Symplectic sum

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.

Let ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ be two symplectic ${\displaystyle 2n}$-manifolds and ${\displaystyle V}$ a symplectic ${\displaystyle (2n-2)}$-manifold, embedded as a submanifold into both ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ via

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