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

Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map \Phi^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded contact homology chain complex for a contact form supported by (S,h). The chain map \Phi^+ commutes up to homotopy with the U-maps defined on both sides and reduces to the quasi-isomorphism \Phi from"The equivalence of Heegaard Floer homology and embedded contact homology I, II"on subcomplexes defining the hat versions. Algebraic considerations then imply that the map \Phi^+ is a quasi-isomorphism.


INTRODUCTION
This is the last paper in the series which proves the isomorphism between certain Heegaard Floer homology and embedded contact homology groups.References from [CGH2] (resp.[CGH3]) will be written as "Section I.x" (resp."Section II.x") to mean "Section x" of [CGH2] (resp.[CGH3]), for example.
Let M be a closed oriented 3-manifold.Let HF (M ) and HF + (M ) be the hat and plus versions of Heegaard Floer homology of M and let ECH(M ) and ECH(M ) be hat and usual versions of the embedded contact homology of M .As usual, embedded contact homology will be abbreviated as ECH.In [CGH1], we introduced the ECH chain group ECC(N, ∂N ) and showed that ECH(N, ∂N ) ≃ ECH(M ).In the papers [CGH2,CGH3], we defined a chain map Φ : CF (−M ) → ECC(N, ∂N ), which induced an isomorphism The goal of this paper is to extend the above result and prove the following theorem: Theorem 1.0.1.If M is a closed oriented 3-manifold, then there is a chain map which is a quasi-isomorphism and which commutes with the U -maps up to homotopy.
We use F = Z/2Z coefficients for both Heegaard Floer homology and ECH.
Given a pair (Σ 0 , h 0 ) consisting of a surface Σ 0 and h 0 ∈ Diff(Σ 0 ), we write the mapping torus of (Σ 0 , h 0 ) as: The map Φ, defined in Section I.6.2, is induced by the cobordism W + which is an S 0 -fibration and which restricts to a half-cylinder over [0, 1] × S 0 at the positive end and to a half-cylinder over the suspension N (S 0 ,h) at the negative end.We say that W + is a cobordism "from [0, 1] × S 0 2 to N (S 0 ,h) ."The map Φ + is induced by a cobordism W + from [0, 1]×Σ to M which extends W + and is described below. 3Although Φ was defined in terms of just one page S 0 , we can no longer ignore the S 1/2 portion of Σ when defining Φ + , since we do not know how to express HF + (−M ) in terms of S 0 .
Step 3. Define the chain map Φ + as a count of I W + = 0 curves in W + and show that Φ + commutes with the U -maps on both sides up to a chain homotopy K.This is done in Section 5.
Step 5.By Theorem 6.4.1, the map Φ alg is a quasi-isomorphism.This is proved by relating Φ alg to the quasi-isomorphism Φ from [CGH2,CGH3].

