The partition conditions for $k$-fold iterates of such elliptic orbits with $k \le 2g$ are particularly simple and have the form $(1, \dots, 1)$ or $(k)$.

**Theorem 4.3.1.** — Suppose $(\dagger\dagger)_{2g}$ holds. If $m \gg 0$, then there exists a chain homotopy

$$K : \mathrm{PFC}_{2g}(\mathbb{N}) \to \mathrm{PFC}_{2g}(\mathbb{N}),$$

such that the following holds:

$$(4.3.1) \qquad \partial_{\mathrm{ECH}} \circ K + K \circ \partial_{\mathrm{ECH}} = \Phi \circ \Psi + \mathrm{id}.$$

*Proof.* — Suppose $m \gg 0$. Fix $\mathfrak{p}(\tau)$ and choose $\{\bar{J}_{\tau}\} \in \bar{\mathcal{I}}^{\mathrm{reg}}$. For sufficiently small $\varepsilon, \delta > 0$ (which depend on the choices of $m$ and $\{\bar{J}_{\tau}\}$), there exists $\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ such that Lemma 4.2.13 holds.

Fix $\mathbf{y}, \mathbf{y}' \in \tilde{\mathcal{O}}_{2g}$ and abbreviate

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

Let $\bar{\mathcal{M}}$ be the SFT compactification of $\mathcal{M}$ and let $\partial\mathcal{M} = \bar{\mathcal{M}} - \mathcal{M}$ be the boundary of $\mathcal{M}$. As in the $\bar{W}_{\tau}^{\pm}$ case, 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. 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 $z_{\infty}$.

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 4.2.12, we may take $\mathcal{M}^{\{\mathrm{K}_{\mathfrak{p}(\tau), \delta}\}}$ to be regular.

In view of the considerations from Claim 3.3.4, we assume that all of $\mathcal{M}$ is regular.

*Step 1 (Breaking at $+\infty$).* The following is proved in Section 4.4.

**Lemma 4.3.2.** — $\partial_{[+\infty]} \mathcal{M} \subset A_1$, where

$$A_1 = \coprod_{\mathbf{y} \in \mathcal{S}_{a, h(a)}} \left( \mathcal{M}_{\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(+\infty))\}}^{I=2, n^*=m} (\mathbf{y}, \mathbf{y}; \bar{\mathfrak{m}}(+\infty)) \times \mathcal{M}_{J_+}^{I=0}(\mathbf{y}, \mathbf{y}') \right).$$

Gluing the pairs in $A_1$ accounts for the term $\Phi \circ \Psi$ in Equation (4.3.1).

*Step 2 (Breaking at $-\infty$).* The following lemmas are proved in Section 4.5.

**Lemma 4.3.3.** — $\partial_{[-\infty]} \mathcal{M} \subset A_2 \sqcup A_3$, where

$$A_2 = \coprod_{\mathbf{y} \in \tilde{\mathcal{O}}_{2g}, \mathfrak{z} \in \mathfrak{z}, r_0=0} \left( \mathcal{M}_{\{\bar{J}_{-\infty, 1}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(-\infty))\}}^{I=0, n^*=0} (\mathbf{y}, \mathbf{y}, \mathfrak{z}) \right).$$
$$ \times \mathcal{M}_{\bar{J}_{-\infty,2}}^{I=4g+2, n^*=m}(\mathfrak{z}; \bar{m}(-\infty)); $$

$$ A_3 = \coprod_{\gamma, \gamma' \in \hat{\mathcal{O}}_{2g}, \mathfrak{z} \in \mathfrak{z}, r_0=1} \left( \mathcal{M}_{\bar{J}_{-\infty,1}^{\diamond}(\epsilon, \delta, \mathfrak{p}(-\infty))}^{I=2g+2, n^*=m}(\gamma, \gamma', \mathfrak{z}) \mathcal{M}_{\bar{J}_{-\infty,2}}^{I=2g, n^*=0}(\mathfrak{z}) \right). $$

Note that if $\bar{v}_2 \in \mathcal{M}_{\bar{J}_{-\infty,2}}^{I=2g, n^*=0}(\mathfrak{z})$, then $\bar{v}_2 = B_{-\infty,2} \times \mathfrak{z}$.

**Lemma 4.3.4.** — Gluing the pairs in $A_2$ accounts for the term id in Equation (4.3.1).

**Lemma 4.3.5.** — Gluing the pairs in $A_3$ gives a total of 0 mod 2.

*Step 3 (Breaking in the middle).* The following is proved in Section 4.6.

**Lemma 4.3.6.** — $\partial_{(-\infty, +\infty)} \mathcal{M} \subset A_4 \sqcup A_5 \sqcup A_6 \sqcup A_7$, where

$$ A_4 = \coprod_{\gamma'' \in \hat{\mathcal{O}}_{2g}} \left( \mathcal{M}_{\bar{J}_{\tau}^{\diamond}(\epsilon, \delta, \mathfrak{p}(\tau))}^{I=1, n^*=m}(\gamma, \gamma''; \bar{m}) \times \mathcal{M}_{J'}^{I=1}(\gamma'', \gamma') \right); $$

$$ A_5 = \coprod_{\gamma'' \in \hat{\mathcal{O}}_{2g}} \left( \mathcal{M}_{J'}^{I=1}(\gamma, \gamma'') \times \mathcal{M}_{\bar{J}_{\tau}^{\diamond}(\epsilon, \delta, \mathfrak{p}(\tau))}^{I=1, n^*=m}(\gamma'', \gamma'; \bar{m}) \right); $$

$$ A_6 = \coprod_{\gamma'', \gamma''' \in \hat{\mathcal{O}}_{2g-1}} \left( \mathcal{M}_{J'}^{I=2, n^*=m-1, \delta_0}(\gamma, \delta_0 \gamma'') \times \mathcal{M}_{\bar{J}_{\tau}^{\diamond}(\epsilon, \delta, \mathfrak{p}(\tau))}^{I=-2, n^*=0}(\delta_0 \gamma'', \delta_0 \gamma''') \right. \\ \left. \times \mathcal{M}_{J'}^{I=2, n^*=1}(\delta_0 \gamma''', \epsilon \gamma''') \right); $$

$$ A_7 = \coprod_{\gamma'', \gamma''' \in \hat{\mathcal{O}}_{2g-2}} \left( \mathcal{M}_{J'}^{I=2, n^*=m-2, \delta_0}(\gamma, \delta_0^2 \gamma'') \times \mathcal{M}_{\bar{J}_{\tau}^{\diamond}(\epsilon, \delta, \mathfrak{p}(\tau))}^{I=-3, n^*=0}(\delta_0^2 \gamma'', \delta_0^2 \gamma''') \right. \\ \left. \times \mathcal{M}_{J'}^{I=3, n^*=2}(\delta_0^2 \gamma''', \epsilon h \gamma''') \right). $$

Gluing the pairs in $A_4$ and $A_5$ accounts for the terms $\partial_{ECH} \circ K$ and $K \circ \partial_{ECH}$ in Equation (4.3.1).

**Lemma 4.3.7.** — Gluing the triples in $A_6$ and $A_7$ gives a total of 0 mod 2.

This completes the proof of Theorem 4.3.1, modulo the results that will be proved in Sections 4.4–4.6. $\square$

**4.4.** Degeneration at $+\infty$. — In this subsection we study the limit of holomorphic maps to $\bar{W}_{\tau}$ as $\tau \to +\infty$, i.e., when $\bar{W}_{\tau}$ degenerates into a concatenation of $\bar{W}_{-}$ with $\bar{W}_{+}$ along the HF-type end. This will prove Lemma 4.3.2.
We assume that $m \gg 0$; $\varepsilon, \delta > 0$ are sufficiently small; and $\{\bar{J}_{\tau}\} \in \bar{\mathcal{I}}^{\text{reg}}$ and $\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))\}$ satisfy Lemma 4.2.13. Fix $\mathbf{y}, \mathbf{y}' \in \bar{\mathcal{O}}_{2g}$ and let

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

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

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

be the limit holomorphic building in order from bottom to top, where each $\bar{v}_*$ is an SFT level, $\bar{v}_{-1,j}, j = 1, \dots, c$, 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 refer to $\bar{v}_+$ as $\bar{v}_{-1,c+1}$ or $\bar{v}_{0,0}$ and $\bar{v}_-$ as $\bar{v}_{0,b+1}$ or $\bar{v}_{1,0}$.

As before, we have the following constraints:

$$ (4.4.1) \qquad n^*(\bar{u}_i) = \sum_{\bar{v}_*} n^*(\bar{v}_*) = m; $$

$$ (4.4.2) \qquad I(\bar{u}_i) = \sum_{\bar{v}_*} I(\bar{v}_*) = 2, $$

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

The following is the analog of Lemma 3.4.21, with a similar proof (omitted):

**Lemma 4.4.1.** — If $\bar{v}_*' \cup \bar{v}_*'' = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$, then $a = b = c = 0$; $I(\bar{v}_+) = 0$; $I(\bar{v}_-) = 2$; $\bar{v}_+$ is a $W_+$-curve; and $\bar{v}_-$ is a $W_-$-curve.

**Lemma 4.4.2.** — If $\bar{v}_*' \cup \bar{v}_*'' \neq \emptyset$ for some level $\bar{v}_*$ of $\bar{u}_\infty$, then:

(1) $p_- = \deg(\bar{v}_') > 0$;

(2) some $\bar{v}_{1,j_0}, j_0 > 0$, has a negative end $\mathcal{E}_-$ that limits to $\delta_0^\beta$ for some $p > 0$ and satisfies $n^*(\mathcal{E}_-) \ge m - p$;

(3) $\bar{u}_\infty$ has no boundary point at $z_\infty$;

(4) $\bar{u}_\infty$ has no fiber components and no components of $\bar{v}_*''$ that intersect the interior of a section at infinity;

(5) each component of $\bar{v}_{0,j}''$, $1 \le j \le b$, is a thin strip;

(6) each component of $\bar{v}_+''$ is an $n^* = 1$, $I = 0$ or 1 section from $z_\infty$ to $h$ or $e$ which is contained in $\bar{W}_+ - W_+$;

(7) each component of $\bar{v}_{-1,j}''$, $0 \le j \le c$, is an $n^* = 1$, $I = 1$ or 2 cylinder from $\delta_0$ to $h$ or $e$ which is contained in $\mathbf{R} \times (\bar{\mathbf{N}} - \mathbf{N})$;
(8) $h$ appears at most once at the negative end of $\bar{v}_{-1,1}^\sharp$.

*Proof.* — The proof is based on Equation (4.4.1). First observe that:

(*) either there is a negative end $\mathcal{E}_-$ that limits to $\delta_0^\flat$ for some $p > 0$ and satisfies $n^*(\mathcal{E}_-) \ge m-p$ by Lemma 3.4.2; or

(**) there is a negative end $\mathcal{E}_-$ that limits to $z_\infty$ and the sum of $n^*(\mathcal{E}_i)$ over all the ends $\mathcal{E}_i$ that limit to $z_\infty$ and $n^*(\mathcal{E}_i')$ over all the neighborhoods $\mathcal{E}_i'$ of the boundary points at $z_\infty$ is $\ge m - 2g$.

(1) If (*) or (**) holds, then $\bar{v}'_- \ne \emptyset$, since otherwise the neighborhood of $\bar{m}(+\infty)$ contributes $m$ towards $n^*(\bar{v}_-)$.

(2) is a consequence of (1) and is a subcase of (*). In particular (**) does not hold. (3)–(7) follow from (2). (8) follows from the definition of the ECH differential. $\square$

**Lemma 4.4.3.** — If $\bar{v}_*' \cup \bar{v}_*^\sharp \ne \emptyset$ for some level $\bar{v}_*$, then $\bar{u}_\infty$ cannot have the following subbuildings:

(1) a degree one component of $\bar{v}_+^\sharp$ from $z_\infty$ to $h$; and

(2) $\bar{v}_+'$ which has degree $p_+$ and $\bar{v}_{-1,1}^\sharp$ which is a union of $p_+$ cylinders from $\delta_0$ to $h$.

*Proof.* — This is due to the positioning of $h$, given in Section 3.2.3. (1) was proved in Lemma I.6.6.5. (2) is due to Lemma 3.7.8: the usual rescaling procedure with fixed $m \gg 0$, together with Lemma 4.4.2, gives rise to an SFT limit $w_+ : \Sigma_+ \to \mathbf{CP}^1$, $\pi_+ : \Sigma_+ \to \text{cl}(B_+)$, where $\Sigma_+$ consists of $p_+$ copies of $\text{cl}(B_+)$ (and hence $\pi_+$ is a trivial branched cover) and the restriction of $w_+$ to each component of $\Sigma_+$ satisfies the conditions of Lemma 3.7.9. $\square$

**Lemma 4.4.4.** — If $\bar{v}_*' \cup \bar{v}_*^\sharp \ne \emptyset$ for some level $\bar{v}_*$, then $\bar{v}_- \ne \emptyset$ and $\bar{u}_\infty$ contains one of the following subbuildings:

(1) A 3-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I=1$ and a negative end $\delta_0\boldsymbol{\gamma}'; \bar{v}_- = \sigma_\infty^-$; and a thin strip of $\bar{v}_{0,1}^\sharp$.

(2) A 3-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I=1$ and a negative end $\delta_0\boldsymbol{\gamma}'; \bar{v}_- = \sigma_\infty^-$; and a component of $\bar{v}_+'$ with $I=1$ from $z_\infty$ to $e$.

(3) A 4-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I=1$ and a negative end $\delta_0\boldsymbol{\gamma}'; \bar{v}_- = \sigma_\infty^-$; $\bar{v}_+' = \sigma_\infty^+$; and a cylinder component of $\bar{v}_{-1,1}^\sharp$ from $\delta_0$ to $e$.

Here we are omitting levels which are connectors.

*Proof.* — The lemma is a consequence of Lemmas 4.4.2 and 4.4.3, the positivity of the ECH indices of all the components with the exception of $\bar{v}_+'$, and the conditions given by Equations (4.4.1) and (4.4.2). $\square$
**Lemma 4.4.5.** — If $m \gg 0$, then there is no $\bar{u}_\infty \in \partial_{(+\infty)} \mathcal{M}$ such that $\bar{v}_*^\prime \cup \bar{v}_*^{\prime\prime} \neq \emptyset$ for some level $\bar{v}_*$.

*Proof.* — The proofs to eliminate Cases (1)–(3) of Lemma 4.4.4 are similar to those of Theorem I.7.10.1 and will be omitted. $\square$

*Proof of Lemma 4.3.2.* — This is a combination of Lemmas 4.4.1 and 4.4.5. $\square$

**4.5.** *Degeneration at $-\infty$.* — In this subsection we study the limit of holomorphic maps to $\bar{W}_\tau$ as $\tau \to -\infty$. This will prove Lemma 4.3.3.

We assume that $m \gg 0$; $\varepsilon, \delta > 0$ are sufficiently small; and $\{\bar{J}_\tau\} \in \bar{\mathcal{I}}^{\prime\mathrm{eg}}$ and $\{\bar{J}_\tau^\diamond(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ satisfy Lemma 4.2.13. Fix $\mathbf{y}, \mathbf{y}' \in \bar{\mathcal{O}}_{2g}$ and let

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

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

$$ \bar{u}_{\infty} = (\bar{v}_{-1,1} \cup \cdots \cup \bar{v}_{-1,c}) \cup \bar{v}_1 \cup \bar{v}_2 \cup (\bar{v}_{1,1} \cup \cdots \cup \bar{v}_{1,a}) $$

be the limit holomorphic building in order from bottom to top, where each $\bar{v}_*$ is an SFT level, $\bar{v}_{-1,j}, j = 1, \dots, c$, maps to $\bar{W}'; \bar{v}_j$ maps to $\bar{W}_{-\infty,j}$; and $\bar{v}_{1,j}, j = 1, \dots, a$ maps to $\bar{W}$. Sometimes we refer to $\bar{v}_1$ as $\bar{v}_{-1,c+1}$ or $\bar{v}_{1,0}$.

We write $\bar{v}_* = \bar{v}'_* \cup \bar{v}''_*$, $\bar{v}''_* = \bar{v}^\sharp_* \cup \bar{v}^\flat_*$, where:

- $\bar{v}'_*$ is the union of branched covers of a section at infinity;

- $\bar{v}^\sharp_*$ is the union of components that are not in $\bar{v}'_*$ and are asymptotic to some multiple of $\delta_0$ or pass through $\mathfrak{z} = z_0^{\delta_0} \zeta_1^{\delta_1} \cdots \zeta_l^{\delta_l}$ with $r_0 > 0$; and

- $\bar{v}^\flat_*$ is the union of the remaining components of $\bar{v}_*$.

**Remark 4.5.1.** — The best way to prove the existence of the limit for $\tau \to -\infty$ is probably by regarding the degeneration of $\bar{W}_\tau$ as a neck stretching as mentioned at the beginning of Section 4. Strictly speaking, in the limit $\tau \to -\infty$, there exist levels between $\bar{v}_1$ and $\bar{v}_2$, i.e., levels that map to $\mathbf{R} \times \mathbf{S}^1 \times \bar{\mathbf{S}}$. By considerations of $n^*$, these levels are connectors (i.e., map to $\mathbf{R} \times \mathbf{S}^1 \times \{pt\}$) and will be ignored until we consider gluing. From the point of view of nodal degenerations, these intermediate levels correspond to curves mapped to the fiber over the node of $B_{-\infty}$, equipped with meromorphic functions à la Ionel and Parker [IP2, Section 5].

The following are the analogs of Lemmas 4.4.1 and 4.4.2:
**Lemma 4.5.2.** — If $\bar{v}_*^\prime \cup \bar{v}_*^\sharp = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$, then $a = c = 0$; $I(\bar{v}_1) = 0$; $I(\bar{v}_2) = 4g + 2$; and $\bar{v}_2$ is a $\bar{W}_{-\infty,2}$-curve.

*Proof.* — Suppose $\bar{v}_*^\prime \cup \bar{v}_*^\sharp = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$. The following are immediate from considerations of $n^*$:

(1) $\bar{v}_2$ is a $\bar{W}_{-\infty,2}$-curve or a degenerate $\bar{W}_{-\infty,2}$-curve; and

(2) $\bar{v}_* = \bar{v}_*''$ and $n^*(\bar{v}_*) = 0$ for $*=(-1,j)$, $1$, and $(1,j)$.

(1) implies that $I(\bar{v}_2) = 4g + 2$ by Lemma 4.2.7. Note that:

(3) a degenerate $\bar{W}_{-\infty,2}$-curve consists of a fiber component with ind = 2 - 2g and 2g components of the type $B_{-\infty,2} \times \{q_l\}$ for some $q_l \in \hat{a}_l$, each with ind = 1, for a total of $I(\bar{v}_2) = 4g + 2$.

Since there are $2g$ codimension two gluing conditions between $\bar{v}_1$ and $\bar{v}_2$, it follows that

$$ (4.5.1) \qquad \sum_* I(\bar{v}_*) = 4g + 2, $$

where the summation is over all the levels $*$. Hence $a = c = 0$, $I(\bar{v}_1) = 0$, and each component $\tilde{v}$ of $\bar{v}_1$ is a branched cover of a trivial cylinder with possibly empty branch locus. (Here we are assuming without loss of generality that the almost complex structure on $\bar{W}_{-\infty,1}$ is $\bar{J}_{-\infty,1}$.)

