The circuit can be used as quotient space with be construed so defines the projection on the second factor , a fiber bundle
.
Conversely, each fiber bundle above the circle can be represented as a mapping torus of a homeomorphism . The mapping is called the monodromy of the fiber bundle.
Homeomorphisms of compact surfaces fall into one of three categories: periodic, reducible, or pseudo-anosov. Thurston has proven that a 3-dimensional mapping torus is hyperbolic if and only if the monodromy is pseudo-Anosov.
In 2012, Ian Agol showed that every compact 3-manifold has a finite overlay , which can be represented as a mapping torus.
Group theory
In group theory, mapping gates are defined for endomorphisms of free groups . Let be the free group created by a set and be an endomorphism. Then the mapping torus is defined by the presentation