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