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

Ib Madsen, Michael Weiss

Research output: Contribution to journalArticle

90 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)843-941
Number of pages99
JournalAnnals of Mathematics
Volume165
Issue number3
DOIs
Publication statusPublished - May 2007

Keywords

  • mapping class group
  • homology
  • classification
  • homotopy
  • bundles
  • theorem
  • sets

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