The link surgery of $S^2\times S^2$ and Scharlemann's manifolds

Abstract

Fintushel-Stern's knot surgery has given many exotic 4-manifolds. We show that if an elliptic fibration has two, parallel, oppositely-oriented vanishing cycles (for example $S^2\times S^2$ or Matsumoto's $S^4$), then the knot surgery does not change its differential structure. We also give a classification of link surgery of $S^2\times S^2$ and a generalization of Akbulut's celebrated result that Scharlemann's manifold is standard.