Introduction to Teichmüller theory, old and new, II (original) (raw)

2009, Handbook of Teichmüller Theory, Volume II

Teichmüller theory is one of those few wonderful subjects which bring together, at an equally important level, fundamental ideas coming from different fields. Among the fields related to Teichmüller theory, one can surely mention complex analysis, hyperbolic geometry, the theory of discrete groups, algebraic geometry, low-dimensional topology, differential geometry, Lie group theory, symplectic geometry, dynamical systems, number theory, topological quantum field theory, string theory, and there are many others. Let us start by recalling a few definitions. Let S g,p be a connected orientable topological surface of genus g ≥ 0 with p ≥ 0 punctures. Any such surface admits a complex structure, that is, an atlas of charts with values in the complex plane C and whose coordinate changes are holomorphic. In the classical theory, one considers complex structures S g,p for which each puncture of S g,p has a neighborhood which is holomorphically equivalent to a punctured disk in C. To simplify the exposition, we shall suppose that the orientation induced on S g,p by the complex structure coincides with the orientation of this surface. Homeomorphisms of the surface act in a natural manner on atlases, and two complex structures on S g,p are said to be equivalent if there exists a homeomorphism of the surface which is homotopic to the identity and which sends one structure to the other. The surface S g,p admits infinitely many non-equivalent complex structures, except if this surface is a sphere with at most three punctures. To say things precisely, we introduce some notation. Let C g,p be the space of all complex structures on S g,p and let Diff + (S g,p) be the group of orientation-preserving diffeomorphisms of S g,p. We consider the action of Diff + (S g,p) by pullback on C g,p. The quotient space M g,p = C g,p /Diff + (S g,p) is called Riemann's moduli space of deformations of complex structures on S g,p. This space was considered by G. F. B. Riemann in his famous paper on Abelian functions, Theorie der Abel'schen Functionen, Crelle's Journal, Band 54 (1857), in which he studied moduli for algebraic curves. In that paper, Riemann stated, without giving a formal proof, that the space of deformations of equivalence classes of conformal structures on a closed orientable surface of genus g ≥ 2 is of complex dimension 3g − 3. The Teichmüller space T g,p of S g,p was introduced in the 1930s by Oswald Teichmüller. It is defined as the quotient of the space C g,p of complex structures by the group Diff + 0 (S g,p) of orientation-preserving diffeomorphisms of S g,p that are isotopic to the identity. The group Diff + 0 (S g,p) is a normal subgroup of Diff + (S g,p), and the quotient group g,p = Diff + (S g,p)/Diff + 0 (S g,p) is called the mapping class group of S g,p (sometimes also called the modular group, or the Teichmüller modular group)