Lagrangian tori in homotopy elliptic surfaces

Abstract

Let E( 1)(K) denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a. bred knot K in S-3 constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in E( 1)(K) and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of K has some irreducible factor which does not divide t(n) - 1 for any positive integer n. We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic 4-manifolds.