(4) $\bar{u}$ converges to a cylinder over a multiple of some $\gamma_j$ near each puncture $p_i$ so that the total multiplicity of $\gamma_j$ over all the $p_i$'s is $m_j$;

(5) the energy of $\bar{u}$ given by Equation (4.3.1) is finite.

Here $\pi_{\mathbb{R}}$ is the projection $\pi_{\mathbb{R}_-}: \overline{\mathbb{W}}_- \to \mathbb{R}_- \subset \mathbb{R} \times \mathbb{S}^1$, followed by the projection to $\mathbb{R}$.

**Remark 5.4.11.** — As in Section 5.3.2, we will assume that $\overline{\mathbb{J}}_-$ is sufficiently close to a Morse-Bott almost complex structure for which all degree $k \le 2g$ multisections of $(\overline{\mathbb{W}}_-, \overline{\mathbb{J}}_−)$ are close to breaking into a Morse-Bott building. This will allow us to use the Morse-Bott formalism to prove topological constraints on the $\overline{\mathbb{J}}_-$-holomorphic multisec- tions. In the context of maps to $\overline{\mathbb{W}}_-$, the definition of a Morse-Bott building is the same as in Definition 5.3.13, with the only difference that the bottommost holomorphic map $u_n$ takes values in $\overline{\mathbb{W}}_-$ and is negatively asymptotic to chords over intersection points in $\mathbf{a} \cap \mathfrak{h}(\mathbf{a})$.

**Definition 5.4.12.** — The section at infinity $\sigma_{\infty}^{-}$ is the restriction of $\sigma'_{\infty} = \mathbb{R} \times \delta_0 \subset \overline{\mathbb{W}}'$ to $\overline{\mathbb{W}}_-$.

Let $\delta_0^{\dagger}$ be the closed orbit of $\overline{\mathbb{R}}_0$ used in Definition 5.3.18. We define $(\sigma_{\infty}^{-})^{\dagger}$ as the restriction of $\mathbb{R} \times \delta_0^{\dagger}$ to $\overline{\mathbb{W}}_-$.
Both $\sigma_{\infty}^{-}$ and $(\sigma_{\infty}^{-})^{\dagger}$ are $\overline{\mathbb{J}}_-$-holomorphic for every almost complex structure $\overline{\mathbb{J}}_- \in \mathcal{J}_{\overline{\mathbb{W}}_-}$.

**Definition 5.4.13.** — Given a multisection $\bar{u}$ of $\overline{\mathbb{W}}_-$, we define $n^{-}(\bar{u}) = \langle \bar{u}, (\sigma_{\infty}^{-})^{\dagger} \rangle$.

The proof of the following is similar to that of Lemma 5.4.8.

**Lemma 5.4.14.** — The intersection number $n^{-}(\bar{u})$ satisfies the following properties:

(1) $n^{-}(\bar{u}) \ge 0$ and $n^{-}(\bar{u}) = 0$ if and only if the image of $\bar{u}$ is disjoint from $(\sigma_{\infty}^{-})^{\dagger}$;

(2) if $\langle \bar{u}, \sigma_{\infty}^{-} \rangle > 0$, then $n^{-}(\bar{u}) \ge m$; moreover $n^{-}(\bar{u}) = m$ if and only if there is a unique transverse intersection point between the image of $\bar{u}$ and $\sigma_{\infty}^{-}$; and

(3) $n^{-}(\bar{u})$ is independent of the choice of $\delta_0^{\dagger}$.

**Definition 5.4.15.** — $A(\overline{\mathbb{W}}_-, \overline{\mathbb{J}}_-)$-curve is a degree 2g multisection of $(\overline{\mathbb{W}}_-, \overline{\mathbb{J}}_-)$ satisfying $n^{-}(\bar{u}) = m$.

Let $\tilde{\overline{\mathbb{W}}}_-$ be $\overline{\mathbb{W}}_-$ with the ends $\{s > 1\}$ and $\{s < -3\}$ removed. We can view $\tilde{\overline{\mathbb{W}}}_-$ as a compactification of $\overline{\mathbb{W}}_-$, obtained by attaching $[0, 1] \times \overline{S}$ at $s = -\infty$ and $\overline{N}$ at $s = +\infty$. Also let

$$Z_{\gamma, \mathbf{y}} = (\mathbb{L}_{\hat{\mathbf{a}}} \cap \tilde{\overline{\mathbb{W}}}) \cup (\{1\} \times \mathbf{y}) \cup (\{-3\} \times [0, 1] \times \mathbf{y}).$$

We write $\mathcal{M}_{\overline{\mathbb{J}}_-}(\mathbf{y}, \mathbf{y})$ for the moduli space of multisections of $(\overline{\mathbb{W}}_-, \overline{\mathbb{J}}_-)$ from $\mathbf{y}$ to $\mathbf{y}$. We denote by $H_2(\overline{\mathbb{W}}_-, \mathbf{y}, \mathbf{y})$ the equivalence classes of continuous degree 2g multisec-
tions in $\overline{W}_-$ satisfying Conditions (1)–(4) of Definition 5.4.10, where two multisections are equivalent if they represent the same element in $H_2(\check{\overline{W}}_-, Z_{\gamma,\mathbf{y}})$. Then

$$ \mathcal{M}_{\overline{J}_-}(\mathbf{y}, \mathbf{y}) = \coprod_{A \in H_2(\overline{W}_-, \mathbf{y}, \mathbf{y})} \mathcal{M}_{\overline{J}_-}(\mathbf{y}, \mathbf{y}, A). $$

Also let $\mathcal{M}_{\overline{J}_-}(\mathbf{y}, \mathbf{y}; \overline{\mathbf{m}})$ be the moduli space of multisections of $(\overline{W}_-, \overline{J}_-)$ from $\mathbf{y}$ to $\mathbf{y}$ that pass through the marked point $\overline{\mathbf{m}}$. This moduli space comes equipped with a forgetful map

$$ \mathcal{M}_{\overline{J}_-}(\mathbf{y}, \mathbf{y}; \overline{\mathbf{m}}) \to \mathcal{M}_{\overline{J}_-}(\mathbf{y}, \mathbf{y}). $$

When we restrict our attention to $\overline{J}_$-holomorphic curves $\overline{u}$ with $n^-(\overline{u}) = m$, the forgetful map $\mathcal{M}_{\overline{J}_-}^{n^-=m}(\mathbf{y}, \mathbf{y}; \overline{\mathbf{m}}) \to \mathcal{M}_{\overline{J}_-}^{n^-=m}(\mathbf{y}, \mathbf{y})$ is injective by Lemma 5.4.14(2) because $\overline{\mathbf{m}} \in \sigma_\infty^-$. This observation allows us to regard $\mathcal{M}_{\overline{J}_-}^{n^-=m}(\mathbf{y}, \mathbf{y}; \overline{\mathbf{m}})$ as a subset of $\mathcal{M}_{\overline{J}_-}^{n^-=m}(\mathbf{y}, \mathbf{y})$.

The following lemma is similar to Lemma 5.4.9:

**Lemma 5.4.16.** — If $\overline{u} \in \mathcal{M}_{\overline{J}_-}^{n^-=0}(\mathbf{y}, \mathbf{y})$, then $\mathrm{Im}(\overline{u}) \subset W_-$.

**5.5. The Fredholm index.** — In this subsection we compute the Fredholm index $\mathrm{ind}_{W_+}$ of a $(W_+, J_+)$-curve from $\mathbf{y} = \{y_1, \dots, y_{2g}\} \in S_{a,h(a)}$ to $\mathbf{y} = \prod_j \gamma_j^{m_j} \in \mathcal{O}_{2g}$ and the Fredholm index $\mathrm{ind}_{\overline{W}_-}$ of a $(\overline{W}_-, \overline{J}_-)$-curve from $\mathbf{y}$ to $\mathbf{y}$.

The Fredholm indices are computed using the doubling technique of Hofer-Lizan-Sikorav [HLS], which we quickly review, referring to the original paper for the details.

**5.5.1. Doubling.** — Let $(F, j)$ be a compact Riemann surface with boundary, $\mathbf{p}$ and $\mathbf{q}$ be finite sets of interior and boundary punctures of $F$, and $\dot{F} = F - \mathbf{p} - \mathbf{q}$. We form the *double* $(2\dot{F}, 2j)$ of $(\dot{F}, j)$ by gluing two copies of $\dot{F}$ with opposite complex structures $j$ and $-j$ along their boundary $\partial\dot{F} = \partial F - \mathbf{q}$. By the Schwarz reflection principle, the two complex structures match and the doubled surface becomes a punctured Riemann surface.

Let $E \to \dot{F}$ be a complex vector bundle with fiberwise complex structure $i$ and let $L \to \partial\dot{F}$ be a totally real subbundle of maximal rank. Let $\overline{E} \to \dot{F}$ be a complex vector bundle whose fiber $\overline{E}_p$ at $p \in \dot{F}$ is $E_p$ with complex structure $-i$. We then construct the doubled complex vector bundle $2E \to 2\dot{F}$ by gluing $E \to \dot{F}$ and $\overline{E} \to \dot{F}$ along $\partial\dot{F}$ such that at each $p \in \partial\dot{F}$ the gluing map identifies the fibers $(E_p, i) \simeq (\overline{E}_p, -i)$ via an involution which fixes $L_p$ pointwise. Let $\sigma : 2\dot{F} \to 2\dot{F}$ be the anti-holomorphic involution $\sigma$ which switches $(\dot{F}, j)$ and $(\dot{F}, -j)$ and let $\tilde{\sigma} : 2E \to 2E$ be the anti-$\mathbf{C}$-linear bundle isomorphism which projects to $\sigma$ and identifies $E_p \simeq \overline{E}_{\sigma(p)}$ by the identity map, where $p \in (\dot{F}, j)$. Finally,
given a linear Cauchy-Riemann type operator D on E, we can define the doubled operator 2D on 2E with the property that 2D is $\tilde{\sigma}$-invariant and its restriction to E $\to \Sigma$ is D.

One of the results of [HLS] is the following:

**Theorem 5.5.1.** — Suppose that both D and 2D are Fredholm operators in some suitable Sobolev spaces. Then:

- ind(D) = $\frac{1}{2}$ ind(2D); and

- if 2D is surjective, then D is also surjective.

Our situation is slightly more general than that considered by [HLS], since we are considering boundary punctures and exponential weights. The proof, however, remains largely unmodified.

**5.5.2. The W<sub>+</sub> case.** — For the purposes of computing indices, we replace W<sub>+</sub> by $\tilde{W}_+$, as defined in Section 5.4.2. The tangent space $T\tilde{W}_+$ splits into the vertical and horizontal subbundles via the symplectic connection. In this section we slightly abuse notation and write TS (resp. TB<sub>+</sub>) for the vertical (resp. horizontal) subbundle over $\tilde{W}_+$.

We define a trivialization $\tau$ of TS along $Z_{\mathbf{y},\mathbf{y}}$ as follows: First define $\tau$ along $L_{\mathbf{a}}^{+} \cap \tilde{W}_+$ by orienting all $\mathbf{a}$-arcs arbitrarily as in Section 4.4.2 and extending the trivialization by parallel transport along the vertical boundary $\partial_{\nu}W_+$. We then extend the trivialization of TS$|_{L_{\mathbf{a}}^{+} \cap \tilde{W}_+}$ in an arbitrary manner to a trivialization of TS along $\{3\} \times [0, 1] \times \mathbf{y}$ and along $\{-1\} \times \mathbf{y}$.

Let $u: (\tilde{F}, j) \to (W_+, J_+)$ be a $W_+$-curve. Suppose the negative ends of $u$ partition $m_j$ into $(m_{j1}, m_{j2}, \dots)$. We then write:

$$ \mu_{\tau}(\gamma, u) = \sum_{j} \sum_{i} \mu_{\tau}(\gamma_{j}^{m_{ji}}), $$

where $\mu_{\tau}(\gamma_j^{m_{ji}})$ is the Conley-Zehnder index of the $m_{ji}$-fold cover of $\gamma_j$ with respect to $\tau$.

We now define a real rank one subbundle $\mathcal{L}_0$ of TS along

$$ (L_{\mathbf{a}}^{+} \cap \tilde{W}_{+}) \cup (\{3\} \times [0, 1] \times \mathbf{y}). $$

Let $\mathcal{L}_0 = \text{TL}_{\mathbf{a}}^{+} \cap \text{TS}$ on $L_{\mathbf{a}}^{+} \cap \tilde{W}_{+}$. In particular, $\mathcal{L}_0 = \text{T}\mathbf{a}(y_i)$ at $\{3\} \times \{1\} \times \{y_i\}$ and $\mathcal{L}_0 = \text{T}\mathbf{h}(\mathbf{a})(y_i)$ at $\{3\} \times \{0\} \times \{y_i\}$. We then extend $\mathcal{L}_0$ across $\{3\} \times [0, 1] \times \mathbf{y}$ by rotating in the counterclockwise direction from $\text{T}\mathbf{h}(\mathbf{a})$ to $\text{T}\mathbf{a}$ in TS by the minimum amount possible. Assuming orthogonal intersections between $\mathbf{a}$ and $\mathbf{h}(\mathbf{a})$, the angle of rotation is $\frac{\pi}{2}$.

Let $\mu_{\tau}(y_i)$ be the Maslov index of $\mathcal{L}_0$ along $\{3\} \times [0, 1] \times \{y_i\}$ with respect to $\tau$. If $u$ is a $W_+$-curve and $\mathcal{L} = \tilde{u}^*\mathcal{L}_0$, then we define $\mu_{\tau}(\mathbf{y})$ to be the Maslov index of $\mathcal{L}$ with
respect to $\tau$. By the definitions of $\mathcal{L}_0$ and $\tau$, it is immediate that

$$ \mu_{\tau}(\mathbf{y}) = \sum_{i=1}^{2g} \mu_{\tau}(y_i). $$

We then have the following Fredholm index formula for $W_+$-curves:

**Proposition 5.5.2 (Fredholm index formula for $W_+$-curves).** — The Fredholm index of a $(W_+, J_+)$-curve $u: (\dot{\mathbb{F}}, j) \to (W_+, J_+)$ from $\mathbf{y}$ to $\mathbf{y}$ is given by the formula:

$$ (5.5.1) \qquad \mathrm{ind}_{W_+}(u) = -\chi(\dot{\mathbb{F}}) - 2g + \mu_{\tau}(\mathbf{y}) - \mu_{\tau}(\mathbf{y}, u) + 2c_1(u^*\mathrm{TS}, \tau). $$

*Proof.* — We double the surface $\dot{\mathbb{F}}$ along $\partial \dot{\mathbb{F}} = \partial \mathbb{F} - \mathbf{q}$, where $\mathbf{q}$ is the set of boundary punctures, and double the pullback bundle $u^*\mathrm{TW}_+$ along the real subbundle $(u|_{\partial \dot{\mathbb{F}}})^*\mathrm{TL}_a^+$ on $\partial \dot{\mathbb{F}}$, to obtain the doubled surface $2\dot{\mathbb{F}}$ with a complex vector bundle $2u^*\mathrm{TW}_+ \to 2\dot{\mathbb{F}}$. The boundary punctures $q_i$ of $\dot{\mathbb{F}}$ are doubled to give positive interior punctures $2q_i$ on $2\dot{\mathbb{F}}$.

Since the trivialization $\tau$ of $\mathrm{TS}|_{Z_{\mathbf{y},\mathbf{y}}}$ was chosen to be tangent to $\mathrm{TL}_a^+ \cap \mathrm{TS}$, its pullback to $u^*\mathrm{TW}_+$ is compatible with the doubling operation and gives a trivialization $2\tau$ of $2u^*\mathrm{TS}$ over a neighborhood of the punctures of $2\dot{\mathbb{F}}$. Also let $\tau'$ be a partially defined trivialization of $\mathrm{TB}_+$ which is given by $\partial_i$ at the positive and negative ends of $B_+$. Then $\tau'$ can be doubled to $2\tau'$ on $2u^*\mathrm{TB}_+$ which is defined over a neighborhood of the punctures of $2\dot{\mathbb{F}}$.

The linearized $\bar{\partial}$-operator at $u$ splits as a sum $D_u = D_u^0 \oplus K_u$, where $D_u^0$ is a Cauchy-Riemann type operator on $W^{1,\rho}$-sections of $u^*\mathrm{TW}_+$ with exponential weights and $K_u$ is a direct sum of finite-dimensional operators, one for each puncture of $\dot{\mathbb{F}}$, with index 2 for interior punctures and index 1 for boundary punctures; see [Dr, Section 2.3] for the definition of the operator $K_u$. We define $2D_u = 2D_u^0 \oplus K'_u$, where $K'_u$ is a finite-dimensional operator and contributes by 2 to the index for each puncture of $2\dot{\mathbb{F}}$. Technically speaking, $K'_u$ is not the double of $K_u$; nevertheless $\mathrm{ind}(K_u) = \frac{1}{2}\mathrm{ind}(K'_u)$. By applying Theorem 5.5.1 to $D_u^0$, we obtain $\mathrm{ind}(D_u) = \frac{1}{2}\mathrm{ind}(2D_u)$.

We now apply the standard Fredholm index formula (see for example Dragnev [Dr]) for holomorphic curves in symplectizations — with slight modifications — to obtain:

$$ (5.5.2) \qquad \mathrm{ind}(2D_u) = -\chi(2\dot{\mathbb{F}}) + \mu_{2\tau}(2\mathbf{y}) - \mu_{2\tau}(2\mathbf{y}, 2u) \\ \qquad \qquad + 2c_1(2u^*\mathrm{TB}_+, 2\tau') + 2c_1(2u^*\mathrm{TS}, 2\tau). $$

Here $2\mathbf{y}$ and $2\mathbf{y}$ stand for the doubles of $\mathbf{y}$ and $\mathbf{y}$. Also $\mu_{2\tau}(2\mathbf{y})$ is the sum of the Conley-Zehnder indices, computed with respect to the trivialization $2\tau$, of the paths of symplectic matrices arising from the asymptotic operators at $2y_i$. The last term is the relative first Chern class of the double of the pullback bundle $u^*\mathrm{TS}$ with respect to $2\tau$. The one
term that is not present in the Fredholm index formula for J-holomorphic curves in a symplectization is the penultimate term $2c_1(2u^*TB_+, 2\tau')$.

We then compute the following:

(a) $\chi(2\dot{F}) = 2\chi(\dot{F}) - 2g;$

(b) $\mu_{2\tau}(2y_i) = 2\mu_{\tau}(y_i) - 1;$

(c) $\mu_{2\tau}(2\mathbf{y}, 2u) = 2\mu_{\tau}(\mathbf{y}, u);$

(d) $c_1(2u^*TB_+, 2\tau') = -2g;$

(e) $c_1(2u^*\mathrm{TS}, 2\tau) = 2c_1(u^*\mathrm{TS}, \tau).$

(a), (c) and (e) are straightforward. (b) will be proved in Lemma 5.5.3 below. (d) By arguing in a manner similar to that of Claim 4.5.10 we have $c_1(2u^*TB_+, 2\tau') = 2g\chi(2B_+) = -2g$.

Summarizing, we obtain:

$$ \begin{aligned} \mathrm{ind}_{W_+}(u) &= (-\chi(\dot{F}) + g) + (\mu_{\tau}(\mathbf{y}) - g) - \mu_{\tau}(\mathbf{y}, u) - 2g + 2c_1(u^*\mathrm{TS}, \tau) \\ &= -\chi(\dot{F}) - 2g + \mu_{\tau}(\mathbf{y}) - \mu_{\tau}(\mathbf{y}, u) + 2c_1(u^*\mathrm{TS}, \tau). \end{aligned} $$

This completes the proof of the proposition.

<figure><img src="image_7.png" /></figure>

**Lemma 5.5.3.** — $\mu_{2\tau}(2y_i) = 2\mu_{\tau}(y_i) - 1$.

*Proof.* — Let $q_i \in \partial\dot{F}$ be a (positive) boundary puncture and let $\mathcal{E} = [0, \infty) \times [0, 1] \subset \dot{F}$ be a strip-like end with coordinates $(s, t)$ which parametrizes a neighborhood of $q_i$. Suppose $u$ maps the end asymptotically to $y_i$. Fix a symplectic trivialization

$$ \Theta : u|_{\mathcal{E}}^* \mathrm{TS} \to \mathbf{R}^2 \times \mathcal{E} $$

such that $u|_{\partial\mathcal{E}}^* (\mathrm{TL}_a^+ \cap \mathrm{TS})$ corresponds to $(\mathbf{R} \oplus 0) \times \partial\mathcal{E}$. Here $\partial\mathcal{E} = \partial\dot{F} \cap \mathcal{E}$.

Let $2\mathcal{E} = ([0, \infty) \times [0, 2])/(s, 0) \sim (s, 2)$ be the end of the doubled surface $2\dot{F}$, which corresponds to the interior puncture $2q_i$ and is obtained by doubling $\mathcal{E}$. The convolution $\sigma$ is given by $\sigma(s, t) = (s, 2-t)$ with respect to these coordinates. Similarly, we have a symplectic trivialization

$$ 2\Theta : 2u|_{\mathcal{E}}^* \mathrm{TS} \to \mathbf{R}^2 \times 2\mathcal{E} $$

and $\tilde{\sigma}$ is given by $\tilde{\sigma}((x_1, x_2), s, t) = ((x_1, -x_2), s, 2-t)$.

Let $J_0\partial_t + S_i(t)$ be the asymptotic operator of $D_u^0$ corresponding to $q_i$ with respect to the trivialization $\Theta$. Here $J_0$ is the standard complex structure on $\mathbf{R}^2$ and $S_i(t)$ is a symmetric $2 \times 2$ matrix with real coefficients. For the definition of the asymptotic operator and its relation with the Fredholm theory of linearized $\bar{\partial}$-operators, see [Dr, Section 3] or [HT2].<fnref n="6" /> The solution of the Cauchy problem

$$ \dot{\Phi}(t) = J_0 S_i(t) \Phi(t), \quad \Phi(0) = \mathrm{id} $$

<footnote n="6">⁶ What we call $S_i$ here is written as $C_{2\infty}^i$ in [Dr].</footnote>
is a path of symplectic matrices which represents the linearized Reeb flow along the chord $y_i$, expressed with respect to the trivialization $\tau$. In our setting, we may assume that $\Phi(t)$ is a path of unitary matrices. By identifying $(\mathbb{R}^2, J_0) = (\mathbf{C}, i)$, we write $\Phi(t) = e^{i\alpha(t)}$ for some function $\alpha: [0, 1] \to \mathbb{R}$. Then

$$ \mu_{\tau}(y_i) = \lfloor \frac{\alpha(1) - \alpha(0)}{\pi} \rfloor + 1, $$

where $\lfloor x \rfloor$ is the greatest integer $\le x$.

The double of the asymptotic operator, i.e., the asymptotic operator of the doubled operator $2D_u^0$ at the interior puncture $2q_i$, can be written as $J_0\partial_t + \tilde{S}_i(t)$, where:

$$ \tilde{S}_i(t) = \begin{cases} S_i(t), & t \in [0, 1]; \\ CS_i(2-t)C^{-1}, & t \in [1, 2], \end{cases} $$

and $C = \text{diag}(1, -1)$. The solution of the corresponding Cauchy problem is

$$ \tilde{\Phi}(t) = \begin{cases} \Phi(t), & t \in [0, 1]; \\ C\Phi(2-t)\Phi(1)^{-1}C^{-1}\Phi(1), & t \in [1, 2]. \end{cases} $$

Hence can write $\tilde{\Phi}(t) = e^{i\tilde{\alpha}(t)}$, where $\tilde{\alpha}: [0, 2] \to \mathbb{R}$ is given by:

$$ \tilde{\alpha}(t) = \begin{cases} \alpha(t), & t \in [0, 1]; \\ -\alpha(2-t) + 2\alpha(1), & t \in [1, 2]. \end{cases} $$

The Conley-Zehnder index of the path $\tilde{\Phi}$ is

$$ \mu_{2\tau}(2y_i) = 2 \lfloor \frac{\tilde{\alpha}(2) - \tilde{\alpha}(0)}{2\pi} \rfloor + 1. $$

Since

$$ \begin{aligned} \lfloor \frac{\tilde{\alpha}(2) - \tilde{\alpha}(0)}{2\pi} \rfloor &= \lfloor \frac{(\tilde{\alpha}(2) - \tilde{\alpha}(1)) + (\tilde{\alpha}(1) - \tilde{\alpha}(0))}{2\pi} \rfloor \\ &= \lfloor \frac{2(\tilde{\alpha}(1) - \tilde{\alpha}(0))}{2\pi} \rfloor = \lfloor \frac{\alpha(1) - \alpha(0)}{\pi} \rfloor, \end{aligned} $$

we obtain $\mu_{2\tau}(2y_i) = 2\mu_{\tau}(y_i) - 1$, as desired. $\square$

<figure><img src="image_8.png" /></figure>

**5.5.3.** The $\overline{W}_-$ case. — The trivialization $\tau$ of $\overline{\text{T}\overline{S}}_{\overline{W}_-}$ is defined on $Z_{\gamma, \gamma}$ in a manner similar to that of $W_+$, except that the positive and negative ends are reversed. The definition of the real rank one subbundle $\mathcal{L}$ of $\overline{\text{T}\overline{S}}$ along $(\overline{L}_{\tilde{\mathfrak{a}}} \cap \tilde{\overline{W}}_-) \cup (\{-3\} \times [0, 1] \times \mathbf{y})$ is also similar and will be omitted.

If $\bar{u}$ is a $\overline{W}_-$-curve from $\gamma$ to $\mathbf{y}$, then the Fredholm index $\text{ind}_{\overline{W}_-}(\bar{u})$ is defined as the expected dimension of the moduli space $M_{\bar{J}_-}(\gamma, \mathbf{y})$ near $\bar{u}$.

**Remark 5.5.4.** — The expected dimension of $M_{\bar{J}_-}(\gamma, \mathbf{y}; \overline{\mathfrak{m}})$ is $\text{ind}_{\overline{W}_-}(\bar{u}) - 2$.
We then have the following Fredholm index formula:

**Proposition 5.5.5 (Fredholm index formula for $\overline{W}_-$-curves).** — The Fredholm index of a $(\overline{W}_-, \overline{J}_-)$-curve $\overline{u} : (\dot{\overline{F}}, j) \to (\overline{W}_-, \overline{J}_-)$ from $\mathbf{y}$ to $\mathbf{y}$ is given by the formula:

$$ (5.5.3) \qquad \mathrm{ind}_{\overline{W}_-}(\overline{u}) = -\chi(\dot{\overline{F}}) + \mu_{\tau}(\mathbf{y}, \overline{u}) - \mu_{\tau}(\mathbf{y}) + 2c_1(\overline{u}^*\overline{\mathrm{T}\bar{S}}, \tau). $$

*Proof.* — By a calculation similar to that of Proposition 5.5.2, we obtain

$$ \mathrm{ind}_{\overline{W}_-}(\overline{u}) = (-\chi(\dot{\overline{F}}) + g) + \mu_{\tau}(\mathbf{y}, \overline{u}) - (\mu_{\tau}(\mathbf{y}) - g) - 2g + 2c_1(\overline{u}^*\overline{\mathrm{T}\bar{S}}, \tau), $$

which simplifies to the desired result. $\square$

**Remark 5.5.6 (Reason for considering $\overline{W}_-$).** — We will give a rough explanation of the reason for considering holomorphic curves in $\overline{W}_-$ which pass through $\overline{m}$. Suppose we have a $W_+$-curve $u$ from $\mathbf{y}$ to $\mathbf{y}$ and a $W_-$-curve $v$ from $\mathbf{y}$ to $\mathbf{y}$ (i.e., $\mathrm{Im}(v) \subset W_-$). Then, by taking the sum of Equations (5.5.1) and (5.5.3), we compute the Fredholm index of the glued curve $u\#v$, corresponding to the stacking of $W_+$ at the top ($s > 0$) and $W_-$ at the bottom ($s < 0$), to be:

$$ \mathrm{ind}(u\#v) = -\chi(\dot{\overline{F}}) + 2c_1((u\#v)^*\mathrm{TS}, \tau) - 2g, $$

where $\dot{\overline{F}}$ is obtained from gluing the domains of $u$ and $v$. The stacking gives rise to a chain map $\widehat{CF} \to \widehat{CF}$, which we expect to map $\mathbf{y} \mapsto \mathbf{y}$ via restrictions of trivial cylinders (modulo chain homotopy). This would mean $\chi(\dot{\overline{F}}) = 0$ and $c_1((u\#v)^*\mathrm{TS}, \tau) = 0$. This leaves us with a deficiency of $2g$. Introducing the point constraint at $\overline{m}$, from the perspective of Fredholm indices, is basically equivalent to applying a multiple connected sum to the holomorphic curve $v$ and the fiber $\overline{S}$ which passes through $\overline{m}$. The multiple connected sum is performed at the $2g$ intersection points between $v$ and $\overline{S}$. We effectively increase the Fredholm index by $2g + 2$, obtained by adding up the following contributions:

(i) the Fredholm index of the fiber $\overline{S}$, which is $\chi(\overline{S}) = 2 - 2g$;

(ii) $2g$ intersection points, each of which contributes +2 to minus the Euler characteristic.

The point constraint then cuts the expected dimension of the moduli space by $-2$, for a net gain of $2g$.

**5.6. The ECH index.** — In this section we present the ECH index $I_{W_+}$, the relative adjunction formula, and the ECH index inequality for $W_+$. The situation for $\overline{W}_-$ is analogous, and will not be discussed explicitly, except to point out some differences.
**5.6.1. Definitions.** — Let $\mathbf{y} = \{y_1, \dots, y_{2g}\} \in S_{\mathbf{a}, \mathbf{h}(\mathbf{a})}$ and $\mathbf{y} = \prod_{j=1}^l \gamma_j^{m_j} \in \mathring{O}_{2g}$. Let $\tau$ be a trivialization of $TS_{\mathring{W}_+}$ along $Z_{\mathbf{y}, \mathbf{y}}$, as given in Section 5.5.2. Using $\tau$, for each simple orbit $\gamma_j$ of $\mathbf{y}$, we choose an identification of a sufficiently small neighborhood $\nu(\gamma_j)$ of $\gamma_j$ with $\gamma_j \times \mathbb{D}^2$, where $\mathbb{D}^2$ has polar coordinates $(r, \theta)$.

**Definition 5.6.1 (τ-trivial representative).** — A τ-trivial representative $\check{\mathbf{C}}$ of $\mathcal{A} \in \mathrm{H}_2(\mathrm{W}_+, \mathbf{y}, \mathbf{y})$ is an oriented immersed compact surface in the class $\mathcal{A}$ which satisfies the following:

(1) $\check{\mathbf{C}}$ is embedded on $\mathring{W}_+ - \{s = -1\}$;

(2) $\check{\mathbf{C}}$ is positively transverse to the fibers $\{(s, t)\} \times \bar{S}$ along all of $\partial\check{\mathbf{C}}$;

(3) $\check{\mathbf{C}}$ is τ-trivial in the sense of Definition 4.5.2 at the HF end;

(4) $\check{\mathbf{C}}$ is τ-trivial at the ECH end, i.e., for all sufficiently small $\varepsilon > 0$, $\check{\mathbf{C}} \cap \{s = -1 + \varepsilon\}$ consists of $m_j$ disjoint circles $\{r = \varepsilon, \theta = \theta_{j\varepsilon}\}$, $i = 1, \dots, m_j$, in $\nu(\gamma_j)$ for all $j$. (See [Hu1, Definition 2.3].)

Let $\partial_+\check{\mathbf{C}} = \partial\check{\mathbf{C}} \cap \{s > 0\}$ and $\partial_-\check{\mathbf{C}} = \partial\check{\mathbf{C}} \cap \{s < 0\}$.

**Definition 5.6.2 (Relative intersection form).** — Let $\mathcal{A} \in \mathrm{H}_2(\mathrm{W}_+, \mathbf{y}, \mathbf{y})$ be a homology class which is realized by a τ-trivial representative $\check{\mathbf{C}}$. Then the relative intersection form $\mathcal{Q}_{\tau}(\mathcal{A})$ is given by $\langle \check{\mathbf{C}}, \check{\mathbf{C}}' \rangle$, where $\check{\mathbf{C}}'$ is a pushoff of $\check{\mathbf{C}}$ which satisfies the following:

(1) $\check{\mathbf{C}}$ is pushed off in the $\mathcal{J}_+\tau$-direction along $\partial_+\check{\mathbf{C}}$; and

(2) for small $\varepsilon > 0$, $\check{\mathbf{C}}' \cap \{s = -1 + \varepsilon\}$ consists of $m_j$ disjoint circles $\{r = \varepsilon, \theta = \theta_{j\varepsilon} + \varepsilon'\}$, $i = 1, \dots, m_j$, in $\nu(\gamma_j)$ for all $j$, where $\varepsilon' > 0$ is a sufficiently small constant.

