In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the sub-field of geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, a torus bundle is a kind of surface bundle over the circle

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 mapping tori....

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

Construction

To obtain a torus bundle: let be an
orientation

Orientability

In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

-preserving homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

 of the
two-dimensional torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

  to itself.
Then the three-manifold is obtained by

• taking the Cartesian product
  Cartesian product
  In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
  
   of and the unit interval
  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...
  
   and
• gluing one component of the boundary
  Boundary (topology)
  In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
  
   of the resulting manifold to the other boundary component via the map .

Then is the torus bundle with monodromy

Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...

 .

Examples

For example, if is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle is the three-torus: the Cartesian product of three circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

s.

Seeing the possible kinds of torus bundles in more detail
requires an understanding of William Thurston

William Thurston

William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...

's
geometrization program.
Briefly, if is finite order

Glossary of group theory

A group is a set G closed under a binary operation • satisfying the following 3 axioms:* Associativity: For all a, b and c in G, • c = a • ....

,
then the manifold has Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

.
If is a power of a Dehn twist

Dehn twist

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...

 then has
Nil geometry. Finally, if 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 on the
homology

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

of the torus: either less than two, equal to two,
or greater than two.