HEEGAARD FLOER CHAIN COMPLEXES
The goal of this section is to introduce some notation and recall the definition of the chain complex CF + (Σ, α, β, z f , J), whose homology is HF + (−M ).
2.1.Heegaard data.Let M be a closed oriented 3-manifold and let (S, h) be an open book decomposition for M .
We use the following notation, which is similar to that of Section I.4.9.1: • Σ = S 0 ∪ −S 1/2 is the associated genus 2g Heegaard surface of M ; • a = {a 1 , ..., a 2g } is a basis of arcs for S and b is a small pushoff of a as given in Figure I.1; • x i and x ′ i are the endpoints of a i in ∂S 0 that correspond to the coordinates of the contact class and ) are the collections of compressing curves on the Heegaard surface Σ; • z f is a point in the large (i.e., non-thin-strip) component of S 1/2 − α − β and (z ′ ) f is a point which is close but not equal to z f .We say that the pointed Heegaard diagram (Σ, α, β, z f ) is compatible with (S, h).
2.2.Symplectic data.The stable Hamiltonian structure on [0, 1] × Σ with coordinates (t, x) is given by (λ, ω), where λ = dt and ω is an area form on Σ which makes (α, β, z f ) weakly admissible with respect to ω, i.e., each periodic domain has zero ω-area.The plane field ξ = ker λ is equal to the tangent plane field of {t} × Σ and the Hamiltonian vector field is R = ∂ ∂t .We introduce the "symplectization" Let J be an Ω-admissible almost complex structure on W ; we assume that J is regular (cf.Lemma I.4.7.2 and [Li, Proposition 3.8]).We also define the Lagrangian submanifolds 2.3.The chain complex CF + (Σ, α, β, z f , J).In this subsection we recall the definition of the chain complex CF + (Σ, α, β, z f , J), whose homology group is isomorphic to HF + (−M ).This definition is due to Lipshitz [Li], with one modification: we are using the ECH index I HF from Definition I.4.5.11.We will often suppress J from the notation.
Let S = S α,β be the set of 2g-tuples y = {y 1 , . . ., y 2g } of intersection points of α and β for which there exists some permutation σ ∈ S 2g such that y j ∈ α j ∩β σ(j) for all j.Then CF + (Σ, α, β, z f , J) is generated over F by pairs [y, i], where y ∈ S and i ∈ N.
The differential ∂ = ∂ HF is given by where the coefficient ∂[y, i], [y ′ , j] is the count of index I HF = 1 finite energy holomorphic multisections in (W, J) with Lagrangian boundary L α ∪ L β from y to y ′ , whose algebraic intersection with the holomorphic strip R × [0, 1] × {(z ′ ) f } is (i − j).We will often refer to such curves as curves from . The goal of this section is to give a geometric definition of the U -map which is analogous to that of ECH.
Let z f , (z ′ ) f be as before and let z = (z b , z f ) ∈ W = B × Σ, where z b ∈ int(B).Let J ♦ be a generic C l -small perturbation of J such that J ♦ = J away from a small neighborhood N (z) ⊂ W of z and such that In particular, we assume that there are no J ♦ -holomorphic curves that are homologous to {pt} × Σ and pass through z.
Remark 3.1.1.When we refer to "C l -close" almost complex structures, etc., we assume that l > 0 is sufficiently large.Definition 3.1.2(Geometric U -map).The geometric U -map with respect to the point z is the map: where the coefficient U z ([y, i]), [y ′ , j] is the count of index I HF = 2 finite energy holomorphic curves in (W, J ♦ ) with Lagrangian boundary By our choice of J ♦ , "passing through z" is a generic codimension 2 condition, i.e., if u is a simple curve from [y, i] to [y ′ , j] that passes through z, then I(u) ≥ 2.Moreover, by a simple ECH index count, an I(u) ≤ 3 curve that passes through z cannot have a fiber component.
Proof.This follows from standard arguments in symplectic geometry by using analytical results proved in [Li].
The rest of this section is devoted to the proof of Theorem 3.1.4.Let J = j D × j Σ be a product complex structure on M and J ♦ be a generic C l -small perturbation of J such that J ♦ = J away from a small neighborhood of z.The key feature of J ♦ is that all the J ♦ -holomorphic curves that pass through z are regular.

A model
We then define the moduli space M A (M, J * ), * = ∅ or ♦, of stable maps , such that ∂F has k connected components and each component of ∂F maps to a distinct Lagrangian ∂D × β i , i = 1, . . ., k.We choose points w i ∈ β i , i = 1, . . ., k, and define Then let M A (M, J * ; z, w) ⊂ M A (M, J * ) be the subset of curves that pass through z and w.We use the modifier "irr" to denote the subset of irreducible curves.

