Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors
Baptiste Chantraine Georgios Dimitroglou Rizell Paolo Ghiggini Roman Golovko
math.SG math.DG math.GT 53D37, 53D40, 57R17



B. Chantraine]Baptiste Chantraine
G. Dimitroglou Rizell]Georgios Dimitroglou Rizell
P. Ghiggini]Paolo Ghiggini
R. Golovko]Roman Golovko

Université de Nantes, France.
baptiste.chantraine@univ-nantes.fr

Uppsala University, Sweden
georgios.dimitroglou@math.uu.se

Université de Nantes, France.
paolo.ghiggini@univ-nantes.fr


Charles University, Czech Republic.
golovko@karlin.mff.cuni.cz

[2010]Primary 53D37; Secondary 53D40, 57R17.



The wrapped Fukaya category of Weinstein sectors]Geometric generation of the
wrapped Fukaya category of Weinstein manifolds and sectors
 Engendrement géométrique des catégories de Fukaya enroulées des variétés et secteurs de Weinstein
[
    [
    December 30, 2023
=====================




We prove that the wrapped Fukaya category of any 2n-dimensional 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 cohomology and
the fairly simple observation that Floer cohomology 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. We work mainly in the “linear
setup” for the wrapped Fukaya category, but we also extend the proofs to the “quadratic” and “localisation” setup. This is necessary for dealing with Weinstein sectors and for the applications.



  Nous démontrons que la catégorie de Fukaya enroulée d'une variété (ou plus généralement d'un secteur) de Weinstein W de dimension 2n est engendrée par les variétés instables des points critiques d'indice n de son champ de Liouville. Notre preuve, de nature géométrique, repose sur une formule pour la cohomologie de Floer d'une chirurgie et sur l'observation relativement simple que la cohomologie de Floer d'une lagrangienne disjointe du squelette isotrope de la variété de Weinstein s'annule (aucune condition supplémentaire n'est demandée au squelette). En appliquant le critère d'engendrement au produit W× W nous obtenons en corollaire que l'application ouverte-fermée de la cohomologie de Hochschild de la catégorie de Fukaya enroulée de W vers sa cohomologie symplectique est un isomorphisme, prouvant une conjecture de Seidel. Nous travaillons principalement avec la définition “linéaire” de la catégorie de Fukaya enroulée mais nous étendons les preuves aux définitions “quadratique” et “par localisation”. Ces modifications sont nécessaires pour traiter secteurs de Weinstein et pour certaines applications.






§ 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 polygons 
with Lagrangian boundary conditions. Wrapped Floer cohomology was defined by A. Abbondandolo and M. Schwarz <cit.>, at least for cotangent fibres, but the general definition and the chain level construction needed to define an A_∞-category is due to M. Abouzaid and P. Seidel  <cit.>. 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 <cit.>, and by S. Ganatra, J. Pardon and V. Shende, who later introduced the similar notion of Liouville sector in <cit.>.

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

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
<ref> for the definition of the terminology used in the statement.

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).


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 <cit.> 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,  which is
sufficient for most applications, but not for all; see for example
<cit.>. Moreover,  Bourgeois, Ekholm
and Eliashberg's proposed proof relies on their Legendrian surgery
formula, whose analytic details  are not complete (see
<cit.> for recent development in that direction).

Most generation results so far, including that of
Bourgeois, Ekholm and Eliashberg, rely on Abouzaid's split-generation criterion <cit.>. On the contrary, our proof is more direct and similar in
spirit to Seidel's proof in <cit.> 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 <cit.>.  Theorems <ref> and <ref> have been proved independently also by Ganatra, Pardon and Shende in <cit.>.

A product of Weinstein manifolds is a Weinstein manifold. Therefore, by applying Theorem <ref> to the diagonal in a twisted product, and using
results of S. Ganatra <cit.> and Y. Gao <cit.>, we obtain the following result.

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

    𝒪C HH_*(𝒟, 𝒟) → SH^*(W)

is an isomorphism.

In Equation (<ref>) HH_* denotes Hochschild homology, SH^* denotes
symplectic cohomology and 𝒪C is the open-closed map defined in <cit.>. Theorem <ref> in particular proves that

    𝒪C HH_*(𝒲F(W, θ), 𝒲F(W, θ)) → SH^*(W)

is an isomorphism. This proves a conjecture of Seidel in <cit.> 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 <cit.>.

The above result implies in particular that Abouzaid's generation criterion <cit.> 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 <ref>, 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 <cit.> 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,  allows for an identification of the Floer complexes and the open-closed map  with the original complexes and map tensored with the Novikov field.  The generalisation of Abouzaid's generation criterion to the monotone setting established in <cit.> thus shows that

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.


The strategy employed in the proof of Theorem <ref> for showing generation fails for non-exact Lagrangian submanifolds in two crucial steps: in Section <ref> and Section <ref>. 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 ×, 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.





 §.§ Comparison of setups

There are three “setups” in which the wrapped Fukaya category is defined: the “linear setup”, where the Floer equations are perturbed by Hamiltonian functions with linear growth at infinity and the wrapped Floer chain complexes are defined as homotopy limits over Hamitonians with higher and higher slope, the “quadratic setting”, where the Floer equations are perturbed by Hamiltonian functions with quadratic growth at infinity, and the “localisation setting”, where the Floer equations are unperturbed and the wrapped Fukaya category is defined by a categorical construction called  localisation. The linear setup was introduced by Abouzaid and Seidel in <cit.> and the quadratic setup by  Abouzaid in <cit.>. The latter is used also in Sylvan's definition of the partially wrapped Fukaya category and in the work of Ganatra <cit.> and Gao <cit.> which we use in the proof of Theorem <ref>. The localisation setup is used in <cit.> because the linear and quadratic setups are not available on sectors for technical reasons.
All three setups are expected to produce equivalent A_∞ categories on Liouville manifolds.
We chose to work in the linear setup for the proof of Theorem <ref>, but our proof can be adapted fairly easily to the other setups. Moreover these extensions are necessary to prove Theorem <ref> and Theorem <ref>. We will describe the small modifications we need in the localisation setup in Section <ref> and those we need in the quadratic setup in Section <ref>.



 §.§ Strategy of the proof of Theorem 1.1

The strategy of the proof of Theorem <ref> 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, for dimensional reasons,  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.
We assume that  the Lagrangian planes are all pairwise disjoint. The unstable manifolds of the index n critical points are what we call the Lagrangian cocore planes.


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 cohomology in <cit.>. 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 <cit.>. 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 <ref> we recall some generalities about Weinstein manifolds and sectors. In Section <ref> we recall the definition and the basic properties of Legendrian contact cohomology. In Sections <ref> and <ref> we define the version of Floer cohomology 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 <ref> we define wrapped Floer cohomology for Lagrangian immersions using the constructions of the previous two sections. In Section <ref> we prove that an immersed Lagrangian submanifold which is disjoint from the skeleton is Floer homologically trivial. In Section <ref> we prove that Lagrangian surgeries correspond to twisted complexes in the wrapped Fukaya category.
In Section <ref> we finish the proof of Theorem <ref> and in particular we construct the bounding cochain for L.  In Section <ref> we prove Theorem <ref>.  We briefly recall  the construction of the wrapped Fukaya category for sectors in the localisation setup from <cit.> and show how all previous arguments adapt in this setting. Finally, in Section <ref> we prove the isomorphism between Hochschild homology and symplectic cohomology.  This requires that we adapt the proof of Theorem <ref> to the quadratic setting.







 §.§ 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. We also thank an anonymous referee for pointing out that it was necessary to use the localisation setup to define the wrapped Fukaya category of Liouville sectors.

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 EPSRC 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”,  GAČR EXPRO Grant 19-28628X and 4EU+/20/F3/21 Minigrant. 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.


 §.§ Liouville manifolds

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

    ι_ℒ d θ = 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 (ℒ_w) >0 if (w)>R_0, and

  (ii) d _w(ℒ_w)=1 if (w) ≥ R_0+1.

In particular, R_0 is the unique critical value of , which is of course highly
nondegenerate, and every other  level set is a contact type hypersurface.

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

For every R ∈ [R_0, + ∞) we denote W_R= ^-1([R_0, R]), W_R^e =
W ∖int(W_R) and V_R = ^-1(R).

The subsets W_R^e will be called the ends of W. The Liouville flow of
(W, θ) induces an identification

    ([R_0+1, + ∞) × V, e^r α) ≅ (W_R_0+1^e, 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
(<ref>) is given by (r, v) ↦ϕ_r(v). Let ξ = α
be the contact structure defined by α. Every V_R, for R>R_0, is contactomorphic
to (V, ξ).  Under the identification (<ref>), the
function 𝔯, restricted to W_R_0+1^e, corresponds to the projection to [R_0+1,+ ∞) in the sense that the following diagram commutes

    ([R_0+1, + ∞) × V [rr]^ϕ[dr]       W_R_0+1^e [dl]_𝔯.
        [R_0+1, + ∞)



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 →. Flows of Hamiltonian
vector fields are, of course, the main source of exact symplectomorphisms. Given a function
H  [- t_-, t_+] × W →, where t_±≥ 0 and are allowed to be infinite, we
define the Hamiltonian vector field X_H by

    ι_X_H d θ = -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.

Let H  [- t_-, t_+] × W → 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=  ∫_0^t (- H_σ + θ(X_H_σ)) ∘φ_σ  dσ.



We compute

    φ_t^*θ - θ = ∫_0^t d/dσ(φ_σ^* θ)
    dσ =  ∫_0^t φ_σ^*(L_X_H_σθ) dσ =  
       ∫_0^t
    φ_σ^* (ι_X_H_σ d θ + d ι_X_H_σθ)
    dσ = ∫_0^t φ_σ^*  (- dH_σ + d (θ(X_H_σ))) dσ.




 §.§ Weinstein manifolds

In this article we will be concerned mostly with Weinstein manifolds of finite type.
We recall their definition, referring to <cit.> for further details.

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 d θ is a symplectic form and the
Liouville vector field ℒ associated to θ is complete, and

  (iii) a proper Morse function 𝔣 W → bounded from below such that
ℒ is a pseudogradient of 𝔣 in the sense of
<cit.>:
i.e.
    d 𝔣(ℒ)   ≥δ(ℒ^2 +  d 𝔣^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 <ref>, 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
Δ_p  and an unstable manifold D_p which are both exact Lagrangian submanifolds.
We will call the unstable manifolds Δ_p of the critical points of index n the Lagrangian cocore planes.


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>0ϕ^-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 <cit.>
and <cit.> 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
d θ 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 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 <cit.> for the description of the standard model, and <cit.> for how to produce the Weinstein homotopy.

We could easily modify 𝔣 so that it agrees with  on W_0^e. However,
this will not be necessary.

The core of the Weinstein handle ℋ_i is the Lagrangian disc C_i=
Δ_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 <cit.> characterised, roughly speaking, by retracting over a Lagrangian skeleton with boundary. In Section <ref> below we will then show that any Weinstein pair as introduced in <cit.> can be completed  to a Weinstein sector.


A Weinstein sector (S,θ, I, 𝔣) consists of:


  * an even dimensional smooth manifold with boundary S;

  * a one-form θ on S such that d θ is a symplectic form and the associated Liouville vector field ℒ is complete and everywhere tangent to ∂ S;

  * a smooth function I ∂ S →
    which satisfies


  * dI(ℒ)= α I for some function α∂ S →_+ which is constant outside of a compact set and

  * dI(C)>0, where C is a tangent vector field on ∂ S such that ι_C d θ|_∂ S=0 and dθ(C,N)>0 for an outward pointing normal vector field N;


  * a proper Morse function 𝔣 S → bounded from below having finitely many critical points, such that ℒ is a pseudogradient of 𝔣 and satisfying moreover


  * d 𝔣(C)>0 on { I >0 } and d 𝔣(C)<0 on { I <0 },

  * the Hessian of a critical point of 𝔣 on ∂ S evaluates negatively on
the normal direction N, and

  * there is a constant c ∈ whose sublevel set satisfies {𝔣≤ c }⊂ S ∖∂ S and contains all interior critical points of 𝔣.




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 <cit.>.

 
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[We abuse the notation by denoting the pull back by the inclusion as a restriction.] and 𝔣_F= 𝔣|_F. To show that the belt is actually a Weinstein manifold it is enough to observe that d θ_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) = α I, and therefore the Liouville vector field of θ_F is ℒ_F=ℒ|_F.

Let κ∈ 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 <ref>, 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 <ref>. We then write S_R 𝔯^-1[R_0,R] and S_R^e = S ∖int(S_R).



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 >0ϕ^-t(S_0).





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:


  * critical points of 𝔣 on ∂ S are also critical points of 𝔣|_∂ S and vice versa,

  * any critical point p ∈∂ S of 𝔣 lies inside {I=0} = F  and is also a critical point of 𝔣_F,
 
  * the Morse indices of the two functions satisfy
  the relation

    ind_𝔣(p)=ind_𝔣_F(p)+1,


  * the skeleton satisfies S^sk∩∂ S = F^sk.



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 Δ_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   Δ_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 Δ_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 <ref> 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).



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.




 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.



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.

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.




 §.§ 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 <cit.>.


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

    ((1-3ϵ, 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 (<ref>).
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_F^-1([R_0, 0]),

  * d 𝔯_F (ℒ_F) > 0 holds inside 𝔯_F^-1(R_0,+∞), and

  * d 𝔯_F(ℒ_F) = 1 holds inside 𝔯_F^-1(R_0+1,+∞).

For simplicity of notation we  also assume that


  * F_1=𝔯_F^-1([R_0, 1]),

where F_1  denotes the manifold appearing in Formula (<ref>). 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  [-3 δ, 3 δ] × F_1 →,
    
    r(u,x)  2-  (u/3δ)^2 - 𝔯_F(x)-c

for some small number c>0.

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 ϵ, 1] → such that:


  * ρ(s)=0 for s ∈ [1-3 ϵ, 1-2 ϵ],

  * ρ(s)= 2s-1 for s ∈ [1-ϵ, 1], and

  * ρ'(s) ≥ 0 for s ∈ [1-3 ϵ, 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.

Let ρ [1-3 ϵ, 1] → 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)/sℒ_F.


We define the Liouville form θ on S_0 as θ|_𝒞= θ_𝒰 and θ|_S_0 ∖𝒞=θ_0. By Lemma <ref> 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/sℒ_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 → 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 →_+ for ℒ can be obtained by interpolating between 𝔣_0 on S_0 ∖𝒞 and u^2 - (s-1)^2 + 𝔣_F+C on 𝒞∩{ s ∈ [1-ϵ, 1] } for sufficiently large C. The Lyapunov function 𝔣 can then be extended to a Lyapunov function 𝔣 S →_+ in a straightforward way. The easy verification that (S, θ, I, 𝔣) is a Weinstein sector is left to the reader.




 §.§ Exact Lagrangian immersions with cylindrical end


  Let (W, θ) be a 2n-dimensional Liouville manifold. An exact Lagrangian immersion with cylindrical end (or, alternately, an immersed exact
Lagrangian submanifold with cylindrical end) is an immersion ι L → W such that:


  * L is an n-dimensional manifold and ι is a proper immersion which is an
embedding outside finitely many points,

  * ι^* θ = df for some function f  L →, called the potential of
(L, ι), and

  * the image of ι is tangent to the Liouville vector field of (W, θ) outside of
a compact set of L.


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).

Let (W, θ, 𝔣) be a Weinstein manifold. The Lagrangian cocore planes D_p
introduced in Section <ref> 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_R^e corresponds to [R, + ∞) ×Λ under the identification

    (W_R^e, θ) ≅ ([R, + ∞) × V, e^r α),

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.

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 → 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].


If f_0  L → is the potential of (L, ι_0), by Lemma <ref>
we can choose

    f_1=f_0 +  ∫_0^1(- H_σ + θ(X_H_σ)) ∘φ_σ  dσ.

as potential for (L, ι_1).


The weakest one is exact Lagrangian regular homotopy.

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.


