ON THE HOMOTOPY OF THE STABLE MAPPING CLASS GROUP

By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, B Gamma(infinity)(+), has the homotopy type of an infinite loop space. The main new tool is a generalized group co mpletion theorem for simplicial categories. The first deloop of B Gamm a(infinity)(+) coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping c lass group onto Siegel's modular group is shown to induce a map of inf inite loop spaces from B Gamma(infinity)(+) to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there i s a space Y such that B Gamma(infinity)(+) similar or equal to ImJ((1/ 2)) x Y, where ImJ((1/2)) is the image of J localized away from the pr ime 2.