Surgery on a smooth manifold ($S^{\lambda -1}\times D^{m+1-\lambda }⇝D^{\lambda }\times S^{m-\lambda }$)

A framed embedding of the left-hand region in an $m$-dimensional manifold specifies an index-$\lambda$ surgery. Remove its interior and glue the right-hand region along their common boundary $S^{\lambda -1}\times S^{m-\lambda }$. The cobordism realizing this change attaches an index-$\lambda$ handle of dimension $m+1$. On an oriented surface, index-two surgery cuts out an annular neighbourhood of a simple closed curve and caps its two boundary circles.