Teichmuller space

Teichmuller space $ \mathcal{T}_g$ for genus $ g$ parametrizes pairs $ (C,\phi)$ of a genus $ g$ Riemann surface $ C$ and a homeomorphism $ \phi:C \rightarrow C_{0}$ to a fixed surface of genus $ g$, up to isotopy of $ \phi$. This is equivalent to parametrizing Riemann surfaces $ C$ with a choice of normalized generators of $ \pi_{1}(C)$. It is naturally an open subset of $ \mathbb{C}^{3g-3}$, homeomorphic to a ball. It admits an action of the mapping class group $ \Gamma_g$ with finite stabilizers and with quotient $ \mathcal{M}_g$.

Jeffrey Herschel Giansiracusa 2005-06-27