The simple type conjecture for mod 2 Seiberg-Witten invariants

TL;DR

This paper proves that under certain cohomological conditions, all smooth structures on a broad class of 4-manifolds exhibit mod 2 Seiberg-Witten simple type, revealing new insights into their topological and geometric properties.

Contribution

It establishes the mod 2 Seiberg-Witten simple type for all closed 4-manifolds satisfying a cohomology condition, expanding understanding of their invariants.

Findings

All smooth structures on certain 4-manifolds have mod 2 simple type.

Existence of non-vanishing mod 2 Seiberg-Witten invariants in some cases.

Derived adjunction inequalities and vanishing results for geometrically simply connected 4-manifolds.

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.