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

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

Roman Golovko
email: golovko.roman@renyi.mta.hu

FLOER THEORY FOR LAGRANGIAN COBORDISMS

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several long exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.

Contents

Chantraine, Dimitroglou Rizell, Ghiggini, Golovko 6.1. The Cthulhu differential 49 6.2. The proof of d ε - 0 ,ε - 1 2 = 0 51 7. The transfer and co-transfer map for concatenations of cobordisms 55 7.1. Concatenations and stretching of the neck 55 7.2. The complex after a neck stretching procedure 56 7.3. Definition of the transfer and co-transfer maps 58 7.4. An auxiliary complex 61 8. Proof of the acyclicity (the invariance) 63 8.1. Wrapping the ends 63 8.2. Invariance under compactly supported Hamiltonian isotopies 65 8.3. The proof of the acyclicity 68 9. Proof of the main theorem 69 10. Long exact sequences induced by an exact Lagrangian cobordism 70 10.1. Various push-offs of an exact Lagrangian cobordism 70 10.2. Computing the Floer homology of a Hamiltonian push-off 74 10.3. Seidel's isomorphism 76 11. Twisted coefficients, L 2 -completions, and applications 77 11.1. Floer homology with twisted coefficients. 77 11.2. Augmentations in finite-dimensional non-commutative algebras. 81 11.3. The fundamental class and twisted coefficients 82 11.4. A brief introduction to homology with L 2 -coefficients. 88 11.5. Estimating the first L 2 -Betti number of a tower 90 12. Applications and examples 91 12.1. The homology of an endocobordism 91 12.2. Characteristic classes of endocobordisms 93 12.3. Restrictions on the fundamental group of an endocobordism between simply connected Legendrians 95 12.4. Explicit examples of Lagrangian cobordisms 96 References

The first author is partially supported by the ANR project COSPIN (ANR-13-JS01-0008-01). The second author is supported by the grant KAW 2013.0321 from the Knut and Alice Wallenberg Foundation. The fourth author is supported by the ERC Advanced Grant LDTBud and the ESF Short Visit Grant.

Introduction

Lagrangian cobordism is a natural relation between Legendrian submanifolds, and it is a crucial ingredient in the definition of the functorial properties of invariants of Legendrian submanifolds in the spirit of symplectic field theory as introduced by Eliashberg, Givental and Hofer in [START_REF] Eliashberg | Introduction to symplectic field theory[END_REF]. This relation is at the heart of many recent developments in the study of Legendrian submanifolds and its properties have been investigated by the authors and many others over the past years.

In the present paper we study rigidity phenomena in the topology of exact Lagrangian cobordisms in the symplectisation of the contactisation of a Liouville manifold. In [START_REF] Eliashberg | Lagrangian caps[END_REF], Eliashberg and Murphy showed that exact Lagrangian cobordisms are flexible when their negative ends are loose (in the sense of Murphy [START_REF] Murphy | Loose Legendrian Embeddings in High Dimensional Contact Manifolds[END_REF]). On the contrary, we will show that they become rigid if we restrict our attention to cobordisms whose negative ends admit augmentations (or more generally finitedimensional representations) of their Chekanov-Eliashberg algebras.

In order to study the topology of such cobordisms, we introduce a version of Lagrangian Floer homology (originally defined for closed Lagrangian submanifolds by Floer in [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF]) for pairs of exact Lagrangian cobordisms. This construction finds its inspiration in the work of Ekholm in [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF], which gives a symplectic field theory point of view on wrapped Floer homology of Abouzaid and Seidel from [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF].

The definition of this new Floer theory requires the use of augmentations of the Chekanov-Eliashberg algebras of the negative ends as bounding cochains in order to algebraically cancel certain "bad" degenerations of the holomorphic curves at the negative ends of the cobordisms. Bounding cochains have been introduced, in the closed case, by Fukaya, Oh, Ohta and Ono in [START_REF] Fukaya | Lagrangian intersection Floer theory: anomaly and obstruction. Part I[END_REF], while augmentations, which play a similar role in the context of Legendrian contact homology, have been introduced by Chekanov in [START_REF] Yu | Differential algebra of Legendrian links[END_REF].

For a pair of exact Lagrangian cobordisms obtained by a suitable small Hamiltonian push-off, our invariant gives rise to various long exact sequences relating the singular homology of the cobordism with the Legendrian contact homology of its ends. We then use these long exact sequences to give restrictions on the topology of exact Lagrangian cobordisms under various hypotheses on the topology of the Legendrian ends. In the context of generating family homology for Legendrian submanifolds in jet spaces, Sabloff and Traynor in [START_REF] Sabloff | Obstructions to Lagrangian cobordisms between Legendrians via generating families[END_REF] describe exact sequences similar to ours for cobordisms which admit compatible generating families.

The notion of Lagrangian cobordism between Legendrian submanifolds studied in this article is (in general) different from the notion of Lagrangian cobordisms between Lagrangian submanifolds introduced by Arnol'd in [START_REF] Arnol ′ D | Lagrange and Legendre cobordisms[END_REF] and [START_REF] Arnol ′ D | Lagrange and Legendre cobordisms[END_REF] and recently popularised by Biran and Cornea in [START_REF] Biran | Lagrangian cobordism I[END_REF] and [START_REF] Biran | Lagrangian cobordism and Fukaya categories[END_REF]. Lagrangian cobordisms in the sense of Arnol'd between Lagrangian submanifolds of a symplectic manifold M are Lagrangian submanifolds of M × C which project to horizontal half-lines of C outside of a compact set. The main difference between the two theories is that Arnol'd-type cobordisms do not distinguish between positive and negative ends and therefore are closer in spirit to the notion of cobordism in classical topology. Despite the differences, for Lagrangian cobordisms between Legendrian submanifolds with no Reeb chords, some of the results we obtain resemble some of the results obtained by Biran and Cornea [START_REF] Biran | Lagrangian cobordism I[END_REF][START_REF] Biran | Lagrangian cobordism and Fukaya categories[END_REF] and Suárez [START_REF] Suárez | Exact Lagrangian cobordism and pseudo-isotopy[END_REF].

Remark 1.1. In fact, under the strong assumption that the Legendrian submanifolds Λ ± ⊂ (P × R, dz + θ) have no Reeb chords, an exact Lagrangian cobordism from Λ -to Λ + inside the symplectisation (R × P × R, d(e t (dz + θ))) ∼ = (P × C, dθ ⊕ d(xdy)) can be deformed to yield an exact Lagrangian cobordism between the exact Lagrangian embeddings Π Lag (Λ -), Π Lag (Λ + ) ⊂ (P, dθ) in the sense of Arnol'd, and vice versa. In some sense, these two notions of cobordisms thus coincide in this case. For readers familiar with the language of [START_REF] Ekholm | Legendrian knots and exact Lagrangian cobordisms[END_REF], this can be explained as follows: one can go between Lagrangian cobordism in the sense studied here and so-called Morse cobordisms. The latter are embedded in the case when the Legendrian ends have no Reeb chords, and they are thus Lagrangian cobordisms in the sense of Arnol'd.

1.1. Main results. Let (P, θ) be a Liouville manifold and (Y, α) := (P × R, dz + θ) its contactisation. We consider a pair of exact Lagrangian embeddings Σ 0 , Σ 1 ֒→ X, where (X, ω) = (R × Y, d(e t α)) is the symplectisation of (Y, α). We assume that the positive and negative ends of Σ i i = 0, 1 are cylindrical over Legendrian submanifolds Λ - i and Λ + i respectively, and thus Σ i is a Lagrangian cobordisms from Λ - i to Λ + i ; see Figure 1 for a schematic representation and Section 2 for the precise formulation of our geometrical setup. We assume that Σ 0 and Σ 1 intersect transversely and that their Legendrian ends are chord-generic in the sense of Section 2.1. The first rigidity phenomena for Lagrangian submanifolds in this setting were proven by Gromov [START_REF] Gromov | Pseudo holomorphic curves in symplectic manifolds[END_REF], who showed that there are no closed exact Lagrangian submanifolds in a symplectisation as above (note that they automatically would be displaceable). Differently put, this means that an exact Lagrangian cobordism must have at least one non-empty end.

t Λ + 1 Λ + 0 Σ 0 Σ 1 Λ - 1 Λ - 0 Figure 1.
Two Lagrangian cobordisms inside a symplectisation R × Y , where the vertical axis corresponds to the R-coordinate.

We denote by R(Λ ± i ) the set of Reeb chords of Λ ± i for i = 0, 1, and by R(Λ ± 1 , Λ ± 0 ) the set of Reeb chords from Λ ± 1 to Λ ± 0 . Let R be a field of characteristic 2 or, if all Σ i 's and Λ ± i 's are (relatively) spin, any commutative ring. (See Section 1.1.1.) We denote by C(Λ ± 0 , Λ ± 1 ) the free R-module spanned by R(Λ ± 1 , Λ ± 0 ). Remark 1.2. In fact the commutativity of R can be dropped -Chekanov's linearisation can be generalised to arbitrary rings, and our long exact sequences exist in this setting as well. See Section 11.2 for more details as well as [START_REF] Chantraine | Noncommutative augmentation categories[END_REF]. However, we point out that in order to use arguments involving ranks, one must impose additional requirements on R, for example commutativity, or finite dimensionality as an algebra over a field.

We assume that the Chekanov-Eliashberg algebra A(Λ - i ; R) of Λ - i admits an augmentation ε - i over R for i = 0, 1 (see Section 5 for the definitions). It follows from the results of Ekholm, Honda and Kálmán in [START_REF] Ekholm | Legendrian knots and exact Lagrangian cobordisms[END_REF] that A(Λ + i ; R) also admits an augmentation ε + i = ε - i • Φ Σi , where Φ Σi : A(Λ + i ; R) → A(Λ - i ; R) is the unital DGA morphism induced by the cobordism Σ i . Thus the bilinearised contact cohomologies LCH ε ± 0 ,ε ± 1 (Λ ± 0 , Λ ± 1 ) are defined. See Chekanov [START_REF] Yu | Differential algebra of Legendrian links[END_REF] and Bourgeois and Chantraine [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF] for the notions of linearisation and bilinearisation of a differential graded algebra.

We denote by CF (Σ 0 , Σ 1 ) the free R-module spanned by the intersection points Σ 0 ∩ Σ 1 . In Section 3.3.2 we define the notion of the action of an intersection point and use it to filter CF (Σ 0 , Σ 1 ): we denote by CF ± (Σ 0 , Σ 1 ) the submodule of CF (Σ 0 , Σ 1 ) generated by intersection points of positive (respectively, negative) action.

The main construction in this article provides a differential on the modules CF ± (Σ 0 , Σ 1 ), leading to homology groups HF ± (Σ 0 , Σ 1 ) of Floer type. The differentials and the resulting homology groups depend on the choice of the augmentations ε - i . In order to define a graded theory, we need that 2c 1 (X) = 0 and that all Lagrangian cobordisms have vanishing Maslov classes. This implies that all Lagrangian cobordisms admit Maslov potentials (as defined in Section 4.2); a particular choice of such a potential leads to the notion of a graded Lagrangian cobordisms, for which HF (Σ 0 , Σ 1 ) has a well-defined grading in Z. In general the grading must be taken in a (possibly trivial) cyclic group.

Our main result is the following relation between the Floer homology of a pair (Σ 0 , Σ 1 ) of exact Lagrangian cobordisms and the bilinearised Legendrian contact homologies of their ends; see Section 9.

Theorem 1.3. Let Σ i , i = 0, 1, be a graded exact Lagrangian cobordisms from the Legendrian submanifold Λ - i to Λ + i inside the symplectisation of the contactisation of a Liouville manifold, and assume that there are augmentations ε - i of A(Λ - i ) for i = 0, 1. Then there exists a spectral sequence whose first page is

E 3,• 1 ⊕ E 2,• 1 ⊕ E 1,• 1 ⊕ E 0,• 1 (1)
LCH •-2 ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) ⊕ HF • + (Σ 0 , Σ 1 ) ⊕ LCH •-1 ε - 0 ,ε - 1 (Λ - 0 , Λ - 1 ) ⊕ HF • -(Σ 0 , Σ 1
) and which collapses to 0 at the fourth page.

This theorem follows from the acyclicity of a complex (Cth(Σ 0 , Σ 1 ), d ε - 0 ,ε - 1 ) associated to a pair of Lagrangian cobordisms which we call the Cthulhu complex (see Section 6). Its underlying R-module is Cth(Σ 0 , Σ 1 ) = C(Λ + 0 , Λ + 1 ) ⊕ CF + (Σ 0 , Σ 1 ) ⊕ C(Λ - 0 , Λ - 1 ) ⊕ CF -(Σ 0 , Σ 1 ). The spectral sequence is induced by the filtration of length four given by C(Λ + 0 , Λ + 1 ) > CF + (Σ 0 , Σ 1 ) > C(Λ - 0 , Λ - 1 ) > CF -(Σ 0 , Σ 1 ), and the acyclicity of the complex (Cth(Σ 0 , Σ 1 ), d ε - 0 ,ε - 1 ) follows from its invariance properties with respect to a large class of Hamiltonian deformations which, in the contactisation of a Liouville manifold, allow us to displace any pair of Lagrangian cobordisms.

When the negative ends are empty, this complex recovers the wrapped Floer cohomology complex as described by Ekholm in [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF]. When the positive ends are empty and there are no homotopically trivial Reeb chords of both Λ - i 's, this complex is similar to the Floer complex sketched in the work of Akaho in [5, Section 8].

Remark 1.4. The latter situation cannot occur in the symplectisation of a contactisation of a Liouville manifold: an exact Lagrangian cobordism with no positive end cannot have a negative end admitting an augmentation by Corollary 1.9 below. This also follows from an even stronger result due to the second author in [START_REF] Rizell | Exact Lagrangian caps and non-uniruled Lagrangian submanifolds[END_REF], where it is shown that such a Legendrian submanifold must have an acyclic Chekanov-Eliashberg algebra.

We will introduce two classes of pairs (Σ 0 , Σ 1 ) of exact Lagrangian cobordisms for which this filtration is of length three: directed and V -shaped pairs (see Section 9). A pair is directed when there are no intersection points of positive action, and V -shaped when there are no intersection points of negative action (see Section 9 for more details). In these situations, the spectral sequence collapses to 0 at the third page, giving rise to the following long exact sequences.

Corollary 1.5. Let (Σ 0 , Σ 1 ) be a pair of exact Lagrangian cobordisms satisfying the assumptions of Theorem 1.3.

• If (Σ 0 , Σ 1 ) is directed, then there exists a long exact sequence (2)

• • • / / LCH k-1 ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) HF k -(Σ 0 , Σ 1 ) / / LCH k ε - 0 ,ε - 1 (Λ - 0 , Λ - 1 ) / / LCH k ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) / / • • • • If (Σ 0 , Σ 1
) is V -shaped, then there exists a long exact sequence (3)

• • • / / LCH k ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) / / LCH k ε - 0 ,ε - 1 (Λ - 0 , Λ - 1 ) / / HF k+2 + (Σ 0 , Σ 1 ) LCH k+1 ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) / / • • • 1.1.1.
Remarks about grading and orientation. Most of the result here are stated for graded Lagrangian cobordisms. However, our methods apply in the ungraded cases as well. The only difference is that the long exact sequences in Corollary 1.5 and in Section 1.2 become exact triangles (the maps are ungraded or, alternatively, graded modulo the Maslov number).

For the results to holds using coefficients in a ring R different from a field of characteristic two, as well for the results in Section 1.3, we need to be able to define the theory using integer coefficients. In order to perform counts with signs, one has to define coherent orientations for the relevant moduli spaces of pseudoholomorphic curves. This can be done in the case when the Legendrian submanifolds and Lagrangian cobordisms are relatively pin (following Ekholm, Etnyre and Sullivan in [START_REF] Ekholm | Orientations in Legendrian contact homology and exact Lagrangian immersions[END_REF] and Seidel in [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF]Section 11]).

Long exact sequences for LCH induced by a Lagrangian cobordism.

If Σ 1 is a Hamiltonian deformation of Σ 0 for some suitable and sufficiently small Hamiltonian, the Floer homology groups HF ± (Σ 0 , Σ 1 ) can be identified with Morse homology groups of Σ 0 . Similarly, the bilinearised Legendrian contact homology groups LCH ε ± 0 ,ε ± 1 (Λ ± 0 , Λ ± 1 ) can be identified with the bilinearised contact homology groups LCH ε ± 0 ,ε ± 1 (Λ ± 0 ) (as defined in Subsection 5.3) following [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF]. Thus the long exact sequences in Corollary 1.5 can be reinterpreted as long exact sequences relating the singular homology of a Lagrangian cobordism and the Legendrian contact homology of its ends. These results are proved in Section 10.1.

Analogous long exact sequences have previously been found by Sabloff and Traynor in [START_REF] Sabloff | Obstructions to Lagrangian cobordisms between Legendrians via generating families[END_REF] in the setting of generating family homology under the additional assumption that the cobordism admits a compatible generating family, and by the fourth author in [START_REF] Golovko | A note on Lagrangian cobordisms between Legendrian submanifolds of R 2n+1[END_REF] in the case when the negative end of the cobordism admits an exact Lagrangian filling. The latter results have been put in a much more general framework in recent work by Cieliebak-Oancea [START_REF] Cieliebak | Symplectic homology and the Eilenberg-Steenrod axioms[END_REF].

In the rest of this introduction, Λ + and Λ -will always denote closed Legendrian submanifolds of dimension n in the contactisation of a Liouville manifold, and every Lagrangian cobordism between them, as well as any Lagrangian filling of them, will always live in the corresponding symplectisation. We will denote by Σ the natural compactification of Σ obtained by adjoining its Legendrian ends Λ ± . Note that Σ is diffeomorphic to Σ ∩ [-T, +T ] × Y for some T ≫ 0 sufficiently large. We will also use the notation ∂ ± Σ := Λ ± ⊂ Σ, which implies that ∂Σ = ∂ + Σ ⊔ ∂ -Σ.

1.2.1.

A generalisation of the long exact sequence of a pair. The first exact sequence we produce from a Lagrangian cobordism (see Section 10.1.1) is given by the following: Theorem 1.6. Let Σ be a graded exact Lagrangian cobordism from Λ -to Λ + and let ε - 0 and ε - 1 be two augmentations of A(Λ -) inducing augmentations ε + 0 , ε + 1 of A(Λ + ). There is a long exact sequence

(4) • • • / / LCH k-1 ε + 0 ,ε + 1 (Λ + ) H n+1-k (Σ, ∂ -Σ; R) / / LCH k ε - 0 ,ε - 1 (Λ -) / / LCH k ε + 0 ,ε + 1 (Λ + ) / / • • • , .
where the map Φ

ε - 0 ,ε - 1 Σ : LCH k ε - 0 ,ε - 1 (Λ -) → LCH k ε + 0 ,ε + 1 (Λ +
) is the adjoint of the bilinearised DGA morphism Φ Σ induced by Σ (see Section 5.3).

When the negative end Λ -= ∅ is empty, i.e. when Σ is an exact Lagrangian filling of Λ + , and ε + i , i = 0, 1 both are augmentations induced by this filling, the resulting long exact sequence simply becomes the isomorphism

LCH k-1 ε + 0 ,ε + 1 (Λ + ) ≃ -→ H n+1-k (Σ; R)
appearing in the work of Ekholm in [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF]. This isomorphism was first observed by Seidel, and is sometimes called Seidel's isomorphism. (See the map G ε - 0 ,ε - 1 Σ in Section 10.2 for another incarnation.) Its proof was completed by the second author in [START_REF] Rizell | Lifting pseudoholomorphic polygons to the symplectisation of P × R and applications[END_REF], also see [START_REF] Chantraine | Floer homology and Lagrangian concordance[END_REF] by the authors for an analogous isomorphism induced by a pair of fillings.

1.2.2.

A generalisation of the duality long exact sequence and fundamental class. A Legendrian submanifold Λ is horizontally displaceable if there exists a Hamiltonian isotopy φ t of (P, dθ) which displaces the Lagrangian projection Π Lag (Λ) ⊂ P from itself. In Section 10.1.1 we obtain the following: Theorem 1.7. Let Σ be an exact graded Lagrangian cobordism from Λ -to Λ + and let ε - 0 and ε - 1 be two augmentations of A(Λ -) inducing augmentations ε + 0 , ε + 1 of A(Λ + ). Assume that Λ -is horizontally displaceable; then there is a long exact sequence

(5) • • • / / LCH k ε + 0 ,ε + 1 (Λ + ) / / LCH ε - 0 ,ε - 1 n-k-1 (Λ -) / / H n-k-1 (Σ; R) LCH k+1 ε + 0 ,ε + 1 (Λ + ) / / • • • ,
where the map G

ε - 0 ,ε - 1 Σ : H n-k-1 (Σ; R) → LCH k+1 ε + 0 ,ε + 1 (Λ +
) is defined in Section 10.3.

When Σ = R × Λ then H • (Σ) = H • (Λ), and hence the above long exact sequence recovers the duality long exact sequence for Legendrian contact homology, which was proved by Sabloff in [START_REF] Sabloff | Duality for Legendrian contact homology[END_REF] for Legendrian knots and later generalised to arbitrary Legendrian submanifolds in [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF] by Ekholm, Etnyre and Sabloff. In the bilinearised setting, the duality long exact sequence was introduced by Bourgeois and the first author in [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF]. In Section 11.3 we use Exact Sequence [START_REF] Akaho | Morse homology and manifolds with boundary[END_REF] to prove that the fundamental class in LCH defined by Sabloff in [START_REF] Sabloff | Duality for Legendrian contact homology[END_REF] and Ekholm, Etnyre and Sabloff in [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF] is functorial with respect to the maps induced by exact Lagrangian cobordisms.

1.2.3.

A generalisation of the Mayer-Vietoris long exact sequence. The last exact sequence that we will extract from Corollary 1.5 generalises the Mayer-Vietoris exact sequence (see Section 10.1.3).

Theorem 1.8. Let Σ be an exact graded Lagrangian cobordism from Λ -to Λ + and let ε - 0 and ε - 1 be two augmentations of A(Λ -) inducing augmentations ε + 0 , ε + 1 of A(Λ + ). Then there is a long exact sequence (6)

• • • / / LCH k-1 ε + 0 ,ε + 1 (Λ + ) H n-k (∂ -Σ; R) / / LCH k ε - 0 ,ε - 1 (Λ -) ⊕ H n-k (Σ; R) / / LCH k ε + 0 ,ε + 1 (Λ + ) / / • • • ,
where the component H n-k (∂ -Σ; R) → H n-k (Σ; R) of the left map is induced by the topological inclusion of the negative end.

If ε - 0 = ε - 1 = ε, it moreover follows that the image of the fundamental class under the component H n (∂ -Σ; R) → LCH 0 ε,ε (Λ -) of the above morphism vanishes. Under the additional assumption that Λ -is horizontally displaceable, it is moreover the case that the image of a generator under H 0 (∂ -Σ; R) → LCH n ε,ε (Λ -) is equal to the fundamental class in Legendrian contact homology.

In particular we get that the fundamental class in H n (∂ -Σ; R) either is non-zero in H n (Σ), or is the image of a class in LCH -1 ε + 0 ,ε + 1 (Λ + ). In both cases, Λ + = ∅. Thus we obtain a new proof of the following result.

Corollary 1.9 ( [START_REF] Rizell | Exact Lagrangian caps and non-uniruled Lagrangian submanifolds[END_REF]). If Λ ⊂ P × R admits an augmentation, then there is no exact Lagrangian cobordism from Λ to ∅, i.e. there is no exact Lagrangian "cap" of Λ.

Remark 1.10. Assume that Λ -admits an exact Lagrangian filling L inside the symplectisation, and that ε -is the augmentation induced by this filling. It follows that ε + is the augmentation induced by the filling L ⊙ Σ of Λ + obtained as the concatenation of L and Σ. Using Seidel's isomorphisms

LCH k ε -,ε -(Λ -) ≃ H n-k (L; R), LCH k ε + ,ε + (Λ + ) ≃ H n-k (L ⊙ Σ; R)
to replace the relevant terms in the long exact sequences (4) and (6), we obtain the long exact sequence for the pair (L ⊙ Σ, L) and the Mayer-Vietoris long exact sequence for the decomposition L ⊙ Σ = L ∪ Σ, respectively. This fact was already observed and used by the fourth author in [START_REF] Golovko | A note on Lagrangian cobordisms between Legendrian submanifolds of R 2n+1[END_REF].

1.3. Topological restrictions on Lagrangian cobordisms. Using the long exact sequences from the previous subsection and their refinements to coefficients twisted by the fundamental group, as defined in Section 11, we find strong topological restrictions on exact Lagrangian cobordisms between certain classes of Legendrian submanifolds.

1.3.1. The homology of an exact Lagrangian cobordism from a Legendrian submanifold to itself. We recall that Λ always will denote a Lagrangian submanifold of the contactisation of a Liouville manifold. In the following results we study the homology of any exact Lagrangian cobordism from Λ to itself in the symplectisation. One of the consequences of Theorem 1.8 is the following theorem, proved in Section 12.1. A similar statement has been proven by the second and the fourth author in [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF]Theorem 1.6] under the more restrictive assumption that Λ bounds an exact Lagrangian filling.

Theorem 1.11. Let Σ be an exact Lagrangian cobordism from Λ to Λ and F a field (of characteristic two if Λ is not spin). If the Chekanov-Eliashberg algebra A(Λ; F) admits an augmentation, then:

(i) There is an equality

dim F H • (Σ; F) = dim F H • (Λ; F); (ii) The map (i - * , i + * ) : H • (Λ; F) → H • (Σ; F) ⊕ H • (Σ; F) is injective; and (iii) The map i + * ⊕ i - * : H • (Λ ⊔ Λ) → H • (Σ) is surjective.
Here i + is the inclusion of Λ as the positive end of Σ, while i -is the inclusion of Λ as the negative end of Σ.

Remark 1.12. The above equalities hold for the Z-graded singular homology groups without assuming that the cobordism Σ is graded.

An immediate corollary of Theorem 1.6 is the following result, which had already appeared in [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF]Theorem 1.7] under the stronger assumption that the negative end is fillable. Theorem 1.13. If Λ is a homology sphere which admits an augmentation over Z, then any exact Lagrangian cobordism Σ from Λ to itself is a homology cylinder (i.e.

H • (Σ, Λ) = 0).
Inspired by the work of Capovilla-Searle and Traynor [START_REF] Capovilla-Searle | Non-orientable Lagrangian cobordisms between Legendrian knots[END_REF], in Section 12.2 we prove the following restriction on the characteristic classes of an exact Lagrangian cobordism from a Legendrian submanifold to itself. Given a manifold M , we denote by w i (M ) the i-th Stiefel-Whitney class of T M . Theorem 1.14. Let Σ be an exact Lagrangian cobordism from Λ to itself, and F = Z/2Z. Assume that A(Λ; F) admits an augmentation. If, for some i ∈ N, w i (Λ) = 0, then w i (Σ) = 0.

If Λ is spin, the same holds for the Pontryagin classes.

By specialising to w 1 we obtain the following corollary, which extends the main result in [START_REF] Capovilla-Searle | Non-orientable Lagrangian cobordisms between Legendrian knots[END_REF]; in particular we partially answer Question 6.1 of the same article.

Corollary 1.15. If Λ is an orientable Legendrian submanifold admitting an augmentation, then any exact Lagrangian cobordism from Λ to itself is orientable.

1.3.2.
Restrictions on the fundamental group of certain exact Lagrangian fillings and cobordisms. Since Theorem 1.13 shows that an exact Lagrangian cobordism from a Legendrian homology sphere to itself is a homology cylinder, it is natural to ask under what conditions this cobordism in fact is an h-cobordism. We therefore need to incorporate the fundamental group in our constructions. To that end, following ideas of Sullivan in [START_REF] Sullivan | K-theoretic invariants for Floer homology[END_REF] and Damian in [START_REF] Damian | Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds[END_REF], we define a "twisted" version of the Floer homology groups HF ± (Σ 0 , Σ 1 ) with coefficient ring R[π 1 (Σ, * )] in Section 11.1. We also establish a result analogous to Theorem 1.5, as well as the long exact sequences in Section 1.2, with twisted coefficients in R[π 1 (Σ, * )]. In the setting of Legendrian contact homology, these techniques were introduced by Eriksson-Östman in [START_REF] Östman | Legendrian contact homology with homotopy coefficients[END_REF].

Using generalisations of the long exact sequence from Theorem 1.6 and the functoriality of the fundamental class from Proposition 11.7 (see Section 12.3.1) we prove the following theorem: Theorem 1.16. Let Σ be a graded exact Lagrangian cobordism from Λ -to Λ + . Assume that A(Λ -; R) admits an augmentation and that Λ + has no Reeb chords in degree zero. If Λ -and Λ + both are simply connected, then Σ is simply connected as well.

Remark 1.17. The seemingly unnatural condition that Λ + has no Reeb chords in degree zero is used to ensure that the Chekanov-Eliashberg algebra A(Λ + ; A) has at most one augmentation in A for every unital R-algebra A. (This algebraic condition does not ensure the existence of an augmentation, but rather it states that Λ + admits exactly one augmentation in the case when Λ -admits an augmentation.) This condition is clearly not invariant under Legendrian isotopy, but the conclusion of Theorem 1.16 can be extended to every Legendrian submanifold which is Legendrian isotopic to Λ + because Legendrian isotopies induce Lagrangian cylinders by [39, 4.2.5] (also, see [START_REF] Chantraine | Lagrangian concordance of Legendrian knots[END_REF]). Ideally, one should replace the algebraic condition with one that only depends on the DGA homotopy type.

We now present another result which imposes constraints on the fundamental group of an exact Lagrangian cobordism from a Legendrian submanifold to itself (see Section 12.3.2). Its proof uses an L 2 -completion of the Floer homology groups with twisted coefficients and the L 2 -Betti numbers of the universal cover (using results of Cheeger and Gromov in [START_REF] Cheeger | Bounds on the von Neumann dimension of L 2 -cohomology and the Gauss-Bonnet theorem for open manifolds[END_REF]).

Theorem 1.18. Let Λ be a simply connected Legendrian submanifold which is spin, and let Σ be an exact Lagrangian cobordism from Λ to itself. If A(Λ; C) admits an augmentation, then Σ is simply connected as well.

Combining Theorem 1.13 with Theorem 1.18, we get the following result.

Corollary 1.19. Let Σ be an n-dimensional Legendrian homotopy sphere and assume that A(Λ; Z) admits an augmentation. Then any exact Lagrangian cobordism Σ from Λ to itself is an h-cobordism. In particular:

(1) If n = 3, 4, then Σ is diffeomorphic to a cylinder;

(2) If n = 3, then Σ is homeomorphic to a cylinder; and

(3) If n = 4 and Λ is diffeomorphic to S 4 , then Σ is diffeomorphic to a cylinder.

When n = 1, a stronger result is known. Namely, in [START_REF] Chantraine | Floer homology and Lagrangian concordance[END_REF]Section 4] we proved that any exact Lagrangian cobordism Σ from the standard Legendrian unknot Λ 0 to itself is compactly supported Hamiltonian isotopic to the trace of a Legendrian isotopy of Λ 0 which is induced by the complexification of a rotation by kπ, k ∈ Z. This classification makes use of the uniqueness of the exact Lagrangian filling of Λ 0 up to compactly supported Hamiltonian isotopy, which was proved in [START_REF] Eliashberg | Local Lagrangian 2-knots are trivial[END_REF] by Eliashberg and Polterovich. In contrast, the methods we develop in this article give restrictions only on the smooth type of the cobordisms and little information is known about their symplectic knottedness in higher dimension. 1.3.3. Obstructions to the existence of a Lagrangian concordance. A Lagrangian concordance from Λ -to Λ + is a symplectic cobordism from Λ -to Λ + which is diffeomorphic to a product. In particular this implies that Λ -and Λ + are diffeomorphic as smooth manifolds. Note that a Lagrangian concordance automatically is exact.

If Σ is a Lagrangian concordance, then H • (Σ, ∂ -Σ; R) = 0, and thus Theorem 1.6 implies the following corollary.

Corollary 1.20. Let Σ be an exact Lagrangian concordance from

Λ -to Λ + . If, for i = 0, 1, ε - i is an augmentation of A(Λ -; R) and ε + i is the pull-back of ε - i under the DGA morphism induced by Σ, then the map Φ ε - 0 ,ε - 1 Σ : LCH • ε - 0 ,ε - 1 (Λ -) → LCH • ε + 0 ,ε + 1 (Λ + )
is an isomorphism. Consequently, there is an inclusion

{LCH • ε - 0 ,ε - 1 (Λ -)}/isom. ֒→ {LCH • ε + 0 ,ε + 1 (Λ + )}/isom.
of the sets consisting of isomorphism classes of bilinearised Legendrian contact cohomologies, for all possible pairs of augmentations.

This corollary can be used to obstruct the existence of Lagrangian concordances. For example, it can be applied to the computation of the linearised Legendrian contact homologies given by Chekanov in [START_REF] Yu | Differential algebra of Legendrian links[END_REF]Theorem 5.8] to prove that there is no exact Lagrangian concordance from either of the two Chekanov-Eliashberg knots to the other. We also use Corollary 1.20 to deduce new examples of non-symmetric concordances in the spirit of the example given by the first author in [START_REF] Chantraine | Lagrangian concordance is not a symmetric relation[END_REF]. We refer to Section 12.4.3 for a simply connected example in high dimensions.

We recall that a Legendrian isotopy induces a Lagrangian concordance. Since Legendrian isotopies are invertible, two isotopic Legendrian submanifolds thus admit Lagrangian concordances going in either direction. On the other hand, we have now many examples of non-symmetric Lagrangian concordances, and hence the following natural question can be asked. Question 1.21. Assume that there exists Lagrangian concordances from Λ 0 to Λ 1 as well as from Λ 1 to Λ 0 . Does this imply that the Legendrian submanifolds Λ 0 and Λ 1 are Legendrian isotopic? Are such Lagrangian concordances moreover Hamiltonian isotopic to one induced by a Legendrian isotopy (as constructed by [39, 4.2.5])?

We argue that this question will not be easily answered by Legendrian contact homology. Chekanov showed in [START_REF] Yu | Differential algebra of Legendrian links[END_REF] that the set of isomorphism classes of all linearised Legendrian contact homology groups is invariant under Legendrian isotopy. Later Bourgeois and the first author in [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF] extended this result to bilinearised Legendrian contact homology. However, Corollary 1.20 says that bilinearised Legendrian contact homology in fact is an invariant of Lagrangian concordances, rather than of Legendrian isotopies. This means that, every time two Legendrian submanifolds have been proved not to be Legendrian isotopic by exhibiting two non-isomorphic bilinearised Legendrian contact homology groups, what has in fact been proved is that there cannot exist Lagrangian concordances between the Legendrian submanifolds going both directions.

1.4. Remarks about the hypotheses.

1.4.1.

Restrictions on the ambient manifolds. The reasons for restricting our attention to Lagrangian cobordisms in the symplectisation of the contactisation of a Liouville manifold are two-fold. First, the analytic framework to have a well defined complex (Cth(Σ 0 , Σ 1 ),

d ε - 0 ,ε - 1
) is vastly simplified from the fact that the Reeb flow has no periodic Reeb orbits. Using recent work of Pardon in [START_REF] Pardon | Contact homology and virtual fundamental cycles[END_REF] (or the polyfold technology being developed by Hofer, Wysocki and Zehnder), it is possible to extend the construction of the complex (Cth(Σ 0 , Σ 1 ), d ε - 0 ,ε -

1

) to more general symplectic cobordisms. Second, our applications use exact sequences arising from the acyclicity of the complex (Cth(Σ 0 , Σ 1 ), d ε -

0 ,ε - 1
), which is a consequence of the fact that that any Lagrangian cobordism can be displaced in the symplectisation of a contactisation. Floer theory for Lagrangian cobordisms in more general symplectic cobordisms will be investigated in a future article. 1.4.2. Restrictions on the Lagrangian submanifolds. Now we describe some examples showing that many of the hypotheses we made in Subsection 1.3 are in fact essential, and not merely artefacts of the techniques used. First, an exact Lagrangian cobordism having a negative end whose Chekanov-Eliashberg algebra admits no augmentation can be a quite flexible object: in fact Eliashberg and Murphy proved in [START_REF] Eliashberg | Lagrangian caps[END_REF] that exact Lagrangian cobordisms with a loose negative end satisfy an hprinciple, and therefore one cannot hope for a result in the spirit of Theorem 1.13 to hold in complete generality. Indeed, we refer to the work of the second and fourth authors in [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF] for examples of exact Lagrangian cobordisms from a loose Legendrian sphere to itself having arbitrarily big Betti numbers.

