We study symplectic invariants of the open symplectic manifolds X Γ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree Γ. For any tree Γ, we calculate (DG-)algebra models of the Fukaya category F(X Γ ) of closed exact Lagrangians in X Γ and the wrapped Fukaya category W(X Γ ). When Γ is a Dynkin tree of type A n or D n (and conjecturally also for E 6 , E 7 , E 8 ), we prove that these models for the Fukaya category F(X Γ ) and W(X Γ ) are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of X Γ for Γ = A n , D n , based on the Legendrian surgery formula of [14]. In the case that Γ is non-Dynkin, we obtain a spectral sequence that converges to symplectic cohomology whose E 2 -page is given by the Hochschild cohomology of the preprojective algebra associated to the corresponding Γ. It is conjectured that this spectral sequence is degenerate if the ground field has characteristic zero.

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

ABSTRACT. Recently, Honda, Kazez and Matić [9] described an adapted partial open book decomposition of a compact contact 3-manifold with convex boundary by generalizing the work of Giroux in the closed case. They also implicitly established a one-to-one correspondence between isomorphism classes of partial open book decompositions modulo positive stabilization and isomorphism classes of compact contact 3-manifolds with convex boundary. In this expository article we explicate the relative version of Giroux correspondence. INTRODUCTIONA sutured manifold (M, Γ) is a compact oriented 3-manifold with nonempty boundary, together with a compact subsurface Γ = A(Γ) ∪ T (Γ) ⊂ ∂M, where A(Γ) is a union of pairwise disjoint annuli and T (Γ) is a union of tori. Moreover each component of ∂M \ Γ is oriented, subject to the condition that the orientation changes every time we nontrivially cross A(Γ). Let R + (Γ) (resp. R − (Γ)) be the open subsurface of ∂M \ Γ on which the orientation agrees with (resp. is the opposite of ) the boundary orientation on ∂M. A sutured manifold (M, Γ) is balanced if M has no closed components, π 0 (A(Γ)) → π 0 (∂M) is surjective, and χ(R + (Γ)) = χ(R − (Γ)) on every component of M. It turns out that if (M, Γ) is balanced, then Γ = A(Γ) and every component of ∂M nontrivially intersects Γ. Since all the sutured manifolds that we will deal with in this paper are balanced, we will think of Γ as a set of oriented curves on ∂M by identifying each annulus in Γ with its core circle. Here we orient Γ as the boundary of R + (Γ).Let ξ be a contact structure on a compact oriented 3-manifold M whose dividing set on the convex boundary ∂M is denoted by Γ. Then it is fairly easy to see that (M, Γ) is a balanced sutured manifold (with annular sutures) via the identification we mentioned above. Conversely, given a balanced sutured manifold (M, Γ), there exists a contact structure ξ on M which makes ∂M convex and realizes Γ as its diving set on ∂M. One should keep in mind, however, that such a contact structure is neither tight nor unique in general.Let (M, Γ) be a balanced sutured 3-manifold and let ξ be a contact structure on M with convex boundary whose dividing set on ∂M is isotopic to Γ. Recently, Honda, Kazez and Matić [9] introduced an invariant of the contact structure ξ which lives in the sutured Floer Key words and phrases. partial open book decomposition, contact three manifold with convex boundary, sutured manifold, compatible contact structure.

We consider product 4-manifolds S 1 × M , where M is a closed, connected and oriented 3-manifold. We prove that if S 1 × M admits a complex structure or a Lefschetz or Seifert fibration, then the following statement is true: admits a symplectic structure if and only if M fibers over S 1 ,under the additional assumption that M has no fake 3-cells. We also discuss the relationship between the geometry of M and complex structures and Seifert fibrations on S 1 × M .

Abstract. For any pair of integers n ≥ 1 and q ≥ 2, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class q[F ] of an elliptic surface E(n), where [F ] is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedded into a more general class of symplectic 4-manifolds.