Mapping torus

In mathematics, the mapping torus in topology of a homeomorphism "f" of some topological space "X" to itself is a particular geometric construction with "f". Take the cartesian product of "X" with a closed interval "I", and glue the boundary components together by the static homeomorphism:
The result is a fiber bundle whose base is a circle and whose fiber is the original space "X". 
If "X" is a manifold, "Mf" will be a manifold of dimension one higher, and it is said to "fiber over the circle". 
Mapping tori of surface homeomorphisms play a key role in the theory of 3-manifolds and have been intensely studied. If "S" is a closed surface of genus "g" ≥ 2 and if "f" is a self-homeomorphism of "S", the mapping torus "Mf" is a closed 3-manifold that fibers over the circle with fiber "S". A deep result of Thurston states that in this case the 3-manifold "Mf" is hyperbolic if and only if "f" is a pseudo-Anosov homeomorphism of "S".