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In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable 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 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 (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 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 ''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: * When ''g'' is periodic, there is an element of its mapping class that is an isometry 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 singular foliations of ''S'', stretching the leaves of one (the ''unstable'' foliation) while contracting the leaves of the other (the ''stable'' foliation). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Nielsen–Thurston classification」の詳細全文を読む スポンサード リンク
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