What are the best known results for the stable homotopy groups of spheres?

Question

There are a number of proposed ways to compute the stable homotopy groups of spheres. One can rather peculiarly consider stable (co)homotopy of an Eilenberg Maclane spectrum as a generalised (co)homology theory and use the Atiyah–Hirzebruch spectral sequence (in the same way one sometimes uses the Serre spectral sequence knowing information about the $E_{\infty}$ page to deduce it about the $E_{2}$ page). Another approach is to use the Adams spectral sequence. Here one plays off the rigidity of the cohomology of generalised Eilenberg Maclane spectra against this failing in general. This leads to a spectral sequence which converges to the p-part (where $p$ refers to taking cohomology in $\mathbb{Z}/p\mathbb{Z}$) of the stable homotopy group of spheres. A variant is to do this with some (nice enough I guess) generalised cohomology theory which leads to the Adams–Novikov spectral sequence. I think there are probably quite a few other methods which are used to calculate these. My questions are:

1. What are the best results on this? I see here it says that the best known result as of 2007 was up to the 64th stem.

2. Which method gives the best known results?

3. What stops us here? Do we simply not yet know the differentials around 64 in the Adams spectral sequence?

Answer 1

Computing $\pi_\ast(S)$ is a tedious business that to this day can only be done "by hand", i.e. by humans. The $p=2$ computation up to dimension 64 was completed by Kochman (see his SLNM book) with later corrections by Kochman/Mahowald. This was mainly (but not exclusively) based on the Atiyah-Hirzebruch spectral sequence $$H_\ast(BP,\pi_\ast(S)) => \pi_\ast(BP).$$

The available approximations to $\pi_\ast(S)$ try to decompose the problem into two steps:

1. computation of the approximation, e.g. the $E_2$ term of a spectral sequence.

2. computation of the differentials.

It's probably not suprising that step 2 requires human intervention; but often even the first step is a difficult computational challenge: for example, nobody seems to know how to compute the $E_2$-term of the Novikov spectral sequence efficiently.

Since this $E_2$-term is the cohomology of the moduli stack of one-dimensional formal groups, this problem should appeal to number theorists as well. And although number theory has a strong computational branch it seems that not much has been done here.

Answer 2

In some ways, the "best" approach to calculating the homotopy groups of spheres is to identify patterns in the homotopy groups of spheres, rather than trying to make a complete calculation of all homotopy groups up to a certain dimension. This "best" approach, led by Ravenel and collaborators, is to try to determine which families of elements in $ Ext = E_2 $ survive to $ E_\infty $ and therefore detect non-zero elements in homotopy. Often these families of elements in homotopy groups have an infinite number of elements, but the recent work of Hill-Hopkins-Ravenel on the Kervaire invariant is a significant example of a family of elements shown to be infinite in number in Ext but finite in number in homotopy (i.e., there are infinitely many non-zero differentials in the classical Adams spectral sequence on this family).

A complete calculation of the homotopy groups of spheres at any prime is difficult for many reasons. Since Ext calculations (using any suitable generalized homology theory) tend to be very large, neither humans nor machines can make a complete calculation up to a large dimension in a short amount of time. Also, calculating differentials often requires topological information not present in a single algebraic Ext object. It is common to compare an Ext calculation for one generalized homology theory (e.g., mod p homology to obtain the classical Adams spectral sequence) to an Ext calculation for another generalized homology theory (e.g., Brown-Peterson theory to obtain the Adams-Novikov spectral sequence) and see if any differentials are forced by comparing the two spectral sequences, but results using this approach are not guaranteed. Of course, there are other methods for calculating differentials, but they tend to be ad-hoc or context specific.

The best overall summary of results would be Doug Ravenel's book on the homotopy groups of spheres, and I would also recommend Kochman's book. Read works of Mark Mahowald for results using the Adams spectral sequence, and Doug Ravenel for the Adams-Novikov spectral sequence. Complete or nearly complete calculations for the homotopy groups of spheres that have been localized at a particular Morava K-theory have been made by Toda, Goerss-Henn-Mahowald-Rezk, and Mark Behrens. If you're interested in computer calculations of Ext, you should contact Robert Bruner or Christian Nassau. Many others have contributed to the calculation of homotopy groups of spheres and probably deserve to be mentioned (if I omitted someone, it was unintentional).

On an unrelated and personal note: I would like to publicly thank Torsten Ekedahl (who recently passed away) for everything he has done to help me.

Answer 3

You could look at this talk: http://neil-strickland.staff.shef.ac.uk/talks/oslo_talk.pdf

About the first third of the talk is a map of various different approaches to the stable and unstable homotopy groups of spheres. However, several approaches are missing:

• Kochman's approach via the AHSS for $\pi_*(BP)$, as in Christian Nassau's answer.
• The similar approach that you mentioned, via the AHSS for $\pi_*(H/p)$
• The superstable EHPSS, which is essentially the AHSS for the pro-spectrum $\mathbb{R}P_{-\infty}^\infty$ together with the theorem that that pro-spectrum is $2$-adically equivalent to $S^{-1}$. (There is also an analog at odd primes using $B\Sigma_p$ in place of $\mathbb{R}P^\infty$.) This should enable one to build a bridge between Kochman-style calculations and the EHP sequence and the Goodwillie tower, but I'm not aware that anyone has worked that out.

(I should mention that Mark Behrens has independently worked out most of the ideas mentioned in the remainder of the talk that I linked to, with much more success than I had.)