Nielsen–Thurston classification
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In mathematics, Thurston's classification theorem characterizes
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 ...
s of a compact orientable surface.
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 theorem completes the work initiated by . Given a homeomorphism ''f'' : ''S'' → ''S'', there is a map ''g'' isotopic to ''f'' such that at least one of the following holds: * ''g'' is periodic, i.e. some power of ''g'' is the identity; * ''g'' preserves some finite union of disjoint simple closed curves on ''S'' (in this case, ''g'' is called ''reducible''); or * ''g'' is pseudo-Anosov. The case where ''S'' is a
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 ...
(i.e., a surface whose
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 ...
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 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 ...
, 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 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 ...
or is not
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
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 union of simple closed curves ''Γ''. Each of the resulting compact surfaces ''with
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 ...
'' is acted upon by some power (i.e. iterated composition) of ''g'', and the classification can again be applied to this homeomorphism.


The mapping class group for surfaces of higher genus

Thurston's classification applies to homeomorphisms of orientable surfaces of genus ≥ 2, but the type of a homeomorphism only depends on its associated element of the
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
''Mod(S)''. In fact, the proof of the classification theorem leads to a
canonical The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
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
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
of a hyperbolic structure on ''S''. * When ''g'' is pseudo-Anosov, there is an element of its mapping class that preserves a pair of
transverse Transverse may refer to: *Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle *Transverse flute, a flute that is held horizontally * Transverse force (or ''Euler force''), the tangen ...
singular
foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
s of ''S'', stretching the leaves of one (the ''unstable'' foliation) while contracting the leaves of the other (the ''stable'' 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 conjecture 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 ...
. The mapping torus ''Mg'' of a homeomorphism ''g'' of a surface ''S'' is 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 ...
obtained from ''S'' × ,1by gluing ''S'' × to ''S'' × using ''g''. If S has genus at least two, 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 ''H''2 × R structure; * If ''g'' is reducible, then ''Mg'' has
incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
tori, and should be cut along these tori to yield pieces that each have geometric structures (the
JSJ decomposition In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: : Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isot ...
); * If ''g'' is pseudo-Anosov, then ''Mg'' has a
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 ...
(i.e. ''H''3) 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). The hyperbolic 3-manifolds that arise in this way are called ''fibered'' because they are
surface bundles 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 ...
, and these manifolds are treated separately in the proof of Thurston's geometrization theorem for
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. Sometimes one considers only orientable Haken manifolds, in w ...
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 In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their ...
has
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. Limit sets are important because they ca ...
which is a sphere-filling curve.


Fixed point classification

The three types of surface homeomorphisms are also related to the dynamics of the mapping class group Mod(''S'') on the
Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüll ...
''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''): * If ''g'' is periodic, then there is a fixed point within ''T''(''S''); this point corresponds to a hyperbolic structure on ''S'' whose
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
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 reducible mapping classes ''g'', there is a single fixed point on the Thurston boundary; an example is a multi-twist 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 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 ...
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
into ''elliptic'', ''parabolic'', and ''hyperbolic'' types (which have fixed point structures similar to the ''periodic'', ''reducible'', and ''pseudo-Anosov'' types listed above).


See also

*
Train track map In the mathematical subject of geometric group theory, a train track map is a continuous map ''f'' from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iter ...


References

* * * ''Travaux de Thurston sur les surfaces'', Astérisque, 66-67, Soc. Math. France, Paris, 1979 * * * * {{DEFAULTSORT:Nielsen-Thurston classification Geometric topology Homeomorphisms Surfaces Theorems in topology