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

Vincent Colin, Paolo Ghiggini, Ko Honda

► To cite this version:

Vincent Colin, Paolo Ghiggini, Ko Honda. The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I. Publications Mathématiques de L'IHÉS, 2024, 139 (1), pp.13-187. 10.1007/s10240-024-00145-x. hal-04752834

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Submitted on 6 Jun 2025

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

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

## ABSTRACT

Given an open book decomposition $(S, h)$ adapted to a closed, oriented 3-manifold $M$, we define a chain map $\Phi$ from a certain Heegaard Floer chain complex associated to $(S, h)$ to a certain embedded contact homology chain complex associated to $(S, h)$, as defined in (Colin et al. in Geom. Topol., 2024), and prove that it induces an isomorphism on the level of homology. This implies the isomorphism between the hat version of Heegaard Floer homology of $-M$ and the hat version of embedded contact homology of $M$.

## CONTENTS

<table><tbody><tr><td>Notation common to [I]-[III]</td><td>14</td></tr><tr><td>1. Introduction and main results</td><td>18</td></tr><tr><td>1.1. Some background</td><td>20</td></tr><tr><td>1.2. Main result</td><td>21</td></tr><tr><td>1.3. Outline of proof</td><td>22</td></tr><tr><td>1.4. Organization of the paper</td><td>23</td></tr><tr><td>2. Adapting ECH to an open book decomposition</td><td>24</td></tr><tr><td>2.1. The first return map</td><td>24</td></tr><tr><td>2.2. ECH(N, ∂N, α) and ECH̄(N, ∂N, α)</td><td>25</td></tr><tr><td>2.3. Splitting of ECH according to homology classes</td><td>26</td></tr><tr><td>2.4. Twisted coefficients in ECH</td><td>27</td></tr><tr><td>2.5. Elimination of elliptic orbits</td><td>28</td></tr><tr><td>3. Periodic Floer homology</td><td>33</td></tr><tr><td>3.1. Interpolating between Reeb and stable Hamiltonian vector fields</td><td>33</td></tr><tr><td>3.2. Definitions</td><td>33</td></tr><tr><td>3.3. The flux</td><td>35</td></tr><tr><td>3.4. Compactness</td><td>35</td></tr><tr><td>3.5. Transversality</td><td>37</td></tr><tr><td>3.6. The equivalence of certain ECH and PFH groups</td><td>37</td></tr><tr><td>3.7. Remarks about Morse-Bott theory</td><td>38</td></tr><tr><td>4. A variation of H̄F(M) adapted to open book decompositions</td><td>39</td></tr><tr><td>4.1. Heegaard data</td><td>39</td></tr><tr><td>4.2. Almost complex structures</td><td>40</td></tr><tr><td>4.3. Holomorphic curves and moduli spaces</td><td>41</td></tr><tr><td>4.4. The Fredholm index</td><td>42</td></tr><tr><td>4.5. The ECH-type index</td><td>44</td></tr><tr><td>4.6. Compactness</td><td>50</td></tr><tr><td>4.7. Transversality</td><td>51</td></tr><tr><td>4.8. Definition of the Heegaard Floer homology groups</td><td>52</td></tr><tr><td>4.9. Restricting the complex to a page</td><td>53</td></tr><tr><td>4.10. Spin'-structures</td><td>57</td></tr><tr><td>4.11. Twisted coefficients in Heegaard Floer homology</td><td>58</td></tr><tr><td>5. Moduli spaces of multisections</td><td>59</td></tr><tr><td>5.1. Symplectic cobordisms</td><td>60</td></tr><tr><td>5.2. Lagrangian boundary conditions</td><td>64</td></tr></tbody></table>

<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>
<table>
  <tbody>
    <tr>
      <td>5.3. Almost complex structures and moduli spaces for W, W̄, W', and W̄'</td>
      <td>65</td>
    </tr>
    <tr>
      <td>5.4. Almost complex structures and moduli spaces for W+, W+, W-, and W̄-</td>
      <td>71</td>
    </tr>
    <tr>
      <td>5.5. The Fredholm index</td>
      <td>76</td>
    </tr>
    <tr>
      <td>5.6. The ECH index</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.7. Holomorphic curves with ends at z∞</td>
      <td>85</td>
    </tr>
    <tr>
      <td>5.8. Transversality</td>
      <td>97</td>
    </tr>
    <tr>
      <td>6. The chain map from H̄F to PFH</td>
      <td>102</td>
    </tr>
    <tr>
      <td>6.1. Compactness for W+, curves</td>
      <td>102</td>
    </tr>
    <tr>
      <td>6.2. Definition of Φ</td>
      <td>108</td>
    </tr>
    <tr>
      <td>6.3. Spin'-structures</td>
      <td>110</td>
    </tr>
    <tr>
      <td>6.4. Twisted coefficients</td>
      <td>111</td>
    </tr>
    <tr>
      <td>6.5. Gluing</td>
      <td>112</td>
    </tr>
    <tr>
      <td>6.6. The variant Φ̄</td>
      <td>118</td>
    </tr>
    <tr>
      <td>7. The chain map from PFH to H̄F</td>
      <td>121</td>
    </tr>
    <tr>
      <td>7.1. Definition of Ψ</td>
      <td>121</td>
    </tr>
    <tr>
      <td>7.2. Outline of proof of Theorem 7.1.3</td>
      <td>124</td>
    </tr>
    <tr>
      <td>7.3. SFT compactness</td>
      <td>126</td>
    </tr>
    <tr>
      <td>7.4. Intersection numbers</td>
      <td>139</td>
    </tr>
    <tr>
      <td>7.5. Some restrictions on ū∞</td>
      <td>142</td>
    </tr>
    <tr>
      <td>7.6. Compactness theorem</td>
      <td>147</td>
    </tr>
    <tr>
      <td>7.7. Asymptotic eigenfunctions</td>
      <td>150</td>
    </tr>
    <tr>
      <td>7.8. The rescaled function</td>
      <td>156</td>
    </tr>
    <tr>
      <td>7.9. Involution lemmas</td>
      <td>168</td>
    </tr>
    <tr>
      <td>7.10. Elimination of some cases</td>
      <td>171</td>
    </tr>
    <tr>
      <td>7.11. Proof of Lemma 7.2.3</td>
      <td>173</td>
    </tr>
    <tr>
      <td>7.12. Gluing</td>
      <td>174</td>
    </tr>
    <tr>
      <td>Acknowledgements</td>
      <td>183</td>
    </tr>
    <tr>
      <td>References</td>
      <td>184</td>
    </tr>
  </tbody>
</table>

## Notation common to [I]-[III]

(α<sub>5</sub>, ω) Stable Hamiltonian structures on N<sub>(S, Φ)</sub>

γ An orbit set; from Section 5.3 onwards, γ ⊂ N and does not contain multiples of δ<sub>0</sub>

δ<sub>0</sub> Closed orbit in N̄ over z<sub>∞</sub>

(ε, U) or (ε, δ, p) Perturbation data for J̄<sup>Δ</sup>; cf. Convention 5.8.12

μ̄<sub>τ</sub>(γ) Symmetric Conley-Zehnder index

π<sub>B</sub>: X → B or W → B Symplectic fibration with fiber Σ or S used in the definition the Heegaard Floer complex

π<sub>B'</sub>: W' → B' Symplectic fibration with fiber S used in the definition of ECC̄

π<sub>B+</sub>: W+ → B+ Symplectic fibration with fiber S used in the definition of Φ

π̄<sub>*</sub> with * = B, B', B+, B̄ Symplectic fibrations with fiber S̄ extending π<sub>*</sub>

π̄<sub>D<sup>2</sup></sub>: W̄<sub>*</sub> ∩ (R × V) → D<sup>2</sup> Projection with respect to balanced coordinates, where * = + or -

π̄<sub>D<sup>2</sup><sub>ρ<sub>0</sub></sub></sub>: R × (R/2Z) × D<sup>2</sup><sub>ρ<sub>0</sub></sub> → D<sup>2</sup><sub>ρ<sub>0</sub></sub>, also π̄ Projection with respect to balanced coordinates

(ρ, φ) Polar coordinates on D<sup>2</sup>

(Σ, α, β, z) Pointed Heegaard diagram for M

σ<sub>∞</sub>, σ'<sub>∞</sub>, σ<sup>+</sup><sub>∞</sub>, σ<sup>-</sup><sub>∞</sub> Sections at infinity of W̄, W', W+, W-

σ<sup>†</sup><sub>∞</sub>, (σ'<sub>∞</sub>)<sup>†</sup>, (σ<sup>+</sup><sub>∞</sub>)<sup>†</sup>, (σ<sup>-</sup><sub>∞</sub>)<sup>†</sup> “Pushoff” of m-fold cover of section at ∞ for W̄, W', W+, W-
Φ Chain map from CF̄(S, a, ℏ(a)) to PFC<sub>2g</sub>(N)
$\Phi'$ Map from $\widehat{\text{CF}}'(S, \mathbf{a}, \hbar(\mathbf{a}))$ to $\text{PFC}_{2g}(\mathbb{N})$

$\tilde{\Phi}$ Variant of $\Phi$ from $\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$ to $\text{PFC}_{2g}(\mathbb{N})$

$\Psi' = \Psi'_{j^\diamond}(m, \bar{m})$ Map from $\text{PFC}_{2g}(\mathbb{N})$ to $\widehat{\text{CF}}'(S, \mathbf{a}, \hbar(\mathbf{a}))$

$\Psi = \Psi_{j^\diamond}(m, \bar{m})$ Chain map from $\text{PFC}_{2g}(\mathbb{N})$ to $\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$

$\omega$ Area form on $S$ invariant under $\hbar$; also viewed as a 2-form on $W_+, W_-$

$\bar{\omega}$ Area form on $\bar{S}$ which restricts to $\omega$ on $S$ and is invariant under $\bar{\hbar}$; also viewed as a 2-form on $\bar{W}_+, \bar{W}_-$

$\Omega_X$ Symplectic form on $X$

$\Omega$ and $\Omega'$ Symplectic forms on $W$ and $W'$ given by $ds \wedge dt + \omega$

$\Omega_+$ and $\Omega_-$ Symplectic forms on $W_+$ and $W_-$

$\bar{\Omega}, \bar{\Omega}', \bar{\Omega}_+, \bar{\Omega}_-$ Extensions of $\Omega, \Omega', \Omega_+, \Omega_-$

* = † Modifier “$\bar{u}' = \emptyset$”

* = ext Modifier “extended moduli space”; cf. Definition 5.7.24

* = $f_{\delta_0}$ Modifier “the normalized asymptotic eigenfunction at the negative end $\delta_0$ is $f_{\delta_0}$”

* = K Modifier “passes through the compact set K”

* = $(l_1, \dots, l_\lambda)$ Modifier “curve with $\lambda$ ends at $\delta_0$ with covering multiplicities $l_1, \dots, l_\lambda$”

* = nc Modifier “without connector components”

* = reg Modifier “regular”

* = s Modifier “somewhere injective”

**a** = **a**(m) Basis {$a_1, \dots, a_{2g}$} of arcs for $S$; depends on $m$ starting from Section 5

$a_i = a_i(m)$ Basis arc in $S$; depends on $m$ starting from Section 5

**$\bar{a}$** Collection {$\bar{a}_1, \dots, \bar{a}_{2g}$}

$\bar{a}_i$ Extension of $a_i$ to $\bar{S}$

$\bar{a}_{i,j}$ (for $j = 0, 1$) Rays from $x_i$ and $x'_i$ to $z_\infty$ in $\bar{a}_i - a_i$

$\bar{a}_{i,j}$ (for $j = 0, 1$) Extension of $\bar{a}_{i,j}$ past $z_\infty$

**$\hat{a}$** Collection {$\hat{a}_1, \dots, \hat{a}_{2g}$}

$\hat{a}_i \bar{a}_i - \{z_\infty\}$

$\partial_\hbar W_+, \partial_v W_+$ Horizontal and vertical boundaries of $W_+$

B Alternate notation for $\mathbf{R} \times [0, 1]$

B' Alternate notation for $\mathbf{R} \times S^1$

$B_+, B_-$ Bases of the symplectic fibrations $\pi_{B_+}: W_+ \to B_+$ and $\bar{\pi}_{B_-}: \bar{W}_- \to B_-$

**C** Data $\mathcal{C}$ as in Definition 5.7.1

$\mathbf{c}^+, \mathbf{c}^-$ Groomings at the positive and negative ends as in Definition 5.7.5

$\widehat{\text{CF}}(\Sigma, \alpha, \beta, z)$ Heegaard Floer hat complex for pointed Heegaard diagram $(\Sigma, \alpha, \beta, z)$

$\widehat{\text{CF}}'(S, \mathbf{a}, \hbar(\mathbf{a}))$ Subcomplex of Heegaard Floer chain complex $\widehat{\text{CF}}(\Sigma, \beta, \alpha, z)$ obtained from the open book decomposition $(S, \hbar)$

$\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$ Heegaard Floer chain complex isomorphic to $\widehat{\text{HF}}(-M)$, obtained by quotienting $\widehat{\text{CF}}'(S, \mathbf{a}, \hbar(\mathbf{a}))$

$\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$ Variant of $\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$ given by Definition 6.6.1

$cl(B_+)$ and $cl(B_-)$ Compactifications of $B_+$ and $B_-$
$\vec{D}, D^{lo}, D^{from}, D$ Data at $z_\infty^p$ with initial points $D^{from}$ and terminal points $D^{lo}$. $D = (D^{lo}, D^{from})$

$D^2, D_\delta^2$ Unit disk $\bar{S} - int(S)$ with polar coordinates $(\rho, \phi)$, from Section 5 onwards; $D_\delta^2 = \{\rho \le \delta\}$ for $\delta > 0$ small

e Elliptic orbit of Conley-Zehnder index $-1$ on $\partial N_{(S, \hat{h})}$

$E(u)$ Hofer energy of the holomorphic curve $u$

$E_\omega(u)$ $E_\omega$-area of the holomorphic curve $u$

ECC(N, $\alpha$) ECH chain complex for (N, $\alpha$)

ECC$_j$(N, $\alpha$) Subcomplex of ECC(N, $\alpha$) generated by orbit sets which intersect a page $j$ times

g Genus of page S

h Hyperbolic orbit of Conley-Zehnder index 0 on $\partial N_{(S, \hat{h})}$; assumed to satisfy Convention 6.6.4 from Section 6.6 onwards

$H_2(M, \mathbf{y}^+, \mathbf{y}^-)$ Homology classes of chains in $N$ with boundary in $\mathbf{y}^-$ and $\mathbf{y}^+$

$H_2(\mathbf{y}, \mathbf{y}')$ Homology classes of maps in HF (denoted $\pi_2(\mathbf{y}, \mathbf{y}')$ elsewhere in the literature)

$H_2(W_+, \mathbf{y}, \mathbf{y})$ Homology classes of continuous multisections for $\Phi$

$H_2(\overline{W}_-, \mathbf{y}, \mathbf{y})$ Homology classes of continuous multisections for $\Psi$

$\hat{h}$ Monodromy $h: S \to S$

$\overline{\hat{h}} = \hat{h}_m$ Extension of $h$ to $\bar{S}$ which depends on $m$

$\mathcal{J}_j$ ECH chain maps given by Equation (2.2.1)

$I_{\text{ECH}}$ ECH index

$I_{\text{HF}}$ ECH-type index for Heegaard Floer homology

$I_{W_+}$, $I_{\overline{W}_-}$ ECH indices for $W_+$, $\overline{W}_-$

ind Fredholm index

$J_\zeta$ Almost complex structure on $\mathbf{R} \times N$ adapted to $(\alpha_\zeta, \omega)$

$J' = J_0$ Another name for $J_\zeta$ when $\zeta = 0$

$\overline{J}_0$ Almost complex structure on $\overline{W}$ which restricts to the standard one on $D^2$ on each fiber

$J$ Almost complex structure on $W$ that is the restriction of $\overline{J}$

$J_+, J_-$ Almost complex structures on $W_+$, $W_-$ that are restrictions of $\overline{J}_+, \overline{J}_-$

$\overline{\overline{J}}, \overline{\overline{J}}, \overline{J}_+, \overline{J}_-$ Almost complex structures on $\overline{W}, \overline{W}', \overline{W}_+, \overline{W}_-$ that restrict to $J, J', J_+, J_-$ on $W, W', W_+, W_-$

$\overline{\overline{J}}^\diamond = \overline{\overline{J}}^\diamond(\varepsilon, U)$ Almost complex structure on $\overline{W}_-$ that is $(\varepsilon, U)$-close to $\overline{J}_-$ as in Definition 5.8.11 and Convention 5.8.12

$\mathcal{J}_X$ Space of $C^\infty$-smooth admissible almost complex structures on $X$

$\mathcal{J}_*^*$ Subspace of $\mathcal{J}_*$ satisfying *

$\mathcal{J}_W$ Space of $C^\infty$-smooth $\Omega$-admissible almost complex structures on $W$

$\mathcal{J}_{\overline{W}}$ Space of $C^\infty$-smooth $\overline{\Omega}$-admissible almost complex structures $\overline{J}$ on $\overline{W}$ satisfying Definition 5.3.2

$\mathcal{J}_{W'}$ Space of $C^\infty$-smooth $(\alpha_0, \omega)$-adapted almost complex structures on $W'$
$\mathcal{J}_{\overline{W}}$ Space of $C^{\infty}$-smooth $(\overline{\alpha}_0, \overline{\omega})$-adapted almost complex structures on $\overline{W}$ satisfying
Definition 5.3.14

$\mathcal{J}_{W_+}$, $\mathcal{J}_{W_-}$, $\mathcal{J}_{\overline{W}_+}$, $\mathcal{J}_{\overline{W}_-}$ Spaces of $C^\infty$-smooth admissible almost complex structures on $W_+, W_-$, $\overline{W}_+, \overline{W}_-$

$K_{\rho, 2\delta} \pi_{\overline{B}_-}^{-1}(\rho) - \{\rho < 2\delta\}$ with $\delta > 0$ small

$\mathcal{R}_j$ ECH chain maps given by Equation (2.5.5)

$L_\alpha$, $L_\beta$ Lagrangian submanifolds used in the definition of the Heegaard Floer complex
$\overline{\mathcal{C}\mathcal{F}}(\Sigma, \alpha, \beta, z)$

$L_a$, $L_{fi(a)}$ Lagrangian submanifolds of W used in the definition of $\overline{\mathcal{C}\mathcal{F}}(S, a, h(a))$

$L_a^+$, $L_{a_i}^+$ Lagrangian submanifold of $W_+$ constructed from **a** and a connected component of $L_a^+$ corresponding to the arc $a_i$

$L_{\bar{a}}^-$, $L_{\bar{a}_i}^-$ Singular Lagrangian in $\overline{W}_-$ constructed from $\overline{a}$ and a connected component of $L_{\bar{a}}$ corresponding to the arc $\bar{a}_i$

$L_{\bar{a}_i}^{\pm}$, $L_{\bar{a}_i}^{\pm}$, $L_{\bar{a}_{i,j}}^{\pm}$, $L_{\bar{a}_i \cup \bar{a}_{i,j}}^{\pm}$ Lagrangian submanifolds of $\overline{W}_{\pm}$ constructed from the arcs $\tilde{a}_i$, $\bar{a}_i$, $\tilde{a}_{i,j}$ and $\bar{a}_i \cup \tilde{a}_{i,j}$

m Positive integer $\gg 2g$; $\bar{h}$ and $\bar{a}$ depend on m starting from Section 5

$\overline{m} = (\overline{m}^b, \overline{m}^f) = ((0, \frac{3}{2}), z_\infty)$ Marked point in $\overline{W}_-$

$\mathcal{M}_j^X(\mathbf{y}, \mathbf{y}')$ Moduli space of Heegaard Floer curves in $(X, J)$ from $\mathbf{y}$ to $\mathbf{y}'$

$\overline{\mathcal{M}}_j^X(\mathbf{y}, \mathbf{y}')$ Moduli space of Heegaard Floer curves in $(X, J)$ from $\mathbf{y}$ to $\mathbf{y}'$ that do not cross the basepoint

$\mathcal{M}_j(\mathbf{y}, \mathbf{y}')$ Moduli space of Heegaard Floer curves in $(\overline{W}, \overline{J})$ from $\mathbf{y}$ to $\mathbf{y}'$

$\mathcal{M}_J(\mathbf{y}, \mathbf{y}')$ Moduli space of $J$-holomorphic curves in $W' = \mathbf{R} \times \mathbf{N}$ from $\mathbf{y}$ to $\mathbf{y}'$

$\mathcal{M}_{\overline{J}}(\delta_0^\beta \mathbf{y}, \delta_0^\beta \mathbf{y}')$ Moduli space of $(\overline{W}', \overline{J})$-holomorphic maps from $\delta_0^\beta \mathbf{y}$ to $\delta_0^\beta \mathbf{y}'$ without fiber components

$\mathcal{M}_{\overline{J}_+}(\mathbf{y}, \mathbf{y})$ Moduli space of multisections of $(\overline{W}_+, \overline{J}_+)$ from $\mathbf{y}$ to $\mathbf{y}$

$\mathcal{M}_{\overline{J}_-}(\mathbf{y}, \mathbf{y})$ Moduli space of multisections in $(\overline{W}_-, \overline{J}_-)$ from $\mathbf{y}$ to $\mathbf{y}$

$\mathcal{M}_{\overline{J}_-}^*(\star_1, \star_2; \overline{m})$ Moduli space of $\overline{J}_-$-holomorphic almost multisections that pass through $\overline{m}$

$\mathcal{N}$ Negative Morse-Bott family of Reeb orbits on $\partial N$; from Section 5 onwards, is a negative Morse-Bott family of orbits of the Hamiltonian vector field $\overline{R}_0$ on $\partial N$

$n_{\tilde{z}}$ (A) Intersection number given by Equation (4.5.1)

$n(\bar{u}), n'(\bar{u}), n^+(\bar{u}), n^-(\bar{u})$ Intersection of a curve $\bar{u}$ in $\overline{W}, \overline{W}', \overline{W}_+, \overline{W}_-$ with the “pushoff” of the section at $\infty$

$N = N_{(S,h)}$ Mapping torus of $h: S \to S$; from Section 5 onwards, is given by $N = S \times [0, 2]/(x, 2) \sim (h(x), 0)$

$\overline{N} = \overline{N}_m$ Mapping torus of $\overline{h}: \overline{S} \to \overline{S}$

$\overline{\mathcal{O}}_k, \overline{\mathcal{O}}_k$ Orbit sets constructed from $\overline{\mathcal{P}}$ and $\overline{\mathcal{P}}$ which intersect $\overline{S} \times \{0\}$ exactly $k$ times

$\mathcal{P}$ Set of simple Reeb orbits of $\alpha$ in $\text{int}(N)$; from Section 5 onwards, is the set of simple orbits of the Hamiltonian vector field $\overline{R}_0$ in $\text{int}(N)$

$\overline{\mathcal{P}}$ $\mathcal{P} \cup \{e, h\}$
$\overline{P} \hat{P} \cup \{\delta_0\}$

PFC$_j$(N, $\alpha_0$, $\omega$) Periodic Floer homology chain complex on N analogous to ECC$_j$(N, $\alpha$)

$\mathcal{R}_{\phi_0}$ Radial ray $\{\phi = \phi_0, \rho \ge 0\} \subset \mathbf{C}$

$\overline{R}_0, \overline{R}_0$ Hamiltonian vector field on $\overline{N}$ and its restriction to N

$\mathcal{S}_{\alpha, \beta}$ Set of generators of $\widehat{\mathrm{CF}}(\Sigma, \alpha, \beta, z)$

$\mathcal{S}_{\mathbf{a}, \hbar(\mathbf{a})}$ 2g-tuples of intersection points of $\mathbf{a}$ and $\hbar(\mathbf{a})$

$\mathcal{S}_{\overline{\mathbf{a}}, \overline{\hbar}(\overline{\mathbf{a}})}$ 2g-tuples of intersection points $\{z_{\infty, i}\}_{i \in I} \cup \mathbf{y}'$ of $\overline{\mathbf{a}}$ and $\overline{\hbar}(\overline{\mathbf{a}})$

$\mathcal{S}(R, R')$ Counterclockwise sector in $D_{\rho_0}^2$ from radial ray R to R'

S Compact oriented connected surface of genus g with connected boundary

(S, $\hbar$) Open book decomposition of M with connected binding, page S, and monodromy $\hbar$

$\overline{S}$ Capped-off surface $S \cup D^2$

$\overline{u} = \overline{u}' \cup \overline{u}''$ Decomposition of a holomorphic multisection $\overline{u}$ in $\overline{W}, \overline{W}', \overline{W}_+, \text{ or } \overline{W}_-$ into a possibly disconnected branched cover $\overline{u}'$ of the section at $\infty$ and the rest $\overline{u}''$

$\overline{u}_{\infty} = \overline{v}_{-b} \cup \cdots \cup \overline{v}_a$ Limit of $\overline{W}_-$-curves in Section 7

V $\overline{N} - \mathrm{int}(N)$ (starting from Section 5.1.2)

$\overline{v}_j^{\beta}$ Union of non-fiber components of $\overline{v}_j^{\prime\prime}$ not asymptotic to any multiple of $\delta_0$ or $z_{\infty}$

$\overline{v}_j^{\tau}$ Union of irreducible components of $\overline{v}_j^{\prime\prime}$ asymptotic to a multiple of $\delta_0$ or $z_{\infty}$ at either end

$\mathbf{R} \times [0, 1] \times S$ and $\mathbf{R} \times N$

$\mathbf{W}_+$ Symplectic fibration with fiber S used in the definition of $\Phi$

$\mathbf{W}_-$ Symplectic fibration with fiber S; $\overline{\mathbf{W}}_-$ is used in the definition of $\Psi$

$\overline{W}, \overline{W}', \overline{W}_+, \overline{W}_-$ Extensions of $\mathbf{W}, \mathbf{W}', \mathbf{W}_+, \mathbf{W}_-$ with fibers $\overline{S}$

$\mathbf{R} \times [0, 1] \times \Sigma$

$x_i, x'_i$ Intersection points on $\partial S$ forming the contact class

(y, $\theta, t$) Coordinates of a collared neighborhood of $\partial N_{(S, \hbar)}$

y A tuple of $\mathbf{a} \cap \hbar(\mathbf{a})$; from Section 5.3 onwards, $\mathbf{y} \subset S$ and does not contain multiples of $z_{\infty}$

$z_{\infty}$ Center of the disk $D^2 = \overline{S} - \mathrm{int}(S)$

$\mathbf{z} = \{z_{\infty}^{\rho}(\vec{D})\} \cup \mathbf{y}$ A tuple of $\overline{\mathbf{a}} \cap \overline{\hbar}(\overline{\mathbf{a}})$, where $z_{\infty}$ has multiplicity $\rho$ and $\mathbf{y} \subset S$

# 1. Introduction and main results

There is a plethora of Floer-type homology theories that can be associated to a three-manifold. This paper and its sequels are concerned with three specific Floer-type homology theories: *monopole Floer homology*, *embedded contact homology*, and *Heegaard Floer homology*. Monopole Floer homology, constructed by Kronheimer and Mrowka [KrM], counts solutions of the Seiberg-Witten equations. Since its definition requires an auxiliary Riemannian metric, it has strong connections with geometry (e.g., positive curvature). Embedded contact homology, abbreviated ECH, is due to Hutchings [Hu1, Hu2] and Hutchings-Taubes [HT1, HT2] and is dynamical in nature. It counts periodic orbits
of a Reeb vector field associated with a contact form. Finally, Heegaard Floer homology, due to Ozsváth and Szabó [OSz1, OSz2], is defined in the most topological way: from a Heegaard diagram. Of the three homologies, it is the easiest to compute and admits a combinatorial description [SW]. These theories have had spectacular applications over the last decade, including a proof of the Gordon conjecture by Kronheimer, Mrowka, Ozsváth and Szabó [KMOS] and progress on the exceptional surgery problem due to Ghiggini [Gh] and Ni [Ni].

In 2006, Taubes obtained a breakthrough result which extended his celebrated correspondence between solutions of the Seiberg-Witten equations and J-holomorphic curves to the relative case. This enabled him to prove the Weinstein conjecture in dimension three [T1] and to establish the equivalence between monopole Floer cohomology and ECH shortly thereafter [T2]. A byproduct of this equivalence was the proof of Arnold's chord conjecture in dimension three [HT3, HT4].

The goal of our series of papers [0, I, II, III] is to prove the equivalence of Heegaard Floer homology and ECH. The first paper [0] can be viewed as a stand-alone paper. In this paper [I] and the sequel [II], we establish an isomorphism between the hat versions of the Heegaard Floer homology and ECH groups associated to a closed, oriented 3-manifold M. This isomorphism is compatible with the splitting of Heegaard Floer homology according to Spin$^c$-structures and of ECH according to first homology classes. For simplicity we will use $F = \mathbb{Z}/2\mathbb{Z}$ coefficients (or coefficients in a module $\Lambda$ over $F[H_2(M; \mathbb{Z})]$) for both Heegaard Floer homology and ECH.

**Remark 1.0.1.** — We expect the isomorphism to hold over the integers. To this end, we would need to define coherent orientations for all the moduli spaces used in this paper. This is a little subtle for the following reasons:

(1) The orientations from [OSz1, OSz2] are not the standard ones in Lagrangian Floer homology, defined using spin structures: if M is a closed 3-manifold with a genus $g$ Heegaard splitting, then there are $2^g$ spin structures on each Heegaard torus, whilst there are $2^{b_1(M)}$ orientation systems in [OSz1, OSz2]. On the other hand, on ECH there is only one orientation system. A preferred orientation system for Heegaard Floer homology is chosen using $HF^\infty$. It would be an interesting future project to relate the preferred orientation systems in Heegaard Floer homology and ECH using the fact that the ECH analog of $HF^\infty$ is algebraically given as the inverse limit of the U-map.

(2) The chain map $\Psi$ from ECH to Heegaard Floer homology works because there are certain cancellations that occur in pairs; see Theorem 7.1.3. One needs to verify that each pair comes with opposite signs. An analogous statement also needs to be proven for the chain homotopies.

The isomorphism between the plus version of Heegaard Floer homology and the usual version of ECH will be proved in [III].
The results of this paper were announced in [CGH1]. An isomorphism between monopole Floer homology and Heegaard Floer homology was proved, more or less simultaneously, by Kutluhan, Lee and Taubes [KLT1, KLT2, KLT3, KLT4, KLT5].

