# The equivalence of Heegaard Floer homology and
embedded contact homology via open book
decompositions II

Vincent Colin, Paolo Ghiggini, Ko Honda

► To cite this version:

Vincent Colin, Paolo Ghiggini, Ko Honda. The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II. Publications Mathématiques de L'IHÉS, 2024, 139 (1), pp.189-348. 10.1007/s10240-024-00146-w. hal-04752836

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Submitted on 6 Jun 2025

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Distributed under a Creative Commons Attribution 4.0 International License
THE EQUIVALENCE OF HEEGAARD FLOER HOMOLOGY
AND EMBEDDED CONTACT HOMOLOGY VIA OPEN BOOK
DECOMPOSITIONS II

by VINGENT COLIN*, PAOLO GHIGGINI**, and KO HONDA***

ABSTRACT

This paper is the sequel to (Colin et al. in Publ. Math. Inst. Hautes Études Sci., 2024), and is devoted to proving
some of the technical parts of the HF=ECH isomorphism.

**Notation specific to [II]**

$\sigma_{\infty}^{\tau}, \sigma_{\infty}^{-\infty, i}$ Sections at infinity of $\overline{W}_{\tau}$ and $\overline{W}_{-\infty, i}$

$\pi_{B_{\tau}} : \overline{W}_{\tau} \to B_{\tau}$ Symplectic fibration with fiber S used in the definition of the chain ho-motopy

$\bar{\mathbf{b}} = \{\bar{b}_1, \dots, \bar{b}_{2g}\}$ Pushoff of $\bar{\mathbf{a}} = \{\bar{a}_1, \dots, \bar{a}_{2g}\}$

$B_{\tau}$ Base of the projection $\pi_{B_{\tau}} : \overline{W}_{\tau} \to B_{\tau}$

$B_{-\infty, i}$, $i = 1, 2$ Base of $\overline{W}_{-\infty, i}$

$\overline{\mathcal{I}}$ Space of all admissible $\{\overline{J}_{\tau} \in \mathcal{J}_{\overline{W}_{\tau}}\}_{\tau \in \mathbb{R}}$

$\mathcal{J}_{\overline{W}_{\tau}}$ Space of all admissible $\overline{J}_{\tau}$

$\mathcal{J}_{\overline{W}_{-\infty}}$ Space of admissible $\overline{J}_{-\infty}$

$\overline{J}_{\tau}$ Almost complex structure on $\overline{W}_{\tau}$

$\overline{J}_{-\infty} = \overline{J}_{-\infty, 1} \cup \overline{J}_{-\infty, 2}$ Almost complex structure on $\overline{W}_{-\infty} = \overline{W}_{-\infty, 1} \cup \overline{W}_{-\infty, 2}$

$\mathcal{L}_{t_0}$ The locus $\{t=t_0\}$, viewed as a subset of $B_+$, $B_-$, $B_\tau$, as appropriate

$L_{\overline{\mathbf{a}}, 1}^{-\infty, 1}$, $L_{\overline{\mathbf{b}}, 2}^{-\infty, 1}$, $L_{\overline{\mathbf{h}}(\overline{\mathbf{b}}), 3}^{-\infty, 1}$, $L_{\overline{\mathbf{h}}(\overline{\mathbf{a}}), 4}^{-\infty, 1}$ Lagrangian submanifolds on $\overline{W}_{-\infty, 1}$

$L_{\overline{\mathbf{a}}+, 1}^{-\infty, 2}$, $L_{\overline{\mathbf{b}}-, 1}^{-\infty, 2}$ Lagrangian submanifolds on $\overline{W}_{-\infty, 2}$

$L_{\overline{\mathbf{a}}, +}^{\tau,+}$, $L_{\overline{\mathbf{b}}, -}^{\tau,-}$ Lagrangian submanifolds on $\overline{W}_{\tau}$

$\overline{\mathbf{m}}(+\infty), \overline{\mathbf{m}}(-\infty)$ Marked points on $\overline{W}_{-}$ and $\overline{W}_{-\infty, 2}$, respectively

$\overline{\mathbf{m}}(\tau) = (\overline{\mathbf{m}}^b(\tau), \overline{\mathbf{m}}^f(\tau))$ 1-parameter family of marked points on $\overline{W}_{\tau}$

$n^*(\overline{u}), n^{*,\text{alt}}(\overline{u})$ Intersection numbers defined in Equation (3.2.3)

$\overline{W}_{\tau}$ Total space of the projection $\pi_{B_{\tau}} : \overline{W}_{\tau} \to B_{\tau}$

$\overline{W}_{-\infty} = \overline{W}_{-\infty, 1} \cup \overline{W}_{-\infty, 2}$ Limit of $\overline{W}_{\tau}$ as $\tau \to -\infty$

$x_{i1}^{\#}, x_{i2}^{\#}, x_{i3}^{\#}$ Intersection points of $\overline{a}_i$ and $\overline{b}_i$ besides $z_{\infty}$

<footnote>* VC supported by the Institut Universitaire de France, ANR Symplexe, ANR Floer Power, and ERC Geodycon.</footnote>

<footnote>** PG supported by ANR Floer Power and ANR TCGD.</footnote>

<footnote>*** KH supported by NSF Grants DMS-0805352, DMS-1105432, and DMS-1406564.</footnote>
# 1. Introduction

This paper is the sequel to [I] and is devoted to proving some of the technical parts of the isomorphism between $\widehat{HF}(-M)$ and $\widehat{ECH}(M)$. References from [I] will be written as "Section I.$x$" to mean "Section $x$" of [I], for example. The notation from [I] carries over to this paper and is summarized in the "Notation common to [I]–[III]" that appears at the beginning of [I].

In [I] we defined the chain maps

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

$$ \Psi : \mathrm{PFC}_{2g}(N) \to \widehat{CF}(S, \mathbf{a}, \mathcal{h}(\mathbf{a})). $$

In this paper we prove the following two results:

**Theorem 1.0.1 (Quasi-isomorphism).** — The chain maps $\Phi$ and $\Psi$ are quasi-isomorphisms, provided the monodromy map $\mathcal{h}$ does not have any elliptic periodic points of period $\le 2g$ and and rotation number $\theta \notin [-\frac{1}{2g}, \frac{1}{2g}]$.

Theorem 1.0.1 is quite involved and takes up most of this paper. The condition on $\mathcal{h}$ is a technical assumption which simplifies gluing and one should be able to remove this condition with more work.

**Theorem 1.0.2 (Stabilization).** — $\widehat{ECH}(M) \simeq \mathrm{PFH}_{2g}(N)$.

By successively stabilizing the open book $(S, \mathcal{h})$ and applying Theorem 1.0.1 to the stabilizations, we obtain a sequence of stable Hamiltonian structures $(\alpha_0, \omega^{2k})$ with contact perturbations $\alpha^{2k}$, for $k \ge g$, such that

$$ \mathrm{ECH}_{2k}(N, \alpha^{2k}) \cong \mathrm{PFH}_{2k}(N, \alpha_0, \omega^{2k}) $$

and the stabilization maps

$$ \mathrm{ECC}_{2k}(N, \alpha^{2k}) \to \mathrm{ECC}_{2k+2}(N, \alpha^{2k+2}), \quad \gamma \mapsto e^2 \mathfrak{K}_{2k}(\gamma) $$

are quasi-isomorphisms, where

$$ \mathfrak{K}_{2k} : \mathrm{ECC}_{2k}(N, \alpha^{2k}) \to \mathrm{ECC}_{2k}(N, \alpha^{2k+2}) $$

is a chain map inducing the continuation map. Then Theorem 1.0.2 follows from Theorem I.2.5.6.

Theorem 1.0.2 is proved in Section 5.

**Notation 1.0.3 (Sub/superscripts \*).** — In this paper, as in [I], \* is often used to denote a variety of possible subscripts/superscripts (e.g., intersection numbers $n^*(\bar{u})$ in
Equation (3.2.3), curves $\bar{v}_*$ and their domains $\dot{F}_*$ as in Notation 3.4.1, and moduli spaces $\mathcal{M}^*(\star)$ as in Notation 3.2.12).

*Organization of this paper.* In Sections 3 and 4 we prove the chain homotopy between the chain maps $\Psi \circ \Phi$ and a quasi-isomorphism, as well as the chain homotopy between the chain maps $\Phi \circ \Psi$ and id. The necessary Gromov-Witten type calculations are carried out in Section 2. Finally, Section 5 is devoted to proving Theorem 1.0.2.

## 2. Gromov-Witten type computations

This section is devoted to Gromov-Witten type calculations which are used in the proof of Theorem 1.0.1. After a brief review of relative Gromov-Witten invariants in Section 2.1, we treat a slightly simpler model situation in Section 2.2. We then tackle the specific situations of interest in this paper in Sections 2.3 and 2.4.

**2.1. Relative Gromov-Witten invariants.** — Relative Gromov-Witten invariants were introduced independently by Ionel and Parker in [IP1] and Li and Ruan in [LR]. The reader can see also McDuff [M] for a review of the topic. Although we were heavily inspired by their work, our definition is different from the original one.

Let $X$ be a four-dimensional symplectic manifold and $V \subset X$ a codimension two symplectic submanifold with trivial self-intersection.<fnref n="1" /> Throughout this section we will fix an almost complex structure which is integrable (and therefore locally split) on a neighborhood of $V$ and generic elsewhere. We fix a primitive homology class $A \in H_2(X; \mathbb{Z})$ such that $A \cdot V = d > 0$ and denote by $\mathcal{M}_{g,n}^J(A)$ the moduli space of $J$-holomorphic maps $u: (F, j, \mathbf{x}) \to (X, J)$, where $(F, j)$ is a genus $g$ closed Riemann surface and $\mathbf{x} = (x_1, \dots, x_n)$ is an ordered $n$-tuple of marked points. In practice, we will also allow $F$ to have boundary or constrain the evaluation at the marked points on some fixed cycles in $X$ which are not contained in $V$; straightforward variations on the theme are left to the reader. Whenever there is no risk of confusion, the almost complex structure will be dropped from the notation.

Since $A$ is a primitive class, every $J$-holomorphic map in the class $A$ is simple, and moreover its image is not contained in $V$ because $A \cdot V > 0$. Therefore the standard Fredholm theory for $J$-holomorphic curves applies and the moduli space $\mathcal{M}_{g,n}(A)$ is a smooth manifold of dimension

$$ (\mathbf{2.1.1}) \qquad \dim \mathcal{M}_{g,n}(A) = 2g - 2 + 2\langle c_1(\mathrm{TX}), A \rangle + 2n. $$

*Definition 2.1.1.* — We define a map $ev_V: \mathcal{M}_{g,n}(A) \to \mathrm{Sym}^d(V)$ such that $ev_V(u)$ is the set of intersection points between $u(F)$ and $V$ counted with multiplicity: that is, $ev_V(u) =$

<footnote n="1">¹ This simplifying condition will be satisfied in all our examples but is not strictly necessary.</footnote>
*(z₁, d₁), ..., (z_l, d_l)* such that {z₁, ..., z_l} = u(F) ∩ V and d_l is the multiplicity of the intersection point z_l.

This definition makes sense because of the local properties of J-holomorphic maps. Note that $ev_V$ is not sensitive to the cardinality of the preimage, e.g., it does not distinguish between a unique point of F mapped to V with order two and two points of F mapped to V with order one. This is undesirable, but in the examples the possibility of multiple points of F mapped to the same point of V will be excluded by topological considerations.

In order to show that $ev_V$ is a smooth map, we need the following weaker version of the Weierstrass preparation theorem. Let $D_\epsilon^n$ denote the ball of radius $\epsilon$ in $\mathbb{R}^n$ centered at 0.

**Lemma 2.1.2.** — If for some $\epsilon, \delta > 0$ a smooth map $F : D_\epsilon^n \times D_\delta^2 \to \mathbf{C}$ satisfies the following properties:

- for every $t \in D_\epsilon^n$ the map $f_t(z) = F(t, z)$ is holomorphic, and

- $f_0(0) = 0$ is a zero of order $k$,

then there exist $0 < \epsilon' < \epsilon$ and $0 < \delta' < \delta$ such that on $D_{\epsilon'}^n \times D_{\delta'}^2$ we can decompose $F(t, z) = P(t, z)Q(t, z)$, where $P$ and $Q$ are smooth maps which satisfy

- $p_t(z) = P(t, z)$ is a monic polynomial of degree $k$ in $z$ for every $t \in D_{\epsilon'}^n$, and

- $Q(t, z) \neq 0$ for every $(t, z) \in D_{\epsilon'}^n \times D_{\delta'}^2$.

In particular $p_t$ and $p_t$ have the same zeros with the same multiplicities for every $t$ and the coefficients of $p_t$ depend smoothly on $t$. For the proof of this lemma see the proof of [Hör, Theorem 7.51].

**Lemma 2.1.3.** — If every J-holomorphic map in $M_{g,n}(A)$ is graphical over V near its intersection points with V, then the map $ev_V : M_{g,n}(A) \to \mathrm{Sym}^d(V)$ is smooth.

The hypothesis that every J-holomorphic map is strong and probably unnecessary, but it will hold for topological reasons in all our applications.

*Proof.* — Take $u_0 \in M_{g,n}(A)$ with $ev_V(u_0) = \{(z_1, d_1), \dots, (z_l, d_l)\}$ and fix a smooth parametrization $D_\epsilon^m \to M_{g,n}(A)$, $t \mapsto u_t$, of a neighborhood of $u_0$. We will show that the composition

$$D_\epsilon^m \to \mathrm{Sym}^d(V), \quad t \mapsto ev_V(u_t)$$

is smooth. Choose pairwise disjoint neighborhoods $U_1, \dots, U_l \subset V$ around the points $z_1, \dots, z_l$ and identify each of them holomorphically with a disk $D_\delta^2 \subset \mathbf{C}$ in such a way that $z_i$ is identified with the origin. Since the almost complex structure $J$ is locally split and the J-holomorphic maps are graphical near $V$, after possibly making $\epsilon$ smaller, in a neighborhood of $V$ we can identify the image of the J-holomorphic maps $u_t$ for $t \in D_\epsilon^m$
with graphs of holomorphic functions $f_i^i: \mathcal{U}_i \cong D_\delta^2 \to \mathbf{C}$ for $i=1, \dots, l$ such that $z_i$ is a zero of order $d_i$ for $f_0^i$ and the zeros of $f_i^i$ correspond to the intersections of the image of $u_i$ with V near $z_i$. Then we apply Lemma 2.1.2 to obtain monic polynomials $p_i^j$ of degree $d_i$ with the same zeros as $f_i^i$ and whose coefficients are smooth functions of $t$, after possibly shrinking the neighborhoods $\mathcal{U}_i$.

The zeros of the functions $f_i^i$ (and the polynomials $p_i^j$) give a map

$$t \mapsto \mathrm{Sym}^{d_1}(\mathcal{U}_1) \times \cdots \times \mathrm{Sym}^{d_l}(\mathcal{U}_l) \subset \mathrm{Sym}^d(V).$$

The holomorphic identifications $\mathcal{U}_i \cong D_\delta^2$ give embeddings

$$\mathrm{Sym}^d(\mathcal{U}_i) \hookrightarrow \mathrm{Sym}^{d_i}(\mathbf{C}) \cong \mathbf{C}^{d_i}.$$

The identification $\mathrm{Sym}^{d_i}(\mathbf{C}) \cong \mathbf{C}^{d_i}$ comes from the correspondence between roots and coefficients of degree $d_i$ monic polynomials and induces the smooth structure on the symmetric product. Since the coefficients of $p_i^j$ define a smooth map $D_\epsilon^m \to \mathbf{C}^{d_i}$, the map $D_\epsilon^m \to \mathrm{Sym}^d(\mathbf{C})$ defined by the zeros is also smooth. $\square$

Let $ev_X : \mathcal{M}_{g,n}(A) \to X^n$ be the evaluation at the marked points. The two maps combine into an evaluation map

$$ev = (ev_X, ev_V) : \mathcal{M}_{g,n}(A) \to X^n \times \mathrm{Sym}^d(V).$$

**Definition 2.1.4.** — If $\mathcal{M}_{g,n}(A)$ is compact, the relative Gromov-Witten invariant $GW_{X,\Lambda,g,n}$ is the class

$$ev_*[\mathcal{M}_{g,n}(A)] \in \mathrm{H}_*(X^n \times \mathrm{Sym}^d(V)).$$

In order to obtain numerical invariants, we will intersect $ev_*[\mathcal{M}_{g,n}(A)]$ with topology classes of the appropriate codimension. In practice $\mathcal{M}_{g,n}(A)$ is rarely compact, so we will need to either replace it by a moduli space with point constraints, or compactify it by adding nodal J-holomorphic curves. Instead of developing a general theory, we prefer to deal with this issue case by case in the applications.

**2.2. First relative Gromov-Witten calculation.** — This first calculation is a warm-up for the following more involved ones, and will never be used in the paper. Let $\Sigma$ be a surface of genus $g \ge 1$. We consider $(X, \omega) = \Sigma \times \mathbf{CP}^1$ with a product symplectic form. Let $\pi_1$ and $\pi_2$ be the projections of $X$ onto $\Sigma$ and $\mathbf{CP}^1$, respectively. Let $V = \Sigma \times \{\infty\}$, $A = [\Sigma] + 2g[\mathbf{CP}^1]$, $g = g(\Sigma)$ and $n = g + 1$. The moduli space $\mathcal{M}_{g,g+1}(A)$ then has virtual dimension $8g + 4$ by Lemma 2.2.2 below; this is equal to the dimension of $X^{g+1} \times \mathrm{Sym}^{2g}(V)$. Hence, given generic $\mathbf{x} = ((z_1, y_1), \dots, (z_{g+1}, y_{g+1})) \in X^{g+1}$ and $\mathfrak{z} = \{(z'_1, \infty), \dots, (z'_{2g}, \infty)\} \in \mathrm{Sym}^{2g}(V)$ we define

$$G_1(\Sigma, \mathbf{x}, \mathfrak{z}) = \#ev^{-1}(\mathbf{x}, \mathfrak{z}),$$
where # denotes the cardinality modulo 2.

**Theorem 2.2.1.** — For generic $(\mathbf{x}, \mathfrak{z})$, $G_1(\Sigma, \mathbf{x}, \mathfrak{z}) = 1$.

In the proof of Theorem 2.2.1 we will use a product complex structure $J = j_{\Sigma} \times j_{CP^1}$ to carry out our calculations. This is legitimate because of certain automatic transversality results, as explained in the next lemma.

First recall that $M_g(A) = M_{g,0}(A)$ is the moduli space of $J$-holomorphic maps $u : (F,j) \to (X,J)$ in the class A, modulo automorphisms of the domain. Here $F$ is a closed surface of genus $g$ and $j$ ranges over all complex structures on $F$.

**Lemma 2.2.2.** — The moduli space $M_{g,g+1}(A)$ is regular and

$$ \dim M_{g,g+1}(A) = 8g + 4. $$

*Proof.* — Every curve $u \in M_g(A)$ is embedded, since the projection $\pi_1 \circ u$ is a holomorphic map of degree one, and therefore a biholomorphism. Then, by the adjunction formula,

$$ (\mathbf{2.2.1}) \qquad \langle c_1(\mathrm{TX}), A \rangle = \chi(\Sigma) + A \cdot A = 2g + 2. $$

Hence the expected dimension of $M_{g,g+1}(A)$ is $8g + 4$ by Equation (2.1.1). Since every $u \in M_g(A)$ is embedded and $\langle c_1(\mathrm{TX}), A \rangle > 0$, Hofer, Lizan and Sikorav's automatic transversality theorem [HLS, Theorem 1] (first suggested by Gromov [Gr]) implies that $M_g(A)$ is regular. Since adding free marked points has no effect on regularity, $M_{g,g+1}(A)$ is also regular and is a manifold of the dimension $8g + 4$. $\square$

Let $u \in M_g(A)$. Since $\pi_1 \circ u : F \to \Sigma$ has degree one, it follows that $(F,j)$ is biholomorphic to our chosen $(\Sigma, j_{\Sigma})$. Hence there is a one-to-one correspondence between maps $u \in M_g(A)$ and meromorphic functions (i.e., branched covers)

$$ v = \pi_2 \circ u : \Sigma \to CP^1 $$

of degree $2g$. Meromorphic functions can be studied using the classical Riemann-Roch theorem, which states that:

$$ l(D) - l(K - D) = \deg(D) + 1 - g, $$

where $D$ is a divisor on $\Sigma$, $K$ is the canonical divisor, and $l(D)$ is the complex dimension of the space of meromorphic functions on $\Sigma$ with poles at most at $D$.

**Lemma 2.2.3.** — If $\deg(D) = 2g - 1 + m$ with $m \ge 0$, then $l(D) = g + m$.

*Proof.* — Since $\deg(K) = 2g - 2$, we have $\deg(K - D) < 0$ and $l(K - D) = 0$. Hence $l(D) = \deg(D) + 1 - g = (2g - 1 + m) + 1 - g = g + m$. $\square$
Given $(\mathbf{x}, \mathfrak{z}) \in X^{g+1} \times \text{Sym}^{2g}(V)$ we write $\mathcal{M}_{g,g+1}(A, \mathbf{x}, \mathfrak{z}) = ev^{-1}(\mathbf{x}, \mathfrak{z})$.

**Lemma 2.2.4.** — If $(\mathbf{x}, \mathfrak{z})$ is generic, then $\mathcal{M}_{g,g+1}(A, \mathbf{x}, \mathfrak{z})$ is compact.

*Proof.* — Arguing by contradiction, let $u_i \in \mathcal{M}_{g,g+1}(A, \mathbf{x}, \mathfrak{z})$ be a sequence that converges to a limit nodal curve $u : F \to X$. Since $u$ is $J$-holomorphic, $\pi_1 \circ u$ is holomorphic and one component of $F$ must be biholomorphic to $\Sigma$. Hence $F$ is obtained from $\Sigma$ by attaching spheres which are mapped by $u$ to $\{z_i''\} \times \mathbf{CP}^1$, $1 \le i \le k$. Moreover, $k \le 2g$ by the positivity of intersections, since $u(F)$ represents the homology class $A = [\Sigma] + 2g[\mathbf{CP}^1]$ and $A \cdot [\Sigma] = 2g$.

(1) Suppose that $u(\Sigma) \neq V = \Sigma \times \{\infty\}$. Since $A \cdot [V] = 2g$ and the intersection points of $u(F)$ and $V$ are $(z'_1, \infty), \dots, (z'_{2g}, \infty)$, we must have

$$ \{z'_1, \dots, z'_k\} \subset \{z'_1, \dots, z'_{2g}\}. $$

We claim that the map $u|_{\Sigma}$ cannot exist. Let us write $v = \pi_2 \circ u|_{\Sigma} : \Sigma \to \mathbf{CP}^1$. Then $v$ is a meromorphic function with poles at $\{z'_1, \dots, z'_{2g}\} \setminus \{z''_1, \dots, z''_k\}$, subject to $v(z_i) = y_i$ for all $i = 1, \dots, g+1$. Hence $v \in l(D' - D'')$, where $D' = z'_1 + \dots + z'_{2g}$ and $D'' = z''_1 + \dots + z''_k$. By Lemma 2.2.3, $l(D' - D'') \le g$. [Apply the lemma to $D' - z''_1$, which has degree $2g-1$. This gives us $l(D' - z''_1) = g$. Then observe that $l(D' - z''_1) \ge l(D' - D'')$, more or less by definition.] On the other hand, there are more constraints $v(z_i) = y_i$ than there are linearly independent functions. Hence $v$ cannot exist provided $\{z_1, \dots, z_{g+1}\}$ and $\{y_1, \dots, y_{g+1}\}$ are generic, a contradiction.

(2) Now suppose that $u(\Sigma) = V$. Then $u$ consists of $u|_{\Sigma}$, together with $2g$ bubbles $\{z_i''\} \times \mathbf{CP}^1$, $i = 1, \dots, 2g$. The set $\{z''_1, \dots, z''_{2g}\}$ contains $\{z_1, \dots, z_{g+1}\}$ and may also contain $k \le g-1$ elements of $\{z'_1, \dots, z'_{2g}\}$. (Recall that the points $z_i$, $z'_i$ and $y_i$ are in generic position.) We then apply the renormalization procedure of [IP1, Proposition 6.6] to the sequence $u_i$ restricted to a neighborhood of $V$. After choosing suitable restrictions and rescalings, the sequence $u_i$ converges to a nonconstant meromorphic function $\xi : \Sigma \to \mathbf{CP}^1$ which encodes how the curves $u_i(\Sigma)$ approach $V$. The function $\xi$ has poles at most at $D'' = z''_1 + \dots + z''_{2g}$ and zeros at $2g-k \ge g+1$ points of $\{z'_1, \dots, z'_{2g}\}$. The details of this argument are left to the reader. Since $l(D'') = g+1$ by Lemma 2.2.3, we have at least as many constraints as linearly independent functions. This implies that $\xi = 0$, which is again a contradiction. $\square$

*Proof of Theorem 2.2.1.* — By Lemma 2.2.2 and Lemma 2.2.4, $G_1(\mathbf{x}, \mathfrak{z})$ is the number of meromorphic functions $v : \Sigma \to \mathbf{CP}^1$ with poles at $z'_1, \dots, z'_{2g}$ such that $v(z_i) = y_i$ for all $i = 1, \dots, g+1$. By Lemma 2.2.3, $l(z'_1 + \dots + z'_{2g}) = g+1$. On the other hand, since there are $g+1$ constraints $v(z_i) = y_i$, there is a unique solution for generic $(\mathbf{x}, \mathfrak{z})$. This shows that $G_1(\mathbf{x}, \mathfrak{z}) = 1$. $\square$
**2.3. Second relative Gromov-Witten calculation.**

**2.3.1. Definition of the invariant.** — Let $\Sigma = \bar{S}$, where $\bar{S} = S \cup D^2$ is obtained by capping off a page S of an open book decomposition with connected binding as in Section I.5.1.2. Also recall the “point at infinity” $z_\infty = \{\rho = 0\} \in \bar{S}$.

Consider $(X, \omega)$, where $X = \Sigma \times CP^1$ and $\omega$ is a product symplectic form. Let $V = V_1 \cup V_\infty$, where $V_* = \Sigma \times \{\ast\}$ and $\ast = 1, \infty$. We take $A = [\Sigma] + 2g[CP^1]$, $g = g(\Sigma) = g(S)$ and $n = 1$. We say that piecewise smooth simple closed curves $\bar{a}'_i$, $i = 1, \dots, 2g$, in $\Sigma$ homotopic to $\bar{a}_i$, $i = 1, \dots, 2g$ are in good position if $\Sigma \setminus (\bar{a}'_1 \cup \cdots \cup \bar{a}'_{2g})$ is connected and either of the following conditions is satisfied:

(P1) each $\bar{a}'_i$ is smoothly embedded, $z_\infty \notin \bar{a}'_1 \cup \cdots \cup \bar{a}'_{2g}$, each pair $\bar{a}'_i, \bar{a}'_j$, $i \neq j$, intersects transversely in at most one point, and no point in $\Sigma$ belongs to more than two curves; or

(P2) the curves $\bar{a}'_i$, $i = 1, \dots, 2g$, all intersect at $z_\infty$ and at no other point, and are smoothly embedded away from $z_\infty$, where they are allowed to have a corner.

We will need Condition (P2) in the proof of Lemma 4.3.4 and Condition (P1) in the computation. Throughout this section we will assume that the curves $\bar{a}'_1, \dots, \bar{a}'_{2g}$ are in good position and write $\bar{a}' = (\bar{a}'_1, \dots, \bar{a}'_{2g})$. We will denote by $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]$ the image of $\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}$ in $\text{Sym}^{2g}(V_\infty)$ using the identification $\Sigma \cong V_\infty$ given by the projection $\Sigma \times CP^1 \to \Sigma$.

**Lemma 2.3.1.** — If $\bar{a}'$ satisfies Condition (P1), then $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]$ is a smooth submanifold of $\text{Sym}^{2g}(V_\infty)$.

*Proof.* — By Condition (P1) and after taking complex charts near the double points, it is enough to consider the case of two smooth simple curves $\gamma_i : (-\varepsilon, \varepsilon) \to \mathbf{C}$, $i = 1, 2$, such that $\gamma_1(0) = \gamma_2(0) = 0$ and $(\dot{\gamma}_1(0), \dot{\gamma}_2(0))$ is a basis of $\mathbf{C}$. We define

$$ \Gamma : (-\varepsilon, \varepsilon)^2 \to \text{Sym}^2(\mathbf{C}), \quad \Gamma(x,y) = \{\gamma_1(x), \gamma_2(y)\}. $$

Recall the identification

$$ \Phi : \text{Sym}^2(\mathbf{C}) \to \mathbf{C}^2, \quad \Phi(\{z, w\}) = (z + w, zw), $$

which defines the smooth structure on $\text{Sym}^2(\mathbf{C})$. We have

$$ \Phi \circ \Gamma(x, y) = (\gamma_1(x) + \gamma_2(y), \gamma_1(x)\gamma_2(y)). $$

Computing the differential of this map and checking that it is injective at $(0, 0)$ is straightforward and therefore $\Gamma : (-\varepsilon, \varepsilon)^2 \to \text{Sym}^2(\mathbf{C})$ parametrizes a regular surface in $\text{Sym}^2(\mathbf{C})$. $\square$
We will use a product complex structure $J = j_{\Sigma} \times j_{CP^1}$ on $\Sigma \times CP^1$. We recall the moduli space $M_{g,1}(A)$ and the evaluation map

$$ ev = (ev_X, ev_{V_1}, ev_{V_{\infty}}) : M_{g,1}(A) \to X \times \mathrm{Sym}^{2g}(V_1) \times \mathrm{Sym}^{2g}(V_{\infty}) $$

defined in the previous section.

**Definition 2.3.2.** — For every $\mathfrak{z} = \{(z_1, d_1), \dots, (z_l, d_l)\} \in \mathrm{Sym}^{2g}(V_1)$ and generic $\bar{\mathbf{a}}'$ in good position, we define the relative Gromov-Witten invariant

$$ G_2(\Sigma, \mathfrak{z}) = \#ev^{-1}(\{(z_{\infty}, 0)\} \times \{\mathfrak{z}\} \times [\bar{a}'_1 \times \dots \times \bar{a}'_{2g}]), $$

where $\#$ denotes the cardinality modulo 2.

In the definition, the genericity condition on $\bar{\mathbf{a}}'$ is in the space of curves in good position and depends on $\mathfrak{z}$. Before proceeding, a couple of remarks are necessary. First, neither the complex structure nor the placement of the point $(z_{\infty}, 0)$ is generic. The meaningfulness of this definition will follow from automatic transversality results for the moduli spaces and the evaluation maps. Second, when Condition (P2) holds, the constraint $[\bar{a}'_1 \times \dots \times \bar{a}'_{2g}]$ may be singular when one or more coordinates are equal to $z_{\infty}$. While, strictly speaking, a cycle would be enough to define the relative Gromov-Witten invariant (and the set in question surely is a cycle), a number of arguments are simpler if we are allowed to use the machinery of smooth topology. The key observation here is that we can still reason as if $[\bar{a}'_1 \times \dots \times \bar{a}'_{2g}]$ were smooth because topological constraints prevent the $J$-holomorphic curves in $M_{g,1}(A)$ from passing both through $(z_{\infty}, 0)$ and the singular locus of $[\bar{a}'_1 \times \dots \times \bar{a}'_{2g}]$. The main result of the section is the following.

**Theorem 2.3.3.** — For every $\mathfrak{z} \in \mathrm{Sym}^{2g}(V_1 \setminus \{z_{\infty}\})$ we have $G_2(\Sigma, \mathfrak{z}) = 1$.

Recall from the previous section that the moduli space $M_g(A)$ is regular and can be identified with the space of meromorphic functions $\Sigma \to CP^1$ with poles of total degree $2g$.

**Lemma 2.3.4.** — The evaluation map

$$ (ev_X, ev_{V_1}) : M_{g,1}(A) \to X \times \mathrm{Sym}^{2g}(V_1) $$

is a submersion.

