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

Ib Madsen and Michael Weiss

AMS classification numbers 57R50; 14H15, 32G15, 57R45, 57M99 

Submitted to arXiv: math.AT/0212321

Institute for the Mathematical Sciences
Aarhus University
8000 Aarhus C
Denmark

Department of Mathematics 
University of Aberdeen 
Aberdeen AB24 3UE
United Kingdom

imadsen@imf.au.dk  m.weiss@maths.abdn.ac.uk

The main result of this paper amounts to a complete evaluation 
of the integral cohomological structure of the stable mapping class
group (i.e, the group of isotopy classes of automorphisms 
of a connected oriented surface of "large" genus).
In particular it verifies the conjecture of D.Mumford 
about the rational cohomology of the stable mapping class group.
It is part of a more recent development in the field which 
began with Ulrike Tillmann's result (Invent. Math., 1997) that the 
plus construction makes the classifying space of the stable 
mapping class group into an infinite loop space. This led to a 
stable homotopy theory version of Mumford's conjecture, stronger than 
the original (Madsen and Tillmann, Invent. Math., 2001). 
We prove the extended version of Mumford's conjecture by a mixture 
of techniques from singularity theory and from homotopy theory. The 
stability theorem of J.Harer (Annals of Math., 1985) and the 
"First Main theorem" of V.Vassiliev ("Complements of Discriminants 
of smooth maps: Topology and Applications", Trans. of Math. Monographs Vol.98, 
revised edition, Amer. Math. Soc. 1994) are prominent components 
of our proof.