In this paper we give a rigorous definition of cylindrical contact homology for contact 3-manifolds that admit nondegenerate contact forms with no contractible Reeb orbits, and show that the cylindrical contact homology is an invariant of the contact structure. ContentsThe curves v 0 and v 1 are also equipped with asymptotic markers at the positive and negative ends; this will be described more precisely later. Let M = M J+ be the moduli space of curves v 1 from γ to γ satisfying the above. We want to glue v 0 to M/R (or its compactification M/R); see Figure 1.By a slight modification of [17], there is an obstruction bundle, † More precisely, 'tame', which is defined in Section 3.1. ‡ This will be our usual convention. § By a curve from γ + to γ − we mean a curve which is asymptotic to γ + at the positive end and to γ − at the negative end.Remark 6.5. The proof is modeled on but is substantially easier than that of [17, Proposition 3.2]. This is due to the fact that the J-holomorphic equation is linear near each R × γ. This allows us to dispense with the quadratic estimates. Proof. Let J J be the subset of J

We develop Lagrangian Floer Theory for exact, graded, immersed Lagrangians with clean self-intersection using Seidel's setup [21]. A positivity assumption on the index of the self intersection points is imposed to rule out certain (but not all) disc bubbles. This allows the Lagrangians to be included in the exact Fukaya category. We also study quasi-isomorphism of Lagrangians under certain exact deformations which are not Hamiltonian.

Symplectic Field Theory studies J-holomorphic curves in almost complex manifolds with cylindrical ends. One natural generalization is to replace "cylindrical" by "asymptotically cylindrical". In this article, we generalize the asymptotic results about the behavior of J-holomorphic curves near infinity in [Ho, HWZ1, Bo, BEHWZ] to the asymptotically cylindrical setting. We also sketch how these asymptotic results allow the main compactness theorems of [BEHWZ] to be extended to the asymptotically cylindrical case.

We define the contact homology algebra for any contact manifold and show that it is an invariant of the contact manifold. More precisely, given a contact manifold (M, ξ) and some auxiliary data D, we define an algebra HC(D). If D1 and D2 are two choices of auxiliary data for (M, ξ), then HC(D1) and HC(D2) are isomorphic. We use a simplified version of Kuranishi perturbation theory, consisting of semi-global Kuranishi charts. CONTENTS1. Introduction 1 2. Orbifolds and multisections 3 3. Almost complex structures and moduli spaces 7 4. Fredholm theory 15 5. Semi-global Kuranishi charts 23 6. Gluing 32 7. Details of gluing 36 8. Construction of semi-global Kuranishi structures 59 9. Contact homology 79 Appendix 89 References 94

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