The simple type conjecture for smooth four-manifolds

Let $X$ be a closed, connected, oriented and smooth four-manifold, and let $\mathfrak{s}$ be a $\operatorname{spin}^c$ structure on $X$. The structure $\mathfrak{s}$ is a basic class if its Seiberg–Witten invariant satisfies $SW(\mathfrak{s})\ne 0$. Let $d(\mathfrak{s})$ denote the virtual dimension of the corresponding Seiberg–Witten moduli space. The four-manifold $X$ is of simple type if $d(\mathfrak{s})=0$ whenever $\mathfrak{s}$ is a basic class.

Simple type conjecture. Every closed, connected, oriented and smooth four-manifold with $b_2^+\ge 2$ is of simple type.

The conjecture asserts that all basic classes on such four-manifolds have zero-dimensional Seiberg–Witten moduli spaces. It is a fundamental conjecture concerning the structure of Seiberg–Witten invariants; the source paper studies upper bounds for virtual dimensions under additional hypotheses, rather than resolving the conjecture in full.