An exact regular Lagrangian homotopy ι_t induces a smooth family of functions G_t  L → 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 → TW 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 → generates an exact Lagrangian isotopy
ι_t  L → (W, θ) for any initial choice of exact immersion ι=ι_0.

 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 cohomology for exact Lagrangian immersions require that we work with a restricted class of exact immersed Lagrangian submanifolds.

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^-).



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 cohomology, 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.

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.


 §.§ Contactisation and  Legendrian lifts

We define a contact manifold (M, β), where M=W ×, with a coordinate z
on , 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 → 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 → is the function defined in Lemma <ref>.

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

    ι^+  L → W ×,   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 COHOMOLOGY

In this section we provide an overview of Legendrian contact cohomology. We recall the notion of augmentation and exaplin how they are used to define bilinearized Legendrian contact cohomology. 



 §.§ The Chekanov-Eliashberg algebra

In view of the correspondence between Legendrian submanifolds of (M, β) and exact
Lagrangian immersions in (W, θ), Floer cohomology for Lagrangian immersions will be
a variation on the theme of Legendrian contact cohomology. The latter was proposed by
Eliashberg and Hofer and later defined rigorously by Chekanov, combinatorially, in ^3
with its standard contact structure in <cit.>, and by Ekholm,
Etnyre and Sullivan, analytically, in the contactisation of any Liouville manifold in
<cit.>. 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

    Δ_ζ = D^2 ∖{ζ_0, …, ζ_d }.

Following <cit.>, 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^+.

Putting both positive and negative strip-like ends near each puncture could be useful to
compare wrapped Floer cohomology and contact cohomology, which use different conventions for positive and negative punctures.



Let (V, α) be a contact manifold with contact structure ξ and Reeb vector field
ℛ. An almost complex structure J on × V is cylindrical if


  * J is invariant under translations in ,

  * J(∂_r)= ℛ, where r is the coordinate in ,

  * J(ξ) ⊂ξ, and J|_ξ is compatible with d α|_ξ.




An almost complex structure J on a Liouville manifold W is compatible with
θ if it is compatible with d θ and, outside a compact set, corresponds to
a cylindrical almost complex structure under the identification (<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 Δ_ζ→ W be a J-holomorphic map with boundary in
L. If u has finite area and no puncture at which the lift of u|_∂Δ_ζ to L has a continuous extension, then lim_z →ζ_i u(z)=p_i for some p_i ∈ D. Since the boundary of u switches branch near p_i, the following dichotomy thus makes sense:

We
say that ζ_i is a positive puncture at p_i if

    lim_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. 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:


  * ζ∈R^d+1 and
u Δ_ζ→ W is a J-holomorphic map,

  * u(∂Δ_ζ) ⊂ L, and

  * 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) 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.

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.

It 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 <cit.>:

For a generic J satisfying the condition <ref>,  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
<cit.>.

The boundary of the compactification of the moduli space
𝔑^1_L(p_0; p_1, …, p_d; J) is

    _q ∈ D_0 ≤ i < j ≤ d𝔑^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 <cit.>.



We say that an almost complex structure J on W is L-regular 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 cohomology 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
/ 2 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 <cit.>
instead.

The differential 𝔡 is defined on the generators as:
  
    𝔡(p_0) = ∑_d ≥ 0∑_p_1, …, p_d#𝔑_L^0(p_0; p_1,
    …, p_n;J) p_1 … p_d.

According to <cit.>, the Chekanov-Eliashberg algebra is a
Legendrian invariant:
 
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 stable tame isomorphic.

The definition of stable tame isomorphism of DGAs was introduced by Chekanov in
<cit.>, and then discussed by Ekholm, Etnyre and Sullivan
in <cit.>. We will not use it in this article but note that on the homological level a stable tame isomorphism induces an isomorphism.



 §.§ 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.

Let 𝔄 be a differential graded algebra over a commutative ring . An
augmentation of 𝔄 is a unital differential graded algebra morphism
ε𝔄→.


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 → and
ε_1 𝔄_1 → 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_1^0, …, p_l_0^0) ∈ D_0^l_0,
𝐩^1= (p_1^1, …, p_l_1^1) ∈ D_1^l_1 and q_±∈𝒞,
we denote 
    𝔑_L^i(q_+; 𝐩^0 , q_-, 𝐩^1 ;J):=
    𝔑_L^i(q_+;  p_1^0, …, p_l_0^0, q_-, p_1^1, …, p_l_1^1; 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_1^i) ⋯ε_i(p_l_i^i).

As an -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

    ∂_ε_0, ε_1 (q_-) = ∑_q_+ ∈𝒞∑_l_0, l_1 ∈∑_𝐩^i ∈ D_i^l_i#𝔑_L^0(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 cohomology groups is independent of
the choice of J and is a Legendrian isotopy invariant by the
adaptation of Chekanov's argument from
<cit.> due to the first author and Bourgeois
<cit.>.



§ FLOER COHOMOLOGY FOR EXACT LAGRANGIAN IMMERSIONS


In this section we define a version of Floer cohomology 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 <cit.>, <cit.> and
<cit.>.


 §.§ Cylindrical Hamiltonians


Let W be a Liouville manifold.
A Hamiltonian function H  [0,1] × W → is cylindrical if there is a
function h ^+ → such that H(t, w)=h(e^(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 (<ref>).

Let (V, α) be a contact manifold with Reeb vector field ℛ and let
(× V, d(e^r α)) be its symplectisation. Given a smooth function
h ^+ →, we define the autonomous Hamiltonian H × V →
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 →, we denote
by 𝒞_H — or simply 𝒞 when there is no risk of confusion —
the set of Hamiltonian chords 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.

A cylindrical Hamiltonian H  [0,1] × W →, with Hamiltonian 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, and

  (iv) all time-one Hamiltonian chords from L_0 to L_1 are contained in a compact
region.


Condition (iv) is equivalent to asking that λ should not be the
length of a Reeb chord from Λ_0 to Λ_1.

If cylindrical Hamiltonian H is compatible with L_0 and L_1, then
𝒞_H is a finite set.





 §.§ Obstructions



If one tries to define Floer cohomology 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 → by

    𝔡^0 (p) = #𝔑_L^0(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 cohomology 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.


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.




An exact immersed Lagrangian (L, ι) is unobstructed if (𝔇,
𝔡) admits an augmentation.

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 <ref> (this fact will not be needed). However, we will need the invariance statement from the following proposition.

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).



The safe isotopy between L_0 and L_1 induces a Legendrian isotopy between the Legendrian lifts L_0^+ and L_1^+ without birth or death of Reeb chords.
By <cit.> the Chekanov-Eliashberg algebras of L_0^+ and L_1^+ are stably tame isomorphic, and moreover stabilisation occurs only at the birth or death of a Reeb chord.

Proposition <ref> 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.


 §.§ The differential

We denote by Z the strip × [0,1] with coordinates (s,t). Let
ℛ^l_0|l_1≅Conf^l_0() ×Conf^l_1()
be the set of pairs (ζ^0, ζ^1) such that
ζ^0= {ζ_1^0, …, ζ_l_0^0 }⊂×{ 0 } and
ζ^1= {ζ_l_1^1, …, ζ_1^1 }⊂×{ 1 }. We
assume that the s-coordinates of ζ_j^i are increasing in j for ζ_j^0 and
decreasing for ζ_j^1.  We define

    Z_ζ^0, ζ^1:= Z ∖{ζ_1^0, …,
    ζ_l_0^0, ζ_1^1, …, ζ_l_1^1  }.

The group Aut(Z)= 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 <ref> for L_0 and  J_1 satisfies <ref> 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_j^i, so that
we can apply standard analytical results.

For the same reason we fix once and for all a nondecreasing function χ→ [0,1]
such that

    χ(t) = 
    0    for  t ∈ [0, 1/4], 
    
    1    for  t ∈ [3/4, 1]

and χ'(t) ≤ 3 for all t. It is easy to see that, if φ_t is the Hamiltonian
flow of a Hamiltonian function H,  then φ_χ(t) is the Hamiltonian flow
of  the Hamiltonian function H' such that H'(t,w)=χ'(t) H(χ(t), w). We will
use χ' to cut off the Hamiltonian vector field in the Floer equation to ensure that
it has the right invariance properties in the strip-like ends corresponding to the boundary
punctures ζ_j^i.

Given Hamiltonian chords
x_+, x_- ∈𝒞 and self-intersections p_1^0, …,
p_l_0^0 of L_0 and p_1^1, …, p_l_1^1 of L_1 we define the moduli space

    𝔐_L_0, L_1( p_1^1, …, p_l_1^1, x_- , p_1^0,
    …, p_l_0^0, 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

    ∂ 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_j^i u(z) = p_j^i, 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)= acts on 𝔐_L_0, L_1( p_1^1, …,
p_l_1^1, x_- , p_1^0, …, p_l_0^0, x_+; H, J_∙) by
reparametrisations. We will denote the quotient by

    𝔐_L_0, L_1( p_1^1, …, p_l_1^1, x_- , p_1^0, …, p_l_0^0,
    x_+; H, J_∙).

For u ∈𝔐_L_0, L_1( p_1^1, …, p_l_1^1,
x_- , p_1^0, …, p_l_0^0, 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_l_1^1, x_- , p_1^0,
    …, p_l_0^0, x_+; H, J_∙)

the subset of 𝔐_L_0, L_1( p_1^1, …, p_l_1^1, x_- , p_1^0, …,
p_l_0^0, 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
<cit.>, where the regularity
statement appears in <cit.>
and compactness is discussed in <cit.>.
In addition, the corresponding statement in the case of Legendrian contact cohomology in
P× was proven by Ekholm, Etnyre and Sullivan, see
<cit.>. The following proposition translates those compactness
and regularity statements to the settings under consideration.


For a generic time-dependent almost complex structure J_∙ satisfying <ref>,
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_l_1^1, x_- , p_1^0, …, p_l_0^0, x_+; H,
    J_∙)

is a transversely cut-out manifold of dimension k-1. If k=1 it is compact, 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_l_1^1, x_- , p_1^0, …, p_l_0^0, x_+; H,
J_∙) is

    _y ∈𝒞_H_0 ≤ h_i ≤ l_i   𝔐^1_L_0, L_1(p_h_1+1^1, …, p_l_1^1, y, p_1^0,
    …, p_h_0^0, x_+; H, J_∙) 
       ×𝔐^1_L_0, L_1 (p_1^1,
    …, p_h_1^1, 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_i^1, q, p_j+1^1, … p_l_1^1, x_-, p_1^0, …, p_l_0^0, x_+; H, J_∙) 
       ×𝔑^0_L_1(q; p_i+1^1, …, p_j^1; J_1) 
    _q ∈ D_0_0 ≤ i ≤ j ≤ l_0   𝔐^1_L_0, L_1( p_1^1, …,
    p_l_1^1, x_-, p_1^0, …, p_i^0, q, p_j+1^0, … p_l_0^0, x_+; H, J_∙) 
       ×𝔑^0_L_0(q; p_i+1^0, …, p_j^0; 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.


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 cohomology. Unfortunately
this can cause some confusion, but it is necessary to remain consistent with the standard
conventions in the literature.



We say that a time-dependent almost complex structure J_∙ on W is
(L_0, L_1)-regular if
it satisfies <ref> for L_0 and L_1 and all moduli spaces

    𝔐^k_L_0, L_1( p_1^1, …, p_l_1^1, x_- , p_1^0, …, p_l_0^0, 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_1^i, …, 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_l_1^1, x_- , p_1^0, …, p_l_0^0, 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 

    CF((L_0, ε_0), (L_1, ε_1); H, J_∙) = ⊕_x ∈𝒞 x

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

    ∂ x_+ = ∑_x_- ∈𝒞_H∑_l_0, l_1 ∈∑_𝐩^i ∈ D_i^l_i𝔪(𝐩^1, x_- , 𝐩^0, x_+) x_-.



The algebraic interpretation of the Gromov-Floer compactification of the one-dimensional
moduli spaces in Proposition <ref> 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 <ref>.



 §.§ Comparison with bilinearised Legendrian contact cohomology

In this subsection we compare the Lagrangian Floer cohomology 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 → 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_t^-1, where φ_1 ∘φ_t^-1∘φ_1^-1 can be generated by the Hamiltonian -H(t,φ_t^-1∘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 = d ϕ_χ(t)∘ J_t ∘ d ϕ_χ(t)^-1.  The time
rescaling by χ ensures that ϕ_*J_∙
satisfies <ref> for ϕ_1(L_0) and L_1 if and only if
J_∙ does for L_0 and L_1.


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

    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)-regular.


Since

    ∂ v/∂ s(s,t)     = d φ_t 
    (∂ u/∂ s
    (s,t) )   and
    ∂ v/∂ t(s,t)     = d φ_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 d φ intertwines the linearised
Floer operators at u and v.



Given two Legendrian submanifolds Λ_0 and Λ_1 in (W
×, 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.

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_t^-1)_*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).


By Lemma <ref> 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^+. As 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 it remains only to match the differentials.

For any i = 1, …, d and ζ= {ζ_0,
…, ζ_d}∈ℛ^d there is a
biholomorphism Δ_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).





 §.§ 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

    ℛ^l_0| … |l_d= Conf^l_0+ … + l_d +d+1(∂ D^2)

where d+1 points ζ_0^m, …, ζ_d^m (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_i^m to ζ_i+1^m)
are labelled as pure.
Given
ζ∈ℛ^l_0| … |l_d, we denote
Δ_ζ = D^2 ∖ζ. For i=0, …, d let
∂_i Δ_ζ be the subset of ∂Δ_ζ whose closure in ∂ D^2 is the counterclockwise arc from
ζ_i^m to ζ_i+1^m.

We will consider also a (generic) domain dependent almost complex structure J_∙ such
that every J_z, z ∈Δ_z satisfies <ref>, and moreover is of the
form <ref> at the strip-like ends of the mixed punctures and is constant in a
neighbourhood of the arcs ∂_i Δ_ζ outside those strip-like ends.

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 Δ_ζ→ W satisfies du + J_∙∘ du ∘ i =0,

  * u(∂_i Δ_ζ) ⊂ L_i,

  * lim_z →ζ_i^m u(z) = x_i,

  * lim_z →ζ_j^i u(z)= p_j^i, and

  * p_j^i is a negative puncture at ζ_j^i for i=0, …, d and j = 1, …, l_i.

As usual, we denote by 𝔐_L_0, …, L_d^0(𝐩^d, x_0,
𝐩^0, x_1, …, 𝐩^d-1, x_d; J_∙) the
zero-dimensional 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_L_0, …, L_d^0(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

    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_j^i ∈ D_i^l_i𝔪(𝐩^d, x_0, 𝐩^0, x_1, …, 𝐩^d-1, x_d) x_0.


The operations μ^d satisfy the A_∞ relations.


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

Let (L, ι) be an exact Lagrangian immersion. We extend ι to a symplectic immersion ι_*  (D_δ T^*L, d𝐪∧ d𝐩) → (W, d θ). Let (L, ι') be safe isotopic to (L, ι) and, moreover, assume that


  * there exists a sufficiently C^1-small Morse function g  L → 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 <ref>.

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.


Under the hypotheses of Lemma <ref>, 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.


The assumption that the augmentation ε' is identified canonically with the augmentation ε implies that e is a cycle by the count of  pseudoholomorphic discs  with a negative puncture at e supplied by Lemma <ref>.

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^+) →
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.


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^+) →

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^+.


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.




 §.§ 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 push-off from <cit.>; when the latter push-off becomes sufficiently small, these discs converge to pseudoholomorphic discs on the single Lagrangian with gradient-flow 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_ϵ f, which we require to be again an exact immersed Lagrangian submanifold with cylindrical end, which is identified with the graph of d(ϵ f) for a Morse function f  L → 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_ϵ f^+ for which (L_ϵ f)^+ is above L^+. For ϵ>0 sufficiently small, it is the case that L ∪ L_ϵ f again has only transverse double points. Moreover, the Reeb chords on the Legendrian lift can be classified as follows, using the notation from <cit.>:


  * Reeb chords 𝒬(L) ≅𝒬(L_ϵ f) on the lifts of L and L_ϵ 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_ϵ 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 <ref>.






