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 $\psi_\pm\in \mathcal{H}_\pm^{\perp,\psi_\pm,T}$.

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 $\psi_\pm\in \mathcal{H}_\pm^{\perp,\psi_\pm,T}$.

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> and <ref>. Then $(\psi_-,\psi_+=P_+(\psi_-))$ is the unique solution to $\Theta_\pm(\psi_-,\psi_+)=0$.

In more practical terms, the fixed point can be obtained by starting with

applying the iteration

and taking the limit $i\to\infty$. By Claim <ref> 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 $T\to\infty$.

The above remark implies the following lemma:

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 $T\to \infty$.

We can finally define the gluing map  as

$$G:V_-/\mathbb R \times V_+/\mathbb R \times [T_0,\infty)\to \widetilde{\mathcal{G}}^{E'}_\delta(V_-/\R, V_+/\R)$$

$$(u_-,u_+,T)\mapsto u= \exp_{u_*}(\beta_{+,T}\psi_{+,T} +\beta_{-,T}\psi_{-,T}),$$

where $(\psi_-,\psi_+)$ is the solution from Lemma <ref>.

§.§ $C^1$-smoothness and injectivity of the gluing map

The goal of this subsection is to prove:

The gluing map $G$ is a $C^1$-smooth embedding.

We start by observing that:

The solution $(\psi_-,\psi_+)$ from Lemma <ref> is $C^\infty$-smooth.

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 [27]: First observe that $(\psi_-,\psi_+)$ satisfies a nonlinear Cauchy-Riemann type operator. We can then view $(\psi_-,\psi_+)$ as satisfying a linear Cauchy-Riemann type operator whose coefficients depend on $(\psi_-,\psi_+)$. Differentiating Equation (<ref>) with respect to $s$, we also see that $({\bdry\psi_-\over \bdry s},{\bdry \psi_+\over \bdry s})$ 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 ${\bdry \psi_\pm\over \bdry t}$.  With the above conditions on the coefficients, the relevant elliptic estimates are given by the last two inequalities on <cit.> to improve $\nabla \psi_\pm$ from Sobolev class $W^{0,2}_\delta$ to $W^{1,2}_\delta$ and <cit.> to improve $\nabla\psi_\pm$ from $W^{1,2}_\delta$ to $\mathcal{H}_{1,\delta}$.

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

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 $\R^{2l^\pm}$.

We assume that:

(T$_1$) the slice $\widetilde{\mathcal{U}}_\pm$ is smooth; and

(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 $\dot F_\pm -D_\pm$.

We may additionally assume that, for any ${\frak a}$ in the summand $\R^2$ of $\R^{2l^+}$ (resp. $\R^{2l^-}$) corresponding to the end that is glued, ${\frak a}$ is a linear combination of $\bdry_s$ and $\bdry_t$ on $s\leq 0$ for $u_+$ (resp. on $s\geq 0$ for $u_-$).

We say that $\phi_\pm=(\phi_\pm^\circ,{\frak a}_\pm)\in \mathcal{H}_\pm\oplus \R^{2l^\pm}$ or $\phi_\pm^\circ\in \mathcal{H}_\pm$ satisfies ($\#_k$) if the following holds:

($\#_k$) $\phi_\pm^\circ$ 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 $C\|\phi_\pm^\circ\|_{*,\delta}$.

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 $\R^{2l^\pm}$. 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>.  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 $\op{Im}(\mathcal{J})$ is bounded and each $(\psi_-,\psi_+)\in \op{Im}(\mathcal{J})$ 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 $e_{+,T}=\overline\bdry u_{\pm,T}$.
$\mathcal{I}^{\mathcal{J}}_-$ is defined analogously.  We will later apply an iteration scheme similar to the one described in Remark <ref>; see the proof of Theorem <ref> at the end of this subsection.

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 $D_{+,T}\psi_{+,T}$.

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 $e_{+,T}(\phi_+):= \overline\bdry \exp_{u_{+,T}}(\phi_{+,T})$.

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>.  The derivatives of the gluing map with respect to $j'_\pm$ are straightforward to control in view of (T$_1$) and (T$_2$).

Assuming $\mathcal{S}_\pm$ are sufficiently small, all $\phi_\pm\in \mathcal{S}_\pm$ satisfy $(\#_k)$.

This follows from the elliptic bootstrapping estimates used in the proof of Lemma <ref>.

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

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

This uses the arguments used in Claim <ref> as well as Lemma <ref>. 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 ($\#_k$).

If $\mathcal{S}_\pm$ and $\widetilde\varepsilon>0$ are sufficiently small, $T,r\gg 0$, $\mathcal{J}$ is $C^1$-smooth, and $\op{Im}(\mathcal{J})$ is bounded, then $\mathcal{I}^{\mathcal{J}}$ is $C^1$-smooth. Moreover, if $\op{Im}(\mathcal{J})$ 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.

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 ${\frak a}_\pm\in \R^{2l^\pm}$. 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{\bdry\beta_{-,T}}{\bdry s}(\zeta_{-,T}+{\frak a}_{-,T}) + {\frak l}(\eta_{+,T})\tfrac{\bdry\zeta_{-,T}}{\bdry t} + {\frak l}(\psi_{+,T}+\phi_{+,T})\tfrac{\bdry\zeta_{-,T}}{\bdry t}),
\end{align}

using the assumption that ${\frak a}_-$ is constant for $s\geq 0$.
Estimates similar to those of Claim <ref> 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

$$\|D^\flat_1 \mathcal{I}_+^{\mathcal{J}} (\phi_-,\phi_+,T)(\zeta_-+{\frak a}_-) \|_{*,\delta} \leq C\cdot \|\zeta_{-}+{\frak a}_-\|_{*,\delta}.$$

\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{\bdry\beta_{-,T}}{\bdry s}(\zeta_{-,T}+{\frak a}_{-,T}) + {\frak l}(\eta_{+,T})\tfrac{\bdry\zeta_{-,T}}{\bdry t} + {\frak l}(\psi_{+,T}+\phi_{+,T})\tfrac{\bdry\zeta_{-,T}}{\bdry t}) \\
& - B(\psi_+ +\phi_+', T') (\tfrac{\bdry\beta_{-,T'}}{\bdry s}(\zeta_{-,T'}+{\frak a}_{-,T'}) + {\frak l}(\eta_{+,T'})\tfrac{\bdry\zeta_{-,T'}}{\bdry t} + {\frak l}(\psi_{+,T'}+\phi'_{+,T'})\tfrac{\bdry\zeta_{-,T'}}{\bdry t}),
\end{align*}

and in view of Lemma <ref> 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{\bdry(\psi_{-,T}+\phi_{-,T})}{\bdry t}) (\zeta_{+,T}+{\frak a}_{+,T})  + {\frak l}(\eta_{+,T})\tfrac{\bdry\zeta_{+,T}}{\bdry 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 (<ref>), (<ref>), and (<ref>) 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{\bdry(\psi_{-,T}+\phi_{-,T})}{\bdry t}) (\zeta_{+,T}+{\frak a}_{+,T})  + {\frak l}(\eta_{+,T})\tfrac{\bdry\zeta_{+,T}}{\bdry 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> 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 (<ref>) and (<ref>) 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*}

\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{\bdry(\psi_{-,T}+\phi_{-,T})}{\bdry t}) (\zeta_{+,T}+{\frak a}_{+,T}) + {\frak l}(\eta_{+,T})\tfrac{\bdry\zeta_{+,T}}{\bdry 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{\bdry(\psi_{-,T'} +\phi'_{-,T'})}{\bdry t}) (\zeta_{+,T'}+{\frak a}_{+,T'}) + {\frak l}(\eta_{+,T'})\tfrac{\bdry\zeta_{+,T'}}{\bdry 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{\bdry(\psi_{-,T}+\phi_{-,T})}{\bdry t})& (\zeta_{+,T}+{\frak a}_{+,T})  - {\frak l}(\tfrac{\bdry(\psi_{-,T'} +\phi'_{-,T'})}{\bdry 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{\bdry\zeta_{+,T}}{\bdry t} -  {\frak l}(\eta_{+,T'})\tfrac{\bdry\zeta_{+,T'}}{\bdry 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> 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 (<ref>) 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 $\op{Im}(\mathcal{J})$ 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.

We are now in a position to prove Theorem <ref>.

$C^1$-smoothness. Recall that we are assuming that $\mathcal{S}_\pm$ are sufficiently small neighborhoods of $u_\pm$. By Lemmas <ref>–<ref> 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}

\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> 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 (<ref>) 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> 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 (<ref>) consists of six terms since Equation (<ref>) 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 (<ref>) is bounded above by

$$ c(\|\psi_-^1\|_{*,\delta}+\|\psi_+^1\|_{*,\delta})(\|\zeta_-\|_{*,\delta}+\|\zeta_+\|_{*,\delta}).$$

\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

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>.

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

$$(\zeta_-+{\frak a}_-,{\frak t})\mapsto \zeta_{-,T}+{\frak a}_{-,T}-{\frak t}\tfrac{\bdry }{\bdry s}$$

when the right-hand side is restricted to $s\leq -(T+T_0)$.  Since $T_{\phi_-}\mathcal{S}_-$ and $\tfrac{\bdry}{\bdry s}$ are independent, we see that $D\widetilde G$ is injective, which in turn implies that $G$ is a local $C^1$-embedding.

§.§ Surjectivity of the gluing map

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 $G(\phi_-,\phi_+,T)=u_0$.

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 $\R$-direction and slightly modifying $T'$, there exists a decomposition of the domain $(\dot F_0,j_0)$ of $u_0$ as:

$$\dot F_0 = (F_0)^\circ_+ \cup (F_0)^\circ_- \cup A_{[-T',T']},$$

* $\mathcal{A}_\varepsilon$ 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;

* $A_{[-T',T']}= u_0^{-1}(\{-T'\leq s\leq T'\})\cap \mathcal{A}_\varepsilon$, $int ((F_0)^\circ_+)\cup int( (F_0)^\circ_-)=\dot F_0- A_{[-T',T']}$, and  $(F_0)^\circ_+$, $(F_0)^\circ_-$ correspond to the top and bottom levels;

* $s'_{-,1}(u_0)$, 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 $s'_{\pm, 1}(u_0)= \pm (T'+T_0)$.

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>, 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\bdry \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>, where $\Phi$ is the parallel transport to $\wedge^{0,1}(u^{(1)}_*)^* T(\R\times M)$.

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>.

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

* $\phi_{+,T^{(1)}}^{(1)}=\phi^{(1)}$ on $(F')^\circ_+$, $\phi_{+,T^{(1)}}^{(1)}=0$ on $(F')^\circ_-$, and
on $A_{[-T^{(1)},T^{(1)}]}$;

* $\phi_{-,T^{(1)}}^{(1)}=0$ for $(F')^\circ_+$, $\phi_{-,T^{(1)}}^{(1)}=\phi$ on $(F')^\circ_-$, and
on $ A_{[-T^{(1)},T^{(1)}]}$.

Here $\phi_\pm^{(1)}\in \mathcal{H}_\pm\oplus \R^{2l^\pm}$ and we are writing $\|\phi_\pm^{(1)}\|_{*,\delta}$ for the sum of $\|\cdot\|_{*,\delta}$ on $\mathcal{H}_\pm$ and the standard norm on $\R^{2l^\pm}$ as before.

We then solve for $(\phi_\pm')^{(1)}$ in

$$\overline\bdry\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)} +(\phi_\pm')^{(1)})\in\Phi_\pm^{\phi_\pm^{(1)}+(\phi_\pm')^{(1)}}E_{\phi_\pm^{(1)}+(\phi'_\pm)^{(1)}},$$

where $(\phi_\pm')^{(1)}\in \mathcal{H}_\pm$, $\Phi_\pm^{\phi_\pm^{(1)}+(\phi_\pm')^{(1)}}$ is the parallel transport

$$\wedge^{0,1}(\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)} +(\phi_\pm')^{(1)}))^* T(\R\times M)\stackrel\sim\longrightarrow \wedge^{0,1}(u_\pm^{(1)})^* T(\R\times M)$$

and $E_{\phi_\pm^{(1)} +(\phi'_\pm)^{(1)}}$ is the obstruction bundle for $\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)} +(\phi_\pm')^{(1)})$.

