Lagrangian Cobordism I

This paper discusses the cobordism of Lagrangian submanifolds (in the monotone setting) and structures it as a category that is related in a functorial way to an appropriate (derived) Fukaya category. Are also discussed obstructions to cobordism based on properties of Lagrangian quantum homology, relations to Lagrangian surgery, as well as examples of non-isotopic but cobordant Lagrangians. This is a revision of our earlier preprint from September 2011.

Early versions of this work have been presented by one or both authors at an MSRI workshop (Spring 2010), Oberwolfach meeting (Summer 2010), Kyoto Symplectic wokshop (February 2011) and the Geometry and Physics conference at the Fields Institute (May 2011). We thank the organizers of all these meetings for giving us the opportunity to present our work.

Introduction
Lagrangian cobordism is a natural notion, initially introduced by Arnold [Arn1,Arn2] at the beginnings of symplectic topology. This notion was studied by Eliashberg [Eli] and Audin [Aud] who showed that, in full generality, this is a very flexible notion that can be translated to purely algebraic topological constraints. By contrast, the work of Chekanov [Che] points out a certain form of rigidity valid in the case of monotone cobordisms.
The present paper is a review of work in progress on the properties of Lagrangian cobordism. Ideas of proof are only sometimes sketched, detailed arguments will appear in a more ample manuscript currently in preparation.
Our focus here is to explain how cobordism naturally organizes the Lagrangian submanifolds of a fixed symplectic manifold (M, ω) in a category and to describe some functors relating this cobordism category to the derived Fukaya category. The data is organized in the following diagram and most of the paper will be spent explaining the different objects in this diagram. We start by describing the right side: F uk * (M) stands for the Fukaya category of M, DF uk * (M) stands for the derived Fukaya category of M, the decoration * restricts the Lagrangians considered to a special class to be described in more detail later (see Theorem A). We assume the reader is familiar with the definition of the Fukaya category as described in Seidel's book [Sei3] but we will shortly recall some of the relevant properties later in the paper. The category DF uk * (M) is triangulated and ΣDF uk * (M) is the stabilization of DF uk * (M) in the sense that the morphisms of DF uk * (M) are enriched by those morphisms that shift "degree" (see §2.4).
Interesting computations are generally easier to understand in the derived Fukaya category, as this is a genuine triangulated category and not only an A ∞ one as the Fukaya category itself. However, to actually complete these computations one often needs to first work at the level of the Fukaya category and deduce the results of actual interest by passage to homology. One of the main points of the current paper is that in some interesting cases the existence of the functor F provides a shortcut by going directly from a geometric category, the cobordism category Cob * (M), to the derived Fukaya category. In this sense, the fundamental property of the functor F is its factorization indicated at the bottom of diagram (1). Here the category T S DF uk * (M) is obtained from the category DF uk * (M) by a general construction (apparently new) that associates to any triangulated category C a new category T S C -the category of (stable) triangular (or cone) resolutions over C. The morphisms sets hom(x, −) in this category parametrize the ways in which x can be resolved by iterated exact triangles (or cone attachments). The category T S C comes with a canonical projection P : T S C → ΣC. The morphisms [V ] in the cobordism category Cob * (M) are represented by a certain type of Lagrangian cobordisms V with possibly multiple ends. The factorization of F through F is of interest because it associates to each such morphism [V ] ∈ Mor Cob * (M ) a series of exact triangles in DF uk * (M) that relate the ends of V .
Here are three consequences of our constructions: i) If two connected Lagrangians L, L ′ are monotone cobordant, then up to shift they represent the same object in the appropriate derived Fukaya category. In particular, for any third Lagrangian N, the two Floer homologies HF (N, L) and HF (N, L ′ ) are isomorphic (if defined). ii) A monotone Lagrangian cobordism (V ; L, L ′ ) with L and L ′ connected is a quantum h-cobordism in the sense that quantum homology verifies QH(V, L) = QH(V, L ′ ) = 0 .
In particular, such a cobordism does not exist for dim L = 2. See §5.3 for more details.
The paper also contains a discussion of some examples. In particular, we notice that the trace of Lagrangian surgery gives rise to a Lagrangian cobordism. We use this fact to produce examples of connected Lagrangians that are monotone cobordant but not isotopic even in the smooth category.
To conclude, Lagrangian cobordism -even in its monotone version -is somewhat more flexible than Hamiltonian isotopy even if, remarkably, it preserves Floer homology and all similar invariants. Thus, this notion appears to have the potential to shed new light both on these invariants as well as on the geometry of Lagrangians.
Here is an outline of the restrictions required for the constructions above (see §2). First, all Lagrangians have to verify a monotonicity condition in a uniform way. This is crucial for the transversality issues involving bubbling of disks to be approachable by the methods in [BC2] and [BC4]. Assuming monotonicity, Floer homology is defined over the universal Novikov ring, denoted in the paper by A. Another version applies to Lagrangians L ⊂ M so that all elements in the image of the morphism H 1 (L; Z) → H 1 (M; Z) induced by inclusion are torsion (see [Oh1]). With this additional constraint, Floer homology is defined over Z 2 . In both cases the resulting homology theory is in general not graded. The universal Novikov ring and Z 2 will be the two coefficient rings that will be used in the paper. It is over Z 2 that the functoriality results above are valid as stated (assuming that the preceding condition on the H 1 's is imposed on all Lagrangians as well as on cobordisms). At the same time, our geometric constructions will be performed over A.
The fact that the ring A is a large ring is a significant advantage at times and it plays a crucial role in the proof of the points ii) and iii) above. However, the Floer chains and structures defined over A are highly sensitive to perturbations and as a consequence functoriality takes a much more complicated form, one that we will not attempt to establish or further discuss here. The point i) above exemplifies this issue: the first part of the statement (concerning the identifications of objects in the Fukaya category) is claimed over Z 2 (under the assumptions on the H 1 's) but the second part of the statement, concerning the isomorphism of HF (N, L) and HF (N, L ′ ), is true over A without these assumptions. However, over A the isomorphism between HF (N, L) and HF (N, L ′ ) is determined by the cobordism V relating L to L ′ only up to multiplication by a unit in A. In contrast, over Z 2 , this isomorphism is completely determined by V .
The structure of the paper is as follows. In §2 we fix our setting and recall some basic facts about Floer homology and the Fukaya category. All of this can be safely skipped by experts except for §2.4 which contains the definition of the category of cone resolutions over a triangulated category. Section 3 contains the main geometric definitions that are at the center of our study of cobordism as well as the definition of the cobordism category that is the central subject of interest in the paper. Section 4 is dedicated to discussing the functor F. It starts with the basic ideas behind the construction of this functor and it pursues with a description of the main technical ingredients needed to prove the commutativity in (1). As mentioned before, the full details of the proof will appear in a later publication. In §5 we analyze some obstructions to the existence of morphisms in our cobordisms categories that are based on the interplay between the constructions involved in the definition of the functor F and the Lagrangian quantum homology. Finally, §6 contains the analysis of surgery through the prism of cobordism as well as the related examples of non-isotopic cobordant Lagrangians.
Mohammed Abouzaid, Denis Auroux, François Charette, Yasha Eliashberg, Paul Gauthier and Clément Hyvrier for useful discussions as well as the MSRI for its hospitality during the Fall of 2009, when the work presented here was initiated.

Preliminaries
In this paper (M 2n , ω) is a connected symplectic manifold. We assume that M is compact but the constructions described have immediate adaptations to the case when M is only tame (see [ALP]). Lagrangian submanifolds L n ⊂ M 2n will be generally assumed to be closed unless otherwise indicated.
2.1. Monotonicity. Given a Lagrangian submanifold L ⊂ M there are two canonical morphisms ω : π 2 (M, L) → R , µ : π 2 (M, L) → Z the first given by integration of ω and the second being the Maslov index. We will say that the Lagrangian L is monotone if there exists a positive constant ρ > 0 so that for all α ∈ π 2 (M, L) we have ω(α) = ρµ(α) and moreover the minimal Maslov number In what follows we will use Z 2 as the ground ring for homological considerations. We remark that most of the discussion below generalizes under additional assumptions on the Lagrangians to arbitrary rings. We therefore denote the ground ring by K, keeping in mind that in this paper K = Z 2 .
To each connected closed, monotone Lagrangian L there is an associated basic Gromov-Witten type invariant d L ∈ K. In short, d L is the number (in K) of J-holomorphic disks of Maslov index 2 going through a generic point P ∈ L (for J a generic almost complex structure that is tamed by ω). Here is a more precise definition. where G acts as follows σ · (u, z) = (u • σ −1 , σ(z)), for σ ∈ G). Assume now that µ(B) = 2 and that J is generic. Then, by standard arguments, (see for instance [BC4]) it follows that ( M(B, J) × ∂D)/G is a smooth compact manifold without boundary and of (real) dimension n. Consider the evaluation map We denote by ν(B) ∈ K the degree of this map. Standard arguments then show that ν(B) does not depend on J but only on B. Moreover, there can be at most a finite number of classes B ∈ H D 2 with ν(B) = 0 and we put: It is easy to see by the Gromov compactness theorem that this number is independent of the almost complex structure J. A family of Lagrangian submanifolds L i , i ∈ I is called uniformly monotone if each L i is monotone and the following condition is satisfied: there exists d ∈ K so that for all i ∈ I we have d L i = d and, if d = 0, then there exists a positive real constant ρ so that the monotonicity constant of L i equals ρ for all i ∈ I.
In the absence of other indications, all the Lagrangians used in the paper will be assumed monotone and, similarly, the Lagrangian families will be assumed uniformly monotone.

