### 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 language | English |
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Pages (from-to) | 843-941 |

Number of pages | 99 |

Journal | Annals of Mathematics |

Volume | 165 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 2007 |

### Keywords

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

### Cite this

*Annals of Mathematics*,

*165*(3), 843-941. https://doi.org/10.4007/annals.2007.165.843