Symplectic sum

A symplectic sum removes tubular neighborhoods of symplectic submanifolds whose normal bundles have opposite Euler classes and glues their boundaries by a fiber-reversing identification.

A symplectic sphere of self-intersection $-4$ can be rationally blown down by taking the symplectic sum with ${\mathbb{CP}}^2$ along the sphere and a smooth conic of self-intersection $+4$. Equivalently, its disk-bundle neighborhood is replaced by the rational ball ${\mathbb{CP}}^2\setminus \nu (Q)$ whose boundary is the lens space $L(4,1)$.