2.2.
A quick review of Lagrangian Floer theory. In the sequel we will use various versions of Lagrangian Floer homology as well as Lagrangian quantum homology for monotone Lagrangian submanifolds which we now briefly recall. We refer the reader to [Oh1,Oh2,Oh3] for the foundations of Floer homology for monotone Lagrangians, and to [FOOO2,FOOO3] for the general case. For Lagrangian quantum homology see [BC2, BC4, BC3, BC1].
2.2.1. Lagrangian Floer homology. Let L 0 , L 1 ⊂ M be two monotone Lagrangian submanifolds with d L 0 = d L 1 = d. In case d = 0 we assume in addition that L 0 and L 1 have the same monotonicity constant (or in other words that the pair (L 0 , L 1 ) is uniformly monotone).
Denote by A the universal Novikov ring, i.e.
Fix η ∈ π 0 (P(L 0 , L 1 )) and let H : M × [0, 1] → R be a Hamiltonian function with Hamiltonian flow ψ H t . We assume that ψ H 1 (L 0 ) is transverse to L 1 . (We generally view H as a mean of possible perturbation of L 0 , and when not needed we will often use H = 0.) We denote by O η (H) the set of paths γ ∈ P η (L 0 , L 1 ) which are orbits of the flow ψ H t . Finally, we choose also a generic 1-parametric family of almost complex structures J = {J t } t∈[0,1] compatible with ω.
Using this data one can define in a standard way the Floer complex CF (L 0 , L 1 ; η; H, J) with coefficients in A. Recall that the underlying module of this complex is generated over A by the elements of O η (H). The Floer differential ∂ : Here M 0 (γ − , γ + ; H, J) stands for the 0-dimensional components of the space of Floer strips u : R × [0, 1] −→ M connecting γ − to γ + , modulo the R-action coming from translation in the s coordinate. The strips u are assumed to have finite energy and we denote by ω(u) = R×[0,1] u * ω the symplectic area of u. Finally, each such strip u comes with a sign ε(u) = ±1 ∈ K. As mentioned before, in this paper we will mostly work with K = Z 2 hence the sings ε(u) are irrelevant. However, under additional assumptions on L 0 , L 1 (such as orientability and prescribed spin structures) the moduli spaces of Floer strips can be canonically oriented which imply the sings ε(u) ∈ Z, hence one can work with any ring K. See [FOOO3] for a detailed account on how to define orientations in Floer theory.
Remark 2.2.1. In the general context of the paper, with CF defined over A, the chain complex CF is not graded and hence HF has no grading too. In special situations one can endow CF with some grading though not always over Z (e.g. when L 0 and L 1 are both oriented, then there is a Z 2 -grading). See [Sei1] for a systematic approach to these grading issues. Even if in this paper we will stick geometrically to the general, non-graded situation, we will perform most algebraic constructions in such a way that adaptation to one of these graded settings is simple.
Standard arguments show that the homology HF (L 0 , L 1 ; η; H, J) is independent of the additional structures H and J up to canonical isomorphisms. We will therefore omit H and J from the notation.
(1) When L 0 and L 1 are mutually transverse we can take H = 0 in CF (L 0 , L 1 ; H, J) in which case the generators of the complex are the intersection points L 0 ∩ L 1 and Floer trajectories M 0 (γ − , γ + ; 0, J) are genuine holomorphic strips connecting intersection points γ − , γ + ∈ L 0 ∩ L 1 . When H = 0 we will omit it from the notation and just write CF (L 0 , L 1 ; J). We will sometimes omit J too when its choice is obvious.
(2) The use of families of almost complex structures J = {J t } t∈[0,1] is needed for transversality reasons typically occurring in the construction of Floer homology. However, it is still possible to work with almost complex structure J that do not depend on t, provided the Hamiltonian H is chosen to be generic (see [FHS]).
The definition of the isomorphism c ϕ involves some subtleties due to our use of the universal Novikov ring A as base ring: given that the symplectic area of the strips with moving boundaries can vary inside a one parametric moduli space it follows that the naive definition of the morphism c ϕ -so that each strip is counted with a weight given by its symplectic area -does not provide a chain map. We explain here in more detail the construction of the map c ϕ .
Let β : R → [0, 1] be a smooth function so that β(s) = 0 for s ≤ 0, β(s) = 1 for s ≥ 1 and β is strictly increasing on (0, 1). Let x be a generator of C 1 . We write with y going over the generators of C 2 and u going over all the elements of a zero dimensional moduli space of solutions to Floer's homogeneous equation ∂ J u = 0 that start at x and end at y and verify the boundary conditions Here the element v u : is a strip with boundary conditions on L 0 and L 1 . It is easy to check that with this definitionc ϕ is a chain map. Note that the quantity |ω(v u ) − ω(u)| is bounded by the variation of H so thatc ϕ is well defined over A. Further, the map c ϕ induced in homology byc ϕ , depends only on the homotopy class with fixed end points of ϕ. Similar constructions can be used to adapt the rest of the usual Floer theoretic machinery to this moving boundary situation. They show in particular that c ϕ induces an isomorphism in homology.
Remark 2.2.3. This argument also applies without modification to cases when M is not compact (but e.g. tame), if we have some control which insures that all solutions u of finite energy as above have their image inside a fixed compact set K ⊂ M.
2.2.3. The pearl complex and Lagrangian quantum homology. Next we briefly describe the version of Lagrangian quantum homology that will be use later in the paper. Let L ⊂ M be a monotone Lagrangian with minimal Maslov number N L . Denote by Λ = K[t −1 , t] the ring of Laurent polynomials in t, graded so that |t| = −N L . (In case L is weakly exact, i.e. ω(A) = 0 for every A ∈ π 2 (M, L) we put Λ = K.) The chain complex used to define the Lagrangian quantum homology QH(L) is denoted by C(D) and called the pearl complex. It is associated to a triple of auxiliary structures D = (f, (·, ·), J) where f : L −→ R is a Morse function on L, (·, ·) is a Riemannian metric on L and J is an ω-compatible almost complex structure on M. With these structures fixed we have This complex is Z-graded with grading combined from both factors. The grading on the left factor is defined by Morse indices of the critical points. The differential d on this complex is defined by counting so called pearly trajectories (which are combined from negative gradient flow lines of f with J-holomorphic disks). The homology H * (C(D), d) is independent of D (up to canonical isomorphisms) and is denoted by QH * (L). Note that this homology is Z-graded. We refer the reader to [BC2,BC4,BC3] for the precise construction of this homology.
In what follows we will actually need also to enrich the coefficients of QH(L) to the Novikov ring A. This is done as follows. Denote by the minimal positive area of a disk with boundary on L. We use the convention that min ∅ = ∞. The Novikov ring A becomes an algebra over Λ via the ring morphism induced by Λ ∋ t → T A L ∈ A. (If L is weakly exact we have Λ = K and we view A as an algebra over Λ in the usual way.) Consider now The homology of this complex will be denoted by QH(L; A). Note that in contrast to C(D) and QH(L) their analogues over A, C(D; A) and QH(L; A) are not graded.
To avoid confusion between Λ and A we will sometimes write QH(L; Λ) for QH(L). coming from a chain morphism P SS η 0 : C(D; A) −→ CF (L, L; η 0 ; H, J). The construction of P SS η 0 is very similar to the one described in [Alb,BC4,BC3] over the ring Λ. The only needed modification when working with A is to incorporate the total areas of the connecting trajectories that appear in the morphism P SS η 0 .
The map P SS η 0 : QH(L; A) −→ HF (L, L; η 0 ) induced in homology by P SS η 0 is an isomorphism. The isomorphism P SS is now defined as i η 0 • P SS η 0 .
There also exists a version of the PSS morphism which is defined using moving boundary conditions. Specifically, assume that ϕ is a Hamiltonian isotopy and let ϕ 1 (L) = L ′ . Then we have an isomorphism Its definition is straightforward in view of the standard case recalled above and §2.2.2.

