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

This paper is the sequel to"The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I"and is devoted to proving some of the technical parts of the HF=ECH isomorphism.

This is the second in a series of papers devoted to proving the equivalence of Heegaard Floer homology and embedded contact homology. The goal of this paper and its sequel [CGH2] is to establish an isomorphism between the hat versions of the Heegaard Floer homology and embedded contact homology (abbreviated ECH) groups associated to a closed, oriented 3-manifold M . The results of this paper were announced in [CGH0]. The isomorphism between the plus version of Heegaard Floer homology and the usual version of ECH will be given in [CGH3].
Heegaard Floer homology, defined by Ozsváth-Szabó [OSz1,OSz2], a priori depends on the choice of a Heegaard surface Σ for M , a basepoint z ∈ Σ, totally real tori T α , T β in Sym g (Σ), and an almost complex structure on Sym g (Σ). However, it was shown to be independent of those choices, i.e., is a topological invariant of M . On the other hand, ECH, defined by Hutchings [Hu1, Hu2, HT1, HT2] a priori depends on the choice of a contact form α on M and an adapted almost complex structure J on the symplectization R × 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 α, 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.
In this paper we will use F = Z/2Z coefficients (or coefficients in a module Λ over F[H 2 (M ; Z)]) for both Heegaard Floer homology and ECH.
A natural setting for relating Heegaard Floer homology and ECH is that of open book decompositions. In the foundational work [Gi2], Giroux  Here the equivalence relation ∼ is generated by (x, 1) ∼ (h(x), 0) for x ∈ S and (y, t) ∼ (y, t ′ ) for y ∈ ∂S and t, t ′ ∈ [0, 1]. Let ξ (S,h) be the contact structure that corresponds to (S, h) under the Giroux correspondence. In this paper we assume that ∂S is connected, unless stated otherwise.

Main result.
The main result of this paper and its sequel [CGH2] is the following: On the other hand, Taubes [T2] has proven that Seiberg-Witten Floer cohomology and ECH are isomorphic. Let HM (M ) be the homology of the mapping cone of An alternate proof of the HF=ECH correspondence has recently been given by Kutluhan-Lee-Taubes [KLT1]- [KLT5].
In the first paper of our series [CGH1], we introduced the ECH group ECH(N, ∂N ), given as a direct limit The map Φ 0 is, roughly speaking, the composition of induced by a symplectic cobordism W + , followed by the natural map induced by the maps (I j ) * . Here g is the genus of S. We prove that the map Φ * is an isomorphism, and then prove that the maps (I j ) * are isomorphisms for j ≥ 2g.
(Strictly speaking, we need to use a "perturbed" version of lim j→∞ 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.) 1.3. Organization of the paper. References from [CGH2] will be written as "Section II.x" to mean "Section x" of [CGH2], for example.
In Section 2 we recall some results of [CGH1], including the definition of ECH(N, ∂N ). In Section 3 we replace the ECH chain complexes ECC j (N ) by the periodic Floer homology chain complexes P F C 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 Lipshitz' reformulation of Heegaard Floer homology, (ii) restrict the Heegaard Floer chain complex to a page S as in [HKM1] and obtain the chain group CF (S, a, h(a)) whose homology is isomorphic to HF (−M ), and (iii) introduce an ECH-type index I 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 Φ and Ψ between the Heegaard Floer chain complex CF (S, a, h(a)) and the periodic Floer homology chain complex P F C 2g (N ). Then in Sections 6 and 7 we show that Φ and Ψ are indeed chain maps. The proof that Ψ is a chain map is substantially more involved than the proof that Φ is a chain map.
The proofs of the chain homotopies between the chain maps Ψ • Φ and id, and between the chain maps Φ • Ψ and id, are rather involved and occupy almost all (Sections II.2-II.4) of the sequel [CGH2]. The necessary Gromov-Witten type calculations 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 the close cousins of the maps (I j ) * are isomorphisms for j ≥ 2g.

ADAPTING ECH TO AN OPEN BOOK DECOMPOSITION
In this section we briefly recall the results of [CGH1]. The reader is referred to [CGH1] for a more complete discussion; the notation here is the same as that of [CGH1], unless indicated otherwise.
Its first return map then satisfies (2).
From now on, we assume that h = h 0 as given by Lemma 2.1.1.
2.2. ECH(N, ∂N, α) and ECH(N, ∂N, α). Let N = N (S,h) and α be as in the previous subsection. We recall the definitions of the variants ECH(N, ∂N, α) and ECH(N, ∂N, α) and the main result concerning them from [CGH1]. In particular, we carry over the Morse-Bott terminology from [CGH1, Section 5]. We will assume that the almost complex structure J on R × N is Morse-Bott regular. The boundary ∂N is foliated by a Morse-Bott family N of simple orbits of R α of the form θ = const. We may assume without loss of generality that α is nondegenerate away from ∂N , after a C k -small perturbation for k ≫ 0. We pick two orbits from 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 α. The Morse-Bott family N is negative. Since N is a Morse-Bott family on ∂N , this means that N plays the role of a sink and that no holomorphic curve (besides a trivial cylinder) is asymptotic to an orbit of N at the positive end.
2.2.1. ECH(N, ∂N, α). Let P be the set of simple Reeb orbits of α in int(N ). We write ECC ♭ j (N, α) for the chain complex generated by orbit sets γ constructed from P ∪ {e}, whose homology class [γ] intersects the page S × {t} exactly j times. The differential for ECC ♭ j (N, α) counts ECH index 1 Morse-Bott buildings in (R × N, J) between orbit sets which are constructed from P ∪ {e}; for more details see [CGH1,Section 5]. In particular, ifũ is an ECH index 1 Morse-Bott building which is counted in the differential, then e can appear only at a negative end ofũ and no single end ofũ can multiply cover e with multiplicity > 1. (It is still possible that there are many ends ofũ which simply cover e.) There are inclusions of chain complexes: where we are using multiplicative notation for orbit sets. Let us write ECH ♭ j (N, α) for the homology of the chain complex ECC ♭ j (N, α). We then define ECH(N, ∂N, α) = lim j→∞ ECH ♭ j (N, α).
The following was the main result of [CGH1]: Then, given m ∈ Z + and ε > 0, there exists a smooth function f : N → (0, +∞) which is ε-close to 1 with respect to a fixed C 1 -norm and whose Reeb vector field R f α has no elliptic orbits γ in int(N ) satisfying F(γ) ≤ m. with the contact structure ξ 0 = ker α 0 , where α 0 = dz + r 2 dθ. We write D z 0 = {z = z 0 } ⊂ V and T r 0 = {r = r 0 } ⊂ V .

Lemma 2.5.4 (Modification lemma).
There exists a function f 2 : V → (0, +∞) such that f 2 α 0 is arbitrarily C 1 -close to f 0 α 0 and the Reeb vector field R f 2 α 0 is equal to R f 0 α 0 near ∂V , is transverse to D z for all z ∈ S 1 , and has only one orbit γ satisfying F(γ) = 1. Moreover the orbit γ is hyperbolic.
By taking δ ′ > 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) ≈ f 0 (r) for all r ∈ [0, δ], in view of (1).
We now claim there exists a C k -small perturbation f 2 of f 1 for any k ≫ 0 such that the linearized first return map dΦ f 2 α 0 (0) of the corresponding orbit has eigenvalues −λ and − 1 λ with λ ∈ R >0 − {1}. This follows from a local model on D 2 × [0, 1] with coordinates (x, y, z): Suppose the Reeb vector field is R = ∂ z . Then the contact 1-form can be written as α = dz + β, where β is a 1-form on D 2 . We consider R hα , where h(x, y) = x 2 − y 2 . The component of R hα in the xy-direction is parallel to y∂ x − x∂ y . Hence the derivative at zero of the holonomy map D 2 × {0} D 2 × {1}, 1 obtained by flowing along R hα , has eigenvalues λ 0 and 1 λ 0 with λ 0 ∈ R >0 − {1}. By appropriately damping εh out to zero outside a small neighborhood of (0, 0, 0) ∈ D 2 × [0, 1] (here ε > 0 is sufficiently small), the above model can be grafted into V to give f 2 . This procedure does not introduce any extra F = 1 orbits, since the graph of Φ f 1 α 0 in D 0 ×D 0 intersects the diagonal transversely in one point and this property is stable under a C k -small perturbation of Φ f 1 α 0 for any k ≫ 0.
Remark 2.5.5. The modification in Lemma 2.5.4 introduces many elliptic orbits satisfying F > 1.

2.5.2.
Proof of Theorem 2.5.2. We now prove Theorem 2.5.2. Starting with the contact form α on N , we make a C 2 -small perturbation of α relative to ∂N such that the resulting Reeb vector field -also called R α -satisfies the following: (1) R α is nondegenerate away from ∂N ; (2) for each F = 1 elliptic orbit γ ⊂ int(N ), the first return map is a rotation by an irrational angle and α is of the form C 0 f 0 α 0 on a tubular neighborhood V γ of γ. Here f 0 and α 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 γ to replace the F = 1 elliptic orbits by hyperbolic orbits plus F > 1 orbits. Next, we perturb the form so that the F = 1 orbits and the F = 2 hyperbolic orbits are left unchanged and the F = 2 elliptic orbits satisfy (2), with F = 1 replaced by F = 2. Using Lemma 2.5.4 again, we replace the F = 2 elliptic orbits by hyperbolic orbits and F > 2 orbits. Continuing in this manner, we obtain α without any F ≤ m elliptic orbits in int(N ).

Direct limits.
Starting with (S, h) and α from Section 2.1, we define f j : N → (0, +∞), j ∈ N, inductively as follows. Let f 0 = 1. Suppose we have chosen up to f j so that R f j α has no elliptic orbits with F ≤ j. Let V j+1 be a small tubular neighborhood of the elliptic orbits of R f j α with F = j + 1. Then we choose f j+1 such that the following hold: (1) f j+1 is C 1 -close to f j ; (2) f j+1 = f j on N − V j+1 ; (3) R f j+1 α has no elliptic orbits with F ≤ j.
The existence of f j+1 is given by Theorem 2.5.2. Next we consider the ECH chain maps (2.5.5) given by composing two maps: J ′ j : ECC j (N, f j α) → ECC j (N, f j+1 α) and ECC j (N, f j+1 α) → ECC j+1 (N, f j+1 α) given by γ → eγ. The map J ′ j is defined by suitably completing f j α and f j+1 α to M and applying the ECH cobordism map given by [HT3,Theorem 2.4]. It is important to remember that the ECH cobordism map is defined through Seiberg-Witten Floer cohomology.
Then we have: Theorem 2.5.6. ECH(M ) ≃ lim j→∞ ECH j (N, f j α), where direct limit is taken with respect to the maps J j .
The proof is omitted, since it is similar to that of Theorem 1.2.1.

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], 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 α = f t dt + β t on S × [0, 1], as defined in Section 2.1. We may assume that R α is parallel to ∂ t on S × [0, 1]. Since where d S is the exterior derivative in the S-direction, it follows that dβt dt = −d S f t . (Hence d S β t is an area form which does not depend on t.) Also, the form α is a contact form as long as d S β t > 0. Hence, for C ≫ 0, the form (C + f t )dt + β t is a contact form with Reeb vector field parallel to ∂ t . Now consider the 1-parameter family of 1-forms τ ∈ [0, 1], on N . It interpolates between the contact form α 1 = Cdt + (f t dt + β t ) and the stable Hamiltonian form α 0 = Cdt. The Reeb vector fields R τ = R ατ are directed by ∂ t and hence are parallel for all τ > 0.
The pair (α τ , ω = d S β t ) is a stable Hamiltonian structure on N . When τ = 0, the Hamiltonian vector field R 0 equals 1 C ∂ t and hence is parallel to all the R τ , τ > 0. Also let ξ τ be the 2-plane field on N given by the kernel of α τ . The closed 2-form ω can either be viewed as an area form on S or as a (maximally nondegenerate) 2-form on N .
Our goal is to replace the ECH chain complexes ECC j (N, α τ , J τ ), ECC ♭ j (N, α τ , J τ ) for τ > 0, by the analogously defined PFH chain complexes P F C j (N, α 0 , ω, J 0 ), P F H ♭ j (N, α 0 , ω, J 0 ). The orbit sets of the ECH chain groups are constructed using the Reeb vector fields R τ and the orbit sets of the PFH chain groups are constructed using the Hamiltonian vector field R 0 .
3.3. The flux. For more details, see for example [CHL,Section 2]. Let be the map on homology induced by h : (S, ω) ∼ → (S, ω) and let K be the kernel of h * − id. Then the flux F h : K → R is defined as follows: Since the map h is the first return map of a Reeb vector field R τ , it has zero flux (cf. [CHL,Lemma 2.2]). This implies that [C] where ω is now viewed as a closed 2-form on N . Indeed, [C] can be represented by a surface of the form (δ × [0, 1]) ∪ (S × {0}), where the relevant boundary components are glued. Hence, if γ and γ ′ are orbit sets of P F C j (N, α 0 , ω 0 , J 0 ), then the ω-area of any Z ∈ H 2 (N, γ, γ ′ ) only depends on γ and γ ′ .
3.4. Compactness. The vanishing of the flux is an important ingredient in establishing that M J 0 (γ, γ ′ ) admits a compactification in the sense of [BEHWZ].
We briefly outline the argument from [Hu1, Section 9]: (1) There is a bound on the ω-area for all elements of M J 0 (γ, γ ′ ). This was done above.
(2) Given a sequence of holomorphic curves u i ∈ M J 0 (γ, γ ′ ), i ∈ N, there is a subsequence which converges weakly as currents to a holomorphic building u ∞ . 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 ] ∈ H 2 (N, γ, γ ′ ) 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 p + and p − be disjoint finite sets of punctures of F . Then let Here the punctures of p ± are asymptotic to the ± ends of u. The positive ends of u partition m i into (m i1 , m i2 , . . . ) and the negative ends of u partition n i into (n i1 , n i2 , . . . ). (We ignore the partition terms that are zero.) Pick a trivialization τ of T S in a neighborhood of all the γ i , and let µ τ (γ k i ) be the usual Conley-Zehnder index of the k-fold cover of γ i with respect to τ . 2 Then we define the total Conley-Zehnder indices at the positive and negative ends of u as follows: The trivialization τ , used in Section 3.4, is not to be confused with the parameter τ , used in the rest of Section 3. and also write µ τ (u) = µ + τ (u) − µ − τ (u). The symmetric Conley-Zehnder index of Hutchings [Hu1], so called because of its motivation from studying the "symplectomorphism" Sym k (h) of Sym k (S) induced by h, 3 is defined as: and does not depend on the choice of u from γ to γ ′ . We write µ τ (u) = µ τ (γ) − µ τ (γ ′ ). We also recall the writhe where w + τ (u) is the total writhe of braids u(Ḟ ) ∩ ({s} × N ), s ≫ 0, viewed in the union of solid torus neighborhoods of γ i and computed with respect to the framing τ ; and w − τ (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: where Q τ (u) is the relative intersection pairing with respect to τ and δ(u) is a nonnegative integer which is a count of the singularities. In particular, δ(u) = 0 if and only if u is an embedding (see [M1,MW]). Together with the writhe bounds [Hu2,Lemma 4.20], we obtain: (See [Hu1, Theorem 10.1].) Since all of the terms on the right-hand side are either homological quantities or depend on the data near γ and γ ′ , we have a lower bound on χ(Ḟ ), which implies an upper bound on the genus ofḞ .
3.5. Transversality. Let J τ be the space of almost complex structures J τ on R × N in the class C ∞ which are adapted to (α τ , ω).
Proof. This follows from [Hu1, Lemma 9.12(b)], which states that a generic J 0 ∈ J 0 is regular away from holomorphic curves which have a fiber {(s, t)} × S as an irreducible component. (Observe that the fibers are holomorphic for any J 0 ∈ J 0 .) In our case, the fibers are not closed and cannot occur as irreducible components of curves in M s J 0 (γ, γ ′ ).
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 ∈ J reg,j 0 and τ 0 = τ 0 (j, J 0 ) > 0 such that there are isomorphisms of chain complexes 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 P F C j (N, α 0 , ω) and ECC j (N, α τ ), we have as F-vector spaces, but not necessarily as chain complexes. In other words, we may view any orbit set γ for R τ as an orbit set of any other R τ ′ .
Let J 0 be an almost complex structure in J reg,j 0 . By Lemma 3.5.2, J reg,j 0 is a dense subset of J 0 , and in particular is nonempty. The moduli spaces M ′ J 0 (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 τ , τ ∈ [0, 1], be a smooth family of (α τ , ω)-adapted almost complex structures which extend J 0 .
We claim that the ECH index 1 moduli spaces M ′ Jτ (γ, γ ′ , Z) are transversely cut out and diffeomorphic to , then the moduli space is regular at u τ . Hence it suffices to prove that, if τ > 0 is sufficiently small, then every u τ ∈ M ′ Jτ (γ, γ ′ , Z) is sufficiently close to M ′ J 0 (γ, γ ′ , Z). Indeed, this follows from the compactness argument from Section 3.4. Let u i ∈ M ′ Jτ i (γ, γ ′ , Z) be a sequence of ECH index 1 holomorphic curves with τ i → 0. By the compactness theorem and incoming/outgoing partition considerations, u i converges to u ∈ M J 0 (γ, γ ′ , Z) with I ECH (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 0 over γ 0 , then u i must also have connector components u 0 i over γ 0 , a contradiction. Hence u ∈ M ′ J 0 (γ, γ ′ , Z), which proves the claim. Since the chain groups are isomorphic as vector spaces and the differentials agree for sufficiently small τ > 0, the theorem follows.

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 HF and define the Heegaard Floer groups in terms of I HF . We then use the work of [HKM1] 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 (Σ, α, β, z) which consists of the following: • a closed oriented surface Σ of genus k; • two collections α = {α 1 , . . . , α k } and β = {β 1 , . . . , β k } of k pairwise disjoint simple closed curves in Σ; and where each of α and β forms a basis of H 1 (Σ; Z) and α and β intersect transversely in Σ.
Let ω be an area form on Σ. We consider [0, 1] × Σ with the stable Hamiltonian structure (dt, ω), where t is the [0, 1]-coordinate. The Hamiltonian vector field is ∂ t , and the 2-plane field ker dt will be written as T Σ, at the risk of some confusion.
If the map f : X → Σ is understood, then we write T Σ X or T Σ for the pullback bundle f * T Σ, e.g., T Σ Σ or T Σ Terminology. We will often write elements of S as y = {y 1 , . . . , y k } and refer to y as a k-tuple of intersection points. Also, if l ≤ k, then an l-tuple of chords/intersection points y = {y 1 , . . . , y l } is a collection of points in α ∩ β where each α i is used at most once and each β i is used at most once.
Definition 4.2.1 (Ω-admissibility). An almost complex structure J on W is Ωadmissible (or simply admissible) if it satisfies the following: (1) J is s-invariant; (2) J(∂ s ) = ∂ t and J(T Σ) = T Σ; (3) J is tamed by the symplectic form Ω; We write J Σ for the space of C ∞ -smooth Ω-admissible almost complex structures J on W .
4.3. Holomorphic curves and moduli spaces. Let (F, j) be a compact Riemann surface, possibly disconnected, with two sets of punctures q + = {q + 1 , . . . , q + k } and q − = {q − 1 , . . . , q − k } on ∂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 q + and q − , and (iii) the punctures on q + and q − alternate around each boundary component. We writeḞ which is a degree l multisection of π B : W → B = R × [0, 1] and which additionally satisfies the following: (2) for each i ∈ {1, ..., k}, u −1 (L α i ) (resp. u −1 (L β i )) consists of exactly one component of ∂Ḟ , which we call α * i (resp. β * i ); (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 (W, 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 → ±∞. By the work of Abbas [Ab], an HF curve u converges exponentially to cylinders over Reeb chords near the ends. Components of u may map to R × [0, 1] × {y i }; such components will be called trivial strips.
We now define some moduli spaces of HF curves with respect to J ∈ J Σ . Let y = {y 1 , . . . , y k } and y ′ = {y ′ 1 , . . . , y ′ k } be k-tuples of α ∩ β. Let M J (y, y ′ ) be the moduli space of HF curves u which are asymptotic to R × [0, 1] × {y i } near q + i and to R × [0, 1] × {y ′ i } near q − i . 4 Such a curve u is said to be an HF curve from y to y ′ . Also let M J (y, y ′ ) ⊂ M J (y, y ′ ) be the subset consisting of curves u which additionally satisfy , obtained by attaching [0, 1] × Σ at the positive and negative ends, and letĽ α = [−1, 1] × {1} × α andĽ β = [−1, 1] × {0} × β be the compactifications of L α and L β . We then define Z y,y ′ ⊂W as the subset Similarly define The exponential decay of HF curves in R × [0, 1] × Σ implies that an HF curve u :Ḟ → W from y to y ′ can be compactified to a continuous map HereF is obtained fromḞ by performing a real blow-up at its boundary punctures. By some abuse of notation, let π 2 (y, y ′ ) be the set of homology classes of continuous maps u :Ḟ → W which satisfy (1), (2) and (3) of Definition 4.3.1 and are positively asymptotic to [0, 1] × y and negatively asymptotic to [0, 1] × y ′ ; here two maps u 1 and u 2 are equivalent in π 2 (y, y ′ ) if their compactificationsǔ 1 anď u 2 are homologous in H 2 (W , Z y,y ′ ). To any HF curve from y to y ′ we can then associate a class in π 2 (y, y ′ ). If we consider moduli spaces of HF curves u in the homology class A ∈ π 2 (y, y ′ ), we will write M J (y, y ′ , A) or M J (y, y ′ , A).

