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

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). Another approach is to use the Adams spectral sequence. Here one takes a so-called Adams resolution of the sphere (it is more sensible to do this with spectra as we then get a genuine free resolution of $\mathbb{Z}/p\mathbb{Z}$ over the Steenrod algebra). One gets a spectral sequence which converges to the p-part of the stable homotopy group. A variant is to do this with some (nice enough I guess) generalised cohomlogy theory which leads to the Adams–Novikov spectral sequence. I have a few different questions:

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. In relation to the (classical) Adams spectral sequence one has that the $E_{2}$ terms (mod 2) are given by $\mathrm{Ext}_{A}(\mathbb{Z}/2,\mathbb{Z}/2\mathbb{Z})$. Now this is rather difficult on the face of it to compute as one must find a workable free resolution of $\mathbb{Z}/2\mathbb{Z}$. There is in fact a certain differential graded algebra called that lambda algebra whose cohomology is precisely this. Does anyone know of good source where the details are worked out for this?

4. Following the last question I wonder if anyone knows any good sources on differentials in the Adams spectral sequence?

[I guess an answer to the last 2 questions is probably just Ravenel's book, but if anyone knows some other fairly readable stuff then that would be more than welcome.]

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wouldn't that be more appropriate for MathOverflow ? – Glougloubarbaki Nov 24 '11 at 21:37

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I'm not a specialist, but I was under the impression that even nowadays nothing beats Ravenel's book. – PseudoNeo Nov 24 '11 at 22:16

## 3 Answers

1) This Question is a little vague. You could be asking about how far out we know these stable homotopy groups and what the best up to date reference is. This is not the way people are approaching this problem currently. I believe Christian Nassau has the most extensive Adams Spectral Sequence computations. Is that what you are interested in?

2) The Adams Spectral Sequence is not easy to improve upon at the prime 2. However, at odd primes it is a result of Haynes Millers that the Adams-Novikov SS is a strict improvement over the Adams SS. The $E_2$ term of the ANSS is very hard to compute, it is a huge obstruction to progress. In fact, the Chromatic SS was developed in order to compute the $E_2$ term of the ANSS.

There are other methods though. The current approach seems to be studying the $K(n)$ and $E_n$ local homotopy groups of spheres by means of a Descent/Adams/Homotopy Fixed Point SS. These are what people are working on now, the main obstruction is, again, the $E_2$ term. For these we need to know the cohomology of certain Profinite groups with coefficients in some non-trivial modules.

3) I don't know about the Lambda algebra, but you should be able to write down a minimal resolution in a small range.

4) There is an approach to a large number off differentials in the Adams SS due to Bob Bruner. You take advantage of the highly multiplicative structure on the Adams filtration. This allows you to propagate differentials in low dimensions to get new ones. Its pretty cool.

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Thanks for the answer! With the first question I guess I was coming from the perspective of the ASS and wondering specifically what is the largest stable homotopy group that we know (for example in Mahowold and Tangora's paper "Some differentials in the Adams spectral sequence" they work out the first 45 stems), but you obviously have a point that this approach is limited and not really getting to the heart of the problem. Do you know some good sources in relation to Bob Bruner's work? – CPM Nov 26 '11 at 23:38
– Sean Tilson Nov 28 '11 at 16:07

All your answers are somewhere in the green book!

In fact in Chapter 7 in the book Ravenel uses the Adams-Novikov spectral sequence to calculate the first thousand (stable) stems for $p=5$

The best method seems to be the Adams-Novikov spectral sequence with Brown-Peterson (co)homology.

The Lambda algebra can be found in Chapter 3 of the green book.

You could also try having a read of McCleary's 'A User's Guide to Spectral Sequences' - the classical Adams spectral sequence is treated in that (Chapter 9), and a brief introduction to the Adams-Novikov spectral sequence (Chapter 11)

Edit: One further comment. You don't really need the Lambda algebra to find a (minimal) free resolutions of $\mathbb{Z}/2\mathbb{Z}$ over the Steenrod algebra. Have a look in Mosher and Tangora's book, or in Allen Hatcher's spectral sequence's book

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Well I know you can just write down naively and think Adams did precisely that. It gets quite demanding though as you need deep knowledge about the Adem relations to calculate the necessary number of generators. As such doing this for large values I believe becomes very difficult and so it is more sensible computationally to consider the lambda algebra. – CPM Nov 26 '11 at 4:50

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I think you are correct! It all depends on how far you wish to push the calculation. You might want to read about the May spectral sequence as well – Juan S Nov 26 '11 at 12:38

If you are looking into Lambda algebra, the green book has some introduction to it. For some extensive examples you can checkout two papers: W.H. Lin's $\rm Ext^{4,\ast}_A(\Bbb Z/2,\Bbb Z/2)$ and $\rm Ext^{5,\ast}_A(\Bbb Z/2,\Bbb Z/2)$, and T.W. Chen's Determination of $\rm Ext^{5,*}_\scr A(\Bbb Z/2,\Bbb Z/2)$, where they worked out the $E_2$ terms of the Adams spectral sequence of spheres for filtration 4 and 5, completely with all relations. The results are rather complicated, but you can get the idea how it works.

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– CPM Jan 9 at 13:56
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