Families of Simply Connected 4-Manifolds with the Same Seiberg-Witten Invariants

Abstract

This article presents the constructions of new infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic and, from their construction, seem to be quite different, but cannot be distinguished by Seiberg-Witten invariants. Whether these manifolds are, or are not, actually diffeomorphic seems to be a very difficult question to answer. The most interesting of these constructions is a surgery that from certain symplectic 4-manifolds will produce nonsymplectic 4-manifolds. This is detected by calculations of Seiberg-Witten invariants. The surgery in question can be performed on any 4-manifold which contains as a codimension 0 submanifold a punctured surface bundle over a punctured surface and a nontrivial loop in the base which has trivial monodromy. A starting point for another class of examples in this paper is a family of examples which show that the Parshin-Arakelov theorem for holomorphic Lefschetz fibrations is false in the symplectic category. Such families are constructed by means of knot surgery on elliptic surfaces. It is shown that for a fixed homeomorphism type X (of a simply connected elliptic surface) and a fixed integer $g \ge 3$, there are infinitely many genus g Lefschetz fibrations on nondiffeomorphic 4-manifolds, all homeomorphic to X.