ECH index.
We briefly indicate the definition of the ECH index I of a homology class B ∈ H 2 (M, ∂D × β) which admits a representative F such that each component of ∂F maps to a distinct ∂D × β i .Although we call I the "ECH index", what we are defining here is a relative version of Taubes' index from [T4].
Let τ be a trivialization of T Σ along β, given by a nonsingular tangent vector field X 1 along β, and let τ ′ be a trivialization of T D along ∂D, given by an outward-pointing radial vector field X 2 along ∂D.Let Q (τ,τ ′ ) (B) be the intersection number between an embedded representative u of B and its pushoff, where the boundary of u is pushed off in the direction given by J(X 1 ).Definition 3.2.1.The ECH index of the homology class B is: The following is the relative version of the adjunction inequality: where δ(u) ≥ 0 is an integer count of the singularities.
Proof.Similar to the proof of Theorem I.4.5.13.
We now calculate some ECH and Fredholm indices: Proof.We compute that Here , denotes the algebraic intersection number.
Proof.Follows from Lemma 3.2.3 and the index inequality.
Hence #M A (M, J ♦ ; z, w) is a certain relative Gromov-Witten invariant [IP1] which is computed to be 1 mod 2. (What we are really computing here is a relative Gromov-Taubes invariant [T4], although the two invariants coincide in this case.) Proof.(1) Let us write M = M A (M, J ♦ ; z, w).Arguing by contradiction, suppose u ∈ M − M irr .Then u consists of an irreducible component u 0 which passes through z and k 0 < k points of w, together with k − k 0 copies of D × {pt}.By Lemma 3.2.4,ind(u 0 ) ≤ 2 − 2k + 3k 0 .On the other hand, the point constraints are (k 0 + 2)-dimensional.Hence u 0 does not exist for generic J ♦ , which is a contradiction.
(2),(3) The compactness follows from the usual Gromov compactness theorem.The regularity of M is immediate from the genericity of J ♦ and (1).Lemma 3.2.4implies the dimension calculation, as well as (3).
(4) We degenerate the fiber Σ into a union of k tori which are successively attached to one another.We perform this pinching away from the curves β and make sure that each torus contains exactly one component of β.Then by a degeneration/gluing argument as in Section II.2.4.4, it suffices to prove the proposition for k = 1.The case k = 1 is proved in Lemmas 3.2.8 and 3.2.9.3.2.3.Doubling.We now explain how to double u ∈ M A (M, J ♦ ; z, w).For technical reasons we will assume that (Σ, j Σ ) admits an anti-holomorphic involution σ Σ so that the curve β is contained in the fixed point set of σ Σ .
We then define the almost complex structure D(J ♦ ) on D(M ) by taking J ♦ on M 1 and S(−J ♦ ) on M 2 .
Given u ∈ M A (M, J ♦ ; z, w), let D(u) be its double, obtained by gluing u and S(u).The map D(u) is holomorphic by the Schwarz reflection principle.Therefore it is an element of because it represents the class and passes through 3 points: (1, w), z, and S(z), where z Thus all the curves of M D,J ♦ come in pairs, except those that are S-invariant, and the S-invariant curves are those obtained by the doubling procedure.Summarizing, we have: 3.2.4.The k = 1 case.For the rest of the subsection we assume k = 1.We first consider the case where J = j D × j Σ is a product complex structure.
(1) follows from the homological constraint for generic pt will be larger than deg(π (2) is similar.
Let J ♦ be an almost complex structure which is C l -close to J. By Gromov compactness and Lemma 3.2.7,all the curves of M A (M, J ♦ ; z, w) and M D,J ♦ are close to the degenerate curves described in Lemma 3.2.7.Lemma 3.2.8.If k = 1 and J ♦ , w, and β are generic, then the following hold: (1) M D,J ♦ = M irr D,J ♦ ; (2) the curves of M D,J ♦ are embedded; and (3) M D,J ♦ is compact, regular, and 0-dimensional.

Proof. (1) Arguing by contradiction, suppose
On the other hand, if v does not have a component S 2 × {pt}, then v must have a fiber component v which is close to some ({0} × Σ) i .In either case, such a fiber component v cannot exist by the genericity of J ♦ .This proves (1).
(2) Let v ∈ M irr D,J ♦ .The proof is similar to that of Lemma 3.2.5(3)and follows from the adjunction inequality [M1, M2] (compare with Lemma 3.2.2):If where F is the domain of v with χ(F ) = −2.Hence v is embedded by the adjunction inequality.
(3) Since v is embedded by (2) and c 1 (v * T D(M ))) = 2, the regularity of v without the point constraints follows from automatic transversality (cf.Hofer-Lizan-Sikorav [HLS, Theorem 1]).The regularity with point constraints is the consequence of the genericity of J ♦ , w, and β.The rest of the assertion is immediate.
Proof.By Lemma 3.2.6,Lemma 3.2.8(1),and Theorem 3.2.5(1), We reduce the calculation of M D,J ♦ to a calculation in McDuff-Salamon [MS,Example 8.6.12] by degenerating the base S 2 into two spheres along the curve corresponding to the boundary of the two copies of D.
More precisely, let let I be a product complex structure on S 2 × Σ, and let I ♦ be a C l -small perturbation of I such that I = I ♦ away from a neighborhood of (0, z f ).Here we are viewing be the moduli spaces of I ♦ -curves in the class B that pass through (0, z f ) and (∞, w), and let M ′ B be the subset of curves in M B that are contained in a neighborhood of (S 2 × {w}) ∪ ({0} × Σ).By McDuff-Salamon [MS, Example 8.6.12],#M ′ B = 1.By the symplectic sum theorem of Ionel-Parker [IP2, p. 940] (also see Section II.2.4.4),

