simple coordinates for $\gamma$} where
$$\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} \times D \to \mathbb{R}/\mathcal{A}_\alpha(\gamma^s)\mathbb{Z}\times D$$
is the $m(\gamma)$-fold covering map.
\end{defn}

Given an $L$-simple $\alpha$, we can construct an $L$-simple $J$ as follows: Let $\xi'\subset TM$ be a $2n$-plane field such that $\xi'=\mathbb{R}\langle\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1},\dots, \frac{\partial}{\partial y_n}\rangle$ on a small neighborhood $V_\gamma$ of each $\gamma\in \mathcal P_\alpha^L$; $\xi'=\xi$ outside a slightly larger neighborhood $V'_\gamma$ of each $\gamma\in \mathcal P_\alpha^L$; and $\xi'$ interpolates between $\mathbb{R}\langle\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_n}, \frac{\partial}{\partial y_1},\dots, \frac{\partial}{\partial y_n}\rangle$ and $\xi$ on the remaining annular regions.  Let $J$ be a complex structure of $\xi'$ such that $d\alpha(v,Jv)>0$ for all $0\neq v\in \xi'$ and $J(\frac{\partial}{\partial x_i})= \frac{\partial}{\partial y_i}$ for $1\leq i \leq n$ on $V_\gamma$. We then extend $J$ to $\mathbb R\times M$ by $J(\partial_s)=g\mathbf R_\alpha$, where $g$ is given by (g) on $V_\gamma$, $g=1$ outside $V'_\gamma$, and $g$ interpolates between then two in the remaining annular regions.

The main reason for using $L$-simple $J$ is the following: Let $u$ be a $J$-holomorphic curve in $\mathbb{R} \times M$ and suppose $u$ has an end which converges to $\mathbb{R} \times \gamma$ with coordinates $(s,t)$. Then, with respect to the supersimple coordinates, this end of $u$ admits a lift to $\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} \times D$ which can be written as $(s,t,\eta(s,t))$, where $\eta$ satisfies
$$\frac{\partial \eta }{\partial s}+j_0\frac{\partial \eta}{\partial t}+S\eta=0,$$
$j_0$ is the standard complex structure, and $S:=-j_0X$ is a $2n\times 2n$ constant symmetric matrix. In other words, {\em the Cauchy-Riemann equations become linear near the ends for curves that are graphical over $\mathbb{R}\times \gamma$.}

\begin{convention}
We are using the convention that when a matrix (e.g., $S$) acts on $\eta$ or its derivatives from the left, $\eta$ or its derivatives are viewed as column vectors.
\end{convention}

More generally, let $(W,\alpha)$ be a $(2n+2)$-dimensional compact exact symplectic cobordism from $(M_+,\alpha_+)$ to $(M_-,\alpha_-)$, i.e., $\partial W= M_+-M_-$, $\alpha$ is a $1$-form on $W$ such that $d\alpha$ is symplectic, and $\alpha_\pm=\alpha|_{M_\pm}$ is a contact form on $M_\pm$.  Let $(\widehat{W},\widehat\alpha)$ be the completion of $(W,\alpha)$, obtained by smoothly attaching the symplectization ends $[1,\infty)\times M_+$ and $(-\infty,-1]\times M_-$.  Let $s$ be the $[1,\infty)$- or $(-\infty,-1]$-coordinate at the positive and negative ends of $\widehat{W}$.

\begin{defn}[$(L_+,L_-)$-simple for a cobordism]
A contact form $\widehat \alpha$ on $\widehat W$ is said to be {\em $(L_+,L_-)$-simple} if it restricts to $L_\pm$-simple contact form $\alpha_\pm$ at the positive and negative ends.
Given an $(L_+,L_-)$-simple $\widehat \alpha$, an almost complex structure $J$ on $\widehat{W}$ said to be  {\em $(L_+,L_-)$-simple for $\widehat \alpha$},
if it tames $\widehat{\alpha}$ and restricts to $J_\pm$ which is $L_\pm$-simple for $\alpha_\pm$ at the positive and negative ends.
A pair $(\widehat \alpha, J)$ is said to be $(L_+,L_-)$-simple, if $\widehat \alpha$ is $(L_+,L_-)$-simple and $J$ is $(L_+,L_-)$-simple for $\widehat \alpha$.
\end{defn}

{\em In this paper we will assume all almost complex structures $J$'s in symplectizations (resp. cobordisms) are $L$-simple (resp. $(L_+,L_-)$-simple)  for some appropriate $L$ (resp. $(L_+,L_-)$) and some appropriate form $\alpha$ (resp. $\widehat \alpha$). }

\begin{rmk}
Strictly speaking, it is not necessary to use simple $J$ in this paper, as most of our discussion carries over to the general case.  Simple almost complex structures allow for better control of the ends and also simplify gluing.
\end{rmk}

\subsection{Riemann surfaces and holomorphic maps}

Let $(F,j)$ be a closed connected Riemann surface.

For much of this paper we do not need to assume that the genus $g(F)$ of $F$ is zero.

\begin{defn}
A {\em marked Riemann surface}
$$\mathcal{F}=(F,j,\mathbf{p},\mathbf{r})$$
is a quadruple which additionally consists of an {\em ordered} tuple $\mathbf{p}=\mathbf{p}_+ \sqcup \mathbf{p}_-$, $\mathbf{p}_\pm=(p_{\pm,1},\dots,p_{\pm,l_\pm})$, of marked points on $F$, where each point of $F$ is used at most once, and an {\em ordered} tuple $\mathbf{r} = \mathbf{r}_+ \sqcup \mathbf{r}_-$, $\mathbf{r}_\pm=(r_{\pm,1},\dots,r_{\pm,l_\pm})$ of asymptotic markers.
Here an {\em asymptotic marker} $r_{\pm,i}$ at a puncture $p_{\pm,i}$ is an element of $(T_{p_{\pm,i}} F-\{0\})/\mathbb{R}^+$.
We denote $\dot F=F-\mathbf{p}$.
\end{defn}

\begin{defn}[Equivalence relation $\sim_0$] \label{defn: equiv relation sim zero}
Given two marked Riemann surfaces $\mathcal{F}=(F,j,\mathbf{p},\mathbf{r})$ and $\mathcal{F}'=(F',j',\mathbf{p}',\mathbf{r}')$,  we write $\mathcal{F}\sim_0\mathcal{F}'$ if there is a diffeomorphism $\phi:F\stackrel\sim \to F'$ satisfying $\phi_*(\mathcal{F})=\mathcal{F}'$ (that is, $\phi_{*}j=j'$, $\phi(\mathbf{p}_\pm)=\mathbf{p}'_\pm,$ and $\phi_{*}(\mathbf{r}_\pm)=\mathbf{r}'_\pm$, where the latter two maps are maps of ordered tuples).
\end{defn}

Let $(W,\alpha)$ be the exact symplectic cobordism from Section~\ref{subsection: almost complex structures}.

\begin{defn}[Equivalence relation $\sim$]\label{defn: equiv relation sim}
Given marked Riemann surfaces $\mathcal{F}=(F,j,\mathbf{p},\mathbf{r})$, $\mathcal{F}'=(F',j',\mathbf{p}',\mathbf{r}')$ and maps $u: \dot F\to \widehat{W}$, $u': \dot F'\to \widehat{W}$, we write $(\mathcal{F},u)\sim (\mathcal{F}',u')$ if there is a diffeomorphism $\phi :  F\stackrel\sim \to F'$ satisfying $\phi_*(\mathcal{F},u)=(\mathcal{F}',u')$ (that is, $\phi_{*}j=j'$, $\phi(\mathbf{p}_\pm)=\mathbf{p}'_\pm$ and $\phi_{*}(\mathbf{r}_\pm)=\mathbf{r}'_\pm$, again as ordered tuples, and $u' \circ \phi=u$).
\end{defn}

\begin{defn}[Equivalence classes]
The equivalence class of $(\mathcal{F},u)$ under $\sim$ will be denoted by $[\mathcal{F},u]$. If $\widehat{W}=\mathbb{R}\times M$, then the equivalence class of $(\mathcal{F},u)$ under $\sim$ and $\mathbb{R}$-translations will be denoted by $\llbracket \mathcal{F},u\rrbracket$.
\end{defn}

Let $\boldsymbol\gamma_+=(\gamma_{+,1},\dots,\gamma_{+,l_+})$ and $\boldsymbol\gamma_-=(\gamma_{-,1},\dots,\gamma_{-,l_-})$ be ordered tuples of Reeb orbits for $\alpha_+$ and $\alpha_-$.  Then let $\mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ be the space of equivalence classes $[\mathcal{F},u]$ of pairs, where $u$ is a $J$-holomorphic map $\dot F \to \widehat{W}$ of Fredholm index $\operatorname{ind}(u)=k$ from $\boldsymbol\gamma_+$ to $\boldsymbol\gamma_-$ (i.e., $u$ is asymptotic to $\gamma_{+,i}$ at the positive end near $p_{+,i}$ and to $\gamma_{-,i}$ at the negative end near $p_{-,i}$) and $u$ ``maps" the asymptotic markers $r_{\pm,i}$ to the markers $x_{\gamma^s_{\pm,i}}$.

If $\widehat{W}=\mathbb{R}\times M$, then $\mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\mathbb{R}\times M;\boldsymbol\gamma_+;\boldsymbol\gamma_-)/\mathbb{R}$ is the usual quotient of $\mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\mathbb{R}\times M;\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ by $\mathbb{R}$-translations.

\begin{defn}
The {\em $\alpha$-energy} of a map $u$ from $\boldsymbol\gamma_+$ to $\boldsymbol\gamma_-$ is given by $$E_\alpha(u):=\mathcal{A}_\alpha(\boldsymbol\gamma_+)-\mathcal{A}_\alpha(\boldsymbol\gamma_-),$$
where $\mathcal{A}_\alpha(\boldsymbol\gamma_\pm)=\sum_{i=1}^{l_\pm}\mathcal{A}_\alpha(\gamma_{\pm,i})$.
\end{defn}

\subsection{Sorting}

The following definition is useful when keeping track of the combinatorics in the proof of $\partial^2=0$ in Section~\ref{subsubsection: d squared is zero}.

Fix an ordering $\vartheta$ of $\mathcal P^{L}_\alpha$.  For example, if the action spectrum is discrete, we can order the orbits by increasing action.

\begin{defn}
An ordered tuple of Reeb orbits $\boldsymbol \gamma$ is {\em sorted} if
$$\boldsymbol \gamma=(\underbrace{\gamma_{1},\dots, \gamma_{1}}_{i_{1} \text{ copies }},\dots, \underbrace{\gamma_{k},\dots, \gamma_{k}}_{i_{k} \text{ copies }}),$$
where $\gamma_{i} \neq \gamma_{j}$ as long as $i\neq j$.  We also allow $k=0$, i.e., $\boldsymbol \gamma=\varnothing$.

If $\boldsymbol \gamma$ is sorted according to $\vartheta$, then $\boldsymbol \gamma$ is said to be {\em $\vartheta$-sorted}.
\end{defn}

{\em From now on we assume that all the positive (and likewise all the negative) ends of all the moduli spaces are $\vartheta$-sorted, unless stated otherwise.}

\subsection{(Symp) vs (Cob)}  \label{subsection: symp vs cob}

Consider the moduli space
$$\mathcal{M}=\mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\widehat W; \boldsymbol\gamma_+;\boldsymbol\gamma_-).$$
There will always be two cases to consider:
\begin{enumerate}
\item[(Symp)] $(\widehat W=\mathbb{R}\times M,J)$ is $\mathbb{R}$-invariant and is viewed as a symplectization.  This is the case when defining the differential $d$ and proving $d^2=0$. We usually take $\mathcal{M}/\mathbb{R}$.
\item[(Cob)] $(\widehat W,J)$ is viewed as a cobordism.  This is the case when defining chain maps.  We allow $(\widehat W,J)$ to be $\mathbb{R}$-invariant (e.g., when $(\widehat W,J)$ is the ``identity cobordism''). In the case of a cobordism there is no quotienting by $\mathbb{R}$, even if there is an $\mathbb{R}$-action.
\end{enumerate}
We will write $(\widehat W,J)\in \operatorname{(Symp)}$ or $(\widehat W,J)\in\operatorname{(Cob)}$ to indicate whether $(\widehat W,J)$ is viewed as a symplectization or a cobordism. There are two reasons for this distinction:
\begin{itemize}
\item We do not quotient by the $\mathbb{R}$-action for (Cob).
\item Trivial cylinders are unperturbed (i.e., are more or less ignored) when considered in (Symp), but are perturbed (i.e., we construct an obstruction bundle $\mathbb{E}\to \mathbb{V}$ where the trivial cylinder is in $\mathbb{V}$) when considered in (Cob).
\end{itemize}

For the $\operatorname{(Cob)}$ case, we also require $J$ to be

\begin{defn} [$(L_+,L_-)$-end-generic]
Let $J$ be an almost complex structure on $(\widehat W,\widehat \alpha)$ that is $(L_+,L_-)$-simple for $\widehat \alpha$.
We say $J$ is {\em $(L_+,L_-)$-end-generic},
if for all $\boldsymbol\gamma_\pm$ with $\mathcal A_{\alpha_\pm}(\boldsymbol\gamma_\pm)\leq L_\pm$ and all $k\in \mathbb Z$,
there is no $J$-holomorphic curve $(\mathcal{F}, u)\in \mathcal M^{\operatorname{ind}=k}_{J} (\dot F, \widehat W; \boldsymbol\gamma_+, \boldsymbol\gamma_-)$  that agrees with the trivial half cylinder near some puncture of $\dot F$ in the region $\widehat W_{s\geq T}$  or $ \widehat W_{s\leq -T}$, for some $T>1$.
\end{defn}

\begin{rmk}
We require $J$ to be $(L_+,L_-)$-end-generic, because with it we can add additional marked points to stabilize the domain of a $J$-holomorphic cylinder or plane in a more canonical way.
\end{rmk}

\begin{lemma}\label{end generic}
Given an $(L_+,L_-)$-simple pair $(\widehat \alpha, J)$, we can perturb $J$ to $J'$ such that $(\widehat \alpha, J')$ is still $(L_+,L_-)$-simple and $J'$ is $(L_+,L_-)$-end-generic.
\end{lemma}
\begin{proof}[Sketch of proof]
For each simple $\gamma_+\in \mathcal P_{\alpha_+}^{L_+}$,
we first perturb $J$ in $[1,\infty) \times \gamma_+$.

Suppose $\boldsymbol\gamma_+$ is a tuple of Reeb orbits of $\alpha_+$ that has a component $\gamma_+$ and satisfies $\mathcal A_{\alpha_+}(\boldsymbol\gamma_+)\leq L_+$.
For each $\boldsymbol\gamma_-$, we consider $(\mathcal{F}, u)\in \mathcal M(\boldsymbol\gamma_+;\boldsymbol\gamma_-):=\sqcup_k \mathcal M^{\operatorname{ind}=k}(\dot F, \widehat W;\boldsymbol\gamma_+;\boldsymbol\gamma_-)$. Suppose that near a puncture $p_+$ of $\dot F$, $u$ converges to $\gamma_+$.
Then near $p_+$, we have $u(s,t)=(s,t,\eta(s,t))$, where $\eta$ satisfies $$\frac{\partial \eta}{\partial s}+j_0\frac{\partial \eta}{\partial t}+S\eta=0.$$
Define an asymptotic operator $$A:W^{1,2}(S^1,\mathbb R^{2n})\to L^2(S^1,\mathbb R^{2n})$$ by  $$A:=-j_0\frac{\partial}{\partial t}-S.$$ Then $\eta$ satisfies $$\frac{\partial \eta}{\partial s}=A \eta.$$
Let $$\cdots,g_{-2},g_{-1},g_1,g_2,\cdots$$ be the eigenfunctions of $A$ with respect to the eigenvalues $$\cdots \theta_{-2},\theta_{-1},\theta_1,\theta_2, \dots$$ with
$$\cdots\leq \theta_{-2}<\theta_{-1}<0<\theta_1<\theta_2\leq \dots.$$
Therefore, $$\eta=\sum_{i \leq -1}c_i e^{\theta_i s}g_i,$$ where the $c_i$ are constants.
There is a continuous evaluation map $$ev_+:\mathcal M(\boldsymbol\gamma_+;\boldsymbol\gamma_-)\to \mathbb R^N$$ for any $N\in \mathbb N$, defined by $$(\mathcal{F},u)\mapsto (c_{-1},\dots,c_{-N}).$$
Note that $ev_+$ can be continuously extended to $\overline{\mathcal M}:=\bigsqcup \overline{\mathcal M}_{J}(\boldsymbol\gamma_+;\boldsymbol\gamma_-)$,
where the disjoint union is take over all tuples of Reeb orbits $\boldsymbol\gamma_\pm$ with $\mathcal A_{\alpha_\pm}(\boldsymbol\gamma_\pm)\leq L_\pm$ and $\boldsymbol\gamma_+$ having $\gamma_+$ as a component.

Since $\overline{\mathcal M}$ is compact, there exists an $N\gg 0$ such that the point $(0,\dots,0,1)\in \mathbb R^N$ does not lie in  $ ev_+(\overline{\mathcal M})$.
Now we can perturb $J$ in the region $[s_0,s_0+1]\times \gamma_+ \times D_\delta \subset \widehat W$ for some  $s_0\gg0$ and  small $\delta>0$ using the eigenfunction $g_{-N}$ as in the proof of Theorem 6.06 of \cite{BH} and show that there is no $J$-holomorphic curve that converges to $\gamma_+$ near some puncture $p_+$ and agrees with the trivial half cylinder over $\gamma_+$ near $p_+$.

We also need to deal with the case when $\boldsymbol\gamma_+$ has a component $\gamma'_+$ that is a $k$-fold cover of $\gamma_+$.
For this, we take the same perturbation to $J$ using $g_{-N}$, but we need to make $N$ even larger.
Now we explain how we pick such $N$.
As before, we can define an evaluation map $ev_+':\overline{\mathcal M}' \to \mathbb R^{N'}$ using $\gamma'_+$ in lieu of $\gamma_+$. Suppose $(\mathcal{F}, u)$ converges to $\gamma_+$ near puncture $p_+$, and $ev_+(\mathcal{F}, u)=(c_{-1},\dots,c_{-N})$.
Let $(\mathcal{F}',u')$ be a $k$-fold (branched) cover of $(\mathcal{F}, u)$,
and then $$ev_+'(\mathcal{F}', u')=(\underbrace{c_{-1},0,\dots,0}_{k \text{ copies }},\dots,\underbrace{c_{-N},0,\dots,0}_{ k \text{ copies}}),$$
if we choose $N'=kN$.
By picking $N$ large, we can require $$(\!\!\!\!\underbrace{0,\dots,0}_{(N-1)k \text{ copies}}\!\!\!\!\!,1,*,\dots,*)\notin ev_+'(\overline{\mathcal M}').$$
With such choice of $N$ one can follow Theorem 6.06 in \cite{BH} and show that there is no $J$-holomorphic curve that converges to $\gamma_+'$ near some puncture $p_+'$ and agrees with the half trivial cylinder over $\gamma_+'$ near $p_+'$.
We can pick a sufficiently large $N$, such that the above is true for all $k\in \mathbb N$ with $\mathcal A_{\alpha_+} (\gamma_+')\leq L_+$.

Finally, we carry out similar perturbation in the negative end of $\widehat W$.
\end{proof}

{\em In this paper, we always assume $J$ is $(L_+,L_-)$-end-generic for some appropriate $(L_+,L_-)$ when $(\widehat W, J)\in (\operatorname{Cob})$.}
\subsection{Teichm\"uller space and mapping class group}

In what follows we fix $(F,\mathbf{p})$ and hence $\dot F$.

Assume $\dot F$ has negative Euler characteristic.  Let $\operatorname{Teich}(\dot F)$ be the Teichm\"uller space of $\dot F$.  Given $[j]\in \operatorname{Teich}(\dot F)$, where $[j]$ is viewed as an equivalence class of complex structures on $\dot F$, let $\mathcal{U}_{[j]}\subset \operatorname{Teich}(\dot F)$ be an open neighborhood of $[j]$. Given an open set $\mathcal{U}\subset \operatorname{Teich}(\dot F)$, let $\widetilde{\mathcal{U}}$ be a {\em Teichm\"uller slice} over $\mathcal{U}$, i.e., a smooth choice of complex structure on $\dot F$ in each equivalence class $x\in \mathcal{U}$.

We make the following definition which is slightly nonstandard but adapted to our situation:

\begin{defn}
Let $\text{Mod}(F,{\bf p})$ be the quotient of the group $\operatorname{Diff}(F,{\bf p})$ of diffeomorphisms of $F$ which fix ${\bf p}_\pm$ pointwise by the subgroup $\operatorname{Diff}_0(F,{\bf p})$, which is the identity component of $\operatorname{Diff}(F,{\bf p})$.
\end{defn}

It is well-known that $\operatorname{Mod}(F,{\bf p})$ acts properly discontinuously on $\operatorname{Teich}(\dot F)$ and hence its quotient is an orbifold, the moduli space $\mathcal{M}(F,{\bf p})$ of complex structures on $F$ with positive marked points ${\bf p}_+$ and negative marked points ${\bf p}_-$. Let
$$\text{Mod}(F,{\bf p})_{[j]}=\{ [g]\in \text{Mod}(F,{\bf p})~|~ [g]([j])=[j]\}$$
be the stabilizer of $[j]$.  If we choose a representative $j\in[j]$, then it is well-known that the group $\operatorname{Aut}(F,{\bf p}, j)$ of automorphisms of $(F,j)$ which take ${\bf p}_\pm$ to ${\bf p}_\pm$ pointwise satisfies
$$\operatorname{Aut}(F,{\bf p}, j)\simeq \operatorname{Mod}(F,{\bf p})_{[j]}.$$

\begin{defn}
An open set $\mathcal{U}\subset \operatorname{Teich}(\dot F)$ is {\em good} if $\mathcal{U}=\mathcal{U}_{[j]}$ for some $[j]$ and, for any $[g]\in \text{Mod}(F,{\bf p})$, one of the following holds:
\begin{itemize}
\item $[g](\mathcal{U})\cap \mathcal{U}=\emptyset $;
\item $[g]\in \text{Mod}(F,{\bf p})_{[j]}$ and $[g](\mathcal{U})=\mathcal{U}.$
\end{itemize}
\end{defn}

The orbifold charts for $\mathcal{M}(F,{\bf p})$ are
$$\mathcal{U}_{[j]}\to \mathcal{U}_{[j]}/\operatorname{Mod}(F,{\bf p})_{[j]},$$
where $\mathcal{U}_{[j]}$ is a sufficiently small good open set about $[j]$.

Also given $(\mathcal{F},u)=((F,j,{\bf p},{\bf r}),u)$, let $\text{Mod}(F,{\bf p})_{[\mathcal{F},u]}$ be the subgroup of $\text{Mod}(F,{\bf p})$ defined by
$$\text{Mod}(F,{\bf p})_{[\mathcal{F},u]}=\{[g]\in \text{Mod}(F,{\bf p})~|~  g_*(\mathcal{F},u)=(\mathcal{F},u) \mbox{ for some $g\in[g]$}\}.$$

\section{Fredholm theory} \label{section: Fredholm theory}

In this section we fix $(F,{\bf p})$ and hence $\dot F$.

\subsection{Canonical cylindrical coordinates}

Let $(\mathcal{F},u)$ be a pair where $u:\dot F\to  \widehat{W}$ is a sufficiently differentiable map which is asymptotic to $\gamma_{+,i}$ at the positive end near $p_{+,i}$ and to $\gamma_{-,i}$ at the negative end near $p_{-,i}$. If $u$ can be written as a graph $(s,t)\mapsto (s,t,\eta_{\pm,i}(s,t))$ on $\mathbb{R}\times (\mathbb{R}/\mathcal{A}(\gamma_{\pm,i}))\times D$ near the end corresponding to the puncture $p_{\pm,i}\in \mathbf{p}$, then $(s,t)$ is said to be the {\em canonical cylindrical coordinates on $\dot F$ near $p_{\pm,i}$ with respect to $u$}.

By the supersimplicity of $J$, if $[\mathcal{F},u]\in \mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)$, then the canonical cylindrical coordinates for a representative $(\mathcal{F},u)$ near the punctures are holomorphic coordinates on $\dot F$.  On the other hand, $(s,t)$ is not necessarily holomorphic if $u$ is not $J$-holomorphic.

If $\llbracket\mathcal{F},u\rrbracket\in \mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\mathbb{R}\times M;\boldsymbol\gamma_+,\boldsymbol\gamma_-)/\mathbb{R}$, the coordinates $(s,t)$ for $(\mathcal{F},u)$ are canonical up to translations in the $s$-direction.

\subsection{Fredholm setup} \label{subsection: Fredholm setup}

Fix an $L$-supersimple $J$ on $\widehat{W}$ in the class $C^\infty$.

Recall the coordinate $s$ at the ends of $\widehat{W}$. Fix a Riemannian metric $g_0$ on $\widehat{W}$ which is $s$-invariant at the ends. For simplicity we assume that $g_0$ restricts to the standard flat metric
$$g_0=ds^2+dt^2+\sum_idx_i^2+\sum_i dy_i^2$$
on the $L$-supersimple coordinate charts (and hence $g_0=g_0^L$, i.e., depends on $L$). Let $\exp: T\widehat W \to \widehat{W}$ be the exponential map with respect to $g_0$, $\epsilon>0$ be a constant smaller than the injectivity radius of $g_0$, and
$$D_\epsilon\subset T\widehat W= \{(w,\xi)~|~w\in W, |\xi|_{g_0}< \epsilon\}$$
be the $\epsilon$-disk bundle of $T\widehat{W}$.

{\em For the rest of the paper we assume that $\boldsymbol\gamma_+$ satisfies $\mathcal{A}_\alpha(\boldsymbol\gamma_+)<L$, unless indicated otherwise.}

Let $\delta>0$ be a small positive number.  We define the ``weighted Sobolev space'' $W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)$, following Dragnev~\cite[Definition 2.6]{Dr}:

\begin{defn}[Weighted Sobolev space of maps $\dot F\to\widehat{W}$]
Let $\mathcal{C}$ be the space of smooth maps $u:\dot F\to \widehat{W}$ that agree with holomorphic maps parametrizing trivial holomorphic half-cylinders $[1,+\infty)\times \gamma_{+,i}$ near $p_{+,i}$ and to $(-\infty,-1]\times\gamma_{-,i}$ near $p_{-,i}$.
Then we define the {\em weighted Sobolev space of maps $\dot F\to\widehat{W}$} by
$$W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-):=\{\exp(u,\xi)~|~u\in \mathcal{C},\xi\in W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)\},$$
where $W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)$ is the usual weighted Sobolev space of sections of $u^*D_\epsilon$, defined using $g_0$.
\end{defn}

Here in the definition of $W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)$ we are using a smooth weight function $f_\delta:\widehat{W}\to \mathbb{R}^+$ which agrees with $e^{\delta |s|}$ at the ends of $\widehat{W}$. Note that the Sobolev norm depends on $g_0$ and $f_\delta$, but the topology on this Sobolev space does not depend on the particular choices of $g_0$ and $f_\delta$.

\begin{rmk}[Remark on values of $k$ and $p$]
By the Sobolev embedding theorem, if $k>r+{m\over p}$, then $W^{k,p}(\Omega)\subset C^r(\Omega)$ for any compact domain $\Omega$ of $\mathbb{R}^m$ with smooth boundary; see Remark~\ref{rmk:sobolev embedding} for obtaining the Sobolev embedding theorem for $\dot F$.  In our case $m=\dim \dot F= 2$ and we take $r\geq 1$, since we want at least one continuous derivative (this will be used for defining $s_{\pm,i}$ in Step 1B in the proof of Theorem~\ref{thm: construction of L-transverse subbundle}). Hence from now on we assume that  $k> r+1$ and $p= 2$.
\end{rmk}

The following is well-known:

\begin{rmk} \label{rmk:sobolev embedding}
We can use the Sobolev embedding theorem for compact domains to show that there exists a global constant $C>0$ such that for all
$$\xi\in W_\delta^{k+1,p}(\dot F, u^*D_\epsilon)$$
we have
\begin{equation}
|f_\delta\xi|_{C^r}\leq C\|f_\delta\xi\|_{W^{k+1,p}}=C \| \xi\|_{W^{k+1,p}_\delta}.
\end{equation}
Suppose for simplicity that $\dot F$ has only one end and $\dot F$ is written as the union $\Omega_0\cup\Omega_1\cup\Omega_2\dots$ of compact domains, where each $\Omega_i\cap \Omega_{i+1}$ is a circle, $\Omega_0$ is the ``thick part'', and $\Omega_i$, $i>0$, is biholomorphic to the standard $S^1\times[0,1]$.  Then we apply the Sobolev embedding theorem for each $\Omega_i$, noting that
$$\| \xi\|_{W^{k+1,p}_\delta(\Omega_i)}\leq \| \xi\|_{W^{k+1,p}_\delta(\dot F)}$$
and that the Sobolev spaces $W^{k+1,p}_\delta(\Omega_i)$ and $W^{k+1,p}_\delta(\Omega_{i+1})$ can be identified for $i\gg 0$ because $u^*D_\epsilon$ is asymptotically cylindrical at the ends of $\dot F$.  Hence the same constant $C>0$ can be used when we apply the Sobolev embedding theorem to all $\Omega_i$.
\end{rmk}

Let $W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W})$ be the usual weighted Sobolev space of sections of
$$\wedge^{0,1}u^*T\widehat{W}=\wedge^{0,1}\dot F \otimes_J u^{*}T\widehat W\to\dot F$$
in the class $(k,p)$ with weight $f_\delta$.

\begin{rmk}
In the functional analysis setup we use $W^{k+1,p}_\delta$ and $W^{k,p}_\delta$ spaces for large $k$ to extract a collection of finite-dimensional orbibundles $\mathbb{E}\to \mathbb{V}$. Once this is done we throw away the Sobolev space setup and work in the category of finite-dimensional orbifolds.  (This is analogous to the \cite{FO} and \cite{FO3} setup and differs from that of \cite{HWZ3}.)
\end{rmk}

\subsubsection{$\chi(\dot F)<0$} \label{subsubsection: negative Euler char}

We first treat the case where $\dot F$ has negative Euler characteristic.
Given a Teichm\"uller slice $\widetilde{\mathcal{U}}$ of $\mathcal{U}$, define
\begin{align*}
\mathcal{B}_{\widetilde{\mathcal{U}}}= &\mathcal{B}_{\widetilde{\mathcal{U}}}(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\\
=&\left\{ (\mathcal{F},u)=((F,j,\mathbf{p},\mathbf{r}),u) \left|
\begin{array}{cc}
j\in \widetilde{\mathcal{U}},u\in W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-), \\  u(r_{\pm, i})=x_{\gamma_{\pm,i}^s}
\end{array}
\right.\right\},
\end{align*}
where $(F,\mathbf{p})$ are fixed.
We define the bundle
\begin{equation}
\pi_{\widetilde{\mathcal{U}}}:\mathcal{E}_{\widetilde{\mathcal{U}}}=\mathcal{E}_{\widetilde{\mathcal{U}}}(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\to \mathcal{B}_{\widetilde{\mathcal{U}}},
\end{equation}
whose fiber over $(\mathcal{F},u)$ is
$$(\mathcal{E}_{\widetilde{\mathcal{U}}})_{(\mathcal{F},u)} = W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W}).$$
Then $\overline{\partial}_J^{\widetilde{\mathcal{U}}}$ is a section of $\mathcal{E}_{\widetilde{\mathcal{U}}}$ defined by
$$\overline{\partial}_J^{\widetilde{\mathcal{U}}} (\mathcal{F}, u)=(\mathcal{F},u,\tfrac{1}{2}(du+J(u)du\circ j)).$$
Also let
\begin{equation} \label{eqn: proj to U}
\Pi_{\widetilde{\mathcal{U}}}: \mathcal{B}_{\widetilde{\mathcal{U}}} \to \widetilde{\mathcal{U}}, \quad (\mathcal{F},u)=((F,j,{\bf p},{\bf r}),u)\mapsto j
\end{equation}
be the projection to $\widetilde{\mathcal{U}}$.
{\em When the Teichm\"uller slice $\widetilde{\mathcal{U}}$ is understood, it will often be omitted from the notation, e.g., $\overline{\partial}_J$, $\mathcal{B}$, $\mathcal{E}$.}

The following is immediate:

\begin{lemma}
If $\mathcal{U}=\mathcal{U}_{[j]}$ is a good open set, then
$$\mathcal{M}_J(\dot F,\widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\cap (\mathcal{B}_{\widetilde{\mathcal{U}}}/\sim ) =(\overline{\partial}_J^{\widetilde{\mathcal{U}}})^{-1}(0)/\operatorname{Mod}(F,{\bf p})_{[j]}$$
as topological spaces (but not necessarily as orbifolds).
\end{lemma}

Let $\nabla$ be the Levi-Civita connection on $T\widehat W$ with respect to the metric $g_0$.  Let $L_{(\mathcal{F},u)}$ be the differential
$$(\overline{\partial}_J)_*: T_{(\mathcal{F},u)} \mathcal{B}\to T_{(\mathcal{F},u,\overline\partial_J u)}\mathcal{E}$$
postcomposed with the projection to $W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W}).$  Then $L_{(\mathcal{F},u)}$ can be written as:
\begin{equation} \label{eqn: linearization}
L_{(\mathcal{F},u)}: T\widetilde{\mathcal{U}}(j)\oplus\mathbb{R}^{2(l_++l_-)}\oplus W^{k+1,p}_\delta (\dot F,u^{*}T\widehat{W}) \to W^{k,p}_\delta (\dot F,\wedge ^{0,1} u^{*}T\widehat{W}),
\end{equation}
$$(\mathfrak{j},\mathfrak{a},\xi)\mapsto Y(\mathfrak{j})+D_u(\mathfrak{a}+\xi),$$
where $Y(\mathfrak{j})$ is induced from the variation of $j$ in the direction on $\mathfrak{j}$, $D_u$ is the usual linearized $\overline\partial_J$-operator (which depends on the choice of $\nabla$ if $u$ is not $J$-holomorphic), and $\mathfrak{a}$ is obtained by damping out $\partial_s$ and $\partial_t$ at each end (away from $\pm \infty$).
Here $l_++l_-$ is the total number of punctures of $\dot F$.

\subsubsection{$\chi(\dot F)=0$ or $1$} \label{subsubsection: nonnegative Euler char}

We now consider the case where $\chi(\dot F)=0$ or $1$.

Consider $(\mathcal{F},u)$ where $u:(\dot F,j)\to (\widehat{W},J)$ is a finite energy holomorphic map. We assume that $u$ is not a trivial cylinder $\mathbb{R}\times \gamma$ over a Reeb orbit $\gamma$. (In our approach, we ignore trivial cylinders, since they will automatically be placed as ``necks'' during gluing.  The only case which requires further analysis is when $\widehat{W}=\mathbb{R}\times M$, viewed as a cobordism with an $\mathbb{R}$-invariant almost complex structure.) By Euler characteristic reasons, $u$ is also not a branched cover of a trivial cylinder and $E_\alpha(u)>0$.

Consider the positive end asymptotic to $\gamma\in \boldsymbol\gamma_+$ (actually there is only one positive end since $\dot F$ is a cylinder or a plane). Then on this end $u$ is graphical, i.e., can be written in canonical cylindrical coordinates on $\dot F$ and supersimple coordinates on the target as
$$u(s,t)=(s,t,\eta(s,t)),$$
after lifting to the $m(\gamma)$-fold cover $\mathbb{R}\times (\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z})\times D$. We assume that the ray $\{s\in [R,\infty), t=0\}\subset \dot F$ corresponds to the asymptotic marker $r_{+}$ which ``maps'' to $\gamma$.

For each representative of $(\mathcal{F}_0,u_0)$ of each $[\mathcal{F}_0,u_0]\in \mathcal{M}_J(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$, we consider a small neighborhood $\mathcal{B}'_{(\mathcal{F}_0,u_0)}$ of $(\mathcal{F}_0,u_0)$ inside
\begin{align*}
\mathcal{B}'(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)=&\left\{ (\mathcal{F},u)=((F,\mathbf{p},\mathbf{r}),u) \left|
\begin{array}{cc}
u\in W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-), \\  u(r_{\pm, i})=x_{\gamma_{\pm,i}^s}
\end{array}
\right.\right\}
\end{align*}
and define a map
$$I: \mathcal B'_{(\mathcal F_0, u_0)} \times [T',+\infty) \to \mathbb R$$
by
$$ I((\mathcal F, u),s)=\int_{s}^{+\infty} \int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds',$$
where $T'$ is a constant such that we can write $u(s,t)=(s,t,\eta(s,t))$ for any $(\mathcal F, u)\in \mathcal B'_{(\mathcal F_0, u_0)}$ and $s\geq T'$.
Clearly, $I$ is independent of $(\mathcal F_0, u_0)$, and $I((\mathcal F, u),\cdot)$ is strictly increasing if $u$ is not a (branched cover of) a trivial cylinder.

\begin{lemma}\label{shenyang}
$I$ is a $C^1$-map.
\end{lemma}

Lemma~\ref{shenyang} will be proved in the Appendix.

\begin{lemma}\label{tenren}
In the $\operatorname{(Symp)}$ case, suppose $\boldsymbol\gamma_+\neq \boldsymbol\gamma_-$. There exists $\varepsilon'>0$ small such that for each representative of $(\mathcal{F},u)$ of each $[\mathcal{F},u]\in \mathcal{M}_J(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ there exists a value
\begin{align*}
s'_+(\mathcal{F},u) :=  \max\left\{ {s\in \mathbb R}\left|
\int_s^{+\infty}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'.
\right.\right\}.
\end{align*}
Moreover, if $u_T$ is $u$ translated up by $s=T$ units, then
$$s'_+(\mathcal{F},u_T)= s'_+(\mathcal{F},u)+T.$$
The same holds for $(\mathcal{F}',u')$ which is $C^1$-close to some representative $(\mathcal{F},u)$ of $\mathcal{M}_J(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$.
\end{lemma}

\begin{proof}
Arguing by contradiction, suppose there exist sequences $(\mathcal{F}_i,u_i)$ and $\varepsilon'_i\to 0$, $i\in \mathbb{N}$, such that:
\begin{itemize}
\item $(\mathcal{F}_i,u_i)$ represents $[\mathcal{F}_i,u_i]\in\mathcal{M}_J(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$; and
\item $s'_+(\mathcal{F}_i,u_i)$ does not exist for $\varepsilon_i'>0$.
\end{itemize}
Then one can verify that $(\mathcal{F}_i,u_i)$ limits to a possible branched cover of a trivial cylinder over a Reeb orbit.  In the (Symp) case a finite energy $J$-holomorphic map that is close to a possible branched cover of a trivial cylinder must be a possible branched cover of a trivial cylinder. This is impossible, because that $\boldsymbol\gamma_+\neq \boldsymbol\gamma_-$ and $\dot F$ is connected imply that $\mathcal{M}_J(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ contains no possible branched cover of a trivial cylinder.  The second and third assertions are immediate.
\end{proof}

\begin{lemma} \label{tenren2}
In the $\operatorname{(Cob)}$ case, there exist a small compactly supported perturbation $J'$ of $J$ and $\varepsilon'>0$ small such that for each representative of $(\mathcal{F},u)$ of each $[\mathcal{F},u]\in \mathcal{M}_{J'}(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ there exists a value
$$s'_+(\mathcal{F},u):= \max\Big\{ {s\in [1,+\infty)}~\Big|~\int_s^{+\infty}\hspace{-0.3cm}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'\Big\}.$$
The same holds for $(\mathcal{F}',u')$ which is $C^1$-close to some representative $(\mathcal{F},u)$ of $\mathcal{M}_{J'}(\dot F,\widehat{W};\boldsymbol\gamma_+;\boldsymbol\gamma_-)$.
\end{lemma}

\begin{proof}
We argue by contradiction as in Lemma~\ref{tenren}. We only need to ensure that there are no holomorphic cylinders of $(\widehat{W},J')$ that agree with trivial half-cylinders at the positive end of $\widehat{W}$. This is done by taking an end-generic perturbation $J'$ of $J$ given in Lemma \ref{end generic}.
\end{proof}

From now on we assume that in the (Cob) case $J$ is sufficiently generic so that the conclusion of Lemma~\ref{tenren2} holds.

We now define the unordered set ${\bf q}$ of punctures:
\begin{enumerate}
\item[(*)] Identify $u^{-1}(\{s=s'_+(\mathcal{F},u)\})\simeq \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z}$ by projecting to the $t$-coordinate, where we are still using supersimple coordinates for $\gamma$. Pick the point $t=0\in \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z}$ corresponding to the asymptotic marker and then define
    $${\bf q}=\{(s,t)=(s'_+(\mathcal{F},u),\tfrac{k}{2m(\gamma)}\mathcal A_\alpha(\gamma))~|~k=1,\dots,2m(\gamma)\}.$$
\end{enumerate}
Let us write $\ddot F=\dot F-{\bf q}$; since the cardinality of $\bf q$ is $2m(\gamma)$, we have $\chi(\ddot F)<0$.

\vskip.1in\noindent
{\em Change of notation.} From now on, we write
$$(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u),$$
with the understanding that ${\bf q}=\varnothing$ when $\chi(\dot F)< 0$, ${\bf q}$ is an unordered set, ${\bf p}$ and ${\bf q}$ are disjoint, and ${\bf q}$ are removable punctures.  We modify Definitions~\ref{defn: equiv relation sim zero} and \ref{defn: equiv relation sim} so that the equivalence relations
$$\mathcal{F}=(F,j,{\bf p},{\bf q},{\bf r})\sim_0 \mathcal{F}'=(F,j',{\bf p},{\bf q}',{\bf r}')$$
and $(\mathcal{F},u)\sim (\mathcal{F}',u')$ require $\phi({\bf q})={\bf q}'$ as ordered sets in addition. Also we modify the definitions of $\operatorname{Mod}(F,{\bf p})$ and $\operatorname{Aut}(F,{\bf p},{\bf q},j)$ to $\operatorname{Mod}(F,{\bf p},{\bf q})$ and $\operatorname{Aut}(F,{\bf p},{\bf q},j)$ so that the diffeomorphism of $F$ takes ${\bf q}$ to ${\bf q}$ setwise in addition.

\vskip.1in The above uniform choice helps stabilize the automorphism group of $(\mathcal{F},u)$. The following is a slight variation of the definition that was given for $\chi(\dot F)<0$:

\begin{align*}
\mathcal{B}= &\mathcal{B}(\dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)\\
=&\left\{ (\mathcal{F},u)=((F,j,\mathbf{p},{\bf q}, \mathbf{r}),u) \left|
\begin{array}{cc}
j\in \widetilde{\mathcal{U}}, u\in W_\delta ^{k+1,p}( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-), u(r_{\pm, i})=x_{\gamma_{\pm,i}^s}, \\
u \mbox{ is $\varepsilon''$-close to $u'$ holomorphic in $W^{k+1,p}_\delta( \dot F, \widehat{W};\boldsymbol\gamma_+,\boldsymbol\gamma_-)$}\\
{\bf q} \mbox{ satisfies (*)}
\end{array}
\right.\right\},
\end{align*}
where $(F,j,{\bf p},{\bf q})$ and $\varepsilon''>0$ are fixed, and $\widetilde{\mathcal U}$ is a Teichm\"uller slice of $\mathcal U\subset \operatorname{Teich}(\dot F)$ (not $\operatorname{Teich}(\ddot F)$).

The fibers $\mathcal{E}_{(\mathcal{F},u)}$ of the bundle $\pi: \mathcal{E}\to \mathcal{B}$ are defined by
$$\mathcal{E}_{(\mathcal{F},u)}=W^{k,p}_\delta(\dot F, \wedge^{0,1}u^* T\widehat{W})$$
as before and $\overline\partial_J$ is the same as before.  The linearization $L_{(\mathcal{F},u)}$ is the same as that of Equation~\eqref{eqn: linearization} with the term $T\widetilde{\mathcal{U}}(j)$ removed (or viewed as the zero vector space).

\subsection{Description of $D_u$}
By McDuff-Salamon~\cite[Proposition~3.1.1]{MS},
$$D_u: W^{k,p}_{\delta} (\dot F, u^* T\widehat W)\to W^{k-1,p}_{\delta} (\dot F, \wedge^{0,1} u^* T\widehat W)$$
is given by
\begin{equation} \label{Du}
D_u \xi=\frac{1}{2}(\nabla \xi +J(u)\nabla \xi \circ j) -\frac{1}{2}J(u)(\nabla_\xi J)(u)\partial_J(u).
\end{equation}
Here, by abuse of notation, we do not distinguish between sections of $u^*T\widehat W$ and sections of $T\widehat W$ along $u$.

Suppose $(s,t)\in \mathbb{R}^\pm\times \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} $ is a holomorphic coordinate around a puncture $p\in {\bf p}_\pm$ of $\dot F$, $p$ corresponds to the Reeb orbit $\gamma$, and $u(s,t)=(s,t,\eta (s,t))$, written in supersimple coordinates for $\gamma$, is $J$-holomorphic. Then, near this puncture,
\begin{equation} \label{Dulocal}
D_u \xi=\frac{1}{2} (\nabla_s \xi +J(u)\nabla_t \xi + \widetilde S^\gamma \xi)\otimes (ds -i dt),
\end{equation}
where
$$J(u)=\left(\begin{array}{ccc}
0 & -1 & 0\\
1 & 0 & 0\\
X\eta & -j_{0}X\eta & j_{0}
\end{array}\right) \quad \text{and  }\quad \widetilde S^\gamma=\left(\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -j_0X
\end{array}\right).$$

Note that if $u$ is $J$-holomorphic, then $D_u u_s=D_u u_t=0$.  If, instead of using the basis
$$\{\partial_s, \partial_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n}\},$$
we use the basis
$$\{ u_s, u_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n} \},$$
then Equation \eqref{Dulocal} becomes
\begin{equation} \label{Dusimple}
D_u \xi=\frac{1}{2}(\partial_s \xi +J_0 \partial_t \xi+\widetilde S^\gamma\xi)\otimes (ds-idt),
\end{equation}
where
$$J_0=\left(\begin{array}{ccc}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & j_{0}
\end{array}\right)$$
is the standard complex structure on $\mathbb{R}^{2n+2}$.

If $u$ is immersed, we denote by $N$ the normal bundle of $u$ such that
$$N=\mathbb{R} \langle\partial_{x_1},\dots,\partial_{x_{2n}},\partial_{y_1},\dots,\partial_{y_{2n}}\rangle$$
near the ends.  We define
$$D^N_u:W_{\delta}^{k,p}(\dot F, u^*N)\to W_{\delta}^{k-1,p}(\dot F, \wedge^{0,1}u^*N)$$
by projecting $D_u$ along the splitting $u^*T\widehat W=u^*N\oplus T\dot F$.

