In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of mathematics, a Dehn surgery, named after
Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
, is a construction used to modify
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s. The process takes as input a 3-manifold together with a
link. It is often conceptualized as two steps: ''drilling'' then ''filling''.
Definitions
* Given a
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
and a
link , the manifold
drilled along
is obtained by removing an open
tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the ...
of
from
. If
, the drilled manifold has
torus boundary components
. The manifold ''
drilled along
'' is also known as the
link complement, since if one removed the corresponding closed tubular neighborhood from
, one obtains a manifold diffeomorphic to
.
* Given a 3-manifold whose boundary is made of 2-tori
, we may glue in one
solid torus by a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
(resp.
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
) of its boundary to each of the torus boundary components
of the original 3-manifold. There are many inequivalent ways of doing this, in general. This process is called Dehn filling.
* Dehn surgery on a 3-manifold containing a link consists of ''drilling out'' a tubular neighbourhood of the link together with ''Dehn filling'' on all the components of the boundary corresponding to the link.
In order to describe a Dehn surgery (see ), one picks two oriented simple closed
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s
and
on the corresponding boundary torus
of the drilled 3-manifold, where
is a meridian of
(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
) and
is a longitude of
(a curve travelling once along
or, equivalently, a curve on
such that the algebraic intersection
is equal to +1).
The curves
and
generate the
fundamental group of the torus
, and they form a basis of its first
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
. This gives any simple closed curve
on the torus
two coordinates
and
, so that