Second, the condition that Λ is a homology sphere in the statement of Theorem 1.13 was shown to be essential already in [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF]Section 2.3].

Finally, the importance of the condition on the Reeb chords of the positive end in Theorem 1.16 is emphasised by the following example, which will be detailed in Section 12.4.

Proposition 1.22.

There exists a non-simply connected exact Lagrangian cobordism from the two-dimensional standard Legendrian sphere to a Legendrian sphere inside the symplectisation of standard contact (R 5 , ξ std ).

As a converse to Theorem 1.16, the existence of a non-simply connected exact Lagrangian cobordism can be used to show the existence of degree zero Reeb chords on the positive end of the cobordism. 1.5. Outline of the article. This article is organised as follows. In sections from 2 to 5 we collect some basic material: in Section 2 we describe our geometric setup and give a precise definition of the class of Lagrangian cobordisms that we consider, in Section 3 we introduce the moduli spaces used to define d ε - 0 ,ε - 1 and discuss their compactification, in Section 4 we study their Fredholm theory (e.g. indices and transversality), and in Section 5 we briefly review the definitions of Legendrian contact homology and augmentations.

Sections 6 to 8 are the theoretical core of the paper: in Section 6 we define the complex (Cth(Σ 0 , Σ 1 ), d ε -

0 ,ε - 1 ), in Section 7 we show how (Cth(Σ 0 , Σ 1 ), d ε - 0 ,ε - 1 )
behaves under concatenations of cobordisms, and in Section 8 we show that the complex (Cth(Σ 0 , Σ 1 ),

d ε - 0 ,ε - 1
) is acyclic. The last part of the article is devoted to applications. In Section 9 we prove Theorem 1.3 and in Section 10 we deduce the long exact sequences of Section 1.2. In Section 11 we lift the coefficient ring to the group ring R[π 1 (Σ)] and to its L 2completion. Finally, in Section 12 we prove the results concerning the topology of Lagrangian cobordisms stated in Section 1.3, and then describe examples which exhibit the necessity of their hypotheses.

While this research was conducted, the authors benefited from the hospitality of various institutions in addition to their own; in particular they are indebted to CIRGET and CRM in Montréal, the Centro De Giorgi in Pisa, and the institute Mittag-Leffler in Stockholm. We warmly thank Stefan Friedl for suggesting that we should look at L 2 -homology theory to gain information on fundamental groups.

Geometric preliminaries

Basic definitions.

A contact manifold (Y, ξ) is a (2n+1)-manifold Y equipped with a smooth maximally non-integrable field of hyperplanes ξ ⊂ T Y , which is called a contact structure. Non-integrability implies that locally ξ can be written as the kernel of a 1-form α satisfying α ∧ (dα) n = 0. We will be interested only in coorientable contact structures, which are globally the kernel of a 1-form α called a contact form. A contact form α defines a canonical vector field R α , called the Reeb vector field, via the equations

i Rα dα = 0, α(R α ) = 1.
We will use φ t : (Y, α) → (Y, α) to denote the flow of the Reeb vector field R α , which can be seen to preserve α. Also, in the following we will always assume the contact form to be fixed.

An n-dimensional submanifold Λ ⊂ Y which is everywhere tangent to ξ is called Legendrian. The Reeb chords of Λ are the trajectories of the Reeb flow starting and ending on Λ. We denote the set of the Reeb chords of Λ by R(Λ).

Let γ be a periodic orbit of R α of length T . It is non-degenerate if dφ T q -Id is invertible for one (and thus all) q ∈ γ. Let γ be a Reeb chord of length T . The flow φ t of the Reeb vector field preserves ξ, and therefore dφ T γ(0) (T γ(0) Λ) ⊂ ξ γ(T ) . We say that a Reeb chord γ is non-degenerate if dφ T γ(0) (T γ(0) Λ) is transverse to T γ(T ) Λ in ξ γ(T ) . We say that Λ is chord-generic if all its Reeb chords are non-degenerate. From now on we assume that all Legendrian submanifolds are chord-generic. This is not a restrictive assumption because chord genericity is a property which can be achieved by a generic Legendrian perturbation of Λ, provided that all periodic Reeb orbits are non-degenerate.

We will here restrict ourselves to the case when (Y, α) is the contactisation of a Liouville manifold. We recall that a Liouville manifold is a pair (P, θ), where P is a 2n-dimensional open manifold and θ is a one-form on P such that dθ is symplectic. The Liouville vector filed v, which is defined by the equation

ι v dθ = θ,
is moreover required to be a pseudo-gradient for an exhausting function f : P → R ≥0 outside of a compact set. For simplicity we will assume that f has a finite number of critical points. We define the contactisation (Y, α) of (P, θ) to be Y = P × R and α := dz + θ, where z is the coordinate on the R-factor. Note that in this case R α = ∂ z and then there are no periodic Reeb orbits. This implies, in particular, that if Λ is a chord-generic closed Legendrian submanifold, then |R(Λ)| < ∞. The prototypical example of contactisation of a Liouville manifold is the standard contact structure on R 2n+1 defined by the contact form

α 0 = dz - n i=1 y i dx i .
There is a natural projection Π Lag : P × R → P defined by Π Lag (x, z) := x which is called the Lagrangian projection. Given a Legendrian submanifold Λ ⊂ P × R, Π Lag | Λ : Λ → P is an exact Lagrangian immersion. In this situation, there is a one-to-one correspondence between the Reeb chords of Λ and the double points of Π Lag (Λ). Furthermore, Λ is chord-generic if and only if the only self-intersections of Π Lag (Λ) are transverse double points.

Lagrangian cobordisms.

The main object of study in this article are exact Lagrangian cobordisms in the symplectisation of a contactisation. Recall that, for a general contact manifold (Y, α), its symplectisation is the exact symplectic manifold (X, ω) := (R × Y, d(e t α)), where t denotes the standard coordinate on the R-factor. In the case when dim Y = 2n + 1, a (n + 1)-dimensional submanifold of the above symplectisation is exact Lagrangian if the pull-back of the one-form e t α is exact. Definition 2.1. Let Λ -and Λ + be two closed Legendrian submanifolds of (Y, α). An exact Lagrangian cobordism from Λ -to Λ + in (R×Y, d(e t α)) is a properly embedded submanifold Σ ⊂ R × Y without boundary satisfying the following conditions:

(1) for some

T ≫ 0, (a) Σ ∩ ((-∞, -T ) × Y ) = (-∞, -T ) × Λ -, (b) Σ ∩ ((T, +∞) × Y ) = (T, +∞) × Λ + , and (c) Σ ∩ ([-T, T ] × Y ) is compact; (2) There exists a smooth function f Σ : Σ → R for which (a) e t α| T Σ = df Σ , (b) f Σ | (-∞,-T )×Λ -is constant, and (c) f Σ | (T,∞)×Λ + is constant.
We will call (T, +∞) × Λ + ⊂ Σ and (-∞, -T ) × Λ -⊂ Σ the positive end and the negative end of Σ, respectively. We will call a cobordism from a submanifold to itself an endocobordism.

Conditions (2b) and (2c) are equivalent to saying that for any smooth paths γ -:

([0, 1], {0, 1}) → (Σ, (-∞, -T ) × Λ -) and γ + : ([0, 1], {0, 1}) → (Σ, (T, ∞) × Λ + ),
we have γ± e t α = 0. Condition (2b) will later be used to rule out certain bad breakings of pseudoholomorphic curves. Condition (2c) is used to ensure that the concatenation of two exact Lagrangian cobordisms (as in Defintion 2.3) still is an exact Lagrangian cobordism. If one does not care about concatenations, then this condition can be dropped.

Example 2.2. If Λ is a closed Legendrian submanifold of (Y, ξ), then R × Λ is an exact Lagrangian cobordism inside (R × Y, d(e t α)) from Λ to itself. Cobordisms of this type are called (trivial) Lagrangian cylinders.

In the case when there exists an exact Lagrangian cobordism from Λ -to Λ + we say that Λ -is exact Lagrangian cobordant to Λ + . If Σ is an exact Lagrangian cobordism from the empty set to Λ, we call Σ an exact Lagrangian filling of Λ. In the latter case we also say that Λ is exactly fillable.

The group R acts on R × Y by translations in the first factor. For any s ∈ R we define

τ s : R × Y → R × Y, τ s (t, p) = (t + s, p).
It is easy to check that the translate of an exact Lagrangian cobordism still is an exact Lagrangian cobordism. Definition 2.3. Given exact Lagrangian cobordisms Σ a from Λ -to Λ and Σ b from Λ to Λ + , their concatenation Σ a ⊙ Σ b is defined as follows.

First, translate Σ a and Σ b so that

Σ a ∩ ((-1, +∞) × Y ) = (-1, +∞) × Λ, Σ b ∩ ((-∞, 1) × Y ) = (-∞, 1) × Λ.
Then we define

Σ a ⊙ Σ b := (Σ a ∩ ((-∞, 0] × Y )) ∪ (Σ b ∩ ([0, +∞) × Y )).
Conditions (2b) and (2c) of Definition 2.1 imply that Σ a ⊙ Σ b is an exact Lagrangian cobordism from Λ -to Λ + . Lemma 2.4. The compactly supported Hamiltonian isotopy class of Σ a ⊙ Σ b is independent of the above choices of translations.

Proof. In order to prove the lemma, it is enough to show that any translation of an exact Lagrangian cobordism can be realised by a compactly supported Hamiltonian isotopy. It is a standard fact that a smooth isotopy of exact Lagrangian submanifolds can be realised by a Hamiltonian isotopy. In this case, the exact Lagrangian submanifolds considered are obviously non-compact and translations act on them in a non compactly supported way. However, given an exact symplectic cobordism Σ ⊂ R × Y and S ≥ 0, we can find a compactly supported smooth isotopy ψ s : R × Y → R × Y such that ψ s (Σ) = τ s (Σ) for all s ∈ [-S, S]. The isotopy ψ s can be defined by integrating the vector field χ∂ t , where χ : R × Y → R is the pull-back of a bump function χ : R → R such that χ(t) = 1 for t ∈ [-S -T, S + T ], while χ(t) = 0 for t ∈ [-S -T -1, S + T + 1]. (Here T ≥ 0 is the constant in Definition 2.1.) 2.3. Deligne-Mumford spaces and labels.

R d+1 = (a 0 , . . . , a d ) : a j = e iθj with θ 0 < . . . < θ d < θ 0 + 2π / Aut(D 2 ),
where the action of Aut(D 2 ) is defined as f • (a 0 , . . . , a d ) := (f (a 0 ), . . . , f (a d )) for f ∈ Aut(D 2 ).

We denote by S d+1 → R d+1 the universal curve over R d+1 . One of its realisations is

S d+1 = (z, a 0 , . . . , a d ) : z ∈ D 2 , a j = e iθj , θ 0 < . . . < θ d < θ 0 + 2π / Aut(D 2 ),
where the action of Aut(D 2 ) is defined as f •(z, a 0 , . . . , a d ) := (f (z), f (a 0 ), . . . , f (a d )) for f ∈ Aut(D 2 ). The map S d+1 → R d+1 is induced by the projection (z, a 0 , . . . , a d ) → (a 0 , . . . , a d ), and therefore is a fibre bundle with d + 1 canonical sections σ 0 , . . . , σ d ,

σ i : R d+1 → S d+1 ,
where σ i is induced by (a 0 , . . . , a d ) → (a i , a 0 , . . . , a d ). By an abuse of notation, when r is clear from the context, we will denote σ i (r) = a i .

For r ∈ R d+1 , we denote by S r ⊂ S d+1 its preimage under the projection and write S r = S r \ {a 0 , . . . , a d }. The connected components of ∂S r are oriented arcs, and we let ∂ i S r , i = 0, . . . , d, denote the arc whose closure has a i as a starting point.

2.3.2.

Strip-like ends. We denote by Z + and Z -the Riemann surfaces (0, +∞) × [0, 1] and (-∞, 0) × [0, 1] respectively, with coordinates (s, t) and conformal structure induced by the complex coordinate s + it.

A universal choice of strip-like ends is given by d + 1 disjoint neighbourhoods ν i of the images of the sections σ i together with identifications

ε 0 : R d+1 × Z + → ν 0 \ σ 0 and ε i : R d+1 × Z -→ ν i \ σ i , for i ≥ 1,
such that for each r ∈ R d+1 we have:

• The maps

ε i | {r}×Z ± : {r} × Z ± → ν i ∩ S r are holomorphic, • lim s→±∞ ε i (s, t) = a i ,

and

The point a 0 is called an incoming end, all the other a i 's are called outgoing ends (see Remark 2.5). See Figure 2. We still denote by (s, t) the coordinates on the strip-like ends S r ∩(ν i \σ i ) induced by the identifications ε i and the coordinates (s, t) on Z ± . In [64, Section 9a], it is shown that such universal choices of strip-like ends exist. 2.3.3. Deligne-Mumford compactification of R d+1 . In this Section, we describe a compactification of the space R d+1 into a manifold with corners R d+1 . We describe the faces of this compactification using the language of stable trees following [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF]Section 9]. A stable rooted tree is a tree with one distinguished exterior vertex (i.e. of valence one), called the root, and whose interior vertices all have valence at least three. Exterior vertices other than the root are called leaves.

For a stable rooted tree with d leaves T , we denote by R T the product Π v∈Ve(T ) R |v| , where Ve(T ) is the set of interior vertices and |v| denotes the valence of the vertex v, which is at least 3 by the stability condition.

Remark 2.5. The root induces a natural orientation of each edges of a rooted tree T by the following convention: the edge at the root is oriented so that it leaves the root. And at each other vertex there is exactly one incoming edge.

If a stable rooted tree T ′ is obtained from the stable rooted tree T by collapsing k edges, in [64, Sections 9e and 9f] there is a description of a gluing map [START_REF] Arnol ′ D | Lagrange and Legendre cobordisms[END_REF] γ T,T ′ : R T × (-1, 0

) k → R T ′ .
Naively, for an element (r v ) v∈Ve(T ) of R T and a k-tuple (l e ) of positive numbers indexed by the collapsed edges, gluing is performed by cutting the strip-like ends of S rv 1 and S rv 2 corresponding to a collapsed edge e from v 1 to v 2 at time |t| = e -πle , identifying the remaining part of the strip-like ends, and then uniformising the resulting disc.

Remark 2.6. Actually, [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF] defines only the map γ T,T d , where T d is the tree with only one interior vertex, which corresponds to R d+1 .

The gluing maps satisfy the cocycle relation:

γ T ′ ,T ′′ (γ T,T ′ ({r v }, ρ 1 , . . . , ρ k1 ), ρ k1+1 , . . . , ρ k1+k2 ) (8) = γ T,T ′′ ({r v }, ρ 1 , . . . , ρ k1+k2 ).
We define R d+1 = ⊔ T R T (as a set). The gluing maps in Equation [START_REF] Arnol ′ D | Lagrange and Legendre cobordisms[END_REF] allow us to build an atlas on R d+1 , and therefore one can prove the following. Lemma 2.7. [64, Lemma 9.2] The set R d+1 has a structure of manifold with corners which is induced by the gluing maps [START_REF] Arnol ′ D | Lagrange and Legendre cobordisms[END_REF]. For every stable rooted tree T with d leaves, R T is a stratum of the stratification of R d+1 defined as follows: if T ′ is obtained from T by collapsing some interior edges, then R T is a face of the compactification of R T ′ .

Elements of R d+1 are called nodal stable (d + 1)-punctured discs.

Remark 2.8. Note that the definition of the maps γ T,T ′ depends on the choice of strip-like ends for all R d ′ for d ′ < d. However, the structure of a manifold with corners on R d+1 turns out to be independent of these choices.

An element {r v } of R T is called a nodal stable (d + 1)-punctured disc. In this situation, marked points of r v corresponding to interior edges are called boundary nodes, while those corresponding to edges connected to leaves or to the root are called ends. A nodal stable (d + 1)-punctured disc in R T ′ which is in the image of a map γ T,T ′ inherits two strip-like ends: one coming from the universal choices of strip-like ends in the moduli spaces R |v| for v ∈ Ve(T ′ ), and one coming from the universal choices of strip-like ends in the moduli spaces R |v| for v ∈ Ve(T ). If there exists an ǫ > 0 such that, for all gluing parameters smaller than ǫ and for all nodal stable (d + 1)-punctured discs, those two strip-like ends agree, then we call the universal choices of strip-like ends in the moduli spaces R d+1 consistent. Consistent universal choices of strip-like ends exist because the space of strip-like ends is contractible, and the relation [START_REF] Atiyah | Elliptic operators, discrete groups and von Neumann algebras[END_REF] allows an inductive argument. From now on, we assume that such consistent choices have been made.

2.3.4.

Asymptotics labels for Deligne-Mumford spaces. In the present paper we will consider various moduli spaces of punctured holomorphic discs whose punctures are asymptotic to either Reeb chords or intersection points between Lagrangian cobordisms. We therefore make the following definitions.

Let Σ 0 and Σ 1 be two Lagrangian cobordisms from Legendrian submanifolds Λ - 0 and Λ - 1 at the negative ends to Legendrian submanifolds Λ + 0 and Λ + 1 at the positive ends, respectively. A Lagrangian label L for R d+1 is a map from Z/(d + 1)Z to {Σ 0 , Σ 1 }. If L(i -1) = L(i), then the marked point a i for i ∈ {0, . . . , d} is called a jump of L. Note that a Lagrangian label has an even number of jumps.

A Lagrangian label is simple if it satisfies the following conditions:

• If a 0 is not a jump, then none of the a i 's are jumps.

• It has at most two jumps.

• If a 0 is a jump, then L(0) = Σ 0 .

Remark 2.9. A Lagrangian label for R d+1 induces compatible Lagrangian labels for all faces of R d+1 . If the Lagrangian label is simple, then the induced labels are also simple.

The set of asymptotics of a given pair of Lagrangian cobordisms Σ 0 , Σ 1 is the union

A(Σ 0 , Σ 1 ) := (Σ 0 ∩ Σ 1 ) ∪ R(Λ + 0 ⊔ Λ + 1 ) ∪ R(Λ - 0 ⊔ Λ - 1
). The main definition in this section is the following. Definition 2.10.

• An asymptotics label for R d+1 is an assignment of an asymptotic in A(Σ 0 , Σ 1 ) to each section σ i , i = 0, . . . , d (thus inducing asymptotics for each marked point a i given any r ∈ R d+1 ).

• A Lagrangian label is compatible with a given asymptotics label if:

-Every marked point asymptotic to a double point is a jump; -In the case when the asymptotic of a i is a Reeb chord starting on Λ ± j and ending on Λ ± k (here {j, k} ⊂ {0, 1}), it follows that

L(i -1) = Σ j while L(i) = Σ k .
Remark 2.11. It is an obvious fact that there can be at most one simple Lagrangian label compatible with a given asymptotics label, thus we will always only specify a set of asymptotics assuming that the implicit compatible label is chosen.

2.3.5.

Semi-stable nodal discs. In the paper we will also consider more general domains called semi stable whose combinatorics is still described by decorated trees similarly as in Section 2.3.3, but possibly containing semi-stable vertices. We use the convention that R d+1 consists of a single point when d = 0, 1. Elements of S 0+1 or S 1+1 is called a semi-stable punctured disc. In other word a punctured disc is semi-stable if it has either one or two punctures on the boundary. After removing the punctures, they can be uniformised either as the strip Z = R × [0, 1] or the half-plane H = {im z > 0} ∈ C with their standard Riemann structures. We call (s, t) the global coordinates on Z, and we require that any strip-like end on Z is a restriction of these coordinates.

A semi-stable rooted tree with d leaves is a tree T with one chosen valence one vertex (the root) and d chosen valence one vertices (the leaves). All other vertices, including the remaining valence one vertices, will be called internal vertices.

The stabilisation of T , denoted S(T ), is the stable tree obtained by performing the following two operations:

• promoting valence one interior vertices to leaves, and • suppressing all vertices of valence 2 and merging the corresponding edges. Note that the interior vertices of S(T ) are in natural bijection with the vertices of T of valence at least three and, therefore, S(T ) is empty in the case when every vertex of T has valence one or two.

To any semi-stable tree T together with an element of R S(T ) , we associate a punctured nodal disc as follows.

• To any interior vertex of T we associate a disc;

• For any edge of T we put a boundary node between the discs corresponding to the vertices joined by the edge; and • For any exterior vertex we put a marked point in the disc associated to the closest interior vertex. The element of R S(T ) specifies a conformal structure for any disc associated to a vertex of valence at least three (there are no moduli for discs with one or two punctures). Finally, the boundary marked points associated to the root will become an incoming puncture and the marked points associated to the leaves will become outgoing punctures.

For any semi-stable tree T with k interior vertices and d leaves, there is a gluing map γ T,T d : R S(T ) → R d+1 similar to the one described in Section 2.3.3, but since H and Z have nontrivial automorphism groups, these gluing maps are not local embeddings. Lagrangian and asymptotics labels can be defined for unstable curves in the same way. Obviously, the Lagrangian label on H must have no jumps.

To a (stable or semi-stable) nodal discs S = {r v } ∈ R T one associates its normalisation which is simply the quadruple ( S, m, n, ι) with

• S = ⊔ k v∈Ve(T ) S rv .
• m is the union of the marked points of S corresponding to edges of T connecting to leaves or to the root of T . • n is the union of the marked point of S corresponding to interior edges of T . It comes equipped with a fixed point free involution ι determined by ι(n 0 ) = n 1 if n 0 and n 1 are connected by an edge e. Elements of m are called ends of S and elements of n are called nodes. Note that one can rebuild the gluing tree out of the data ( S, m, n, ι).

Analytic preliminaries

3.1. Almost complex structures. Before defining the moduli spaces of pseudoholomorphic curves that are relevant for the theory (see Section 3.2), it is necessary to describe the almost complex structures. For technical reasons it will be necessary to make certain additional assumptions on them that we here describe.

3.1.1. Cylindrical almost complex structures. Let (Y, α) be a contact manifold with the choice of a contact form. We denote by J cyl (Y ) the set of cylindrical almost complex structures on the symplectisation (R × Y, d(e t α)), i.e. almost complex structures J satisfying the following conditions:

• J is invariant under the natural (symplectically conformal) action of R on R × Y ; • J ∂ ∂t = R α ; • J(ξ) = ξ, where ξ := ker α ⊂ T Y ; and • J is compatible with dα| ξ , i.e. dα ξ (•, J•) is a metric on ξ.
We will say that an almost complex structure defined on a subset of the symplectisation of the form I × Y (where I is an interval) comes from J cyl (Y ) if it is the restriction of an almost complex structure in J cyl (Y ).

3.1.2. Almost complex structures on Liouville manifolds. Let (P, θ) be a Liouville manifold. Recall that there is a subset P ∞ ⊂ P that is exact symplectomorphic to half a symplectisation

([0, +∞) × V, d(e τ α V )),
and where P \ P ∞ ⊂ P is pre-compact. We say that an almost complex structure J P compatible with dθ is admissible if the almost complex structure J P on P comes from J cyl (V ) outside of a compact subset of P ∞ . We denote by J adm (P ) the set of these almost complex structures.

3.1.3. Cylindrical lifts. We will now restrict our attention to the manifolds that we will be considering here; namely, the contactisation of a Liouville manifold (P, θ), i.e.

(Y, α) := (P × R, dz + θ), and its symplectisation (X, ω) := (R × P × R, d(e t (dz + θ))).

From now on (Y, α) and (X, ω) will always denote manifolds of this type.

Given a compatible almost complex structure J P in J adm (P ) as defined above, there is a unique cylindrical almost complex structure J P on (X, d(e t α)) which makes the projection π : X = R × P × R → P a ( J P , J P )-holomorphic map. We will call this almost complex structure the cylindrical lift of J P . An important feature of the cylindrical lift is that the diffeomorphisms (t, p, z) → (t, p, z + z 0 ), z 0 ∈ R, induced by the Reeb flow all are Jholomorphic.

We denote by J cyl π (Y ) ⊂ J cyl (Y ) the set of cylindrical lifts of almost complex structures in J adm (P ).

3.1.4. Compatible almost complex structures with cylindrical ends. Let J + and J - be almost complex structures in J cyl π (Y ), and let T ∈ R + . We will require that both almost complex structures J ± coincide outside of R × K for some compact subset K ⊂ Y . We denote by J adm J -,J + ,T (X) the set of almost complex structures on X = R × Y that tame d(e t α) and satisfy the following.

(a) The almost complex structure J is equal to the cylindrical almost complex structures J -and J + on subsets of the form

(-∞, -T ] × P × R, [T, +∞) × P × R,
respectively; and (b) outside of R×K, the almost complex structure J coincides with a cylindrical lift in J cyl π (X). Condition (b) is needed in order to deal with compactness issues. Recall that the contact manifold P × R as as well as the Liouville manifold P are non-compact.

In the case when we do not care about the parameter T > 0, we will simply write J adm J -,J + (X). The union of all J adm J -,J + (X) over all J -, J + ∈ J cyl (Y ) is denoted J adm (X), almost complex structure in this set will be called admissible.

3.1.5. Domain dependent almost complex structures. Let J ± ∈ J cyl (Y ) be two cylindrical almost complex structures and let {J t } be a smooth path in J adm J -,J + (X) which is locally constant near t = 0 and t = 1.

Let Σ 0 and Σ 1 be two Lagrangian cobordisms in the symplectisation (X, d(e t α)) of (Y, ξ) with Legendrian ends Λ ± i i = 0, 1. To a punctured disc S (either stable or semi-stable) with simple Lagrangian label L we associate a domain-dependent almost complex structure J (S,L) : S → J adm J -,J + (X) using the path {J t }, as described below.

For every punctured disc S, if L has no jumps, then J (S,L) is constant and has value J 0 or J 1 depending if the Lagrangian label L is constant at Σ 0 or Σ 1 . Note that S = H (i.e. a half-plane) always falls into this case.

If L has two jumps (which means, in particular, that S has d + 1 boundary punctures with d ≥ 1), we uniformise S to a strip R × [0, 1] (with coordinates (s, t)) with d -1 punctures in the boundary, such that the jumps of L correspond to the ends of the strip, and moreover the incoming puncture of S corresponds to the end of the strip at s ≪ 0. This uniformisation, which is unique up to translations in the s-coordinate of the strip, defines a map t : S → [0, 1] by composition with the tcoordinate on the strip, and we define J (S,L) : S → J adm J -,J + (X) by J (S,L) (z) = J t(z) . Note that J (S,L) is constantly equal to J 0 near the boundary components associated to Σ 0 and constantly equal to J 1 near the boundary components associated to Σ 1 .

When S is a stable (d + 1)-punctured disc, the maps J (S,L) fit into smooth maps

J (d,L) : S d+1 → J adm J -,J +
which are obviously compatible with the degenerations of the Deligne-Mumford moduli spaces. In order to uniformise the notation we will denote J (2,L) := J (Z,L) and J (1,L) := J (H,L) , The collection of these maps is called a universal choice of domain dependent almost complex structures induced by (J + , J -, J t ). For a point z ∈ S d+1 we denote by J(z) the corresponding almost complex structure on X induced by this construction.

3.2.

Moduli space of holomorphic discs. We describe now the moduli space of discs used in Section 6 in order to define the differential of the Cthulhu complex.

We begin with the general definition of these moduli spaces, and then detail some of them with particular asymptotics.

3.2.1. General definitions. In this subsection we set up some terminology which will be used in all subsequent discussions about moduli spaces. Let S be a d + 1punctured disc (d ≥ 0) and L a simple Lagrangian label for S with values in a pair of Lagrangian cobordisms (Σ 0 , Σ 1 ). Suppose we have a (possibly domain dependent) almost complex structure J on X. We say that a map u :

S → X is J-holomorphic with boundary conditions in L if ∀z ∈ S (9) d z u • j = J(z) • d z u,
where j denotes the standard complex structure on D 2 , and u(∂ i S) ⊂ L(i). If the Lagrangian label L is constant, we will say that u is pure; if it has jumps, we will say that u is mixed. Here we will remain vague about the almost complex structure J, because it will depend on the specific moduli space that we will consider. Given an intersection point p ∈ Σ 0 ∩ Σ 1 , we say that u is asymptotic to p at a i if • the marked point a i is a jump, and

• lim z→ai u(z) = p. Let γ be a Reeb chord of Λ ± 0 ⊔ Λ ± 1 of length T . The map u = (a, v) into R × Y has a positive asymptotic to γ at a i if: • lim s→+∞ v(ε i (s, t)) = γ(T t) and lim s→+∞ a(ε i (s, t)) = +∞, given that a i = a 0 is the incoming puncture, or • lim s→-∞ v(ε i (s, t)) = γ(T (1 -t)) and lim s→-∞ a(ε i (s, t)) = +∞, given that i = 0. Let γ be a Reeb chord of Λ ± 0 ⊔Λ ± 1 of length T . The map u = (a, v) has a negative asymptotic to γ at a i , i = 0, given that • lim s→-∞ a(ε i (s, t)) = -∞ and lim s→-∞ v(ε i (s, t)) = γ(T t).
We will never consider holomorphic curves which have a negative asymptotic at the incoming end. Associated to a pair of cobordisms there are three types of possible targets for Lagrangian labels that will be considered here:

(Σ 0 , Σ 1 ), (R × Λ ± 0 , R × Λ ± 1 ) and R × Λ ±
i for i = 0, 1; note that the asymptotics of the latter two are subsets of those of the first pair. We will use L to denote any of those labels. Given x 0 , . . . , x d in A(Σ 0 , Σ 1 ) and r ∈ R d+1 , we denote by M r L (x 0 ; x 1 , . . . , x d ; J) the space of J-holomorphic maps from S r to X with asymptotics to x i at a i modulo reparametrisations of S r . In other words, x 0 is the asymptotic of the incoming puncture. We denote by M L (x 0 ; x 1 , . . . , x d ; J) the union of the M r L (x 0 ; x 1 , . . . , x d ; J) over all r ∈ R d+1 . Note that once the asymptotics for the moduli space is fixed, the actual Lagrangian label is uniquely determined and, hence, we do not need to specify it.

In the case when both L and J are invariant under translations of the symplectisation coordinate, there is an induced R-action on M L (x 0 ; x 1 , . . . , x d ; J). We use M L (x 0 ; x 1 , . . . , x d ; J)

to denote the quotient of the moduli space by this action.

In the following subsections we describe the moduli spaces with simple Lagrangian labels that appear in the definition of the Cthulhu differential in Section 6.

Strips and half-planes.

As already seen, the pseudoholomorphic discs considered here will have a number of different types of asymptotics. However, it will be useful to make a distinction between the following two types of discs considered.

• A pseudoholomorphic disc with no jumps is called a (punctured) half-plane; while • A pseudoholomorphic disc having precisely two jumps will be called a (punctured) strip. The puncture corresponding to the unique incoming end will be called the output while the puncture corresponding to the unique outgoing end at which a jump occurs will be called the input.

Remark 3.1. The fact that outgoing ends are inputs and incoming ends are outputs might seem confusing. The incoming/outgoing dichotomy comes from the notion of incoming and outgoing edges in a rooted tree (see Remark 2.5) and refers to particular coordinates of the domain (the strip like ends). This follows the convention of [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF]. The dichotomy input/output refers to what will belong to the domain/codomain of the differential as defined in Section 6.

The above two types of pseudoholomorphic discs (i.e. halfplanes and strips) will play radically different roles in our theory. Roughly speaking, punctures halfplanes appear when defining Chekanov-Eliashberg DGA as well as DGA morphisms, while punctured strips appear when defining the Floer homology, the bilinearised Legendrian contact cohomology, and the Cthulhu homology.

LCH moduli spaces.

The LCH moduli spaces are the moduli spaces appearing in the definition of linearised contact homology. We will consider two types of LCH moduli spaces, depending on whether the involved Reeb chords are mixed or not. We fix two cylindrical almost complex structures J ± ∈ J cyl (Y ) on X = R × Y and a path {J t } of almost complex structures in J adm J + ,J -(X). We denote J (d,L) the corresponding universal choice of domain dependent almost complex structures.

Denote Σ = Σ i and Λ

± = Λ ± i for i = 0, 1. Let γ + , δ + i , . . . , δ + d be Reeb chords of Λ + and γ -, δ - 1 , . . . , δ - d Reeb chords of Λ -.
Throughout the paper we will write a d-tuple of pure Reeb chords as a word; this notation is reminiscent of the multiplicative structure of the Chekanov-Eliashberg algebra. Therefore, we set δ ± = δ ± 1 . . . δ ± d . We will consider three types of pure LCH moduli spaces:

M R×Λ + (γ + ; δ + ; J + ), M R×Λ -(γ -; δ -; J -), and M Σ (γ + ; δ -; J (d,L) ).
The first two moduli spaces will be called pure cylindrical LCH moduli spaces and are used to define the Legendrian contact homology differential of Λ ± . The third moduli space will be called pure cobordism LCH moduli space and is used in [START_REF] Ekholm | Legendrian knots and exact Lagrangian cobordisms[END_REF] to define maps between Legendrian contact homology algebras. Since the Lagrangian labels are constant for these moduli spaces, the almost complex structures J (d,L) ) are actually domain independent. However, their value depends on whether Σ = Σ 0 or Σ = Σ 1 . Finally, recall that in the cylindrical moduli spaces, we take a quotient by the R-action, while such an operation is not possible (nor desirable) for the cobordism moduli space. Now we describe mixed LCH moduli spaces. Let

δ ± := δ ± 1 . . . δ ± i-1 be Reeb chords of Λ ± 1 and ζ ± = ζ ± i+1 . . . ζ ± d be Reeb chords of Λ ± 0 .
We will consider three types of mixed LCH moduli spaces:

M R×Λ ± 0 ,R×Λ ± 1 (γ + ; δ ± , γ -, ζ ± ; J ± ) and M Σ0,Σ1 (γ + ; δ -, γ -, ζ -; J (d,L) ), where γ ± ∈ R(Λ + 1 , Λ + 0 ) for M R×Λ + 0 ,R×Λ + 1 (γ + ; δ + , γ -, ζ + ; J + ), γ ± ∈ R(Λ - 1 , Λ - 0 ) for M R×Λ - 0 ,R×Λ - 1 (γ + ; δ -, γ -, ζ -; J -), and γ + ∈ R(Λ + 1 , Λ + 0 ), γ -∈ R(Λ - 1 , Λ - 0 ) for M Σ0,Σ1 (γ + ; δ -, γ -, ζ -; J (d,L) ).
The first two moduli spaces will be called mixed cylindrical LCH moduli spaces and are used to define the bilinearised Legendrian contact homology differential of (Λ ± 0 , Λ ± 1 ) (see Section 5.3 or [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF]). The third moduli space will be called mixed cobordism LCH moduli space and is used to define maps between bilinearised Legendrian contact homology groups. An illustration of a curve in the mixed cobordism LCH moduli spaces is shown in Figure 3.

γ + δ 1 δ 2 γ - ζ 1 ζ 2 Σ 0 Σ 1 Figure 3. A mixed cobordism LCH curve. 3.2.4. Floer moduli space. Let p, q ∈ Σ 0 ∩Σ 1 be intersection points, δ -= δ - 1 . . . δ - i-1
a word of Reeb chords on Λ - 0 , and

ζ -= ζ - i+1 , . . . ζ - d a word of Reeb chords on Λ - 1 . J-holomorphic curves in the moduli space M Σ0,Σ1 (p; δ -, q, ζ -; J (d,L) )
will be called Floer strips. (Note that this is an abuse of terminology as the actual domain is not a strip unless δ -and ζ -are empty.) See Figure 4.

