GEORGIOS Baptiste Chantraine
email: baptiste.chantraine@univ-nantes.fr

Dimitroglou Rizell
email: georgios.dimitroglou@math.uu.se

Paolo Ghiggini
email: paolo.ghiggini@univ-nantes.fr

Roman Golovko
email: rgolovko@ulb.ac.be

GEOMETRIC GENERATION OF THE WRAPPED FUKAYA CATEGORY OF WEINSTEIN MANIFOLDS AND SECTORS

Keywords: 2010 Mathematics Subject Classification. Primary 53D37; Secondary 53D40, 18E30 Weinstein manifold, wrapped Fukaya category, generation, Lagrangian cocore planes

We prove that the wrapped Fukaya category of any 2ndimensional Weinstein manifold (or, more generally, Weinstein sector) W is generated by the unstable manifolds of the index n critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer homology and the fairly simple observation that Floer homology vanishes for Lagrangian submanifolds that can be disjoined from the isotropic skeleton of the Weinstein manifold. Note that we do not need any additional assumptions on this skeleton. By applying our generation result to the diagonal in the product W × W , we obtain as a corollary that the open-closed map from the Hochschild homology of the wrapped Fukaya category of W to its symplectic cohomology is an isomorphism, proving a conjecture of Seidel.

Introduction

The wrapped Fukaya category is an A ∞ -category associated to any Liouville manifold. Its objects are exact Lagrangian submanifolds which are either compact or cylindrical at infinity, possibly equipped with extra structure, the morphism spaces are wrapped Floer chain complexes, and the A ∞ operations are defined by counting perturbed holomorphic polygones with Lagrangian boundary conditions. Wrapped Floer cohomology was defined by A. Abbondandolo and M. Schwarz [START_REF] Abbondandolo | On the Floer homology of cotangent bundles[END_REF], at least for cotangent fibres, but the general definition and the chain level construction needed to define an A ∞ -category is due to M. Abouzaïd and P. Seidel [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]. The definition of the wrapped Fukaya category was further extended to the relative case by Z. Sylvan, who introduced the notions of stop and partially wrapped Fukaya category in [START_REF] Sylvan | On partially wrapped Fukaya categories[END_REF], and by S. Ganatra, J. Pardon and V. Shende, who later introduced the similar notion of Liouville sector in [START_REF] Ganatra | Covariantly functorial Floer theory on Liouville sectors[END_REF].

In this article we study the wrapped Fukaya category of Weinstein manifolds and sectors. In the absolute case our main result is the following.

Theorem 1.1. If (W, θ, f) is a 2n-dimensional Weinstein manifold of finite type, then its wrapped Fukaya category WF(W, θ) is generated by the Lagrangian cocore planes of the index n critical points of f. In the relative case (i.e. for sectors) our main result is the following. We refer to Section 2.3 for the definition of the terminology used in the statement.

Theorem 1.2. The wrapped Fukaya category of the Weinstein sector (S, θ, f) is generated by the Lagrangian cocore planes of its completion (W, θ W , f W ) and by the spreading of the Lagrangian cocore planes of its belt (F, θ F , f F ).

Remark 1.3. Exact Lagrangian submanifolds are often enriched with some extra structure: Spin structures, grading or local systems. We ignore them for simplicity, but the same arguments carry over also when that extra structure is considered.

Generators of the wrapped Fukaya category are known in many particular cases. We will not try to give a comprehensive overview of the history of this recent but active subject because we would not be able to make justice to everybody who has contributed to it. However, it is important to mention that F. Bourgeois, T. Ekholm and Y. Eliashberg in [START_REF] Bourgeois | Effect of Legendrian surgery[END_REF] sketch a proof that the Lagrangian cocore discs split-generate the wrapped Fukaya category of a Weinstein manifold of finite type. Split-generation is a weaker notion than generation, but is sufficient for most applications. Unfortunately however, Bourgeois, Ekholm and Eliashberg's proposed proof relies on their Legendrian surgery formula, whose analytic details are still in progress. Most generation results so far, including that of Bourgeois, Ekholm and Eliashberg, rely on Abouzaïd's split-generation criterion [START_REF] Abouzaid | A geometric criterion for generating the Fukaya category[END_REF]. On the contrary, our proof is more direct and similar in spirit to Seidel's proof in [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF] that the Lagrangian thimbles generate the Fukaya-Seidel category of a Lefschetz fibration or to Biran and Cornea's cone decomposition of Arnol'd type Lagrangian cobordisms [START_REF] Biran | Lagrangian cobordism and Fukaya categories[END_REF].

A product of Weinstein manifolds is a Weinstein manifold. Therefore, by applying Theorem 1.1 to the diagonal in a twisted product, and using results of S. Ganatra [START_REF] Ganatra | Symplectic cohomology and duality for the wrapped Fukaya category[END_REF] and Y. Gao [START_REF] Gao | Functors of wrapped Fukaya categories from Lagrangian correspondences[END_REF], we obtain the following result. is an isomorphism.

In Equation 1HH * denotes Hochschild homology, SH * denotes symplectic cohomology and OC is the open-closed map defined in [START_REF] Abouzaid | A geometric criterion for generating the Fukaya category[END_REF]. Theorem 1.4 in particular proves that

(2) OC : HH * (WF (W, θ), WF (W, θ)) → SH * (W ) is an isomorphism. This proves a conjecture of Seidel in [START_REF] Seidel | Symplectic homology as Hochschild homology[END_REF] for Weinstein manifolds of finite type. Note that a proof of this conjecture, assuming the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg was given by S. Ganatra and M. Maydanskiy in the appendix of [START_REF] Bourgeois | Effect of Legendrian surgery[END_REF].

The above result implies in particular that Abouzaïd's generation criterion [START_REF] Abouzaid | A geometric criterion for generating the Fukaya category[END_REF] is satisfied for the subcategory consisting of the cocore planes of a Weinstein manifold, from which one can conclude that the cocores split-generate the wrapped Fukaya category. In the exact setting under consideration this of course follows a fortiori from Theorem 1.1, but there are extensions of the Fukaya category in which this generation criterion has nontrivial implications. Notably, this is the case for the version of the wrapped Fukaya category for monotone Lagrangians, as we proceed to explain.

The wrapped Fukaya category as well as symplectic cohomology were defined in the monotone symplectic setting in [START_REF] Ritter | The monotone wrapped Fukaya category and the open-closed string map[END_REF] using coefficients in the Novikov field. When this construction is applied to exact Lagrangians in an exact symplectic manifold, a change of variables x → t -A(x) x, where A(x) is the action of the generator x and t is the formal Novikov parameter, makes the Floer complexes and the open-closed map become identified with the original complexes and map tensored with the Novikov field. The generalisation of Abouzaïd's generation criterion to the monotone setting established in [START_REF] Ritter | The monotone wrapped Fukaya category and the open-closed string map[END_REF] thus shows that Corollary 1.5. The wrapped Fukaya category of monotone Lagrangian submanifolds of a Weinstein manifold which are unobstructed in the strong sense (i.e. with µ 0 = 0, where µ 0 is the number of Maslov index two holomorphic discs passing through a generic point) is split-generated by the Lagrangian cocore planes of the Weinstein manifold.

Remark 1.6. The strategy employed in the proof of Theorem 1.1 for showing generation fails for non-exact Lagrangian submanifolds in two crucial steps: in Section 7 and Section 8. First, there are well known examples of unobstructed monotone Lagrangian submanifolds in a Weinstein manifold which are Floer homologically nontrivial even if they are disjoint from the skeleton. Second, our treatment of Lagrangian surgeries requires that we lift the Lagrangian submanifolds in W to Legendrian submanifolds of W × R, and this is possible only for exact Lagrangian submanifolds. It is unclear to us whether it is true that the cocores generate (and not merely split-generate) the µ 0 = 0 part of the monotone wrapped Fukaya category.

The strategy of the proof of Theorem 1.1 is the following. Given a cylindrical Lagrangian submanifold L, by a compactly supported Hamiltonian isotopy we make it transverse to the stable manifolds of the Liouville flow. Thus, by dimension reason, it will be disjoint from the stable manifolds of the critical points of index less than n and will intersect the stable manifolds of the critical points of index n in finitely many points a 1 , . . . , a k . For each a i we consider a Lagrangian plane D a i passing through a i , transverse both to L and to the stable manifold, and Hamiltonian isotopic to the unstable manifold of the same critical point. The Lagrangian planes D a 1 , . . . , D a k are what we call the Lagrangian cocore planes. We can assume that they are all pairwise disjoint.

At each a i we perform a Lagrangian surgery between L and D a i so that the resulting Lagrangian L is disjoint from the skeleton of W . Since L will be in general immersed, we have to develop a version of wrapped Floer cohomology for immersed Lagrangian submanifolds. To do that we borrow heavily from the construction of Legendrian contact homology in [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF]. In particular our wrapped Floer cohomology between immersed Lagrangian submanifolds uses augmentations of the Chekanov-Eliashberg algebras of the Legendrian lifts as bounding cochains. A priori there is no reason why such a bounding cochain should exists for L, but it turns out that we can define it inductively provided that D a 1 , . . . , D a k are isotoped in a suitable way. A large part of the technical work in this paper is devoted to the proof of this claim.

Then we prove a correspondence between twisted complexes in the wrapped Fukaya category and Lagrangian surgeries by realising a Lagrangian surgery as a Lagrangian cobordism between the Legendrian lifts and applying the Floer theory for Lagrangian cobordisms we defined in [START_REF] Chantraine | Floer theory for lagrangian cobordisms[END_REF]. This result can have an independent interest. Then we can conclude that L is isomorphic, in an appropriated triangulated completion of the wrapped Fukaya category, to a twisted complex L built from L, D a 1 , . . . , D a k .

Finally, we prove that the wrapped Floer cohomology of L with any other cylindrical Lagrangian is trivial. This is done by a fairly simple action argument based on the fact that the Liouville flow displaces L from any compact set because L is disjoint from the skeleton of W . Then the twisted complex L is a trivial object, and therefore some simple homological algebra shows that L is isomorphic to a twisted complex built from D a 1 , . . . , D a k .

This article is organised as follows. In Section 2 we recall some generalities about Weinstein manifolds and sectors. In Section 3 we recall the definition and the basic properties of Legendrian contact homology. In Sections 4 and 5 we define the version of Floer homology for Lagrangian immersions that we will use in the rest of the article. Despite their length, these sections contain mostly routine verifications and can be skipped by the readers who are willing to accept that such a theory exists. In Section 6 we define wrapped Floer cohomology for Lagrangian immersions using the constructions of the previous two sections. In Section 7 we prove that an immersed Lagrangian submanifold which is disjoint from the skeleton is Floer homologically trivial.

In Section 8 we prove that Lagrangian surgeries correspond to twisted complexes in the wrapped Fukaya category. In Section 9 we finish the proof of the generation results, and in particular we construct the bounding cochain for L. Finally, in Section 10 we prove the isomorphism between Hochschild homology and symplectic cohomology.

Acknowledgements. We thank Tobias Ekholm for pushing us to extend our original result to Weinstein sectors and Yasha Eliashberg for a useful conversation about Weinstein pairs. The third author would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program Homology theories in low dimensional topology when work on this paper was undertaken. This work was supported by EP-SRC Grant Number EP/K032208/1. The first author is partially supported by the ANR project COSPIN (ANR-13-JS01-0008-01). The second author is supported by the grant KAW 2016.0198 from the Knut and Alice Wallenberg Foundation. In addition, he would like to express his gratitude to Université de Nantes where he was professeur invité in October 2016 when this project was started. The fourth author is supported by the ERC Consolidator Grant 646649 "SymplecticEinstein". Finally, the first and third authors are partially supported by the ANR project QUANTACT (ANR-16-CE40-0017).

Geometric setup

In this section we revise some elementary symplectic geometry with the purpose of fixing notation and conventions.

2.1. Liouville manifolds. Let (W, θ) be a Liouville manifold of finite type, from now on called simply a Liouville manifold. This means that dθ is a symplectic form, the Liouville vector field L defined by the equation

ι L dθ = θ
is complete and, for some R 0 < 0, there exists a proper smooth function r : W → [R 0 , +∞) such that, for w ∈ W , (i) d w r(L w ) > 0 if r(w) > R 0 , and (ii) dr w (L w ) = 1 if r(w) ≥ R 0 + 1.

In particular, R 0 is the unique critical value of r, which is of course highly nondegenerate, and every other level set is a contact type hypersurface.

We use the function r to define some useful subsets of W .

Definition 2.1.

For every R ∈ [R 0 , +∞) we denote W R = r -1 ([R 0 , R]), W e R = W \ int(W R ) and V R = r -1 (R).
The subsets W e R will be called the ends of W . The Liouville flow of (W, θ) induces an identification [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF] ([R 0 + 1, +∞) × V, e r α) ∼ = (W e R 0 +1 , θ), where V = V 0 and α is the pull-back of θ to V 0 . More precisely, if φ denotes the flow of the Liouville vector field, the identification (3) is given by (r, v) → φ r (v). Let ξ = ker α be the contact structure defined by α. Every V R , for R > R 0 , is contactomorphic to (V, ξ). Under the identification (3), the function r, restricted to W e R 0 +1 , corresponds to the projection to [R 0 + 1, +∞) in the sense that the following diagram commutes

([R 0 + 1, +∞) × V φ / / ) ) W e R 0 +1 . r w w [R 0 + 1, +∞)
Remark 2.2. The choice of R 0 in the definition of r is purely arbitrary because the Liouville flow is complete. In fact, for every map r : W → [R 0 , +∞) as above and for any R 0 < R 0 there is a map r : W → [R 0 , +∞) satisfying (i) and (ii), which moreover coincides with r on r -1 ([R 0 +1, +∞)).

A diffeomorphism ψ : W → W is an exact symplectomorphism if ψ * θ = θ+dq for some function q : W → R. Flows of Hamiltonian vector fields are, of course, the main source of exact symplectomorphisms. Given a function H : [-t -, t + ] × W → R, where t ± ≥ 0 and are allowed to be infinite, we define the Hamiltonian vector field X H by ι X H dθ = -dH.

Here dH denotes the differential in the directions tangent to W , and therefore X H is a time-dependent vector field on W .

We spell out the change in the Liouville form induced by a Hamiltonian flow because it is a computation that will be needed repeatedly.

Lemma 2.3. Let H : [-t -, t + ] × W → R be a Hamiltonian function and ϕ t its Hamiltonian flow. Then, for all t ∈ [t -, t + ], we have ϕ * t θ = θ + dq t , where

q t = t 0 (-H σ + θ(X Hσ )) • ϕ σ dσ.
Proof. We compute

ϕ * t θ -θ = t 0 d dσ (ϕ * σ θ)dσ = t 0 ϕ * σ (L X Hσ θ)dσ = t 0 ϕ * σ (ι X Hσ dθ + dι X Hσ θ)dσ = t 0 ϕ * σ (-dH σ + d(θ(X Hσ )))dσ.

Weinstein manifolds.

In this article we will be concerned mostly with Weinstein manifolds of finite type. We recall their definition, referring to [START_REF] Cieliebak | From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds[END_REF] for further details.

Definition 2.4. A Weinstein manifold (W, θ, f) consists of:

(i) an even dimensional smooth manifold W without boundary, (ii) a one-form θ on W such that dθ is a symplectic form and the Liouville vector field L associated to θ is complete, and (iii) a proper Morse function f : W → R bounded from below such that L is a pseudogradient of f in the sense of [START_REF] Cieliebak | From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds[END_REF]Equation (9.9)]: i.e.

df(L) ≥ δ( L 2 + df 2 ),
where δ > 0 and the norms are computed with respect to some Riemannian metric on W .

The function f is called a Lyapunov function (for L).

If f has finitely many critical points, then (W, θ, f) is a Weinstein manifold of finite type. From now on, Weinstein manifold will always mean Weinstein manifold of finite type.

Given a regular value M of f the compact manifold {f ≤ M } is called a Weinstein domain. Any Weinstein domain can be completed to a Weinstein manifold in a canonical way by adding half a symplectisation of the contact boundary.

By Condition (iii) in Definition 2.4, the zeroes of L coincide with the critical points of f. If W has dimension 2n, the critical points of f have index at most n. For each critical point p of f of index n, there is a stable manifold ∆ p and an unstable manifold D p which are both exact Lagrangian submanifolds. We will call the unstable manifolds ∆ p of the critical points of index n the Lagrangian cocore planes.

Definition 2.5. Let W 0 ⊂ W be a Weinstein domain containing all critical points of f. The Lagrangian skeleton of (W, θ, f) is the attractor of the negative flow of the Liouville vector field on the compact part of W , i.e.

W sk := t>0 φ -t (W 0 ),
where φ denotes the flow of the Liouville vector field L. Alternatively, W sk can be defined as the union of unstable manifolds of all critical points of f.

The stable manifolds of the index n critical points form the top dimensional stratum of the Lagrangian skeleton.

A Morse function gives rise to a handle decomposition. In the case of a Weinstein manifold (W, θ, f), the handle decomposition induced by f is compatible with the symplectic structure and is called the Weinstein handle decomposition of (W, θ, f). By the combination of [START_REF] Cieliebak | From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds[END_REF]Lemma 12.18] and [START_REF] Cieliebak | From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds[END_REF]Corollary 12.21] we can assume that L is Morse-Smale. This implies that we can assume that handles of higher index are attached after handles of lower index. The deformation making L Morse-Smale can be performed without changing the symplectic form dθ and so that the unstable manifolds corresponding to the critical points of index n before and after such a deformation are Hamiltonian isotopic.

We will denote the union of the handles of index strictly less than n by W sc . This will be called the subcritical subdomain of W . By construction, ∂W sc is a contact type hypersurface in W .

We choose r : W → [R 0 , +∞), and we homotope the Weinstein structure so that W R 0 = W sc ∪ H 1 ∪ . . . ∪ H l , where H 1 , . . . H l all are standard Weinstein handles corresponding to the critical points p 1 , . . . , p l of f of Morse index n; see [START_REF] Weinstein | Contact surgery and symplectic handlebodies[END_REF] for the description of the standard model, and [START_REF] Cieliebak | From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds[END_REF]Section 12.5] for how to produce the Weinstein homotopy.

Remark 2.6. We could easily modify f so that it agrees with r on W e 0 . However, this will not be necessary.

The core of the Weinstein handle H i is the Lagrangian disc

C i = ∆ p i ∩ H i . Let D δ T * C i denote the disc cotangent bundle of C i of radius δ > 0.
By the Weinstein neighbourhood theorem, there is a symplectic identification H i ∼ = D δ T * C i for some δ. However, the restriction of θ to H i does not correspond to the restriction of the canonical Liouville form to D δ T * C i .

Weinstein sectors.

In this section we introduce Weinstein sectors. These will be particular cases of Liouville sectors as defined in [START_REF] Ganatra | Covariantly functorial Floer theory on Liouville sectors[END_REF] characterised, roughly speaking, by retracting over a Lagrangian skeleton with boundary. In Section 2.4 below we will then show that any Weinstein pair as introduced in [19, Section 2] can be completed to a Weinstein sector. Definition 2.7. A Weinstein sector (S, θ, I, f) consists of:

(1) an even dimensional smooth manifold with boundary S;

(2) a one-form θ on S such that dθ is a symplectic form and the associated Liouville vector field L is complete and everywhere tangent to ∂S; For simplicity we will often drop part of the data from the notation. We will always assume that S is a Weinstein sector of finite type, i.e. that f has only finitely many critical points. A Weinstein sector is a particular case of an exact Liouville sector in the sense of [START_REF] Ganatra | Covariantly functorial Floer theory on Liouville sectors[END_REF].

Example 2.8. After perturbing the canonical Liouville form, the cotangent bundle of a smooth manifold Q with boundary admits the structure of a Weinstein sector.

To a Weinstein sector (S, θ, I, f) we can associate two Weinstein manifolds in a canonical way up to deformation: the completion and the belt. The completion of S is the Weinstein manifold (W, θ W , f W ) obtained by completing the Weinstein domain W 0 = {f ≤ c}, which contains all interior critical points of f. The belt of S is the Weinstein manifold (F, θ F , f F ) where

F = I -1 (0), θ F = θ| F 1 and f F = f| F .
To show that the belt is actually a Weinstein manifold it is enough to observe that dθ F is a symplectic form because F is transverse to the vector field C, and that the Liouville vector field L is tangent to F because dI(L) = αI, and therefore the Liouville vector field of θ F is L F = L| F .

Let κ ∈ R be a number such that all critical points of f are contained in {f ≤ κ}. We denote S 0 = {f ≤ κ} and F 0 = F ∩ S 0 = {f F ≤ κ}. By Condition (4a) of Definition 2.7, the boundary ∂S 0 is a contact manifold with convex boundary with dividing set ∂F 0 . Moreover S \ S 0 can be identified to a half symplectisation. Thus, given R 0 0, we can define a function r : S → [R 0 , +∞) satisfying the properties analogous to those in Section 2.1. We then write S R := r -1 [R 0 , R] and S e R = S \ int(S R ). Definition 2.9. Let φ be the flow of L. The skeleton S sk ⊂ S of a Weinstein sector (S, θ, f) is given by

S sk := t>0 φ -t (S 0 ).
Remark 2.10. Let W and F the completion and the belt, respectively, of the Weinstein sector S. To understand the skeleton S sk it is useful to note the folowing:

(1) critical points of f on ∂S are also critical points of f| ∂S and vice versa, The top stratum of the skeleton of (S, θ, f) is given by the union of the stable manifolds of the critical points of f of index n, where 2n is the dimension of S. Those are of two types: the stable manifolds ∆ p where p is an interior critical point of f, which are also stable manifolds for f W in the completion, and the stable manifolds Θ p where p is a boundary critical point of f, for which ∆ p = Θ p ∩ ∂S is the stable manifold of p for f F in F .

Thus the Weinstein sector S can be obtained by attaching Weinstein handles, corresponding to the critical points of f in the interior of ∂S, and Weinstein half-handles, corresponding to the critical points of f in the boundary ∂S. We denote by S sc the subcritical part of S, i.e. the union of the handles and half-handles of index less than n (where 2n is the dimension of S), by {H i } the critical handle corresponding to ∆ i and by {H j } the half-handle corresponding to Θ j . Finally we also choose the function r : S → [R 0 , ∞) as in Section 2.1 which furthermore satisfies

S R 0 = S sc ∪ H 1 ∪ . . . ∪ H l ∪ H 1 ∪ . . . ∪ H l
and modify the Liouville form θ so that H 1 , . . . , H l , H 1 , . . . , H l are standard Weinstein handles.

It follow the from the symplectic standard neighbourhood theorem that a collar neighbourhood of ∂S is symplectomorphic to

(F × T * (-2ε, 0], dθ F + dp ∧ dq).
Definition 2.11. Let S be a Weinstein sector and let L be a Lagrangian submanifold of its belt F . The spreading of L is

spr(L) = L × T * -ε (-2ε, 0] ⊂ F × T * (-2ε, 0) ⊂ S. Remark 2.
12. The spreading of L depends on the choice of symplectic standard neighbourhood of the collar. However, given two different choices, the corresponding spreadings are Lagrangian isotopic. Furthermore, if L is exact in F , then spr(L) is exact in S, and thus two different spreadings are Hamiltonian isotopic.

Example 2.13. When the Weinstein sector is the cotangent bundle of a manifold with boundary, the spreading of a cotangent fibre of T * ∂Q is simply a cotangent fibre of T * Q.

The proof of the following lemma is immediate.

Lemma 2.14. The cocore planes of the index n half-handles of S are the spreading of the cocore planes of the corresponding index n -1 handles of F .

