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
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.