Furthermore, the punctured disc is here required to makes a jump from Σ 1 to Σ 0 at the puncture asymptotic to p (this is the incoming puncture), while it makes a jump from Σ 0 to Σ 1 at the puncture asymptotic to q. This follows from our convention for Lagrangian labels; see Remark 2.11.

p δ 1 δ 2 q ζ 1 ζ 2 Σ 0 Σ 1 Figure 4. A Floer strip. 3.2.5. LCH to Floer moduli space. Let γ -∈ R(Λ - 1 , Λ - 0 ) be a mixed chord, p ∈ Σ 0 ∩ Σ 1 an intersection point, δ -= δ - 1 . . . δ - i-1 a word of Reeb chords on Λ - 0 , and ζ -= ζ - i+1 . . . ζ - d a word of Reeb chords on Λ - 1 . Curves in the moduli space M Σ0,Σ1 (p; δ -, γ -, ζ -; J (d,L) )
will be called holomorphic Cthulhus. See Figure 5.

p δ 1 δ 2 γ - ζ 1 ζ 2 Σ 0 Σ 1 Figure 5. A holomorphic Cthulhu. 3.2.6. Floer to LCH moduli space. Let γ + ∈ R(Λ + 1 , Λ + 0 ) be a mixed chord, p ∈ Σ 0 ∩ Σ 1 an intersection point, δ -= δ - 1 . . . δ - i-1 Reeb chords of Λ - 0 and ζ -= ζ - i+1 . . . ζ - d Reeb chords of Λ - 1 . Curves in the moduli space M Σ0,Σ1 (γ + ; δ -, p, ζ -; J (d,L) )
will be called J-holomorphic cultists. See Figure 6.

γ + δ 1 δ 2 q ζ 1 ζ 2 Σ 0 Σ 1 Figure 6.
Curve contributing to the second cultist map.

3.2.7. Bananas moduli space. Let γ 1,0 ∈ R(Λ + 1 , Λ + 0 ) and γ 0,1 ∈ R(Λ + 0 , Λ + 1 ) be mixed Reeb chords (and note that they go in opposite directions), let

δ -= δ - 1 . . . δ - i-1 be Reeb chords of Λ - 0 and ζ -= ζ - i+1 . . . ζ - d Reeb chords of Λ - 1 . Curves in the moduli space M Σ0,Σ1 (γ 1,0 ; δ -, γ 0,1 , ζ -; J (d,L) )
will be called J-holomorphic bananas. See Figure 7.

γ 1,0 δ 1 δ 2 γ 0,1 ζ 1 ζ 2 Σ 0 Σ 1 Figure 7. A J-holomorphic banana. When Σ i = R × Λ i , we again use M(γ 1,0 ; δ, γ 0,1 , ζ -)
to denote the quotient of the moduli space by the natural R-action.

Note that, by our our definition of an exact Lagrangian cobordism, there are no non-constant pseudoholomorphic curves with boundary on Σ 0 ∪ Σ 1 with all punctures having negative asymptotics to Reeb chords; see Section 3.3.2 for more details.

Energy and compactness.

In this section, we recall the notion of the Hofer energy for holomorphic curves in the symplectisation of a contact manifold as introduced in [START_REF] Hofer | Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three[END_REF] and [START_REF] Bourgeois | Compactness results in symplectic field theory[END_REF]. We also give estimates for this energy in terms of the asymptotics of the curves appearing in the moduli spaces of Section 3.2. The goal is to formulate the compactness theorem for pseudoholomorphic curves in the present setting.

3.3.1. The Hofer energy. Assume that we are given two exact Lagrangian cobordisms Σ 0 and Σ 1 in the symplectisation (X = R × Y, d(e t α)) of (Y, α). Let f i : Σ i → R be primitives of e t α| Σi which are constant at the cylindrical ends. (By a slight abuse of notation, we will denote by •| Σi the pull-back of e t α under the inclusion of Σ i .) Without loss of generality we will assume that both constants are 0 on the negative ends, while the constants on the positive end of Σ i will be denoted by c i , i = 0, 1. Here we rely on Definition 2.1 of an exact cobordism.

Take any T > 0 and T > ǫ > 0 for which

Σ i ∩ ((-∞, -T + ε) × Y ) = (-∞, -T + ε) × Λ - i Σ i ∩ ((T -ε, +∞) × Y ) = (T -ε, +∞) × Λ + i , for i = 0, 1. Now, we let φ : R → [e -T , e T ] be a smooth function satisfying: • φ(±t) = e ±T for t > T ; • φ(t) = e t for t ∈ [-T + ǫ, T -ǫ]; • φ ′ (t) ≥ 0.
In the case when both Σ 0 and Σ 1 are trivial cylinders over Legendrian submanifolds, we will also allow the case T = ǫ = 0, and φ ≡ 1.

By construction we have φα| Σi = e t α| Σi for the reason that α| Σi = 0 in the subset where φ is not equal to e t . A primitive of e t α| Σi (which exists by exactness) is hence also a primitive of φα| Σi .

Let C -be the set of compactly supported smooth functions

w -: (-∞, -T + ǫ) → [0, +∞) satisfying -T +ǫ -∞
w -(s)ds = e -T , and let C + be the set of compactly supported smooth functions

w + : (T -ǫ, +∞) → [0, +∞) satisfying +∞ T -ǫ w + (s)ds = e T .
We are now ready to introduce different versions of energies for pseudoholomorphic curves in the symplectisation, all which are standard. We refer to [START_REF] Bourgeois | Compactness results in symplectic field theory[END_REF] for the absolute case, and [START_REF] Ekholm | Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms[END_REF] as well as [START_REF] Abbas | Pseudoholomorphic strips in symplectisations. I. Asymptotic behavior[END_REF] for the relative case. Definition 3.2. Let S be a punctured disc and u = (a, v) : S → R × Y be a smooth map.

• The d(φα)-energy of u is given by

E d(φα) (u) = S u * (d(φα)).
• The α-energy of u is given by

E α (u) = sup (w-,w+)∈C -×C + S (w -• a)da ∧ v * α + S (w + • a)da ∧ v * α .
• The total energy, or the Hofer energy, of u is given by

E(u) = E α (u) + E d(φα) (u).
In the case when u is a proper map for which E(u) < ∞, we say that u is a finite energy pseudoholomorphic disc.

Non-constant holomorphic curves have positive total energy, as stated in the following simple lemma. We leave the proof to the reader. Lemma 3.3. If u is non-constant punctured pseudoholomorphic disc with boundary on a pair of exact Lagrangian cobordisms, and if the almost complex structure is cylindrical outside of

[-T +ǫ, T -ǫ]×Y , then E(u) > 0, E α (u) ≥ 0, and E d(φα) (u) ≥ 0. Moreover, E d(φα) (u) = 0 implies that u is contained inside a trivial cylinder over a Reeb orbit.
The techniques in [START_REF] Hofer | Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics[END_REF] can be to the applied to the current setting with nonempty boundary, similarly to what was done in [START_REF] Abbas | Pseudoholomorphic strips in symplectisations. I. Asymptotic behavior[END_REF], in order to show that Proposition 3.4. Assume that we endow R × P × R with an admissible complex structure in the sense of Section 3.1.4. A proper punctured pseudoholomorphic disc inside the symplectisation having boundary on a Lagrangian cobordism Σ 0 ∪ Σ 1 is of finite energy if and only if all of its punctures are contained in the boundary, and such that the disc is exponentially converging to either trivial strips over Reeb chords on Λ ± 0 ∪ Λ ± 1 or intersection points Σ 0 ∩ Σ 1 at each of its boundary punctures.

Action and energy. Consider a pair of exact Lagrangian cobordisms Σ

i , i = 0, 1, from Λ - i to Λ + i .
Here we define the action of intersection points and Reeb chords and relate it to the different energies of holomorphic discs having boundary on this pair of cobordisms.

For a Reeb chord c we define

ℓ(c) := c α.
Recall that we are given a choice of T ≥ 0 in the construction of the E d(φα) -energy above, where equality only is possible under the assumption that both cobordisms are trivial cylinders. The action of a mixed Reeb chord γ on Λ ± 1 ∪ Λ ± 0 is defined by

a(γ) := e T ℓ(α) + (c i -c j ) if γ is a chord from Λ + i to Λ + j , and a(γ) := e -T ℓ(α) if γ is a chord on Λ - 0 ∪ Λ - 1 .
In particular, we observe that the action of a pure Reeb chord γ on Λ ± i is defined by a(γ) := e ±T ℓ(γ).

The action of an intersection point

p ∈ Σ 0 ∩ Σ 1 is defined by a(p) := f 1 (p) -f 0 (p).
Applications of Stoke's theorem gives the following proposition (see [START_REF] Rizell | Legendrian ambient surgery and Legendrian contact homology[END_REF] for details), where we heavily rely on the fact that each cobordism Σ i , i = 0, 1, is exact.

Proposition 3.5. Let γ ± ∈ R(Λ ± 1 , Λ ± 0 ) be mixed Reeb chords, δ -= δ - 1 . . . δ - i-1
and ζ -= ζ i+1 . . . ζ d words of pure Reeb chords on Λ - 1 and Λ - 0 , respectively, and

p, q ∈ Σ 0 ∩ Σ 1 intersection points. We denote a(δ -) := i-1 k=1 a(δ - k ) and a(ζ -) := d k=i+1 a(ζ - k ). • If u ∈ M L (γ + ; δ -, γ -, ζ -), then E d(φα) (u) = a(γ + ) -a(γ -) -a(δ -) + a(ζ -) . (10) E α (u) ≤ 2a(γ + ) • If u ∈ M L (γ + ; δ -, p, ζ -), then E d(φα) (u) = a(γ + ) -a(p) -a(δ -) + a(ζ -) . (11) E α (u) ≤ 2a(γ + ) • If u ∈ M L (p; δ -, γ -, ζ -), then E d(φα) (u) = a(p) -a(γ -) -a(δ -) + a(ζ -) . (12) E α (u) ≤ a(p) • If u ∈ M L (p; δ -, q, ζ -), then E d(φα) (u) = a(p) -a(q) -a(δ -) + a(ζ -) . (13) E α (u) ≤ a(p) • If u ∈ M L (γ 1,0 ; δ -, γ 0,1 , ζ -), then E d(φα) (u) = a(γ 1,0 ) + a(γ 0,1 ) -a(δ -) + a(ζ -) . (14) E α (u) ≤ 2a(γ 1,0 ) + 2a(γ 0,1 )
For the above inequalities involving the α-energy we must assume that the almost complex structure is cylindrical outside of [-T + ǫ, Tǫ] × Y .

Holomorphic buildings with boundary on Lagrangian cobordisms.

The moduli space of punctured pseudoholomorphic discs with boundary on a Lagrangian cobordism can be compactified by the space of pseudoholomorphic buildings, which we now proceed to define. Definition 3.6. Let S be a (stable or semi-stable) nodal disc with normalisation ( S, m ∪ n, ι) as described in Section 2.3.5. By a level function for S we mean a locally constant map

f : S → {-k -, -k -+ 1, . . . , k + -1, k + } for some k -, k + ∈ Z ≥0 such that for any node n ∈ n we have |f (n) -f (ι(n))| ≤ 1, while f (n) = f (ι(n)) is allowed only if f (n) = 0.
If S is a nodal disc with level function f , a Lagrangian label L on (S, f ) will be a collection of Lagrangian labels L i on each connected component S i of the normalisation such that:

• L i takes values in {R × Λ + 0 , R × Λ + 1 } if f (S i ) > 0, in {Σ 0 , Σ 1 } if f (S i ) = 0 and in {R × Λ - 0 , R × Λ - 1 } if f (S i )
< 0; and • At a node n ∈ n the asymptotic corresponding to n agrees with the asymptotic corresponding to ι(n).

In particular, performing boundary connected sums at all nodes of S, one can canonically assign a Lagrangian label on the resulting nodal disc taking values in {Σ 0 , Σ 1 }. A Lagrangian label on (S, f ) is simple if, after performing boundary connected sums at all nodes, the resulting Lagrangian label is simple. Let Σ 0 , Σ 1 be two admissible exact Lagrangian cobordisms, and {J t } a path in J adm J + ,J -inducing domain dependent almost complex structures J (d,L) for any d ≥ 0 and any Lagrangian label L for elements in R d+1 . Definition 3.7. Let S be a nodal disc with Lagrangian label L and level function

f mapping onto {-k -, -k -+ 1, . . . , k + -1, k + } for some integers k ± ≥ 0. A holomorphic building of height k -|1|k + with
domain S is given by a family {u i | i = 1, . . . , k} of punctured pseudo-holomorphic discs, where the connected component S i of the normalisation of S corresponds to u i , such that the following conditions moreover are satisfied:

• All discs satisfying f (S i ) = l for some fixed l are said to live in level l ∈ Z. We require that each level different from zero contains at least one component which is not a trivial strip over a Reeb chord.

• If f (S i ) > 0, then u i ∈ M ri R×Λ + 0 ,R×Λ + 1
(y 0 ; y 1 , . . . , y di ; J + ). These discs are said to live in a top level.

• If f (S i ) < 0, then u i ∈ M ri R×Λ - 0 ,R×Λ - 1 (y 0 ; y 1 , . . . , y di ; J -). These discs are said to live in a bottom level. • If f (S i ) = 0, then u i ∈ M ri Σ0,Σ1 (y 0 ; y 1 , . . . , y di ; J (di,L i ) )
. These discs are said to live in the middle level.

• For any node n whose asymptotic corresponds to a Reeb chord, we require that |f

(n) -f (ι(n))| = 1; • For any node n such that 0 ≤ f (n) < f (ι(n))
, let y n and y ι(n) be the corresponding asymptotics. Then

y n = y ι(n) is a Reeb chord on R × (Λ + 0 ⊔ Λ + 1
), such that y n and y ι(n) is a positive and negative asymptotic, respectively.

• For any node n such that f (n) < f (ι(n)) ≤ 0 let y n and y ι(n) be the corresponding asymptotics. Then

y n = y ι(n) is a Reeb chord on R × (Λ - 0 ⊔ Λ - 1 )
, such that y n and y ι(n) is a positive and negative asymptotic, respectively.

• For any node n such that f (n) = f (ι(n)) (and thus is equal to 0) the corresponding asymptotics y n and y ι(n) correspond to a given intersection point in Σ 0 ∩ Σ 1 . • The positive (negative) punctures asymptotic to Reeb chords which are not nodes correspond precisely to the positive (negative) punctures in the k + :th (k -:th) level. Furthermore, we identify two buildings consisting of the same nodal domain whenever the images of the discs in level l = 0 coincide, while the images in level l = 0 differ by a translation of the symplectisation coordinate by a number that only depends on the level l.

By M

k -|1|k + Σ0,Σ1 (x 0 ; x 1 , . . . , x d ) we denote the buildings having a single incoming puncture x 0 which is not a node (necessarily contained in level k + ), and whose outgoing punctures which are not nodes correspond to x 1 , . . . , x d (respecting the order of the punctures induced by the boundary orientation after a boundary connected sum). Observe that, by definition, we have

M Σ0,Σ1 (x 0 ; x 1 , . . . , x d ) ⊂ M 0|1|0 Σ0∪Σ1 (x 0 ; x 1 , . . . , x d ).
The latter subspace consists of honest one-component pseudoholomorphic curves, and they will sometimes be referred to as unbroken solutions, while the pseudoholomorphic buildings not being of this form will be referred to as broken solutions.

The definition of a building is analogous in the case when the boundary condition is a single Lagrangian cobordism Σ, and we use

M k -|1|k + Σ (x 0 ; x 1 , . . . , x d )
to denote the space of such buildings.

Remark 3.8. Observe that the dichotomy between punctured pseudo-holomorphic half-planes and strips (see Section 3.2.2) also applies to the above pseudo-holomorphic buildings. We will call a pseudoholomorphic building a broken punctured pseudoholomorphic half-plane (resp. strip) in the case when taking a boundary connected sum at each node produces a topological punctured half-plane (resp. strip).

The space of buildings has a topology as defined in [START_REF] Bourgeois | Compactness results in symplectic field theory[END_REF] and [START_REF] Abbas | An introduction to compactness results in symplectic field theory[END_REF], where it also is shown that a sequence of buildings with fixed outgoing punctures has a subsequence converging to another building of the same type (but with possibly additional levels). Observe that, here we also rely on the compactness results for Lagrangian intersection Floer homology [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF]. We do not reproduce the definition of the topology used in order to formulate this convergence, and we refer the reader to the relevant sections of the papers above. The following result is the main compactness result that we will need, obtained by combining the above mentioned compactness results in the SFT and Floer settings. Theorem 3.9. For an admissible almost complex structure, the disjoint unions of all buildings

k + ,k -≥0 M k -|1|k + Σ0,Σ1 (x 0 ; x 1 , . . . , x d ) k + ,k -≥0 M k -|1|k + Σ (x 0 ; x 1 , . . . , x d )
are compact and, in particular, these unions are finite.

The crucial property needed in order to apply the compactness theorems is that the above buildings consist of components having an a priori upper bound on the sum of their total energies. Indeed, this upper bound can be expressed solely in terms of the action a(x 0 ) of the asymptotic x 0 . This can be readily seen to follow from Proposition 3.5, using the fact that there is only a finite number of Reeb chords and intersection points below a fixed action. (In fact, in the setting considered here it is even the case that the totality of Reeb chords and intersection points for a given pair (Σ 0 , Σ 1 ) is finite.) 4. Fredholm theory and transversality 4.1. Linear Cauchy-Riemann operators.

4.1.1. The Grassmannian of Lagrangian planes. Let (V, ω) be a symplectic vector space of dimension 2n+2. We denote by Gr(V, ω) the space of Lagrangian subspaces of (V, ω). The choice of a symplectic basis of V leads to an identification Gr(V, ω) ∼ = U (n + 1)/O(n + 1). Given a path λ : [0, 1] → Gr(V, ω), for |s -r| small enough there is a family of linear isomorphisms φ r,s : λ(s) → λ(r) such that φ s,s = Id. If λ ∈ Gr(V, ω), the crossing form q λ,λ,s is the quadratic form on λ(s) ∩ λ defined by [START_REF] Chantraine | Lagrangian concordance of Legendrian knots[END_REF] q λ,λ,s (v

) := - d dr r=s ω(φ r,s (v), v).
Example 4.1. Let η ∈ Gr(V, ω) be a Lagrangian plane and f : [0, 1] → R a smooth function. We fix a complex structure on V compatible with ω and define the path λ : [0, 1] → Gr(V, ω) as λ(t) := e if (t) η. Then, at any point s ∈ [0, 1], the crossing form q λ,λ(s),s is

q λ,λ(s),s (v) = f ′ (s) v 2 .
We denote by P -(Gr(V, ω)) the set of paths λ : [0, 1] → Gr(V, ω) such that λ(0) ⋔ λ(1) and the crossing form q λ,λ(1),1 is negative definite. For a generic path λ ∈ P -(Gr(V, ω)) the spaces λ(s) ∩ λ(1) has positive dimension only at at finite set of s, and the crossing forms on the positive dimensional intersections is nondegenerate. (We can also assume, generically, that λ(s) ∩ λ(1) has dimension at most one for s = 1.) If λ is a generic path, we define ( 16)

I(λ) := s<1 ind(q λ,λ(1),s ),
where ind(q) denotes the number of negative eigenvalues of a non-degenerate quadratic form q.

Let det : U (n + 1) → S 1 denote the complex determinant. The map

A → det(A) 2
descends to a well-defined map α : Gr(V, ω) → S 1 which induces an isomorphism α * : π 1 (Gr(W, ω)) → π 1 (S 1 ). We regard α * as a cohomology class µ ∈ H 1 (Gr(V, ω); Z) called the Maslov class. We denote by Gr # (V, ω) → Gr(V, ω) the universal covering of Gr(V, ω). The elements of Gr # (V, ω) are called graded Lagrangian planes. The map α : Gr(V, ω) → S 1 lifts, in a non-unique way, to a map α # : Gr # (V, ω) → R such that α = e 2πiα # . Different choices for α # differ by an integer constant. We can think of a graded Lagrangian plane λ # as a pair (λ, α # (λ # )).

Similarly, we denote by

P -(Gr # (V, ω)) the set of paths λ # : [0, 1] → Gr # (V, ω) that project to paths λ ∈ P -(Gr(V, ω)). Given λ # 0 , λ # 1 ∈ Gr # (V, ω)
whose projections λ 0 and λ 1 intersect transversely, we chose a path λ # ∈ P -(Gr # (V, ω)) such that λ # (0) = λ # 0 and λ # (1) = λ # 1 and define the absolute index i(λ # 0 , λ # 1 ) := I(λ). Since λ # 0 and λ # 1 determine the homotopy class of λ and I is invariant under homotopies relative to the boundary, the absolute index is well defined. Now we want to extend the absolute index to a map

i : Gr # (V, ω) × Gr # (V, ω) → Z.
Let λ # 0 and λ # 1 be graded Lagrangian planes. If λ 0 ∩ λ 1 = 0, then i(λ # 0 , λ # 1 ) is defined as above. Otherwise, let (λ 0 ∩ λ 1 ) ω be the symplectic orthogonal to λ 0 ∩ λ 1 and define V := (λ 0 ∩ λ 1 ) ω /(λ 0 ∩ λ 1 ). The symplectic form ω induces, by linear symplectic reduction, a symplectic form ω on V . If λ ⊂ (λ 0 ∩ λ 1 ) ω is a Lagrangian subspace of (V, ω), then λ = λ/(λ 0 ∩ λ 1 ) is a Lagrangian subspace of ( V , ω). This construction gives an embedding Gr( V , ω) ֒→ Gr(V, ω) which lifts to an embedding Gr # ( V , ω) ֒→ Gr # (V, ω). Let λ # 0 and λ # 1 be the preimages of λ # 0 and λ # 1 in Gr # ( V , ω). Since λ 0 and λ 1 intersect transversely in V , we define

(17) i(λ # 0 , λ # 1 ) := i( λ # 0 , λ # 1 )
.

Next we study how the index changes for small perturbation of the Lagrangian planes. From now on we fix a complex structure on V which is compatible with ω. For a Lagrangian plane λ # and a real number δ = 0, we define e iδ λ # as the end point of the path [0, 1] → Gr # (V, ω) which starts at λ # and lifts the path t → e iδt λ.

In particular α # (e iδ λ # ) = α # (λ # ) + (n+1)δ π . Given two graded Lagrangian planes λ # 0 and λ # 1 , we say that δ ∈ R * is "small" if λ 0 ⋔ e iδt λ 1 for all t ∈ (0, 1].
Lemma 4.2. Let λ # 0 and λ # 1 be graded Lagrangian planes and δ = 0 a small real number.

(

1) If δ > 0, then i(λ # 0 , e iδ λ # 1 ) = i(λ # 0 , λ # 1 ) + dim(λ 0 ∩ λ 1 ), and (2) If δ < 0, then i(λ # 0 , e iδ λ # 1 ) = i(λ # 0 , λ # 1 ).
Proof. There is an equivalent definition of the absolute index which is less intrinsic, but more practical for this computation. Let η = λ 0 ∩ λ 1 , and choose a symplectic subspace (V ′ , ω ′ ) ⊂ (V, ω) containing η as a Lagrangian subspace. Let (V ′′ , ω ′′ ) be the symplectic orthogonal of (V ′ , ω ′ ); then the subspaces

ζ i = λ i ∩ V ′′ , for i = 0, 1, are Lagrangian in V ′′ . Let ζ ∈ P -(Gr(V ′′ , ω ′′ ) be a path from ζ 0 to ζ 1 such that λ = η ⊕ ζ lifts to a path in Gr # (V, ω) from λ # 0 to λ # 1 .
Here η is regarded as a constant path. The path ζ exists because (V ′′ , ω ′′ ) is symplectically isomorphic to ( V , ω). Then we have [START_REF] Chantraine | Noncommutative augmentation categories[END_REF] i(λ # 0 , λ # 1 ) = I(ζ).

Choose a complex structure on V which is compatible with ω and preserves the direct sum decomposition

V = V ′ ⊕ V ′′ , and define ζ δ (t) := e iδt ζ(t) and η δ (t) := e iδt η. Since δ is small, ζ δ ∈ P -(Gr(V ′′ , ω ′′ )) and I(ζ δ ) = I(ζ) = i(λ # 0 , λ # 1 ). If δ < 0, then by Example 4.1 the crossing form q η δ ,η δ (1),1 is negative definite, so η δ ∈ P -(Gr(V ′ , ω ′ )) and I(η δ ) = 0. Then λ δ = η δ ⊕ ζ δ ∈ P -(Gr(V, ω)), and therefore i(λ # 0 , e iδt λ # 1 ) = I(λ δ ) = i(λ # 0 , λ # 1
). If δ > 0, then by Example 4.1 the crossing form q η δ ,η δ (1),1 is positive definite, so we have to modify η δ to obtain a path in P -(Gr(V ′ , ω ′ )). One possible modification consists in replacing the path η δ with the path η δ defined as η δ (t) := e iδ(-2t 2 +3t) η. If δ is small enough, η δ (t) = e iδ η only for t = 1 and t = 1 2 . Moreover, by Example 4.1, the crossing form q η δ , η δ (1),1/2 is positive definite and the crossing form q η δ , η δ (1),1 is negative definite. Then η δ ∈ P -(Gr(V ′ , ω ′ ) and

I( η δ ) = dim η. Therefore i(λ # 0 , e iδt λ # 1 ) = I(λ δ ) = i(λ # 0 , λ # 1 ) + dim(λ 0 ∩ λ 1 ).
4.1.2. Cauchy-Riemann operators over punctured discs. Let r ∈ R d+1 . We assume that the punctured disc Σ r is equipped with strip-like ends ε i as described in Section 3.1 such that ε 0 is an incoming end and ε i , for i = 1, . . . , d, are outgoing ends. For i = 0, . . . , d we choose orientation preserving parametrisations l i : R → ∂Σ r of the connected components ∂ i Σ r of ∂Σ r , which we assume to be cyclically ordered according to the natural boundary orientation. Given λ = (λ 0 , . . . , λ d ), where each λ i is a map λ i : R → Gr(C n , ω 0 ) which is constant outside a compact set, for every p > 2 we define the Banach space

W 1,p (Σ r , λ) := {ζ ∈ W 1,p (Σ r , C n ) | ∀i ζ(l i (s)) ∈ λ i (s)}.
We denote by D r,λ : W 1,p (Σ r , λ) → L p (Σ r , C n ) the standard linear Cauchy-Riemann operator. Here we have identified

T 0,1 Σ ⊗ C C n with C n .
We denote

λ + i = lim s→+∞ λ i (s) and λ - i = lim s→-∞ λ i (s).
It is well known (see [START_REF] Floer | A relative Morse index for the symplectic action[END_REF]Proposition 4.1] for the case of strips; the general case is similar) that D r,λ is a Fredholm operator if λ + i ⋔ λ - i+1 for all i = 0, . . . , d (with the convention that λ d+1 = λ 0 ). If this is the case, the index of D r,λ can be computed as follows.

For each path λ i : R → Gr(C n , ω 0 ) we make the choice of a continuous lift

λ # i : R → Gr(C n , ω 0 ) # . Then it follows from [64, Proposition 11.13] that (19) ind(D r,λ ) = i((λ - 0 ) # , (λ + d ) # ) - d-1 i=0 i((λ + i ) # , (λ - i+1 ) # ).
Note that the above expression is independent of the choice of lifts made. We will need to consider also more general boundary conditions which do not intersect transversely on the strip-like ends. It is well known that the Cauchy-Riemann operator with these boundary conditions is not Fredholm, unless we use weighted Sobolev spaces, which we are now going to describe briefly.

We define χ : Σ r → R to be a smooth function which is equal to 0 outside the strip-like ends (i.e. on Σ r \ (ν 0 ∪ . . . ∪ ν d )), is equal to 1 on the outgoing strip-like end ε 0 (where s ≥ 1), and is equal to -1 on the incoming strip-like ends ε i , for i = 1, . . . , d (where s ≤ -1). Given δ > 0, we introduce the Banach spaces

W 1,p δ (Σ r , λ) := {ζ : e δχ(z)s ζ ∈ W 1,p (Σ r , λ)} and L p δ (Σ r ) := {ζ : e δχ(z)s ζ ∈ L p (Σ r , C n )};
note that the function χ(z)s is well defined as χ(z) = 0 outside the neighbourhoods where s is a well defined coordinate. Moreover the function e δχ(z)s goes to +∞ as s → ±∞, and this forces the functions ζ in the weighted Sobolev spaces to go to zero sufficiently fast in the strip-like ends. We will denote by D δ r,λ the Cauchy-Riemann operator from W 1,p δ (Σ r , λ) to L p δ (Σ r ). We define the Lagrangian label λ δ = (λ δ 0 , . . . , λ δ d ) by the isomorphism

W 1,p δ (Σ r , λ) → W 1,p (Σ r , λ δ ), ζ → e δχ(z)(s+it) ζ.
In order to understand the Fredholm properties of D δ r,λ , we introduce the operator D r,λ δ defined by the following commutative diagram:

W 1,p δ (Σ r , λ) (e δχ(z)(s+it) )• D δ r,λ / / L p δ (Σ r , C n ) (e δχ(z)(s+it) )• W 1,p (Σ r , λ δ ) D r,λ δ / / L p (Σ r , C n ).
In other words,

D r,λ δ (ζ) = e δχ(z)(s+it) D δ r,λ (e -δχ(z)(s+it) ζ) = D r,λ δ (ζ) -δ∂(χ(z)(s + it))ζ for ζ ∈ W 1,p (Σ r , λ δ )
, and thus D r,λ δ is a compact deformation of D r,λ δ because χ is constant outside a compact set. For a generic choice of δ, the asymptotic Lagrangian labels associated to λ δ are transverse, and therefore D δ r,λ is Fredholm. Standard Fredholm theory implies that D δ r,λ also is Fredholm and that Ind(

D δ r,λ ) = Ind( D r,λ δ ) = Ind(D r,λ δ ). Choose lifts (λ δ 0 ) # , . . . , (λ δ d ) # ; then by Equation (19) the index of D δ r,λ is (20) ind(D δ r,λ ) = i((λ δ,- 0 ) # , (λ δ,+ d ) # ) - d-1 i=0 i((λ δ,+ i ) # , (λ δ,- i+1 ) # ).
In order to make Equation ( 20) more explicit, we relate the absolute indices of the asymptotic Lagrangian labels before and after the perturbation. At the incoming puncture (λ δ,- 0 ) # = e iδ (λ - 0 ) # , and (λ δ,+ d ) # = (λ + d ) # , while at the outgoing punctures , for i = 0, . . . , d -1,

(λ δ,+ i ) # = e -iδ (λ + i ) # , and (λ δ,- i+1 ) # = (λ - i+1 ) # . Since i(e iδ (λ - 0 ) # , (λ + d ) # ) = i((λ - 0 ) # , e -iδ (λ + d ) # ) and i(e -iδ (λ + i ) # , (λ - i+1 ) # ) = i((λ + i ) # , e iδ (λ - i+1 ) # ), Example 4.1 implies that the index of D δ r,λ is (21) Ind(D δ r,λ ) = i((λ - 0 ) # , (λ + d ) # ) - d-1 i=0 (i((λ + i ) # , (λ - i+1 ) # ) + dim(λ + i ∩ λ - i+1 )).
4.2. Grading. We are now ready to define the gradings of the generators that will appear in the Cthulhu complex described Section 6. For intersection points, the grading follows Seidel [START_REF] Seidel | Fukaya categories and Picard-Lefschetz theory[END_REF], while for Reeb chords we recover the definition in [START_REF] Ekholm | The contact homology of Legendrian submanifolds in R 2n+1[END_REF] by Ekholm-Etnyre-Sullivan.

The Maslov potential of a Lagrangian cobordism. We assume that 2c 1 (P ) = 0, which is equivalent to saying that Λ n C (T X) ⊗2 ≃ X × C. Let ν be such a trivialisation. Note that since Σ is Lagrangian for any basis (v 1 , . . . , v n+1 ) of T p L, the value ν(v 1 , . . . , v n+1 ) := ν(v 1 , . . . , v n+1 , v 1 , . . . , v n+1 ) is non-zero and, moreover,

ν(v 1 , . . . , v n+1 ) ν(v 1 , . . . , v n+1 ) ∈ S 1
does not depend on the choice of the basis. This defines a function α :

L → S 1 whose homotopy class [α] ∈ [L, S 1 ] ≃ H 1 (L) is mapped to the Maslov class µ ∈ H 2 (M, L)
through the connecting homomorphisms ρ : H 1 (L) → H 2 (M, L). We now make the assumption that the Maslov class vanishes, from which it follows that there exists a (not uniquely determined) function α # : L → R satisfying α = e 2iπα # . Note that, for any p ∈ Σ, the pair (T p Σ, α # (p)) can be identified with an element of Gr # (T p X, ω p ).

Remark 4.3. In the case when the Maslov does not vanish one defines the potential to take values in the cover of S 1 associated to im[α] ⊂ H 1 (S 1 ) = Z; this leads to a cyclic grading defined modulo im[α].

Let us now assume that we have fixed choices of functions α # 0 and α # 1 as above for both of the cobordisms Σ 0 and Σ 1 ; these functions are called Maslov potentials for the cobordisms. Using the same notation as in Section for p ∈ Σ i , we let

α # i (p) denote (T p Σ i ) # .
Grading of intersection points. Let p be an intersection point between Σ 0 and Σ 1 . Given a choice of ordering (Σ 0 , Σ 1 ), we define the grading of p by

gr(p) = i(α # 0 (p), α # 1 (p)). Grading of Reeb chords. Let γ be a Reeb chord of the link Λ ± := Λ ± 0 ∪ Λ ± 1 with starting point p -∈ Λ ± i -, endpoint p + ∈ Λ ± i +
, and length ℓ. The Maslov potential on Σ 0 ∪ Σ 1 restricts to a Maslov potential on the cylindrical ends R × Λ ± . Let φ t be the flow of the Reeb vector field ∂ ∂z . Observe that a Hamiltonian isotopy acts on the space of graded Lagrangian planes by the homotopy lifting property. Let α # γ,t be the lift of the path of Lagrangian planes dφ t (T (t0,p -) (R × Λ ± )) starting at α i (t 0 , p -) # for any t 0 ∈ R. We define gr(γ) = i(α i + (t 0 , p + ) # , α # γ,ℓ ) -1. Remark 4.4. This grading coincides with the grading of the Reeb chord generators of the Chekanov-Eliashberg algebra used in [START_REF] Ekholm | Legendrian contact homology in P × R[END_REF], which are defined in terms of the Conley-Zehnder index.

Note that the above Lagrangian subspaces do not intersect transversely. Here we must use the definition that utilises the symplectic reduction as in Section 4.1.1. In this case, this symplectic reduction can be seen to geometrically correspond to taking the canonical projection R × P × R → P . 4.2.1. Virtual dimension of the moduli spaces. We now turn our attention to the general settings and prove the virtual dimension formulae for the moduli spaces of holomorphic discs M L (x 0 , x 1 , . . . , x j + +j -+l ) for a Lagrangian label L taking values in {Σ 0 , Σ 1 }. Let p and {q k } l k=1 be intersection points of Σ 0 ∩ Σ 1 . In addition, we let {γ + 0 } and {γ

+ k } j + k=1 be Reeb chords of Λ + 0 ∪ Λ + 1 , and {γ - k } j - k=1 be Reeb chords of Λ - 0 ∪ Λ - 1 .
We require that the Lagrangian label satisfies the property that: • The asymptotic x 0 at the incoming puncture is either an intersection point p 0 , or a positive puncture asymptotic to the Reeb chord γ + 0 . In either case, this puncture is required to be a jump from Σ 1 to Σ 0 ; and • The asymptotics of the other punctures x 1 , . . . , x j + +j -+l correspond bijec-

tively to {γ + k } j + k=1 ∪ {γ - k } j - k=1 ∪ {q k } l k=1 .
Here we make no requirements regarding the jumps made at the different punctures. (In the case of a Reeb chord, such a jump is uniquely determined. The jumps at intersection points are determined by the order of the asymptotics (x 1 , . . . , x j + +j -+l ) since the jump at x 0 is fixed.) For an intersection point, we define the function (σ q k (0), σ q k (1)) := (0, 1) in case the jump is from Σ 0 to Σ 1 , while we define (σ q k (0), σ q k (1)) := (1, 0) in case the jump is from Σ 1 to Σ 0 . Note that here we do not insist on the Lagrangian label being simple.

For a punctured pseudoholomorphic disc u : S r → X inside the above moduli space M L (x 0 , x 1 , . . . , x j + +j -+l ) we let F u be the linearisation of Equation ( 9).

The following result provides a computation of the Fredholm index of F u . Recall that this Fredholm index is equal to the dimension of the moduli space of such discs modulo reparametrisation in the case when this moduli space is transversely cut out.

