Subcritical contact surgeries and the topology of symplectic fillings

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 $\Omega S{O}^{\ast }(W)$ 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 $H^{\ast }(W)$. 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.

42 pages, 6 figures. (accepted by Journal de l'École polytechnique)

References in corpus (2)

• The plastikstufe - a generalization of the overtwisted disk to higher dimensions
• The topology of Stein fillable manifolds in high dimensions II

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• On the symplectic fillings of standard real projective spaces
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• Disjoinable Lagrangian tori and semisimple symplectic cohomology