A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of

surface bundle over the circle In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mappi ...

, which in turn is a class of three-manifolds.

Construction

To obtain a torus bundle: let

f

be an

orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...

-preserving

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

of the two-dimensional

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

T

to itself. Then the three-manifold

M(f)

is obtained by * taking the Cartesian product of

T

and the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...

and * gluing one component of the

boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...

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 three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, \mathbb^3 = S^1 \times S^1 \times S^1. In contrast, the usual torus is the Cartesian product of only two ...

: the Cartesian product of three

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

s. Seeing the possible kinds of torus bundles in more detail requires an understanding of

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thursto ...

's geometrization program. Briefly, if

f

finite order In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the sub ...

, then the manifold

M(f)

has

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

. If

f

is a power of a

Dehn twist In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold). Definition Suppose that ''c'' is a simple closed curve in a closed, orientable surface ''S''. Let ...

then

M(f)

has Nil geometry. Finally, if

f

is an

Anosov map In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...

then the resulting three-manifold has

Sol geometry In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...

. 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

*{{cite book , author=Jeffrey R. Weeks , title=The Shape of Space , url=https://archive.org/details/shapeofspace0000week , url-access=registration , year=2002 , publisher=Marcel Dekker, Inc. , edition=Second , ISBN=978-0824707095 Fiber bundles Geometric topology 3-manifolds