(See [Hu1, Definition 2.4].)

**5.6.2. Relative adjunction formula.** — Let $u : \dot{\mathcal{F}} \to \mathrm{W}_+$ be a $\mathrm{W}_+$-curve and let $\tilde{u} : \check{\mathcal{F}} \to \mathring{W}_+$ be its compactification. Then we write $w_\tau^-(u)$ for the total writhe of the braids $u(\dot{\mathcal{F}}) \cap \{s = s_0\}$, $s_0 \ll 0$, with respect to $\tau$. Similarly, if $\bar{u} : \dot{\mathcal{F}} \to \bar{\mathrm{W}}_-$ is a $\bar{\mathrm{W}}$-curve, then we write $w_\tau^+(\bar{u})$ for the total writhe of the braids $\bar{u}(\dot{\mathcal{F}}) \cap \{s = s_0\}$, $s_0 \gg 0$, with respect to $\tau$.

**Lemma 5.6.3 (Relative adjunction formula).** — Let $u : \dot{\mathcal{F}} \to \mathrm{W}_+$ be a $\mathrm{W}_+$-curve in the homology class $\mathcal{A} \in \mathrm{H}_2(\mathrm{W}_+, \mathbf{y}, \mathbf{y})$. Then

$$ (5.6.1) \qquad c_1(u^*\mathrm{TW}_+, (\tau, \partial_t)) = \chi(\dot{\mathcal{F}}) - w_\tau^-(u) + \mathcal{Q}_\tau(\mathcal{A}) - 2\delta(u), $$

where $\partial_t$ trivializes $\mathrm{TB}_+$.

For a $\bar{\mathrm{W}}$-curve $\bar{u}$, we replace $-w_\tau^-(u)$ by $w_\tau^+(\bar{u})$.

*Proof.* — Suppose $u$ is immersed. If $\nu$ is the normal bundle of $u$, then we have the formula:

$$ (5.6.2) \qquad c_1(u^*\mathrm{TW}_+, (\tau, \partial_t)) = c_1(\mathrm{T}\check{\mathcal{F}}, \partial_t) + c_1(\nu, \tau). $$
In the general case, we combine the calculations of Lemma 4.5.8 and [Hu1, Proposition 3.1] to obtain the equation

$$c_1(u^*\mathrm{TW}_+, (\tau, \partial_t)) = c_1(\mathrm{T}\check{\mathrm{F}}, \partial_t) - w_\tau^-(u) + \mathrm{Q}_\tau(\mathrm{A}) - 2\delta(u).$$

The analog of Claim 4.5.10 for the present situation is

$$\textbf{(5.6.3)} \qquad c_1(\mathrm{T}\check{\mathrm{F}}, \partial_t) = \chi(\mathrm{\check{F}}).$$

The difference in the shape of the base (i.e., B vs. $B_+$) accounts for the discrepancy between Equation (5.6.3) and Claim 4.5.10. $\square$

*Remark 5.6.4.* — Since $c_1(u^*\mathrm{TB}_+, \partial_t) = 0$, we have

$$c_1(u^*\mathrm{TW}_+, (\tau, \partial_t)) = c_1(u^*\mathrm{TS}, \tau) + c_1(u^*\mathrm{TB}_+, \partial_t) = c_1(u^*\mathrm{TS}, \tau).$$

**5.6.3. ECH index.** — We now define the ECH indices for $W_+$ and $\overline{W}_-$.

*Definition 5.6.5 (ECH index for $W_+$).* — Given a class $A \in \mathrm{H}_2(W_+, \mathbf{y}, \mathbf{y})$ which admits a $\tau$-trivial representative $\check{\mathcal{C}}$, we define

$$\textbf{(5.6.4)} \qquad \mathrm{I}_{W_+}(\mathrm{A}) = c_1(\mathrm{T}\check{\mathrm{W}}_+|_A, (\tau, \partial_t)) + \mathrm{Q}_\tau(\mathrm{A}) + \mu_\tau(\mathbf{y}) - \tilde{\mu}_\tau(\mathbf{y}) - 2g,$$

where $\tilde{\mu}_\tau(\mathbf{y})$ is the symmetric Conley-Zehnder index at the negative (ECH) end.

*Definition 5.6.6 (ECH index for $\overline{W}_-$).* — Given a class $A \in \mathrm{H}_2(\overline{\mathrm{W}}_-, \mathbf{y}, \mathbf{y})$ which admits a $\tau$-trivial representative $\check{\mathcal{C}}$, we define

$$\textbf{(5.6.5)} \qquad \mathrm{I}_{\overline{W}_-}(\mathrm{A}) = c_1(\mathrm{T}\check{\mathrm{W}}_-|_A, (\tau, \partial_t)) + \mathrm{Q}_\tau(\mathrm{A}) + \tilde{\mu}_\tau(\mathbf{y}) - \mu_\tau(\mathbf{y}).$$

As usual, the ECH indices $\mathrm{I}_{W_+}(\mathrm{A})$ and $\mathrm{I}_{\overline{W}_-}(\mathrm{A})$ are independent of the choice of trivialization $\tau$.

*Remark 5.6.7.* — To obtain a finite count of $\overline{W}_-$-curves which pass through the point $\overline{\mathfrak{m}}$, we count curves $\overline{u}$ with ECH index $\mathrm{I}_{\overline{W}_-}(\overline{u}) = 2$.

**5.6.4. Additivity of indices.**

*Lemma 5.6.8 (Additivity of indices).* — If $u \in \mathcal{M}_J(\mathbf{y}, \mathbf{y}')$, $v \in \mathcal{M}_{J_+(\mathbf{y}', \mathbf{y})}$, and $u\#v$ is a pre-glued curve, then

$$\textbf{(5.6.6)} \qquad \mathrm{ind}_{W_+}(u\#v) = \mathrm{ind}_{\mathrm{HF}}(u) + \mathrm{ind}_{W_+}(v),$$

$$\textbf{(5.6.7)} \qquad \mathrm{I}_{W_+}(u\#v) = \mathrm{I}_{\mathrm{HF}}(u) + \mathrm{I}_{W_+}(v).$$
Similarly, if $u \in \mathcal{M}_{J_+}(\mathbf{y}, \mathbf{\gamma})$, $v \in \mathcal{M}_{J_+}(\mathbf{\gamma}, \mathbf{\gamma}')$, and $u\#v$ is a pre-glued curve, then

$$ (5.6.8) \qquad \mathrm{ind}_{W_+}(u\#v) = \mathrm{ind}_{W_+}(u) + \mathrm{ind}_{\mathrm{ECH}}(v), $$

$$ (5.6.9) \qquad \mathrm{I}_{W_+}(u\#v) = \mathrm{I}_{W_+}(u) + \mathrm{I}_{\mathrm{ECH}}(v). $$

*Proof.* — The additivity for ind is well-known and the additivity for I is immediate from the definitions. □

**5.6.5. Index inequality.** — Although we will not define it here, given an integer $m_k > 0$ and a simple orbit $\gamma_k$, we can define the incoming partition $\mathcal{P}_{\gamma_k}^{\mathrm{in}}(m_k)$ and the outgoing partition $\mathcal{P}_{\gamma_k}^{\mathrm{out}}(m_k)$ as in [Hu2, Definition 4.14].

We have the following index inequality, which is analogous to [Hu1, Theorem 1.7] (also see [Hu2, Theorem 4.15] which is applicable to symplectic cobordisms). Note that a $W_+$-curve is automatically simply-covered.

**Theorem 5.6.9 (Index inequality).** — Let $u$ be a $W_+$-curve from $\mathbf{y}$ to $\mathbf{\gamma} = \prod_{k=1}^l \gamma_k^{m_k}$. If the negative ends of $u$ partition $m_k$ into $(m_{k1}, m_{k2}, \dots)$, then

$$ \mathrm{ind}_{W_+}(u) + 2\delta(u) \le \mathrm{I}_{W_+}(u). $$

Moreover, if equality holds, then $\mathcal{P}_{\gamma_k}^{\mathrm{in}}(m_k) = (m_{k1}, m_{k2}, \dots)$ for all $k$. Similarly, if $\bar{u}$ is a $\bar{W}_-$-curve from $\mathbf{\gamma} = \prod_{k=1}^l \gamma_k^{m_k}$ to $\mathbf{y}$ and the positive ends of $u$ partition $m_k$ into $(m_{k1}, m_{k2}, \dots)$, then

$$ \mathrm{ind}_{W_-}(\bar{u}) + 2\delta(\bar{u}) \le \mathrm{I}_{W_-}(\bar{u}). $$

Moreover, if equality holds, then $\mathcal{P}_{\gamma_k}^{\mathrm{out}}(m_k) = (m_{k1}, m_{k2}, \dots)$ for all $k$.

Here $\delta(u)$ is the signed count of interior singularities of $u$, where each singularity contributes positively to $\delta(u)$. There are no boundary singularities because $\pi_{B_+} \circ u$ has no branch point at the boundary.

The following are immediate: (i) if $\mathrm{ind}_{W_+}(u) = \mathrm{I}_{W_+}(u)$, then $u$ is embedded; and (ii) if $\mathrm{I}_{W_+}(u) = 0$ or 1, then $u$ is embedded.

*Proof.* — If we plug Equation (5.5.1), Equation (5.6.4), and the relative adjunction formula (5.6.1) into $\mathrm{I}_{W_+}(u) - \mathrm{ind}_{W_+}(u)$, we obtain:

$$ \mathrm{I}_{W_+}(u) - \mathrm{ind}_{W_+}(u) = \mu_{\tau}(\mathbf{\gamma}, u) + w_{\tau}^{-}(u) - \tilde{\mu}_{\tau}(\mathbf{\gamma}) + 2\delta(u). $$

The statement for $W_+$ then follows from the writhe inequality

$$ w_{\tau}^{-}(u) \ge \tilde{\mu}_{\tau}(\mathbf{\gamma}) - \mu_{\tau}(\mathbf{\gamma}, u), $$

where equality holds if and only if the partition $(m_{k1}, m_{k2}, \dots)$ of the negative end of $u$ coincides with the incoming partition $\mathcal{P}_{\gamma_k}^{\mathrm{in}}(m_k)$. (See [Hu2, Lemma 4.20].) The proof for $\bar{W}_-$ is similar. □
**5.7.** Holomorphic curves with ends at $z_\infty$. — In this subsection we explain how to extend the definitions of the Fredholm and ECH indices to holomorphic curves which have ends at multiples of $z_\infty$. The novelty is that the Lagrangian boundary condition is singular at $z_\infty$ and that the chord over $z_\infty$ can be used more than once. We will treat in detail the case of a curve $\bar{u}: \dot{\bar{F}} \to \bar{W}$ which is a multisection of $\bar{W} \to \mathbf{R} \times [0, 1]$; multisections of $\bar{W}_+$ and $\bar{W}_-$ can be treated similarly.

**5.7.1.** Data at $z_\infty$. — We define the data $\overrightarrow{D}$ at $z_\infty$ as a $p$-tuple of matchings

$$ \{(i'_1, j'_1) \to (i_1, j_1), \dots, (i'_p, j'_p) \to (i_p, j_p)\}, $$

where $i_\ell, i'_\ell \in \{1, \dots, 2g\}, j_\ell, j'_\ell \in \{0, 1\}$ for $\ell = 1, \dots, p$ and $i_\ell \neq i_{\ell'}, i'_\ell \neq i_{\ell'}$ for $\ell \neq \ell'$. To the data $\overrightarrow{D}$ we associate its set of initial points $\mathcal{D}^{\text{from}} = \{(i'_1, j'_1), \dots, (i'_p, j'_p)\}$ and its set of terminal points $\mathcal{D}^{\text{to}} = \{(i_1, j_1), \dots, (i_p, j_p)\}$, and define $\mathcal{D} = (\mathcal{D}^{\text{to}}, \mathcal{D}^{\text{from}})$.

We write $\mathbf{z} = \{z_\infty^p(\overrightarrow{D})\} \cup \mathbf{y}$ for a tuple of points of $\bar{a} \cap \bar{h}(\bar{a})$, where $\mathbf{y} \subset S$, $z_\infty$ has multiplicity $p$, $\overrightarrow{D}$ is the data at $z_\infty^p$, and each arc of $\{\bar{a}_i, \bar{h}(\bar{a}_i)\}_{i=1}^{2g}$ is used at most once. Here $z_\infty^p(\overrightarrow{D})$ is viewed as a collection of chords $z_\infty$ from $\bar{h}(\bar{a}_{i'_\ell j'_\ell})$ to $\bar{a}_{i_\ell j_\ell}$, where $\ell = 1, \dots, p$.

**5.7.2.** Multisections. — In this subsection we define multisections $\bar{u}: (\dot{\bar{F}}, j) \to (\bar{W}, \bar{J})$ with irreducible components which branched cover $\sigma_\infty$. The notation is complicated by the fact that there may be branch points of $\bar{\pi}_B \circ \bar{u}$ along $\partial B$.

For the rest of Section 5.7, $(\bar{F}, j)$ will be a compact nodal Riemann surface, possibly disconnected, with two sets of punctures $\mathbf{q}^+ = \{q_1^+, \dots, q_\ell^+\}$ and $\mathbf{q}^- = \{q_1^-, \dots, q_\ell^-\}$ on $\partial \bar{F}$ and a set $\mathbf{r} = \{r_1, \dots, r_l\}$ of nodes on $\partial \bar{F}$, such that

(i) the nodes $\mathbf{r}$ are disjoint from $\mathbf{q}^+$ and $\mathbf{q}^-$;

(ii) each connected component of $\bar{F}$ has nonempty boundary;

(iii) on each oriented loop of $\partial \bar{F}$ whose orientation agrees with that of $\partial \bar{F}$, there is at least one puncture from each of $\mathbf{q}^+$ and $\mathbf{q}^-$; and

(iv) the punctures on $\mathbf{q}^+$ and $\mathbf{q}^-$ alternate along each oriented loop of $\partial \bar{F}$.

Here, by a *loop or path on* $\partial \bar{F}$ we mean a loop or path on $\partial \bar{F}$ which is a concatenation of subarcs of $\partial \bar{F}$ with endpoints on $\mathbf{q}^+ \cup \mathbf{q}^- \cup \mathbf{r}$. An oriented loop (or path) is a loop (or path) formed from subarcs of $\partial \bar{F}$ which all point in the same direction, when oriented as the boundary of $\bar{F}$.

We write $\dot{\bar{F}} = \bar{F} - \mathbf{q}^+ - \mathbf{q}^-$. Given a holomorphic multisection $\bar{u}: (\dot{\bar{F}}, j) \to (\bar{W}, \bar{J})$, we decompose $\bar{F} = \bar{F}' \cup \bar{F}''$ such that

- $\bar{u}' = \bar{u}|_{\dot{\bar{F}}}$ is a possibly disconnected branched cover of $\sigma_\infty$ and

- $\bar{u}'' = \bar{u}|_{\dot{\bar{F}}''}$ is the union of irreducible components which do not branch cover $\sigma_\infty$.
By abuse of notation, we write

$$ (5.7.1) \qquad \bar{u} = \bar{u}' \cup \bar{u}'' . $$

We additionally assume the following:

(v) all the nodes of $\partial F$ lie on $\partial F'$.

Observe that the nodes of $\partial F'$ correspond to branch points of $\pi_B \circ \bar{u}$ along $\partial B$. If $\dot{F}'$ is a Riemann surface that branched covers B with some branch points along $\partial B$, then let $\dot{F}'_{\text{ext}}$ be the Riemann surface that branched covers B, such that the branch points on $\partial B$ have been pushed into $\text{int}(B)$. The cases of $\bar{W}_+$ and $\bar{W}_-$ are analogous.

The following is the local model near a branch point along the boundary: Consider the map $\mathbf{C} \to \mathbf{C}$, $z \mapsto z^n$. If we write $z = re^{i\theta}$, then the preimage of the upper half plane is a union of sectors $\theta \in [\frac{2k\pi}{n}, \frac{(2k+1)\pi}{n}]$ for $k = 0, \dots, n-1$. The local model of the modification from $F'$ to $\dot{F}'_{\text{ext}}$ is obtained by going from $z \mapsto z^n$ to $z \mapsto z^n + i\epsilon$ for $\epsilon > 0$ small, and taking the preimage of the upper half plane.

**Definition 5.7.1 (Data C).** — A maximal path $d$ of $\partial \dot{F}'$ is an oriented connected path from $\mathbf{q}^+$ to $\mathbf{q}^-$ along $\partial \dot{F}'$, where the orientation either agrees with that of $\partial F$ on all the subarcs of $d$ or is opposite that of $\partial F$ on all the subarcs of $d$. A set $\{d_1, \dots, d_r\}$ is a decomposition of $\partial \dot{F}'$ into maximal paths, if $\partial \dot{F}' = \cup_{i=1}^r d_i$, $d_i$ is a maximal path of $\partial \dot{F}'$, and $d_i, d_j, i \neq j$, intersect only at points of $\mathbf{q}^+ \cup \mathbf{q}^- \cup \mathbf{r}$. The canonical decomposition $\Delta = \{d_1, \dots, d_r\}$ of $\partial \dot{F}'$ is the decomposition which corresponds to $\partial \dot{F}'_{\text{ext}}$. A data C is a map

$$ \Delta \to \{\bar{a}_{i,j}, \bar{h}(\bar{a}_{i,j})\}_{i,j}. $$

In words, $\bar{u}'$ can be viewed as mapping $\Delta$ to the set of Lagrangians $L_{\bar{a}_{i,j}}$ or $L_{\bar{h}(\bar{a}_{i,j})}$. We then make the following definition which agrees with Definition 4.3.1 when $\mathbf{z}_+ = \mathbf{y}_+$ and $\mathbf{z}_- = \mathbf{y}_-$:

**Definition 5.7.2.** — Let $\mathbf{z}_+ = \{z_\infty^{p_+}(\vec{D}_+)\} \cup \mathbf{y}_+$ and $\mathbf{z}_- = \{z_\infty^{p_-}(\vec{D}_-)\} \cup \mathbf{y}_-$ be $k$-tuples of points of $\bar{a} \cap \bar{h}(\bar{a})$, where $z_\infty$ is counted with multiplicities $p_+$, $p_-$. A degree $k$ multisection of $(\bar{W}, \bar{J})$ from $\mathbf{z}_+$ to $\mathbf{z}_-$ is a pair $(\bar{u}, C)$ consisting of a holomorphic map

$$ \bar{u} : (\dot{F}, j) \to (\bar{W}, \bar{J}), $$

where $\bar{u}$ is a degree $k$ multisection of $\pi_B : \bar{W} \to B$ and there are decompositions $\dot{F} = \dot{F}' \cup \dot{F}''$, $\bar{u} = \bar{u}' \cup \bar{u}''$ as above, and the data C for $\bar{u}'$, and which additionally satisfies the following:

(1) $\bar{u}''(\partial \dot{F}'') \subset L_{\hat{a}} \cup L_{\bar{h}(\hat{a})}$;

(2) there is a canonical decomposition $\Delta$ of $\partial \dot{F}'$ such that $\bar{u}$ maps each connected component of $\partial \dot{F}''$ and each irreducible component of $\Delta$ to a different $L_{\bar{a}_i}$ or $L_{\bar{h}(\bar{a}_i)}$ (here we are using C to assign some $L_{\bar{a}_i}$ or $L_{\bar{h}(\bar{a}_i)}$ to each irreducible component of $\Delta$);
(3) $\lim_{w \to q_i^+} \bar{\pi}_R \circ \bar{u}(w) = +\infty$ and $\lim_{w \to q_i} \bar{\pi}_R \circ \bar{u}(w) = -\infty$;

(4) $\bar{u}$ converges to a trivial strip over $[0, 1] \times \mathbf{z}_+$ near $\mathbf{q}^+$ and to a trivial strip over $[0, 1] \times \mathbf{z}_-$ near $\mathbf{q}^-$;

(5) the positive and negative ends of $\bar{u}$ which limit to $z_\infty$ are described by $\vec{D}_+$ and $\vec{D}_-$;

(6) the energy of $\bar{u}$ is finite.

**Remark 5.7.3.** — It will always be assumed that a multisection $\bar{u}$ of $\overline{W}$ from $\{z_\infty^+\} \cup \mathbf{y}_+$ to $\{z_\infty^-\} \cup \mathbf{y}_-$ comes with data $\mathcal{C}$. Indeed, in this paper, such a curve $\bar{u}$ only appears as the SFT limit of curves $\bar{u}_i$ without components which branch cover $\sigma_\infty$ and hence naturally inherits $\mathcal{C}$ and $\vec{D}_\pm$.

**5.7.3. Multivalued trivializations.** — We define a class of trivializations $\tau = \tau_{\vec{D}_+, \vec{D}_-}$ which will be used in the definition of the Fredholm index of a map $\bar{u}$ from $\{z_\infty^+(\vec{D}_+)\} \cup \mathbf{y}_+$ to $\{z_\infty^-(\vec{D}_-)\} \cup \mathbf{y}_-$. The trivialization $\tau = \tau_{\vec{D}_+, \vec{D}_-}$ is multivalued near $z_\infty$ and depends on $\vec{D}_+$ and $\vec{D}_-$.

Let $\tau' = \partial_\rho$ be a trivialization of $\overline{TS}$ on $D_\delta^2 - \{z_\infty\} = \{0 < \rho \le \delta\} \subset \overline{S}$, where $\delta > 0$ is small. Let $\tilde{W} = [-1, 1] \times [0, 1] \times \overline{S}$. We extend $\tau'$ arbitrarily to a trivialization of $\overline{TS}$ along $\bar{a}$ and pull it back to $\tau_{\vec{D}_+, \vec{D}_-}$ on $\overline{TS}_{\tilde{W}}$ along $[-1, 1] \times \{1\} \times \bar{a}$; similarly we extend $\tau'$ arbitrarily to a trivialization of $\overline{TS}$ along $\bar{h}(\bar{a})$ and pull it back to $\tau_{\vec{D}_+, \vec{D}_-}$ on $\overline{TS}_{\tilde{W}}$ along $[-1, 1] \times \{0\} \times \bar{h}(\bar{a})$. Then we extend $\tau_{\vec{D}_+, \vec{D}_-}$ along $(\{1\} \times [0, 1] \times \mathbf{y}_+) \cup (\{-1\} \times [0, 1] \times \mathbf{y}_-)$, while keeping the same notation. Finally, for each $(i'_{\pm, \ell}, j'_{\pm, \ell}) \to (i_{\pm, \ell}, j_{\pm, \ell})$ in $\vec{D}_\pm$, we choose an extension $\tau_{\vec{D}_+, \vec{D}_-}$ to $\{\pm 1\} \times [0, 1] \times \{z_\infty\}$ in an arbitrary manner.

A multivalued trivialization $\tau = \tau_{\vec{D}_+, \vec{D}_-}$ as constructed in the previous paragraph is said to be compatible with $(\vec{D}_+, \vec{D}_-)$. We also say that $\tau$ is compatible with $(D_+, D_-)$ if it is compatible with some $(\vec{D}_+, \vec{D}_-)$ whose initial/terminal point sets are $(D_+, D_-)$.

**Remark 5.7.4.** — Note that the extensions to $\bar{a}$ and to $\bar{h}(\bar{a})$ might conflict, but it does not matter here. In the cobordism cases (i.e., for $\overline{W}_+$ and $\overline{W}_-$), when $\bar{a}$ and $\bar{h}(\bar{a})$ are connected by the Lagrangian $L_\pm$, we extend the trivialization $\tau'$ to $\bar{a}$ in an arbitrary manner and then sweep it around using the symplectic connection along the boundary of the cobordism.

**5.7.4. Groomed multivalued trivializations.** — Let $\tau$ be a multivalued trivialization. Let

$$A_\epsilon = [0, 1] \times \partial D_\epsilon^2 \subset [0, 1] \times \overline{S},$$
where $\varepsilon > 0$ is small. We use coordinates $(t, \phi)$ on

$$A_{\varepsilon} \simeq ([0, 1] \times [0, 2\pi])/(t, 0) \sim (t, 2\pi).$$

The branches of the trivialization $\tau$ along $[0, 1] \times \{z_{\infty}\}$ can be used to push off $[0, 1] \times \{z_{\infty}\}$ into a collection of arcs $c_{\ell}^{\pm}, \ell = 1, \dots, p_{\pm}$, in $A_{\varepsilon}$, subject to some conditions: If we denote the initial point of $c_{\ell}^{\pm}$ by $p_{\ell,0}^{\pm}$ and the terminal point by $p_{\ell,1}^{\pm}$, then $p_{\ell,0}^{\pm}$ lies on $\{0\} \times \bar{h}(\bar{a}_{\ell, \ell', j, j', \ell'})$ and $p_{\ell,1}^{\pm}$ lies on $\{1\} \times \bar{a}_{i_{\pm}, \ell, j_{\pm}, \ell}$. We also assume that each $c_{\ell}^{\pm}$ is linear with respect to the identification of the universal cover of $A_{\varepsilon}$ with $[0, 1] \times \mathbf{R}$.

We then write

$$P_0^{\pm} = \{p_{\ell,0}^{\pm}\}_{\ell=1}^{p_{\pm}} \quad \text{and} \quad P_1^{\pm} = \{p_{\ell,1}^{\pm}\}_{\ell=1}^{p_{\pm}}.$$

**Definition 5.7.5.** — The multivalued trivialization $\tau$ is groomed if the arcs $\{c_{\ell}^{\pm}\}_{\ell=1}^{p_{\pm}}$ are pairwise disjoint for both $*=+$ and $*=-$.
The collections $\mathbf{c}^{+} = \{c_{\ell}^{+}\}_{\ell=1}^{p_{+}}$ and $\mathbf{c}^{-} = \{c_{\ell}^{-}\}_{\ell=1}^{p_{-}}$ are the groomings at the positive and negative ends.

Note that every groomed multivalued trivialization induces data $\vec{D}_{\pm}$, but not every $\vec{D}_{\pm}$ is compatible with a groomed multivalued trivialization.

**5.7.5. Index formulas.** — We first give the Fredholm index of a multisection.

**Proposition 5.7.6.** — Let $\bar{u}: \dot{\mathbb{F}} \to \bar{W}$ be a degree $k$ multisection with data $\vec{D}_{\pm}$ at $z_{\infty}$, and let $\tau = \tau_{\vec{D}_{+}, \vec{D}_{-}}$. Then the Fredholm index of $\bar{u}$ is given by the formula:

$$(5.7.2) \qquad \mathrm{ind}(\bar{u}) = -\chi(\mathcal{F}) + k + \mu_{\tau}(\bar{u}) + 2c_1(\bar{u}^*\mathrm{T}\bar{S}, \tau).$$

The proof is a computation in the pullback bundle $\bar{u}^*\mathrm{T}\bar{W}$, which is completely analogous to that yielding Equation (4.4.4).

Next we discuss the ECH index of a multisection $(\bar{u}, \mathcal{C})$ with ends

$$\mathbf{z}_{+} = \{z_{\infty}^{p_{+}}(\vec{D}_{+})\} \cup \mathbf{y}_{+}, \quad \mathbf{z}_{-} = \{z_{\infty}^{p_{-}}(\vec{D}_{-})\} \cup \mathbf{y}_{-}.$$

Let $Z_{\mathbf{z}_{+}, \mathbf{z}_{-}}$ be the subset of $\tilde{\mathbb{W}}$ given by

$$Z_{\mathbf{z}_{+}, \mathbf{z}_{-}} = \check{\mathcal{L}}_{\hat{\mathbf{a}}} \cup \check{\mathcal{L}}_{\bar{h}(\hat{\mathbf{a}})} \cup (\{1\} \times [0, 1] \times \mathbf{z}_{+}) \cup (\{-1\} \times [0, 1] \times \mathbf{z}_{-}),$$

where we are viewing $\mathbf{z}_{\pm}$ as sets of points (i.e., we are forgetting the data $\vec{D}_{\pm}$),

$$\check{\mathcal{L}}_{\hat{\mathbf{a}}} = [-1, 1] \times \{1\} \times \hat{\mathbf{a}}, \quad \text{and} \quad \check{\mathcal{L}}_{\bar{h}(\hat{\mathbf{a}})} = [-1, 1] \times \{0\} \times \bar{h}(\hat{\mathbf{a}}).$$

We then define $H_2(\bar{W}, \mathbf{z}_{+}, \mathbf{z}_{-}) \subset H_2(\check{\bar{W}}, Z_{\mathbf{z}_{+}, \mathbf{z}_{-}})$ in a manner analogous to that of $H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ in Section 4.3. Observe that $H_2(\bar{W}, \mathbf{z}_{+}, \mathbf{z}_{-})$ and $Z_{\mathbf{z}_{+}, \mathbf{z}_{-}}$ depend on the end-points $\mathcal{D}_{\pm}$ but not on the matchings in $\vec{D}_{\pm}$.
τ-trivial representative. Let $\tau$ be a groomed multivalued trivialization which is compatible with $(\mathcal{D}_+, \mathcal{D}_-)$ and let $c^\pm = \{c_\ell^\pm\}$ be the groomings of $\tau$. The matchings defined by $c^\pm$ do not need to coincide with the matchings in $\tilde{\mathcal{D}}_\pm$; only their endpoints do. The construction of a $\tau$-trivial representative $\check{\mathcal{C}} \subset \check{\mathcal{W}}$ of $[\bar{u}] \in H_2(\bar{\mathcal{W}}, \mathbf{z}_+, \mathbf{z}_-)$ is as follows:

Step 1. Replace $\bar{u}' : \check{\mathcal{F}}' \to \check{\mathcal{W}}$ by $\bar{u}'_{\text{ext}} : \check{\mathcal{F}}'_{\text{ext}} \to \check{\mathcal{W}}$ so that there are no nodes along $\partial \check{\mathcal{F}}_{\text{ext}}$ and each component of $\partial \check{\mathcal{F}}_{\text{ext}}$ is mapped to some $L_{\bar{a}_{i,j}}$ or $L_{\bar{h}(\bar{a}_{i,j})}$ in a manner consistent with the data $\mathcal{C} : \Delta \to \{\bar{a}_{i,j}, \bar{h}(\bar{a}_{i,j})\}_{i,j}$.

Step 2. Compactify the ends of $\bar{u}_{\text{ext}} = \bar{u}'_{\text{ext}} \cup \bar{u}''$ to $\check{u}_{\text{ext}}$ as in Section 4.3.

Step 3. Perturb $\check{u}_{\text{ext}}$ so that the resulting representative $\check{\mathcal{C}}$ is immersed, $\partial \check{\mathcal{C}} \subset Z_{\mathbf{z}_+, \mathbf{z}_-, \tau}$, each component of $\partial \check{\mathcal{C}} \cap \{t=0, 1\}$ is mapped to $\check{L}_{\bar{a}_{i,j}}$ or $\check{L}_{\bar{h}(\bar{a}_{i,j})}$ as specified by $\mathcal{C}$, and $\check{\mathcal{C}}$ satisfies

$$ \pi(\check{\mathcal{C}}|_{s=\pm 1}) \cap ([0, 1] \times \text{int}(\mathcal{D}^2)) = \mathbf{c}^{\pm} \subset \mathcal{A}_{\epsilon} $$

and the conditions in Definition 4.5.2 at all the other ends. Here

$$ \pi : [-1, 1] \times [0, 1] \times \bar{S} \to [0, 1] \times \bar{S} $$

is the projection onto the second and third factors. Then resolve the self-intersections to make $\check{\mathcal{C}}$ embedded.

The quadratic form $Q_\tau$. Let $\check{\mathcal{C}}$ be a $\tau$-trivial representative of $\bar{u}$, where $\tau$ is viewed as a nonsingular vector field. Let $c_\ell^\pm(\delta)$ be the $\phi = \delta$ translate of $c_\ell^\pm$, where $\delta > 0$ is small. We then take a pushoff $\check{\mathcal{C}}'$ of $\check{\mathcal{C}}$ such that $\partial \check{\mathcal{C}}$ is pushed in the direction of $J\tau + Y$, where Y is $C^0$-small, and

$$ \pi(\check{\mathcal{C}}'|_{s=\pm 1}) \cap ([0, 1] \times \text{int}(\mathcal{D}^2)) = \bigcup_{\ell=1}^{\rho} c_\ell^\pm(\delta) \subset \mathcal{A}_\epsilon. $$

Then $Q_\tau([u]) = \langle \check{\mathcal{C}}, \check{\mathcal{C}}' \rangle$ as in Definition 4.5.5.

