A splitting for the stable mapping class group

The main result of [15] is that the classifying space of the stable mapping class group after plus construction B Gamma(infinity)(+) is an infinite loop space. This result is used to show that, localized away from two, a connected component of the stable homotopy groups of spheres QS(0) splits off B Gamma(infinity)(+). The splitting is a splitting of infinite loop spaces. It follows immediately that the homology with coefficients in Z[1/2] of the infinite symmetric group is a direct summand of the homology of the stable mapping class group.