Torus bundle

In mathematics, in the sub-field of geometric topology, the torus bundle is a kind of three-dimensional manifold.

To obtain a torus bundle: let ${\displaystyle f}$ be an orientation-preserving homeomorphism of the two-dimensional torus ${\displaystyle T}$ to itself. Then the three-manifold ${\displaystyle M(f)}$ is obtained by

• taking the Cartesian product of ${\displaystyle T}$ and the unit interval and
• gluing one component of the boundary of the resulting manifold to the other boundary component via the map ${\displaystyle f}$.

Then ${\displaystyle M(f)}$ is the torus bundle with monodromy ${\displaystyle f}$.

For example, if ${\displaystyle f}$ is the identity map (i.e., the map which fixes every point of the torus) then the torus bundle ${\displaystyle M(f)}$ 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 ${\displaystyle f}$ is finite order, then the manifold ${\displaystyle M(f)}$ has Euclidean geometry. If ${\displaystyle f}$ is a power of a Dehn twist then ${\displaystyle M(f)}$ has Nil geometry. Finally, if ${\displaystyle f}$ 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 ${\displaystyle f}$ on the homology of the torus: either less than two, equal to two, or greater than two.