The Fredholm index.
In this subsection and the next, we fix J ∈ J Σ .
In this subsection we discuss the Fredholm index of an HF curve u :Ḟ → W , which is the expected dimension of a neighborhood U of u ∈ M J (y, y ′ ), modulo reparametrizations of the domain. Since the curve u cannot be multiply-covered, the regularity of M J (y, y ′ ) is straightforward; see Lemma 4.7.2. The Fredholm index of u will be denoted by ind(u) = ind HF (u).
4.4.1. The Fredholm index, first version. We start with Lipshitz's formula [Li,Equation 5] for the Fredholm index of u: where k is the genus of Σ, χ(F ) = χ(Ḟ ) is the Euler characteristic of F orḞ , and α * i and β * i are as in Definition 4.3.1. We now define the Maslov indices µ(α * i ) and µ(β * i ) which appear in Equation (4.4.1): Choose a trivialization τ ′ 0 of T Σ ≃ C in a neighborhood of the points w ∈ α∩β so that R corresponds to T w β and iR corresponds to T w α. Then let τ 0 be a trivialization of u * T Σ W which coincides with the one already given near p and q by pulling back τ ′ 0 . Along each component of ∂Ḟ -called α * i or β * i depending on whether it is mapped to α or to β, and oriented in the same way as ∂F -we have a loop of real lines in C, given by the pullback of T α or T β. The Maslov index µ(α * i ) (resp. µ(β * i )) is twice the degree of the loop along α * i (resp. β * i ) with respect to τ 0 . 4.4.2. The Fredholm index, second version. For the purposes of computing indices, we replace Recall Z α,β ⊂W which was given by Equation (4.3.3). We define a trivialization τ of T ΣW along Z α,β ⊂W as follows: First choose a nonsingular tangent vector field along each component of α and β. This induces a trivializa- Given an HF curve u :Ḟ → W , we define its Maslov index µ τ (u) as follows: Letǔ be the compactification of u. We then construct a (not necessarily oriented) real rank one subbundle L ofǔ * T Σ on ∂F . The bundle L is given byǔ * T α andǔ * T β along ∂Ḟ . We extend L to ∂F − ∂Ḟ by rotating in the counterclockwise direction fromǔ * T β toǔ * T α by the minimum amount possible. (Assuming orthogonal intersections, this is a π 2 -rotation.) Then µ τ (u) is the sum of the Maslov indices of L with respect to the trivialization τ , where the sum is over all the connected components of ∂F .
Proof. By standard Maslov index theory, we have where τ 0 denotes the trivialization of u * T Σ from Section 4.4.1. We immediately obtain c 1 (u * T Σ, τ 0 ) = 0 since τ 0 is a trivialization on all ofḞ . Hence it suffices to prove that: The difference between the two sides of Equation (4.4.3) is the total amount of rotation of the real lines introduced at α ∩ β in the definition of µ τ 0 : if we go from β to α we rotate by π 2 , while if we go from α to β we rotate by − π 2 ; hence the total amount of rotation is 0.
We can now rephrase the Fredholm index as follows: Remark 4.4.2. Since the Maslov index of L with respect to τ is an integer along each chord [0, 1] × {y i }, it makes sense to write µ τ (y i ) ∈ Z. If we let In particular, µ τ (u) only depends on y, y ′ , and the choice of τ .
4.5. The ECH-type index. In this subsection we define an ECH-type index I HF (u) and prove an index inequality which is analogous to the ECH index inequality of [Hu1].
, and induces a framing which agrees with τ along [0, 1] × {y i } (resp. [0, 1] × {y ′ i }). Let A be a homology class in π 2 (y, y ′ ). Then we define where z j ∈ Σ − α − β are given in Definition 4.2.1 and ·, · is the signed intersection number. We say that A ∈ π 2 (y, y ′ ) is positive if n z j (A) ≥ 0 is nonnegative for all z j .
Proof. Let A be a positive homology class in π 2 (y, y ′ ). Then we can glue closures of connected components of Σ − α − β with multiplicity n z j (A) as in Rasmussen [Ra,Lemma 9.3] (also see [Li,Lemma 4.1]) to construct a continuous map u 2 : F → Σ which is smooth onḞ and satisfies the following: . . , k, is used exactly once; • u 2 | ∂F switches from β to α (resp. α to β) in a neighborhood of y i (resp. y ′ i ) as we move in the direction given by the orientation of ∂F . The map u 2 can be pulled back toǔ 2 :F → Σ, whereF is the real blow-up of F given in Section 4.3. Now take a branched coverǔ 1 :F → [−1, 1] × [0, 1] with interior branch points, and formǔ = (ǔ 1 ,ǔ 2 ) :F →W . Condition (1) of Definition 4.5.1 is immediately satisfied. We can resolve all the (interior) singularities to makeǔ embedded. It is a local exercise to modifyǔ in a neighborhood of ∂F − ∂Ḟ so thatǔ becomes τ -trivial.
Lemma 4.5.4. If u :Ḟ → W is an HF curve andČ is the image of the compactificationǔ :F →W , then the following hold: (1) π B • u :Ḟ → B has no branch points along ∂B.
In other words,Č satisfies all the conditions of a τ -trivial representative for some τ , with the exception of the embeddedness ofČ. Proof.
(1) follows from the fact that π B • u is a k-fold branched cover of B. Let H = {Im(z) ≥ 0} be the upper half-plane and U ⊂ H be an open subset which contains 0. If f is a holomorphic map U → R × [0, 1] which maps 0 to (0, 0) and U ∩ ∂H to R × {0}, then it can be extended to a holomorphic map f : If df (0) = 0, then f is locally a composition of z → z l for some integer l > 1 and a biholomorphism. This contradicts the requirement that f (H) stay on one side of R × {0}.
4.5.2. The relative intersection form. We now define the relative intersection form Q τ (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 τ (A)). Let A ∈ π 2 (y, y ′ ) be a positive homology class and letČ be a τ -trivial representative of A. Let ψ be a section of the normal bundle ν toČ such that ψ| ∂Č = Jτ , and letČ ′ be a pushoff ofČ in the direction of ψ. Then the relative intersection form Q τ (A) is given by: where ·, · denotes the algebraic count of intersection points.
Note that, since a representativeČ is positively transverse to the fibers {(s, t)}× Σ along all of ∂Č, we may take the normal bundle ν toČ to satisfy ν| ∂Č = T Σ| ∂Č . Also, since J is Ω-admissible, it takes T Σ to itself. Hence (τ, Jτ ) is a trivialization of ν| ∂Č . Although τ and Jτ are homotopic, we will often use Jτ due to its appearance in the definition of Q τ (A).
4.5.4. The relative adjunction formula. In this subsection we prove the relative adjunction formula (Lemma 4.5.9). Let δ(Č) be the signed count of singularities ofČ in its interior. In particular, ifČ is immersed, then δ(Č) is the signed count of transverse double points ofČ. We will also use the notation δ(u) or δ(ǔ).
Suppose now thatČ 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 B ⊂W be a small ball centered at d. ThenČ ∩ ∂B is a Hopf link, andČ ∩ B is the union of two slice disks for the components which intersect at d. We can construct a new surfaceČ sm by replacing the two disks with a Hopf band connecting the two components of the Hopf link. By definition, we have On the other hand, if ν sm is the normal bundle toČ sm , then c 1 (ν sm , Jτ ) = c 1 (ν, Jτ ) + 2.
This can be seen easily by embedding B into S 2 × S 2 and using the properties of the intersection product for closed 4-manifolds.
In general, , and the lemma follows.
We can now state and prove the relative adjunction formula: Lemma 4.5.9 (Relative adjunction formula). If u :Ḟ → W is an HF curve in the homology class A ∈ π 2 (y, y ′ ), then Here δ(u) is a nonnegative integer which equals 0 if and only if u is an embedding, and ∂ t is the pullback to ∂F of the trivialization ∂ t on [−1, 1] × [0, 1].
Proof. By [M1,MW], there exists a modification v :Ḟ → W of u :Ḟ → W in a neighborhood of its finitely many singular points so that v is symplectic with only transverse double points. Since the modification is purely local and is away from ∂Ḟ , it follows thatǔ andv belong to the same homology class in π 2 (y, y ′ ) and c 1 (ǔ * T Σ, τ ) = c 1 (v * T Σ, τ ). Hence Equation (4.5.2) for u is equivalent to Equation (4.5.2) for v, and we may assume without loss of generality that u is immersed with positive transverse double points. The vector field ∂ t is a global trivialization of the complex line bundle On the other hand, c 1 (ǔ * TW , (τ, ∂ t )) = c 1 (TF , ∂ t ) + c 1 (ν, Jτ ).
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.
Proof. Let τ ∂F be the trivialization of T F | ∂F which is given by an oriented nonsingular vector field tangent to ∂F . We then have where deg(∂ t , τ ∂F ) is the degree of ∂ t with respect to τ ∂F . By an easy direct calculation we obtain deg(∂ t , τ ∂ F ) = −k. 4.5.5. 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).
Proof. Each of the terms c 1 (T Σ| A , τ ), Q τ (A), and µ τ (y)−µ τ (y ′ ) in the definition of I HF (A) is additive under stacking; see Lemma 4.5.7 for the additivity of Q τ (A).
The following index inequality is analogous to (but much easier than) the ECH index inequality, due to Hutchings [Hu1, 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 :Ḟ → W be an HF curve in the class A ∈ π 2 (y, y ′ ). Then  Proof. We calculate: The first line is Equation (4.4.4). The equivalence of the first and second lines follows from the relative adjunction formula. Hence ind(u) + 2δ(u) = I HF (A).
4.6. Compactness. We now discuss the requisite compactness issues. The key notion is that of weak admissibility, which is analogous to the vanishing of the flux in the PFH situation (see Section 3.3). Let (Σ, α, β, z) be a weakly admissible Heegaard diagram, i.e., for every Spin c -structure s and nontrivial periodic domain Q which satisfies c 1 (s), Q = 0, there exist j 1 and j 2 for which n z j 1 (Q) > 0 and n z j 2 (Q) < 0. Equivalently, by [OSz1, Lemma 4.12], (Σ, α, β, z) is weakly admissible if and only if there is an area form ω on Σ such that each periodic domain has total signed ω-area zero. Let N > 0 be a fixed constant. We consider the subset π N 2 (y, y ′ ) consisting of homology classes of π 2 (y, y ′ ) which intersect [−1, 1] × [0, 1] × {z} at most N times. (This is sufficient for CF and CF + , defined in the next subsection.) The difference of two homology classes A 1 , A 2 ∈ π 0 2 (y, y ′ ) is a periodic domain Q and has zero ω-area. This implies that the ω-areas of any two A 1 , A 2 ∈ π N 2 (y, y ′ ) differ by i · ω(Σ) where 0 ≤ i ≤ N . Let φ 1 , . . . , φ r be the connected components of Σ − α − β. If A is represented by a holomorphic curve, then the projection of A to Σ can be written as j n j (A)φ j with n j (A) ≥ 0. Since each φ i has finite area, there must only be a finite number of homology classes A ∈ π N 2 (y, y ′ ) for which the moduli space M J (y, y ′ , A) is nonempty.
We now prove the existence of a compactification of M J (y, y ′ , A)/R. It suffices to show that if u :Ḟ → W is an element of M J (y, y ′ , A), then the genus ofḞ 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 (y, y ′ , A)/R. Lemma 4.6.1. There is an upper bound on the genus of a holomorphic curve u : F → W in a fixed homology class A ∈ π 2 (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 δ(u), we have (4.6.1) The lemma follows by observing that the terms on the right-hand side depend only on the homology class A.

Transversality.
Definition 4.7.1. An almost complex structure J ∈ J Σ is regular if the moduli spaces M J (y, y ′ ) for all y, y ′ ∈ S = S α,β are transversely cut out.
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)} × Σ as irreducible components.
We write J reg Σ ⊂ J Σ for the subset of regular almost complex structures J. For J ∈ J reg Σ , the dimension of M J (y, y ′ ) near u is equal to the Fredholm index ind(u). The moduli space M J (y, y ′ ) carries a natural R-action given by translations in the s-direction, and the quotient M J (y, y ′ )/R is a manifold. Proof. This follows from [Li, Proposition 3.8], 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 (y, y ′ ) consist of simple curves. We will use the notation M I=r J (y, y ′ ) to denote the moduli space of HF curves from y to y ′ with ECH index I = r. Proof. This follows from Equation (4.5.4) by observing that the term 2δ(u) is even and nonnegative and that ind(u) ≥ 0 since J is regular and u is not multiplycovered. 4.8. Definition of the Heegaard Floer homology groups. Let (Σ, α, β, z) be a weakly admissible Heegaard diagram and let J ∈ J reg Σ . We define the Heegaard Floer chain complexes ( CF (Σ, α, β, z, J), ∂) and (CF + (Σ, α, β, z, J), ∂ + ), whose corresponding homology groups are HF (Σ, α, β, z, J) and HF + (Σ, α, β, z, J).
The hat group CF (Σ, α, β, z, J) is the F-vector space generated by S = S α,β and the plus group CF + (Σ, α, β, z, J) is the F-vector space generated by S ×Z ≥0 . Elements of S will be written as y and elements of S ×Z ≥0 will be written as [y, i].
We now define the differentials ∂ and ∂ + . The differential ∂ is given by where ∂y, y ′ is the count of M I=1 J (y, y ′ )/R. The differential ∂ + is given by where ∂ + ([y, i]), [y ′ , j] is the count of M I=1 J (y, y ′ )/R whose representatives have intersection number i − j with R × [0, 1] × {z}. By Theorem 4.5.13, the count of I 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.
A basis of S is a collection of properly embedded pairwise disjoint arcs a = {a 1 , ..., a 2g } of S such that S − a is a connected 8g-gon. Given a basis a of S, there is a natural collection of compression curves where the presence of † indicates a copy of an arc in S 1/2 and the absence indicates a copy of an arc in S 0 . Recall the monodromy h maps (y, θ) → (y, θ − y) near ∂S. We then construct a collection of compression where b i is the simplest arc (= fewest number of intersections with the a j ) in S 1/2 which is parallel to a i and extends h(a i ) to smooth curve in Σ. See Figure 1.
The arcs a i and h(a i ) intersect at their endpoints x i and x ′ i by the definition of h near ∂S, and the arcs a i and b i intersect at a unique point x ′′ i in int(S 1/2 ). This means that all the intersection points of α i ∩β j lie in S 0 , except for one intersection point x ′′ i of α i ∩ β 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 Σ − α − β which nontrivially intersect S 1/2 are the following: • the "forbidden region" containing the basepoint z; • for each i = 1, . . . , 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 (Σ, β, α, z). (&) J is a product complex structure on R × [0, 1] × S 1/2 . All the holomorphic curves and moduli spaces in this subsection are for the Heegaard diagram (Σ, β, α, z). Claim 4.9.2. Let u ∈ M J (y, y ′ ) for some y, y ′ ∈ S β,α . Then the following hold: (1) If u is not asymptotic to any x ′′ i , then its image is contained in R × [0, 1] × 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 Σ 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 . The only nontrivial components of u which intersect R × [0, 1] × S 1/2 are the "thin strips" in (2) and are easily seen to satisfy automatic transversality. Hence a generic J which satisfies (&) is in J reg Σ .
4.9.3. The variant CF (S, a, h(a)). Let J ∈ J reg Σ which satisfies (&). We now define the chain complex CF (S, a, h(a), J), which can be defined on a page of the open book (S, h) and whose homology is isomorphic to HF (Σ, β, α, z, J). The almost complex structure J will usually be suppressed from the notation.
The following theorem allows us to restrict from the Heegaard surface Σ to the page S: Proof. Let us write CF for CF (Σ, β, α, z). Also let CF k be the subgroup of 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 ∂ on CF as ∂ = ∂ 0 + ∂ 1 , where ∂ 0 : CF k → CF k counts I HF = 1 curves whose nontrivial part is contained in S 0 and ∂ 1 : CF k → CF k−1 counts I HF = 1 curves whose nontrivial part is contained in S 1/2 . In particular, ∂ 1 counts HF curves which correspond to the domains D i and D ′ i . Since ∂ 2 = 0, it follows that ∂ 2 0 = ∂ 2 1 = ∂ 0 ∂ 1 + ∂ 1 ∂ 0 = 0, i.e., CF becomes a double complex.
The ∂ 1 -homology of CF is: 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 HF (Σ, β, α, z) such that: Since E 2 is concentrated in degree k = 0, the spectral sequence degenerates at the second step and HF (Σ, β, α, z) ∼ = E 2 . This proves the theorem.
Before proceeding to Lemma 4.9.5, let us introduce some notation. Let I be a k-element subset of {1, . . . , 2g} and let I c be its complement. Then let S I c be the set of (2g − k)-tuples y 1 of chords from h(a) to a such that each a j and h(a j ), j ∈ I c , is used exactly once and no x j , x ′ j , x ′′ j is in y 1 . In particular, a i and h(a i ) remain unoccupied for all i ∈ I. Lemma 4.9.5. The homology of ( CF , ∂ 1 ) is: Proof. Let ( CF (y 1 ), ∂ 1 ) ⊂ ( CF , ∂ 1 ) be the subcomplex generated by 2g-tuples of chords of the form y 0 ∪ y 1 , where y 0 is a k-tuple of chords consisting of one of x j , x ′ j , x ′′ j for each j ∈ I and y 1 ∈ S I c . Since ( CF , ∂ 1 ) is the direct sum of chain complexes of the form ( CF (y 1 ), ∂ 1 ), it suffices to treat each ( CF (y 1 ), ∂ 1 ) separately.
Consider the chain complexes (C(j), d) = (C 0 (j) ⊕ C 1 (j), d), where The homology groups of those complexes are: By Claim 4.9.2(2), we have By the Künneth formula, H * ( CF (y 1 ), ∂ 1 ) is generated by the equivalence class {y ′ 0 ∪ y 1 }, where y 1 is fixed and y ′ 0 ranges over all k-tuples of chords which consist of one of x j , x ′ j for each j ∈ I. The lemma then follows. 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 β , the difference between the Spin c -structures corresponding to y and y ′ is given by: where a cycle representing ǫ(y, y ′ ) can be constructed on the Heegaard diagram as follows: For each i = 1, . . . , k, choose an arc α ⋆ i on α i from y i to y ′ i , where y i , y ′ i ∈ α i . Similarly, we choose arcs β ⋆ i on β i , i = 1, . . . , k, which connect y ′ to y. Then ǫ(y, y ′ ) is the homology class of which is a union of closed curves; see [OSz1, Definition 2.11 and Lemma 2.19]. It is easy to verify that ǫ(y, y ′ ) does not depend on the choice of arcs α ⋆ i and β ⋆ i and provides a topological obstruction to the existence of HF curves connecting y and y ′ .
The Heegaard Floer chain complex CF (Σ, α, β, z) therefore splits into a direct sum where the subgroup CF (Σ, α, β, z, s) is generated by y ∈ S α,β with s z (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 CF (S, a, h(a)) which is generated by the set S a,h(a) of 2g-tuples of intersection points of a and h(a), i.e., we are restricting to a page S. The homology groups H 1 (M ; Z) ≃ H 1 (N, ∂N ; Z) are identified via the isomorphism ̟, given in Lemma 2.3.1.
We then define the map by assigning a cycle h(y) to y = {y i } 2g i=1 ∈ S a,h(a) as follows: Suppose y i ∈ a i ∩ h(a σ(i) ) for some σ ∈ S 2g . On [0, 1] × S, we consider the union of the following oriented arcs: • [0, 1] × {y i }, i = 1, . . . , 2g, where the orientation is given by ∂ t ; • {0} × c i , i = 1, . . . , 2g, where c i is a subarc of h(a i ) which goes from h(y σ(i) ) to y i .
With the identification (x, 1) ∼ (h(x), 0), the arcs glue to give a cycle in N which represents h(y). Proof. The equality holds for any 2g-tuple x 0 which represents the contact class. In fact, s z (x 0 ) = s ξ by the definition of the contact class and h(x 0 ) = 0 since the cycle representing it is parallel to ∂N . Hence, in order to prove the proposition, it suffices to prove that h(y) − h(y ′ ) = ǫ(y, y ′ ) for all y, y ′ ∈ S a,h(a) . One can check that h(y)−h(y ′ ) is homologous to the union δ of the following types of arcs: • subarcs of {1} × a i connecting from y to y ′ ; and • subarcs of {0} × h(a i ) connecting from y ′ to y.
By homotoping δ to a page S, we see that [δ] = ǫ(y, y ′ ) with respect to the Heegaard diagram (Σ, β, α, z) given in Section 4.9.1. 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 HF ; the construction for HF +which will be used in [CGH3] -can be treated in a similar way.
Fix a Spin c -structure s and a k-tuple of intersection points y 0 such that s z (y 0 ) = s. A complete set of paths for s based at y 0 is the choice, for each k-tuple of intersection points y such that s z (y) = s, of a surface C y which is the projection to [0, 1] × Σ of a surface representing an element of π 2 (y, y 0 ). 5 A complete set of paths determines maps for all y and y ′ such that s z (y) = s z (y ′ ) = s by This map is compatible with the action of H 2 (M ) on π 2 (y, y ′ ) and with the concatenation of chains with matching ends. We define The homology of this complex is the Heegaard Floer homology with twisted coefficients HF (M, s). Consider the special Heegaard diagram constructed in Section 4.9.1. For every Spin c -structure s ∈ Spin c (M ) we define the complex with the differential induced by the differential on CF (Σ, β, α, z, s). Proof. Fix a distinguished 2g-tuple of generators y 0 such that s z (y 0 ) = s. We choose a complete set of paths C y with the following property: if y = y ∪ {x i }, 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 a complete set of paths, we have A(D i ) = A(D ′ i ) = 0 for all i, so the proof of Theorem 4.9.4 goes through unchanged.

MODULI SPACES OF MULTISECTIONS
The goal of this section is to introduce the moduli spaces which will be used to define the chain maps The definition of these chain maps can be viewed as a melding of ideas of Seidel [Se1,Se2] and Donaldson-Smith [DS].
Intuitively, y is mapped to an element of P F C 2g (N ) through an intermediary called a broken closed string γ y . It is a union of closed curves in N = S × [0, 1]/ ∼, obtained by taking the union of y i × [0, 1], i = 1, . . . , 2g, and c i × {0}, i = 1, . . . , 2g, where c i is a subarc of h(a i ) which connects h(y σ(i) ) to y i . Note that there is a unique homotopy class of arcs from h(y σ(i) ) to y i , since h(a i ) is an arc (and not a closed curve). The arcs y i ×[0, 1], c i ×{0}, i = 1, . . . , 2g, glue up to give a union of closed curves since (h(y σ(i) ), 0) ∼ (y σ(i) , 1).

