# The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Vincent Colin, Paolo Ghiggini, Ko Honda

► To cite this version:

Vincent Colin, Paolo Ghiggini, Ko Honda. The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus. Publications Mathématiques de L'IHÉS, 2024, 139 (1), pp.349-385. 10.1007/s10240-024-00147-9. hal-04752833

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# THE EQUIVALENCE OF HEEGAARD FLOER HOMOLOGY AND EMBEDDED CONTACT HOMOLOGY III: FROM HAT TO PLUS

by V*INCENT COLIN*, P*AOLO GHIGGINI**, and K*O HONDA***

## ABSTRACT

Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF⁺(−M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S, ℏ) of M, we construct a chain map Φ⁺ from a Heegaard Floer chain complex associated to (S, ℏ) to an embedded contact homology chain complex for a contact form supported by (S, ℏ). The chain map Φ⁺ commutes up to homotopy with the U-maps defined on both sides and reduces to the quasi-isomorphism Φ from (Colin et al. in Publ. Math. Inst. Hautes Études Sci., 2024a, 2024b) on subcomplexes defining the hat versions. Algebraic considerations then imply that the map Φ⁺ is a quasi-isomorphism.

## 1. Introduction

This is the last paper in the series which proves the isomorphism between certain Heegaard Floer homology and embedded contact homology groups. References from [I] (resp. [II]) will be written as “Section I. x” (resp. “Section II. x”) to mean “Section x” of [I] (resp. [II]), for example.

Let M be a closed oriented 3-manifold. Let $\widehat{HF}(M)$ and $HF^+(M)$ be the hat and plus versions of Heegaard Floer homology of M and let $\widehat{ECH}(M)$ and ECH(M) be the hat and usual versions of the embedded contact homology of M. As usual, embedded contact homology will be abbreviated as ECH. In [0], we introduced the ECH chain group $\widehat{ECC}(N, \partial N)$ and showed that $\widehat{ECH}(N, \partial N) \simeq \widehat{ECH}(M)$. In the papers [I, II], we defined a chain map

$$ \Phi : \widehat{CF}(-M) \to \widehat{ECC}(N, \partial N), $$

which induced an isomorphism

$$ \Phi_* : \widehat{HF}(-M) \xrightarrow{\sim} \widehat{ECH}(M). $$

The goal of this paper is to extend the above result and prove the following theorem:

**Theorem 1.0.1.** — If M is a closed oriented 3-manifold, then there is a chain map

$$ \Phi^+ : CF^+(M) \xrightarrow{\sim} ECC(M) $$

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

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

<footnote>*** KH supported by NSF Grants DMS-0805352, DMS-1105432, and DMS-1406564.</footnote>
which is a quasi-isomorphism and which commutes with the U-maps up to homotopy. On the level of homology $\Phi^+$ maps the contact class to the contact class.

We use $\mathbf{F} = \mathbf{Z}/2\mathbf{Z}$ coefficients for both Heegaard Floer homology and ECH. As is the case for the hat versions, we expect Theorem 1.0.1 to hold over the integers; see Remark I.1.0.1.

**Remark 1.0.2.** — The construction of $\Phi^+$ can be carried out with twisted coefficients as in Sections I.6.4 and I.7.1.

Let $(S, \hbar)$ be an open book decomposition for $M$, where $S$ is a genus $g \ge 2$ bordered surface with connected boundary and $\hbar \in \mathrm{Diff}(S, \partial S)^1$. In particular we identify

$$M \simeq (S \times [0, 1]) / \sim,$$

where $(x, 1) \sim (\hbar(x), 0)$ for all $x \in S$ and $(x, t) \sim (x, t')$ for all $x \in \partial S$ and $t, t' \in [0, 1]$. We write $S_t = S \times \{t\}$ for $t \in [0, 1]$. Let $\Sigma = S_0 \cup -S_{1/2}$ be the Heegaard surface corresponding to $(S, \hbar)$.

Given a pair $(\Sigma_0, \hbar_0)$ consisting of a surface $\Sigma_0$ and $\hbar_0 \in \mathrm{Diff}(\Sigma_0)$, we write the mapping torus of $(\Sigma_0, \hbar_0)$ as:

$$N_{(\Sigma_0, \hbar_0)} = (\Sigma_0 \times [0, 2]) / (x, 2) \sim (\hbar_0(x), 0).$$

The map $\Phi$, defined in Section I.6.2, is induced by the cobordism $W_+$ which is an $S_0$-fibration and which restricts to a half-cylinder over $[0, 1] \times S_0$ at the positive end and to a half-cylinder over the mapping torus $N_{(S_0, \hbar)}$ at the negative end. We say that $W_+$ is a cobordism “from $[0, 1] \times S_0$ to $N_{(S_0, \hbar)}$.”

**Remark 1.0.3.** — We will interchangeably write $[0, 1] \times S_0$ and $S_0 \times [0, 1]$. This is partly due to the fact that the open book is usually written as $(S \times [0, 1]) / \sim$ and the positive end of $W_+$ is a “symplectization” $\mathbf{R} \times [0, 1] \times S_0$.

The map $\Phi^+$ is induced by a cobordism $X_+$ from $[0, 1] \times \Sigma$ to $M$ which extends $W_+$ and is described below. Although $\Phi$ was defined in terms of just one page $S_0$, we can no longer ignore the $S_{1/2}$ portion of $\Sigma$ when defining $\Phi^+$, since we do not know how to express $\mathrm{HF}^+(\mathrm{-M})$ in terms of $S_0$.

A symplectic cobordism similar to $X_+$ is constructed by Wendl in [We].

**1.0.1.** *The cobordism $X_+$. —* We give a description of $X_+ = X_+^0 \cup X_+^1 \cup X_+^2$ and $W_+ = W_+^0 \cup W_+^1 \cup W_+^2$ as topological spaces, where $W_+^i \subset X_+^i$ for $i = 0, 1, 2$. See Figure 1. The description given here is the simplified version of the actual construction, and the notation of Section 1.0.1 is not used outside of Section 1.0.1.

<footnote n="1">¹ The condition $g \ge 2$ is a technical condition which will used in the definition of $\Phi^+$.</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. — Schematic diagram for $X_+^0 \cup X_+^1$ which indicates the fibers over each subsurface</figcaption></figure>

First extend $\hbar \in \mathrm{Diff}(S_0, \partial S_0)$ to $\hbar^+ \in \mathrm{Diff}(\Sigma)$ so that $\hbar^+|_{S_{1/2}} = \mathrm{id}$. Let $N_{(\Sigma, \hbar^+)}$ and $N_{(S_0, \hbar)}$ be the mapping tori of $\hbar^+$ and $\hbar$ and let

$$ \pi : [0, \infty) \times N_{(\Sigma, \hbar^+)} \to [0, \infty) \times \mathbf{R}/2\mathbf{Z} $$

be the projection $(s, x, t) \mapsto (s, t)$. Then define $B_+^0 = ([0, \infty) \times \mathbf{R}/2\mathbf{Z}) - B_+^c$, where $B_+^c$ is the subset $[2, \infty) \times [1, 2]$ with the corners rounded. We then set

$$ X_+^0 := \pi^{-1}(B_+^0), \quad W_+^0 := \pi^{-1}(B_+^0) \cap ([0, \infty) \times N_{(S_0, \hbar)}). $$

Observe that $W_+^0$ is the “top half” of $W_+$ defined in Section I.5.1. Next we set

$$ X_+^1 := S_{1/2} \times D^2, \quad W_+^1 := \emptyset $$

and identify $\{0\} \times S_{1/2} \times \mathbf{R}/2\mathbf{Z} \subset \partial X_+^0$ with $S_{1/2} \times \partial D^2 \subset \partial X_+^1$ via the map $(0, x, t) \mapsto (x, e^{\pi i t})$. Then one component of $\partial(X_+^0 \cup X_+^1)$ is given by $M = (\{0\} \times N_{(S_0, \hbar)}) \cup (\partial S_0 \times D^2)$.

Finally we set

$$ X_+^2 := (-\infty, 0] \times M, \quad W_+^2 := (-\infty, 0] \times (\{0\} \times N_{(S_0, \hbar)}), $$

where $\{0\} \times M$ is identified with $M$.

**1.0.2. Sketch of proof.** — The proof of Theorem 1.0.1 proceeds as follows:

*Step 1.* Express the U-map on $\mathrm{HF}^+(\mathrm{-M})$ as a count of $I_{\mathrm{HF}} = 2$ curves that pass through a point, in analogy with the definition of U in ECH. This is given by Theorem 3.1.4.

*Step 2.* Construct a symplectic cobordism $(X_+, \Omega_{X_+})$ from $[0, 1] \times \Sigma$ to $M$, together with stable Hamiltonian and contact structures on $[0, 1] \times \Sigma$ and $M$. This is the goal of Section 4.

*Step 3.* Define the chain map $\Phi^+$ as a count of $I_{X_+} = 0$ curves in $X_+$ and show that $\Phi^+$ commutes with the U-maps on both sides up to a chain homotopy $K$. This is done in Section 5.
*Step 4.* By an algebraic theorem (Theorem 6.1.5), $\Phi^+$ is a quasi-isomorphism if a map

$$ \Phi_{alg}: \widehat{CF}(-M) \to \widehat{ECC}(M), $$

defined using $\Phi^+$ and $K$, is a quasi-isomorphism.

*Step 5.* By Theorem 6.4.1, the map $\Phi_{alg}$ is a quasi-isomorphism. This is proved by relating $\Phi_{alg}$ to the quasi-isomorphism $\Phi$ from [I, II].

## 2. Heegaard Floer chain complexes

The goal of this section is to introduce some notation and recall the definition of the chain complex $CF^+(\Sigma, \alpha, \beta, z^f, J)$, whose homology is $HF^+(−M)$.

**2.1. Heegaard data.** — Let $M$ be a closed oriented 3-manifold and let $(S, \tilde{h})$ be an open book decomposition for $M$.

We use the following notation, which is similar to that of Section I.4.9.1:

- $\Sigma = S_0 \cup -S_{1/2}$ is the associated genus $2g$ Heegaard surface of $M$;

- $\mathbf{a} = \{a_1, \dots, a_{2g}\}$ is a basis of arcs for $S$ and $\mathbf{b}$ is a small pushoff of $\mathbf{a}$ as given in Figure I.1;

- $x_i$ and $x'_i$ are the endpoints of $a_i$ in $\partial S_0$ that correspond to the coordinates of the contact class and $x''_i$ is the unique point of $a_i \cap b_i \cap \text{int}(S_{1/2})$;

- $\alpha = (\mathbf{a} \times \{\frac{1}{2}\}) \cup (\mathbf{a} \times \{0\})$ and $\beta = (\mathbf{b} \times \{\frac{1}{2}\}) \cup (\tilde{h}(\mathbf{a}) \times \{0\})$ are the collections of compressing curves on the Heegaard surface $\Sigma$;

- $z^f$ is a point in the large (i.e., non-thin-strip) component of $S_{1/2} - \alpha - \beta$ and $(z')^f$ is a point which is close but not equal to $z^f$.

We say that the pointed Heegaard diagram $(\Sigma, \alpha, \beta, z^f)$ is compatible with $(S, \tilde{h})$. We let $\mathbf{x} = \{x_1, \dots, x_{2g}\}$ and consider the contact element $[\mathbf{x}, 0]$. In the definition of $\mathbf{x}$ we could replace any component $x_i$ with $x'_i$.

**Remark 2.1.1.** — The orientation for $\Sigma$ is opposite to that of Section I.4.9.1. This is done so that the triple $(S, \mathbf{a}, \tilde{h}(\mathbf{a}))$, used in [I, II], embeds in $(\Sigma, \alpha, \beta)$ in an orientation-preserving manner.

**2.2. Symplectic data.** — The stable Hamiltonian structure on $[0, 1] \times \Sigma$ with coordinates $(t, x)$ is given by $(\lambda, \omega)$, where $\lambda = dt$ and $\omega$ is an area form on $\Sigma$ which makes $(\alpha, \beta, z^f)$ weakly admissible with respect to $\omega$, i.e., each periodic domain has zero $\omega$-area. The plane field $\xi = \ker \lambda$ is equal to the tangent plane field of $\{t\} \times \Sigma$ and the Hamiltonian vector field is $R = \frac{\partial}{\partial t}$.

We introduce the “symplectization”

$$ (X, \Omega) = (\mathbf{R} \times [0, 1] \times \Sigma, ds \wedge dt + \omega), $$
where $(s, t)$ are coordinates on $\mathbf{R} \times [0, 1]$. Let $\pi_B: X \to B = \mathbf{R} \times [0, 1]$ be the projection along the fibers $\{(s, t)\} \times \Sigma$.

Let $J$ be an $\Omega_X$-admissible almost complex structure on $X$; we assume that $J$ is regular (cf. Lemma I.4.7.2 and [Li, Proposition 3.8]). We also define the Lagrangian submanifolds

$$L_{\alpha} = \mathbf{R} \times \{1\} \times \alpha, \quad L_{\beta} = \mathbf{R} \times \{0\} \times \beta.$$

**2.3.** *The chain complex* $\mathrm{CF}^+(\Sigma, \alpha, \beta, z^f, J)$. — In this subsection we recall the definition of the chain complex $\mathrm{CF}^+(\Sigma, \alpha, \beta, z^f, J)$, whose homology group

$$\mathrm{HF}^+(\Sigma, \alpha, \beta, z^f, J)$$

is isomorphic to $\mathrm{HF}^+(\mathbb{M})$. This definition is due to Lipshitz [Li], with one modification: we are using the ECH index $I_{\mathrm{HF}}$ from Definition I.4.5.11. We will often suppress $J$ from the notation.

Let $S = S_{\alpha, \beta}$ be the set of $2g$-tuples $\mathbf{y} = \{y_1, \dots, y_{2g}\}$ of intersection points of $\alpha$ and $\beta$ for which there exists some permutation $\sigma \in \mathfrak{S}_{2g}$ such that $y_j \in \alpha_j \cap \beta_{\sigma(j)}$ for all $j$. Then $\mathrm{CF}^+(\Sigma, \alpha, \beta, z^f, J)$ is generated over $\mathbf{F}$ by pairs $[\mathbf{y}, i]$, where $\mathbf{y} \in S$ and $i \in \mathbf{N}$, with the French convention that $0 \in \mathbf{N}$.

The differential $\partial = \partial_{\mathrm{HF}}$ is given by

$$\partial[\mathbf{y}, i] = \sum_{[\mathbf{y}', j] \in S \times \mathbf{N}} \langle \partial[\mathbf{y}, i], [\mathbf{y}', j] \rangle \cdot [\mathbf{y}', j],$$

where the coefficient $\langle \partial[\mathbf{y}, i], [\mathbf{y}', j] \rangle$ is the count of index $I_{\mathrm{HF}} = 1$ finite energy holomorphic multisections in $(X, J)$ with Lagrangian boundary $L_{\alpha} \cup L_{\beta}$ from $\mathbf{y}$ to $\mathbf{y}'$, whose algebraic intersection with the holomorphic strip $\mathbf{R} \times [0, 1] \times \{(z')^f\}$ is $(i-j)$. We will often refer to such curves as curves from $[\mathbf{y}, i]$ to $[\mathbf{y}', j]$.

Let us write $\partial = \sum_{k=0}^{\infty} \partial_k$, where $\partial_k$ only counts curves whose algebraic intersection with $\mathbf{R} \times [0, 1] \times \{(z')^f\}$ is $k$.

**Lemma 2.3.1.** — The contact element $[\mathbf{x}, 0]$ is a cycle and its homology class does not depend on the choice of $x_i$ or $x'_i$ as its coordinates.

*Proof.* — The proof of the first statement is the same as that for the contact element $\mathbf{x}$ in the hat version since curves from $[\mathbf{x}, 0]$ cannot intersect $\mathbf{R} \times [0, 1] \times \{z^f\}$. The second statement follows from Claim I.4.9.2. $\square$

**3. The geometric U-map**

**3.1. Introduction.** — In [OSz, Li], the U-map

$$U: \mathrm{CF}^+(\Sigma, \alpha, \beta, z^f) \to \mathrm{CF}^+(\Sigma, \alpha, \beta, z^f),$$
is defined algebraically as $U([\mathbf{y}, i]) = [\mathbf{y}, i - 1]$ if $i > 0$ and $U([\mathbf{y}, 0]) = 0$. The goal of this section is to give a geometric definition of the U-map which is analogous to that of ECH.