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

This is proved using the contraction mapping theorem as in Section <ref> and relies on the fact that $\overline\bdry\exp_{u_\pm^{(1)}}(\phi_\pm^{(1)})$ 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 $\overline\bdry \exp_{u_+^{(1)}}(\phi_+^{(1)})|_{s\geq -T_0}$ is the $s=-(T^{(1)}+T_0)$ translate of $\overline\bdry \exp_{u_*^{(1)}}(\phi_{+,T^{(1)}}^{(1)})|_{s\geq T^{(1)}}$ and that

\begin{equation} \label{eqn: dbar}
\overline\bdry \exp_{u_*^{(1)}}(\phi_{+,T^{(1)}}^{(1)}+\psi_{+,T^{(1)}}^{(1)})=0,\quad \overline\bdry \exp_{u_*^{(1)}}(\psi_{+,T^{(1)}}^{(1)})=0,  \quad \overline\bdry \exp_{u_*^{(1)}}(0)=0.
\end{equation}

We claim that

\begin{equation} \label{eqn: bandaid}
\|\overline\bdry \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

($\dagger$) $\exp_u(\phi)=u+\phi$, $\exp_u(\psi)=u+\psi$, $\exp_u(\phi+\psi)=u+\phi+\psi$,

e.g., the target metric is flat.
Using Equation (<ref>) and writing $v=u+\psi$ we obtain:

\begin{align}
\label{mouse1} \overline\bdry (u+\phi) &= \tfrac{\bdry\phi}{\bdry \sigma} +J(u)\tfrac{\bdry \phi}{\bdry \tau} + \nabla J(u)(\phi)\tfrac{\bdry(u+\phi)}{\bdry \tau} + Q(\phi)\tfrac{\bdry(u+\phi)}{\bdry \tau},\\
\label{mouse2} 0=\overline\bdry (v+\phi) &= \tfrac{\bdry\phi}{\bdry \sigma} +J(v)\tfrac{\bdry \phi}{\bdry \tau} + \nabla J(v)(\phi)\tfrac{\bdry(v+\phi)}{\bdry \tau} + Q(\phi)\tfrac{\bdry(v+\phi)}{\bdry \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
$\|\overline\bdry (u+\phi)\|$ 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 $\|\overline\bdry (u+\phi)\|\leq c \|\psi\|_* \|\phi\|_*$.  The standard exponential decay estimates then yield Equation (<ref>).

Next we bound $\overline\bdry \exp_{u_+^{(1)}}(\phi_+^{(1)})$ on $-(T^{(1)}+T_0)-hr\leq s\leq -T_0$, which is the $s=-(T^{(1)}+T_0)$ translate of $\overline\bdry \exp_{u_*^{(1)}}(\phi_{+,T^{(1)}}^{(1)})|_{-hr\leq s\leq T^{(1)}}$.  Applying the same procedure as above[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 <cit.> we obtain

\begin{equation} \label{eqn: bandaid2}
\|\overline\bdry \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 $\overline\bdry \exp_{u_+^{(1)}}(\phi_+^{(1)})$ using Estimates (<ref>) and (<ref>) and the contraction mapping theorem.  This proves the claim.

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>.

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 $\V_\pm/\R$). 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 (<ref>), 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> and Estimate (<ref>) imply that the error between $u_0$ and $u^{(i)}$ is of order $C^i$ after the $i$th iteration.  The proof then follows.

§.§ Sketch of proof of Theorem <ref>

In this subsection we sketch the proof of Theorem <ref>.  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 $V_i/\R$. For ease of notation we assume that $u_2$ and $u_3$ only have one negative end.

§.§.§ 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>. We solve for $\psi_1^{(123)},\psi_2^{(123)},\psi_3^{(123)}$ which are analogs of $\psi_-,\psi_+$ in the equations

which are analogs of $\Theta_\pm(\psi_-,\psi_+)=0$. This yields

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> and is obtained by using the linearized $\overline\bdry$-operator to invert the errors that arise from the pregluing of $u_1,u_2,u_3$. Estimates similar to those of Sections <ref> 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 $T\to\infty$.

§.§.§ 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

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),
%\label{epsilon} \|\psi_3^{(123),\dagger}- \psi_3^{((12)3),\dagger}\|_{*,\delta} &\leq \widetilde C(T)\|\psi_3^{((12)3),\dagger}\|_{*,\delta},
\end{align}

where $\widetilde C(T), \widetilde D(T) \to 0$ as $T\to \infty$.

§.§.§ Conclusion

The $C^0$-closeness follows from Estimates (<ref>), (<ref>), and (<ref>). 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 (<ref>), (<ref>), and (<ref>) from Section <ref>. In words, the derivatives of the errors from the pregluing go to zero as $T_{12},T_{23}\to \infty$ as well.

§ CONSTRUCTION OF SEMI-GLOBAL KURANISHI STRUCTURES

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 $\bdry^2=0$.  In Sections <ref> and <ref> we explain the modifications needed for chain maps and chain homotopy.

§.§ Trimming

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

We start by taking $\V$ (and also $\K$) to be invariant under automorphisms that relabel the negative punctures while preserving the $\vartheta$-sorting.  In other words, $\V$ and $\K$ satisfy:

(Inv) Given $(\F,u)\in \V$ (resp. $\K$), we require that $(\F',u)\in \V$ (resp. $\K$), where $\F'$ is the same Riemann surface as $\F$ but with a different ordering of the negative punctures, subject to the condition that the negative ends of $(\F',u)$ be $\vartheta$-sorted.

This can be achieved by slightly shrinking $\V$ and $\K$ if necessary, while keeping the same names.

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

$$(\F,u)=((F,j,{\bf p},{\bf q},{\bf r}),u)\in \widetilde{\mathcal{G}}_\delta(V_1/\R,...,V_m/\R),$$

let $g$ be the complete finite volume hyperbolic metric on $\ddot F$. If there is an annular component $A$ of $\op{Thin}_\varepsilon(F,g)$ such that $u|_A$ is close to a trivial cylinder over $\gamma$ , then its neck length (with respect to $\gamma$) $\nl(A)$ is the value $C$ such that $A$ is conformally equivalent to $(\R/\mathcal A(\gamma)\Z)\times [0,C]$ with the standard complex structure.

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

The boundary $\bdry \V$ will be decomposed into two parts, the vertical boundary $\bdry_v \V=\V_{=\mathcal{L}}$ and the horizontal boundary $\bdry_h \V$ which is the closure in $\bdry \V$ of $\bdry\V-\bdry_h\V$.

§.§ CH trees

Let  $\mathcal{M}_1,\mathcal{M}_2,\dots,\mathcal{M}_\rho$ be a sequence of moduli spaces

$$\mathcal{M}_i=\mathcal{M}_{J}^{\op{ind}=k_i}(\dot F_i,\R\times M;\gamma_{i,+},\bs\gamma_{i,-}),$$

such that each component of each level of a building in $\bdry\mathcal{M}_i$ is either a trivial cylinder or in $\mathcal{M}_j$ with $j<i$. (A specific choice will be given in Section <ref>.)   The semi-global Kuranishi chart for $\mathcal{M}_i/\R$ will be denoted by $\mathcal{C}'_i=(\mathcal{C}_i,{\frak s}_i)$, where

$$\mathcal{C}_i=(\K_i,\pi_i: \E_i\to \V_i)$$

and $\K_i\subset \mathcal{M}_i/\R$ is a large compact subset.

§.§.§ Definition of CH tree

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

A contact homology tree of $\op{(Symp)}$ type (or simply a 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:

* The set of vertices $V(T)$ is labeled using the vertex labeling function

which is not necessarily injective or surjective.

* The set of edges $E(T)$ consists of the glued edges $G(T)$ and the 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.

* 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).

* 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 $F(T)_v=E(T)_v\cap F(T)$.

* For each $v\in V(T)$, $E(T)_{v}$ is labeled using a bijective edge labeling function

where $m$ is the number of negative punctures of $\mathbb{V}_{v}.$  The labeling function $l_{E_v}$ 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 $l_{E_{v}}(e_{1})<l_{E_{v}}(e_{2}).$

* Each edge $e\in G(T)$ corresponds to a single gluing of a nonempty $\V_{l_V(i(e))}$ on the upper level with a nonempty $\V_{l_V(t(e))}$ on the lower level and the gluing occurs along the $l_{E_{i(e)}}(e)$th negative end.

* There is a unique topmost vertex and each vertex except for the topmost vertex has one incoming edge and $\geq 0$ outgoing edges.

* The reordering function

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 $l_{E_{v}}(e_{1})<l_{E_{v}}(e_{2})$.

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 $O(T)$.

§.§.§ Isomorphisms of CH trees

An isomorphism $\theta:T\to T'$ of CH trees satisfies the following:

* $\theta$ is an isomorphism of directed trees which sends $V(T), G(T), F(T)$ to $V(T'),G(T'), F(T')$;

* $\theta$ preserves the labels of vertices, i.e., $l_{V(T)}=l_{V(T')}\circ \theta|_{V(T)}$;

* $\theta$ maps $O(T)$ to $O(T')$, i.e., $O(T)=O(T')\circ\theta|_{F(T)}.$

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*}
\V_{l_V(v)} &\stackrel\sim \to & \V_{l_V(\theta(v))},\\
(\F,u) & \mapsto & (\F',u),
\end{eqnarray*}

where $\F'$ is the same Riemann surface as $\F$, 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 $\F$ becomes the $\theta(l_{E_{v}}(e))$th negative puncture of $\F'$.  Putting the above isomorphisms together, $\theta:T\stackrel\sim\to T'$ gives an isomorphism

$$\times_{v\in V(T)}\V_{l_V(v)} \stackrel\sim\to\times _{w\in V(T')}\V_{l_V(w)}.$$

Let $\op{Aut}(T)$ 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.

§.§.§ Contraction of CH trees

The 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 $\mathbb V_{\tau(T)}$.

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 contraction $T'=T\git\sqcup_i S_i$ of $T$ along $\sqcup_i S_i$ is the CH tree given as follows:

* 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.

* 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 $O(T')\circ c=O(T).$

* 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 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 $O(T')$.

§.§ Overview of the construction

§.§.§ Complexity

The semi-global Kuranishi structure is constructed by induction on a triple which we call the complexity

\begin{equation}\label{eqn: definition of complexity}
c(\F,u)=(c_1(\F,u),c_2(\F,u),c_3(\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 $\bs\gamma_-$.

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> holds for all moduli spaces with $\mathcal{A}_\alpha(\gamma_+)< L$.

Fix $c_1(\F,u)$ and assume by induction that the semi-global Kuranishi charts for all the strata with smaller $c_1(\F,u)$ have been constructed as in Section <ref>.    The strata with the smallest $c_2(\F,u)$ are branched covers of trivial cylinders, i.e., satisfy $E_\alpha(u)=0$. 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

$$\mathcal{M}_i=\mathcal{M}_{J}^{\op{ind}=k_i}(\dot F_i,\R\times M;\gamma_{i,+},\bs\gamma_{i,-}),$$

where the moduli spaces are ordered according to nondecreasing complexity. The semi-global Kuranishi chart for $\mathcal{M}_i/\R$ is

$$\mathcal{C}'_i=(\mathcal{C}_i,{\frak s}_i), \quad \mathcal{C}_i=(\K_i,\pi_i: \E_i\to \V_i),$$

where $\K_i\subset \mathcal{M}_i/\R$ is a large compact subset.

