A-infinity category of Lagrangian cobordisms in the symplectization of PxR
aa r X i v : . [ m a t h . S G ] D ec A ∞ -category of Lagrangian cobordisms in thesymplectization of P × R . Noémie Legout
Abstract
We define a unital A ∞ -category F uk( R × Y ) whose objects are exact Lagrangian cobor-disms in the symplectization of Y = P × R , with negative cylindrical ends over Legendriansequipped with augmentations. The morphism spaces hom F uk( R × Y ) (Σ , Σ ) are given interms of Floer complexes Cth + (Σ , Σ ) which are versions of the Rabinowitz Floer complexdefined by Symplectic Field Theory (SFT) techniques. Contents
Cth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Cth + Continuation element 478 An A ∞ -category of Lagrangian cobordisms 58 This paper deals with Lagrangian cobordisms in the symplectization ( R × Y, d ( e t α )) of a contactmanifold ( Y, α ) . These cobordisms are properly embedded Lagrangian submanifolds admittingcylindrical ends on Legendrian submanifolds of Y , and here Y will be the contactization ( P × R , dz + β ) of a Liouville manifold ( P, β ) .Our goal is to define through SFT-techniques introduced in [EGH00] a unital A ∞ -category F uk( R × Y ) whose objects are Lagrangian cobordisms equipped with an augmentation of theChekanov-Eliashberg algebra of its negative end, and whose morphism spaces are given by cer-tain Floer-type complexes Cth + (Σ , Σ ) . In particular, when Σ is a cylinder over a Legendrianequipped with an augmentation and Σ a parallel copy the complex Cth + (Σ , Σ ) is an SFT-formulation of the Lagrangian Rabinowitz Floer complex due to Merry [Mer14]. There already ex-ists several versions of Fukaya categories whose objects are (non compact) exact Lagrangians, no-tably the (partially) wrapped Fukaya categories of a Liouville domain (see [AS10, Abo10, Syl19])and more recently of Liouville sectors [GPS20]. In this paper we instead consider Lagrangiansubmanifolds in a trivial Liouville cobordism, meaning a trivial cylinder over a contact mani-fold. The main difference between this and the case of Liouville domains is that we have a nonempty concave end. It is known that additional assumptions are necessary in order to defineFloer complexes in this setting (Lagrangian cobordisms with loose negative ends are known tosatisfy some flexibility results). For that reason, we impose some restriction on the Lagrangians.More precisely, we consider only exact Lagrangian cobordisms with negative cylindrical endsover Legendrian submanifolds whose CE-algebra admit an augmentation. In particular, an exactLagrangian filling implies the existence of an augmentation [EHK16].Generalizing the structures we define in this paper to the case of Lagrangians in a moregeneral Liouville cobordism should be possible using the latest techniques of virtual perturba-tions in [Par19] or the polyfold technology developed in [HWZ17]. Note that Cieliebak-Oanceain [CO18] have defined a version of Rabinowitz Floer homology for Lagrangians in a Liouvillecobordism under the assumption that this cobordism admits a filling. The Floer complex wedefine in this paper are similar to the ones defined by Cieliebak-Oancea; for instance, there is anidentification on the level of generators. The main difference is that their differential is definedby Floer strips with a Hamiltonian perturbation term that corresponds to wrapping, while thedifferential considered here is defined in terms of honest SFT-type pseudo-holomorphic discs. Itis expected that the two theories give quasi-isomorphic complexes.We start by contrasting the complex considered here with the Floer-type complex for pairsof Lagrangian cobordisms considered in [CDRGG20]. Namely, given a pair of transverse ex-act Lagrangian cobordisms (Σ , Σ ) where Σ i has positive and negative cylindrical ends overLegendrians Λ + i and Λ − i respectively and Λ − i are equipped with augmentations, the authors in[CDRGG20] define the Floer complex (Cth(Σ , Σ ) , d ) whose underlying vector space is given by Cth(Σ , Σ ) = C (Λ +0 , Λ +1 ) ⊕ CF (Σ , Σ ) ⊕ C (Λ − , Λ − ) C (Λ ± , Λ ± ) is generated by Reeb chords from Λ ± to Λ ± and CF (Σ , Σ ) is generated byintersection points in Σ ∩ Σ . This complex is actually the cone of a map f : CF −∞ (Σ , Σ ) := CF (Σ , Σ ) ⊕ C (Λ − , Λ − ) → C (Λ +0 , Λ +1 ) In [Leg] the author defined a product structure on CF −∞ (Σ , Σ ) , as well as higher order mapssatisfying the A ∞ -equations. Moreover, in the same paper it is proved that the map f generalizesto a family of maps { f d } d ≥ , f d : Cth(Σ d − , Σ d ) ⊗ · · · ⊗ Cth(Σ , Σ ) → C (Λ +0 , Λ + d ) defined for a ( d + 1) -tuple of pairwise transverse exact Lagrangian cobordisms (Σ , . . . , Σ d ) ,and satisfying the A ∞ -functor equations; where the A ∞ -structure maps on C (Λ + d − , Λ + d ) ⊗ · · · ⊗ C (Λ +0 , Λ +1 ) are given by the structure maps of the augmentation category A ug − (Λ +0 ∪· · ·∪ Λ + d ) , see[BC14]. However, there exists no non trivial A ∞ -structure on the whole complex Cth(Σ , Σ ) for example for degree reasons: the grading of Reeb chord generators in the positive end in Cth(Σ , Σ ) is given by the Conley-Zehnder index plus (see Subsections 2.5 and 2.8) so acount of rigid pseudo-holomorphic discs with boundary on the positive cylindrical ends, with twonegative Reeb chord asymptotics and one positive Reeb chord asymptotic would not provide adegree order map for example.In this article, we use similar techniques for constructing a version of the Rabinowitz complex (Cth + (Σ , Σ ) , m ) , on which it will be possible to define higher order structure maps. Theunderlying vector space is: Cth + (Σ , Σ ) = C (Λ +1 , Λ +0 ) ⊕ CF (Σ , Σ ) ⊕ C (Λ − , Λ − ) so the only difference with Cth(Σ , Σ ) is the generators we consider in the positive end; unlikein the complex Cth(Σ , Σ ) these generators consist of the chords which start at Λ +0 and end at Λ +1 . The differential is defined by a count of pseudo-holomorphic discs with boundary on thecobordisms and asymptotic to Reeb chords and intersection points such that• C (Λ − , Λ − ) is a subcomplex which is the linearized Legendrian contact cohomology complexof Λ − ∪ Λ − restricted to chords from Λ − to Λ − ,• C (Λ +1 , Λ +0 ) is a quotient complex which is the linearized Legendrian contact homologycomplex of Λ +0 ∪ Λ +1 restricted to chords from Λ +0 to Λ +1 .In the case Σ = R × Λ and Σ is a cylinder over a perturbed copy of Λ translated far in thepositive Reeb direction, the complex Cth + (Σ , Σ ) is the complex of the -copy of Λ consideredin [EES09].Then we proceed to investigate the properties of this complex. Some of them resemble prop-erties satisfied by the complex Cth(Σ , Σ ) but there are also some significant differences. Acyclicity:
Contrary to
Cth(Σ , Σ ) , the complex Cth + (Σ , Σ ) is not always acyclic. For ex-ample if Y = J M for a closed manifold M , Σ is a cylinder over the -section in J M and Σ a cylinder over a Morse perturbation of the -section, the homology of Cth + (Σ , Σ ) does notvanish but equals instead the Morse homology of M . However, in the case Y = P × R whereany compact subset of P is Hamiltonian displaceable, for example Y = R n +1 , the complex Cth + (Σ , Σ ) is always acyclic. It is also always acyclic whenever Λ − = Λ − = ∅ as in this casethe complex is actually the same as the dual complex of Cth(Σ , Σ ) .3 tructure maps and continuation element: The new Cthulhu complex carries structuremaps which satisfy the A ∞ -equations. More precisely, for any ( d + 1) -tuple (Σ , . . . , Σ d ) ofpairwise transverse exact Lagrangian cobordisms, we define a map m d : Cth + (Σ d − , Σ d ) ⊗ · · · ⊗ Cth + (Σ , Σ ) → Cth + (Σ , Σ d ) by counts of SFT-buildings consisting of pseudo-holomorphic discs with boundary on the Σ i ’sand asymptotic to Reeb chords and intersection points. Then, for any ≤ k ≤ d and sub-tuple (Σ i , . . . , Σ i k ) with ≤ i < · · · < i k ≤ d , one has k X j =1 k − j X n =0 m k − j +1 (cid:0) id ⊗ k − j +1 ⊗ m j ⊗ id ⊗ n (cid:1) = 0 (1)where the inner m j has domain Cth + (Σ i n + j − , Σ i n + j ) ⊗ · · · ⊗ Cth + (Σ i n , Σ i n +1 ) and m k − j +1 hasdomain Cth + (Σ i k − , Σ i k ) ⊗ · · · ⊗ Cth + (Σ i n , Σ i n + j ) ⊗ · · · ⊗ Cth + (Σ i , Σ i ) .In the case when Σ is a suitable small Hamiltonian perturbation of Σ one establishes theexistence of a continuation element in Cth + (Σ , Σ ) (see Section 7 for a precise description ofthe Hamiltonian perturbation Σ ): Theorem 1.
There exists an element e Σ , Σ ∈ Cth + (Σ , Σ ) satisfying that for any exact La-grangian cobordism Σ transverse to Σ and Σ the map m ( · , e Σ , Σ ) : Cth + (Σ , Σ ) → Cth + (Σ , Σ ) is a quasi-isomorphism. Finally we use these ingredients to construct a unital A ∞ -category F uk( R × Y ) via localiza-tion, in the same spirit as the construction of the wrapped Fukaya category of Liouville sectorsin [GPS20]: Theorem 2.
There exists a unital A ∞ -category F uk( R × Y ) whose objects are Lagrangiancobordisms equipped with augmentations of its negative ends and whose morphism spaces in thecohomological category satisfy H ∗ hom F uk( R × Y ) (Σ , Σ ) ∼ = H ∗ (Cth + (Σ , Σ ) , m ) whenever Σ and Σ are transverse. The homology of the Rabinowitz complex
Cth + (Σ , Σ ) is invariant under cylindrical at in-finity Hamiltonian isotopies (in particular under Legendrian isotopies of its ends). This impliesthat the quasi-equivalence class of the category F uk( R × Y ) does not depend on choices of repre-sentatives of Hamiltonian isotopy classes of Lagrangian cobordisms involved in its constructionby localization (see Section 8.2). Behaviour under concatenation:
Given a pair of concatenated cobordisms ( V ⊙ W , V ⊙ W ) ,we describe the complex Cth + ( V ⊙ W , V ⊙ W ) in terms of the complexes Cth + ( V , V ) and Cth + ( W , W ) and some transfer maps fitting into a diagram Cth + ( V , V ) ∆ W ←−−− Cth + ( V ⊙ W , V ⊙ W ) b V −−→ Cth + ( W , W ) We prove that ∆ W and b V are chain maps which induce a Mayer-Vietoris sequence and moreoverpreserve the continuation element in homology.4n addition, the transfer maps generalize also to families of maps { ∆ d } d ≥ and { b d } d ≥ satisfying the A ∞ -functor equations. That is to say, for a ( d +1) -tuple of concatenated cobordisms ( V ⊙ W , . . . , V d ⊙ W d ) there are maps m V ⊙ Wd : Cth + ( V d − ⊙ W d − , V d ⊙ W d ) ⊗ · · · ⊗ Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V ⊙ W , V d ⊙ W d ) ∆ Wd : Cth + ( V d − ⊙ W d − , V d ⊙ W d ) ⊗ · · · ⊗ Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V , V d ) b Vd : Cth + ( V d − ⊙ W d − , V d ⊙ W d ) ⊗ · · · ⊗ Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( W , W d ) such that for all ≤ k ≤ d and sub-tuple ( V i ⊙ W i , . . . , V i k ⊙ W i k ) with ≤ i < · · · < i k ≤ d , themaps { m V ⊙ Wk } ≤ k ≤ d satisfy the A ∞ -equations (1), and the maps { ∆ Wk } ≤ k ≤ d and { b Vk } ≤ k ≤ d satisfy k X s =1 X j + ··· + j s = k m Vs (cid:0) ∆ Wj s ⊗ · · · ⊗ ∆ Wj (cid:1) + k X j =1 j X n =0 ∆ Wk − j +1 (cid:0) id ⊗ k − j +1 ⊗ m V ⊙ Wj ⊗ id ⊗ n (cid:1) = 0 k X s =1 X j + ··· + j s = k m Ws (cid:0) b Vj s ⊗ · · · ⊗ b Vj (cid:1) + k X j =1 j X n =0 b Vk − j +1 (cid:0) id ⊗ k − j +1 ⊗ m V ⊙ Wj ⊗ id ⊗ n (cid:1) = 0 . Acknowledgements
The author warmly thanks Baptiste Chantraine, Georgios Dimitroglou-Rizell and Paolo Ghiggini for helpful discussions and comments on earlier versions of the paper,as well as Alexandru Oancea for stimulating discussions. The author was partly supported bythe grant KAW 2016.0198 from the Knut and Alice Wallenberg Foundation and the SwedishResearch Council under the grant no. 2016-03338.
Throughout the paper we will be working with a contact manifold ( Y, α ) given by the contacti-zation of a Liouville manifold . We briefly recall the definition of these terms. A
Liouville domain ( b P , θ ) is the data of a n -dimensional manifold with boundary b P as well as a -form θ on b P suchthat dθ is symplectic, and the Liouville vector field V defined by ι V dθ = θ is required to pointoutward on the boundary ∂ b P . In particular, θ | ∂ b P is a contact form on ∂ b P . The completion of ( b P , θ ) is the exact symplectic manifold ( P = b P ∪ ∂ b P [0 , ∞ ) × ∂ b P , ω = dβ ) , where β equals θ in b P and e τ θ | ∂ b P on [0 , ∞ ) × ∂ b P where τ is the coordinate on [0 , ∞ ) . The Liouville vector fieldsmoothly extends to the whole manifold P . We call ( P, β ) a Liouville manifold .The contactization of a Liouville manifold ( P, β ) is the contact manifold ( Y, α ) where Y isthe n + 1 -dimensional manifold Y = P × R and α = dz + β , where z is the R -coordinate. TheReeb vector field of α is given by R α = ∂ z so in particular there are no closed Reeb orbits in Y . A Legendrian submanifold of ( Y, α ) is a n -dimensional submanifold Λ satisfying α | T Λ = 0 ,and Reeb chords of Λ are trajectories of the Reeb flow starting and ending on Λ . We consideronly Legendrian with a finite number of isolated Reeb chords, and denote R (Λ) the set of Reebchords of Λ . These are called pure Reeb chords. Given two Legendrian Λ and Λ , we denote R (Λ , Λ ) the set of Reeb chords starting on Λ and ending on Λ , these are called mixed Reebchords.The main objects under consideration in this article are exact Lagrangian cobordisms betweenLegendrian submanifolds of Y . These are Lagrangian submanifolds in the symplectization of ( Y, α ) which is the symplectic manifold ( R × Y, d ( e t α )) where t is the R -coordinate.5 efinition 1. Given Λ − , Λ + ⊂ Y Legendrian, an exact Lagrangian cobordism from Λ − to Λ + is a submanifold Σ ⊂ R × Y such that there exists• T > such that1. Σ ∩ [ T, ∞ ) × Y = [ T, ∞ ) × Λ + ,2. Σ ∩ ( −∞ , − T ] × Y = ( −∞ , − T ] × Λ − ,3. Σ ∩ [ − T, T ] × Y is compact.• f : Σ → R a smooth function called a primitive of Σ , satisfying1. e t α | T Σ = df ,2. f is constant on [ T, ∞ ) × Λ + and ( −∞ , − T ] × Λ − .In all the paper, we will assume that the coefficient field is Z . Moreover, we assume that c ( P ) = 0 , and that the Legendrian submanifolds and Lagrangian cobordisms between themhave Maslov number . This will ensure a well-defined Z grading for the various complexes thatwill appear. Given a family of pairwise transverse Lagrangian cobordisms (Σ , . . . , Σ d ) with Legendrian cylin-drical ends R × Λ ± i , ≤ i ≤ d , we consider several types of moduli spaces of pseudo-holomorphicdiscs with boundary on those Lagrangian cobordisms. Those discs are asymptotic to intersectionpoints and/or Reeb chords of the links Λ ± ∪ · · · ∪ Λ ± d . First, let us describe briefly the almostcomplex structure we consider on R × Y , in order to define the moduli spaces mentioned aboveand achieve transversality.An almost complex structure J on ( R × Y, d ( e t α )) is called cylindrical if• it is compatible with d ( e t α ) ,• J ( ∂ t ) = R α ,• J ( ξ ) = ξ ,• J is invariant by translation along the t -coordinate axis.We denote J cyl ( R × Y ) the set of cylindrical almost complex structures on R × Y . An almostcomplex structure on P is called admissible if it is cylindrical on P \ b P outside of a compact set.The cylindrical lift of an admissible almost complex structure J P on P is the unique cylindricalalmost complex structure e J P on R × ( P × R ) making the projection π P : R × ( P × R ) → P holomorphic.Let J + and J − be two cylindrical almost complex structures which coincide outside of R × K for some compact K ⊂ Y . Assuming that the cobordisms we consider are all cylindrical outsideof [ − T, T ] × Y for some fixed T > , we take an almost complex structure J which is equalto J − on ( −∞ , − T ) × Y , to J + on ( T, + ∞ ) × Y , and to the cylindrical lift of an admissiblecomplex structure J P on [ − T, T ] × ( Y \ K ) . We denote J J + ,J − ( R × Y ) this class of almost complexstructures on R × Y .In order to achieve transversality for the moduli spaces later on, we will finally need domaindependent almost complex structures with values in J J + ,J − ( R × Y ) , i.e. families of almostcomplex structures in J J + ,J − ( R × Y ) parametrized by the domains of the pseudo-holomorphicdiscs (punctured Riemann discs), which is part of a universal choice of perturbation data , see[Sei08, Section (9h)]. 6 .3 Moduli spaces of curves with boundary on Lagrangian cobordisms Let R d +1 be the space of d + 1 cyclically ordered points y = ( y , . . . , y d ) ∈ ( S ) d +1 quotientedby the automorphisms of the unit disc D . This is the Deligne-Mumford space. For y ∈ R , letus denote S y = D \{ y , . . . , y d } . In a sufficiently small neighborhood of the punctures y i in thedisc, we have strip-like end coordinates [ s i , t i ] ∈ (0 , + ∞ ) × [0 , , ≤ i ≤ d .Let us denote Σ ...d = (Σ , . . . , Σ d ) a d + 1 -tuple of Lagrangian cobordisms satisfying thefollowing:• if Σ = Σ d , then Σ i = Σ for all ≤ i ≤ d ,• if Σ = Σ d , then the ordered family Σ , . . . , Σ d is of the form Σ i , . . . , Σ i , Σ i , . . . , Σ i , Σ i , . . . , Σ i k , with Σ i := Σ and Σ i k := Σ d , and such that Σ i , Σ i , . . . , Σ i d are pairwise transverse. Inother words, we allow only consecutive repetition of a given Lagrangian.The set of asymptotics A (Σ i − , Σ i ) associated to the pair (Σ i − , Σ i ) consists of Reeb chords in R (Λ ± i − ∪ Λ ± i ) , and intersection points in Σ i − ∩ Σ i when the cobordisms are transverse. Considera d + 1 -tuple of asymptotics ( a , . . . , a d ) , with a i ∈ A (Σ i − , Σ i ) , Σ − := Σ d and Λ ±− := Λ ± d . If Σ i − = Σ i , then a i is called a pure asymptotic , and it is a pure Reeb chord of Λ ± i − = Λ ± i , whileif Σ i − = Σ i , then a i is called a mixed asymptotic . Given J an almost complex structure on R × Y , we denote M Σ ,...,d ,J ( a ; a , . . . , a d ) the set of pairs ( y , u ) where1. y ∈ R d +1 ,2. u : ( S y , j ) → ( R × Y, J ) is a pseudo-holomorphic map (with j standard a.c.s. on D ),3. u maps the boundary of S r contained between y i and y i +1 for ≤ i ≤ d ( y d +1 := y ) to Σ i ,4. lim z → y i u ( z ) = a i .Let us specify the condition (4) in the case a i is a Reeb chord, for which we also denote a i :[0 , → Y a parametrization. We say that• u has a positive asymptotic to a i at y i if lim s i → + ∞ u ( s i , t i ) = a i ( t i ) ,• u has a negative asymptotic to a i at y i if lim s i → + ∞ u ( s i , t i ) = a i (1 − t i ) . Remark 1.
Note that the fact that a mixed Reeb chord asymptotic is a positive or a negativeasymptotic is entirely determined by the “jump” of the chord. Namely, positive mixed Reebchord asymptotics are mixed chords of Λ + i ∪ Λ + i +1 from Λ + i to Λ + i +1 , and negative mixed Reebchord asymptotics are mixed Reeb chords of Λ − i ∪ Λ − i +1 from Λ − i +1 to Λ − i . Notations 1.
From now on, we denote the Lagrangian boundary condition for discs only by thefamily (Σ i , Σ i , . . . , Σ i k ) , even though the pseudo-holomorphic discs we will consider can havepure Reeb chords asymptotics too.In the following two subsections we describe the several types of moduli spaces we will makeuse of later. 7 .3.1 Moduli spaces of curves with cylindrical boundary conditions The Lagrangian boundary conditions we consider here are trivial cylinders over Legendrians, andwe take an almost complex structure J ∈ J cyl ( R × Y ) . If the boundary conditions consists ofonly one cylinder R × Λ then we denote M R × Λ ,J ( γ + ; γ , . . . , γ d ) the moduli space of discs with boundary on R × Λ , with a positive asymptotic to γ + andnegative asymptotics at γ i for ≤ i ≤ d . We call discs in such moduli spaces pure , as allasymptotics are pure. In case the Lagrangian conditions is a family of distinct transverse cylinders R × Λ ...d := ( R × Λ , . . . , R × Λ d ) with d > , we consider the:1. Banana-type moduli spaces: M R × Λ ...d ,J ( γ d, ; δ , γ , δ , . . . , γ d , δ d ) where γ d, ∈ R (Λ , Λ d ) is a mixed Reeb chord from Λ d to Λ , γ i ∈ R (Λ i − , Λ i ) ∪R (Λ i , Λ i − ) are mixed chords of Λ i − ∪ Λ i and δ i are words of Reeb chords of Λ i and are negativeasymptotics. Note that according to Remark 1, γ d, is a positive Reeb chord asymptoticand then γ i is a positive asymptotic if it is in R (Λ i , Λ i − ) and a negative one if it is in R (Λ i − , Λ i ) .2. ∆ -type moduli spaces: M R × Λ ...d ,J ( γ ,d ; δ , γ , δ , . . . , γ d , δ d ) where γ ,d ∈ R (Λ d , Λ ) is a negative Reeb chord asymptotic and with the same conditionas above on asymptotics γ i and δ i .The discs in moduli spaces of type (1) and (2) are called mixed as d + 1 asymptotics are mixed.There is a R -action by translation on moduli spaces with cylindrical Lagrangian boundary con-dition, we use the notation f M to denote the quotient of the moduli space M by R . The Lagrangian boundary conditions consist of Lagrangian cobordisms (Σ , . . . , Σ d ) such thatat least one is not a trivial cylinder. Denote J a domain dependent almost complex structurewith values in J J + ,J − ( R × Y ) . If d = 0 , Σ := Σ is a non trivial cobordism from Λ − to Λ + andwe denote M Σ , J ( γ + ; γ , . . . , γ d ) the moduli space of discs where γ ∈ R (Λ + ) is a positive Reeb chord asymptotic and γ i ∈ R (Λ − ) are negative Reeb chord asymptotics. We call again those discs pure . If the Lagrangian boundarycondition consists of several distinct Lagrangians Σ ...d = (Σ , . . . , Σ d ) where d > and Σ i is acobordism from Λ − i to Λ + i , then we consider the following mixed moduli spaces:1. Banana-type moduli space: M Σ ...d , J ( γ d, ; δ , a , δ , . . . , a d , δ d ) ,2. m -type moduli space: M Σ ...d , J ( x ; δ , a , δ , . . . , a d , δ d ) ,3. ∆ -type moduli space M Σ ...d , J ( γ ,d ; δ , a , δ , . . . , a d , δ d ) ,where γ d, ∈ R (Λ +0 , Λ + d ) is a positive Reeb chord asymptotic, γ ,d ∈ R (Λ − d , Λ − ) is a negativeReeb chord asymptotic, a i are intersection points in Σ i − ∩ Σ i or mixed Reeb chord asymptoticsin R (Λ + i , Λ + i − ) ∪ R (Λ − i − , Λ − i ) , and δ i are words of pure Reeb chords of Λ − i .8 .4 Action and energy Consider a d + 1 -tuple of pairwise disjoint cobordisms (Σ , . . . , Σ d ) with cylindrical ends over Λ ± i . Let T > and ε > such that all cobordisms are cylindrical outside of [ − T + ε, T − ε ] × Y .The length of a chord γ is defined by ℓ ( γ ) := R γ α . Then, the action of asymptotics is defined asfollows: a ( γ ) = e T ℓ ( γ ) + c j − c i for γ ∈ R (Λ + i , Λ + j ) , a ( x ) = f j ( x ) − f i ( x ) for x ∈ Σ i ∩ Σ j , i < j, a ( γ ) = e − T ℓ ( γ ) for γ ∈ R (Λ − i , Λ − j ) , a ( γ ) = e ± T ℓ ( γ ) for γ ∈ R (Λ ± ) . Given a function χ : R → R satisfying for some ε > χ ( t ) = e T for t ≥ Te t for − T + ε ≤ t ≤ T − εe − T for t ≤ − T and χ ′ ( t ) ≥ , one defines the energy of a pseudo-holomorphic disc u to be: E ( u ) = Z u d ( χ ( t ) α ) This energy is always positive and vanishes if and only if the disc is constant. The energy of thepseudo-holomorphic discs considered in this paper is finite and can be expressed in terms of theaction of the asymptotics.
Proposition 1.
