# Torus bundle

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A **torus bundle**, in the sub-field of [geometric topology](/source/Geometric_topology) in [mathematics](/source/Mathematics), is a kind of [surface bundle over the circle](/source/Surface_bundle_over_the_circle), which in turn is a class of [three-manifolds](/source/Three-manifold).

## Construction

To obtain a torus bundle: let f be an [orientation](/source/Orientability)-preserving [homeomorphism](/source/Homeomorphism) of the two-dimensional [torus](/source/Torus) T to itself. Then the three-manifold M(f) is obtained by

- taking the [Cartesian product](/source/Cartesian_product) of T and the [unit interval](/source/Unit_interval) and
- gluing one component of the [boundary](/source/Boundary_(topology)) of the resulting manifold to the other boundary component via the map f.

Then M(f) is the torus bundle with [monodromy](/source/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](/source/Three-torus): the Cartesian product of three [circles](/source/Circle).

Seeing the possible kinds of torus bundles in more detail requires an understanding of [William Thurston](/source/William_Thurston)'s [geometrization](/source/Thurston's_geometrization_conjecture) program. Briefly, if f is [finite order](/source/Glossary_of_group_theory), then the manifold M(f) has [Euclidean geometry](/source/Euclidean_geometry). If f is a power of a [Dehn twist](/source/Dehn_twist) then M(f) has [Nil geometry](/source/Nil_geometry). Finally, if f is an [Anosov map](/source/Anosov_map) then the resulting three-manifold has [Sol geometry](/source/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](/source/Homology_(mathematics)) of the torus: either less than two, equal to two, or greater than two.

## References

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

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Adapted from the Wikipedia article [Torus bundle](https://en.wikipedia.org/wiki/Torus_bundle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Torus_bundle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.