We eliminate degenerate $\bar{W}_{-\infty,2}$-curves as follows: Since we may assume that $\hat{\mathcal{O}}_{2g}$ is disjoint from $\bar{a} \times \{1\} \subset \bar{S} \times \{1\}$ by genericity, the gluing condition for $\bar{v}_1$ and $\bar{v}_2$ is not satisfied in view of (3). $\square$

**Lemma 4.5.3.** — If $\bar{v}_*^\prime \cup \bar{v}_*^\sharp \neq \emptyset$ for some level $\bar{v}_*$ of $\bar{u}_\infty$, then:

(1) $p_2 = \deg(\bar{v}_2') > 0$;

(2) $\bar{u}_\infty$ has no boundary point at $z_\infty$;

(3) $\bar{u}_\infty$ has no fiber components and no components of $\bar{v}_*''$ that intersect the interior of a section at infinity; and

(4) $\bar{v}_2''$ is the union of $2g - p_2$ copies of $B_{-\infty,2} \times \{pt\}$, and $I(\bar{v}_2) = 2g$.

*Proof.* — (1)–(3) are analogs of Lemma 4.4.2 and (4) is a consequence of Lemma 4.2.8. $\square$

The following is the analog of Lemma 4.4.4.

**Lemma 4.5.4.** — If $\bar{v}_*^\prime \cup \bar{v}_*^\sharp \neq \emptyset$ for some level $\bar{v}_*$, then $\bar{v}_2^\prime \neq \emptyset$ and $\bar{v}_2''$ is a union of components $B_{-\infty,2} \times \{pt\}$, and $\bar{u}_\infty$ contains one of the following subbuildings:

(1i) A 2-level building consisting of $\bar{v}_1^\sharp$ with $I = i + (\deg(\bar{v}_1^\sharp) - 1)$, $i = 2, 3$, which passes through $\beta$ with multiplicity $r_0 = 1$; and $\bar{v}_2^\prime = \sigma_\infty^{-\infty,2}$ with $I = 1$.
<figure><img src="image_29.png" /><figcaption>FIG. 18. — Schematic diagrams for the possible types of degenerations. The dotted lines indicate the section at infinity. The quantities −2 and −4 along the intersections of $\bar{v}_1$ and $\bar{v}_2$ indicate the reduction of $I$ due to the gluing conditions. For simplicity we are assuming that $\deg \bar{v}_1^2 = 1$ in (1$_i$) and $\deg \bar{v}_1^2 = 2$ in (2).</figcaption></figure>

(2) A 2-level building consisting of $\bar{v}_1^\sharp$ with $I = 4 + (\deg(\bar{v}_1^\sharp) - 2)$ which passes through $\zeta$ with multiplicity $r_0 = 2$; and $\bar{v}_2'$ with $I = 2$ which is a branched double cover of $\sigma_\infty^{-\infty,2}$.

(3$_i$) A 4-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I = 1$ or 2 and a negative end at $\delta_0$; $\bar{v}_1' = \sigma_\infty^{-\infty,1}$; $\bar{v}_2' = \sigma_\infty^{-\infty,2}$; and a cylinder of $\bar{v}_{-1,1}^\sharp$ from $\delta_0$ to $h$ or $e$ with $I = 1$ or 2.

(4) A 4-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I = 1$ and a negative end at $\delta_0^2$; $\bar{v}_1' = \sigma_\infty^{-\infty,1}$; a cylinder of $\bar{v}_1^\sharp$ from $\delta_0$ to $h$ or $e$ with $I = 1, 2$; $\bar{v}_2' = \sigma_\infty^{-\infty,2}$; a cylinder of $\bar{v}_{-1,1}^\sharp$ from $\delta_0$ to $h$ with $I = 1$; and a cylinder of $\bar{v}''_{-1,1}$ from $h$ or $e$ to $e$ with $I = 1, 0$.

(5) A 4-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I = 1$ and a negative end at $\delta_0^2$; $\bar{v}_1'$ with $I = 0$ which is a branched double cover of $\sigma_\infty^{-\infty,1}$; $\bar{v}_2'$ with $I = 2$ which is a branched double cover of $\sigma_\infty^{-\infty,2}$; and two cylinder components of $\cup_{j=1}^c \bar{v}_{-1,j}^\sharp$ from $\delta_0$ to $h$ or $e$, each with $I = 1$ or 2.

(6) A 5-level building consisting of $\bar{v}_{1,2}^\sharp$ with $I = 1$ and a negative end at $\delta_0^2$; $\bar{v}_{1,1}' = \mathbf{R} \times \delta_0$; a component of $\bar{v}_{1,1}^\sharp$ with $I = 1$ which is a cylinder from $\delta_0$ to $h$; $\bar{v}_1' = \sigma_\infty^{-\infty,1}$; $\bar{v}_2' = \sigma_\infty^{-\infty,2}$; and a cylinder of $\bar{v}_{-1,1}^\sharp$ from $\delta_0$ to $h$ with $I = 1$.

We are omitting levels which are branched covers of trivial cylinders. Moreover, each gluing condition reduces the sum of ECH indices by 2.

See Figure 18.

*Proof.* — The proof is similar to that of Lemma 4.4.4 and we only provide a sketch. We have $\bar{v}_2' \neq \emptyset$ by Lemma 4.5.3(1). By Lemma 4.5.3(4), $\bar{v}_2''$ is a union of components $B_{-\infty,2} \times \{q_l\}$, $q_l \in \hat{a}_l$, and $I(\bar{v}_2) = 2g$.

First suppose that $\bar{v}'_1 = \emptyset$. Then $\bar{v}_1^\sharp$ passes through $((0, \frac{3}{2})$, $z_\infty$) exactly once and $\pi_{B_{-\infty,1}} \circ \bar{v}_1^\sharp$ has a $k$-fold branch point (here a “1-fold branch point” is a regular point) at
(0, 3/2) for some k > 0, where $\pi_{B_{-\infty,1}} : \overline{W}_{-\infty,1} \to B_{-\infty,1}$ is the projection along $\overline{S}$. The k-fold branch point at ((0, 3/2), z_\infty) contributes 2k towards ind($\overline{v}_1^\sharp$) and each of the remaining 2g - k intersection points with {(0, 3/2)} × $\overline{S}$ passes through some $\hat{a}_l$ and contributes +1 towards ind($\overline{v}_1^\sharp$).

We restate the ind calculation from the previous paragraph in terms of ECH indices $I'(\overline{v}_1)$ and $I'(\overline{v}_2)$ when:

(I₁) $\overline{v}_1, \overline{v}_2$ are viewed as curves in the fibrations

$$ \begin{aligned} \overline{W}_{-\infty,1} - (\{(0, 3/2)\} \times \overline{S}) &\to B_{-\infty,1} - \{(0, 3/2)\}, \\ \overline{W}_{-\infty,2} - (\{0\} \times \overline{S}) &\to B_{-\infty,2} - \{0\}; \end{aligned} $$

(I₂) $\{(0, 3/2)\} \times \overline{S}$ is in one-to-one correspondence with a Morse-Bott family of orbits, viewed as orbits at the *positive end of* $\overline{v}_1$;

(I₃) the orbit corresponding to $z_\infty$ is viewed as a negative elliptic orbit, i.e., an elliptic orbit with Conley-Zehnder index $-1$ with respect to the framing coming from the Morse-Bott fibration, and the orbits corresponding to $q_l$ are viewed as hyperbolic orbits.

Then $I'(\overline{v}_1) \ge 0$, $I'(\overline{v}_2') = k$, and $I'(\overline{v}_2'') = 0$, and the ECH index of the curve $\tilde{v}$ obtained by (pre-)gluing $\overline{v}_1$ and $\overline{v}_2$ is given by:

$$ I(\tilde{v}) \ge I'(\overline{v}_1) + I'(\overline{v}_2) \ge 0 + (k+0) = k. $$

This implies that $k \le 2$, giving us $(1_i), i = 2, 3$, or (2).

Next suppose that $\overline{v}'_1 \ne \emptyset$. Then the ECH index $I(\tilde{v})$ of the curve $\tilde{v}$ obtained by (pre-)gluing $\overline{v}'_1$ and $\overline{v}'_2$ is $-k$, where $k = \deg(\overline{v}'_1) = \deg(\overline{v}'_2)$. Hence $(3_i), i = 1, 2$, and (4)–(6) follow from enumerating all the possibilities, subject to the condition that $h$ appear only once at the negative end of $\overline{v}_{-1,1}$. $\square$

**Lemma 4.5.5.** — If $m \gg 0$, $\overline{u}_\infty \in \partial_{\{-\infty\}} \mathcal{M}$, and $\overline{v}'_* \cup \overline{v}^\sharp_* \ne \emptyset$ for some level $\overline{v}_*$, then the only possibility is Case (1₃).

*Proof.* — Cases (1₂) and (2). We will eliminate Case (1₂); Case (2) is similar. We apply the usual rescaling argument with $m \gg 0$ fixed to obtain a function $w_2 : B_{-\infty,2} \to \mathbf{CP}^1$ satisfying the following:

$$ (i_2) w_2(0) = \infty; $$

$$ (ii_2) w_2(\frac{i}{2}) = 0; $$

$$ (iii_2) w_2(\partial B_{-\infty,2}) \subset \mathbf{R}^+ \cdot e^{i\phi(\bar{a}_{k,l})} \text{ for some } (k, l); $$

$$ (iv_2) w_2 \text{ is a biholomorphism away from } \partial B_{-\infty,2}. $$

We now observe that $w_2$ is uniquely determined by $(i_2)-(iv_2)$, up to multiplication by a positive real constant. This implies that:
(v₂) $w_2$ maps the asymptotic marker $\partial_y$ at $0 \in B_{-\infty, 2}$ to the asymptotic marker $\dot{\mathcal{R}}_{\pi+\phi(\bar{a}_{k,l})}(\infty)$.

Here $z = x + iy$ is the complex coordinate on $B_{-\infty, 2}$. (v₂) translates into an asymptotic condition for $\bar{v}_1^\sharp$ at $((0, \frac{3}{2})$, $z_\infty)$. Hence $I(\bar{v}_1^\sharp)$ is at least 3.

Cases (3i), (4)–(6). We will eliminate Case (3i); the remaining cases are similar. The rescaling argument gives $w_2 \cup w_1$, where $w_2$ is as in the previous paragraph and $w_1 : cl(B_{-\infty, 1}) \to \mathbf{CP}^1$ satisfies the following:

$$ (i_1) \quad w_1(0, \frac{3}{2}) = 0 \text{ and } w_1(+\infty) = +\infty; $$

(ii₁) $w_1$ is a biholomorphism.

(v₂) implies the following asymptotic condition for $w_1$:

(iii₁) $w_1$ maps the marker $\partial_s$ at $(0, \frac{3}{2})$ to the marker $\dot{\mathcal{R}}_{\pi+\phi(\bar{a}_{k,l})}(0)$.

Hence $w_1$ is uniquely determined by (i₁)–(iii₁) up to multiplication by a positive real constant.

As a consequence of the uniqueness of $w_1$ up to multiplication by a positive real constant, the following are uniquely determined:

(a) the asymptotic eigenfunction of $\bar{v}_{1,1}^\sharp$ at the negative end $\delta_0$;

(b) the asymptotic eigenfunction of $\bar{v}_{-1,1}^\sharp$ at the positive end $\delta_0$;

(a) is determined by the image of the asymptotic marker $\dot{\mathcal{L}}_{3/2}(+\infty)$ at $+\infty \in cl(B_{-\infty, 1})$ and (b) by the radial ray that contains $w_1(-\infty)$. (a) and (b) give rise to one constraint each on $\bar{v}_{1,1}^\sharp$ and $\bar{v}_{-1,1}^\sharp$. Hence $I(\bar{v}_{1,1}^\sharp) \ge 2$ and $I(\bar{v}_{-1,1}^\sharp) \ge 2$, which is a contradiction. $\square$

*Proof of Lemma 4.3.3.* — This is a combination of Lemmas 4.5.2 and 4.5.5. $\square$

*Proof of Lemma 4.3.4.* — We first claim that the mod 2 cardinality of

$$ \mathcal{M}_{\bar{J}_{-\infty, 2}}^{I=4g+2, n^*=m}(\mathfrak{z}; \bar{\mathfrak{m}}(-\infty)) $$

is 1 when $r_0 = 0$ (i.e., $\mathfrak{z}$ does not contain $z_\infty$) and the arcs $\bar{a}$ are chosen generically. This is proved by reducing to the calculation of Theorem 2.3.3 as follows: Degenerate $B_{-\infty, 2}$ into a sphere $B_{-\infty, 21}$ and a disk $B_{-\infty, 22}$ which are identified at one point and degenerate $\bar{W}_{-\infty, 2} = B_{-\infty, 2} \times \bar{S}$ into $(B_{-\infty, 21} \times \bar{S}) \cup (B_{-\infty, 22} \times \bar{S})$. We assume that the marked point is in $B_{-\infty, 21} \times \bar{S}$. Then a curve

$$ \bar{v}_2 \in \mathcal{M}_{\bar{J}_{-\infty, 2}}^{I=4g+2, n^*=m}(\mathfrak{z}; \bar{\mathfrak{m}}(-\infty)) $$

degenerates into a pair $(\bar{v}_{21}, \bar{v}_{22})$, where $\bar{v}_{2i}$, $i = 1, 2$, maps to $B_{-\infty, 2i} \times \bar{S}$ and $\bar{v}_{22}$ is a union of constant sections $B_{-\infty, 22} \times \{q_i\}$, where $q_i \in \hat{a}_i$. Hence $\bar{v}_{21}$ is a curve in $B_{-\infty, 21} \times \bar{S}$ with exactly the same type of constraints as in Theorem 2.3.3. This implies the claim.
In the rest of the proof we discuss how to glue pairs $(\bar{v}_1, \bar{v}_2) \in A_2$. For simplicity we work with $\bar{J}_{-\infty,1}$ instead of $\bar{J}_{-\infty,1}^\diamond(\varepsilon, \delta, \mathfrak{p}(-\infty))$. Since $I(\bar{v}_1) = 0$, each component $\tilde{v}$ of $\bar{v}_1$ is a branched cover of a trivial cylinder with possibly empty branch locus. If $\tilde{v}$ is simple, then it glues to $\bar{v}_2$ in the usual manner. On the other hand, if $\tilde{v}$ is multiply covered, then by $(\dagger\dagger)_{2g}$ it is a branched cover of degree $k \le 2g$ of the cylinder over an elliptic orbit $\gamma$ with $\mu_{\tau^*}(\gamma^l) = \pm 1$ for all $l = 1, \dots, k$ (here $\gamma^l$ denotes the $l$th cover of the orbit $\gamma$) and partitions $(k)$ and $(1, \dots, 1)$; this is the same partition condition as that of $\bar{u}_i$ for $i \gg 0$ where $\bar{u}_i \to \bar{u}_\infty$. If $\mu_{\tau^*}(\gamma^i) = 1$ the partition $(1, \dots, 1)$ is at the positive end, and if $\mu_{\tau^*}(\gamma^i) = -1$ it is at the negative end. The partition $(k)$ is at the other end. The component $\tilde{v}$ intersects $\zeta_1$ at $(0, \frac{3}{2})$ with total multiplicity $r_1 = k$.

Suppose the elliptic orbit $\gamma$ has partition $(k)$ at the positive end and $(1, \dots, 1)$ at the negative end; the other case is analogous. Then $\tilde{v}$ has 1 positive end, $k$ negative ends, and $b \ge k - 1$ branch points (in the sense of Definition 3.4.26). Since the image of $\bar{v}_2$ in $\bar{W}_{-\infty,2} = B_{-\infty,2} \times \bar{S}$ is graphical over $\bar{S}$, there are local holomorphic coordinates $(w, z)$ about $(0, \zeta_1) \in \bar{W}_{-\infty,2}$ with respect to which $\mathrm{Im}(\bar{v}_2)$ has the form $w = z^k$. Hence $\tilde{v}$ must have a branch point at $(0, \frac{3}{2})$ which contributes $k - 1$ towards $b$.

Next we define $\mathrm{ind}'(\bar{v}_1)$ and $\mathrm{ind}'(\bar{v}_2)$ as the Fredholm indices when $(I_1)$ and $(I_2)$ from the proof of Lemma 4.5.4 and $(I'_3)$ hold, where:

$\mathrm{Ind}'(\bar{v}_1)$ the ends of $\bar{v}_1$ corresponding to $\zeta \in \{(0, \frac{3}{2})\} \times \bar{S}$ are viewed as limiting to positive elliptic orbits, i.e., elliptic orbits with Conley-Zehnder index 1 with respect to the framing coming from the Morse-Bott fibration.

By the usual Fredholm index calculation, we obtain $\mathrm{ind}'(\tilde{v}) \ge 0$, with equality if and only if $b = k - 1$. We also compute $\mathrm{ind}'(\bar{v}_2) = 2$. When we use $\mathrm{ind}'$ instead of $\mathrm{ind}$, the gluing conditions between $\bar{v}_1$ and $\bar{v}_2$ are 0-dimensional instead of $2g$-dimensional. This implies that $\mathrm{ind}'(\tilde{v}) = 0$. Since the index calculation is topological, this also implies that $\bar{u}_\infty$ has no ghost bubble at $(\zeta_1, (0, \frac{3}{2}))$.

Finally we discuss the automatic transversality of $\tilde{v}$. Recall from [We2, Theorem 1] that automatic transversality holds if

$$ (4.5.2) \qquad \mathrm{ind}'(\tilde{v}) \ge 2g + \#\Gamma_0 - 1, $$

where $g$ is the genus of $\tilde{v}$ and $\#\Gamma_0$ is the number of punctures (of $\tilde{v}$ with $(0, \frac{3}{2})$ removed) with even Conley-Zehnder index. Observe that the Conley-Zehnder index of the Morse-Bott ends is odd by [We2, Section 3.2] because they are constrained. Since the right-hand side of Equation (4.5.2) is equal to $-1$, automatic transversality holds and $\tilde{v}$ glues in the usual manner (without any concerns of inserting branch-covered cylinders) to $\bar{v}_2$. This implies the lemma. $\square$

*Sketch of proof of Lemma 4.3.5.* — We use some considerations of Section I.7.12. For simplicity we assume we are gluing degree one curves and the curve in $\bar{W}_{-\infty,2}$ is $\sigma_\infty^{-\infty,2}$,
the section at $\infty$; this is justified by noting that the gluing of $\sigma_{\infty}^{-\infty,2}$ to $\bar{v}_1$ can be done essentially independently of the gluings of the remaining components of $\bar{W}_{-\infty,2}$ to $\bar{v}_1$.

