torus bundle

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 $f$.

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.

• {{cite book |author=Jeffrey R. Weeks |title=The Shape of Space |url=https://archive.org/details/shapeofspace0000week |url-access=registration |year=2002 |publisher=Marcel Dekker, Inc. |edition=Second |ISBN=978-0824707095}}

Category:Fiber bundles

Category:Geometric topology

Category:3-manifolds