**1.1. Some background.** — Before stating the main result of this paper and describing the ideas involved in its proofs, we will give a brief summary of the definitions of Heegaard Floer homology and ECH.

We first describe Heegaard Floer homology in its “cylindrical reformulation” due to Lipshitz [Li]. The starting point of its construction is a pointed Heegaard diagram $(\Sigma, \alpha, \beta, z)$ which describes a three-manifold M. Here $\Sigma$ is a genus $g > 0$ Heegaard surface associated to some self-indexing Morse function with a unique maximum and a unique minimum, $\alpha$ is the collection of the attaching circles for the index one critical points, $\beta$ is the collection of the attaching circles for the index two critical points, and $z$ is a basepoint in the complement of $\alpha$ and $\beta$. The Heegaard Floer complex $\widehat{CF}(\Sigma, \alpha, \beta, z)$ is generated by $g$-tuples of intersection points between the $\alpha$-curves and the $\beta$-curves and the differential counts certain J-holomorphic curves in $R \times [0, 1] \times \Sigma$ with boundary on $R \times \{0\} \times \beta$ and $R \times \{1\} \times \alpha$; see Section 4 or [Li] for a more detailed exposition. While Heegaard Floer homology *a priori* depends on the choice of a pointed Heegaard diagram $(\Sigma, \alpha, \beta, z)$ and an almost complex structure on $R \times [0, 1] \times \Sigma$, it was shown to be independent of those choices, i.e., is a topological invariant of M.

The starting point for embedded contact homology is a contact form $\alpha$ on M. The contact form determines the *Reeb vector field* R by

$$ \iota_{R} d\alpha = 0 \quad \text{and} \quad \alpha(R) = 1. $$

The complex ECC(M, $\alpha$) is generated by finite sets of simple Reeb orbits with finite multiplicities and its differential counts certain J-holomorphic curves in the symplectization $(R \times M, d(e^*\alpha))$; see [Hu1, Hu2, Hu3] for more details. The hat version of ECH is defined as the homology of the mapping cone of a chain map $U : \text{ECC}(M, \alpha) \to \text{ECC}(M, \alpha)$. ECH *a priori* depends on the choice of a contact form $\alpha$ on M and an adapted almost complex structure J on the symplectization $R \times M$. There is currently no direct proof of the fact that the ECH groups are topological invariants of M (or even invariants of the contact structure $ker \alpha$, for that matter); the only known proof is due to Taubes [T1, T2], and is a consequence of the equivalence between Seiberg-Witten Floer cohomology and ECH. A direct proof of the invariance would provide, combined with the present work and computations in Heegaard Floer homology, an alternative proof of the Weinstein conjecture.

A natural setting for relating Heegaard Floer homology and ECH is that of *open book decompositions* (see Definition 2.1.1) because an open book decomposition determines both a Heegaard splitting and a contact structure. The Heegaard splitting is obtained by taking as Heegaard surface the union of two opposite pages. The contact structure is provided by the Thurston-Winkelnkemper construction [TW]. In the foundational work
[Gi], Giroux proved the equivalence of contact structures up to isomorphism and (abstrac) open book decompositions modulo positive stabilization. While we are not using its full strength, Giroux's correspondence was an important source of inspiration for this work.

We use open book decompositions and their relationship with both Heegaard splittings and contact structures to build symplectic cobordisms relating the geometric setting of Heegaard Floer homology to that of embedded contact homology. Then the isomorphisms are defined by counting certain J-holomorphic maps in those cobordisms. Their definition is similar in spirit to the definition of the open-closed and closed-open maps in [Abo]. The similarity is more direct for the open-closed map, while our closed-open map require some extra twist, as we will see.

**1.2. Main result.** — Let M be a closed, connected, oriented three-manifold. We fix an open book decomposition of M with connected binding, page S, and monodromy $h$, and denote by $\xi_{(S,h)}$ the contact structure supported by it. The contact structure $\xi_{(S,h)}$ determines a Spin$^c$-structure which we denote by $\mathfrak{s}_{(S,h)}$.

The main result of [I] and [II] is the following:

**Theorem 1.2.1.** — There is an isomorphism

$$ \Phi_0 : \widehat{HF}(-M, \mathfrak{s}_{(S,h)} + PD(A)) \xrightarrow{\sim} \widehat{ECH}(M, \xi_{(S,h)}, A), $$

where $A \in H_1(M; \mathbb{Z})$, defined via the open book decomposition $(S, h)$ of M. Moreover, $\Phi_0$ sends the Heegaard Floer contact invariant for $\xi_{(S,h)}$ to the ECH contact invariant for $\xi_{(S,h)}$.

The definition of $\Phi_0$ depends on many choices, and first of all on the choice of the open book decomposition. In this series of papers we will not address the question of its naturality, but we conjecture that it only depends on the contact structure $\xi_{(S,h)}$. Even the dependence on the contact structure should be very mild, and due only to the fact that ECH groups defined from different contact structures are isomorphic after a shift in their decompositions according to homology classes of orbit sets.

There is a similar map for the so-called twisted coefficients. Let $\Lambda$ be any module over the group ring $\mathbf{F}[H_2(M; \mathbb{Z})]$. We denote the versions of Heegaard Floer homology and ECH with twisted coefficients in $\Lambda$ by $\widehat{HF}(-M, \mathfrak{s}; \Lambda)$ and $\widehat{ECH}(M, \xi, A; \Lambda)$. Twisted coefficients are defined in [OSz2, Section 8] for Heegaard Floer homology and in [HS2, Section 11] for ECH.

**Theorem 1.2.2.** — There is an $\mathbf{F}[H_2(M; \mathbb{Z})]$-module isomorphism

$$ \Phi_0 : \widehat{HF}(-M, \mathfrak{s}_{(S,h)} + PD(A); \Lambda) \xrightarrow{\sim} \widehat{ECH}(M, \xi_{(S,h)}, A; \Lambda), $$

where $A \in H_1(M; \mathbb{Z})$, defined via an open book decomposition $(S, h)$ of M. Moreover, $\Phi_0$ sends the Heegaard Floer contact invariant for $\xi_{(S,h)}$ to the ECH contact invariant for $\xi_{(S,h)}$.
On the other hand, Taubes [T2] has proven that Seiberg-Witten Floer cohomology and ECH are isomorphic. Let $\overline{HM}(M)$ be the homology of the mapping cone of $U_+ : \tilde{\mathcal{C}}(M) \to \tilde{\mathcal{C}}(M)$, where $\tilde{\mathcal{C}}(M)$ is the chain complex for $\overline{HM}(M)$. Combining Taubes' theorem with Theorem 1.2.1, we obtain the following “corollary”:

**Corollary 1.2.3.** — $\overline{HF}(M, s) \simeq \overline{HM}(M, s)$ for any $s \in Spin^c(M)$.

An alternate proof of Corollary 1.2.3 is due to Kutluhan, Lee and Taubes [KLT1, KLT2, KLT3, KLT4, KLT5].

**1.3. Outline of proof.** — Fix an open book decomposition $(S, \tilde{h})$ for $M$. The first step in the construction of the map $\Phi_0$ is to adapt the definitions of $\overline{HF}(-M)$ and $\overline{ECH}(M)$ to the open book decomposition $(S, \tilde{h})$. For Heegaard Floer homology this is achieved in Section 4, essentially by pushing all interesting intersection points between the $\alpha$- and $\beta$-curves to one side of the Heegaard surface obtained from $(S, \tilde{h})$.

On the ECH side this was achieved in the first paper [0] of our series, where we introduced the group $\overline{ECH}(N, \partial N)$ for the mapping torus $N$ of $(S, \tilde{h})$. This group was defined as a direct limit

$$ \widehat{ECH}(N, \partial N) = \lim_{j \to \infty} ECH_j(N), $$

where $j$ is the number of intersections of an orbit set in $N$ with a page and the direct limit is taken with respect to maps

$$ (\mathcal{J}_j)_* : ECH_j(N) \to ECH_{j+1}(N), $$

defined by increasing the multiplicity of an elliptic orbit on $\partial N$ which can be regarded intuitively as a receptacle for the $J$-holomorphic curves in $\mathbf{R} \times M$ which intersect the cylinder over the binding. The following was proved in [0]:

**Theorem 1.3.1.** — $\widehat{ECH}(M) \simeq \widehat{ECH}(N, \partial N)$.

The map $\Phi_0$ is, roughly speaking, the composition of a map

$$ \Phi_* : \overline{HF}(-M) \to ECH_{2g}(N), $$

induced by a symplectic cobordism $W_+$, followed by the natural map

$$ ECH_{2g}(N) \to \lim_{j \to \infty} ECH_j(N), $$

induced by the maps $(\mathcal{J}_j)_*$. Here $g$ is the genus of $S$. (Strictly speaking, we need to use a “perturbed” version of $ECH_j(N)$, as explained in Section 2.5, and replace the ECH groups by certain periodic Floer homology groups, as explained in Section 3.)
The cobordism $W_+$ is a symplectic fibration with fiber S and monodromy $\hbar$ over a Riemann surface with a strip-like end and a cylindrical end, which is biholomorphic to a disk with a puncture at the center and a puncture on the boundary. On the boundary of $W_+$ there is a (disconnected) Lagrangian submanifold $\Lambda$ which, over the strip-like end, roughly speaking coincides with the Lagrangians used in the definition of Heegaard Floer homology (restricted to the nontrivial half of the Heegaard splitting). Then $\Phi_*$ is defined by counting degree $2g$ J-holomorphic multisections of $W_+$ with boundary on $\Lambda$ which converge to generators of the Heegaard Floer complex over the strip-like end and to generators of the ECH complex over the cylindrical end.

We also define a map $\Psi_* : \text{ECH}_{2g}(N) \to \widehat{\text{HF}}(-M)$ by counting degree $2g$ J-holomorphic multisections of a symplectic cobordism $\overline{W}_-$ obtained by closing all fibers of $W_+$ with a disk and turning it upside down. The Lagrangian boundary condition on $\overline{W}_-$ becomes singular, and this leads to many more potential degenerations of J-holomorphic sections. For this reason, the proof that $\Psi_*$ is defined is the longest and most difficult part of this paper.

In the next paper [II] we prove that $\Phi_*$ and $\Psi_*$ are inverses of each other by composing the two cobordisms and degenerating them in a different way. The proof that $\Phi_* \circ \Psi_*$ and $\Psi_* \circ \Phi_*$ are the identity is thus reduced to a computation of some relative Gromov-Witten (or relative Gromov-Taubes) invariants. Finally we prove that the maps $(\mathfrak{J}_j)_*$ are isomorphisms for $j \ge 2g$ by an argument based on stabilizing the open book decomposition.

**1.4. Organization of the paper.** — Papers [I] and [II] should be read as a single paper which has been split for practical reasons. References from [II] will be written as “II.x”; for example “Section II.x” will mean “Section x” of [II].

In Section 2 we recall some results of [0], including the definition of $\widehat{\text{ECH}}(N, \partial N)$. In Section 3 we replace the ECH chain complexes $\text{ECC}_j(N)$ by the periodic Floer homology chain complexes $\text{PFC}_j(N)$, which are technically a little easier to use when defining chain maps to and from Heegaard Floer homology. Then in Section 4 we (i) review Lipsitz’ reformulation of Heegaard Floer homology, (ii) restrict the Heegaard Floer chain complex to a page S as in [HKM] and obtain the chain group $\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$ whose homology is isomorphic to $\widehat{\text{HF}}(-M)$, and (iii) introduce an ECH-type index $I_{\widehat{\text{HF}}}$ for Heegaard Floer homology. Section 5 is devoted to describing the moduli spaces of multisections which are used in the definitions of the chain maps $\Phi$ and $\Psi$ between the Heegaard Floer chain complex $\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$ and the periodic Floer homology chain complex $\text{PFC}_{2g}(N)$. Then in Sections 6 and 7 we show that $\Phi$ and $\Psi$ are indeed chain maps. The proof that $\Psi$ is a chain map is substantially more involved than the proof that $\Phi$ is a chain map.

The proofs of the chain homotopies between the chain maps $\Psi \circ \Phi$ and $id$, and between the chain maps $\Phi \circ \Psi$ and $id$, are rather involved and occupy almost all (Sections II.2–II.4) of [II]. The necessary Gromov-Witten (or Gromov-Taubes) type calcula-
tions which are used in the proof of the chain homotopy are carried out in Section II.2.
Finally, in Section II.5 we prove that (variants of) the maps $(\mathcal{J}_j)_*$ are isomorphisms for
$j \ge 2g$.

**2. Adapting ECH to an open book decomposition**

In this section we briefly recall the results of [0]. The reader is referred to [0] for
a more complete discussion; the notation here is the same as that of [0], unless indicated
otherwise.

**2.1. The first return map.** — We start by recalling the definition of an open book
decomposition in order to set the notation. Let $(S, \hbar)$ be a pair consisting of a compact
oriented surface $S$ with nonempty boundary (sometimes called a *bordered surface*) and a
diffeomorphism $\hbar: S \to S$ which restricts to the identity on $\partial S$. In this paper we will always
assume that $S$ is connected and $\partial S$ is connected, unless stated otherwise.

We define the mapping torus

$$
N_{(S, \hbar)} = S \times [0, 1] / \sim,
$$

where $(x, 1) \sim (\bar{h}(x), 0)$.

**Definition 2.1.1.** — Let $K \subset M$ be a link. Then $M$ admits an open book decomposition $(S, \mathcal{h})$ with binding $K$ if there is a diffeomorphism

$$
M \cong N_{(S,h)} \cup V,
$$

where $V \cong D^2 \times S^1$ is a tubular neighborhood of $K$ and, if we parametrize $\partial D^2$ by $t \in [0, 1]$, then
$\partial S \times \{t\} \subset \partial N_{(S,h)}$ is glued to $\{t\} \times S^1$.

The decomposition we give here is the standard one and is slightly different from
the one in [0], where we introduced a “no man’s land” diffeomorphic to $T^2 \times [0, 1]$
between $N_{(S, \hbar)}$ and $V$. Of course the two decompositions are equivalent.

Let $S$ be a bordered surface. Let $\nu(\partial S) \simeq [-\epsilon, 0] \times \mathbf{R}/\mathbf{Z}$ be a neighborhood of $\partial S$
with coordinates $(y, \theta)$ so that $\partial S = \{y=0\}$. (The slight difference with [0] is that, in [0,
Section 9.3], $\partial S = \{y=1\}$ instead of $\{y=0\}$.)

The following was essentially proved in [0]:

**Lemma 2.1.2.** — Given $\hbar \in \operatorname{Diff}(S, \partial S)$, there exists $\hbar_0 \in \operatorname{Diff}(S, \partial S)$ in the same connected component as $\hbar$, and which satisfies the following:

(1) there exists a contact form $\alpha$ on $N_{(S, \hbar)}$ such that the Reeb vector field $R_\alpha$ of $\alpha$ is transverse to $S \times \{t\}$, $t \in [0, 1]$, and $\hbar_0$ is the first return map of $R_\alpha$ on $S \times \{0\}$; and

(2) the diffeomorphism $\hbar_0$ restricts to $(y, \theta) \mapsto (y, \theta - y)$ on $\nu(\partial S)$.

Moreover, $\alpha = f_i dt + \beta_t$, where $f_i$ is a positive function on $S$ and $\beta_t$ is a Liouville $1$-form on $S$.
*Proof.* — (1) and the last sentence of the lemma follow from combining Lemmas 9.3.2 and 9.3.3 from [0]. To verify (2), we refer to Section 9.3.1 and the discussion after the proof of Lemma 9.3.3 in [0] and take a slight modification of $\alpha$ of the form

$$\alpha = g(y)d\theta + f(y)dt$$

on a neighborhood $\nu(\partial N_{(S,\bar{h})})$ of $\partial N_{(S,\bar{h})}$. Here $\nu(\partial N_{(S,\bar{h})})$ is the quotient of

$$\nu(\partial S) \times [0, 1] \simeq [-\epsilon, 0] \times (\mathbf{R}/\mathbf{Z}) \times [0, 1]$$

with coordinates $(y, \theta, t)$, by the equivalence relation $(y, \theta, 1) \sim (y, \theta, 0)$. If we take

$$(f(y), g(y)) = (f(0) + y^2/2, g(0) + y),$$

with $(f(0), g(0))$ in the interior of the first quadrant, then the Reeb vector field $R_\alpha$ is parallel to

$$(2.1.1) \qquad -f'(y)\partial_\theta + g'(y)\partial_t = -y\partial_\theta + \partial_t.$$

Its first return map then satisfies (2). $\square$

*From now on, we assume that $\bar{h} = h_0$ as given by Lemma 2.1.2.*

**2.2.** ECH(N, $\partial$N, $\alpha$) and $\widehat{\text{ECH}}(N, \partial N, \alpha)$. — Let $N = N_{(S,\bar{h})}$ and $\alpha$ be as in the previous subsection. We recall the definitions of the variants ECH(N, $\partial$N, $\alpha$) and $\widehat{\text{ECH}}(N, \partial N, \alpha)$ and the main result concerning them from [0]. In particular, we carry over the Morse-Bott terminology from [0, Section 4]. We will assume that the almost complex structure $J$ on $\mathbf{R} \times N$ is Morse-Bott regular.

The boundary $\partial N$ is foliated by a Morse-Bott family $\mathcal{N}$ of simple orbits of $R_\alpha$ of the form $\theta = \text{const}$. We may assume without loss of generality that $\alpha$ is nondegenerate away from $\partial N$, after a $C^k$-small perturbation for $k \gg 0$. We pick two orbits from $\mathcal{N}$ and label them $h$ and $e$. The orbits $h$ and $e$ are meant to become hyperbolic and elliptic after a small, controlled perturbation of $\alpha$. The Morse-Bott family $\mathcal{N}$ is *negative*. Since $\mathcal{N}$ is a Morse-Bott family on $\partial N$, this means that $\mathcal{N}$ plays the role of a sink and that no holomorphic curve (besides a trivial cylinder) is asymptotic to an orbit of $\mathcal{N}$ at the positive end.

**2.2.1.** ECH(N, $\partial$N, $\alpha$). — Let $\mathcal{P}$ be the set of simple Reeb orbits of $\alpha$ in $\text{int}(N)$. We write $\text{ECC}_j^\beta(N, \alpha)$ for the chain complex generated by orbit sets $\gamma$ constructed from $\mathcal{P} \cup \{e\}$, whose homology class $[\gamma]$ intersects the page $S \times \{t\}$ exactly $j$ times. The differential for $\text{ECC}_j^\beta(N, \alpha)$ counts ECH index 1 Morse-Bott buildings in $(\mathbf{R} \times N, J)$ between orbit sets which are constructed from $\mathcal{P} \cup \{e\}$; for more details see [0, Section 7.3]. In particular, if $\tilde{u}$ is an ECH index 1 Morse-Bott building which is counted in the differential,
then $e$ can appear only at a negative end of $\tilde{u}$ and no single end of $\tilde{u}$ can multiply cover $e$ with multiplicity $> 1$. (It is still possible that there are many ends of $\tilde{u}$ which simply cover $e$.)

There are inclusions of chain complexes:

$$ \mathcal{I}_j^b : \mathrm{ECC}_j^b(N, \alpha) \to \mathrm{ECC}_{j+1}^b(N, \alpha), $$

$$ \gamma \mapsto e\gamma, $$

where we are using multiplicative notation for orbit sets. Let us write $\mathrm{ECH}_j^b(N, \alpha)$ for the homology of the chain complex $\mathrm{ECC}_j^b(N, \alpha)$. We then define

$$ \mathrm{ECH}(N, \partial N, \alpha) = \lim_{j \to \infty} \mathrm{ECH}_j^b(N, \alpha). $$

**2.2.2.** $\widehat{\mathrm{ECH}}(N, \partial N, \alpha)$. — Let $\mathrm{ECC}_j(N, \alpha)$ be the chain complex generated by orbit sets $\gamma$ constructed from $\widehat{\mathcal{P}} = \mathcal{P} \cup \{e, h\}$, whose homology class $[\gamma]$ intersects $S \times \{t\}$ exactly $j$ times. The differential for $\mathrm{ECC}_j(N, \alpha)$ counts ECH index 1 Morse-Bott buildings in $(\mathbb{R} \times N, J)$. There are inclusions:

$$ (2.2.1) \qquad \mathcal{I}_j : \mathrm{ECC}_j(N, \alpha) \to \mathrm{ECC}_{j+1}(N, \alpha), $$

$$ \gamma \mapsto e\gamma, $$

as before. Writing $\mathrm{ECH}_j(N, \alpha)$ for the homology of $\mathrm{ECC}_j(N, \alpha)$, we define

$$ \widehat{\mathrm{ECH}}(N, \partial N, \alpha) = \lim_{j \to \infty} \mathrm{ECH}_j(N, \alpha). $$

The following was the main result of [0]:

**Theorem 2.2.1.** — We have the isomorphisms:

$$ \mathrm{ECH}(M) \simeq \mathrm{ECH}(N, \partial N, \alpha), $$

$$ \widehat{\mathrm{ECH}}(M) \simeq \widehat{\mathrm{ECH}}(N, \partial N, \alpha). $$

**2.3.** Splitting of ECH according to homology classes. — Given an orbit set $\gamma = \prod_{j=1}^l \gamma_j^{m_j}$ in $M$ or $N$, its total homology class $[\gamma]$ is defined as

$$ [\gamma] = \sum_{j=1}^{l} m_j [\gamma_j], $$
where $[\gamma_j] \in H_1(M; \mathbb{Z})$ if M is a closed manifold and $[\gamma_j] \in H_1(N; \mathbb{Z})$ or $H_1(N, \partial N; \mathbb{Z})$ (as appropriate) if N has torus boundary. We then have the direct sum decomposition:

$$ \mathrm{ECC}(M) = \bigoplus_{A \in H_1(M)} \mathrm{ECC}(M, A), $$

where $\mathrm{ECC}(M, A)$ is the subcomplex of $\mathrm{ECC}(M)$ generated by orbit sets with total homology class A. The direct sum of chain groups descends to a direct sum of homology groups $\mathrm{ECH}(M, A)$.

There is an analogous splitting for $\mathrm{ECC}(N, \partial N, \alpha)$. In fact,

$$ \mathrm{ECC}_j^b(N, \alpha) = \bigoplus_{A \in H_1(N, \partial N)} \mathrm{ECC}_j^b(N, \alpha, A), $$

and the inclusion $\gamma \mapsto e\gamma$ respects this splitting since $[e] = 0$ in $H_1(N, \partial N)$.

**Lemma 2.3.1.** — If M has an open book decomposition with binding K and N is the mapping torus of a page, then there is an isomorphism

$$ \varpi : H_1(M) \xrightarrow{\sim} H_1(N, \partial N). $$

*Proof.* — We use the long exact sequence for the pair (M, K):

$$ H_1(K) \to H_1(M) \xrightarrow{i} H_1(M, K) \to H_0(K). $$

Since $[K] = 0$ in $H_1(M)$ and K is connected, the map $i$ is an isomorphism. By excision and homotopy invariance, we obtain the isomorphism $H_1(M, K) \xrightarrow{\sim} H_1(N, \partial N)$. Combining the two isomorphisms gives us $\varpi$. $\square$

The isomorphism $\mathrm{ECH}(M) \cong \mathrm{ECH}(N, \partial N, \alpha)$ respects the splitting into total homology classes. In fact

$$ \mathrm{ECH}(M, A) \cong \mathrm{ECH}(N, \partial N, \alpha, \varpi(A)). $$

The same also holds for the hat versions.

**2.4. Twisted coefficients in ECH.** — In this subsection we describe the construction of ECH with twisted coefficients. We will adapt the analogous construction in Heegaard Floer homology from [OSz2, Section 8] instead of following the original construction in [HS2, Section 11].

Fix a homology class A and a closed curve $\Gamma \subset M$ such that $[\Gamma] = A$. Let

$$ \gamma^+ = \prod_{j=1}^{l_+} \gamma_{j,+}^{m_{j,+}}, \quad \gamma^- = \prod_{j=1}^{l_-} \gamma_{j,-}^{m_{j,-}} $$
be orbit sets with $[\boldsymbol{\gamma}^+] = [\boldsymbol{\gamma}^-] = A$. Here we are assuming that all the multiplicities $m_{j,\pm}$ are positive. We denote by $H_2(M, \boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-)$ the set of relative homology classes [C] such that $\partial C = \boldsymbol{\gamma}^+ - \boldsymbol{\gamma}^-$, namely $[C] \in H_2(M, \mathcal{O}(\boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-))$, such that $\mathcal{O}(\boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-) = (\cup_{j=1}^l \gamma_{j,+}) \cup (\cup_{j=1}^l \gamma_{j,-})$ and $[\partial C] = \sum_{j=1}^l m_{j,+}[\gamma_{j,+}] - \sum_{j=1}^l m_{j,-}[\gamma_{j,-}]$. We also denote by $M^{l=1}(\boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-)$ the moduli spaces of $I=1$ holomorphic curves in $\mathbf{R} \times \mathbf{M}$ from $\boldsymbol{\gamma}^+$ to $\boldsymbol{\gamma}^-$ representing the homology class of C.

A complete set of paths for A based at $\Gamma$ is the choice, for every orbit set $\boldsymbol{\gamma}$ such that $[\boldsymbol{\gamma}] = A$, of a surface $C_{\boldsymbol{\gamma}} \subset \mathbf{M}$ such that $\partial C_{\boldsymbol{\gamma}} = \boldsymbol{\gamma} - \Gamma$, defined in the same way as the previous paragraph. A complete set of paths for A induces maps

$$ \mathfrak{A}' : H_2(M, \boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-) \to H_2(M) $$

for all $\boldsymbol{\gamma}^+$ and $\boldsymbol{\gamma}^-$ in A by $\mathfrak{A}'(C) = [C_{\boldsymbol{\gamma}^-} \cup C \cup -C_{\boldsymbol{\gamma}^+}]$. This map is compatible with the action of $H_2(M)$ on $H_2(M, \boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-)$ and with the concatenation of chains with matching ends.

We denote the group ring of $H_2(M; \mathbf{Z})$ by $\mathbf{F}[H_2(M; \mathbf{Z})]$ and the generator corresponding to $c \in H_2(M; \mathbf{Z})$ by $e^c$. We define

$$ \underline{\mathrm{ECC}}(M, \alpha, A) = \mathrm{ECC}(M, \alpha, A) \otimes_{\mathbf{F}} \mathbf{F}[H_2(M; \mathbf{Z})] $$

as an $\mathbf{F}[H_2(M; \mathbf{Z})]$-module, with differential

$$ \partial \boldsymbol{\gamma}^+ = \sum_{\boldsymbol{\gamma}^-} \sum_{C \in H_2(M, \boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-)} \# (M^{l=1}(\boldsymbol{\gamma}^+, \boldsymbol{\gamma}^-, C)/\mathbf{R}) e^{\mathfrak{A}'(C)} \boldsymbol{\gamma}^- $$

The homology of this complex is ECH with twisted coefficients $\underline{\mathrm{ECH}}(M, A)$. The U-map can be defined in a similar manner and $\widehat{\mathrm{ECH}}(M, A)$ is the homology of its mapping cone. The construction of $\underline{\mathrm{ECH}}(N, \partial N, \omega(A))$ with coefficient ring $\mathbf{F}[H_2(N, e; \mathbf{Z})] \cong \mathbf{F}[H_2(M; \mathbf{Z})]$ is similar, and there are isomorphisms

$$ (2.4.1) \qquad \underline{\mathrm{ECH}}(M, A) \simeq \underline{\mathrm{ECH}}(N, \partial N, \omega(A)). $$

$$ (2.4.2) \qquad \widehat{\mathrm{ECH}}(M, A) \simeq \widehat{\mathrm{ECH}}(N, \partial N, \omega(A)). $$

Moreover, by considering only orbit sets that intersect a page $j$ times we can define $\underline{\mathrm{ECH}}_j(N)$ and we have

$$ \widehat{\mathrm{ECH}}(N, \partial N, \alpha, \omega(A)) = \lim_{j \to \infty} \underline{\mathrm{ECH}}_j(N, \omega(A)). $$

**2.5. Elimination of elliptic orbits.** — The goal of this subsection is to show how to locally replace elliptic orbits by hyperbolic orbits with the same parity (i.e., with negative eigenvalues). The main result, which is used in [II] but is also of independent interest, is Theorem 2.5.2 below. Let us first give the following definition:
**Definition 2.5.1 (Filtration F).** — If $N = N_{(S,h)}$ is the mapping torus of $(S, h)$ and $\gamma \subset N$ is a link which is everywhere transverse to $S \times \{t\}$, $t \in [0, 1]$, then we define $\mathcal{F}(\gamma) = \langle \gamma, S \times \{0\} \rangle$, where $\langle \cdot, \cdot \rangle$ is the algebraic intersection number.

**Theorem 2.5.2 (Elimination of elliptic orbits).** — Let $\alpha$ be a contact form on the mapping torus $N = N_{(S,h)}$, $h \in \mathrm{Diff}(S, \partial S)$, such that the Reeb vector field $R_\alpha$ is transverse to $S \times \{t\}$, $t \in [0, 1]$, and $h$ is the first return map of $R_\alpha$ on $S \times \{0\}$. Then, given $m \in \mathbb{Z}^+$ and $\varepsilon > 0$, there exists a smooth function $f: N \to (0, +\infty)$ which is $\varepsilon$-close to 1 with respect to a fixed $\mathbb{C}^1$-norm and whose Reeb vector field $R_{f\alpha}$ has no elliptic orbits $\gamma$ in $\mathrm{int}(N)$ satisfying $\mathcal{F}(\gamma) \le m$.

**2.5.1. Model situation on the solid torus.** — Fix a constant $\delta > 0$. Consider the solid torus

$$V = D^2 \times S^1 = D^2 \times (\mathbf{R}/\mathbb{Z}) = \{(r, \theta, z) \mid r \le \delta\}$$

with the contact structure $\xi_0 = \ker \alpha_0$, where $\alpha_0 = dz + r^2 d\theta$. We write $D_{z_0} = \{z = z_0\} \subset V$ and $T_{r_0} = \{r = r_0\} \subset V$.