\subsection{Asymptotic operator}

Let $\gamma$ be a Reeb orbit of $\boldsymbol{R}_\alpha$ and $S^1=\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z}$. Let
$$\widetilde A^\gamma=-J_0\frac{\partial}{\partial t}-\widetilde S^\gamma:W^{1,2}(S^1,\mathbb{R}^{2n+2})\to L^{2}(S^1,\mathbb{R}^{2n+2})$$
be the asymptotic operator for $\gamma$. Then $\widetilde A^{\gamma}$ is self-adjoint. Let
$$\cdots \leq \lambda_{-2}^\gamma < \lambda_{-1}^\gamma=0=\lambda_1^\gamma < \lambda_2^\gamma\leq \cdots$$ be eigenvalues of $\widetilde A^{\gamma}$ and
$$\cdots,f_{-2}^\gamma,f_{-1}^\gamma,f_1^\gamma,f_2^\gamma,\cdots$$
be the associated complete set of orthonormal eigenfunctions. Here the self-adjoint\-ness and orthonormality are defined with respect to the standard inner product in $\mathbb{R}^{2n+2}$.

We will write $\widetilde A$ and $\widetilde S$ for $\widetilde A^\gamma$ and $\widetilde S^\gamma$ if $\gamma$ is understood; similarly we may suppress the superscript $\gamma$ in $\lambda_i^\gamma, f_i^\gamma,\dots$ below, when they are understood.

\begin{rmk}
$\widetilde A$ is an ``enlargement'' of the operator $A: W^{1,2}(S^1,\mathbb{R}^{2n})\to L^{2}(S^1,\mathbb{R}^{2n})$ in the proof of Lemma \ref{end generic} which takes into account only the transverse directions (i.e., the $\partial_{x_i}$- and $\partial_{y_i}$-directions).
\end{rmk}

Summarizing the above discussion, we have:

\begin{lemma} \label{Dukernel}
Suppose $(s,t)\in \mathbb{R}^\pm\times \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z} $ is a holomorphic coordinate around a puncture $p\in {\bf p}_\pm$ of $\dot F$, $p$ corresponds to the Reeb orbit $\gamma$, and near $p$, $u$ admits a lift to $\mathbb{R}\times \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z}\times D$ which can be written as $(s,t)\mapsto (s,t,\eta (s,t))$ with respect to supersimple coordinates for $\gamma$ and is $J$-holomorphic. Then, near $p$, we have
$$D_u \xi =\frac{1}{2}(\partial_s \xi -\widetilde A \xi )\otimes (ds-idt),$$
with respect to the basis
$$\frak{B}:=\{ u_s, u_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n} \}.$$
(Here the tensor product $\otimes$ is taken to be $\mathbb{C}$-linear.)

Hence, for any $\xi \in \ker D_u$, near $p\in {\bf p}_+$ we have
$$\xi(s,t)=\sum_{\lambda_i<0} c_{+,i} e^{\lambda_i s} f_i(t),$$
and near $p\in {\bf p}_-$
we have
$$\xi(s,t)=\sum_{\lambda_i>0} c_{-,i} e^{\lambda_i s} f_i(t),$$
where the $c_{\pm,i}$'s are constant.
\end{lemma}

\subsection{The adjoint operator}

Let
$$D_u^*: W^{k,p}_\delta (\dot F,\wedge ^{0,1} u^{*}T\widehat{W})\to W^{k-1,p}_\delta (\dot F,u^{*}T\widehat{W})$$
be the adjoint operator of $D_u$ defined by
\begin{equation} \label{Du*}
\int_{\dot F} \langle \zeta, D_u \xi \rangle_1 dvol_{\dot F} =\int_{\dot F} \langle D_u^* \zeta, \xi \rangle_0 dvol_{\dot F} ,
\end{equation}
where $\langle, \rangle_1$ is an inner product on $W_\delta^{k,p}(\dot F,\wedge^{0,1}u^{*}T\widehat{W})$, $\langle, \rangle_0$ is an inner product on $ W^{k+1,p}_\delta (\dot F,u^{*}T\widehat{W})$, and $dvol_{\dot F}$ is defined using a metric $g_{\dot F}$ on $\dot F$. We additionally assume that, near $p\in {\bf p}_\pm$:
\begin{itemize}
\item[(i)] $g_{\dot F}$ restricts to $ds^2+dt^2$;
\item[(ii)]$\langle, \rangle_0$ is defined by requiring the basis ${\frak B}$ to be an orthonormal basis; and
\item[(iii)] $\langle, \rangle_1$ is induced by $\langle, \rangle_0$ and $g_{\dot F}$.
\end{itemize}

\begin{rmk}
Recall that the adjoint operator $D_u^*$ involve some choices.  We are not using the Riemannian metric $g_0$ in the definition of $D_u^*$.
\end{rmk}

The following lemma is almost evident:

\begin{lemma}
With the assumptions of Lemma~\ref{Dukernel}, if $\zeta\in W^{k,p}_\delta(\dot F,\wedge^{0,1}u^*T\widehat{W})$ and $\zeta=\zeta^0\otimes (ds-idt)$ near $p\in {\bf p}_\pm$, then we have
$$D_u^* (\zeta^0\otimes (ds-idt)) =\frac{1}{2}(-\partial_s \zeta^0 -\widetilde A \zeta^0),$$
with respect to the basis ${\frak B}$. Hence, for any $\zeta\in \ker D_u^*$, near $p\in{\bf p}_+$ we have
$$\zeta^0(s,t)=\sum_{\lambda_i>0} c_{+,i} e^{-\lambda_i s} f_i(t),$$
whereas near $p\in{\bf p}_-$ we have
$$\zeta^0(s,t)=\sum_{\lambda_i<0} c_{-,i} e^{-\lambda_i s} f_i(t),$$
where the $c_{\pm,i}$'s are constant.
\end{lemma}

\section{Semi-global Kuranishi charts} \label{section: semi-global Kuranishi charts}

Consider the moduli space
$$\mathcal{M}=\mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\widehat W; \boldsymbol\gamma_+;\boldsymbol\gamma_-).$$
As discussed in Section~\ref{subsection: symp vs cob}, there are two slightly different cases to consider: (Symp) and (Cob).

We would like to construct a quadruple
$$(\mathbb{K},\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V},\overline\partial_J,{\frak s}),$$
which we call a {\em semi-global Kuranishi chart}. In the (Symp) case this consists of:
\begin{enumerate}
\item a large compact subset $\mathbb{K}\subset \mathcal{M}/\mathbb{R}$; if $\mathcal{M}/\mathbb{R}$ is compact, then we take $\mathbb{K}=\mathcal{M}/\mathbb{R}$;
\item an orbibundle $\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V}$ (called an {\em obstruction orbibundle}) and a section $\overline\partial_J:\mathbb{V}\to \mathbb{E}$ such that
\begin{itemize}
\item $\mathbb{V}$ is an open set containing $\mathbb{K}$,
\item elements of $\mathbb{V}$ are equivalence classes $\llbracket\mathcal{F},u\rrbracket$ of maps
    $$(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u)$$
    for the quadruple $(\dot F,\widehat W,\boldsymbol\gamma_+;\boldsymbol\gamma_-)$,
\item the fibers $\mathbb{E}_{\llbracket\mathcal{F},u\rrbracket}=\pi^{-1}_{\mathbb{V}}(\llbracket\mathcal{F},u\rrbracket)$ are naturally identified with finite quotients of finite-dimensional subspaces $$E_{(\mathcal{F},u)}\subset W^{k,p}_\delta(\dot F,\wedge^{0,1}u^*T\widehat W)$$
    that are transverse to the linearization $L_{(\mathcal{F},u)}$,
\item $\overline\partial_J(\llbracket \mathcal{F},u\rrbracket)\in \mathbb{E}_{\llbracket\mathcal{F},u\rrbracket}$ is represented by $\overline\partial_J(\mathcal{F},u)\in E_{(\mathcal{F},u)}$ and
    $$\overline\partial_J^{-1}(0)= \mathbb{V}\cap (\mathcal{M}/\mathbb{R});$$
\end{itemize}
\item a decomposition of $\partial \mathbb{V}$ into the {\em horizontal part $\partial_h \mathbb{V}$} and the {\em vertical part $\partial_v\mathbb{V}$} which intersect along their mutual boundary and such that $\overline\partial_J$ is never zero on $\partial_h\mathbb{V}$;
\item a multisection ${\frak s}: \mathbb{V}\to \mathbb{E}$ of bounded degree (called an {\em obstruction multisection} over $\mathbb{V}$) such that ${\frak s}=0$ on a neighborhood of $\partial_h\mathbb{V}$, ${\frak s}$ is close to zero, and ${\frak s}\pitchfork \overline\partial_J$.
\end{enumerate}
In the (Cob) case, $\mathcal{M}/\mathbb{R}$ is replaced by $\mathcal{M}$ in (1)--(4).

In this section we do most of the construction, proving (1), (2), and (4) with ``${\frak s}=0$ on a neighborhood of $\partial_h\mathbb{V}$'' replaced by ``${\frak s}=0$ on a neighborhood of $\partial \mathbb{V}$'', and leaving (3) and in particular the definitions of the horizontal and vertical boundary to Section~\ref{subsection: trimming}.

\subsection{Construction of $\overline{\partial}_J$-transverse subbundles}

Consider the Banach bundle $\pi: \mathcal{E}\to \mathcal{B}$ and the section $\overline{\partial}_J: \mathcal{B}\to \mathcal{E}$ corresponding to the quadruple $(\dot F,\widehat W,\boldsymbol\gamma_+;\boldsymbol\gamma_-)$, as defined in Section~\ref{subsubsection: negative Euler char} or \ref{subsubsection: nonnegative Euler char}.

\begin{defn}
Let $\mathcal{V}$ be a subset of $\mathcal{B}$.
A subbundle $E\to \mathcal{V}$ of $\mathcal{E}|_{\mathcal{V}}$ is {\em $\overline{\partial}_J$-transverse} if $E$ has finite rank and, for each $(\mathcal{F},u)\in \mathcal{V}$,
\begin{equation} \label{eqn: transverse}
\operatorname{Im} L_{(\mathcal{F},u)} + E_{(\mathcal{F},u)} =\mathcal{E}_{(\mathcal{F},u)},
\end{equation}
where $E_{(\mathcal{F},u)}$ and $\mathcal{E}_{(\mathcal{F},u)}$ refer to the fibers of $E$ and $\mathcal{E}$ over $(\mathcal{F},u)$. Here $+$ in Equation~\eqref{eqn: transverse} means span and {\em does not mean direct sum.}
\end{defn}

Let $Z=\overline\partial_J^{-1}(0)$.
The goal of this section is to prove the following:

\begin{thm} \label{thm: construction of L-transverse subbundle}
Let $K\subset Z$ be a compact subset and $\mathcal{N}(K)$ be a sufficiently small open neighborhood of $K$ in $\mathcal{B}$.  Then there exist a positive integer $\ell_0=\ell_0(K)>0$ and a positive real number $\varepsilon_0=\varepsilon_0(K)>0$ (or $\ell_0=\ell_0(K,\varepsilon')$ and $\varepsilon_0=\varepsilon_0(K,\varepsilon')$ if $\chi(\dot F)=0$ or $1$ and $\varepsilon'>0$ is as in Section~\ref{subsubsection: negative Euler char}) such that for all integers $\ell\geq \ell_0$ and $0<\varepsilon\leq \varepsilon_0$, there exists a $\overline{\partial}_J$-transverse subbundle
$$E=E^{\ell,\varepsilon}\to \mathcal{N}(K)$$
of $\mathcal{E}|_{\mathcal{N}(K)}$ satisfying:
\begin{enumerate}
\item if $(\mathcal{F},u), (\mathcal{F}',u')\in \mathcal{N}(K)$ and there exists a $C^\infty$-diffeomorphism $\phi: (\mathcal{F},u)\stackrel\sim\to (\mathcal{F}',u')$, then there is a canonical identification $\phi_*E_{(\mathcal{F},u)}= E_{(\mathcal{F}',u\circ \phi^{-1})}$ which is induced from
$$\phi_*:\wedge^{0,1} T\dot F\otimes u^* T\widehat{W} \stackrel\sim\longrightarrow  \wedge ^{0,1} T\dot F\otimes (u\circ\phi^{-1})^{*}T\widehat{W};$$

\item  if $(\widehat{W}=\mathbb{R}\times M,J)\in \operatorname{(Symp)}$ and $(\mathcal{F},u), (\mathcal{F},u_T)\in \mathcal{N}(K)$, where $u_T=\Phi_T\circ u$ is $u$ translated up by $s=T$ and
$$\Phi_T: \mathbb{R}\times M\stackrel\sim\longrightarrow \mathbb{R}\times M, \quad (s,x)\mapsto (s+T,x)$$
is the $s=T$ translation map, then $(\Phi_T)_*E_{(\mathcal{F},u)}= E_{(\mathcal{F},u_T)}$.
\end{enumerate}
\end{thm}

\begin{rmk}
When $\operatorname{dim}(M)=3$, it is possible to obtain an effective bound on $\ell$ using the positivity of intersections and winding numbers (cf.\ \cite{HLS,HWZ2,We}).
\end{rmk}

\begin{proof}
The most important feature of the construction of $E^{\ell,\varepsilon}$ is the following:
\begin{enumerate}
\item[($\#$)] given $(\mathcal{F},u)\in\mathcal{N}(K)$, the essential choices only depend on $\varepsilon$, $\ell$, the image of $u$, and the hyperbolic geometry of a complete finite volume hyperbolic metric $g_{(\dot F,j)}$ on $\dot F$ which is compatible with $j$ (or on $g_{(\ddot F,j)}$, where $\ddot F=\dot F-{\bf q}$ and ${\bf q}$ depends on the image of $u$, in the case when $\chi(\dot F)=0$ or $1$).
\end{enumerate}
A nonessential choice for example is a choice of a cutoff function $\beta:\mathbb{R}\to [0,1]$ which is fixed throughout. This implies (1) and (2).

The proof consists of the following five steps.

\vskip.1in\noindent
{\em Step 1.} (Definition of $s_{\pm,i}$)
Let $K\subset Z$ be a compact subset.  The goal of this step is to define continuous maps $s_{\pm,i}:K \to \mathbb{R}$
with differentiable extensions $s_{\pm,i}: \mathcal{N}(K)\to \mathbb{R}$, where the differentiability depends on the choice of $W^{k+1,p}$. Here each $s_{\pm,i}$ corresponds to $\gamma_{\pm,i}$.

The collection $\{s_{\pm,i}\}$ will be invariant under group actions as follows:
\begin{itemize}
\item[(a)] If $(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\sim (\mathcal{F}',u')=((F,j',{\bf p},{\bf q}',{\bf r}'),u')\in \mathcal{N}(K)$ via the automorphism $\phi:\dot F\stackrel\sim\to\dot F$, then  $\{s_{\pm,i}(\mathcal{F},u)\}=\{s_{\pm,i}(\mathcal{F}',u')\}$.
\item[(b)] If $(\mathcal{F},u), (\mathcal{F},u_T)\in \mathcal{N}(K)$, then $\{s_{\pm,i}(\mathcal{F},u_T)\}=\{s_{\pm,i}(\mathcal{F},u)+T\}$.
\end{itemize}

\vskip.1in\noindent
{\em Step 1A.} Assume first that $\chi(\dot F)<0$.  Let $g=g_{(\dot F,j)}$.  For $\varepsilon>0$ sufficiently small, consider the $\varepsilon$-thin part $\operatorname{Thin}_\varepsilon(\dot F,g)$ of $(\dot F,g)$, which is the union of cusps
\begin{equation}\label{equation: cusps}
C_{+,1},\dots,C_{+,l_+},C_{-,1},\dots,C_{-,l_-},
\end{equation}
where each $\partial C_{\pm,i}$ is a circle which corresponds to $\gamma_{\pm,i}$.
We assume that $\varepsilon=\varepsilon(K)$ is sufficiently small so that, for all $(\mathcal{F},u)\in K$, $\operatorname{Im}(u|_{C_{\pm,i}})$ is $\kappa$-close (in the Hausdorff metric derived from the Riemannian metric $g_0$ on $\widehat{W}$ which is $s$-invariant at the ends) to $[s_{+,i},\infty)\times \gamma_{+,i}^s$ or $(-\infty,s_{-,i}]\times \gamma_{-,i}^s$ for some $s_{\pm,i}$. (The same holds for all $(\mathcal{F},u)\in \mathcal{N}(K)$, where $\mathcal{N}(K)$ is a sufficiently small neighborhood of $K$.) Here $\kappa>0$ is a small, predetermined constant which is independent of $K$.  The existence of $\varepsilon>0$ relies on the exponential decay estimates of the ends of $u$ from \cite{HWZ1}.

We then choose a specific $s_{\pm,i}=s_{\pm,i}(\mathcal{F},u)$ as follows: First consider the $m(\gamma_{\pm,i})$-fold covering map
$\mathbb{R}/\mathcal{A}_\alpha(\gamma_{\pm,i})\mathbb{Z}\to \mathbb{R}/\mathcal{A}_\alpha(\gamma_{\pm,i}^s)\mathbb{Z}$. We then write $u$ on $C_{\pm,i}$ in terms of supersimple coordinates as
$$u(s,t)=(s,t,\eta(s,t)),$$
after lifting to the $m(\gamma_{\pm,i})$-fold cover $\mathbb{R}\times (\mathbb{R}/\mathcal{A}_\alpha(\gamma_{\pm,i})\mathbb{Z})\times D$, subject to the condition that $t=0$ corresponds to the asymptotic marker $r_{\pm,i}$. Then we define $s_{\pm,i}$ as the $s$-coordinate of the point on $u|_{\partial C_{\pm,i}}$ with $t=0$.

\begin{defn} \label{defn: somewhat informal}
The $c_{\pm,i}$ will be called the {\em point on $\partial C_{\pm,i}$ corresponding to $r_{\pm,i}$.}
\end{defn}

\vskip.1in\noindent
{\em Step 1B.} Next assume that $\chi(\dot F)=0$ or $1$.  In view of the discussion from Section~\ref{subsubsection: nonnegative Euler char} (with $\varepsilon'>0$ fixed but small), we take $\ddot F= \dot F-{\bf q}$ with $\chi(\ddot F)<0$. Let $g=g_{(\ddot F,j)}$ and apply the considerations from Step 1A with $\dot F$ replaced by $\ddot F$ to obtain $s_{\pm,i}=s_{\pm,i}(\mathcal{F},u)$.

(a) and (b) are immediate from the construction, both in Steps 1A and 1B.

\begin{rmk}[Trivial cylinders]
We do not construct Kuranishi charts for trivial cylinders when $(\widehat W,J)$ is in the (Symp) case. On the other hand, when a symplectization is viewed in (Cob), we will construct Kuranishi charts around trivial cylinders.  We note that trivial (possibly multiply-covered) cylinders are automatically transverse.
\end{rmk}

\vskip.1in\noindent
{\em Step 2.} (Definition of $E^{\ell,\varepsilon}$) Let $\varepsilon=\varepsilon(K)$ be the constant from Step 1.

Let $\beta:\mathbb{R}\to [0,1]$ be a nondecreasing smooth cutoff function such that $\beta(s)=0$ for $s\leq 0$ and $\beta(s)=1$ for $s\geq 1$ and let $\beta^+_{s_0}(s)=\beta(s-s_0)$ and $\beta^-_{s_0}(s)=-\beta(s_0-s)$, where $s_0>0$ is a constant.

Given $(\mathcal{F},u)\in K$ and $s_{\pm,i}=s_{\pm,i}(\mathcal{F},u)$ from Step 1, let
\begin{equation} \label{f tilde}
\widetilde{f}^{\gamma_{\pm,i}}_p(s,t):=\frac{\partial \beta^\pm_{s_{\pm,i}}}{\partial s}(s)f^{\gamma_{\pm,i}}_p(t)\otimes \pi_j (ds-idt)\in \Gamma (\dot F ,\wedge^{0,1} u^{*}T\widehat{W}).
\end{equation}
Here $\pi_j:W^{k,p}(\dot F, \wedge ^1 T^{*} \dot F)\to W^{k,p}(\dot F, \wedge^{0,1}T^{*} \dot F)$ is the projection with respect to the hyperbolic metric and $\widetilde{f}^{\gamma_{\pm,i}}_j(s,t)$ has support on a single end of $\dot F$.
We then define
\begin{equation} \label{equation: def of each component of E}
E_{(\mathcal{F},u),{\gamma_{\pm,i}}}^{\ell,\varepsilon}:=\mathbb{R} \langle \widetilde{f}^{\gamma_{\pm,i}}_1,\dots,\widetilde{f}^{\gamma_{\pm,i}}_\ell \rangle
\end{equation}
for ${\gamma_{\pm,i}}\in \boldsymbol\gamma_\pm$ and define
\begin{equation}
E_{(\mathcal{F},u)}^{\ell,\varepsilon}:=\bigoplus_{{\gamma_{\pm,i}}\in\boldsymbol{\gamma}_+\cup\boldsymbol\gamma_-} E_{(\mathcal{F},u),{\gamma_{\pm,i}}}^{\ell,\varepsilon}.
\end{equation}

This gives us a bundle $E^{\ell,\varepsilon} \to \mathcal{N}(K)$ with fiber $E_{(\mathcal{F},u)}^{\ell,\varepsilon}$. The following will be proven in the Appendix:

\begin{lemma} \label{lemma: in Wkp and C1}
$E^{\ell,\varepsilon}_{(\mathcal{F},u)}\subset \mathcal{E}_{(\mathcal{F},u)}$ for all $(\mathcal{F},u)\in \mathcal{N}(K)$ and $E^{\ell,\varepsilon} \to \mathcal{N}(K)$ is of class $C^1$ if $k\geq 3$ and $p>2$.
\end{lemma}

\vskip.1in\noindent
{\em Step 3.} ($\overline{\partial}_J$-transversality)  We first prove the following transversality lemma, which is applicable for a single $(\mathcal{F},u)\in K$.

\begin{lemma}[Transversality lemma] \label{lemma: transversality lemma}
For each $(\mathcal{F}, u)\in K$, there exist $\ell_0 \in \mathbb{N}$ and $\varepsilon_0>0$ such that
\begin{equation} \label{eqn: trans lemma}
E_{(\mathcal{F},u)}^{{\ell,\varepsilon}} + \operatorname{Im} D_u=W^{k,p}_\delta (\wedge ^{0,1}\dot F \otimes u^{*}T\widehat W)
\end{equation}
for all $\ell\geq \ell_0$ and $0<\varepsilon\leq \varepsilon_0$.
\end{lemma}

\begin{rmk}
Observe that we are actually proving a slightly stronger statement than
$$E_{(\mathcal{F},u)}^{{\ell,\varepsilon}} + \operatorname{Im} L_{(\mathcal{F},u)}=W^{k,p}_\delta (\wedge ^{0,1}\dot F \otimes u^{*}T\widehat W).$$
\end{rmk}

\begin{proof}
Observe that, if Equation~\eqref{eqn: trans lemma} does not hold, then there exists $0\not=\zeta\in \ker D_u^*$ which is $L^2$-orthogonal to $E_{(\mathcal{F},u)}^{\ell,\varepsilon}$. To see this, consider the projection of $E_{(\mathcal{F},u)}^{\ell,\varepsilon}$ under the map $\pi:\ker D_u^*\oplus \operatorname{Im} D_u\to \ker D_u^*$. If $\zeta\in \ker D_u^*$ is orthogonal to $\pi(E_{(\mathcal{F},u)}^{\ell,\varepsilon})$, then $\zeta\perp E_{(\mathcal{F},u)}^{\ell,\varepsilon}$ since $\zeta\perp \operatorname{Im} D_u$.

Let $\varepsilon_0>0$ be sufficiently small so that on the $\varepsilon_0$-thin neighborhood of a puncture of $\dot F$ corresponding to the Reeb orbit $\gamma\in \boldsymbol\gamma_+$ ($\gamma\in \boldsymbol\gamma_-$ is similar), $\zeta\in \ker D_u^*$ can be written with respect to the basis $\frak{B}$ as
$$\zeta(s,t)=\sum_{\lambda_i>0} c_i e^{-\lambda_i^\gamma s} f_i^\gamma(t) \otimes (ds-idt),$$
and let $0<\varepsilon\leq \varepsilon_0$. The key point is that, when $\varepsilon_0$ is sufficiently small, $\frak{B}$ is arbitrarily close to $\{\partial_s,\partial_t,\partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n}\}$ and, without loss of generality, we may assume $\zeta(s,t)$ is written with respect to the latter.  There exists a basis $\{\zeta_1,\dots, \zeta_k\}$ for $\ker D_u^*$ for which the coefficients $c_{ji}$ (this means $c_i$ for $\zeta_j$) form a matrix in row echelon form; in particular, there are no rows with all $c_{ji}=0$ since the corresponding $\zeta_j=0$ by unique continuation, which is a contradiction.

Let $\ell_0>0$ be larger than the smallest $i$ for which $c_{ki}\not=0$.  Now observe that if $\zeta\in \ker D_u^*$ is $L^2$-orthogonal to $E_{(\mathcal{F},u)}^{\ell,\varepsilon}$, then its coefficients $c_i=0$ for all $i\leq \ell_0$, since the $L^2$-inner product of $\zeta$ and $\widetilde f_i^\gamma$ is nonzero if $c_i\not=0$.  This implies that $\zeta=0$.  The lemma then follows.
\end{proof}

We then improve Lemma~\ref{lemma: transversality lemma} as follows:

\begin{lemma}[Family transversality lemma] \label{lemma: family transversality lemma}
There exist $\ell_0 \in \mathbb{N}$ and $\varepsilon_0>0$ such that
$$E_{(\mathcal{F},u)}^{{\ell,\varepsilon}} + \operatorname{Im} D_u=W^{k,p}_\delta (\wedge ^{0,1}\dot F \otimes u^{*}T\widehat W).$$
for all $(\mathcal{F}, u)\in \mathcal{N}(K)$, $\ell\geq \ell_0$, and $0<\varepsilon\leq \varepsilon_0$
\end{lemma}

\begin{proof}
This follows from Lemma~\ref{lemma: transversality lemma}, the compactness of $K$, and the following well-known property of Fredholm maps: Let $L,L':V\to W$ be Fredholm maps between Banach spaces. If $L$ and $L'$ are close in the operator norm and $\operatorname{Im}(L)+W'=W$ for a finite-dimensional subspace $W'$, then $\operatorname{Im}(L')+W'=W$.

For $(\mathcal{F}_0,u_0), (\mathcal{F}_1,u_1)\in K$ that are close, we identify the Banach spaces involved in $L_{(\mathcal{F}_i,u_i)}$, $i=0,1$, by defining an isomorphism $\mathcal P: u_0^*T\widehat W \stackrel\sim\to u_1^*T\widehat W$ as follows:
\begin{itemize}
\item[(i)] on a large compact region of $\dot F$, $\mathcal P$ is given by parallel translation along the shortest geodesics between $u_0(z)$ and $u_1(z)$;
\item[(ii)] on the ends of $\dot F$ (assumed to be sufficiently small), $\mathcal P$ is given by identifying the bases
$$\{(u_0)_s,(u_0)_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n}\}$$
$$\{(u_1)_s, (u_1)_t, \partial_{x_1},\dots,\partial_{x_n},\partial_{y_1},\dots,\partial_{y_n} \};$$
\item[(iii)] $\mathcal{P}$ interpolates between (i) and (ii) in the intermediate region.
\end{itemize}
Then $L_{(\mathcal{F}_0,u_0)}$ and $L_{(\mathcal{F}_1,u_1)}$ are close in the operator norm; the details are left to the reader.
\end{proof}

Lemma~\ref{lemma: family transversality lemma} then implies the proposition.
\end{proof}

\subsection{The bundles $\pi_V:E|_V\to V$}

\subsubsection{Specific choices for $K$} \label{subsubsection: specific choices}

Recall the projection $\Pi=\Pi_{\widetilde{\mathcal{U}}}: \mathcal{B}_{\widetilde{U}}\to\widetilde{\mathcal{U}}$ given by $(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\mapsto j$.

Let $\mathcal{K}=\mathcal{K}_{\varepsilon,L}\subset \widetilde{\mathcal{U}}$ be a compact domain with smooth boundary.  We assume that $K$ is
\begin{enumerate}
\item[(i)] the compact subset of $Z=\overline\partial_J^{-1}(0)$ given by $Z\cap \Pi^{-1}(\mathcal{K})$ in the $\operatorname{(Cob)}$ case;
\item[(ii)] a compact subset of $Z=\overline\partial_J^{-1}(0)$ which projects onto $(Z\cap \Pi^{-1}(\mathcal{K}))/\mathbb{R}$ in the $\operatorname{(Symp)}$ case.
\end{enumerate}

\subsubsection{Definition of $\pi_V:E|_V\to V$}

With the bundle $E=E^{\ell,\varepsilon}\to \mathcal{N}(K)$ constructed in Theorem~\ref{thm: construction of L-transverse subbundle}, we define
\begin{equation} \label{eqn: V and V prime}
V:= \overline\partial_J^{-1}(E)\subset \mathcal{N}(K).
\end{equation}
Since $E$ is $\overline\partial_J$-transverse and of class $C^1$, $V\supset K$ is a finite-dimensional submanifold of $\mathcal{N}(K)$ of class $C^1$.  Once we have extracted $V$, since the elements of $V$ are smooth by elliptic regularity, $V$ can be ``upgraded to" a smooth finite-dimensional manifold of the expected dimension.

Observe that if $(\mathcal{F},u)\in V$, then $u:\dot F\to \widehat{W}$ is a smooth map whose ends are $J$-holomorphic and asymptotic to $\boldsymbol\gamma_+\cup \boldsymbol\gamma_-$. Let
\begin{equation}
\pi_V:E|_V\to V
\end{equation}
be the restriction of $E\to \mathcal{N}(K)$ to $V$ and let $\overline\partial_J:  V\to E|_V$ be the restriction of $\overline\partial_J$ to $V$. The vector bundle $\pi_V$ is a smooth vector bundle over $V$.

\subsubsection{Patching together the bundles $\pi_V: E|_V\to V$} \label{subsubsection: patching together}

We would like to patch together the bundles $\pi_V: E|_V\to V$ to obtain an orbibundle.  To this end we prove Lemmas~\ref{lemma: canonical identification}, \ref{lemma: canonical identification 2}, and \ref{lemma: canonical identification 3} below, which we can roughly state as:
\begin{enumerate}
\item independence of the choice of Teichm\"uller slice;
\item $G$-equivariance; and
\item invariance under $\mathbb{R}$-translations.
\end{enumerate}

We first consider (1). Suppose we are in the (Cob) case.  (An analogous result holds in the (Symp) case in view of (3) below, but will not be stated explicitly.)  Let $\widetilde{\mathcal{U}}_0$ and $\widetilde{\mathcal{U}}_1$ be two Teichm\"uller slices of $\mathcal{U}\subset \operatorname{Teich}(\dot F)$. Then there is a smooth family of diffeomorphisms $\phi_x: \dot F\stackrel\sim\to \dot F$ isotopic to the identity parame\-trized by $x\in \mathcal{U}$ such that $(\phi_x)_* j_x = j'_x$, where $j_x\in\widetilde{\mathcal{U}}_0$ and $j_x'\in\widetilde{\mathcal{U}}_1$ are lifts of $x\in \mathcal{U}$.

Let $\pi_{V_i}:E_i|_{V_i}\to V_i$, $i=0,1$, be the bundle $\pi_V:E|_V\to V$ corresponding to $\widetilde{\mathcal{U}}_i$ and let $K_i$, $\mathcal{K}_i$ be $K$, $\mathcal{K}$ corresponding to $\widetilde{\mathcal{U}}_i$ such that $\mathcal{K}_0$, $\mathcal{K}_1$ are Teichm\"uller slices of the same compact subset of $\mathcal{U}$.

\begin{lemma}[First patching lemma] \label{lemma: canonical identification}
Suppose we are in the $\operatorname{(Cob)}$ case. After possibly shrinking $V_i$ subject to the condition $V_i\supset K_i$, $V_0$ and $V_1$ can be made canonically diffeomorphic and the bundles $\pi_{V_i}:E_i|_{V_i}\to V_i$ can also be made canonically isomorphic.
\end{lemma}

\begin{proof}
Suppose $(\mathcal{F},u)=(F,j,\bf p, \bf q, \bf r, u) \in V_0$, $(\mathcal{F}',u')=(F, j',\bf p, \bf q', \bf r', u') \in \mathcal{N}(K_1)$, and $\phi=\phi_x$ is the diffeomorphism satisfying
$$\phi_*((F,j,{\bf p},{\bf q},{\bf r}),u)= ((F,j',{\bf p},{\bf q}',{\bf r}'),u').$$
Then $\phi$ induces a natural identification $\phi_* E_{(\mathcal{F},u)}\simeq E_{(\mathcal{F}',u')}$ by ($\#$) in the proof of Theorem~\ref{thm: construction of L-transverse subbundle}, which applies equally well to our case, provided
$$E_i=E_i^{\ell,\varepsilon}\to \mathcal{N}(K_i), \quad i=0,1$$ are constructed as in Theorem~\ref{thm: construction of L-transverse subbundle} with fixed $\ell$ and $\varepsilon$.  Hence $(\mathcal{F}',u')\in V_1$, provided $(\mathcal{F},u)\in V_0$ is sufficiently close to $K_0$.

The family $\{\phi_x\}, x\in \mathcal{U},$ gives a smooth map $\Phi$ from $E_0|_{V_0}\to V_0$ to $E_1|_{V_1}\to V_1$, after possibly shrinking $V_0$.  Since $\Phi$ is a diffeomorphism onto its image and the image of $V_0$ contains $K_1$, by the invariance of domain, $\Phi$ is a diffeomorphism $E_0|_{V_0}\to V_0$ to $E_1|_{V_1}\to V_1$, after possibly shrinking $V_1$.
\end{proof}

\begin{rmk}
We pass to $\pi_{V_i}:E_i|_{V_i}\to V_i$ at an early stage to avoid the well-known ``loss of derivative" issue when dealing with Sobolev spaces.  In particular, we avoid identifying $E_0\to \mathcal{N}(K_0)$ and $E_1\to \mathcal{N}(K_1)$.
\end{rmk}

(2) is stated more precisely as follows: Suppose we are in the $\operatorname{(Cob)}$ case.  Let $\mathcal{U}=\mathcal{U}_{[j]}$ be a sufficiently small good neighborhood of $[j]$.

Let $\widetilde {\mathcal{U}}_0=\widetilde {\mathcal{U}}$ be a Teichm\"uller slice of $\mathcal{U}$ containing $j$. Then $\widetilde {\mathcal{U}}_1=g(\widetilde{\mathcal{U}})$ is also a Teichm\"uller slice of $\mathcal{U}$ where $g\in \operatorname{Aut}(F,{\bf p},{\bf q}, j)$. As in (1), let $\{\phi_x\}_{x\in\mathcal{U}}$ be the family of diffeomorphisms taking $\widetilde{\mathcal{U}}_0$ to $\widetilde{\mathcal{U}}_1$. Also let $\pi_{V_i}:E_i|_{V_i}\to V_i$, $i=0,1$, be the bundle $\pi_V:E|_V\to V$ corresponding to $\widetilde{\mathcal{U}}_i$ and let $K_i$, $\mathcal{K}_i$ be $K$, $\mathcal{K}$ corresponding to $\widetilde{\mathcal{U}}_i$ such that $\mathcal{K}_0$, $\mathcal{K}_1$ are Teichm\"uller slices of the same $\operatorname{Mod}(F,{\bf p})_{[j]}$-invariant compact subset of $\mathcal{U}$.

\begin{lemma}[Second patching lemma]\label{lemma: canonical identification 2}
Suppose we are in the $\operatorname{(Cob)}$ case. After possibly shrinking $V_i$ subject to the condition $V_i\supset K_i$, $V_0$ and $V_1$ can be made diffeomorphic via the automorphism $g$ and the family $\{\phi_x\}_{x\in\mathcal{U}}$ of diffeomorphisms and the bundles $\pi_{V_i}: E_i|_{V_i}\to V_i$ can be made isomorphic.
\end{lemma}

\begin{proof}
Similar to that of Lemma~\ref{lemma: canonical identification}.
\end{proof}

(3) is stated more precisely as follows:

\begin{lemma}[Third patching lemma] \label{lemma: canonical identification 3}
Suppose we are in the $\operatorname{(Symp)}$ case and $K_0$ and $K_1$ are compact subsets for $\mathcal{K}\subset \widetilde{\mathcal{U}}$ such that $(\mathcal{F},u)\in \mathcal{N}(K_0)$ if and only if $(\mathcal{F},u_T)\in \mathcal{N}(K_1)$, where $u_T$ is $u$ translated up by $s=T$. Then, after possibly shrinking $V_i$ subject to the condition $V_i\supset K_i$, the bundles $\pi_{V_i}: E_i|_{V_i}\to V_i$ are canonically isomorphic.
\end{lemma}

\begin{proof}
Similar to that of Lemma~\ref{lemma: canonical identification}.
\end{proof}

\subsection{Semi-global Kuranishi chart} \label{subsection: semi-global Kuranishi chart}

In this subsection we construct a semi-global Kuranishi chart
$$(\mathbb{K},\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V},\overline\partial_J,{\frak s})$$
for $\mathcal{M}=\mathcal{M}_J^{\operatorname{ind}=k}(\dot F,\mathbb{R}\times M;\boldsymbol\gamma_+;\boldsymbol\gamma_-)$ or $\mathcal{M}/\mathbb{R}$ such that (1), (2), and (4) (with ``${\frak s}=0$ on a neighborhood of $\partial_h\mathbb{V}$'' replaced by ``${\frak s}=0$ on a neighborhood of $\partial \mathbb{V}$'') from the beginning of this section hold.  The existence of the section $\overline\partial_J$ is automatic from the construction.

Let $\mathbb{K}\subset \mathcal{M}/\mathbb{R}$ be a large compact subset; in particular, if $\mathcal{M}/\mathbb{R}$ is compact, then $\mathbb{K}=\mathcal{M}/\mathbb{R}$.  Consider a compact subset $K\subset Z=\overline\partial_J^{-1}(0)$ for the appropriate Banach bundle $\mathcal{E}\to \mathcal B$ described in Section~\ref{subsection: Fredholm setup}, such that the projection of $K$ to $\mathbb{K}$ is onto.  Theorem~\ref{thm: construction of L-transverse subbundle} gives a $\overline\partial_J$-transverse subbundle $E\to \mathcal{N}(K)$. If we let $V=\overline\partial_J^{-1}(E)$, then $\pi_V: E|_V\to V$ is a finite rank vector bundle over a finite-dimensional manifold.

\subsubsection{Construction of $\pi_{\mathbb{V}}:\mathbb{E}\to \mathbb{V}$}

Let $\mathbb{V}$ be the quotient of $V$ by the equivalence relation $\sim$ and the $\mathbb{R}$-translations.  Also let $\mathbb{E}$ be the quotient of $E$ by the induced equivalences
$$((\mathcal{F}=(F,j,{\bf p},{\bf q},{\bf r}),u),\zeta)\sim ((\mathcal{F}'=(F,j',{\bf p},{\bf q},{\bf r}'),u'),\zeta'),$$
where $\phi: F\stackrel\sim\to F$ is a diffeomorphism such that $\phi_*j=j'$, $\phi({\bf p}_\pm)={\bf p}_\pm$, $\phi({\bf q})={\bf q}$, $\phi_*({\bf r}_\pm)={\bf r}'_\pm$, $u'\circ\phi=u$, and $\phi_*\zeta=\zeta'$, and
$$((\mathcal{F},u),\zeta)\sim ((\mathcal{F},u_T),\zeta_T),$$
where $u_T$ is $u$ translated up by $s=T$ and $\zeta_T$ is $\zeta$ translated up by $s=T$.

\begin{prop}\label{prop: semi-global orbibundle}
After possibly shrinking $\mathbb{V}$ subject to the condition $\mathbb{V}\supset \mathbb{K}$, there is an orbibundle structure on $\pi_{\mathbb{V}}:\mathbb{E}\to \mathbb{V}\supset \mathbb{K}$.
\end{prop}

\begin{proof}
The proposition follows from Section~\ref{subsubsection: patching together} and Lemmas~\ref{lemma: canonical identification}, \ref{lemma: canonical identification 2}, and \ref{lemma: canonical identification 3}. The orbibundle charts are constructed by taking a sufficiently small good open set $\mathcal{U}_{[j]}$ about each $[j]\in \operatorname{Teich}(\dot F)$, a slice $\widetilde{\mathcal{U}}_{[j]}$ of $\mathcal{U}_{[j]}$, and a compact domain $\mathcal{K}\subset \widetilde{\mathcal{U}}_{[j]}$ which is the closure of a good open set about $[j]\in\operatorname{Teich}(\dot F)$. The group $\operatorname{Mod}(F,{\bf p})_{[j]}$ can be made to act on the resulting $\pi_V: E\to V$; one may need to take a quotient if the action of $\operatorname{Mod}(F,{\bf p})_{[j]}$ is not effective.

It remains to verify that $\mathbb{V}$ is Hausdorff: If there exist $x\not=y\in \mathbb{V}$ such that arbitrarily small neighborhoods $V_x$ and $V_y$ of $x$ and $y$ intersect nontrivially, then there exist lifts $\widetilde x\in V$ of $x$ and $\widetilde y\in V$ of $y$ and sequences $\widetilde x_i\to \widetilde x$ and $\widetilde y_i\to \widetilde y$ in $V$ such that $\widetilde x_i\sim \widetilde y_i$.  On the other hand, by Gromov-Hofer compactness, $\widetilde x\sim \widetilde y$, a contradiction.
\end{proof}

\subsubsection{Construction of ${\frak s}$} \label{subsubsection: obstruction multisection}

Next we choose a generic smooth multisection ${\frak s}$ of $\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V}$ which is a perturbation of the zero (multi-)section, subject to the conditions ${\frak s}=0$ on a neighborhood of $\partial \mathbb{V}$ and ${\frak s}\pitchfork \overline\partial_J$.   By the multisection perturbation lemma (Lemma~\ref{lemma: multisection perturbation lemma}) or the relative version thereof, there exists ${\frak s}$ which satisfies the desired conditions. Note that each (local) branch of ${\frak s}$ is assigned a weight by the discussion in Section~\ref{subsection: multisections}.

We then consider $\overline\partial_J^{-1}({\frak s})$. Since $\overline\partial_J(\mathcal{F},u)=0$ if and only if $\llbracket\mathcal{F},u\rrbracket\in \mathcal{M}/\mathbb{R}$,  we have $\overline\partial_J^{-1}({\frak s})\cap \partial \mathbb{V}=\varnothing$, i.e., $\overline\partial_J^{-1}({\frak s})$ stays away from $\partial \mathbb{V}$.

\subsection{Dependence on $\varepsilon$ and $\ell$} \label{subsection: stabilization}

In this subsection we briefly discuss the effect of increasing $\ell>0$ and shrinking $\varepsilon>0$.

\subsubsection{Stabilizations}

\begin{defn}
Let $E\to \mathcal{N}(K)$ be a $\overline\partial_J$-transverse subbundle. A subbundle $E'\subset \mathcal{E}|_{\mathcal{N}(K)}\to \mathcal{N}(K)$ is a {\em stabilization of $E$} if $E'\simeq E\oplus \mathbb{R}^a$ for some $a\in \mathbb{Z}^+$.
\end{defn}

An example of a stabilization of $E^{\ell,\varepsilon}\to \mathcal{N}(K)$ is $E^{\ell+1,\varepsilon}\to \mathcal{N}(K)$.

Given a stabilization $E'$ of $E\to\mathcal{N}(K)$, we have the following diagram:
\begin{equation} \label{equation: diagram}
\begin{diagram}
E & \rInto^{j} & E'=E\oplus \mathbb{R}^a\\
\dTo^{\pi_{V}} &  & \dTo_{\pi_{V'}} \\
V & \rInto^{i} & V'
\end{diagram}
\end{equation}
where $V=\overline\partial_J^{-1}(E)$ and $V'=\overline\partial_J^{-1}(E')$.

After shrinking $V'$ subject to $V'\supset K$ if necessary, we may assume that:
\begin{enumerate}
\item[(StV)] $i:V\hookrightarrow V'$ extends to a diffeomorphism
$$i: V\times (-1,1)^a\stackrel\sim \longrightarrow V'$$
so that $\overline{\partial}_J(i((\mathcal{F},u),x))=(\overline{\partial}_J(\mathcal{F},u),x)$, where $x$ is viewed both as an element of $(-1,1)^a$ and as an element of $\mathbb{R}^a$.
\end{enumerate}

\begin{defn}
A bundle $\pi_{V'}:E'|_{V'}\to V'$ is a {\em stabilization of $\pi_V:E|_V\to V$} if
\begin{enumerate}
\item $V=\overline\partial_J^{-1}(E)$ and $V'=\overline\partial_J^{-1}(E')$, where $E'\to \mathcal{N}(K)$ is a stabilization of $E\to \mathcal{N}(K)$;
\item $\pi_{V'}$ and $\pi_V$ satisfy Diagram~\eqref{equation: diagram}; and
\item $V$ and $V'$ are related by (StV).
\end{enumerate}
We also say that ``$(\pi_{V'},\pi_V)$ is a stabilization''.
\end{defn}

\begin{defn}
An orbibundle $\pi_{\mathbb{V}'}: \mathbb{E}'\to \mathbb{V}'$ is a {\em stabilization of the orbibundle $\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V}$} if
\begin{enumerate}
\item for each $p\in \mathbb{V}$ there exist orbibundle charts $\pi_{V'}:E'|_{V'}\to V'$ and $\pi_{V}:E|_V\to V$ for $\pi_{\mathbb{V}'}$ and $\pi_{\mathbb{V}}$ about $p$ such that $(\pi_{V'},\pi_V)$ is a stabilization;
\item given any two stabilizations $(\pi_{V'_1},\pi_{V_1})$ and $(\pi_{V'_2},\pi_{V_2})$ that overlap and any $p\in \mathbb{V}$ in the overlap, there exists a stabilization $(\pi_{V'_0},\pi_{V_0})$ about $p$ such that, for $i=1,2$, $\pi_{V'_0}$ embeds in $\pi_{V'_i}$, $\pi_{V_0}$ embeds in $\pi_{V_i}$, and we have a commuting cube consisting of two copies of Diagram~\eqref{equation: diagram} (for $(\pi_{V'_0},\pi_{V_0})$ and $(\pi_{V'_i},\pi_{V_i})$) and the embedding maps between them.
\end{enumerate}
\end{defn}

Given a stabilization $\pi_{V'}:E'|_{V'}\to V'$ of $\pi_V:E|_V\to V$ and a section $s: V\to E$, we construct the {\em stabilization $s': V'\to E'|_{V'}$ of $s$} as follows: Assume $E'=E\oplus \mathbb{R}$. (The case of $E'=E\oplus \mathbb{R}^a$ can be done inductively.)
Let $g:(-1,1)\to [0,1]$ be a smooth even function such that $g(x)=1$ on $0\leq x\leq {1\over 3}$ and $g(x)=0$ on $x\geq {2\over 3}$.
We then define
\begin{equation} \label{eqn: stabilization}
s'(i((\mathcal{F},u),x))= (g(x)s(\mathcal{F},u),0).
\end{equation}

\begin{defn}
The pair $(E',s')$ consisting of a stabilization $E'$ of $E$ and a stabilization $s'$ of $s$ is called a {\em stabilization of $(E,s)$.}
\end{defn}

\begin{lemma} \label{lemma: peaches}
If $(E',s')$ is a stabilization of $(E,s)$, then $\overline\partial^{-1}_J(s)=\overline\partial^{-1}_J(s')$.
\end{lemma}

\begin{proof}
Immediate from the construction of $(E',s')$.
\end{proof}

Analogously, if ${\frak s}$ is a $\overline\partial$-transverse multisection of $\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V}$ constructed in Section~\ref{subsubsection: obstruction multisection}, then we can construct the {\em stabilization ${\frak s}'$ of ${\frak s}$} which is a multisection of $\pi_{\mathbb{V}'}: \mathbb{E}'\to \mathbb{V}'$ such that ${\frak s}'$ is $\overline\partial$-transverse, ${\frak s}'=0$ on $\partial \mathbb{V}'$, and $\overline\partial^{-1}_J({\frak s})=\overline\partial^{-1}_J({\frak s}')$ as weighted branched manifolds.

