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.
ConstructionEdit
To obtain a torus bundle: let <math>f</math> be an orientation-preserving homeomorphism of the two-dimensional torus <math>T</math> to itself. Then the three-manifold <math>M(f)</math> is obtained by
- taking the Cartesian product of <math>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>f</math>.
Then <math>M(f)</math> is the torus bundle with monodromy <math>f</math>.
ExamplesEdit
For example, if <math>f</math> is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle <math>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>f</math> is finite order, then the manifold <math>M(f)</math> has Euclidean geometry. If <math>f</math> is a power of a Dehn twist then <math>M(f)</math> has Nil geometry. Finally, if <math>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>f</math> on the homology of the torus: either less than two, equal to two, or greater than two.