The Stable Mapping Class Group and Stable Homotopy Theory

Abstract

This overview is intended as a lightweight companion to the long article [20]. One of the main results there is the determination of the rational cohomology of the stable mapping class group, in agreement with the Mumford conjecture [26]. This is part of a recent development in surface theory which was set in motion by Ulrike Tillmann’s discovery [34] that Quillen’s plus construction turns the classifying space of the stable mapping class group into an infinite loop space. Tillmann’s discovery depends heavily on Harer’s homological stability theorem [15] for mapping class groups, which can be regarded as one of the high points of geometric surface theory.