\subsubsection{Changing $\varepsilon$}

The effect of changing $\varepsilon>0$ can be summarized as follows:

\begin{rmk}[Changing $\varepsilon>0$]\label{rmk: shrinking varepsilon}
There exist $\varepsilon_0>0$ and $\ell_0>0$ such that Theorem~\ref{thm: construction of L-transverse subbundle} holds for all $\varepsilon>0$ satisfying $\varepsilon_0\geq\varepsilon$ and $\ell$ satisfying $\ell\geq \ell_0$.  If we fix $\ell$, then for any two $0<\varepsilon_1,\varepsilon_2<\varepsilon_0$, there exists a $1$-parameter family of $\overline\partial_J$-transverse bundles $E^{\ell,\varepsilon_t}\to \mathcal{N}(K)$, $t\in[1,2]$.  In the process of extracting weighted branched manifolds in later sections, it follows that the weighted branched manifolds corresponding to $\varepsilon_1$ and $\varepsilon_2$ are cobordant.
\end{rmk}

\section{Gluing} \label{section: gluing}

The goal of this section is to define the terms that appear in the gluing theorems and also state the gluing theorems that we need.

Suppose we are considering (Symp) case; in this case we need to quotient by the $\mathbb{R}$-translations. A key observation is that $E_{(\mathcal{F},u)}$ is equivariant under $\mathbb{R}$-translations by the construction in Theorem~\ref{thm: construction of L-transverse subbundle}.  The necessary modifications can be made for the (Cob) case.

\subsection{The setup}

We will explain a simplified situation, but the general case is similar. Let
$$\mathcal{M}=\mathcal{M}^{\operatorname{ind}=k}_J(\dot F,\mathbb{R}\times M;\boldsymbol\gamma_+;\boldsymbol\gamma_-).$$
Let $\overline{\mathcal{M}/\mathbb{R}}$ be the compactification of $\mathcal{M}/\mathbb{R}$ and let $\partial (\mathcal{M}/\mathbb{R})$ be the boundary of $\mathcal{M}/\mathbb{R}$.

We assume the following:

\begin{assumption} \label{assumption: simple}
There exist
$$\mathcal{M}_1=\mathcal{M}^{\operatorname{ind}=k_1}_J(\dot F_1,\mathbb{R}\times M;\boldsymbol\gamma;\boldsymbol\gamma_-),$$
$$\mathcal{M}_2=\mathcal{M}^{\operatorname{ind}=k_2}_J(\dot F_2,\mathbb{R}\times M;\boldsymbol\gamma_+;\boldsymbol\gamma),$$
with $k_1+k_2=k$ such that $\dot F_1$ and $\dot F_2$ are connected, $\mathcal{M}_1/\mathbb{R}$ and $\mathcal{M}_2/\mathbb{R}$ are compact, and
$$\partial (\mathcal{M}/\mathbb{R})\subset \mathcal{M}_1/\mathbb{R}\times \mathcal{M}_2/\mathbb{R}.$$
\end{assumption}

Here and in the rest of the paper will use the convention that the lower subscript corresponds to the lower SFT level.

For $i=1,2$, $(F_i,{\bf p}_i,{\bf q}_i)$ and hence $\ddot F_i$ are fixed. Let $\varepsilon_i'>0$ be the constant that appears in the definition of $s'_+(\mathcal{F}_i,u_i)$ in Section~\ref{subsubsection: negative Euler char}, in case $\chi(\dot F)=0$ or $1$. We take a Teichm\"uller slice $\widetilde{\mathcal{U}}_i$ such that each $[\mathcal{F}_i,u_i]\in \mathcal{M}_i$ admits a representative $(\mathcal{F}_i,u_i)=((F_i,j_i,{\bf p}_i,{\bf q}_i,{\bf r}_i),u_i)$ with $j_i\in \widetilde{\mathcal{U}}_i$. Define
$$\overline\partial^{\widetilde{\mathcal{U}}_i}_J: \mathcal{E}_{\widetilde{\mathcal{U}}_i}\to \mathcal{B}_{\widetilde{\mathcal{U}_i}}$$
as before.  Let $Z_i=(\overline\partial_J^{\widetilde{\mathcal{U}}_i})^{-1}(0)$, $K_i\subset Z_i$ be a compact subset which projects onto $\mathcal{M}_i/\mathbb{R}$, and let $\mathcal{N}(K_i)$ be a sufficiently small neighborhood of $K_i$ in $\mathcal{B}_{\widetilde{\mathcal{U}_i}}$.

We are also assuming that
$${\bf p}=({\bf p}_1\cup {\bf p}_2)-({\bf p}_{+,1}\cup{\bf p}_{-,2}),\quad {\bf r}=({\bf r}_1\cup {\bf r}_2)-({\bf r}_{+,1}\cup{\bf r}_{-,2}),$$
and $F=F_1\# F_2$, where $\#$ is a multiple connected sum and the gluing occurs along subsets ${\bf p}_{+,1}\subset {\bf p}_1$, ${\bf p}_{-,2}\subset {\bf p}_2$ that correspond to ${\bf r}_{+,1}\subset {\bf r}_1$, ${\bf r}_{-,2}\subset {\bf r}_2$.

We apply Theorem~\ref{thm: construction of L-transverse subbundle} to $K_i\subset Z_i$ to obtain
$$E_i=E^{\ell,\varepsilon}_i\to V_i/\mathbb{R},$$
where $V_i=(\overline\partial^{\widetilde{\mathcal{U}}_i}_J)^{-1}(E_i)\subset \mathcal{N}(K_i)$ as in Equation~\eqref{eqn: V and V prime}.
By abuse of notation, we are still denoting the total spaces of the vector bundles by $E_i$ since the fibers are still the same.

{\em In what follows $\varepsilon>0$ is a very small constant which we may shrink (but never let be zero) from step to step.}

We fix a slice $V_i/\mathbb{R}\to V_i$ and write $(\mathcal{F}_i,u_i)\in V_i/\mathbb{R}$ to indicate the representative of its equivalence class in $V_i/\mathbb{R}$ in that slice. We also assume that
$$\min_j\{s_{-,j}(\mathcal{F}_2,u_2)\}\gg 0, \quad \max_j \{s_{+,j}(\mathcal{F}_1,u_1)\}\ll 0$$
for all $(\mathcal{F}_i,u_i)\in V_i/\mathbb{R}$.

\subsection{Close to breaking} \label{subsection: close to breaking}

Let $\delta>0$ be a small constant\footnote{Unrelated to the weight $\delta$ used in the Sobolev spaces earlier.} satisfying $\varepsilon_i'\gg \delta$, $i=1,2$. Given a constant $T\in \mathbb{R}$, let $u_{i,T}$ (resp.\ $u_T$) be $u_i$ (resp. $u$) translated up by $s=T$ units.

We define the notion of {\em close to breaking} in the spirit of \cite[Definition 7.1]{HT2} as follows:

\begin{defn}[Close to breaking] \label{defn: close to breaking}
A map $(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u)$ is {\em $\delta$-close to a building}
$$(\mathcal{F}_1,u_1)\cup (\mathcal{F}_2,u_2),\quad (\mathcal{F}_i,u_i)\in V_i/\mathbb{R},\quad i=1,2,$$
if the following hold:  after possibly translating $u$,
\begin{itemize}
\item there exist $T_2> T_2'>0$ such that the domain $\ddot F_2'$ of $u_{2,T_2}|_{s\geq T_2'}$ is obtained from $\ddot F_2$ by removing ends $C_\gamma$ corresponding to $\gamma\in\boldsymbol\gamma$; each end $C_\gamma$ is contained in the $\varepsilon$-thin part of $g_{(\ddot F_2,j_2)}$ and admits a biholomorphism
    $$(C_\gamma,j_2)\simeq ((-\infty,T_2']\times \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z},j_{12})$$
    with coordinates $(s,t)$ and standard complex structure $j_{12}:\partial_s\mapsto \partial_t$, such that $u_{2,T_2}|_{C_\gamma}$ is $\delta$-close in the $C^0$-norm to the map
    $$C_\gamma\to \mathbb{R}\times (\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z})\times D\subset \mathbb{R}\times M,\quad (s,t)\mapsto (s,t,0),$$
    where we are using supersimple coordinates for $\gamma$;
\item there exist $T_1> T_1'>0$ such that the domain $\ddot F_1'$ of $u_{1,-T_1}|_{s\leq -T_1'}$ and the ends satisfy analogous conditions;
\item let $r(\partial C_\gamma)\in \partial C_\gamma$ be the point corresponding to the asymptotic marker for $C_\gamma$ as in Definition~\ref{defn: somewhat informal}; writing
    $$A_\gamma=([-T_1,T_2']\times \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z},j_{12})$$
    with $\gamma\in \boldsymbol\gamma$, coordinates $(s,t)$, and standard complex structure $j_{12}:\partial_s\mapsto \partial_t$, the Riemann surface $(\ddot F,j)$ is $\delta$-close in the Teichm\"uller metric to the Riemann surface $(\ddot F,j')$\footnote{We are identifying $\ddot F$ with the surface obtained by gluing $\ddot F'_1$, $\sqcup_\gamma([-T_1,T_2']\times \mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z})$, and $\ddot F'_2$.} obtained by gluing
    $$(\ddot F'_1,j_1), \quad \sqcup_{\gamma\in \boldsymbol \gamma} A_\gamma, \quad \mbox{and} \quad (\ddot F'_2,j_2),$$
    along their appropriate boundaries such that each $r(\partial C_\gamma)$ is mapped to $(s,t)=(-T_1,0)$ or $(T_2',0)$;
\item $u_{2,T_2}|_{\ddot F'_2}$ is $\delta$-close in the $C^0$-norm to the map $u|_{\ddot F'_2}$;
\item $u_{1,-T_1}|_{\ddot F'_1}$ is $\delta$-close in the $C^0$-norm to the map $u|_{\ddot F'_1}$; and
\item for each $\gamma\in\boldsymbol\gamma$, $u|_{A_\gamma}$ is $\delta$-close in the $C^0$-norm to the map
    $$A_\gamma\to \mathbb{R}\times (\mathbb{R}/\mathcal{A}_\alpha(\gamma)\mathbb{Z})\times D\subset \mathbb{R}\times M,\quad (s,t)\mapsto (s,t,0),$$
    where we are using supersimple coordinates for $\gamma$.
\end{itemize}
\end{defn}

We make our choice of ${\bf q}$ more precise for $(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u)$ which is $\delta$-close to $(\mathcal{F}_1,u_1)\cup (\mathcal{F}_2,u_2)$, following Section~\ref{subsubsection: nonnegative Euler char}: Let us write the ends (or truncated ends) of $u|_{s\geq T_2'}$ and $u|_{s\leq -T_1'}$ by $u(s,t)=(s,t,\eta(s,t))$ using supersimple coordinates. For each $s\in (-T'_1, T'_2)$, we have
$$\eta(s)=\sum_{i}c_ie^{\theta_i s}g_i(t),$$ as in the proof of Lemma \ref{end generic}. Hence,
 $$\int_0^{\mathcal A_{\alpha}(\gamma)}|\eta(s,t)|^2dt=\sum_{i} c_i^2e^{2\theta_i s}.$$
 Since $\sum_{\theta_i>0} c_i^2e^{2\theta_i s}$ is strictly increasing with respect to $s$ and $\sum_{\theta_i<0} c_i^2e^{2\theta_i s}$ is strictly decreasing with respect to $s$, there exists a unique $s=s_0\in (-T'_1,T'_2)$ that minimizes $\int_0^{\mathcal A_{\alpha}(\gamma)}|\eta(s,t)|^2dt$.
 As in Lemma~\ref{tenren} let us define
$$s'_+(u|_{s\geq T_2'},\gamma)=\min\Big\{ {s\in [s_0,+\infty)}~\Big|~\int_{s_0}^{s}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon_2'\Big\},$$
$$s'_+(u|_{s\leq -T_1'},\gamma)=\max\Big\{ {s\in (-\infty,s_0]}~\Big|~\int_s^{s_0}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'_1\Big\},$$
where $\gamma\in\boldsymbol\gamma_+$ in the former case and $\gamma\in\boldsymbol\gamma$ in the latter case.  Then ${\bf q}={\bf q}'_2\cup {\bf q}'_1$ is an unordered collection of removable punctures where ${\bf q}'_2$ and ${\bf q}'_1$ are lists of punctures defined using $s'_+(u|_{s\geq T_2'},\gamma)$ and $s'_+(u|_{s\leq -T_1'},\gamma)$ and (*) in Section~\ref{subsubsection: negative Euler char}.
If $\chi(\dot F_i)<0$, then $\bf q_i'=\varnothing$, where $i=1,2$.
Note that ${\bf q}'_2$ and ${\bf q}'_1$ are close to ${\bf q}_2$ and ${\bf q}_1$ for $(\mathcal{F}_2,u_{2,T_2})$ and $(\mathcal{F}_1,u_{1,-T_1})$ under the identifications of $\ddot F_2'$ and $\ddot F_1'$ with subsurfaces of $\ddot F$ as in Definition~\ref{defn: close to breaking}.

We observe that for $(\mathcal{F},u)$ close to breaking into $(\mathcal{F}_1,u_1)\cup (\mathcal{F}_2,u_2)$, the data $(\mathcal{F}_1,u_1)\cup (\mathcal{F}_2,u_2)$ and $((F,j,{\bf p},{\bf r}),u)$ are sufficient to recover ${\bf q}$. In other words, the forgetting map
$$((F,j,{\bf p},{\bf q},{\bf r}),u)\mapsto ((F,j,{\bf p},{\bf r}),u)$$
presents no loss of information.

Let $\mathcal{G}_\delta(V_1/\mathbb{R},V_2/\mathbb{R})$ be the set of equivalence classes of maps
$$(\mathcal{F},u)=((F,j,{\bf p},{\bf q}, {\bf r}),u)$$
modulo $\mathbb{R}$-translations which have representatives that are $\delta$-close to some building $(\mathcal{F}_1,u_1)\cup (\mathcal{F}_2,u_2)$ with $(\mathcal{F}_i,u_i)\in V_i/\mathbb{R}$ and let $\widetilde{\mathcal{G}}_\delta(V_1/\mathbb{R},V_2/\mathbb{R})$ be a slice of $\mathcal{G}_\delta(V_1/\mathbb{R},V_2/\mathbb{R})$ whose elements are maps which are $\delta$-close to breaking.

\subsection{The bundle $E'$}

We define the bundle
$$E'\to \widetilde{\mathcal{G}}_\delta(V_1/\mathbb{R},V_2/\mathbb{R})$$
for $\delta>0$ small as follows:  Given $(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\in \widetilde{\mathcal{G}}_\delta(V_1/\mathbb{R},V_2/\mathbb{R})$, consider Step 1 of the proof of Theorem~\ref{thm: construction of L-transverse subbundle}.  The $\varepsilon$-thin part $\operatorname{Thin}_\varepsilon(\ddot F,g)$ for $g=g_{(\ddot F,j)}$ is now a union of cusps (as in Equation~\eqref{equation: cusps}) and annuli. We now apply Step 1 of the proof of Theorem~\ref{thm: construction of L-transverse subbundle} and use the boundaries of the cusps to obtain $\{s_{+,i}(\mathcal{F},u)\}$ and $\{s_{-,i}(\mathcal{F},u)\}$ (corresponding to the $\pm$ ends of $(\mathcal{F},u)$) and the boundaries of the annuli to obtain $\{s_{-,i}'(\mathcal{F},u)\}$ (corresponding to the negative ends of the top level) and $\{s_{+,i}'(\mathcal{F},u)\}$ (corresponding to the positive ends of the bottom level). This gives us $E^{\ell,\varepsilon}_{(\mathcal{F},u),\gamma}$ as in Equation~\eqref{equation: def of each component of E}, where $\gamma$ is viewed as an element of
$$\boldsymbol\gamma_+\sqcup \boldsymbol \gamma \sqcup \boldsymbol \gamma\sqcup \boldsymbol \gamma_-.$$
We then set
\begin{equation} \label{equation: defn of E prime}
E'_{(\mathcal{F},u)}:=\bigoplus_{\gamma\in\boldsymbol\gamma_+\sqcup \boldsymbol \gamma \sqcup \boldsymbol \gamma\sqcup \boldsymbol \gamma_-}E^{\ell,\varepsilon}_{(\mathcal{F},u),\gamma}.
\end{equation}

\subsection{Statements of gluing theorems} \label{subsection: statements of gluing theorems}

In this subsection we collect the gluing theorems that we need to construct the semi-global Kuranishi structures.  They are standard gluing results and will be proven in the next section.

In general, the $2$-level building condition in Assumption~\ref{assumption: simple} is replaced by an analogous $m$-component condition: We are gluing $m$ non-trivial (not trivial cylinder) components, one from each $V_i/\mathbb{R}$, and we can analogously define
$$E'\to \widetilde{\mathcal{G}}_\delta(V_1/\mathbb{R},\dots, V_m/\mathbb{R}).$$
Let us define
$$\widetilde{\mathcal{G}}^{E'}_\delta(V_1/\mathbb{R},\dots,V_m/\mathbb{R}):= \overline\partial_J^{-1}(E')\subset \widetilde{\mathcal{G}}_\delta(V_1/\mathbb{R},\dots,V_m/\mathbb{R}).$$
We also introduce the notation $V_i^{\text{sh}}$ (resp.\ $V_i^{\text{en}}$) to denote a slight $\mathbb{R}$-invariant shrinking (resp.\ enlargement) of $V_i$ which contains $K_i$ and is still open.

\begin{thm}[Gluing] \label{thm: gluing}
For sufficiently large $R>0$, there exists a gluing map
\begin{equation}
G_{(1,\dots,m)}:(V_1/\mathbb{R}) \times\dots \times (V_m/\mathbb{R}) \times [R,\infty)^{m-1}\to \widetilde{\mathcal{G}}^{E'}_\delta(V_1/\mathbb{R},\dots, V_m/\mathbb{R})
\end{equation}
which is a $C^1$-diffeomorphism onto its image and whose image contains
$$\widetilde{\mathcal{G}}^{E'}_{\delta'}(V_1^{\text{\em sh}}/\mathbb{R},\dots,V_m^{\text{\em sh}}/\mathbb{R})$$
for sufficiently small $\delta'>0$.
\end{thm}

Writing $V_{(i,\dots,j)}/\mathbb{R}=\widetilde{\mathcal{G}}^{E'_{(i,\dots,j)}}_{\delta'}(V_i^{\text{sh}}/\mathbb{R},\dots,V_j^{\text{sh}}/\mathbb{R})$, where $E'_{(i,\dots,j)}$ is $E'$ for $V_i/\mathbb{R},\dots,$ $V_j/\mathbb{R}$, we also have the following:

\begin{thm}[Iterated gluing] \label{thm: iterated gluing}
For sufficiently large $R>0$, there is a gluing map
\begin{align*}
G_{(1,\dots, (i,\dots,j),\dots m)}:&  (V_1/\mathbb{R})\times\dots\times (V_{i-1}/\mathbb{R})\times (V_{(i,\dots,j)}/\mathbb{R}) \\
& \times (V_{j+1}/\mathbb{R})\times  \dots\times (V_m/\mathbb{R}) \to \widetilde{\mathcal{G}}^{E'_{(1,\dots,m)}}_\delta(V_1/\mathbb{R},\dots, V_m/\mathbb{R}),
\end{align*}
such that $G_{(1,\dots, (i,\dots,j),\dots m)}\circ (id,\dots, G_{(i,\dots,j)},\dots, id)$ is $C^1$-close to the gluing map $G_{(1,\dots,m)}$.
\end{thm}

\section{Details of gluing}\label{section: details of gluing}

We prove Theorem~\ref{thm: gluing} with $m=2$, following the general outline of \cite{HT2} and \cite{BH}.  The case for general $m$ only differs in notation.  The proof of Theorem~\ref{thm: iterated gluing} also uses the same type of estimates as in Theorem~\ref{thm: gluing}  and is left to the reader.

We are assuming that $\varepsilon>0$ is small, that $(\alpha,J)$ is an $L$-simple pair (cf.\ Definition~\ref{defn: L-supersimple 1}), and that $J$ is smooth.

We write $E_+\to V_+$ for $E_2\to V_2$ and $E_-\to V_-$ for $E_1\to V_1$.  Let $\llbracket\mathcal{F}_\pm,u_\pm \rrbracket \in \mathbb{V}_\pm/\mathbb{R}$, where $\mathcal{F}_\pm=(F_\pm,j_\pm,{\bf p}_\pm,{\bf r}_\pm)$. For simplicity we assume that $(\mathcal{F}_+,u_+)$ is a curve from $\gamma_+$ to $\boldsymbol\gamma=(\gamma_1,\dots,\gamma_k)$ and $(\mathcal{F}_-,u_-)$ is a curve from $\gamma_1$ to $\boldsymbol\gamma_-=(\gamma_{-,1},\dots,\gamma_{-,l})$, i.e., we are gluing the first negative end of $[(\mathcal{F}_+,u_+)]$ to the positive end of $[(\mathcal{F}_-,u_-)]$.  We fix representatives $(\mathcal{F}_\pm,u_\pm)$ of $\mathbb{V}_\pm/\mathbb{R}$ such that $s_{\mp,1}(\mathcal{F}_\pm,u_\pm)=0$. Here $s_{\pm,i}(\mathcal{F},u)$ is as defined in the proof of Theorem 5.1.2.  We write ``$(\mathcal{F}_\pm,u_\pm)\in \mathbb{V}_\pm/\mathbb{R}$'' to mean $(\mathcal{F}_\pm,u_\pm)$ is the chosen representative in its equivalence class.  Note that the elements of $(E_\pm)_{(\mathcal{F}_\pm,u_\pm)}$ are supported on $-1\leq s\leq 0$ for $(\mathcal{F}_+,u_+)$ and on $0\leq s\leq 1$ for $(\mathcal{F}_-,u_-)$.

\begin{notation}
In general, if we decorate $(\mathcal{F},u)$ with subscripts and superscripts as in $(\mathcal{F}^*_\star,u^*_\star)$, then $\mathcal{F}^*_\star=(F^*_\star, j^*_\star, {\bf p}^*_\star,{\bf r}^*_\star)$ and $\dot F^*_\star=F^*_\star-{\bf p}^*_\star$.
\end{notation}

In the next several subsections we retrace the steps of Sections 8.1--8.4 of \cite{BH}, adapted to the current setting.

\subsection{Pregluing}\label{subsection:pregluing}

Fix a constant $T_0\gg 0$.  Also let $T\gg T_0$, which is allowed to vary.

\begin{notation}
If $\tau_T: \mathbb{R}\times M \stackrel\sim\to \mathbb{R}\times M$ is  given by the translation $(s,x)\mapsto (s+T,x)$, then let $u_{\pm,T}=\tau_{\pm (T+T_0)}\circ u_\pm$.  In general a subscript or superscript $+,T$ (resp.\ $-,T$) indicates the result of translating up (resp.\ down) by $s=T+T_0$.
\end{notation}

Recall the simple coordinates $(s,t,x=(x_1,\dots,x_n),y=(y_1,\dots,y_n))$ on a sufficiently small neighborhood $\mathbb{R}\times\mathbb{R}/\mathcal{A}_\alpha(\gamma_1)\mathbb{Z} \times D$ of $\mathbb{R}\times \gamma_1$; here we are slightly abusing notation and passing to a finite cover if $\gamma_1$ is not simple. We assume that $T_0>0$ is sufficiently small so that $u_-|_{s\geq -T_0}$ (resp.\ $u_+|_{s\leq T_0}$) can be written as a graph $(s,t,\eta_-(s,t))$ (resp.\ $(s,t,\eta_+(s,t))$) over $[0,\infty)\times \gamma_1$ (resp.\ $(-\infty,0]\times\gamma_1$).

Fix constants $0<h<1$ and $r\gg h^{-1}$.  We take $T_0> 5r$.
Choose a cutoff function $\beta: \mathbb{R} \to [0,1]$ such that $\beta(s)=0$ for $s\leq 0$ and $\beta(s)=1$ for $s\geq 1$.  Let $\beta_{-,T}(s)=\beta({T-s\over hr})$ and $\beta_{+,T}(s)=\beta({T+s\over hr})$.

For $T'\leq T+T_0$, let $\mathcal{E}_{+, T'}$ be the component of
$$u_{+,T}^{-1}(\{s< T'\})\simeq u_+^{-1}(\{s< T'-(T+T_0)\})$$
corresponding to $\gamma_1$, let $\mathcal{E}_{-,T'}$ be
$$u_{-,T}^{-1}(\{s> -T'\})\simeq u_-^{-1}(\{s> -T'+(T+T_0)\}),$$
and let $A_{[T_1,T_2]}$ be the annulus $[T_1,T_2]\times \mathbb{R}/\mathcal{A}_\alpha(\gamma_1)\mathbb{Z}$ with coordinates $(s,t)$ and the standard complex structure.

\begin{figure}[ht]
\begin{center}
\psfragscanon
\psfrag{s}{\tiny $s$}
\psfrag{A}{\tiny $-T$}
\psfrag{B}{\tiny $-T+hr$}
\psfrag{C}{\tiny $T-hr$}
\psfrag{D}{\tiny $T$}
\psfrag{E}{\tiny $\beta_{+,T}$}
\psfrag{F}{\tiny $\beta_{-,T}$}
\psfrag{G}{\tiny $-T-T_0$}
\psfrag{H}{\tiny $T+T_0$}
\includegraphics[width=13cm]{cutoff-functions.eps}
\end{center}
\caption{The cutoff functions $\beta_{\pm,T}$.  The little bump functions indicate the support of $(E_\pm)_{(\mathcal{F}_\pm,u_{\pm,T})}$ } \label{fig: cutoff functions}
\end{figure}

We define the {\em pregluing of $u_+$ and $u_-$ with gluing parameter $T$} as follows: Let $F^\circ_+:=\dot F_+ -\mathcal{E}_{+,T}$ and $F^\circ_-:=\dot F_- -\mathcal{E}_{-,T}$.  The domain $(\dot F =F-{\bf p},j)$ is then obtained by gluing $(F^\circ_+,j_+|_{F^\circ_+})$, $(F^\circ_-,j_-|_{F^\circ_-})$, and $A_{[-T,T]}$ in the expected way. The preglued map $u_*=u_*(u_+,u_-,T)$ agrees with $u_{+,T}$ on $F^\circ_+$ and with $u_{-,T}$ on $F^\circ_-$, and is given by
\begin{equation}\label{eqn: pregluing}
u_*(s,t)= (s,t, \beta_{+,T}(s)\eta_{+,T}(s,t)+\beta_{-,T}(s)\eta_{-,T}(s,t) )
\end{equation}
on $A_{[-T,T]}$.

\subsection{Gluing of domain complex structures} \label{subsection: domain complex structures}

We consider the following domain gluing, which is close to (but not quite the same as) the gluing of the domains in Section~\ref{subsection:pregluing}.  The reason for the slight discrepancy is that each of $F^\circ_+$ and $F^\circ_-$ is obtained from $\dot F_+$ and $\dot F_-$ by removing a cusp that is close to but not necessarily the same as a $\widetilde \varepsilon_\pm$-thin cusp for some $\widetilde \varepsilon_\pm$.

Fix $\dot F_-$, $\dot F_+$, and $\varepsilon''>0$ small. We are gluing $\dot F_-$ and $\dot F_+$ along one ``positive end'' of $\dot F_-$ and one ``negative end'' of $\dot F_+$. Let $\widetilde{\mathcal{U}}_\pm$ be a Teichm\"uller slice for a small open set $\mathcal{U}_\pm\subset \operatorname{Teich}(\dot F_\pm)$ and let $g(j_\pm)$ be the complete finite-volume hyperbolic metric compatible with $j_\pm\in \widetilde{\mathcal{U}}_\pm$. We also fix $j^0_\pm\in \widetilde{\mathcal{U}}$. Let $\mathcal{E}_\pm^{\varepsilon''}(j_\pm)$ be the (interior of the) $\varepsilon''$-thin component of $g(j_\pm)$ corresponding to the end that is glued and let $F_\pm^\bullet(j_\pm):= \dot F_\pm - \mathcal{E}_\pm^{\varepsilon''}(j_\pm)$.  Also choose $r_\pm(j_\pm)\in \partial F_\pm^\bullet(j_\pm)$ smoothly with respect to $j_\pm$.

Given $(T,\tau)\in [R,\infty)\times \mathbb{R}$, we glue
$$(F_-^\bullet(j_-),j_-) \cup ([0,T]\times \mathbb{R}/\mathbb{Z},j_{std}) \cup (F_+^\bullet(j_+),j_+)$$
so that $r_-(j_-)$ is identified with $(0,0)\in [0,T]\times \mathbb{R}/\mathbb{Z}$ and $r_+(j_+)$ is identified with $(T,\tau)\in [0,T]\times \mathbb{R}/\mathbb{Z}$.
We denote the resulting Riemann surface by $(\dot F(j_-,j_+,T,\tau),j(j_-,j_+,T,\tau))$. We also write $\dot F=\dot F(j_-^0,j_+^0,R,0)$; this is our reference surface.  (Note that when $(T,\tau_1), (T,\tau_2)\in [R,\infty)\times \mathbb{R}$ and $\tau_2-\tau_1\in \mathbb{Z}$, the complex structures are diffeomorphic, but differ by a Dehn twist about a separating surface and correspond to distinct points in Teichm\"uller space.)

This defines a map
$$\kappa:\widetilde{\mathcal{U}}_-\times \widetilde{\mathcal{U}}_+\times[R,\infty)\times \mathbb{R}\to \operatorname{Teich}(\dot F),$$
$$(j_-,j_+,T,\tau)\mapsto (\dot F(j_-,j_+,T,\tau),j(j_-,j_+,T,\tau)).$$

\begin{thm} \label{thm: Kazdan-Warner}
For $R\gg 0$, $\kappa$ is a $C^1$-embedding.
\end{thm}

\begin{proof}[Idea of proof.]
This follows from Kazdan-Warner~\cite{KW1,KW2}. For $R\gg 0$, we can down a metric with negative curvature on $\dot F(j_-,j_+,T,\tau)$ which is conformally equivalent to $j(j_-,j_+,T,\tau)$, agrees with $g(j_\pm)$ on $F_\pm^\bullet(j_\pm)$, and whose curvature is very close to $-1$ on $[0,T]\times \mathbb{R}/\mathbb{Z}$. In order to find a hyperbolic metric in the same conformal class we solve an equation of type $\Delta u = 1-he^u$, where $u:\dot F\to \mathbb{R}$, $\Delta$ is a Laplacian, and $h:\dot F\to \mathbb{R}$ is close to $1$ (and approaches $1$ as $R\to \infty$); see \cite[Section 2]{KW1}.  When $h$ is close to $1$, this can be solved using the inverse function theorem; see \cite{KW2}.  We remark that negative curvature implies that the linearized operator is invertible and also that solving the equation can be done in a differentiable family.

The above discussion implies that the Fenchel-Nielsen coordinates for $j_\pm\in \widetilde{\mathcal{U}}_\pm$ are very close to the corresponding Fenchel-Nielsen coordinates for $\kappa(j_-,j_+,T,\tau)\in \operatorname{Teich}(\dot F)$ when $R\gg 0$, implying the theorem.
\end{proof}

\subsection{Banach spaces} \label{subsection: banach spaces}

The function spaces that we use are {\em Morrey spaces}, following \cite[Section 5.5]{HT2}.\footnote{This is rather nonstandard and we chose to adopt it to avoid redoing some work in \cite{HT2} for $W^{k,p}$-spaces.} Let $u:\dot F\to \mathbb{R}\times M$ be a finite energy holomorphic curve. On $\dot F$ we choose a Riemannian metric so that the ends are cylindrical and  we use the $\mathbb{R}$-invariant Riemannian metric on $\mathbb{R}\times M$.

The {\em Morrey space} $\mathcal{H}_0(\dot F, \wedge^{0,1} T^*\dot F\otimes  u^*T(\mathbb{R}\times M))$ is the Banach space which is the completion of the compactly supported sections of $\wedge^{0,1} T^*\dot F\otimes u^*T(\mathbb{R}\times M)$ with respect to the norm
$$\| \xi\|= \left( \int_{\dot F} |\xi|^2  \right)^{1/2} + \left( \sup_{x\in \dot F} \sup_{\rho\in(0,1]} \rho^{-1/2} \int_{B_\rho(x)} |\xi|^2  \right)^{1/2},$$
where $B_\rho(x)\subset \dot F$ is the ball of radius $\rho$ about $x$.  Similarly, $\mathcal{H}_1(\dot F, u^*T(\mathbb{R}\times M))$ is the completion of the compactly supported sections of $u^*T(\mathbb{R}\times M)$ with respect to $$\| \xi \|_*= \|\nabla \xi\| + \|\xi\|.$$

The analog of the usual Sobolev embedding theorem is the following:

\begin{lemma} \label{lemma: sobolev embedding}
There is a bounded linear map
$$\mathcal{H}_1(\dot F, u^*T(\mathbb{R}\times M))\to C^{0,1/4}(\dot F, u^*T(\mathbb{R}\times M)), \quad \xi\mapsto \xi,$$
where $C^{0,1/4}$ denotes the space of H\"older continuous functions with exponent ${1\over 4}$.
\end{lemma}

We also define the weighted Morrey spaces $\mathcal{H}_{i,\delta}(*)=\mathcal{H}_{0,\delta}(\dot F_\pm, \wedge^{0,1} T^*\dot F_\pm\otimes  (u_\pm)^*T(\mathbb{R}\times M))$ and $\mathcal{H}_{1,\delta}(\dot F_\pm, (u_\pm)^*T(\mathbb{R}\times M))$ with weight $\delta>0$ as the space of sections $\psi$ such that $\psi\cdot f_\delta^\pm\in \mathcal{H}_i (*)$. Here $f_\delta^\pm: \mathbb{R}\times M\to \mathbb{R}^+$ is the weight function for $u_\pm$ which agrees with $e^{\delta |s|}$ at the ends and is normalized so that $f_\delta^\pm(s,x)= e^{\delta|s|}$ for $\mp s\geq 0$. Their norms are denoted by $\|\cdot \|_\delta$ and $\|\cdot\|_{*,\delta}$.  We will also write $\|\cdot\|_\delta^{\pm,T}$ and $\|\cdot\|_{*,\delta}^{\pm,T}$ with respect to the translates $f_\delta^{\pm,T}$.

\subsection{The setup}\label{subsection: gluing}

Let $\psi_{\pm}$ be sections of $u_{\pm}^*T(\mathbb{R}\times M)$ and let $\psi_{\pm,T}=\tau_{\pm (T+T_0)}\circ \psi_\pm$ be sections of $u^*_{\pm,T} T(\mathbb{R}\times M)$.  The exponential maps $\exp_{u_{\pm,T}}$ can be chosen such that
$$\exp_{u_{\pm,T}}\psi_{\pm,T}= (s,t,\eta_{\pm,T})+\psi_{\pm,T}$$
on $\mathcal{E}_{\pm,T+T_0}$, by taking the Riemannian metric $g_0$ on $\mathbb{R}\times M$ to be $\mathbb{R}$-invariant and equal to the standard flat metric on $\mathbb{R}\times\mathbb{R}/\mathcal{A}_\alpha(\gamma_1)\mathbb{Z} \times D$.

We then deform $u_*$ to
\begin{equation}\label{equation: deform u*}
u=\exp_{u_*} (\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T}),
\end{equation}
where $\exp_{u_*}$ agrees with $\exp_{u_{\pm,T}}$ on $\dot F_\pm -\mathcal{E}_{\pm,T+T_0}$ and with
$$\exp_{u_*} (\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T})=(s,t,\eta_*)+\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T}$$
on $A_{[-(T+T_0), T+T_0]}$.
Here $\eta_*=\beta_{+,T}\eta_{+,T} +\beta_{-,T}\eta_{-,T}$.  We want to solve for $\psi_+$ and $\psi_-$ in the equation
\begin{equation}
\overline{\partial}u\in E'_{(\mathcal{F},u)},
\end{equation}
with $(\mathcal{F},u)=(F,j,{\bf p}, u)$.  Here $J$ is understood and $E'$ is as given in Equation~(6.3.1).

We identify
$$\Phi=\Phi_T:\wedge^{0,1}u^*  T(\mathbb R\times M)\stackrel\sim\longrightarrow\wedge^{0,1}(u_*)^*T(\mathbb{R}\times M)$$
as follows: away from the neck region $A_{[-(T+2T_0), T+2T_0]}$ we parallel transport using a complex linear connection induced by the Levi-Civita connection of an adapted Riemannian metric (cf.\ \cite[p.39]{MS}); on $A_{[-(T+T_0),T+T_0]}$ we map $\eta\otimes (ds-idt)\mapsto \eta\otimes (ds-idt)$; and on $A_{[-(T+2T_0), -(T+T_0)]}$ and $A_{[T+T_0,T+2T_0]}$ we interpolate between the two identifications.
Similarly, we identify
$$\Phi_\pm=\Phi_{\pm}^{\psi_{\pm,T}}: \wedge^{0,1}(\exp_{u_{\pm,T}}\psi_{\pm,T})^{*}T(\mathbb{R}\times M)\stackrel\sim\longrightarrow\wedge^{0,1}u_{\pm,T}^{*}T(\mathbb{R}\times M).$$

The equation $\overline{\partial} u\in E'_{(\mathcal{F},u)}$, or more precisely the equation
\begin{equation}\label{d-bar2}
{\Phi} \overline{\partial} u\in {\Phi } E'_{(\mathcal{F},u)},
\end{equation}
can then be written as
\begin{equation} \label{eqn: d-bar rewritten}
\beta_{-,T} \left(D_{-,T} \psi_{-,T} + e_{-,T} +\mathcal{L}_- +\mathcal{R}_-\right)+\beta_{+,T}\left(D_{+,T}\psi_{+,T} + e_{+,T}+\mathcal{L}_+ +\mathcal{R}_+\right)\in {\Phi} E_{(\mathcal F, u)}',
\end{equation}
where $D_\pm$ (resp.\ $D_{\pm,T}$) is the linearization of $\overline{\partial}$ for $u_{\pm}$ (resp.\ $u_{\pm,T}$), $e_\pm:=\overline{\partial} u_\pm$ (resp.\ $e_{\pm,T}:=\overline{\partial} u_{\pm,T}$), and $\mathcal{L}_\pm$ and $\mathcal{R}_\pm$ are described below. (The descriptions of $\mathcal{L}_\pm$ and $\mathcal{R}_\pm$ on $A_{[-T,T]}$ are obtained by expanding $\tfrac{\partial u}{\partial s} +J(u) \tfrac{\partial u}{\partial t}$, where we write
$$u(s,t)=\begin{pmatrix} s \\ t \\ \beta_{+,T}\eta_{+,T} +\beta_{-,T}\eta_{-,T}\end{pmatrix}+\beta_{+,T}\psi_{+,T}+\beta_{-,T}\psi_{-,T}$$
and
$$J(u)=
\begin{pmatrix}
0 & -1 & 0\\
1 & 0 & 0\\
X\eta' & -j_0 X\eta'  & j_0
\end{pmatrix},$$
where:
\begin{itemize}
\item we are using simple coordinates $(s,t,(x,y))$ written as column vectors;
\item $\eta'= \beta_{+,T}(\eta_{+,T}+\psi_{+,T}^N) +\beta_{-,T}(\eta_{-,T}+\psi_{-,T}^N)$;
\item $X$ is the constant matrix defined in the paragraph after Definition 3.1.1;
\item $j_0$ is the standard complex structure on $\mathbb R^{2n}$;
\item $\psi^N_{\pm,T}$ is the $(x,y)$-component of $\psi_{\pm,T}$.
\end{itemize}
The details are left to the reader.)

\vskip.1in\noindent
1. $\mathcal L_\pm=\mathcal L_\pm(\psi_{-,T},\psi_{+,T})$ has support on $A_{[-T,T]}$ and is given by
\begin{align} \label{equation: mathcal L}
2\mathcal L_\pm= &
\left(
\begin{array}{ccc}
0\\0\\ \frac{\partial \beta_{\mp,T}}{\partial s}\eta_{\mp,T}
\end{array}
\right)+\frac{\partial \beta_{\mp,T}}{\partial s}\psi_{\mp,T}\\
\nonumber &
+\widetilde\beta_{\pm,T}\beta_{\mp,T}\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0 \\
 X \eta_{\pm,T} &  -j_0 X \eta_{\pm,T} & 0
\end{array}
\right)\frac{\partial \psi_{\mp,T}}{\partial t}
\\
\nonumber &+\widetilde\beta_{\pm,T}\beta_{\mp,T}\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0 \\
 X \psi^N_{\pm,T} &  -j_0 X \psi^N_{\pm,T} & 0
\end{array}
\right)\frac{\partial \psi_{\mp,T}}{\partial t}\\
\nonumber &+
\widetilde\beta_{\pm,T}(\beta_{\pm,T}-1)\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0 \\
X\eta_{\pm,T} & -j_0X\eta_{\pm,T} & 0
\end{array}
\right)
\frac{\partial \psi_{\pm,T}}{\partial t}.
\end{align}
Here:
\begin{itemize}
\item the factor $\otimes (ds-idt)$ is omitted from each term;
\item $\widetilde\beta_{-,T}(s)=\beta({T+hr-s\over hr})$ and $\widetilde\beta_{+,T}(s)=\beta({T+hr+s\over hr})$;
\end{itemize}

\begin{terminology} $\mbox{}$
\begin{enumerate}
\item We say that a function $F(\psi)$ is ``linear''  if there exists a constant $c>0$ such that $|F(\psi)(x)|<c|\psi(x)|$ at every point $x$ of the domain of $\psi$.  We will use the shorthand ${\frak l}(\psi)$ to denote an unspecified ``linear" map.
\item Following \cite[Definition~5.1]{HT2} we say that $F(\psi)$ is {\em type $1$ quadratic} if it can be written as
$$F(\psi)=P(\psi) + Q(\psi)\cdot \nabla\psi,$$
where there exists a constant $c>0$ such that $|P(\psi)(x)|\leq c|\psi(x)|^2$ and $|Q(\psi)(x)|\leq c|\psi(x)|$ at every point $x$ of the domain of $\psi$.
\end{enumerate}
\end{terminology}

\vskip.1in\noindent
2. The remaining terms of ${\Phi} \overline{\partial}u$ are grouped into $\beta_-\mathcal{R}_-$ and $\beta_+\mathcal{R}_+$.
One can see that $\mathcal{R}_{\pm}= \mathcal{R}_{\pm} (\psi_{\pm,T})$ is type $1$ quadratic. On $A_{[-T,T]}$, we can explicitly write
\begin{equation}
\mathcal R_\pm={1\over 2} \beta_{\pm,T}
\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
 X\psi_{\pm,T}^N & - j_0X\psi_{\pm,T}^N & 0
\end{array}
\right)
\frac{\partial \psi_{\pm,T}}{\partial t}.
\end{equation}

\vskip.1in

We also write $E'_{(\mathcal{F},u)}=E'_{\psi_\pm,T}$ and decompose
$$E_{\psi_\pm,T}'=(E'_{\psi_\pm,T})_+\oplus (E'_{\psi_\pm,T})_-,$$
where $(E'_{\psi_\pm,T})_\pm$ corresponds to $u_{\pm,T}$. In order to solve Equation \eqref{eqn: d-bar rewritten} we solve for $(\psi_-,\psi_+)$ in the pair of equations
\begin{equation} \label{theta+-}
D_{\pm,T} \psi_{\pm,T} + e_{\pm,T} + \mathcal{L}_{\pm}(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_{\pm}(\psi_{\pm,T})\in \Phi_{\pm}^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm.
\end{equation}
Let
$$\Pi_\pm^{\psi_\pm,T}:\mathcal{H}_{0,\delta}(\dot F, \wedge^{0,1}u_{\pm,T}^{*}T(\mathbb{R}\times M))\to  (\Phi_\pm^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm)^\perp$$
be the projection onto the $L^2$-orthogonal complement of $\Phi_\pm^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm$.
Then Equation~\eqref{theta+-} is equivalent to
\begin{equation} \label{Theta pm}
\Theta_\pm(\psi_-,\psi_+) :=\Pi_\pm^{\psi_\pm,T}(D_{\pm,T} \psi_{\pm,T} + e_{\pm,T} + \mathcal L_\pm(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_\pm(\psi_{\pm,T}))=0.
\end{equation}

Since $D_{\pm,T}$ is transverse to $E_\pm$ and $\Phi_{\pm}^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm$ is close to $E_\pm$ when $T\gg 0$ and $\|\psi_\pm\|_{*,\delta}$ is small, it follows that $D_{\pm,T}$ is transverse to $\Phi_{\pm}^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm$, i.e.,
$$\Pi_\pm^{\psi_\pm,T} D_{\pm,T}: \mathcal{H}_{1,\delta}(\dot F, u_{\pm,T}^{*}T(\mathbb{R}\times M))\to (\Phi_\pm^{\psi_{\pm,T}} (E'_{\psi_\pm,T})_\pm)^\perp$$
is surjective.

\subsection{Definition of gluing map} \label{subsection: defn of gluing map}

In this subsection we make some estimates and define the gluing map at the end.  We use the convention that constants such as $C$ may change from line to line when making estimates.

Let $\lambda= \min \{\lambda_1, |\lambda_{-1}|\}$. Suppose that $\delta$ satisfies $0<100\cdot\delta<\lambda$.  Let $\mathcal{H}_\pm =\mathcal{H}_{1,\delta}(\dot F,u_\pm^{*}T(\mathbb{R}\times M))$, let $\mathcal{H}_\pm^{\perp,\psi_{\pm},T}$ be the $L^2$-orthogonal complement of
$$\ker \left( \Pi_\pm^{\psi_\pm,T}\circ D_{\pm,T}\circ \tau_{\pm(T+T_0)}\right)$$
in $\mathcal{H}_{\pm}$, where $\tau_{\pm(T+T_0)}$ is the map induced by $\tau_{\pm(T+T_0)}$, and let $\mathcal{B}_{\pm}$ be the closed ball of radius $\widetilde\varepsilon$ in $\mathcal{H}_{\pm}$ centered at $0$.

