Dehn surgery

Let M be a smooth 3-manifold, and KM a smooth knot. Since K is an embedded submanifold, by the tubular neighborhood theorem there is a closed neighborhoodMathworldPlanetmathPlanetmath U of K diffeomorphicMathworldPlanetmath to the solid torus D2×S1. We let U denote the interior of U. Now, let φ:UU be an automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the torus, and consider the manifold M=M\UφU, which is the disjoint unionMathworldPlanetmath of M\U and U, with points in the boundary of U identified with their images in the boundary of M\U under φ.

It’s a bit hard to visualize how this actually results in a different manifold, but it generally does. For example, if M=S3, the 3-sphere, K is the trivial knotMathworldPlanetmath, and φ is the automorphism exchanging meridians and parallelsMathworldPlanetmathPlanetmath (i.e., since UD2×S1, get an isomorphismPlanetmathPlanetmathPlanetmath US1×S1, and φ is the map interchanging to the two copies of S1), then one can check that MS1×S2 (S3\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 D2 paste along the edges to make S2).

Every compactPlanetmathPlanetmath 3-manifold can obtained from the S3 by surgery around finitely many knots.

Title Dehn surgeryMathworldPlanetmath
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