nLab
Teichmüller theory

Contents

Idea

The Teichmüller space 𝒯 Σ\mathcal{T}_{\Sigma} of a (closed) 2-dimensional manifold Σ\Sigma is the moduli space of complex structures on Σ\Sigma, where two complex structures are identified if they are taken into each other by a homeomorphism ϕ:ΣΣ\phi \colon \Sigma \to \Sigma which is isotopic to the identity.

Since the diffeomorphism class of a closed manifold of dimension 2 is given by its genus gg\in \mathbb{Z}, typically one speaks of the Teichmüller space 𝒯 g\mathcal{T}_g for each gg. More generally one considers Σ\Sigmas equipped with nn punctures/boundary circle and then writes 𝒯 g,n\mathcal{T}_{g,n}.

Teichmüller space is a covering space for the moduli space of curves over the complex numbers (see below). When also the singular nodal curve is included one speaks of the augmented Teichmüller space which in turn is a cover of the Deligne-Mumford compactification of the moduli space of complex curves.

Properties

Complex structure on Teichmüller space

Teichmüller space itself carries a complex structure.

This was envisioned in (Teichmüller 44) and proven in (Ahlfors 60, Bers 60).

Review includes (Schumacher, section 2)

Relation to moduli stack of complex curves / Riemann surfaces

Quotienting the space of complex structures on Σ\Sigma by all homeomorphisms produces what is called the moduli space of curves (over the complex numbers). Closely related to the moduli space of conformal structures hence of Riemann surfaces.

Hence the Riemann moduli space is the orbifold quotient of Teichmüller space by the mapping class group of Σ\Sigma. See (Hubbard-Koch 13).

References

The original articles are

  • Oswald Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Preußische Akademie der Wissenschaften, nat. Kl. 22 1-197 (1939)

  • Oswald Teichmüller, Veränderliche Riemannsche Flächen, Deutsche Math. 7 344-359 (1944)

The complex structure on Teichmüller spaces was fully established in

  • L. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, in Analytic Functions, Princeton University Press (1960)

  • Lipman Bers, Spaces of Riemann surfaces, Proc. Int. Cong. of Mathematics 1958, Cambridge 1960

Reviews include

  • Georg Schumacher, The theory of Teichmüller spaces – A view towards moduli spaces of Kähler manifolds (pdf)

  • Wikipedia, Teichmüller space

Further developments include

  • Alexander Grothendieck, Techniques de construction en géométrie analytique. I. Description axiomatique de l’espace de Teichmüller et de ses variantes Séminaire Henri Cartan 13 no. 1, Exposés No. 7 and 8 (Paris: Secrétariat Mathématique). (1960/1961) (Numdam)

  • John Hubbard, Sarah Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves (arXiv:1301.0062)

Revised on September 8, 2014 19:45:42 by David Corfield (91.125.67.210)