  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 $b_2^-\not\equiv 1\pmod{4}$
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.