For the moduli spaces described in Sections 2.3.1 and 2.3.2, we have the follow-ing:1. If u ∈ M R × Λ ( γ + ; γ , . . . , γ d ) , then E ( u ) = a ( γ + ) − P i a ( γ i ) .2. Assume { γ , . . . , γ d } = { γ +1 , . . . , γ + j + , γ − , . . . , γ − j − } where γ + i are positive Reeb chord asymp-totics and γ − i are negative Reeb chord asymptotics, then(a) if u ∈ M R × Λ ...d ( γ d, ; δ , γ , δ , . . . , γ d , δ d ) , E ( u ) = a ( γ d, ) + j + X i =1 a ( γ + i ) − j − X i =1 a ( γ − i ) − d X i =0 a ( δ i ) , (b) if u ∈ M R × Λ ...d ( γ ,d ; δ , γ , δ , . . . , γ d , δ d ) , E ( u ) = − a ( γ ,d ) + j + X i =1 a ( γ + i ) − j − X i =1 a ( γ − i ) − d X i =0 a ( δ i ) .
3. If u ∈ M Σ ( γ + ; γ , . . . , γ d ) , then E ( u ) = a ( γ + ) − P i a ( γ i ) .4. Assume { a , . . . , a d } = { γ +1 , . . . , γ + j + , γ − , . . . , γ − j − , q , . . . , q l } , where γ + i are positive mixedReeb chords of S Λ + i , γ − i are negative mixed Reeb chords of S Λ − i , p i are intersection pointsasymptotics, then a) if u ∈ M Σ ...d ( γ d, ; δ , a , δ , . . . , a d , δ d ) , E ( u ) = a ( γ d, ) + j + X i =1 a ( γ + i ) − j − X i =1 a ( γ − i ) − l X i =1 a ( q i ) − d X i =0 a ( δ i ) , (b) if u ∈ M Σ ...d ( x ; δ , a , δ , . . . , a d , δ d ) , E ( u ) = a ( x ) + j + X i =1 a ( γ + i ) − j − X i =1 a ( γ − i ) − l X i =1 a ( q i ) − d X i =0 a ( δ i ) , (c) if u ∈ M Σ ...d ( γ ,d ; δ , a , δ , . . . , a d , δ d ) , E ( u ) = − a ( γ ,d ) + j + X i =1 a ( γ + i ) − j − X i =1 a ( γ − i ) − l X i =1 a ( q i ) − d X i =0 a ( δ i ) , Given cobordisms (Σ , . . . , Σ d ) as above, we associate a grading to the asymptotics of pseudo-holomorphic discs with boundary on Σ ...d using the Conley-Zehnder index. We refer for exampleto [EES05] for the definition of this index.1. Grading of Reeb chords:
Consider Λ ⊂ Y a connected Legendrian, then the grading ofa Reeb chord γ ∈ R (Λ) is defined to be | γ | = CZ( γ ) − where CZ( γ ) denotes the Conley-Zehnder index of a capping path for γ . Note that itdoes not depend on the choice of capping path as by hypothesis we consider Maslov Legendrians, and it does not depend neither on a choice of symplectic trivialization of
T P along the capping path, as c ( P ) = 0 . If the Legendrian Λ is not connected, there areno capping paths for Reeb chords connecting two distinct components so some additionalchoices are needed (see [DR16]). If γ is a chord from Λ j to Λ i , one fixes points p j ∈ Λ j and p i ∈ Λ i and a path Γ i,j from p i to p j as well as a path of Lagrangians from T p i π P (Λ i ) to T p j π P (Λ j ) . Then, one takes as capping path for γ a path from the ending point of γ to p i , followed by Γ i,j , followed by a path from p j to the starting point of γ . The grading ofmixed chords depends on those additional paths but the difference in grading of two chordsdoes not.2. Grading of intersection points:
Let p ∈ Σ i ∩ Σ j , for i < j . Generically, the immersedLagrangian Σ i ∪ Σ j lifts to an embedded Legendrian submanifold e Σ i ∪ e Σ j ⊂ (cid:0) ( R × Y ) × R u , du + e t α (cid:1) and p is the projection of a Reeb chord γ p of e Σ i ∪ e Σ j . If γ p is a chord from e Σ j to e Σ i , then | p | = CZ( γ p ) . If γ p is a chord from e Σ i to e Σ j , then | p | = n + 1 − CZ( γ p ) .These Conley-Zehnder indices are computed after a choice of path connecting e Σ i and e Σ j as explained above for the non-connected case. Again, the vanishing of the Maslov numberfor Lagrangian cobordisms, and of the first Chern class of P imply that the grading ofintersection points does not depend on the choices made, except paths to connect any twodistinct components of the Legendrian lift.10he expected dimension of the moduli spaces described in Sections 2.3.1 and 2.3.2 can thenbe expressed in terms of the grading of asymptotics. Consider the set of asymptotics ( a , . . . , a d ) ,and assume again that it decomposes as follows { a , . . . , a d } = { γ + j } ≤ j ≤ j + ∪ { q j } ≤ j ≤ l ∪{ γ − j } ≤ j ≤ j − where γ + j are positive Reeb chord asymptotics, q j are intersection points asymp-totics and γ − j are negative Reeb chord asymptotics. For cylindrical boundary conditions, wehave obviously l = 0 . Moreover, we assume that negative asymptotics to pure Reeb chords δ i have degree . Proposition 2.
Under the decomposition of the set of asymptotics as above, we have: dim f M R × Λ ( γ + ; γ , . . . , γ d ) = | γ + | − X | γ i | − , dim f M R × Λ ...d ( γ d, ; δ , γ , δ , . . . , γ d , δ d ) = | γ d, | + X | γ + j | − X | γ − j | + (2 − n ) j + − , dim f M R × Λ ...d ( γ ,d ; δ , γ , δ , . . . , γ d , δ d ) = −| γ ,d | + X | γ + j | − X | γ − j | + (2 − n )( j + − − , dim M Σ ( γ + ; γ , . . . , γ d ) = | γ + | − X | γ i | , dim M Σ ...d ( γ d, ; δ , a , δ , . . . , a d , δ d ) = | γ d, | + X | γ + j | − X | q j | − X | γ − j | + (2 − n ) j + + l, dim M Σ ...d ( x ; δ , a , δ , . . . , a d , δ d ) = | x | + X | γ + j | − X | q j | − X | γ − j | + (2 − n ) j + + l − , dim M Σ ...d ( γ ,d ; δ , a , δ , . . . , a d , δ d ) , = −| γ ,d | + X | γ + j | − X | q j | − X | γ − j | + n + (2 − n ) j + + l − . Notations 2.
Given a moduli space M Σ ...d ( a ; a , . . . , a d ) , we add an exponent indicating the(expected) dimension of it as a smooth manifold: M i Σ ...d ( a ; a , . . . , a d ) . This dimension is equalto the index of the Fredholm operator obtained by linearizing ¯ ∂ at a pseudo-holomorphic disc u . This section sums up what will be used to prove almost all the results in this paper. Namely,once transversality is achieved simultaneously for all moduli spaces considered above, which ispossible using a domain dependent almost complex structure, -dimensional (eventually afterquotienting by an R -action) moduli spaces are compact manifolds. We call discs in these modulispaces rigid discs . Then, -dimensional moduli spaces are not necessarily compact and can becompactified by adding broken discs . The goal of this section is to describe the types of brokendiscs one can find in the boundary of the compactification of the moduli spaces in Sections 2.3.1and 2.3.2.Consider a -dimensional moduli space M L ...d ( a ; δ , a , δ , . . . , δ d − , a d , δ d ) of curves where L ...d = R × Λ ...d or Σ ...d (in the first case M L ...d = g M L ...d ), with mixed asymptotics a i and asymptotics to words of pure Reeb chords δ i . By results in [BEH + pseudo-holomorphic building (see the cited references for a precise definition) consisting of several pseudo-holomorphic discs together with nodes connecting these components and choices of asymptoticsfor these nodes, satisfying the following:• each disc in the pseudo-holomorphic building has positive energy, so in particular a com-ponent with only Reeb chords asymptotics must have at least one positive Reeb chordasymptotic,• each disc has a non negative Fredholm index because of the regularity of the almost complexstructure, 11 if the building consists of the discs u , . . . , u k , then the glued solution u has index ind( u ) = ν + P i ind( u i ) , where ν is the number of nodes asymptotic to intersection points.Let us now precise what these conditions imply in particular in the case of moduli spacesdescribed in the previous sections.1. Cylindrical boundary condition:
Consider a -dimensional moduli space g M R × Λ ...d ( γ ; δ , γ , δ , . . . , δ d − , γ d , δ d ) The con-ditions described above imply that a sequence of discs in this moduli space admits a sub-sequence which limits to a pseudo-holomorphic building consisting two index pseudo-holomorphic discs with boundary on cylinders, and which glue together along a nodeasymptotic to a Reeb chord. Remark that this Reeb chord can be pure or mixed, wewill later be interested only in the case of nodes asymptotic to mixed Reeb chords, seeRemark 3.2. Non-cylindrical boundary conditions:
Consider a -dimensional moduli space M ...d ( a ; δ , a , δ , . . . , δ d − , a d , δ d ) of curveswith boundary on the cobordisms Σ , . . . , Σ d , with mixed asymptotics a i and asymptoticsto words of pure Reeb chords δ i . A sequence of discs in such a moduli space admits asubsequence converging to a pseudo-holomorphic building which is(a) either a pseudo-holomorphic building with two index components (which are nottrivial strips) with boundary on non cylindrical parts of the cobordisms and connectedby node asymptotic to an intersection point, or(b) a pseudo-holomorphic building consisting of some index components with boundaryon the non cylindrical parts of the cobordisms, and one index component withboundary on the positive or negative cylindrical ends of the cobordisms, connected toeach index component via a node asymptotic to a Reeb chord. Notations 3.
We denote ∂ M R × Λ ...d := ∂ g M R × Λ ...d , and ∂ M ...d the set of pseudo-holomorphic buildings arising at the boundary of the compactification of the correspondingmoduli spaces. Consider a compact Legendrian submanifold Λ ⊂ Y . We denote by C (Λ) the Z -vector spacegenerated by Reeb chords of Λ . The Legendrian contact homology of Λ is an invariant of Λ up to Legendrian isotopy, introduced by Chekanov in [Che02] and Eliashberg [Eli98] (see also[EES05, EES07]). It is the homology of a differential graded algebra (DGA) ( A (Λ) , ∂ ) associatedto Λ . The algebra A (Λ) , called the Chekanov-Eliashberg algebra of Λ , is the unital tensor algebraof C (Λ) , i.e. A (Λ) = L i ≥ C (Λ) ⊗ i with C (Λ) ⊗ := Z . The grading of Reeb chords is as definedin Section 2.5. Given a cylindrical almost complex structure, the differential ∂ is defined asfollows. For γ ∈ R (Λ) we have ∂γ = X d ≥ X γ i ∈R (Λ) g M R × Λ ( γ + ; γ , . . . , γ d ) γ γ . . . γ d The differential ∂ extends to the whole algebra by Leibniz rule, and satisfies ∂ = 0 . Weconsider the SFT definition of the differential here, which has been proved in [DR16] to givethe same invariant as the original version with a differential defined by a count of discs in P π P (Λ) . For a generic cylindrical almost complex structure, the moduli spaces g M R × Λ ( γ ; γ , . . . , γ d ) are compact -dimensional manifolds. The Legendrian contact homologyof Λ , denoted LCH ∗ (Λ) , does not depend on a generic choice of almost complex structure andis invariant under Legendrian isotopy.Consider now an almost complex structure J ∈ J J + ,J − ( R × Y ) . It is proved in [EHK16] thatan exact Lagrangian cobordism Σ from Λ − to Λ + induces a DGA-map Φ Σ : A (Λ + ) → A (Λ − ) ,defined by Φ Σ ( γ ) = X d ≥ X γ i ∈R (Λ) M ( γ + ; γ , . . . , γ d ) γ γ . . . γ d When Σ is a Lagrangian filling of Λ , i.e. Λ − = ∅ , then Φ Σ is a map from A (Λ) to A ( ∅ ) := Z .It is an instance of an augmentation of A (Λ) . More generally, we have the following definition. Definition 2.
An augmentation of ( A (Λ) , ∂ ) over Z is a map ε : A (Λ) → Z satisfying• ε ◦ ∂ = 0 • ε ( γ ) = 0 if | γ | 6 = 0 ,• ε (1) = 1 ,• ε ( γ γ ) = ε ( γ ) ε ( γ ) ,In other words, it is a unital DGA-map when considering Z as a DGA with a vanishing differ-ential.Chekanov made use of augmentations to linearize the DGA ( A (Λ) , ∂ ) , leading to finite di-mensional invariants called linearized Legendrian contact homologies . Bourgeois and Chantraine[BC14] generalized this idea using two augmentations instead of one: assume A (Λ) admits aug-mentations ε , ε , then there is a complex ( LCC ε ,ε ∗ (Λ) , ∂ ε ,ε ) where LCC ε ,ε ∗ (Λ) := C (Λ) andfor a Reeb chord γ , ∂ ε ,ε ( γ ) = X d ≥ X γ ,...,γ d ∈R (Λ) d X i =1 g M R × Λ ( γ + ; γ , . . . , γ d ) ε ( γ . . . γ i − ) ε ( γ i +1 . . . γ d ) γ i This map satisfies ( ∂ ε ,ε ) = 0 . The Legendrian contact homology of Λ bilinearized by ( ε , ε ) is the homology of this complex, denoted LCH ε ,ε ∗ (Λ) . The dual complex ( LCC ∗ ε ,ε (Λ) , µ ε ,ε ) is the complex of the bilinearized Legendrian contact cohomology of Λ , LCH ∗ ε ,ε (Λ) . For Reebchords γ, β ∈ R (Λ) , if we denote by h ∂ ε ,ε ( β ) , γ i the coefficient of γ in ∂ ε ,ε ( β ) , then we have µ ε ,ε ( γ ) = X β ∈R (Λ) h ∂ ε ,ε ( β ) , γ i β When ε = ε , these complexes correspond to the linearized Legendrian contact (co)homologycomplexes defined by Chekanov. Finally, given an augmentation ε − of A (Λ − ) and an exactLagrangian cobordism Λ − ≺ Σ Λ + , it induces an augmentation ε + := ε − ◦ Φ Σ of A (Λ + ) . Cth
The Cthulhu homology is the homology of a Floer-type complex defined in [CDRGG20] for a pair Λ − ≺ Σ Λ +0 and Λ − ≺ Σ Λ +1 of transverse exact Lagrangian cobordisms in R × Y such that the13lgebras A (Λ − ) and A (Λ − ) admit augmentations ε − and ε − respectively. The Cthulhu complex (cid:0) Cth(Σ , Σ ) , d ε − ,ε − (cid:1) has three types of generators, Cth(Σ , Σ ) = C (Λ +0 , Λ +1 )[2] ⊕ CF (Σ , Σ ) ⊕ C (Λ − , Λ − )[1] where C (Λ +0 , Λ +1 )[2] denotes the Z -vector space generated by Reeb chords from Λ +1 to Λ +0 witha grading shift, namely if γ ∈ C (Λ +0 , Λ +1 )[2] then | γ | Cth(Σ , Σ ) = | γ | + 2 , CF (Σ , Σ ) is the Z -vector space generated by intersection points in Σ ∩ Σ , and C (Λ − , Λ − ) is generated by Reebchords from Λ − to Λ − . The differential is given by the matrix d ε − ,ε − = d ++ d +0 d + − d d − d − d −− It is a degree map defined by a count of rigid pseudo-holomorphic discs with boundary on thecobordisms, as schematized on Figure 1. The study of broken discs arising at the boundary of thecompactification of -dimensional moduli spaces gives that d ε − ,ε − squares to , see [CDRGG20,Theorem 4.1]. out out out out out outin in in in in in Figure 1: Curves contributing to the differential ∂ ε − ,ε − , " in " stands for input and " out " foroutput. The " " and " " indicate the Fredholm index of the respective discs.Denote by (cid:0) CF −∞ (Σ , Σ ) , m −∞ ) the quotient complex of the Cthulhu complex Cth(Σ , Σ ) ,with CF −∞ (Σ , Σ ) = CF (Σ , Σ ) ⊕ C (Λ − , Λ − )[1] and m −∞ = (cid:18) d d − d − d −− (cid:19) In [Leg] the author proved that given a triple of pairwise transverse cobordisms Σ , Σ and Σ ,there is a non trivial map: m −∞ : CF −∞ (Σ , Σ ) ⊗ CF −∞ (Σ , Σ ) → CF −∞ (Σ , Σ ) satisfying the Leibniz rule m −∞ ( m −∞ ⊗ id) + m −∞ (id ⊗ m −∞ ) + m −∞ ◦ m −∞ = 0 , see Section5.1 for more details. Cth + In this section, we define the Rabinowitz complex
Cth + (Σ , Σ ) for a pair of transverse exactLagrangian cobordisms Λ − ≺ Σ Λ +0 and Λ − ≺ Σ Λ +1 . We assume again that A (Λ − i ) admit14ugmentations ε − i for i = 0 , , inducing augmentations ε + i of A (Λ + i ) . The complex Cth + (Σ , Σ ) is generated by three types of generators: Cth + (Σ , Σ ) = C (Λ +1 , Λ +0 ) † [ n − ⊕ CF (Σ , Σ ) ⊕ C (Λ − , Λ − )[1] If we denote | · |
Cth + the grading of generators in Cth + (Σ , Σ ) , then we have | γ | Cth + = n − − | γ | , for γ ∈ C (Λ +1 , Λ +0 ) † [ n − | x | Cth + = | x | , for x ∈ CF (Σ , Σ ) | ξ | Cth + = | ξ | + 1 , for ξ ∈ C (Λ − , Λ − )[1] The difference on generators between
Cth + (Σ , Σ ) and the original Cthulhu complex Cth(Σ , Σ ) in [CDRGG20] is that the generators that are Reeb chords in the positive end are chords from Λ +0 to Λ +1 in Cth + (Σ , Σ ) , whereas they are chords from Λ +1 to Λ +0 in Cth(Σ , Σ ) . The differentialon Cth + (Σ , Σ ) is then given by m ε − ,ε − = ∆ +1 d d d − b − ◦ ∆ Σ1 b − ◦ ∆ Σ1 b − where:1. ∆ +1 : C (Λ +1 , Λ +0 ) † [ n − → C (Λ +1 , Λ +0 ) † [ n − is defined by ∆ +1 ( γ +01 ) = X γ − X ζ i g M R × Λ +01 ( γ − ; ζ , γ +01 , ζ ) · ε +0 ( ζ ) ε +1 ( ζ ) · γ − and is of degree according to Proposition 2.2. m := d + d + d − : Cth + (Σ , Σ ) → CF (Σ , Σ ) with d ( γ +01 ) = X x X δ i M , Σ ( x ; δ , γ +01 , δ ) · ε − ( δ ) ε − ( δ ) · xd ( q ) = X x X δ i M , Σ ( x ; δ , q, δ ) · ε − ( δ ) ε − ( δ ) · xd − ( γ − ) = X x X δ i M , Σ ( x ; δ , γ − , δ ) · ε − ( δ ) ε − ( δ ) · x is also of degree .3. ∆ Σ1 : Cth ∗ + (Σ , Σ ) → C n − −∗ (Λ − , Λ − ) is defined, for a ∈ Cth + (Σ , Σ ) , by: ∆ Σ1 ( a ) = X γ X δ i M , Σ ( γ ; δ , a, δ ) · ε − ( δ ) ε − ( δ ) · γ so in particular it vanishes for energy reasons on C (Λ − , Λ − ) . This map is of degree , i.e. | γ | = n − − | a | Cth + where | γ | is as defined in Section 2.5.4. Let us denote C ∗ (Λ − , Λ − ) = C n − −∗ (Λ − , Λ − ) ⊕ C ∗− (Λ − , Λ − ) , one finally defines the map b − : C ∗ (Λ − , Λ − ) → C ∗− (Λ − , Λ − ) by b − ( γ ) = X γ +10 X δ i g M R × Λ − ( γ ; δ , γ, δ ) · ε − ( δ ) ε − ( δ ) · γ where γ is a positive asymptotic if it is in C (Λ − , Λ − ) and a negative asymptotic if it is in C (Λ − , Λ − ) . This map is of degree . 15n Figure 2 are schematized the pseudo-holomorphic curves contributing to the differential m ε − ,ε − . out out out out out outoutin in in in in in in Figure 2: Curves contributing to the differential m ε − ,ε − . Remark 2.
In the definition of m ε − ,ε − , all components are related to components of the differ-ential of Cth(Σ , Σ ) or Cth(Σ , Σ ) as follows:• the map ∆ +1 is the dual of d ++ in Cth(Σ , Σ ) , and it is the differential of the bilinearizedLegendrian contact homology of Λ +0 ∪ Λ +1 restricted to C (Λ +1 , Λ +0 ) ,• the map d is the dual of d +0 in Cth(Σ , Σ ) ,• the map ∆ Σ1 restricted to the positive Reeb chords is the dual of d + − in Cth(Σ , Σ ) andrestricted to intersection points it is the dual of d − in Cth(Σ , Σ ) ,• b − restricted to C (Λ − , Λ − ) is the banana map in Cth(Σ , Σ ) , and restricted to C (Λ − , Λ − ) it is the map d −− in Cth(Σ , Σ ) that is to say the differential of the Legendrian contactcohomology of Λ − ∪ Λ − restricted to C (Λ − , Λ − ) .In particular, the Floer complex ( CF −∞ (Σ , Σ ) , m −∞ ) is a subcomplex of Cth + (Σ , Σ ) . Theorem 3. m ε − ,ε − is a degree map satisfying m ε − ,ε − ◦ m ε − ,ε − = 0 . Remark 3 ( ∂ -breaking) . Before we prove the theorem, let us give some precision about sometypes of pseudo-holomorphic buildings arising as limit of a sequence of pseudo-holomorphic discsin a -dimensional moduli space. Namely, the buildings containing a non-trivial pure disc are abit special.Consider again the case 1. in Section 2.6, i.e. the limit of a sequence of discs with boundaryon trivial cylinders. As we said, it consists of two index discs connected by a Reeb chord node.If this node is a pure Reeb chord γ ∈ R (Λ) , then the (non trivial) disc u in the building forwhich this node is a positive Reeb chord asymptotic is a pure disc, given the condition we takeon the Lagrangian boundary conditions in Section 2.3, and thus contributes to the differential of γ in the Legendrian contact homology DGA of Λ . Then, applying an augmentation ε of A (Λ) toall the negative pure Reeb chord asymptotics of u results in a pseudo-holomorphic curve count16ontributing to ε ◦ ∂ ( γ ) . The sum over all possible negative pure Reeb chord asymptotics of apure disc with positive asymptotic γ leads in a curve count giving the whole term ε ◦ ∂ ( γ ) whichvanishes by definition of an augmentation.Consider now the case 2.(b) in Section 2.6, and the subcase where the index curve that wedenote u has boundary on the negative ends of the cobordisms and has one positive Reeb chordasymptotic which is a pure Reeb chord γ ∈ R (Λ − ) . In this case, for the same reason as above, u is a pure disc, and the contribution of such discs vanish once we apply an augmentation to purenegative Reeb chords.In the subcase of 2.(b) where the index curve u has boundary on the positive ends of thecobordisms, u can not have a positive asymptotic to a pure Reeb chord because this is not anode and thus would imply that the sequence of discs we started with had a positive pure Reebchord asymptotic. However, there can be one (or several) index pure curve with a positiveasymptotic to a pure chord γ of Λ + , and with boundary on a cobordism Λ − ≺ Σ Λ + . Such anindex curve, call it v , contributes thus to Φ Σ ( γ ) where Φ Σ : A (Λ + ) → A (Λ − ) is the mapinduced by the cobordism. Applying the augmentation ε − to negative Reeb chord asymptoticsof v leads to a curve count contributing to ε − ◦ Φ Σ ( γ ) = ε + ( γ ) by definition of ε + . Fixing γ andsumming over all possible negative Reeb chords of Λ − leads in a curve count giving the term ε + ( γ ) .In this paper, every time we define a map via a count of mixed pseudo-holomorphic discs insome moduli spaces, we sum over all possible pure negative Reeb chord asymptotics, to whichwe apply then the given augmentations of the Legendrian negative ends. Thus,(A) the contribution of broken discs having a non trivial pure disc component with boundaryon cylindrical ends will vanish,(B) applying the augmentation ε − i to negative pure chords in R (Λ − i ) corresponds to applying ε + i to the potential pure chords asymptotics in R (Λ + i ) of a disc with boundary on thepositive cylindrical ends.This being said, we will now ignore the broken discs of case (A), and the use of the inducedaugmentations ε + i when describing the boundary of the compactification of moduli spaces refersto breakings in case (B). Proof of Theorem 3.