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_ϵ 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,

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 <ref>. In the second case, it is a Floer strip corresponding to the negative gradient flow-line connecting the saddle point and the local minimum.


This is an immediate application of Parts (3) and (4) of <cit.>. 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_ϵ f transversely. For ϵ>0 sufficiently small there is a bijection between the intersection points L ∩ L' and L_ϵ f∩ L'.


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


We need to apply Lemma <ref> in the case when L^+ is taken to be the Legendrian lift (L ∪ L_ϵ f)^+, where L_ϵ f^+ is above L^+, and the push-off is taken to be (L ∪ L')_ϵ F^+ for a Morse function F  L ∪ L' → that restricts to f on L.




§ CONTINUATION MAPS

In this section we analyse what happens to the Floer cohomology when we change J, H
(in some suitable way) or move the Lagrangian submanifolds by a compactly supported safe
exact isotopy.



 §.§ Changing the almost complex structure

Following <cit.> (see also <cit.>) we will use the bifurcation
method to prove invariance of Floer cohomology 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 <ref> 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_-).


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 zero-dimensional
parametrised moduli spaces. The analogous statement in the setting of Lagrangian Floer
homology (for Lagrangian submanifolds) appears in <cit.>.
In the case of Legendrian contact cohomology, the corresponding construction appears in
<cit.>.

For a generic J_∙^∙ there is a finite set Δ⊂ (δ_-, δ_+)
such that
for δ∈Δ exactly one of the following cases holds:


  (i)  there is a unique nonempty moduli space 𝔑^-1_L_0(q_0^0;
q_1^0, …, q_d^0; J_0^δ) and all other moduli spaces are transversely cut out,

  (ii)   there is a unique nonempty moduli space 𝔑^-1_L_1(q_0^1;
q_1^1, …, q_d^1; J_1^δ)  and all other moduli spaces are transversely cut out, or

  (iii) there is a unique nonempty  moduli space
𝔐_L_0, L_1^0(𝐪^1, y_-, 𝐪^0, y_+; H, J_∙^δ) and
all other moduli spaces are transversely cut out,

while for every δ∈ [δ_-, δ_+] ∖Δ the moduli spaces of
negative virtual dimension are empty.

(Of course, the self-intersection points q_j^i 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∉Δ: for
example, if δ is a critical value of the projection

    𝔐_L_0, L_1^1(𝐪^1, y_-, 𝐪^0, y_+;
    H, J_∙^∙) → [δ_-, δ_+],

then J_∙^δ is not (L_0, L_1)-regular, but δ∉Δ.


If Δ = ∅, then the 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_+) 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_- and J_∙^δ_+.

Next we describe what happens when we cross δ∈Δ of type (i) or (ii). Since the
cases are symmetric, we will describe only (i)  and assume, without loss of generality, that δ=0.

Suppose that Δ = { 0 } and that the unique nontransversely cut out moduli space for J_∙^0 is 𝔑^-1_L_0(q_0^0; q_1^0, …, q_d^0;
J_0^0). Then for δ>0 the  differentials of
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) are equal. 



We adapt the cobordism method of <cit.> (see also
<cit.>). Given a positive Morse function f → with local minima at ± 1 satisfying f(1)= f(-1)=1, a local maximum at 0 and no other critical point, we define exact Lagrangian immersions ι_i × L_i → T^*× W following <cit.>. We denote
L_i the image of ι_i.

Let (r, ρ) be the canonical coordinates on T^*. We consder the Hamiltonian function

    H [0, 1] × T^*× W →

such that H(t, r, ρ, w)=H(t,w).
Then its Hamiltonian vector field X_H is tangent to {(r, ρ) }× W for all (r, ρ) ∈ T^*.
Each double point q of L_i gives rise to three double points q[-1], q[0], q[+1] of L_i and each each Hamiltonian chord x from L_0 to L_1 gives rise to three Hamiltonian chords x[-1], x[-1], x[+1] from L_0 to L_1.

Let J_∙ be the (time dependent) almost complex structure on T^*× W such that on T_(r, ρ, w)≅^2 ⊕ T_wW is j_0 ⊕ J^α_δ(r)_∙ where j_0 is the standard almost complex structure on ^2 and α_δ→ [- δ, δ] is as in Equation (4.9) of <cit.> for δ small. Proposition <ref> provides isomorphisms

    𝔜_i  (𝔇_i^-, 𝔡_i^-) → (𝔇_i^+,
    𝔡_i^+)

such that ε_i^- = ε_i^+ ∘𝔜_i. Those isomorphisms are the ones constructed in <cit.> and denoted by ψ there. Let 𝔇_i be the obstruction algebra of
L_i and define ε_i 𝔇_i → as

    ε_i(q[-1])=ε_-(q), 
    ε_i(q[0])= 0, 
    ε_i(q[+1])=ε_+(q)

for every double point q of L_i.
From the structure of the differential of the Chekanov-Eliashberg algebras of L_i described in  <cit.> it follows that ε_i is an augmentation of 𝔇_i, and therefore the Floer chain complex CF((L_0, ε_0), (L_1, ε_1); H, J_∙) is well defined.

Lemmas 4.14, 4.14, 4.18 and 4.19 of <cit.> have direct counterparts for Floer solutions because the projection of a solution of the Floer equation with Hamiltonian H to T^* is
a holomorphic map. This implies that the differential ∂ on CF((L_0, ε_0), (L_1, ε_1); H, J_∙) has the following form

    ∂(x[-1])=(∂_- x)[-1], 
    ∂(x[0])= x[+1]-x[-1]+ ∑_y ∈𝒞_H a_y y[0], 
    ∂(x[+1])=(∂_+ x)[+1]

where ∂_± denotes the differentials of CF((L_0, ε_0^±), (L_1, ε_1^±); H, J_∙^±δ).
The crucial point here is that no chord y[± 1] contributes to ∂(x[0]) if y  x.  This is a consequence of the assumption that there is no Floer strip of index 0 for J^∙_∙ and of the Hamiltonian version of <cit.>. From ∂^2=0 it is easy to see that ∂_+=∂_-.



Now we analyse how the complex changes when we cross δ∈Δ of type (iii).

Suppose that Δ = {δ_0 } and that the unique nontransversely cut out moduli
space for J_∙^δ_0 is 𝔐_L_0, L_1^0(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_L_0, L_1^0(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.


The proof is the same as in <cit.>.
However,  the proof in <cit.> holds only in the case of _2-coefficients.
For more general coefficients, we rely on the discussion in <cit.>.

Given a generic homotopy J_∙^∙, we split it into pieces containing only one point of
Δ and compose the maps obtained in Lemma <ref>
and <ref>.



 §.§ 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) ∂_sh'_s(e^𝔯(w)) ≤ 0 if 𝔯(w) is sufficiently large,

we will define a continuation map

    Φ_H_sCF((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_1^0, …, p_l_0^0) of L_0 and 𝐩^1 = (p_1^1, …, p_l_1^1) 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

    ∂ u/∂ s + J_t  (∂ u/∂ t - χ'(t)
    X_H_s(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_j^i is a negative puncture at p_j^i for i=0,1 and j=1, …, l_i.

Note the only difference between Equation (<ref>) and
Equation (<ref>) is that we made X_H_s depend on s in
Equation (<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_L_0, L_1^k(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, <cit.>. A similar boundary degeneration statement in the case of Legendrian contact cohomology appears in <cit.>.

Given H_s, for a generic time-dependent almost complex structure J_∙ satisfying
<ref> with respect to both H_+ and H_- , the moduli space
𝔐_L_0, L_1^k (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 of the moduli space (see <cit.>).


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_L_0,L_1^0( 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

    Φ_H_s(x_+) = ∑_x_- ∈𝒞_-∑_l_0, l_1 ∈∑_𝐩^i ∈ D_i^l_i𝔪(𝐩^1, x_-  𝐩^0, x_+; H_s) x_-.

The Gromov-Floer compactification of the one-dimensional moduli spaces implies the
following lemma.

The map Φ_H_s 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_H_s — soon it will be apparent that the notation is
justified.
As it happens in the more standard Floer cohomology for Lagrangian submanifolds, the
continuation maps satisfy the following properties.



  * Up to homotopy, Φ_H_s depends only on the endpoints H_+ and H_- of
H_s, 

  * Φ^*_H,H is the identity for every H, and 

  * Φ^*_H_-,H∘Φ^*_H, H_+ = Φ^*_H_-, H_+. 



In order to prove (<ref>), we follow the standard procedure for defining chain homotopies
in Floer theory; see <cit.> for more details. Given a homotopy H_s^δ,
δ∈ [0,1], between s-dependent Hamiltonian functions H_s^0 and H_s^1 with
H_s^δ≡ H_- for s ≪ 0 and H_s^δ≡ H_+ for s ≫ 0, we define
the parametrised moduli spaces
𝔐_L_0, L_1^k(𝐩^1, x_-, 𝐩^0, x_+; H_s^∙, J_∙)
of pairs (δ, u) such δ∈ [0,1] and u ∈𝔐_L_0, L_1^k
(𝐩^1, x_-, 𝐩^0, x_+; H_s^δ, J_∙).
We define the weighted count

    𝔪(𝐩^1, x_-,  𝐩^0, x_+; H_s^∙) =  #𝔐_L_0,L_1^-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_s^0 and Φ_H_s^1 is defined as

    K(x_+) = ∑_x_- ∈𝒞_-∑_l_0, l_1 ∈∑_𝐩^i ∈ D_i^l_i𝔪(𝐩^1, x_-,  𝐩^0, x_+; H_s^∙)x_-.


In order to prove (<ref>) we can choose H_s ≡ H: then the moduli space
𝔐_L_0, L_1^0(𝐩^0, x_-, 𝐩^1, x_+; H_s, J_∙) consists
of constant strips.

We 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 (<ref>) we introduce the family of Hamiltonian functions

    H_s^R =  H_s-R^+    for  s ≥ 0,  and 
    
                                                  H_s+R   for  s ≤ 0

with R>0. By (<ref>), Φ_H_s^R induces Φ^*_H_+, H_- for all R.
For R ≫ 0 there is an identification

    𝔐_L_0,L_1^0(p_1^1, …, p_l_1^1, x_-,  p_1^0, …, p_l_0^0, x_+;
    H_s^R,J_∙) ≅
    _x ∈𝒞_H_0 ≤ h_i ≤ l_i   𝔐_L_0,L_1^0( p_h_1+1^1, …, p_l_1^1,x, p_1^0,
    …, p_h_0^0, x_+;
    H_s^+,J_∙) ×
       𝔐_L_0,L_1^0(p_1^1, …, p_h_1^1, x_-, p_1+h_0^0,
    …, p_l_0^0, 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_L_0,L_1^0
    (𝐩^1, x_- 𝐩^0, x_+; H_s^R,J_∙),

then ζ_j^i ∉[-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_s^R_n, J_∙)
 and for every n
there is some ζ_j^i ∈ [-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.  For
index reasons  this level must have index zero, but it cannot be constant
because of the boundary puncture. This is a contradiction.


With lemma <ref> at hand, we can prove the following invariance
property in the usual formal way.

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.




 §.§ 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 <ref> 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 ∘ d ψ_t^-1,    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.


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'.


The obstruction algebras 𝔇_1 of (L_1, J_1) and 𝔇_1' of
(L_1', (ψ_1)_*J_1) are tautologically isomorphic because ψ_1 matches the generators and the holomorphic curves contributing to the differentials, 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.

Recall the sets 
    Conf^n() = { (ζ_1, …, ζ_n) ∈^n |  ζ_1 < … <
    ζ_n }  
       Conf^n() = { (ζ_1, …, ζ_n) ∈^n |  ζ_1
    ≤…≤ζ_n}.
 (Note that this is not how configuration spaces are usually
compactified.)
Given n ∈, we denote 𝐧 = {1, …, n} and for m < n
we denote hom(𝐧, 𝐦) the set of nondecreasing and
surjective function ϕ𝐧→𝐦. Every ϕ∈hom(𝐧, 𝐦) induces an embedding ϕ^* Conf^m() →Conf^n() defined by

    ϕ^*(ζ_1,
    …, ζ_m) = (ζ_ϕ(1), …, ζ_ϕ(m)).

The boundary of
Conf^n() is a stratified space with dimension m stratum

    _ϕ∈hom(𝐧, 𝐦)ϕ^*(Conf^m()).


The embeddings ϕ defined above extend to diffeomorphisms

    ϕ^* Conf^m() ×_+^n-m→Conf^n()

such that

    (ζ_1', …, ζ_n')= ϕ^*((ζ_1, …, ζ_m), (ϵ_1, …,
    ϵ_n-m))

if 
    ζ_i' = ζ_ϕ(i) + ∑_k=0^i - ϕ(i)ϵ_k,

where ϵ_0=0 for the sake of the formula.



Fix δ >0. There is a family of constants κ_n >0 and smooth functions

    ν_n Conf^n() ×→ [0,1]

such that, denoting by s the coordinate in the second factor,


  * ν_n(ζ_1, …, ζ_n, s) =0 for s > κ_n,

  * ν_n(ζ_1, …, ζ_n, s) =1 for s < - κ_n,

  * ∂_s ν_n(ζ_1, …, ζ_n, s) ∈ [-2, 0] for all (ζ_1, …, ζ_n, s)
∈Conf^n() ×,

  * ∂_s ν_n(ζ_1, …, ζ_n, s)=0 if |s- ζ_i| ≤δ/2 for
some i=1, …, n, and

  * ν_n ∘ϕ^*= ν_m for all m <n and all ϕ∈hom(𝐧,
𝐦).



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.


Given ζ = (ζ_1, …, ζ_n) ∈Conf^n(), we define
ν_ζ→ [0,1] by ν_ζ(s)=
ν_n(ζ_1, …, ζ_n, s).


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

    J̃_n Conf^n() × Z →𝒥(θ)

such that


  * J̃_n(ζ, s,t) = J_t^+ if s > κ_n+1,

  * J̃_n(ζ, s,t) = J_t^- if s < - κ_n-1,

  * J̃_n(ζ, s,t)=J_0 if t ∈ [0, 1/4],

  * J̃_n(ζ, s,t)= dψ_ν_ζ(s)∘ J_1 ∘
dψ_ν_ζ(s)^-1 if t ∈ [3/4, 1], and

  * for all ϕ∈hom(𝐧, 𝐧-1),
J̃_n(ϕ^*(ζ), s, t)= J̃_n-1(ζ, s,t).



We build J̃_n inductively on n. At each step, the map J̃ is determined in
the complement of  Conf^n() × [- κ_n-1, κ_n+1] × [1/4, 3/4].
We can extend it to Conf^n() × Z because 𝒥(θ)
is contractible.

Given ζ∈Conf^n(), we will denote
J̃_ζ the s- and t-dependent almost
complex structure obtained by restricting J̃ to {ζ}× Z. Given (ζ^0, ζ^1) ∈ℛ^l_0|l_1, we will not distinguish between ζ^1_j ∈×{ 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 J̃_ζ^1.
For simplicity, the s- and t-dependence of  J̃_ζ^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' }.


Given x_+ ∈𝒞_H, x_- ∈𝒞_H', and
𝐩^i ∈ D_i^l_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, J̃, 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

    ∂ u/∂ s + J̃_ζ^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_j^0 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 × L →.) We denote by
𝔐_L_0, L_1^k(𝐩^1, x_-, 𝐩^0, x_+; H, G,J̃, ν) the
set of triples (ζ^0, ζ^1, u) where ind(u)=k.


 
For a generic J̃ as in Lemma <ref>, the moduli space

    𝔐_L_0, L_1^k(𝐩^1, x_-, 𝐩^0, x_+; H, G,J̃, 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 admits a compactification 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 of the moduli space (see <cit.>).