Theorem 4.5. Let (Σ 0 , Σ 1 ) be an ordered pair of Lagrangian cobordisms of dimension n + 1, and let u be a punctured disc with boundary on this pair of Lagrangian submanifolds as described above. The Fredholm index of F u given by

Ind(F u ) = gr(p 0 ) + j + k=1 gr(γ + k ) - σq k (0)=1 (n + 1 -gr(q k ))+ (22) - σq k (0)=0 gr(q k ) - j - k=1 gr(γ - k )+ + (2 -n)j + + l -2 if x 0 = p 0 , Ind(F u ) = gr(γ + 0 ) + j + k=1 gr(γ + k ) - σq k (0)=1 (n + 1 -gr(q k ))+ (23) - σq k (0)=0 gr(q k ) - j - k=1 gr(γ - k )+ + (2 -n)j + + l if x 0 = γ + 0 ,
where the gradings of intersection points are defined using the order (Σ 0 , Σ 1 ) as in Section 4.2.

Proof. The operator F u decomposes as D u ⊕ K u ⊕ τ m where D u acts on W 1,p δ , K u : R j → 0 with j = j + + j -(resp. j = j + + j -+ 1) in the case when x 0 = p 0 (resp. x 0 = γ + 0 ), and τ j + +j -+l+1 is the operator associated to the deformation of the Teichmüller space at r. The latter operator has index equal to the dimension of R j + +j -+l+1 , minus the dimension of the automorphisms group of the curve S r .

The index of K u is j + + j -if x 0 = p 0 and j + + j -+ 1 if x 0 = γ + 0 . Moreover, Ind τ m = j + + j -+ l -2. For k ∈ {1 . . . j ± } we denote by p ± k the end of the chord γ ± k . From Equation ( 21)we get that

Ind(D u ) = i((T p0 Σ 0 ) # , (T p0 Σ 1 ) # ) - j + k=1 i(α # γ + k ,ℓ k , T (0,p + k ) (R × Λ + ) # ) + 1 - k i(T q k Σ σq k (0) ) # , (T q k Σ σq k (1) ) # ) - j - k=1 i(T (0,p - k ) (R × Λ -) # ), α # γ - k ,ℓ k ) + 1 or Ind(D u ) = i(T (0,p + 0 ) (R × Λ + ) # , α # γ + 0 ,ℓ0 ) - j + k=1 i(α # γ + k ,ℓ k , T (0,p + k ) (R × Λ + ) # ) + 1 - k i(T q k Σ σq k (0) ) # , (T q k Σ σq k (1) ) # ) - j - k=1 i(T (0,p - k ) (R × Λ -) # , α # γ - k ,ℓ k ) + 1 .
Depending on whether x 0 is p 0 or γ + 0 . Note that for q ∈ Σ 0 ∩ Σ 1 , we have

i((T q Σ 0 ) # , (T q Σ 1 ) # ) = n + 1 -i((T q Σ 1 ) # , (T q Σ 0 ) # )
whereas that for a Reeb chord γ with end point p + we have that

i(α # γ,ℓ , T (0,p + ) (R × Λ ± ) # ) = n -i(T (0,p + ) (R × Λ ± ) # , α # γ,ℓ ). Thus we obtain Ind(D u ) = gr(p 0 ) - j + i=1 (n -gr(γ + k )) - σ(q k )(0)=0 gr(q k ) - σq k (0)=1 (n + 1 -gr(q k )) - j - 1 (gr(γ - k ) + 2) or Ind(D u ) = gr(γ + 0 ) + 1 - j + i=1 (n -gr(γ + k )) - σ(q k )(0)=0 gr(q k ) - σq k (0)=1 (n + 1 -gr(q k )) - j - 1 (gr(γ - k ) + 2) or Since Ind(F u ) = Ind(D u ) + Ind(K u ) + Ind(τ m ) the result follows.
We apply Equation ( 22) to compute the virtual dimension of the moduli spaces defined in Section 3.2. For (n + 1)-dimensional Lagrangian cobordisms, we obtain

dim( M(γ + ; δ -, γ -, ζ -)) = dim(M(γ + ; δ -, γ -, ζ -)) -1 (24) = gr(γ + ) -gr(γ -) -gr(δ) -gr(ζ) -1, dim(M(γ + ; δ -, q, ζ -)) = gr(γ + ) -gr(q) -gr(δ) -gr(ζ) + 1, (25) dim(M(p; δ -, q, ζ -)) = gr(p) -gr(q) -gr(δ) -gr(ζ) -1, (26) dim(M(p; δ -, γ -, ζ -)) = gr(p) -gr(γ -) -gr(δ) -gr(ζ) -2, ( 27
) dim( M(γ 2,1 ; δ -, γ 1,2 , ζ -)) = dim(M(γ 2,1 ; δ -, γ 1,2 , ζ -)) -1 (28)
= gr(γ 2,1 ) + gr(γ 1,2 )gr(δ)gr(ζ)n + 1.

4.3.

Transversality. Here we have gather various results from the literature that are needed in order to achieve transversality for the relevant moduli spaces of punctured pseudoholomorphic discs inside R × P × R. Recall that the transversality of a moduli space in particular implies that it is a smooth manifold of dimension equal to the Fredholm index of the corresponding linearised problem (see Section 4.1.2). As usual, an almost complex structure for which the relevant moduli spaces are transversely cut out is called regular.

4.3.1.

Punctured discs and strips with a cylindrical boundary condition. Consider the moduli spaces for a cylindrical boundary condition, and where all punctures are asymptotic to Reeb chords of which precisely one is positive with asymptotic γ + . In other words, we are interested in moduli spaces of the form

M R×Λ (γ + ; γ - 1 , . . . , γ - d ), M R×Λ0,R×Λ1 (γ + ; γ - 1 , . . . , γ - d ), γ + ∈ R(Λ 1 , Λ 0
), as described in Section 3.2.3. Note that solutions in the latter moduli space also are solutions (of a special kind) inside the former moduli space for Λ = Λ 0 ∪ Λ 1 . We also consider moduli spaces of the form

M R×Λ0,R×Λ1 (γ 1,0 ; δ -, γ 0,1 , ζ -),
as considered in Section 3.2.7. Recall that here all punctures have negative asymptotics to Reeb chords except precisely two, where these two punctures moreover have positive asymptotics to two distinct Reeb chords γ 1,0 ∈ R(Λ 1 , Λ 0 ), γ 0,1 ∈ R(Λ 0 , Λ 1 ), and where δ -and ζ -denote words of chords on Λ 0 and Λ 1 , respectively.

The result [24, Proposition 3.13] can be applied to all of these moduli spaces, showing that they are transversely cut-out for a Baire first category subset of almost complex structures J cyl (P × R). This result can be seen as an adaptation of Dragnev's result in [START_REF] Dragnev | Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations[END_REF] to the case of a cylindrical boundary condition. Note that, even if the aforementioned result assumes that there is precisely one positive puncture, in this case its proof can still be applied, since the two positive punctures of the solution have different asymptotics.

For technical reasons it will be necessary to achieve transversality also for the subset J cyl π (P × R) ⊂ J cyl (P × R) of cylindrical lifts of almost complex structures (see Section 3.1.3). In this case, [START_REF] Rizell | Lifting pseudoholomorphic polygons to the symplectisation of P × R and applications[END_REF]Lemma 8.2] shows that a J P -holomorphic disc u in R × P × R is transversely cut out if and only if its J P -holomorphic projection u := π • u is transversely cut out. That the latter discs are transversely cut out for a Baire first category subset of admissible almost complex structures was shown in [34, Proposition 2.3] under the following additional assumption:

(A) Fix a chord-generic Legendrian submanifold Λ ⊂ P × R. The pair (Λ, J P ) said to be admissible if the following is satisfied. The almost complex structure J P is integrable in some neighbourhood of each double point of the Lagrangian projection Π Lag (Λ), where the two sheets of this Lagrangian immersion moreover are real-analytic. Note that this condition imposes no restriction on the Legendrian submanifold. Namely, every transverse Lagrangian intersection is symplectomorphic to the intersection of the real and imaginary parts in (C n , ω 0 ) by a version of Weinstein's Lagrangian neighbourhood theorem.

Combining the above results, we obtain:

Proposition 4.6. The above moduli spaces are transversely cut out for a Baire first category subset of the cylindrical almost complex structures J cyl (P × R), as well as for a Baire first category subset of the admissible cylindrical lifts J cyl π (P × R) of almost complex structures on J P satisfying condition (A).

4.3.2.

Discs with boundary on a general embedded Lagrangian cobordism. We are here interested in the case when the boundary condition is a general embedded Lagrangian cobordism Σ from Λ -to Λ + , and when all asymptotics are Reeb chords. In other words, we consider moduli spaces of the form

M Σ (γ + ; γ - 1 , . . . , γ - d ) as described in Section 3.2.3.
Proposition 4.7. Assume that we are given cylindrical almost complex structures J ± ∈ J cyl (P × R) which are regular for M R×Λ ± (γ + ; γ - 1 , . . . , γ - d ), as well as a number T > 0 for which Σ \ ([-T, T ] × P × R) is cylindrical. The above moduli spaces of the form M Σ (γ + ; γ - 1 , . . . , γ - d ) are transversely cut out for a Baire first category subset of J adm J -,J + ,T (R × P × R). Proof. From [17, Theorem 2.8] it follows that the discs in these moduli spaces are simple, and a standard transversality argument [START_REF] Mcduff | J-holomorphic curves and symplectic topology[END_REF]Chapter 3] can thus be applied. To that end, observe that it suffices to find an injective point of the disc contained inside (-T, T ) × P × R, where the sought perturbation of the almost complex structure will be supported. (In the case when the disc is disjoint from this subset, it is in fact already transversely cut out by the assumptions of the proposition.) 4.3.3. Strips with at least one asymptotic to an intersection point. Let Σ i , i = 0, 1, be two transversely intersecting exact Lagrangian cobordisms from Λ - i to Λ + i . We now consider moduli spaces of strips being of the form

M Σ0,Σ1 (x 0 ; δ -, x 1 , ζ -),
where δ -and ζ -denote words of Reeb chords on Λ - 1 and Λ - 0 , respectively, and where at least one of x 0 and x 1 is an intersection point of Σ 0 ∩ Σ 1 . See Sections 3.2.4, 3.2.5, and 3.2.6. Recall that exactness implies that x 0 = x 1 holds in either of the cases. It is here important to note that Σ 0 ∪ Σ 1 is not an embedded boundary condition.

One option for achieving transversality for these moduli spaces is via the technique in [START_REF] Ekholm | Legendrian contact homology in P × R[END_REF], while imposing a condition analogous to (A) near the intersection points. In other words, one has to consider almost complex structures which are integrable in a neighbourhood of Σ 0 ∩ Σ 1 , which moreover make the Lagrangian submanifolds real-analytic inside the same neighbourhood.

Alternatively, transversality is also possible to achieve when using so-called "time-dependent" almost complex structures as described in Section 3.1.5. This technique goes back to Floer [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF]. Recall that the strips in the above moduli space are maps of the form

u : (R × [0, 1], R × {0}, R × {1}) → (R × P × R, Σ 0 , Σ 1 ),
where the domain is endowed with the conformal structure coming from the standard holomorphic coordinate s + it on R × [0, 1] ⊂ R 2 = C (this parametrisation is determined up to a translation of the s coordinate).

We consider sufficiently regular one-parameter families {J t } t∈[0,1] of almost complex structures J t ∈ J adm J -,J + (R × P × R), and define the moduli spaces for the corresponding time-dependent version of the Cauchy-Riemann operation on a strip as in Section 3.1.5. The somewhere injectivity result in e.g. [START_REF] Oh | On the structure of pseudo-holomorphic discs with totally real boundary conditions[END_REF]Theorem 5.1] or [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF]Section 8.6] also holds in the current setting, showing that: Proposition 4.8. Assume that we are given cylindrical almost complex structures J ± ∈ J cyl (P × R) which are regular for M R×Λ ± (γ + ; γ - 1 , . . . , γ - d ), as well as a number T > 0 for which Σ \ ([-T, T ] × P × R) is cylindrical. The above moduli spaces of the form

M Σ0,Σ1 (x 0 ; δ -, x 1 , ζ -)
are transversely cut out for a Baire first category subset of the time-dependent almost complex structures J t : I → J adm J -,J + ,T (R × P × R). 4.3.4. Strips with only Reeb chord asymptotics. We now consider moduli spaces of strips being of the form M Σ0,Σ1 (γ

+ ; δ -, γ -, ζ -)
as above, but where γ ± both are Reeb chords; see Sections 3.2.3 and 3.2.7. Observe that this strip has a puncture with a positive asymptotic to a Reeb chord γ + from Λ + 1 to Λ + 0 , while γ -is either a Reeb chord from Λ - 1 to Λ - 0 , which hence is a negative asymptotic, or a Reeb chord from Λ + 0 to Λ + 1 , which hence is a positive asymptotic. In either case, we have γ + = γ -. Proposition 4.9. Assume that we are given cylindrical almost complex structures J ± ∈ J cyl (P × R) which are regular for every moduli space of the form

M R×Λ ± 0 ∪R×Λ ± 1 (γ + ; γ - 1 , . . . , γ - d ), M R×Λ ± 0 ,R×Λ ± 1 (γ + 1 ; γ - 1 , . . . , γ - d1 , γ + 2 , γ - d1+1 , . . . , γ - d )
, as well as a number T > 0 for which Σ \ ([-T, T ] × P × R) is cylindrical. Any almost complex structure J ∈ J adm J -,J + ,T (R × P × R) can be perturbed to an almost complex structure J ′ ∈ J adm J -,J + ,T ′ (R × P × R) which is regular for the moduli spaces of the form M Σ0,Σ1 (γ

+ ; δ -, γ -, ζ -),
under the assumption that T ′ ≫ T was chosen sufficiently large.

Proof. The argument is done by induction on the energy of the discs in the moduli spaces. To that end, the following feature of the SFT compactness theorem is crucial:

• In our setting, the space of solutions of d(φα)-energy at most E > 0 has also a uniform bound on the total energy. Hence, the space M E of these solutions can be compactified to M E by adding broken configurations consisting of pseudoholomorphic buildings; We also note that, in our setting, the possible d(φα)-energies attained by a nontrivial pseudoholomorphic curve having at most one output puncture is a finite set of numbers E k > E k-1 > . . . > E 1 > 0, as follows from the finite possibilities of asymptotics in the current setting.

The base case of the induction is as follows. The J-holomorphic discs in M E1 of lowest d(φα)-energy cannot break by the above, and must thus be compact as a space of maps. Using the asymptotic properties of finite energy discs, all solutions must have injective points in some subset of the form [T ′′ , T ′′ + 1] × P × R, for T ′′ ≫ T sufficiently big. After a perturbation of J supported inside [T ′′ , T ′′ + 1] × P × R, we may thus assume that the moduli space M E1 = M E1 of lowest energy is transversely cut out. By induction we assume that all moduli spaces of energy strictly less than E i > 0, i > 1, are transversely cut out. Pseudoholomorphic gluing as in Theorem 4.11 can be upgraded to the following statement: Gluing of transversely cut out solutions can always be performed and, a priori, produces transversely cut out solutions. Since the boundary strata of M

Ei consists of transversely cut out components by the induction hypothesis together with the assumptions of the proposition, the moduli space M Ei thus also consists of transversely cut out solutions in some neighbourhood of its boundary strata. We are thus left to achieve transversality for a subset of solutions contained in the complement of some neighbourhood of the boundary of M Ei . By the above, these solutions form a compact space of maps. An argument as in the base case can again be used to show the following. After a suitable perturbation of the almost complex structure J in a region containing the positive end (where these maps all have injective points by the asymptotic properties), the full moduli space M Ei can be assumed to be transversely cut out.

We also give the following alternative approach to transversality, using timedependent almost complex structures. Proposition 4.10. Assume that we are given a pair J ± ∈ J cyl (P ×R) of cylindrical almost complex structures which are regular for all moduli spaces of the form

M R×Λ ± 0 ∪R×Λ ± 1 (γ + ; γ - 1 , . . . , γ - d ), M R×Λ ± 0 ,R×Λ ± 1 (γ + 1 ; γ - 1 , . . . , γ - d1 , γ + 2 , γ - d1+1 , . . . , γ - d ), as well as a number T > 0 for which Σ \ ([-T, T ] × P × R) is cylindrical. The above moduli spaces of the form M Σ0,Σ1 (γ + ; δ -, γ -, ζ -)
are transversely cut out for a Baire first category subset of the time-dependent almost complex structures J t : I → J adm J -,J + ,T (R × P × R). Proof. The somewhere injectivity result in [START_REF] Audin | Théorie de Morse et homologie de Floer[END_REF]Section 8.6] can be extended to the case when all asymptotics of the strip are Reeb chords. Observe that the strips that do not pass through [-T, T ] × P × R are transversely cut out by assumption. Consequently, it suffices to turn on the time-dependence in that region. 4.4. Gluing. In this section we explain how transversely cut-out buildings as in Section 3.3.3 can be glued to give a family of holomorphic disc which converges to the original building. In other words, the moduli space of buildings, which can be compactified by Theorem 3.9, will under transversality assumptions be a compact manifold whose boundary strata consists of broken solutions.

We assume from now on that all almost complex structures are regular (see Section 4.3). To a nodal curve S with normalisation S, we associate a tree T S as in Section 2.3.3 whose vertices correspond to connected components of S, interior edges connecting two vertices correspond to nodes, and whose leaves correspond to ends.

The following theorem is a standard consequence of the techniques used in [32, Section 8].

Theorem 4.11. Assume that the almost complex structure is regular and admissible, and let $ denote either "Σ" or "Σ 0 , Σ 1 ". Consider a holomorphic building in M

k -|1|k + $ (x 0 ; x 1 , • • • , x d )
associated a nodal curve S, a tree T S , and consisting of the pseudoholomorphic discs {u i }. For ǫ 0 > 0 sufficiently small, let ǫ 0 > ρ 1 , . . . , ρ ν > 0 be numbers associated to each pairs of nodes asymptotic to an intersection point, and let ǫ 0 > ρ ν+1 , . . . , ρ ν+k > 0, k = k + + k -≥ 0, be numbers associated to each non-empty level of the building except the 0:th level. Then, there exists a uniquely determined punctured pseudoholomorphic disc

u ρ ∈ M $ (x 0 ; x 1 , • • • , x d ), ρ = (ρ 1 , . . . , ρ ν+k ),
the so-called glued solution, satisfying the property that u ρ i , i → ∞, converges to the original building whenever ρ i = (ρ 1,i , . . . , ρ ν+k,i ) satisfies lim i→∞ ρ j,i = 0, j = 1, . . . , k + ν.

Remark 4.12. The expected dimension of a glued solution in M $ (x 0 ; x 1 , • • • , x d ) produced by the above theorem is given by the sum

ν + i Ind(u i ),
in terms of the expected dimensions of all involved components u 1 , . . . , u m in the original building, where ν is the total number of the involved pairs of nodes asymptotic to intersection points.

Together with the compactness result in Theorem 3.9 we obtain the following crucial result. Corollary 4.13. Assume that the almost complex structure is regular and admissible, and let $ denote either "Σ" or "Σ 0 , Σ 1 ". The compactification M $ (x 0 ; x 1 , . . . , x d ) of a moduli space M $ (x 0 ; x 1 , . . . , x d ) of punctured pseudoholomorphic discs being of index one is a transversely cut out one-dimensional manifold with boundary. Moreover, its boundary points are in bijective correspondence with the broken solutions inside M

k -|1|k + $ (x 0 ; x 1 , • • • , x d ) whose components u 1 , . . . , u m satisfy ν + i Ind(u i ) = 1,
and where ν denotes the total number of pairs of nodes asymptotic to intersection points.

A geometric analysis of the a priori possibilities of building leads to the following structure of the boundary of the relevant moduli spaces of dimension one.

Let a, b be asymptotics and δ and ζ be sets of Reeb chords of Λ - 0 and Λ - 1 respectively such that for regular J the moduli space M(a; δ, b, ζ) is a 1-dimensional manifold. By Corollary 4.13 it follows that the compactified moduli space M(a; δ, b, ζ) is a compact 1-dimensional manifold whose boundary consists of broken solutions.

For words w, w ′ and w ′′ in a free group such that w ′ is a subword of w (denoted w ′ ⊂ w) we denote by w w ′ the word obtain removing w ′ from w and by w w ′ (w ′′ ) the word obtain replacing w ′ by w ′′ .

We proceed to explicitly describe the boundary of M(a; δ, b, ζ) for different incoming and outgoing ends. Note that, by the energy estimates of Section 3.3.2 (which use the exactness assumptions), there are no components all whose punctures have negative asymptotics. Also, since every possible component of a broken configuration has non-negative index by the regularity assumptions, we have a restriction on the number of breakings involved in these compactification. In the following lists all unions are assumed to be over 0-dimensional moduli spaces.

For

a = p ∈ Σ 0 ∩ Σ 1 and b = γ -∈ R(Λ - 1 , Λ - 0 )
we conclude that (see Figure 8) Observe that the latter two types of boundary points in the above union, i.e. involving pseudoholomorphic half-planes, always can appear in the boundary of a 1-dimensional moduli space. We call them ∂-breakings and denote them by

∂M(p; δ, γ -, ζ) = (29) c,δ ′ δ ′′ =δ,ζ ′ ζ ′′ =ζ M(p; δ ′ , c, ζ ′′ ) × M(c; δ ′′ , γ -, ζ ′ ) δ ′ ⊂δ,δ0 M(p; δ δ ′ (δ 0 ), γ -, ζ) × M(δ 0 ; δ ′ ) ζ ′ ⊂ζ,ζ0 M(p; δ, γ -, ζ ζ ′ (ζ 0 )) × M(ζ 0 ; ζ ′ )
M ∂ (a; δ, b, ζ).
The upshot of the constructions of the Cthulhu complex will be that, when counting the boundary points of the moduli space weighted by an augmentation, the boundary points corresponding to ∂-breakings will give a total contribution by 0; see (II) below for more details. For a = p, b = q ∈ Σ 0 ∩ Σ 1 we conclude that

∂M(p; δ, q, ζ) = (30) r,δ ′ δ ′′ =δ,ζ ′ ζ ′′ =ζ M(p; δ ′′ , r, ζ ′ ) × M(r; δ ′ , q, ζ ′′ ) M(p; δ ′ , γ 10 , ζ ′′′ ) × M(γ 10 ; δ ′′ , γ 01 , ζ ′′ ) × M(γ 01 ; δ ′′′ , q, ζ ′ ) M ∂ (p; δ, q, ζ).
Where the second union is over:

• γ 01 ∈ R(Λ - 0 , Λ - 1 ), γ 10 ∈ R(Λ - 1 , Λ - 0 ); and • δ ′ δ ′′ δ ′′′ = δ, ζ ′ ζ ′′ ζ ′′′ = ζ.
The second type of breaking here is depicted in Figure 9.

For a = γ + ∈ R(Λ + 1 , Λ + 0 ) and b = γ -∈ R(Λ - 1 , Λ - 0 ), we have ∂M(γ + ; δ, γ -, ζ) = (31) M(γ + ; δ + 1 • • • δ + l , γ 0 , ζ + 1 • • • ζ + m ) × M(δ + 1 ; δ - 1 ) ⊔ • • • ⊔ M(δ + m ; δ - m ) ⊔ M(γ 0 ; δ - l+1 , γ -, ζ - 0 ) ⊔ M(ζ + 1 ; ζ - 1 ) ⊔ • • • ⊔ M(ζ + m ; ζ - m ) M(γ + ; δ ′ , c, ζ ′′ ) × M(c; δ ′′ , γ -, ζ ′ ) M ∂ (γ + ; δ, γ -, ζ)
with the first union being over

• γ + ∈ R(Λ + 1 , Λ + 0 ); • δ + 1 , . . . , δ + l ∈ R(Λ + 0 ); • ζ + 1 , . . . , ζ + m ∈ R(Λ + 1 ); and • δ - 1 δ - 2 • • • δ - l+1 = δ, ζ - 0 ζ - 1 • • • ζ - m = ζ, and the second being over • c ∈ Σ 0 ∩ Σ 1 ∪ R(Λ - 1 , Λ - 0 ); and • δ ′ δ ′′ = δ, ζ ′ ζ ′′ = ζ.
The first type of breaking is depicted in Figure 10.

Finally, for a = γ + ∈ R(Λ + 1 , Λ + 0 ) and b = p ∈ Σ 0 ∩ Σ 1 , we have ∂M(γ + ; δ, p, ζ) = (32) M(γ + ; δ + 1 • • • δ + l , γ 0 , ζ + 1 • • • ζ + m ) × M(δ + 1 ; δ - 1 ) ⊔ • • • ⊔ M(δ + m ; δ - m ) ⊔ M(γ 0 ; δ - l+1 , p, ζ - 0 ) ⊔ M(ζ + 1 , ζ - 1 ) ⊔ • • • ⊔ M(ζ + m , ζ - m ) q∈Σ0∩Σ1 M(γ + ; δ ′ , q, ζ ′′ ) × M(q; δ ′′ , p, ζ ′ ) M(γ + ; δ ′ , γ 10 , ζ ′′′ ) × M(γ 10 ; δ ′′ , γ 01 , ζ ′′ ) × M(γ 01 , δ ′′′ , p, ζ ′ ) M ∂ (γ + ; δ, p, ζ)
The union being over the same sets as in ( 31) and (30).

Preliminaries of Legendrian contact homology

Here we give a quick review of the theory of Legendrian contact homology, and we direct the reader to [START_REF] Ekholm | Legendrian contact homology in P × R[END_REF] and [START_REF] Ekholm | Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms[END_REF] for more details. Legendrian contact homology is a Legendrian isotopy invariant associated to a Legendrian submanifold which was defined by Chekanov for Legendrian knots in the standard R 3 [START_REF] Yu | Differential algebra of Legendrian links[END_REF] and then extended to Legendrian submanifolds of contactisations of Liouville manifolds by Ekholm, Etnyre and Sullivan [START_REF] Ekholm | Legendrian contact homology in P × R[END_REF].

5.1. The Chekanov-Eliashberg DGA. Let (Y, ξ) be the contactisation of a Liouville manifold (P, λ) and Λ ⊂ (Y, ξ) a chord-generic Legendrian submanifold. Fix a unital commutative ring R which, most often, we will be R = Z or R = Z/2Z. However, recall that we have to assume Λ to be spin and fix a spin structure in order to use R different from a unital algebra over a field of characteristic 2. The Legendrian contact homology differential graded algebra (DGA) of Λ, also called the Chekanov-Eliashberg algebra and denoted by (A(Λ), ∂), is a unital tensor R-algebra freely generated by the Reeb chords of Λ.

The grading of a Reeb chord generator a ∈ R(Λ) is given by |γ| := gr(γ) ∈ Z/Z µΛ(H2(P ×R,Λ))

as defined in Section 4.2. Here µ Λ : H 2 (P × R, Λ) → Z is the Maslov class of Λ associated to the contact planes. We define the degree of the unit to be zero.

Its differential ∂ is of degree -1 and counts certain punctured holomorphic discs in the symplectisation of Y which have boundary on R × Λ and are asymptotic to Reeb chords. More precisely, ∂ is defined on a generator δ 0 ∈ R(Λ) as

∂(δ 0 ) := dim M R×Λ (δ0;δ)=1 # M R×Λ (δ 0 ; δ)δ,
where δ = δ 1 . . . δ m(δ) is a formal product of Reeb chords and M R×Λ (δ 0 ; δ) is the moduli space defined in Section 3.2.3. Here (and from now on) the symbol '#' indicates the signed count of 0-dimensional moduli spaces and we use the convention that it gives 0 for moduli spaces of dimension different from zero. Then we extend ∂ to all of A(Λ) by R-linearity together with the Leibniz rule

∂(δδ ′ ) = ∂(δ)δ ′ + (-1) |δ| δ∂(δ ′ ).
Remark 5.1. By the results in [START_REF] Rizell | Lifting pseudoholomorphic polygons to the symplectisation of P × R and applications[END_REF], we may equivalently define ∂ by counting holomorphic polygons in P having boundary on the Lagrangian projection of Λ. For us, however, the above perspective will turn out to be more useful, since it fits better with the SFT framework.

The fact that ∂ 2 = 0 and that the homology is independent of the choice of almost complex structure and invariant under Legendrian isotopy was shown in [START_REF] Ekholm | Legendrian contact homology in P × R[END_REF].

The DGA morphism induced by an exact Lagrangian cobordism.

Given an exact Lagrangian cobordism Σ ⊂ R × P × R from Λ -to Λ + , there is an induced unital DGA morphism

Φ Σ : (A(Λ + ), ∂ Λ + ) → (A(Λ -), ∂ Λ -),
constructed in [START_REF] Ekholm | Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms[END_REF] and [START_REF] Ekholm | Legendrian knots and exact Lagrangian cobordisms[END_REF], following the general philosophy of SFT. This DGA morphism is defined on a generator δ + of A(Λ + ) by the J-holomorphic disc count

Φ Σ (δ + ) := dim MΣ(δ + ;δ -)=0 #M Σ (δ + ; δ -)δ -,
and then extended as a unital algebra map. Here δ -= δ - 1 . . . δ - m(δ -) is an element of A(Λ -), and the moduli space M Σ (a; b) is defined in Section 3.2.3. Observe that, in order to define the above map with coefficients in Z, we must fix a spin structure on each end Λ ± which can be extended to a spin structure on Σ.

Example 5.2. For the trivial (cylindrical) cobordism R × Λ, we get Φ R×Λ = id A(Λ) .

The fact that Φ Σ is a chain map was shown in [START_REF] Ekholm | Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms[END_REF]. In fact, based upon the abstract perturbations in the same paper, it follows that the DGA-homotopy class (as defined in [36, Lemma 3.13]) of Φ Σ is independent of the choices of almost complex structure and Hamiltonian isotopy class of Σ.

Remark 5.3. The invariance properties of our Floer theory does not rely on the above abstract perturbation argument in the cases under consideration here, i.e. that of a symplectisation of a contactisation. The reason is that, in these cases, our invariance statement boils down to showing that the complex Cth • (Σ 0 , Σ 1 ) is acyclic. 5.3. Augmentations and bilinearised Legendrian contact homology. An (R-valued) augmentation is a unital DGA morphism ε : A(Λ) → R, where R is regarded as a DGA with trivial differential. In general, the Chekanov-Eliashberg algebra need not admit any augmentations. However, in the case when the Legendrian submanifold Λ ⊂ Y has an exact Lagrangian filling Σ ⊂ R × Y , the above unital DGA morphism ε Σ := Φ Σ is indeed an augmentation (also see [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF]). In fact, one can think of the trivial DGA R as the Chekanov-Eliashberg DGA of the empty set.

On the other hand, there are plenty of examples of Legendrian submanifolds which do not admit any exact Lagrangian filling in the symplectisation, but whose Chekanov-Eliashberg algebra still admits augmentations. For instance, this is the case for the Legendrian twist knots constructed in [START_REF] Etnyre | Legendrian and transverse twist knots[END_REF]; see the discussion in [63, Section 10.1].

Augmentations are important for the following reason. Loosely speaking, they should be seen as "obstruction cocycles" for the Chekanov-Eliashberg algebra. Given an augmentation, Chekanov defined linearised Legendrian contact homology in [START_REF] Yu | Differential algebra of Legendrian links[END_REF], which is a Legendrian isotopy invariant in the form of a chain complex spanned by the Reeb chords on Λ. Linearised Legendrian contact homology is computationally tractable at the expense of discarding non-linear (and noncommutative) information.

Recently, Bourgeois and the first author have introduced a generalisation of linearised LCH which is called bilinearised Legendrian contact homology [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF]. It is constructed using a pair of augmentations ε 0 and ε 1 of (A(Λ), ∂). We proceed to give a brief description of this complex.

The bilinearised Legendrian contact homology complex is the free module

LCC ε0,ε1 • (Λ) := R R(Λ)
spanned by the Reeb chords with the above grading, whose differential is of degree -1 and defined by

∂ ε0,ε1 (δ) = dim M R×Λ (δ;δ)=1 m δ i=1 # M R×Λ (δ; δ)ε 0 (δ 1 . . . δ i-1 )ε 1 (δ i+1 . . . δ m δ )δ i .
The corresponding homology groups will be denoted by

LCH ε0,ε1 • (Λ).
The set of augmentations of the Chekanov-Eliashberg algebra (A(Λ), ∂) is not a Legendrian isotopy invariant of Λ. However, as shown in [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF], the invariance proof of [START_REF] Yu | Differential algebra of Legendrian links[END_REF] can be generalised to show that the isomorphism classes

{LCH ε0,ε1 • (Λ)}/ ∼
of the graded modules, where the union is over all pairs of augmentations, is a Legendrian isotopy invariant of Λ. Observe that, when ε 0 = ε 1 , we simply recover Chekanov's linearised LCH. We will also be interested in the dual complex of (LCC ε0,ε1

• (Λ), ∂ ε0,ε1
), the socalled bilinearised Legendrian cohomology complex, where the differential ∂ ε0,ε1 is of degree 1. This complex will be denoted by (LCC • ε0,ε1 (Λ), d ε0,ε1 ), while we write LCH • ε0,ε1 (Λ) for the corresponding cohomology group.

The bilinearised LCH is a stronger invariant compared to linearised LCH, due to the fact that it remembers some of the "non-commutativity" of A(Λ). In [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF] it was shown that these homology groups are the morphisms spaces of an A ∞ -category Aug -(Λ) called the augmentation category, whose objects are the augmentations of the Chekanov-Eliashberg algebra of Λ.

Given an exact Lagrangian cobordisms Σ from Λ -to Λ + , and two augmentations ε - 0 ε - 1 of Λ -, the DGA map Φ Σ described in the previous section induces a linear chain map from LCC

• ε - 0 ε - 1 (Λ -) to LCC • ε + 0 ε + 1 (Λ + ) via the formula Φ ε - 0 ε - 1 Σ (γ -) = γ + ,δ -,ζ - #M Σ (γ + ; δ -, γ -, ζ -)ε - 0 (δ -)ε - 1 (ζ -)γ + ,
where

ε + i = ε - i • Φ Σ for i = 0, 1.

The Cthulhu complex

Let Σ 0 and Σ 1 be two exact Lagrangian cobordisms inside the symplectisation (R × P × R, d(e t α)) of a contactisation. We assume that:

• Σ 0 ⋔ Σ 1 (in particular this implies that Λ ± 0 ∩ Λ ± 1 = ∅), • The links Λ ± 0 ⊔ Λ ± 1
are chord-generic. The Cthulhu complex of the pair (Σ 0 , Σ 1 ) is the complex whose underlying graded R-module

Cth • (Σ 0 , Σ 1 ) := C • (Λ + 0 , Λ + 1 )[2] ⊕ C • (Σ 0 , Σ 1 ) ⊕ C • (Λ - 0 , Λ - 1 )[1] for a unital ring R. Here C • (Λ + 0 , Λ + 1 )
is the free graded module spanned by the Reeb chords from Λ ± 1 to Λ ± 0 and C • (Σ 0 , Σ 1 ) is the free graded module spanned by the intersection points Σ 0 ∩ Σ 1 . The gradings are taken as described in Section 4.2, depending on the choice of a Maslov potential.

6.1. The Cthulhu differential. Fix two augmentations ε - 0 and ε - 1 of the Chekanov-Eliashberg algebras of Λ - 0 and Λ - 1 , respectively, both of which are defined using a cylindrical almost complex structure J -. We will define the Cthulhu differential

d ε - 0 ,ε - 1
, which is a differential of degree 1 on the above graded module. With respect to the above decomposition, this differential takes the form

(33) d ε - 0 ,ε - 1 =   d ++ d +0 d +- 0 d 00 d 0- 0 d -0 d --   .
Loosely speaking, every non-zero entry in this matrix is given by a count of rigid punctured pseudoholomorphic strips of appropriate type, as described in Section 3.2, where the counts are "weighted by" the above augmentations. First, however, we need to fix the choice of an admissible almost complex structure J ∈ J adm J -,J + ,T (R × P × R), i.e. a compatible almost complex structure on R × P × R satisfying the assumptions in Section 3.1.4. In particular, J coincides with the cylindrical almost complex structures J -and J + in subsets of the form (-∞, -T ] × P × R and [T, +∞) × P × R, respectively. In order for these moduli spaces to be transversely cut out, we moreover assume that the almost complex structure was generically chosen as in Section 4.3.

