Seiberg–Witten invariant of a four-manifold ($S{W}_X(\mathfrak{s})$)

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.

A closed symplectic manifold of real dimension four with $b_2^{+}>1$ has a nonzero Seiberg–Witten invariant of a four-manifold in its canonical structure, with value $\pm 1$. The positive index of the intersection form hypothesis avoids chamber dependence. This provides the symplectic connected-sum obstruction.

A closed symplectic manifold of real dimension four cannot be a connected sum of two oriented smooth four-manifolds each with positive positive index of the intersection form. The total $b_2^{+}$ exceeds one; Taubes nonvanishing theorem and connected-sum vanishing of the Seiberg–Witten invariant would force the same invariant to be both nonzero and zero.