Symplectic cobordisms.
Recall the stable Hamiltonian structure (α 0 , ω) on N from Section 3, where α 0 is given by Equation (3.1.1). For simplicity we assume that α 0 = dt. The fibration N is given by: where h is the first return map of the stable Hamiltonian vector field R 0 = ∂ t with zero flux. Here we make one slight modification: the interval [0, 1] in Section 2 is now replaced by [0, 2]. We may assume that R 0 is Morse-Bott nondegenerate -i.e., nondegenerate in the interior of N and Morse-Bott along ∂N -after a C k -small perturbation for k ≫ 0.
Remark 5.1.1. Indeed, the stable Hamiltonian vector field R 0 on N has the same first return map as a Reeb vector field R τ , τ > 0, by construction, and we could have taken R τ to be Morse-Bott nondegenerate.
× × Similarly, we define The symplectic form Ω + (resp. Ω − ) is the restriction of to W + (resp. W − ). By this we mean the following: On R × S × [0, 2], we take the symplectic form ds ∧ dt + ω. Then the symplectic form glues under the identification (s, x, 2) ∼ (s, h(x), 0). We also write cl(B + ), cl(B − ) for the compactifications of B + , B − , obtained by adjoining the points at infinity p + corresponding to s = +∞, and p − corresponding to s = −∞. 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 cobordisms (W + , Ω + ) and (W − , Ω − ). We now extend (W + , Ω + ) to (W + , Ω + ), which corresponds to capping off each fiber S by a disk; the definition of (W − , Ω − ) is analogous.
We first define the capped-off surface S: Let D 2 = {ρ ≤ 1} be a disk with polar coordinates (ρ, φ). We write z ∞ for the origin ρ = 0. Let (y, θ) be the coordinates Let m be an integer > 2g, which we take to be arbitrarily large. We define h m : S ∼ → S to be a smooth extension of h : S ∼ → S, depending on m, such that: where ν m : [0, 1] → R is a smooth function which satisfies the following: In particular, ν ∞ (ρ) = 0 for ρ ≤ 1 2 . Taking the limit m → ∞ becomes important starting from Section 7.8, but until then we just need m ≫ 0 and we simply write h = h m and ν = ν m .
We then define the suspension Note that, although N m depends on m and ν m , all the N m are diffeomorphic. The closed manifold N is obtained from M by a 0-surgery along the binding of the open book. Let ω be an area form on S which extends ω and equals ρdρ ∧ dφ on D 2 . We then extend the stable Hamiltonian structure (α 0 = dt, ω) on N to the stable Hamiltonian structure it is called δ 0 since it lies on the level set ρ = 0. The 2-plane field of the stable Hamiltonian structure is ker α 0 = T S.

Let us write
be the projection of W * ∩ (R × V ) to V , followed by the projection of V to D 2 via the identification ϕ, i.e., with respect to the balanced coordinates.
The marked point will play a crucial role in the definition of the chain map Ψ.

Lagrangian boundary conditions for W ± . The symplectic fibration
admits a symplectic connection, defined as the Ω + -orthogonal of the tangent plane to the fibers. The symplectic connection is spanned by ∂ s and ∂ t if we consider We first place a copy of the basis a on the fiber π −1 B + (3, 1) and take its parallel transport along ∂B + using the symplectic connection. The parallel transport sweeps out a Lagrangian submanifold L + a of (W + , Ω + ). Let L + a i be the connected component of L + a given by parallel transport of a i . Since the symplectic connection is spanned by ∂ s and ∂ t on R × S × [0, 2], over the strip {s ≥ 3, t ∈ [0, 1]} we have: . Similarly, the Lagrangian submanifold L − a on the vertical boundary of (W − , Ω − ) is obtained by taking the parallel transport of a copy of a -placed on the fiber π −1 B − (−3, 1) -by the symplectic connection.

Extended Lagrangian boundary conditions.
In what follows we assume that the basis arcs a i , i = 1, . . . , 2g, depend on m ≫ 0 and satisfy some additional conditions. Let E ⊂ ∂D 2 be the set of endpoints of ∪ i=1,...,2g a i and let y 1 (m), . . . , y 4g (m) be the points of E in counterclockwise order. Then we assume the following: 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 a i be the (oriented) extension of a i ⊂ S to S, obtained by attaching two radial rays Here a i,0 (resp. a i,1 ) is the initial (resp. terminal) segment of a i . We also define the extension a i, We write a = {a 1 , . . . , a 2g }. Then L ± a is the extension of L ± a , obtained by the parallel transport of a copy of a, placed at π −1 Definition 5.2.1. A bigon (with acute angles) contained in D 2 and bounded by a i,j and h(a i,j ) will be called a thin strip. The portion of a thin strip contained in D 1/2 = {ρ ≤ 1 2 } will be called a thin wedge. See In this subsection we specify the almost complex structures and moduli spaces for W , W , W ′ = R × N , and W ′ = R × N . The notation may conflict with the older ones, namely those used in Sections 3 and 4; in the case of conflict, the new notation supersedes the older ones.
Convention 5.3.1. When we write y or γ (with possible superscripts, subscripts and other decorations), it is assumed that y ⊂ S and γ ⊂ N . In particular, y and γ do not contain any multiples of z ∞ or δ 0 .
The analogous space J for W is slightly more complicated: Definition 5.3.2. Fix ε, δ > 0 sufficiently small and k > 0 sufficiently large. Then J is the set of C ∞ -smooth Ω-admissible almost complex structures J on W which satisfy the following: (1) there exists an Ω-admissible almost complex structure J 0 which restricts to the standard complex structure on the subsurface D 2 ⊂ S of each fiber; We denote by J the restriction of J to W . Let S = S a,h(a) be the set of 2g-tuples of intersection points on a ∩ h(a). Given y, y ′ ∈ S and J ∈ J , let M J (y, y ′ ) be the moduli space of HF curves from y to y ′ with respect to J which are contained We next discuss holomorphic curves in W .
Definition 5.3.3. Let y, y ′ be k-tuples of a ∩ h(a) and J ∈ J . Then a degree k ≤ 2g multisection u from y to y ′ in (W , J ) is a holomorphic map u :Ḟ → W which is a degree k multisection of π B : W → B, satisfies the conditions of Definition 4.3.1 with L α and L β replaced by L a = R × {1} × a and L h(a) = R × {0} × h(a), and is asymptotic to y and y ′ at the positive and negative ends.
The section at infinity is J-holomorphic for every J ∈ J . We will be sloppy with the notation and routinely identify σ ∞ with its image.
Let z † ∞ ∈ D 1/2 ⊂ S be a point with ρ-coordinate less than 1 2 and in the complement of all arcs a i,j and h( a i,j ). The orbit O(z † ∞ ) of z † ∞ under the action of h consists of m points, and each thin wedge in D 1/2 between a i,j and h(a i,j ) con- . (Note that this multisection does not have Lagrangian boundary conditions; it will only be used to impose topological constraints on the HF curves.) Definition 5.3.5. Given a degree k multisection u of W , we define n(u) = u, σ † ∞ , where ·, · denotes the algebraic intersection between the images.
The intersection number n(u) is a homological invariant since z † ∞ was chosen so that O(z † ∞ ) is disjoint from the Lagrangian arcs. Lemma 5.3.6. The intersection number n(u) satisfies the following properties: Modifiers. For any moduli space M ⋆ 1 (⋆ 2 ) we may place modifiers * as in M * ⋆ 1 (⋆ 2 ) to denote the subset of M ⋆ 1 (⋆ 2 ) satisfying * . Typical self-explanatory modifiers are I = i, n = m, and deg = k. (Note however that the degree can be inferred from ⋆ 2 .) The following lemma is an easy consequence of Lemma 5.3.6, in particular of points (1) and (4).
The definition of the analogous space J ′ for W ′ = R × N is slightly more complicated, but completely analogous to Definition 5.3.2: Definition 5.3.9. Fix ε, δ > 0 sufficiently small and k > 0 sufficiently large. Then J ′ is the set of C ∞ -smooth (α 0 , ω)-adapted almost complex structures J ′ on R × N which satisfy the following: (1) there exists an (α 0 , ω)-adapted almost complex structure J ′ 0 which restricts to the standard complex structure on the subsurface D 2 ⊂ S of each fiber; Let P be the set of orbits of R 0 in int(N ), together with e and h on ∂N , and let P = P ∪{δ 0 }. Let O k and O k be the set of orbit sets constructed respectively from P or P which intersect S × {0} exactly k times. A PFH curve is a J ′ -holomorphic Morse-Bott building in R × N without fiber components, which connect two orbit sets in O 2g . If we perturb the Morse-Bott family, then the perturbed PFH curve (if it exists) can be viewed as a degree 2g J ′ -holomorphic multisection of the holomorphic fibration R × N → R × S 1 .
We denote the section at infinity R × δ 0 by σ ′ ∞ . Choose a point z † ∞ ∈ S which is sufficiently close to z ∞ . Then there is a periodic orbit δ † 0 of R 0 with period m which passes through z † ∞ . We then write Definition 5.3.10. Given a PFH curve u in W ′ , we define n ′ (u) = u, (σ ′ ∞ ) † . The proof of the following is similar to that of Lemma 5.3.6. Lemma 5.3.11. The intersection number n ′ (u) satisfies the following properties: (1) n ′ (u) ≥ 0 and n ′ (u) = 0 if and only if the image of u is disjoint from We also use the modifier * = s to indicate the subset of somewhere injective curves.
oriented as the boundary of the solid torus {ρ ≤ ρ 0 }. The torus T ρ 0 is foliated by orbits of the Hamiltonian vector field R 0 and, for a dense set of ρ 0 , those orbits are closed. If δ ρ 0 is one such orbit, then the homology class . This implies that u, R × δ ρ 0 = 0 for ρ 0 in a dense set. The positivity of intersections then implies that the image of u is contained in N .

Almost complex structures.
Definition 5.4.1. An almost complex structure J + on W + is admissible if it is the restriction to W + of an almost complex structure J ′ ∈ J ′ on W ′ . If J + agrees with J (resp. J ′ ) at the positive (resp. negative) end, then J + is compatible with J (resp. J ′ ).
An admissible almost complex structure J + on W + is the restriction of an admissible almost complex structure on W + . The admissibility criteria for J − and J − on W − and W − are analogous.
The restrictions of σ ′ ∞ = R × δ 0 ⊂ W ′ to W + and W − , respectively, are holomorphic sections at infinity, written as σ + ∞ and σ − ∞ . Remark 5.4.2. If J + ∈ J + , then the projection π B + : W + → B + is holomorphic. This is due to the fact that the fibers {(s, t)} × S are holomorphic and J + takes ∂ s to ∂ t . The same holds for J + , J − , and J − .

5.4.2.
Moduli spaces for W + . Let (F, j) be a compact Riemann surface, possibly disconnected, with an l-tuple of punctures p = {p 1 , ..., p l } in the interior of F and a k-tuple of punctures q = {q 1 , ..., q k } on ∂F , such that (i) each component of F has nonempty boundary and at least one interior puncture and (ii) each component of ∂F has at least one boundary puncture. We writeḞ = F − p − q and ∂Ḟ = ∂F − q.
Let y be a k-tuple of a ∩ h(a) and let γ = j γ m j j be an orbit set in O k .
Definition 5.4.3. Let J + ∈ J + . Then a degree k ≤ 2g multisection of (W + , J + ) from y to γ is either (i) a holomorphic map which is a degree k multisection of π B + : W + → B + and which additionally satisfies the following: (3) u converges to a strip over [0, 1] × y near q; (4) u converges to a cylinder over a multiple of some γ j near each puncture p i so that the total multiplicity of γ 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 which, after perturbing R 0 using an appropriate Morse function and perturbing J + , becomes a degree k multisection of π B + : W + → B + which additionally satisfies (1)-(5). Here π R is the projection π B + : W + → B + ⊂ R × S 1 , followed by the projection to R.
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 y and γ. Moreover, since the orbits are nondegenerate, the convergence is exponential by the work of Abbas [Ab] 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 ∂N has been perturbed into a pair h, e of nondegenerate orbits and that J + ∈ J + with respect to the perturbed Reeb vector field.
We write M J + (y, γ) for the moduli space of multisections of (W + , J + ) from y to γ. We denote by H 2 (W + , y, γ) the equivalence classes of continuous degree 2g multisections 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 (W + , Z y,γ ).
Then a degree k ≤ 2g multisection of (W + , J + ) from y to γ is defined as in 6 We write M J + (y, γ) for the moduli space of multisections of (W + , J + ) from y to γ.
Let δ † 0 be the closed orbit of R 0 used in Definition 5.3.10. We define (σ + ∞ ) † as the restriction of R × δ † 0 to W + . Definition 5.4.5. Given a multisection u of W + , we define n + (u) = u, (σ + ∞ ) † . Lemma 5.4.6. The intersection number n + (u) satisfies the following properties: (1) n + (u) ≥ 0 and n + (u) = 0 if and only if the image of u is disjoint from

and only if there is a unique transverse intersection point between the image of u and σ + ∞ ; and
(3) n + (u) is independent of the choice of δ † 0 . Proof. The proof is similar to that of Lemma 5.3.6. The only difference is in (2), which we discuss in more detail. Consider u :Ḟ → W + and let p ∈Ḟ be a point such that u(p) ∈ σ + ∞ . If π D 2 is the projection of a neighborhood of u(p) ∈ W + to D 2 ⊂ S along the symplectic connection, then π D 2 • u is holomorphic and nonconstant, and therefore maps an open neighborhood of p inḞ to an open neighborhood of z ∞ . This implies that n + (u) ≥ d · m, where d is the multiplicity of the intersection between the image of u and σ + ∞ .
The proof is similar to that of Lemma 5.3.13 and will be omitted.

5.4.4.
Moduli spaces for W − . The moduli space of holomorphic maps which is used to define the map Ψ from P F C to CF is of a slightly different type from the moduli space which is used to define the map Φ from CF to P F C. In particular, the target of the holomorphic maps is 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 p = {p 1 , . . . , p l } in the interior of F and a k-tuple of punctures q = {q 1 , . . . , q k } on ∂F , such that (i) each component of F has nonempty boundary and has at least one interior puncture and (ii) each component of ∂F has at least one boundary puncture. We writeḞ = F − p − q and ∂Ḟ = ∂F − q.
Let y be a k-tuple of a ∩ h(a) and let γ = j γ which is a degree k multisection of π B − : W − → B − and which additionally satisfies the following: (3) u converges to strip over [0, 1] × y near q; (4) u converges to a cylinder over a multiple of some γ j near each puncture p i so that the total multiplicity of γ 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 which, after perturbing R 0 using an appropriate Morse function and perturbing J − , becomes a degree k multisection of π B − : W − → B − which additionally satisfies (1)-(5). Here π R is the projection π B − : W − → B − ⊂ R × S 1 , followed by the projection to R.
Let δ † 0 be the closed orbit of R 0 used in Definition 5.3.10. We define (σ − ∞ ) † as the restriction of R × δ † 0 to W − . Definition 5.4.9. Given a multisection u of W − , we define n − (u) = u, (σ − ∞ ) † . The proof of the following is similar to that of Lemma 5.4.6.
Lemma 5.4.10. The intersection number n − (u) satisfies the following properties: (1) n − (u) ≥ 0 and n − (u) = 0 if and only if the image of u is disjoint from

and only if there is a unique transverse intersection point between the image of u and
LetW − be W − with the ends {s > 1} and {s < −3} removed. We can view W − as a compactification of W − , obtained by attaching [0, 1] × S at s = −∞ and N at s = +∞. Also let We write M J − (γ, y) for the moduli space of multisections of (W − , J − ) from γ to y. We denote by H 2 (W − , γ, y) the equivalence classes of continuous degree 2g multisections in W − satisfying Conditions (1)-(4) of Definition 5.4.8, where two multisections are equivalent if they represent the same element in H 2 (W − , Z γ,y ).
be the subset of curves which pass through the marked point m.
The following lemma is similar to Lemma 5.4.7:

The Fredholm index. In this subsection we compute the Fredholm index
The Fredholm indices are computed using the doubling technique of Hofer-Lizan-Sikorav [HLS], which we quickly review, referring to the original paper for the details. 5.5.1. Doubling. Let (F, j) be a compact Riemann surface with boundary, p and q be finite sets of interior and boundary punctures of F , andḞ = F −p−q. We form the double (2Ḟ , 2j) of (Ḟ , j) by gluing two copies ofḞ with opposite complex structures j and −j along their boundary ∂Ḟ = ∂F − q. By the Schwarz reflection principle, the two complex structures match and the doubled surface becomes a punctured Riemann surface.
Let E →Ḟ be a complex vector bundle with fiberwise complex structure i and let L → ∂Ḟ be a totally real subbundle of maximal rank. Let E →Ḟ be a complex vector bundle whose fiber E p at p ∈Ḟ is E p with complex structure −i. We then construct the doubled complex vector bundle 2E → 2Ḟ by gluing E →Ḟ and E →Ḟ along ∂Ḟ such that at each p ∈ ∂Ḟ the gluing map identifies the fibers be the anti-holomorphic involution σ which switches (Ḟ , j) and (Ḟ , −j) and let σ : 2E ∼ → 2E be the anti-C-linear bundle isomorphism which projects to σ and identifies E p ≃ E σ(p) by the identity map, where p ∈ (Ḟ , j). Finally, given a linear Cauchy-Riemann type operator D on E, we can define the doubled operator 2D on 2E with the property that 2D isσ-invariant and its restriction to E → Σ is D.
One of the results of [HLS] is the following: Theorem 5.5.1. Suppose that both D and 2D are Fredholm operators in some suitable Sobolev spaces. Then: • ind(D) = 1 2 ind(2D); and • if 2D is surjective, then D is also surjective.
Our situation is slightly more general than that considered by [HLS], since we are considering boundary punctures and exponential weights. The proof, however, remains largely unmodified. 5.5.2. The W + case. For the purposes of computing indices, we replace W + by W + , as defined in Section 5.4.2. The tangent space TW + splits into the vertical and horizontal subbundles T S and T B + via the symplectic connection. (In this section we slightly abuse notation and write T S for the vertical subbundle oveř We define a trivialization τ of T S along Z y,γ as follows: First define τ along L + a ∩W + by orienting all a-arcs arbitrarily as in Section 4.4.2 and extending the trivialization by parallel transport along ∂ v W + . We then extend the trivialization of T S| L + a ∩W + in an arbitrary manner to a trivialization of T S along {3}×[0, 1]×y and along {−1} × γ. Let u : (Ḟ , j) → (W + , J + ) be a W + -curve. Suppose the negative ends of u partition m j into (m j1 , m j2 , . . . ). We then write: We now define a real rank one subbundle L 0 of T S along .
We then extend L 0 across {3} × [0, 1] × y by rotating in the counterclockwise direction from T h(a) to T a in T S by the minimum amount possible. Assuming orthogonal intersections between a and h(a), the angle of rotation is π 2 . Let µ τ (y i ) be the Maslov index of L 0 along {3} × [0, 1] × {y i } with respect to τ . If L =ǔ * L 0 , then we define µ τ (y) to be the Maslov index of L with respect to τ . By the definitions of L 0 and τ , it is immediate that We then have the following Fredholm index formula for W + -curves: Proposition 5.5.2 (Fredholm index formula for W + -curves). The Fredholm index of a (W + , J + )-curve u : (Ḟ , j) → (W + , J + ) from y to γ is given by the formula: Proof. We double the surfaceḞ along ∂Ḟ = ∂F − q, where q is the set of boundary punctures, and double the pullback bundle u * T W + along the real subbundle (u| ∂Ḟ ) * T L + a on ∂Ḟ , to obtain the doubled surface 2Ḟ with a complex vector bundle 2u * T W + → 2Ḟ . The boundary punctures q i ofḞ are doubled to give positive interior punctures 2q i on 2Ḟ .
Since the trivialization τ of T S| Zy,γ was chosen to be tangent to T L + a ∩ T S, its pullback to u * T W + is compatible with the doubling operation and gives a trivialization 2τ of 2u * T S over a neighborhood of the punctures of 2Ḟ . Also let τ ′ be a partially defined trivialization of T B + which is given by ∂ s at the positive and negative ends of B + . Then τ ′ can be doubled to 2τ ′ on 2u * T B + which is defined over a neighborhood of the punctures of 2Ḟ .
The linearized ∂-operator at u splits as a sum is a Cauchy-Riemann type operator on W 1,p -sections of u * T W + with exponential weights and K u is a direct sum of finite-dimensional operators, one for each puncture ofḞ , with index 2 for interior punctures and index 1 for boundary punctures; see [Dr, Section 2.3] for the definition of the operator K u . We define u is a finite-dimensional operator and contributes by 2 to the index for each puncture of 2Ḟ . Technically speaking, . We now apply the standard Fredholm index formula (see for example Dragnev [Dr]) for holomorphic curves in symplectizations -with slight modifications -to obtain: Here µ 2τ (2y) is the sum of the Conley-Zehnder indices, computed with respect to the trivialization 2τ , of the paths of symplectic matrices arising from the asymptotic operators at 2y i . The last term is the relative first Chern class of the double of the pullback bundle u * T S with respect to 2τ . The one term that is not present in the Fredholm index formula for J-holomorphic curves in a symplectization is the penultimate term 2c 1 (2u * T B + , 2τ ′ ).
We then compute the following: Summarizing, we obtain: This completes the proof of the proposition.
Proof. Let q i ∈ ∂F be a (positive) boundary puncture and let E = [0, ∞)×[0, 1] ⊂ F be a strip-like end with coordinates (s, t) which parametrizes a neighborhood of q i . Suppose u maps the end asymptotically to y i . Fix a symplectic trivialization be the end of the doubled surface 2Ḟ , which corresponds to the interior puncture 2q i and is obtained by doubling E. The involution σ is given by σ(s, t) = (s, 2 − t) with respect to these coordinates. Similarly, we have a symplectic trivialization be the asymptotic operator of D 0 u corresponding to q i with respect to the trivialization Θ. Here J 0 is the standard complex structure on R 2 and S i (t) is a symmetric 2 × 2 matrix with real coefficients. For the definition of the asymptotic operator and its relation with the Fredholm theory of linearized ∂-operators, see [Dr, Section 3] or [HT2]. 7 The solution of the Cauchy probleṁ is a path of symplectic matrices which represents the linearized Reeb flow along the chord y i , expressed with respect to the trivialization τ . In our setting, we may assume that Φ(t) is a path of unitary matrices. By identifying (R 2 , J 0 ) = (C, i), we write Φ(t) = e iα(t) for some function α : [0, 1] → R. Then The double of the asymptotic operator, i.e., the asymptotic operator of the doubled operator 2D 0 u at the interior puncture 2q i , can be written as −1). The solution of the corresponding Cauchy problem is Hence can write Φ(t) = e iα(t) , whereα : [0, 2] → R is given by: The Conley-Zehnder index of the path Φ is 7 What we call Si here is written as C i 2∞ in [Dr].
is also similar and will be omitted.
If u is a W − -curve from γ to y, then the Fredholm index ind W − (u) is defined as the expected dimension of the moduli space M J − (γ, y) near u.
We then have the following Fredholm index formula: to y is given by the formula: Proof. By a calculation similar to that of Proposition 5.5.2, we obtain which simplifies to the desired result.
Remark 5.5.6 (Reason for considering W − ). We will give a rough explanation of the reason for considering holomorphic curves in W − which pass through m. Suppose we have a W + -curve u from y to γ and a W − -curve v from γ to y (i.e., Im(v) ⊂ W − ). Then, by taking the sum of Equations (5.5.1) and (5.5.3), we compute the Fredholm index of the glued curve u#v, corresponding to the stacking of W + at the top (s > 0) and W − at the bottom (s < 0), to be: whereḞ is obtained from gluing the domains of u and v. The stacking gives rise to a chain map CF → CF , which we expect to map y → y via restrictions of trivial cylinders (modulo chain homotopy). This would mean χ(Ḟ ) = 0 and c 1 ((u#v) * T S, τ ) = 0. This leaves us with a deficiency of 2g. Introducing the point constraint at m, from the perspective of Fredholm indices, is basically equivalent to applying a multiple connected sum to the holomorphic curve v and the fiber S which passes through m. The multiple connected sum is performed at the 2g intersection points between v and S. We effectively increase the Fredholm index by 2g + 2, obtained by adding up the following contributions: (i) the Fredholm index of the fiber S, which is χ(S) = 2 − 2g; (ii) 2g intersection points, each of which contributes +2 to minus the Euler characteristic.
The point constraint then cuts the expected dimension of the moduli space by −2, for a net gain of 2g.
5.6. The ECH index. In this section we present the ECH index I W + , the relative adjunction formula, and the ECH index inequality for W + . The situation for W − is analogous, and will not be discussed explicitly, except to point out some differences.
5.6.1. Definitions. Let y = {y 1 , . . . , y 2g } ∈ S a,h(a) and γ = l j=1 γ m j j ∈ O 2g . Let τ be a partial trivialization of T SW + along Z y,γ as given in Section 5.5.2. Using the trivialization τ , for each simple orbit γ j of γ, we choose an identification of a sufficiently small neighborhood N (γ j ) of γ j with γ j ×D 2 , where D 2 has polar coordinates (r, θ).
Lemma 5.6.3 (Relative adjunction formula). Let u :Ḟ → W + be a W + -curve in the homology class A ∈ H 2 (W + , y, γ). Then If ν is the normal bundle of u, then we have the formula: In the general case, we combine the calculations of Lemma 4.5.8 and [Hu1, Proposition 3.1] to obtain the equation . The analog of Claim 4.5.10 for the present situation is The difference in the shape of the base (i.e., B vs. B + ) accounts for the discrepancy between Equation (5.6.3) and Claim 4.5.10.