Below we give a careful description of each term in the above matrix, where the degrees mentioned are the degrees as maps between the above summands without the shifts in grading as appearing in the above definition of Cth • (Σ 0 , Σ 1 ).

In the cases when we want to emphasise to which pair of Lagrangian cobordisms the differential belongs, we use the superscript "Σ 0 , Σ 1 ", i.e. d Σ0,Σ1

ε + 0 ,ε - 1 , d Σ0,Σ1
+,-,..., etc.

6.1.1. The bilinearised LCH differential. We define ε + i := ε - i • Φ Σi,J , i = 0, 1, for the pull-backs of the above augmentations to augmentations of the Chekanov-Eliashberg algebras of Λ + 0 and Λ + 1 , respectively. The term d ±± is the bilinearised Legendrian cohomology differential for (Λ ± 0 , Λ ± 1 ) induced by the pair (ε ± 0 , ε ± 1 ) of augmentations as defined in [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF] and described in Section 5.3. In other words, it is given by

d ±± (γ ± 2 ) := d ε ± 0 ,ε ± 1 (γ ± 2 ) = (34) = γ ± 1 δ ± ,ζ ± # M R×Λ ± 0 ,R×Λ ± 1 (γ ± 1 ; δ ± , γ ± 2 , ζ ± ; J ± ) • ε ± 0 (δ ± )ε ± 1 (ζ ± ) • γ ± 1 .
It follows can be computed using Equation ( 24) that this term is of degree 1.

6.1.2. The Floer differential. The differential d 00 can be seen as a modification of the differential in Lagrangian Floer homology as introduced by Floer in [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF], where the version defined here has found its inspiration in [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF]. For an intersection point q, it is defined by the count

d 00 (q) := p δ -,ζ - #M Σ0,Σ1 (p; δ -, q, ζ -; J) • ε - 0 (δ -)ε - 1 (ζ -) • p. ( 35
)
From Equation [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF] we deduce that this map is of degree 1. 6.1.3. The Cultist maps. The maps d +0 and d 0-are defined using the moduli spaces described in Section 3.2.6 and in Section 3.2.5, respectively. A version of the map d +0 appears in [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF] in the case when the negative ends of the cobordisms are empty. More precisely, we define

d 0-(γ -) := p δ -,ζ - #M Σ0,Σ1 (p; δ -, γ -, ζ -) • ε - 0 (δ -)ε - 1 (ζ -) • p, (36)
d +0 (q) := γ + δ -,ζ - #M Σ0,Σ1 (γ + ; δ -, q, ζ -) • ε - 0 (δ -)ε - 1 (ζ -) • γ + . ( 37
)
Equations (25) (resp. ( 27)) show that d 0-(resp. d +0 ) is of degree 2 (resp. -1).

6.1.4. The LCH map. The map d +-is defined analogously to the bilinearised map in LCH induced by an exact Lagrangian cobordism. It is given as follows:

d +-(γ -) := γ + δ -,ζ - #M Σ0,Σ1 (γ + ; δ -, γ -, ζ -) • ε - 0 (δ -)ε - 1 (ζ -) • γ + . ( 38
)
As follows from Equation [START_REF] Rizell | Legendrian ambient surgery and Legendrian contact homology[END_REF], this map is of degree 0.

The Nessie map. Let

C • (Λ ± 1 , Λ ± 0 ) be the dual of C • (Λ ± 1 , Λ ± 0 ) with δ ±± the induced adjoint of the differential d Σ1,Σ0
±± . These differentials are entries in the Cthulhu differential d Σ1,Σ0 ε - 1 ,ε - 0 . Further, let CF • (Σ 0 , Σ 1 ) be the dual of CF • (Σ 0 , Σ 1 ). Observe that, since all the above spaces are endowed with a canonical basis, we are free to identify any such space with its dual. Moreover, with a fixed choice of Maslov potentials for the two cobordisms, there is also a canonical identification

CF • (Σ 0 , Σ 1 ) = CF • (Σ 0 , Σ 1 ) = CF n+1-• (Σ 1 , Σ 0 ),
when reversing the ordering of the pair of cobordisms.

The count of "banana" pseudoholomorphic strips gives rise to a map

b : C n-1-• (Λ ± 1 , Λ ± 0 ) → C • (Λ ± 0 , Λ ± 1 ), γ 01 → γ10 δ,ζ #M Σ0,Σ1 (γ 10 ; δ, γ 01 , ζ) • ε 0 (δ)ε 1 (ζ) • γ 10 ,
where the degree of the map follows from Equation [START_REF] Dragnev | Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations[END_REF].

Using δ -0 : CF

• (Σ 1 , Σ 0 ) → C •-2 (Λ - 1 , Λ - 0 ) to denote the adjoint of the map d Σ1,Σ0 0- in the entry of d Σ1,Σ0
ε1,ε0 , we are finally ready to define

d -0 := b • δ -0 : CF • (Σ 0 , Σ 1 ) = CF n+1-• (Σ 1 , Σ 0 ) → C • (Λ - 0 , Λ - 1 )
, which is of degree 0.

The moduli spaces above were previously considered in the Floer theory involving concave ends due to Akaho in [START_REF] Akaho | Floer's chain complexes for Lagrangian submanifolds in symplectic manifolds with concave ends[END_REF]. Also, see [START_REF] Akaho | Morse homology and manifolds with boundary[END_REF] where Morse homology in the presence of a non-empty boundary was considered by the same author.

The proof of d ε -

0 ,ε - 1 2 = 0. We are now ready to present and prove the following central result. Theorem 6.1. Let Σ i ⊂ R × P × R, i = 0, 1, be a pair of exact Lagrangian cobordisms from Λ - i to Λ + i as above. Given a generic admissible almost complex structure J ∈ J adm J -,J + (R × P × R) in the sense of Section 3.1.4, and augmentations ε - i of the Chekanov-Eliashberg algebras of Λ - i defined using J -, then

• d ε - 0 ,ε - 1 is well-defined, and • d ε - 0 ,ε - 1 2 = 0,
under the assumption that the entries above have been defined using J. In the case when Σ i , i = 0, 1, both are spin, the above counts can moreover be defined by signed counts with coefficients in Z given the choice of a spin structure on each cobordism. In general, the count can always be performed in Z 2 .

In order to prove d ε -0 ,ε - 1 2 = 0, we need to study the boundary points of onedimensional moduli spaces of pseudoholomorphic strips. As usual in Floer theory, this composition is defined by counting broken pseudoholomorphic strips that correspond to the boundary points of these moduli spaces. Recall that, in our setting, every strip is punctured, and that all counts are weighted by the augmentations chosen. For this reason, we start by prescribing two important points that need to be taken into account when performing these counts:

(I) Recall that the punctured pseudoholomorphic strips used in the definition of d ++ live in the top level and are allowed to have negative punctures. When adjoining the rigid punctured strips in the definition of d ++ and d + * , where * = 0, -, i.e. the glued configurations corresponding to the compositions d ++ d + * , we do not necessarily obtain a broken pseudoholomorphic strip.

In order to obtain a broken strip, we will adjoin the pseudoholomorphic half-planes that appear in the count defining the right-hand side of

ε + i = ε - i • Φ Σi , i = 0, 1
, to the middle level. From the latter equality, it also follows that the composition d ++ d + * indeed is obtained by counting buildings of precisely this form.

(II) Not all broken punctured pseudoholomorphic strips correspond to two glued pseudoholomorphic strips. Namely, as shown in Section 4.4, there are socalled ∂-breakings that consist of a punctured strip together with a punctured half-plane of index 1 having boundary on R × Λ ± i , i = 0, 1. Recall the fact that the counts of the latter half-planes define the differential ∂ ± of the Chekanov-Eliashberg algebras of Λ ± 0 ∪ Λ ± 1 , and that the equality ε ± i • ∂ ± = 0, i = 0, 1, holds by definition. It thus follows that the counts of the totality of the broken strips of this kind must vanish in the case when this count is weighted by the augmentations.

Proof. The fact that the map is well-defined follows form the compactness result Theorem 3.9 together with the transversality results in Section 4.3. Namely, all moduli spaces of index 0 having fixed asymptotics are compact (since there is a uniform upper bound on its total energy) 0-dimensional manifolds.

We now make a term-by-term argument for the matrix

d ε - 0 ,ε - 1 2 =   d ++ 2 d ++ d +0 + d +0 d 00 + d +-d -0 d ++ d +-+ d +0 d 0-+ d +-d -- 0 d 00 d 00 + d 0-d -0 d 00 d 0-+ d 0-d -- 0 d -0 d 00 + d --d -0 d -0 d 0-+ d --d --  
in order to show that all entries vanish.

• d 2 ++ = 0. The term d ++ is the standard bilinearised Legendrian contact cohomology differential [START_REF] Bourgeois | Bilinearized Legendrian contact homology and the augmentation category[END_REF] restricted to mixed chords from Λ + 1 to Λ + 0 . More precisely, the subspace generated by these chords form a subcomplex of the linearised cohomology complex of the link Λ + 0 ⊔ Λ + 1 , under the assumption that we use an augmentation of the link which vanishes on mixed chords, while it takes the value ε + i (γ i ) on a chord γ i ∈ R(Λ + i ). This term thus vanishes, as mentioned in Section 5.3.

• d ++ d +0 + d +0 d 00 + d +-d -0 = 0.
We must study the boundary of a moduli space M Σ0,Σ1 (γ; δ, p, ζ) being of dimension 1. The possibilities for the breakings involved are schematically depicted in Figure 11. We claim that, when counting these boundary points weighted by the augmentations ε - i , i = 0, 1, we get the contribution

(d ++ d +0 + d +0 d 00 + d +-d -0 )(p), γ .
Indeed, inspecting the boundary components of different types as described in Equation ( 32) we obtain the terms d ++ d +0 (p), γ (here we use

ε + i = ε - i • Φ Σi ; see (I)), d +0 d 00 (p)
, γ , and d +-d -0 )(p), γ , together with the ∂breakings which can be seen to contribute to zero (here we use ε

± i • ∂ ± = 0; see (II)). Λ + 0 ∪ Λ + 1 Σ 0 ∪ Σ 1 Λ - 0 ∪ Λ - 1 0 0 0 1 0 0 0 1 0 0 1 0 Figure 11. Breakings involved in d ++ d +0 +d +0 d 00 +d +-d -0 = 0.
The number on each component denotes its Fredholm index.

• d ++ d +-+ d +0 d 0-+ d +-d --= 0.
The argument is similar to the argument above, but where Equation ( 31) has been used. The possibilities for the breakings involved are schematically depicted in Figure 12 (also, see Figure 10 for some of the breakings with explicit negative ends). • d 00 d 00 + d 0-d -0 = 0. Again this follows as above, but while using Equation [START_REF] Ekholm | Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms[END_REF]. The possibilities for the breakings involved are schematically depicted in Figure 13 (also, see Figure 9). • d 00 d 0-+ d 0-d --= 0. This follows similarly as above, but while using Equation [START_REF] Eckmann | Introduction to l 2 -methods in topology: reduced l 2 -homology, harmonic chains, l 2 -Betti numbers[END_REF]. The possibilities for the breakings involved are schematically depicted in Figure 14. 15), where δ --again denotes the adjoint of d Σ1,Σ0 --. Hence, we get that

Λ + 0 ∪ Λ + 1 Σ 0 ∪ Σ 1 Λ - 0 ∪ Λ -
d -0 d 00 + d --d -0 = bδ -0 d 00 + d --bδ -0 = b δ -0 d 00 + δ --δ -0 , Σ 0 ∪ Σ 1 Λ - 0 ∪ Λ -
Λ + 0 ∪ Λ + 1 Σ 0 ∪ Σ 1 Λ - 0 ∪ Λ -
+ d Σ1,Σ0 0- d Σ1,Σ0
--. Since the latter term vanishes by the previous case, the claim now follows. See Figure 9.

Λ - 0 ∪ Λ - 1 Λ - 0 ∪ Λ - 1 1 0 1 1 1 0 1 0 Figure 15. Breakings involved in b • δ --= d --• b. • d -0 d 0-+ d --d --= 0.

The transfer and co-transfer map for concatenations of cobordisms

Recall that two exact Lagrangian cobordisms having a common end can be concatenated; see Section 2.2. In this section we will provide formulas which relate the Floer homologies of the different pieces of such a concatenation. This will be done by introducing a relative version of Viterbo's transfer map, originally defined in [START_REF] Viterbo | Functors and computations in Floer homology with applications[END_REF] for symplectic (co)homology. (Recall that Viterbo's transfer map concerns concatenations of symplectic cobordisms.) For the Hamiltonian formulation of wrapped Floer homology, the transfer map was constructed and treated in [START_REF] Abouzaid | An open string analogue of Viterbo functoriality[END_REF].

In the following we will consider the exact Lagrangian cobordisms V 0 , V 1 , W 0 , W 1 ⊂ R × P × R inside the symplectisation of a contactisation. We assume that V i is an exact Lagrangian cobordism from Λ - i to Λ i , and that W i is an exact Lagrangian cobordism from Λ i to Λ + i , i = 0, 1. It follows that we can form the concatenations

V i ⊙ W i ⊂ R × P × R, i = 0, 1,
being exact Lagrangian cobordisms from Λ - i to Λ + i . Under the further assumption that the negative ends of V i , i = 0, 1, are empty, a transfer map

Φ W0,W1 : Cth • (V 0 , V 1 ) → Cth • (V 0 ⊙ W 0 , V 1 ⊙ W 1 )
was constructed in [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF]Section 4.2.2]; recall that the analytic set-up of the latter article is the same as the one used here. Our construction of the transfer map will be a straight-forward generalisation of this construction to the case when the negative ends are non-empty.

We will also construct a map that we call a co-transfer map

Φ V0,V1 : Cth • (V 0 ⊙ W 0 , V 1 ⊙ W 1 ) → Cth • (W 0 , W 1 ).
This map should be thought of as a quotient projection associated to a transfer map.

7.1. Concatenations and stretching of the neck. Recall that the Hamiltonian isotopy class of a concatenation is unique. However, for us it will be necessary to concatenate the cobordism together with an almost complex structure, i.e. keeping track of conformal data as well, thus breaking this symmetry. In order to pin-point the almost complex structure obtained it will be useful to introduce a parameter keeping track of how the concatenation was performed. We start with the following hypotheses. Assume that we are given almost complex structures J a and J b on R×P ×R which are cylindrical in the subsets {t ≥ -1} and {t ≤ 1}, respectively, and which moreover agree in the subset {-1 ≤ t ≤ 1}. We also assume that V and W are cylindrical in the subsets {t ≥ -1} and {t ≤ 1}, respectively, where they coincide. For each N ≥ 0 we define

V ⊙ N W := (V ∩ {t ≤ 0}) ∪ (τ N (W ) ∩ {t ≥ 0}), (J a ⊙ N J b )(t, p, z) := J a (t, p, z) t ≤ 0 J b (t -N, p, z) t ≥ 0,
where we recall that τ T is the translation of the t-coordinate by T ∈ R. We also write J a ⊙ J b := J a ⊙ 0 J b .

The Hamiltonian isotopy class of V ⊙ N W ⊂ R × P × R is independent of N ≥ 0. In the case when J a and J b are cylindrical outside of a compact subset we have produced a family of boundary-value problems of J a ⊙ N J b -holomorphic curves in R × P × R having boundary on V ⊙ N W , N ≥ 0. It can be seen that this family of boundary-value problems in fact is conformally equivalent to the family which "stretches the neck" along the contact-type hypersurface {0} × P × R ⊂ R × P × R with boundary condition V ⊙ W ; see [START_REF] Bourgeois | Compactness results in symplectic field theory[END_REF]Section 3.4] as well as [START_REF] Eliashberg | Introduction to symplectic field theory[END_REF]Section 1.3] for more details. This fact will be important below.

There is a compactness theorem analogous to Theorem 3.9 in the case of a neckstretching sequence of almost complex structure; see [START_REF] Bourgeois | Compactness results in symplectic field theory[END_REF]Section 10] for the precise formulation. The key fact is that a sequence of J a ⊙ N J b -holomorphic discs, with N → +∞, has a subsequence converging to a building consisting of several levels whose components satisfy non-cylindrical boundary conditions. In the case under consideration, the limit buildings consist of:

• An upper level containing punctured J b -holomorphic discs with boundary on W 0 ∪ W 1 ; and • A lower level containing punctured J a -holomorphic discs with boundary on

V 0 ∪ V 1 .
A priori there can also be intermediate levels consisting of pseudoholomorphic discs for a cylindrical almost complex structure satisfying a cylindrical boundary condition. Since we are only interested in rigid configurations, and since the latter solutions will have positive dimension (unless they are trivial strips), they can be omitted from our breaking analysis (given the assumption that transversality is achieved for every level). The gluing result Theorem 4.11 also generalises to this setting (see [START_REF] Ekholm | Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms[END_REF]Lemma 3.14]), giving a bijection between buildings of the above type where all components are of Fredholm index zero, and punctured J a ⊙ N J b -holomorphic discs for each N ≫ 0 sufficiently large. Figure 16 schematically depicts two such buildings.

W 0 ∪ W 1 V 0 ∪ V 1 (1) (2)
Figure 16. A holomorphic buildings appearing after stretching the neck along Λ 7.2. The complex after a neck stretching procedure. The first goal is to find a description of the complex

(Cth • (V 0 ⊙ N W 0 , V 1 ⊙ N W 1 ), d V ⊙W ε - 0 ε - 1 )
when the almost complex structure is given by J a ⊙ N J b for N ≫ 0 sufficiently large. Here we assume that the almost complex structures J a and J b satisfy the properties described in Section 7.1, i.e. so that their concatenations can be taken.

We first consider the complex defined using J a

(Cth • (V 0 , V 1 ) = C •-2 (Λ 0 , Λ 1 ) ⊕ C • (V 0 , V 1 ) ⊕ C •-1 (Λ - 0 , Λ - 1 ), d V ε - 0 ε - 1 ), d V ε - 0 ε - 1 =   -d V0,V1 ++ d V0,V1 +0 d V0,V1 +- 0 d V0,V1 00 d V0,V1 0- 0 d V0,V1 -0 d V0,V1 --   .
Consider the entries d V1,V0 +0 , d V1,V0 +-, and

d V0,V1
±± in the differential of the complex Cth • (V 1 , V 0 ), where again the almost complex structure J a has been used. We will need their adjoints

δ V0,V1 0+ := (d V1,V0 +0 ) * : C • (Λ 1 , Λ 0 ) → C •-2 (V 0 , V 1 ), δ V0,V1 -+ := (d V1,V0 +-) * : C • (Λ 1 , Λ 0 ) → C •-1 (Λ - 1 , Λ - 0 ), δ V0,V1 ++ := (d V1,V0 ++ ) * : C • (Λ 1 , Λ 0 ) → C •-1 (Λ 1 , Λ 0 ), δ V0,V1 -- := (d V1,V0 --) * : C • (Λ - 1 , Λ - 0 ) → C •-1 (Λ - 1 , Λ - 0 ), δ V0,V1 -+ := (d V1,V0 +-) * : C • (Λ 1 , Λ 0 ) → C •-1 (Λ - 1 , Λ - 0 )
, where the canonical basis of Reeb chords and double points has been used in order to identify the modules and their duals. Observe that, exploiting the same notation, we also get δ V0,V1 00 = d V0,V1 00 . We write ε ′ i := ε i • Φ Vi,Ja , i = 0, 1, for the pull-backs of the augmentations under the DGA morphisms induced by the respective cobordisms. These augmentations now give rise to a complex

(Cth • (W 0 , W 1 ) = C •-2 (Λ + 0 , Λ + 1 ) ⊕ C • (W 0 , W 1 ) ⊕ C •-1 (Λ 0 , Λ 1 ), d W ε ′ 0 ε ′ 1 ), d W ε ′ 0 ε ′ 1 =   -d W0,W1 ++ d W0,W1 +0 d W0,W1 +- 0 d W0,W1 00 d W0,W1 0- 0 d W0,W1 -0 d W0,W1 --   ,
where the moduli spaces are defined using the almost complex structure J b . Note that d W0,W1 --

= d V0,V1
++ . In addition we will also need the map

b V0,V1 : C • (Λ 1 , Λ 0 ) → C n-2-• (Λ 0 , Λ 1 ) which is defined similarly to b Λ0,Λ1 : C • (Λ 1 , Λ 0 ) → C n-1-• (Λ 0 , Λ 1 )
as defined in Section 6.1.5, but which instead counts rigid "bananas" having boundary on V 0 ∪ V 1 , and two punctures with positive asymptotics to Reeb chords.

Recall that the compactness theorem for a neck-stretching sequence together with pseudoholomorphic gluing shows the following. For N ≫ 0 sufficiently large, the rigid

J a ⊙ N J b -holomorphic curves in R × P × R having boundary on (V 0 ⊙ N W 0 )∪(V 1 ⊙ N W 1
) are in bijective correspondence with pseudoholomorphic buildings of the form described in Section 7.1, in which every involved component is rigid.

Analysing the possible such pseudoholomorphic buildings, we obtain the following. When N ≫ 0 is sufficiently large, the differential of the complex for the concatenated cobordisms, defined using the almost complex structure J a ⊙ N J b as in the previous paragraph, is given by

(Cth • (V 0 ⊙ N W 0 , V 1 ⊙ N W 1 ) = C •-2 (Λ + 0 , Λ + 1 ) ⊕ C • (W 0 , W 1 ) ⊕ C • (V 0 , V 1 ) ⊕ C •-1 (Λ - 0 , Λ - 1 ), d V ⊙W ε - 0 ε - 1 ), d V ⊙W ε - 0 ε - 1 =      -d W0,W1 ++ d W0,W1 +0 + d W0,W1 +- b V0,V1 δ W0,W1 -0 d W0,W1 +- d V0,V1 +0 d W0,W1 +- d V0,V1 +- 0 d W0,W1 00
+ d W0,W1 0- b V0,V1 δ W0,W1 -0 d W0,W1 0- d V0,V1 +0 d W0,W1 0- d V0,V1 +- 0 d V0,V1 0+ δ W0,W1 -0 d V0,V1 00 d V0,V1 0- 0 b Λ - 0 ,Λ - 1 δ V0,V1 -+ δ W0,W1 -0 b Λ - 0 ,Λ - 1 δ V0,V1 -0 d V0,V1 --      ,
in terms of pseudoholomorphic strips on V 0 ∪ V 1 and W 0 ∪ W 1 for each N ≫ 0 sufficiently large. (For instance the term d W0,W1

+- b V0,V1 δ W0,W1
-0 corresponds to the breaking (1) in Figure 16 and the term d W0,W1

+- d V0,V1
+0 corresponds to the breaking (2) in the same figure .)

We have here relied on the exactness assumptions in Definition 2.1 and action considerations from Section 3.3.2 in order to rule out certain configurations. 7.3. Definition of the transfer and co-transfer maps. The transfer and cotransfer maps on the chain level are defined for a very "stretched" almost complex structure on a concatenated cobordism (i.e. when the parameter N ≫ 0 in Section 7.1 is sufficiently large), so that the complexes take the form as described in Section 7.2 above.

Definition 7.1. The transfer map is defined by

Φ W0,W1 : Cth • (V 0 , V 1 ) → Cth • (V 0 ⊙ N W 0 , V 1 ⊙ N W 1 ), Φ W0,W1 =     d W0,W1 +- 0 0 d W0,W1 0- 0 0 0 id 0 0 0 id     ,
while the co-transfer map is defined by

Φ V0,V1 : Cth • (V 0 ⊙ N W 0 , V 1 ⊙ N W 1 ) → Cth • (W 0 , W 1 ), Φ V0,V1 =   id 0 0 0 0 id 0 0 0 b V0,V1 δ W0,W1 -0 d V0,V1 +0 d V0,V1 +-   . Lemma 7.2. Under the assumption that d V ε - 0 ε - 1 , d W ε ′ 0 ε ′ 1 and d V ⊙W ε - 0 ε - 1 are computed
using J a , J b and J a ⊙ N J b (for N ≫ 0 as before) respectively, the transfer and co-transfer maps defined above are chain maps.

Proof. We start by showing the claim for the transfer map; i.e. we have to establish the equality

Φ W0,W1 • d V ε - 0 ε - 1 = d V ⊙W ε ′ 0 ε ′ 1 • Φ W0,W1 .
The matrices on the left and right hand sides become

    -d W0,W1 +- d V0,V1 ++ d W0,W1 +- d V0,V1 +0 d W0,W1 +- d V0,V1 +- -d W0,W1 0- d V0,V1 ++ d W0,W1 0- d V0,V1 +0 d W0,W1 0- d V0,V1 +- 0 d V0,V1 00 d V0,V1 0- 0 d V0,V1 -0 d V0,V1 --     and     -d W0,W1 ++ d W0,W1 +- + d W0,W1 +0 d W0,W1 0- + A 1 d W0,W1 +- d V0,V1 +0 d W0,W1 +- d V0,V1 +- d W0,W1 00 d W0,W1 0- + A 2 d W0,W1 0- d V0,V1 +0 d W0,W1 0- d V0,V1 +- A 3 d V0,V1 00 d V0,V1 0- A 4 d V0,V1 -0 d V0,V1 --     ,
respectively, where

A 1 := d W0,W1 +- b V0,V1 δ W0,W1 -0 d W0,W1 0- , A 2 := d W0,W1 0- b V0,V1 δ W0,W1 -0 d W0,W1 0- , A 3 := d V0,V1 0+ δ W0,W1 -0 d W0,W1 0- , A 4 := b Λ - 0 ,Λ - 1 δ V0,V1 -+ δ W0,W1 -0 d W0,W1 0- . As in Section 6.2 δ W0,W1 -0 d W0,W1
0-= 0 by action reasons, and hence A i = 0, i = 1, 2, 3, 4. What now remains is showing the equalities

-d W0,W1 ++ d W0,W1 +- + d W0,W1 +- d V0,V1 ++ + d W0,W1 +0 d W0,W1 0- = 0, d W0,W1 00 d W0,W1 0- + d W0,W1 0- d V0,V1 ++ = 0. Recall that d V0,V1 ++ = d W0,W1
--and, hence, both these equalities hold since the lefthand sides are defined by the signed counts of the number of boundary components of certain one-dimensional moduli spaces of pseudoholomorphic strips with boundary on (W 0 , W 1 ). The first one comes from the boundary of M W0,W1 (γ + ; δ, γ -, ζ) for γ ± a chord from Λ ± 1 to Λ ± 0 . The second comes from M(q; δ, γ -, ζ) for q ∈ W 0 ∩ W 1 . Also, see Figures 12 and14.

We now continue with the co-transfer map; i.e. we have to establish the equality

Φ V0,V1 • d V ⊙W ε - 0 ε - 1 = d W ε ′ 0 ε ′ 1 • Φ V0,V1
. The matrices on the left and right hand side become

  -d W0,W1 ++ d W0,W1 +0 + d W0,W1 +- b V0,V1 δ W0,W1 -0 d W0,W1 +- d V0,V1 +0 d W0,W1 +- d V0,V1 +- 0 d W0,W1 00
+ d W0,W1 0- b V0,V1 δ W0,W1 -0 d W0,W1 0- d V0,V1 +0 d W0,W1 0- d V0,V1 +- 0 A 1 + B 1 A 2 + B 2 A 3 + B 3   ,
where

A 1 = b V0,V1 δ W0,W1 -0 d W0,W1 0- b V0,V1 δ W0,W1 -0 , A 2 = b V0,V1 δ W0,W1 -0 d W0,W1 0- d V0,V1 +0 , A 3 = b V0,V1 δ W0,W1 -0 d W0,W1 0- d V0,V1 +-, B 1 = b V0,V1 δ W0,W1 -0 d W0,W1 00 + d V0,V1 +-b Λ - 0 ,Λ - 1 δ V0,V1 -+ δ W0,W1 -0 + d V0,V1 +0 δ V0,V1 0+ δ W0,W1 -0 , B 2 = d V0,V1 +0 d V0,V1 00
+ d V0,V1 +-d V0,V1 -0 , B 3 = d V0,V1 +0 d V0,V1 0- + d V0,V1 +-d V0,V1 --, and   -d W0,W1 ++ d W0,W1 +0 + d W0,W1 +- b V0,V1 δ W0,W1 -0 d W0,W1 +- d V0,V1 +0 d W0,W1 +- d V0,V1 +- 0 d W0,W1 00
+ d W0,W1 0- b V0,V1 δ W0,W1 -0 d W0,W1 0- d V0,V1 +0 d W0,W1 0- d V0,V1 +- 0 d W0,W1 -0 + d W0,W1 -- b V0,V1 δ W0,W1 -0 d W0,W1 -- d V0,V1 +0 d W0,W1 -- d V0,V1 +-   ,
respectively. Again, we have A i = 0, i = 1, 2, 3, by action reasons. The equalities remaining to be shown are

d V0,V1 +0 d V0,V1 00
+ d V0,V1 +-d V0,V1 -0 -d V0,V1 ++ d V0,V1 +0 = 0 d V0,V1 +0 d V0,V1 0- + d V0,V1 +-d V0,V1 ---d V0,V1 ++ d V0,V1 +- = 0 (recall again that d W0W1 -- = d V0V1 ++ ) together with the equality (b Λ0,Λ1 + d V0,V1 ++ b V0,V1 )δ W0,W1 -0 = = b V0,V1 δ W0,W1 -0 d W0,W1 00
+ (d V0,V1 +-b Λ - 0 ,Λ - 1 δ V0,V1 -+ + d V0,V1 +0 d V0,V1 0+ )δ W0,W1 -0 . (Recall that d W0,W1 -0 = b Λ0,Λ1 δ W0,W1 -0
.) The first two equalities again follow from the fact that the left-hand sides correspond to counts of number of boundary component of appropriate one-dimensional moduli spaces of pseudoholomorphic discs (M(γ + ; δ, p, ζ) for the first one, the second follows by an analysis similar to the one for the transfer map). Also, see Figures 14 and12.

The third equality finally follows from

δ W0,W1 -0 d W0,W1 00 -δ W0,W1 -- δ W0,W1 -0 = 0 b Λ0,Λ1 + d V0,V1 ++ b V0,V1 = b V0,V1 δ W0,W1 -- + d V0,V1 +-b Λ - 0 ,Λ - 1 δ V0,V1 -+ + d V0,V1 +0 δ V0,V1
0+ , both which can be shown by arguments similar to the above (more precisely, studying M(q; δ, γ 1 , ζ) for the first, and degeneration of bananas on (V 0 , V 1 ) for the second). Also, see Figures 11 and22.

Remark 7.3. In the special case when there are no Reeb chords from Λ 1 to Λ 0 (recall that these are the Legendrian submanifolds along which the concatenations are performed), the corresponding transfer and co-transfer maps take the following particularly simple form. Since C(Λ 0 , Λ 1 ) = 0, the transfer map Φ W0,W1 is simply the inclusion of a subcomplex, while the co-transfer map Φ V0,V1 becomes the corresponding quotient projection. In fact, as will be shown below in Section 8.1, this situation can always be achieved after the application of a Hamiltonian isotopy that "wraps" the positive and negative ends of V 1 and W 1 , respectively.

The following lemma is standard. It follows from the fact that, in the cylindrical situation, regular 0-dimensional moduli space are trivial, together with a stretchingthe-neck argument.

Lemma 7.4. The (co)transfer map satisfies the following properties • In the case when W i = R × Λ i , i = 0, 1, and J b is a cylindrical almost complex structure we have

Φ W0,W1 = Φ R×Λ0,R×Λ1 = id.
• In the case when V i = R × Λ i , i = 0, 1, and J a is a cylindrical almost complex structure we have

Φ V0,V1 = Φ R×Λ0,R×Λ1 = id.
• In the case when

W i = U i ⊙ M U ′ i , i = 0, 1, and J b = J c ⊙ M J d we have Φ W0,W1 = Φ U ′ 0 ,U ′ 1 • Φ U0,U1
in the case when M ≫ 0 is sufficiently large

• In the case when V i = U i ⊙ M U ′ i , i = 0, 1, and J a = J c ⊙ M J d we have Φ V0,V1 = Φ U ′ 0 ,U ′ 1 • Φ U0,U1
in the case when M ≫ 0 is sufficiently large.

In the case when W 1 = R × Λ, we write

Φ W0 := Φ W0,W1 = Φ W0,R×Λ and, similarly, when V 1 = R × Λ ⊂ R × Y , we write Φ V0 := Φ V0,V1 = Φ V0,R×Λ .
7.4. An auxiliary complex. Here we try to shed some light on the algebraic relationship between the transfer and the co-transfer map. We assume that all differentials are computed using almost complex structures such that Lemma 7.2 holds.

Starting with the above complex

(C • := Cth • (V 0 ⊙ N W 0 , V 1 ⊙ N W 1 ), d V ⊙W ε - 0 ε - 1 ),
we construct the complex

( C • := C • ⊕ C • (Λ 0 , Λ 1 ) ⊕ C •-1 (Λ 0 , Λ 1 ), d V ⊙W ε - 0 ε - 1 ), d V ⊙W ε - 0 ε - 1 :=   d V ⊙W 0 0 0 d W0,W1 -- 0 0 id d V0,V1 ++   . (Again, d W0W1 -- = d V0V1 ++ .) This complex is clearly homotopic to (C • , d V ⊙W ε - 0 ε - 1
), since the canonical inclusion and projection maps

ι : (C • , d V ⊙W ε - 0 ε - 1 ) → ( C • , d V ⊙W ε - 0 ε - 1 ), π : ( C • , d V ⊙W ε - 0 ε - 1 ) → (C • , d V ⊙W ε - 0 ε - 1 ),
are homotopy inverses of each other.

We have a canonical identification

C • = Cth • (V 0 , V 1 ) ⊕ Cth • (W 0 , W 1 )
on the level of modules. We let

ι V0,V1 : Cth • (V 0 , V 1 ) ֒→ C • , ι W0,W1 : Cth • (W 0 , W 1 ) ֒→ C • ,
be the canonical inclusions of the respective summands. Even though ι V0,V1 is not a chain-map in general, after a change of coordinates it will be apparent that (

C • , d V ⊙W ε - 0 ε - 1
) in fact is the mapping-cone of a chain map

δ V W : Cth • (W 0 , W 1 ) → Cth • (V 0 , V 1 ).
To that end, we start by considering the map

Ψ : C • ⊕ C • (Λ 0 , Λ 1 ) ⊕ C •-1 (Λ 0 , Λ 1 ) → C • ⊕ C • (Λ 0 , Λ 1 ) ⊕ C •-1 (Λ 0 , Λ 1 ), Ψ =   id 0 d W0,W1 +- + d W0,W1 0- -(d V0,V1 +-+ d V0,V1 +0 + b V0,V1 δ W0,W1 -0 ) id 0 0 0 id   .
The property that δ W0,W1 -0 d W0,W1 0-= 0, which holds by action considerations, implies the equality

(d V0,V1 +-+ d V0,V1 +0 + b V0,V1 δ W0,W1 -0 )(d W0,W1 +- + d W0,W1 0
- ) = 0.
From this it follows that Ψ in fact is an isomorphism of modules, with inverse given by

Ψ -1 =   id 0 -(d W0,W1 +- + d W0,W1 0
- ) d V0,V1 +-+ d V0,V1 +0 + b V0,V1 δ W0,W1 -0 id 0 0 0 id   .
Using Ψ we define the complex

Ψ : ( C • , d V ⊙W ε - 0 ε - 1 ) → ( C • , d V ⊙W ε - 0 ε - 1 ), d V ⊙W ε - 0 ε - 1 := Ψ -1 • d V ⊙W ε - 0 ε - 1 • Ψ,
obtained by applying a coordinate change to the original complex. The upshot is that the differential of ( C • , d

V ⊙W ε - 0 ε - 1
) has the following particularly nice and apparent cone structure. The proof follows from a breaking analysis similar to the one in the proof of Lemma 7.2.