**Definition 5.7.7 (ECH index).** — Given $A \in H_2(\bar{\mathcal{W}}, \mathbf{z}_+, \mathbf{z}_-)$ and a groomed multivalued trivialization $\tau$ compatible with $(\mathcal{D}_+, \mathcal{D}_-)$, we define

$$ (5.7.3) \qquad I_\tau(A) = Q_\tau(A) + \tilde{\mu}_\tau(\partial A) + c_1(\bar{T}\bar{S}|_A, \tau), $$

where the analog $\tilde{\mu}_\tau(\partial A)$ of the symmetric Conley-Zehnder index will be given in Section 5.7.6.

Sometimes we will write $I_\tau(u)$ to mean $I_\tau(A)$, where A is the relative homology class defined by u.

**5.7.6. Definition of $\tilde{\mu}_\tau(\partial A)$.** — We first define the Conley-Zehnder index $\mu_\tau(\partial A)$. The groomed multivalued trivialization $\tau$ determines the matchings $\mathcal{D}_\pm^{\text{from}} \to \mathcal{D}_\pm^{\text{to}}$. Pick a cycle $\zeta$ which represents $\partial A$ and respects the matchings along $\{\pm 1\} \times [0, 1] \times \{z_\infty\}$. The pullback bundle $\zeta^*\bar{T}\bar{S}$ is trivialized by the pullback of $\tau$. We now define a multivalued
real rank one subbundle $\mathcal{L}_0$ of $\overline{T\bar{S}}$ along $Z_{\overline{\mathfrak{a}}, \overline{h}(\overline{\mathfrak{a}})}$ by setting $\mathcal{L}_0 = \overline{T\mathcal{L}}_{\overline{\mathfrak{a}}} \cap \overline{T\bar{S}}$ on $\mathcal{L}_{\overline{\mathfrak{a}}}$ and $\overline{T\mathcal{L}}_{\overline{\mathcal{h}}(\overline{\mathfrak{a}})} \cap \overline{T\bar{S}}$ on $\mathcal{L}_{\overline{\mathcal{h}}(\overline{\mathfrak{a}})}$ and extending $\mathcal{L}_0$ across $\{\pm 1\} \times [0, 1] \times \mathbf{z}_{\pm}$ by rotating in the counterclockwise direction from $\overline{T\mathcal{h}}(\overline{\mathfrak{a}})$ to $\overline{T\mathfrak{a}}$ in $\overline{T\bar{S}}$ by the minimum amount possible. We then define $\mu_{\tau}(\partial A)$ as the Maslov index of $\mathcal{L} = \zeta^*\mathcal{L}_0$ with respect to $\tau$.

Now we define the corrections to add to $\mu_{\tau}(\partial A)$ to obtain the symmetric Conley-Zehnder index $\tilde{\mu}_{\tau}(\partial A)$. Let $\tau$ be a groomed multivalued trivialization which is compatible with $(\mathcal{D}_+, \mathcal{D}_-)$ and which induces the groomings $\mathfrak{c}^+ = \{c_{\ell}^+\}_{\ell=1}^{p_+}$ and $\mathfrak{c}^- = \{c_{\ell}^-\}_{\ell=1}^{p_-}$.

**Definition 5.7.8.** — Given a grooming $\mathfrak{c} = \{c_{\ell}\}_{\ell=1}^{p_+}$ from $P_0$ to $P_1$, its winding number is given by:

$$ (5.7.4) \qquad w(\mathfrak{c}) = \sum_{\ell} w(c_{\ell}), \quad w(c_{\ell}) = \langle c_{\ell}, \{\phi = \pi\} \rangle, $$

where $\langle \cdot, \cdot \rangle$ is the algebraic intersection number in $A_{\varepsilon}$, the arcs $c_{\ell}$ and $\{\phi = \pi\}$ are oriented from $t=0$ to $t=1$, and $A_{\varepsilon}$ is oriented by $(\partial_{\phi}, \partial_{t})$.

**Remark 5.7.9.** — A grooming $\mathfrak{c}$ is determined by its endpoints $P_0$ and $P_1$ and its winding number $w(\mathfrak{c})$.

Given $P_0$, $P_1$, and $q \in \mathbb{Z}$, choose a grooming $\mathfrak{c} = \{c_{\ell}\}_{\ell=1}^{p_+} = \mathfrak{c}(P_0, P_1, q)$ from $P_0$ to $P_1$ such that $w(\mathfrak{c}) = q$. Let $c_{\ell}^{\beta}$ be the linear arc in $A_{\varepsilon}$ which is disjoint from $\{\phi = \pi\}$ and has the same endpoints as $c_{\ell}$. (Note that the collection $\{c_{\ell}^{\beta}\}$ is usually not groomed.)

**Definition 5.7.10.** — $\alpha_{(P_0, P_1, q)}$ (or simply $\alpha_q$ if $P_0$, $P_1$ are understood) is the number of arcs in $\{c_{\ell}^{\beta}\}$ whose $\phi$-coordinate decreases as $t$ increases (in the universal cover).

If $P_0$, $P_1$ are fixed, then the number $\alpha_q$ only depends on $q$ (modulo $p$), through the bijection $P_0 \to P_1$.

Let $q_{\pm} = w(\mathfrak{c}^{\pm})$ and $P_0^{\pm}, P_1^{\pm}$ be the endpoints of $\mathfrak{c}^{\pm}$.

**Definition 5.7.11.** — The discrepancies $d^{\pm}$ at the positive and negative ends are:

$$ (5.7.5) \qquad d^{\pm} = -(\alpha_{(P_0^{\pm}, P_1^{\pm}, q_{\pm})} - \alpha_{(P_0^{\pm}, P_1^{\pm}, 0)}) - q_{\pm}(p_{\pm} - 1). $$

**Remark 5.7.12.** — Observe that $d^{\pm} = 0$ if the grooming $\mathfrak{c}^{\pm}$ satisfies $w(\mathfrak{c}^{\pm}) = q_{\pm} = 0$; in other words, we are viewing $q_{\pm} = 0$ as a reference point and comparing the discrepancy that arises from choosing another grooming $\mathfrak{c}^{\pm}$.

**Definition 5.7.13.** — The symmetric Conley-Zehnder index $\tilde{\mu}_{\tau}(\partial A)$ is given by:

$$ (5.7.6) \qquad \tilde{\mu}_{\tau}(\partial A) = \mu_{\tau}(\partial A) + d^{+} - d^{-}. $$
The discrepancy term $d^+ - d^-$ was (somewhat artificially) added to make Lemma 5.7.15 hold.

**Remark 5.7.14.** — Let us view $P_0^*$, $P_1^*$ as points on $(-\pi, \pi)$, where $*=\pm$. In the special case where the points of $P_0^*$ and $P_1^*$ alternate along $(-\pi, \pi)$, we can write:

$$ (5.7.7) \qquad d^* = \begin{cases} -p_*(q_* - \|\frac{q_*}{p_*}\|), & \text{if } \min_{x \in P_0} x < \min_{x \in P_1} x; \\ -p_*(q_* - \|\frac{q_*}{p_*}\|), & \text{if } \min_{x \in P_0} x > \min_{x \in P_1} x. \end{cases} $$

Here $\|x\|$ is the greatest integer $\le x$ and $\lceil x \rceil$ is the smallest integer $\ge x$.

**5.7.7. ECH indices of branched covers of sections at infinity.** —

**Lemma 5.7.15.** — If $A \in H_2(\overline{W}, \{z_\infty^p(\vec{D}_+), \{z_\infty^p(\vec{D}_-)\})$ is the relative homology class of a p-fold branched cover of $\sigma_\infty$ with possibly empty branch locus, then $I_\tau(A) = 0$ for all groomed multivalued trivializations $\tau$ compatible with $(D_+, D_-)$.

*Proof.* — Let $c^+ = \{c_\ell^+\}$ and $c^- = \{c_\ell^-\}$ be the groomings of $\tau$ and let $q_\pm = w(c^\pm)$. Here $P_i = P_i^+ = P_i^-$, $i = 0, 1$. Let us also assume that $\tau'$ which appears in the definition of $\tau_{\vec{D}_+, \vec{D}_-}$ in Section 5.7.3 satisfies $\tau' = \partial_\rho$ on $D_2^2 - \{z_\infty\}$.

In order to compute $Q_\tau(A)$, we choose a $\tau$-trivial representative $\check{C}$ of A such that:

$$ - \check{C} \cap ([0, 1] \times [0, 1] \times \overline{S}) = [0, 1] \times c^+; $$

$$ - \check{C} \cap ((-1, 0) \times [0, 1] \times \overline{S}) \subset (-1, 0) \times [0, 1] \times \text{int}(D_\epsilon^2); $$

$$ - \check{C} \cap (\{-1\} \times [0, 1] \times \overline{S}) = \{-1\} \times c^-; $$

and a representative $\check{C}'$ such that:

$$ - \check{C}' \cap (\{1\} \times [0, 1] \times \overline{S}) = \{1\} \times c^+(\delta); $$

$$ - \check{C}' \cap ((0, 1) \times [0, 1] \times \overline{S}) \subset (0, 1) \times [0, 1] \times \text{int}(D_\epsilon^2); $$

$$ - \check{C}' \cap ([\overline{-1}, 0] \times [0, 1] \times \overline{S}) = [-1, 0] \times c^-(\delta). $$

Recall that $c^\pm \subset A_\epsilon = [0, 1] \times \partial D_\epsilon^2$. Since all the intersections between $\check{C}$ and $\check{C}'$ are contained in the level $s = 0$,

$$ Q_\tau(A) = c^+ \cdot c^-(\delta) = p(q_+ - q_-). $$

Next we claim that

$$ \mu_\tau(\partial A) = (\alpha_{q_+} - 2q_+) - (\alpha_{q_-} - 2q_-). $$

Indeed, given the end which corresponds to the strand $c_\ell^\pm$, the Maslov index of the end is given by $-2w(c_\ell^\pm)$ if the endpoints of $c_\ell^\pm$ satisfy $0 < p_{\ell,0}^\pm < p_{\ell,1}^\pm < 2\pi$, and is given by $1 - 2w(c_\ell^\pm)$ if $0 < p_{\ell,1}^\pm < p_{\ell,0}^\pm < 2\pi$. On the other hand, the number of strands for which
$p_{\ell,0}^{\pm} > p_{\ell,1}^{\pm}$ holds is exactly $\alpha_{q_{\pm}}$. Since the Maslov indices of the portions of $\partial A$ that map to $[-1, 1] \times \{1\} \times \bar{\mathbf{a}}$ and $[-1, 1] \times \{0\} \times \bar{h}(\bar{\mathbf{a}})$ are zero, the claim follows.

Hence,

$$ \begin{aligned} \tilde{\mu}_{\tau}(\partial A) &= (\alpha_{q_+} - 2q_+) - (\alpha_{q_+} - \alpha_0) - q_+(p-1) \\ &\quad - (\alpha_{q_-} - 2q_-) + (\alpha_{q_-} - \alpha_0) + q_-(p-1) \\ &= -(q_+ - q_-)(p+1). \end{aligned} $$

We also have $c_1(\overline{\mathrm{TS}}|_{\Lambda}, \tau) = q_+ - q_-$. Putting everything together, we obtain:

$$ I_{\tau}(A) = p(q_{+} - q_{-}) - (q_{+} - q_{-})(p + 1) + (q_{+} - q_{-}) = 0. $$

This proves the lemma.

<figure><img src="image_9.png" /></figure>

We also state the following lemmas without proof:

**Lemma 5.7.16.** — If $\bar{u} : \dot{F} \to \overline{W}_-$ is a degree $p \le 2g$ multisection which branch covers $\sigma_{\infty}^{-}$ with possibly empty branch locus, then $I_{\tau}(\bar{u}) = 0$ for all groomed multivalued trivializations $\tau$ that are compatible with $\mathcal{D}_-$ for the negative ends of $\bar{u}$.

**Lemma 5.7.17.** — If $\bar{u} : \dot{F} \to \overline{W}_-$ — int($\overline{W}_-$) is a degree $p \le 2g$ multisection with positive ends at $\delta_0$ with total multiplicity $p$ and negative ends at a $p$-element subset of $\{x_1, \dots, x_{2g}, x'_1, \dots, x'_{2g}\}$, then $I(\bar{u}) = p$.

**5.7.8. Additivity of indices and independence of the trivialization.** — The Fredholm and the ECH index are additive with respect to concatenation. The proofs of the following lemmas are straightforward.

**Lemma 5.7.18.** — Let $\bar{u}_1$ be a multisection with positive end $\{z_{\infty}^{p}(\vec{D})\} \cup \mathbf{y}$, and let $\bar{u}_2$ be a multisection with negative end $\{z_{\infty}^{p'}(\vec{D})\} \cup \mathbf{y}$. If $\bar{u}_1 \# \bar{u}_2$ is the multisection obtained by gluing $\bar{u}_1$ and $\bar{u}_2$ along their common end, then

$$ \mathrm{ind}(\bar{u}_1 \# \bar{u}_2) = \mathrm{ind}(\bar{u}_1) + \mathrm{ind}(\bar{u}_2). $$

**Lemma 5.7.19.** — Let $\tau_2$ and $\tau_1$ be groomed multivalued trivializations compatible with $(\mathcal{D}_2, \mathcal{D}_1)$ and $(\mathcal{D}_1, \mathcal{D}_0)$, respectively, and let $\tau$ be obtained by concatenating $\tau_2$ and $\tau_1$. Given relative homology classes

$$ A_2 \in \mathrm{H}_2(\overline{W}, \{z_{\infty}^{p_2}(\vec{D}_2)\} \cup \mathbf{y}_2, \{z_{\infty}^{p_1}(\vec{D}_1)\} \cup \mathbf{y}_1), $$

$$ A_1 \in \mathrm{H}_2(\overline{W}, \{z_{\infty}^{p_1}(\vec{D}_1)\} \cup \mathbf{y}_1, \{z_{\infty}^{p_0}(\vec{D}_0)\} \cup \mathbf{y}_0), $$
we can form the concatenation

$$A_2\#A_1 \in H_2(\overline{W}, \{z_{\infty}^{p_2}(\overrightarrow{D}_2)\} \cup \mathbf{y}_2, \{z_{\infty}^{p_0}(\overrightarrow{D}_0)\} \cup \mathbf{y}_0).$$

Then we have:

$$I_{\tau}(A_2\#A_1) = I_{\tau_2}(A_2) + I_{\tau_1}(A_1).$$

In view of the following lemma, we can suppress $\tau$ from $I_{\tau}$.

**Lemma 5.7.20.** — $I_{\tau}(A)$ is independent of the choice of groomed multivalued trivialization $\tau$.

*Proof.* — Let $\tau$ and $\tau'$ be two groomed multivalued trivializations adapted to the same data $(D_+, D_-)$. It suffices to consider the particular cases when $\tau$ and $\tau'$ differ only either at some $y_i$ or at $z_\infty$. In the first case, the argument is analogous to the proof of Lemma 4.5.6. In the second case, we can glue branched covers of $\sigma_\infty$ to switch groomings. Then the statement follows from Lemma 5.7.15 and the additivity of the ECH index. $\square$

**5.7.9. The ECH index inequality.** — Let $\bar{u}$ be a degree $k$ multisection of $\overline{W}$ from $\mathbf{z}_+ = \{z_\infty^{p_+(\overrightarrow{D}_+)}\} \cup \mathbf{y}_+$ to $\mathbf{z}_- = \{z_\infty^{p_-(\overrightarrow{D}_-)}\} \cup \mathbf{y}_-$ such that $\bar{u} = \bar{u}''$. Let $c_\ell^\pm$, $\ell = 1, \dots, p_\pm$ be the intersections of $A_\epsilon$ with the $\pi$-projections of the $\pm$ ends of $\bar{u}$ that limit to $z_\infty$. Here $\epsilon > 0$ is small and depends on $\bar{u}$, and the map $\pi$ projects out the $s$-direction. Let $c^\pm = \{c_\ell^\pm\}_{\ell=1,\dots,p_\pm}$ and let $P_0^\pm$ (resp. $P_1^\pm$) be the set of the initial (resp. terminal) points of the arcs $c_\ell^\pm$.

**Lemma 5.7.21.** — If $c^\pm$ are groomings and $A \in H_2(\overline{W}, \mathbf{z}_+, \mathbf{z}_-)$ is the relative homology class of $\bar{u}$, then

$$(5.7.8) \qquad I(A) \ge \mathrm{ind}(\bar{u}) + (d^+ - d^-).$$

In particular, if the points of $P_0^*$ and $P_1^*$ alternate along $(0, 2\pi)$ for both $* = +$ and $-$, then $I(A) \ge \mathrm{ind}(\bar{u})$.

*Proof.* — Let $\tau$ be a groomed multivalued trivialization which is compatible with $(D_+, D_-)$ and $c^\pm$. If we use $\tau$ to compute both the Fredholm and ECH indices, then the proof of the relative adjunction formula (Lemma 4.5.9) goes through unmodified. Hence Equation (5.7.2) and Lemma 4.5.9 imply:

$$\begin{align*}
\mathrm{ind}(\bar{u}) &= -\chi(F) + k + \mu_{\tau}(\bar{u}) + 2c_1(\bar{u}^*T\bar{S}, \tau) \\
&= Q_{\tau}(A) + \mu_{\tau}(\bar{u}) + c_1(\bar{u}^*T\bar{S}, \tau) - 2\delta(\bar{u}).
\end{align*}$$
Since $\tau$ is compatible with $c^{\pm}$, we have $\mu_{\tau}(\bar{u}) = \mu_{\tau}(\partial A)$. Comparing with the definition of $I(\bar{u})$, we have:

$$ (5.7.9) \qquad \begin{aligned} I(\bar{u}) &= \mathrm{ind}(\bar{u}) + (\tilde{\mu}_{\tau}(\partial A) - \mu_{\tau}(\partial A)) + 2\delta(\bar{u}) \\ &= \mathrm{ind}(\bar{u}) + (d^{+} - d^{-}) + 2\delta(\bar{u}). \end{aligned} $$

(Note that $\bar{u}$ has no boundary singularities because $\bar{\pi}_B \circ \bar{u}$ has no branch point at the boundary: in fact $\bar{u}$ has boundary values in $L_{\hat{a}} \cup L_{\bar{u}(\hat{a})}$, and therefore the images of different boundary components of $\bar{F}$ are mutually disjoint.) Equation (5.7.8) follows from observing that $\delta(\bar{u}) \ge 0$ since $\bar{u}$ is J-holomorphic.

Next suppose that the points of $P_0^*$ and $P_1^*$ alternate along $(0, 2\pi)$ for both $* = +$ and $-$. We claim that $d^+ \ge 0$. Let $\pi : \mathbf{R} \times [0, 1] \times \bar{S} \to [0, 1] \times \bar{S}$ be the projection onto the second and third factors. By the positivity of intersections, $\pi(\bar{u})$ is positively transverse to the Hamiltonian vector field $\partial_i$; hence $q_+ \le 0$. By Equation (5.7.7), if $q_+ \le 0$, then $d^+ \ge 0$ (in both cases). $d^- \le 0$ is proved similarly. $\square$

**5.7.10. ECH index calculation.** — In this subsection we compute the ECH index of a multisection of $\bar{W}$ which is the union of a branched cover of the section at infinity $\sigma_\infty$ and a curve which limits to a multiple of $z_\infty$. This calculation will be used in Section II.3.7.2. For simplicity we consider the cases of positive ends and negative ends at $z_\infty$ separately. The general statement can easily be deduced from the two separate cases.

Let $\bar{u} = \bar{u}' \cup \bar{u}''$ be a degree $p_1 + p_2 + l$ multisection of $\bar{W}$, where $\deg \bar{u}' = p_1$, $\bar{u}''$ is a multisection from $\mathbf{y}$ to $\{z_\infty^{p_2}(\bar{D}_2)\} \cup \mathbf{y}'$, and $l$ is the cardinality of $\mathbf{y}'$. Suppose that the data $\bar{D}_1$ and $\bar{D}_2$ for $\bar{u}'$ and $\bar{u}''$ at the negative ends $z_\infty^{p_1}$ and $z_\infty^{p_2}$ satisfy:

$$ (D_1) \quad \bar{D}_1 = \{(i_1, j_1) \to (i_1, j_1), \dots, (i_{p_1}, j_{p_1}) \to (i_{p_1}, j_{p_1})\}; $$

$$ (D_2) \quad \bar{D}_2 = \{(i'_1, j'_1) \to (i'_1, j'_1), \dots, (i'_{p_2}, j'_{p_2}) \to (i'_{p_2}, j'_{p_2})\}; $$

$$ (D_3) \quad \mathcal{D}_1^{\text{from}} = \mathcal{D}_1^{\text{to}} \text{ and the sets } \mathcal{D}_2^{\text{from}} \text{ and } \mathcal{D}_2^{\text{to}} \text{ are disjoint from } \mathcal{D}_1^{\text{from}} = \mathcal{D}_1^{\text{to}}. $$

Let $\mathcal{E}_{-, \ell}$ be the end of $\bar{u}''$ at $z_\infty$ corresponding to $(i'_\ell, j'_\ell) \to (i''_\ell, j''_\ell)$.

Suppose there exist groomings $c_1^- \subset A_{\varepsilon/2} = \partial D_{\varepsilon/2}^2 \times [0, 1]^7$ and $c_2^- \subset A_\varepsilon = \partial D_\varepsilon^2 \times [0, 1]$ which correspond to $\bar{D}_1$ and $\bar{D}_2$ and satisfy the following:

$$ (G_1) \quad c_1^- \text{ has winding number } q_1 := w(c_1^-) = 0; $$

$$ (G_2) \quad c_2^- = \pi(\cup_{i=1}^{p_2} \mathcal{E}_{-, \ell}) \cap A_\varepsilon \text{ and } q_2 := w(c_2^-) = 0 \text{ or } 1; $$

$$ (G_3) \quad \text{the points of } P_0^2 \text{ and } P_1^2 \text{ alternate along } (0, 2\pi), \text{ i.e., the projection of } c_2^- \text{ to } \partial D_\varepsilon^2 \text{ is injective.} $$

Here $P_0^i$ (resp. $P_1^i$) is the set of initial (resp. terminal) points of $c_i^-$.

<footnote n="7">⁷ Here we are writing $A_{\varepsilon/2} = \partial D_{\varepsilon/2}^2 \times [0, 1]$ instead of $[0, 1] \times \partial D_{\varepsilon/2}^2$ to indicate the orientation of $A_{\varepsilon/2}$.</footnote>
Let $\pi$ be the projection of $\tilde{W} = [-1, 1] \times [0, 1] \times \bar{S}$ to $[0, 1] \times \bar{S}$ and let $\check{C}'$ and $\check{C}''$ be representatives of $\bar{u}'$ and $\bar{u}''$ in $\tilde{W}$ such that $\pi(\check{C}'|_{s=-1}) = \mathfrak{c}_1^-$ and $\pi(\check{C}''|_{s=-1}) = \mathfrak{c}_2^-$.

Let $\pi_\rho : [\frac{\epsilon}{2}, \epsilon] \times \partial D^2 \times [0, 1] \to \partial D^2 \times [0, 1]$ be the projection along the $\rho$-direction. Let $w$ be the signed number of crossings of $\pi_\rho(\mathfrak{c}_1^- \cup \mathfrak{c}_2^-)$, i.e., the writhe. Observe that all the crossings of $\pi_\rho(\mathfrak{c}_1^- \cup \mathfrak{c}_2^-)$ are positive as a consequence of (G1) and (G2), and $p_1 \ge w$ as a consequence of (G3).

**Lemma 5.7.22.** — Let $\bar{u} = \bar{u}' \cup \bar{u}''$ be a degree $p_1 + p_2 + l$ multisection of $\bar{W}$, where $\deg \bar{u}' = p_1$, $\bar{u}''$ is a multisection from $\mathbf{y}$ to $\{z_\infty^{p_2}(\tilde{D}_2)\} \cup \mathbf{y}'$, and $l$ is the cardinality of $\mathbf{y}'$. If the data $\tilde{D}_1, \tilde{D}_2$ at the negative end satisfy $(D_1)-(D_3)$ and $(G_1)-(G_3)$, then

$$ (5.7.10) \qquad I(\bar{u}) \ge I(\bar{u}') + I(\bar{u}'') + \begin{cases} 2w, & \text{if } q_2 = 0; \\ p_1 + w, & \text{if } q_2 = 1. \end{cases} $$

If $\bar{u}' \cap \bar{u}'' = \emptyset$ in addition, then equality holds.

*Proof.* — It suffices to prove the equality, assuming $\bar{u}' \cap \bar{u}'' = \emptyset$, since the extra intersections contribute positively towards the ECH index. The representatives $\check{C}'$ and $\check{C}''$ can be taken to be disjoint; in fact we can assume that $\check{C}''$ is disjoint from the $\frac{\epsilon}{2}$-neighborhood $\mathbf{R} \times [0, 1] \times D_{\frac{\epsilon}{2}}^2$ of $\sigma_\infty$.

If $q_2 = 0$, then we resolve the positive crossings of $\pi_\rho(\mathfrak{c}_1^- \cup \mathfrak{c}_2^-)$. This is equivalent to appending a union of disks $\check{D} \subset [-2, -1] \times [0, 1] \times \bar{S}$ to $\check{C}' \cup \check{C}''$ such that $\check{D}|_{s=-1} = (\check{C}' \cup \check{C}'')|_{s=-1}$ and $\pi(\check{D}|_{s=-2})^8$ is a grooming on $A_\epsilon$ which satisfies $w(\pi(\check{D}|_{s=-2})) = 0$. A quick calculation shows that $\check{D}$ contributes $w$ to $\mathcal{Q}$, $0$ to $c_1$, and $w$ to $\mu$. Since the discrepancy for $\pi(\check{D}|_{s=-2})$ is zero, Equation (5.7.10) follows for $q_2 = 0$.

If $q_2 = 1$, then we first switch the crossings of $\pi_\rho(\mathfrak{c}_1^- \cup \mathfrak{c}_2^-)$ by appending a union of disks $\check{D}_{-1} \subset [-2, -1] \times [0, 1] \times \bar{S}$ such that $\pi(\check{D}_{-1}|_{s=-2})$ consists of $\mathfrak{c}_1^-$ on $A_\epsilon$ and $\mathfrak{c}_2^-$ on $A_{\epsilon/2}$. This contributes $2w$ to $\mathcal{Q}$ and $0$ to $c_1$ and $\mu$. Next append $\check{D}_{-2} \subset [-3, -2] \times [0, 1] \times \bar{S}$ so that $\pi(\check{D}_{-2}|_{s=-3})$ consists of $\mathfrak{c}_1^-$ on $A_\epsilon$ and $\mathfrak{d}_2^-$ on $A_{\epsilon/2}$, where $\mathfrak{d}_2^-$ is groomed and $w(\mathfrak{d}_2^-) = 0$. The surface $\check{D}_{-2}$ is similar to that used in the proof of Lemma 5.7.15 and has zero ECH index. Finally, we resolve the positive crossings of $\pi_\rho(\mathfrak{c}_1^- \cup \mathfrak{d}_2^-)$ by appending $\check{D}_{-3} \subset [-4, -3] \times [0, 1] \times \bar{S}$. Since there are $p_1 - w$ crossings, $\check{D}_{-3}$ contributes $p_1 - w$ to $\mathcal{Q}$, $0$ to $c_1$, and $0$ to $\mu$. Equation (5.7.10) then follows for $q_2 = 1$. $\square$

Next we consider the variant where the multiple of $z_\infty$ is at the positive end. Let $\bar{u} = \bar{u}' \cup \bar{u}''$ be a degree $p_1 + p_2 + l$ multisection of $\bar{W}$, where $\deg \bar{u}' = p_1$, $\bar{u}''$ is a multisection from $\{z_\infty^{p_2}(\tilde{D}_2)\} \cup \mathbf{y}'$ to $\mathbf{y}$, and $l$ is the cardinality of $\mathbf{y}'$. We use the same notation as above, with $-$ replaced by +.

We make the following assumptions:

<footnote n="8">⁸ Here we are taking $\pi$ to be the projection of the appropriate space to $[0, 1] \times \bar{S}$.</footnote>
(G'_1) $\mathfrak{c}_1^+$ has winding number $q_1 := w(\mathfrak{c}_1^+) = 0$;

(G'_2) $\mathfrak{c}_2^+ = \pi(\cup_{i=1}^{p_2} \mathcal{E}_{+,\ell}) \cap \mathcal{A}_\varepsilon$ and $q_2 := w(\mathfrak{c}_2^+) = 0$ or $-1$; and

(G'_3) the projection of $\mathfrak{c}_2^+$ to $\partial D_\varepsilon^2$ is injective except on $\kappa \ge 0$ short intervals of $\partial D_\varepsilon^2$ which correspond to thin sectors between $\bar{a}_{i,j}$ and $\bar{h}(\bar{a}_{i,j})$.

Let $\mathfrak{w}$ be the signed number of crossings of $\pi_\rho(\mathfrak{c}_1^+ \cup \mathfrak{c}_2^+)$ in this case, all the crossings of $\pi_\rho(\mathfrak{c}_1^+ \cup \mathfrak{c}_2^+)$ are negative. The analog of Lemma 5.7.22, stated without proof, is:

**Lemma 5.7.23.** — Let $\bar{u} = \bar{u}' \cup \bar{u}''$ be a degree $p_1 + p_2 + l$ multisection of $\overline{W}$, where $\deg \bar{u}' = p_1$, $\bar{u}''$ is a multisection from $\{z_\infty^{p_2}(\overline{D}_2)\} \cup \mathbf{y}'$ to $\mathbf{y}$, and $l$ is the cardinality of $\mathbf{y}'$. If the data $\overline{D}_1, \overline{D}_2$ at the positive end satisfy $(D_1)-(D_3)$ and $(G'_1)-(G'_3)$, then

$$ (5.7.11) \qquad I(\bar{u}) \ge I(\bar{u}') + I(\bar{u}'') + \begin{cases} -\mathfrak{w}, & \text{if } q_2 = 0; \\ -2\mathfrak{w}, & \text{if } q_2 = -1. \end{cases} $$

If $\bar{u}' \cap \bar{u}'' = \emptyset$ in addition, then equality holds.

We also compute the discrepancy $d_+$ of $\mathfrak{c}_2^+$ when $q_2 = -1$. Since $\alpha_{-1} = p_2 - 1$ and $\alpha_0 = \kappa$,

$$ (5.7.12) \qquad \begin{aligned} d_+ &= -(\alpha_{-1} - \alpha_0) - q_2(p_2 - 1) \\ &= -((p_2 - 1) - \kappa) - (-1)(p_2 - 1) = \kappa. \end{aligned} $$

By Lemma 5.7.21,

$$ (5.7.13) \qquad I(\bar{u}'') \ge \mathrm{ind}(\bar{u}'') + \kappa. $$

**5.7.11. Extended moduli spaces.** — We now describe the extended moduli spaces which involve multiples of $z_\infty$ at the ends. Details will be given for $\overline{W}$; the $\overline{W}_+$ and $\overline{W}_-$ cases are analogous.

Let $\mathcal{M} = \mathcal{M}_J(\mathbf{z}_+, \mathbf{z}_-)$ be the moduli space of multisections of $(\overline{W}, \overline{J})$ from $\mathbf{z}_+ = \{z_\infty^{p_+}(\overline{D}_+)\} \cup \mathbf{y}_+$ to $\mathbf{z}_- = \{z_\infty^{-p_+}(\overline{D}_-)\} \cup \mathbf{y}_-$. A map $\bar{u} \in \mathcal{M}$ has boundary conditions on $\overline{L}_{\bar{a}} \cup \overline{L}_{\bar{h}(\bar{a})}$. Let $\dagger$ be the modifier “$\bar{u}' = \emptyset$”. We now describe an enlargement of the moduli space $\mathcal{M}^\dagger$.

Recall that $\bar{a}_{i,j} \subset D^2$ is of the form $\{-1 < \rho \le 1, \phi = \phi_{i,j}\}$ for some constant $\phi_{i,j}$.