The degree of m ε − ,ε − follows from Proposition 2. Then, we have m ε − ,ε − ◦ m ε − ,ε − = ∆ +1 ◦ ∆ +1 d ◦ ∆ +1 + d ◦ d + d − ◦ b − ◦ ∆ Σ1 d + d − ◦ b − ◦ ∆ Σ1 d ◦ d − + d − ◦ b − b − ◦ ∆ Σ1 ◦ ∆ +1 + b − ◦ ∆ Σ1 ◦ d + ( b − ) ◦ ∆ Σ1 b − ◦ ∆ Σ1 ◦ d + ( b − ) ◦ ∆ Σ1 b − ◦ ∆ Σ1 ◦ d − + ( b − ) ∆ +1 ◦ ∆ +1 vanishes because for any γ ∈ C (Λ +1 , Λ +0 ) , the discs contributing to ∆ +1 ◦ ∆ +1 ( γ ) are in one-to-one correspondence with broken curves in the boundary of the compactifi-cation of moduli spaces g M R × Λ +01 ( ξ ; ζ , γ, ζ ) , for all possible chord ξ ∈ C (Λ +1 , Λ +0 ) and words of pure Reeb chords ζ i . Observe that ∆ +1 is in fact the bilinearized Legendrianhomology differential of Λ +0 ∪ Λ +1 restricted to the subcomplex C (Λ +1 , Λ +0 ) .2. For γ ∈ C (Λ +1 , Λ +0 ) , the term (cid:0) d ◦ ∆ +1 + d ◦ d + d − ◦ b − ◦ ∆ Σ1 (cid:1) ( γ ) is given exactlyby the count of broken curves in ∂ M , Σ ( p ; δ , γ , δ ) for all p ∈ Σ ∩ Σ and words δ i .17. For γ ∈ C (Λ +1 , Λ +0 ) , the term (cid:0) b − ◦ ∆ Σ1 ◦ ∆ +1 + b − ◦ ∆ Σ1 ◦ d + ( b − ) ◦ ∆ Σ1 (cid:1) ( γ ) is givenby the count of curves in g M R × Λ − ( ξ ; δ ′ , β , δ ′ ) × ∂ M , Σ ( β ; δ , γ , δ ) and ∂ M R × Λ − ( ξ ; δ ′ , β , δ ′ ) × M , Σ ( β ; δ , γ , δ ) for all ξ ∈ C (Λ − , Λ − ) , β ∈ C (Λ − , Λ − ) and words of pure Reeb chords δ i , δ ′ i of Λ − i . In-deed, the study of ∂ M R × Λ − ( ξ ; δ ′ , β , δ ′ ) gives that the map b − restricted to C (Λ +1 , Λ +0 ) satisfies ( b − ) + b − ◦ ∆ − = 0 , where ∆ − is the obvious analogue of ∆ +1 but defined on C (Λ − , Λ − ) . Then one can write (cid:0) b − ◦ ∆ Σ1 ◦ ∆ +1 + b − ◦ ∆ Σ1 ◦ d + ( b − ) ◦ ∆ Σ1 (cid:1) ( γ ) = b − (cid:0) ∆ Σ1 ◦ ∆ +1 + ∆ Σ1 ◦ d + ∆ − ◦ ∆ Σ1 (cid:1) ( γ ) and there is a one-to-one correspondence between broken discs contributing to ∆ Σ1 ◦ ∆ +1 +∆ Σ1 ◦ d + ∆ − ◦ ∆ Σ1 and broken discs in ∂ M , Σ ( β ; δ , γ, δ ) .4. (cid:0) d + d − ◦ b − ◦ ∆ Σ1 (cid:1) ( x ) , for x ∈ CF (Σ , Σ ) , counts broken curves in ∂ M , Σ ( p ; δ , x, δ ) ,for all p ∈ Σ ∩ Σ and words δ i .5. (cid:0) b − ◦ ∆ Σ1 ◦ d + ( b − ) ◦ ∆ Σ1 (cid:1) ( x ) counts broken curves in g M R × Λ − ( ξ ; δ ′ , β , δ ′ ) × ∂ M , Σ ( β ; δ , x, δ ) and ∂ M R × Λ − ( ξ ; δ ′ , β , δ ′ ) × M , Σ ( β ; δ , x, δ ) for all ξ , β and words of Reeb chords δ i and δ ′ i as above.6. (cid:0) d ◦ d − + d − ◦ b − (cid:1) ( γ ) , for γ ∈ C (Λ − , Λ − ) , counts broken curves in ∂ M , Σ ( p ; δ , γ , δ ) .7. (cid:0) b − ◦ ∆ Σ1 ◦ d − + ( b − ) (cid:1) ( γ ) = ( b − ) ( γ ) for energy reasons, and vanishes as b − restrictedto C (Λ − , Λ − ) is the bilinearized Legendrian contact cohomology differential of Λ − ∪ Λ − restricted to the subcomplex C (Λ − , Λ − ) (observe otherwise that the broken discs con-tributing to ( b − ) ( γ ) are exactly the one appearing in ∂ M R × Λ − ( ξ ; δ , γ , δ ) , for all ξ , δ , δ ). Notations 4.
A few remarks about notations of maps:1. Very rigorously, we should write the augmentations involved in the definition of each mapall the time, but we drop it to enlighten the notation.2. Given a pair of cobordisms (Σ , Σ ) , we will then write m Σ1 for the differential on Cth + (Σ , Σ ) ,so without specifying the augmentations, and “ Σ ” stands for the ordered pair (Σ , Σ ) . Ifwe want to explicit the order, we will sometimes write m Σ , Σ or m Σ . Similarly, we write ∆ Σ1 instead of ∆ Σ , Σ , and finally b − is a short notation for b Λ − , Λ − and ∆ +1 is a shortnotation for ∆ Λ +0 , Λ +1 .3. We write m Σ , +1 , m Σ , and m Σ , − (or simply m +1 , m and m − when the pair of cobordisms isclear from the context) the components of the differential with values in C (Λ +1 , Λ +0 ) † [ n − , CF (Σ , Σ ) and C (Λ − , Λ − )[1] respectively, and then m Σ ,ij := m Σ ,i + m Σ ,j , for i, j ∈{ + , , −} distinct. 18. We will sometimes denote CF + ∞ (Σ , Σ ) := C (Λ +1 , Λ +0 ) † [ n − ⊕ CF (Σ , Σ ) , but observethat contrary to CF −∞ (Σ , Σ ) , this is not a complex.Considering the notations above, the components m +1 and m − of m Σ1 can be expressed as: m +1 = ∆ +1 (2) m − = b − ◦ ∆ Σ1 (3)where ∆ Σ1 : Cth + (Σ , Σ ) → C n − −∗ (Λ − , Λ − ) ⊕ C ∗− (Λ − , Λ − ) is defined by: ∆ Σ1 ( a ) = (cid:26) a if a ∈ C (Λ − , Λ − )∆ Σ1 ( a ) otherwise Example 1 (Case of concordances) . Consider a compact non-degenerate Legendrian submani-fold Λ ⊂ Y , admitting augmentations ε , ε . Consider a 2-copy of Λ which we denote Λ (2) andconsisting of Λ ∪ Λ , such that Λ := Λ and Λ a push-off of Λ in the positive Reeb directionlying entirely above Λ (the smallest z -coordinate of a point in Λ is greater than the biggest z coordinate of a point in Λ ) and then perturbed by a Morse function. This is the 2-copy of Λ considered in [EES09]. We have in this case (cid:0) Cth + ( R × Λ , R × Λ ) , m ε ,ε (cid:1) = ( C (Λ , Λ ) † [ n − , ∆ +1 ) and this complex is the complex of the bilinearized Legendrian contact homology of Λ ∪ Λ restricted to mixed chords (chords from Λ to Λ ), but with a choice of grading making thedifferential a degree map. For an horizontally displaceable Λ , this complex is acyclic and givesthe duality long exact sequence for the Legendrian Λ as proved in [EES09].Figure 3: Left: -copy Λ (2) of Λ ; right: -copy Λ (2) .Consider another -copy Λ (2) of Λ consisting of the components Λ and Λ such that Λ is acopy of Λ := Λ perturbed by a small negative Morse function f , i.e. Λ is identified with j ( f ) in a neigborhood of Λ identified with a neighborhood of the -section of J (Λ) , as schematizedon Figure 3. In this case we have: Cth + ( R × Λ , R × Λ ) = C (Λ , Λ ) † [ n − ⊕ C (Λ , Λ )[1] = C ∗ (Λ , Λ ) with differential: m ε ,ε = (cid:18) ∆ +1 b − ◦ ∆ Σ1 b − (cid:19) = (cid:18) ∆ +1 b − b − (cid:19) because the cobordisms are trivial cylinders so the map ∆ Σ1 is the identity map.19here is actually a canonical isomorphism of complexes (cid:0) Cth + ( R × Λ , R × Λ ) , (cid:18) ∆ +1 b − b − (cid:19) (cid:1) ≃ −→ (Cth + ( R × Λ , R × Λ ) , ∆ +1 ) sending chords in C (Λ , Λ ) to their corresponding long chords in C (Λ , Λ ) . The other part ofthe isomorphism is obtained by considering pseudo-holomorphic strips with mixed Reeb chordsasymptotics and inverting the role of the input and output, see for example [NRS + , Proposition5.4]. Example 2 ( -section of a jet space) . Consider the jet space J ( M ) = T ∗ M × R of a smoothmanifold M , endowed with the standard contact form dz − λ where z is the R coordinate and λ the canonical form on T ∗ M . Then the -section is a Legendrian Λ := M . Take a small push-offof Λ in the positive Reeb ( ∂ z ) direction and perturb it by a small Morse function f : Λ → R .Denote this Legendrian Λ . Consider the trivial augmentations ε i , i = 0 , . Then, the complex Cth + ( R × Λ , R × Λ ) , m ε − ,ε − ) is just the complex ( C (Λ , Λ ) † [ n − , ∆ +1 ) which is canonicallyidentified with the Morse complex of f . Cth + ( V ⊙ W , V ⊙ W ) Consider Legendrian submanifolds Λ − i , Λ i , Λ + i for i = 0 , , and cobordisms Λ − i ≺ V i Λ i and Λ i ≺ W i Λ + i . As the positive end of V i is a cylinder over Λ i , as well as the negative end of W i ,one can perform the concatenation of V i and W i denoted V i ⊙ W i , which is an exact Lagrangiancobordism from Λ − i to Λ + i , see for example [CDRGG20, Section 5.1]. Assume that A (Λ − i ) admitaugmentations ε − , ε − . These augmentations induce augmentations ε and ε of A (Λ ) and A (Λ ) respectively, and augmentations ε +0 and ε +1 of A (Λ +0 ) and A (Λ +1 ) . Assuming that the cobordisms V ⊙ W and V ⊙ W intersect transversely, the Cthulhu complex of the pair ( V ⊙ W , V ⊙ W ) has four types of generators Cth + ( V ⊙ W , V ⊙ W ) = C (Λ +1 , Λ +0 ) ⊕ CF ( W , W ) ⊕ CF ( V , V ) ⊕ C (Λ − , Λ − )= CF + ∞ ( W , W ) ⊕ CF −∞ ( V , V ) and the differential is given by m V ⊙ W = m W, +1 m W, (id + b V ◦ ∆ W ) m W, (id + b V ◦ ∆ W ) m W, ◦ b V m W, ◦ b V m V, ◦ ∆ W m V, ◦ ∆ W m V, m V, m V, − ◦ ∆ W m V, − ◦ ∆ W m V, − m V, − where b V : Cth + ( V , V ) → C (Λ , Λ )[1] is the degree map defined by b V ( a ) = X γ X δ i M V ,V ( γ ; δ , a, δ ) · ε − ( δ ) ε − ( δ ) · γ We will extend the definitions of the maps b V and ∆ V to Cth + ( V ⊙ W , V ⊙ W ) in order to obtaina compact formula for m V ⊙ W . Namely, we define b V : Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( W , W ) by b V ( a ) = (cid:26) a + b V ◦ ∆ W ( a ) for a ∈ CF + ∞ ( W , W ) b V ( a ) for a ∈ CF −∞ ( V , V ) ∆ W : Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V , V ) by ∆ W ( a ) = (cid:26) ∆ W ( a ) for a ∈ CF + ∞ ( W , W ) a for a ∈ CF −∞ ( V , V ) This may seem confusing because we have already defined a map ∆ Σ1 for the case of a pair ofcobordisms (Σ , Σ ) in the previous section. However, this map ∆ Σ1 can be recovered from themap ∆ W for the pair ( V ⊙ W , V ⊙ W ) where ( V , V ) = ( R × Λ − , R × Λ − ) and ( W , W ) =(Σ , Σ ) , see Section 3.2.3 for more details. In the remaining of this section, to make it clear wewrite ∆ W ⊂ W when we consider the map for the pair ( W , W ) not in the concatenation.One can thus write the product in the following more compact way: m V ⊙ W = m W, +01 ◦ b V + m V, − ◦ ∆ W Let us now check that m V ⊙ W is indeed a differential. Using the definition of b V and ∆ W , wehave: (cid:0) m V ⊙ W (cid:1) = (cid:16) m W, +01 ◦ b V + m V, − ◦ ∆ W (cid:17) ◦ (cid:16) m W, +01 ◦ b V + m V, − ◦ ∆ W (cid:17) = m W, +01 ◦ m W, +01 ◦ b V + m W, +01 ◦ (cid:0) b V ◦ ∆ W (cid:1) ◦ m W, +01 ◦ b V + m W, +01 ◦ b V ◦ m V, − ◦ ∆ W + m V, − ◦ ∆ W ◦ m W, +01 ◦ b V + m V, − ◦ m V, − ◦ ∆ W where by definition the term m W, +1 ◦ b V ◦ ∆ W vanishes but we keep it in the formula to make itmore homogeneous. We use then the following Lemma 1.
The maps1. ∆ W ◦ m W, +01 ◦ b V + m V, +1 ◦ ∆ W : Cth + ( V ⊙ W , V ⊙ W ) → C n − −∗ (Λ , Λ ) , and2. b V ◦ m V ◦ ∆ W + b Λ1 ◦ ∆ W ⊂ W ◦ b V : Cth + ( V ⊙ W , V ⊙ W ) → C ∗− (Λ , Λ ) vanish.Proof. 1. For a ∈ CF −∞ ( V , V ) we have ∆ W ◦ m W, +01 ◦ b V ( a ) + m V, +1 ◦ ∆ W ( a ) = ∆ W ◦ m W, +01 ◦ b V ( a ) + m V, +1 ( a ) and the first term vanishes for energy reason and the second one by definition. Then, for a ∈ CF + ∞ ( W , W ) we have ∆ W ◦ m W, +01 ◦ b V ( a ) + m V, +1 ◦ ∆ W ( a ) = ∆ W ◦ m W, +01 ( a + b V ◦ ∆ W ( a )) + m V, +1 ◦ ∆ W ( a )= ∆ W ◦ m W, +01 ( a ) + m V, +1 ◦ ∆ W ( a ) because ∆ W ◦ m W, +01 ◦ b V ◦ ∆ W ( a ) vanishes for energy reason. Consider the boundary of theone-dimensional moduli space M W ,W ( β ; δ , a, δ ) for β ∈ C (Λ , Λ ) . The broken discsarising in the boundary (schematized on Figure 4 for the case a = γ ∈ C (Λ +1 , Λ +0 ) ) contributeexactly to (cid:10) (∆ W ◦ m W, +01 + m V, +1 ◦ ∆ W )( a ) , β (cid:11) (4)21 Figure 4: Types of broken discs in the boundary of M W ,W ( β ; δ , γ , δ ) . For a ∈ CF −∞ ( V , V ) we have b V ◦ m V ◦ ∆ W ( a ) + b Λ1 ◦ ∆ W ⊂ W ◦ b V ( a ) = b V ◦ m V ( a ) + b Λ1 ◦ b V ( a ) and for a ∈ CF + ∞ ( W , W ) we have b V ◦ m V ◦ ∆ W ( a ) + b Λ1 ◦ ∆ W ⊂ W ◦ b V ( a ) = b V ◦ m V ◦ ∆ W ( a ) + b Λ1 ◦ ∆ W ⊂ W ( a + b V ◦ ∆ W ( a ))= b V ◦ m V ◦ ∆ W ( a ) + b Λ1 ◦ ∆ W ( a ) + b Λ1 ◦ b V ◦ ∆ W ( a ) To conclude that this map vanishes, one has to consider the broken curves in the boundary ofthe compactification of moduli spaces of bananas with boundary on V , namely the boundary of M V ,V ( γ ; δ , a, δ ) for γ ∈ C (Λ , Λ ) , and a ∈ Cth + ( V ⊙ W , V ⊙ W ) , see Figure 5 forthe case a = x ∈ CF ( W , W ) .
111 1 1 1 00 00 0 0 0 00 000 00
Figure 5: Types of broken discs in ∂ M V ,V ( γ ; δ , γ , δ ) .22hus, using part . of the lemma we can rewrite (cid:0) m V ⊙ W (cid:1) = m W, +01 ◦ m W, +01 ◦ b V + m W, +01 ◦ b V ◦ m V, +1 ◦ ∆ V + m W, +01 ◦ b V ◦ m V, − ◦ ∆ W + m V, − ◦ m V, +1 ◦ ∆ V + m V, − ◦ m V, − ◦ ∆ W = m W, +01 ◦ m W, +01 ◦ b V + m W, +01 ◦ b V ◦ m V ◦ ∆ V + m V, − ◦ m V ◦ ∆ V Now, using m W, +01 ◦ m W, +01 = m W, +01 ◦ m W, − = m W, +01 ◦ b Λ1 ◦ ∆ W ⊂ W , part . of Lemma 1, and thefact that m V, − ◦ m V = 0 , one gets that (cid:0) m V ⊙ W (cid:1) = 0 . The maps b V and ∆ W defined in the previous section are in fact what we will call transfer maps .In particular, they are chain maps as we prove now. Proposition 3. b V : Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( W , W ) is a chain map.Proof. We need to prove that b V ◦ m V ⊙ W + m W ◦ b V = 0 (5)By definition of m V ⊙ W we have that the left-hand side of (5) is equal to b V ◦ m W, +01 ◦ b V + b V ◦ m V, − ◦ ∆ W + m W ◦ b V = m W, +01 ◦ b V + b V ◦ ∆ W ◦ m W, +01 ◦ b V + b V ◦ m V, − ◦ ∆ W + m W ◦ b V = b V ◦ ∆ W ◦ m W, +01 ◦ b V + b V ◦ m V, − ◦ ∆ W + m W, − ◦ b V Using the first part of Lemma 1 on the first term and the second part of the lemma on the secondand third terms, recalling that m W, − = b Λ1 ◦ ∆ W ⊂ W , we get that the sum above vanishes. Proposition 4. ∆ W : Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V , V ) is a chain map.Proof. We have to prove that ∆ W ◦ m V ⊙ W + m V ◦ ∆ W = 0 (6)The left-hand side of the equation is ∆ W ◦ m W, +01 ◦ b V + ∆ W ◦ m V, − ◦ ∆ W + m V ◦ ∆ W whose first term equals m V, +1 ◦ ∆ W by Lemma 1 and the second term equals m V, − ◦ ∆ W bydefinition of ∆ W . Thus the sum vanishes. Let us have a look at the two following cases for the pair ( V ⊙ W , V ⊙ W ) :1. ( W , W ) = ( R × Λ , R × Λ ) ,2. ( V , V ) = ( R × Λ , R × Λ )
23n the first case, one has
Cth + ( V ⊙ ( R × Λ ) , V ⊙ ( R × Λ )) = C (Λ , Λ ) † [ n − ⊕ CF ( V , V ) ⊕ C (Λ − , Λ − )[1] = Cth + ( V , V ) and we actually have an equality of complexes as ∆ W on C (Λ , Λ ) is the identity map (as itcounts index discs with boundary on Lagrangian cylinders so it can only count trivial strips).Thus, the map b V defined for a general pair of concatenated cobordisms before gives in this casea map b V : Cth + ( V , V ) → Cth + ( R × Λ , R × Λ ) = C ∗ (Λ , Λ ) satisfying b V ( a ) = a + b V ( a ) for a ∈ C (Λ , Λ ) and b V ( a ) = b V ( a ) for a ∈ CF −∞ ( V , V ) . In thesecond case, one has Cth + (( R × Λ ) ⊙ W , ( R × Λ ) ⊙ W ) = C (Λ +1 , Λ +0 ) † [ n − ⊕ CF ( W , W ) ⊕ C (Λ , Λ )[1] = Cth + ( W , W ) and again this equality holds in terms of complexes as b V is the identity map on C (Λ , Λ ) andvanishes on C (Λ , Λ ) (no index banana with boundary on R × (Λ ∪ Λ ) and two positive Reebchord asymptotics), and ∆ V is the identity map on C (Λ , Λ ) . For such a pair of concatenatedcobordisms, we get the map ∆ W : Cth + ( W , W ) → C ∗ (Λ , Λ ) satisfying ∆ W ( a ) = ∆ W ( a ) for a ∈ CF + ∞ ( W , W ) and ∆ W ( a ) = a for a ∈ C (Λ , Λ ) , whichrecovers exactly the definition we gave at the end of the Section 3.1. Notations 5.
From now on, we use the maps b V and ∆ W without specifying if we are in thecase of a pair ( V ⊙ W , V ⊙ W ) , (cid:0) V ⊙ ( R × Λ ) , V ⊙ ( R × Λ ) (cid:1) or (cid:0) ( R × Λ ) ⊙ W , ( R × Λ ) ⊙ W (cid:1) . From the previous special cases, one deduces a Mayer-Vietoris sequence. Consider a pair ofconcatenations ( V ⊙ W , V ⊙ W ) . By definition, we have Lemma 2. b V ◦ ∆ W + ∆ W ◦ b V : Cth + ( V ⊙ W , V ⊙ W ) → C ∗ (Λ , Λ ) vanishes.Proof. First, remember that ∆ V : Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V , V ) , so the term b V ◦ ∆ W should be read as b V ⊂ V ◦ ∆ W ⊂ V ⊙ W . Similarly, b V : Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( W , W ) ,thus the term ∆ W ◦ b V should be read as being ∆ W ⊂ W ◦ b V ⊂ V ⊙ W . Then we have for a ∈ CF + ∞ ( W , W ) : b V ◦ ∆ W ( a ) = b V ◦ ∆ W ( a ) = ∆ W ( a ) + b V ◦ ∆ W ( a ) ∆ W ◦ b V ( a ) = ∆ W ( a + b V ◦ ∆ W ( a )) = ∆ W ( a ) + ∆ W ◦ b V ◦ ∆ W ( a ) = ∆ W ( a ) + b V ◦ ∆ W ( a ) and for a ∈ CF −∞ ( V , V ) , b V ◦ ∆ W ( a ) = b V ( a ) = b V ( a ) ∆ W ◦ b V ( a ) = ∆ W ◦ b V ( a ) = b V ( a ) → Cth ∗ + ( V ⊙ W , V ⊙ W ) ( ∆ W , b V ) −−−−−−→ Cth ∗ + ( V , V ) ⊕ Cth ∗ + ( W , W ) b V + ∆ W −−−−−→ Cth ∗ + ( R × Λ , R × Λ ) → which gives rise to a Mayer-Vietoris sequence · · · → H ∗ Cth + ( V ⊙ W , V ⊙ W ) → H ∗ Cth + ( V , V ) ⊕ H ∗ Cth + ( W , W ) → H ∗ Cth + ( R × Λ , R × Λ ) g ∗ −→ H ∗ +1 Cth + ( V ⊙ W , V ⊙ W ) . . . The connecting morphism g ∗ is given on the chain level by m W, ◦ b V + m V, − on C (Λ , Λ ) † [ n − ⊂ Cth + ( R × Λ , R × Λ ) and by m W, on C (Λ , Λ )[1] ⊂ Cth + ( R × Λ , R × Λ ) . Below wecheck that g is indeed a chain map so induces a well-defined map in homology, and that thesequence is exact.We need to prove that g ◦ m R × Λ1 = m V ⊙ W ◦ g . Instead of writing big matrices, let us prove itfor the two types of generators separately. Consider γ ∈ C (Λ , Λ ) , we have g ◦ m R × Λ1 ( γ )+ m V ⊙ W ◦ g ( γ )= g (cid:0) ∆ Λ1 ( γ ) + b Λ1 ( γ ) (cid:1) + (cid:0) m W, +01 ◦ b V + m V, − ◦ ∆ W (cid:1) ◦ (cid:0) m W, ◦ b V + m V, − (cid:1) ( γ )= (cid:0) m W, ◦ b V + m V, − (cid:1) ◦ ∆ Λ1 ( γ ) + m W, ◦ b Λ1 ( γ ) + m W, ◦ m W, ◦ b V ( γ )+ m W, ◦ b V ◦ m V, − ( γ ) + m V, − ◦ m V, − ( γ ) where we have removed terms vanishing for energy reasons. Now, observe that m V, − ◦ ∆ Λ1 ( γ ) + m V, − ◦ m V, − ( γ ) = 0 as ∆ Λ1 ( γ ) = m V, +1 ( γ ) and m V is a differential. Finally, the remainingterms are the algebraic contributions of the broken curves arising in the boundary of productsof moduli spaces of the following type M W ,W ( x, ξ , γ , ξ ) × M V ,V ( γ , δ , γ , δ ) M W ,W ( x, ξ , γ , ξ ) × M V ,V ( γ , δ , γ , δ ) for x ∈ CF ( W , W ) , γ ∈ C (Λ , Λ ) , δ i words of pure Reeb chords of Λ − i and ξ i words of pureReeb chords of Λ i .Then, consider γ ∈ C (Λ , Λ ) , we have g ◦ m R × Λ1 ( γ )+ m V ⊙ W ◦ g ( γ )= g ◦ b Λ1 ( γ ) + (cid:0) m W, +01 ◦ b V + m V, − ◦ ∆ W (cid:1) ◦ m W, ( γ )= m W, ◦ b Λ1 ( γ ) + m W, ◦ m W, ( γ ) where we have removed terms vanishing for energy reason. Then the two remaining terms arealgebraic contributions of the broken configurations in the boundary of the compactification ofmoduli spaces M W ,W ( x, ξ , γ , ξ ) .Now we check the exactness of · · · → H ∗− Cth + ( R × Λ , R × Λ ) g ∗− −−−→ H ∗ Cth + ( V ⊙ W , V ⊙ W ) ( ∆ W , b V ) −−−−−−→ H ∗ Cth + ( V , V ) ⊕ H ∗ Cth + ( W , W ) → . . . Given a cycle γ + γ ∈ Cth + ( R × Λ , R × Λ ) , we need to check that in homology ∆ W ◦ g ∗ ( γ + γ ) = b V ◦ g ∗ ( γ + γ ) = 0 . We have ∆ W ◦ g ∗ ( γ + γ ) = ∆ W (cid:16) m W, ( b V ( γ ) + γ ) + m V, − ( γ ) (cid:17) = ∆ W ◦ m W, (cid:0) b V ( γ ) + γ (cid:1) + m V, − ( γ )= m V, − ( γ ) m V, − ( γ ) = m V ( γ ) because m V, +1 ( γ ) = ∆ Λ1 ( γ ) = 0 by as-sumption. Thus ∆ W ◦ g ∗ ( γ + γ ) ∈ Cth + ( V , V ) is a boundary so vanishes in homology.Then b V ◦ g ∗ ( γ + γ ) = b V (cid:16) m W, (cid:0) b V ( γ ) + γ (cid:1) + m V, − ( γ ) (cid:17) = m W, (cid:0) b V ( γ ) + γ (cid:1) + b V ◦ ∆ W ◦ m W, (cid:0) b V ( γ ) + γ (cid:1) + b V ◦ m V, − ( γ )= m W, (cid:0) b V ( γ ) + γ (cid:1) + b V ◦ m V, − ( γ ) Then, by the study of index bananas with boundary on V ∪ V as above, one has b V ◦ m V, − ( γ ) = b Λ1 ◦ b V ( γ ) + b V ◦ ∆ Λ1 ( γ ) + b Λ1 ( γ ) . But ∆ Λ1 ( γ ) = 0 by assumption, as well as b Λ1 ( γ ) = b Λ1 ( γ ) = m W, − ( γ ) . Thus, we get b V ◦ g ∗ ( γ + γ ) = m W, ◦ b V ( γ ) + m W, ( γ ) + b Λ1 ◦ b V ( γ ) + m W, − ( γ ) Finally, b Λ1 ◦ b V ( γ ) = m W, − ◦ b V ( γ ) by definition, and one can add the terms m W, +1 ◦ b V ( γ ) and m W, +1 ( γ ) which vanish to obtain that b V ◦ g ∗ ( γ + γ ) = m W (cid:0) b V ( γ )+ γ (cid:1) is a boundaryin Cth + ( W , W ) .This proves one part of the exactness of the Mayer-Vietoris sequence, the other part is provedin an analogous way and the details are left to the reader. The acyclicity of the complex
Cth + (Σ , Σ ) is proved in the same way as the acyclicity of thecomplex Cth(Σ , Σ ) in [CDRGG20]. However, in the case of Cth + we need some horizontaldisplaceability assumption of at least one of the two Legendrian ends to achieve acyclicity. Definition 3.