ECH index.
We now define the ECH indices for W + and W − .
Definition 5.6.6 (ECH index for W − ). Given a class A ∈ H 2 (W − , γ, y) which admits a τ -trivial representativeČ, we define (5.6.5) As usual, the ECH indices I W + (A) and I W − (A) are independent of the choice of trivialization τ .
Remark 5.6.7. To obtain a finite count of W − -curves which pass through the point m, we count curves u with ECH index I W − (u) = 2.

Additivity of indices.
Lemma 5.6.8 (Additivity of indices). If u ∈ M J (y, y ′ ), v ∈ M J + (y ′ , γ), and u#v is a pre-glued curve, then , and u#v is a pre-glued curve, then Proof. The additivity for ind is well-known and the additivity for I is immediate from the definitions. 5.6.5. Index inequality. Although we will not define it here, given an integer m k > 0 and a simple orbit γ k , we can define the incoming partition P in γ k (m k ) and the outgoing partition P out γ k (m k ) as in [Hu2,Definition 4.14]. We have the following index inequality, which is analogous to [Hu1, Theorem 1.7] (also see [Hu2,Theorem 4.15] which is applicable to symplectic cobordisms). Note that a W + -curve is automatically simply-covered.
Theorem 5.6.9 (Index inequality). Let u be a W + -curve from y to γ = l k=1 γ m k k . If the negative ends of u partition m k into (m k1 , m k2 , . . . ), then Here δ(u) is the signed count of interior singularities of u, where each singularity contributes positively to δ(u). The following are immediate: (i) if ind W + (u) = I W + (u), then u is embedded; and (ii) if I W + (u) = 0 or 1, then u is embedded.
Proof. If we plug Equation (5.5.1), Equation (5.6.4), and the relative adjunction formula (5.6.1) into I W + (u) − ind W + (u), we obtain: . The statement for W + then follows from the writhe inequality .) The proof for W − is similar. 5.7. Holomorphic curves with ends at z ∞ . In this subsection we explain how to extend the definitions of the Fredholm and ECH indices to holomorphic curves which have ends at multiples of z ∞ . The novelty is that the Lagrangian boundary condition is singular at z ∞ and that the chord over z ∞ can be used more than once. We will treat in detail the case of a curve u :Ḟ → W which is a degree l multisection of W → R × [0, 1]; multisections of W + and W − can be treated similarly.
We write {z p ∞ ( − → D )} ∪ y for the 2g-tuple of points of a ∩ h(a), where y ⊂ S, z ∞ has multiplicity p, − → D is the data at z p ∞ , and each arc of

5.7.2.
Multisections. In this subsection we define multisections u : (Ḟ , j) → (W , J ) with irreducible components which branched cover σ ∞ . The notation is complicated by the fact that there may be branch points of π B • u along ∂B.
Notation. Let (F, j) be a compact nodal Riemann surface, possibly disconnected, with two sets of punctures q + = {q + 1 , . . . , q + k } and q − = {q − 1 , . . . , q − k } on ∂F and a set p = {p 1 , . . . , p l } of nodes on ∂F , such that (i) the nodes p are disjoint from q + and q − , (ii) each connected component of F has nonempty boundary, (iii) on each oriented loop of ∂F whose orientation agrees with that of ∂F , there is at least one puncture from each of q + and q − , and (iv) the punctures on q + and q − alternate along each oriented loop of ∂F . Here, by a loop or path on ∂F we mean a loop or path which is a concatenation of subarcs of ∂F with endpoints on Given a holomorphic multisection u : (Ḟ , j) → (W , J ), we write where u ′ is a possibly disconnected branched cover of σ ∞ and u ′′ is the union of irreducible components which do not branch cover σ ∞ . We also decompose F = F ′ ⊔ F ′′ such thatḞ ′ is the domain of u ′ ,Ḟ ′′ is the domain of u ′′ , and all the nodes are on ∂F ′ . The case of W + and W − are analogous.
The data C. The nodes of ∂F ′ correspond to branch points of π B • u along ∂B. If we viewḞ ′ as a branched cover of B with some branch points along ∂B, then leṫ F ′ ext be the branched cover of B obtained by pushing the boundary branch points to int(B).
Definition 5.7.1 (Data C). An irreducible component d of ∂Ḟ ′ is an oriented path from q + to q − along ∂Ḟ ′ , where the orientation either agrees with that of ∂F on all the subarcs of d or is opposite that of ∂F on all the subarcs of d. A set In words, u ′ can be viewed as mapping ∆ to the set of Lagrangians L a i,j or L h(a i,j ) . Observe that C determines the data − → D + and − → D − at the positive and negative ends.
We then make the following definition which agrees with Definition 4.3.1 when z + = y + and z − = y − : ∩ h(a). A degree k multisection of (W , J) from z + to z − is a pair (u, C) consisting of a holomorphic map which is a degree k multisection of π B : W → B and data C for u ′ , and which additionally satisfies the following: ( (2) there is a canonical decomposition ∆ of ∂Ḟ ′ such that u maps each connected component of ∂Ḟ ′′ and each irreducible component of ∆ to a different L a i or L h(a i ) (here we are using C to assign some L a i or L h(a i ) to each irreducible component of ∆); (4) u converges to a trivial strip over [0, 1] × z + near q + and to a trivial strip over [0, 1] × z − near q − ; (5) the positive and negative ends of u which limit to z ∞ are described by − → D + and − → D − ; (6) the energy of u is finite.
Remark 5.7.3. It will always be assumed that a multisection u of W from {z Indeed, in this paper, such a curve u only appears as the SFT limit of curves u i without components which branch cover σ ∞ and hence naturally inherits C and − → D ± . similarly we extend τ ′ arbitrarily to a trivialization of T S along h(a) and pull it back in an arbitrary way. If − → D ± are the data induced by a multivalued trivialization τ , then we say that τ is compatible with − → D ± .
Remark 5.7.4. Note that the extensions to a and to h(a) might conflict, but it does not matter here. In the cobordism cases (i.e., for W + and W − ), when a and h(a) are connected by the Lagrangian L ± a , we extend the trivialization τ ′ to a in an arbitrary manner and then sweep it around using the symplectic connection along the boundary of the cobordism. 5.7.4. Groomed multivalued trivializations. Let τ be a multivalued trivialization.
The branches of the trivialization τ along [0, 1] × {z ∞ } give rise to a family of arcs c ± k in A ε . Without loss of generality we may assume that c ± k , k = 1, . . . , p ± , is linear with respect to the identification of the universal cover of A ε with [0, 1] × R. For each arc c ± k we denote its initial point by p ± k,0 and its terminal point by p ± k,1 . We finally define . The point p ± k,0 belongs to the Lagrangian subarc h(a i ′ ±,k ,j ′ ±,k ) and the point p ± k,1 belongs to the Lagrangian subarc a i ±,k ,j ±,k . Note that every groomed multivalued trivialization induces data − → D ± , but not every − → D ± is compatible with a groomed multivalued trivialization. 5.7.5. Index formulas. We first give the Fredholm index of a multisection.
Proposition 5.7.6. Let u :Ḟ → W be a degree l multisection with data − → D ± at z ∞ , and let τ = τ− → . Then the Fredholm index of u is given by the formula: The proof is a computation in the pullback bundle u * T W , which is completely analogous to that yielding Equation (4.4.4).
Next we discuss the ECH index of a multisection (u, C) with ends τ -trivial representative. Define π 2 (z + , z − , τ ) ⊂ H 2 (W , Z z + ,z − ,τ ) in analogy to the definition of π 2 (y, y ′ ) in Section 4. We now describe the construction of a τ -trivial representativeČ ⊂W of the class [u] in π 2 (z + , z − , τ ): Step 1. Replace u ′ :Ḟ ′ → W by u ′ ext :Ḟ ′ ext → W so that there are no nodes along ∂Ḟ ext and each component of ∂Ḟ ext is mapped to some L a i,j or L h(a i,j ) in a manner consistent with the data C : Step 2. Compactify the ends of u ext = u ′ ext ∪ u ′′ toǔ ext as in Section 4.3.
Step 3. Perturbǔ ext so that the resulting representativeČ is immersed, The quadratic form Q τ . LetČ be a τ -trivial representative of u. Let c ± k (δ) be the φ = δ translate of c ± k , where δ > 0 is small. We then take a pushoffČ ′ ofČ such that ∂Č is pushed in the direction of Jτ and Then Q τ (A) = Č ,Č ′ as in Definition 4.5.5.
We remark that, in passing from u to [u], we only record D ± (and not − → D ± ), because the matching is not a homological invariant.
Sometimes we will write I τ (u) to mean I τ (A), where A is the relative homology class defined by u. 5.7.6. Definition of µ τ (∂A). We first define µ τ (∂A). The groomed multivalued trivialization τ determines the matchings D f rom ± → D to ± , and we pick a cycle ζ which represents ∂A and respects the matchings along {±1} × [0, 1] × {z ∞ }. The pullback bundle ζ * T S is trivialized by the pullback of τ . We now define a multivalued real rank one subbundle L 0 of T S along Z a,h(a) by setting L 0 = TĽ a ∩ T S onĽ a and TĽ h(a) ∩ T S onĽ h(a) and extending L 0 across {±1} × [0, 1] × z ± by rotating in the counterclockwise direction from T h(a) to T a in T S by the minimum amount possible. We then define µ τ (∂A) as the Maslov index of L = ζ * L 0 with respect to τ . Now we define the corrections to add to µ τ (∂A) to obtain µ τ (∂A). Let τ be a groomed multivalued trivialization which is compatible with (D + , D − ) and which induces the groomings c Definition 5.7.9. Given a grooming c = {c k } p k=1 from P 0 to P 1 , its winding number is given by: where the arcs c k and {φ = π} are oriented from t = 0 to t = 1 and A ε is oriented by (∂ φ , ∂ t ).
Remark 5.7.10. The grooming c is determined by its endpoints P 0 and P 1 and its winding number w(c).
Let w(c) = q. Let c ♭ k be the linear arc in A ε which is disjoint from {φ = π} and has the same endpoints as c k . (Note that the collection {c ♭ k } is usually not groomed.) Then we define α q as the number of arcs in {c ♭ k } whose φ-coordinate decreases as t increases (in the universal cover). The number α q only depends on q (modulo p), through the bijection P 0 ∼ → P 1 . We finally define µ τ (∂A) as follows: Let q + = w(c + ) and q − = w(c − ). Then we define the "discrepancies" d ± at the positive and negative ends as: and we set: (This means that we are using q − = 0 as the reference point.) The discrepancy d + − d − was (somewhat artificially) added to make Lemma 5.7.12 hold.
Remark 5.7.11. Let us view P 0 and P 1 as points on (−π, π). In the special case where the points of P 0 and P 1 alternate along (−π, π), we can write: Here x is the greatest integer ≤ x and x is the smallest integer ≥ x. Proof. Let c + = {c + k } and c − = {c − k } be the groomings of τ and let q ± = w(c ± ). In order to compute Q τ (A), we choose a τ -trivial representativeČ of A such that:

ECH indices of branched covers of sections at infinity.
and a representativeČ ′ such that: Since all the intersections betweenČ andČ ′ are contained in the level s = 0, Next we claim that Indeed, given the end which corresponds to a strand c ± k , the Maslov index of the end is given by −2w(c ± k ) if the endpoints of c ± k satisfy 0 < p ± k,0 < p ± k,1 < 2π, and is given by On the other hand, the number of strands for which p ± k,0 > p ± k,1 holds is exactly α q . The claim follows. Hence, We also have c 1 (T S| A , τ ) = q + − q − . Putting everything together, we obtain: This proves the lemma.
We also state the following lemmas without proof: Lemma 5.7.13. If u :Ḟ → W − is a degree p ≤ 2g multisection which branch covers σ − ∞ with possibly empty branch locus, then I(u) = 0.
Lemma 5.7.14. Lemma 5.7.16. Let τ 2 and τ 1 be groomed multivalued trivializations compatible with (D 2 , D 1 ) and (D 1 , D 0 ), respectively, and let τ be obtained by concatenating τ 2 and τ 1 . Given relative homology classes we can form the concatenation Then we have: In view of the following lemma, we can suppress τ from I τ .
Lemma 5.7.17. I τ (A) is independent of the choice of groomed multivalued trivialization τ .
Proof. Let τ and τ ′ be two groomed multivalued trivializations adapted to the same data (D + , D − ). It suffices to consider the particular cases when τ and τ ′ differ only either at some y i or at z ∞ . In the first case, the argument is analogous to the proof of Lemma 4.5.6. In the second case, we can glue branched covers of σ ∞ to switch groomings. Then the statement follows from Lemma 5.7.12 and the additivity of the ECH index. 5.7.9. The ECH index inequality. Let u be a degree l multisection of W from We define c ± k , k = 1, . . . , p ± as the intersections of A ε with the π-projections of the ± ends of u which limit to z ∞ . Here ε > 0 is small and depends on u, and the map π projects out the s-direction. Let c ± = {c ± k } k=1,...,p ± and let P ± 0 (resp. P ± 1 ) be the set of the initial (resp. terminal) points of the arcs c ± k . Lemma 5.7.18. If τ is a groomed multivalued trivialization which is compatible with c ± and A ∈ π 2 (z + , z − , τ ) is the relative homology class of u, then In particular, if the points of P * 0 and P * 1 alternate along (0, 2π) for both * = + and −, then I(A) ≥ ind(u).
Next suppose that the points of P * 0 and P * 1 alternate along (0, 2π) for both * = + and −. We claim that d + ≥ 0. Let π : R × [0, 1] × S → [0, 1] × S be the projection onto the second and third factors. By the positivity of intersections, π(u) is positively transverse to the Hamiltonian vector field ∂ t ; hence q + ≤ 0. By Equation (5.7.6), if q + ≤ 0, then d + ≥ 0 (in both cases). d − ≤ 0 is proved similarly. 5.7.10. ECH index calculation. In this subsection we compute the ECH index of a multisection of W which is the disjoint union of a branched cover of the section at infinity σ ∞ and a curve which limits to a multiple of z ∞ . This calculation will be used in Section II.3.7.2.
Next we consider the variant where the multiple of z ∞ is at the positive end. Let u = u ′ ∪ u ′′ be a degree p 1 + p 2 + l multisection of W , where deg u ′ = p 1 , u ′′ is a multisection from {z p 2 ∞ ( − → D 2 )} ∪ y ′ to y, and l is the cardinality of y ′ . We use the same notation as above, with − replaced by +.
We make the following assumptions: (G ′ 1 ) c + 1 has winding number q 1 := w(c + 1 ) = 0; ∩ A ε and q 2 := w(c + 2 ) = 0 or −1; and (G ′ 4 ) the projection of c + 2 to ∂D 2 ε is injective except on κ ≥ 0 short intervals of ∂D 2 ε which correspond to thin sectors of type S(a i,j , h(a i,j )). Let w be the signed number of crossings of π ρ (c + 1 ∪ c + 2 ). In this case, all the crossings of π ρ (c + 1 ∪ c + 2 ) are negative. The analog of Lemma 5.7.19, stated without proof, is: We also compute the discrepancy d + of c + 2 when q 2 = −1. Since α −1 = p 2 − 1 and α 0 = κ, By Lemma 5.7.18, (5.7.12) I(u ′′ ) ≥ ind(u ′′ ) + κ. 5.7.11. Extended moduli spaces. We now describe the extended moduli spaces which involve multiples of z ∞ at the ends. Details will be given for W ; the W + and W − cases are analogous. Let M = M J (z + , z − ) be the moduli space of multisections of (W , J ) from Let † be the modifier "u ′ = ∅". We now describe an enlargement of the moduli space M † .
Definition 5.7.21. An extended W -curve u from z + to z − is a multisection of (W , J ) which satisfies the conditions of Definition 5.7.2 withḞ ′ = ∅ and (1) and (2) replaced by the following: (1 ′ ) There exists positive and negative ends E +,i and E −,i ofḞ that limit to z ∞ such that: • the two components of u(∂Ḟ ∩ E +,i ) (resp. u(∂Ḟ ∩ E −,i )) are subsets The moduli space of extended W -curves from z + to z − is denoted by M †,ext The extended moduli spaces M †,ext Notation. A sector of D 2 is large if it has angle π < φ and small if it has angle 0 < φ < π. If R, R ′ ⊂ D 2 are distinct radial rays and I ⊂ R is an interval, then let S(R, R ′ ; I) be a counterclockwise sector in D 2 from R to R ′ such that the angle of the sector lies in I; if we do not specify I, we write S(R, R ′ ).
Role of extended moduli spaces. We briefly explain the role of the extended moduli space − → D + corresponds to an end E +,i of u, whose projection π D 2 to a small neighborhood of z ∞ in S sweeps out a sector S = S(a i k ,j k , h(a i ′ k ,j ′ k )). If S is large, then π D 2 (E +,i ) has no slits by the definition of u ∈ M † (i.e., π D 2 maps no boundary , and the neighborhood of u in M † is generically a codimension ≥ 1 submanifold of an extended moduli space M †,ext . 5.8. Transversality. We first discuss the regularity of almost complex structures on W , W ′ , W + and W − . (1) all moduli spaces M J + (y, γ) with y a k-tuple of a ∩ h(a) and γ ∈ O k , k ≤ 2g, are transversely cut out (in the Morse-Bott sense in the case of a Morse-Bott building); and (2) the restrictions J and J ′ of J + to the positive and negative ends belong to J reg and (J ′ ) reg , respectively.
Here every u ∈ M J + (y, γ) is somewhere injective due to the presence of the HF end. In other words, M J + (y, γ) = M s J + (y, γ).
Definition 5.8.6. The almost complex structure J + ∈ J + is regular if the following hold: (1) all moduli spaces M †,ext Note that M †,ext The regularity of J − ∈ J − is defined similarly.
We write J reg + for the space of regular J + ∈ J + and J reg ± for the space of regular J ± ∈ J ± . Without loss of generality we may assume that the regular J + of interest are the restrictions of regular J + . Proof. We first treat the W + case. The proposition follows from a standard transversality argument along the lines of [MS, Theorem 3.1.5], with some modifications. The necessary modifications for almost complex structures J ′ ∈ J ′ , defined on R × N , were described in [Hu1, Lemma 9.12(b)], and our situation is almost identical since J + ⊂ J + is the restriction to W + of some J ′ ∈ J ′ .
The key observation is that each irreducible component of a W + -curve u :Ḟ → W + is somewhere injective, since each [0, 1]×{y i }, y i ∈ y, is used exactly once as a positive asymptotic limit. Let π N : W + → N be the restriction of the projection π N : R × N → N onto the second factor. We then observe that there is a dense open set of points p ∈Ḟ which are π N -injective, i.e., (i) d(π N • u)(p) has rank 2; and (ii) (π N • u)(p) = (π N • u)(q) implies p = q.
The perturbations to J + can then be carried out in a neighborhood of a π N -injective point p ∈Ḟ as in [Hu1, Lemma 9.12(b)].
The regularity of the almost complex structures J and J ′ at the ends was already treated, i.e., J reg ⊂ J and (J ′ ) reg ⊂ J ′ are dense by Lemmas 3.5.2 and 4.7.2.
In the W + case, the perturbations of J + are allowed on the subset U = W + ∩ ((R×N )−{ρ ≤ ε}) for some small ε > 0. We simply observe that all the curves u in the moduli spaces M †,ext J + (z, δ r 0 γ ′ ) in Definition 5.8.6 pass through U , and pick a π N -injective point p ∈Ḟ such that u(p) ∈ U . The W − case is similar. 5.8.3. Some automatic transversality results. We collect some automatic transversality results.
We now use the doubling technique from Theorem 5.5.1. The double ofḞ -a sphere with three punctures -is denoted by 2Ḟ and the double of u is denoted by 2u. The index of the doubled operator 2D u is ind(2u) = 2 ind(u) = 0 and D u is surjective if and only if 2D u is surjective. Now, by Wendl's automatic transversality theorem [We3, Theorem 1], 2D u is surjective if where g is the genus of 2Ḟ and #Γ 0 is the count of punctures with even Conley-Zehnder index. In the present situation, g = 0 and #Γ 0 = 0, so Equation (5.8.1) becomes ind(2u) ≥ −1, which is satisfied.
The following is easier, and is stated without proof: Here the ε-closeness is measured with respect to a metric g on W − which is the restriction of an s-invariant metric on R × N .
which is defined in the same way as a degree k multisection of (W − , J − ), except that u is a degree k multisection of from δ r 0 γ to z ′ . The regularity of J − and the closeness of J ♦ − to J − imply: (δ r 0 γ, z ′ ). Note that we can still refer to n * (u) since it is a homological quantity.
Let K ∋ m be a compact set of W − . We define the modifier K to mean that u passes through K. 6. THE CHAIN MAP FROM HF TO P F H 6.1. Compactness for W + . In this subsection we treat the compactness of holomorphic curves in W + which will be used to establish the chain map Φ in Section 6.2.
Suppose J + ∈ J + and J, J ′ are the restrictions of J + to the positive and negative ends. Let y = {y 1 , . . . , y 2g } ∈ S a,h(a) and γ = l k=1 γ m k k ∈ O 2g . In this section, we may pass to a subsequence of a sequence of holomorphic curves without specific mention. 6.1.1. Euler characteristic bounds. We first state a preliminary lemma: Lemma 6.1.1. Let u i : (Ḟ i , j i ) → (W + , J + ), i ∈ N, be a sequence of W + -curves from y to γ. Then there is a subsequence such that all theḞ i are diffeomorphic to a fixedḞ .
Proof. The proof is given in two steps.
Step 1 (ω-area bounds). This is a consequence of the vanishing of the flux F h of h (cf. Section 3.3). View the broken closed string γ y corresponding to y as a collection of curves in Then γ y is uniquely determined up to a homotopy which is supported on L + a ∩W + . Let u i :Ḟ i → W + , i = 1, 2, be two W + -curves from y to γ and letǔ i :F →W + be their compactifications. Thenǔ i (∂ +F ) is homotopic to γ y , i = 1, 2, where ∂ +F is the union of boundary components ofF which map to the positive (s > 0) part ofW + . Henceǔ 1 −ǔ 2 can be viewed as a closed surface Z ∈ H 2 (W + ) ≃ H 2 (W + ) ≃ H 2 (N ). Since the flux F h vanishes, the ω-area of a W + -curve u only depends on y and γ.
Step 2 (Genus bounds). Now that we have an ω-area bound on the sequence {u i }, we can apply the Gromov-Taubes compactness theorem [T3, Proposition 3.3], which is a local result and carries over to the symplectic cobordism (W + , Ω + ) without difficulty. As explained in [Hu1, Lemma 9.8], the Gromov-Taubes compactness theorem implies the weak convergence of u i as currents to a holomorphic building u ∞ . In particular, we may assume that [u i ] ∈ H 2 (W + , y, γ) is fixed for all i.
We now use the fact that [u i ] is fixed to bound the genus ofḞ i . The relative adjunction formula (Lemma 5.6.3) gives: [Hu2,Lemma 4.20] and the nonnegativity of δ(u i ), we obtain: . This bounds χ(Ḟ i ) from below, since all the terms on the right-hand side either depend on the homology class of u i or the data of the ends. Hence we may assume that all theḞ i are diffeomorphic to a fixedḞ .

