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

be an orientation-preserving homeomorphism of the two-dimensional torus is obtained by

• taking the Cartesian product and the unit interval and

• gluing one component of the boundary.

is the torus bundle with monodromy.

Examples

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 is finite order has Euclidean geometry is a power of a Dehn twist has Nil geometry is an Anosov map then the resulting three-manifold has Sol geometry.

on the homology of the torus: either less than two, equal to two, or greater than two.

References