The Stable Mapping Class Group and Stable Homotopy Theory
Jørgen Ellegaard AndersenUniversity of Aarhus, Denmark
Michael WeissUniversity of Aberdeen, United Kingdom
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This overview is intended as a lightweight companion to the long article . 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 . This is part of a recent development in surface theory which was set in motion by Ulrike Tillmann’s discovery  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  for mapping class groups, which can be regarded as one of the high points of geometric surface theory.