Mapping torus

In mathematics Abbildungstori are topological spaces, which topological maps are described.

Definition

Be a topological space and a homeomorphism. The Abbildungstorus of f is defined as the quotient

Of respect to the equivalence relation for all.

Fiber bundle over the circle

The circuit can be regarded as the ratio of space, so that the projection defines the first factor, a fiber bundle

Conversely, each fiber bundle is displayed above the circle as Abbildungstorus a homeomorphism. The Figure is referred to as monodromy the fiber bundle.

Abbildungstori in the 3-dimensional topology

Abbildungstori play an important role in Thurston's geometrization of access to 3-manifolds.

Homeomorphisms of compact surfaces fall into one of three categories: periodic, reducible or pseudo - Anosov. Thurston proved that a 3 -dimensional hyperbolic Abbildungstorus is exactly then, when the pseudo- Anosov monodromy is.

Ian Agol 2012 has shown that every compact 3-manifold has a finite superposition, which can be represented as Abbildungstorus.

Group Theory

In group theory one defines Abbildungstori for endomorphisms of free groups. Be the free group generated by a set and an endomorphism. Then Abbildungstorus defined by the presentation