Family of cobordisms.
We now describe a family of marked points z τ ∈ W and a family of almost complex structures J ♦ τ on W for τ ∈ [0, 1), as well as their limits for τ = 1.These families give rise to the chain homotopy In the limit τ = 1, the base B 1 is (B ⊔D)/ ∼, where D = {|z| ≤ 1} ⊂ C and ∼ identifies (0, 0) ∈ B with −1 ∈ D, and the total space W 1 is (W ⊔ (D × Σ))/ ∼, where ((0, 0), x) ∼ (−1, x) for all x ∈ Σ. See Figure 2. We write w b for the node The limit z 1 of z τ is in D × Σ and we assume that z b 1 = 0 ∈ int(D).When τ = 1, the almost complex structure J ♦ 1 restricts to the complex structure J on W and to the almost complex structure (J 1,D ) ♦ , where J 1,D is a product complex structure on D×Σ and (J 1,D ) ♦ is a C l -small perturbation of J 1,D such that (J 1,D ) ♦ = J 1,D away from a small neighborhood N (z 1 ) of z 1 and that pass through z τ i .Applying Gromov compactness, we obtain the limit u 1 = u B ∪ u D , where u B ⊂ W , u D ⊂ D × Σ, and u D passes through z 1 .Components of u 1 that map to the fiber {w b } × Σ will be viewed as components of u D .Lemma 3.4.1. (1) (2) is a consequence of (1) and computations as in the proof of Lemma 3.2.3.We remind the reader that the genus of Σ is 2g.The first sentence of Theorem 3.1.4follows from the usual construction of chain homotopies in Floer theory.By Lemma 3.4.2,U z is chain homotopic to aU , where a is the count of holomorphic curves u D in (D × Σ, (J 1,D ) ♦ ) that pass through z 1 and w = ((1, y 1 ), . . ., (1, y 2g )), where y = {y 1 , . . ., y 2g }.Since a = 1 modulo 2 by Theorem 3.2.5,U z is chain homotopic to U .
Next we prove the second sentence of Theorem 3.1.4.For all y ∈ S, H([y, 0]) is obtained by counting I HF = 1 curves that pass through z τ for some τ ∈ (0, 1) and that do not cross the holomorphic strip is strictly positive by the positivity of intersections, and so is its intersection with R × [0, 1]× {(z ′ ) f }.