**Proof.** — The first step of the proof is to show that $ev_{V_1}$ is a submersion. Each fiber $\pi^{-1}(\mathfrak{z})$, where $\mathfrak{z} = \{(z_1, d_1), \dots, (z_l, d_l)\} \in \mathrm{Sym}^{2g}(V_1)$, can be viewed, after identifying $V_1$ with $\Sigma$, as the space of meromorphic functions $\Sigma \to CP^1$ with a pole of order $d_i$ at $z_i$ for $i = 1, \dots, l$. This space is an open dense subset of $H^0(\Sigma, \mathcal{O}(\mathfrak{z}))$, where we identify $V_1$ with $\Sigma$ and regard $\mathfrak{z}$ as the divisor $d_1z_1 + \dots + d_lz_l$. By Lemma 2.2.3, $H^0(\Sigma, \mathcal{O}(\mathfrak{z}))$ is a
complex vector space of dimension $l(\mathfrak{z}) = g + 1$ for each $\mathfrak{z} \in \text{Sym}^{2g}(\Sigma)$; the denseness of $\pi^{-1}(\mathfrak{z})$ is a consequence of the fact that $l(\mathfrak{z}') = g$ if $\mathfrak{z}' \in \text{Sym}^{2g-1}(\Sigma)$ and $\mathfrak{z}' \le \mathfrak{z}$ (i.e., $\mathfrak{z} - \mathfrak{z}'$ is effective). Hence $ev_{V_1}$ is the restriction of a holomorphic vector bundle

$$ \pi : \mathcal{E} \to \text{Sym}^{2g}(\Sigma), \quad \pi^{-1}(\mathfrak{z}) = \text{H}^0(\Sigma, \mathcal{O}(\mathfrak{z})) $$

to a dense open subset of $\mathcal{E}$. In particular, $\pi$ is a submersion.

For $(u, x) \in \mathcal{M}_{g,1}(A)$ (here $x$ is the marked point on the domain of $u$, which we identify with $\Sigma$), let $N_u$ be the normal bundle of $u$ and $D_u$ the linearized normal Cauchy-Riemann operator on $N_u$. Since $u$ is an embedding, there is an identification $T_{(u,x)}\mathcal{M}_{g,1}(A) = \ker D_u \oplus T_x\Sigma$ and $d_{(u,x)}ev_X(\xi, v) = \xi(x) + d_xu(v)$, while $d_{(u,x)}ev_{V_1}|_{T_x\Sigma} = 0$.

Since $ev_{V_1}$ is a submersion, then by the discussion above, the pair $(ev_X, ev_{V_1})$ is a submersion if for every $(u, x)$ there exists $\xi \in \ker D_u$ such that $\xi(x) \neq 0$ and $d_{(u,x)}ev_{V_1}(\xi, 0) = 0$.

We define an action of $\mathbf{C}$ on $\mathcal{M}_{g,1}(A)$ as follows. For every $(u, x) \in \mathcal{M}_{g,1}(A)$, we identify $u$ with a meromorphic function $\Sigma \to \mathbf{CP}^1$ with poles at $ev_{V_1}(u)$, and we define

$$ \mathbf{C} \times \mathcal{M}_{g,1}(A) \ni (a, (u, x)) \mapsto (u + a, x). $$

Let $\xi \in \ker D_u$ be the infinitesimal generator of this action. It is clear that $\xi(x) \neq 0$ (and in fact $\xi$ has no zeros) and that $d_{(u,x)}ev_{V_1}(\xi, 0) = 0$ because $ev_{V_1}(u) = ev_{V_1}(u+a)$ for every $a \in \mathbf{C}$. $\square$

For $\mathfrak{z} \in \text{Sym}^{2g}(V_1)$ let us write

$$ \mathcal{M}_{g,1}(A, z_{\infty}, \mathfrak{z}) = ev^{-1}(\{(z_{\infty}, 0)\} \times \{\mathfrak{z}\} \times [\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]). $$

The following statement is a corollary of the automatic transversality for the moduli space $\mathcal{M}_{g,1}(A)$ (see Lemma 2.2.2), Lemma 2.3.4, and the dimension formula (2.1.1).

**Lemma 2.3.5.** — For every $\mathfrak{z} \in \text{Sym}^{2g}(V_1)$ and generic $\bar{a}'$ in good position, the moduli space $\mathcal{M}_{g,1}(A, z_{\infty}, \mathfrak{z})$ is regular and zero-dimensional.

*Proof.* — For every $\mathfrak{z} \in \text{Sym}^{2g}(V_1)$ the moduli space

$$ \tilde{\mathcal{M}}_{g,1}(A, z_{\infty}, \mathfrak{z}) = (ev_X, ev_{V_1})^{-1}(\{(z_{\infty}, 0)\} \times \{\mathfrak{z}\}) $$

is regular by Lemma 2.3.4. We consider the restriction of the evaluation map

$$ ev_{V_{\infty}} : \tilde{\mathcal{M}}_{g,1}(A, z_{\infty}, \mathfrak{z}) \to \text{Sym}^{2g}(V_{\infty}). $$

If $\bar{a}'$ satisfies Condition (P1), generically $ev_{V_\infty}$ is transverse to $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]$ by the standard transversality theorem since $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]$ is a smooth manifold.

The only issue which remains to be considered is the nonsmoothness of $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]$ when $\bar{a}'$ satisfies Condition (P2). In that case, the nonsmoothness is concentrated in
the locus where at least one of the coordinates is equal to $(z_\infty, \infty)$. However, it is not possible that $u \in \mathcal{M}_{g,1}(A)$ passes through $(z_\infty, 0)$ and $(z_\infty, \infty)$ at the same time because $A \cdot \{[z_\infty] \times \mathbf{CP}^1\} = 1$. Therefore, for every $u \in \mathcal{M}_{g,1}(A)$, $ev_{V_\infty}(u)$ is disjoint from the singular part of $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]$, and therefore we can still apply the standard transversality theorem. $\square$

Given $\bar{a}'$ in good position, we define subsets $\mathfrak{G}_1(\bar{a}')$, $\mathfrak{G}_2(\bar{a}') \subset \mathrm{Sym}^{2g}(\Sigma)$ as follows: Let $\mathfrak{G}_1(\bar{a}')$ be the set of elements $\mathfrak{z} = \{(z_1, d_1), \dots, (z_l, d_l)\} \in \mathrm{Sym}^{2g}(\Sigma)$ such that $z_i \in \bar{a}'_i \cup \cdots \cup \bar{a}'_{2g}$ or $z_i = z_\infty$ for some $i \in \{1, \dots, l\}$.

If $\bar{a}'$ satisfies Condition $(\mathfrak{P}_1)$, then we set $\mathfrak{G}_2(\bar{a}') = \emptyset$. In order to define $\mathfrak{G}_2(\bar{a}')$ when $\bar{a}'$ satisfies Condition $(\mathfrak{P}_2)$, we first introduce the set $D(\bar{a}')$ of divisors $w_1 + \cdots + w_{2g}$ such that $w_i \in \bar{a}'_i$ for $i = 1, \dots, 2g$ and $w_i = z_\infty$ for at least one $i$. Then we define $\mathfrak{G}_2(\bar{a}')$ as the set of divisors $\mathfrak{z}$ of degree $2g$ which are linearly equivalent to a divisor of $D(\bar{a}')$, which means that there exist a divisor $\mathfrak{w} \in D(\bar{a}')$ and a meromorphic function $f : \Sigma \to \mathbf{CP}^1$ whose associated divisor is $\mathfrak{w} - \mathfrak{z}$. Finally, we set

$$ \mathfrak{G}(\bar{a}') = \mathfrak{G}_1(\bar{a}') \cup \mathfrak{G}_2(\bar{a}') $$

**Lemma 2.3.6.** — For every $\mathfrak{z} \in \mathrm{Sym}^{2g}(\Sigma \setminus \{z_\infty\})$ we have $\mathfrak{z} \notin \mathfrak{G}(\bar{a}')$ for a generic $\bar{a}'$.

*Proof.* — If $\mathfrak{z} \in \mathrm{Sym}^{2g}(\Sigma \setminus \{z_\infty\})$, then $\mathfrak{z} \notin \mathfrak{G}_1(\bar{a}')$ is immediate for generic $\bar{a}'$. Next we show that $\mathfrak{z} \notin \mathfrak{G}_2(\bar{a}')$ for generic $\bar{a}'$: Let $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]_\infty$ be the subset of $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]$ where at least one of the coordinates is equal to $z_\infty$. Then $\mathfrak{z} \notin \mathfrak{G}_2(\bar{a}')$ if and only if the linear equivalence class of $\mathfrak{z}$, seen as a divisor, is disjoint from $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]_\infty$. The linear equivalence class of $\mathfrak{z}$ can be identified with the quotient of the space of nonzero meromorphic functions on $\Sigma$ with poles at most at $\mathfrak{z}$ by the action of $\mathbf{C}^*$ by multiplication. By Lemma 2.2.3 this quotient is $\mathbf{CP}^g$, so it has real dimension $2g$. On the other hand, $[\bar{a}'_1 \times \cdots \times \bar{a}'_{2g}]_\infty$ is a union of submanifolds of dimension at most $2g - 1$, and therefore generically these two subsets do not intersect in $\mathrm{Sym}^{2g}(\Sigma)$. $\square$

**Lemma 2.3.7.** — If $\mathfrak{z} \in \mathrm{Sym}^{2g}(V_1) \setminus \mathfrak{G}(\bar{a}')$, then $\mathcal{M}_{g,1}(A, z_\infty, \mathfrak{z})$ is compact.

*Proof.* — Arguing by contradiction, let $u_i \in \mathcal{M}_{g,1}(A, z_\infty, \mathfrak{z})$ be a sequence that converges to a nodal J-holomorphic curve $u : F' \to X$. Recall the holomorphic projections $\pi_1 : X \to \Sigma$ and $\pi_2 : X \to \mathbf{CP}^1$. Since $u(F)$ represents the homology class $A = [\Sigma] + 2g[\mathbf{CP}^1]$ and $\pi_1 \circ u$ is holomorphic, one of the irreducible components of $F'$ must be biholomorphic to $\Sigma$ and all other irreducible components must be spheres which $u$ maps biholomorphically onto $\{w_i\} \times \mathbf{CP}^1$ for $i = 1, \dots, k$. Moreover, since $A \cdot V_1 = A \cdot V_\infty = 2g$, the positivity of intersections implies that $k \le 2g$.

(1) Suppose that $\bar{a}'$ satisfies Condition $(\mathfrak{P}_1)$.
If $w_i \notin \{z_1, \dots, z_l\}$ for some $i$, then the image of $u$ must contain $V_1$; otherwise, by the positivity of intersections, the total multiplicity of the intersections between $u(F')$ and $V_1$ would be at least equal to $2g + 1$, contradicting $A \cdot V_1 = 2g$. Hence the image of $u$ consists of $V_1$ and $2g$ bubbles $\{w_i\} \times \mathbf{CP}^1$ for $i = 1, \dots, 2g$. Moreover one bubble must pass through $(z_\infty, 0)$ and the total number of bubbles intersecting $\bar{a}'_1 \cup \dots \cup \bar{a}'_{2g}$, counted with multiplicity, is $2g$, which is impossible.

If $w_i \in \{z_1, \dots, z_l\}$ for some $i$, then there is an intersection between $u(F')$ and $V_\infty$ which does not belong to any of $\bar{a}'_1, \dots, \bar{a}'_{2g}$ since $\mathfrak{z} \notin \mathfrak{G}_1(\bar{\mathbf{a}}')$ (i.e., no $z_i$ lies on $\bar{a}'_1 \cup \dots \cup \bar{a}'_{2g}$). Hence the image of $u$ consists of $V_\infty$ with some bubbles attached that map to $\{z_i\} \times \mathbf{CP}^1$ with multiplicity $d_i$ for $i = 1, \dots, l$. It is impossible by intersection reasons that such a curve passes through $(z_\infty, 0)$.

(2) Suppose that $\bar{\mathbf{a}}'$ satisfies Condition (P2).

If $w_i \in \{z_1, \dots, z_l\}$ for some $i$, then again there is an intersection between $u(F')$ and $V_\infty$ which does not belong to any of $\bar{a}'_1, \dots, \bar{a}'_{2g}$, and this gives a contradiction as in (1).

If $w_i \notin \{z_1, \dots, z_l\}$ for some $i$, then, as in (1), the image of $u$ consists of $V_1$ and $2g$ bubbles $\{w_1\} \times \mathbf{CP}^1$ with $w_i \in \bar{a}_1$. Moreover $w_i = z_\infty$ for at least one $i$ because the image of $u$ must contain $(z_\infty, 0)$. Define the divisor $\mathfrak{w} = w_1 + \dots + w_{2g}$ on $\Sigma$. Applying the renormalization procedure of [IP1, Proposition 6.6] as in Lemma 2.2.4, we obtain a meromorphic function $\xi : \Sigma \to \mathbf{CP}^1$ with divisor $D(\xi) = \mathfrak{w} - \mathfrak{z}$. Clearly $\mathfrak{w} \in D(\bar{\mathbf{a}}')$, and therefore the existence of $\xi$ contradicts $\mathfrak{z} \notin \mathfrak{G}_2(\bar{\mathbf{a}}')$. $\square$

**Proposition 2.3.8.** — For every $\mathfrak{z} \in \text{Sym}^{2g}(\Sigma \setminus \{z_\infty\})$ and generic $\bar{\mathbf{a}}'$ in good position, the relative Gromov-Witten invariant $G_2(\Sigma, \mathfrak{z})$ is well-defined and its value does not depend on the choice of $\bar{\mathbf{a}}'$ in good position, complex structure on $\Sigma$ and $\mathfrak{z}$.

*Proof.* — Fix $\mathfrak{z} \in \text{Sym}^{2g}(\Sigma \setminus \{z_\infty\})$. By Lemma 2.3.6, $\mathfrak{z} \notin \mathfrak{G}(\bar{\mathbf{a}}')$ for a generic $\bar{\mathbf{a}}'$ in good position and therefore by Lemmas 2.3.5 and 2.3.7 the moduli spaces $M_{g,1}(A, z_\infty, \mathfrak{z})$ are regular, zero-dimensional, and compact. This proves that $G_2(\Sigma, \mathfrak{z})$ is well-defined.

At first we prove invariance when Condition (P1) holds: in this case $\mathfrak{G}(\bar{\mathbf{a}}') = \mathfrak{G}_1(\bar{\mathbf{a}}')$ and therefore $\text{Sym}^{2g}(V_1) \setminus \mathfrak{G}(\bar{\mathbf{a}}')$ is connected because

$$ \text{Sym}^{2g}(V_1) \setminus \mathfrak{G}_1(\bar{\mathbf{a}}') \cong \text{Sym}^{2g}(\Sigma \setminus (\bar{a}'_1 \cup \dots \cup \bar{a}'_{2g} \cup \{z_\infty\})). $$

Then, if we have pairs

(i) $\bar{\mathbf{a}}'_0$ and $\bar{\mathbf{a}}'_1$ of generic $2g$-tuples of curves in good position satisfying Condition (P1),

(ii) $j_0$ and $j_1$ of complex structures on $\Sigma$, and

(iii) $\mathfrak{z}_0$ and $\mathfrak{z}_1$ of points in $\text{Sym}^{2g}(V_1)$ such that $\mathfrak{z}_i \notin \mathfrak{G}(\bar{\mathbf{a}}'_i)$, $i = 1, 2$,
then we can find generic paths<fnref n="2" /> $\bar{\mathfrak{a}}'_t$, $j_t$ and $\mathfrak{z}_t$ interpolating between them, where $\bar{\mathfrak{a}}'_t$ is in good position and satisfies Condition (P1), and $\mathfrak{z}_t \in \text{Sym}^{2g}(V_1) \setminus \mathfrak{G}(\bar{\mathfrak{a}}'_t)$.

Since we can repeat the proofs of the Lemmas 2.3.5 and 2.3.7 for paths, the invariance follows in a standard manner.

If, instead, $\bar{\mathfrak{a}}'$ satisfies Condition (P2) and $\mathfrak{z} \in \text{Sym}^{2g}(V_1) \setminus \mathfrak{G}(\bar{\mathfrak{a}}')$, then we can find an open neighborhood $U$ of $\mathfrak{z}$ and a small perturbation $\bar{\mathfrak{a}}'_t$ of $\bar{\mathfrak{a}}'$ such that the good position condition is preserved all along the deformation, the final 2g-tuple of curves satisfies Condition (P1), and $U \cap \mathfrak{G}(\bar{\mathfrak{a}}'_t) = \emptyset$ all along the deformation. Then we can prove the analogues of Lemmas 2.3.5 and 2.3.7 for paths where $\mathfrak{z}$ remains in $U$, and again the invariance follows in a standard manner. This concludes the proof of the proposition. $\square$

**2.3.2. Case of $T^2$.** — We consider the situation where $\Sigma = T^2$. We suppose that $\bar{\mathfrak{a}}'$ satisfies Condition (P1). By the correspondence between J-holomorphic curves in $T^2 \times \mathbf{CP}^1$ representing the homology class $A = [T^2] + 2[\mathbf{CP}^1]$ and degree 2 meromorphic functions on $T^2$ for a generic $\mathfrak{z} = \{z_1, z_2\} \subset \text{Sym}^2(V_1) \setminus \mathfrak{G}(\bar{\mathfrak{a}}')$ with $z_1 \neq z_2$, the relative Gromov-Witten invariant $G_2(T^2, \mathfrak{z})$ is equal to the number of meromorphic functions $v: T^2 \to \mathbf{CP}^1$ of degree 2 such that $v(z_\infty) = 0$, $v(z_1) = v(z_2) = 1$ and the poles are on $\bar{\mathfrak{a}}'$.

**Lemma 2.3.9.** — If $z_1 \neq z_2$, then every meromorphic function on $T^2$ with poles at $z_1$ and $z_2$ of order 1 can be written as $\alpha_1 + \alpha_2 f$, where $\alpha_1, \alpha_2 \in \mathbf{C}$, $\alpha_2 \neq 0$, and $f$ is a fixed function on $T^2$ with poles at $z_1$ and $z_2$ of order 1.

In other words, $f$ is unique up to a postcomposition by a fractional linear transformation $\mathbf{CP}^1 \to \mathbf{CP}^1$ which fixes $\infty$.

*Proof.* — By Lemma 2.2.3, $l(z_i) = 1$ and is given by the constants, and $l(z_1 + z_2) = 2$ and one dimension is taken by the constants. The lemma follows. $\square$

**Lemma 2.3.10.** — $G_2(T^2, \mathfrak{z}) = 1$.

*Proof.* — Degree two meromorphic functions on $T^2$ are branched double covers $T^2 \to \mathbf{CP}^1$. According to Lemma 2.3.9, a branched double cover $v: T^2 \to \mathbf{CP}^1$ satisfying $v(z_0) = 0$ and $v(z_1) = v(z_2) = 1$ is determined up to postcomposition by $\eta \in \text{PSL}(2, \mathbf{C})$ which fixes 0 and 1; see Figure 1.

<footnote n="2">² A complication here is that the smooth structure on $\text{Sym}^{2g}(\Sigma)$ depends on the complex structure on $\Sigma$, and therefore for different values of $t$ the symmetric product $\text{Sym}^{2g}(\Sigma)$ could have different smooth structures, which are all diffeomorphic but not through the identity. However they can all be put in a smooth fiber bundle over $[0, 1]$ such that the fiber over $t$ has the smooth structure defined by $j_t$. The necessary modifications to the proof are standard but notationally heavy, so we prefer to leave them to the reader.</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. — The branched double cover $v$ of the torus $T^2$. The branch points $w_1, \dots, w_4$ of $v$ are indicated by ◦. (Color figure online)</figcaption></figure>

Let $w_1, \dots, w_4 \in \mathbf{CP}^1$ be the branch points of $v$. Since $z_1$ and $z_2$ can be chosen generic, we may assume that $w_i \notin \{0, 1, \infty\}$ for $i = 1, \dots, 4$ and are all distinct. Let $C_{ij}$ be a simple closed curve which bounds a small neighborhood of an arc from $w_i$ to $w_j$. We may assume that $C_{12}$ and $C_{23}$ intersect only at 2 points $y_1, y_2$. Let $\bar{a}'_1, \bar{a}'_2$ be lifts of $C_{12}$, $C_{23}$ to $T^2$ such that $\#(\bar{a}'_1 \cap \bar{a}'_2) = 1$ and $v(\bar{a}'_1 \cap \bar{a}'_2) = y_1$. We now impose the condition $\eta \circ v(z_3) = \eta \circ v(z_4) = \infty$, where $z_3 \in \bar{a}'_1, z_4 \in \bar{a}'_2$, and $\eta$ is a fractional linear transformation which fixes 0, 1. This is possible only if $z_3, z_4 \in v^{-1}(y_i)$ for $i = 1$ or 2. On the other hand, since $y_1, y_2$ are not branch points, $z_3$ and $z_4$ must be distinct. Hence the only possibility is $\{z_3, z_4\} = v^{-1}(y_2)$ and $\eta \in \mathrm{PSL}(2, \mathbf{C})$ is the uniquely determined by $\eta(0) = 0, \eta(1) = 1, \eta(y_2) = \infty$. This completes the proof of the lemma. $\square$

**2.3.3. Reduction to a torus.** — Now we reduce the computation of $G_2(\Sigma, \beta)$ to the case $\Sigma = T^2$ by a degeneration argument. In this section we assume again that the curves $\bar{a}'$ satisfy Condition (P$_1$).

Let $\Sigma_\tau = (\bar{S}, j_{\Sigma_\tau})$, $\tau \in [0, \infty)$, be a 1-parameter family of Riemann surfaces which degenerate into a nodal Riemann surface

$$ (\Sigma_\infty, \mathbf{n}) = ((\Sigma_-, \mathbf{n}_-) \sqcup (\Sigma_+, \mathbf{n}_+)) / \mathbf{n}_- \sim \mathbf{n}_+, $$
where $\Sigma_-$ has genus one, $\Sigma_+$ has genus $g-1$ and $\mathfrak{n} = \{\mathfrak{n}_-, \mathfrak{n}_+\} / \sim$. Let $j_{\Sigma_+}$ and $j_{\Sigma_-}$ be the complex structures on $\Sigma_+$ and $\Sigma_-$ respectively. Topologically $\Sigma_\infty$ is obtained as follows: Let $\gamma$ be a separating curve in $\bar{S}$ and let $\sim_0$ be an equivalence relation on $\bar{S}$ such that $x \sim_0 y$ if $x, y \in \gamma$. Then $\Sigma_\infty = \bar{S} / \sim_0$. We assume that $z_\infty$ is disjoint from $\gamma$ and is mapped to $\Sigma_-$. We also assume that the curves in $\bar{\alpha}'$ are disjoint from $\gamma$ and that $\bar{\alpha}'_1, \bar{\alpha}'_2$ are mapped to $\Sigma_-$, while $\bar{\alpha}'_3, \dots, \bar{\alpha}'_{2g}$ are mapped to $\Sigma_+$. Finally we choose $\mathfrak{z} = \{z_1, \dots, z_{2g}\} \in \text{Sym}^{2g}(V_1) \setminus \mathfrak{G}(\bar{\alpha}')$ such that $z_i \notin \gamma$ for $i=1, \dots, 2g$, $\{z_1, z_2\}$ are mapped to $\Sigma_-$ and $\{z_3, \dots, z_{2g}\}$ are mapped to $\Sigma_+$. We denote $\mathfrak{z}_- = \{z_1, z_2\}$ and $\mathfrak{z}_+ = \{z_3, \dots, z_{2g}\}$.

Define product complex structures $J_\tau = j_{\Sigma_\tau} \times j_{CP^1}$ on $\Sigma_\tau \times CP^1$ and $J_\pm = j_{\Sigma_\pm} \times j_{CP^1}$ on $\Sigma_\pm \times CP^1$. We denote $P_n = \{\mathfrak{n}\} \times CP^1$, which we see as a submanifold of both $\Sigma_+ \times CP^1$ and $\Sigma_- \times CP^1$. We define the parametric moduli space<fnref n="3" />

$$ \mathcal{M}_{g,1}^*(\mathfrak{A}, z_\infty, \mathfrak{z}) = \{ (u, \tau) \mid \tau \in [0, \infty), u \in \mathcal{M}_{g,1}^{J_\tau}(\mathfrak{A}, z_\infty, \mathfrak{z}) \}. $$

**Proposition 2.3.11.** — Let $\overline{\mathcal{M}}_{g,1}^*(\mathfrak{A}, z_\infty, \mathfrak{z})$ be the compactification of $\mathcal{M}_{g,1}^*(\mathfrak{A}, z_\infty, \mathfrak{z})$.

Then

$$ \overline{\mathcal{M}}_{g,1}^*(\mathfrak{A}, z_\infty, \mathfrak{z}) - \mathcal{M}_{g,1}^*(\mathfrak{A}, z_\infty, \mathfrak{z}) = \mathcal{M}_{1,1}^{J_-(\mathfrak{A}_-, z_\infty, \mathfrak{z}_-)} \times_{P_n} \mathcal{M}_{g-1}^{J_+(\mathfrak{A}_+, \mathfrak{z}_+)} $$

where $A_+ = [\Sigma_+] + 2(g-1)[CP^1]$ and $A_- = [\Sigma_-] + 2[CP^1]$.

*Proof.* — By Gromov compactness, up to passing to a subsequence, $(u_n, \tau_n)$ converges to $(u_+, u_0, u_-)$, where $u_\pm$ maps to $\Sigma_\pm \times CP^1$ and $u_0$ maps to $\{\mathfrak{n}\} \times CP^1$. Let $A_\pm$ be the homology class in $H_2(\Sigma_\pm \times CP^1)$ represented by $u_\pm$, and let $k_0[CP^1]$ be the homology class represented by $u_0$. A simple computation shows that $A_\pm = [\Sigma_\pm] + k_\pm[CP^1]$, where $k_+ + k_- + k_0 = 2g$. The point constraints for $u_\pm$ imply that $k_+ \ge 2g-2$ and $k_- \ge 2$ and, on the other hand, $k_0 \ge 0$ and $k_0 = 0$ if and only if $u_0$ is constant. This implies that $k_+ = 2g-2$, $k_- = 2$, and $u_0$ is constant. By Lemma 2.3.7, $u_-$ is irreducible, and therefore $u_- \in \mathcal{M}_{1,1}^{J_-(\mathfrak{A}_-, z_\infty, \mathfrak{z}_-)}$.

By the positivity of intersections, the image of $u_-$ must intersect $P_n$ at a single point $w$. Without loss of generality we can assume that $w \notin \bar{a}'_3 \cup \dots \cup \bar{a}'_{2g}$, since $u_-$ is the unique element of $\mathcal{M}_{1,1}^{J_-(\mathfrak{A}_-, z_\infty, \mathfrak{z}_-)}$ by Lemma 2.3.10. Since the Gromov limit of a sequence of connected holomorphic curves is connected, $u_+$ must also pass through $w$, once we have identified $\{\mathfrak{n}_+\} \times CP^1$ with $\{\mathfrak{n}_-\} \times CP^1$. Then we can repeat the proof of Lemma 2.3.7 and show that $u_+$ is also irreducible, and therefore $u_+ \in \mathcal{M}_{g-1,1}^{J_+(\mathfrak{A}_+, w, \mathfrak{z}_+)}$. This shows that

$$ \overline{\mathcal{M}}_{g,1}^*(\mathfrak{A}, z_\infty, \mathfrak{z}) - \mathcal{M}_{g,1}^*(\mathfrak{A}, z_\infty, \mathfrak{z}) \subseteq \mathcal{M}_{1,1}^{J_-(\mathfrak{A}_-, z_\infty, \mathfrak{z}_-)} \times_{P_n} \mathcal{M}_{g-1}^{J_+(\mathfrak{A}_+, w, \mathfrak{z}_+)} $$

<footnote n="3">³ We do not know if $\bar{\alpha}'$ is generic for every $\tau$. The proper way to proceed would be to choose generic paths $\tau \mapsto \mathfrak{z}_\tau$ and $\tau \mapsto \bar{\alpha}'_\tau$ such that $\mathfrak{z}_\tau \in \text{Sym}^{2g}(V_1) \setminus \mathfrak{G}(\bar{\alpha}')$ for all $\tau$. The caveat about the smooth structure of the symmetric product explained in Footnote 2 also applies here.</footnote>
The reverse inclusion follows from what Seidel in [Se, Proposition 2.7] calls an average specimen of the “gluing theorem” type. □

*Proof of Theorem 2.3.3.* — We prove the theorem by induction on the genus of $\Sigma$. The case $g=1$ follows from Proposition 2.3.8 and Lemma 2.3.10. If $\Sigma$ has genus $g$ and we assume that the statement is true for genus $g-1$, we use Proposition 2.3.8 to modify the curves $\bar{a}'$ and the points $\zeta$ so that we can apply Proposition 2.3.11. Then

$$G_2(\Sigma, \zeta) = G_2(\Sigma_-, \zeta_-)G_2(\Sigma_+, \zeta_+)$$

and $G_2(\Sigma, \zeta) = 1$ by Lemma 2.3.10 and the inductive hypothesis. □

**2.4. Third relative Gromov-Witten calculation.**

**2.4.1. Definitions.** — Let $D^2$ be the closed unit disk $\{|z| \le 1\} \subset \mathbb{C}$ and let $\Sigma = \bar{S}$. We consider $(X, \omega)$, where $X = \Sigma \times D^2$ and $\omega$ is a product symplectic form.

Recall $\hat{a} \subset \bar{S}$ as well as its components $\hat{a}_i$, $i=1, \dots, 2g$, from Section I.5.2.2; here $\hat{a}_i$ is an open arc, $\{z_\infty\} = \bar{a}_i - \hat{a}_i$, and $z_\infty$ is the point at infinity. Then $L_{\hat{a}} = \hat{a} \times \partial D^2 \subset \partial X$ is a Lagrangian submanifold of $(X, \omega)$.<fnref n="4" /> We write $\pi_i$, $i=1, 2$, for the projection of $X$ onto the $i$th factor. Let $A$ be the relative homology class

$$A = [\Sigma] + 2g \cdot [D^2] \in \mathcal{H}_2(X, L_{\hat{a}}),$$

and let $(F, j, x_0, \mathbf{x})$ be a compact Riemann surface $(F, j)$ of genus $g$ with $2g$ boundary components, together with an interior marked point $x_0$ and boundary marked points $\mathbf{x} = (x_1, \dots, x_{4g})$, where $x_i, x_{2g+i}$ are on the $i$th boundary component $\partial_i F$.

Again, we work with a product almost complex structure $J = j_\Sigma \times i_{D^2}$ on $X$. Let $\mathcal{M}_{g,1,4g}(A)$ (resp. $\mathcal{M}_{g,0,0}(A)$) be the moduli space of holomorphic maps

$$u : (F, j, x_0, \mathbf{x}) \to (X, J) \quad (\text{resp. } u : (F, j) \to (X, J))$$

such that $u(\partial_i F) \subset L_{\hat{a}_i}$ and $u_*([F]) = A$, modulo automorphisms of the domain. We consider the evaluation maps

$$(2.4.1) \qquad ev_X : \mathcal{M}_{g,1,4g}(A) \to X, \quad (u, x_0, \mathbf{x}) \mapsto u(x_0);$$

$$(2.4.2) \qquad ev_\partial : \mathcal{M}_{g,1,4g}(A) \to L_{\hat{a}_1} \times \dots \times L_{\hat{a}_{2g}} \times L_{\hat{a}_1} \times \dots \times L_{\hat{a}_{2g}}, \\ (u, x_0, \mathbf{x}) \mapsto (u(x_1), \dots, u(x_{4g})).$$

and $ev = (ev_X, ev_\partial)$. Let $\mathbf{z} = (z_1, \dots, z_{4g})$ be a $4g$-tuple of points in $\Sigma$ such that $z_i, z_{2g+i} \in \hat{a}_i$, $i=1, \dots, 2g$. Then we write

$$\mathcal{M}_{g,1,4g}(A, z_\infty) = ev_X^{-1}(z_\infty, 0) \quad \text{and}$$

<footnote n="4">⁴ We also could have used $L_{\hat{a}} = \bar{a} \times \partial D^2 \subset \partial X$, but we write $L_{\hat{a}}$ to emphasize that the holomorphic maps never intersect $\{z_\infty\} \times \partial D^2$.</footnote>
$$
\begin{align*}
\mathcal{M}_{g,1,4g}(\mathbf{A}, z_{\infty}, \mathbf{z}) \\
&= ev^{-1}((z_{\infty}, 0), (z_1, 1), \dots, (z_{2g}, 1), (z_{2g+1}, -1), \dots, (z_{4g}, -1)).
\end{align*}
$$

Definition 2.4.1. — The relative Gromov-Witten type invariant G$_3$(Σ, **z**) is the cardinality modulo two of M$_{g,1,4g}$(A, z$_{∞}$, **z**) for **z** generic.