We define 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, J̃, ν)
    ε_0(𝐩^0) ε_1(𝐩^1)

and then we define Ψ_G as

    Ψ_G(x_+)= ∑_x_- ∈𝒞_H∑_l_0, l_1 ∈∑_𝐩^i ∈ D_i^l_i𝔪(𝐩^1, x_-, 𝐩^0, x_+;
    H, G)x_-.



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.



A consideration of Proposition <ref> together with a count of solutions of index -1 which arise in a one-parametric family of moduli spaces implies the following:


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.


Let G^0,G^1  L_1 → be local Hamiltonian functions generating the safe isotopies
ψ_t^0 ∘ι_1 and ψ_t^1 ∘ψ_1^0 ∘ι_1 respectively, and let G^2  L_1 → be a local Hamiltonian function generating

    ψ_t^2 = 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 <ref> is analogous to the proof of
Lemma <ref>. 

If G × L → satisfies dG_t=0 outside a compact subset of (0,1) × L, then the map
Ψ_G induces an isomorphism in homology.

If J̅_∙^+ is one (L_0, L_1)-regular almost complex structure and J̅_∙^- is another (L_0, ψ_1(L_1))-regular almost complex structure, instead of
repeating the construction of J̃ with  J̅_∙^± as starting point, we
prefer to consider J̃ 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, J̅_∙^+) →CF((L_0, ε_0), (L_1', ε_1); H, J̅_∙^-)

as the composition Υ_- ∘Ψ_G ∘Υ_+, where

    Υ_+    CF((L_0, ε_0), (L_1, ε_1); H, J̅_∙^+) →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, J̅_∙^-)

are the maps defined in Section <ref>.



 §.§ Continuation maps for almost complex structures and Hamiltonian functions commute

Let H_s be a homotopy from H_- to H_+ as in Section <ref> and
J_∙^∙ a homotopy from J_∙^-1 to J_∙^+1 as in Section <ref>. 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.

The diagram

    CF(H_+, J_∙^+) [r]^Φ_+[d]_Υ_+   CF(H_-, J_∙^+)
    [d]^Υ_-
    CF(H_+, J^-) [r]_Φ_-   CF(H_-, J_∙^-)

commutes up to homotopy.


Proposition <ref> 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 Δ of bifurcation points such that, for all δ∈Δ, there is a unique nonempty moduli space of one of the following types:


  (i) 𝔑_L_0^-1(q_0^0; q_1^0, …, q_d^0; J_0^δ) or
𝔑_L_1^-1(q_0^1; q_1^1, …, q_d^1; J_1^δ),

  (ii) 𝔐_L_0,L_1^0(𝐪^1, y_-, 𝐪^0, y_+; H_-,
J_∙^δ) or 𝔐_L_0,L_1^0(𝐪^1, y_-,𝐪^0, y_+; H_+,
J_∙^δ),

  (iii) 𝔐_L_0,L_1^-1(𝐪^1, y_-, 𝐪^0, y_+; H_s,
J_∙^δ).

To these moduli spaces correspond four types of boundary configuration for
the compactification of the one-dimensional parametrised moduli spaces
𝔐_L_0,L_1^0(𝐩^1, x_-, 𝐩^0, x_+; H_s, J_∙^∙),
which we write schematically as:


  (i)  𝔑_L_0^-1(J_0^δ) ×𝔐^0(H_s,
J_∙^δ) or 𝔑_L_1^-1(J_1^δ) ×𝔐^0
(H_s, J_∙^δ),

  (ii) 𝔐^0(H_-, J_∙^δ) ×𝔐^0(H_s, J_∙^δ) or 𝔐^0(H_s,
J_∙^δ) ×𝔐^0(H_+, J_∙^δ),

  (iii) 𝔐^1(H_-, J_∙^δ) ×𝔐^-1(H_s, J_∙^δ) or 𝔐^-1(H_s,
J_∙^δ) ×𝔐^1(H_+, J_∙^δ),

  (iii)' 𝔑^ 0_L_0(J_0^δ) ×𝔐^-1(H_s, J_∙^δ) or 𝔑^ 0_L_1(J_1^δ) ×𝔐^-1(H_s, J_∙^δ).

There is also one fifth type of boundary configuration:


  (iv) 𝔐^0(H_s, J_∙^-1) or
𝔐^0(H_s, J_∙^1).

In order to prove Proposition <ref> we split the homotopy J_∙^∙ into pieces, each of which contains only
one element of Δ, and we prove that for each piece the corresponding
diagram (<ref>) commutes up to
homotopy. Putting all pieces 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. 


Let Δ = { 0 } be of type (i). Then Diagram (<ref>) commutes.


We have proved in Lemma <ref> that
Υ_± are the identity maps. Here we will prove that, under the hypothesis of the lemma, Φ_+=Φ_-. We use the same construction, and the same notation, as in the proof of Lemma <ref>.

The homotopy of Hamiltonians H_s on W induces a homotopy of Hamiltonians H_s on T^*× W. Let

    ΦCF((L_0, ϵ_0), (L_1, ε_1); H_+, J_∙) →CF((L_0, ϵ_0), (L_1, ε_1); H_-, J_∙)

be the continuation map associated to the homotopy H_s.

Solutions of the Floer equation perturbed by H_s are still holomorphic when projected to T^*, and therefore the counterpart of Lemmas 4.14 and 4.19 of <cit.> give that, for κ=-1,0,+1,  Φ(x[κ]) is a linear combination of chords y[κ].  The statement for κ=0 is a consequence of the assumption that there is no Floer strip for H_s of index  -1. (Unlike in the proof of Lemma <ref> we do not have the two strips between from
x[0] to x[± 1] because in the definition of Φ we consider solutions of index zero.) Then ∂∘Φ= Φ∘∂ and Equation <ref> give Φ_+= Φ_-.




Let Δ = { 0 } be of type (ii). Then Diagram (<ref>) commutes.


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 <ref> the continuation maps for the change of almost complex structure are Υ_- =
Id and

    Υ_+ (x) = 
    x    if  x  y_+, 
    
    y_+  + #𝔐_L_0, L_1^0(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^1𝐩^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 (<ref>) 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.

Let Δ = { 0 } be of type (iii). Then Diagram (<ref>) commutes up to homotopy.


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^1𝐩^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_∙), 
       #𝔐^0(𝐪^1𝐩^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 (<ref>) 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 × L_1 → 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

    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.

The diagram

    CF((L_0, ε_0), (L_1, ε_1); H_+, J_∙^+) [r]^Φ[d]_Ψ_G^+   CF((L_0, ε_0), (L_1, ε_1); H_-, J_∙^+) [d]^Ψ_G^+
    CF((L_0, ε_0), (L_1', ε_1'); H_+, J_∙^-) [r]^Φ'   CF((L_0, ε_0), (L_1', ε_1'); H_-, J_∙^-)

commutes up to homotopy.


For R ∈ we define H_s^R= H_s-R. We define the moduli spaces

𝔐_L_0, L_1^k(𝐩^1, x_-, 𝐩^0, x_+; H_s^R, G, J̃,
ν)
as in Definition <ref> by replacing the Hamiltonian H by the
s-dependent Hamiltonian H_s^R. Counting pairs (R, u) where R ∈ and

    u ∈𝔐_L_0, L_1^-1 (𝐩^1, x_-, 𝐩^0, x_+; H_s^R, G,
    J̃, ν),
 weighted by the augmentations, we obtain a homotopy between Ψ_G^-
∘Φ and Φ' ∘Ψ_G^+.






§ 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.


 §.§ 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 sketch of the chain level
construction, following <cit.>, 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∈ we denote by h_λ^+ → the function

    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∈^+.
We define time-independent cylindrical Hamiltonians H_λ W → by

    H_λ(w)= h_λ(e^ 𝔯(w)).

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_λ_n is dense, and we pick an element J_∙.



By Subsection <ref>, for every  m ≥ n there are continuation
maps

    Φ_λ_n, λ_mHF((L_0, ε_0), (L_1, ε_1);
    H_λ_n, J_∙) →HF((L_0, ε_0), (L_1, ε_1); H_λ_m,
    J_∙)

forming a direct system.

 The wrapped Floer cohomology
of (L_0, ε_0) and (L_1, ε_1) is defined as

    HW((L_0, ε_0), (L_1, ε_1); J_∙) = HF((L_0, ε_0),
    (L_1, ε_1); H_λ_n, 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_λ_n and of the sequence λ_n. Invariance of the almost
complex structure follows form Proposition <ref>. Invariance of the smoothing of H_λ_n follows from Lemma <ref> and Corollary <ref>. 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

    HF((L_0, ε_0), (L_1, ε_1); H_λ_n”, J_∙) ≅   HF((L_0, ε_0), (L_1, ε_1); H_λ_n, J_∙) 
    ≅   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.

Let G × L_1 → 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_∙').



It is enough to observe that, for every n, the isomorphisms

    CF((L_0, ε_0), (L_1, ε_1); H_λ_n, J_∙) ≅CF((L_0, ε_0), (L_1', ε_1'); H_λ_n, J_∙')

defined in Subsection <ref> commute with the
continuation maps Φ_λ_n, λ_m and therefore define isomorphisms
of direct systems. This follows from Lemma <ref>
and Proposition <ref>.



 §.§ 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 (<ref>) such that, for every w ∈, the Hamiltonian
wH is compatible with L_0 and L_1 (in the sense of Definition <ref>). 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 <cit.> we define the
wrapped Floer chain complex as the [q]/(q^2)-module

    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 <ref>.

The endomorphism ι_q (denoted ∂_q in <cit.>) defined as

    ι_q(x+yq)=y

is a chain map. However, its action in homology is trivial.



The direct sum (<ref>) starts from
w=1 in <cit.>. It is equivalent to start from w=0, when possible, by
<cit.>. 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 (<ref>) by
<cit.>.


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_𝒲FCW(L_d-1, L_d) ⊗…⊗CW(L_0, L_1) →CW(L_0, L_d)

coincides with the operation μ^d defined in Equation (<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 <cit.>
but also follows from e.g. <cit.>. Note that the statement is false
in the more general case when the Lagrangian is only assumed to be monotone. In this section we extend this
classical vanishing result to our setting of exact Lagrangian immersions.



 §.§ 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

    ∫_Δ_ζ u^*d θ = 𝔞(p_0) - ∑_i=1^d
    𝔞(p_i).

Since ∫_Δ_ζ u^*d θ >0 for a nonconstant J-holomorphic
map, if

    𝔑_L(p_0; p_1, …, p_d; J) ≠∅,

we obtain

    𝔞(p_0) - ∑_i=1^d
    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

    𝒜(x) = ∫_0^1 x^* θ - ∫_0^1 H(x(t))dt +
    f_0(x(0))-f_1(x(1)).

Note that this is the negative of the action used in <cit.>.

Let H  W → be a cylindrical Hamiltonian such that H(w)= h(e^𝔯(w)),
where h ^+ →.
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

    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 →, we denote f _∞^+ := sup_a
∈ Amax{f(a), 0 }.

Let H_s × [0,1] × W → be an s-dependent cylindrical Hamiltonian
function satisfying conditions (i), (ii), and (iii) of Subsection <ref>.
We make the simplifying assumption that ∂_sH_s ≡ 0 if s ∉[-1, 1]. If
𝔐_L_0, L_1(𝐩^1, x_-, 𝐩^0, x_+; H_s, J_∙) ∅, then

    𝒜(x_-) ≤𝒜(x_+) + 6 ∂_sH_s _∞^+.


Note that Equation (<ref>) is far from being sharp, but there will be no need
for a sharper estimate.

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^*TW induced by d θ and J_∙,

    ∫_- ∞^+∞∫_0^1 | ∂_s u |^2 dt ds     = ∫_- ∞^+∞∫_0^1 d θ (∂_s u, ∂_t u - χ'(t) X_H_s ( χ(t) ,u)) dt ds 
        = ∫_Z_ζ^0,   ζ^1 u^* d θ - ∫_- ∞^+∞∫_0^1 χ'(t) dH_s( χ(t) ∂_su) dt ds.

Using Stokes's theorem we obtain

    ∫_Z_ζ^0,   ζ^1 u^* d θ   = f_1(x_-(1))
    -  f_1(x_+(1)) -
        f_0(x_-(0)) + f_0(x_+(0))  - ∑_i=0^1 ∑_j=1^l_i𝔞(p_j^i).


Using the equality ∂_s (H_s ∘ u)= (∂_s H_s) ∘ u + dH_s(∂_s u)
we obtain

    ∫_- ∞^+∞∫_0^1 χ'(t) dH_s(χ(t), (∂_s u)) dt ds = 
       ∫_- ∞^+∞∫_0^1 χ'(t) ∂_s (H_s (χ(t),u(s,t)))dt ds - 
       ∫_- ∞^+∞∫_0^1 χ'(t) (∂_s H_s)(χ(t), u(s,t)) dt ds.

We can compute

    ∫_- ∞^+∞∫_0^1 χ'(t) ∂_s (H_s (χ(t),u(s,t)))dt ds = 
       ∫_0^1 χ'(t) H_+ (χ(t), x_+(χ(t))) dt - ∫_0^1 χ'(t) H_- (χ(t), x_-(χ(t))) dt=
    
       ∫_0^1 H_+ (t, x_+(t)) dt - ∫_0^1 H_- (t, x_-(t)) dt.



Thus, rearranging the equalities, we have

    𝒜(x_+) - 𝒜(x_-) =    ∫_- ∞^+∞∫_0^1 | ∂_s u |^2 dt ds + ∑_i=0^1 ∑_j=1^l_i𝔞(p_j^i) -
       ∫_- ∞^+∞∫_0^1 χ'(t) ∂_sH_s(χ(t) u(s,t)) dsdt.

Finally, we estimate

    ∫_- ∞^+∞∫_0^1 χ'(t) (∂_s H_s)(χ(t), u(s,t)) dt ds ≤ 6 ∂_sH_s _∞^+

and obtain Equation (<ref>).


The differential in CF(L_0, L_1; H, J_∙) decreases the action. If ∂_sH_s ≤ 0, then the continuation map Φ_H_s also decreases the action.


Now we turn our attention at the continuation map Ψ_G defined in
Subsection <ref>. Let
G × L_1 → 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.
 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



  * ψ_t(L) is a safe isotopy, and

  * 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 → 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_1^s. Fixing a choice of of local Hamiltonian G ×
L_1 → 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 <ref>.


For every chords 𝐱_- ∈𝒞_H' and 𝐱_+ ∈𝒞_H and for every sets of self-intersection points 𝐩^0 = (p_1^0,
…, p_l_0^0) of L_0 and 𝐩^1 = (p_1^1, …, p_l_1^1) of L_1,
if 𝔐_L_0, L_1(𝐩^1, x_-, 𝐩^0, x_+; H, G, J̃, ν) ∅, then

    𝒜(x_-) ≤𝒜(x_+)+ 2 μ G _∞,

where G _∞ is the supremum norm of G and μ≥ 0 is the measure of the subset {s ∈} for which G_s  L → is not constantly zero.


Consider (ζ^0, ζ^1, u) ∈𝔐_L_0, L_1(𝐩^1, x_-, 𝐩^0, x_+; H, G, J̃, ν).
Observe that the map u  Z_ζ^0, ζ^1→ W
extends to a continuous map u  Z → W. We have:

    ∫_- ∞^+∞∫_0^1 | ∂_s u |^2 dt ds =    ∫_Z d u^*θ - ∫_0^1  (∫_- ∞^+∞ (
    ∂/∂ s(H ∘ u)  ) ds  ) dt =   
       ∫_Zu^*d θ  - ∫_0^1 H(x_+(t))dt + ∫_0^1 H(x_-(t))dt.


We denote by u_i → W, for
i=0,1, the continuous and piecewise smooth maps  u_i(s)=u(s,i) and use Stokes theorem:

    ∫_Z u^*dθ = ∫_0^1 x_+^* θ - ∫_0^1 x_-^* θ + ∫_ u_0^*θ -
    ∫_  u_1^*θ.

Let f_0 and f_1 be the potentials of L_0 and L_1 respectively, and f̃_1
the potential of ψ_1(L_1).
The map u_0 takes values in L_0, and therefore

    ∫_ u_0^*θ = f_0(x_+(0)) - ∑_j=1^l_0𝔞(p_j^0) -
    f_0(x_-(0)).