SFT compactness.
Proposition 6.1.2. Let u i : (Ḟ i , j i ) → (W + , J + ), i ∈ N, be a sequence of W +curves from y to γ. Then there is a subsequence which converges in the sense of SFT to a level a + b + 1 holomorphic building where v j is a holomorphic map to W j = W for j > 0, W 0 = W + for j = 0, and W j = R × N for j < 0, and the levels W −b , . . . , W a are arranged in order from lowest to highest.
Here "convergence in the SFT sense" means convergence with respect to the topology described in [BEHWZ]. We will also write π j : W j → B j for the projection of W j to the appropriate base B j = B, B + , or B ′ .
Proof. By Lemma 6.1.1, we may assume thatḞ i =Ḟ as smooth surfaces. We can then apply the SFT compactness theorem from [BEHWZ]. Since we are dealing with holomorphic curves with Lagrangian boundary, we sketch some of the standard details of SFT compactness.
Let h be a Riemannian metric on W + which is compatible with J + and is cylindrical at both ends. By [BEHWZ,Lemma 10.7], we can add a bounded number of interior marked points Z i toḞ so that the following holds: Let g i be a hyperbolic metric onḞ − Z i which is compatible with j i and which has geodesic boundary and cusps at the boundary/interior punctures. Then we have a bound on ρ ∇u i , the so-called "gradient bound", where the norm · and the gradient are measured with respect to the metrics g i and h, and ρ is the injectivity radius of the surface doubled along ∂Ḟ .
The sequence u i converges on the thick part by the gradient bound. If g i degenerates as i → ∞, then there is a finite collection of homotopy classes of properly embedded arcs and closed curves onḞ − Z i , whose geodesic representatives are mutually disjoint, and which are pinched as i → ∞, i.e., the lengths of the geodesic representatives go to zero. Let δ be any such homotopy class and let Thin ε (δ, g i ) be the ε-thin annulus ⊂ (Ḟ − Z i , g i ) whose core is homotopic to δ. Here ε > 0 is sufficiently small. Then u i (Thin ε (δ, g i )) converges to a "holomorphic sausage" as in [BEHWZ, Figure 14], i.e., a stack of holomorphic cylinders or strips whose ends are either removable or are asymptotic to chords or closed orbits, and whose successive components have ends which are identified. (The same holds for u i of cusp pieces of the thin part.) The limiting curve u ∞ can be written as a level a + b + 1 holomorphic building v −b ∪ · · · ∪ v a , where each v j is not necessarily irreducible and may have nodes. Here (i) a, b are nonnegative integers, (ii) v j is a holomorphic map to W j , and (iii) the levels v j are ordered from the negative end to the positive end as j increases. As usual, if the level v j is just a union of trivial cylinders, then it will be elided. 6.1.3. Main theorem. Suppose J + ∈ J reg + and J, J ′ are the restrictions of J + to the positive and negative ends.
The following is the main theorem of this subsection: Theorem 6.1.3. Let u i : (Ḟ i , j i ) → (W + , J + ), i ∈ N, be a sequence of W + -curves from y to γ. If I W + (u i ) = 1 for all i, then a subsequence of u i converges in the sense of SFT to one of the following: (1) an I W + = 1 curve; (2) a building with two levels consisting of an I HF = 1 curve and an I W + = 0 curve; or (3) a building with multiple levels consisting of an I W + = 0 curve, an I ECH = 1 curve, and possible I ECH = 0 connectors in between. Similarly, if I W + (u i ) = 0 for all i, then a subsequence of u i converges to an I W + = 0 curve.
We write "an I # = i curve" as shorthand for "a #-curve with ECH index I # = i".
We postpone the proof of the main theorem after a more detailed discussion of the structure of the SFT limit. Let u ∞ be the SFT limit of the sequence {u i }, given by Proposition 6.1.2. A ghost component of u ∞ is an irreducible component of u ∞ which maps to a point. By the SFT compactness theorem, the domain of a ghost component is necessarily a stable Riemann surface. We recall that a Riemann surface F with k int interior marked points and k bdr boundary marked points is stable if: where v j has no ghost components and u g ∞ is the union of ghost components. We will also write u ng ∞ = v −b ∪ · · · ∪ v a . Remark 6.1.4. The ECH index of a curve depends only on its relative homology class and therefore ghost components do not contribute to it. Hence, by the additivity of ECH indices (Lemma 5.6.8), if u i is a sequence of J + -holomorphic maps with constant ECH index, then: Lemma 6.1.5. Each level v j , j = −b, . . . , a, is a degree 2g multisection of π j : W j → B j with no branch points along ∂B j .
Proof. Since u i is a degree 2g multisection of π B + : W + → B + for all i, it follows that, with the exception of finitely many p ∈ B j , every level v j intersects a fiber π −1 j (p) exactly 2g times. We show that on any level v j there are no irreducible components which lie in a fiber π −1 j (p). Arguing by contradiction, suppose v : F → W j is an irreducible component which maps to a fiber π −1 j (p). If p ∈ int(B j ), then v is a holomorphic map from a closed Riemann surface F to π −1 j (p). Since π −1 j (p) is a Riemann surface with nonempty boundary, v must be constant. On the other hand, if p ∈ ∂B j , it is also possible that F is a compact Riemann surface with nonempty boundary and v(∂ F ) ⊂ a. However, since S − a is connected and nontrivially intersects ∂S, v must also be constant. Since ghost components are excluded from v j by definition, we have a contradiction.
Finally, if j ≥ 0, then we claim that π j •v j has no branch points along ∂B j . This is due to the fact that v 0 uses each component of L + a exactly once and v j , j > 0, uses each component of R × {1} × a and each component of R × {0} × h(a) exactly once. Lemma 6.1.6. Let u ∞ be the SFT limit of a sequence of J + -holomorphic multisections u i with constant ECH index. If J + is regular, then: • I HF (v j ) > 0 for j > 0, Proof. Since v j , j ≥ 0, is a degree 2g multisection by Lemma 6.1.5 and uses each connected component of the boundary Lagrangian exactly once, it is somewhere injective. Also since J and J + are regular, it follows that the curves v j , j ≥ 0, are regular. Hence ind W (v j ) ≥ 0 for j > 0 and ind W + (v 0 ) ≥ 0. Moreover, since ind W (v j ) = 0, j > 0, if and only if v j is a union of trivial strips, we may assume that ind W (v j ) > 0 for all j > 0. By the index inequality (Theorems 4.5.13 and 5.6.9) we have I HF (v j ) > 0 for j > 0 and I W + (v 0 ) ≥ 0. On the other hand, if j < 0, then I ECH (v j ) ≥ 0 by [HT1, Proposition 7.15(a)].
If I W + (u i ) ≤ 1 in addition, then I ECH (v j ) ≤ 1 by the additivity of the ECH index. Hence [Hu1, Lemma 9.5] implies that v j is somewhere injective and ind(v j ) ≥ 0 since J ′ is regular.
Lemma 6.1.7. Let u ∞ be the SFT limit of a sequence of J + -holomorphic multisections u i with I W + (u i ) ≤ 1 for all i. If J + is regular, then u g ∞ = ∅. Proof. Let ind(u g ∞ ) and ind(u ng ∞ ) be the sum of the Fredholm indices of the irreducible components of u g ∞ and u ng ∞ , respectively. Also let F ng and F g be the domains of u ng ∞ and u g ∞ , respectively. Then ind(u g ∞ ) = −χ(F g ). If u ∞ has k int interior nodes and k bdr boundary nodes, then ind(u i ) = ind(u ng ∞ ) + ind(u g ∞ ) + 2k int + k bdr (6.1.3) = ind(u ng ∞ ) − χ(F g ) + 2k int + k bdr ≤ 1. In fact interior nodes are codimension two phenomena, and boundary nodes are codimension one phenomena. Next, if F g = ∅, then the stability of the Riemann surface implies that (6.1.5) −χ(F g ) + 2k 1 + k 2 ≥ 2.
Equations (6.1.4) and (6.1.5) together imply that ind(u ng ∞ ) ≤ −1. Hence we have ind(v j ) ≤ −1 for some v j , which is a contradiction of Lemma 6.1.6 for a regular J + . The contradiction came from assuming that u g ∞ = ∅. We now finish the proof of Theorem 6.1.3.
Proof of Theorem 6.1.3. The proof is based on a classification of the types of allowable buildings u ∞ = v −b ∪ · · · ∪ v a ∪ u g ∞ . We consider the situation of I W + (u i ) = 1, leaving the easier I W + (u i ) = 0 case to the reader.
By Lemmas 6.1.5 and 6.1.7, the limit u ∞ consists of a building of degree 2g multisections v j . Moreover, by Lemma 6.1.6, I W j (v j ) ≥ 0 for all j. The additivity of ECH indices gives three possibilities for the limit: (1) u ∞ = v 0 , where v 0 is a multisection of W + and I W + (v 0 ) = 1; is a multisection of W + , all but one of v −b , . . . , v 0 have I = 0, and the remaining level has I = 1.
What is left to prove is that if there is a Fredholm index zero branched cover of R × γ k with positive ends which partition m k into (a 1 , . . . , a l 1 ) and negative ends which partition m k into (b 1 , . . . , b l 2 ). By [HT1, Lemma 7.5], the incoming partition P out γ k (m k ) -the partition which corresponds to γ m k We will usually suppress J, J ′ , and J + from the notation. We first define an approximation of Φ: Definition 6.2.1. Let CF ′ (S, a, h(a)) be the chain complex generated by S a,h(a) , before quotienting by the equivalence relation ∼ given in Section 4.9.3. We define the map where Φ ′ (y), γ is the mod 2 count of M I=0 J + (y, γ). The count is meaningful since M I=0 J + (y, γ) is compact by Theorem 6.1.3.

Proposition 6.2.2. The map
is a chain map.
Proof. By Theorem 6.1.3 and Lemma 5.4.7, ∂M I=1 Here we have omitted the potential contributions of connector components for simplicity.
We examine the corresponding gluings of the holomorphic buildings. The first gluing is that of (v 1 , v 0 ) ∈ A. This type of gluing was treated by Lipshitz; see Propositions A.1 and A.2 in [Li, Appendix A]. Observe that there are no multiplycovered curves to glue, since each Reeb chord of a 2g-tuple is used exactly once.
The second type of gluing is that of (v 0 , v −b ) ∈ B, with I ECH = 0 connectors v −1 , . . . , v −b+1 in between. The curve v 0 is simply-covered since it has an HF end and the curve v −b is simply-covered since I ECH = 1 (see [HT1,Proposition 7.15]). This type of gluing was treated carefully in [HT1,HT2]. Although the setting there was the gluing for ∂ 2 = 0, in fact most of the work goes towards properly counting I ECH = 0 connectors, i.e., branched covers of trivial cylinders. Their treatment of gluing/counting the I ECH = 0 connectors carries over with little modification to our case. See Section 6.5 for more details. The composition π S • v is holomorphic. Let (r, θ) be polar coordinates on a small neighborhood N ( If v is not the restriction of a trivial cylinder, then π S • v must map a neighborhood of the puncture ofḞ corresponding to x i to a sector {0 ≤ θ ≤ kπ − π 4 , r > 0} or {π ≤ θ ≤ (k + 1)π − π 4 , r > 0}, where k ≥ 1. Since such a sector cannot be contained in S, the map π S • v| s≥C must map identically to x i , where C ≫ 0. By the unique continuation property, v = (R × δ x i ) + . Proof. Let y = {y 1 , . . . , y 2g } ∈ S a,h(a) . Assume without loss of generality that y 1 = x 1 . Then y is equivalent to y ′ = {x ′ 1 , y 2 , . . . , y 2g }. Let u be an I W + = 0 Morse-Bott building from y to a generator γ of P F C 2g (N ). Then (R × δ x 1 ) + is an irreducible component of u by Lemma 6.2.3 and γ can be written as eγ ′ , where e is the elliptic orbit of the negative Morse-Bott family N . Now, by replacing (R × δ x 1 ) + by (R × δ x ′ 1 ) + and the augmenting gradient trajectory from δ x 1 to e by the augmenting trajectory from δ x ′ 1 to e, we obtain an I W + = 0 Morse-Bott building from y ′ to γ = eγ ′ . Hence there is a one-to-one correspondence between W + -curves from y to γ and W + -curves from y ′ to γ. Since Φ ′ (y) = Φ ′ (y ′ ), the map Φ ′ descends to CF (S, a, h(a)).
The Heegaard Floer contact invariant is given by the equivalence class of x = {x 1 , . . . , x 2g } and the ECH contact invariant is given by e 2g . By Lemma 6.2.3, the only I W + = 0 Morse-Bott building from x consists of (R × δ x i ) + , i = 1, . . . , 2g, augmented by connecting gradient trajectories from δ x i to e in the Morse-Bott family N . Hence Φ ′ (x) = e 2g . Then Φ maps the equivalence class of x to e 2g . 6.3. Spin c -structures. Let S a,h(a) be the set of 2g-tuples of a ∩ h(a) and let We define a map h ′ + : S a,h(a) → H 1 (W + , ∂ h W + ) as follows: Given y = {y 1 , . . . , y 2g } ∈ S a,h(a) , where y i ∈ a i ∩ h(a σ(i) ) for some permutation S 2g , we define h ′ + (y) as the homology class of the broken closed string obtained by concatenating the following oriented arcs: • The following lemma is an immediate consequence of the definition of W +curves. There are natural isomorphisms With respect to these isomorphisms, the total homology class in H 1 (W + , ∂ h W + ) of an orbit set γ at the negative end of W + corresponds to the usual total homology class of γ in H 1 (M ). Moreover, h + (y) = h(y), where h is as defined in Section 4.10.
Combining Proposition 4.10.1 and Lemma 6.3.1, we obtain the following theorem: given in the following paragraphs. Choose a 2g-tuple of intersection points y 0 such that s z (y 0 ) = s ξ + P D(A) and a complete set of paths {C y } for s ξ + P D(A) based at y 0 .
Let π N : W + → N be the restriction of the projection R × N → N , (s, x) → x, and let Γ ⊂ N be the projection by π N of a broken closed string associated to y 0 . By Lemma 6.3.1, [Γ] = A. We choose a complete set of paths {C γ } for A based at Γ.
The projection π N associates a well-defined homology class in H 2 (N ) ∼ = H 2 (M ) to any 2-chain inW + whose boundary consists of a broken closed string corresponding to y 0 on one side and Γ on the other side. Then we can define maps for all y such that s z (y 0 ) = s ξ + P D(A) and γ such that by

Theorem 6.4.2. The map Φ A is a chain map.
Proof. The proof is the same as that of Theorem 6.2.4, plus some bookkeeping of the homology classes of the holomorphic maps involved.
6.5. Gluing. We explain here how to glue the pair (v 0 , v −1 ) consisting of a W +curve v 0 with I(v 0 ) = 0 and an ECH curve v −1 with I(v −1 ) = 1, by inserting branched covers of trivial cylinders. The procedure of gluing two I ECH = 1 curves, as explained in Hutchings-Taubes [HT1,HT2], applies with very little modification.
6.5.1. Close to breaking. We first make precise what we mean by a holomorphic curve which is "close to breaking". We treat W ′ , leaving the analogous definitions for W , W + , and W − to the reader. Choose an s-invariant Riemannian metric g on W ′ . Definition 6.5.1. Given κ, ν > 0, two curves u i : where the distance | · | is measured with respect to g.
The following is similar to [HT1, Definition 1.10], although the phrasing is slightly different.
such that, after a suitable translation of u in the s-direction which we still call u, each of the pairs below is (κ, ν)-close: (2) and the restriction of a collection of branched covers of (2) +R (3) +R (4) and the restriction of a collection of branched covers of trivial cylinders to R (1) + 2R (2) + R (3) ≤ s ≤ R (1) + 2R (2) + R (3) + R (4) ; and so on.
Note that, in the case of u in W + or W − , we do not need to (and indeed we cannot) translate u in the s-direction. [HT1,HT2]. We now summarize the Hutchings-Taubes proof of ∂ 2 = 0 and discuss the small changes that need to be carried out.

Review of
Let (Y, ξ = ker λ) be a closed 3-manifold with a nondegenerate contact form λ and corresponding Reeb vector field R = R λ . Let (R × Y, d(e s λ)) be the symplectization of Y , where s denotes the R-coordinate, and let J be an adapted almost complex structure on R × Y .
Let α = (α 1 , . . . , α k ) and β = (β 1 , . . . , β l ) be ordered sets of Reeb orbits of R. Here the Reeb orbits may not be embedded and may also be repeated. Then let M J (α, β) be the moduli space of finite energy J-holomorphic curves in (R×Y, J) from α to β. Let {γ 1 , γ 2 , . . . } be the list of simple orbits of R. For each simple orbit γ i , we tally the total multiplicity m i (α) of γ i in α. In this way we can assign an orbit set γ(α) = i γ m i (α) i to α. We want to glue the pair (u + , u − ), where u + ∈ M I=1 J (α + , β + ) and u − ∈ M I=1 J (β − , α − ), provided γ(β + ) = γ(β − ). Since I(u + ) = I(u − ) = 1, the curves u + and u − satisfy the partition condition at β + and β − for a generic J (cf. Definition 7.11 and Proposition 7.14 of [HT1]). In particular, for each simple orbit γ i , the total multiplicity m i (β + ) = m i (β − ) completely determines the number of ends of u + or u − going to a cover of γ i , together with their individual multiplicities. For u + (resp. u − ), this is encoded by the incoming partition (resp. outgoing partition) and is denoted by (a 1 , . . . , a r ), then u + has r ends which go to a cover of γ i with covering multiplicities a 1 , . . . , a r .
Since P in γ i (m i (β + )) = P out γ i (m i (β − )) in general, we need to insert branched covers of cylinders R × γ i in order to be able to glue u + to u − . Such a branched cover π : Σ → R × S 1 must have β + at the positive end and β − at the negative end. A Fredholm index count ([HT1, Lemma 1.7]) implies that Σ must have genus zero; moreover, the partition condition is equivalent to saying that the Fredholm index of the composition (id, γ i )•π : Σ → R×Y is zero, where (id, γ i ) : R×S 1 → R×Y is the holomorphic map for the trivial cylinder R × γ i .
(2) Let M be the moduli space of connected, genus zero branched covers π : Σ → R × S 1 which have positive and negative ends determined by P in γ 1 (m 1 ) and P out γ 1 (m 1 ). With r, h fixed, choose the "gluing parameters" which consist of π ∈ M and real numbers T + , T − ≥ 5r.
(4) Fix κ > 0 sufficiently small. We choose a representative u + in [u + ] ∈ M I=1 J (α + , β + )/R such that each component of u + | s≤0 is (κ, 0)-close to a cylinder over some multiple cover of γ 1 ; here the multiplicities are given by P in γ 1 (m 1 ). Similarly, choose u − so that each component of u − | s≥0 is (κ, 0)-close to a cylinder over some multiple cover of γ 1 .
(5) Let u +T be the (s + + T + )-translate of u + in the R-direction and let u ′ +T be the restriction of u +T to s ≥ s + + T + . Similarly, let u −T be the (s − − T − )-translate of u − in the R-direction and let u ′ −T be the restriction of ±T , modulo the identifications along their boundary components. (To do this correctly, we need asymptotic markers at the ends.) The preglued map u * is defined explicitly (see [HT2,Equations (5.5) and (5.6)]) via a cutoff function which allows us to interpolate between u ±T and Σ in the regions s + ≤ s ≤ s + +T + and s − − T − ≤ s ≤ s − . This cutoff function involves the constants h and r.
Step 2: Deform the preglued curve (cf. [HT2, Section 5.3]). We choose "exponential maps" e − , e, e + which are obtained by flowing in the directions "normal" to u − , u Σ , u + in R × Y . (Here u Σ is the composition of π with R × S 1 → R × Y as a trivial cylinder.) The exponential maps can be glued to give the exponential map e * which maps to a small tubular neighborhood of u * in R × Y . Also choose the cutoff function β + : C * → [0, 1] 10 which equals 1 on C ′ +T , interpolates between 0 and 1 on the cylinders s + ≤ s ≤ s + + T + in Σ ′ and equals 0 elsewhere. Similarly define β Σ and β − .
Let (ψ + , ψ Σ , ψ − ) be triple, where ψ ± is a section of the normal bundle of u ±T and ψ Σ is a function Σ → C. The deformation of u * with respect to (ψ + , ψ Σ , ψ − ) is a map C * → R × Y given by Step 3: We now consider the equation for the deformation to be J-holomorphic. This equation has the form: See Equations (5.11), (5.12) and (5.13) of [HT2] for explicit expressions of Θ − , Θ + and Θ Σ . The strategy is to solve three equations separately: In [HT2, Proposition 5.6] it is shown that for sufficiently small ψ Σ (in a suitable Banach space) there exist maps ψ ± such that ψ ± = ψ ± (ψ Σ ) solves Equations (6.5.2) and (6.5.3).
Step 5: We then deform the section s to a linearized section s 0 with the same count of zeros. Here s 0 only depends on Σ and the linearized ∂-operator D Σ (which in turn depends on a neighborhood of γ 1 ), but not on the actual choices of u + and u − . The key point to check is that, during the deformation (s t ) t∈ 6.5.3. The Φ-map. We now turn to gluing the pair (v 0 , v −1 ), where v 0 is a W +curve with I(v 0 ) = 0 and v −1 is an ECH curve with I(v −1 ) = 1. Suppose the negative end of v 0 is given by β + , the positive end of v −1 is given by β − , and γ(β + ) = γ(β − ) = γ m 1 1 . In our case, there are a few things to check: (1) The curves v 0 and v −1 must satisfy the partition conditions at their negative and positive ends, respectively. This is a consequence of the ECH index inequality, i.e., Theorem 5.6.9.
(2) Since the W + -curve v 0 is not s-translation invariant, we pick s 0 so that each component of v 0 | s≤s 0 is (κ, 0)-close to a cylinder over some multiple cover of γ 1 . (We may still assume that v −1 satisfies the condition that v −1 | s≥0 is (κ, 0)-close to a cylinder.) Given the gluing parameters (T + , T − , Σ), we take • v 0 | s≥s 0 ; Definition 6.6.1. We define the chain complex ( CF (S, a, h(a), J ), ∂) generated by the 2g-tuples {z ∞,i } i∈I ∪ y ′ , where I ⊂ {1, . . . , 2g}, y ′ = {y ′ i } i∈I c , and the following hold: • z ∞,i is viewed as an intersection point of a i and h(a i ) and • y ′ i ∈ a i ∩ h(a σ(i) ) for some σ ∈ S I c . The differential ∂ counts I = 1 multisections u in W with n(u) ≤ 1, which satisfy one extra condition, i.e., if we write u = u ′ ∪ u ′′ (according to the notation introduced in Section 5.7.2), then u ′ has empty branch locus. The homology of ( CF (S, a, h(a)), ∂) is denoted by HF (S, a, h(a)).
In the differential ∂, with the exception of trivial strips, we are counting the following curves: (1) thin strips from z ∞,i to either x i or x ′ i ; and (2) I = 1 curves whose projections to S have image in S. Proposition 6.6.2. There is an isomorphism of chain complexes: S, a, h(a)), ∂), β, α, z) is as given in Section 4.9.1.
Recall from Section 4.9.3 that ( CF (S, a, h(a)), ∂ ′ ) = ( CF ′ (S, a, h(a)), ∂ ′ )/ ∼ is the E 1 -term of the spectral sequence for ( CF (Σ, β, α, z), ∂) (viewed as a double complex) in Theorem 4.9.4. (Here we are writing ∂ ′ for the differential of CF (S, a, h(a)) to distinguish it from the differential ∂ of CF (Σ, β, α, z).) In this paragraph y and y ′ i denote linear combinations of 2g-tuples. By tracing the zigzags in the double complex we obtain the isomorphism ν : HF (S, a, h(a)) ∼ → HF (Σ, β, α, z), which is defined as follows: Let y ∈ CF ′ (S, a, h(a)) be a cycle in CF (S, a, h(a)), i.e., Then ν maps the equivalence class [y] to the equivalence class of where "h.o." means terms with more x ′′ i components. Composing with the map induced by κ in homology, we obtain an isomorphism ν = κ * • ν : HF (S, a, h(a)) ∼ → HF (S, a, h(a)).