**Definition 5.7.24.** — An extended $\overline{W}$-curve $\bar{u}$ from $\mathbf{z}_+$ to $\mathbf{z}_-$ is a multisection of $(\overline{W}, \overline{J})$ which satisfies the conditions of Definition 5.7.2 with $\dot{\mathbf{F}}' = \emptyset$ and (1) and (2) replaced by the following:

(1') There exist either positive or negative ends (or both) $\mathcal{E}_{+,i}$ and $\mathcal{E}_{-,i}$ of $\dot{\mathbf{F}}$ that limit to $z_\infty$ such that:

$$ - \bar{u}(\partial \dot{\mathbf{F}} - \cup_i \mathcal{E}_{+,i} - \cup_i \mathcal{E}_{-,i}) \subset \overline{L}_{\hat{a}} \cup \overline{L}_{\bar{h}(\hat{a})}; $$
— $\bar{u}$ maps one component of $\partial \dot{\mathbf{F}} \cap \mathcal{E}_{+,i}$ (resp. $\partial \dot{\mathbf{F}} \cap \mathcal{E}_{-,i}$) to $L_{\vec{a}_{i\ell}, j\ell}$ and the other to $L_{\vec{h}(\vec{a}_{i\ell}, j\ell)}$, where $(i_\ell, j_\ell) \to (i_\ell, j_\ell) \in \vec{D}_+$ (resp. $\in \vec{D}_-$).

(2') $\bar{u}$ maps each connected component of $\partial \dot{\mathbf{F}} - \cup_i \mathcal{E}_{+,i} - \cup_i \mathcal{E}_{-,i}$ to a different $L_{\hat{a}_i}$ or $L_{\bar{h}(\hat{a}_i)}$.

The moduli space of extended $\overline{\mathbf{W}}$-curves from $\mathbf{z}_+$ to $\mathbf{z}_-$ is denoted by $M_{\vec{J}}^{\dagger, \text{ext}}(\mathbf{z}_+, \mathbf{z}_-)$.

The extended moduli spaces $M_{\vec{J}}^{\dagger, \text{ext}}(\mathbf{z}_+, \delta_0^r \mathbf{y}')$ and $M_{\vec{J}}^{\dagger, \text{ext}}(\delta_0^r \mathbf{y}', \mathbf{z}_-)$ are defined similarly.

*Terminology.* A sector $\mathcal{S}$ of $D_{\rho_0}^2$, $0 < \rho_0 \le 1$, from $\phi_0$ to $\phi_1$ is the map

$$[0, \rho_0] \times [\phi_0, \phi_1] \to D_{\rho_0}^2, \quad (\rho, \phi) \mapsto \rho e^{i\phi}.$$

By abuse of notation, we also refer to the image of $\mathcal{S}$ as a sector.

If $R, R' \subset D_{\rho_0}^2$ are distinct radial rays and $I \subset \mathbf{R}$ is an interval, then $\mathcal{S}(R, R'; I)$ is a sector of $D_{\rho_0}^2$ from $\phi_0$ to $\phi_1$, where $R$ and $R'$ can be written as $\{\phi = \phi_0, \rho \ge 0\}$ and $R' = \{\phi = \phi_1, \rho \ge 0\}$ and $[\phi_0, \phi_1] \subset I$. If we do not specify $I$, then we write $\mathcal{S}(R, R')$ and assume that $\phi_1 - \phi_0$ is the smallest positive angle.

A sector is *large* if it has angle $\phi_1 - \phi_0 > \pi$ and *small* if it has angle $0 < \phi_1 - \phi_0 < \pi$.

Let $\mathcal{M}^* = \mathcal{M}^*(\mathbf{z}_+, \mathbf{z}_-)$, where $*$ denotes any set of modifiers, and let

$$\pi_{\mathcal{D}^2} : \mathcal{D}^2 \times [0, 1] \to \mathcal{D}^2$$

be the projection onto the first factor. If $\bar{u} \in \mathcal{M}^\dagger$, then $(i'_\ell, j'_\ell) \to (i_\ell, j_\ell)$ in $\vec{D}_+$ corresponds to an end $\mathcal{E}_{+,\ell}$ of $\bar{u}$ such that the restriction of $\pi_{\mathcal{D}^2} \circ \bar{u}(\mathcal{E}_{+,\ell})$ to $D_\varepsilon^2$ for $\varepsilon > 0$ small is a sector $\mathcal{S} = \mathcal{S}(\vec{a}_{i_\ell, j_\ell}, \bar{h}(\vec{a}_{i'_\ell, j'_\ell}))$. If $\mathcal{S}$ is large, then $\pi_{\mathcal{D}^2} \circ \bar{u}(\mathcal{E}_{+,\ell})$ has a slit of length 0 by the definition of $\bar{u} \in \mathcal{M}^\dagger$ (i.e., $\pi_{\mathcal{D}^2} \circ \bar{u}$ maps no boundary point of $\mathcal{E}_{+,i}$ to $\vec{a}_{i_\ell, j_\ell} - \vec{a}_{i'_\ell, j'_\ell}$ or $\bar{h}(\vec{a}_{i'_\ell, j'_\ell} - \vec{a}_{i_\ell, j_\ell})$). The ends of $\bar{u}$ can cover at most one large sector because $n(\bar{u}) = m$, and therefore $\bar{u}$ is an extended curve. The neighborhood of $\bar{u}$ in $\mathcal{M}^\dagger$ is generically a codimension $\ge 1$ submanifold of an extended moduli space $\mathcal{M}^{\dagger, \text{ext}}$.

**5.8. Transversality.** — We first discuss the regularity of almost complex structures on $\overline{\mathbf{W}}, \overline{\mathbf{W}}', \overline{\mathbf{W}}_+$ and $\overline{\mathbf{W}}_-$.

**5.8.1. Transversality for $\overline{\mathbf{W}}$ and $\overline{\mathbf{W}}'$. —**

**Definition 5.8.1.** — The almost complex structure $\mathcal{J} \in \mathcal{J}_{\overline{\mathbf{W}}}$ is regular if $M_{\mathcal{J}}(\mathbf{y}, \mathbf{y}')$ is transversely cut out for all tuples $\mathbf{y}$ and $\mathbf{y}'$ in $\mathbf{a} \cap \mathcal{h}(\mathbf{a})$. The almost complex structure $\mathcal{J} \in \mathcal{J}_{\overline{\mathbf{W}}}$ is regular if $M_{\mathcal{J}}^{\dagger, \text{ext}}(\mathbf{z}, \mathbf{z}')$ is transversely cut out for all tuples $\mathbf{z} = \{z_\infty^p(\mathcal{D})\} \cup \mathbf{y}$ and $\mathbf{z}' = \{z_\infty^q(\mathcal{D}')\} \cup \mathbf{y}'$.
Note that $M_J(\mathbf{y}, \mathbf{y}') = M_J^s(\mathbf{y}, \mathbf{y}')$ and $M_J^{\dagger, \text{ext}}(\mathbf{z}, \mathbf{z}') = M_J^{\dagger, \text{ext}, s}(\mathbf{z}, \mathbf{z}')$ because the positive ends have boundaries on distinct Lagrangian submanifolds. This implies that a generic $J \in \mathcal{J}_W$ is regular; see Lemma 4.7.2. The same proof also gives:

**Lemma 5.8.2.** — A generic $\bar{J} \in \mathcal{J}_{\bar{W}}$ is regular.

We write $\mathcal{J}_W^{\text{reg}} \subset \mathcal{J}_W$ for the subset of all regular $J$ and $\mathcal{J}_{\bar{W}}^{\text{reg}} \subset \mathcal{J}_{\bar{W}}$ for the subset of all regular $\bar{J}$.

**Definition 5.8.3.** — The almost complex structure $J' \in \mathcal{J}_{W'}$ is regular if $M_J^s(\mathbf{y}, \mathbf{y}')$ is transversely cut out (in the Morse-Bott sense) for all $\mathbf{y}$ and $\mathbf{y}'$ in $\hat{\mathcal{O}}_k$ and all $k \le 2g$. The almost complex structure $\bar{J}' \in \mathcal{J}_{\bar{W}'}$ is regular if $M_J^s(\delta^p\mathbf{y}, \delta^q\mathbf{y}')$ is transversely cut out for all $\delta^p\mathbf{y}, \delta^q\mathbf{y}' \in \bar{\mathcal{O}}_k$ and for all $k \le 2g$.

Recall that a generic $J' \in \mathcal{J}_{W'}$ is regular by Lemma 3.5.2. The same proof also gives:

**Lemma 5.8.4.** — A generic $\bar{J}' \in \mathcal{J}_{\bar{W}'}$ is regular.

We write $\mathcal{J}_W^{\text{reg}} \subset \mathcal{J}_W$ for the subset of all regular $J'$ and $\mathcal{J}_{\bar{W}'}^{\text{reg}} \subset \mathcal{J}_{\bar{W}'}$ for the subset of all regular $\bar{J}'$.

**5.8.2. Transversality for $W_+, \bar{W}_+$ and $\bar{W}_-$.—**

**Definition 5.8.5.** — The almost complex structure $J_+ \in \mathcal{J}_{W_+}$ is regular if the following hold:

(1) all moduli spaces $M_{J_+}(\mathbf{y}, \mathbf{y})$ with $\mathbf{y}$ a $k$-tuple of $\mathbf{a} \cap \mathfrak{h}(\mathbf{a})$ and $\mathbf{y} \in \hat{\mathcal{O}}_k$, for all $k \le 2g$, are transversely cut out (in the Morse-Bott sense in the case of a Morse-Bott building); and

(2) the restrictions $J$ and $J'$ of $J_+$ to the positive and negative ends belong to $\mathcal{J}_W^{\text{reg}}$ and $\mathcal{J}_{\bar{W}'}^{\text{reg}}$, respectively.

Here every $u \in M_{J_+}(\mathbf{y}, \mathbf{y})$ is somewhere injective due to the presence of the HF end. In other words, $M_{J_+}(\mathbf{y}, \mathbf{y}) = M_{J_+}^s(\mathbf{y}, \mathbf{y})$.

**Definition 5.8.6.** — The almost complex structure $\bar{J}_+ \in \mathcal{J}_{\bar{W}_+}$ is regular if the following hold:

(1) all moduli spaces $M_{\bar{J}_+}^{\dagger, \text{ext}}(\mathbf{z}, \delta_0^r \mathbf{y}')$ with $\mathbf{z} = \{z_\infty^p(\mathcal{D})\} \cup \mathbf{y}, \mathbf{y}' \in \hat{\mathcal{O}}_k$ for some $k \le 2g$, and $\mathbf{y}$ a tuple of $\mathbf{a} \cap \mathfrak{h}(\mathbf{a})$, are transversely cut out; and

(2) the restrictions $\bar{J}$ and $\bar{J}'$ of $\bar{J}_+$ to the positive and negative ends belong to $\mathcal{J}_{\bar{W}}^{\text{reg}}$ and $\mathcal{J}_{\bar{W}'}^{\text{reg}}$, respectively.

Note that $M_{\bar{J}_+}^{\dagger, \text{ext}}(\mathbf{z}, \delta_0^r \mathbf{y}') = M_{\bar{J}_+}^{\dagger, \text{ext}, s}(\mathbf{z}, \delta_0^r \mathbf{y}')$. The regularity of $\bar{J}_- \in \mathcal{J}_{\bar{W}_-}$ is defined similarly.
We write $\mathcal{J}_{W_+}^{reg}$ for the space of regular $J_+ \in \mathcal{J}_{W_+}$ and $\mathcal{J}_{\bar{W}_+}^{reg}$ for the space of regular $\bar{J}_+ \in \mathcal{J}_{\bar{W}_+}$. Without loss of generality we may assume that the regular $J_+$ of interest are the restrictions of regular $\bar{J}_+$.

**Remark 5.8.7.** — The vertical fibers $\{(s, t)\} \times S$ and $\{(s, t)\} \times \bar{S}$ are holomorphic, but are not transversely cut out.

**Proposition 5.8.8.** — A generic admissible $J_+$ (resp. $\bar{J}_+$) is regular.

*Proof.* — We first treat the $W_+$ case. The proposition follows from a standard transversality argument along the lines of [MS, Theorem 3.1.5], with some modifications. The necessary modifications for almost complex structures $J' \in \mathcal{J}_W$, defined on $\mathbf{R} \times \mathbf{N}$, were described in [Hul, Lemma 9.12(b)], and our situation is almost identical since $J_+ \in \mathcal{J}_{W_+}$ is the restriction to $W_+$ of some $J' \in \mathcal{J}_W$.

The key observation is that each irreducible component of a $W_+$-curve $u: \dot{\mathbb{F}} \to W_+$ is somewhere injective, since each $[0, 1] \times \{y_i\}, y_i \in \mathbf{y}$, is used exactly once as a positive asymptotic limit. Let $\pi_N: W_+ \to \mathbb{N}$ be the restriction of the projection $\pi_N: \mathbf{R} \times \mathbb{N} \to \mathbb{N}$ onto the second factor. We then claim that there is a dense open set of points $p \in \dot{\mathbb{F}}$ which are $\pi_N$-injective, i.e.,

(i) $d(\pi_N \circ u)(p)$ has rank 2; and

(ii) $(\pi_N \circ u)(p) = (\pi_N \circ u)(q)$ implies $p=q$.

First note that if the claim does not hold, then there exist open sets $U, U' \subset \dot{\mathbb{F}}$ such that $u(U)$ and $u(U')$ differ by some translation $T_{s_0}$ by $s_0$ in the $s$-direction. By repeated application of $T_{s_0}$ or $T_{-s_0}$, there is an infinite sequence $U, U'', U'''$, $\cdots \subset \dot{\mathbb{F}}$ that have the same $\pi_N \circ u$ image. This contradicts the finite energy condition, and hence proves the claim. The perturbations to $J_+$ can then be carried out in a neighborhood of a $\pi_N$-injective point $p \in \dot{\mathbb{F}}$ as in [Hul, Lemma 9.12(b)].

The regularity of the almost complex structures $J$ and $J'$ at the ends was already treated, i.e., $\mathcal{J}_W^{reg} \subset \mathcal{J}_W$ and $\mathcal{J}_{\bar{W}}^{reg} \subset \mathcal{J}_{\bar{W}}$ are dense by Lemmas 3.5.2 and 4.7.2.

In the $\bar{W}_+$ case, the perturbations of $\bar{J}_+$ are allowed on the subset $U = \bar{W}_+ \cap ((\mathbf{R} \times \bar{\mathbb{N}}) - \{\rho \le \varepsilon\})$ for some small $\varepsilon > 0$. We simply observe that all the curves $\bar{u}$ in the moduli spaces $M_{\bar{J}_+}^{\dagger, \text{ext}}(\mathbf{z}, \delta_0^\dagger \mathbf{y}')$ in Definition 5.8.6 pass through $U$, and pick a $\pi_N$-injective point $p \in \dot{\mathbb{F}}$ such that $\bar{u}(p) \in U$. The $\bar{W}_-$ case is similar. $\square$

**5.8.3.** *Some automatic transversality results.* — We collect some automatic transversality results.

**Lemma 5.8.9.** — The curve $\sigma_\infty^- \subset \bar{W}_-$, viewed as having Lagrangian boundary $L_{a_{ij}}^-$, is a regular holomorphic curve with $\text{ind}(\sigma_\infty^-) = 0$.
*Proof.* — We first calculate the Fredholm index of the holomorphic embedding $\bar{u}: \dot{\bar{F}} \to \bar{W}_-$ with image $\sigma_\infty^- \subset \bar{W}_-$. Here $\dot{\bar{F}}$ is a disk with a boundary puncture and an interior puncture. Let $\tau$ be a groomed trivialization whose grooming $c$ corresponds to the matching $(i,j) \to (i,j)$ and satisfies $w(c) = 0$. Then we compute that $-\chi(\dot{\bar{F}}) = 0$, $\mu_\tau(\delta_0, \bar{u}) = 1$, $\mu_\tau(z_\infty) = 1$, $c_1(\bar{u}^*T\bar{S}, \tau) = 0$. Hence $\mathrm{ind}(\bar{u}) = 0$ by Equation (5.5.3), which is still valid in the current situation.

We now use the doubling technique from Theorem 5.5.1. The double of $\dot{\bar{F}}$ — a sphere with three punctures — is denoted by $2\dot{\bar{F}}$ and the double of $\bar{u}$ is denoted by $2\bar{u}$. The index of the doubled operator $2D_{\bar{u}}$ is $\mathrm{ind}(2\bar{u}) = 2\mathrm{ind}(\bar{u}) = 0$ and $D_{\bar{u}}$ is surjective if and only if $2D_{\bar{u}}$ is surjective.

Now, by Wendl's automatic transversality theorem [We2, Theorem 1], $2D_{\bar{u}}$ is surjective if

$$ (5.8.1) \qquad \mathrm{ind}(2\bar{u}) \ge 2g + \#\Gamma_0 - 1, $$

where $g$ is the genus of $2\dot{\bar{F}}$ and $\#\Gamma_0$ is the count of punctures with even Conley-Zehnder index. We have $g=0$. Recall that doubling a chord gives a closed orbit with odd Conley-Zehnder index by Lemma 5.5.3. Hence $\#\Gamma_0 = 0$ and Equation (5.8.1) becomes $\mathrm{ind}(2\bar{u}) \ge -1$, which is satisfied since $\mathrm{ind}(2\bar{u}) = 0$. $\square$

**Lemma 5.8.10.** — Let $\bar{J}$ and $\bar{J}_0$ be almost complex structures as in Definition 5.3.2. If $\bar{J}$ is sufficiently close to $\bar{J}_0$, then $M_{\bar{J}}^{\dagger,n}=1(\{z_\infty\} \cup \mathbf{y}, \{y_0\} \cup \mathbf{y})/\mathbf{R}$ is transversely cut out and consists of a unique curve which is represented by a thin strip in $\mathcal{D}^2$ from $z_\infty$ to $y_0 = x_i$ or $x'_i$, together with trivial cylinders.

*Proof.* — The proof of transversality is similar to but easier than that of Lemma 5.8.9 and is omitted. The uniqueness of the curve (modulo $\mathbf{R}$-translation in the target) from $z_\infty$ to $y_0$ is immediate from projecting the curve to $\mathcal{D}^2$ and using the fact that $\bar{J}$ is sufficiently close to $\bar{J}_0$. $\square$

**5.8.4. Marked points and transversality.** — In the definition of the $\Psi$-map in Section 7, we consider multisections of $\bar{W}_-$ which pass through the marked point $\bar{m} = ((0, \frac{3}{2}), z_\infty)$. The marked point $\bar{m}$, however, is *nongeneric*. In order to ensure the regularity of such moduli spaces with respect to $\bar{m}$, we need to enlarge the class of $\bar{J}_- \in \mathcal{J}_{\bar{W}_-}^{\mathrm{reg}}$ to the class of $\bar{J}_-^\diamond$, which we now define.

**Definition 5.8.11.** — Let $\varepsilon > 0$ and let $U \not\ni \bar{m}$ be an open set of $\bar{W}_-$. Then an almost complex structure $\bar{J}_-^\diamond$ on $\bar{W}_-$ is $(\varepsilon, U)$-close to $\bar{J}_-$ if:

$$ - \bar{J}_-^\diamond = \bar{J}_- \text{ on } \bar{W}_- - U; $$

$$ - \bar{J}_-^\diamond \text{ is } \varepsilon\text{-close to } \bar{J}_- \text{ on } U; \text{ and} $$
$\bar{\nabla} \bar{J}_{-}^{\diamond}$ is $\varepsilon$-close to $\nabla \bar{J}_{-}$ on U.

Here the $\varepsilon$-closeness is measured with respect to a metric $g$ on $\overline{W}_{-}$ which is the restriction of an $s$-invariant metric on $\mathbf{R} \times \overline{\mathbb{N}}$ and $\nabla$ is the Levi-Civita connection of $g$.

**Convention 5.8.12.** — Unless stated otherwise:

- $U = U_{p,2\delta} = \bar{\pi}_{B_{-}}^{-1}(B_{\delta}(p)) - \{\rho \le \delta\}$ is an open neighborhood of $K_{p,2\delta} = \bar{\pi}_{B_{-}}^{-1}(p) - \{\rho < 2\delta\}$, where $\delta > 0$ is arbitrarily small, $\bar{m}^{b} \neq p \in B_{-}$, and $B_{\delta}(p) \subset B_{-}$ is an open ball of radius $\delta$ about $p$.

- $\bar{J}_{-}^{\diamond}$ is $(\varepsilon, U)$-close to $\bar{J}_{-} \in \mathcal{J}_{\overline{W}_{-}}^{\text{reg}}$, where $\varepsilon > 0$ is arbitrarily small.

Observe that $U$ is disjoint from the section at infinity. When we want to emphasize $(\varepsilon, U)$ or $(\varepsilon, \delta, p)$, we write $\bar{J}_{-}^{\diamond}(\varepsilon, U)$ or $\bar{J}_{-}^{\diamond}(\varepsilon, \delta, p)$.

**Definition 5.8.13.** — A degree $k$ almost multisection of $(\overline{W}_{-}, \bar{J}_{-}^{\diamond})$ from $\delta_{0}^{r}y$ to $z' = \{z_{\infty}^{g}(D')\} \cup y'$ is a pair $(\bar{u}, C)$ which is defined in the same way as a degree $k$ multisection of $(\overline{W}_{-}, \bar{J}_{-})$, except that $\bar{u}$ is a degree $k$ multisection of

$$ \bar{\pi}_{B_{-}} : \overline{W}_{-} - \bar{\pi}_{B_{-}}^{-1}(B_{\delta}(p)) \to B_{-} - B_{\delta}(p). $$

$A(\overline{W}_{-}, \bar{J}_{-}^{\diamond})$-curve is a degree 2g almost multisection of $(\overline{W}_{-}, \bar{J}_{-}^{\diamond})$ satisfying $n^{-}(\bar{u}) = m$ if $r = q = 0$ and $0 < n^{-}(\bar{u}) < m$ if $r > 0$ or $q > 0$.

Let $\mathcal{M}_{\bar{J}_{-}^{\diamond}}(\delta_{0}^{r}y, z')$ be the moduli space of almost multisections of $(\overline{W}_{-}, \bar{J}_{-}^{\diamond})$ from $\delta_{0}^{r}y$ to $z'$. The regularity of $\bar{J}_{-}$ and the closeness of $\bar{J}_{-}^{\diamond}$ to $\bar{J}_{-}$ imply:

(i) $\mathcal{M}_{\bar{J}_{-}^{\diamond}}^{\dagger}(\delta_{0}^{r}y, z')$ is regular;

(ii) $\mathcal{M}_{\bar{J}_{-}^{\diamond}}^{\dagger}(\delta_{0}^{r}y, z')$ is close to $\mathcal{M}_{\bar{J}_{-}}^{\dagger}(\delta_{0}^{r}y, z')$; and

(iii) all the boundary strata of $\mathcal{M}_{\bar{J}_{-}^{\diamond}}^{\dagger}(\delta_{0}^{r}y, z')$ are close to the corresponding boundary strata of $\mathcal{M}_{\bar{J}_{-}}^{\dagger}(\delta_{0}^{r}y, z')$.

Note that we can still refer to $n^{*}(\bar{u})$ since it is a homological quantity.

Let $K \not\supset \bar{m}$ be a compact set of $\overline{W}_{-}$. We define the modifier $K$ to mean that $\bar{u}$ passes through $K$.

**Definition 5.8.14.** — The almost complex structure $\bar{J}_{-}^{\diamond}$ on $\overline{W}_{-}$ is $K$-regular with respect to $\bar{m}$ if all the moduli spaces $\mathcal{M}_{\bar{J}_{-}^{\diamond}}^{\dagger, \text{ext}, K}(\delta_{0}^{r}y, z'; \bar{m})$ are transversely cut out.

**Lemma 5.8.15.** — A generic $\bar{J}_{-}^{\diamond}$ is $K_{p,2\delta}$-regular with respect to $\bar{m}$.
*Proof.* — The proof is similar to that of [MS, Theorem 3.4.1], with modifications as in Proposition 5.8.8. □

**6. The chain map from $\widehat{HF}$ to PFH**

**6.1. Compactness for $W_+$-curves.** — In this subsection we treat the compactness of holomorphic curves in $W_+$ which will be used to establish the chain map $\Phi$ in Section 6.2.

Suppose $J_+ \in \mathcal{J}_{W_+}$ and $J, J'$ are the restrictions of $J_+$ to the positive and negative ends. Let $\mathbf{y} = \{y_1, \dots, y_{2g}\} \in S_{a, \mathcal{h}(a)}$ and $\mathbf{y} = \prod_{k=1}^l \gamma_k^{m_k} \in \mathcal{O}_{2g}$.

*In this section, we may pass to a subsequence of a sequence of holomorphic curves without specific mention.*

**6.1.1. Euler characteristic bounds.** — We first state a preliminary lemma:

**Lemma 6.1.1.** — Let $u_i : (\dot{\mathbb{F}}_i, j_i) \to (W_+, J_+)$, $i \in \mathbf{N}$, be a sequence of $W_+$-curves from $\mathbf{y}$ to $\mathbf{y}$ with index $I_{W_+}(u_i) = n$ for some integer $n$. Then there is a subsequence such that all the $\dot{\mathbb{F}}_i$ are diffeomorphic to a fixed $\dot{\mathbb{F}}$.

*Proof.* — The proof is given in two steps.

**Step 1 (ω-area bounds).** We define the $\omega$-area of $u_i$ as

$$E_{\omega}(u_i) = \int_{\dot{\mathbb{F}}_i} u_i^* \omega,$$

where $\omega$ is the 2-form as in Section 5.1. The boundedness of $E_{\omega}(u_i)$ is a consequence of the vanishing of the flux $F_{\mathcal{h}}$ of $\mathcal{h}$ (cf. Section 3.3). View the broken closed string $\gamma_{\mathbf{y}}$ corresponding to $\mathbf{y}$ as a collection of curves in

$$(L_{\mathbf{a}}^{+} \cap \check{W}_{+}) \cup (\{3\} \times [0, 1] \times \mathbf{y}) \subset \check{W}_{+}.$$

Then $\gamma_{\mathbf{y}}$ is uniquely determined up to a homotopy which is supported on $L_{\mathbf{a}}^{+} \cap \check{W}_{+}$. Let $u_i : \dot{\mathbb{F}}_i \to W_+$, $i = 1, 2$, be two $W_+$-curves from $\mathbf{y}$ to $\mathbf{y}$ and let $\check{u}_i : \check{\mathbb{F}}_i \to \check{W}_+$ be their compactifications. Then $\check{u}_i(\partial_+ \check{\mathbb{F}}_i)$ is homotopic to $\gamma_{\mathbf{y}}$, $i = 1, 2$, where $\partial_+ \check{\mathbb{F}}_i$ is the union of boundary components of $\check{\mathbb{F}}_i$ which map to the positive $(s > 0)$ part of $\check{W}_+$. Hence $\check{u}_1 - \check{u}_2$ can be viewed as a closed surface $Z \in H_2(W_+) \simeq H_2(\check{W}_+) \simeq H_2(\mathbb{N})$. Since the flux $F_{\mathcal{h}}$ vanishes, the integral of $\omega$ over $Z$ vanishes, and therefore the $\omega$-area of a $W_+$-curve $u$ only depends on $\mathbf{y}$ and $\mathbf{y}$.

**Step 2 (Genus bounds).** The $\omega$-area bound on the sequence $\{u_i\}$ and the fact that all $u_i$ have the same asymptotics imply an energy bound on the $u_i$ (for the energy defined as in Equation (4.3.1)).

We then apply the Gromov-Taubes compactness theorem [T3, Proposition 3.3], whose proof carries over to the symplectic cobordism $(W_+, \Omega_+)$ with little change: The
energy bound implies a local $\Omega_+$-area bound, and we note the statement of [T3, Proposition 3.3] is local, i.e., works on a fixed ball in $W_+$. To take care of the Lagrangian boundary condition, we take local doubles of $u_i$ using Conditions (SR) that appear in Definitions 5.3.2 and 5.3.14.

More precisely, given any point $p \in L_a^+$, there is a sufficiently small neighborhood of $p$ in $\bar{W}_+$ of the form $\mathcal{U} \times D$ and a homeomorphism

$$ \phi = (\phi_1, \phi_2) : H \times D^2 \to \mathcal{U} \times D, $$

where:

(1) $H = \{(x_1, y_1) \in \mathbb{R}^2 \mid x_1^2 + y_1^2 \le 1, y_1 \ge 0\}$ and $D^2 = \{(x_2, y_2) \in \mathbb{R}^2 \mid x_2^2 + y_2^2 \le 1\}$ with the standard complex structures, $\mathcal{U} \subset B_+$ is a closed neighborhood of $\pi_{W_+}(p)$, and $D \subset \bar{S}$ is a closed disk;

(2) $\phi_1 : H \to \mathcal{U}$ is a homeomorphism which is holomorphic on $\text{int}(H)$ and is a diffeomorphism away from the two corner points of $H$;

(3) $\phi_2 : D^2 \to D$ is a biholomorphism; and

(4) $\phi^{-1}(p) = (0, 0, 0, 0)$ and $\phi^{-1}(L_a^+ \cap (\mathcal{U} \times D)) = \{y_1 = y_2 = 0\} \subset H \times D^2$.

The existence of the map $\phi_1$ is proved as follows: Let $\mathcal{U}$ be the intersection of $B_+$ with a closed ball in $\mathbb{R} \times S^1$ centered at $\pi_{W_+}(p)$. If the radius of the ball is sufficiently small, then $\mathcal{U}$ is a closed half-disk whose boundary is continuous and piecewise smooth with two corner points only at $\mathcal{U} \cap \partial B_+$. By the Riemann mapping theorem there exists a biholomorphism $\phi_1' : \text{int}(H) \to \text{int}(\mathcal{U})$, which, by Carathéodory's theorem, extends to a homeomorphism $\phi_1 : H \to \mathcal{U}$; see [Ru, Theorem 14.18]. Away from the two corner points, the extension is smooth by [Wa1, Theorem IV] and has nonvanishing derivative by [Wa2, Theorem 1]. Note that we cannot assume that $\phi_1$ is holomorphic up to the boundary because $\partial B_+$ was not defined to be real analytic. The existence of the map $\phi_2$ is similar but easier. Condition (SR) from Definitions 5.3.2 implies that the pullback of $J_+$ to $\{y_1 < 0\} \times D$ by $\phi$ is the standard split complex structure.

Denote $G_i := (u_i)^{-1}(\mathcal{U} \times D) \subset \tilde{F}$ and $v_i = \phi^{-1} \circ u_i|_{G_i}$. By the Schwarz reflection principle we can double $G_i$ and $H$ along $G_i \cap \partial \tilde{F}$ and $\partial H \cap \{y_1 = 0\}$ and obtain maps $\tilde{v}_i : \tilde{G}_i \to \tilde{H} \times D^2$ which extend the maps $v_i$ and are holomorphic on $\tilde{G}_i \setminus \partial \tilde{F}$. Then the $\tilde{v}_i$ are holomorphic on $\tilde{G}_i$ by an application of Morera's theorem; see [Ru, Theorem 16.8]. The maps $\tilde{v}_i$ also satisfy energy bounds and [T3, Proposition 3.3] carries over to $\tilde{H} \times D^2$, hence yielding a holomorphic limit $\tilde{v}_\infty$. It remains to note that half of $\tilde{v}_\infty$ corresponding to the limit of $v_i$ will have boundary on $L_a^+$ since during the construction of $\text{Im} \tilde{v}_\infty$ as a set in the proof of [T3, Lemma 3.5], we take small balls centered at points of $\text{Im} \tilde{v}_i$ and covering $\text{Im} \tilde{v}_i$ and we can choose a subset of them to be centered at points of $v_i(\partial \tilde{F}_i)$ and to cover $v_i(\partial \tilde{F}_i)$.

