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.