The map Φ.
Definition 6.6.3. Let J + ∈ J reg + which restricts to J ∈ J reg above. We define the map Φ as follows: In our analysis of Φ we will use balanced coordinates (cf. Section 5.1.2) for N − int(N ) = D 2 × (R/2Z). The Morse-Bott family N can be identified with ∂D 2 and we write γ φ for the orbit in N corresponding to e iφ . So far in this paper h ∈ N was an arbitrary point. Convention 6.6.4. From now on we specialize h so that h = γ φ h is generic and φ h is close to − 2π m , where m is as defined in Section 5.2.2. In particular, the radial ray corresponding to φ h does not lie on the thin wedges from a i to h(a i ) for all i. There are no restrictions on the orbit e except that e = h.
We then set Z + s,φ = Z s,φ ∩ W + and examine the intersections Z + s,φ ∩ u ′′ . Observe that K = π N (Z + s,φ ) ∩ π N (u ′′ ) = ∅ for a suitable choice of φ which is close to but not equal to φ h ; this is possible by the positioning of h given by Convention 6.6.4. Hence Z + s 0 ,φ ∩ u ′′ = ∅ for some s 0 . On the other hand, since K is compact, Z + s 0 +s 1 ,φ ∩ u ′′ = ∅ for a sufficiently large s 1 . Finally, since Z + s,φ and u ′′ have no boundary intersections and no intersections near their ends for all s ∈ R, we have a contradiction.
Theorem 6.6.6. Φ is a chain map.
Proof. Since Φ is a chain map by Theorem 6.2.4, it suffices to verify that On the one hand, Φ({z ∞,i } I ∪ y ′ ) = 0 by Lemma 6.6.5. On the other hand, if |I| = 1, then and if |I| > 1, then each term of ∂({z ∞,i } i∈I ∪ y ′ ) contains some copy of z ∞ , and Φ maps the term to zero by Lemma 6.6.5. This proves the theorem.
A corollary of Lemma 6.6.5 is the following: (1) We define the map where Ψ ′ (γ), y is the mod 2 count of M I=2,n * =m J ♦ − (γ, y; m).
(2) We define the map Ψ = Ψ J ♦ − (m, m) as the composition where q is the quotient map of chain complexes.
The count Ψ ′ (γ), y is meaningful because of the following theorem: Proof. This follows from Theorems 7.7.3 and 7.11.1(i) and Corollary 7.12.1.
Let ∂ ′ HF be the differential for CF ′ (S, a, h(a)) and let ∂ P F H be the differential for P F C 2g (N ). If y ′ is a (2g−1)-tuple of chords from h(a) to a such that {x i }∪y ′ (and also {x ′ i } ∪ y ′ ) is in S a,h(a) , then we define (7.1.1) The proof of the following theorem will occupy the rest of Section 7.
Theorem 7.1.3. If m ≫ 0, then The maps U m−1 and Ψ 0 are defined as follows: Let γ ∈ O 2g and γ ′ ∈ O 2g−p , p = 1, . . . , 2g. Then where A is the count of I = 2 multisections u of W ′ from γ to δ 0 γ ′ which satisfy n * (u) = m − 1 and a certain asymptotic condition near δ 0 (the precise definition will be given in Section 7.8.3), and where B is the count of degree 2g − 1, I = 0 almost multisections u of W − from γ ′ to y ′ which satisfy n * (u) = 0, together with the section at infinity σ − ∞ from δ 0 to z ∞ .
Assuming Theorem 7.1.3 for the moment, we have the following: Corollary 7.1.4. The map Ψ is a chain map.
Proof. Similar to the proof of Theorem 6.2.4 and based on Equation (7.1.2). There is one major difference: Ψ ′ is not a chain map. However, since ker q, by composing Equation (7.1.2) with q we obtain: where ∂ HF is the differential for CF (S, a, h(a)).
We can also define a map which is F[H 2 (M ; Z)]-linear. As for Φ in Section 6.4, the key point is the definition of maps Step 1. By the SFT compactness theorem (Proposition 7.3.1), a sequence u i ∈ M 3 m admits a subsequence which converges to a holomorphic building where v j is a holomorphic map to W j = R × N for j > 0, W 0 = W − for j = 0, and W j = W for j < 0, and the levels W −b , . . . , W a are arranged in order from lowest to highest. As before, we write where v ′ j branch covers the section at infinity σ * ∞ and v ′′ j is the union of irreducible components which do not branch cover σ * ∞ . Here * = ∅, ′ , or −. There are two cases: v ′ 0 = ∅ or v ′ 0 = ∅. The latter case is harder, and will be treated first. By analyzing the two constraints n − (u i ) = m and I(u i ) = 3, we obtain Theorem 7.7.1, which gives a preliminary list of possibilities for u ∞ when v ′ 0 = ∅. Many of the possibilities in Theorem 7.7.1 actually do not occur. Theorem 7.11.1 eliminates Cases (2)-(6) from the list. This is done by a finer analysis of the behavior of u i in the vicinity of the component σ − ∞ and is similar in spirit to the layer structures of Ionel-Parker [IP1, Section 7].
Summarizing, we have: Here we have omitted the potential contributions of connector components for simplicity.
Here the only remaining Case (1) in Theorem 7.7.1 corresponds to A 1 .
Step 2. We now glue the triples (v 1 , v 0 , v −1 ) in A 1 , subject to the constraint m. This gluing accounts for the term ∂ 1 • Ψ 0 • U m−1 and is a bit involved. Let us abbreviate Here f δ 0 is a nonzero normalized asymptotic eigenfunction of δ 0 at the negative end which, used as a modifier, stands for "the normalized asymptotic eigenfunction at the negative end δ 0 is f δ 0 ." See Section 7.8 for more details on asymptotic eigenfunctions.
We first observe . We may assume that all the multiplicities of γ ′ are 1, since the Hutchings-Taubes gluing of branched covers is essentially independent of the present gluing problem. Next we introduce the extended moduli space Here the modifier ext means that u : (Ḟ , j) → (W − , J − ) is a multisection which maps all the connected components of ∂Ḟ but one to a different L − a i and the last connected component to some L − a i ∪ a i,j . There is a gluing map which is a diffeomorphism onto its image for r ≫ 0. Here T ± is shorthand for the pair T + , T − . The map G is defined in a manner similar to that of Section 6.5.2 and is described in more detail in Section 7.13.3. Let r 0 ≫ 0, let P (r 0 ) ⊂ P be the where v j is a holomorphic map to W j = R × N for j > 0, W 0 = W − for j = 0, and W j = W for j < 0, and the levels W −b , . . . , W a are arranged in order from lowest to highest. The same also holds when {u i } is a sequence of (W − , J ♦ − )-curves.
Proof. The ω-area bound for the W − case is similar to that of the W + case. We take the difference of two W − -curves u 1 and u 2 from γ to y to obtain Z ∈ H 2 (W − ). Since they both intersect σ − ∞ once, Z can be represented by a surface ⊂ W − , and hence can be viewed as a class in H 2 (N ). The zero flux condition implies that the ω-area of Z is zero.
We then use the Gromov-Taubes compactness theorem as before to extract a subsequence u i which converges in the sense of currents to a holomorphic building. After passing to a subsequence, we may assume that the homology class [u i ] ∈ H 2 (W − , Z γ,y ) is fixed for all i. Once the homology class [u i ] is fixed, we use the relative adjunction formula in the same way as in Lemma 6.1.1 to obtain a bound on χ(Ḟ i ).
Finally, we use the SFT compactness theorem as in Proposition 6.1.2 to obtain a level a + b + 1 holomorphic building u ∞ ; the proof carries over even though L − a is a singular Lagrangian submanifold. (We still are left with the task of analyzing the limit more precisely.) 7.4. Novelty of the W − case. For the next several subsections (Sections 7.4-7.6), J − ∈ J − and u ∞ = v −b ∪ · · · ∪ v a is the SFT limit of a sequence u i :Ḟ → (W − , J − ) of W − -curves from γ to y. We assume that we have already passed to a subsequence so that the domain of u i is independent of i. Let D 2 = {ρ ≤ 1} ⊂ S with polar coordinates (ρ, φ) and let z ∞ = (0, 0), as in Section 5.1.2.
The main novelty of the W − case, as opposed to the W + case, is that there may exist a sequence of points z i ∈ ∂Ḟ such that u i (z i ) approaches the section at infinity σ − ∞ as i → ∞. On the HF end, there is an extra Reeb chord [0, 1] × {z ∞ } which connects any h(a k ) to any a l . On the ECH end, there is an extra closed orbit δ 0 = {ρ = 0} ⊂ N . Notice that the only orbit in N − N which intersects S × {0} at most 2g times is δ 0 . Therefore the levels v j may be asymptotic to [0, 1] × {z ∞ } or δ 0 at the ends.
By the SFT compactness argument from Proposition 6.1.2, the map v 0 :Ḟ 0 → W − satisfies one of the following on a neighborhood of a boundary puncture p oḟ We will use the convention that v 0 takes each component of ∂Ḟ 0 to a single L − a k ; hence removable punctures of type (ii) are eliminated.
An analogous convention will be used for v j :Ḟ j → W , j < 0, and a removable puncture of type (iii) in this case is: (iii') p is a removable puncture and the extension v j : Definition 7.4.2. Let v be an irreducible component of v j :Ḟ j → W or W j , j ≤ 0, which does not branch cover σ * ∞ . Then a boundary puncture p of v is removable at z ∞ if it is of type (iii). 7.5. Intersection numbers. In order to analyze the SFT limit u ∞ , we use the intersection numbers n * (u i ) and n * (v j ) to constrain the behavior of holomorphic maps which are asymptotic to δ 0 or to [0, 1] × {z ∞ }.
We briefly recall the definition of the intersection numbers n * (u) given in Section 5. Let ρ 0 > 0 be sufficiently small. Consider the torus T ρ 0 = {ρ = ρ 0 } ⊂ N , oriented as the boundary of {ρ ≤ ρ 0 }. We take an oriented identification T ρ 0 ≃ R 2 /Z 2 such that the meridian has slope 0 and the closed orbits of R 0 on T ρ 0 have slope m. We pick a closed orbit δ † 0 ⊂ T ρ 0 and consider the parallel sections (σ * ∞ ) † determined by δ † 0 . We assume we have chosen δ † 0 so that (σ * ∞ ) † is disjoint from the relevant Lagrangian submanifold. Then n * (u) = u, (σ * ∞ ) † . Since n − (u i ) = m ≫ 2g for an W − -curve u i , we have (1) If v ′′ j has a positive end E + which converges to δ p 0 , then j has a negative end E − which converges to δ p 0 , then j has multiple ends at covers of δ 0 , then their contributions to n ′ (v j ) are summed.
Since E + , (σ ′ ∞ ) † > 0 by the positivity of intersections in dimension four and m ≫ p, we must have q ≤ 0. We then obtain: Let E − be a negative end which converges to δ p 0 . As above, π N (E − ) ∩ T ρ 0 determines a homology class (q, −p) ∈ H 1 (T ρ 0 ) ≃ R 2 /Z 2 such that by the positivity of intersections in dimension four and m ≫ p, we must have q ≤ −1. We then obtain: j has multiple ends at covers of δ 0 , then the total intersection of v ′′ j with (σ ′ ∞ ) † is bounded below by the sum of the contributions of each end. Next we consider v j when j < 0. Let π S : R × [0, 1] × S → S be the projection along the Hamiltonian vector field ∂ t . Also let R φ 0 be the radial ray Lemma 7.5.2 (Intersection with v j , j < 0). Suppose j < 0. Then the following hold for ρ 0 > 0 sufficiently small: (1) If v ′′ j has a positive end E + which converges to [0, 1] × {z ∞ }, then (7.5.4) E + , σ † ∞ ≥ 1, and the relevant sector is a thin wedge if and only if equality holds. Moreover, if the sector is not a thin wedge, then where 0 ≤ l < m and φ l = φ 0 + l(2π/m). Let E + (resp. E − ) be a positive (resp. negative) end of v ′′ j which converges to [0, 1] × {z ∞ } and let S(E ± ) be the corresponding sector in D 2 ε . By the assumptions on the a i given in Section 5.2.2, the angles of the thin wedges are 2π m and the other sectors of D 2 ε − ∪ i a i − ∪ i h(a i ) have angles greater than 2π(2g) m . This implies that: Similarly, we have the following lemma, whose proof is the same as those of Lemmas 7.5.1 and 7.5.2 and will be omitted.
Lemma 7.5.3 (Intersection with v 0 ). The following hold for ρ 0 > 0 sufficiently small: (1) If v ′′ 0 has a positive end E + which converges to δ p , then (7.5.6) j is a boundary puncture which is removable at z ∞ , then, for any sufficiently small neighborhood Here the constant k 0 is as given in Section 5.2.2.
Proof. We will prove the case j < 0, leaving j = 0 for the reader. Let p be a removable boundary puncture of v ′′ j which maps to a point on R × {1} × {z ∞ } (without loss of generality). We consider the projection π S : R × [0, 1] × S → S. By Definition 5.3.2, the projection π S is holomorphic when restricted to π −1 S (D 2 ε ), where ε > 0 is sufficiently small. Hence π S • v ′′ j is holomorphic when restricted to (π S • v ′′ j ) −1 (D 2 ε ) and some sector of D 2 ε − a must be contained in Im(π S • v ′′ j ) because holomorphic maps are open. This in turn shows that v ′′ j (N (p)), (σ * ∞ ) † ≥ k 0 − 1 > 2g by assumption, provided ρ 0 is sufficiently small. 7.6. Some restrictions on u ∞ . We now present some lemmas in preparation for Theorems 7.7.1 and 7.7.3. Lemmas 7.6.1-7.6.3 give restrictions on u ∞ , which arise from intersection number calculations from Section 7.5, and Lemma 7.6.5 gives a lower bound on the ECH index of the levels v j .
In what follows, "component" is shorthand for "irreducible component". We first discuss the "fiber components", i.e., components v : F → W j of v j which map to fibers of W j . There are three types of fiber components: (i) ghosts, i.e., v is constant; (ii) "closed fiber components", i.e., F is closed and v is nonconstant; (iii) "boundary fiber components", i.e., ∂F = ∅ and v is nonconstant. If v is a closed fiber component, then v is a branched cover of a fiber of W j . Next let v : F → W j be a boundary fiber component. If j < 0 and v maps to π −1 B (p), This implies that d = 1.
We now prove that v is a closed fiber component passing through m. Arguing by contradiction, if m ∈ Im( v), then there is a componentv of u ∞ such that m ∈ Im(v). Ifv is a cover of σ − ∞ , then there must be some component of v ′′ j for j > 0, which has a negative end at some δ p 0 . Therefore n − (v ′′ j ) > 0 by Lemma 7.5.1(2). Ifv is not a cover of σ − ∞ , thenv has a nonzero intersection with σ − ∞ and hence n − (v) > 0. In either case we have n * (u ∞ ) > m, which is a contradiction. This proves that m ∈ Im( v).
Finally boundary fiber components are eliminated because they project to a point in ∂B − , and therefore cannot pass through m.

