Torus bundle

In mathematics, in the sub-field of geometric topology, a torus bundle is a kind of surface bundle over the circle, which in turn are a class of three-manifolds. In mathematics, in the sub-field of geometric topology, a torus bundle is a kind of surface bundle over the circle, which in turn are a class of three-manifolds. To obtain a torus bundle: let f {displaystyle f} be an orientation-preserving homeomorphism of the two-dimensional torus T {displaystyle T} to itself. Then the three-manifold M ( f ) {displaystyle M(f)} is obtained by Then M ( f ) {displaystyle M(f)} is the torus bundle with monodromy f {displaystyle f} . For example, if f {displaystyle f} is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle M ( f ) {displaystyle 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 {displaystyle f} is finite order, then the manifold M ( f ) {displaystyle M(f)} has Euclidean geometry. If f {displaystyle f} is a power of a Dehn twist then M ( f ) {displaystyle M(f)} has Nil geometry. Finally, if f {displaystyle 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 {displaystyle f} on the homology of the torus: either less than two, equal to two, or greater than two.