On Grothendieck’s construction of Teichmüller space

Abstract

Abstract. In his 1944 paper Veränderliche Riemannsche Flächen, Teichmüller 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 Teichmüller space. In the same paper, Teichmüller introduced the so-called universal Teichmüller curve -a space over Teichmüller space where the fiber above each point is a Riemann surface representing that point. In fact, Teichmüller proved the existence of the Teichmüller 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 Teichmüller's explicit parameters by a general construction of fiber bundles whose base is an arbitrary analytic space. This work on Teichmüller space led him to recast the bases of analytic geometry using the language of categories and functors. In Grothendieck's words, the Teichmüller 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 Teichmüller 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 Story of Reimann’s Moduli Space

“…According to Courant [64, p. 178], "If today we are able to build on the work of Riemann, it is thanks to Klein." 3 In the book [45], Klein tried to explain some work of Riemann in ways he thought that Riemann did it. For example, the title of Chapter 19 of Part III of this book is On the moduli of Algebraic Equations.…”

On the early history of moduli and Teichmüller spaces

“…The notion of analytic continuation was known to Weierstrass before Riemann, and is present in unpublished papers he wrote in 1841 through 1842. 9 Riemann was aware of this concept when he wrote his thesis in 1851, but it is unlikely that he had access to Weierstrass' notes. 10 7 The sphere is, in Riemann's words: "die ganze unendliche Ebene A."…”

“…Pringsheim and Faber [198]. 9 These papers are reported on in [27]. Weyl, in [267] p. 1, also refers to an 1842 article by Weierstrass [247].…”

Actions of the absolute Galois group

“…Taking infinite covers forces one to deal with schemes which are not of finite type. 3 Let us also note by the way that the notion of "tower" is dear to the twentieth century algebraic geometers, and several towers existed before Grothendieck introduced some very important ones. We shall review for instance the Postnikov tower in Section 3 of this paper, a tower which is also related to the actions of the absolute Galois group.…”

“…From the one-to-one correspondence between dessins d'enfants and covers of the Riemann sphere S 2 with ramification points at {0, 1, ∞}, we get an action of Γ Q on the algebraic fundamental group of S 2 − {0, 1, ∞}. But this fundamental group is the first level of the Teichmüller tower, since it is the fundamental group of the 3 In the paper [88], it is shown that infinite coverings, although no more algebraic, do convey essential arithmetic information. What is studied in that paper is a version of the "annular uniformization" of the thrice-punctured sphere that is developed in Strebel's book on quadratic differentials [78].…”