Given a function $f: [0, \delta] \to (0, +\infty)$, the Reeb vector field $R_{f(r)\alpha_0}$ is given by:

$$(2.5.1) \qquad R_{f\alpha_0} = \frac{1}{2rf^2}((r^2f' + 2rf)\partial_z - f'\partial_\theta).$$

In particular, $R_{f\alpha_0}$ is transverse to each $D_z$, provided $r^2f' + 2rf > 0$, and $R_{f\alpha_0}$ is tangent to each $T_r$. The first return map $\Phi_{f\alpha_0}: D_0 \to D_0$ is a rotation on each circle $\{r = r_0\}$ and can be written as $(r, \theta) \mapsto (r, \theta + \phi_{f\alpha_0}(r))$.

**Claim 2.5.3.** — Let $C: [0, \delta] \to (0, +\infty)$ be a smooth function. If

$$(2.5.2) \qquad f(r) = A \exp \left( - \int_0^r \frac{2sC(s)}{1+C(s)s^2} ds \right),$$

where $A$ is a positive constant, then the first return map of $R_{f\alpha_0}$ satisfies $\phi_{f\alpha_0}(r) = C(r)$ for each $r \in (0, \delta]$.

Here we are ignoring the differentiability at $r = 0$.

**Proof.** — In view of Equation (2.5.1), $R_{f\alpha_0}$ is parallel to $\partial_z + C(r)\partial_\theta$ if and only if

$$(2.5.3) \qquad C(r)(r^2f' + 2rf) = -f'$$

is satisfied.

□
Let $f_0 : [0, \delta] \to (0, +\infty)$ be a function such that $\phi_{f_0\alpha_0}(r) = \phi_0$, where $\phi_0 \in (0, 2\pi)$. By Claim 2.5.3, we may take

$$f_0(r) = \exp\left(-\int_{\delta}^{r} \frac{2s\phi_0}{1 + \phi_0 s^2} ds\right) = \frac{1 + \phi_0 \delta^2}{1 + \phi_0 r^2}.$$

In particular, $\gamma_0 = \{0\} \times S^1$ is the only orbit $\gamma$ of $R_{f_0\alpha_0}$ satisfying $\mathcal{F}(\gamma) = 1$, where $\mathcal{F}(\gamma) = \langle\gamma, D_0\rangle$, and the orbit $\gamma_0$ is elliptic and nondegenerate.

**Lemma 2.5.4 (Modification lemma).** — There exists a function $f_2 : V \to (0, +\infty)$ such that $f_2\alpha_0$ is arbitrarily $C^1$-close to $f_0\alpha_0$ and the Reeb vector field $R_{f_2\alpha_0}$ is equal to $R_{f_0\alpha_0}$ near $\partial V$, is transverse to $D_z$ for all $z \in S^1$, and has only one orbit $\gamma$ satisfying $\mathcal{F}(\gamma) = 1$. Moreover the orbit $\gamma$ is hyperbolic.

*Proof.* — Without loss of generality we take $\phi_0 \in (0, \pi]$; the case $\phi_0 \in [\pi, 2\pi)$ is similar.

Let $0 < \delta' \ll \delta$ and let $C_{\delta'} : [0, \delta] \to [\phi_0, \pi]$ be a smooth function such that:

- $C_{\delta'}(r) = \pi$ on $[0, \frac{\delta'}{2}]$; and

- $C_{\delta'}(r) = \phi_0$ on $[\delta', \delta]$.

Let $f_1 : [0, \delta] \to (0, +\infty)$ be the smooth function

$$f_1(r) = \exp\left(-\int_{\delta}^{r} \frac{2sC_{\delta'}(s)}{1 + C_{\delta'}(s)s^2} ds\right).$$

The function $f_1$ satisfies the following:

(1) $f_1=f_0$ and $R_{f_1\alpha_0} = R_{f_0\alpha_0}$ for $r \in [\delta', \delta]$;

(2) $R_{f_1\alpha_0} \cap D_z$ for all $z \in S^1$;

(3) $\phi_{f_1\alpha_0}(r) = C_{\delta'}(r)$ for all $r \in [0, \delta]$.

We compute that

$$(2.5.4) \qquad \frac{f_1(r)}{f_0(r)} = \exp\left(\int_{\delta}^{r} \left(\frac{2s\phi_0}{1 + \phi_0 s^2} - \frac{2sC_{\delta'}(s)}{1 + C_{\delta'}(s)s^2}\right) ds\right).$$

By taking $\delta' > 0$ to be arbitrarily small, the absolute value of the integrand of Equation (2.5.4) can be made arbitrarily small. Hence $f_1(r) \approx f_0(r)$ for all $r \in [0, \delta]$, in view of (1). (Here $f \approx g$ means $|f - g|$ is bounded above by a continuous function of $\delta'$ which approaches 0 as $\delta' \to 0$.)

Next we consider $\partial_{\phi}f_i = \frac{\phi_{f_i'(r)}}{r}$ and $\partial_{\phi}f_i = \frac{\phi_{f_i'(r)}}{r}$, where $r = \sqrt{x^2 + y^2}$. We appeal to Equation (2.5.3) and write

$$f_1'(r) = \frac{2\phi f_1(r)}{1 + C_{\delta'}(r)r^2}, \quad f_0'(r) = \frac{2\phi f_0(r)}{1 + \phi_0 r^2}.$$
For $\delta' > 0$ sufficiently small, $f'_1(r)$ and $f'_0(r)$ are both arbitrarily close to 0. Moreover $f'_1(r) = f'_0(r)$ for $r \in [\delta', \delta]$. Hence $f'_1 \approx f'_0$ for all $r \in [0, \delta]$. This implies that $\partial_*f_1 \approx \partial_*f_0$ and $\partial_*(f_1/f_0) \approx 0$ for $*=x,y$.

The Reeb vector field $R_{f_1\alpha_0}$ has exactly one orbit $\gamma_0$ satisfying $\mathcal{F}(\gamma_0) = 1$. The linearized first return map $d\Phi_{f_1\alpha_0}(0)$ of $\gamma_0$ has eigenvalues $-1$.

We now claim there exists a $C^k$-small perturbation $f_2$ of $f_1$ for any $k \gg 0$ such that the linearized first return map $d\Phi_{f_2\alpha_0}(0)$ of the corresponding orbit has eigenvalues $-\lambda$ and $-\frac{1}{\lambda}$ with $\lambda \in \mathbf{R}^{>0} - \{1\}$. This follows from a local model on $D^2 \times [0, 1]$ with coordinates $(x, y, z)$: Suppose the Reeb vector field is $\mathbf{R} = \partial_z$. Then the contact 1-form can be written as $\alpha = dz + \beta$, where $\beta$ is a 1-form on $D^2$. We consider $R_{\mathcal{G}\alpha}$, where $\mathcal{G}(x,y) = 1 + \epsilon(x^2 - y^2)$ and $\epsilon > 0$ is sufficiently small. The component of $R_{\mathcal{G}\alpha}$ in the $xy$-direction is parallel to $y\partial_x - x\partial_y$. Hence the derivative at zero of the holonomy map $D^2 \times \{0\} \to D^2 \times \{1\},^1$ obtained by flowing along $R_{\mathcal{G}\alpha}$, has eigenvalues $\lambda_0$ and $\frac{1}{\lambda_0}$ with $\lambda_0 \in \mathbf{R}^{>0} - \{1\}$. By appropriately damping $\mathcal{G}$ out to 1 outside a small neighborhood of $(0, 0, 0) \in D^2 \times [0, 1]$, the above model can be grafted into $V$ to give $f_2$. This procedure does not introduce any extra $\mathcal{F} = 1$ orbits, since the graph of $\Phi_{f_1\alpha_0}$ in $D_0 \times D_0$ intersects the diagonal transversely in one point and this property is stable under a $C^k$-small perturbation of $\Phi_{f_1\alpha_0}$ for any $k \gg 0$. $\square$

**Remark 2.5.5.** — The modification in Lemma 2.5.4 introduces many elliptic orbits satisfying $\mathcal{F} > 1$.

**2.5.2.** *Proof of Theorem 2.5.2.* — Starting with the contact form $\alpha$ on $N$, we make a $C^2$-small perturbation of $\alpha$ relative to $\partial N$ such that the resulting Reeb vector field — also called $R_\alpha$ — satisfies the following:

(1) $R_\alpha$ is nondegenerate away from $\partial N$;

(2) for each $\mathcal{F} = 1$ elliptic orbit $\gamma \subset \text{int}(N)$, the first return map is a rotation by an irrational angle and $\alpha$ is of the form $C_{0f_0\alpha_0}$ on a tubular neighborhood $V_\gamma$ of $\gamma$.

Here $f_0$ and $\alpha_0$ are as in Section 2.5.1 and $C_0$ is some constant. We then use Lemma 2.5.4 on the tubular neighborhoods $V_\gamma$ to replace the $\mathcal{F} = 1$ elliptic orbits by hyperbolic orbits plus $\mathcal{F} > 1$ orbits. Next, we perturb the form so that the $\mathcal{F} = 1$ orbits and the $\mathcal{F} = 2$ hyperbolic orbits are left unchanged and the $\mathcal{F} = 2$ elliptic orbits satisfy (2), with $\mathcal{F} = 1$ replaced by $\mathcal{F} = 2$. Using Lemma 2.5.4 again, we replace the $\mathcal{F} = 2$ elliptic orbits by hyperbolic orbits and $\mathcal{F} > 2$ orbits. Continuing in this manner, we obtain $\alpha$ without any $\mathcal{F} \le m$ elliptic orbits in $\text{int}(N)$.

<footnote n="1">¹ The dashed arrow indicates that the map is only partially defined.</footnote>
**2.5.3. Direct limits.** — Starting with $(S, \mathfrak{h})$ and $\alpha$ from Section 2.1, by Theorem 2.5.2 there exists a sequence of contact 1-forms $\alpha_j, j \in \mathbb{N}$, on $\mathbb{N}$ such that:

(1) $\alpha$ and $\alpha_j$ agree on a small neighborhood of $\partial N$ and $\ker \alpha$ and $\ker \alpha_j$ are isotopic relative to $\partial N$;

(2) all the orbit sets of $R_{\alpha_j}$ that satisfy $\mathcal{F} = i, i \le j$, have $\alpha_j$-action between $i - \epsilon$ and $i + \epsilon$, where $0 < \epsilon \ll 1$;

(3) $R_{\alpha_j}$ and $R_{\alpha_{j+1}}$ are parallel outside of a small neighborhood $V_{j+1}$ of the elliptic orbits of $R_{\alpha_j}$ with $\mathcal{F} = j + 1$;

(4) $R_{\alpha_{j+1}}$ has no elliptic orbits with $\mathcal{F} \le j + 1$.

We take suitable extensions of $\alpha_j$ and $\alpha_{j+1}$ to $M$ as in [0, Section 9.3], which we still call $\alpha_j$ and $\alpha_{j+1}$. Then

$$ \widehat{\mathrm{ECH}}(M) \simeq \lim_{j \to \infty} \mathrm{ECH}^{\le L_j}(M, \alpha_j), $$

where $L_j = j + \epsilon$, the superscript $\le L_j$ indicates orbit sets of action $\le L_j$, and the direct limit is the limit of the induced maps of ECH chain maps

$$ \bar{\mathcal{L}}_j : \mathrm{ECC}^{\le L_j}(M, \alpha_j) \to \mathrm{ECC}^{\le L_{j+1}}(M, \alpha_{j+1}), $$

given by [HT3, Theorem 2.4]. It is important to remember that the ECH cobordism maps are defined by using cobordism maps in Seiberg-Witten Floer cohomology.

Next we consider the ECH chain maps

$$ (2.5.5) \qquad \mathcal{R}_j : \mathrm{ECC}_j(N, \alpha_j) \to \mathrm{ECC}_{j+1}(N, \alpha_{j+1}), $$

given by composing the restriction

$$ (2.5.6) \qquad \mathcal{R}'_j : \mathrm{ECC}_j(N, \alpha_j) \to \mathrm{ECC}_j(N, \alpha_{j+1}) $$

of $\bar{\mathcal{L}}_j$ to $N$ and $\mathfrak{J}_j : \mathrm{ECC}_j(N, \alpha_{j+1}) \to \mathrm{ECC}_{j+1}(N, \alpha_{j+1})$ given by $\gamma \mapsto \epsilon\gamma$. The methods of [0, Section 9] that are used to prove Theorem 1.3.1 show that

$$ \mathrm{ECH}^{\le L_j}(M, \alpha_j) \simeq \mathrm{ECH}_j(N, \alpha_j), $$

and $(\bar{\mathcal{L}}_j)_* = (\mathcal{R}_j)_*$ under this identification.

Hence we have the following theorem which will be used in [II]:

**Theorem 2.5.6.** — $\widehat{\mathrm{ECH}}(M) \simeq \lim_{j \to \infty} \mathrm{ECH}_j(N, f_j \alpha)$, where the direct limit is taken with respect to the maps $\mathcal{R}_j$.
### 3. Periodic Floer homology

In order to simplify some technicalities, we would like to replace the ECH groups by the periodic Floer homology groups of Hutchings [Hu1, Hu2, HS1], abbreviated PFH in this paper. The PFH groups are defined in a manner completely analogous to the ECH groups, with stable Hamiltonian vector fields replacing the Reeb vector fields.

If M is a closed manifold which fibers over the circle, then the PFH groups of M are equivalent to the Seiberg-Witten Floer cohomology groups of M by the work of Lee-Taubes [LT].

**3.1. Interpolating between Reeb and stable Hamiltonian vector fields.** — Consider the contact form $\alpha = f_i dt + \beta_t$ on $S \times [0, 1]$, as defined in Section 2.1. We may assume that $R_\alpha$ is parallel to $\partial_t$ on $S \times [0, 1]$, after transforming $S \times [0, 1]$ by a smooth family of diffeomorphisms parametrized by $t \in [0, 1]$ and isotopic to the identity. Since

$$d\alpha = d_S f_t \wedge dt + d_S \beta_t + dt \wedge \frac{d\beta_t}{dt},$$

where $d_S$ is the exterior derivative in the S-direction, it follows that $\frac{d\beta_t}{dt} = d_S f_t$. (Hence $d_S \beta_t$ is an area form which does not depend on $t$.) Also, the form $\alpha$ is a contact form as long as $d_S \beta_t > 0$. Hence, for $C \gg 0$, the form $(C + f_t) dt + \beta_t$ is a contact form with Reeb vector field parallel to $\partial_t$.

Now consider the 1-parameter family of 1-forms

$$(3.1.1) \qquad \alpha_{\zeta} = Cdt + \zeta(f_i dt + \beta_t),$$

$\zeta \in [0, 1]$, on $N$. It interpolates between the contact form $\alpha_1 = Cdt + (f_i dt + \beta_t)$ and the stable Hamiltonian form $\alpha_0 = Cdt$. The Reeb vector fields $R_\zeta = R_{\alpha_\zeta}$ are directed by $\partial_t$ and hence are parallel for all $\zeta > 0$.

The pair $(\alpha_\zeta, \omega = d_S \beta_t)$ is a stable Hamiltonian structure on $N$, i.e., $\omega$ is closed, $\alpha_\zeta \wedge \omega > 0$, and $\ker \alpha_\zeta \supset \ker \omega$; see for example [CV]. When $\zeta = 0$, the Hamiltonian vector field $R_0$ equals $\frac{1}{C}\partial_t$ and hence is parallel to all the $R_\zeta$, $\zeta > 0$. Also let $\xi_\zeta$ be the 2-plane field on $N$ given by the kernel of $\alpha_\zeta$. The closed 2-form $\omega$ can either be viewed as an area form on S or as a (maximally nondegenerate) 2-form on N.

**3.2. Definitions.** — Consider the infinite cylinder $\mathbf{R} \times N$ with coordinates $(s, x)$. We will also use the notation $W' = \mathbf{R} \times N$.

**Definition 3.2.1.** — An almost complex structure $J_\zeta$ on $\mathbf{R} \times N$ is adapted to the stable Hamiltonian structure $(\alpha_\zeta, \omega)$ if $J_\zeta$ satisfies the following:

- $J_\zeta$ is s-invariant;

- $J_\zeta(\partial_s) = R_\zeta$ and $J_\zeta(\xi_\zeta) = \xi_\zeta$;
– $J_\zeta$ is tamed by the symplectic form $\Omega' = ds \wedge dt + \omega$.

Our goal is to replace the ECH chain complexes

$$ \mathrm{ECC}_j(N, \alpha_\zeta, J_\zeta), \quad \mathrm{ECC}_j^\flat(N, \alpha_\zeta, J_\zeta) $$

for $\zeta > 0$, by the analogously defined PFH chain complexes

$$ \mathrm{PFC}_j(N, \alpha_0, \omega, J_0), \quad \mathrm{PFC}_j^\flat(N, \alpha_0, \omega, J_0). $$

The orbit sets of the ECH chain groups are constructed using the Reeb vector fields $R_\zeta$ and the orbit sets of the PFH chain groups are constructed using the Hamiltonian vector field $R_0$.

We introduce some notation. Let $J_\zeta$, $\zeta \in [0, 1]$, be a smooth family of $(\alpha_\zeta, \omega)$-adapted almost complex structures, let $\gamma = \prod_i \gamma_i^{m_i}$, $\gamma' = \prod_i \gamma_i^{m'_i}$ be orbit sets of $\mathrm{PFC}_j(N, \alpha_0, \omega)$ or $\mathrm{ECC}_j(N, \alpha_\zeta)$, and let $Z \in \mathrm{H}_2(N, \gamma, \gamma')$.

A $J_\zeta$-holomorphic curve (or map) from $\gamma$ to $\gamma'$ is a holomorphic map

$$ u : (\dot{\mathbb{F}}, j) \to (\mathbf{R} \times \mathbb{N}, J_\zeta) $$

satisfying the following:

(1) $(\dot{\mathbb{F}}, j)$ is a closed Riemann surface with a finite number of punctures removed;

(2) the neighborhoods of the punctures are mapped asymptotically to cylinders over Reeb orbits;

(3) at the positive end of $\mathbf{R} \times \mathbb{N}$, $u$ is asymptotic to $\mathbf{R} \times \gamma_i$ with total multiplicity $m_i$ (more precisely, if we list the positive ends of $u$ that are asymptotic to some multiple cover of $\mathbf{R} \times \gamma_i$ and the covering degrees are $m_{i1}, \dots, m_{ij_i}$, then $m_i = m_{i1} + \dots + m_{ij_i}$); and

(4) at the negative end of $\mathbf{R} \times \mathbb{N}$, $u$ is asymptotic to $\mathbf{R} \times \gamma_i$ with total multiplicity $m'_i$.

We write:

– $M_{J_\zeta}(\gamma, \gamma')$ for the moduli space of $J_\zeta$-holomorphic curves from $\gamma$ to $\gamma'$;

– $M_{J_\zeta}(\gamma, \gamma', Z) \subset M_{J_\zeta}(\gamma, \gamma')$ for the subset of curves in the class $Z$;

– $M_{J_\zeta}^s(\gamma, \gamma') \subset M_{J_\zeta}(\gamma, \gamma')$ for the subset of simply-covered curves; and

– $M_{J_\zeta}^{ac}(\gamma, \gamma', Z) \subset M_{J_\zeta}(\gamma, \gamma', Z)$ for the subset of curves without connector components.

**Convention 3.2.2.** — The curves in all the moduli spaces are defined only up to automorphisms of the domain, the complex structures of their domains are not fixed, and the curves are allowed to be disconnected.
A connector over an orbit set $\delta = \prod_i \delta_i^{m_i}$ is a collection of branched covers of trivial cylinders where the branching is optional and the total covering degree over $\mathbf{R} \times \delta_i$ is $m_i$.

Given two orbit sets $\delta = \prod_i \delta_i^{m_i}$ and $\delta' = \prod_i \delta_i^{m'_i}$, we set $\delta/\delta' = \prod_i \delta_i^{m_i - m'_i}$ if $m'_i \le m_i$ for all $i$; otherwise we set $\delta/\delta' = 0$. Here some $m_i$ and $m'_i$ may be zero. We then write $u \in \mathcal{M}_{J_5}(\mathbf{\gamma}, \mathbf{\gamma}', \mathbf{Z})$ as $u^0 \cup u^1$, where $u^0$ is a connector over the orbit set $\mathbf{\gamma}_0$ and $u^1 \in \mathcal{M}_{J_5}^{nc}(\mathbf{\gamma}/\mathbf{\gamma}_0, \mathbf{\gamma}/\mathbf{\gamma}_0)$. By [HT1, Proposition 7.5], if $\Gamma_{\text{ECH}}(\mathbf{\gamma}, \mathbf{\gamma}', \mathbf{Z}) = 1$ (resp. 2), then $u^1 \in \mathcal{M}_{J_5}^s(\mathbf{\gamma}/\mathbf{\gamma}_0, \mathbf{\gamma}/\mathbf{\gamma}_0)$ and has Fredholm index 1 (resp. 2).

**3.3. The flux.** — For more details, see for example [CHL, Section 2]. Let

$$ \mathcal{h}_* : H_1(\mathbf{S}; \mathbf{Z}) \to H_1(\mathbf{S}; \mathbf{Z}) $$

be the map on homology induced by $\mathcal{h} : (\mathbf{S}, \omega) \to (\mathbf{S}, \omega)$ and let $K$ be the kernel of $\mathcal{h}_* - \text{id}$. Then the flux $F_{\mathcal{h}} : K \to \mathbf{R}$ is defined as follows: Let $[\zeta] \in K$. Since $[\zeta] = [\mathcal{h}(\zeta)] \in H_1(\mathbf{S}; \mathbf{Z})$, there is a 2-chain $\mathcal{S} \subset \mathbf{S}$ such that $\partial\mathcal{S} = \zeta - \mathcal{h}(\zeta)$. We then define:

$$ F_{\mathcal{h}}(\zeta) = \int_{\mathcal{S}} \omega. $$

Since the map $\mathcal{h}$ is the first return map of a Reeb vector field $R_\zeta$, it has zero flux (cf. [CHL, Lemma 2.2]). This implies that $\int_{[C]} \omega = 0$ for every $[C] \in H_2(\mathbf{N})$, where $\omega$ is now viewed as a closed 2-form on $\mathbf{N}$. Indeed, $[C]$ can be represented by a surface of the form $(\zeta \times [0, 1]) \cup (\mathcal{S} \times \{0\})$, where the relevant boundary components are glued. Hence, if $\mathbf{\gamma}$ and $\mathbf{\gamma}'$ are orbit sets of $\text{PFC}_j(\mathbf{N}, \alpha_0, \omega_0, J_0)$, then the $\omega$-area of any $Z \in H_2(\mathbf{N}, \mathbf{\gamma}, \mathbf{\gamma}')$ only depends on $\mathbf{\gamma}$ and $\mathbf{\gamma}'$.

**3.4. Compactness.** — The vanishing of the flux is an important ingredient in establishing that $\mathcal{M}_{J_0}(\mathbf{\gamma}, \mathbf{\gamma}')$ admits a compactification in the sense of [EGH, BEHWZ].

We briefly outline the argument from [Hu1, Section 9]: (1) There is a bound on the $\omega$-area for all elements of $\mathcal{M}_{J_0}(\mathbf{\gamma}, \mathbf{\gamma}')$. This was done above. (2) Given a sequence of holomorphic curves $u_i \in \mathcal{M}_{J_0}(\mathbf{\gamma}, \mathbf{\gamma}')$, $i \in \mathbf{N}$, there is a subsequence which converges weakly as currents to a holomorphic building $u_\infty$. This is due to the Gromov-Taubes compactness theorem [T3], which works in dimension four and does not require any a priori bound on the genus of $u_i$. The extraction of the holomorphic building is treated in some detail in [Hu1, Lemma 9.8]. Hence we may assume that the homology classes $[u_i] \in H_2(\mathbf{N}, \mathbf{\gamma}, \mathbf{\gamma}')$ are fixed. (3) There is a bound on the genus of the curve, provided the homology classes $[u_i]$ are fixed. This follows from the adjunction inequality and will be discussed below. (4) Once there is a genus bound, apply the SFT compactness theorem of [BEHWZ].

We now explain in some detail how to obtain genus bounds from bounds on the homology classes $[u_i]$, especially since similar arguments will appear in later sections. But first let us introduce some notation.
Let $(F,j)$ be a closed Riemann surface and $\mathbf{p}^+$ and $\mathbf{p}^-$ be disjoint finite sets of punctures of $F$. Then let

$$u: \dot{F} = F - \mathbf{p}^{+} - \mathbf{p}^{-} \to \mathbf{R} \times \mathbb{N},$$

be a $(j, J_0)$-holomorphic map from $\mathbf{\gamma} = \prod_i \gamma_i^{m_i}$ to $\mathbf{\gamma}' = \prod_i \gamma_i^{n_i}$. Here the punctures of $\mathbf{p}^{\pm}$ are asymptotic to the $\pm$ ends of $u$. The positive ends of $u$ partition $m_i$ into $(m_{i1}, m_{i2}, \dots)$ and the negative ends of $u$ partition $n_i$ into $(n_{i1}, n_{i2}, \dots)$. (We ignore the partition terms that are zero.) Pick a trivialization $\tau$ of TS in a neighborhood of all the $\gamma_i$, and let $\mu_{\tau}(\gamma_i^k)$ be the usual Conley-Zehnder index of the $k$-fold cover of $\gamma_i$ with respect to $\tau$. Then we define the *total Conley-Zehnder indices* at the positive and negative ends of $u$ as follows:

$$\mu_{\tau}^{+}(u) = \sum_{i} \sum_{r} \mu_{\tau}(\gamma_{i}^{m_{ir}}),$$

$$\mu_{\tau}^{-}(u) = \sum_{i} \sum_{r} \mu_{\tau}(\gamma_{i}^{n_{ir}}),$$

and also write $\mu_{\tau}(u) = \mu_{\tau}^{+}(u) - \mu_{\tau}^{-}(u)$. The *symmetric Conley-Zehnder index* of Hutchings [Hu1], so called because of its motivation from studying the “symplectomorphism” $\mathrm{Sym}^k(\mathcal{H})$ of $\mathrm{Sym}^k(S)$ induced by $\mathcal{H}$,<fnref n="2" /> is defined as:

$$\tilde{\mu}_{\tau}(\mathbf{\gamma}) = \sum_{i} \sum_{r=1}^{m_i} \mu_{\tau}(\gamma_i^r),$$

and does not depend on the choice of $u$ from $\mathbf{\gamma}$ to $\mathbf{\gamma}'$. We write $\tilde{\mu}_{\tau}(u) = \tilde{\mu}_{\tau}(\mathbf{\gamma}) - \tilde{\mu}_{\tau}(\mathbf{\gamma}')$. We also recall the *writhe*

$$w_{\tau}(u) = w_{\tau}^{+}(u) - w_{\tau}^{-}(u),$$

where $w_{\tau}^{+}(u)$ is the total writhe of braids $u(\dot{F}) \cap (\{s\} \times \mathbb{N})$, $s \gg 0$, viewed in the union of solid torus neighborhoods of $\gamma_i$ and computed with respect to the framing $\tau$; and $w_{\tau}^{-}(u)$ is defined similarly.

The key ingredient in establishing genus bounds is the relative adjunction formula from [Hu1, Equation (18)] for simple curves $u$ with a finite number of singularities and no connector components:

$$(3.4.1) \qquad c_1(u^*\mathrm{TS}, \tau) = \chi(\dot{F}) + w_{\tau}(u) + Q_{\tau}(u) - 2\delta(u),$$

where $Q_{\tau}(u)$ is the relative intersection pairing with respect to $\tau$ and $\delta(u)$ is a nonnegative integer which is a count of the singularities. In particular, $\delta(u) = 0$ if and only if $u$ is an

<footnote n="2">² $\mathrm{Sym}^k(\mathcal{H})$ is a symplectomorphism away from the (multi)-diagonal.</footnote>
embedding (see [M, MW]). Together with the writhe bounds

$$
\begin{equation} \tag{3.4.2}
\begin{aligned}
w_{\tau}^{+}(u) &\le \tilde{\mu}_{\tau}(\boldsymbol{\gamma}) - \mu_{\tau}^{+}(u), \\
w_{\tau}^{-}(u) &\ge \tilde{\mu}_{\tau}(\boldsymbol{\gamma}') - \mu_{\tau}^{-}(u),
\end{aligned}
\end{equation}
$$

from [Hu2, Lemma 4.20], we obtain:

$$
(3.4.3) \qquad \chi(\dot{\mathbf{F}}) \ge c_1(u^*\mathrm{TS}, \tau) + \mu_{\tau}(u) - \tilde{\mu}_{\tau}(u) - Q_{\tau}(u) + 2\delta(u).
$$

(See [Hu1, Theorem 10.1].) Since all of the terms on the right-hand side are either ho-
mological quantities or depend on the data near **y** and **y'**, we have a lower bound on
**χ**(**F**), which implies an upper bound on the genus of **F**.

**3.5. Transversality.** — Let $\mathcal{J}_{W',\zeta}$ be the space of almost complex structures $J_\zeta$ on $W' = \mathbf{R} \times \mathbf{N}$ in the class $C^\infty$ which are adapted to $(\alpha_\zeta, \omega)$.

*Definition 3.5.1.* — An almost complex structure $J_\zeta \in \mathcal{J}_{W',\zeta}$ is *j*-regular if, for all orbit sets $\mathbf{y}$ and $\mathbf{y}'$ of $\mathbf{R}_\zeta$ which intersect $S \times \{0\}$ at most $j$ times, the moduli space $\mathcal{M}_{J_\zeta}^s(\mathbf{y}, \mathbf{y}')$ is transversely cut out.

Let $\mathcal{J}_{W',\zeta}^{reg,j} \subset \mathcal{J}_{W',\zeta}$ be the subset of $j$-regular $J_\zeta$. In order to simplify the notation, from now on we will write $\mathcal{J}_{W'}$ for $\mathcal{J}_{W',0}$ and $\mathcal{J}_{W'}^{reg,j}$ for $\mathcal{J}_{W',0}^{reg,j}$. The following lemma states that $\mathcal{J}_{W'}^{reg,j} \subset \mathcal{J}_{W'}$ is dense.

