Dehn surgery

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".
Definitions.
In order to describe a Dehn surgery (see ), one picks two oriented simple closed curves formula_16 and formula_17 on the corresponding boundary torus formula_15 of the drilled 3-manifold, where formula_16 is a meridian of formula_20 (a curve staying in a small ball in formula_1 and having linking number +1 with formula_20 or, equivalenty, a curve that bounds a disc that intersects once the component formula_20) and formula_17 is a longitude of formula_15 (a curve travelling once along formula_20 or, equivalenty, a curve on formula_15 such that the algebraic intersection formula_28 is equal to +1). 
The curves formula_16 and formula_17 generate the fundamental group of the torus formula_15, and they form a basis of its first homology group. This gives any simple closed curve formula_32 on the torus formula_15 two coordinates formula_34 and formula_35, so that formula_36. These coordinates only depend on the homotopy class of formula_32.
We can specify a homeomorphism of the boundary of a solid torus to formula_15 by having the meridian curve of the solid torus map to a curve homotopic to formula_32. As long as the meridian maps to the surgery slope formula_40, the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio formula_41 is called the surgery coefficient of formula_20.
In the case of links in the 3-sphere or more generally an oriented integral homology sphere, there is a canonical choice of the longitudes formula_17: every longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a Seifert surface. 
When the ratios formula_44 are all integers (note that this condition does not depend on the choice of the longitudes, since it corresponds to the new meridians intersecting exactly once the ancient meridians), 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 Andrew H. Wallace in 1960 and independently by W. B. R. 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, a theorem originally proved by 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.