Four-manifolds up to connected sum with complex projective planes

Based on results of Kreck, we show that closed, connected $4$-manifolds up to connected sum with copies of the complex projective plane are classified in terms of the fundamental group, the orientation character and an extension class involving the second homotopy group. For fundamental groups that are torsion free or have one end, we reduce this further to a classification in terms of the homotopy 2-type.