2.2.5.
Products and other structures. The Lagrangian and ambient quantum homologies as well as the Floer homologies are all related via several compatible algebraic structures endowed with ring and module operations. We will typically denote these operations by the same notation * since they are algebraically compatible. We refer the reader to [BC2,BC4,BC3] for more details.
The quantum homologies QH(L; Λ) and QH(L; A) can be endowed with a quantum product * which turns them into associative rings with unity (note that they are in general not commutative). We denote the unity by [L] ∈ QH n (L; Λ) (in analogy to the fundamental class in classical homology).
For a uniformly monotone pair Lagrangians (L 1 , L 2 ) the Floer homology HF (L 1 , L 2 ) is a left module over QH(L 1 ; A) and a right module over QH(L 2 ; A). We denote these module operations by α 1 * x and x * α 2 , for x ∈ HF (L 1 , L 2 ), α 1 ∈ QH(L 1 ; A), α 2 ∈ QH(L 2 ; A). The two module structures are mutually compatible in the sense that associativity holds: For a uniformly monotone triple of Lagrangians L 1 , L 2 , L 3 there is an associative product called the triangle or Donaldson product. When L = L 1 = L 2 = L 3 this product can be identified via the PSS isomorphism with the quantum product on QH(L; A).
There is a duality isomorphism relating HF (L 1 , L 2 ) and hom A (HF (L 2 , L 1 ), A). In case L = L 1 = L 2 is exact this duality reduces to Poincaré duality.
Finally, the Floer homology HF (L 1 , L 2 ) is also a module over the ambient quantum homology QH(M). We denote this operation by * too.
2.2.6. Floer homology over Z 2 . Assume that, in addition to the monotonicity conditions discussed in §2.1, every Lagrangian L considered in the Floer theoretic constructions satisfies: where i * is induced by the inclusion L ⊂ M. In this case an observation due to Oh [Oh1] shows that all the structures defined in the previous paragraphs are defined (at the chain level) with coefficients in the "polynomial" ring A 0 = n k=0 a k T λ k | a k ∈ K, n ∈ Z (i.e. those elements in A formed by finite sums). There is an obvious ring map A 0 → K obtained by sending T → 1 and this allows to change the coefficients in all the structures described by specializing to T = 1. All the results mentioned in the case of A remain valid.
In this paper we will use only these two coefficient rings for Floer-theoretic constructions: A and K = Z 2 . Whenever Z 2 is used, the assumption (6) is supposed to be in force without additional mention.
To shorten notation, K will denote below one of these two rings. We will use the same notation as the one in the sections above but with K replacing A.
2.3. The Fukaya category, derived and not. It is beyond the scope of this section to recall the actual definition and construction of the Fukaya category. Fortunately, this is not really needed as we can refer to Seidel [Sei3] for a careful description of the construction. However, we would like to indicate some of the points that are significant from our point of view. Below we will follow closely Seidel's notation. We first fix some notation that will be used throughout the paper. A simplified version of the Fukaya category is the Donaldson category: its objects are the elements of L * d (M) and the morphisms are Mor(L 1 , L 2 ) = HF (L 1 , L 2 ). Composition is given by the triangle product. The existence of the identity morphism in Mor(L, L) for each L ∈ L * d (M) is insured by the fact that L is not narrow. The main drawback of the Donaldson category can be seen by analogy with usual topology. It is well known that if X f −→ Y is a map of CW -complexes, then the H * f does not determine the homotopy type of the (homotopy) cofiber of f . We recall that this homotopy type coincides with that of Y ∪ f CX, where CX is the cone over X. For instance, the Hopf map S 3 → S 2 has as cofiber CP 2 but is homologically trivial. In topology it is said that a sequence X → Y → Z with Z ≃ Y ∪ f CX is a homotopy cofibration sequence, or that Z is obtained from Y by a cone-attachment (over X). In a more general categorical setting, the analogues of the cofibration sequences from the homotopy world are called exact triangles and the proper setting to study cone-decompositions is that of triangulated categories [Wei]. In this language, the problem with the Donaldson category is that it is not triangulated and there is no natural way to complete it to a triangulated category without taking into account more structure. The Fukaya category solves this problem in what first appears to be a simple way: its objects are still the elements of L * d (M) but one intends to now take as morphisms the Floer complexes CF (L 1 , L 2 ) instead of the Floer homology. This change however leads to considerable difficulties. First, Floer complexes actually depend on additional data, almost complex structures, Hamiltonian perturbations etc. Secondly, the composition of morphisms is now defined as a chain-level triangle product and as such is associative only up to homotopy (and similarly for the unit). The first problem can be dealt with by fixing some coherent system of additional data, defining the relevant category associated to it and then remarking that the dependence on the additional data can be controlled by a version of the usual continuation technique. The second issue on the other hand shows that this process does not actually lead to a category but rather to an A ∞ -category -called the Fukaya category of M, F uk d (M). An A ∞ -category is roughly a structure where morphisms compose associatively only up to homotopy, the respective homotopies are themselves included in the structure and related by higher homotopies and so forth (and there are additional identities concerning units). While algebraically A ∞ -categories are complicated objects, the advantage is that the A ∞ -category defined as above admits a reasonably natural completion to a triangulated A ∞ -category. The resulting completion is denoted by F uk d (M) ∧ (this is not a standard notation). In fact, as explained in [Sei3] there are multiple such completions but all are equivalent from the point of view of the present paper. The precise version that we use here (see Remark 3.21 in [Sei3]) is obtained by first using the Yoneda embedding to view F uk d (M) as a functor category over itself with values into chain complexes and then making use of the usual cone construction at the level of chain complexes to build a triangulated closure of the image of the embedding. In all cases, the advantage of completing at the A ∞ -level is that A ∞ -categories are sufficiently rich in structure so as to recover cofibration sequences like the one associated above to the Hopf map. However, the actual construction of the triangulated completion is rather non-geometric and is not immediate. Finally, the derived Fukaya category, denoted by DF uk d (M), has the same objects as F uk d (M) ∧ but its morphisms are obtained by applying the homology functor to the morphisms in F uk d (M) ∧ . This category is naturally triangulated, the exact triangles being the image of those in F uk d (M) ∧ and it contains the Donaldson category.
Remark 2.3.1. There is a further possible completion of F uk d (M) with respect to idempotents (or split factors). This is very useful for computations but we will not make use of it in this paper.
2.4. Cone decompositions over a triangulated category. In this subsection we will discuss a construction valid in any triangulated category. The purpose of the construction is to parametrize the various ways to decompose an object by iterated exact triangles. The morphisms in this category have a certain formal resemblance with the morphisms in the cobordism category that will be introduced in §3.3. This analogy will be rendered precise later in the paper when we describe the functor F. Let C be a triangulated category. We recall [Wei] that this is an additive category together with a translation automorphism T : C → C and a class of triangles called exact triangles that satisfy a number of axioms due to Verdier and to Puppe (see e.g. [Wei]).
A cone decomposition of length k of an object A ∈ C is a sequence of exact triangles: To such a cone decomposition we associate the family l(A) = (X 1 , X 2 , . . . , X k ) and we call it the linearization of the cone decomposition. This definition is an abstractization of the familiar iterated cone construction in case C is the homotopy category of chain complexes. In that case T is the shift functor T X = X[−1] and the cone decomposition simply means that each chain complex Y i+1 is obtained from Y i as the mapping cone of a morphism coming from some chain complex, in other words We will now define a category T S C. The construction of this category starts with the stabilization category of C, ΣC: ΣC has the same objects as C and the morphisms in ΣC from a to b ∈ Ob(C) are morphisms in C of the form a → T s b for some integer s. Next, the free monoidal isomorphism category F * ΣC over ΣC has as objects finite families (x 1 , . . . , x k ) where the x i 's are objects in C. The monoidal addition, denoted by +, is concatenation. The morphisms are corresponding families of isomorphisms in ΣC (thus this category differs from the free monoidal category over ΣC because that category has as morphisms families of morphisms and not only isomorphisms).
The category T S C, called the category of (stable) triangle (or cone) resolutions over C is obtained from F * ΣC by enriching the morphisms with the elements constructed as follows. Given x ∈ Ob(C) and (y 1 , . . . , y q ) ∈ Ob(F * ΣC) a morphism Ψ : x −→ (y 1 , . . . , y q ) is a triple (φ, a, η), where a ∈ Ob(C), φ : x → T s a is an isomorphism for some index s and η is a cone decomposition of the object a with linearization (T s 1 y 1 , T s 2 y 2 , . . . , T s q−1 y q−1 , y q ) for some family of indices s 1 , . . . , s q−1 . Below we will also sometimes use a shift index s q attached to the last element y q with the understanding that s q = 0. Thus, not only a admits a cone decomposition of length q but such a decomposition is part of the data defining the morphism Ψ.
i. For x ∈ C the union of all the morphisms Mor T S C (x, −) parametrizes all the possible cone decompositions of x. Of course, the same cone-decomposition appears infinitely many times by changing the x i 's into T l x i 's and adjusting accordingly all the indexes s, s i . ii. The presence of the indexes s, s i might appear as an unnecessary distraction here and be slightly disconcerting. However, their presence is explained by the fact that in the geometric example that is of interest in this paper the morphisms shift degrees (in Floer homology). Moreover, is it easy to see that it is natural for geometric reasons to allow for the index s to appear in the map φ and, on the other hand, not include any shift in front of y q (or, equivalently, take s q = 0).
We will now define Φ ′′ = Φ • Φ ′ . We will assume for simplicity that the families of shifting degrees indices for both Φ ′ and Φ are 0 (the general case is a straightforward generalization of the argument below). From the morphism Φ ′ we get an isomorphism φ ′ : x → a ′ and a cone decomposition of a ′ with linearization y 1 , . . . , y k ), i.e. objects a ′ 1 = 0, a ′ 2 = y 1 , a ′ 3 , . . . , a ′ k , a ′ k+1 = a ′ and exact triangles: For i = h − 1, h, h + 1 we have: Similarly, from the morphism Φ we obtain an isomorphism φ : y h → a, a sequence of objects a 1 = 0, a 2 = z 1 , a 3 , . . . , a l , a l+1 = a and exact triangles: We will construct below new objects and exact triangles: is defined by taking the cone decomposition η ′ (i.e. (8)) and replacing the line i = h in it (i.e. the middle one in (9)) by the list of triangles from (11).
We now turn to the construction of the objects a ′ h,q and the triangles (11). Given 1 ≤ q ≤ l, let β q : a q → a be the morphism obtained by successive composition of the middle arrows of (10) for j = q, . . . , l. Consider now the composition α q = u ′ h •(T −1 φ −1 )β q of the following three morphisms: l here is the first arrow in the middle line of (9). By the axioms of "triangulated category" the morphism α q can be completed into an exact triangle, i.e. there exists an object a ′ h,q ∈ Ob(C) and an exact triangle: By the octahedral axiom and standard results on triangulated categories (see e.g. [Wei]) the triangles in (12) (for q and q + 1) and those in (10) (for j = q and after a shift) fit into the following diagram: (13) in which all rows and columns are exact triangles. (In fact, the diagram is determined by the upper left square, from which the upper two triangles and left two triangles are extended. The existence of the third column follows from octahedral axioms applied a few times.) Moreover, all squares in the diagram, except of the lower right, are commutative and the lower right one commutes up to sign. Note that the third column is precisely the triangle that we needed in (11) in order to complete the construction of the composition in (7).
It remains to define the composition Φ • Φ ′ of more general morphisms than (7). The case when the domain of Φ ′ consists of a tuple of objects in C and Φ is as in (7) is an obvious generalization of the preceding construction. Next, the case when the rest of the components of Φ are not necessarily id (but rather general cone decompositions too) is done by reducing to the case discussed above by successive compositions. Namely, assume that Φ ′ : x → (y 1 , . . . , y k ) and Φ = Ψ 1 + · · · + Ψ k with Ψ j : noting that each step of this composition is of the type already defined.
This completes the definition of composition of morphisms in the category T S C. It is not hard to see that the this composition of morphisms is associative. Note also that, by construction, T S C is a monoidal category.
To conclude this discussion we remark that there is a projection functor (14) P : T S C −→ ΣC that is defined by P(x 1 , . . . x k ) = x k and whose value on morphisms is induced by associating to Φ ∈ Mor T S C (x, (x 1 , . . . , x k )), Φ = (φ, a, η), the composition: with p : a → x k defined by the last exact triangle in the cone decomposition η of a, A straightforward verification shows the P is indeed a functor.
Remark 2.4.2. Let H : C → A be a covariant cohomological functor with values in the abelian category A (see [Wei]). Let Φ : x −→ (x 1 , x 2 ) be a morphism in T S C consisting of a cone decomposition of x of length 2. Then the cohomology H(x) is related to H(x 1 ), H(x 2 ), by the following long exact sequence: Here s, s 1 , s 2 , are the shift-indices coming from the morphism Φ. More generally, a morphism Φ ∈ Mor T S C (x, (x 1 , . . . , x k )) gives rise to a spectral sequence relating the cohomology H(T * x i ) and H(T * x) and whose E 2 -term can be computed in terms of H(T * x i ).

Cobordism: definitions and geometric aspects
In what follows we fix a symplectic manifold M endowed with a symplectic structure ω. The plane R 2 as well as domains in R 2 will be endowed with the symplectic structure ω R 2 = dx ∧ dy, (x, y) ∈ R 2 . We endow the product R 2 × M with the symplectic form ω R 2 ⊕ ω. We denote by π : 1≤j≤k + be two families of closed, Lagrangian submanifolds of M. We say that that these two (ordered) families are Lagrangian cobor- The manifold V is called a Lagrangian cobordism from the Lagrangian family (L ′ j ) to the family (L i ). We will denote such a cobordism by V : The values of the y coordinate of V (i.e. 1, . . . , k ± ) is not really important. (This convention is fixed for categorical purposes that will become clear later.) What really matters is the order in which each family of Lagrangians (L i ) and (L ′ j ) is placed (with respect to the y-coordinate) at the ends of the cobordism. Sometimes we will shift the numbering of the y-coordinates to start with other values of i. It is important to remark that the Lagrangians in the family (L i ) (or (L ′ j )) are not assumed to be mutually disjoint inside M. In fact, one of the roles of the y coordinate at the ends is to "make them disjoint" (inside 0 × R × M) thus enabling the cobordisms V to be embedded. In this respect our setting is somewhat different than in [Che].
A few variants of this notion will be useful later in the paper. An elementary cobordism is a cobordism (which might be connected or not) so that the number of negative ends k − as well as the number of positive ends k + in Definition 3.1.1 both have value at most one. Finally, V is called a pseudo-isotopy if it is smoothly trivial which means that there is a monotone Lagrangian submanifold. Other types of restricted cobordism are, for instance, oriented and spin -this means that an orientation, respectively a spin structure, is defined on the ends of the cobordism and is preserved by the cobordism itself. Clearly, as in the smooth case, there are many other possible variants of cobordism depending on additional structures. In practice, particularly when studying one cobordism at a time, it is often more convenient to view cobordisms as embedded in R 2 ×M. Given a cobordism V ⊂ ([0, 1]×R 2 )×M as in Definition 3.1.1 we can extend trivially its negative ends towards −∞ and its positive ends to +∞ thus getting a Lagrangian V ⊂ R 2 × M. We will in general not distinguish between V and V but if this distinction is needed we will call The R-extension of V .

