# 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, *M _{f}* 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 *M _{f}* 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

*M*is hyperbolic if and only if

_{f}*f*is a pseudo-Anosov homeomorphism of

*S*.

^{[1]}

## References[edit]

**^**W. Thurston,*On the geometry and dynamics of diffeomorphisms of surfaces*, Bulletin of the American Mathematical Society, vol. 19 (1988), pp. 417–431