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

@article{Madsen2002TheSM,
  title={The stable moduli space of Riemann surfaces: Mumford's conjecture},
  author={Ib Henning Madsen and Michael Weiss},
  journal={Annals of Mathematics},
  year={2002},
  volume={165},
  pages={843-941}
}
D.Mumford conjectured in (30) that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes i of di- mension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by B 1, where 1 is the group of isotopy classes of automorphisms of a smooth oriented connected surface of "large" genus. Tillmann's insight (41) that the plus construction makes B 1 into an infinite loop… 
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References

SHOWING 1-10 OF 100 REFERENCES
On the homotopy of the stable mapping class group
Abstract. By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ∞+, has the homotopy
A fibre bundle description of Teichmüller theory
(A) In this paper we prove the theorems which we announced in [14] concerning the diffeomorphism groups of a closed surface, and, in addition, the corresponding theorems for the diffeomorphism groups
Stability of the homology of the mapping class groups of orientable surfaces
The mapping class group of F = Fgs r is F = rgs = wo(A) where A is the topological group of orientation preserving diffeomorphisms of F which are the identity on dF and fix the s punctures. When r =
On axiomatic homology theory.
provide a protective representation of H(X) as a direct product. It is easily verified that the singular homology and cohomology theories are additive. Also the Cech theories based on infinite
The stable mapping class group and Q(ℂP∞+)
Abstract.In [T2] it was shown that the classifying space of the stable mapping class groups after plus construction ℤ×BΓ+∞ has an infinite loop space structure. This result and the tools developed in
THE HOMOLOGY OF THE MAPPING CLASS GROUP
(1.2) H*{BTg:A) = H*(Mg:Q). In this paper we will show that M g , BTg, and BΌiϊl +(Sg) get more and more complicated as the genus g tends to infinity. More precisely, we will prove: Theorem 1.1. Let
Configuration-spaces and iterated loop-spaces
The object of this paper is to prove a theorem relating "configurationspaces" to iterated loop-spaces. The idea of the connection between them seems to be due to Boardman and Vogt [2]. Part of the
Mod p homology of the stable mapping class group
...
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