We recall the notation
where v ′ j denotes the union of all components of v j which cover the section at infinity σ * ∞ and v ′′ j denotes the union of all other components of v j . The covering degree of v ′ j will always be denoted by p j . We also define v ♯ j to be the union of the components of v ′′ j which are asymptotic to a multiple of δ 0 or z ∞ at either end and v ♭ j to be the union of the remaining non-fiber components of v ′′ j .
Lemma 7.6.2. If v ′ 0 = ∅, then, with the exception of ghost components: (1) v ′ j = ∅ and v ♯ j = ∅ for all j; (2) no v ′′ j , j ≤ 0, has a boundary puncture which is removable at z ∞ ; (3) every level v j , j = 0, has image inside W ′ or W ; and (4) v 0 is a W − -curve or a degenerate W − -curve, i.e., v 0 is the union of a W − -curve and a fiber π −1 B − (p) which passes through m. Proof.
Since u i passes through m for all i, the level v 0 :Ḟ 0 → W − must also pass through m. By Lemma 5.4.10, 11 n − (v 0 ) ≥ m because m ∈ σ − ∞ . Equation (7.5.1) and the nonnegativity of n * then imply that n − (v 0 ) = m and n * (v j ) = 0 for j = 0.
(4) Since n − (v 0 ) = m, v 0 intersects σ − ∞ only at m and the intersection is transverse by Lemma 5.4.10(2). By (1) and (2) and Lemmas 7.6.1 and 5.4.12, if v 0 is not a W − -curve, then it must be a degenerate W − -curve. This completes the proof of the lemma.
Lemma 7.6.3. If v ′ 0 = ∅, then, with the exception of ghost components: (1) there is only one negative end of v ♯ j , j > 0, (say v ♯ a ′ ) which is asymptotic to a multiple of δ 0 ; (2) u ∞ has no fiber components; (3) no v ′′ j , j ≤ 0, has a boundary puncture which is removable at z ∞ ; and has δ r 0 0 , for some r 0 > 0, at the positive end and some subset of {x 1 , . . . , x 2g , x ′ 1 , . . . , x ′ 2g } of cardinality p at the negative end; and has δ r j 0 , for some r j > 0, at the positive end and h r ′ j e r ′′ j with r ′ j + r ′′ j = r j , at the negative end. Proof.
(1) By Lemma 7.5.1, each negative end of v ♯ j , for j > 0, which is asymptotic to a multiple of δ 0 contributes at least m−2g to a j=1 n * (v j ), where m ≫ 2g, because the total multiplicity of δ 0 is ≤ 2g. If there are at least two such negative ends of v ♯ j , then the total contribution to a j=1 n * (v j ) is at least 2(m − 2g) > m, which is a contradiction. Suppose v ♯ a ′ , 0 < a ′ ≤ a, has a negative end at a multiple at δ 0 .
(2) By Lemma 7.6.1, a fiber component of u ∞ is a fiber π −1 B − (p) which passes through m and additionally contributes m to a j=−b n * (v j ), a contradiction. (3) By Lemma 7.5.4, a boundary puncture which is removable at z ∞ additionally contributes ≥ 2g to a j=−b n * (v j ). This is again a contradiction. (4) By (3), v ♭ j does not have any boundary punctures which are removable at z ∞ . Since the ends of v ♭ j are contained in N or [0, 1] × S and n * (v ♭ j ) = 0, we conclude that Im(v ♭ j ) ⊂ W , W − , or W by Lemmas 5.3.8, 5.4.12 and 5.3.13. (5), (6) Let p j = deg(v ′ j ) be the covering degree of v ′ j over σ ′ ∞ . Then p 0 ≤ p 1 ≤ · · · ≤ p a ′ −1 and p a ′ = 0. This follows from (1) since a negative end of v ♯ j+1 is required to decrease p j .
Let p − a ′ be the multiplicity of δ 0 at the negative end of v ♯ a ′ . The negative end of v ♯ a ′ contributes at least m − p − a ′ to n * by Equation (7.5.3) and the positive ends of v ♯ j , for j = 0, . . . , a ′ − 1, give a total contribution of at least p − a ′ − p 0 to n * by Equations (7.5.2) and (7.5.6). Hence, On the other hand, by Equation (7.5.4), the contributions of the positive ends of v ♯ j , j = −b, . . . , −1, add up to Equations (7.6.1) and (7.6.2) give: Equality holds by Equation (7.5.1). This in turn implies that: (i) equality holds in both Equations (7.6.1) and (7.6.2); and (ii) v ♯ j , j ≤ 0, has no negative end which limits to z ∞ . Since p 0 ≤ 2g, each v ♯ j , j < 0, must be a union of thin strips by Lemma 7.5.2. This gives (5). Moreover, v ♯ 0 has δ p 1 −p 0 0 at the positive end and no negative ends at z ∞ by (ii) and (7.6.4) n − (v ♯ 0 ) = p 1 − p 0 by Lemma 7.5.3 and (i).
In order to prove (6), we consider C ρ = π N (v ♯ 0 ) ∩ T ρ , where 0 < ρ < 1. The argument is similar to the proof of the blocking lemma in [CGH1], and the presence of the Lagrangian boundary condition for v ♯ 0 does not change the proof in any essential way. Suppose ρ 1 > 0 is small. Let ∪ i E i be the union of positive ends of v ♯ 0 that limit to multiples of δ 0 , and let C ′ By Equation (7.6.4), q = 0 and C ρ 1 = C ′ ρ 1 , since any intersections besides those coming from ∪ i E i would give some extra contribution to n + (v ♯ 0 ). Next consider C ρ 2 , where ρ 2 = 1 − ε, ε > 0 small. Since there are no ends of v ♯ 0 between T ρ 1 and T ρ 2 , it follows that C ρ 2 = (0, p 1 − p 0 ) ∈ H 1 (T ρ 2 ). Negative ends can approach x i or x ′ i either from the interior of V = N − int(N ) or from the exterior of V . A negative end of v ♯ 0 approaching x i or x ′ i from the interior contributes (0, 1) to the homology class of C ρ 2 . This means that at most p 1 − p 0 ends of v ♯ 0 approach x i or x ′ i from the interior. Finally let ρ 3 = 1 + ε, ε > 0 small, and let C ′ where v is obtained from v ♯ 0 by truncating the negative ends which limit to x i or x ′ i from the exterior of V . Then C ′ ρ 3 = (0, r) ∈ H 1 (T ρ 3 ) for some r ≥ 0. We can eliminate the possibility r > 0 by examining the intersection of C ′ ρ 3 and the Hamiltonian vector field on T ρ 3 and applying the positivity of intersections. Since ρ 3 was arbitrarily close to 1, it follows that the image of v ♯ 0 cannot escape W − − int(W − ), which implies (6).
(7) This is similar to (6) and is left to the reader.
Proof. Recall that the restrictions J and J ′ are also regular by definition. By [HT1, Proposition 7.15(a)], I ECH (v j ) ≥ 0 for j > 0 since J ′ is regular.
Then v ′ j = ∅ and v ♯ j = ∅ for all j < 0 by Lemma 7.6.2, and we are left with v ♭ j , which is simply-covered and has nonnegative Fredholm index by regularity. By the index inequality (Theorem 4.5.13), Then v ♯ j is a union of thin strips from z ∞ to some x i or x ′ i by Lemma 7.6.3 and each thin strip has ECH index 1. We also have I(v ′ j ) = 0 by Lemma 5.7.12 and I(v ♭ j ) ≥ 0 by the previous paragraph. We claim that (7.6.5) Note that, although v ′ j , v ♯ j and v ♭ j are disjoint, v ′ j and v ♯ j are both asymptotic to (a multiple of) z ∞ at the positive end and the additivity of the ECH indices of v ′ j and v ♯ j needs to be verified. For that purpose, recall from Section 5.7 that each v ′ j comes equipped with data D ′ j = ((D ′ ) to j , (D ′ ) f rom j ) at the positive and negative ends, since u ∞ is the limit of the sequence {u i }. The key observation here is that (D ′ ) to j = (D ′ ) f rom j , since all the components of v ♯ j ′ , j ′ < j, are thin strips whose data (D ♯ Hence we can choose a simultaneous grooming c + = {c + k } for both v ′ j and v ♯ j at the positive end z ∞ such that c + k has w(c + k ) = 0 and connects h(a i k ,j k ) to a i k ,j k . If we choose a groomed multivalued trivialization τ compatible with c + , then , which immediately implies Equation (7.6.5).
Case j = 0. Suppose that v ′ 0 = ∅. Then, v 0 is a W − -curve or a degenerate W −curve by Lemma 7.6.2. If v 0 is a W − -curve, then it is simply-covered and satisfies If v 0 is a degenerate W − -curve, then v 0 can be written as a union of a fiber C and a W − -curve v ♭ 0 . The Fredholm index of v ♭ 0 is nonnegative since v ♭ 0 is simply-covered and hence is regular. The Fredholm index of C is given by: The algebraic intersection number C, v ♭ 0 is equal to 2g and by Theorem 5.6.9. Next suppose that v ′ 0 = ∅. We have I(v ′ 0 ) = 0 by Lemma 5.7.13. Next, by and has δ p 1 −p 0 0 , p 1 −p 0 > 0, at the positive end and some (p 1 −p 0 )-element subset of {x 1 , . . . , x 2g , x ′ 1 , . . . , x ′ 2g } at the negative end. Hence I(v ♯ 0 ) = p 1 − p 0 by Lemma 5.7.14. Finally, since and v ♭ 0 are additive. This completes the proof of the lemma. 7.7. Compactness theorem. Let J ♦ − (ε, δ, p) be a generic almost complex structure which is (ε, U )-close to J − ∈ J reg − , where U and K p,2δ are as in Convention 5.8.12. We write: As before, u ∞ ∈ ∂M 3 m (ε, δ, p) is written as u ∞ = v −b ∪ · · · ∪ v a and each v j is written as We now prove the following compactness theorem, which is basically a consequence of two constraints: I(u i ) = 3 and n * (u i ) = m. The list of possibilities should be viewed as a preliminary list, since we will subsequently eliminate Cases (2)-(6) in Theorem 7.11.1. Theorem 7.7.1. Let J ♦ − be (ε, U )-close to J − ∈ J reg − and K p,2δ -regular with respect to m, and let u ∞ ∈ ∂M 3 m (ε, δ, p). If v ′ 0 = ∅, then u ∞ contains one of the following subbuildings: and v ♯ −1 and v ♯ −2 which are both thin strips.
(4) A 3-level building consisting of v 1 with I = 1 from γ to δ 2 0 γ ′ ; v ′ 0 which is a branched double cover of σ − ∞ ; and v ♯ −1 which is the union of two thin strips.
(6) A 4-level building consisting of v 2 with I = 1 from γ to  Proof. By Proposition 7.3.1, u i converges in the sense of SFT to a holomorphic building u ∞ = v a ∪ · · · ∪ v −b . We have three constraints: (i) comes from the additivity of ECH indices, (ii) comes from Lemma 7.6.5, and (iii) comes from Equation (7.5.1).
Suppose that v ′ 0 = ∅, i.e., we are in the situation of Lemma 7.6.3. We have the following restrictions: • the top level v a is nontrivial and satisfies I(v a ) ≥ 1; • ∪ j<0 v ♯ j consists of p 0 ≥ 1 thin strips and contributes j<0 I(v ♯ j ) = p 0 ≥ 1 to the total ECH index.
This immediately implies p − a ′ ≤ 2 because p − a ′ is also the number of nontrivial curves with a positive end at δ 0 and each of them has ECH index I ≥ 1 by Lemma 7.6.3. We also have p 0 ≤ p − a ′ , and therefore we can divide the proof into three cases: • Case I: p − a ′ = p 0 = 1. • Case II': p − a ′ = 2 and p 0 = 1. • Case II": p − a ′ = 2 and p 0 = 2.
Case I. In this case v ′ 0 = σ − ∞ and v ♯ −1 is a thin strip. This leaves two possibilities for v 1 : either I(v 1 ) = 2 and we are in Case (1), or I(v 1 ) = 1 and we are in Case (2).
Case II'. In this case we have v ♯ j 0 = 0 for some j 0 ≥ 0. Since I(v ♯ j 0 ) = 1, v −1 consists of a single thin strip and there are no other levels with j < 0. If j 0 = 0 we are in Case (5), and if j 0 > 0 we are in Case (6). In Case (6), v ♯ 1 is a cylinder connecting δ 0 with h by Lemma 7.6.3 and Remark 7.6.4.
Case II". In this case ∪ j<0 v j consists of two thin strips. If they are on the same level we are in Case (4) and if they are on different levels we are in Case (3).
This completes the proof of Theorem 7.7.1.
Remark 7.7.2. In Cases (3)-(6), the total number of branch points of ∪ a j=−b v ′ j is one, where we are not ignoring connector components that cover σ * ∞ : Assume without loss of generality that the only nontrivial v ′ j is v ′ 0 and that v ′ 0 double covers σ − ∞ . Let b be the number of branch points of v ′ 0 . Then ind(v ′ 0 ) = b − 1 by Proposition 5.5.5, the Riemann-Hurwitz formula, and the proof of Lemma 5.8.9. The index inequality, the additivity of the indices, and the condition I(u i ) = 3 force b = 1.
The proof of the following theorem is similar and will be omitted.
Definition 7.8.2. We define the asymptotic operator We remark that the asymptotic operator which appears in [HWZ1] is A l , whereas the asymptotic operator in [HT2] is −A l . The eigenfunctions of A l are the asymptotic eigenfunctions, and are given by ce πint/l , c ∈ C, n ∈ Z, with corresponding eigenvalues πn l − ε. An asymptotic eigenfunction ce πint/l is said to be normalized if |c| = 1. Let E πn/l−ε be the eigenspace of A l corresponding to the eigenvalue πn l − ε.
Proof. The Fredholm theory for holomorphic curves with Morse-Bott asymptotics uses Sobolev spaces with exponential weights. The regularity of simply-covered moduli spaces in this setting is treated in Wendl [We4].
As before, we write (J ′ ∞ ) reg ⊂ J ′ ∞ for the dense subset of regular almost complex structures and (J ′ ∞ ) reg ⋆ ⊂ (J ′ ∞ ) reg for the dense subset of almost complex structures which satisfy (⋆). Note that, if J ′ ∞ ∈ (J ′ ∞ ) reg , then nearby almost complex structures J ′ ∈ J ′ for m ≫ 0 are in (J ′ ) reg ⋆ . (γ, δ l 0 γ ′ ) has a subsequence which converges to a generalized Morse-Bott building by the Morse-Bott compactness theorem [Bo2,Theorem 4.16]. The limit must be negatively asymptotic to δ l 0 γ ′ because there is no negative gradient trajectory of f on D 2 that goes to z ∞ . Moreover the limit can have only one nontrivial level, because any further level must have I = 0.
By the compactness and regularity of moduli spaces, there are only finitely many curves C 1 , . . . , C r , modulo R-translation, such that for some orbit sets γ i , γ ′ i ∈ O * and partition (l i1 , . . . , l iλ i ) of l i . Here if u is a curve with an end at δ p 0 , then its ECH index is computed using the Conley-Zehnder index of δ p 0 with respect to h m , m ≫ 0, and n(u) is defined as the intersection number of u and the section at infinity. Let be the asymptotic eigenfunction corresponding the end δ l ij 0 of C i . The condition c ij = 0 follows from Lemma 7.8.5. We may therefore assume without loss of generality that all the f ij are normalized. 7.8.5. Radial rays.
A radial ray R φ 0 which is not bad is said to be good.
A good radial ray must exist and, as it is explained in the following remark, we can assume it is R π .
Remark 7.8.8. The set of bad radial rays is determined by (R × N , J ′ ∞ ). Strictly speaking, we should choose the set of endpoints E ⊂ ∂D 2 as in Section 5.2.2 such that R φ 0 +π is a good radial ray and φ 0 < φ(y j (m)) < φ 0 + c(m) where c(m) → 0 as m → ∞. After a rotational coordinate change of D 2 , we may assume that R π is a good radial ray and 0 < φ(y j (m)) < c(m).
7.9. The rescaled function. In this subsection we will be using limiting arguments in which m → ∞ and h m → h ∞ ; see Section 7.8.4. Hence many of the almost complex structures and moduli spaces will have an additional subscript m, where m = ∞ is also a possibility. For example, J ′ m and J −,m refer to J ′ and J − with respect to m. Let J ′ ∞ ∈ (J ′ ∞ ) reg ⋆ and let J ′ m ∈ (J ′ m ) reg ⋆ be a nearby almost complex structure with respect to the integer m ≫ 0. Let J −,m ∈ J reg −,m be an almost complex structure which restricts to J ′ m and let J ♦ −,m be (ε, U )-close to J −,m .
We write π : V → D ρ 0 for the projection which was denoted π Dρ 0 in previous sections.
Let m i and u ij be sequences satisfying the following properties: for some l; and (S3) for all i ∈ N and κ, ν > 0, there exists j i,κ,ν such that, if j ≥ j i,κ,ν , then u ij is (κ, ν)-close (cf. Definition 6.5.2) to a J ♦ −,m i -holomorphic building u i∞ = ∪ l v l,i with v ′ 0,i = ∅. Moreover, assume that one of the following holds: (S4') the first negative end at δ 0 in the building u i∞ has multiplicity one, or (S4") the first negative end at δ 0 in the building u i∞ has multiplicity two. Property (S3) is a consequence of the fact that the sequence u ij converges in the SFT sense to the building u i∞ for each fixed i. Property (S4') corresponds to Cases (1) and (2) in Theorem 7.7.1 and Property (S4") corresponds to the remaining cases.
⋆ and let v : (−∞, s 0 ] → R × N be a negative end of v which is asymptotic to a multiple cover of δ 0 . Then for every ε > 0 there exist positive constants R and κ ′ , κ < ε such that κ ′ κ > 1 ε and for all t, where f is the normalized asymptotic eigenfunction forṽ.
Proof. This is a rephrasing of Lemma 7.8.3, in view of Lemma 7.8.4. • F is a subsurface of F and u is the restriction of u to F ; • e + (resp. e − ) is the union of components of ∂ F − p −1 (∂B − ) for which ds(n) > 0 (resp. < 0), where n is the outward normal vector field along ∂ F .
Lemma 7.9.4. Let m i and u ij be sequences satisfying (S1)-(S3) and either (S4') or (S4"). Then there is a sequence j(i) such that the following hold for all j ≥ j(i): (1) u ij admits a good truncation (2) lim Proof. We prove the lemma in Case (S4'), where the notation is simpler, i.e., we can use (s, t) as global coordinates on F ij . In this case a = 2. Case (S4") is conceptually the same. Fix a sequence ε i → 0. Then, by Lemma 7.9.1, for each i there exist κ ′ i , κ i and R i which satisfy: where f i is the normalized asymptotic eigenfunction for a negative end v i of a J ′ m i -holomorphic curve v i of ECH index I ≤ 2 that limits to δ 0 .
On the other hand, by (S3) and Definition 6.5.2, for all i there exists j(i) such that if j > j(i) then there exists R ′ ij such that Similar considerations also hold at the negative truncated end.
Remark 7.9.5. As i → ∞, the curves v i approach one of the C a from Section 7.8.4, modulo R-translation, by Lemma 7.8.6. Hence the normalized asymptotic eigenfunctions f i limit to the normalized eigenfunction f ab for C a at the negative end δ 0 as i → ∞. 7.9.3. Ansatz. We define z i = π • u ij(i) and, to simplify computations, we will make the ansatz (7.9.1) Lemma 7.9.6. The functions w i : F i → C defined by Equation (7.9.1) are holomorphic with respect to the standard complex structure on C.
Proof. We give the proof for Case (S4"); Case (S4') is similar but simpler. The functions (s, t) give local conformal coordinates on F i outside the branch locus because p i : F i → B − is holomorphic. Then z i : F i → C satisfies Equation (7.8.1): If we plug in Equation (7.9.1) into Equation (7.8.1) we obtain: Hence ∂ s w + i∂ t w i = 0, so w i is a holomorphic map to the standard complex line (C, i) in the complement of the branch point. Then it is holomorphic everywhere by the removal of singularities.
By passing to a subsequence we may assume that the topology of K i and F i − K i are constant. In order to simplify the exposition, we will assume that this is the case for all i. Definition 7.9.7. We define C i = sup z∈K i | w i (z)| and w i = w i /C i .
Properties of w i . The holomorphic maps w i : F i → C satisfy the following: (1) sup z∈K i |w i (z)| = 1; (2) there is a unique point m i ∈ F i such that w i ( m i ) = 0; the zero m i is a simple zero; (3) if x ∈ ∂ F i and p i (x) = (s, t) ∈ ∂B − , then w i (x) ∈ e i(ε i t+φ i ) R + with ε i = π m i and lim i→∞ φ i = 0 for some φ i depending on the boundary component of ∂ F i containing x; (4) at the truncated ends the inequality holds, where f i is a (not necessarily normalized) asymptotic eigenfunction of δ or z ∞ on each component of e ± i , |f i | ≥ κ ′ i , and κ i and κ ′ i satisfy κ ′ i / κ i → ∞ as i → ∞. Remark 7.9.8. The loops of w i | e ± i have winding number 1 around the origin when i is sufficiently large: in fact κ ′ i > κ i , so the linear homotopy between w i | e ± i and f i (t) is contained in C × . Lemma 7.9.9. After passing to a subsequence, w i | K i converges to a nonconstant function w ∞ | K 0 .
Proof. After passing to a subsequence we may assume that K i converges to a Riemann surface K 0 with boundary. Since w i | K i is uniformly bounded, the lemma follows from Montel's theorem.
From now on we assume that we have passed to a subsequence so that Lemma 7.9.9 holds. In the following subsections we analyze the convergence of w i , not just on the uniformly bounded part K i . This involves techniques of SFT compactness. 7.9.5. Energy bound. Following Hofer [Ho1], we define an energy for holomorphic functions on Riemann surfaces with boundary and punctures. Definition 7.9.10 (Energy). Let C be the set of smooth functions ϕ : [0, ∞) → [0, 1] such that ϕ(0) = 0 and ϕ ′ (r) ≥ 0 for all r ∈ [0, ∞). If F is a Riemann surface and u : F → C a holomorphic function, we define the energy of u as where (r, θ) are polar coordinates on C.
Remark 7.9.11. If we identify C × ∼ → R × S 1 using the log map, then agrees with the usual expression for the Hofer energy of log •u : Lemma 7.9.12. The sequence w i has uniformly bounded energy.
Proof. By Stokes' theorem, if c is a circle and c η ≈ π m if c is an arc. The argument works because we may take w i | c to be C l -close (not just C 0 -close) to an asymptotic eigenfunction, with l ≥ 1. This follows from the exponential decay estimates of [HWZ1]; also see [HT2, Lemma 2.3]. Moreover, for k = 0, 1, f ik η = 0 since w| f ik , k = 0, 1, projects to a radial ray. Finally, f i2 η < 0 since w| f i2 always has a component in the negative θ-direction; here it is important to remember that we are projecting using balanced coordinates. This proves the lemma.
Remark 7.9.13. Observe that E(cu) = E(u) where c ∈ C × . 7.9.6. Bubbling. The goal of this subsection is to eliminate certain types of bubbling.
Let ( F i , g i ), i ∈ N, be a sequence of Riemannian manifolds which are compatible with the complex structures and have injectivity radii which are uniformly bounded below. For example, one could obtain the metrics g i by scaling the compatible hyperbolic metrics by a conformal factor so that the thick parts remain hyperbolic, while the thin parts become flat cylinders. We will use these metrics to compute the pointwise norm of the differential of the functions w i which are denoted by |w ′ i |. Lemma 7.9.14. For every compact set K ⊂ B − , there is a constant C K > 0 such that |w i (z)| < C K and |w ′ i (z)| < C K for all z ∈ p −1 i (K) and all i ∈ N. Proof. The lemma holds for K ∩ K i by Lemma 7.9.9. Hence it suffices to consider Since w i (K c i ) ⊂ C × , we compose with log : C × → R × (R/2πZ) and consider ζ i = log •w i | K c i . The sequence ζ i has uniformly bounded energy by Lemma 7.9.12. If |ζ i ′ |, i ∈ N, is not uniformly bounded on K c i , then the usual bubbling analysis yields a nonconstant finite energy Suppose we are in Case (S4'). Recall the compactification cl(B − ) of B − defined in Section 5.1.1, obtained by adjoining the points at infinity p + , p − . Here cl(B − ) is isomorphic to the closed unit disk and p + is a marked point in the interior and p − is a marked point on the boundary under this identification.
Theorem 7.9.15. The sequence (w i ) i∈N converges uniformly on compact subsets to a holomorphic map w ∞ : B − → C such that: (1) w ∞ (∂B − ) ⊂ R + and w ∞ (m b ) = 0; (2) lim s→+∞ |w ∞ (s, t)| = +∞ and lim (3) lim (4) w ∞ | int(B − ) is biholomorphism onto its image. In particular m b is the unique zero of w ∞ | cl(B − ) and is simple; Proof. The uniform convergence on compact subsets is a consequence of the bounds from Lemma 7.9.14.
(1) is immediate from the convergence, together with the observation that the angles between the arcs in a tend to 0 as m i → ∞.
(2) Let K + i (resp. K − i ) be the component of F i − int(K i ) with positive (resp. negative) s-coordinates. We expand w i and w ∞ in Fourier series on K + i : Since we are in Case (S4'), {R i }, i.e., a = 2. By Equation (7.9.2), for all n ∈ Z. On the other hand, if we write f i (t) = c i e πit , where f i (t) is not necessarily normalized, then where δ in is the Kronecker delta. Hence (a) |a i n | · e πnR (2) i ≤ κ i for all n = 1; and We prove (ii). Arguing by contradiction, suppose that lim i→∞ |a i n | = 0 for some n ≥ 2. By (a), there exists C > 0 such that Ce 2πR (2) i < κ i for all i. Since R (2) i → ∞, this implies that |a i 1 e πR (2) i | < κ i for all i. On the other hand, |c i | ≥ κ ′ i and lim i→∞ κ ′ i κ i = +∞, which contradicts (b). Next we prove (i). We claim that lim i→∞ a i 1 = 0 for topological reasons. Indeed, if lim i→∞ a i 1 = 0, then lim i→∞ a i n = 0 for all n ≥ 1 by (ii) and the curve w i | s=R 0 has nonpositive winding number around 0 when i ≫ 0. On the other hand, w i | s=R 0 is homotopic to the curve w i | s=R (2) i in C × , a contradiction. This proves the claim.
(3) This is similar to (2). We expand w i in Fourier series on K − i : By the uniform convergence, lim s→−∞ a i n = a ∞ n and we can similarly prove that a ∞ n = 0 for all n < 0. This implies (3) because the normalized eigenfunctions converge to a constant as i → +∞.
7.9.8. Case (S4"). We give a brisk treatment of the construction of the SFT limit, mostly pointing out the differences with (S4'). The main difference is that the projection p i : F i → B − is a branched double cover over its image with a single branch point b i ∈ B − , and we must analyze different cases depending on the behavior of the branch point as i → ∞. There are five cases: The sequence p i : F i → B − converges in the SFT sense to a 1or 2-level holomorphic building ±,j is mapped to p * ± . In Case (a), F ∞ is an annulus with a puncture q + in the interior and a puncture q −,j on each boundary component. In Case (b), F ∞ is a disk with one puncture q + in the interior, two punctures q −,j , j = 1, 2, on the boundary, and two boundary points b ∞,j , j = 1, 2, which are glued together to give b ∞ . The points q −,j and b ∞,j alternate along the boundary. In Case (c), F m ∞ is a disk with a puncture q m + in the interior and a puncture q m − on the boundary, and F b ∞ is a sphere with three punctures q b + , q b −,j , j = 1, 2. In Case (d'), F m ∞ is a disk with a puncture q m + in the interior and two punctures q m −,j on the boundary and F b ∞ is a disk with four punctures q b ±,j on the boundary. In Case (d"), F m ∞ is a disk with a puncture q m + in the interior and two punctures q m −,j on the boundary and F b ∞ is a disk with four punctures q b ±,j on the boundary and two boundary points b ∞,j , j = 1, 2, identified. Cases (a) and (b) are similar to Case (S4'), while the situation in Cases (c), (d') and (d") is complicated by the fact that the limit is a 2-level holomorphic building.
(4a) As in the proof of Theorem 7.9.15(4), we can define the degree for maps of pairs (cl( F ∞ ), ∂cl( F ∞ )) → (CP 1 , R + ). The degree of w ∞ is 1 because it has a unique pole of order 1. The order of the pole at the positive puncture can be computed from the winding number of f ab , which is the smallest one for a positive eigenvalue by Lemma 7.8.4. Then w ∞ can have no branch points in the interior of cl( F ∞ ). (4b) is similar. (5) follows from (4a) and (4b).
Proof. Similar to Lemma 7.9.14. Lemma 7.9.17 implies that the limits w m ∞ and w b ∞ exist. The following lemma gives the behavior of w m ∞ and w b ∞ near the punctures. Lemma 7.9.18.
(1) Let u : R + × (R/πZ) → C × be a finite energy holomorphic map. If the map t → u(s, t) has degree one for some (and therefore all) s ∈ R + , then (2) Let u : R − × (R/πZ) → C × be a finite energy holomorphic map. If the map t → u(s, t) has degree one for some (and therefore all) s ∈ R − , then lim Proof.
(1) Let us view u as a map R + × (R/πZ) → R × (R/πZ). As in the proof of Lemma 7.9.14, since u has finite energy, it has bounded derivative. Let u n (s, t) = u(s + k n , t), where k n ∈ R + and lim n→+∞ k n = +∞. The sequence u n has uniformly bounded derivative and converges to a finite energy holomorphic map Such a holomorphic map is of the form u ∞ (s, t) = (s + a, t + b), where a, b are constants. This implies (1).
(2) is similar and is left to the reader.

