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By a result of Eliashberg, every symplectic filling of a three-dimensionalcontact connected sum is obtained by performing a boundary connected sum onanother symplectic filling. We prove a partial generalization of this resultfor subcritical contact surgeries in higher dimensions: given any contactmanifold that arises from another contact manifold by subcritical surgery, itsbelt sphere is null-bordant in the oriented bordism group $\Omega SO^*(W)$ ofany symplectically aspherical filling $W$, and in dimension five, it will evenbe nullhomotopic. More generally, if the filling is not aspherical but issemipositive, then the belt sphere will be trivial in $H^*(W)$. Using the samemethods, we show that the contact connected sum decomposition for tight contactstructures in dimension three does not extend to higher dimensions: inparticular, we exhibit connected sums of manifolds of dimension at least fivewith Stein fillable contact structures that do not arise as contact connectedsums. The proofs are based on holomorphic disk-filling techniques, withfamilies of Legendrian open books (so-called "Lobs") as boundary conditions.

Subcritical contact surgeries and the topology of symplectic fillings