##

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