Thurston's classification theorem

In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact surface to itself. William Thurston's theorem completes the work initiated by Nielsen in the 1930s.

Given a homeomorphism f : S → S, there is a map g isotopic to f such that either:

• g is periodic;
• g preserves some multi-curve 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 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.

(More to come on how Teichmuller theory is used.)