Fix $\gamma, \gamma' \in \hat{O}_1$, $k \in \{1, \dots, 2g\}$, and $l \in \{0, 1\}$. We consider the gluing parameter space

$$ \mathfrak{P}_{k,l} := \coprod_{\mathfrak{z}_0 \in \mathfrak{u}_{k,l}} ((-\infty, -r] \times \mathcal{M}_1(\mathfrak{z}_0) \times \mathcal{M}_2(\mathfrak{z}_0)), $$

where $\mathfrak{u}_{k,l} \subset \bar{a}_{k,l}$ is a small open interval containing $z_{\infty}$, $\mathfrak{z}_0$ is a formal product consisting of a single point on $\mathfrak{u}_{k,l}$,

$$ \mathcal{M}_1(\mathfrak{z}_0) := \mathcal{M}_{\bar{J}_{-\infty,1}^{\diamond}(\varepsilon, \delta, \mathbf{p}(-\infty))}^{1=3, n^*=m}(\gamma, \gamma', \mathfrak{z}_0), $$

$$ \mathcal{M}_2(\mathfrak{z}_0) := \mathcal{M}_{\bar{J}_{-\infty,2}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{1=1, n^*=0, \text{ext}}(\mathfrak{z}_0), $$

$\text{ext}$ means the boundary of the holomorphic curve is mapped to $L_{\bar{a}_k \cup \bar{a}_{k,l}}^{-\infty,2}$, and $(-\infty, -r]$ is viewed as a subset of $(-\infty, \infty)$ with parameter $\tau$. Note that $\mathcal{M}_2(\mathfrak{z}_0)$ consists of a single point $B_{-\infty,2} \times \{\mathfrak{z}_0\}$. For each $\mathfrak{z}_0 \in \mathfrak{u}_{k,l}$ there is a covering map

$$ \pi_{\mathfrak{z}_0} : \mathcal{M}_1(\mathfrak{z}_0) \to S^1 = \mathbf{R}/2\pi\mathbf{Z} \simeq T_{\mathfrak{z}_0}\bar{S}/\mathbf{R}^+, $$

such that at $((0, \frac{3}{2}), \mathfrak{z}_0)$, $\bar{v}_1 \in \mathcal{M}_1(\mathfrak{z}_0)$ is tangent to $\frac{\partial}{\partial s} + \pi_{\mathfrak{z}_0}(\bar{v}_1)$ (where the second term is defined up to a positive real constant). The maps $\pi_{\mathfrak{z}_0}$ are continuous in $\mathfrak{z}_0$.

Let

$$ G_{k,l} : \mathfrak{P}_{k,l} \to \coprod_{\tau \in (-\infty, r]} \mathcal{M}_{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{1=2, n^*=m, \text{ext}}(\gamma, \gamma'), \quad \mathfrak{d} = (\tau, \bar{v}_1, \bar{v}_2) \mapsto \bar{u}(\mathfrak{d}) $$

be the gluing map, defined as usual. Also let $\mathfrak{P}'_{k,l} \subset \mathfrak{P}_{k,l}$ be the subset of gluing parameters $\mathfrak{d}$ such that $G_{k,l}(\mathfrak{d}) \in \coprod_{\tau \in (-\infty, r]} \mathcal{M}_{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathbf{p}(\tau))}^{1=2, n^*=m}(\gamma, \gamma')$.

We now apply the rescaling argument from Section I.7.2.2: if a pair in $A_3$ is the limit of a sequence $\bar{u}_1, \bar{u}_2, \dots$ of curves in $\mathcal{M}, \bar{u}_i \in \mathcal{M}_{\tau_i}$ (in particular $\bar{u}_i$ passes through the marked point $\bar{m}(\tau_i)$), and $\tau_i \to -\infty$, then there is a unique transverse limit profile, which is encoded by an element of $\tilde{\mathcal{N}}/\mathbf{R}^+$ that we describe now.

Let $\tilde{\mathcal{N}}$ be the space of holomorphic maps $w_2 : B_{-\infty,2} = \bar{D} \to \mathbf{CP}^1$ satisfying $(i_2)$ and $(iv_2)$ of Lemma 4.5.5 and

$$ (\text{iii}'_2) \ w_2(\partial B_{-\infty,2}) \subset \mathbf{R} \cdot e^{i\phi(\bar{a}_{k,l})} \text{ for some } (k, l). $$

For simplicity we assume that $\phi(\bar{a}_{k,l}) = 0$. We leave it to the reader to verify the following:

**Claim 4.5.6.** — $\tilde{\mathcal{N}} = \{w_2(z) = az + b + \frac{\bar{a}}{z} | a \in \mathbf{C}^*, b \in \mathbf{R}\}$. Hence $\dim \tilde{\mathcal{N}} = 3$ and $\tilde{\mathcal{N}}$ admits an $\mathbf{R}^*$-action which is multiplication by $c \in \mathbf{R}^*$.
Let $D(\overline{\mathfrak{m}}^b(-\infty)) \subset B_{-\infty,2}$ be a small disk about $\frac{i}{2}$ and let $\tilde{\mathcal{N}}_x := \{w_2 \in \tilde{\mathcal{N}} | w_2(x) = 0\}$, where $x \in D(\overline{\mathfrak{m}}^b(-\infty))$. The total space of the bundle

$$ \tilde{\mathcal{N}}_{D(\overline{\mathfrak{m}}^b(-\infty))} := \bigsqcup_{x \in D(\overline{\mathfrak{m}}^b(-\infty))} \tilde{\mathcal{N}}_x \to D(\overline{\mathfrak{m}}^b(-\infty)) $$

smoothly embeds into $\tilde{\mathcal{N}}$. We will call this the “transversality of the constraint $\overline{\mathfrak{m}}^b(-\infty)$”. Also, the transverse limit profile is an element of $\tilde{\mathcal{N}}_{\overline{\mathfrak{m}}^b(-\infty)}/\mathbf{R}^+$.

For $\tilde{\tau} \ll 0$, the restrictions of $\tilde{u}(\mathfrak{d})$ to a neighborhood of the section at infinity are approximated by elements of $\tilde{\mathcal{N}}$. More precisely, for $\tilde{\tau} \ll 0$ we define:

$$ g_{\tilde{\tau},k,l} : \mathfrak{P}_{k,l} \cap \{\tau = \tilde{\tau}\} \to \tilde{\mathcal{N}}/\mathbf{R}^+, $$

as follows: Given $\mathfrak{d} \in \mathfrak{P}_{k,l} \cap \{\tau = \tilde{\tau}\}$, restrict $G_{k,l}(\mathfrak{d})$ to a neighborhood of the section at infinity so that the domain of $G_{k,l}(\mathfrak{d})$ is

$$ \overline{\mathbf{D}}^{\varepsilon(\tilde{\tau})} := \{\varepsilon(\tilde{\tau}) \leq |z| \leq 1\} \subset \overline{\mathbf{D}}, \quad \varepsilon(\tilde{\tau}) > 0 \quad \text{small.} $$

Let us write $\pi_{\overline{S}} \circ G_{k,l}(\mathfrak{d})|_{\overline{\mathbf{D}}^{\varepsilon(\tilde{\tau})}}$ as a Laurent series $\sum_{i=-\infty}^{\infty} c_i(\mathfrak{d})z^i$. Then we set

$$ g_{\tilde{\tau},k,l}(\mathfrak{d}) = \left[ \overline{c_{-1}(\mathfrak{d})} \cdot z + \mathrm{Re}(c_0(\mathfrak{d})) + \frac{c_{-1}(\mathfrak{d})}{z} \right]. $$

The definition makes sense in view of the following consequence of Gromov-Hofer compactness:

**Claim 4.5.7.**

(1) $\lim_{\tilde{\tau} \to -\infty} \frac{c_i(\mathfrak{d})}{c_{-1}(\mathfrak{d})} = 0$ if $i > 1$ or $i < -1$ and $\lim_{\tilde{\tau} \to -\infty} \frac{\overline{c_1(\mathfrak{d})}}{c_{-1}(\mathfrak{d})} = 1$, where the convergence is uniform in $\bigsqcup_{\mathfrak{z}_0 \in \overline{\mathfrak{u}}_{k,l}} (M_1(\mathfrak{z}_0) \times M_2(\mathfrak{z}_0))$.

(2) If $\overline{v}_2 = B_{-\infty,2} \times \{\mathfrak{z}_0\}$, where $\mathfrak{z}_0$ is a fixed point $\neq z_\infty$, then

$$ \lim_{\tilde{\tau} \to -\infty} \frac{c_{-1}(\mathfrak{d})}{c_0(\mathfrak{d})} = 0 \quad \text{and} \quad \lim_{\tilde{\tau} \to -\infty} \frac{\mathrm{Im}(c_0(\mathfrak{d}))}{\mathrm{Re}(c_0(\mathfrak{d}))} = 0, $$

where the convergence is uniform in $M_1(\mathfrak{z}_0)$.

What Claim 4.5.7(2) is saying is that when $\mathfrak{z}_0 \neq z_\infty$ and $\tilde{\tau} = \tilde{\tau}(\mathfrak{z}_0) \ll 0$, then $\pi_{\overline{S}} \circ G_{k,l}(\mathfrak{d})$ maps $\partial B_{-\infty,2}$ to a small slit on $\mathbf{R}$ which is disjoint from 0 but close to $\mathfrak{z}_0$.

Now define the evaluation map:

$$ ev_{\tilde{\tau},k,l} : \mathfrak{P}_{k,l} \cap \{\tau = \tilde{\tau}\} \to \mathbf{C}, $$

which sends $\mathfrak{d}$ to $g_{\tilde{\tau},k,l}(\mathfrak{d})(\frac{i}{2})$. By Gromov-Hofer compactness $\frac{c_{-1}(\mathfrak{d})}{|c_{-1}(\mathfrak{d})|}$ and $\pi_{\mathfrak{z}_0}(\overline{v}_2)$ are close, up to an overall rotation and possibly a reflection. Together with Claim 4.5.7(2) and the
transversality of the constraint $\bar{\mathfrak{m}}^b(-\infty)$, the local degree of $(ev_{\tau,k,l})|_{\mathfrak{P}'_{k,l} \cap \{\tau=\bar{\tau}\}}$ near $0 \in \mathbf{C}$ is deg $\pi_{\mathfrak{3}_0}$. We will informally say that "$\mathcal{M}$ has deg $\pi_{\mathfrak{3}_0}$ ends near $\mathfrak{P}'_{k,l}$".

Finally observe that an element of $M_2(\mathfrak{3}_0)$ can be viewed as a map to $\bar{a}_{i,0}$ or to $\bar{a}_{i,1}$. Hence $\mathcal{M}$ has deg $\pi_{\mathfrak{3}_0}$ ends near $\mathfrak{P}'_{k,l}$ and deg $\pi_{\mathfrak{3}_0}$ ends near $\mathfrak{P}'_{k,1-l}$ for a total of $2 \cdot \deg \pi_{\mathfrak{3}_0}$ ends mod 2. This proves Lemma 4.3.5. $\square$

**4.6.** *Breaking in the middle.* — In this subsection we study the limit of holomorphic maps to $\bar{W}_{\tau}$ as $\tau \to T'$ for some $T' \in (-\infty, \infty)$. This will prove Lemma 4.3.6.

We assume that $m \gg 0$; $\varepsilon, \delta > 0$ are sufficiently small; and $\{\bar{J}_{\tau}\} \in \bar{\mathcal{I}}^{\mathrm{reg}}$ and $\{\bar{J}_{\tau}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(\tau))\}$ satisfy Lemma 4.2.13. Fix $\mathbf{y}, \mathbf{y}' \in \bar{\mathcal{O}}_{2g}$ and let

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

We will analyze $\partial_{(-\infty, \infty)}\mathcal{M}$.

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 $\lim_{i \to \infty} \tau_i = T'$, and let

$$ \bar{u}_{\infty} = (\bar{v}_{-1,1} \cup \dots \cup \bar{v}_{-1,c}) \cup \bar{v}_0 \cup (\bar{v}_{1,1} \cup \dots \cup \bar{v}_{1,a}) $$

be the limit holomorphic building in order from bottom to top, where each $\bar{v}_*$ is an SFT-type level, $\bar{v}_{-1,j}$ and $\bar{v}_{1,j}$ map to $\bar{W}'$ and $\bar{v}_0$ maps to $\bar{W}_{T'}$. Sometimes we refer to $\bar{v}_0$ as $\bar{v}_{-1,c+1}$ or $\bar{v}_{1,0}$.

The following is the analog of Lemma 4.4.1, and is stated without proof.

**Lemma 4.6.1.** — If $\bar{v}_* \cup \bar{v}_*^2 = \emptyset$ for all levels $\bar{v}_*$ of $\bar{u}_\infty$, then $\bar{u}_\infty$ is one of the following:

(1) $a=0, c=1; \bar{v}_0$ is a $\bar{W}_{T'}$-curve with $I=1$ which passes through $\bar{\mathfrak{m}}(T')$; and $\bar{v}_{-1,1}$ is a $W'$-curve with $I=1$; or

(2) $a=1, c=0; \bar{v}_{1,1}$ is a $W'$-curve with $I=1$; and $\bar{v}_0$ is a $\bar{W}_{T'}$-curve with $I=1$ which passes through $\bar{\mathfrak{m}}(T')$.

Here either $T' \in \mathcal{T}_2$ and a component of $\bar{v}_0$ is in

$$ \mathcal{M}_{\{\bar{J}_{T'}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(T'))\}}^{\dagger, s, \mathrm{irr}, \mathrm{ind}=1, n^*=m}(\mathbf{y}_1, \mathbf{y}_2; \bar{\mathfrak{m}}(T')) $$

from Lemma 4.2.13(2), for some $\mathbf{y}_1, \mathbf{y}_2$; or $T' \in \mathcal{T}_1$ and a component of $\bar{v}_0$ does not pass through $\bar{\mathfrak{m}}(T')$ but is in

$$ \mathcal{M}_{\{\bar{J}_{T'}^{\diamond}(\varepsilon, \delta, \mathfrak{p}(T'))\}}^{\dagger, s, \mathrm{irr}, \mathrm{ind}=-1, n^*=0}(\mathbf{y}_1, \mathbf{y}_2) $$

from Lemma 4.2.13(1).

The following is the analog of Lemma 4.4.2 and is stated without proof.
**Lemma 4.6.2.** — If $\bar{v}_*^\sharp \cup \bar{v}_*^{\sharp'} \neq \emptyset$ for some level $\bar{v}_* \text{ of } \bar{u}_\infty$, then:

(1) $p_0 = \deg(\bar{v}'_0) > 0$;

(2) some $\bar{v}_{1,j_0}, j_0 > 0$, has a negative end $\mathcal{E}_-$ that limits to $\delta_0^\flat$ for some $p > 0$ and satisfies $n^*(\mathcal{E}_-) \ge m - p$;

(3) $\bar{u}_\infty$ has no boundary point at $z_\infty$;

(4) $\bar{u}_\infty$ has no fiber components and no components of $\bar{v}_*^{\prime\prime}$ that intersect the interior of a section at infinity;

(5) each component of $\bar{v}_{-1,j}^\sharp$, $0 \le j \le c$, is an $n^* = 1$, $I = 1$ or 2 cylinder from $\delta_0$ to $h$ or $e$ which is contained in $\mathbf{R} \times (\bar{\mathbb{N}} - \bar{\mathbb{N}})$;

(6) $h$ appears at most once at the negative end of $\bar{v}_{-1,1}^\sharp$.

The following is the analog of Lemma 4.4.4 and is stated without proof.

**Lemma 4.6.3.** — If $\bar{v}_*^\prime \cup \bar{v}_*^{\prime\prime} \neq \emptyset$ for some level $\bar{v}_*$, then $\bar{v}'_0 \neq \emptyset$ and $\bar{u}_\infty$ contains one of the following subbuildings, subject to two conditions:

- the sum of the ECH indices of the components of the subbuildings is at most 3;

- the total multiplicity of $h$ at the negative end of $\bar{v}_{-1,1}^\sharp$ is at most 1.

(1$_i$) A 3-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I = i$, $i = 1, 2$, and a negative end $\delta_0 \boldsymbol{\gamma}'; \bar{v}'_0 = \sigma_\infty^{\mathrm{T}'}$; and a cylinder component of $\bar{v}_{-1,1}^\sharp$ with $I = 1$ or 2 from $\delta_0$ to $h$ or $e$.

(2$_i$) A 4-level building consisting of $\bar{v}_{1,2}^\sharp$ with $I = i$, $i = 1, 2$, and a negative end $\delta_0^2 \boldsymbol{\gamma}'; \bar{v}'_{1,1} = \mathbf{R} \times \delta_0$; a cylinder component of $\bar{v}_{1,1}^\sharp$ with $I = 1$ or 2 from $\delta_0$ to $h$ or $e$; $\bar{v}'_0 = \sigma_\infty^{\mathrm{T}'}$; and a cylinder component of $\bar{v}_{-1,1}^\sharp$ with $I = 1$ or 2 from $\delta_0$ to $h$ or $e$.

(3$_i$) A 3-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I = i$, $i = 1, 2$, and a negative end $\delta_0^2 \boldsymbol{\gamma}'; \bar{v}'_0 = \sigma_\infty^{\mathrm{T}'}$; a component of $\bar{v}_0^\sharp$ with $I = 0$ or 1 from $\delta_0$ to $h$ or $e$; and a cylinder component of $\bar{v}_{-1,1}^\sharp$ with $I = 1$ or 2 from $\delta_0$ to $h$ or $e$.

(4$_i$) A 3-level building consisting of $\bar{v}_{1,1}^\sharp$ with $I = i$, $i = 1, 2$, and a negative end $\delta_0^2 \boldsymbol{\gamma}'; \bar{v}'_0$ with $I = -2$ which is a degree 2 branched cover of $\sigma_\infty^{\mathrm{T}'}$; and two cylinder components of $\cup_{j=1}^c \bar{v}_{-1,j}^\sharp$ from $\delta_0$ to $h$ or $e$, each with $I = 1$ or 2.

(5) A 5-level building consisting of $\bar{v}_{1,3}^\sharp$ with $I = 1$ and a negative end $\delta_0^3 \boldsymbol{\gamma}'; \bar{v}'_{1,2}$ which is a degree 2 branched cover of $\mathbf{R} \times \delta_0$; a cylinder component of $\bar{v}_{1,2}^\sharp$ with $I = 1$ from $\delta_0$ to $h$; $\bar{v}'_{1,1} = \mathbf{R} \times \delta_0$; a cylinder component of $\bar{v}_{1,1}^\sharp$ with $I = 1$ from $\delta_0$ to $h$; $\bar{v}'_0 = \sigma_\infty^{\mathrm{T}'}$; and a cylinder component of $\bar{v}_{-1,1}^\sharp$ with $I = 1$ from $\delta_0$ to $h$.

(6) A 4-level building consisting of $\bar{v}_{1,2}^\sharp$ with $I = 1$ and a negative end $\delta_0^3 \boldsymbol{\gamma}'; \bar{v}'_{1,1} = \mathbf{R} \times \delta_0$; two cylinder components of $\bar{v}_{1,1}^\sharp$ from $\delta_0$ to $h$, each with $I = 1$; $\bar{v}'_0 = \sigma_\infty^{\mathrm{T}'}$; and a cylinder component of $\bar{v}_{-1,1}^\sharp$ with $I = 1$ from $\delta_0$ to $h$.

(7) A 4-level building consisting of $\bar{v}_{1,2}^\sharp$ with $I = 1$ and a negative end $\delta_0^3 \boldsymbol{\gamma}'; \bar{v}_{1,1}^\sharp$ which is a degree 2 branched cover of $\mathbf{R} \times \delta_0$; a cylinder component of $\bar{v}_{1,1}^\sharp$ with $I = 1$ or 2 from $\delta_0$
to $h$ or $e$; $\bar{v}'_0 = \sigma_{\infty}^{T'}$; a component of $\bar{v}_0^{\sharp}$ with $I=0$ or $1$ from $\delta_0$ to $h$ or $e$; and a cylinder component of $\bar{v}_{-1,1}^{\sharp}$ with $I=1$ or $2$ from $\delta_0$ to $h$ or $e$.

