Cohomology of the stable mapping class group

Abstract

The stable mapping class group is the group of isotopy classes of automorphisms of a connected oriented surface of “large” genus. The Mumford conjecture postulates that its rational cohomology is a polynomial ring generated by certain classes κi of dimension 2i, for i > 0. Tillmann’s insight [38] that the plus construction makes the classifying space of the stable mapping class group into an infinite loop space led to a stable homotopy theory version of Mumford’s conjecture, stronger than the original [25]. This stronger form of the conjecture was recently proved by Ib Madsen and myself [26]. I will describe some of the ideas which led to the proof, and some retrospective thoughts, rather than trying to condense large portions of [26].