§.§.§ Boundary strata

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

Given $\mathcal{M}_{i_0}/\R$, its boundary $\bdry (\mathcal{M}_{i_0}/\R)$ is described as follows:

\begin{equation}
\bdry (\mathcal{M}_{i_0}/\R)\subset \left(\coprod_{T} \mathcal{M}_T'\right)/\sim, \quad \mathcal{M}_T': =\times_{v\in V(T)}(\mathcal{M}_{l_V(v)}/\R),
\end{equation}

where the disjoint union is over all CH trees $T$ such that $T\git T$ 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

$$(\mathcal{C}_{i},{\frak s}_{i})=(\K_{i},\pi_{i}:\E_{i}\to \V_{i},{\frak s}_{i}), \quad i\in V(T),$$

have already been constructed inductively, our goal is to construct their product

$$\mathcal{C}'_T=(\mathcal{C}_T,{\frak s}_T)=(\K_T,\pi_T: \E_T\to \V_T,{\frak s}_T).$$

The chart $\mathcal{C}'_T$ “covers” $\K_T\subset \mathcal{M}_{i_0}/\R$ and the collection of all charts $\mathcal{C}'_T$, where $T$ ranges over all the strata of $\bdry(\mathcal{M}_{i_0}/\R)$, together with the main stratum $T_0$ consisting of one vertex $i_0$, covers all of $\mathcal{M}_{i_0}/\R$.

Even if $\mathcal{M}_{i_0}/\R=\varnothing$ 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

* $T\git T$ is a one-vertex CH tree whose vertex is labeled $i_0$ (for $\mathcal{M}_{i_0}$); and

* $\times_{i\in V(T)}(\mathcal{M}_{i}/\R)$ is nonempty.

There are examples, due to Michael Hutchings, where omitting $\mathcal{C}'_T$ leads to some inconsistencies.

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

We construct $\K_T$ and $\pi_T:\E_T\to \V_T$ in this subsection under the following assumption:

(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 $\chi (\dot F_{l_V(v)})<0$,

* the definition of the multisection ${\frak s}_T$ to Sections <ref> and <ref>; and

* the modifications necessary when (H) does not hold to Section <ref>.

To this end we consider the appropriate gluing map

$$G_T: (\times_{i\in V(T)}\V_i)\times [R,\infty)^{|G(T)|}\to \widetilde{\mathcal{G}}_\delta^{E'}(\{\V_i\}_{i\in V(T)}),$$

which is obtained from Theorem <ref> by passing to the orbifold.
Note that we are assuming that, for each $e\in G(T)$, $\V_{i(e)}$ is glued to $\V_{t(e)}$ and the gluing occurs along a connected neck.

We define the neck length functions

$$\nl_e(\F,u):\widetilde{\mathcal{G}}_\delta^{E'}(\{\V_i\}_{i\in V(T)}) \to \R^+, \quad e\in G(T),$$

where $\nl_e(\F,u)$ is the neck length of the annular component corresponding to $e$ of the $\varepsilon$-thin part of $g=g_{(\ddot F,j)}$.

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

Let $\K_T$ be a trimming of $(\mathcal{M}_{i_0}/\R)\cap \op{Im}(G_T)$ and $\V_T$ be a trimming of $\op{Im}(G_T)$, where we trim $(\mathcal{M}_{i_0}/\R)\cap \op{Im}(G_T)$ (resp. $\op{Im}(G_T)$) so that $\nl_e(u)\geq \mathcal{L}-{\varepsilon''\over 2}$ (resp. $\nl_e(u)> \mathcal{L}-\varepsilon''$) for all $e\in E(T)$. Here $\mathcal{L}$ and $\varepsilon''$ are from Section <ref>. Observe that $\K_T$ is not necessarily compact, although it admits a compactification.

Finally, let $\pi_T: \E_T\to \V_T$ be the orbibundle which, in an orbibundle chart, is of the type $E'$ as defined in Equation (<ref>) and is isomorphic to the product bundle $\times_{i\in V(T)} E_i$.

§.§ Multisections on products

Consider the “product” of two charts

$$\mathcal{C}'_{i_j}=(\mathcal{C}_{i_j},{\frak s}_{i_j})=(\K_{i_j},\pi_{i_j}: \E_{i_j}\to \V_{i_j}, {\frak s}_{i_j}),\quad j=1,2.$$

The goal of this subsection is to define ${\frak s}_T$ on $\E_T\to \V_T$, where $T=(V(T),E(T))$, $V(T)=\{i_1,i_2\}$, $G(T)=\{(i_2,i_1)\}$.

First let us define a multisection

$${\frak s}_T^\circ: \V_{i_1}\times \V_{i_2}\to \E_{i_1}\times \E_{i_2},$$

as follows: There exists a smooth multifunction $h_{i_j}$ on $\V_{i_j}$ (i.e., a multisection of the trivial $\R$-bundle over $\V_{i_j}$) 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 $x_{i_j}\in \V_{i_j}$ for which some branch of ${\frak s}_{i_j}\not=0$) and $h_{i_j}(x_{i_j})=0$ near $\bdry_h\V_{i_j}$. (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

* ${\frak s}^\circ_T=0$ on $(\bdry_h \V_{i_1}\times \V_{i_2})\cup (\V_{i_1}\times \bdry_h \V_{i_2})$;

* $(\overline\bdry_J^{-1},\overline\bdry_J^{-1})({\frak s}^\circ_T)=\overline\bdry_J^{-1}({\frak s}_{i_1})\times \overline\bdry_J^{-1}({\frak s}_{i_2})$.

We then define the multisection ${\frak s}_T$ on $\E_T\to \V_T$ as follows: We first view $\E_{i_1}\times \E_{i_2}$ as an orbibundle over $\V_{i_1}\times \V_{i_2}\times[R,\infty)$ by pulling back via the projection $p:\V_{i_1}\times \V_{i_2}\times[R,\infty)\to \V_{i_1}\times \V_{i_2}$ and identify:

\begin{equation}\label{diagram for pushing forward}
\begin{diagram}
\E_{i_1}\times \E_{i_2} & \rTo{} & \V_{i_1}\times \V_{i_2}\times [R,\infty)\\
\dTo^{\simeq} &  & \dTo_{\simeq} \\
\E_T & \rTo &  G_{T}(\V_{i_1}\times \V_{i_2}\times [R,\infty))
\end{diagram}
\end{equation}

We trim $G_{T}(\V_{i_1}\times \V_{i_2}\times [R,\infty))$ to obtain $\V_T$.   On $\V_{T}\cap\{\nl_{(i_2,i_1)}\geq\mathcal{L}+\varepsilon''\}$, 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 $\V_T\cap \{\nl_{(i_2,i_1)}\geq \mathcal{L}+\varepsilon''\}$. On $\V_T\cap \{\nl_{(i_2,i_1)}<\mathcal{L}\}$, ${\frak s}_T$ is the stabilization of ${\frak s}_{T\git T}$ of the main stratum and, on $\V_{T}\cap\{\mathcal{L}+\varepsilon''>\nl_{(i_2,i_1)}\geq \mathcal{L}\}$, we interpolate between the two.

In Section <ref> 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.

§.§ Transition functions

Let $T$ be a CH tree. Given the charts

$$\mathcal{C}'_{T\git T}=(\K_{T\git T},\pi_{T\git T}:\E_{T\git T}\to \V_{T\git T},{\frak s}_{T\git T}),$$

$$\mathcal{C}'_T=(\K_T,\pi_T: \E_T\to \V_T,{\frak s}_T),$$

we define the morphism $\mathcal{C}'_{T\git T}\to \mathcal{C}'_T$ (not a map) by restriction-inclusion: first restrict $\mathcal{C}'_{T\git T}$ to $\V_{T\git T}\cap\{\mathcal{L}>\min_{e\in E(V)}\nl_e > \mathcal{L}-\varepsilon''\}$ and then include into $\V_{T}\cap\{\mathcal{L}>\min_{e\in E(V)} \nl_e>\mathcal{L}-\varepsilon''\}$ as a stabilization.

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>):

E$\V_{T\git T}$
F$\V_{T\git S}$
G$\V_{T\git S'}$

Schematic diagram of $\V_{T\git T}$, $\V_{T\git S}$, $\V_{T\git S'}$, and $\V_{T}$. The horizontal and vertical lines represent the $\nl_{e_1}$- and $\nl_{e_2}$-coordinate axes and we are considering the projections of $\V_*$ to the $(\nl_{e_1},\nl_{e_2})$-coordinate plane.

$$\mathcal{C}'_{T\git T}, \quad \mathcal{C}'_{T\git S}, \quad \mathcal{C}'_{T\git S'}, \quad \mathcal{C}'_{T},$$

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\git T} & \rTo & \mathcal{C}'_{T\git S} \\
\dTo & & \dTo\\
\mathcal{C}'_{T\git 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.

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}

\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 $\op{Aut}(T)$ is nontrivial.

§.§ 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.

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

(1) On $\{\mathcal{L}>\nl_{e_1}>\mathcal{L}-\varepsilon'',\nl_{e_2}>\mathcal{L}-\varepsilon''\}$, ${\frak s}_T$ is the stabilization of ${\frak s}_{T\git S}$. Similarly, on $\{\nl_{e_1}>\mathcal{L}-\varepsilon'',\mathcal{L}>\nl_{e_2}>\mathcal{L}-\varepsilon''\}$, ${\frak s}_T$ is the stabilization of ${\frak s}_{T\git S'}$.  The two definitions agree on the overlap.

(2) On $\{\nl_{e_1},\nl_{e_2}\geq \mathcal{L}+\varepsilon''\}$, ${\frak s}_T$ 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

$$p:\V_{i_1}\times\V_{i_2}\times \V_{i_3}\times [R,\infty)^2 \to \V_{i_1}\times\V_{i_2}\times \V_{i_3}$$

and pushing forward as in Diagram (<ref>).

(3) Consider the region $\{\mathcal{L}+\varepsilon''\geq \nl_{e_1}\geq \mathcal{L}, \nl_{e_2}\geq \mathcal{L}+\varepsilon''\}$.   On the subregion $\{\mathcal{L}+\varepsilon''-\varepsilon'''\geq \nl_{e_1}\geq \mathcal{L}, \nl_{e_2}\geq \mathcal{L}+\varepsilon''\}$, where $0<\varepsilon'''\ll \varepsilon''$, we define ${\frak s}_T$ to be the product of ${\frak s}_{S}$ restricted to $\{\mathcal{L}+\varepsilon''\geq \nl_{e_1}\geq \mathcal{L}\}$ and ${\frak s}_{i_3}$, as in Section <ref>.

To define ${\frak s}_T$ on $\{\mathcal{L}+\varepsilon''\geq \nl_{e_1}\geq \mathcal{L}+\varepsilon''-\varepsilon''', \nl_{e_2}\geq \mathcal{L}+\varepsilon''\}$, we rephrase Theorem <ref> as follows:

The triple gluing map

$$G_{T}: \V_{i_1}\times \V_{i_2}\times \V_{i_3}\times [R,\infty)^2\to \widetilde{\mathcal{G}}^{E'}_\delta(\V_{i_1},\V_{i_2},\V_{i_3}),$$

\begin{align*}
((\F_1,u_1),(\F_2,u_2),(\F_3,u_3), & T_1,T_2)\mapsto\\
& G_{T}((\F_1,u_1),(\F_2,u_2),(\F_3,u_3),T_1,T_2),
\end{align*}

and the iterated gluing map

$$G_{T\git S}\circ G_{S}: \V_{i_1}\times \V_{i_2}\times \V_{i_3}\times [R,\infty)^2\to \widetilde{\mathcal{G}}^{E'}_\delta(\V_{i_1},\V_{i_2},\V_{i_3}),$$

\begin{align*}
((\F_1,u_1),(\F_2,u_2), &(\F_3,u_3), T_1,T_2)\mapsto\\
& G_{T\git S}(G_{S}((\F_1,u_1),(\F_2,u_2),T_1),(\F_3,u_3), T_2),
\end{align*}

do not necessarily agree but are $C^1$-close for $T_1,T_2\gg 0$.