Theorem 2.4.2. — If **z** is generic, then G<sub>3</sub>(Σ, **z**) = 1.

Proof. — The regularity and compactness for the moduli space $\mathcal{M}_{g,1,4g}(\mathbf{A}, z_{\infty}, \mathbf{z})$ are proved in Corollary 2.4.10 and Lemma 2.4.12, respectively. The independence of $G_3(\Sigma, \mathbf{z})$ of the almost complex structure and $\mathbf{z}$ is proved in Lemma 2.4.13, and finally $G_3(\Sigma, \mathbf{z})$ is computed in Sections 2.4.4–2.4.6. $\square$

**2.4.2. Transversality.** — We recall the following automatic transversality result from [HLS]; see Theorem 2 and Remark (1) following the statement of Theorem 2 in [HLS]:

Theorem 2.4.3 (Hofer-Lizan-Sikorav). — Let (F, j) be a compact Riemann surface with nonempty boundary, (M, J) be an almost complex 4-manifold, and L ⊂ (M, J) be a totally real surface. Let

$$
u_0 : (\mathrm{F}, \partial \mathrm{F}, j) \to (\mathrm{M}, \mathrm{L}, \mathrm{J})
$$

be a holomorphic map which sends $\partial F$ to $L$. If $u_0$ is immersed and the sum of the Maslov indices of $u_0|_{\partial F}$ with respect to any unitary trivialization of $u_0^*\text{TM}$ is positive, then the set of holomorphic maps $u: (F, \partial F, j) \to (M, L, J)$ near $u_0$ is regular.

**Remark 2.4.4.** — Strictly speaking, Theorem 2.4.3 applies to the case where (M, J)
has no boundary. However, we note that Theorem 2.4.3 is a local result, i.e., only uses
an open neighborhood of the image of $u_0$. The complex manifold $(X = \Sigma \times D^2, J)$ can
then be slightly enlarged to a product complex manifold $\Sigma \times D_{1+\epsilon}^2$, where $D_{1+\epsilon}^2$ is a disk
of radius $1 + \epsilon$, so that the theorem applies. We then note that any nearby curve in the
enlargement with boundary on $L_{\hat{a}}$ has image in X by considering the projection to $D_{1+\epsilon}^2$.

Using Theorem 2.4.3, we prove the following key result:

**Lemma 2.4.5.** — The moduli space $\mathcal{M}_{g,0,0}(\mathbf{A})$ is regular and has dimension

$$
\dim \mathcal{M}_{g,0,0}(\mathbf{A}) = 4g + 2.
$$

*Proof.* — Let $u \in \mathcal{M}_{g,0,0}(\mathbf{A})$, where $u: (\mathrm{F},j) \to (\mathrm{X},\mathrm{J})$. Since $u$ maps each $\partial_i \mathrm{F}$ to a distinct $L_{\hat{a}_i}$, there are no branch points of $\pi_2 \circ u$ along $\partial \mathrm{F}$ and the curve $u$ is immersed
near $\partial F$. On the other hand, the restriction of $\pi_1 \circ u$ to $\text{int}(F)$ is a biholomorphism onto its image. Hence $u$ is an immersion on all of $F$.

The total Maslov index $\mu(u)$ relative to the Lagrangian $L_{\hat{a}}$, with respect to any unitary trivialization of $u^*TX$ is:

$$ \mu(u) = 2g \cdot \mu(\{pt\} \times D^2) + 2c_1(T\Sigma), $$

where $\{pt\} \times D^2$ has boundary on $L_{\hat{a}}$ and $\mu(\{pt\} \times D^2)$ is computed with respect to a trivialization of $TX|_{\{pt\} \times D^2}$. An easy calculation gives $\mu(\{pt\} \times D^2) = 2$ and $c_1(T\Sigma) = \chi(\Sigma) = 2 - 2g$. Hence

$$ (2.4.3) \qquad \mu(u) = 2g(2) + 2(2 - 2g) = 4 > 0. $$

The regularity of $M_{g,0,0}(A)$ then follows from Theorem 2.4.3, in view of Remark 2.4.4. Using the usual index formula for holomorphic curves with Lagrangian boundary, we obtain:

$$ \dim M_{g,0,0}(A) = -\chi(F) + \mu(u) = -(2 - 4g) + 4 = 4g + 2, $$

This completes the proof of Lemma 2.4.5. $\square$

**Corollary 2.4.6.** — The moduli space $M_{g,1,4g}(A)$ is regular and has dimension

$$ \dim M_{g,1,4g}(A) = 8g + 4. $$

Since the point $z_\infty$ is not generic, we also need an automatic transversality result for the evaluation map.

**Lemma 2.4.7.** — The evaluation map $ev_X : M_{g,1,4g}(A) \to X$ is a submersion.

*Proof.* — For every $(u, x_0, \mathbf{x}) \in M_{g,1,4g}(A)$, the differential

$$ d_{(u,x_0,\mathbf{x})} ev_X : T_{(u,x_0,\mathbf{x})} M_{g,1,4g}(A) \to T_{u(x_0)} X $$

is surjective thank to the following two local deformations of $(u, x_0, \mathbf{x})$: (i) postcomposition of $u$ by a fractional linear transformation and (ii) variation of the point $x$. $\square$

**Corollary 2.4.8.** — The moduli space $M_{g,1,4g}(A, z_\infty)$ is regular and has dimension $\dim M_{g,1,4g}(A, z_\infty) = 8g$.

Next we consider the evaluation map at the boundary points.

**Lemma 2.4.9.** — For a generic $\mathbf{z} = (z_1, \dots, z_{4g})$, the point

$$ \mathbf{z}_* = ((z_1, 1), \dots, (z_{2g}, 1), (z_{2g+1}, -1), \dots, (z_{4g}, -1)) $$
is a regular value of the evaluation map

$$ev_{\partial}: \mathcal{M}_{g,1,4g}(A, z_{\infty}) \to L_{\hat{a}_1} \times \cdots \times L_{\hat{a}_{2g}} \times L_{\hat{a}_1} \times \cdots \times L_{\hat{a}_{2g}},$$

defined as restriction of the map from Equation (2.4.2).

*Proof.* — We take the generic **z** to be a regular value of the composition of $ev_{\partial}$ with the projection

$$L_{\hat{a}_1} \times \cdots \times L_{\hat{a}_{2g}} \times L_{\hat{a}_1} \times \cdots \times L_{\hat{a}_{2g}} \xrightarrow{\pi_1} \hat{a}_1 \times \cdots \times \hat{a}_{2g} \times \hat{a}_1 \times \cdots \times \hat{a}_{2g};$$

by Sard's theorem the set of such **z** is open and dense.

We claim that if **z** is generic, then a regular value of $\pi_1 \circ ev_{\partial}$ is a regular value of $ev_{\partial}$: In fact we decompose

$$T_{(z_j, \pm 1)} L_{\hat{a}_j} = T_{z_j} \hat{a}_j \oplus T_{\pm 1} \partial D^2$$

and observe that the directions tangent to $\partial D^2$ are covered by moving the boundary marked points (since $u$ is an embedding) and the directions tangent to the arcs $\hat{a}_j$ are covered by moving the J-holomorphic curve in the moduli space as long as **z** is a regular value of $\pi_1 \circ ev_{\partial}$. $\square$

**Corollary 2.4.10.** — The moduli space $\mathcal{M}_{g,1,4g}(A, z_{\infty}, \mathbf{z})$ is regular and has dimension $\dim \mathcal{M}_{g,1,4g}(A, z_{\infty}, \mathbf{z}) = 0$.

We consider now a slightly different moduli space. Let

$$A' = [\Sigma] + (2g - 1)[D^2] \in H_2(X, L_{\hat{a}}),$$

and let $\mathbf{z}' = (z_1, \dots, z_{2g-1}, z_{2g+1}, \dots, z_{4g-1})$ be a $(4g-2)$-tuple of points such that $z_i, z_{2g+i} \in \hat{a}_i, i = 1, \dots, 2g-1$. We define the moduli space $\mathcal{M}_{g,1,4g-2}(A', z_{\infty}, \mathbf{z}')$ in a manner analogous to $\mathcal{M}_{g,1,4g}(A, z_{\infty}, \mathbf{z})$.

**Lemma 2.4.11.** — For generic **z** the moduli space $\mathcal{M}_{g,1,4g-2}(A', z_{\infty}, \mathbf{z}')$ is empty.

*Proof.* — For generic $\mathbf{z}'$ the moduli space $\mathcal{M}_{g,1,4g-2}(A', z_{\infty}, \mathbf{z}')$ is regular; the proof is the same as the proof of Corollary 2.4.10. One can compute from the dimension formula that the virtual dimension of $\mathcal{M}_{g,1,4g-2}(A', z_{\infty}, \mathbf{z}')$ is $-1$. This proves the lemma. $\square$

Observe that the lemma is still valid if the unoccupied arc is any $\hat{a}_i$ instead of $\hat{a}_{2g}$.
### 2.4.3. *Compactness.*

**Lemma 2.4.12.** — The moduli space $\mathcal{M}_{g,1,4g}(A, z_\infty, \mathbf{z})$ is compact for generic $\mathbf{z}$.

*Proof.* — Let $u_i : F_i \to X$ be a sequence of curves in $\mathcal{M}_{g,1,4g}(A, z_\infty, \mathbf{z})$ that limits to $u : F \to X$. The compactness of holomorphic curves with this kind of singular Lagrangian boundary condition has already been discussed in detail in Section I.7.3 in the SFT setting, but here the existence of the limit can be justified more easily as follows. The Lagrangian boundary condition is the product of a Lagrangian submanifold in $D^2$ (i.e., $\partial D^2$) and an immersed Lagrangian submanifold in $\Sigma = \bar{S}$ (i.e., the arcs $\bar{\mathbf{a}}$). Since we work with split almost complex structures, we can apply the Gromov compactness theorem on the two components separately. On the component where the Lagrangian boundary condition is smooth, this is of course classical, and on the component where the Lagrangian boundary condition is immersed we apply [EES, Theorem 9.2]. It is true that the reference only considers disks, but the only new ingredient in the higher genus case are domain degenerations, which happen at the source, and the necessary adjustments are straightforward.

We will eliminate all possible degenerations in the limit to show that $u \in \mathcal{M}_{g,1,4g}(A, z_\infty, \mathbf{z})$. Note in particular that we are assuming that all the points of $\mathbf{z}$ are distinct.

(i) We first claim that the domain of $u$ can have no irreducible component which is a disk. Arguing by contradiction, suppose that $u$ has a disk component $u'$. Since $\pi_1 \circ u_i|_{int(F_i)}$ is a biholomorphism onto its image for all $i$, $\text{Im}(u') = \{z\} \times D^2$ for some $z$. Also there is no fiber component of $u$ of the form $\Sigma \times \{pt\}$ where $pt \in \partial D^2$, since otherwise $u$ cannot pass through $(z_\infty, 0)$ and be in the class A. The point $z$ must then be an element of $\{z_1, \dots, z_{2g}\}$ by considering the intersections of $u$ with $\Sigma \times \{1\}$. Similarly $z \in \{z_{2g+1}, \dots, z_{4g}\}$. This is contradicts the assumption that all points in $\mathbf{z}$ are distinct.

(ii) Next we claim that $u(\partial F)$ does not intersect $\{z_\infty\} \times \partial D^2$. Arguing by contradiction, suppose that $u(\partial F) \cap (\{z_\infty\} \times \partial D^2) \neq \emptyset$. Since $u$ cannot have a disk component $\{z_\infty\} \times D^2$ by (i), there exists a point $z \in \Sigma - \bar{\mathbf{a}}$ close to $z_\infty$, such that $u$ intersects $\{z\} \times D^2$ at two points: once near $(z_\infty, 0)$ and once near $\{z_\infty\} \times \partial D^2$. Hence $\langle u, \{z\} \times D^2 \rangle \ge 2$, which is a contradiction. This also implies that $u$ has no nodes along the boundary: since each $u_i$ maps each component of $\partial F_i$ to a different $L_{\hat{u}_i}$, such a node could only be at a point of $\{z_\infty\} \times \partial D^2$.

(iii) Finally we eliminate interior nodes. Since $u$ has no disk components and sphere components (the latter easily follows from $\pi_2(X) = 0$), if $u$ has an interior node, it must have a component $u' : F' \to X$ of genus $< g$. Since $\pi_1 \circ u'|_{int(F')}$ cannot map onto the genus $g$ surface $\Sigma - (\hat{\mathbf{a}} - \text{N}(z_\infty))$, we have a contradiction. $\square$
Corollary 2.4.10 and Lemma 2.4.12 imply that $M_{g,1,4g}(A, z_\infty, \mathbf{z})$ is a finite set, and therefore the count modulo two

$$G_3(\Sigma, \mathbf{z}) = \#M_{g,1,4g}(A, z_\infty, \mathbf{z})$$

is well-defined, when $\mathbf{z}$ is generic.

**Lemma 2.4.13.** — The count $G_3(\Sigma, \mathbf{z})$ is independent of the choice of product complex structure $J$ and $\mathbf{z}$ generic.

*Proof.* — Given $J_t$ and $\mathbf{z}_t$ generic, $i = 0, 1$, we can connect them by generic paths $J_t$ and $\mathbf{z}_t$, $t \in [0, 1]$. We further assume that the path $\mathbf{z}_t$ satisfies the following (generic) properties:

(1) the number of parameters $t \in [0, 1]$ for which two points in $\mathbf{z}_t$ coincide is finite,

(2) at a given $t \in [0, 1]$ at most two points coincide, and

(3) if for some $t \in [0, 1]$ two points of $\mathbf{z}_t$ coincide, the remaining $4g-2$ are generic in the sense of Lemma 2.4.11.

We define the parametric moduli space

$$M_{g,1,4g}^{(J_t)}(A, z_\infty, \{\mathbf{z}_t\}) := \{(t, u) \mid t \in [0, 1], u \in M_{g,1,4g}^{J_t}(A, z_\infty, \mathbf{z}_t)\}.$$

We prove the compactness of $M_{g,1,4g}^{(J_t)}(A, z_\infty, \{\mathbf{z}_t\})$. The proof is similar to that of Lemma 2.4.12 and the only difference is that, when two points of $\mathbf{z}_t$ (say, without loss of generality, $z_{2g,t}$ and $z_{4g,t}$) come together for some isolated $t_0 \in (0, 1)$, a family of $J_t$-holomorphic curves $u_t$ could degenerate, as $t \to t_0$, into a nodal curve with a disk component $\{z_{2g,t_0}\} \times \mathbb{D}^2$ and one irreducible component $u' : F' \to X$ with $2g-1$ boundary components which passes through $(0, z_\infty)$ and $(z_1, 1), \dots, (z_{2g-1}, 1), (z_{2g+1}, -1), \dots, (z_{4g-1}, -1)$. The component $u'$ however cannot exist by Lemma 2.4.11 and Property (3) of the path $\{\mathbf{z}_t\}$.

The regularity of $M_{g,1,4g}^{(J_t)}(A, z_\infty, \{\mathbf{z}_t\})$ follows from Corollary 2.4.10 adapted to a generic 1-parameter family. The compactness and regularity then imply the lemma. $\square$

**2.4.4. Reduction to a torus.** — We now explain how to reduce to the case of a torus. As in Section 2.3.3, we degenerate $\Sigma_\tau = (\Sigma, j_{\Sigma_\tau})$, $\tau \in [0, \infty)$, into a nodal Riemann surface

$$(\Sigma_\infty, \mathbf{n}) = ((\Sigma_-, \mathbf{n}_-) \sqcup (\Sigma_+, \mathbf{n}_+)) / \mathbf{n}_- \sim \mathbf{n}_+,$$

where $\Sigma_-$ has genus one and $\Sigma_+$ has genus $g-1$ by pinching along a separating curve $\gamma$. We denote $D_n = \{\mathbf{n}\} \times \mathbb{D}^2$. Let $j_{\Sigma_+}$ and $j_{\Sigma_-}$ be the complex structures on $\Sigma_+$ and $\Sigma_-$, and define product complex structures $J_\tau = j_{\Sigma_\tau} \times i_{\mathbb{D}^2}$ and $J_\pm = j_{\Sigma_\pm} \times i_{\mathbb{D}^2}$. Here we assume the following:
(i) $z_\infty$, $\mathbf{z}$, and $\hat{\mathbf{a}}$ are fixed for all $\tau \in [0, \infty)$;

(ii) $\gamma$ is disjoint from $\mathbf{z}$, $\hat{a}_1$, $\hat{a}_2$, but intersects $\hat{a}_3, \dots, \hat{a}_{2g}$ at two points each;

(iii) in the limit $z_\infty$, $z_1$, $z_2$, $z_{2g+1}$, $z_{2g+2} \in \Sigma_-$ and $z_3, \dots, z_{2g}$, $z_{2g+3}, \dots, z_{4g} \in \Sigma_+$.

Let $\tilde{\mathbf{a}}$ be the quotient of $\hat{\mathbf{a}}$ at $\tau = \infty$. Let $v_\tau = \pi_2 \circ u_\tau$, where $\pi_2$ is the projection to $D^2$. We write $\mathbf{z}_- = (z_1, z_2, z_{2g+1}, z_{2g+2})$ and $\mathbf{z}_+ = (z_3, \dots, z_{2g}, z_{2g+3}, \dots, z_{4g})$.

**Proposition 2.4.14.** — Let $\overline{\mathcal{M}}_{g,1,4g}^{(\cup\tau)}(A, z_\infty, \mathbf{z})$ be the compactification of $\mathcal{M}_{g,1,4g}^{(\cup\tau)}(A, z_\infty, \mathbf{z})$. Then

$$ \overline{\mathcal{M}}_{g,1,4g}^{(\cup\tau)}(A, z_\infty, \mathbf{z}) - \mathcal{M}_{g,1,4g}^{(\cup\tau)}(A, z_\infty, \mathbf{z}) = \mathcal{M}_{1,1,4}^{J-}(A_-, z_\infty, \mathbf{z}_-) \times_{\mathrm{D}_n} \mathcal{M}_{g-1,0,4g-4}^{J+}(A_+, \mathbf{z}_+). $$

*Proof.* — Consider a sequence $(u^{\tau_i}, x_{0,\tau_i}, \mathbf{x}_{\tau_i}) \in \mathcal{M}_{g,1,4g}^{J_{\tau_i}}(A, z_\infty, \mathbf{z})$. We first note that $u^{\tau_i}$ cannot degenerate as $\tau_i$ approaches a finite $\tau_*$. This is as argued in Lemma 2.4.12. Hence we may assume that $\tau_i \to \infty$.

Let $u = u_- \cup u_0 \cup u_+$ be the limit of $u^{\tau_i}$ as $\tau_i \to \infty$, where $\mathrm{Im}(u_\pm) \subset \Sigma_\pm \times D^2$, $u_\pm$ has no components in $\mathrm{D}_n$, and $u_0$ maps to $\mathrm{D}_n$, and let $v_\pm = \pi_2 \circ u_\pm$. By the argument of Part (i) of the proof of Lemma 2.4.12, $u$ cannot have disk components $\{z\} \times D^2$, since otherwise they would introduce extra intersection points with

$$ \Sigma_\infty \times \{w\} = ((\Sigma_+ \cup \Sigma_-)/\sim) \times \{w\}, $$

where $w \in \mathrm{int}(D^2)$. In particular, this implies that $u_0 = \emptyset$.

Let $F_\pm$ be the domain of $u_\pm$. By Part (ii) of the proof of Lemma 2.4.12, $u_-(\partial F_-)$ does not intersect $\{z_\infty\} \times \partial D^2$. Let $\#(\partial F_\pm)$ be the number of boundary components of $F_\pm$.

We claim that $\#(\partial F_-) = 2$ and $\#(\partial F_+) = 2g - 2$. First observe that $\#(\partial F_-) \ge 2$, since two boundary components are needed to map to $\tilde{a}_i$, $i = 1, 2$; similarly $\#(\partial F_+) \ge 2g - 2$ since $z_3, \dots, z_{2g}, z_{2g+3}, \dots, z_{4g} \in \Sigma_+$.

The restriction of $v_-$ to each component $C$ of $\partial F_-$ is either a positive degree map $C \to \partial D^2$ or is a constant map to a point $w \in \partial D^2$. If $v_-$ maps $C$ to a point $w \in \partial D^2$, then $u_-$ maps an irreducible component of $F_-$ to a fiber $\Sigma_- \times \{w\}$. This in turn implies that $u_-$ has disk components, a contradiction. Hence the restriction of $v_-$ to each $C$ is a positive degree map. If $\#(\partial F_-) > 2$, then $\mathrm{deg}(v_-|_{\partial F_-}) > 2$. Similarly we obtain that $\mathrm{deg}(v_+|_{\partial F_+}) \ge 2g - 2$. This implies that $\mathrm{deg}((v_+ \cup v_-)|_{\partial F}) > 2g$, which is a contradiction. Hence $\#(\partial F_0) = 2$ and $\#(\partial F_-) = 2g - 2$.

The above claim implies that

$$ u_- \in \mathcal{M}_{1,1,4}^{J-}(A_-, z_\infty, \mathbf{z}_-), \quad u_+ \in \mathcal{M}_{g-1,0,4g-4}^{J+}(A_+, \mathbf{z}_+). $$
Moreover, the images of $u_-$ and $u_+$ intersect $D_n$ at the same point because the Gromov limit is compact. This proves the inclusion

$$ \overline{M}_{g, 1, 4g}^{(J_{\tau})}(A, z_{\infty}, \mathbf{z}) - M_{g, 1, 4g}^{(J_{\tau})}(A, z_{\infty}, \mathbf{z}) \\ \subseteq \mathcal{M}_{1, 1, 4}^{J_{-}}(A_{-}, z_{\infty}, \mathbf{z}_{-}) \times_{D_n} \mathcal{M}_{g-1, 0, 4g-4}^{J_{+}}(A_{+}, \mathbf{z}_{+}). $$

The opposite inclusion follows from the usual gluing argument. □

By the usual gluing argument we obtain the following:

**Corollary 2.4.15.**

$$ G_3(\Sigma, \mathbf{z}) = G_3(\Sigma_{-}, \mathbf{z}_1) \times G_3(\Sigma_{+}, \mathbf{z}_{+}). $$

*Proof.* — By Proposition 2.4.14 we have

$$ G_3(\Sigma, \mathbf{z}) = \# \left( \mathcal{M}_{1, 1, 4}^{J_{-}}(A_{-}, z_{\infty}, \mathbf{z}_{-}) \times_{D_n} \mathcal{M}_{g-1, 0, 4g-4}^{J_{+}}(A_{+}, \mathbf{z}_{+}) \right). $$

We have, by definition, $\#\mathcal{M}_{1, 1, 4}^{J_{-}}(A_{-}, z_{\infty}, \mathbf{z}_{-}) = G_3(T^2, \mathbf{z}_{-})$. Moreover, every curve $u \in \mathcal{M}_{1, 1, 4}^{J_{-}}(A_{-}, z_{\infty}, \mathbf{z}_{-})$ intersects $D_n$ at a unique point $(n, w) \in \Sigma_{-} \times D^2$. Now we observe that $n$ plays the role of $z_{\infty}$ for the arcs $\tilde{a}$ in $\Sigma_{+}$ (i.e., it is the unique intersection point of all the arcs). Moreover, we can assume $w = 0$ after an automorphism of $CP^1$ that fixes +1 and -1. This means that, for every curve $u_{-} \in \#\mathcal{M}_{1, 1, 4}^{J_{-}}(A_{-}, z_{\infty}, \mathbf{z}_{-})$, there are $G_3(\Sigma_{+}, \mathbf{z}_{+})$ curves in $\mathcal{M}_{g-1, 0, 4g-4}^{J_{+}}(A_{+}, \mathbf{z}_{+})$ which intersect $D_n$ at the same point as $u_{-}$. □

**2.4.5.** *Two calculations on $CP^1 \times D^2$. — We now calculate two model situations which are key ingredients in the proof of Lemma 2.4.19 below. Note that all the holomorphic curves on $CP^1 \times D^2$ that are considered below satisfy automatic transversality; see Hofer-Lizan-Sikorav [HLS, Theorem 2'.]*

Fix real numbers $a > b > 0$. Let $S_1$ be the set of pairs $(v_1, w)$, where $v_1$ is a degree 1 holomorphic map $D^2 \to CP^1$ (more precisely, is a biholomorphism onto its image when restricted to $int(D^2)$) such that $v_1(\partial D^2) \subset R^+$, $v_1(1) = a$, $v_1(-1) = b$, $v_1(0) = \infty$, and $w$ is a point in $D^2$ such that $v_1(w) = 0$, and let $C_1$ be the set of points $w$ for which there is some $v_1$ with $(v_1, w) \in S_1$. Similarly, let $S_2$ be the set of pairs $(v_2, w)$, where $v_2$ is a degree 1 holomorphic map $D^2 \to CP^1$ such that $v_2(\partial D^2) \subset R$, $v_2(1) = a$, $v_2(-1) = b$, $v_2(0) = \infty$, and $w$ is a point on $D^2$ such that $v_2(w) = -i$, and let $C_2$ be the set of points $w$ for which there is some $v_2$ with $(v_2, w) \in S_2$.

Let $\mathcal{R}_{\theta}$ be the restriction to $D^2$ of the radial ray which passes through $0 \in D^2$ and makes an angle of $\theta$ with the positive real axis.
<figure><img src="image_3.png" /><figcaption>FIG. 2. — Rough pictures of curves $C_1$ and $C_2$ appearing in Lemmas 2.4.16 and 2.4.17. (Color figure online)</figcaption></figure>

**Lemma 2.4.16.** — There exists $0 < \theta_0 < \frac{\pi}{2}$ such that $C_1 \subset D^2$ can be written as the image of a curve which is parametrized by $\theta \in (\pi - \theta_0, \pi + \theta_0)$, $C_1(\theta) = C_1 \cap \text{int}(\mathcal{R}_{\theta})$ (here we are abusing notation and using $C_1$ for both the curve and its image), and

$$ \lim_{\theta \to (\pi+\theta_0)^-} C_1(\theta) = e^{i(\pi+\theta_0)}, \quad \lim_{\theta \to (\pi-\theta_0)^+} C_1(\theta) = e^{i(\pi-\theta_0)}. $$

See Figure 2 for a rough picture of the curve $C_1$.

*Proof.* — By the Schwarz reflection principle, a map $v_1$ with $(v_1, w) \in S_1$ extends to a degree 2 branched cover $\mathbf{CP}^1 \to \mathbf{CP}^1$ with two branch points which lie on $\mathbf{R}^+ \subset \mathbf{CP}^1$. Hence $v_1$ admits a factorization

$$ D^2 \xrightarrow{f} \mathbf{H} \xrightarrow{\tilde{v}_1} \mathbf{CP}^1 \xrightarrow{g} \mathbf{CP}^1, $$

where $\mathbf{H} = \{\text{Im}(z) \ge 0\}$ is the upper half plane, $f: D^2 \to \mathbf{H}$ is a fractional linear transformation, $\tilde{v}_1(z) = z^2$, and $g \in \text{PSL}(2, \mathbf{R})$. In order for $v_1(0) = \infty$ and $v_1(w) = 0$ to hold, $f$ must map the ray $\mathcal{R}_{\theta}$ through 0 and $w$ to the line $\{\text{Re}(z) = 0\} \subset \mathbf{H}$. We may set $f(e^{i\theta}) = 0$ and $f(0) = i$; these conditions uniquely determine $f=f_{\theta}$. We leave it to the reader to verify that for each $\theta$, there is a one-to-one correspondence between $(v_1, w) \in S_1$ with $w \in \mathcal{R}_{\theta}$ and $w \in \mathcal{R}_{\theta}$ satisfying the following equality of cross ratios:

$$ (2.4.4) \qquad (\tilde{v}_1 \circ f_{\theta}(w), \tilde{v}_1 \circ f_{\theta}(1); \tilde{v}_1 \circ f_{\theta}(-1), -1) = (0, b; a, \infty). $$

Note that $\tilde{v}_1 \circ f_{\theta}(\mathcal{R}_{\theta}) = [-1, 0]$ and that there is at most one $w \in \mathcal{R}_{\theta}$ such that Equation (2.4.4) holds.

When $\theta = \pi$, $\tilde{v}_1 \circ f_{\pi}(1) = \infty$ and $\tilde{v}_1 \circ f_{\pi}(-1) = 0$. Hence there is a unique $w \in \mathcal{R}_{\pi}$ satisfying Equation (2.4.4). As $\theta$ moves from $\pi$ to $\frac{3\pi}{2}$, the points $\tilde{v}_1 \circ f_{\theta}(1)$ and $\tilde{v}_1 \circ f_{\theta}(-1)$ approach each other and become equal when $\theta = \frac{3\pi}{2}$. Hence there exists $0 < \theta_0 < \frac{\pi}{2}$ such that there is no $w \in \mathcal{R}_{\theta}$ for $\pi + \theta_0 < \theta < \frac{3\pi}{2}$ and there is a unique $w \in \mathcal{R}_{\theta}$ for $\pi \le \theta < \pi + \theta_0$. The situation of $\theta \in (\frac{\pi}{2}, \pi]$ is symmetric. The lemma then follows. $\square$
**Lemma 2.4.17.** — There exists a parametrization of $C_2 \subset D^2$ by $\theta \in (\frac{\pi}{2}, \frac{3\pi}{2})$ such that

$$ \lim_{\theta \to \frac{\pi}{2}+} C_2(\theta) = 1, \quad \lim_{\theta \to \frac{3\pi}{2}-} C_2(\theta) = -1. $$

See Figure 2 for a rough picture of the curve $C_2$.

*Proof.* — As in the proof of Lemma 2.4.16, $v_2$ can be factored as

$$ D^2 \xrightarrow{f_0} \mathbf{H} \xrightarrow{\tilde{v}_2} \mathbf{CP}^1 \xrightarrow{g} \mathbf{CP}^1, $$

where $\tilde{v}_2(z) = z^2$ and $f_0$ maps 0 to $i$ and $\mathcal{R}_\theta$ to $\{\mathrm{Re}(z) = 0, 0 \le \mathrm{Im}(z) \le 1\} \subset \mathbf{H}$. Note that there is a unique fractional linear transformation $g$ such that $g(-1) = \infty$, $g(0) = b$, and $g(\infty) = a$. For each $\theta \in (\frac{\pi}{2}, \frac{3\pi}{2})$, there exists $(v_2, C_2(\theta)) \in S_2$ such that $C_2(\theta)$ is in one of the half-disks of $D^2$ divided by $\mathcal{R}_\theta \cup \mathcal{R}_{\theta+\pi}$. As a reference point, $C_2(\pi)$ is in the upper half-disk. As $\theta$ approaches $\frac{\pi}{2}$ from above, the corresponding $v_2$ sends $-1$ and $1$ to arbitrarily close points. Hence $\lim_{\theta \to \frac{\pi}{2}+} C_2(\theta) = 1$. Similarly, $\lim_{\theta \to \frac{3\pi}{2}-} C_2(\theta) = -1$. $\square$

The following lemma is immediate from Lemmas 2.4.16 and 2.4.17; the key ingredient is that the endpoints of $C_1$ and $C_2$ alternate on $\partial D^2$:

**Lemma 2.4.18.** — The total count of intersection points between $C_1$ and $C_2$ is 1 modulo 2.

**2.4.6. Reduction to $CP^1 \times D^2$.** — We now explain how to further reduce from $T^2$ to $S^2$. We pinch $T^2$ along three parallel, disjoint, essential closed curves $\gamma_1, \gamma_2, \gamma_3$ to obtain a “sausage”

$$ (2.4.5) \qquad ((\Sigma_1, w_1, w'_2) \sqcup (\Sigma_2, w_2, w'_3) \sqcup (\Sigma_3, w_3, w'_1)) / \sim, $$

where $\Sigma_i \simeq S^2$, $i = 1, 2, 3$, and $w_i \sim w'_i$, $i = 1, 2, 3$. More precisely, pick an oriented identification $T^2 \simeq \mathbf{R}^2/\mathbf{Z}^2$ with coordinates $(x, y)$ so that $\bar{a}_1 = \{y=0\}$ and $\bar{a}_2 = \{x=0\}$. Then $\gamma_1 = \{x=\frac{1}{4}\}$, $\gamma_2 = \{x=\frac{1}{2}\}$, and $\gamma_3$ is obtained from $\{x=\frac{3}{4}\}$ by applying a finger move along the arc $[\frac{3}{4}, 1+\varepsilon] \times \{\frac{1}{2}\}$ so that $\gamma_3$ has two intersections $y = \frac{1}{2}-\varepsilon, \frac{1}{2}+\varepsilon$ with $\hat{a}_2$. We also assume that $z_1, z_3 \in \hat{a}_1$ lie on $\{\frac{1}{4} < x < \frac{1}{2}\}$ and $z_2, z_4 \in \hat{a}_2$ lie on $\{\frac{1}{2}-\varepsilon < y < \frac{1}{2}+\varepsilon\}$. Then $\Sigma_1$ is obtained from the closure of the connected component of $T^2 - \cup_{i=1}^3 \gamma_i$ which is bounded by (copies of) $\gamma_1$ and $\gamma_2$, by identifying all of $\gamma_1$ to $w_1$ and all of $\gamma_2$ to $w'_2$. The other components $\Sigma_2$ and $\Sigma_3$ are defined similarly. In particular, $z_\infty \in \Sigma_3$. See Figure 3.

