---
tag: ZZGC6BZNJ5
kind: concept
name: Teichmüller space
msc: [30F60, 32G15]
sources:
  - url: https://en.wikipedia.org/wiki/Teichmüller_space
same_as:
  wikidata: Q2400539
  wikipedia: Teichmüller space
---

## Summary

The Teichmüller space $T_g$ parametrises the complex structures on a fixed closed surface of genus $g$.
Each point also carries a marking, which remembers the topology up to isotopy.
For $g \ge 2$ it is a complex manifold of dimension $3g - 3$ and is homeomorphic to $\R^{6g - 6}$.
The [[mapping class group|JV5NYG913V]] acts on it properly discontinuously, and the quotient is the [[moduli space of curves|NK9ATH4WB6]] of genus $g$.

## Explanations

### L1 human

A closed surface of genus $g$, like a doughnut with $g$ holes, can be given many different shapes as a [[Riemann surface|5K0N8QMV6Q]].
The moduli space lists these shapes up to isomorphism, but surfaces with extra symmetries make it an orbifold rather than a manifold.
Teichmüller space adds a marking, a way of naming the loops on the surface, which removes this problem.
For genus $1$ a marked torus is $\C / (\Z + \tau \Z)$ with $\tau$ in the [[upper half-plane|X6XGGH5FPW]], and $T_1$ is the upper half-plane itself.