By Theorem <ref>, on $\{\mathcal{L}+\varepsilon''\geq \nl_{e_1}\geq \mathcal{L}+\varepsilon''-\varepsilon''', \nl_{e_2}\geq \mathcal{L}+\varepsilon''\}$, (i) the pushforward of the product of ${\frak s}_{S}$ and ${\frak s}_{i_3}$ via $G_{T\git S}$ 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 $\{\mathcal{L}+\varepsilon''\geq \nl_{e_1}\geq \mathcal{L}+\varepsilon''-\varepsilon''', \nl_{e_2}\geq \mathcal{L}+\varepsilon''\}$ so that ${\frak s}_T$ is given by (i) along $\nl_{e_1}= \mathcal{L}+\varepsilon''-\varepsilon'''$ and by (ii) along $\nl_{e_1}=\mathcal{L}+\varepsilon''$.

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

(4) We finally extend arbitrarily to $\{\mathcal{L}+\varepsilon''\geq \nl_{e_1}\geq \mathcal{L},\mathcal{L}+\varepsilon''\geq \nl_{e_2}\geq \mathcal{L} \}$, subject to the conditions that the multisection be small and vanish away from $\mathcal{M}_{T\git T}/\R$.

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

§.§ Modifications when (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.

(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 $(\F_{i_0},u_{i_0})=(F_{i_0},j_{i_0},{\bf p}_{i_0},{\bf q}_{i_0}, {\bf r}_{i_0})$ which is close to breaking into

$$(\F_{i_1},u_1)\cup (\F_{i_2},u_2)\in \V_{i_1}\times \V_{i_2}$$

with $(\F_{i_j},u_{i_j})=(F_{i_j},j_{i_j},{\bf p}_{i_j},{\bf q}_{i_j}, {\bf r}_{i_j})$, $j=1,2$, we recall from Section <ref> 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 $\nl_e=\nl_e^{g_e}$ 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 $E_{(\F_{i_0},u_{i_0})}$ using $\widetilde{s}_{\pm,j}(\F_{i_0},u_{i_0})$, which in turn is defined as:

* $s_{\pm,j}^{g}(\F_{i_0},u_{i_0})$ on $\{\nl_e\leq \mathcal{L}'-\varepsilon'''\}$;

* $s_{\pm,j}^{g_e}(\F_{i_0},u_{i_0})$ on $\{\nl_e\geq \mathcal{L}'\}$; and

* an interpolation between the two on $\{\mathcal{L}'-\varepsilon'''\leq \nl_e\leq \mathcal{L}'\}$.

On $\V_{i_0}$ the orbibundle $\E_{i_0}$ agrees with $E_{(\F_{i_0},u_{i_0})}$; on $\V_{T}$ the orbibundle $\E_T$ agrees with the stabilization $E'_{(\F_{i_0},u_{i_0})}$ of $E_{(\F_{i_0},u_{i_0})}$, constructed as in Equation (<ref>) using $g_e$.

(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 $(\F_{i_0},u_{i_0})=(F_{i_0},j_{i_0},{\bf p}_{i_0},{\bf q}_{i_0}, {\bf r}_{i_0})$ which is close to breaking into

\begin{equation}\label{eqn: triple}
(\F_{i_1},u_1)\cup (\F_{i_2},u_2)\cup (\F_{i_3},u_3)\in \V_{i_1}\times \V_{i_2}\times \V_{i_3}
\end{equation}

with $(\F_{i_j},u_{i_j})=(F_{i_j},j_{i_j},{\bf p}_{i_j},{\bf q}_{i_j}, {\bf r}_{i_j})$, $j=1,2,3$, 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 $S\git S$ (resp. $S'\git S'$) is a $1$-vertex CH tree whose vertex we label $i_4$ (resp. $i_5$).

We will use $\nl_{e_1}=\nl_{e_1}^{g_{e_1e_2}}$ and $\nl_{e_2}=\nl_{e_2}^{g_{e_1e_2}}$ and define $E_{(\F_{i_0},u_{i_0})}$ using $\widetilde{s}_{\pm,j}(\F_{i_0},u_{i_0})$, which is given by:

* $s_{\pm,j}^g(\F_{i_0},u_{i_0})$ on $\{\nl_{e_1}\leq \mathcal{L}'-\varepsilon''',\nl_{e_2}\leq \mathcal{L}'-\varepsilon'''\}$;

* $s_{\pm,j}^{g_{e_2}}(\F_{i_0},u_{i_0})$ on $\{\nl_{e_1}\leq \mathcal{L}'-\varepsilon''',\nl_{e_2}\geq \mathcal{L}'\}$;

* $s_{\pm,j}^{g_{e_1}}(\F_{i_0},u_{i_0})$ on $\{\nl_{e_2}\leq \mathcal{L}'-\varepsilon''',\nl_{e_1}\geq \mathcal{L}'\}$;

* $s_{\pm,j}^{g_{e_1e_2}}(\F_{i_0},u_{i_0})$ on $\{\nl_{e_1}\geq \mathcal{L}',\nl_{e_2}\geq \mathcal{L}'\}$;

* an interpolation on $\{\nl_{e_1}\leq \mathcal{L}'-\varepsilon''', \mathcal{L}'-\varepsilon'''\leq \nl_{e_2}\leq \mathcal{L}'\}$ which “only depends on $\nl_{e_2}$” and analogous interpolations for

* $\{\nl_{e_2}\leq \mathcal{L}'-\varepsilon''', \mathcal{L}'-\varepsilon'''\leq \nl_{e_1}\leq \mathcal{L}'\}$,

* $\{\nl_{e_1}\geq \mathcal{L}',\mathcal{L}'-\varepsilon'''\leq \nl_{e_2}\leq \mathcal{L}'\}$, and

* $\{\nl_{e_2}\geq \mathcal{L}',\mathcal{L}'-\varepsilon'''\leq \nl_{e_1}\leq \mathcal{L}'\}$;

* an arbitrary interpolation on $\{\mathcal{L}'-\varepsilon'''\leq \nl_{e_1}\leq \mathcal{L}',\mathcal{L}'-\varepsilon'''\leq \nl_{e_2}\leq \mathcal{L}'\}$.

By “only depending on $\nl_{e_2}$" we mean the following: For $a<b$, let $\lambda_{a,b}:\R\to [0,1]$ be a smooth nondecreasing function such that $\lambda_{a,b}(a)=0$ and $\lambda_{a,b}(b)=1$. We then take

$$(1-\lambda(\nl_{e_2}))s_{\pm,j}^g+\lambda(\nl_{e_2}) s_{\pm,j}^{g_{e_2}},$$

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 $\Z/2$-action interchanging $e_1$ and $e_2$, we require the interpolation in (6) to respect the group action.) See Figure <ref>.

The horizontal and vertical lines represent the $\nl_{e_1}$- and $\nl_{e_2}$-coordinate axes and we are considering the projections of $\V_*$ to the $(\nl_{e_1},\nl_{e_2})$-coordinate plane. The numbers indicate the regions appearing in the definition of $\widetilde{s}_{\pm,j}$.

We assume that the interpolations have been chosen so that, as $\nl_{e_1}\to \infty$, the restriction of the Kuranishi structure to the $\nl_{e_1}=const$ slices limit to a stabilization of the Kuranishi structure already constructed for $S'$.  This is possible because of the following:

As we take $\nl_{e_1}\to\infty$, $g_{e_1e_2}$ for $(\F_{i_0},u_{i_0})$ converges to a $2$-component hyperbolic metric, one of which is $g''_{e_2}$ for $(\F_{i_5},u_{i_5})$; and $g_{e_1}$ for $(\F_{i_0},u_{i_0})$ converges to a $2$-component hyperbolic metric, one of which is $g''$ for the same $(\F_{i_5},u_{i_5})$.   The situation for $\nl_{e_2}\to\infty$ is analogous.

The stabilizations are constructed as before with little change: On $\V_{i_0}$, the orbibundle $\E_{i_0}$ agrees with $E_{(\F_{i_0},u_{i_0})}$.  On $\V_{T\git S}$, the orbibundle $\E_{T\git S}$ agrees with the stabilization $E'_{(\F_{i_0},u_{i_0})}$ of $E_{(\F_{i_0},u_{i_0})}$, constructed as in Equation (<ref>) using $s'_{\pm,j}(\F_{i_0},u_{i_0})$ with respect to $g_{e_2}$ on $\nl_{e_1}\leq \mathcal{L}'-\varepsilon'''$, using $s'_{\pm,j}(\F_{i_0},u_{i_0})$ with respect to $g_{e_1e_2}$ on $\nl_{e_1}\geq \mathcal{L}'$, and interpolating between the two on $\mathcal{L}'-\varepsilon'''\leq\nl_{e_1}\leq \mathcal{L}'$; 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 $\E_{i_0}$ to $\E_{T\git S'}$ and take a common stabilization when passing from $\E_{i_0}$ to $\E_{T}$.

§.§ The 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 semi-global Kuranishi structure $\Kur^L(\alpha,J)$ which is a category consisting of the following:

* objects which are charts $\mathcal{C}'_T=(\mathcal{C}_T,{\frak s}_T)=(\K_T,\pi_T:\E_T\to \V_T,\overline\bdry_J,{\frak s}_T)$, where $T$ ranges over all CH trees;

* morphisms $\mathcal{C}'_{T\git\sqcup_i S_i}\to \mathcal{C}'_T$ given by restriction-inclusion as in Section <ref>;

* commutativity of diagrams of the form

\begin{equation}
\begin{diagram}
\mathcal{C}'_{T\git(S'\sqcup S''\sqcup (\sqcup_i S_i)} & \rTo & \mathcal{C}'_{T\git (S'\sqcup (\sqcup_i S_i))} \\
\dTo & & \dTo\\
\mathcal{C}'_{T\git (S''\sqcup (\sqcup _i S_i))} & \rTo & \mathcal{C}'_{T\git(\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\git(S\sqcup(\sqcup_i S_i))} & \rTo & \mathcal{C}'_{T\git (S'\sqcup(\sqcup_i S_i))} \\
\dTo & & \dTo\\
\mathcal{C}'_{T\git(S''\sqcup(\sqcup_i S_i))} & \rTo & \mathcal{C}'_{T\git (\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 $S$.

Given a semi-global Kuranishi structure $\Kur^L(\alpha,J)$, we define

$$\mathcal{Z}(\Kur^L(\alpha,J))=\left(\coprod_T \overline\bdry^{-1}_J({\frak s}_T)\right)/\sim_{\Kur},$$

where $T$ ranges over all CH trees and $\sim_{\Kur}$ is the identification given by the morphisms.

$\mathcal{Z}(\Kur^L(\alpha,J))$ is a weighted branched manifold.

Each $\overline\bdry^{-1}_J({\frak s}_T)$ is a weighted branched manifold since ${\frak s}_T$ is transverse to $\overline\bdry_J$. 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 $\overline\bdry_J$.

In order to verify that $\left(\coprod_T \overline\bdry^{-1}_J({\frak s}_T)\right)/\sim_{\Kur}$ 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

$$Z_1=\overline\bdry^{-1}_J({\frak s}_{T\git T}), \quad Z_2=\overline\bdry^{-1}_J({\frak s}_{T})$$

corresponding to two charts $\mathcal{C}'_{T\git T}$ and $\mathcal{C}'_{T}$. Since $\sim_{\Kur}$ 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 $\nl_e$ since $\nl_e(z_2)\geq \mathcal{L}$ and $\nl_e(z_1)< \mathcal{L}$ (or there is no $\varepsilon$-thin annulus corresponding to $\nl_e$).

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

§.§ Cobordisms

In this subsection we describe the modifications needed for the cobordism maps in Section <ref>.

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>. 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^{\op{ind}=k_i}(\dot F_i, \widehat{W}; \gamma_{i,+},\bs\gamma_{i,-}),\\ \mathcal{M}_i^{J_\pm}&=\mathcal{M}_{J_\pm}^{\op{ind}=k_i}(\dot F_i,\R\times M_\pm;\gamma_{i,+},\bs\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 $\bdry \mathcal{M}_i$ is either a trivial cylinder or in $\mathcal{M}_j$ with $j<i$.

