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 completion theorem for simplicial categories. The first deloop of B Gamma(infinity)(+) coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite 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 is 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 prime 2.