The following lemma is modeled on \cite[Proposition 5.6]{HT2}, but the estimates are slightly different.

\begin{lemma} \label{azalea}
There exist $r\gg 0$ and $C,\widetilde\varepsilon>0$ such that for $T\gg 0$ and $\|e_+\|_{*,\delta}$ small the following holds:
\begin{enumerate}
\item There is a map $P_+:\mathcal{B}_-\to \mathcal{H}_+$ such that $\Theta_+(\psi_-,P_+(\psi_-))=0$ and $\psi_+=P_+(\psi_-)\in \mathcal{H}_+^{\perp,\psi_+,T}$.
\item $\| P_+(\psi_-)\|_{*,\delta}\leq C (\|e_{+}\|_{*,\delta}+ r^{-1/2}e^{-2(\lambda-\delta) T}+(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta})$.
\item $\| P_+(\psi_-) - P_+(\psi'_-)\|_{*,\delta} \leq C \| \psi_- - \psi_-'\|_{*,\delta} (e^{-2\delta T} + r^{-1}  +\widetilde\varepsilon)$.
\end{enumerate}
\end{lemma}

Let $\mathcal{I}_+: \mathcal{B}_-\times \mathcal{B}_+\to \mathcal{H}_+$ be the map given by:
\begin{align} \label{def of I}
\mathcal{I}_+(\psi_-,\psi_+) =&  -\tau_{+(T+T_0)}^{-1}(\Pi_+^{\psi_+,T}D_{+,T})^{-1}\Pi_+^{\psi_+,T}(e_{+,T}  + \mathcal L_+(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_+(\psi_{+,T})).
\end{align}
Here $(\Pi_+^{\psi_+,T}D_{+,T})^{-1}$ is the bounded inverse of $\Pi_+^{\psi_+,T}D_{+,T}$.

\begin{proof}
(1) We are trying to solve for $\psi_+$ in
$$\Pi_+^{\psi_+,T} (D_{+,T} \psi_{+,T}+e_{+,T}+\mathcal L_+(\psi_{-,T},\psi_{+,T})+\mathcal R_+(\psi_{+,T}))=0.$$
Using the function $\mathcal{I}_+$ we define $P(\psi_-)$ as the unique fixed point, i.e., $\psi_+$ satisfying
\begin{equation} \label{fixed point}
\mathcal{I}_+(\psi_-,\psi_+)=\psi_+.
\end{equation}

The unique fixed point is guaranteed by the contraction mapping theorem, which follows from Claims~\ref{claim 1} and \ref{claim 2}.

\begin{claim} \label{claim 1}
\begin{align*}
\|\mathcal{I}_+(\psi_-,\psi_+)\|_{*,\delta}  \leq & (\widetilde C_1(\|\psi_+\|_{*,\delta})+\widetilde C_2(T))\|e_{+}\|_{*,\delta} \\
\nonumber & + \widetilde C(\|\psi_+\|_{*,\delta}) ( r^{-1/2}e^{-2(\lambda-\delta) T}+(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta}\\
\nonumber &\quad  +\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta}  +e^{-2\lambda T}\|\psi_+\|_{*,\delta}+\|\psi_+\|^2_{*,\delta}),
\end{align*}
where $\widetilde C(\sharp)$, $\widetilde C_1(\sharp)$, $\widetilde C_2(\sharp)$ are continuous functions of $\sharp$;  $\widetilde C_1(\sharp)\to 0$ as $\sharp\to 0$; and $\widetilde C_2(\sharp)\to 0$ as $\sharp\to\infty$.
\end{claim}

\begin{proof}[Proof of Claim~\ref{claim 1}]
We will be using Lemma~\ref{lemma: sobolev embedding} (more precisely $|\zeta|_{C^0}\leq C \|\zeta\|_{*,\delta}$) several times without explicit mention.

We first obtain
\begin{align} \label{one}
\left\| \mathcal L_+(\psi_{-,T},\psi_{+,T}) \right\|_\delta^{+,T}\leq &  C(r^{-1/2}e^{-2(\lambda-\delta) T}+r^{-1}\|\psi_-\|_{*,\delta}+e^{-2\delta T}\|\psi_-\|_{*,\delta}\\
\nonumber & \quad +\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta} +e^{-2\lambda T}\|\psi_+\|_{*,\delta}),
\end{align}
where each term on the right-hand side corresponds to the terms on the right-hand side in Equation~\eqref{equation: mathcal L}.  The first term has bound
\begin{align*}
\| \tfrac{\partial \beta_{-,T}}{\partial s}& \eta_{-,T}\|_\delta^{+,T}  = \| \tfrac{\partial \beta_{-,T}}{\partial s} \eta_{-,T} f_\delta^{+,T}\|
=\| \tfrac{\partial \beta_{-,T}}{\partial s} \eta_{-,T} e^{-\delta(s-(T+T_0))}\|  \\
& \leq C(r^{-1}\cdot e^{-\lambda\cdot 2T}\cdot e^{\delta\cdot 2T_0}\cdot \sqrt{hr}\cdot  + r^{-1}\cdot e^{-\lambda \cdot 2T}\cdot e^{\delta\cdot 2T_0})\\
& \leq C(r^{-1/2}e^{-2(\lambda-\delta) T}),
\end{align*}
where we are using the bound $|{\partial\beta_-\over \partial s}|< cr^{-1}$ for some $c>0$ and exponential decay bounds for $\eta_-$. The second term has bound
\begin{align*}
\|\tfrac{\partial\beta_{-,T}}{\partial s}\psi_{-,T} \cdot f_\delta^{+,T}\|& = \|\tfrac{\partial\beta_{-,T}}{\partial s}\psi_{-,T} \cdot e^{-\delta (s-(T+T_0))} \| \leq\|\tfrac{\partial\beta_{-,T}}{\partial s}\psi_{-,T} \cdot e^{2\delta T}\| \\
& \leq Cr^{-1} \|\psi_-\|_\delta\leq Cr^{-1}\|\psi_-\|_{*,\delta}.
\end{align*}
The third term has bound
\begin{align*}
C\|\widetilde\beta_{+,T}\beta_{-,T} & e^{\lambda (s-(T+T_0))}\tfrac{\partial \psi_{-,T}}{\partial t} e^{-\delta (s-(T+T_0))}\| \\
& = C \|\widetilde\beta_{+,T}\beta_{-,T} e^{ (\lambda-2\delta)s-\lambda(T+T_0)}  \tfrac{\partial \psi_{-,T}}{\partial t}e^{\delta( s+ (T+T_0))}  \| \\
& \leq  C  e^{-2\delta(T+T_0)}\|\psi_-\|_{*,\delta} \leq C e^{-2\delta T}\|\psi_-\|_{*,\delta}.
\end{align*}
The fourth term has bound
\begin{align*}
C\|\widetilde\beta_{+,T}\beta_{-,T} & \psi_{+,T} \psi_{-,T} e^{-\delta(s-(T+T_0))}\| \\
& \leq  C |\psi_{+,T} e^{-\delta(s-(T+T_0))}|_{C^0}\cdot \|\widetilde\beta_{+,T}\beta_{-,T} \psi_{-,T}\|\\
& \leq C \|\psi_+\|_{*,\delta} \|\psi_-\|_{\delta}\leq C \|\psi_+\|_{*,\delta} \|\psi_-\|_{*,\delta}.
\end{align*}
The last term has bound
\begin{align*}
C\| (\widetilde \beta_{+,T})(\beta_{+,T}-1)\eta_{+,T} \tfrac{\partial\psi_{+,T}}{\partial t} e^{-\delta(s-(T+T_0))}\| & \leq C e^{-2\lambda T} \|\psi_+\|_{*,\delta}.
\end{align*}

Also we obtain
\begin{align}\label{two}
\| \mathcal{R}_+(\psi_{+,T})\|^{+,T}_\delta \leq C\|\psi_+\|^2_{*,\delta},
\end{align}
since $\mathcal{R}_+(\psi_{+,T})$ is type 1 quadratic.

Next we consider the $L^2$-projection $\Pi_+$ to $(\Phi_+ E'_+)^\perp$, where we are suppressing ${\psi_\pm,T}$ from the notation. Let $e_1(\psi_+,T),\dots,e_k(\psi_+,T)$ be an orthonormal basis for $\Phi_+ E'_+$; it is not hard to see that $e_i(\psi_+,T)$ can be taken to be continuous with respect to $\psi_+\in \mathcal{B}_+$.
Then
\begin{align*}
\Pi_+(\zeta)=\zeta - \sum_i \langle e_i(\psi_+,T),\zeta\rangle e_i(\psi_+,T),
\end{align*}
where $\langle\cdot,\cdot\rangle$ is the $L^2$-inner product.
This allows us to estimate
\begin{align} \label{three}
\|\Pi_+(\zeta)\|_\delta \leq   \|\zeta\|_\delta +\sum_i\|\zeta\|_\delta\cdot \|e_i(\psi_+,T)\|_\delta\leq \widetilde C(\|\psi_+\|_{*,\delta}) \|\zeta\|_\delta.
\end{align}
The first inequality follows from $|\langle e_i(\psi_+,T),\zeta\rangle|\leq \|e_i(\psi_+)\|_{L^2} \cdot\|\zeta\|_{L^2} \leq \|\zeta\|_\delta$, where $\| \cdot \|_{L^2}$ is the $L^2$-norm, and the second inequality follows from the continuity of $e_i(\psi_+,T)$ with respect to $\psi_+$.

\begin{claim} \label{claim: continuous}
$(\Pi_+ D_+)^{-1}$ depends continuously on $\psi_+\in\mathcal{B}_+$.
\end{claim}

Observe that the domain of $(\Pi_+ D_+)^{-1}$ uses the norm $\|\cdot \|_\delta$ and the range of $(\Pi_+ D_+)^{-1}$ uses $\|\cdot \|_{*,\delta}$.

\begin{proof}[Sketch of proof of Claim~\ref{claim: continuous}]
$\Pi_+$ depends continuously on $\psi_+\in\mathcal{B}_+$ and hence so does $\Pi_+ D_+$.  The inverse hence also depends continuously on $\psi_+$.
\end{proof}

We also obtain
\begin{align} \label{four}
\|\tau_{+(T+T_0)}^{-1}(\Pi_+^{\psi_+,T}D_{+,T})^{-1}\Pi_+^{\psi_+,T}(e_{+,T})\|_\delta \leq (\widetilde C_1(\|\psi_+\|_{*,\delta})+\widetilde C_2(T))\|e_{+}\|_{*,\delta}.
\end{align}
Claim~\ref{claim 1} then follows from combining Estimates~\eqref{one}, \eqref{two}, \eqref{three}, \eqref{four}, and Claim~\ref{claim: continuous}.
\end{proof}

If $r,T\gg 0$, $\widetilde\varepsilon>0$ is small, and $\|e_+\|_{*,\delta}$ is small, then $\|\mathcal I (\psi_-,\psi_+)\|_{*,\delta}<\widetilde\varepsilon$ whenever $\|\psi_-\|_{*,\delta}, \|\psi_+\|_{*,\delta}<\widetilde\varepsilon$. Hence $\mathcal{I}_+(\psi_-,\cdot)$ maps a radius $\widetilde\varepsilon$ ball into itself.

\begin{claim}\label{claim 2}
\begin{align*}
\|\mathcal{I}_+(&\psi_-,\psi_+) -\mathcal{I}_+(\psi_-, \psi_+')\|_{*,\delta}\\
\leq &  C ( \|\psi_-\|_{*,\delta}+e^{-2\lambda T}+\|\psi_+\|_{*,\delta} +\|\psi_+'\|_{*,\delta})\cdot \| \psi_+-\psi'_+\|_{*,\delta}\\
& + C\|\psi_+-\psi_+'\|_{*,\delta}\cdot (\|e_{+}\|_{*,\delta} + r^{-1/2}e^{-2(\lambda-\delta) T} +(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta}\\
& \qquad\qquad \qquad +\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta}  +e^{-2\lambda T}\|\psi_+\|_{*,\delta}+\|\psi_+\|^2_{*,\delta}).
\end{align*}
\end{claim}

\begin{proof}[Proof of Claim~\ref{claim 2}]
This follows from the estimates
\begin{align*}
\| \mathcal{L}_+(\psi_{-,T},\psi_{+,T})-\mathcal{L}_+(\psi_{-,T},\psi_{+,T}')\|_\delta^{+,T} & \leq C (\|\psi_-\|_{*,\delta}+e^{-2\lambda T})\|\psi_+-\psi_+'\|_{*,\delta},\\
\|\mathcal{R}_+(\psi_{+,T})-\mathcal{R}_+(\psi_{+,T}')\|_\delta^{+,T} & \leq C( \|\psi_+\|_{*,\delta} +\|\psi_+'\|_{*,\delta}) \|\psi_+ - \psi_+'\|_{*,\delta},
\end{align*}
as well as the calculations from Claim~\ref{claim 1}.
\end{proof}

Hence if $r,T\gg 0$, $\widetilde\varepsilon>0$ is small, and $\|e_+\|_{*,\delta}$ is small, then $\mathcal{I}_+(\psi_-,\cdot)$ gives a contraction mapping, provided $\|\psi_+\|_{*,\delta}, \|\psi_+'\|_{*,\delta}\leq \widetilde\varepsilon$. This proves (1).  We also prove the following, which is used in (3):

\begin{claim}\label{claim 3}
\begin{align*}
\|\mathcal{I}_+(\psi_-,\psi_+') &-\mathcal{I}_+(\psi_-',\psi_+')\|_{*,\delta} \\
& \leq   \widetilde C(\|\psi_+'\|_{*,\delta} ) (e^{-2\delta T} + r^{-1} + \|\psi_+'\|_{*,\delta})\|\psi_- - \psi_-'\|_{*,\delta}.
\end{align*}
\end{claim}

\begin{proof}[Proof of Claim~\ref{claim 3}]
This follows from the estimate
\begin{align*}
\| \mathcal{L}_+(\psi_{-,T},\psi_{+,T}')-\mathcal{L}_+(\psi_{-,T}',\psi_{+,T}')\|^{+,T}_\delta & \leq C (e^{-2\delta T}+ r^{-1} + \|\psi_+'\|_{*,\delta})\|\psi_- - \psi_-'\|_{*,\delta}.
\end{align*}
as well as the calculations from Claim~\ref{claim 1}.
\end{proof}

\vskip.1in\noindent
Now we continue the proof of Lemma~\ref{azalea}.

(2)
Since $\psi_+=P_+(\psi_-)$ satisfies $\mathcal{I}_+(\psi_-,\psi_+)=\psi_+$, Claim~\ref{claim 1} gives:
\begin{align*}
\|\psi_+\|_{*,\delta}  \leq & \widetilde C(\| \psi_+\|_{*,\delta}) (\|e_{+}\|_{*,\delta} + r^{-1/2}e^{-2(\lambda-\delta) T} +(e^{-2\delta T}+r^{-1})\|\psi_-\|_{*,\delta}\\
&+\|\psi_-\|_{*,\delta}\|\psi_+\|_{*,\delta} +e^{-2\lambda T}\|\psi_+\|_{*,\delta}+\|\psi_+\|^2_{*,\delta}).
\end{align*}
By moving the last three terms on the right-hand side to the left, we obtain (2) since $\widetilde C(\| \psi_+\|_{*,\delta})(\|\psi_-\|_{*,\delta}+e^{-2\lambda T}+ \|\psi_+\|_{*,\delta})<1 $ for $\widetilde\varepsilon>0$ small and $r>0$ large.

\vskip.1in
(3) Letting $\psi_+=P_+(\psi_-)$ and $\psi_+'= P_+(\psi_-')$,
\begin{align*}
\| P_+(\psi_-)- & P_+(\psi_-')\|_{*,\delta} =  \| \mathcal{I}_+ (\psi_-,\psi_+)- \mathcal{I}_+ (\psi_-',\psi_+')\|_{*,\delta}\\
\leq & \| \mathcal{I}_+ (\psi_-,\psi_+)- \mathcal{I}_+ (\psi_-,\psi'_+)\|_{*,\delta} + \| \mathcal{I}_+ (\psi_-,\psi'_+) - \mathcal{I}_+ (\psi_-',\psi_+')  \|_{*,\delta},
\end{align*}
and the two terms on the right-hand side are bounded using Claims~\ref{claim 2} and \ref{claim 3}.  The terms from the right-hand side of the inequality in Claim~\ref{claim 2} can be moved to the left.  $\|\psi_-\|_{*,\delta}\leq \widetilde \varepsilon$ by assumption. If $r,T\gg 0$, $\widetilde\varepsilon>0$ is small, and $\|e_{+}\|_{*,\delta}$ is small, then $\|\psi_+\|_{*,\delta}=\|P_+(\psi_-)\|_{*,\delta}\leq \widetilde \varepsilon$ by (2). (3) then follows.
\end{proof}

Analogously, we have the following for $P_-$:

\begin{lemma} \label{Tsutsuji}
There exist $r\gg 0$ and $C, \widetilde\varepsilon>0$ such that for $T\gg 0$ and $\|e_-\|_{*,\delta}$ small the following holds:
\begin{enumerate}
\item There is a map $P_-:\mathcal{B}_+\to \mathcal{H}_-$ such that $\Theta_-(P_-(\psi_+),\psi_+))=0$ and $\psi_-=P_-(\psi_+)\in \mathcal{H}_-^{\perp,\psi_-,T}$.
\item $\| P_-(\psi_+)\|_{*,\delta}\leq C (\|e_{-}\|_{*,\delta}+ r^{-1/2}e^{-2(\lambda-\delta) T}+(e^{-2\delta T}+r^{-1})\|\psi_+\|_{*,\delta})$.
\item $\| P_-(\psi_+) - P_-(\psi'_+)\|_{*,\delta} \leq C \| \psi_+ - \psi_+'\|_{*,\delta} (e^{-2\delta T} + r^{-1} +\widetilde\varepsilon)$.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{estimate for psi and tau}
There exist $r\gg 0$, $\widetilde\varepsilon>0$, and $\widetilde\varepsilon_0>0$ such that for $T\gg 0$ there is a unique solution $(\psi_-,\psi_+)\in \mathcal{B}_-\times \mathcal{B}_+$ to the equations $\Theta_+(\psi_-,\psi_+)=0$ and $\Theta_-(\psi_-,\psi_+)=0$ subject to the constraints $\psi_\pm\in \mathcal{H}_\pm^{\perp,\psi_\pm,T}$.
\end{lemma}

Alternatively, we can view the desired $(\psi_-,\psi_+)$ as the unique fixed point of the map
$$\mathcal{I}= (\mathcal{I}_-,\mathcal{I}_+): \mathcal{B}_-\times\mathcal{B}_+ \to \mathcal{H}_-\times\mathcal{H}_+,$$
$$(\psi_-,\psi_+)\mapsto (\mathcal{I}_-(\psi_-,\psi_+),\mathcal{I}_+(\psi_-,\psi_+)),$$
subject to the constraints $\psi_\pm\in \mathcal{H}_\pm^{\perp,\psi_\pm,T}$.

\begin{proof}
We are looking for the unique solution $\psi_-$ to $P_-\circ P_+(\psi_-)=\psi_-$ subject to the constraints.  The existence follows from the contraction mapping principle and Lemmas~\ref{azalea} and \ref{Tsutsuji}. Then $(\psi_-,\psi_+=P_+(\psi_-))$ is the unique solution to $\Theta_\pm(\psi_-,\psi_+)=0$.
\end{proof}

\begin{rmk} \label{rmk: iteration}
In more practical terms, the fixed point can be obtained by starting with
$$(\psi_-^{(0)},\psi_+^{(0)})=(0,0),$$
applying the iteration
$$(\psi_-^{(i+1)},\psi_+^{(i+1)})=\mathcal{I}(\psi_-^{(i)},\psi_+^{(i)}),$$
and taking the limit $i\to\infty$. By Claim~\ref{claim 1} and its analog for $\mathcal{I}_-$, $\|\psi_\pm^{(1)}\|_{*,\delta}\to 0$ as $T\to\infty$ and similarly $\|\psi_\pm^{(i)}\|_{*,\delta}\to 0$ as $T\to\infty$.
\end{rmk}

The above remark implies the following lemma:

\begin{lemma} \label{lemma: T to infty}
If $(\psi_-,\psi_+)\in \mathcal{B}_-\times \mathcal{B}_+$ is the unique solution to $\Theta_+(\psi_-,\psi_+)=0$ and $\Theta_-(\psi_-,\psi_+)=0$ subject to the constraints $\psi_\pm\in \mathcal{H}_\pm^{\perp,\psi_\pm,T}$, then $\|\psi_\pm\|_{*,\delta}\to 0$ as $T\to \infty$.
\end{lemma}

We can finally define the gluing map  as
$$G:V_-/\mathbb R \times V_+/\mathbb R \times [T_0,\infty)\to \widetilde{\mathcal{G}}^{E'}_\delta(V_-/\mathbb{R}, V_+/\mathbb{R})$$
$$(u_-,u_+,T)\mapsto u= \exp_{u_*}(\beta_{+,T}\psi_{+,T} +\beta_{-,T}\psi_{-,T}),$$
where $(\psi_-,\psi_+)$ is the solution from Lemma~\ref{estimate for psi and tau}.

\subsection{$C^1$-smoothness and injectivity of the gluing map} \label{subsection: C1 smoothness}

The goal of this subsection is to prove:

\begin{thm} \label{thm: gluing map smooth}
The gluing map $G$ is a $C^1$-smooth embedding.
\end{thm}

We start by observing that:

\begin{lemma} \label{lemma: psi pm smooth}
The solution $(\psi_-,\psi_+)$ from Lemma~\ref{estimate for psi and tau} is $C^\infty$-smooth.
\end{lemma}

\begin{proof}[Sketch of proof]
The solution $(\psi_-,\psi_+)$ is in Morrey class $\mathcal{H}_{1,\delta}$ and hence is in Sobolev class $W^{1,2}_\delta$ and in $C^0$.  We apply the usual elliptic bootstrapping technique, where the necessary estimates for Morrey spaces are given in \cite{Mo}: First observe that $(\psi_-,\psi_+)$ satisfies a nonlinear Cauchy-Riemann type operator. We can then view $(\psi_-,\psi_+)$ as satisfying a {\em linear} Cauchy-Riemann type operator whose coefficients depend on $(\psi_-,\psi_+)$. Differentiating Equation~\eqref{theta+-} with respect to $s$, we also see that $({\partial\psi_-\over \partial s},{\partial \psi_+\over \partial s})$ satisfies a linear Cauchy-Riemann type operator such that the coefficients of the first order part are in $\mathcal{H}_{1,\delta}$ and the coefficients of the zeroth order part are in $\mathcal{H}_{0,\delta}$; the same holds for ${\partial \psi_\pm\over \partial t}$.  With the above conditions on the coefficients, the relevant elliptic estimates are given by the last two inequalities on \cite[p.~145]{Mo} to improve $\nabla \psi_\pm$ from Sobolev class $W^{0,2}_\delta$ to $W^{1,2}_\delta$ and \cite[Theorem 5.4.1]{Mo} to improve $\nabla\psi_\pm$ from $W^{1,2}_\delta$ to $\mathcal{H}_{1,\delta}$.
\end{proof}

Let $\mathcal{S}_\pm\subset V_\pm/\mathbb{R}$ be small neighborhoods of $u_\pm$ in $V_\pm/\mathbb{R}$. Each point of $\mathcal{S}_\pm$ is given by the pair $(\exp_{u_\pm } \phi_\pm ,j_\pm')$, where $\phi_\pm=(\phi^\circ_\pm,\frak{a}_\pm)\in \mathcal{H}_\pm\oplus \mathbb{R}^{2l^\pm}$, $j_\pm'\in \widetilde{\mathcal{U}}_\pm$, $l^\pm$ is the total number of punctures of $\dot F_\pm$ (cf.\ Section 4.2.1), and $\widetilde{\mathcal{U}}_\pm$ is a Teichm\"uller slice containing $j_\pm$ (the complex structure for $u_\pm$). We will usually write $(\phi_\pm,j_\pm')\in \mathcal{S}_\pm$.

{\em By abuse of notation we write $\|\phi_\pm^\circ+{\frak a}_\pm\|_{*,\delta}$ or $\| \phi_\pm \|_{*,\delta}$ for the sum of $\|\phi_\pm^\circ\|_{*,\delta}$ on $\mathcal{H}_\pm$ and the standard norm $|{\frak a}_\pm|$ on $\mathbb{R}^{2l^\pm}$.}

We assume that:
\begin{itemize}
\item[(T$_1$)] the slice $\widetilde{\mathcal{U}}_\pm$ is smooth; and
\item[(T$_2$)] there exists a small disk $D_\pm\subset \dot F_\pm- \mathcal{E}_{\pm,T+T_0}$ such that all $j'_\pm\in \widetilde{\mathcal{U}}_\pm$ agree on $\dot F_\pm -D_\pm$.
\end{itemize}
We may additionally assume that, for any ${\frak a}$ in the summand $\mathbb{R}^2$ of $\mathbb{R}^{2l^+}$ (resp.\ $\mathbb{R}^{2l^-}$) corresponding to the end that is glued, ${\frak a}$ is a linear combination of $\partial_s$ and $\partial_t$ on $s\leq 0$ for $u_+$ (resp.\ on $s\geq 0$ for $u_-$).

We say that $\phi_\pm=(\phi_\pm^\circ,{\frak a}_\pm)\in \mathcal{H}_\pm\oplus \mathbb{R}^{2l^\pm}$ or $\phi_\pm^\circ\in \mathcal{H}_\pm$ {\em satisfies} ($\#_k$) if the following holds:
\begin{enumerate}
\item[($\#_k$)] $\phi_\pm^\circ$ is in class $C^k$ and $\|\nabla^i \phi_\pm^\circ|_{-2(T+T_0)-1<\pm s<1}\|_\delta$, $\forall i\leq k$, is bounded above by $C\|\phi_\pm^\circ\|_{*,\delta}$.
\end{enumerate}
Here $C$ is a constant that does not depend on $\phi_\pm^\circ$. Note that ($\#_k$) for $\phi_\pm^\circ$ implies ($\#_k$) for $\phi_\pm^\circ+{\frak a}_\pm$, since ${\frak a}_\pm$ lives in the finite-dimensional vector space $\mathbb{R}^{2l^\pm}$. In what follows we assume that $k\gg 0$.

Our gluing setup is slightly more complicated than one initially expects: this is to get around the loss of one derivative as explained in Lemma~\ref{lemma: sufficiently differentiable}.  Let
$$\mathcal{J}=(\mathcal{J}_-,\mathcal{J}_+):\mathcal{S}_-\times \mathcal{S}_+ \times [T_0,\infty) \to\mathcal{B}_-\times\mathcal{B}_+$$
be a $C^1$-smooth function such that $\operatorname{Im}(\mathcal{J})$ is bounded and each $(\psi_-,\psi_+)\in \operatorname{Im}(\mathcal{J})$ satisfies ($\#_k$).  We then consider the function
$$\mathcal{I}^{\mathcal{J}}=(\mathcal{I}^{\mathcal{J}}_-,\mathcal{I}^{\mathcal{J}}_+): \mathcal{S}_-\times \mathcal{S}_+ \times [T_0,\infty) \to \mathcal{H}_-\times \mathcal{H}_+,$$
\begin{align*}
\mathcal{I}^{\mathcal{J}}_+&((\phi_-,j_-'),(\phi_+,j_+'),T)\\
=& -\tau^{-1}_{\pm(T+T_0)}(\Pi_+^{\psi_+ + \phi_+,j_+', T}D_{+,T}^{j_+'})^{-1}\Pi_+^{\psi_+ + \phi_+,j_+',T} [D_{+,T}^{j_+'}(\psi_{+,T} + \phi_{+,T}) +  e_{+,T} \\
& \qquad \qquad +\mathcal L_+(\psi_{-,T} +\phi_{-,T},\psi_{+,T}+\phi_{+,T}) +\mathcal{R}_+(\psi_{+,T}+\phi_{+,T})],
\end{align*}
where $\psi_\pm=\mathcal{J}_\pm((\phi_-,j_-'),(\phi_+,j_+'),T)$, $D_{\pm,T}^{j_\pm'}$ is $D_{\pm,T}$ with respect to $j_\pm'$, $j'_+$ in $\Pi_+^{\psi_+ + \phi_+,j_+',T}$ indicates the dependence on $j_+'$, and $e_{+,T}=\overline\partial u_{\pm,T}$.
$\mathcal{I}^{\mathcal{J}}_-$ is defined analogously.  We will later apply an iteration scheme similar to the one described in Remark~\ref{rmk: iteration}; see the proof of Theorem~\ref{thm: gluing map smooth} at the end of this subsection.

\begin{rmk}
Observe that $\mathcal{I}^{\mathcal{J}}_+$ has the term $D_{+,T}^{j_\pm'}(\psi_{+,T} + \phi_{+,T})$ unlike the expression for $\mathcal{I}_+$ which does not have $D_{+,T}\psi_{+,T}$.
\end{rmk}

The expression inside the brackets $[\cdot]$ can be written as
\begin{align*}
D_{+,T}^{j_\pm'}\psi_{+,T} + e_{+,T}(\phi_+) +   \mathcal L_+(\psi_{-,T} +\phi_{-,T},\psi_{+,T}+\phi_{+,T}) +\mathcal{R}'_+(\psi_{+,T}+\phi_{+,T}),
\end{align*}
where $\mathcal{R}'_+(\psi_{+,T}+\phi_{+,T})= \mathcal{R}_+(\psi_{+,T}+\phi_{+,T})- \mathcal{R}_+(\phi_{+,T})$
and $e_{+,T}(\phi_+):= \overline\partial \exp_{u_{+,T}}(\phi_{+,T})$.

\begin{rmk}
In order to avoid the (already cumbersome) notation, in the rest of the subsection we omit $j'_\pm$ from the notation and also will not explicitly treat variations of $j_\pm$ in the proof of Theorem~\ref{thm: gluing map smooth}.  The derivatives of the gluing map with respect to $j'_\pm$ are straightforward to control in view of (T$_1$) and (T$_2$).
\end{rmk}

\begin{lemma} \label{lemma: sharp}
Assuming $\mathcal{S}_\pm$ are sufficiently small, all $\phi_\pm\in \mathcal{S}_\pm$ satisfy $(\#_k)$.
\end{lemma}

\begin{proof}
This follows from the elliptic bootstrapping estimates used in the proof of Lemma~\ref{lemma: psi pm smooth}.
\end{proof}

Next let us write $B(\psi_++\phi_+,T) = (\Pi_+^{\psi_+ + \phi_+,T} D_+)^{-1}\Pi_+^{\psi_++\phi_+,T}.$

\begin{lemma} \label{lemma: sufficiently differentiable}
With the above conditions on $\mathcal{J}$, $B(\psi_++\phi_+,T)$ is sufficiently differentiable with respect to $\phi_+\in \mathcal{S}_+$ and $T\in [T_0,\infty)$.
\end{lemma}

\begin{proof}
This uses the arguments used in Claim~\ref{claim: continuous} as well as Lemma~\ref{lemma: sharp}. The reason for requiring ($\#_k$) is that when we differentiate $e_i(\psi_+ + \phi_+,T)$ with respect to $\phi_+$ or $T$, we lose one derivative, which must be recovered using ($\#_k$).
\end{proof}

\begin{lemma}\label{lemma: I smoothness}
If $\mathcal{S}_\pm$ and $\widetilde\varepsilon>0$ are sufficiently small, $T,r\gg 0$, $\mathcal{J}$ is $C^1$-smooth, and $\operatorname{Im}(\mathcal{J})$ is bounded, then $\mathcal{I}^{\mathcal{J}}$ is $C^1$-smooth. Moreover, if $\operatorname{Im}(\mathcal{J})$ lies in a sufficiently small ball about the origin and the derivative $D\mathcal{J}$ is small, then the derivative $D\mathcal{I}^{\mathcal{J}}$ is small.
\end{lemma}

\begin{proof}
We write $\psi_\pm=\mathcal{J}_\pm(\phi_-,\phi_+,T)$.   Suppose that $\mathcal{J}_\pm$ is a constant function; we later explain how to modify the proof in the general case.

We first consider the partial derivative $D^\flat_1\mathcal{I}_+^{\mathcal{J}}$, where the superscript $\flat$ indicates that we are assuming that $\mathcal{J}_\pm$ is constant. Let $\zeta_\pm +{\frak a}_\pm \in T_{u_\pm}\mathcal{S}_\pm$, where $\zeta_\pm\in\mathcal{H}_\pm$ and ${\frak a}_\pm\in \mathbb{R}^{2l^\pm}$. We compute
\begin{align} \label{I}
D^\flat_1 & \mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T)(\zeta_-+{\frak a}_-) \\
\nonumber & = \tfrac{d}{d\tau}|_{\tau=0}B(\psi_+ +\phi_+,T) (\mathcal{L}_+(\psi_{-,T}+\phi_{-,T}+\tau(\zeta_{-,T}+{\frak a}_{-,T}),\psi_{+,T}+\phi_{+,T}))\\
\nonumber &= B(\psi_+ +\phi_+,T)  (\tfrac{\partial\beta_{-,T}}{\partial s}(\zeta_{-,T}+{\frak a}_{-,T}) + {\frak l}(\eta_{+,T})\tfrac{\partial\zeta_{-,T}}{\partial t} + {\frak l}(\psi_{+,T}+\phi_{+,T})\tfrac{\partial\zeta_{-,T}}{\partial t}),
\end{align}
using the assumption that ${\frak a}_-$ is constant for $s\geq 0$.
Estimates similar to those of Claim~\ref{claim 1} imply that
\begin{align} \label{result of I}
\|D^\flat_1 \mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T)(\zeta_-+{\frak a}_-) \|_{*,\delta} \leq c(r^{-1} + \widetilde{C}(T) + \|\psi_+\|_{*,\delta} +\|\phi_+\|_{*,\delta})\cdot \|\zeta_{-}+{\frak a}_-\|_{*,\delta},
\end{align}
where $\widetilde{C}(T)\to 0$ as $T\to\infty$. Hence the partial derivative $D^\flat_1$ exists. We are assuming that $\|\phi_+\|_{*,\delta}$ is sufficiently small and $T,r\gg 0$. If we assume that $\|\psi_+\|_{*,\delta}$ in addition, then there exists $0<C\ll 1$ such that
$$\|D^\flat_1 \mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T)(\zeta_-+{\frak a}_-) \|_{*,\delta} \leq C\cdot \|\zeta_{-}+{\frak a}_-\|_{*,\delta}.$$

Similarly,
\begin{align*}
(D^\flat_1 &\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)- D^\flat_1 \mathcal{I}_+^{\mathcal{J}}(\phi_-',\phi_+',T')) (\zeta_-+{\frak a}_-) \\
= & B(\psi_+ +\phi_+,T) (\tfrac{\partial\beta_{-,T}}{\partial s}(\zeta_{-,T}+{\frak a}_{-,T}) + {\frak l}(\eta_{+,T})\tfrac{\partial\zeta_{-,T}}{\partial t} + {\frak l}(\psi_{+,T}+\phi_{+,T})\tfrac{\partial\zeta_{-,T}}{\partial t}) \\
& - B(\psi_+ +\phi_+', T') (\tfrac{\partial\beta_{-,T'}}{\partial s}(\zeta_{-,T'}+{\frak a}_{-,T'}) + {\frak l}(\eta_{+,T'})\tfrac{\partial\zeta_{-,T'}}{\partial t} + {\frak l}(\psi_{+,T'}+\phi'_{+,T'})\tfrac{\partial\zeta_{-,T'}}{\partial t}),
\end{align*}
and in view of Lemma~\ref{lemma: sufficiently differentiable} and the ($\#_k$)-condition applied to $\zeta_-$ there exists a constant $C>0$ such that:
\begin{align*}
\|((D^\flat_1 &\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)- D^\flat_1 \mathcal{I}_+^{\mathcal{J}}(\phi_-',\phi_+',T')) (\zeta_-+{\frak a}_-)\|_{*,\delta} \\
& \leq C (\|\phi_+-\phi_+'\|_{*}^c + |T-T'|) \cdot \|\zeta_{-}+{\frak a}_-\|_{*,\delta}.
\end{align*}
(Moreover, $0<C\ll 1$.) Here the superscript $c$ means restriction of the function to
$$-(\max(T,T')+T_0) \leq s\leq (\max(T,T') + T_0).$$
This proves that $D^\flat_1\mathcal{I}_+^{\mathcal{J}}$ is in $C^0$.

Next we compute
\begin{align} \label{main}
D^\flat_2  \mathcal{I}_+^{\mathcal{J}} (\phi_-,&\phi_+,T)(\zeta_++{\frak a}_+) =   \tfrac{dB(\psi_++\phi_++\tau(\zeta_++{\frak a}_+),T)}{d\tau}|_{\tau=0}(v (\psi_-+\phi_-,\psi_++\phi_+,T)) \\
\nonumber &  +B(\psi_+ +\phi_+,T) (D^\flat_{+,T}(\zeta_{+,T}+{\frak a}_{+,T}) \\
\nonumber & \quad + D^\flat_2 \mathcal{L}_+(\psi_{-,T}+\phi_{-,T},\psi_{+,T}+\phi_{+,T})(\zeta_{+,T}+{\frak a}_{+,T}) \\
\nonumber & \quad +\tfrac{d}{d\tau}|_{\tau=0} e_{+,T}(\phi_+ +\tau(\zeta_++{\frak a}_+)) + D^\flat\mathcal{R}'_+(\psi_{+,T}+\phi_{+,T}) (\zeta_{+,T}+{\frak a}_{+,T})),
\end{align}
where we are writing
\begin{align*}
v(\psi_-+\phi_-,\psi_++\phi_+,T)& =D^\flat_{+,T}\psi_{+,T} +e_{+,T}(\phi_+) + \mathcal{L}_+(\psi_{-,T}+\phi_{-,T},\psi_{+,T}+\phi_{+,T}) \\
& \qquad\qquad+ \mathcal{R}'_+(\psi_{+,T}+\phi_{+,T}).
\end{align*}
We compute
\begin{align} \label{L}
D^\flat_2 \mathcal{L}_+(&\psi_{-,T}+\phi_{-,T}, \psi_{+,T}+\phi_{+,T})(\zeta_{+,T}+{\frak a}_{+,T}) \\
\nonumber & = {\frak l}(\tfrac{\partial(\psi_{-,T}+\phi_{-,T})}{\partial t}) (\zeta_{+,T}+{\frak a}_{+,T})  + {\frak l}(\eta_{+,T})\tfrac{\partial\zeta_{+,T}}{\partial t},
\end{align}
\begin{align}\label{R}
D^\flat\mathcal{R}'_+  (\psi_{+,T}&+\phi_{+,T})  (\zeta_{+,T}+{\frak a}_{+,T}) \\
\nonumber = & {\frak l}(\psi_{+,T}+\phi_{+,T}) (\zeta_{+,T}+{\frak a}_{+,T}) + {\frak l}(\psi_{+,T}+\phi_{+,T})(\nabla \zeta_{+,T}) \\
\nonumber &  + {\frak l}(\nabla (\psi_{+,T}+\phi_{+,T}))(\zeta_{+,T}+{\frak a}_{+,T}).
\end{align}
Combining Equations~\eqref{main}, \eqref{L}, and \eqref{R} we obtain
\begin{align}\label{II}
D^\flat_2 & \mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T)(\zeta_++{\frak a}_+) \\
\nonumber =&  \tfrac{dB(\psi_++\phi_++\tau(\zeta_++{\frak a}_+),T)}{d\tau}|_{\tau=0}(v(\psi_- +\phi_-,\psi_+ + \phi_+,T)) \\
\nonumber & + B(\psi_+ +\phi_+,T)  [\tfrac{d}{d\tau}|_{\tau=0} e_{+,T}(\phi_+ +\tau(\zeta_++{\frak a}_+)) \\
\nonumber & \quad +{\frak l}(\tfrac{\partial(\psi_{-,T}+\phi_{-,T})}{\partial t}) (\zeta_{+,T}+{\frak a}_{+,T})  + {\frak l}(\eta_{+,T})\tfrac{\partial\zeta_{+,T}}{\partial t} + {\frak l}(\psi_{+,T}+\phi_{+,T})(\zeta_{+,T}+{\frak a}_{+,T}) \\
\nonumber & \quad + {\frak l}(\psi_{+,T}+\phi_{+,T})(\nabla \zeta_{+,T}) + {\frak l}(\nabla (\psi_{+,T}+\phi_{+,T}))(\zeta_{+,T}+{\frak a}_{+,T})].
\end{align}

Let $[\cdot]$ denote the expression inside the brackets. We first bound
$$\| \tfrac{d}{d\tau}|_{\tau=0} e_{+,T}(\phi_+ +\tau(\zeta_++{\frak a}_+))\|_\delta^{+,T}\leq c \|e_+\|_{*,\delta}\cdot \|\zeta_++{\frak a}_+\|_{*,\delta},$$
where $c>0$.  The remainder of the terms can be estimated as in Claim~\ref{claim 1} and:
\begin{align}\label{result of II}
\|[\cdot]\|_{\delta}^{+,T} & \leq   c(\|e_+\|_{*,\delta}+\widetilde C(T)+\|\psi_+\|_{*,\delta}+\|\psi_-\|_{*,\delta}\\
\nonumber & \qquad\qquad +\|\phi_+\|_{*,\delta} + \|\phi_-\|_{*,\delta} )\cdot \|\zeta_++{\frak a}_+\|_{*,\delta},
\end{align}
where $\widetilde{C}(T)\to 0$ as $T\to \infty$.

We also compute
\begin{align*}
\tfrac{d}{d\tau}|_{\tau=0} &\Pi_+^{\psi_+ + \phi_+ + \tau (\zeta_++{\frak a}_+),T}(v) \\
= &  \sum_i\langle \tfrac{d}{d\tau}|_{\tau=0}e_i(\psi_++\phi_+ + \tau (\zeta_++{\frak a}_+),T),v\rangle e_i(\psi_++\phi_+,T)  \\
& + \sum_i \langle e_i(\psi_++\phi_+,T),v \rangle \tfrac{d}{d\tau}|_{\tau=0} e_i(\psi_++\phi_+ + \tau (\zeta_++{\frak a}_+),T),
\end{align*}
where $\tfrac{d}{d\tau}|_{\tau=0}e_i(\psi_+ +\phi_+ + \tau (\zeta_++{\frak a}_+),T)$ has terms of the form
$$\langle \nabla e_i(\psi_++\phi_+,T)\cdot  (\zeta_++{\frak a}_+), e_j(\psi_+ +\phi_+,T)\rangle.$$
We then bound
\begin{align} \label{result of II prime}
\|\tfrac{d}{d\tau}|_{\tau=0}& \Pi_+^{\psi_+ + \phi_+ + \tau (\zeta_++{\frak a}_+),T}(v)\|_\delta^T \leq c \|v\|_\delta^T \cdot \|\zeta_++{\frak a}_+\|_{*,\delta}\\
\nonumber &\leq c(\|\psi_+\|_{*,\delta}+\|\psi_-\|_{*,\delta} + \|\phi_+\|_{*,\delta} + \|\phi_-\|_{*,\delta})\|\zeta_++{\frak a}_+\|_{*,\delta}.
\end{align}
Combining Equations~\eqref{result of II} and \eqref{result of II prime} we obtain
\begin{align} \label{result of II main}
\| D^\flat_2\mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T) (\zeta_++{\frak a}_+)\|_{*,\delta} &\leq c(\|e_+\|_{*,\delta}+\widetilde C(T)+\|\psi_+\|_{*,\delta}+\|\psi_-\|_{*,\delta}\\
\nonumber &\qquad +\|\phi_+\|_{*,\delta} + \|\phi_-\|_{*,\delta} )\cdot\|\zeta_++{\frak a}_+\|_{*,\delta}.
\end{align}
Hence the partial derivative $D^\flat_2\mathcal{I}_+$ exists.  Assuming that $\|\psi_\pm\|_{*,\delta}$, $\|\phi_\pm\|_{*,\delta}$, and $\|e_+\|_{*,\delta}$ are sufficiently small and $T\gg 0$, it follows that there exists $0<C\ll 1$ such that
\begin{align*}
\| D^\flat_2\mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T) (\zeta_++{\frak a}_+)\|_{*,\delta} & \leq C \|\zeta_++{\frak a}_+\|_{*,\delta}.
\end{align*}

Similarly,
\begin{align*}
(D^\flat_2  &\mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T)  -  D^\flat_2  \mathcal{I}_+^{\mathcal{J}} (\phi'_-,\phi'_+,T')) (\zeta_++{\frak a}_+)  \\
=&\tfrac{dB(\psi_++\phi_++\tau (\zeta_++{\frak a}_+),T)}{d\tau}|_{\tau=0}(v(\psi_-+\phi_-,\psi_++\phi_+,T))\\
&- \tfrac{dB(\psi_+ +\phi_+' + \tau (\zeta_++{\frak a}_+),T')}{d\tau}|_{\tau=0}(v(\psi_-+\phi_-',\psi_++\phi_+',T'))\\
&+  B(\psi_+ +\phi_+,T)  ({\frak l}(\tfrac{\partial(\psi_{-,T}+\phi_{-,T})}{\partial t}) (\zeta_{+,T}+{\frak a}_{+,T}) + {\frak l}(\eta_{+,T})\tfrac{\partial\zeta_{+,T}}{\partial t} \\
& \quad + {\frak l}(\psi_{+,T}+\phi_{+,T}) (\zeta_{+,T}+{\frak a}_{+,T})  + {\frak l}(\psi_{+,T}+\phi_{+,T})(\nabla \zeta_{+,T}) \\
& \quad + {\frak l}(\nabla (\psi_{+,T}+\phi_{+,T})) (\zeta_{+,T}+{\frak a}_{+,T}) )\\
&- B(\psi_+ +\phi'_+,T')  ({\frak l}(\tfrac{\partial(\psi_{-,T'} +\phi'_{-,T'})}{\partial t}) (\zeta_{+,T'}+{\frak a}_{+,T'}) + {\frak l}(\eta_{+,T'})\tfrac{\partial\zeta_{+,T'}}{\partial t}\\
&\quad + {\frak l}(\psi_{+,T'}+\phi'_{+,T'}) (\zeta_{+,T'}+{\frak a}_{+,T'})  + {\frak l}(\psi_{+,T'}+\phi_{+,T'}')(\nabla \zeta_{+,T'})\\
&\quad + {\frak l}(\nabla (\psi_{+,T'}+\phi'_{+,T'})) (\zeta_{+,T'}+{\frak a}_{+,T'}) ).
\end{align*}
We have bounds
\begin{align*}
\|{\frak l}(\tfrac{\partial(\psi_{-,T}+\phi_{-,T})}{\partial t})& (\zeta_{+,T}+{\frak a}_{+,T})  - {\frak l}(\tfrac{\partial(\psi_{-,T'} +\phi'_{-,T'})}{\partial t}) (\zeta_{+,T'}+{\frak a}_{+,T'}) \|_\delta \\
& \leq C (\| \phi_- - \phi_-'\|_{*,\delta}^c +|T-T'|) \cdot \|\zeta_++{\frak a}_+\|_{*,\delta},
\end{align*}
\begin{align*}
\| {\frak l}(\eta_{+,T})\tfrac{\partial\zeta_{+,T}}{\partial t} -  {\frak l}(\eta_{+,T'})\tfrac{\partial\zeta_{+,T'}}{\partial t}\|_\delta & \leq  C|T-T'| \cdot \|\zeta_+\|_{*,\delta},
\end{align*}
\begin{align*}
\|{\frak l}(\psi_{+,T}+\phi_{+,T})  & (\zeta_{+,T}+{\frak a}_{+,T})  -{\frak l}(\psi_{+,T'}+\phi'_{+,T'})  (\zeta_{+,T'}+{\frak a}_{+,T'})  \|_\delta \\
& \leq C (\| \phi_+ - \phi_+'\|_{*,\delta}^c +|T-T'|) \cdot \|\zeta_++{\frak a}_+\|_{*,\delta},
\end{align*}
\begin{align*}
\|{\frak l}(\psi_{+,T}+\phi_{+,T})& (\nabla \zeta_{+,T})  -{\frak l}(\psi_{+,T'}+\phi_{+,T'}')(\nabla \zeta_{+,T'})\|_\delta \\
& \leq C (\| \phi_+ - \phi_+'\|_{*,\delta}^c +|T-T'|) \cdot \|\zeta_+\|_{*,\delta},
\end{align*}
\begin{align*}
\| {\frak l}(\nabla (\psi_{+,T}+\phi_{+,T}))& (\zeta_{+,T}+{\frak a}_{+,T}) - {\frak l}(\nabla (\psi_{+,T'}+\phi'_{+,T'})) (\zeta_{+,T'}+{\frak a}_{+,T'}) )\|_\delta  \\
& \leq C (\| \phi_+ - \phi_+'\|_{*,\delta}^c +|T-T'|) \cdot  \|\zeta_++{\frak a}_+\|_{*,\delta}.
\end{align*}
Together with the bounds for
$$B(\psi_++\phi_+,T)-B(\psi_++\phi_+,T') ~ \mbox{ and } ~ \tfrac{dB(\psi_++\phi_++\tau(\zeta_++{\frak a}_+),T)}{d\tau}|_{\tau=0},$$
we obtain
\begin{align*}
\|(D^\flat_2  &\mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T)  -  D^\flat_2  \mathcal{I}_+^{\mathcal{J}} (\phi'_-,\phi'_+,T'))(\zeta_++{\frak a}_+)\|_{*,\delta} \\
&\leq C(\| \phi_+ - \phi_+'\|_{*,\delta}^c   + \| \phi_- - \phi_-'\|_{*,\delta}^c + |T-T'|)\cdot  \|\zeta_++{\frak a}_+\|_{*,\delta},
\end{align*}
where $C>0$. Hence $D^\flat_2\mathcal{I}_+^{\mathcal{J}}$ is in $C^0$.

