Exotic non-orientable four-manifolds with prescribed fundamental group

Abstract We show that any finitely presented group with an index two subgroup is realised as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilise after adding a single copy of the complex projective plane.