Nielsen-Thurston classification
Encyclopedia
In mathematics
, Thurston's classification theorem characterizes homeomorphism
s of a compact surface. William Thurston
's theorem completes the work initiated by .
Given a homeomorphism f : S → S, there is a map g isotopic to f such that either:
The case where S is a torus
(i.e., a surface whose genus
is one) is handled separately (see torus bundle
) and was known before Thurston's work. If the genus of S is two or greater, then S is naturally hyperbolic
, and the tools of Teichmüller theory
become useful. In what follows, we assume S has genus at least two, as this is the case Thurston considered. (Note, however, that the cases where S has boundary
or is not orientable are definitely still of interest.)
The three types in this classification are not mutually exclusive, though a pseudo-Anosov homeomorphism is never periodic or reducible. A reducible homeomorphism g can be further analyzed by cutting the surface along the preserved multi-curve Γ. Each of the resulting compact surfaces with boundary
is acted upon by some power (i.e. iterated composition
) of g, and the classification can again be applied to this homeomorphism.
Mod(S). In fact, the proof of the classification theorem leads to a canonical
representative of each mapping class with good geometric properties. For example:
. The mapping torus
Mg of a homeomorphism g of a surface S is the 3-manifold
obtained from S × [0,1] by gluing S × {0} to S × {1} using g. The geometric structure of Mg is related to the type of g in the classification as follows:
The first two cases are comparatively easy, while the existence of a hyperbolic structure on the mapping torus of a pseudo-Anosov homeomorphism is a deep and difficult theorem (also due to Thurston
). The hyperbolic 3-manifolds that arise in this way are called fibered because they are surface bundles over the circle
, and these manifolds are treated separately in the proof of Thurston's geometrization theorem
for Haken manifold
s. Fibered hyperbolic 3-manifolds have a number of interesting and pathological properties; for example, Cannon and Thurston showed that the surface subgroup of the arising Kleinian group
has limit set
which is a sphere-filling curve
.
of the mapping class group Mod(S) on the Teichmüller space
T(S). Thurston introduced a compactification
of T(S) that is homeomorphic to a closed ball, and to which the action of Mod(S) extends naturally. The type of an element g of the mapping class group in the Thurston classification is related to its fixed points when acting on the compactification of T(S):
This is reminiscent of the classification of hyperbolic
isometries into elliptic, parabolic, and hyperbolic types (which have fixed point structures similar to the periodic, reducible, and pseudo-Anosov types listed above).
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...
, Thurston's classification theorem characterizes 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...
s of a compact surface. 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 theorem completes the work initiated by .
Given a homeomorphism f : S → S, there is a map g isotopic to f such that either:
- g is periodicPeriodic functionIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
; - g preserves some multi-curve on S (in this case, g is called reducible); or
- g is pseudo-AnosovPseudo-Anosov mapIn mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus...
.
The case where S is a 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...
(i.e., a surface whose genus
Genus
In biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
is one) is handled separately (see torus bundle
Torus bundle
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.-Construction:To obtain a torus bundle: let f be an...
) and was known before Thurston's work. If the genus of S is two or greater, then S is naturally hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, and the tools of Teichmüller theory
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
become useful. In what follows, we assume S has genus at least two, as this is the case Thurston considered. (Note, however, that the cases where S has 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...
or is not orientable are definitely still of interest.)
The three types in this classification are not mutually exclusive, though a pseudo-Anosov homeomorphism is never periodic or reducible. A reducible homeomorphism g can be further analyzed by cutting the surface along the preserved multi-curve Γ. Each of the resulting compact surfaces with boundary
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
is acted upon by some power (i.e. iterated composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
) of g, and the classification can again be applied to this homeomorphism.
The mapping class group
Thurston's classification applies to homeomorphisms of surfaces, but the type of a homeomorphism only depends on its associated element of the mapping class groupMapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
Mod(S). In fact, the proof of the classification theorem leads to a canonical
Canonical form
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....
representative of each mapping class with good geometric properties. For example:
- When g is periodic, there is an element of its mapping class that is an isometryIsometryIn mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
of a hyperbolic structureHyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
on S. - When g is pseudo-Anosov, there is an element of its mapping class that preserves a pair of transverse singular foliationFoliationIn mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
s of S, stretching the leaves of one (the stable foliation) while contracting the leaves of the other (the unstable foliation).
Mapping tori
Thurston's original motivation for developing this classification was to find geometric structures on mapping tori of the type predicted by the Geometrization conjectureGeometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
. The mapping torus
Mapping torus
In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f...
Mg of a homeomorphism g of a surface S is the 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
obtained from S × [0,1] by gluing S × {0} to S × {1} using g. The geometric structure of Mg is related to the type of g in the classification as follows:
- If g is periodic, then Mg has an H2 × R structure;
- If g is reducibleReduction (mathematics)In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible is called "reducing a fraction"...
, then Mg has incompressibleIncompressible surfaceIn mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
toriTorusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
, and should be cut along these tori to yield pieces that each have geometric structures (the JSJ decompositionJSJ decompositionIn mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson...
); - If g is pseudo-AnosovPseudo-Anosov mapIn mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus...
, then Mg has a hyperbolicHyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
(i.e. H3) structure.
The first two cases are comparatively easy, while the existence of a hyperbolic structure on the mapping torus of a pseudo-Anosov homeomorphism is a deep and difficult theorem (also due to 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...
). The hyperbolic 3-manifolds that arise in this way are called fibered because they are surface bundles 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....
, and these manifolds are treated separately in the proof of Thurston's geometrization theorem
Geometrization conjecture
Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
for Haken manifold
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
s. Fibered hyperbolic 3-manifolds have a number of interesting and pathological properties; for example, Cannon and Thurston showed that the surface subgroup of the arising Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
has limit set
Limit set
In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time...
which is a sphere-filling curve
Space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square...
.
Fixed point classification
The three types of surface homeomorphisms are also related to the dynamicsDynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
of the mapping class group Mod(S) on the Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
T(S). Thurston introduced a compactification
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of T(S) that is homeomorphic to a closed ball, and to which the action of Mod(S) extends naturally. The type of an element g of the mapping class group in the Thurston classification is related to its fixed points when acting on the compactification of T(S):
- If g is periodic, then there is a fixed point within T(S); this point corresponds to a hyperbolic structureHyperbolic geometryIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
on S whose isometry groupIsometry groupIn mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
contains an element isotopic to g; - If g is pseudo-Anosov, then g has no fixed points in T(S) but has a pair of fixed points on the Thurston boundary; these fixed points correspond to the stable and unstable foliations of S preserved by g.
- For some reducibleReduction (mathematics)In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible is called "reducing a fraction"...
mapping classes g, there is a single fixed point on the Thurston boundary; an example is a multi-twistDehn twistIn geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...
along a pants decomposition Γ. In this case the fixed point of g on the Thurston boundary corresponds to Γ.
This is reminiscent of the classification of hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
isometries into elliptic, parabolic, and hyperbolic types (which have fixed point structures similar to the periodic, reducible, and pseudo-Anosov types listed above).