(8) A 4-level building consisting of $\bar{v}_{1,2}^{\sharp}$ with $I=1$ and a negative end $\delta_0^3\boldsymbol{\gamma}'$; $\bar{v}_{1,1}$ which is a degree 2 branched cover of $\mathbf{R} \times \delta_0$; a cylinder component of $\bar{v}_{1,1}^{\sharp}$ with $I=1$ from $\delta_0$ to $h$; $\bar{v}'_0$ with $I=-2$ which is a degree 2 branched cover of $\sigma_{\infty}^{T'}$; and two cylinder components of $\cup_{j=1}^c \bar{v}_{-1,j}^{\sharp}$ from $\delta_0$ to $h$ or $e$, each with $I=1$ or $2$.

(9) A 3-level building consisting of $\bar{v}_{1,1}^{\sharp}$ with $I=1$ and a negative end $\delta_0^3\boldsymbol{\gamma}'$; $\bar{v}'_0 = \sigma_{\infty}^{T'}$; two components of $\bar{v}_0^{\sharp}$ from $\delta_0$ to $h$ or $e$, each with $I=0$ or $1$; and a cylinder component of $\bar{v}_{-1,1}^{\sharp}$ with $I=1$ or $2$ from $\delta_0$ to $h$ or $e$.

(10) A 3-level building consisting of $\bar{v}_{1,1}^{\sharp}$ with $I=1$ and a negative end $\delta_0^3\boldsymbol{\gamma}'$; $\bar{v}'_0$ with $I=-2$ which is a degree 2 branched cover of $\sigma_{\infty}^{T'}$; a component of $\bar{v}_0^{\sharp}$ with $I=0$ or $1$ from $\delta_0$ to $h$ or $e$; and two cylinder components of $\cup_{j=1}^c \bar{v}_{-1,j}^{\sharp}$ from $\delta_0$ to $h$ or $e$, each with $I=1$ or $2$.

(11) A 3-level building consisting of $\bar{v}_{1,1}^{\sharp}$ with $I=1$ and a negative end $\delta_0^3\boldsymbol{\gamma}'$; $\bar{v}'_0$ with $I=-3$ which is a degree 3 branched cover of $\sigma_{\infty}^{T'}$; and three cylinder components of $\cup_{j=1}^c \bar{v}_{-1,j}^{\sharp}$ from $\delta_0$ to $h$ or $e$, each with $I=1$ or $2$.

We are omitting levels which are connectors.

See Figure 19. We will write $(1_{i,e})$, $(1_{i,h})$, etc. to indicate that we are in Case (1) and the negative ends of the lowest level are $e$, $h$, etc.

**Lemma 4.6.4.** — If $m \gg 0$, $\bar{u}_{\infty} \in \partial(-\infty, +\infty)\mathcal{M}$, and $\bar{v}'_* \cup \bar{v}_*^{\sharp} \neq \emptyset$ for some level $\bar{v}_*$, then the only possibilities are (1$_2$) with a cylinder component of $\bar{v}_{-1,1}^{\sharp}$ from $\delta_0$ to $e$ and (4$_2$) with two cylinder components of $\bar{v}_{-1,1}^{\sharp}$, one from $\delta_0$ to $h$ and another from $\delta_0$ to $e$.

*Proof.* — Cases (1$_i$), (2$_i$), (3$_i$), (5), (6), (7) and (9). We will treat Case (1$_i$); the rest of the cases are similar and can be eliminated. The key observation here is that $\deg(\bar{v}'_0) = 1$ and $\bar{v}_{-1,1}^{\sharp}$ is a cylinder from $\delta_0$ to $h$ or $e$. Applying the usual rescaling argument with $m \gg 0$ fixed, we obtain a holomorphic map $w_0 : \mathrm{cl}(\mathbf{B}_{T'}) \to \mathbf{CP}^1$ which satisfies the following:

(i) $w_0(+\infty) = \infty$ and $w_0(\bar{m}^b(\Gamma')) = 0$;

(ii) $w_0(s, t) \in \mathrm{int}(\mathcal{R}_{\eta(t)})$ for all $(s, t) \in \partial\mathbf{B}_{T'}$;

(iii) $\deg(w_0) = 1$ away from $\mathfrak{S}(\bar{a}_{i_1,j_1}, \bar{h}(\bar{a}_{i_1,j_1}))$ for some $(i_1, j_1)$.

Here $\eta(t) = \phi_0 + \frac{\pi}{m}(t-1)$, where $\phi_0$ is the $\phi$-coordinate of $\bar{a}_{i_1,j_1}$. (Recall that we are projecting to $\pi_{\mathcal{D}_{\rho_0}^2}$ using balanced coordinates.)

We now observe that $w_0 : \mathrm{cl}(\mathbf{B}_{T'}) \to \mathbf{CP}^1$ is uniquely determined by (i)–(iii), up to multiplication by a positive real constant; this is argued in the same way as in Lemma 3.7.9. Using the same method as in Case (3$_i$) of Lemma 4.5.5, we obtain that $I(\bar{v}_{1,1}^{\sharp}) \ge 2$ and $I(\bar{v}_{-1,1}^{\sharp}) \ge 2$. Hence the only possibility is Case (1$_2$) with a cylinder component of $\bar{v}_{-1,1}^{\sharp}$ from $\delta_0$ to $e$.
<figure><img src="image_30.png" /><figcaption>FIG. 19. — Schematic diagrams for the possible types of degenerations. For simplicity we have drawn only one level $\bar{v}_{-1,1}$ to indicate the cylindrical components of $\cup_{j=1}^f \bar{v}_{-1,j}$ from $\delta_0$ to $h$ or $e$</figcaption></figure>

Cases (4$_i$), (8), (10), and (11). We will treat Case (4$_i$). In this case we first apply the rescaling argument with $m \to \infty$ to obtain a holomorphic map $w_0 : \Sigma_0 \to \mathbf{CP}^1$ and a branched double cover $\pi_0 : \Sigma_0 \to \mathrm{cl}(\mathbf{B}_{T'})$ such that:

(i) $w_0(z_0) = \infty$, where $\pi_0^{-1}(+\infty) = \{z_0\}$;

(ii) $w_0(z_1) = 0$ for some $z_1 \in \pi_0^{-1}(\bar{m}^b(T'))$;

(iii) $w_0(\pi_0^{-1}(\partial\mathbf{B}_{T'})) \subset \{\phi = 0, \rho > 0\}$;

(iv) $w_0|_{\mathrm{int}(\Sigma_0)}$ is a biholomorphism onto its image $\mathbf{CP}^1 - ([a_1, a_2] \cup [a_3, a_4])$ with $0 < a_1 < a_2 \le a_3 < a_4$.

Here an interval $[a, b]$ stands for $\{\phi = 0, a \le \rho \le b\}$.

By the Involution Lemma I.7.9.3,

(v) $\pi_0 \circ w_0^{-1}$ maps both $(-\infty, a_1]$ and $[a_4, \infty)$ to $\mathcal{L}_{3/2} \cap \{s \ge \frac{l(T')}{2}\}$ and $[a_2, a_3]$ to $\mathcal{L}_{1/2} \cup (\mathcal{L}_{3/2} \cap \{s \le \frac{-l(T')}{2}\})$.

In particular this constrains the asymptotic behavior of $\bar{v}_{1,1}^\sharp$ and hence $\mathrm{I}(\bar{v}_{1,1}^\sharp) \ge 2$. Hence the only possibility is Case (4$_2$) with two cylinder components of $\bar{v}_{-1,1}^\sharp$, one from $\delta_0$ to $h$ and another from $\delta_0$ to $e$. $\square$
*Proof of Lemma 4.3.6.* — This is a combination of Lemmas 4.6.1 and 4.6.4. □

*Proof of Lemma 4.3.7.*

*Case* (1$_2$). We glue triples in $A_6$, i.e., we are in Case (1$_2$). Note that the middle level has $I = -2$: there is a component $\sigma_{\infty}^{T'}$ (the section at infinity) which has $I = \text{ind} = -1$ and another $I = -1$ component corresponding to $T' \in \mathcal{T}_1$.

We make the following simplification:

- the top level is $M_+ = M_{\bar{y}}^{I=2, n^*=m-1, \delta_0}(\gamma, \delta_0)$ consisting of $\bar{v}_{1,1} = \bar{v}_{1,1}^{\#}$ and such that $\partial M_+ = \emptyset$;

- the middle level is $\sigma_{\infty}^{T'}$; and

- the bottom level is $M_- = M_{\bar{y}}^{I=2, n^*=1}(\delta_0, e)$ consisting of cylinders $\bar{v}_{-1,1} = \bar{v}_{-1,1}^{\#}$,

and the gluing is for any fixed $\tau = T'$. This simplification is justified by noting that the gluing of the $I = -1$ component $\neq \sigma_{\infty}^{T'}$ can be done essentially independently of the above gluing.

The gluing argument is similar to that of Theorem I.7.2.2 but has an additional twist since $I(\sigma_{\infty}^{T'}) = \text{ind}(\sigma_{\infty}^{T'}) = -1$ and we need to do an obstruction bundle gluing. The key observation is that the gluing of triples in $A_6$ come in pairs $(i_1, 0)$ and $(i_1, 1)$, where $\sigma_{\infty}^{T'}$ is viewed as having boundary mapping to $L_{\bar{a}_{i_1,0}}^{T'}$ or $L_{\bar{a}_{i_1,1}}^{T'}$. Since we can identify the obstruction bundles of each pair in such a way that the obstructions sections are close, the count corresponding to $A_6$ will be 0 mod 2.

As in Section I.7.7.1, the normal linearized $\bar{\partial}$-operator for $\sigma_{\infty}^{T'}$ acts on the appropriate Banach space of maps $B_{T'} \to \mathbf{C}$ and has the form $D = \partial_s - A$, where $A = -i\partial_t - \varepsilon$ and the coordinates are $s, t$. Without loss of generality we may assume that the boundary condition for $\partial B_{T'}$ is $\mathbf{R} \subset \mathbf{C}$, since the actual boundary condition is very close to $\mathbf{R}$. Its adjoint is $D^* = \partial_s + A$ with boundary condition $\mathbf{R}\langle dx - idy \rangle$ if a collar neighborhood of $\partial B_{T'}$ is given by $\mathbf{R} \times [0, 1]$ with holomorphic coordinates $x + iy$ and $\partial B_{T'} = \{y=0\}$.

*Claim* **4.6.5.** — $\ker D^*$ is 1-dimensional and if we write a nonzero element $\xi \in \ker D^*$ as $f(ds - idt)$ where $f$ is a map $B_{T'} \to \mathbf{C}$, then, for $R \gg 0$, $f|_{s=R}$ has winding wind$(f|_{s=R}, 0) = 1$ about 0 and likewise wind$(f|_{s=-R}, 0) = 0$.

*Proof of Claim* **4.6.5.** — $\ker D^*$ is 1-dimensional since $\ker D = 0$, where $D$ is the normal linearized $\bar{\partial}$-operator. This is because the normal bundle to $\sigma_{\infty}^{T'}$ is trivial.

The asymptotic decay conditions for $f$ imply that wind$(f|_{s=R}, 0) \geq 1$ and wind$(f|_{s=-R}, 0) \leq 0$. On the other hand, the considerations of Wendl [We2] (the methods of Lemma 1.6 and Theorem 4.1 of [BH] using the negativity of zeros of $\xi$ are more directly applicable here) imply that wind$(f|_{s=R}, 0) \leq 1$ and wind$(f|_{s=-R}, 0) \geq 0$. The claim then follows. □

We choose smooth slices $\tilde{M}_+$ and $\tilde{M}_-$ of $M_+$ and $M_-$ transverse to the $\mathbf{R}$-translation as in Section I.7.12.3 so that if $u_+ \in \tilde{M}_+$ (resp. $u_- \in \tilde{M}_-$), then for $s \le 0$
(resp. $s \ge 0$) the negative end of $u_+$ (resp. the positive end of $u_-$) has image in a small neighborhood of the section at $\infty$ and is dominated by the term of the form $c(u_+)e^{\lambda_+ s}f_+(t)$ (resp. $c(u_-)e^{\lambda_- s}f_-(t)$) when projected to the **C**-direction. Here $f_+(t)$ (resp. $f_-(t)$) is a fixed $L^2$-normalized leading asymptotic eigenfunction of $A = -i\partial_t - \epsilon$ for the negative end of $u_+$ (resp. positive end of $u_-$) with eigenvalue $\lambda_+ > 0$ (resp. $\lambda_- < 0$) and $c(u_\pm) \in \mathbf{C}$ satisfy $|c(u_\pm)| = \epsilon_0$ for $\epsilon_0 > 0$ small.

For $R \gg 0$, $\xi$ is dominated by $d_+e^{-\lambda_+R}f_+(t)(ds - idt)$ for $s = R$ and is dominated by $d_-e^{\lambda_-R}f_-(t)(ds - idt)$ for $s = -R$, where $d_+$ and $d_-$ are both nonzero.

As in [HT2], there exist an obstruction section $\mathfrak{s}$ and a linearized obstruction section $\mathfrak{s}_0$ which are both maps

$$ (5r, \infty)^2 \times \tilde{\mathcal{M}}_+ \times \tilde{\mathcal{M}}_- \to \mathcal{O} := \operatorname{Hom}(\ker D^*, \mathbf{R}), $$

such that the set of gluable $(T_+, T_-, u_+, u_-)$ — by this we mean we are gluing $u_+$, $\sigma_\infty^{T'}$, and $u_-$ with gluing parameters $T_\pm$ that indicate how much $u_\pm$ is translated up/down when pregluing — is given by $\mathfrak{s}^{-1}(0)$. Following [HT2, Definition 8.1], $\mathfrak{s}_0$ has the form:

$$ \begin{align*} \mathfrak{s}_0(T_+, T_-, u_+, u_-)(\xi) &= \langle c(u_+)e^{\lambda_+(R-T_+)}f_+(t), d_+e^{-\lambda_+R}f_+(t) \rangle \\ &\quad + \langle c(u_-)e^{\lambda_-(-R+T_-)}f_-(t), d_-e^{\lambda_-R}f_-(t) \rangle \\ &= e^{-\lambda_+T_+} \langle c(u_+)f_+(t), d_+f_+(t) \rangle \\ &\quad + e^{-|\lambda_-|T_-} \langle c(u_-)f_-(t), d_-f_-(t) \rangle, \end{align*} $$

where the brackets are **R**-linear inner products in $L^2(\mathbf{R}/2\mathbb{Z}, \mathbf{C})$. The specific form of $\mathfrak{s}_0$ is not that important; what matters here is that $\mathfrak{s}_0$ is transverse to the zero section since $d_\pm \ne 0$ and $c(u_\pm) \ne 0$.

On the other hand, the rescaling argument from Section I.7.2.2 implies that, if a triple in $A_6$ is the limit of a sequence $\bar{u}_1, \bar{u}_2, \dots$ of curves in $\mathcal{M}$ and $\bar{u}_i \in \mathcal{M}_{\tau_i}$ (in particular $\bar{u}_i$ passes through the marked point $\bar{m}(\tau_i)$), then there is a unique transverse limit profile $w_0 : \mathbf{B}_{T'} \to \mathbf{C}$ modulo multiplication by $\mathbf{R}^+$ such that:

(i) $w_0(+\infty) = \infty$;

(ii) $w_0(\bar{m}^b(T')) = 0$; and

(iii) $w_0(\partial \mathbf{B}_{T'}) \subset \mathbf{R}^+$.

This in turn implies that there exist:

(1) a finite subset $Z_{\bar{m}^b(T')} \subset \tilde{\mathcal{M}}_+ \times \tilde{\mathcal{M}}_-$; and

(2) a 1-dimensional subset

$$ \begin{align*} \mathcal{Z}_{\bar{m}^b(T')} &= \{(T_+, T_-(T_+, u_+, u_-), u_+, u_-) \mid T_+ \ge T'_+, \\ &\qquad (u_+, u_-) \in Z_{\bar{m}^b(T')} \} \\ &\subset (5r, \infty)^2 \times \tilde{\mathcal{M}}_+ \times \tilde{\mathcal{M}}_- \quad \text{for some } T'_+ \gg 0, \end{align*} $$
such that if $(T_+, T_-, u_+, u_-) \in \mathcal{Z}_{\overline{\mathfrak{m}}^b(T')}$, then there exists a constant $\tilde{c} > 0$ such that:

- the negative end of the $T_+$-translate of $u_+$ projected to the **C**-direction and the positive end of $\tilde{c}e^{-\varepsilon s}w_0$ — recall the factor $e^{-\varepsilon s}$ in the ansatz given by Equation (I.7.8.1) — agree up to first order (i.e., on the level of leading asymptotic eigenfunctions) on their overlap; and

- the same holds for the positive end of the $T_-$-translate of $u_-$ projected to the **C**-direction and the negative end of $\tilde{c}e^{-\varepsilon s}w_0$.

In particular, all the $\bar{u}_i$, $i \gg 0$, are close to the pregluing of elements in $\mathcal{Z}_{\overline{\mathfrak{m}}^b(T')}$.

We can analogously define $\mathcal{Z}_x$ for $x$ in a small disk $D(\overline{\mathfrak{m}}^b(T'))$ about $\overline{\mathfrak{m}}^b(T') \in \sigma_\infty^{T'}$, form the bundle

$$ (4.6.3) \qquad \mathcal{Z}_{D(\overline{\mathfrak{m}}^b(T'))} := \bigsqcup_{x \in D(\overline{\mathfrak{m}}^b(T'))} \mathcal{Z}_x \to D(\overline{\mathfrak{m}}^b(T')) $$

with 3-dimensional total space, and show that the natural map

$$ (4.6.4) \qquad \mathcal{Z}_{D(\overline{\mathfrak{m}}^b(T'))} \to (5r, \infty)^2 \times \tilde{\mathcal{M}}_+ \times \tilde{\mathcal{M}}_- $$

is an embedding. The details are left to the reader.

Note that, for $c \ge 0$ and sufficiently large $T'_+$, the simultaneous translation

$$ \operatorname{Tr}_c : (5r, \infty)^2 \times \tilde{\mathcal{M}}_+ \times \tilde{\mathcal{M}}_- \to (5r, \infty)^2 \times \tilde{\mathcal{M}}_+ \times \tilde{\mathcal{M}}_-, \\ (T_+, T_-, u_+, u_-) \mapsto (T_+ + c|\lambda_-|, T_- + c\lambda_+, u_+, u_-) $$

takes $\mathcal{Z}_x$ to $\mathcal{Z}_x \cap \{T_+ \ge T'_+ + c|\lambda_-|\}$, i.e., is $\operatorname{Tr}_c$-invariant.

Let $\mathcal{N}$ be a small $\operatorname{Tr}_c$-invariant neighborhood of a connected component of $\mathcal{Z}_{\overline{\mathfrak{m}}^b(T')}$ in $(5r, \infty)^2 \times \tilde{\mathcal{M}}_+ \times \tilde{\mathcal{M}}_-$ which nontrivially intersects $\mathfrak{s}_0^{-1}(0)$. By the form of Equation (4.6.1), $\mathfrak{s}_0$ is transverse to the zero section and, by the estimate [HT2, Lemma 8.7], there exists a $\operatorname{Tr}_c$-invariant tubular neighborhood $(\mathfrak{s}_0^{-1}(0) \cap \mathcal{N}) \times (-\delta, \delta)$ of $\mathfrak{s}_0^{-1}(0) \cap \mathcal{N}$ (after slightly enlarging $\mathcal{N}$ if necessary) such that on $(\mathfrak{s}_0^{-1}(0) \cap \mathcal{N}) \times \{\pm\delta\}$:

(1) $|\mathfrak{s} - \mathfrak{s}_0| \ll |\mathfrak{s}_0|$ and

(2) as $T'_+ \to \infty$, the ratio $|\mathfrak{s} - \mathfrak{s}_0|/|\mathfrak{s}_0|$ goes to zero.

