The simple type conjecture for mod 2 Seiberg–Witten invariants

Abstract

We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg–Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have mod 2 simple type, and yet some have non-vanishing (mod 2) Seiberg–Witten invariants. As corollaries, we obtain adjunction inequalities and show that, under a mild topological condition, every geometrically simply connected closed 4-manifold has the vanishing mod 2 Seiberg–Witten invariant for at least one orientation.