THE COBORDISM W +
In this section we give the construction of the symplectic cobordism (W + , Ω + ) from [0, 1] × Σ to M , together with the Lagrangian submanifold L α ⊂ ∂W + .4.1.Construction of (W + , Ω + ).We describe the construction of W + , leaving some key details for later4 : First we construct fibrations π 0 : with the corners rounded.We then glue W + 0 and W + 1 and smooth a boundary component Let δ > 0 be a small irrational number and N a large positive number which depends on δ and whose dependence will be described later.
Lemma 4.1.1.There exists a symplectic manifold (W + , Ω + ) which depends on δ > 0 and which satisfies the following: (1) There is a symplectic surface where ω is an area form on Σ.Moreover, where β ′ is isotopic to β. (5) On the negative end W + 2 of W + , Ω + restricts to the negative symplectization of a contact form λ − on B ≃ M which is adapted to the open book decomposition (S, h).
(6) The manifold B ≃ M admits a decomposition into three disjoint pieces: the suspension N (S 0 ,h 0 ) , where h 0 is isotopic to h relative to ∂S 0 , a closed neighborhood N (K) of the binding K, and an open thickened torus N in between that we refer to as the "no man's land".(7) All the orbits of the Reeb vector field R
The S 1 -family P + (resp.P − ) of simple orbits of T + (resp.T − ) can be viewed equivalently as a pair e ′ , h ′ (resp.e, h) consisting of an elliptic orbit and a hyperbolic orbit.The proof of Lemma 4.1.1 will be given in Section 4.3. Let where we are using polar coordinates (r ′ , θ ′ ).
The actual construction of (W + , Ω + ) is a bit involved, and consists of several steps.
Step 1.The following lemma is a rephrasing of Lemma I.2.1.1 and its proof.
, for ε > 0 sufficiently small and C > 0. In particular, f t and β t are independent of t and R λ is parallel to Here ε > 0 depends on δ > 0, d 2 is the differential in the S 0 -direction, and the C 0 -norm is with respect to a fixed Riemannian metric on S 0 .
Step 2. We then extend h and the contact form λ to the contact form λ + = f t dt+β t to N (Σ−N (z f ),h + 0 ) , all of which depend on δ > 0, as follows: and ω is an area form on Σ which agrees with Without loss of generality we may assume that α × {1} is Legendrian with respect to λ + .This is an easy consequence of the Legendrian realization principle; see for example [H,Theorem 3.7].
Let ω D 2 be an area form on D 2 satisfying: An easy calculation shows that ω D 2 = ds ∧ dt (and hence Ω + 1 = Ω + 0 ) on their overlap.

Lemma 4.1.3. There exists r
with respect to coordinates (r 1 , θ 1 , t).The Reeb vector field R λ − is parallel to , which imply the lemma.
Construction of B 1 .Let ζ : [0, 1] → R be a smooth map such that: We also define N (K) ⊂ B as the closed neighborhood of the binding K = {r 2 = 0} that is bounded by the torus {r 1 = r * 1 }.The region N = {0 < r 1 < r * 1 } ⊂ B will be called "no man's land".

Hamiltonian structure on
. Then the Hamiltonian structure on Σ × [0, 1] at the positive end of W + is given by (dt, ω| Σ×[0,1] ).Let h + 2 be the flow of the corresponding Hamiltonian vector field from Σ×{0} to Σ×{1}.Note that we do not necessarily have h + 2 = id by construction.Lagrangian submanifold L α .As in Section I.5.2.1, we define the Lagrangian submanifold L α ⊂ ∂W + by placing a copy of α on the fiber π −1 (3, 1) over (3, 1) ∈ ∂B 0 + and using the symplectic connection Ω + to parallel transport α along the boundary component Proof.First observe that h + 1 and h + 2 are isotopic to the identity.Then h + is isotopic to h + 0 where h + 0 | S 1/2 = id and h + 0 | S 0 is isotopic to h.The lemma then follows.
Definition of W + .Let W + be the closure of the component of W + − H which is disjoint from S (z ′ ) f .In particular, the restriction π 1 : . The cobordism W + is diffeomorphic to the cobordism used to define the map Φ in Section I.5.1.
(7) By Lemma 4.1.4,the Reeb vector field R λ − has no closed orbits in B 1 since δ > 0 is irrational.By Lemma 4.1.3and Equation (4. , where C > 0 is independent of δ.The second sentence of (7) is immediate from the construction of λ − .

