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

Abstract

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 space led to a stable homo- topy version of Mumford's conjecture, stronger than the original (22). We prove the stronger version, relying on Harer's stability theorem (15), Vassiliev's theorem concerning spaces of functions with moderate singularities (43), (42) and methods from homotopy theory.