### 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 |
---|---|

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

**The stable moduli space of Riemann surfaces : Mumford's conjecture.** / Madsen, Ib; Weiss, Michael.

Research output: Contribution to journal › Article

*Annals of Mathematics*, vol. 165, no. 3, pp. 843-941. https://doi.org/10.4007/annals.2007.165.843

}

TY - JOUR

T1 - The stable moduli space of Riemann surfaces

T2 - Mumford's conjecture

AU - Madsen, Ib

AU - Weiss, Michael

PY - 2007/5

Y1 - 2007/5

N2 - 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.

AB - 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.

KW - mapping class group

KW - homology

KW - classification

KW - homotopy

KW - bundles

KW - theorem

KW - sets

U2 - 10.4007/annals.2007.165.843

DO - 10.4007/annals.2007.165.843

M3 - Article

VL - 165

SP - 843

EP - 941

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 3

ER -