Hence $\mathfrak{s}^{-1}(0) \cap \mathcal{N}$ is arbitrarily close to and limits to $\mathfrak{s}_0^{-1}(0) \cap \mathcal{N}$ as $T'_+ \to +\infty$, and there is a degree 1 map

$$ \mathfrak{s}^{-1}(0) \cap \mathcal{N} \cap \{T_+ = T'_+\} \to \mathfrak{s}_0^{-1}(0) \cap \mathcal{N} \cap \{T_+ = T'_+\}. $$

By the construction of the transverse limit profile, $\mathcal{Z}_{D(\overline{\mathfrak{m}}^b(T'))} \cap \mathcal{N} \cap \{T_+ = T'_+\}$ approaches $\mathfrak{s}^{-1}(0) \cap \mathcal{N} \cap \{T_+ = T'_+\}$ as $T'_+ \to \infty$ and hence must agree with $\mathfrak{s}_0^{-1}(0) \cap \mathcal{N} \cap \{T_+ = T'_+\}$ by the $\operatorname{Tr}_c$-invariance. Hence there is a degree 1 map

$$ \mathfrak{s}^{-1}(0) \cap \mathcal{N} \cap \{T_+ = T'_+\} \to D(\overline{\mathfrak{m}}^b(T')) $$
and the condition of passing through $\bar{\mathfrak{m}}^b(T')$ is a transverse condition. The gluings in $A_6$ therefore occur in pairs.

Case (4$_2$). Next we treat $A_7$, i.e., Case (4$_2$) with two cylinder components of $\bar{v}_{-1,1}^\sharp$, one from $\delta_0$ to $h$ and another from $\delta_0$ to $e$.

As in Case (4$_2$), there is an obstruction bundle

$$ \mathcal{O} := \operatorname{Hom}(\ker D^*, \mathbf{R}) \to (5r, \infty)^3 \times \tilde{\mathcal{M}}_+ \times \mathcal{M}_0 \times \tilde{\mathcal{M}}_{-,h} \times \tilde{\mathcal{M}}_{-,e}, $$

where $\mathcal{M}_0$ is the moduli space of branched covers of $\sigma_\infty^{T'}$ with one branch point, $\tilde{\mathcal{M}}_{-,h}$ is a one-point set consisting of a bottom level cylinder from $\delta_0$ to $h$, and $\tilde{\mathcal{M}}_{-,e}$ is a 1-dimensional set consisting of bottom level cylinders from $\delta_0$ to $e$, $\mathcal{O}$ has rank 4, and the base has dimension 7. There are also obstruction sections $\mathfrak{s}$ and $\mathfrak{s}_0$, where $\mathfrak{s}_0$ is defined as in Equation (4.6.1) and is the sum of three terms, each given by pairing the cokernel elements with the first term (possibly nonzero) of the Fourier expansions of the ends. A winding condition for any nonzero cokernel element $\xi$ analogous to that of Claim 4.6.5 implies that $\mathfrak{s}_0$ is transverse to the zero section. The verification of the claims in this paragraph are left to the reader.

Next we consider the moduli space $\mathcal{M}_\infty$ of pairs $(w_0, \pi_0)$ consisting of a holomorphic map $w_0: \Sigma_0 \to \mathbf{CP}^1$ and a branched double cover $\pi_0: \Sigma_0 \to \mathrm{cl}(\mathbf{B}_{T'})$ satisfying (i)–(iv) in the proof of Case (4$_2$) of Lemma 4.6.4 for some $a_1, \dots, a_4$. There is an action of $\mathbf{R}^+$ on $\mathcal{M}_\infty$ given by $(c, (w_0, \pi_0)) \mapsto (cw_0, \pi_0)$.

**Claim 4.6.6.** — The mod 2 count of transverse limit profiles $[(w_0, \pi_0)] \in \mathcal{M}_\infty/\mathbf{R}^+$, i.e., those that arise when taking the limit of a sequence $\bar{u}_1, \bar{u}_2, \dots$ of curves in $\mathcal{M}$ such that $\bar{u}_i$ passes through the marked point $\bar{\mathfrak{m}}(\tau_i)$, is 1.

*Proof of Claim 4.6.6.* — One can verify that:

(a) $\dim \mathcal{M}_\infty/\mathbf{R}^+ = 1$ and is parametrized by moving a branched point of $w_0$ along $\mathcal{L}_{1/2} \cup (\mathcal{L}_{3/2} \cap \{s \le \frac{-l(T')}{2}\})$;

(b) either $w_0(\pi_0^{-1}(-\infty)) \subset [a_2, a_3]$ or is on a line orthogonal to $[a_2, a_3]$ at $a^* \in [a_2, a_3]$, which maps to a branch point of $\pi_0$.

Let $\tilde{\mathcal{M}}_\infty$ be the set of $(w_0, \pi_0) \in \mathcal{M}_\infty$ together with an ordering of the points of $\pi^{-1}(-\infty)$; if $\pi^{-1}(-\infty)$ is a single point, we view it as a duplicate pair of points. The forgetful map $\tilde{\mathcal{M}}_\infty \to \mathcal{M}_\infty$ is generically a double cover. Then consider the following evaluation map:

$$ ev_\infty: \tilde{\mathcal{M}}_\infty/\mathbf{R}^+ \to \mathbf{S}^1 \times \mathbf{S}^1, $$

$$ (w_0, \pi_0) \mapsto \arg(w_0(\pi^{-1}(-\infty))). $$

Here $\arg$ refers to the projection to the $\phi$-coordinate, and we are using coordinates $(\phi_1, \phi_2)$ on $\mathbf{S}^1 \times \mathbf{S}^1$.
Let $Z = \{\phi_1 = c\} \subset S^1 \times S^1$, where $c$ is a small negative constant. (b) implies that $ev_\infty$ has intersection number 1 (mod 2) with $Z$. The claim then follows from noting that specifying $c$ corresponds to choosing a radial direction corresponding to the hyperbolic orbit $h$. $\square$

The same conclusion also holds if we take $m \gg 0$ fixed and if replace (ii) in the proof of Case (4$_2$) of Lemma 4.6.4 by “$w_0(z_1) = 0$ for some $z_1 \in \pi_0^{-1}(x)$”, where $x \in D(\bar{m}^b(T'))$. We can define the 3-dimensional manifold $\mathcal{Z}_{D(\bar{m}^b(T'))}$ as in Equation (4.6.3) such that the analog of Equation (4.6.4) is an embedding; the embedding condition implies the transversality of the point condition for $\bar{m}^b(T')$.

Once we have the obstruction bundle, the obstruction sections $\mathfrak{s}_0$ and $\mathfrak{s}$, and the embedding property of $\mathcal{Z}_{D(\bar{m}^b(T'))}$, the rest of the argument proceeds as in Case (4$_2$). Again we note that the branched double cover of $\sigma_\infty^{T'}$ can be viewed as having boundary mapping to $L_{\bar{a}_{k,0}}^{T'}$ or $L_{\bar{a}_{k,1}}^{T'}$ for $k=1,2$. Hence the count corresponding to $A_7$ is 0 mod 2. $\square$

## 5. Stabilization

The goal of this section is to prove Theorem 1.0.2. Let $N = N_{(S, \tilde{h})}$ be the mapping torus of $(S, \tilde{h})$, where $S$ is a bordered surface of genus $g$ with connected boundary and $\tilde{h}: (S, \omega) \tilde{\to} (S, \omega)$ is a symplectomorphism which has zero flux and restricts to the identity on $\partial S$.

The strategy of the proof is to apply two positive stabilizations to $(S, \tilde{h})$ — corresponding to the connected sum with a trefoil knot — to obtain $(S', \tilde{h}')$, where $S'$ has connected boundary and genus $g+1$. We then compare $\Phi_{(S, \tilde{h})}$ and $\Phi_{(S', \tilde{h}')}$, which both induce isomorphisms on the level of homology. Here $\Phi_{(S, \tilde{h})}$ is the $\Phi$ map for $(S, \tilde{h})$.

**5.1. The setup.** — Let $T$ be a genus one surface with connected boundary and let $\eta_0, \eta_1$ be two essential simple closed curves on $T$ which intersect transversely in one point. A positive Dehn twist along a closed curve $\eta$ will be denoted by $\tau_\eta$. Let $\tilde{h}_T: T \tilde{\to} T$ be the first return map of a Reeb vector field on the mapping torus

$$N_T = N_{(T, \tilde{h}_T)} = (T \times [0, 1]) / ((x, 1) \sim (\tilde{h}_T(x), 0))$$

so that the following hold:

(1) $\tilde{h}_T$ is isotopic to $\tau_{\eta_0} \circ \tau_{\eta_1}$ relative to the boundary;

(2) all the Reeb orbits in the interior of $N_T$ which intersect $T \times \{0\}$ at most $2g+2$ times are nondegenerate;

(3) $\tilde{h}_T|_{\partial T} = \text{id}$ and $\partial N_T$ is foliated by a negative Morse-Bott family of slope $\infty$.

In view of the discussion in Section I.3, $\tilde{h}_T$ will be viewed interchangeably as (i) the first return map of a Reeb vector field or (ii) the time-1 map of a stable Hamiltonian vector field with zero flux.
<figure><img src="image_31.png" /><figcaption>FIG. 20. — The page S'. The gradient trajectories of the Morse function f are given</figcaption></figure>

Now let $S'$ be the boundary connected sum of S and T. More precisely, if P is a pair-of-pants with boundary $∂P = ∂_1P ∪ ∂_2P ∪ ∂_3P$, then $S'$ is obtained from $S ∪ T ∪ P$ by identifying $∂S \simeq -∂_1P$ and $∂T \simeq -∂_2P$. Observe that $∂S' = ∂_3P$ is connected and $g(S') = g + 1$. See Figure 20.

We now define a symplectomorphism $\tilde{h}' : (S', \omega') \to (S', \omega')$ with zero flux (with respect to some $\omega'$) as follows: First set $\tilde{h}'|_S = \tilde{h}$ and $\tilde{h}'|_T = \tilde{h}_T$. Then define $\tilde{h}'|_P$ as the first return map of a Reeb flow $R_α$ on

$$N_P = N_{(P,id)} = (P \times [0, 1])/((x, 1) \sim (x, 0)).$$

The contact form $α$ is given by $f_0dt + β$, where $f_0$ is a function and $β$ is a 1-form on P, and both $f_0$ and $β$ do not depend on $t$. We choose a Morse-Bott function $f_0: P \to R$ which satisfies the following:

(1) $f_0$ is $C^k$-close to 1 for $k \gg 0$;

(2) $f_0$ attains its minimum along the Morse-Bott family $∂P$;

(3) the critical points of $f_0$ in $int(P)$ are isolated and consist of the maximum $e_P$ and two saddles $h_{1P}, h_{2P}$.

The Reeb orbits corresponding to $e_P, h_{1P}, h_{2P}$ will also be denoted by $e_P, h_{1P}, h_{2P}$.

Let $N_i, i = 1, 2, 3$, be the negative Morse-Bott family of Reeb orbits corresponding to $∂_iP$. If $f_0$ is $C^k$-close to 1 for $k \gg 0$, then the only orbits that intersect $P \times \{0\}$ at most $2g + 2$ times are: $e_P, h_{1P}$ and the orbits of $N_i$. We pick two orbits in each $N_i$ and label them $e_i, h_i$; they will become elliptic and hyperbolic when the Morse-Bott function $f_0$ is perturbed into a Morse function, which we call $f$. For convenience we write $N_S$ for the mapping torus of $(S, \tilde{h})$, $N_{S'}$ for the mapping torus of $(S', \tilde{h}')$, etc.

Since $(S', \tilde{h}')$ is obtained from $(S, \tilde{h})$ by applying two positive stabilizations, the corresponding contact structures $\xi_{(S, \tilde{h})}$ and $\xi_{(S', \tilde{h}')}$ are isomorphic. As a special case, ob-
serve that if $(S, \tilde{h}) = (D^2, id)$, then $(S', \tilde{h}') = (T, \tilde{h}_T)$ and $(T, \tilde{h}_T)$ is an open book decomposition for the standard tight contact structure on $S^3$.

**5.2. Morse-Bott theory.** — Let $J$ be an adapted almost complex structure on $\mathbf{R} \times \mathbb{N}_{S'}$ and let $\pi : \mathbf{R} \times \mathbb{N}_{S'} \to \mathbb{N}_{S'}$ be the projection onto the second factor. Also let $u$ be a finite energy Morse-Bott building in $\mathbf{R} \times \mathbb{N}_{S'}$ from $\mathbf{\gamma}$ to $\mathbf{\gamma}'$, where both orbit sets intersect $S' \times \{0\}$ in $2g + 2$ points.

The following lemma partitions the irreducible components of $u$ into three regions. When $u$ is a Morse-Bott building, then we say that $u$ is “irreducible” if the holomorphic curve, obtained by perturbing the Morse-Bott contact form to a nondegenerate one and correspondingly gluing up the levels of the Morse-Bott building, is irreducible.

**Lemma 5.2.1.** — Every irreducible component of the Morse-Bott building $u$ in $\mathbf{R} \times \mathbb{N}_{S'}$ has image in one of $\mathbf{R} \times \mathbb{N}_S$, $\mathbf{R} \times \mathbb{N}_T$, or $\mathbf{R} \times \mathbb{N}_P$.

*Proof.* — The lemma is an application of the winding number $\text{wind}_\pi$ of [HWZ1] and the positivity of intersections. (For example, see [0, Lemma 5.5.1].)

Suppose without loss of generality that $u : \dot{\mathbf{F}} \to \mathbf{R} \times \mathbb{N}_{S'}$ is a single-level Morse-Bott building and that $u(\dot{\mathbf{F}})$ nontrivially intersects $\mathbf{R} \times \mathbb{N}_P$; the case of a multiple-level Morse-Bott building only differs in notation. If $u$ corresponds to a gradient trajectory from $h_i$ to $e_i$, then the lemma holds. Otherwise let $P_\epsilon \subset \mathbf{P}$ be a slight retraction of $\mathbf{P}$ so that $\partial P_\epsilon = \partial_1^\epsilon \mathbf{P} \sqcup \partial_2^\epsilon \mathbf{P} \sqcup \partial_3 \mathbf{P}$. Let $\Pi : \mathbb{N}_P = \mathbf{P} \times \mathbb{S}^1 \to \mathbf{P}$ be the projection onto the first factor, let $T_i^\epsilon = \Pi^{-1}(\partial_i^\epsilon \mathbf{P})$, $i = 1, 2$, and let $N_{P_\epsilon} = \Pi^{-1}(P_\epsilon)$. We may assume that $T_i^\epsilon$ is foliated by (not necessarily closed) Reeb orbits. We consider the intersection $\delta_i = \pi(u(\dot{\mathbf{F}})) \cap T_i^\epsilon$, where $\delta_i$ is given the boundary orientation of $\pi(u(\dot{\mathbf{F}})) \cap \mathbb{N}_{P_\epsilon}$. The curve $\delta_i$ is transverse to the Reeb vector field away from a finite number of points by [0, Lemma 5.5.1]. If $\delta_i$ is not homologous to a multiple of $\{pt\} \times \mathbb{S}^1$ in $H_1(T_i^\epsilon)$, then $\Pi(\delta_i)$ is in the homology class $k[\partial_i P_\epsilon] \in H_1(P_\epsilon)$, $k > 0$, by the positivity of intersections in dimension four. This implies that $\Pi(\pi(u(\dot{\mathbf{F}})) \cap \partial \mathbb{N}_{S'})$ is in the class $k[\partial_3 \mathbf{P}] \in H_1(\mathbf{P})$ for $k > 0$. Since this is a contradiction, we must have $[\delta_i] = l[\{pt\} \times \mathbb{S}^1] \in H_1(T_i^\epsilon)$, for some $l \ge 0$. This implies that $u$ has negative ends along $\mathcal{N}_i$, $i = 1, 2$. Hence $u$ has image in $\mathbf{R} \times \mathbb{N}_P$. $\square$

We now use Morse-Bott theory (cf. Bourgeois [Bo1, Bo2]) to analyze holomorphic curves on $\mathbf{R} \times \mathbb{N}_P$. In particular, we consider a perturbation of the Morse-Bott family of orbits on $\mathbb{N}_P$, perturbed by the Morse function $f : \mathbf{P} \to \mathbf{R}$ as described above.

**Lemma 5.2.2.** — Let $f : \mathbf{P} \to \mathbf{R}$ be $\mathbb{C}^k$-close to 1 for $k \gg 0$. Then there is an adapted almost complex structure $J$ on $\mathbf{R} \times \mathbb{N}_P$ such that every $\text{I}_{\text{ECH}} = 1$ finite energy $J$-holomorphic curve $u$ whose image is in $\mathbf{R} \times \mathbb{N}_P$, is a simply-covered cylinder which corresponds to a gradient flow line between critical points of $f$ of adjacent index. The complete list is as follows:

(1) one cylinder from $e_P$ to $h_i$, $i = 1, 2, 3$, and two cylinders from $e_P$ to $h_{iP}$, $i = 1, 2$;
(2) two cylinders from $h_i$ to $e_i$, $i = 1, 2, 3$;

(3) one cylinder from $h_{1P}$ to $e_1$ and one cylinder from $h_{1P}$ to $e_3$;

(4) one cylinder from $h_{2P}$ to $e_2$ and one cylinder from $h_{2P}$ to $e_3$.

Moreover, all the $I_{\text{ECH}} = 1$ curves above are regular.

See Figure 20 for the gradient trajectories of $f$ which correspond to the above holomorphic cylinders.

*Proof.* — The proof is very similar to that of [0, Lemma 9.2.2]. We use the following fact, which can be proved using Morse-Bott theory or by a direct computation:

**Fact.** There is an adapted almost complex structure $J$ on $\mathbf{R} \times \mathbb{N}_P$ such that there is a one-to-one correspondence between (parametrized) gradient trajectories $\delta : \mathbf{R} \to \mathbf{P}$ of $f$ and finite energy $J$-holomorphic cylinders $Z_\delta$ in $\mathbf{R} \times \mathbb{N}_P$ which intersect each $\{(s, t)\} \times \mathbf{P}$ exactly once and project to $\text{Im}(\delta)$ under the projection $\Pi \circ \pi : \mathbf{R} \times \mathbb{N}_P \to \mathbf{P}$, $(s, x, t) \mapsto x$. Moreover, the cylinders $Z_\delta$, together with the trivial cylinders over the orbits corresponding to the Morse critical points, give a finite energy foliation of $\mathbf{R} \times \mathbb{N}_P$.

Fix $f$ and $J$ as above. We will use the notation $s\delta : \mathbf{R} \to \mathbf{P}$ for the translation $(s\delta)(\tau) = \delta(\tau + s)$.

