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I have heard the perspective that chromatic homotopy theory is meant to break apart the stable homotopy groups of spheres into manageable pieces, and that this perspective has led to various insights.

I cannot help but wonder: Have chromatic techniques actually been used to compute more stable homotopy groups of spheres?

I have seen the motivic ASS used to compute differentials in the standard ASS, but I have yet to see the anything like $E^{hG_m}_{m, p}$ (with the full Morava stabilizer group) used to compute further stems. The closest I've seen is the use of $E^{hC_8}_4$ in HHR.

It is unclear to me that the identity $\pi_*(L_{K(n)}\mathbb{S}_p) = E^{hG_n}_{n, p}$ has helped with computations of $\pi_*(\mathbb{S})$ at all. Indeed, it seems that working with the $K(n)$-local categories (where $n > 2$) is just as complicated as working with the full stable category. It is therefore unclear to me how to philosophically think of chromatic techniques as furthering the pursuit of more stems. I am hoping someone here can enlighten me with examples I may not have seen.

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    $\begingroup$ Not quite computing stems, but chromatic techniques (among other things) are crucial to the solution of the Kervaire invariant one problem, which is in a sense "computing homotopy groups". I don't think that "computing stems" is a useful metric though. Chromatic techniques are very useful to understand the structure of the homotopy groups, which is far more interesting. $\endgroup$
    – Denis Nardin
    Commented May 3, 2017 at 0:14
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    $\begingroup$ Yes. Look at Ravenel's method of infinite descent, for example. $\endgroup$
    – Dylan Wilson
    Commented May 3, 2017 at 2:22
  • $\begingroup$ As I understand it, the purpose of chromotopy is to organize infinite families of elements in pi_* S. For instance, there's Mahowald's infinite eta_j family; does that count? $\endgroup$
    – skd
    Commented Jun 24, 2017 at 15:57

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