Symplectic fillings and cobordisms of lens spaces

We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling X of a virtually overtwisted contact structure on L(p,q) has another symplectic structure that fills the universally tight contact structure on L(p,q). Moreover, we show that the Stein filling of L(p,q) with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.