2.4.

Going from a Weinstein pair to a Weinstein sector. In this section we describe how to associate a Weinstein sector to a Weinstein pair. We recall the definition of Weinstein pair, originally introduced in [START_REF] Eliashberg | Weinstein manifolds revisited[END_REF]. Definition 2.15. A Weinstein pair (W 0 , F 0 ) is a pair of Weinstein domains (W 0 , θ 0 , f 0 ) and (F 0 , θ F , f F ) together with a codimension one Liouville embedding of F 0 into ∂W 0 .

We denote the completions of (W 0 , θ 0 , f 0 ) and (F 0 , θ F , f F ) by (W, θ 0 , f 0 ) and (F, θ F , f F ) respectively. Let F 1 ⊂ F be a Weinstein domain retracting on F 0 If F 1 is close enough to F 0 , the symplectic standard neighbourhood theorem provides us with a Liouville embedding

(4) ((1 -3 , 1] × [-3δ, 3δ] × F 1 , sdu + sθ F ) → (W 0 , θ 0 ).
Here s and u are coordinates on the first and second factors, respectively, and we require that the preimage of ∂W 0 is {s = 1} and F 1 ⊂ ∂W 0 is identified with {(1, 0, x) : x ∈ F 1 }. We denote U the image of the embedding (4).

After deforming f 0 we may assume that it is of the form f 0 (s, u, x) = s in the same coordinates.

Let L F be the Liouville vector field of (F, θ F ). We choose a smooth function

r F : F → [R 0 , +∞), R 0 0, such that • F 0 = r -1 F ([R 0 , 0]),
• dr F (L F ) > 0 holds inside r -1 F (R 0 , +∞), and

• dr F (L F ) = 1 holds inside r -1 F (R 0 + 1, +∞).
For simplicity of notation we also assume that

• F 1 = r -1 F ([R 0 , 1]),
where F 1 denotes the manifold appearing in Formula (4). This condition is apparently a loss of generality because it cannot be satisfied for every Liouville form on F . However, the general case can be treated with minimal changes.

Consider the smooth function

r : [-3δ, 3δ] × F 1 → R, r(u, x) := 2 - u 3δ 2 -r F (x) -c
for some small number c > 0.

Lemma 2.16. There exists a Weinstein domain W 0 ⊂ W containing all critical points of f 0 and which intersects some collar neighbourhood of W 0 containing U precisely in the subset

C := {s ≤ r(u, x)} ⊂ U.
The goal is now to deform the Liouville form θ 0 on

S 0 := W 0 ∩ W 0
to obtain a Liouville form θ so that the completion of (S 0 , θ) is the sought Weinstein sector. The deformation will be performed inside C. Given a smooth function ρ : [1 -3 , 1] → R such that:

• ρ(s) = 0 for s ∈ [1 -3 , 1 -2 ], • ρ(s) = 2s -1 for s ∈ [1 -, 1], and • ρ (s) ≥ 0 for s ∈ [1 -3 , 1],
we define a Liouville form θ U on U by

θ U = s(du + θ F ) -d(ρ(s)u).
The proof of the following lemma is a simple computation.

Lemma 2.17. Let ρ : [1 -3 , 1] → R be a smooth function such that:

The Liouville vector field L U corresponding to the Liouville form θ U on U is equal to

L U = (s -ρ(s))∂ s + ρ (s)u∂ u + ρ(s) s L F .
We define the Liouville form θ on S 0 as θ| C = θ U and θ| S 0 \C = θ 0 . By Lemma 2.17 the Liouville vector field L of θ is transverse to ∂S 0 \ ∂W 0 and is equal to

(1 -s)∂ s + 2u∂ u + 2s -1 s L F in a neighbourhood of ∂S 0 \ ∂W 0 ; in particular it is tangent to ∂S 0 \ ∂W 0 .
Thus we can complete (S 0 , θ) to (S, θ) by adding a half-symplectisation of ∂S 0 \ int(∂W 0 ). We define a function I : ∂S → R by setting I = u on ∂S 0 ∩ ∂W 0 and extending it to ∂S so that dI(L) = 2I everywhere. It is easy to check that (S, θ, I) is an exact Liouville sector.

A Lyapunov function f : S 0 → R + for L can be obtained by interpolating between f 0 on S 0 \ C and (1) L is an n-dimensional manifold and ι is a proper immersion which is an embedding outside finitely many points, (2) ι * θ = df for some function f : L → R, called the potential of (L, ι), and (3) the image of ι is tangent to the Liouville vector field of (W, θ) outside of a compact set of L.

u 2 -(s -1) 2 + f F + C on C ∩ {s ∈ [1
In the rest of the article, immersed exact Lagrangian submanifold will always mean immersed exact Lagrangian submanifold with cylindrical end. Note that L is allowed to be compact, and in that case Condition (3) is empty: a closed immersed Lagrangian submanifold is a particular case of immersed Lagrangian submanifolds with cylindrical ends. With an abuse of notation, we will often write L either for the pair (L, ι) or for the image ι(L).

Example

f 1 = f 0 + 1 0 (-H σ + θ(X Hσ )) • ϕ σ dσ.
as potential for (L, ι 1 ).

The weakest one is exact Lagrangian regular homotopy.

Definition 2.22. Two exact Legendrian immersions (L, ι 0 ) and (L, ι 1 ) with cylindrical ends are exact Lagrangian regular homotopic if there exists a smooth path of immersions ι t : L → W for t ∈ [0, 1] such that (L, ι t ) is an exact Lagrangian immersion with cylindrical ends for every t ∈ [0, 1].

We recall that any exact regular homotopy ι t : L → W can be generated by a local Hamiltonian defined on L in the following sense.

Lemma 2.23. An exact regular Lagrangian homotopy ι t induces a smooth family of functions G t : L → R determined uniquely, up to a constant depending on t, by the requirement that the equation ι * t (dθ(•, X t )) = dG t be satisfied, where X t : L → T W is the vector field along the immersion that generates ι t . When ι t has compact support, then dG t has compact support as well.

Conversely, any Hamiltonian G t : L → R generates an exact Lagrangian isotopy ι t : L → (W, θ) for any initial choice of exact immersion ι = ι 0 . Remark 2.24. If ι t is generated by an ambient Hamiltonian isotopy, then H extends to a single-valued Hamiltonian on W itself. However, this is not necessarily the case for an arbitrary exact Lagrangian regular homotopy.

The limitations of our approach to define Floer homology for exact Lagrangian immersions require that we work with a restricted class of exact immersed Lagrangian submanifolds. Definition 2.25. We say that a Lagrangian immersion (L, ι) is nice if the singularities of ι(L) are all transverse double points, and for every double point p the values of the potential at the two points in the preimage of p are distinct. Then, given a double point p, we will denote ι -1 (p) = {p + , p -}, where f (p + ) > f (p -).

Remark 2.26. If L is not connected, we can shift the potential on different connected components by independent constants. If ι -1 (p) is contained in a connected component of L, then f (p + )-f (p -) is still well defined. However, if the points in ι -1 (p) belong to different connected components, the choice of p + and p -in ι -1 (p), and f (p + )-f (p -), depend of the choice of potential. For technical reasons related to our definition of Floer homology, it seems useful, although unnatural, to consider the potential (up to shift by an overall constant) as part of the data of an exact Lagrangian immersion.

For nice immersed exact Lagrangian submanifolds we define a stronger form of exact Lagrangian regular homotopy. Definition 2.27. Let (L, ι 0 ) and (L, ι 1 ) be nice exact Lagrangian immersed submanifolds with cylindrical ends. An exact Lagrangian regular homotopy (L, ι t ) is a safe isotopy if (L, ι t ) is nice for every t ∈ [0, 1].

Niceness can always be achieved after a C 1 -small exact Lagrangian regular homotopy. In the rest of this article exact Lagrangian immersions will always be assumed nice.

2.6. Contactisation and Legendrian lifts. We define a contact manifold (M, β), where M = W × R, with a coordinate z on R, and β = dz + θ. We call (M, β) the contactisation of (W, θ). A Hamiltonian isotopy ϕ t : W → W which is generated by a Hamiltonian function H : [0, 1] × W → R lifts to a contact isotopy ψ + t : M → M such that ψ + t (x, z) = (ψ t (x), z -q t (x)), where q t : W → R is the function defined in Lemma 2.3.

An immersed exact Lagrangian (L, ι) with potential f : L → R uniquely defines a Legendrian immersion

ι + : L → W × R, ι + (x) = (ι(x), -f (x)).
Moreover ι + is an embedding when (L, ι) is nice. We denote the image of ι + by L + and call it the Legendrian lift of L. On the other hand, any Legendrian submanifold of (M, β) projects to an immersed Lagrangian in W . This projection is called the Lagrangian projection of the Legendrian submanifold.

Double points of L correspond to Reeb chords of L + , and the action (i.e. length) of the Reeb chord projecting to a double point p is f (p + ) -f (p -). If L is connected, different potentials induce Legendrian lifts which are contact isotopic by a translation in the z-direction. In particular, the action of Reeb chords is independent of the lift. On the other hand, if L is disconnected, different potentials can induce non-contactomorphic Legendrian lifts and the action of Reeb chords between different connected components depends on the potential.

Legendrian contact homology

3.1. The Chekanov-Eliashberg algebra. In view of the correspondence between Legendrian submanifolds of (M, β) and exact Lagrangian immersions in (W, θ), Floer homology for Lagrangian immersions will be a variation on the theme of Legendrian contact homology. The latter was proposed by Eliashberg and Hofer and later defined rigorously by Chekanov, combinatorially, in R 3 with its standard contact structure in [START_REF] Yu | Differential algebra of Legendrian links[END_REF], and by Ekholm, Etnyre and Sullivan, analytically, in the contactisation of any Liouville manifold in [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF]. In this subsection we summarise the analytical definition.

For d > 0, let R d+1 = Conf d+1 (∂D 2 ) be the space of parametrised discs with d + 1 punctures on the boundary. The automorphism group Aut(D 2 ) acts on R d+1 and its quotient is the Deligne-Mumford moduli space R d+1 . Given ζ = (ζ 0 , . . . , ζ d ) ∈ R d+1 , we will denote

∆ ζ = D 2 \ {ζ 0 , . . . , ζ d }.
Following [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF], near every puncture ζ i we will define positive and negative universal striplike ends with coordinates (σ + i , τ + i ) ∈ (0, +∞) × [0, 1] and (σ - i , τ - i ) ∈ (-∞, 0) × [0, 1] respectively. We will assume that σ - i = -σ + i and τ - i = 1 -τ + i . Remark 3.1. Putting both positive and negative strip-like ends near each puncture could be useful to compare wrapped Floer cohomology and contact homology, which use different conventions for positive and negative punctures. Definition 3.2. Let (V, α) be a contact manifold with contact structure ξ and Reeb vector field R.

An almost complex structure J on R × V is cylindrical if (1) J is invariant under translations in R, (2) J(∂ r ) = R, where r is the coordinate in R, (3)
J(ξ) ⊂ ξ, and J| ξ is compatible with dα| ξ . Definition 3.3. An almost complex structure J on a Liouville manifold W is compatible with θ if it is compatible with dθ and, outside a compact set, corresponds to a cylindrical almost complex structure under the identification [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]. We denote by J (θ) the set of almost complex structures on W which are compatible with θ.

It is well known that J (θ) is a contractible space.

Given an exact Lagrangian immersion (L, ι) in W , we will consider almost complex structures J on W which satisfy the following ( †) J is compatible with θ, integrable in a neighbourhood of the double points of (L, ι), and for which L moreover is real-analytic near the double points.

We will denote the set of double points of (L, ι) by D.

Let u : ∆ ζ → W be a J-
σ + i →+∞ (ι -1 • u)(σ + i , 0) = p + i and that ζ i is a negative puncture at p i if lim σ - i →-∞ (ι -1 • u)(σ - i , 0) = p + i .
Let L be an immersed exact Lagrangian. N 0 L (p 0 ; p 1 , . . . , p i , q, p j+1 , . . . , p d ; J)×N 0 L (q; p i+1 , . . . , p j ; J).

If L is spin, the moduli spaces are orientable and a choice of spin structure induces a coherent orientation of the moduli spaces; see [START_REF] Ekholm | Orientations in Legendrian contact homology and exact Lagrangian immersions[END_REF].

Definition 3.6. We say that an almost complex structure J on W is Lregular if it satisfies ( †) for L and all moduli spaces N L (p 0 ; p 1 , . . . , p d ; J) are transversely cut out.

To a Legendrian submanifold L + of (M, β) we can associate a differential graded algebra (A, d) called the Chekanov-Eliashberg algebra (or Legendrian contact homology algebra) of L + . As an algebra, A is the free unital noncommutative algebra generated by the double points of the Lagrangian projection L or, equivalently, by the Reeb chords of L + . The grading takes values in Z/2Z and is simply given by the self-intersection of the double points. If 2c 1 (W ) = 0 and the Maslov class of L + vanishes, it can be lifted to an integer valued grading by the Conley-Zehnder index. We will not make explicit use of the integer grading, and therefore we will not describe it further, referring the interested reader to [17, Section 2.2] instead.

The differential d is defined on the generators as:

d(p 0 ) = d≥0 p 1 ,...,p d #N 0 L (p 0 ; p 1 , . . . , p n ; J)p 1 . . . p d .
According to [17, Proposition 2.6], the Chekanov-Eliashberg algebra is a Legendrian invariant: [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF]). If L + 0 and L + 1 are Legendrian isotopic Legendrian submanifolds of (M, β), then their Chekanov-Eliashberg algebras (A 0 , d 0 ) and (A 1 , d 1 ) are stably tame isomorphic.

Theorem 3.7 ([
The definition of stable tame isomorphism of DGAs was introduced by Chekanov in [START_REF] Yu | Differential algebra of Legendrian links[END_REF], and then discussed by Ekholm, Etnyre and Sullivan in [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF]. We will not use it in this article but note that on the homological level a stable tame isomorphism induces an isomorphism.

3.2.

Bilinearised Legendrian contact cohomology. Differential graded algebras are difficult objects to manipulate, and therefore Chekanov introduced a linearisation procedure. The starting point of this procedure is the existence of an augmentation. Definition 3.8. Let A be a differential graded algebra over a commutative ring F. An augmentation of A is a unital differential graded algebra morphism ε : A → F.

Let L + 0 and L + 1 be Legendrian submanifolds of (M, β) with Lagrangian projections L 0 and L 1 with potentials f 0 and f 1 respectively. We recall that the potential is the negative of the z coordinate. We will assume that L + 0 and L + 1 are chord generic, which means in this case that L + 0 ∩ L + 1 = ∅ and all singularities of L 0 ∪ L 1 are transverse double points.

Let A 0 and A 1 be the Chekanov-Eliashberg algebras of L + 0 and L + 1 respectively. Let ε 0 : A 0 → F and ε 1 : A 1 → F be augmentations. Now we describe the construction of the bilinearised Legendrian contact cohomology complex LCC ε 0 ,ε 1 (L + 0 , L + 1 ; J). First, we introduce some notation. We denote by D i the set of double points of L i (for i = 1, 2) and by C the intersection points of L 0 and L 1 such that f 0 (q) < f 1 (q). Double points in D i correspond to Reeb chords of L + i , and double points in C corresponds to Reeb chords from L + 1 to L + 0 (note the order!). We define the exact immersed Lagrangian L = L 0 ∪ L 1 and, for p 0 = (p 0 1 , . . . , p 0 l 0 ) ∈ D l 0 0 , p 1 = (p 1 1 , . . . , p 1 l 1 ) ∈ D l 1 1 and q ± ∈ C, we denote N i L (q + ; p 0 , q -, p 1 ; J) := N i L (q + ; p 0 1 , . . . , p 0 l 0 , q -, p 1 1 , . . . , p 1 l 1 ; J), where J is an L-regular almost complex structure.

If ε i is an augmentation of A i , we denote ε i (p i ) := ε i (p i 1 ) • • • ε i (p i l i ).
As an F-module, LCC ε 0 ,ε 1 (L + 0 , L + 1 ; J) is freely generated by the set C and the differential of a generator q -∈ C is defined as

(7) ∂ ε 0 ,ε 1 (q -) = q + ∈C l 0 ,l 1 ∈N p i ∈D l i i #N 0 L (q + ; p 0 , q -, p 1 ; J)ε 0 (p 0 )ε 1 (p 1 )q + .
The bilinearised Legendrian contact cohomology LCH ε 0 ,ε 1 (L + 0 , L + 1 ) is the homology of this complex. The set of isomorphism classes of bilinearised Legendrian contact homology groups is independent of the choice of J and is a Legendrian isotopy invariant by the adaptation of Chekanov's argument from [START_REF] Yu | Differential algebra of Legendrian links[END_REF] due to the first author and Bourgeois [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF].

Floer homology for exact Lagrangian immersions

In this section we define a version of Floer homology for exact Lagrangian immersions. Recall that our exact Lagrangian immersions are equipped with choices of potentials making their Legendrian lifts embedded. This is not new material; similar or even more general accounts can be found, for example, in [START_REF] Akaho | Immersed Lagrangian Floer theory[END_REF], [START_REF] Alston | Exact, graded, immersed Lagrangians and Floer theory[END_REF] and [START_REF] Mak | Dehn twists exact sequences through Lagrangian cobordism[END_REF]

: [0, 1] × W → R is cylindrical if there is a function h : R + → R such that H(t, w) = h(e r(w) ) outside a compact set of W .
The following example describes the behaviour of the Hamiltonian vector field of a cylindrical Hamiltonian in an end of W , after taking into account the identification (3).

Example 4.2. Let (V, α) be a contact manifold with Reeb vector field R and let (R × V, d(e r α)) be its symplectisation. Given a smooth function h : R + → R, we define the autonomous Hamiltonian H : R × V → R by H(r, v) = h(e r ). Then the Hamiltonian vector field of H is

X H (r, v) = h (e r )R(r, v).
Let (L 0 , ι 0 ) and (L 1 , ι 1 ) be two immersed exact Lagrangian submanifolds of W with cylindrical ends over Legendrian submanifolds Λ 0 and Λ 1 of (V, α). Given a cylindrical Hamiltonian H : [0, 1] × W → R, we denote by C H -or simply C when there is no risk of confusion -the set of Hamiltonian chords and (iv) all time-one Hamiltonian chords from L 0 to L 1 are contained in a compact region.

x : [0, 1] → W of H such that x(0) ∈ L 0 and x(1) ∈ L 1 . If ϕ t denotes the Hamiltonian flow of H, then C H is in bijection with ϕ 1 (L 0 ) ∩ L 1 . Definition 4.3. A cylindrical Hamiltonian H : [0, 1] × W → R, with Hamil- tonian flow ϕ t , is compatible with L 0 and L 1 if (i) no starting point or endpoint of a chord x ∈ C H is a double point of (L 0 , ι 0 ) or (L 1 , ι 1 ), (ii) ϕ 1 (L 0 ) intersects L 1 transversely, (iii) for ρ large enough h (ρ) = λ is constant,
Condition (iv) is equivalent to asking that λ should not be the length of a Reeb chord from Λ 0 to Λ 1 .

Remark 4.4. If cylindrical Hamiltonian H is compatible with L 0 and L 1 , then C H is a finite set.
For Liouville sectors we need to modify the definition of a cylindrical Hamiltonian in order to prevent the trajectories from hitting the boundary in finite time. To that aim, the crucial notion is that of excellent dynamics, for which we refer to [23, Section 2.9]. We introduce the following terminology.

Definition 4.5. Let X be a vector field on a manifold V . We say that a subset N ⊂ V is coconvex (for X) if every finite time trajectory of the flow of X with initial and final point in

V \ N is contained in V \ N .
Excellent dynamics implies that ∂V has an coconvex collar neighbourhood N , but the converse is not true. The following lemma is proved in [23, Section 2.9]. Also, see [13, Section 4.2] for a related construction.

Lemma 4.6. Given a contact manifold (V, α), it is possible to find a function g : V → R ≥0 such that

(1) g > 0 outside the boundary ∂V , and

g ≡ 1 outside a collar neigh- bourhood ∂V × [0, δ), (2) g = t 2 G on ∂V × [0, δ)
, where G > 0 and t is the coordinate of [0, δ), and (3) the contact vector field X g with contact Hamiltonian g has excellent dynamics.

Note that X g vanishes along ∂V . It is called a cut off Reeb vector field in [START_REF] Ganatra | Covariantly functorial Floer theory on Liouville sectors[END_REF] because it is the Reeb vector field of the contact form g -1 α on int(V ).

Definition 4.7. Let S be a Liouville sector. A Hamiltonian function

H : [0, 1]× S → R is cylindrical if there is a function h : R + → R such that H(t, w) = g(w)h(e r(w) ) outside a compact set of S.
The definition of cylindrical Hamiltonian compatible with two immersed exact Lagrangian submanifolds in the case of sectors is the same as Definition 4.3. Condition (iv) is equivalent to asking that λ should not be the length of a chord of the cut off Reeb vector field.

4.2.

Obstructions. If one tries to define Floer homology for immersed Lagrangian submanifolds by extending the usual definition naively, one runs into the problem that the "differential" might not square to zero because of the bubbling of teardrops in one-dimensional families of Floer strips. Thus, if (L, ι) is an immersed Lagrangian submanifold and J is L-regular, we define a map d 0 : D → Z by d 0 (p) = #N 0 L (p; J) and extend it by linearity to the free module generated by D. The map d 0 is called the obstruction of (L, ι). If d 0 = 0 we say that (L, ι) is uncurved.

Typically, asking that an immersed Lagrangian submanifold be unobstructed is too much, and a weaker condition will ensure that Floer homology can be defined. We observe that d 0 is a component of the Chekanov-Eliashberg algebra of the Legendrian lift of L, and make the following definition. Definition 4.8. Let (L, ι) be an immersed exact Lagrangian submanifold. The obstruction algebra (D, d) of (L, ι) -or of (L, ι, J) when the almost complex structure is not clear from the context -is the Chekanov-Eliashberg algebra of the Legendrian lift L + .

If L is connected, its obstruction algebra (D, d) does not depend on the potential. On the other hand, if L is disconnected, the potential differences at the double points of L involving different connected components, and therefore what holomorphic curves are counted in (D, d), depend on the choice of the potential. Definition 4.9. An exact immersed Lagrangian (L, ι) is unobstructed if (D, d) admits an augmentation.

Weak unobstructedness does not depend on the choice of L-regular almost complex structure and is invariant under general Legendrian isotopies as a consequence of Theorem 3.7 (this fact will not be needed). However, we will need the invariance statement from the following proposition. Proposition 4.10. If L 0 and L 1 are safe isotopic exact Lagrangian immersions, and J 0 and J 1 are L 0 -regular and L 1 -regular almost complex structures respectively, then the obstruction algebras (D 0 , d 0 ) of L 0 and (D 1 , d 1 ) of L 1 are isomorphic. In particular, there is a bijection between the augmentations of (D 0 , d 0 ) and the augmentations of (D 1 , d 1 ).

If in addition J 1 = (ϕ 1 ) * J 0 is satisfied for a diffeomorphism ϕ 1 , then the bijection of double points of L 0 and L 1 induced by ϕ 1 moreover induces an isomorphism of the obstruction algebras. Sketch of proof. Let ϕ t be a smooth isotopy such that ϕ t (L 0 ) = L t is the projection of a Legendrian isotopy. For any t and > 0 there is a bijection between the double points of L t and the double points of L t+ = ψ(L t ), where ψ is a diffeomorphism which may be assumed to be arbitrarily C 2small for sufficiently small > 0. This diffeomorphism induces an isomorphism between the obstruction algebra (D 1 , d 1 ) and the obstruction algebra ( D 0 , d 0 ) of L 0 defined using the pulled-back almost complex structure dψ -1 • J 1 • dψ, which is tame since ψ is C 2 -small. Thus, to prove the proposition, it is enough to compare the obstruction algebras of L defined using different L-regular almost complex structures.

