Teichmüller space

Remarks

Riemann surfaces carry their orientation from the complex structure, which is why the marking is required to preserve orientation. $T_g$ carries a natural complete metric, the Teichmüller metric, defined from the least dilatation of quasiconformal maps between marked surfaces. For $g \ge 2$, $T_g$ is a complex manifold of dimension $3g - 3$, homeomorphic to $\R^{6g - 6}$; $T_0$ is a point. The mapping class group of $S$ acts on $T(S)$ by changing the marking, and the quotient is the moduli space $M_g$.

Example: $g = 1$. For $\tau$ in the upper half-plane let $X_\tau = \C / (\Z + \tau \Z)$. Mark it by the homeomorphism $f_\tau$ from the standard torus $\R^2/\Z^2$ induced by the real linear map sending $(1, 0)$ to $1$ and $(0, 1)$ to $\tau$. Every marked torus is equivalent to exactly one such $(X_\tau, f_\tau)$, so $T_1$ is identified with the upper half-plane.

Non-example: a pair $(X, f)$ with $f$ orientation-reversing is not a marked Riemann surface, so it is not a point of $T(S)$.