Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 4.

Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 4.

A surgery on an $m$-dimensional manifold $M$ is the procedure of constructing a new $m$-dimensional manifold

$$M^{\prime} =cl.(M\setminus S^n\times D^{m-n})\cup_{S^n\times S^{m-n-1}}D^{n+1}\times S^{m-n-1}$$

by cutting out $S^n\times D^{m-n}\subset M$ and replacing it by $D^{n+1}\times S^{m-n-1}$.