A torus bundle, in the sub-field of
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, is a kind of
surface bundle over the circle, which in turn is a class of
three-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds l ...
s.
Construction
To obtain a torus bundle: let
be an
orientation-preserving
homeomorphism of the two-dimensional
torus to itself. Then the three-manifold
is obtained by
* taking the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
of
and the
unit interval and
* gluing one component of the
boundary of the resulting manifold to the other boundary component via the map
.
Then
is the torus bundle with
monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
.
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
circles.
Seeing the possible kinds of torus bundles in more detail requires an understanding of
William Thurston's
geometrization program. Briefly, if
is
finite order, then the manifold
has
Euclidean geometry. If
is a power of a
Dehn twist then
has
Nil 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-dimensi ...
. 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 may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
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