Fiber sum of four-manifolds ($X{\#}_{C}Y$)

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.