I’d like to use the next couple of posts to compute the first three stable stems, using the Adams spectral sequence. Recall from the linked post that, for a connective spectrum with appropriate finiteness hypotheses, we have a first quadrant spectral sequence

where the groups are computed in the category of comodules over (the dual of the Steenrod algebra), and the convergence is to the -adic completion of the homotopy groups of . In the case of the sphere spectrum, we thus get a spectral sequence

converging to the 2-torsion in the stable stems. In this post and the next, we’ll compute the first couple of groups of , or equivalently of (this is usually called the cohomology of the Steenrod algebra), and thus show:

- , generated by the Hopf map (coming from the Hopf fibration ).
- , generated by the square of the Hopf map.
- , generated by the Hopf map (coming from the Hopf fibration ). We have . (This is actually true only mod odd torsion; there is also a , so the full thing is a .)

In fact, we’ll be able to write down the first four columns of the Adams spectral sequence by direct computation. There are numerous fancier tools which let one go further. (more…)