**Lemma 3.5.2 (Transversality).** — A generic $J_0 \in \mathcal{J}_{W'}$ is $j$-regular.

*Proof.* — This follows from [Hu1, Lemma 9.12(b)], which states that a generic $J_0 \in \mathcal{J}_{W'}$ is regular away from holomorphic curves which have a fiber $\{(s, t)\} \times S$ as an irreducible component. (Observe that the fibers are holomorphic for any $J_0 \in \mathcal{J}_{W'}$.) In our case, the fibers are not closed and cannot occur as irreducible components of curves in $\mathcal{M}_{J_0}^s(\mathbf{y}, \mathbf{y}')$. $\square$

**3.6. The equivalence of certain ECH and PFH groups.** — In this section we prove the following theorem:

**Theorem 3.6.1.** — Given $j > 0$, there exist $J_0 \in \mathcal{J}_{W'}^{reg,j}$ and $\zeta_0 = \zeta_0(j, J_0) > 0$ such that there are isomorphisms of chain complexes

$$
\mathrm{PFC}_j(\mathbf{N}, \alpha_0, \omega, \mathbf{J}_0) \simeq \mathrm{ECC}_j(\mathbf{N}, \alpha_\zeta, \mathbf{J}_\zeta),
$$

$$
\mathrm{PFC}_j^{\mathrm{b}}(\mathbf{N}, \alpha_0, \omega, \mathbf{J}_0) \simeq \mathrm{ECC}_j^{\mathrm{b}}(\mathbf{N}, \alpha_\zeta, \mathbf{J}_\zeta),
$$

for all $0 < \zeta \le \zeta_0$. Here $J_\zeta \in \mathcal{J}_{W',\zeta}^{reg,j}$ is sufficiently close to $J_0$. Similar isomorphisms hold with twisted coefficients.
*Proof.* — We will prove the first equivalence, leaving the second to the reader.

Since there is a one-to-one correspondence between the generators of the chain groups PFC<sub>j</sub>(N, α<sub>0</sub>, ω) and ECC<sub>j</sub>(N, α<sub>ζ</sub>), we have

$$PFC_j(N, \alpha_0, \omega) \cong ECC_j(N, \alpha_\zeta)$$

as **F**-vector spaces, but not necessarily as chain complexes. In other words, we may view any orbit set **γ** for R<sub>ζ</sub> as an orbit set of any other R<sub>ζ'</sub>.

Let J<sub>0</sub> be an almost complex structure in J<sub>W'</sub><sup>reg,j</sup>. By Lemma 3.5.2, J<sub>W'</sub><sup>reg,j</sup> is a dense subset of J<sub>0</sub>, and in particular is nonempty. The moduli spaces M<sub>J<sub>0</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z) of ECH index 1 and 2 are transversely cut out since they are simple by the ECH index inequality. Now let J<sub>ζ</sub>, ζ ∈ [0, 1], be a smooth family of (α<sub>ζ</sub>, ω)-adapted almost complex structures which extend J<sub>0</sub>.

We claim that the ECH index 1 moduli spaces M<sub>J<sub>ζ</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z) are transversely cut out and diffeomorphic to M<sub>J<sub>0</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z), when ζ > 0 is sufficiently small. Indeed, if u<sub>ζ</sub> ∈ M<sub>J<sub>ζ</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z) is sufficiently close to M<sub>J<sub>0</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z), then the moduli space is regular at u<sub>ζ</sub>. Hence it suffices to prove that, if ζ > 0 is sufficiently small, then every u<sub>ζ</sub> ∈ M<sub>J<sub>ζ</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z) is sufficiently close to M<sub>J<sub>0</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z). Indeed, this follows from the compactness argument from Section 3.4. Let u<sub>i</sub> ∈ M<sub>J<sub>ζ<sub>i</sub></sub></sub><sup>nc</sup>(**γ**, **γ'**, Z) be a sequence of ECH index 1 holomorphic curves with ζ<sub>i</sub> → 0. By the compactness theorem and incoming/outgoing partition considerations, u<sub>i</sub> converges to u ∈ M<sub>J<sub>0</sub></sub>(**γ**, **γ'**, Z) with I<sub>ECH</sub>(u) = 1, after possibly taking a subsequence. In particular, the limit u is not a holomorphic building with multiple levels. If u has connector components u<sup>0</sup> over **γ<sub>0</sub>**, then u<sub>i</sub> must also have connector components u<sub>i</sub><sup>0</sup> over **γ<sub>0</sub>**, a contradiction. Hence u ∈ M<sub>J<sub>0</sub></sub><sup>nc</sup>(**γ**, **γ'**, Z), which proves the claim.

Since the chain groups are isomorphic as vector spaces and the differentials agree for sufficiently small ζ > 0, the theorem follows. □

**3.7. Remarks about Morse-Bott theory.** — There is an equivalent approach to the definition of the groups PFC<sub>j</sub>(N, α<sub>0</sub>, ω) which does not use the Morse-Bott language. A similar construction is possible also for the groups PFC<sub>j</sub><sup>β</sup>(N, α<sub>0</sub>, ω), but will be left to the reader.

We add a small collar [0, η] × ∂S to S and extend the monodromy such that ℏ restricts to (y, θ) ↦→ (y, θ - y) on [0, η] × ∂S. (See Lemma 2.1.2). We denote by (S<sup>+</sup>, ℏ<sup>+</sup>) the extensions. The mapping torus of (S<sup>+</sup>, ℏ<sup>+</sup>) is a manifold N<sup>+</sup> such that N ⊂ N<sup>+</sup>. The Hamiltonian structure (α<sub>0</sub>, ω) can be extended in an obvious way from N to N<sup>+</sup>, and the extension will still be denoted by (α<sub>0</sub>, ω). The groups PFC<sub>j</sub>(N<sup>+</sup>, α<sub>0</sub>, ω) are defined because ∂N<sup>+</sup> is foliated by orbits of the Hamiltonian vector field. If η is small enough, all Hamiltonian orbits in N<sup>+</sup> - N intersect a fiber more than j times, and therefore PFC<sub>j</sub>(N<sup>+</sup>, α<sub>0</sub>, ω) ≅ PFC<sub>j</sub>(N, α<sub>0</sub>, ω) and PFC<sub>j</sub><sup>β</sup>(N<sup>+</sup>, α<sub>0</sub>, ω) ≅ PFC<sub>j</sub><sup>β</sup>(N, α<sub>0</sub>, ω).

*Remark 3.7.1.* — In the above construction, η depends on j.
Given a function $f: N^+ \to \mathbf{R}$ and a real number $\lambda > 0$, we define a new stable Hamiltonian structure

$$ \alpha'_{0} = \alpha_{0}, \quad \omega' = \omega + \lambda df \wedge \alpha_{0}. $$

Note that $(\alpha'_0, \omega')$ is stable Hamiltonian since $d\alpha_0 = 0$. If $R_0$ is the Hamiltonian vector field of $(\alpha_0, \omega)$ and $R'_0$ of $(\alpha'_0, \omega')$, then $R'_0 = R_0 + \lambda X$, where $X$ is a vector field satisfying the equations

$$ \begin{cases} \alpha_0(X) = 0, \\ \iota_X \omega = df - df(R_0)\alpha_0. \end{cases} $$

We construct a function $f$ on $N^+$ as follows: We fix a Morse function on $\partial S$ with a unique minimum and a unique maximum, pull it back to $\partial N$, and extend it to a nonpositive function $f$ supported on a small neighborhood of $\partial N$ using a bump function centered at $\partial N$. We choose $f$ to be supported away from the (finitely many and nondegenerate) orbits of $R_0$ in $\text{int}(N)$ with $\mathcal{F} \le j$, so that those orbits will also be orbits of $R'_0$.

On $\partial N$, the vector field $R'_0$ has a pair of nondegenerate orbits $e$ and $h$, where $e$ corresponds to the minimum of the Morse function on $\partial S$, and $h$ to the maximum. We will choose $\lambda$ small enough so that the perturbation creates no new orbits with $\mathcal{F} \le j$. Since $\partial N^+$ is foliated by the flow of $R'_0$, we can define the periodic Floer homology group $\text{PFC}_j(N^+, \alpha'_0, \omega')$ and, as explained in [0, Section 4], there is an isomorphism $\text{PFC}_j(N^+, \alpha_0, \omega) \cong \text{PFC}_{2g}(N^+, \alpha'_0, \omega')$. ([0, Section 4] is mostly concerned with contact structures, but there is no problem in applying those arguments here.) The isomorphism holds already on the level of complexes because there is no direct limit involved in its definition. This is a consequence of the topological constraint $\mathcal{F} \le j$ on the Hamiltonian orbits, which implies that only finitely many of them enter the definition of the groups $\text{PFH}_j$.

## 4. A variation of HF(M) adapted to open book decompositions

In this section we recall the cylindrical reformulation of Heegaard Floer homology. This reformulation was suggested by Eliashberg and worked out in detail by Lipshitz [Li]. The discussion is slightly different from that of [Li] in that we introduce an ECH-type index $I_{\text{HF}}$ and define the Heegaard Floer groups in terms of $I_{\text{HF}}$. We then use the work of [HKM] to restrict the Heegaard Floer data to the page S of an open book decomposition.

**4.1. Heegaard data.** — A pointed Heegaard diagram is a quadruple $(\Sigma, \alpha, \beta, z)$ which consists of the following:

- a closed oriented surface $\Sigma$ of genus $k$;
- two collections $\alpha = \{\alpha_1, \dots, \alpha_k\}$ and $\beta = \{\beta_1, \dots, \beta_k\}$ of $k$ pairwise disjoint simple closed curves in $\Sigma$; and

- a point $z \in \Sigma - \alpha - \beta$;

where each of $\alpha$ and $\beta$ forms a basis of $H_1(\Sigma; \mathbb{Z})$ and $\alpha$ and $\beta$ intersect transversely in $\Sigma$.

If $M$ is a closed oriented 3-manifold, then $(\Sigma, \alpha, \beta, z)$ is a pointed Heegaard diagram for $M$ if $\Sigma$ decomposes $M$ into two handlebodies, i.e., $M = H_\alpha \cup_\Sigma H_\beta$ and $\Sigma = \partial H_\alpha = -\partial H_\beta$, where the $\alpha$-curves (resp. $\beta$-curves) bound compression disks in the handlebody $H_\alpha$ (resp. $H_\beta$).

Let $\omega$ be an area form on $\Sigma$. We consider $[0, 1] \times \Sigma$ with the stable Hamiltonian structure $(dt, \omega)$, where $t$ is the $[0, 1]$-coordinate. The Hamiltonian vector field is $\partial_t$, and the 2-plane field $\ker dt$ will be written as $T\Sigma$, at the risk of some confusion. (If $Y$ is a topological space and $f: Y \to \Sigma$ is a continuous map, then we write $T\Sigma_Y$ or $T\Sigma$ for the pullback bundle $f^*T\Sigma$.)

Let $S_{\alpha, \beta}$ be the set of $k$-tuples of chords $\{[0, 1] \times \{y_1\}, \dots, [0, 1] \times \{y_k\}\}$ in $[0, 1] \times \Sigma$, where there exists a permutation $\sigma \in \mathfrak{S}_k$ for which $y_i \in \alpha_i \cap \beta_{\sigma(i)}$, $i = 1, \dots, k$. Here the chords $[0, 1] \times \{y_i\}$ are orbits of the Hamiltonian vector field $\partial_t$ which connect from $\{0\} \times \beta$ to $\{1\} \times \alpha$.

*Terminology.* We will often write elements of $S_{\alpha, \beta}$ as $\mathbf{y} = \{y_1, \dots, y_k\}$ and refer to $\mathbf{y}$ as a $k$-tuple of intersection points. Also, if $l \le k$, then an $l$-tuple of chords/intersection points $\mathbf{y} = \{y_1, \dots, y_l\}$ is a collection of points in $\alpha \cap \beta$ where each $\alpha_i$ is used at most once and each $\beta_i$ is used at most once.

**4.2. Almost complex structures.** — Consider the natural projection $\pi_B : X \to B$, where $X = \mathbf{R} \times [0, 1] \times \Sigma$ and $B = \mathbf{R} \times [0, 1]$. We also write $\pi_{\mathbf{R}}$, $\pi_{[0,1]\times\Sigma}$ and $\pi_{\Sigma}$ for the natural projections of $X$ onto $\mathbf{R}$, $[0, 1] \times \Sigma$ and $\Sigma$. Let $(s, t)$ be the coordinates on the base $B = \mathbf{R} \times [0, 1]$. We then define the symplectic form

$$ \Omega_X = ds \wedge dt + \omega $$

on $X$. The submanifolds $L_\alpha = \mathbf{R} \times \{1\} \times \alpha$ and $L_\beta = \mathbf{R} \times \{0\} \times \beta$ are Lagrangian submanifolds of the symplectic manifold $(X, \Omega_X)$. We denote by $L_{\alpha_i}$ or $L_{\beta_i}$ their connected components.

**Definition 4.2.1 (Ω_X-admissibility).** — An almost complex structure $J$ on $X$ is $\Omega_X$-admissible (or simply admissible) if it satisfies the following:

(1) $J$ is $s$-invariant;

(2) $J(\partial_s) = \partial_t$ and $J(T\Sigma) = T\Sigma$;

(3) $J$ is tamed by the symplectic form $\Omega_X$;

(4) there is a point $z_i$ in each component of $\Sigma - \alpha - \beta$ such that $J$ is a product complex structure $j_B \times j_\Sigma$ in a small neighborhood of $\mathbf{R} \times [0, 1] \times \{z_i\}$ in $X$; some $z_i$ coincides with the basepoint $z$.
**Remark 4.2.2.** — The fibers ${\{(s, t)\} \times \Sigma}$ of an admissible $J$ on $X$ are holomorphic and the projection $\pi_B$ is $(J, j_B)$-holomorphic, where $j_B$ is the standard complex structure on $B = \mathbf{R} \times [0, 1]$.

We write $\mathcal{J}_X$ for the space of $\mathbb{C}^\infty$-smooth $\Omega_X$-admissible almost complex structures $J$ on $X$.

**4.3. Holomorphic curves and moduli spaces.** — Let $(F, j)$ be a compact Riemann surface, possibly disconnected, with two sets of punctures $\mathbf{q}^+ = \{q_1^+, \dots, q_k^+\}$ and $\mathbf{q}^- = \{q_1^-, \dots, q_k^-\}$ on $\partial F$, such that (i) each component of $F$ has nonempty boundary, (ii) on each boundary component there is at least one puncture from each of $\mathbf{q}^+$ and $\mathbf{q}^-$, and (iii) the punctures on $\mathbf{q}^+$ and $\mathbf{q}^-$ alternate around each boundary component. We write $\dot{F} = F - \mathbf{q}^+ - \mathbf{q}^-$ and $\partial \dot{F} = \partial F - \mathbf{q}^+ - \mathbf{q}^-$.

**Definition 4.3.1.** — Let $J \in \mathcal{J}_X$. A degree $l \le k$ multisection of $(X, J)$ is a holomorphic map

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

which is a degree $l$ multisection of $\pi_B : X \to B = \mathbf{R} \times [0, 1]$ and which additionally satisfies the following:

(1) $u(\partial \dot{F}) \subset L_\alpha \cup L_\beta$;

(2) for each $i \in \{1, \dots, k\}$, $u^{-1}(L_{\alpha_i})$ (resp. $u^{-1}(L_{\beta_i})$) consists of at most one component of $\partial \dot{F}$, which we call $\alpha_i^*$ (resp. $\beta_i^*$);

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

(4) the energy of $u$ (see Definition 4.3.2 below) is finite.

A Heegaard Floer curve (or HF curve) is a degree $k$ multisection of $(X, J)$.

By the compactness theorem of [BEHWZ] (adapted to the Lagrangian case), a holomorphic curve $u$ satisfying (1), (2) and (4) converges to cylinders over Reeb chords as $s \to \pm\infty$. By the work of Abbas [Abb1], an HF curve $u$ converges exponentially to cylinders over Reeb chords near the ends. Components of $u$ may map to $\mathbf{R} \times [0, 1] \times \{y_i\}$; such components will be called *trivial strips*.

**Definition 4.3.2.** — The energy of $u$ is the quantity

$$E(u) = \int_{\dot{F}} u^* \omega + \sup_{\phi \in C} \int_{\dot{F}} u^* d(\phi(s)) dt,$$

where $C$ is the set of nondecreasing smooth functions $\phi : \mathbf{R} \to [0, 1]$.
We now define some moduli spaces of HF curves with respect to $J \in \mathcal{J}_X$. Let $\mathbf{y} = \{y_1, \dots, y_k\}$ and $\mathbf{y}' = \{y'_1, \dots, y'_k\}$ be $k$-tuples of $\alpha \cap \beta$. Let $M_j^X(\mathbf{y}, \mathbf{y}')$ be the moduli space of HF curves $u$ which are asymptotic to $\mathbf{R} \times [0, 1] \times \{y_i\}$ near $q_i^+$ and to $\mathbf{R} \times [0, 1] \times \{y'_i\}$ near $q_i^-$.<fnref n="3" /> Such a curve $u$ is said to be an *HF curve from $\mathbf{y}$ to $\mathbf{y}'$. Also let $\tilde{M}_j^X(\mathbf{y}, \mathbf{y}') \subset M_j^X(\mathbf{y}, \mathbf{y}')$ be the subset consisting of curves $u$ which additionally satisfy $\pi_\Sigma \circ u(\tilde{F}) \cap \{z\} = \emptyset$.

Let $\tilde{X} = [-1, 1] \times [0, 1] \times \Sigma$ be the compactification of $X = \mathbf{R} \times [0, 1] \times \Sigma$, obtained by attaching $[0, 1] \times \Sigma$ at the positive and negative ends, and let $\tilde{L}_\alpha = [-1, 1] \times \{1\} \times \alpha$ and $\tilde{L}_\beta = [-1, 1] \times \{0\} \times \beta$ be the compactifications of $L_\alpha$ and $L_\beta$. We then define $Z_{\mathbf{y}, \mathbf{y}'} \subset \tilde{X}$ as the subset

$$ (4.3.2) \qquad Z_{\mathbf{y}, \mathbf{y}'} = \tilde{L}_{\alpha} \cup \tilde{L}_{\beta} \cup (\{1\} \times [0, 1] \times \mathbf{y}) \cup (\{-1\} \times [0, 1] \times \mathbf{y}'). $$

Similarly define

$$ (4.3.3) \qquad Z_{\alpha, \beta} = \tilde{L}_{\alpha} \cup \tilde{L}_{\beta} \cup (\{-1, 1\} \times [0, 1] \times (\alpha \cap \beta)). $$

The exponential decay of HF curves in $\mathbf{R} \times [0, 1] \times \Sigma$ implies that an HF curve $u: \tilde{F} \to X$ from $\mathbf{y}$ to $\mathbf{y}'$ can be compactified to a continuous map

$$ \tilde{u}: (\tilde{F}, \partial\tilde{F}) \to (\tilde{X}, Z_{\mathbf{y}, \mathbf{y}'}). $$

Here $\tilde{F}$ is obtained from $\tilde{F}$ by performing a real blow-up at its boundary punctures.

We define $H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ as the set of *homology* classes of continuous maps $u: \tilde{F} \to X$ which satisfy (1), (2) and (3) of Definition 4.3.1 and are positively asymptotic to $[0, 1] \times \mathbf{y}$ and negatively asymptotic to $[0, 1] \times \mathbf{y}'$; here two maps $u_1$ and $u_2$ are equivalent in $H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ if their compactifications $\tilde{u}_1$ and $\tilde{u}_2$ are homologous in $H_2(\tilde{X}, Z_{\mathbf{y}, \mathbf{y}'})$. To any HF curve from $\mathbf{y}$ to $\mathbf{y}'$ we can then associate a class in $H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$. If we consider moduli spaces of HF curves $u$ in the homology class $A \in H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$, we will write $M_j^X(\mathbf{y}, \mathbf{y}', A)$ or $\tilde{M}_j^X(\mathbf{y}, \mathbf{y}', A)$.

**Remark 4.3.3.** — The sets $H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ are denoted $\pi_2(\mathbf{y}, \mathbf{y}')$ in the Heegaard Floer literature. We opted for this change of notation in order to be consistent with the notation we introduced for ECH and for the maps to the open-closed cobordisms introduced in Section 5.

**4.4. The Fredholm index.** — In this subsection and the next, we fix $J \in \mathcal{J}_X$. Now we discuss the Fredholm index of an HF curve $u: \tilde{F} \to X$, which is the expected dimension of a neighborhood $U$ of $u \in M_j^X(\mathbf{y}, \mathbf{y}')$, modulo reparametrizations of the domain. The Fredholm index of $u$ will be denoted by $\mathrm{ind}(u) = \mathrm{ind}_{\mathrm{HF}}(u)$.

<footnote n="3">³ In [Li], W has the opposite orientation, $\mathbf{y}$ is at $-\infty$, and $\mathbf{y}'$ is at $+\infty$. The moduli spaces, however, are diffeomorphic.</footnote>
**4.4.1. The Fredholm index, first version.** — We start with Lipshitz’s formula [Li, Equation 5] for the Fredholm index of $u$:

$$ (4.4.1) \qquad \mathrm{ind}(u) = -\chi(\mathbf{F}) + k + \sum_{i=1}^{k} \mu(\alpha_i^*) - \sum_{i=1}^{k} \mu(\beta_i^*), $$

where $k$ is the genus of $\Sigma$, $\chi(\mathbf{F}) = \chi(\dot{\mathbf{F}})$ is the Euler characteristic of $\mathbf{F}$ or $\dot{\mathbf{F}}$, and $\alpha_i^*$ and $\beta_i^*$ are as in Definition 4.3.1.

We now define the Maslov indices $\mu(\alpha_i^*)$ and $\mu(\beta_i^*)$ which appear in Equation (4.4.1): Choose a trivialization $\tau_0'$ of $T\Sigma \simeq \mathbf{C}$ on a neighborhood of the points $w \in \alpha \cap \beta$ so that $\mathbf{R}$ corresponds to $T_w\beta$ and $i\mathbf{R}$ corresponds to $T_w\alpha$. Then let $\tau_0$ be a trivialization of $u^*T\Sigma_X$ which coincides with the one already given near $\mathbf{p}$ and $\mathbf{q}$ by pulling back $\tau_0'$. Along each component of $\partial\dot{\mathbf{F}}$ — called $\alpha_i^*$ or $\beta_i^*$, depending on whether it is mapped to $\alpha_i$ or to $\beta_i$, and oriented in the same way as $\partial\mathbf{F}$ for $\alpha_i$ and in the opposite way for $\beta_i$ — we have a loop of unoriented real lines in $\mathbf{C}$, given by the pullback of $T\alpha_i$ or $T\beta_i$. The Maslov index $\mu(\alpha_i^*)$ (resp. $\mu(\beta_i^*)$) is the degree of the loop along $\alpha_i^*$ (resp. $\beta_i^*$) with respect to $\tau_0$.

**4.4.2. The Fredholm index, second version.** — For the purposes of computing indices, we replace $X = \mathbf{R} \times [0, 1] \times \Sigma$ by the compactification $\tilde{X} = [-1, 1] \times [0, 1] \times \Sigma$ from Section 4.3.

Recall $Z_{\alpha, \beta} \subset \tilde{X}$ given by Equation (4.3.3). We define a trivialization $\tau$ of $T\Sigma_{\tilde{X}}$ along $Z_{\alpha, \beta} \subset \tilde{X}$ as follows: First choose nonzero sections of $T\alpha \subset T\Sigma|_{\alpha}$ and $T\beta \subset T\Sigma|_{\beta}$. The nonzero sections induce a trivialization $\tau'$ of $T\Sigma_{[0,1]\times\Sigma}$ on $(\{0\} \times \beta) \cup (\{1\} \times \alpha)$. We then extend $\tau'$ arbitrarily to $[0, 1] \times (\alpha \cap \beta)$. Finally we pull $\tau'$ back to $Z_{\alpha, \beta} \subset \tilde{X}$ using the projection $\pi_{[0,1]\times\Sigma}: \tilde{X} \to [0, 1] \times \Sigma$ to obtain $\tau$.

Given an HF curve $u: \dot{\mathbf{F}} \to X$, we define its Maslov index $\mu_{\tau}(u)$ as follows: Let

$$ \check{u}: (\check{\mathbf{F}}, \partial\check{\mathbf{F}}) \to (\check{X}, Z_{\alpha, \beta}) $$

be the compactification of $u$. We then construct a (not necessarily oriented) real rank one subbundle $\mathcal{L}$ of $\check{u}^*T\Sigma$ on $\partial\check{\mathbf{F}}$. The bundle $\mathcal{L}$ is given by $\check{u}^*T\alpha$ and $\check{u}^*T\beta$ along $\partial\check{\mathbf{F}}$. We extend $\mathcal{L}$ to $\partial\check{\mathbf{F}} - \partial\dot{\mathbf{F}}$ by rotating in the counterclockwise direction from $\check{u}^*T\beta$ to $\check{u}^*T\alpha$ by the minimum amount possible. (Assuming orthogonal intersections, this is a $\pi/2$-rotation.) Then $\mu_{\tau}(u)$ is the sum of the Maslov indices of $\mathcal{L}$ with respect to the trivialization $\tau$, where the sum is over all the connected components of $\partial\check{\mathbf{F}}$.

**Lemma 4.4.1.** — If $u: \dot{\mathbf{F}} \to X$ is an HF curve, then

$$ (4.4.2) \qquad \mu_{\tau}(u) + 2c_1(u^*T\Sigma, \tau) = \sum_{i=1}^{k} \mu(\alpha_i^*) - \sum_{i=1}^{k} \mu(\beta_i^*). $$
*Proof.* — By standard Maslov index theory, we have

$$ \mu_{\tau}(u) + 2c_{1}(u^{*}\mathrm{T}\Sigma, \tau) = \mu_{\tau_{0}}(u) + 2c_{1}(u^{*}\mathrm{T}\Sigma, \tau_{0}), $$

where $\tau_0$ denotes the trivialization of $u^*\mathrm{T}\Sigma$ from Section 4.4.1. We immediately obtain $c_1(u^*\mathrm{T}\Sigma, \tau_0) = 0$ since $\tau_0$ is a trivialization on all of $\mathbb{F}$. Hence it suffices to prove that:

$$ (4.4.3) \qquad \mu_{\tau_0}(u) = \sum_{i=1}^{k} \mu(\alpha_i^*) - \sum_{i=1}^{k} \mu(\beta_1^*). $$

The difference between the two sides of Equation (4.4.3) is the total amount of rotation of the real lines introduced at $\alpha \cap \beta$ in the definition of $\mu_{\tau_0}$: if we go from $\beta$ to $\alpha$ we rotate by $\frac{\pi}{2}$, while if we go from $\alpha$ to $\beta$ we rotate by $-\frac{\pi}{2}$; hence the total amount of rotation is 0. $\square$

We can now rephrase the Fredholm index as follows:

$$ (4.4.4) \qquad \mathrm{ind}(u) = -\chi(\mathbb{F}) + k + \mu_{\tau}(u) + 2c_{1}(u^{*}\mathrm{T}\Sigma, \tau). $$

*Remark 4.4.2.* — Since the Maslov index of $\mathcal{L}$ with respect to $\tau$ is an integer along each chord $[0, 1] \times \{y_i\}$, it makes sense to write $\mu_{\tau}(y_i) \in \mathbb{Z}$. If we let

$$ \mu_{\tau}(\mathbf{y}) = \sum_{i=1}^{k} \mu_{\tau}(y_i), $$

then $\mu_{\tau}(u) = \mu_{\tau}(\mathbf{y}) - \mu_{\tau}(\mathbf{y}')$. In particular, $\mu_{\tau}(u)$ only depends on $\mathbf{y}, \mathbf{y}'$, and the choice of $\tau$.

**4.5. The ECH-type index.** — In this subsection we define an ECH-type index $I_{\mathrm{HF}}(u)$ and prove an index inequality which is analogous to the ECH index inequality of [Hu1].

**4.5.1. The relative intersection form.** — Let $\tau$ be a trivialization of $\mathrm{T}\Sigma_{\tilde{X}}$ along $Z_{\alpha, \beta}$ as defined in Section 4.4.2. We will define the notions of a *representative* and a $\tau$-trivial representative of a homology class $A \in \mathrm{H}_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$, where $\mathbf{y} = \{y_1, \dots, y_k\}$ and $\mathbf{y}' = \{y'_1, \dots, y'_k\}$ are $k$-tuples in $S_{\alpha, \beta}$.

*Definition 4.5.1.* — An oriented immersed compact surface

$$ \check{C} \subset \check{X} = [-1, 1] \times [0, 1] \times \Sigma $$

in the homology class $A \in \mathrm{H}_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ is an immersed representative of A if:

(1) $\check{C}$ is positively transverse to the fibers $\{(s, t)\} \times \Sigma$ along all of $\partial\check{C}$.
(2) $(\check{\pi}_{[0,1]\times\Sigma})|_{\check{\mathcal{C}}}$ is an embedding near $\partial\check{\mathcal{C}} \cap (\{-1, 1\} \times [0, 1] \times \Sigma)$, where $\check{\pi}_{[0,1]\times\Sigma}$ is the projection $\check{X} \to [0, 1] \times \Sigma$.

If $\check{\mathcal{C}}$ is embedded in addition, then $\check{\mathcal{C}}$ is a representative of A.

**Definition 4.5.2 (τ-trivial representative).** — A representative $\check{\mathcal{C}}$ of A is τ-trivial if, for all sufficiently small $\varepsilon > 0$, $\check{\mathcal{C}} \cap \{s = \pm(1 - \varepsilon)\}$ is the union of single-stranded braids $\zeta_i^{\pm}$, $i = 1, \dots, k$, where $\zeta_i^{+}$ (resp. $\zeta_i^{-}$) lies in a tubular neighborhood of $[0, 1] \times \{y_i\}$ (resp. $[0, 1] \times \{y'_i\}$), is disjoint from $[0, 1] \times \{y_i\}$ (resp. $[0, 1] \times \{y'_i\}$), and induces a framing which agrees with $\tau$ along $[0, 1] \times \{y_i\}$ (resp. $[0, 1] \times \{y'_i\}$).

