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 {X} with appropriate finiteness hypotheses, we have a first quadrant spectral sequence

\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}_2^{\vee}}(\mathbb{Z}/2, H_*( X; \mathbb{Z}/2)) \implies \widehat{\pi_{t-s} X} ,

where the {\mathrm{Ext}} groups are computed in the category of comodules over {\mathcal{A}_2^{\vee}} (the dual of the Steenrod algebra), and the convergence is to the {2}-adic completion of the homotopy groups of {X}. In the case of {X} the sphere spectrum, we thus get a spectral sequence

\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}_2^{\vee}}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \widehat{\pi_{t-s} S^0},

converging to the 2-torsion in the stable stems. In this post and the next, we’ll compute the first couple of {\mathrm{Ext}} groups of {\mathcal{A}_2^{\vee}}, or equivalently of {\mathcal{A}_2} (this is usually called the cohomology of the Steenrod algebra), and thus show:

  1. {\pi_1 S^0 = \mathbb{Z}/2}, generated by the Hopf map {\eta} (coming from the Hopf fibration {S^3 \rightarrow S^2}).
  2. {\pi_2 S^0 = \mathbb{Z}/2}, generated by the square {\eta^2} of the Hopf map.
  3. {\pi_3 S^0 = \mathbb{Z}/8}, generated by the Hopf map {\nu} (coming from the Hopf fibration {S^7 \rightarrow S^4}). We have {\eta^3 = 4 \nu}. (This is actually true only mod odd torsion; there is also a {\mathbb{Z}/3}, so the full thing is a {\mathbb{Z}/24}.)

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…)