Two Legendrian submanifolds Λ , Λ ⊂ Y = P × R are horizontally displaceable if there exists an Hamiltonian isotopy ϕ t of P which displace the Lagrangian projections Π P (Λ ) and Π P (Λ ) , i.e. Π P (Λ ) and ϕ (Π P (Λ )) are contained in two disjoint balls. A Legendrian iscalled horizontally displaceable if it can be displaced from itself.The goal of the next subsections is to prove the following: Theorem 4.
Let Λ − , Λ − , Λ +0 , Λ +1 ⊂ Y be Legendrian submanifolds such that Λ − and Λ − , or Λ +0 and Λ +1 are horizontally displaceable. Assume moreover that A (Λ − ) and A (Λ − ) admit augmen-tations ε − and ε − . Then, for any pair of transverse exact Lagrangian cobordisms Λ − ≺ Σ Λ +0 and Λ − ≺ Σ Λ +1 , the complex (cid:0) Cth + (Σ , Σ ) , m ε − ,ε − (cid:1) is acyclic. The hypothesis of horizontal displaceability is necessary. Indeed, in the setting of Example 2the -section of a jet space is not horizontally displaceable, and in fact the complex is not acyclic.When Cth + (Σ , Σ ) is acyclic, one recovers long exact sequences obtained in [CDRGG20].Indeed, the complex Cth + (Σ , Σ ) is the cone of the degree map d + b − ◦ ∆ Σ1 : C (Λ +1 , Λ +0 ) † [ n − → CF −∞ (Σ , Σ ) , which is then a quasi-isomorphism since the complex is acyclic, i.e. we have H ∗ (cid:0) C (Λ +1 , Λ +0 ) † [ n − (cid:1) ≃ HF ∗ +1 −∞ (Σ , Σ ) (7)Assume first that the Legendrian submanifolds Λ +0 and Λ +1 are horizontally displaceable. Then,the acyclicity of Cth + ( R × Λ +0 , R × Λ +1 ) yields H ∗ (cid:0) C (Λ +1 , Λ +0 ) † [ n − (cid:1) ≃ H ∗ ( C (Λ +0 , Λ +1 )) (8)26s there are no intersection point generators, and the Legendrians in the negative end arealso Λ +0 and Λ +1 . When (Σ , Σ ) is a directed pair, then d − = 0 and CF ∗−∞ (Σ , Σ ) is thecone of d − : CF ∗ (Σ , Σ ) → C ∗ (Λ − , Λ − ) . When (Σ , Σ ) is a V-shaped pair, then d − = 0 and CF ∗−∞ (Σ , Σ ) is the cone of d − : C ∗− (Λ − , Λ − ) → CF ∗ +1 (Σ , Σ ) . The long exact se-quence of a cone, together with the isomorphisms (7) and (8), and the fact that by definition H ∗ ( C (Λ +0 , Λ +1 )) = LCH ∗ ε +0 ,ε +1 (Λ +0 , Λ +1 ) and H ∗ ( C (Λ − , Λ − )) = LCH ∗ ε − ,ε − (Λ − , Λ − ) , give · · · → LCH k − ε +0 ,ε +1 (Λ +0 , Λ +1 ) → HF k (Σ , Σ ) ↓ LCH kε − ,ε − (Λ − , Λ − ) → LCH kε +0 ,ε +1 (Λ +0 , Λ +1 ) → . . . for a directed pair, and · · · → LCH k − ε +0 ,ε +1 (Λ +0 , Λ +1 ) → LCH k − ε − ,ε − (Λ − , Λ − ) ↓ HF k +1 (Σ , Σ ) → LCH kε +0 ,ε +1 (Λ +0 , Λ +1 ) → . . . for a V-shaped pair. These are the long exact sequences in [CDRGG20, Corollary 1.3].In the case where Λ +0 and Λ +1 are not horizontally displaceable but Λ − and Λ − are, one getsthe same exact sequences from the acyclicity of the dual complex Cth dual + (Σ , Σ ) : Cth dual + (Σ , Σ ) = C ∗ (Λ +1 , Λ +0 ) † [ n − ⊕ CF ∗ (Σ , Σ ) ⊕ C ∗ (Λ − , Λ − ) with the degree − differential b +1 b Σ , Σ b Σ , Σ ◦ b − d Σ , Σ d Σ , Σ − Σ , Σ ∆ − However as we do not especially need this dual complex in this article we will not give moredetails here.
Given a pair of cobordisms (Σ , Σ ) cylindrical outside [ − T, T ] × Y , we will wrap the positiveand negative ends of Σ in order to get a pair of cobordisms such that the associated Cth + complex has only intersection points generators. The wrapping is done by Hamiltonian isotopy.A smooth function h : R → R gives rise to a Hamiltonian H : R × P × R → R defined by H ( t, p, z ) = h ( t ) . The corresponding Hamiltonian vector field X h is defined through the equation d ( e t α )( X h , · ) = − dH , and its Hamiltonian flow ϕ sh takes the following simple form ϕ sh ( t, p, z ) = ( t, p, z + se − t h ′ ( t )) Moreover, the image of an exact Lagrangian cobordism Σ with primitive f Σ by an Hamiltonianisotopy ϕ sh as above is still an exact Lagrangian cobordism Σ s = ϕ sh (Σ) , with a primitive f Σ s given by f Σ s = f Σ + s ( h ′ − h ) ◦ π R (9)27here π R : R × Y → R is the projection on the symplectization coordinate t . Given N > T ,consider a function h + T,N : R → R satisfying: h + T,N ( t ) = 0 for t ≤ T + N,h + T,N ( t ) = − e t for t ≥ T + N + 1 , ( h + T,N ) ′ ( t ) ≤ , such that the Hamiltonian vector field takes the form ρ + T,N ( t ) ∂ z where ρ + T,N : R → R satisfies ρ + T,N ( t ) = 0 for t < T + N,ρ + T,N ( t ) = − for t > T + N + 1 , ( ρ + T,N ) ′ ( t ) ≤ . T+N+1T+NT-T
Figure 6: Wrapping the positive end of Σ .Let S + > greater than the length of the longest Reeb chord from Λ +0 to Λ +1 . We set W := ϕ S + h + T,N ( R × Λ +1 ) , and W := R × Λ +0 , and consider the pair (Σ ⊙ W , Σ ⊙ W ) , where infact Σ ⊙ W = Σ , see Figure 6. The complex Cth + (Σ ⊙ W , Σ ⊙ W ) has only three types ofgenerators, namely Cth + (Σ ⊙ W , Σ ⊙ W ) = CF ( W , W ) ⊕ CF (Σ , Σ ) ⊕ C (Λ − , Λ − )[1] Under this decomposition, the transfer map ∆ W : Cth + (Σ ⊙ W , Σ ⊙ W ) → Cth + (Σ , Σ ) isequal to the matrix ∆ W = ∆ W We have then the following
Proposition 5.
The transfer map ∆ W : Cth + (Σ ⊙ W , Σ ⊙ W ) → Cth + (Σ , Σ ) is anisomorphism.Proof. The proof is the same as the proof of [CDRGG20, Proposition 8.2]. After wrapping,each Reeb chord from Λ +0 to Λ +1 creates an intersection point in W ∩ W , and observing that28he wrapping in the positive end makes the Conley-Zehnder index increasing by , there is acanonical identification of graded modules: CF ∗ ( W , W ) = C ∗ (Λ +1 , Λ +0 ) † [ n − ⊂ Cth + (Σ , Σ ) (10)If p ∈ CF ( W , W ) we denote by γ p ∈ C ∗ (Λ +1 , Λ +0 ) † [ n − the corresponding Reeb chord. Thegoal is to prove that this identification also applies at the level of complexes. We will show thatunder the identification (10), the map ∆ W is the identity map.We consider the component ∆ W : CF ∗ ( W , W ) → C n − −∗ (Λ +1 , Λ +0 ) of ∆ W which is ofdegree . Let u ∈ M W ,W ( γ ; δ , p, δ ) where p ∈ W ∩ W , γ ∈ R (Λ +1 , Λ +0 ) is a negativeReeb chord asymptotic, and δ i are words of degree pure Reeb chords which are also negativeasymptotics. This disc contributes to ∆ W ( p ) . By rigidity of u , we have n − − | γ | − | p | Cth + ( W ,W ) = 0 Now, the projection of u to P is a pseudo-holomorphic map in M π P (Λ +0 ) ,π P (Λ +1 ) ( γ ; δ , π P ( p ) , δ ) which has dimension | π P ( p ) | − | γ | − | γ p | − | γ | − , but we have n − − | γ | − | p | Cth + ( W ,W ) = n − − | γ | − ( n − − | γ p | ) = | γ p | − | γ | where we have used the identification (10). This implies that π P ( u ) is in a moduli space ofdimension − so it must be constant. Hence, γ = γ p . On the other side, for each intersectionpoint p ∈ W ∩ W a strip over γ p lifts to a disk in M W ,W ( γ p ; δ , p, δ ) . We obtain that ∆ W is the identity map.Next, we wrap the negative end of Σ ⊙ W as schematized on Figure 7, using a Hamiltoniandefined by a function h − T,N : R → R satisfying: h − T,N ( t ) = e t for t < − T − N − ,h − T,N ( t ) = D for t > − T − N, ( h − T,N ) ′ ( t ) ≥ , for some positive constant D ≥ e − T − N , such that the Hamiltonian vector field is given by ρ − T,N ( t ) ∂ z where ρ − T,N : R → R satisfies ρ − T,N ( t ) = 1 for t ≤ − T − N , ρ − T,N ( t ) = 0 for t ≥ − T ,and ( ρ − T,N ) ′ ( t ) ≤ . Let S − > be greater than the length of the longest chord from Λ − to Λ − and define V := ϕ S − h − T,N ( R × Λ − ) and set V := R × Λ − . After concatenation, we obtain apair ( V ⊙ Σ ⊙ W , V ⊙ Σ ⊙ W ) = (Σ , V ⊙ Σ ⊙ W ) . The Cthulhu complex of the pair (cid:0) Σ , V ⊙ (Σ ⊙ W ) (cid:1) has only intersection points generators, we have Cth + (cid:0) Σ , V ⊙ (Σ ⊙ W ) (cid:1) = CF (Σ , Σ ⊙ W ) ⊕ CF ( V , V ) Under this decomposition, the map b V : Cth + (cid:0) Σ , V ⊙ (Σ ⊙ W ) (cid:1) → Cth + (Σ , Σ ⊙ W ) isgiven by b V = (cid:18) id 0 b V ◦ ∆ Σ ⊙ W b V (cid:19) Proposition 6.
The map b V above is an isomorphism. +N+1T+NT-T-T-N-T-N-1 Figure 7: Wrapping the negative end of Σ ⊙ W . Proof.
It is the same kind of proof as for Proposition 5. In this case we have a canonicalidentification: CF ( V , V ) = C (Λ − , Λ − )[1] ⊂ Cth + (Σ , Σ ⊙ W ) (11)Let us consider the component b V : CF ∗ ( V , V ) → C ∗− (Λ − , Λ − ) of b V which is of degree .Let u ∈ M V ,V ( γ ; δ , p, δ ) where p ∈ V ∩ V , γ ∈ R (Λ − , Λ − ) is a positive Reeb chordasymptotic, and δ i are words of degree pure Reeb chords which are also negative asymptotics,contributing to this component. By rigidity, we have ( | γ | + 1) − | p | CF ( V ,V ) = 0 The projection of u to P is a pseudo-holomorphic map in M π P (Λ − ) ,π P (Λ − ) ( γ ; δ , π P ( p ) , δ ) which has dimension | γ | − | γ p | − . Using the identification (11) we have | γ | − | p | CF ( V ,V ) + 1 = | γ | − ( | γ p | + 1) + 1 = | γ | − | γ p | So the disk π P ( u ) must be constant and γ = γ p . Let us consider a pair (Σ , Σ ) of exact Lagrangian cobordisms and a path of exact Lagrangiancobordisms Σ s for s ∈ [0 , induced by a compactly supported Hamiltonian isotopy, with Σ :=Σ . In particular, for all s ∈ [0 , , Σ s have positive and negative cylindrical ends over Λ ± . Proposition 7.
The complexes (cid:0)
Cth + (Σ , Σ ) , m ε − ,ε − (cid:1) and (cid:0) Cth(Σ , Σ ) , m ε − ,ε − (cid:1) are homo-topy equivalent.Proof. First, wrap the positive and negative ends of Σ in the negative and positive Reeb directionrespectively, as done in the previous section. One gets the pair of cobordisms (Σ , V ⊙ Σ ⊙ W ) ,whose Cthulhu complex is isomorphic to that of the pair (Σ , Σ ) by Propositions 5 and 6. Then,all along the isotopy the complex (Σ s , V ⊙ Σ ⊙ W ) as only intersection point generators andthe bifurcation analysis explained in [CDRGGa, Proposition 8.4] (see also [Ekh12] for the caseof fillings) proves that the complexes Cth + (Σ , V ⊙ Σ ⊙ W ) and Cth + (Σ , V ⊙ Σ ⊙ W ) are homotopy equivalent. Finally, unwrapping the ends of Σ leads again to an isomorphism ofcomplexes. 30 .3 Proof of Theorem 4 Consider a pair of Lagrangian cobordisms (Σ , Σ ) satisfying the hypothesis of the Theorem. Weassume without loss of generality that Λ − and Λ − are horizontally displaceable (in the case Λ + i are horizontally displaceable but Λ − i are not, the same type of argument works but moving thewrapping in the positive end instead of the negative end, see below). By wrapping the cylindricalends of Σ we get the pair (Σ , V ⊙ Σ ⊙ W ) such that:1. Σ and V ⊙ Σ ⊙ W are cylindrical outside [ − T − N, T + N ] × Y .2. Cth + (Σ , V ⊙ Σ ⊙ W ) has only intersection points generators.By a Hamiltonian isotopy ϕ h c compactly supported in [ − T − N, T + N ] × Y , we perturb V ⊙ Σ ⊙ W in such a way that all the intersection points are in fact contained in [ − T − N, − T ] × Y ,and are in bijective correspondence with mixed chords of Λ − ∪ Λ − , as schematized on Figure8. For this purpose we use for example the Hamiltonian H c ( t, p, z ) = h c ( t ) with h c : R → R satisfying h c ( t ) = − e t + C for t ∈ [ − T, T ] , ( −∞ , − T − N ) ∪ ( T + N, ∞ ) ⊂ ( h ′ c ) − (0) h ′ c ( t ) ≤ with C > constant such that h c ( t ) = 0 for t ≤ − T − N , to ensure the primitive of the perturbedcobordism to still vanish on the negative cylindrical end. The Hamiltonian vector field is givenby ρ c ( t ) ∂ z with ρ c ( t ) = − on [ − T, T ] and on ( −∞ , − T − N ) ∪ ( T + N, ∞ ) . Let us denote e Σ = Figure 8: Deformation by a compactly supported Hamiltonian isotopy. ϕ Sh c ( V ⊙ Σ ⊙ W ) , with S big enough so that there are no intersection points in [ − T, T + N ] × Y anymore. This S exists as Σ ∩ [ − T − N, T + N ] × Y and V ⊙ Σ ⊙ W ∩ [ − T − N, T + N ] × Y are compact. By Proposition 7, the complexes Cth + (Σ , Σ ) and Cth + (Σ , e Σ ) have the samehomology. Now we prove that Cth + (Σ , e Σ ) is acyclic. Given the Hamiltonian we used to perturb V ⊙ Σ ⊙ W , we have the canonical identification: Cth + (Σ , e Σ ) = Cth + ( R × Λ − , ϕ Sh c ( V )) Then, we unwrap the negative end of ϕ Sh c ( V ) , and thus Cth + ( R × Λ − , ϕ Sh c ( V )) is isomorphic to Cth + ( R × Λ − , R × e Λ − ) where e Λ − is a translation of Λ − in the negative Reeb direction and lies31ntirely below Λ − , see Figure 9. In this case, we have Cth + ( R × Λ − , R × e Λ − ) = C (Λ − , e Λ − )[1] with the differential b − being the Legendrian contact cohomology differential bilinearized by ε − and ε − . But as the pair (Λ − , Λ − ) is a pair of horizontally displaceable Legendrians, so thiscomplex is acyclic (observe that π P (Λ − ) = π P ( e Λ − ) ).Figure 9: Left: pair of concordances (cid:0) R × Λ − , ϕ Sh c ( V ) (cid:1) ; right: pair ( R × Λ − , R × e Λ − ) . Given Λ − i ≺ Σ i Λ + i for i = 0 , , three exact Lagrangian cobordisms that are pairwise transversesuch that A (Λ − i ) admit augmentations ε − , ε − and ε − , we will define a map m : Cth + (Σ , Σ ) ⊗ Cth + (Σ , Σ ) → Cth + (Σ , Σ ) and we prove that it satisfies the Leibniz rule. Let us denote the components of the product m by m kij , with i, j, k ∈ { + , , −} such that m kij takes as arguments a generator of type i in Cth + (Σ , Σ ) , a generator of type j in Cth + (Σ , Σ ) and has for output a generator of type k in Cth + (Σ , Σ ) . For example, m − is the component C (Λ +2 , Λ +1 ) † [ n − ⊗ C (Λ − , Λ − )[1] → CF (Σ , Σ ) . We define m as follows. First, the eight components corresponding to the map CF −∞ (Σ , Σ ) ⊗ CF −∞ (Σ , Σ ) → CF −∞ (Σ , Σ ) are the same components as those defining the product m −∞ in [Leg], we start by recall-ing its definition (see also Figure 10). For a pair of asymptotics ( a , a ) which is equal to ( x , x ) , ( x , γ ) , ( γ , x ) or ( γ , γ ) in CF −∞ (Σ , Σ ) ⊗ CF −∞ (Σ , Σ ) ( x ij is an inter-section point in Σ i ∩ Σ j and γ ij is a chord from Λ − i to Λ − j ) we have m ( a , a ) = X p ∈ Σ ∩ Σ , δ i M ( p ; δ , a , δ , a , δ ) ε − ( δ ) ε − ( δ ) ε − ( δ ) · p where the sum is over all intersection points p ∈ Σ ∩ Σ and words δ i of pure Reeb chords of Λ − i . Then, for a pair ( x , x ) ∈ CF (Σ , Σ ) ⊗ CF (Σ , Σ ) , we have m − ( x , x ) = X γ ,γ δ i , δ ′ i g M R × Λ − ( γ ; δ , γ , δ ) M ( γ ; δ ′ , x , δ ′ , x , δ ′ ) · ε − i ( δ i δ ′ i ) · γ + X γ ,γ ,γ δ i , δ ′ i , δ ′′ i g M R × Λ − ( γ ; δ , γ , δ , γ , δ ) M ( γ ; δ ′ , x , δ ′ ) M ( γ ; δ ′′ , x , δ ′′ ) · ε − i ( δ i δ ′ i δ ′′ i ) · γ ε − i ( δ i ) stands for the product of the augmentations applied to the corresponding purechords. For a pair ( x , γ ) ∈ CF (Σ , Σ ) ⊗ C (Λ − , Λ − ) , we have m − − ( x , γ ) = X γ ,γ δ i , δ ′ i g M R × Λ − ( γ ; δ , γ , δ ) M ( γ ; δ ′ , γ , δ ′ , x , δ ′ ) · ε − i ( δ i δ ′ i ) · γ + X γ ,γ δ i , δ ′ i g M R × Λ − ( γ ; δ , γ , δ , γ , δ ) M ( γ ; δ ′ , x , δ ′ ) · ε − i ( δ i δ ′ i ) · γ and the obvious symmetric formula for a pair ( γ , x ) , and finally for a pair of Reeb chords ( γ , γ ) ∈ C (Λ − , Λ − ) ⊗ C (Λ − , Λ − ) , m −−− ( γ , γ ) = X γ , δ i g M R × Λ − ( γ ; δ , γ , δ , γ , δ ) · ε − i ( δ i δ ′ i ) · γ Figure 10: Pseudo-holomorphic discs contributing to m −∞ .Then, let us define the remaining components of the map m , involving Reeb chords in thepositive end. First, the components m +00 , m +0 − , m + − , and m + −− vanish. It remains to define m k ++ , m k + i , and m ki + for i ∈ { , −} and k ∈ { + , , −} . Given a pair ( γ , γ ) ∈ C (Λ +2 , Λ +1 ) ⊗ C (Λ +1 , Λ +0 ) ,we have first m +++ ( γ , γ ) = X γ , ζ i g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) ε + i ( ζ i ) · γ + X γ ,γ ζ i , δ i g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) M ( γ ; δ , γ , δ ) · ε + i ( ζ i ) ε − i ( δ i ) · γ + X γ ,γ ζ i , δ i g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) M ( γ ; δ , γ , δ ) · ε + i ( ζ i ) ε − i ( δ i ) · γ summing over γ ij ∈ C (Λ + j , Λ + i ) , ζ i words of Reeb chords of Λ + i , for i = 0 , , , and δ i words ofReeb chords of Λ − i , for i = 0 , , . Then we have m ( γ , γ ) = X p , δ i M ( p ; δ , γ , δ , γ , δ ) ε − i ( δ i ) · p summing over p ∈ Σ ∩ Σ and δ i as above. And finally the last component of the product forthis pair of generators is: m − ++ ( γ , γ ) = X γ ,ξ δ i , δ ′ i g M R × Λ − ( γ ; δ , ξ , δ ) M ( ξ ; δ ′ , γ , δ ′ , γ , δ ′ ) ε − i ( δ i ) ε − i ( δ ′ i ) · γ + X γ ,ξ ,ξ δ i , δ ′ i , δ ′′ i g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) M ( ξ ; δ ′ , γ , δ ′ ) M ( ξ ; δ ′′ , γ , δ ′′ ) · ε − i ( δ i δ ′ i δ ′′ i ) · γ γ ∈ C (Λ − , Λ − ) , ξ ij ∈ C (Λ − j , Λ − i ) , and δ i , δ ′ i words of Reeb chords of Λ − i . Then,for a pair of generators ( γ , x ) ∈ C (Λ +2 , Λ +1 ) ⊗ CF (Σ , Σ ) we define: m ++0 ( γ , x ) = X γ ,γ ζ i , δ i g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) M ( γ ; δ , x , δ ) · ε + i ( ζ i ) ε − i ( δ i ) · γ m ( γ , x ) = X p , δ i M ( p ; δ , x , δ , γ , δ ) ε − i ( δ i ) · p m − +0 ( γ , x ) = X γ ,ξ δ i , δ ′ i g M R × Λ − ( γ ; δ , ξ , δ ) M ( ξ ; δ ′ , x , δ ′ , γ , δ ′ ) ε − i ( δ i ) ε − i ( δ ′ i ) · γ + X γ ,ξ ,ξ δ i , δ ′ i , δ ′′ i g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) M ( ξ ; δ ′ , x , δ ′ ) M ( ξ ; δ ′′ , γ , δ ′′ ) · ε − i ( δ i δ ′ i δ ′′ i ) · γ We finish by defining the product for a pair ( γ , γ ) ∈ C (Λ +2 , Λ +1 ) ⊗ C (Λ − , Λ − ) as follows: m ++ − ( γ , γ ) = X γ ,ξ ζ i , δ i g M R × Λ +012 ( γ ; ζ , ξ , ζ , γ , ζ ) M ( ξ ; δ , γ , δ ) · ε + i ( ζ i ) ε − i ( δ i ) · γ m − ( γ , γ ) = X p , δ i M ( p ; δ , γ , δ , γ , δ ) ε − i ( δ i ) · p m − + − ( γ , γ ) = X γ ,ξ δ i , δ ′ i g M R × Λ − ( γ ; δ , ξ , δ ) M ( ξ ; δ ′ , γ , δ ′ , γ , δ ′ ) ε − i ( δ i ) ε − i ( δ ′ i ) · γ + X γ ,ξ δ i , δ ′ i g M R × Λ − ( γ ; δ , γ , δ , ξ , δ ) M ( ξ ; δ ′ , γ , δ ′ ) · ε − i ( δ i δ ′ i ) · γ The components m k and m k − + for k = + , , − are defined analogously as m k +0 and m k + − . SeeFigures 11, 12 and 13. Figure 11: Pseudo-holomorphic discs contributing to m k ++ , k = + , , − . Theorem 5.
The map m satisfies the Leibniz rule, i.e. given three exact pairwise transverseLagrangian cobordisms Λ − i ≺ Σ i Λ + i with augmentations ε − i of A (Λ − i ) for i = 0 , , , we have: m ( m ε − ,ε − , · ) + m ( · , m ε − ,ε − ) + m ε − ,ε − ◦ m ( · , · ) = 0 Remark 4.