Let $z^f, (z')^f$ be as before and let $z = (z^b, z^f) \in X = B \times \Sigma$, where $z^b \in \text{int}(B)$. Let $J^\diamond$ be a generic $C^\ell$-small perturbation of $J$ such that $J^\diamond = J$ away from a small neighborhood $N(z) \subset X$ of $z$ and such that $N(z) \cap (R \times [0, 1] \times \{(z')^f\}) = \emptyset$. In particular, we assume that there are no $J^\diamond$-holomorphic curves that are homologous to $\{pt\} \times \Sigma$ and pass through $z$.

**Remark 3.1.1.** — When we refer to “$C^\ell$-close” almost complex structures, etc., we assume that $\ell > 0$ is sufficiently large.

Let $\mathcal{M}_{J^\diamond}^{I=k}([\mathbf{y}, i], [\mathbf{y}', j])$ (resp. $\mathcal{M}_{J^\diamond}^{I=k}([\mathbf{y}, i], [\mathbf{y}', j], z)$) be the moduli space of $I_{\text{HF}} = k$ finite energy holomorphic curves in $(X, J^\diamond)$ with Lagrangian boundary $L_\alpha \cup L_\beta$ from $[\mathbf{y}, i]$ to $[\mathbf{y}', j]$ (resp. from $[\mathbf{y}, i]$ to $[\mathbf{y}', j]$ that pass through $z$). There is a natural forgetful map

$$ \mathcal{M}_{J^\diamond}^{I=k}([\mathbf{y}, i], [\mathbf{y}', j], z) \to \mathcal{M}_{J^\diamond}^{I=k}([\mathbf{y}, i], [\mathbf{y}', j]), $$

which is an injection when $I \le 3$: If a curve $u$ passes through $z$ twice (or passes through $z$ once with a singularity at $z$), then the nodal or singular point contributes 2 to $I$. Also, by our choice of $J^\diamond$, “passing through $z$” is a generic codimension 2 condition, and therefore $\text{ind}(u) \ge 2$. Hence, by the index inequality (I.4.5.5), $I(u) \ge 4$, a contradiction.

Also note that, by a simple count of $I$ and the ECH index inequality for $I$ as in Equation (I.7.5.6), an $I(u) \le 3$ curve that passes through $z$ cannot have a fiber component.

**Definition 3.1.2 (Geometric U-map).** — The geometric U-map with respect to the point $z$ is the map:

$$ U_z([\mathbf{y}, i]) = \sum_{[\mathbf{y}', j] \in \mathcal{S} \times \mathbf{N}} \# \mathcal{M}_{J^\diamond}^{I=2}([\mathbf{y}, i], [\mathbf{y}', j], z) \cdot [\mathbf{y}', j]. $$

**Proposition 3.1.3.** — $U_z$ is a chain map.

**Proof.** — Since we are using almost complex structures of type $J^\diamond$, the transversality of $\mathcal{M}_{J^\diamond}^{I=3}([\mathbf{y}, i], [\mathbf{y}', j], z)$ follows from the combination of Theorems 3.1.7 and 3.4.1 of [MS], with modifications as in Proposition I.5.8.8. The compactness follows from Lemma I.4.6.1 and the usual SFT compactness; also see [Li, Corollary 7.2]. Fiber bubbling was already eliminated. Finally, gluing is as in Propositions A.1 and A.2 of [Li, Appendix A]. $\square$
**Theorem 3.1.4.** — There exists a chain homotopy

$$H: CF^{+}(\Sigma, \alpha, \beta, z') \to CF^{+}(\Sigma, \alpha, \beta, z')$$

such that

$$(3.1.1) \qquad U_z - U = H \circ \partial_{HF} + \partial_{HF} \circ H.$$

Moreover, for all $\mathbf{y} \in S$, one has $H([\mathbf{y}, 0]) = 0$.

The rest of this section is devoted to the proof of Theorem 3.1.4.

**3.2. A model calculation.** — Let $\Sigma$ be a closed surface of genus $k$. We consider the manifold $D \times \Sigma$, where $D = \{|z| \le 1\} \subset C$. Let $\pi_D: D \times \Sigma \to D$ and $\pi_\Sigma: D \times \Sigma \to \Sigma$ be the projections of $D \times \Sigma$ onto the first and second factors. Let $\beta = \{\beta_1, \dots, \beta_k\}$ be the set of $\beta$-curves for $\Sigma$. Choose $z' \in \Sigma - \beta$ and let $z = (0, z') \in D \times \Sigma$.

Let $J = j_D \times j_\Sigma$ be a product complex structure on $D \times \Sigma$ and $J^\diamond$ be a generic $C^\ell$-small perturbation of $J$ such that $J^\diamond = J$ away from a small neighborhood of $z$. The key feature of $J^\diamond$ is that all the $J^\diamond$-holomorphic curves that pass through $z$ are regular.

We then define the moduli space $M_\Lambda(D \times \Sigma, J^*)$, $*=\emptyset$ or $\diamond$, of stable maps

$$u: (F, j) \to (D \times \Sigma, J^*)$$

in the class $A = [\{pt\} \times \Sigma] + k[D \times \{pt\}] \in H_2(D \times \Sigma, \partial D \times \beta)$, such that $\partial F$ has $k$ connected components and each component of $\partial F$ maps to a distinct Lagrangian $\partial D \times \beta_i$, $i = 1, \dots, k$. We choose points $w_i \in \beta_i$, $i = 1, \dots, k$, and define

$$\mathbf{w} = \{(1, w_1), \dots, (1, w_k)\} \subset D \times \Sigma.$$

Let $M_\Lambda(D \times \Sigma, J^*; z, \mathbf{w})$ be the moduli space of stable maps $u$ as above, with the extra data of an interior puncture and $k$ boundary punctures that map to $z$ and $\mathbf{w}$. There is a forgetful map

$$M_\Lambda(D \times \Sigma, J^*; z, \mathbf{w}) \to M_\Lambda(D \times \Sigma, J^*),$$

which is an injection when we restrict to curves that pass through $z$ only once and there is no singularity at $z$. This will be the case in our setting. The points of $\mathbf{w}$ are distinct and there is no risk of passing through the same point of $\mathbf{w}$ twice. We use the modifier “irr” to denote the subset of irreducible curves.

**3.2.1. ECH index.** — We briefly indicate the definition of the ECH index $I$ of a homology class $B \in H_2(D \times \Sigma, \partial D \times \beta)$ which admits a representative $F$ such that each component of $\partial F$ maps to a distinct $\partial D \times \beta_i$. Although we call $I$ the “ECH index”, what we are defining here is a relative version of Taubes’ index from [T].
Let $\tau$ be a trivialization of $T\Sigma$ along $\beta$, given by a nonsingular tangent vector field $Y_1$ along $\beta$, and let $\tau'$ be a trivialization of $TD$ along $\partial D$, given by an outward-pointing radial vector field $Y_2$ along $\partial D$. Let $Q_{(\tau, \tau')}(B)$ be the intersection number between an embedded representative $u$ of $B$ and its pushoff, where the boundary of $u$ is pushed off in the direction given by $J(Y_1)$.

**Definition 3.2.1.** — The ECH index of the homology class $B$ is:

$$I(B) = c_1(T(D \times \Sigma)|_B, (\tau, \tau')) + \mu_{(\tau, \tau')}(\partial B) + Q_{(\tau, \tau')}(B).$$

The following is the relative version of the adjunction inequality:

**Lemma 3.2.2 (Index inequality).** — Let $u: (F, j) \to (D \times \Sigma, \partial D \times \beta)$. Then

$$\mathrm{ind}(u) + 2\delta(u) = I(B),$$

where $\delta(u) \ge 0$ is an integer count of the singularities.

*Proof.* — Similar to the proof of Theorem I.4.5.13. $\square$

<figure><img src="image_3.png" /></figure>

We now calculate some ECH and Fredholm indices:

**Lemma 3.2.3.** — If $B = [\{pt\} \times \Sigma] + k_0[D \times \{pt\}]$ with $k_0 \le k$, then

$$I(B) = 2 - 2k + 3k_0.$$

*Proof.* — We compute that

$$\begin{align*}
I(B) &= I([\{pt\} \times \Sigma] + k_0[D \times \{pt\}]) \\
&= I([\{pt\} \times \Sigma]) + k_0 \cdot I([D \times \{pt\}]) + 2k_0 \cdot \langle [\{pt\} \times \Sigma], [D \times \{pt\}] \rangle \\
&= (2 - 2k) + k_0 \cdot 1 + 2k_0 = 2 - 2k + 3k_0.
\end{align*}$$

Here $\langle, \rangle$ denotes the algebraic intersection number. $\square$

<figure><img src="image_4.png" /></figure>

**Lemma 3.2.4.** — If $B = [\{pt\} \times \Sigma] + k_0[D \times \{pt\}]$ with $k_0 \le k$ and $u$ is an irreducible $J^\diamond$-holomorphic curve in the class $B$, then

$$\mathrm{ind}(u) = 2 - 2k + 3k_0 - \delta(u).$$

*Proof.* — Follows from Lemma 3.2.3 and the index inequality. $\square$

<figure><img src="image_5.png" /></figure>
**3.2.2. Main result.** — The following is the main result of this subsection:

*Theorem 3.2.5.* — If $J^\diamond$ is generic, then the following hold:

(1) $\mathcal{M}_\Lambda(D \times \Sigma, J^\diamond; z, \mathbf{w}) = \mathcal{M}_\Lambda^{irr}(D \times \Sigma, J^\diamond; z, \mathbf{w})$;

(2) $\mathcal{M}_\Lambda(D \times \Sigma, J^\diamond; z, \mathbf{w})$ is compact, regular, and 0-dimensional;

(3) the curves of $\mathcal{M}_\Lambda(D \times \Sigma, J^\diamond; z, \mathbf{w})$ are embedded; and

(4) $\#\mathcal{M}_\Lambda(D \times \Sigma, J^\diamond; z, \mathbf{w}) \equiv 1 \mod 2$.

Hence $\#\mathcal{M}_\Lambda(D \times \Sigma, J^\diamond; z, \mathbf{w})$ is a certain relative Gromov-Witten invariant [IP] which is computed to be 1 mod 2. (What we are really computing here is a relative Gromov-Taubes invariant [T], although the two invariants coincide in this case.)

*Proof.* — (1) Let us write $\mathcal{M} = \mathcal{M}_\Lambda(D \times \Sigma, J^\diamond; z, \mathbf{w})$. Arguing by contradiction, suppose $u \in \mathcal{M} - \mathcal{M}^{irr}$. Then $u$ consists of an irreducible component $u_0$ which passes through $z$ and $k_0 < k$ points of $\mathbf{w}$, together with $k-k_0$ copies of $D \times \{pt\}$. By Lemma 3.2.4, $\mathrm{ind}(u_0) \le 2-2k+3k_0$. On the other hand, the point constraints are $(k_0+2)$-dimensional. Hence $u_0$ does not exist for generic $J^\diamond$, which is a contradiction.

(2), (3) The compactness follows from the usual Gromov compactness theorem: We have already specified the homology class $A$ and the genus bound is a consequence of Lemma 3.2.2, from which we see that the Euler characteristic term that appears in the formula for $\mathrm{ind}(u)$ is controlled by the homology class $A$. The regularity of $\mathcal{M}$ is immediate from the genericity of $J^\diamond$ and (1). Lemma 3.2.4 implies the dimension calculation, as well as (3).

(4) We degenerate $\Sigma$ along the union $C$ of $k-1$ separating curves into a nodal surface $\tilde{\Sigma}$ whose irreducible components are $k$ tori which are successively attached to one another; let $J_\tau^\diamond, \tau \in [0, \infty)$, be the family of almost complex structures corresponding to the degeneration. We choose $C$ so that they are disjoint from $\beta$ and each irreducible component contains exactly one component of $\beta$ (and hence exactly one $w_i$). Since the basepoint $z$ remains in one component, the almost complex structure on $D \times \tilde{\Sigma}$ is a product almost complex structure in all but one of the irreducible components of $D \times \tilde{\Sigma}$. In order to attain transversality, we need to further perturb $J_\tau^\diamond$ to $J_\tau^\diamond$ on a compact subset $K \subset \mathrm{int}(D) \times (\Sigma - C)$ such that each component of $K \cap (D \times (\Sigma - C))$ nontrivially intersects each curve of $\mathcal{M}_\Lambda^{irr}(D \times \Sigma, J_\tau^\diamond; z, \mathbf{w})$. By a standard continuation argument,

$$ \#\mathcal{M}_\Lambda^{irr}(D \times \Sigma, J_\tau^\diamond; z, \mathbf{w}) = \#\mathcal{M}_\Lambda^{irr}(D \times \Sigma, J_\tau^\diamond; z, \mathbf{w}); $$

from now on we will work with the latter almost complex structure.

As $\Sigma$ degenerates into $\tilde{\Sigma}$, a sequence $u^\tau \in \mathcal{M}_\Lambda^{irr}(D \times \Sigma, J_\tau^\diamond; z, \mathbf{w})$ of holomorphic curves with $\tau \to \infty$ (after passing to a subsequence) degenerates into a nodal holomorphic curve $u_1 \cup \dots \cup u_k$ in $D \times \tilde{\Sigma}$, where each $u_i$ lies on a separate level and $u_i$ is attached to $u_{i+1}$ for $i=1, \dots, k-1$. Starting with the component $u_1$ that passes through $z$, the incidence condition between $u_1$ and $u_2$ is analogous to a point constraint for $u_2$, and so on.
Hence it suffices to prove Theorem 3.2.5(4) for $k = 1$; this is the content of Lemma 3.3.4 in Section 3.3. See Section II.2.4.4 for a similar argument. □

**Remark 3.2.6.** — The section $\{\infty\} \times \Sigma$ is not regular, and thus neither $J_S^{\diamond}$ nor $J_S^{\heartsuit}$ are generic almost complex structures. What we are computing here is a simple instance of relative Gromov-Witten invariant in the sense of [IP].

**3.3. Computation of $\#M_\Lambda(D \times \Sigma, J^\diamond; z, \mathbf{w})$ when $k=1$ and $\Sigma$ is a torus.** — The first step is to degenerate $D$ into $D \cup S^2$, where $0 \in D$ is identified with $\infty \in S^2 \cong \mathbf{C} \cup \{\infty\}$ (we will refer to the identified point by $\mathbf{n}$) and $z = (0, z') \in S^2 \times \Sigma$; equivalently, we are taking a 1-parameter family $J_\kappa^\diamond$, $\kappa \in [0, \infty)$, and taking the limit $\kappa \to \infty$. Let $J_D^\diamond \cup J_{S^2}^\diamond$ denote the limit almost complex structure on $(D \times \Sigma) \cup (S^2 \times \Sigma)$, which we assume to be a small perturbation of a product almost complex structure $J_D \cup J_{S^2}$ in a small neighborhood of $z$.

Let $v_1 \cup v_2$ be a limit of a sequence $u^\kappa \in M_\Lambda(D \times \Sigma, J_\kappa^\diamond; z, \mathbf{w})$ of curves with $\kappa \to \infty$. Then $v_1$ is the trivial multisection $D \times \{w_1\}$ in $D \times \Sigma$ and

$$v_2 \in \mathcal{M}_B^{\diamond} := \mathcal{M}_B(S^2 \times \Sigma, J_{S^2}^{\diamond}; z, \mathbf{w} = \{(\infty, w_1)\}),$$

where $\mathcal{M}_B^{\diamond}$ is the moduli space of $J_{S^2}^{\diamond}$-holomorphic curves in $S^2 \times \Sigma$ representing the homology class $B = [S^2] + [\Sigma]$ and passing through $z = (0, z')$ and $(\infty, w_1)$.

In order to analyze $\mathcal{M}_B^{\diamond}$, we first describe $\mathcal{M}_B := \mathcal{M}_B(S^2 \times \Sigma, J_{S^2}; z, \mathbf{w})$ for a product complex structure $J_{S^2}$:

**Lemma 3.3.1.** — If $k=1$, then:

(1) $M_\Lambda(D \times \Sigma, J; z, \mathbf{w})$ is a one-element set consisting of a degenerate curve $(D \times \{w_1\}) \cup (\{0\} \times \Sigma)$; and

(2) $M_B$ is a two-element set consisting of degenerate curves $v_{21} := (S^2 \times \{w_1\}) \cup (\{0\} \times \Sigma)$ and $v_{22} := (S^2 \times \{z'\}) \cup (\{\mathbf{n}\} \times \Sigma)$.

