The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in our forthcoming paper [7].In the Appendix written by S Ganatra and M Maydanskiy it is shown that the results of this paper imply P Seidel's conjecture from [42]. 53D05, 53D42, 57R17 IntroductionWe study how attaching of a Lagrangian handle in the sense of Weinstein [46], Eliashberg [22] and Cieliebak and Eliashberg [13] to a symplectic manifold with contact boundary affects symplectic invariants of the manifold and contact invariants of its boundary. We establish several surgery exact triangles for these invariants. As explained in Section 8 below, symplectic handlebody presentations and Lefschetz fibration presentations of Liouville symplectic manifolds are closely related. In this sense our results can be viewed as generalizations of P Seidel's exact triangles for symplectic Dehn twists; see [38; 37; 39]. In particular, as shown in the Appendix written by S Ganatra and M Maydanskiy, our results imply P Seidel's conjecture from [42]. This is the first paper in a series devoted to this subject. In order to make the results more accessible and the algebraic formalism not too heavy, we make here a number of simplifying assumptions (like vanishing of the first Chern class). Though proofs are only sketched, we indicate the main ideas and provide some details which should help specialists to reconstruct the proofs. The general setup and detailed proofs will appear in the forthcoming paper [7]. Corollary 5.7, which gives a closed form formula for symplectic homology; Theorem 5.10, which relates the Legendrian homology algebras of a Legendrian submanifold before and after surgery; Theorem 5.8, which provides a formula for the linearized Legendrian homology of the so-called cocore (the meridian of the handle) Legendrian sphere after surgery.Section 7 is devoted to first examples and applications. It is worthwhile to point out that already quite primitive computations yield interesting geometric applications. In particular, we show that Legendrian surgery on Y Chekanov's two famous Legendrian .5; 2/-knots in S 3 [10] give noncontactomorphic 3-manifolds and that attaching a Lagrangian handle to the ball along stabilized trivial Legendrian knots produce examples of exotic Weinstein symplectic structures on T S n (exotic structures on T S n were first constructed by M McLean in [35] and also by M Maydanskiy and P Seidel in [34]).In Section 8 we explain the relation between the Weinstein handlebody and the Lefschetz fibration formalisms. In the Appendix, written by M Maydanskiy and S Ganatra, this description is used to deduce P Seidel's conjecture [42] from the results of the current paper.Acknowledgements This paper was conceived several years ago and over this period we benefited a lot from discussions with several mathematicians. We are especially thankful (in alphabetical order) to

ABSTRACT. We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X, L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L ⊂ X which agrees with cylinders over Legendrian links Λ+ and Λ− at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of Λ+ to that of Λ−. We give a gradient flow tree description of the DGA maps for certain pairs (X, L), which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

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Abstract. A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × R, where P is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of R n and, more generally, invariants of self transverse immersions into R n up to restricted regular homotopies. When n = 3, this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.