# 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 $\phi :\partial U\to \partial U$ be an automorphism^{} of the torus, and consider the manifold ${M}^{\prime}=M\backslash {U}^{\prime}{\coprod}_{\phi}U$, 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 $\phi $.

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 $\phi $ 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 $\phi $ 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^{} |
---|---|

Canonical name | DehnSurgery |

Date of creation | 2013-03-22 13:56:07 |

Last modified on | 2013-03-22 13:56:07 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 4 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 57M99 |