Lemma 7.5. The complex ( C • , d V ⊙W ε - 0 ε - 1
) is equal to the mapping cone

( C • = Cth • (W 0 , W 1 ) ⊕ Cth • (V 0 , V 1 ), d V ⊙W ε - 0 ε - 1 ), d V ⊙W ε - 0 ε - 1 = d W ε ′ 0 ε ′ 1 δ V W 0 d V ε - 0 ε - 1 ,
where

C •-1 (Λ + 0 , Λ + 1 ) ⊕ C • (W 0 , W 1 ) ⊕ C • (Λ 0 , Λ 1 ) ↓ δ V W C •-1 (Λ 0 , Λ 1 ) ⊕ C • (V 0 , V 1 ) ⊕ C • (Λ - 0 , Λ - 1 ), is of the form δ V W =    0 -b V0,V1 δ W0,W1 -0 id C(Λ0,Λ1) 0 d V0,V1 0+ δ W0,W1 -0 0 0 b Λ - 0 ,Λ - 1 δ V0,V1 -+ δ W0,W1 -0 0    .
In particular, the canonical inclusion

ι V0,V1 : (Cth • (V 0 , V 1 ), d V ε - 0 ε - 1 ) ֒→ ( C • , d V ⊙W ε - 0 ε - 1 )
as well as the canonical projection

π W0,W1 : ( C • , d V ⊙W ε - 0 ε - 1 ) → (Cth • (W 0 , W 1 ), d W ε ′ 0 ε ′ 1 )
, are both chain maps. Furthermore, the transfer and co-transfer map can be expressed as

Φ W0,W1 = π • Ψ • ι V0,V1 , Φ V0,V1 = π W0,W1 • Ψ -1 • ι,
where all factors are chain maps.

Proof. The claim that

ι V0,V1 : (Cth • (V 0 , V 1 ), d V ε - 0 ε - 1 ) ֒→ ( C • , d V ⊙W ε - 0 ε - 1 )
is a chain map can be seen by considering the relations

d W ε ′ 0 ,ε ′ 1 (d W0,W1 +- + d W0,W1 0
- ) + (d W0,W1 +- + d W0,W1 0
- )d Λ0,Λ1 = 0, d Λ0,Λ1 (d V0,V1 +-+ d V0,V1 +0 ) = (d V0,V1 +-+ d V0,V1 +0 )d V ε - 0 ,ε - 1 , which follow from the fact that (d W ε ′ 0 ε ′ 1 ) 2 = 0 and (d V ε - 0 ε - 1 ) 2 = 0, respectively, together with the expressions of d V ⊙W ε - 0 ε - 1 , d W ε ′ 0 ε ′ 1 and d V ε - 0 ε - 1
as given in Section 7.2. To see that the corresponding quotient complex is as claimed, one must use the identity

(-d Λ0,Λ1 + d V0,V1 +-+ d V0,V1 +0 )δ V W + b V0,V1 δ W0,W1 -0 )d W0,W1 00 = b Λ0,Λ1 δ W0,W1 -0
, which in turn follows from

-d Λ0,Λ1 b V0,V1 + b V0,V1 δ V0V1 ++ + d V0,V1 +-b Λ - 0 ,Λ - 1 δ V0,V1 -+ + δ V0,V1 +0 d V0,V1 0+ = b Λ0,Λ1 , δ W0,W1 -0 d W0,W1 00 = δ W0,W1 -- δ W0,W1 -0 .
The latter identities can be seen by analysing the possible boundaries of the appropriate one-dimensional moduli spaces.

Proof of the acyclicity (the invariance)

In this section we establish the invariance result for our Floer theory. In fact, in our context, the invariance is simply the fact the complex Cth(Σ 0 , Σ 1 ) is acyclic (actually null-homotopic). The naive reason for this is that the symplectisation of a contactisation (R × P × R, d(e t (dz + θ))) is "subcritical": it is symplectomorphic to (P × R 2 , dθ ⊕ ω 0 ). More precisely, the main feature that will be used is that one can use the Reeb flow (which is Hamiltonian) in order to displace an exact Lagrangian cobordism from any other given cobordism.

In order to circumvent certain technical difficulties, we will here restrict our attention to almost complex structures on the symplectisation R × P × R that are admissible, as defined in Section 3.1.4, and moreover coinciding with cylindrical lifts of almost complex structures in P outside of a compact subset; see Section 3.1.3. 8.1. Wrapping the ends. Let Σ i , i = 0, 1, be exact Lagrangian cobordisms from Λ - i to Λ + i , i = 0, 1. We assume that Σ i , i = 0, 1, both are cylindrical in the subset {|t| ≥ T } for some T > 0.

Fix a smooth non-decreasing cut-off function ρ : R → [0, 1] satisfying ρ(t) = 0 for t ≤ 1 and ρ(t) = 1 for t ≥ 2. Consider the smooth real-valued function

h N : R × P × R → R, (t, p, z) → e t ρ N (t),
where ρ N : R → [0, 1] is defined uniquely by the relations ρ N (-t) = ρ N (t), and ρ N (t) = ρ(t -(T + N )) for all t ≥ 0. We also consider the smooth real-valued functions

h N,± : R × P × R → R, (t, p, z) → e t ρ(±t -(T + N )),
and set h ± := h 0,± . The above functions are then used to construct the Hamiltonian isotopies φ s hN ∂z (t, p, z) = (t, p, z + sρ N (t)), φ s hN,±∂z (t, p, z) = (t, p, z + sρ(±t -(T + N ))), which should be thought of as "wrapping" the ends of R×P ×R. For instance, after applying the isotopy φ -S hN ∂z to Σ 0 for S ≫ 0 sufficiently large, we get additional double points φ -S hN ∂z (Σ 0 ) ∩ Σ 1 , all which correspond to the Reeb chords going from the ends of Σ 1 to the corresponding end of Σ 0 . More precisely, the following is true:

Lemma 8.1. When S ≥ S 0 := 2 max c∈R(Λ - 1 ,Λ - 0 )∪R(Λ + 1 ,Λ + 0 ) ℓ(c)
there is a canonical bijective correspondence

w ± : φ -S hN,±∂z (R × Λ ± 0 ) ∩ (R × Λ ± 1 ) → R(Λ ± 1 , Λ ± 0 )
induced by the Lagrangian projection, i.e. by identifying elements on both sides with a double point in Π Lag (Λ ± 0 ∪ Λ ± 1 ) ⊂ P . On the level of gradings this bijection moreover satisfies

|w -(p)| = |p| and |w + (p)| = |p| -1.
In particular, there is a canonical identification

Cth • (Σ 0 , Σ 1 ) = Cth • (φ -S hN ∂z (Σ 0 ), Σ 1
) on the level of graded modules. After taking N ≫ 0 sufficiently large, we may assume that the action of a generator in

C(Λ - 0 , Λ - 1 ) ⊂ Cth • (φ -S hN ∂z (Σ 0 ), Σ 1 )
, is arbitrarily small, the action of a generator in

C(Λ + 0 , Λ + 1 ) ⊂ Cth • (φ -S hN ∂z (Σ 0 ), Σ 1
) is arbitrarily large, while the action of a generator in

C(Σ 0 , Σ 1 ) ⊂ Cth • (φ -S
hN ∂z (Σ 0 ), Σ 1 ) coincides with its original action.

The first goal will be showing that the identification given by the above lemma, in fact may be assumed to hold on the level of complexes as well. Proposition 8.2. For each N ≫ 0 sufficiently large and S ≥ S 0 as defined above, there is a canonical identification of complexes

(Cth • (Σ 0 , Σ 1 ), d ε - 0 ε - 1 ) = (Cth • (φ -S hN ∂z (Σ 0 ), Σ 1 ), d ε - 0 ε - 1 ),
under the assumption that J ∈ J adm J -,J + (R × P × R) is a regular admissible almost complex structure and where, moreover, J ± = J P both are the cylindrical lift of an almost complex structure on (P, dθ). Remark 8.3.

• Recall that J is equal to J P outside of a compact set by assumption, where the latter is a cylindrical almost complex structure which is invariant under the Reeb flow. Since the negative ends of Σ 0 and φ -S hN ∂z (Σ 0 ) differ by the time-S Reeb flow, it follows that these Legendrian submanifolds have canonically isomorphic Chekanov-Eliashberg algebras when defined by J P and, in particular, we can identify their augmentations.

• For a general choice of almost complex structure, there should again exist an analogous isomorphism, albeit non-canonical.

Proof. Consider the transfer and co-transfer maps

Φ φ -S h + ∂z (R×Λ + 0 ) : Cth • (Σ 0 , Σ 1 ) → Cth • (φ -S hN,+∂z (Σ 0 ), Σ 1 ), Φ φ -S h -∂z (R×Λ - 0 ) : Cth • (φ -S hN ∂z (Σ 0 ), Σ 1 ) → Cth • (φ -S hN,+∂z (Σ 0 ), Σ 1 ),
defined by counting J P -holomorphic strips having boundary on

φ -S hN,+∂z (R × Λ + 0 ) ∪ (R × Λ + 1 ) and φ -S hN,-∂z (R × Λ - 0 ) ∪ (R × Λ - 1 )
, respectively. In order to identify the domains and codomains of the above maps we have used the fact that, J ⊙ N J P = J P ⊙ N J = J, N ≥ 0, which holds by the assumptions made on J, as well as the facts that

φ -S hN,+∂z (Σ 0 ) = Σ 0 ⊙ N φ -S h+∂z (R × Λ + 0 ), φ -S hN ∂z (Σ 0 ) = φ -S h-∂z (R × Λ - 0 ) ⊙ N φ -S hN,+∂z (Σ 0 ),
hold for every N ≥ 0 by construction.

Recall that the transfer and co-transfer maps are chain maps by Lemma 7.2, assuming that N ≫ 0 has been chosen sufficiently large. The proposition will follow from the claim that Φ φ -S h N,+ ∂z (R×Λ + 0 ) and Φ φ -S h N,-∂z (R×Λ - 0 ) both are isomorphisms that, moreover, induce the respective canonical identifications of graded modules described in Lemma 8.1. The latter facts follows by the explicit disc count performed in the proof of [START_REF] Chantraine | Floer homology and Lagrangian concordance[END_REF]Theorem 2.15]; also, see [START_REF] Rizell | Lifting pseudoholomorphic polygons to the symplectisation of P × R and applications[END_REF]Proposition 5.11] for a similar argument. Roughly speaking, it is there shown that every J P -holomorphic disc of index zero in the definition of the above (co)transfer map is a transversely cut-out strip having one positive puncture and one negative puncture, and whose image under the canonical projection to P is constant. Conversely, there is an explicitly defined such strip for every double point in P corresponding to a Reeb chord. In particular, under an appropriate choice of basis, the matrices of both maps Φ φ -S h N,+ ∂z (R×Λ + 0 ) and Φ φ -S h N,-∂z (R×Λ - 0 ) are equal to the identity matrices.

Invariance under compactly supported Hamiltonian isotopies.

The following is the core of the invariance result that we need in order to deduce the acyclicity of the Cthulhu complex. (As we need to compare the complexes computed with two different almost structures, here we denote by d ε - 0 ε - 1 (J) the differential on Cth • (Σ 0 , Σ 1 ) computed using the almost complex structure J.) Proposition 8.4. Let (Σ s 0 , Σ 1 ), s ∈ [0, 1], be a compactly supported one-parameter family of pairs of exact Lagrangian cobordisms from Λ - i to Λ + i , i = 0, 1. Also, consider a one-parameter family {J s } s∈[0,1] of admissible almost complex structures which agree outside of a compact set. There is an induced homotopy equivalence

Ψ {(Σ s 0 ,Js)} : (Cth • (Σ 0 0 , Σ 1 ), d ε - 0 ε - 1 (J 0 )) → (Cth • (Σ 1 0 , Σ 1 ), d ε - 0 ε - 1 (J 1 )).
(note that the Lagrangian cobordisms considered are of dimension at least two by assumption). Moreover, all J s0 -holomorphic strips as above whose punctures are asymptotic to transverse intersection points are transversely cut out.

Handleslide: The intersection Σ s0 0 ∩ Σ 1 is transverse, and all J s -holomorphic strips with boundary on Σ s0 0 ∪ Σ 1 are regular except a single solution. Moreover, the latter strip has index -1, and is generic when considered in the one-parameter family of boundary value problems.

Observe that, it suffices to consider each case above separately in order to deduce the sought invariance. Namely, for a generic family in which none of the above cases occur, all involved complexes are naturally isomorphic.

The case of a birth/death moment is treated analogously as in the closed case in [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF], a careful account appearing also in [START_REF] Sullivan | K-theoretic invariants for Floer homology[END_REF]. Note that the difference of action between the newly created intersection points is small near the birth point and, thus, the involved holomorphic curves cannot have negative asymptotics to Reeb chords.

In the case of a handle slide we construct an isomorphism of complexes by the following formula. Assume that the unique strip u of index -1 has the intersection points x, y as incoming and outgoing ends, respectively, and that it has additional negative punctures asymptotic to the words a, b of Reeb chords on Λ - 0 and Λ - 1 , respectively. On the intersection point generators we define the isomorphism

p → p + K(p), K(p) = ε 0 (a)ε 1 (b)x, p = y, 0, p = y, of graded modules. 0 -1 -1 1 -1 0 1 Λ + 0 ∪ Λ + 1 Σ 0 ∪ Σ 1 Λ - 0 ∪ Λ - 1 Figure 17.
The a priori possibilities for the boundary points of a one-parameter family of rigid strips with both incoming and outgoing ends asymptotic to intersection points. The number on each component denotes its Fredholm index.

We must show that this isomorphism in fact is a chain map. Again, this is shown by gluing of pseudoholomorphic curves. The only difference between this case and the case when a = b = ∅ comes from the gluing of ∂-breakings. More precisely, the difference between the differential d s0-ǫ before the handle slide and d s0+ǫ come from glued rigid configuration of index 0 curve to u such configuration are of two types (see Figure 17): The first type of gluing does not involve pure negative Reeb chords, and correspond to the term K in the definition of the isomorphism. The second type of gluing involves a pure Reeb chord, and corresponds to that of a ∂-breaking. The totality of the glued solutions of this kind thus contributes to 0 when counted utilising the augmentation (see the proof of Theorem 6.1). Finally, note that breaking involving an index -1 curve with a negative asymptotic to a mixed chord cannot occur; such a breaking would involve a holomorphic banana and, one of its two ends would necessarily be a chord from Λ - 1 to Λ - 0 (which does not exist by assumption).

Finally, in order to deduce the last claim of the proposition, it will be necessary to use the following additional property of the identifications of complexes described in the proof of Proposition 8.2. Consider the subset C ⊂ φ -S hN ∂z (Σ s 0 )∩Σ 1 of intersection points corresponding to the Reeb chords from Λ + 1 to Λ + 0 ; these intersection points are contained in a subset of the form (N, +∞) × P × R, which may be assumed to be fixed in the one-parameter family φ -S hN ∂z (Σ s 0 ) of cobordisms. Even though these intersection points are fixed, their actions will in general vary with s ∈ [0, 1]. We use M to denote the minimum of the action of a intersection point in C ⊂ φ -S hN ∂z (Σ s 0 ) ∩ Σ 1 taken over all s ∈ [0, 1] . For N ≫ 0 sufficiently large, Lemma 8.1 shows that any intersection point (φ -S hN ∂z (Σ s 0 ) ∩ Σ 1 ) \ C has action strictly less than M . In conclusion, we must have

K(C(Λ + 0 , Λ + 1 )) ⊂ C(Λ + 0 , Λ + 1 )
for the map K as defined above, thus implying the claim. 8.3. The proof of the acyclicity. We are finally ready to prove the main result of this section. Recall that we are in the special case of a symplectisation of a contactisation. In this setting, the idea is to show that a complex having no Reeb chord generators is acyclic by alluding to Proposition 8.4, together with an explicitly defined compactly supported Hamiltonian displacement. Finally, Proposition 8.2 shows that we can replace every complex with a complex induced by a pair of Lagrangian cobordisms having no Reeb chord generators (which is obtained by wrapping the cylindrical ends of one of the Lagrangian cobordisms). Theorem 8.6. Assume that we are given a choice of regular admissible almost complex structure coinciding with a cylindrical lifts outside some subset of the form [-T, T ] × Y . For any pair Σ 0 , Σ 1 ⊂ R × P × R of exact Lagrangian cobordisms from Λ - i to Λ + i , and choices ε - i of augmentations of the Chekanov-Eliashberg algebra of

Λ - i , i = 0, 1, the complex (Cth • (Σ 0 , Σ 1 ), d ε - 0 ε - 1 )
is homotopic to the trivial complex, Proof. We use the fact that, for S ≫ 0 sufficiently large, there is an exact Lagrangian cobordism Σ ′ 0 ⊂ R × (P × R) satisfying the properties that, first, it is isotopic to φ -S hN ∂z (Σ 0 ) by a compactly supported Hamiltonian isotopy and, second, the complex

(Cth(Σ ′ 0 , Σ 1 ), d ε - 0 ε - 1 ) = 0
has no generators. To that end, observe that the complex Cth(φ -S hN ∂z (Σ 0 ), Σ 1 ) has no Reeb chord generators for S ≫ 0 sufficiently large by Lemma 8.1. Moreover, the compactly supported Hamiltonian isotopy φ -s e t • φ

-(S-s)
hN ∂z (Σ 0 ) can be seen to remove all intersection points, whenever s ≫ 0 is sufficiently large.

The invariance result for a compactly supported Hamiltonian isotopy provided by Proposition 8.4 thus implies that the complex Cth • (φ -S hN ∂z (Σ 0 ), Σ 1 ) is nullhomotopic. The fact that the same is true for the complex Cth • (Σ 0 , Σ 1 ) is now an immediate consequence of Proposition 8.2, for N ≫ 0 that have been chosen sufficiently large.

Proof of the main theorem

Now we proceed to prove the main result of the paper. In Section 10, this will then be used to deduce several long exact sequences.

We denote the quotient complex

C(Σ 0 , Σ 1 ) ⊕ C(Λ - 0 , Λ - 1 ) of Cth(Σ 0 , Σ 1 ) by CF -∞ (Σ 0 , Σ 1 ) with differential given by d ∞ = d 00 d 0- d -0 d --
, its homology is denoted by HF -∞ (Σ 0 , Σ 1 ). The complex Cth(Σ 0 , Σ 1 ) is the cone of the chain map

d +0 + d +-: CF ∞ (Σ 0 , Σ 1 ) → C(Λ + 0 , Λ + 1
). The acyclicity of the complex (Cth(Σ 0 , Σ 1 ), d ε -

0 ε - 1
) implies that this map is a quasi-isomorphism, and hence that (39)

LCH k ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) ≃ HF k+1 -∞ (Σ 0 , Σ 1 ).
Let f i , i = 0, 1, be primitives of e t α| Λi as in Section 3.

3. A point p ∈ Σ 0 ∩ Σ 1 is positive (resp. negative) if f 1 (p) -f 0 (p) is positive (resp. negative). This leads to a decomposition CF (Σ 0 , Σ 1 ) = CF + (Σ 0 , Σ 1 ) ⊕ CF -(Σ 0 , Σ 1 ). Proposition 9.1. With respect to the decomposition CF -∞ (Σ 0 , Σ 1 ) = CF + (Σ 0 , Σ 1 ) ⊕ C(Λ - 0 , Λ - 1 ) ⊕ CF -(Σ 0 , Σ 1 ), the differential takes the form (40) d -∞ =   d 0 + 0 + d 0 + -d 0 + 0 - 0 d --d -0 - 0 0 d 0 -0 -  
of an upper-triangular matrix.

Proof. This follows from the energy estimates of Section 3.3.

This decomposition allows us to prove Theorem 1.3.

Proof of Theorem 1.3. From Equations ( 33) and ( 40), the decomposition

Cth(Σ 0 , Σ 1 ) = C(Λ + 0 , Λ + 1 ) ⊕ CF + (Σ 0 , Σ 1 ) ⊕ C(Λ - 0 , Λ - 1 ) ⊕ CF -(Σ 0 , Σ 1 ) induces a filtration for the differential d ε - 0 ,ε - 1 .
The first page thus takes the prescribed form by construction. The fact that (Cth(Σ 0 , Σ 1 ),

d ε - 0 ,ε - 1
) is acyclic implies that the associated spectral sequence collapses on the fourth page.

Note that if either all intersection points have positive actions or they all have negative actions, then the differential described in Formula ( 40) is a mapping cone from or to C(Λ - 0 , Λ - 1 ). This motivates the following definition for a pair (Σ 0 , Σ 1 ) of cobordisms. Definition 9.2. We say that (Σ

0 , Σ 1 ) is directed if CF + (Σ 0 , Σ 1 ) = 0, while it is called V -shaped if CF -(Σ 0 , Σ 1 ) = 0.
We are now able to prove Corollary 1.5.

Proof of Corollary 1.5. For directed cobordisms, from Equation [START_REF] Eliashberg | Lagrangian caps[END_REF] we get that the differential on CF -∞ (Σ 0 , Σ 1 ) is the cone of the map d -0 : CF (Σ 0 , Σ 1 ) → C(Λ - 0 , Λ - 1 ). Thus we get a long exact sequence

• • • / / HF k -∞ (Σ 0 , Σ 1 ) HF k -(Σ 0 , Σ 1 ) / / LCH k ε - 0 ,ε - 1 (Λ - 0 , Λ - 1 ) / / HF k+1 -∞ (Σ 0 , Σ 1 ) / / • • •
For the case of V -shaped cobordisms we get that the differential on CF -∞ (Σ 0 , Σ 1 ) is the cone of the map d 0-: C(Λ - 0 , Λ - 1 ) → CF (Σ 0 , Σ 1 ) which leads to the following exact sequence 39) concludes the proof.

• • • / / HF k+1 -∞ (Σ 0 , Σ 1 ) / / LCH k ε - 0 ,ε - 1 (Λ - 0 , Λ - 1 ) / / HF k+2 + (Σ 0 , Σ 1 ) HF k+2 -∞ (Σ 0 , Σ 1 ) / / • • • In both cases the isomorphism LCH k ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) ≃ HF k+1 -∞ (Σ 0 , Σ 1 ) of equation (

Long exact sequences induced by an exact Lagrangian cobordism

Here we establish the long exact sequences described in Section 1.2 associated to a Lagrangian cobordism Σ. In order to do this, we need to make use of the invariance result proven in the former section. More precisely, the sought long exact sequence will be induced by Corollary 1.5 applied to a pair (Σ 0 , Σ 1 ) of exact Lagrangian cobordisms, where Σ 0 := Σ and Σ 1 is obtained from Σ by a suitable Hamiltonian perturbation. 10.1. Various push-offs of an exact Lagrangian cobordism. In the following, we assume that we are given an exact Lagrangian cobordism Σ ⊂ R × P × R from Λ -to Λ + inside the symplectisation of a contactisation. We furthermore assume that Σ is cylindrical outside of the set [A, B] × P × R for some A < B. We shall write

Σ := Σ ∩ {t ∈ [A, B]}, ∂ -Σ := Σ ∩ {t = A}, ∂ + Σ := Σ ∩ {t = B}, so that clearly ∂Σ = ∂ -Σ ∪ ∂ + Σ.
We will consider different push-offs constructed via autonomous Hamiltonians h : R×P ×R → R induced by a function h(t) only depending on the symplectisation coordinate. We write Σ h := φ 1 h (Σ) and observe that the Hamiltonian flow takes the particularly simple form φ s h (t, p, z) = (t, p, se -t h ′ (t) + z). In particular, the one-form e t α pulls back to e t α + (h ′′h ′ )dt under φ 1 h . So, if Σ 0 is an exact Lagrangian cobordism with a primitive f 0 : Σ 0 → R of Σ * 0 e t α, then the primitive of (φ 1 h (Σ 0 )) * (e t α) is given by [START_REF] Eliashberg | Local Lagrangian 2-knots are trivial[END_REF] f 0 (q) = f 0 (q) + (h ′h)(a(Σ 0 (q))),

where a is the canonical projection to R t , and Σ 0 : Σ 0 ֒→ R × P × R denotes the inclusion. Assuming that h coincides with λe t near -∞ for a constant λ ∈ R (i.e. h ′ = h), the primitive f 0 vanishes at -∞ if and only if f 0 vanishes there.

We are now ready to define the Hamiltonians h : Σ → R needed for the different perturbations needed. The long exact sequences of Theorems 1.6, 1.7 and 1.8 are then derived by combining Corollary 1.5 applied to (Σ, Σ h ) with Theorems 10.3 and 10.5 proven below.

h dir (t) A B 1 1 t t e -t h ′ dir (t) A B
Figure 18. The Hamiltonian h dir : R × P × R → R applied to an exact Lagrangian cobordism Σ produces a directed pair. The corresponding Hamiltonian vector field is given by e -t h ′ dir (t)∂ z .

10.1.1. The push-off inducing the long exact sequence of a pair. Consider the Hamiltonian h dir : R × P × R → R depending only on the t-coordinate and satisfying • h dir (t) = e t for t ≤ 0;

• h dir (t) = e t -C for all t ≥ B + 1, and some C ≥ 0; and

• (h ′ dir ) -1 (0) is a connected interval containing [A, B] (in particular h dir is constant on [A, B].
See Figure 18 for a schematic picture of h dir (t) and the corresponding Hamiltonian vector field e -t h ′ dir (t)∂ z . Let Σ ′ := Σ ǫh dir , for ǫ > 0 sufficiently small, and take Σ 1 to be a generic sufficiently small and compactly supported Hamiltonian perturbation of Σ ′ . Let f 0 be the primitive of Σ * (e t α) which vanishes at -∞. From Equation [START_REF] Eliashberg | Local Lagrangian 2-knots are trivial[END_REF] and from the fact that h > 0 and h ′ = 0 on [A, B] we get that the primitive of (Σ ′ ) * (e t α) is smaller than f 0 on [A, B] × Y . Writing Σ 0 := Σ, for ǫ > 0 sufficiently small, all intersections between Σ 0 and Σ 1 are contained inside [A, B] × Y . Thus, if the Hamiltonian perturbation of Σ ′ is small enough, we get that CF + (Σ 0 , Σ 1 ) = 0 or, using the terminology of Definition 9.2, that the pair of cobordisms (Σ 0 , Σ 1 ) is directed.

Furthermore, under the assumption that Σ 1 is a sufficiently C 1 -small perturbation of Σ 0 , and that the almost complex structures are chosen appropriately, Theorem 1.6 now follows by applying Theorems 10.3 and 10.5 to the terms in the long exact sequence (2) of Corollary 1.5 (also see Remark 10.6).

h V (t) A B 1 -1 -1 t t e -t h ′ V (t) A B
Figure 19. The Hamiltonian h V : R × P × R → R applied to an exact Lagrangian cobordism Σ produces a V -shaped pair. The corresponding Hamiltonian vector field is given by e -t h ′ V (t)∂ z .

10.1.2. The push-off inducing the duality long exact sequence. Consider a Hamiltonian h V : R × P × R depending only on the t-coordinate and satisfying • h V (t) = -e t for t ≤ A -1;

• h V (t) = e t -C for all t ≥ B + 1, and some C > 0; and • (h ′ V ) -1 (0); is a connected interval containing [A, B]. See Figure 18 for a schematic picture of h V (t) as well as of the corresponding Hamiltonian vector field e -t h ′ V (t)∂ z . Let Σ ′ := Σ ǫhV , for ǫ > 0 sufficiently small, and let Σ 1 be a generic, sufficiently C 1 -small, and compactly supported Hamiltonian perturbation of Σ ′ . Again, from Equation (41), we deduce that the primitive of e t α on Σ ′ is equal to

f 0 +(h ′ V -h V )• a • Σ,
which is this time greater than f 0 on [A, B]. In conclusion, CF -(Σ 0 , Σ 1 ) = 0 or, using the terminology of Definition 9.2, (Σ 0 , Σ 1 ) is a V -shaped pair.

When Σ 1 is a sufficiently small perturbation of Σ 0 , and the almost complex structures are chosen appropriately, Theorem 1.7 follows by applying Proposition 10.1 and Theorems 10.3 to the terms in the long exact sequence (3) of Corollary 1.5 (also see Remark 10.6). 10.1.3. The push-off inducing the Mayer-Vietoris long exact sequence. The Hamiltonian h V considered above also gives rise to the long exact sequence in Theorem 1.8. It is simply a matter of applying Theorem 1.3 to a different filtration of the same complex.

Consider the filtration

C(Λ + 0 , Λ + 1 ) > C + (Λ - 0 , Λ - 1 ) ⊕ CF + (Σ 0 , Σ 1 ) > C 0 (Λ - 0 , Λ - 1 )
where the decomposition

C(Λ - 0 , Λ - 1 ) = C + (Λ - 0 , Λ - 1 ) ⊕ C 0 (Λ - 0 , Λ - 1 )
has been made so that the left summand is generated by those Reeb chords corresponding to Reeb chords on Λ -(being of length bounded from below by the minimal length of a Reeb chord on Λ -, i.e. of some significant length), while the right summand is generated by those Reeb chords corresponding to the critical points of a Morse function on Λ -(being of length roughly equal to ǫ > 0, which thus may be assumed to be arbitrarily small). To see this, observe that we have Λ - 0 = Λ -, and that Λ - 1 is obtained as the one-jet j 1 g of a negative Morse function g : Λ -→ (-ǫ, 0) in some standard contact neighbourhood of Λ -, where the latter is identified with a neighbourhood of the zero-section in J 1 Λ -. For simplicity, we here make the assumption that there is a unique local minimum and maximum of g.

The above filtration induces a spectral sequence whose first page is given by

E 2,• 1 ⊕ E 1,• 1 ⊕ E 0,• 1 LCH •-2 ε + 0 ,ε + 1 (Λ + 0 , Λ + 1 ) ⊕ (LCH •-1 ε - 0 ,ε - 1 (Λ -) ⊕ HF + • (Σ 0 , Σ 1 )) ⊕ H • (Λ -),
where we rely on [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF] together with Proposition 10.1 in order to make the identifications

H i (C • + (Λ - 0 , Λ - 1 )) = LCH i ε - 0 ,ε - 1 (Λ -), H i (C • -(Λ - 0 , Λ - 1 )) = H Morse n-1-i (g).
Note that the middle term on the first page of the above spectral sequence really is a direct sum. This follows from the fact that the positive intersection points may be assumed to have arbitrarily small action and hence, in particular, action smaller than the length of any Reeb chord generator of C + (Λ - 0 , Λ - 1 ). Consequently, there are no pseudoholomorphic strips as in the definition of the differential with input being a Reeb chord in C + (Λ - 0 , Λ - 1 ) and output being an intersection point contained in CF + (Σ 0 , Σ 1 ), or vice versa.

The technique in the proof of [START_REF] Rizell | Lifting pseudoholomorphic polygons to the symplectisation of P × R and applications[END_REF]Theorem 6.2(ii)] shows that the Cthulhu differential restricts to the natural map

H n-1-i (Λ -) ≃ H i (C • -(Λ - 0 , Λ - 1
)) → H n-1-i (Σ) in homology induced by the topological inclusion Λ -֒→ Σ. To that end, we start by considering a perturbation h V (t) of h V (t) being of the form as shown in Figure 20. Note that any sufficiently C 1 -small perturbation f : Σ → R of the restriction h V | Σ has the property that the map H • (Λ -) → H • (Σ) can be realised as the inclusion of the Morse homology complex generated by the critical points near {t = a} into the full Morse homology complex of f . Also, see Section 11.3.2 for a similar analysis. It then suffices to show that the differential is equal to the identity map

H n-1-i (Λ -) ≃ H i (C • -(Λ - 0 , Λ - 1 )) → H n-1-i (Λ -) under the appropriate natural identifications.
Finally, the statements concerning the fundamental classes is a consequence of [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF]Theorem 5.5], a result which is only valid if we are working with ε - 0 = ε - 1 = ε. Namely, the latter result shows that the minimum of -g defines a nonvanishing cycle inside LCH -1 ε,ε (Λ - 0 , Λ - 1 ). Similarly, under the additional assumption that Λ -

h V (t) a A B 1 -1 -1 t t e -t h ′ V (t) a A B
Figure 20. The Hamiltonian h V : R × P × R → R applied to an exact Lagrangian cobordism Σ produces a V -shaped pair. The corresponding Hamiltonian vector field is given by e -t h ′ V (t)∂ z .

is horizontally displaceable, the differential of the maximum of -g is a non-zero class in With the above considerations, we have now managed to establish Theorem 1.8.

H n (C • + (Λ - 0 , Λ - 1 )) ≃ LCH n ε,ε (Λ -) called the

10.2.

Computing the Floer homology of a Hamiltonian push-off. We now consider the pair (Σ 0 , Σ 1 ), where Σ 0 = Σ and Σ 1 is equal to Σ h for a C 1 -small smooth function h : R → R as constructed above.

To that end, we will assume that J P is the cylindrical lift of a regular almost complex structure J P on P , i.e. the unique cylindrical almost complex structure on R × P × R for which the canonical projection to P is ( J P , J P )-holomorphic.

Recall that the time-s Hamiltonian flow generated by the Hamiltonian e t on the symplectisation R × P × R simply is a translation of the z-coordinate by s. Recall that this is the same as the time-s flow φ s : P × R of the Reeb vector field induced by the standard contact form.

Proposition 10.1. For the cylindrical lift J P of an almost complex structure, we have a canonical isomorphism

LCC • ε0,ε1 (Λ, φ ǫ (Λ ′ )) ≃ LCC • ε0,ε1 (
Λ) of complexes for any sufficiently small epsilon > 0, and generic and sufficiently C 1 -small Legendrian perturbation Λ ′ of Λ.

Under the additional assumption that Λ is horizontally displaceable, we moreover have a quasi-isomorphism

LCC • ε0,ε1 (Λ, φ -ǫ (Λ ′ )) ∼ LCC ε0,ε1
n-1-• (Λ) of complexes, where n is the dimension of Λ.

Theorem 10.5. For an appropriately chosen Maslov potential and for any generic smooth function f : Σ → R obtained as a sufficiently C 1 -small and compactly supported perturbation of h| Σ ′ : Σ → R, there is an induced Hamiltonian perturbation Σ ′ of Σ inducing a canonical identification of complexes

CF • (Σ, φ ǫ h (Σ ′ )) = C Morse n+1-• (f ).
Here we require that the almost complex structure is regular, admissible, and induced by a Riemannian metric on g for which (f, g) is a Morse-Smale pair in some compact neighbourhood of the non-cylindrical part of Σ. If Σ is pin, then the Hamiltonian perturbation admits an pin structure for which the above identification holds with coefficients in Z.

Proof. Observe that the Weinstein Lagrangian neighbourhood theorem implies that there is a symplectomorphism identifying a neighbourhood of Σ with a neighbourhood of T * Σ, such that Σ moreover gets identified with the zero-section; see Section 11.3.2. For an appropriate such identification, we may assume that φ ǫ h (Σ) = φ 1 ǫh (Σ) is contained in this neighbourhood and that this Lagrangian is identified with the section -ǫd(h| Σ ). In particular, every intersection point φ 1 ǫh (Σ) ∩ Σ corresponds to a critical point of ǫh and may be assumed to have action corresponding to ǫh. After replacing ǫh with ǫf , we can choose Σ ′ to be the exact Lagrangian cobordism given as the graph of ǫdf , while using the above Weinstein neighbourhood.

The isomorphism of Morse and Floer complexes is standard, going back to the original work made by Floer [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF]. For the current setting, the analogous computation made in [START_REF] Rizell | Lifting pseudoholomorphic polygons to the symplectisation of P × R and applications[END_REF]Theorem 6.2] is also relevant.

Remark 10.6. Consider the autonomous Hamiltonian h in Theorem 10.5 above, together with the induced perturbation f of h| Σ .

(1) If h = h dir as defined in Section 10.1.1, then

H(C Morse n+1-• (f )) = H n+1-• (Σ, ∂ -Σ; R) ≃ H • (Σ, ∂ + Σ; R).
(2) If h = h V as defined in 10.1.2, then

H(C Morse n+1-• (f )) = H n+1-• (Σ; R).
G ε - 0 ,ε - 1 Σ : C Morse n+1-• (f ) → LCC •-1 ε + 0 ,ε + 1 (Λ + , φ ǫ (Λ + ); R) = LCC •-1 ε + 0 ,ε + 1 (Λ + ; R),
identified with the term d +0 in the differential of (Cth(Σ,

φ ǫ f (Σ)), d ε - 0 ,ε - 1
). Observe that, since the pair of cobordisms is V -shaped, we can conclude that Lemma 10.7. The above map G

ε - 0 ,ε - 1 Σ
is a chain map which, in the case when the negative end of Σ is empty, is a quasi-isomorphism.