*Proof.* — (1) follows from the homological constraint

$$A = [\{pt\} \times \Sigma] + [D \times \{pt\}].$$

If $u: (F, j) \to (D \times \Sigma, J)$ is a stable map in $M_\Lambda(D \times \Sigma, J; z, \mathbf{w})$, then $\pi_D \circ u$ and $\pi_\Sigma \circ u$ are degree 1 maps. This implies that $F$ consists of two components $F_1, F_2$ and $\pi_D \circ u|_{F_1}$ and $\pi_\Sigma \circ u|_{F_2}$ are biholomorphisms. On the other hand, $\pi_\Sigma \circ u|_{F_1}$ maps to a point since $F_1$ is a disk and $\pi_D \circ u|_{F_2}$ maps to a point since otherwise the cardinality of $(\pi_D \circ u)^{-1}(pt)$ for generic $pt$ will be larger than $\deg(\pi_D \circ u) = 1$.

(2) is similar and follows from the fact that there are no degree 1 holomorphic maps from the torus $\Sigma$ to $S^2$. □
By Gromov compactness and Lemma 3.3.1(2), all the curves of $\mathcal{M}_B^{\diamond}$ are close to the degenerate curves in $\mathcal{M}_B$ described in Lemma 3.3.1(2). Note that elements in $\mathcal{M}_B^{\diamond}$ can be reducible and only the irreducible component passing through $z$ has to be regular. Simple considerations taking into account the homological and point constraints imply:

**Lemma 3.3.2.** — If $k = 1$ and $J^{\diamond}$, $w$, and $\beta$ are generic, then the only element $v'_{22} \in \mathcal{M}_B^{\diamond} - \mathcal{M}_B^{\diamond, \text{irr}}$ is close to $v_{22}$ and consists of $\{\mathfrak{n}\} \times \Sigma$ together with one sphere in the class $[S^2]$ passing through $z$.

We also have:

**Lemma 3.3.3.** — If $k = 1$ and $J^{\diamond}$, $w$, and $\beta$ are generic, then:

(1) the curves of $\mathcal{M}_B^{\diamond, \text{irr}}$ are embedded; and

(2) $\mathcal{M}_B^{\diamond, \text{irr}}$ is compact, regular, and 0-dimensional.

*Proof.* — (1) The proof is similar to that of Lemma 3.2.5(3) and follows from the adjunction inequality [M1, M2] (compare with Lemma 3.2.2): If $v \in \mathcal{M}_B^{\diamond, \text{irr}}$, then

$$I(v) = c_1(v^*T(S^2 \times \Sigma)) + Q(v),$$

where $Q(v)$ is the self-intersection number of $v$, and

$$\text{ind}(v) + 2\delta(v) = I(v),$$

where $\delta(v) \ge 0$ is an integer count of the singularities. Since $c_1(v^*T(S^2 \times \Sigma)) = 2$ and $Q(v) = 2$, it follows that $I(v) = 4$. On the other hand,

$$\text{ind}(v) = -\chi(F) + 2c_1(v^*T(S^2 \times \Sigma)) = -0 + 2(2) = 4,$$

where $F$ is the domain of $v$ with $\chi(F) = 0$. Hence $v$ is embedded by the adjunction inequality.

(2) Since $v$ is embedded and $c_1(v^*T(S^2 \times \Sigma)) = 2$, the regularity of $v$ without the point constraints follows from automatic transversality (cf. Hofer-Lizan-Sikorav [HLS, Theorem 1]). The regularity with point constraints is the consequence of the genericity of $J^{\diamond}$, $w$, and $\beta$. The rest of the assertion is immediate. $\square$

Next we argue that $v_1 \cup v'_{22}$ cannot appear as the limit of $u^k$. This can be proved by an analysis of the limit in the SFΓ sense (or equivalently in the relative Gromov-Witten sense): in brief, we can view the component $\{\mathfrak{n}\} \times \Sigma$ of $v'_{22}$ as an intermediate irreducible level with image in $S^2 \times \Sigma$, is in the class $[\{pt\} \times \Sigma] + [S^2 \times \{pt\}]$, and passes through $(\infty, w_1)$ and $(0, z')$. Such a curve does not exist since there are no degree 1 holomorphic maps from the torus $\Sigma$ to $S^2$. Therefore,

$$\#\mathcal{M}_{\Lambda}(D \times \Sigma, J^{\diamond}; z, w) \equiv \#\mathcal{M}_{B}^{\diamond, \text{irr}} \mod 2.$$
The following lemma then completes the proof of Theorem 3.1.4.

**Lemma 3.3.4.** — $\#\mathcal{M}_{\mathcal{B}}^{\diamond, \text{irr}} \equiv 1 \mod 2$.

*Proof.* — The lemma follows from [MS, Example 8.6.12], but one can also argue more explicitly by degenerating $\Sigma = T^2$ into a nodal surface $\Sigma_0 \cup \Sigma_1$, where the sphere $\Sigma_0$ contains $z$, the sphere $\Sigma_1$ contains $w_1$, and $n_1$ and $n_2$ are the two nodes.

Consider a limit $u_0 \cup u_1$ of $u^\tau \in \mathcal{M}_{\mathcal{B}, \tau}^{\diamond, \text{irr}}$ as $\tau \to \infty$, where we are using $J^\diamond$ instead of $J^\diamond$ and the subscript $\tau$ indicates the dependence of $J^\diamond$ on $\tau \in [0, \infty)$ as we degenerate $\Sigma$. Here $u_0$ has image in $S^2 \times \Sigma_0$ and passes through $(0, z')$, and $u_1$ has image in $S^2 \times \Sigma_1$ and passes through $(\infty, w_1)$. Since $u^\tau$ is $C^0$-close to $v_{21}$ for $\tau$, the curve $u_0$ represents the homology class $[\Sigma_0]$, while the curve $u_1$ represents the homology class $[S^2] + [\Sigma_1]$. Moreover the images of $u_0$ and $u_1$ match at $S^2 \times \{n_1, n_2\}$. The image of $u_0$ is a small perturbation of the graph of a degree zero holomorphic map $\Sigma_0 \to S^2$ and the image of $u_1$ is a small perturbation of the graph of a degree one holomorphic map $\Sigma_1 \to S^2$. Then by elementary complex analysis there is a unique choice for $u_0$, while the choice for $u_1$ becomes unique once the intersection of its image with $S^2 \times \{n_1, n_2\}$ is fixed. Hence $\#\mathcal{M}_{\mathcal{B}}^{\diamond, \text{irr}} \equiv 1 \mod 2$. $\square$

**3.4. Family of cobordisms.** — We now describe a family of marked points $z_\tau \in X$ and a family of almost complex structures $J_\tau^\diamond$ on $X$ for $\tau \in [0, 1)$, as well as their limits for $\tau = 1$. These families give rise to the chain homotopy $H$ of Theorem 3.1.4.

Let $z_\tau^b \in \text{int}(\mathcal{B})$, $\tau \in [0, 1)$, be a family of points such that $z_0^b = z^b$, $\lim_{\tau \to 1} z_\tau^b = (0, 0)$, and $z_\tau^b \in \{s=0\}$ for $\tau \in [1/2, 1)$. Then let $z_\tau = (z_\tau^b, z^f) \in X$.

Assume that the almost complex structure $J$ on $X$ is a product complex structure on $\mathbf{R} \times [0, \varepsilon] \times \Sigma$ for $\varepsilon > 0$ small. We then define a family of $C^\ell$-small perturbations $J_\tau^\diamond$, $\tau \in [0, 1)$, of $J$ such that $J_\tau^\diamond = J$ away from a small neighborhood $N(z_\tau)$ of $z_\tau$ and

$$N(z_{\tau}) \cap (\mathbf{R} \times [0, 1] \times \{(z')^{f}\}) = \emptyset.$$

In the limit $\tau = 1$, the base $\tilde{\mathcal{B}}$ is $(\mathcal{B} \sqcup \mathcal{D})/\sim$, where $\mathcal{D} = \{|z| \le 1\} \subset \mathbf{C}$ and $\sim$ identifies $(0, 0) \in \mathcal{B}$ with $-1 \in \mathcal{D}$, and the total space $\tilde{\mathcal{X}}$ is $(\mathcal{X} \sqcup (\mathcal{D} \times \Sigma))/\sim$, where $((0, 0), x) \sim (-1, x)$ for all $x \in \Sigma$. See Figure 2. We write $w^b$ for the node $[(0, 0)] = [-1] \in \tilde{\mathcal{B}}$. Let $\pi_{\mathcal{B}}: \mathcal{X} \to \mathcal{B}$ and $\pi_{\mathcal{D}}: \mathcal{D} \times \Sigma \to \mathcal{D}$ be the projections onto the first factors.

The limit $z_1$ of $z_\tau$ is in $\mathcal{D} \times \Sigma$ and we assume that $z_1^b = 0 \in \text{int}(\mathcal{D})$. When $\tau = 1$, the almost complex structure $J_1^\diamond$ restricts to the complex structure $J$ on $X$ and to the almost complex structure $J_D^\diamond$, where $J_D$ is a product complex structure on $\mathcal{D} \times \Sigma$ and $J_D^\diamond$ is a $C^\ell$-small perturbation of $J_D$ such that $J_D^\diamond = J_D$ away from a small neighborhood $N(z_1)$ of $z_1$ and

$$N(z_1) \cap (\mathcal{D} \times \{(z')^f\}) = \emptyset.$$
<figure><img src="image_6.png" /><figcaption>FIG. 2. — The degeneration of the base B together with the marked point $z_{\tau}^b$ as $\tau \to 1$. (Color figure online)</figcaption></figure>

The Lagrangian boundary condition for $\tau \in [0, 1)$ is $L_\alpha \cup L_\beta$. In the limit $\tau = 1$, we use $L_\alpha \cup L_\beta$ for X and $\partial D \times \beta$ for D $\times \Sigma$.

The degeneration for $\tau \to 1$ can be described in an equivalent way as a neck-stretching along a stable Hamiltonian hypersurface $\gamma \times \Sigma$, where $\gamma$ is a boundary-parallel arc in the base B which separates a disk containing the $z_{\tau}^b$.

**3.5. Proof of Theorem 3.1.4.** — Let $u_{\tau_i}$, $\tau_i \to 1$, be a sequence of $I_{\text{HF}} = 2$ curves in $(X, J_{\tau_i}^\diamond)$ from $[\mathbf{y}, i]$ to $[\mathbf{y}', i-k]$ that pass through $z_{\tau_i}$. Applying SFT compactness in the neck-stretching setting and transferring the result to the nodal degeneration picture, we obtain the limit $\tilde{u} = u_B \cup u_D$, where $u_B \subset X$, $u_D \subset D \times \Sigma$, and $u_D$ passes through $z_1$. Components of $\tilde{u}$ that map to the fiber $\{w^b\} \times \Sigma$ will be viewed as components of $u_D$.

**Lemma 3.5.1.**

$$ (1) \quad [u_D] = k_0[\{pt\} \times \Sigma] + 2g[D \times \{pt\}] \in H_2(D \times \Sigma) \text{ for some } 0 < k_0 \le k. $$

$$ (2) \quad I(u_D) = 2k_0 + 2g \ge 2g + 2. $$

*Proof.* — (1) $\deg(\pi_D \circ u_D) = 2g$, since $u_{\tau_i}$ is a degree $2g$ multisection of X for each $\tau_i$, away from a neighborhood of $z_{\tau_i}^b$. Also, since $\langle u_{\tau_i}, B \times \{(z^f)'\} \rangle = k$ for all $\tau_i$, it follows that $\langle u_D, D \times \{(z^f)'\} \rangle = k_0$, where $0 < k_0 \le k$. Here $k_0 > 0$ since $u_D$ passes through $z_1$.

(2) is a consequence of (1) and computations as in the proof of Lemma 3.2.3. We remind the reader that the genus of $\Sigma$ is $2g$. $\square$

**Lemma 3.5.2.** — $I(u_D) = 2g + 2$ and $I_{\text{HF}}(u_B) = 0$. In particular, $\mathbf{y} = \mathbf{y}', u_B$ consists of 2g trivial strips, and $k_0 = k = 1$.

*Proof.* — The gluing constraints give $I_{\text{HF}}(u_\tau) = I(u_D) + I_{\text{HF}}(u_B) - 2g = 2$. Strictly speaking, if there are (possibly multiply-covered) fiber components over $z = -1$ in D, then we should view $\tilde{u}$ as an SFT limit, in which case there will be intermediate levels with image in $D \times \Sigma$, where D has nodes at $z = \pm 1$, and there are no fiber components over $z = \pm 1$. We can then view $u_D$ as the union of all the levels besides $u_B$, to which one can apply gluing constraints. By the regularity of J and the index inequality, we
have $I_{\text{HF}}(u_B) \ge 0$. The first sentence of the lemma then follows from Lemma 3.5.1(2); the second sentence is a consequence of the first. □

The first sentence of Theorem 3.1.4 follows from the usual construction of chain homotopies in Floer theory: By Lemma 3.5.2, $U_z$ is chain homotopic to $aU$, where $a$ is the count of holomorphic curves $u_D$ in $(D \times \Sigma, J_D^\flat)$ that pass through $z_1$ and $w = \{(0, y_1), \dots, (0, y_{2g})\}$, where $y = \{y_1, \dots, y_{2g}\}$. Since $a = 1$ modulo 2 by Theorem 3.2.5, $U_z$ is chain homotopic to $U$.

Next we prove the second sentence of Theorem 3.1.4. For all $y \in S$, $H([y, 0])$ is obtained by counting $I_{\text{HF}} = 1$ curves that pass through $z_\tau$ for some $\tau \in (0, 1)$ and that do not cross the holomorphic strip $R \times [0, 1] \times \{(z')^f\}$. There are no such curves since $R \times [0, 1] \times \{z'\}$ is holomorphic and homologous to $R \times [0, 1] \times \{(z')^f\}$: if a curve passes through $z_\tau$, its intersection with $R \times [0, 1] \times \{z'\}$ is strictly positive by the positivity of intersections, and so is its intersection with $R \times [0, 1] \times \{(z')^f\}$.

## 4. The cobordism $X_+$

In this section we give the construction of the symplectic cobordism $(X_+, \Omega_{X_+})$ from $[0, 1] \times \Sigma$ to $M$, together with the Lagrangian submanifold $L_\alpha \subset \partial X_+$.

**4.1. Construction of $(X_+, \Omega_{X_+})$.** — We describe the construction of $X_+$, leaving some key details for later:<fnref n="2" /> First we construct fibrations $\pi_0 : X_+^0 \to B_+^0$ and $\pi_1 : X_+^1 \to D^2$ with fibers diffeomorphic to $\Sigma$ and $S_{1/2}$. Here $B_+^0 = ([0, \infty) \times \mathbf{R}/2\mathbf{Z}) - B_+^c$ with coordinates $(s, t)$ and $B_+^c$ is the subset $[2, \infty) \times [1, 2]$ with the corners rounded. We then glue $X_+^0$ and $X_+^1$ and smooth a boundary component $\mathcal{B}$ of $X_+^0 \cup X_+^1$ to obtain $\tilde{\mathcal{B}} \simeq M$. Finally we attach the negative end $X_+^2 = (-\infty, 0] \times \tilde{\mathcal{B}}$ to obtain $X_+$.

Let $\delta > 0$ be a small irrational number and $N$ a large positive number which depends on $\delta$ and whose dependence will be described later.

**Lemma 4.1.1.** — There exists a symplectic manifold $(X_+, \Omega_{X_+})$ which depends on $\delta > 0$ and which satisfies the following:

(1) There is a symplectic surface $S_{\delta} := \{z^f\} \times (B_+^0 \cup D^2)$, obtained by gluing sections $\{z^f\} \times B_+^0 \subset X_+^0$ and $\{z^f\} \times D^2 \subset X_+^1$.

(2) $\Omega_{X_+} = d\Theta^+$ for some 1-form $\Theta^+$ on $X_+ - N(S_{\delta})$, where $N(S_{\delta})$ is a small neighborhood of $S_{\delta}$.