**Lemma 2.4.19.** — $G_3(T^2, \mathbf{z}) = 1$, where $\mathbf{z} = \{z_1, \dots, z_4\}$.

*Proof.* — We degenerate the Riemann surfaces $\Sigma^\tau = (T^2, i_\tau)$, $\tau \to \infty$, by pinching along $\gamma_1 \cup \gamma_2 \cup \gamma_3$. Then a sequence of holomorphic maps

$$ u^\tau : (F, j_\tau) \to (\Sigma^\tau \times D^2, J_\tau) $$
<figure><img src="image_4.png" /><figcaption>FIG. 3. — The top diagram is the torus $T^2$, where the sides are identified and the top and the bottom are identified. The arrow indicates the projection of $T^2$ onto $\Sigma_1 \sqcup \Sigma_2 \sqcup \Sigma_3 / \sim$, given in (2.4.5). The components of $\tilde{a}_i = \tilde{a}_i / \sim$ which do not contain points in the set $\{z_1, \dots, z_4\}$ are not drawn in $\Sigma_i$. Also the two disks on the left and the right of the bottom diagram are glued into $\Sigma_3$. (Color figure online)</figcaption></figure>

in $M_{1,4}^{J_\tau}(A, z_\infty, \mathbf{z})$ converges to a nodal curve $(u_1, u_2, u_3)$, where

$$u_i : F_i \to \Sigma_i \times D^2, \quad i = 1, 2, 3,$$

and $F_1 = F_1 = D^2$, $F_3 = CP^1$. This is because $z_1, z_3 \in \Sigma_1$ and $z_2, z_4 \in \Sigma_2$, and the total number of boundary components $\sum_{i=1}^3 \#(\partial F_i)$ is equal to two by the argument in Section 2.4.4. Now, $u_3$ must have image $\Sigma_3 \times \{0\}$ since $z_\infty \in \Sigma_3$. The sets $S_1$ and $S_2$ for $v_i = \pi_2 \circ u_i, i = 1, 2$, were determined in Section 2.4.5. The gluing of intersecting curves $u_1$ and $u_2$ is given by the signed intersection number of $C_1$ and $C_2$, which is 1 modulo 2 by Lemma 2.4.18. $\square$

*Proof of Theorem 2.4.2.* — By Lemma 2.4.13 we can write $G_3(\Sigma) = G_3(\Sigma, \mathbf{z})$ for generic $\mathbf{z}$. By Corollary 2.4.15, if $\Sigma$ has genus $g$, then $G(\Sigma) = G(T^2)^g$. Then Lemma 2.4.19 implies that $G_3(\Sigma) = 1$. $\square$

### 3. Homotopy of cobordisms I

In this section and the next we prove Theorem 1.0.1. The chain homotopies that appear in the proof of Theorem 1.0.1 are induced by homotopies of cobordisms $\overline{W}_\tau^\pm$ and $\overline{W}_\tau^\mp$ which are parametrized by $\tau \in \mathbb{R}$. In this section we treat $\overline{W}_\tau^\pm$, leaving $\overline{W}_\tau^\mp$ for Section 4. If $\pm$ is understood (as it will be in the rest of this section), then it will be omitted.

We now give a brief informal description of $\overline{W}_\tau$, leaving precise definitions for later. The base $B_\tau$ of $\overline{W}_\tau$ is biholomorphic to an annulus with one puncture on each
boundary component; the neighborhoods of the punctures are viewed as strip-like ends. As $\tau \to +\infty$, $\overline{W}_\tau$ degenerates to the stacking of $\overline{W}_+$ “on top of” $\overline{W}_-$, where $\overline{W}_+$ and $\overline{W}_-$ are used in the definitions of the chain maps $\Phi$ and $\Psi$. (This is the reason why there is a $\pm$ in $\overline{W}_\tau^\pm$.) On the other hand, as $\tau \to -\infty$, $\overline{W}_\tau$ degenerates to $\overline{W}_{-\infty}$, whose base $B_{-\infty}$ is (more or less) given by:

$$ (3.0.1) \qquad B_{-\infty} = ((\mathbf{R} \times [0, 1]) \sqcup D) / \sim, $$

where $D = \{|z| \le 1\} \subset \mathbf{C}$ and $\sim$ identifies $(0, 1) \in \mathbf{R} \times [0, 1]$ with $1 \in D$ and $(0, 0) \in \mathbf{R} \times [0, 1]$ with $-1 \in D$.

**3.1. Construction of the homotopy of cobordisms for $\Psi \circ \Phi$**

**3.1.1. Recollections.** — In this subsection we recall some notation from [I].

Recall that $S$ is a compact oriented surface of genus $g$ with connected boundary (a page of an open book $(S, \tilde{h})$), $\overline{S} = S \cup D^2$ is a closed surface obtained by capping off $S$, $\overline{h} = \overline{h}_m : \overline{S} \tilde{\to} \overline{S}$ is an extension of $\tilde{h}$ which is dependent on the integer $m \gg 0$ as in Section I.5.1.2, and $\overline{\omega}$ is the area form on $\overline{S}$ from Section I.5.1.2 which is invariant under $\overline{h}$. Also $z_\infty$ is the origin $\rho = 0$ of $D^2 = \{\rho \le 1\}$ with polar coordinates $(\rho, \phi)$.

The mapping tori

$$ N = (S \times [0, 2])/(x, 2) \sim (\tilde{h}(x), 0), \quad \overline{N} = (\overline{S} \times [0, 2])/(x, 2) \sim (\overline{h}(x), 0) $$

were defined in Section I.5.1. Let $W = \mathbf{R} \times [0, 1] \times S$, $\overline{W} = \mathbf{R} \times [0, 1] \times \overline{S}$, $W' = \mathbf{R} \times N$, and $\overline{W}' = \mathbf{R} \times \overline{N}$; they admit symplectic fibrations with fibers diffeomorphic to $S$, $\overline{S}$, $S$, and $\overline{S}$, respectively. We also have the symplectic fibrations $\pi_{B_+} : W_+ \to B_+$, $\overline{\pi}_{B_+} : \overline{W}_+ \to B_+$, and $\overline{\pi}_{B_-} : \overline{W}_- \to B_-$ from Sections I.5.1.1 and I.5.1.2, with fibers diffeomorphic to $S$, $\overline{S}$, and $\overline{S}$, respectively.

The fibration $W$ (or $\overline{W}$) was used in the definition of $\widehat{CF}(S, \mathbf{a}, \tilde{h}(\mathbf{a}))$ and the fibration $W'$ (or $\overline{W}'$) in the definition of $PFC_{2g}(N)$. The positive end of $\overline{W}_+$ and the negative end of $\overline{W}_-$ agree with those of $\overline{W}$ and the negative end of $\overline{W}_+$ and the positive end of $\overline{W}_-$ agree with those of $\overline{W}'$. The fibrations $\pi_{B_+}$ and $\overline{\pi}_{B_-}$ were used in the definitions of the chain maps

$$ \Phi : \widehat{CF}(S, \mathbf{a}, \tilde{h}(\mathbf{a})) \to PFC_{2g}(N), $$

$$ \Psi : PFC_{2g}(N) \to \widehat{CF}(S, \mathbf{a}, \tilde{h}(\mathbf{a})). $$

**3.1.2. Definition of the family $(\overline{W}_\tau, \overline{m}(\tau))$.** — For each $r \in [2, \infty)$, consider the fibration

$$ \pi_r : \mathbf{R} \times \overline{N}_r \to \mathbf{R} \times (\mathbf{R}/r\mathbb{Z}), $$
where

$$ \overline{N}_r = (\overline{S} \times [0, r])/(x, r) \sim (\overline{h}(x), 0) $$

and $(s, t)$ are coordinates on $\mathbf{R} \times (\mathbf{R}/r\mathbf{Z})$. For each $l, r \in [2, \infty)$, define $\overline{W}_{l,r} = \pi_r^{-1}(B_{l,r})$, where the base $B_{l,r}$ is obtained by smoothing the corners of

$$ \{-l \le s \le l\} \cup \{0 \le t \le 1\} \subset \mathbf{R} \times (\mathbf{R}/r\mathbf{Z}). $$

Next choose a function

$$ (3.1.1) \qquad \eta = (l, r) : \mathbf{R} \to [2, \infty) \times [2, \infty), $$

which is obtained by smoothing

$$ \eta_0(\tau) = \begin{cases} (\tau + 2, 2), & \text{for } \tau \ge 0; \\ (2, 2 - \tau), & \text{for } \tau \le 0; \end{cases} $$

near $\tau = 0$. We then let $\overline{W}_\tau = \overline{W}_{\eta(\tau)}$ and $B_\tau = B_{\eta(\tau)}$. Let $\pi_{B_\tau} : \overline{W}_\tau \to B_\tau$ be the projection along $\{(s, t)\} \times \overline{S}$.

**3.1.3. Neck-stretching.** — As $\tau \to +\infty$, the cobordism $\overline{W}_\tau$ approaches the concatenation of $\overline{W}_+$ and $\overline{W}_-$; see Figure 4. On the other hand, for $\tau \ll 0$ we can view the rectangle $[-2, 2] \times [\frac{3}{2}, \frac{3}{2} - \tau] \subset B_\tau$ as a neck and by taking $\tau \to -\infty$ we are stretching along this neck and degenerating the cobordism $\overline{W}_\tau$ into a 2-component manifold $\overline{W}_{-\infty} = \overline{W}_{-\infty,1} \cup \overline{W}_{-\infty,2}$, which we describe now; see Figure 5.

The base of $\overline{W}_{-\infty} = \overline{W}_{-\infty,1} \cup \overline{W}_{-\infty,2}$ is $B_{-\infty} = B_{-\infty,1} \cup B_{-\infty,2}$, where $B_{-\infty,1}$ is obtained from $\{-2 \le s \le 2\} \cup \{0 \le t \le 1\} \subset \mathbf{R}^2$ by smoothing the corners and $B_{-\infty,2} = [-2, 2] \times \mathbf{R}$. Here both $\mathbf{R}^2$ and $[-2, 2] \times \mathbf{R}$ have coordinates $(s, t)$. The component $B_{-\infty,1}$ has four strip-like ends: the ends $s \to +\infty$, $s \to -\infty$, $t \to +\infty$, $t \to -\infty$ will be referred to as the top, bottom, left, and right ends. The component $B_{-\infty,2}$ has two strip-like ends: the ends $t \to +\infty$ and $t \to -\infty$ will be referred to as the left and right ends. As usual, $B_{-\infty}$ is endowed with identifications of the compactifications of the strip-like ends. More precisely, if the compactification $\tilde{B}_{-\infty,1}$ is obtained from $B_{-\infty,1}$ by attaching $\{\pm\infty\} \times [0, 1]$ and $[-2, 2] \times \{\pm\infty\}$ and the compactification $\tilde{B}_{-\infty,2}$ is obtained from $B_{-\infty,2}$ by attaching $[-2, 2] \times \{\pm\infty\}$, then we identify $(s, \pm\infty) \in \tilde{B}_{-\infty,1}$ with $(s, \mp\infty) \in \tilde{B}_{-\infty,2}$.

Let $\overline{W}_{-\infty,i} = B_{-\infty,i} \times \overline{S}$ for $i = 1, 2$. The 2-component building $\overline{W}_{-\infty} = \overline{W}_{-\infty,1} \cup \overline{W}_{-\infty,2}$ is endowed with identifications of compactifications of the ends. The compactification $\overline{\tilde{W}}_{-\infty,1}$ is obtained from $\overline{W}_{-\infty,1}$ by attaching $\{\pm\infty\} \times [0, 1] \times \overline{S}$ and $[-2, 2] \times \{\pm\infty\} \times \overline{S}$, the compactification $\overline{\tilde{W}}_{-\infty,2}$ is obtained from $\overline{W}_{-\infty,2}$ by attaching $[-2, 2] \times \{\pm\infty\} \times \overline{S}$, and we identify $(s, +\infty, x) \in \overline{\tilde{W}}_{-\infty,1}$ with $(s, -\infty, x) \in \overline{\tilde{W}}_{-\infty,2}$.
<figure><img src="image_5.png" /><figcaption>FIG. 4. — The bases of the family $\overline{W}_{\tau}$. The parameter $\tau$ increases as we go to the right. The sides are identified in this picture, as indicated. The location of $\overline{\mathfrak{m}}^{b}(\tau)$ is indicated by $\times$</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5. — The bases of $\overline{W}_{-\infty,2}$ to the left and $\overline{W}_{-\infty,1}$ to the right</figcaption></figure>

and $(s, +\infty, x) \in \overline{W}_{-\infty,2}$ with $(s, -\infty, \bar{h}(x)) \in \overline{W}_{-\infty,1}$. We write $\pi_{B_{-\infty,i}} : \overline{W}_{-\infty,i} \to B_{-\infty,i}$ for the projection along $\overline{S}$.

We write $cl(B_{\tau})$, $cl(B_{+})$, $cl(B_{-})$ to denote the compactifications of $B_{\tau}$, $B_{+}$, $B_{-}$, obtained by adjoining a point at infinity for each end $s = \pm\infty$. Similarly, we write $cl(B_{-\infty,1})$ for the compactification of $B_{-\infty,1}$, obtained by adding 4 points $s = \pm\infty$ and $t = \pm\infty$, and $cl(B_{-\infty,2})$ for the compactification of $B_{-\infty,2}$, obtained by adding 2 points $t = \pm\infty$.

**3.1.4. Marked points.** — We choose a 1-parameter family of marked points

$$ \overline{\mathfrak{m}}(\tau) = (\overline{\mathfrak{m}}^b(\tau), \overline{\mathfrak{m}}^f(\tau)) = ((-l(\tau) + 2, (r(\tau) + 1)/2), z_{\infty}) \in \overline{W}_{\tau}. $$

Observe that the map $\tau \mapsto \overline{\mathfrak{m}}^b(\tau)$ is a smoothing of the function

$$ \tau \mapsto \begin{cases} (-\tau, \frac{3}{2}), & \text{for } \tau \ge 0; \\ (0, \frac{3-\tau}{2}), & \text{for } \tau \le 0, \end{cases} $$

and therefore $\overline{\mathfrak{m}}^b(\tau)$ is always on the axis $\{t = (r(\tau) + 1)/2\}$ of involution for the involution $(s, t) \mapsto (s, 1-t)$ of $B_{\tau}$ and always at a fixed distance from the lower component of $\partial B_{\tau}$. See Figure 4 for a schematic representation of $B_{\tau}$ and the placement of $\overline{\mathfrak{m}}^b(\tau)$.

The following also hold:
(i) as $\tau \to +\infty$, $\bar{m}(\tau)$ limits to $\bar{m}(+\infty) = (\bar{m}^b(+\infty), \bar{m}^f(+\infty))$, where $\bar{m}^b(+\infty) = (0, \frac{3}{2}) \in B_-$ and $\bar{m}^f(+\infty) = z_\infty$;

(ii) as $\tau \to -\infty$, $\bar{m}(\tau)$ limits to $\bar{m}(-\infty) = (\bar{m}^b(-\infty), \bar{m}^f(-\infty))$, where $\bar{m}^b(-\infty) = (0, 0) \in B_{-\infty,2}$ and $\bar{m}^f(-\infty) = z_\infty$.

**Convention 3.1.1.** — In this section and the next, $\bar{m}$ will denote a 1-parameter family (as opposed to a single marked point).

Let us write $\mathcal{L}_{t_0}$ for the locus $\{t = t_0\}$, viewed as a subset of $B_+$, $B_-$, $B_\tau$, as appropriate. Of particular interest is $\mathcal{L}_{(r(\tau)+1)/2}$, which passes through $\bar{m}^b(\tau) \in B_\tau$ or $\bar{m}^b(+\infty) \in B_-$.

**3.1.5. Stable Hamiltonian structures and symplectic forms.** — We first consider $\bar{W}_\tau$. The stable Hamiltonian structure on $\bar{N}_{r(\tau)} = (\bar{S} \times [0, r(\tau)]) / \sim$ is obtained from $(dt, \bar{\omega})$ on $\bar{S} \times [0, r(\tau)]$ by passing to the quotient, where $\bar{\omega}$ is the area form on $\bar{S}$ from Section I.5.1.2. The 2-plane field is $\xi_\tau = \ker dt = T\bar{S}$ and the Hamiltonian vector field is $\bar{R}_\tau = \partial_\tau$. The symplectic form $\bar{\Omega}_\tau$ is obtained from the symplectic form $ds \wedge dt + \bar{\omega}$ on $\mathbf{R} \times \bar{S} \times [0, r(\tau)]$ by passing to the quotient $\mathbf{R} \times \bar{N}_{r(\tau)}$ and then restricting to $\bar{W}_\tau$.

Next we consider $\bar{W}_{-\infty} = \bar{W}_{-\infty,1} \cup \bar{W}_{-\infty,2}$. Let $\omega_{-\infty,1}$ be the restriction of the area form $ds \wedge dt$ on $\mathbf{R}^2$ to $B_{-\infty,1}$ and let $\omega_{-\infty,2} = ds \wedge dt$ on $B_{-\infty,2} = [-2, 2] \times \mathbf{R}$. Then we set $\bar{\Omega}_{-\infty,i} = \omega_{-\infty,i} + \bar{\omega}$. The stable Hamiltonian structure at the $s \to \pm\infty$ ends of $\bar{W}_{-\infty,1}$ are given by $(dt, \bar{\omega})$ and the stable Hamiltonian structure at the $t \to \pm\infty$ ends of $\bar{W}_{-\infty,1}$ are given by $(ds, \bar{\omega})$.

**3.2. Holomorphic curves and moduli spaces.**

**3.2.1. Lagrangian boundary conditions.** — Recall that the monodromy map $\bar{h} = \bar{h}_m : \bar{S} \to \bar{S}$ depends on the integer $m \gg 0$. Also $\bar{a} = \{\bar{a}_1, \dots, \bar{a}_{2g}\}$ is the extension of the basis $\mathbf{a} = \{a_1, \dots, a_{2g}\}$ to $\bar{S}$ so that $\bar{a}_i = a_i \cup \bar{a}_{i,0} \cup \bar{a}_{i,1}$, as described in Section I.5.2.2. We also note that $\bar{a}$ depends on $m$.

We first describe the pushoff $\bar{b}$ of $\bar{a}$ which also depends on $m$: Let

$$ \varepsilon_0 = \varepsilon_0(m) = \frac{2\pi}{mK(m)} > 0, $$

where $K(m)$ is a positive integer such that $\lim_{m \to \infty} K(m) = \infty$, and let $\bar{b}_i$ be a $\varepsilon_0$-close transverse pushoff of $\bar{a}_i$ which satisfies the following:

- in a neighborhood of $z_\infty$, $\bar{b}_i$ is obtained from $\bar{a}_i$ by a $-\varepsilon_0$-rotation; and

- $\bar{a}_i$ and $\bar{b}_i$ intersect at three points $x_{i1}^{\#}, x_{i2}^{\#}$ and $x_{i3}^{\#}$ (besides at $z_\infty$); $x_{i2}^{\#} \in \text{int}(S)$ and $x_{i1}^{\#}, x_{i3}^{\#} \in \bar{S} - S$.
<figure><img src="image_7.png" /><figcaption>FIG. 6. — The arcs $\bar{a}_i$, $\bar{b}_i$, $\bar{h}(\bar{a}_i)$, and $\bar{h}(\bar{b}_i)$ near $z_\infty$</figcaption></figure>

See Figure 6. We also write $\bar{b}_{i,j}$ for the portion of $\bar{b}_i$ analogous to $\bar{a}_{i,j}$. When we want to signal that $z_\infty$ is an intersection point of $\bar{a}_i$ and $\bar{b}_i$, then we write it as $z_\infty^#$.

Let us write $\phi(\bar{a}_{i,j})$ for the $\phi$-coordinate of the portion of $\bar{a}_{i,j}$ near $z_\infty$, subject to the condition $0 \le \phi(\bar{a}_{i,j}) < 2\pi$; similarly define $\phi(\bar{b}_{i,j})$, etc. Then, near $z_\infty$,

$$ (3.2.1) \qquad 0 < \phi(\bar{b}_{i,j}) < \phi(\bar{a}_{i,j}) < \phi(\bar{h}(\bar{b}_{i,j})) < \phi(\bar{h}(\bar{a}_{i,j})) < c(m), $$

where $c(m) \to 0$ as $m \to \infty$; cf. Section I.5.2.2.

*Remark 3.2.1.* — In view of the choices of $\bar{a}_{i,j}$ and $\bar{h}(\bar{a}_{i,j})$ from Section I.5.2.2 and the choices of $\bar{b}_{i,j}$ above, as $m \to \infty$:

- $\phi(\bar{a}_{i,j}) - \phi(\bar{b}_{i,j}) \to 0$ the fastest;

- $\phi(\bar{h}(\bar{a}_{i,j})) - \phi(\bar{a}_{i,j}) \to 0$ the next fastest;

- $\phi(\bar{a}_{i,j}) - \phi(\bar{a}_{i',j'}) \to 0$ the slowest if $(i,j) \ne (i',j')$.

The symplectic fibration

$$ \pi_{B_{\tau}} : (\overline{W}_{\tau}, \overline{\Omega}_{\tau}) \to (B_{\tau}, ds \wedge dt) $$

induces a symplectic connection, defined as the $\overline{\Omega}_{\tau}$-orthogonal of the tangent plane to the fibers. We place a copy of $\overline{a}$ on the fiber $\pi_{B_{\tau}}^{-1}(s, 1)$ with $s > l(\tau)$ and use the symplectic connection to parallel transport $\overline{a}$ along $\partial \overline{W}_{\tau}$. (Note that $\partial \overline{W}_{\tau}$ is equal to the vertical boundary $\partial_v \overline{W}_{\tau} := \pi_{B_{\tau}}^{-1}(\partial B_{\tau})$.) This gives us a singular Lagrangian submanifold $L_{\overline{a}}^{\tau,+}$. (Note that $\overline{a}$ is a singular Lagrangian submanifold of $\overline{S}$ with singularity $z_\infty$ and is a union of radial rays in a neighborhood of $z_\infty$. Hence the singular set of $L_{\overline{a}}^{\tau,+}$ and its neighborhood in $L_{\overline{a}}^{\tau,+}$ are obtained from those of $\overline{a}$ by multiplying with the upper boundary of $B_{\tau}$.) Similarly, we place a copy of $\overline{b}$ on the fiber $\pi_{B_{\tau}}^{-1}(s, 1)$ with $s < -l(\tau)$ and use the symplectic connection to parallel transport $\overline{b}$ along $\partial \overline{W}_{\tau}$ to construct the singular Lagrangian submanifold $L_{\overline{b}}^{\tau,-}$. The Lagrangian submanifolds $L_{\overline{a}}^{\tau,+}$, $L_{\overline{a}_i}^{\tau,+}$, $L_{\overline{b}}^{\tau,-}$, $L_{\overline{b}_i}^{\tau,-}$, etc. are defined
similarly, where $\hat{\mathbf{a}} = \{\hat{a}_1, \dots, \hat{a}_{2g}\}$ and $\hat{a}_i = \bar{a}_i - \{z_i\}$ were defined in Section I.5.2.2 and $\hat{\mathbf{b}}$ and $\hat{\mathbf{b}}_i$ are defined analogously.

On $(\overline{W}_{-\infty}, \overline{\Omega}_{-\infty})$, we define the singular Lagrangian submanifolds as follows: Let us write

$$ \partial B_{-\infty,1} = \bigsqcup_{i=1}^{4} \partial_i B_{-\infty,1}, $$

where the boundary components, in order from $i=1$ to $i=4$, satisfy

$$ s > 0, t > 1/2; \quad s < 0, t > 1/2; \quad s < 0, t < 1/2; \quad \text{and} \quad s > 0, t < 1/2. $$

Also let $\partial_i \overline{W}_{-\infty,1}$ be the component of $\partial \overline{W}_{-\infty,1}$ corresponding to $\partial_i B_{-\infty,1}$. Then we define:

$$ - L_{\bar{a},1}^{-\infty,1} = \partial_1 B_{-\infty,1} \times \bar{a} \text{ on } \partial_1 \overline{W}_{-\infty,1}; $$

$$ - L_{\bar{b},2}^{-\infty,1} = \partial_2 B_{-\infty,1} \times \bar{b} \text{ on } \partial_2 \overline{W}_{-\infty,2}; $$

$$ - L_{\bar{h}(\bar{b}),3}^{-\infty,1} = \partial_3 B_{-\infty,1} \times \bar{h}(\bar{b}) \text{ on } \partial_3 \overline{W}_{-\infty,1}; $$

$$ - L_{\bar{h}(\bar{a}),4}^{-\infty,1} = \partial_4 B_{-\infty,1} \times \bar{h}(\bar{a}) \text{ on } \partial_4 \overline{W}_{-\infty,1}; $$

$$ - L_{\bar{a},+}^{-\infty,2} = \{2\} \times \mathbf{R} \times \bar{a} \text{ on } \partial \overline{W}_{-\infty,2}; $$

$$ - L_{\bar{b},-}^{-\infty,2} = \{-2\} \times \mathbf{R} \times \bar{b} \text{ on } \partial \overline{W}_{-\infty,2}. $$

**3.2.2. Almost complex structures.** — Recall the space $\mathcal{J}_{\overline{W}}$ of admissible almost complex structures $\overline{J}$ on $\overline{W}$ from Definition I.5.3.2, the space $\mathcal{J}_{\overline{W}'}$ of adapted almost complex structures $\overline{J}'$ on $\overline{W}'$ from Definition I.5.3.14, and the spaces $\mathcal{J}_{\overline{W}_+}$ and $\mathcal{J}_{\overline{W}_-}$ of admissible almost complex structures $\overline{J}_+$ and $\overline{J}_-$ on $\overline{W}_+$ and $\overline{W}_-$ from Definition I.5.4.1.

**Definition 3.2.2.** — An almost complex structure $\overline{J}_{-\infty,2}$ on $\overline{W}_{-\infty,2}$ is admissible if the following hold:

(1) $\overline{J}_{-\infty,2}$ is $t$-invariant, $\overline{J}_{-\infty,2}(\partial_t) = -\partial_s$, and $\overline{J}_{-\infty,2}(T\overline{S}) = T\overline{S}$; and

(2) there exists $\varepsilon > 0$ such that $\overline{J}_{-\infty,2}$ restricts to the standard complex structure on the subsurface $D_\varepsilon^2 = \{\rho \le \varepsilon\} \subset \overline{S}$ of each fiber.

The space of all admissible $\overline{J}_{-\infty,2}$ will be denoted by $\mathcal{J}_{\overline{W}_{-\infty,2}}$.

**Definition 3.2.3.** — An almost complex structure $\overline{J}_{-\infty,1}$ on $\overline{W}_{-\infty,1}$ is admissible if the following hold:

(1) the projection $\pi_{B_{-\infty,1}}$ is $(\overline{J}_{-\infty,1}, j_{-\infty,1})$-holomorphic with respect to the standard complex structure $j_{-\infty,1}$ on $B_{-\infty,1}$;

(2) there exists $\varepsilon > 0$ such that $\overline{J}_{-\infty,1}$ restricts to the standard complex structure on the subsurface $D_\varepsilon^2 = \{\rho \le \varepsilon\} \subset \overline{S}$ of each fiber;
(3) there exist $\bar{J} \in \mathcal{J}_{\overline{W}}$ and $\bar{J}_{-\infty,2} \in \mathcal{J}_{\overline{W}_{-\infty,2}}$ such that $\bar{J}_{-\infty,1}$ agrees with $\bar{J}$ on $\overline{W} = \mathbf{R} \times [0, 1] \times \overline{S}$, with $\bar{J}_{-\infty,2}$ on $[-2, 2] \times [3, +\infty) \times \overline{S}$, and with $(\mathrm{id} \times \bar{h})_* (\bar{J}_{-\infty,2})$ on $[-2, 2] \times [-2, -\infty) \times \overline{S}$.

If (3) holds, we say that $\bar{J}_{-\infty,1}$ is compatible with $\bar{J} \in \mathcal{J}_{\overline{W}}$ and $\bar{J}_{-\infty,2} \in \mathcal{J}_{\overline{W}_{-\infty,2}}$. The space of admissible $\bar{J}_{-\infty,1}$ will be denoted by $\mathcal{J}_{\overline{W}_{-\infty,1}}$.

**Definition 3.2.4.** — An almost complex structure $\bar{J}_{-\infty} = \bar{J}_{-\infty,1} \cup \bar{J}_{-\infty,2}$ on $\overline{W}_{-\infty} = \overline{W}_{-\infty,1} \cup \overline{W}_{-\infty,2}$ is admissible if $\bar{J}_{-\infty,i} \in \mathcal{J}_{\overline{W}_{-\infty,i}}$ for $i = 1, 2$ and $\bar{J}_{-\infty,1}$ is compatible with $\bar{J}_{-\infty,2}$. The space of admissible $\bar{J}_{-\infty}$ will be denoted by $\mathcal{J}_{\overline{W}_{-\infty}}$.

**Definition 3.2.5.** — An almost complex structure $\bar{J}_{\tau}$ on $\overline{W}_{\tau}$ is admissible if the following hold:

(1) the projection $\pi_{B_{\tau}}$ is $(\bar{J}_{\tau}, j_{\tau})$-holomorphic with respect to the standard complex structure $j_{\tau}$ on $B_{\tau}$;

(2) there exists $\varepsilon > 0$ such that $\bar{J}_{\tau}$ restricts to the standard complex structure on the subsurface $D_{\varepsilon}^2 = \{\rho \le \varepsilon\} \subset \overline{S}$ of each fiber;

(3) if $\tau \ge 0$, then $\bar{J}_{\tau}$ is the restriction of some $\bar{J}' \in \mathcal{J}_{\overline{W}'}$;

(4) if $\tau \le 0$, then $\bar{J}_{\tau}$ agrees with some $\bar{J} \in \mathcal{J}_{\overline{W}}$ on $\overline{W} = \mathbf{R} \times [0, 1] \times \overline{S}$ and with some $\bar{J}_{-\infty,2} \in \mathcal{J}_{\overline{W}_{-\infty,2}}$ on $[-2, 2] \times [3, r(\tau) - 2] \times \overline{S}$ (provided $r(\tau) \ge 5$).

If (3) holds, we say that $\bar{J}_{\tau}$ is compatible with $\bar{J}' \in \mathcal{J}_{\overline{W}}$ and if (4) holds, we say that $\bar{J}_{\tau}$ is compatible with $\bar{J} \in \mathcal{J}_{\overline{W}}$ and $\bar{J}_{-\infty,2} \in \mathcal{J}_{\overline{W}_{-\infty,2}}$. The space of all admissible $\bar{J}_{\tau}$ on $\overline{W}_{\tau}$ will be denoted by $\mathcal{J}_{\overline{W}_{\tau}}$.