Lagrangian submanifolds with cylindrical ends.
In what follows it is convenient to make use of a slightly more general notion of Lagrangian submanifold with cylindrical ends that we discuss here. To simplify notation we will write from now on M = R 2 × M endowed with the split form ω R 2 ⊕ ω. We will also identify in the standard way R 2 ∼ = C endowed with the standard complex structures i.
By a Lagrangian submanifold with cylindrical ends we mean a Lagrangian submanifold V ⊂ M without boundary that has the following properties: (1) For every a < b the subset V | [a,b]×R is compact.
(2) There exists R + such that for some a + 1 < · · · < a + k + and some Lagrangian submanifolds L + 1 , . . . , . . , L ′ r ) and (L 1 , . . . , L s ) then its Rextension W is a Lagrangian submanifold of M with cylindrical ends. Vice versa, if W is a Lagrangian submanifold with cylindrical ends then by an obvious modification of the ends (and a possible symplectomorphism on the R 2 component) it is easy to obtain a Lagrangian cobordism between the families of Lagrangians corresponding to the positive and negative ends of W .
In order to simply terminology, we will say that a Lagrangian with cylindrical ends V is cylindrical outside of a compact subset . We will also need the following notion.
Definition 3.1.2. Two Lagrangians with cylindrical ends V , W ⊂ M are said to be cylindrically distinct at infinity if there exists R > 0 such that Finally, let us describe a class of Hamiltonian isotopies that will be useful in the following.
iii. There is a constant K > 0 so that for all In other words, the Hamiltonian flow ψ t moves tangentially along the cylindrical ends of V 0 and at bounded speed. Of course, the ends of all the Lagrangians V t coincide at infinity. We say that two Lagrangians V , V ′ ⊂ M with cylindrical ends are horizontally isotopic if there exists an isotopy as above {V t } t∈[0,1] with V 0 = V and V 1 = V ′ . Finally, we will sometime say that an ambient Hamiltonian isotopy {ψ t } t∈[0,1] as above is horizontal with respect to V 0 . a. If L ⊂ M is a Lagrangian submanifold and γ ∈ C is any curve so that outside a compact set γ agrees with R×{y}, then γ×L ∈ M is an elementary cobordism. If L is monotone, then so is the cobordism γ×L, with the same minimal Maslov number and monotonicity constant. More generally, a possibly non-connected curve γ that coincides with R × {j} outside a compact set gives rise to cobordisms L × γ. In particular, this shows that the Lagrangian family (L, L) is null-bordant. b. If the connected Lagrangians L, L ′ ⊂ M are Hamiltonian isotopic it is easy to construct an elementary cobordism joining them (notice however that the projection of this cobordism on R 2 will in general not be a curve). c. Let (V ; (L i ), (L ′ j )) be an immersed Lagrangian cobordism between two families of embedded Lagrangians. This is a cobordism as in Definition 3.1.1 with the exception that V → ([0, 1] × R) × M is not a Lagrangian embedding but only a Lagrangian immersion. Such a cobordism can be transformed into an embedded one by first changing the self intersection points of V into generic double points and then resolving these double points by Lagrangian surgery (see for instance [Pol]). It is important to note that by resolving these singularities various properties that the initial V might have verified are in general lost. Monotonicity, for instance, is in general not preserved, nor is orientability. However, if we do not keep track of these additional structures we see that immersed Lagrangian cobordism implies embedded cobordism (as noticed by Chekanov [Che]). d. Finally, a less immediate verification shows that the trace of surgery is also a Lagrangian cobordism. In other words, given two transverse Lagrangians L 1 , L 2 by applying surgery at each of their intersection points one can obtain (a possibly disconnected) Lagrangian L that is cobordant to the family (L 1 , L 2 ), the cobordism being given by the composition of the traces of the surgeries. We will elaborate more on this construction in §6.3.
Remark 3.2.1. i. It is not difficult to see that cobordism is an equivalence relation among Lagrangian families: reflexivity is of course obvious as well as transitivity. For symmetry a little argument is required. Assume V is a cobordism between (L 1 , L 2 , . . . L h ) and (L ′ 1 , . . . , L ′ k ). The transformation a : C × M → C × M given by a(z, m) = (−z, m) is symplectic and, after adjusting the ends of the cobordism a(V ), it provides a cobordism from (L ′ k , . . . , L ′ 1 ) to (L h , . . . , L 1 ). This cobordism can be easily adjusted at the ends to an immersed cobordism between (L ′ 1 , . . . , L ′ k ) and (L 1 , L 2 , . . . , L h ). By point c. above this can be transformed into an embedded Lagrangian cobordism. (Note that this construction fails for monotone cobordisms, hence being monotone cobordant does not seem to be an equivalence relation.) ii. Given two Lagrangian families L = (L 1 , . . . , L h ) and In view of the properties described above it is easy to see that this operation defines a group structure on the set of equivalence classes of Lagrangian families of M. By applying appropriate surgeries it is easy to see that this group is commutative. (In contrast, there is no apriori reason why L + L ′ should be monotone cobordant to L ′ + L.) iii. It is easy to see that elementary cobordism is also an equivalence relation among the Lagrangians of M (surgery is not needed for this argument). iv. Special elementary cobordism of any of the three following types -monotone, oriented, or spin -is an equivalence relation. Again reflexivity is obvious and symmetry follows as in Remark 3.2.1 i. without any need to perform surgeries. Transitivity is obvious too in the orientable and spin cases. In the monotone case, it follows from the Van Kampen theorem for relative π 2 (−, −)'s viewed as cross-modules (see [BH]) that gluing two monotone cobordisms with the same monotonicity constant along a connected monotone end produces a monotone cobordism. However, as already mentioned earlier, non-elementary monotone cobordism is not necessarily an equivalence relation.
3.3. The category Cob d (M). The purpose of this subsection is to set up Lagrangian cobordism as a category. We first introduce an auxiliary category Cob d (M), d ∈ K. Its objects are families (L 1 , L 2 , . . . , L r ) with r ≥ 1, L i ∈ L d (M). (Recall that L d (M) stands for the class of uniformly monotone Lagrangians L with d L = d, and when d = 0 with the same monotonicity constant ρ which is omitted from the notation.) To describe the morphisms in this category we proceed in two steps. First, for any two horizontal isotopy classes of cobordisms [V ] and [U] with V : (L ′ j ) ❀ (L i ) (as in Definition 3.1.1) and U : (K ′ s ) ❀ (K r ) we define the sum [V ] + [U] to be the isotopy class of a cobordism family W : a cobordism horizontally isotopic to U and so that U is disjoint from V (to insure embeddedness we can not simply take V and U in the disjoint union.) Notice that the sum [V ] + [U] is not commutative.
The morphisms in Cob d (M) are now defined as follows. A morphism a cobordism from the Lagrangian family formed by the single Lagrangian L ′ j and a subfamily (L r(j) , . . . , L r(j)+s(j) ) of the (L i )'s, and do that r(j)+s(j)+1 = r(j +1). In other words, V decomposes as a union of V i 's each with a single positive end but with possibly many negative ones. We will often denote such a morphism by V : (L ′ j ) −→ (L i ). The composition of morphisms is induced by concatenation followed by a rescaling to reduce the "length" of the cobordism to the interval [0, 1]. It is an easy exercise to see that this is well defined precisely because our morphisms are (horizontal) isotopy classes of cobordisms and because morphisms are represented by sums of cobordisms with a single positive end.
We will consider here the void set as a Lagrangian of arbitrary dimension. With the above conventions one would expect that the sum operation described above gives the category Cob d (M) the structure of a strict monoidal category with neutral element ∅. However, it is easy to see that L + ∅ = ∅ + L simply because the two families (L, ∅) and (∅, L) are different. We now intend to factor both the objects and the morphisms in this category by equivalence relations that will transform this category in a strict monoidal one. For the objects the equivalence relation is induced by the relations We now construct the category There is a subcategory of Cob d (M), that will be denoted by Cob d 0 (M), whose objects belong to L * d (M) (see §2.3) and whose morphisms are represented by Lagrangian cobordisms V verifying the analogous condition to (6), but in R 2 × M. This is again a strict monoidal category.