Let $u : \dot{\mathbf{F}} \to \mathbf{R} \times \mathbb{N}_P$ be a finite energy $J$-holomorphic curve. Let $D_\varepsilon \subset \mathbf{P}$ be an arbitrarily small disk centered at the point $e_\varepsilon$ and let $N(e_\varepsilon) = \Pi^{-1}(D_\varepsilon)$ be a solid torus neighborhood of the orbit $e_\varepsilon$. We assume that $\partial N(e_\varepsilon)$ is foliated by (not necessarily closed) orbits of the Reeb vector field. We identify $\partial N(e_\varepsilon) \simeq \mathbf{R}^2/\mathbf{Z}^2$ so that the meridian has slope zero and a fiber $\{pt\} \times \mathbb{S}^1$ has slope $\infty$. Consider $\eta = \pi(u(\dot{\mathbf{F}})) \cap \partial N(e_\varepsilon)$, where $\eta$ is given the boundary orientation of $\pi(u(\dot{\mathbf{F}})) \cap \Pi^{-1}(\mathbf{P} - \text{int}(D_\varepsilon))$. If the projection $[\Pi(\eta)] \in H_1(\mathbf{P} - \text{int}(D_\varepsilon))$ is nonzero, then $[\Pi(\eta)] = -k[\partial D_\varepsilon]$, $k > 0$, by the positivity of intersections. This is a contradiction as in the proof of Lemma 5.2.1. Hence $[\eta] = l[\{pt\} \times \mathbb{S}^1] \in H_1(\partial N(e_\varepsilon))$ for some $l \ge 0$, i.e., has slope $\infty$. In other words, $u$ cannot intersect $\mathbf{R} \times e_\varepsilon$ and can only have $e_\varepsilon$ at the positive end. Similar considerations hold for $N(e_\varepsilon)$, $i = 1, 2, 3$, where $D_{i,\varepsilon} \subset \mathbf{P}$ is a half-disk centered at $e_\varepsilon$ and $N(e_\varepsilon) = \Pi^{-1}(D_{i,\varepsilon})$.

We now claim that $u$ is some multiple cover of some $Z_\delta$ with multiplicity $\ge 1$. Arguing by contradiction, suppose $u$ does not multiply cover any $Z_\delta$. Let us first consider the case where $\Pi \circ \pi(u(\dot{\mathbf{F}}))$ does not equal $\text{Im}(\delta)$ for any $\delta$. Then there is some $Z_\delta$ from $e_\varepsilon$ to some $e_i$ such that the intersection $u(\dot{\mathbf{F}}) \cap Z_\delta$ is nonempty; moreover, in view of the asymptotics on $N(e_\varepsilon)$ and $N(e_i)$, we may assume that $K = \pi(u(\dot{\mathbf{F}}) \cap Z_\delta)$ is compact. This implies that $u(\dot{\mathbf{F}})$ and $Z_{s\delta}$ do not intersect for sufficiently large $s$. On the other hand, since the intersection pairing $\langle u(\dot{\mathbf{F}}), Z_{s\delta} \rangle$ is a homological quantity and does not depend on $s$ due to the asymptotics, it follows that $K = \emptyset$, which is a contradiction. This implies that $\Pi \circ \pi(u(\dot{\mathbf{F}})) = \text{Im}(\delta)$ for some $\delta$. Now $\mathbf{R} \times \Pi^{-1}(\delta)$ is a 3-manifold which is foliated by $Z_{s\delta}$, $s \in \mathbf{R}$, and if $u$ does not multiply cover any $Z_{s\delta}$, then $u$ intersects some $Z_{s\delta}$ along a 1-manifold, a contradiction. We conclude that $u$ is a multiple cover of some $Z_\delta$.
Now Fredholm index one cylinders $Z_\delta$ — i.e., those that correspond to $\delta$ connecting two Morse critical points of adjacent index — are regular by the automatic transversality results of Wendl [We1, We2]. Moreover, $u$ cannot multiply cover $Z_\delta$ with multiplicity $> 1$ by [HT1, Proposition 7.15], since otherwise $I(u) > 1$. This implies that $u$ is equal to some $Z_\delta$, thereby completing the proof of the lemma. □

**5.3. Computation of PFH(N<sub>S</sub>).** — In this section we compute PFH(N<sub>S</sub>) in terms of PFH(N<sub>S</sub>) and PFH(N<sub>T</sub>). From now on we will write PFC(S, *h*) for PFC(N<sub>S</sub>), with the understanding that the periodic Floer homology group is defined using a stable Hamiltonian structure induced by the fibration, and whose stable Hamiltonian vector field has first return map *h*. Similar notation will be used for PFC(N<sub>T</sub>) and PFC(N<sub>S</sub>). Sometimes we will even drop the monodromy from the notation for the periodic Floer homology groups.

**5.3.1. Description of the differential of PFH(S', h').** — Given two orbit sets $\gamma' = \prod \gamma_i^{m'_i}$ and $\gamma = \prod \gamma_i^{m_i}$, we set $\gamma/\gamma' = \prod \gamma_i^{m_i-m'_i}$ if $m'_i \le m_i$ for all $i$; otherwise we set $\gamma/\gamma' = 0$. The chain group PFC<sub>k</sub>(S') can be written as:

$$ \mathrm{PFC}_k(S') = \bigoplus_{m+i+j=k} \mathbf{F}[h_{1P}, h_{2P}, h_3, e_P, e_3]_m \otimes \mathrm{PFC}_i(S) \otimes \mathrm{PFC}_j(T). $$

$\mathbf{F}[h_{1P}, h_{2P}, h_3, e_P, e_3]$ is a polynomial ring where $h_{1P}, h_{2P}, h_3$ (resp. $e_P, e_3$) are considered as Grassmann variables of odd degree (resp. even degree) and the subscript $m$ indicates the subspace spanned by monomials with total exponent $m$.

Let us write a generator of $\mathrm{PFC}_k(S')$ as $\gamma \otimes \Gamma_1 \otimes \Gamma_2$, where $\Gamma_1 \in \mathrm{PFC}_i(S)$, $\Gamma_2 \in \mathrm{PFC}_j(T)$, and $\gamma$ is constructed from orbits passing through $S' - S - T$. Using Lemma 5.2.1 and the description of ECH index one curves in $\mathbf{R} \times N_P$ from Lemma 5.2.2, we write the differential $\partial$ of $\mathrm{PFC}_k(S')$ as follows:

$$ \begin{aligned} \partial(\gamma \otimes \Gamma_1 \otimes \Gamma_2) &= \gamma \otimes (\partial_S \Gamma_1) \otimes \Gamma_2 + \gamma \otimes \Gamma_1 \otimes (\partial_T \Gamma_2) \\ &\quad + (\gamma/e_P)(h_3 \otimes \Gamma_1 \otimes \Gamma_2 + 1 \otimes h_1 \Gamma_1 \otimes \Gamma_2 + 1 \otimes \Gamma_1 \otimes h_2 \Gamma_2) \\ &\quad + (\gamma/h_{1P})(e_3 \otimes \Gamma_1 \otimes \Gamma_2 + 1 \otimes e_1 \Gamma_1 \otimes \Gamma_2) \\ &\quad + (\gamma/h_{2P})(e_3 \otimes \Gamma_1 \otimes \Gamma_2 + 1 \otimes \Gamma_1 \otimes e_2 \Gamma_2). \end{aligned} $$

Here $\partial_S$ and $\partial_T$ are the differentials on PFC(S) and PFC(T).

**5.3.2. Spectral sequence calculation.** — In this subsection we use spectral sequences to prove the following:
Lemma 5.3.1. — The inclusion

$$ \bigoplus_{i+j=k} \mathrm{PFC}_i(S) \otimes \mathrm{PFH}_j(T) \to \mathrm{PFC}_k(S') $$

induces an isomorphism between the quotient $(\bigoplus_{i+j=k} \mathrm{PFH}_i(S) \otimes \mathrm{PFH}_j(T)) / \sim$ and $\mathrm{PFH}_k(S')$, where the equivalence relation $\sim$ is given by $e_1 \Gamma_1 \otimes \Gamma_2 \sim \Gamma_1 \otimes e_2 \Gamma_2$.

*Proof.* — Let $\mathcal{F}$ be a filtration on $(\mathrm{PFC}_k(S'), \partial_t)$ which, on the generators, counts the multiplicity of $h_{1P}$. This means that $\mathcal{F}$ takes values in $\{0, 1\}$. We write $(E'(\mathcal{F}), \partial_t)$ for the $E'$-term and the $E'$-differential of the spectral sequence for $\mathcal{F}$. Each page $E'(\mathcal{F})$ has a grading coming from $\mathcal{F}$ and $E'_\ell(\mathcal{F})$ is the degree $\ell$ component of $E'(\mathcal{F})$ with respect to this grading. We remark that the spectral sequence associated to $\mathcal{F}$ is nothing but a long exact sequence in homology, and its use is motivated by our wish to give a parallel treatment of the cases where we filter by the multiplicity of a hyperbolic orbit or by the multiplicity of an elliptic orbit.

Next let $\mathcal{G}$ be a filtration on $(E^0(\mathcal{F}), \partial_0)$ which counts the multiplicity of $h_{2P}$. Again, $\mathcal{G}$ takes values in $\{0, 1\}$. We write $(E'(\mathcal{G}), \partial_{0r})$ for the $E'$-term and the $E'$-differential of the spectral sequence for $\mathcal{G}$. Finally, let $\mathcal{H}$ be the filtration on $(E^0(\mathcal{G}), \partial_{00})$ which counts the multiplicity of $e_P$, and let $(E'(\mathcal{H}), \partial_{00r})$ be the $E'$-term and the $E'$-differential of the spectral sequence for $\mathcal{H}$.

We first consider $(E^0(\mathcal{H}), \partial_{000})$, where:

$$ \partial_{000}(\boldsymbol{\gamma} \otimes \boldsymbol{\Gamma}_1 \otimes \boldsymbol{\Gamma}_2) = \boldsymbol{\gamma} \otimes (\partial_S \boldsymbol{\Gamma}_1) \otimes \boldsymbol{\Gamma}_2 + \boldsymbol{\gamma} \otimes \boldsymbol{\Gamma}_1 \otimes (\partial_T \boldsymbol{\Gamma}_2). $$

By the Künneth formula, we have:

$$ E^1(\mathcal{H}) = \bigoplus_{m+i+j=k} \mathbf{F}[h_{1P}, h_{2P}, h_3, e_P, e_3]_m \otimes \mathrm{PFH}_i(S) \otimes \mathrm{PFH}(T)_j. $$

Next consider $(E^1(\mathcal{H}), \partial_{001})$, where:

$$ \begin{aligned} \partial_{001}(e_P^\nu \boldsymbol{\gamma} \otimes \boldsymbol{\Gamma}_1 \otimes \boldsymbol{\Gamma}_2) = & e_P^{n-1} h_3 \boldsymbol{\gamma} \otimes \boldsymbol{\Gamma}_1 \otimes \boldsymbol{\Gamma}_2 + e_P^{n-1} \boldsymbol{\gamma} \otimes (h_1 \boldsymbol{\Gamma}_1) \otimes \boldsymbol{\Gamma}_2 \\ & + e_P^{n-1} \boldsymbol{\gamma} \otimes \boldsymbol{\Gamma}_1 \otimes (h_2 \boldsymbol{\Gamma}_2), \end{aligned} $$

Here $\boldsymbol{\Gamma}_1 \in \mathrm{PFH}_i(S)$, $\boldsymbol{\Gamma}_2 \in \mathrm{PFH}_j(T)$, and $\boldsymbol{\gamma}$ has no $e_P$ term. Note that any $e_P^{n-1} h_3 \boldsymbol{\gamma} \otimes \boldsymbol{\Gamma}_1 \otimes \boldsymbol{\Gamma}_2$ is homologous (with respect to the differential $\partial_{001}$) to a linear combination of $e_P^{n-1} \boldsymbol{\gamma}' \otimes \boldsymbol{\Gamma}'_1 \otimes \boldsymbol{\Gamma}'_2$, where $\boldsymbol{\gamma}'$ does not have any $e_P$ and $h_3$ terms, $\boldsymbol{\Gamma}'_1$ is in some $\mathrm{PFH}_i(S)$, and $\boldsymbol{\Gamma}'_2$ is in some $\mathrm{PFH}(T)_j$. Hence every element of $E^1(\mathcal{H}) / \mathrm{Im}(\partial_{001})$ can be represented by $w = \sum_{i \ge 0} e_P^i w_i$, where $w_i$ has no $e_P$ and $h_3$ terms, and we can write

$$ \partial_{001}(w) = \sum_{i>0} e_P^{i-1} h_3 w_i + w', $$
where $w'$ has no terms which contain $h_3$. If $\partial_{001}(w) = 0$, then all the $w_i$, $i > 0$, must be zero. We therefore obtain:

$$E^2(\mathcal{H}) = \bigoplus_{m+i+j=k} \mathbf{F}[h_{1P}, h_{2P}, e_3]_m \otimes \mathrm{PFH}_i(\mathcal{S}) \otimes \mathrm{PFH}_j(\mathcal{T}).$$

Since $E^2(\mathcal{H})$ is supported in degree 0, the spectral sequence for $\mathcal{H}$ converges at the $E^2$-term and we have $E^1(\mathcal{G}) \simeq E^2(\mathcal{H})$. Moreover, $E^2(\mathcal{H}) = E_0^2(\mathcal{H})$ is naturally isomorphic to $E^1(\mathcal{G})$ since 0 is the lowest filtration level.

Now consider $(E^1(\mathcal{G}), \partial_{01})$, where

$$\partial_{01}(\gamma \otimes \Gamma_1 \otimes \Gamma_2) = e_3(\gamma/h_{2P}) \otimes \Gamma_1 \otimes \Gamma_2 + (\gamma/h_{2P}) \otimes \Gamma_1 \otimes e_2 \Gamma_2.$$

The calculation of $E^2(\mathcal{G})$ is similar to the calculation of $E^2(\mathcal{H})$ in the previous paragraph. Any $e_3^{n+1}\gamma \otimes \Gamma_1 \otimes \Gamma_2$ is homologous to a linear combination of $e_3''\gamma' \otimes \Gamma_1' \otimes \Gamma_2'$, where $\gamma, \gamma'$ do not have any $h_{2P}$ and $e_3$ terms. Hence every element of $E^1(\mathcal{G})/\mathrm{Im}(\partial_{01})$ can be represented by $w = w' + \sum_{i=0}^k h_{2P}e_3^i w_i$, where $w'$ and $w_i$ have no $h_{2P}$ or $e_3$ terms, and we can write

$$\partial_{01}w = \sum_{i=0}^{k} e_3^{i+1}w_i + \sum_{i=1}^{k} e_3^i e_2 w_i.$$

If $\partial_{01}w = 0$, then all the $w_i$ must be zero. Only the $w'$ term remains, and we have:

$$E^2(\mathcal{G}) = \bigoplus_{m+i+j=k} \mathbf{F}[h_{1P}]_m \otimes \mathrm{PFH}_i(\mathcal{S}) \otimes \mathrm{PFH}_j(\mathcal{T}),$$

which we can write as a direct sum $\mathcal{L}_0 \oplus \mathcal{L}_1$, where:

$$\mathcal{L}_0 = \bigoplus_{i+j=k} \mathrm{PFH}_i(\mathcal{S}) \otimes \mathrm{PFH}_j(\mathcal{T}),$$

$$\mathcal{L}_1 = \bigoplus_{i+j=k-1} \mathbf{F}\{h_{1P}\} \otimes \mathrm{PFH}_i(\mathcal{S}) \otimes \mathrm{PFH}_j(\mathcal{T}).$$

Since $E^2(\mathcal{G})$ is supported in the lowest degree 0, the spectral sequence for $\mathcal{G}$ converges at the $E^2$-term and $E^2(\mathcal{G}) = E_0^2(\mathcal{G})$ is naturally isomorphic to $E^1(\mathcal{F})$.

Finally, $E^1(\mathcal{F}) \simeq \mathcal{L}_0 \oplus \mathcal{L}_1$ has differential $\partial_1$ given by:

$$(5.3.1) \qquad \partial_1(\Gamma_1 \otimes \Gamma_2) = 0,$$

$$(5.3.2) \qquad \partial_1(h_{1P} \otimes \Gamma_1 \otimes \Gamma_2) = e_1 \Gamma_1 \otimes \Gamma_2 + \Gamma_1 \otimes e_2 \Gamma_2,$$

since $\partial(h_{1P} \otimes \Gamma_1 \otimes \Gamma_2) = e_1 \Gamma_1 \otimes \Gamma_2 + e_3 \otimes \Gamma_1 \otimes \Gamma_2$ and $\partial(h_{2P} \otimes \Gamma_1 \otimes \Gamma_2) = \partial_0(h_{2P} \otimes \Gamma_1 \otimes \Gamma_2) = e_3 \otimes \Gamma_1 \otimes \Gamma_2 + \Gamma_1 \otimes e_2 \Gamma_2$. Viewing the spectral sequence as a long exact sequence
with connecting homomorphism $\mathcal{L}_1 \xrightarrow{\delta} \mathcal{L}_0$, where $\delta$ is given by Equation (5.3.2), we see that $E^2(\mathcal{F})$ is isomorphic to $\mathcal{L}_0/\mathrm{Im}(\delta) = \mathcal{L}_0/\sim$, where the equivalence relation is given by $e_1\Gamma_1 \otimes \Gamma_2 \sim \Gamma_1 \otimes e_2\Gamma_2$. As argued previously, the spectral sequence converges at the $E^2$-term and $E^2(\mathcal{F})$ is naturally isomorphic to $\mathrm{PFH}_k(S')$. $\square$

**5.4. Stabilization maps.** — For every integer $k \ge g$ we define

$$i_k : \mathrm{PFC}_{2k}(S, \mathfrak{h}) \to \mathrm{PFC}_{2k}(S', \mathfrak{h}')$$

as the map induced by the inclusion, and

$$j_k : \mathrm{PFC}_{2k}(S, \mathfrak{h}) \to \mathrm{PFC}_{2k+2}(S, \mathfrak{h})$$

as $j_k(\gamma) = e_1^2\gamma$ for every orbit set $\gamma \in \mathrm{PFC}_{2k}(S, \mathfrak{h})$. We also define the stabilization map $\mathfrak{S}_k = i_{k+1} \circ j_k$.

The maps $j_k$ are chain maps by [0, Lemma 5.3.2] and the maps $i_k$ are chain maps by Lemma 5.2.1; therefore the maps $\mathfrak{S}_k$ are also chain maps.

**Lemma 5.4.1.** — If $(\mathfrak{S}_k)_*$ is an isomorphism, then $(j_k)_*$ is an isomorphism.

*Proof.* — If $(\mathfrak{S}_k)_*$ is an isomorphism, then $(j_k)_*$ : $\mathrm{PFH}_{2g}(S) \to \mathrm{PFH}_{2g+2}(S)$ is injective. It remains to show that it is surjective.

First note that $\mathrm{PFH}_0(T) = \mathbf{F}\{1\}$. We also have $\mathrm{PFH}_2(T) = \mathbf{F}\{e_2^2\}$, since $\widehat{\mathrm{HF}}(S^3) \simeq \mathbf{F}$, generated by the contact class $c(\xi_{std})$ of the standard tight contact structure $\xi_{std}$ on $S^3$, and $e_2^2$ is the image of $c(\xi_{std})$ under the isomorphism

$$(\Phi_{(T, \mathfrak{h}_T)})_* : \widehat{\mathrm{HF}}(S^3) = \widehat{\mathrm{HF}}(T, \mathfrak{h}_T) \xrightarrow{\sim} \mathrm{PFH}_2(T).$$

Now, since the isomorphism $\mathrm{PFH}_0(T) \xrightarrow{\sim} \mathrm{PFH}_2(T)$, $1 \mapsto e_2^2$, factors as

$$\mathrm{PFH}_0(T) \to \mathrm{PFH}_1(T) \to \mathrm{PFH}_2(T), \quad 1 \mapsto e_2 \mapsto e_2^2,$$

it follows that $\mathrm{PFH}_1(T) = \mathbf{F}\{e_2\} \oplus \mathbf{W}$ for some $\mathbf{F}$-vector space $\mathbf{W}$.

