How are p-primary parts determined for odd p?

Question

When looking at surveys of computations of the homotopy groups of spheres there is a common theme. All the odd primary parts are thrown away.

How are odd primary part calculations done in relation to the 2 primary part ones?

I know you can put these groups together after decomposing them but how do you know you have all the pieces?

What’s the point of decomposing if we don’t know which prime to stop at?

Now these questions are the ones circling my head when reading about such things, I would like to have some references or helpful tips about which results let us reason like this etc.

I have seen a common theme on MO which is to simply look at Ravenels book. I don’t have much time to sift through a large book, I am sure it has the answers but I don’t know where or how to find them.

I should note I am aware of the EHP spectral sequence and how Toda has fibrations for odd pets of spheres. I just don’t understand how this is put together and how you can overall determine the full group.

Answer 1

The reason that most of the focus is on the prime $2$ is that this prime is, by far, the hardest. Major techniques in this area are the Adams spectral sequence and the Adams-Novikov spectral sequence, both of which have the same derivations at even primes and at odd primes as for the prime $2$. Roughly, the complexity to calculate out to dimension $n$ is roughly something like the log-base-$p$ of $n$.¹

As Dylan Wilson commented, the first $p$-torsion in the stable groups is a copy of $\Bbb Z/p$ that occurs in dimension $2p-3$; this grows as $p$ grows, and so for any individual group there is a finite list of primes that need to be checked. However, the situation is even a little better than that: there is a range where the stable homotopy groups of spheres consist entirely of something called the "image of $J$". The image of $J$ is nonzero (at odd primes) precisely in degrees congruent to $(-1)$ mod $2p-2$, it is cyclic, and its order in dimension $(2p-2)k + 2p-3$ is governed by the knowing the $p$-adic valuation of $k$. The first element not in the image of $J$ is called $\beta_1$ and it occurs in dimension $2p^2-2p-2$.

The current state of the art at the prime $2$ is roughly somewhere in dimensions 70-80 (though this is moving rapidly, due to people like Dan Isaksen, Zhouli Xu, Guozhen Wang, and others). The primes $3$ and $5$ have work invested that goes out reasonably far. Ravenel's book, for example, includes computation of the stable homotopy groups of spheres at $p=5$ that goes out out to dimension 1000; unless I'm mistaken, nobody else has gone nearly as far. By contrast, a very non-state-of-the-art homotopy theorist like myself could get by just knowing the image of $J$ up to dimension

• 82 at the prime 7

• 218 at the prime 11

• 310 at the prime 13

• 542 at the prime 17

• 682 at the prime 19

• and 1010 at the prime 23.

This is to say nothing of the fact that knowing a little bit more gets you much farther.

So far as your comments about Ravenel's book, you essentially understand the issues already. It takes work to get through, and your ability to get into some depth will depend on your motivation for knowing more. It is also one of the shortest ways to learn the details.

[1] This should be interpreted in the same way as the Richter scale. The jumps from 1 to 2 to 3 at 4 are all extremely significant.2

[2] Yes, I'm aware that saying "take the log of $n$, then interpret it on the Richter scale" sounds really stupid. It's for cross-comparison across primes, not absolute interpretation.