Let A be a homology class in $H_2(X, \mathbf{y}, \mathbf{y}')$. Then we define

$$ (\mathbf{4.5.1}) \qquad n_{z_j}(A) = \langle A, [-1, 1] \times [0, 1] \times \{z_j\} \rangle, $$

where $z_j \in \Sigma - \alpha - \beta$ are given in Definition 4.2.1 and $\langle \cdot, \cdot \rangle$ is the signed intersection number. We say that $A \in H_2(X, \mathbf{y}, \mathbf{y}')$ is positive if $n_{z_j}(A) \ge 0$ is nonnegative for all $z_j$.

**Lemma 4.5.3.** — A positive $A \in H_2(X, \mathbf{y}, \mathbf{y}')$ admits a τ-trivial representative $\check{\mathcal{C}}$.

*Proof.* — Let A be a positive homology class in $H_2(X, \mathbf{y}, \mathbf{y}')$. Then we can glue closures of connected components of $\Sigma - \alpha - \beta$ with multiplicity $n_{z_j}(A)$ as in Rasmussen [Ra, Lemma 9.3] (also see [Li, Lemma 4.1] and its correction [Li2, Lemma 4.1']) to construct a smooth map $u = (u_1, u_2)$, where $u_1 : \check{F} \to \mathbf{R} \times [0, 1]$ is a branched cover with interior branch points and $u_2 : \check{F} \to \Sigma$ admits an extension $u_2 : F \to \Sigma$ such that

- $u_2(q_i^+) = y_i$ and $u_2(q_i^-) = y'_i$;

- each component of $\partial\check{F}$ is mapped to some $\alpha_i$ or $\beta_i$ so that each $\alpha_i, \beta_i$, $i = 1, \dots, k$, is used exactly once; and

- $u_2$ is holomorphic on a neighborhood of $\partial\check{F}$ and is locally given by $w \mapsto w^{\ell(q_i^{\pm})/2}$ for some positive odd integer $\ell(q_i^{\pm})$ near $q_i^{\pm}$, provided $q_i^{\pm}$ is not the end of a trivial strip of $u_2$.

We extend $u$ to $\check{u} : \check{F} \to X$, where $\check{F}$ is the real blow-up of $F$ given in Section 4.3. Condition (1) of Definition 4.5.1 is immediately satisfied. We can resolve all the (interior) singularities to make $\check{u}$ embedded. It is a local exercise to modify $\check{u}$ in a neighborhood of $\partial\check{F} - \partial\check{F}$ so that $\check{u}$ becomes τ-trivial. $\square$

**Lemma 4.5.4.** — If $u : \check{F} \to X$ is an HF curve and $\check{\mathcal{C}}$ is the image of the compactification $\check{u} : \check{F} \to \check{X}$, then the following hold:

(1) $\pi_B \circ u : \check{F} \to B$ has no branch points along $\partial B$.

(2) $\check{u}$ is positively transverse to the fibers $\{(s, t)\} \times \Sigma$ along all of $\partial\check{F}$ and $\check{\pi}_{[0,1]\times\Sigma}|_{\check{\mathcal{C}}}$ is an embedding near $\partial\check{\mathcal{C}}$.
In other words, $\check{\mathcal{C}}$ satisfies all the conditions of a $\tau$-trivial representative for some $\tau$, with the exception of the embeddedness of $\check{\mathcal{C}}$.

*Proof.* — (1) follows from the fact that $\pi_B \circ u$ is a $k$-fold branched cover of B. Let $\mathbf{H} = \{\mathrm{Im}(z) \ge 0\}$ be the upper half-plane and $U \subset \mathbf{H}$ be an open subset which contains 0. If $f$ is a holomorphic map $U \to \mathbf{R} \times [0, 1]$ which maps 0 to (0, 0) and $U \cap \partial \mathbf{H}$ to $\mathbf{R} \times \{0\}$, then it can be extended to a holomorphic map $f: U \cup \bar{U} \to \mathbf{R} \times [-1, 1]$ by Schwarz reflection, where $\bar{U} = \{\bar{z} | z \in U\}$. If $df(0) = 0$, then $f$ is locally a composition of $z \mapsto z'$ for some integer $l > 1$ and a biholomorphism. This contradicts the requirement that $f(\mathbf{H})$ stay on one side of $\mathbf{R} \times \{0\}$.

(2) follows from (1), together with the asymptotics of $u$ as $s \to \pm\infty$. $\square$

We now define the relative intersection form $Q_\tau(A)$, which is analogous to the relative intersection form which appears in the definition of the ECH index $I_{ECH}$, but is easier.

*Definition 4.5.5 (Relative intersection form $Q_\tau(A)$).* — Let $A \in \mathrm{H}_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ be a positive homology class and let $\check{\mathcal{C}}$ be a $\tau$-trivial representative of A. Let $\psi$ be a section of the normal bundle $\nu$ to $\check{\mathcal{C}}$ such that $\psi|_{\partial\check{\mathcal{C}}} = J\tau$, and let $\check{\mathcal{C}}'$ be a pushoff of $\check{\mathcal{C}}$ in the direction of $\psi$. Then the relative intersection form $Q_\tau(A)$ is given by:

$$Q_\tau(A) = \langle \check{\mathcal{C}}, \check{\mathcal{C}}' \rangle$$

where $\langle \cdot, \cdot \rangle$ denotes the algebraic count of intersection points.

Note that, since a representative $\check{\mathcal{C}}$ is positively transverse to the fibers $\{(s, t)\} \times \Sigma$ along all of $\partial\check{\mathcal{C}}$, we may take the normal bundle $\nu$ to $\check{\mathcal{C}}$ to satisfy $\nu|_{\partial\check{\mathcal{C}}} = T\Sigma|_{\partial\check{\mathcal{C}}}$. Also, since $J$ is $\Omega_X$-admissible, it takes $T\Sigma$ to itself. Hence $(\tau, J\tau)$ is a trivialization of $\nu|_{\partial\check{\mathcal{C}}}$. Although $\tau$ and $J\tau$ are homotopic, we will often use $J\tau$ due to its appearance in the definition of $Q_\tau(A)$.

Let $\tau'$ and $\tau$ be trivializations of $T\Sigma_{\check{\mathcal{X}}}$ along $Z_{\alpha, \beta}$ which differ only on $\{\pm 1\} \times [0, 1] \times (\alpha \cap \beta)$. Let $\mathrm{deg}(\tau, \tau', y_i)$ (resp. $\mathrm{deg}(\tau, \tau', y'_i)$) be the degree of $\tau$ with respect to the trivialization $\tau'$ along $[0, 1] \times \{y_i\}$ (resp. $[0, 1] \times \{y'_i\}$), oriented in the $\partial_t$-direction. We then have the following:

*Lemma 4.5.6 (Change of trivialization).* — If $A \in \mathrm{H}_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ is positive, then

$$(4.5.2) \qquad Q_\tau(A) - Q_{\tau'}(A) = \sum_{i=1}^k \mathrm{deg}(\tau, \tau', y_i) - \sum_{i=1}^k \mathrm{deg}(\tau, \tau', y'_i).$$

*Proof.* — Let $\check{\mathcal{C}}_{\tau'}$ be a $\tau'$-trivial representative of A. Let $\varepsilon > 0$ be small and let

$$\check{\mathcal{C}}_{\tau', 0} = \check{\mathcal{C}}_{\tau'} \cap ([-1 + \varepsilon, 1 - \varepsilon] \times [0, 1] \times \Sigma).$$
We can extend $\check{\mathcal{C}}_{\tau',0}$ to $\check{\mathcal{C}}_{\tau}$ in $\check{\mathcal{X}}$ by gluing disks $D_i, D'_i, i = 1, \dots, k$, corresponding to $y_i, y'_i$, so that $\check{\mathcal{C}}_{\tau}$ becomes $\tau$-trivial.

Let $\psi$ be a section of the normal bundle to $\check{\mathcal{C}}_{\tau}$ such that $\psi|_{\partial\check{\mathcal{C}}_{\tau}} = J\tau$ and $\psi|_{\partial\check{\mathcal{C}}_{\tau',0}} = J\tau'$. (Here we are assuming that $\tau'$ has been extended to a neighborhood of the $[0, 1] \times \{y_i\}$.) Then

$$Q_{\tau}(A) - Q_{\tau'}(A) = \sum_{i=1}^{k} \#(\psi|_{D_i})^{-1}(0) + \sum_{i=1}^{k} \#(\psi|_{D'_i})^{-1}(0),$$

where $\#$ is a signed count. A local calculation gives

$$\#(\psi|_{D_i})^{-1}(0) = \deg(\tau, \tau', y_i), \quad \#(\psi|_{D'_i})^{-1}(0) = -\deg(\tau, \tau', y'_i),$$

which proves the lemma. $\square$

The following is immediate from the definition of $Q_{\tau}(A)$.

**Lemma 4.5.7 (Additivity).** — If $A_1 \in H_2(X, \mathbf{y}, \mathbf{y}')$ and $A_2 \in H_2(X, \mathbf{y}', \mathbf{y}'')$ are positive, then

$$Q_{\tau}(A_1 \# A_2) = Q_{\tau}(A_1) + Q_{\tau}(A_2),$$

where $A_1 \# A_2 \in H_2(X, \mathbf{y}, \mathbf{y}'')$ is obtained from stacking two copies of $\check{\mathcal{X}}$.

**4.5.2. The relative adjunction formula.** — In this subsection we prove a relative adjunction formula for HF curves (Lemma 4.5.9).

If $\check{\mathcal{C}} \subset \check{\mathcal{X}}$ is a properly immersed surface with only transverse double points in its interior, we denote by $\delta(\check{\mathcal{C}})$ the signed count of the double points of $\check{\mathcal{C}}$.

**Lemma 4.5.8.** — If $\check{\mathcal{C}}$ is an immersed representative of $A \in H_2(X, \mathbf{y}, \mathbf{y}')$ with only transverse double points in its interior, then

$$c_1(v, J\tau) = Q_{\tau}(A) - 2\delta(\check{\mathcal{C}}).$$

*Proof.* — Let us first assume that $\check{\mathcal{C}}$ is embedded. If $\check{\mathcal{C}}$ is $\tau$-trivial, then there is a section $\psi$ of the normal bundle to $\check{\mathcal{C}}$ such that $\psi|_{\partial\check{\mathcal{C}}} = J\tau$, and

$$Q_{\tau}(A) = \#\psi^{-1}(0) = c_1(v, J\tau).$$

Next let $\tau'$ and $\tau$ be trivializations of $T\Sigma_{\check{\mathcal{X}}}$ along $Z_{\alpha,\beta}$, which differ only on $\{\pm 1\} \times [0, 1] \times (\alpha \cap \beta)$. By Equation (4.5.2), together with an analogous equation for $c_1(v, J\tau) - c_1(v, J\tau')$, we have

$$Q_{\tau}(A) - Q_{\tau'}(A) = c_1(v, J\tau) - c_1(v, J\tau'),$$

which proves the lemma for embedded $\check{\mathcal{C}}$.
Suppose now that $\check{\mathcal{C}}$ has a single positive transverse double point $d$. (The case of a negative double point is similar.) We resolve the intersection in the following way: Let $\mathcal{B} \subset \check{\mathcal{X}}$ be a small ball centered at $d$. Then $\check{\mathcal{C}} \cap \partial\mathcal{B}$ is a Hopf link, and $\check{\mathcal{C}} \cap \mathcal{B}$ is the union of two slice disks for the components which intersect at $d$. We can construct a new surface $\check{\mathcal{C}}_{sm}$ by replacing the two disks with a Hopf band connecting the two components of the Hopf link. By definition, we have $Q_\tau(A) = \langle \check{\mathcal{C}}_{sm}, \check{\mathcal{C}}'_{sm} \rangle$. On the other hand, if $v_{sm}$ is the normal bundle to $\check{\mathcal{C}}_{sm}$, then

$$c_1(v_{sm}, J\tau) = c_1(v, J\tau) + 2.$$

This can be seen easily by embedding $\mathcal{B}$ into $S^2 \times S^2$ and using the properties of the intersection product for closed 4-manifolds.

In general,

$$c_1(v, J\tau) + 2\delta(\check{\mathcal{C}}) = c_1(v_{sm}, J\tau) = \langle \check{\mathcal{C}}_{sm}, \check{\mathcal{C}}'_{sm} \rangle = Q_\tau(A),$$

and the lemma follows. $\square$

<figure><img src="image_2.png" /></figure>

Let $u: \check{\mathcal{F}} \to X$ be an HF curve. Then $u$ has no singular points on $\partial\check{\mathcal{F}}$ because $\pi_B \circ u$ has no branch point on the boundary. By [M, MW] (see also [MS, Appendix E]), there exists a modification $v: \check{\mathcal{F}} \to X$ of $u: \check{\mathcal{F}} \to X$ in a neighborhood of its finitely many singular points so that $v$ is an immersion with only positive transverse double points. We define $\delta(u)$ the number of positive double points of $v$, which depends only on $u$ and not on the modification $v$ used to define it. Then $\delta(u) \ge 0$ and $\delta(u) = 0$ if and only if $u$ is an embedding. We can now state and prove the relative adjunction formula:

**Lemma 4.5.9 (Relative adjunction formula).** — If $u: \check{\mathcal{F}} \to X$ is an HF curve in the homology class $A \in H_2(X, \mathbf{y}, \mathbf{y}')$, then

$$ \begin{aligned} (4.5.3) \qquad c_1(\check{u}^*T\Sigma, \tau) &= c_1(T\check{\mathcal{F}}, \partial_t) + Q_\tau(A) - 2\delta(u) \\ &= \chi(\check{\mathcal{F}}) - k + Q_\tau(A) - 2\delta(u). \end{aligned} $$

Here $\partial_t$ is the pullback to $\partial\check{\mathcal{F}}$ of the trivialization $\partial_t$ on $[-1, 1] \times [0, 1]$.

*Proof.* — Let $v: \check{\mathcal{F}} \to X$ be the immersion with transverse double points obtained by modifying $u: \check{\mathcal{F}} \to X$. Since the modification is purely local and is away from $\partial\check{\mathcal{F}}$, it follows that $\check{u}$ and $\check{v}$ belong to the same homology class in $H_2(X, \mathbf{y}, \mathbf{y}')$ and $c_1(\check{u}^*T\Sigma, \tau) = c_1(\check{v}^*T\Sigma, \tau)$. Hence Equation (4.5.3) for $u$ is equivalent to Equation (4.5.3) for $v$, and we may assume without loss of generality that $u$ is immersed with positive transverse double points.

The vector field $\partial_t$ is a global trivialization of the complex line bundle $T([-1, 1] \times [0, 1])$ over $\check{\mathcal{X}} = [-1, 1] \times [0, 1] \times \Sigma$. Hence

$$c_1(\check{u}^*T\check{\mathcal{X}}, (\tau, \partial_t)) = c_1(\check{u}^*T\Sigma, \tau).$$
On the other hand,

$$c_1(\check{u}^*T\check{X}, (\tau, \partial_t)) = c_1(T\check{F}, \partial_t) + c_1(v, J\tau).$$

The first line of the relative adjunction formula now follows from Lemma 4.5.8. The equivalence of the first and second lines is a consequence of Claim 4.5.10, proved below. □

$$\textbf{Claim 4.5.10.} \quad c_1(T\check{F}, \partial_t) = \chi(F) - k.$$

*Proof.* — Let $\tau_{\partial F}$ be the trivialization of $TF|_{\partial F}$ which is given by an oriented non-singular vector field tangent to $\partial F$. We then have

$$c_1(T\check{F}, \partial_t) = \chi(F) + \deg(\partial_t, \tau_{\partial F}),$$

where $\deg(\partial_t, \tau_{\partial F})$ is the degree of $\partial_t$ with respect to $\tau_{\partial F}$. By an easy direct calculation we obtain $\deg(\partial_t, \tau_{\partial F}) = -k$. □

**4.5.3. The index $I_{HF}$ and the index inequality.** — We are now ready to define the ECH-type index $I_{HF}$ and prove the ECH-type index inequality (Theorem 4.5.13).

*Definition 4.5.11 (ECH-type index).* — Let $A \in H_2(X, \mathbf{y}, \mathbf{y}')$ be a positive homology class. Then the ECH-type index $I_{HF}$ of A is given as follows:

$$(4.5.4) \qquad I_{HF}(A) = c_1(T\Sigma|_{\Lambda}, \tau) + Q_{\tau}(A) + \mu_{\tau}(\mathbf{y}) - \mu_{\tau}(\mathbf{y}').$$

We observe that $I_{HF}(A)$ does not depend on the choice of $\tau$: Suppose $\tau$ and $\tau'$ differ only at $y_i$ and $\deg(\tau, \tau', y_i) = 1$. Then we compute that (i) $Q_{\tau}(A) = Q_{\tau'}(A) + 1$ by Lemma 4.5.6, (ii) $\mu_{\tau}(\mathbf{y}) = \mu_{\tau'}(\mathbf{y}) - 2$ and (iii) $c_1(T\Sigma|_{\Lambda}, \tau) = c_1(T\Sigma|_{\Lambda}, \tau') + 1$. Next we reverse the direction of $\tau$ (viewed as a vector field) along a single component $[-1, 1] \times \{1\} \times \alpha_i$ of $[-1, 1] \times \{1\} \times \alpha$, keeping in mind that $\tau$ is tangent to $\{s\} \times \{1\} \times \alpha$ and $\{s\} \times \{0\} \times \beta$ for all $s \in [-1, 1]$. More precisely, let

- $\tau' = -\tau$ on $[-1, 1] \times \{1\} \times \alpha_i$,

- on each $\{-1, 1\} \times [0, 1] \times y_j \subset Z_{\alpha, \beta}$ such that $y_j \in \alpha_i$, $\tau'$ is obtained by adding an extra $\pi$-rotation to $\tau$ near $\{-1, 1\} \times \{1\} \times y_j$, and

- $\tau' = \tau$ elsewhere.

Then it is easy to verify that (i) $Q_{\tau}(A) = Q_{\tau'}(A)$, (ii) $\mu_{\tau}(\mathbf{y}) = \mu_{\tau'}(\mathbf{y})$, and (iii) $c_1(T\Sigma|_{\Lambda}, \tau) = c_1(T\Sigma|_{\Lambda}, \tau')$. Any two trivializations $\tau$ and $\tau'$ can be related to each other by a sequence of differences of the above two types.

The index $I_{HF}$ satisfies the following additivity property under the stacking operation $\# : H_2(X, \mathbf{y}, \mathbf{y}') \times H_2(X, \mathbf{y}', \mathbf{y}'') \to H_2(X, \mathbf{y}, \mathbf{y}'')$.
**Lemma 4.5.12 (Additivity of I<sub>HF</sub>).* — If A<sub>1</sub> ∈ H<sub>2</sub>(X, y, y') and A<sub>2</sub> ∈ H<sub>2</sub>(X, y', y'') are positive, then

$$
I_{\mathrm{HF}}(A_1 \# A_2) = I_{\mathrm{HF}}(A_1) + I_{\mathrm{HF}}(A_2).
$$

*Proof.* — Each of the terms $c_1(\Gamma\Sigma|_A, \tau)$, $Q_\tau(A)$, and $\mu_\tau(\mathbf{y}) - \mu_\tau(\mathbf{y}')$ in the definition of $I_{\mathrm{HF}}(A)$ is additive under stacking; see Lemma 4.5.7 for the additivity of $Q_\tau(A)$. $\square$

The following index inequality is analogous to (but much easier than) the ECH index inequality, due to Hutchings [Hul, Theorem 1.7]. We remark that *u* is required to be simply-covered in the statement of the usual ECH index inequality. This is automatically satisfied for HF-curves.

**Theorem 4.5.13 (ECH-type index inequality).** — Let $u : \dot{\mathbb{F}} \to X$ be an HF curve in the class $A \in H_2(X, y, y')$. Then

$$
(4.5.5) \qquad \mathrm{ind}(u) + 2\delta(u) = \mathrm{I}_{\mathrm{HF}}(A),
$$

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

$$
(4.5.6) \qquad \mathrm{ind}(u) \leq \mathrm{I}_{\mathrm{HF}}(A),
$$

with equality if and only if *u* is an embedding.

Note that *u* has no boundary singularities because $\pi_B \circ u$ has no branch points at the boundary.

*Proof.* — We calculate:

$$
\begin{align*}
\mathrm{ind}(u) &= -\chi(\mathbf{F}) + k + \mu_{\tau}(\mathbf{y}) - \mu_{\tau}(\mathbf{y}') + 2c_1(\check{u}^*\Gamma\Sigma, \tau) \\
&= c_1(\check{u}^*\Gamma\Sigma, \tau) + Q_{\tau}(A) + \mu_{\tau}(\mathbf{y}) - \mu_{\tau}(\mathbf{y}') - 2\delta(u).
\end{align*}
$$

The first line is Equation (4.4.4). The equivalence of the first and second lines follows
from the relative adjunction formula. Hence

$$
\mathrm{ind}(u) + 2\delta(u) = \mathrm{I}_{\mathrm{HF}}(A).
$$

The index inequality (4.5.6) follows immediately. $\square$

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

**4.6. Compactness.** — We now discuss the requisite compactness issues. The key notion is that of weak admissibility from [OSz1, Definition 4.10], which is analogous to the vanishing of the flux in the PFH situation (see Section 3.3). A pointed Heegaard diagram (Σ, α, β, z) is weakly admissible if, for every Spin'-structure s and nontrivial periodic
domain $\mathcal{Q}$ which satisfies $\langle c_1(s), \mathcal{Q} \rangle = 0$, there exist $j_1$ and $j_2$ for which $n_{z_{j_1}}(\mathcal{Q}) > 0$ and $n_{z_{j_2}}(\mathcal{Q}) < 0$. (The points $z_{j_1}$ and $z_{j_2}$ are the points from Definition 4.2.1(4).) Equivalently, by [OSz1, Lemma 4.12], $(\Sigma, \alpha, \beta, z)$ is weakly admissible if and only if there is an area form $\omega$ on $\Sigma$ such that each periodic domain has total signed $\omega$-area zero.

Let $N > 0$ be a fixed constant. We consider the subset $H_2^N(X, y, y')$ consisting of homology classes of $H_2(X, y, y')$ which intersect $[-1, 1] \times [0, 1] \times \{z\}$ at most $N$ times. (This is sufficient for $\overline{CF}$ and $CF^+$, defined in Section 4.8.) The difference of two homology classes $A_1, A_2 \in H_2^0(X, y, y')$ is a periodic domain $\mathcal{Q}$ and has zero $\omega$-area. This implies that the $\omega$-areas of any two $A_1, A_2 \in H_2^N(X, y, y')$ differ by $i \cdot \omega(\Sigma)$ where $0 \le i \le N$. Let $\phi_1, \dots, \phi_i$ be the connected components of $\Sigma - \alpha - \beta$ with $z_j \in \phi_j$. If $A$ is represented by a holomorphic curve, then the projection of $A$ to $\Sigma$ can be written as $\sum_j n_{z_j}(A)\phi_j$ with $n_{z_j}(A) \ge 0$. Since each $\phi_i$ has finite area, there must only be a finite number of homology classes $A \in H_2^N(X, y, y')$ for which the moduli space $M_j^X(y, y', A)$ is nonempty.

We now prove the existence of a compactification of $M_j^X(y, y', A)/\mathbf{R}$. It suffices to show that if $u: \dot{\mathbf{F}} \to X$ is an element of $M_j^X(y, y', A)$, then the genus of $\dot{\mathbf{F}}$ is bounded as long as $A$ is fixed. This will be carried out in Lemma 4.6.1. Once we have a genus bound, the SFT compactness theorem from [BEHWZ] can be applied to give a compactification of $M_j^X(y, y', A)/\mathbf{R}$.

**Lemma 4.6.1.** — There is an upper bound on the genus of a holomorphic curve $u: \dot{\mathbf{F}} \to X$ in a fixed homology class $A \in H_2(X, y, y')$.

*Proof.* — The proof is analogous to the proof in the PFH case. In view of the relative adjunction formula (Lemma 4.5.9) and the nonnegativity of $\delta(u)$, we have

$$ (4.6.1) \qquad \chi(\dot{\mathbf{F}}) \ge c_1(\tilde{u}^*T\Sigma, \tau) + k - Q_{\tau}(A). $$

The lemma follows by observing that the terms on the right-hand side depend only on the homology class $A$. $\square$

**4.7. Transversality.**

*Definition 4.7.1.* — An almost complex structure $J \in \mathcal{J}_X$ is regular if the moduli spaces $M_j^X(y, y')$ are transversely cut out for all $y, y' \in S_{\alpha, \beta}$.

Note that if $u$ is an HF curve, then it does not have any closed irreducible components by definition. In particular, $u$ cannot have any fibers $\{(s, t)\} \times \Sigma$ as irreducible components.

We write $\mathcal{J}_X^{\text{reg}} \subset \mathcal{J}_X$ for the subset of regular almost complex structures $J$. For $J \in \mathcal{J}_X^{\text{reg}}$, the dimension of $M_j^X(y, y')$ near $u$ is equal to the Fredholm index $\text{ind}(u)$. The moduli space $M_j^X(y, y')$ carries a natural $\mathbf{R}$-action given by translations in the $s$-direction, and the quotient $M_j^X(y, y')/\mathbf{R}$ is a manifold.
**Lemma 4.7.2.** — A generic $J \in \mathcal{J}_X$ is regular.

*Proof.* — This follows from [Li, Proposition 3.8] (see also [Li2, Section 2]), by noting that an HF curve $u$ does not have any fibers as irreducible components. Lemma 4.7.2 can also be proved in the same way as in [Hu1, Lemma 9.12(b)]. Note that the transversality theory is relatively straightforward because HF curves are never multiply-covered, i.e., all the moduli spaces $M_{J}^{X}(\mathbf{y}, \mathbf{y}')$ consist of simple curves. $\square$

We will use the notation $M_{J}^{X,I=r}(\mathbf{y}, \mathbf{y}')$ to denote the moduli space of HF curves from $\mathbf{y}$ to $\mathbf{y}'$ with ECH index $I_{\text{HF}} = r$.

*Corollary 4.7.3 (Corollary of Theorem 4.5.13).* — If $I_{\text{HF}}(A) = 0, 1$ and $J \in \mathcal{J}_X^{\text{reg}}$, then every HF curve $u$ in the class A satisfies $\text{ind}(u) = I_{\text{HF}}(A)$ and is therefore embedded.

*Proof.* — This follows from Equation (4.5.5) by observing that the term $2\delta(u)$ is even and nonnegative and that $\text{ind}(u) \ge 0$ since $J$ is regular and $u$ is not multiply-covered. $\square$

**4.8. Definition of the Heegaard Floer homology groups.** — Let $(\Sigma, \alpha, \beta, z)$ be a weakly admissible Heegaard diagram and let $J \in \mathcal{J}_X^{\text{reg}}$. We define the Heegaard Floer chain complexes

$$(\widehat{\text{CF}}(\Sigma, \alpha, \beta, z, J), \widehat{\partial}) \text{ and } (\text{CF}^{+}(\Sigma, \alpha, \beta, z, J), \partial^{+}),$$

whose corresponding homology groups are

$$\widehat{\text{HF}}(\Sigma, \alpha, \beta, z, J) \text{ and } \text{HF}^{+}(\Sigma, \alpha, \beta, z, J).$$

The hat group $\widehat{\text{CF}}(\Sigma, \alpha, \beta, z, J)$ is the **F**-vector space generated by $S_{\alpha, \beta}$ and the plus group $\text{CF}^{+}(\Sigma, \alpha, \beta, z, J)$ is the **F**-vector space generated by $S_{\alpha, \beta} \times \mathbf{Z}^{\ge 0}$. Elements of $S_{\alpha, \beta}$ will be written as $\mathbf{y}$ and elements of $S_{\alpha, \beta} \times \mathbf{Z}^{\ge 0}$ will be written as $[\mathbf{y}, i]$.

We now define the differentials $\widehat{\partial}$ and $\partial^{+}$. The differential $\widehat{\partial}$ is given by

$$\widehat{\partial}\mathbf{y} = \sum_{\mathbf{y}' \in S_{\alpha, \beta}} \langle \widehat{\partial}\mathbf{y}, \mathbf{y}' \rangle \cdot \mathbf{y}',$$

where $\langle \widehat{\partial}\mathbf{y}, \mathbf{y}' \rangle$ is the count of $\widehat{M}_{J}^{X,I=1}(\mathbf{y}, \mathbf{y}')/\mathbf{R}$. The differential $\partial^{+}$ is given by

$$\partial^{+}([\mathbf{y}, i]) = \sum_{[\mathbf{y}', j] \in S_{\alpha, \beta} \times \mathbf{Z}^{\ge 0}} \langle \partial^{+}([\mathbf{y}, i]), [\mathbf{y}', j] \rangle \cdot [\mathbf{y}', j],$$

where $\langle \partial^{+}([\mathbf{y}, i]), [\mathbf{y}', j] \rangle$ is the count of $M_{J}^{X,I=1}(\mathbf{y}, \mathbf{y}')/\mathbf{R}$ whose representatives have intersection number $i-j$ with $\mathbf{R} \times [0, 1] \times \{z\}$. By Theorem 4.5.13, the count of $I_{\text{HF}}(u) = 1$
curves is equivalent to the count of embedded ind($u$) = 1 curves. Hence our definition is the same as that of Lipshitz.

The differentials $\hat{\partial}$ and $\partial^+$ indeed satisfy $\hat{\partial}^2 = 0$ and $(\partial^+)^2 = 0$ by [Li]. A tricky issue which arises for $\partial^+$ (but not for $\hat{\partial}$) is that an element $u$ of the boundary of $M_j^{X,1=2}(\mathbf{y}, \mathbf{y}')/\mathbf{R}$ might a priori have a fiber $\{(s, t)\} \times \Sigma$ as an irreducible component. (In that case, $u$ consists of one copy of $\Sigma$, together with $k$ trivial strips.) This possibility is eliminated in [Li, Lemma 8.2].

Both $\widehat{\mathrm{HF}}(\Sigma, \alpha, \beta, z, J)$ and $\widehat{\mathrm{HF}}^+(\Sigma, \alpha, \beta, z, J)$ are independent of the choices and can be written as $\widehat{\mathrm{HF}}(M)$ and $\widehat{\mathrm{HF}}^+(M)$. In this paper, we are interested in $\widehat{\mathrm{HF}}(-M)$, where $-M$ is the manifold $M$ with the opposite orientation. The group $\widehat{\mathrm{HF}}(-M)$ is the homology of the chain complex $\widehat{\mathrm{CF}}(\Sigma, \beta, \alpha, z, J)$, i.e., the complex for which the roles of the $\alpha$- and $\beta$-curves are exchanged.