Construction of F
The purpose of this section is to discuss the proof of the following theorem (more details will appear in a more ample paper currently in preparation). Recall from (1) that the functor F is actually determined by F by F = P • F . Nevertheless we will construct F independently in §4.1 below.
By recalling the construction of the category T S (−) from §2.4 it is immediate that the existence of F implies: In slightly different terms: The notion of generation of a subcategory here is the one commonly used in the theory of triangulated categories. Namely, if C is a triangulated category and S ⊂ Ob(C) is a collection of objects, then the subcategory generated by S is the minimal full category C S that contains S and is closed under isomorphisms and formation of exact triangles (i.e. if A is isomorphic to B ∈ C S then A ∈ Ob(C S ), and if ϕ : A → B with A, B ∈ Ob(C S ) then the objects in the completion of ϕ into an exact triangle also belong to C S ). In this paper we do not complete with respect to idempotents.
Theorem A and its Corollaries above are formulated for K = Z 2 and thus assume that condition (6) is satisfied for all Lagrangians and cobordisms involved. Nevertheless, the methods used in the proof also apply when working over A and thus without the assumption (6), but in this case they lead to less precise results. Still, the following Corollary is true for K = A as well as for K = Z 2 (in this second case it is a simplified version of Corollary 4.0.1). We recall that K is not graded so we have omitted any grading/suspension etc in this statement.
Note that Corollary 4.0.3 together with Example d. in §3.2 (expanded in §6.3) imply the existence of exact sequences associated to surgery over K. An exact sequence of Floer homologies corresponding to Lagrangian surgery (of two Lagrangian submanifolds) has been previously obtained in [FOOO1] by other methods. See also [Sei2].
Before passing to the proof of Theorem A we will discuss the definition of the functor F as this already indicates what are the basic technical ingredients of the whole machinery. Moreover, the definition of this functor is relatively simple on the algebraic side.
4.1. The functors F and F. In this subsection we will work over the ring K that can be either A or Z 2 . We will indicate in each relevant statement the differences between these two cases. Each time a statement is claimed over Z 2 , the condition (6) is assumed for all involved Lagrangians in M and in R 2 × M.
4.1.1. The functor F . The construction of F is relatively easy. However, we emphasize it is only the existence of the functor F that implies the exact sequences and isomorphisms in the Corollaries 4.0.1 and 4.0.2 above.
At the level of the objects, the functor F is defined as follows: We now describe the functor F on morphisms. We will show how to associate to an appropriate cobordism V : L ❀ (L 1 , . . . , L k ) with a single positive end, a homology class where [−l] is the degree shift by l ∈ Z and T is the shift in the triangulated category DF uk d (M). We then put F ([V ]) = α V . For a more general morphism, W = V 1 + · · · + V k with each V i with a single positive end, we put F (W ) = F (V k ) = α V k . We will not verify here the properties of F with respect to composition. We only notice here that they follow from those of the functor F .
In the sequel we will make use of Floer homology for pairs of Lagrangian submanifolds with cylindrical ends (recall the definition of Lagrangians with such ends from §3.1.2). This is analogous to usual Floer homology (for closed Lagrangians) with a few modifications that are described in §4.2.2 below.
We will make use of the following result whose proof is postponed. Returning to the construction of F we will now construct a particular family of cobordisms that will be useful for our considerations. Let a ≥ 0, q, r, s ∈ R. Consider a smooth function σ a;q,r,s : R → R, with the following properties: i. σ a;q,r,s (t) = q for t ≤ −a, σ a;q,r,s (t) = s for t ≥ 3. ii. σ a;q,r,s (t) = r for t ∈ [−a + 1, 2]. iii. σ a;q,r,s is strictly monotone on (−a, −a + 1) and strictly monotone on (2, 3). We denote by γ a;q,r,s ⊂ R 2 the graph of σ a;q,r,s . To simplify the notation we also put λ k = graph(σ −2;k−1/2,k+1,1/2 ).
Let V : L ❀ (L 1 , . . . , L k ) be a cobordisms with one positive end. Let L ′ ⊂ M be a small Hamiltonian perturbation of L which is transverse to L as well as to L 1 , . . . , L k . By a possible isotopy of V we may assume that V ⊂ ([0, 1] × (−∞, k]) × M. Consider now the Lagrangian λ k × L ′ . Clearly λ k × L ′ intersects V transversely and It follows from Theorem 4.1.1 that the Floer homology HF (V , λ k × L ′ ) is well defined and depend only on the horizontal isotopy classes of V and λ k × L ′ . We now need another variant of Theorem 4.1.1 whose proof will also be postponed. (1) For every t, J t is tamed by the symplectic structure ω R 2 ⊕ ω of R 2 × M. Moreover, for a generic such J the Floer complex (CF (λ k × L ′ , V ; J), d) is well defined and has the form: for some l 1 , l 2 ∈ Z and differential where d 1 and d 2 are, up to sign, the Floer differentials of CF (L ′ , L k ; J b ) and CF (L ′ , L; J c ) respectively.
Since D 2 = 0 it follows that φ V,λ k : CF (L ′ , L; J c ) −→ s l 1 −l 2 CF (L ′ , L k ; J b ) is a chain morphism. (See figure 4.) Next recall that to a monotone Lagrangian L we can associate its quantum homology QH(L; K) (see §2.2.3, and also [BC3,BC4,BC2]). This is a ring with a unit [L] ∈ QH(L; K). Recall also that there is a canonical PSS isomorphism (see §2.  −→ HF (L, L k ) by using the moving boundaries continuation isomorphisms c ϕ as described in §2.2.2 with ϕ a (small) Hamiltonian isotopy with ϕ 1 (L) = L ′ ; H(φ V,λ k ) stands for the map induced in homology by φ V,λ k ; H is an auxiliary generic Hamiltonian. It is not difficult to see that H(φ V,λ k ) depends possibly on V and λ k but not on the other choices made in its construction and the same is true for the class α V,λ k .
We now restrict to the case K = Z 2 . Put We claim that over Z 2 this defines indeed a functor as desired.
Remark 4.1.3. Although we will not prove this claim here it is useful to understand why the restriction to K = Z 2 is necessary. The reason is that, for this construction to produce a functor with domain Cob, the class α V,λ k , or equivalently the morphism H(φ V,λ k ), has to be independent of the choice of λ k (at least for k large enough) and also has to be invariant with respect to horizontal isotopies of V . Over A the invariance with respect to λ k is not verified simply because in the expression of φ V,λ k the areas of the green strips in Figure 4 are taken into account. Thus, if λ k × L ′ is moved by a horizontal isotopy induced from a planar Hamiltonian vector field to λ ′ k × L ′ for some other curve λ ′ k -as the one in Figure 4 -the new morphism φ V,λ ′ k will differ from φ V,λ k because the areas of these strips will change. In fact, the exact difference is that φ V,λ k and T a φ V,λ ′ k are chain homotopic for some real constant a. This is easily seen by using the moving boundary formulas in §2.2.2 (the adaptation to the setting of Lagrangians cylindrical at infinity is discussed in §4.2.2). Clearly, over K = Z 2 this dependence disappears and the independence on horizontal isotopies of V is also easily established. To describe the functor F on morphisms we first mention that this will be by definition a monoidal functor so that it is enough to describe F(Φ) where Φ ∈ Mor Cob d 0 (M ) (L, (L 1 , . . . , L k )) .
A part of the difficulty of the construction resides in the fact that we now need to use explicitly the triangulated structure of DF uk d (M) and as that is induced from the structure of F uk d (M) ∧ we need to deal with the triangulated structure of this last A ∞category.
To simplify notation we will denote A = F uk d (M). We denote by Ch the category of chain complexes viewed as an A ∞ -category (with operations of order ≥ 3 being trivial). We also recall from [Sei3] the Yoneda embedding: where F (A, B) denotes the (non-unitary) A ∞ -functor category from the category A to the category B, and A opp is the A ∞ -category opposite to A. The definition of the functor Y is so that, in our context, Y(L) = CF (−, L) where CF (−, L) represents the Floer chains over K = Z 2 viewed as a functor in the first variable (and actually defined as a directed system because the actual definition of the Floer chain complex also depends on the almost complex structure and on various perturbation data). The Yoneda embedding is faithful and the actual triangulated structure of F uk d (M) ∧ is provided by the triangulated completion of the image of Y. In turn, the exact triangles in F (A opp , Ch) are induced from the exact triangles in Ch. In short, we need to show how to associate to V : L ❀ (L 1 , . . . , L k ) a sequence η V of exact triangles in F (A opp , Ch): where Z 1 = 0, s k = 1. Moreover, writing A V := Z k+1 , we need to provide also an isomorphism φ V : CF (−, L) −→ T r A V . We then define More specifically, we will construct for each Lagrangian N ∈ L * d (M) the following: a sequence of chain complexes Z N i , 1 ≤ i ≤ k + 1, with Z N 1 = 0, and chain morphisms Of course, the above is not sufficient in order to prove that F is a functor as one needs to also verify that the exact sequences (19) are functorial in N, that they respect all the higher structures of an A ∞ -category and that F so defined respects the composition of morphisms. We will omit these verifications here -they are postponed to the more complete version of this work that is in preparation. Still, the basic idea behind these verifications is rather obvious once the construction of the triple (φ V , A V , η V ) needed in order to define F (Φ) is made explicit. This is the construction that we will describe below. (19). In this subsection we will work over K which can be either A or Z 2 . The geometric constructions described here imply Corollary 4.0.3. However, we emphasize that for reasons similar to the discussion in Remark 4.1.3, the functoriality properties required for the construction of F are satisfied only over Z 2 .

Construction of the mapping cones
The construction below is another corollary of Theorem 4.1.1. We will use the family of functions and the notation introduced just before the statement of Corollary 4.1.2. First, to simplify the construction we will assume that N is transverse to L as well as to L i for all 1 ≤ i ≤ k. By a possible isotopy of V we may assume that and that for some large enough a > 2 we have We now consider two Lagrangians in R 2 × M: The intersections between these Lagrangians and V is: where p ∈ (2, 3), σ a;−1,−2,2 (p) = 1.
The following is another variant of Theorem 4.1.1. (1) For every t, J t is tamed by ω R 2 ⊕ ω. Assuming this corollary we will now construct a cone decomposition of CF (N ∧ , V ) with linearization (CF (N, L 1 ), . . . , CF (N, L k )). Define (graded) vector spaces Z N 1 = 0, We endow each Z N q with an operator δ q : Z N q −→ Z N q which is the upper left q × q sub-matrix of the differential D (which appears in Corollary 4.1.5). Define also maps Figure 5. The cobordisms V , N ∧ and N ∨ together with, in green, some of the J-holomorphic strips relevant for the iterated cone structure (everything projected to R 2 ).
Since D is upper triangular and D 2 = 0 it easily follows that: (1) each δ q is a differential.
(2) each u q is a chain map.
(3) (Z N q+1 , δ q+1 ) = cone (CF (N, L q )[−s q ], D qq ) uq −→ (Z N q , δ q ) for every q. The chain mapφ N V can be used to produce φ N V after adjusting the relevant degree shifts. Note that in this construction there are some signs that we have omitted (in the matrix D as well as in u q and δ q ). Since we work with the ground field Z 2 we just ignore them.

Some technical ingredients.
The purpose of this subsection is to summarize the main ingredients needed to prove the technical Theorem 4.1.1 and its Corollaries 4.1.2 and 4.1.5.
There are essentially three ingredients that are important in these proofs: a compactness argument, a definition of Floer complexes and homology for Lagrangian with cylindrical ends, and finally a method to use plane curve combinatorics to deduce algebraic properties of the differential in such Floer complexes. We will briefly discuss each of them below. Variants of these constructions appear in slightly different settings in the literature (see for instance the works of Seidel [Sei3], Abouzaid [Abo], Auroux [Aur], as well as older work of Oh [Oh4]). At the same time, each of these ingredients is sufficiently non-standard so that we feel it useful to discuss all three of them here. Besides this, standard techniques together with the methods in [BC2], [BC4] are sufficient to deal with transversality issues. 4.2.1. Compactness. Given that cobordisms are viewed as Lagrangians with cylindrical ends and thus non-compact, it is clear that compactness for pseudo-holomorphic curves with boundaries on such Lagrangians is the first main technical issue that one has to deal with.
In this subsection we will discuss the main idea that insures compactness in this setting. Variants of this idea will appear in all the constructions contained later in the section.
For this discussion we will fix two Lagrangians with cylindrical ends W and W ′ as in Theorem 4.1.1 -see also Figure 6. In contrast to (16) we do not assume that they are cylindrically horizontal outside of [0, 1] × R × M, but rather that they are cylindrically horizontal outside a compact subset K ⊂ R 2 as described in §3.1.2. We also fix a compact region in the plane B ⊂ C ∼ = R 2 and we will only consider almost complex structures J so that π is ( J, i)-holomorphic outside B × M. Moreover, outside B each of the cobordisms coincide for the negative ends with products γ − i × L − i between certain planar curves γ − i and Lagrangians L i ⊂ M and similarly for the positive ends, they are products γ + j × L + j with Lagrangians L + j ⊂ M and γ + j curves in C. Moreover, we assume that the negative planar curves of W and those of W ′ intersect transversely, and similarly for the positive planar curves of the two cobordisms. Further, two curves that correspond to positive (respectively, negative) ends of W do not intersect outside B and similarly for W ′ . Further, we will also assume that the Lagrangians in M corresponding to the positive ends of W and those corresponding to the positive ends of W ′ are two-by-two transverse in M and similarly for the negative ends. We will assume also that there is a compact set B ′ ⊂ C with B ⊂ B ′ and so that outside B ′ the curves γ + i and γ − j coincide with horizontal lines, and similarly for the curves of W ′ . The basic argument here appeared already in Chekanov's work [Che] and is as follows. Assume that u : Σ → C × M is a J-holomorphic curve where Σ is either the disk D 2 , the strip R × [0, 1] or the sphere S 2 . In case Σ is the disk we assume that u maps the boundary ∂Σ either to W or to W ′ , and if Σ is the strip, we assume Lemma 4.2.1. Assume that the symplectic energy of u is finite. Then either π • u is constant or π • u(Σ) ⊂ B ′ .
Proof. The first remark is that π • u(Σ) is bounded. Indeed, this is clear for Σ = D 2 , S 2 . If Σ = R × [0, 1] then due to the finite energy condition we get that u(Σ) converges at ±∞ to some point in W ∩ W ′ . But as π(W ∩ W ′ ) ⊂ B ′ we get that π • u(Σ) is bounded in this case too. Now assume that π • u(Σ) ⊂ B ′ . Notice that C\(B ′ ∪ π(W ) ∪ π(W ′ )) is a union of unbounded domains in C. As Im(π • u) is bounded it follows that π • u is constant. Indeed, otherwise an application of the open mapping theorem to the holomorphic map π • u implies that the image of π • u| Int Σ contains an unbounded region.

