In
mathematics, the mapping torus in
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of a
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 ...
''f'' of some
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' to itself is a particular geometric construction with ''f''. Take the
cartesian product of ''X'' with a
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
''I'', and glue the boundary components together by the static homeomorphism:
:
The result is a
fiber bundle whose base is a circle and whose fiber is the original space ''X''.
If ''X'' is a
manifold, ''M
f'' will be a manifold of dimension one higher, and it is said to
"fiber over the circle".
As a simple example, let
be the circle, and
be the inversion
, then the mapping torus is the Klein bottle.
Mapping tori of surface homeomorphisms play a key role in the theory of
3-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 lo ...
s and have been intensely studied. If ''S'' is a closed surface of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''g'' ≥ 2 and if ''f'' is a self-homeomorphism of ''S'', the mapping torus ''M
f'' is a
closed 3-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 lo ...
that
fibers over the
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 ...
with fiber ''S''. A
deep result
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in ...
of
Thurston states that in this case the
3-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 lo ...
''M
f'' is
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
if and only if ''f'' is a
pseudo-Anosov homeomorphism of ''S''.
[W. Thurston, ''On the geometry and dynamics of diffeomorphisms of surfaces'', ]Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...
, vol. 19 (1988), pp. 417–431
References
{{reflist
General topology
Geometric topology
Homeomorphisms