**Definition 3.2.6.** — A family $\{\bar{J}_{\tau} \in \mathcal{J}_{\overline{W}_{\tau}}\}_{\tau \in \mathbb{R}}$ of almost complex structures is admissible if there exist $\bar{J}' \in \mathcal{J}_{\overline{W}}$, $\bar{J} \in \mathcal{J}_{\overline{W}}$, $\bar{J}_{+} \in \mathcal{J}_{\overline{W}_{+}}$, $\bar{J}_{-} \in \mathcal{J}_{\overline{W}_{-}}$ and $\bar{J}_{-\infty} = \bar{J}_{-\infty,1} \cup \bar{J}_{-\infty,2} \in \mathcal{J}_{\overline{W}_{-\infty}}$ such that the following hold:

(1) $\bar{J}_{\tau}$ converges to $\bar{J}_{-\infty}$ as $\tau \to -\infty$;

(2) $\bar{J}_{\tau}$ converges to $\bar{J}_{+}$ and $\bar{J}_{-}$ as $\tau \to +\infty$;

(3) $\bar{J}_{+}$ and $\bar{J}_{-}$ are compatible with $\bar{J}'$ and $\bar{J}$; and

(4) $\bar{J}_{\tau}$ is compatible with $\bar{J}$ and $\bar{J}_{-\infty,2}$ for $\tau \le 0$ and with $\bar{J}'$ for $\tau \ge 0$.

The convergence of almost complex structures is to be understood in the sense of neck-stretching as in [BEHWZ, Section 3.4]. The space of all admissible $\{\bar{J}_{\tau} \in \mathcal{J}_{\overline{W}_{\tau}}\}_{\tau \in \mathbb{R}}$ will be denoted by $\overline{\mathcal{I}}$.

**3.2.3. Some notation and conventions.** — We now collect some notation and conventions.

**Notation 3.2.7 (Tuples and orbit sets).** — When we write a tuple of $\overline{\mathbf{a}} \cap \overline{\mathfrak{h}}(\overline{\mathbf{a}})$ as $\mathbf{y}$ or an orbit set of $\overline{\mathbb{N}}$ as $\mathbf{\gamma}$ (with possible superscripts, subscripts and other decorations), it is
assumed that $\mathbf{y} \subset \mathcal{S}$ and $\mathbf{y} \subset \mathcal{N}$. In particular, $\mathbf{y}$ and $\mathbf{y}$ do not contain any multiples of $z_\infty$ or $\delta_0$.

**Notation 3.2.8 (Sections at $\infty$).** — The sections $\{\rho = 0\}$ of $\overline{\mathcal{W}}$, $\overline{\mathcal{W}'} = \mathbf{R} \times \overline{\mathcal{N}}$, $\overline{\mathcal{W}}_\tau$, $\overline{\mathcal{W}}_+$, $\overline{\mathcal{W}}_-$ and $\overline{\mathcal{W}}_{-\infty,i}$ are holomorphic with respect to almost complex structures in $\mathcal{J}_{\overline{\mathcal{W}}}$, $\mathcal{J}_{\overline{\mathcal{W}}'}$, $\mathcal{J}_{\overline{\mathcal{W}}_\tau}$, etc. They are called sections at $\infty$ and are denoted by $\sigma_\infty, \sigma'_\infty, \sigma_\tau^\infty, \sigma_\infty^+, \sigma_\infty^-$ and $\sigma_{-\infty,i}^-$.

**Notation 3.2.9 (The intersection numbers $n^*(\bar{u})$ and $n^{*,alt}(\bar{u})$).** — Let $\delta_{\rho_0, \phi_0}$ be a closed orbit of the Hamiltonian vector field $\partial_t$ which lies on the torus

$$ \{\rho = \rho_0\} \subset \overline{\mathcal{N}}_r = (\bar{S} \times [0, r])/(x, 1) \sim (\bar{h}(x), 0) $$

for appropriate $r$ and $\rho_0 > 0$ sufficiently small and which passes through the point $(t, \rho, \phi) = (0, \rho_0, \phi_0)$. Since $\bar{h} = \bar{h}_m$ is a $\frac{2\pi}{m}$-rotation on $D_{1/2}^2 = \{\rho \le 1/2\} \subset D^2$, the orbit $\delta_{\rho_0, \phi_0}$ winds $m$ times in the longitudinal direction and once in the meridian direction. The point $(0, \rho_0, \phi_0)$ is with respect to balanced coordinates on $\overline{\mathcal{N}}_r$; see Section I.5.1.2. We assume additionally that $\delta_{\rho_0, \phi_0}$ does not intersect the projections of the Lagrangians of $\overline{\mathcal{W}}_\tau$, $\overline{\mathcal{W}}_+$ and $\overline{\mathcal{W}}_-$ to $\overline{\mathcal{N}}_r$.

Recall from Section 3.2.1 that $\phi(\bar{a}_{i,j})$ is the $\phi$-coordinate of $\bar{a}_{i,j}$ near $z_\infty$ such that $0 \le \phi(\bar{a}_{i,j}) < 2\pi$. Also let $\varepsilon_0 = \frac{2\pi}{nK(m)}$ be the constant appearing in the definition of $\bar{b}$ and let $\varepsilon_1$ be a constant satisfying $0 < \varepsilon_1 < \varepsilon_0$. We consider two possibilities for $\phi_0$:

$$ \phi_0^{\pm} = \phi(\bar{a}_{i,j}) \pm \varepsilon_1. $$

Comparing with Equation (3.2.1), we obtain:

$$ (3.2.2) \qquad 0 < \phi(\bar{b}_{i,j}) < \phi_0^- < \phi(\bar{a}_{i,j}) < \phi_0^+ < \phi(\bar{h}(\bar{b}_{i,j})) < \phi(\bar{h}(\bar{a}_{i,j})) < c(m). $$

We write $(\sigma_\infty^*)^{\dagger, \pm}$ for the restriction of $\mathbf{R} \times \delta_{\rho_0, \phi_0^\pm}$ to $\overline{\mathcal{W}}_*$, where $* = \emptyset, \', \tau, +, -$ or $-$.

$$ (\sigma_{\infty}^{-\infty,i})^{\dagger, \pm} = \mathcal{B}_{-\infty,i} \times \{\rho = \rho_0, \phi = \phi_0^{\pm} + 2\pi k/m, k \in \mathbb{Z}\}. $$

Finally we define:

$$ (3.2.3) \qquad n^*(\bar{u}) = \langle \bar{u}, (\sigma_\infty^*)^{\dagger,+} \rangle, \quad n^{*,alt}(\bar{u}) = \langle \bar{u}, (\sigma_\infty^*)^{\dagger,-} \rangle, $$

where $* = \emptyset, \', \tau, +, -, \text{ or } (-\infty, i)$. The two quantities $n^*$ and $n^{*,alt}$ can be used interchangeably, except when $\tau = -\infty$; for the most part we will use $n^*$.

**Notation 3.2.10 (Components of a holomorphic curve $\bar{u}$).** — Given a holomorphic curve $\bar{u}$ in $\overline{\mathcal{W}}$, $\overline{\mathcal{W}}'$, etc., we write

$$ \bar{u} = \bar{u}' \cup \bar{u}'' = \bar{u}' \cup \bar{u}'' \cup \bar{u}^{\sharp} \cup \bar{u}^{\flat} \cup \bar{u}'', $$
where

- $\bar{u}'$ is a possibly disconnected branched cover of $\sigma_{\infty}^*$;

- $\bar{u}''$ is the union of irreducible components which do not branch cover $\sigma_{\infty}^*$;

- $\bar{u}'''$ is the union of components of $\bar{u}''$ which are asymptotic to a multiple of $\delta_0$ or $z_\infty$ at one or more ends;

- $\bar{u}''$ is the union of the remaining non-fiber components of $\bar{u}''$; and

- $\bar{u}'$ is the union of fiber components of $\bar{u}$, including ghosts.

If $\bar{u}$ is a multisection, then $\deg \bar{u}$ is the degree of $\bar{u}$ as a multisection.

*The choice of hyperbolic orbit.* By the definition of the monodromy map $h$, $\partial N$ is a negative Morse-Bott torus; we denote the negative Morse-Bott family of simple orbits on $\partial N$ by $\mathcal{N}$. Let $\phi_\gamma$ be the $\phi$-coordinate of $\gamma \in \mathcal{N}$. Also recall the orbit $\delta_0 = \{z_\infty\} \times [0, 2]/\sim$ of $\overline{\mathbb{N}}$.

Let $\overline{J'} \in \mathcal{J}_{\overline{W'}}$. Without loss of generality, we may assume that there is only one holomorphic cylinder $Z_\gamma$ in $(\overline{W'} = \mathbf{R} \times \overline{\mathbb{N}}, \overline{J'})$ from $\delta_0$ to any orbit $\gamma \in \mathcal{N}$, modulo $\mathbf{R}$-translation. Each $Z_\gamma$ corresponds to a radial ray $\mathcal{R}_{\phi_\gamma} = \{\phi = \phi_\gamma, \rho \ge 0\} \subset \mathbb{D}^2$, which is the asymptotic direction of $\pi_{\mathbb{D}^2}(Z_\gamma)$ at the positive end. Here $\pi_{\mathbb{D}^2}: \overline{\mathbb{N}} - \text{int}(\mathbb{N}) \to \mathbb{D}^2$ is the projection with respect to the balanced coordinates; see Section I.5.1.2.

We now choose a hyperbolic orbit $h$ and an elliptic orbit $e$ in $\mathcal{N}$. The choice of $h = \gamma_{\phi_h}$ is the same as that of Convention I.6.6.4: $h$ is generic and $\phi_h$ is close to $-\frac{2\pi}{m}$, where the integer $m$ which appears in the definition of $\bar{h}_m$ additionally satisfies the conditions of Section I.5.2.2. In particular, the radial ray $\mathcal{R}_{\phi_h}$ 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 $e$ except that $e \neq h$.

**3.2.4.** *Holomorphic maps to $\overline{W}_\tau$. —* Let $(F,j)$ be a compact Riemann surface, possibly disconnected, with two $k$-tuples of boundary punctures $\mathbf{q}^+ = \{q_1^+, \dots, q_k^+\}$ and $\mathbf{q}^- = \{q_1^-, \dots, q_k^-\}$ on $\partial F = \partial_+F \cup \partial_-F$, such that:

(i) each component of $F$ nontrivially intersects $\partial_+F$ and $\partial_-F$;

(ii) each of $\partial_+F$ and $\partial_-F$ is a union of connected components of $\partial F$; and

(iii) on each component of $\partial_+F$ (resp. $\partial_-F$) there is at least one puncture from $\mathbf{q}^+$ (resp. $\mathbf{q}^-$) and none from $\mathbf{q}^-$ (resp. $\mathbf{q}^+$).

We write $\dot{F} = F - \mathbf{q}^+ - \mathbf{q}^-$, $\partial_+\dot{F} = \partial_+F - \mathbf{q}^+$ and $\partial_-\dot{F} = \partial_-F - \mathbf{q}^-$.

Let $\mathbf{z} = \{z_\infty^\beta(\overrightarrow{D})\} \cup \mathbf{y}$, $p \ge 0$, be a $k$-tuple of points of $\bar{a} \cap \bar{h}(\bar{a})$, where $k \le 2g$, $z_\infty$ has multiplicity $p$, and $\overrightarrow{D}$ is the data at $z_\infty^\beta$ with respect to $\bar{a} \cap \bar{h}(\bar{a})$. The definition of $\mathbf{z}$ and the notion of data at $z_\infty^\beta$ are given in Section I.5.7. In particular, by definition, each arc of $\{\bar{a}_i, \bar{h}(\bar{a}_i)\}_{i=1}^{2g}$ is used at most once. Also let $\mathbf{z}' = \{z_\infty^q(\overrightarrow{D}')\} \cup \mathbf{y}'$, $q \ge 0$, be a $k$-tuple of points of $\bar{b} \cap \bar{h}(\bar{b})$, where $z_\infty$ has multiplicity $q$ and $\overrightarrow{D}'$ is the data at $z_\infty^q$ with respect to $\bar{b} \cap \bar{h}(\bar{b})$.
Let $\overline{J}_{\tau} \in \mathcal{J}_{\overline{W}_{\tau}}$. If $\bar{u}' : \dot{F} \to (\overline{W}_{\tau}, \overline{J}_{\tau})$ is a branched cover of $\sigma_{\infty}^{\tau}$, then $\bar{u}'$ comes equipped with data $\mathcal{C}$ (cf. Definition I.5.7.1), which is a map from $\pi_0(\partial_+ \dot{F})$ (resp. $\pi_0(\partial_- \dot{F})$) to the set of arcs $\bar{a}_{i,j}$ (resp. $\bar{b}_{i,j}$). In words, $\bar{u}'$ is viewed as mapping each component of $\partial_+ \dot{F}$ (resp. $\partial_- \dot{F}$) to some $L_{\bar{a}_{i,j}}^{\tau, +}$ (resp. $L_{\bar{b}_{i,j}}^{\tau, -}$). Then $\mathcal{C}$ determines the data $\overrightarrow{\mathcal{D}}$ and $\overrightarrow{\mathcal{D}}'$ at the positive and negative ends.

We then make the following definition:

**Definition 3.2.11.** — Let $\overline{J}_{\tau} \in \mathcal{J}_{\overline{W}_{\tau}}$, $\mathbf{z} = \{z_{\infty}^{\rho}(\overrightarrow{\mathcal{D}})\} \cup \mathbf{y}$ be a $k$-tuple of $\bar{\mathbf{a}} \cap \bar{h}(\bar{\mathbf{a}})$ and $\mathbf{z}' = \{z_{\infty}^{q}(\overrightarrow{\mathcal{D}}')\} \cup \mathbf{y}'$ be a $k$-tuple of $\bar{\mathbf{b}} \cap \bar{h}(\bar{\mathbf{b}})$.

A degree $k$ multisection of $(\overline{W}_{\tau}, \overline{J}_{\tau})$ from $\mathbf{z}$ to $\mathbf{z}'$ is a pair $(\bar{u}, \mathcal{C})$ consisting of a holomorphic map

$$ \bar{u} = \bar{u}' \cup \bar{u}'' : (\dot{F} = \dot{F}' \cup \dot{F}''; j) \to (\overline{W}_{\tau}, \overline{J}_{\tau}) $$

which is a degree $k$ multisection of $\pi_{B_{\tau}} : \overline{W}_{\tau} \to B_{\tau}$ and data $\mathcal{C}$ for $\bar{u}'$, and which additionally satisfies the following:

(1) $\bar{u}''(\partial_+ \dot{F}'') \subset L_{\bar{\mathbf{a}}}^{\tau, +}$ and $\bar{u}''(\partial_- \dot{F}'') \subset L_{\bar{\mathbf{b}}}^{\tau, -}$;

(2) $\bar{u}$ maps each connected component of $\partial_+ \dot{F}$ to a different $L_{\bar{a}_i}^{\tau, +}$ and each connected component of $\partial_- \dot{F}$ to a different $L_{\bar{b}_i}^{\tau, -}$ (here we are using $\mathcal{C}$ to assign some $L_{\bar{a}_i}^{\tau, +}$ or $L_{\bar{b}_i}^{\tau, -}$ to each component of $\partial_\pm \dot{F}$);

(3) $\lim_{w \to q_i^+} \pi_{\mathbf{R}} \circ \bar{u}(w) = +\infty$ and $\lim_{w \to q_i^-} \pi_{\mathbf{R}} \circ \bar{u}(w) = -\infty$;

(4) $\bar{u}$ converges to a strip over $[0, 1] \times \mathbf{z}$ near $\mathbf{q}^+$ and to a 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 $\overrightarrow{\mathcal{D}}$ and $\overrightarrow{\mathcal{D}}'$.

Here $\pi_{\mathbf{R}} : \overline{W}_{\tau} \to \mathbf{R}$ is the projection to the s-coordinate.

A $(\overline{W}_{\tau}, \overline{J}_{\tau})$-curve from $\mathbf{y}$ to $\mathbf{y}'$ is a degree 2g multisection of $(\overline{W}_{\tau}, \overline{J}_{\tau})$ satisfying $n^*(\bar{u}) = m$. (Recall that the integer $m$ is the integer on which the monodromy map $\bar{h} = \bar{h}_m$ depends.)

Let $\mathcal{M}_{\overline{J}_{\tau}}(\mathbf{z}, \mathbf{z}')$ be the moduli space of degree $k$ multisections of $(\overline{W}_{\tau}, \overline{J}_{\tau})$ from $\mathbf{z}$ to $\mathbf{z}'$ and let $\mathcal{M}_{\overline{J}_{\tau}}(\mathbf{z}, \mathbf{z}'; \overline{\mathfrak{m}}(\tau))$ be the moduli space of degree $k$ multisections of $(\overline{W}_{\tau}, \overline{J}_{\tau})$ from $\mathbf{z}$ to $\mathbf{z}'$ and with a marked point mapped to $\overline{\mathfrak{m}}(\tau)$. By a slight abuse of terminology we call this the moduli space of multisections passing through $\overline{\mathfrak{m}}(\tau)$.

We write

$$ \mathcal{M}_{\overline{J}_{\tau}}(\mathbf{z}, \mathbf{z}') := \{(\tau, u) \mid \tau \in \mathbf{R}, u \in \mathcal{M}_{\overline{J}_{\tau}}(\mathbf{z}, \mathbf{z}')\}, $$

$$ \mathcal{M}_{\overline{J}_{\tau}}(\mathbf{z}, \mathbf{z}'; \overline{\mathfrak{m}}) := \{(\tau, u) \mid \tau \in \mathbf{R}, u \in \mathcal{M}_{\overline{J}_{\tau}}(\mathbf{z}, \mathbf{z}'; \overline{\mathfrak{m}}(\tau))\}. $$

**Notation 3.2.12 (Modifiers).** — For any moduli space $\mathcal{M}_{\star_1}(\star_2)$, we may place modifiers $*$ as in $\mathcal{M}_{\star_1}^*(\star_2)$ to denote the subset of $\mathcal{M}_{\star_1}(\star_2)$ satisfying $*$. Typical self-explanatory modifiers are $I = i$, $n^* = m$, and $\deg = k$. Note that the degree can be inferred from $\star_2$.
The following is a list of non-self-explanatory modifiers:

† = no component of $\bar{u}$ branch covers $\sigma_{\infty}^*$ with possibly empty branch locus.

$s$ = all the components of $\bar{u}$ are simply covered.

irr = the curve $\bar{u}$ is irreducible.

**3.2.5. Holomorphic maps to $\bar{W}_{-\infty}$.** — We first discuss holomorphic curves without ends at $z_{\infty}$.

**Definition 3.2.13.** — Let $\bar{J}_{-\infty,1} \in \mathcal{J}_{\bar{W}_{-\infty,1}}$, $\mathbf{y}_1 \in S_{\mathbf{a},\mathfrak{h}(\mathbf{a})}$, $\mathbf{y}_2 \in S_{\mathbf{b},\mathbf{a}}$, $\mathbf{y}_3 \in S_{\mathbf{b},\mathfrak{h}(\mathbf{b})}$ and $\mathbf{y}_4 \in S_{\mathfrak{h}(\mathbf{a}),\mathfrak{h}(\mathbf{b})}$. (Recall that an element of $S_{\mathbf{a},\mathfrak{h}(\mathbf{a})}$ is a tuple of intersection points $a_i \cap \mathfrak{h}(a_j)$, where each $a_i$ and each $\mathfrak{h}(a_j)$ is used at most once; the other $S_{*,*}$ are defined analogously. In this definition all the tuples are $k$-tuples.)

A degree $k \le 2g$ multisection $\bar{u}$ of $(\bar{W}_{-\infty,1}, \bar{J}_{-\infty,1})$ with ends $\mathbf{y}_1, \mathbf{y}_2, \mathbf{y}_3, \mathbf{y}_4$ is a holomorphic map

$$ \bar{u} : (\dot{\mathbf{F}}, j) \to (\bar{W}_{-\infty,1}, \bar{J}_{-\infty,1}) $$

which is degree $k$ multisection of $\pi_{B_{-\infty,1}} : \bar{W}_{-\infty,1} \to B_{-\infty,1}$ and which additionally satisfies the following:

(1) $\bar{u}(\partial \dot{\mathbf{F}}) \subset L_{\bar{\mathbf{a}},1}^{-\infty,1} \cup L_{\bar{\mathbf{b}},2}^{-\infty,1} \cup L_{\bar{\mathfrak{h}}(\bar{\mathbf{b}}),3}^{-\infty,1} \cup L_{\bar{\mathfrak{h}}(\bar{\mathbf{a}}),4}^{-\infty,1}$ and $\bar{u}$ maps each component of $\partial \dot{\mathbf{F}}$ to a different $L_{\bar{a}_i,1}^{-\infty,1}$, $L_{\bar{b}_i,2}^{-\infty,1}$, $L_{\bar{h}(\bar{b}_i),3}^{-\infty,1}$, or $L_{\bar{h}(\bar{a}_i),4}^{-\infty,1}$;

(2) $\bar{u}$ converges to a strip over $[0, 1] \times \mathbf{y}_1$ as $s \to +\infty$; $[-2, 2] \times \mathbf{y}_2$ as $t \to +\infty$; $[0, 1] \times \mathbf{y}_3$ as $s \to -\infty$; and $[-2, 2] \times \mathbf{y}_4$ as $t \to -\infty$.

$A(\bar{W}_{-\infty,1}, \bar{J}_{-\infty,1})$-curve with ends $\mathbf{y}_1, \dots, \mathbf{y}_4$ is a degree 2g multisection of $(\bar{W}_{-\infty,1}, \bar{J}_{-\infty,1})$ satisfying $n^*(\bar{u}) = 0$.

We will use the convention to list the ends of a multisection $\bar{u}$ of $\bar{W}_{-\infty,1}$ in counterclockwise order, starting with the top end.

**Definition 3.2.14.** — Let $\bar{J}_{-\infty,2} \in \mathcal{J}_{\bar{W}_{-\infty,2}}$ and $\mathbf{y}, \mathbf{y}' \in S_{\mathbf{b},\mathbf{a}}$. A degree $k \le 2g$ multisec- tion $\bar{u}$ of $(\bar{W}_{-\infty,2}, \bar{J}_{-\infty,2})$ with ends $\mathbf{y}$ and $\mathbf{y}'$ is a holomorphic map

$$ \bar{u} : (\dot{\mathbf{F}}, j) \to (\bar{W}_{-\infty,2}, \bar{J}_{-\infty,2}) $$

which is degree $k$ multisection of $\pi_{B_{-\infty,2}} : \bar{W}_{-\infty,2} \to B_{-\infty,2}$ and which additionally satisfies the following:

(1) $\bar{u}(\partial \dot{\mathbf{F}}) \subset L_{\bar{\mathbf{a}},+}^{-\infty,2} \cup L_{\bar{\mathbf{b}},-}^{-\infty,2}$ and $\bar{u}$ maps each component of $\partial \dot{\mathbf{F}}$ to a different $L_{\bar{a}_i,+}^{-\infty,2}$ or a different $L_{\bar{b}_i,-}^{-\infty,2}$;

(2) $\bar{u}$ converges to a cylinder over $[-2, 2] \times \mathbf{y}$ as $t \to +\infty$ and to a cylinder over $[-2, 2] \times \mathbf{y}'$ as $t \to -\infty$.
$A(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$-curve with ends $\mathbf{y}$ and $\mathbf{y}'$ is a degree 2g multisection of $(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$ satisfying $n^*(\bar{u}) = m$.

Note that the definition is not symmetric in $\mathbf{y}$ and $\mathbf{y}'$; the $t = +\infty$ end is always written first.

Next we discuss holomorphic curves with ends at $z_\infty$. For $s = 1, \dots, 4$, the data $\overrightarrow{D}_s$ at $z_\infty$ (cf. Section I.5.7) corresponding to the $s$th end is given by

$$ \overrightarrow{D}_s = \{(i'_{s,\ell}, j'_{s,\ell}) \to (i_{s,\ell}, j_{s,\ell})\}_{\ell=1}^{p_s} $$

When $s = 1$, $D_{s=1}^{\text{from}} = \{(i'_{1,\ell}, j'_{1,\ell})\}_{\ell=1}^{p_1}$ and $D_{s=1}^{\text{to}} = \{(i_{1,\ell}, j_{1,\ell})\}_{\ell=1}^{p_1}$ specify the initial points on $\overline{h}(\bar{a}_{i_1,\ell,j_{1,\ell}})$ and terminal points on $\bar{a}_{i_1,\ell,j_{1,\ell}}$, respectively; the cases $s = 2, 3, 4$ are analogous.

We then extend the definition of a degree $k$ multisection $\bar{u}$ of $\overline{W}_{-\infty,1}$ to include those with ends $\mathbf{z}_s = \{z_\infty^{p_s}(\overrightarrow{D}_s)\} \cup \mathbf{y}_s$, $s = 1, \dots, 4$, by attaching data $\mathcal{C}$ to $\bar{u}'$ (cf. Definition I.5.7.1) and modifying Definition 3.2.13 in the same way Definition I.5.7.2 modifies Definition I.4.3.1. Degree $k$ multisections of $\overline{W}_{-\infty,2}$ with ends $\mathbf{z} = \{z_\infty^p(\overrightarrow{D})\} \cup \mathbf{y}$ and $\mathbf{z}' = \{z_\infty^q(\overrightarrow{D}')\} \cup \mathbf{y}'$ are defined similarly.

Let $M_{\overline{J}_{-\infty,1}}(\mathbf{z}_1, \mathbf{z}_2, \mathbf{z}_3, \mathbf{z}_4)$ be the moduli space of degree $k$ multisections of $(\overline{W}_{-\infty,1}, \overline{J}_{-\infty,1})$ with ends $\mathbf{z}_s$, $s = 1, \dots, 4$, and let $M_{\overline{J}_{-\infty,2}}(\mathbf{z}, \mathbf{z}')$ be the moduli space of degree $k$ multisections of $(\overline{W}_{-\infty,2}, \overline{J}_{-\infty,2})$ with ends $\mathbf{z}$ and $\mathbf{z}'$. Also let $M_{\overline{J}_{-\infty,2}}(\mathbf{z}, \mathbf{z}'; \overline{\mathfrak{m}}(-\infty))$ be the moduli space of multisections as above passing through $\overline{\mathfrak{m}}(-\infty)$).

We define the extended moduli spaces $M_{\overline{J}_{-\infty,1}}^{\dagger, \text{ext}}(\mathbf{z}_1, \mathbf{z}_2, \mathbf{z}_3, \mathbf{z}_4)$ and $M_{\overline{J}_{-\infty,2}}^{\dagger, \text{ext}}(\mathbf{z}, \mathbf{z}')$ in a manner similar to that of Section I.5.7.11. The precise definitions will be omitted.

**3.2.6. Indices.** — We now discuss the Fredholm index $\text{ind}(\bar{u})$ and the ECH index $\text{I}(\bar{u})$ of a $\overline{W}_\tau$-curve $\bar{u}: \dot{\mathcal{F}} \to \overline{W}_\tau$ from $\mathbf{y}$ to $\mathbf{y}'$ (i.e., when $\bar{u}' = \emptyset$). The discussion will be brief since all the key ingredients have already been discussed in Sections I.5.5 and I.5.6.

We remark that, once again, $\text{ind}(\bar{u})$ and $\text{I}(\bar{u})$ do not take into account the point constraint $\overline{\mathfrak{m}}(\tau)$ and that the condition “passing through $\overline{\mathfrak{m}}(\tau)$” is a codimension 2 condition; more precisely, adding a marked point increases the dimension of the moduli space by 2 and constraining the marked point to $\overline{\mathfrak{m}}(\tau)$ reduces the dimension by 4.

Let $\tilde{\overline{W}}_\tau$ be the compactification of $\overline{W}_\tau$ that we identify with $\overline{W}_\tau - \{s > l(\tau) + 1\} - \{s < -l(\tau) - 1\}$, where $l(\tau)$ is given in Equation (3.1.1). Let $\tilde{\bar{u}}: \dot{\mathcal{F}} \to \tilde{\overline{W}}_\tau$ be the compactification of $\bar{u}$, where $\dot{\mathcal{F}}$ is obtained by performing a real blow-up of $\mathcal{F}$ at its boundary punctures. We also define

$$ Z_{\mathbf{y},\mathbf{y}'} = (\{l(\tau) + 1\} \times [0, 1] \times \mathbf{y}) \cup (\{-l(\tau) - 1\} \times [0, 1] \times \mathbf{y}') \\ \cup ((L_{\tilde{\mathbf{a}}}^{\tau,+} \cup L_{\tilde{\mathbf{b}}}^{\tau,-}) \cap \tilde{\overline{W}}_{\tau}). $$
The trivialization $\tau^*$ of $\overline{T\bar{S}}$ along $Z_{y,y'}$ is defined in a manner similar to that of Section I.4.4.2:<fnref n="5" /> First we define the trivialization $\tau^*$ of $\overline{T\bar{S}}|_{\pi_{\overline{B}_\tau}^{-1}(l(\tau)+1,1)}$ (resp. $\overline{T\bar{S}}|_{\pi_{\overline{B}_\tau}^{-1}(-l(\tau)-1,1)}$) along the $\hat{a}_i$ (resp. $\hat{b}_i$) by choosing a nonsingular tangent vector field along $\hat{a}_i$ (resp. $\hat{b}_i$). We then parallel transport $\tau^*$ along $\partial\overline{W}_\tau$ and extend $\tau^*$ arbitrarily to $\{l(\tau)+1\} \times [0, 1] \times \mathbf{y}$ and $\{-l(\tau)-1\} \times [0, 1] \times \mathbf{y'}$.

Let $\mathcal{Q}_{\tau^*}(\tilde{u})$ be the relative intersection form given by intersecting $\tilde{u}$ and a pushoff of $\tilde{u}$ in the direction of $J\tau^*$ along $\partial\overline{W}_\tau$. Then

$$ (3.2.4) \qquad \mathrm{I}(\bar{u}) = c_1(\tilde{u}^* \overline{T\bar{S}}, \tau^*) + \mathcal{Q}_{\tau^*}(\tilde{u}) + \mu_{\tau^*}(\mathbf{y}) - \mu_{\tau^*}(\mathbf{y}') - 2g, $$

$$ (3.2.5) \qquad \mathrm{ind}(\bar{u}) = -\chi(\dot{\mathbf{F}}) + 2c_1(\tilde{u}^* \overline{T\bar{S}}, \tau^*) + \mu_{\tau^*}(\mathbf{y}) - \mu_{\tau^*}(\mathbf{y}') - 2g, $$

These index formulas are obtained by adding the index formulas for holomorphic curves in $\overline{W}_\pm$ of [I]. The index inequality holds as usual:

$$ (3.2.6) \qquad \mathrm{ind}(\bar{u}) + 2\delta(\bar{u}) \le \mathrm{I}(\bar{u}), $$

where $\delta(\bar{u}) \ge 0$ and equals zero if and only if $\bar{u}$ is an embedding.

In the general case when $\tilde{u} \ne \emptyset$, modifications can be made as in Section I.5.7.5 and one can easily verify the following:

$$ (3.2.7) \qquad \mathrm{I}(\sigma_{\infty}^{+}) = \mathrm{ind}(\sigma_{\infty}^{+}) = -1; $$

$$ (3.2.8) \qquad \mathrm{I}(\sigma_{\infty}^{-}) = \mathrm{ind}(\sigma_{\infty}^{-}) = 0; $$

$$ (3.2.9) \qquad \mathrm{I}(\sigma_{\infty}^{\tau}) = \mathrm{ind}(\sigma_{\infty}^{\tau}) = -1. $$

The Fredholm and ECH indices for $\overline{W}_{-\infty,i}$-curves can be defined and computed similarly. We now prove the following:

**Lemma 3.2.15.** — If there exists a $\overline{W}_{-\infty,2}$-curve $\bar{u}$ with $I = 2$ and ends $\mathbf{y}$ and $\mathbf{y}'$, then $\mathbf{y} = \{x_{ij(i)}^{\#}\}_{i=1}^{2g}$ and $\mathbf{y}' = \{x_{ik(i)}^{\#}\}_{i=1}^{2g}$, where $j(i)$ is odd and $k(i)$ is even for all $i$.