Floer homology for Lagrangians with cylindrical ends.
Here we explain the necessary modifications needed for the constructions and structures from §2.2 to work for Lagrangian cobordisms (rather than just closed Lagrangian submanifolds). In particular, the precise type of Floer homology in Theorem 4.1.1 and the Floer complexes in the Corollaries 4.1.2 and 4.1.5 appear as a special cases of the construction below.
Let W and W ′ be two uniformly monotone Lagrangians with cylindrical ends. We will not assume for now that they are cylindrically distinct at ∞.
We intend to define the Floer complex CF (W , W ′ ; η; (H, f ); J) with coefficients in K with K either A or Z 2 (see §2.2) and we now describe the data involved in this definition. (1) For every t, J t is an ω-tamed almost complex structure on M .
If B = ∅ we simply write J . B. The component η ∈ π 0 (P(W , W ′ ) is fixed as in §2.2.1. C. The perturbation (H, f ). To describe these perturbations we first fix the notation for the ends of W (see §3.1.2). Thus for R + and R − sufficiently big we assume for some a + 1 < · · · < a + k + and (1) The support of H is compact.
(2) The function f : R 2 → R verifies: 1. The support of f is contained in the union of the sets and where the positive constants ǫ ± i are small enough (and the numbers R + and R − are big enough) so that the all the sets U ± i above are pairwise disjoint.

The restriction of f on each set
R as well as on [R + , ∞) × R and we assume that ǫ ± i is sufficiently small so that for all indexes i and similarly We denote the space of pairs (H, f ) as above by H(W , W ′ ). The role of the function f is to perturb the Lagrangian W so as to render it cylindrically distinct at infinity from W ′ by using the Hamiltonian flow associated to the composition e = f • π with π : M = R 2 × M → R 2 the projection. This is precisely the meaning of the requirements in equations (20) (H, f ) ∈ H(W , W ′ ) generic and J ∈ J B for some compact set B is also generic.
We put: where η ′ is the path component that corresponds to η under the isotopy φ f •π t . Of course, we still have to justify the right term in equation (22). In view of the fact that H is compactly supported and due to our choice of J it is immediate to see that the (standard) construction of the Floer complex -recalled in §2.2.1 -carries over to this setting. This is true because compactness for the finite energy solutions of Floer's equation drically distinct at infinity, then one can also take f = 0 in the construction above. The purpose of the rather complicated definition above is to provide a notion of Floer homology that is sufficiently robust so that the standard constructions familiar from the compact case admit analogues in the "cylindrical at infinity" setting.
The key statement is the following: We also pick a compact set B ⊂ R 2 as well as a generic J ∈ J B . Let ν : R → [0, 1] be an increasing C ∞ function so that ν(τ ) = 0 for τ ≤ 0 and ν(τ ) = 1 for τ ≥ 1. Define We now define a morphism: by a sum like in Equation (5) running over the elements of zero dimensional moduli spaces consisting of finite energy solutions to Floer's equation (23) subject to the boundary conditions The only difficulty in checking that this morphism is well-defined and verifies the expected properties in standard Floer theory (i.e. it induces a canonical isomorphism in homology as in §2.2.2) reside in insuring that the moduli spaces of finite energy Floer trajectories with moving boundary conditions as above verify the usual compactness properties. But this follows immediately by noticing that, because [f ] = [f ′ ] we have that W τ and W ′ are cylindrically distinct at infinity for all τ ∈ R. This implies that Lemma 4.2.1 can still be applied and it shows that the image of a finite energy solution of equation (23) subject to (24) is either constant or it has its image contained in a compact set K ⊂ M that contains the support of H and whose projection on R 2 contains B as well as the rectangle i for all i. The argument showing invariance with respect to horizontal isotopies is similar.
Remark 4.2.4. Given a pair of Lagrangians with cylindrical ends that are cylindrically distinct at infinity the fact that the fibration π : C × M → C is trivial provides a variety of possible horizontal isotopies. Each of them leads to some relation among appropriate Floer complexes. An example appears in Figure 5 where the horizontal isotopy between N ∧ and N ∨ produces a chain homotopy equivalence CF (N ∨ , V ) −→ CF (N ∧ , V ).

4.2.3.
Non-existence of certain holomorphic strips. The arguments here are needed to justify the particular form of the Floer complexes as described for instance in Corollaries 4.1.2 and 4.1.5, in particular the upper triangular from of the matrix of the differential D. To describe them we return to the setting in §4.2.1: in other words W , W ′ are Lagrangians in M with cylindrical ends and J ∈ J B for some compact set B ⊂ R 2 . We also recall the particular choices for the ends of W and W ′ from §4.2.1. Namely, outside B each of these two cobordisms coincides -for the negative ends -with products γ − i × L − i between certain planar curves γ − i and Lagrangians L i ⊂ M and similarly for the positive ends, they are products γ + j × L + j with Lagrangians L + j ⊂ M and γ + j curves in C. We also assume the transversality conditions mentioned in §4.2.1.
The actual argument reduces to the following remark. Assume that u : This means that a is of the form a = (p, x), with p an intersection point of an end of W with an end of W ′ and x is an intersection point of two of the curves, say γ i and γ ′ j . Moreover, in the neighborhood of x, the map v = π•u is a C-valued holomorphic map with transverse (embedded) Lagrangian boundary conditions. The intersection pattern of γ i and γ ′ j together with an application of the open mapping theorem limit the possible choices for the point x. More precisely, after an orientation preserving change of coordinates in the neighborhood U of x we may assume that γ i coincides with the real axis and γ ′ j with the imaginary axis. Then U \ (γ i ∪ γ ′ j ) is divided into four quadrants. Then the image of v can possibly lie only in two of the four quadrants, namely in the first or the third. Moreover, if the connected component of one of these admissible quadrants inside C \ (π(W ) ∪ π(W ′ )) is unbounded then v cannot have its image inside that quadrant. See Figure 7. Similarly, in Figure 6 the only possible choices for such an x are the points R, Q, P but not S. The same type of arguments also apply to the positive end of the strip u, i.e. b = lim s→+∞ u(s, t). In case b ∈ (W ∩ W ′ ) \ π −1 (B) then b = (q, y) where again q is an intersection of an end of W and one of W ′ and y is an intersection point of two of the curves γ i or γ ′ j . Then the only admissible quadrants are the second and fourth (see Figure 7). And again, if one of these quadrants is inside an unbounded component of the complement of π(W ) ∪ π(W ′ ) then this quadrant is ruled out. In Figure 6 for instance, this means that the choices for y are S, Q, P but not R. A little more detailed analysis shows that a strip starting at Q can end at S but not at P or Q itself (except, in this last case, if v is the constant strip at Q). The reason is that, by the open mapping theorem, a strip starting from Q has to have S either as end or inside its image. But if it is inside the image this image is not bounded which is not possible. Thus S is the end of the strip. This also explains Figure 5 where each intersection point of N ∧ and V can not be the exit point of holomorphic strips ending at any of the intersection points in N ∧ ∩ V that have a strictly bigger vertical coordinate. This shows that in Corollary 4.1.5 the differential D is upper triangular and it also follows that the diagonal elements have the form claimed. The same argument can also be applied to Figure 4 and implies the structure of the complex CF (λ k × L ′ , V ) claimed in Corollary 4.1.2.

Quantum homology and the morphisms in Cob d (M)
The purpose of this section is to discuss some of the properties of the morphisms in Cob d (M) using as main tool the Lagrangian quantum homology (as introduced in [BC4,BC3]). The Floer homology constructions in this section will be done over K = A thus assumption (6) is not required here. The key fact for the applications of this section is that the ring Λ from §2.2.3 injects into the field A.
In the first part of the section we first make more precise the version of Lagrangian quantum homology that is appropriate in our context. We pursue in the second subsection by analyzing elementary cobordisms and in the third subsection we study morphisms of the type L ❀ (L 1 , L 2 ). 5.1. Quantum homology for Lagrangians with cylindrical ends. We first discuss the definition of quantum homology in this context and then will see how the PSStype comparison morphisms between quantum homology and Floer homology (recalled in §2.2.4) adapt to this setting. 5.1.1. Quantum Homology. Let W ⊂ M be a monotone Lagrangian with cylindrical ends and let S be a union of some of its ends. In other words, assume the ends of W are The quantum homology QH(W , S) is defined as follows. Fix ǫ > 0 and put W = W | [R − −ǫ,R + +ǫ]×R , so that W is a compact manifold with boundary Let S ′ be the part of the boundary of W that corresponds to S: Choose a Morse function f : W −→ R together with a Riemannian metric (·, ·) and an almost complex structure J on M . We require the function f to be so that its negative gradient −∇ f is transverse to ∂W and moreover it points outside of W along S ′ and inside W along ∂W \ S ′ . We also require J to be so that the projection π is holomorphic outside a compact set K ⊂ [R − − ǫ/2, R + + ǫ/2] × R × M. Denote by D S = ( f , (·, ·), J) our data.
Proposition 5.1.1. If the data D S is generic, then the pearl complex C(D S ) is well-defined by the same construction as the one recalled in §2.2.3. The resulting quantum homology does not depend, up to canonical isomorphism, on the choice of data D S nor on the choice of ǫ and R + , R − above. We denote the resulting homology by QH(W , S).
Similarly to the conventions in §2.2.3 we will denote by QH(W , S; A) the homology of the complex C(D S ) ⊗ Λ A.
Sketch of the proof of Proposition 5.1.1. Recall that the relevant pearly trajectories are composed of flow lines of −∇ f and J-holomorphic disks. By Lemma 4.2.1 and our restriction on J , there are no pseudo-holomorphic disks with boundary on W with nonconstant projection to R 2 that reach the complement of K. In view of the fact that −∇ f is transverse to ∂W we deduce that all pearly trajectories that originate and end at critical points of f can not reach the boundary of W . This immediately implies that the complex C(D S ) is well defined and indeed a chain complex. The same argument also applies to show the rest of the statement.
The notation QH(W , S) is coherent with the standard notation in Morse (or singular) homology. Moreover, the standard manipulations with Morse functions relative to boundaries adapt to the "pearly" setting and induce exact sequences that are quantum homology versions of the respective Morse theoretic sequences. The following lemma will be useful later in the paper.
Lemma 5.1.2. Assume W is as in Proposition 5.1.1. Pick a union of the ends of W and denote it by A. Take also some other union B of the ends of W so that A ∩ B = ∅. There is a long exact sequence: A similar exact sequence also exists with coefficients in A.
Proof. We put S = A ∪ B and we intend to construct a particular function f as the one appearing in the definition of QH(W , S) but with a number of additional properties. We use below the same notation as the one fixed before the statement of Proposition 5.1.1. In particular, J + ⊂ {1, . . . , k + }, J − ⊂ {1, . . . , k − } are so that is the part of the boundary of W corresponding to S. We also denote by J ′ The functions σ + i , σ + j are also Morse, each with a single critical point and are required to satisfy the following conditions: (1) σ + i (x) is a non-constant linear function for x ∈ [R + + 3ǫ/4, R + + ǫ]. Moreover, in this interval σ + i is decreasing if i ∈ J + and increasing if i ∈ J ′ + . Further, σ + i has a single critical point at R + + ǫ/2 and this is of index 1 if i ∈ J + and of index 0 if j has a single critical point at R − − ǫ/2 and this is of index 1 if j ∈ J − and of index 0 if j ∈ J ′ − . A function f with these properties will be called adapted to the exit region S.
Let now I − , I + be index sets so that corresponds to A and let U(A ′ ) be a tubular neighborhood of A ′ in W given by We now let V = W \ U(A ′ ) and also denote We assume the various choices made are generic so that the pearl complexes C(W, f , J), C(A ′′ , f | A ′′ , J) and C(V, f | V , J) are well defined. These three complexes are related by an obvious short exact sequence: Remark 5.1.3. We will mainly apply the construction above to Lagrangians V that are the R-extensions of Lagrangian cobordisms V . In this case we denote QH(V , S) by QH(V, S) and similarly when working over A.

