= Smooth fiber sum along unknotted null-homologous spheres
{title2=$X\#_{S^2}Y\cong X\#Y\#(S^1\times S^3)$}
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>.
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