Next consider
\begin{align} \label{III}
D^\flat_3\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)(\frak t)= & \tfrac{d}{d\tau}|_{\tau=0}B(\psi_+ + \phi_+, T+\tau{\frak t})(v(\psi_- + \phi_-, \psi_+ +\phi_+,T))\\
\nonumber &+ B(\psi_+ +\phi_+,T) \tfrac{d}{d\tau}|_{\tau=0} v(\psi_- + \phi_-, \psi_+ +\phi_+,T+\tau {\frak t}).
\end{align}
The bound
\begin{align} \label{result of III}
\|D^\flat_3\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)(\frak t)\|_{*,\delta} \leq c (\|\psi_+\|_{*,\delta}+\|\psi_-\|_{*,\delta} +\|\phi_+\|_{*,\delta} + \|\phi_-\|_{*,\delta} )\cdot |{\frak t}|,
\end{align}
follows from applying Lemma~\ref{lemma: sufficiently differentiable} to the first line and ($\#_k$) to the second.  Its differentiability also follows in a similar manner and is left to the reader.

Consider the general case where $\mathcal{J}_\pm$. One can then verify that
\begin{align} \label{difference}
D_1&\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)(\zeta_- +{\frak a}_-) - D_1^\flat\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)(\zeta_- +{\frak a}_-)\\
\nonumber &= D_1^\flat\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)\circ D\mathcal{J}_-(\phi_-,\phi_+,T)(\zeta_-+{\frak a}_-).
\end{align}
The second row of Equation~\eqref{difference} can be bounded above by $C\cdot\|\zeta_- +{\frak a}_-\|_{*,\delta}$ with $0<C\ll 1$ if  $\|\psi_\pm\|_{*,\delta}$ and $D\mathcal{J}$ are sufficiently small.  The situation for $D_2\mathcal{I}_+^{\mathcal{J}}$ and $D_3\mathcal{I}_+^{\mathcal{J}}$ are similar.

This completes the proof that $\mathcal{I}_+^{\mathcal{J}}$ and hence $\mathcal{I}^{\mathcal{J}}$ are in $C^1$.  Moreover, if $\operatorname{Im}(\mathcal{J})$ lies in a sufficiently small ball about the origin and the derivative $D\mathcal{J}$ is small, then $D\mathcal{I}^{\mathcal{J}}$ is small.
\end{proof}

We are now in a position to prove Theorem~\ref{thm: gluing map smooth}.

\begin{proof}[Proof of Theorem~\ref{thm: gluing map smooth}]$\mbox{}$

\vskip.1in\noindent
{\em $C^1$-smoothness.} Recall that we are assuming that $\mathcal{S}_\pm$ are sufficiently small neighborhoods of $u_\pm$. By Lemmas~\ref{azalea}--\ref{estimate for psi and tau} we see that the unique solution $(\psi_-,\psi_+)$ to $\Theta_\pm(\psi_-,\psi_+)=0$ for $(\phi_-,\phi_+,T)\in \mathcal{S}_-\times\mathcal{S}_+\times[T_0,\infty)$ is obtained as follows:  Let $\mathcal{J}_0=0$, $\mathcal{J}_1=\mathcal{I}^{\mathcal{J}_0}$, and $\mathcal{J}_i=\mathcal{I}^{\sum_{j=1}^{i-1} \mathcal{J}_j}$.  Also let us write $(\mathcal{J}_i)_\pm$ as the $\mathcal{H}_\pm$ component of $\mathcal{J}_i$. Then
\begin{equation}\label{eqn: psi pm}
(\psi_-,\psi_+)=\sum_{j=1}^\infty \mathcal{J}_j(\phi_-,\phi_+,T),
\end{equation}
and
\begin{equation}\label{eqn: defn of G}
G(\psi_-,\psi_+,T):= \exp_{u_*}(\beta_{-,T}(\phi_{-,T}+\psi_{-,T}) +\beta_{+,T}(\phi_{+,T}+\psi_{+,T})).
\end{equation}
Each $\mathcal{J}_j(\phi_-,\phi_+,T)$ is in $C^1$ by Lemma~\ref{lemma: I smoothness} and it remains to show a bound of type $\|D\mathcal{J}_i\|\leq C^i$, where $0<C\ll 1$.

We will check the easiest case $\mathcal{J}_2=\mathcal{I}^{\mathcal{J}_1}$, leaving the higher $\mathcal{J}_i$ to the reader. Let $\psi^1_\pm=(\mathcal{J}_1)_\pm(\phi_-,\phi_+,T)$. Then
\begin{align} \label{uno}
\Pi_+^{\phi_{+},T}D_{+,T}(\psi_{+,T}^1) =-\Pi_+^{\phi_{+},T}(&D_{+,T}(\phi_{+,T}) +  e_{+,T}   \\
\nonumber & +\mathcal L_+(\phi_{-,T},\phi_{+,T}) +\mathcal{R}_+(\phi_{+,T})).
\end{align}
Using Equation~\eqref{uno} we can write:
\begin{align} \label{eqn: J 2}
(\mathcal{J}_2)_\pm & (\phi_-,\phi_+,T) \\ \nonumber=&(\Pi_+^{\psi_+^1+\phi_+,T}D_{+,T})^{-1}(\Pi_+^{\psi_+^1+\phi_+,T}-\Pi_+^{\phi_+,T})(D_{+,T}(\psi_{+,T}^1+\phi_{+,T})+e_{+,T}\\
\nonumber &\qquad \qquad +\mathcal L_+(\psi_{-,T}^1 +\phi_{-,T},\psi_{+,T}^1+\phi_{+,T}) + \mathcal{R}_+(\psi_{+,T}^1+\phi_{+,T}) )\\
\nonumber & +(\Pi_+^{\psi_+^1+\phi_+,T}D_{+,T})^{-1}\Pi_+^{\phi_+,T} ( \mathcal L_+(\psi_{-,T}^1 +\phi_{-,T},\psi_{+,T}^1+\phi_{+,T})- \mathcal L_+(\phi_{-,T},\phi_{+,T})\\
\nonumber & \qquad \qquad +\mathcal{R}_+(\psi_{+,T}^1+\phi_{+,T}) -\mathcal{R}_+(\phi_{+,T})).
\end{align}

Calculations similar to those of Lemma~\ref{lemma: I smoothness} imply that:
\begin{align*}
\| D \mathcal{J}_2&(\phi_-,\phi_+,T) (\zeta_-+{\frak a}_-,\zeta_++{\frak a}_+,{\frak t})\|_{*,\delta}\\
& \leq c (\|\psi_-^1\|_{*,\delta}+\|\psi_+^1\|_{*,\delta})(\|\zeta_-+{\frak a}_-\|_{*,\delta}+ \|\zeta_++{\frak a}_+\|_{*,\delta}+|{\frak t}|).
\end{align*}
[Sketch of calculation: The derivative of Equation~\eqref{eqn: J 2} consists of six terms since Equation~\eqref{eqn: J 2} is the sum of two terms, each of which is a product of three terms. The six terms can all be estimated in the same manner and we calculate some representative terms. We estimate
\begin{align}
\label{temp1} \tfrac{d}{d\tau}|_{\tau=0}( \mathcal L_+(\psi_{-,T}^1 +\phi_{-,T}+&\tau \zeta_{-,T},\psi_{+,T}^1+\phi_{+,T}+\tau\zeta_{+,T})\\
\nonumber &- \mathcal L_+(\phi_{-,T}+\tau\zeta_{-,T},\phi_{+,T}+\tau\zeta_{+,T}))
\end{align}
as follows.  Observe that $\mathcal L_+$ is bilinear in each variable.  Hence we have an expression of the form
\begin{align*}
(\psi_- + \phi_-+\tau\zeta_-)(\psi_+ + \phi_+ + \tau\zeta_+)-(\phi_-+\tau\zeta_-)(\phi_+ +\tau\zeta_+)\\
= (\psi_-+\phi_-)(\psi_+ + \phi_+) -\phi_-\phi_+ + \tau(\zeta_-\psi_+ + \zeta_+\psi_-) +\mbox{h.o.}
\end{align*}
Hence Equation~\eqref{temp1} is bounded above by
$$ c(\|\psi_-^1\|_{*,\delta}+\|\psi_+^1\|_{*,\delta})(\|\zeta_-\|_{*,\delta}+\|\zeta_+\|_{*,\delta}).$$
Similarly,
\begin{align}
\mathcal L_+(\psi_{-,T}^1 +\phi_{-,T},\psi_{+,T}^1+\phi_{+,T})- \mathcal L_+(\phi_{-,T},\phi_{+,T})
\end{align}
is bounded above by
$$c(\|\psi_-^1\|_{*,\delta}+\|\psi_+^1\|_{*,\delta})(\|\phi_-\|_{*,\delta}+\|\phi_+\|_{*,\delta}).$$
This concludes the sketch.]
We also calculate:
\begin{align*}
\|(D \mathcal{J}_2&(\phi_-,\phi_+,T)-  D \mathcal{J}_2(\phi_-',\phi_+',T'))(\zeta_-+{\frak a}_-,\zeta_++{\frak a}_+,{\frak t})\|_{*,\delta}\\
& \leq c (\|\psi_-^1\|_{*,\delta}+\|\psi_+^1\|_{*,\delta})(\|\phi_+-\phi'_+\|^c_* + \|\phi_--\phi_-'\|^c_*+|T-T'|)\\
& \qquad \cdot (\|\zeta_-+{\frak a}_-\|_{*,\delta} + \|\zeta_++{\frak a}_+\|_{*,\delta}+|{\frak t}|).
\end{align*}
This completes the proof of Theorem~\ref{thm: gluing map smooth}.

\vskip.1in\noindent
{\em Local embedding.} It suffices to show that $G$ is a local $C^1$-embedding, i.e., $DG$ is an isomorphism at any $(\phi_-,\phi_+,T)$, provided $T_0\gg 0$. (It is not hard to see that, for any $d>0$, there exists $T_0\gg 0$ such that if $(\phi_-,\phi_-,T)$ and $(\phi_-',\phi_+',T')$ are a distance $d$ apart, then $G(\phi_-,\phi_+,T)$ and $G(\phi_-',\phi_+',T')$ cannot be equal.)

First consider the map
$$H:(\phi_-,\phi_+,T)\mapsto (\phi_-,\phi_+,T)+\sum_{j=1}^\infty(\mathcal{J}_j(\phi_-,\phi_+,T),0).$$
Since its leading term is the identity map and the subsequent terms $\mathcal{J}_j$  have derivatives bounded by a constant $C^j$ with $0<C\ll 1$, it follows that $DH$ is invertible at any $(\phi_-,\phi_+,T)$, provided $\widetilde\varepsilon>0$ is sufficiently small and $T_0>0$ is sufficiently large.

Next, instead of $G$ we consider the map
$$\widetilde G: (\phi_-,\phi_+,T)\mapsto \beta_{-,T}(\phi_{-,T}+\psi_{-,T})+\beta_{+,T}(\phi_{+,T}+\psi_{+,T})$$
without the $\exp$.  If $T_0\gg 0$, then from the considerations of the previous paragraph $D\widetilde G(\phi_-,\phi_+,T)|_{T_{\phi_-}\mathcal{S}_-\times T_T[T_0,\infty)}$ is very close to
$$(\zeta_-+{\frak a}_-,{\frak t})\mapsto \zeta_{-,T}+{\frak a}_{-,T}-{\frak t}\tfrac{\partial }{\partial s}$$
when the right-hand side is restricted to $s\leq -(T+T_0)$.  Since $T_{\phi_-}\mathcal{S}_-$ and $\tfrac{\partial}{\partial s}$ are independent, we see that $D\widetilde G$ is injective, which in turn implies that $G$ is a local $C^1$-embedding.
\end{proof}

\subsection{Surjectivity of the gluing map} \label{subsection: surjectivity}

\begin{thm} \label{thm: surjectivity}
Given compact subsets $\mathcal{K}_\pm \subset \mathcal{S}_\pm$, there exist $T_0\gg 0$ and $\widetilde \delta>0$ such that, for all curves $u_0$ that are $\widetilde\delta$-close to breaking into $(u'_-,u'_+,T')\in \mathcal{K}_-\times \mathcal{K}_+\times[2T_0,\infty)$, there exists a triple $(\phi_-,\phi_+,T)\in \mathcal{S}_-\times\mathcal{S}_+\times[T_0,\infty)$ such that $G(\phi_-,\phi_+,T)=u_0$.
\end{thm}

\begin{proof}
Let $T_0>0$ be sufficiently large and $\widetilde\delta>0$ be sufficiently small. Let $u_0$ be $\widetilde\delta$-close to breaking into $(u'_-,u'_+,T')$. After possibly translating $u_0$ in the $\mathbb{R}$-direction and slightly modifying $T'$, there exists a decomposition of the domain $(\dot F_0,j_0)$ of $u_0$ as:
$$\dot F_0 = (F_0)^\circ_+ \cup (F_0)^\circ_- \cup A_{[-T',T']},$$
where
\begin{itemize}
\item $\mathcal{A}_\varepsilon$ is the $\varepsilon$-thin annular part of $\dot F_0$ (with respect to the complete finite-volume hyperbolic metric compatible with $j_0$) corresponding to the neck that is being stretched;
\item $A_{[-T',T']}= u_0^{-1}(\{-T'\leq s\leq T'\})\cap \mathcal{A}_\varepsilon$, $int ((F_0)^\circ_+)\cup int( (F_0)^\circ_-)=\dot F_0- A_{[-T',T']}$, and  $(F_0)^\circ_+$, $(F_0)^\circ_-$ correspond to the top and bottom levels;
\item $s'_{-,1}(u_0)$, corresponding to the first negative end of the top level, and $s'_{+,1}(u_0)$, corresponding to the positive end of the bottom level, are as defined in Section 6.3 and we assume that $s'_{\pm, 1}(u_0)= \pm (T'+T_0)$.
\end{itemize}

Next consider
$$(\dot F'_+,j'_+):= ((F_0)^\circ_+,j_0)\cup (A_{(-\infty,T']},j_{std}),$$
$$(\dot F'_-,j'_-):= ((F_0)^\circ_-,j_0)\cup (A_{[-T',\infty)},j_{std}),$$
such that $\dot F'_\pm$ extends $(F_0)^\circ_\pm\cup A_{[-T',T']}$.
For $\widetilde \delta>0$ small and $T_0\gg 0$, $j'_\pm$ is close to the domain complex structure of $u'_\pm$.  We may replace $u'_\pm$ by $u''_\pm\in \mathcal{S}_\pm$ with domain $(\dot F'_\pm, j'_\pm)$ which is close to $u'_\pm$ and such that $u_0$ is $\widetilde\delta$-close to breaking into $(u''_-,u''_+,T')$, after possibly slightly enlarging $\widetilde \delta$.

Let $u^{(1)}_*$ be the pregluing of $u^{(1)}_-=u''_-$ and $u^{(1)}_+=u''_+$ with gluing parameter $T^{(1)}=T'$, as defined in Section~\ref{subsection:pregluing}, and let
$$\dot F^{(1)}=(F^{(1)})^\circ _+\cup (F^{(1)})^\circ_-\cup A_{[-T^{(1)},T^{(1)}]}$$
be the domain of $u^{(1)}_*$, defined analogously.

We first solve for $\psi_\pm^{(1)}$ in
\begin{align} \label{eqn: first step}
\Phi \overline\partial \exp_{u^{(1)}_*}(\beta_{-,T^{(1)}}\psi_{-,T^{(1)}}^{(1)} +\beta_{+,T^{(1)}}\psi_{+,T^{(1)}}^{(1)})\in \Phi E'_{u_0},
\end{align}
as in Section~\ref{subsection: defn of gluing map}, where $\Phi$ is the parallel transport to $\wedge^{0,1}(u^{(1)}_*)^* T(\mathbb{R}\times M)$.
Let us write $u^{(1)}= \exp_{u^{(1)}_*}(\beta_{-,T^{(1)}}\psi^{(1)}_{-,T^{(1)}} +\beta_{+,T^{(1)}}\psi^{(1)}_{+,T^{(1)}})$.  Here the superscript $(1)$ indicates that we are in the first round of an iterative scheme. Note that $\|\psi_\pm^{(1)}\|_{*,\delta}\to 0$ as $T_0\to\infty$ by Lemma~\ref{lemma: T to infty}.

Next suppose $\phi^{(1)}$ satisfies
$$u_0=\exp_{u^{(1)}_*}(\beta_{-,T^{(1)}}\psi_{-,T^{(1)}}^{(1)} +\beta_{+,T^{(1)}}\psi_{+,T^{(1)}}^{(1)} +\phi^{(1)}).$$
We decompose $\phi^{(1)}=\phi_{-,T^{(1)}}^{(1)} + \phi_{+,T^{(1)}}^{(1)}$, such that
\begin{itemize}
\item $\phi_{+,T^{(1)}}^{(1)}=\phi^{(1)}$ on $(F')^\circ_+$, $\phi_{+,T^{(1)}}^{(1)}=0$ on $(F')^\circ_-$, and $$\phi_{+,T^{(1)}}^{(1)}(s,t)=\phi^{(1)}(s,t)\beta(\tfrac{s+hr}{2hr})$$ on $A_{[-T^{(1)},T^{(1)}]}$;
\item $\phi_{-,T^{(1)}}^{(1)}=0$ for $(F')^\circ_+$, $\phi_{-,T^{(1)}}^{(1)}=\phi$ on $(F')^\circ_-$, and $$\phi_{-,T^{(1)}}^{(1)}(s,t)=\phi^{(1)}(s,t)(1-\beta(\tfrac{s+hr}{2hr}))$$ on $ A_{[-T^{(1)},T^{(1)}]}$.
\end{itemize}
Here $\phi_\pm^{(1)}\in \mathcal{H}_\pm\oplus \mathbb{R}^{2l^\pm}$ and we are writing $\|\phi_\pm^{(1)}\|_{*,\delta}$ for the sum of $\|\cdot\|_{*,\delta}$ on $\mathcal{H}_\pm$ and the standard norm on $\mathbb{R}^{2l^\pm}$ as before.

We then solve for $(\phi_\pm')^{(1)}$ in
$$\overline\partial\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)} +(\phi_\pm')^{(1)})\in\Phi_\pm^{\phi_\pm^{(1)}+(\phi_\pm')^{(1)}}E_{\phi_\pm^{(1)}+(\phi'_\pm)^{(1)}},$$
where $(\phi_\pm')^{(1)}\in \mathcal{H}_\pm$, $\Phi_\pm^{\phi_\pm^{(1)}+(\phi_\pm')^{(1)}}$ is the parallel transport
$$\wedge^{0,1}(\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)} +(\phi_\pm')^{(1)}))^* T(\mathbb{R}\times M)\stackrel\sim\longrightarrow \wedge^{0,1}(u_\pm^{(1)})^* T(\mathbb{R}\times M)$$
and $E_{\phi_\pm^{(1)} +(\phi'_\pm)^{(1)}}$ is the obstruction bundle for $\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)} +(\phi_\pm')^{(1)})$.

\begin{claim} \label{first}
$\|(\phi_\pm')^{(1)}\|_{*,\delta}\leq C \|\phi_\pm^{(1)}\|_{*,\delta}$, where $0<C\ll 1$.
\end{claim}

\begin{proof}[Proof of Claim~\ref{first}.]
This is proved using the contraction mapping theorem as in Section~\ref{subsection: defn of gluing map} and relies on the fact that $\overline\partial\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)})$ is close to an element in $\Phi_\pm^{\phi_\pm^{(1)}}E_{\phi_\pm^{(1)}}$.  We will indicate a proof of this fact under the simplified assumption that $E_{\phi_\pm^{(1)}}=0$; the general case is only more complicated in notation.

First observe that $\overline\partial \exp_{u_+^{(1)}}(\phi_+^{(1)})|_{s\geq -T_0}$ is the $s=-(T^{(1)}+T_0)$ translate of $\overline\partial \exp_{u_*^{(1)}}(\phi_{+,T^{(1)}}^{(1)})|_{s\geq T^{(1)}}$ and that
\begin{equation} \label{eqn: dbar}
\overline\partial \exp_{u_*^{(1)}}(\phi_{+,T^{(1)}}^{(1)}+\psi_{+,T^{(1)}}^{(1)})=0,\quad \overline\partial \exp_{u_*^{(1)}}(\psi_{+,T^{(1)}}^{(1)})=0,  \quad \overline\partial \exp_{u_*^{(1)}}(0)=0.
\end{equation}
We claim that
\begin{equation} \label{eqn: bandaid}
\|\overline\partial \exp_{u_*^{(1)}}(\phi_{+,T^{(1)}}^{(1)})|_{s\geq -T_0}\|_\delta^{+,T^{(1)}}\leq c\|\psi_{+,T^{(1)}}^{(1)}\|_{*,\delta}^{+,T^{(1)}} \cdot \|\phi_{+,T^{(1)}}^{(1)}\|_{*,\delta}^{+,T^{(1)}}
\end{equation}
for some constant $c>0$. We write $u,\psi,\phi$ for $u_*^{(1)},\psi_{+,T^{(1)}}^{(1)},  \phi_{+,T^{(1)}}^{(1)}$ and use local coordinates $(\sigma,\tau)$ for the domain.  Assume that
\begin{enumerate}
\item[($\dagger$)] $\exp_u(\phi)=u+\phi$, $\exp_u(\psi)=u+\psi$, $\exp_u(\phi+\psi)=u+\phi+\psi$,
\end{enumerate}
e.g., the target metric is flat.
Using Equation~\eqref{eqn: dbar} and writing $v=u+\psi$ we obtain:
\begin{align}
\label{mouse1} \overline\partial (u+\phi) &= \tfrac{\partial\phi}{\partial \sigma} +J(u)\tfrac{\partial \phi}{\partial \tau} + \nabla J(u)(\phi)\tfrac{\partial(u+\phi)}{\partial \tau} + Q(\phi)\tfrac{\partial(u+\phi)}{\partial \tau},\\
\label{mouse2} 0=\overline\partial (v+\phi) &= \tfrac{\partial\phi}{\partial \sigma} +J(v)\tfrac{\partial \phi}{\partial \tau} + \nabla J(v)(\phi)\tfrac{\partial(v+\phi)}{\partial \tau} + Q(\phi)\tfrac{\partial(v+\phi)}{\partial \tau},
\end{align}
where we are writing $J(u+\phi)=J(u) + \nabla J(u)(\phi) + Q(\phi)$ and $Q(\phi)$ is a quadratic term. Taking the difference between the two, we can locally bound
$\|\overline\partial (u+\phi)\|$ by terms of the form $c \|\psi\|_* \|\phi\|_*$. In general, when ($\dagger$) does not hold, we may take $\exp_u(\phi+\psi)=u+\phi+\psi + Q(\phi,\psi)$, where $Q(\phi,\psi)$ is a pointwise function of $\phi$ and $\psi$ times a pointwise bilinear function of $\phi$ and $\psi$, and we obtain the local bound $\|\overline\partial (u+\phi)\|\leq c \|\psi\|_* \|\phi\|_*$.  The standard exponential decay estimates then yield Equation~\eqref{eqn: bandaid}.

Next we bound $\overline\partial \exp_{u_+^{(1)}}(\phi_+^{(1)})$ on $-(T^{(1)}+T_0)-hr\leq s\leq -T_0$, which is the $s=-(T^{(1)}+T_0)$ translate of $\overline\partial \exp_{u_*^{(1)}}(\phi_{+,T^{(1)}}^{(1)})|_{-hr\leq s\leq T^{(1)}}$.  Applying the same procedure as above\footnote{Note that we are dealing with a flat metric here, so ($\dagger$) holds.} with
$$u=u_*^{(1)}, \quad v= u_*^{(1)}+ \beta_{-,T^{(1)}}\psi_{-,T^{(1)}}^{(1)} +\beta_{+,T^{(1)}}\psi_{+,T^{(1)}}^{(1)},$$
$$\psi=\beta_{-,T^{(1)}}\psi_{-,T^{(1)}}^{(1)} +\beta_{+,T^{(1)}}\psi_{+,T^{(1)}}^{(1)}, \quad \phi=\phi_{+,T^{(1)}}^{(1)}$$
as well as exponential bounds on annuli from \cite[Lemma 2.3]{HT2} we obtain
\begin{equation} \label{eqn: bandaid2}
\|\overline\partial \exp_{u_+^{(1)}}(\phi_+^{(1)})|_{-(T^{(1)}+T_0)-hr\leq s\leq -T_0}\|_\delta^{+,T^{(1)}}\leq C(T^{(1)})\cdot \|\phi_{+,T^{(1)}}^{(1)}\|_{*,\delta}^{+,T^{(1)}},
\end{equation}
where $C(T^{(1)})\to 0$ as $T^{(1)}\to\infty$.  Note that $\phi_+^{(1)}|_{s\leq -(T^{(1)}+T_0) -hr}=0$ by definition.

We then invert the error term $\overline\partial \exp_{u_+^{(1)}}(\phi_+^{(1)})$ using Estimates~\eqref{eqn: bandaid} and \eqref{eqn: bandaid2} and the contraction mapping theorem.  This proves the claim.
\end{proof}

Also observe that $\|\phi_\pm^{(1)}\|_{*,\delta}$ is bounded above by a fixed constant times $\widetilde \delta$; this follows from elliptic bootstrapping as in Lemma~\ref{lemma: psi pm smooth}.

We iterate the above procedure with $(u^{(1)}_-,u^{(1)}_+,T^{(1)})$ replaced by
$$(u^{(2)}_-,u^{(2)}_+,T^{(2)})=(\exp_{u^{(1)}_-}^\circ(\phi_-^{(1)}+(\phi_-')^{(1)}), \exp_{u^{(1)}_+}^\circ(\phi_+^{(1)} +(\phi_+')^{(1)}),T^{(1)}+T_-+T_+).$$
Here $\exp_{u^{(1)}_\pm}^\circ(\phi_\pm^{(1)}+(\phi_\pm')^{(1)})$ is $\exp_{u^{(1)}_\pm}(\phi_\pm^{(1)}+(\phi_\pm')^{(1)})$ normalized by shifting $T_\pm$ units in the $\pm s$-direction; the amount that we shift is determined by the choice of slice (i.e., the choice of representative $\exp_{u^{(1)}_\pm}^\circ(\phi_\pm^{(1)}+(\phi_\pm')^{(1)})$ of $\mathbb{V}_\pm/\mathbb{R}$). Let $u_*^{(2)}$ be the resulting pregluing.

We can similarly verify that, if we replace $u_*^{(1)}$ and $\psi_\pm^{(1)}$ by $u_*^{(2)}$ and $\psi_\pm^{(1)}+\psi_\pm^{(2)}$ in Equation~\eqref{eqn: first step}, then:
\begin{equation} \label{third}
\|\psi_\pm^{(2)}\|_{*,\delta}\leq C \|\phi_\pm^{(1)}\|_{*,\delta},
\end{equation}
where $0<C\ll 1$.
The bounds from Claim~\ref{first} and Estimate \eqref{third} imply that the error between $u_0$ and $u^{(i)}$ is of order $C^i$ after the $i$th iteration.  The proof then follows.
\end{proof}

\subsection{Sketch of proof of Theorem~\ref{thm: iterated gluing}} \label{subsection: multiple gluing}

In this subsection we sketch the proof of Theorem~\ref{thm: iterated gluing}.  Without loss of generality assume that $m=3$.  We are comparing $G_{(1,2,3)}$ and $G_{((1,2),3)}\circ (G_{(1,2)},id)$. We are gluing $u_1\cup u_2\cup u_3$, where $(\mathcal{F}_i,u_i)$, $i=1,2,3$, is a representative of $V_i/\mathbb{R}$. For ease of notation we assume that $u_2$ and $u_3$ only have one negative end.

\subsubsection{Description of $G_{(1,2,3)}$}

Let $u_*^{(123)}$ be the pregluing of $u_1,u_2,u_3$ with gluing parameters $T_{12}, T_{23}$; this is defined in the same way as in Section~\ref{subsection:pregluing}. We solve for $\psi_1^{(123)},\psi_2^{(123)},\psi_3^{(123)}$ which are analogs of $\psi_-,\psi_+$ in the equations
$$\Theta_1^{(123)}(\psi_1^{(123)},\psi_2^{(123)})=0,$$
$$\Theta_2^{(123)}(\psi_1^{(123)},\psi_2^{(123)},\psi_3^{(123)})=0,$$
$$\Theta_3^{(123)}(\psi_2^{(123)},\psi_3^{(123)})=0,$$
which are analogs of $\Theta_\pm(\psi_-,\psi_+)=0$. This yields
$$u_{(123)}=G_{(1,2,3)}(u_1,u_2,u_3,T_{12},T_{23}).$$

We also write $\psi_i^{(123),\dagger}$, $i=1,2,3$, for the first approximation of $\psi_i^{(123)}$ when applying the contraction mapping principle; this corresponds to $\mathcal{J}_1$ in the proof of Theorem~\ref{thm: gluing map smooth} and is obtained by using the linearized $\overline\partial$-operator to invert the errors that arise from the pregluing of $u_1,u_2,u_3$. Estimates similar to those of Sections~\ref{subsection: defn of gluing map} imply the existence of functions $\widetilde C(T)$ and $\widetilde D(T)$, $T:=\min(T_{12},T_{23})$, such that
\begin{align} \label{alpha}
\|\psi_i^{(123)}-\psi_i^{(123),\dagger}\|_{*,\delta}& \leq \widetilde C(T) \|\psi_i^{(123),\dagger}\|_{*,\delta},\\
\nonumber \|\psi_i^{(123),\dagger}\|_{*,\delta} & \leq \widetilde D(T),
\end{align}
and $\widetilde C(T),\widetilde D(T)\to 0$ as $T\to\infty$.

\subsubsection{Description of $G_{((1,2),3)}\circ (G_{(1,2)},id)$}

Let $u_*^{(12)}$ be the pregluing of $u_1,u_2$ with gluing parameter $T_{12}$.  We solve for $\psi_1^{(12)},\psi_2^{(12)}$ in
$$\Theta_1^{(12)}(\psi_1^{(12)},\psi_2^{(12)})=0, \quad \Theta_2^{(12)}(\psi_1^{(12)},\psi_2^{(12)})=0,$$
which yields $u_{(12)}=G_{(1,2)}(u_1,u_2,T_{12})$. Similarly we define $\psi_i^{(12),\dagger}$, $i=1,2$.  We have
\begin{align}\label{beta}
\|\psi_i^{(12)}-\psi_i^{(12),\dagger}\|_{*,\delta}& \leq \widetilde C(T_{12}) \|\psi_i^{(12),\dagger}\|_{*,\delta},\\
\nonumber \|\psi_i^{(12),\dagger}\|_{*,\delta} &\leq \widetilde D(T_{12}),
\end{align}
where $\widetilde C(T_{12}), \widetilde D(T_{12})\to 0$ as $T_{12}\to\infty$.

Next let $u_*^{((12)3)}$ be the pregluing of $u_{(12)},u_3$ with gluing parameter $T_{23}$. We solve for $\psi_{(12)}^{((12)3)}, \psi_3^{((12)3)}$ in
$$\Theta_{(12)}^{((12)3)}(\psi_{(12)}^{((12)3)},\psi_3^{((12)3)})=0,\quad \Theta_{3}^{((12)3)}(\psi_{(12)}^{((12)3)},\psi_3^{((12)3)})=0,$$
which yields
$$u_{((12)3)}=G_{((1,2),3)}(u_{(12)},u_3,T_{23}).$$
Similarly we define $\psi_{(12)}^{((12)3),\dagger}, \psi_3^{((12)3),\dagger}$. For $i=(12)$ and $i=3$ we have
\begin{align}
\label{delta} \|\psi_i^{((12)3)}-\psi_i^{((12)3),\dagger}\|_{*,\delta}& \leq \widetilde C(T) \|\psi_i^{((12)3),\dagger}\|_{*,\delta}, \\
\nonumber \|\psi_i^{((12)3),\dagger}\|_{*,\delta} &\leq \widetilde D(T),
\end{align}
where $\widetilde C(T), \widetilde D(T) \to 0$ as $T\to \infty$.

\subsubsection{Conclusion}

The $C^0$-closeness follows from Estimates~\eqref{alpha}, \eqref{beta}, and \eqref{delta}. Roughly speaking, the errors from the pregluing go to zero as $T_{12},T_{23}\to \infty$.  The $C^1$-closeness is left to the reader and follows from Estimates~\eqref{result of I}, \eqref{result of II main}, and \eqref{result of III} from Section~\ref{subsection: C1 smoothness}. In words, the derivatives of the errors from the pregluing go to zero as $T_{12},T_{23}\to \infty$ as well.

\section{Construction of semi-global Kuranishi structures} \label{section: construction of semi-global}

{\em Starting from this section we specialize to contact homology.} For the most part we will describe the semi-global Kuranishi structures that are involved in proving $\partial^2=0$.  In Sections~\ref{subsection: cob case} and \ref{subsection: chain homotopy case} we explain the modifications needed for chain maps and chain homotopy.

\subsection{Trimming} \label{subsection: trimming}

Choose $\mathcal{L}\gg 0$ and $\varepsilon''>0$ small. Consider the semi-global Kuranishi chart $(\mathbb{K},\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V},\overline\partial_J,{\frak s})$ for $\mathcal{M}/\mathbb{R}$ (with $\mathbb{K}$ sufficiently large in response to $\mathcal{L}$) constructed in Section~\ref{subsection: semi-global Kuranishi chart}.  {\em From now on we suppress $\overline\partial_J$ from the notation since it appears in every chart.}

We start by taking $\mathbb{V}$ (and also $\mathbb{K}$) to be invariant under automorphisms that relabel the negative punctures while preserving the $\vartheta$-sorting.  In other words, $\mathbb{V}$ and $\mathbb{K}$ satisfy:
\begin{enumerate}
\item[(Inv)] Given $(\mathcal{F},u)\in \mathbb{V}$ (resp.\ $\mathbb{K}$), we require that $(\mathcal{F}',u)\in \mathbb{V}$ (resp.\ $\mathbb{K}$), where $\mathcal{F}'$ is the same Riemann surface as $\mathcal{F}$ but with a different ordering of the negative punctures, subject to the condition that the negative ends of $(\mathcal{F}',u)$ be $\vartheta$-sorted.
\end{enumerate}
This can be achieved by slightly shrinking $\mathbb{V}$ and $\mathbb{K}$ if necessary, while keeping the same names.

Next we explain how to further trim $(\mathbb{K},\pi_{\mathbb{V}}: \mathbb{E}\to \mathbb{V},{\frak s})$.  But first we make the following definition:

\begin{defn}[Neck length]
Given $$(\mathcal{F},u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\in \widetilde{\mathcal{G}}_\delta(V_1/\mathbb{R},...,V_m/\mathbb{R}),$$
let $g$ be the complete finite volume hyperbolic metric on $\ddot F$. If there is an annular component $A$ of $\operatorname{Thin}_\varepsilon(F,g)$ such that $u|_A$ is close to a trivial cylinder over $\gamma$ , then its {\em neck length} (with respect to $\gamma$) $\mathfrak{nl}(A)$ is the value $C$ such that $A$ is conformally equivalent to $(\mathbb{R}/\mathcal A(\gamma)\mathbb{Z})\times [0,C]$ with the standard complex structure.
\end{defn}

Let $\mathbb{V}_{\geq \mathcal{L}}$ (resp.\ $\mathbb{V}_{> \mathcal{L}}$) be the subset of $\mathbb{V}$ such that $g=g_{(\ddot F,j)}$ has an annular component $A$ in $\operatorname{Thin}_\varepsilon(\ddot F,g)$ such that $\mathfrak{nl}(A)\geq \mathcal{L}$ (resp.\ $\mathfrak{nl}(A)>\mathcal{L}$) and let $\mathbb{V}_{<\mathcal{L}}=\mathbb{V}-\mathbb{V}_{\geq \mathcal{L}}$ and $\mathbb{V}_{\leq \mathcal{L}}=\mathbb{V}-\mathbb{V}_{> \mathcal{L}}$. We then restrict $\mathbb{V}$ to $\mathbb{V}_{< \mathcal{L}}$. By resetting notation, we assume $\mathbb{V}=\mathbb{V}_{< \mathcal{L}}$, $\mathbb{K}=\mathbb{K}\cap \mathbb{V}_{\leq\mathcal{L}-\varepsilon''}$, and denote the resulting restrictions of $\pi_{\mathbb{V}}$ and ${\frak s}$ by the same names.  Note that $\mathbb{V}$ and $\mathbb{K}$ still satisfy (Inv).

The boundary $\partial \mathbb{V}$ will be decomposed into two parts, the {\em vertical boundary} $\partial_v \mathbb{V}=\mathbb{V}_{=\mathcal{L}}$ and the {\em horizontal boundary} $\partial_h \mathbb{V}$ which is the closure in $\partial \mathbb{V}$ of $\partial\mathbb{V}-\partial_h\mathbb{V}$.

\subsection{CH trees} \label{subsection: CH trees}

Let  $\mathcal{M}_1,\mathcal{M}_2,\dots,\mathcal{M}_\rho$ be a sequence of moduli spaces
$$\mathcal{M}_i=\mathcal{M}_{J}^{\operatorname{ind}=k_i}(\dot F_i,\mathbb{R}\times M;\gamma_{i,+},\boldsymbol\gamma_{i,-}),$$
such that each component of each level of a building in $\partial\mathcal{M}_i$ is either a trivial cylinder or in $\mathcal{M}_j$ with $j<i$. (A specific choice will be given in Section~\ref{subsubsection: complexity}.)   The semi-global Kuranishi chart for $\mathcal{M}_i/\mathbb{R}$ will be denoted by $\mathcal{C}'_i=(\mathcal{C}_i,{\frak s}_i)$, where
$$\mathcal{C}_i=(\mathbb{K}_i,\pi_i: \mathbb{E}_i\to \mathbb{V}_i)$$
and $\mathbb{K}_i\subset \mathcal{M}_i/\mathbb{R}$ is a large compact subset.

\subsubsection{Definition of CH tree}

Contact homology (or CH) trees are combinatorial objects that encode the data for the boundary strata.

\begin{defn}[CH trees]
A {\em contact homology tree of $\operatorname{(Symp)}$ type} (or simply a {\em CH tree}) is a connected directed tree $T=(V(T), E(T)=G(T)\sqcup F(T),O(T))$, with $|V(T)|$ and $|E(T)|$ finite, which satisfies the following:
\begin{enumerate}
\item The set of vertices $V(T)$ is labeled using the {\em vertex labeling function}
$$l_{V}:V(T)\to\{1,\dots,\rho\},$$
which is not necessarily injective or surjective.
\item The set of edges $E(T)$ consists of the {\em glued edges} $G(T)$ and the {\em free edges} $F(T)$.  Both types of edges are directed and the edges of $G(T)$ have initial and terminal points but the edges of $F(T)$ only have an initial point.
\item Given an edge $e\in E(T)$, we write $i(e)$ for the initial point of $e$ and $t(e)$ for the terminal point of $e$ (if it exists).
\item For each vertex $v\in V(T)$, we write $E(T)_{v}$ for the set of all edges $e$ with $i(e)=v$, $G(T)_{v}=E(T)_v\cap G(T)$, and $F(T)_v=E(T)_v\cap F(T)$.
\item For each $v\in V(T)$, $E(T)_{v}$ is labeled using a bijective {\em edge labeling function}
    $$l_{E_{v}}:E(T)_{v}\to\{1,2,\dots,m\},$$
    where $m$ is the number of negative punctures of $\mathbb{V}_{v}.$  The labeling function $l_{E_v}$ {\em respects the ordering $\vartheta$}, i.e., for any $e_{1},e_{2}\in E(T)_{v},$ if the curves in $\mathbb{V}_{v}$ limit to $\gamma_{1}$ near the $l_{E_{v}}(e_{1})$th puncture and limit to $\gamma_{2}$ near the $l_{E_{v}}(e_{2})$th puncture, and if $\vartheta(\gamma_{1})<\vartheta(\gamma_{2}),$ then $l_{E_{v}}(e_{1})<l_{E_{v}}(e_{2}).$
\item Each edge $e\in G(T)$ corresponds to a single gluing of a nonempty $\mathbb{V}_{l_V(i(e))}$ on the upper level with a nonempty $\mathbb{V}_{l_V(t(e))}$ on the lower level and the gluing occurs along the $l_{E_{i(e)}}(e)$th negative end.
\item There is a unique topmost vertex and each vertex except for the topmost vertex has one incoming edge and $\geq 0$ outgoing edges.
\item The {\em reordering function}
    $$O(T):F(T)\to\{1,\dots,|F(T)|\},$$
    is bijective, respects the ordering $\vartheta$, and, for each vertex $v,$ $O(T)|_{F(T)_{v}}$ is consistent with the ordering of $F(T)_{v}$ induced by $l_{E_{v}},$ i.e., for any $e_{1},e_{2}\in F(T)_{v},$ we have $O(T)(e_{1})<O(T)(e_{2})$ if and only if $l_{E_{v}}(e_{1})<l_{E_{v}}(e_{2})$.
\end{enumerate}
\end{defn}

Note that $F(T)$ has a natural ``lexicographic ordering" $O^{lex}$: For each $e\in F(T)$, consider the sequence $(e_1,\dots, e_s=e)$, where $i(e_1)$ is the topmost vertex and $t(e_j)=i(e_{j+1})$, $j=1,\dots,s-1$.  We then take $(l_{E_{i(e_1)}}(e_1), l_{E_{i(e_2)}}(e_2),\dots)$ and order the $e$ lexicographically using these tuples. Here  $O^{lex}$ does not need to be the same as $O(T)$.

\subsubsection{Isomorphisms of CH trees}

\begin{defn}
An {\em isomorphism $\theta:T\to T'$ of CH trees} satisfies the following:
\begin{enumerate}
\item $\theta$ is an isomorphism of directed trees which sends $V(T), G(T), F(T)$ to $V(T'),G(T'), F(T')$;
\item $\theta$ preserves the labels of vertices, i.e., $l_{V(T)}=l_{V(T')}\circ \theta|_{V(T)}$;
\item $\theta$ maps $O(T)$ to $O(T')$, i.e., $O(T)=O(T')\circ\theta|_{F(T)}.$
\end{enumerate}
\end{defn}

By (3), $\theta$ preserves the ordering of the free edges at each vertex. On the other hand, we do not require $\theta$ to preserve the ordering of the glued edges.

Given an isomorphism $\theta:T\stackrel\sim\to T'$ of CH trees, for each $v\in V(T)$, $\theta$ induces an isomorphism
\begin{eqnarray*}
\mathbb{V}_{l_V(v)} &\stackrel\sim \to & \mathbb{V}_{l_V(\theta(v))},\\
(\mathcal{F},u) & \mapsto & (\mathcal{F}',u),
\end{eqnarray*}
where $\mathcal{F}'$ is the same Riemann surface as $\mathcal{F}$, but with a possibly different ordering of the negative punctures. More precisely, for any $e$ with $i(e)=v$, the $l_{E_{v}}(e)$th negative puncture of $\mathcal{F}$ becomes the $\theta(l_{E_{v}}(e))$th negative puncture of $\mathcal{F}'$.  Putting the above isomorphisms together, $\theta:T\stackrel\sim\to T'$ gives an isomorphism
$$\times_{v\in V(T)}\mathbb{V}_{l_V(v)} \stackrel\sim\to\times _{w\in V(T')}\mathbb{V}_{l_V(w)}.$$

Let $\operatorname{Aut}(T)$ be the group of automorphisms of $T$.
If $\theta:T\stackrel\sim\to T$ is a nontrivial automorphism of a CH tree, i.e., $\theta$ does not preserve the labels of the glued edges, then passing to the quotient is the source of orbifold points.

\subsubsection{Contraction of CH trees}

\begin{defn}[Contraction function $\tau$]
The {\em contraction function} $\tau$ is the map from the set of CH trees to $\{1,\dots,\rho\}$ such that, if we glue $\{\mathbb V_i\}_{i\in V(T)}$ according to the prescription given by $T$, we get an element of $\mathbb V_{\tau(T)}$.
\end{defn}

\begin{defn}[Contraction] \label{defn: contraction}
Let $T$ be a CH tree and $\sqcup_i S_i$ be a disjoint union of connected subtrees $S_i\subset T$ with no free edges. Then the {\em contraction $T'=T/\!\!/\sqcup_i S_i$ of $T$ along $\sqcup_i S_i$} is the CH tree given as follows:
\begin{enumerate}
\item Each subtree $S_i$ is replaced by a vertex labeled by $\tau(S_i)$. For the vertices not in $S_i$, the vertex labeling function $l_V$ remains the same.
\item The glued edges not in $S_i$ remain glued edges in $T'$.  There exists a natural bijection $c:F(T)\stackrel\sim\to F(T')$ and $O(T')\circ c=O(T).$
\item For each $v'\in V(T')$, define the edge labeling function on $E(T')_{v'}$ as follows: Start with the lexicographic ordering for $E(T')_{v'}$ given by $T$. Define an {\em adjacent swap} of an ordering $l$ for $E(T')_{v'}$ to be $l'$ such that if $l(e_2)= l(e_1)+1$ for some $e_1,e_2\in E(T')_{v'}$, then $l'(e_1)=l(e_2)$ and $l'(e_2)=l(e_1)$. Apply the minimum number of adjacent swaps so that the resulting ordering $l$ respects the orderings $\vartheta$ and $O(T')$.
\end{enumerate}
\end{defn}

\subsection{Overview of the construction} \label{subsection: overview of construction}

\subsubsection{Complexity} \label{subsubsection: complexity}

The semi-global Kuranishi structure is constructed by induction on a triple which we call the {\em complexity}
\begin{equation}\label{eqn: definition of complexity}
c(\mathcal{F},u)=(c_1(\mathcal{F},u),c_2(\mathcal{F},u),c_3(\mathcal{F},u))=(\mathcal{A}_\alpha(\gamma_+),E_\alpha(u),-\chi(\dot F)),
\end{equation}
where we are using lexicographic ordering. Here $u$ is a map from $\gamma_+$ to $\boldsymbol\gamma_-$.

