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- taking the Cartesian product of and the unit interval and
- gluing one component of the boundary of the resulting manifold to the other boundary component via the map .
Then is the torus bundle with monodromy .
Seeing the possible kinds of torus bundles in more detail requires an understanding of William Thurston's geometrization program. Briefly, if is finite order, then the manifold has Euclidean geometry. If is a power of a Dehn twist then has Nil geometry. Finally, if 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 on the homology of the torus: either less than two, equal to two, or greater than two.