A CH tree of $\op{(Cob)}$ type is defined in the same way as a CH tree of (Symp) type, with the following exceptions:

* there is a splitting of $V(T)$ into $V^{\op{Cob}}(T) \sqcup V^{\op{Symp}}(T)$, where elements of $V^{\op{Cob}}(T)$ are the (Cob) vertices and the elements of $V^{\op{Symp}}(T)$ are the (Symp) vertices;

* referring to a maximal oriented path of the oriented tree (somewhat nonstandardly) as a branch, each branch has precisely one (Cob) vertex;

* the (Cob) vertices are labeled with indices corresponding to a moduli space of curves with image in $\widehat{W}$.

An edge is an upper (resp. lower) $\op{(Symp)}$ edge if it is an edge between two (Symp) vertices lying above (resp. below) a (Cob) vertex, and is an upper (resp. lower) $\op{(Cob)}$ edge if it is an edge from (resp. to)  a (Symp) vertex to (resp. from) a (Cob) vertex.

An isomorphism of CH trees of $\op{(Cob)}$ 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.

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:

($\alpha$) $S_i$ 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>.

($\beta$) $S_i$ is not  a CH tree of (Symp) or (Cob) type (i.e., Condition (2) of Definition <ref> is not satisfied).  In this case $\tau(S_i)$ is just viewed as a symbol.

The contraction $T\git\sqcup_i S_i$ is defined as in Definition <ref> 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, $\nl_e$ 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, $\nl_{\gamma_1}$ and $\nl_{\gamma_2}$ are dependent, where

\begin{equation}
\nl_\gamma:= \sum_{e\in G(\gamma)} \nl_e.
\end{equation}

Let $T$ be a CH tree of (Cob) type and let $T\git T$ 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 $\nl_{e_1},\dots, \nl_{e_l}$, where $G(T)=\{e_1,\dots,e_l\}$.

Whenever we pass from $\nl_{e_j}\leq \mathcal{L}-\varepsilon''$ to $\nl_{e_j}>\mathcal{L}-\varepsilon''$, 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 $T'=T\git T_{\mathcal{C}}$. Then on the region

$$\{\nl_e\geq \mathcal{L}-\varepsilon''\}_{e\in\mathcal{C}}\cap \{\nl_e < \mathcal{L}\}_{e\in G(T)-\mathcal{C}},$$

the obstruction orbibundle

$$\pi_{T'}: \E_{T'}\to \V_{T'}\quad  \mbox{ or } \quad \pi_{T'}^{J}: \E_{T'}^J\to V_{T'}^J,$$

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.

(S0) The multisections are invariant under isomorphisms of CH trees of (Symp) type or (Cob) type, as appropriate.

(S1) If $S=T$ or $S\subset T$ is a connected subtree of type ($\beta$) without free edges and $Cl(S)=T$, then:

* The multisection $\frak{s}_{T\git S}$ is defined arbitrarily (but generically), subject to consistency with adjacent regions and (S0).

* The multisection $\frak{s}_T$ is defined arbitrarily (but generically) on the region

$$\{\nl_e\geq\mathcal{L}+\varepsilon''\}_{e\in G(T)-G(S)}\cap \{ \mathcal{L}\leq \nl_e\leq \mathcal{L}+\varepsilon''\}_{e\in G(S)},$$

subject to consistency with adjacent regions and (S0).

(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

$$\{\nl_e\geq\mathcal{L}+\varepsilon''\}_{e\in \mathcal{C}}\cap \{ \mathcal{L}\leq \nl_e\leq \mathcal{L}+\varepsilon''\}_{e\in\mathcal{C}'}\cap  \{\nl_e< \mathcal{L}\}_{e\in G(T)-\mathcal{C}-\mathcal{C}'},$$

the multisection ${\frak s}_{T'}$, $T'=T\git (\cup_{e\in G(T)} -\sqcup_{e\in\mathcal{C}\cup\mathcal{C}'} int(e) ))$, 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.

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.

Although strictly speaking not necessary, we give some examples which illustrate the combinatorics of the corners (see Figure <ref>), and then explain the general case.    Since the projection of $\mathcal{M}_{i_0}$ to the $(\nl_{e_1},\dots,\nl_{e_l})$-coordinate plane (when applicable) lies within a fixed distance from the locus

$$\{\nl_{\gamma_1}=\nl_{\gamma_2}~|~ \mbox{ $\gamma_1$, $\gamma_2$ oriented paths from top vertex to a (Cob) vertex}\},$$

for the purposes of combinatorics we may assume that $\nl_{\gamma_1}=\nl_{\gamma_2}$ for any two $\gamma_1$, $\gamma_2$.  Given a collection of edges $e_1,\dots,e_k$, let

\begin{equation}
\nl_{e_1\dots e_k}:=\min(\nl_{e_1},\dots,\nl_{e_k}).
\end{equation}

The (Symp) vertices are given by solid dots and the (Cob) vertices by open squares.

Case (A).
Let $T$ be a CH tree of (Cob) type such that

$$V^{\op{Symp}}(T)=\{i_{k+1}\},~~ V^{\op{Cob}}(T)=\{i_1,\dots, i_k\},$$

$$G(T)=\{e_1=(i_{k+1},i_1), \dots,e_k=(i_{k+1},i_k)\}.$$

The graph for $k=2$ is depicted in Figure <ref>(A).  Refer to Figure <ref> which shows the $(\nl_{e_1},\nl_{e_2})$-coordinates.

d$\nl_{e_1}=\mathcal L-\varepsilon''$
g$\nl_{e_2}=\mathcal L-\varepsilon''$

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 $\nl_{e_1}=\nl_{e_2}$.

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 $T\git S_1$ and $T\git S_2$, 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 $\nl_{e_1},\dots,\nl_{e_k}$ are related on $\mathcal{M}_{i_0}$; in fact, $\nl_{e_1}=\nl_{e_2}=\dots =\nl_{e_k}$ “in the large”.  This motivates (S1) in the construction of the multisections.

Case (B).
Let $T$ be a CH tree of (Cob) type such that

$$V^{\op{Symp}}(T)=\{i_4,i_5\},~~ V^{\op{Cob}}(T)=\{i_1,i_2,i_3\},$$

$$G(T)=\{e_1=(i_4,i_1), e_2=(i_4,i_2), e_3=(i_5,i_4), e_4=(i_5,i_3)\}.$$

The graph $T$ is given in Figure <ref>(B). We first map $\mathcal{M}_{i_0}$ to the $(\nl_{e_1},\dots,\nl_{e_4})$-coordinate plane.  Note that, “in the large” we have relations $\nl_{e_1}=\nl_{e_2}$ and $\nl_{e_1}+\nl_{e_3}=\nl_{e_4}$. Hence we may project to coordinates $\nl_{e_3}$ and any one of $\nl_{e_1}, \nl_{e_2},\nl_{e_4}$, or $\nl_{e_1e_2e_4}$. Note that $\nl_{e_4}> \nl_{e_1e_2}$ when we are sufficiently close to the corner and hence $\nl_{e_1e_2e_4}=\nl_{e_1e_2}$.

Case (C). Let $T$ be a CH tree of (Cob) type such that

$$V^{\op{Symp}}(T)=\{i_3,i_4,i_5\},~~ V^{\op{Cob}}(T)=\{i_1,i_2\},$$

$$G(T)=\{e_1=(i_3,i_1), e_2=(i_4,i_2), e_3=(i_5,i_3), e_4=(i_5,i_4)\}.$$

First map $\mathcal{M}_{i_0}$ to the $(\nl_{e_1},\dots,\nl_{e_4})$-coordinate plane.  The image of $\mathcal{M}_{i_0}$ is a bounded distance away from $\nl_{e_1}+\nl_{e_3}=\nl_{e_2}+\nl_{e_4}$. It is not hard to verify that the neighborhood of the corner (i.e., $\nl_{e_1}+\nl_{e_3}=\nl_{e_2}+\nl_{e_4}$ with $\nl_{e_j}$ 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 $\nl_{e_i}$-coordinate axes. The boundary faces are the $(\nl_{e_1},\nl_{e_2})$-, $(\nl_{e_2},\nl_{e_3})$-, $(\nl_{e_3},\nl_{e_4})$-, and $(\nl_{e_4},\nl_{e_1})$-coordinate planes which intersect the closure of $\{\nl_{e_i}=\mathcal{L}-\varepsilon''\}_{i=1,\dots,5}$ in a pattern similar to that of Figure <ref>.

General Case.
We will inductively describe the convex polytope $\zeta(T)$ such that the $\nl_{e_1},\dots,\nl_{e_l}\to\infty$ 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

$$\bdry \zeta(T)=\bdry \zeta(T_1)* \bdry\zeta(T_2).$$

We then “recover” $\zeta(T)$ by taking the convex hull of the vertices of $\bdry\zeta(T)$, where we are assuming that $\zeta(T)$ lies on the hyperplane determined by $\nl_{\gamma}=\nl_{\gamma'}$ for all pairs $\gamma,\gamma'$ of oriented paths from the top vertex to a (Cob) vertex.

§.§ Chain homotopy

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>. 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> holds simultaneously for all $J_\tau$, $\tau\in[0,1]$. If $T$ is a CH tree of (Cob) type, then let

$$\pi_T^{\{J_\tau\}}: \E_T^{\{J_\tau\}}\to \V_T^{\{J_\tau\}}$$

be the $1$-parametric version of the obstruction orbibundle $\pi_T^{J_\tau}:\E_T^{J_\tau}\to \V_T^{J_\tau}$.  The obstruction multisection $\frak{s}_T^{\{J_\tau\}}$ on $\V_T^{\{J_\tau\}}$ is also the $1$-parametric version of $\frak{s}^{J_\tau}$ on $\V_T^{J_\tau}$. We assume that $\pi_T^{\{J_\tau\}}$ and $\frak{s}_T^{\{J_\tau\}}$ are generic subject to the following conditions:

($\tau_1$) they are independent of $\tau$ on $[0,\epsilon]$ and on $[1-\epsilon,1]$; and

($\tau_2$) 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 $J_1$.

To define the chain homotopy $\Omega_K$ we consider disconnected curves; more formally we take the multiple fibered products

$$\V_{T_1\dots T_k}^{\{J_\tau\}}:=\V_{T_1}^{\{J_\tau\}}\times_{[0,1]}\V_{T_2}^{\{J_\tau\}}\times_{[0,1]} \dots \times_{[0,1]} \V_{T_k}^{\{J_\tau\}},$$

$$\E_{T_1\dots T_k}^{\{J_\tau\}}:=\E_{T_1}^{\{J_\tau\}}\times_{[0,1]}\E_{T_2}^{\{J_\tau\}}\times_{[0,1]} \dots \times_{[0,1]} \E_{T_k}^{\{J_\tau\}},$$

where $\bs\gamma_+=(\gamma_{+,1},\dots,\gamma_{+,k})$ and curves in $\V_{T_i}^{\{J_\tau\}}$ limit to $\gamma_{+,i}$ at the positive end.  We denote the corresponding obstruction orbibundle by

$$\pi_{T_1\dots T_k}^{\{J_\tau\}}:\E_{T_1\dots T_k}^{\{J_\tau\}}\to \V_{T_1\dots T_k}^{\{J_\tau\}}.$$

§.§.§ Basic perturbation scheme

For the purposes of Proposition <ref>, 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 $\V_{T_i}^{\{J_\tau\}}$ as follows:

(*) For each pair $S\subset T$ satisfying (S1) in Section <ref> and for each occurrence of $\V_{T\git S}^{\{J_\tau\}}$ or $\V_{T}^{\{J_\tau\}}$ restricted to

$$\{\nl_e\geq\mathcal{L}+\varepsilon''\}_{e\in G(T)-G(S)}\cap \{ \mathcal{L}\leq \nl_e\leq \mathcal{L}+\varepsilon''\}_{e\in G(S)}$$

as in (S1)(a) or (S1)(b), we choose a different perturbation of the multisection ${\frak s}_{T\git S}^{\{J_\tau\}}$ 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 $[0,\epsilon]\cup [1-\epsilon,1]$.

