Atoroidal

In the three-dimensional topology Atoroidalität describes a relationship between the edge of a manifold and the manifold itself

An irreducible manifold is called geometrically atoroidal when each can be moved by isotopy to an edge component of in incompressible embedded 2- torus. This means that no embedded tori contains, except such as must exist obvious.

A manifold is called irreducible homotopisch atoroidal if each figure, the injective maps the fundamental group of the torus in the fundamental group of is homotopic to a map into the margin. This corresponds to the property of the fundamental group of that every subset of the form is conjugate to the fundamental group of a torus boundary component.

It can be shown that " geometric atoroidal " off " homotopisch atoridal " follows. However, the converse is not true.

The Hyperbolisierungvermutung of Thurston states that every irreducible homotopisch atoroidale manifold with infinite fundamental group carries a hyperbolic structure.

• Geometric topology