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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Thurston's
classification theorem In mathematics, a classification theorem answers the classification problem: "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues rela ...
characterizes
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
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. Thurst ...
'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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
(i.e., a surface whose
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
is one) is handled separately (see
torus bundle A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three-manifolds. Construction To obtain a torus bundle: let f be an orientability, orientation-preserv ...
) and was known before Thurston's work. If the genus of ''S'' is two or greater, then ''S'' is naturally
hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
, 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 union of simple closed curves ''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.


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 exampl ...
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 topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ...
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 (now a theorem) 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 theor ...
. The mapping torus ''Mg'' of a homeomorphism ''g'' of a surface ''S'' is the
3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
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 Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...
tori, and should be cut along these tori to yield pieces that each have geometric structures (the JSJ decomposition); * If ''g'' is pseudo-Anosov, then ''Mg'' has a
hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
(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, and these manifolds are treated separately in the proof of Thurston's geometrization theorem for Haken manifolds. 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 (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
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 c ...
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. Teichmülle ...
''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 (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element ...
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 may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
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


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