Euclidean division

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.

Construction

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 and
• gluing 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 $f$.

Examples

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.

References

Anyone seeking more information on this subject, presented in an elementary way, may consult Jeff Weeks' book The Shape of Space.