In other words, $\mathbf{y}$ is a summand of the top generator $\Theta_{\bar{a},\bar{b}} \in \widehat{\mathrm{HF}}(\bar{a}, \bar{b})$ and $\mathbf{y}'$ is a summand of the top generator $\Theta_{\bar{b},\bar{a}} \in \widehat{\mathrm{HF}}(\bar{b}, \bar{a})$. We remind the reader that our convention is that $\mathbf{y}$ and $\mathbf{y}'$ do not contain $z_\infty$.

**Proof.** — The proof is similar to the index calculation of Lemma 2.2.2. Let $\bar{u}$ be a $\overline{W}_{-\infty,2}$-curve; note that $n^*(\bar{u}) = m$ by definition. First consider the situation where $\bar{u}$ is in the homology class consisting of a copy $\{pt\} \times \overline{S}$ of the fiber and $2g$ trivial strips. Then

$$ \mathrm{ind}(\{pt\} \times \overline{S}) = -\chi(\overline{S}) + 2c_1(\overline{T\bar{S}}) = -(2-2g) + 2(2-2g) = 2-2g. $$

<footnote n="5">⁵ Here we are writing $\tau^*$ instead of $\tau$ due to the notational conflict with $\tau \in \mathbb{R}$.</footnote>
The strips contribute 0 to ind and there are $2g$ intersection points. Hence

$$I(\bar{u}) = (2 - 2g) + 0 + 2(2g) = 2g + 2$$

when $\mathbf{y}' = \mathbf{y}$. The only way to lower $I(\bar{u})$ to 2 is to take $\mathbf{y} = \{x_{ij(i)}^{\#}\}_{i=1}^{2g}$ and $\mathbf{y}' = \{x_{ik(i)}^{\#}\}_{i=1}^{2g}$ so that all the $j(i)$ are odd and all the $k(i)$ are even. $\square$

**3.2.7. Regularity.** — We now discuss the regularity of the family $\{\bar{J}_{\tau}\}_{\tau \in \mathbb{R}}$.

**Definition 3.2.16.**

(1) $\bar{J}_{-\infty, 2} \in \mathcal{J}_{\bar{W}_{-\infty, 2}}$ is regular if all the moduli spaces $M_{\bar{J}_{-\infty, 2}}^{\dagger, \text{ext}}(\mathbf{z}, \mathbf{z}')$ are transversely cut out.

(2) $\bar{J}_{-\infty, 1} \in \mathcal{J}_{\bar{W}_{-\infty, 1}}$ is regular if all the moduli spaces $M_{\bar{J}_{-\infty, 1}}^{\dagger, \text{ext}}(\mathbf{z}_1, \dots, \mathbf{z}_4)$ are transversely cut out and the restrictions $\bar{J}_{-\infty, 2}$ and $\bar{J}$ of $\bar{J}_{-\infty, 1}$ to the ends are regular.

(3) $\bar{J}_{-\infty} = \bar{J}_{-\infty, 1} \cup \bar{J}_{-\infty, 2} \in \mathcal{J}_{\bar{W}_{-\infty}}$ is regular if $\bar{J}_{-\infty, i}$ are regular for $i = 1, 2$.

**Definition 3.2.17.** — The family $\{\bar{J}_{\tau}\}_{\tau \in \mathbb{R}} \in \bar{\mathcal{I}}$ is regular if:

(1) all the moduli spaces $M_{\bar{J}_{\tau}}^{\dagger, \text{ext}}(\mathbf{z}, \mathbf{z}')$ are transversely cut out;

(2) the restriction $\bar{J}$ of $\bar{J}_{\tau}$ to the positive and negative ends is regular;

(3) $\bar{J}_+$ and $\bar{J}_-$ in the limit $\tau \to +\infty$ are regular; and

(4) $\bar{J}_{-\infty}$ in the limit $\tau \to -\infty$ is regular.

Let $\bar{\mathcal{I}}^{\text{reg}}$ be the space of regular $\{\bar{J}_{\tau}\}_{\tau \in \bar{\mathcal{I}}}$. As usual, we have:

**Lemma 3.2.18.** — The generic $\{\bar{J}_{\tau}\}_{\tau \in \bar{\mathcal{I}}}$ is regular, provided no points of $(\bar{\mathbf{a}} \cap \bar{h}(\bar{\mathbf{a}})) - \{z_{\infty}\}$ are fixed points of $\bar{h}$.<fnref n="6" />

*Proof.* — This is analogous to the proof of regularity for $\bar{J}_+$, with one caveat: the family $\{\bar{J}_{\tau}\}$ is not sufficiently generic to achieve the transversality of negative index horizontal sections, where by a horizontal section we mean the restriction of a cylinder over an orbit that passes through a fixed point $y$ of $\bar{h}$. The additional assumption on the fixed points of $\bar{h}$ eliminates the horizontal sections besides $\sigma_{\infty}^{\tau}$, which we do not consider because of the superscript $\dagger$. $\square$

Next we discuss the regularity of moduli spaces passing through $\bar{\mathfrak{m}}$. Since the point constraints $\bar{\mathfrak{m}}(\tau)$ are nongeneric, we need to introduce a perturbation of the family $\{\bar{J}_{\tau}\}$:

<footnote n="6">⁶ Here (and henceforth) we make this additional assumption on $\bar{h}$, which we are free to do.</footnote>
**Definition 3.2.19.** — Let $\varepsilon > 0$ and let $\{U_\tau\}_{\tau \in \mathbb{R}}$, $U_\tau \subset \overline{W}_\tau$, be a family of open sets such that $U_\tau \not\supset \overline{m}(\tau)$. Then a family $\{\overline{J}_\tau^\diamond\}_{\tau \in \mathbb{R}}$ of almost complex structures on $\{\overline{W}_\tau\}$ is $(\varepsilon, \{U_\tau\})$-close to a regular $\{\overline{J}_\tau\}$ if:

$$- \overline{J}_\tau^\diamond = \overline{J}_\tau \text{ on } \overline{W}_\tau - U_\tau;$$

$$- \overline{J}_\tau^\diamond \text{ is } \varepsilon\text{-close to } \overline{J}_\tau \text{ on } U_\tau; \text{ and}$$

$$- \nabla \overline{J}_\tau^\diamond \text{ is } \varepsilon\text{-close to } \nabla \overline{J}_\tau \text{ on } U_\tau.$$

Here the $\varepsilon$-closeness is measured with respect to a family $\{g_\tau\}_{\tau \in \mathbb{R}}$ of Riemannian metrics which is defined as follows: Let $h$ be an $s$-invariant metric on $\mathbf{R} \times \overline{\mathbb{N}}_2$ such that $h$, viewed as a metric on $\mathbf{R} \times [0, 2] \times \overline{S}$, is also $t$-invariant on $\mathbf{R} \times [1, 2] \times \overline{S}$. There is an extension of $h$ to $h_\tau$ on $\mathbf{R} \times [0, r(\tau)] \times \overline{S}$ which is $s$- and $t$-invariant on $\mathbf{R} \times [2, r(\tau)] \times \overline{S}$. We then view $h_\tau$ as a metric on $\mathbf{R} \times \overline{\mathbb{N}}_{r(\tau)}$ and define $g_\tau$ as the restriction of $h_\tau$ to $\overline{W}_\tau \subset \mathbf{R} \times \overline{\mathbb{N}}_{r(\tau)}$.

Let $\mathfrak{p}(\tau) \subset \mathrm{int}(B_\tau)$, $\tau \in [-\infty, \infty]$, be a family of points, where:

$$- \text{ the cardinality } \#\mathfrak{p}(\tau) \text{ is finite and independent of } \tau \in [-\infty, \infty];$$

$$- \mathfrak{p}(\tau) \text{ is smooth for } \tau \in (-\infty, \infty);$$

$$- \lim_{\tau \to +\infty} \mathfrak{p}(\tau) \text{ exists and equals } \mathfrak{p}(+\infty);$$

$$- \lim_{\tau \to -\infty} \mathfrak{p}(\tau) \text{ exists and equals } \mathfrak{p}(-\infty).$$

In order for $\lim_{\tau \to +\infty} \mathfrak{p}(\tau)$ to be defined we require the existence of $C > 0$ such that, for all $\tau \gg 0$, $\mathfrak{p}(\tau)$ is contained in a C-neighborhood of $\partial B_\tau$. Then $\mathfrak{p}(\tau)$ can be viewed as a subset of $B_+ \cup B_-$ and we are asking $\lim_{\tau \to +\infty} \mathfrak{p}(\tau) = \mathfrak{p}(+\infty)$ in $B_+ \cup B_-$. $\lim_{\tau \to -\infty} \mathfrak{p}(\tau)$ is defined analogously.

$$- \mathfrak{p}(+\infty) \text{ is a nontrivial union of points of } \mathrm{int}(B_+) \text{ and } \mathrm{int}(B_-); \text{ similarly, } \mathfrak{p}(-\infty) \text{ is a nontrivial union of points of } \mathrm{int}(B_{-\infty,1}) \text{ and } \mathrm{int}(B_{-\infty,2});$$

$$- \text{ for each } \tau \in [-\infty, \infty], \overline{\mathfrak{m}}^b(\tau) \notin \mathfrak{p}(\tau).$$

We will use the following specific open sets $U_\tau$: For $\tau \in \mathbf{R}$, let $U_\tau$ be an open $\delta$-neighborhood of $K_\tau = \pi_{B_\tau}^{-1}(\mathfrak{p}(\tau)) - \{\rho < 2\delta\}$, where $\delta > 0$ is arbitrarily small. Then let $U_{\pm\infty}$ and $K_{\pm\infty}$ be the limits of $U_\tau$ and $K_\tau$ as $\tau \to \pm\infty$. When we want to emphasize $(\varepsilon, U_\tau)$ or $(\varepsilon, \delta, \mathfrak{p}(\tau))$, we write $\overline{J}_\tau^\diamond(\varepsilon, U_\tau)$ or $\overline{J}_\tau^\diamond(\varepsilon, \delta, \mathfrak{p}(\tau))$ for $\overline{J}_\tau^\diamond, U_{\varepsilon, \delta, \mathfrak{p}(\tau)}$ for $U_\tau$, and $K_{\mathfrak{p}(\tau), \delta}$ for $K_\tau$.

We define a degree $k$ almost multisection $\bar{u}$ of $(\overline{W}_\tau, \overline{J}_\tau^\diamond)$ in the same way as a degree $k$ multisection of $(\overline{W}_\tau, \overline{J}_\tau)$, with the following difference: $\bar{u}$ is a degree $k$ multisection when restricted to

$$\pi_{B_\tau} : \overline{W}_\tau - \pi_{B_-}^{-1}(V_\tau) \to B_\tau - V_\tau, \quad V_\tau = \pi_{B_-}(U_\tau),$$

but $\bar{u}$ is just a degree $k$ map when restricted to $\pi_{B_-}^{-1}(V_\tau) \to V_\tau$. The moduli spaces of almost multisections are defined in the same way as the moduli spaces of multisections,
with $\bar{J}_{\tau}^{\diamond}$ replacing $\bar{J}_{\tau}$. If $\{K_{\tau} \neq \bar{m}(\tau)\}$ is a family of compact sets of $\bar{W}_{\tau}$, then the modifier $\{K_{\tau}\}$ means that the image of $\bar{u}$ in $\bar{W}_{\tau}$ intersects $K_{\tau}$. We stress the fact that, unlike the basepoint $\bar{m}$, the modifier $\{K_{\tau}\}$ is not meant to define a moduli space with a point constraint at $K_{\tau}$, but only to select a subset of the moduli space.

Almost complex structures $\bar{J}_{-\infty,i}^{\diamond}$, almost multisections on $(\bar{W}_{-\infty,i}, \bar{J}_{-\infty,i}^{\diamond})$, and moduli spaces of almost multisections are defined similarly.

**Definition 3.2.20.** — The family $\{\bar{J}_{\tau}^{\diamond}\}$ is $\{K_{\tau}\}$-regular with respect to $\bar{m}$ if all the $\{\bar{J}_{\tau}^{\diamond}\}$-holomorphic maps in $M_{\{\bar{J}_{\tau}^{\diamond}\}}^{\dagger, \text{ext}, \{K_{\tau}\}}(\mathbf{z}, \mathbf{z}'; \bar{m})$ are regular, i.e., their linearized Cauchy-Riemann operator, taking into account the variation of domain complex structure and the parameter $\tau$, is surjective.

If $\{\bar{J}_{\tau}^{\diamond}\}$ is $\{K_{\tau}\}$-regular with respect to $\bar{m}$, then $M_{\{\bar{J}_{\tau}^{\diamond}\}}^{\dagger, \text{ext}, \{K_{\tau}\}}(\mathbf{z}, \mathbf{z}'; \bar{m})$ is an open subset of $M_{\{\bar{J}_{\tau}^{\diamond}\}}^{\dagger, \text{ext}}(\mathbf{z}, \mathbf{z}'; \bar{m})$ and is a transversely cut out manifold.

**Lemma 3.2.21.** — A generic family $\{\bar{J}_{\tau}^{\diamond}\}$ is $\{K_{\tau}\}$-regular with respect to $\bar{m}$.

*Proof*. — The proof is similar to the combination of Theorems 3.1.7 and 3.4.1 of [MS], with modifications as in Proposition I.5.8.8. $\square$

The following can also be proved using a standard regularity and compactness argument:

**Lemma 3.2.22.** — If $\{\bar{J}_{\tau}\}$ is a generic family, then for $\varepsilon, \delta > 0$ sufficiently small, there exist a generic family $\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ which is $\{K_{\mathfrak{p}(\tau), \delta}\}$-regular with respect to $\bar{m}$ and disjoint finite subsets $\mathcal{T}_1, \mathcal{T}_2 \subset \mathbb{R}$ with the following properties:

(1) $\tau \in \mathcal{T}_1$ if and only if there exists $\bar{v}_{\tau} \in M_{\bar{J}_{\tau}}^{\dagger, \text{irr}, \text{ind}=-1}(\mathbf{z}, \mathbf{z}')$ for some $\mathbf{z}$ and $\mathbf{z}'$.

(2) $\tau \in \mathcal{T}_2$ if and only if there exists $\bar{v}_{\tau} \in M_{\bar{J}_{\tau}}^{\dagger, \text{irr}, \{K_{\tau}\}, \text{ind}=1}(\mathbf{z}, \mathbf{z}'; \bar{m})$ for some $\mathbf{z}$ and $\mathbf{z}'$.

Moreover, for each $\tau \in \mathcal{T}_i$ there is a unique such irreducible curve $\bar{v}_{\tau}$.

*Sketch of proof.* — The existence of discrete sets $\mathcal{T}_1$ and $\mathcal{T}_2$ as in the lemma follows from Lemmas 3.2.18 and 3.2.21. Note that the only section which is not transversely cut out is the section at infinity $\sigma_{\infty}^*$, which is excluded from the moduli space by the modifier $\dagger$.

The finiteness of $\mathcal{T}_1$ and $\mathcal{T}_2$ relies on an SFT compactness result which will be discussed in the next section; see Equations (3.4.1) and (3.5.1). If either $\mathcal{T}_1$ or $\mathcal{T}_2$ is infinite, then there exists a limit $\bar{J}_{\pm\infty}$-holomorphic building of the same total index by SFT compactness. However, since $\bar{J}_{\pm\infty}$ are regular, at least one level in the building will have index which is too low to exist. $\square$
By shrinking $\overline{\mathcal{I}}$, we assume that all $\{\overline{J}_\tau\} \in \overline{\mathcal{I}}$ satisfy Lemma 3.2.22.

**3.3.** *Proof of half of Theorem 1.0.1.* — In this subsection we prove that $\Psi \circ \Phi$ is an isomorphism on the level of homology.

In the next several paragraphs we briefly recall the chain complexes

$$(\widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a})), \partial'), \quad (\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})), \partial), \quad (\widetilde{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})), \widetilde{\partial}),$$

from Sections I.4.9.3 and I.6.6, where the first and third have isomorphic homology groups; the quotient map

$$q : \widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}));$$

and the maps $\tilde{\Phi}$, $\Psi$, and $\Psi'$ from Sections I.6.6 and I.7.1.

The chain complex $\widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ is generated by the set $S_{\mathbf{a},\mathbf{h}(\mathbf{a})}$ of 2g-tuples of intersection points of $\mathbf{a}$ and $\mathbf{h}(\mathbf{a})$, where each $\mathbf{y} \in S_{\mathbf{a},\mathbf{h}(\mathbf{a})}$ intersects $a_i$ and $\mathbf{h}(a_i)$ exactly once, and the differential $\partial'$ counts $I=1$, degree 2g multisections of $W = \mathbf{R} \times [0, 1] \times S$. The chain complex $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ is the quotient of $\widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ under the equivalence relation $\sim$ which identifies $\mathbf{y} \sim \mathbf{y}'$ if $\mathbf{y}'$ can be obtained from $\mathbf{y}$ by successively replacing $x_i$ by $x'_i$ or $x'_i$ by $x_i$ for any $i=1, \dots, 2g$; the map $q$ is the corresponding quotient map, which is a chain map. Here $x_i$ and $x'_i$ are intersection points of $a_i$ and $\mathbf{h}(a_i)$ on $\partial S$ given in Section I.4.9.1.

The chain complex $\widetilde{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ is a variant of $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ (with isomorphic homology groups) generated by 2g-tuples of intersection points $\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'$ of $\bar{\mathbf{a}}$ and $\bar{\mathbf{h}}(\bar{\mathbf{a}})$ with $\mathcal{I} \subset \{1, \dots, 2g\}$, where:

- $z_\infty$ can be used more than once, but is always viewed as an intersection point of $\bar{a}_i$ and $\bar{\mathbf{h}}(\bar{a}_i)$ and hence is written as $z_{\infty,i}$;

- writing $\mathbf{y}' = \{y'_i\}_{i \in \mathcal{I}'}$ where $\mathcal{I}' = \{1, \dots, 2g\} - \mathcal{I}$, there exists a permutation $\sigma$ of $\mathcal{I}'$ such that $y'_i \in a_i \cap \mathbf{h}(a_{\sigma(i)})$ for all $i \in \mathcal{I}'$.

The differential $\widetilde{\partial}$ counts $I=1$, degree 2g multisections $\bar{u} = \bar{u}' \cup \bar{u}''$ of $\overline{W}$ with $n^*(\bar{u}) \le 1$ such that $\bar{u}'$ has empty branch locus; see Definition I.6.6.1.

The map

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

defined in Section I.6.6, is a variant of the map $\Phi : \widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \mathrm{PFC}_{2g}(\mathbb{N})$ and

$$\langle \tilde{\Phi}(\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'), \mathbf{y} \rangle,$$

counts $I=0$, degree 2g multisections of $\overline{W}_+$ with $n^* \le |\mathcal{I}|$ from $\{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}'$ to $\mathbf{y}$.

The map

$$\Psi' : \mathrm{PFC}_{2g}(\mathbb{N}) \to \widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a})),$$
defined in Section I.7.1, counts $I = 2$, degree $2g$ almost multisections of $(\overline{W}_{-}, \overline{J}_{-}^{\diamond})$ (cf. Definition I.5.8.3) with $n^* = m$ that pass through $\overline{m}(+\infty)$. The map

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

is then given as the composition $q \circ \Psi'$.

The following is proved in this subsection:

**Theorem 3.3.1.** — Suppose $m \gg 0$. Then there exist maps

$$ \begin{align*} H', \Theta'_0 : \widetilde{\mathrm{CF}}(\mathrm{S}, \mathbf{a}, \hbar(\mathbf{a})) &\to \widehat{\mathrm{CF}}'(\mathrm{S}, \mathbf{b}, \hbar(\mathbf{b})), \\ \widetilde{V} : \widetilde{\mathrm{CF}}(\mathrm{S}, \mathbf{a}, \hbar(\mathbf{a})) &\to \widehat{\mathrm{CF}}(\mathrm{S}, \mathbf{b}, \hbar(\mathbf{b})), \end{align*} $$

which satisfy the following:

$$ (3.3.1) \qquad \Psi' \circ \widetilde{\Phi} - \Theta'_0 = (\partial'H' + H'\widetilde{\partial}) + \widetilde{\partial}_1 \circ \widetilde{V}, $$

where $\widetilde{\partial}_1$, defined in Equation (I.7.1.1), satisfies:

$$ \widetilde{\partial}_1(\{z_{\infty,i}\} \cup \mathbf{y}') = \{x_i\} \cup \mathbf{y}' + \{x'_i\} \cup \mathbf{y}' $$

for all $i = 1, \dots, 2g$ and is zero for any other generator. (Here $z_{\infty,i}$, $x_i$, and $x'_i$ are for the basis **b**.)
Postcomposing with $q$ (for the basis **b**) and writing $H = q \circ H'$ and $\Theta_0 = q \circ \Theta'_0$, we obtain the chain homotopy

$$ (3.3.2) \qquad \Psi \circ \widetilde{\Phi} - \Theta_0 = \partial H + H \widetilde{\partial}, $$

where $\Theta_0$ induces an isomorphism on homology.

To go from Equation (3.3.1) to Equation (3.3.2), we observe that $q \circ \widetilde{\partial}_1 = 0$ and $q \circ \partial' = \partial \circ q$.

Since $\Theta_0$ is an isomorphism on the level of homology, so is $\Psi \circ \widetilde{\Phi}$. In view of Corollary I.6.6.7, $\Psi \circ \Phi$ is also an isomorphism on the level of homology.

*Proof.* — We prove Theorem 3.3.1, assuming the results of Sections 3.4–3.9. In Steps 1–3 we consider the situation where the holomorphic curves $\bar{u}$ in $\overline{W}_{\tau}$ are asymptotic to some $\mathbf{y} \in S_{\mathbf{a},\hbar(\mathbf{a})}$ at the positive end. In Step 4 we describe the modifications needed for the situation where $\bar{u}$ is asymptotic to some $\mathbf{z} = \{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}$ at the positive end and $\mathcal{I}$ is a subset of $\{1, \dots, 2g\}$ with $|\mathcal{I}| > 0$. Steps 1–4 prove Equation (3.3.1) and hence Equation (3.3.2). In Step 5 we prove that $\Theta_0$ induces an isomorphism on homology by further degenerating $\overline{W}_{-\infty,1}$.

Suppose $m \gg 0$. Choose $\mathfrak{p}(\tau)$ and $\{\overline{J}_{\tau}\} \in \overline{\mathcal{I}}^{\mathrm{reg}}$. For sufficiently small $\varepsilon, \delta > 0$ (which depend on the choices of $m$ and $\{\overline{J}_{\tau}\}$), there exists $\{\overline{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ so that Lemma 3.2.22 holds.
Fix $\mathbf{y} \in S_{\mathbf{a},\mathbf{h}(\mathbf{a})}$, $\mathbf{y}' \in S_{\mathbf{b},\mathbf{h}(\mathbf{b})}$ and abbreviate

$$ \mathcal{M} = \mathcal{M}_{\{\mathcal{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}}^{I=2, n^*=m} (\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}), \quad \mathcal{M}^{(\mathbf{K}_{\mathbf{p}(\tau), \delta})} = \mathcal{M}_{\{\mathcal{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}}^{I=2, n^*=m, (\mathbf{K}_{\mathbf{p}(\tau), \delta})} (\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}). $$

Let $\overline{\mathcal{M}}$ be the SFT compactification of $\mathcal{M}$. The limit SFT buildings can be described in a manner analogous to Definition I.7.3.1 and the proof of existence follows the same steps as that of Proposition I.7.3.2. This is because:

- the ends of $\overline{W}_{\tau}$ and $\overline{W}_{-\infty,i}$ are basically the same as the strip-like ends of $\overline{W}_{+}$ and $\overline{W}_{-}$ and

- the Lagrangian boundaries of $\overline{W}_{\tau}$ and $\overline{W}_{-\infty,i}$ are locally identical to those of $\overline{W}_{+}$ and $\overline{W}_{-}$.

The main point is that for each component of the limit SFT building the boundary punctures either map to points on the singular Lagrangian or to Reeb chords, including chords over $z_{\infty}$.

Let $\partial \mathcal{M} = \overline{\mathcal{M}} - \mathcal{M}$ be the boundary of $\mathcal{M}$. If $U \subset [-\infty, +\infty]$, then we write $\partial_U \mathcal{M}$ for the set of $\bar{u}_{\infty} \in \partial \mathcal{M}$ where $\bar{u}_{\infty}$ is a building which corresponds to some $\tau \in U$. By Lemma 3.2.21, we may take $\mathcal{M}^{(\mathbf{K}_{\mathbf{p}(\tau), \delta})}$ to be regular.

Step 1 (Breaking at $+\infty$). Recall the definition of a bad radial ray $\mathcal{R}_{\phi}$ from Definition I.7.7.10. We now enlarge the class of bad radial rays as follows: Let $(\bar{J}_{+})_{\infty}$ be the limit of $(\bar{J}_{+})_{m}$ as $m \to \infty$. Let

$$ \coprod_{\mathbf{y}, \delta_0^{l_i} \mathbf{y}'} \mathcal{M}_{(\bar{J}_+)_\infty}^{I=0, (l_i)} (\mathbf{y}, \delta_0^{l_i} \mathbf{y}') = \{C_1, \dots, C_r\}, $$

where the disjoint union is over all $\mathbf{y}$ and $\delta_0^{l_i} \mathbf{y}'$ with $l_i > 0$ and $(l_i)$ is the “trivial” partition of $l_i$.<fnref n="7" /> Let $f_i: \mathbf{R}/2l_i\mathbf{Z} \to \mathbf{C}$ be the asymptotic eigenfunction corresponding to the end $\delta_0^{l_i}$ of $C_i$. (We remark that $C_1, \dots, C_r$ and $f_i$ also appear in Section I.7.7.2, but denote similar but different things.) We then add the radial rays which pass through

$$ \{f_i(t) \mid i = 1, \dots, r; 0 < t < 2l_i; t \equiv 3/2 \mod 2\} $$

to the class of bad radial rays. We can still assume that $\mathcal{R}_{\pi}$ is a good radial ray.

Recall the set $\tilde{\mathcal{O}}_k$ of orbit sets constructed from $\tilde{\mathcal{P}}$ (the set of simple Reeb orbits in $\mathrm{int}(\mathcal{N})$, together with $h$ and $e$) which intersect $\bar{\mathcal{S}} \times \{0\}$ exactly $k$ times. We then have the following:

<footnote n="7">⁷ Note that the orbit $\delta_0$ is degenerate when $m = \infty$ and the moduli spaces with $\delta_0^{l_i}$ at the negative end are treated in the same way as in Section I.7.7.4.</footnote>
Lemma 3.3.2. — $\partial_{(+\infty)} \mathcal{M} \subset A_1 \sqcup A_2$,<fnref n="8" /> where

$$
A_1 = \coprod_{\mathbf{y} \in \hat{O}_{2g}} \left( \mathcal{M}_{\hat{J}_+^{\diamond}(\varepsilon, \delta, \mathbf{p}(+\infty))}^{I=0} (\mathbf{y}, \mathbf{y}) \times \mathcal{M}_{\hat{J}_-^{\diamond}(\varepsilon, \delta, \mathbf{p}(+\infty))}^{I=2, n^*=m} (\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}(+\infty)) \right);
$$

$$
A_2 = \coprod_{\delta_0 \mathbf{y}, \{z_\infty\} \cup \mathbf{y}''} \left( \mathcal{M}_{\hat{J}_+^{\diamond}(\varepsilon, \delta, \mathbf{p}(+\infty))}^{I=1, n^*=m-1, f_{\delta_0}} (\mathbf{y}, \delta_0 \mathbf{y}) \times \mathcal{M}_{\hat{J}_-^{\diamond}(\varepsilon, \delta, \mathbf{p}(+\infty))}^{I=0, n^*=0} (\delta_0 \mathbf{y}, \{z_\infty\} \cup \mathbf{y}'') \right. \\
\left. \times \mathcal{M}_{\hat{J}^{\diamond}}^{I=1, n^*=1} (\{z_\infty\} \cup \mathbf{y}'' , \mathbf{y}') \right),
$$

if $\mathbf{y}' = \{x_i^j\} \times \mathbf{y}''$ for some $x_i^j$ and $A_2 = \emptyset$ otherwise. The disjoint union for $A_2$ ranges over all $\delta_0 \mathbf{y}$ such that $\mathbf{y} \in \hat{O}_{2g-1}$ and all $\{z_\infty\} \cup \mathbf{y}''$ such that $\mathbf{y}'$ can be written as $\{x_i^j\} \cup \mathbf{y}''$ for some $x_i^j$. Here we have omitted the potential contributions of connector components and we are writing $x_i^0 := x_i$ and $x_i^1 := x_i'$.

We will explain the moduli spaces that are involved in $A_2$: $f_{\delta_0}$ is a nonzero normal-
ized asymptotic eigenfunction of $\delta_0$ at the negative end such that $f_{\delta_0}(\frac{3}{2})$ lies on the good
radial ray $\mathcal{R}_{\pi}$. Used as a modifier, $f_{\delta_0}$ stands for “the normalized asymptotic eigenfunction
at the negative end $\delta_0$ is $f_{\delta_0}$”. If

$$
\bar{u} = \bar{u}' \cup \bar{u}'' \in \mathcal{M}_{\bar{J}^{\diamond}(\varepsilon, \delta, \mathbf{p}(+\infty))}^{I=0, n^*=0} (\delta_0 \mathbf{y}, \{z_{\infty}\} \cup \mathbf{y}''),
$$

then $\bar{u}$ consists of $\bar{u}' = \sigma_{\infty}^{-}$ and a curve $\bar{u}''$ from $\mathbf{y}$ to $\mathbf{y}''$ which is arbitrarily close to a curve
with image in $W_{-}$. If

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

then $\mathbf{y}' = \{x_i^j\} \cup \mathbf{y}''$ for some $i,j$ and $\bar{u}$ consists of one thin strip from $z_\infty$ to $x_i^j$ and $2g - 1$ trivial strips.

Remark 3.3.3. — As in the proof of the chain map property for Ψ from Sec-
tion I.7.2, the point constraint of passing through $\bar{\mathfrak{m}}(+\infty)$ is converted to an asymptotic
constraint of the normalized asymptotic eigenfunction being $f_{\delta_0}$ when a section at infinity
$\sigma_{\infty}^{-}$ is present.

Lemma 3.3.2 will be proved in Section 3.4. Gluing the pairs in A₁ using the Hutchings-Taubes gluing theorem [HT1, HT2] (see Section I.6.5) accounts for the term Ψ' ◦ ˜Φ in Equation (3.3.1).

Gluing the triples in $A_2$ accounts for the term $\partial_1 \circ \tilde{V}$. This is similar to Section I.7.12, and the details will be omitted. For each triple $(\bar{v}_+, \bar{v}_-, \bar{v}_{-1,1}) \in A_2$, where

<footnote n="8">⁸ Lemma 3.3.2 and its later analogs are compactness statements, which is why we write “⊂” even when equality
holds. The equality will follow from the discussion of gluing, which appears later in this step.</footnote>
the nontrivial component of $\bar{v}_{-1,1}$ is a thin strip from $z_{\infty,i,0}$ to $x_i$, there is another triple $(\bar{v}_+, \bar{v}_-, \bar{v}_{-1,1}^*) \in A_2$, where the nontrivial component of $\bar{v}_{-1,1}^*$ is a thin strip from $z_{\infty,i,1}$ to $x_i'$.

Since $\mathcal{M}^{[K_{p(\tau),\delta})}$ is regular but $\mathcal{M} - \mathcal{M}^{[K_{p(\tau),\delta})}$ is not a priori regular, it remains to verify the following:

**Claim 3.3.4.** — For $\varepsilon, \delta > 0$ sufficiently small, then there exists a truncation $\mathcal{M}'$ of $\mathcal{M}$ containing $\mathcal{M}^{[K_{p(\tau),\delta})}$ and whose boundary, when restricted to a neighborhood of $\tau = +\infty$, is $A_1 \cup \tilde{A}_2$, where $\#A_2 = \#\tilde{A}_2 \bmod 2$.

Note that there are two types of boundary points: (i) points of the SFT compactification and (ii) boundary points of a manifold with boundary. (i) gives $A_1$ and (ii) gives $\tilde{A}_2$.