(3) $\Theta^+$ is exact on the Lagrangian submanifold $L_\alpha \subset \partial X_+$.

(4) On the positive end

$$ \pi_0^{-1}([3, \infty) \times [0, 1]) = [3, \infty) \times \Sigma \times [0, 1] \subset X_+^0 $$

<footnote n="2">² Compare with the description in Section 1.0.1, keeping in mind that the notation will be slightly different.</footnote>
of $X_+$, $\Omega_{X_+}$ restricts to $\tilde{\omega} + ds \wedge dt$, where $\tilde{\omega}$ is an area form on $\Sigma$. Moreover,

$$L_{\alpha} \cap \{s \ge 3\} = ([3, \infty) \times \{0\} \times \beta') \cup ([3, \infty) \times \{1\} \times \alpha),$$

where $\beta'$ is isotopic to $\beta$.

(5) On the negative end $X_+^2$ of $X_+$, $\Omega_{X_+}$ restricts to the negative symplectization of a contact form $\lambda_-$ on $\tilde{B} \simeq M$ which is adapted to the open book decomposition $(S, \hbar)$.

(6) The manifold $\tilde{B} \simeq M$ admits a decomposition into three disjoint pieces: the mapping torus $N_{(S_0, \hbar)}$, a closed neighborhood $N(K)$ of the binding $K$, and an open thickened torus $N$ in between that we refer to as the “no man’s land”.

(7) All the orbits of the Reeb vector field $R_{\lambda_-}$ of $\lambda_-$ in $\text{int}(N(K)) \cup N$ have $\lambda_-$-action $\ge \frac{1}{2\delta} - \kappa$, where $\kappa > 0$ is independent of $\delta$. Moreover, $T_+ = \partial N(K)$ (resp. $T_- = \partial N_{(S_0, \hbar)}$) is a positive (resp. negative) Morse-Bott torus of meridian orbits.

(8) There is an embedding of $W_+$, defined in Section I.5.1.1, into $X_+$ such that the restriction $\pi_1: W_+ \cap X_+^0 \to B_+^0$ is a fibration with fiber $S_0$, $W_+ \cap X_+^1 = \emptyset$, $W_+ \cap X_+^2 = (-\infty, 0] \times N_{(S_0, \hbar)}$, and $W_+ \cap N(S_0) = \emptyset$.

Here $X_+$, $\Omega_{X_+}$, $\Theta^+$, $L_\alpha$, and $\lambda_-$ depend on $\delta > 0$.

The $S^1$-family $P_+$ (resp. $P_-$) of simple orbits of $T_+$ (resp. $T_-$) can be viewed equivalently as a pair $e'$, $h'$ (resp. $e$, $h$) consisting of an elliptic orbit and a hyperbolic orbit. The proof of Lemma 4.1.1 will be given in Section 4.3.

Let $A_{[-1,N]} \simeq [-1, N] \times S^1$ be a small neighborhood of $\partial S_0 = \{0\} \times S^1$ in $\Sigma$ with coordinates $(r_1, \theta_1)$, such that $z' \notin A_{[-1,N]}$, $A_{[-1,0]} \subset S_0$ and $A_{[0,N]} \subset S_{1/2}$. Here we write $A_\mathfrak{J} = \mathfrak{J} \times S^1$ if $\mathfrak{J}$ is a subset of $[-1, N]$. Also let $N(z') \subset S_{1/2} - A_{[0,N]} - \alpha - \beta$ be a small ball $D_\tau = \{r' \le \tau\}$ about $z'$, where we are using polar coordinates $(r', \theta')$.

The actual construction of $(X_+, \Omega_{X_+})$ is a bit involved, and consists of several steps. **Step 1.** The following lemma is a rephrasing of Lemma I.2.1.2 and its proof.

**Lemma 4.1.2.** — After possibly isotoping $h$ relative to $\partial S_0$, there exists a factorization $h = h_0 \circ h_1$ and a contact form $\lambda = f_1(x)dt + \beta_1(x)$, $(x, t) \in S_0 \times [0, 2]$, on $N_{(S_0, h_0)}$ with Reeb vector field $R_\lambda$, such that the following hold:

(1) $h: S_0 \times \{0\} \to S_0 \times \{0\}$ is the first return map of $R_\lambda$.

(2) $h$ has no elliptic periodic point of period $\le 2g$ in $\text{int}(S_0)$, as required for technical reasons in II.1.0.1.

(3) $h_0 = \text{id}$ on $A_{[-1/2,0]}$.

(4) $h_1$ is the flow of $R_\lambda$ from $S_0 \times \{0\}$ to $S_0 \times \{2\}$.

(5) $R_\lambda$ is parallel to $\partial_t$ on $(S_0 - A_{[-1,0]}) \times [0, 2]$. In particular, $h_1 = \text{id}$ on $S_0 - A_{[-1,0]}$.

<footnote n="3">³ In a departure from the stable Hamiltonian vector field $R_0 = \partial_t$ from Section I.5.1, we are not assuming $R_\lambda$ to be parallel to $\partial_t$ on all of $S_0 \times [0, 2]$.</footnote>
(6) $f_t(r_1, \theta_1) = 1 + \epsilon r_1^2/2$ and $\beta_t(r_1, \theta_1) = (C + r_1)d\theta_1$ on $A_{[-1/2,0]}$, for $\epsilon > 0$ sufficiently small and $C > 0$. In particular, $f_t$ and $\beta_t$ are independent of $t$ and $R_\lambda$ is parallel to $\partial_t - \epsilon r_1 \partial_\theta$ on $A_{[-1/2,0]}$.

(7) $|d_2f_t|_{A_{[-1/2,0]}}|_{C^0} \le \delta$ and $\frac{1}{2} \le f_t \le 2$.

Here $\epsilon > 0$ depends on $\delta > 0$, $d_2$ is the differential in the $S_0$-direction, and the $C^0$-norm is with respect to a fixed Riemannian metric on $S_0$.

**Step 2.** We then extend $h_0$, $h_1$, $\bar{h} \in \text{Diff}(S_0, \partial S_0)$ to $h_0^+$, $h_1^+$, $\bar{h}^+ = h_0^+ \circ h_1^+ \in \text{Diff}(\Sigma)$ and the contact form $\lambda$ to the contact form $\lambda_+ = f_t dt + \beta_t$ to $N_{(\Sigma-N(z'),h_0^+)}$, all of which depend on $\delta > 0$, as follows:

(3') $h_0^+ = id$ on $S_{1/2}$.

(4') $h_1^+|_{\Sigma-N(z')}$ is the flow of $R_{\lambda_+}$ from $(\Sigma - N(z')) \times \{0\}$ to $(\Sigma - N(z')) \times \{2\}$ and $h_1^+|_{N(z')} = id$.

(5') $f_t$ and $\beta_t$ are independent of $t$ on $S_{1/2} - N(z')$. Hence $R_{\lambda_+}$ is parallel to $\partial_t + X_f$, where $X_f$ is the Hamiltonian vector field satisfying $i_{X_f} \omega = d_2f$ and $\omega$ is an area form on $\Sigma$ which agrees with $d_2\beta_t$ on $\Sigma - N(z')$.

(6a') $f_t(r', \theta') = \text{const} > 0$ and $\beta_t(r', \theta') = (-C' + r')d\theta'$ near $\partial N(z')$, for $-C' > 0$. In particular, $R_{\lambda_+}$ is parallel to $\partial_t$ near the mapping torus of $\partial N(z')$.

(6b') $f_t(r_1, \theta_1) = 1 + \epsilon r_1^2/2$ near $A_{[0,1]}$ and $\beta_t(r_1, \theta_1) = (C + r_1)d\theta_1$ on $A_{[0,N]}$.

(7') $|d_2f_t|_{S_{1/2}-N(z')}|_{C^0} \le \delta$ and $\frac{1}{2} \le f_t|_{S_{1/2}-N(z')} \le 2$.

Without loss of generality we may assume that $\alpha \times \{1\}$ is Legendrian with respect to $\lambda_+$. This is an easy consequence of the Legendrian realization principle; see for example [H, Theorem 3.7].

**Step 3 (Construction of $(X_+^0, \Omega_{X_+}^0)$).** Let

$$ \tilde{X}_+^0 = ([0, \infty) \times \Sigma \times [0, 2])/(s, x, 2) \sim (s, h_0^+(x), 0) $$

and let $\pi_0: \tilde{X}_+^0 \to [0, \infty) \times \mathbf{R}/2\mathbf{Z}$ be the projection $(s, x, t) \mapsto (s, t)$. We then set

$$ X_+^0 := \pi_0^{-1}(\mathbf{B}_+^0). $$

Let $g: [0, \frac{1}{2}] \to \mathbf{R}$ be a smooth function such that $g(r) = 1 + \epsilon r^2/2$ near $r=0$, $0 < g'(r) \le \delta$ for $r \in (0, \frac{1}{2})$, $g'(r)$ is monotonically decreasing for $r \in (\frac{1}{4}, \frac{1}{2})$, $g'(\frac{1}{2}) = 0$, and $g(\frac{1}{2}) = 1 + \epsilon$. In particular, this requires $2\epsilon < \delta$. Then let

$$ \lambda_{+,s} = f_{s,t} dt + \beta_t, \quad s \in [0, \infty), $$

be a 1-parameter family of contact forms<fnref n="4" /> on $N_{(\Sigma-N(z'),h_0^+)}$ such that the following hold:

(a) $\lambda_{+,s} = \lambda_+$ if $s \ge \frac{3}{2}$ or $(x, t) \in N_{(S_0, h_0)}$.

<footnote n="4">⁴ Note that $\beta_t$ does not depend on $s$.</footnote>
(b) $\lambda_{+,s}$ is independent of $s$ if $s \in [0, \frac{1}{2}]$.

(c) $f_{0,t}(r_1, \theta_1) = g(r_1)$ on $A_{[0,1/2]}$.

(d) $f_{0,t}|_{S_{1/2}-A_{[0,1/2]}-N(z')} = 1 + \varepsilon$. In particular, $d\lambda_{+,0} = d_2\beta_t$ and $R = \partial_t$ on the mapping torus of $S_{1/2} - A_{[0,1/2]} - N(z')$.

(e) $f_{s,t}$ is a constant $C_s > 0$ near $\partial N(z')$.

(f) $|d_2f_{s,t}|_{A_{[-1/2,0]} \cup S_{1/2} - N(z')}|_{C^0} \le \delta$, $|\partial_s f_{s,t}|_{A_{[-1/2,0]} \cup S_{1/2} - N(z')}|_{C^0} \le \delta$ and $\frac{1}{2} \le f_{s,t}|_{\Sigma - N(z')} \le 2$ for all $s, t$.

We then define:

$$ \Omega_{X_+}^0 := \tilde{\omega} + ds \wedge dt, $$

where

$$ \tilde{\omega} = \begin{cases} d\lambda_{+,s} & \text{on } X_+^0 - (N(z') \times B_+^0); \\ \omega & \text{on } N(z') \times B_+^0; \end{cases} $$

and $\omega$ is an area form on $\Sigma$ which agrees with $d_2\beta_t$ on $\Sigma - N(z')$. The 2-form $\Omega_{X_+}^0$ is symplectic by an easy calculation which uses (f).

**Step 4** (Construction of $(X_+^1, \Omega_{X_+}^1)$ and primitives $\Theta_0^+, \Theta_1^+$). Let

$$ X_+^1 := S_{1/2}^1 \times D^2, \quad S_{1/2}^1 := S_{1/2} - A_{[0,1/2]}. $$

We use polar coordinates $(r_2, \theta_2)$ on $D^2 = \{r_2 \le 1\}$. We identify neighborhoods of $\{0\} \times S_{1/2}^1 \times \mathbf{R}/2\mathbf{Z} \subset \partial X_+^0$ and $S_{1/2}^1 \times \partial D^2 \subset \partial X_+^1$ as follows:

$$ \phi_{01}: [-\varepsilon', \varepsilon'] \times S_{1/2}^1 \times \mathbf{R}/2\mathbf{Z} \xrightarrow{\sim} S_{1/2}^1 \times \{(r_2, \theta_2) \mid e^{-\pi\varepsilon'} \le r_2 \le e^{\pi\varepsilon'}\}, \\ (s, x, t) \mapsto (x, e^{\pi s}, \pi t), $$

where $\varepsilon' > 0$ is sufficiently small.

Let $\omega_{D^2}$ be an area form on $D^2$ satisfying:

$$ \omega_{D^2} = \begin{cases} r_2 dr_2 d\theta_2 & \text{near } r_2 = 0; \\ \frac{1}{\pi^2 r_2^2} dr_2 d\theta_2 & \text{near } r_2 = 1. \end{cases} $$

We then define:

$$ \Omega_{X_+}^1 := \tilde{\omega}|_{S_{1/2}^1} + \omega_{D^2}. $$

An easy calculation shows that $\omega_{D^2} = ds \wedge dt$, and hence $\Omega_{X_+}^1 = \Omega_{X_+}^0$, on their overlap.

We write $\omega_{D^2} = d(\phi(r_2)d\theta_2)$, where $\phi: [0, 1] \to \mathbf{R}$ satisfies

$$ \phi(r_2) = \begin{cases} r_2^2/2 & \text{near } r_2 = 0; \\ \frac{1}{\pi^2}\log r_2 + \frac{1}{10} & \text{near } r_2 = 1. \end{cases} $$
<figure><img src="image_7.png" /><figcaption>FIG. 3. — Schematic diagram for rounding the corner of $\mathcal{B}$. The diagram shows a neighborhood $N(\mathcal{B})$ of $\mathcal{B}$, where we are projecting $X_+^0 \cap N(\mathcal{B})$ to coordinates $(s, r_1)$ and $X_+^1 \cap N(\mathcal{B})$ to coordinates $(r_2, r_1)$. (Color figure online)</figcaption></figure>

Then $\phi(r_2)d\theta_2 = (s + \frac{\pi}{10})dt$ on their overlap. The choice of the constant $\frac{\pi}{10} < 1$ will be used in the proof of Lemma 5.4.2. We then define primitives $\Theta_i^+$ of $\Omega_{X_+}^i$, $i = 0, 1$, as follows:

$$ (4.1.1) \qquad \Theta_0^+ = \lambda_{+,s} + (s + \frac{\pi}{10})dt \quad \text{on } X_+^0 - (N(z') \times B_+^0); $$

$$ (4.1.2) \qquad \Theta_1^+ = \lambda_{+,0} + \phi(r_2)d\theta_2 \quad \text{on } X_+^1 - (N(z') \times D^2). $$

We have $\Theta_0^+ = \Theta_1^+$ on their overlap.

**Step 5** (Corner smoothing). We now have a 4-manifold $X_+^0 \cup X_+^1$ with a concave corner along $(\partial S_{1/2}') \times \partial D^2$. The component $\mathcal{B}$ of $\partial(X_+^0 \cup X_+^1)$ that contains the corner is homeomorphic to $M$ and $(\partial S_{1/2}') \times D^2$ is a neighborhood of the binding $(\partial S_{1/2}') \times \{0\}$.

In this step we round the corner of $\mathcal{B}$ to obtain the smoothing $\tilde{\mathcal{B}} \subset X_+^0 \cup X_+^1$. We write $\tilde{\mathcal{B}}_i = \tilde{\mathcal{B}} \cap X_+^i$, $i = 0, 1$. We define the contact form $\lambda_-$ on $\tilde{\mathcal{B}}$ so that $\lambda_-|_{\tilde{\mathcal{B}}_i} = \Theta_i^+|_{\tilde{\mathcal{B}}_i}$, $i = 0, 1$. Here the notation $|_A$ refers to the pullback to $A$. See Figure 3.

