Torus bundle
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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.
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 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
Text derived from https://en.wikipedia.org/w/index.php?oldid=934944893 under CC BY-SA 4.0; modifications have been made.