The Mumford conjecture
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque no. 307  (2006), Talk no. 945, p. 247-282

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 BΓ derived from the mapping class groups. The conjecture admits a topological generalization, inspired by Tillmann’s theorem that BΓ 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.

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 BΓ construit à partir des groupes modulaires de Teichmüller. La conjecture admet une généralisation topologique, inspirée du théorème de Tillmann que BΓ 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.

Classification:  32G15,  57R20,  55R40,  55R65,  55P15
Keywords: conjecture de Mumford, espace de modules des courbes, groupe modulaire de Teichmüller, théorie de Morse, stratification
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     author = {Powell, Geoffrey},
     title = {The Mumford conjecture},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:945},
     pages = {247-282},
     zbl = {1126.14032},
     mrnumber = {2296421},
     language = {en},
     url = {http://www.numdam.org/item/SB_2004-2005__47__247_0}
}
Powell, Geoffrey. The Mumford conjecture, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 945, pp. 247-282. http://www.numdam.org/item/SB_2004-2005__47__247_0/

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