Surface bundle over the circle

In general, fiber bundles over the circle are a special case of mapping tori.

Here is the construction: take the Cartesian product of a surface with the unit interval.

This construction is an important source of examples both in the field of low-dimensional topology as well as in geometric group theory.

This is the fibered part of William Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen–Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups.

A simple special case of this construction (considered in Henri Poincaré's foundational paper) is that of a torus bundle.