THE CHAIN MAP Φ +
The goal of this section is to define the chain map which is induced by the symplectic cobordism (W + , Ω + ) and an admissible almost complex structure J + .We take , in view of Equation (4.2.1) and Lemma 4.2.1 and the fact that h + 2 is the flow of the Hamiltonian vector field of ω| s=s 0 , s 0 ≫ 0, from Σ × {0} to Σ × {1}.
Let J, J ′ be the adapted almost complex structures that agree with J + at the positive and negative ends.
Note that (4) imposes additional conditions on Ω + and λ − .In practice, the order in which we construct Ω + and J + is a little convoluted: (i) choose regular J 0 + , (ii) choose τ > 0 sufficiently small and J + sufficiently close to J 0 + , (iii) construct Ω + using λ τ in place of λ, and (iv) extend J + to the rest of W + .
Let J + be the set of all (W + , Ω + )-admissible almost complex structures.
5.2.The ECH index.Let P = P λ − be the set of simple orbits of R λ − and let O = O λ − be the set of orbit sets constructed from P. Let J + ∈ J + be an admissible almost complex structure.Let M J + (y, γ) be the set of holomorphic maps u : (F, j) → (W + , J + ) from y ∈ S α,β to γ ∈ O, such that ∂F is mapped to a distinct component of L α and each component is used exactly once.Elements of M J + (y, γ) will be called W + -curves.
Let W + be W + with the ends {s > 3} and {s < −1} removed and let as in Section I.5.4.2.The class [u] of u ∈ M J + (y, γ) is the relative homology class of the compactification ǔ in H 2 ( W + , Z y,γ ).Given A ∈ H 2 ( W + , Z y,γ ), we write M J + (y, γ, A) ⊂ M J + (y, γ) for the subset of W + -curves u in the class A.
Definition 5.2.1 (Filtration F).Given a W + -curve u, we define where , is the algebraic intersection number.Since The definition of the ECH index given in Section I.5.6 also extends directly to our case.The ECH index of a W + -curve from y to γ in the class A is denoted by I W + (γ, A).