The Gromov-Taubes compactness theorem and the argument of [Hu1, Lemma 9.8] then imply the weak convergence of $u_i$ as currents to a holomorphic building $u_\infty$. In particular, we may assume that $[u_i] \in H_2(W_+, \mathbf{y}, \mathbf{y})$ is fixed for all $i$.
We now use the fact that $[u_i]$ is fixed to bound the genus of $\dot{F}_i$. The relative adjunction formula (Lemma 5.6.3) gives:

$$c_1(\tilde{u}_i^* \mathrm{TW}_+, (\tau, \partial_i)) = \chi(\dot{F}_i) - w_\tau^-(u_i) + Q_\tau(u_i) - 2\delta(u_i).$$

In view of the writhe bound

$$w_\tau^-(u_i) \geq \tilde{\mu}_\tau(\gamma) - \mu_\tau^-(u_i)$$

from [Hu2, Lemma 4.20] and the nonnegativity of $\delta(u_i)$, we obtain:

$$(6.1.1) \qquad \chi(\dot{F}_i) \geq c_1(\tilde{u}_i^* \mathrm{TW}_+, (\tau, \partial_i)) + \tilde{\mu}_\tau(\gamma) - \mu_\tau^-(u_i) - Q_\tau(u_i).$$

This bounds $\chi(\dot{F}_i)$ from below, since all the terms on the right-hand side either depend on the homology class of $u_i$ or the data of the ends. Hence we may assume that all the $\dot{F}_i$ are diffeomorphic to a fixed $\dot{F}$. $\square$

**6.1.2. SFT compactness.**

*Definition 6.1.2.* — A holomorphic $W_+$-building

$$u_\infty = v_{-b} \cup \dots \cup v_a, \quad a, b \in \mathbb{Z}^{\ge 0}$$

consists of the following data:

(B1) For each $j = -b, \dots, a$, a compact nodal Riemann surface $G_j$ (possibly with boundary), a nodal set $\mathfrak{n}_j$, disjoint sets of interior punctures $\mathbf{p}_j^+$ and $\mathbf{p}_j^-$ if $j \le 0$, and disjoint sets of boundary punctures $\mathbf{q}_j^+$ and $\mathbf{q}_j^-$ if $j \ge 0$. The nodes may be interior or boundary nodes and are disjoint from $\mathbf{p}_j^\pm$, $\mathbf{q}_j^\pm$, and the nodal set $\mathfrak{n}_j$ may be empty.

(B2) For each $j = -b, \dots, a$, a holomorphic map $v_j : \dot{G}_j \to W_j$, where $W_j = W$ for $0 < j \le a$, $W_0 = W_+$ for $j = 0$, $W_j = W'$ for $-b \le j < 0$, and $\dot{G}_j = G_j - \mathbf{p}_j^+ - \mathbf{p}_j^- - \mathbf{q}_j^+ - \mathbf{q}_j^-$ for all $j$. (Here some sets of punctures may be empty.)

(B3) For each $j = 0, \dots, a$, $\partial \dot{G}_j$ is mapped to the appropriate Lagrangian submanifold $L_a \sqcup L_{fi(a)}$ or $L_a^+$.

(B4) For each $j = -b, \dots, a$, $v_j$ converges to a strip over a “Reeb chord” at the positive (resp. negative) end near each boundary puncture of $\mathbf{q}_j^+$ (resp. $\mathbf{q}_j^-$) and to a cylinder over a closed orbit at the positive (resp. negative) end near each interior puncture of $\mathbf{p}_j^+$ (resp. $\mathbf{p}_j^-$).

(B5) for each $j = -b, \dots, a-1$, there is an identification between $\mathbf{p}_j^+$ and $\mathbf{p}_{j+1}^-$ and an identification between $\mathbf{q}_j^+$ and $\mathbf{q}_{j+1}^-$ such that the pairs that are identified are asymptotic to the same Reeb chord or closed orbit.

(B6) No level $v_j$ is a union of trivial cylinders or trivial strips and no constant sphere has fewer than three nodes.

The levels $v_j$ are arranged in order from lowest to highest.
**Proposition 6.1.3.** — Let $u_i : (\dot{F}_i, j_i) \to (W_+, J_+)$, $i \in \mathbf{N}$, be a sequence of $W_+$-curves from $\mathbf{y}$ to $\mathbf{y}$ with index $I_{W_+}(u_i) = n$ for some integer $n$. Then there is a subsequence which converges in the sense of SFT to a level $a + b + 1$ holomorphic $W_+$-building $u_\infty = v_{-b} \cup \cdots \cup v_a$.

Here “convergence in the SFT sense” means convergence with respect to the topology described in [BEHWZ].

*Proof.* — By Lemma 6.1.1, we may assume that $\dot{F}_i = \dot{F}$ as smooth surfaces. We can then apply the SFT compactness theorem from [BEHWZ]; note that SFT compactness in the presence of Lagrangian boundary conditions is sketched in [BEHWZ, Section 11.3] and [Abb2, Theorem 3.20]. More details can be extracted from the proof of the less standard compactness theorem for $\overline{W}_-$-curves; see Proposition 7.3.2. $\square$

The limiting curve $u_\infty$ can be written as a level $a + b + 1$ holomorphic building $v_{-b} \cup \cdots \cup v_a$, where each $v_j$ is not necessarily irreducible and may have nodes. Here (i) $a, b$ are nonnegative integers, (ii) $v_j$ is a holomorphic map to $W_j$, and (iii) the levels $v_j$ are ordered from the negative end to the positive end as $j$ increases. As usual, if the level $v_j$ is just a union of trivial cylinders, then it will be elided.

**6.1.3. Main theorem.** — Suppose $J_+ \in \mathcal{J}_{W_+}^{\text{reg}}$ and $J, J'$ are the restrictions of $J_+$ to the positive and negative ends.

The following is the main theorem of this subsection:

**Theorem 6.1.4.** — Let $u_i : (\dot{F}_i, j_i) \to (W_+, J_+)$, $i \in \mathbf{N}$, be a sequence of $W_+$-curves from $\mathbf{y}$ to $\mathbf{y}$. If $I_{W_+}(u_i) = 1$ for all $i$, then a subsequence of $u_i$ converges in the sense of SFT to one of the following:

(1) an $I_{W_+} = 1$ curve;

(2) a building with two levels consisting of an $I_{\text{HF}} = 1$ curve and an $I_{W_+} = 0$ curve; or

(3) a building with multiple levels consisting of an $I_{W_+} = 0$ curve, an $I_{\text{ECH}} = 1$ curve, and possible $I_{\text{ECH}} = 0$ connectors in between.

Similarly, if $I_{W_+}(u_i) = 0$ for all $i$, then a subsequence of $u_i$ converges to an $I_{W_+} = 0$ curve.

We write “an $I_\# = i$ curve” as shorthand for “a \#-curve with ECH index $I_\# = i$”.

We postpone the proof of the main theorem until after a more detailed discussion of the structure of the SFT limit. Let $u_\infty$ be the SFT limit of the sequence $\{u_i\}$, given by Proposition 6.1.3. A *ghost component* of $u_\infty$ is an *irreducible component* of $u_\infty$ which maps to a point. By the SFT compactness theorem, the domain of a ghost component is necessarily a stable Riemann surface. We recall that a Riemann surface is stable if the following holds for each component F with $k_{\text{int}}$ interior marked points and $k_{\text{bdr}}$ boundary marked points:

$$ -\chi(F) + k_{\text{int}} \ge 1 \text{ when } \partial F = \emptyset, \text{ or} $$
$$ -2\chi(\mathbf{F}) + 2k_{\text{int}} + k_{\text{bdr}} \geq 1 \text{ if } \partial\mathbf{F} \neq \emptyset. $$

Let us write $u_{\infty} = v_{-b} \cup \cdots \cup v_a \cup u_{\infty}^g$, where $v_j$ has no ghost components and $u_{\infty}^g$ is the union of ghost components. We will also write $u_{\infty}^{ng} = v_{-b} \cup \cdots \cup v_a$.

**Remark 6.1.5.** — The ECH index of a curve depends only on its relative homology class and therefore ghost components do not contribute to it. Hence, by the additivity of ECH indices (Lemma 5.6.8), if $u_i$ is a sequence of $J_+$-holomorphic maps with constant ECH index, then:

$$ (6.1.2) \qquad \sum_{j=1}^{a} \mathrm{I}_{\mathrm{HF}}(v_j) + \mathrm{I}_{W_+}(v_0) + \sum_{j=1}^{b} \mathrm{I}_{\mathrm{ECH}}(v_{-j}) = \mathrm{I}_{W_+}(u_i). $$

**Lemma 6.1.6.** — Each level $v_j, j = -b, \dots, a$, is a degree $2g$ multisection of $\pi_j : W_j \to B_j$ with no branch points along $\partial B_j$.

*Proof.* — Since $u_i$ is a degree $2g$ multisection of $\pi_{B_+} : W_+ \to B_+$ for all $i$, it follows that, with the exception of finitely many $p \in B_j$, every level $v_j$ intersects a fiber $\pi_j^{-1}(p)$ exactly $2g$ times.

We show that on any level $v_j$ there are no irreducible components which lie in a fiber $\pi_j^{-1}(p)$. Arguing by contradiction, suppose $\tilde{v} : \tilde{\mathbf{F}} \to W_j$ is an irreducible component which maps to a fiber $\pi_j^{-1}(p)$. If $p \in \mathrm{int}(B_j)$, then $\tilde{v}$ is a holomorphic map from a closed Riemann surface $\tilde{\mathbf{F}}$ to $\pi_j^{-1}(p)$. Since $\pi_j^{-1}(p)$ is a Riemann surface with nonempty boundary, $\tilde{v}$ must be constant. On the other hand, if $p \in \partial B_j$, it is also possible that $\tilde{\mathbf{F}}$ is a compact Riemann surface with nonempty boundary and $\tilde{v}(\partial \tilde{\mathbf{F}}) \subset \mathbf{a}$. However, since $S - \mathbf{a}$ is connected and nontrivially intersects $\partial S$, $\tilde{v}$ must also be constant. Since ghost components are excluded from $v_j$ by definition, we have a contradiction.

Finally, if $j \geq 0$, then we claim that $\pi_j \circ v_j$ has no branch points along $\partial B_j$. This is due to the fact that $v_0$ uses each component of $L_{\mathbf{a}}^+$ exactly once and $v_j, j > 0$, uses each component of $\mathbf{R} \times \{1\} \times \mathbf{a}$ and each component of $\mathbf{R} \times \{0\} \times \mathcal{H}(\mathbf{a})$ exactly once. $\square$

**Lemma 6.1.7.** — Let $u_{\infty}$ be the SFT limit of a sequence of $J_+$-holomorphic multisections $u_i$ with constant ECH index. If $J_+$ is regular, then:

$$ - \mathrm{I}_{\mathrm{HF}}(v_j) > 0 \text{ for } j > 0, $$

$$ - \mathrm{I}_{W_+}(v_0) \geq 0, \text{ and} $$

$$ - \mathrm{I}_{\mathrm{ECH}}(v_j) \geq 0 \text{ for } j < 0. $$

Moreover, all the $v_j, j \geq 0$, are somewhere injective and satisfy $\mathrm{ind}(v_j) \geq 0$. If $\mathrm{I}_{W_+}(u_i) \leq 1$ in addition, then $v_j, j < 0$, is somewhere injective and satisfies $\mathrm{ind}(v_j) \geq 0$.

*Proof.* — Since $v_j, j \geq 0$, is a degree $2g$ multisection by Lemma 6.1.6 and uses each connected component of the boundary Lagrangian exactly once, it is somewhere
injective. Also since J and J_+ are regular, it follows that the curves v_j, j ≥ 0, are regular. Hence ind_W(v_j) ≥ 0 for j > 0 and ind_{W_+}(v_0) ≥ 0. Moreover, since ind_W(v_j) = 0, j > 0, if and only if v_j is a union of trivial strips, we may assume that ind_W(v_j) > 0 for all j > 0. By the index inequality (Theorems 4.5.13 and 5.6.9) we have I_{HF}(v_j) > 0 for j > 0 and I_{W_+}(v_0) ≥ 0. On the other hand, if j < 0, then I_{ECH}(v_j) ≥ 0 by [HT1, Proposition 7.15(a)].

If $I_{W_+}(u_i) \le 1$ in addition, then $I_{ECH}(v_j) \le 1$ by the additivity of the ECH index. Hence [Hu1, Lemma 9.5] implies that $v_j$ is somewhere injective and $\mathrm{ind}(v_j) \ge 0$ since J' is regular. □

**Lemma 6.1.8.** — Let $u_\infty$ be the SFT limit of a sequence of $J_+$-holomorphic multisections $u_i$ with $I_{W_+}(u_i) \le 1$ for all $i$. If $J_+$ is regular, then $u_\infty^g = \emptyset$.

*Proof.* — Let $\mathrm{ind}(u_\infty^g)$ and $\mathrm{ind}(u_\infty^{ng})$ be the sum of the Fredholm indices of the irreducible components of $u_\infty^g$ and $u_\infty^{ng}$, respectively. Also let $F^g$ and $F^{ng}$ be the domains of $u_\infty^g$ and $u_\infty^{ng}$, respectively. Then $\mathrm{ind}(u_\infty^g) = -\chi(F^g)$. If $u_\infty^g$ and $u_\infty^{ng}$ are attached along $k_{int}$ interior nodes and $k_{bdr}$ boundary nodes, then

$$ (6.1.3) \qquad \mathrm{ind}(u_i) = \mathrm{ind}(u_\infty^{ng}) + \mathrm{ind}(u_\infty^g) + 2k_{int} + k_{bdr} $$

$$ (6.1.4) \qquad = \mathrm{ind}(u_\infty^{ng}) - \chi(F^g) + 2k_{int} + k_{bdr} \le 1. $$

Next, if $F^g \ne \emptyset$, then the stability of the Riemann surface and the observation that each component of $\partial F^g$ has a marked point (which in turn follows from Definition 5.4.3(1)) imply that

$$ (6.1.5) \qquad -\chi(F^g) + 2k_{int} + k_{bdr} \ge 2. $$

Equations (6.1.4) and (6.1.5) together imply that $\mathrm{ind}(u_\infty^{ng}) \le -1$. Hence we have $\mathrm{ind}(v_j) \le -1$ for some $v_j$, which is a contradiction of Lemma 6.1.7 for a regular $J_+$. The contradiction came from assuming that $u_\infty^g \ne \emptyset$. □

We now finish the proof of Theorem 6.1.4.

*Proof of Theorem 6.1.4.* The proof is based on a classification of the types of allowable buildings $u_\infty = v_{-b} \cup \cdots \cup v_a \cup u_\infty^g$. We consider the situation of $I_{W_+}(u_i) = 1$, leaving the easier $I_{W_+}(u_i) = 0$ case to the reader.

By Lemmas 6.1.6 and 6.1.8, the limit $u_\infty$ consists of a building of degree $2g$ multisections $v_j$. Moreover, by Lemma 6.1.7, $I_{W_j}(v_j) \ge 0$ for all $j$. The additivity of ECH indices gives three possibilities for the limit:

(1) $u_\infty = v_0$, where $v_0$ is a multisection of $W_+$ and $I_{W_+}(v_0) = 1$;

(2) $u_\infty = v_0 \cup v_1$, where $v_0$ is a multisection of $W_+$ with $I_{W_+}(v_0) = 0$, and $v_1$ is a multisection of W with $I_{HF}(v_1) = 1$;
(3) $u_\infty = v_{-b} \cup \cdots \cup v_0$, where $v_{-b}, \dots, v_{-1}$ are multisections of $W'$, $v_0$ is a multi-section of $W_+$, all but one of $v_{-b}, \dots, v_0$ have $I=0$, and the remaining level has $I=1$.

What is left to prove is that $I_{ECH}(v_{-b}) = 1$ in Case (3). This follows from considerations of incoming partitions as in [HT1, Proof of Lemma 7.23]. Let $m_k$ be the multiplicity of the elliptic orbit $\gamma_k$ in the orbit set $\mathbf{\gamma}$ and let $\theta_k$ be the rotation of the first return map of $\gamma_k$. By [HT1, Definition 1.8], there is a partial order $\geq_{\theta_k}$ on the set of partitions of $m_k$, where

$$ (a_1, \dots, a_{l_1}) \geq_{\theta_k} (b_1, \dots, b_{l_2}) $$

if there is a Fredholm index zero branched cover of $\mathbf{R} \times \gamma_k$ with positive ends which partition $m_k$ into $(a_1, \dots, a_{l_1})$ and negative ends which partition $m_k$ into $(b_1, \dots, b_{l_2})$. By [HT1, Lemma 7.5], the incoming partition $P_{\gamma_k}^{\text{out}}(m_k)$ — the partition which corresponds to $\gamma_k^{m_k}$ at the negative end of $v_{-b}$ — is maximal. Hence $I_{ECH}(v_{-b}) = 0$ implies that $v_{-b}$ is a trivial cylinder, which we have already eliminated. $\square$

**6.2. Definition of $\Phi$.** — Suppose $J_+ \in \mathcal{J}_{W_+}^{\text{reg}}$ and $J, J'$ are the restrictions of $J_+$ to the positive and negative ends. In this subsection we define the chain map

$$ (6.2.1) \qquad \Phi_{J_+} : \widehat{CF}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}), J) \to \text{PFC}_{2g}(N, J'). $$

We will usually suppress $J, J'$, and $J_+$ from the notation.

We first define an approximation of $\Phi$:

**Definition 6.2.1.** — Let $\widehat{CF}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ be the chain complex generated by $S_{\mathbf{a}, \mathbf{h}(\mathbf{a})}$, before quotienting by the equivalence relation $\sim$ given in Section 4.9.3. We define the map

$$ \Phi' : \widehat{CF}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \text{PFC}_{2g}(N), $$

$$ \Phi'(\mathbf{y}) = \sum_{\mathbf{y} \in \widehat{\mathcal{O}}_{2g}} \langle \Phi'(\mathbf{y}), \mathbf{y} \rangle \cdot \mathbf{y}, $$

where $\langle \Phi'(\mathbf{y}), \mathbf{y} \rangle$ is the mod 2 count of $M_{J_+}^{I=0}(\mathbf{y}, \mathbf{y})$. The count is meaningful since $M_{J_+}^{I=0}(\mathbf{y}, \mathbf{y})$ is compact by Theorem 6.1.4.

**Proposition 6.2.2.** — The map

$$ \Phi' : \widehat{CF}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \text{PFC}_{2g}(N), $$

is a chain map.
*Proof.* — By Theorem 6.1.4 and Lemma 5.4.9, $\partial \mathcal{M}_{J_+}^{I=1}(\mathbf{y}, \mathbf{\gamma}) = A \sqcup B$, where

$$A = \coprod_{\mathbf{y}' \in S_{a, \hbar(a)}} \left( \left( \mathcal{M}_{J}^{I=1}(\mathbf{y}, \mathbf{y}')/\mathbf{R} \right) \times \mathcal{M}_{J_+}^{I=0}(\mathbf{y}', \mathbf{\gamma}) \right),$$

$$B = \coprod_{\mathbf{\gamma} \in \tilde{\mathcal{O}}_{2g}} \left( \mathcal{M}_{J_+}^{I=0}(\mathbf{y}', \mathbf{\gamma}') \times \left( \mathcal{M}_{J'}^{I=1}(\mathbf{\gamma}', \mathbf{\gamma})/\mathbf{R} \right) \right).$$

Here we have omitted the connector components for simplicity.

We examine the corresponding gluings of the holomorphic buildings. The first gluing is that of $(v_1, v_0) \in A$. This type of gluing was treated by Lipshitz; see Propositions A.1 and A.2 in [Li, Appendix A]. Observe that there are no multiply-covered curves to glue, since each Reeb chord of a $2g$-tuple is used exactly once.

The second type of gluing is that of $(v_0, v_{-b}) \in B$, with $I_{ECH} = 0$ connectors $v_{-1}, \dots, v_{-b+1}$ in between. The curve $v_0$ is simply-covered since it has an HF end and the curve $v_{-b}$ is simply-covered since $I_{ECH} = 1$ (see [HT1, Proposition 7.15]). This type of gluing was treated carefully in [HT1, HT2]. Although the setting there was the gluing for $\partial^2 = 0$, in fact most of the work goes towards properly counting $I_{ECH} = 0$ connectors, i.e., branched covers of trivial cylinders. Their treatment of gluing/counting the $I_{ECH} = 0$ connectors carries over with little modification to our case. See Section 6.5 for more details. $\square$

Let $\delta_x = ([0, 2] \times \{x\})/(2, x) \sim (0, x)$, where $x \in \partial S$; it is a Reeb orbit in the negative Morse-Bott family $\mathcal{N}$ which foliates $\partial N$. Also let $(\mathbf{R} \times \delta_x)^+ = \text{the restriction of } \mathbf{R} \times \delta_x$ to $W_+$.

**Lemma 6.2.3.** — A $W_+$-curve $u$ which has $x_i$ or $x'_i$ at the positive end must have $(\mathbf{R} \times \delta_{x_i})^+$ or $(\mathbf{R} \times \delta_{x'_i})^+$ as an irreducible component.

Recall that the intersection points $x_i, x'_i$ are components of the Heegaard Floer contact class of $\xi_{(S, \hbar)}$.

*Proof.* — Let $v : \dot{\mathbf{F}} \to W_+$ be the irreducible component of $u$ which has $x_i$ at the positive end. The component $v$ may *a priori* have other positive ends besides $x_i$. We will show that $v = (\mathbf{R} \times \delta_{x_i})^+$. Let $\pi_S : \{s \ge 3\} \cap W_+ \to S$ be defined by identifying $\{s \ge 3\} \cap W_+ = [3, \infty) \times [0, 1] \times S$ and projecting to the third factor.

Let $(r, \theta)$ be polar coordinates on a small neighborhood $v(x_i) \subset S$ of $x_i$ so that $\hbar(a_i) = \{\theta = -\frac{\pi}{4}\}, a_i = \{\theta = 0\}$, and $v(x_i) = \{-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}, r > 0\}$. By Condition (1) in Definition 5.3.2, the composition $\pi_S \circ v$ is holomorphic when restricted to the end that limits to $x_i$. If $v$ is not the restriction of a trivial cylinder, then $\pi_S \circ v$ must map a neighborhood of the puncture of $\dot{\mathbf{F}}$ corresponding to $x_i$ to a sector $\{0 \le \theta \le k\pi - \frac{\pi}{4}, r > 0\}$ or $\{\pi \le \theta \le (k+1)\pi - \frac{\pi}{4}, r > 0\}$, where $k \ge 1$. Since such a sector cannot
be contained in S, the map $\pi_S \circ v|_{S \ge C}$ must map identically to $x_i$, where $C \gg 0$. By the unique continuation property, $v = (\mathbf{R} \times \delta_{x_i})^+$. $\square$

**Theorem 6.2.4.** — The map $\Phi'$ descends to a chain map

$$ \Phi: \widehat{CF}(S, \mathbf{a}, \hbar(\mathbf{a})) \to \mathrm{PFC}_{2g}(N) $$

which maps the Heegaard Floer contact invariant for $\xi_{(S, \hbar)}$ to the ECH contact invariant for $\xi_{(S, \hbar)}$.

*Proof.* — Let $\mathbf{y} = \{y_1, \dots, y_{2g}\} \in S_{\mathbf{a}, \hbar(\mathbf{a})}$. Assume without loss of generality that $y_1 = x_1$. Then $\mathbf{y}$ is equivalent to $\mathbf{y}' = \{x'_1, y_2, \dots, y_{2g}\}$. Let $u$ be an $I_{W_+} = 0$ Morse-Bott building from $\mathbf{y}$ to a generator $\mathbf{y}$ of $\mathrm{PFC}_{2g}(N)$. Since $(\mathbf{R} \times \delta_{x_1})^+ = \mathrm{PFC}_{2g}(N)$ is an irreducible component of $u$ by Lemma 6.2.3 and is automatically transverse in the Morse-Bott sense, $\mathbf{y}$ can be written as $e\mathbf{y}'$, where $e$ is the elliptic orbit of the negative Morse-Bott family $\mathcal{N}$. Now, by replacing $(\mathbf{R} \times \delta_{x_1})^+$ by $(\mathbf{R} \times \delta_{x'_1})^+$ and the augmenting gradient trajectory from $\delta_{x_1}$ to $e$ by the augmenting trajectory from $\delta_{x'_1}$ to $e$, we obtain an $I_{W_+} = 0$ Morse-Bott building from $\mathbf{y}'$ to $\mathbf{y} = e\mathbf{y}'$. Hence there is a one-to-one correspondence between $W_+$-curves from $\mathbf{y}$ to $\mathbf{y}$ and $W_+$-curves from $\mathbf{y}'$ to $\mathbf{y}$. Since $\Phi'(\mathbf{y}) = \Phi'(\mathbf{y}')$, the map $\Phi'$ descends to $\widehat{CF}(S, \mathbf{a}, \hbar(\mathbf{a}))$.

The Heegaard Floer contact invariant is given by the equivalence class of $\mathbf{x} = \{x_1, \dots, x_{2g}\}$ and the ECH contact invariant is given by $e^{2g}$. By Lemma 6.2.3 and some ECH index considerations, the only $I_{W_+} = 0$ Morse-Bott building from $\mathbf{x}$ consists of $(\mathbf{R} \times \delta_{x_i})^+$, $i = 1, \dots, 2g$, augmented by connecting gradient trajectories from $\delta_{x_i}$ to $e$ in the Morse-Bott family $\mathcal{N}$. Hence $\Phi'(\mathbf{x}) = e^{2g}$. Then $\Phi$ maps the equivalence class of $\mathbf{x}$ to $e^{2g}$. $\square$

**6.3. Spin<sup>*e*</sup>-structures.** — Let $S_{\mathbf{a}, \hbar(\mathbf{a})}$ be the set of 2g-tuples of $\mathbf{a} \cap \hbar(\mathbf{a})$ and let $\bar{S}_{\mathbf{a}, \hbar(\mathbf{a})} = S_{\mathbf{a}, \hbar(\mathbf{a})} / \sim$, where $\{x_i\} \cup \mathbf{y}' \sim \{x'_i\} \cup \mathbf{y}'$ for all $(2g-1)$-tuples $\mathbf{y}'$. We define a map

$$ \mathfrak{h}'_+ : S_{\mathbf{a}, \hbar(\mathbf{a})} \to \mathrm{H}_1(W_+, \partial_{\hbar}W_+) $$

as follows: Given $\mathbf{y} = \{y_1, \dots, y_{2g}\} \in S_{\mathbf{a}, \hbar(\mathbf{a})}$, where $y_i \in a_i \cap \hbar(a_{\sigma(i)})$ for some permutation $\sigma \in \mathfrak{S}_{2g}$, we define $\mathfrak{h}'_+(\mathbf{y})$ as the homology class of the broken closed string obtained by concatenating the following oriented arcs:

- for each $i \in \{1, \dots, 2g\}$, the arc $\{3\} \times [0, 1] \times \{y_i\}$, with orientation given by $\partial_i$;

- for each $i \in \{1, \dots, 2g\}$, an arc from $\{3\} \times \{1\} \times \{y_i\}$ to $\{3\} \times \{0\} \times \{y_{\sigma^{-1}(i)}\}$ contained in $L_{a_i}^+$.

The homology class $\mathfrak{h}'_+(\mathbf{y})$ is well-defined since the Lagrangians $L_{a_i}^+$ are simply-connected.
The map $\mathfrak{h}'_+$ descends to a map

$$ \mathfrak{h}_+ : \bar{S}_{\mathbf{a}, \mathfrak{h}(\mathbf{a})} \to \mathrm{H}_1(\mathrm{W}_+, \partial_{\mathfrak{h}}\mathrm{W}_+), $$

since the components $x_i$ and $x'_i$ of the contact class are both converted into broken closed strings which are nullhomologous in $\mathrm{H}_1(\mathrm{W}_+, \partial_{\mathfrak{h}}\mathrm{W}_+)$.

The following lemma is an immediate consequence of the definition of $\mathrm{W}_+$-curves.

**Lemma 6.3.1.** — Let $\mathbf{\gamma}$ be an orbit set at the negative end of $\mathrm{W}_+$ with total homology class $[\mathbf{\gamma}] \in \mathrm{H}_1(\mathrm{W}_+, \partial_{\mathfrak{h}}\mathrm{W}_+)$. Then $\mathcal{M}_{\mathrm{H}_+(\mathbf{y}, \mathbf{\gamma})} \neq \emptyset$ implies that $\mathfrak{h}_+(\mathbf{y}) = [\mathbf{\gamma}]$.

There are natural isomorphisms

$$ \mathrm{H}_1(\mathrm{W}_+, \partial_{\mathfrak{h}}\mathrm{W}_+) \cong \mathrm{H}_1(\mathrm{N}, \partial\mathrm{N}) \cong \mathrm{H}_1(\mathrm{M}). $$

With respect to these isomorphisms, the total homology class in $\mathrm{H}_1(\mathrm{W}_+, \partial_{\mathfrak{h}}\mathrm{W}_+)$ of an orbit set $\mathbf{\gamma}$ at the negative end of $\mathrm{W}_+$ corresponds to the usual total homology class of $\mathbf{\gamma}$ in $\mathrm{H}_1(\mathrm{M})$. Moreover, $\mathfrak{h}_+(\mathbf{y}) = \mathfrak{h}(\mathbf{y})$, where $\mathfrak{h}$ is as defined in Section 4.10.

Combining Proposition 4.10.1 and Lemma 6.3.1, we obtain the following theorem:

**Theorem 6.3.2.** — The chain map $\Phi$ respects the splitting according to Spin$^c$-structures, i.e., $\Phi$ is the direct sum of maps

$$ \Phi_{\Lambda} : \widehat{\mathrm{CF}}(\mathrm{S}, \mathbf{a}, \mathfrak{h}(\mathbf{a}), \mathfrak{s}_{\xi} + \mathrm{PD}(\mathrm{A})) \to \mathrm{PFC}_{2g}(\mathrm{N}, \mathrm{A}), $$

with $A \in \mathrm{H}_1(\mathrm{M}; \mathbf{Z})$.

**6.4. Twisted coefficients.** — Let $\widehat{\mathrm{PFH}}_{2g}(\mathrm{N}, \mathrm{A})$ be the twisted coefficient version of $\mathrm{PFH}_{2g}(\mathrm{N}, \mathrm{A})$, defined as in Section 2.4. For any homology class $A \in \mathrm{H}_1(\mathrm{M})$ we can define a map

$$ \Phi_{\Lambda} : \widehat{\mathrm{CF}}(\mathrm{S}, \mathbf{a}, \mathfrak{h}(\mathbf{a}), \mathfrak{s}_{\xi} + \mathrm{PD}(\mathrm{A})) \to \widehat{\mathrm{PFC}}_{2g}(\mathrm{N}, \mathrm{A}), $$

given in the following paragraphs. Choose a $2g$-tuple of intersection points $\mathbf{y}_0$ such that $s_z(\mathbf{y}_0) = \mathfrak{s}_{\xi} + \mathrm{PD}(\mathrm{A})$ and a complete set of paths $\{C_{\mathbf{y}}\}$ for $\mathfrak{s}_{\xi} + \mathrm{PD}(\mathrm{A})$ based at $\mathbf{y}_0$.

Let $\pi_N : \mathrm{W}_+ \to \mathrm{N}$ be the restriction of the projection $\mathbf{R} \times \mathrm{N} \to \mathrm{N}$, $(s, x) \mapsto x$, and let $\Gamma \subset \mathrm{N}$ be the projection by $\pi_N$ of a broken closed string associated to $\mathbf{y}_0$. By Lemma 6.3.1, $[\Gamma] = A$. We choose a complete set of paths $\{C_{\mathbf{y}}\}$ for $A$ based at $\Gamma$.

