Geography of irreducible 4-manifolds with order two fundamental group

TL;DR

This paper investigates the conditions under which certain 4-manifolds with order two fundamental groups admit irreducible smooth structures, expanding the understanding of their geometric and topological properties.

Contribution

It introduces new methods and constructions, including equivariant fiber summing, to build 4-manifolds with specified fundamental groups and smooth structures.

Findings

Most 4-manifolds with given invariants admit irreducible smooth structures

New techniques like equivariant fiber summing are developed

Conditions on Euler characteristic and signature are established

Abstract

Let $R$ be a closed, oriented topological 4-manifold whose Euler characteristic and signature are denoted by $e$ and $\sigma$. We show that if $R$ has order two $\pi_1$, odd intersection form, and $2e + 3\sigma \geq 0$, then for all but seven $(e, \sigma)$ 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 $\pi_1$. 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.