*Construction of* $\tilde{\mathcal{B}}_0$. There exist $\varepsilon_0, \varepsilon_1 > 0$ small with $\frac{\varepsilon_1}{2\varepsilon_0} < \delta$ and $\varepsilon_1 < \varepsilon'$ and a smooth map $\psi : [0, \frac{1}{2} + \varepsilon_0] \to \mathbb{R}$ such that:

$$ - \psi(r_1) = \varepsilon_1 \text{ on } [0, \frac{1}{2} - \varepsilon_0]; $$

$$ - \psi'(r_1) \text{ is monotonically decreasing and } -\delta \le \psi'(r_1) < 0 \text{ on } (\frac{1}{2} - \varepsilon_0, \frac{1}{2} + \varepsilon_0); \text{ and} $$

$$ - \psi(\frac{1}{2} + \varepsilon_0) = 0 \text{ and } \psi'(\frac{1}{2} + \varepsilon_0) = -\delta. $$

We then let $\tilde{\mathcal{B}}_0 = \tilde{\mathcal{B}}_{00} \cup \tilde{\mathcal{B}}_{01}$, where

$$ \tilde{\mathcal{B}}_{00} = \{s = \varepsilon_1\} \times N_{(S_0, h_0)}, $$
$$ \tilde{\mathcal{B}}_{01} = \{s = \psi(r_1), r_1 \in [0, \frac{1}{2} + \varepsilon_0]\} \times \mathbf{R}/2\mathbf{Z} \times \mathbf{S}^1. $$

Here $\mathbf{R}/2\mathbf{Z} \times \mathbf{S}^1$ has coordinates $(t, \theta_1)$.

**Lemma 4.1.3.** — There exists a unique $r_1^* \in (0, \frac{1}{2} + \varepsilon_0)$ such that each orbit in $\tilde{\mathcal{B}}_{01} \cap \{r_1 \neq r_1^*\}$ is directed by some $\partial_t + \delta'\partial_{\theta_1}$, where $0 < |\delta'| \le \delta$ and $\delta'$ depends on the orbit. Also $\tilde{\mathcal{B}}_{01} \cap \{r_1 = \frac{1}{2} + \varepsilon_0\}$ is directed by $\partial_t + \delta\partial_{\theta_1}$.

*Proof.* — The 1-form $\lambda_-|_{\tilde{\mathcal{B}}_{00}}$ is clearly a contact form and

$$ (4.1.3) \qquad \lambda_-|_{\tilde{\mathcal{B}}_{01}} = (\psi(r_1) + f_{0,t}(r_1, \theta_1) + \pi/10)dt + (C + r_1)d\theta_1, $$

with respect to coordinates $(r_1, \theta_1, t)$. The Reeb vector field $R_{\lambda_-}$ is parallel to $\partial_t - \frac{\partial}{\partial r_1}(\psi + f_{0,t})\partial_{\theta_1}$. Let $r_1^* \in [0, \frac{1}{2} + \varepsilon_0]$ be the point where $\frac{\partial}{\partial r_1}(\psi + f_{0,t}) = 0$. Then $0 < -\frac{\partial}{\partial r_1}(\psi + f_{0,t}) \le \delta$ for $r_1 \in [r_1^*, \frac{1}{2} + \varepsilon_0]$, $-\frac{\partial}{\partial r_1}(\psi + f_{0,t})(\frac{1}{2} + \varepsilon_0) = \delta$, and $0 < \frac{\partial}{\partial r_1}(\psi + f_{0,t}) \le \delta$ for $r_1 \in (0, r_1^*)$, which imply the lemma. $\square$

*Construction of $\tilde{\mathcal{B}}_1$.* Let $\zeta : [0, 1] \to \mathbf{R}$ be a smooth map such that:

$$ - \zeta(r_2) = k_0 - k_1 r_2^2/2 \text{ near } r_2 = 0, \text{ where } k_0, k_1 \gg 0; $$

$$ - \zeta'' < 0 \text{ on } (0, 1]; $$

$$ - \zeta(1) = \frac{1}{2} + \varepsilon_0. $$

We then define $\tilde{\mathcal{B}}_1 = \{r_1 = \zeta(r_2)\}$.

**Lemma 4.1.4.** — There exist $k_0, k_1 \gg 0$, $N = N(k_0, k_1) \gg 0$, and $\zeta$ such that $R_{\lambda_-}|_{\tilde{\mathcal{B}}_1}$ is directed by $\pi\partial_{\theta_2} + \delta\partial_{\theta_1}$, which agrees with $\partial_t + \delta\partial_{\theta_1}$ on $\tilde{\mathcal{B}}_0$.

*Proof.* — $\lambda_-|_{\tilde{\mathcal{B}}_1}$ is given by

$$ (4.1.4) \qquad \lambda_-|_{\tilde{\mathcal{B}}_1} = (\phi(r_2) + (1+\varepsilon)/\pi)d\theta_2 + (C + \zeta(r_2))d\theta_1, $$

with respect to coordinates $(\theta_1, r_2, \theta_2)$. The Reeb vector field $R_{\lambda_-}$ is parallel to $\pi\partial_{\theta_2} - \pi\frac{\phi'}{\zeta'}\partial_{\theta_1}$. By choosing $k_0, k_1 \gg 0$, $N(k_0, k_1) \gg 0$, and $\zeta$ suitably, we may assume that $-\frac{\phi'}{\zeta'}(r_2) = \frac{\delta}{\pi}$ for all $r_2 \in (0, 1]$. $\square$

We also define $N(K) \subset \tilde{\mathcal{B}}$ as the closed neighborhood of the binding $K = \{r_2 = 0\}$ that is bounded by the torus $\{r_1 = r_1^*\}$. The region $\mathcal{N} = \{0 < r_1 < r_1^*\} \subset \tilde{\mathcal{B}}$ will be called “no man’s land”.

**Step 6** (Construction of $(X_+^2, \Omega_{X_+}^2)$). Let $X_+^{01} \subset X_+^0 \cup X_+^1$ be the closure of the component of $(X_+^0 \cup X_+^1) - \tilde{\mathcal{B}}$ that does not contain $\mathcal{B}$. We then glue the negative cylindrical end

$$ (X_+^2, \Omega_{X_+}^2) := ((-\infty, 0] \times \tilde{\mathcal{B}}, d(e^{\lambda_-})) $$
to $X_+^{01}$ along $\tilde{\mathcal{B}}$, where $s'$ is the coordinate for $(-\infty, 0]$. This concludes the construction of $(X_+, \Omega_{X_+})$.

**4.2. Further definitions.**

Hamiltonian structure on $\Sigma \times [0, 1]$. Let $\bar{\omega} = \tilde{\omega}|_{s=\frac{3}{2}}$. The Hamiltonian structure on $\Sigma \times [0, 1]$ at the positive end of $X_+$ is given by $(dt, \bar{\omega}|_{\Sigma \times [0,1]})$. Let $\tilde{h}_2^+$ be the flow of the corresponding Hamiltonian vector field from $\Sigma \times \{0\}$ to $\Sigma \times \{1\}$; this is different from $h_1^+$, which is the flow from $\Sigma \times \{0\}$ to $\Sigma \times \{2\}$. Also note that we do not necessarily have $\tilde{h}_2^+ = id$ by construction. Lagrangian submanifold $L_\alpha$. As in Section I.5.2.1, we define the Lagrangian submanifold $L_\alpha \subset \partial X_+$ by placing a copy of $\alpha$ on the fiber $\pi^{-1}(3, 1)$ over $(3, 1) \in \partial B_+^0$ and using the symplectic connection $\Omega_{X_+}$ to parallel transport $\alpha$ along the boundary component $(\partial B_+^0) \cap \{s \ge 1\}$ of $B_+^0$. Observe that

$$ (4.2.1) \qquad L_\alpha \cap \{s \ge 3\} = ([3, \infty) \times \{0\} \times \tilde{h}_0^+(\alpha)) \cup ([3, \infty) \times \{1\} \times \alpha). $$

**Lemma 4.2.1.** — $\beta' := \tilde{h}^+ \circ (\tilde{h}_2^+)^{-1}(\alpha)$ is isotopic to $\beta$.

*Proof.* — Observe that $\tilde{h}_1^+$ and $\tilde{h}_2^+$ are isotopic to the identity. Then $\tilde{h}^+$ is isotopic to $\tilde{h}_0^+$ where $\tilde{h}_0^+|_{S_{1/2}} = id$ and $\tilde{h}_0^+|_{S_0}$ is isotopic to $\tilde{h}$. The lemma then follows. $\square$

Submanifolds $S_z$, $C_\theta$, and $\mathcal{H}$. Given $z \in N(z')$, let

$$ S_z = \{z\} \times (B_+^0 \cup D^2), $$

where $\{z\} \times B_+^0 \subset X_+^0$ and $\{z\} \times D^2 \subset X_+^1$. Also let

$$ C_{\theta} = (\{\theta\} \times B_{+}^{0}) \cup (\{\theta\} \times (-\infty, 0]_{s'} \times \mathbf{R}/2\mathbf{Z}), $$

where $\theta \in \partial S_0$, and let $\mathcal{H} = \bigcup_{\theta \in \partial S_0} C_\theta$.

*Definition of* $W_+$. Let $W_+$ be the closure of the component of $X_+ - \mathcal{H}$ which is disjoint from $S_{(z')s}$. In particular, the restriction $\pi_1: W_+ \cap X_+^0 \to B_+^0$ is a fibration with fiber $S_0$, $W_+ \cap X_+^1 = \emptyset$, and $W_+ \cap X_+^2 = (-\infty, 0] \times N_{(S_0, \emptyset)}$. The cobordism $W_+$ is diffeomorphic to the cobordism used to define the map $\Phi$ in Section I.5.1.

**4.3.** *Proof of Lemma 4.1.1.* — (1), (5), (6), (8) are clear from the construction.

(2) follows by letting $\Theta^+ = \Theta_i^+$, $i = 0, 1, 2$, where defined.

(3) By construction, $L_\alpha$ is Lagrangian and $d\Theta^+|_{L_\alpha} = 0$. It then suffices to observe that $\Theta^+ = 0$ on $L_\alpha \cap \pi^{-1}(3, 1)$. This follows from the fact that $\alpha \times \{1\}$ is a Legendrian submanifold of $(N_{(\Sigma-N(z'), \tilde{h}_0^+)}, \lambda_+)$.

(4) The first sentence follows from the construction and the second sentence follows from Lemma 4.2.1.

(7) By Lemma 4.1.4, the Reeb vector field $R_{\lambda_-}$ has no closed orbits in $\tilde{\mathcal{B}}_1$ since $\delta > 0$ is irrational. By Lemma 4.1.3 and Equation (4.1.3), each orbit of $R_{\lambda_-}$ in $\tilde{\mathcal{B}}_{01} \cap \{r_1 \neq r_1^*\}$
has $\lambda_-$-action $\ge \frac{1}{2\delta} - (C + \frac{1}{2})$, where $C > 0$ is independent of $\delta$. The second sentence of (7) is immediate from the construction of $\lambda_-$.

## 5. The chain map $\Phi^+$

The goal of this section is to define the chain map

$$ \Phi^+ : \mathrm{CF}^+(\Sigma, \alpha, \beta, z') \to \mathrm{ECC}(M, \lambda_-), $$

which is induced by the symplectic cobordism $(X_+, \Omega_{X_+})$ and an admissible almost complex structure $J^+$. We can write $\beta = \tilde{h}_2^+ \circ \tilde{h}^+ \circ (\tilde{h}_2^+)^{-1}(\alpha)$, in view of Equation (4.2.1) and Lemma 4.2.1 and the fact that $\tilde{h}_2^+$ is the flow of the Hamiltonian vector field of $\tilde{\omega}|_{s=s_0}$, $s_0 \gg 0$, from $\Sigma \times \{0\}$ to $\Sigma \times \{1\}$ before normalization.

For simplicity we identify $X_+ \cap \{s \ge s_0\} \simeq [s_0, \infty) \times [0, 1] \times \Sigma$ with coordinates $(s, t, x)$ so that $\tilde{h}_2^+ = \mathrm{id}$ and the Hamiltonian vector field is $\partial_t$.

**5.1. Almost complex structures.** — Let $\bar{\omega} = \tilde{\omega}|_{s=3/2}$.

**Lemma 5.1.1.** — There exists a family $(\bar{\lambda}_\tau, \bar{\omega})$, $\tau \in [0, 1]$, of stable Hamiltonian structures on $N_{(S_0, \tilde{h}_0)}$ such that $\bar{\lambda}_1 = \lambda$, $\bar{\lambda}_\tau$ is a contact form for $\tau > 0$, and $\bar{\lambda}_0 = \mathrm{dt}$. The 1-forms $\bar{\lambda}_\tau = f_{\tau, \tau} dt + \beta_{\tau, \tau}$ can be normalized so that $\frac{1}{2} < |f_{\tau, \tau}| \le 2$.

*Proof.* — Follows from the discussion of Section I.3.1. $\square$

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

**Definition 5.1.2.** — An almost complex structure $J^+$ on $X_+$ is $(X_+, \Omega_{X_+})$-admissible if the following hold:

(1) $J^+$ is tamed by $\Omega_{X_+}$;

(2) $J^+$ is s-invariant for $\{s \ge \frac{3}{2}\} \cap X_+^0$ and is adapted to the stable Hamiltonian structure $(\mathrm{d}t, \bar{\omega}|_{\Sigma \times [0, 1]})$ at the positive end;

(3) $J^+$ is s'-invariant for $\{s' \le -\frac{1}{2}\} \cap X_+^2$ and is adapted to the contact form $\lambda_-$ at the negative end;

(4) the restriction $J_+$ of $J^+$ to $W_+$ is $C^\ell$-close to a regular admissible almost complex structure $J_+^0$ on $W_+$ with respect to $(\bar{\lambda}_0, \bar{\omega})$ (cf. Definitions I.5.4.1 and I.5.8.5);

(5) the surfaces $S_{(z')z}$ and $C_\theta$ are $J^+$-holomorphic for all $\theta \in \partial S_0$.

Let $J, J'$ be the adapted almost complex structures that agree with $J^+$ at the positive and negative ends.

Note that (4) imposes additional conditions on $\Omega_{X_+}$ and $\lambda_-$. In practice, the order in which we construct $\Omega_{X_+}$ and $J^+$ is a little convoluted: (i) choose a regular $J_+^0$, (ii) choose $\tau > 0$ sufficiently small and $J_+$ sufficiently close to $J_+^0$, (iii) construct $\Omega_{X_+}$ using $\bar{\lambda}_\tau$ in place of $\lambda$, and (iv) extend $J^+$ to the rest of $X_+$.

Let $\mathcal{J}_{X_+}$ be the set of all $(X_+, \Omega_{X_+})$-admissible almost complex structures.
**5.2. The ECH index.** — Let $\mathcal{P} = \mathcal{P}_{\lambda_{-}}$ be the set of simple orbits of $\mathbb{R}_{\lambda_{-}}$ and let $\mathcal{O} = \mathcal{O}_{\lambda_{-}}$ be the set of orbit sets constructed from $\mathcal{P}$.

Let $J^+ \in \mathcal{J}_{X_+}$ be an admissible almost complex structure. Let $\mathcal{M}_{J^+}(\mathbf{y}, \mathbf{y})$ be the set of holomorphic maps $u : (\dot{\mathbb{F}}, j) \to (X_+, J^+)$ from $\mathbf{y} \in \mathcal{S}_{\alpha, \beta}$ to $\mathbf{y} \in \mathcal{O}$, such that each component of $\partial \dot{\mathbb{F}}$ is mapped to a distinct component of $L_{\alpha}$ and each component of $L_{\alpha}$ is used exactly once. Here $(\mathbb{F}, j)$ is a compact Riemann surface with boundary, $\dot{\mathbb{F}} = \mathbb{F} - \mathbf{q}_+ - \mathbf{q}_-$, $\mathbf{q}_+$ is the set of boundary punctures, and $\mathbf{q}_-$ is the set of interior punctures. Elements of $\mathcal{M}_{J^+}(\mathbf{y}, \mathbf{y})$ will be called $X_+$-curves.

Let $\tilde{X}_+$ be $X_+$ with the ends $\{s > 3\}$ and $\{s' < -1\}$ removed and let

$$Z_{\mathbf{y},\mathbf{y}} = (L_{\alpha} \cap \tilde{X}_+) \cup (\{3\} \times [0, 1] \times \mathbf{y}) \cup (\{-1\} \times \mathbf{y})$$

as in Section I.5.4.2. The class $[u]$ of $u \in \mathcal{M}_{J^+}(\mathbf{y}, \mathbf{y})$ is the relative homology class of the compactification $\tilde{u}$ in $H_2(\tilde{X}_+, Z_{\mathbf{y},\mathbf{y}})$. Given $A \in H_2(\tilde{X}_+, Z_{\mathbf{y},\mathbf{y}})$, we write $\mathcal{M}_{J^+}(\mathbf{y}, \mathbf{y}, A) \subset \mathcal{M}_{J^+}(\mathbf{y}, \mathbf{y})$ for the subset of $X_+$-curves $u$ in the class A.