A “complete” notation for the product would be something like m Σ , Σ , Σ ε − ,ε − ,ε − as itdepends on the choice of cobordisms and on the choice of augmentations of the negative ends.However, to simplify, we will just write it m as the choices mentioned are clear from the context.34 ut out out out0 0 0 0 01 1 1 Figure 12: Pseudo-holomorphic discs contributing to m k +0 , k = + , , − . Figure 13: Pseudo-holomorphic discs contributing to m k + − , k = + , , − .As for m , we can write the components m +2 and m − as a composition of maps, it will beconvenient when describing the boundary of the compactification of 1-dimensional moduli spaces.First, we introduce the maps ∆ +2 : C ∗ (Λ +1 , Λ +2 ) ⊗ C ∗ (Λ +0 , Λ +1 ) → C n − −∗ (Λ +2 , Λ +0 )∆ Σ2 : Cth + (Σ , Σ ) ⊗ Cth + (Σ , Σ ) → C n − −∗ (Λ − , Λ − ) b − : C ∗ (Λ − , Λ − ) ⊗ C ∗ (Λ − , Λ − ) → C ∗− (Λ − , Λ − ) defined by ∆ +2 ( γ , γ ) = X γ , ζ i g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) ε + i ( ζ i ) · γ ∆ Σ2 ( a , a ) = X γ , δ i M ( γ ; δ , a , δ , a , δ ) ε − i ( δ i ) · γ b − ( γ , γ ) = X γ , δ i g M R × Λ − ( γ ; δ , γ , δ , γ , δ ) ε − i ( δ i ) · γ and observe that ∆ Σ2 vanishes on C (Λ , Λ ) ⊗ C (Λ , Λ ) for energy reasons. Using these maps,we have m +2 = ∆ +2 ( b Σ1 ⊗ b Σ1 ) (12) m − = b − ◦ ∆ Σ2 + b − ( ∆ Σ1 ⊗ ∆ Σ1 ) (13)where the maps b − , ∆ Σ1 are defined in Section 3.1 and b Σ1 in Section 3.2 (see also Section 3.2.3).35 .2 Leibniz rule The map m restricted to CF −∞ (Σ , Σ ) ⊗ CF −∞ (Σ , Σ ) satisfies the Leibniz rule because m −∞ satisfies it with respect to the differential m −∞ (see [Leg]) and there is no component ofthe differential m ε − ,ε − from the subcomplex CF −∞ (Σ , Σ ) to C (Λ +0 , Λ +1 ) . It remains to checkthe Leibniz rule for each pair of generators containing at least one Reeb chord in the positiveend:(a) ( γ , γ ) ∈ C (Λ +2 , Λ +1 ) ⊗ C (Λ +1 , Λ +0 ) ,(b) ( γ , x ) ∈ C (Λ +2 , Λ +1 ) ⊗ CF (Σ , Σ ) and ( x , γ ) ∈ CF (Σ , Σ ) ⊗ C (Λ +1 , Λ +0 ) ,(c) ( γ , γ ) ∈ C (Λ +2 , Λ +1 ) ⊗ C (Λ − , Λ − ) and ( ξ , γ ) ∈ C (Λ − , Λ − ) ⊗ C (Λ +1 , Λ +0 ) ,As usual, the Leibniz rule will follow from the study of the boundary of the compactificationof some (product of) moduli spaces. Recall that we described in Section 2.6 the different typesof broken discs arising in this boundary. We focus now on some particular moduli spaces andspecify the algebraic contribution of each broken disc. Leibniz rule for a pair of type (a):
For the pair of generators of type (a), we will show thatthe following three relations are satisfied: m +2 ( m ε − ,ε − ( γ ) , γ ) + m +2 ( γ , m ε − ,ε − ( γ )) + m +1 ◦ m +2 ( γ , γ ) = 0 (14) m ( m ε − ,ε − ( γ ) , γ ) + m ( γ , m ε − ,ε − ( γ )) + m ◦ m ( γ , γ ) (15) m − ( m ε − ,ε − ( γ ) , γ ) + m − ( γ , m ε − ,ε − ( γ )) + m − ◦ m ( γ , γ ) (16)After adding in (14) the vanishing terms m +1 ◦ m ( γ , γ ) = m +1 ◦ m − ( γ , γ ) = 0 , the sumof these three relations gives the Leibniz rule for the pair ( γ , γ ) ∈ C (Λ +1 , Λ +2 ) ⊗ C (Λ +0 , Λ +1 ) .First, we see that relation (14) follows from the study of the boundary of the compactificationof the following products of moduli spaces: g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) (17) g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) × M ( γ ; δ , γ , δ ) (18) g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) × M ( γ ; δ , γ , δ ) (19) g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) × M ( γ ; δ , γ , δ ) (20) g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) × M ( γ ; δ , γ , δ ) (21)The broken discs in ∂ M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) are schematized on Figure 14. The sumof their algebraic contributions vanishes, and thus gives: ∆ +2 ( m +1 ( γ ) , γ ) + ∆ +2 ( γ , m +1 ( γ )) + m +1 ◦ ∆ +2 ( γ , γ ) (22) + ∆ +2 ( b +1 ( γ ) , γ ) + ∆ +2 ( γ , b +1 ( γ )) = 0 The boundary of the compactification of (18), see Figure 15, gives the algebraic relation: ∆ +2 ( b Σ1 ( γ ) , m +1 ( γ )) + m +1 ◦ ∆ +2 ( b Σ1 ( γ ) , γ ) + ∆ +2 ( b +1 ◦ b Σ1 ( γ ) , γ ) = 0 (23)36 Figure 14: Types of broken discs in the boundary of M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) . Figure 15: Broken discs in ∂ M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) × M ( γ ; δ , γ , δ ) .One gets the symmetric relation ∆ +2 ( m +1 ( γ ) , b Σ1 ( γ )) + m +1 ◦ ∆ +2 ( γ , b Σ1 ( γ )) + ∆ +2 ( γ , b +1 ◦ b Σ1 ( γ )) = 0 (24)by studying the boundary of (20). Finally, one gets the relation ∆ +2 ( b Σ1 ◦ m +1 ( γ ) , γ ) + ∆ +2 ( b +1 ◦ b Σ1 ( γ ) , γ ) + ∆ +2 ( b +1 ( γ ) , γ ) (25) + ∆ +2 ( b Σ1 ◦ m ( γ ) , γ ) + ∆ +2 ( b Σ1 ◦ m − ( γ ) , γ ) = 0 and the symmetric ∆ +2 ( γ , b Σ1 ◦ m +1 ( γ )) + ∆ +2 ( γ , b +1 ◦ b Σ1 ( γ )) + ∆ +2 ( γ , b +1 ( γ )) (26) + ∆ +2 ( γ , b Σ1 ◦ m ( γ )) + ∆ +2 ( γ , b Σ1 ◦ m − ( γ )) = 0 by studying first (19) and then (21) (see Figure 16). Observe that for these last two, we con-sider the boundary of the compactification of moduli spaces of bananas with boundary on noncylindrical parts of the cobordisms and with two positive Reeb chord asymptotics, as we havedone already in the proof of Lemma 1. Summing (22), (23), (24), (25) and (26), cancelling termsappearing twice and using the definition of b Σ1 and m +2 given in (12), one obtains relation (14).Then, the study of the boundary of the compactification of M ( p ; δ , γ , δ , γ , δ ) gives relation (15), see Figure 17 for a description of broken discs. Indeed, the algebraic contri-butions of those discs are (from left to right and top to bottom on the figure): m ( m +1 ( γ ) , γ ) + m ( γ , m +1 ( γ )) + m ◦ ∆ +2 ( γ , γ ) + m ◦ ∆ +2 ( b Σ1 ( γ ) , γ )+ m ◦ ∆ +2 ( γ , b Σ1 ( γ )) + m ( m ( γ ) , γ ) + m ( γ , m ( γ )) + m ◦ m ( γ , γ )+ m ( m − ( γ ) , γ ) + m ( γ , m − ( γ )) + m ◦ b − ◦ ∆ Σ2 ( γ , γ ) + m ◦ b − (∆ Σ1 ( γ ) , ∆ Σ1 ( γ )) = 0
10 1001 0 1 10 0 10 011
Figure 16: Broken discs in g M R × Λ +012 ( γ ; ζ , γ , ζ , γ , ζ ) × ∂ M ( γ ; δ , γ , δ ) .And using the definitions of m +2 and m − given in (12) and (13) one deduces relation (15). Finally,
101 00 00 1 10 0 0 1 0 0000 0 0 0 Figure 17: Broken discs in ∂ M ( p ; δ , γ , δ , γ , δ ) .analogously to the previous cases, the broken curves in the boundary of the compactification of g M R × Λ − ( γ ; δ , γ , δ ) × M ( γ ; δ ′ , γ , δ ′ , γ , δ ′ ) (27) g M R × Λ − ( γ ; δ , γ , δ ) × M ( γ ; δ ′ , γ , δ ′ , γ , δ ′ ) (28) g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) × M ( ξ ; δ ′ , γ , δ ′ ) × M ( ξ ; δ ′′ , γ , δ ′′ ) (29) g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) × M ( ξ ; δ ′ , γ , δ ′ ) × M ( ξ ; δ ′′ , γ , δ ′′ ) (30) g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) × M ( ξ ; δ ′ , γ , δ ′ ) × M ( ξ ; δ ′′ , γ , δ ′′ ) (31)give relation (16). First, there are two types of broken discs arising in ∂ M R × Λ − ( γ ; δ , γ , δ ) giving the algebraic relation b − ◦ b − ( γ ) + b − ◦ ∆ − ( γ ) = 0 ∂ M ( γ ; δ ′ , γ , δ ′ , γ , δ ′ ) are schematized on Figure18. From this, the broken discs in ∂ M R × Λ − ( γ ; δ , γ , δ ) × M ( γ ; δ ′ , γ , δ ′ , γ , δ ′ ) contribute algebraically to b − ◦ b − ◦ ∆ Σ2 ( γ , γ ) + b − ◦ ∆ − ◦ ∆ Σ2 ( γ , γ ) (32)and the broken discs in the boundary of g M R × Λ − ( γ ; δ , γ , δ ) × ∂ M ( γ ; δ ′ , γ , δ ′ , γ , δ ′ ) to (from top to bottom and left to right on Figure 18) b − ◦ ∆ Σ2 ( m +1 ( γ ) , γ ) + b − ◦ ∆ Σ2 ( γ , m +1 ( γ )) + m − ◦ ∆ +2 ( γ , γ ) + m − ◦ ∆ +2 ( b Σ1 ( γ ) , γ ) + m − ◦ ∆ +2 ( γ , b Σ1 ( γ ))+ b − ◦ ∆ Σ2 ( m ( γ ) , γ ) + b − ◦ ∆ Σ2 ( γ , m ( γ )) + m − ◦ m ( γ , γ )+ b − ◦ ∆ Σ2 ( m − ( γ ) , γ ) + b − ◦ ∆ Σ2 ( γ , m − ( γ )) + b − ◦ ∆ − (∆ Σ1 ( γ ) , ∆ Σ1 ( γ )) + b − ◦ ∆ − ◦ ∆ Σ2 ( γ , γ ) Note that the three last terms on the first line give m − ◦ m +2 ( γ , γ ) by definition of m +2 .
101 1 0 111 100 0 0 0 00000 00 0 0 0 0 0 011
Figure 18: Broken discs in M ( γ ; δ ′ , γ , δ ′ , γ , δ ′ ) .Then all the terms starting with b − ◦ ∆ Σ2 contribute to m − ( m ( γ ) , γ ) + m − ( γ , m ( γ )) because recall that m − = b − ◦ ∆ Σ2 + b − ( ∆ Σ1 ⊗ ∆ Σ1 ) . Then, the configuration contributing to b − ◦ ∆ − (∆ Σ1 ( γ ) , ∆ Σ1 ( γ )) appears also in the boundary of the compactification of (29). Fi-nally, summing with (32) the last term b − ◦ ∆ − ◦ ∆ Σ2 ( γ , γ ) appears twice (so disappears!) andwe get the remaining term b − ◦ b − ◦ ∆ Σ2 ( γ , γ ) which contributes to m − ◦ m − ( γ , γ ) . Thestudy of the boundary of (29), (30) and (30) gives the other terms of the Leibniz rule. Leibniz rule for a pair of type (b):
Let us consider a pair ( γ , x ) of generators of type (b). The Leibniz rule for such a pairdecomposes into the following three relations: m +2 (∆ +1 ( γ ) , x ) + m +2 ( γ , m ( x )) + m +1 ◦ m +2 ( γ , x ) = 0 (33) m ( m ( γ ) , x ) + m ( γ , m ( x )) + m ◦ m ( γ , x ) (34) m − ( m ( γ ) , x ) + m − ( γ , m ( x )) + m − ◦ m ( γ , x ) (35)39here for (33) we make use of the fact that m +2 ( m ( γ ) , x )) and m +2 ( m − ( γ ) , x ) vanish bydefinition ( m +00 = m + − = 0 ). The study of the boundary of the compactification of the products g M R × Λ +012 ( γ ; ζ , ξ , ζ , γ , ζ ) × M ( ξ ; δ , x , δ ) g M R × Λ +012 ( γ ; ζ , ξ , ζ , γ , ζ ) × M ( ξ ; δ , x , δ ) gives relation (33). In order to get relation (34) we need to study the boundary of M ( p ; δ , x , δ , γ , δ ) and finally for relation (35), we study g M R × Λ − ( γ ; δ , ξ , δ ) × M ( ξ ; δ ′ , x , δ ′ , γ , δ ′ ) g M R × Λ − ( γ ; δ , ξ , δ ) × M ( ξ ; δ ′ , x , δ ′ , γ , δ ′ ) g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) × M ( ξ ; δ ′ , x , δ ′ ) × M ( ξ ; δ ′′ , γ , δ ′′ ) g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) × M ( ξ ; δ ′ , x , δ ′ ) × M ( ξ ; δ ′′ , γ , δ ′′ ) g M R × Λ − ( γ ; δ , ξ , δ , ξ , δ ) × M ( ξ ; δ ′ , x , δ ′ ) × M ( ξ ; δ ′′ , γ , δ ′′ ) Leibniz rule for a pair of type (c):
Finally, for a pair ( γ , γ ) of generators of type (c), we decompose the Leibniz rule into: m +2 (∆ +1 ( γ ) , γ ) + m +2 ( γ , m ( γ )) + m +1 ◦ m +2 ( γ , γ ) = 0 (36) m ( m ( γ ) , γ ) + m ( γ , m ( γ )) + m ◦ m ( γ , γ ) (37) m − ( m ( γ ) , γ ) + m − ( γ , m ( γ )) + m − ◦ m ( γ , γ ) (38)and observe that one of the two terms contributing to m − ( γ , m ( γ )) , namely b − (∆ Σ1 ( γ ) , ∆ Σ1 ◦ m ( γ )) , vanishes for energy reasons. Relations (36), (37) and (38) are obtained respectively bystudying the boundary of the compactification of g M R × Λ +012 ( γ ; ζ , ξ , ζ , γ , ζ ) × M ( ξ ; δ , γ , δ ) g M R × Λ +012 ( γ ; ζ , ξ , ζ , γ , ζ ) × M ( ξ ; δ , γ , δ ) of M ( p ; δ , γ , δ , γ , δ ) , and of g M R × Λ − ( γ ; δ , ξ , δ ) × M ( ξ ; δ ′ , γ , δ ′ , γ , δ ′ ) g M R × Λ − ( γ ; δ , ξ , δ ) × M ( ξ ; δ ′ , γ , δ ′ , γ , δ ′ ) g M R × Λ − ( γ ; δ , γ , δ , ξ , δ ) × M ( ξ ; δ ′ , γ , δ ′ ) g M R × Λ − ( γ ; δ , γ , δ , ξ , δ ) × M ( ξ ; δ ′ , γ , δ ′ ) Given a pair of concatenation ( V ⊙ W , V ⊙ W ) , we denote m V , m W the differentials of thecomplexes Cth + ( V , V ) and Cth + ( W , W ) respectively. Given a third concatenation V ⊙ W ,40e denote again m V and m W the differentials on complexes Cth + ( V i , V j ) and Cth + ( W i , W j ) respectively, for ≤ i = j ≤ , without specifying the pair of cobordisms when it is clear from thecontext. Moreover, we will use the transfer maps b V i ,V j : Cth + ( V i ⊙ W i , V j ⊙ W j ) → Cth + ( W i , W j ) and ∆ W i ,W j : Cth + ( V i ⊙ W i , V j ⊙ W j ) → Cth + ( V i , V j ) and will shorten the notations to b V and ∆ W as there should not be any risk of confusion about which pair of cobordisms is involved in thedomain and codomain. Finally, we denote m V , m W the products Cth + ( V , V ) ⊗ Cth + ( V , V ) → Cth + ( V , V ) and Cth + ( W , W ) ⊗ Cth + ( W , W ) → Cth + ( W , W ) respectively. We now definea product: m V ⊙ W : Cth + ( V ⊙ W , V ⊙ W ) ⊗ Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V ⊙ W , V ⊙ W ) Using maps we already defined before, as well as the two inputs banana b V with boundary on V ∪ V ∪ V (we encountered in Section 5.1 the two inputs banana b − with boundary on cylindricalends), defined by b V : Cth + ( V , V ) ⊗ Cth + ( V , V ) → C ∗− (Λ , Λ ) b V ( a , a ) = X γ , δ i M V ( γ ; δ , a , δ , a , δ ) · ε − i ( δ i ) · γ we set: m V ⊙ W = m W, +02 (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, − (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) where m W, +0 i = m W, + i + m W, i , i = 1 , is the component of m Wi with values in C (Λ +2 , Λ +0 ) ⊕ CF ( W , W ) , and m V, − i = m V, i + m V, − i , i = 1 , , is the component of m Vi with values in CF ( V , V ) ⊕ C (Λ − , Λ − ) . Observe that m W, +1 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) and m W, +1 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) vanish, but we keep it in the formula to make it look more homogeneous, which helps a bit tocheck the Leibniz rule in the next section. This section is dedicated in proving that the map m V ⊙ W satisfies the Leibniz rule with respectto m V ⊙ W . This is just computation. We want to show m V ⊙ W ( m V ⊙ W ⊗ id) + m V ⊙ W (id ⊗ m V ⊙ W ) + m V ⊙ W ◦ m V ⊙ W = 0 We will actually decompose it into two equations: m V ⊙ W, +0 W ( m V ⊙ W ⊗ id) + m V ⊙ W, +0 W (id ⊗ m V ⊙ W ) + m V ⊙ W, +0 W ◦ m V ⊙ W = 0 (39) m V ⊙ W, V − ( m V ⊙ W ⊗ id) + m V ⊙ W, V − (id ⊗ m V ⊙ W ) + m V ⊙ W, V − ◦ m V ⊙ W = 0 (40)The first one corresponds to the components of the Leibniz rule taking values in C (Λ +2 , Λ +0 ) ⊕ CF ( W , W ) , and the second one to the components taking values in CF ( V , V ) ⊕ C (Λ − , Λ − ) .In the proof of the Leibniz rule, we will refer to the following equations: m W, +02 ( m W ⊗ id) + m W, +02 (id ⊗ m W ) + m W, +01 ◦ m W = 0 (41) m V, − ( m V ⊗ id) + m V, − (id ⊗ m V ) + m V, − ◦ m V = 0 (42) ∆ W ( m W ⊗ id) + ∆ W (id ⊗ m W ) + ∆ W ◦ m W +∆ Λ2 ( ∆ W ⊗ ∆ W ) + ∆ Λ1 ◦ ∆ W = 0 (43) b V ( m V ⊗ id) + b V (id ⊗ m V ) + b V ◦ m V + b Λ2 ( b V ⊗ b V ) + b Λ1 ◦ b V = 0 (44) b V ◦ ∆ W = ∆ W ◦ b V (45)41quations (41) and (42) come from the fact that m W and m V satisfy the Leibniz rule. Equations(43) and (44) (for other Lagrangian boundary conditions) appear implicitly in Section 5.2: theycome respectively from the study the boundary of the compactification of moduli spaces M W ( γ ; δ W , a W , δ W , a W , δ W ) and M V ( γ , δ V , a V , δ V , a V , δ V ) , for γ ∈ C (Λ , Λ ) , γ ∈ C (Λ , Λ ) , ( a W , a W ) ∈ Cth + ( W , W ) ⊗ Cth + ( W , W ) , ( a V , a V ) ∈ Cth + ( V , V ) ⊗ Cth + ( V , V ) , δ Wi words of pure Reeb chords of Λ i , δ Vi words of pure Reeb chordsof Λ − i . Finally, Equation (45) is the content of Lemma 2. (39)Let us write the left-hand side of Equation (39) as (LR1), i.e. (39) ⇔ (LR1)=0. We start bydeveloping the first term of (LR1), using the definition of m V ⊙ W and the fact that b V and ∆ W are chain maps: m V ⊙ W, +0 W ( m V ⊙ W ⊗ id)= (cid:0) m W, +02 + m W, +01 ◦ b V ◦ ∆ W (cid:1)h b V ◦ m V ⊙ W ⊗ b V i + m W, +01 ◦ b V h ∆ W ◦ m V ⊙ W ⊗ ∆ W i = (cid:0) m W, +02 + m W, +01 ◦ b V ◦ ∆ W (cid:1)h m W ◦ b V ⊗ b V i + m W, +01 ◦ b V h m V ◦ ∆ W ⊗ ∆ W i = m W, +02 (cid:0) m W ⊗ id (cid:1)(cid:2) b V ⊗ b V (cid:3) + m W, +01 ◦ b V ◦ ∆ W (cid:0) m W ⊗ id (cid:1)(cid:2) b V ⊗ b V (cid:3) + m W, +01 ◦ b V (cid:0) m V ⊗ id (cid:1)(cid:2) ∆ W ⊗ ∆ W (cid:3) One decomposes similarly the symmetric term m V ⊙ W, +0 W (id ⊗ m V ⊙ W ) . Now let us take a lookat m V ⊙ W, +0 W ◦ m V ⊙ W . We have m V ⊙ W, +0 W ◦ m V ⊙ W = m W, +01 ◦ b V ◦ m V ⊙ W = m W, +01 ◦ b V (cid:0) m V ⊙ W, +0 W + m V ⊙ W, V − (cid:1) = m W, +01 (cid:0) m V ⊙ W, +0 W + b V ◦ ∆ W ◦ m V ⊙ W, +0 W + b V ◦ m V ⊙ W, V − (cid:1) = (cid:0) m W, +01 + m W, +01 ◦ b V ◦ ∆ W (cid:1)h m W, +02 (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1)i + m W, +01 ◦ b V h m V, − (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1)i The term m W, +01 ◦ b V ◦ ∆ W ◦ m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) vanishes for energy reasons, as well as m W, +01 ◦ b V ◦ ∆ W ◦ m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) , hence we finally get: m V ⊙ W, +0 W ◦ m V ⊙ W = m W, +01 ◦ m W, +02 (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V ◦ ∆ W ◦ m W, +02 (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) + m W, +01 ◦ b V ◦ m V, − (cid:0) ∆ W ⊗ ∆ W (cid:1) + m W, +01 ◦ b V ◦ m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) = (cid:2) m W, +01 ◦ m W, +02 + m W, +02 ( m W ⊗ id) + m W, +02 (id ⊗ m W ) (cid:3)(cid:0) b V ⊗ b V (cid:1) (L1) + (cid:2) m W, +01 ◦ b V (cid:3)(cid:2) ∆ W ◦ m W, +02 +∆ W ( m W ⊗ id) + ∆ W (id ⊗ m W ) (cid:3)(cid:0) b V ⊗ b V (cid:1) (L2) + m W, +01 ◦ m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (L3) + m W, +01 ◦ m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) (L4) + m W, +01 (cid:2) b V ◦ m V, − + b V ( m V ⊗ id) + b V (id ⊗ m V ) (cid:3)(cid:0) ∆ W ⊗ ∆ W (cid:1) (L5) + m W, +01 ◦ b V ◦ m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (L6)Now we use Equation (41) on (L1), Equation (43) on (L2), the fact that m W, +01 ◦ m W, +01 = m W, +01 ◦ m W, − on (L3) and (L4) and finally Equation (44) on (L5), to write(LR1) = m W, +01 ◦ m W, − (cid:0) b V ⊗ b V (cid:1) (L1’) + m W, +01 ◦ b V (cid:2) ∆ W ◦ m W, − +∆ Λ2 (cid:0) ∆ W ⊗ ∆ W ) + ∆ Λ1 ◦ ∆ W (cid:3)(cid:0) b V ⊗ b V (cid:1) (L2’) + m W, +01 ◦ m W, − ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (L3’) + m W, +01 ◦ m W, − ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) (L4’) + m W, +01 (cid:2) b V ◦ m V, +2 + b Λ2 ( b V ⊗ b V ) + b Λ1 ◦ b V (cid:3)(cid:0) ∆ W ⊗ ∆ W (cid:1) (L5’) + m W, +01 ◦ b V ◦ m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (L6’)We apply then the following modifications:1. on (L1’) we write m W, − = b Λ1 ◦ ∆ W + b Λ2 ( ∆ W ⊗ ∆ W ) ,2. on (L2’) observe that ∆ W ◦ m W, − vanishes for energy reasons,3. on (L3’) and (L4’) we have m W, − ◦ b V = b Λ1 ◦ ∆ W ◦ b V = b Λ1 ◦ b V and m W, − ◦ b V = b Λ1 ◦ ∆ W ◦ b V = b Λ1 ◦ b V ,4. on (L5’), we write m V, +2 = ∆ Λ2 ( b V ⊗ b V ) ,5. finally, in the last term of (L2’) we have ∆ Λ1 = m V, +1 so adding it to (L6’) gives m W, +01 ◦ b V ◦ m V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) . But observe that by definition of ∆ W one has b V ◦ m V ◦ ∆ W = b V ◦ m V ◦ ∆ W ◦ ∆ W , which gives by Lemma 1 b Λ1 ◦ ∆ W ◦ b V ◦ ∆ W . We thus get m W, +01 ◦ b V ◦ m V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) = m W, +01 ◦ b Λ1 ◦ ∆ W ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) which, using Equation (45), is equal to: m W, +01 ◦ b Λ1 ◦ b V ◦ ∆ W ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) = m W, +01 ◦ b Λ1 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) = (cid:2) m W, +01 ◦ b Λ1 (cid:3)(cid:2) ∆ W + b V ◦ ∆ W (cid:3)(cid:0) b V ⊗ b V (cid:1)
43o finally we have:(LR1) = m W, +01 ◦ b Λ1 ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (R1) + m W, +01 ◦ b Λ2 (cid:0) ∆ W ◦ b V ⊗ ∆ W ◦ b V (cid:1) (R2) + m W, +01 ◦ b V ◦ ∆ Λ2 (cid:0) ∆ W ◦ b V ⊗ ∆ W ◦ b V (cid:1) (R3) + m W, +01 ◦ b Λ1 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (R4) + m W, +01 ◦ b Λ1 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) (R5) + m W, +01 ◦ b V ◦ ∆ Λ2 (cid:0) b V ◦ ∆ W ⊗ b V ◦ ∆ W (cid:1) (R6) + m W, +01 ◦ b Λ2 (cid:0) b V ◦ ∆ W ⊗ b V ◦ ∆ W (cid:1) (R7) + m W, +01 ◦ b Λ1 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) (R8) + m W, +01 ◦ b Λ1 ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b Λ1 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (R9)We have (R1)+(R4)+(R9)=0 and (R5)+(R8)=0. Then, using Equation (45) gives (R2)+(R7)=0and (R3)+(R6)=0. Thus (LR1)=0. (40)Denote (LR2) the left-hand side of Equation (40) so that this equation is equivalent to (LR2)=0.Using again the fact that b V and ∆ W are chain maps, the first term of (LR2) is: m V ⊙ W, V − ( m V ⊙ W ⊗ id)= m V, − (cid:0) ∆ W ◦ m V ⊙ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W (cid:0) b V ◦ m V ⊙ W ⊗ b V (cid:1) = m V, − (cid:0) m V ◦ ∆ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W (cid:0) m W ◦ b V ⊗ b V (cid:1) = m V, − (cid:0) m V ⊗ id (cid:1)(cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W (cid:0) m W ⊗ id (cid:1)(cid:0) b V ⊗ b V (cid:1) One writes analogously the symmetric term m V ⊙ W, V − (id ⊗ m V ⊙ W ) . Now let us consider thethird term of (LR2): m V ⊙ W, V − ◦ m V ⊙ W = m V, − ◦ ∆ W ◦ m V ⊙ W = m V, − ◦ ∆ W ◦ m V ⊙ W, +0 W + m V, − ◦ m V ⊙ W, V − = m V, − ◦ ∆ W h m W, +02 (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1)i + m V, − ◦ m V, − (cid:2) ∆ W ⊗ ∆ W (cid:3) + m V, − ◦ m V, − ◦ ∆ W (cid:2) b V ⊗ b V (cid:3) The term m V, − ◦ ∆ W h m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1)i vanishes for energyreasons. Then, observe that m V, − ◦ m V, − = m V, − ◦ m V, +1 and ∆ W ◦ m W, +02 = ∆ W ◦ m W because ∆ W ◦ m W, − = 0 , so we have: m V ⊙ W, V − ◦ m V ⊙ W = m V, − ◦ ∆ W ◦ m W (cid:0) b V ⊗ b V (cid:1) + m V, − ◦ m V, − (cid:2) ∆ W ⊗ ∆ W (cid:3) + m V, − ◦ m V, +1 ◦ ∆ W (cid:2) b V ⊗ b V (cid:3) Summing all together gives(LR2) = m V, − (cid:2) ∆ W ◦ m W +∆ W ( m W ⊗ id) + ∆ W (id ⊗ m W ) (cid:3)(cid:0) b V ⊗ b V (cid:1) (L1) + (cid:2) m V, − ◦ m V, − + m V, − ( m V ⊗ id) + m V, − (id ⊗ m V ) (cid:3)(cid:0) ∆ W ⊗ ∆ W (cid:1) (L2) + m V, − ◦ m V, +1 ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) = m V, − (cid:2) ∆ Λ2 ( ∆ W ⊗ ∆ W ) + ∆ Λ1 ◦ ∆ W (cid:3)(cid:0) b V ⊗ b V (cid:1) (L1’) + m V, − ◦ m V, +2 (cid:2) ∆ W ⊗ ∆ W (cid:3) (L2’) + m V, − ◦ m V, +1 ◦ ∆ W (cid:2) b V ⊗ b V (cid:3) (T3)But remark that m V, +1 = ∆ Λ1 and m V, +2 = ∆ Λ2 ( b V ⊗ b V ) by definition so then by Equation (45),we get (LR2)=0. In this section, we prove that the product structures behave well under the transfer maps b V and ∆ W . Namely, we have first the following: Proposition 8.
