# Dehn surgery

Let $M$ be a smooth 3-manifold, and $K\subset M$ a smooth knot. Since $K$ is an embedded submanifold, by the tubular neighborhood theorem there is a closed neighborhood $U$ of $K$ diffeomorphic to the solid torus $D^{2}\times S^{1}$. We let $U^{\prime}$ denote the interior of $U$. Now, let $\varphi:\partial U\to\partial U$ be an automorphism of the torus, and consider the manifold $M^{\prime}=M\backslash U^{\prime}\coprod_{\varphi}U$, which is the disjoint union of $M\backslash U^{\prime}$ and $U$, with points in the boundary of $U$ identified with their images in the boundary of $M\backslash U^{\prime}$ under $\varphi$.

It’s a bit hard to visualize how this actually results in a different manifold, but it generally does. For example, if $M=S^{3}$, the 3-sphere, $K$ is the trivial knot, and $\varphi$ is the automorphism exchanging meridians and parallels (i.e., since $U\cong D^{2}\times S^{1}$, get an isomorphism $\partial U\cong S^{1}\times S^{1}$, and $\varphi$ is the map interchanging to the two copies of $S^{1}$), then one can check that $M^{\prime}\cong S^{1}\times S^{2}$ ($S^{3}\backslash U$ is also a solid torus, and after our automorphism, we glue the two solid tori, meridians to meridians, parallels to parallels, so the two copies of $D^{2}$ paste along the edges to make $S^{2}$).

Every compact 3-manifold can obtained from the $S^{3}$ by surgery around finitely many knots.

Title Dehn surgery DehnSurgery 2013-03-22 13:56:07 2013-03-22 13:56:07 bwebste (988) bwebste (988) 4 bwebste (988) Definition msc 57M99