5.3.
Homology of W + .The goal of this subsection is to compute H 2 (W + ).We introduce some notation which will be used only in this subsection: where the extra Z factor is generated by a meridian of the binding.
Proof.The lemma follows from the exact sequence of the pair (M, N ).
0 is homotopy equivalent to N .We compute H 2 (N ) using the Mayer-Vietoris sequence: Since i = 0 and ker j = Z ∂S 0 = Z ∂S 1/2 , the lemma follows.
1 is homotopy equivalent to S 1/2 , the Mayer-Vietoris sequence becomes: ) by the Künneth formula, the restriction j : ) is an isomorphism, and the restriction j : ) is injective because the image of the generator of H 1 (S 1 ) is dual to the fiber Σ.Then the energy of a W + -curve u : F → W + from [y, i] to γ is given by: (5.4.1) where the supremum is taken over all pairs (φ, The condition imposed on the intersection with S (z ′ ) f gives an energy bound: Lemma 5.4.2 (Energy bound).For all k ∈ N, there exists N k > 0 such that E(u) ≤ N k for all y ∈ S α,β , γ ∈ O, and u ∈ M F =k J + (y, γ).
Let v : Since the energy is obtained by integrating a closed form, By Equations (4.1.1)and (4.1.2),Θ + 1 can be written as λ +,s + (s + π 10 )dt on Recall that λ +,s = λ + for s ≥ 3 2 .In the above calculation, We then obtain which is the desired bound.
5.5.Regularity.Define the subset M h J + (y, γ, A) ⊂ M J + (y, γ, A) consisting of holomorphic curves without vertical fiber components.As in Lemma I.5.8.2, the set J + reg of regular J + ∈ J + for which all the moduli spaces M h J + (y, γ, A) are transversally cut out is a dense subset of J + .We can restrict attention to M h J + (y, γ, A) for the following reason: Lemma 5.5.1.
By a combination of Lemma 5.4.2 and the Gromov-Taubes compactness theorem (cf.Section I.3.4), the sum in the definition of Φ + is finite.Hence Φ + is well-defined.
Proof.Similar to that of Theorem I.6.2.4, with slight modifications in view of Lemmas 5.6.2 and 5.6.3.
Remark 5.7.2.One can define the twisted coefficient analog of Φ + , taking into account Lemma 5.3.3.5.8.Restriction to Φ.In this subsection δ still denotes the constant that appears in the construction of λ − .Let P| N be the subset of P consisting of orbits that are contained in N .Also let γ θ ∈ P − be the orbit corresponding to θ ∈ ∂S 0 .
Proof.Let u ∈ M F =0 J + (y, γ) such that u(F ) ⊂ W + .Suppose that u is not a multi-level Morse-Bott building.Then u(F ) ∩ C θ 0 = ∅ for some θ 0 ∈ ∂S 0 − α − β, and moreover we may assume that γ θ 0 is not an asymptotic limit of u at −∞.Since J + is admissible, all the curves C θ are holomorphic.Hence u(F ), C θ 0 > 0 by the positivity of intersections.
If u is a multi-level Morse-Bott building, then we need to make the appropriate modifications (left to the reader), but the same argument goes through.For example, we need to replace C θ 0 by a multi-level building Note that if u is a Morse-Bott building, then it could have a component u 1 with a negative end that limits to some γ θ 1 , followed by a gradient trajectory from θ 1 to θ 2 , and then by a component u 2 with a positive end that limits to γ θ 2 .
Proof.The almost complex structure J + is sufficiently close to J 0 + .For J 0 + , the analogous chain map was shown to be a quasi-isomorphism (Theorem II.1.0.1).Considerations similar to those of Theorem I.3.6.1 imply that Φ is a quasi-isomorphism.
5.9.Commutativity with the U -map.Let z b be a point in R × [0, 1] with tcoordinate 1 2 and let z = (z b , z f ) ∈ W .Let U z be the geometric U -map with respect to z on the HF side.On the ECH side, let z ′ = (s, z M ) be a generic point in R × int(N (K)) near the binding K. We define U ′ = U ′ z ′ so that U ′ (γ), γ ′ is the count of I ECH = 2 curves in the symplectization (R × M, J ′ ) from γ to γ ′ that pass through z ′ .
Theorem 5.9.1.There exists a chain homotopy Proof.The commutativity of Φ + with the U -maps up to homotopy is obtained by moving the point constraint in the cobordism W + from s = +∞ to s = −∞.
The 1-parameter family of points (z τ ) τ ∈R is chosen as follows: For τ ≥ 0, let z τ = (z b τ , z f ), where z b τ approaches (s, t) = (+∞, 1 2 ) as τ → +∞ and z b 0 is near the center of the disk , where z M ∈ M = B is a point near the binding.Finally, we consider a small perturbation of (z τ ) τ ∈R to make it generic (without changing its name).
We define the 1-parameter family of almost complex structures (J + τ ) τ ∈R so that J + τ is C l -close to J + and agrees with J + outside a small neighborhood of z τ .The rest of the chain homotopy argument is standard, with the exception of the obstruction theory that was carried out in [HT1,HT2].
Proof.Similar to the proof of Theorem 5.8.3.The coefficient K([y, 0]), γ is given by the count of I W + = 1 curves from y to γ that pass through z τ for some τ and do not intersect S (z ′ ) f .If such a curve u exists, then Im(u) ⊂ W + .This is not possible by Theorem 5.8.3.6. PROOF OF THEOREM 1.0.1 In this section we prove Theorem 1.0.1.In Section 6.1 we prove an algebraic result (Theorem 6.1.5)which is sufficient to prove that Φ + is a quasi-isomorphism.The conditions of Theorem 6.1.5are verified in Section 6.4.6.1.Some algebra.Definition 6.1.1.Let (A, d) be a chain complex.We say that a chain map f : Prototypical examples of han maps are the U -maps in HF + and ECH.Let (A, d A ) and (B, d B ) be chain complexes with han maps U A : A → A and U B : B → B and let Φ + : A → B be a chain map such that the diagram commutes up to a chain homotopy K.We form a chain complex D = A ⊕ A ⊕ B ⊕ B with differential Given a chain map f , we denote its mapping cone by C(f ).Lemma 6.1.2.There is an exact triangle: y y r r r r r r r r r r

H(D)
e e ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ where Lemma 6.1.3.There is an exact triangle: y y r r r r r r r r r r
Proof.Consider the following commutative diagram with exact rows: Proof.If Φ alg is a quasi-isomorphism, then H(D) = 0 by Exact Triangle (6.1.1).This in turn implies that U Φ + is a quasi-isomorphism by Exact Triangle (6.1.2).However the han map U Φ + cannot be a quasi-isomorphism, unless H(C(Φ + )) = 0. Finally, the triangle y y r r r r r r r r r r
We finish this subsection with a lemma which compares the homology of C(U ) with that of ker U .Lemma 6.1.6.Let (C, d) be a chain complex and let U be a chain map.If U is surjective, then the inclusion Proof.Let U : C/ ker U → C be the map induced by U .We have a short exact sequence of complexes which induces the exact triangle: / / H(C(U )) x x q q q q q q q q q q H(C(U )) Since U is surjective, U is an isomorphism.Hence H(C(U )) = 0 and the lemma follows.

