Geography of irreducible 4-manifolds with order two fundamental group

Abstract

Let $R$ be a closed, oriented topological 4-manifold whose Euler characteristic and signature are denoted by $e$ and $σ$. We show that if $R$ has order two $π_1$, odd intersection form, and $2e + 3σ\geq 0$, then for all but seven $(e, σ)$ coordinates, $R$ admits an irreducible smooth structure. We accomplish this by performing a variety of operations on irreducible simply-connected 4-manifolds to build 4-manifolds with order two $π_1$. These techniques include torus surgeries, symplectic fiber sums, rational blow-downs, and numerous constructions of Lefschetz fibrations, including a new approach to equivariant fiber summing.