If (X, ω) is a closed symplectic manifold, and Σ is a smooth symplectic submanifold Poincaré dual to a positive multiple of ω, then X \ Σ can be completed to a Liouville manifold (W, dλ).Under monotonicity assumptions on X and on Σ, we construct a chain complex whose homology computes the symplectic homology of W . We show that the differential is given in terms of Morse contributions, Gromov-Witten invariants of X relative to Σ and Gromov-Witten invariants of Σ.We use a Morse-Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to Σ. Contents
We introduce a chain complex associated to a Liouville domain pW , dλq whose boundary Y admits a Boothby-Wang contact form (i.e. is a prequantization space). The differential counts Floer cylinders with cascades in the completion W of W , in the spirit of Morse-Bott homology [Bou02,Fra04,BO09b]. The homology of this complex is the symplectic homology ofLet X be obtained from W by collapsing the boundary Y along Reeb orbits, giving a codimension 2 symplectic submanifold Σ. Under monotonicity assumptions on X and Σ, we show that for generic data, the differential in our chain complex counts elements of moduli spaces of cascades that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential. Contents arXiv:1804.08013v2 [math.SG] 21 Nov 2018Definition 2.6. An admissible almost complex structure J X on X is compatible with ω and its restriction to ϕpUq is the push-forward by ϕ of a bundle almost complex structure on N Σ.An almost complex structure J W on pW, dλq is admissible if J W " ψ˚J X for an admissible J X . In particular, such almost complex structures are cylindrical and Reeb-invariant on W zW .A compatible almost complex structure J Y on the symplectization RˆY is admissible if J Y is cylindrical and Reeb-invariant.In the following, we will identify W with XzΣ by means of the diffeomorphism ψ and identify the corresponding almost complex structures. By an abuse of notation, 6 LUÍS DIOGO AND SAMUEL T. LISI we will both write π Σ : Y Ñ Σ to denote the quotient map that collapses the Reeb fibres, and π Σ : RˆY Ñ Σ to denote the composition of this projection with the projection to Y .Definition 2.7. Denote the space of almost complex structures in Σ that are compatible with ω Σ by J Σ .Let J Y denote the space of admissible almost complex structures on RˆY . Then, the projection dπ Σ induces a diffeomorphism between J Y and J Σ .Let J W denote the space of admissible almost complex structures on W . By Proposition 2.3, for any J W P J W , we obtain an almost complex structure J Σ .Denote this map by P : J W Ñ J Σ . This map is surjective and open by Proposition 2.3. For any given J Σ P J Σ , P´1pJ Σ q consist of almost complex structures on W that differ in W , or equivalently, can be identified (using ψ) with almost complex structures on X that differ in V " XzϕpUq. The chain complexWe will now describe the chain complex for the split symplectic homology associated to W .
A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e.g. on the boundary of a symplectic filling with a Lefschetz fibration over any compact oriented surface with boundary. In this first paper of a two-part series, we introduce the basic notions relating spinal open books to contact structures and symplectic or Stein structures on Lefschetz fibrations, leading to the definition of a new symplectic cobordism construction called spine removal surgery, which generalizes previous constructions due to Eliashberg [Eli04], Gay-Stipsicz [GS12] and the third author [Wen13b]. As an application, spine removal yields a large class of new examples of contact manifolds that are not strongly (and sometimes not weakly) symplectically fillable. This paper also lays the geometric groundwork for a theorem to be proved in part II, where holomorphic curves are used to classify the symplectic and Stein fillings of contact 3-manifolds admitting a spinal open book with a planar page.
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