We are left with the problem of estimating ∫_ 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 ×× with the Liouville form Θθ + τ dσ,
where σ is the coordinate of the first copy of  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 ×^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 → W to ũ_1 → W ×× by defining

    ũ_1(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 <ref>, together with the
Lagrangian condition satisfied by Σ, we obtain

    ∫_-∞^+∞u_1^* Θ= f'_1(x_+(1)) - ∑_j=1^l_1𝔞(p_j^1) - 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_j^1)>0 for the action computed with respect to the induced potential function on ψ^ν_s(L_1) for the corresponding value of s ∈.

Since
ν_ζ^1' _∞≤ 2, we finally obtain

    𝒜(x_-) ≤𝒜(x_+) + 2 μ G _∞,

where μ≥ 0 is as required.




 §.§ Pushing up

In this subsection we prove the following proposition.

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 to after a couple of lemmas. Given Λ > λ >0 and L>0,
we define a function
h_λ, Λ, 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 <ref> 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,
λ e^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 → 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 generality, that L_0 and L_1 intersect transversely, that L_i
∩ W_1^e 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, λ
e^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.

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.


If x is a Hamiltonian chord contained in ∂ W_r, then the action of x is

    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 (<ref>) 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) ≥ R λ_+ - λ R + 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 <ref> 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 <ref> is the following lemma.

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.


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 × L → for which dG_t=0 outside of a compact subset of (0,1) × L →; see Lemma <ref>. Since the Liouville form is conformally symplectic, it actually preserves the space of compatible almost complex structures which are 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 <ref>
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 <ref>, 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, 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.


For every λ and Λ there are continuation maps

    Φ_λ, Λ^(1)   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_λ, Λ^(2)   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_∙)

such that there is a chain homotopy between Φ_λ, Λ and
Φ_λ, Λ^(2)∘Φ_λ, Λ^(1).


We can assume that Φ_λ, Λ, Φ_λ, Λ^(1) and
Φ_λ, Λ^(2) 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_λ, Λ^(1)
is contained in 
    CF^I((L_0, ε_0), (L_1, ε_1); H_λ, Λ, R,
    J_∙)

(here we use Lemma <ref>) and therefore it follows from Lemma <ref> 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.






§ FLOER COHOMOLOGY AND LAGRANGIAN SURGERY


Lalonde and Sikorav in <cit.> and then
Polterovich in <cit.> 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 <cit.>, Fukaya, Oh, Ohta and Ono in
<cit.> and by Biran and Cornea in <cit.>. After a first version of this article had appeared, Palmer and Woodward gave a more comprehensive treatment of Lagrangian surgery in <cit.>. Our goal in this section is to establish Proposition <ref>, 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 <ref>. This turns out to be a very useful perspective, since it enables us to apply techniques from Legendrian contact cohomology in order to study them.



 §.§ The Cthulhu complex

In this subsection we recall, and slightly generalise, the definition of Floer cohomology for
Lagrangian cobordisms we defined in <cit.>.

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⊂ (× M,d(e^tβ))=(× W ×,d(e^t(dz+θ)))

such that, for C
and R sufficiently large,


  * Σ∩ (- ∞, -C] × W × =  (- ∞, -C] × L_-^+,

  * Σ∩ [C, + ∞) × W × =  [C, + ∞) × L_+^+,

  * Σ∩ (× W_R^e ×) is tangent to both ∂_s and the lift of
the Liouville vector field ℒ of (W, θ), and  

  * e^s α |_Σ = dh for a function h Σ→ which is constant on
Σ∩ (- ∞, -C] × W ×.

The intersection

    Σ∩ (× W_R^e ×)=× (L^+_±∩ (W_R^e ×))

of (<ref>) is called the lateral end of Σ.

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
-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 -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_R^e=∅, 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 × 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 <cit.> for the detailed definition. The cobordisms
considered in <cit.> 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 <cit.> can be extended to the present situation thank to
the following maximum principle.
 
Let J and J be almost complex structures on × W × and W, respectively, each cylindrical inside the respective symplectisation × W_R^e and half-symplectisation W_R^e for some R >0. We moreover require that the the canonical projection

    (× W_R^e ×,J) → (W_R^e,J)

is holomorphic. Then
every J-holomorphic map u Δ→× W × with


  * Δ = D^2 ∖{ζ_0, …, ζ_d } where (ζ_0, …, ζ_d)
∈ Conf^d+1(∂ D^2),

  * u(∂Δ) ⊂Σ^0 ∪Σ^1, and

  * u maps some neighbourhood of the punctures {ζ_0, …, ζ_d } into × W_R ×,

has its entire image contained inside × W_R ×.


By the assumptions the image of the curve u|_u^-1(× W_R^e ×) under the canonical projection

    (× W_R^e ×,J) → (W_R^e,J)

is compact with boundary on ×∂ W_R^e ×. The statement is now a consequence of the maximum principle for
pseudoholomorphic curves inside W_R^e ≅ [R, + ∞) × V which


  * satisfy a cylindrical boundary condition, and

  * are pseudoholomorphic for a cylindrical almost complex structure.

Namely, by e.g. <cit.>, the symplectisation coordinate
𝔯 W_R^e → [R, + ∞) restricted to such a curve cannot have a local
maximum.

With Lemma <ref> at hand, the arguments of <cit.> go
through, and therefore we have the following result.

The map 𝔡_ε_-^0, ε_-^1 is a differential and 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.
 
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.


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.



 §.§ The surgery cobordism


 In this subsection we describe the Lagrangian surgery of
<cit.> and <cit.> 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
<cit.> 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/2ηs^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→ given by

    F_η,δ,ζ^+(q,ξ)=ξ^3/3-g^+(|q|)ξ,

where

    g^+(s)=(3/2f_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)=(3/2f_η,δ,ζ(s))^2/3 for
s>3η/4,

  (ii) g^-(s)<0 for s<η/2, and

  (iii) 0<(g^-)'(s)<2δη/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 <ref>). 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 <ref> 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.




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 (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_iη_i/16.



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. (Note that 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

    Δ_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.


 §.§ Effect of surgery on Floer cohomology


In this subsection we use Σ(a_1, …, a_k) and our
Floer theory for Lagrangian cobordisms to relate the Floer cohomology 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_Σ𝔄(𝕃^+) →𝔄(𝕃^+(a_1, …, a_k)).

If follows from  <cit.> 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     i=1, …, k,; Φ_Σ(c)  = c+ 𝐰        c  a_i,;                ]

where 𝐰 is a linear combination of products c_1 … c_m with

    𝔞(c_1) + … + 𝔞(c_m) < 𝔞(c).


 
If ε𝔄(𝕃^+) → is an augmentation such that
ε(a_i)=1 for i=1, …, k, then there is an augmentation
ε𝔄(𝕃^+(a_1, …, a_k)) →
such that ε= ε∘Φ_Σ.


Let  ℑ be the bilateral ideal generated by a_i-1, …, a_k-1: then
ε induces an augmentation 
    ε𝔄(𝕃^+)/
    ℑ→.
 By Equation (<ref>) 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 ε𝔄(𝕃^+)/
ℑ→ induces an augmentation on 𝔄(𝕃^+(a_1, …, a_k)),
which we still denote by ε.

The construction of ε is not explicit because the isomorphism
𝔄(𝕃^+)/ ℑ≅𝔄(𝕃^+(a_1, …, a_k)) is not explicit.

 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 <ref>.


We denote by Σ_T the trivial cobordism Σ_T = × T^+ ⊂× 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 <ref> implies that the map d_+- in the Cthulhu
differential for the cobordisms Σ_T and Σ(a_1, …, a_k) is a
quasi-isomorphism.




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 <ref> is applied to an augmentation

    ε_c 𝔄(𝕃^+ ∪ (𝕃')^+) →

induced by the cycle

    c ∈CF((𝕃,ε),(𝕃',ε'))

as in Lemma <ref>, then the push-forward of the augmentation under the DGA morphism

    ε_c=ε_c𝔄(𝕃^+(a_1,…,a_k) ∪ (𝕃')^+) →

is induced by a cycle

    c∈CF((𝕃(a_1,…,a_k),ε),(𝕃',ε'))

which moreover is mapped to c under the quasi-isomorphism from Proposition <ref> .


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 <ref> 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 <ref>.


Now assume that 𝕃' is a push off of 𝕃 as constructed in Lemma <ref>, 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.


The cycle

    e∈LCC_ε,ε'(𝕃^+(a_1,…,a_k),(𝕃')^+)

provided by Lemma <ref> (which is mapped to e under the quasi-isomorphism by Proposition <ref>) satisfies the property that

    μ^2(e,·) CF(T, (𝕃(a_1,…,a_k),ε)) →CF(T, (𝕃',ε'))

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



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')^+, is specified uniquely by the
requirement that it coincides with the first lift outside of a compact subset.

Recall that e is closed by Lemma <ref> and by Lemma <ref> 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 <ref> to the Legendrian 𝐋^+(a_1,…,a_k) obtained by surgery on 𝐋, yields a quasi-isomorphism

    LCC_ε_e(T^+, 𝐋^+(a_1, …, a_k)) ≅⟶LCC_ε_e(T^+, 𝐋^+).

(Here we use that ε_e=ε_e by Lemma <ref>.) The complex on the right-hand side is acyclic by Lemma <ref>, 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

    LCC_ε_e(T^+, 𝐋^+(a_1, …, a_k))

is equal to the mapping cone of

    μ^2(e,·) CF(T, (𝕃(a_1,…,a_k),ε)) →CF(T, (𝕃',ε')).






 §.§ 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
Tw𝒜 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.


  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

  * a matrix X= (x_ij)_0 ≤ i, j ≤ k such that
x_ij∈_𝒜(L_i,L_j)
and x_ij=0 if i ≥ j, which satisfies the Maurer-Cartan equation

    ∑_d=1^k μ^d_𝒜(X,…,X_d times)=0.

  

The integers κ_i are degree shifts and are part of the definition only if the morphism
spaces _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 
    _Tw𝒜(𝔏,𝔏'):=
    ⊕_i,j_𝒜(L_i,L_j)[κ_i- κ_j']

and, given d+1 twisted complexes ℒ_0, …, ℒ_d, we define
A_∞ operations

    μ^d_Tw𝒜_Tw𝒜(𝔏_d-1,𝔏_d)
    ⊗…⊗_Tw𝒜(𝔏_0,𝔏_1) →_Tw𝒜 (𝔏_0, 𝔏_d)

by

    μ^d_Tw𝒜(q_d,…,q_1)=
       ∑_k_1,…, k_d ≥ 0μ_𝒜^k_1+… +k_d+d(X_d, …,
    X_d_k_d, q_d, X_d-1, …, X_1,q_1,X_0…,X_0_k_0).


It is shown in <cit.> that the set
of twisted complexes with operations μ^d_Tw𝒜 constitutes an A_∞-category
Tw𝒜 which contains 𝒜 as a full subcategory. Furthermore it is
shown in <cit.> that  Tw𝒜 is
the triangulated envelope of 𝒜 and thus H^0Tw(𝒜) is the derived
category of 𝒜.
 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
Tw𝒜 to a twisted complex built from the object L_i's.




If there is a twisted complex 𝔏 built from L_0, …, L_k such that, for
every object T of 𝒜 we have H _Tw𝒜(T, 𝔏)=0,
then L_0 is quasi-isomorphic in Tw𝒜 to a twisted complex built from L_1, …,
L_k.


  This follows from the iterated cone description of twisted complexes
  from <cit.>. More precisely, from the
  definition of twisted complexes, for any object T we have that
  _𝒜(T,L_0) is a quotient complex of hom_Tw𝒜(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 _Tw𝒜(T,
𝔏)  implies then that

    H_𝒜(T,L_0)≅ H_Tw𝒜(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.


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. We 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.



Let (L_i, ϵ_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 ε𝔄(𝕃^+) → is an
augmentation such that:


  * ε(p) = ε_i(p) for every pure chord p of L_i^+,
and

  * 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_jε(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 <ref>).  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_Tw𝒲F(T, 𝔏)= HW(T, (𝕃,
    ε)).



Denote by ε_0 the augmentation of 𝔄(𝕃^i) 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 <ref>, 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_L_0, …, L_d^0(𝐩^d, a_0, 𝐩^1, a_1, …,
𝐩^d-1, a_d; J) as in Section <ref>.
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
<ref>). Condition (2) in the definition of
ε however implies that #𝔐_L_0, …, L_d^0
(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 _Tw𝒲F(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 <ref> implies the following result, which is the main result of this section:


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 <ref>, and where 𝔏 is a twisted complex built from the L_i with i=1,…, m.



 Conditions (2) and (3) of
Proposition <ref> 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 <ref> imply that if a_k is a
self-intersection point of L_i, then augmentation ε_i evaluates to 1 on a_k.



We consider the twisted complex 𝔏 built from L_i, i=1,…,m, that is constructed by an application of Lemma <ref> 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 <ref>. We proceed to give the details.

Let 𝕃' be the push-off of L_1∪…∪ L_m as considered in Lemma <ref>.

Consider the cycle e∈CW((𝕃(a_1,…,a_k),ε),(𝕃',ε')) supplied by Corollary <ref>. As above, e is  an element in the summand

    CF((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 <ref> combined with Lemma <ref> then establishes the sought quasi-isomorphism. Indeed, every 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 <ref>. (Here 𝐋=𝕃(a_1,…,a_k) ∪𝕃' as in the proof of the latter corollary.)

Here we remind the reader that the components of 𝕃(a_1,…,a_k) typically are only immersed, as opposed to embedded. The statement that 
    CW(T, (𝕃(a_1,…,a_k),ε)) ≅hom(T,𝔏),

 is hence established on the level of twisted complexes on the pre-category level; c.f. Remark <ref>.




§ GENERATING THE WRAPPED FUKAYA CATEGORY


In this section we prove Theorem <ref>.





 §.§ Geometric preparations


Before proving the main theorems we need some geometric preparations which will be used in the technical work of Section <ref>.
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,d θ) and
(D_δ T^* C_i, d 𝐩∧ d 𝐪) maps the new
Liouville form θ_c to 𝐩 d 𝐪. 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 <ref>.) 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 <ref>.

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 <cit.>. 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 <ref> 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 <ref>, 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 ≥ 0ψ_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,𝐩 d 𝐪)

with the standard symplectic cotangent bundles.

Recall Conditions (i) and (ii) from Subsection <ref>. 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

    𝔯^-1(R_0+1) ∩ℋ_i=S^*_r_0T^*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

    𝔯(𝐩,𝐪)=log𝐩- logr_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^*_aC_i ⊂ D_δ T^*C_i under the identification ℋ_i ≅ D_δ T^*C_i.

For every i= 1, …, l and a ∈ C_i, the Lagrangian plane D_a is isotopic
to D_i by a cylindrical Hamiltonian isotopy.


Recall that (ℋ_i, θ_c) is isomorphic to (T^*C_i, 𝐩
d 𝐪) 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.

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

    𝒟_a_i⊂ (ℋ_i,θ_c) ≅ (T^*C_i,𝐩
    d𝐪)

of D_a_i≅ T^*_a_iC_i and (D_η T^*D_a_i,-d𝐩̃∧ d 𝐪̃)
for some η >0 small, where (𝐩̃, 𝐪̃) are the canonical coordinates on T^*D_a_i.
It is clearly possible to make this
identification so that

    𝔯(𝐩̃,𝐪̃)=log𝐪̃- logr_0+R_0+1,   𝐪̃≥ r_0,

is satisfied.

We redefine 𝔯 as in Remark <ref>, 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→ 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∩{𝐪̃≤ e^k+2r_0}),

where it is described by the graph of the differential of a function g_a_i D_a_i∩{𝐪̃≤ e^k+2r_0}→ 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≤ϵ' for i=1, …, k and ϵ'>0 small which will be specified in Lemma <ref>.