Lemma 6.2.1.
There is an isomorphism j : Proof.This follows from the discussion of Theorem I.4.9.4.Note that the natural candidate CF (S 0 , a, h(a)) → CF (Σ, α, β, z f ), [Z] → Z for a chain map is not a well-defined map.

ECH chain complexes.
We describe several ECH chain complexes that are related to (ECC(M, λ − ), ∂ ′ ) and are constructed from certain subsets S of the set P = P λ − of simple orbits of R λ − .Many of these appeared in [CGH1, Section 10] (some with different names).Let U ′ be the U -map of ECC(M, λ − ) with respect to (s 0 , z M ) ∈ R × M , where z M is a generic point which is sufficiently close to the binding.
Let O S be the set of orbit sets that are constructed from S. Then S is closed if γ ′ ∈ O S , whenever γ ∈ O S , γ ′ ∈ O P , and ∂ ′ γ, γ ′ = 0 or U ′ γ, γ ′ = 0.If S is closed, then let (A S , ∂ ′ S ) be the subcomplex of ECC(M, λ − ) generated by O S and let U ′ S be the restriction of U ′ to A S .Let P| N ⊂ P be the set of orbits in the suspension N .The subsets S 1 = P| N ∪ {e ′ , h ′ }, S 2 = P| N ∪N ∪ {e ′ , h ′ }, P| N ∪ {h ′ }, P| N ∪N ∪ {h ′ }, P| N are closed and we write A i = A S i , ∂ ′ i = ∂ ′ S i , and U ′ i = U ′ S i for i = 1, 2, as well as ECC ♮ (N ) = A P| N ∪{h ′ } , ECC ♮♮ (N ) = A P| N∪N ∪{h ′ } , ECC(N ) = A P| N .
where γ i ∈ O| N (K) and Γ i ∈ O| N ∪N .Let F ♮♮ : ECC ♮♮ (N ) → Z ≥0 be its restriction to ECC ♮♮ (N ).(Note that F ♮♮ is a trivial filtration.)Next define the filtration F : C(U ′ ) → Z ≥0 such that The map i is an (F ♮♮ , F )-filtered chain map.The induced map on the E 1 -level agrees with the isomorphism (p 2 ) * ; the proof is similar to that of [CGH1, Section 10].If a filtered chain map between filtered chain complexes which are bounded below is an isomorphism on the E r -level, then it is a quasiisomorphism.This implies that i is a quasi-isomorphism.
Observe that there is a discrepancy between the algebra and the geometry: the map Φ alg which we are using here is not the map Φ, and we need to reconcile the two.

FIGURE 1 .
FIGURE 1. Schematic diagram for W + 0 ∪W + 1 which indicates the fibers over each subsurface.
Lemma 3.4.2.I(u D ) = 2g + 2 and I HF (u B ) = 0.In particular, y = y ′ , u B consists of 2g trivial strips, and k 0 = k = 1.Proof.The gluing constraints give I HF (u τ ) = I(u D ) + I HF (u B ) − 2g = 2.By the regularity of J and the index inequality, we have I HF (u B ) ≥ 0. The first sentence of the lemma then follows from Lemma 3.4.1(2); the second sentence is a consequence of the first.
FIGURE 3. Schematic diagram for rounding the corner of B. The diagram shows a neighborhood N (B) of B, where we are projecting W + 0 ∩ N (B) to coordinates (s, r 1 ) and W + 1 ∩ N (B) to coordinates (r 0 , r 1 ).

Lemma 6.3. 3 .
The inclusion i :ECC ♮♮ (N ) → C(U ′ ) given by Γ → Γ 0 is a quasi-isomorphism.Proof.This is similar to the argument in [CGH1, Section 10].Choose an identification η : H 1 (N (K); Z)∼ → Z such that the homology class of the binding is 1.Define the filtration F : ECC(M ) → Z ≥0 such that