5.1.2.
The PSS isomorphism for Lagrangians with cylindrical ends. Let W ⊂ M be a Lagrangian with cylindrical ends and assume that S is a union of some of its ends as in §5.1.1. The choice of S determines a path component c S ∈ π 0 (H(W , W )) in the following way. Consider a perturbation function f , as at point C in §4.2.2, so that: (1) for each positive end i of W , the constant α + i is negative if the end is in S and is negative if the end i is not in S.
(2) for each negative end j, the constant α − j is positive if the end is in S and is negative if the end j is not in S and put c S := [f ].
The purpose of this subsection is to discuss the proof of the following result. After multiplying f by a small positive constant we may assume that f has a small differential d f so that the graph of d f fits inside of U. Recall that f has a linear horizontal component along the ends. Extend the function f first to a function on U using the identification of U with a neighborhood of V ⊂ T * W (making it constant along each cotangent fibre), and then to the rest of M so that the resulting function H f vanishes outside a closed neighborhood U ′ of U.
Pick a generic autonomous almost complex structure J ∈ J B with B a compact set sufficiently large so that W is cylindrical outside B. We will also assume R + and |R − | sufficiently big so that [R + , ∞) × R as well as (−∞, R − ] × R are both outside B. The linearity of the function f at infinity immediately shows that W 1 := φ H f 1 (W ) and W are cylindrically distinct at infinity, that for a generic choice of f the Floer complex CF (W 1 , W ; J) is well defined and that, by Proposition 4.2.3, its homology is canonically identified with HF (W , W ; c S ) (see §4.2.2 and in particular (22)).
We will also need below another function f ′ : W → R with the same properties from Lemma 5.1.2 as f except that the value of ǫ used to construct f ′ is fixed to be ǫ ′ = ǫ/2.
We also fix a metric (·, ·) on W that is adapted to the ends of W (in the sense indicated in the proof of Lemma 5.1.2) and so that the pearl complex C(D S ) is defined for D S = ( f ′ , (·, ·), J). We will work in this proof only over A so that the homology computed by C(D S ) is QH(W , S; A).
We now intend to consider the moving boundaries PSS -chain morphism -see §2.2.4: In fact, the only issue that is specific to our cylindrical at infinity setting is again whether the necessary compactness is satisfied by the moduli spaces used to define this map. If this is the case, the rest of the construction takes place like in the compact setting. In particular, we also obtain that this morphism induces an isomorphism in homology. Thus, our focus will now be to describe the relevant moduli spaces and indicate the reason why compactness hols.
It is easy to see that the needed compactness properties for the definition of P SS as well as that of its (homological) inverse and all the other relevant properties are an immediate consequences of the following result.
Lemma 5.1.5. With the notation above Proof of Lemma 5.1.5. Put P = {R + + ǫ/2} × a + i and notice that this is a point of intersection of l s = π(W s ) and l = π(W ) for all s, the intersection is transverse for s > 0. This is because P is a critical point for the function σ + i . Let now u ′ = π • u and let z ∈ {0} × R ∪ {1} × R. The key remark for proving the lemma is that it is not possible that u ′ (z) = P . The reason for this is again the open mapping theorem: if u ′ (z) = P then the image of u would contain a small open quadrant around P as in Figure 8 which is not possible. A similar statement holds also for the negative ends of W . Figure 8. The cobordisms W 1 and W . The quadrants Q 1 , Q 2 around P . Also appears he image of u ′ , the points Q = π a and π(b) as well as the direction of the flow −∇ f ′ when projected on R 2 . By using again the open mapping theorem we deduce that if the image of u does not verify the claim, then the whole image of u ′ is contained in ((−∞, To simplify the discussion assume that we are in the first case, the second one is treated in a perfectly similar fashion. The fact that π(a) = Q implies that the strip u ′ "arrives" at Q and this is easily seen to imply that ind σ + i 0 (Q) = 0. Moreover, π(b) can be written as π(b) = (b ′ , a + i 0 ) with b ′ ≥ R + + ǫ/2. At this point we use the particular form of the function f ′ : as the function σ + i 0 used in the construction of f ′ is increasing on the interval [R + + 3ǫ/8, +∞) (because ǫ ′ = ǫ/2) and, as the metric (·, ·) is adapted to the ends of W , we deduce that there can not be any flow lines of −∇( f ′ ) that come from the interior of the region W ∩ π −1 ([R − − ǫ/2, R + + ǫ/2]) and reach the point b. Clearly, by Lemma 4.2.1, there can not be any J-holomorphic disk with boundary on W reaching b either. Taken together, these two facts contradict our assumption on the image of u and this concludes the proof of the lemma.
The proof of Proposition 5.1.4 follows now by standard arguments. Proof. By Proposition 5.1.4 the Floer homology associated to the two Lagrangian cobordisms,V andV ′ , in Figure 9 verifies: where c L ∈ π 0 (H(V ,V )) is defined as at the beginning of §5.1.2. Our main result is valid under a strong additional assumption.
Theorem 5.3.1. Assume that L 1 and L 2 are not narrow (over Λ) and that QH(L) is a field (in other words, each element in QH(L) admits an inverse with respect to the quantum multiplication, and moreover the unity [L] ∈ QH(L) is different from 0). Then the inclusion QH(L) → QH(V ) is injective. Moreover, for each k we have the inequality: The next step is to notice the commutativity of the diagram (31) QH(L; A) up to multiplication by T a for some a ∈ R. Here φ V is the morphism from Corollary 4.1.2 and P SS and P SS ′ are both isomorphisms as explained below. Recall from Remark 4.1.3 that over A the map φ V is in fact unique only up to multiplication by a real power of T . This explains why we can expect the commutativity of (31) only up to multiplication by T α . The morphism P SS is just the Piunikin-Salamon-Schwarz-type isomorphism QH(L; A) → HF (L, L) as recalled in §2.2.4. The morphism P SS ′ is given by the composition: Here the first morphism P SS L 1 is the PSS-type isomorphism discussed in Proposition 5.1.4. The second isomorphism, η, follows from the definition of HF (−, −) in §4.2.2 and Figure 10. The cobordismV ′ obtained by a Hamiltonian deformation associated to a small function f :V → R adapted to the exit region L 1 . We have QH(V, L 1 ; A) ∼ = HF (V ′ ,V ).
Proposition 4.2.3. The third isomorphism, ξ, is itself a composition of two isomorphisms Here ξ ′ is provided (again via Proposition 4.2.3 ) by the fact thatV ′ is horizontally isotopic to the cobordismV ′′ in Figure 11. As for ξ ′′ , it is an identification ξ ′′ : HF (V ′′ ,V ) = HF (L, L 2 ) that follows from the fact that π(V ′′ ) and π(V ) intersect in a single point in the region where bothV ′′ andV are just products between curves in the plane and, respectively, L and L 2 . The next step is to justify the commutativity of Diagram 31. This verification is based on a slight strengthening of the argument used to justify the isomorphism P SS ′ . The purpose is to identify geometrically the maps φ V and i 1 and to relate them to the construction of P SS ′ . The geometric part of this argument consists in composing the two isotopic cobordismsV ′ andV ′′ from the Figures 10 and 11 with a cobordism of the form γ × L as in the Figure 12. To be more precise, assume, without loss of generality, that the cylindrical positive end of bothV ′ andV ′′ coincide with [1, +∞) × {2} × L. Assume also that the positive end ofV coincides with [1, +∞) × {1} × L. Now take the curve γ to be the graph of a function g : [1, +∞) → R so that g is smooth, g(t) = 2 for t ∈ [1, 2]∪[4, +∞), g attains its minimum at the point 3 with minimal value g(3) = −1 and 3 is the single critical point of g in the interval (2, 4). The curve γ intersects (transversely) the curve y = 1 in two points P = (p, 1) and Q = (q, 1) with p < q. Finally, we put Certainly,W ′ is horizontally isotopic toW ′′ (and both are horizontally isotopic withV ′ andV ′′ ). We will use the fact that the isotopy fromW ′ toW ′′ may be assumed constant on π −1 ([1, +∞) × R).
We use the two cobordismsV andW ′ to deduce the commutativity of the following diagram: (33) QH(L; A) where j is the map induced in homology by the inclusion of the subcomplex of CF (W ′ ,V ) generated by the intersection points ofW ′ andV that project onto Q; P SS ′′ is a composition like η • P SS L 1 from Equation (32) only withW ′ instead ofV ′ . We now use the cobordismsW ′′ andV . The fact that the horizontal isotopy fromW ′ tō W ′′ may be assumed constant π −1 ([1, +∞) × R) implies the commutativity of the triangle below up to multiplication with a term of the form T a : T  T  T  T  T  T  T  T  T  T  T  T  T  T HF (W ′ ,V ) To summarize what was shown till now, we proved that Diagram (31) commutes and that P SS and P SS ′ are isomorphisms. The next important point is that the morphism φ V is a QH(L; A)-module morphism. This is easy to show by a direct argument (of course, by using a specific choice of function λ k as in Corollary 4.1.2 and acting on the left with QH(L ′ )). As QH(L) is a field this means that either φ V is null or it is an injection. Thus, the same is true for i 1 and it is easy to see that a similar argument can be applied to the morphism i 2 from Diagram (30). The exactness of (30) together with the duality between the i j 's and the η r 's implies that one of the i j 's has to vanish and the other is injective. We will assume that i 1 is injective and that i 2 vanishes. It is immediate to see that injectivity of i 1 with coefficients in A immediately implies that the corresponding morphism i Λ 1 : QH(L) → QH(V, L 1 ) is also injective. Moreover, the vanishing of i 2 with coefficients in A also implies the vanishing of i 2 over Λ. To shorten notation we will not indicate the coefficients in the notation for these morphisms i 1 , i 2 , etc as long as there is no risk of confusion.
The first claim of the Theorem now follows easily. Indeed i 1 (now taken over Λ) factors: and thus QH(L) → QH(V ) is injective. The rank inequality (29) follows immediately if we can show that if I i = Im(QH(L i ) → QH(V )) and I 0 = Im(QH(L) → QH(V )), then I 1 ⊕ I 0 ⊂ I 2 and QH(L 1 ) → QH(V ) is injective.
To do this we go back to the Diagram (30) and we start by noticing that the vanishing of i 2 implies that k 1 vanishes. This is seen as follows. First, by an argument similar to that applied to i 1 and i 2 we see that, over A, k 1 is a QH(L 1 ; A)-module map. Thus it suffices to show that k 1 ([L 1 ]) = 0 ([L 1 ] is the fundamental class and is the unit in QH(L 1 ; A)). Secondly, by using explicitly the form of the pearl complexes associated to a function f : V → R adapted to the exit region L ∪ L 1 it is easy to see that i 2 ([L]) = k 1 ([L 1 ]) and thus k 1 ([L 1 ]) = 0. This means that k 1 vanishes over A. But this implies that it also vanishes over Λ. Now k 1 and s 1 are dual so the vanishing of k 1 implies that of s 1 which means that l 1 : QH(L 1 ) → QH(V, L) is injective. But this implies that QH(L 1 ) → QH(V ) is injective.
We now show that I 0 , I 1 ⊂ I 2 . This follows from the exact sequence: i. The inequality (29) shows that often a Lagrangian with QH(L) a field can not be split in two non-narrow parts by a Lagrangian cobordism. For instance, it is easy to see that the inequality (29) can never be verified in dimension n = 2 if L 0 and L 1 are non-narrow. ii. Concerning the condition that QH(L) be a field we mention that, when working under enough assumptions so that the base field can be taken to be Q, a 2-torus has its quantum homology a field as soon as its discriminant -as defined in [BC1] -is not a perfect square.