We choose $L\gg 0$ and only consider $\gamma_+$ satisfying $\mathcal{A}_\alpha(\gamma_+)< L$.  (Eventually we will take direct limits as $L\to \infty$.)
We also choose $\varepsilon>0$ sufficiently small and $\ell$ sufficiently large for which Theorem~\ref{thm: construction of L-transverse subbundle} holds for all moduli spaces with $\mathcal{A}_\alpha(\gamma_+)< L$.

Fix $c_1(\mathcal{F},u)$ and assume by induction that the semi-global Kuranishi charts for all the strata with smaller $c_1(\mathcal{F},u)$ have been constructed as in Section~\ref{subsection: semi-global Kuranishi chart}.    The strata with the smallest $c_2(\mathcal{F},u)$ are branched covers of trivial cylinders, i.e., satisfy $E_\alpha(u)=0$. {\em We do not construct Kuranishi charts about trivial cylinders.}  The choice of $\gamma_+$ implies an upper bound on the number of punctures,
which in turn gives an upper bound on the number of branch points.  Similarly, for $E_\alpha(u)>0$, we start with the stratum with the smallest $E_\alpha(u)$ and use the fact that there is an upper bound on the number of punctures.

We then choose a finite sequence $\mathcal{M}_1,\mathcal{M}_2,\dots,\mathcal{M}_\rho$ of moduli spaces
$$\mathcal{M}_i=\mathcal{M}_{J}^{\operatorname{ind}=k_i}(\dot F_i,\mathbb{R}\times M;\gamma_{i,+},\boldsymbol\gamma_{i,-}),$$
where the moduli spaces are ordered according to nondecreasing complexity. The semi-global Kuranishi chart for $\mathcal{M}_i/\mathbb{R}$ is
$$\mathcal{C}'_i=(\mathcal{C}_i,{\frak s}_i), \quad \mathcal{C}_i=(\mathbb{K}_i,\pi_i: \mathbb{E}_i\to \mathbb{V}_i),$$
where $\mathbb{K}_i\subset \mathcal{M}_i/\mathbb{R}$ is a large compact subset.

\subsubsection{Boundary strata}

We now explain how to construct charts corresponding to the boundary strata.

Given $\mathcal{M}_{i_0}/\mathbb{R}$, its boundary $\partial (\mathcal{M}_{i_0}/\mathbb{R})$ is described as follows:
\begin{equation}
\partial (\mathcal{M}_{i_0}/\mathbb{R})\subset \left(\coprod_{T} \mathcal{M}_T'\right)/\sim, \quad \mathcal{M}_T': =\times_{v\in V(T)}(\mathcal{M}_{l_V(v)}/\mathbb{R}),
\end{equation}
where the disjoint union is over all CH trees $T$ such that $T/\!\!/ T$ is a one-vertex CH tree whose vertex is labeled $\mathcal{M}_{i_0}$ and the equivalence relation $\sim$ is induced by isomorphisms $\theta: T\stackrel \sim\to T'$ of CH trees. (To simplify notation, we will often write ``$i\in V(T)$'' with $i\in \{1,\dots, \rho\}$ to mean that there is $v\in V(T)$ with $l_V(v)=i$.)

Assuming that the semi-global Kuranishi charts
$$(\mathcal{C}_{i},{\frak s}_{i})=(\mathbb{K}_{i},\pi_{i}:\mathbb{E}_{i}\to \mathbb{V}_{i},{\frak s}_{i}), \quad i\in V(T),$$
have already been constructed inductively, our goal is to construct their product
$$\mathcal{C}'_T=(\mathcal{C}_T,{\frak s}_T)=(\mathbb{K}_T,\pi_T: \mathbb{E}_T\to \mathbb{V}_T,{\frak s}_T).$$
The chart $\mathcal{C}'_T$ ``covers'' $\mathbb{K}_T\subset \mathcal{M}_{i_0}/\mathbb{R}$ and the collection of all charts $\mathcal{C}'_T$, where $T$ ranges over all the strata of $\partial(\mathcal{M}_{i_0}/\mathbb{R})$, together with the main stratum $T_0$ consisting of one vertex $i_0$, covers all of $\mathcal{M}_{i_0}/\mathbb{R}$.

\begin{warning}
Even if $\mathcal{M}_{i_0}/\mathbb{R}=\varnothing$ and the semi-global chart $(\mathcal{C}_{i_0},{\frak s}_{i_0})$ is empty, we need to construct $\mathcal{C}'_T$ if there exists a CH tree $T$ such that
\begin{itemize}
\item $T/\!\!/ T$ is a one-vertex CH tree whose vertex is labeled $i_0$ (for $\mathcal{M}_{i_0}$); and
\item $\times_{i\in V(T)}(\mathcal{M}_{i}/\mathbb{R})$ is nonempty.
\end{itemize}
There are examples, due to Michael Hutchings, where omitting $\mathcal{C}'_T$ leads to some inconsistencies.
\end{warning}

\begin{rmk}
For our purposes it is not necessary to cover the SFT compactification $\overline{\mathcal{M}_{i_0}/\mathbb{R}}$ of $\mathcal{M}_{i_0}/\mathbb{R}$.
\end{rmk}

We construct $\mathbb{K}_T$ and $\pi_T:\mathbb{E}_T\to \mathbb{V}_T$ in this subsection under the following assumption:
\begin{enumerate}
\item[(H)]  $\chi(\dot F_{i_0})<0$ and all the domains $\dot F_{l_V(v)}$ of $\mathcal{M}_{l_V(v)}$, $v\in V(T)$, satisfy $\chi (\dot F_{l_V(v)})<0$,
\end{enumerate}
leaving
\begin{itemize}
\item the definition of the multisection ${\frak s}_T$ to Sections~\ref{subsection: multisections on products} and \ref{subsection: multisections on higher products}; and
\item the modifications necessary when (H) does not hold to Section~\ref{subsection: H does not hold}.
\end{itemize}

To this end we consider the appropriate gluing map
$$G_T: (\times_{i\in V(T)}\mathbb{V}_i)\times [R,\infty)^{|G(T)|}\to \widetilde{\mathcal{G}}_\delta^{E'}(\{\mathbb{V}_i\}_{i\in V(T)}),$$
which is obtained from Theorem~\ref{thm: gluing} by passing to the orbifold.
Note that we are assuming that, for each $e\in G(T)$, $\mathbb{V}_{i(e)}$ is glued to $\mathbb{V}_{t(e)}$ and the gluing occurs along a connected neck.

\begin{defn}[Neck length functions]
We define the {\em neck length functions}
$$\mathfrak{nl}_e(\mathcal{F},u):\widetilde{\mathcal{G}}_\delta^{E'}(\{\mathbb{V}_i\}_{i\in V(T)}) \to \mathbb{R}^+, \quad e\in G(T),$$
where $\mathfrak{nl}_e(\mathcal{F},u)$ is the neck length of the annular component corresponding to $e$ of the $\varepsilon$-thin part of $g=g_{(\ddot F,j)}$.
\end{defn}

We observe that when $\mathfrak{nl}_e(\mathcal{F},u)$ is large, it is very close to $T_e(\mathcal{F},u)+ D_e$, where $T_e(\mathcal{F},u)\in [R,\infty)$ is the gluing parameter corresponding to $e$ which yields $(\mathcal{F},u)$ and  $D_e$ is a fixed constant.

Let $\mathbb{K}_T$ be a trimming of $(\mathcal{M}_{i_0}/\mathbb{R})\cap \operatorname{Im}(G_T)$ and $\mathbb{V}_T$ be a trimming of $\operatorname{Im}(G_T)$, where we trim $(\mathcal{M}_{i_0}/\mathbb{R})\cap \operatorname{Im}(G_T)$ (resp.\ $\operatorname{Im}(G_T)$) so that $\mathfrak{nl}_e(u)\geq \mathcal{L}-{\varepsilon''\over 2}$ (resp.\ $\mathfrak{nl}_e(u)> \mathcal{L}-\varepsilon''$) for all $e\in E(T)$. Here $\mathcal{L}$ and $\varepsilon''$ are from Section~\ref{subsection: trimming}. {\em Observe that $\mathbb{K}_T$ is not necessarily compact,} although it admits a compactification.

Finally, let $\pi_T: \mathbb{E}_T\to \mathbb{V}_T$ be the orbibundle which, in an orbibundle chart, is of the type $E'$ as defined in Equation~\eqref{equation: defn of E prime} and is isomorphic to the product bundle $\times_{i\in V(T)} E_i$.

\subsection{Multisections on products} \label{subsection: multisections on products}

Consider the ``product'' of two charts
$$\mathcal{C}'_{i_j}=(\mathcal{C}_{i_j},{\frak s}_{i_j})=(\mathbb{K}_{i_j},\pi_{i_j}: \mathbb{E}_{i_j}\to \mathbb{V}_{i_j}, {\frak s}_{i_j}),\quad j=1,2.$$
The goal of this subsection is to define ${\frak s}_T$ on $\mathbb{E}_T\to \mathbb{V}_T$, where $T=(V(T),E(T))$, $V(T)=\{i_1,i_2\}$, $G(T)=\{(i_2,i_1)\}$.

First let us define a multisection
$${\frak s}_T^\circ: \mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\to \mathbb{E}_{i_1}\times \mathbb{E}_{i_2},$$
as follows: There exists a smooth multifunction $h_{i_j}$ on $\mathbb{V}_{i_j}$ (i.e., a multisection of the trivial $\mathbb{R}$-bundle over $\mathbb{V}_{i_j}$) with image in $[0,1]$ such that $h_{i_j}(x_{i_j})=1$ on the support of ${\frak s}_{i_j}$ (the closure of the set of all $x_{i_j}\in \mathbb{V}_{i_j}$ for which some branch of ${\frak s}_{i_j}\not=0$) and $h_{i_j}(x_{i_j})=0$ near $\partial_h\mathbb{V}_{i_j}$. (The proof of the existence of $h_{i_j}$ is not trivial, but is left to the reader.) We then set
\begin{equation} \label{bilochun}
{\frak s}_T^\circ(x_{i_1},x_{i_2})= (h_{i_2}(x_{i_2}) {\frak s}_{i_1}(x_{i_1}),h_{i_1}(x_{i_1}) {\frak s}_{i_2}(x_{i_2})),
\end{equation}
where the multiplications of a multifunction and a multisection are done in all possible ways (this is the usual convention).
The multisection ${\frak s}^\circ_T$ is chosen so that
\begin{enumerate}
\item ${\frak s}^\circ_T=0$ on $(\partial_h \mathbb{V}_{i_1}\times \mathbb{V}_{i_2})\cup (\mathbb{V}_{i_1}\times \partial_h \mathbb{V}_{i_2})$;
\item $(\overline\partial_J^{-1},\overline\partial_J^{-1})({\frak s}^\circ_T)=\overline\partial_J^{-1}({\frak s}_{i_1})\times \overline\partial_J^{-1}({\frak s}_{i_2})$.
\end{enumerate}

We then define the multisection ${\frak s}_T$ on $\mathbb{E}_T\to \mathbb{V}_T$ as follows: We first view $\mathbb{E}_{i_1}\times \mathbb{E}_{i_2}$ as an orbibundle over $\mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times[R,\infty)$ by pulling back via the projection $p:\mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times[R,\infty)\to \mathbb{V}_{i_1}\times \mathbb{V}_{i_2}$ and identify:
\begin{equation}\label{diagram for pushing forward}
\begin{diagram}
\mathbb{E}_{i_1}\times \mathbb{E}_{i_2} & \rTo{} & \mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times [R,\infty)\\
\dTo^{\simeq} &  & \dTo_{\simeq} \\
\mathbb{E}_T & \rTo &  G_{T}(\mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times [R,\infty))
\end{diagram}
\end{equation}
We trim $G_{T}(\mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times [R,\infty))$ to obtain $\mathbb{V}_T$.   On $\mathbb{V}_{T}\cap\{\mathfrak{nl}_{(i_2,i_1)}\geq\mathcal{L}+\varepsilon''\}$, the multisection ${\frak s}_T$ is obtained by pulling back ${\frak s}^\circ_T$ via $p$, pushing forward via the above diagram, and restricting to $\mathbb{V}_T\cap \{\mathfrak{nl}_{(i_2,i_1)}\geq \mathcal{L}+\varepsilon''\}$. On $\mathbb{V}_T\cap \{\mathfrak{nl}_{(i_2,i_1)}<\mathcal{L}\}$, ${\frak s}_T$ is the stabilization of ${\frak s}_{T/\!\!/ T}$ of the main stratum and, on $\mathbb{V}_{T}\cap\{\mathcal{L}+\varepsilon''>\mathfrak{nl}_{(i_2,i_1)}\geq \mathcal{L}\}$, we interpolate between the two.

In Section~\ref{subsection: multisections on higher products} we explain how to define ${\frak s}_T$ when the cardinality $|I|\geq 3$, subject to some compatibility conditions.  But first we need to discuss the transition functions between charts and the compatibility conditions that arise.

\subsection{Transition functions} \label{subsection: transition functions}

Let $T$ be a CH tree. Given the charts
$$\mathcal{C}'_{T/\!\!/ T}=(\mathbb{K}_{T/\!\!/ T},\pi_{T/\!\!/ T}:\mathbb{E}_{T/\!\!/ T}\to \mathbb{V}_{T/\!\!/ T},{\frak s}_{T/\!\!/ T}),$$
$$\mathcal{C}'_T=(\mathbb{K}_T,\pi_T: \mathbb{E}_T\to \mathbb{V}_T,{\frak s}_T),$$
we define the morphism $\mathcal{C}'_{T/\!\!/ T}\to \mathcal{C}'_T$ (not a map) by restriction-inclusion: first restrict $\mathcal{C}'_{T/\!\!/ T}$ to $\mathbb{V}_{T/\!\!/ T}\cap\{\mathcal{L}>\min_{e\in E(V)}\mathfrak{nl}_e > \mathcal{L}-\varepsilon''\}$ and then include into $\mathbb{V}_{T}\cap\{\mathcal{L}>\min_{e\in E(V)} \mathfrak{nl}_e>\mathcal{L}-\varepsilon''\}$ as a stabilization.

\vskip.1in\noindent
Case (A).  Suppose that
\begin{equation} \label{eqn: triple}
V(T)=\{i_1,i_2,i_3\},\quad G(T)=\{e_1=(i_2,i_1),e_2=(i_3,i_2)\}.
\end{equation}
We would like to reconcile the different ways of taking the product of three charts $\mathcal{C}'_{i_j}$, $j=1,2,3$. Let
\begin{equation}\label{eqn: triple2}
S=(\{i_1,i_2\},\{e_1\}), \quad S'=(\{i_2,i_3\},\{e_2\})
\end{equation}
be subtrees of $T$. There are four charts to consider (see Figure~\ref{fig: corner}):
\begin{figure}[ht]
\begin{center}
\psfragscanon
\psfrag{A}{\tiny $\mathfrak{nl}_{e_1}$}
\psfrag{B}{\tiny $\mathfrak{nl}_{e_2}$}
\psfrag{E}{\tiny $\mathbb{V}_{T/\!\!/ T}$}
\psfrag{F}{\tiny $\mathbb{V}_{T/\!\!/ S}$}
\psfrag{G}{\tiny $\mathbb{V}_{T/\!\!/ S'}$}
\psfrag{H}{\tiny $\mathbb{V}_{T}$}
\psfrag{c}{\tiny $\mathcal{L}-\varepsilon''$}
\psfrag{d}{\tiny $\mathcal{L}$}
\psfrag{e}{\tiny $\mathcal{L}+\varepsilon''$}
\psfrag{f}{\tiny $\mathcal{L}-\varepsilon''$}
\psfrag{g}{\tiny $\mathcal{L}$}
\psfrag{h}{\tiny $\mathcal{L}+\varepsilon''$}
\psfrag{i}{\tiny $\infty$}
\includegraphics[width=5cm]{corner.eps}
\end{center}
\caption{Schematic diagram of $\mathbb{V}_{T/\!\!/ T}$, $\mathbb{V}_{T/\!\!/ S}$, $\mathbb{V}_{T/\!\!/ S'}$, and $\mathbb{V}_{T}$. The horizontal and vertical lines represent the $\mathfrak{nl}_{e_1}$- and $\mathfrak{nl}_{e_2}$-coordinate axes and we are considering the projections of $\mathbb{V}_*$ to the $(\mathfrak{nl}_{e_1},\mathfrak{nl}_{e_2})$-coordinate plane.} \label{fig: corner}
\end{figure}

$$\mathcal{C}'_{T/\!\!/ T}, \quad \mathcal{C}'_{T/\!\!/ S}, \quad \mathcal{C}'_{T/\!\!/ S'}, \quad \mathcal{C}'_{T},$$
where ${\frak s}_{T}$ has not been defined yet, and we want to establish the commutativity of:
\begin{equation} \label{eqn: four charts}
\begin{diagram}
\mathcal{C}'_{T/\!\!/ T} & \rTo & \mathcal{C}'_{T/\!\!/ S} \\
\dTo & & \dTo\\
\mathcal{C}'_{T/\!\!/ S'} & \rTo & \mathcal{C}'_{T}.
\end{diagram}
\end{equation}
By construction the diagram commutes on the level of $\mathcal{C}$, i.e., we do not take into consideration the multisections.

\vskip.1in\noindent
Case (B).   Suppose that
\begin{equation} \label{eqn: triple 3}
V(T)=\{i_1,i_2,i_3\},\quad G(T)=\{e_1=(i_1,i_2),e_2=(i_1,i_3)\}.
\end{equation}
Let
\begin{equation}\label{eqn: triple 4}
S=(\{i_1,i_2\},\{e_1\}), \quad S'=(\{i_1,i_3\},\{e_2\})
\end{equation}
be subtrees of $T$. Case (B) is similar to Case (A) except that $\mathcal{M}_{i_2}$ may equal $\mathcal{M}_{i_3}$, in which case $\operatorname{Aut}(T)$ is nontrivial.

\subsection{Multisections on higher products} \label{subsection: multisections on higher products}

It remains to construct ${\frak s}_T$ when the cardinality $|V(T)|\geq 3$ and to show the commutativity for multisections. We consider $T$ with $|V(T)|=3$ and subtrees $S, S'\subset T$ with $|V(S)|=|V(S')|=2$ as in Cases (A) or (B); the general case is analogous and is left to the reader.

\vskip.1in
Let us first suppose that $\operatorname{Aut}(T)=1$. There are several regions to consider:

\vskip.1in\noindent
(1) On $\{\mathcal{L}>\mathfrak{nl}_{e_1}>\mathcal{L}-\varepsilon'',\mathfrak{nl}_{e_2}>\mathcal{L}-\varepsilon''\}$, ${\frak s}_T$ is the stabilization of ${\frak s}_{T/\!\!/ S}$. Similarly, on $\{\mathfrak{nl}_{e_1}>\mathcal{L}-\varepsilon'',\mathcal{L}>\mathfrak{nl}_{e_2}>\mathcal{L}-\varepsilon''\}$, ${\frak s}_T$ is the stabilization of ${\frak s}_{T/\!\!/ S'}$.  The two definitions agree on the overlap.

\vskip.1in\noindent
(2) On $\{\mathfrak{nl}_{e_1},\mathfrak{nl}_{e_2}\geq \mathcal{L}+\varepsilon''\}$, ${\frak s}_T$ is obtained in a manner analogous to the $|V(T)|=2$ case from the similarly defined
\begin{align*}
{\frak s}_T^\circ(x_{i_1},x_{i_2},x_{i_3}) = & (h_{i_2}(x_{i_2})h_{i_3}(x_{i_3}){\frak s}_{i_1}(x_{i_1}), h_{i_1}(x_{i_1})h_{i_3}(x_{i_3}){\frak s}_{i_2}(x_{i_2}),\\
& \qquad\qquad \qquad h_{i_1}(x_{i_1}) h_{i_2}(x_{i_2}){\frak s}_{i_3}(x_{i_3})),
\end{align*}
by pulling back via the similarly defined projection
$$p:\mathbb{V}_{i_1}\times\mathbb{V}_{i_2}\times \mathbb{V}_{i_3}\times [R,\infty)^2 \to \mathbb{V}_{i_1}\times\mathbb{V}_{i_2}\times \mathbb{V}_{i_3}$$
and pushing forward as in Diagram~\eqref{diagram for pushing forward}.

\vskip.1in\noindent
(3) Consider the region $\{\mathcal{L}+\varepsilon''\geq \mathfrak{nl}_{e_1}\geq \mathcal{L}, \mathfrak{nl}_{e_2}\geq \mathcal{L}+\varepsilon''\}$.   On the subregion $\{\mathcal{L}+\varepsilon''-\varepsilon'''\geq \mathfrak{nl}_{e_1}\geq \mathcal{L}, \mathfrak{nl}_{e_2}\geq \mathcal{L}+\varepsilon''\}$, where $0<\varepsilon'''\ll \varepsilon''$, we define ${\frak s}_T$ to be the product of ${\frak s}_{S}$ restricted to $\{\mathcal{L}+\varepsilon''\geq \mathfrak{nl}_{e_1}\geq \mathcal{L}\}$ and ${\frak s}_{i_3}$, as in Section~\ref{subsection: multisections on products}.

To define ${\frak s}_T$ on $\{\mathcal{L}+\varepsilon''\geq \mathfrak{nl}_{e_1}\geq \mathcal{L}+\varepsilon''-\varepsilon''', \mathfrak{nl}_{e_2}\geq \mathcal{L}+\varepsilon''\}$, we rephrase Theorem~\ref{thm: iterated gluing} as follows:

\begin{thm} \label{thm: comparison of two gluing maps}
The triple gluing map
$$G_{T}: \mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times \mathbb{V}_{i_3}\times [R,\infty)^2\to \widetilde{\mathcal{G}}^{E'}_\delta(\mathbb{V}_{i_1},\mathbb{V}_{i_2},\mathbb{V}_{i_3}),$$
\begin{align*}
((\mathcal{F}_1,u_1),(\mathcal{F}_2,u_2),(\mathcal{F}_3,u_3), & T_1,T_2)\mapsto\\
& G_{T}((\mathcal{F}_1,u_1),(\mathcal{F}_2,u_2),(\mathcal{F}_3,u_3),T_1,T_2),
\end{align*}
and the iterated gluing map
$$G_{T/\!\!/ S}\circ G_{S}: \mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times \mathbb{V}_{i_3}\times [R,\infty)^2\to \widetilde{\mathcal{G}}^{E'}_\delta(\mathbb{V}_{i_1},\mathbb{V}_{i_2},\mathbb{V}_{i_3}),$$
\begin{align*}
((\mathcal{F}_1,u_1),(\mathcal{F}_2,u_2), &(\mathcal{F}_3,u_3), T_1,T_2)\mapsto\\
& G_{T/\!\!/ S}(G_{S}((\mathcal{F}_1,u_1),(\mathcal{F}_2,u_2),T_1),(\mathcal{F}_3,u_3), T_2),
\end{align*}
do not necessarily agree but are $C^1$-close for $T_1,T_2\gg 0$.
\end{thm}

By Theorem~\ref{thm: comparison of two gluing maps}, on $\{\mathcal{L}+\varepsilon''\geq \mathfrak{nl}_{e_1}\geq \mathcal{L}+\varepsilon''-\varepsilon''', \mathfrak{nl}_{e_2}\geq \mathcal{L}+\varepsilon''\}$, (i) the pushforward of the product of ${\frak s}_{S}$ and ${\frak s}_{i_3}$ via $G_{T/\!\!/ S}$ is very close to (ii) the pushforward of the product of ${\frak s}_{i_1}$, ${\frak s}_{i_2}$, and ${\frak s}_{i_3}$ via $G_{T}$.  We then interpolate between the two on the region $\{\mathcal{L}+\varepsilon''\geq \mathfrak{nl}_{e_1}\geq \mathcal{L}+\varepsilon''-\varepsilon''', \mathfrak{nl}_{e_2}\geq \mathcal{L}+\varepsilon''\}$ so that ${\frak s}_T$ is given by (i) along $\mathfrak{nl}_{e_1}= \mathcal{L}+\varepsilon''-\varepsilon'''$ and by (ii) along $\mathfrak{nl}_{e_1}=\mathcal{L}+\varepsilon''$.

The situation for $\{\mathfrak{nl}_{e_1}\geq \mathcal{L}+\varepsilon'',\mathcal{L}+\varepsilon\geq \mathfrak{nl}_{e_2}\geq \mathcal{L}\}$ is similar.

\vskip.1in\noindent
(4) We finally extend arbitrarily to $\{\mathcal{L}+\varepsilon''\geq \mathfrak{nl}_{e_1}\geq \mathcal{L},\mathcal{L}+\varepsilon''\geq \mathfrak{nl}_{e_2}\geq \mathcal{L} \}$, subject to the conditions that the multisection be small and vanish away from $\mathcal{M}_{T/\!\!/ T}/\mathbb{R}$.

\vskip.1in
If $\operatorname{Aut}(T)\not = 1$ (this means that we are in Case (B)), then $\operatorname{Aut}(T)$ acts on the disjoint union of the four charts given in Equation~\eqref{eqn: four charts}, and we take the quotient; in fact this is the only way orbifold points arise in the semi-global Kuranishi charts.

\subsection{Modifications when (H) does not hold} \label{subsection: H does not hold}

In this subsection we describe the modifications that we need to make when (H) does not hold.

Choose $\mathcal{L}'\ll \mathcal{L}$ but still sufficiently large and $\varepsilon'''>0$ small.

\vskip.1in\noindent
(A) First consider the case where
$$V(T)=\{i_1,i_2\}, \quad G(T)=\{e=(i_2,i_1)\},$$
and at least one of $\chi(\dot F_{i_0}), \chi(\dot F_{i_1}), \chi(\dot F_{i_2})\geq 0$.

For $(\mathcal{F}_{i_0},u_{i_0})=(F_{i_0},j_{i_0},{\bf p}_{i_0},{\bf q}_{i_0}, {\bf r}_{i_0})$ which is close to breaking into
$$(\mathcal{F}_{i_1},u_1)\cup (\mathcal{F}_{i_2},u_2)\in \mathbb{V}_{i_1}\times \mathbb{V}_{i_2}$$
with $(\mathcal{F}_{i_j},u_{i_j})=(F_{i_j},j_{i_j},{\bf p}_{i_j},{\bf q}_{i_j}, {\bf r}_{i_j})$, $j=1,2$, we recall from Section~\ref{subsection: close to breaking} the sets of punctures ${\bf q}'_{i_1}$ and ${\bf q}'_{i_2}$ on $F_{i_0}$ which are close to ${\bf q}_{i_1}$ and ${\bf q}_{i_2}$ when $u_1$ and $u_2$ are preglued.

Consider the two hyperbolic metrics
$$g:=g_{(\dot F_{i_0}-{\bf q}_{i_0},j)}, \quad g_e:=g_{(\dot F_{i_0} - {\bf q}'_{i_1}-{\bf q}'_{i_2},j)}.$$
Let $\mathfrak{nl}_e=\mathfrak{nl}_e^{g_e}$ be the neck length function defined using $g_e$. Let us write $s_{\pm,j}^g$ for $s_{\pm,j}$ defined using $g$.

We then define $E_{(\mathcal{F}_{i_0},u_{i_0})}$ using $\widetilde{s}_{\pm,j}(\mathcal{F}_{i_0},u_{i_0})$, which in turn is defined as:
\begin{enumerate}
\item $s_{\pm,j}^{g}(\mathcal{F}_{i_0},u_{i_0})$ on $\{\mathfrak{nl}_e\leq \mathcal{L}'-\varepsilon'''\}$;
\item $s_{\pm,j}^{g_e}(\mathcal{F}_{i_0},u_{i_0})$ on $\{\mathfrak{nl}_e\geq \mathcal{L}'\}$; and
\item an interpolation between the two on $\{\mathcal{L}'-\varepsilon'''\leq \mathfrak{nl}_e\leq \mathcal{L}'\}$.
\end{enumerate}
On $\mathbb{V}_{i_0}$ the orbibundle $\mathbb{E}_{i_0}$ agrees with $E_{(\mathcal{F}_{i_0},u_{i_0})}$; on $\mathbb{V}_{T}$ the orbibundle $\mathbb{E}_T$ agrees with the stabilization $E'_{(\mathcal{F}_{i_0},u_{i_0})}$ of $E_{(\mathcal{F}_{i_0},u_{i_0})}$, constructed as in Equation~\eqref{equation: defn of E prime} using $g_e$.

\vskip.1in\noindent
(B) Next consider the case where
$$V(T)=\{i_1,i_2,i_3\},\quad G(T)=\{e_1=(i_2,i_1), e_2=(i_3,i_2)\},$$
and at least one of $\chi(\dot F_{i_j})$, $j=0,\dots,3$, is $\geq 0$.

For $(\mathcal{F}_{i_0},u_{i_0})=(F_{i_0},j_{i_0},{\bf p}_{i_0},{\bf q}_{i_0}, {\bf r}_{i_0})$ which is close to breaking into
\begin{equation}\label{eqn: triple}
(\mathcal{F}_{i_1},u_1)\cup (\mathcal{F}_{i_2},u_2)\cup (\mathcal{F}_{i_3},u_3)\in \mathbb{V}_{i_1}\times \mathbb{V}_{i_2}\times \mathbb{V}_{i_3}
\end{equation}
with $(\mathcal{F}_{i_j},u_{i_j})=(F_{i_j},j_{i_j},{\bf p}_{i_j},{\bf q}_{i_j}, {\bf r}_{i_j})$, $j=1,2,3$, we similarly define ${\bf q}'_{i_j}$, $j=1,2,3$.

Let us write
\begin{align*}
g:=g_{(\dot F_{i_0}-{\bf q}_{i_0},j)}, & \quad g_{e_1}:= g_{(\dot F_{i_0} - {\bf q}'_{i_3}-{\bf q}'_{i_1},j)},\\
g_{e_2}:= g_{(\dot F_{i_0} - {\bf q}'_{i_3}-{\bf q}'_{i_2},j)}, & \quad g_{e_1e_2}:=g_{(\dot F_{i_0} - {\bf q}'_{i_3}-{\bf q}'_{i_2}-{\bf q}'_{i_1},j)}.
\end{align*}
Also let $g'$ and $g_{e_1}'$ be $g$ and $g_{e_1}$ for
$$V(S)=\{i_1,i_2\},\quad G(S)=\{e_1=(i_2,i_1)\}$$
and let $g''$ and $g_{e_2}''$ be $g$ and $g_{e_2}$ for
$$V(S')=\{i_2,i_3\},\quad G(S')=\{e_2=(i_3,i_2)\},$$
both as defined in (A). Here $S/\!\!/ S$ (resp.\ $S'/\!\!/ S'$) is a $1$-vertex CH tree whose vertex we label $i_4$ (resp.\ $i_5$).

We will use $\mathfrak{nl}_{e_1}=\mathfrak{nl}_{e_1}^{g_{e_1e_2}}$ and $\mathfrak{nl}_{e_2}=\mathfrak{nl}_{e_2}^{g_{e_1e_2}}$ and define $E_{(\mathcal{F}_{i_0},u_{i_0})}$ using $\widetilde{s}_{\pm,j}(\mathcal{F}_{i_0},u_{i_0})$, which is given by:
\begin{enumerate}
\item $s_{\pm,j}^g(\mathcal{F}_{i_0},u_{i_0})$ on $\{\mathfrak{nl}_{e_1}\leq \mathcal{L}'-\varepsilon''',\mathfrak{nl}_{e_2}\leq \mathcal{L}'-\varepsilon'''\}$;
\item $s_{\pm,j}^{g_{e_2}}(\mathcal{F}_{i_0},u_{i_0})$ on $\{\mathfrak{nl}_{e_1}\leq \mathcal{L}'-\varepsilon''',\mathfrak{nl}_{e_2}\geq \mathcal{L}'\}$;
\item $s_{\pm,j}^{g_{e_1}}(\mathcal{F}_{i_0},u_{i_0})$ on $\{\mathfrak{nl}_{e_2}\leq \mathcal{L}'-\varepsilon''',\mathfrak{nl}_{e_1}\geq \mathcal{L}'\}$;
\item $s_{\pm,j}^{g_{e_1e_2}}(\mathcal{F}_{i_0},u_{i_0})$ on $\{\mathfrak{nl}_{e_1}\geq \mathcal{L}',\mathfrak{nl}_{e_2}\geq \mathcal{L}'\}$;
\item an interpolation on $\{\mathfrak{nl}_{e_1}\leq \mathcal{L}'-\varepsilon''', \mathcal{L}'-\varepsilon'''\leq \mathfrak{nl}_{e_2}\leq \mathcal{L}'\}$ which ``only depends on $\mathfrak{nl}_{e_2}$'' and analogous interpolations for
\begin{itemize}
\item $\{\mathfrak{nl}_{e_2}\leq \mathcal{L}'-\varepsilon''', \mathcal{L}'-\varepsilon'''\leq \mathfrak{nl}_{e_1}\leq \mathcal{L}'\}$,
\item $\{\mathfrak{nl}_{e_1}\geq \mathcal{L}',\mathcal{L}'-\varepsilon'''\leq \mathfrak{nl}_{e_2}\leq \mathcal{L}'\}$, and
\item $\{\mathfrak{nl}_{e_2}\geq \mathcal{L}',\mathcal{L}'-\varepsilon'''\leq \mathfrak{nl}_{e_1}\leq \mathcal{L}'\}$;
\end{itemize}
\item an arbitrary interpolation on $\{\mathcal{L}'-\varepsilon'''\leq \mathfrak{nl}_{e_1}\leq \mathcal{L}',\mathcal{L}'-\varepsilon'''\leq \mathfrak{nl}_{e_2}\leq \mathcal{L}'\}$.
\end{enumerate}
By ``only depending on $\mathfrak{nl}_{e_2}$" we mean the following: For $a<b$, let $\lambda_{a,b}:\mathbb{R}\to [0,1]$ be a smooth nondecreasing function such that $\lambda_{a,b}(a)=0$ and $\lambda_{a,b}(b)=1$. We then take
$$(1-\lambda(\mathfrak{nl}_{e_2}))s_{\pm,j}^g+\lambda(\mathfrak{nl}_{e_2}) s_{\pm,j}^{g_{e_2}},$$
where $\lambda=  \lambda_{\mathcal{L}'-\varepsilon''',\mathcal{L}'}$.
(In a slightly different case where $G(T)=\{e_1=(i_3,i_1), e_2=(i_3,i_2)\}$ and there is a $\mathbb{Z}/2$-action interchanging $e_1$ and $e_2$, we require the interpolation in (6) to respect the group action.) See Figure~\ref{fig: corner2}.

\begin{figure}[ht]
\begin{center}
\psfragscanon
\psfrag{A}{\tiny $\mathfrak{nl}_{e_1}$}
\psfrag{B}{\tiny $\mathfrak{nl}_{e_2}$}
\psfrag{E}{\tiny (1)}
\psfrag{F}{\tiny (2)}
\psfrag{G}{\tiny (3)}
\psfrag{H}{\tiny (4)}
\psfrag{I}{\tiny (5)}
\psfrag{J}{\tiny (6)}
\psfrag{c}{\tiny $\mathcal{L}'-\varepsilon'''$}
\psfrag{d}{\tiny $\mathcal{L}'$}
\psfrag{e}{\tiny $\mathcal{L}'+\varepsilon'''$}
\psfrag{f}{\tiny $\mathcal{L}'-\varepsilon'''$}
\psfrag{g}{\tiny $\mathcal{L}'$}
\psfrag{h}{\tiny $\mathcal{L}'+\varepsilon'''$}
\psfrag{i}{\tiny $\infty$}
\psfrag{j}{\tiny $\mathcal{L}$}
\psfrag{k}{\tiny $\mathcal{L}$}
\includegraphics[width=5cm]{corner2.eps}
\end{center}
\caption{The horizontal and vertical lines represent the $\mathfrak{nl}_{e_1}$- and $\mathfrak{nl}_{e_2}$-coordinate axes and we are considering the projections of $\mathbb{V}_*$ to the $(\mathfrak{nl}_{e_1},\mathfrak{nl}_{e_2})$-coordinate plane. The numbers indicate the regions appearing in the definition of $\widetilde{s}_{\pm,j}$.} \label{fig: corner2}
\end{figure}

We assume that the interpolations have been chosen so that, as $\mathfrak{nl}_{e_1}\to \infty$, the restriction of the Kuranishi structure to the $\mathfrak{nl}_{e_1}=const$ slices limit to a stabilization of the Kuranishi structure already constructed for $S'$.  This is possible because of the following:
As we take $\mathfrak{nl}_{e_1}\to\infty$, $g_{e_1e_2}$ for $(\mathcal{F}_{i_0},u_{i_0})$ converges to a $2$-component hyperbolic metric, one of which is $g''_{e_2}$ for $(\mathcal{F}_{i_5},u_{i_5})$; and $g_{e_1}$ for $(\mathcal{F}_{i_0},u_{i_0})$ converges to a $2$-component hyperbolic metric, one of which is $g''$ for the same $(\mathcal{F}_{i_5},u_{i_5})$.   The situation for $\mathfrak{nl}_{e_2}\to\infty$ is analogous.

The stabilizations are constructed as before with little change: On $\mathbb{V}_{i_0}$, the orbibundle $\mathbb{E}_{i_0}$ agrees with $E_{(\mathcal{F}_{i_0},u_{i_0})}$.  On $\mathbb{V}_{T/\!\!/ S}$, the orbibundle $\mathbb{E}_{T/\!\!/ S}$ agrees with the stabilization $E'_{(\mathcal{F}_{i_0},u_{i_0})}$ of $E_{(\mathcal{F}_{i_0},u_{i_0})}$, constructed as in Equation~\eqref{equation: defn of E prime} using $s'_{\pm,j}(\mathcal{F}_{i_0},u_{i_0})$ with respect to $g_{e_2}$ on $\mathfrak{nl}_{e_1}\leq \mathcal{L}'-\varepsilon'''$, using $s'_{\pm,j}(\mathcal{F}_{i_0},u_{i_0})$ with respect to $g_{e_1e_2}$ on $\mathfrak{nl}_{e_1}\geq \mathcal{L}'$, and interpolating between the two on $\mathcal{L}'-\varepsilon'''\leq\mathfrak{nl}_{e_1}\leq \mathcal{L}'$; note that the stabilizations defined using $g_{e_2}$ and $g_{e_1e_2}$ are close when $\mathcal{L}\gg 0$. We apply an analogous stabilization when we pass from $\mathbb{E}_{i_0}$ to $\mathbb{E}_{T/\!\!/ S'}$ and take a common stabilization when passing from $\mathbb{E}_{i_0}$ to $\mathbb{E}_{T}$.

\subsection{The semi-global Kuranishi structure} \label{subsection: semi-global Kuranishi structure}

We now summarize the above discussion.

Given $L\gg 0$ and an $L$-simple pair $(\alpha,J)$ in the (Symp) case, we have a {\em semi-global Kuranishi structure} $\mathscr{K}^L(\alpha,J)$ which is a category consisting of the following:
\begin{enumerate}
\item objects which are charts $\mathcal{C}'_T=(\mathcal{C}_T,{\frak s}_T)=(\mathbb{K}_T,\pi_T:\mathbb{E}_T\to \mathbb{V}_T,\overline\partial_J,{\frak s}_T)$, where $T$ ranges over all CH trees;
\item morphisms $\mathcal{C}'_{T/\!\!/\sqcup_i S_i}\to \mathcal{C}'_T$ given by restriction-inclusion as in Section~\ref{subsection: transition functions};
\item commutativity of diagrams of the form
\begin{equation}
\begin{diagram}
\mathcal{C}'_{T/\!\!/(S'\sqcup S''\sqcup (\sqcup_i S_i)} & \rTo & \mathcal{C}'_{T/\!\!/ (S'\sqcup (\sqcup_i S_i))} \\
\dTo & & \dTo\\
\mathcal{C}'_{T/\!\!/ (S''\sqcup (\sqcup _i S_i))} & \rTo & \mathcal{C}'_{T/\!\!/(\sqcup_iS_i)},
\end{diagram}
\end{equation}
where $S'$ and $S''$ are connected disjoint subtrees of $T$, and $S_i$'s are connected subtrees of $T$ that are disjoint from the $S'\sqcup S''$,
and of the form
\begin{equation}
\begin{diagram}
\mathcal{C}'_{T/\!\!/(S\sqcup(\sqcup_i S_i))} & \rTo & \mathcal{C}'_{T/\!\!/ (S'\sqcup(\sqcup_i S_i))} \\
\dTo & & \dTo\\
\mathcal{C}'_{T/\!\!/(S''\sqcup(\sqcup_i S_i))} & \rTo & \mathcal{C}'_{T/\!\!/ (\sqcup_i S_i)},
\end{diagram}
\end{equation}
where $S$ is a connected subtree of $T$ with $3$ vertices, $S'$ and $S''$ are connected subtrees of $S$ with $2$ vertices,
and $S_i$'s are subtrees of $T$ that are disjoint from $S$.
\end{enumerate}

Given a semi-global Kuranishi structure $\mathscr{K}^L(\alpha,J)$, we define
$$\mathcal{Z}(\mathscr{K}^L(\alpha,J))=\left(\coprod_T \overline\partial^{-1}_J({\frak s}_T)\right)/\sim_{\mathscr{K}},$$
where $T$ ranges over all CH trees and $\sim_{\mathscr{K}}$ is the identification given by the morphisms.

\begin{lemma}
$\mathcal{Z}(\mathscr{K}^L(\alpha,J))$ is a weighted branched manifold.
\end{lemma}

\begin{proof}
Each $\overline\partial^{-1}_J({\frak s}_T)$ is a weighted branched manifold since ${\frak s}_T$ is transverse to $\overline\partial_J$. The proof is the same as the usual manifold case since ${\frak s}_T$ is liftable by assumption.  The weights of the branches are equal to the weights of the branches of ${\frak s}_T$ that are intersected with $\overline\partial_J$.

In order to verify that $\left(\coprod_T \overline\partial^{-1}_J({\frak s}_T)\right)/\sim_{\mathscr{K}}$ is a weighted branched manifold, it remains to verify the Hausdorff property.  We explain the idea in the simplest case when $V(T)=\{i_1,i_2\}$, $E(T)=\{e=(i_2,i_1)\}$, and we are gluing
$$Z_1=\overline\partial^{-1}_J({\frak s}_{T/\!\!/ T}), \quad Z_2=\overline\partial^{-1}_J({\frak s}_{T})$$
corresponding to two charts $\mathcal{C}'_{T/\!\!/ T}$ and $\mathcal{C}'_{T}$. Since $\sim_{\mathscr{K}}$ homeomorphically identifies open subsets $U_1\subset Z_1$ and $U_2\subset Z_2$, and $Z_1$ and $Z_2$ are already Hausdorff, it suffices to find disjoint open sets for $z_1\in Z_1$ and $z_2\in Z_2-U_2$ (or vice versa). In the former case, such open sets clearly exist by using the neck length function $\mathfrak{nl}_e$ since $\mathfrak{nl}_e(z_2)\geq \mathcal{L}$ and $\mathfrak{nl}_e(z_1)< \mathcal{L}$ (or there is no $\varepsilon$-thin annulus corresponding to $\mathfrak{nl}_e$).
\end{proof}

When we want to specify the orbit $\gamma_+$ at the positive end of the topmost level and the orbits $\boldsymbol\gamma_-$ at the negative end of the bottommost level, we write $\mathscr{K}^L(\alpha,J;\gamma_+;\boldsymbol\gamma_-)$ and $\mathcal{Z}(\mathscr{K}^L(\alpha,J;\gamma_+;\boldsymbol\gamma_-))$.

\subsection{Cobordisms} \label{subsection: cob case}

In this subsection we describe the modifications needed for the cobordism maps in Section~\ref{chain map}.

Let $(\widehat W,\widehat \alpha)$ be the completion of the exact symplectic cobordism $(W,\alpha)$ from $(M_+,\alpha_+)$ to $(M_-,\alpha_-)$ as in Section \ref{subsection: almost complex structures}. Given $L_+\leq L_-$, let $J$ be an almost complex structure on $\widehat W$ which restricts to $J_+$ at the positive end and to $J_-$ at the negative end.
Suppose $(\widehat \alpha, J)$ is an $(L_+,L_-)$-simple pair and $J$ is $(L_+,L_-)$-end-generic.

As before, we choose an ordering $\mathcal{M}_1,\mathcal{M}_2,\dots,\mathcal{M}_\rho$ of moduli spaces where each $\mathcal{M}_i$ is one of
\begin{align*}
\mathcal{M}_i^J&=\mathcal{M}_J^{\operatorname{ind}=k_i}(\dot F_i, \widehat{W}; \gamma_{i,+},\boldsymbol\gamma_{i,-}),\\ \mathcal{M}_i^{J_\pm}&=\mathcal{M}_{J_\pm}^{\operatorname{ind}=k_i}(\dot F_i,\mathbb{R}\times M_\pm;\gamma_{i,+},\boldsymbol\gamma_{i,-}).
\end{align*}
Here $\dot F_i$ is connected, $\mathcal{A}_{\alpha_\pm}(\gamma_{i,+})\leq L$ with respect to the appropriate $\alpha_\pm$,  and each component of each level of a building of $\partial \mathcal{M}_i$ is either a trivial cylinder or in $\mathcal{M}_j$ with $j<i$.

\begin{defn} \label{defn: cob type}
A {\em CH tree of $\operatorname{(Cob)}$ type} is defined in the same way as a CH tree of (Symp) type, with the following exceptions:
\begin{enumerate}
\item there is a splitting of $V(T)$ into $V^{\operatorname{Cob}}(T) \sqcup V^{\operatorname{Symp}}(T)$, where elements of $V^{\operatorname{Cob}}(T)$ are the (Cob) vertices and the elements of $V^{\operatorname{Symp}}(T)$ are the (Symp) vertices;
\item referring to a maximal oriented path of the oriented tree (somewhat nonstandardly) as a {\em branch}, each branch has precisely one (Cob) vertex;
\item the (Cob) vertices are labeled with indices corresponding to a moduli space of curves with image in $\widehat{W}$.
\end{enumerate}
An edge is an {\em upper (resp.\ lower) $\operatorname{(Symp)}$ edge} if it is an edge between two (Symp) vertices lying above (resp.\ below) a (Cob) vertex, and is an {\em upper (resp.\ lower) $\operatorname{(Cob)}$ edge} if it is an edge from (resp.\ to)  a (Symp) vertex to (resp.\ from) a (Cob) vertex.
\end{defn}

\begin{defn}
An {\em isomorphism} of CH trees of $\operatorname{(Cob)}$ type is defined in the same way as an isomorphism of CH trees of (Symp) type, except that we additionally require that (Symp) vertices be taken to (Symp) vertices and (Cob) vertices to (Cob) vertices.
\end{defn}

We now consider a contraction of a CH tree $T$ of (Cob) type along a disjoint union $\sqcup_i S_i$ of subtrees of $T$ with no free edges.  For each $S_i$ there are two cases to consider:
\begin{enumerate}
\item[($\alpha$)] $S_i$ is a CH tree of (Symp) or (Cob) type. In this case we can replace $S_i$ by $\tau(S_i)$ as in Definition~\ref{defn: contraction}.
\item[($\beta$)] $S_i$ is not  a CH tree of (Symp) or (Cob) type (i.e., Condition (2) of Definition~\ref{defn: cob type} is not satisfied).  In this case $\tau(S_i)$ is just viewed as a symbol.
\end{enumerate}
The contraction $T/\!\!/\sqcup_i S_i$ is defined as in Definition~\ref{defn: contraction} with the above changes.  Let $Cl(\sqcup_i S_i)$ be the smallest disjoint union of subtrees of $T$ of type ($\alpha$) without free edges which is obtained from $\sqcup_i S_i$ by adding gluing edges.

