Irreducible 4-manifolds with abelian non-cyclic fundamental group of small rank

In this paper we construct several irreducible 4-manifolds, both small and arbitrarily large, with abelian non-cyclic fundamental group. The manufacturing procedure allows us to fill in numerous points in the geography plane of symplectic manifolds with π 1 = Z ⊕ Z , Z ⊕ Z p and Z q ⊕ Z p ( gcd ( p , q ) ≠ 1 ). We then study the botany of these points for π 1 = Z p ⊕ Z p .