We obtain several results for (iterated) planar contact manifolds in higher dimensions. (1) Iterated planar contact manifolds are not weakly symplectically co-fillable. This generalizes a 3D result of Etnyre [ 14] to a higher-dimensional setting, where the notion of weak fillability is that due to Massot-Niederkrüger-Wendl [ 38]. (2) They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds. This generalizes a result by Albers-Bramham-Wendl [ 4]. (3) They satisfy the Weinstein conjecture, that is, every contact form admits a closed Reeb orbit. This is proved by an alternative approach as that of [ 2] and is a higher-dimensional generalization of a result of Abbas-Cieliebak-Hofer [ 1]. The results follow as applications from a suitable symplectic handle attachment, which bears some independent interest.

In this article, we provide an introduction to an algorithm for constructing Weinstein handlebodies for complements of smoothed toric divisors using explicit coordinates and a simple example. This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirby calculus. Finally, we include one complicated example at the end of the article to showcase the algorithm and the types of Weinstein Kirby diagrams it produces.

Abstract. In this paper, we examine mapping class group relations of some symplectic manifolds. For each n ě 1 and k ě 1, we show that the 2n-dimensional Weinstein domain W " tf " δu X B 2n`2 , determined by the degree k homogeneous polynomial f P Crz0, . . . , zns, has a Boothby-Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case n " 2 recovering the classical chain relation for the torus with two boundary components.

We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centered condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface F , where the 2-handles are attached along the co-oriented conormal lifts of curves on F . We discuss how to use these diagrams to calculate invariants and provide numerous examples applying this procedure. For example, we provide Weinstein handlebody diagrams for the complements of the smooth and nodal cubics in CP 2 . Contents 1. Introduction 2.1. Background on Weinstein domains 2.2. Weinstein Kirby calculus 2.3. Weinstein structures on the cotangent bundle 2.4. Morsification of the canonical Weinstein structure 3. Toric transformations 3.1. Algebraic viewpoint 3.2. Toric 4-manifolds 4. Weinstein structure on the complement of a partially smoothed toric divisor 4.1. Smoothing 4.2. The topology of the complements 4.3. The core, co-core, and attaching sphere from the toric perspective 4.4. An almost toric point of view 4.5. The global Weinstein structure on the complement 5. Combinatorial Possibilities for Delzant Polytopes and resulting Weinstein divisor complements 5.1. On the richness of Delzant family 5.2. Non-uniqueness of Weinstein manifolds from Delzant polytopes 5.3. The restrictiveness of the centeredness condition 6. Non-centered toric manifolds and obstructions 6.1. Exactness obstructions 6.2. Convexity obstructions 7. Cotangent bundles and handlebody diagrams 8. Lifting co-normals to Kirby diagrams 8.1. The procedure to obtain a standard Weinstein handlebody diagram

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