Spherical classes in some finite loop spaces of spheres

We completely determine spherical classes in single, double and triple loop spaces of spheres. We also show that for l⩾4 and n>1, there exists no spherical class in H⁎ΩlSl+n in dimensions more than 2ln+2l−1(l−2)+2. This bound may not be optimal, but reduces the computation of spherical classes in H⁎ΩlSl+n to verifying finite number of cases. We also provide a table for possible spherical classes in H⁎ΩlSl+n with 3