Teichmüller space
/ TYKH-mool-er /
A closed surface of genus g ≥ 2 can be given hyperbolic geometry, but unlike its three-dimensional cousins it can be given that geometry in continuously many genuinely different ways — squeeze a handle here, fatten one there. Teichmüller space is the space of all those distinct shapes: every point is one hyperbolic (equivalently, one complex) structure on the surface, recorded together with a 'marking' that pins down how the surface is labeled so we do not accidentally call two shapes the same just because some self-symmetry relates them.
Precisely, for a closed orientable surface S of genus g, the Teichmüller space T(S) is the set of pairs (X, f) where X is a hyperbolic (or complex) surface and f: S -> X is a homeomorphism (the marking), modulo the equivalence (X1, f1) ~ (X2, f2) when there is an isometry (or biholomorphism) h: X1 -> X2 with h after f1 isotopic to f2. The marking is essential: it remembers which loop of S became which loop of X. Teichmüller proved T(S) is homeomorphic to an open ball; for genus g ≥ 2 it has real dimension 6g-6 (and complex dimension 3g-3), and it is a complete metric space under the Teichmüller metric measuring the least quasiconformal distortion between marked structures.
Teichmüller space is the universal deformation space of geometric structures on a surface — exactly the moduli that Mostow rigidity forbids in dimension ≥3. The mapping class group (isotopy classes of self-homeomorphisms of S) acts on T(S) by changing the marking, properly discontinuously; the quotient is moduli space M(S), the space of unmarked hyperbolic structures (an orbifold, not a manifold, because some surfaces have symmetries). Coordinates such as Fenchel-Nielsen lengths-and-twists parametrize T(S) explicitly, and the Weil-Petersson metric makes it a (incomplete) Kähler manifold of negative curvature — the playground where surface dynamics, complex analysis, and geometry meet.
For a genus-2 surface, decompose it into two pairs of pants by cutting along 3 disjoint simple closed curves. Each cut curve contributes one length parameter (the length of the geodesic in its homotopy class, any positive real) and one twist parameter (how much you rotate before regluing, any real). That is 3 lengths and 3 twists: 6 = 6g-6 real coordinates with g=2, freely chosen, and they give an explicit homeomorphism T(S) ≅ (0,∞)^3 x R^3 ≅ R^6.
Fenchel-Nielsen coordinates on genus-2 Teichmüller space: 3 lengths and 3 twists give a global chart onto R^6.
Teichmüller space is NOT moduli space — the marking is what separates them. T(S) is a smooth ball; moduli space M(S) = T(S) / MCG(S) is its quotient and is an orbifold with singular points wherever a surface has extra symmetry. Conflating the two is the single most common error here.