Disc theorem
Encyclopedia
In the area of mathematics known as differential topology
, the disc theorem of states that two embedding
s of a closed k-disc into a connected
n-manifold
are ambient isotopic
provided that if k = n the two embeddings are equioriented.
The disc theorem implies that the connected sum
of smooth oriented manifolds is well defined.
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, the disc theorem of states that two embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
s of a closed k-disc into a connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
n-manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
are ambient isotopic
Ambient isotopy
In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an "ambient space", a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the...
provided that if k = n the two embeddings are equioriented.
The disc theorem implies that the connected sum
Connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...
of smooth oriented manifolds is well defined.