Geometrically simply connected $4$–manifolds and stable cohomotopy Seiberg–Witten invariants

Abstract

We show that every positive definite closed $4$–manifold with $b_2^{+}>1$ and without $1$–handles has a vanishing stable cohomotopy Seiberg–Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented $4$–manifold with $b_2^{+}\not\equiv 1$ and $b_2^{-}\not\equiv 1\left(mod4\right)$ and without $1$–handles admits no symplectic structure for at least one orientation of the manifold. In fact, relaxing the $1$–handle condition, we prove these results under more general conditions which are much easier to verify.