**Definition 5.2.1 (Filtration $\mathcal{F}$).** — Given a $X_+$-curve $u$ that limits to $\mathbf{y}$ at the positive end and $\mathbf{y}$ at the negative end, we define

$$\mathcal{F}(u) = \langle [u], S_{(z')f} \rangle,$$

where $\langle, \rangle$ is the algebraic intersection number. Since $S_{(z')f}$ is a holomorphic divisor, $\mathcal{F}(u) \ge 0$. We will also refer to $u$ as an $X_+$-curve from $[\mathbf{y}, \mathcal{F}(u)]$ to $\mathbf{y}$.

The definition of the ECH index given in Section I.5.6 also extends directly to our case. The ECH index of a $X_+$-curve from $\mathbf{y}$ to $\mathbf{y}$ in the class A is denoted by $I_{X_+}(\mathbf{y}, A)$.

**5.3. Homology of $X_+$.** — The goal of this subsection is to compute $H_2(X_+)$. Let us write $N = N_{(S_0, \mathfrak{h})}$, $N_0 = N_{(S_{1/2}, \mathfrak{h}^+|_{S_{1/2}})}$ and $\bar{N} = N_{(\Sigma, \mathfrak{h}^+)}$.

**Lemma 5.3.1.** — $H_2(N) \cong H_2(M)$ and $H_1(N) \cong H_1(M) \oplus \mathbf{Z}$, where the extra $\mathbf{Z}$ factor is generated by a meridian of the binding.

*Proof.* — The lemma follows from the exact sequence of the pair $(M, N)$. $\square$

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

$$\textbf{Lemma 5.3.2.} \quad - H_2(X_+^0) \cong H_2(N) \oplus H_2(N_0) \oplus H_2(\Sigma).$$

*Proof.* — Observe that $X_+^0$ is homotopy equivalent to $\bar{N}$. We compute $H_2(\bar{N})$ using the Mayer-Vietoris sequence:

$$\begin{aligned} H_2(N \cap N_0) &\xrightarrow{i} H_2(N) \oplus H_2(N_0) \to H_2(\bar{N}) \to H_1(N \cap N_0) \\ &\xrightarrow{j} H_1(N) \oplus H_1(N_0). \end{aligned}$$
Since $i=0$ and $\ker j = \mathbf{Z}\langle\partial S_0\rangle = \mathbf{Z}\langle\partial S_{1/2}\rangle$, the lemma follows. $\square$

**Lemma 5.3.3.** — $H_2(X_+) \cong H_2(M) \oplus H_2(\Sigma)$.

*Proof.* — $X_+$ is homotopy equivalent to $X_+^0 \cup X_+^1$ and $X_+^0 \cap X_+^1 \cong N_0$. Since $X_+^1$ is homotopy equivalent to $S_{1/2}$, the Mayer-Vietoris sequence becomes:

$$H_2(N_0) \xrightarrow{i} H_2(X_+^0) \to H_2(X_+) \to H_1(N_0) \xrightarrow{j} H_1(X_+^0) \oplus H_1(S_{1/2}).$$

The map $i$ surjects onto the factor $H_2(N_0)$ in the decomposition of $H_2(X_+^0)$ coming from Lemma 5.3.2. The map $j$ is injective, since $H_1(N_0) \cong H_1(S_{1/2}) \oplus H_1(S^1)$ by the Küneth formula, the restriction $j: H_1(S_{1/2}) \to H_1(S_{1/2})$ is an isomorphism, and the restriction $j: H_1(S^1) \to H_1(\overline{N}) \cong H_1(X_+^0)$ is injective because the image of the generator of $H_1(S^1)$ is dual to the fiber $\Sigma$. The lemma then follows from Lemma 5.3.1. $\square$

**5.4. Energy bound.**

*Definition 5.4.1.* — Let $C_+$ be the set of nondecreasing functions $\phi: [0, +\infty) \to [0, 1]$ such that $\phi(s) = s + \frac{\pi}{10}$ near $s = 0^5$ and let $C_-$ be the set of nondecreasing functions $\psi: (-\infty, 0] \to [0, 1]$ such that $\psi(s') = e^s$ near $s' = 0$. Let

$$\Omega_{\phi, \psi}^{+} := \begin{cases} \tilde{\omega} + d\phi(s) \wedge dt & \text{on } X_{+}^{0} \cap X_{+}^{01}; \\ \Omega_{X_{+}}^{1} & \text{on } X_{+}^{1} \cap X_{+}^{01}; \\ d(\psi(s')\lambda_{-}) & \text{on } X_{+}^{2}, \end{cases}$$

where $(\phi, \psi) \in C_+ \times C_-$.<fnref n="6" /> Then the energy of an $X_+$-curve $u: \dot{F} \to X_+$ from $[\mathbf{y}, k]$ to $\mathbf{y}$ is given by:

$$(5.4.1) \qquad E(u) = \sup_{\phi, \psi} \int_{\mathcal{F}} u^* \Omega_{\phi, \psi}^+,$$

where the supremum is taken over all pairs $(\phi, \psi) \in C_+ \times C_-.$

The condition imposed on the intersection with $S_{(z')f}$ gives an energy bound:

**Lemma 5.4.2 (Energy bound).** — For all $k \in \mathbf{N}$, there exists $N_k > 0$ such that $E(u) \le N_k$ for all $\mathbf{y} \in S_{\alpha, \beta}$, $\mathbf{y} \in \mathcal{O}$, and $u \in \mathcal{M}_{J^+}^{F=k}(\mathbf{y}, \mathbf{y})$.

*Proof.* — Let $u: (\dot{F}, j) \to (X_+, J^+)$ be an element of $\mathcal{M}_{J^+}^{F=k}(\mathbf{y}, \mathbf{y})$. By (2) and (3) of Lemma 4.1.1, $\Omega_{X_+} = d\Theta^+$ on $X_+^\circ := X_+ - N(S_{z'})$ and $\Theta^+$ is exact on the Lagrangian

<footnote n="5">⁵ See the discussion in the second paragraph of the proof of Lemma 5.4.2 which justifies this definition.</footnote>

<footnote n="6">⁶ $\phi, \psi$ used here are not to be confused with $\phi, \psi$ which appeared in Section 4.1.</footnote>
L<sub>α</sub>. Hence ∫<sub>∂F</sub> u<sup>*</sup>Θ<sup>+</sup> only depends on **y**. Since Θ<sup>+</sup> = (s + π/10)dt + λ<sub>+</sub> along Im u(∂F) by Equation (4.1.1) and Section 4.1, Step 3, Item (a), there exists a constant C(**y**) such that

$$
(5.4.2) \qquad \int_{\partial \tilde{F}} u^* \lambda_+ < C(\mathbf{y}).
$$

Let $v : \dot{F}' \to X_+^\circ$ be a representative of the homology class $[u] - k[\Sigma] \in$
$H_2(\tilde{X}_+, Z_{\mathbf{y},\mathbf{y}})$. Since the energy is obtained by integrating a closed form,

$$
(5.4.3) \qquad E(u) = E(v) + k \int_{\Sigma} \tilde{\omega}.
$$

Now $\Omega_{\phi, \psi}^{+} = d\Theta_{\phi, \psi}^{+}$ on $X_{+}^{\circ}$, where

$$
\Theta_{\phi, \psi}^{+} = \begin{cases} \lambda_{+,s} + \phi(s)dt & \text{on } X_{+}^{0} \cap X_{+}^{\circ} \cap X_{+}^{01}; \\ \Theta_{1}^{+} & \text{on } X_{+}^{1} \cap X_{+}^{\circ} \cap X_{+}^{01}; \\ \psi(s')\lambda_{-} & \text{on } X_{+}^{2}. \end{cases}
$$

By Equations (4.1.1) and (4.1.2), Θ₁⁺ can be written as λ<sub>+,s</sub> + (s + π/10)dt on X₀⁺ ∩ X₁⁺ ∩ X₀⁺ ∩ X₁<sup>01</sup>. Observe that, since π/10 < 1, there exist φ ∈ C⁺ such that φ(s) = s + π/10 near s = 0; the compatibility with Θ₁⁺ justifies the definition of C⁺.

By Stokes’ theorem,

$$
\begin{align*}
(5.4.4) \quad E(v) \le \int_{\{s\}\times[0,1]\times\mathbf{y}, s\ge 3/2} \lambda_+ + \sup_{\phi\in C_+} \lim_{s\to\infty} \int_{\{s\}\times[0,1]\times\mathbf{y}} \phi dt \\
\phantom{(5.4.4) \quad E(v) \le} + \int_{\partial\dot{F}'} v^*\lambda_+ + \sup_{\phi\in C_+} \int_{\partial\dot{F}'} \phi dt - \inf_{\psi\in C_-} \int_{\mathbf{y}} \psi\lambda_- \\
\phantom{(5.4.4) \quad E(v) \le} \le 4g + \int_{[0,1]\times\mathbf{y}} \lambda_+ + \int_{\partial\dot{F}'} v^*\lambda_+.
\end{align*}
$$

Recall that $\lambda_{+,s} = \lambda_+$ for $s \ge \frac{3}{2}$. In the above calculation,

$$
\sup_{\phi \in C_+} \lim_{s \to \infty} \int_{\{s\} \times [0,1] \times \mathbf{y}} \phi dt = 2g, \quad \sup_{\phi \in C_+} \int_{\partial \dot{F}'} \phi dt = 2g, \quad \inf_{\psi \in C_-} \int_{\mathbf{y}} \psi \lambda_- = 0.
$$

Combining Equations (5.4.2), (5.4.3), and (5.4.4), we obtain

$$
E(u) \le 4g + C(\mathbf{y}) + \int_{[0,1]\times\mathbf{y}} \lambda_+ + k \int_{\Sigma} \tilde{\omega},
$$

which is the desired bound.

<figure><img src="image_10.png" /></figure>
**5.5. Regularity.** — Define the subset $\mathcal{M}_{J^+}^h(\mathbf{y}, \mathbf{\gamma}, A) \subset \mathcal{M}_{J^+}(\mathbf{y}, \mathbf{\gamma}, A)$ consisting of holomorphic curves without vertical fiber components. As in Lemma I.5.8.2, the set $\mathcal{J}_{X_+}^{reg}$ of regular $J^+ \in \mathcal{J}_{X_+}$ for which all the moduli spaces $\mathcal{M}_{J^+}^h(\mathbf{y}, \mathbf{\gamma}, A)$ are transversally cut out is a dense subset of $\mathcal{J}_{X_+}$. We can restrict attention to $\mathcal{M}_{J^+}^h(\mathbf{y}, \mathbf{\gamma}, A)$ for the following reason:

**Lemma 5.5.1.** — If $J^+ \in \mathcal{J}_{X_+}^{reg}$ and $u \in \mathcal{M}_{J^+}(\mathbf{y}, \mathbf{\gamma}, A) - \mathcal{M}_{J^+}^h(\mathbf{y}, \mathbf{\gamma}, A)$, then $\mathrm{I}_{X_+}(u) \ge 2+2g$.

*Proof.* — Suppose $u = u_1 \cup u_2$, where $u_1$ is regular and $u_2$ is homologous to $k \ge 1$ times a fiber. Since $\langle u_1, u_2 \rangle = k \cdot 2g$,

$$ 
\begin{aligned}
\mathrm{I}(u) &= \mathrm{I}(u_1) + \mathrm{I}(u_2) + 2k(2g) \\
&\ge 0 + k(2 - 2g) + 4kg \ge k(2 + 2g).
\end{aligned}
$$

Here $\mathrm{I}(u_1) \ge 0$ since $\mathrm{I}(u_1) \ge \mathrm{ind}(u_1)$ by the index inequality and $\mathrm{ind}(u_1) \ge 0$ by the regularity of $u_1$. $\square$

**5.6. Holomorphic curves in $X_+$ without positive ends.** — In this subsection and the next, we make essential use of the assumption $g(S) \ge 2$.

Let $S'' = S_{1/2} - A_{[0,N]}$ and let $\bar{S}'' = S'' \cup \{\infty\}$ be the one-point compactification of $S''$. We define the “projection” $\pi_{\bar{S}''}: X_+ \to \bar{S}''$ as follows:

- on $X_+^0$, $\pi_{\bar{S}''}(s, x, t) = x$ if $x \in S''$ and $\pi_{\bar{S}''}(s, x, t) = \infty$ if $x \notin S''$;

- on $X_+^1$, $\pi_{\bar{S}''}(x, r_2, \theta_2) = x$ if $x \in S''$ and $\pi_{\bar{S}''}(x, r_2, \theta_2) = \infty$ if $x \notin S''$;

- $\pi_{\bar{S}''}(X_+^2) = \{\infty\}$.

**Lemma 5.6.1.** — If $u : \dot{\mathbb{F}} \to (X_+, J^+)$ is a holomorphic map without positive ends, then $g(\mathbb{F}) \ge 2$.

*Proof.* — The map $\pi_{\bar{S}''} \circ u$ can be extended to a continuous map $f : \mathbb{F} \to \bar{S}''$. Observe that the curve $u$ must intersect $S_{(z')f}$ because the symplectic form is exact on $X_+ - S_{(z')f}$. Hence $\mathrm{deg} f > 0$. Now we use the following fact: If $f : \Sigma_1 \to \Sigma_2$ is a positive degree map between closed oriented surfaces, then $g(\Sigma_1) \ge g(\Sigma_2)$. Since $g(S) = g(\bar{S}'') \ge 2$, it follows that $g(\mathbb{F}) \ge 2$. $\square$

**Lemma 5.6.2.** — There are no $\mathrm{I}=0$ closed holomorphic curves in $(X_+, J^+)$.

*Proof.* — We argue by contradiction. Let $A = [u_*(\mathbb{F})]$. By Lemma 5.3.3, the intersection form on $\mathbb{H}_2(X_+)$ is trivial. Hence $A \cdot A = 0$. If $\mathrm{I}(A) = A \cdot A + c_1(A) = 0$, then it follows that $c_1(A) = 0$.
Suppose that $u$ is simple. Then $\chi(F) \ge 0$ by the adjunction formula. This contradicts Lemma 5.6.1. In particular $I(u) > 0$ by the regularity of $u$ and the index inequality. If $v$ is a degree $d$ branched cover of $u$ in the class $A$, then $I(v) = I(dA) = dI(A) \ge d$ using the formula

$$ (5.6.1) \qquad I(dA) = dI(A) + (d^2 - d)A \cdot A. $$

**Lemma 5.6.3.** — A multiply-covered holomorphic curve $u$ with only negative ends has $I(u) > 0$.

*Proof.* — This follows from the inequality

$$ (5.6.2) \qquad I(dC) \ge dI(C) + \frac{(d^2 - d)}{2}(2g(C) - 2 + \mathrm{ind}(C) + h) $$

from [Hu, Section 5.1], where $C$ is a simple curve, $\mathrm{ind}(C)$ is the Fredholm index of $C$ (which is nonnegative), and $h$ is the number of hyperbolic ends. Here $2g(C) - 2 > 0$ by Lemma 5.6.1. $\square$

**5.7.** The map $\Phi^+$. — Let $J^+ \in \mathcal{J}_{X_+}^{\mathrm{reg}}$. The chain map $\Phi^+$ is given as follows:

$$ \Phi^+ : (\mathrm{CF}^+(\Sigma, \alpha, \beta, z'), \partial) \to (\mathrm{ECC}(M, \lambda_-, \partial'), \partial') $$

$$ [\mathbf{y}, i] \mapsto \sum_{\mathbf{\gamma}, A} \#\mathcal{M}_{J^+}^{\mathcal{F}=i, I_{X_+}=0}(\mathbf{y}, \mathbf{\gamma}, A) \cdot \mathbf{\gamma}, $$

where the summation is over all $\mathbf{\gamma} \in \mathcal{O}_{\lambda_-}$ and $A \in H_2(\tilde{X}_+, Z_{\mathbf{y},\mathbf{y}})$. Here $\partial'$ is the usual ECH differential on ECC(M, $\lambda_-$).

By a combination of Lemma 5.4.2 and the Gromov-Taubes compactness theorem (cf. Section I.3.4), the sum in the definition of $\Phi^+$ is finite. Hence $\Phi^+$ is well-defined.