Let (D, d) and (D, d) be obstruction algebras of L defined using L-regular almost complex structures J and J respectively. We will sketch a bifurcation analysis argument to define an isomorphism

Y : (D, d) → (D, d);
for more details see [START_REF] Ekholm | Orientations in Legendrian contact homology and exact Lagrangian immersions[END_REF]Lemma 4.21]). We fix a path J • of almost complex structures satisfying ( †) such that J 0 = J and J 1 = J, and define the

parametrised moduli spaces N k L (p o ; p 1 , . . . , p d ; J • ) consisting of pairs (δ, u) such that δ ∈ [0, 1] and u ∈ N k L (p o ; p 1 , . . . , p d ; J δ ).
For a generic path J • there is a finite set of "bifurcation instants" δ 1 < . . . < δ n and intersection points q i 0 , . . . , q i d i (for i = 1, . . . , n) such that the unique nonempty moduli spaces of negative index are N -1 L (q i 0 ; q i 1 , . . . , q i d i ; J δ i ) for i = 1, . . . , n, each of which consists of a finite number of points.

First, assume that there is a unique bifurcation point δ ∈ (0, 1) in the path J • . Then Y is defined on the generators as ( 8)

Y(p) = p if p = q 0 , q 0 + #N -1 L (q 0 ; q 1 , . . . , q d ; J δ )q 1 . . . q d if p = q 0 .
If J • has more that one bifurcation instant, we split it into pieces, each of which has a unique bifurcation instant, and we compose the isomorphisms induced by each piece.

Proposition 4.10 is the main reason why we have made the choice of distinguishing between the obstruction algebra of a Lagrangian immersion and the Chekanov-Eliashberg algebra of its Legendrian lift. In fact Legendrian submanifolds are more naturally considered up to Legendrian isotopy. However, in this article we will consider immersed Lagrangian submanifolds only up to the weaker notion of safe isotopy.

Remark 4.11. Presumably one can prove that, up to homotopy, Y is independent of the choice of path between J and J by doing bifurcation analysis for homotopies of paths. However, this might require some form of abstract perturbations as in [START_REF] Ekholm | A version of rational SFT for exact Lagrangian cobordisms in 1-jet spaces[END_REF].

4.3. The differential. We denote by Z the strip R × [0, 1] with coordinates (s, t). Let R l 0 |l 1 ∼ = Conf l 0 (R) × Conf l 1 (R) be the set of pairs (ζ 0 , ζ 1 ) such that ζ 0 = {ζ 0 1 , . . . , ζ 0 l 0 } ⊂ R × {0} and ζ 1 = {ζ 1 l 1 , . . . , ζ 1 1 } ⊂ R × {1}.
We assume that the s-coordinates of ζ i j are increasing in j for ζ 0 j and decreasing for ζ 1 j . We define

Z ζ 0 ,ζ 1 := Z \ {ζ 0 1 , . . . , ζ 0 l 0 , ζ 1 1 , . . . , ζ 1 l 1 }. The group Aut(Z) = R acts on R l 0 |l 1 .
In the rest of this section we will assume that H is compatible with (L 0 , ι 0 ) and (L 1 , ι 1 ). We will consider time-dependent almost complex structure J • on W which satisfy the following ( † †) (i) J t is compatible with θ for all t, (ii) J t = J 0 for t ∈ [0, 1/4] in a neighbourhood of the double points of L 0 and J t = J 1 for t ∈ [3/4, 1] in a neighbourhood of the double points of L 1 , (iii) J 0 satisfies ( †) for L 0 and J 1 satisfies ( †) for L 1 .

Condition (ii) is necessary to ensure that J • is independent of the coordinate σ - i,j in some neighbourhoods of the boundary punctures ζ i j , so that we can apply standard analytical results.

For the same reason we fix once and for all a nondecreasing function χ : R → [0, 1] such that Given Hamiltonian chords x + , x -∈ C and self-intersections p 0 1 , . . . , p 0 l 0 of L 0 and p 1 1 , . . . , p 1 l 1 of L 1 we define the moduli space

χ(t) = 0 for t ∈ [0, 1/4], 1 for t ∈ [3/4,
M L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) of triples (ζ 0 , ζ 1 , u) such that • (ζ 0 , ζ 1 ) ∈ R l 0 |l 1 and u : Z ζ 0 ,ζ 1 → W is a map satisfying the Floer equation (9) ∂u ∂s + J t ∂u ∂t -χ (t)X H (χ(t), u) = 0, • lim s→±∞ u(s, t) = x ± (χ(t)) uniformly in t, • u(s, 0) ∈ L 0 for all (s, 0) ∈ Z ζ 0 ,Z ζ 1 , • u(s, 1) ∈ L 1 for all (s, 1) ∈ Z ζ 0 ,Z ζ 1 ,
• lim

z→ζ i j u(z) = p i j ,

and

• ζ i j is a negative puncture at p i j for i = 0, 1 and j = 1, . . . , l i .

The group Aut(Z) = R acts on M L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) by reparametrisations. We will denote the quotient by

M L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ). For u ∈ M L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • )
, we denote by F u the linearisation of the Floer operator

F(u) = ∂u ∂s + J t ∂u ∂t -χ (t)X H (χ(t), u)
at u. By standard Fredholm theory, F u is a Fredholm operator with index ind(F u ). We define ind(u) = ind(F u ) + l 0 + l 1 . The index is locally constant, and we denote by

M k L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) the subset of M L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) consisting of classes of maps u with index ind(u) = k.
Observe that similar construction for closed, exact, graded, immersed Lagrangian submanifolds was considered by Alston and Bao in [START_REF] Alston | Exact, graded, immersed Lagrangians and Floer theory[END_REF], where the regularity statement appears in [5, Proposition 5.2] and compactness is discussed in [5, Section 4]. In addition, the corresponding statement in the case of Legendrian contact homology in P ×R was proven by Ekholm, Etnyre and Sullivan, see [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF]Proposition 2.3]. The following proposition translates those compactness and regularity statements to the settings under consideration. Proposition 4.12. For a generic time-dependent almost complex structure J • satisfying ( † †), for which moreover J 0 is L 0 -regular and J 1 is L 1 -regular, the moduli space

M k L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) is a transversely cut-out manifold of dimension k -1. If k = 1 it is com- pact,
and therefore consists of a finite number of points. If k = 2 it can be compactified in the sense of Gromov-Floer.

The boundary of the compactification of the one-dimensional moduli space

M 2 L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) is y∈C H 0≤h i ≤l i M 1 L 0 ,L 1 (p 1 h 1 +1 , . . . , p 1 l 1 , y, p 0 1 , . . . , p 0 h 0 , x + ; H, J • ) (10) × M 1 L 0 ,L 1 (p 1 1 , . . . , p 1 h 1 , x -, p 0 h 0 +1 , . . . , p 0 l 0 , y; H, J • ) q∈D 1 0≤i≤j≤l 1 M 1 L 0 ,L 1 (p 1 1 , . . . , p 1 i , q, p 1 j+1 , . . . p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) × N 0 L 1 (q; p 1 i+1 , . . . , p 1 j ; J 1 ) q∈D 0 0≤i≤j≤l 0 M 1 L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 i , q, p 0 j+1 , . . . p 0 l 0 , x + ; H, J • ) × N 0 L 0 (q; p 0 i+1 , . . . , p 0 j ; J 0 ).
If both L 0 and L 1 are spin the moduli spaces are orientable, and a choice of spin structure on each Lagrangian submanifold induces a coherent orientation on the moduli spaces.

Remark 4.13. We use different conventions for the index of maps involved in the definition of the obstruction algebra and for maps involved in the definition of Floer homology. Unfortunately this can cause some confusion, but it is necessary to remain consistent with the standard conventions in the literature.

Definition 4.14. We say that a time-dependent almost complex structure J • on W is (L 0 , L 1 )-regular if it satisfies ( † †) for L 0 and L 1 and all moduli spaces M k L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) are transversely cut out.

Note that, strictly speaking, the condition of being (L 0 , L 1 )-regular depends also on the Hamiltonian, even if we have decided to suppress it from the notation.

Suppose that the obstruction algebras D 0 and D 1 admit augmentations ε 0 and ε 1 respectively. To simplify the notation we write

p i = (p i 1 , . . . , p i l i ), M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, J • ) = M k L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H, J • ) and ε i (p i ) = ε i (p 1 1 ) . . . ε i (p i l i ) for i = 0, 1.
We also introduce the weighted count

m(p 1 , x -, p 0 , x + ) = #M 0 L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, J • )ε 0 (p 0 )ε 1 (p 1 ).
Then we define the Floer complex over the commutative ring

F CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J • ) = x∈C Fx with differential ∂ : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J • )
defined on the generators by ( 11)

∂x + = x -∈C H l 0 ,l 1 ∈N p i ∈D l i i m(p 1 , x -, p 0 , x + )x -.
The algebraic interpretation of the Gromov-Floer compactification of the one-dimensional moduli spaces in Proposition 4.12 is that ∂ 2 = 0. We will denote the homology by HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H). The suppression of J • from the notation is justified by Subsection 5.1.

Comparison with bilinearised Legendrian contact cohomology.

In this subsection we compare the Lagrangian Floer homology of a pair of immersed exact Lagrangian submanifolds with the bilinearised Legendrian contact cohomology of a particular Legendrian lift of theirs.

Let L 0 and L 1 be exact Lagrangian immersions, H : [0, 1]×W → R a Hamiltonian function compatible with L 0 and L 1 with Hamiltonian flow ϕ t , and J • an (L 0 , L 1 )-regular almost complex structure. We will introduce the "backward" isotopy

ϕ t = ϕ 1 • ϕ -1 t , where ϕ 1 • ϕ -1 t • ϕ -1 1 can be generated by the Hamiltonian -H(t, ϕ -1 t • ϕ -1 1 ).
Given an almost complex structure J • and an arbitrary path φ t of symplectomorphisms, we denote by φ * J • the almost complex structure defined as

φ * J t = dφ χ(t) •J t •dφ -1 χ(t)
. The time rescaling by χ ensures that φ * J • satisfies ( † †) for φ 1 (L 0 ) and L 1 if and only if J • does for L 0 and L 1 .

Lemma 4.15. Denote by 0 the constantly zero function on W and set q ± = x ± (1) ∈ ϕ 1 (L 0 ) ∩ L 1 , regarded as Hamiltonian chords of 0. There is a bijection Given two Legendrian submanifolds Λ 0 and Λ 1 in (W × R, dz + θ) = (M, β), we say that Λ 0 is above Λ 1 if the z-coordinate of any point of Λ 0 is larger than the z-coordinate of any point of Λ 1 .

ϕ * : M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, J • ) → M ϕ 1 (L 0 ),L 1 (p 1 , q -, ϕ 1 (p 0 ), q + ; 0, ϕ * J • ) defined by (ϕ * u)(s, t) = ϕ t (u(s, t)), and moreover J • is (L 0 , L 1 )-regular if and only if ϕ * J • is (ϕ 1 (L 0 ), L 1 )
Lemma 4.16. Let L 0 and L 1 be immersed exact Lagrangian submanifolds of (W, θ) and H a cylindrical Hamiltonian compatible with L 0 and L 1 . We denote by ϕ t the Hamiltonian flow of H and L 0 = ϕ 1 (L 0 ). We choose Legendrian lifts of L 0 and L 1 to Legendrian submanifolds L + 0 and

L + 1 of (M, β) such that L + 0 is above L + 1 . If J is an L-regular almost complex structure on W for L = L 0 ∪ L 1 , let J • = (ϕ -1 t ) * J.
For every pair of augmentations ε 0 and ε 1 of the obstruction algebras of (L 0 , J 0 ) and (L 1 , J 1 ) respectively, there is an isomorphism of complexes

CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J • ) ∼ = LCC ε 0 ,ε 1 ( L + 0 , L + 1 ; J) where ε 0 = ε 0 • ϕ -1
1 is an augmentation of the obstruction algebra of ( L 0 , J).

Proof. By Lemma 4.15 there is an isomorphism of complexes

CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J • ) ∼ = CF (( L 0 , ε 0 ), (L 1 , ε 1 ); 0, J).
By definition the obstruction algebras of L 0 and L 1 are isomorphic to the Chekanov-Eliashberg algebras of L + 0 and L + 1 . Moreover, the chain complexes CF (( L 0 , ε 0 ), (L 1 , ε 1 ); 0, J) and LCC ε 0 ,ε 1 ( L + 0 , L + 1 ; J) are both generated by intersection points between L 0 and L 1 . Thus it remains only to match the differentials.

For any i = 1, . . . , d and

ζ = {ζ 0 , . . . , ζ d } ∈ R d there is a biholomor- phism ∆ ζ ∼ = Z ζ 0 ,ζ 1 , where ζ 0 = {ζ 1 , . . . , ζ i-1 }, ζ 1 = {ζ i+1 , . . . , ζ d }, ζ i is mapped to s = +∞ and ζ 0 is mapped to s = -∞.
Such biholomorphisms induce bijections between the moduli spaces defining the boundary of CF (( L 0 , ε 0 ), (L 1 , ε 1 ); 0, J) and the moduli spaces defining the boundary of LCC ε 0 ,ε 1 ( L + 0 , L + 1 ; J).

4.5.

Products. After the work done for the differential, the higher order products can be easily defined. For simplicity, in this section we will consider immersed exact Lagrangian submanifolds L 0 , . . . , L d which are pairwise transverse and cylindrical over chord generic Legendrian submanifolds. Thus the generators of the Floer complexes will be intersection points, which we will assume to be disjoint from the double points. The routine modifications needed to introduce Hamiltonian functions into the picture are left to the reader.

Given d ≤ 2 and l i ≥ 0 for i = 0, . . . , d we define i to ζ m i+1 . We will consider also a (generic) domain dependent almost complex structure J • such that every J z , z ∈ ∆ z satisfies ( †), and moreover is of the form ( † †) at the strip-like ends of the mixed punctures and is constant in a neighbourhood of the arcs ∂ i ∆ ζ outside those strip-like ends.

R l 0 |...|l d = Conf l 0 +...

Finally we define the moduli spaces

M L 0 ,...,L d (p d , x 0 , p 0 , x 1 , . . . , p d-1 , x d ; J • ) of pairs (u, ζ) (up to action of Aut(D 2 ))
, where:

• ζ ∈ R l 0 |...|l d and u : ∆ ζ → W satisfies du + J • • du • i = 0, • u(∂ i ∆ ζ ) ⊂ L i , • lim z→ζ m i u(z) = x i , • lim z→ζ i j u(z) = p i j ,

and

• p i j is a negative puncture at ζ i j for i = 0, . . . , d and j = 1, . . . , l i .

As usual, we denote by M 0 L 0 ,...,L d (p d , x 0 , p 0 , x 1 , . . . , p d-1 , x d ; J • ) the zerodimensional part of the moduli spaces.

If ε 0 , . . . , ε d are augmentations for the corresponding Lagrangian immersions, we define the weighted count

m(p d , x 0 , p 0 , x 1 , . . . , p d-1 , x d ) = #M 0 L 0 ,...,L d (p d , x 0 , p 0 , x 1 , . . . , p d-1 , x d ; J • )ε 0 (p 0 ) . . . ε d (p d ) and define a product (12) µ d : CF (L d-1 , L d ) ⊗ . . . ⊗ CF (L 0 , L 1 ) → CF (L 0 , L d ),
where we wrote CF (L i , L j ) instead of CF ((L i , ε i ), (L j , ε j )) for brevity sake, via the formula

µ d (x 1 , . . . , x d ) = x 0 ∈L 0 ∩L d l i ≥0 p i j ∈D l i i m(p d , x 0 , p 0 , x 1 , . . . , p d-1 , x d )x 0 .
Remark 4.17. The operations µ d satisfy the A ∞ relations.

The following lemma will be useful in Section 8. It is a straight forward corollary of the existence of pseudoholomorphic triangles supplied by Corollary 4.22 below. The only point where we need to take some care is due to the fact that the Weinstein neighbourhood considered is immersed.

Lemma 4.18. Let (L, ι) be an exact Lagrangian immersion. We extend ι to a symplectic immersion ι * : (D δ T * L, dq ∧ dp) → (W, dθ). Let (L, ι ) be safe isotopic to (L, ι) and, moreover, assume that

• there exists a sufficiently C 1 -small Morse function g : L → R with local minima e i , all whose critical points are disjoint from the double points of L, such that ι = ι * • dg, and • outside of some compact subset L is obtained by a small perturbation of L by the negative Reeb flow.

We will denote L = (L, ι) and L = (L, ι ). Then, if L admits an augmentation ε and ε is the corresponding augmentation of L , for every cylindrical exact Lagrangian submanifold T such that ι -1 * (T ) is a union of cotangent disc fibres, the map

µ 2 (e, •) : CF (T, (L, ε)) → CF (T, (L , ε )), e := i e i ∈ CF ((L, ε), (L , ε )),
is an isomorphism of complexes for a suitable almost complex structure on W as in Corollary 4.22.

In the case when L is closed and embedded, the element e is always a cycle which is nontrivial in homology as was shown by Floer (it is identified with the minimum class in the Morse cohomology of L). In general the following holds. Proof. The assumption that the augmentation ε is identified canonically with the augmentation ε implies that e is a cycle by the count of pseudoholomorphic dics with a negative puncture at e supplied by Lemma 4.21.

Assume that ∂E = e. The last property is then an algebraic consequence of the Leibniz rule ∂µ 2 (E, x) = µ 2 (∂E, x) in the case when ∂x = 0, combined with the fact that µ 2 (e, •) is a quasi-isomorphism as established by the previous lemma.

Later it will be useful to switch perspectives slightly, and instead of with the chain e, work with an augmentation induced by that chain. In general, given (L 0 , ε 0 ), (L 1 , ε 1 ) and a chain c ∈ CF ((L 0 , ε 0 ), (L 0 , ε 1 )) we can consider Legendrian lifts L + 0 and L + 1 such that L + 0 is above L + 1 and the unital algebra morphism ε c :

A(L + 0 ∪ L + 1 ) → F uniquely determined ε c (x) := ε i (x), if x ∈ A(L i ), c, x , if x ∈ L 0 ∩ L 1 ,
where

•, • is the coefficient of x in c. Lemma 4.20. The element c ∈ CF ((L 0 , ε 0 ), (L 1 , ε 1 ); 0, J)
is a cycle if and only if

ε c : A(L + 0 ∪ L + 1 )
→ F is an augmentation, where the almost complex structure J has been used to define the latter algebra as well, and the Legendrian lifts have been chosen so that no Reeb chord starts on L + 0 and ends on L + 1 .

Proof. Note that the Floer complex under consideration has a differential which counts J-pseudoholomorphic strips, and that the obstruction algebra has a differential counting pseudoholomorphic discs with at least one boundary puncture. Identifying the appropriate counts of discs, the statement can be seen to follow by pure algebra, together with the fact that the differential of the DGA counts punctured pseudoholomorphic discs, and thus respects the filtration induced by the different components. The crucial property that is needed here is that, under the assumptions made on the Legendrian lifts, the differential of the Chekanov-Eliashberg algebra applied to a mixed chord is a sum of words, each of which contains precisely one mixed chord.

4.6. Existence of triangles. In this section we prove an existence result for small pseudoholomorphic triangles with boundary on an exact Lagrangian cobordism and a small push-off. The existence of these triangles can be deduced as a consequence of the fact that the wrapped Fukaya category is homologically unital. Here we take a more direct approach based upon the adiabatic limit of pseudoholomorphic discs on a Lagrangian and its pushoff from [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF]; when the latter push-off becomes sufficiently small, these discs converge to pseudoholomorphic discs on the single Lagrangian with gradientflow lines attached (called generalised pseudoholomorphic discs in the same paper).

Let L ⊂ W be an exact immersed Lagrangian submanifold with cylindrical end. We recall that, as usual, we assume that every immersed Lagrangian submanifold is nice. Consider the Hamiltonian push-off L f , which we require to be again an exact immersed Lagrangian submanifold with cylindrical end, which is identified with the graph of d( f ) for a Morse function

f : L → R inside a Weinstein neighbourhood (T * δ L, -d(pdq)) (W, dθ) of L.
We further assume that df (L) > 0 outside of a compact subset. (The assumption that the push-off is cylindrical at infinity does of course impose additional constraints on the precise behaviour of the Morse function outside of a compact subset.) Now consider a Legendrian lift L + ∪ L + f for which (L f ) + is above L + . For > 0 sufficiently small, it is the case that L ∪ L f again has only transverse double points. Moreover, the Reeb chords on the Legendrian lift can be classified as follows, using the notation from [18, Section 3.1]:

• Reeb chords Q(L) ∼ = Q(L f )
on the lifts of L and L f respectively, which stand in a canonical bijection; • Reeb chords C being in a canonical bijection with the critical points of f ; and • two sets Q and P of Reeb chords from L to L f , each in canonical bijection with Q(L), and where the lengths of any Reeb chord in Q is greater than the length of any Reeb chord in P.

See the aforementioned reference for more details, as well as Figure 1.

Lemma 4.21 ([18]). For a suitable generic Riemannian metric g on L for which (f, g) constitutes a Morse-Smale pair and associated almost complex structure, which can be made to coincide with an arbitrary cylindrical almost complex structure outside of a compact subset, there is a bijection between the set of pseudoholomorphic discs which have

• boundary on L + ∪ L + f and precisely one positive puncture, • at least one negative puncture at a local minimum e ∈ C of f , and • form a moduli space of expected dimension zero,

c q c e max p c c L f L x x L + (L f ) + y z c c q c p c
Figure 1. The small triangles on the two-copy living near gradient flow-lines of -∇f shown in red. The upper copy with respect to the z-coordinate is shown in blue, while the lower copy is in black. The contact form used here is dz-ydx.

and the set of negative gradient flow-lines on (L, g) that connect either the starting point or the end point of a Reeb chord c ∈ Q(L) with the local minimum e, together with the set of negative gradient flow-lines that connect some critical point of index one with e.

More precisely, each such pseudoholomorphic disc lives in an small neighbourhood of the aforementioned flow-line. In the first case, it is a triangle with a positive puncture at the Reeb chord q c ∈ Q corresponding to c, and its additional negative punctures at e and either c (for the flow-line from the starting point of c) or c (for the flow-line from the endpoint of c); see Figure 1. In the second case, it is a Floer strip corresponding to the negative gradient flow-line connecting the saddle point and the local minimum.

Proof. This is an immediate application of Parts (3) and ( 4) of [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF]Theorem 5.5]. A generalised pseudoholomorphic disc with a negative puncture at a local minimum can be rigid only if it a flow-line connecting a saddle point to the minimum, or consists of a constant pseudoholomorphic disc located at one of the Reeb chords at c ∈ Q(L) together with a flow-line from that double point to the local minimum. The aforementioned result gives a bijection between such generalised pseudoholomorphic discs and pseudoholomorphic strips and triangles on the two-copy.

Now consider an auxiliary exact immersed Lagrangian L intersecting L∪L f transversely. For > 0 sufficiently small there is a bijection between the intersection points L ∩ L and L f ∩ L . Proof. We need to apply Lemma 4.21 in the case when L + is taken to be the Legendrian lift (L ∪ L f ) + , where L + f is above L + , and the push-off is taken to be (L ∪ L ) + F for a Morse function F : L ∪ L → R that restricts to f on L.