**4.9. Restricting the complex to a page.** — In this subsection we describe a pointed Heegaard diagram for $-M$ which is adapted to an open book and which has the property that $\widehat{\mathrm{HF}}(-M)$ can be computed from a single page. The Heegaard diagram constructed here is a slight modification of the Heegaard diagram constructed by Honda, Kazez and Matić in [HKM].

Let $S$ be a bordered surface of genus $g$ and connected boundary, and let $(S, \hbar)$ be an open book decomposition of $M$ with binding $K$. Then there is a homeomorphism

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

where $(x, 1) \sim (\hbar(x), 0)$ for $x \in S$ and $(y, t) \sim (y, t')$ for $y \in \partial S$ and $t, t' \in [0, 1]$.

**4.9.1. A Heegaard diagram compatible with $(S, \hbar)$.** — We define a pointed Heegaard diagram $(\Sigma, \beta, \alpha, z)$ for $-M$ which is compatible with $(S, \hbar)$; this means that $L_\beta = \mathbf{R} \times \{1\} \times \beta$ and $L_\alpha = \mathbf{R} \times \{0\} \times \alpha$. Recall from above that switching the roles of $\alpha$ and $\beta$ has the effect of reversing the orientation of the ambient manifold.

The open book decomposition $(S, \hbar)$ gives a natural Heegaard decomposition of $M$ into two handlebodies $H_1 = (S \times [0, \frac{1}{2}]) / \sim$ and $H_2 = (S \times [\frac{1}{2}, 1]) / \sim$. The Heegaard surface $\Sigma$ is $S_{1/2} \cup -S_0$ and has genus $2g$. Here we abbreviate $S \times \{t\}$ by $S_t$.

A basis of $S$ is a collection of properly embedded pairwise disjoint arcs $\mathbf{a} = \{a_1, \dots, a_{2g}\}$ of $S$ such that $S - \mathbf{a}$ is a connected $8g$-gon. Given a basis $\mathbf{a}$ of $S$, there is a natural collection of compression curves $\alpha_i = \partial(a_i \times [0, \frac{1}{2}])$ for $H_1$. We write $\alpha_i = a_i^\dagger \cup a_i$, where the presence of $\dagger$ indicates a copy of an arc in $S_{1/2}$ and the absence indicates a copy of the arc in $S_0$. Recall the monodromy $\hbar$ maps $(y, \theta) \mapsto (y, \theta - y)$ near $\partial S$. We then construct a collection of compression curves $\beta_i = b_i^\dagger \cup \hbar(a_i)$ for $H_2$, where $b_i$ is the simplest arc (i.e., with the fewest number of intersections with the $a_j$) in $S_{1/2}$ which is parallel to $a_i$ and extends $\hbar(a_i)$ to smooth curve in $\Sigma$. See Figure 1.

The arcs $a_i$ and $\hbar(a_i)$ intersect at their endpoints $x_i$ and $x'_i$ by the definition of $\hbar$ near $\partial S$, and the arcs $a_i^\dagger$ and $b_i^\dagger$ intersect at a unique point $x''_i$ in $\mathrm{int}(S_{1/2})$. This means that
all the intersection points of $\alpha_i \cap \beta_j$ lie in $S_0$, except for one intersection point $x_i''$ of $\alpha_i \cap \beta_i$ for each $i$. We then place the basepoint $z$ on the binding, away from all the intersection points $x_i$, $x_i'$. The regions of $\Sigma - \alpha - \beta$ which nontrivially intersect $S_{1/2}$ are the following:

- the “forbidden region” containing the basepoint $z$;

- for each $i = 1, \dots, 2g$, a bigon $D_i$ from $x_i''$ to $x_i$ and a bigon $D'_i$ from $x_i''$ to $x'_i$.

By the placement of the basepoint $z$, it is clear that any periodic domain must have terms of the form $k(D_i - D'_i)$, where $k$ is an integer. This implies the weak admissibility of the Heegaard diagram $(\Sigma, \beta, \alpha, z)$.

**Remark 4.9.1.** — The point $x_i$ or $x'_i$ (either one) is a component of the contact class $c(\xi_{(S,\hat{h})}) \in \hat{\text{HF}}(\Sigma, \beta, \alpha, z)$, where $\xi_{(S,\hat{h})}$ is the contact structure which corresponds to $(S, \hat{h})$.

**4.9.2.** *Holomorphic curves in the region* $\mathbf{R} \times [0, 1] \times S_{1/2}$. — Let $J \in \mathcal{J}_{\mathbf{R}}$ with the additional property:

(&) $J$ is a product complex structure on $\mathbf{R} \times [0, 1] \times S_{1/2}$.

All the holomorphic curves and moduli spaces in this subsection are for the Heegaard diagram $(\Sigma, \beta, \alpha, z)$.

*Claim 4.9.2.* — Let $u \in \hat{\mathcal{M}}_J^X(\mathbf{y}, \mathbf{y}')$ for some $\mathbf{y}, \mathbf{y}' \in S_{\beta, \alpha}$. Then the following hold:

(1) If $u$ is not asymptotic to any $x_i''$, then its image is contained in $\mathbf{R} \times [0, 1] \times S_0$.

(2) If $u$ is asymptotic to some $x_i''$, then $u$ has $x_i''$ at the positive end and a component of $u$ is either (i) a trivial strip over $x_i''$ or (ii) a “thin strip” from $x_i''$ to $x_i$ or $x'_i$, whose projection to $\Sigma$ is $D_i$ or $D'_i$.

(3) If $u$ is asymptotic to $x_i$ or $x'_i$ at the positive end, then a component of $u$ is a trivial strip over $x_i$ or $x'_i$.

<figure><img src="image_4.png" /><figcaption>FIG. 1. — A portion of $S_{1/2}$ whose normal orientation points out of the page. The shaded regions are the disks $D_i$ and $D'_i$</figcaption></figure>
The only nontrivial components of $u$ which intersect $\mathbf{R} \times [0, 1] \times S_{1/2}$ are the “thin strips” in (2) and are easily seen to be regular since their projections to $\mathbf{R} \times [0, 1]$ and $S_{1/2}$ are regular; see for example [Se2, Section (13a)] for the automatic transversality of holomorphic curves with boundary punctures in dimension 2. Hence a generic $J$ which satisfies $(\&)$ is in $\mathcal{J}_X^{\text{reg}}$.

**4.9.3.** *The variant $\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}))$. — Let $J \in \mathcal{J}_X^{\text{reg}}$ which satisfies $(\&)$. We now define the chain complex $\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a}), J)$, which can be defined on a page of the open book $(S, \hbar)$ and whose homology is isomorphic to $\widehat{\text{HF}}(\Sigma, \beta, \alpha, z, J)$. The almost complex structure $J$ will usually be suppressed from the notation.*

Let $S_{\mathbf{a},\hbar(\mathbf{a})}$ be the set of $2g$-tuples of intersection points of $\mathbf{a}$ and $\hbar(\mathbf{a})$; equivalently, $S_{\mathbf{a},\hbar(\mathbf{a})} = \{\mathbf{y} \in S_{\beta,\alpha} \mid \mathbf{y} \subset S_0\}$. Then we define $(\widehat{\text{CF}}'(S, \mathbf{a}, \hbar(\mathbf{a})), \partial')$ as the subcomplex of $(\widehat{\text{CF}}(\Sigma, \beta, \alpha, z), \partial)$ generated by $S_{\mathbf{a},\hbar(\mathbf{a})}$. The differential $\partial$ restricts to $\partial'$ by Claim 4.9.2(1).

Next define an equivalence relation $\sim$ on $\widehat{\text{CF}}'(S, \mathbf{a}, \hbar(\mathbf{a}))$ as follows: if we write $\mathbf{y} = \mathbf{y}_0 \cup \mathbf{y}_1$, where $\mathbf{y}_0$ consists of chords of type $x_i, x'_i$, $i = 1, \dots, 2g$, and $\mathbf{y}_1$ does not contain any $x_i, x'_i$, $i = 1, \dots, 2g$, then $\mathbf{y} \sim \mathbf{y}'$ if and only if $\mathbf{y} = \mathbf{y}_0 \cup \mathbf{y}_1$, $\mathbf{y}' = \mathbf{y}'_0 \cup \mathbf{y}'_1$ and $\mathbf{y}_1 = \mathbf{y}'_1$. We then take the quotient complex

$$ \widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a})) = \widehat{\text{CF}}'(S, \mathbf{a}, \hbar(\mathbf{a}))/\sim, $$

with the differential $\widehat{\partial}$ induced from $\partial'$. The differential $\partial'$ descends to the quotient $\widehat{\partial}$ by Claim 4.9.2(3).

**Remark 4.9.3.** — Since $\Sigma$ and $S = S_0$ have opposite orientations, the order $(\beta, \alpha)$ is switched to $(\mathbf{a}, \hbar(\mathbf{a}))$.

The following theorem allows us to restrict from the Heegaard surface $\Sigma$ to the page $S$:

**Theorem 4.9.4.** — $H_*(\widehat{\text{CF}}(S, \mathbf{a}, \hbar(\mathbf{a})), \widehat{\partial}) \simeq \widehat{\text{HF}}(\Sigma, \beta, \alpha, z)$.

*Proof.* — Let us write $\widehat{\text{CF}}$ for $\widehat{\text{CF}}(\Sigma, \beta, \alpha, z)$. Also let $\widehat{\text{CF}}_k$ be the subgroup of $\widehat{\text{CF}}$ generated by $2g$-tuples of chords, exactly $k$ of which are of the form $x_i''$. Using Claim 4.9.2, we can write the differential $\partial$ on $\widehat{\text{CF}}$ as $\partial = \partial_0 + \partial_1$, where $\partial_0 : \widehat{\text{CF}}_k \to \widehat{\text{CF}}_k$ counts $I_{\text{HF}} = 1$ curves whose nontrivial part is contained in $S_0$ and $\partial_1 : \widehat{\text{CF}}_k \to \widehat{\text{CF}}_{k-1}$ counts $I_{\text{HF}} = 1$ curves whose nontrivial part is contained in $S_{1/2}$. In particular, $\partial_1$ counts HF curves which correspond to the domains $D_i$ and $D'_i$. Since $\partial^2 = 0$, it follows that

$$ \partial_0^2 = \partial_1^2 = \partial_0 \partial_1 + \partial_1 \partial_0 = 0, $$

i.e., $\widehat{\text{CF}}$ becomes a double complex.
The $\partial_1$-homology of $\widehat{CF}$ is:

$$ H_k(\widehat{CF}, \partial_1) = \begin{cases} \widehat{CF}(S, \mathbf{a}, \hbar(\mathbf{a})), & \text{if } k=0; \\ 0, & \text{if } k>0. \end{cases} $$

This claim will be proved in Lemma 4.9.5. For the moment we assume it to finish the proof of the theorem. The double complex gives rise to a spectral sequence converging to $\widehat{HF}(\Sigma, \beta, \alpha, z)$ such that:

$$ E^1 = H_*(\widehat{CF}, \partial_1) = \widehat{CF}(S, \mathbf{a}, \hbar(\mathbf{a})) $$

$$ E^2 = H_*(H_*(\widehat{CF}, \partial_1), [\partial_0]) = H_*(\widehat{CF}(S, \mathbf{a}, \hbar(\mathbf{a})), \widehat{\partial}). $$

Since $E^2$ is concentrated in degree $k=0$, the spectral sequence degenerates at the second step and $\widehat{HF}(\Sigma, \beta, \alpha, z) \cong E^2$. This proves the theorem. $\square$

Before proceeding to Lemma 4.9.5, let us introduce some notation. Let $\mathcal{I}$ be a $k$-element subset of $\{1, \dots, 2g\}$ and let $\mathcal{I}^c$ be its complement. Then let $S_{\mathcal{I}^c}$ be the set of $(2g-k)$-tuples $\mathbf{y}_1$ of chords from $\hbar(\mathbf{a})$ to $\mathbf{a}$ such that each $a_j$ and $\hbar(a_j)$, $j \in \mathcal{I}^c$, is used exactly once and no $x_j, x'_j, x''_j$ is in $\mathbf{y}_1$. In particular, $a_i$ and $\hbar(a_i)$ remain unoccupied for all $i \in \mathcal{I}$.

**Lemma 4.9.5.** — The homology of $(\widehat{CF}, \partial_1)$ is:

$$ H_k(\widehat{CF}, \partial_1) = \begin{cases} \widehat{CF}(S, \mathbf{a}, \hbar(\mathbf{a})), & \text{if } k=0; \\ 0, & \text{if } k>0. \end{cases} $$

*Proof.* — Let $(\widehat{CF}(\mathbf{y}_1), \partial_1) \subset (\widehat{CF}, \partial_1)$ be the subcomplex generated by 2g-tuples of chords of the form $\mathbf{y}_0 \cup \mathbf{y}_1$, where $\mathbf{y}_0$ is a $k$-tuple of chords consisting of one of $x_j, x'_j, x''_j$ for each $j \in \mathcal{I}$ and $\mathbf{y}_1 \in S_{\mathcal{I}^c}$. Since $(\widehat{CF}, \partial_1)$ is the direct sum of chain complexes of the form $(\widehat{CF}(\mathbf{y}_1), \partial_1)$, it suffices to treat each $(\widehat{CF}(\mathbf{y}_1), \partial_1)$ separately.

Consider the chain complexes $(C(j), d) = (C_0(j) \oplus C_1(j), d)$, where

$$ C_1(j) = \mathbf{F}\{x_j''\}, \quad C_0(j) = \mathbf{F}\{x_j, x_j'\}, \quad d(x_j'') = x_j - x_j' $$

The homology groups of those complexes are:

$$ (\mathbf{4.9.1}) \qquad H_k(C(j), d) = \begin{cases} \langle x_j, x'_j \rangle / \langle x_j - x'_j \rangle, & \text{if } k=0; \\ 0, & \text{if } k=1. \end{cases} $$

By Claim 4.9.2(2), we have

$$ (\widehat{CF}(\mathbf{y}_1), \partial_1) \cong \bigotimes_{j \in \mathcal{I}} (C(j), d). $$
By the Künneth formula, $H_* (\widehat{\mathrm{CF}}(\mathbf{y}_1), \partial_1)$ is generated by the equivalence class $\{\mathbf{y}'_0 \cup \mathbf{y}_1\}$, where $\mathbf{y}_1$ is fixed and $\mathbf{y}'_0$ ranges over all $k$-tuples of chords which consist of one of $x_j$, $x'_j$ for each $j \in \mathcal{I}$. The lemma then follows. $\square$

**4.10. Spin<sup>*c*</sup>-structures.** — Let $S_{\alpha, \beta}$ be the set of $k$-tuples of intersection points of the pointed Heegaard diagram $(\Sigma, \alpha, \beta, z)$ and let Spin<sup>*c*</sup>(M) be the set of Spin<sup>*c*</sup>-structures on M. In [OSz1, Section 2.6], Ozsváth and Szabó defined a map

$$s_z : S_{\alpha, \beta} \to \mathrm{Spin}^c(M).$$

Although the precise definition of $s_z$ will not be given here, we review an important property of $s_z$ which is more or less equivalent to the definition. Given $\mathbf{y} = \{y_i\}_{i=1}^k$, $\mathbf{y}' = \{y'_i\}_{i=1}^k \in S_{\alpha, \beta}$, the difference between the Spin<sup>*c*</sup>-structures corresponding to $\mathbf{y}$ and $\mathbf{y}'$ is given by:

$$\epsilon(\mathbf{y}, \mathbf{y}') = \mathrm{PD}(s_z(\mathbf{y}) - s_z(\mathbf{y}')) \in H_1(M),$$

where a cycle representing $\epsilon(\mathbf{y}, \mathbf{y}')$ can be constructed on the Heegaard diagram as follows: For each $i = 1, \dots, k$, choose an arc $\alpha_i^*$ on $\alpha_i$ from $y_i$ to $y_i'$, where $y_i, y_i' \in \alpha_i$. Similarly, we choose arcs $\beta_i^*$ on $\beta_i$, $i = 1, \dots, k$, which connect $\mathbf{y}'$ to $\mathbf{y}$. Then $\epsilon(\mathbf{y}, \mathbf{y}')$ is the homology class of $\cup_{i=1}^k (\alpha_i^* \cup \beta_i^*)$, which is a union of closed curves; see [OSz1, Definition 2.11 and Lemma 2.19]. It is easy to verify that $\epsilon(\mathbf{y}, \mathbf{y}')$ does not depend on the choice of arcs $\alpha_i^*$ and $\beta_i^*$ and provides a topological obstruction to the existence of HF curves connecting $\mathbf{y}$ and $\mathbf{y}'$.

The Heegaard Floer chain complex $\widehat{\mathrm{CF}}(\Sigma, \alpha, \beta, z)$ therefore splits into a direct sum

$$\widehat{\mathrm{CF}}(\Sigma, \alpha, \beta, z) = \bigoplus_{s \in \mathrm{Spin}^c(M)} \widehat{\mathrm{CF}}(\Sigma, \alpha, \beta, z, s),$$

where the subgroup $\widehat{\mathrm{CF}}(\Sigma, \alpha, \beta, z, s)$ is generated by $\mathbf{y} \in S_{\alpha, \beta}$ with $s_z(\mathbf{y}) = s$ and is a subcomplex.

We now interpret the above discussion in a way which relates more easily to the splitting of ECH in terms of homology classes of orbit sets. Consider the chain complex $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathfrak{h}(\mathbf{a}))$ which is generated by the set $S_{\mathbf{a}, \mathfrak{h}(\mathbf{a})}$ of $2g$-tuples of intersection points of $\mathbf{a}$ and $\mathfrak{h}(\mathbf{a})$, i.e., we are restricting to a page $S$. The homology groups $H_1(M; \mathbb{Z}) \simeq H_1(N, \partial N; \mathbb{Z})$ are identified via the isomorphism $\bar{\omega}$, given in Lemma 2.3.1.

We then define the map

$$\mathfrak{h} : S_{\mathbf{a}, \mathfrak{h}(\mathbf{a})} \to H_1(M)$$

by assigning a cycle $\mathfrak{h}(\mathbf{y})$ to $\mathbf{y} = \{y_i\}_{i=1}^{2g} \in S_{\mathbf{a}, \mathfrak{h}(\mathbf{a})}$ as follows: Suppose $y_i \in a_i \cap \mathfrak{h}(a_{\sigma(i)})$ for some $\sigma \in \mathfrak{S}_{2g}$. On $[0, 1] \times S$, we consider the union of the following oriented arcs:

$$- [0, 1] \times \{y_i\}, i = 1, \dots, 2g, \text{ where the orientation is given by } \partial_i;$$
– $\{0\} \times c_i$, $i = 1, \dots, 2g$, where $c_i$ is a subarc of $\mathcal{h}(a_i)$ which goes from $\mathcal{h}(y_{\sigma(i)})$ to $y_i$.

With the identification $(x, 1) \sim (\mathcal{h}(x), 0)$, the arcs glue to give a cycle in $\mathcal{N}$ which represents $\mathcal{h}(\mathbf{y})$.

**Proposition 4.10.1.** — Let $\xi$ be the contact structure supported by the open book decomposition $(S, \mathcal{h})$ of $\mathcal{M}$, and let $\mathfrak{s}_{\xi}$ be the canonical Spin$^c$-structure determined by $\xi$. Then for any $\mathbf{y} \in \mathcal{S}_{\mathbf{a}, \mathcal{h}(\mathbf{a})}$ we have

$$s_{\xi}(\mathbf{y}) = \mathfrak{s}_{\xi} + \mathrm{PD}(\mathcal{h}(\mathbf{y})).$$

*Proof.* — The equality holds for any $2g$-tuple $\mathbf{x}_0$ which represents the contact class. In fact, $s_{\xi}(\mathbf{x}_0) = \mathfrak{s}_{\xi}$ by the definition of the contact class and $\mathcal{h}(\mathbf{x}_0) = 0$ since the cycle representing it is parallel to $\partial\mathcal{N}$. Hence, in order to prove the proposition, it suffices to prove that

$$\mathcal{h}(\mathbf{y}) - \mathcal{h}(\mathbf{y}') = \epsilon(\mathbf{y}, \mathbf{y}')$$

for all $\mathbf{y}, \mathbf{y}' \in \mathcal{S}_{\mathbf{a}, \mathcal{h}(\mathbf{a})}$. One can check that $\mathcal{h}(\mathbf{y}) - \mathcal{h}(\mathbf{y}')$ is homologous to the union $\delta$ of the following types of arcs:

- $[0, 1] \times \{y_i\}$ with orientation $\partial_i$;

- $[0, 1] \times \{y_i'\}$ with orientation $-\partial_i$;

- subarcs of $\{1\} \times a_i$ connecting from $\mathbf{y}$ to $\mathbf{y}'$; and

- subarcs of $\{0\} \times \mathcal{h}(a_i)$ connecting from $\mathbf{y}'$ to $\mathbf{y}$.

By homotoping $\delta$ to a page $\mathcal{S}$, we see that $[\delta] = \epsilon(\mathbf{y}, \mathbf{y}')$ with respect to the Heegaard diagram $(\Sigma, \beta, \alpha, z)$ given in Section 4.9.1. $\square$

**4.11.** *Twisted coefficients in Heegaard Floer homology.* — In this subsection we review the definition of Heegaard Floer homology with twisted coefficients, originally defined in [OSz2, Section 8], and prove a twisted coefficient analog of Theorem 4.9.4. We describe the construction for $\widehat{\mathrm{HF}}$; the construction for $\mathrm{HF}^+$ — which will be used in [III] — can be treated in a similar manner.

Fix a Spin$^c$-structure $\mathfrak{s}$ and a $k$-tuple of intersection points $\mathbf{y}_0$ such that $s_{\xi}(\mathbf{y}_0) = \mathfrak{s}$. A complete set of paths for $\mathfrak{s}$ based at $\mathbf{y}_0$ is the choice, for each $k$-tuple of intersection points $\mathbf{y}$ such that $s_{\xi}(\mathbf{y}) = \mathfrak{s}$, of a surface $C_{\mathbf{y}}$ which is the projection to $[0, 1] \times \Sigma$ of a surface representing an element of $H_2(\mathcal{X}, \mathbf{y}, \mathbf{y}_0)^4$.

A complete set of paths determines maps

$$\mathfrak{A}: H_2(\mathcal{X}, \mathbf{y}, \mathbf{y}') \to H_2([0, 1] \times \Sigma, \{0\} \times \beta \cup \{1\} \times \alpha) \simeq H_2(\mathcal{M})$$

<footnote n="4">⁴ We are identifying relative homology classes in $\mathcal{X}$ with relative homology classes in $[0, 1] \times \Sigma$ as necessary.</footnote>
for all $\mathbf{y}$ and $\mathbf{y}'$ such that $s_z(\mathbf{y}) = s_z(\mathbf{y}') = \mathfrak{s}$ by $\mathfrak{A}(\mathfrak{A}) = [C_{\mathbf{y}'} \cup A \cup -C_{\mathbf{y}}]$. This map is compatible with the action of $H_2(\mathbf{M})$ on $H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')$ and with the concatenation of chains with matching ends.

We define

$$ \widehat{\mathcal{C}\mathcal{F}}(\Sigma, \alpha, \beta, z, \mathfrak{s}) = \widehat{\mathcal{C}\mathcal{F}}(\Sigma, \alpha, \beta, z, \mathfrak{s}) \otimes_{\mathbf{F}} \mathbf{F}[H_2(\mathbf{M}; \mathbf{Z})] $$

as an $\mathbf{F}[H_2(\mathbf{M}; \mathbf{Z})]$-module, with differential

$$ \partial_{\mathbf{y}} = \sum_{s_z(\mathbf{y}') = \mathfrak{s}} \sum_{A \in H_2(\mathbf{X}, \mathbf{y}, \mathbf{y}')} \#(\widehat{\mathcal{M}}^{X, I=1}(\mathbf{y}, \mathbf{y}', A)/\mathbf{R}) e^{\mathfrak{A}(\mathfrak{A})\mathbf{y}'} $$

The homology of this complex is the Heegaard Floer homology with twisted coefficients $\widehat{\mathcal{H}}(\mathbf{M}, \mathfrak{s})$.

Consider the special Heegaard diagram constructed in Section 4.9.1. For every Spin$^c$-structure $\mathfrak{s} \in \mathrm{Spin}^c(\mathbf{M})$ we define the complex

$$ \widehat{\mathcal{C}\mathcal{F}}(\mathbf{S}, \mathbf{a}, \mathcal{h}(\mathbf{a}), \mathfrak{s}) = \widehat{\mathcal{C}\mathcal{F}}(\mathbf{S}, \mathbf{a}, \mathcal{h}(\mathbf{a}), \mathfrak{s}) \otimes_{\mathbf{F}} \mathbf{F}[H_2(\mathbf{M}; \mathbf{Z})] $$

with the differential induced by the differential on $\widehat{\mathcal{C}\mathcal{F}}(\Sigma, \beta, \alpha, z, \mathfrak{s})$.

**Theorem 4.11.1.** — $H_*(\widehat{\mathcal{C}\mathcal{F}}(\mathbf{S}, \mathbf{a}, \mathcal{h}(\mathbf{a}), \mathfrak{s}))$ is isomorphic to $\widehat{\mathcal{H}}(\Sigma, \beta, \alpha, z, \mathfrak{s})$ as $\mathbf{F}[H_2(\mathbf{M}; \mathbf{Z})]$-modules.

*Proof.* — Fix a distinguished $2g$-tuple of generators $\mathbf{y}_0$ such that $s_z(\mathbf{y}_0) = \mathfrak{s}$. We choose a complete set of paths $C_{\mathbf{y}}$ with the following property: if $\mathbf{y} = \tilde{\mathbf{y}} \cup \{x_i\}, \mathbf{y}' = \tilde{\mathbf{y}} \cup \{x'_i\}$ and $\mathbf{y}'' = \tilde{\mathbf{y}} \cup \{x''_i\}$, then $C_{\mathbf{y}} = C_{\mathbf{y}''} \cup D_i$ and $C_{\mathbf{y}'} = C_{\mathbf{y}''} \cup D'_i$, where $D_i$ and $D'_i$ are the surfaces corresponding to the thin strips connecting $x''_i$ to $x_i$ and $x'_i$, respectively; see Figure 1. With this choice of complete set of paths, we have $\mathfrak{A}(D_i) = \mathfrak{A}(D'_i) = 0$ for all $i$, so the proof of Theorem 4.9.4 goes through unchanged. $\square$

# 5. Moduli spaces of multisections

The goal of this section is to introduce the moduli spaces which will be used to define the chain maps

$$ \Phi : \widehat{\mathcal{C}\mathcal{F}}(\mathbf{S}, \mathbf{a}, \mathcal{h}(\mathbf{a})) \to \mathrm{PFC}_{2g}(\mathbb{N}, \alpha_0, \omega) $$

and

$$ \Psi : \mathrm{PFC}_{2g}(\mathbb{N}, \alpha_0, \omega) \to \widehat{\mathcal{C}\mathcal{F}}(\mathbf{S}, \mathbf{a}, \mathcal{h}(\mathbf{a})). $$

The definition of these chain maps can be viewed as a melding of ideas of Seidel [Se1, Se2] and Donaldson-Smith [DS].
Let $\mathbf{y} = \{y_1, \dots, y_{2g}\}$ be a generator of $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathcal{h}(\mathbf{a}))$, where $y_i \in a_i \cap \mathcal{h}(a_{\sigma(i)})$. Intuitively, $\mathbf{y}$ is mapped to an element of $\mathrm{PFC}_{2g}(\mathbb{N})$ through an intermediary called a *broken closed string* $\gamma_{\mathbf{y}}$. It is a union of closed curves in $\mathbb{N} = S \times [0, 1]/\sim$, obtained by taking the union of $y_i \times [0, 1]$, $i = 1, \dots, 2g$, and $c_i \times \{0\}$, $i = 1, \dots, 2g$, where $c_i$ is a subarc of $\mathcal{h}(a_i)$ which connects $\mathcal{h}(y_{\sigma(i)})$ to $y_i$. Note that there is a unique homotopy class of arcs from $\mathcal{h}(y_{\sigma(i)})$ to $y_i$, since $\mathcal{h}(a_i)$ is an arc (and not a closed curve). The arcs $y_i \times [0, 1]$, $c_i \times \{0\}$, $i = 1, \dots, 2g$, glue up to give a union of closed curves since $(\mathcal{h}(y_{\sigma(i)}), 0) \sim (y_{\sigma(i)}, 1)$.

**5.1. Symplectic cobordisms.** — We recall the stable Hamiltonian structure $(\alpha_0, \omega)$ on $\mathbb{N}$ from Section 3. The mapping torus $\mathbb{N}$ is given by:

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

where $\mathcal{h}$ is an area-preserving map of $S$ with zero flux. The projection $S \times [0, 2] \to [0, 2]$ defines a projection $\pi_{S^1}: \mathbb{N} \to S^1$, where we identify $S^1 \cong [0, 2]/0 \sim 2$. Here we made a slight modification to the definition of $\mathbb{N}$: the interval $[0, 1]$ in Section 2.1 is now replaced by $[0, 2]$.

The area form $\omega$ on $S$, the form $dt$ on $[0, 2]$, and the vector field $\partial_t$ on $S \times [0, 2]$ give well-defined objects on $\mathbb{N}$ which we still denote by $\omega$, $dt$, and $\partial_t$. By Equation (3.1.1) the form $\alpha_0$ is a positive multiple of $dt$. For simplicity we assume that $\alpha_0 = dt$. Then the stable Hamiltonian vector field is $R_0 = \partial_t$ and, by construction, the monodromy $\mathcal{h}$ is the first return map of $R_0$. In view of Lemma 2.1.2 we may assume that:

- all orbits of $R_0$ in $\mathrm{int}(\mathbb{N})$ satisfying $\mathcal{F} \le 2g$ are nondegenerate; and

- $R_0$ is negative Morse-Bott along $\partial \mathbb{N}$.

**Remark 5.1.1.** — Indeed, the stable Hamiltonian vector field $R_0$ on $\mathbb{N}$ has the same first return map as a Reeb vector field $R_\tau$, $\tau > 0$, by construction, and we could have taken $R_\tau$ to be Morse-Bott nondegenerate.