Conditions (L1)—(L3) provides sufficient control of the intersections of L and the Lagrangian skeleton. Later in Lemma <ref> we will use this in order to perform a deformation of the immersed Lagrangian submanifold

    L ∪ D_a_1∪…∪ D_a_k

by 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 <ref>) will then give us the twisted complex which exhibits L as an object built out of the different D_a_i.



 §.§ Proof of Theorem <ref>

In this section we prove Theorem <ref> assuming the results of Section <ref>.
The result is a corollary of the following proposition.

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 Tw𝒲ℱ(W, θ) to a twisted complex built from the objects D_a_1, …, D_a_k.


We assume that L satisfies Conditions (L1), (L2) and (L3) from the previous section. Then by Lemma <ref> combined with Lemma <ref> there exist Lagrangian planes D_a_1^w, …, D_a_k^w satisfying the
following properties. First D_a_i^w 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_a_1^w ∪…∪ D_a_k^w to (W ×, θ + dz) such that the intersection point a_i lifts to a Reeb chord from (D_a_i^w)^+ to L^+ of
length ϵ >0 for i=1, …, k — see Lemma <ref> for more details — there exists an augmentation ε𝔄(𝕃^+)
→ for which


  * ε(a_i)=1 for i=1, …, k, and

  * ε(d)=0 if d is a chord from L^+ to (D_a_i^w)^+ for i=1,
…, k, or a chord from (D_a_i^w)^+ to (D_a_j^w)^+ with i>j.

Moreover, using Property (L3) above for ϵ'>0 sufficiently small, it follows that the Reeb chords a_i all are contractible (c.f. Definition <ref>).

By Proposition <ref> the augmentation ε induces a twisted complex 𝔏 built from L_1=D_a_1^w, …, L_k=D_a_k^w,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 <ref>. Using this acyclicity, Proposition <ref> implies that L is quasi-isomorphic to a twisted complex built from the different D_a_i^w ≅ D_a_i (this last isomorphism follows from the invariance properties for wrapped Floer cohomology under cylindrical Hamiltonian isotopy; see e.g. <cit.>).



We can therefore complete the proof of Theorem <ref>:


Lemma <ref> and Proposition <ref> imply that L is isomorphic to a twisted complex built out of the Lagrangia planes D_a_i. Lemma <ref> and the fact that Hamiltonian isotopies generated by cylindrical Hamiltonians induce isomorphisms in the wrapped Fukaya category (see e.g. <cit.>) imply that each D_a_i is isomorphic to one of the cocore of W.
  





 §.§ Constructing the augmentation
 
We start by assuming that the modifications from Section <ref> 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 → 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∩{𝐪̃≤ e^k+2r_0 }→
from (L1) can be assumed to be sufficiently small by (L3), so that df is almost zero inside
L ∩ W_R_0+k+1.



We fix functions 𝔥_i ^+ → such that

    𝔥_i(ρ) = 
    0,    ρ≤ R_0+i,
    
    - ρ +  R_0 +i+ 1/2,    ρ≥  R_0+i+1,

and 𝔥_i”(ρ) ≤ 0 for all ρ∈^+.
Then we define the cylindrical
Hamiltonians  H^i  W →, i=1, …, k, by

    H^i(w)= 𝔥_i(e^𝔯(w)).

The graph of 𝔥_i appears in Figure <ref>.


We will denote by ϕ^i_t the flow of the Hamiltonian vector field of H^i.
Given T_i ∈, we denote D_a_i^w = ϕ^i_T_i(D_a_i).

We fix ϵ >0, and on each Lagrangian plane D_a_i we choose the
potential function f_i  D_a_i→ such that

    f_i = f(a_i)+ ϵ.

Note that the functions f_i indeed are constant, since the Liouville vector field is tangent to the planes D_a_i. Let  f_i^w  D_a_i^w → be the potential function on
D_a_i^w induced by f_i using Equation (<ref>).

We denote by 𝕃= L ∪ D_a_1^w ∪…∪ D_a_k^w, which we
regard as an exact Lagrangian immersion, and by 𝕃^+ the Legendrian lift of
𝕃 to (W ×, θ + dz) defined using the potential functions
f, f_1^w, … ,  f_k^w. Note that an intersection point d ∈ D_a_i^w
∩ D_a_j^w lifts to a chord starting on D_a_i^w and ending on D_a_j^w if and only if f_i^w(d)
> f_j^w(d), and similarly if one of the two discs is replaced by L and its potential is
replaced by f.

 




There exist real numbers 0 < T_k < … < T_1 and ϵ, ϵ' >0 such that,
if L satisfies (L1)–(L3), then each chord of 𝕃^+
is of one of the following types:


  * type a: the chords a_i, going from (D_a_i^w)^+ to L^+ for i=1, …, k, of length
ϵ,

  * type b: chords b_ij^m consisting of all other chords from (D_a_i^w)^+ to L^+ for 1 ≤ i < j ≤ k and
1 ≤ m ≤ m_0(i,j) for some m_0(i,j),

  * type c: chords c_ij^m from (D_a_i^w)^+ to (D_a_j^w)^+ for  1 ≤ i < j
≤ k and 1 ≤ r ≤ m_0(i,j), and

  * “order-reversing” type: chords from L^+ to (D_a_i^w)^+ for i=1, …, k or
chords  from (D_a_i^w)^+ to (D_a_j^w)^+ for i=1, …, k and i>j.

(see Figure <ref>).
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_a_i^w ∪ D_a_j^w, 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_ij^m} could be empty for some i,j. In that case, we say that m_0(i,j)=0.


Recall that Properties (L1)–(L3) from Subsection <ref> 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_a_i^w and D_a_j, as opposed to the intersection points of D_a_i^w 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_a_1^w, …, D_a_k^w are embedded exact Lagrangian
submanifolds, and therefore there is no Reeb chord either from L^+ to L^+ or from
(D_a_i^w)^+ to (D_a_i^w)^+ for any i=1, …, k.

From Equation (<ref>), the potential of D_a_i^w is

    f_i^w = f(a_i)+ ϵ + T_i(𝔥_i'(e^𝔯) e^𝔯 -
    𝔥_i(e^𝔯)).

Note that the quantity T_i(𝔥_i'(e^r)e^r - 𝔥_i(e^r)) is nonincreasing
in r because 𝔥_i”≤ 0. Therefore f_i^w satisfies

    [                                 f_i^w(w)  = f(a_i) + ϵ                             if  w ∈ D_a_i^w ∩ W_R_0+i,; f_i^w(w)  ∈ [f(a_i) + ϵ - T_i(R_0+i+ 1/2), f(a_i) + ϵ]             if  w ∈  D_a_i^w ∩ ( W_R_0+i+1∖  W_R_0+i),;              f_i^w(w)   = f(a_i) + ϵ - T_i(R_0+i+ 1/2)                        if  w ∈ D_a_i^w ∩
     W_R_0+i+1^e.;                                                        ]


Note that D_a_i^w ∩ W_R_0+i = D_a_i∩ W_R_0+i and that D_a_i^w ∩
W_R_0+i+1^e is a cylinder over a Legendrian submanifold.

We choose positive numbers 0<T_k< …< T_1 such that


  * f(a_1)+ ϵ -T_1(R_0+1+ 1/2) < … < f(a_k) + ϵ - T_k (R_0+k+
1/2),

  * f(a_i) + ϵ - T_i (R_0+i+ 1/2) < min_L f for all i=1, …, k,


  * there are no intersection points between L, D_a_1, …, D_a_k, D_a_1^w, …
D_a_k^w in their cylindrical parts, and

  * at every intersection point between L, D_a_1, … D_a_k, D_a_1^w, …
D_a_k^w 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.

We observe that, for any point c ∈ D_a_i^w ∩ D_a_j and any i,j = 1, …, k,
the quantity 𝔞(c)= |f_i^w(c)-f_j(c)| is independent of ϵ. Then we choose ϵ >0 sufficiently small so that

    ϵ < min{𝔞(c) : c ∈ D_a_i^w ∩ D_a_j and 𝔞(c)  0  for  i,j = 1, …, k }.

This implies that, for all c ∈ D_a_i^w ∩ D_a_j such that 𝔞(c)  0,
the signs of f_i^w(c)- f_j(c) and of f_i^w(c)- f(c) are equal. (Recall that f=f_j-ϵ holds there by construction.)

Consider the set of points c_ij^m ∈ D_a_i^w ∩ D_a_j with positive action difference

    0<f_i^w(c_ij^m) - f_j(c_ij^m) =f_i^w(c_ij^m) - (f(a_j)+ϵ).

(Here m is an index distinguishing the various points with the
required property.) Then i <j and c_ij^m ∈ W_R_0+i+1∖ W_R_0+i; in
particular c_ij^m ∈ D^w_a_i∩ D^w_a_j. See Figure <ref>.




The intersection points b^m_ij now coincide with c^m_ij, but seen as intersections of L = D_a_j and D_a_i^w. 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≤ϵ' for ϵ'>0 sufficiently small. In particular, this means that each intersection point
c_ij^m still corresponds to a unique intersection point b_ij^m ∈ D_a_i^w ∩ L, such that moreover
f_i^w(b_ij^m) - f(b_ij^m) >0 is satisfied. Conversely, any intersection point d ∈ D_a_i^w ∩ L with f_i^w(d) - f(d) >0 is either a_i or one of the b_ij^m. This is the case
because the only intersection point in D_a_i^w ∩ L ∩ W_R_0+i is a_i and for any intersection point d ∈ D_a_i^w ∩ L ∩ W_R_0+i+1^e we must have f_i^w(d) - f(d) < 0.

The existence of the triangle  follows now by applying Corollary <ref> to L ∪ D_a_i^w ∪ D_a_j^w 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 can 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^+).

The Chekanov-Eliashberg algebra (𝔄(𝕃^+), 𝔡) of
𝕃^+ admits an augmentation 
    ε𝔄(𝕃^+)
    →
 such that ε(a_i)=1 for all i=1, …, k. Moreover, this
augmentation vanishes on the order reversing chords.


Set L_i:=D_a_i^w 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 <ref>.

 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^→ with 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

    = 𝔄^→_k+1⊂𝔄^→_k ⊂…⊂𝔄^→_0 = 𝔄^→,

where 𝔄^→_i is generated by all chords
a_s, b_sj^m, c_sj^m 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_ij^m and
c_ij^m are pairwise distinct and,

  (ii) for all i,j,m, the actions
𝔞(b_ij^m) and 𝔞(c_ij^m) are close enough that,
whenever 𝔞(c_i_- j_-^m_-) <  𝔞(c_ij^m) <
𝔞(c_i_+ j_+^m_+), we also have  𝔞(c_i_- j_-^m_-) <
𝔞(b_i j^m) < 𝔞(c_i_+ j_+^m_+).

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

For each fixed i we define a total order on the pairs (j,m) by declaring that (h,l) ≺_i (j,m) if
𝔞(c_ih^l) < 𝔞(c_ij^m). When the index i is clear from the context, we will simply write ≺.

We know that 𝔡^→ a_i=0 for action reasons and ⟨𝔡^→ b_ij^m,
a_jc_ij^m ⟩ =1 by the last part of Lemma <ref>.
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:

    𝔡^→ a_i     =0, 
    𝔡^→ b_ij^m     = α_j^m a_i + ∑_(h,l) ≺_i (j,m)β_jl^mh b_ih^l
    + a_j c_ij^m + ∑_(l,h) ≺_i (j,m) w_lj^hmc_ih^l , 
    𝔡^→ c_ij^m     = ∑_(l,h) ≺_i (j,m)w_lj^hmc_ih^l

with α_j^m, β_jl^mh∈ and w_lj^hm, w_lj^hm∈𝔄^→_i+1.

Then the filtration (<ref>) is preserved by 𝔡^→.
We want to define an
augmentation ε𝔄^→→ 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 =.

Suppose now we have defined an augmentation
ε𝔄^→_i+1→. We will extend it
to an augmentation ε𝔄^→_i → by
an inductive argument over the action of the chords c_ij^m. For this reason in the following discussion i will be fixed.

We define ε(a_i)=1 and ε(b_ij^m)=0 for all
j and m. To define ε on c_ij^m we work inductively
with respect to the order ≺ induced by the action.  Suppose that we
have defined ε(c_ih^l) for all c_il^h such that
(h,l) ≺ (j,m). Then we can achieve
ε(𝔡^→ b_ij^m)=0 by prescribing an appropriate value to

    ε(c_ij^m)=ε(a_jc_ij^m),

since the values of ε on all other chords appearing in the expression of 𝔡^→ b_ij^m 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_ij^m. We will prove
that in fact ε(𝔡^→ c_ij^m)=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_ij^m.

If (j,m) is the minimal element for the order ≺, then
𝔡^→ c_ij^m=0 and therefore
ε(𝔡^→ c_ij^k)=0. Suppose now that we have
verified that ε(𝔡^→ c_ih^l)=0 for all
(h,l) ≺ (j,m). From 𝔡^→ (𝔡^→ b_ij^m)=0 and
ε (𝔡^→ ( a_jc_ij^m))= ε(𝔡^→
c_ij^m) we obtain

    ε(𝔡^→ c_ij^m) + ∑_(h,l) ≺ (j,m)β_jl^mhε(𝔡^→ b_ih^l) + ∑_(l,h) ≺ (j,m)ε(𝔡^→(w_lj^hmc_ih^l))=0.

We have ε(𝔡^→ b_ih^l)=0 by construction and
ε(𝔡^→(w_lj^hmc_ih^l))=0 by the induction
hypothesis.
From this we conclude that ε(𝔡^→ c_ij^m)=0.

Finally we simply precompose ε with the projection 𝔄(𝕃) →𝔄^→ and
obtain an augmentation of 𝔄(𝕃) satisfying the required conditions.





§ GENERATION OF THE WRAPPED FUKAYA CATEGORY OF WEINSTEIN SECTORS.


In this section we prove Theorem <ref>. We recall that the “linear setup”, introduced by Abouzaïd and Seidel in <cit.> and used in the proof of Theorem <ref>, is not available for sectors; instead, Ganatra, Pardon and Shende in <cit.> define the wrapped Fukaya category of a Liouville sector by a localisation procedure. However, the strategy of the proof of Theorem <ref> applies to
the “localisation setup” as well, with only minor modifications of some technical details.
The goal of this section is to explain those modifications, which in most cases will be simplications. Before proceeding, we recall that the proof of Theorem <ref> had four main steps:


  * an extension of the construction of wrapped Floer cohomology to certain exact immersed Lagrangian submanifolds,

  * triviality of wrapped Floer cohomology for immersed exact Lagrangian submanifolds which are disjoint from the skeleton (“trivial triviality”),

  * identification of certain twisted complexes in the wrapped Fukaya category with Lagrangian surgeries, and

  * construction of the bounding cochain after a suitable modification of the Lagrangian cocores.





 §.§ The wrapped Fukaya category for sectors

In this subsection we recall briefly the definition of the wrapped Floer cohomology and the wrapped Fukaya category for sectors following <cit.> and show that
our construction of the wrapped Floer cohomology of an exact immersed Lagrangian submanifold
can be carried over to this setting.

Given ϵ >0 we, denote _0 ≤ < ϵ= { x+iy ∈ : 0 ≤ x < ϵ}. If (S, θ, I) is a Liouville sector, by <cit.> there is an identification

    (Nbd(∂ S), θ) ≅(F ×_0 ≤ < ϵ, θ_F + 1/2 (x dy -y dx) + df)

where f  F ×_0 ≤ < ϵ→ satisfies the following properties:


  * the support of f is contained in F_0 ×_0 ≤ < ϵ for some Liouville
domain F_0 ⊂ F, and

  * f coincides with f_±∞ F → for |y| ≫ 0.

We denote πNbd(∂ S) →_0 ≤ < ϵ the projection induced by
the identification (<ref>).

We will consider almost complex structures J on S which are cylindrical with respect to the Liouville vector field ℒ of θ and make the projection π holomorphic (where, of course, we endow  with its standard complex structure). It is easy to see that this choice of almost complex structures constrains the holomorphic curves with boundary in int(S) so that they stay away from the boundary ∂ S; see <cit.>. Thus, if L_0, L_1 are two transversely intersecting exact Lagrangian submanifolds with cylindrical ends, the Floer chain complex with zero Hamiltonian CF(L_0, L_1) is defined.

