Torus bundle

A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three-manifolds.

Construction
To obtain a torus bundle: let $$f$$ be an orientation-preserving homeomorphism of the two-dimensional torus $$T$$ to itself. Then the three-manifold $$M(f)$$ is obtained by
 * taking the Cartesian product of $$T$$ and the unit interval and
 * gluing one component of the boundary of the resulting manifold to the other boundary component via the map $$f$$.

Then $$M(f)$$ is the torus bundle with monodromy $$f$$.

Examples
For example, if $$f$$ is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle $$M(f)$$ is the three-torus: the Cartesian product of three circles.

Seeing the possible kinds of torus bundles in more detail requires an understanding of William Thurston's geometrization program. Briefly, if $$f$$ is finite order, then the manifold $$M(f)$$ has Euclidean geometry. If $$f$$ is a power of a Dehn twist then $$M(f)$$ has Nil geometry. Finally, if $$f$$ is an Anosov map then the resulting three-manifold has Sol geometry.

These three cases exactly correspond to the three possibilities for the absolute value of the trace of the action of $$f$$ on the homology of the torus: either less than two, equal to two, or greater than two.