= Symplectic sum
{wiki}
Suppose <symplectic manifolds> $(M_1^{2n},\omega_1)$ and $(M_2^{2n},\omega_2)$ contain compact codimension-two <symplectic submanifolds> $V_1,V_2$. A <symplectomorphism> $V_1\to V_2$ identifies their restricted forms. To form their symplectic sum, their <symplectic normal bundles>, viewed as <complex line bundles>, must have opposite <First Chern classes> under this identification. Choosing a fiber-reversing normal-bundle identification then permits removal of tubular neighborhoods and gluing of their circle boundaries.<Lefschetz pencils, branched covers and symplectic invariants by Denis Auroux and Ivan Smith>[Section 1, symplectic-sum construction, manuscript page 8]{ref}
The local smoothing resembles $xy=\varepsilon$ in the two normal directions. After adjusting the forms near the gluing locus, the joined manifold carries a <symplectic form>. The identification is part of the construction, so the result is not determined solely by the two ambient manifolds.<A new construction of symplectic manifolds by Robert E. Gompf>[Symplectic-sum construction]{ref}
In real <dimension> four, the surfaces must have matching <genus> and <symplectic area>, and their <self-intersection numbers> must sum to zero. For square-zero surfaces the normal bundles are trivial, giving the <symplectic fiber sum along a square-zero surface>.
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