The projection $\pi_N$ associates a well-defined homology class in $\mathrm{H}_2(\mathrm{N}) \cong \mathrm{H}_2(\mathrm{M})$ to any 2-chain in $\tilde{\mathrm{W}}_+$ whose boundary consists of a broken closed string corresponding to $\mathbf{y}_0$ on one side and $\Gamma$ on the other side. Then we can define maps

$$ \mathfrak{A}_+ : \mathrm{H}_2(\mathrm{W}_+, \mathbf{y}, \mathbf{\gamma}) \to \mathrm{H}_2(\mathrm{M}) $$

for all $\mathbf{y}$ such that $s_z(\mathbf{y}_0) = \mathfrak{s}_{\xi} + \mathrm{PD}(\mathrm{A})$ and $\mathbf{\gamma}$ such that $[\mathbf{\gamma}] = A$ by

$$ \mathfrak{A}_+(C) = (\pi_N)_*[C_{\mathbf{y}} \cup C \cup -C_{\mathbf{y}}]. $$
**Definition 6.4.1.** — We define the $\mathbf{F}[H_2(M; \mathbf{Z})]$-linear map

$$ \Phi_{\Lambda} : \widehat{\mathbf{C}\mathbf{F}}(S, \mathbf{a}, h(\mathbf{a}), \mathfrak{s}_{\xi} + \mathrm{PD}(A)) \to \underline{\mathrm{PFC}}_{2g}(N, A) $$

by

$$ \Phi_{\Lambda}(\mathbf{y}) = \sum_{\mathbf{y} \in \mathcal{O}_{2g}} \sum_{C \in H_2(W_+, \mathbf{y}, \mathbf{y})} \# \mathcal{M}_{J_+}^{I=0}(\mathbf{y}, \mathbf{y}, C) e^{\mathfrak{a}_+(C)} \mathbf{y}. $$

**Theorem 6.4.2.** — The map $\Phi_{\Lambda}$ is a chain map.

*Proof.* — The proof is the same as that of Theorem 6.2.4, plus some bookkeeping of the homology classes of the holomorphic maps involved. $\square$

**6.5. Gluing.** — We explain here how to glue the pair $(v_0, v_{-1})$ consisting of a $W_+$-curve $v_0$ with $I(v_0) = 0$ and an ECH curve $v_{-1}$ with $I(v_{-1}) = 1$, by inserting a branched cover of a trivial cylinder (a *connector*). The connector is needed because, while the negative ends of $v_0$ and the positive ends of $v_{-1}$ converge to the same orbit sets, they usually do not match; in fact the ends of $v_0$ must satisfy the outgoing partition, while the ends of $v_{-1}$ must satisfy the incoming partition. The same situation appears in the definition of ECH and the procedure of gluing two $I_{\mathrm{ECH}} = 1$ curves in the symplectization, as explained in Hutchings-Taubes [HT1, HT2], applies with very little modification.

**6.5.1. Close to breaking.** — We first make precise what we mean by a holomorphic curve which is “close to breaking”. We rephrase the definition for $W'$ (which appears in [HT1]), leaving the analogous definitions for $W$, $W_+$, and $W_-$ to the reader.

Choose an $s$-invariant Riemannian metric $g$ on $W'$ and denote the induced distance by $d_g$.

**Definition 6.5.1.** — Given $\kappa, \nu > 0$, two curves $u_i : (F_i, j_i) \to W'$, $i = 1, 2$, are $(\kappa, \nu)$-close if there exists a $(1 + \nu)$-quasiconformal homeomorphism $\phi : F_1 \to F_2$ with respect to $(j_1, j_2)$ such that

$$ \sup_{x \in F_1} d_g(u_1(x), u_2 \circ \phi(x)) \le \kappa. $$

Given a $J$-holomorphic curve $u$ in $\mathbf{R} \times \mathbf{Y}$, by an abuse of notation we will denote the restriction of $u$ to the preimage of $[C, +\infty) \times \mathbf{M}$ by $u|_{s \le C}$. (Self-explanatory variations of this notations will also be used). The following is similar to [HT1, Definition 1.10], although the phrasing is slightly different.

**Definition 6.5.2.** — Let $\kappa > 0$ and $\nu \ge 0$. A curve $u$ in $W'$ is $(\kappa, \nu)$-close to an $(a+b+1)$-level building $u_{\infty} = v_{-b} \cup \dots \cup v_a$ if there exist $R^{(-2b+1)}$, $\dots$, $R^{(2a)} > \frac{1}{\kappa}$ such that, after a suitable translation of $u$ in the $s$-direction which we still call $u$, each of the pairs below is $(\kappa, \nu)$-close:
$$- u|_{-R^{(0)} \le s \le R^{(1)}} \text{ and } v_0|_{-R^{(0)} \le s \le R^{(1)}};$$

$$- u|_{R^{(1)} \le s \le R^{(1)} + R^{(2)}} \text{ and the restriction of a collection of branched covers of trivial cylinders to } R^{(1)} \le s \le R^{(1)} + R^{(2)};$$

$$- u|_{R^{(1)}+R^{(2)} \le s \le R^{(1)}+2R^{(2)}+R^{(3)}} \text{ and an } s \mapsto s+R^{(1)}+2R^{(2)} \text{ translate of } v_1|_{-R^{(2)} \le s \le R^{(3)}};$$

$$- u|_{R^{(1)}+2R^{(2)}+R^{(3)} \le s \le R^{(1)}+2R^{(2)}+R^{(3)}+R^{(4)}} \text{ and the restriction of a collection of branched covers of trivial cylinders to } R^{(1)}+2R^{(2)}+R^{(3)} \le s \le R^{(1)}+2R^{(2)}+R^{(3)}+R^{(4)};$$

and so on.

Note that, in the case of $u$ in $W_+$ or $W_-$, we do not need to (and indeed we cannot) translate $u$ in the $s$-direction.

**6.5.2. Review of [HT1, HT2].** — We now summarize the Hutchings-Taubes proof of $\partial^2 = 0$ and discuss the small changes that need to be carried out.

Let $(Y, \xi = \ker \lambda)$ be a closed 3-manifold with a nondegenerate contact form $\lambda$ and corresponding Reeb vector field $R = R_\lambda$. Let $(\mathbf{R} \times Y, d(e^\xi \lambda))$ be the symplectization of $Y$, where $s$ denotes the **R**-coordinate, and let $J$ be an adapted almost complex structure on $\mathbf{R} \times Y$.

Let $\alpha = (\alpha_1, \dots, \alpha_k)$ and $\beta = (\beta_1, \dots, \beta_l)$ be ordered sets of Reeb orbits of $\mathbf{R}$. This notation is confined to Section 6.5 and therefore should not create confusion with the similar notation denoting the Heegaard Floer multicurve. Here the Reeb orbits may not be embedded and may also be repeated. Then let $M_J(\alpha, \beta)$ be the moduli space of finite energy $J$-holomorphic curves in $(\mathbf{R} \times Y, J)$ from $\alpha$ to $\beta$. Let $\{\gamma_1, \gamma_2, \dots\}$ be the list of simple orbits of $\mathbf{R}$. For each simple orbit $\gamma_i$, we tally the total multiplicity $m_i(\alpha)$ of $\gamma_i$ in $\alpha$. In this way we can assign an orbit set $\mathbf{\gamma}(\alpha) = \prod_i \gamma_i^{m_i(\alpha)}$ to $\alpha$.

We want to glue the pair $(u_+, u_-)$, where $u_+ \in M_J^{I=1}(\alpha_+, \beta_+)$ and $u_- \in M_J^{I=1}(\alpha_-, \beta_-)$, provided $\mathbf{\gamma}(\beta_+) = \mathbf{\gamma}(\beta_-)$. Since $I(u_+) = I(u_-) = 1$, the curves $u_+$ and $u_-$ satisfy the partition condition at $\beta_+$ and $\beta_-$ for a generic $J$ (cf. Definition 7.11 and Proposition 7.14 of [HT1]). In particular, for each simple orbit $\gamma_i$, the total multiplicity $m_i(\beta_+) = m_i(\beta_-)$ completely determines the number of ends of $u_+$ or $u_-$ going to a cover of $\gamma_i$, together with their individual multiplicities. For $u_+$ (resp. $u_-$), this is encoded by the incoming partition (resp. outgoing partition) and is denoted by $P_{\gamma_i}^{\text{in}}(m_i(\beta_+))$ (resp. $P_{\gamma_i}^{\text{out}}(m_i(\beta_-))$). If $P_{\gamma_i}^{\text{in}}(m_i(\beta_+)) = (a_1, \dots, a_r)$, then $u_+$ has $r$ negative ends which go to a cover of $\gamma_i$ with covering multiplicities $a_1, \dots, a_r$. A similar statement holds for the positive ends of $u_-$.

Since $P_{\gamma_i}^{\text{in}}(m_i(\beta_+)) \neq P_{\gamma_i}^{\text{out}}(m_i(\beta_-))$ in general, we need to insert branched covers of cylinders $\mathbf{R} \times \gamma_i$ in order to be able to glue $u_+$ to $u_-$. Such branched covers $\pi : \Sigma \to \mathbf{R} \times \gamma_i$ must have $\beta_+$ at the positive end and $\beta_-$ at the negative end. A Fredholm index count ([HT1, Lemma 1.7]) implies that $\Sigma$ must have genus zero; moreover, the partition condition is equivalent to saying that the Fredholm index of the composition $u_\Sigma = \iota \circ \pi : \Sigma \to \mathbf{R} \times Y$ is zero, where $\iota : \mathbf{R} \times \gamma_i \to \mathbf{R} \times Y$ is the inclusion of the trivial cylinder $\mathbf{R} \times \gamma_i$.
Given $u_+ \in \mathcal{M}_J^{1=1}(\alpha_+, \beta_+)$ and $u_- \in \mathcal{M}_J^{1=1}(\beta_-, \alpha_-)$ with $\mathbf{\gamma}(\beta_+) = \mathbf{\gamma}(\beta_-)$, we define $\mathcal{G}_{\kappa,\nu}(u_+, u_-)$ as the set of curves in $\mathcal{M}_J^{1=2}(\alpha_+, \alpha_-)$ which are $(\kappa, \nu)$-close to breaking to $(u_+, u_-)$. This is a rephrasing of [HT1, Definition 1.10]. We define $G(u_+, u_-)$ as the modulo 2 count of the boundary points of $\mathcal{G}_{\kappa,\nu}(u_+, u_-)/\mathbf{R}$ for $\kappa$ and $\nu$ sufficiently small.

The main result of [HT1, HT2] implies the following.

**Theorem 6.5.3.** — Let $u_+ \in \mathcal{M}_J^{1=1}(\alpha_+, \beta_+)$ and $u_- \in \mathcal{M}_J^{1=1}(\beta_-, \alpha_-)$ be J-holomorphic curves with $\mathbf{\gamma}(\beta_+) = \mathbf{\gamma}(\beta_-)$. Then $G(u_+, u_-) = 1$.

It is clear that Theorem 6.5.3 is sufficient to define ECH; the original Hutchings-Taubes gluing theorem is more general, but for simplicity we have stated only the relevant part.

We now give the steps of the Hutchings-Taubes gluing construction. For simplicity we assume that $\mathbf{\gamma}(\beta_+) = \mathbf{\gamma}(\beta_-) = \mathbf{\gamma}^m$ and $\mathbf{\gamma}$ is elliptic. The case of more than one orbit $\mathbf{\gamma}_i$ only differs in notation and the case of hyperbolic orbits is simpler since it can be treated with standard techniques; see [HT1, Section 1.5].

*Step 1: Form a preglued curve* (cf. [HT2, Section 5.2]). First we fix some notation. Let $\mathcal{M}$ be the moduli space of connected, genus zero branched covers $\pi : \mathcal{Sigma} \to \mathbf{R} \times \mathbf{gamma}$ which have positive and negative ends determined by $P_{\mathbf{gamma}}^{\text{in}}(m)$ and $P_{\mathbf{gamma}}^{\text{out}}(m)$. Abusing the notation, we will often write only $\mathcal{Sigma}$. The preglued curve is constructed as follows:

(1) Choose gluing constants $0 < h < 1$ and $r \gg 1/h$.

(2) With $r, h$ fixed, choose the “gluing parameters” which consist of $\mathcal{Sigma} \in \mathcal{M}$ and real numbers $T_+, T_- \ge 5r$.

(3) Given $\pi : \mathcal{Sigma} \to \mathbf{R} \times \mathbf{gamma}$, let $b \subset \mathbf{R} \times \mathbf{gamma}$ be the set of branch points of $\pi$. Then let $s_+ = \max_{b \in b} s(b) + 1$ and $s_- = \min_{b \in b} s(b) - 1$. Let $\mathcal{Sigma}'$ be the preimage of $[s_- - T_-, s_+ + T_+] \times \mathbf{gamma}$. The preimages of $[s_+, s_+ + T_+] \times \mathbf{gamma}$ are cylinders of “height” $T_+$ and the preimages of $[s_- - T_-, s_-]$ are cylinders of “height” $T_-$.

(4) Fix $\kappa > 0$ sufficiently small. We choose a representative $u_+$ in $[u_+] \in \mathcal{M}_J^{1=1}(\alpha_+, \beta_+)/\mathbf{R}$ such that each component of $u_+|_{s \le 0}$ is $(\kappa, 0)$-close to a cylinder over some multiple cover of $\mathbf{gamma}$; here the multiplicities are given by $P_{\mathbf{gamma}}^{\text{in}}(m)$. Similarly, choose $u_-$ so that each component of $u_-|_{s \ge 0}$ is $(\kappa, 0)$-close to a cylinder over some multiple cover of $\mathbf{gamma}$.

(5) Let $u_{+T}$ be the $(s_+ + T_+)$-translate of $u_+$ in the $\mathbf{R}$-direction and let $u'_{+T} = u_{+T}|_{s \ge s_+ + T_+}$. Similarly, let $u_{-T}$ be the $(s_- - T_-)$-translate of $u_-$ in the $\mathbf{R}$-direction and let $u'_{-T} = u_{-T}|_{s \le s_- - T_-}$. The domain $C_*$ of the preglued curve is $C'_{+T} \cup \mathcal{Sigma}' \cup C'_{-T}$, where $C'_{\pm T}$ are domains of $u'_{\pm T}$, modulo the identifications along their boundary components. (To do the identifications correctly, we need asymptotic markers at the ends.)

(6) The preglued map $u_*$ is defined explicitly (see [HT2, Equations (5.5) and (5.6)]) via a cutoff functions which allow us to interpolate between $u_{\pm T}$ and $\mathcal{Sigma}$ in the preimages of the regions $s_+ \le s \le s_+ + T_+$ and $s_- - T_- \le s \le s_-$. These cutoff functions involve the constants $h$ and $r$.

*Step 2: Deform the preglued curve* (cf. [HT2, Section 5.3]). We choose “exponential maps” $e_-, e, e_+$ which are obtained by flowing in the directions “normal” to $u_-$, $u_\mathcal{Sigma}$, $u_+$ in $\mathbf{R} \times \mathbf{Y}$.
The exponential maps can be glued to give the exponential map $e_*$ which maps to a small tubular neighborhood of $u_*$ in $\mathbf{R} \times \mathbf{Y}$. Also choose the cutoff function $\tilde{\beta}_+ : C_* \to [0, 1]^9$ which equals 1 on $C'_{+T}$, interpolates between 0 and 1 on the cylinders $s_+ \le s \le s_+ + T_+$ in $\Sigma'$ and equals 0 elsewhere. Similarly define $\tilde{\beta}_\Sigma$ and $\tilde{\beta}_-$.

Let $(\psi_+, \psi_\Sigma, \psi_-)$ be a triple, where $\psi_\pm$ is a section of the normal bundle of $u_{\pm T}$ and $\psi_\Sigma$ is a section of the normal bundle of $u_\Sigma$ (which is trivial, so that $\psi_\Sigma$ can be identified with a function $\Sigma \to \mathbf{C}$). The deformation of $u_*$ with respect to $(\psi_+, \psi_\Sigma, \psi_-)$ is a map $C_* \to \mathbf{R} \times \mathbf{Y}$ given by

$$x \mapsto e_*(x, \tilde{\beta}_-\psi_- + \tilde{\beta}_\Sigma\psi_\Sigma + \tilde{\beta}_+\psi_+).$$

*Step 3:* We now consider the equation for the deformation to be J-holomorphic. This equation has the form:

$$(6.5.1) \qquad \tilde{\beta}_{-} \Theta_{-}(\psi_{-}, \psi_{\Sigma}) + \tilde{\beta}_{\Sigma} \Theta_{\Sigma}(\psi_{-}, \psi_{\Sigma}, \psi_{+}) + \tilde{\beta}_{+} \Theta_{+}(\psi_{\Sigma}, \psi_{+}) = 0.$$

See Equations (5.11), (5.12) and (5.13) of [HT2] for explicit expressions of $\Theta_-$, $\Theta_+$ and $\Theta_\Sigma$.

The strategy is to solve three equations separately:

$$(6.5.2) \qquad \Theta_{-}(\psi_{-}, \psi_{\Sigma}) = 0 \quad \text{on all of } u_{-T};$$

$$(6.5.3) \qquad \Theta_{+}(\psi_{\Sigma}, \psi_{+}) = 0 \quad \text{on all of } u_{+T};$$

$$(6.5.4) \qquad \Theta_{\Sigma}(\psi_{-}, \psi_{\Sigma}, \psi_{+}) = 0 \quad \text{on all of } \Sigma.$$

In [HT2, Proposition 5.6] it is shown that for sufficiently small $\psi_\Sigma$ (in a suitable Banach space) there exist maps $\psi_\pm$ such that $\psi_\pm = \psi_\pm(\psi_\Sigma)$ solves Equations (6.5.2) and (6.5.3).

We can then view Equation (6.5.4) as an equation in the variable $\psi_\Sigma$ on all of $\Sigma$:

$$(6.5.5) \qquad \Theta_{\Sigma}(\psi_{-}(\psi_{\Sigma}), \psi_{\Sigma}, \psi_{+}(\psi_{\Sigma})) = 0.$$

Equation (6.5.5) can be written as

$$(6.5.6) \qquad D_{\Sigma} \psi_{\Sigma} + \mathcal{F}_{\Sigma}(\psi_{\Sigma}) = 0,$$

where $D_\Sigma$ is the normal component of the linearized Cauchy-Riemann operator at $u_\Sigma$ and $\mathcal{F}_\Sigma$ is a nonlinear term which is small in some suitable sense.

*Step 4:* If $N$ denotes the normal bundle to $\mathbf{R} \times \gamma$ in $\mathbf{R} \times \mathbf{M}$, then

$$D_{\Sigma}: L_{1}^{2}(\pi^{*}\mathbf{N}) \to L^{2}(\pi^{*}\mathbf{N} \otimes \mathbf{T}^{0,1}\Sigma).$$

<footnote n="9">⁹ The notation in [HT2] is $\beta_+$. Here write $\tilde{\beta}_+$ to distinguish it from the sets of orbits in Section 6.5.2.</footnote>
The operator $D_Σ$ is injective and moreover $\dim \mathrm{coker}(D_Σ) = \dim \mathcal{M}$; see [HT1, Lemma 2.14] and [HT1, Lemma 2.15]. The explicit form of $D_Σ$ is given in [HT1, Definition 2.3].

We introduce the orthogonal projection

$$ \Pi : L^2(\pi^*\mathbb{N} \otimes T^{0,1}\Sigma) \to \ker(D_\Sigma^*) \cong \mathrm{coker}(D_\Sigma) $$

and decompose Equation (6.5.6) into two equations (cf. Equations (5.37) and (5.38) in [HT2]):

$$ (6.5.7) \qquad D_\Sigma \psi_\Sigma + (1 - \Pi) \mathcal{F}_\Sigma(\psi_\Sigma) = 0, $$

$$ (6.5.8) \qquad \Pi \mathcal{F}_\Sigma(\psi_\Sigma) = 0 $$

Equation (6.5.7) has a unique solution by [HT2, Proposition 5.7]. Hence the problem reduces to solving Equation (6.5.8).

This is shown to be equivalent to finding the zeros of a section of the associated obstruction bundle, defined in [HT1, Section 2]. Briefly, the obstruction bundle $O' \to [5r, \infty)^2 \times \mathcal{M}$ has fiber

$$ O'_{(T_+, T_-, \Sigma)} = \mathrm{Hom}(\mathrm{Coker}(D_\Sigma), \mathbb{R}), $$

and the section $\mathfrak{s} : [5r, \infty)^2 \times \mathcal{M} \to O'$ is given by:

$$ \mathfrak{s}(T_+, T_-, \Sigma)(\sigma) = \langle \sigma, \mathcal{F}_\Sigma(\psi_\Sigma) \rangle, $$

where $\sigma \in \mathrm{Coker}(D_\Sigma)$ and $\psi_\Sigma$ is the solution to Equation (6.5.7) corresponding to the gluing parameters $(T_+, T_-, \Sigma)$.

*Step 5:* Recall that $\mathcal{G}_{\kappa,\nu}(u_+, u_-)$ is the set of J-holomorphic maps of index $I=2$ which are $(\kappa, \nu)$-close to breaking into $(u_+, u_-)$ (see [HT2, Section 7.1]). Let $\mathcal{U}_{\kappa,\nu} \subset [5r, \infty)^2 \times \mathcal{M}$ be the set of $(T_+, T_-, \Sigma)$ such that the corresponding pre-glued curve $u_*(T_+, T_-, \Sigma)$ (as given in Step 1) is $(\kappa, \nu)$-close to breaking. It remains to show that $\mathcal{G}_{\kappa,\nu}(u_+, u_-)$ is homeomorphic to $\mathfrak{s}^{-1}(0) \cap \mathcal{U}_{\kappa,\nu}$ for $r > 0$ sufficiently large and $\kappa, \nu$ sufficiently small. This is the content of [HT2, Theorem 7.3].

*Step 6:* We define the linearized section $\mathfrak{s}_0$ which depends only on the asymptotic eigenfunctions of the negative ends of $u_+$ and the positive ends of $u_-$ but not on the actual choices of $u_+$ and $u_-$. The explicit formula for $\mathfrak{s}_0$ is given in [HT1, Definition 8.1]. We consider the linear deformation $\mathfrak{s}_t = t\mathfrak{s} + (1-t)\mathfrak{s}_0$.

*Step 7:* Consider the action of $\mathbb{R}$ on $[5r, +\infty)^2 \times \mathcal{M}$ that fixes the $[5r, +\infty)$ factors and translates the $s$-coordinate on $\mathcal{M}$. This action extends to the obstruction bundle and the sections $\mathfrak{s}_t$ are invariant under this action. Hence the obstruction bundle can be viewed

<footnote n="10">¹⁰ In [HT2] the obstruction bundle is denoted by $O$. However, in [HT1], $O$ denotes a related, but different, bundle and this causes confusion. We reserve the notation $O$ for the latter bundle.</footnote>
as a bundle over $[5r, +\infty)^2 \times \mathcal{M}/\mathbb{R}$ which we still denote by $\mathcal{O}'$ by abuse of notation, and each $\mathfrak{s}_t$ can be viewed as a section of it.

Given $R \ge 10r$, we define the “slice” $\mathcal{V}_R$ which consists of triples

$$ (T_{-}, T_{+}, \Sigma) \in [5r, +\infty)^2 \times \mathcal{M}, \quad T_{+} + s_{+} - s_{-} + T_{-} = \mathbb{R}. $$

The action of $\mathbb{R}$ on $[5r, +\infty)^2 \times \mathcal{M}$ preserves $\mathcal{V}_R$ and the boundary of $\mathcal{G}_{\kappa, \nu}(u_+, u_-)$ is in bijection with $\mathfrak{s}^{-1}(0) \cap \mathcal{V}_R/\mathbb{R}$; in other words

$$ G(u_+, u_-) = \#(\mathfrak{s}^{-1}(0) \cap \mathcal{V}_R/\mathbb{R}). $$

It turns out that the section $\mathfrak{s}_0$ has the same number of zeroes as $\mathfrak{s}$ on $\mathcal{V}_R/\mathbb{R}$. The key point to check is that, during the deformation $(\mathfrak{s}_t)_{t \in [0,1]}$ from $\mathfrak{s}_1 = \mathfrak{s}$ to $\mathfrak{s}_0$, the zeros of $\mathfrak{s}_t$ do not cross the boundary of $\mathcal{V}_R/\mathbb{R}$. This is guaranteed by [HT2, Proposition 8.2].

*Step 8: Define*

$$ \mathcal{M}_R = \{ \Sigma \in \mathcal{M} | -\frac{R}{2} + 5r \le s_-, s_+ \le \frac{R}{2} - 5r \}. $$

Then $\mathcal{V}_R/\mathbb{R} \simeq \mathcal{M}_R$ and the bundle $\mathcal{O} \to \mathcal{M}_R$ with fiber $\mathcal{O}_\Sigma = \mathrm{Hom}(\mathrm{Coker}(\mathrm{D}_\Sigma), \mathbb{R})$ is isomorphic to the bundle $\mathcal{O}' \to \mathcal{V}_R/\mathbb{R}$. The explicit formula for $\mathfrak{s}_0$, regarded as a section of $\mathcal{O} \to \mathcal{M}_R$, is given in [HT1, Definition 3.2]. See [HT2, Remark 8.5] for more details about the correspondence between $\mathcal{O}'$ and $\mathcal{O}$. Finally, the combinatorial formula for the algebraic count of zeros of $\mathfrak{s}_0$ (as a section of $\mathcal{O}$) is given by [HT1, Theorem 1.13].

**6.5.3. The $\Phi$-map.** — We now turn to gluing the pair $(v_0, v_{-1})$, where $v_0$ is a $W_+$-curve with $I(v_0) = 0$ and $v_{-1}$ is an ECH curve with $I(v_{-1}) = 1$. Suppose the negative end of $v_0$ is given by $\beta_+$, the positive end of $v_{-1}$ is given by $\beta_-$, and $\gamma(\beta_+) = \gamma(\beta_-) = \gamma^m$ for some elliptic orbit $\gamma$. Again, the case of more than one orbit $\gamma_i$ only differs in notation and the case of hyperbolic orbits can be treated by standard techniques. In our case, there are a few things to check:

(1) The curves $v_0$ and $v_{-1}$ must satisfy the partition conditions at their negative and positive ends, respectively. This is a consequence of the ECH index inequality, i.e., Theorem 5.6.9.

(2) Since the $W_+$-curve $v_0$ is not $s$-translation invariant, we pick $s_0$ so that each component of $v_0|_{s \le s_0}$ is $(\kappa, 0)$-close to a cylinder over some multiple cover of $\gamma$. (We may still assume that $v_{-1}$ satisfies the condition that $v_{-1}|_{s \ge 0}$ is $(\kappa, 0)$-close to a cylinder.) Given $(T_+, T_-, \Sigma) \in [5r, +\infty)^2 \times \mathcal{M}$, we take

$$ -v_0|_{s \ge s_0}; $$

$$ -\Sigma' \text{ shifted by } s_0 - (s_+ + T_+); \text{ and} $$

$$ -v_{-1}|_{s \le 0} \text{ shifted by } (s_- - T_-) + s_0 - (s_+ + T_+); $$

and preglue. The preglued curve only depends on $(T_+, T_-, [\Sigma])$, where $[\Sigma] \in \mathcal{M}/\mathbb{R}$.
(3) [HT2, Proposition 5.6] allows us to solve Equations (6.5.2) and (6.5.3) in terms of $\psi_{\Sigma}$. The inputs for [HT2, Proposition 5.6] are [HT2, Lemmas 5.3 and 5.4], which are consequences of the fact that the linearized $\bar{\partial}$-operators corresponding to $v_0$ and $v_{-1}$ are Fredholm and surjective; this also holds in our case with Lagrangian boundary conditions. Hence Step 3 extends easily to our setting, yielding a section $\mathfrak{s}$ of the obstruction bundle $\mathcal{O}' \to [5r, +\infty)^2 \times \mathcal{M}/\mathbb{R}$. The linearized section $\mathfrak{s}_0$ and the “slice” $\mathcal{V}_{\mathbb{R}}$ are defined similarly and, after identifying $\mathcal{O}' \to \mathcal{V}_{\mathbb{R}}/\mathbb{R}$ with the bundle $\mathcal{O} \to \mathcal{M}_{\mathbb{R}}$, the remaining steps carry over without change.

**6.6.** The variant $\tilde{\Phi}$. — In this subsection we define a complex $\tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ which is closely related to $\tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ and a variant

$$ \tilde{\Phi} : \tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \mathrm{PFC}_{2g}(\mathbb{N}) $$

of $\Phi$. This will be useful in the proof of Theorem II.3.3.1. As a vector space, $\tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ will also be used in Section 7.

**6.6.1.** The chain complex $\tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$. — Let $\bar{J} \in \mathcal{J}_{\bar{W}}^{\mathrm{reg}}$. We recall some notation introduced in Section 4.9.3: Let $\mathcal{I} \subset \{1, \dots, 2g\}$ be a subset, $\mathcal{I}^c$ its complement, and $\mathcal{S}_{\mathcal{I}^c}$ the group of permutations of $\mathcal{I}^c$.

**Definition 6.6.1.** — We define the chain complex $(\tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}), \bar{J}), \tilde{\partial})$ generated by the 2g-tuples $\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'$, where $\mathcal{I} \subset \{1, \dots, 2g\}$, $\mathbf{y}' = \{y'_i\}_{i \in \mathcal{I}^c}$, and the following hold:

- $z_{\infty,i}$ is viewed as an intersection point of $\bar{a}_i$ and $\bar{h}(\bar{a}_i)$ and

- $y'_i \in a_i \cap \mathbf{h}(a_{\sigma(i)})$ for some $\sigma \in \mathcal{S}_{\mathcal{I}^c}$.

We denote the set of intersection points $\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'$ as above by $S_{\bar{a},\bar{h}(\bar{a})}$.

The datum $z_{\infty,i}$ is equivalent to “$z_{\infty}$ with either of the two matchings $(i, 0) \to (i, 0)$ or $(i, 1) \to (i, 1)$” and therefore, using the notation of Section 5.7, $\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'$ is equivalent to the equivalence class of elements $z_{\infty}^{\beta}(\vec{D}) \cup \mathbf{y}'$, where $\mathcal{I} = \{i_1, \dots, i_p\}$ and $\vec{D} = \{(i_1, j_1) \to (i_1, j_1), \dots, (i_p, j_p)\}$ for any choice of $(j_1, \dots, j_p) \in \{0, 1\}^p$.

The differential $\tilde{\partial}$ counts $\mathcal{I} = 1$ multisections $\bar{u}$ in $\bar{W}$ with $n(\bar{u}) \le 1$, which satisfy one extra condition, i.e., if we write $\bar{u} = \bar{u}' \cup \bar{u}''$ (according to the notation introduced in Section 5.7.2), then $\bar{u}'$ has empty branch locus. The homology of $(\tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})), \tilde{\partial})$ is denoted by $\tilde{\mathcal{HF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$.

In the differential $\tilde{\partial}$, with the exception of trivial strips, we are counting the following curves:

(1) thin strips from $z_{\infty,i}$ to either $x_i$ or $x'_i$; and

(2) $\mathcal{I} = 1$ curves whose projections to $\bar{S}$ have image in S.

**Lemma 6.6.2.** — There is an isomorphism of chain complexes:

$$ \kappa : (\tilde{\mathcal{CF}}(\Sigma, \beta, \alpha, z), \tilde{\partial}) \to (\tilde{\mathcal{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})), \tilde{\partial}), $$
$$ \{x_i''\}_{i \in \mathcal{I}} \cup \mathbf{y}' \mapsto \{z_{\infty, i}\}_{i \in \mathcal{I}} \cup \mathbf{y}', $$

where $(\Sigma, \beta, \alpha, z)$ is as given in Section 4.9.1.

*Proof.* — This is immediate from the definitions.

<figure><img src="image_10.png" /></figure>

Recall from Section 4.9.3 that

$$ (\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a})), \widehat{\partial}') = (\widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathcal{h}(\mathbf{a})), \partial') / \sim $$

is the $E^1$-term of the spectral sequence for $(\widehat{\mathrm{CF}}(\Sigma, \beta, \alpha, z), \widehat{\partial})$ (viewed as a double complex) in Theorem 4.9.4. (Here we are writing $\widehat{\partial}'$ for the differential of $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a}))$ to distinguish it from the differential $\widehat{\partial}$ of $\widehat{\mathrm{CF}}(\Sigma, \beta, \alpha, z)$.)

In this paragraph $\mathbf{y}$ and $\mathbf{y}'$ denote linear combinations of tuples. By tracing the zigzags in the double complex we obtain the isomorphism

$$ \nu : \widehat{\mathrm{HF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a})) \xrightarrow{\sim} \widehat{\mathrm{HF}}(\Sigma, \beta, \alpha, z), $$

which is defined as follows: Let $\mathbf{y} \in \widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathcal{h}(\mathbf{a}))$ be a cycle in $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a}))$, i.e.,

$$ \partial' \mathbf{y} = \sum_i (\{x_i\} \cup \mathbf{y}'_i + \{x'_i\} \cup \mathbf{y}'_i). $$

Then $\nu$ maps the equivalence class $[\mathbf{y}]$ to the equivalence class of

$$ \mathbf{y} + \sum_i \{x_i''\} \cup \mathbf{y}'_i + \text{h.o.}, $$

where “h.o.” means terms with more $x_i''$ components. Composing with the map induced by $\kappa$ in homology, we obtain an isomorphism

$$ \tilde{\nu} = \kappa_* \circ \nu : \widehat{\mathrm{HF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a})) \xrightarrow{\sim} \widehat{\mathrm{HF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a})). $$

**6.6.2.** *The map $\tilde{\Phi}$. — Let*

$$ \mathcal{M}_{\bar{J}_+} (\{z_{\infty, i}\}_{i \in \mathcal{I}} \cup \mathbf{y}', \mathbf{y}) = \coprod_{\vec{D}} \mathcal{M}_{\bar{J}_+} (\{z_{\infty}^{\#\mathcal{I}}(\vec{D})\} \cup \mathbf{y}', \mathbf{y}), $$

where the moduli spaces on the right-hand side are defined in a manner analogous to Definition 5.7.2 and $\vec{D}$ ranges over all matchings $\{(i, j_i) \to (i, j_i)\}_{i \in \mathcal{I}, j_i \in \{0, 1\}}$.

