Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds (Q1340628)

The longstanding conjecture that every smooth simply-connected 4-manifold is the connected sum of complex surfaces was recently disproved by Gompf and Mrowka. They constructed infinite families of such 4-manifolds which are homeomorphic to the K3 surface but admit no complex structure. In the same spirit, the authors of this paper provide infinite families of irreducible noncomplex 4-manifolds homotopy equivalent to minimal elliptic surfaces respectively various classes of surfaces of general type. These examples are then combined to obtain simply connected irreducible 4-manifolds which are not homotopy equivalent to any complex surface.