Continuation maps

In this section we analyse what happens to the Floer homology when we change J, H (in some suitable way) or move the Lagrangian submanifolds by a compactly supported safe exact isotopy. 5.1. Changing the almost complex structure. Following [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF] (see also [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF]) we will use the bifurcation method to prove invariance of Floer homology for Lagrangian intersection under change of almost complex structure. It seems, in fact, that the more usual continuation method is not well suited to describe how the obstruction algebras change when the almost complex structure changes.

Let us fix Lagrangian immersions (L 0 , ι 0 ) and (L 1 , ι 1 ) and a cylindrical Hamiltonian H compatible with L 0 and L 1 . For a generic one-parameter family of time-dependent almost complex structure J •

• parametrised by an interval [δ -, δ + ] such that

• the extrema J δ - • and J δ +
• are (L 0 , L 1 )-compatible, and

• J δ • satisfies ( † †) for all δ ∈ [δ -, δ + ]
we define the parametrised moduli spaces

M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, J • • ) consisting of pairs (δ, u) such that δ ∈ [δ -, δ + ] and u ∈ M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, J δ • ).
Using the zero-dimensional parametrised moduli spaces, we will define a continuation map

Υ J • • : LCC((L 0 , ε + 0 ), (L 1 , ε + 1 ); H, J δ + • ) → LCC((L 0 , ε - 0 ), (L 1 , ε - 1 ); H, J δ - •
). Proposition 5.1. For a generic one-parameter family J •

• of time-dependent almost complex structures as above, the parametrised moduli space

M k L 0 ,L 1 (q 1 , y -, q 0 , y + ; H, J • • ) is a transversely cut-out manifold of dimension k. If k = 0 it
is compact, and therefore consists of a finite number of points. If k = 1 it can be compactified in the sense of Gromov-Floer.

If both L 0 and L 1 are spin, the moduli spaces are orientable, and a choice of spin structure on each Lagrangian submanifold induces a coherent orientation on the parametrised moduli spaces.

In the following lemma we look more closely at the structure of the zerodimensional parametrised moduli spaces. The analogous statement in the setting of Lagrangian Floer homology (for Lagrangian submanifolds) appears in [20, Section 3]. In the case of Legendrian contact homology, the corresponding construction appears in [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF]Section 2.4].

Lemma 5.2. For a generic J •

• there is a finite set ∆ ⊂ (δ -, δ + ) such that for δ ∈ ∆ exactly one of the following cases holds:

(i) there is a unique nonempty moduli space N -1 L 0 (q 0 0 ; q 0 1 , . . . , q 0 d ; J δ 0 ) and all other moduli spaces are transversely cut out, (ii) there is a unique nonempty moduli space N -1 L 1 (q 1 0 ; q 1 1 , . . . , q 1 d ; J δ 1 ) and all other moduli spaces are transversely cut out, or (iii) there is a unique nonempty moduli space M 0 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H, J δ • ) and all other moduli spaces are transversely cut out, while for every δ ∈ [δ -, δ + ] \ ∆ the moduli spaces of negative virtual dimension are empty.

(Of course, the self-intersection points q i j appearing in the three cases above have nothing to do with each other.) Note that the lemma does not claim that J δ

• is (L 0 , L 1 )-regular for δ ∈ ∆: for example, if δ is a critical value of the projection Next we describe what happens when we cross δ ∈ ∆ of type (i) or (ii). Since the cases are symmetric, we will describe only (i).

M 1 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H, J • • ) → [δ -, δ + ], then J δ • is not (L 0 , L 1 )-regular, but δ ∈ ∆. Remark 5.3. If ∆ = ∅, then the complexes CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J δ - • ) and CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J
Lemma 5.4. Suppose that ∆ = {δ 0 } and that the unique nontransversely cut out moduli space for J δ 0

• is N -1 L 0 (q 0 0 ; q 0 1 , . . . , q 0 d ; J δ 0 0 ). Then the identity is an isomorphism of complexes between CF ((L 0 , ε - 0 ), (L 1 , ε - 1 ); H, J

δ - • ) and CF ((L 0 , ε + 0 ), (L 1 , ε + 1 ); H, J δ + • ).
Proof. By Proposition 4.10 there are isomorphisms

Y i : (D - i , d - i ) → (D + i , d + i ) and, given augmentations ε + i for (D + i , d + i ), we define ε - i = ε + i • Y i .
For brevity, we denote

CF ± = CF ((L 0 , ε ± 0 ), (L 1 , ε ± 1 ); H, J δ ± • ). We decompose the differential ∂ ε ± 0 ,ε ± 1 : CF ± → CF ± as ∂ ε ± 0 ,ε ± 1 = (ε ± 0 ⊗ Id ⊗ ε ± 1 ) • ∂ ± ,
where

∂ ± : CF ± → D ± 0 ⊗ CF ± ⊗ D ±
1 is the antecedent of the differential defined by counting the same moduli spaces as in Equation ( 11), but without augmenting the self-intersection points. The lemma then follows from the commutativity of the following diagram

CF - ∂ -/ / Id D - 1 ⊗ CF -⊗ D - 0 Y 1 ⊗Id⊗Y 0 ε - 1 ⊗Id⊗ε - 0 / / CF - Id CF + ∂ -/ / D + 1 ⊗ CF + ⊗ D + 0 ε + 1 ⊗Id⊗ε + 0 / / CF +
where the first square is commutative because of the structure of the boundary of the one-dimensional parametrised moduli spaces for J • • , while the second square commutes because of the relationship between the augmentations.

Finally, we analyse how the complex changes when we cross δ ∈ ∆ of type (iii). Lemma 5.5. Suppose that ∆ = {δ 0 } and that the unique nontransversely cut out moduli space for J δ 0

• is M 0 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H, J δ • ). Then the map

Υ J • • : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J δ + • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J δ - • ) defined as Υ J • • (x) = x if x = y + , y + + #M 0 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H, J • • )ε 0 (q 0 )ε 1 (q 1 )y -if x = y + is an isomorphism of complexes.
Proof. The proof is the same as in [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF]. However, the proof in [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF] holds only in the case of Z 2 -coefficients. For more general coefficients, we rely on the discussion in [START_REF] Ekholm | Orientations in Legendrian contact homology and exact Lagrangian immersions[END_REF][START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF].

Given a generic homotopy J •

• , we split it into pieces containing only one point of ∆ and compose the maps obtained in Lemma 5.4 and 5.5.

Changing the Hamiltonian.

In this section we will keep the almost complex structure fixed. Let H -and H + be time-dependent cylindrical Hamiltonian functions which are compatible with immersed Lagrangian submanifolds L 0 and L 1 . From a one-parameter family of cylindrical Hamiltonians H s such that (i) H s = H -for s 0, (ii) H s = H + for all s 0, and (iii) ∂ s h s (e r(w) ) ≤ 0 if r(w) is sufficiently large, we will define a continuation map

Φ Hs : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H + ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H -).
Given a time-dependent almost complex structure J • , an H --Hamiltonian chord x -, an H + -Hamiltonian chord x + and double points p 0 = (p 0 1 , . . . , p 0 l 0 ) of L 0 and p 1 = (p 1 1 , . . . , p 1 l 1 ) of L 1 , we define the moduli spaces

M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J • )
as the set of triples (ζ 0 , ζ 1 , u) such that:

• (ζ 0 , ζ 1 ) ∈ R l 0 |l 1 and u : Z ζ 0 ,ζ 1 → W is a map satisfying the Floer equation (13) ∂u ∂s + J t ∂u ∂t -χ (t)X Hs (χ(t), u) = 0, • lim s→±∞ u(s, t) = x ± (χ(t)), • u(s, 0) ∈ L 0 for all (s, 0) ∈ Z ζ 0 ,ζ 1 , • u(s, 1) ∈ L 1 for all (s, 1) ∈ Z ζ 0 ,ζ 1 ,

and

• each ζ i j is a negative puncture at p i j for i = 0, 1 and j = 1, . . . , l i .

Note the only difference between Equation (13) and Equation ( 9) is that we made X Hs depend on s in Equation [START_REF] V. Colin | Sutures and contact homology I[END_REF]. For this reason there is no action of Aut(Z) on the moduli spaces

M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J • ).
Let F u be the linearisation at u of the Floer operator with s-dependent Hamiltonian. We define

ind(u) = ind(F u ) + l 0 + l 1 ,
and define

M k L 0 ,L 1 (p 0 , x -, p 1 , x + ; H s , J • ) as the subset of M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J • )
consisting of the maps u with ind(u) = k.

The following statement is analogous to the statement in Morse theory, [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF]Section 3]. A similar boundary degeneration statement in the case of Legendrian contact homology appears in [17, Section 2.4].

Proposition 5.6. Given H s , for a generic time-dependent almost complex structure J • satisfying ( † †) with respect to both H + and H -, the moduli space M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J • ) is a transversely cut-out manifold of dimension k. If k = 0 it is compact, and therefore consists of a finite set of points. If k = 1 it can be compactified in the sense of Gromov-Floer.

If both L 0 and L 1 are spin, the choice of a spin structure on each induces a coherent orientation.

We denote C -the set of Hamiltonian chords of H -and C + the set of Hamiltonian chords of H + . We also introduce the weighted count

m(p 1 , x -p 0 , x + ; H s ) = #M 0 L 0 ,L 1 (p 1 , x -p 0 , x + ; H s , J • )ε 0 (p 0 )ε 1 (p 1
). Given x + ∈ C + , we define the continuation map [START_REF] Rizell | Legendrian ambient surgery and Legendrian contact homology[END_REF] Φ Hs (x

+ ) = x -∈C -l 0 ,l 1 ∈N p i ∈D l i i m(p 1 , x -p 0 , x + ; H s )x -.
The Gromov -Floer compactification of the one-dimensional moduli spaces implies the following lemma.

Lemma 5.7. The map Φ Hs is a chain map.

We denote by Φ * H -,H + : HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H + ) → HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H -) the map induced in homology by Φ Hs -soon it will be apparent that the notation is justified. As it happens in the more standard Floer homology for Lagrangian submanifolds, the continuation maps satisfy the following properties.

Lemma 5.8.

(1) Up to homotopy, Φ Hs depends only on the endpoints H + and H -of H s , (2) Φ * H,H is the identity for every H, and

(3) Φ * H -,H • Φ * H,H + = Φ * H -,H + .
Sketch of proof. In order to prove (1), we follow the standard procedure for defining chain homotopies in Floer theory; see [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF] for more details. Given a homotopy H δ s , δ ∈ [0, 1], between s-dependent Hamiltonian functions H 0 s and H 1 s with H δ s ≡ H -for s 0 and H δ s ≡ H + for s 0, we define the parametrised moduli spaces

M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H • s , J • ) of pairs (δ, u) such δ ∈ [0, 1] and u ∈ M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H δ s , J • ). We define the weighted count m(p 1 , x -, p 0 , x + ; H • s ) = #M -1 L 0 ,L 1 (p 1 , x -, p 0 , x + ; H • s , J • )ε 0 (p 0 )ε 1 (p 1
). Then the chain homotopy

K : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H + , J • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H -, J • ) between Φ H 0 s and Φ H 1 s is defined as K(x + ) = x -∈C -l 0 ,l 1 ∈N p i ∈D l i i m(p 1 , x -, p 0 , x + ; H • s )x -.
In order to prove (2) we can choose H s ≡ H: then the moduli space M 0 L 0 ,L 1 (p 0 , x -, p 1 , x + ; H s , J • ) consists of constant strips.

We

M 0 L 0 ,L 1 (p 1 1 , . . . , p 1 l 1 , x -, p 0 1 , . . . , p 0 l 0 , x + ; H R s , J • ) ∼ = (15) x∈C H 0≤h i ≤l i M 0 L 0 ,L 1 (p 1 h 1 +1 , . . . , p 1 l 1 , x, p 0 1 , . . . , p 0 h 0 , x + ; H + s , J • )× M 0 L 0 ,L 1 (p 1 1 , . . . , p 1 h 1 ,
x -, p 0 1+h 0 , . . . , p 0 l 0 , x; H - s , J • ) which follows from standard compactness and gluing techniques, once we know that, for any R > 0, there is R 0 such that, for all R ≥ R 0 , if

(ζ 0 , ζ 1 , u) ∈ M 0 L 0 ,L 1 (p 1 , x -p 0 , x + ; H R s , J • ), then ζ i j ∈ [-R , R
] for i = 0, 1 and j = 1, . . . , l i .

This follows from a simple compactness argument: if there is R and a sequence

R n with (ζ 0 n , ζ 1 n , u n ) ∈ M 0 L 0 ,L 1 (p 1 ,
x -, p 0 , x + ; H Rn s , J • ) and for every n there is some ζ i j ∈ [-R , R ], then the limit for n → ∞ has one level which is a solution of a Floer equation with s-invariant data and at least one boundary puncture. By index reasons this level must have index zero, but it cannot be constant because of the boundary puncture. This is a contradiction.

With lemma 5.8 at hand, we can prove the following invariance property in the usual formal way. Corollary 5.9. If H 0 and H 1 are cylindrical Hamiltonian functions which are compatible with L 0 and L 1 and such that h 0 (r(w)) = h 1 (r(w)) for w outside of a compact set, then the continuation map Φ * H 0 ,H 1 : HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H 0 ) → HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H 1 ) is an isomorphism.

5.3.

Compactly supported safe isotopies. Let ψ t : W → W be a compactly supported smooth isotopy such that ι t = ψ t • ι 1 : L 1 → W is a safe isotopy. By Lemma 2.23 there exists a local Hamiltonian G t defined on L 1 which generates the ι t and for which dG t has compact support. (Recall that G t may not extend to a single-valued function on W .)

In the following we will make the further assumption that the path

(ψ t ) * J 1 = dψ t • J 1 • dψ -1 t , t ∈ [0, 1
], consists of compatible almost complex structures. This will cause no restriction, since we only need the case when ψ t is equal to the Liouville flow, which is conformally symplectic.

Remark 5.10. In the following manner more general safe isotopies can be considered. Since it is possible to present any smooth isotopy as a concatenation of C 2 -small isotopies, it then suffices to carry out the constructions here for each step separately. Namely, since tameness is an open condition, sufficiently C 2 -small isotopies may be assumed to preserve any given tame almost complex structure. Further control near the double points can then be obtained by assuming that ψ t actually is conformally symplectic there, which can be assumed without loss of generality.

Denote by L 1 = ψ 1 • ι 1 (L 1 ) the image. By the usual abuse of notation, we will write L 1 or ψ 1 (L 1 ) instead of (L 1 , ψ 1 • ι). From now on we will assume that the Hamiltonian H is compatible both with L 0 and L 1 and with L 0 and L 1 .

By Proposition 4.10 the obstruction algebras D 1 of (L 1 , J 1 ) as well as D 1 of (L 1 , (ψ 1 ) * J 1 ) are isomorphic by a canonical isomorphism induced by ψ 1 , and therefore any augmentation ε 1 of D 1 corresponds to an augmentation ε 1 of D 1 .

We fix time-dependent almost complex structures J +

• and J - • such that

• J ± t = J 0 for t ∈ [0, 1/4], • J + t = J 1 and J - t = (ψ 1 ) * J 1 for t ∈ [3/4, 1], • J + • is (L 0 , L 1 )-regular and J - • is (L 0 , L 1 )-regular.
Given augmentations ε 0 for L 0 and ε 1 for L 1 , we will define a chain map

Ψ G : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J + • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1
); H, J - • ) using a Floer equation with moving boundary conditions. The presence of self-intersection points of L 1 makes the construction of the moduli spaces more subtle than in the usual case because, in order to have strip-like ends, we need to make the moving boundary conditions constant near the boundary punctures, and therefore domain dependent.

(1) Jn (ζ, s, t) = J + t if s > κ n + 1, (2) Jn (ζ, s, t) = J - t if s < -κ n -1, (3) Jn (ζ, s, t) = J 0 if t ∈ [0, 1/4], (4) Jn (ζ, s, t) = dψ ν ζ (s) • J 1 • dψ -1 ν ζ (s) if t ∈ [3/4, 1]
, and (5) for all φ ∈ hom(n, n -1), Jn (φ * (ζ), s, t) = Jn-1 (ζ, s, t).

Proof. We build Jn inductively on n. At each step, the map J is determined

in the complement of Conf n (R) × [-κ n -1, κ n + 1] × [1/4, 3/4]. We can extend it to Conf n (R) × Z because J (θ) is contractible.
Given ζ ∈ Conf n (R), we will denote Jζ the s-and t-dependent almost complex structure obtained by restricting J to {ζ} × Z. Given (ζ 0 , ζ 1 ) ∈ R l 0 |l 1 , we will not distinguish between ζ 1 j ∈ R×{1} and its s-coordinate, and by this abuse of notation, to (ζ 0 , ζ 1 ) ∈ R l 0 |l 1 we will associate ν ζ 1 and Jζ 1 . For simplicity, the s-and t-dependence of Jζ 1 will be omitted in writing the Floer equation.

Consider the sets

C H = {x : [0, 1] → W : x(0) ∈ L 0 , x(1) ∈ L 1 }, C H = {x : [0, 1] → W : x(0) ∈ L 0 , x(1) ∈ L 1 }.
Definition 5.13. Given x + ∈ C H , x -∈ C H , and p i ∈ D l i i for i = 0, 1 and l i ≥ 0 we define the moduli space

M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, G, J, ν)
as the set of triples (ζ 0 , ζ 1 , u) such that:

• (ζ 0 , ζ 1 ) ∈ R l 0 |l 1 and u : Z ζ 0 ,ζ 1 → W satisfies the Floer equation (16) ∂u ∂s + Jζ 1 ∂u ∂t -χ (t)X H (χ(t), u) = 0, • lim s→±∞ u(s, t) = x ± (χ(t)), • u(s, 0) ∈ L 0 for all (s, 0) ∈ Z ζ 0 ,ζ 1 , • u(s, 1) ∈ ψ ν ζ 1 (s) (L 1 ) for all (s, 1) ∈ Z ζ 0 ,ζ 1 ,

and

• each ζ 0 j is a negative puncture at p 0 j and each ζ 1 j is a negative puncture at

ψ ν ζ 1 (ζ 1 j ) (L 1 ).
(Recall that ψ t : W → W here is a smooth isotopy satisfying the assumptions made in the beginning of this section, whose restriction to L 1 in particular is the compactly supported safe isotopy generated by G : R × L → R.)

We denote by M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, G, J, ν) the set of triples (ζ 0 , ζ 1 , u) where ind(u) = k. Proposition 5.14. For a generic J as in Lemma 5.12, the moduli space M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, G, J, ν) is a transversely cut out manifold of dimension k. If k = 0 it is compact, and therefore consists of a finite set of points. If k = 1 it can compactified in the Gromov-Floer sense.

If L 0 and L 1 are spin, a choice of a spin structure on each induces a coherent orientation.

Definition 5.15. We denote the weighted count

m(p 1 , x -, p 0 , x + ; H, G) = #M 0 L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, G, J, ν)ε 0 (p 0 )ε 1 (p 1 ) and define Ψ G as Ψ G (x + ) = x -∈C H l 0 ,l 1 ∈N p i ∈D l i i m(p 1 , x -, p 0 , x + ; H, G)x -.
Remark 5.16. The word p 1 consists of double points living on the different Lagrangian immersions ψ ν ζ 1 (ζ 1 j ) (L 1 ). However, when using the pushed forward almost complex structures (ψ ν ζ 1 (ζ 1 j ) ) * J 1 , their obstruction algebras all become canonically identified with (A(L 1 ), d) defined using J 1 . This motivates our abuse of notation ε 1 for an augmentation induced by these canonical identifications.

Lemma 5.17. The map Ψ G is a chain map. Moreover, up to chain homotopy, it does not depend on the choice of ν and on the homotopy class of ψ t relative to the endpoints.

Proof. Proposition 5.14 implies that Ψ G is a chain map. The chain homotopies between the continuation maps defined using different choices of ν and ψ t are defined, as usual, by counting index -1 elements in parametric moduli spaces. We leave the standard details to the reader. Lemma 5.18. Let G 0 , G 1 : L 1 → R be local Hamiltonian functions generating the safe isotopies ψ 0 t • ι 1 and ψ 1 t • ψ 0 1 • ι 1 respectively, and let G 2 : L 1 → R be a local Hamiltonian function generating

ψ 2 t = ψ 1 2t for t ∈ [0, 1/2], ψ 2 2t-1 • ψ 1 1 for t ∈ [1/2, 1]. Then Ψ G 2 is chain homotopic to Ψ G 0 • Ψ G 1 .
The proof of Lemma 5.18 is analogous to the proof of Lemma 5.8 and is therefore left to the reader.

Corollary 5.19. If G : R × L → R satisfies dG t = 0 outside a compact subset of (0, 1) × L, then the map Ψ G induces an isomorphism in homology.

If J+

• is one (L 0 , L 1 )-regular almost complex structure and J-• is another (L 0 , ψ 1 (L 1 ))-regular almost complex structure, instead of repeating the construction of J with J±

• as starting point, we prefer to consider J assigned once and for all to the triple (L 0 , L 1 , G) and define the continuation map

CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J+ • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J- • ) as the composition Υ -• Ψ G • Υ + , where Υ + : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J+ • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J + • ), Υ -: CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J - • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H, J- • )
are the maps defined in Section 5.1.

5.4.

Commutation between continuation maps. Let H s be a homotopy from H -to H + as in Section 5.2 and J •

• a homotopy from J -1 • to J +1

• as in Section 5.1. For simplicity we will denote

CF (H ± , J ± • ) = CF ((L 0 , ε ± 0 ), (L 1 , ε ± 1 ); H ± , J ±1 • ). If ε + i = ε - i • Y i , we have defined there continuation maps Υ ± : CF (H ± , J + • ) → CF (H ± , J - • ), Φ ± : CF (H + , J ± • ) → CF (H -, J ± •
) and now we will to prove that they are compatible in the following sense.

Proposition 5.20. The diagram [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF] CF (H + , J + • )

Φ + / / Υ + CF (H -, J + • ) Υ - CF (H + , J -) Φ - / / CF (H -, J - • )
commutes up to homotopy.

Proposition 5.20 will be proved by applying the bifurcation method to the definition of the continuation maps Φ ± : i.e. we will study the parametrised moduli spaces

M 0 L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J • • ) consisting of pairs (δ, u) such that δ ∈ [0, 1] and u ∈ M 0 L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J δ • )
. For a generic homotopy J •

• , these parametrised moduli spaces are transversely cut out manifolds of dimension one. As before, there is a finite set ∆ of bifurcation points such that, for all δ ∈ ∆, there is a unique nonempty moduli space of one of the following types:

(i) N -1
L 0 (q 0 0 ; q 0 1 , . . . , q 0 d ; J δ 0 ) or N -1 L 1 (q 1 0 ; q 1 1 , . . . , q 1 d ; J δ 1 ), (ii) M 0 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H -, J δ • ) or M 0 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H + , J δ • ), (iii) M -1 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H s , J δ • ).

continuation maps for the change of almost complex structure are Υ -= Id and

Υ + (x) = x if x = y + , y + + #M 0 L 0 ,L 1 (q 1 , y -, q 0 , y + ; H + , J δ • )ε 0 (q 0 )ε 1 (q 1 )y -if x = y + .
The structure of the compactification of one-dimensional parametrised moduli spaces implies that #M 0 (q 1 p 1 , x -, p 0 q 0 , y + ; H s , J 1 • ) -#M 0 (q 1 p 1 , x -, p 0 q 0 , y + ; H s , J 0 • ) = #M 0 (q 1 , y -, q 0 , y + ; H + , J δ

• )#M 0 (p 1 , x -, p 0 , y -; H s , J 0 • ), while the cardinality of all other moduli spaces remains unchanged. This implies that Diagram (17) commutes.