**Definition 6.6.3.** — Let $\bar{J}_+ \in \mathcal{J}_{\bar{W}_+}^{\mathrm{reg}}$ which restricts to $\bar{J} \in \mathcal{J}_{\bar{W}}^{\mathrm{reg}}$ and $\bar{J}' \in \mathcal{J}_{\bar{W}}^{\mathrm{reg}}$. We define the map $\tilde{\Phi} : \tilde{\mathrm{CF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a})) \to \mathrm{PFC}_{2g}(\mathbb{N})$ as follows:

$$ \langle \tilde{\Phi}(\{z_{\infty, i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'), \mathbf{y} \rangle = \# \mathcal{M}_{\bar{J}_+}^{I=0, n^* \le |\mathcal{I}|} (\{z_{\infty, i}\}_{i \in \mathcal{I}} \cup \mathbf{y}', \mathbf{y}). $$
In our analysis of $\tilde{\Phi}$ we will use balanced coordinates (cf. Section 5.1.2) for $\overline{\mathbb{N}} - \text{int}(\mathbb{N}) = \mathbb{D}^2 \times (\mathbf{R}/2\mathbb{Z})$. The Morse-Bott family $\mathcal{N}$ can be identified with $\partial\mathbb{D}^2$ and we write $\gamma_\phi$ for the orbit in $\mathcal{N}$ corresponding to $e^{i\phi}$. So far in this paper $h \in \mathcal{N}$ was an arbitrary point.

**Convention 6.6.4.** — From now on we specialize $h$ so that $h = \gamma_{\phi_h}$ is generic and $\phi_h$ is close to $-\frac{2\pi}{m}$, where $m$ is as defined in Section 5.2.2. In particular, the radial ray corresponding to $\phi_h$ does not lie on the thin wedges from $\bar{a}_i$ to $\bar{h}(\bar{a}_i)$ for all $i$. There are no restrictions on the orbit $e$ except that $e \neq h$.

**Lemma 6.6.5.** — $\tilde{\Phi}(\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}') = 0$ if $\mathcal{I} \neq \emptyset$ and $\tilde{\Phi}(\mathbf{y}') = \Phi(\mathbf{y})$ if $\mathcal{I} = \emptyset$.

*Proof.* — Let us fix $\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'$ with $|\mathcal{I}| \ge 1$ and write

$$ \mathcal{M}^0(\mathbf{y}) := \mathcal{M}_{\bar{J}_+}^{I=0, n^* \le |\mathcal{I}|} (\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}', \mathbf{y}), $$

where $\mathbf{y} \in \tilde{\mathcal{O}}_{2g}$.

We claim that $\mathcal{M}^0(\mathbf{y}) = \emptyset$. Arguing by contradiction, suppose there exists $\bar{u} \in \mathcal{M}^0(\mathbf{y})$. By our assumptions on $\bar{J}_+$, $\bar{u}$ is close to breaking into a Morse-Bott building. We will show that such a Morse-Bott building cannot exist using the positivity of intersections, thus establishing the contradiction. For the rest of the proof, $\bar{u}$ will denote the holomorphic part of the Morse-Bott building instead of the original curve.

We consider the projection $\pi_{\overline{\mathbb{N}}} : \overline{W}_+ \to \overline{\mathbb{N}}$, obtained by restricting the projection $\overline{W}' = \mathbf{R} \times \overline{\mathbb{N}} \to \overline{W}_+$. Let $T_\rho$ be the boundary of the neighborhood $V_\rho = \mathbb{S}^1 \times \mathbb{D}_\rho^2$ of the orbit $\delta_0$. We choose $\rho > 1$, but arbitrarily close to 1, so that $T_\rho$ is contained in $\overline{\mathbb{N}}$ and parallel to $\partial\mathbb{N}$. Then $T_\rho$ is a pre-Lagrangian torus and we can assume without loss of generality that it is foliated by closed orbits.

We identify $H_1(T_\rho) \cong \mathbb{Z}^2$ such that — writing vectors as rows — (1, 0) corresponds to the homology class of the meridian (i.e., the closed curve which bounds a disk in $V_\rho$ intersecting $\delta_0$ once positively) and (0, 1) corresponds to the class of the orbits in $\partial\mathbb{N}$ (i.e., those which are perturbed into $e$ and $h$) under the obvious identification $H_1(\partial\mathbb{N}) \cong H_1(\mathbb{N})$. Then the Hamiltonian orbits on $T_\rho$ represent a homology class $(-p, q)$ with $p, q \ge 1$.

By abuse of notation, we do not distinguish $\bar{u}$ from its image, e.g., we write $\pi_{\overline{\mathbb{N}}}(\bar{u})$ to denote the image of the composition $\pi_{\overline{\mathbb{N}}} \circ \bar{u}$. The intersection $\pi_{\overline{\mathbb{N}}}(\bar{u}) \cap T_\rho$, oriented as the boundary of $\pi_{\overline{\mathbb{N}}}(\bar{u}) \cap (\overline{\mathbb{N}} - V_\rho)$, represents a class $C \in H_1(T_\rho, \pi_{\overline{\mathbb{N}}}(\mathbb{L}_{\bar{a}}^+)) \cap T_\rho$. We recall that $\pi_{\overline{\mathbb{N}}}(\mathbb{L}_{\bar{a}}^+) \cap T_\rho$ consists of 2g pairwise disjoint segments parallel to the Hamiltonian flow. The condition $n^*(\bar{u}) \le |\mathcal{I}|$ implies that C consists of at most $|\mathcal{I}|$ arcs; moreover each arc has both endpoints on the same connected component of $\pi_{\overline{\mathbb{N}}}(\mathbb{L}_{\bar{a}}^+) \cap T_\rho$ and represents the image of the class $(0, 1) \in H_1(T_\rho)$ under the map $H_1(T_\rho) \to H_1(T_\rho, \pi_{\overline{\mathbb{N}}}(\mathbb{L}_{\bar{a}}^+))$. This can be seen by considering the relation of the asymptotic eigenfunctions of $\bar{u}$ at $z_{\infty,i}$ with $n^*(\bar{u})$ and the possible ends of $\bar{u}$ in $V_\rho$.
The algebraic intersection between a Hamiltonian orbit of $T_\rho$ and the image of $\bar{u}$ is the same as the algebraic intersection in $T_\rho$ of a Hamiltonian orbit with C, which is negative. This contradicts the positivity of intersections, unless $\pi_{\overline{N}}(\bar{u})$ is disjoint from $T_\rho$. Since this holds for $\rho > 1$ and arbitrarily close to 1, there is a decomposition $\bar{u} = \bar{u}'' \cup \bar{u}'''$, where $I(\bar{u}'') = I(\bar{u}'') = 0$, $\bar{u}''$ has image in $\overline{W}_+ - \text{int}(\overline{W}_+)$, and $\bar{u}'''$ has image in $\overline{W}_+$.

Then $\bar{u}''$ is a curve from $z_\infty$ to $h$ since $I(\bar{u}'') = 0$. The proof of the nonexistence of $\bar{u}''$ is modeled on the proof of [0, Lemma 8.4.8]. By [We1, Section 4.2], $\mathbf{R} \times (\overline{\mathbb{N}} - \mathbb{N} - \delta_0)$ is foliated by finite energy cylinders $Z_{s_0, \phi_0}$, $(s_0, \phi_0) \in \mathbf{R} \times (\mathbf{R}/2\pi\mathbf{Z})$, from $\delta_0$ to $\gamma_{\phi_0}$ such that:

- the image of $Z_{s_0, \phi_0}$ under the projection $\pi_{\overline{N}}: \mathbf{R} \times \overline{\mathbb{N}} \to \overline{\mathbb{N}}$ is the open annulus $\{\phi = \phi_0, 0 < \rho < 1\}$;

- $Z_{s_0+s_1, \phi_0}$ is the $s_1$-translate of $Z_{s_0, \phi_0}$.

We then set $Z_{s,\phi}^+ = Z_{s,\phi} \cap \overline{W}_+$ and examine the intersections $Z_{s,\phi}^+ \cap \bar{u}''$. Observe that $K = \pi_{\overline{N}}(Z_{s,\phi}^+) \cap \pi_{\overline{N}}(\bar{u}'') \neq \emptyset$ for a suitable choice of $\phi$ which is close to but not equal to $\phi_h$; this is possible by the positioning of $h$ given by Convention 6.6.4. Hence $Z_{s_0,\phi}^+ \cap \bar{u}'' \neq \emptyset$ for some $s_0$. On the other hand, since $K$ is compact, $Z_{s_0+s_1,\phi}^+ \cap \bar{u}'' = \emptyset$ for a sufficiently large $s_1$. Finally, since $Z_{s,\phi}^+$ and $\bar{u}''$ have no boundary intersections and no intersections near their ends for all $s \in \mathbf{R}$, we have a contradiction. $\square$

**Theorem 6.6.6.** — $\tilde{\Phi}$ is a chain map.

*Proof.* — Since $\Phi$ is a chain map by Theorem 6.2.4, it suffices to verify that

$$ \partial_{\text{PFH}} \circ \tilde{\Phi}(\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}') = \tilde{\Phi} \circ \tilde{\partial}(\{z_{\infty,i}\}_{\mathcal{I}} \cup \mathbf{y}'), $$

whenever $\mathcal{I} \neq \emptyset$. On the one hand, $\tilde{\Phi}(\{z_{\infty,i}\}_{\mathcal{I}} \cup \mathbf{y}') = 0$ by Lemma 6.6.5. On the other hand, if $|\mathcal{I}| = 1$, then

$$ \tilde{\Phi} \circ \tilde{\partial}(\{z_{\infty,i_1}\} \cup \mathbf{y}') = \tilde{\Phi}(\{x_{i_1}\} \cup \mathbf{y}' + \{x'_{i_1}\} \cup \mathbf{y}') = 0; $$

and if $|\mathcal{I}| > 1$, then each term of $\tilde{\partial}(\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}')$ contains some copy of $z_\infty$, and $\tilde{\Phi}$ maps the term to zero by Lemma 6.6.5. This proves the theorem. $\square$

A corollary of Lemma 6.6.5 is the following:

**Corollary 6.6.7.** — $\Phi = \tilde{\Phi} \circ \kappa \circ \nu$ on the level of homology.

## 7. The chain map from PFH to $\widehat{\text{HF}}$

**7.1. Definition of $\Psi$.** — Fix the integer $m \gg 0$ which appears in the definition of the monodromy map $\bar{h} = \bar{h}_m$ from Section 5.1.2. The condition $m \gg 0$ will be useful when applying limiting arguments in Sections 7.8 and 7.10.
Suppose $\bar{J}_- \in \mathcal{J}_{\bar{W}_-}^{\text{reg}}$, $\bar{J}'$ and $\bar{J}$ are restrictions of $\bar{J}_-$ to the positive and negative ends, and $\bar{J}^{\diamond}$ is $K_{p,2\delta}$-regular with respect to $\bar{\mathfrak{m}} = ((0, \frac{3}{2}), z_\infty)$. We assume that the constants $\varepsilon, \delta > 0$ that go into the definition of $\bar{J}^{\diamond}$ (cf. Section 5.8.4) are arbitrarily small.

**Definition 7.1.1 (Definitions of $\Psi'$ and $\Psi$).**

(1) We define the map

$$ \Psi' = \Psi'_{\bar{J}^{\diamond}}(m, \bar{\mathfrak{m}}) : \mathrm{PFC}_{2g}(\mathbb{N}) \to \widehat{\mathrm{CF}}'(\mathbb{S}, \mathbf{a}, \mathfrak{h}(\mathbf{a})) \\ \gamma \mapsto \sum_{\mathbf{y} \in \mathcal{S}_{\mathbf{a}, \mathfrak{h}(\mathbf{a})}} \langle \Psi'(\mathbf{y}), \mathbf{y} \rangle \cdot \mathbf{y}, $$

where $\langle \Psi'(\mathbf{y}), \mathbf{y} \rangle$ is the mod 2 count of $\mathcal{M}_{\bar{J}^{\diamond}}^{I=2, n^*=m}(\mathbf{y}, \mathbf{y}; \bar{\mathfrak{m}})$.

(2) We define the map $\Psi = \Psi_{\bar{J}^{\diamond}}(m, \bar{\mathfrak{m}})$ as the composition

$$ \mathrm{PFC}_{2g}(\mathbb{N}) \xrightarrow{\Psi'} \widehat{\mathrm{CF}}'(\mathbb{S}, \mathbf{a}, \mathfrak{h}(\mathbf{a})) \xrightarrow{q} \widehat{\mathrm{CF}}(\mathbb{S}, \mathbf{a}, \mathfrak{h}(\mathbf{a})), $$

where $q$ is the quotient map of chain complexes.

The count $\langle \Psi'(\mathbf{y}), \mathbf{y} \rangle$ is meaningful because of the following theorem:

**Theorem 7.1.2.** — $\mathcal{M}_{\bar{J}^{\diamond}}^{I=2, n^*=m}(\mathbf{y}, \mathbf{y}; \bar{\mathfrak{m}})$ is compact.

*Proof.* — This follows from Theorems 7.6.3 and 7.10.1(i) and Corollary 7.11.1. $\square$

<figure><img src="image_11.png" /></figure>

Now we define the maps $\tilde{U}_{m-1}$, $\tilde{\Psi}_0$, and $\tilde{\partial}_1$ which measure the failure of $\Psi'$ to be a chain map. The subscripts in $\tilde{U}_{m-1}$, $\tilde{\Psi}_0$, and $\tilde{\partial}_1$ indicate that we are counting curves $\bar{u}$ satisfying $n^*(\bar{u}) = m-1, 0$, and $1$.

First we define $\tilde{U}_{m-1} : \mathrm{ECC}_{2g}(\mathbb{N}) \to \mathrm{ECC}_{2g}(\bar{\mathbb{N}})$: Let $\mathbf{y} \in \mathcal{O}_{2g}$ and $\mathbf{y}' \in \mathcal{O}_{2g-p}, p = 0, \dots, 2g$. Then

$$ \langle \tilde{U}_{m-1}(\mathbf{y}), \delta_0^p \mathbf{y}' \rangle = \begin{cases} \mathbf{A}, & \text{if } p=1; \\ 0, & \text{if } p \neq 1, \end{cases} $$

where $\mathbf{A}$ is the count of $I=2$ multisections $\bar{u}$ of $\bar{W}'$ from $\mathbf{y}$ to $\delta_0 \mathbf{y}'$ which satisfy $n^*(\bar{u}) = m-1$ and a certain asymptotic condition near $\delta_0$ (the precise definition will be given in Section 7.7.3).

Next we define $\tilde{\Psi}_0 : \mathrm{ECC}_{2g}(\bar{\mathbb{N}}) \to \widehat{\mathrm{CF}}(\mathbb{S}, \mathbf{a}, \mathfrak{h}(\mathbf{a}))$: Let $\mathbf{y}' \in \mathcal{O}_{2g-p}, p = 0, \dots, 2g$, and $\{z_{\infty,i}\}_{i \in I} \cup \mathbf{y}' \in \mathcal{S}_{\mathbf{a}, \mathfrak{h}(\mathbf{a})}$. Then

$$ \langle \tilde{\Psi}_0(\delta_0^p \mathbf{y}'), \{z_{\infty,i}\}_{i \in I} \cup \mathbf{y}' \rangle = \begin{cases} \mathbf{B}, & \text{if } p=1 \text{ and } |I|=1; \\ 0, & \text{if } p \neq 1 \text{ or } |I| \neq 1, \end{cases} $$
where B is the count of degree $2g-1$, $I=0$ almost multisections $\bar{u}$ of $\overline{W}_-$ from $\mathbf{y}'$ to $\mathbf{y}'$
which satisfy $n^*(\bar{u}) = 0$, together with the section at infinity $\sigma_\infty^-$ from $\delta_0$ to $z_\infty$.

Finally we define $\tilde{\partial}_1 : \tilde{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \tilde{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$. If $\{z_{\infty,i}\}_{I \cup \mathbf{y}' \in S_{\bar{\mathbf{a}}, \bar{\mathbf{h}}(\bar{\mathbf{a}})}}$, then we define

$$
(7.1.1) \quad \tilde{\partial}_1(\{z_{\infty,i}\}_{I \cup \mathbf{y}'} \cup \mathbf{y}') = \begin{cases} \{x_i\} \cup \mathbf{y}' + \{x_i'\} \cup \mathbf{y}', & \text{if } I = \{i\}; \\ 0, & \text{if } |I| \neq 1. \end{cases}
$$

We can see $\tilde{\partial}_1$ as the composition of two maps: a map from $\tilde{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ to itself induced by the count of $n^*(\bar{u}) = 1$ curves in $\overline{W}$ (which are necessarily thin wedges), followed by the projection $\tilde{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \tilde{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ sending every $2g$-tuple of chords involving $z_\infty$ to zero.

Let $\partial'_{\mathrm{HF}}$ be the differential for $\tilde{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ and let $\partial_{\mathrm{PFH}}$ be the differential for $\mathrm{PFC}_{2g}(N)$. The proof of the following theorem will occupy the rest of Section 7.

Theorem **7.1.3.** — If $m \gg 0$, then

$$
(7.1.2) \quad \partial'_{\mathrm{HF}} \circ \Psi' + \Psi' \circ \partial_{\mathrm{PFH}} = \tilde{\partial}_1 \circ \tilde{\Psi}_0 \circ \tilde{U}_{m-1}.
$$

Assuming Theorem 7.1.3 for the moment, we have the following:

Corollary **7.1.4.** — For all $m \gg 0$, $\Psi$ is a chain map.

*Proof.* — Similar to the proof of Theorem 6.2.4 and based on Equation (7.1.2).
There is one major difference: $\Psi'$ is not a chain map. However, since the image of $\tilde{\partial}_i$ is
contained in $\ker q$, by composing Equation (7.1.2) with $q$ we obtain:

$$
(7.1.3) \quad \hat{\partial}_{\mathrm{HF}} \circ \Psi + \Psi \circ \partial_{\mathrm{PFH}} = 0,
$$

where $\hat{\partial}_{\mathrm{HF}}$ is the differential for $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$. $\square$

<figure><img src="image_12.png" /></figure>

We can also define a map

$$
(7.1.4) \quad \underline{\Psi} : \underline{\mathrm{PFH}}_{2g}(N) \to \widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))
$$

which is $\mathbf{F}[H_2(\mathbf{M}; \mathbf{Z})]$-linear. As for $\underline{\Phi}$ in Section 6.4, the key point is the definition of maps

$$
\mathfrak{A}_- : H_2(\overline{W}_-, \mathbf{\gamma}, \mathbf{y}) \to H_2(\mathbf{M}).
$$

The main difference is that $H_2(\overline{W}_-) \cong H_2(\overline{N}) \cong H_2(\mathbf{M}) \oplus \langle \overline{S} \rangle$. Hence we define

$$
\mathfrak{A}_-(C) = p((\pi_{\overline{N}})_*[C_y \cup C \cup -C_y]),
$$

where $\pi_{\overline{N}}$ is the projection $\overline{W}_- \to \overline{N}$ and $p: H_2(\overline{N}) \to H_2(\mathbf{M})$ is the projection given by
$p = id - \langle \delta_0, \cdot \rangle[\overline{S}]$.
**7.2.** *Outline of proof of Theorem 7.1.3.* — In this subsection we outline the proof of Theorem 7.1.3.

Let $\bar{J}_- \in \mathcal{J}_{\bar{W}_-}^{\text{reg}}$, with restrictions $\bar{J}$ and $\bar{J}$ to the positive and negative ends, and let $\bar{J}_-^\diamond$ be $K_{p,2\delta}$-regular with respect to $\bar{m}$. Suppose that the constants $\epsilon, \delta > 0$ that go into the definition of $\bar{J}_-$ are arbitrarily small.

We abbreviate:

$$ \mathcal{M}^i := \mathcal{M}_{\bar{J}_-^\diamond}^{I=i,n^*=m}(\mathbf{y}, \mathbf{y}), \quad \mathcal{M}_{\bar{m}}^i := \mathcal{M}_{\bar{J}_-^\diamond}^{I=i,n^*=m}(\mathbf{y}, \mathbf{y}; \bar{m}). $$

Let $\overline{\mathcal{M}}_{\bar{m}}^i$ be the SFT compactification of $\mathcal{M}_{\bar{m}}^i$, obtained by adding holomorphic $\overline{W}_-$-buildings as in Definition 7.3.1, and let $\partial \mathcal{M}_{\bar{m}}^i = \overline{\mathcal{M}}_{\bar{m}}^i - \mathcal{M}_{\bar{m}}^i$.

**Step 1.** By SFT compactness for $\overline{W}_-$-curves (Proposition 7.3.2), a sequence $\bar{u}_i \in \mathcal{M}_{\bar{m}}^3$ admits a subsequence which converges to a holomorphic $\overline{W}_-$-building

$$ \bar{u}_{\infty} = \bar{v}_{-b} \cup \dots \cup \bar{v}_a. $$

As before, we write $\bar{v}_j = \bar{v}'_j \cup \bar{v}''_j$, where $\bar{v}'_j$ branch covers the section at infinity $\sigma_\infty^*$ and $\bar{v}''_j$ is the union of irreducible components which do not branch cover $\sigma_\infty^*$. Here $*=\emptyset,>$, or $-$.

There are two cases: $\bar{v}'_0 = \emptyset$ or $\bar{v}'_0 \neq \emptyset$. The latter case is harder, and will be treated first. The former one will be treated in Step 3. By analyzing the two constraints $n^-(\bar{u}_i) = m$ and $I(\bar{u}_i) = 3$, we obtain Theorem 7.6.1, which gives a preliminary list of possibilities for $\bar{u}_\infty$ when $\bar{v}'_0 \neq \emptyset$. Many of the possibilities in Theorem 7.6.1 actually do not occur. Theorem 7.10.1 eliminates Cases (2)–(6) from the list. This is done by a finer analysis of the behavior of $\bar{u}_i$ in the vicinity of the component $\sigma_\infty^-$ and is similar in spirit to the *layer structures* of Ionel-Parker [IP, Section 7].

Summarizing, we have:

**Lemma 7.2.1.** — If $\bar{u}_\infty \in \partial \mathcal{M}_{\bar{m}}^3$ and $\bar{v}'_0 \neq \emptyset$, then $\bar{u}_\infty \in A_1$, where

$$ A_1 = \coprod_{\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}'} \left( \mathcal{M}_{\bar{J}'}^{I=2, n^*=m-1}(\mathbf{y}, \delta_0 \mathbf{y}') \times \mathcal{M}_{\bar{J}_-^\diamond}^{I=0, n^*=0}(\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}') \times \left( \mathcal{M}_{\bar{J}'}^{I=1, n^*=1}(\{z_\infty\} \cup \mathbf{y}', \mathbf{y}) / \mathbf{R} \right) \right), $$

if $\mathbf{y} = \{x_i\} \times \mathbf{y}'$ or $\{x'_i\} \times \mathbf{y}'$ and $A_1 = \emptyset$ otherwise. Here we have omitted the potential contributions of connector components for simplicity.

Case (1) in Theorem 7.6.1 corresponds to $A_1$.

**Step 2.** We now glue the triples $(\bar{v}_1, \bar{v}_0, \bar{v}_{-1})$ in $A_1$, subject to the constraint $\bar{m}$. This gluing accounts for the term $\partial_1 \circ \tilde{\Psi}_0 \circ \tilde{U}_{m-1}$ and is a bit involved. Let us abbreviate

$$ \mathcal{M}_1 := \mathcal{M}_{\bar{J}'}^{I=2, n^*=m-1}(\mathbf{y}, \delta_0 \mathbf{y}'); $$
$$ \mathcal{M}'_1 := \mathcal{M}_{\bar{J}}^{I=2, n^*=m-1, f_{\delta_0}}(\mathbf{y}, \delta_0 \mathbf{y}'); $$

$$ \mathcal{M}_0 := \mathcal{M}_{\bar{J}_{-}}^{I=0, n^*=0}(\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}'); $$

$$ \mathcal{M}_{-1} := \mathcal{M}_{\bar{J}}^{I=1, n^*=1}(\{z_\infty\} \cup \mathbf{y}', \mathbf{y}), $$

where $\mathbf{y} = \{x_i\} \cup \mathbf{y}''$ or $\{x'_i\} \cup \mathbf{y}''$. Here $f_{\delta_0}$ is a nonzero normalized asymptotic eigenfunction of $\delta_0$ at the negative end which, used as a modifier, stands for "the normalized asymptotic eigenfunction at the negative end $\delta_0$ is $f_{\delta_0}$." See Section 7.7 for more details on asymptotic eigenfunctions.

We first observe that $\bar{v}_0 = \bar{v}'_0 \cup \bar{v}''_0 \in \mathcal{M}_0$ is regular: $\bar{v}'_0$ is regular by Lemma 5.8.9 and $\bar{v}''_0$ is regular since $\bar{J}_- \in \mathcal{J}_{\bar{W}_-}^{\text{reg}}$ and $\bar{J}_-^\diamond$ is $(\varepsilon, \text{U})$-close to $\bar{J}_-$. The moduli spaces $\mathcal{M}_1$ and $\mathcal{M}_{-1}$ are also regular since $\bar{J}_-$ is regular. Hence we can glue triples $([\bar{v}_1], \bar{v}_0, [\bar{v}_{-1}])$ in $(\mathcal{M}_1/\mathbf{R}) \times \mathcal{M}_0 \times (\mathcal{M}_{-1}/\mathbf{R})$. More precisely, consider the gluing parameter space

$$ (7.2.1) \qquad \mathcal{P} := \coprod_{\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}'} \mathcal{P}_{\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}'}, $$

where

$$ (7.2.2) \qquad \mathcal{P}_{\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}'} = (5r, \infty)^2 \times (\mathcal{M}_1/\mathbf{R}) \times \mathcal{M}_0 \times (\mathcal{M}_{-1}/\mathbf{R}), $$

$0 < h < 1$ and $r \gg 1/h$ are gluing constants, and $\mathcal{M}_1, \mathcal{M}_0, \mathcal{M}_{-1}$ correspond to the pair $(\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}')$. We may assume that all the multiplicities of $\mathbf{y}'$ are 1, since the Hutchings-Taubes gluing of branched covers is essentially independent of the present gluing problem.

We recall the extended moduli space

$$ \mathcal{M}^{\text{ext}} := \mathcal{M}_{\bar{J}_-}^{I=3, n^*=m, \text{ext}}(\mathbf{y}, \mathbf{y}) $$

from Definition 5.7.24, where $\mathbf{y} = \{x_i\} \cup \mathbf{y}''$ or $\{x'_i\} \cup \mathbf{y}''$. Here the modifier ext means that $\bar{u} : (\dot{\mathbf{F}}, j) \to (\bar{\mathbf{W}}_-, \bar{J}_-)$ is a multisection which maps all the connected components of $\partial \dot{\mathbf{F}}$ but one to a different $L_{\bar{a}_i}^{-}$ and the last connected component to some $L_{\bar{a}_i \cup \bar{a}_{i,j}}^{-}$.

There is a gluing map

$$ G : \mathcal{P} \to \mathcal{M}^{\text{ext}}, \quad \mathfrak{d} = (T_{\pm}, \bar{v}_1, \bar{v}_0, \bar{v}_{-1}) \mapsto \bar{u}(\mathfrak{d}), $$

which is a diffeomorphism onto its image for $r \gg 0$. Here $T_\pm$ is shorthand for the pair $T_+$, $T_-$. The map $G$ is defined in a manner similar to that of Section 6.5.2 and is described in more detail in Section 7.12.3. Let $r_0 \gg 0$, let $\mathcal{P}_{(r_0)} \subset \mathcal{P}$ be the subset
`{T_+ ≥ r_0}`<fnref n="11" /> and let

$$ \mathcal{M}_{\bar{m}}^{3,(r_0)} := \mathcal{M}_{\bar{m}}^3 - G(\mathfrak{P}_{(r_0)}) $$

be the truncated moduli space. For generic $r_0 \gg 0$,

$$ \partial_0 \mathcal{M}_{\bar{m}}^{3,(r_0)} := G(\partial \mathfrak{P}_{(r_0)}) \cap \mathcal{M}_{\bar{m}}^3 $$

is a transverse intersection.

The following theorem is proved in Section 7.12:

**Theorem 7.2.2.** — For generic $r_0 \gg 0$,

$$ (7.2.3) \qquad \#(\partial_0 \mathcal{M}_{\bar{m}}^{3,(r_0)} \cap G(\mathfrak{P}_{\delta_0 \mathbf{y}', \{z_\infty\} \cup \mathbf{y}'})) = \#(\mathcal{M}'_1/\mathbf{R}) \cdot \#\mathcal{M}_0 \cdot \#(\mathcal{M}_{-1}/\mathbf{R}) \mod 2. $$

Hence the contributions from $A_1$ account for the term $\tilde{\partial}_1 \circ \tilde{\Psi}_0 \circ \tilde{U}_{m-1}$.

**Step 3.** Let $\partial_1 \mathcal{M}_{\bar{m}}^{3,(r_0)} = \partial \mathcal{M}_{\bar{m}}^{3,(r_0)} - \partial_0 \mathcal{M}_{\bar{m}}^{3,(r_0)}$. The following lemma is proved in Section 7.11.

**Lemma 7.2.3.** — $\partial_1 \mathcal{M}_{\bar{m}}^{3,(r_0)} \subset A_2 \sqcup A_3$, where

$$ A_2 = \coprod_{\mathbf{y}'' \in S_{a, \tilde{h}(a)}} \left( \mathcal{M}_{\mathcal{J}_{\perp}^{\uparrow}}^{I=2, n^*=m}(\mathbf{y}, \mathbf{y}''; \bar{m}) \times (\mathcal{M}_{\mathcal{J}_{\perp}^{\uparrow}}^{I=1, n^*=0}(\mathbf{y}'', \mathbf{y})/\mathbf{R}) \right); $$

$$ A_3 = \coprod_{\mathbf{y}' \in \tilde{O}_{2g}} \left( \left( \mathcal{M}_{\mathcal{J}'_{\perp}}^{I=1, n^*=0}(\mathbf{y}, \mathbf{y}')/\mathbf{R} \right) \times \mathcal{M}_{\mathcal{J}_{\perp}^{\uparrow}}^{I=2, n^*=m}(\mathbf{y}', \mathbf{y}; \bar{m}) \right). $$

*Here we have omitted the potential contributions of connector components for simplicity.*

On the other hand, gluing the pairs in $A_2$ (resp. $A_3$) using the usual (resp. Hutchings-Taubes) gluing theorem implies that $A_2 \cup A_3 \subset \partial_1 \mathcal{M}_{\bar{m}}^{3,(r_0)}$ and accounts for the term $\partial'_{HF} \circ \Psi'$ (resp. $\Psi' \circ \partial_{\text{PFH}}$). This proves Theorem 7.1.3.

*Organization of Section 7.* In Sections 7.3 we describe the necessary modifications to apply the proof of SFT compactness to the nonstandard setting of $\bar{W}_{-}$-curves. The signed intersection numbers $n^*(\bar{u})$ are analyzed in Sections 7.4 and 7.5. Theorem 7.6.1 is then proved in Section 7.6. We then discuss asymptotic eigenfunctions in Section 7.7, rescaling in Section 7.8, and the “involution lemmas” in Section 7.9, on our way to proving Theorem 7.10.1 in Section 7.10. Lemma 7.2.3 is proved in Section 7.11. Finally, the gluing is discussed in Section 7.12.

