Abstract
By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is null-bordant in the oriented bordism group $\Omega SO^*(W)$ of any symplectically aspherical filling $W$, and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in $H^*(W)$. Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called "Lobs") as boundary conditions.
Highlights
Abstract. — By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling
Wendl the Eliashberg-Floer-McDuff theorem, is essentially a classification of fillings up to homotopy type
Given an oriented (2n − 1)-dimensional manifold M, a contact structure on M is a hyperplane distribution of the form ξ = ker α, where the contact form α is a smooth 1-form satisfying α ∧n−1 > 0, and the co-orientation of ξ is determined by α > 0
Summary
We modify slightly the proof of the Eliashberg-Floer-McDuff theorem [McD91, Th. 1.5] in order to illustrate the methods that will be applied in the rest of the article. Complex multiplication gives i · ∂r = ∂φ, dy[1] Du · ∂r = dy[1] Du · (−i · ∂φ) = −dy[1] i · Du · ∂φ = −dx[1] Du · ∂φ = 0 It follows that the outward derivative of the y1-coordinate vanishes at the point eiφ0 ∈ D2, so that according to the boundary point lemma, y1 must equal the constant 1 on the whole disk, and as a consequence u lies entirely in SA. X = X × D2 where D2 is contractible, B can be represented by a cycle in X × {p} for any point p ∈ D2; in particular we are free to choose p ∈ ∂D2, B is represented by a cycle in ∂X This implies that the class A is homologous to a cycle in the sphere ∂Y , which shows that A must be trivial.
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