We have dropped the Lagrangian labels from the notation in order to keep the formulas compact. We will do the same in the proofs of the following lemma.

Lemma 5.23. Let ∆ = {0} be of type (iii). Then Diagram (17) commutes up to homotopy.

Proof. Let M -1 L 0 ,L 1 (q 1 , y -, q + , y + ; H s , J 0 • ) be the nonempty moduli space of negative formal dimension. In this case Υ ± = Id and we define a linear map

K : CF (H + , J +1 • ) → CF (H -, J -1 • ) by K(x) = 0 if x = y + , #M -1
L 0 ,L 1 (q 1 , y -, q + , y + ; H s , J δ • )ε 0 (q 0 )ε 1 (q 1 )y -if x = y + . The structure of the boundary of the compactification of the one-dimensional parametrised moduli spaces implies that #M 0 (q 1 p 1 , x -, p 0 q 0 , y + ; H s , J 1

• ) -#M 0 (q 1 p 1 , x -, p 0 q 0 , y + ; H s , J 0

• ) = #M -1 (q 1 , y -, q 0 , y + ; H s , J δ • )#M 1 (p 1 , x -, p 0 , y -; H -, J 0 • ), and #M 0 (q 1 p 1 , y -, p 0 q 0 , x + ; H s , J 1 • ) -#M 0 (q 1 p 1 , y -, p 0 q 0 , x + ; H s , J 0 • ) = #M -1 (q 1 , y -, q 0 , y + ; H s , J δ • )#M 1 (p 1 , x -, p 0 , y -; H -, J 0 • ). From this it follows that Φ + -Φ -= ∂K + K∂.
The degenerations of type (iii)' are cancelled algebraically by the augmentations, and therefore we obtain the commutativity of the diagram (17) for a generic homotopy J •

• . Now we compare the continuation maps Φ for the change of Hamiltonian and the continuation maps Ψ for compactly supported safe isotopies of L 1 . Let G : R × L 1 → R be a local Hamiltonian function satisfying dG t = 0 outside a compact subset of (0, 1) × L 1 which generates the safe isotopy ψ t • ι 1 , and denote L 1 = ψ 1 (L 1 ). If H + and H -are two Hamiltonian functions which are compatible both with L 0 and L 1 and with L 0 and L 1 , then we have continuation maps L 0 ,L 1 (p 1 , x -, p 0 , x + ; H R s , G, J, ν), weighted by the augmentations, we obtain a homotopy between Ψ - G • Φ and Φ • Ψ + G .

Ψ ± G : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H ± , J + • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H ± , J - • ) and continuation maps Φ : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H + , J + • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H -, J + • ), Φ : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H + , J + • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H -, J + • ) induced by a homotopy of Hamiltonians H s with H s = H + for s ≥ 1 and H s = H -for s ≤ -1. Lemma 5.24. The diagram CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H + , J + • ) Φ / / Ψ + G CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H -, J + • ) Ψ - G CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H + , J + • ) Φ / / CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H -, J + • ) commutes up to homotopy. Sketch of proof. For R ∈ R we define H R s = H s-R . We define the moduli spaces M k L 0 ,L 1 (p 1 , x -, p 0 , x + ; H R s , G, J, ν)

Wrapped Floer cohomology for exact Lagrangian immersions

In this section we define wrapped Floer cohomology for unobstructed immersed exact Lagrangian submanifolds. With the preparation of the previous sections in place, the definition is not different from the usual one for Lagrangian submanifolds.

6.1. Wrapped Floer cohomology as direct limit. We start by defining wrapped Floer cohomology as a direct limit. This point of view will be useful in the vanishing theorem of the following section. A sketched of the chain level construction, following [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF], will be given in the next subsection.

Let (L 0 , ι 0 ) and (L 1 , ι 1 ) be immersed exact Lagrangian submanifolds with augmentations ε 0 and ε 1 respectively. We assume that all intersection points between L 0 and L 1 are transverse, L 0 and L 1 are cylindrical over Legendrian submanifolds Λ 0 and Λ 1 respectively, and all Reeb chords between Λ 0 and Λ 1 are nondegenerate.

For every λ ∈ R we denote by h λ : R + → R the function

(18) h λ (ρ) = 0 if ρ ∈ (0, 1], λρ -λ if ρ ≥ 1.
We smooth h λ inside the interval [4/5, 6/5] (or any sufficiently small neighbourhood of 1 independent of λ) and, by abuse of notation, we still denote the resulting function by h λ . We assume also that the resulting smooth function satisfies h (ρ) ≥ 0 for all ρ ∈ R + . We define time-independent cylindrical Hamiltonians H λ : W → R by [START_REF] Eliashberg | Weinstein manifolds revisited[END_REF] H λ (w) = h λ (e r(w) ).

For sectors, we multiply h λ (e r(w) ) by the function g as in Definition 4.7.

Hamiltonian functions of this form will be called wrapping Hamiltonian functions.

We fix a sequence of positive real number λ n such that lim n→+∞ λ n = +∞ and, for any n, λ n is not the length of a Reeb chord from Λ 0 to Λ 1 . The set of (L 0 , L 1 )-regular almost complex structures for every H λn is dense, and we pick an element J • .

By Subsection 5.2, for every m ≥ n there are continuation maps Φ λn,λm : HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λn , J • ) → HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λm , J • ) forming a direct system. Definition 6.1. The wrapped Floer cohomology of (L 0 , ε 0 ) and (L 1 , ε 1 ) is defined as

(20) HW ((L 0 , ε 0 ), (L 1 , ε 1 ); J • ) = lim -→ HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λn , J • ).
Wrapped Floer cohomology is well defined, in the sense that it is independent of the choice of the almost complex structure J • , and of the smoothing of the piecewise linear functions H λn and of the sequence λ n . Invariance of the almost complex structure follows form Proposition 5.20. Invariance of the smoothing of H λn follows from Lemma 5.8 and Corollary 5.9. Finally, if λ n → +∞ is another sequence such that no λ n is not the length of a Reeb chord from Λ 0 to Λ 1 , we can make both λ n and λ n subsequences of a diverging sequences λ n and standard properties of the direct limit give canonical isomorphisms

lim -→ HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ n , J • ) ∼ = lim -→ HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λn , J • ) ∼ = lim -→ HF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ n , J • ).
Therefore HW ((L 0 , ε 0 ), (L 1 , ε 1 )) does not depend on the sequence λ n up to isomorphism. It can also be proved that it is invariant under safe isotopies, but we will need, and prove, only invariance under compactly supported ones.

Lemma 6.2. Let G : R × L 1 → R be a local Hamiltonian function which satisfies dG t = 0 outside a compact subset of (0, 1) × L 1 and let ψ t • ι 1 be the exact regular homotopy it generates, which is assumed to be a safe isotopy.

If L 1 = ψ 1 (L 1 ), J • is an (L 0 , L 1 )-regular almost complex structure and ε 1 is the augmentation for L 1 with respect to J • corresponding to ε 1 , then there is an isomorphism

HW ((L 0 , ε 0 ), (L 1 , ε 1 ); J • ) ∼ = HW ((L 0 , ε 0 ), (L 1 , ε 1 ); J • ).
Proof. It is enough to observe that, for every n, the isomorphisms

CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λn , J • ) ∼ = CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λn , J • )
defined in Subsection 5.3 commute with the continuation maps Φ λn,λm and therefore define isomorphisms of direct systems. This follows from Lemma 5.24 and Proposition 5.20.

6.2.

A sketch of the chain level construction. Here we recall very briefly the definition of the wrapped Floer complex and the A ∞ -operations. Since Lagrangian immersions will appear only in an intermediate step of the proof of the main theorem, we will not try to make them objects of an enlarged wrapped Fukaya category. Presumably this can be done as in the embedded case, but we have not checked the details of the construction of the necessary coherent Hamiltonian perturbations.

Let L 0 and L 1 be exact Lagrangian immersions which intersect transversely and are cylindrical over chord generic Legendrian submanifolds. We fix a wrapping Hamiltonian H ≥ 0 as in Equation ( 18) such that, for every w ∈ N, the Hamiltonian wH is compatible with L 0 and L 1 (in the sense of Definition 4.3). We also fix an (L 0 , L 1 )-regular almost complex structure J • .

Let ε 0 and ε 1 be augmentations of the obstruction algebras of (L 0 , J 0 ) and (L 1 , J 1 ) respectively. Following [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF] we define the wrapped Floer chain complex as the F[q]/(q 2 )-module

(21) CW ((L 0 , ε 0 ), (L 1 , ε 1 ); J • ) = ∞ w=0 CF ((L 0 , ε 0 ), (L 1 , ε 1 ); wH, J • )[q]
with a differential µ 1 such that, on x+yq ∈ CF ((L 0 , ε 0 ), (L 1 , ε 1 ); wH, J • )[q], it is defined as

µ 1 (x + yq) = ∂x + y + Φ w (y) + (∂y)q where Φ w : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); wH, J • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); (w + 1)H, J • )
is the continuation map for the change of Hamiltonian defined in Subsection 5.2.

Remark 6.3. The endomorphism ι q (denoted ∂ q in [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]) defined as

ι q (x + yq) = y
is a chain map. However, its action in homology is trivial.

Remark 6.4. The direct sum [START_REF] Fukaya | Lagrangian intersection floer theoryanomaly and obstruction -chapter 10[END_REF] starts from w = 1 in [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]. It is equivalent to start from w = 0, when possible, by [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]Lemma 3.11]. The homology of CW ((L 0 , ε 0 ), (L 1 , ε 1 ); J • ) is isomorphic to HW ((L 0 , ε 0 ), (L 1 , ε 1 )) defined as the direct limit in Equation ( 20) by [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]Lemma 3.12].

The A ∞ -operations between wrapped Floer complexes are defined by counting pseudoholomorphic polygons with carefully constructed Hamiltonian perturbations. In the immersed case, those polygons will be allowed to have boundary punctures at double points and, as usual, must be counted with a weight coming from the augmentations. The only thing we need to know about the operations between wrapped Floer cohomology groups is that the component

µ d : CF (L d-1 , L d ) ⊗ . . . ⊗ CF (L 0 , L 1 ) → CF (L 0 , L d )
of the operation

µ d WF : CW (L d-1 , L d ) ⊗ . . . ⊗ CW (L 0 , L 1 )
→ CW (L 0 , L d ) coincides with the operation µ d defined in Equation [START_REF] Cieliebak | Symplectic homology and the Eilenberg-Steenrod axioms[END_REF]. For simplicity of notation we have dropped the augmentations from the above formulas.

A trivial triviality result

An exact Lagrangian embedding with cylindrical ends which is disjoint from the skeleton is known to have vanishing wrapped Floer cohomology. This was proven in [START_REF] Cieliebak | Symplectic homology and the Eilenberg-Steenrod axioms[END_REF]Theorem 9.11(b)] but also follows from e.g. [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]Section 5.1]. Note that the statement is false in the more general case when the Lagrangian merely is monotone. In this section we extend this classical vanishing result to our setting of exact Lagrangian immersions. We will write the details of the proof for a Liouville manifold and explain the necessary modifications for Liouville sectors in Remark 7.9. 7.1. Action and energy. In this subsection we define an action for double points of immersed exact Lagrangian submanifolds and for Hamiltonian chords and prove action estimates for various continuation maps. Let p ∈ W be a double point of a Lagrangian immersion (L, ι) with potential f . We recall that there are points p ± ∈ L characterised by ι -1 (p) = {p + , p -} and f (p + ) > f (p -). We define the action of p as

a(p) = f (p + ) -f (p -).
If L is disconnected and p is in the intersection between the images of two connected components, then a(p) depends on the choice of the potential function f , otherwise it is independent of it.

Given a holomorphic map (ζ, u) ∈ N L (p 0 ; p 1 , . . . , p d ; J), Stokes's theorem immediately yields

∆ ζ u * dθ = a(p 0 ) - d i=1 a(p i ). Since ∆ ζ u * dθ > 0 for a nonconstant J-holomorphic map, if N L (p 0 ; p 1 , . . . , p d ; J) = ∅,
we obtain [START_REF] Ganatra | Symplectic cohomology and duality for the wrapped Fukaya category[END_REF] a(p 0 ) -

d i=1 a(p i ) > 0.
Given two Lagrangian submanifolds L 0 and L 1 with potentials f 0 and f 1 and a Hamiltonian function H, we define the action of a Hamiltonian chord

x : [0, 1] → W from L 0 to L 1 as (23) A(x) = 1 0 x * θ - 1 0 H(x(t))dt + f 0 (x(0)) -f 1 (x(1)).
Note that this is the negative of the action used in [START_REF] Ritter | Topological quantum field theory structure on symplectic cohomology[END_REF].

Example 7.1. Let H : W → R be a cylindrical Hamiltonian such that H(w) = h(e r(w) ), where h : R + → R. Then a Hamiltonian chord x : [0, 1] → W from L 0 to L 1 is contained in a level set r -1 (r) and has action (24) A(x) = h (e r )e r -h(e r ) + f 0 (x(0)) -f 1 (x(1)).

The following lemma, which we prove in the more general case of the moduli spaces of Floer solutions with an s-dependent Hamiltonian, applies equally to the particular case of moduli spaces used in the definition of the Floer differential. We introduce the following notation. Given a set A and a function f : A → R, we denote f + ∞ := sup a∈A max{f (a), 0}.

Lemma 7.2. Let H s : R × [0, 1] × W → R be an s-dependent cylindrical Hamiltonian function satisfying conditions (i), (ii), and (iii) of Subsection 5.2. We make the simplifying hypothesis that

∂ s H s ≡ 0 if s ∈ [-1, 1]. If M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J • ) = ∅, then (25) A(x -) ≤ A(x + ) + 6 ∂ s H s + ∞ .
Note that Equation ( 25) is far from being sharp, but there will be no need for a sharper estimate.

Proof. Let (u, ζ 0 , ζ 1 ) ∈ M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H s , J • ).
Then, in a metric on u * T W induced by dθ and J • ,

+∞ -∞ 1 0 |∂ s u| 2 dtds = +∞ -∞ 1 0 dθ(∂ s u, ∂ t u -χ (t)X Hs (χ(t), u))dtds = Z ζ 0 ,ζ 1 u * dθ - +∞ -∞ 1 0 χ (t)dH s (χ(t)∂ s u)dtds.
Using Stokes's theorem we obtain

Z ζ 0 ,ζ 1 u * dθ = f 1 (x -(1)) -f 1 (x + (1))- f 0 (x -(0)) + f 0 (x + (0)) - 1 i=0 l i j=1 a(p i j ).
Using the equality

∂ s (H s • u) = (∂ s H s ) • u + dH s (∂ s u) we obtain +∞ -∞ 1 0 χ (t)dH s (χ(t), (∂ s u))dtds = +∞ -∞ 1 0 χ (t)∂ s (H s (χ(t), u(s, t)))dtds- +∞ -∞ 1 0 χ (t)(∂ s H s )(χ(t), u(s, t))dtds.
We can compute

+∞ -∞ 1 0 χ (t)∂ s (H s (χ(t), u(s, t)))dtds = 1 0 χ (t)H + (χ(t), x + (χ(t)))dt - 1 0 χ (t)H -(χ(t), x -(χ(t)))dt = 1 0 H + (t, x + (t))dt - 1 0 H -(t, x -(t))dt.
Thus, rearranging the equalities, we have

A(x + ) -A(x -) = +∞ -∞ 1 0 |∂ s u| 2 dtds + 1 i=0 l i j=1 a(p i j )- +∞ -∞ 1 0 χ (t)∂ s H s (χ(t)u(s, t))dsdt.
Finally, we estimate

+∞ -∞ 1 0 χ (t)(∂ s H s )(χ(t), u(s, t))dtds ≤ 6 ∂ s H s + ∞
and obtain Equation ( 25).

Corollary 7.3. The differential in CF (L 0 , L 1 ; H, J • ) decreases the action.

If ∂ s H s ≤ 0, then the continuation map Φ Hs also decreases the action. Now we turn our attention at the continuation map Ψ G defined in Subsection 5.3. Let G : R × L 1 → R be a local Hamiltonian function such that dG t = 0 outside a compact subset of (0, 1) × L 1 and let ψ t • ι 1 be the compactly supported exact regular homotopy it generates. Now assume that ψ t is a safe isotopy. Denote, as usual, L 1 = ψ 1 (L 1 ).

First we make the following remark about a special type of safe isotopy and the action of the image of the double points.

Remark 7.4. Let (L, ι) be an exact Lagrangian immersion and

ψ t : W → W a smooth isotopy. If ψ * t θ = e c(t) θ, then (1)
ψ t (L) is a safe isotopy, and (2) if p is a double point of (L, ι), then ψ t (p) is a double point of (L, ψ t •ι) whose action satisfies a(p) = e c(t) a(ψ t (p)).

Given a Hamiltonian function H : [0, 1] × W → R which is compatible both with L 0 and L 1 , as well as with L 0 and L 1 , let C H be the set of Hamiltonian chords of H from L 0 to L 1 and let C H be the set of Hamiltonian chords of H from L 0 to L 1 .

Observe that any safe Lagrangian isotopy from L 1 to L 1 induces a continuous family of potentials f s 1 . Fixing a choice of of local Hamiltonian G : R × L 1 → R generating the safe isotopy makes the potential f 1 on L 1 determined by the choice of potential f 1 on L 1 via a computation as in the proof of Lemma 2.3. Lemma 7.5. For every chords x -∈ C H and x + ∈ C H and for every sets of self-intersection points p 0 = (p 0 1 , . . . , p 0 l 0 ) of L 0 and

p 1 = (p 1 1 , . . . , p 1 l 1 ) of L 1 , if M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, G, J, ν) = ∅, then A(x -) ≤ A(x + ) + 2µ G ∞ ,
where G ∞ is the supremum norm of G and µ ≥ 0 is the measure of the subset {s ∈ R} for which

G s : L → R is not constantly zero. Proof. Consider (ζ 0 , ζ 1 , u) ∈ M L 0 ,L 1 (p 1 , x -, p 0 , x + ; H, G, J, ν). Observe that the map u : Z ζ 0 ,ζ 1 → W extends to a continuous map u : Z → W . We have: +∞ -∞ 1 0 |∂ s u| 2 dtds = Z du * θ - 1 0 +∞ -∞ ∂ ∂s (H • u) ds dt = Z u * dθ - 1 0 H(x + (t))dt + 1 0 H(x -(t))dt.
We denote by u i : R → W , for i = 0, 1, the continuous and piecewise smooth maps u i (s) = u(s, i) and use Stokes theorem:

Z u * dθ = 1 0 x * + θ - 1 0 x * -θ + R u * 0 θ - R u * 1 θ.
Let f 0 and f 1 be the potentials of L 0 and L 1 respectively, and f1 the potential of ψ 1 (L 1 ). The map u 0 takes values in L 0 , and therefore

R u * 0 θ = f 0 (x + (0)) - l 0 j=1 a(p 0 j ) -f 0 (x -(0)).
We are left with the problem of estimating R u * 1 θ, which is slightly more complicated here because u 1 (s) ∈ ψ ν ζ 1 (s) (L 1 ). We will denote ψ ν s := ψ ν ζ 1 (s) . Recall that ψ ν s is a smooth isotopy inducing a safe isotopy of L 1 generated by the local Hamiltonian function

G ν (s, w) = ν ζ 1 (s)G(ν ζ 1 (s), w), w ∈ L 1 .
We use the following trick. Consider W × R × R with the Liouville form Θ := θ + τ dσ, where σ is the coordinate of the first copy of R and τ is the coordinate in the second copy. The notation here conflicts with the use of (σ, τ ) as coordinates in the strip-like ends near the boundary punctures, but this will not cause confusion.

Consider the symplectic suspension Σ := {(x, s, t) ∈ W × R 2 ; x ∈ ψ ν s (y), y ∈ L 1 , t = -G ν (s, y).} of the isotopy ψ ν s (L 1 ), which is an exact Lagrangian immersion. This should be seen as a corrected version of the trace of the isotopy, in order to make it Lagrangian.

Lift u 1 : R → W to ũ1 : R → W × R × R by defining

ũ1 (s) = (u 1 (s), s, -G ν (s, u 1 (s))) ∈ Σ,
and where u 1 is the lift of u 1 to L which is smooth away from the punctures.

Using the computation in the proof of Lemma 2.3, together with the Lagrangian condition satisfied by Σ, we obtain

+∞ -∞ u * 1 Θ = f 1 (x + (1)) - l 1 j=1 a(p 1 j ) -f 1 (x -(1)),
as well as

u * 1 θ -u * 1 Θ = G ν (s, u 1 (s))dσ.
Observe that we here abuse notation, and use a(p 1 j ) > 0 for the action computed with respect to the induced potential function on ψ ν s (L 1 ) for the corresponding value of s ∈ R.

Since ν ζ 1 ∞ ≤ 2, we finally obtain A(x -) ≤ A(x + ) + 2µ G ∞ ,
where µ ≥ 0 is as required.

ρ 1 e R λe R -λ h λ,Λ,R (ρ) ≡ Λ h λ,Λ,R (ρ) ≡ λ h λ,Λ,R (ρ)
Figure 2. The graph of h λ,Λ,R .

7.2. Pushing up. In this subsection we prove the following proposition.

Proposition 7.6. Let (L 0 , ι 0 ) and (L 1 , ι 1 ) be exact Lagrangian immersions in a Liouville manifold (W, θ) and let J • be an (L 0 , L 1 )-regular almost complex structure. If the Liouville flow of (W, θ) displaces L 1 from any compact set, then, for all pair of augmentations ε 0 and ε 1 of the obstruction algebras of (L 0 , J 0 ) and (L 1 , J 1 ) respectively,

HW ((L 0 , ε 0 ), (L 1 , ε 1 ), J • ) = 0.
We postpone the proof after a couple of lemmas. Given Λ > λ > 0 and L > 0, we define a function h λ,Λ,R : R + → R such that

h λ,Λ,R (ρ) =      0 for ρ ≤ 1, λρ -λ for ρ ∈ [1, e R ], Λρ -(Λ -λ)e R -λ for ρ ≥ e R .
See Figure 2 for the graph of h λ,Λ,R .

The function h λ,Λ,R has two corners: one at (1, 0) and one at (e R , λe R -λ). We smooth h λ,Λ,R in a small neighbourhood of these corners so that the new function (which we still denote by h λ,Λ,R ) satisfies h λ,Λ,R (ρ) ≥ 0 for all ρ. We define the (time independent) cylindrical Hamiltonian H λ,Λ,R : W → R by H λ,Λ,R (w) = h λ,Λ,R (e r(w) ). We make the assumption that there is no Hamiltonian time-1 chord from L 0 to L 1 on ∂W r when r satisfies either h λ,Λ,R (e r ) = λ or h λ,Λ,R (e r ) = Λ. This is equivalent to assuming that there is no Reeb chord from Λ 0 to Λ 1 of length either λ or Λ.

We assume, without loss of generation, that L 0 and L 1 intersect transversely, that L i ∩ W e 1 is a cylinder over a Legendrian submanifold Λ i and that all Reeb chords from Λ 0 to Λ 1 are nondegenerate. Then we have three types of chords:

• constant chords, i.e. intersection points between L 0 and L 1 , which are contained in W 0 , • chords coming from smoothing the corner of h λ,Λ,R , which are concentrated around ∂W 1 , and • chords coming from smoothing the corner of h λ,Λ,R at (e R , λe R -λ), which are concentrated around ∂W L .

