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 formula_1 be an orientation-preserving homeomorphism of the two-dimensional torus formula_2 to itself. Then the three-manifold formula_3 is obtained by
Then formula_3 is the torus bundle with monodromy formula_1. 
Examples.
For example, if formula_1 is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle formula_3 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 formula_1 is finite order, then the manifold formula_3 has Euclidean geometry. If formula_1 is a power of a Dehn twist then formula_3 has Nil geometry. Finally, if formula_1 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 formula_1 on the homology of the torus: either less than two, equal to two, or greater than two.