The stable moduli space of Riemann surfaces: Mumford's conjecture
type: Item
instance of: scholarly article
title: The stable moduli space of Riemann surfaces: Mumford's conjecture
zbMATH Open document ID: 1156.14021
author: Michael S. Weiss
author: Ib Madsen
published in: Annals of Mathematics. Second Series
publication date: 2007-11-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0212321
review text: The moduli space \(\mathcal M_g\) of Riemann surfaces of positive genus \(g\) is a quasi-projective variety of complex dimension \(3g-3\). This is the space whose points correspond bijectively to isomorphism classes of nonsingular complex projective curves of genus \(g\). Understanding the geometry or cohomology of these spaces for various \(g\) continues to be an attractive and challenging problem. Most challenging has been a conjecture of David Mumford dating from 1983 [\textit{D. Mumford}, Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271-328 (1983; Zbl 0554.14008)] to the effect that the rational cohomology of the \textsl{stable} moduli space of Riemann surfaces is a polynomial algebra generated by certain classes \(\kappa_i\) of dimension \(2i\) (the Mumford-Miller-Morita classes). To state and explain the conjecture, one first considers the mapping class group \(\Gamma_g\) which consists of all isotopy classes of orientation preserving diffeomorphisms of a fixed compact oriented smooth surface \(F_g\) of positive genus \(g\). This is a discrete group that acts properly discontinuously on the space of conformal structures on \(F_g\) up to isotopy (i.e. Teichmuller space \({\mathcal T}_g\)). The quotient \({\mathcal T}_g/\Gamma_g\) is identified with \({\mathcal M}_g\) the moduli space. Because \(\mathcal T_g\) is contractible and the action of \(\Gamma_g\) has finite stabilizers, it follows that \(\mathcal M_g\) is rationally homotopy equivalent to the classifying space \(B\Gamma_g\). From the topologist point of view, and at least with rational coefficients, it is enough to study \(B\Gamma_g\). By introducing boundary components and the associated mapping class groups \(\Gamma_{g,b}\) where \(b\) is the number of boundary circles, it becomes possible to define maps \(\Gamma_{g,b}\rightarrow \Gamma_{g+1,b}\) and \(\Gamma_{g,b}\rightarrow \Gamma_{g,b-1}\) which at the level of classifying spaces induce isomorphisms in…
reviewed by: Sadok Kallel
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 5214865
zbMATH Keywords: moduli space of curves
zbMATH Keywords: mapping class group
zbMATH Keywords: classifying spaces
zbMATH Keywords: thom spectra.
MaRDI profile type: Publication
OpenAlex ID: W2164439626
Wikidata QID: Q122888370
arXiv ID: math/0212321
DOI: 10.4007/ANNALS.2007.165.843
description: scientific article