Constant chords and chords coming from smoothing the first corner will be called type I chords, while chords coming from smoothing the second corner will be called type II chords. We say that a chord of H Λ,λ,L appears at slope s if it is contained in ∂W r for r such that h Λ,λ,L (e r ) = s. By abuse of terminology, we will consider the intersection points between L 0 and L 1 as chords appearing at slope zero.

Lemma 7.7. Given λ > 0, there exists C > 0 such that, for every Λ > λ and every R ≥ C, every chord of type II of H λ,Λ,R has larger action than any chord of type I.

Proof. If x is a Hamiltonian chord contained in ∂W r , then the action of x is ( 26)

A(x) = h λ,Λ,R (e r )e r -h λ,Λ,R (e r ) + f 0 (x(0)) -f 1 (x (1)).
Observe that |f 0 (x(0))-f 1 (x(1))| is uniformly bounded because f 0 and f 1 are locally constant outside of a compact set. The Hamiltonian chords of type I appear at slope λ -< λ and near ∂W 0 , and therefore r in Equation ( 26) is close to zero. Then, there is a constant C -, depending on f 0 , f 1 and the smoothing procedure at the first corner such that, if x is a chord of type I, then A(x) ≤ λ -+ C -.

On the other hand, if x is a chord of type II, then it appears at slope λ + > λ and around r = log(R). Then there is a constant C + , depending on f 0 , f 1 and the smoothing procedure at the second corner such that A(x) ≥

Rλ + -λR + λ -C + = R(λ + -λ) + λ -C + .
The lemma follows from λ + -λ > 0 and the fact that chords arise at a discrete set of slopes.

From now on we will always take R ≥ C. The consequence of Lemma 7.7 is that the chords of type I generate a subcomplex of

CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • )
which we will denote by CF I ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • ). The main ingredient in the proof of Proposition 7.6 is the following lemma.

Lemma 7.8. If the Liouville flow of (W, θ) displaces L 1 from any compact set, then the inclusion map

CF I ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • )
is trivial in homology whenever Λ 0 is sufficiently large.

Proof. The Liouville flow applied to L 1 gives rise to a compactly supported safe isotopy from L 1 to L 1 , and is generated by the time-dependent local Hamiltonian G : R × L → R for which dG t = 0 outside of a compact subset of (0, 1) × L → R; see Lemma 2.23. Since the Liouville form is conformally symplectic, it actually preserves the space of compatible almost complex structure cylindrical at infinity.

We will choose to apply the Liouville flow so that L 1 ⊂ {ρ ≥ e R }; recall that this is possible by our assumptions.

The continuation map

Ψ G : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • )
defined in Subsection 5.3 induces an isomorphism in homology if J • is an (L 0 , L 1 )-regular almost complex structure such that J 0 = J 0 and J 1 = (ψ 1 ) * J 1 , and ε 1 is the augmentation of the obstruction algebra of (L 1 , J 1 ) defined by

ε 1 = ε 1 • ψ -1 1 .
By Lemma 7.5, there is a constant C, independent of Λ, such that Ψ G (x) is a linear combination of chords of action at most C whenever x is a chord from L 0 to L 1 of type I. On the other hand, the complex CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R ) is generated by Hamiltonian chords x of action )).

A(x) = (Λ -λ)e R + λ + f 0 (x(0)) -f 1 (x( 1
Here we have used L 1 ⊂ {ρ ≥ e R }, together with the particular form of H λ,Λ,R in the same subset. This implies that, for Λ large enough, Ψ G (x) = 0 for all chords x of type I.

Proof of Proposition 7.6. For every λ and Λ there are continuation maps Φ

λ,Λ : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ , J • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • ), Φ (2) λ,Λ : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H Λ , J • ) and Φ λ,Λ : CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ , J • ) → CF ((L 0 , ε 0 ), (L 1 , ε 1 ); H Λ , J • ) (1)
such that there is a chain homotopy between Φ λ,Λ and Φ

(2) λ,Λ • Φ (1) λ,Λ .
We can assume that Φ λ,Λ , Φ

λ,Λ and Φ

(2) λ,Λ are defined using s-dependent Hamiltonians H * s ( * = ∅, (1), ( 2)) such that ∂ s H * s ≤ 0, and therefore they decrease the action. Hence, the image of Φ (1)

λ,Λ is contained in CF I ((L 0 , ε 0 ), (L 1 , ε 1 ); H λ,Λ,R , J • )
(here we use Lemma 7.7) and therefore it follows from Lemma 7.8 that Φ λ,Λ = 0 in homology. By the definition of wrapped Floer cohomology as a direct limit, this implies that HW ((L 0 , ε 0 ), (L 1 , ε 1 ), J • ) = 0. Remark 7.9. The arguments of this section apply equally well to Liouville sectors, provided that the following precautions are taken. If the cylindrical Hamiltonian is chosen as in Definition 4.7, then its Hamiltonian vector field has excellent dynamics on V . We will assume that the Legendrian end of L 0 and L 1 are disjoint from the coconvex neighbourhood N of Definition 4.5. This condition is easy to obtain up to safe isotopy. Then the excellent dynamics of the Hamiltonian vector filed implies that Hamiltonian chords between L 0 and L 1 are contained in a compact subset of int(W ) . Then we can use the contact form g -1 α instead of α to estimates actions, and the proof goes through.

Floer homology and Lagrangian surgery

Lalonde and Sikorav in [START_REF] Lalonde | Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents[END_REF] and then Polterovich in [START_REF] Polterovich | The surgery of Lagrange submanifolds[END_REF] defined a surgery operation on Lagrangian submanifolds. It is expected that Lagrangian surgery should correspond to a twisted complex (i.e. an iterated mapping cone) in the Fukaya category. Results in this direction have been proved by Seidel in [START_REF] Seidel | A long exact sequence for symplectic Floer cohomology[END_REF], Fukaya, Oh, Ohta and Ono in [START_REF] Fukaya | Lagrangian intersection floer theoryanomaly and obstruction -chapter 10[END_REF] and by Biran and Cornea in [START_REF] Biran | Lagrangian cobordism and Fukaya categories[END_REF]. Our goal in this section is to establish Proposition 8.16, which provides us with a result along these lines in the generality that we need.

The difficult point in handling the Lagrangian surgery from the Floer theoretic perspective is that, except in very favourable situations, the Lagrangian submanifolds produced are not well behaved from the point-of-view of pseudoholomorphic discs. In our situation, we turn out to be lucky, since only surgeries that preserve exactness are needed. Nevertheless, there still is a complication stemming from the fact that the resulting Lagrangian only is immersed (as opposed to embedded). This is the main reason for the extra work needed, and here we rely on the theory developed in the previous sections.

The bounding cochains that we will consider in this exact immersed setting are those corresponding to augmentations of the corresponding obstruction algebras introduced in Section 4.2. This turns out to be a very useful perspective, since it enables us to apply techniques from Legendrian contact homology in order to study them. 8.1. The Cthulhu complex. In this subsection we recall, and slightly generalise, the definition of Floer homology for Lagrangian cobordisms we defined in [START_REF] Chantraine | Floer theory for lagrangian cobordisms[END_REF]. Definition 8.1. Given cylindrical exact Lagrangian immersions L + and L -in (W, θ) which coincide outside of a compact set, an exact Lagrangian cobordism Σ from

L + -to L + + is a properly embedded submanifold Σ ⊂ (R × M, d(e t β)) = (R × W × R, d(e t (dz + θ)))
such that, for C and R sufficiently large,

(1) Σ ∩ (-∞, -C] × W × R = (-∞, -C] × L + -, (2) Σ ∩ [C, +∞) × W × R = [C, +∞) × L + + , (3) Σ ∩ (R × W e R × R
) is tangent to both ∂ s and the lift of the Liouville vector field L of (W, θ), and (4) e s α| Σ = dh for a function h 3) is called the lateral end of Σ.

: Σ → R which is constant on Σ ∩ (-∞, -C] × W × R. The intersection Σ ∩ (R × W e R × R) = R × (L + ± ∩ (W e R × R)) of (
The surgery cobordism Σ(a 1 , . . . , a k ) defined in the previous subsection clearly satisfies all these properties when L(a 1 , . . . , a k ) is connected.

Given two exact Lagrangian cobordisms Σ 0 and Σ 1 from (L 0 -) + to (L 0 + ) + and from (L 1 -) + to (L 1 + ) + with augmentations ε 0 -and ε 1 -of (L 0 -) + and (L 1 -) + respectively, we define the Cthulhu complex Cth ε 0 -,ε 1 -(Σ 0 , Σ 1 ) which, as an F-module, splits as a direct sum

Cth ε 0 -,ε 1 -(Σ 0 , Σ 1 ) = LCC ε 0 + ,ε 1 + ((L 0 + ) + , (L 1 + ) + ) ⊕ CF ε 0 -,ε 1 -(Σ 0 , Σ 1 ) ⊕ LCC ε 0 -,ε 1 -((L 0 -) + , (L 1 -) + ),
where ε i + is the augmentations of A((L i + ) + ) induced by ε i -and Σ i , and

CF ε 0 -,ε 1 -(Σ 0 , Σ 1
) is the F-module freely generated by the intersection points Σ 0 ∩ Σ 1 , which we assume to be transverse. Furthermore, we assume that L 0 ± ∩ L 1 ± ∩ W e R = ∅, which is not a restriction since the ends are cylinders over Legendrian submanifolds. The differential on the Cthulhu complex can be written as a matrix

d ε 0 -,ε 1 -=   d ++ d +0 d +- 0 d 00 d 0- 0 d -0 d --  
where d ++ and d --are the differentials of LCC ε 0

+ ,ε 1 + ((L 0 + ) + , (L 1 + ) + ) and LCC ε 0 -,ε 1 -((L 0 -) + , (L 1 -) +
) respectively, and the other maps are defined by counting J-holomorphic discs in R × M with boundary on Σ 0 ∪ Σ 1 and boundary punctures asymptotic to Reeb chords from (L ± 1 ) + to (L ± 1 ) + and intersection points between Σ 0 and Σ 1 . See [START_REF] Chantraine | Floer theory for lagrangian cobordisms[END_REF]Section 6] for the detailed definition. The cobordisms considered in [START_REF] Chantraine | Floer theory for lagrangian cobordisms[END_REF] have the property that Σ i ∩ [-C, C] × M is compact for every C > 0, while here we consider cobordisms with a lateral end. The theory developed in [START_REF] Chantraine | Floer theory for lagrangian cobordisms[END_REF] can be extended to the present situation thank to the following maximum principle. Lemma 8.2. Let J and J be almost complex structures on R × W × R and W , respectively, each cylindrical inside the respective symplectisation R × W e R and half-symplectisation W e R for some R > 0. We moreover require that the the canonical projection

(R × W e R × R, J) → (W e R , J) is holomorphic. Then every J-holomorphic map u : ∆ → R × W × R with • ∆ = D 2 \ {ζ 0 , . . . , ζ d } where (ζ 0 , . . . , ζ d ) ∈ Conf d+1 (∂D 2 ), • u(∂∆) ⊂ Σ 0 ∪ Σ 1 ,
and • u maps some neighbourhood of the punctures {ζ 0 , . . . ,

ζ d } into R × W R × R, has its entire image contained inside R × W R × R.
Proof. By the assumptions the image of the curve u| u -1 (R×W e R ×R) under the canonical projection With Lemma 8.2 at hand, the arguments of [START_REF] Chantraine | Floer theory for lagrangian cobordisms[END_REF] go through, and therefore we have the following result. Theorem 8.3 ([9]). The map

(R × W e R × R, J) → (W e R , J) is compact with boundary on R × ∂W e R × R
d ε 0 -,ε 1 -is a differential ad the Cthulhu complex (Cth ε 0 -,ε 1 -(Σ 0 , Σ 1 ), d ε 0 -,ε 1 -) is acyclic.
The consequence of interest for us is the following.

Corollary 8.4. If Σ 0 ∩ Σ 1 = ∅, then the map d +-: LCC ε 0 -,ε 1 -((L 0 -) + , (L 1 -) + ) → LCC ε 0 + ,ε 1 + ((L 0 + ) + , (L 1 + ) + ) is a quasi-isomorphism.
Proof. If Σ 0 ∩ Σ 1 = ∅, the Cthulhu differential simplify as follows:

d ε 0 -,ε 1 -=   d ++ 0 d +- 0 0 0 0 0 d --  
and thus the Chthulhu complex becomes the cone of d +-. Since it is acyclic, it follows that d +-is a quasi-isomorphism.

8.2. The surgery cobordism. In this subsection we describe the Lagrangian surgery of [START_REF] Lalonde | Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents[END_REF] and [START_REF] Polterovich | The surgery of Lagrange submanifolds[END_REF] from the Legendrian viewpoint. In particular, we interpret it as as a Lagrangian cobordism between the Legendrian lifts of the Lagrangian submanifolds before and after the surgery. We refer to [START_REF] Rizell | Legendrian ambient surgery and Legendrian contact homology[END_REF] for more details.

We first describe the local model for Lagrangian surgery. Given η, δ > 0, we consider the open subset

V η,δ := {|q| < η, |p| < 2δ, z ∈ R} of J 1 (R n ).
Given ζ > 0, we denote by Λ + η,δ,ζ the (disconnected) Legendrian submanifold of V η,δ given by the two sheets

{(q, ±df η,δ,ζ (|q|), ±f η,δ,ζ (|q|)) : |q| < η}, where f η,δ,ζ (s) = δ 2η s 2 + ζ 2 .
This is a Legendrian submanifold with a single Reeb chord of length ζ. Note that Λ + η,δ,ζ is described by the generating family F + η,δ,ζ : R n × R → R given by

F + η,δ,ζ (q, ξ) = ξ 3 3 -g + (|q|)ξ,
where

g + (s) = 3 2 f η,δ,ζ (s) 2 3
.

Note that g + is smooth because g + (s) > 0 holds for every s. Let g -: R + → R be a function such that

(i) g -(s) = 3 2 f η,δ,ζ (s) 2
3 for s > 3η/4, (ii) g -(s) < 0 for s < η/2, and (iii) 0 < (g -) (s) < 2 δη δη+ζ .

Note that Condition (iii) can be achieved if ζ < 7δη 16 . The Legendrian submanifold Λ - η,δ,ζ of V δ,η generated by

F - η,δ,ζ (q, ξ) = ξ 3 3 -g -(|q|)ξ
coincides with Λ + η,δ,ζ near |q| = ε and has no Reeb chords (see Figure 3). Note that indeed Λ - η,δ,ζ ⊂ V δ,η because Condition (iii) ensures that the p coordinates of Λ - η,δ,ζ , given by

∂F - η,δ,ζ ∂p i
along critical values of F (q, •), are smaller than 2δ.

On Figure 4 we see the front and Lagrangian projections of the one dimensionnal version of Λ + and Λ -. Let L be an exact Lagrangian immersion in (W, θ) with double points a 1 , . . . , a k , and let L + be a Legendrian lift of L. The double points of L lift to Reeb chords of L + which we will denote with the same name by an abuse of notation. Definition 8.5. A set of Reeb chords {a 1 , . . . , a k } on L + is called contractible if, for all i = 1, . . . , k, there is a neighbourhood U i of the Reeb chord a i in the contactisation (M, β) of (W, θ) and a strict contactomorphism

Λ - η,δ,ζ ⊂ V η,δ Λ + η,δ,ζ ⊂ V η,δ
(U i , U i ∩ L + ) ∼ = (V η i ,δ i , Λ 1 η i ,δ i ,ζ i ) for numbers η i , δ i , ζ i satisfying ζ i < 7δ i η i
16 . Remark 8.6. This is a restrictive assumption because, in general, the lengths of the chords a 1 , . . . , a k cannot be modified independently. An example when this is possible, and which will be the case in our main theorem, is when L + is a link with k +1 components, all a i are mixed chords, and each component contains either the starting point or the end point of at least one of the a i . In this situation we can indeed modify the Legendrian link by Legendrian isotopies of each his components so that its Lagrangian projection is unchanged and all the previous conditions on the neighbourhoods are satisfied. (Warning: this might not be an isotopy of the Legendrian link.)

In the following we assume that {a 1 , . . . , a k } is a set of contractible Reeb chords on L + . We denote by L + (a 1 , . . . , a k ) the Legendrian submanifold of (M, β) obtained by replacing each of the Λ + η i ,δ i ,ζ i by the corresponding Λ - η i ,δ i ,ζ i and by L(a 1 , . . . , a k ) the Lagrangian projection of L + (a 1 , . . . , a k ). Observe that we here need to make use of the identifications with the standard model, which exist by the contractibility condition.

Then L(a 1 , . . . , a k ) is an exact Lagrangian immersion in (W, θ) which is the result of Lagrangian surgery on L along the self-intersection points a 1 , . . . , a k . It is evident from the construction that L(a 1 , . . . , a k ) coincides with L outside of a neighbourhood of the a i 's and has k self-intersection points removed. The latter fact follows from the fact that since ζ i can be chosen arbitrarily small, no Reeb chords are created when going from Λ

+ η i ,δ i ,ζ i to Λ - η i ,δ i ,ζ i .
Next we construct an exact Lagrangian cobordism Σ(a 1 , . . . , a k ) in the symplectisation of (M, β) with L at the positive end and L(a 1 , . . . , a k ) at the negative end. Fix T > 0 and choose a function G : (0, ε) × R + → R such that:

• G(t, s) = g -(s) for t < 1/T , • G(t, s) = g + (s) for t > T ,

• ∂G ∂t (t, 0) > 0, and

• G(t, s) = g + (s) = g -(s) for s > 3η/4.
We consider the Lagrangian submanifold of T * (R + × B n (η)) described by the generating family

F (t, q, ξ) = t • ξ 3 3 + G(t, |q|)ξ ,
which is mapped by the symplectomorphism

T * (R + ×B n (η)) ∼ = R×J 1 (B n (η))
to a Lagrangian cobordisms Σ η,δ,ζ in the symplectisation of (M, β) from Λ - η k ,δ k ,ζ k at the negative end to Λ + η k ,δ k ,ζ k at the positive end. Self-intersections of Σ η,δ,ζ are given by the critical points of the function ∆ F (t, q, ξ 1 , ξ 2 ) = F (t, q, ξ 1 ) -F (t, q, ξ 2 ) with non 0 critical value, and such points do not exist because of the third condition on G. Thus this cobordism is embedded.

In the trivial cobordism R × L + we replace R × (U i ∩ L + ) with Σ η i ,δ i ,ζ i , for all i = 1, . . . , k, to get a a cobordism Σ(a 1 , . . . , a k ) from L + (a 1 , . . . , a k ) at the negative end to L + at the positive end. 8.3. Effect of surgery on Floer homology. In this subsection we use Σ(a 1 , . . . , a k ) and our Floer theory for Lagrangian cobordisms to relate the Floer homology of L with the Floer homology of L(a 1 , . . . , a k ). The Lagrangian cobordism Σ(a 1 , . . . , a k ) induces a dga morphism Φ Σ : A(L + ) → A(L + (a 1 , . . . , a k )).

If follows from [14, Theorem 1.1] that, for a suitable almost complex structure on the cobordism that has been obtained by perturbing an arbitrary cylindrical almost complex structure, we have

Φ Σ (a i ) = 1 for i = 1, . . . , k, Φ Σ (c) = c + w if c = a i , ( 27
)
where w is a linear combination of products c 1 . . . c m with a(c 1 ) + . . . + a(c m ) < a(c). Lemma 8.7. If ε : A(L + ) → F is an augmentation such that ε(a i ) = 1 for i = 1, . . . , k, then there is an augmentation ε :

A(L + (a 1 , . . . , a k )) → F such that ε = ε • Φ Σ .
Proof. Let I be the bilateral ideal generated by a i -1, . . . , a k -1: then ε induces an augmentation ε : A(L + )/I → F. By Equations ( 27) Φ Σ is surjective and its kernel is I. Surjectivity is proved by a sort of Gauss elimination using the action filtration. Then there is an isomorphism between A(L + )/I and A(L + (a 1 , . . . , a k )), and therefore the augmentation ε : A(L + )/I → F induces an augmentation on A(L + (a 1 , . . . , a k )), which we still denote by ε.

Note that the construction of ε is not explicit because the isomorphism A(L + )/I ∼ = A(L + (a 1 , . . . , a k )) is not explicit. Proposition 8.8. For any immersed cylindrical exact Lagrangian submanifold T ⊂ W with augmentation ε there is a quasi-isomorphism

LCC ε ,ε (T + , L + (a 1 , . . . , a k )) -→ LCC ε ,ε (T + , L + ),
under the assumption that the augmentations ε and ε are as in Lemma 8.7.

Proof. We denote by Σ T the trivial cobordism Σ T = R × T + ⊂ R × M . Recall that the surgery cobordism goes from L + (a 1 , . . . , a k ) to L + . Since the surgery is localised in a neighbourhood of the intersection points a 1 , . . . , a k , by a Hamiltonian isotopy we can assume that Σ T ∩ Σ(a 1 , . . . , a k ) = ∅.

Then Corollary 8.4 implies that the map d +-in the Cthulhu differential for the cobordisms Σ T and Σ(a 1 , . . . , a k ) is a quasi-isomorphism. by µ d T wA (q d , . . . , q 1 ) =

k 1 ,...,k d ≥0 µ k 1 +...+k d +d A (X d , . . . , X d k d , q d , X d-1 , . . . , X 1 , q 1 , X 0 . . . , X 0 k 0 ). ( 28
)
It is shown in [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF]Section 3.k] that the set of twisted complexes with operations µ d T wA constitutes an A ∞ -category T wA which contains A as a full subcategory. Furthermore it is shown in [32, Lemma 3.32 and Lemma 3.33] that T wA is the triangulated envelope of A and thus H 0 T w(A) is the derived category of A. Definition 8.12. We say that a collection of objects L 1 , . . . , L k of A generates A if and only if any object L of A is quasi-isomorphic in T wA to a twisted complex built from the object L i 's. Lemma 8.13. If there is a twisted complex L built from L 0 , . . . , L k such that, for every object T of A we have H hom T wA (T, L) = 0, then L 0 is quasi-isomorphic in T wA to a twisted complex built from L 1 , . . . , L k .

Proof. This follows from the iterated cone description of twisted complexes from [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF]Lemma 3.32]. More precisely, from the definition of twisted complexes, for any object T we have that hom A (T, L 0 ) is a quotient complex of hom T wA (T, L) by the twisted complex L built from L starting at L 1 (i.e. "chopping" out L 0 from the twisted complex L), and thus those three objects fit in an exact triangle. The vanishing of H hom T wA (T, L) implies then that H hom A (T, L 0 ) ∼ = H hom T wA (T, L ). The result follows now because the map from L 0 to L , which is given by the maps (x 0j ), is a map of twisted complexes.

We now relate twisted complexes in the wrapped Fukaya category with certain augmentations of the Chekanov-Eliashberg algebra of the Legendrian lift of the involved Lagrangian submanifolds. Remark 8.14. In the following lemma we will make a slight abuse of notation by building twisted complexes from immersed exact Lagrangian submanifolds: to our knowledge, the wrapped Fukaya category has not yet been extended to include also exact immersed Lagrangian submanifolds. However, since the statements and proofs only concern transversely intersecting Lagrangian submanifolds, there are no additional subtleties arising when considering the A ∞ operations. In other words, we only consider morphisms between different objects in the category. The worried reader can thus think of twisted complexes in the "Fukaya pre-category". Of course if all Lagrangian submanifolds L i involved are embedded, the statements make sense also in the ordinary wrapped Fukaya category. Lemma 8.15. Let (L i , i ), for i = 0, . . . , k, be unobstructed exact immersed Lagrangian submanifolds which are assumed to be equipped with fixed potentials f i . We denote L = L 1 ∪ . . . ∪ L k and L + its Legendrian lift determined by the given potentials. We assume that L + is embedded.