Case (c).
Theorem 7.9.19. Suppose lim i→∞ s(b i ) = +∞. Then (w m i ) i∈N converges to a holomorphic map w m ∞ : F m ∞ → C such that: where f is a normalized eigenfunction of the asymptotic operator at δ 0 with winding number one; is a biholomorphism; and (5) m b is the unique zero of w m ∞ | int(cl( F m ∞ )) and is simple.
(7) at the negative puncture q b −,1 that connects to q m + , lim (9) at the punctures q m (10) w b ∞ extends to a biholomorphism w b ∞ : CP 1 → CP 1 ; and (11) q b −,1 is the unique zero of w b ∞ and is simple. Proof. The proof of Theorem 7.9.15 goes through with little modification, in view of Lemma 7.9.18. We remark that (8) is a consequence of Convention 6.6.4 and the proof technique of Lemma 6.6.5.
10. Involution lemmas. In this subsection we collect some lemmas on holomorphic maps between Riemann surfaces with anti-holomorphic involutions. These lemmas, collectively referred to as the involution lemmas, will play an important role in Section 7.11 and in the sequel [CGH2].
Our starting point is the following observation, whose proof is straightforward.
Observation 7.10.1. Let Σ 1 , Σ 2 be Riemann surfaces with anti-holomorphic involutions ι 1 , ι 2 , respectively. If f : Σ 1 → Σ 2 is a holomorphic map, then f := There are four versions of the involution lemma; the first two will be used in this paper and the last two only in the sequel [CGH2]. We start by introducing some common notation which will be used in all four versions: For i = 1, 2, the Riemann surface Σ i is an open subset of CP 1 which is invariant under complex conjugation and has finitely generated fundamental group; moreover no component of CP 1 −Σ i is a single point. The complex conjugation on CP 1 restricts to an anti-holomorphic involution ι i : Σ i → Σ i . On each Σ i we fix "radial rays" The asymptotic markerṘ i (0) is the connected component of T 0 RP 1 − {0} (i.e., a tangent half-line) consisting of vectors with negative ∂ x -component; similarly, the asymptotic markerṘ i (∞) is the component of T ∞ RP 1 − {0} that is mapped tȯ R i (0) under the inversion z → 1 z . The radial rays R i and their related asymptotic markers are invariant under the involution ι i . In this section we will use the notation D for the open unit disk in C, considered as a Riemann surface.
Lemma 7.10.2. Given Σ i as above, there is a compact Riemann surface with boundary Σ i with a biholomorphism Σ i ∼ → int(Σ i ). Moreover there is an antiholomorphic involution ι i : Σ i → Σ i such that the diagram Sketch of proof. We outline the proof of the first statement. Use the uniformization theorem to identify the universal cover of Σ = Σ 1 or Σ 2 with the open upper half space H. Let G ⊂ P SL(2, R) be the deck transformation group of H such that H/G = Σ. If G is finitely generated, then (∂H − L)/G is a collection of boundary circles of Σ, where L ⊂ ∂H is the limit set of G. Hence Σ = (H − L)/G.
Note that Σ i will not necessarily be the closure of Σ i in CP 1 . However, when referring to points in the interior of Σ i , we denote them by the corresponding point in Σ i . In the same way we view the radial rays R i as subsets of Σ i , and the asymptotic markers as tangent half-lines to Σ i . Lemma 7.10.2 implies that they are invariant by the involution on Σ i .
The image of an asymptotic marker by a holomorphic function is defined by the differential at regular points. At singular points the local behavior of a holomorphic map still allows us to define the image of a tangent ray.
For the first version of the involution lemma, let where a i ∈ R + and a 1 < · · · < a 2p , and let Σ 2 = D. We write Lemma 7.10.3 (Involution Lemma, Version 1). Let f : Σ 1 → Σ 2 be a holomorphic map which is a q-fold branched cover with q ≥ p, such that: Then f maps Fix(ι 1 ) to Fix(ι 2 ) andṘ 1 (∞) toṘ 2 (0). We briefly sketch the idea of the proof. In all the cases that are eliminated by Theorem 7.11.1, the (unique) component of ∪ a j=1 v ♯ j ⊂ u ∞ which is asymptotic to a multiple of δ 0 at the negative end has ECH index I = 1.
Suppose that, for m ≫ 0, there is a sequence of holomorphic curves which converges to a configuration u ∞ that we want to exclude. In Section 7.9, we applied a rescaling argument to construct a holomorphic building which keeps track of how the limit u ∞ is approached; this is similar to the layer structures of Ionel-Parker [IP1, Section 7]. In the simplest case, this building is a holomorphic map w ∞ : B − → C which satisfies the asymptotic condition lim s→+∞ w ∞ (s, t) |w ∞ (s, t)| = f ab (t), where f ab (t) is a normalized asymptotic eigenfunction of an I = 1 curve with a negative end asymptotic to δ 0 . The condition I = 1 is used as follows: since there are only finitely many I = 1 curves with negative ends asymptotic to δ 0 , we may assume that −1 ∈ {f ab ( 3 2 )} as explained in Remark 7.8.8. Remark 7.11.2. In this subsection we identify cl(B − ) ≃ D so that p + corresponds to 0 and p − corresponds to 1. There is an anti-holomorphic involution ι on B − that fixes the half-line {t = 3 2 }, and {t = 3 2 } corresponds to the radial ray R = D∩R ≤0 by Observation 7.10.1. In particular, m b is mapped to a point on R.
Similarly we identify cl(B ′ ) ≃ CP 1 so that p + corresponds to 0, p − corresponds to ∞, and {t = 3 2 } corresponds to the radial ray R ≤0 , We now use the involutions lemmas from Section 7.10 to obtain a contradiction. By the involution lemmas and the symmetric placement of the basepoint m b , we obtain that w ∞ • ι = i • w ∞ . Hence lim s→+∞ w ∞ (s, 3 2 ) |w ∞ (s, 3 2 )| = −1, which contradicts −1 ∈ {f ab ( 3 2 )}. Elimination of Case (2). Suppose for each i the sequence u ij converges a building u i∞ satisfying Case (2). By Theorem 7.9.15, we obtain a holomorphic map w ∞ : cl(B − ) → CP 1 , whose restriction to int(cl(B − )) is a biholomorphism onto its image.
We apply the Involution Lemma 7.10.3 to obtain a contradiction: Let Σ 1 be the compactification of Σ 1 = CP 1 − [a 1 , a 2 ] and let Σ 2 = cl(B − ) be identified with D as in Remark 7.11.2. Let f : Σ 1 → Σ 2 be the extension of (w ∞ | int(cl(B − )) ) −1 . Such an extension exists because Σ 1 is biholomorphic to the open unit disk and biholomorphisms of the open unit disk extend continuously to the boundary. By Lemma 7.10.3, f mapsṘ 1 (∞) toṘ 2 (0) and, conversely, w ∞ mapsṘ 2 (0) toṘ 1 (∞). Since the asymptotic markerṘ 2 (0) in Σ 2 corresponds to the asymptotic marker {t = 3 2 } for p + ∈ cl(B − ) by Remark 7.11.2,Ṙ 1 (∞) is a bad radial ray (in the sense of Definition 7.8.7) by Theorem 7.9.15(2). This contradicts Remark 7.8.8, so we have eliminated Case (2). (3) and (4). We will treat Case (4); Case (3) is almost identical. Suppose for each i the sequence u ij converges to a building u i∞ satisfying Case (4). By Remark 7.7.2, for each i, the total number of branched points of ∪ a j=−b v ′ j,i is one. If we exercise some care in choosing the diagonal sequence in Lemma 7.9.4, we can divide the argument for Case (4) further into Subcases (a), (b), (c), (d') and (d") as in Section 7.9.8, depending on the behavior of the branch points of the maps π B − • v ′ 0,i . Subcases (a) and (b). By Theorem 7.9.16, we obtain holomorphic maps

Elimination of Cases
where w ∞ | int(cl( F∞)) is a biholomorphism onto its image Σ 1 = CP 1 − ([a 1 , a 2 ] ∪ [a 3 , a 4 ]) and p ∞ is a branched double cover with one branch point. Let Σ 1 be the compactification of Σ 1 as in Lemma 7.10.2, and let Σ 2 = cl(B − ) be identified with D as in Remark 7.11.2. We define f : Σ 1 → Σ 2 as the extension of p ∞ • (w ∞ | int(cl( F∞)) ) −1 . Such an extension exists because Σ 1 is biholomorphic to an open annulus, and biholomorphisms of the unit annulus always extend to the boundary. At this point we apply Lemma 7.10.3 as in Case (2) to obtain a contradiction. Case (b) is completely analogous and can be excluded in the same way.

Subcases (d') and (d")
. By Theorem 7.9.20, we obtain a holomorphic map w m ∞ : cl( F m ∞ ) → CP 1 which restricts to a biholomorphism of int(cl( F m ∞ )) with CP 1 − [a 1 , a 2 ]. Then the proof proceeds as in Case (2). (5) and (6). The limit configurations of Cases (5) and (6) must contain a connector over δ 0 . This implies that Theorem 7.9.19 applies, and we obtain pairs of holomorphic maps
This completes the proof of Theorem 7.11.1. 7.12. Proof of Lemma 7.2.3. We begin with the following corollary of Theorem 7.11.1.
Multiplication by a real constant gives an R + -action on N . Even though N is the space we are interested in, it will be convenient for technical reasons to regard N as an open subset of a vector space N obtained by relaxing properties (N 1 )-(N 3 ).
In order to compute the dimension of N , we identify B − with D − {−1, 0} and N with the space of the holomorphic sections of a holomorphic line bundle E → D with values in a real rank one subbundle F along ∂D − {−1}.
We construct the bundles E and F as follows. Consider a cover of D by three open sets Over each open set we take a trivial line bundle E i = C × U i → U i and define the bundle E by gluing the bundles E i via the transition maps ψ 1 : E 0 | U 0 ∩U 1 → E 1 | U 0 ∩U 1 , ψ 1 (z, v) = (z, zv), ψ 2 : E 0 | U 0 ∩U 2 → E 2 | U 0 ∩U 2 , ψ 2 (z, v) = z, i z + 1 −z + 1 v .

The linear Cauchy-Riemann operator
is Fredholm for p > 2 and its kernel consists of smooth holomorphic functions. We denote by H 0 (E, F ) its kernel and by H 1 (E, F ) its cokernel.
Lemma 7.13.3. There is an identification H 0 (E, F ) ∼ = N for every choice of η ε .
Proof. The isomorphism H 0 (E, F ) ∼ = N associates to a holomorphic section ξ ∈ H 0 (E, F ) the holomorphic function π E 0 • ξ : Σ → C, i.e., π E 0 • ξ is obtained by writing ξ| U 0 with respect to the trivialization of E 0 . On the negative end Z we can write π E 2 • ξ(s, t) = n≥1 c n e (nπ−ε)(s+it)+iε , since π E 2 • ξ is holomorphic. By applying the transition function ψ −1 2 , we obtain that the leading term of π E 0 • ξ on Z is e −ε(s+it)+iε , which is condition ( N 2 ) in Definition 7.13.2. For a similar reason π E 0 • ξ has a pole at 0 of order at most 1.
By the doubling argument of Theorem 5.5.1, Lemma 5.5.3 (or, rather, its proof) and the formula for the index of the Cauchy-Riemann operator on line bundles over punctured surfaces (see for example [We3, Formula 2.1]), the index of D is ind D = χ(D) + µ τ (F ) + 2c 1 (Ě, τ ) − 1.
Lemma 7.13.5. The operator D is surjective for every choice of η ε .
Proof. In view of Theorem 5.5.1, the surjectivity of D follows from [We3, Formula 2.5] and [We3, Proposition 2.2(2)] applied to the double of D.
Corollary 7.13.6. The real dimension of N is 3 for every choice of η ε .
7.13.2. The maps E and F. We define an R-linear map: where c 1 , c 2 are the coefficients from ( N 1 ) and ( N 2 ) of Definition 7.13.2.
Lemma 7.13.7. The map E is an isomorphism.
Hence E (N ) is an open positive cone contained in R + × C × .
Proof. It suffices to check that E is injective. This follows from the winding number argument of [Se2, Lemma 11.5], which we sketch: First observe that ker E consists of holomorphic functions u : D → C such that u(−1) = 0 and u(e iθ ) ∈ R · e ηε(θ) for all θ ∈ (−π, π). Suppose that u ∈ ker E and u = 0. Given z ∈ D, let ν(z) be the order of zero at z, with the convention that ν(z) = 0 if u(z) = 0. Then, by analyzing the boundary and asymptotic conditions of u, we obtain (7.13.1) z∈int(Σ) ν(z) + 1 2 z∈∂Σ ν(z) < 0, where the right-hand side comes from half the Maslov index of F , suitably extended across the boundary puncture.
Next we prove that N is nonempty for any choice of η ε .
Proof. The only nontrivial part of the statement is that u 0 has no zeros along the boundary. In this case the analog of Equation (7.13.1) is: which implies that u 0 has no zeros.
Corollary 7.13.9. N is nonempty for any choice of η ε .
Proof. Take any u ∈ N with c 2 = 0. Then u + cu 0 ∈ N for any c sufficiently large.
Lemma 7.13.10. If c 1 , c 2 = 0, then E −1 (c 1 , c 2 ) either has a simple zero in the interior or has two zeros (counted with multiplicity) along the boundary.
In particular the maps in N have a unique zero in the interior. We denote PN = N /R + , where R + acts on N by multiplication and we define the maps Proof. We first prove injectivity. Let w 0 and w 1 be maps in N such that w −1 0 (0) = w −1 1 (0). Then ω = w 0 w 1 is a holomorphic map on D such that ω(∂D) ⊂ R, so it is constant. Then w 0 and w 1 represent the same element in PN .
Next we prove surjectivity. Fix an element w 0 ∈ N and let z 0 ∈ D such that w 0 (z 0 ) = 0. For any z ∈ D − {0, z 0 } we look for a holomorphic function ω z : D → CP 1 such that ω z (z) = 0, ω z (z 0 ) = ∞, and ω z (∂D) ⊂ R + . By the argument of the previous paragraph, if such ω z exists, it is unique up to multiplication by a positive real constant. Moreover, w = ω z w 0 ∈ N and w(z) = 0.
Finally we prove that F is a diffeomorphism. We define Ξ(z) = ω z w 0 ∈ N ; then F −1 (z) is the class of Ξ(z) in PN . The map Ξ is smooth because the maps f , g, and hence ω z are rational maps whose coefficients depend smoothly on z. Then in order to prove that F is a diffeomorphism it suffices to prove that dΞ(z) is injective for any z ∈ D − {0}.
In order to distinguish the differential of Ξ at z ∈ D − {0} from the differential of the function Ξ(z) : D − {0, −1} → C, we will use the notation dΞ(z) for the former and ∂Ξ(z) ∂ζ for the latter. We define the map F : N × (D − {0}) → C, F (w, z) = w(z).
We observe that ∂F ∂z (Ξ(z),z) = ∂Ξ(z) ∂ζ ζ=z , which is invertible because z is a simple zero of Ξ(z). Hence dΞ(z) is injective. 7.13.3. Proof of Theorem 7.2.2. We refer to Section 7.2 for the definition of the moduli spaces M 1 , M 0 , and M −1 , and of the gluing parameter space P. In order to simplify the exposition we will assume (without loss of generality) that all the multiplicities of γ ′ are 1 and that each of M 1 /R, M 0 , and M −1 /R is connected; in particular, both M 0 and M −1 /R are single points. We choose a smooth slice M 1 of the R-action on M 1 such that the following hold for some κ ′ 0 , κ 0 > 0, K 0 > π − ε, and for all v 1 ∈ M 1 : • each component of v 1 | s≤0 is (κ ′ 0 + κ 0 , 0)-close to a cylinder over a component of δ 0 γ ′ ; and • the component v 1 of v 1 | s≤0 which is close to σ ′ ∞ satisfies (7.13.2) π • v 1 − κ ′ 0 e (π−ε)s (ce πit ) ≤ κ 0 e K 0 s , where ce πit is the normalized asymptotic eigenfunction corresponding to the negative end δ 0 of v 1 and π = π Dρ 0 is the projection to D ρ 0 with respect to balanced coordinates. By a slight abuse of notation, we will refer to κ ′ 0 e (π−ε)s (ce πit )| s=0 = κ ′ 0 ce πit as the asymptotic eigenfunction of v 1 at δ 0 . Similarly, we choose a smooth slice M −1 of the R-action on M −1 such that the following hold for some κ ′ 1 , κ 1 > 0, K 1 > 2ε: • each component of v −1 | s≥0 is (κ ′ 1 +κ 1 , 0)-close to a strip over a component of {z ∞ } ∪ y ′ ; and • the component v −1 of v −1 | s≥0 which is close to σ ∞ satisfies (7.13.3) π • v −1 − κ ′ 1 e −2εs (de −εit ) ≤ κ 1 e −K 1 s , where de −εit is a normalized asymptotic eigenfunction corresponding to the positive end z ∞ of v −1 . Note that d is completely determined by the data {(i, j) → (i, j)} at the positive end z ∞ . Without loss of generality, we assume that a i,j = R + , h(a i,j ) = R + ·e iε , so that d = e iε . Similarly, we refer to κ ′ 1 e −2εs de −εit | s=0 = κ ′ 1 de −εit as the asymptotic eigenfunction of v −1 at z ∞ .
In the rest of this section, when we write v i , i = 1, −1, we will assume that v i is in the slice M i . Also, for T ∈ R and i = ±1, let v i,T be the T -translates of v i in the R-direction; i.e., if s : W * → R, * = ∅, ′ , is the R-coordinate, then Let f (t) = βe πit be the asymptotic eigenfunction of v 1 at δ 0 with eigenvalue π − ε. Then the asymptotic eigenfunction of v 1,T at δ 0 is e −(π−ε)T f . Similarly, let g(t) = α 0 e iε(1−t) be the asymptotic eigenfunction of v −1 at z ∞ with eigenvalue 2ε. Then the asymptotic eigenfunction of v −1,T at z ∞ is e 2εT g.
Proof. By Lemma 7.13.13 we can extract a subsequence (which we still call d i ) such that u(d i ) converges to u ∞ = (v 1,∞ , v 0 , v −1 ). By the SFT convergence of u(d i ) to u ∞ we obtain a sequence of good truncations satisfying the estimates of Lemma 7.9.4. The proof of Theorem 7.9.15 then goes through essentially unchanged.
Let w 0 ∈ N be a map such that w 0 (m b ) = 0 and let U δ ⊂ N be the subset consisting of maps w such that w(m) = 0 for some m ∈ D δ (m b ). Note that U δ is an R + -invariant open neighborhood of w 0 . We now have the following diagram: The map Π is defined as follows: Π(T ± , v 1 , v 0 , v −1 ) = (α, β), where αe iε(1−t) is the asymptotic eigenfunction of v −1,−2T − at z ∞ and βe iπt is the asymptotic eigenfunction of v 1,2T + at δ 0 . Let us write P δ := Π −1 (E(U δ )). Then P δ ⊂ P is an open set since E is a diffeomorphism.
Also define (7.13.7) The map Υ ′′ is well-defined by the definitions of P 2δ/3 and U δ . It is also proper and the local degree is well-defined. The maps Υ ′ and Υ ′′ are sufficiently close in the following sense.
Lemma 7.13.16. For any k, Υ ′ and Υ ′′ can be made arbitrarily C 0 -close by choosing r 0 sufficiently large.
In particular, the local degrees of Υ ′ and Υ ′′ near m b agree. The local degree of Υ ′ is equal to the left-hand side of Equation (7.2.3), while the local degree of Υ ′′ is equal to the right-hand side of Equation (7.2.3). (We were assuming that each of M 1 /R, M 0 , and M −1 /R is connected.) This completes the proof of Theorem 7.2.2.