In the (Symp) case, $\mathfrak{nl}_e$ for all the glued edges $e\in G(T)$ are independent.  The main difference with the (Cob) case is that for any two oriented paths $\gamma_1$, $\gamma_2$ from the top vertex to a (Cob) vertex, $\mathfrak{nl}_{\gamma_1}$ and $\mathfrak{nl}_{\gamma_2}$ are dependent, where
\begin{equation}
\mathfrak{nl}_\gamma:= \sum_{e\in G(\gamma)} \mathfrak{nl}_e.
\end{equation}

Let $T$ be a CH tree of (Cob) type and let $T/\!\!/ T$ be a $1$-vertex tree labeled by $\mathcal{M}_{i_0}$. We describe the obstruction orbibundles which ``cover'' $\mathcal{M}_{i_0}$ near the corner given by $T$.  We use coordinates $\mathfrak{nl}_{e_1},\dots, \mathfrak{nl}_{e_l}$, where $G(T)=\{e_1,\dots,e_l\}$.
Whenever we pass from $\mathfrak{nl}_{e_j}\leq \mathcal{L}-\varepsilon''$ to $\mathfrak{nl}_{e_j}>\mathcal{L}-\varepsilon''$, we stabilize the obstruction orbibundle using the ``first $\ell$" asymptotic eigenfunctions that correspond to the edge $e_j$. Let $\mathcal{C}\subset G(T)$, $T_{\mathcal{C}}=\cup_{e\in G(T)-\mathcal{C}} e$, and $T'=T/\!\!/ T_{\mathcal{C}}$. Then on the region
$$\{\mathfrak{nl}_e\geq \mathcal{L}-\varepsilon''\}_{e\in\mathcal{C}}\cap \{\mathfrak{nl}_e < \mathcal{L}\}_{e\in G(T)-\mathcal{C}},$$
the obstruction orbibundle
$$\pi_{T'}: \mathbb{E}_{T'}\to \mathbb{V}_{T'}\quad  \mbox{ or } \quad \pi_{T'}^{J}: \mathbb{E}_{T'}^J\to V_{T'}^J,$$
is stabilized using the asymptotic eigenfunctions corresponding to all $e\in \mathcal{C}$.

The obstruction multisections are constructed inductively as before.  Here we are assuming that $\chi<0$; the modifications used in the $\chi\geq 0$ case are analogous.
\begin{enumerate}
\item[(S0)] The multisections are invariant under isomorphisms of CH trees of (Symp) type or (Cob) type, as appropriate.
\item[(S1)] If $S=T$ or $S\subset T$ is a connected subtree of type ($\beta$) without free edges and $Cl(S)=T$, then:
\begin{enumerate}
\item The multisection $\frak{s}_{T/\!\!/ S}$ is defined arbitrarily (but generically), subject to consistency with adjacent regions and (S0).
\item The multisection $\frak{s}_T$ is defined arbitrarily (but generically) on the region
    $$\{\mathfrak{nl}_e\geq\mathcal{L}+\varepsilon''\}_{e\in G(T)-G(S)}\cap \{ \mathcal{L}\leq \mathfrak{nl}_e\leq \mathcal{L}+\varepsilon''\}_{e\in G(S)},$$
    subject to consistency with adjacent regions and (S0).
\end{enumerate}
\item[(S2)] Let $\mathcal{C}$ and $\mathcal{C}'$ be disjoint subsets of $G(T)$,
    $$\sqcup_i S_i=\cup_{e\in G(T)} e-\sqcup_{e\in \mathcal{C}} int(e)$$
    (here $S_i$ may be a single vertex), and $Cl(\sqcup_i S_i)=\sqcup_j \widetilde S_j$.  On the region
    $$\{\mathfrak{nl}_e\geq\mathcal{L}+\varepsilon''\}_{e\in \mathcal{C}}\cap \{ \mathcal{L}\leq \mathfrak{nl}_e\leq \mathcal{L}+\varepsilon''\}_{e\in\mathcal{C}'}\cap  \{\mathfrak{nl}_e< \mathcal{L}\}_{e\in G(T)-\mathcal{C}-\mathcal{C}'},$$
    the multisection ${\frak s}_{T'}$, $T'=T/\!\!/ (\cup_{e\in G(T)} -\sqcup_{e\in\mathcal{C}\cup\mathcal{C}'} int(e) ))$, is the pushforward of the product of the multisections ${\frak s}_{\widetilde S_j\cap (\cup_{e\in G(T)} -\sqcup_{e\in\mathcal{C}\cup\mathcal{C}'} int(e) ))}$, via the simultaneous gluing map.
\end{enumerate}
Again we use the fact that the iterated gluing maps are $C^1$-close to the simultaneous gluing maps and make $C^1$-small modifications on their overlaps.

\vskip.1in
Although strictly speaking not necessary, we give some examples which illustrate the combinatorics of the corners (see Figure~\ref{fig: trees}), and then explain the general case.    Since the projection of $\mathcal{M}_{i_0}$ to the $(\mathfrak{nl}_{e_1},\dots,\mathfrak{nl}_{e_l})$-coordinate plane (when applicable) lies within a fixed distance from the locus
$$\{\mathfrak{nl}_{\gamma_1}=\mathfrak{nl}_{\gamma_2}~|~ \mbox{ $\gamma_1$, $\gamma_2$ oriented paths from top vertex to a (Cob) vertex}\},$$
for the purposes of combinatorics we may assume that $\mathfrak{nl}_{\gamma_1}=\mathfrak{nl}_{\gamma_2}$ for any two $\gamma_1$, $\gamma_2$.  Given a collection of edges $e_1,\dots,e_k$, let
\begin{equation}
\mathfrak{nl}_{e_1\dots e_k}:=\min(\mathfrak{nl}_{e_1},\dots,\mathfrak{nl}_{e_k}).
\end{equation}

\begin{figure}[ht]
\begin{center}
\psfragscanon
\psfrag{a}{\tiny $\operatorname{(A)}$}
\psfrag{b}{\tiny $\operatorname{(B)}$}
\psfrag{c}{\tiny $\operatorname{(C)}$}
\psfrag{A}{\tiny $e_1$}
\psfrag{B}{\tiny $e_2$}
\psfrag{C}{\tiny $e_3$}
\psfrag{D}{\tiny $e_4$}
\includegraphics[width=9cm]{treesbis.eps}
\end{center}
\caption{The (Symp) vertices are given by solid dots and the (Cob) vertices by open squares.} \label{fig: trees}
\end{figure}

\vskip.1in\noindent {\bf Case (A).}
Let $T$ be a CH tree of (Cob) type such that
$$V^{\operatorname{Symp}}(T)=\{i_{k+1}\},~~ V^{\operatorname{Cob}}(T)=\{i_1,\dots, i_k\},$$
$$G(T)=\{e_1=(i_{k+1},i_1), \dots,e_k=(i_{k+1},i_k)\}.$$
The graph for $k=2$ is depicted in Figure~\ref{fig: trees}(A).  Refer to Figure~\ref{fig: corner3bis} which shows the $(\mathfrak{nl}_{e_1},\mathfrak{nl}_{e_2})$-coordinates.
\begin{figure}[ht]
\begin{center}
\psfragscanon
\psfrag{A}{\tiny $\mathfrak{nl}_{e_1}$}
\psfrag{B}{\tiny $\mathfrak{nl}_{e_2}$}
\psfrag{d}{\tiny $\mathfrak{nl}_{e_1}=\mathcal L-\varepsilon''$}
\psfrag{g}{\tiny $\mathfrak{nl}_{e_2}=\mathcal L-\varepsilon''$}
\psfrag{E}{\tiny $\varnothing$}
\psfrag{F}{\tiny $e_2$}
\psfrag{G}{\tiny $e_1$}
\psfrag{H}{\tiny $e_1,e_2$}
\psfrag{i}{\tiny $\infty$}
\includegraphics[width=4.5cm]{corner3bis.eps}
\end{center}
\caption{The shaded region is the image of $\mathcal{M}_{i_0}$. Note that the image of $\mathcal{M}_{i_0}$ is a bounded distance from $\mathfrak{nl}_{e_1}=\mathfrak{nl}_{e_2}$.} \label{fig: corner3bis}
\end{figure}
The labels for the regions indicate that we have stabilized using the asymptotic eigenfunctions that correspond to the given edges. The regions labeled $e_1$ and $e_2$ correspond to $T/\!\!/ S_1$ and $T/\!\!/ S_2$, respectively, where $G(S_1)=e_2$, $G(S_2)=e_1$, and both $S_1$ and $S_2$ are of type ($\beta$).

Observe that $\mathfrak{nl}_{e_1},\dots,\mathfrak{nl}_{e_k}$ are related on $\mathcal{M}_{i_0}$; in fact, $\mathfrak{nl}_{e_1}=\mathfrak{nl}_{e_2}=\dots =\mathfrak{nl}_{e_k}$ ``in the large''.  This motivates (S1) in the construction of the multisections.

\vskip.1in\noindent {\bf Case (B).}
Let $T$ be a CH tree of (Cob) type such that
$$V^{\operatorname{Symp}}(T)=\{i_4,i_5\},~~ V^{\operatorname{Cob}}(T)=\{i_1,i_2,i_3\},$$
$$G(T)=\{e_1=(i_4,i_1), e_2=(i_4,i_2), e_3=(i_5,i_4), e_4=(i_5,i_3)\}.$$
The graph $T$ is given in Figure~\ref{fig: trees}(B). We first map $\mathcal{M}_{i_0}$ to the $(\mathfrak{nl}_{e_1},\dots,\mathfrak{nl}_{e_4})$-coordinate plane.  Note that, ``in the large'' we have relations $\mathfrak{nl}_{e_1}=\mathfrak{nl}_{e_2}$ and $\mathfrak{nl}_{e_1}+\mathfrak{nl}_{e_3}=\mathfrak{nl}_{e_4}$. Hence we may project to coordinates $\mathfrak{nl}_{e_3}$ and any one of $\mathfrak{nl}_{e_1}, \mathfrak{nl}_{e_2},\mathfrak{nl}_{e_4}$, or $\mathfrak{nl}_{e_1e_2e_4}$. Note that $\mathfrak{nl}_{e_4}> \mathfrak{nl}_{e_1e_2}$ when we are sufficiently close to the corner and hence $\mathfrak{nl}_{e_1e_2e_4}=\mathfrak{nl}_{e_1e_2}$.

\vskip.1in\noindent
{\bf Case (C).} Let $T$ be a CH tree of (Cob) type such that
$$V^{\operatorname{Symp}}(T)=\{i_3,i_4,i_5\},~~ V^{\operatorname{Cob}}(T)=\{i_1,i_2\},$$
$$G(T)=\{e_1=(i_3,i_1), e_2=(i_4,i_2), e_3=(i_5,i_3), e_4=(i_5,i_4)\}.$$
First map $\mathcal{M}_{i_0}$ to the $(\mathfrak{nl}_{e_1},\dots,\mathfrak{nl}_{e_4})$-coordinate plane.  The image of $\mathcal{M}_{i_0}$ is a bounded distance away from $\mathfrak{nl}_{e_1}+\mathfrak{nl}_{e_3}=\mathfrak{nl}_{e_2}+\mathfrak{nl}_{e_4}$. It is not hard to verify that the neighborhood of the corner (i.e., $\mathfrak{nl}_{e_1}+\mathfrak{nl}_{e_3}=\mathfrak{nl}_{e_2}+\mathfrak{nl}_{e_4}$ with $\mathfrak{nl}_{e_j}$ large for all $j$) is a cone over a solid quadrilateral $\zeta(T)$ with vertices $e_1,e_2,e_3,e_4$ (in that order around the boundary) which correspond to the $\mathfrak{nl}_{e_i}$-coordinate axes. The boundary faces are the $(\mathfrak{nl}_{e_1},\mathfrak{nl}_{e_2})$-, $(\mathfrak{nl}_{e_2},\mathfrak{nl}_{e_3})$-, $(\mathfrak{nl}_{e_3},\mathfrak{nl}_{e_4})$-, and $(\mathfrak{nl}_{e_4},\mathfrak{nl}_{e_1})$-coordinate planes which intersect the closure of $\{\mathfrak{nl}_{e_i}=\mathcal{L}-\varepsilon''\}_{i=1,\dots,5}$ in a pattern similar to that of Figure~\ref{fig: corner}.

\vskip.1in\noindent{\bf General Case.}
We will inductively describe the convex polytope $\zeta(T)$ such that the $\mathfrak{nl}_{e_1},\dots,\mathfrak{nl}_{e_l}\to\infty$ corner  is a cone over $\zeta(T)$. Let $\Delta(f_0,\dots,f_n)$ be the $n$-dimensional simplex whose vertices are $f_0,\dots,f_n$; we are assuming that the vertices are linearly independent.

(1) A tree $T$ with one edge $e$ has $\zeta(T)=\Delta(e)$.

(2) If $T$ is obtained from $T'$ by adding an edge $e$ such that $t(e)$ is the topmost vertex of $T'$, then $\zeta(T)= \zeta(T')* \Delta(e)$, where $*$ is the join.

(3) Suppose $T$ is obtained from two connected CH trees $T_1$ and $T_2$ of (Cob) type by identifying the topmost vertices of $T_1$ and $T_2$.  If $T_1$ (resp.\ $T_2$) has only one edge $e$, then we take $\zeta(T)=\zeta(T_1)$ (resp.\ $\zeta(T_2)$).  Otherwise, we set
$$\partial \zeta(T)=\partial \zeta(T_1)* \partial\zeta(T_2).$$
We then ``recover'' $\zeta(T)$ by taking the convex hull of the vertices of $\partial\zeta(T)$, where we are assuming that $\zeta(T)$ lies on the hyperplane determined by $\mathfrak{nl}_{\gamma}=\mathfrak{nl}_{\gamma'}$ for all pairs $\gamma,\gamma'$ of oriented paths from the top vertex to a (Cob) vertex.

\subsection{Chain homotopy} \label{subsection: chain homotopy case}

Let $\{(\widehat W^\tau=\widehat W,\widehat \alpha^\tau)\}_{0\leq \tau \leq1}$  be a family of completed exact symplectic cobordisms from $(M_+,\alpha_+)$ to $(M_-,\alpha_-)$. For each $\tau \in[0,1]$, let $J^\tau$ be an almost complex structure on $\widehat W^\tau$ that restricts to $J_\pm$ at the positive/negative ends of $\widehat W^\tau$. We assume that $(\widehat \alpha^\tau, J^\tau)$ is an $(L_+,L_-)$-simple pair, $J^\tau$ is $(L_+,L_-)$-end-generic, and $(\widehat \alpha^\tau, J^\tau)$ is independent of $\tau$ on each of $\tau\in[0,\epsilon]$ and $\tau\in[1-\epsilon,1]$, where $\epsilon>0$ is small.

We use the ordering of moduli spaces from Section~\ref{subsection: cob case}. We write
$$\mathcal{M}_i^{\{J_\tau\}}=\coprod_{\tau\in[0,1]} \mathcal{M}_i^{J_\tau}$$
and let $\Pi_i:\mathcal{M}_i^{\{J_\tau\}}\to [0,1]$ be the projection that sends $\mathcal{M}_i^{J_\tau}$ to $\tau$.  We use $\varepsilon>0$ sufficiently small and $\ell>0$ sufficiently large so that Theorem~\ref{thm: construction of L-transverse subbundle} holds simultaneously for all $J_\tau$, $\tau\in[0,1]$. If $T$ is a CH tree of (Cob) type, then let
$$\pi_T^{\{J_\tau\}}: \mathbb{E}_T^{\{J_\tau\}}\to \mathbb{V}_T^{\{J_\tau\}}$$
be the $1$-parametric version of the obstruction orbibundle $\pi_T^{J_\tau}:\mathbb{E}_T^{J_\tau}\to \mathbb{V}_T^{J_\tau}$.  The obstruction multisection $\frak{s}_T^{\{J_\tau\}}$ on $\mathbb{V}_T^{\{J_\tau\}}$ is also the $1$-parametric version of $\frak{s}^{J_\tau}$ on $\mathbb{V}_T^{J_\tau}$. We assume that $\pi_T^{\{J_\tau\}}$ and $\frak{s}_T^{\{J_\tau\}}$ are generic subject to the following conditions:
\begin{enumerate}
\item[($\tau_1$)] they are independent of $\tau$ on $[0,\epsilon]$ and on $[1-\epsilon,1]$; and
\item[($\tau_2$)] they agree with $\pi_T^{J_0}$, $\frak{s}_T^{J_0}$, $\pi_T^{J_1}$, and $\frak{s}_T^{J_1}$, which we assume have already been constructed for the chain maps corresponding to $J_0$ and $J_1$.
\end{enumerate}

To define the chain homotopy $\Omega_K$ we consider disconnected curves; more formally we take the multiple fibered products
$$\mathbb{V}_{T_1\dots T_k}^{\{J_\tau\}}:=\mathbb{V}_{T_1}^{\{J_\tau\}}\times_{[0,1]}\mathbb{V}_{T_2}^{\{J_\tau\}}\times_{[0,1]} \dots \times_{[0,1]} \mathbb{V}_{T_k}^{\{J_\tau\}},$$
$$\mathbb{E}_{T_1\dots T_k}^{\{J_\tau\}}:=\mathbb{E}_{T_1}^{\{J_\tau\}}\times_{[0,1]}\mathbb{E}_{T_2}^{\{J_\tau\}}\times_{[0,1]} \dots \times_{[0,1]} \mathbb{E}_{T_k}^{\{J_\tau\}},$$
where $\boldsymbol\gamma_+=(\gamma_{+,1},\dots,\gamma_{+,k})$ and curves in $\mathbb{V}_{T_i}^{\{J_\tau\}}$ limit to $\gamma_{+,i}$ at the positive end.  We denote the corresponding obstruction orbibundle by
$$\pi_{T_1\dots T_k}^{\{J_\tau\}}:\mathbb{E}_{T_1\dots T_k}^{\{J_\tau\}}\to \mathbb{V}_{T_1\dots T_k}^{\{J_\tau\}}.$$

\subsubsection{Basic perturbation scheme} \label{subsubsection: basic}

For the purposes of Proposition~\ref{prop: chain homotopy}, we use an inductive perturbation scheme. The multisection $\frak{s}_{T_1\dots T_k}^{\{J_\tau\}}$ is obtained from the product of multisections $\frak{s}_{T_i}^{\{J_\tau\}}$ on each $\mathbb{V}_{T_i}^{\{J_\tau\}}$ as follows:
\begin{enumerate}
\item[(*)] For each pair $S\subset T$ satisfying (S1) in Section~\ref{subsection: cob case} and for each occurrence of $\mathbb{V}_{T/\!\!/ S}^{\{J_\tau\}}$ or $\mathbb{V}_{T}^{\{J_\tau\}}$ restricted to
$$\{\mathfrak{nl}_e\geq\mathcal{L}+\varepsilon''\}_{e\in G(T)-G(S)}\cap \{ \mathcal{L}\leq \mathfrak{nl}_e\leq \mathcal{L}+\varepsilon''\}_{e\in G(S)}$$
as in (S1)(a) or (S1)(b), we choose a different perturbation of the multisection ${\frak s}_{T/\!\!/ S}^{\{J_\tau\}}$ or ${\frak s}_{T}^{\{J_\tau\}}$, subject to Conditions ($\tau_1$) and ($\tau_2$), so that all the perturbed multisections are mutually generic away from $[0,\epsilon]\cup [1-\epsilon,1]$.
\end{enumerate}

\subsubsection{Refined perturbation scheme} \label{subsubsection: refined}

For the purposes of Proposition~\ref{prop: chain homotopy ver 2}, we use the following refined perturbation scheme, which is that of Ekholm-Oancea~\cite{EO} in our setting. Choose sufficiently small constants $0<\epsilon_1\ll \epsilon_2 \ll \dots \ll \epsilon_\rho \ll\epsilon$ corresponding to the ordering $\mathcal{M}_1,\dots,\mathcal{M}_\rho$ of moduli spaces as before. Then $\frak{s}_{T_1\dots T_k}^{\{J_\tau\}}$ is inductively obtained from the product of perturbations of $\frak{s}_{T_i}^{\{J_\tau\}}$:
\begin{enumerate}
\item[(**)] For the $j$th occurrence of $\mathbb{V}_{T_i}^{\{J_\tau\}}$ in $\mathbb{V}_{T_1\dots T_k}^{\{J_\tau\}}$, we perturb ${\frak s}_{T_i}^{\{J_\tau\}}$ so that its intersection with $\overline\partial_{J_\tau}$ is obtained from $\overline\partial_{J_\tau}^{-1}({\frak s}_{T_i}^{\{J_\tau\}})$ by shifting via $\tau\mapsto \chi_j(\tau)$.  Here $\chi_j:[0,1]\stackrel\sim\to[0,1]$ is a diffeomorphism such that
\begin{itemize}
\item $\chi_j'(\tau)$ is close to $1$;
\item $\chi_j(\tau)=\tau$ on $[0,\epsilon]\cup[1-\epsilon,1]$ and $\chi_j(\tau)=\tau+ j\epsilon_{i(T_i/\!\!/ T_i)}$ on $[2\epsilon, 1-2\epsilon]$;
\item $i(T_i/\!\!/ T_i)$ is the index corresponding to $T_i/\!\!/ T_i$.
\end{itemize}
\end{enumerate}

\section{Contact homology} \label{section: contact homology}

\subsection{Chain complex} \label{chain complex}

Let $(M^{2n+1},\xi)$ be a closed contact manifold. Given $L>0$, let $\alpha$ be a contact form for $\xi$ and $J$ be an almost complex structure on $\mathbb R \times M$.
Suppose that $(\alpha,J)$ is an $L$-simple pair.  By Section~\ref{section: construction of semi-global} there exists a semi-global Kuranishi structure
$$\mathscr{K}^L(\alpha,J)=\{ \mathscr{K}^L(\alpha, J; \gamma_+; \boldsymbol \gamma_-)~|~ \mathcal A ( \gamma_+),\mathcal A(\boldsymbol \gamma_-)\leq L  \}.$$

Let
$$\mathfrak A^L=\mathfrak A^L(M,\alpha, J, \mathscr{K}^L(\alpha,J), \mathcal{S}^{L}_{\alpha})$$
be the unital graded commutative algebra freely generated by the good Reeb orbits in $\mathcal P_\alpha^{L}$ over the group algebra $\mathbb{Q}[H_2(M;\mathbb{Z})]$.  The generators of the group algebra $\mathbb{Q}[H_2(M;\mathbb{Z})]$ will be written as $e^A$, where $A\in H_2(M;\mathbb{Z})$.  In Section~\ref{subsubsection: grading} we explain how to assign a $\mathbb{Q}$-grading $|\cdot |$ to each Reeb orbit in $\mathcal P_\alpha^{L}$ and to each $e^A$, $A\in H_2(M;\mathbb{Z})$, and give a definition of $\mathcal{S}^L_\alpha$.

We also define a $\mathbb{Z}/2$-grading $|\cdot|_0$ on $\mathcal{P}_\alpha^L$:  Given $\gamma\in  \mathcal{P}_\alpha^L$, we set
$$|\gamma|_0\equiv \mu_\tau(\gamma)+n-3 \mbox{ mod } 2,$$
where $\mu_\tau(\gamma)$ is the Conley-Zehnder index with respect to the trivialization $\tau$.  Observe that the parity of $\mu_\tau(\gamma)$ does not depend on the choice of $\tau$.

\begin{defn}
An ordered tuple of Reeb orbits $\boldsymbol \gamma$ is {\em good} if all of its entries are good Reeb orbits. Otherwise, $\boldsymbol\gamma$ is {\em bad}.
\end{defn}

Next we define the differential $\partial: \mathfrak A^L \to \mathfrak A^L$. This makes $\mathfrak A^L$ into a dga and we denote its homology by $HC(M,\alpha, J, \mathscr{K}^L(\alpha,J),\mathcal{S}^L_\alpha)$ or $HC^L$. Fix an ordering $\vartheta$ of $\mathcal P^{L}_\alpha$.  We then set
\begin{equation} \label{eqn: partial}
\partial \gamma_+=\sum_{\boldsymbol\gamma_-}\sum_{A\in H_2(M;\mathbb{Z})} \frac{d_{\gamma_+,\boldsymbol \gamma_-,A}}{m_{\boldsymbol \gamma_-}} e^A \gamma_{-,1}^{i_1}\dots \gamma_{-,k}^{i_k},
\end{equation}
where the sums are taken over all good $\vartheta$-sorted $\boldsymbol \gamma_-$ and $A\in H_2(M;\mathbb{Z})$ with  $|\gamma_+|-(|e^A|+|\boldsymbol \gamma_-|)=1$.  A sorted $\boldsymbol\gamma_-$ is written as
\begin{equation}\label{eqn: gamma-}
\boldsymbol \gamma_-=(\underbrace{\gamma_{-,1},\dots, \gamma_{-,1}}_{i_{1} \text{ copies }},\dots, \underbrace{\gamma_{-,k},\dots, \gamma_{-,k}}_{i_{k} \text{ copies }}),
\end{equation}
$d_{\gamma_+,\boldsymbol \gamma_-, A}$ is the weighted signed count of elements in $\mathcal Z(\mathscr{K}^L(\alpha, J;\gamma_+; \boldsymbol \gamma_-))$ in the class $A$, and
$$m_{\boldsymbol \gamma_-}:=\Pi_{j=1}^k ( i_j ! m(\gamma_{-,j})^{i_j}).$$
(If $|\gamma_+|-(|e^A|+|\boldsymbol \gamma_-|)\neq 1$, then we set $d_{\gamma_+,\boldsymbol \gamma_-,A}=0$.) In Section~\ref{subsubsection: grading} we explain how to define the class $A$ corresponding to an approximate $J$-holomorphic map $u$ from $\gamma_+$ to $\boldsymbol\gamma_-$ and in Section~\ref{subsubsection: orientations} we explain how to orient the moduli spaces. Then we extend $\partial$ to $\mathfrak A^L$ using the graded Leibniz rule, namely,
$$\partial (\gamma \gamma')=(\partial \gamma) \gamma'+(-1)^{|\gamma|_0}\gamma (\partial  \gamma').$$

\subsubsection{Grading and $\mathbb{Q}[H_2(M;\mathbb{Z})]$-coefficients} \label{subsubsection: grading}

We follow the discussion in \cite{Bo}.  Let us write $H_1(M;\mathbb{Z})= F\oplus T$, where $F$ is the free part and $T$ is the torsion part.  Pick representatives $C_1,\dots, C_a$ of a basis for $F$ and representatives $D_1,\dots, D_b$ of a minimal generating set for $T$, and fix a trivialization $\tau$ for $\xi$ along each $C_i$ and $D_i$. Given $\gamma\in \mathcal{P}^L_\alpha$, let us write $[\gamma]=[\sum c_i C_i +\sum d_i D_i]\in H_1(M;\mathbb{Z})$, where $d_i$ are the smallest nonnegative coefficients for the $D_i$.  Then choose a surface $S_\gamma$ such that $\partial S_\gamma=\gamma-\sum_i c_i C_i -\sum_i d_i D_i$ and use $S_\gamma$ to extend the trivializations $\tau$ along ${C_i}$ and ${D_i}$ to all of $S_\gamma$ and hence to $\gamma$.  If $[D_i]$ has order $m_i$, then choose a spanning surface $S_{m_i D_i}$ for $m_i D_i$. This gives a trivialization $\tau'$ along $m_i D_i$ and we let $w_i$ be the rotation number of $\tau$ with respect to $\tau'$ along $m_i D_i$.  We then set
$$|\gamma|= \mu_\tau(\gamma)+2\sum_{i=1}^b{w_i d_i\over m_i} + n-3\in \mathbb{Q} ,$$
where $\mu_\tau$ is the Conley-Zehnder index with respect to $\tau$.  We also set
$$|e^A|=-2\langle c_1(\xi),A\rangle.$$

The collection
$$\mathcal{S}^L_\alpha=\{S_\gamma\}_{\gamma\in \mathcal{P}^L_\alpha}\cup \{S_{m_i D_i}\}_{i=1,\dots, b}$$
will be called a {\em complete set of trivializing surfaces for $\mathcal{P}^L_\alpha$}.

Given a map $u$ from $\gamma$ to $\boldsymbol\gamma_-$, we can cap off the projection $\pi_M\circ u$ of $u$ to $M$ along $\gamma$ by $S_\gamma$ and along $\gamma_{-,j}$ by $S_{\gamma_{-,j}}$ and further cap off any extraneous $m_i D_i$ by $S_{m_i D_i}$, which gives a closed surface $A$. Note that if
$$\partial S_\gamma=\gamma-\sum_i c_i C_i -\sum_i d_i D_i,$$
where the $d_i$ are the smallest nonnegative coefficients, then
$$\sum_j \partial S_{\gamma_{-,j}}=\sum_j\gamma_{-,j} -\sum_i c_i C_i -\sum_i d_i D_i + \zeta,$$
where $\zeta$ is a sum of $m_i D_i$.

\subsubsection{Orientations} \label{subsubsection: orientations}

We explain how to assign a sign to
$$[\mathcal{F}, u]\in \mathcal Z(\mathscr{K}^L(\alpha, J;\gamma_+; \boldsymbol \gamma_-))$$
as we range over all $\mathcal Z(\mathscr{K}^L(\alpha, J;\gamma_+; \boldsymbol \gamma_-))$ in $\mathscr{K}^L(\alpha,J)$.

For any $(\mathcal{F}, u)\in \mathcal{B}_{\widetilde{\mathcal U}}$, let $L_{(\mathcal{F},u)}$ be the full linearized $\overline\partial_J$-operator and
$$\operatorname{det}L_{(\mathcal{F}, u)}=\wedge^{\operatorname{top}}\ker L_{(\mathcal{F}, u)} \otimes \wedge^{\operatorname{top}}(\operatorname{coker}L_{(\mathcal{F}, u)})^*$$
be its determinant line. An {\em orientation} $\mathfrak o(\mathcal{F}, u)$ of $\operatorname{det}L_{(\mathcal{F}, u)}$ is an equivalence class of nonzero vectors of $\operatorname{det}L_{(\mathcal{F}, u)}$, where the equivalence relation is $v\sim v'$ if there exists $c>0$ such that $v=cv'$.

According to \cite{BM}, an orientation $\mathfrak o(\mathcal{F}, u)$ can be chosen in a continuous and coherent way for all $(\mathcal{F},u)$ in all the $\mathcal{B}_{\widetilde{\mathcal U}}$, where ``coherent" means:
\begin{itemize}
\item[(O1)] if we exchange the $i$th and $(i+1)$st negative punctures of $(\mathcal{F},u)$, then the sign changes by $(-1)^{|\gamma_{-,i}|_0\cdot|\gamma_{-,i+1}|_0}$, where $u$ is asymptotic to $\gamma_{-,j}$ near the $j$th negative puncture $p_{-,j}$ (\cite[Theorem 2]{BM});
\item[(O2)] the gluing/pregluing and disjoint union maps preserve orientations up to a specific sign change which arises from reordering the punctures; in particular, if $(\mathcal{F}_2,u_2)$ above is glued/preglued to $(\mathcal{F}_1,u_1)$ below along the last puncture of $u_2$, and the negative ends of the (pre-)glued curve are ordered by using the ordering for $u_2$ first, followed by the ordering of $u_1$, then there is no sign correction (\cite[Corollary 10]{BM});
\item[(O3)] if $u$ is asymptotic to $\gamma$ near the  puncture $p$ and $m(\gamma)>1$, then cyclically rotating the asymptotic marker at $p$ through an angle of ${2\pi\over m(\gamma)}$ preserves the orientation if and only if $\gamma$ is good (\cite[Theorem 3]{BM});
\item[(O4)] precomposing with an automorphism of the domain which preserves the punctures and asymptotic markers preserves the orientation  (\cite[Proposition 11]{BM}).
\end{itemize}

Note that the coherent orientation exists for {\em all maps $(\mathcal{F},u)$ in all $\mathcal{B}_{\widetilde{\mathcal{U}}}$} and results of \cite{BM}, namely Theorems 2 and 3 and Corollaries 10 and 11, while stated only for moduli spaces of $J$-holomorphic maps, easily generalize to the case of semi-global Kuranishi structures.

Given a semi-global Kuranishi chart $(\mathbb K, \pi_{\mathbb V}:\mathbb E\to \mathbb V, \overline \partial_J, \mathfrak s)$, we explain how the coherent orientation induces an orientation of the total space of $\mathbb{E}\to \mathbb{V}$.  We consider the slightly easier (Cob) case, leaving the (Symp) case to the reader.  Let $(E\to V,\Gamma)$ be an orbibundle chart for $\mathbb{E}\to\mathbb{V}$. Given $(\mathcal{F}, u)\in  V$, let $L_{(\mathcal{F},u)}$ be the full linearized $\overline \partial_J$-operator at $(\mathcal{F},u)$ and $L_{(\mathcal{F},u)}^E : T_{(\mathcal{F},u)}V\to E_{(\mathcal{F},u)}$ its restriction to $T_{(\mathcal{F},u)}V=L^{-1}_{(\mathcal{F},u)}(E_{(\mathcal{F},u)})$. By \cite[Section 8.1.1]{FO3}, the orientation $\mathfrak o(L_{(\mathcal{F},u)})$ induces an orientation $\mathfrak o(\det L_{(\mathcal{F}, u)}^E)$ and, equivalently, an orientation ${\frak o}(E)$ on the total space of $E$.  By Proposition (O4), the group action $\Gamma$ on $E$ preserves ${\frak o}(E)$.  Hence the orientation descends to the total space of $\mathbb{E}\to \mathbb{V}$.

Finally we explain how $\mathcal{Z}(\mathbb{K}^L(\alpha,J; \gamma_+;\boldsymbol\gamma_-))$ is oriented: Let ${\frak s}=[s_1,\dots,s_m]$ be a liftable multisection of $(E\to V,\Gamma)$ that is transverse to $\overline \partial _J$.  For $(\mathcal{F}, u)\in \overline \partial_J^{-1}(s_i)$, $i=1,\dots,m$, we orient $T_{(\mathcal{F}, u)} V$ arbitrarily and orient $\partial_J^{-1}(s_i)$ as the transverse intersection of the sections $\partial_J$ and $s_i$, where both $\partial_J$ and $s_i$ are given the orientation induced from $T_{(\mathcal{F}, u)} V$.  Observe that the orientation of $\partial_J^{-1}(s_i)$ does not depend on the choice of orientation for $T_{(\mathcal{F}, u)}V$.

\subsubsection{Examples}

To illustrate the definition of the differential, we consider two examples (here we use $\mathbb{Q}$-coefficients):

1. Suppose that $\gamma_+, \gamma_{-,1}, \gamma_{-,2}\in \mathcal P_\alpha^L$ are good and that $\gamma_{-,1}, \gamma_{-,2}$ are simple.
Also suppose that
$$\mathcal M:= \mathcal M (\gamma_+;\underbrace{\gamma_{-,1},\dots,\gamma_{-,1}}_{3 \text{ copies}}, \underbrace{\gamma_{-,2},\dots,\gamma_{-,2}}_{5 \text{ copies}})$$
is regular and that $\llbracket\mathcal{F},u\rrbracket\in \mathcal M/\mathbb R$ is simple and is counted as $+1$ with respect to the coherent orientation.
Then by relabeling the punctures in the domain of $u$ we get $3!5!$ elements in $ \mathcal M /\mathbb R$ and $3!5!$ elements in
$$\mathcal M(\gamma_+;\underbrace{\gamma_{-,2},\dots,\gamma_{-,2}}_{5 \text{ copies}}, \underbrace{\gamma_{-,1},\dots,\gamma_{-,1}}_{3 \text{ copies}})/\mathbb R.$$
Depending on whether $|\gamma_{-,1}|_0$ and $|\gamma_{-,2}|_0$ are both even, the total contribution to $\partial \gamma_+$ from the image of $u$ (modulo $\mathbb{R}$-translations) is either $\gamma_{-,1}^3\gamma_{-,2}^5$ or zero.

2. Suppose that $\gamma_\pm,\gamma'_+ \in \mathcal P_\alpha^{L}$ are good and that $\gamma_-$ is simple. Also suppose that $\mathcal M(\gamma_+;\gamma_-,\gamma_-)$ is regular and that $(\mathcal{F}, u)\in \mathcal M(\gamma_+;\gamma_-,\gamma_-)$ is a possibly branched double cover of a simple curve $(\mathcal{F}', u')\in \mathcal M(\gamma'_+;\gamma_-)$. Since reordering the negative punctures of $(\mathcal{F},u)$ gives the same curve but changes the sign by $(-1)^{|\gamma_-|_0\cdot |\gamma_-|_0}$, the existence of the coherent orientation system implies that $|\gamma_-|_0$ must be even and that $\llbracket\mathcal{F}, u\rrbracket$ contributes $\pm 1$ (depending on the coherent orientation) to $\mathcal M(\gamma_+;\gamma_-,\gamma_-)/\mathbb R$ and contributes $\pm \frac{1}{2} \gamma_-^2$ to $\partial \gamma_+$.

\subsubsection{$\partial^2=0$} \label{subsubsection: d squared is zero}

\begin{prop} \label{dga}
$\partial$ is well-defined and $\mathfrak A^L $ is a dga with differential given by $\partial$.
\end{prop}

\begin{proof}
By the construction of the semi-global Kuranishi structure and the above discussion on coherent orientations, if $|\gamma_+|-(|e^A|+|\boldsymbol\gamma_-|)=1$ and $\gamma_+, \boldsymbol\gamma_-$ are good, then $\mathcal Z(\mathscr{K}^L(\alpha, J;\gamma_+; \boldsymbol \gamma_-;A))$  (this means the subset of $\mathcal Z(\mathscr{K}^L(\alpha, J;\gamma_+; \boldsymbol \gamma_-))$ in the class $A$) is an oriented compact weighted branched $0$-dimensional manifold.  Hence $\partial$ is well-defined. In this proof we will abbreviate
$$\mathcal Z(\mathscr{K}(\gamma_+;\boldsymbol\gamma_-))=\mathcal Z(\mathscr{K}^L(\alpha, J;\gamma_+;\boldsymbol \gamma_-;A)),$$
assume that the homology classes have been chosen appropriately, and suppress homology classes from the notation.

To show $\partial^2=0$, as usual we identify the terms in $\partial^2$ as the signed weighted count of the boundary components of certain oriented compact weighted branched $1$-dimensional manifolds.

{\em In this proof we write an ordered tuple of Reeb orbits multiplicatively. Moreover, when there is no confusion, we do not distinguish an ordered tuple of Reeb orbits from the monomial associated to it in $\mathfrak A^L$.}

For any $\vartheta$-sorted $\boldsymbol \gamma=\gamma_1^{c_1}\dots\gamma_m^{c_m}$ with $c_1,\dots,c_m\geq 0$, we obtain
\begin{align}\label{eqn: coefficient}
m_{\boldsymbol\gamma}\langle \partial^2 \gamma_+, \boldsymbol\gamma \rangle=&\sum_{\boldsymbol \beta\subseteq \boldsymbol\gamma}\sum_{k=1}^m\sum_{q=0}^{c_k-b_k}(-1)^{a+b} \frac{d_{\gamma_+;(\boldsymbol\gamma-\boldsymbol\beta)\sqcup \gamma_k}}{m_{(\boldsymbol\gamma-\boldsymbol\beta)\sqcup \gamma_k}}\frac{d_{\gamma_k;\boldsymbol \beta}}{m_{\boldsymbol \beta}}m_{\boldsymbol\gamma}\\
\label{eqn: coefficient 2} =&\sum_{\boldsymbol \beta\subseteq \boldsymbol\gamma}\sum_{k=1}^m\sum_{q=0}^{c_k-b_k}(-1)^{a+b}
{c_1\choose b_1}\dots{c_m \choose b_m}\frac{d_{\gamma_+;(\boldsymbol\gamma-\boldsymbol\beta)\sqcup \gamma_k}d_{\gamma_k;\boldsymbol \beta}}{m(\gamma_k)(c_k-b_k+1)},
\end{align}
where the first sum is taken over all $\vartheta$-sorted subtuples $\boldsymbol \beta$ of $\boldsymbol \gamma$, which we write as $\boldsymbol\beta=\gamma_1^{b_1}\dots\gamma_m^{b_m}$ with $0\leq b_i\leq c_i$ for $i=1,\dots,m$, and {\em the summands on the right-hand side depend on the variable $q$ only through $a$ and $b$.}
Now we explain the rest of the notation in the above formula.
We define
$$(\boldsymbol \gamma-\boldsymbol\beta)\sqcup \gamma_k:=\gamma_1^{c_1-b_1}\dots \gamma_{k}^{c_k-b_{k}+1}\dots\gamma_{m}^{c_m-b_m},$$
$$(\boldsymbol\gamma-\boldsymbol\beta) \sqcup_{\gamma_k,q}\boldsymbol\beta:= \gamma_1^{c_1-b_1}\dots\gamma_k^q(\gamma_1^{b_1}\dots\gamma_m^{b_m}) \gamma_k^{c_k-b_k-q}\dots\gamma_m^{c_m-b_m},$$
and then $(-1)^b$ is given by the equation $$\boldsymbol\gamma=(-1)^b(\boldsymbol\gamma-\boldsymbol\beta) \sqcup_{\gamma_k,q} \boldsymbol\beta$$ as supercommutative monomials and $a=|\gamma_1^{c_1-b_1}\dots\gamma_k^q|_0$.

We claim that, for each $\boldsymbol\beta\subseteq\boldsymbol\gamma$ and $k\in \{1,\dots,m\}$ with $|\gamma_k|-(|e^{A_1}|+|\boldsymbol\beta|)=1$ and $|\gamma_+|-(|e^{A_2}|+|(\boldsymbol \gamma-\boldsymbol\beta)\sqcup \gamma_k|)=1$ (for some $A_1+A_2=A$) and $q\in \{0,1,\dots,c_k-b_k\}$, there exists a truncation $T(\mathcal Z(\mathscr{K}(\gamma_+;\boldsymbol \gamma)))$ of the ends of $\mathcal Z(\mathscr{K}(\gamma_+;\boldsymbol \gamma))$ and a map
$$G_{\boldsymbol\beta,k,q}:  \mathcal Z(\mathscr{K}(\gamma_k;\boldsymbol\beta)) \times  \mathcal Z(\mathscr{K}(\gamma_+;(\boldsymbol\gamma-\boldsymbol\beta)\sqcup \gamma_k))  \times O\to  \partial T( \mathcal Z(\mathscr{K}(\gamma_+;\boldsymbol \gamma))), $$
where $|O|={c_1 \choose b_1}\dots{c_m \choose b_m}$.

Before delving into the combinatorics, we explain why each end of the $1$-dimen\-sional branched manifold $\mathcal{Z}(\mathscr{K}(\gamma_+;\boldsymbol\gamma))$ corresponds to the gluing of an element of some $\mathcal{Z}(\mathscr{K}(\gamma_k;\boldsymbol\beta))$ and an element of some $\mathcal Z(\mathscr{K}(\gamma_+;(\boldsymbol\gamma-\boldsymbol\beta)\sqcup \gamma_k))$. The key fact to remember is that the dimension of any $\mathcal{Z}(\mathscr{K}(\gamma',\boldsymbol\gamma''))$ is the same as the expected dimension given by $\operatorname{ind}-1$ in the (Symp) case. By construction, each end of $\mathcal{Z}(\mathscr{K}(\gamma_+;\boldsymbol\gamma))$ corresponds to an element of some product
$$\times_{v\in V(T)} \mathcal{Z}(\mathscr{K}(\gamma_{v,+};\boldsymbol\gamma_{v,-})),$$
where $T/\!\!/ T$ is a one-vertex CH tree whose vertex is labeled $i_0$ for the moduli space $\mathcal{M}_{i_0}$ corresponding to $\mathcal{Z}(\mathscr{K}(\gamma_+;\boldsymbol\gamma))$, and $\mathcal{Z}(\mathscr{K}(\gamma_{v,+};\boldsymbol\gamma_{v,-}))$ corresponds to the moduli space $\mathcal{M}_{l_V(v)}$.  Since the $\operatorname{ind}$ (resp.\ the dimension) of each $\mathcal{Z}(\mathscr{K}(\gamma_{v,+};\boldsymbol\gamma_{v,-}))$ is $\geq 1$ (resp.\ $\geq 0$), it follows that $|V(T)|=2$, i.e., the gluing consists of exactly two nontrivial components.

Next we work out the combinatorics to prove the claim. For any $k\in\{1,\dots,m\}$, $q\in \{0,1,\dots,c_m-b_m\}$, and
$$(\llbracket\mathcal{F}_1,u_1\rrbracket,\llbracket\mathcal{F}_2, u_2\rrbracket)\in \mathcal Z(\mathscr{K}(\gamma_k;\boldsymbol \beta))\times \mathcal Z(\mathscr{K}(\gamma_+; (\boldsymbol\gamma-\boldsymbol \beta) \sqcup \gamma_k)),$$
after fixing a gluing parameter we can glue $(\mathcal{F}_1,u_1)$ and $(\mathcal{F}_2, u_2)$ so that the positive puncture of $\mathcal{F}_1$ is glued to the $(q+1)$st negative puncture of $\mathcal{F}_2$ among those that converge to $\gamma_k$ and denote the glued curve by $(\mathcal{F}', u')$. Note that the negative punctures of $\mathcal{F}'$ have a natural ordering  $\tilde o$ coming from the gluing so that $\llbracket\mathcal{F}', u'\rrbracket\in  \mathcal Z(\mathscr{K}(\gamma_+;(\boldsymbol\gamma-\boldsymbol\beta) \sqcup_{\gamma_k,q} \boldsymbol\beta))$. Next let $o$ be an ordering of the negative punctures of $\mathcal{F}'$ such that:
\begin{enumerate}
\item the induced orderings of the negative punctures of $\mathcal{F}_1$ (minus the negative end that is glued) and $\mathcal{F}_2$ agree with the initial orderings of the negative punctures of $\mathcal{F}_1$ and $\mathcal{F}_2$; and
\item the surface $\mathcal{F}'$ with the ordering $o$, denoted by $\mathcal{F}'_o$, has $\vartheta$-sorted negative punctures;
\end{enumerate}
and let $O$ be the set of such orderings $o$. It is immediate from the definition that $|O|={c_1 \choose b_1}\dots{c_m \choose b_m}$.  We then define
$$G_{\boldsymbol\beta,k,q}(\llbracket\mathcal{F}_1,u_1\rrbracket,\llbracket\mathcal{F}_2, u_2\rrbracket, o):=\llbracket\mathcal{F}'_o, u=u'\rrbracket;$$
by choosing the gluing parameter and truncations with some care, one proves the claim.

