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 ${E_2}$-page of the Adams spectral sequence for the sphere look like:

Furthermore, we have the relation ${h_0^2 h_2 = h_1^3}$. 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 ${h_0}$ corresponds to multiplication by 2 in the stable stems, this corresponds to the fact that ${\pi_3(S^0) = \mathbb{Z}/8}$: in fact, we find that ${\pi_3(S^0)}$ has a three-term filtration with successive quotients ${\mathbb{Z}/2}$, and that passage down each step of the filtration is given by multiplication by ${2}$. The relation ${h_0^2 h_2 = h_1^3}$ corresponds to the fact that the Hopf map ${\nu \in \pi_3(S^0)}$ (which corresponds to the element of Hopf invariant one in ${\pi_3(S^0)}$) satisfies

$\displaystyle 4 \nu = \eta^3,$

for ${\eta}$ the element of Hopf invariant one in ${\pi_1(S^0)}$ represented by ${h_1}$. (more…)

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…)