**Theorem 5.7.1.** — If $g(S) \ge 2$, then $\Phi^+$ is a chain map.

*Proof.* — Similar to that of Theorem I.6.2.4, with slight modifications in view of Lemmas 5.6.2 and 5.6.3. $\square$

**Remark 5.7.2.** — One can define the twisted coefficient analog of $\Phi^+$, taking into account Lemma 5.3.3.

**5.8. Restriction to $\Phi$.** — In this subsection $\delta$ still denotes the constant that appears in the construction of $\lambda_-$, Let $\mathcal{P}|_N$ be the subset of $\mathcal{P}$ consisting of orbits that are contained in $N = N_{(S_0,h)}$. Also let $\gamma_\theta \in \mathcal{P}_-$ be the orbit corresponding to $\theta \in \partial S_0$.
**Lemma 5.8.1.** — For $\delta > 0$ sufficiently small, if $u \in \mathcal{M}_{J^+}^{\mathcal{F}=0}(\mathbf{y}, \mathbf{\gamma})$, $\mathbf{y} \in \mathcal{S}_{\alpha, \beta}$, $\mathbf{y} \subset S_0$, and $\mathbf{\gamma} \in \mathcal{O}$, then $\mathbf{\gamma}$ is constructed from $\mathcal{P}|_N \cup \{e', h'\}$.

*Proof.* — If $\mathcal{M}_{J^+}^{\mathcal{F}=0}(\mathbf{y}, \mathbf{\gamma})$ is nonempty, then by considerations similar to those of Lemma 5.4.2:

$$4g + C(\mathbf{y}) + \int_{[0,1]\times\mathbf{y}} \lambda_+ \geq \mathcal{A}_{\lambda_+}(\mathbf{y}),$$

where $\mathcal{A}_{\lambda_+}(\mathbf{y})$ is the action of $\mathbf{y}$ with respect to $\lambda_+$. By taking the maximum of the left-hand side over all $\mathbf{y}$, we obtain an upper bound for $\mathcal{A}_{\lambda_+}(\mathbf{y})$ which is independent of $\mathbf{y}$ and $\delta$. By Lemma 4.1.1(7), all the orbit sets $\mathbf{y}$ in $\text{int}(\text{N}(K)) \cup \mathcal{N}$ satisfy $\mathcal{A}_{\lambda_+}(\mathbf{y}) \geq \frac{1}{2\delta} - \kappa$. Hence, for $\delta > 0$ sufficiently small, no negative end of $u$ is asymptotic to an orbit in $\text{int}(\text{N}(K)) \cup \mathcal{N}$. $\square$

**Lemma 5.8.2.** — If $u \in \mathcal{M}_{J^+}^{\mathcal{F}=0}(\mathbf{y}, \mathbf{\gamma})$, where $\mathbf{y} \in \mathcal{S}_{\alpha, \beta}$, $\mathbf{y} \subset S_0$, and $\mathbf{y}$ is constructed from $\mathcal{P}|_N \cup \{e', h'\}$, then $\text{Im}(u) \subset W_+$ and $\mathbf{y} \in \mathcal{O}|_N$.

*Proof.* — Let $u \in \mathcal{M}_{J^+}^{\mathcal{F}=0}(\mathbf{y}, \mathbf{\gamma})$ such that $u(\dot{\mathbf{F}}) \not\subset W_+$.

Suppose that $u$ is not a multi-level Morse-Bott building. Then $u(\dot{\mathbf{F}}) \cap C_{\theta_0} \neq \emptyset$ for some $\theta_0 \in \partial S_0 - \alpha - \beta$, and moreover we may assume that $\gamma_{\theta_0}$ is not an asymptotic limit of $u$ at $-\infty$. Since $J^+$ is admissible, all the curves $C_{\theta}$ are holomorphic. Hence $\langle u(\dot{\mathbf{F}}), C_{\theta_0} \rangle > 0$ by the positivity of intersections.

Let $D_{\theta}, \theta \in \partial S_0$, be a meridian disk of the solid torus $\mathcal{N} \cup \text{N}(K)$ that is bounded by $\{\theta\} \times \mathbf{R}/2\mathbf{Z}$ and is disjoint from $e'$ and $h'$, and let $D_{\theta, s'} = \{s'\} \times D_{\theta} \subset X_+^2$, where $s' < 0$ and $\theta \in \partial S_0$. We then define

$$C_{\theta, s'_0} := (C_{\theta} - \{s' < s'_0\}) \cup D_{\theta, s'_0},$$

where $s'_0 < 0$. When $s'_0$ is sufficiently negative, the curve $u(\dot{\mathbf{F}})$ intersects $C_{\theta_0, s'_0}$ only in the region $C_{\theta_0} - \{s' < s'_0\}$, since $\mathbf{y}$ is constructed from $\mathcal{P}|_N \cup \{e', h'\}$ and $D_{\theta_0}$ does not intersect $e'$ and $h'$. Hence $\langle u(\dot{\mathbf{F}}), C_{\theta_0, s'_0} \rangle > 0$. Now, since $[S_{(z')\gamma}] = [C_{\theta_0, s'_0}]$ in $H_2(\tilde{X}_+, \partial\tilde{X}_+ - Z_{\mathbf{y},\mathbf{y}})$, we have

$$\mathcal{F}(u) = \langle [u], S_{(z')\gamma} \rangle = \langle [u], C_{\theta_0, s'_0} \rangle > 0.$$

This contradicts our assumption that $\mathcal{F}(u) = 0$.

If $u$ is a multi-level Morse-Bott building, then we need to make the appropriate modifications (left to the reader), but the same argument goes through. For example, we need to replace $C_{\theta_0}$ by a multi-level building $C_{\theta_0} \cup (\mathbf{R} \times \gamma_{\theta_0}) \cup \cdots \cup (\mathbf{R} \times \gamma_{\theta_0})$. Note that if $u$ is a Morse-Bott building, then it could have a component $u_1$ with a negative end that limits to some $\gamma_{\theta_1}$, followed by a gradient trajectory from $\theta_1$ to $\theta_2$, and then by a component $u_2$ with a positive end that limits to $\gamma_{\theta_2}$. $\square$
**Theorem 5.8.3.** — For $\delta > 0$ sufficiently small, if $u \in \mathcal{M}_{J^+}^{\mathcal{F}=0}(\mathbf{y}, \mathbf{\gamma})$, $\mathbf{y} \in S_{\alpha, \beta}$, $\mathbf{y} \subset S_0$, and $\mathbf{\gamma} \in \mathcal{O}$, then $\mathrm{Im}(u) \subset W_+$ and $\mathbf{\gamma} \in \mathcal{O}|_N$.

*Proof.* — Follows from Lemmas 5.8.1 and 5.8.2. $\square$

**Corollary 5.8.4.** — $\Phi^+([\mathbf{x}, 0]) = e^{2g}$, where $e$ is the elliptic orbit of the negative Morse-Bott family on $T_- = \partial N_{(S_0, \bar{\mathcal{H}})}$.

*Proof.* — By Theorem 5.8.3, any curve $u \in \mathcal{M}_{J^+}^{\mathcal{F}=0}(\mathbf{x}, \mathbf{\gamma})$ must have image in $W_+$. Then, by Lemma I.6.2.3 and its consequence in Theorem I.6.2.4, the only curves from $\mathbf{x}$ that do not intersect $S_{(z')/\mathcal{I}}$ are curves of type $C_\theta$. $\square$

The restriction $\Phi$ of $\Phi^+$ to $(W_+, J_+)$ is given as follows:

$$ \Phi: \widehat{\mathrm{CF}}(\Sigma, \alpha, \beta, z') \to \mathrm{ECC}_{2g}(M, \lambda_-), \\ [\mathbf{y}, 0] \mapsto \sum_{\mathbf{\gamma}, A} \# \mathcal{M}_{J^+}^{\mathrm{Iw}_+ = 0}(\mathbf{y}, \mathbf{\gamma}, A) \cdot \mathbf{\gamma}, $$

where $\mathcal{M}_{J^+}^{\mathrm{Iw}_+ = 0}(\mathbf{y}, \mathbf{\gamma}, A)$ is the subset of $\mathcal{M}_{J^+}(\mathbf{y}, \mathbf{\gamma}, A)$ consisting of curves with image in $W_+$.

**Theorem 5.8.5.** — $\Phi$ is a quasi-isomorphism.

*Proof.* — The almost complex structure $J_+$ is sufficiently close to $J_+^0$. For $J_+^0$, the analogous chain map was shown to be a quasi-isomorphism (Theorem II.1.0.1). Considerations similar to those of Theorem I.3.6.1 imply that $\Phi$ is a quasi-isomorphism. $\square$

**5.9.** *Commutativity with the U-map.* — Let $z^b$ be a point in $\mathbf{R} \times [0, 1]$ with $t$-coordinate $\frac{1}{2}$ and let $z = (z^b, z') \in \mathcal{X}$. Let $U_z$ be the geometric U-map with respect to $z$ on the HF side. On the ECH side, let $z' = (s, z^M)$ be a generic point in $\mathbf{R} \times \mathrm{int}(\mathcal{N}(\mathcal{K}))$ near the binding $\mathcal{K}$. We define $U' = U'_z$ so that $\langle U'(\mathbf{y}), \mathbf{\gamma}' \rangle$ is the count of $I_{\mathrm{ECH}} = 2$ curves in the symplectization $(\mathbf{R} \times \mathcal{M}, J')$ from $\mathbf{\gamma}$ to $\mathbf{\gamma}'$ that pass through $z'$.

**Theorem 5.9.1.** — There exists a chain homotopy

$$ K: \mathrm{CF}^+(\Sigma, \alpha, \beta, z') \to \mathrm{ECC}(M, \lambda_-) $$

which satisfies

$$ U' \circ \Phi^+ - \Phi^+ \circ U_z = \partial' \circ K + K \circ \partial. $$

*Proof.* — The commutativity of $\Phi^+$ with the U-maps up to homotopy is obtained by moving the point constraint in the cobordism $X_+$ from $s = +\infty$ to $s = -\infty$.
The 1-parameter family of points $(z_\tau)_{\tau \in \mathbb{R}}$ is chosen as follows: For $\tau \ge 0$, let $z_\tau = (z_\tau^b, z^f)$, where $z_\tau^b$ approaches $(s, t) = (+\infty, \frac{1}{2})$ as $\tau \to +\infty$ and $z_0^b$ is near the center of the disk $D^2 = \{r_2 \le 1\}$. Next, for $\tau \in [-1, 0]$, let $z_\tau = (z_0^b, z_\tau^f)$ so that $(z_0^b, z_{-1}^f) \in \{0\} \times \tilde{B}$ is near the binding K. For $\tau \le -1$, let $z_\tau = (\tau + 1, z^M) \in (-\infty, 0] \times \mathbb{M}$, where $z^M \in \mathbb{M} = \tilde{B}$ is a point near the binding. Finally, we consider a small perturbation of $(z_\tau)_{\tau \in \mathbb{R}}$ to make it generic (without changing its name).

We define the 1-parameter family of almost complex structures $(J_\tau^+)_{\tau \in \mathbb{R}}$ so that $J_\tau^+$ is $C^\ell$-close to $J^+$ and agrees with $J^+$ outside a small neighborhood of $z_\tau$.

The rest of the chain homotopy argument is standard, with the exception of the obstruction theory that was carried out in [HT1, HT2]. □

**Theorem 5.9.2.** — For $\delta > 0$ sufficiently small, if $y \in S_{\alpha, \beta}$ and $y \subset S_0$, then $K([y, 0]) = 0$.

*Proof.* — The coefficient $\langle K([y, 0]), y \rangle$ is given by the count of $I_{X_+} = 1$ curves from $y$ to $y$ that pass through $z_\tau$ for some $\tau$ and do not intersect $S_{(z^f)\tau}$. If such a curve $u$ exists, then $\mathrm{Im}(u) \not\subset W_+$. This is not possible by the proof of Theorem 5.8.3. □

**6. Proof of Theorem 1.0.1**

In this section we prove Theorem 1.0.1. In Section 6.1 we prove an algebraic result (Theorem 6.1.5) which is sufficient to prove that $\Phi^+$ is a quasi-isomorphism. The conditions of Theorem 6.1.5 are verified in Section 6.4.

**6.1. Some algebra.**

*Definition 6.1.1.* — Let $(A, d)$ be a chain complex. We say that a chain map $f: A \to A$ is homologically almost nilpotent (abbreviated *han*) if for every $x \in H(A)$ there exists $n \in \mathbb{N}$ such that $(f_*)^n(x) = 0$.

Prototypical examples of *han* maps are the U-maps in $HF^+$ and ECH.

Let $(A, d_A)$ and $(B, d_B)$ be chain complexes with *han* maps $U_A: A \to A$ and $U_B: B \to B$ and let $\Phi^+: A \to B$ be a chain map such that the diagram

<figure><img src="image_11.png" /></figure>
is the cone of $U_{\Phi^+}$. Moreover $f: D \to D'$ where

$$f = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

is an isomorphism of complexes. □

**Lemma 6.1.4.** — $U_{\Phi^+}$ is a *han map*.

*Proof.* — Consider the following commutative diagram with exact rows:

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
H(B) \arrow[r, "i_*"] \arrow[d, "U_B^n"] & H(C(\Phi^+)) \arrow[r, "j_*"] \arrow[d, "U_{\Phi^+}^n"] & H(A) \\
U_B^n \arrow[d] & U_{\Phi^+}^n \arrow[d] & U_{\Lambda}^n \arrow[d] \\
H(B) \arrow[r, "i_*"] \arrow[d, "U_B^m"] & H(C(\Phi^+)) \arrow[r, "j_*"] \arrow[d, "U_{\Phi^+}^m"] & H(A) \\
U_B^m \arrow[d] & U_{\Phi^+}^m \arrow[d] & U_{\Lambda}^m \arrow[d] \\
H(B) \arrow[r, "i_*"] \arrow[d, "U_B^{m+m}"] & H(C(\Phi^+)) \arrow[r, "j_*"] \arrow[d, "U_{\Phi^+}^{m+m}"] & H(A)
\end{tikzcd}$$

Given $x \in H(C(\Phi^+))$, we choose $n \in \mathbb{N}$ sufficiently large so that $U_{\Lambda}^n(j_*(x)) = j_*(U_{\Phi^+}^n(x)) = 0$. Then $U_{\Phi^+}^n(x) = i_*(y)$ for some $y \in H(B)$. Next choose $m \in \mathbb{N}$ sufficiently large so that $U_B^m(y) = 0$. Then $U_{\Phi^+}^{n+m}(x) = U_{\Phi^+}^m(i_*(y)) = i_*(U_B^m(y)) = 0$. □

**Theorem 6.1.5.** — If $\Phi_{alg}$ is a quasi-isomorphism, then $\Phi^+$ is a quasi-isomorphism.

*Proof.* — If $\Phi_{alg}$ is a quasi-isomorphism, then $H(D) = 0$ by Exact Triangle (6.1.1). This in turn implies that $U_{\Phi^+}$ is a quasi-isomorphism by Exact Triangle (6.1.2). However the *han map* $U_{\Phi^+}$ cannot be a quasi-isomorphism, unless $H(C(\Phi^+)) = 0$. Finally, the triangle

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
H(A) \arrow[r, "\Phi_*"] & H(B) \\
& \\
H(C(\Phi^+)) \arrow[u, "\Phi_*"] \arrow[r] & \\
&
\end{tikzcd}$$

implies that $\Phi^+$ is a quasi-isomorphism. □

We finish this subsection with a lemma which compares the homology of $C(U)$ with that of $ker U$.
**Lemma 6.1.6.** — Let (C, d) be a chain complex and let $U : C \to C$ be a chain map. If U is surjective, then the inclusion

$$i : \ker U \to C(U)$$

$$x \mapsto \begin{pmatrix} x \\ 0 \end{pmatrix}$$

is a quasi-isomorphism.

