This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [4]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in [8] as well as compactness theorems in Floer homology theory, [6,7], and in contact geometry, [9,19].

First, I would like to thank my academic advisor, Professor Yasha Eliashberg, for introducing me to the complex world of contact homology, as well as for his guidance and his support during my work. During my graduate studies, I had the opportunity to have very interesting conversations with several researchers at Stanford. In particular, I particularly enjoyed working jointly with Klaus Mohnke on some of the aspects of Symplectic Field Theory. I also thank John Etnyre, Kai Cieliebak, Tobias Ekholm, Edward Goldstein and Michael Hutchings for their interest in my work. Special thanks go to Francisco Presas, for his review of and his remarks about this thesis. I enjoyed the stimulating environment provided by contemporary students of Yasha,

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

We study the coherent orientations of the moduli spaces of holomorphic curves in Symplectic Field Theory, generalizing a construction due to Floer and Hofer. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. The orientations are determined by a certain choice of orientation at each closed Reeb orbit, that is similar to the orientation of the unstable tangent spaces of critical points in finite-dimensional Morse theory.

We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling.2000 Mathematics Subject Classification: 53D40.We denote by P(H) the set of 1-periodic orbits of X θ H and by P a (H) the set of 1-periodic orbits in a given free homotopy class a in W .Let J denote the set of admissible almost complex structureswhich are compatible with ω and have the following standard form for t large enough:An almost complex structure J ∈ J is called regular for u ∈ M A (γ, γ; H, J) if D u is surjective, and it is called regular if D u is surjective for all γ, γ ∈ P(H), A ∈ H 2 (W ; Z) and u ∈ M A (γ, γ; H, J). It is proved in [15] that the space J reg (H) of regular almost complex structures is of the second category in J . For every J ∈ J reg (H) the space M A (γ, γ; H, J) is a smooth manifold of dimension µ(γ) − µ(γ) + 2 c 1 (T W ), A . From now on we fix some J ∈ J reg (H). According to Floer [12] we have ∂ 2 = 0. We define the symplectic homology groups of the pair (H, J) by SH a * (H, J) := H * (SC a * (H), ∂).Remark 2.3. In view of condition (1) the Novikov ring Λ ω can be replaced by Z[H 2 (W ; Z)], or even by Z at the price of losing the grading. Indeed, the energy of a Floer trajectory depends only on its endpoints, hence the moduli spaces M(γ, γ; H, J) := A M A (γ, γ; H, J) are compact. Therefore the sum (14) involves only a finite number of classes A.By a standard argument [12] the groups SH a * (H, J) do not depend on J ∈ J reg (H). Nevertheless, they do depend on H and, in order to obtain an invariant of (W, ω), we need an additional algebraic limit construction. We define an