By Lemma 5.3.1,

$$\mathrm{PFH}_{2k+2}(S') \simeq \left( \bigoplus_{i+j=2k+2} \mathrm{PFH}_i(S) \otimes \mathrm{PFH}_j(T) \right) / \sim.$$

Moreover, $\mathrm{PFH}_{2k}(S) \otimes \mathrm{PFH}_2(T) = \mathrm{PFH}_{2k}(S) \otimes \mathbf{F}\{e_2^2\}$ is the image of $\mathrm{PFH}_{2k}(S)$ under the map $(\mathfrak{S}_k)_*$. Since this map is an isomorphism, every element of $\mathrm{PFH}_{2k+2}(S) \otimes \mathrm{PFH}_0(T) = \mathrm{PFH}_{2k+2}(S) \otimes \mathbf{F}\{1\}$ is equivalent to some element of $\mathrm{PFH}_{2k}(S) \otimes \mathrm{PFH}_2(T)$,
i.e., if $v_{2k+2} \in \text{PFH}_{2k+2}(S)$, then $v_{2k+2} \otimes 1 \sim v_{2k} \otimes e_2^2$ for some $v_{2k} \in \text{PFH}_{2k}(S)$. More explicitly,

$$v_{2k+2} \otimes 1 + v_{2k} \otimes e_2^2 = (e_1 v'_{2k+1} \otimes 1 + v'_{2k+1} \otimes e_2) + (e_1 v'_{2k} \otimes e_2 + v'_{2k} \otimes e_2^2) \\ + \sum_i (e_1 v''_{2k,i} \otimes w_i + v''_{2k,i} \otimes e_2 w_i) + \dots,$$

where $v'_{2k+1} \in \text{PFH}_{2g+1}(S)$; $v'_{2k}$, $v''_{2k,i} \in \text{PFH}_{2g}(S)$; $\{w_i\}$ is a basis for W; and $\dots$ is a linear combination of terms which are not in $\text{PFH}_{2k+2}(S) \otimes \text{PFH}_0(T)$ and $\text{PFH}_{2k+1}(S) \otimes \text{PFH}_1(T)$. A term-by-term comparison gives $v_{2k+2} = e_1 v'_{2k+1}$ and $v'_{2k+1} = e_1 v'_{2k}$. Hence $v_{2k+2} = e_1^2 v'_{2k}$ and $v_{2k+2}$ is in the image of $(j_k)_*$. $\square$

**Corollary 5.4.2.** — If $(\mathfrak{S}_k)_*$ is an isomorphism, then $(i_{k+1})_*$ is an isomorphism.

**5.5. Isomorphisms from stabilisation.** — In this section we prove the following:

**Proposition 5.5.1.** — If $k \ge g$ and $h$ satisfies Condition $(\dagger\dagger)_{2k+2}$, then the map

$$(\mathfrak{S}_k)_* : \text{PFH}_{2k}(S, h) \to \text{PFH}_{2k+2}(S', h')$$

is an isomorphism.

The proof will be by induction on $k-g$. First we consider the base step $g=k$, which proved by comparing the stabilization maps in Heegaard Floer homology and periodic Floer homology.

Let $\mathbf{a} = \{a_1, \dots, a_{2g}\}$ be a basis of S. We extend $\mathbf{a}$ to $\bar{\mathbf{a}} = \{\bar{a}_1, \dots, \bar{a}_{2g}\}$ so that $\bar{a}_i$ is a properly embedded arc with boundary on $\partial S'$ and then complete $\bar{\mathbf{a}}$ to a basis $\mathbf{a}'$ of S' by adding the extensions $\bar{a}_{2g+1}$ and $\bar{a}_{2g+2}$ of the basis arcs $a_{2g+1}$ and $a_{2g+2}$ of T, subject to the following conditions:

- $\bar{a}_i - a_i$ and $h'(\bar{a}_j - a_j)$ are disjoint for $1 \le i \ne j \le 2g+2$;

- $\bar{a}_i$ and $h'(\bar{a}_i)$ have two extra pairs of canceling intersections $\bar{x}_i, \theta_i$ and $\bar{x}'_i, \theta'_i$ in $S' - (S \cup T)$ for $i = 1, \dots, 2g+2$;

- $\bar{x}_1, \bar{x}'_1, \dots \bar{x}_{2g+2}, \bar{x}'_{2g+2}$ lie on $\partial S'$.

We then define

$$\mathfrak{S}'_{\text{HF}} : \widehat{\text{CF}}'(S, \mathbf{a}, h(\mathbf{a})) \to \widehat{\text{CF}}'(S', \mathbf{a}', h'(\mathbf{a}'))$$

as $\mathfrak{S}_{\text{HF}}(\mathbf{y}) = \mathbf{y} \cup \{\bar{x}_{2g+1}, \bar{x}_{2g+2}\}$. It is easy to see that $\mathfrak{S}_{\text{HF}}$ is a chain map. (Compare with the gluing map from [HKM].)

**Lemma 5.5.2.** — The map $\mathfrak{S}'_{\text{HF}}$ induces a map

$$(\mathfrak{S}_{\text{HF}})_* : \widehat{\text{HF}}(S, \mathbf{a}, h(\mathbf{a})) \to \widehat{\text{HF}}(S', \mathbf{a}', h'(\mathbf{a}))$$
which is an isomorphism.

*Proof.* — Let $S^e$ and $T^e$ be extensions of S and T such that:

- $\mathbf{a}'_S = \{\bar{a}_1, \dots, \bar{a}_{2g}\}$ and $\mathbf{h}'(\mathbf{a}'_S) = \mathbf{h}'(\{\mathbf{h}'(\bar{a}_1), \dots, \mathbf{h}'(\bar{a}_{2g})\})$ are bases of arcs of $S^e$,

- $\mathbf{a}'_T = \{\bar{a}_{2g+1}, \bar{a}_{2g+2}\}$ and $\mathbf{h}'(\mathbf{a}'_T) = \{\mathbf{h}'(\bar{a}_1), \mathbf{h}'(\bar{a}_{2g+2})\}$ are bases of arcs of $T^e$,

- $S'$ can be seen as a boundary connected sum of $S^e$ and $T^e$.

It is easy to see that

$$ \widehat{\mathrm{CF}}(S'\mathbf{a}', \mathbf{h}'(\mathbf{a}')) \cong \widehat{\mathrm{CF}}(S^e\mathbf{a}'_S, \mathbf{h}'(\mathbf{a}'_S)) \otimes \widehat{\mathrm{CF}}(T^e, \mathbf{a}'_T, \mathbf{h}'(\mathbf{a}'_T)) $$

and $\widehat{\mathrm{HF}}(T^e, \mathbf{a}'_T, \mathbf{h}'(\mathbf{a}'_T)) \cong \mathbf{F}$ and is generated by the class of $\{\bar{x}_{2g+1}, \bar{x}_{2g+2}\}$ because $(T^e, \mathbf{a}'_T, \mathbf{h}'(\mathbf{a}'_T))$ represents $S^3$. Hence the map

$$ \widehat{\mathrm{CF}}(S^e, \mathbf{a}'_S, \mathbf{h}'(\mathbf{a}'_S)) \to \widehat{\mathrm{CF}}(S'\mathbf{a}', \mathbf{h}'(\mathbf{a}')), \quad \mathbf{y} \mapsto \mathbf{y} \cup \{\bar{x}_{2g+1}, \bar{x}_{2g+2}\} $$

induces an isomorphism in homology. In view of the discussion above, it remains to prove that the inclusion $\widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \widehat{\mathrm{CF}}(S^e, \mathbf{a}'_S, \mathbf{h}'(\mathbf{a}'_S))$ induces a map $\widehat{\mathrm{HF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \widehat{\mathrm{HF}}(S^e, \mathbf{a}'_S, \mathbf{h}'(\mathbf{a}'_S))$ which is an isomorphism. Note that the inclusion does not factor through a map $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \widehat{\mathrm{CF}}(S^e, \mathbf{a}'_S, \mathbf{h}'(\mathbf{a}'_S))$; we will need a slightly more complicated argument.

Let $\mathcal{Q} \subset \widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$ be the subspace generated by elements of the form $\mathbf{y} \cup \{x_i\} + \mathbf{y} \cup \{x'_i\}$ for some $i = 1, \dots, 2g$, where $\mathbf{y}$ is a $(2g-1)$-uple of intersection points none of which is in $a_i$ or $\mathbf{h}(a_i)$. Then $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) = \widehat{\mathrm{CF}}'(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) / \mathcal{Q}$. Similarly, we define $\mathcal{Q} \subset \widehat{\mathrm{CF}}'(S^e, \mathbf{a}'_S, \mathbf{h}(\mathbf{a}'_S))$ as the subspace generated by elements of the form $\mathbf{y} \cup \{\bar{x}_i\} + \mathbf{y} \cup \{\bar{x}'_i\}$. Then, as before, $\widehat{\mathrm{CF}}(S^e, \mathbf{a}'_S, \mathbf{h}(\mathbf{a}'_S)) = \widehat{\mathrm{CF}}'(S^e, \mathbf{a}'_S, \mathbf{h}(\mathbf{a}'_S)) / \mathcal{Q}$.

We also define $\mathcal{R} \subset \widehat{\mathrm{CF}}'(S^e, \mathbf{a}'_S, \mathbf{h}(\mathbf{a}'_S))$ as the subspace generated by elements of the form $\mathbf{y} \cup \{\bar{x}_i\} + \mathbf{y} \cup \{\bar{x}'_i\}, \mathbf{y} \cup \{x_i\} + \mathbf{y} \cup \{x'_i\}$ and $\mathbf{y} \cup \{\theta_i\} + \mathbf{y} \cup \{\theta'_i\}$ for some $i = 1, \dots, 2g$. Since both $\mathcal{Q}$ and $\mathcal{Q}$ are subcomplexes of $\mathcal{R}$, we have chain maps

$$ (5.5.1) \qquad \widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a})) \to \widehat{\mathrm{CF}}'(S^e, \mathbf{a}'_S, \mathbf{h}(\mathbf{a}'_S)) / \mathcal{R}, $$

$$ (5.5.2) \qquad \widehat{\mathrm{CF}}(S^e, \mathbf{a}'_S, \mathbf{h}'(\mathbf{a}'_S)) \to \widehat{\mathrm{CF}}'(S^e, \mathbf{a}'_S, \mathbf{h}(\mathbf{a}'_S)) / \mathcal{R}. $$

The map (5.5.2) induces an isomorphism in homology because $H(R/\mathcal{Q}) \cong 0$. This can be proved in a similar manner to Theorem I.4.9.4. The map $(\mathcal{S}_{\mathrm{HF}})_*$ is then the composition of map induced by (5.5.1) with the inverse of the map induced by (5.5.2).

It remains to show that the map induced by (5.5.1) is an isomorphism. We define $\tilde{\mathbb{R}} \subset \widehat{\mathrm{CF}}'(S^e, \mathbf{a}'_S, \mathbf{h}(\mathbf{a}'_S))$ as the subspace generated by intersection points of the form $\mathbf{y} \cup \{\bar{x}_i\} \mathbf{y} \cup \{\bar{x}'_i\}, \mathbf{y} \cup \{\theta_i\}$ and $\mathbf{y} \cup \{\theta'_i\}$ for $i = 1, \dots, 2g$. If we apply the identity of vector spaces

$$ X/(A+B) \cong \frac{X/A}{B/(B \cap A)} $$
to $X = \widehat{CF}'(S^e, \mathbf{a}'_S, \mathcal{h}'(\mathbf{a}'_S))$, $A = \widehat{CF}'(S, \mathbf{a}, \mathcal{h}(\mathbf{a}))$, $B = R$ and observe that $X/A \cong \tilde{R}$ and $A \cap B \cong Q$, we obtain the short exact sequence

$$0 \to \widehat{CF}(S, \mathbf{a}, \mathcal{h}(\mathbf{a})) \to \widehat{CF}'(S^e, \mathbf{a}'_S, \mathcal{h}'(\mathbf{a}'_S))/R \to \tilde{R}/(R/Q) \to 0.$$

Since both $\tilde{R}$ and $R/Q$ are acyclic (by an argument similar to the proof of Theorem I.4.9.4), it follows that the map (5.5.1) induces an isomorphism in homology. $\square$

**Lemma 5.5.3.** — The diagram of chain complexes

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
(5.5.3) \arrow[r, "\widehat{CF}'(S, \mathbf{a}, \mathcal{h}(\mathbf{a}))"] & \widehat{CF}'(S', \mathbf{a}', \mathcal{h}'(\mathbf{a}')) \arrow[r, "\mathcal{S}'_{\text{HF}}"] & \widehat{CF}'(S', \mathbf{a}', \mathcal{h}'(\mathbf{a}')) \arrow[d, "\Phi'_{(S, \mathcal{h})}"] \\
& PFC_{2g}(S, \mathcal{h}) \arrow[r, "\mathcal{S}_{2g}", "\cong"] & PFC_{2g+2}(S', \mathcal{h}') \arrow[d, "\Phi'_{(S', \mathcal{h}')}"]
\end{tikzcd}$$

commutes up to homotopy.

*Proof.* — Since $x_{2g+1}$ and $x_{2g+2}$ are components of the contact class, there are no holomorphic curves in $W_+$, besides restrictions of trivial cylinders, which have a positive limit to $x_{2g+1}$ or $x_{2g}$. The argument is the same as that of Lemma I.6.2.3.

The restriction of the trivial cylinder over $x_i$, $i = 2g + 1, 2g + 2$ has its negative end to a generic point $p_i$ of the Morse-Bott family $\mathcal{N}_3$ and is concatenated with a cylinder corresponding to a gradient trajectory from $p_i$ to $e_3$ to give a Morse-Bott building. Moreover, by automatic transversality [We1, Theorem 4.5.36] and the discussion from Lemma I.5.8.9, the above Morse-Bott building is Morse-Bott regular. Hence the pair $\{x_{2g+1}, x_{2g+2}\}$ is “mapped” to $e_3^2$.

By arguments similar to the proof of Lemma 5.2.1, every $W_+$-curve for $(S', \mathcal{h}')$ which is positively asymptotic to $\mathbf{y} \cup \{x_{2g+1}, x_{2g+2}\}$ is the disjoint union of a $W_+$-curve for $(S, \mathcal{h})$ which is positively asymptotic to $\mathbf{y}$ with the Morse-Bott buildings from $x_{2g+1}$ and $x_{2g+2}$ described above. This implies that

$$\Phi'_{(S', \mathcal{h}')}(\mathbf{y} \cup \{x_{2g+1}, x_{2g+2}\}) = e_3^2 \cdot \Phi'_{(S, \mathcal{h})}(\mathbf{y}).$$

Then the map $\mathbf{y} \mapsto \mathcal{h}_{1P}(e_1 + e_3)\Phi'_{(S, \mathcal{h})}(\mathbf{y})$ is a chain homotopy between $\Phi'_{(S', \mathcal{h}')} \circ \mathcal{S}_{\text{HF}}$ and $\mathcal{S}_g \circ \Phi'_{(S, \mathcal{h})}$. $\square$
**Corollary 5.5.4.** — *The diagram*

$$ (5.5.4) \qquad \begin{array}{ccc} \widehat{\mathrm{HF}}(S, \mathbf{a}, \mathfrak{h}(\mathbf{a})) & \xrightarrow{(\mathfrak{S}_{\mathrm{HF}})_*} & \widehat{\mathrm{HF}}(S', \mathbf{a}', \mathfrak{h}'(\mathbf{a}')) \\ \Phi_{(S, \mathfrak{h})}_* \downarrow & & \Phi_{(S', \mathfrak{h}')}_* \downarrow \\ \mathrm{PFH}_{2g}(S, \mathfrak{h}) & \xrightarrow{(\mathfrak{S}_g)_*} & \mathrm{PFH}_{2g+2}(S', \mathfrak{h}') \end{array} $$

commutes.

*Proof.* — Since we are working over a field, the map (5.5.2) has an inverse up to homotopy. □

The following lemma provides the base step of the induction.

**Lemma 5.5.5.** — If $\mathfrak{h}$ satisfies Condition $(\dagger\dagger)_{2g+2}$, then the map

$$ (\mathfrak{S}_g)_* : \mathrm{PFH}_{2g}(S, \mathfrak{h}) \to \mathrm{PFH}_{2g+2}(S', \mathfrak{h}') $$

is an isomorphism.

*Proof.* — Since $\mathfrak{h}$ satisfies Condition $(\dagger\dagger)_{2g+2}$, we can arrange the construction of $\mathfrak{h}'$ so that it also satisfies Condition $(\dagger\dagger)_{2g+2}$. Then by Theorem 1.0.1 the vertical arrows of Diagram (5.5.4) are isomorphisms. The map $(\mathfrak{S}_{\mathrm{HF}})_*$ is also an isomorphism since $\widehat{\mathrm{HF}}(S', \mathbf{a}', \mathfrak{h}'(\mathbf{a}'))$ can be computed as the tensor product of the S and T sides and the T side gives $\widehat{\mathrm{HF}}(S^3)$, which is generated by $\{x_{2g+1}, x_{2g+2}\}$. Then $(\mathfrak{S}_g)_*$ is an isomorphism. □

The next lemma provides the inductive step of the induction, which in turn completes the proof of Proposition 5.5.1.

**Lemma 5.5.6.** — If there exists $k_0 \ge 1$ such that, for all $g > 0$, surfaces S of genus g, and monodromies $\mathfrak{h}: S \to S$ satisfying Condition $(\dagger\dagger)_{2g+2k_0}$, the map

$$ (\mathfrak{S}_{g+k_0-1})_* : \mathrm{PFH}_{2g+2k_0-2}(S, \mathfrak{h}) \to \mathrm{PFH}_{2g+2k_0}(S', \mathfrak{h}') $$

is an isomorphism, then for all $g > 0$ and monodromies $\mathfrak{h}: S \to S$ satisfying Condition $(\dagger\dagger)_{2g+2k_0+2}$ the map

$$ (\mathfrak{S}_{g+k_0})_* : \mathrm{PFH}_{2g+2k_0}(S, \mathfrak{h}) \to \mathrm{PFH}_{2g+2k_0+2}(S', \mathfrak{h}') $$

is an isomorphism.
*Proof.* — Let $(S'', \mathfrak{h}'')$ be a stabilization of $(S', \mathfrak{h}')$, i.e., a double stabilization of $(S, \mathfrak{h})$. If $\mathfrak{h}$ satisfies Condition $(\dagger\dagger)_{2g+2k_0+2}$, then we can arrange $\mathfrak{h}'$ and $\mathfrak{h}''$ so that they also satisfy Condition $(\dagger\dagger)_{2g+2k_0+2}$. It is easy to see that the diagram

$$ (5.5.5) \qquad \begin{array}{ccc} \text{PFH}_{2g+2k_0}(S, \mathfrak{h}) & \xrightarrow{(\mathfrak{S}'_{g+k_0})_*} & \text{PFH}_{2g+2k_0+2}(S', \mathfrak{h}') \\ \Big\downarrow_{(i_{g+k_0})_*} & & \Big\downarrow_{(i'_{g+k_0})_*} \\ \text{PFH}_{2g+2k_0}(S', \mathfrak{h}') & \xrightarrow{(\mathfrak{S}'_{g+k_0})_*} & \text{PFH}_{2g+2k_0+2}(S'', \mathfrak{h}'') \end{array} $$

commutes. Here $\mathfrak{S}'_{g+k_0}$ and $i'_{g+k_0}$ are the obvious analogues of $\mathfrak{S}_{g+k_0}$ and $i_{g+k_0}$. The proof that the corresponding chain level diagram commutes up to homotopy is similar to the proof that Diagram (5.5.3) commutes up to homotopy.

