## Teichmüller's lemma and the complex structure of Teichmüller spaces

Consider the Fricke homeomorphism (given by Fricke coordinates on Teichmüller space) and let be a lift of .

Theorem 2.24 (Teichmüller's Lemma)   The set

is the kernel of , the differential of at the origin .

This theorem can be used to give a complex structure to the Teichmüller space . For that we need some general facts on mappings between complex and real manifolds, and when such mappings induce complex structures (on the real manifold). Let be a submersion from a complex Banach manifold onto a real manifold . We say that induces a well-defined almost complex structure on if for all the space inherits via the differential a unique complex structure, independent of the choice of .

Theorem 2.25   Let be a surjective submersion from a complex manifold onto a -dimensional manifold . Then there is a complex structure on , compatible with the structure, making holomorphic if and only if induces a well-defined almost complex structure on . The complex structure, if it exists,is unique.

We can apply the above result to the mapping , define from the unit ball of to , where is the Fricke mapping. To check that induces a well-defined almost complex structure one needs to use the right-translation mappings, , from the unit ball of to itself, defined by

Theorem 2.26   For a Riemann surface of type , with , there is a unique complex structure of a )-dimensional complex manifold on such that it is compatible with the real-analytic structure of the open domain and that makes a holomorphic submersion.

The details of the proof of this result can be found in [39, Chapter 3].

The Fricke homeomorphism gives then a complex structure to Teichmüller space . The mapping becomes biholomorphic and the projection is then a holomorphic submersion.

The discontinuous action of the modular group on Teichmüller space gives us some information about the moduli space of surfaces.

Theorem 2.27   The moduli space of compact surfaces of type ( ) is a normal complex space of dimension . The space is simply connected.

Pablo Ares Gastesi 2005-08-31