June 17, 2012

The third stable homotopy group of spheres

Posted by Akhil Mathew under topology | Tags: , , |
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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…)

 

June 16, 2012

Computing the first three stable stems I

Posted by Akhil Mathew under topology | Tags: , , , |
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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…)