The Mumford conjecture

Powell, Geoffrey

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 945, pp. 247-282.

La conjecture de Mumford affirme que la cohomologie à coefficients rationnels de l’espace de modules stable des surfaces de Riemann est une algèbre de polynômes sur les classes de Mumford-Morita-Miller ; on peut la reformuler en termes de la cohomologie de l’espace classifiant $B{\Gamma }_{\infty }$ construit à partir des groupes modulaires de Teichmüller. La conjecture admet une généralisation topologique, inspirée du théorème de Tillmann que $B{\Gamma }_{\infty }$ devient un espace de lacets infinis après application de la construction plus de Quillen. Le texte présente la démonstration par Madsen et Weiss de la conjecture de Mumford généralisée.

The Mumford Conjecture asserts that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra on the Mumford-Morita-Miller characteristic classes; this can be reformulated in terms of the classifying space $B{\Gamma }_{\infty }$ derived from the mapping class groups. The conjecture admits a topological generalization, inspired by Tillmann’s theorem that $B{\Gamma }_{\infty }$ admits an infinite loop space structure after applying Quillen’s plus construction. The text presents the proof by Madsen and Weiss of the generalized Mumford conjecture.

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