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.
To obtain a torus bundle: let
be an orientation-preserving homeomorphism of the two-dimensional torus
is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle
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.
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.