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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 $B\Gamma_{\infty}$, where $\Gamma_\infty$ 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 $B\Gamma_{\infty}$ 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.
@article{10_4007_annals_2007_165_843,
author = {Ib Madsen and Michael Weiss},
title = {The stable moduli space of {Riemann} surfaces: {Mumford{\textquoteright}s} conjecture},
journal = {Annals of mathematics},
pages = {843--941},
publisher = {mathdoc},
volume = {165},
number = {3},
year = {2007},
doi = {10.4007/annals.2007.165.843},
mrnumber = {2335797},
zbl = {1156.14021},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.843/}
}
TY - JOUR AU - Ib Madsen AU - Michael Weiss TI - The stable moduli space of Riemann surfaces: Mumford’s conjecture JO - Annals of mathematics PY - 2007 SP - 843 EP - 941 VL - 165 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.843/ DO - 10.4007/annals.2007.165.843 LA - en ID - 10_4007_annals_2007_165_843 ER -
%0 Journal Article %A Ib Madsen %A Michael Weiss %T The stable moduli space of Riemann surfaces: Mumford’s conjecture %J Annals of mathematics %D 2007 %P 843-941 %V 165 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2007.165.843/ %R 10.4007/annals.2007.165.843 %G en %F 10_4007_annals_2007_165_843
Ib Madsen; Michael Weiss. The stable moduli space of Riemann surfaces: Mumford’s conjecture. Annals of mathematics, Tome 165 (2007) no. 3, pp. 843-941. doi: 10.4007/annals.2007.165.843
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