If ε : A(L + ) → F is an augmentation such that:

(1) ε(p) = ε i (p) for every pure chord p of L + i , and (2) ε(a) = 0 for every mixed chord a from L + i to L + j such that i > j, we define

x ij :=    a∈L i ∩L j ε(a)a if i < j, 0 if i ≥ j,
and X = (x ij ) 0≤i,j≤k , where the double point a is considered as an element in the summand with wrapping parameter w = 0 (see Section 6.2). Then (ignoring the degrees for simplicity) the pair L = ({(L i , ε i )}, X) is a twisted complex in the wrapped Fukaya category. Moreover, for any test Lagrangian submanifold T , H hom T wWF (T, L) = HW (T, (L, ε)).

Proof. Denote by ε 0 the augmentation of A(L) which vanishes on the mixed chords, while taking the value ε i on the generators living on the component L i . Recall the chain model for wrapped Floer complex described in Subsection 6.2, where the homotopy direct limit CW ((L, ε 0 ), (L, ε 0 ); J • ) is an infinite direct sum starting with the term with wrapping parameter w = 0, i.e. the complex

CF ((L, ε 0 ), (L, ε 0 )); 0, J • ) ⊕ CF ((L, ε 0 ), (L, ε 0 ); 0, J • )q.
The bounding cochain X can be identified to a sum of elements in the leftmost summand by definition.

Note that L, of course, is only immersed. However, in the case when it consists of a union of embeddings, it still represents an object in the twisted complexes of the ordinary wrapped Fukaya category; namely, it is the "direct sum" of the Lagrangian submanifolds L i , i = 1, . . . , m.

First we prove that X satisfies the Maurer-Cartan equation. The Maurer-Cartan equation involves a count of holomorphic polygons in moduli spaces M 0 L 0 ,...,L d (p d , a 0 , p 1 , a 1 , . . . , p d-1 , a d ; J) as in Section 4.5. On the other hand, the equation ε • d = 0 counts holomorphic polygons in the moduli spaces N L (a 0 ; p 1 , a 1 , . . . , a d , p d ), which are the subset of the previous moduli spaces consisting of those holomorphic polygons which satisfy the extra requirement that the intersection points a 1 , . . . , a d should be negative punctures (in the sense of Definition 3.4). Condition (2) in the definition of ε however implies that #M 0 L 0 ,...,L d (p d , a 0 , p 1 , a 1 , . . . , p d-1 , a d ; J) is multiplied by a nonzero coefficient only if a 1 , . . . a d are negative punctures. This proves that X satisfies the Maurer-Cartan equation.

For the second part, note that the differential in hom T wWF (T, L) counts the same holomorphic polygons (with Hamiltonian perturbations) as the differential in CW (T, (L, ε)) because the Maurer-Cartan element X involves only elements in the Floer complexes defined with wrapping parameter w = 0, and hence vanishing Hamiltonian term.

The previous lemma together with Proposition 8.8 implies the following result, which is the main result of this section: Proposition 8.16. Let (L 1 , ε 1 ), . . . , (L m , ε m ) be unobstructed immersed exact Lagrangian submanifolds with preferred choices of potentials f i , and let a 1 , . . . , a k be a set of intersection points lifting to contractible Reeb chords on the induced Legendrian lift L + , where L := L 1 ∪ . . . ∪ L m . Assume that there is an augmentation ε of the Chekanov-Eliashberg algebra of L + such that:

(

1) ε(c) = ε i (c) if c is a double point of L i , (2) ε(a i ) = 1 for i = 1, . . . , k, and (3)
ε(q) = 0 if q ∈ L i ∩ L j is an intersection point, with i > j, at which f i (q) > f j (q) (i.e. q corresponds to a Reeb chord from L + i to L + j ).

Then for any other exact Lagrangian submanifold T there is a quasi-isomorphism CW (T, (L(a 1 , . . . , a k ), ε)) ∼ = hom(T, L), with ε induced by ε as in Lemma 8.7, and where L is a twisted complex built from the L i with i = 1, . . . , m.

Remark 8.17. Conditions (2) and (3) of Proposition 8.16 imply that if a k is an intersection point between different Lagrangians L i and L j for i < j, then f i (a k ) < f j (a k ). Conditions (1) and (2) of Proposition 8.16 imply that if a k is a self-intersection point of L i , then augmentation ε i evaluates to 1 on a k .

Proof of Proposition 8.16. We consider the twisted complex L built from L i , i = 1, . . . , m, that is constructed by an application of Lemma 8.15 with the augmentation ε. In other words, the twisted complex is defined using the Maurer-Cartan element

X := a 1 + . . . + a k ∈ CF ((L, ε 0 ), (L, ε 0 ); 0, J • ) ⊂ CW ((L, ε 0 ), (L, ε 0 ); J • )
living in the summand with wrapping parameter w = 0. The quasi-isomorphism

hom(T, L) ∼ = CW (T, (L, ε))
is then a consequence of the same lemma.

What remains is constructing a quasi-isomorphism

CW (T, (L(a 1 , . . . , a k ), ε)) ∼ = hom(T, L)
for all test Lagrangian submanifolds T . This is done by considering the twisted complex corresponding to the cone of the "unit" e from Corollary 8.10. We proceed to give the details.

Let L be the push-off of L 1 ∪. Here we remind the reader that the components of L(a 1 , . . . , a k ) typically are only immersed, as opposed to embedded. The latter statement established is hence on the level of twisted complexes on the pre-category level; c.f. Remark 8.14.

Generating the Wrapped Fukaya category

In this section we prove Theorem 1.1 and Theorem 1.2. Most of the proof is the same in the two cases, and for simplicity we will write the details for a Weinstein manifold. In Section 9.4 we present the modifications needed to extend the proof to Weinstein sectors. 9.1. Geometric preparations. Before proving the main theorems we need some geometric preparations which will be used in the technical work of Section 9.3. Recall that the Liouville form θ has been modified in order to make (H 1 ∪ . . . ∪ H l , θ, f) into a union of standard critical Weinstein handles. After adding the differential of a function supported in a small neighbourhood of H 1 ∪ . . . ∪ H l , we change the Liouville form once again so that the symplectomorphism between (H i , dθ) and (D δ T * C i , dp ∧ dq) maps the new Liouville form θ c to pdq. We make the modification so that the new Liouville vector field L c is still positively transverse to ∂W 0 , has no zeros outside W 0 , and so that the new and old Lagrangian skeleta coincide. (On the other hand L c it is no longer a pseudo-gradient vector field for f, but this will not impair the proof of Theorem 1.1.) Note that the above identification maps the core of a handle to the zero section and the cocore into a cotangent fibre. Further, we perform the construction of the new Liouville form so that the corresponding Liouville vector field is still everywhere tangent to D i . The reason for changing θ to θ c is to simplify the arguments of Subsection 9.3.

The set of cylindrical exact Lagrangian submanifolds of (W, θ) coincide with that of (W, θ c ) and the wrapped Floer cohomology between any two such Lagrangian submanifolds is unaffected by the modification of θ by the invariance properties of wrapped Floer homology; see [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]Section 5]. This means that WF(W, θ) is quasi-equivalent to WF(W, θ c ).

With a new Liouville vector field we will choose a new function r : W → [R 0 , +∞) satisfying Conditions (i) and (ii) of Section 2.1 for R 0 0 such that, on ∂W 0 , the old and new r coincide. From now on, r will always be defined using the new Liouville vector field L c . Later in the proof of Proposition 9.3, we will modify r so that the new R 0 0 becomes sufficiently small, while keeping r fixed outside of a compact subset. Let ψ t be the Liouville flow of (W, θ c ) and let

H i := t≥0 ψ t (H i ).
It follows that H i ⊂ W are pairwise disjoint, embedded codimension zero manifolds. Moreover, there are exact symplectomorphisms

( H i , θ c ) ∼ = (T * C i , pdq)
with the standard symplectic cotangent bundles.

Recall Conditions (i) and (ii) from Subsection 2.1. In particular, r -1 (R 0 ) = W sc ∪ H 1 ∪ . . . ∪ H l , while r| r -1 [R 0 +1,+∞) is a symplectisation coordinate induced by the hypersurface r -1 (R 0 + 1) of contact type. In the following we make the further assumption that (29) r -1 (R 0 + 1) ∩ H i = S * r 0 T * C i for some r 0 > 0, where the latter radius-r spherical cotangent bundle is induced by the flat metric on C i . This means that [START_REF] Ritter | The monotone wrapped Fukaya category and the open-closed string map[END_REF] r(p, q) = log p -log r 0 + R 0 + 1, p ≥ r 0 , holds in the above canonical coordinates.

Given a point a ∈ C i (for some i = 1, . . . , l), we denote by D a the Lagrangian plane which satisfies D a ∩ C i = {a} while being everywhere tangent to the Liouville vector field. In particular, D a ∩ H i corresponds to the cotangent fibre D δ T * a C i ⊂ D δ T * C i under the identification H i ∼ = D δ T * C i . Lemma 9.1. For every i = 1, . . . , l and a ∈ C i , the Lagrangian plane D a is isotopic to D i by a cylindrical Hamiltonian isotopy.

Proof. Recall that ( H i , θ c ) is isomorphic to (T * C i , pdq) as a Liouville manifold and D a and D i correspond to two cotangent fibres. Therefore they are clearly isotopic by a cylindrical Hamiltonian isotopy.

In particular, D a and D i are isomorphic objects in the wrapped Fukaya category when a ∈ C i .

The next lemma is immediate. Lemma 9.2. Let L ⊂ W be a cylindrical exact Lagrangian submanifold. Then, up to a (compactly supported) Hamiltonian isotopy, we can assume that L ∩ (C 1 ∪ . . . ∪ C l ) = {a 1 , . . . , a k }, the intersections are transverse and L ∩ W sc = ∅. Now we are going to normalise the intersections between L and the planes D a i . For every a i we choose the natural symplectomorphism between a neighbourhood

a i ⊂ ( H i , θ c ) ∼ = (T * C i , pdq) of D a i ∼ = T * a i C i and (D η T * D a i , -dp ∧ dq)
for some η > 0 small, where (p, q) are the canonical coordinates on T * D a i . It is clearly possible to make this identification so that [START_REF] Seidel | A long exact sequence for symplectic Floer cohomology[END_REF] r(p, q) = log q -log r 0 + R 0 + 1, q ≥ r 0 , is satisfied.

We redefine r as in Remark 2.2, without deforming it outside of a compact subset. After making R 0 0 sufficiently small in this manner, we may assume that:

• R 0 + k + 3 ≤ 0, • L∩W R 0 +k+3
is the union of k disjoint discs with centres at a 1 , . . . , a k , and

• the connected component of L ∩ W R 0 +k+3 containing a i is identified inside D a i ∼ = D η T * D a i with the graph of the differential of a function g a i : D a i → R for i = 1, . . . , k.
Then we modify L by a compactly supported Hamiltonian isotopy so that it satisfies the following properties: (L1) The connected component of L∩W R 0 +k+3 containing a i is contained inside the Weinstein neighbourhood

D a i ∩ W R 0 +k+3 ∼ = D η T * (D a i ∩ { q ≤ e k+2 r 0 }),
where it is described by the graph of the differential of a function g a i : D a i ∩ { q ≤ e k+2 r 0 } → R with a nondegenerate minimum at a i and no other critical points, (L2) the connected components of L ∩ W R 0 +k+3 \ W R 0 +k+2 are cylinders which are disjoint from all the cocores D a i ; moreover, these cylinders are tangent to the Liouville vector field L c in the same subset; and (L3) g a i C 2 ≤ for i = 1, . . . , k and > 0 small which will be specified in Lemma 9.4. Proof. We assume that L satisfies Conditions (L1), (L2) and (L3) from the previous section. Then by Lemma 9.4 combined with Lemma 9.5 there exist Lagrangian planes D w a 1 , . . . , D w a k satisfying the following properties. First D w a i is Hamiltonian isotopic to D a i (possibly after re-indexing the a 1 , . . . , a k ) by a cylindrical Hamiltonian isotopy supported in W \W R 0 +i . Second, for an appropriate Legendrian lift L + of L = L ∪ D w a 1 ∪ . . . ∪ D w a k to (W × R, θ + dz) such that the intersection point a i lifts to a Reeb chord from (D w a i ) + to L + of length > 0 for i = 1, . . . , k -see Lemma 9.4 for more details -there exists an augmentation ε : A(L + ) → F for which (1) ε(a i ) = 1 for i = 1, . . . , k, and (2) ε(d) = 0 if d is a chord from L + to (D w a i ) + for i = 1, . . . , k, or a chord from (D w a i ) + to (D w a j ) + with i > j.

Conditions

Moreover, using Property (L3) above for > 0 sufficiently small, it follows that the Reeb chords a i all are contractible (c.f. Definition 8.5).

By Proposition 8.16 the augmentation ε induces a twisted complex L built from L 1 = D w a 1 , . . . , L k = D w a k , L k+1 = L, for which hom(T, L) ∼ = CW (T, (L(a 1 , . . . , a k ), ε)).

The right-hand side is an acyclic complex by Proposition 7.6. Using this acyclicity, the aforementioned proposition implies the sought statement that L is quasi-isomorphic to a twisted complex built from the different D w a i ∼ = D a i . For the last statement, we have used the invariance properties for wrapped Floer cohomology under cylindrical Hamiltonian isotopy; see e.g. [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF]Section 5].

The proof of Theorem 1.1 is now a simple combination of Lemma 9.2, Proposition 9.3 and Lemma 9.1. To that end, we again need the fact that Hamiltonian isotopies by cylindrical Hamiltonians induce isomorphisms in the wrapped Fukaya category; see e.g. [3, Section 5]. 9.3. Constructing the augmentation. We start by assuming that the modifications from Section 9.1 have been performed, so that in particular (L1)-(L3) are satisfied. When considering potentials in this subsection, recall that we have modified the Liouville form from θ to θ c . Let f : L → R be a potential function for L. We order the intersection points a 1 , . . . , a k such that f (a k ) ≤ . . . ≤ f (a 1 ).

The Morse function g a i : D a i ∩ {q ≤ e k+2 r 0 } → R from (L1) can be assumed to be sufficiently small by (L3), so that df is almost zero inside L∩W R 0 +k+1 . ρ R 0 + i R 0 + i + 1 We fix functions h i : R + → R such that The graph of h i appears in Figure 5. We will denote by φ i t the flow of the Hamiltonian vector field of H i . Given T i ∈ R, we denote D w a i = φ i T i (D a i ). We fix > 0, and on each Lagrangian plane D a i we choose the potential function f i : D a i → R such that

h i (ρ) = 0, if ρ ≤ R 0 + i, -ρ + R 0 + i + 1 2 , if ρ ≥ R 0 + i + 1,
f i = f (a i ) + .
Note that the functions f i indeed are constant, since the Liouville vector field is tangent to the planes D a i . Let f w i : D w a i → R be the potential function on D w a i induced by f i using Equation ( 5). We denote by L = L∪D w a 1 ∪. . .∪D w a k , which we regard as an exact Lagrangian immersion, and by L + the Legendrian lift of L to (W × R, θ + dz) defined using the potential functions f, f w 1 , . . . , f w k . Note that an intersection point d ∈ D w a i ∩ D w a j lifts to a chord starting on D w a i and ending on D w a j if and only if f w i (d) > f w j (d), and similarly if one of the two discs is replaced by L and its potential is replaced by f . Lemma 9.4. There exist real numbers 0 < T k < . . . < T 1 and , > 0 such that, if L satisfies (L1)-(L3), then each chord of L + is of one of the following types:

(1) type a: the chords a i , going from (D w a i ) + to L + for i = 1, . . . , k, of length , (2) type b: chords b m ij consisting of all other chords from (D w a i ) + to L + for 1 ≤ i < j ≤ k and 1 ≤ m ≤ m 0 (i, j) for some m 0 (i, j), (3) type c: chords c m ij from (D w a i ) + to (D w a j ) + for 1 ≤ i < j ≤ k and 1 ≤ r ≤ m 0 (i, j), and (4) "order-reversing" type: chords from L + to (D w a i ) + for i = 1, . . . , k or chords from (D w a i ) + to (D w a j ) + for i = 1, . . . , k and i > j.

(see Figure 6). Moreover, for every i < j and m, there exists a unique rigid and transversely cut out pseudoholomorphic triangle in W having boundary on L ∪ D w a i ∪ D w a j , a positive puncture at b m ij , and negative punctures at a j and c m ij , in the order following the boundary orientation. (Positivity and negativity is determined by our choice of Legendrian lift.)

Note that the set {c m ij } could be empty for some i, j. In that case, we say that m 0 (i, j) = 0.

Proof. Recall that Properties (L1)-(L3) from Subsection 9.1 have been made to hold; in particular L ∩ W R 0 +k+2 consists of a k number of discs which may be assumed to be close to the discs D a i , i = 1, . . . , k.

The proof of the lemma at hand is easier to see if one starts by Hamiltonian isotoping L to make it coincide with D a 1 ∪ . . . ∪ D a k inside W R 0 +k+2 . (Thus, we can argue about the intersection points of the deformations D w a i and D a j , as opposed to the intersection points of D w a i and the different parts of L.) By Property (L2) it suffices to deform L in such a way that it becomes the graph dg a i for a function satisfying g a i ≡ 0 inside the subsets D a i ∩W R 0 +k+2 . Note that, in the case when L and D a j ∩ W R 0 +k+2 coincide, the intersection points b m ij and c m ij coincide as well. First, we observe that L, D w a 1 , . . . , D w a k are embedded exact Lagrangian submanifolds, and therefore there is no Reeb chord either from L + to L + or from (D w a i ) + to (D w a i ) + for any i = 1, . . . , k. From Equation ( 5), the potential of D w a i is f w i = f (a i ) + + T i (h i (e r )e rh i (e r )). Note that the quantity T i (h i (e r )e rh i (e r )) is nonincreasing in r because h i ≤ 0. Therefore f w i satisfies

f w i (w) = f (a i ) + if w ∈ D w a i ∩ W R 0 +i , f w i (w) ∈ [f (a i ) + -T i (R 0 + i + 1 2 ), f (a i ) + ] if w ∈ D w a i ∩ (W R 0 +i+1 \ W R 0 +i ), f w i (w) = f (a i ) + -T i (R 0 + i + 1 2 ) if w ∈ D w a i ∩ W e R 0 +i+1 .
Note that D w a i ∩ W R 0 +i = D a i ∩ W R 0 +i and that D w a i ∩ W e R 0 +i+1 is a cylinder over a Legendrian submanifold.

We choose positive numbers 0 < T k < . . . < T 1 such that (1) f (a 1 ) + -T 1 (R 0 + 1 + 1 2 ) < . . . < f (a k ) + -T k (R 0 + k + 1 2 ), (2) f (a i ) + -T i (R 0 + i + We observe that, for any point c ∈ D w a i ∩ D a j and any i, j = 1, . . . , k, the quantity a(c) = |f w i (c) -f j (c)| is independent of . Then we choose > 0 sufficiently small so that < min{a(c) : c ∈ D w a i ∩ D a j and a(c) = 0 for i, j = 1, . . . , k}. This implies that, for all c ∈ D w a i ∩ D a j such that a(c) = 0, the signs of f w i (c) -f j (c) and of f w i (c) -f (c) are equal. (Recall that f = f j -holds there by construction.)

Consider the set of points c m ij ∈ D w a i ∩ D a j with positive action difference 0 < f w i (c m ij ) -f j (c m ij ) = f w i (c m ij ) -(f (a j ) + ). (Here m is an index distinguishing the various points with the required property.) Then i < j and c m ij ∈ W R 0 +i+1 \ W R 0 +i ; in particular c m ij ∈ D w a i ∩ D w a j . See Figure 7. The intersection points b m ij now coincide with c m ij , but seen as intersections of L = D a j and D w a i . We now perturb L back to make it coincide with the graph of dg a i of a sufficiently small Morse function g a i near each D a i having a unique critical point consisting of a global minimum. Recall that this global minimum corresponds to the intersection point a i ∈ L ∩ D a i .

We can make the Morse function satisfy g a i C 2 ≤ for > 0 sufficiently small. In particular, this means that each intersection point c m ij still corresponds to a unique intersection point b m ij ∈ D w a i ∩ L, such that moreover f w i (b m ij ) -f (b m ij ) > 0 is satisfied. Conversely, any intersection point d ∈ D w a i ∩ L with f w i (d) -f (d) > 0 is either a i or one of the b m ij . This is the case because the only intersection point in D w a i ∩ L ∩ W R 0 +i is a i and for any intersection point d ∈ D w a i ∩ L ∩ W e R 0 +i+1 we must have f w i (d) -f (d) < 0. The existence of the triangle follows now by applying Corollary 4.22 to L ∪ D w a i ∪ D w a j intersected with the subset W R 0 +j ⊂ W . Note that, inside this Liouville subdomain, our deformed Lagrangian L is given as the graph of the differential of a small Morse function g a j on D a j (using a Weinstein neighbourhood of the latter); hence the lemma indeed applies. Here the monotonicity property for the symplectic area of a pseudoholomorphic disc must be used in order to deduce that the triangles of interest can be a priori confined to the same Liouville subdomain. The triangle provided by the previous lemma is the stepping stone in the inductive construction of an augmentation for A(L + ).

f (a 1 ) + f (a 2 ) + f (a 3 ) + f (a 1 ) + -T 1 (R 0 + 3 2 ) f (a 2 ) + -T 2 (R 0 + 5 2 ) f (a 3 ) + -T 3 (R 0 + 7 2 ) R 0 R 0 + 1 R 0 + 2 R 0 + 3 R 0 + 4 f 1 f 2 f 3
Lemma 9.5. The Chekanov-Eliashberg algebra (A(L + ), d) of L + admits an augmentation ε : A(L + ) → F such that ε(a i ) = 1 for all i = 1, . . . , k. Moreover, this augmentation vanishes on the order reversing chords.

Proof. Set L i := D w a i and L k+1 := L. recall that each of the L i is embedded, and therefore there is no Reeb chord from L + i to itself for any i. Thus all Reeb chords go between different connected components of L + and are as described in Lemma 9.4.

The bilateral ideal of A(L) generated by the order reversing chords is preserved by the differential, and therefore the quotient algebra, which we will denote by A → , inherits a differential d → . We can identify A → to the subalgebra of A generated by the chords of type a, b and c, and d → to the portion of the differential of A(L) involving only generators of A → .

On A → we define a filtration of algebras (32)

Z = A → k+1 ⊂ A → k ⊂ . . . ⊂ A → 0 = A → ,
flow of H i for sufficiently large time. The flow can push D w a i close to ∂W , and therefore we can no longer assume that the wrapped planes D w a i are safe. However, L and the planes D a i were safe, and therefore the excellent dynamics of the Hamiltonian vector fields of H i implies that all intersection points of type a, b, and c between L and the planes D w a 1 , . . . , D w a k correspond to Hamiltonian chords contained in the complement of N . Since g = 1 outside N , the proof of Theorem 1.2 is concluded in the same way as the proof of Theorem 1.1.

Hochschild homology and symplectic cohomology

In this section using the work of Ganatra [START_REF] Ganatra | Symplectic cohomology and duality for the wrapped Fukaya category[END_REF] we prove that Weinstein manifolds are non-degenerate. Recall from [22, Definition 1.1] that a Liouville manifold is called non-degenerate if it admits a finite collection of objects B such that the map [OC] : HH n- * (B, B) → SH * (W ) has the unit in the symplectic cohomology in its image. Here B denotes also the full subcategory of WF(W, θ) generated by B and HH * denotes the Hochschild homology. Such a collection B is called an essential collection of (W, θ). More precisely, we prove the following result: Theorem 10.1. Let (W, θ) be a Weinstein manifold and B its collection of cocore discs. Then B is an essential collection of (W, θ).

