Subcritical contact surgeries and the topology of symplectic fillings

TL;DR

This paper investigates the topology of symplectic fillings after subcritical contact surgeries, revealing null-bordant belt spheres in higher dimensions and demonstrating limitations of contact connected sum decompositions beyond dimension three.

Contribution

It generalizes Eliashberg's result to higher dimensions for subcritical surgeries and shows the non-extension of contact connected sum decompositions in dimensions five and above.

Findings

Belt spheres are null-bordant in symplectically aspherical fillings.

In dimension five, belt spheres are nullhomotopic.

Contact connected sum decomposition does not extend to higher dimensions.

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.