Geometrically simply connected 4-manifolds and stable cohomotopy Seiberg-Witten invariants

Kouichi Yasui

Published 2018-07-30Version 1

We show that every positive definite closed oriented 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 $b_2^+\equiv b_2^-\equiv 3\pmod{4}$ and without 1-handles admits no symplectic structure for at least one orientation of the manifold. In fact, we prove these results for non-simply connected 4-manifolds as well, by relaxing the condition `without 1-handles'.