*Proof.* — Let $\bar{U} : C/\ker U \to C$ be the map induced by $U$. We have a short exact sequence of complexes

$$0 \to \ker U \to C(U) \to C(\bar{U}) \to 0,$$

which induces the exact triangle:

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

Since $U$ is surjective, $\bar{U}$ is an isomorphism. Hence $H(C(\bar{U})) = 0$ and the lemma follows. $\square$

<figure><img src="image_13.png" /></figure>

**6.2.** Heegaard Floer chain complexes. — Recall the subcomplex $\widehat{CF}'(S_0, \mathbf{a}, \hbar(\mathbf{a}))$ of $\widehat{CF}(\Sigma, \alpha, \beta, z^f)$ from Section I.4.9.3, which is generated by $S_{\alpha, \hbar(\mathbf{a})}$; let

$$j' : \widehat{CF}'(S_0, \mathbf{a}, \hbar(\mathbf{a})) \to \widehat{CF}(\Sigma, \alpha, \beta, z^f)$$

be the natural inclusion map. We are viewing

$$\widehat{CF}(\Sigma, \alpha, \beta, z^f) \subset \widehat{CF}^+(\Sigma, \alpha, \beta, z^f)$$

as the subcomplex generated by elements of the form $[y, 0]$. The chain complex $\widehat{CF}(S_0, \mathbf{a}, \hbar(\mathbf{a}))$ is the quotient $\widehat{CF}'(S_0, \mathbf{a}, \hbar(\mathbf{a}))/\sim$, defined in Section I.4.9.3.

**Lemma 6.2.1.** — There is an isomorphism $j : \widehat{HF}(S_0, \mathbf{a}, \hbar(\mathbf{a})) \to \widehat{HF}(\Sigma, \alpha, \beta, z^f)$ given by $[Z] \mapsto [Z]$.

*Proof.* — This follows from the discussion of Theorem I.4.9.4. Note that the natural candidate

$$\widehat{CF}(S_0, \mathbf{a}, \hbar(\mathbf{a})) \to \widehat{CF}(\Sigma, \alpha, \beta, z^f), \quad [Z] \to Z$$
for a chain map is not a well-defined map.

□

**Lemma 6.2.2.** — The inclusion $i: \widehat{CF}(\Sigma, \alpha, \beta, z') \to C(U)$ given by $\mathbf{y} \mapsto \begin{pmatrix} [\mathbf{y}, 0] \\ 0 \end{pmatrix}$ is a quasi-isomorphism.

*Proof.* — This follows from Lemma 6.1.6, since $U([\mathbf{y}, i]) = [\mathbf{y}, i-1]$ for $i \ge 1$ and $\ker U \simeq \widehat{CF}(\Sigma, \alpha, \beta, z')$. □

**6.3. ECH chain complexes.** — We describe several ECH chain complexes that are related to (ECC(M, $\lambda_-$), $\partial'$) and are constructed from certain subsets $S$ of the set $\mathcal{P} = \mathcal{P}_{\lambda_-}$ of simple orbits of $\mathbb{R}_{\lambda_-}$. Many of these appeared in [0, Section 9]. Let $U'$ be the U-map of ECC(M, $\lambda_-$) with respect to $(s_0, z^M) \in \mathbb{R} \times \mathbb{M}$, where $z^M$ is a generic point which is sufficiently close to the binding.

Let $\mathcal{O}_S$ be the set of orbit sets that are constructed from $S$. Then $S$ is closed if $\mathbf{y}' \in \mathcal{O}_S$, whenever $\mathbf{y} \in \mathcal{O}_S$, $\mathbf{y}' \in \mathcal{O}_P$, and $\langle \partial'\mathbf{y}, \mathbf{y}' \rangle \neq 0$ or $\langle U'\mathbf{y}, \mathbf{y}' \rangle \neq 0$. If $S$ is closed, then let $(A_S, \partial'_S)$ be the subcomplex of ECC(M, $\lambda_-$) generated by $\mathcal{O}_S$ and let $U'_S$ be the restriction of $U'$ to $A_S$. Let $\mathcal{P}|_N \subset \mathcal{P}$ be the set of orbits in the mapping torus $N$. The subsets

$$S_1 = \mathcal{P}|_N \cup \{e', h'\}, \quad S_2 = \mathcal{P}|_{N \cup \mathcal{N}} \cup \{e', h'\}, \quad \mathcal{P}|_N \cup \{h'\}, \quad \mathcal{P}|_{N \cup \mathcal{N}} \cup \{h'\}, \quad \mathcal{P}|_N$$

are closed and we write $A_i = A_{S_i}$, $\partial'_i = \partial'_{S_i}$, and $U'_i = U'_{S_i}$ for $i = 1, 2$, as well as

$$\widehat{\mathrm{ECC}}^{\sharp}(N) = A_{\mathcal{P}|_{N \cup \{h'\}}}, \quad \widehat{\mathrm{ECC}}^{\sharp\sharp}(N) = A_{\mathcal{P}|_{N \cup \mathcal{N} \cup \{h'\}}}, \quad \mathrm{ECC}(N) = A_{\mathcal{P}|_N}.$$

Also let $\mathrm{ECC}_{2g}(N) \subset \mathrm{ECC}(N)$ be the subcomplex generated by orbit sets $\mathbf{y}$ satisfying $\langle \mathbf{y}, S \times \{t\} \rangle = 2g$. Let

$$q_1 : \mathrm{ECC}_{2g}(N) \to \widehat{\mathrm{ECC}}^{\sharp}(N), \quad q_2 : \mathrm{ECC}_{2g}(N) \to \widehat{\mathrm{ECC}}^{\sharp\sharp}(N)$$

be the chain maps given by the natural inclusion. Then we have the following:

**Lemma 6.3.1.** — The chain maps $q_1$ and $q_2$ are quasi-isomorphisms.

*Proof.* — The chain map $q_1$ is a quasi-isomorphism by Section II.5 and Section 0.9.9. By a direct limit argument similar to that of Proposition 0.7.2.1, there is a quasi-isomorphism $r: \widehat{\mathrm{ECC}}^{\sharp\sharp}(N) \to \widehat{\mathrm{ECC}}^{\sharp}(N)$ such that $r \circ q_2 = q_1$. This implies that $q_2$ is also a quasi-isomorphism. □

**Lemma 6.3.2.** — The inclusions $p_1: \widehat{\mathrm{ECC}}^{\sharp}(N) \to C(U'_1)$ and $p_2: \widehat{\mathrm{ECC}}^{\sharp\sharp}(N) \to C(U'_2)$ given by $\Gamma \mapsto \begin{pmatrix} \Gamma \\ 0 \end{pmatrix}$ are quasi-isomorphisms.
*Proof.* — This follows from Lemma 6.1.6. The map $U'_i$, $i=1,2$, is given by:

$$ (6.3.1) \qquad U'_i((e')^k(h')^l\Gamma) = (e')^{k-1}(h')^l\Gamma, $$

where $\Gamma \in \mathcal{O}|_{\mathbb{N}}$ or $\mathcal{O}|_{\mathbb{N}\cup\mathcal{N}}$; see Claim 0.9.9.3 for a similar calculation. Hence $U'_i$ is surjective, $\ker U'_1 = \widehat{\mathrm{ECC}}^{\mathbb{Z}}(\mathbb{N})$, and $\ker U'_2 = \widehat{\mathrm{ECC}}^{\mathbb{Z}^2}(\mathbb{N})$. This implies the lemma. $\square$

**Lemma 6.3.3.** — The inclusion $\mathfrak{i} : \widehat{\mathrm{ECC}}^{\mathbb{Z}}(\mathbb{N}) \to \mathrm{C}(\mathfrak{U}')$ given by $\Gamma \mapsto \begin{pmatrix} \Gamma \\ 0 \end{pmatrix}$ is a quasi-isomorphism.

*Proof.* — This is similar to the argument in [0, Section 9].

Choose an identification $\eta : \mathrm{H}_1(\mathrm{N}(\mathbb{K}); \mathbb{Z}) \xrightarrow{\sim} \mathbb{Z}$ such that the homology class of the binding is 1. Define the filtration $\mathcal{F} : \mathrm{ECC}(\mathbb{M}) \to \mathbb{Z}^{\ge 0}$ such that

$$ \mathcal{F}\left(\sum_i \mathbf{\gamma}_i \otimes \Gamma_i\right) = \max_i \eta([\mathbf{\gamma}_i]), $$

where $\mathbf{\gamma}_i \in \mathcal{O}|_{\mathbb{N}(\mathbb{K})}$ and $\Gamma_i \in \mathcal{O}|_{\mathbb{N}\cup\mathcal{N}}$. Let $\mathcal{F}^{\mathbb{Z}} : \widehat{\mathrm{ECC}}^{\mathbb{Z}}(\mathbb{N}) \to \mathbb{Z}^{\ge 0}$ be its restriction to $\widehat{\mathrm{ECC}}^{\mathbb{Z}}(\mathbb{N})$. (Note that $\mathcal{F}^{\mathbb{Z}}$ is a trivial filtration.) Next define the filtration $\widehat{\mathcal{F}} : \mathrm{C}(\mathfrak{U}') \to \mathbb{Z}^{\ge 0}$ such that

$$ \widehat{\mathcal{F}}\left(\sum_i \mathbf{\gamma}_i \otimes \Gamma_i \atop \sum_j \mathbf{\gamma}_j' \otimes \Gamma_j'\right) = \max_{i,j}\{\eta([\mathbf{\gamma}_i]), \eta([\mathbf{\gamma}_j'])\}. $$

The map $\mathfrak{i}$ is an $(\mathcal{F}^{\mathbb{Z}}, \widehat{\mathcal{F}})$-filtered chain map. The induced map

$$ \mathrm{E}^1(\mathfrak{i}) : \mathrm{E}^1(\mathcal{F}^{\mathbb{Z}}) \to \mathrm{E}^1(\widehat{\mathcal{F}}) $$

on the $\mathrm{E}^1$-level agrees with the isomorphism $(p_2)_*$; the proof is similar to that of Section 0.9. If a filtered chain map between filtered chain complexes which are bounded below is an isomorphism on the $\mathrm{E}^r$-level, then it is a quasi-isomorphism. This implies that $\mathfrak{i}$ is a quasi-isomorphism. $\square$

**6.4.** Completion of proof of Theorem 1.0.1. — By Theorems 3.1.4, 5.7.1, and 5.9.1, the map

$$ \Phi^+ : \mathrm{CF}^+(\Sigma, \alpha, \beta, z') \to \mathrm{ECC}(\mathbb{M}, \lambda_-) $$

is a chain map which commutes with $\mathfrak{U}$ and $\mathfrak{U}'$ up to the chain homotopy $\mathbb{K}^+ = \mathbb{K} + \Phi^+ \circ \mathfrak{H}$, where $\mathfrak{H}$ is given in Theorem 3.1.4 and $\mathbb{K}$ is given in Theorem 5.9.1. Here $\mathfrak{U}$ is the original algebraically-defined $\mathfrak{U}$-map on $(\mathrm{CF}^+(\Sigma, \alpha, \beta, z'), \partial)$ and $\mathfrak{U}'$ is the $\mathfrak{U}$-map on $(\mathrm{ECC}(\mathbb{M}, \lambda_-), \partial')$.
In view of Theorem 6.1.5, the quasi-isomorphism statement of Theorem 1.0.1 immediately follows from:

**Theorem 6.4.1.** — The algebraic map $\Phi_{alg}$ is a quasi-isomorphism.

Let $\Phi' : \widehat{CF}'(S_0, \mathbf{a}, \mathfrak{h}(\mathbf{a})) \to \mathrm{ECC}_{2g}(\mathbb{N})$ be the map from Definition I.6.2.1. The map $\Phi'$ descends to $\Phi : \widehat{CF}(S_0, \mathbf{a}, \mathfrak{h}(\mathbf{a})) \to \mathrm{ECC}_{2g}(\mathbb{N})$, which was shown to be a quasi-isomorphism in [I, II]. Here we are using $\mathrm{ECC}_{2g}(\mathbb{N})$ instead of $\mathrm{PFC}_{2g}(\mathbb{N})$, but there is no substantial difference; see Theorem I.3.6.1.

Observe that there is a discrepancy between the algebra and the geometry: the map $\Phi_{alg}$ which we are using here is not the map $\Phi$, and we need to reconcile the two.

*Proof.* — If $Z \in \widehat{CF}'(S_0, \mathbf{a}, \mathfrak{h}(\mathbf{a}))$, then $\Phi^+(Z) = \Phi'(Z)$ by Theorem 5.8.3. We observed in Theorem 3.1.4 that $H(Z) = 0$. Moreover, $K(Z) = 0$ by Theorem 5.9.2 and thus $K^+(Z) = 0$. Hence

$$ \Phi_{alg} \begin{pmatrix} Z \\ 0 \end{pmatrix} = \begin{pmatrix} \Phi^+(Z) \\ K^+(Z) \end{pmatrix} = \begin{pmatrix} \Phi'(Z) \\ 0 \end{pmatrix}, $$

and the following diagram is commutative:

<figure><img src="image_14.png" /></figure>

This gives rise to the following commutative diagram of homology groups:

<figure><img src="image_15.png" /></figure>

Since $j, i_*, \Phi_*, (q_2)_*$, and $i_*$ are isomorphisms by Lemma 6.2.1, Lemma 6.2.2, [I, II], Lemma 6.3.1, and Lemma 6.3.3, $\Phi_{alg}$ itself is a quasi-isomorphism. $\square$

Finally, the statement about $\Phi^+$ mapping the contact class to the contact class follows from Corollary 5.8.4.
**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 Tobias Ekholm, Dusa McDuff, 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.

**Declarations:**

**Competing Interests**

The authors declare no competing interests.

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**REFERENCES**

[0] V. COLIN, P. GHIGGINI and K. HONDA, Embedded contact homology and open book decompositions, (2024), *Geom. Topol.*, to appear.

[I] V. COLIN, P. GHIGGINI and K. HONDA, The equivalence of Heegaard Floer homology and embedded contact homology I, *Publ. Math. Inst. Hautes Études Sci.*, (2024). https://doi.org/10.1007/s10240-024-00145-x.

[II] V. COLIN, P. GHIGGINI and K. HONDA, The equivalence of Heegaard Floer homology and embedded contact homology II, *Publ. Math. Inst. Hautes Études Sci.*, (2024). https://doi.org/10.1007/s10240-024-00146-w.

[HLS] H. HOFER, V. LIZAN and J.-C. SIKORAV, On genericity for holomorphic curves in four-dimensional almost complex manifolds, *J. Geom. Anal.*, **7** (1997), 149–159.

[H] K. HONDA, On the classification of tight contact structures I, *Geom. Topol.*, **4** (2000), 309–368.
[Hu] M. HUTCHINGS, The embedded contact homology index revisited, new perspectives and challenges in symplectic field theory, in *CRM Proc. Lecture Notes*, vol. 49, pp. 263–297, Am. Math. Soc., Providence, 2009.

[HT1] M. HUTCHINGS and C. TAUBES, Gluing pseudoholomorphic curves along branched covered cylinders I, *J. Symplectic Geom.*, **5** (2007), 43–137.

[HT2] M. HUTCHINGS and C. TAUBES, Gluing pseudoholomorphic curves along branched covered cylinders II, *J. Symplectic Geom.*, **7** (2009), 29–133.

[IP] E. IONEL and T. PARKER, Relative Gromov-Witten invariants, *Ann. Math. (2)*, **157** (2003), 45–96.

[Li] R. LIPSHTZ, A cylindrical reformulation of Heegaard Floer homology, *Geom. Topol.*, **10** (2006), 955–1097.

[M1] D. McDUFF, The local behaviour of holomorphic curves in almost complex 4-manifolds, *J. Differ. Geom.*, **34** (1991), 143–164.

[M2] D. McDUFF, Singularities and positivity of intersections of J-holomorphic curves, in *Holomorphic Curves in Symplectic Geometry*, Progr. Math., vol. 117, pp. 191–215, Birkhäuser, Basel, 1994, With an appendix by Gang Liu.

[MS] D. McDUFF and D. SALAMON, *J-Holomorphic Curves and Symplectic Topology*, American Mathematical Society Colloquium Publications, vol. 52, Am. Math. Soc., Providence, 2004.

[OSZ] P. OZSVÁTH and Z. SZABÓ, Holomorphic disks and topological invariants for closed three-manifolds, *Ann. Math. (2)*, **159** (2004), 1027–1158.

[T] C. TAUBES, *Counting Pseudo-Holomorphic Submanifolds in Dimension 4*, Seiberg Witten and Gromov Invariants for Symplectic 4-Manifolds, First Int. Press Lect. Ser., 2, pp. 99–161, Int. Press, Somerville, 2000.

[We] C. WENDL, Non-exact symplectic cobordisms between contact 3-manifolds, *J. Differ. Geom.*, **95** (2013), 121–182.

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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.