Twisted coefficients, L 2 -completions, and applications

In order to deduce information about the fundamental group of an exact Lagrangian cobordism it is necessary to introduce a version of our Floer complex coefficients twisted by the fundamental group, analogous to that defined for Lagrangian Floer homology in [START_REF] Sullivan | K-theoretic invariants for Floer homology[END_REF] by Sullivan and in [START_REF] Damian | Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds[END_REF] by Damian. Since it is not possible to make sense of the rank of a general module with group ring coefficients, it will also be necessary to introduce a version of an L 2 -completion of this complex. So-called L 2 -coefficients were first considered by Atiyah in [START_REF] Atiyah | Elliptic operators, discrete groups and von Neumann algebras[END_REF]. We start by describing the version of the complex with twisted coefficients, and we also introduce a version of the fundamental class in this setting. The fundamental class will be crucial for the proof of Theorems 1.16 (see Section 12.3.1). We then continue by defining the L 2 -completion of this complex, for which we recall some basic properties. The proof of Theorem 1.18 will use this theory (see Section 12.3.2). 11.1. Floer homology with twisted coefficients. First, we introduce the algebraic setup needed in order to define the Chekanov-Eliashberg algebra, along with its linearisations, in the setting of twisted coefficients. We refer the reader to [START_REF] Chantraine | Noncommutative augmentation categories[END_REF] by the authors for a more detailed treatment. Observe that M ⊗ A N has the structure of a A -A-bimodule, where left and right multiplications are given by the formula a(m ⊗ n)a ′ = (am) ⊗ (na ′ ) with m ∈ M , n ∈ N , and a, a ′ ∈ A. In order to avoid confusion, we would like to emphasise that in the bimodule M ⊗ A N elements of the form m ⊗ n and m ⊗ an are in general not in relation to each other.

Given a A -A-bimodule M , the tensor ring of M is the graded ring

T A (M ) = ∞ k=0 M ⊗Ak ,
where M ⊗A0 = A, and

M ⊗Ak = M ⊗ A M ⊗Ak-1 . Multiplication in T A (M ) is
induced by the natural isomorphism from M ⊗Al ⊗ A M ⊗Am to M ⊗Al+m . Observe that T A (M ) contains a subring M ⊗A0 = A as well as a A-A-bimodule M ⊗A1 = M , and that T A (M ) is the universal ring satisfying these properties. (Note that, as A is not necessarily commutative, T A (M ) is not an algebra according to standard terminology. This is the reason why we refer to it as the tensor ring).

If M is freely generated by elements {γ 1 , . . . , γ k } as a bimodule, homogeneous elements of T A (M ) are generated by elements of the form

a 1 γ i1 ⊗ a 2 γ i2 ⊗ • • • ⊗ a j γ ij a j+1 ,
where a 1 , . . . , a j+1 ∈ A. In most cases A will be the group ring R[π] over a commutative ring R. In this situation, the tensor ring of M over A = R[π] will be denoted by T π (M ) for simplicity. 11.1.2. The Chekanov-Eliashberg algebra with twisted coefficients. Legendrian contact homology with twisted coefficients has previously been considered in [START_REF] Ekholm | Exact Lagrangian immersions with one double point revisited[END_REF], and a detailed account is currently under development in [START_REF] Östman | Legendrian contact homology with homotopy coefficients[END_REF]. Here we consider a version of the Chekanov-Eliashberg algebra for a Legendrian submanifold Λ ⊂ P × R with twisted coefficients, constructed as tensor ring over the group ring R[π 1 (Λ)], as defined above, where R is a commutative ring.

Fix a base point * ∈ Λ and write π 1 (Λ) := π 1 (Λ, * ) for short. Let A be a unital, not necessarily commutative, ring for which:

• There is a ring homomorphism i : R[π 1 (Λ)] → A. This induces an R[π 1 (Λ)]-R[π 1 (Λ)]-bimodule structure on A; • There is an augmentation homomorphism a : A → R such that Π := a • i is the standard augmentation Π : R[π 1 (Λ)] → R.

By abuse of notation we will see any element a in R[π 1 (Λ)] as an element of A by identifying it with its image under the ring homomorphism i. For example, take any group homomorphism π 1 (Λ) → G, this induces a ring homomorphism R[π 1 (Λ)] → R[G] and the augmentation corresponds to the standard ring homomorphism R[G] → R. When G = {1}, the construction we describe below will recover the standard Chekanov-Eliashberg DGA.

For any Reeb chord γ of Λ, we fix a capping path ℓ e γ (resp. ℓ s γ ) on Λ which connects the end point (resp. starting point) of γ to the base point * . (Such paths exist because we assume that Λ is connected.) Let C(Λ) be the free A -Abimodule generated by the Reeb chords of Λ. A punctured pseudoholomorphic disc u ∈ M r (γ + ; γ - 1 , γ - 2 , . . . , γ - k ) determines an element c u of C(Λ) ⊗Ak via the following procedure. Let ∂ 0 S r , . . . , ∂ k S r be the connected components of ∂S r ordered as in Section 2.3.1. We denote by p the canonical projection R × P × R → P × R from the symplectisation to the contact manifold.

• For j ∈ {1, . . . , k -1}, we denote by a j the based loop (ℓ e γ - j+1

) -1 * (p • u| ∂j ) * ℓ s γ - j ;
• For j = 0, we denote by a j the based loop ℓ e γ - 1 * (p • u| ∂0 ) * (ℓ e γ + ) -1 ; and • For j = k, we denote by a j the based loop

ℓ s γ + * (p • u| ∂ k ) * (ℓ s γ - k ) -1 .
The element c u is then given by

c u = a 0 γ - 1 a 1 ⊗ γ - 2 a 2 ⊗ • • • ⊗ γ - k a k . The Chekanov-Eliashberg differential is defined on T A (C(Λ)) by the formula ∂(γ + ) = γ1,...,γ k u∈M(γ + ;γ - 1 ,γ - 2 ,...,γ - k )
sign(u)c u on generators, where the sum is taken over the rigid components of the moduli spaces. The differential is then extended as a bimodule homomorphism to C(Λ), and ultimately to the whole tensor ring using the Leibniz rule. The DGA obtained will be denoted by

A A (Λ) (or A G (Λ) if A = R[G]).
If in this notation we omit the subscript A, then we just mean the standard Chekanov-Eliashberg algebra of Λ, which is the particular case A(Λ) := A R (Λ). By an augmentation of the Chekanov-Eliashberg DGA we mean a homomor-

phism of R[π 1 (Λ)] -R[π 1 (Λ)]-bimodules ε : T A (C(Λ)) → A being a unital ring homomorphism satisfying ε • ∂ = 0.
Remark 11.1.

(1) An augmentation in this setting is still determined by its values on the Reeb chord generators.

(2) Any homomorphism G → H of groups induces a unital DGA morphism r : A G (Λ) → A H (Λ). In particular, when H is the trivial group, we get a canonical DGA homomorphism r : A G (Λ) → A(Λ). The pre-composition ε := ε • r of an augmentation ε of A(Λ) is clearly an augmentation of A G (Λ); this augmentation will be called the lift of ε.

(3) Similarly to the definition of the differential of the DGA with twisted coefficients, an exact Lagrangian cobordism Σ from Λ -to Λ + can be seen to induce a unital DGA homomorphism Φ Σ : A π1(Σ) (Λ + ) → A π1(Σ) (Λ -) with twisted coefficients. In particular, an exact Lagrangian filling induces an augmentation in the group ring of its fundamental group.

For any pair of augmentations, the linearisation procedure gives rise to a differential d ε0,ε1 : LCC ε0,ε1

• (Λ; A) → LCC ε0,ε1
• (Λ; A) on the free A -A-bimodule spanned by the Reeb chords in the usual way. The map d ε0,ε1 is in this situation a bimodule homomorphism. We again denote the resulting homology by LCH ε0,ε1 • (Λ; A) called the bilinearised Legendrian contact homology with twisted coefficients.

In the case when Λ = Λ 0 ⊔ Λ 1 and ε i comes from an augmentation of Λ i for i = 0, 1, we can again define the sub-complex LCC • ε0,ε1 (Λ 0 , Λ 1 ; A) which is the free right A-module spanned by the Reeb chords starting on Λ 1 and ending on Λ 0 , as well as the corresponding cohomology groups LCH • ε0,ε1 (Λ 0 , Λ 1 ; A). The result [17, Proposition 2.7] carries over immediately to this setting, and thus the identification

LCH • ε0,ε1 (Λ, Λ ′ ; A) = LCH • ε0,ε1 (
Λ; A) holds on the level of homology (again, for a suitable small push-off Λ ′ of Λ, together with a suitable lifted almost complex structure).

Remark 11.2. Note that, if ε 1 takes values in R, then the left action of ε 1 (γ) on the free module is the same as the right action. In this case, the whole module structure factors through a complex defined as a free left R[π 1 (Σ 0 )]-module generated by the Reeb chords. This is relevant for the next section, where we will twist coefficients using R[π 1 (Σ 0 )] in a way so that the augmentation ε 1 still will take values in R.

11.1.3. The Floer complex with twisted coefficients. Let R be a unital commutative ring. We are now ready to define our Floer complex for a pair (Σ 0 , Σ 1 ) of exact Lagrangian cobordisms with twisted coefficients, defined as an R[π 1 (Σ 0 )] -R[π 1 (Σ 0 )] bimodule. For Lagrangian intersection Floer homology, such a construction has previously been carried out in [START_REF] Sullivan | K-theoretic invariants for Floer homology[END_REF] and [START_REF] Damian | Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds[END_REF]. This was subsequently generalised to the case of Wrapped Floer homology in [17, Section 4.2], i.e. for a pair of Lagrangian cobordisms having empty negative ends. Here we will define this theory for a pair of exact Lagrangian cobordisms having non-empty negative ends whose Chekanov-Eliashberg algebras admit augmentations.

As before, we let Σ i ⊂ R × P × R be exact Lagrangian cobordisms from Λ - i to Λ + i , i = 0, 1, where both Σ 0 and Λ - 0 are assumed to be connected. We consider the non-free R

[π 1 (Λ ± 0 )] -R[π 1 (Λ ± 0 )]-bimodule R[π 1 (Σ 0 )
] with structure coming from the ring homomorphism induced by the inclusion maps {±T } × Λ ± 0 → Σ 0 . In order to obtain this bimodule structure, the base point of Σ 0 need to be chosen of the form (-T, * ), where * is the based point of Λ - 0 . We then choose, once and for all, a path connecting (-T, * ) and (T, * ′ ), where * ′ is the based point of Λ + 0 .

makes sense, since the latter modules are of dimension (dim F A 0 )(dim F A 1 ) times the number of generators.

To construct the differentials in this setting one proceeds as in [START_REF] Dimitroglou Rizell | Estimating the number of Reeb chords using a linear representation of the characteristic algebra[END_REF], which also is analogous to the construction of the complexes in Section 11.1 with twisted coefficients. In particular, the differentials are of the form

d((a 0 ⊗ a 1 )x) = y δ -,ζ - #M(y; δ -, x, ζ -) • ε 0 (δ -)a 0 ⊗ a 1 ε 1 (ζ -) • y,
where x and y denote either intersection points or Reeb chords, and where a i ∈ A i , i = 0, 1.

Remark 11.5. This convention tells us that the differential is defined by multiplication of A 0 ⊗ A op 1 from the left, and is hence a morphism of right modules. The long exact sequences in homology for these bimodules now follow verbatim from the proofs in the case when the augmentation is taken into F. It is important to notice that all the complexes above are of finite dimension over F. More precisely,

dim F LCC • (Λ ± 0 , Λ ± 1 ) = |R(Λ ± 1 , Λ ± 0 )| • dim F A 0 • dim F A 1 , while dim F H • (X, Y ; A 0 ⊗ F (A 1 ) op ) = dim F H • (X, Y ; F) dim F (A 0 ) dim F (A 1 )
holds by the universal coefficients theorem.

Example 11.6. There are examples of Legendrian submanifolds which admit augmentations into finite-dimensional non-commutative unital algebras, but which do not admit any augmentation into any commutative unital algebra. We refer to Part (1) of Example 12.4 below for such Legendrian torus knots found by Sivek in [START_REF] Sivek | The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant[END_REF], which admit augmentations into the matrix algebra M 2 (Z 2 ). The second and the fourth author later used these examples in order to construct plenty of Legendrian submanifolds inside contact spaces (R 2n+1 , ξ std ) for arbitrary n ∈ N whose Chekanov-Eliashberg algebras admit augmentations into M 2 (Z 2 ), but not into any commutative algebra. 11.3. The fundamental class and twisted coefficients. In this section we will introduce the fundamental class in the setting of twisted coefficients. We will prove that this class coincides with the fundamental class introduced in [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF] in the general twisted coefficients setting. We will also prove that this class is functorial under exact Lagrangian cobordisms. In Section 12.3.1 we will use the naturality of this class to prove Theorem 1.16. In the following we let Σ be a connected exact Lagrangian cobordism from Λ -to Λ + , where the latter Legendrian submanifolds are connected as well.

ε - 0 ,ε - 1 Σ : H • (Σ; R) → LCH n-• ε + 0 ,ε + 1 (Λ + , Λ + 1 ; R)
in homology constructed in Section 10.3, whose underlying chain map is defined by a count of punctured strips with boundary on Σ 1 ∪Σ. Here Σ 1 is an exact Lagrangian cobordism from Λ - 1 to Λ + 1 obtained by the Hamiltonian push-off of Σ as defined in Section 10.1.2. Roughly speaking, Σ 1 is obtained by a small perturbation using the positive and negative Reeb flows at the positive and negative end of Σ, respectively.

there is an identification

c ε + 0 ,ε + 1 Λ + ,m + = c ε - 0 ,ε - 1 Σ,m ∈ LCH n ε + 0 ,ε + 1 (Λ + ; R[π 1 (Σ)])
of fundamental classes.

In Section 12.3.1 we will see that Theorem 1.16 is a direct consequence of the above proposition. Its proof is postponed until Section 11.3.3, and it relies on studying a specific push-off of Σ to be describe in the subsequent subsection.

11.3.2.

A more careful version of the push-off defined in Section 10.1.2. In order to facilitate the proof of Proposition 11.7 we will choose a push-off Σ 1 of Σ of a very special form, where Σ 1 will be an exact Lagrangian cobordism from Λ - 1 to Λ + 1 . We will suppose that Σ is cylindrical outside of the subset [-T, T ] × P × R.

First we need to construct a Weinstein neighbourhood of Σ of a particular form.

Begin by fixing contact-form preserving contactomorphisms φ ± :

U ± → J 1 Λ ± identifying a neighbourhood U ± ⊂ P × R of Λ ± with a neighbourhood φ ± (U ± ) of Λ ± ⊂ (J 1 Λ ± , dz -θ Λ ± ), where φ ± (Λ ± ) = 0 Λ ± ⊂ J 1 Λ ± is the zero-section. We also get induced exact symplectomorphisms (Id R , φ ± ) : R × U ± → R × φ ± (U ± ) ⊂ R × J 1 Λ ± .
Pre-composing (Id R , (φ ± ) -1 ) with the (non-exact) symplectomorphism

ψ ± : (T * (I ± × Λ ± ), d(p i dq i )) → (R × J 1 (Λ ± ), d(e t (dz -y i dx i ))), ( (q
, q), (p, p)) → (log q, (q, -p/q, -p)), where I + = [e T , +∞) and I -= (0, e -T ], we get an induced Weinstein neighbourhood of the cylindrical ends of Σ parametrised by (Id R , (φ ± ) -1 ) • ψ ± .

Fix a proper open embedding

I -× Λ -⊔ I + × Λ + ֒→ Σ.
An adaptation of the proof of Weinstein's Lagrangian neighbourhood theorem (see e.g. [START_REF] Mcduff | Introduction to symplectic topology. Oxford Mathematical Monographs[END_REF]) shows that we can construct a symplectic identification Ψ : (D * δ Σ, d(p i dq i )) ֒→ (R × P × R, d(e t (dz + θ))) of a co-disc bundle of some small radius δ > 0, for which the zero-section is identified with Ψ(0 Σ ) = Σ ⊂ R × P × R, and whose restriction to

(D * δ (I -× Λ -⊔ I + × Λ + ), d(p i dq i )) ⊂ (D * δ Σ, d(p i dq i ))
, coincides with the symplectomorphisms (Id R , (φ ± ) -1 ) • ψ ± constructed above. For the definition of the above disc bundle we will pick a metric on the cotangent bundle T * Σ which is induced by a Riemannian metric on Σ being of the form dq⊗dq+q -1 g ± on the ends I ± × Λ ± , where g ± denote Riemannian metrics on Λ ± .

We are now ready to describe the construction of the Hamiltonian push-off Σ 1 of Σ, which is done by performing the push-off in Section 10.1.2 while taking extra care.

First, we choose a C 1 -small push-off Σ ′ 1 of Σ by applying the time-ǫ flow of the Hamiltonian vector-field e -t h ′ Λ (t)∂ z defined in Section 10.1.2. Here ǫ > 0 is chosen smaller than the shortest Reeb chords on both ends Λ ± . (Recall that this flow coincides with the positive and negative Reeb flow at the positive and negative end, respectively.) Second, we perform a non-compact perturbation of the cylindrical ends of Σ ′ 1 induced by cylindrically extending a perturbation of the Legendrian ends (where this perturbation has been appropriately cut off). Finally, we perform a compactly supported Hamiltonian perturbation of Σ ′ 1 yielding the sought exact Lagrangian cobordism Σ 1 . These perturbations will moreover be performed so that the following properties are satisfied.

(1) Under the above identifications of standard contact neighbourhoods of Λ ± , we require that φ ± (Λ ± 1 ) = ±j 1 f ± ⊂ J 1 Λ ± for positive Morse functions f ± : Λ ± → (0, ǫ]. We moreover require that f + has a unique local minimum m + . (This is possible since Λ + is connected by assumption.) (2) Under the above identification of a Weinstein neighbourhood of Σ, we require that Ψ -1 (Σ 1 ) = -dF ⊂ T * Σ for a Morse function F : Σ → R. We moreover require that F has a unique local minimum m (note the sign!).

(This is possible since Σ is connected and since F increases as |t| → +∞ along either of the cylindrical ends.) (3) Above I -× Λ -, the function F is required to restrict to a function of the form

F | I-×Λ -= -qf -,
where q denotes the standard coordinate on I -= (0, e -T ] ⊂ R. (4) Above I + × Λ + ⊂ Σ, the function F restricts to a function of the form

F | I+×Λ + = qf + + C(q)
by construction, where q is the standard coordinate on I + = [e T , +∞) ⊂ R and C(q) is constant outside of a compact set. We moreover require that C : R → R satisfies: (a) C(q) = -2(max Λ + f + ) • q < min I-×Λ -F < 0 holds near q = e T ; (b) C(q) ≡ 0 for all q ≫ 0 sufficiently large; (c) C ′ (q) is non-decreasing; (d) C ′ (e T ′ ) = -f + (m + ), i.e. (e T ′ , m + ) ∈ I + × Λ + is a critical point being a local minimum for some T ′ > T ; and (e) C ′′ (e T ′ ) > 0, i.e. this critical point is a non-degenerate local minimum. Note that, in particular, we require that the unique local minimum of F is given by the above critical point m = (e T ′ , m + ) ∈ I + × Λ + ⊂ Σ. Also, see Figure 21. In order to see that we indeed can find the above function F , we observe that a function defined on the subsets I -× Λ -∪ I + × Λ + ⊂ Σ satisfying the above requirements (3) and ( 4) can be extended to a Morse function F : Σ → R without introducing additional local minima, as follows by considering property (4a).

f + (m + ) f + (m + ) -2 max Λ + f + q f + (m + ) + C ′ (q) e T e T ′ Figure 21. The graph of the differential ∂ q F along [e T , +∞) × {m + } ⊂ I + × Λ + .
Observe that F | {q}×Λ + has a non-degenerate local minimum at (q, m + ) for each q ∈ I + .

Lemma 11.8. Let Σ 1 be the Hamiltonian push-off of Σ described by all the previous conditions. For a suitable choice of almost complex structure J there is a unique and transversely cut out J-holomorphic disc having boundary on Σ ∪ Σ 1 and a single positive puncture asymptotic to γ m + . This disc is moreover a rigid strip having precisely two punctures, where the second puncture maps to the intersection point m = (e T ′ , m + ) ∈ Σ ∩ Σ 1 .

Proof. We will choose an admissible almost complex structure J on R × P × R which, on the subset [T, +∞) × P × R, will be the cylindrical lift of a compatible almost complex structure J P on (P, dθ) as described in Section 3.1.3. First we show that a punctured disc as in the assumption cannot pass through the hypersurface {t = T } for a suitable choice of almost complex structure J. Namely, consider a neck-stretching limit around this hypersurface, and observe that

{t = T } ∩ (Σ ∪ Σ 1 ) = R × (Λ + ∪ φ -2 max Λ + f + (Λ + 1
)) holds by property (4a) above. (Here φ s (t, x, z) = (t, x, z + s) denotes the Reeb flow as usual.) By action reasons it thus follows that no such strip can pass through this hypersurface for a sufficiently stretched almost complex structure, since otherwise we would get a component of negative energy in the SFT limit. Observe that stretching the neck in this setting can be equivalently performed by fixing the almost complex structure J, but while changing the boundary condition in a way so that T ′ ≫ 0 above becomes arbitrarily large.

We may hence assume that any disc as in the assumption is contained in {t ≥ T }. The canonical (J, J P )-holomorphic projection [T, +∞) × P × R → P maps {t ≥ T } ∩ (Σ ∪ Σ 1 ) to the Lagrangian projection Π Lag (Λ + ∪ Λ + 1 ) ⊂ (P, dθ). The fact that the positive puncture of the projection of the disc in the assumption is mapped to m + ∈ Π Lag (Λ + ) ∩ Π Lag (Λ + 1 ) implies that this disc must have a constant projection to P . In other words, the disc is contained inside the J-holomorphic plane [T, +∞) × {m + } × R ⊂ R × P × R.

Finally, it can be checked by hand that there exists a unique J-holomorphic strip contained in the above plane [T, +∞) × {m + } × R having boundary on Σ ∪ Σ 1 ; see Figure 21 for Λ + ,m + (using the canonical basis of the Reeb chord generators). Recall that this coefficient is given by the count of rigid punctured strips inside the moduli spaces of the form M(γ; δ, m + , ζ), where each strip is counted with the weight ε + 0 (δ)ε + 1 (ζ). Consider punctured Floer strips with boundary on Σ 0 ∪ Σ 1 having precisely two positive punctures asymptotic to Reeb chords: one being the Reeb chord m + from Λ + 0 to Λ + 1 corresponding to the minimum of the Morse function f , and one corresponding to the above Reeb chord γ from Λ + 1 to Λ + 0 . By the non-negativity of the Fredholm index for a generic almost complex structure, together with the positivity of the energy, the compactification of this moduli space a priori consists of pseudoholomorphic buildings of the following form:

(1) (2) (3) (4) (5) 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 Λ + ∪ Λ + 1 Σ ∪ Σ 1 Λ -∪ Λ - 1 Figure 22.
The pseudoholomorphic buildings (1)-( 5) described in the proof of Proposition 11.7. The number on each component denotes its Fredholm index.

Not that the construction allows us to give a proof of the fact (already pointed out in [START_REF] Ekholm | Legendrian knots and exact Lagrangian cobordisms[END_REF]) that the fundamental class is functorial with respect to exact Lagrangian cobordisms. Indeed, stretching the neck in the slice {t = -T } decomposes the map G

ε - 0 ,ε - 1 Σ into Φ ε - 0 ,ε - 1 Σ • G ε - 0 ,ε - 1 R×Λ -
, where Φ Σ is the DGA morphism induced by the cobordism. Alternatively, one can also use the long exact sequence produced by Theorem 1.8 together with Proposition 11.7 in order to deduce this. In either case, we have: Theorem 11.9. [36, Theorem 7.7] Let Σ be a connected exact Lagrangian cobordism from Λ -to Λ + , and let ε - i , i = 0, 1, be augmentations of the Chekanov-Eliashberg algebra of Λ -which pull back to augmentations ε + i under the DGA morphism Φ Σ induced by Σ. It follows that

Φ ε - 0 ,ε - 1 Σ (c ε - 0 ,ε - 1 Λ -,m -) = c ε + 0 ,ε + 1
Λ + ,m + , i.e. the fundamental class is preserved under the bilinearised dual of the DGA morphism induced by Σ, under the additional assumption that the images of m ± under the natural maps H 0 (Λ ± , F) → H 0 (Σ) agree.

11.4.

A brief introduction to homology with L 2 -coefficients. We use the technology of L 2 -Betti numbers, introduced by Atiyah in [START_REF] Atiyah | Elliptic operators, discrete groups and von Neumann algebras[END_REF], as a tool to study rank properties of Legendrian contact cohomology when the coefficient ring is a group ring C[π] for a group π. In the following, π denotes the fundamental group of the cobordism. Observe that in this case π is countable. The main idea is to replace C[π], which is not a priori a Noetherian ring, with a more manageable module. Namely, we consider its L 2 -completion ℓ 2 (π) defined by the set of functions f : π → C satisfying g∈π |f (g)| 2 < ∞, endowed with its natural structure of a Hilbert space.

We do not intend to give a comprehensive introduction to the subject of L 2ranks and refer the reader to the book of Lück [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF] and the introductory paper of Eckmann [START_REF] Eckmann | Introduction to l 2 -methods in topology: reduced l 2 -homology, harmonic chains, l 2 -Betti numbers[END_REF] as the main references for the results used, but we still try to give an understandable overview of the needed techniques. The main result that we will need is a version of the snake lemma for L 2 -cohomology, due to Cheeger and Gromov [START_REF] Cheeger | Bounds on the von Neumann dimension of L 2 -cohomology and the Gauss-Bonnet theorem for open manifolds[END_REF] (also, see [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Theorem 1.21]), which applies since all our complexes are finitely generated as ℓ 2 (π)-modules.

A Hilbert π-module V is a Hilbert space on which π acts by isometry. It is said to be of finite type if it can be realised as a closed subspace of ℓ 2 (π) ⊗ C C m for a certain m ∈ N. Morphisms of Hilbert π-modules are bounded linear maps which are π-equivariant. Given an endomorphism f : V → V of a Hilbert module of finite type, we define its von Neumann trace by [START_REF] Etnyre | Legendrian and transverse twist knots[END_REF] tr

L 2 (f ) := m i=1 f (1 ⊗ e i ), 1 ⊗ e i .
Here

f := i • f • p, i : V → ℓ 2 (π) ⊗ C C m is the inclusion, p : ℓ 2 (π) ⊗ C C m → V
is the orthogonal projection, and {e i } is the standard basis of C m . A simple computation shows that this trace only depends on f and not on the particular choice of the embedding.

The von Neumann dimension of V is rk L 2 (V ) = tr L 2 Id, which clearly is a nonnegative number bounded from above by m, under the assumption that V can be embedded in ℓ 2 (π) ⊗ C C m . Note that the von Neumann dimension can take non-integer values. The following basic properties will be crucial: Lemma 11.10 (Theorem 1.12 in [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]).

(1) V = 0 if and only if rk L 2 (V ) = 0; and (2) If 0 → U → V → W → 0 is weakly exact, i.e. im i = ker p, then rk L 2 (V ) = rk L 2 (U ) + rk L 2 (W ).

We will be interested in applying this theory to a complex (C • , ∂) which is the L 2completion of a G-equivariant complex (C • , ∂) consisting of finitely generated free C[G]-modules. In this case, the L 2 -modules are all of finite type. The corresponding L 2 -homology will be denoted by H Lemma 11.11. In the above situation, we have

rk L 2 H (2) i (C • , ∂) ≤ rk C[G] C i .
Furthermore, for a finite-dimensional complex (C ′ • , ∂ ′ ) over C, we have

H (2) i (C ′ • ⊗ C[G], ∂ ′ ⊗ Id C[G] ) = H i (C ′ • , ∂ ′ ) ⊗ C ℓ 2 (
G), and thus, in particular,

rk L 2 H (2) i (C ′ • ⊗ C[G], ∂ ′ ⊗ Id C[G] ) = dim C H i (C ′ • , ∂ ′ ).
Proof. The first statement follows from Lemma 11.10 together with the Hodge decomposition in [START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF]Lemma 1.18]. The second statement follows by a direct computation.

For a pair of CW complexes (X, Y ), Y ⊂ X and a choice of homomorphism ϕ : π 1 (X) → G, there is an induced covering ( X, Y ) → (X, Y ) with fibre G, monodromy described by ϕ, and where there is a natural free G-action on the covering. In the case when (C • , ∂) is the G-equivariant cellular complex associated to such a covering, we will write the corresponding L 2 -homology groups by H

• (X, Y ; G). Observe that the covering Σ ⊙k → Σ ⊙k obtained is induced by a group epimorphism

π 1 (Σ ⊙k ) ≃ π * . . . * π k → π.
Lemma 11.12. In the case when π is finite, we have

dim F (H 1 ( Σ ⊙k , ∂ -Σ ⊙k ; F)) = (|π| -1)k, k ≥ 1.
If π := π 1 (Σ) is infinite, the L 2 -homology H

Applications and examples

In this section we deduce all applications mentioned in the introduction of the paper. In addition, we provide explicit examples of Lagrangian cobordisms: both examples to which our results apply, but also examples showing the importance of the different hypotheses used.

12.1. The homology of an endocobordism. The following proofs of Theorems 1.11 and 1.13 are similar to the proofs given in [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF].

Proof of Theorem 1.11. We begin by showing the result in the case when F = Z 2 .

(i): First, recall the elementary fact from algebraic topology that dim F H(Σ; F) ≥ dim F H(Λ; F) [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF] is satisfied, which follows by studying the long exact sequence of the pair (Σ, ∂Σ) together with Poincaré duality (see [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF]Lemma 2.1]).

We proceed to prove the opposite inequality dim where ε + is the augmentation of A(Λ; F) obtained as the pull-back ε + := ε • Φ Σ . Formula (47) implies that dim F H(Σ; F)dim F H(Λ; F) ≤ 0. Together with inequality [START_REF] Floer | Morse theory for Lagrangian intersections[END_REF], we obtain dim F H(Σ; F) = dim F H(Λ; F) [START_REF] Fukaya | Lagrangian intersection Floer theory: anomaly and obstruction. Part I[END_REF] of the dimension of the total homology.

In order to show that dim F H i (Σ; F) = dim F H i (Λ; F) for all i, we argue by contradiction, assuming that d i0 (Σ) := dim F H i0 (Σ; F)dim F H i0 (Λ; F) > 0 Proof. Let Σ ⊙k , k ≥ 1, be the k-fold concatenation of Σ with itself, which again is an exact Lagrangian cobordism from Λ to Λ. Since Λ is a homology sphere it is spin and, hence, Σ ⊙k is spin for all k ≥ 1 by Corollary 12.3.

We fix an augmentation ε of A(Λ; F) and let ε k be the augmentation of A(Λ; F) obtained by the pull-back of ε under the unital DGA morphism induced by Σ ⊙k .

The (ungraded version of the) long exact sequence in Theorem 1.6 becomes (49)

LCH • ε (Λ) / / LCH • ε k (Λ) v v ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ H • (Σ ⊙k , ∂ -Σ ⊙k ; F) h h ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗
Observe that dim H i (Σ ⊙k , ∂ -Σ ⊙k ; F) = 0, i = 0, n + 1, k dim H i (Σ, ∂ -Σ; F), 0 < i < n + 1, as follows from the Mayer-Vietoris long exact sequence together with the assumption that Λ is a F-homology sphere.

Since the linearised contact cohomology satisfies the bound

dim F LCH ε ′ (Λ) ≤ |R(Λ)|
for any ε ′ , we get the inequality

k dim H i (Σ, ∂ -Σ; F) = dim H i (Σ ⊙k , ∂ -Σ ⊙k ; F) ≤ 2|R(Λ)|, 0 < i < n + 1,
for each k, where the exactness of the above triangle has been used to show the last inequality. In conclusion, we have established dim H i (Σ, ∂ -Σ; F) = 0, 0 < i < n + 1, which finishes the proof.

Proof of Theorem 1.13. Since Λ is assumed to have an augmentation over Z it admits an augmentation over Q as well. And thus it follows from Theorem 12.1 that H • (Σ, Λ; Q) = 0 and thus that H • (Σ, Λ; Z) is torsion. The augmentation over Z also induces an augmentation over any finite field, and thus Theorem 12.1 implies that H • (Σ, Λ; Z) has no p-torsion for any prime p. Thus H • (Σ, Λ; Z) = 0.

Remark 12.2. Following the discussion in Section 11.2 we get that Theorem 1.13 holds under the weaker assumption that the Chekanov-Eliashberg algebra admits a non-commutative augmentation in a finite-dimensional F-algebra. (The proof is a verbatim reproduction of the precedent.)

12.2. Characteristic classes of endocobordisms. Recall that from Section 1.1.1, that Theorem 1.11 still applies in the case when the cobordism Σ is not orientable and of Maslov number one i.e. when the Cthulhu complexes involving Σ necessarily are ungraded. In this case, we obtain exact triangles instead of long exact sequences.

Assume now that we are given a chord-generic orientable Legendrian submanifold Λ ⊂ P ×R whose Chekanov-Eliashberg algebra admits an augmentation over Z 2 (or, more generally, a linear m-dimensional representation over Z 2 ). The dual statement of Part (iii) of Theorem 1.11 reads as follows. Let Σ be an exact Lagrangian endocobordism Σ of Λ. The map (i * + , i * -) : H * (Σ, Z 2 ) → H * (Λ ⊔ Λ, Z 2 ) is injective.

Theorem 1.14 is an immediate corollary of this and of the naturality of characteristic classes. In turn, it implies the following:

Corollary 12.3. If Λ is orientable (respectively, spin), and that it admits an augmentation into a finite-dimensional algebra, then any exact endocobordism Σ of Λ is orientable (respectively, spin) as well.

This result can be seen as a generalisation of the result of Capovilla-Searle and Traynor, see [START_REF] Capovilla-Searle | Non-orientable Lagrangian cobordisms between Legendrian knots[END_REF]Theorem 1.2]. The proof of Theorem 1.14 for Pontryagin classes follows similarly assuming that Λ is spin. Example 12.4. Recall that a Legendrian knot in the standard contact R 3 for which the Kauffman bound on tb is not sharp does not admit an augmentation in a commutative ring [START_REF] Rutherford | Thurston-Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond[END_REF].

(1) Consider a family of the Legendrian representatives of torus (p, -q)-knots Λ (p,-q) ⊂ R 3 with q > p ≥ 3 and p odd; see Figure 23. Following Sivek [START_REF] Sivek | The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant[END_REF], we observe that tb(Λ (p,-q) ) = -pq and, hence, from the classification result of Etnyre and Honda [START_REF] Etnyre | Knots and contact geometry. I. Torus knots and the figure eight knot[END_REF] it follows that Λ (p,-q) is tb-maximising. Recall that Sivek [START_REF] Sivek | The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant[END_REF] proved that the Chekanov-Eliashberg algebra of Λ (p,-q) admits a 2-dimensional representation over Z 2 , but for which the Kauffman bound on tb is not sharp. Therefore, these Legendrian knots do not admit non-orientable exact Lagrangian endocobordisms. (2) Consider Λ (p,-q) #Λ, where p is odd, q > p ≥ 3, and let Λ be a tbmaximising Legendrian knot of R 3 whose Chekanov-Eliashberg algebra admits an augmentation (or, more generally, m-dimensional linear representation) over Z 2 . Then, following the discussion in [27, Lemma 4.3], we see that the Kauffman bound for Λ (p,-q) #Λ is not sharp and that the Chekanov-Eliashberg algebra of Λ (p,-q) #Λ admits a finite-dimensional linear representation over Z 2 . In addition, from the fact that Λ (p,-q) and Λ are tb-maximising, together with [44, Corollary 3.5] (or [68, Theorem 1.1]), it follows that Λ (p,-q) #Λ also is tb-maximising. This leads us to many other examples, besides Λ (p,-q) , which do not admit non-orientable exact Lagrangian endocobordisms. (3) There is also an example due to Sivek, see [ It remains to show that |π 1 (Σ)| is finite. Assuming the contrary, we use the (weakly) long exact sequence obtained from (50) by taking the L 2 -completions of the above C[π 1 (Σ)]-equivariant complexes (since the complexes are freely and finitely generated we can again apply Cheeger and Gromov's result in [START_REF] Cheeger | Bounds on the von Neumann dimension of L 2 -cohomology and the Gauss-Bonnet theorem for open manifolds[END_REF]), establishing the exact triangle

LCH (2)• ε (Λ) / / LCH (2)• ε k (Λ) u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ H (2) • (Σ ⊙k , ∂ -Σ ⊙k ; π 1 (Σ)) i i ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙
The inequality rk L 2 (H 12.4. Explicit examples of Lagrangian cobordisms. We start by recalling a few general constructions of Legendrian submanifolds and exact Lagrangian cobordisms. Below these will be used in order to construct explicit examples of Lagrangian cobordisms. 12.4.1. A Legendrian ambient surgery on the front-spin. The front S m -spinning construction described in [START_REF] Golovko | A note on the front spinning construction[END_REF] by the fourth author constructs a Legendrian embedding Σ S m Λ ⊂ (R 2(m+n)+1 , ξ std ) of S m × Λ, given a Legendrian embedding Λ ⊂ (R 2n+1 , ξ std ). In the same article, it was also shown that the same construction can be applied to an exact Lagrangian cobordism Σ ⊂ R × R 2n+1 from Λ -to Λ + inside the symplectisation, producing an exact Lagrangian cobordism Σ S m Σ ⊂ R × R 2(n+m)+1 from Σ S m Λ -to Σ S m Λ + that is diffeomorphic to S m × Σ.

