Homotopy K3's with several symplectic structures

In this note we prove that, for any integer n, there exist a smooth 4-manifold, homotopic to a K3 surface, defined by applying the link surgery method of Fintushel-Stern to a certain 2-component graph link, which admits n inequivalent symplectic structures Keywords: Symplectic topology of 4-manifolds; Seiberg-Witten theoryComment: Version 2: a few minor corrections from version 1. Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper8.abs.htm

On the topology of Symplectic Calabi-Yau 4-manifolds

Let M be a 4-manifold with residually finite fundamental group G having b_1(G) > 0. Assume that M carries a symplectic structure with trivial canonical class K = 0 in H^2(M). Using a theorem of Bauer and Li, together with some classical results in 4-manifold topology, we show that for a large class of groups $is determined up to homotopy and, in favorable circumstances, up to homeomorphism by its fundamental group. This is analogous to what was proven by Morgan-Szabo in the case of b_1 = 0 and provides further evidence to the conjectural classification of symplectic 4-manifolds with K = 0$. As a side, we obtain a result that has some independent interest, namely the fact that the fundamental group of a surface bundle over a surface is large, except for the obvious cases.Comment: 13 pages. Final version, to appear in the Journal of Topolog