In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling (also known as Dehn-tistry).
Contents
Definitions
We can pick two oriented simple closed curves m and ℓ on the boundary torus of the 3-manifold that generate the fundamental group of the torus. This gives any simple closed curve
We can specify a homeomorphism of the boundary of a solid torus to T by having the meridian curve of the solid torus map to a curve homotopic to
In the case of links in the 3-sphere or more generally an oriented homology sphere, there is a canonical choice of the meridians and longitudes of T. The longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a Seifert surface. The meridian is the curve that bounds a disc in the tubular neighbourhood of the link. When the ratios p/q are all integers, the surgery is called an integral surgery. Such surgeries are closely related to handlebodies, cobordism and Morse functions.
Results
Every closed, orientable, connected 3-manifold is obtained by performing Dehn surgery on a link in the 3-sphere. This result, the Lickorish-Wallace theorem, was first proven by Wallace in 1960 and independently by Lickorish in a stronger form in 1962. Via the now well-known relation between genuine surgery and cobordism, this result is equivalent to the theorem that the oriented cobordism group of 3-manifolds is trivial, originally due to Vladimir Abramovich Rokhlin in 1951.
Since orientable 3-manifolds can all be generated by suitably decorated links, one might ask how distinct surgery presentations of a given 3-manifold might be related. The answer is called the Kirby calculus.