Since $g + k_0 = g(S') + k_0 - 1$, the map $(\mathfrak{S}'_{g+k_0})_*$ is an isomorphism by hypothesis. The map $(\mathfrak{S}_{g+k_0-1})_*$ is also an isomorphism because $\mathfrak{h}$ satisfies Condition $(\dagger\dagger)_{2g+2k_0}$ a fortiori. Then, by Corollary 5.4.2, the maps $(i_{g+k_0})_*$ and $(i'_{g+k_0})_*$ are isomorphisms. Therefore, the commutativity of Diagram (5.5.5) implies that $(\mathfrak{S}_{g+k_0})_*$ is an isomorphism. $\square$

**5.6.** *Proof of Theorem 1.0.2.* — The following is similar to Theorem I.2.5.2 with a slightly simpler proof which is left to the reader:

*Lemma 5.6.1.* — Let $(\alpha_0 = dt, \omega = d\beta)$ be a stable Hamiltonian structure on $N$ with stable Hamiltonian vector field $R$ such that all closed orbits of $R$ that intersect $\text{int}(S)$ at most $k$ times are hyperbolic. Then, for every $\delta > 0$ sufficiently small, there is a stable Hamiltonian structure $(\alpha_0, \omega_\delta = d\beta_\delta)$ with stable Hamiltonian vector field $R_\delta$ such that:

(1) $\beta$ and $\beta_\delta$ (and hence also $R$ and $R_\delta$) coincide outside of a $\delta$-neighborhood $V_\delta$ of the orbits of $R$ which intersect $\text{int}(S)$ exactly $k+1$ times,

(2) $\beta_\delta \to \beta$ in the $C^1$-topology, and hence $R_\delta \to R$ in the $C^0$ topology, as $\delta \to 0$, and

(3) all orbits of $R_\delta$ that intersect $\text{int}(S)$ at most $k+1$ times are hyperbolic.

We perturb the stable Hamiltonian structures $(\alpha_0, \omega)$ and $(\alpha_0, \omega_\delta)$ to contact structures

$$ \alpha_{\zeta} = dt + \zeta\beta, \quad \alpha_{\delta, \zeta} = dt + \zeta\beta_{\delta} $$

for $\zeta > 0$ sufficiently small as in Equation (I.3.1.1). The $C^1$-convergence $\beta_\delta \to \beta$ ensures that $\zeta$ can be chosen independently of $\delta$. Note that, in the notation of Section I.3.1, $\beta = \beta_t + ftdt$. The Reeb vector fields $R_\zeta$ of $\alpha_\zeta$ and $R_{\delta,\zeta}$ of $\alpha_{\delta,\zeta}$ are parallel to $R$ and $R_\delta$ respectively. By Theorem I.3.6.1, for every $\delta > 0$ sufficiently small there exists $\zeta'_\delta > 0$ such that for all $0 < \zeta_\delta < \zeta'_\delta$, $\alpha_{\delta,\zeta_\delta}$ is a contact form and there is an isomorphism of chain complexes

$$ (5.6.1) \qquad \text{PFC}_j(N, \alpha_0, \omega_\delta) \cong \text{ECC}_j(N, \alpha_{\delta, \zeta_\delta}) $$
for all $j \le k + 1$. Moreover, there is $\zeta'_0 > 0$ such that for all $0 < \zeta < \zeta'_0$, $\alpha_{\zeta_0}$ is a contact form and there is an isomorphism of chain complexes

$$ (5.6.2) \qquad \mathrm{PFC}_j(\mathbb{N}, \alpha_0, \omega) \cong \mathrm{ECC}_j(\mathbb{N}, \alpha_{\zeta_0}) $$

for all $j \le k$.

In the next two lemmas we study the continuation maps between the various contact forms appearing in Lemma 5.6.1.

**Lemma 5.6.2.** — There exists $\epsilon > 0$ such that for every $0 < \zeta, \zeta' < \epsilon$ the forms $\alpha_\zeta$ and $\alpha_{\zeta'}$ are contact forms, the identifications from Equation (5.6.2) hold, and the continuation map

$$ (\mathcal{R}'_j)_* : \mathrm{ECH}_j(\mathbb{N}, \alpha_\zeta) \to \mathrm{ECH}_j(\mathbb{N}, \alpha_{\zeta'}) $$

given by Equation (I.2.5.6) with $\alpha_j$ and $\alpha_{j'}$ replaced by $\alpha_\zeta$ and $\alpha_{\zeta'}$, coincides with the map induced by the identifications from Equation (5.6.2) for $j \le k$.

*Proof.* — By the Holomorphic Curves Property in [HT3, Theorem 2.4], the continuation map $(\mathcal{R}'_j)_*$ is induced by a noncanonical chain map $\mathcal{R}'_j$ which is supported on $J_{\zeta, \zeta'}$-holomorphic buildings of ECH index $I = 0$ in an exact symplectic cobordism $(\mathbb{R} \times \mathbb{N}, \Omega_{\zeta, \zeta'})$ between $\alpha_\zeta$ and $\alpha_{\zeta'}$ for a generic almost complex structure $J_{\zeta, \zeta'}$ which is compatible with $\Omega_{\zeta, \zeta'}$. This means that, if $(\mathcal{R}'_j(\boldsymbol{\gamma}), \boldsymbol{\gamma}') \neq 0$, then there is an $I = 0$ $J_{\zeta, \zeta'}$-holomorphic building in $\mathbb{R} \times \mathbb{N}$ from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$. Strictly speaking, the continuation maps are defined by passing to a closure $\overline{M}$ of $\mathbb{N}$, but the positivity of intersections implies that the holomorphic buildings in $\mathbb{R} \times \overline{M}$ from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$ are contained in $\mathbb{R} \times \mathbb{N}$; see [0, Lemma 5.2.3].

Now we choose sequences $\zeta_\ell, \zeta'_\ell \to 0$ as $\ell \to +\infty$ and generic almost complex structures $J_\ell = J_{\zeta_\ell, \zeta'_\ell}$ such that $J_\ell \to J$ as $\ell \to +\infty$, where $J$ is a generic almost complex structure on $\mathbb{R} \times \mathbb{N}$ which is adapted to the stable Hamiltonian structure $(\alpha_0, \omega)$. Then $I = 0$ $J_\ell$-holomorphic buildings converge to $I = 0$ $J$-holomorphic buildings. Since $J$ is generic and cylindrical, the only $I = 0$ $J$-holomorphic buildings are trivial cylinders. This implies that, for $\zeta$ and $\zeta'$ small enough, there is no $I = 0$ $J_{\zeta, \zeta'}$-holomorphic building between $\boldsymbol{\gamma}$ and $\boldsymbol{\gamma}'$ unless $\boldsymbol{\gamma} = \boldsymbol{\gamma}'$. Then the Holomorphic Curves Property implies that $\mathcal{R}'_j(\boldsymbol{\gamma}) = \epsilon(\boldsymbol{\gamma})\boldsymbol{\gamma}$.

Since we are working over $\mathbb{Z}/2\mathbb{Z}$, this implies that (after identifying $\mathrm{ECC}_j(\mathbb{N}, \alpha_\zeta)$ with $\mathrm{ECC}_j(\mathbb{N}, \alpha_{\zeta'})$ via Equation (5.6.2)), $(\mathcal{R}'_j)^2 = \mathcal{R}'_j$. Since $(\mathcal{R}'_j)_*$ is an isomorphism, we obtain that $(\mathcal{R}'_j)_*$ coincides with the map induced by the identification from Equation (5.6.2). $\square$

**Remark 5.6.3.** — A similar results holds for the continuation maps

$$ (\mathcal{R}'_j)_* : \mathrm{ECH}_j(\mathbb{N}, \alpha_{\delta, \zeta}) \to \mathrm{ECH}_j(\mathbb{N}, \alpha_{\delta, \zeta'}) $$

for $j \le k + 1$, where $\epsilon$ is allowed to depend on $\delta$.
**Lemma 5.6.4.** — For $j \le k$ and $\delta, \zeta > 0$ sufficiently small, the continuation map

$$ (\mathcal{R}_j'')_* : \mathrm{ECH}_j(N, \alpha_\zeta) \to \mathrm{ECH}_j(N, \alpha_{\delta, \zeta}) $$

given by Equation (I.2.5.6) with $\mathcal{R}_j'$, $\alpha_i$, and $\alpha_j$ replaced by $\mathcal{R}_j''$, $\alpha_\zeta$, and $\alpha_{\delta, \zeta}$, is induced by the chain map $\mathcal{R}_j''$ satisfying $\mathcal{R}_j''(\gamma) = \gamma$, and is therefore a quasi-isomorphism.

*Proof.* — Let $\delta_\ell$ and $\zeta_\ell$ be sequences converging to 0 as $\ell \to +\infty$. We choose the following generic almost complex structures on $\mathbf{R} \times \mathbf{N}$:

- an almost complex structure $J$ adapted to the stable Hamiltonian structure $(\alpha_0, \omega)$;

- almost complex structures $J_\ell$ adapted to the contact forms $\alpha_{\zeta_\ell}$ such that $J_\ell \to J$ in the $C^\infty$-topology on $\mathbf{R} \times \mathbf{N}$ as $\ell \to +\infty$;

- almost complex structures $J'_\ell$ adapted to the contact forms $\alpha_{\delta_\ell, \zeta_\ell}$ which coincide with $J_\ell$ on $\mathbf{R} \times (N - V_{\delta_\ell})$ and converge to $J_0$ in the $C^0$-topology on $\mathbf{R} \times \mathbf{N}$ as $\ell \to +\infty$;

- almost complex structures $\tilde{J}_\ell$ which are compatible with an exact symplectic cobordism $(\mathbf{R} \times \mathbf{N}, \Omega_\ell)$ between $\alpha_{\zeta_\ell}$ and $\alpha_{\delta_\ell, \zeta_\ell}$, interpolate between $J_\ell$ at the positive end and $J'_\ell$ at the negative end, coincides with $J_\ell$ and $J'_\ell$ on $\mathbf{R} \times (N \setminus V_{\delta_\ell})$, and converge to $J$ in the $C^0$-topology on $\mathbf{R} \times \mathbf{N}$ as $\ell \to +\infty$.

If $u_\ell$ is a sequence of $\tilde{J}_\ell$-holomorphic curves in $\mathbf{R} \times \mathbf{N}$ from an orbit set $\gamma'$ to $\gamma$ where both $\gamma$ and $\gamma'$ consist of orbits which intersect a fiber of $\mathbf{N}$ at most $k$ times, then, up to passing to a subsequence, $u_\ell$ converges to a $J$-holomorphic building. This follows from Gromov compactness for $C^0$-convergence of almost complex structures, due to Ivashkovich-Shevchishin [IS].

Alternatively, we can argue as follows, using the usual Gromov compactness for $C^\infty$-convergence of almost complex structures. By the compactness argument from Section I.3.4, we may assume that $[u_\ell] \in H_2(\mathbf{N}, \gamma', \gamma)$ and the topological type of the domains $F_\ell$ of $u_\ell$ are fixed. We then restrict $u_\ell$ to the preimage $G_\ell$ of $\mathbf{R} \times (N - V_{\delta_\ell})$. We may assume that $G_\ell$ is obtained from a compact Riemann surface with boundary by removing interior punctures. (This is possible by taking $\delta_\ell$ to be generic.)

We claim that $|\chi(G_\ell)|$ is bounded above. This is equivalent to an upper bound on the number of disk components of $F_\ell - G_\ell$. Let $\gamma''$ be the union of core orbits of $V_{\delta_\ell}$, which is independent of $\ell$. The number of disk components is bounded above by the intersection number with $\mathbf{R} \times \gamma''$, which in turn is controlled by the homology class $[u_\ell] \in H_2(\mathbf{N}, \gamma', \gamma)$.

Since $\tilde{J}_\ell \to J$ in $C_{\mathrm{loc}}^\infty$ on $\mathbf{R} \times (N - V_{\delta_\ell})$, and the ends $\gamma, \gamma'$ are contained in $N - V_{\delta_\ell}$ for all $\ell$, we can take the limit of $u_\ell|_{G_\ell}$ to obtain $u|_G$. For simplicity assume that $u$ has only one level. Then $G$ is a punctured surface and the punctures which are not mapped to $\gamma$ or $\gamma'$ are removable.
Next we use the Holomorphic Curves Property in [HT3, Theorem 2.4], which states that $\langle \mathcal{R}_j''(\boldsymbol{\gamma}'), \boldsymbol{\gamma} \rangle$ is supported on $\tilde{J}_\ell$-holomorphic buildings of ECH index $I=0$. The convergence discussed above implies that, for $\ell$ large enough, the only $\tilde{J}_\ell$-holomorphic buildings of index $I=0$ between orbit sets $\boldsymbol{\gamma}'$ and $\boldsymbol{\gamma}$, when both orbit sets consist of Reeb orbits intersecting a fiber of $N$ at most $k$ times, are covers of trivial cylinders contained in a product region. Then [HT3, Theorem 2.4] implies that $\mathcal{R}_j''(\boldsymbol{\gamma}) = \boldsymbol{\gamma}$ when $j \le k$. $\square$

**Lemma 5.6.5.** — There are sequences of stable Hamiltonian structures $(\alpha_0, \omega^k)$ and real numbers $\zeta_k$ (both indexed by $k$) satisfying the following:

(1) The stable Hamiltonian vector field $\mathbf{R}_k$ of $(\alpha_0, \omega^k)$ has no elliptic Reeb orbit in int($N$) that intersects a fiber at most $k$ times.

(2) $\mathbf{R}_k$ and $\mathbf{R}_{k+1}$ coincide outside of a neighborhood of the orbits of $\mathbf{R}_k$ which intersect a fiber of $N$ exactly $k+1$ times.

(3) $\alpha_\zeta^k$ is a contact form for every $0 < \zeta \le \zeta_k$.

(4) For every $j \le k$ and $0 < \zeta \le \zeta_k$ there are canonical identifications

$$ \mathrm{PFC}_j(\mathrm{N}, \alpha_0, \omega^k) \cong \mathrm{ECC}_j(\mathrm{N}, \alpha_\zeta^k). $$

(5) For every $j \le k$ and $0 < \zeta, \zeta' \le \zeta_k$ the continuation maps

$$ (\mathcal{R}_j')_* : \mathrm{ECC}_j(\mathrm{N}, \alpha_\zeta^k) \to \mathrm{ECC}_j(\mathrm{N}, \alpha_{\zeta'}^{k'}) $$

are induced by the identifications of (4).

(6) For every $j \le k$ the continuation maps

$$ (\mathcal{R}_j)_* : \mathrm{ECH}_j(\mathrm{N}, \alpha_{\zeta_k}^k) \to \mathrm{ECH}_j(\mathrm{N}, \alpha_{\zeta_{k+1}}^{k+1}) $$

are induced by chain maps $\mathcal{R}_j$ such that $\mathcal{R}_j(\boldsymbol{\gamma}) = \boldsymbol{\gamma}$ for every orbit set $\boldsymbol{\gamma}$ of $\mathbf{R}_k$ that intersects a fiber of $N$ $j$ times.

*Proof.* — The proof is by induction on $k$. For $k=0$ we choose a stable Hamiltonian structure $(\alpha_0, \omega)$ and define $(\alpha_0, \omega^0) = (\alpha_0, \omega)$ and $\alpha^0 = \alpha_\zeta$ for $\zeta > 0$ sufficiently small such that the identification of Equation (5.6.2) holds. Suppose we have constructed the sequences up to some $k_0$. By Lemma 5.6.1, Equation (5.6.2), and Lemma 5.6.2 there exist a stable Hamiltonian structure $(\alpha_0, \omega^{k_0+1})$ and a real number $\zeta_{k_0+1}$ such that (1)–(5) hold for $k \le k_0 + 1$, and moreover

$$ \mathcal{R}_j'' : \mathrm{ECC}(\mathrm{N}, \alpha_{\zeta_{k_0+2}}^{k_0+1}) \to \mathrm{ECC}(\mathrm{N}, \alpha_{\zeta_{k_0+2}}^{k_0+2}) $$

satisfies $\mathcal{R}_j''(\boldsymbol{\gamma}) = \boldsymbol{\gamma}$ as in Lemma 5.6.4. Without loss of generality we can assume that $\zeta_{k_0+2} < \zeta_{k_0+1}$. Since $(\mathcal{R}_j)_* = (\mathcal{R}_j'')_* \circ (\mathcal{R}_j')_*$ by [HT3, Theorem 2.4], (6) is satisfied up to $k_0 + 1$. $\square$
Let us write $\alpha^k = \alpha_{5k}^k$. In view of Theorem I.2.5.6, we can write

$$ \widehat{ECH}(M) \simeq \lim_{k \to \infty} ECH_{2k}(N, \alpha^{2k}), $$

where the limit is taken with respect to the maps

$$ ECH_{2k}(N, \alpha^{2k}) \to ECH_{2k+2}(N, \alpha^{2k+2}) $$

induced by the chain map $\gamma \mapsto e^2 \mathcal{R}_{2k}(\gamma)$. The diagram

$$ \begin{tikzcd} (5.6.3) \arrow[r, "(\mathcal{R}_{2k})_*"] & ECH_{2k}(N, \alpha^{2k}) \arrow[r, "(j_k)_*"] & ECH_{2k+2}(N, \alpha^{2k+2}) \arrow[d, "\simeq"] \\ & PFH_{2k}(N, \alpha_0, \omega^{2k+2}) \arrow[r, "(j_k)_*", "(j_k)_*"'] & PFH_{2k+2}(N, \alpha_0, \omega^{2k+2}) \arrow[d, "\simeq"'] \end{tikzcd} $$

commutes. Therefore, Proposition 5.5.1 and Lemma 5.4.1 imply that $(j_k)_*$ is an isomorphism. This proves Theorem 1.0.2.

## Acknowledgements

We are indebted to Michael Hutchings for many helpful conversations and for our previous collaboration which was a catalyst for the present work. We also thank Denis Auroux, Thomas Brown, Tobias Ekholm, Dusa McDuff, Roberto Pignatelli, Ivan Smith and Jean-Yves Welschinger for illuminating exchanges. Part of this work was done while KH and PG visited MSRI during the academic year 2009–2010. We are extremely grateful to MSRI and the organizers of the “Symplectic and Contact Geometry and Topology” and the “Homology Theories of Knots and Links” programs for their hospitality; this work probably would never have seen the light of day without the large amount of free time which was made possible by the visit to MSRI. KH also thanks the Simons Center for Geometry and Physics for their hospitality during his visit in May 2011.

## Declarations:

### Competing Interests

The authors declare no competing interests.
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## Publisher's Note

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V. C.
CNRS, Laboratoire de Mathématiques Jean Leray
Nantes Université,
Nantes, F-44000, France
Vincent.Colin@univ-nantes.fr

P. G.
CNRS, IF,
Univ. Grenoble Alpes,
Grenoble, 38000, France
paolo.ghiggini@univ-grenoble-alpes.fr

K. H.
University of California, Los Angeles,
Los Angeles, CA, 90095, USA
honda@math.ucla.edu

Manuscrit reçu le 7 août 2012

Version révisée le 10 avril 2017

Manuscrit accepté le 28 février 2024

publié en ligne le 2 avril 2024.