On Grothendieck's construction of Teichm\"uller space
Norbert A'Campo, Lizhen Ji, Athanase Papadopoulos (IRMA)

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…
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Taxonomy
TopicsHistory and Theory of Mathematics · Homotopy and Cohomology in Algebraic Topology