§.§.§ Refined perturbation scheme

For the purposes of Proposition <ref>, we use the following refined perturbation scheme, which is that of Ekholm-Oancea [9] 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 $\frak{s}_{T_i}^{\{J_\tau\}}$:

(**) For the $j$th occurrence of $\V_{T_i}^{\{J_\tau\}}$ in $\V_{T_1\dots T_k}^{\{J_\tau\}}$, we perturb ${\frak s}_{T_i}^{\{J_\tau\}}$ so that its intersection with $\overline\bdry_{J_\tau}$ is obtained from $\overline\bdry_{J_\tau}^{-1}({\frak s}_{T_i}^{\{J_\tau\}})$ by shifting via $\tau\mapsto \chi_j(\tau)$.  Here $\chi_j:[0,1]\stackrel\sim\to[0,1]$ is a diffeomorphism such that

* $\chi_j'(\tau)$ is close to $1$;

* $\chi_j(\tau)=\tau$ on $[0,\epsilon]\cup[1-\epsilon,1]$ and $\chi_j(\tau)=\tau+ j\epsilon_{i(T_i\git T_i)}$ on $[2\epsilon, 1-2\epsilon]$;

* $i(T_i\git T_i)$ is the index corresponding to $T_i\git T_i$.

§ CONTACT HOMOLOGY

§.§ 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> there exists a semi-global Kuranishi structure

$$\Kur^L(\alpha,J)=\{ \Kur^L(\alpha, J; \gamma_+; \bs \gamma_-)~|~ \mathcal A ( \gamma_+),\mathcal A(\bs \gamma_-)\leq L  \}.$$

$$\mathfrak A^L=\mathfrak A^L(M,\alpha, J, \Kur^L(\alpha,J), \mathcal{S}^{L}_{\alpha})$$

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

We also define a $\Z/2$-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 $\tau$.

An ordered tuple of Reeb orbits $\bs \gamma$ is good if all of its entries are good Reeb orbits. Otherwise, $\bs\gamma$ is bad.

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 $HC(M,\alpha, J, \Kur^L(\alpha,J),\mathcal{S}^L_\alpha)$ or $HC^L$. Fix an ordering $\vartheta$ of $\mathcal P^{L}_\alpha$.  We then set

\begin{equation} \label{eqn: partial}
\partial \gamma_+=\sum_{\bs\gamma_-}\sum_{A\in H_2(M;\Z)} \frac{d_{\gamma_+,\bs \gamma_-,A}}{m_{\bs \gamma_-}} e^A \gamma_{-,1}^{i_1}\dots \gamma_{-,k}^{i_k},
\end{equation}

where the sums are taken over all good $\vartheta$-sorted $\bs \gamma_-$ and $A\in H_2(M;\Z)$ with  $|\gamma_+|-(|e^A|+|\bs \gamma_-|)=1$.  A sorted $\bs\gamma_-$ is written as

\begin{equation}\label{eqn: gamma-}
\bs \gamma_-=(\underbrace{\gamma_{-,1},\dots, \gamma_{-,1}}_{i_{1} \text{ copies }},\dots, \underbrace{\gamma_{-,k},\dots, \gamma_{-,k}}_{i_{k} \text{ copies }}),
\end{equation}

$d_{\gamma_+,\bs \gamma_-, A}$ is the weighted signed count of elements in $\mathcal Z(\Kur^L(\alpha, J;\gamma_+; \bs \gamma_-))$ in the class $A$, and

$$m_{\bs \gamma_-}:=\Pi_{j=1}^k ( i_j ! m(\gamma_{-,j})^{i_j}).$$

(If $|\gamma_+|-(|e^A|+|\bs \gamma_-|)\neq 1$, then we set $d_{\gamma_+,\bs \gamma_-,A}=0$.) In Section <ref> we explain how to define the class $A$ corresponding to an approximate $J$-holomorphic map $u$ from $\gamma_+$ to $\bs\gamma_-$ and in Section <ref> we explain how to orient the moduli spaces. Then we extend $\partial$ to $\mathfrak A^L$ using the graded Leibniz rule, namely,

$$\partial (\gamma \gamma')=(\partial \gamma) \gamma'+(-1)^{|\gamma|_0}\gamma (\partial  \gamma').$$

§.§.§ Grading and $\Q[H_2(M;\Z)]$-coefficients

We follow the discussion in [3].  Let us write $H_1(M;\Z)= F\oplus T$, 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 $[\gamma]=[\sum c_i C_i +\sum d_i D_i]\in H_1(M;\Z)$, where $d_i$ are the smallest nonnegative coefficients for the $D_i$.  Then choose a surface $S_\gamma$ such that $\bdry S_\gamma=\gamma-\sum_i c_i C_i -\sum_i d_i D_i$ 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

$$|\gamma|= \mu_\tau(\gamma)+2\sum_{i=1}^b{w_i d_i\over m_i} + n-3\in \Q ,$$

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 complete set of trivializing surfaces for $\mathcal{P}^L_\alpha$.

Given a map $u$ from $\gamma$ to $\bs\gamma_-$, 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

$$\bdry S_\gamma=\gamma-\sum_i c_i C_i -\sum_i d_i D_i,$$

where the $d_i$ are the smallest nonnegative coefficients, then

$$\sum_j \bdry S_{\gamma_{-,j}}=\sum_j\gamma_{-,j} -\sum_i c_i C_i -\sum_i d_i D_i + \zeta,$$

where $\zeta$ is a sum of $m_i D_i$.

§.§.§ Orientations

We explain how to assign a sign to

$$[\F, u]\in \mathcal Z(\Kur^L(\alpha, J;\gamma_+; \bs \gamma_-))$$

as we range over all $\mathcal Z(\Kur^L(\alpha, J;\gamma_+; \bs \gamma_-))$ in $\Kur^L(\alpha,J)$.

For any $(\F, u)\in \mathcal{B}_{\widetilde{\mathcal U}}$, let $L_{(\F,u)}$ be the full linearized $\overline\bdry_J$-operator and

$$\op{det}L_{(\F, u)}=\wedge^{\op{top}}\ker L_{(\F, u)} \otimes \wedge^{\op{top}}(\op{coker}L_{(\F, u)})^*$$

be its determinant line. An orientation $\mathfrak o(\F, u)$ of $\op{det}L_{(\F, u)}$ is an equivalence class of nonzero vectors of $\op{det}L_{(\F, u)}$, where the equivalence relation is $v\sim v'$ if there exists $c>0$ such that $v=cv'$.

According to [5], an orientation $\mathfrak o(\F, u)$ can be chosen in a continuous and coherent way for all $(\F,u)$ in all the $\mathcal{B}_{\widetilde{\mathcal U}}$, where “coherent" means:

(O1) if we exchange the $i$th and $(i+1)$st negative punctures of $(\F,u)$, 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 $p_{-,j}$ (<cit.>);

(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 (<cit.>);

(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 (<cit.>);

(O4) precomposing with an automorphism of the domain which preserves the punctures and asymptotic markers preserves the orientation  (<cit.>).

Note that the coherent orientation exists for all maps $(\mathcal{F},u)$ in all $\mathcal{B}_{\widetilde{\mathcal{U}}}$ and results of [5], 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 $\E\to \V$.  We consider the slightly easier (Cob) case, leaving the (Symp) case to the reader.  Let $(E\to V,\Gamma)$ be an orbibundle chart for $\E\to\mathbb{V}$. Given $(\F, u)\in  V$, 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 $T_{(\F,u)}V=L^{-1}_{(\F,u)}(E_{(\F,u)})$. By <cit.>, the orientation $\mathfrak o(L_{(\F,u)})$ induces an orientation $\mathfrak o(\det L_{(\F, u)}^E)$ 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 $\E\to \V$.

Finally we explain how $\mathcal{Z}(\K^L(\alpha,J; \gamma_+;\bs\gamma_-))$ 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 $(\F, u)\in \overline \partial_J^{-1}(s_i)$, $i=1,\dots,m$, we orient $T_{(\F, u)} V$ arbitrarily and orient $\bdry_J^{-1}(s_i)$ as the transverse intersection of the sections $\bdry_J$ and $s_i$, where both $\bdry_J$ and $s_i$ are given the orientation induced from $T_{(\F, u)} V$.  Observe that the orientation of $\partial_J^{-1}(s_i)$ does not depend on the choice of orientation for $T_{(\F, u)}V$.

§.§.§ Examples

To illustrate the definition of the differential, we consider two examples (here we use $\Q$-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 $\llbracket\F,u\rrbracket\in \mathcal M/\mathbb R$ 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 $\R$-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 $(\F, u)\in \mathcal M(\gamma_+;\gamma_-,\gamma_-)$ is a possibly branched double cover of a simple curve $(\F', u')\in \mathcal M(\gamma'_+;\gamma_-)$. Since reordering the negative punctures of $(\F,u)$ 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 $\llbracket\F, u\rrbracket$ 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 $\partial \gamma_+$.

§.§.§ $\bdry^2=0$

$\partial$ is well-defined and $\mathfrak A^L $ is a dga with differential given by $\partial$.

By the construction of the semi-global Kuranishi structure and the above discussion on coherent orientations, if $|\gamma_+|-(|e^A|+|\bs\gamma_-|)=1$ and $\gamma_+, \bs\gamma_-$ are good, then $\mathcal Z(\Kur^L(\alpha, J;\gamma_+; \bs \gamma_-;A))$  (this means the subset of $\mathcal Z(\Kur^L(\alpha, J;\gamma_+; \bs \gamma_-))$ in the class $A$) is an oriented compact weighted branched $0$-dimensional manifold.  Hence $\partial$ is well-defined. In this proof we will abbreviate

$$\mathcal Z(\Kur(\gamma_+;\bs\gamma_-))=\mathcal Z(\Kur^L(\alpha, J;\gamma_+;\bs \gamma_-;A)),$$

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.

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 $\bs \gamma=\gamma_1^{c_1}\dots\gamma_m^{c_m}$ with $c_1,\dots,c_m\geq 0$, we obtain

\begin{align}\label{eqn: coefficient}
m_{\bs\gamma}\langle \partial^2 \gamma_+, \bs\gamma \rangle=&\sum_{\bs \beta\subseteq \bs\gamma}\sum_{k=1}^m\sum_{q=0}^{c_k-b_k}(-1)^{a+b} \frac{d_{\gamma_+;(\bs\gamma-\bs\beta)\sqcup \gamma_k}}{m_{(\bs\gamma-\bs\beta)\sqcup \gamma_k}}\frac{d_{\gamma_k;\bs \beta}}{m_{\bs \beta}}m_{\bs\gamma}\\
\label{eqn: coefficient 2} =&\sum_{\bs \beta\subseteq \bs\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_+;(\bs\gamma-\bs\beta)\sqcup \gamma_k}d_{\gamma_k;\bs \beta}}{m(\gamma_k)(c_k-b_k+1)},
\end{align}

where the first sum is taken over all $\vartheta$-sorted subtuples $\bs \beta$ of $\bs \gamma$, which we write as $\bs\beta=\gamma_1^{b_1}\dots\gamma_m^{b_m}$ with $0\leq b_i\leq c_i$ for $i=1,\dots,m$, and 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

$$(\bs \gamma-\bs\beta)\sqcup \gamma_k:=\gamma_1^{c_1-b_1}\dots \gamma_{k}^{c_k-b_{k}+1}\dots\gamma_{m}^{c_m-b_m},$$

$$(\bs\gamma-\bs\beta) \sqcup_{\gamma_k,q}\bs\beta:= \gamma_1^{c_1-b_1}\dots\gamma_k^q(\gamma_1^{b_1}\dots\gamma_m^{b_m}) \gamma_k^{c_k-b_k-q}\dots\gamma_m^{c_m-b_m},$$

and then $(-1)^b$ is given by the equation
$$\bs\gamma=(-1)^b(\bs\gamma-\bs\beta) \sqcup_{\gamma_k,q} \bs\beta$$
as supercommutative monomials and $a=|\gamma_1^{c_1-b_1}\dots\gamma_k^q|_0$.

