Inmathematics,Thurston'sclassification theorem characterizeshomeomorphisms of acompact orientable surface.William Thurston's theorem completes the work initiated byJakob Nielsen (1944).
Given a homeomorphismf : S → S, there is a mapgisotopic tof such that at least one of the following holds:
The case whereS is atorus (i.e., a surface whosegenus is one) is handled separately (seetorus bundle) and was known before Thurston's work. If the genus ofS is two or greater, thenS is naturallyhyperbolic, and the tools ofTeichmüller theory become useful. In what follows, we assumeS has genus at least two, as this is the case Thurston considered. (Note, however, that the cases whereS hasboundary or is notorientable are definitely still of interest.)
The three types in this classification arenot mutually exclusive, though apseudo-Anosov homeomorphism is neverperiodic orreducible. Areducible homeomorphismg can be further analyzed by cutting the surface along the preserved union of simple closed curvesΓ. Each of the resulting compact surfaceswithboundary is acted upon by some power (i.e.iterated composition) ofg, and the classification can again be applied to this homeomorphism.
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 themapping class groupMod(S). In fact, the proof of the classification theorem leads to acanonical representative of each mapping class with good geometric properties. For example:
Thurston's original motivation for developing this classification was to find geometric structures onmapping tori of the type predicted by theGeometrization conjecture. Themapping torusMg of a homeomorphismg of a surfaceS is the3-manifold obtained fromS × [0,1] by gluingS × {0} toS × {1} usingg. If S has genus at least two, the geometric structure ofMg is related to the type ofg 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 toThurston). The hyperbolic 3-manifolds that arise in this way are calledfibered because they aresurface bundles over the circle, and these manifolds are treated separately in the proof of Thurston'sgeometrization theorem forHaken 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 arisingKleinian group haslimit set which is asphere-filling curve.
The three types of surface homeomorphisms are also related to thedynamics of the mapping class group Mod(S) on theTeichmüller spaceT(S). Thurston introduced acompactification ofT(S) that is homeomorphic to a closed ball, and to which the action of Mod(S) extends naturally. The type of an elementg of the mapping class group in the Thurston classification is related to its fixed points when acting on the compactification ofT(S):
This is reminiscent of the classification ofhyperbolicisometries intoelliptic,parabolic, andhyperbolic types (which have fixed point structures similar to theperiodic,reducible, andpseudo-Anosov types listed above).