In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum. The symplectic sum is the inverse of the symplectic cut, which decomposes a given manifold into two pieces. Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to deformation to the normal cone in algebraic geometry. The symplectic sum has been used to construct previously unknown families of symplectic manifolds, and to derive relationships among the Gromov–Witten invariants of symplectic manifolds. Definition Let and be two symplectic -manifolds and a symplectic -manifold, embedded as a submanifold into both and via such that the Euler classes of the normal bundles are opposite: In the 1995 paper that defined the symplectic sum, Robert Gompf proved that for any orientation-reversing isomorphism there is a canonical isotopy class of symplectic structures on the connected sum meeting several conditions of compatibility with the summands . In other words, the theorem defines a symplectic sum operation whose result is a symplectic manifold, unique up to isotopy. To produce a well-defined symplectic structure, the connected sum must be performed with special attention paid to the choices of various identifications. Loosely speaking, the isomorphism is composed with an orientation-reversing symplectic involution of the normal bundles of (or rather their corresponding punctured unit disk bundles); then this composition is used to glue to along the two copies of . Generalizations In greater generality, the symplectic sum can be performed on a single symplectic manifold containing two disjoint copies of , gluing the manifold to itself along the two copies. The preceding description of the sum of two manifolds then corresponds to the special case where consists of two connected components, each containing a copy of . Additionally, the sum can be performed simultaneously on submanifolds of equal dimension and meeting transversally. Other generalizations also exist. However, it is not possible to remove the requirement that be of codimension two in the , as the following argument shows. A symplectic sum along a submanifold of codimension requires a symplectic involution of a -dimensional annulus. If this involution exists, it can be used to patch two -dimensional balls together to form a symplectic -dimensional sphere. Because the sphere is a