We claim that, for each $\bs\beta\subseteq\bs\gamma$ and $k\in \{1,\dots,m\}$ with $|\gamma_k|-(|e^{A_1}|+|\bs\beta|)=1$ and $|\gamma_+|-(|e^{A_2}|+|(\bs \gamma-\bs\beta)\sqcup \gamma_k|)=1$ (for some $A_1+A_2=A$) and $q\in \{0,1,\dots,c_k-b_k\}$, there exists a truncation $T(\mathcal Z(\Kur(\gamma_+;\bs \gamma)))$ of the ends of $\mathcal Z(\Kur(\gamma_+;\bs \gamma))$ and a map

$$G_{\bs\beta,k,q}:  \mathcal Z(\Kur(\gamma_k;\bs\beta)) \times  \mathcal Z(\Kur(\gamma_+;(\bs\gamma-\bs\beta)\sqcup \gamma_k))  \times O\to  \bdry T( \mathcal Z(\Kur(\gamma_+;\bs \gamma))), $$

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 $1$-dimensional branched manifold $\mathcal{Z}(\Kur(\gamma_+;\bs\gamma))$ corresponds to the gluing of an element of some $\mathcal{Z}(\Kur(\gamma_k;\bs\beta))$ and an element of some $\mathcal Z(\Kur(\gamma_+;(\bs\gamma-\bs\beta)\sqcup \gamma_k))$. The key fact to remember is that the dimension of any $\mathcal{Z}(\Kur(\gamma',\bs\gamma''))$ is the same as the expected dimension given by $\op{ind}-1$ in the (Symp) case. By construction, each end of $\mathcal{Z}(\Kur(\gamma_+;\bs\gamma))$ corresponds to an element of some product

$$\times_{v\in V(T)} \mathcal{Z}(\Kur(\gamma_{v,+};\bs\gamma_{v,-})),$$

where $T\git T$ is a one-vertex CH tree whose vertex is labeled $i_0$ for the moduli space $\mathcal{M}_{i_0}$ corresponding to $\mathcal{Z}(\Kur(\gamma_+;\bs\gamma))$, and $\mathcal{Z}(\Kur(\gamma_{v,+};\bs\gamma_{v,-}))$ corresponds to the moduli space $\mathcal{M}_{l_V(v)}$.  Since the $\op{ind}$ (resp. the dimension) of each $\mathcal{Z}(\Kur(\gamma_{v,+};\bs\gamma_{v,-}))$ 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

$$(\llbracket\F_1,u_1\rrbracket,\llbracket\F_2, u_2\rrbracket)\in \mathcal Z(\Kur(\gamma_k;\bs \beta))\times \mathcal Z(\Kur(\gamma_+; (\bs\gamma-\bs \beta) \sqcup \gamma_k)),$$

after fixing a gluing parameter we can glue $(\F_1,u_1)$ and $(\F_2, u_2)$ so that the positive puncture of $\F_1$ is glued to the $(q+1)$st negative puncture of $\F_2$ among those that converge to $\gamma_k$ and denote the glued curve by $(\F', u')$. Note that the negative punctures of $\F'$ have a natural ordering  $\tilde o$ coming from the gluing so that $\llbracket\F', u'\rrbracket\in  \mathcal Z(\Kur(\gamma_+;(\bs\gamma-\bs\beta) \sqcup_{\gamma_k,q} \bs\beta))$. Next let $o$ be an ordering of the negative punctures of $\mathcal{F}'$ such that:

* 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

* the surface $\mathcal{F}'$ with the ordering $o$, denoted by $\mathcal{F}'_o$, has $\vartheta$-sorted negative punctures;

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

$$G_{\bs\beta,k,q}(\llbracket\F_1,u_1\rrbracket,\llbracket\F_2, u_2\rrbracket, o):=\llbracket\F'_o, u=u'\rrbracket;$$

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 $T(\mathcal Z(\Kur(\gamma_+;\bs \gamma)))$ is given by the right-hand side of Equation (<ref>). This follows by considering

\begin{align*}
\coprod_{\bs\beta,k,q} G_{\bs\beta,k,q}:  \coprod_{\bs\beta,k,q}( \mathcal Z(\Kur(\gamma_k;\bs\beta)) \times &  \mathcal Z(\Kur(\gamma_+;(\bs\gamma-\bs\beta)\sqcup \gamma_k)) \times O)\\
& \to  \bdry T( \mathcal Z(\Kur(\gamma_+;\bs \gamma))),
\end{align*}

keeping track of weights, and combining the following observations:

* By Property (O3), the components of $ \mathcal Z(\Kur(\gamma_+;\bs \gamma))$ that involve bad orbits do not contribute to the count.

* By Properties (O1) and (O2), the sign difference between $(\mathcal{F}'_o,u')$ and $(\mathcal{F}',u')$ is $(-1)^{a+b}$ if $\bs\gamma\not=0$.  (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 (<ref>) becomes zero but on the other hand $\bs\gamma=0$ by supercommutativity.)

* $\op{Im}G_{\bs\beta,k,q}=\op{Im}G_{\bs\beta,k,q'}$ 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 (<ref>).

* There are $m(\gamma_k)$ ways to simultaneously change the asymptotic markers of $\F_1$ and $\F_2$ such that $G_{\bs\beta,k,q}(\llbracket\F_1,u_1\rrbracket,\llbracket\F_2,u_2\rrbracket,o)$ does not change; this cancels the term $m(\gamma_k)$ in the  denominator of Equation (<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>. 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 $\Kur^{L_\pm}:=\Kur^{L_\pm}(\alpha_\pm,J_\pm)$ 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

$$\mathfrak A^{L_\pm}:=\mathfrak A^{L_\pm}(M_\pm,\alpha_\pm,J_\pm,\Kur^{L_\pm}, \mathcal{S}^{L_\pm}_{\alpha_\pm})$$

as in Section <ref>.  Here we are also using orderings $\vartheta_\pm$ of $\mathcal{P}^{L_\pm}_{\alpha_\pm}$.
We can also construct a semi-global Kuranishi structure

$$\Kur^{(L_+,L_-)}:=\Kur^{(L_+,L_-)}(\widehat \alpha,J)$$

that is compatible with  $\Kur^{L_\pm}$. We denote by the above collection of data by:

$$\mathcal C:=(M_\pm, L_\pm, \alpha_\pm, J_\pm, \Kur^{L_\pm}, \mathcal{S}^{L_\pm}_{\alpha_\pm}, \widehat W, \widehat \alpha, J, \Kur^{(L_+,L_-)}).$$

We now define the dga morphism

$$\Phi:\mathfrak A^{L_+} \otimes_{\Q[H_2(M_+;\Z)]}\Q[H_2(W;\Z)]\to \mathfrak A^{L_-}\otimes_{\Q[H_2(M_-;\Z)]}\Q[H_2(W;\Z)].$$

Here we are viewing $\mathfrak{A}^{L_\pm}$ as $\Q[H_2(W;\Z)]$-modules via the algebra maps

$$\Q[H_2(M_\pm;\Z)]\to \Q[H_2(W;\Z)]$$

induced by $H_2(M_\pm;\Z)\to H_2(W;\Z)$.  For any $\gamma_+ \in \mathcal P_{\alpha_+}^{L_+}$, let

\begin{equation}\label{eqn: chain map}
\Phi(\gamma_+)=\sum_{\bs \gamma_-}\sum_{A\in H_2(W;\Z)} \frac{p_{\gamma_+,\bs \gamma_-}}{m_{\bs\gamma_-}} e^A \gamma_{-,1}^{i_1}\dots \gamma_{-,k}^{i_k},
\end{equation}

where the sums are taken over all good $\vartheta_-$-sorted $\bs\gamma_-$ written in the form of Equation (<ref>) and $A\in H_2(W;\Z)$ with $|\gamma_+|-(|e^A|+|\bs\gamma_-|)=0$, and

$p_{\gamma_+,\gamma_-}$ is the weighted signed count of elements in $\mathcal Z(\Kur^{(L_+,L_-)}(\widehat \alpha, J; \gamma_+;\bs \gamma_-))$ in the class $A$. (If $|\bs \gamma_-|\neq |e^A|+|\bs\gamma_-|$, then we set $p_{\gamma_+,\bs\gamma_-}=0$.) The homology class $A$ can be obtained from $(\mathcal{F},u)\in \mathcal{Z}(\Kur^{(L_+,L_-)}(\widehat \alpha, J; \gamma_+;\bs \gamma_-))$ by capping off using $\mathcal{S}^{L_\pm}_{\alpha_\pm}$ as in Section <ref>.  $\Phi$ can then be extended to $\mathfrak A^{L_+}$ as an algebra homomorphism.

$\Phi$ is a well-defined dga morphism.

The proof is similar to that of Proposition <ref>.

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

§.§ 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

$$\mathcal D=(\alpha, \{  L_i, \varphi_i, J_i, \Kur^{L_i}(\alpha_i,J_i),\mathcal{S}^{L_i}_{\alpha_i},\widehat \alpha_{i,i+1},J_{i,i+1}, \Kur^{(L_i,L_{i+1})}(\widehat \alpha_{i,i+1},J_{i,i+1})\}_{i\in \mathbb N})$$

be a collection that consists of the following data:

* a nondegenerate contact form $\alpha$ for $\xi$;

* an increasing sequence $\{L_i\}_{i\in \mathbb N}$ such that $L_i\to \infty$;

and, for each $i\in \mathbb N$,

* 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 $\varphi_i<\varphi_{i+1}$;

* an almost complex structure $J_i$ on $\mathbb R\times M$ such that $(\alpha_i,J_i)$ is an $L_i$-simple pair;

* a semi-global Kuranishi structure $\Kur^{L_i}(\alpha_i,J_i)$;

* a complete set $\mathcal{S}^{L_i}_{\alpha_i}$ of trivializing surfaces for $\mathcal{P}^{L_i}_{\alpha_i}$;

* a completed exact symplectic cobordism $(\widehat W_{i,i+1}=\R\times M,\widehat \alpha_{i,i+1})$ 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 $\varphi_{i+1}$;

* 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;

* a semi-global Kuranishi structure $\Kur^{(L_i,L_{i+1})}(\widehat \alpha_{i,i+1},J_{i,i+1})$ that is compatible with $\Kur^{L_i}(\alpha_i,J_i)$ and $\Kur^{L_{i+1}}(\alpha_{i+1},J_{i+1})$.

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 $\Kur^{(L_i,L_{i+1})}(\widehat \alpha_{i,i+1}, J_{i,i+1}))$ by

$$\Phi_{i*}: HC^{L_i}(\alpha_i,J_i; \Kur^{L_i}(\alpha_i,J_i)) \to HC^{L_{i+1}}(\alpha_{i+1},J_{i+1};\Kur^{L_{i+1}}(\alpha_{i+1},J_{i+1})).$$

Finally, we define the contact homology algebra by

$$H(\mathcal D):=\lim_{\longrightarrow} HC^{L_i}(\alpha_i,J_i; \Kur^{L_i}(\alpha_i,J_i)),$$

where the directed system is constructed using $\Phi_{i*}$.

§.§ Invariance of contact homology

In this section we prove Propositions <ref> and <ref> 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 $\Kur^{L_\pm}:=\Kur^{L_\pm}(\alpha_\pm,J_\pm)$ be the semi-global Kuranishi structures constructed for $(M_\pm,\alpha_\pm, J_\pm)$ and let $\mathfrak A^{L_\pm}=\mathfrak A^{L_\pm}(M_\pm,\alpha_\pm,J_\pm,\Kur^{L_\pm})$.