Let L_∙ = { L_t }_t ∈ I be an isotopy of exact Lagrangian submanifolds which are cylindrical at infinity over Legendrian submanifolds Λ_t in the contact manifold (V, α) which is the boundary at infinity of (S, θ, I). Let X_t be a vector field along L_t directing the isotopy. We can choose this vector field so that, where the isotopy is cylindrical, it is the lift of a vector field along Λ_t which we denote by X_t^∞.  We say that the Lagrangian isotopy is positive if α(X_t^∞) ≥ 0 everywhere. We say that the  isotopy is short if its trace
⋃_t ∈ IΛ_t is embedded. This implies that Legendrian links Λ_0 ∪Λ_t are embedded and thus Legendrian isotopic to each other for all t ∈ I ∖{0}.

Following <cit.>, to any positive isotopy L_∙ of exact Lagrangian submanifolds with cylindrical ends, we associate a continuation element c(L_∙) ∈HF(L_1, L_0) as follows. If the isotopy is small, there is a map H^*(L_0) →HF(L_1, L_0), and we define c(L_∙) as the image of the unit in H^*(L_0) under this map. If L_∙ is not small, then we decompose it into a concatenation of small isotopies and define c(L_∙) as the composition (by the triangle product) of the continuation elements of the small isotopies. Then, for any Lagrangian submanifold K which is transverse with both L_0 and L_1, we define the continuation map

    HF(L_0, K) HF(L_1, K).

See <cit.> for the properties of the continuation element.

Given a Lagrangian submanifold L with cylindrical end, following <cit.> we consider its wrapping category
(L → -)^+, which is the category whose objects are isotopies of Lagrangian submanifolds ϕ L → L^w and morphisms from (ϕ L → L^w) to (ϕ'  L → L^w') are homotopy classes of positive isotopies ψ L^w → L^w' such that ϕ#ψ= ϕ'.

With all this in place, wrapped Floer cohomology is defined as

    HW(L, K) = _(L → L^w)^+HF(L^w, K)

where the maps in the direct system are the continuation maps defined above.

Now suppose that K is immersed and ε is an augmentation of its obstruction algebra. Then CF(L, (K, ϵ)) is defined as in Section <ref>, as long as we use the trivial Hamiltonian H= 0 in the definition — being in a Liouville sector makes no difference in any other aspect of the construction.
The definition of HW(L, (W, ε)) is then the same as in Equation (<ref>) using the product μ_2 defined in Section <ref>.


This definition is sufficient for our needs because in the proof of Theorem <ref> we only need wrapped Floer cohomology with immersed Lagrangian submanifolds in the right entry. However, it is possible to extend the definition to the case of immersed Lagrangian submanifolds on the left by identifying augmentations of L with augmentations of L^w and defining the continuation element for small isotopies using Lemmas <ref> and <ref>. Note that these lemmas are stated for Floer cohomology with trivial Hamiltonian, and therefore they extend immediately to Liouville sectors.


Now we sketch the construction of the wrapped Fukaya category following <cit.>. We recall that we do not need to extend the definition so that it includes immersed Lagrangian submanifolds, even if it would probably not be too difficult. We fix a countable set I of exact Lagrangian submanifolds with cylindrical ends so that any cylindrical Hamiltonian isotopy class has at least one representative and, for every L ∈ I, we fix a a cofinal sequence L=L^(0)→ L^(1)→… in (L → -)^+. We denote by 𝒪 the set of all these
Lagrangian submanifolds.  We assume that we have chosen the elements in 𝒪 so that all finite subsets { L_1^i_1, …, L_k^i_k} with i_1 < … < i_k consist of mutually transverse Lagrangian submanifolds.

We make 𝒪 into a strictly unital A_∞-category by defining

    hom_𝒪 (L^(i), K^(j))=
    CF(L^(i), K^(j))    if  i > j, 
       if  L^(i)= K^(j), 
      0    otherwise.

If i>j, the continuation element of the positive isotopy L^(j)→ L^(i) belongs to H_𝒪(L^(i), L^(j))= HF(L^(i), L^(j)). We will write L^(i) > K^(j) if i>j.

We denote by C the set of all closed morphisms of 𝒪 which represent a continuation element. Thus we define the wrapped Fukaya category of (S, θ, I) as 𝒲F(S, θ)=𝒪_C^-1, where 𝒪_C^-1 is the A_∞-category obtained by dividing 𝒪 by all cones of morphisms in C: i.e. 𝒪_C^-1 has the same objects as 𝒪 and its morphisms are defined as the morphisms of the image of 𝒪 in the quotient of the triangulated closure of 𝒪 by its full subcategory of cones of elements of C. This construction has the effect of turning all elements of C into quasi-isomorphisms.
See <cit.>, and in particular Definition 3.1 therein, for a precise definition of the localisation of an A_∞-category. In the following lemma we summarise the properties of the localisation that we will need.

The categories 𝒪 and 𝒲F(S, θ) are related as follows:


  * 𝒲F(S, θ) and 𝒪 have the same objects,

  * H(hom_𝒲F(L, K)) ≅HW(L, K),

  * the category 𝒲F(S, θ) is independent of all choices up to quasi-equivalence, and

  * The localisation functor 𝒪→𝒲F(S, θ) is the identity on objects and has trivial  higher order terms (i.e. it matches A_∞ operations on the nose). Moreover, when hom_𝒪(L, K)= CF(L, K), it induces the natural map HF(L, K) →HW(L, K).



(1) follows from the definition of localisation. (2) is the statement of <cit.>. (3) is the statement of <cit.>. (4) follows from the definition of 𝒜_∞-products in 𝒜_∞-quotients; see <cit.>.



 §.§ Trivial triviality.


In this section we prove Proposition <ref> for sectors. The proof is the same as in Section <ref> except for few details which must be adjusted because we need to use geometric wrapping of the Lagrangian submanifolds instead of Hamiltonian perturbations of the Floer equation and the continuation maps from Equation (<ref>) instead of those from Subsection <ref>.

For Liouville sectors we  need to modify the definition of a cylindrical Hamiltonian in order to have a complete flow. To that aim, the crucial  notion is that of a coconvex set.
 
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.

The following lemma is a rewording of <cit.>.

Given a contact manifold (V, α) with convex boundary, it is possible to find a function g  V →_≥ 0
such that


  * g > 0 outside the boundary ∂ V and g ≡ 1 outside a collar neighbourhood ∂ V × [0, δ) on which α = dt+ β where β is a one-form on ∂ V,

  * g=t^2G on ∂ V × [0, δ), where G>0 and
t is the coordinate of [0, δ), and

  * there is a collar neighbourhood 𝒩 of ∂ V, contained in ∂ V × [0, δ), which is coconvex for the contact Hamiltonian X_g of g.


Note that X_g vanishes along ∂ V. It is called a cut off Reeb
vector field in <cit.> because it is the Reeb vector field of the
contact form g^-1α on int(V). From now on we will assume that g and 𝒩 are fixed once and for all for the contact manifold (V, α) arising as
boundary at infinity of (S, θ, I). We will also extend g to the complement of a compact set of S so that it is invariant under the Liouville flow.


Let S be a Liouville sector. A Hamiltonian function H  [0,1] × S → is
cylindrical if there is a function h ^+ → such that
H(t, w)=g(w)h(e^(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 <ref>.
Condition (iv) in the latter definition becomes equivalent to asking that λ should not be the length of a chord of the cut off Reeb vector field. In this section, cylindrical Hamiltonian will be used to define positive Hamiltonian isotopies of Lagrangian submanifolds and not to deform the Floer equation.

We say that an exact Lagrangian submanifold of S (possibly immersed) is safe if it  is
cylindrical over a Legendrian submanifold contained in V ∖𝒩. Since
𝒩 is strictly contained in an invariant neighbourhood of ∂ V,
every cylindrical exact Lagrangian submanifold of S is Hamiltonian isotopic to one which is safe by a cylindrical Hamiltonian isotopy. We will assume that all Lagrangian submanifolds with cylindrical end are safe unless we explicitly state the contrary.

Fix an exact Lagrangian submanifold L in S with cylindrical end. Given λ, Λ, R ∈ such that 0<λ≤Λ and 0<R, let h_λ, Λ, R be the function defined in Equation (<ref>) and consider the cylindrical Hamiltonian H_λ, Λ, R induced by h_λ, Λ, R as in definition <ref>. Note that, when Λ = λ, we obtain the Hamiltonian function H_λ induced by the function h_λ of Equation (<ref>)  independently of R. We denote by L_∙^λ,Λ,R= { L_t^λ,Λ,R}_t ∈ the positive Hamiltonian isotopy generated by H_λ, Λ, R such that L_0^λ,Λ,R=L. When Λ= λ we write L_∙^λ instead.


Let λ_n → + ∞ be an increasing sequence. Then the Lagrangian sumbanifolds
L^λ_n_1 form a cofinal collection in (L → -)^+.


Since the Hamiltonian H_λ is autonomous, for every κ∈ we have L_1^κλ = L_κ^λ. The family { L^λ_t }_t  ≥ 0 is cofinal by <cit.> because the Hamiltonian vector field of H_λ in the cylindrical end of S is the lift of a (strictly) positive multiple of the Reeb vector field of the contact form g^-1α.



Let K be an immersed exact Lagrangian submanifold with cylindrical end, and let ε be an augmentation of the obstruction algebra of K. Often we will drop ε from the notation: even if the Floer complex depends on it, the arguments in this subsection do not. Given λ<Λ, we call the intersection points in L_1^λ, Λ, R∩ K ∩𝔯^-1((- ∞, R/2)) intersection points of type I and the intersection points in L_1^λ, Λ, R∩ K ∩𝔯^-1((R/2, + ∞) intersection points of type II. They correspond to the Hamiltonian chords of type I or II in Subsection <ref>. We denote by CF^I(L_1^λ, Λ, R, K) the subcomplex of  CF(L_1^λ, Λ, R, K) generated by the intersection points of type
I. The following lemma is the equivalent of Lemma <ref> in this context.

If the Liouville flow of (S, θ) displaces K from every compact set, then the inclusion

CF^I(L_1^λ, Λ, R, K) ↪CF(L_1^λ, Λ, R, K) is trivial in homology when Λ and R are sufficiently large.


The proof is the same as that of Lemma <ref>, whose main ingredients are Equation (<ref>), which computes the action of the generators of the Floer complex, and Lemma <ref> which estimates the action shift of the  continuation maps for compactly supported safe isotopies from Subsection <ref>. Both ingredients are still available for Liouville sectors: in fact intersection points between L_1^λ, Λ, R and K are in bijection with Hamiltonian chords of H_λ, Λ, R and the action of an intersection point is the same as the action of the corresponding chord by Equation (<ref>) and and Equation (<ref>). Thus (<ref>) still gives bounds on the action of the generators of  CF(L_1^λ, Λ, R, K), after taking into account the fact that the extra factor involving g coming from Definition  <ref> is uniformly bounded because L_1^λ, Λ, R and K are safe.

Moreover, the definition of the continuation maps for compactly supported safe isotopies from Subsection <ref> and the proof of Lemma <ref> do not depend on the Hamiltonian deformation in the Floer equation and therefore, setting H ≡ 0 in the Floer equations, they hold also for sectors.


Nowe we can finish the proof of the equivalent of Proposition <ref> for sectors.

Let (S, θ, I) be a Liouville sector and let K and L be exact Lagrangian submanifolds of S with cylindrical ends. We allow K to be immersed, and in that case we assume its obstruction algebra admits an augmentation ε. If the Liouville flow of (S, θ) displaces K from every compact set of S, then HW(L, (K, ε))=0.


For any fixed λ<Λ there is a natural homotopy class of positive isotopies L_1^∙ from L_1^λ to L_1^Λ. We need to show that, for Λ large enough with respect to λ, the continuation map associated to this class is trivial.

We represent this class by a concatenation of positive isotopies L_1^λ, ∙, R from L_1^λ to L_1^λ,Λ,R and L_1^∙, Λ, R from L_1^λ,Λ,R to L_1^Λ which lead to continuation maps

    [μ^2(·,c^λ,∙, R)]    HF(L_1^λ,K)→HF(L_1^λ,Λ,R,K) 
    
    [μ^2(·,c^∙, Λ, R)]    HF(L_1^λ,Λ,R,K)→HF(L_1^Λ,K)

To prove the proposition, it is sufficient to prove that, for any fixed λ. [μ^2(·,c^λ,∙, R)] is trivial if Λ and R are large enough.



It follows from <cit.> and the definition of H_λ, Λ, R that the map μ^2(·,c(L_1^λ, ∙, R) is the natural inclusion CF(L_1^λ ,K)⊂CF(L_1^λ,Λ,R,K) whose image is the subcomplex  CF^I(L_1^λ,Λ,R,K). Thus, the triviality of [μ^2(·,c(L_1^λ, ∙, R)] follows from Lemma <ref>.



In view of Remark <ref>, we expect Proposition <ref> to hold also when L is immersed, as long as its obstruction algebra admits an augmentation. However, we haven't checked the details.




 §.§ Twisted complexes.


In this subsection we extend to sectors the results of Section <ref> identifying certain Lagrangian surgeries with twisted complexes.


The first step is to observe that the constructions in Subsecton <ref> can be extended to Lagrangian cobrdisms in the symplectisation of the contactisation of a Liouville sector. In fact, we can work with almost complex structures on × S × satisfying the hypothesis of Lemma <ref> and such that the projection × S ×→ S is holomorphic near ×∂ S ×. Then, by <cit.>, the holomorphic curves appearing in the definition of the Cthulhu complex do not approach ×∂ S ×.

In Subsection <ref> we work only with Floer complexes with trivial Hamiltonian, and therefore the results of that subsection extend to Liouville sectors without effort. Moreover, some contortions which were needed to apply those results in the linear setup are no longer necessary in the localisation setup. Let L_1, … L_m be exact Lagrangian submanifolds with cylindrical ends and denote 𝕃 = L_1 ∪…∪ L_m. Unlike in Section <ref>, here we do not need to consider the case of immersed L_i.

We recall some notation from Section <ref>. Given a set of intersection points { a_1, …, a_k } which corresponds to a set of contractible chords (see Definition <ref>) for a Legendrian lift 𝕃^+ of 𝕃 to the contactisation of S, we denote by 𝕃(a_1, …, a_k) the result of Lagrangian surgery performed on a_1, …, a_k as explained  in <ref>.  If 𝕃^+ admits an augmentation ε such that ε(a_i)=1 for i=1, …, k, then by Lemmas <ref> and <ref> there exists an augmentation ε of 𝕃(a_1, …, a_k)^+ such that, for any exact Lagrangian submanifold T with cylindrical ends, there is an isomorphism

    Φ_* HF(T, (𝕃(a_1, …, a_k), ε)) HF(T, (𝕃, ε)).

Moreover, this isomorphism preserves the triangle products in the sense that, given two exact Lagrangian submanifolds T_0 and T_1, the diagram

    HF(T_0, (𝕃(a_1, …, a_k), ε))⊗HF(T_1, T_0) [r]^-[μ_2][d]_Φ⊗ Id   HF(T_1, (𝕃(a_1, …, a_k), ε)) [d]^Φ
    HF(T_0, (𝕃, ε))⊗HF(T_1, T_0) [r]^-[μ_2]   HF(T_1, (𝕃, ε))

commutes. This is a particular case of <cit.>. Then the isomorphisms (<ref>) induce isomomorphisms

    HW(T, (𝕃(a_1, …, a_k), ε)) ≅HW(T, (𝕃, ε))

for every exact Lagrangian submanifold T with cylindrical ends.

