The stable moduli space of Riemann surfaces: Mumford's conjecture

• 成果类型: Article
• 署名作者: Madsen, Ib; Weiss, Michael
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2007.165.843
• 发表日期: 2007
• 页码: 843-941
• 关键词: mapping class group; HOMOLOGY; CLASSIFICATION; Homotopy; bundles; THEOREM; sets

摘要

D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes kappa(i) of dimension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by B Gamma infinity, where Gamma infinity is the group of isotopy classes of automorphisms of a smooth oriented connected surface of large genus. Tillmann's theorem [44] that the plus construction makes B Gamma infinity into an infinite loop space led to a stable homotopy version of Mumford's conjecture, stronger than the original [24]. We prove the stronger version, relying on Harer's stability theorem [17], Vassiliev's theorem concerning spaces of functions with moderate singularities [46], [45] and methods from homotopy theory.