Theorem 10.1 follows directly from our geometric construction, [22, Proposition 14.1] applied to the product Weinstein manifold W × W , together with [START_REF] Gao | Functors of wrapped Fukaya categories from Lagrangian correspondences[END_REF]Theorem 1.1]. In order to conclude this, we first briefly recall the construction of the category W 2 from [START_REF] Ganatra | Symplectic cohomology and duality for the wrapped Fukaya category[END_REF].

First we recall some facts about the version of the wrapped Fukaya category for the product Liouville manifold (W × W, θ ⊕ (-θ)).

Lemma 10.2. The product Weinstein manifold W × W has a Weinstein handle decomposition for which the Lagrangian cocores are precisely the products of the cocores D i × D j , where D i denotes a Lagrangian cocore in the Weinstein decomposition of (W, θ, f).

Let W := WF(W, θ). Further, let W 2 be a version of the wrapped Fukaya category for the product Liouville manifold (W × W, θ ⊕ (-θ)) as defined in [START_REF] Ganatra | Symplectic cohomology and duality for the wrapped Fukaya category[END_REF]. This category contains a full subcategory W 2 split ⊂ W 2 generated by "split" Lagrangian products L i × L j ⊂ W × W. Ganatra also constructs a version W 2 of the latter category, in which the split Lagrangian submanifolds form a strictly unital subcategory W 2 split ⊂ W2 ,

Theorem 1 . 4 .

 14 Let (W, θ, f) be a Weinstein manifold of finite type. Let D the full A ∞ subcategory of WF (W, θ) whose objects are the Lagrangian cocore planes. Then the open-closed map (1) OC : HH * (D, D) → SH * (W )

  ) a smooth function I : ∂S → R which satisfies (a) dI(L) = αI for some function α : ∂S → R + which is constant outside of a compact set and (b) dI(C) > 0, where C is a tangent vector field on ∂S such that ι C dθ| ∂S = 0 and dθ(C, N ) > 0 for an outward pointing normal vector field N ; (4) a proper Morse function f : S → R bounded from below having finitely many critical points, such that L is a pseudogradient of f and satisfying moreover (a) df(C) > 0 on {I > 0} and df(C) < 0 on {I < 0}, (b) the Hessian of a critical point of f on ∂S evaluates negatively on the normal direction N , and (c) there is a constant c ∈ R whose sublevel set satisfies {f ≤ c} ⊂ S \ ∂S and contains all interior critical points of f.

( 2 )

 2 any critical point p ∈ ∂S of f lies inside {I = 0} = F and is also a critical point of f F , (3) the Morse indices of the two functions satisfy the relation ind f (p) = ind f F (p) + 1, (4) the skeleton satisfies S sk ∩ ∂S = F sk .

  .

4. 1 .

 1 Cylindrical Hamiltonians. Definition 4.1. Let W be a Liouville manifold. A Hamiltonian function H

  +l d +d+1 (∂D 2 ) where d + 1 points ζ m 0 , . . . , ζ m d (ordered counterclockwise) are labelled as mixed and the other ones ζ i j , with i = 0, . . . , d and j = 1, . . . , l i (ordered counterclockwise and contained in the sector from ζ m i to ζ m i+1 ) are labelled as pure. Given ζ ∈ R l 0 |...|l d , we denote ∆ ζ = D 2 \ ζ. For i = 0, . . . , d let ∂ i ∆ ζ be the subset of ∂∆ ζ whose closure in ∂D 2 is the counterclockwise arc from ζ m

Lemma 4 . 19 .

 419 Under the hypotheses of Lemma 4.18, the element e ∈ CF ((L, ε), (L , ε )) is a cycle. Furthermore, e is a boundary if and only if CF (T, (L, ε)) = 0 for every Lagrangian T.

Corollary 4 . 22 .

 422 For a suitable Morse-Smale pair (f, g) and almost complex structure as in Lemma 4.21 there is a unique rigid and transversely cut out pseudoholomorphic triangle with corners at e, c ∈ L ∩ L f and the corresponding double point c ∈ L f for any connected gradient flow-line from c ∈ L ∩ L f to the local minimum e ∈ C. The triangle is moreover contained inside a small neighbourhood of the same flow-line.

•

  ) are isomorphic. In fact the one-dimensional parametrised moduli spaces have boundary points only at J δ - • and J δ +• , and this implies that the algebraic count of elements of the zero-dimensional moduli spaces is the same for J δ -

  as in Definition 5.13 by replacing the Hamiltonian H by the s-dependent Hamiltonian H R s . Counting pairs (R, u) where R ∈ R and u ∈ M -1

.

  The statement is now a consequence of the maximum principle for pseudoholomorphic curves insideW e R ∼ = [R, +∞) × V which• satisfy a cylindrical boundary condition, and • are pseudoholomorphic for a cylindrical almost complex structure. Namely, by e.g. [25, Lemma 5.5], the symplectisation coordinate r : W e R → [R, +∞) restricted to such a curve cannot have a local maximum.

Figure 3 .Figure 4 .

 34 Figure 3. The front projections of Λ + and Λ -.

Proposition 9 . 3 .

 93 Let L ⊂ W be an exact Lagrangian submanifold with cylindrical end. If L ∩ W sk = L ∩ (C 1 ∪ . . . ∪ C l ) = {a 1 , . . . , a k } and the intersections are transverse, then L is isomorphic in T wWF(W, θ) to a twisted complex built from the objects D a 1 , . . . , D a k .

Figure 5 .

 5 Figure 5. The graph of h i .

e r 0 +j e r 0 +j- 1 0Figure 6 .

 16 Figure 6. A schematic picture of the wrapping and of the small triangle with i < j. (Also c.f. Equation (30) combined with Figure5.)

1 2 )

 2 < min L f for all i = 1, . . . , k, (3) there are no intersection points between L, D a 1 , . . . , D a k , D w a 1 , . . . D w a k in their cylindrical parts, and (4) at every intersection point between L, D a 1 , . . . D a k , D w a 1 , . . . D w a k the respective potential functions are different, except for intersection points p ∈ L ∩ D w a i where H i (p) = h i (e r ) = 0. The last two conditions are achieved by choosing T 1 , . . . , T k generically.

Figure 7 .

 7 Figure 7. The profiles of f w i .

  R with Hamiltonian flow ϕ t such that ι 1 = ϕ 1 • ι 0 , and moreover ι t = ϕ t • ι 0 has cylindrical ends for all t ∈ [0, 1].

	where Λ is a Legendrian submanifold of (V, ξ). Then we say that L is
	cylindrical over Λ. Here Λ can be empty (if L is compact) or disconnected.
	There are different natural notions of equivalence between immersed exact
	Lagrangian submanifolds. The stronger one is Hamiltonian isotopy.
	Definition 2.20. Two exact Legendrian immersions (L, ι 0 ) and (L, ι 1 ) with
	cylindrical ends are Hamiltonian isotopic if there exists a function H : [0, 1]×
	W → Remark 2.21. If f 0 : L → R is the potential of (L, ι 0 ), by Lemma 2.3 we
	can choose
	(5)
	2.19. Let (W, θ, f) be a Weinstein manifold. The Lagrangian
	cocore planes D p introduced in Section 2.2 are Lagrangian submanifolds
	with cylindrical ends.
	Properness of ι and Condition (3) imply that for every immersed exact
	Lagrangian submanifold with cylindrical end ι : L → W there is R > 0
	sufficiently large such that ι(L) ∩ W e R corresponds to [R, +∞) × Λ under the
	identification
	(W e R , θ) ∼ = ([R, +∞) × V, e r α),

  holomorphic map with boundary in L. If u has finite area and no puncture at which the lift of u| ∂∆ ζ to L has a continuous extension, then lim = p i for some p i ∈ D. Since the boundary of u switches branch near p i , the following dichotomy thus makes sense: Definition 3.4. We say that ζ i is a positive puncture at p i if lim

	z→ζ i	u(z)

  ) acts on N L (p 0 ; p 1 , . . . , p d ; J) by reparametrisations; the quotient is the moduli space N L (p 0 ; p 1 , . . . , p d ; J). Note that the set p 1 , . . . , p d can be empty. In this case, the elements of the moduli spaces N(p 0 ; J) are called teardrops.

	Proposition 3.5. For a generic J satisfying the condition ( †), the moduli
	space N k L (p 0 ; p 1 , . . . , p d ; J) is a transversely cut out manifold of dimension
	k. In particular, if k < 0 it is empty; if k = 0 it is compact, and therefore
	consists of a finite number of points; and if k = 1, it can be compactified in
	the sense of Gromov, see [17, Section 2.2].
	The boundary of the compactification of the moduli space N 1 L (p 0 ; p 1 , . . . , p d ; J)
	is
	(6)
	q∈D 0≤i<j≤d
	If p 1 , . . . , p d are double points of
	L (possibly with repetitions), we denote by N L (p 0 ; p 1 , . . . , p d ; J) the set of
	pairs (ζ, u) where:
	(1) ζ ∈ R d+1 and u : ∆ ζ → W is a J-holomorphic map,
	(2) u(∂∆ ζ ) ⊂ L, and
	(3) ζ 0 is a positive puncture at p 0 and ζ i , for i = 1, . . . , d, is a negative
	puncture at p i .
	The group Aut(D 2 Given u ∈ N L (p 0 ; p 1 , . . . , p d ; J), let D u be the linearisation of the Cauchy-
	Riemann operator at u. By standard Fredholm theory, D u is a Fredholm
	operator with index ind(D u ). We define the index of u as
	ind(u) = ind(D u ) + d -2.
	The index is locally constant, and we denote by N k L (p 0 ; p 1 , . . . , p d ; J) the
	subset of N L (p 0 ; p 1 , . . . , p d ; J) consisting of classes of maps u with ind(u) =
	k.
	The following proposition is a version of [17, Proposition 2.3]:

  -regular.

	Proof. Since
	∂v ∂s	(s, t) = dϕ t	∂u ∂s	(s, t)	and
	∂v ∂t	(s, t) = dϕ

t ∂u ∂t (s, t) -χ (t)X(χ(t), u(s, t)) , u satisfies the Floer equation with Hamiltonian H if and only if ϕ * u satisfies the Floer equation with Hamiltonian 0. The map ϕ * is invertible because ϕ t is for each t. Finally, we observe that dϕ intertwines the linearised Floer operators at u and v.

  fix s-dependent Hamiltonian function H + s and H - s such that H + s = H + for s ≥ 1 and H + s = H for s ≤ 0, and H - s = H for s ≥ 0 and H - s = H - for s ≤ -1. In order to prove (3) we introduce the family of Hamiltonian functions

	H R s =	H + s-R for s ≥ 0, and H s+R for s ≤ 0
	with R > 0. By (1), Φ H R s induces Φ * H

+ ,Hfor all R. For R 0 there is an identification

  . .∪L m as considered in Lemma 4.18. Consider the cycle e ∈ CW ((L(a 1 , . . . , a k ), ε), (L , ε )) supplied by Corollary 8.10. As above, e is an element in the summandCF ((L(a 1 , . . . , a k ), ε), (L , ε ); 0, J • )⊂ CW ((L(a 1 , . . . , a k ), ε), (L , ε )); J

• )

with wrapping parameter w = 0. Then {(L(a 1 , . . . , a k ), ε), (L , ε )}, e is a twisted complex L corresponding to the cone of µ 2 (e, •). The last part of Corollary 8.10 combined with Lemma 8.13 then establishes the sought quasi-isomorphism. Indeed, summand CF (T, (L, ε e ); w • H, J • ) ⊂ CW (T, (L, ε e ); J • ) in the homotopy direct limit which computes the homology of the cone is acyclic by Corollary 8.10. (Here L = L(a 1 , . . . , a k ) ∪ L as in the proof of the latter corollary.)

  (L1)-(L3) provides sufficient control of the intersections of L and the Lagrangian skeleton. Later in Lemma 9.4 we will use this in order to perform a deformation of the immersed Lagrangian submanifoldL ∪ D a 1 ∪ . . . ∪ D a kby Hamiltonian isotopies applied to the different components D a i . The goal is to obtain an exact Lagrangian immersion admitting a suitable augmentation; the corresponding bounding cochain (see Lemma 8.15) will then give us the twisted complex which exhibits L as an object built out of the different D a i . 9.2. Proof of Theorem 1.1. In this section we prove Theorem 1.1 assuming the results of Section 9.3. The proof of Theorem 1.2 follows similarly with just a bit of additional work; see Subsection 9.4. The result is a corollary of the following proposition.

We abuse the notation by denoting the pull back by the inclusion as a restriction.

B. CHANTRAINE, G. DIMITROGLOU RIZELL, P. GHIGGINI, AND R. GOLOVKO

φ * (Conf m (R)).

The embeddings φ defined above extend to diffeomorphisms

if

where 0 = 0 for the sake of the formula.

for some i = 1, . . . , n, and (5) ν n • φ * = ν m for all m < n and all φ ∈ hom(n, m).

Proof. We can construct the sequences κ n and ν n inductively over n using the fact that the set of functions satisfying (1)-( 4) is convex.

Lemma 5.12. For every n, there is a contractible set of smooth maps

To these moduli spaces correspond four types of boundary configuration for the compactification of the one-dimensional parametrised moduli spaces

, which we write schematically as:

There is also one fifth type of boundary configuration:

In order to prove Proposition 5.20 we split the homotopy J • • into pieces, each of which contains only one element of ∆ and we prove that, for each pieces, the corresponding diagram [START_REF] Ekholm | Legendrian contact homology in P ×R[END_REF] commutes up to homotopy. Putting every piece together, we will obtain the result. We rescale each piece of homotopy so that it is parametrised by [-1, 1] and the bifurcation point is 0. Proof. Assume without loss of generality that the nonempty moduli space of negative formal dimension is N -1 L 1 (q 1 0 ; q 1 1 , . . . , q 1 d ; J 0 1 ). We have proved in Lemma 5.4 that Υ ± are the identity map. We define a map

The first square commutes because of the structure of the compactification of the one-dimensional parametrised moduli spaces and the second by the choice of augmentations. (Recall that Υ ± simply are the identity maps here.)

Lemma 5.22. Let ∆ = {0} be of type (ii). Then Diagram (17) commutes.

Proof. We assume, without loss of generality, that the moduli space of negative formal dimension is M 0 L 1 ,L 1 (q 1 , y -, q 0 , y + ; H + , J 0 • ). By Lemma 5.5 the Lemma 8.9. Let L be an immersed exact Lagrangian submanifold with an augmentation ε and let L + , (L ) + be Legendrian lifts such that L + is above (L ) + . When Lemma 8.7 is applied to an augmentation

induced by the cycle c ∈ CF ((L, ε), (L , ε )) as in Lemma 4.20, then the push-forward of the augmentation under the DGA morphism

which moreover is mapped to c under the quasi-isomorphism from Proposition 8.8 .

Proof. There is no Reeb chord starting on either L + or L + (a 1 , . . . , a k ) and ending on (L ) + , so the pushed-forward augmentation is automatically of the form ε c . Lemma 4.20 then implies that c is a cycle.

The last statement is an algebraic consequence of the fact that the discs counted by the DGA morphism Φ Σ induced by the surgery cobordism can be identified with the discs counted by the quasi-isomorphism from Proposition 8.8. Now assume that L is a push off of L as constructed in Lemma 4.18, and let e ∈ CF (L, L ) be the "unit" defined by the sum of the local minima e i of the Morse function on the connected components of L; i.e. e = e i .

Corollary 8.10. The cycle e ∈ LCC ε,ε (L + (a 1 , . . . , a k ), (L ) + ) provided by Lemma 8.9 (which is mapped to e under the quasi-isomorphism by Proposition 8.8) satisfies the property that

is a quasi-isomorphism for any exact Lagrangian submanifold T with cylindrical end.

Proof. Consider the Legendrian lift L + = L + ∪ (L ) + such that L + is above (L ) + . Then let the lift L + (a 1 , . . . , a k ) = L + (a 1 , . . . , a k ) ∪ (L ) + , be specified uniquely by the requirement that it coincides with the first lift outside of a compact subset.

Recall that e is closed by Lemma 4.19 and by Lemma 4.20 there is thus an induced augmentation ε e of A(L + ). Recall that this augmentation coincides with ε and ε when restricted to the generators on the components L + and (L ) + , respectively, while ε e (e i ) = 1 holds for any chord corresponding to a local minimum and ε e (c) = 0 for every other chord c between L + and (L ) + .

Applying Proposition 8.8 to the Legendrian L + (a 1 , . . . , a k ) obtained by surgery on L, yields a quasi-isomorphism

(Here we use that ε e = ε e by Lemma 8.9.) The complex on the right-hand side is acyclic by Lemma 4.18, and hence so is the complex on the left-hand side. The sought statement is now a consequence of the straight-forward algebraic fact that the complex

is equal to the mapping cone of µ 2 (e, •) : CF (T, (L(a 1 , . . . , a k ), ε)) → CF (T, (L , ε )). 8.4. Twisted complexes. The aim of this section is to relate the geometric notion of Lagrangian surgery to the algebraic notion of twisted complex in the wrapped Fukaya category. We first recall the definition of a twisted complex in an A ∞ -category.

Given a unital A ∞ -category A, we describe the category T wA of twisted complexes over A and recall its basic properties. We introduce the following notation: given a number d of matrices A i with coefficients in the morphism spaces of an A ∞ -algebra, we denote by µ d A (A d , . . . , A 1 ) the matrix whose entries are obtained by applying µ d A to the entries of the formal product of the A i 's. Definition 8.11. A twisted complex over A is given by the following data:

• a finite collection of objects L 0 , . . . , L k of A for some k, • integers κ i for i = 0, . . . , k, and

The integers κ i are degree shifts and are part of the definition only if the morphism spaces hom A (L i , L j ) are graded, and otherwise are suppressed.

Given two twisted complexes L = ({L i }, {κ i }, X) and L = ({L i }, {κ i }, X ) we define hom T wA (L, L ) := i,j hom A (L i , L j )[κ i -κ j ] and, given d + 1 twisted complexes L 0 , . . . , L d , we define A ∞ operations

where A → i is generated by all chords a s , b m sj , c m sj with s ≥ i. Given a chord c of L + , we denote its action by a(c). The differential d → preserves the action filtration on A → (and on all its subalgebras). We assume that (i) the actions of all chords b m ij and c m ij are pairwise distinct and, (ii) for all i, j, m, the actions a(b m ij ) and a(c m ij ) are close enough that, whenever a(c

The first is a generic assumption, and the second is achieved by choosing > 0 sufficiently small in Lemma 9.4.

For each fixed i we define a total order on the pairs (j, m) by declaring that (h, l)

When the index i is clear from the context, we will simply write ≺.

We know that d → a i = 0 by action reasons and d → b m ij , a j c m ij = 1 by the last part of Lemma 9.4. Combining this partial information on the differential d → and the assumptions (i) and (ii) above with the action filtration, we obtain the following structure for the differential:

w hm lj c l ih with α m j , β mh jl ∈ Z and w hm lj , w hm lj ∈ A → i+1 . Then the filtration (32) is preserved by d → . We want to define an augmentation ε : A → → Z such that ε(a i ) = 1 for all i = 1, . . . , k working by induction on i.

For i = k + 1, there is nothing to prove since A → k+1 = Z. Suppose now we have defined an augmentation ε : A → i+1 → Z. We will extend it to an augmentation ε : A → i → Z by an inductive argument over the action of the chords c m ij . For this reason in the following discussion i will be fixed. We define ε(a i ) = 1 and ε(b m ij ) = 0 for all j and m. To define ε on c m ij we work inductively with respect to the order ≺ induced by the action. Suppose that we have defined ε(c l ih ) for all c h il such that (h, l) ≺ (j, m). Then we can achieve ε(d → b m ij ) = 0 by prescribing an appropriate value to ε(c m ij ) = ε(a j c m ij ), since the values of ε on all other chords appearing in the expression of d → b m ij already have been determined. Now we have defined ε on all generators of A → i and, by construction, ε(d → d) = 0 for every chord d in A → i except possibly for the chords c m ij . We will prove that in fact ε(d → c m ij ) = 0 holds as well, and thus show that ε is an augmentation on A → i . Once again we will argue by induction on the action of the chords c m ij . If (j, m) is the minimal element for the order ≺, then d → c m ij = 0 and therefore ε(d → c k ij ) = 0. Suppose now that we have verified that ε(d → c l ih ) = 0 for all (h, l) ≺ (j, m).

We have ε(d → b l ih ) = 0 by construction and ε(d → (w hm lj c l ih )) = 0 by the induction hypothesis. From this we conclude that ε(d → c m ij ) = 0. Finally we simply precompose ε with the projection A(L) → A → and obtain an augmentation of A(L) satisfying the required conditions. 9.4. Proof of Theorem 1.2. In this section we prove the generation theorem for Weinstein sectors. If (S, θ, I, f) is a Weinstein sector which is cylindrical at infinity over the contact manifold (V, α) with convex boundary, we fix once and for all a function g : V → R as in Definition 4.7 and a corresponding convex collar N of ∂V as in Definition 4.5.

Let H i ∼ = D δ T * H i be a half-handle of the Weinstein handle decomposition of S (using the notation of Section 2.3). We denote ∂ + H i ∼ = S δ T * H i . The Reeb vector field on ∂ + H i induced by the canonical Liouville form is the cogeodesic flow of H i for the flat metric, and therefore ∂ + H i ∩ ∂W has a cocompact collar. This is an important observation, because it allows us to choose N such that g ≡ 1 on H i \ N for all Weinstein half-handles H i .

We say that a cylindrical exact Lagrangian submanifold of W is safe if it is cylindrical over a Legendrian submanifold contained in V \ N . Every cylindrical exact Lagrangian submanifold of W is Hamiltonian isotopic to one which is safe by a cylindrical Hamiltonian isotopy. Therefore, from now on we will assume that all Lagrangian submanifolds are safe unless we explicitly state the contrary.

As in the previous case, we isotope L so that it is disjoint from the subcritical part of S sk and intersects the cores of the critical Weinstein handles and halfhandles transversely in a finite number of points a i , . . . , a k , and for each point a i we consider the cocore plane D a i passing through it. The wrapping of the cocore planes taking place in the proof of Lemma 9.4 is the point where the proof requires a little more work than the case of a Weinstein manifold.

We define the Hamiltonian functions H i of Lemma 9.4 to be H i (w) = g(w)h i (e r(w) ) and denote by D w a i the image of D a i under the Hamiltonian and where the canonical inclusion W 2 ⊂ W 2 is a quasi-isomorphism.

Let B be a collection of objects in WF(W, θ) and B 2 denotes the collection of objects of W 2 which are products of objects in B. In addition, let ∆ be the diagonal of W × W . Observe that [START_REF] Ganatra | Symplectic cohomology and duality for the wrapped Fukaya category[END_REF]Proposition 14.1] implies that:

Proposition 10.3. If ∆ is generated by B 2 in W 2 , then B is an essential collection.

From Lemma 10.2 it follows that Theorem 10.1 is a direct corollary of Propositions 9.3 and 10.3. Note that the caveat of considering a product of objects and wrapping using a product Hamiltonian as in [START_REF] Ganatra | Symplectic cohomology and duality for the wrapped Fukaya category[END_REF] is resolved by the recent result [24, Theorem 1.1] of Gao.