Assume that ε(q)=0 for all Reeb chord from L^+_i to L^+_j with i>j. We add enough objects to 𝒪 so that L_1, …, L_m are objects of 𝒪 and L_m > … > L_1. By Lemma <ref>(3) this operation does not change 𝒲F(S, θ) up to quasi-isomorphism. Then X=(x_ij)_0 ≤ i,j ≤ m∈⊕_0 ≤ i,j ≤ m_𝒪(L_j, L_i) defined as

    x_ij= ∑_a ∈ L_i ∩ L_jε(a)a    if  i<j 
    
    0     if  i ≥ j

satisfies the Maurer-Cartan equation in 𝒪 by the same argument as in the proof of Lemma <ref> and our choice of ordering of the objects L_1, …, L_m. Thus X satisfies the Maurer-Cartan equation also in 𝒲F(S, θ) by Lemma <ref>(4). We denote by 𝔏=({ L_i }, X) the corresponding twisted complex both in 𝒪 and in 𝒲F(S, θ). For any object T of 𝒪 with T>L_m there is a tautological identification of chain complexes _Tw𝒪(T, 𝔏) = CF(T, (𝕃, ε)) which, moreover, respects the triangle products; this follows from a direct comparison of the holomorphic  polygons counted by the differentials on the left and on the right as in the proof of Lemma <ref>. Thus the
isomorphisms are compatible with the multiplication by continuation elements and in the limit we obtain a map

    HW(T, (𝕃, ε)) → H_Tw𝒲F(T, 𝔏)

which is an isomorphism by <cit.> and a simple spectral sequence argument.
We can summarise these results in the following lemma, which is the analogue of Proposition <ref> in the context of sectors.


Let L_1,…,L_m, be embedded exact Lagrangian submanifolds with cylindrical ends. If there exist a Legendrian lift 𝕃^+ of 𝕃=L_1 ∪…∪ L_m,
an augmentation ε of the Chekanov-Eliashberg algebra of 𝕃^+ and a set of contractible Reeb chord { a_1, …, a_k } such that:


  (1) ε(a_i)=1 for i=1,…,k, and

  (2) ε(q)=0 if q is a Reeb chord from L_i^+ to L_j^+,

then there exist a twisted complex 𝔏 built from L_1, …, L_m and an augmentation ε of the Chekanov–Eliashberg algebra of 𝕃(a_1, …, a_k)^+ such that, for any other exact Lagrangian submanifold with cylindrical end T there is an isomorphism

    HW(T, (𝕃(a_1,…,a_k),ε)) ≅ H_𝒲F(T,𝔏).





 §.§ Construction of the augmentation


In this section we finish the proof of Theorem <ref>.
Let (S, θ, I, 𝔣) be a Weinstein sector. We modify θ so that it coincides with the standard Liouville form on the handles ℋ_i and half-handles ℋ_i'.

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 coconvex collar. This is an important observation, because it allows us to choose, once and for all, g and a corresponding coconvex collar 𝒩 for ∂ V as in Definition <ref> on the contact manifold (V, α) which is the boundary at infinity of (S, θ, I, 𝔣) such that g ≡ 1 on ℋ_i' ∖𝒩 for all Weinstein half-handles  ℋ_i'.

As in Section <ref>, 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 half-handles
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 <ref> 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 <ref> to be
H^i(w)= g(w)𝔥_i(e^𝔯(w)) and denote by D_a_i^w the image of
D_a_i under the Hamiltonian flow of H^i for a sufficiently large time. The flow can push
D_a_i^w close to ∂ W, and therefore we can no longer assume that the wrapped
planes D_a_i^w are safe. However, L and the planes D_a_i were safe, and therefore all intersection points of type a, b, and c (defined in Lemma <ref>) between L  and the planes D_a_1^w, …, D_a_k^w correspond to Hamiltonian chords contained in the complement of 𝒩 because the Hamiltonian vector field of H^i is a negative multiple of the cut off Reeb vector field outside of a compact set. Since g=1 outside 𝒩, the
energy estimates of Subsection <ref> still hold in the case of sectors, and that allows us to construct an augmentation ε for a suitable Legendrian lift 𝕃^+ of 𝕃=L ∪ D_a_i^w ∪…∪ D_a_k^w as in Lemma <ref>. At this point the proof of
Theorem <ref> proceeds in the same way as the proof of Theorem <ref>. 



§ HOCHSCHILD HOMOLOGY AND SYMPLECTIC COHOMOLOGY


In this section we use the work of Ganatra <cit.> and Gao <cit.> to derive Theorem <ref> from Theorem <ref>. Since Ganatra and Gao work in the quadratic setup of the wrapped Fukaya category, we must extend the proof of Theorem <ref> to that setup first.



 §.§ Wrapped Floer cohomology in the quadratic setup


On the level of complexes, wra­pped Floer cohomology in the quadratic setup is in some sense the simplest one to define; in this case the wrapped Floer complex CW(L_0,L_1) is the Floer complex CF(L_0,L_1;H) for a quadratic Hamiltonian H  W →, by which we mean that H=C· e^2𝔯 is satisfied outside of some compact subset of W for some constant C > 0. This construction of the wrapped Floer complexes can be generalised to the case when L_1 has transverse double points in the same manner as in the linear case, by using the obstruction algebra.

In the following we assume that all Lagrangians are cylindrical inside the noncompact cylindrical end W^e_R-1⊂ W for some R ≫ R_0. Denote by ψ_t  (W,θ) → (W,e^-tθ) the Liouville flow of (W,θ) and recall that 𝔯∘ψ_t=𝔯+t in W^e_R-1 and hence ψ_t(W^e_r)=W^e_r+t for all r ≥ R-1.




 §.§ Trivial triviality

In this subsection we prove Proposition <ref>, i.e. “trivial triviality”, in the quadratic setting. In fact, since the quadratic wrapped Floer cohomology complex does not involve continuation maps the, the proof becomes even simpler here.

Assume that L_1 ⊂ W is disjoint from the skeleton, and thus that ψ_t(L_1) is a safe exact isotopy that displaces L_1 from any given compact subset. The Liouville flow ψ_t is conformally symplectic with the conformal factor e^t, i.e. (ψ_t)^*dθ=e^t· dθ. Since L_1 is cylindrical outside of a compact subset, the aforementioned safe exact isotopy is generated by a locally defined Hamiltonian function G_t which satisfies the following action estimate.


The locally defined generating Hamiltonian G_t ψ_t(L_1) → from Section <ref> is of the form G_t=e^t· G_0∘ψ^-t for a function G_0  W → which vanishes outside of a compact set, and thus satisifes the bound G_t_∞≤ C_L_1e^t for some constant C_L_1 that only depends on L_1. In particular, the Hamiltonian isotopy whose image of L_1 at time t=1 is equal to ψ_t(L_1) can be generated by a compactly supported G_t that satisfies G_t_∞≤ tC_L_1e^t.

The conformal symplectic property implies that the primitive of θ satisfies θ|_Tψ_t(L_1)=e^td(f_1∘ψ_t) for the primitive f_1 L_1→ of θ|_TL_1 that vanishes in the cylindrical end. The formula for the action of the Reeb chords (<ref>), in particular c.f. (<ref>) in the example thereafter applied with h(x)=x^2, now readily implies that:

The generators of CF(L_0,ψ_t(L_1);H) have action bounded from below by C'_L_1e^2t for some constant C'_L_1 >0 whenever t ≫ 0 is sufficiently large.


A construction using moving boundary conditions (as in  Section <ref>) yields quasi-isomorphisms

    Ψ_G_tCF(L_0,L_1;H) →CF(L_0,ψ_t(L_1);H).

By an action estimate (c.f. Lemma <ref>) in conjunction with the above two lemmas,
 any subcomplex of CF(L_0,L_1;H) that is spanned by the generators below some given action level is contained in the kernel of Ψ_G_t whenever t ≫ 0 is taken sufficiently large.
In conclusion, we obtain our sought result:

If ψ_t(L_1) displaces L_1 from any given compact subset, then HW(L_0,L_1)=0.




 §.§ Twisted complexes and surgery formula 


In order to obtain the surgery formula in Proposition <ref> in the quadratic setting one needs to take additional care. The reason is that, since the products and higher A_∞-operations in this case are defined using a trick that involves rescaling by the Liouville flow, it is a priori not so clear how to relate these operations to operations defined by counts of ordinary pseudoholomorphic polygons, as in the differential of the Chekanov–Eliashberg algebra. (Recall that our surgery formula is proven by identifying bounding cochains with augmentations for the Chekanov–Eliashberg algebra of an exact Lagrangian immersion.) Our solution to this problem is to amend the construction of the A_∞-structure to yield a quasi-isomorphic version, for which the compact part of the Weinstein manifold is left invariant by the rescaling (while in the cylindrical end we still apply the Liouville flow, as is necessary for compactness issues). The upshot is that the new A_∞-structure is given by counts of ordinary pseudoholomorphic discs (together with small Hamiltonian perturbations) inside the compact part.

Here we show how to modify the definition of the A_∞-operations of the wrapped Fukaya category in a way that allows us to apply our strategy for proving the surgery formula. It will follow that the cones constructed can be quasi-isomorphically identified with cones in the original formulation of the quadratic wrapped Fukaya category, which is sufficient for establishing the generation result.

In order to modify the definition of the A_∞-operations we begin by constructing a family ψ_s,t W → W of diffeomorphisms parametrised by t ≥ 0 and s ∈ [0,1] that satisfies the following:


  * ψ_s,t|_W_R-1=ψ_st inside W_R-1;

  * ψ_s,t|_∂ W_r=ψ_(s+(1-s)β(r))t for any r ∈ [R-1,R]; and

  * ψ_s,t|_W^e_R=ψ_t for any r ≥ R,

where β(r) is a smooth  cut off function that satifies


  * β(r)=0 near r=R-1;

  * β(r)=1 near r=R; and

  * ∂_rβ(r) ≥ 0.

The above flow ψ_s,t is a conformal symplectomorphism only when s=1, in which case ψ_1,t=ψ_t is the Liouville flow. For general values of s it is the case that ψ_s,t is a conformal symplectomorphism only outside of a compact subset. Notwithstanding, it is the case that:

  


  * If L is an exact Lagrangian immersion that is cylindrical inside W^e_R-1, it follows that L_τ := ψ_s(τ),t(τ)^-1(L) is a safe exact isotopy, which moreover is fixed setwise inside W^e_R-1;

  * Any compatible almost complex structure J_t which is cylindrical inside W^e_R-1+st satisfies the property that ψ_s,t^*J_t is a compatible almost complex structure which is cylindrical inside W^e_R and, moreover, equal to J_t inside W^e_R-1+st; and

  * Conjugation ψ_s_0,t_0^-1∘φ_t^H ∘ψ_s_0,t_0 with the diffeomorphism ψ_s_0,t_0 induces a bijective correspondance between Hamiltonian isotopies of W generated by Hamiltonians which depend only on 𝔯 inside W^e_R-1+s_0t_0 and Hamiltonian isotopies of W generated by Hamiltonians which only depend on 𝔯 inside W^e_R-1. More precisely, if the former Hamiltonian is given by H  W → then the latter is given by (f_s_0,t_0· H) ∘ψ_s_0,t_0 for a smooth function f_s_0,t_0 W →_>0 which only depends on 𝔯 inside W^e_R-1+s_0t_0, while f_s_0,t_0|_W_R-1+s_0t_0≡ e^-s_0t_0.




The function f_s_0,t_0 W →_>0 in Part (3) of the previous lemma can be determined as follows. First note that f_1,t≡ e^-t, while for general s ∈ [0,1] the equality f_s,t≡ e^-t still holds outside of a compact subset. Inside W^e_R-1+s_0t_0 the simple ordinary differential equation

    e^-𝔯∂_𝔯((f_s_0,t_0(𝔯) · H)∘ψ_s_0,t_0)=(e^-𝔯∂_𝔯H(𝔯))∘ψ_s_0,t_0

then determines f_s,t.


By the properties described in Lemma <ref>, it follows that we can use ψ_s_0,t for any fixed s_0 instead of the Liouville flow ψ_t in Abouzaid's construction of the wrapped Fukaya category <cit.>. We illustrate this in the case of the product μ^2. One first defines a morphism

    CF(L_1,L_2;H,J_t) ⊗CF(L_0,L_1;H,J_t) →
    CF   ((ψ_s_0,log 2)^-1(L_0), (ψ_s_0,log 2)^-1(L_2);(f_s_0,log 2· H)∘ψ_s_0,log 2,ψ_s_0,log 2^*J_t )

for any fixed s_0 ∈ [0,1] that is defined by a count of three-punctured discs with a suitable moving boundary condition, and where

    f_s_0,log 2 W →_>0

is the function from Part (2) of Lemma <ref>. (In particular, 
    f_s_0,log 2≡ e^-log 2
 holds outside of a compact subset.)

The fact that (f_s_0,log w(𝔯)· H)∘ψ_s_0,log w coincides with 1/we^2(t+log w)=we^2t outside of a compact subset is crucial for the maximum principle (and thus compactness properties) of the Floer curves involved in the definition of the morphism of Equation (<ref>).

It is immediate that an analogous version of <cit.> now also holds in the present setting, giving rise to a canonical identification between the Floer complexes

    CF((ψ_s_0,log 2)^-1(L_0),(ψ_s_0,log 2)^-1(L_2);(f_s_0,log 2· H)∘ψ_s_0,log 2,ψ_s_0,log 2^*J_t )

and 
    CF(L_1,L_2;H,J_t).

In this manner we obtain the operation μ^2. The general case follows similarly, by an adaptation of the construction <cit.> based upon ψ_s_0,t.

Finally, to compare the A_∞ structures defined by different values of the paramter s_0 ∈ [0,1], one can use <cit.>. Here Part (1) of Lemma <ref> is crucial. Also, note that the family (f_s_0,log w· H)∘ψ_s_0,log w of Hamiltonians is independent of the paramter s_0 outside of a compact subset for any fixed value of w.

Finally, since the intersection points of type a, b,c of Lemma <ref> belong to the region where we have turned off the rescaling by the Liouville flow, the construction of the augmentation in Lemma <ref> remains unchanged. From this point, the proof of Theorem <ref> in the therefore quadratic setup procedes as in the linear setup.



 §.§ Proof of Theorem <ref>

The proof of Theorem <ref> is based on Theorem <ref> and the following trivial observation. If (W, θ, 𝔣) is a Weinstein manifold and π_i  W × W → W, for i=1,2, are the projections to the factors, we consider the Weinstein manifold (W × W,
π_1^* θ - π_2^* θ, 𝔣∘π_1 + 𝔣∘π_2).
Note that the sign in the Liouville form has been chosen so that the diagonal Δ⊂ W × W is an exact Lagrangian submanifold with cylindrical end. The following lemma is a direct consequence of the definition.


The Weinstein manifold (W × W, π_1^* θ - π_2^* θ, 𝔣∘π_1 + 𝔣∘π_2) 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^2 be the version of the wrapped Fukaya category for the product Liouville manifold (W × W, π_1^*θ - π_2^*θ) defined in <cit.>,  where the wrapping is performed by a split Hamiltonian. If ℬ is a full subcategory of 𝒲ℱ(W,θ), we denote by ℬ^2 the full subcategory of 𝒲^2 whose objects are products of objects in ℬ. Then <cit.> implies the following.

  If Δ is generated by ℬ^2 in 𝒲^2, then the map

    [𝒪𝒞]  HH_n-*(ℬ,ℬ) → SH^*(W)

has the unit in the symplectic cohomology in its image.



Let (W, θ, 𝔣) be a Weinstein manifold and let 𝒟 be the colection of the Lagrangian cocore discs of W. Then, by Lemma <ref> and Theorem <ref> the collection 𝒟^2 of products of cocore discs of W generates 𝒲F(W × W, π_1^*θ - π_2^*θ) and so, in particular,
generates the diagonal Δ. By <cit.> the category 𝒲F(W × W, π_1^*θ - π_2^*θ) is equivalent to the category 𝒲^2 defined above, and therefore the collection 𝒟^2 generates the diagonal also in 𝒲^2. Thus Proposition<ref> implies that the image of the open-closed map (<ref>) contains the unit and therefore Theorem <ref> follows from <cit.>.

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