For $\tau=0,1$, we have two collections of data

$$\mathcal C^\tau=(M_\pm, L_\pm, \alpha_\pm, J_\pm, \Kur^{L_\pm},  \mathcal{S}^{L_\pm}_{\alpha_\pm},\widehat W^\tau, \widehat \alpha^\tau, J^\tau, \Kur^{(L_+,L_-)})$$

as in Section <ref> and chain maps

$$\Phi^\tau:\mathfrak A^{L_+} \otimes_{\Q[H_2(M_+;\Z)]}\Q[H_2(W;\Z)]\to \mathfrak A^{L_-}\otimes_{\Q[H_2(M_-;\Z)]}\Q[H_2(W;\Z)],$$

defined using the data $\mathcal C^\tau$.

There is a degree $1$ $\Q[H_2(W;\Z)]$-module map

$$\Omega_K: \mathfrak A^{L_+} \otimes_{\Q[H_2(M_+;\Z)]}\Q[H_2(W;\Z)]\to \mathfrak A^{L_-}\otimes_{\Q[H_2(M_-;\Z)]}\Q[H_2(W;\Z)]$$

such that $\Phi^0-\Phi^1=\Omega_K\circ\partial+\partial \circ\Omega_K.$ In particular, $\Phi^0_*=\Phi^1_*$.

We construct a semi-global Kuranishi structure

$$\Kur^{(L_+,L_-)}(\{(\widehat \alpha^\tau,J^\tau)\}_{\tau\in[0,1]})$$

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 $\Kur^{(L_+,L_-)}(\widehat \alpha^\tau,J^\tau)$ 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 $\bs\gamma_+$ of  good Reeb orbits, we have

\begin{equation}
\Omega_K(\bs\gamma_+)=\sum_{\bs \gamma_-}\sum_{A\in H_2(W;\Z)} \frac{\kappa_{\bs\gamma_+,\bs \gamma_-}}{m_{\bs\gamma_{-,1}}\cdots m_{\bs\gamma_{-,k}}} e^A \bs\gamma_{-,1}\dots \bs\gamma_{-,k},
\end{equation}

where the sums are taken over all $\bs\gamma_-$ which is a $k$-tuple of $\vartheta_-$-sorted tuples of good Reeb orbits, i.e.,

\begin{equation}
\bs\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}

$A\in H_2(W;\Z)$ with $|\bs\gamma_+|-(|e^A|+|\bs\gamma_-|)=-1$, and $\kappa_{\bs\gamma_+,\bs\gamma_-}$ is the weighted signed count of the elements in

$$\mathcal{Z}^{[0,1]}(\Kur(\bs\gamma_+,\bs\gamma_-)):=\mathcal Z( \Kur^{(L_+,L_-)}(\{(\widehat \alpha^\tau, J^\tau)\}_{\tau\in[0,1]};\bs\gamma_+;\bs\gamma_-))$$

in the class $A$.

Let $\Phi^0,\Phi^1$ be two dga morphisms

$$\mathfrak A^{L_+} \otimes_{\Q[H_2(M_+;\Z)]}\Q[H_2(W;\Z)]\to \mathfrak A^{L_-}\otimes_{\Q[H_2(M_-;\Z)]}\Q[H_2(W;\Z)],$$

induced by the cobordism $(\widehat W^\tau, \widehat \alpha^\tau, J^\tau)$, $\tau=0,1$, using the semi-global Kuranishi structure $ \Kur^{(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>.   The key point is to observe that, with the perturbation from Section <ref>, each end of a $1$-dimensional (and hence $\op{ind}=0$) branched manifold $\mathcal{Z}^{[0,1]}(\Kur(\gamma_+,\bs\gamma_-))$ corresponds to (i) a term contributing to $\Phi^0$ or $\Phi^1$, (ii) the gluing of an element of some $\mathcal{Z}(\Kur(\gamma_+,\bs\gamma_+'))$ of index $1$ and an element of some $\mathcal{Z}^{[0,1]}(\Kur(\bs\gamma_+',\bs\gamma_-))$ of index $-1$, or (iii) the gluing of an element of some $\mathcal{Z}^{[0,1]}(\Kur(\gamma_+,\bs\gamma_-'))$ of index $-1$ and an element of some $\mathcal{Z}(\Kur(\bs\gamma_-',\bs\gamma_-))$ where all but one of the components is a trivial cylinder.  (In particular, if $\op{ind}< -1$, then $\mathcal{Z}^{[0,1]}(\Kur(\bs\gamma_+',\bs\gamma_-))=\varnothing$ and $\mathcal{Z}^{[0,1]}(\Kur(\gamma_+,\bs\gamma_-'))=\varnothing$.)

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

There is a degree $1$ map

$$K: \mathcal{P}_{\alpha_+}^{L_+}\to \mathfrak A^{L_-}\otimes_{\Q[H_2(M_-;\Z)]}\Q[H_2(W;\Z)]$$

such that

\begin{equation} \label{equation: exp}
\Phi^1=\Phi^0\exp(K\partial+\partial K)=\Phi^0\left( 1+ (K\bdry + \bdry K) + {(K\bdry+\bdry K)^2\over 2!} +\dots\right).
\end{equation}

We need to explain what the right-hand side of Equation (<ref>) means.
Given a monomial $\bs\gamma=\gamma_1\dots \gamma_k$ where $\gamma_i\in\mathcal{P}_{\alpha_+}^{L_+}\cup \mathcal{P}_{\alpha_-}^{L_-}$,  we define

$$K(\bs\gamma)=\sum_{i=1}^k(-1)^{|\gamma_1|+\dots + |\gamma_{i-1}|} \gamma_1\dots \gamma_{i-1}K(\gamma_i)\gamma_{i+1}\dots\gamma_k,$$

where $K(\gamma_i)=0$ if $\gamma_i\in \mathcal{P}_{\alpha_-}^{L_-}$, and define

$$\Phi^0(\bs\gamma)= \Phi^0(\gamma_1)\dots \Phi^0(\gamma_k),$$

where $\Phi^0(\gamma_i)=\gamma_i$ if $\gamma_i\in \mathcal{P}_{\alpha_-}^{L_-}$.

Follows from the refined perturbation scheme from Section <ref> and Section 5.4 (and in particular the proof of Lemma 5.6) of Ekholm-Oancea [9].

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

$$\mathcal C_{ij}=(M_i,M_j, L_i,L_j, \alpha_i,\alpha_j, J_i,J_j, \Kur^{L_i},\Kur^{L_j},  \mathcal{S}^{L_i}_{\alpha_i}, \mathcal{S}^{L_j}_{\alpha_j}, \widehat W_{ij}, \widehat \alpha_{ij}, J_{ij}, \Kur^{(L_+, L_-)}_{ij} )$$

be a collection of data as in Section <ref> for $(\widehat W_{ij}, \widehat \alpha_{ij})$.  Using the data $\mathcal C_{ij}$ we obtain a chain map

$$\Phi_{ij}: \mathfrak A^{L_i}\otimes_{\Q[H_2(M_i;\Z)]}\Q[H_2(W_{ij};\Z)]\to \mathfrak A^{L_j}\otimes_{\Q[H_2(M_j;\Z)]}\Q[H_2(W_{ij};\Z)].$$

There exists a degree 1 map

$$\Omega_K:\mathfrak A^{L_1}\otimes_{\Q[H_2(M_1;\Z)]}\Q[H_2(W_{13};\Z)]\to \mathfrak A^{L_3} \otimes_{\Q[H_2(M_3;\Z)]}\Q[H_2(W_{13};\Z)]$$

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, $\Phi_{23*}\Phi_{12*}=\Phi_{13*}.$

The proof is almost the same as that of Proposition <ref>.

Finally we are ready to prove the main theorem.

The contact homology algebra $HC(\mathcal D)$ is independent of the choices $\mathcal D$ and hence  is an invariant of $(M,\xi)$.

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, \Kur^{L_i}(\alpha_i,J_i)),\\
HC(\mathcal D')&=\lim_{\longrightarrow} HC^{L'_i}(\alpha'_i,J'_i, \Kur^{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, \Kur^{L_i}(\alpha_i,J_i)),\\
HC^{L'_i}&:=HC^{L'_i}(\alpha'_i,J'_i, \Kur^{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 $\Kur^{L_i}(\alpha_i, J_i)$ and $\Kur^{L_{j(i)}}(\alpha_{j(i)},J_{j(i)})$ 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

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 $\Theta_* \Psi_*=\op{id}$; similarly $\Psi_* \Theta_*=\op{id}$.

§ APPENDIX

In the Appendix we prove Lemma <ref> and Lemma <ref>.
We first start with some basic facts about Sobolev spaces.

Suppose that $p=2$ and $k\geq3$.  If $g,h\in W^{k,p}_{\op{loc}}(\R^2)$, then $g\cdot h \in W^{k,p}_{\op{loc}}(\R^2)$.

We prove that $ (g\cdot h)^{(m)}\in L^p_{\op{loc}}(\R^2)$ for any $0\leq m \leq k$. Let $\Omega\subset \R^2$ be a domain with smooth boundary and compact closure. For $m=0$, we have $g,h\in C^0(\Omega)$[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 $\|(g\cdot h)^{(m)}\|_{L^p(\Omega)}<\infty.$

Suppose that $p=2$ and $k\geq 3$. If $G\in C^{\infty}(\mathbb R^2, \mathbb R)$ and $H\in W^{k,p}_{\op{loc}}(\mathbb R^2, \mathbb R^2)$,
then $G\circ H\in W^{k,p}_{\op{loc}}(\mathbb R^2, \mathbb R)$.

By the Sobolev embedding theorem, $H\in C^{k-1}(\Omega,\R^2)$, 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>.  The lemma then follows.

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 $W^{k+1,p}_{\op{loc}}.$

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 Hölder 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 $C^1$.

In order to prove Lemma <ref>, we first treat the case $\chi(\dot F)<0$.
To simplify the notation in Equation (<ref>), 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 (<ref>) 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 $W^{k+1,p}_{\op{loc}}$ by Lemma <ref>. Next

the projection $\pi_j$ is smooth, so $\pi_j(ds-idt)\in W_{\op{loc}}^{k+1,p}(\dot F, \wedge^{0,1}T^{*}\dot F)$. Hence $\widetilde f\in W^{k+1,p}_{\op{loc}}(\dot F, \wedge^{0,1}u^{*}T\widehat W)$ by Lemma <ref>, and this proves the first part of Lemma <ref>, namely $ E^{\ell,\varepsilon}_{(\F,u)}\subset \mathcal{E}_{(\F,u)}$ for all $(\F,u)\in \mathcal{N}(K)$.

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

$$\mathcal E_{(j',u)}=W^{k,p}_\delta (\dot F, \wedge^{0,1}_{j'} u^{*}T\widehat W)$$

$$\mathcal E_{(j,v)}=W^{k,p}_\delta (\dot F, \wedge ^{0,1}_j v^{*}T\widehat W)$$

for any $(j',u)\in \mathcal D$ via the exponential map. More precisely, we define

$$\Psi: W^{k,p}_\delta (\dot F, \wedge^{0,1}_{j'} u^{*}T\widehat W) \to W^{k,p}_\delta (\dot F, \wedge ^{0,1}_j v^{*}T\widehat W)$$

$$\zeta=\eta \otimes \alpha  \mapsto \op{Par}_{\xi} \eta \otimes \pi_j \alpha,$$

where $\xi=(\exp_v)^{-1} u$, $\eta\in W^{k,p}_\delta(\dot F, u^{*}T\widehat W)$, $\alpha\in W^{k,p}_\delta(\dot F, \wedge^{0,1}T^{*}F),$

$$\op{Par}_\xi: W^{k,p}_\delta(\dot F, u^{*}T\widehat W)\to W^{k,p}_\delta(\dot F, v^{*}T\widehat W)$$