The map induced by b V in homology preserves the product structures, in otherwords we have: b V ◦ m V ⊙ W = m W (cid:0) b V ⊗ b V (cid:1) in homology.Proof. Given a triple ( V ⊙ W , V ⊙ W , V ⊙ W ) , we define a map b V : Cth + ( V ⊙ W , V ⊙ W ) ⊗ Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( W , W ) by b V = b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + b V (cid:0) ∆ W ⊗ ∆ W (cid:1) In order to prove the proposition, we prove that the following relation is satisfied: b V (cid:0) m V ⊙ W ⊗ id (cid:1) + b V (cid:0) id ⊗ m V ⊙ W (cid:1) + b V ◦ m V ⊙ W + m W (cid:0) b V ⊗ b V (cid:1) + m W ◦ b V = 0 (46)Let us first consider b V (cid:0) m V ⊙ W ⊗ id (cid:1) . We have b V (cid:0) m V ⊙ W ⊗ id (cid:1) = b V ◦ ∆ W (cid:2) b V ◦ m V ⊙ W ⊗ b V (cid:3) + b V (cid:2) ∆ W ◦ m V ⊙ W ⊗ ∆ W (cid:3) = b V ◦ ∆ W (cid:0) m W ◦ b V ⊗ b V (cid:1) + b V (cid:0) m V ◦ ∆ W ⊗ ∆ W (cid:1) = b V ◦ ∆ W (cid:0) m W ⊗ id (cid:1)(cid:0) b V ⊗ b V (cid:1) + b V (cid:0) m V ⊗ id (cid:1)(cid:0) ∆ W ⊗ ∆ W (cid:1) Then we consider b V ◦ m V ⊙ W . Observe that we have already computed this term in Section 6.2.1when considering the term m V ⊙ W, +0 W ◦ m V ⊙ W . So recall that we have b V ◦ m V ⊙ W = m W, +02 (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) + b V ◦ ∆ W ◦ m W, +02 (cid:0) b V ⊗ b V (cid:1) + b V ◦ m V, − (cid:0) ∆ W ⊗ ∆ W (cid:1) + b V ◦ m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) The left-hand side of (46), rearranging terms according to the decompositions above is thus givenby b V (cid:0) ∆ W (cid:0) m W ⊗ id (cid:1) + ∆ W (cid:0) id ⊗ m W (cid:1) + ∆ W ◦ m W, +02 (cid:1)(cid:0) b V ⊗ b V (cid:1) (L1) + (cid:0) b V (cid:0) m V ⊗ id (cid:1) + b V (cid:0) id ⊗ m V (cid:1) + b V ◦ m V, − (cid:1)(cid:0) ∆ W ⊗ ∆ W (cid:1) (L2) + (cid:0) m W, +02 + m W (cid:1)(cid:0) b V ⊗ b V (cid:1) (L3) + (cid:0) m W, +01 ◦ b V ◦ ∆ W + m W ◦ b V ◦ ∆ W (cid:1)(cid:0) b V ⊗ b V (cid:1) (L4) + (cid:0) m W, +01 ◦ b V + m W ◦ b V (cid:1)(cid:0) ∆ W ⊗ ∆ W (cid:1) (L5) + b V ◦ m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (L6)45e use now Equation (43) on (L1), Equation (44) on (L2) and the same modification as thepoint 5. on Section 6.2.1 on line (L6) to rewrite: (cid:0) b V ◦ ∆ Λ2 (cid:0) ∆ W ⊗ ∆ W (cid:1) + b V ◦ ∆ Λ1 ◦ ∆ W (cid:1)(cid:0) b V ⊗ b V (cid:1) + (cid:0) b V ◦ m V, +2 + b Λ2 ( b V ⊗ b V ) + b Λ1 ◦ b V (cid:1)(cid:0) ∆ W ⊗ ∆ W (cid:1) + m W, − (cid:0) b V ⊗ b V (cid:1) + m W, − ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m W, − ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) + (cid:0) b V ◦ m V, +1 ◦ ∆ W + b Λ1 ◦ b V ◦ ∆ W (cid:1)(cid:0) b V ⊗ b V (cid:1) Finally, using1. m V, +2 = ∆ Λ2 ( b V ⊗ b V ) and m V, +1 = ∆ Λ1 ,2. m W, − ◦ b V = b Λ1 ◦ ∆ W ◦ b V = b Λ1 ◦ b V , and also m W, − ◦ b V = b Λ1 ◦ b V ,3. b Λ1 ◦ b V ◦ ∆ W = b Λ1 ◦ ∆ W + b Λ1 ◦ b V ◦ ∆ W we rewrite (cid:0) b V ◦ ∆ Λ2 (cid:0) ∆ W ⊗ ∆ W (cid:1) + b V ◦ ∆ Λ1 ◦ ∆ W (cid:1)(cid:0) b V ⊗ b V (cid:1) + (cid:0) b V ◦ ∆ Λ2 ( b V ⊗ b V ) + b Λ2 ( b V ⊗ b V ) + b Λ1 ◦ b V (cid:1)(cid:0) ∆ W ⊗ ∆ W (cid:1) + (cid:0) b Λ1 ◦ ∆ W + b Λ2 ( ∆ W ⊗ ∆ W ) (cid:1)(cid:0) b V ⊗ b V (cid:1) + b Λ1 ◦ b V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + b Λ1 ◦ b V (cid:0) ∆ W ⊗ ∆ W (cid:1) + (cid:0) b V ◦ ∆ Λ1 ◦ ∆ W + b Λ1 ◦ ∆ W + b Λ1 ◦ b V ◦ ∆ W (cid:1)(cid:0) b V ⊗ b V (cid:1) and making use of Equation (45), all the terms in the sum cancel by pair.The same functorial property applies for the map ∆ W : Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V , V ) .Indeed, we have Proposition 9.
The map induced by ∆ W in homology preserves the product structures, that isto say: ∆ W ◦ m V ⊙ W = m V (cid:0) ∆ W ⊗ ∆ W (cid:1) in homology.Proof. Given a triple ( V ⊙ W , V ⊙ W , V ⊙ W ) , we define a map ∆ W : Cth + ( V ⊙ W , V ⊙ W ) ⊗ Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V , V ) by ∆ W = ∆ W (cid:0) b V ⊗ b V (cid:1) where the map ∆ W was defined in Section 5.1 for the case of three pairwise transverse Lagrangiancobordisms. In order to prove the proposition, we prove that the following relation is satisfied: ∆ W (cid:0) m V ⊙ W ⊗ id (cid:1) + ∆ W (cid:0) id ⊗ m V ⊙ W (cid:1) + ∆ W ◦ m V ⊙ W + m V (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V ◦ ∆ W = 0 (47)First, we have ∆ W (cid:0) m V ⊙ W ⊗ id (cid:1) =∆ W (cid:0) b V ◦ m V ⊙ W ⊗ b V (cid:1) = ∆ W (cid:0) m W ⊗ id (cid:1)(cid:0) b V ⊗ b V (cid:1) ∆ W ◦ m V ⊙ W when considering m V ⊙ W, V − ◦ m V ⊙ W in Section 6.2.2. Recall that we have ∆ W ◦ m V ⊙ W =∆ W ◦ m W, +02 (cid:0) b V ⊗ b V (cid:1) + m V, − (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) Hence, the left-hand side of Equation (47) is equal to: (cid:0) ∆ W ( m W ⊗ id) + ∆ W (id ⊗ m W ) + ∆ W ◦ m W, +02 (cid:1)(cid:0) b V ⊗ b V (cid:1) (L1) + m V, − (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (L2) + m V (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) (L3)Using Equation (43) on line (L1) and summing (L2) and (L3) gives ∆ W ◦ m W, − (cid:0) b V ⊗ b V (cid:1) + ∆ Λ2 (cid:0) ∆ W ◦ b V ⊗ ∆ W ◦ b V (cid:1) + ∆ Λ1 ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) + m V, +2 (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, +1 ◦ ∆ W (cid:0) b V ⊗ b V (cid:1) Observe that ∆ W ◦ m W, − (cid:0) b V ⊗ b V (cid:1) = 0 for energy reasons. Then, m V, +1 = ∆ Λ1 and m V, +2 =∆ Λ2 (cid:0) b V ⊗ b V (cid:1) , so using Equation (45) one gets that the terms sum to .Observe that given the maps b V and ∆ V defined in the proofs of Propositions 8 and 9, wecan rewrite the formula of the product m V ⊙ W as follows: m V ⊙ W = m W, +02 (cid:0) b V ⊗ b V (cid:1) + m W, +01 ◦ b V + m V, − (cid:0) ∆ W ⊗ ∆ W (cid:1) + m V, − ◦ ∆ W Moreover, if we restrict again to the special cases where the triples ( W , W , W ) or ( V , V , V ) are trivial cylinders, one has1. ( W , W , W ) = ( R × Λ , R × Λ , R × Λ ) : The map b V becomes a map b V : Cth + ( V , V ) ⊗ Cth + ( V , V ) → C (Λ , Λ ) which is equal to the map b V for the case of three pairwise transverse Lagrangian cobordisms ( V , V , V ) , and ∆ W vanishes.2. ( V , V , V ) = ( R × Λ , R × Λ , R × Λ ) : in this case the map b V vanishes and ∆ W is a map ∆ W : Cth + ( W , W ) ⊗ Cth + ( W , W ) → C (Λ , Λ ) which is actually equal to the map ∆ W for the case of three pairwise transverse Lagrangiancobordisms ( W , W , W ) . Again let Σ be an exact Lagrangian cobordism from Λ − to Λ +0 with A (Λ − ) admitting anaugmentation ε − . In this section we prove that there is a continuation element e ∈ Cth + (Σ , Σ ) ,where Σ is a suitable small Hamiltonian perturbation of Σ . Assume Σ is cylindrical outside [ − T, T ] × Y . Fix η > smaller than the length of any chord of Λ − and Λ +0 , and N > . Then47e set Σ := ϕ η e H (Σ ) for a Hamiltonian e H : R × ( P × R ) → R being a small perturbation of H ( t, p, z ) = h T,N ( t ) for h T,N : R → R satisfying h T,N ( t ) = − e t for t < − T − Nh T,N ( t ) = − e t + C for t > T + Nh ′ T,N ( t ) ≤ − T, T ] ⊂ ( h ′ ) − (0) for a positive constant C , and whose corresponding Hamiltonian vector field is given by ρ T,N ∂ z ,with ρ T,N : R → R satisfying ρ ( t ) = − for t ≤ − T − N and t ≥ T + N , ρ ( t ) = 0 for t ∈ [ − T, T ] , ρ ′ ≥ in [ − T − N, − T ] and ρ ′ ≤ in [ T, T + N ] . Moreover, under an appropriate identificationof a tubular neighborhood of Σ with a standard neighborhood of the -section in T ∗ Σ , see[DR16, Section 6.2.2], we assume that Σ is given by the graph of dF in T ∗ Σ , with F : Σ → R Morse function satisfying under this identification the following properties:1. the critical points of F (in one-to-one correspondence with intersection points in Σ ∩ Σ )are all contained in ( − T, T ) × Y .2. on the cylindrical ends of Σ , F is equal to e t ( f ± − η ) , for f ± : Λ ± → R Morse functionssuch that the C -norm of f ± is much smaller than η . In other words, it means that thecylindrical ends of Σ ∪ Σ are cylinders over the -copy Λ ± ∪ Λ ± where Λ ± is a Morseperturbation of Λ ± − η∂ z (translation of Λ ± by η in the negative Reeb direction). Moreover,we assume that f − admits a unique minimum on each connected component.3. F admits a unique minimum on each filling component of Σ and has no minimum on eachcomponent of Σ with a non empty negative end.See Figure 19 for a schematization of the -copy Σ ∪ Σ . The Chekanov-Eliashberg algebras T-T-T-NT+N
Figure 19: Schematization of the perturbation Σ of Σ . A (Λ − ) and A (Λ − ) are canonically identified and thus an augmentation ε − of A (Λ − ) can be seenas an augmentation of A (Λ − ) . Moreover, for η small enough and Morse functions F, f ± such48hat Σ is sufficiently C -close to Σ , one has ε − ◦ Φ Σ = ε − ◦ Φ Σ , see [CDRGG15, Theorem 2.15].Let us denote e = e + e − , where e = P e i is the sum of the minima e i of F and e − = P e − i isthe sum of the minima e − i of f − , where the sum is indexed over the connected components of Σ .Each e − i corresponds to a Reeb chord from Λ − to Λ − . Proposition 10.
We have m Σ ( e ) = 0 , i.e. e is a cycle.Proof. We develop m Σ ( e ) = P m ( e i ) + P m ( e − i ) + P m − ( e i ) + m − ( e − i ) .First, for all i m − ( e i ) = 0 for energy reasons because given the Hamiltonian perturbation wechoose all intersection points have positive action (we assume the perturbation from H to e H is C -small).Then, we prove that P m − ( e − i ) = 0 . This results from the analysis of pseudo-holomorphicdiscs with boundary on a -copy of a Legendrian done in [EES09], as well as the fact that whenthe almost complex structure is the cylindrical lift of an admissible complex structure on P ,there is a bijection between rigid pseudo-holomorphic discs with boundary on π P (Λ) and rigidpseudo-holomorphic discs with boundary on R × Λ , as proved in [DR16]. Recall that m − ( e − i ) is defined by a count of rigid pseudo-holomorphic discs with boundary on R × (Λ − ∪ Λ − ) , witha negative asymptotic to e − i and a positive asymptotic to an output Reeb chord from Λ − to Λ − . In particular we distinguish two cases: either the output Reeb chord, call it β is a Morsechord, or it is not. If it is Morse, then such a disc has no pure Reeb chords asymptotics foraction reasons and corresponds to a gradient flow line of the Morse function f − flowing downfrom the critical point corresponding to β to the one corresponding to e − i . There are exactlytwo such flow lines. If β is not a Morse chord, we refer to [EES09, Theorem 5.5]: a rigid discwith positive asymptotic at β and negative asymptotic at e − i corresponds to a rigid generalizeddisc with boundary on R × Λ − . By rigidity, this generalized disc consists of a constant disc u at β (pure chord of Λ − corresponding to the chord β of the -copy) together with a gradient flowline starting on the boundary of u and flowing down to e − i . There are two ways this descendinggradient flow line can be attached to u : either on the starting point of β , or on its ending point.Thus we get that the contribution of β to P m − ( e − i ) is given by ε − ( β ) + ε − ( β ) = 0 .Finally, we prove P m ( e i ) + m ( e − i ) = 0 . Wrap the negative end of Σ slightly in thepositive Reeb direction using the Hamiltonian vector field ρ − T + N,N ∂ z (see Section 4.1). Let V be the image of R × Λ − by the corresponding time- s − flow where s − is bigger than the longestMorse chord from Λ − to Λ − but much smaller than the shortest non Morse chord from Λ − to Λ − . We set V = R × Λ − . Observe that each Morse chord becomes an intersection point in V ∩ V . We denote m i the intersection point corresponding to e − i , see Figure 20. Consider thepair of concatenations ( V ⊙ Σ , V ⊙ Σ ) . By projecting curves on P as done in Section 4.1, onecan prove that b V ( m i ) = e − i . Then, by definition of the differential in a concatenation, m V ⊙ Σ1 ( m i ) = m Σ , +01 ◦ b V ( m i ) + m V, − ◦ ∆ Σ1 ( m i ) which gives m V ⊙ Σ , Σ ( m i ) = m Σ , ◦ b V ( m i ) = m Σ , ( e − i ) . The Hamiltonian used to wrap thenegative end of Σ is assumed to be sufficiently small so that V ⊙ Σ is a small Morse perturbationof V ⊙ Σ by a function e F which equals F on [ − T, T ] × Y ∩ Σ . Then, there is a one-to-onecorrespondence between curves contributing to m V ⊙ Σ , Σ ( m i ) and gradient flow lines of e F from acritical point in Σ to the critical point of e F corresponding to m i . Now, each intersection pointin Σ ∩ Σ which corresponds to an index critical point of F and so also of e F , is the startingpoint of two descending gradient flow lines flowing down to a minimum e i or m i . Thus, modulo we have P m ( e i ) + m ( e − i ) = 0 . 49 -T-T-NT+N-T -2N-1 Figure 20: Schematization of the wrapping of the negative of Σ . Theorem 6.
Consider Σ and Σ as above. Take Λ − ≺ Σ Λ +2 another exact Lagrangian cobor-dism such that the intersection of Σ with a small standard neighborhood of Σ identified with D ε T ∗ Σ and containing also Σ , consists of a union of fibres, then we have: m Σ ( · , e ) : Cth + (Σ , Σ ) → Cth + (Σ , Σ ) (48) is an isomorphism. Remark 5.
Given Σ and a transverse cobordism Σ , one can always find a sufficiently smallperturbation Σ of Σ such that the intersection of Σ with D ε T ∗ Σ which contains Σ , consistsof a union of fibres. This way, there is a canonical identification of vector spaces Cth + (Σ , Σ ) ∼ =Cth + (Σ , Σ ) , and for a generator γ , x or γ in Cth + (Σ , Σ ) , one denotes respectively γ , x or γ the corresponding generator in Cth + (Σ , Σ ) . Remark 6.
Proposition 10 states that the element e is a cycle. Given Theorem 6 one gets thatit is a boundary if and only if Cth + (Σ , Σ ) is acyclic for every cobordism Σ satisfying thehypothesis of the theorem, as proved in [CDRGGb, Lemma 4.17]. Proof of Theorem 6.
