Smooth four-manifold

A smooth four-manifold is a four-dimensional smooth manifold. Its intersection form and gauge-theoretic invariants impose strong constraints on its possible smooth and symplectic structures.

Remove tubular neighborhoods of diffeomorphic embedded oriented surfaces with opposite normal Euler numbers, and glue their boundary circle bundles by a fiber-reversing identification. The resulting smooth manifold is a fiber sum. The square-zero case uses trivial normal bundles. Additional symplectic hypotheses give a symplectic fiber sum along a square-zero surface.

For standard unknotted spheres contained in four-balls in $X$ and $Y$, the standard fiber-reversing gluing gives the displayed connected sum. The complement of an unknotted $S^2\times D^2$ in $S^4$ is $S^1\times D^3$, and its double is $S^1\times S^3$. The portions of $X$ and $Y$ outside the four-balls remain as connected summands. Taking both ambient manifolds to be the Complex projective plane gives a smooth fiber sum with no symplectic form, by the symplectic connected-sum obstruction.

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.