**7.3. SFT compactness.**

<footnote n="11">¹¹ Restricting to {$T_+ \ge r_0$} seems a little asymmetrical, but it will become clear in Section 7.12 that $T_-$ depends on $(T_+, \bar{v}_1, \bar{v}_0, \bar{v}_{-1})$ if $\bar{u}(\partial) \in \mathcal{M}_{\bar{m}}^3$.</footnote>
**7.3.1. Statement of the compactness theorem.**

*Definition 7.3.1.* — A holomorphic $\overline{W}_-$-building

$$ \bar{u}_{\infty} = \bar{v}_{-b} \cup \dots \cup \bar{v}_{a}, \quad a, b \in \mathbb{Z}^{\ge 0}, $$

consists of the following data:

(B1') For each $j = -b, \dots, a$, a compact nodal Riemann surface $G_j$, possibly with boundary and possibly disconnected. The nodal Riemann surface $G_j$ is obtained from a Riemann surface $\tilde{G}_j$ by identifying pairs of interior points or pairs of boundary points. The set of identified points is the nodal set $n_j$, which may be empty. Let $\mathbf{p}_j^+$, $\mathbf{p}_j^-$ be sets of interior punctures for $j \ge 0$ and let $\mathbf{q}_j^+$, $\mathbf{q}_j^-$ be sets of boundary punctures for $j \le 0$, subject to $\mathbf{p}_0^-, \mathbf{q}_0^+ = \emptyset$. The sets $n_j$, $\mathbf{p}_j^{\pm}$, $\mathbf{q}_j^{\pm}$ are mutually disjoint.

(B2') For each $j = -b, \dots, a$, a holomorphic map $\bar{v}_j : \dot{G}_j \to \overline{W}_j$, where $\overline{W}_j = \overline{W}'$ for $0 < j \le a$, $\overline{W}_0 = \overline{W}_-$ for $j = 0$, $\overline{W}_j = \overline{W}$ for $-b \le j < 0$, and $\dot{G}_j = G_j - \mathbf{p}_j^+ - \mathbf{p}_j^- - \mathbf{q}_j^+ - \mathbf{q}_j^-$ for all $j$. (Here some sets of punctures may be empty.)

(B3') For each $j = -b, \dots, 0$, $\partial \dot{G}_j$ is mapped to the appropriate Lagrangian submanifold $L_{\bar{a}} \sqcup L_{\bar{v}_i(\bar{a})}$ or $L_{\bar{a}}$.

(B4') For each $j = -b, \dots, a$, $\bar{v}_j$ converges to a trivial strip over a “Reeb chord” at the positive (resp. negative) end near each boundary puncture of $\mathbf{q}_j^+$ (resp. $\mathbf{q}_j^-$) and to a trivial cylinder over a closed orbit at the positive (resp. negative) end near each interior puncture of $\mathbf{p}_j^+$ (resp. $\mathbf{p}_j^-$).

(B5') For each $j = -b, \dots, a-1$, there is an identification between $\mathbf{p}_j^+$ and $\mathbf{p}_{j+1}^-$ and an identification between $\mathbf{q}_j^+$ and $\mathbf{q}_{j+1}^-$ such that the pairs that are identified are asymptotic to the same Reeb chord or closed orbit.

(B6') No level $\bar{v}_j$ is a union of trivial cylinders or trivial strips and no constant sphere has fewer than three nodes.

The levels $\bar{v}_j$ are arranged in order from lowest to highest.

**Proposition 7.3.2.** — Let $\bar{J}_- \in \mathcal{J}_{\overline{W}_-}$ and let $\bar{u}_i : (\dot{F}_i, j_i) \to (\overline{W}_-, \bar{J}_-)$, $i \in \mathbf{N}$, be a sequence of $\overline{W}_-$-curves from $\mathbf{Y}$ to $\mathbf{Y}$. Then there is a subsequence which converges in the sense of SFT to a level $a+b+1$ holomorphic $\overline{W}_-$-building

$$ \bar{u}_{\infty} = \bar{v}_{-b} \cup \dots \cup \bar{v}_{a}. $$

The same also holds when $\{\bar{u}_i\}$ is a sequence of $(\overline{W}_-, \bar{J}_-)\text{-curves}$ (see Definition 5.8.13).

**Remark 7.3.3.** — Our setting is more general than that of [BEHWZ] in some ways, e.g., the presence of a singular Lagrangian boundary condition. However, the nonexistence of contractible Hamiltonian orbits or chords and the existence of a holomorphic projection $\pi_{\overline{W}_-} : \overline{W}_- \to \mathcal{B}_-$ simplify the proofs considerably to the point that
they are often simpler than the corresponding proofs in [BEHWZ]. Other important simplifying features are the absence of “teardrops” (i.e., punctured holomorphic disks with boundary in $L_{\bar{a}}$ which, at the puncture, are asymptotic to a double point of $L_{\bar{a}}$) and the fact that different connected components of the boundary of the domain are mapped to different smooth components of $L_{\bar{a}}$.

**7.3.2. Preliminary lemmas.** — Here we prove some lemmas which will be used in the proof of Proposition 7.3.2. First we recall some terminology. Let $\overline{W}_*$ denote one of $\overline{W}$, $\overline{W'}$, $\overline{W}_{\pm}$ and $\overline{\omega}$ the 2-form on $\overline{W}_*$ defined in Section 5.1.2. We recall that the $\overline{\omega}$-area of a map $w: \dot{F} \to \overline{W}_*$ is defined as

$$E_{\overline{\omega}}(w) = \int_{\dot{F}} w^* \overline{\omega}.$$

We also introduce the energy $E(w)$ in the spirit of Definition 4.3.2. More precisely, let $C$ be the set of nondecreasing smooth functions $\phi: \mathbf{R} \to [0, 1]$; for all $\phi \in C$ we define the 1-form $\lambda_{\phi} = \bar{\pi}_{B_*}^*(\phi(s)dt)$, where as usual $B_*$ can stand for $B$, $B'$ or $B_{\pm}$. Then the energy of a map $w: \dot{F} \to \overline{W}_*$ is

$$(7.3.1) \qquad E(w) = \int_{\dot{F}} w^* \overline{\omega} + \sup_{\phi \in C} \int_{\dot{F}} w^* d\lambda_{\phi} = E_{\overline{\omega}}(w) + \sup_{\phi \in C} \int_{\dot{F}} w^* d\lambda_{\phi}.$$

If $w$ is a holomorphic map, then $E_{\overline{\omega}}(w) \ge 0$ and moreover a finite energy holomorphic cylinder $w: [-R, R] \times S^1 \to \overline{W}_*$ (or strip $w: [-R, R] \times [0, 1] \to \overline{W}_*$) with $E_{\overline{\omega}}(w) = 0$ is either a constant map or is part of a trivial cylinder (or strip) over a Hamiltonian orbit (or chord). We recall also that the *action* of a Hamiltonian orbit or chord is the integral, over that orbit or chord, of the 1-form $\overline{\alpha}_0$ defined in Section 5.1.2. By Equation (7.3.1), the energy of $w$ is determined by its $\overline{\omega}$-area and the actions of the orbits and chords at its ends.

In what follows we measure distances with respect to fixed Riemannian metrics $g$ on $\overline{N}$ and $ds^2 + g$ on $\overline{W'}$, as well as their restrictions to $N$ and $\overline{W}_*$.

The following lemma is a by-now standard observation originally due to Hofer.

**Lemma 7.3.4.** — Let $F$ be a finite type Riemann surface with boundary and interior and boundary punctures. Fix a compatible metric on $F$ which is cylindrical near the punctures. Let $F_1 \subset F_2 \subset \dots \subset F$ be a sequence of Riemann surfaces with corners of the same topological type, where $\partial F_n$ admits a decomposition $\partial_0 F_n \cup \partial_1 F_n$ into codimension 0 submanifolds that intersect at the corners, $\partial_0 F_n \subset \partial F$, the number of components of $\partial_i F_n$, $i = 0, 1$, are independent of $n$, each component of $\partial_0 F_n$ is contained in a different component of $\partial_0 F_{n+1}$, and $\text{int}(\partial_1 F_n) \subset \text{int} F$. Let $w_n: F_n \to \overline{W}_*$ be a sequence of holomorphic maps such that:

(A) $\partial_0 F_n$ is mapped to the appropriate Lagrangian boundary $L_{\bar{a}} \sqcup L_{\bar{h}(\bar{a})}$ or $L_{\bar{a}}$,

(B) $E(w_n) \le C_0$ for some constant $C_0$, and
$$ (C) \lim_{n \to +\infty} \mathrm{E}_{\bar{\omega}}(w_n) = 0. $$

Then for any $\epsilon > 0$ there is a constant $C > 0$ such that $\|\nabla w_n|_{F_n^\circ}\|_{C^0} \le C$ for all $n$, where $F_n^\circ = F_n - N_\epsilon(\partial_1 F_n)$ and $N_\epsilon(\partial_1 F_n)$ is an $\epsilon$-neighborhood of $\partial_1 F_n$.

*Sketch of proof.* — The lemma follows from the standard bubbling argument: if there exists a sequence of points $p_n \in F_n^\circ$ such that $\lim_{n \to +\infty} |\nabla w_n(p_n)| = +\infty$, then, by the usual rescaling argument (we may apply a translation in the target if the sequence $w_n$ is not bounded), we have bubbling of a *nonconstant*, $\mathrm{E}_{\bar{\omega}}(\tilde{w}) = 0$ holomorphic plane or half-plane $\tilde{w}$ with Lagrangian boundary.

The $\mathrm{E}_{\bar{\omega}}(\tilde{w}) = 0$ condition implies that $\tilde{w}$ maps to a trivial cylinder or strip over a (not necessarily closed) Hamiltonian orbit or chord. This key fact does not depend on whether the Lagrangian boundary condition is singular. The function $f = \bar{\pi}_* \circ \tilde{w}$ is holomorphic and has finite energy. We explain two cases where the existence of a nontrivial $f$ leads to a contradiction. First consider the case $f: \mathbf{C} \to \mathbf{B}'$, which can be lifted to the universal covering map $\tilde{f}: \mathbf{C} \to \mathbf{C}$. By the Little Picard Theorem, $\tilde{f}$ misses at most one point of $\mathbf{C}$, which contradicts the fact that $f$ has finite energy. Next consider the case $f: \mathbf{H}_+ \to \mathbf{B}_-$ (here $\mathbf{H}_+$ is the upper half-plane), which can be lifted to the universal covering map $\tilde{f}: \mathbf{H}_+ \to \mathbf{H}_+$ such that $\tilde{f}(\partial \mathbf{H}_+) \subset \partial \mathbf{H}_+$. Doubling $\tilde{f}$ to a map $\mathbf{C} \to \mathbf{C}$ using the Schwarz reflection principle and applying the Little Picard Theorem again gives a contradiction. $\square$

**Lemma 7.3.5 (Long Cylinder Lemma, Version 1).** — Let $\gamma_+$ and $\gamma_-$ be closed Hamiltonian orbits in $\overline{\mathbb{N}}$. Fix $E_0$ larger than the actions of both $\gamma_+$ and $\gamma_-$, and $\eta_0 > 0$ such that every closed Hamiltonian orbit $\gamma$ of action at most $E_0$ is at distance at least $2\eta_0$ from $\gamma_+$, if $\gamma \neq \gamma_+$, and from $\gamma_-$, if $\gamma \neq \gamma_-$. Then for all $\eta \in (0, \eta_0)$ there exist $c_0 > 0$, $R_0 > 0$, $\eta' > 0$ and $\delta > 0$ such that, for all $c \ge c_0$ and all $\mathbf{R} > R_0$, the following holds: If

$$ w: [-\mathbf{R} - c, \mathbf{R} + c] \times \mathbf{S}^1 \to \overline{\mathbf{W}}' $$

is a $\overline{\mathcal{J}}'$-holomorphic map which satisfies:

(1) $w|_{[-\mathbf{R}-c, -\mathbf{R}] \times \mathbf{S}^1}$ and $w|_{[\mathbf{R}, \mathbf{R}+c] \times \mathbf{S}^1}$ are $\eta'$-close in the $\mathbf{C}^1$-metric to portions of trivial cylinders over $\gamma_-$ and $\gamma_+$, respectively, and

(2) $\mathrm{E}_{\bar{\omega}}(w) < \delta$,

then $\gamma_- = \gamma_+$ and $w$ is $\eta$-close in the $\mathbf{C}^1$-metric to a portion of the trivial cylinder over $\gamma_-$.

**Lemma 7.3.6 (Long Strip Lemma, Version 1).** — Let $y_+$ and $y_-$ be chords corresponding to intersection points in $\bar{a}_i \cap \bar{h}(\bar{a}_i')$. Fix $\eta_0 > 0$ such that all other chords are a distance at least $2\eta_0$ from $y_+$ and $y_-$. Then for all $\eta \in (0, \eta_0)$ there exist $c_0 > 0$, $R_0 > 0$, $\eta' > 0$ and $\delta > 0$ such that, for all $c \ge c_0$ and all $\mathbf{R} > R_0$, the following holds: If

$$ w: [-\mathbf{R} - c, \mathbf{R} + c] \times [0, 1] \to \overline{\mathbf{W}} $$
is a $\bar{J}$-holomorphic map which satisfies:

(1) $w$ maps $[-R-c, R+c] \times \{1\}$ to $L_{\bar{a}_i}$ and $[-R-c, R+c] \times \{0\}$ to $L_{\bar{a}_j}$;

(2) $w|_{[-R-c, -R] \times [0, 1]}$ and $w|_{[R, R+c] \times [0, 1]}$ are $\eta'$-close in the $C^1$-metric to portions of trivial strips over $y_-$ and $y_+$, respectively, and

(3) $E_{\bar{\omega}}(w) < \delta$,

then $y_- = y_+$ and $w$ is $\eta$-close in the $C^1$-metric to a portion of the trivial strip over $y_-.$

*Proof.* — The proof will not depend on whether the Lagrangian boundary condition is singular because we are only concerned with one smooth component at a time in the lemma.

First we claim that, if the hypotheses of the lemma hold, then the image of $w$ is contained in an $\eta$-neighborhood of the strip over $y_-$. Arguing by contradiction, assume there exist $\eta < \eta_0$, sequences $R_n, c_n, \eta'_n, \delta_n, p_n \in [-R, R] \times [0, 1]$, and $\bar{J}$-holomorphic maps

$$w_n : [-R_n - c_n, R_n + c_n] \times [0, 1] \to \bar{W}$$

such that:

- $R_n \to +\infty, c_n \to +\infty, \eta'_n \to 0$ and $\delta_n \to 0,$

- $w_n$ satisfy (1)–(3) with $R_n, c_n, \eta'_n$ and $\delta_n$ in place of $R, c, \eta'$ and $\delta$, and

- $w_n(p_n)$ is at a distance of $\eta$ from the trivial strip over $y_-.$

Conditions (1)–(3) imply that $E_{\bar{\omega}}(w_n) \to 0$ and $E(w_n) < E_0$ for $n$ sufficiently large. Hence by Lemma 7.3.4 there is a constant $C$ such that $\|\nabla w_n\|_{C^0} \le C$ for all $n$. Then, after translating the maps $w_n$ so that the points $w_n(p_n) \in \bar{W} = \mathbf{R} \times [0, 1] \times \bar{S}$ all have the same $\mathbf{R}$-coordinate and taking a subsequence, the maps $w_n$ converge in $C_{\text{loc}}^\infty$ to a $\bar{J}$-holomorphic map $w_\infty : \mathbf{R} \times [0, 1] \to \bar{W}$ and the points $p_n$ converge to a point $p_\infty = (s_\infty, t_\infty) \in \mathbf{R} \times [0, 1]$ such that $E(w_\infty) < E_0$, $E_{\bar{\omega}}(w_\infty) = 0$, and $w_\infty(p_\infty)$ is at $\eta$-distance from the trivial strip over $y_-$. Since there is no chord at $\eta$-distance from $y_-$, the existence of the map $w_\infty$ is a contradiction. This proves the claim, which in turn implies that $y_- = y_+$.

Next we claim that $w$ is $\eta$-close in the $C^0$-metric to a trivial strip over $y_-$. The composition $\bar{\pi}_B \circ w$ of $w$ with the projection $\bar{\pi}_B : \bar{W} \to \mathbf{R} \times [0, 1]$ is holomorphic. Since $\bar{\pi}_B \circ w$ maps $[-R-c, R+c] \times \{i\}$ to $\mathbf{R} \times \{i\}$, for $i = 0, 1$, and $\pm(R+c) \times [0, 1]$ to arcs that are $\eta'$ close to $s = s_\pm$, where $s_\pm$ are constants, it follows that $\bar{\pi}_B \circ w$ must be a biholomorphism onto the image and moreover must be $C^0$-close to a translation of the map $(s, t) \mapsto (s, t)$, provided $\eta'$ is small. Therefore, after possibly reducing $\eta'$ in the statement of the theorem, $w$ is close to a trivial strip over $y_-.$

Finally, the $C^0$-closeness implies $C^1$-closeness (and in fact $C^k$-closeness for all $k$) by elliptic regularity. This concludes the proof of the lemma. $\square$
Now we consider long cylinders and strips whose extremities are close to constant maps. First we develop a bound on their $\bar{\omega}$-area in terms of the behavior of the extremities, which is a consequence of the topology of $\bar{W}_-$.

**Lemma 7.3.7.** — For every $\delta > 0$ there is $\eta' > 0$ such that, for every $R > 0$, if either

- $w : [-R, R] \times S^1 \to \bar{W}_-$ is a $\bar{J}_-$-holomorphic map, or

- $w : [-R, R] \times [0, 1] \to \bar{W}_-$ is a $\bar{J}_-$-holomorphic map mapping $[-R, R] \times \{0, 1\}$ to $L_{\bar{a}}$,

and $w|_{(\pm R) \times S^1}$ (or $w|_{(\pm R) \times [0, 1]}$) are $\eta'$-close to constant maps in the $C^1$-metric, then $E_{\bar{\omega}}(w) < \delta$.

*Proof.* — We consider strips first since cylinders are easier. We have two cases: either the boundary of $[-R, R] \times [0, 1]$ is mapped to the same component $L_{\bar{a}_i}$ or the two boundary components are mapped to different components $L_{\bar{a}_i}$ and $L_{\bar{a}_j}$.

We embed $[-R, R] \times [0, 1]$ holomorphically in $D^2 = \{|z| \le 1 | z \in C\}$ as the complement of an open neighborhood of $\{1, -1\}$ and extend $w$ to a smooth map $\tilde{w} : D^2 \to \bar{W}_-$ with boundary on $L_{\bar{a}_i}$ or $L_{\bar{a}_i} \cup L_{\bar{a}_j}$, depending on the situation. We can make the extension so that, in the neighborhood of $\{1, -1\}$, $\tilde{w}$ remains $\eta'$-close in the $C^1$-metric to constant maps, and therefore there is a positive constant $C$ depending only on $\bar{\omega}$ such that

$$ (7.3.2) \qquad |E_{\bar{\omega}}(w) - E_{\bar{\omega}}(\tilde{w})| \le C\eta'. $$

In the first case, $\tilde{w}$ defines an element in $\pi_2(\bar{W}_-, L_{\bar{a}_i})$, and, since the latter group is trivial, $E_{\bar{\omega}}(\tilde{w}) = 0$ and then we can chose $\eta'$ so that Equation (7.3.2) implies that $E_{\bar{\omega}}(w) < \delta$.

The second case is similar, but the map $\tilde{w}$ defines an element in the relative homotopy group $\pi_2(\bar{W}_-, L_{\bar{a}_i} \cup L_{\bar{a}_j})$. If $g(\bar{S}) > 1$, then the group is trivial and we can conclude as before.

If $\bar{S}$ is a torus, the first observation is that $\tilde{w}$ is homotopic to a map (denoted by the same symbol) $\tilde{w} : (D^2, \partial D^2) \to (\bar{S}, \bar{a}_i \cup \bar{a}_j)$ such that the preimage of an $\eta'$-neighborhood of $z_\infty$ consists of the two fixed neighborhoods of $+1$ and $-1$, respectively. The homotopy is obtained by taking the universal cover $\tilde{B}_-$ of $B_-$, identifying the pullback of the fibration $\bar{W}_- \to B_-$ with $\tilde{B}_- \times \bar{S}$, lifting $\tilde{w}$ to $\tilde{B}_- \times \bar{S}$, and composing the lift with the projection onto $\bar{S}$.

The universal cover of $\bar{S}$ is diffeomorphic to $\mathbb{R}^2$ and we can assume that the preimage of $\bar{a}_i \cup \bar{a}_j$ is a grid with vertices in the integer points of $\mathbb{R}^2$. If we lift $\tilde{w}$ to the universal cover of $\bar{S}$, we obtain a map from $D^2$ to $\mathbb{R}^2$ with boundary on the grid. If $\tilde{w}$ represents a non-trivial element in $\pi_2(\bar{S}, \bar{a}_i \cup \bar{a}_j)$, then its lift maps $\partial D^2$ surjectively to some square of the grid, and therefore the preimage of an $\eta'$-neighborhood of the grid points consists of at least four disjoint open sets in $D^2$, which is a contradiction.
The case of cylinders is proved in the same way, using the fact that $\pi_2(\overline{W}_-) = 0$. $\square$

**Remark 7.3.8.** — Since we are free to chose the genus of the open book decomposition in Theorem 1.2.1, the simpler case of $g(\overline{S}) > 1$ would have sufficed for the main result of the paper. However we also treated the case of $g(\overline{S}) = 1$ for completeness.

**Lemma 7.3.9 (Long Cylinder Lemma, Version 2).** — For all $\eta > 0$ there exists $\eta' > 0$ such that, for all $R > 0$, if

$$w : [-R, R] \times S^1 \to \overline{W}_-$$

is a $\overline{J}_-$-holomorphic such that $w|_{[-R] \times S^1}$ and $w|_{[R] \times S^1}$ are $\eta'$-close in the $\mathbb{C}^1$-metric to constant maps to points $p_-$ and $p_+$, respectively, then $w$ is $\eta$-close to a constant map in the $\mathbb{C}^1$-metric.

**Lemma 7.3.10 (Long Strip Lemma, Version 2).** — For all $\eta > 0$ there exists $\eta' > 0$ such that for all $R > 0$ the following holds: If

$$w : [-R, R] \times [0, 1] \to \overline{W}_-$$

is a $\overline{J}_-$-holomorphic map which satisfies:

(1) $w$ maps $[-R, R] \times \{1\}$ to $L_{\overline{a}_i}$ and $[-R, R] \times \{0\}$ to $L_{\overline{a}'_j}$,

(2) $w|_{[R] \times [0, 1]}$ and $w|_{[-R] \times [0, 1]}$ are $\eta'$-close in the $\mathbb{C}^1$-norm to constant maps to $p_+$ and $p_-$, respectively,

then $w$ is $\eta$-close to a constant map in the $\mathbb{C}^1$-metric.

*Proof.* — First we show that, for $\eta'$ sufficiently small (and at least smaller than $\eta/2$), the map $\overline{\pi}_{B_-} \circ w$ is $\eta$-close to a constant map. We embed $[-R, R] \times [0, 1]$ holomorphically in $\mathbb{D}^2$ as the complement of an open neighborhood of $\{-1, 1\}$ as in the proof of Lemma 7.3.7. Since $\eta'$ is sufficiently small, we can

- assume without loss of generality that $p_+, p_- \in L_{\overline{a}_i}$;

- extend $w$ to a smooth map $\tilde{w} : \mathbb{D}^2 \to \overline{W}_-$ such that $\partial \mathbb{D}^2$ is mapped to $L_{\overline{a}_i} \cup L_{\overline{a}'_j}$; and

- assume that the extension occurs inside $\eta'$-neighborhoods of $p_+$ and $p_-.

The composition $\overline{\pi}_{B_-} \circ w : [-R, R] \times [0, 1] \to B_-$ is holomorphic and its extension $\overline{\pi}_{B_-} \circ \tilde{w} : \mathbb{D}^2 \to B_-$, viewed as an element of $\pi_2(B_-)$, is contractible. This implies that the image of $\overline{\pi}_{B_-} \circ \tilde{w}$ is contained in the union of the balls of radius $\eta'$ centered at $\overline{\pi}_{B_-}(p_+)$ and $\overline{\pi}_{B_-}(p_-)$, as follows: Assume by contradiction that this is not the case. Then there is a regular value $q \in B_- - \overline{\pi}_{B_-}(p_+) - \overline{\pi}_{B_-}(p_-)$ of $\overline{\pi}_{B_-} \circ \tilde{w}$ such that $(\overline{\pi}_{B_-} \circ \tilde{w})^{-1}(q)$ is contained in an open subset of $\mathbb{D}^2$ on which $\overline{\pi}_{B_-} \circ \tilde{w}$ coincides with $\overline{\pi}_{B_-} \circ w$, and therefore
is holomorphic. This is a contradiction since all preimages of $q$ contribute positively to the degree of $\bar{\pi}_{B_-} \circ \tilde{w}$. The balls of radius $\eta'$ centered at $\bar{\pi}_{B_-}(p_+)$ and $\bar{\pi}_{B_-}(p_-)$ have a nonempty intersection because $[-R, R] \times [0, 1]$ is connected, and therefore the image of $\bar{\pi}_{B_-} \circ w$ is contained in a ball or radius $2\eta' < \eta$ around some point $y \in B_-$. □

Let $\bar{S}_y = \bar{\pi}_{B_-}^{-1}(y)$ be the fiber over $y$, $N_\eta(\bar{S}_y) \subset \bar{W}_-$ the $\eta$-neighborhood of $\bar{S}_y$, and $p: N(\bar{S}_y) \to \bar{S}_y$ the map obtained by projecting out the directions transverse to the fibers after straightening out using the symplectic connection. Since the image of $\tilde{w}$ is contained in $N_\eta(\bar{S}_y)$, Lemma 7.3.7 and its proof imply that the map $p \circ \tilde{w} : D^2 \to \bar{S}_y$, viewed as an element of $\pi_2(\bar{S}_y, \bar{a}_i \cup \bar{a}_{i'})$, is contractible and $E_{\bar{\omega}}(\tilde{w}) \le \delta$ for some $\delta$ which depends only on $\bar{\omega}$. □

The map $p$ is not holomorphic, but its differential $dp(x)$ at any point $x \in N_\eta(\bar{S}_y)$ is the composition of the $\mathbf{C}$-linear projection $T_x \bar{W}_- \to T_x \bar{S}_{\bar{\pi}_{B_-}(x)}$ with the differential of the orientation-preserving diffeomorphism $\bar{S}_{\bar{\pi}_{B_-}(x)} \to \bar{S}_y$ induced by parallel transporting using the symplectic connection. This implies that all the preimages of a regular value of $p \circ w$ come with positive signs for the purposes of computing the degree. We can therefore repeat for $p \circ w$ the argument we used for $\bar{\pi}_{B_-} \circ w$ and this proves the lemma. □

**7.3.3. Proof of the compactness theorem.** — We follow the lines of the proof of Proposition 6.1.3. First we state the analog of Lemma 6.1.1 and sketch its proof.

**Lemma 7.3.11.** — Let $\bar{u}_i : (\dot{\bar{F}}_i, j_i) \to (\bar{W}_-, \bar{J}_-)$, $i \in \mathbf{N}$, be a sequence of $\bar{W}_-$-curves from $\boldsymbol{\gamma}$ to $\boldsymbol{y}$ with index $I_{\bar{W}_-}(\bar{u}_i) = n$ for some integer $n$. Then there is a subsequence such that all the $\dot{\bar{F}}_i$ are diffeomorphic to a fixed $\dot{\bar{F}}$.

*Proof.* — The $\bar{\omega}$-area bound for the $\bar{W}_-$ case is similar to that of the $W_+$ case. We take the difference of two $\bar{W}_-$-curves $\bar{u}_1$ and $\bar{u}_2$ from $\boldsymbol{\gamma}$ to $\boldsymbol{y}$ to obtain $Z \in H_2(\bar{W}_-)$. Since they both intersect $\sigma_\infty^-$ once, $Z$ can be represented by a surface in $W_-$, and hence can be viewed as a class in $H_2(\mathbb{N})$. The zero flux condition implies that the $\bar{\omega}$-area of $Z$ is zero.

We then use the Gromov-Taubes compactness theorem as in Step 2 of Lemma 6.1.1 to extract a subsequence $\bar{u}_i$ which converges in the sense of currents to a holomorphic building. In order to apply Schwarz reflection, we need to slightly extend $\bar{a}$ past $z_\infty$, but the rest of the proof is the same. After passing to a subsequence, we may assume that the homology class $[\bar{u}_i] \in H_2(\bar{W}_-, Z_{\boldsymbol{\gamma},\boldsymbol{y}})$ is fixed for all $i$. Once the homology class $[\bar{u}_i]$ is fixed, we use the relative adjunction formula (Lemma 5.6.3) in the same way as in Lemma 6.1.1 to obtain a bound on $\chi(\dot{\bar{F}}_i)$. □

The geometric setup for the $\bar{W}_-$-curves deviates from the standard SFT setup because the Lagrangian submanifold $L_{\bar{a}}$ is a union of finitely many immersed non-compact Lagrangians with boundary, with the property that each is an embedding when restricted to the interior, and the boundary components all lie in a properly embedded noncompact one-manifold along which the various branches of $L_{\bar{a}}$ have a clean intersection.
The noncompactness of the singular set means that the resulting “Legendrian” boundary condition at the negative end is singular. For this reason we will sketch the proof of Proposition 7.3.2.

*Proof of Proposition 7.3.2.* — By Lemma 7.3.11 we may assume that $\dot{F}_i = \dot{F}$ as smooth surfaces. We recall the main steps of the proof of SFT compactness.

(1) gradient bound,

(2) degeneration of the conformal structure,

(3) convergence on the thick part, and

(4) convergence on the thin part.

*Gradient bound.* Let $h$ be a Riemannian metric on $\overline{W}_-$ which is compatible with $\overline{J}_-$ and is cylindrical at both ends. After passing to a subsequence, we can add finite sets of marked points $Z_i$ to $\dot{F}$, all of the same cardinality, so that the following holds:

**Lemma 7.3.12.** — Let $g_i$ be a complete hyperbolic metric on $\dot{F} - Z_i$ which is compatible with $j_i$ and which has geodesic boundary and cusps at the boundary/interior punctures. Then there is a uniform bound

$$ (7.3.3) \qquad \rho_i \| \nabla \bar{u}_i \|_{C^0} \le C, $$

the so-called “gradient bound”, where $\rho_i$ is the injectivity radius of $(\dot{F} - Z_i, g_i)$ doubled along the boundary, and the norm $\| \nabla \bar{u}_i \|_{C^0}$ is computed with respect to the metrics $g_i$ and $h$.

Lemma 7.3.12 is proved in [BEHWZ, Lemma 10.7] for the case without boundary and the appropriate modifications are outlined in [Abb2, Remark 3.11]. Summarizing the argument, if there is a sequence $p_i \in \dot{F}$ for which $\rho_i(p_i) \| \nabla \bar{u}_i(p_i) \|_{C^0} \to \infty$, then by the usual rescaling argument as in Lemma 7.3.4, we have the bubbling of a nonconstant holomorphic plane or half-plane. The Long Cylinder Lemmas control the ends of the holomorphic plane or half-plane. We can place a pair of points of $Z_i$ in the middle of this bubbling region, which effectively bounds $\rho_i \| \nabla \bar{u}_i \|_{C^0}$ on the bubbling region with respect to the modified hyperbolic metric. Each bubble uses up a quantum of $E_{\bar{\omega}}$ (i.e., there exists $\bar{h} > 0$ such that $E_{\bar{\omega}} > \bar{h}$ on the bubble) — the key point is that this also holds for the singular Lagrangian boundary condition as we explained in the proof of Lemma 7.3.4 — which bounds the number of bubbles and hence the number of marked points.

*Degeneration of the conformal structure.* If $g_i$ degenerates as $i \to \infty$, then after passing to a subsequence there is a finite collection of homotopy classes of properly embedded arcs and closed curves on $\dot{F} - Z_i$, whose geodesic representatives are mutually disjoint, and which are pinched as $i \to \infty$, i.e., the lengths of the geodesic representatives go to zero. Let $\dot{F}_\infty$ be the nodal surface obtained by pinching all these geodesics. The complex structures $j_i$ converge in the $C_{\text{loc}}^\infty$ topology outside the pinching geodesics to a complex structure $j_\infty$ on $\dot{F}_\infty$. Then in the limit we obtain a nodal punctured Riemann surface with boundary $(\dot{F}_\infty, j_\infty)$. Let $F'_\infty$ be $\dot{F}_\infty$ with the nodes removed.