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.
In order to describe a Dehn surgery,[1] one picks two oriented simple closed curves
on the corresponding boundary torus
(a curve staying in a small ball in
and having linking number +1 with
or, equivalently, a curve that bounds a disc that intersects once the component
(a curve travelling once along
or, equivalently, a curve on
such that the algebraic intersection
generate the fundamental group of the torus
, and they form a basis of its first homology group.
This gives any simple closed curve
These coordinates only depend on the homotopy class of
We can specify a homeomorphism of the boundary of a solid torus to
by having the meridian curve of the solid torus map to a curve homotopic to
As long as the meridian maps to the surgery slope
, the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism).
is called the surgery coefficient of
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
: every longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a Seifert surface.
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.
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.