On Grothendieck's construction of Teichm\"uller space

TL;DR

This paper reviews Grothendieck's reformulation of Teichmüller space construction, emphasizing the categorical approach and its impact on the development of analytic geometry and moduli spaces.

Contribution

It explains Grothendieck's categorical reinterpretation of Teichmüller space and the universal Teichmüller curve, connecting complex analysis, algebraic geometry, and category theory.

Findings

Grothendieck's approach recasts Teichmüller space using fiber bundles and functors.

The survey clarifies complex notions for geometers and algebraic geometers.

It highlights the evolution of moduli space concepts in relation to Teichmüller theory.

Abstract

In his 1944 paper Ver\"anderliche Riemannsche Fl\"achen , Teichm\"uller defined a structure of complex manifold on the set of isomorphism classes of marked closed Riemann surfaces of genus g. The complex manifold he obtained is the space called today Teichm\"uller space. In the same paper, Teichm\"uller introduced the so-called universal Teichm\"uller curve -- a space over Teichm\"uller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichm\"uller proved the existence of the Teichm\"uller curve as a space of Riemann surfaces parametrized by an analytic space, with an existence and uniqueness theorem establishing this analytic structure. This result was later reformulated and proved by Grothendieck in a series of ten lectures he gave at Cartan's seminar in 1960-1961. In his approach , Grothendieck replaced Teichm\"uller's explicit parameters by a general construction of fiber bundles whose base is an arbitrary analytic space. This work on Teichm\"uller space led him to recast the bases of analytic geometry using the language of categories and functors. In Grothendieck's words, the Teichm\"uller curve becomes a space representing a functor from the category of analytic spaces into the category of sets. In this survey, we comment on Grothendieck's series of lectures. The survey is primarily addressed to low-dimensional topologists and geometers. In presenting Grothendieck's results, we tried to explain or rephrase in more simple terms some notions that are usually expressed in the language of algebraic geometry. However, it is not possible to short-circuit the language of categories and functors. The survey is also addressed to those algebraic geometers who wish to know how the notion of moduli space evolved in connection with Teichm\"uller theory. Explaining the origins of mathematical ideas contributes in dispensing justice to their authors and it usually renders the theory that is surveyed more attractive. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm\"uller theory. This volume is dedicated to the memory of Alexander Grothendieck.