This is a continuation of yesterday’s post, which used the Adams spectral sequence to compute the first two stable homotopy groups of spheres (only as a toy example for myself: one can use more elementary tools). In this post, I’d like to describe the third stable stem.The claim is that the first four columns of the -page of the Adams spectral sequence for the sphere look like:

Furthermore, we have the relation . This is the complete picture for the first four columns.

Note that there can be no nontrivial differentials in this range for dimensional reasons. Since corresponds to multiplication by 2 in the stable stems, this corresponds to the fact that : in fact, we find that has a three-term filtration with successive quotients , and that passage down each step of the filtration is given by multiplication by . The relation corresponds to the fact that the Hopf map (which corresponds to the element of Hopf invariant one in ) satisfies

for the element of Hopf invariant one in represented by . (more…)