Torus bundle

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

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

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

Then [math]\displaystyle{ M(f) }[/math] is the torus bundle with monodromy [math]\displaystyle{ f }[/math].

Examples

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

References

• Jeffrey R. Weeks (2002). The Shape of Space (Second ed.). Marcel Dekker, Inc.. ISBN 978-0824707095. https://archive.org/details/shapeofspace0000week.