Surgery

In the process of attaching a $k$-Handle to a Manifold $M$, the Boundary of $M$ is modified by a process called $(k-1)$-surgery. Surgery consists of the removal of a Tubular Neighborhood of a $(k-1)$-Sphere $\Bbb{S}^{k-1}$ from the Boundaries of $M$ and the $\dim(M)-1$ standard Sphere, and the gluing together of these two scarred-up objects along their common Boundaries.

See also Boundary, Dehn Surgery, Handle, Manifold, Sphere, Tubular Neighborhood