Examples
This section describes some explicit constructions of Lagrangian cobordisms. They expand on the constructions sketched in §3.2 and are all based on the Lagrangian surgery construction as described for instance by Polterovich in [Pol].
6.1. Topologically non-trivial monotone cobordisms. Assume that L and L ′ are two Lagrangians L, L ′ ∈ L k (M) and suppose that, additionally, L and L ′ are transverse and that they intersect in a single point. Fix a real number 1 < l < 2 and consider the curve C + (l) = {z+l ∈ C : |z| = 1 , Re(z) ≥ 0}∪{z ∈ C : |Im(z)| = r, 0 ≤ Re(z) ≤ l}this is a semicircle of radius 1, centered at (l, 0) and extended horizontally till it reaches the imaginary axis. Let C(l) be a C ∞ curve in C obtained by smoothing C + (l). We now define two Lagrangian cobordisms in M × T * ([0, 2]):L = L × C(l) andL ′ = L ′ × [0, 2] × {0}. Obviously,L andL ′ intersect transversely in a single point P . We now consider the Lagrangian V obtained fromL andL ′ by surgery at P (there are actually two choices of surgeries here but for this example this choice is not relevant). It is easy to see that V is monotone with the same monotonicity constant as L and L ′ and with Notice that V is a monotone, connected, cobordism that is non-trivial topologically. In our language V : L ′ (L, L ′ , L). Of somewhat more interest is the cobordism constructed by doubling V . We consider . As ξ is a symplectic diffeomorphism we see thatV is also a Lagrangian cobordism. The two Lagrangians V andV can be glued along their intersection that equals L×{0}×{1}∪L ′ ×{0}×{0}∪L×{0}×{−1} -see also Figure 13 and we denote by W the resulting Lagrangian.
Notice that the group H 2 (M ×T * [−2, 2], W ) has two additional generators besides those coming from V andV . They are denoted by a and b. The first one, a, corresponds to the cycle in the plane given by {(s, {0}) ∈ C : −1 − l ≤ s ≤ 1 + l} ∪ {z ∈ (C(l) ∪ −C(l)) : Im(z) ≥ 0} and b corresponds to {(s, {0}) ∈ C : −1−l ≤ s ≤ 1+l}∪{z ∈ (C(l)∪−C(l)) : Im(z) ≤ 0}. Endowed with the correct orientation it is easy to see that these two generators are of Maslov index 2 and of positive symplectic area. Moreover, by adjusting appropriately the constant l (and, if needed, also changing the radius of the semi-circle C + (l)) we may insure that a verifies ω 0 (a) = ρµ(a) where ρ is the monotonicity constant of V andV . After possibly a small Lagrangian isotopy supported on the branch of C(l) with Im(C(l)) < 0 we can achieve the relation ω 0 (b) = ρµ(b) and thus the resulting Lagrangian cobordism W ′ is monotone.
Example 6.1.1. The simplest example of Lagrangians verifying the conditions required for this construction are the longitude and the latitude on the torus, L = S 1 × {1} ⊂ S 1 × S 1 , L ′ = {1} × S 1 ⊂ S 1 × S 1 . One can construct many other examples by taking products of this cobordism with other suitable Lagrangians.
6.2. The trace of surgery as Lagrangian cobordism. The purpose here is to show that the trace of a Lagrangian surgery gives rise to a Lagrangian cobordism. As we shall see, this is a bit less obvious than the reader might first expect because Lagrangian cobordism is less flexible than Lagrangian isotopy. We start with the local picture and fix the following two Lagrangians: L 1 = R n ⊂ C n and L 2 = iR n ⊂ C n .
Lemma 6.2.1. The submanifold L ⊂ C n as defined above is Lagrangian and there is a Lagrangian cobordism L ❀ (L 1 , L 2 ).
By a slight abuse of notation (because we omit the handle from the notation) we will denote L = L 1 #L 2 .
Proof. A straightforward calculation shows that L ⊂ C n is Lagrangian (see e.g. [Pol]).
To construct the desired cobordism we now define H = H · S n ⊂ C n+1 .
A similar computation as before shows that H is also Lagrangian.
We now only look to W 0 = W ∩ π −1 ([−2, 0] × [0, 2]). It is not difficult to see that ∂W 0 = L 1 × {(−2, 0)} ∪ L 2 × {(0, 2)} ∪ L × {0, 0}. We would like to be able to say that W 0 is a cobordism W 0 : L ❀ (L 1 , L 2 ). For this however we still need to show that the L-boundary component of W 0 can be continued to be cylindrical. We now describe explicitly this adjustment (the argument here is in fact quite general). Let V L ⊂ C n × C be the Lagrangian given by V L = L × {(x, y) ∈ C : x = −y}.
Note that at each point of L 0 , V L projects 1-1 on the tangent space of H (via p). Thus reducing U(L 0 ) if necessary we can write V L ∩ N as the graph of a 1-form α on U(L 0 ) that vanishes on L 0 . Since V L is Lagrangian the form α is closed. As U(L 0 ) can be chosen so that it contracts to L 0 , we have H 1 (U(L 0 ), L 0 ) = 0 hence α is exact. Let f : U(L 0 ) → R be so that α = df . Using a partition of unity construct g : W ∪ U(L 0 ) → R so that it agrees with f on U(L 0 ) \ W and vanishes outside a neighborhood of U(L 0 ). Then the Lagrangian W ′ obtained by isotoping W 0 by the time-one Hamiltonian diffeomorphism induced by X g provides the cobordism desired between L and (L 1 , L 2 ) -see also Figure  16. i. For further use, we notice that the homotopy type of the total space of the cobordism W ′ : L ❀ L 1 #L 2 constructed above coincides with that of the set L 1 ∪ L 2 . ii. The construction described here also provides a cobordism W ′′ : (L 2 , L 1 ) → L 2 #L 1 by simply using instead of W the region π −1 (S − ) with S − = {(x, y) ∈ R 2 : y ≤ x} .
Going from the local argument above to a global one is easy. Suppose that we have two Lagrangians L ′ and L ′′ that intersect transversely, possibly in more than a single point. At each intersection point we fix symplectic coordinates mapping (locally) L ′ to R n ⊂ C n and mapping (again locally) L ′′ to iR n ⊂ C n . We then apply the construction above at each of these intersection points. This produces a new Lagrangian submanifold L ′# L ′′ as well as a cobordism L ′# L ′′ ❀ (L ′ , L ′′ ) (we use# in the notation as L ′# L ′′ is topologically not a connected sum if there are several intersection points). The homotopy type of V coincides with that of the set L ′ ∪ L ′′ ⊂ M.
Remark 6.2.3. One can easily generalize the previous construction to a configuration of Lagrangian submanifolds (L 1 , . . . , L r ) and the total surgery L of the Lagrangians in the configuration. The result will be a Lagrangian cobordism V : L ❀ (L 1 , . . . , L r ) with one positive end and r negative ends. Of course, monotonicity will in general not be preserved in this case. However, if the intersection diagram of the configuration is a tree, and if (L 1 , . . . , L r ) are uniformly monotone it seems that the Lagrangian L and the cobordism V will be monotone too. An interesting example is when (L 1 , . . . , L r ) is a configuration of Lagrangian spheres corresponding to a simple singularity. The relation between singularity theory and Fukaya categories has been extensively studied in recent years (see e.g. [Sei3]). Thus the constructions above (together with Corollaries 4.0.3, 4.0.1, 4.0.2) suggests that the cobordism category is relevant in this study.
6.3. Cobordant Lagrangians that are not isotopic. In this subsection we will make use of the constructions described in §6.2 to construct an example of non smoothly isotopic, monotone cobordant, connected Lagrangians. A variety of other examples can be constructed following the same ideas.
We now consider the Lagrangians A ′ = γ 2 × A ⊂ C × M and B ′ = γ 1 × B ⊂ C × M. By performing surgery -as explained in §6.2 -at both intersection points A∩B we can extend the union of the two Lagrangians A ′ ∪ B ′ towards the positive end as well as towards the negative end as in the Figure 17 thus obtaining a cobordism V : A#B ❀ B#A.
Put L = A#B and L ′ = B#A. With our choice of handles it is easy to see that L and L ′ look as in Figure 18. Moreover, if the surgeries used in both intersection Figure 17. The projection of V on C; in red the surgery regions; the curves γ 1 (in blue) and γ 2 in gray.  D(B)) (the two handles can also be picked differently and this can modify the areas bounded by these two circles, thus producing a -non-monotone -cobordism relating non-Hamiltonian isotopic, connected Lagrangians).
It is easy to see that V as constructed before is not orientable. Moreover, V is also not monotone. However there is an easy way to transform the cobordism into a monotone one and we describe it now.
Instead of performing all the construction above in M = C we can as well do it in M ′ = C\{P 1 , P 2 , P 3 } where the three points P i are such that P 1 ∈ D(A)\D(B), P 2 ∈ D(A)∩D(B) and P 3 ∈ D(B)\D(A) as in Figure 18. We will explicitly check monotonicity below. We notice for now that in M ′ , L and L ′ are not even smoothly isotopic.
To verify that V is monotone in There are two generators for this group, u and v, each associated to one of the intersection points of A and B . Each of them is represented by a flat lift of the region R bounded by the union of the two curves γ 1 and γ 2 and the two planar projection of the handles at the ends. If we take these generators to be of positive areas we see that they have Maslov class 3: the horizontal loop is of Maslov class 2 and there is also a vertical half loop of Maslov +1. Moreover, by choosing the handles used for the intersection points appropriately we may arrange that these two generators have equal areas. Thus V is monotone of minimal Maslov class equal to 3. Remark 6.3.1. It is possible to compactify M ′ to a surface of high genus, while still keeping V monotone. This can be done by enlarging the punctures around the points P i and adding appropriate handles.