Fiber sum of four-manifolds
= Fiber sum of four-manifolds
{title2=$X\#_C Y$}
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>.