*Proof.* — There exist $\varepsilon, \delta > 0$ sufficiently small such that $\bar{v}_-$ passes through $K_{p(\tau),\delta}$ whenever $(\bar{v}_+, \bar{v}_-) \in A_1$. Hence if $\bar{u} \in \mathcal{M}$ is close to a curve in $A_1$, then $\bar{u} \in \mathcal{M}^{[K_{p(\tau),\delta})}$. This accounts for the term $A_1$ in the claim.

Next suppose that $\bar{u} \in \mathcal{M} - \mathcal{M}^{[K_{p(\tau),\delta})}$ for $\tau$ near $+\infty$. By the argument of Lemma I.7.2.3, $\bar{u}$ is arbitrarily close to a building of type $A_2$ and $\mathcal{M}'$ is obtained from $\mathcal{M}$ by truncating ends that are close to $A_2$. Then, by the adaptation of Theorem I.7.2.2 to our case, $\tilde{A}_2 := \partial \mathcal{M}' - A_1$ satisfies $\#A_2 = \#\tilde{A}_2 \bmod 2$. $\square$

*Step 2 (Breaking at $-\infty$).*

**Lemma 3.3.5.** — $\partial_{(-\infty)} \mathcal{M} \subset A_3$, where

$$A_3 = \coprod_{\mathbf{y}_2, \mathbf{y}_4} \left( \mathcal{M}_{\bar{J}_{-\infty,1}^{\diamond} (\varepsilon, \delta, \mathbf{p}(-\infty))}^{I=0, n^*=0} (\mathbf{y}, \mathbf{y}_2, \mathbf{y}', \bar{h}(\mathbf{y}_4)) \right. \\ \left. \times \mathcal{M}_{\bar{J}_{-\infty,2}^{\diamond} (\varepsilon, \delta, \mathbf{p}(-\infty))}^{I=2, n^*=m} (\mathbf{y}_4, \mathbf{y}_2; \bar{\mathfrak{m}}(-\infty)) \right).$$

Here the union is over all $\mathbf{y}_2, \mathbf{y}_4$ such that $\mathbf{y}_4 = \{x_{ij(i)}^{\#} \}_{i=1}^{2g} \text{ and } \mathbf{y}_2 = \{x_{ik(i)}^{\#} \}_{i=1}^{2g}$, where $j(i)$ is odd and $k(i)$ is even.

Lemma 3.3.5 will be proved in Section 3.5. Gluing the pairs $(\bar{v}_1, \bar{v}_2)$ in $A_3$ accounts for the term $\Theta'_0$ in Equation (3.3.1). The map $\Theta'_0$ is given by:

$$\langle \Theta'_0(\mathbf{y}), \mathbf{y}' \rangle = \sum_{\mathbf{y}_2, \mathbf{y}_4} \# \mathcal{M}_{\bar{J}_{-\infty,1}^{\diamond} (\varepsilon, \delta, \mathbf{p}(-\infty))}^{I=0, n^*=0} (\mathbf{y}, \mathbf{y}_2, \mathbf{y}', \bar{h}(\mathbf{y}_4)),$$

where $\mathbf{y}_2, \mathbf{y}_4$ are as in $A_3$. Write $\Theta_{\bar{b},\bar{a}}$ for the sum of all $\mathbf{y}_2$ from Lemma 3.3.5 and write $\Theta_{\bar{h}(\bar{a}),\bar{h}(\bar{b})}$ for the sum of all $\bar{h}(\mathbf{y}_4)$ from Lemma 3.3.5. Theorem 2.4.2 can be rephrased as (verification left to the reader):
**Theorem 3.3.6.** — Suppose $\mathbf{y}_4 = \{x_{ij}^{\#}(i)\}_{i=1}^{2g}$ and $\mathbf{y}_2 = \{x_{ik}(i)\}_{i=1}^{2g}$, where $j(i)$ is odd and $k(i)$ is even. Then

$$ \# \mathcal{M}_{\tilde{J}_{-\infty,2}^{\diamond}(\varepsilon, \delta, \mathbf{p}(-\infty))}^{I=2, n^*=m} (\mathbf{y}_4, \mathbf{y}_2; \bar{\mathbf{m}}(-\infty)) \equiv 1 \mod 2. $$

The argument of Claim 3.3.4 gives:

**Claim 3.3.7.** — For $\varepsilon, \delta > 0$ sufficiently small, the restriction of $\partial \mathcal{M}'$ to a neighborhood of $-\infty$ is $A_3$.

**Step 3 (Breaking in the middle).**

**Lemma 3.3.8.** — $\partial_{(-\infty, +\infty)} \mathcal{M} \subset A_4 \sqcup A_5$, where:

$$ A_4 = \coprod_{\mathbf{y}'' \in S_{a, \tilde{h}(a)}} \left( \mathcal{M}_{\tilde{J}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1} (\mathbf{y}, \mathbf{y}'') \times \mathcal{M}_{\tilde{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1, n^*=m} (\mathbf{y}'', \mathbf{y}'; \bar{\mathbf{m}}) \right); $$

$$ A_5 = \coprod_{\mathbf{y}''' \in S_{b, \tilde{h}(b)}} \left( \mathcal{M}_{\tilde{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1, n^*=m} (\mathbf{y}, \mathbf{y}'''; \bar{\mathbf{m}}) \times \mathcal{M}_{\tilde{J}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1} (\mathbf{y}''', \mathbf{y}') \right). $$

Lemma 3.3.8 will be proved in Section 3.6. Using the technique of [Li, Prop. A.1 and A.2], we can glue each of the pairs in $A_4$ and $A_5$. This gluing accounts for the term $\partial' \mathbf{H}' + \mathbf{H}' \partial$ in Equation (3.3.1), where the map $\mathbf{H}'$ is given by:

$$ \langle \mathbf{H}'(\mathbf{y}), \mathbf{y}' \rangle = \sum_{\tau \in \mathcal{T}_1 \cup \mathcal{T}_2} \# \mathcal{M}_{\tilde{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1, n^*=m} (\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}). $$

**Claim 3.3.9.** — For $\varepsilon, \delta > 0$ sufficiently small,

$$ \partial \mathcal{M}' = A_1 \cup \tilde{A}_2 \cup A_3 \cup A_4 \cup A_5. $$

**Step 4 (Additional degenerations).** In this step we give the necessary modifications for

$$ \mathcal{M} = \mathcal{M}_{\tilde{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=2, n^*=m+|\mathcal{I}|} (\mathbf{z}, \mathbf{y}'; \bar{\mathbf{m}}), $$

where $\mathbf{z} = \{z_{\infty,i}\}_{i \in \mathcal{I}} \cup \mathbf{y}, \mathbf{y}$ and $\mathbf{y}'$ are tuples in $S$, and $\mathcal{I} \subset \{1, \dots, 2g\}$ with $|\mathcal{I}| > 0$.

The following is proved in Section 3.7:

**Lemma 3.3.10.** — $\partial_{(+\infty)} \mathcal{M} \subset A_2'$, where

$$ A_2' = \coprod_{\delta_0 \mathbf{y}, \{z_\infty\} \cup \mathbf{y}''} \left( \mathcal{M}_{\tilde{J}_+^{\diamond}(\varepsilon, \delta, \mathbf{p}(+\infty))}^{I=1, n^*=m+|\mathcal{I}|-1, \delta_0, \dagger} (\mathbf{z}, \delta_0 \mathbf{y}) \right. \\ \left. \times \mathcal{M}_{\tilde{J}_-^{\diamond}(\varepsilon, \delta, \mathbf{p}(+\infty))}^{I=0, n^*=0} (\delta_0 \mathbf{y}, \{z_\infty\} \cup \mathbf{y}'') \right) $$
$$ \times \mathcal{M}_{\bar{j}}^{I=1, n^*=1} (\{z_{\infty}\} \cup \mathbf{y}''', \mathbf{y}') ), $$

if $\mathbf{y}' = \{x_i^j\} \times \mathbf{y}''$ for some $x_i^j$ and $A_2' = \emptyset$ otherwise. The disjoint union for $A_2'$ ranges over all $\delta_0 \mathbf{y}$ such that $\mathbf{y} \in \tilde{\mathcal{O}}_{2g-1}$ and all $\{z_{\infty}\} \cup \mathbf{y}''$ such that $\mathbf{y}'$ can be written as $\{x_i^j\} \cup \mathbf{y}''$ for some $x_i^j$. Here we have omitted the potential contributions of connector components and we are writing $x_i^0 := x_i$ and $x_i^1 := x_i'$.

If $\mathcal{I} \neq \emptyset$ (i.e., $|\mathcal{I}| \ge 1$), then $\tilde{\Phi}(\mathbf{z}) = 0$ by Lemma I.6.6.5 and we have $\Psi \circ \tilde{\Phi}(\mathbf{z}) = 0$; this is consistent with the analog of $A_1$ being empty. On the other hand, gluing the triples in $A_2'$ accounts for the term $\partial_1 \circ \tilde{V}$.

Next, the following is proved in Section 3.8:

**Lemma 3.3.11.** — $\partial_{[-\infty, \infty]} \mathcal{M} \subset A_3'$, where:

$$ A_3' = \coprod_{\mathbf{y}_2, \mathbf{y}_4} \left( \begin{aligned} & \mathcal{M}_{\tilde{j}^\diamond_{-\infty,1}(\varepsilon, \delta, \mathbf{p}(-\infty))}^{I=0, n^*=|\mathcal{I}|} (\mathbf{z}, \mathbf{y}_2, \mathbf{y}', \bar{h}(\mathbf{y}_4)) \\ & \times \mathcal{M}_{\tilde{j}^\diamond_{-\infty,2}(\varepsilon, \delta, \mathbf{p}(-\infty))}^{I=2, n^*=m} (\mathbf{y}_4, \mathbf{y}_2; \bar{\mathbf{m}}(-\infty)) \end{aligned} \right) $$

and the summation is over $\mathbf{y}_2$ and $\mathbf{y}_4$ as in Lemma 3.3.5.

We define:

$$ \langle \Theta'_0(\mathbf{z}), \mathbf{y}' \rangle = \sum_{\mathbf{y}_2, \mathbf{y}_4} \# \mathcal{M}_{\tilde{j}^\diamond_{-\infty,1}(\varepsilon, \delta, \mathbf{p}(-\infty))}^{I=0, n^*=|\mathcal{I}|} (\mathbf{z}, \mathbf{y}_2, \mathbf{y}', \bar{h}(\mathbf{y}_4)), $$

where $\mathbf{y}_2, \mathbf{y}_4$ are summands of $\Theta_{\bar{\mathbf{b}}, \bar{\mathbf{a}}}$ and $\Theta_{\bar{h}(\bar{\mathbf{a}}), \bar{h}(\bar{\mathbf{b}})}$.

The following is proved in Section 3.9. The corresponding gluing accounts for the term $\partial'H' + H'\tilde{\partial}$ in Equation (3.3.1) when $|\mathcal{I}| \ge 1$. Here the map $H'$ is given by:

$$ \langle \mathbf{H}'(\mathbf{z}), \mathbf{y}' \rangle = \sum_{\tau \in \mathcal{T}_1 \cup \mathcal{T}_2} \# \mathcal{M}_{\tilde{j}^\diamond_{\tau}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1, n^*=m+|\mathcal{I}|} (\mathbf{z}, \mathbf{y}'; \bar{\mathbf{m}}). $$

**Lemma 3.3.12.** — $\partial_{(-\infty, +\infty)} \mathcal{M} \subset A_4' \sqcup A_5'$, where:

$$ A_4' = \coprod_{\mathbf{z}'} \left( \mathcal{M}_{\bar{j}}^{I=1, n^* \le |\mathcal{I}|} (\mathbf{z}, \mathbf{z}') \times \mathcal{M}_{\tilde{j}^\diamond_{\tau}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1, n^* \le m+|\mathcal{I}|} (\mathbf{z}', \mathbf{y}'; \bar{\mathbf{m}}) \right); $$

$$ A_5' = \coprod_{\mathbf{y}''' \in \mathcal{S}_{\mathbf{b}, \tilde{h}(\mathbf{b})}} \left( \mathcal{M}_{\tilde{j}^\diamond_{\tau}(\varepsilon, \delta, \mathbf{p}(\tau))}^{I=1, n^*=m+|\mathcal{I}|} (\mathbf{z}, \mathbf{y}'''; \bar{\mathbf{m}}) \times \mathcal{M}_{\bar{j}}^{I=1} (\mathbf{y}''', \mathbf{y}') \right). $$

Moreover, if $\bar{v}_{1,1} \in \mathcal{M}_{\bar{j}}^{I=1, n^* \le |\mathcal{I}|} (\mathbf{z}, \mathbf{z}')$, then either (i) $\bar{v}_{1,1}^\sharp$ is a thin strip and $\bar{v}_{1,1}^\flat$ is a union of trivial strips, or (ii) $\bar{v}_{1,1}^\sharp = \emptyset$ and $\bar{v}_{1,1}^\flat$ has image in W.

As before, the analog of Claim 3.3.9 holds.
<figure><img src="image_8.png" /><figcaption>FIG. 7. — Degeneration of the base $B_{-\infty,1}$</figcaption></figure>

Step 5 (The map $\Theta_0$). The map $\Theta_0$ is defined as follows:

$$ \Theta_0 : \widetilde{\mathrm{CF}}(\mathbf{a}, \hbar(\mathbf{a})) \to \widehat{\mathrm{CF}}(\mathbf{b}, \hbar(\mathbf{b})), \\ \langle \Theta_0(\mathbf{z}), [\mathbf{y}'] \rangle = \sum_{\mathbf{y}_2, \mathbf{y}_4, \mathbf{y}' \in [\mathbf{y}'] } \# \mathcal{M}_{\bar{J}_{-\infty,1}}^{I=0, n^*=|\mathcal{I}|} (\mathbf{z}, \mathbf{y}_2, \mathbf{y}', \bar{h}(\mathbf{y}_4)). $$

where $\mathbf{y}_2, \mathbf{y}_4$ are summands of $\Theta_{\bar{b},\bar{a}}$ and $\Theta_{\bar{h}(\bar{a}),\bar{h}(\bar{b})}$. It is clear that $\Theta_0 = q \circ \Theta'_0$.

**Lemma 3.3.13.** — The map $\Theta_0$ induces an isomorphism on the level of homology.

*Proof of Lemma 3.3.13.* We degenerate the base $B_{-\infty,1}$ as given in Figure 7. Slightly more precisely, we take $B_{-\infty,1,\tau'}, \tau' \in [0, +\infty)$, which is obtained from

$$ \{-2 \le s \le 2\} \cup \{0 \le t \le 1, s \le 2\} \cup \{\tau' \le t \le \tau' + 1, s \ge -2\} \subset \mathbf{R}^2 = \mathbf{C} $$

by smoothing the corners and use the complex structure $j_{-\infty,1,\tau'}$ induced from the standard complex structure on $\mathbf{C}$; then $B_{-\infty,1,\tau'} = B_{-\infty,1}$ and $j_{-\infty,1,\tau'} = j_{-\infty,1}$. Let $\bar{W}_{-\infty,1,\tau'} = B_{-\infty,1,\tau'} \times \bar{S}$ and define the almost complex structures $\bar{J}_{-\infty,1,\tau'}$ on $\bar{W}_{-\infty,1,\tau'}$ in the same way as on $\bar{W}_{-\infty,1}$ with $j_{-\infty,1}$ replaced by $j_{-\infty,1,\tau'}$.

The 1-parameter family $(\bar{W}_{-\infty,1,\tau'}, \bar{J}_{-\infty,1,\tau'})$ induces a chain homotopy between $\Theta_0$ and the composition $\Theta_B \circ \Theta_T$, where:

$$ \begin{aligned} \Theta_T : \widetilde{\mathrm{CF}}(\mathbf{a}, \hbar(\mathbf{a})) &\to \widehat{\mathrm{CF}}(\mathbf{b}, \hbar(\mathbf{a})), \\ \Theta_B : \widehat{\mathrm{CF}}(\mathbf{b}, \hbar(\mathbf{a})) &\to \widehat{\mathrm{CF}}(\mathbf{b}, \hbar(\mathbf{b})) \end{aligned} $$

are defined by counting holomorphic multisections of $\Delta \times \bar{S}$, where $\Delta$ is a triangle (i.e., a disk with three boundary punctures), which are asymptotic to the top generators $\Theta_{\bar{b},\bar{a}} \in \widehat{\mathrm{CF}}(\bar{b}, \bar{a})$ and $\Theta_{\bar{h}(\bar{a}),\bar{h}(\bar{b})} \in \widehat{\mathrm{CF}}(\bar{h}(\bar{a}), \bar{h}(\bar{b}))$, respectively, at one of the vertices. Usually in the definitions of chain maps such as $\Theta_T$ and $\Theta_B$ we require $\Theta_{\bar{b},\bar{a}}$ and $\Theta_{\bar{h}(\bar{a}),\bar{h}(\bar{b})}$ to be cycles. In our case they are not, but we have workarounds. In the $\Theta_T$ case we must consider one undesirable type of breaking which can a priori occur as we vary $\tau'$: a two-level building $\bar{u}_1 \cup \bar{u}_2$, where

$$ - \bar{u}_1 \text{ is an index } I = -1, n^* = |\mathcal{I}| \text{ curve in } \bar{W}_{-\infty,1,\tau'} \text{ with ends } \mathbf{z}, \mathbf{y}'_2, \mathbf{y}', \bar{h}(\mathbf{y}_4); $$
$$- \bar{u}_2 \in \mathcal{M}_{\bar{J}_{-\infty,2}}^{1=1,n^*=0}(\mathbf{y}_2, \mathbf{y}'_2); \text{ and}$$

$$- \mathbf{y}_2 \text{ and } \mathbf{y}'_2 \text{ differ by replacing one } x_{ik(i)}^{\#} \text{ by } x_{ij(i)}^{\#} \text{ where } j(i) \text{ is odd and } k(i) \text{ is even.}$$

One can verify that the only possible component of $\bar{u}_1$ which has left end $x_{i1}^\#$ and which satisfies $n^* \le |\mathcal{I}|$, projects to the quadrilateral $\mathcal{Q}$ with edges $\bar{a}_i, \bar{b}_i, \bar{h}(\bar{b}_i), \bar{h}(\bar{a}_i)$ in Figure 6. However, the component corresponding to $\mathcal{Q}$ has ECH index $I=0$, which is a contradiction. The index calculation basically follows from the fact that $\mathcal{Q}$ has three angles smaller than $\pi$ and one larger. This implies that undesirable breakings do not exist and that all the breakings contribute toward the chain homotopy.

Both $\Theta_T$ and $\Theta_B$ — and hence $\Theta_0$ — induce isomorphisms on the level of homology. $\square$

This completes the proof of Theorem 3.3.1, assuming the results from Sections 3.4–3.9. $\square$

**3.4.** Degeneration at $+\infty$. — In this subsection we study the limits of holomorphic maps to $\bar{W}_\tau$ as $\tau \to \infty$, i.e., when $\bar{W}_\tau$ degenerates into the concatenation of $\bar{W}_+$ with $\bar{W}_-$ along the ECH-type end, in order to prove Lemma 3.3.2.

We assume that $m \gg 0$; $\varepsilon, \delta > 0$ are sufficiently small; and $\{\bar{J}_\tau\} \in \bar{\mathcal{I}}^{\text{reg}}$ and $\{\bar{J}_\tau^\diamond(\varepsilon, \delta, \mathbf{p}(\tau))\}$ satisfies Lemma 3.2.22. Fix $\mathbf{y} \in S_{\mathbf{a},\mathbf{h}(\mathbf{a})}$, $\mathbf{y}' \in S_{\mathbf{b},\mathbf{h}(\mathbf{b})}$ and let

$$\mathcal{M} = \mathcal{M}_{\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}}^{1=2,n^*=m}(\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}), \quad \mathcal{M}_{\tau} = \mathcal{M}_{\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}}^{1=2,n^*=m}(\mathbf{y}, \mathbf{y}'; \bar{\mathbf{m}}(\tau)).$$

We will analyze $\partial_{(+\infty]} \mathcal{M}$.

Let $\bar{u}_i$, $i \in \mathbf{N}$, be a sequence of curves in $\mathcal{M}$ such that $\bar{u}_i \in \mathcal{M}_{\tau_i}$ and $\lim_{i \to \infty} \tau_i = +\infty$, and let

$$(3.4.1) \qquad \bar{u}_{\infty} = (\bar{v}_{-1,1} \cup \dots \cup \bar{v}_{-1,c}) \cup \bar{v}_{-} \cup (\bar{v}_{0,1} \cup \dots \cup \bar{v}_{0,b}) \cup \bar{v}_{+} \cup (\bar{v}_{1,1} \cup \dots \cup \bar{v}_{1,a})$$

be the limit holomorphic building, where each $\bar{v}_*$ is an SFT level (recall Notation 1.0.3 regarding the use of subscripts *), the levels are ordered from bottom to top as we go from left to right, $\bar{v}_{-1,j}, j = 1, \dots, c$, maps to $\bar{W}$; $\bar{v}_{-}$ maps to $\bar{W}_{-}$; $\bar{v}_{0,j}, j = 1, \dots, b$, maps to $\bar{W}';$ $\bar{v}_{+}$ maps to $\bar{W}_{+}$; and $\bar{v}_{1,j}, j = 1, \dots, a$, maps to $\bar{W}$. Here we are allowing the possibility that $a, b$, or $c = 0$. For notational convenience, sometimes we will refer to $\bar{v}_{+}$ as $\bar{v}_{0,b+1}$ or $\bar{v}_{1,0}$ and $\bar{v}_{-}$ as $\bar{v}_{-1,c+1}$ or $\bar{v}_{0,0}$.

**Notation 3.4.1.** — We will be using the conventions established in Section 3.2.3 (and in particular Notation 3.2.10).

- We write $\dot{F}_*$, $\dot{F}'_*$, $\dot{F}''_*$ for the domains of $\bar{v}_*$, $\bar{v}'_*$, $\bar{v}''_*$.

- We write $p_*$ for the covering degree of $\bar{v}'_*$.
- If $\tilde{v}_{*1}$ is a union of components of a level $\bar{v}_{*1}$ and $\tilde{v}_{*2}$ is a union of components of a possibly different level $\bar{v}_{*2}$, then we write $\tilde{v}_{*1} > \tilde{v}_{*2}$ (resp. $\tilde{v}_{*1} \ge \tilde{v}_{*2}$) to indicate that the level $\bar{v}_{*1}$ is above (resp. equal to or above) the level $\bar{v}_{*2}$.

Since ghost components can be eliminated by the discussion in Lemma I.6.1.8 they will not be explicitly mentioned in the rest of the paper.

We have the following two constraints:

$$ (3.4.2) \qquad n^*(\bar{u}_i) = \sum_{\bar{v}_*} n^*(\bar{v}_*) = m; $$

$$ (3.4.3) \qquad I(\bar{u}_i) = \sum_{\bar{v}_*} I(\bar{v}_*) = 2, $$

where the summations are over all the levels $\bar{v}_*$ of $\bar{u}_\infty$.

*Outline of proof of Lemma 3.3.2.* The proof of Lemma 3.3.2 follows the same general outline of Sections I.7.4–I.7.11: First we calculate the contributions to $n^*$ of the ends that limit to multiples of $z_\infty$ or $\delta_0$ in Section 3.4.1 and obtain lower bounds on the ECH indices of the levels $\bar{v}_*$ in Section 3.4.2, under the assumption that there are no boundary points at $z_\infty$. Boundary points at $z_\infty$ are treated in Sections 3.4.3 and 3.4.4. The main new difficulty is to show that $I(\bar{v}_*) \ge I(\bar{v}_*)' + I(\bar{v}_*)''$ for $\bar{v}_* \ge \bar{v}_+$; this uses the more complicated version of the ECH index inequality given in Lemma I.5.7.21. We then use Equations (3.4.2) and (3.4.3) to obtain Lemma 3.4.21, which describes the case when $\bar{v}_*' \cup \bar{v}_*'' = \emptyset$ for all levels $\bar{v}_*$, and Lemma 3.4.25, which gives a preliminary list when $\bar{v}_*' \cup \bar{v}_*'' \ne \emptyset$ for some $\bar{v}_*$. The renormalization argument from Sections I.7.8–7.10, given in Lemma 3.4.28, eliminates all the possibilities with the exception of Case (2$_1$) of Lemma 3.4.25 when $\bar{v}_*' \cup \bar{v}_*'' \ne \emptyset$ for some $\bar{v}_*$.

*At this point the reader is strongly encouraged to review Section I.5.7 on holomorphic curves with ends at $z_\infty$.*

**3.4.1. Intersection numbers.** — In this subsection we give the analogs of Lemmas I.7.4.1–I.7.4.5 for $\bar{u}_\infty \in \partial_{(+\infty)} \mathcal{M}$:

**Lemma 3.4.2.**

(1) If $\bar{v}_*''$ has a negative end $\mathcal{E}_-$ that converges to $\delta_0^p$, then $n^*(\mathcal{E}_-) \ge m - p$.

(2) If $\bar{v}_*''$ has a positive end $\mathcal{E}_+$ that converges to $\delta_0^p$, then $n^*(\mathcal{E}_+) \ge p$.

*Proof.* — This is analogous to Lemma I.7.4.1 and is proved in the same way. $\square$

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

**Definition 3.4.3.** — An end $\mathcal{E}$ is nontrivial if it is not an end of a trivial cylinder or trivial strip.
*Convention.* In this paper we assume that an end of a holomorphic curve is connected, unless stated otherwise.

Recall the sequence $\bar{u}_i \in \mathcal{M}_{\tau_i}$ with $\tau_i \to +\infty$ and its limit SFT building $\bar{u}_\infty$ given by Equation (3.4.1). Let $\dot{G}_i$ be the domain of $\bar{u}_i$. Fix $k \in \{1, \dots, 2g\}$. The following Lemma 3.4.4 describes the breaking/"SFT limit" of the component of $\bar{u}_i|_{\partial \dot{G}_i}$ which maps to $L_{\bar{a}_k}^{\tau_i, +}$, into a sequence of paths $g_a^1, \dots, g_1^1, g_0, g_1^0, \dots, g_a^0, g_j^* \subset \text{Im}(\bar{v}_{1,j}), j \ge 0, * = 0, 1, \emptyset$, as $\tau_i \to \infty$.

Let $\partial_+ B_\tau$ (resp. $\partial_+ \bar{W}_\tau$) be the $s > 0$ boundary of $B_\tau$ (resp. $\bar{W}_\tau$) and let $C_{1,j}, j \ge 0$, be the data for $\bar{v}_{1,j}$; see Section I.5.7.2 for the definition of the data $C_{1,j}$. Then we have the following, whose proof is immediate.

**Lemma 3.4.4.** — Given a sequence $\bar{u}_i$ and a choice of $k \in \{1, \dots, 2g\}$, the components of $\bar{u}_i|_{\partial \dot{G}_i}$ which map to $L_{\bar{a}_k}^{\tau_i, +}$ converge uniquely (in the "SFT sense") to a sequence of paths

$$ (3.4.4) \qquad g_a^1, \dots, g_1^1, g_0, g_1^0, \dots, g_a^0 $$

which satisfies the following:

(1) $g_j^* = \bar{v}_{1,j}(f_j^*)$ where $j \ge 0, * = 0, 1, \emptyset$, and $f_j^*$ is a (connected) component of $\partial \dot{F}_{1,j}$, where $\dot{F}_*$ is as given in Notation 3.4.1.

(2) If $f_j^*$ is a component of $\partial \dot{F}_{1,j}''$, then $g_j^1 \subset \mathbf{R} \times \{1\} \times \bar{a}_k$ and $g_j^0 \subset \mathbf{R} \times \{0\} \times \bar{h}(\bar{a}_k)$ if $j > 0$ and $g_0 \subset L_{\bar{a}_k}^+$ if $j = 0$.

(3) If $f_j^*$ is a component of $\partial \dot{F}_{1,j}'$, then $f_j^*$ and also $g_j^*$ come with extra data $C_{1,j}$ which assigns:

$$ f_j^1, g_j^1 \mapsto L_{\bar{a}_{k,l}} = \mathbf{R} \times \{1\} \times \bar{a}_{k,l}, $$

$$ f_j^0, g_j^0 \mapsto L_{\bar{h}(\bar{a}_{k,l}')} = \mathbf{R} \times \{0\} \times \bar{h}(\bar{a}_{k,l}), $$

$$ f_0, g_0 \mapsto L_{\bar{a}_{k,l}''}^+ $$

for some $l, l', l'' = 0$ or $1$.

For convenience we write $g_0 = g_0^1 = g_0^0$. We say an element $g_j^*$ is *trivial* if the corresponding $f_j^*$ satisfies (3).

**Definition 3.4.5 (Continuations).** — If $g_j^*$ is any element of Sequence (3.4.4), then Sequence (3.4.4) is the continuation of $g_j^*$ along $\partial_+ B_\tau$ and the terms to the right of $g_j^*$ in Sequence (3.4.4) form the continuation of $g_j^*$ in the direction of $\partial_+ B_\tau$.

Lemma 3.4.4 and Definition 3.4.5 play an important role in the proof of Lemma 3.4.7, which is a refined version of Lemma I.7.4.2, where the ends are considered collectively as well as individually. The proof strategy will usually be referred to as the *continuation argument*.
Let $\rho_0 > 0$ be small and let $\pi_{D_{\rho_0}^2}$ be the projection of

(3.4.5) $\mathfrak{D}_1 := \{\rho \le \rho_0\} \subset \bar{W} - \text{int}(W)$ or $\mathfrak{D}_2 := \{\rho \le \rho_0, |s| \ge l(\tau) + 1\} \subset \bar{W}_{\tau} - \text{int}(W_{\tau})$ to $D_{\rho_0}^2 = \{(\rho, \phi) | \rho \le \rho_0\} \subset \bar{S}$ along the stable Hamiltonian vector field $\bar{R}_{\tau}$ which was defined in Section 3.1.5.

**Remark 3.4.6.** — $\pi_{D_{\rho_0}^2}$ maps the intersection of $\mathbf{R} \times \delta_{\rho_0, \phi_0^+}$ (which appears in the definition of $n^*$ in Equation (3.2.3)) and one of $\mathfrak{D}_1$ or $\mathfrak{D}_2$ to $m$ equally spaced points on $\partial D_{\rho_0}^2$.

**Lemma 3.4.7.** — Suppose $\bar{v}'_{1,j} \cup \bar{v}^\sharp_{1,j} \neq \emptyset$ for some $j > 0$. Let $\mathcal{E}_{-,i}$, $i = 1, \dots, q$, be the negative ends of $\cup_{j=1}^a \bar{v}^\sharp_{1,j}$ that converge to $z_\infty$ and let $\mathcal{E}_{+,i}$, $i = 1, \dots, r$, be the positive ends of $\cup_{j=0}^{a-1} \bar{v}^\sharp_{1,j}$ that converge to $z_\infty$.

(1) For each $i$,

(3.4.6) \qquad $n^*(\mathcal{E}_{-,i}) \ge k_0 - 1 \gg 2g,$

where the constant $k_0$ is as given in Section I.5.2.2.

(2) If there are no boundary points at $z_\infty$, then

(3.4.7) \qquad $n^*((\cup_{i=1}^q \mathcal{E}_{-,i}) \cup (\cup_{i=1}^r \mathcal{E}_{+,i})) \ge m - p_+$,

where $p_+ = \deg(\bar{v}'_+)$, and

$$D_{\rho_0}^2 - (\cup_{i=1}^q \pi_{D_{\rho_0}^2}(\mathcal{E}_{-,i})) \cup (\cup_{i=1}^r \pi_{D_{\rho_0}^2}(\mathcal{E}_{+,i}))$$

consists of at most $p_+$ thin sectors.

Recall that a sector $\mathfrak{S}$ of $D_{\rho_0}^2$ from $\phi_0$ to $\phi_1$ is the map

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

or its image (by abuse of notation). A thin sector (also referred to as a thin wedge in [I]) is the smallest counterclockwise sector from $\bar{a}_{i,j} \cap D_{\rho_0}^2$ to $\bar{h}(\bar{a}_{i,j}) \cap D_{\rho_0}^2$ for some $i, j$ and has angle $\frac{2\pi}{m}$.