First we write
Cth + (Σ , Σ ) = C (Λ +2 , Λ +1 ) † [ n − ⊕ CF − (Σ , Σ ) ⊕ C (Λ − , Λ − ) ⊕ CF + (Σ , Σ ) (49)where CF ± (Σ , Σ ) ⊂ CF (Σ , Σ ) is the sub-vector space generated by positive, resp. neg-ative action intersection points. According to this decomposition, ordering Reeb chords in C (Λ +2 , Λ +1 ) † [ n − from biggest to smallest action and intersection points in CF − (Σ , Σ ) ⊕ CF + (Σ , Σ ) from smallest to biggest action, we will show that the matrix of the map (48) islower triangular with identity terms on the diagonal. For γ ∈ C (Λ +2 , Λ +1 ) , we have m ( γ , e ) = m +2 ( γ , e ) + m ( γ , e ) + m − ( γ , e )= ∆ +2 (cid:0) γ , b Σ1 ( e ) (cid:1) + m ( γ , e ) + b − ◦ ∆ Σ2 ( γ , e ) + b − (cid:0) ∆ Σ1 ( γ ) , e − (cid:1) + b − (cid:0) ∆ Σ1 ( γ ) , ∆ Σ1 ( e ) (cid:1) = ∆ +2 (cid:0) γ , b Σ1 ( e ) (cid:1) + m ( γ , e ) + b − ◦ ∆ Σ2 ( γ , e ) + b − (cid:0) ∆ Σ1 ( γ ) , e − (cid:1) ∆ Σ1 ( e i ) vanishes for energy reasons. On Figure 21 we schema-tized pseudo-holomorphic configurations contributing to m ( γ , e ) . Let i denote the index of theconnected component of Σ containing the starting point of γ . Note that by the hypothesis onthe Morse function F , if this component has a non-empty negative end only the configurationsA, B, C, and D are relevant, whereas if it is a filling component then only the configurations A’,B’ and C’ are. We consider only the first case, the proof for the other one being similar. We willprove that m ( γ , e ) = γ + ζ + y − + ξ + y +02 where γ ∈ C (Λ +2 , Λ +0 ) is the Reeb chord canonically identified to γ , ζ ∈ C (Λ +2 , Λ +0 ) isa linear combination of Reeb chords whose action are smaller than the action of γ , y ± ∈ CF ± (Σ , Σ ) and ξ ∈ C (Λ − , Λ − ) . A B C D0 (cid:0) (cid:1) (cid:2)(cid:3) B(cid:4) (cid:5)(cid:6) Figure 21: Types of curves (potentially) contributing to m ( γ , e ) .Let us consider the configurations of type A. Denote v a rigid disc with boundary on thepositive cylindrical ends, with a positive asymptotic to γ , a negative Reeb chord asymptotic β ∈ C (Λ +0 , Λ +1 ) , and an output negative Reeb chord asymptotic γ out ∈ C (Λ +2 , Λ +0 ) , and u arigid disc with boundary on Σ ∪ Σ with a positive asymptotic to β and a negative asymptoticto a minimum Morse Reeb chord e − i . We distinguish two cases: either β is a Morse chord, orit is not a Morse chord.(a) If β is a Morse chord. First, rigidity implies that | β | = | e − i | = − , and thus β corresponds to the (only one by assumption) minimum of f + on the component of Λ +0 containing the starting point of γ . Then for action reasons the disc u has no pure Reebchords asymptotics. Similarly as in the proof of Proposition 10 we show that the countsuch discs u coincides with the count of some rigid gradient flow lines of a Morse function e F which equals F on Σ ∩ ([ − T, T ] × Y ) . To get this correspondence, wrap the negativeand the positive ends of Σ slightly in the positive Reeb direction: take V , resp W , to bethe image of R × Λ − , resp R × Λ +1 , by the time s − , resp s + , flow of the Hamiltonian vectorfield ρ − T + N,N ∂ z , resp − ρ + T + N,N ∂ z , with s ± bigger than the longest Morse chord from Λ ± to Λ ± but smaller than the shortest non Morse chord from Λ ± to Λ ± . See Figure 22 for aschematization of the perturbation. This way, e − i corresponds canonically an intersectionpoint m i ∈ CF ( V , V ) and β corresponds to an intersection point x β ∈ CF ( W , W ) . Asbefore, by projecting discs on P one can prove that b V ( m i ) = e − i and m W, ( β ) = x β .51 T -2N-1 T(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)
Figure 22: Schematization of wrapping of the negative and positive ends of Σ .Now, by definition of the differential for the pairs of concatenated cobordisms ( V ⊙ (Σ ⊙ W ) , V ⊙ (Σ ⊙ W )) and (Σ ⊙ W , Σ ⊙ W ) one has m V ⊙ (Σ ⊙ W )1 ( m i ) = m Σ ⊙ W, +01 ◦ b V ( m i ) + m V, − ◦ ∆ Σ ⊙ W ( m i ) (50)and m Σ ⊙ W, +01 ◦ b V ( m i ) = m W, +01 ◦ b Σ1 ◦ b V ( m i ) + m Σ , ◦ ∆ W ◦ b V ( m i ) Considering the components with values in CF ( W , W ) on both sides of 50 gives m V ⊙ (Σ ⊙ W ) , W ( m i ) = m W, ◦ b Σ1 ◦ b V ( m i ) = m W, ◦ b Σ1 ◦ b V ( m i ) = m W, ◦ b Σ1 ( e − i ) The coefficient of β in b Σ1 ( e − i ) is thus equal to the coefficient of x β in m V ⊙ (Σ ⊙ W ) , W ( m i ) .The wrapping of Σ being sufficiently small, one can view V ⊙ Σ ⊙ W as a Morseperturbation of Σ by a Morse function e F which is equal to F in the non-cylindricalpart of Σ . Thus, pseudo-holomorphic strips asymptotic to m i and x β are in one-to-onecorrespondence with gradient flow lines of e F from the critical point corresponding to x β tothe one corresponding to m i , and there exists exactly one such gradient flow line. Indeed, x β is the starting point of two gradient flow lines of d e F , but according to the perturbationwe performed one of them has the t coordinate going to + ∞ while the other one flowsdown to m i .It remains to understand the pseudo-holomorphic disc v with boundary on R × (Λ +0 ∪ Λ +1 ∪ Λ +2 ) with a positive asymptotic to γ and negative asymptotics to a minimum Morsechord β and a chord γ out ∈ C (Λ +2 , Λ +0 ) . Again by [EES09, Theorem 5.5], such a rigiddisc corresponds to a rigid generalized disc, which in this case is a disc with boundaryon R × (Λ +0 ∪ Λ +2 ) with a gradient flow line of f + flowing from a point on the boundaryof the disc in R × Λ +0 to the minimum corresponding to the chord β . By rigidity, onehas γ out = γ . Conversely, following the flow of df + from the starting point of γ leadsto the minimum of f + on the corresponding connected component. Such a flow line is ageneralized disc which corresponds to such a disc v with boundary on R × (Λ +0 ∪ Λ +1 ∪ Λ +2 ) .We have proved that the coefficient of γ in m ( γ , e ) is .52b) If β is not a Morse chord. Given R ≥ such that the three cobordisms Σ , Σ and Σ arecylindrical outside of [ − R, R ] × Y , the energy of the disc v with boundary on the positivecylindrical ends is given by E ( v ) = a ( γ ) − a ( β ) − a ( γ out ) − X i =0 a ( δ i ) with a ( γ ) = e R ℓ ( γ )+ c − c , a ( β ) = e R ℓ ( β )+ c − c and a ( γ out ) = e R ℓ ( γ out )+ c − c .One can check that | a ( γ ) − a ( γ ) | ≤ e R (max k f + k C + η ) + c − c | a ( β ) − a ( β ) | ≤ e R (max k f + k C + η ) + c − c and thus a ( γ ) − a ( γ out ) ≥ E ( v ) + a ( β ) − (cid:0) e R (max k f + k C + η ) + c − c (cid:1) and for η sufficiently small, the term on the right hand side is strictly positive, so the actionof γ out is strictly smaller than that of γ .Thus, together with the curves of type B, C, and D (actually one can prove that the configu-ration D never happens, by projecting on P the curve with boundary on the negative cylindricalends), we obtain as expected m ( γ , e ) = γ + ζ + y − + ξ + y +02 For x ∈ CF (Σ , Σ ) = CF + (Σ , Σ ) ⊕ CF − (Σ ∪ Σ ) we have m ( x , e ) = m ( x , e ) + m − ( x , e )= m ( x , e ) + b − ◦ ∆ Σ2 ( x , e ) + b − (cid:0) ∆ Σ1 ( x ) , e − (cid:1) + b − (cid:0) ∆ Σ1 ( x ) , ∆ Σ1 ( e ) (cid:1) = m ( x , e ) + b − ◦ ∆ Σ2 ( x , e ) + b − (cid:0) ∆ Σ1 ( x ) , e − (cid:1) see Figure 23, and we will prove that for x +12 ∈ CF + (Σ , Σ ) and x − ∈ CF − (Σ , Σ ) , one has m ( x +12 , e ) = x +02 + y +02 , and m ( x − , e ) = x − + z − + ξ + z +02 where y +02 , z +02 ∈ CF + (Σ , Σ ) , z − ∈ CF − (Σ , Σ ) and ξ ∈ C (Λ − , Λ − ) , and each intersectionpoint in y +02 , resp. z − , has action strictly bigger than x +02 , resp. x − .As for the previous case, we assume that x is an intersection point of Σ with the i-thconnected component of Σ having a non empty negative end. Then, only configurations E, Fand G are relevant. We consider first configurations of type E. Let u be a pseudo-holomorphic discwith boundary on Σ ∪ Σ ∪ Σ negatively asymptotic to e − i , and asymptotic to x ∈ CF (Σ , Σ ) and an output x out ∈ CF (Σ , Σ ) . The action of intersection points are assumed to be muchsmaller than that of pure Reeb chords, so u has no pure Reeb chord asymptotes.To understand what can be the output of such a disc, we wrap as before the negative endof Σ slightly in the positive Reeb direction to get the pair ( V ⊙ Σ , V ⊙ Σ ) where the MorseReeb chords e − j in C (Λ − , Λ − ) correspond to intersection points m j in CF ( V , V ) and b V ( m j ) =
10 0 0
E F G(cid:12) (cid:13) (cid:14)(cid:15) (cid:16)(cid:17) Figure 23: Curves contributing to m ( x , e ) . b V ( m j ) = e − j . By definition of the product for a pair of concatenated cobordisms (see Section 6)we have: m V ⊙ Σ , Σ ( x , m i )= m Σ , ( b V ( x ) , b V ( m i )) + m Σ , ◦ b V ◦ ∆ Σ2 ( b V ( x ) , b V ( m i )) + m Σ , ◦ b V ( ∆ Σ1 ( x ) , ∆ Σ1 ( m i ))= m Σ , ( b V ( x ) , e − i ) + m Σ , ◦ b V ◦ ∆ Σ2 ( b V ( x ) , e − i ) + m Σ , ◦ b V (∆ Σ1 ( x ) , m i )= m Σ , ( x , e − i ) + m Σ , ( b V ◦ ∆ Σ1 ( x ) , e − i ) + m Σ , ◦ b V ◦ ∆ Σ2 ( x , e − i ) + m Σ , ◦ b V (∆ Σ1 ( x ) , m i ) See Figure 24. All these terms except the first one involve bananas with two positive Reebchord asymptotics and with boundary on V ∪ V ∪ V where V = R × Λ − , V is a wrappingof R × Λ − and V := R × Λ − . These rigid bananas project to rigid discs with boundary on π P (Λ − ∪ Λ − ∪ Λ − ) and for dimension reasons they must be constant. This is not possible asthey all have two distinct positive Reeb chord asymptotics (a constant curve with boundary on π P (Λ − ∪ Λ − ∪ Λ − ) does not lift to a banana with two positive asymptotics but on a trivial strip).So we are left with m V ⊙ Σ , Σ ( x , m i ) = m Σ , ( x , e − i ) .Let us denote again e F the Morse function such that V ⊙ Σ is viewed as a -jet perturbationof Σ by e F , and e F equals F on Σ ∩ ([ − T, T ] × Y ) . The intersection point m i is a minimum of e F and the gradient flow line of e F flowing from x to m i corresponds to a pseudo-holomorphictriangle asymptotic to x , m i and x (if the component of Σ containing x is a filling, thenwe don’t need to wrap the negative end and consider the one-to-one correspondence betweengradient flow lines of F from x to e i and pseudo-holomorphic triangles with vertices x , e i and x ). Thus the coefficient of x in m Σ , ( x , e − i ) is . Note also that the energy of thistriangle is given by E ( u ) = a ( x ) − a ( e − i ) − a ( x ) (51)and by definition of the action one can check that it can be made as small as possible by takingsmaller η .Now suppose there is another pseudo-holomorphic triangle with asymptotics x , e − i and y = x , contributing to the coefficient of y in m ( x , e ) . This triangle necessary leaves asmall neighborhood of the gradient flow line from x to m i and thus according to the relation(51) between the energy of such a triangle and the action of its asymptotics, the action of y isstrictly bigger than the action of x , independently of how small η is.Then, about configurations of type F and G, observe that a disc with boundary on thenon-cylindrical parts in such configurations exists only if the action of x is negative.54 Figure 24: Curves contributing to m V ⊙ Σ , Σ ( x, m i ) .To sum up, the configurations of type E, F, and G (but for the same reasons as in it canbe proved that G configurations never happen) give that m ( x +12 , e ) = x +02 + y +02 , and m ( x − , e ) = x − + z − + ξ + z +02 Finally, for ξ ∈ C (Λ − , Λ − ) we have m ( ξ , e ) = m ( ξ , e ) + m − ( ξ , e )= m ( ξ , e − ) + b − ( ξ , e − ) because the Morse function F as no minima e i on the component of Σ involved as this componenthas a non empty negative end. See Figure 25. For energy reasons, if a disc of type H exists thenthe output intersection point must have positive action. Then a disc of type I is such that ξ out isthe chord in C (Λ − , Λ − ) canonically identified with ξ ∈ C (Λ − , Λ − ) , by [EES09, Theorem 5.5].So we have m ( ξ , e ) = ξ + y +02 . I(cid:18)H
Figure 25: Curves contributing to m ( ξ , e ) Remark 7.
Generalizing the conjectural [Ekh12, Lemma 4.10] to the case of cobordisms, onecould probably prove that with the choice of basis given above the matrix of m ( · , e ) is actuallythe identity matrix. However we don’t need such a strong statement on the chain level here, butwhat we get is that it is the identity in homology (see details at end of the current section).We will apply now the previous theorem to a -copy (Σ , Σ , Σ ) of Σ . By -copy we mean Σ , resp Σ , is viewed as the graph of dF , resp dF , in a standard neighborhood of Σ and Σ is viewed as the graph of dF in a standard neighborhood of Σ , for Morse functions F , F , F satisfying the properties listed at the beginning of the Section. We have the following:55 orollary 1. Given the -copy (Σ , Σ , Σ ) described above, we have m ( e Σ , Σ , e Σ , Σ ) = e Σ , Σ Proof.
It is enough to consider the case where Σ is connected. The case of a filling is alreadyknown, see for example [GPS20]. We recall a proof in our setting. Assume Σ is a connectedfilling of Λ +0 , then we have e Σ , Σ = e , Σ and e Σ , Σ = e , Σ and according to Theorem 6: m ( e , Σ , e , Σ ) = e , Σ + y +02 where y +02 ∈ CF + (Σ , Σ ) , and each element in y +02 has action bigger than the action of e , Σ .Observe then that any triangle asymptotic to y +02 = e , Σ , e , Σ , and e , Σ would have to leavea small neighborhood of a gradient flow line of F from e , Σ to e , Σ . But for a sufficientlysmall perturbation no y +02 = e , Σ has action big enough for such a triangle to exist.Suppose now that Σ is a connected cobordism from Λ − = ∅ to Λ +0 . Then e Σ , Σ = e − Σ , Σ and e Σ , Σ = e − Σ , Σ and according to Theorem 6: m ( e − Σ , Σ , e − Σ , Σ ) = e − Σ , Σ + y +02 The proof is the same as in the filling case after wrapping slightly the negative ends of Σ and Σ in the positive Reeb direction. We wrap so that the negative end becomes a cylinder over Λ − ∪ e Λ − ∪ e Λ where e Λ − is a small push-off of Λ − in the positive Reeb direction and e Λ − is a smallpush-off of e Λ − in the positive Reeb direction. The pseudo-holomorphic disc asymptotic to e − Σ , Σ , e − Σ , Σ and e − Σ , Σ corresponds after wrapping to a triangle asymptotic to the correspondingintersection points. So then, for the same reasons as before a disc asymptotic to y +02 , e − Σ , Σ and e − Σ , Σ can not exist.We end this section by proving that the transfer maps preserve the continuation element.Consider a pair ( V ⊙ W , V ⊙ W ) such that V ⊙ W is a small perturbation of ( V ⊙ W ) the same way as we perturbed Σ to get Σ previously, in particular Λ ± is a perturbation of apush-off of Λ ± in the negative Reeb direction. We assume moreover that the Morse function F used to perturb the compact part of V ⊙ W is such that ( V , V ) and ( W , W ) are also pairsof cobordisms of the same type, so Λ is a perturbation of a push-off of Λ in the negative Reebdirection.Giving this, by what we did previously, there are continuation elements e V ∈ Cth + ( V , V ) , e W ∈ Cth + ( W , W ) and e V ⊙ W ∈ Cth + ( V ⊙ W , V ⊙ W ) , described as follows e V = e V + e − V = X e V,i + X e − V,i e W = e W + e − W = X e W,i + X e − W,i e V ⊙ W = e W + e V + e − V Proposition 11.
The transfer map ∆ W preserves the continuation element.Proof. Directly from the definition, one has ∆ W ( e V ⊙ W ) = ∆ W ( e W + e V + e − V ) = ∆ W ( e W ) + e V + e − V = e V + e − V where the last equality holds for energy reason.56 roposition 12. The transfer map b V preserves the continuation element in homology, i.e. [ b V ( e V ⊙ W )] = [ e W ] .Proof. Observe first that b V ( e V ⊙ W ) = b V ( e W + e V + e − V ) = e W + b V ◦ ∆ W ( e W ) + b V ( e V + e − V ) = e W + b V ( e V + e − V ) for energy reasons. Now, wrapping slightly the positive and negative cylindrical ends of V inthe positive Reeb direction one can prove that b V ( e V + e − V ) = e − W + E − W , where E − W ∈ C (Λ , Λ ) is a linear combination of non Morse chords (same type of argument as in the proof of Theorem6). Now, take a third copy V ⊙ W as in Corollary 1 such that ( V , V , V ) and ( W , W , W ) arealso -copies. We claim that the map m W ( · , b V ( e V ⊙ W )) : Cth + ( W , W ) → Cth + ( W , W ) (52)is a quasi-isomorphism. Again this follows from studying the pseudo-holomorphic curves involved,repeating some arguments of the proof of Theorem 6. As we are working over a field, it admitsan inverse.Consider finally a fourth copy V ⊙ W being a perturbation of V ⊙ W using the same typeof perturbation as before. From the third A ∞ -relation satisfied by m W (see Section 8.1), the factthat m V ⊙ W ( e V ⊙ W , e V ⊙ W ) = e V ⊙ W , and the fact that b V preserves the product structuresin homology, we get that the maps m W (cid:0) m W ( · , b V ( e V ⊙ W )) , b V ( e V ⊙ W ) (cid:1) : Cth + ( W , W ) → Cth + ( W , W ) m W ( · , b V ( e V ⊙ W )) : Cth + ( W , W ) → Cth + ( W , W ) are homotopic. It implies that the map (52) is homotopic to the identity map (after canonicalidentification of the generators of the complexes Cth + ( W , W ) and Cth + ( W , W ) ). Finally, as e W is a continuation element we have m W ( e W , b V ( e V ⊙ W )) = m W ( e W , e W + E − W ) = m W ( e W , e W ) + m W ( e W , E − W )= e W + m W, ( e W , E − W ) + m W, − ( e W , E − W )= e W + m W, − ( e − W , E − W )= e W + E − W where the second to last equality comes from the fact that the connected components of thecobordisms W , W have non empty negative end so there is no minimum of the perturbationMorse function, so m W, − ( e W , E − W ) = 0 , and then m W, ( e − W , E − W ) = 0 for action reasons. Wehave thus m W ( e W , b V ( e V ⊙ W )) = b V ( e V ⊙ W ) . As (52) is the identity in homology, we get [ e W ] =[ b V ( e V ⊙ W )] . Remark 8.
Observe that the same arguments show that m Σ ( · , e ) : Cth + (Σ , Σ ) → Cth + (Σ , Σ ) is the identity in homology as we have proved that it is an isomorphism (Theorem 6) and that m ( e Σ , Σ , e Σ , Σ ) = e Σ , Σ (Corollary 1). 57 emark 9. In Sections 8 and 9 we will extend the algebraic structures we have encounteredto A ∞ ones. In particular we will define an A ∞ -category of cobordisms in R × Y , F uk( R × Y ) , and generalize the transfer maps to families of maps satisfying the A ∞ -functor equations.Once the technical details to extend our algebraic constructions to Lagrangian cobordisms in amore general Liouville cobordism are carried out, the transfer maps will provide A ∞ -functors F uk dec ( X ⊙ X ) → F uk( X i ) from the full subcategory F uk dec ( X ⊙ X ) ⊂ F uk( X ⊙ X ) generated by decomposable Lagrangian cobordisms to the Fukaya category of each cobordism.By Proposition 11 and 12 these functors will be cohomologically unital. A ∞ -category of Lagrangian cobordisms In this section, we extend the differential m Σ1 and the product m Σ2 to families of maps m Σ d definedfor each ( d + 1) -tuple of pairwise transverse exact Lagrangian cobordisms (Σ , . . . , Σ d ) for all d ≥ . Remember that we denote C (Λ ± i , Λ ± j ) = C n − −∗ (Λ ± i , Λ ± j ) ⊕ C ∗− (Λ ± j , Λ ± i ) . We definefirst six families of maps, b + d , b − d , ∆ + d , ∆ − d , b Σ d and ∆ Σ d : b ± d : C (Λ ± d − , Λ ± d ) ⊗ C (Λ ± d − , Λ ± d − ) ⊗ · · · ⊗ C (Λ ± , Λ ± ) → C ∗− (Λ ± , Λ ± d )∆ ± d : C (Λ ± d − , Λ ± d ) ⊗ C (Λ ± d − , Λ ± d − ) ⊗ · · · ⊗ C (Λ ± , Λ ± ) → C n − −∗ (Λ ± d , Λ ± ) b Σ d : Cth + (Σ d − , Σ d ) ⊗ Cth + (Σ d − , Σ d − ) ⊗ · · · ⊗ Cth + (Σ , Σ ) → C ∗− (Λ +0 , Λ + d )∆ Σ d : Cth + (Σ d − , Σ d ) ⊗ Cth + (Σ d − , Σ d − ) ⊗ · · · ⊗ Cth + (Σ , Σ ) → C n − −∗ (Λ − d , Λ − ) as follows: b + d ( a d , . . . , a ) = X γ d, X ζ i g M R × Λ +0 ,...,d ( γ d, ; ζ , a , . . . , a d , ζ d ) · ε + i ( ζ i ) · γ d, b − d ( a d , . . . , a ) = X γ d, X δ i g M R × Λ − ,...,d ( γ d, ; δ , a , . . . , a d , δ d ) · ε − i ( δ i ) · γ d, ∆ + d ( a d , . . . , a ) = X γ ,d X ζ i g M R × Λ +0 ,...,d ( γ ,d ; ζ , a , . . . , a d , ζ d ) · ε + i ( ζ i ) · γ ,d ∆ − d ( a d , . . . , a ) = X γ ,d X δ i g M R × Λ − ,...,d ( γ ,d ; δ , a , . . . , a d , δ d ) · ε − i ( δ i ) · γ ,d b Σ d ( a d , . . . , a ) = X γ d, X δ i M ,...,d ( γ d, ; δ , a , . . . , a d , δ d ) · ε − i ( δ i ) · γ d, ∆ Σ d ( a d , . . . , a ) = X γ ,d X δ i M ,...,d ( γ ,d ; δ , a , . . . , a d , δ d ) · ε − i ( δ i ) · γ ,d Observe that these maps for the case d = 1 have already been considered in Section 3, and ∆ +2 , ∆ Σ2 and b − have been defined in Section 5.1 already.Given these families of maps, we define the higher order maps m d as being the sum m d = + d + m d + m − d , where each component is defined by: m + d ( a d , . . . , a ) = d X j =1 X i + ··· + i j = d ∆ + j (cid:0) b Σ i j ( a d , . . . , a d − i j +1 ) , . . . , b Σ i ( a i +1 , . . . , a ) (cid:1) (53) m d ( a d , . . . , a ) = X x ∈ Σ ∩ Σ d X δ i M ,...,d ( x ; δ , a , δ , . . . , a d , δ d ) · ε − i ( δ i ) · x (54) m − d ( a d , . . . , a ) = d X j =1 X i + ··· + i j = d b − j (cid:0) ∆ Σ i j ( a d , . . . , a d − i j +1 ) , . . . , ∆ Σ i ( a i +1 , . . . , a ) (cid:1) (55)where the maps b Σ1 and ∆ Σ1 are special cases of transfer maps as explained in Section 3.2.3, andfor j ≥ one has b Σ j := b Σ j and ∆ Σ j := ∆ Σ j . In the formulas above, for ≤ j ≤ d fixed and anindex i s , the maps b Σ i s and ∆ Σ i s are defined on (with convention i = − ): Cth + (Σ i s + ··· + i , Σ i s + ··· + i ) ⊗ · · · ⊗ Cth + (Σ i s − + ··· + i , Σ i s − + ··· + i ) and the maps ∆ + j and b − j on Cth + (Σ i j − + ··· + i , Σ d ) ⊗ · · · ⊗ Cth + (Σ , Σ i +1 ) For d = 1 , , the Formulas 53, 54 and 55 recover the definitions of the differential m and theproduct m given in Sections 3.1 and 5.1. Remark 10.
Observe that for energy reasons, depending on the d -tuple of asymptotics, it canhappen that a lot of terms in the Formulas (53) and (55) vanish, but for example, if a i is a Reebchord in C (Λ + i +1 , Λ + i ) for i = 0 , . . . d , then none of them vanish. Remark 11.
The maps b + d and ∆ − d defined previously are not useful to define the maps m d butthey naturally appear in the proof of the A ∞ -equations, see Sections 8.1.1 and 8.1.3 below.Now we want to show that the maps { m d } d ≥ satisfy the A ∞ -equations, i.e. for all k ≥ andall ( k + 1) -tuple of transverse cobordisms (Σ , . . . , Σ k ) , we want to check that for every ≤ d ≤ k and ( d + 1) -sub-tuple (Σ i , . . . , Σ i d ) with i < · · · < i d , we have: d X j =1 d − j X n =0 m d − j +1 (id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n ) = 0 To simplify notations in the following we assume that the ( d +1) -tuple (Σ i , . . . , Σ i d ) is (Σ , . . . , Σ d ) .As usual, we decompose this equation into three equations to check: d X j =1 d − j X n =0 m + d − j +1 (id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n ) = 0 (56) d X j =1 d − j X n =0 m d − j +1 (id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n ) = 0 (57) d X j =1 d − j X n =0 m − d − j +1 (id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n ) = 0 (58)59 .1.1 Proof of Equation (56)Consider the boundary of the compactification of g M R × Λ +0 ,...,d ( γ ,d ; ζ , a , . . . , a d , ζ d ) . Accordingto the compactness results for one dimensional moduli spaces of pseudo-holomorphic discs withcylindrical Lagrangian boundary conditions as recalled in Section 2.6, the non trivial componentsof broken discs in the boundary consist of two index discs glued along a node asymptotic to aReeb chord. If it is a positive asymptotic for the index disc not containing the output puncture,this disc contributes to a map b + j , and if it is a negative asymptotic, this disc contributes to amap ∆ + j . Hence we get the following: Lemma 3.
For all ≤ d ≤ k , we have d P j =1 d − j P n =0 ∆ + d − j +1 (cid:0) id ⊗ d − j − n ⊗ ( b + j + ∆ + j ) ⊗ id ⊗ n (cid:1) = 0 Then, we also have:
Lemma 4.
For all ≤ d ≤ k , we have d X j =1 d − j X n =0 b Σ d − j +1 (cid:0) id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n (cid:1) + d X j =1 X i + ··· + i j = d b + j (cid:0) b Σ i j ⊗ · · · ⊗ b Σ i (cid:1) = 0 Proof.
This time we have to consider the boundary of the compactification of a moduli space M ,...,d ( γ d, ; δ , a , . . . , a d , δ d ) . Again as recalled in Section 2.6, the broken discs are of twotypes. It can first consist of two index discs glued at a common intersection point. In thiscase, the one not containing the output puncture asymptotic to γ d, contributes m j , and thedisc containing the output contributes to a banana b Σ d − j +1 . The other type of possible brokendisc consists of several (possibly 0 !) non trivial index components and an index disc withboundary on the negative or positive cylindrical ends, such that each index disc is connectedto the index one via a Reeb chord. Observe that the output puncture is asymptotic to a chordin the positive end, so there are two subcases:1. the output puncture is contained in the index disc. In this case, this disc has boundaryon the positive ends and contributes to b + j while the index discs must then have at leastone positive Reeb chord asymptotic (connecting it to the index disc) and so each of themcontributes to a banana b Σ . Note that if among the asymptotics a , a , . . . , a d there is achord in the positive end, this chord could be an asymptotic of the index disc or of anindex banana b Σ , this is why we have the bold symbols maps b Σ in the formula.2. the output puncture is contained in an index disc. This disc contributes thus to a map b Σ d − j +1 . Then, if the index disc has boundary in the positive ends, it contributes, withthe index discs not containing the output, to m + j . If the index disc has boundary onthe negative ends, it will contribute, with the index discs not containing the output, to m − j .Summing the algebraic contributions of all the different types of broken discs described abovegives the relation. 60ow we can compute d X j =1 d − j X n =0 m + d − j +1 (id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n )= d X j =1 d − j X n =0 d − j +1 X k =1 k X s =1 X i + ... + i k = d − j +10 ≤ r = n − i − ... − i s − ≤ i s ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ b Σ i s (id ⊗ i s − j − r ⊗ m j ⊗ id ⊗ r ) ⊗ · · · ⊗ b Σ i (cid:1) In this sum, we fix first the number j of entries for the map m j , and then a partition of d − j + 1 for the maps b Σ . Note that if i s < j the terms b Σ i s (id ⊗ i s − j − r ⊗ m j ⊗ id ⊗ r ) vanish. We could alsochoose first a partition of d and then the number of entries for the m "in the middle". Thus, thesum above is equal to: = d X k =1 X i + ... + i k = d k X s =1 i s X j =1 i s − j X n =0 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ b Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) ⊗ · · · ⊗ b Σ i (cid:1) = d X k =1 X i + ... + i k = d k X s =1 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ i s X j =1 i s − j X n =0 b Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) ⊗ · · · ⊗ b Σ i (cid:1) Using the definition of b and then Lemma 4, we have i s X j =1 i s − j X n =0 b Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) = i s X j =1 i s − j X n =0 b Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) + m + i s = i s X u =1 X t + ··· + t u = i s b + u (cid:0) b Σ t u ⊗ · · · ⊗ b Σ t (cid:1) + m + i s Given this, we rewrite d X j =1 d − j X n =0 m + d − j +1 (id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n )= d X k =1 X i + ... + i k = d k X s =1 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ i s X u =1 X t + ··· + t u = i s b + u (cid:0) b Σ t u ⊗ · · · ⊗ b Σ t (cid:1) ⊗ · · · ⊗ b Σ i (cid:1) + d X k =1 X i + ... + i k = d k X s =1 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ m + i s ⊗ · · · ⊗ b Σ i (cid:1) d X j =1 d − j X n =0 m + d − j +1 (id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n )= d X k =1 X i + ... + i k = d k X s =1 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ i s X u =1 X t + ··· + t u = i s ∆ + u (cid:0) b Σ t u ⊗ · · · ⊗ b Σ t (cid:1) ⊗ · · · ⊗ b Σ i (cid:1) + d X k =1 X i + ... + i k = d k X s =1 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ m + i s ⊗ · · · ⊗ b Σ i (cid:1) = d X k =1 X i + ... + i k = d k X s =1 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ m + i s ⊗ · · · ⊗ b Σ i (cid:1) + d X k =1 X i + ... + i k = d k X s =1 ∆ + k (cid:0) b Σ i k ⊗ · · · ⊗ m + i s ⊗ · · · ⊗ b Σ i (cid:1) = 0 (57)This equation is obtained after describing the broken discs in the boundary of the compactifica-tion of M ,...,d ( x ; δ , a , δ , . . . , a d , δ d ) . As well as in the proof of Lemma 4, there are differenttypes of broken discs, depending on if it contains an index component or not, but the totalalgebraic contribution of them gives the relation (57). (58)Finally, to get Equation (58), we study the broken discs in the boundary of the compactificationof the moduli spaces g M R × Λ − ,...,d ( γ d ; δ , a , . . . , a d , δ d ) (59)and M ,...,d ( γ d ; δ , a , . . . , a d , δ d ) (60)This gives us respectively the following lemmas: Lemma 5.