Consider a Legendrian knot Λ ⊂ (R 3 , ξ std ). Its left-most cusp edge in the front projection for a generic representative corresponds to a cusp edge diffeomorphic to S m in the front projection of the front spin Σ S m Λ ⊂ (R 2m+3 , ξ std ). Moreover, this cusp edge bounds an obvious embedding of a Lagrangian (m + 1)-disc D ⊂ (R 2(m+n)+1 , ξ std ) whose interior is disjoint from Σ S m Λ, while its boundary coincides with this cusp edge; see Figure 24.

A Legendrian ambient m-surgery, as described in [START_REF] Rizell | Legendrian ambient surgery and Legendrian contact homology[END_REF] by the second author, can be applied to the sphere S m ֒→ Σ S m Λ corresponding to the cusp edge ∂D, utilising the bounding Legendrian disc D. The Legendrian submanifold Λ + ⊂ (R 2(m+n)+1 , ξ std ) resulting from the surgery has the front projection shown in Figure 25 in the case of m = 1 = dim Λ. Recall that there also is a corresponding elementary Lagrangian (m + 1)-handle attachment, which is an exact Lagrangian cobordism from Σ S m Λ to the Legendrian submanifold Λ + obtained after the surgery. Topologically, this cobordism is simply the handle attachment corresponding to the surgery. 12.4.2. Non-simply connected exact Lagrangian fillings of Legendrian spheres (the proof of Proposition 1.22). Using the constructions in Section 12.4.1 above, the sought examples will not be difficult to produce. We start with a Legendrian knot Λ ⊂ (R 3 , ξ std ) which admits a non-simply connected Lagrangian filling Σ.

For instance, we can take the Legendrian right handed trefoil knot and its exact Lagrangian filling by a punctured torus; see [START_REF] Ekholm | Legendrian knots and exact Lagrangian cobordisms[END_REF]. It follows that Σ S m Λ ⊂ (R 2(m+1)+1 , ξ std ) is a Legendrian S m × S 1 which admits an exact Lagrangian filling Σ S m Σ diffeomorphic to S m × Σ that is not simply connected. The Legendrian ambient surgery along a cusp-edge in the class S m × {p} for p ∈ Λ corresponding to the left-most cusp edge of Λ ⊂ (R 3 , ξ std ) as described above produces a Legendrian sphere, and concatenating Σ S m Σ with the corresponding elementary Lagrangian (m + 1)-handle provides a non-simply connected filling of Σ S m Λ. These are the sought non-simply connected exact Lagrangian cobordisms.

Remark 12.6. Theorem 1.18 is applicable to the constructed sphere in order to rule out non-simply connected endocobordisms. In addition, note that since the conclusion of Theorem 1.16 is not satisfied, the Reeb chords created by the surgery are essential, and the Legendrian sphere admits at least two distinct augmentations. 12.4.3. Non-invertible Lagrangian concordances. Here we will prove the statement that Proposition 12.7. In all contact spaces (R 2n+1 , ξ std ) with n ≥ 1 there exists a Legendrian n-sphere Λ of tb = -1 which is fillable by a Lagrangian disc, but for which there is no Lagrangian concordance to the standard Legendrian sphere Λ 0 of tb = -1. (Recall that the filling induces a Lagrangian concordance from Λ 0 to Λ.)

In [START_REF] Chantraine | Lagrangian concordance is not a symmetric relation[END_REF] the first author proved that the relation of Lagrangian concordance is not symmetric by establishing the above proposition in the case n = 1. In particular, it was shown that the Legendrian representative Λ 946 ⊂ (R 3 , ξ std ) of the knot 9 46 as depicted in Figure 26 (satisfying tb = -1; this is maximal for this smooth knot class), which is fillable by a Lagrangian disc, is not concordant to the standard Legendrian unknot Λ 0 of tb = -1.

Recall that an exact Lagrangian filling by a disc can be used to construct a concordance C from Λ 0 to Λ 946 , which was explicitly described in the same article. One such concordance is described in Figure 27 below. Notice that along the entire concordance the leftmost cusp-edge p is fixed, and so we can assume that the cylinder C coincides with the trivial cylinder R × l for a small arc p ∈ l ⊂ Λ 946 inside a neighbourhood of this cusp. This fact will be important below.

Using the results in the current article, the non-existence of a concordance from Λ 946 to Λ 0 can be reproved by applying Corollary 1.20 together with the calculations in [START_REF] Chantraine | Lagrangian concordance is not a symmetric relation[END_REF]. Namely, in the latter article it is shown that, for an appropriate pair ε 0 , ε 1 of augmentations of the Chekanov-Eliashberg algebra of Λ 0 , we have LCH ε0,ε1 -1 (Λ 946 ) = 0, and no concordance going the other way can thus exist by Corollary 1.20.

The front spinning construction produces exact Lagrangian concordances Σ S m C ⊂ R × R 3+2m , obtained as the front spin of C, from Σ S m Λ 0 ⊂ (R 3+2m , ξ std ) to Σ S m Λ 946 ⊂ (R 3+2m , ξ std ). Here, the latter Legendrian submanifolds are the front spins of Λ 0 and Λ 946 , respectively. In [17, Section 5] the authors proved using the Künneth formula in Floer homology that again LCH ε0, ε1 -1 (Σ S m Λ 946 ) = 0 holds for a suitable pair of augmentations, which together with Corollary 1.20 implies that there is no Lagrangian concordance from Σ S m Λ 946 to Σ S m Λ 0 . Recall that Σ S m Λ 0 ≃ Σ S m Λ 946 ≃ S m × S 1 , while Σ S m C ≃ R × S m × S 1 . We will now perform an explicit modification of the above example to produce an example of Legendrian spheres in all dimensions which admit a concordance from the standard sphere, but which do not admit a concordance to the standard sphere; this establishes Proposition 12.7. Proof of Proposition 12.7. The Legendrian ambient surgery can be performed to the cusp-edge of the front projection of Σ S m Λ 946 corresponding to the left-most cusp edge p ∈ Λ 946 . In this way, a Legendrian sphere Λ + ⊂ (R 2(m+1)+1 , ξ std ) is produced. Since the concordance C moreover may be assumed to be a trivial cylinder over a neighbourhood of p ∈ Λ and, hence, so is Σ S m C, we obtain a Lagrangian concordance from Λ -to Λ + , where Λ -is the Legendrian sphere obtained by performing the corresponding Legendrian ambient surgery on Σ S m Λ 0 . In fact, the latter sphere is the standard Legendrian (m + 1)-sphere of tb = -1.

Recall that the Legendrian ambient surgery also produces an exact Lagrangian handle attachment cobordism from Σ S m Λ 946 to Λ + . Inspecting the long exact sequence induced by Theorem 1.6, we immediately conclude that there are augmentations ε + i , i = 0, 1 for the Chekanov-Eliashberg algebra of the Legendrian sphere Λ + satisfying

LCH ε + 0 ,ε + 1 -1 (Λ + ) ≃ LCH ε0, ε1
-1 (Σ S m Λ 946 ) = 0. Once again, Corollary 1.20 shows that there is no concordance from Λ + to Λ -.

Figure 2 . 2 . 3 . 1 .

 2231 Figure 2. A surface with strip-like ends. The incoming end is on the right, the outgoing ends are on the left.

Figure 8 .

 8 Figure 8. A schematic view of the boundary of M(p; δ, γ -, ζ).

Figure 9 .

 9 Figure 9. A boundary point in M(p; δ, q, ζ)

Figure 10 .

 10 Figure 10. A boundary point in M(γ + ; δ, γ -, ζ)

Figure 12 .

 12 Figure 12. Breakings involved in d ++ d +-+ d +0 d 0-+ d +-d --= 0. The number on each component denotes its Fredholm index.

  • d -0 d 00 + d --d -0 = 0. Analysing the breakings of holomorphic bananas, we get that the map b satisfies b • δ --= d --• b (see Figure

Figure 13 .

 13 Figure 13. Breakings involved in d 00 d 00 + d 0-d -0 = 0. The number on each component denotes its Fredholm index.

Figure 14 .

 14 Figure 14. Breakings involved in d 00 d 0-+ d 0-d --= 0. The number on each component denotes its Fredholm index.

  For action reasons we must have d -0 d 0-= 0. Finally, d --d --= 0 holds since d --is the bilinearised Legendrian contact cohomology differential, i.e. for the same reason to why d ++ d ++ = 0.

  fundamental class in Legendrian contact cohomology (also see Section 11.3).

11. 1 . 1 .

 11 The tensor ring of a bimodule. First we recall a few classical and general algebraic constructions. Let A be a (not necessarily commutative) unital ring. We start with the definition of tensor product of A -A-bimodules. Given two A -Abimodules M and N , their (balanced) tensor product M ⊗ A N is the quotient of the abelian group M ⊗ Z N by the relation ∀m ∈ M, n ∈ N, a ∈ A : ma ⊗ n = m ⊗ an.

11. 3 . 1 .

 31 The definition of the fundamental class. Recall the map

G

  a picture. This strip is transversely cut out by the explicit calculation made in [25, Lemma 8.2]. (This argument is similar to the proof of [17, Theorem 2.15].) 11.3.3. The proof of Proposition 11.7. We now proceed with the proof of Proposition 11.7. Proof of Proposition 11.7. Let l γ denote the coefficient of the Reeb chord generator γ of the fundamental class c ε + 0 ,ε + 1

( 2 )

 2 • (C • , ∂), where we note that H (2) i (C • , ∂) is defined as the quotient of the subspace of cycles by the closure of the subspace of boundaries. It follows that H (2) • (C • , ∂) again is a G-equivariant L 2 -module of finite type.

( 2 )

 2 • (X, Y ; ϕ) or, by abuse of notation, H

11. 5 .

 5 Estimating the first L 2 -Betti number of a tower. Suppose that Σ is a compact (n + 1)-dimensional manifold with boundary∂Σ = ∂ + Σ ⊔ ∂ -Σ, such that ∂ ± Σ ∼ = Λ both are simply connected. Let Σ ⊙k be the quotient of ⊔ k i=1 Σ i , Σ i ∼ = Σ, which identifies ∂ + (Σ) ⊂ Σ i with ∂ -(Σ) ⊂ Σ i+1 . We will write ∂Σ ⊙k = ∂ -Σ ⊙k ⊔ ∂ + Σ ⊙k , where ∂ -Σ ⊙k = ∂ -Σ 1 , ∂ + Σ ⊙k = ∂ + Σ k .Further, consider the covering space Σ ⊙k → Σ ⊙k obtained by gluing the boundary of the universal cover⊔ k i=1 Σ i → ⊔ k i=1 Σ i via the identification of the induced cover Σ i ⊃ ⊔ g∈π1(Σ) ∂ + (Σ) → ∂ + (Σ) ⊂ Σ iwith the induced coverΣ i+1 ⊃ ⊔ g∈π1(Σ) ∂ -(Σ) → ∂ -(Σ) ⊂ Σ i+1 .

  ⊙k , ∂ -Σ ⊙k ; π) with coefficients twisted by the above coveringΣ ⊙k → Σ ⊙k satisfies rk L 2 (H ⊙k , ∂ -Σ ⊙k ; π)) ≥ k.Proof. The first statement is proven by induction on k, using a standard comparison of dimensions obtained via the Mayer-Vietoris long exact sequence. . . → H 1 (∂ -Σ; F) → → H 1 ( Σ; F) ⊕ H 1 ( Σ ⊙(k-1) , ∂ -Σ ⊙(k-1) ; F) → → H 1 ( Σ ⊙k , ∂ -Σ ⊙k ; F) → H 0 (∂ -Σ; F) → . . . .We now show the statement concerning the L 2 -ranks, which follows by analogous computations. Lemma 11.11 implies that H (2) 0 (∂ ± Σ; π) = ℓ 2 (π) and H

2 H

 2 ± Σ; π) = 0, since ∂ ± Σ are simply connected. Observe that we also haveH ⊙k ; π) = 0, k ≥ 1,as follows from[START_REF] Lück | L 2 -invariants: theory and applications to geometry and K-theory[END_REF] Theorem 1.35(8)], using the fact that π is infinite. The (weak) long exact sequence of a pair [54, Theorem 1.21] immediately implies the base case rk L , ∂ -Σ; π) ≥ 1 as well as the vanishingH (2) 0 (Σ ⊙k , ∂ -Σ ⊙k ; π) = 0.The Mayer-Vietoris long (weakly) exact sequence . . . → H

2 H 1 )

 21 -Σ; π) → 0 → . . . , together withH (2) 1 (∂ -Σ; ℓ 2 (π)) = 0 and [54, Theorem 1.12(2)] gives that rk L , ∂ -Σ ⊙(k-1) ; π) + rk L 2 H (2) 0 (∂ -Σ; π). Since rk L 2 H (2)0 (∂ -Σ; π) = 1, the claim now follows by induction.

  F H(Σ; F) ≤ dim F (Λ; F). The linearised Legendrian contact cohomology satisfies the bounddim F LCH ε ′ (Λ) ≤ |R(Λ)| for any ε ′ . Thus we can fix an augmentation ε of A(Λ; F) satisfying dim F LCH ε (Λ; F) = max ε ′ {dim F LCH ε ′ (Λ; F)}.(47)The exact triangle in Theorem 1.8 gives usdim F LCH ε+ (Λ; F) ≥ ≥ dim F LCH ε (Λ; F) + dim F H(Σ; F)dim F H(Λ; F)

Figure 23 .

 23 Figure 23. The front projection of Λ (5,-8) .

  ⊙k , ∂ -Σ ⊙k ; π 1 (Σ))) ≥ k,as shown in Lemma 11.12, together with the universal boundrk L 2 LCH (2)• ε ′ (Λ) ≤ |R(Λ)|, which follows byLemma 11.11, finally gives the sought contradiction, from which it follows that π 1 (Σ) is finite.

Figure 24 .Figure 25 .

 2425 Figure[START_REF] Rizell | Legendrian ambient surgery and Legendrian contact homology[END_REF]. The front projection of the front spin Σ S 1 Λ ⊂ (R 5 , ξ std ) near the left-most cusp of Λ ⊂ (R 3 , ξ std ). The corresponding cusp-edge for the front projection of the front spin bounds an obvious embedded Legendrian disc D intersecting Σ S 1 Λ cleanly along this cusp edge.

Figure 26 .

 26 Figure 26. Front (left) and Lagrangian (right) projections of the maximal TB m(9 46 ) knot.

Figure 27 .

 27 Figure 27. A Lagrangian concordance from Λ 0 to Λ 946 .

•

  The boundary strata of the compactified moduli space M E consists of buildings whose components u 1 , . . . , u k have d(φα)-energies whose sum is at most E; and • The complement of any open neighbourhood of the boundary strata in M

E

is a compact subspace of M E in the appropriate Whitney topology.

  10.3. Seidel's isomorphism. We end this section by recalling the definition of Seidel's isomorphism. Let h : R × P × R be an autonomous Hamiltonian coinciding with e t + C + on the positive end, and with -e t on the negative end; we can take e.g. the Hamiltonian h V constructed in Section 10.1.2. Combining Theorem 10.5 and Proposition 10.1 we obtain a module morphism

  65, Sections 2.2 and 3], of a tbmaximising knot with non-sharp Kauffman bound on tb, whose Chekanov-Eliashberg algebra does not admit a finite-dimensional linear representation over Z 2 . as shown in Lemma 11.12, together with the universal bound dim C LCH • ε0,ε1 (Λ; C[π 1 (Σ)]) ≤ |π 1 (Σ)||R(Λ)| gives a contradiction.

Chantraine, Dimitroglou Rizell, Ghiggini, Golovko

Acknowledgements

Furthermore, the restriction of Ψ {(Σ s 0 ,Js)} to the subcomplex (C • (Λ + 0 , Λ + 1 ), -d ++ ) ⊂ Cth • (Σ 0 , Σ 1 )) induces an isomorphism Ψ {(Σ s 0 ,Js)} | C(Λ + 0 ,Λ + 1 ) : (C(Λ + 0 , Λ + 1 ), -d ++ ) → (C(Λ + 0 , Λ + 1 ), -d ++ ).

Remark 8.5.

• Under the additional assumption that Λ - i = ∅, i = 0, 1, this result was established in [31, Section 4.2.1].

• Our proof of the above proposition does not rely on abstract perturbations. This is possible since we can use Proposition 8.2 in order to replace Reeb chords with intersection points, thereby breaking some of the symmetry. An alternative proof, more closely adapted to the SFT formalism, would be simply to generalise [START_REF] Ekholm | Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology[END_REF]Section 4.2.1] to the current setting. The latter approach depends on the abstract perturbation scheme outlined in [START_REF] Ekholm | Rational symplectic field theory over Z 2 for exact Lagrangian cobordisms[END_REF]Appendix B].

Proof. After an application of Proposition 8.2, we may assume that there are no Reeb chords from Λ ± 1 to Λ ± 0 and, hence, that the involved complexes are generated by intersection points only. More precisely, the aforementioned proposition provides such natural identifications

of complexes for S, N ≫ 0 sufficiently large. Obviously φ -S hN ∂z (Σ s 0 ) still is a compactly supported family of exact Lagrangian cobordisms. This family is moreover fixed in a neighbourhood of the intersection points corresponding to the generators of C(Λ ± 0 , Λ ± 1 ), assuming that N ≫ 0 is chosen sufficiently large. To sum up, we have reduced the invariance problem to the case when there are no Reeb chords from Λ ± 1 to Λ ± 0 . In this case the involved complexes thus have generators corresponding to the intersection points of Σ s 0 ∩ Σ 1 , s = 0, 1, and the one-parameter family Σ s 0 is a compactly supported family of exact Lagrangian cobordisms.

In this case, the bifurcation analysis conducted in [START_REF] Ekholm | The contact homology of Legendrian submanifolds in R 2n+1[END_REF], [START_REF] Ekholm | Legendrian contact homology in P × R[END_REF] can be generalised to produce the sought homotopy equivalence. We proceed to provide a sketch of this argument, highlighting the points where some additional care must be taken to adapt it to the current setting.

Recall that the differential is defined via a count of J s -holomorphic strips having boundary on Σ s 0 ∪Σ 1 , two punctures asymptotic to intersection points, and possibly additional negative punctures asymptotic to Reeb chords on either Λ - 0 or Λ - 1 . For a generic one-parameter family Σ s 0 ∩ Σ 1 and J s s ∈ [0, 1], we can assume that for all but a finite number of instances s ∈ [0, 1],

• the intersection Σ s 0 ∩ Σ 1 is transverse, and • the above spaces of J s -holomorphic strips are transversely cut out. Moreover, in the case when the above two conditions do not hold for some s 0 ∈ [0, 1] we may assume that precisely one of the following cases occur.

Birth/death: All intersection points of Σ s0 0 ∩ Σ 1 are transverse except a single intersection point. The latter intersection point moreover arises in the family as a standard birth/death-intersection point, which was described in [START_REF] Ekholm | The contact homology of Legendrian submanifolds in R 2n+1[END_REF]Section 3] Remark 10.2. The above identifications of augmentations, despite the fact that the DGAs are associated to geometrically different Legendrians, can be justified as follows. For Λ, Λ ′′ ⊂ P ×R being sufficiently C 1 -close together with a fixed choice of compatible almost complex structure J P on P , the invariance theorem in [START_REF] Ekholm | Legendrian contact homology in P × R[END_REF] gives a canonical isomorphism between the Chekanov-Eliashberg algebras (A(Λ), ∂ Λ ) and (A(Λ ′′ ), ∂ Λ ′′ ) induced by the canonical bijection identifying the Reeb chords on Λ with the Reeb chords on Λ ′′ .

Proof. The first isomorphism is simply [START_REF] Chantraine | Floer homology and Lagrangian concordance[END_REF]Proposition 2.7].

The second isomorphism follows by combining this result with the isomorphism [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF]Proposition 4.1]. Here we must use the assumption of horizontal displaceability.

Consider an admissible almost complex structure J on R× P × R which coincides with the cylindrical lifts J + P and J - P of almost complex structures on P in subsets of the form

respectively. In the following we will also assume that h : R×P ×R is an autonomous Hamiltonian being of the form ±e t + C in each of the two latter cylindrical subsets.

Theorem 10.3. Let Σ be an (n + 1)-dimensional exact Lagrangian cobordisms from Λ -to Λ. Assume that there are augmentations ε i , i = 0, 1, of the Chekanov-Eliashberg algebra of Λ -defined using J - P . For ǫ > 0 sufficiently small, and Σ ′ being a sufficiently C 1 -small compactly supported perturbation of Σ, we may assume that J is a regular admissible almost complex structure for which:

(1) There is a natural isomorphism

of complexes; (2) There is an equality

of chain maps, where the former is the map induced by the linearised DGA morphism Φ Σ , as described in Section 5.3, and the latter is the corresponding component of the differential

Remark 10.4. All Legendrian contact homology complexes above are induced by J ± P , while the DGA morphisms, as well as d +-, are induced by J. Proof. Both results follow from Proposition 10.1 together with the bijections provided by [START_REF] Chantraine | Floer homology and Lagrangian concordance[END_REF]Theorem 2.15], where the latter provides the necessary identifications of pseudoholomorphic strips on the cobordism Σ and its two-copy Σ ∪ φ ǫ h (Σ ′ ) that are used in the definitions of the DGA morphisms induced by the respective cobordisms. To that end, an admissible almost complex structure as above must be used, i.e. which coincides with cylindrical lifts in the prescribed subsets.

Fix augmentations ε

, respectively (which by definition take values in R[π 1 (Σ 0 )] and R, respectively). Let u ∈ M r (x; ζ, y, δ) be a pseudoholomorphic strip involved in the Cthulhu differential d ε - 0 ,ε - 1 as defined in Section 3.2. Order the connected components of the boundary of ∂S r which are mapped to Σ 0 starting with the arc adjacent to the incoming puncture (as in Section 2.3.1), and denote them by ∂ 0 S r , . . . , ∂ k S r . We associate to each of these arcs an element a j ∈ π 1 (Σ 0 ) in the same manner as in the definition of the differential of the Chekanov-Eliashberg algebra with twisted coefficients described above. These paths, together with the above augmentations, now determine an element

where the sum is taken over the rigid components of the moduli space. The formula for y being a Reeb chord from Λ + 1 to Λ + 0 is similar, but involves the pull-backs ε - 0 • Φ Σ0 and ε - 1 •Φ Σ1 of the augmentations under the DGA homomorphism induced by the cobordisms with and without twisted coefficients, respectively. The differential is then extended to all of Cth(Σ 0 , Σ

The techniques in Section 8 can be used to prove the following theorem.

Theorem 11.3. The map d ε -

it is a differential, and it gives rise to an acyclic complex,

Proof. The proof is similar to the one in Sections 6.2 and 8.3. To that end, we observe the following important feature of the above construction of the coefficient

. Let u and v be holomorphic strips whose outgoing and incoming punctures, respectively, agree so that one can pre-glue them to a strip u * v. It then follows that

. It now follows from the compactness theorem that that the boundary of a component of a one-dimensional moduli space of holomorphic strips either consists of two broken configurations u * v and u ′ * v ′ , both which contribute with the same coefficient (with opposite signs)

, or at least one boundary point corresponds to a broken configuration involving a pure chord. However, when counted with the augmentations as above, the counts of the latter boundary points cancel for the algebraic reason that the augmentation is a chain map (see (II) in Section 6).

In view of the above theorem, the computations in Section 10 can be carried over immediately to the case of twisted coefficients. We proceed to explicitly describe the long exact sequence analogous to (4) in Theorem 1.6. Let Σ be an exact Lagrangian cobordism from Λ -to Λ + . Let ε - 0 and ε - 1 be two augmentations of A π(Σ) (Λ -) and A(Λ -) into R[π(Σ)] and R, respectively. Further, we consider the pull-backs

It is important to note that ε + 0 need not be the lift of an augmentation into R in general, even in the case when ε - 0 is.

Writing Σ for the universal cover of Σ, and Σ for its compactification to a manifold with boundary, there is a long exact sequence:

The identification of the topological term H n+1-k ( Σ, ∂ -Σ; R) is proven in the same manner as before (see Theorem 10.5), while making the observation that the Morse homology of a manifold with coefficients twisted by its fundamental group computes the homology of its universal cover. Finally, we point out that

is satisfied in the case when Λ -is simply connected.

11.2. Augmentations in finite-dimensional non-commutative algebras. The DGAs with "coefficients" in non-commutative unital algebras as treated in Section 11.1 can also be considered in a setting where the algebra is more general than a group ring, and without using twisted coefficient. Here we describe augmentations in non-commutative unital algebras, as done by the second and fourth authors in [START_REF] Dimitroglou Rizell | Estimating the number of Reeb chords using a linear representation of the characteristic algebra[END_REF], which can be seen to fit into this framework. Since we will be interested in computing the ranks of involved linearised complexes, we will restrict ourselves to the case when the involved algebra is finite-dimensional over the ground field F.

A finite-dimensional augmentation of the Chekanov-Eliashberg algebra is a unital DGA homomorphism

where A is a not necessarily commutative unital algebra which is finite-dimensional over the ground field F. Here F denotes the field that was used as coefficient ring for A(Λ). Recall that the existence of such a (graded) augmentation is equivalent to the existence of a finite-dimensional representation of the so-called characteristic algebra, which is defined as the quotient algebra A(Λ)/ ∂(A) by the two-sided ideal generated by the boundaries (see [START_REF] Ng | Computable Legendrian invariants[END_REF]). Given two such augmentations

), i = 0, 1, we can form all our complexes as free A 0 ⊗ F (A 1 ) op -modules or, differently put, a free A 0 -A 1 bimodule. It is important to note that the dimension over F still The underlying chain map of G

lifts to the corresponding complexes with twisted coefficients. Namely, we define the chain map by the same counts of pseudoholomorphic strips, but where the count takes the homotopy class of the boundary of the strips into account in the manner described above. The lifted map on homology will be denoted by

We will be particularly interested in the restriction to the degree 0 part of

). Observe that this map is linear over R[π 1 (Σ)] or, put differently, it is π 1 (Σ)equivariant. Also, we recall that H • (Σ; R[π 1 (Σ)]) of the twisted complex here computes the homology H • ( Σ, R) of the universal cover Σ → Σ, and that ( 43)

holds by [START_REF] Chantraine | Floer homology and Lagrangian concordance[END_REF]Proposition 2.7]. Later we will be particularly interested in the case when Λ + is simply connected and when ε + i , i = 0, 1, both take values in R. In this situation the universal coefficients theorem gives us an identification ( 44)

Choosing a generator m ∈ H 0 (Σ; R[π 1 (Σ)]), the fundamental class induced by Σ is defined to be the image

where we rely on the identification (43) above. Let Λ + 1 be obtained from Λ by a C 1 -small perturbation of its image under the time-ǫ Reeb flow, where ǫ > 0 is sufficiently small. We moreover assume that Λ + 1 can be identified with the graph j 1 f + ⊂ J 1 Λ + in a standard contact neighbourhood of Λ + , where f + : Λ + → R is a Morse function with a unique local minimum m + ∈ Λ + . Recall the definition of the fundamental class in [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF], which was defined by the following count of pseudoholomorphic strips having boundary on Λ + 1 ∪ Λ + :

Here we have again used identification [START_REF] Etnyre | Knots and contact geometry. I. Torus knots and the figure eight knot[END_REF] above. In the case when ε + 0 = ε + 1 and Λ + is horizontally displaceable, this class has moreover been shown to be nonvanishing; see [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF]Theorem 5.5].

The following proposition shows that the two definitions of the fundamental class given above in fact coincide.

Proposition 11.7. Assume that the natural map

). For appropriate choices of almost complex structures and Hamiltonian perturbations in the constructions,

(1) Pseudoholomorphic buildings with:

• A top level consisting of a single punctured strip with boundary on R × Λ + 0 ∪ R × Λ + 1 (which hence is rigid up to translation); • A middle level consisting of punctured half-planes of index zero having boundary on Σ i , i = 0, 1. (2) Pseudoholomorphic buildings with:

• A top level consisting of one punctured strip of index one having boundary on R × Λ + 0 ∪ R × Λ + 1 (which hence is rigid up to translation) together with a trivial strip over a Reeb chord;

• A middle level consisting of a single punctured strip of index zero having boundary on Σ 0 ∪ Σ 1 .

(3) Pseudoholomorphic buildings with:

• A middle level consisting of two punctured strips of index zero having boundary on Σ 0 ∪ Σ 1 ; • A bottom level consisting of a single punctured strip with boundary on R × Λ - 0 ∪ R × Λ - 1 which is of index one (and hence rigid up to translation). ( 4) Pseudoholomorphic buildings with:

• A middle level consisting of a single punctured strip of index zero having boundary on Σ 0 ∪ Σ 1 ; • A bottom level consisting of a single punctured half-plane of index one with boundary on R × Λ - i , i = 0, 1 (which hence are rigid up to translation), together with additional trivial strips over Reeb chords.

(5) A broken punctured strip having boundary on Σ 0 ∪ Σ 1 .

See Figure 22 for a schematic picture of the above pseudoholomorphic buildings.

A gluing argument implies that the configurations in (1) are in bijection with the configurations contributing to the above coefficient l γ in front of γ of the fundamental class. Furthermore, the count of the configurations in [START_REF] Akaho | Morse homology and manifolds with boundary[END_REF] gives the coefficients of

by Lemma 11.8. We proceed to infer that the signed counts of all buildings of type ( 2)-( 4) is equal to the coefficient of γ in the expression

from which the sought equality on the level of homology now follows. (As usual, all counts above are weighted by the augmentations ε - i , i = 0, 1.) (2): There are two cases: either the non-trivial strip in the top level has a positive puncture asymptotic to m + , or it has positive puncture asymptotic to γ. The former case can be excluded by actions reasons, while the count of the latter configurations corresponds exactly to the coefficient in front of γ of the boundary

: There are no buildings of this type. Namely, by Lemma 11.8, we may assume there are no punctured pseudoholomorphic strips with boundary on Σ 0 ∪ Σ 1 having positive asymptotic to the minimum m + and a negative asymptotic to a Reeb chord from Λ - 0 to Λ - 1 . (4): The sum of these contributions vanishes, as follows from the fact that ε - i , i = 0, 1, vanishes on any boundary of the Chekanov-Eliashberg algebra of Λ - i (see (II) in Section 6). Recall that the latter differential is defined by a count of punctured pseudoholomorphic half-planes of index one having boundary on R×Λ - i .

for some i 0 . By the Mayer-Vietoris sequence we conclude that the inequality

holds. In particular,

which by induction leads to a contradiction with equality [START_REF] Fukaya | Lagrangian intersection Floer theory: anomaly and obstruction. Part I[END_REF].

(ii): The argument is the same as the one in the proof of [START_REF] Dimitroglou Rizell | On homological rigidity and flexibility of exact Lagrangian endocobordisms[END_REF]Theorem 1.6 (ii)], and follows form Part (i) applied to the concatenation Σ ⊙ Σ. Namely the Mayer-Vietoris sequence for the concatenation Σ ⊙ Σ seen as two copies of Σ glued along the boundary component Λ shows that

) and by the above result, we conclude that dim F im(i - * , i + * ) = dim F H(Σ; F) = dim F H(Λ; F), from which the claim follows.

(iii): By contradiction, we assume that i + * ⊕ i - * : H(Λ ⊔ Λ) → H(Σ) is not a surjection. Considering a representative V ⊂ H(Σ) of the cokernel of this map, which hence is of dimension dim F V > 0, the Mayer-Vietoris long exact sequence implies that the image of V ⊕ V under the map

has image being of dimension 2 dim F V > 0. Moreover, V ⊕ V can again be seen to not be contained in the image of i + * ⊕ -i - * : H(Λ ⊔ Λ) → H(Σ ⊙ Σ). Namely, the above inclusion factorises through the canonical maps as

, where the latter morphism is the one from the above Mayer-Vietoris long exact sequence. In conclusion, the cokernel of

Arguing by induction, now we arrive at the sought contradiction with Part (i) above.

The proof is now complete for Z 2 .

Under the additional assumption that Λ is spin, and admitting an augmentation in an arbitrary field F, Corollary 12.3 of Theorem 1.14 implies that any endocobordism of Λ is spin as well. This allows us to repeat the previous argument with coefficients in the field F. Note that Theorem 1.14 relies on Theorem 1.11, which we established above in the needed case F = Z 2 .

We now prove the following Theorem of which Theorem 1.13 is an immediate corollary. Observe that the following result also can be proved by alluding to Theorem 1.11.

Theorem 12.1. Let Λ be a Legendrian homology sphere inside a contactisation, Σ be an exact Lagrangian cobordism from Λ to itself inside the symplectisation, and F a field. If A(Λ; F) admits an augmentation, then H • (Σ, Λ; F) = 0, i.e. Σ is a F-homology cylinder. 12.3. Restrictions on the fundamental group of an endocobordism between simply connected Legendrians. We now prove the results concerning the fundamental groups of endocobordisms between simply connected Legendrian submanifolds.

12.3.1. Proof of Theorem 1.16.

Proof of Theorem 1.16. Recall the construction of the fundamental class in the setting of twisted coefficients carried out in Section 11.3. The proof will be a straightforward consequence of Proposition 11.7 therein.

From the assumptions of the theorem, the Legendrian submanifold Λ + has a unique augmentation. It follows that [START_REF] Ekholm | A duality exact sequence for Legendrian contact homology[END_REF]Theorem 5.5] can be applied, and hence the fundamental class c ε + 0 ,ε + 1 Λ + ,m + is non-vanishing. By Proposition 11.7 we, moreover, conclude that this fundamental class is the image of a generator m of H 0 (Σ; R[π 1 (Σ)]) under the map G ε -,ε -Σ . Since Λ + is simply connected by assumption, it follows from [START_REF] Etnyre | On connected sums and Legendrian knots[END_REF] above that this image is not torsion. In particular

. Thus, m is not torsion either, and since it generates H 0 (Σ; R[π 1 (Σ)]) we conclude that H 0 (Σ; R[π 1 (Σ)]) = R[π 1 (Σ)]. However, since Σ is connected, we know that H 0 (Σ; R[π 1 (Σ)]) = H 0 ( Σ) = R. In other words, π 1 (Σ) is the trivial group, as sought.

12.3.2. Proof of Theorem 1.18.

Proof of Theorem 1.18. Here it will be crucial to use the machinery of L 2 -coefficients as described in Section 11.4.

We will let Σ ⊙k , k ≥ 1, denote the k-fold concatenation of the cobordism Σ from Λ to Λ. Since Λ is spin by assumption, it follows from Corollary 12.3 that the cobordisms Σ ⊙k are spin for all k ≥ 1. We also consider the cover p : Σ ⊙k → Σ ⊙k as constructed in the previous section.

First, we argue that the claim follows from the fact |π 1 (Σ)| < ∞, which will be shown below. Indeed, under these assumptions, the version of the long exact sequence in Theorem 1.6 applied to the system of local coefficients induced by the above covering (see Section 11.1), becomes