Connected-sum vanishing of the Seiberg–Witten invariant

If a closed oriented smooth four-manifold splits as a connected sum $X_1\#X_2$ with both $b_2^{+}(X_i)>0$, its ordinary Seiberg–Witten invariants of a four-manifold vanish. Combined with Taubes nonvanishing theorem, this obstructs a symplectic structure in that orientation.