We now claim that the weighted signed count of the boundary components of the $1$-dimensional branched manifold $T(\mathcal Z(\mathscr{K}(\gamma_+;\boldsymbol \gamma)))$ is given by the right-hand side of Equation \eqref{eqn: coefficient}. This follows by considering
\begin{align*}
\coprod_{\boldsymbol\beta,k,q} G_{\boldsymbol\beta,k,q}:  \coprod_{\boldsymbol\beta,k,q}( \mathcal Z(\mathscr{K}(\gamma_k;\boldsymbol\beta)) \times &  \mathcal Z(\mathscr{K}(\gamma_+;(\boldsymbol\gamma-\boldsymbol\beta)\sqcup \gamma_k)) \times O)\\
& \to  \partial T( \mathcal Z(\mathscr{K}(\gamma_+;\boldsymbol \gamma))),
\end{align*}
keeping track of weights, and combining the following observations:
\begin{enumerate}
\item By Property (O3), the components of $ \mathcal Z(\mathscr{K}(\gamma_+;\boldsymbol \gamma))$ that involve bad orbits do not contribute to the count.
\item By Properties (O1) and (O2), the sign difference between $(\mathcal{F}'_o,u')$ and $(\mathcal{F}',u')$ is $(-1)^{a+b}$ if $\boldsymbol\gamma\not=0$.  (Observe that if $o,o'\in O$ differ by switching two punctures converging to the same $\gamma_i$ and $|\gamma_i|_0$ is odd, then there is negative sign difference between $(\mathcal{F}'_o,u')$ and $(\mathcal{F}'_{o'},u')$ and the total sum on the right-hand side of Equation~\eqref{eqn: coefficient} becomes zero but on the other hand $\boldsymbol\gamma=0$ by supercommutativity.)
\item $\operatorname{Im}G_{\boldsymbol\beta,k,q}=\operatorname{Im}G_{\boldsymbol\beta,k,q'}$ for any $q,q'\in \{0,1,\dots,c_m-b_m\}$ and this cancels the term $c_k-b_k+1$ in the denominator of Equation~\eqref{eqn: coefficient 2}.
\item There are $m(\gamma_k)$ ways to simultaneously change the asymptotic markers of $\mathcal{F}_1$ and $\mathcal{F}_2$ such that $G_{\boldsymbol\beta,k,q}(\llbracket\mathcal{F}_1,u_1\rrbracket,\llbracket\mathcal{F}_2,u_2\rrbracket,o)$ does not change; this cancels the term $m(\gamma_k)$ in the  denominator of Equation~\eqref{eqn: coefficient 2}.
\end{enumerate} \vskip-.2in
\end{proof}

\subsection{Chain map} \label{chain map}

Let $(\widehat W,\widehat \alpha)$ be the completion of the exact symplectic cobordism $(W,\alpha)$ from $(M_+,\alpha_+)$ to $(M_-,\alpha_-)$ as in Section \ref{subsection: almost complex structures}. Given $L_+\leq L_-$, let $J$ be an almost complex structure on $\widehat W$ which restricts to $J_+$ at the positive end and to $J_-$ at the negative end.
Suppose $(\widehat \alpha, J)$ is an $(L_+,L_-)$-simple pair and $J$ is $(L_+,L_-)$-end-generic.

Using the semi-global Kuranishi structures $\mathscr{K}^{L_\pm}:=\mathscr{K}^{L_\pm}(\alpha_\pm,J_\pm)$ and complete sets $\mathcal{S}^{L_\pm}_{\alpha_\pm}$ of trivializing surfaces for $\mathcal{P}^{L_\pm}_{\alpha_\pm}$, we can construct the dga's
$$\mathfrak A^{L_\pm}:=\mathfrak A^{L_\pm}(M_\pm,\alpha_\pm,J_\pm,\mathscr{K}^{L_\pm}, \mathcal{S}^{L_\pm}_{\alpha_\pm})$$
as in Section \ref{chain complex}.  Here we are also using orderings $\vartheta_\pm$ of $\mathcal{P}^{L_\pm}_{\alpha_\pm}$.
We can also construct a semi-global Kuranishi structure
$$\mathscr{K}^{(L_+,L_-)}:=\mathscr{K}^{(L_+,L_-)}(\widehat \alpha,J)$$
that is compatible with  $\mathscr{K}^{L_\pm}$. We denote by the above collection of data by:
$$\mathcal C:=(M_\pm, L_\pm, \alpha_\pm, J_\pm, \mathscr{K}^{L_\pm}, \mathcal{S}^{L_\pm}_{\alpha_\pm}, \widehat W, \widehat \alpha, J, \mathscr{K}^{(L_+,L_-)}).$$

We now define the dga morphism
$$\Phi:\mathfrak A^{L_+} \otimes_{\mathbb{Q}[H_2(M_+;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})]\to \mathfrak A^{L_-}\otimes_{\mathbb{Q}[H_2(M_-;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})].$$
Here we are viewing $\mathfrak{A}^{L_\pm}$ as $\mathbb{Q}[H_2(W;\mathbb{Z})]$-modules via the algebra maps
$$\mathbb{Q}[H_2(M_\pm;\mathbb{Z})]\to \mathbb{Q}[H_2(W;\mathbb{Z})]$$
induced by $H_2(M_\pm;\mathbb{Z})\to H_2(W;\mathbb{Z})$.  For any $\gamma_+ \in \mathcal P_{\alpha_+}^{L_+}$, let
\begin{equation}\label{eqn: chain map}
\Phi(\gamma_+)=\sum_{\boldsymbol \gamma_-}\sum_{A\in H_2(W;\mathbb{Z})} \frac{p_{\gamma_+,\boldsymbol \gamma_-}}{m_{\boldsymbol\gamma_-}} e^A \gamma_{-,1}^{i_1}\dots \gamma_{-,k}^{i_k},
\end{equation}
where the sums are taken over all good $\vartheta_-$-sorted $\boldsymbol\gamma_-$ written in the form of Equation \eqref{eqn: gamma-} and $A\in H_2(W;\mathbb{Z})$ with $|\gamma_+|-(|e^A|+|\boldsymbol\gamma_-|)=0$, and
$p_{\gamma_+,\gamma_-}$ is the weighted signed count of elements in $\mathcal Z(\mathscr{K}^{(L_+,L_-)}(\widehat \alpha, J; \gamma_+;\boldsymbol \gamma_-))$ in the class $A$. (If $|\boldsymbol \gamma_-|\neq |e^A|+|\boldsymbol\gamma_-|$, then we set $p_{\gamma_+,\boldsymbol\gamma_-}=0$.) The homology class $A$ can be obtained from $(\mathcal{F},u)\in \mathcal{Z}(\mathscr{K}^{(L_+,L_-)}(\widehat \alpha, J; \gamma_+;\boldsymbol \gamma_-))$ by capping off using $\mathcal{S}^{L_\pm}_{\alpha_\pm}$ as in Section~\ref{subsubsection: grading}.  $\Phi$ can then be extended to $\mathfrak A^{L_+}$ as an algebra homomorphism.

\begin{prop}
$\Phi$ is a well-defined dga morphism.
\end{prop}

\begin{proof}
The proof is similar to that of Proposition \ref{dga}.
\end{proof}

Let  $\Phi_*: HC^{L_+}\to HC^{L_-}$ be the induced map on homology.

\subsection{Definition of contact homology}

We are now in a position to define the contact homology algebra $HC(\mathcal D)$.  Given a closed cooriented contact manifold $(M,\xi)$,  let
$$\mathcal D=(\alpha, \{  L_i, \varphi_i, J_i, \mathscr{K}^{L_i}(\alpha_i,J_i),\mathcal{S}^{L_i}_{\alpha_i},\widehat \alpha_{i,i+1},J_{i,i+1}, \mathscr{K}^{(L_i,L_{i+1})}(\widehat \alpha_{i,i+1},J_{i,i+1})\}_{i\in \mathbb N})$$
be a collection that consists of the following data:
\begin{itemize}
\item a nondegenerate contact form $\alpha$ for $\xi$;
\item an increasing sequence $\{L_i\}_{i\in \mathbb N}$ such that $L_i\to \infty$;
\end{itemize}
and, for each $i\in \mathbb N$,
\begin{itemize}
\item a positive function $\varphi_i$ on $M$ which is close to $1$, such that $\alpha_i:=\varphi_i \alpha$ is an $L_i$-simple contact form and $\varphi_i<\varphi_{i+1}$;
\item an almost complex structure $J_i$ on $\mathbb R\times M$ such that $(\alpha_i,J_i)$ is an $L_i$-simple pair;
\item a semi-global Kuranishi structure $\mathscr{K}^{L_i}(\alpha_i,J_i)$;
\item a complete set $\mathcal{S}^{L_i}_{\alpha_i}$ of trivializing surfaces for $\mathcal{P}^{L_i}_{\alpha_i}$;
\item a completed exact symplectic cobordism $(\widehat W_{i,i+1}=\mathbb{R}\times M,\widehat \alpha_{i,i+1})$ from $(M,\alpha_i)$ to $(M,\alpha_{i+1})$; here $\widehat \alpha_{i,i+1}:= \varphi_{i,i+1}\alpha$, where $\varphi_{i,i+1}$ interpolates from a positive multiple of $\varphi_i$ to $\varphi_{i+1}$;
\item an almost complex structure $J_{i,i+1}$ on $\widehat W_{i,i+1}$ that tames $d\widehat \alpha_{i,i+1}$, restricts to $J_i$ at the positive end and to $J_{i+1}$ at the negative end, and is $(L_i,L_{i+1})$-end-generic;
\item a semi-global Kuranishi structure $\mathscr{K}^{(L_i,L_{i+1})}(\widehat \alpha_{i,i+1},J_{i,i+1})$ that is compatible with $\mathscr{K}^{L_i}(\alpha_i,J_i)$ and $\mathscr{K}^{L_{i+1}}(\alpha_{i+1},J_{i+1})$.
\end{itemize}

Denote the map induced by the cobordism $(\widehat W_{i,i+1},\widehat \alpha_{i,i+1}, J_{i,i+1})$ and the semi-global Kuranishi structure $\mathscr{K}^{(L_i,L_{i+1})}(\widehat \alpha_{i,i+1}, J_{i,i+1}))$ by
$$\Phi_{i*}: HC^{L_i}(\alpha_i,J_i; \mathscr{K}^{L_i}(\alpha_i,J_i)) \to HC^{L_{i+1}}(\alpha_{i+1},J_{i+1};\mathscr{K}^{L_{i+1}}(\alpha_{i+1},J_{i+1})).$$
Finally, we define the contact homology algebra by
$$H(\mathcal D):=\lim_{\longrightarrow} HC^{L_i}(\alpha_i,J_i; \mathscr{K}^{L_i}(\alpha_i,J_i)),$$
where the directed system is constructed using $\Phi_{i*}$.

\subsection{Invariance of contact homology} \label{chain homotopy}

In this section we prove Propositions~\ref{prop: chain homotopy} and \ref{prop: composition} and use them to prove that $HC(\mathcal D)$ is an invariant of the contact manifold $(M,\xi)$.

Let $\{(\widehat W^\tau=\widehat W,\widehat \alpha^\tau)\}_{0\leq \tau \leq1}$  be a family of completed exact symplectic cobordisms from $(M_+,\alpha_+)$ to $(M_-,\alpha_-)$. For each $\tau \in[0,1]$, let $J^\tau$ be an almost complex structure on $\widehat W^\tau$ that restricts to $J_\pm$ at the positive/negative ends of $\widehat W^\tau$. We assume that $(\widehat \alpha^\tau, J^\tau)$ is an $(L_+,L_-)$-simple pair, $J^\tau$ is $(L_+,L_-)$-end-generic, and $(\widehat \alpha^\tau, J^\tau)$ is independent of $\tau$ on each of $\tau\in[0,\epsilon]$ and $\tau\in[1-\epsilon,1]$, where $\epsilon>0$ is small.

Let $\mathscr{K}^{L_\pm}:=\mathscr{K}^{L_\pm}(\alpha_\pm,J_\pm)$ be the semi-global Kuranishi structures constructed for $(M_\pm,\alpha_\pm, J_\pm)$ and let $\mathfrak A^{L_\pm}=\mathfrak A^{L_\pm}(M_\pm,\alpha_\pm,J_\pm,\mathscr{K}^{L_\pm})$.

For $\tau=0,1$, we have two collections of data
$$\mathcal C^\tau=(M_\pm, L_\pm, \alpha_\pm, J_\pm, \mathscr{K}^{L_\pm},  \mathcal{S}^{L_\pm}_{\alpha_\pm},\widehat W^\tau, \widehat \alpha^\tau, J^\tau, \mathscr{K}^{(L_+,L_-)})$$
as in Section \ref{chain map} and chain maps
$$\Phi^\tau:\mathfrak A^{L_+} \otimes_{\mathbb{Q}[H_2(M_+;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})]\to \mathfrak A^{L_-}\otimes_{\mathbb{Q}[H_2(M_-;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})],$$
defined using the data $\mathcal C^\tau$.

\begin{prop}[Chain homotopy, Version 1] \label{prop: chain homotopy}
There is a degree $1$ $\mathbb{Q}[H_2(W;\mathbb{Z})]$-module map
$$\Omega_K: \mathfrak A^{L_+} \otimes_{\mathbb{Q}[H_2(M_+;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})]\to \mathfrak A^{L_-}\otimes_{\mathbb{Q}[H_2(M_-;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})]$$
such that $\Phi^0-\Phi^1=\Omega_K\circ\partial+\partial \circ\Omega_K.$ In particular, $\Phi^0_*=\Phi^1_*$.
\end{prop}

\begin{proof}
We construct a semi-global Kuranishi structure
$$\mathscr{K}^{(L_+,L_-)}(\{(\widehat \alpha^\tau,J^\tau)\}_{\tau\in[0,1]})$$
as before so that it is independent of $\tau$ on each of $\tau\in[0,\epsilon]$ and $\tau\in[1-\epsilon,1]$ and agrees with $\mathscr{K}^{(L_+,L_-)}(\widehat \alpha^\tau,J^\tau)$ for $\tau=0,1$. We are using orderings $\vartheta_\pm$ for $\mathcal{P}^{L_\pm}_{\alpha_\pm}$.

We then define $\Omega_K$ by $\Omega_K(1)=0$ and for any $k$-tuple $\boldsymbol\gamma_+$ of  good Reeb orbits, we have
\begin{equation}
\Omega_K(\boldsymbol\gamma_+)=\sum_{\boldsymbol \gamma_-}\sum_{A\in H_2(W;\mathbb{Z})} \frac{\kappa_{\boldsymbol\gamma_+,\boldsymbol \gamma_-}}{m_{\boldsymbol\gamma_{-,1}}\cdots m_{\boldsymbol\gamma_{-,k}}} e^A \boldsymbol\gamma_{-,1}\dots \boldsymbol\gamma_{-,k},
\end{equation}
where the sums are taken over all $\boldsymbol\gamma_-$ which is a $k$-tuple of $\vartheta_-$-sorted tuples of good Reeb orbits, i.e.,
$$\boldsymbol\gamma_-=(\boldsymbol\gamma_{-,1},...,\boldsymbol\gamma_{-,k})$$ and
\begin{equation}
\boldsymbol\gamma_{-,j}=(\underbrace{\gamma_{-,j,1},\dots, \gamma_{-,j,1}}_{i_{j,1} \text{ copies }},\dots, \underbrace{\gamma_{-,j,r_j},\dots, \gamma_{-,j,r_j}}_{i_{j,r_j} \text{ copies }}),
\end{equation}
 $A\in H_2(W;\mathbb{Z})$ with $|\boldsymbol\gamma_+|-(|e^A|+|\boldsymbol\gamma_-|)=-1$, and $\kappa_{\boldsymbol\gamma_+,\boldsymbol\gamma_-}$ is the weighted signed count of the elements in
$$\mathcal{Z}^{[0,1]}(\mathscr{K}(\boldsymbol\gamma_+,\boldsymbol\gamma_-)):=\mathcal Z( \mathscr{K}^{(L_+,L_-)}(\{(\widehat \alpha^\tau, J^\tau)\}_{\tau\in[0,1]};\boldsymbol\gamma_+;\boldsymbol\gamma_-))$$
in the class $A$.

Let $\Phi^0,\Phi^1$ be two dga morphisms
$$\mathfrak A^{L_+} \otimes_{\mathbb{Q}[H_2(M_+;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})]\to \mathfrak A^{L_-}\otimes_{\mathbb{Q}[H_2(M_-;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})],$$
induced by the cobordism $(\widehat W^\tau, \widehat \alpha^\tau, J^\tau)$, $\tau=0,1$, using the semi-global Kuranishi structure $ \mathscr{K}^{(L_+,L_-)}( \widehat \alpha^\tau,J^\tau)$. We can verify that $\Omega_K$ is well-defined and satisfies $\Phi^0-\Phi^1=\Omega_K\circ\partial+\partial\circ \Omega_K$ as in the proof of Proposition \ref{dga}.   The key point is to observe that, with the perturbation from Section~\ref{subsubsection: basic}, each end of a $1$-dimensional (and hence $\operatorname{ind}=0$) branched manifold $\mathcal{Z}^{[0,1]}(\mathscr{K}(\gamma_+,\boldsymbol\gamma_-))$ corresponds to (i) a term contributing to $\Phi^0$ or $\Phi^1$, (ii) the gluing of an element of some $\mathcal{Z}(\mathscr{K}(\gamma_+,\boldsymbol\gamma_+'))$ of index $1$ and an element of some $\mathcal{Z}^{[0,1]}(\mathscr{K}(\boldsymbol\gamma_+',\boldsymbol\gamma_-))$ of index $-1$, or (iii) the gluing of an element of some $\mathcal{Z}^{[0,1]}(\mathscr{K}(\gamma_+,\boldsymbol\gamma_-'))$ of index $-1$ and an element of some $\mathcal{Z}(\mathscr{K}(\boldsymbol\gamma_-',\boldsymbol\gamma_-))$ where all but one of the components is a trivial cylinder.  (In particular, if $\operatorname{ind}< -1$, then $\mathcal{Z}^{[0,1]}(\mathscr{K}(\boldsymbol\gamma_+',\boldsymbol\gamma_-))=\varnothing$ and $\mathcal{Z}^{[0,1]}(\mathscr{K}(\gamma_+,\boldsymbol\gamma_-'))=\varnothing$.)
\end{proof}

The following is a slightly improved version which was explained to us by Tobias Ekholm.

\begin{prop}[Chain homotopy, Version 2] \label{prop: chain homotopy ver 2}
There is a degree $1$ map
$$K: \mathcal{P}_{\alpha_+}^{L_+}\to \mathfrak A^{L_-}\otimes_{\mathbb{Q}[H_2(M_-;\mathbb{Z})]}\mathbb{Q}[H_2(W;\mathbb{Z})]$$
such that
\begin{equation} \label{equation: exp}
\Phi^1=\Phi^0\exp(K\partial+\partial K)=\Phi^0\left( 1+ (K\partial + \partial K) + {(K\partial+\partial K)^2\over 2!} +\dots\right).
\end{equation}
\end{prop}

We need to explain what the right-hand side of Equation~\eqref{equation: exp} means.
Given a monomial $\boldsymbol\gamma=\gamma_1\dots \gamma_k$ where $\gamma_i\in\mathcal{P}_{\alpha_+}^{L_+}\cup \mathcal{P}_{\alpha_-}^{L_-}$,  we define
$$K(\boldsymbol\gamma)=\sum_{i=1}^k(-1)^{|\gamma_1|+\dots + |\gamma_{i-1}|} \gamma_1\dots \gamma_{i-1}K(\gamma_i)\gamma_{i+1}\dots\gamma_k,$$
where $K(\gamma_i)=0$ if $\gamma_i\in \mathcal{P}_{\alpha_-}^{L_-}$, and define
$$\Phi^0(\boldsymbol\gamma)= \Phi^0(\gamma_1)\dots \Phi^0(\gamma_k),$$
where $\Phi^0(\gamma_i)=\gamma_i$ if $\gamma_i\in \mathcal{P}_{\alpha_-}^{L_-}$.

\begin{proof}
Follows from the refined perturbation scheme from Section~\ref{subsubsection: refined} and Section 5.4 (and in particular the proof of Lemma 5.6) of Ekholm-Oancea~\cite{EO}.
\end{proof}

For $(i,j)=(1,2), (i,j)=(2,3)$, or $(i,j)=(1,3)$, let $(\widehat W_{ij}, \widehat \alpha_{ij})$  be the completion of an exact symplectic cobordism $(W_{ij}, \alpha_{ij})$ from $(M_i,\alpha_i)$ to $(M_j,\alpha_j)$ such that $(W_{12},\alpha_{12},J_{12})$ and $(W_{23},\alpha_{23},J_{23})$ can be glued to give $(W_{13},\alpha_{13},J_{13})$ and let
$$\mathcal C_{ij}=(M_i,M_j, L_i,L_j, \alpha_i,\alpha_j, J_i,J_j, \mathscr{K}^{L_i},\mathscr{K}^{L_j},  \mathcal{S}^{L_i}_{\alpha_i}, \mathcal{S}^{L_j}_{\alpha_j}, \widehat W_{ij}, \widehat \alpha_{ij}, J_{ij}, \mathscr{K}^{(L_+, L_-)}_{ij} )$$
be a collection of data as in Section \ref{chain map} for $(\widehat W_{ij}, \widehat \alpha_{ij})$.  Using the data $\mathcal C_{ij}$ we obtain a chain map
$$\Phi_{ij}: \mathfrak A^{L_i}\otimes_{\mathbb{Q}[H_2(M_i;\mathbb{Z})]}\mathbb{Q}[H_2(W_{ij};\mathbb{Z})]\to \mathfrak A^{L_j}\otimes_{\mathbb{Q}[H_2(M_j;\mathbb{Z})]}\mathbb{Q}[H_2(W_{ij};\mathbb{Z})].$$

\begin{prop}[Composition of chain maps] \label{prop: composition}
There exists a degree 1 map
$$\Omega_K:\mathfrak A^{L_1}\otimes_{\mathbb{Q}[H_2(M_1;\mathbb{Z})]}\mathbb{Q}[H_2(W_{13};\mathbb{Z})]\to \mathfrak A^{L_3} \otimes_{\mathbb{Q}[H_2(M_3;\mathbb{Z})]}\mathbb{Q}[H_2(W_{13};\mathbb{Z})]$$
such that, after the appropriate base changes (but keeping the same notation),
$$\Phi_{23}\Phi_{12}-\Phi_{13}=\partial\circ \Omega_K+\Omega_K\circ\partial.$$
In particular, $\Phi_{23*}\Phi_{12*}=\Phi_{13*}.$
\end{prop}

\begin{proof}
The proof is almost the same as that of Proposition \ref{dga}.
\end{proof}

Finally we are ready to prove the main theorem.
\begin{thm}
The contact homology algebra $HC(\mathcal D)$ is independent of the choices $\mathcal D$ and hence  is an invariant of $(M,\xi)$.
\end{thm}

\begin{proof}
Given two choices of data $\mathcal D$ and $\mathcal D'$, there exist two direct limits
\begin{align*}
HC(\mathcal D)&=\lim_{\longrightarrow} HC^{L_i}(\alpha_i,J_i, \mathscr{K}^{L_i}(\alpha_i,J_i)),\\
HC(\mathcal D')&=\lim_{\longrightarrow} HC^{L'_i}(\alpha'_i,J'_i, \mathscr{K}^{L'_i}(\alpha'_i,J'_i)),
\end{align*}
where the maps for $\mathcal{D}$ are written as $\Phi_{i*}$ and the maps for $\mathcal{D}'$ are written as $\Phi_{i*}'$.
We abbreviate:
\begin{align*}
HC^{L_i}&:=HC^{L_i}(\alpha_i,J_i, \mathscr{K}^{L_i}(\alpha_i,J_i)),\\
HC^{L'_i}&:=HC^{L'_i}(\alpha'_i,J'_i, \mathscr{K}^{L'_i}(\alpha'_i,J'_i)).
\end{align*}
For each $i$, let $j(i)$ be the smallest $j$ such that $L_i<L'_j$.
An exact symplectic cobordism from $(M,\alpha_i,J_i)$ to $(M,\alpha'_{j(i)},J'_{j(i)})$ with Kuranishi data $\mathscr{K}^{L_i}(\alpha_i, J_i)$ and $\mathscr{K}^{L_{j(i)}}(\alpha_{j(i)},J_{j(i)})$ induces a map $\Psi_{i*}: HC^{L_i} \to HC^{L'_{j(i)}}$ which satisfies
$$\Psi_{i+1*}\Phi_{i*}=\Phi_{j(i+1)-1*}'\cdots \Phi_{j(i)+1*}' \Phi_{j(i)*}' \Psi_{i*}.$$
Similarly, we obtain maps $\Theta_{j*}:HC^{L'_j}\to HC^{L_{i(j)}}$ for each $j\in \mathbb N$ that satisfy $$\Theta_{j+1*}\Phi'_{j*}=\Phi_{i(j+1)-1*}\cdots\Phi_{i(j)+1*}\Phi_{i(j)*}\Theta_{j*}.$$
Hence we obtain two direct limit maps
$$\Psi_*:  HC(\mathcal D)\to HC(\mathcal D'), \quad \Theta_*: HC(\mathcal D') \to HC(\mathcal D).$$
Note that, for each $k\in \mathbb N$, the two maps
$$\Theta_{j(k)*}\Psi_{k*} \text{ and } \Phi_{i(j(k))-1*}\cdots \Phi_{k+1*} \Phi_{k*}: HC^{L_k}\to HC^{L_{i(j(k))}}$$
are the same. Therefore $\Theta_* \Psi_*=\operatorname{id}$; similarly $\Psi_* \Theta_*=\operatorname{id}$.
\end{proof}

\section*{Appendix}

In the Appendix we prove Lemma \ref{shenyang} and Lemma \ref{lemma: in Wkp and C1}.
We first start with some basic facts about Sobolev spaces.

\begin{lemma} \label{fg}
Suppose that $p=2$ and $k\geq3$.  If $g,h\in W^{k,p}_{\operatorname{loc}}(\mathbb{R}^2)$, then $g\cdot h \in W^{k,p}_{\operatorname{loc}}(\mathbb{R}^2)$.
\end{lemma}

\begin{proof}
We prove that $ (g\cdot h)^{(m)}\in L^p_{\operatorname{loc}}(\mathbb{R}^2)$ for any $0\leq m \leq k$. Let $\Omega\subset \mathbb{R}^2$ be a domain with smooth boundary and compact closure. For $m=0$, we have $g,h\in C^0(\Omega)$\footnote{More precisely, $g|_\Omega,h|_\Omega$ are in $C^0(\Omega)$; by abuse of notation we often write $g,h$ for $g|_\Omega,h|_\Omega$.} by the Sobolev embedding theorem, which states that $\|f\|_{C^r(\Omega)}\leq C_\Omega\|f\|_{W^{k,p}(\Omega)}$ for some constant $C_\Omega$ if $k-{n\over p}=k-{2\over p}>r$.  Hence $g\cdot h\in C^0(\Omega)$ and $g\cdot h \in L^p(\Omega)$. For $m>0$, we have:
$$\|(g\cdot h)^{(m)}\|_{L^p(\Omega)}\leq \sum_{i=0}^m \binom{m}{i}\|g^{(i)}\cdot h^{(m-i)}\|_{L^p(\Omega)}.$$
Observe that one of $i$ or $m-i$ is $\leq {m\over 2}\leq {k\over 2}$.  This implies that $g^{(i)}\in C^0(\Omega)$ or $h^{(m-i)}\in  C^0(\Omega)$ since $g,h\in C^{k-1}(\Omega)$ by the Sobolev embedding theorem. Hence $\|(g\cdot h)^{(m)}\|_{L^p(\Omega)}<\infty.$
\end{proof}

\begin{lemma} \label{g(h)}
Suppose that $p=2$ and $k\geq 3$. If $G\in C^{\infty}(\mathbb R^2, \mathbb R)$ and $H\in W^{k,p}_{\operatorname{loc}}(\mathbb R^2, \mathbb R^2)$,
then $G\circ H\in W^{k,p}_{\operatorname{loc}}(\mathbb R^2, \mathbb R)$.
\end{lemma}

\begin{proof}
By the Sobolev embedding theorem, $H\in C^{k-1}(\Omega,\mathbb{R}^2)$, where $\Omega$ is as before, so $G(H)=G\circ H\in C^{k-1}(\Omega)\subset W^{k-1,p}(\Omega)$. This in particular implies that $G(H)\in L^p(\Omega)$. By the same argument, we also have $G'(H)\in W^{k-1,p}(\Omega)$.  Hence $(G(H))'=G'(H)\cdot H' \in W^{k-1,p}(\Omega)$ by Lemma \ref{fg}.  The lemma then follows.
\end{proof}

We assume that $p=2$ and $k\geq 3$. We focus on a positive puncture $p$ of $\dot F$ around which the maps that we are interested in converge to $\gamma_{+}$. Let $\dot D\subset \dot F$ be an open disk about $p$ and let $(\sigma,\tau)$ be smooth coordinates on $\dot D$. For any $u$ of the class $W^{k+1,p}$ that is sufficiently close to a $J$-holomorphic curve, there is a map $$\phi_u:\dot D\to \mathbb R\times S^1$$ defined by
$$(\sigma,\tau)\mapsto (s,t)=(s\circ u(\sigma, \tau), t\circ u(\sigma, \tau)),$$
where $(\sigma,\tau)$ is a cylindrical holomorphic coordinate of $\dot D$,
$s\circ u$ is the $s$-coordinate of $u$, and $t\circ u$ is the $t$-coordinate of $u$. Then $(s,t)$, viewed as a function of $(\sigma, \tau)$, is of class $W^{k+1,p}_{\operatorname{loc}}.$

\begin{proof}[Proof of Lemma \ref{shenyang}]
It suffices to show that $I(\cdot, s_0)$ is $C^1$ for each $s_0>T'$. Given $(\mathcal F,u)\in \mathcal B'_{(\mathcal F_0, u_0)}$ and $\xi \in W^{k+1,p}_{\delta}(\dot F, u^{*}T(\mathbb R \times M))$, we choose a smooth path $\{u^{\ell}\}_{-\epsilon< \ell<\epsilon}$ in $\mathcal B'_{(\mathcal F_0, u_0)}$ with $u^{0}=u$ and $\frac{d}{d\ell}\big|_{\ell=0}u^\ell =\xi$. Then
\begin{align*}
&dI((\mathcal F, u),s_0)(\xi, 0)\\
=&\tfrac{d}{d\ell}\big|_{\ell=0}I((\mathcal F, u_\ell),s_0)\\
=&2\int_{s_0}^\infty \int_0^{\mathcal A_\alpha(\gamma_+)}\big(\langle \xi(z),\eta(z)\rangle -\langle d\eta(z) d\phi_u^{-1}(z)\xi^{s,t}(z), \eta(z)\rangle \big) dtds,
\end{align*}
where $z=(\sigma,\tau)=\phi_u^{-1}(s,t)$ and $\xi^{s,t}(z)$ is the $(s,t)$-component of $\xi$ at $z$.
Using the H\"older inequality and factoring out the $C^1$-norm of $u$, we get
\begin{align*}
|dI((\mathcal F, u),s_0)(\xi,0)|\leq C(\|u\|_{W^{k+1,2}})\cdot \|\xi\|_{L^2},
\end{align*}
where $k\geq 3$ and $C(\|u\|_{W^{k+1,2}})$ is a constant that depends on $\|u\|_{W^{k+1,2}}$ by the Sobolev inequality.
Hence $I$ is differentiable. Similarly, we can show that $I$ is $C^1$.
\end{proof}

In order to prove Lemma \ref{lemma: in Wkp and C1}, we first treat the case $\chi(\dot F)<0$.
To simplify the notation in Equation~\eqref{f tilde}, we write $\widetilde{f}(s,t)$ for $\widetilde{f}^{\gamma_{\pm,i}}_j(s,t)$, $\rho_{s_{+}}(s)$ for $\frac{\partial \beta^\pm_{s_{\pm,i}}}{\partial s}(s)$ (recall $\rho_{s_+}(s)$ has compact support), and $f(t)$ for $f^{\gamma_{\pm,i}}_j(t)$.
With respect to the coordinates $s,t,x_1,\dots,x_n,y_1,\dots,y_n$,
Equation~\eqref{f tilde} can be written as
\begin{align} \label{section}
\widetilde{f}(s,t):=\rho_{s_{+}}(s)f(t)\otimes \pi_j (ds-idt).
\end{align}

First observe that $\rho_{s_{+}}(s)$ and $f(t)$
are smooth functions of $(s,t)$, so viewed as functions of $(\sigma, \tau)$, they are in class $W^{k+1,p}_{\operatorname{loc}}$ by Lemma \ref{g(h)}. Next
the projection $\pi_j$ is smooth, so $\pi_j(ds-idt)\in W_{\operatorname{loc}}^{k+1,p}(\dot F, \wedge^{0,1}T^{*}\dot F)$. Hence $\widetilde f\in W^{k+1,p}_{\operatorname{loc}}(\dot F, \wedge^{0,1}u^{*}T\widehat W)$ by Lemma \ref{fg}, and this proves the first part of Lemma \ref{lemma: in Wkp and C1}, namely $ E^{\ell,\varepsilon}_{(\mathcal{F},u)}\subset \mathcal{E}_{(\mathcal{F},u)}$ for all $(\mathcal{F},u)\in \mathcal{N}(K)$.

For $(j, v)\in \mathcal N(K)$ with $v:\dot F\to \widehat W$ smooth, there exists a small neighborhood $\mathcal D=\mathcal V \times \mathcal W$ of $(j, v)$ inside $\mathcal N(K)$ with $\mathcal V\subset \widetilde{\mathcal U}$ and $\mathcal W\subset \{ \exp (v,\xi)~|~ \xi \in W_\delta^{k+1,p}(\dot F, v^{*}T\widehat W)\}$.
We can trivialize $\mathcal E|_{\mathcal D}$ by identifying
$$\mathcal E_{(j',u)}=W^{k,p}_\delta (\dot F, \wedge^{0,1}_{j'} u^{*}T\widehat W)$$
with
$$\mathcal E_{(j,v)}=W^{k,p}_\delta (\dot F, \wedge ^{0,1}_j v^{*}T\widehat W)$$
for any $(j',u)\in \mathcal D$ via the exponential map. More precisely, we define
$$\Psi: W^{k,p}_\delta (\dot F, \wedge^{0,1}_{j'} u^{*}T\widehat W) \to W^{k,p}_\delta (\dot F, \wedge ^{0,1}_j v^{*}T\widehat W)$$
$$\zeta=\eta \otimes \alpha  \mapsto \operatorname{Par}_{\xi} \eta \otimes \pi_j \alpha,$$
where $\xi=(\exp_v)^{-1} u$, $\eta\in W^{k,p}_\delta(\dot F, u^{*}T\widehat W)$, $\alpha\in W^{k,p}_\delta(\dot F, \wedge^{0,1}T^{*}F),$
$$\operatorname{Par}_\xi: W^{k,p}_\delta(\dot F, u^{*}T\widehat W)\to W^{k,p}_\delta(\dot F, v^{*}T\widehat W)$$ is the parallel transport along the path $\{ \exp(v,(1-t)\xi)\}_{0\leq t \leq 1}$, and $$\pi_j: W^{k,p}_\delta (\dot F, \wedge^{1}T^{*}F)\to W^{k,p}_\delta (\dot F, \wedge^{0,1}_jT^{*}F)$$
is the projection with respect to $g_{(\dot F,j)}$.

Let $\widetilde f$ be the section of $\mathcal E|_{\mathcal{N}(K)} \to \mathcal N(K)$ with
$$\widetilde f (j',u)(s,t)=\rho_{s_{+}}(s)f(t)\otimes \pi_{j'} (ds-idt),$$
where $(s,t)$ are viewed as coordinates of $\dot F$ via $\phi_u$ as usual.   We show that $\widetilde f$ is a $C^1$-section of $\mathcal E|_{\mathcal N(K)}\to \mathcal N(K)$. In the next several paragraphs we calculate the derivative $d\widetilde f(j',u)$.

We first calculate $d\widetilde f (j',u)(0,\xi)$, where $\{u_{l_1}\}_{-\epsilon\leq l_1 \leq \epsilon}$ is a smooth path in $\mathcal W$ with $u_0=u$ and $\xi= \frac{d}{dl_1}|_{l_1=0}u_{l_1}$:
\begin{align} \label{eqn: part 1}
d\widetilde f (j',u)(0,\xi)=&\tfrac{d}{dl}|_{l_1=0} \widetilde f (j',u_{l_1})\\
\label{eqn: part 1A}=& \tfrac{d}{dl_1} \rho_{s_{+}(l_1)}(s_{l_1})|_{l_1=0} f(t) \otimes \pi_{j'}(ds-idt)\\
\label{eqn: part 1B} &+\rho_{s_+(0)}(s) \tfrac{d}{dl_1} f(t_{l_1})|_{l_1=0}\otimes \pi_{j'}(ds-idt)\\
\label{eqn: part 1C} &+\rho_{s_+(0)}(s)f(t)\otimes \pi_{j'}\tfrac{d}{dl_1}(ds_{l_1}-idt_{l_1})|_{l_1=0},
\end{align}
where $(s_{l_1}, t_{l_1})=\phi_{u_{l_1}}(\sigma,\tau)$.

Until the end of the Appendix, $W^{k,p}_\delta$ means $W^{k,p}_\delta$ with domain $\dot D$.  We claim that $d\widetilde f(j',u)(0,\xi)\in W^{k,p}_\delta$. Note that the weight $\delta$ can be ignored since $d\widetilde f(j',u)(0,\xi)$ has compact support. Since $\xi \in W^{k+1,p}_\delta$,
\begin{itemize}
\item $\tfrac{d}{dl_1}s_{l_1}|_{l_1=0},\tfrac{d}{dl_1}t_{l_1}|_{l_1=0},$ $\tfrac{d}{dl_1} f(t_{l_1})|_{l_1=0}\in W^{k+1,p}_\delta$; and
\item $\tfrac{d}{dl_1}(ds_{l_1}-idt_{l_1})|_{l_1=0}\in W^{k,p}_\delta$.
\end{itemize}
Next we consider the term $\tfrac{d}{dl_1} \rho_{s_{+}(l_1)}(s_{l_1})|_{l_1=0}$.  If we write $\rho_{s_+}(s)$ as $\rho(s_+,s)$, then $\rho(\cdot, \cdot)$ is smooth in both variables. Observe that
$$\tfrac{d}{dl_1} \rho(s_{+}(l_1),s_{l_1})|_{l_1=0}=\tfrac{\partial \rho}{\partial s_+}(s_+(0),s)\tfrac{d}{dl_1} s_+(l_1)|_{l_1=0}+\tfrac{\partial \rho}{\partial s}(s_+(0),s)\tfrac{d}{dl_1} s_{l_1} |_{l_1=0}.$$

\begin{lemma} \label{s_+C1}
$s_+:\mathcal N(K)\to \mathbb R$ is a $C^1$-map
\end{lemma}

The proof of this lemma will be postponed until the very end. By Lemma \ref{s_+C1}, $\tfrac{d}{dl_1} \rho(s_{+}(l_1),s_{l_1})|_{l_1=0}\in W^{k+1,p}_\delta$.  It follows that $d\widetilde f (j',u)(0,\xi)\in W^{k,p}_\delta$.

Next we calculate $d\widetilde f (j',u)(\frak j,0)$ with $\mathfrak j\in T_{j'}\widetilde{ \mathcal U}$. Let $\{j_{l_2}\}_{-\epsilon\leq l_2\leq \epsilon}$ be a smooth path in $\widetilde{\mathcal U}$ with $j_0=j'$ and $\frac{d}{dl}|_{l_2=0}j_{l_2}=\mathfrak j$. Then
\begin{align} \label{eqn: part 2}
d\widetilde f (j',u)(\frak j,0)=& \tfrac{d}{dl}|_{l_2=0} \widetilde f (j_{l_2},u)\\
\label{eqn: part 2A} =&\tfrac{d}{dl_2} \rho_{s_{+}(l_2)}(s)|_{l_2=0} f(t) \otimes \pi_{j'}(ds-idt)\\
\label{eqn: part 2B} &+\rho_{s_{+}(0)}(s) f(t) \otimes \tfrac{d}{dl_2} \pi_{j_{l_2}}(ds-idt)|_{l_2=0}.
\end{align}
Since $\pi_j$ depends smoothly on $j$, we have $\tfrac{d}{dl_2}\pi_{j_{l_2}}(ds-idt)|_{l_2=0}\in W^{k,p}_{\operatorname{loc}}$. Again we write $\rho_{s_{+}}(s)$ as $\rho(s_+,s)$ and obtain
$$\tfrac{d}{dl_2} \rho(s_{+}(l_2),s)|_{l_2=0}=\tfrac{\partial \rho}{\partial s_+}(s_+(0), s)\tfrac{d}{dl_2} s_+(l_2)|_{l_2=0}.$$
By Lemma \ref{s_+C1}, $\tfrac{d}{dl_2} \rho(s_{+}(l_2),s)|_{l_2=0}\in W^{k+1,p}_\delta$. Hence $d\widetilde f (j',u)(\frak j,0)\in W^{k,p}_\delta$.

We now bound the $W^{k,p}_\delta$-norm of each term in Equations~\eqref{eqn: part 1} and \eqref{eqn: part 2}:
\begin{itemize}
\item \eqref{eqn: part 1A} is bounded above by $C\|\xi\|_{W^{k+1,p}}\cdot C(j')\cdot \|gu\|_{W^{k+1,p}}$;
\item \eqref{eqn: part 1B} by $C\|\xi\|_{W^{k+1,p}}\cdot C(j')\cdot \|gu\|_{W^{k+1,p}}$;
\item \eqref{eqn: part 1C} by $C \cdot C(j') \cdot \|\xi\|_{W^{k+1,p}}$;
\item \eqref{eqn: part 2A} by $C\|\frak j\|\cdot C(j')\cdot \|gu\|_{W^{k+1,p}}$; and
\item \eqref{eqn: part 2B} by $C\|\frak j\|\cdot \|gu\|_{W^{k+1,p}}$.
\end{itemize}
Here $C(j')$ is a positive continuous function of $j'$ and $g$ is a smooth cutoff function which equals one on $s_+(v)-c\leq s\leq s_+(v)+c$ for some $c>0$ and has compact support on $\dot D$.
Hence
$$\|d \widetilde f (j',u)(\frak j,\xi)\|_{W^{k,p}_\delta} \leq C(C(j') + C(j') \cdot \|gu\|_{W^{k+1,p}}+\|gu\|_{W^{k+1,p}}) (\| \frak j\| + \|\xi\|_{W^{k+1,p}_\delta}),$$
which implies that $\widetilde f$ is a differentiable section of $\mathcal E|_{\mathcal{N}(K)}\to \mathcal N(K)$. Almost the same calculation shows that $\widetilde f$ is in $C^1$.

\begin{proof}[Proof of Lemma~\ref{s_+C1}]
Let $r_{j'}:\dot F\to \mathbb R$ be the injectivity radius map, i.e, $r_{j'}$ maps $x\in \dot F$ to the injectivity radius $\operatorname{inj}(x)$ with respect to $g_{(\dot F,j)}$. The map $r_{j}$ is smooth on the ends of $\dot F$ and also smoothly depends on $j'$.  For any $(j',u)\in \mathcal D$, consider the map
\begin{equation}
R_{(j',u)}: [R,\infty) \to \mathbb R^+, \quad s\mapsto r_{j'}(\phi^{-1}_u(s,0)),
\end{equation}
where $R\gg 0$ and depends on $(j',u)$.
Note that $R_{(j',u)}$ is invertible near the positive end.

When $\chi(\dot F)<0$, we have
\begin{align} \label{s_+ explicit}
s_+(j',u)=R_{(j',u)}^{-1}(\varepsilon).
\end{align}
Differentiating the equation
\begin{equation}
r_{j'}(\phi^{-1}_{u_{l_1}}(s_+(j',u_{l_1}),0))=\varepsilon
\end{equation}
with respect to $l_1$ and rearranging, we obtain:
\begin{align}
ds_+(j',u)(0,\xi)=\tfrac{d}{dl_1}s_+(j',u_{l_1})|_{l_1=0}= -\frac{d r_{j'}(z)\circ (d\phi_u (z))^{-1} (\xi^{s,t}(z)) }{dr_{j'}(z)\circ (d\phi_u(z))^{-1} (\partial_s) },
\end{align}
where $\partial_s$ is a vector field on $\mathbb R^+ \times S^1$, $\xi^{s,t}(z)$ is the $(s,t)$-component of $\xi$ at $z$,  and $z=\phi_u^{-1}(s_+(j',u),0).$  Here ${d\over dl_1} \phi^{-1}_{u_{l_1}}(s_+(j',u),0))|_{l_1=0}=(d\phi_u (z))^{-1} (\xi^{s,t}(z))$.  Note that all terms are evaluated at the point $z$ and $z$ continuously depends on $(j',u)$, so $d r_{j'}(z)\circ (d\phi_u (z))^{-1}$ and $dr_{j'}(z)\circ (d\phi_u(z))^{-1} (\partial_s)$ depend continuously on $(j',u)$.  Hence $ds_+(j',u)(0,\cdot)$ is a bounded linear map from $W_\delta^{k+1,p}(\dot F, v^{*}T\widehat W)$ to $\mathbb R$ which depends continuously on $(j',u)$.

Next, differentiating the equation
\begin{equation}
r_{j_{l_2}}(\phi^{-1}_{u}(s_+(j_{l_2},u),0))=\varepsilon
\end{equation}
with respect to $l_2$, we obtain
\begin{align}
ds_+(j',u)(\frak j,0)=\tfrac{d}{dl_2}s_+(j_{l_2},u)|_{l_2=0}= -\frac{{d\over dl_2} r_{j_{l_2}}(z)|_{l_2=0}}{dr_{j'}(z)\circ(d\phi_u(z))^{-1}(\partial_s)}.
\end{align}
Since ${d\over dl_2} r_{j_{l_2}}(z)|_{l_2=0}$ depends continuously on $(j',u)$,  $ds_+(j',u)(\cdot,0)$ is a bounded linear map from $T_{j'}\mathcal V$ to $\mathbb R$ which depends continuously on $(j',u)$. This proves that $s_+$ is a $C^1$-map when $\chi(\dot F)<0$.

When $\chi(\dot F)=0$ or $1$, $s_+$ is still given by Equation~\eqref{s_+ explicit}, but the metric $g_{(\ddot F,j')}$ is now a function of $u$. More precisely, for any $u=(s,t,\eta)$, define
$$S_u:[R,\infty)\to \mathbb R^+,\quad s\mapsto \int_s^{+\infty}\hspace{-0.3cm}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'$$
with $R\gg 0$ and then
$$s_+'(u)=S_u^{-1}(\varepsilon').$$
Recall that the set $\mathbf{q}=(q_1,\dots,q_{2m(\gamma_{+})})$ of additional punctures that we put on $\dot F$ satisfies
$$q_k=\phi_u^{-1}\big(s_+', \tfrac{k}{2m(\gamma_{+})}\mathcal A_\alpha(\gamma_{+})\big)\in \dot F.$$
Then $g_{(\ddot F,j')}$ is smooth function of $\mathbf q$.  An argument similar to that of the $\chi(\dot F)<0$ case shows that $s_+$ is a $C^1$-map.

\end{proof}