For all d ≥ , we have d P j =1 d − j P n =0 b − d − j +1 (cid:0) id ⊗ d − j − n ⊗ ( b − j + ∆ − j ) ⊗ id ⊗ n (cid:1) = 0 and Lemma 6.
For all d ≥ , we have d X j =1 d − j X n =0 ∆ Σ d − j +1 (cid:0) id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n (cid:1) + d X j =1 X i + ··· + i j = d ∆ − j (cid:0) ∆ Σ i j ⊗ · · · ⊗ ∆ Σ i (cid:1) = 0
62e can now prove Equation (58) for d ≥ in a direct way: d X j =1 d − j X n =0 m − d − j +1 (cid:0) id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n (cid:1) = d X j =1 d − j X n =0 d − j +1 X k =1 k X s =1 X i + ... + i k = d − j +10 ≤ r = n − i − ... − i s − ≤ i s b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ ∆ Σ i s (id ⊗ i s − j − r ⊗ m j ⊗ id ⊗ r ) ⊗ · · · ⊗ ∆ Σ i (cid:1) = d X k =1 X i + ... + i k = d k X s =1 i s X j =1 i s − j X n =0 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ ∆ Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) ⊗ · · · ⊗ ∆ Σ i (cid:1) = d X k =1 X i + ... + i k = d k X s =1 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ i s X j =1 i s − j X n =0 ∆ Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) ⊗ · · · ⊗ ∆ Σ i (cid:1) Observe that using the definition of ∆ , then adding the vanishing term ∆ Σ1 ◦ m − i s , and thenapplying Lemma 6, we have the following consecutive equalities: i s X j =1 i s − j X n =0 ∆ Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n )= i s − X j =1 i s − j X n =0 ∆ Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) + ∆ Σ1 ◦ m + i s +∆ Σ1 ◦ m i s + m − i s = i s X j =1 i s − j X n =0 ∆ Σ i s − j +1 (id ⊗ i s − j − n ⊗ m j ⊗ id ⊗ n ) + m − i s = i s X u =1 X t + ··· + t u = i s ∆ − u (cid:0) ∆ Σ t u ⊗ · · · ⊗ ∆ Σ t (cid:1) + m − i s If we plug it into the expression above, we get: d X j =1 d − j X n =0 m − d − j +1 (cid:0) id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n (cid:1) = d X k =1 X i + ... + i k = d k X s =1 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ (cid:0) i s X u =1 X t + ··· + t u = i s ∆ − u (cid:0) ∆ Σ t u ⊗ · · · ⊗ ∆ Σ t (cid:1)(cid:1) ⊗ · · · ⊗ ∆ Σ i (cid:1) + d X k =1 X i + ... + i k = d k X s =1 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ m − i s ⊗ · · · ⊗ ∆ Σ i (cid:1) d X j =1 d − j X n =0 m − d − j +1 (cid:0) id ⊗ d − j − n ⊗ m j ⊗ id ⊗ n (cid:1) = d X k =1 X i + ... + i k = d k X s =1 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ (cid:0) i s X u =1 X t + ··· + t u = i s b − u (cid:0) ∆ Σ t u ⊗ · · · ⊗ ∆ Σ t (cid:1)(cid:1) ⊗ · · · ⊗ ∆ Σ i (cid:1) + d X k =1 X i + ... + i k = d k X s =1 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ m − i s ⊗ · · · ⊗ ∆ Σ i (cid:1) = d X k =1 X i + ... + i k = d k X s =1 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ m − i s ⊗ · · · ⊗ ∆ Σ i (cid:1) + d X k =1 X i + ... + i k = d k X s =1 b − k (cid:0) ∆ Σ i k ⊗ · · · ⊗ m − i s ⊗ · · · ⊗ ∆ Σ i (cid:1) = 0 We define an A ∞ -category F uk( R × Y ) whose objects are exact Lagrangian cobordisms whosenegative ends are cylinders over Legendrian admitting augmentations. We define this categoryby localization, in the same spirit as the definition of the wrapped Fukaya category of a Liouvillesector in [GPS20] to which we refer for details about quotient and localization, as well as [LO06]. Definition 4.
A Hamiltonian isotopy ϕ sh of R × Y is called cylindrical at infinity if there existsa R > such that ϕ sh does not depend on the symplectization coordinate t in ( −∞ , − R ) × Y and ( R, ∞ ) × Y .Let E be a countable set of exact Lagrangian cobordisms in R × Y , with negative cylindricalends on Legendrian submanifolds of Y admitting an augmentation. Assume that any exactLagrangian cobordism Λ − ≺ Σ Λ + such that Λ − admits an augmentation is isotopic to one in E through a cylindrical at infinity Hamiltonian isotopy. For each cobordism Λ − ≺ Σ Λ + in E , wechoose a sequence Σ • of cobordisms Σ • = (Σ (0) , Σ (1) , Σ (2) , . . . ) as follows. First Σ (0) = Σ , and then we need to make several choices:1. a sequence { η i } i ≥ of real numbers such that P j> η j is strictly smaller than the length ofthe shortest Reeb chord of Λ + ∪ Λ − , and denote τ i = i P j =1 η j ,2. given that Σ is cylindrical outside [ − T, T ] × Y , and given N > , we choose Hamiltonians H i : R × Y → R , i ≥ , being small perturbations of h T,N (see Section 7) and set Σ ( i ) = ϕ τ i H i (Σ) such that Σ ( i ) is the graph of dF i in a standard neighborhood of the -section in T ∗ Σ , for F i : Σ → R Morse function satisfying the following:(a) on Σ ∩ (cid:0) [ T + N, ∞ ) × Y (cid:1) , resp. Σ ∩ (cid:0) ( −∞ , − T − N ] × Y (cid:1) , F i is equal to e t ( f + i − τ i ) ,resp e t ( f − i − τ i ) where f ± i : Λ ± → R are Morse functions such that the C -norm of f ± i is strictly smaller than min { η i , η i +1 } / ,64b) the functions F i − F j , f ± i − f ± j are Morse for i = j ,(c) the functions f − i and f ± i − f ± j admit a unique minimum on each connected componentwhile the functions F i and F i − F j admit a unique minimum on each filling connectedcomponent and no minimum on each connected component of Σ admitting a nonempty negative end.We call such a sequence of cobordisms cofinal . Note that an augmentation of Λ − gives canonicallyan augmentation of the negative end of Σ ( i ) for i ≥ . The construction is inductive and thedifferent choices above are made so that for any ( d + 1) -tuple of cobordisms Σ , Σ , . . . , Σ d in E (not necessarily distinct), and any strictly increasing sequence of integers i < i < · · · < i d , thecobordisms Σ ( i )0 , Σ ( i )1 , . . . , Σ ( i d ) d are pairwise transverse. Let us construct now a strictly unital A ∞ -category O as follows:• Obj( O ): pairs (Σ ( i ) , ε − ) where Σ ∈ E is an exact Lagrangian cobordism from Λ − to Λ + ,and ε − is an augmentation of Λ − ,• hom O (cid:0) (Σ ( i )0 , ε − ) , (Σ ( j )1 , ε − ) (cid:1) = Cth + (cid:0) Σ ( i )0 , Σ ( j )1 (cid:1) if i < j Z e ( i ) ε − if Σ = Σ , i = j and ε − = ε − otherwisewhere e ( i ) ε − is a formal degree element. The A ∞ -operations are given by the maps defined inSection 8.1 for each ( d + 1) -tuple of cobordisms Σ ( i )0 , Σ ( i )1 , . . . , Σ ( i d ) d with i < i < · · · < i d , i.e.for such a tuple we have a map m d : Cth + (Σ ( i d − ) d − , Σ ( i d ) d ) ⊗ · · · ⊗ Cth + (Σ ( i )0 , Σ ( i )1 ) → Cth + (Σ ( i )0 , Σ ( i d ) d ) These maps extend to maps defined for any ( d + 1) -tuple Σ ( i )0 , Σ ( i )1 , . . . , Σ ( i d ) d with the conditionthat the elements e ( i ) ε − j ∈ hom O ((Σ ( i ) j , ε − j ) , (Σ ( i ) j , ε − j )) behave as strict units.We finally define the Fukaya category F uk( R × Y ) of Lagrangian cobordisms in R × Y asa quotient of O by the set of continuation elements, as follows. Consider Σ ∈ E togetherwith an augmentation ε − of Λ − . For all i < j , there is a continuation element e Σ ( i ) , Σ ( j ) ∈ hom O (cid:0) (Σ ( i ) , ε − ) , (Σ ( j ) , ε − ) (cid:1) as described in Section 7, which is a cycle in O . Let T w ( O ) denotethe A ∞ -category of twisted complexes of O and C the full subcategory of T w ( O ) generated bycones of the continuation elements. We define F uk( R × Y ) := O [ C − ] to be the image of O inthe quotient T w ( O ) / C .Defined as follows, the category F uk( R × Y ) depends on various choices, namely:1. the choice for each Σ in E of a cofinal sequence Σ • = (Σ (0) , Σ (1) , . . . ) .2. the choice of the countable set E of representatives of Hamiltonian isotopy classes of exactLagrangian cobordisms with negative end admitting an augmentation,The fact that the quasi-equivalence class of the category does not depend on the choice ofa cofinal sequence for each element in E is purely algebraic. Assume Σ e • is a cofinal sequencefor Σ which is a subsequence of a bigger cofinal sequence Σ • . Then, denote e O the categoryconstructed using the cofinal sequence Σ e • and O the one constructed using Σ • . The inclusionfunctor e O → O is full and faithful, and if e C ⊂
T w e O denotes the full subcategory generated bycones of continuation elements, one gets a cohomologically full and faithful functor e O [ e C − ] → [ C − ] . Moreover, the continuation elements become quasi-isomorphisms in O [ C − ] thus thisfunctor is a quasi-equivalence. Now if Σ • , and Σ • , are two cofinal sequences for Σ , then onecan find a cofinal sequence Σ e • such that Σ • ,i ∪ Σ e • , i = 1 , , are also cofinal sequences. As Σ e • and Σ • ,i are subsequences of Σ • ,i ∪ Σ e • , the Fukaya category constructed using the cofinal sequence Σ e • is quasi-equivalent to the one using Σ • ,i ∪ Σ e • which is quasi-equivalent to the one using Σ • ,i .Then, the fact that the category does not depend (up to quasi-equivalence) to the choice ofrepresentatives of cylindrical at infinity Hamiltonian isotopy classes follows from the invarianceresult: Proposition 13.
Let Σ be an exact Lagrangian cobordism and ( ϕ sh ) s ∈ [0 , a cylindrical at infinityHamiltonian isotopy such that Σ and Σ := ϕ h (Σ ) are transverse. Then, for any T exactLagrangian cobordism transverse to Σ and Σ , the complexes Cth + (Σ , T ) and Cth + (Σ , T ) arehomotopy equivalent.Proof. All the ingredients to prove this proposition already appeared in Section 4. Observe firstthat if Σ is a cobordism from Λ − to Λ +0 then Σ is a cobordism from Λ − to Λ +1 with Λ ± Legendrian isotopic to Λ ± . The isotopy from Σ to Σ can be decomposed as a cylindrical atinfinity isotopy of Σ giving a cobordism e Σ from Λ − to Λ +1 , followed by a compactly supportedHamiltonian isotopy from e Σ to Σ . The proof of the proposition now goes as follows.Start from Σ and wrap its positive, resp. negative, end in the positive, resp negative, Reebdirection to obtain the cobordism Σ s , s ≥ , having cylindrical ends over Λ − , − s and Λ +0 ,s where Λ ± , ± s = Λ ± ± s∂ z . Take s big enough so that the Cthulhu complex Cth + (Σ s , T ) has onlyintersection points generators.Denote also Λ +1 ,a := Λ +1 + a∂ z and Λ − , − b := Λ − − b∂ z for a, b ≥ so that Λ +1 ,a lies entirelyabove Λ + T and Λ − , − b lies entirely below Λ − T . Denote C + a concordance from Λ +0 ,s to Λ +1 ,a and C − a concordance from Λ − , − b to Λ − , − s . We can assume that C + ∩ ( R × Λ + T ) = C − ∩ ( R × Λ − T ) = ∅ .Concatenating the concordances C − and C + with Σ s gives C − ⊙ Σ S ⊙ C + which is an exactLagrangian cobordism from Λ − , − b to Λ +1 ,a , satisfying Cth + ( C − ⊙ Σ s ⊙ C + , T ) = Cth + (Σ s , T ) by construction, where it is an equality of complexes. Finally, wrap the ends of C − ⊙ Σ s ⊙ C + in such a way that it “translates back” Λ +1 ,a to Λ +1 and Λ − , − b to Λ − , to obtain a cobordism e Σ from Λ − to Λ +1 , Hamiltonian isotopic to Σ by a compactly supported Hamiltonian isotopy.Invariance of the Cthulhu complex by wrapping the ends and compactly supported Hamiltonianisotopy ends the proof. In case of a ( d + 1) -tuple of concatenated cobordisms ( V ⊙ W , . . . , V d ⊙ W d ) one can also definehigher order maps m V ⊙ Wd , which will recover the maps m Σ d defined in the previous section in thecase of concatenation of a cobordism with a trivial cylinder. We define recursively, for d ≥ , themaps ∆ Wd : Cth + ( V d − ⊙ W d − , V d ⊙ W d ) ⊗ . . . Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V , V d ) b Vd : Cth + ( V d − ⊙ W d − , V d ⊙ W d ) ⊗ . . . Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( W , W d )
66s follows. First, b V and ∆ W are the transfer maps from Section 3.2, and then for d ≥ onesets: ∆ Wd = d X s =2 X ≤ i ,...,i s i + ··· + i s = d ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ b Vi (cid:1) b Vd = d X s =1 X ≤ i ,...,i s i + ··· + i s = d b Vs (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) Using the maps ∆ Ws , b Vs from Section 8.1. Observe that the maps b V and ∆ W already appearedin Section 6.3. Given this, we define m V ⊙ Wd : Cth + ( V d − ⊙ W d − , V d ⊙ W d ) ⊗ . . . Cth + ( V ⊙ W , V ⊙ W ) → Cth + ( V ⊙ W , V d ⊙ W d ) by m V ⊙ Wd = d X s =1 X ≤ i ,...,i s i + ··· + i s = d m W, +0 s (cid:0) b Vi s ⊗ · · · ⊗ b Vi (cid:1) + m V, − s (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) We will first prove that the maps ∆ Wj and b Vj satisfy the A ∞ -functor equations, and then wewill prove that the maps m V ⊙ Wj satisfy the A ∞ equations. We start by proving the following: Lemma 7.
For all d ≥ , d X j =1 X i + ··· + i j = d b Vj (cid:0) ∆ Wi j ⊗ · · · ⊗ ∆ Wi (cid:1) + ∆ Wj (cid:0) b Vi j ⊗ · · · ⊗ b Vi (cid:1) = 0 Proof.
This holds by definition of the maps. Observe that we have already made use of the case d = 1 in Section 6.2. For d ≥ , one has d X j =1 X i + ··· + i j = d b Vj (cid:0) ∆ Wi j ⊗ · · · ⊗ ∆ Wi (cid:1) + ∆ Wj (cid:0) b Vi j ⊗ · · · ⊗ b Vi (cid:1) = b V ◦ ∆ Wd + ∆ W ◦ b Vd + d X j =2 X i + ··· + i j = d b Vj (cid:0) ∆ Wi j ⊗ · · · ⊗ ∆ Wi (cid:1) + ∆ Wj (cid:0) b Vi j ⊗ · · · ⊗ b Vi (cid:1) Observe that ∆ Wi j ⊗· · ·⊗ ∆ Wi takes values in Cth + ( V d − i j , V d ) ⊗· · ·⊗ Cth + ( V , V i ) and b Vi j ⊗· · ·⊗ b Vi takes values in Cth + ( W d − i j , W d ) ⊗ · · · ⊗ Cth + ( W , W i ) , hence the maps b Vj and ∆ Wj are theone corresponding to one cobordism only and not the concatenation, as defined in Section 8.1.So the sum above equals ∆ Wd + b V ◦ ∆ Wd + b Vd + d X j =2 X i + ··· + i j = d b Vj (cid:0) ∆ Wi j ⊗ · · · ⊗ ∆ Wi (cid:1) + ∆ Wj (cid:0) b Vi j ⊗ · · · ⊗ b Vi (cid:1) = ∆ Wd + b Vd + d X j =1 X i + ··· + i j = d b Vj (cid:0) ∆ Wi j ⊗ · · · ⊗ ∆ Wi (cid:1) + ∆ Wj (cid:0) b Vi j ⊗ · · · ⊗ b Vi (cid:1) where note that in the sum on the right we have ∆ W ◦ b Vd = 0 . Then, this gives by definitionof b Vd and ∆ Wd . 67ow we can prove: Lemma 8.
For all d ≥ d X s =1 X i + ··· + i s = d m Vs (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id | {z } n (cid:1) = 0 and d X s =1 X i + ··· + i s = d m Ws (cid:0) b Vi s ⊗ · · · ⊗ b Vi (cid:1) + d X j =1 d − j X n =0 b Vd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id | {z } n (cid:1) = 0 Proof.
We prove it by recursion on d . For d = 1 , the relations above means that b V and ∆ W are chain maps, which is the content of Proposition 3 and Proposition 4. Note that we have alsoproved the case d = 2 in Section 6.3. For d ≥ , we have d X s =1 X i + ··· + i s = d m Vs (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id | {z } n (cid:1) = d X s =1 X i + ··· + i s = d m Vs (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + ∆ W ◦ m V ⊙ Wd + d − X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id | {z } n (cid:1) but by definition of m V ⊙ Wd : ∆ W ◦ m V ⊙ Wd = ∆ W ◦ (cid:16) d X s =1 X i + ··· + i s = d m W, +0 s (cid:0) b Vi s ⊗ · · · ⊗ b Vi (cid:1) + m V, − s (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1)(cid:17) = d X s =1 X i + ··· + i s = d ∆ W ◦ m W, +0 s (cid:0) b Vi s ⊗ · · · ⊗ b Vi (cid:1) + m V, − s (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) So we get d X s =1 X i + ··· + i s = d m Vs (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id (cid:1) = d X s =1 X i + ··· + i s = d m V, + s (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X s =1 X i + ··· + i s = d ∆ W ◦ m W, +0 s (cid:0) b Vi s ⊗ · · · ⊗ b Vi (cid:1) + d − X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id (cid:1) d − X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id (cid:1) = d − X j =1 d − j X n =0 d − j +1 X s =2 X i + ··· + i s = d − j +1 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ b i (cid:1)(cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id (cid:1) = d − X j =1 d − j X n =0 d − j +1 X s =2 X i + ··· + i s = d − j +1 s X l =1 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ b Vi l (id ⊗ · · · ⊗ m V ⊙ Wj ⊗ . . . id) ⊗ · · · ⊗ b Vi (cid:1) = d X s =2 X i + ··· + i s = d s X l =1 i l X j =1 i l − j X n =0 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ b Vi l (id ⊗ · · · ⊗ m V ⊙ Wj ⊗ . . . id) ⊗ · · · ⊗ b Vi (cid:1) = d X s =2 X i + ··· + i s = d s X l =1 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ i l X j =1 i l − j X n =0 b Vi l (id ⊗ · · · ⊗ m V ⊙ Wj ⊗ . . . id) ⊗ · · · ⊗ b Vi (cid:1) Observe that i l ≤ d − so by recursion we have that d − X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id (cid:1) = d X s =2 X i + ··· + i s = d s X l =1 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ i l X u =1 X t + ··· + t u = i l m Wu ( b Vt u ⊗ · · · ⊗ b Vt ) ⊗ · · · ⊗ b i (cid:1) So we get d X s =1 X i + ··· + i s = d m Vs (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ id ⊗ · · · ⊗ id (cid:1) = d X s =1 X i + ··· + i s = d m V, + s (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X s =1 X i + ··· + i s = d ∆ W ◦ m W, +0 s (cid:0) b Vi s ⊗ · · · ⊗ b Vi (cid:1) + d X s =2 X i + ··· + i s = d s X l =1 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ i l X u =1 X t + ··· + t u = i l m Wu ( b Vt u ⊗ · · · ⊗ b Vt ) ⊗ · · · ⊗ b i (cid:1) = d X s =1 X i + ··· + i s = d m V, + s (cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X s =1 X i + ··· + i s = d s X l =1 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ i l X u =1 X t + ··· + t u = i l m Wu ( b Vt u ⊗ · · · ⊗ b Vt ) ⊗ · · · ⊗ b i (cid:1) ∆ W ◦ m W, − s = 0 . By definition of m V, + s , it gives = d X s =1 X i + ··· + i s = d s X u =1 X t + ··· + t u = s ∆ Λ u (cid:0) b Vt u ⊗ · · · ⊗ b Vt (cid:1)(cid:0) ∆ Wi s ⊗ · · · ⊗ ∆ Wi (cid:1) + d X s =1 X i + ··· + i s = d s X l =1 ∆ Ws (cid:0) b Vi s ⊗ · · · ⊗ i l X u =1 X t + ··· + t u = i l m Wu ( b Vt u ⊗ · · · ⊗ b Vt ) ⊗ · · · ⊗ b i (cid:1) and finally, using Lemma 6 which states in this case that d X j =1 d − j X n =0 ∆ Wd − j +1 (cid:0) id ⊗ d − j − n ⊗ m Wj ⊗ id ⊗ n (cid:1) + d X j =1 X i + ··· + i j = d ∆ Λ j (cid:0) ∆ Wi j ⊗ · · · ⊗ ∆ Wi (cid:1) = 0 and Lemma 7, one obtains that the sum vanishes, and the maps ∆ Wj satisfy the A ∞ -functorequations.One proves analogously that the maps b Vj satisfy the A ∞ -functor equations. Proposition 14.
The maps m V ⊙ Wd satisfy the A ∞ -equations.Proof. For all d ≥ , one has d X j =1 d − j X n =0 m V ⊙ Wd − j +1 (cid:0) id ⊗ · · · ⊗ id ⊗ m V ⊙ Wj ⊗ n z }| { id ⊗ · · · ⊗ id (cid:1) = X j,n (cid:16) d − j +1 X k =1 X i + ··· + i k = d − j +1 m W, +0 k (cid:0) b Vi k ⊗ · · · ⊗ b Vi (cid:1) + m V, − k (cid:0) ∆ Wi k ⊗ · · · ⊗ ∆ Wi (cid:1)(cid:17)(cid:0) id ⊗ · · · ⊗ m V ⊙ Wj ⊗ · · · ⊗ id (cid:1) = d X k =1 X i + ... + i k = d k X s =1 i s X j =1 i s − j X n =0 m W, +0 k (cid:0) b Vi k ⊗ · · · ⊗ b Vi s − j +1 (id ⊗ · · · ⊗ m V ⊙ Wj ⊗ · · · ⊗ id) ⊗ · · · ⊗ b Vi (cid:1) + m V, − k (cid:0) ∆ Wi k ⊗ · · · ⊗ ∆ Vi s − j +1 (id ⊗ · · · ⊗ m V ⊙ Wj ⊗ · · · ⊗ id) ⊗ · · · ⊗ ∆ Wi (cid:1) = d X k =1 X i + ... + i k = d k X s =1 m W, +0 k (cid:0) b Vi k ⊗ · · · ⊗ i s X j =1 i s − j X n =0 b Vi s − j +1 (id ⊗ · · · ⊗ m V ⊙ Wj ⊗ · · · ⊗ id) ⊗ · · · ⊗ b Vi (cid:1) + d X k =1 X i + ... + i k = d k X s =1 m V, − k (cid:0) ∆ Wi k ⊗ · · · ⊗ i s X j =1 i s − j X n =0 ∆ Vi s − j +1 (id ⊗ · · · ⊗ m V ⊙ Wj ⊗ · · · ⊗ id) ⊗ · · · ⊗ ∆ Wi (cid:1) d X k =1 X i + ... + i k = d k X s =1 m W, +0 k (cid:0) b Vi k ⊗ · · · ⊗ i s X u =1 X t + ··· + t u = i s m Wu ( b Vt u ⊗ · · · ⊗ b Vt ) ⊗ · · · ⊗ b Vi (cid:1) + d X k =1 X i + ... + i k = d k X s =1 m V, − k (cid:0) ∆ Wi k ⊗ · · · ⊗ i s X u =1 X t + ··· + t u = i s m Vu ( ∆ Wt u ⊗ · · · ⊗ ∆ Wt ) ⊗ · · · ⊗ ∆ Wi (cid:1) = d X j =1 X r + ··· + r j = d j X u =1 j − u +1 X s =1 m W, +0 j − u +1 (cid:0) id ⊗ · · · ⊗ id | {z } k − s ⊗ m Wu ⊗ id ⊗ · · · ⊗ id | {z } s − (cid:1)(cid:0) b Vr j ⊗ · · · ⊗ b Vr (cid:1) + d X j =1 X r + ··· + r j = d j X u =1 j − u +1 X s =1 m V, − j − u +1 (cid:0) id ⊗ · · · ⊗ id | {z } k − s ⊗ m Vu ⊗ id ⊗ · · · ⊗ id | {z } s − (cid:1)(cid:0) ∆ Vr j ⊗ · · · ⊗ ∆ Vr (cid:1) = 0 as the maps m Wj and m Vj satisfy the A ∞ -equations. References [Abb14] C. Abbas.
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