Ozsvath-Szabo invariants and tight contact three-manifolds, II

Let p and n be positive integers with p > 1, and let E-p,E-n be the oriented 3-manifold obtained by performing p(2)n - pn - 1 surgery on a positive torus knot of type (p,pn + 1). We prove that E-2,E-n does not carry tight contact structures for any n, while Ep,n carries tight contact structures for any n and any odd p. In particular, we exhibit the first infinite family of closed, oriented, irreducible 3-manifolds which do not support tight contact structures. We obtain the nonexistence results via standard methods of contact topology, and the existence results by using a quite delicate computation of contact Ozsvith-Szabo invariants.