Let us write $W = \mathbf{R} \times [0, 1] \times S$ and $W' = \mathbf{R} \times \mathbb{N}$. Let

$$\Omega = ds \wedge dt + \omega, \quad \Omega' = ds \wedge dt + \omega$$

be the symplectic forms on $W$ and $W'$, where $s$ is the **R**-coordinate.

In this section we introduce the symplectic cobordisms $(W_+, \Omega_+)$ and $(W_-, \Omega_-)$, as well as their “compactifications” $(\overline{W_+}, \overline{\Omega_+})$ and $(\overline{W_-}, \overline{\Omega_-})$. The cobordism $(W_+, \Omega_+)$ interpolates from the stable Hamiltonian structure $([0, 1] \times S, (dt, \omega))$ at the positive end to the stable Hamiltonian structure $(\mathbb{N}, (\alpha_0, \omega))$ at the negative end, whereas the cobordism $(W_-, \Omega_-)$ goes from $(\mathbb{N}, (\alpha_0, \omega))$ at the positive end to $([0, 1] \times S, (dt, \omega))$ at the negative end.
<figure><img src="image_5.png" /><figcaption>FIG. 2. — The bases $B_+$ and $B_-$, The sides are identified. The dotted lines on the top and bottom indicate cylindrical ends. Both $B_+$ and $B_-$ are biholomorphic to a disk with an interior puncture and a boundary puncture. Here the location of $\bar{m}^d$ on $B_-$ is indicated by $\times$</figcaption></figure>

**5.1.1.** The symplectic cobordisms $(W_+, \Omega_+)$ and $(W_-, \Omega_-)$. — Consider the infinite cylinder $\mathbf{R} \times S^1 \simeq \mathbf{R} \times (\mathbf{R}/2\mathbf{Z})$ with coordinates $(s, t)$. Let $\pi_{S^1}: N \to S^1$ be the fibration $(x, t) \mapsto t$ and let

$$ \pi_{B'}: \mathbf{R} \times N = W' \to B' = \mathbf{R} \times S^1 $$

be the extension $(s, x, t) \mapsto (s, \pi_{S^1}(x, t))$. Let us write $N_s = \pi_{B'}^{-1}({\{s\} \times S^1})$ for $s \in \mathbf{R}$. We define $W_+ = \pi_{B'}^{-1}(B_+)$, where $B_+ = (\mathbf{R} \times (\mathbf{R}/2\mathbf{Z})) - B_+^c$ and $B_+^c$ is the subset $[2, \infty) \times [1, 2] \subset \mathbf{R} \times (\mathbf{R}/2\mathbf{Z})$ with the corners rounded. See the left-hand side of Figure 2. We write

$$ \pi_{B_+}: W_+ \to B_+ $$

for the restriction of $\pi_{B'}$. Note that the boundary of $W_+$ can be decomposed into two parts that meet along a codimension two corner: the vertical boundary $\partial_v W_+ = \pi_{B_+}^{-1}(\partial B_+)$, and the horizontal boundary $\partial_h W_+$, which is equal to the union of the boundaries of the fibers.

Similarly, we define $W_- = \pi_{B'}^{-1}(B_-)$, where $B_- = (\mathbf{R} \times (\mathbf{R}/2\mathbf{Z})) - B_-^c$ and $B_-^c$ is $(-\infty, -2] \times [1, 2]$ with the corners rounded. The projection

$$ \pi_{B_-}: W_- \to B_- $$

the vertical boundary $\partial_v W_-$, and the horizontal boundary $\partial_h W_-$ are defined analogously.

The symplectic form $\Omega_+$ (resp. $\Omega_-$) is the restriction of

$$ ds \wedge dt + \omega = ds \wedge dt + d_S \beta_t $$

to $W_+$ (resp. $W_-$). By this we mean the following: On $\mathbf{R} \times S \times [0, 2]$, we take the symplectic form $ds \wedge dt + \omega$. Then the symplectic form glues under the identification $(s, x, 2) \sim (s, \hbar(x), 0)$.
We also write $cl(B_+)$, $cl(B_-)$ for the compactifications of $B_+$, $B_-$, obtained by ad-
joining the points at infinity $\mathfrak{p}_+$ corresponding to $s = +\infty$, and $\mathfrak{p}_-$ corresponding to
$s = -\infty$. Therefore $cl(B_+)$ and $cl(B_-)$ are isomorphic to the closed unit disk with one
marked point on the interior and one marked point on the boundary.

**5.1.2.** *The extended cobordisms $(\overline{W}_+, \overline{\Omega}_+)$ and $(\overline{W}_-, \overline{\Omega}_-)$. — We now extend $(W_+$, $\Omega_+$) to $(\overline{W}_+, \overline{\Omega}_+)$, which corresponds to capping off each fiber $S$ by a disk; the definition of $(\overline{W}_-, \overline{\Omega}_-)$ is analogous.

We first define the capped-off surface $\bar{S}$: Let $D^2 = \{\rho \le 1\}$ be a disk with polar coordinates $(\rho, \phi)$. We write $z_\infty$ for the origin $\rho = 0$. Let $(y, \theta)$ be the coordinates on a neighborhood $\nu(\partial S) \simeq [-\epsilon, \epsilon] \times \mathbf{R}/\mathbf{Z}$ of $\partial S = \{0\} \times \mathbf{R}/\mathbf{Z}$ (inside a slight extension of $S$), as before. Then $\bar{S} = (S \sqcup D^2)/\sim$, where $(y, \theta) \in \nu(\partial S)$ is identified with $(\frac{1}{y+1}, -2\pi\theta) \in D^2$.

For every integer $m > 2g$ we define

$$
\bar{h}_m : \bar{S} \xrightarrow{\sim} \bar{S}
$$

as a smooth extension of $\bar{h} : S \xrightarrow{\sim} S$, depending on $m$, such that $\bar{h}|_{D^2}$ is the diffeomorphism of $D^2$ given by

$$
(\rho, \phi) \mapsto (\rho, \phi + \nu_m(\rho)),
$$

where $\nu_m : [0, 1] \to \mathbf{R}$ is a smooth function which satisfies the following:

$$
- \nu_m(\rho) = \frac{2\pi}{m} \text{ for } \rho \le \frac{1}{2};
$$

$$
- \nu_m(\rho) \text{ is increasing for } \frac{1}{2} < \rho < \frac{3}{4};
$$

$$
- \nu_m(\rho) \text{ is decreasing and independent of } m \text{ for } \frac{3}{4} < \rho < 1;
$$

$$
- \nu_m(\frac{3}{4}) \ll \frac{2\pi}{2g} \text{ and } \nu_m(1) = 0;
$$

$$
- \nu_{\infty} := \lim_{m \to \infty} \nu_{m} \text{ exists in the } \mathbb{C}^{k}\text{-topology for } k \gg 0.
$$

In particular, $\nu_\infty(\rho) = 0$ for $\rho \le \frac{1}{2}$. Taking the limit $m \to \infty$ becomes important
starting from Section 7.7, but until then we just need $m \gg 0$ and we simply write $\bar{h} = \bar{h}_m$
and $\nu = \nu_m$.

We then define the mapping torus

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

When we do not need to specify $m$, we will simplify the notation by writing $\overline{N}_m = \overline{N}$.
Note that, although $\overline{N}_m$ depends on $m$ and $\nu_m$, all the $\overline{N}_m$ are diffeomorphic. The closed
manifold $\overline{N}$ is obtained from $M$ by a 0-surgery along the binding of the open book.

Let $\overline{\omega}$ be an area form on $\overline{S}$ which extends $\omega$ and equals $\rho d\rho \wedge d\phi$ on $D^2$. We then extend the stable Hamiltonian structure $(\alpha_0, \omega)$ on $\overline{N}$ to the stable Hamiltonian structure
($\bar{\alpha}_0, \bar{\omega}$) with Hamiltonian vector field $\bar{R}_0$ on $\bar{N}$, where we still have $\bar{\alpha}_0 = dt$ and $\bar{R}_0 = \partial_t$. Moreover, the orbit

$$ \delta_0 = \{z_\infty\} \times [0, 2] / \sim $$

is the only simple closed orbit of $\bar{R}_0$ on $\bar{N} - N$ which intersects $\bar{S}$ at most $2g$ times; it is called $\delta_0$ since it lies on the level set $\rho = 0$. The 2-plane field of the stable Hamiltonian structure is $\ker \bar{\alpha}_0 = T\bar{S}$.

We now define $\bar{W}_+$ and $\bar{W}_-$. First define extensions $\bar{W} = \mathbf{R} \times [0, 1] \times \bar{S}$ of $W = \mathbf{R} \times [0, 1] \times S$ and $\bar{W}' = \mathbf{R} \times \bar{N}$ of $W' = \mathbf{R} \times N$. Let $\bar{\pi}_{S^1}: \bar{N} \to S^1$ be the fibration $(x, t) \mapsto t$ and let

$$ \bar{\pi}_{B'}: \bar{W}' = \mathbf{R} \times \bar{N} \to B' = \mathbf{R} \times S^1 $$

be the extension $(s, x, t) \mapsto (s, \bar{\pi}_{S^1}(x, t))$. For $*=+\text{ or }-$, we set $\bar{W}_* = \bar{\pi}_{B'}^{-1}(B_*)$ and write

$$ \bar{\pi}_{B_*}: \bar{W}_* \to B_* $$

for the restriction of $\bar{\pi}_{B'}$ to $\bar{W}_*$. We also write

$$ \bar{\pi}_B: \bar{W} \to B $$

for the restriction of $\bar{\pi}_{B'}$ to $\bar{W}$.

The symplectic forms $\bar{\Omega}_+$, $\bar{\Omega}_-$, $\bar{\Omega}$, $\bar{\Omega}'$ for $\bar{W}_+$, $\bar{W}_-$, $\bar{W}$, $\bar{W}'$ are obtained from $ds \wedge dt + \bar{\omega}$ by gluing and restricting as necessary.

Let us write

$$ V := \bar{N} - \text{int}(N) = (D^2 \times [0, 2])/(x, 2) \sim (\bar{h}(x), 0). $$

We then identify $\varphi: V \to D^2 \times (\mathbf{R}/2\mathbf{Z})$ via $(\rho e^{i\varphi}, t) \mapsto (\rho e^{i(\varphi+tv(\rho)/2)}, t)$. Note that $\varphi$ relates two coordinate systems:

(i) the “Reeb coordinates” $(x, t)$ on $D^2 \times [0, 2]$ such that $\bar{R}_0 = \partial_t$ and $(x, 2) \sim (\bar{h}(x), 0)$, and

(ii) the “balanced coordinates” on $D^2 \times (\mathbf{R}/2\mathbf{Z})$ such that $\bar{R}_0 = \partial_t + \frac{v(\rho)}{2}\partial_\varphi$ and $(x, 2) \sim (x, 0)$.

For $*=+\text{ or }-$, let

$$ \bar{\pi}_{D^2}: \bar{W}_* \cap (\mathbf{R} \times V) \to D^2 $$

be the projection of $\bar{W}_* \cap (\mathbf{R} \times V)$ to $V$, followed by the projection of $V$ to $D^2$ via the identification $\varphi$, i.e., with respect to the balanced coordinates.

We also choose the marked point $\bar{m} = (\bar{m}^b, \bar{m}') \in \bar{W}_-$, where $\bar{m}^b = (0, \frac{3}{2}) \in B_-$ and $\bar{m}' = z_\infty \in \bar{S}$. The marked point will play a crucial role in the definition of the chain map $\Psi$.
<figure><img src="image_6.png" /><figcaption>FIG. 3. — Extended arcs in $D^2$. The black arcs are portions of the $\bar{a}_i$ and the red ones are portions of the $\bar{h}(\bar{a}_i)$. The dashed arc is one of the $\bar{a}_{i,j} - \bar{a}_i$. (Color figure online)</figcaption></figure>

**5.2. Lagrangian boundary conditions.** — Let us first write

$$L_a = \mathbf{R} \times \{1\} \times \mathbf{a} \quad \text{and} \quad L_{fi}(a) = \mathbf{R} \times \{0\} \times \mathbf{h}(a)$$

for the Lagrangians in $W$ that are used in the definition of $\widehat{\mathrm{CF}}(S, \mathbf{a}, \mathbf{h}(\mathbf{a}))$.

**5.2.1. Lagrangian boundary conditions for $W_\pm$.** — The symplectic fibration

$$\pi_{B_+} : (W_+, \Omega_+) \to (B_+, ds \wedge dt)$$

admits a *symplectic connection*, defined as the $\Omega_+$-orthogonal of the tangent plane to the fibers. The symplectic connection is spanned by $\partial_s$ and $\partial_t$ if we consider $\Omega = ds \wedge dt + \omega$ on $\mathbf{R} \times S \times [0, 2]$ before the identification $(s, x, 2) \sim (s, \mathbf{h}(x), 0)$. (We recall that $W_+$ is defined as a subset of $\mathbf{R} \times N = \mathbf{R} \times (S \times [0, 2]) / \sim$.)

We first place a copy of the basis $\mathbf{a}$ on the fiber $\pi_{B_+}^{-1}(3, 1)$ and take its parallel transport along $\partial B_+$ using the symplectic connection. The parallel transport sweeps out a Lagrangian submanifold $L_a^+$ of $(W_+, \Omega_+)$. Let $L_{\bar{a}_i}^+$ be the connected component of $L_a^+$ given by parallel transport of $a_i$. Since the symplectic connection is spanned by $\partial_s$ and $\partial_t$ on $\mathbf{R} \times S \times [0, 2]$, over the strip $\{s \ge 3, t \in [0, 1]\}$ we have:

$$L_a^+ \cap \{s \ge 3, t = 0\} = \{s \ge 3\} \times \mathbf{h}(a) \times \{0\},$$

$$L_a^+ \cap \{s \ge 3, t = 1\} = \{s \ge 3\} \times \mathbf{a} \times \{1\}.$$

Similarly, the Lagrangian submanifold $L_a^-$ on the vertical boundary of $(W_-, \Omega_-)$ is obtained by taking the parallel transport of a copy of $\mathbf{a}$ — placed on the fiber $\pi_{B_-}^{-1}(-3, 1)$ — by the symplectic connection.

**5.2.2. Extended Lagrangian boundary conditions.** — Fix $k_0 > 2g$. In what follows we assume that the basis arcs $a_i$, $i = 1, \dots, 2g$, depend on an integer $m \gg 0$ (i.e., sufficiently large that the conditions below make sense) and satisfy some additional conditions. Let $E \subset \partial D^2$ be the set of endpoints of $\cup_{i=1,\dots,2g} a_i$ and let $y_1(m), \dots, y_{4g}(m)$ be the points of $E$
in counterclockwise order. We denote by $\phi(y_i(m))$ the $\phi$-coordinate of the endpoint $y_i(m)$. Then we assume the following:

- $a_i = a_i(m)$ is oriented;

- $0 < \phi(y_1(m)) < \frac{2\pi}{m}$;

- for $1 \le j_1, j_2 \le 4g$,

$$ \phi^{j_1, j_2}(m) := \phi(y_{j_1}(m)) - \phi(y_{j_2}(m)) $$

is an integer multiple of $\frac{2\pi}{m}$;

- $\frac{\phi^{j_1, j_2}(m) \cdot m}{2\pi} \ge k_0$ for all $j_1 > j_2$, i.e., the spacing between any two $\phi(y_{j_1}(m))$ and $\phi(y_{j_2}(m))$ with $j_1 > j_2$ is at least $k_0$ times $\frac{2\pi}{m}$;

- $\lim_{m \to +\infty} \frac{\phi^{j_1, j_2}(m) \cdot m}{2\pi} = +\infty$ and $\lim_{m \to +\infty} \phi^{j_1, j_2}(m) = 0$;

- for all quadruples $j_1 > j_2, j_3 > j_4$, $\lim_{m \to +\infty} \frac{\phi^{j_1, j_2}(m)}{\phi^{j_3, j_4}(m)} \ne 1$ unless $(j_1, j_2) = (j_3, j_4)$.

Assume without loss of generality that the initial point of $a_i$ is $x_i$ and the terminal point of $a_i$ is $x'_i$. Let $\bar{a}_i$ be the (oriented) extension of $a_i \subset S$ to $\bar{S}$, obtained by attaching two radial rays $\bar{a}_{i,j} = \{0 \le \rho \le 1, \phi = \phi_{i,j}\}, j = 0, 1$, where $\phi_{i,j}$ is a constant. Here $\bar{a}_{i,0}$ (resp. $\bar{a}_{i,1}$) is the initial (resp. terminal) segment of $\bar{a}_i$. We also define the extension $\bar{a}_{i,j} = \{-1 < \rho \le 1, \phi = \phi_{i,j}\}, j = 0, 1$, of $\bar{a}_{i,j} = \{0 \le \rho \le 1, \phi = \phi_{i,j}\}$. Here $(\rho, \phi)$ with $\rho < 0$ is a notation for $(-\rho, \phi + \pi)$.

We write $\bar{a} = \{\bar{a}_1, \dots, \bar{a}_{2g}\}$. Then $L_{\bar{a}}^{\pm}$ is the extension of $L_{\bar{a}}^{\pm}$, obtained by the parallel transport of a copy of $\bar{a}$, placed at $\bar{\pi}_{B_+}^{-1}(3, 1)$ or $\bar{\pi}_{B_-}^{-1}(-3, 1)$. We similarly define $L_{\hat{a}}^{\pm}$, $L_{\hat{a}_i}^{\pm}$, $L_{\bar{a}_i}^{\pm}$, $L_{\hat{a}_i \cup \bar{a}_i}^{\pm}$, and $L_{\bar{a}_i \cup \bar{a}_i}^{\pm}$, where $\hat{a}_i = \bar{a}_i - \{z_\infty\}$ and $\hat{a} = (\hat{a}_1, \dots, \hat{a}_{2g})$.

**Definition 5.2.1.** — A bigon (with acute angles) contained in $D^2$ and bounded by $\bar{a}_{i,j}$ and $\bar{h}(\bar{a}_{i,j})$ will be called a thin strip. The portion of a thin strip contained in $D_{1/2} = \{\rho \le \frac{1}{2}\}$ will be called a thin wedge. See Figure 3.

**5.3.** *Almost complex structures and moduli spaces for W, $\overline{W}$, W', and $\overline{W'}$.* — In this subsection we specify the almost complex structures and moduli spaces for $W = \mathbf{R} \times [0, 1] \times S$, $\overline{W} = \mathbf{R} \times [0, 1] \times \overline{S}$, $W' = \mathbf{R} \times N$, and $\overline{W'} = \mathbf{R} \times \overline{N}$.

**Convention 5.3.1.** — When we write $\mathbf{y}$ or $\mathbf{\gamma}$ (with possible superscripts, subscripts and other decorations), it is assumed that every intersection point in $\mathbf{y}$ is contained in $S$ and every orbit in $\mathbf{\gamma}$ is contained in $N$. (By abuse of notation, we will write $\mathbf{y} \subset S$ and $\mathbf{\gamma} \subset N$.) In particular, $\mathbf{y}$ and $\mathbf{\gamma}$ do not contain any multiples of $z_\infty$ or $\delta_0$.

**5.3.1.** *Holomorphic maps to W and $\overline{W}$.* — Let $W = \mathbf{R} \times [0, 1] \times S$ and $\overline{W} = \mathbf{R} \times [0, 1] \times \overline{S}$. Also let $\Omega = ds \wedge dt + \omega$ and $\bar{\Omega} = ds \wedge dt + \bar{\omega}$. Then $\mathcal{J}_W$ is defined as the set of $C^\infty$-smooth $\Omega$-admissible almost complex structures on W.
The analogous space $\mathcal{J}_{\overline{W}}$ for $\overline{W}$ is slightly more complicated:

**Definition 5.3.2.** — Fix $\varepsilon, \delta > 0$ sufficiently small and $k \gg 0$. Then $\mathcal{J}_{\overline{W}}$ is the space of $C^{\infty}$-smooth $\overline{\Omega}$-admissible almost complex structures $\overline{J}$ on $\overline{W}$ which satisfy the following:

(1) there exists an $\overline{\Omega}$-admissible almost complex structure $\overline{J}_0$ which restricts to the standard complex structure on the subsurface $D^2 \subset \overline{S}$ of each fiber;

(2) $\overline{J} = \overline{J}_0$ on $\{\rho \le \varepsilon\} \cup W$ and $|\overline{J} - \overline{J}_0|_k < \delta$ over $\overline{W}$;

(SR) $\overline{J}$ is a split complex structure on $\mathbf{R} \times \{0\}$ times a neighborhood of $\overline{h}(\overline{\mathbf{a}})$ and on $\mathbf{R} \times \{1\}$ times a neighborhood of $\overline{\mathbf{a}}$; moreover $\overline{h}(\overline{\mathbf{a}})$ and $\overline{\mathbf{a}}$ are real analytic with respect to the fiber complex structures on $\overline{\pi}_B^{-1}(s, 0)$ and $\overline{\pi}_B^{-1}(s, 1)$, respectively.

Here $|\cdot|_k$ is some fixed $C^k$-norm.

**Remark 5.3.3.** — Condition (SR) and the analogous Condition (SR) for $\overline{J}'$, given later, are technical conditions that allow us to apply Schwarz reflection on a neighborhood of every point of the Lagrangian boundary. It is easily satisfied and the inclusion of this condition does not affect the proofs of regularity.

We denote by $\overline{J}$ the restriction of $\overline{J}$ to $W$. (We will be assuming that $\overline{J} \in \mathcal{J}_W$ and admits an extension to $\overline{J} \in \mathcal{J}_{\overline{W}}$.) Let $S_{\mathbf{a}, \overline{h}(\mathbf{a})}$ be the set of $2g$-tuples of intersection points of $\mathbf{a} \cap \overline{h}(\mathbf{a})$. Given $\mathbf{y}, \mathbf{y}' \in S_{\mathbf{a}, \overline{h}(\mathbf{a})}$, let $M_J(\mathbf{y}, \mathbf{y}')$ be the moduli space of HF curves from $\mathbf{y}$ to $\mathbf{y}'$ with respect to $\overline{J}$ which are contained in $\mathbf{R} \times [0, 1] \times S$.

We next discuss holomorphic curves in $\overline{W}$.

**Definition 5.3.4.** — Let $\mathbf{y}, \mathbf{y}'$ be $k$-tuples of $\mathbf{a} \cap \overline{h}(\mathbf{a})$ and $\overline{J} \in \mathcal{J}_{\overline{W}}$. Then a degree $k \le 2g$ multisection $\overline{u}$ from $\mathbf{y}$ to $\mathbf{y}'$ in $(\overline{W}, \overline{J})$ is a holomorphic map $\overline{u} : \dot{\mathcal{F}} \to \overline{W}$ which is a degree $k$ multisection of $\overline{\pi}_B : \overline{W} \to B$, satisfies the conditions of Definition 4.3.1 with $L_\alpha$ and $L_\beta$ replaced by $L_{\overline{\mathbf{a}}} = \mathbf{R} \times \{1\} \times \overline{\mathbf{a}}$ and $L_{\overline{\overline{h}}(\overline{\mathbf{a}})} = \mathbf{R} \times \{0\} \times \overline{h}(\overline{\mathbf{a}})$ and $(k, l)$ replaced by $(2g, k)$, and is asymptotic to $\mathbf{y}$ and $\mathbf{y}'$ at the positive and negative ends.

Note that we require $\partial \dot{\mathcal{F}}$ to be mapped to $L_{\overline{\mathbf{a}}}, L_{\overline{\overline{h}}(\overline{\mathbf{a}})}$ and be disjoint from $L_{\overline{\mathbf{a}}} - L_{\overline{\mathbf{a}}}, L_{\overline{\overline{h}}(\overline{\mathbf{a}})} - L_{\overline{\overline{h}}(\overline{\mathbf{a}})}$.

**Definition 5.3.5.** — The section at infinity $\sigma_\infty$ is the map $\sigma_\infty : \mathbf{R} \times [0, 1] \to \overline{W}$ such that $\sigma_\infty(s, t) = (s, t, z_\infty)$. By abuse of notation we will also refer to its image as the section at $\infty$.

The section at infinity is $\overline{J}$-holomorphic for every $\overline{J} \in \mathcal{J}_{\overline{W}}$. Let $z_\infty^\dagger \in D_{1/2}^2 \subset \overline{S}$ be a point with $\rho$-coordinate less than $1/2$ and in the complement of all arcs $\overline{a}_{i,j}$ and $\overline{h}(\overline{a}_{i,j})$. The orbit $\mathcal{O}(z_\infty^\dagger)$ of $z_\infty^\dagger$ under the action of $\overline{h}$ consists of $m$ points, and each thin wedge in $D_{1/2}^2$ between $\overline{a}_{i,j}$ and $\overline{h}(\overline{a}_{i,j})$ contains exactly one point of $\mathcal{O}(z_\infty^\dagger)$. Let $\sigma_\infty^\dagger = \mathbf{R} \times [0, 1] \times \mathcal{O}(z_\infty^\dagger)$. (Note that this multisection does not have Lagrangian boundary conditions; it will only be used to impose topological constraints on the HF curves.)
**Definition 5.3.6.** — Given a degree $k$ multisection $\bar{u}$ of $\overline{W}$, we define $n(\bar{u}) = \langle \bar{u}, \sigma_{\infty}^{\dagger} \rangle$, where $\langle \cdot, \cdot \rangle$ denotes the algebraic intersection between the images.

The intersection number $n(\bar{u})$ is a homological invariant since $z_{\infty}^{\dagger}$ was chosen so that $O(z_{\infty}^{\dagger})$ is disjoint from the Lagrangian arcs.

**Lemma 5.3.7.** — The intersection number $n(\bar{u})$ satisfies the following properties:

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

(2) if $\langle \bar{u}, \sigma_{\infty} \rangle > 0$, then $n(\bar{u}) \ge m$;

(3) if $n(\bar{u}) = 1$, then $\bar{u}$ projects onto a thin strip; and

(4) $n(\bar{u})$ is independent of the choice of $z_{\infty}^{\dagger}$.

**Proof.** — (1) follows from the positivity of intersections of pseudo-holomorphic maps in dimension four, (2) is a consequence of the fact that holomorphic maps are open, and (3) is a consequence of the positivity of intersections and of the fact that every thin wedge contains only one point of $O(z_{\infty}^{\dagger})$. Finally, (4) follows from the fact that different choices for $\sigma_{\infty}^{\dagger}$ are connected by a path of multisections which are disjoint from the Lagrangian boundary conditions. $\square$

**Definition 5.3.8.** — Let $\mathbf{y}$ and $\mathbf{y}'$ be $k$-tuples of $\mathbf{a} \cap \mathbf{h}(\mathbf{a})$. Then $M_{\bar{J}}(\mathbf{y}, \mathbf{y}')$ is the moduli space of degree $k$ multisections $\bar{u}$ of $(\overline{W}, \bar{J})$ from $\mathbf{y}$ to $\mathbf{y}'$.

**Modifier.** For any moduli space $M_{\star_1}(\star_2)$ we may place modifiers * as in $M_{\star_1}^*(\star_2)$ to denote the subset of $M_{\star_1}(\star_2)$ satisfying *. Typical self-explanatory modifiers are $I = i$, $n = m$, and $\deg = k$. (Note however that the degree can be inferred from $\star_2$.) $* = s$ means “somewhere injective”.

The following lemma is an easy consequence of Lemma 5.3.7, in particular of points (1) and (4).

**Lemma 5.3.9.** — $M_{\bar{J}}^{n=0}(\mathbf{y}, \mathbf{y}') = M_{\bar{J}}(\mathbf{y}, \mathbf{y}')$ if $\mathbf{y}, \mathbf{y}' \in S_{\mathbf{a}, \mathbf{h}(\mathbf{a})}$.

**5.3.2. Holomorphic maps to $W'$ and $\overline{W}$.** — Recall the stable Hamiltonian structures $(\alpha_0, \omega)$ on $\mathbb{N}$ and $(\bar{\alpha}_0, \bar{\omega})$ on $\overline{\mathbb{N}}$. We will denote by $\mathbb{R}_0$ the Hamiltonian vector field of $(\alpha_0, \omega)$ on $\mathbb{N}$ and by $\overline{\mathbb{R}}_0$ the Hamiltonian vector field of $(\bar{\alpha}_0, \bar{\omega})$ on $\overline{\mathbb{N}}$. Clearly $\mathbb{R}_0$ is the restriction of $\overline{\mathbb{R}}_0$ to $\mathbb{N}$. Since $\partial\mathbb{N}$ is a Morse-Bott torus of $\overline{\mathbb{R}}_0$, we briefly review Morse-Bott theory in the context of periodic Floer homology. A more extensive reference is [0, Section 4], where a completely analogous discussion is given for embedded contact homology.

While the closed orbits of $\overline{\mathbb{R}}_0$ are nondegenerate on $\mathrm{int}(\mathbb{N}) \cup \mathrm{int}(\mathbb{V}) = \overline{\mathbb{N}} - \partial\mathbb{N}$, the orbits on $\partial\mathbb{N}$ form a *negative* Morse-Bott torus, i.e., $\partial\mathbb{N}$ is foliated by closed orbits whose
linearized first return map at any point $p \in \partial N$ is given by a matrix $\begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix}$ with $a < 0$ with respect to an oriented basis ($v_1, v_2$) of the tangent space to the fiber of the fibration $\bar{N} \to S^1$ passing through $p$. In this basis $v_1$ is transverse to $\partial N$ and $v_2$ is tangent to $\partial N$.

We denote the Morse-Bott orbit family associated to $\partial N$ by $\mathcal{N}$, i.e., $\mathcal{N}$ is the quotient of $\partial N$ by the flow. (Previously we had used $\mathcal{N}$ for the Morse-Bott family for the Reeb vector field $R_\alpha$ on $\partial N$; for the rest of the paper $R_\alpha$ is replaced by $\bar{R}_0$.) We choose a Morse function $\bar{g}: \mathcal{N} \to \mathbb{R}$ with two critical points: a maximum $h$ and a minimum $e$. After perturbing the Hamiltonian vector field by $\bar{g}$, the orbits $e$ and $h$ become nondegenerate elliptic and hyperbolic, respectively. For this reason, in constructing orbit sets, $e$ will be treated as an elliptic orbit and $h$ as a hyperbolic orbit.