Following Definition I.7.4.4, a point $p \in \partial \dot{F}_*''$ is a boundary point at $z_\infty$ if $\bar{v}_*(p) \in (L_{\bar{h}(\bar{a})} - L_{\bar{h}(\hat{a})}) \cup (L_{\bar{a}} - L_{\hat{a}}) = \mathbf{R} \times \{0, 1\} \times \{z_\infty\}$ for $*= (1, j), j > 0$, and $\bar{v}_*(p) \in L_{\bar{a}}^+ - L_{\hat{a}}^+$ for $*= +$.

**Proof.** — We consider the $\pi_{D_{\rho_0}^2}$-projections of the positive and negative ends of $\bar{v}_{1,j}$, $j > 0$, and the positive ends of $\bar{v}_{1,0}$ that limit to $z_\infty$.
(1) Given a nontrivial negative end $\mathcal{E}_{-,i}$ limiting to $z_\infty$, its projection can be written as:

$$ \pi_{D_{\rho_0}^2}(\mathcal{E}_{-,i}) = \mathfrak{S}(\bar{h}(\bar{a}_{k,l}), \bar{a}_{k',l'}), $$

where $\mathfrak{S}(A, B)$ denotes the smallest counterclockwise sector from the radial ray A to the radial ray B. Then Equation (3.4.6) follows from Remark 3.4.6 and the fact that the angle between $\bar{h}(\bar{a}_{k,l})$ and $\bar{a}_{k',l'}$ is at least $\frac{2\pi(k_0-1)}{m}$ as defined in Section I.5.2.2. (This is more or less the same as Lemma I.7.4.2(2); in fact "$> 2g$" in the statement can be replaced by "$\ge k_0 - 1 \gg 2g$".)

(2) We start at a nontrivial negative end of some $\bar{v}_{1,j_1}$ limiting to $z_\infty$, which we call $\mathcal{E}_{-,1}$ without loss of generality. Then

$$ \pi_{D_{\rho_0}^2}(\mathcal{E}_{-,1}) = \mathfrak{S}(\bar{h}(\bar{a}_{k_1,l_1}), \bar{a}_{k_2,l_2}). $$

We analyze the continuation

$$ (3.4.8) \qquad g_{j_1-1,1}^1, \dots, g_{1,1}^1, g_{0,1}, g_{1,1}^0, \dots, g_{a,1}^0 $$

of $g_{j_1,1}^1 \supset \partial_1 \mathcal{E}_{-,1}$ in the direction of $\partial_+ B_\tau$. Here $\partial_k \mathcal{E}_{-,k}$, $k=0,1$, is the $t=k$ boundary of $\mathcal{E}_{-,k}$, and $f_{*2}^{*1}$ corresponds to $g_{*2}^{*1}$ as in Definition 3.4.5.

(i) Suppose there is some $0 \le j < j_1$ such that $g_{j_1,1}^1$ is nontrivial. Let $j_2 \ge 0$ be the first such occurrence in the continuation. Then $\bar{v}_{1,j_2}^\sharp$ has some nontrivial end which we call $\mathcal{E}_{+,1}$, such that

$$ \pi_{D_{\rho_0}^2}(\mathcal{E}_{+,1}) = \mathfrak{S}(\bar{a}_{k_2,l_2}, \bar{h}(\bar{a}_{k_3,l_3})). $$

This is because all the terms of Equation (3.4.8) are assigned $L_{\bar{a}_{k_2}}$ or $L_{\bar{a}_{k_2}}^+$ and all the terms between $g_{j_1,1}^1$ and $g_{j_2,1}^1$ correspond to the same $L_{\bar{a}_{k_2,l_2}}$.

(ii) On the other hand, if $g_{j_1,1}^1$ is trivial for all $0 \le j \le j_1$, then we set $j_2 = 0$ and $\bar{h}(\bar{a}_{k_2,l_2}) = \bar{h}(\bar{a}_{k_3,l_3})$ and skip $\mathfrak{S}(\bar{a}_{k_2,l_2}, \bar{h}(\bar{a}_{k_3,l_3}))$, which is a thin sector.

Next we consider the continuation

$$ g_{j_2+1,2}^0, \dots, g_{a,2}^0 $$

of $g_{j_2,2}^0$ in the direction of $\partial_+ B_\tau$. Here $g_{j_2,2}^0 \supset \partial_0 \mathcal{E}_{+,1}$ if (i) holds; and $j_2 = 0, g_{0,2} = g_{0,1}$, and $f_{0,2} = f_{0,1}$ if (ii) holds. There must exist some nontrivial $g_{j_2,2}^0, j > j_2$, and we write $j_3$ for the first such occurrence in the continuation. Then $\bar{v}_{1,j_3}^\sharp$ has some nontrivial end which we call $\mathcal{E}_{-,2}$, such that

$$ \pi_{D_{\rho_0}^2}(\mathcal{E}_{-,2}) = \mathfrak{S}(\bar{h}(\bar{a}_{k_3,l_3}), \bar{a}_{k_4,l_4}). $$
Continuing in the same manner, we eventually return to $\mathcal{E}_{-,1}$, and the sectors

$$ (3.4.9) \qquad \mathcal{S}(\bar{h}(\bar{a}_{k_1,l_1}), \bar{a}_{k_2,l_2}), \mathcal{S}(\bar{a}_{k_2,l_2}, \bar{h}(\bar{a}_{k_3,l_3})), \mathcal{S}(\bar{h}(\bar{a}_{k_3,l_3}), \bar{a}_{k_4,l_4}), \dots, $$
with some thin sectors of type $\mathcal{S}(\bar{a}_{k_2i,l_2i}, \bar{h}(\bar{a}_{k_2i+1,l_2i+1}))$ omitted if they correspond to (ii), sweep out a neighborhood of $z_\infty$ in $D_{\rho_0}^2$, possibly more than once and with the possible exception of $p_+$ thin sectors. This proves (2). $\square$

We remark that, a posteriori, the sectors from Equation (3.4.9) sweep out a neighborhood of $z_\infty$ only once in view of Equation (3.4.7).

**Lemma 3.4.8.** — If $\mathfrak{p} \in \partial F_*''$ is a boundary point of $\bar{v}_*''$ at $z_\infty$, then $n^*(\bar{v}_*''(\mathfrak{N}(\mathfrak{p}))) \ge k_0 - 1 \gg 2g$, where $\mathfrak{N}(\mathfrak{p}) \subset F_*''$ is a sufficiently small neighborhood of $\mathfrak{p}$.

*Proof.* — The proof is the same as that of Lemma I.7.4.5. $\square$

**3.4.2. Bounds on ECH indices.** — The goal of this subsection is to show the nonnegativity of $I(\bar{v}_*)$ except when $\bar{v}_* = \bar{v}_+$, under the assumption that there are no boundary points at $z_\infty$; see Lemma 3.4.14. The main new difficulty is to show that $I(\bar{v}_*) \ge 0$ for $\bar{v}_* > \bar{v}_+$. If $\bar{v}_* > \bar{v}_+$ and $\bar{v}_*' \cup \bar{v}_*^2 \ne \emptyset$, then we need to apply the version of the ECH index inequality in the presence of ends that limit to $z_\infty$ (Lemma I.5.7.21). To apply Lemma I.5.7.21, we need to verify a certain “alternating property” for the ends of $\bar{v}_*$ that limit to $z_\infty$; this is the content of Lemma 3.4.13.

**Remark 3.4.9.** — The ECH index in the presence of ends that limit to $z_\infty$ was given in Definition I.5.7.3. Although the definition of the ECH index involves a groomed multivalued trivialization $\tau$ (Definition I.5.7.5), by Lemma I.5.7.20 it is independent of the choice of groomed multivalued trivialization and we may write $I_\tau(\bar{v}_*)$ or $I(\bar{v}_*)$.

Let $A_\epsilon = \partial D_\epsilon^2 \times [0, 1]$ for $0 < \epsilon < \rho_0$ small and let $\pi_{[0,1]\times\bar{S}}$ be the projection of $\bar{W}$ or the positive end of $\bar{W}_\tau$ to $[0, 1] \times \bar{S}$. The following is a corollary of the proof of Lemma 3.4.7:

**Corollary 3.4.10.** — The intersection $\mathcal{C} := \pi_{[0,1]\times\bar{S}}(\cup_i \mathcal{E}_{-,i}) \cap A_\epsilon$ is groomed and the sets $P_0$ and $P_1$ of initial and terminal points of $\mathcal{C}$ alternate along $(0, 2\pi)$.

*Note that we will often view $P_0$ and $P_1$ (and analogous defined points on $A_\epsilon$) as subsets of $\partial D_\epsilon^2$.*

**Definition 3.4.11.** — A cycle $\mathcal{Z} = (\mathfrak{z}_1 \to \mathfrak{z}_2 \to \dots \to \mathfrak{z}_k \to \mathfrak{z}_1)$ on $\partial D_\epsilon^2 = \mathbf{R}/2\pi\mathbf{Z}$ is a sequence of points $\mathfrak{z}_i \in \mathbf{R}/2\pi\mathbf{Z}$, together with chords in $\mathbf{R}/2\pi\mathbf{Z}$ from $\mathfrak{z}_i$ to $\mathfrak{z}_{i+1}$,<fnref n="9" /> where the indices are taken modulo $k$.

<footnote n="9">⁹ By “chord” we mean a path with positive derivative in the $\mathbf{R}$-direction.</footnote>
The continuation method from Lemma 3.4.7 gives a cycle

$$ \mathcal{Z} = (\mathfrak{z}_{10} \to \mathfrak{z}_{11} \to \dots \to \mathfrak{z}_{k0} \to \mathfrak{z}_{k1} \to \mathfrak{z}_{10}), $$

satisfying the following:

(1) the cycle winds around $\mathbf{R}/2\pi\mathbf{Z}$ once;

(2) $P_i = \{\mathfrak{z}_{1i}, \dots, \mathfrak{z}_{ki}\}$ for $i=0, 1$ and each point of $P_i$ appears only once in $\mathcal{Z}$; and

(3) the chords correspond to the sectors listed in (3.4.9) as well as thin sectors of type $\mathcal{S}(\bar{a}_{k',l}, \bar{h}(\bar{a}_{k',l}))$ that are skipped in Step (ii) of Lemma 3.4.7.

Next let $\vec{D}_{\pm,j}$ be the data at $z_\infty$ for the $\pm$ end of $\bar{v}_{1,j}$ and let $P_{\pm,j,0}$ and $P_{\pm,j,1}$ be the initial and terminal points on $A_\epsilon$ determined by $\vec{D}_{\pm,j}$. Then we write

$$ (3.4.10) \qquad P_{\pm,j,i} = P'_{\pm,j,i} \sqcup P''_{\pm,j,i}, $$

where $P'_{\pm,j,i}$ corresponds to $\bar{v}'_{1,j}$ and $P''_{\pm,j,i}$ corresponds to $\bar{v}''_{1,j}$. Note that

$$ (3.4.11) \qquad P_{+j-1,i} = P_{-j,i} \quad \text{and} \quad P'_{+j-1,i} = P'_{-j-1,i}. $$

**Definition 3.4.12.** — Let $P$ be a finite subset of $S^1 = \mathbf{R}/2\pi\mathbf{Z}$. Denoting an element of $S^1$ by an equivalence class $[c]$, where $c \in \mathbf{R}$, if $[a] \neq [b] \in P$, then we write $[a] \ll_P [b]$ if there exist $a' \in [a]$, $b' \in [b]$ such that $a' < b' < a' + 2\pi$ and there are no representatives of $P$ in the open interval $(a', b')$.

Observe that the relation $\ll_P$ is not symmetric, i.e., $[a] \ll_P [b]$ does not necessarily imply that $[b] \ll_P [a]$.

**Lemma 3.4.13.** — For each $* = (\pm, j)$, $P_{*,0} \subset P_0$ and $P_{*,1} \subset P_1$ and the points of $P_{*,0}$ and $P_{*,1}$ alternate along $(0, 2\pi)$.

**Proof.** — The proof is by induction on the level; see Figure 8 for an example. We will inductively define $P_i^{(0)} \supset P_i^{(1)} \supset P_i^{(2)} \supset \dots$, $i = 0, 1$, and the corresponding cycles $\mathcal{Z}^{(0)}, \mathcal{Z}^{(1)}, \dots$ and show that the following hold for each $j = 0, 1, \dots$:

(j0) the points of $P_0^{(j)}$ and $P_1^{(j)}$ alternate along $(0, 2\pi)$;

(j1) $P_{-j_0-j,i} = P_{+j_0-j-1,i} \subset P_i^{(j)}$, $i = 0, 1$;

(j2) there is a partition of $P_{-j_0-j,0} \cup P_{-j_0-j,1}$ into pairs of type $\{\mathfrak{p}_0, \mathfrak{p}_1\}$, $\mathfrak{p}_i \in P_{-j_0-j,i}$, such that $\mathfrak{p}_0 \ll_{P_0^{(j)} \cup P_1^{(j)}} \mathfrak{p}_1$; in particular, the points of $P_{-j_0-j,0}$ and $P_{-j_0-j,1}$ alternate along $(0, 2\pi)$;

(j3) $P'_{+j_0-j-1,i} = P'_{-j_0-j-1,i} \subset P_i^{(j+1)}$, $i = 0, 1$;

(j4) there is a partition of $P'_{+j_0-j-1,0} \cup P'_{+j_0-j-1,1}$ into pairs of type $\{\mathfrak{p}_0, \mathfrak{p}_1\}$, $\mathfrak{p}_i \in P'_{+j_0-j-1,i}$, such that $\mathfrak{p}_0 \ll_{P_0^{(j+1)} \cup P_1^{(j+1)}} \mathfrak{p}_1$; in particular, the points of $P'_{+j_0-j-1,0}$ and $P'_{+j_0-j-1,1}$ alternate along $(0, 2\pi)$.
<figure><img src="image_10.png" /><figcaption>FIG. 8. — Each rectangular box, with the sides identified, is $A_s$; the top is $\partial D_s^2 \times \{1\}$ and the bottom is $\partial D_s^2 \times \{0\}$. For each figure, $P_{s,0}^*$ and $P_{s,1}^*$ are the sets of initial and terminal points of the arcs drawn, where the arcs are always oriented from bottom to top. The extra dots are drawn only for reference. The $k$th row corresponds to Step $k$. The solid and dashed arcs in the left column together indicate $\mathcal{Z}^{(0)}$ and $\mathcal{Z}^{(1)}$. The dashed arcs are potential arcs whose endpoints give pairs of points in $P_{s,j_0}''$ and $P_{s,j_1}'$.</figcaption></figure>

Note that $P_{-,j_0-j,i} = P_{+,j_0-j-1,i}$ in (j1) and $P'_{+,j_0-j-1,i} = P'_{-,j_0-j-1,i}$ in (j3) by Equation (3.4.11).

*Step 1.* Let $\bar{v}_{1,j_0}$ be the highest level which has a negative end at $z_\infty$, let $P_i^{(0)} = P_i$, $i = 0, 1$, and let $\mathcal{Z}^{(0)} = \mathcal{Z}$. The construction of $\mathcal{Z}^{(0)}$ (following the proof of Lemma 3.4.7) immediately implies (00)–(02).

*Step 2.* We consider the positive ends of $\bar{v}_{1,j_0-1}$ that limit to $z_\infty$. There is a partition of $P''_{+,j_0-1,0} \cup P''_{+,j_0-1,1}$ into pairs of type $\{\mathfrak{p}_0, \mathfrak{p}_1\}$, $\mathfrak{p}_i \in P''_{+,j_0-1,i}$, such that $\mathfrak{p}_1 \subset P_0^{(0)} \cup P_1^{(0)}$, i.e., there is a chord $\mathfrak{p}_1 \to \mathfrak{p}_0$ in $\mathcal{Z}^{(0)}$. Let $P_i^{(1)} = P_i^{(0)} - P''_{+,j_0-1,i}$ and let $\mathcal{Z}^{(1)}$ be obtained from $\mathcal{Z}^{(0)}$ by inductively replacing $\mathfrak{q} \to \mathfrak{p}_1 \to \mathfrak{p}_0 \to \mathfrak{q}'$ by $\mathfrak{q} \to \mathfrak{q}'$, given by concatenation. This makes sense since each point of $P_i^{(0)}$ appears only once in $\mathcal{Z}^{(0)}$. Then $P_0^{(1)} \cup P_1^{(1)}$ is the set of (alternating) points of $\mathcal{Z}^{(1)}$, each point of $P_i^{(1)}$ appears only once in $\mathcal{Z}^{(1)}$, and $\mathcal{Z}^{(1)}$ winds around $\mathbf{R}/2\pi\mathbf{Z}$ once. Hence (10) follows from the description of $\mathcal{Z}^{(1)}$. Similarly, since $P'_{+,j_0-1,i} = P_{+,j_0-1,i} - P''_{+,j_0-1,i}$ and (02) holds, (03) and (04) follow immediately.

*Step 3.* We consider the negative ends of $\bar{v}_{1,j_0-1}$ that limit to $z_\infty$. There is a partition of $P''_{-,j_0-1,0} \cup P''_{-,j_0-1,1}$ into pairs of type $\{\mathfrak{p}_0, \mathfrak{p}_1\}$, $\mathfrak{p}_i \in P''_{-,j_0-1,i}$, such that $\mathfrak{p}_0 \subset P_0^{(1)} \cup P_1^{(1)}$, i.e., there is a chord $\mathfrak{p}_0 \to \mathfrak{p}_1$ in $\mathcal{Z}^{(1)}$. Since $P_{-,j_0-1,i} = P'_{-,j_0-1,i} \cup P''_{-,j_0-1,i}$ and (04) holds, (11) and (12) follow immediately.

Repeated application of the above then implies the lemma. $\square$

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

*Lemma 3.4.14.* — If fiber components are removed from $\bar{u}_\infty$ and there are no boundary points at $z_\infty$, then:
(1) the only components of $\bar{v}'_*$ with negative ECH index are the branched covers of $\sigma_\infty^+$;

(2) the ECH index of each level $\bar{v}_* \neq \bar{v}_+$ is nonnegative; and

(3) $I(\bar{v}_{1,j}) \ge I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j})$ for $0 \le j \le a$, with equality for $j > 0$.

*Proof.* — The proof is analogous to that of Lemma I.7.5.5.

(1) If a component $\tilde{v}$ of $\bar{u}_\infty$ is a branched cover of some $\sigma_\infty^+$ with possibly empty branch locus, then, by Lemmas I.5.7.15 and I.5.7.16, $I(\tilde{v}) = 0$ with the exception of $I(\tilde{v}) = -\deg(\tilde{v})$ if $\tilde{v}$ covers $\sigma_\infty^+$. For all other components $\tilde{v}$ (i.e., those that do not branch cover any $\sigma_\infty^+$), the regularity of $\{\bar{J}_\tau\}$ and the ECH-type index inequalities imply that $I(\tilde{v}) \ge 0$.

(2) for $\bar{v}_{0,j}$, $1 \le j \le b$. By [HT1, Proposition 7.15(a)] and the regularity of $\{\bar{J}_\tau\}$, we have $I(\bar{v}_{0,j}) \ge 0$ for $1 \le j \le b$, where equality holds if and only if $\bar{v}_{0,j}$ is a connector.

(2) and (3) for $\bar{v}_{1,j}$. We claim that $I(\bar{v}_{1,j}) \ge 0$ for $0 < j \le a$ and $I(\bar{v}_{1,j}) \ge I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j})$ for $0 \le j \le a$, with equality for $j > 0$.

Suppose $z_\infty$ does not appear at an end of $\bar{v}_{1,j}$. With the possible exception of fiber components, $\bar{v}_{1,j}$ is simply-covered and regular since there is at least one HF end. Hence the claim follows from the regularity of $\{\bar{J}_\tau\}$ and the usual index inequality (Lemmas I.4.5.13 and I.5.6.9).

Suppose $z_\infty$ appears at an end of $\bar{v}_{1,j}$. Let $c_{\pm,j}$ (resp. $c'_{\pm,j}$, $c''_{\pm,j}$) be the groomings corresponding to the $\pm$ ends of $\bar{v}_{1,j}$ (resp. $\bar{v}'_{1,j}$, $\bar{v}''_{1,j}$) at $z_\infty$, such that:

$$ (\alpha_1) \quad c_{\pm,j} = c'_{\pm,j} \cup c''_{\pm,j}; $$

$$ (\alpha_2) \quad c_{-j} \text{ has winding } w(c_{-j}) \ge 0 \text{ and } c_{+j} \text{ has winding } w(c_{+j}) \le 0; $$

$$ (\alpha_3) \quad c''_{-j} = \pi_{[0,1]\times\bar{S}}(\mathcal{E}_-^j) \cap A_\epsilon \text{ and } c'_{+,j} = \pi_{[0,1]\times\bar{S}}(\mathcal{E}_+^j) \cap A_\epsilon, \text{ where } \mathcal{E}_+^j \text{ (resp. } \mathcal{E}_-^j) \text{ is the set of positive (resp. negative) ends of } \bar{v}_{1,j}^\sharp \text{ converging to } z_\infty. $$

Recall that the matchings defined by $c_{\pm,j}$ do not need to coincide with the matchings in the data $\overrightarrow{D}_{\pm,j}$ at $z_\infty$ for the $\pm$ ends of $\bar{v}_{1,j}$; only their endpoints do. What $(\alpha_3)$ says is that we are requiring $c''_{\pm,j}$ to coincide with the data at $z_\infty$ for $\bar{v}''_{1,j}$. On the other hand, in view of $(\alpha_1)$ the data at $z_\infty$ for $\bar{v}'_{1,j}$ does not necessarily coincide with $c'_{\pm,j}$.

By Lemma 3.4.13, the sets $P_{\pm,j,0}$ and $P_{\pm,j,1}$ of initial and terminal points of $c_{\pm,j}$ alternate along $(0, 2\pi)$. Hence, by the version of the ECH index inequality given by Lemma I.5.7.21,

$$ (3.4.12) \quad I(\bar{v}''_{1,j}) \ge \mathrm{ind}(\bar{v}''_{1,j}) \ge 0. $$

Also, by $(\alpha_1)$ and an easy unwinding of the definition of $I(\bar{v}_{1,j})$,

$$ (3.4.13) \quad I(\bar{v}_{1,j}) \ge I(\bar{v}'_{1,j}) + I(\bar{v}''_{1,j}) \quad \text{for } j \ge 0; $$

moreover, equality holds for $j > 0$ since $\bar{v}'_{1,j}$ and $\bar{v}''_{1,j}$ do not intersect (if they did, then $\sum_{\bar{v}_*} n^*(\bar{v}_*) > m$). Finally, $I(\bar{v}'_{1,j}) = 0$ for $j > 0$ by Lemma I.5.7.15; together with Equations (3.4.12) and (3.4.13), this gives $I(\bar{v}_{1,j}) \ge 0$ for $j > 0$. This proves the claim.
(2) and (3) for $\bar{v}_{-1,j}$ and $\bar{v}_-$. The case of $\bar{v}_{-1,j}$, $1 \le j \le c+1$, (this includes $\bar{v}_-$) is
similar, and the lemma follows.
□

**3.4.3. Boundary points at $z_\infty$.** — In this subsection we describe the necessary modifications when $\bar{u}_\infty$ has boundary points at $z_\infty$.

Let us suppose the following:

(S) There is only one boundary point $\mathbf{r} \in F''_{*0}$ at $z_\infty$ and $\bar{v}''_{*0}(\mathbf{r}) \in \{(0,1)\} \times \bar{\mathbf{a}} \subset \bar{W} = \mathbf{R} \times [0,1] \times \bar{S}$.

We are assuming (S) for notational simplicity; the general case of multiple boundary
points at $z_\infty$ is treated in exactly the same way (except for more complicated notation).

For the purposes of computing the Fredholm and ECH indices, we make the fol-
lowing modifications which allow us to use the considerations from Sections 3.4.1, 3.4.2,
and I.5.7. We view the base $\mathbf{B} = \mathbf{R} \times [0, 1]$ with the preferred point $(0, 1)$ as a two-
level building consisting of a disk $B_1^\circ$ with three boundary punctures and a disk $B_2^\circ$ with
one boundary puncture. (What we are doing here is bubbling off a neighborhood of the
point $(0, 1) \in \mathbf{B}$ when taking the limit $\bar{u}_i \to \bar{u}_\infty$.) All the punctures are viewed as strip-like
ends: the punctures of $B_1^\circ$ are called the positive, left negative, and right negative ends,
corresponding to the positive end, $(0, 1)$, and the negative end of $\mathbf{B}$, and the puncture
of $B_2^\circ$ is called the positive end and is identified with the left negative end of $B_1^\circ$. See
Figure 9. We write $\partial B_1^\circ = \bigsqcup_{i=1}^3 \partial_i B_1^\circ$, arranged in counterclockwise order, such that $\partial_1 B_1^\circ$,
$\partial_2 B_1^\circ$, and $\partial B_2^\circ$ correspond to $\mathbf{R} \times \{1\}$ and $\partial_3 B_1^\circ$ corresponds to $\mathbf{R} \times \{0\}$. The cobordism
$\bar{W} = \mathbf{B} \times \bar{S}$ decomposes into $\bar{W}_1^\circ = B_1^\circ \times \bar{S}$ and $\bar{W}_2^\circ = B_2^\circ \times \bar{S}$, the Lagrangian submanifold
$\mathbf{R} \times \{1\} \times \bar{\mathbf{a}}$ decomposes into $\partial_1 B_1^\circ \times \bar{\mathbf{a}}$, $\partial B_2^\circ \times \bar{\mathbf{a}}$, and $\partial_2 B_1^\circ \times \bar{\mathbf{a}}$, and the Lagrangian
submanifold $\mathbf{R} \times \{0\} \times \bar{h}(\bar{\mathbf{a}})$ corresponds to $\partial_3 B_1^\circ \times \bar{h}(\bar{\mathbf{a}})$. We identify the positive end of
$\bar{W}_1^\circ$ with $[0, \infty) \times [0, 1] \times \bar{S}$, the left negative end with $(-\infty, 0] \times [\frac{1}{2}, 1] \times \bar{S}$, and the
right negative end with $(-\infty, 0] \times [0, \frac{1}{2}] \times \bar{S}$.

Denote the sections at infinity of $\bar{W}_i^\circ$, $i=1,2$, by $\sigma_\infty^{\circ,i} = B_i^\circ \times \{z_\infty\}$. A curve $\bar{v}_{*0}$ in $\bar{W}$ decomposes into $\bar{v}_{*0}^{\circ,1} \cup \bar{v}_{*0}^{\circ,2}$, where $\bar{v}_{*0}^{\circ,i} = \bar{v}_{*0}^{\circ,i'} \cup \bar{v}_{*0}^{\circ,i''}$, $i=1,2$, is a curve in $\bar{W}_i^\circ$, $\bar{v}_{*0}^{\circ,i'}$ is a possibly branched cover of $\sigma_\infty^{\circ,i}$, and $\bar{v}_{*0}^{\circ,2''}$ is a union of constant sections $B_1^\circ \times \{pt\}$, $pt \in \hat{\mathbf{a}}$.

(R) We view the chords at the positive ends of $\bar{v}_{*0}^{\circ,2}$ and the left negative ends of $\bar{v}_{*0}^{\circ,1}$, including $z_\infty$, as Reeb chords in a Morse-Bott family and, for the purposes of calculating Maslov indices, we pretend that the Morse-Bott family has been perturbed so that $z_\infty$ is the bottom generator of each $\bar{a}_i$. In particular, at $z_\infty$ we assume that, when we go around the boundary of $\bar{W}_2^\circ$ in the counterclockwise direction, each $\bar{a}_i$ is rotated slightly in the negative $\phi$-direction by a Hamiltonian isotopy.

We emphasize that $\bar{v}_{*0}^{\circ,1} \cup \bar{v}_{*0}^{\circ,2}$ is essentially the same thing as $\bar{v}_{*0}$, described in a slightly different way.
<figure><img src="image_12.png" /><figcaption>FIG. 9. — The dot in the left-hand picture is the preferred point (0, 1)</figcaption></figure>

The reader can verify that, with our convention (R),

$$
\begin{equation} \tag{3.4.14}
\begin{aligned}
\mathrm{ind}(\sigma_{\infty}^{\circ, i}) &= 0, && i = 1, 2, \quad \mathrm{I}(\bar{v}_{*0}^{\circ, 2,'}) = 0, \\
\mathrm{I}(\bar{v}_{*0}^{\circ, 2,''}) &= \mathrm{ind}(\bar{v}_{*0}^{\circ, 2,''}) = 0, && \mathrm{ind}(\bar{v}_{*0}^{\circ, 1,''}) = \mathrm{ind}(\bar{v}_{*0}^{\circ, 1,''}).
\end{aligned}
\end{equation}
$$

We apply the continuation method to $\bar{v}_{1,j}$, $0 \le j \le a$, $(1,j) \ne *_0 = (1,j_0)$, and $\bar{v}_{*0}^{\circ,i}$, $i=1,2$: We adapt the definition of a continuation along $\partial_+ B_\tau$ in a natural way by replacing $g_{j_0}^1$ by $g_{j_0,1}^1$, $g_{j_0,2}^1$, $g_{j_0,1}^{1/2}$, which correspond to $\partial_1 B_1^\circ$, $\partial_2 B_2^\circ$, $\partial_2 B_1^\circ$, respectively, in Equation (3.4.4). As in Lemma 3.4.7, start with a nontrivial negative end $\mathcal{E}_1$ of some $\bar{v}_{1,j_1}$ or $\bar{v}_{*0}^{\circ,1}$ limiting to $z_\infty$ and continue the corresponding $g_{j_1}^1$, $g_{j_0,1}^1$, or $g_{j_0,1}^{1/2}$ in the direction of $\partial_+ B_\tau$, until some $g_{*2}^{*1} \not\subset \sigma_\infty^*$ is reached. Suppose $\mathcal{E}_2$ is the end of some component of $\bar{u}_\infty$ which “lies between” $g_{*2}^{*1}$ and some $g_{*2}^{*1}$. We then switch from $g_{*2}^{*1}$ to $g_{*2}^{*1}$ and continue. This gives a sequence of sectors of type $\mathcal{S}(\bar{a}_{k,l}, \bar{a}_{k',l'})$ (corresponding to the left negative end of $\bar{W}_1^\circ$; here if $(k,l) = (k',l')$, then we view the sector as the full $2\pi$ sector), $\mathcal{S}(\bar{h}(\bar{a}_{k,l}), \bar{a}_{k',l'})$, or $\mathcal{S}(\bar{a}_{k',l'}, \bar{h}(\bar{a}_{k,l}))$, and hence a unique cycle

$$
\mathcal{Z} = (\mathfrak{z}_1 \to \mathfrak{z}_2 \to \dots \to \mathfrak{z}_k \to \mathfrak{z}_1)
$$

that winds around $\mathbf{R}/2\pi\mathbf{Z}$ once. Note that if there is a boundary point at $z_\infty$, then there
is at least one sector of type $\mathcal{S}(\bar{a}_{k,l}, \bar{a}_{k',l'})$.

We leave it to the reader to make the proper generalizations of $\mathcal{Z}$ when we do not
assume $(S); *_0 = +$ or $(-1, j_0)$, $1 \le j_0 \le c + 1$; and/or the boundary point at $z_\infty$ lies on
$L_{\bar{h}(\bar{a})}$, $L_{\bar{a}}^+$, $L_{\bar{b}}$, $L_{\bar{h}(\bar{b})}$, or $L_{\bar{b}}^-$.

Let $A_{\epsilon}^{[a,b]}$ be the annulus $\{\rho = \epsilon\} \subset [a, b] \times D_{\rho_0}^2$. Writing $*_0 = (1, j_0)$, let $\vec{D}_{+,j_0}$, $\vec{D}_{L-,j_0}$, $\vec{D}_{R-,j_0}$ be the data at $z_\infty$ of the positive, left negative, and right negative ends, and let $P_{L-,j_0,1/2}$ and $P_{L-,j_0,1}$ (resp. $P_{R-,j_0,0}$ and $P_{R-,j_0,1/2}$) be the initial and terminal points of $A_{\epsilon}^{[1/2,1]}$ (resp. $A_{\epsilon}^{[0,1/2]}$) determined by $\vec{D}_{L-,j_0}$ (resp. $\vec{D}_{R-,j_0}$). We also decompose $P_{*,i} =$