In this paper, we construct an A ∞ -category associated to a Legendrian submanifold of a jet space. Objects of the category are augmentations of the Chekanov algebra A(Λ) and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearized Legendrian contact homology. Those are constructed as a generalization of linearized Legendrian contact homology using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order composition maps in the category and generalizes the idea of [6] where an A ∞ -algebra was constructed from one augmentation. This category allows us to define a notion of equivalence of augmentations when the coefficient ring is a field regardless of its characteristic. We use simple examples to show that bilinearized cohomology groups are efficient to distinguish those equivalences classes. We also generalize the duality exact sequence from [12] in our context, and interpret geometrically the bilinearized homology in terms of the Floer homology of Lagrangian fillings (following [8]).

In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of the classical invariants under this relation, namely the Thurston-Bennequin number and the rotation number, and we provide some examples of non-trivial Legendrian knots bounding Lagrangian surfaces in $D^4$. Using these examples, we are able to provide a new proof of the local Thom conjecture.Comment: 18 pages, 4 figures. v2: Minor corrections and a proof of conjecture 6.4 of version 1 (now Theorem 6.4). v3: Several substantial changes notably the proof of theorems 1.2 and 5.1, this is the version accepted for publication in "Algebraic and Geometric Topology" published in January 201

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.In both cases, Λ + = ∅. Thus we obtain a new proof of the following result.Corollary 1.4 ([23]). 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 1.5. 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 isomorphismsto replace the relevant terms in the long exact sequences (1) and (3), 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 [49].

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

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We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian L which has nonzero Morse-Novikov homology for the restriction of the Lee form β cannot be disjoined from itself by a C 0 -small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of β. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry. * (X 0 , X 0 \ Q × B) ≥ 2r: Consider the Mayer-Vietoris sequence:

In this note, we define the notion of collarable slices of Lagrangian submanifolds. These are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact hypersurface. Such a notion arises naturally when studying intersections of Lagrangian submanifolds with contact hypersurfaces. We then give two explicit examples of Lagrangian disks in C 2 transverse to S 3 whose slices are non-collarable.

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds filled by a Liouville domain with non-zero symplectic homology are strongly orderable in the sense of Liu.