We define the following set of orbits. Let $\mathcal{P}$ be the set of simple orbits of $R_0$ in $\text{int}(N)$. All orbits in $\mathcal{P}$ are nondegenerate. We also define

$$ \hat{\mathcal{P}} = \mathcal{P} \cup \{e, h\} \quad \text{and} \quad \bar{\mathcal{P}} = \hat{\mathcal{P}} \cup \{\delta_0\}, $$

together with the sets $\hat{\mathcal{O}}_k$, $\bar{\mathcal{O}}_k$ of orbit sets constructed respectively from $\hat{\mathcal{P}}$ and $\bar{\mathcal{P}}$ which intersect $\bar{S} \times \{0\}$ exactly $k$ times.

Let $\mathcal{J}_W'$ be the space of $C^\infty$-smooth $(\alpha_0, \omega)$-adapted almost complex structures $J'$ on $W' = \mathbb{R} \times N$ satisfying the following local Schwarz reflection condition:

(SR) $J'$ is a split complex structure on $\mathbb{R} \times [1, 2]$ times a neighborhood of $\mathbf{a}$; moreover $\mathbf{a}$ is real analytic with respect to the fiber complex structure on $\bar{\pi}_B^{-1}(s, t)$ for $t \in [1, 2]$.

A $\mathcal{J}_W'$-holomorphic Morse-Bott building is one of the following objects:

(i) a union of negative gradient flow trajectories from $h$ to $e$, or

(ii) a $\mathcal{J}_W'$-holomorphic map $u: \dot{\mathbb{F}} \to W'$ from an orbit set $\tilde{\mathbf{y}}_+ \in \mathcal{P} \cup \mathcal{N}$ to an orbit set $\tilde{\mathbf{y}}_- \in \mathcal{P} \cup \mathcal{N}$, together with half-infinite negative gradient flow trajectories connecting $h$ to positive ends of $u$ at $\mathcal{N} - \{e, h\}$ and negative ends of $u$ at $\mathcal{N} - \{e, h\}$ to $e$, or

(iii) a union of (i) and (ii).

See [0] for more details.

**Definition 5.3.10.** — Given $\boldsymbol{\gamma}, \boldsymbol{\gamma}' \in \hat{\mathcal{O}}_k$ and $J' \in \mathcal{J}_W'$, let $\mathcal{M}_J(\boldsymbol{\gamma}, \boldsymbol{\gamma}')$ be the moduli space of $J'$-holomorphic Morse-Bott buildings in $W'$ from $\boldsymbol{\gamma}$ to $\boldsymbol{\gamma}'$.

**Remark 5.3.11.** — The stable Hamiltonian structure $(\alpha_0, \omega)$ can be perturbed to a stable Hamiltonian structure $(\alpha'_0, \omega')$ as described in Section 3.7; the resulting Hamiltonian vector field has two nondegenerate orbits corresponding to $e$ and $h$ (and denoted by the same names) instead of the Morse-Bott torus $\mathcal{N}$. Any almost complex structure $J' \in \mathcal{J}_W'$ is then perturbed to an almost complex structure adapted to $(\alpha'_0, \omega')$.
Because of the simple nature of the J'-holomorphic Morse-Bott buildings, and since $\tilde{\mathcal{O}}_k$ for $k \le 2g$ are finite sets, for a sufficiently small perturbation $(\alpha'_0, \omega')$ of $(\alpha_0, \omega)$ and for every $\mathbf{\gamma}, \mathbf{\gamma}' \in \mathcal{O}_k$, there is a bijection between the ECH index one moduli spaces $\mathcal{M}_{\mathcal{J}'}^{1}(\mathbf{\gamma}, \mathbf{\gamma}')$ and the index one moduli spaces of holomorphic maps from $\mathbf{\gamma}$ to $\mathbf{\gamma}'$ for the perturbation of $\mathcal{J}'$ adapted to $(\alpha'_0, \omega')$. See [0, Lemma 4.4.5] and [0, Lemma 7.1.2].

Since the Morse-Bott situation and its generic perturbation are completely equiv-
alent, we will go from one point of view to the other as more convenient, often without
explicit mention. For this reason, we will still denote by $\mathcal{J}'$ a perturbation of $\mathcal{J}'$ adapted to $(\alpha'_0, \omega')$,
and by $\mathcal{M}_{\mathcal{J}'}(\mathbf{\gamma}, \mathbf{\gamma}')$ the moduli spaces of holomorphic maps with respect to the perturbation of $\mathcal{J}'$.

The definition of the analogous space $\mathcal{J}_{\overline{W}}$ for $\overline{W}' = \mathbf{R} \times \overline{\mathbf{N}}$ is slightly more complicated, but completely analogous to Definition 5.3.2. We first discuss the Morse-Bott version $\mathcal{J}_{\overline{W}'}^{\text{MB}}$. Let $D_{\varepsilon}^2 = \{|z| \le \varepsilon\} \subset \mathbf{C}$ and, for $\varepsilon \le 1$, let $V_{\varepsilon}$ be the mapping torus of $h|_{D_{\varepsilon}^2}$. For $\varepsilon = 1$ this coincides with the already defined solid torus V.

*Definition 5.3.12.* — Fix $\varepsilon, \delta > 0$ sufficiently small and $k \gg 0$. Then $\mathcal{J}_{\overline{W}'}^{\text{MB}}$ is the space of $C^{\infty}$-smooth $(\overline{\alpha}_0, \overline{\omega})$-adapted almost complex structures $\overline{\mathcal{J}}'$ on $\mathbf{R} \times \overline{\mathbf{N}}$ which satisfy the following:

(1) there exists an $(\bar{\alpha}_0, \bar{\omega})$-adapted almost complex structure $\overline{\mathcal{J}}'_0$ which restricts to the standard complex structure on the subsurface $D^2 \subset \bar{S}$ of each fiber;

(2) $\overline{\mathcal{J}}' = \overline{\mathcal{J}}'_0$ on $\mathbf{R} \times (V_\varepsilon \cup \mathbf{N})$ and $|\overline{\mathcal{J}}' - \overline{\mathcal{J}}'_0|_k < \delta$ over $\mathbf{R} \times \overline{\mathbf{N}}$;

(SR) $\overline{\mathcal{J}}'$ is a split complex structure on $\mathbf{R} \times [1, 2]$ times a neighborhood of $\overline{\mathbf{a}}$; moreover $\overline{\mathbf{a}}$ is real analytic with respect to the fiber complex structure on $\overline{\pi_B^{-1}(s, t)}$ for $t \in [1, 2]$.

Here $| \cdot |_k$ is some fixed $C^k$-norm.

The Morse-Bott theory for $\overline{W}'$ is more subtle than that of $W'$, and therefore we will discuss it in more detail. In the following definition, all $\overline{\mathcal{J}}'$-holomorphic maps are allowed to be nodal and to have disconnected domains. Let $\mathbf{\gamma}_+ = \gamma_{+,1}^{k_1} \dots \gamma_{+,m_+}^{k_{m_+}} h^a$ be an orbit set such that $\mathbf{\gamma}_{+,i} \in \mathcal{P} \cup \{e, \delta_0\}$ and $a \in \{0, 1\}$. Let $\mathbf{\gamma}_- = \gamma_{-,1}^{l_1} \dots \gamma_{-,m_-}^{l_{m_-}} e^b$ be an orbit set such that $\mathbf{\gamma}_{-,i} \in \mathcal{P} \cup \{h, \delta_0\}$ and $b \ge 0$.

*Definition 5.3.13 (Morse-Bott holomorphic buildings)* — Let $\overline{\mathcal{J}}'$ be an almost complex structure in $\mathcal{J}_{\overline{W}'}^{\text{MB}}$ and let $\mathbf{\gamma}_+ = \gamma_{+,1}^{k_1} \dots \gamma_{+,m_+}^{k_{m_+}} h^a$ and $\mathbf{\gamma}_- = \gamma_{-,1}^{l_1} \dots \gamma_{-,m_-}^{l_{m_-}} e^b$ be orbit sets as above. A $\overline{\mathcal{J}}'$-holomorphic Morse-Bott building $u$ from $\mathbf{\gamma}$ to $\mathbf{\gamma}'$ consists of

(1) a set of $\ell = 0$ or 1 negative gradient flow trajectories (with $a \ge \ell$) from $h$ to $e$ in $\mathcal{N}$, and

(2) a (possibly empty) set of $\overline{\mathcal{J}}'$-holomorphic maps $u_i : \dot{\mathcal{F}}_i \to \overline{W}'$, for $i = 1, \dots, n$, such that the following hold:

(a) Positive ends of $u_n$ converge to multiples of $\gamma_{+,i}$ with total multiplicity $k_i$ and to a simple orbit in $\mathcal{N} - \{e\}$ if $a - \ell \neq 0$.

(b) For $i = 1, \dots, n-1$, the negative ends of $u_{i+1}$ are paired with the positive ends of $u_i$. For each pair, the two ends either converge to the same orbit, or they converge to covers of
orbits in $\mathcal{N}$ with the same multiplicity and there is a negative gradient flow line of $\bar{g}$ in $\mathcal{N}$ from the limit of $u_{i+1}$ to the limit of $u_i$.

(c) Negative ends of $u_1$ converge to multiples of $\gamma_{-,i}$ with total multiplicity $l_i$ and to multiple covers of orbits in $\mathcal{N} - \{h\}$ with total multiplicity $b - \ell$.

The stable Hamiltonian structure $(\bar{\alpha}_0, \bar{\omega})$ can be perturbed as described in Section 3.7 — in fact, $(\mathbb{N}^+, \bar{\alpha}_0, \bar{\omega})$ embeds in $(\bar{\mathbb{N}}, \bar{\alpha}_0, \bar{\omega})$. The perturbed stable Hamiltonian structure obtained in this way will be called a *nondegenerate perturbation of* $(\bar{\alpha}_0, \bar{\omega})$. After the perturbation, the Morse-Bott torus $\mathcal{N}$ is replaced by two non-degenerate orbits corresponding to $e$ and $h$ (and still denoted by the same name) and all orbits in $\bar{\mathbb{N}} - \partial\mathbb{N}$ which intersect a fiber at most $2g$ times remain unchanged.

We fix a nondegenerate perturbation $(\bar{\alpha}'_0, \bar{\omega}')$ of $(\bar{\alpha}_0, \bar{\omega})$ once and for all. We define the following space of almost complex structures on $\bar{W}'$ in analogy to Definition 5.3.12.

**Definition 5.3.14.** — Fix $\varepsilon, \delta > 0$ sufficiently small and $k \gg 0$. Then $\mathcal{J}_{\bar{W}'}$ is the space of $C^\infty$-smooth $(\bar{\alpha}'_0, \bar{\omega}')$-adapted almost complex structures $\bar{J}'$ on $\mathbf{R} \times \bar{\mathbb{N}}$ which satisfy the following:

(1) there exists an $(\bar{\alpha}'_0, \bar{\omega}')$-adapted almost complex structure $\bar{J}'_0$ which restricts to the standard complex structure on the subsurface $\bar{D}^2 \subset \bar{S}$ of each fiber;

(2) $\bar{J}' = \bar{J}'_0$ on $\mathbf{R} \times (V_\varepsilon \cup \mathbb{N})$ and $|\bar{J}' - \bar{J}'_0|_k < \delta$ over $\mathbf{R} \times \bar{\mathbb{N}}$;

(3) every $\bar{J}'$-holomorphic map $u: \dot{\mathbf{F}} \to \bar{W}'$ connecting two orbit sets in $\bar{\mathcal{O}}_k$, for $k \le 2g$, is close to breaking to a holomorphic Morse-Bott building for some almost-complex structure in $\mathcal{J}_{\bar{W}'}^{\mathrm{MB}}$;

(SR) $\bar{J}'$ is a split complex structure on $\mathbf{R} \times [1, 2]$ times a neighborhood of $\bar{\mathbf{a}}$; moreover $\bar{\mathbf{a}}$ is real analytic with respect to the fiber complex structure on $\bar{\pi}_B^{-1}(s, t)$ for $t \in [1, 2]$.

Here $|\cdot|_k$ is some fixed $C^k$-norm.

**Remark 5.3.15.** — Condition (3) in Definition 5.3.14 will be used to constrain the topological behavior of $\bar{J}'$-holomorphic maps: to show that a map has a certain behavior, we will show that the corresponding Morse-Bott building does, which is often simpler.

**Definition 5.3.16.** — Given $\delta_0^\ell \mathbf{y}, \delta_0^q \mathbf{y}' \in \bar{\mathcal{O}}_k$ (with $p, q \ge 0$ and $k \le 2g$) and $\bar{J}' \in \mathcal{J}_{\bar{W}'}$, let $\mathcal{M}_{\bar{J}'}(\delta_0^\ell \mathbf{y}, \delta_0^q \mathbf{y}')$ be the moduli space of $\bar{J}'$-holomorphic maps in $\bar{W}'$ from $\delta_0^\ell \mathbf{y}$ to $\delta_0^q \mathbf{y}'$ without fiber components.

Maps in $\mathcal{M}_{\bar{J}'}(\delta_0^\ell \mathbf{y}, \delta_0^q \mathbf{y}')$, with $\delta_0^\ell \mathbf{y}, \delta_0^q \mathbf{y}' \in \bar{\mathcal{O}}_k$, will be called *degree $k$ PFH curve* (or *degree $k \bar{W}'$-curve*). A degree $k$ PFH curve can be viewed as a degree $k$ holomorphic multi-section of the holomorphic fibration $\mathbf{R} \times \bar{\mathbb{N}} \to \mathbf{R} \times \mathbb{S}^1$. A degree $2g$ PFH curve will also be called a *PFH curve* (or *$\bar{W}'$-curve*).

**Definition 5.3.17.** — The section at infinity $\sigma'_\infty$ in $\mathbf{R} \times \bar{\mathbb{N}}$ is $\mathbf{R} \times \delta_0$.
Choose a point $z_{\infty}^{\dagger} \in \bar{S}$ which is sufficiently close to $z_{\infty}$. Then there is a periodic orbit $\delta_{0}^{\dagger}$ of $\bar{R}_{0}$ with period $m$ which passes through $z_{\infty}^{\dagger}$. We then write $(\sigma_{\infty}')^{\dagger} = \mathbf{R} \times \delta_{0}^{\dagger}$. Both $\sigma_{\infty}'$ and $(\sigma_{\infty}')^{\dagger}$ are $\bar{J}'$-holomorphic sections for every $\bar{J}' \in \mathcal{J}_{\bar{W}'}$.

**Definition 5.3.18.** — Given a degree $k$ PFH curve $\bar{u}$ in $\bar{W}'$, we define $n'(\bar{u}) = \langle \bar{u}, (\sigma_{\infty}')^{\dagger} \rangle$.

The proof of the following is similar to that of Lemma 5.3.7.

**Lemma 5.3.19.** — The intersection number $n'(\bar{u})$ satisfies the following properties:

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

(2) if $\langle \bar{u}, \mathbf{R} \times \delta_{0} \rangle > 0$, then $n'(\bar{u}) > 0$; and

(3) $n'(\bar{u})$ is independent of the choice of $z_{\infty}^{\dagger}$.

**Lemma 5.3.20.** — If $\mathbf{y}$ and $\mathbf{y}'$ are contained in $N$, then $M_{\bar{J}'}^{n'=0}(\mathbf{y}, \mathbf{y}') = M_{\bar{J}}(\mathbf{y}, \mathbf{y}')$.

*Proof*. — Let $\bar{u}$ be a $\bar{J}'$-holomorphic map in $M_{\bar{J}}(\mathbf{y}, \mathbf{y}')$ satisfying the constraint $n'(\bar{u}) = 0$. By Condition (3) in Definition 5.3.14, we can assume that $\bar{u}$ is a Morse-Bott building to the purpose of finding homological constraints on it.

Let $T_{\rho_0} \subset V$ be the torus $\{\rho = \rho_0\}$, oriented as the boundary of the solid torus $\{\rho \le \rho_0\}$. The torus $T_{\rho_0}$ is foliated by orbits of the Hamiltonian vector field $\bar{R}_0$ and, for a dense set of $\rho_0$, those orbits are closed. If $\delta_{\rho_0}$ is one such orbit, then the homology class $[\delta_{\rho_0}]$ is a rational multiple of $[\delta_0^\dagger]$ in $H_2(\bar{N} - \mathbf{y} \cup \mathbf{y}')$. This implies that $\langle \bar{u}, \mathbf{R} \times \delta_{\rho_0} \rangle = 0$ for $\rho_0$ in a dense set. The positivity of intersections then implies that the image of $\bar{u}$ is contained in $N$. $\square$

**5.4.** *Almost complex structures and moduli spaces for $W_+, \bar{W}_+, W_-, \text{ and } \bar{W}_-$.* —

**5.4.1.** *Almost complex structures.* —

**Definition 5.4.1.** — An almost complex structure $\bar{J}_+$ on $\bar{W}_+$ is admissible if it is the restriction to $\bar{W}_+$ of an almost complex structure $\bar{J}' \in \mathcal{J}_{\bar{W}'}$ on $\bar{W}'$. If $\bar{J}_+$ agrees with $\bar{J}$ (resp. $\bar{J}'$) at the positive (resp. negative) end, then $\bar{J}_+$ is compatible with $\bar{J}$ (resp. $\bar{J}'$).

An admissible almost complex structure $\bar{J}_+$ on $W_+$ is the restriction of an admissible almost complex structure on $\bar{W}_+$. The admissibility criteria for $\bar{J}_-$ and $\bar{J}'$ on $W_-$ and $\bar{W}_-$ are analogous.

The space of $C^{\infty}$-smooth admissible almost complex structures $\bar{J}_{\pm}$ on $W_{\pm}$ will be denoted by $\mathcal{J}_{W_{\pm}}$. The space of $C^{\infty}$-smooth admissible almost complex structures $\bar{J}_{\pm}$ on $\bar{W}_{\pm}$ will be denoted by $\mathcal{J}_{\bar{W}_{\pm}}^{\text{MB}}$ if $\bar{J}_{\pm}$ is the restriction of $\bar{J}' \in \mathcal{J}_{\bar{W}'}^{\text{MB}}$ and by $\mathcal{J}_{\bar{W}_{\pm}}$ if $\bar{J}_{\pm}$ is obtained from an almost complex structure in $\mathcal{J}_{\bar{W}_{\pm}}^{\text{MB}}$ by modifying its positive (resp. negative) end as in Remark 5.3.11.
**Remark 5.4.2.** — If $J_+ \in \mathcal{J}_{W_+}$, then the projection $\pi_{B_+}: W_+ \to B_+$ is holomorphic. This is due to the fact that the fibers $\{(s, t)\} \times S$ are holomorphic and $J_+$ takes $\partial_s$ to $\partial_t$. The same holds for $\bar{J}_+, \bar{J}_-$, and $\bar{J}_-$.

**5.4.2. Moduli spaces for $W_+$**. — Let $(F, j)$ be a compact Riemann surface, possibly disconnected, with an $l$-tuple of punctures $\mathbf{p} = \{p_1, \dots, p_l\}$ in the interior of $F$ and a $k$-tuple of punctures $\mathbf{q} = \{q_1, \dots, q_k\}$ on $\partial F$, such that (i) each component of $F$ has nonempty boundary and at least one interior puncture and (ii) each component of $\partial F$ has at least one boundary puncture. We write $\dot{\mathbf{F}} = \mathbf{F} - \mathbf{p} - \mathbf{q}$ and $\partial \dot{\mathbf{F}} = \partial \mathbf{F} - \mathbf{q}$.

**Definition 5.4.3.** — Let $J_+ \in \mathcal{J}_{W_+}$, $\mathbf{y}$ be a $k$-tuple of $\mathbf{a} \cap h(\mathbf{a})$, and $\mathbf{\gamma} = \prod_j \gamma_j^{m_j}$ be an orbit set in $\hat{\mathcal{O}}_k$. Then a degree $k \le 2g$ multisection of $(W_+, J_+)$ from $\mathbf{y}$ to $\mathbf{\gamma}$ is either (i) a holomorphic map

$$u: (\dot{\mathbf{F}}, j) \to (W_+, J_+)$$

which is a degree $k$ multisection of $\pi_{B_+}: W_+ \to B_+$ and which additionally satisfies the following:

(1) $u(\partial \dot{\mathbf{F}}) \subset L_+^+$ and $u$ maps each connected component of $\partial \dot{\mathbf{F}}$ to a different $L_{a_i}^+$;

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

(3) $u$ converges to a strip over $[0, 1] \times \mathbf{y}$ near $\mathbf{q}$;

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

(5) the energy of $u$ given by Equation (4.3.1) is finite;

or is (ii) a Morse-Bott building consisting in a $J_+$-holomorphic curve as above, except for some negative ends at Reeb orbits in $\mathcal{N} - \{e, h\}$, followed by gradient flow trajectories from those orbits to $e$. Here $\pi_{\mathbf{R}}$ is the projection $\pi_{B_+}: W_+ \to B_+ \subset \mathbf{R} \times S^1$, followed by the projection to $\mathbf{R}$.

$A(W_+, J_+)$-curve is a degree $2g$ multisection of $(W_+, J_+)$.

The finiteness of the Hofer energy $E(u)$ implies that $u$ is a cylinder over a Reeb chord or a closed orbit in neighborhoods of punctures $p_i$ and $q_i$. Hence (5) implies (3) and (4) for some $\mathbf{y}$ and $\mathbf{\gamma}$. Moreover, since the orbits are nondegenerate, the convergence is exponential by the work of Abbas [Abb1] for chords and Hofer-Wysocki-Zehnder [HWZ1] for closed orbits.

**Remark 5.4.4.** — For all practical purposes, it suffices to assume that the Morse-Bott family on $\partial \mathcal{N}$ has been perturbed into a pair $h, e$ of nondegenerate orbits as in 5.3.11 and that $J_+$ is the restriction of $J' = J_0$ which satisfies Definition 3.2.1 for the perturbed Hamiltonian vector field. Since the two points of view are completely equivalent, we will switch liberally from one to another, often without explicit mention.
Let $\tilde{W}_+$ be $W_+$ with the ends $\{s > 3\}$ and $\{s < -1\}$ removed. We can view $\tilde{W}_+$ as a compactification of $W_+$, obtained by attaching $[0, 1] \times S$ at $s = +\infty$ and $N$ at $s = -\infty$. We define $Z_{y,\gamma} \subset \tilde{W}_+$ as the subset

$$Z_{y,\gamma} = (L_a^+ \cap \tilde{W}_+) \cup (\{3\} \times [0, 1] \times \mathbf{y}) \cup (\{-1\} \times \mathbf{y}).$$

As in Section 4.3, the $W_+$-curve $u: \tilde{F} \to W_+$ can be compactified to a continuous map

$$\check{u}: (\check{F}, \partial \check{F}) \to (\check{W}_+, Z_{y,\gamma}).$$

We write $M_{J_+}(\mathbf{y}, \mathbf{\gamma})$ for the moduli space of multisections of $(W_+, J_+)$ from $\mathbf{y}$ to $\mathbf{\gamma}$. We denote by $H_2(W_+, \mathbf{y}, \mathbf{\gamma})$ the equivalence classes of continuous degree $2g$ multisec- tions in $W_+$ satisfying Conditions (1)–(4) of Definition 5.4.3, where two multisections are equivalent if they represent the same element in $H_2(\tilde{W}_+, Z_{y,\gamma})$, and by $M_{J_+}(\mathbf{y}, \mathbf{\gamma}, A)$ the moduli space of maps $u \in M_{J_+}(\mathbf{y}, \mathbf{\gamma})$ which represent the class $A \in H_2(W_+, \mathbf{y}, \mathbf{\gamma})$. Then

$$M_{J_+}(\mathbf{y}, \mathbf{\gamma}) = \coprod_{A \in H_2(W_+, \mathbf{y}, \mathbf{\gamma})} M_{J_+}(\mathbf{y}, \mathbf{\gamma}, A).$$

**5.4.3.** Moduli spaces for $\overline{W}_+$. — Let $\overline{J}_+ \in \mathcal{J}_{\overline{W}_+}$, $\mathbf{y}$ be a $k$-tuple of $\mathbf{a} \cap \mathcal{h}(\mathbf{a})$ and let $\mathbf{\gamma} \in \hat{\mathcal{O}}_k$. Then a degree $k \le 2g$ multisection of $(\overline{W}_+, \overline{J}_+)$ from $\mathbf{y}$ to $\mathbf{\gamma}$ is defined as in Definition 5.4.3, where $W_+, J_+$, $L_{\mathcal{a}_i}^+$ are replaced by $\overline{W}_+, \overline{J}_+$, $L_{\mathcal{a}_i}^+$.<fnref n="5" /> We write $M_{\overline{J}_+}(\mathbf{y}, \mathbf{\gamma})$ for the moduli space of multisections of $(\overline{W}_+, \overline{J}_+)$ from $\mathbf{y}$ to $\mathbf{\gamma}$.

**Remark 5.4.5.** — As in Section 5.3.2, we will assume that $\overline{J}_+$ is sufficiently close to a Morse-Bott almost complex structure for which all degree $k \le 2g$ multisections of $(\overline{W}_+, \overline{J}_+)$ are close to breaking into a Morse-Bott building. This will allow us to use the Morse-Bott formalism to prove topological constraints on the $\overline{J}_+$-holomorphic multisec- tions. In the context of maps to $\overline{W}_+$, the definition of a Morse-Bott building is the same as in Definition 5.3.13, with the only difference that the topmost holomorphic map $u_1$ takes values in $\overline{W}_+$ and is positively asymptotic to chords over intersection points in $\mathbf{a} \cap \mathcal{h}(\mathbf{a})$.

**Definition 5.4.6.** — The section at infinity $\sigma_\infty^+$ is the restriction of $\sigma'_\infty = \mathbf{R} \times \delta_0 \subset \overline{W}'$ to $\overline{W}_+$.

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

**Definition 5.4.7.** — Given a multisection $\bar{u}$ of $\overline{W}_+$, we define $n^+(\bar{u}) = \langle \bar{u}, (\sigma_\infty^+)^\dagger \rangle$.

<footnote n="5">⁵ In particular, the image of the multisection is disjoint from the section at infinity $\sigma_\infty^+$, defined in Definition 5.4.6.</footnote>
**Lemma 5.4.8.** — The intersection number $n^+(\bar{u})$ satisfies the following properties:

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

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

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

*Proof.* — The proof is similar to that of Lemma 5.3.7. The only difference is in (2), which we discuss in more detail. Consider $\bar{u}: \dot{\mathbb{F}} \to \bar{W}_+$ and let $p \in \dot{\mathbb{F}}$ be a point such that $\bar{u}(p) \in \sigma_\infty^+$. If $\pi_{\mathbb{D}^2}$ is the projection of a neighborhood of $\bar{u}(p) \in \bar{W}_+$ to $\mathbb{D}^2 \subset \bar{S}$ along the symplectic connection (not to be confused with the balanced projection $\bar{\pi}_{\mathbb{D}^2}$ from Section 5.1.2), then $\pi_{\mathbb{D}^2} \circ \bar{u}$ is holomorphic and nonconstant, and therefore maps an open neighborhood of $p$ in $\dot{\mathbb{F}}$ to an open neighborhood of $z_\infty$. This implies that $n^+(\bar{u}) \ge d \cdot m$, where $d$ is the multiplicity of the intersection between the image of $\bar{u}$ and $\sigma_\infty^+$. $\square$

**Lemma 5.4.9.** — If $\bar{u} \in \mathcal{M}_{\bar{J}_+}^{+\dagger}(\mathbf{y}, \mathbf{y})$, then $\mathrm{Im}(\bar{u}) \subset \bar{W}_+$.

The proof is similar to that of Lemma 5.3.20 and will be omitted.

**5.4.4. Moduli spaces for $\bar{W}_-$** — The moduli space of holomorphic maps which is used to define the map $\Psi$ from $\widehat{PFC}$ to $\widehat{CF}$ is of a slightly different type from the moduli space which is used to define the map $\Phi$ from $\widehat{CF}$ to $\widehat{PFC}$. In particular, the target of the holomorphic maps is $\bar{W}_-$ instead of $W_-$. The reason we need to consider more complicated holomorphic curves instead of curves analogous to $W_+$-curves is that the naive $W_-$-curves do not have the desired Fredholm index. See the index calculations in Section 5.5.2 and Remark 5.5.6 for an explanation.

Let $(F,j)$ be a compact Riemann surface, possibly disconnected, with an $l$-tuple of punctures $\mathbf{p} = \{p_1, \dots, p_l\}$ in the interior of $F$ and a $k$-tuple of punctures $\mathbf{q} = \{q_1, \dots, q_k\}$ on $\partial F$, such that (i) each component of $F$ has nonempty boundary and has at least one interior puncture and (ii) each component of $\partial F$ has at least one boundary puncture. We write $\dot{\mathbb{F}} = F - \mathbf{p} - \mathbf{q}$ and $\partial \dot{\mathbb{F}} = \partial F - \mathbf{q}$.

**Definition 5.4.10.** — Let $\bar{J}_- \in \mathcal{J}_{\bar{W}_-}$, $\mathbf{y}$ be a $k$-tuple of $\mathbf{a} \cap \mathfrak{h}(\mathbf{a})$, and $\mathbf{y} = \prod_j \gamma_j^{m_j} \in \hat{\mathcal{O}}_k$. Then a degree $k \le 2g$ multisection of $(\bar{W}_-, \bar{J}_-)$ from $\mathbf{y}$ to $\mathbf{y}$ is a $\bar{J}_$-holomorphic map

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

which is a degree $k$ multisection of $\bar{\pi}_{\mathbb{B}_-}: \bar{W}_- \to \mathbb{B}_-$ and which additionally satisfies the following:

(1) $\bar{u}(\partial \dot{\mathbb{F}}) \subset \mathcal{L}_{\bar{\mathfrak{a}}}^-$ and $\bar{u}$ maps each connected component of $\partial \dot{\mathbb{F}}$ to a different $\mathcal{L}_{\bar{\mathfrak{a}}_i}^-$;

(2) $\lim_{w \to q_i} \bar{\pi}_{\mathbb{R}} \circ \bar{u}(w) = -\infty$ and $\lim_{w \to p_i} \bar{\pi}_{\mathbb{R}} \circ \bar{u}(w) = +\infty$;

(3) $\bar{u}$ converges to a strip over $[0, 1] \times \mathbf{y}$ near $\mathbf{q}$;