Fillability of contact surgeries

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks: what properties (e.g. tightness, fillability, vanishing or non-vanishing of various Floer type invariants) of contact structures are preserved under various types of contact surgeries. The case of the negative contact surgeries is fairly well understood. In this talk, extending an earlier work of the speaker with Conway and Etnyre, I will discuss some new results about symplectic fillability of positive contact surgeries, and in particular I will provide a necessary and sufficient condition for contact (positive) integer surgery along a Legendrian knot to yield a fillable contact manifold. When specialized to knots in the three sphere, this result is rather efficient to find many examples of fillable surgeries along with various obstructions and surprising topological applications. This will report on joint work with Thomas Mark.