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In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are conveniently tabulated, and reproduced on the Wikipedia page Homotopy groups of spheres. The computations involve composition product, Toda brackets and the EHP sequence, as well as cohomology operations for the higher values of $k$.

I am fairly certain that more values of $\pi_{n+k}(S^n)$ have been computed in the intervening years, perhaps with more modern methods such as the unstable Adams spectral sequence.

Does anybody know of an up-to-date table of known unstable homotopy groups of spheres, beyond the range shown in Toda's tables?

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    $\begingroup$ Mark Behrens used the Goodwillie tower and the EHP sequence to reproduce calculations in the Toda range. arxiv.org/abs/1009.1125 $\endgroup$ – Dev Sinha Dec 24 '14 at 15:33
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I don't know the answer to your question, but I asked Fred Cohen. He had this to say:

Most of the computations are in Mahowald's work with the EHP sequence. This gives infinite families at p = 2 with Rob Thompson's extensions to p > 2.

Specific extensions of 2-primary components in fixed stems are in

  • N. Oda, On the 2-components of the unstable homotopy groups of spheres. II, Proc. Japan Acad. 53, Ser. A(1977), 215-218.

  • N. Oda, Unstable homotopy groups of spheres, Bull. of the Inst. for Advanced Re- search of Fukuoka Univ. 44 (1979), 49-152.

  • N. Oda, Some relations in the 18-stem of the homotopy groups of spheres, Bull. Central Res. Inst. Fukuoka Univ., 104, (1988), 75–83. ​

Brayton Gray gave families of elements of order $p^r$ in $\pi_*(S^{2r+1})$ for $p$ an odd prime.

Example: It is not known whether there are elements of order 64 in $\pi_*(S^{11})$.

Similarly, it is not known whether there are elements of order 64 in the stable homotopy of $\mathbb RP^{10}$, thus a potential counterexample to the Freyd conjecture.

Other than that, not much more in terms of specific computations are in the literature as far as I know.

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  • $\begingroup$ Thanks, Ryan. This seems fairly definitive. I'll take a look at these references, and if nobody tells me that the results have been handily tabulated I'll accept this answer. $\endgroup$ – Mark Grant Dec 18 '14 at 11:50
  • $\begingroup$ Does anybody have a reference for Rob Thompson’s extensions? $\endgroup$ – Ali Caglayan Aug 15 '18 at 17:03
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By following Ryan's leads I've been able to find references computing $\pi_{n+k}(S^n)$ for $20\leq k \leq 30$ at the prime $2$ (and in some cases at odd primes as well). I thought I'd post these as an answer for ease of reference.

  • M. Mimura, H. Toda, The (n+20)-th homotopy groups of n-spheres, J. Math. Kyoto Univ. 3 1963 37–58. [Contains $\pi_{n+20}(S^n)$ for all $n$ and at all primes]
  • M. Mimura, On the generalized Hopf homomorphism and the higher composition, Parts I, II. J. Math. Kyoto Univ., 4, 171-190, 301-326 (1964/5). [Contains $\pi_{n+21}(S^n)$ and $\pi_{n+22}(S^n)$ for all $n$ and at all primes. For odd primes the author refers to work of Toda later published as a series of papers On iterated suspensions I--IV, J. Math. Kyoto Univ (1966/68).]
  • M. Mimura, M. Mori, N. Oda, Determination of 2-components of the 23- and 24-stems in homotopy groups of spheres, Mem. Fac. Sci. Kyushu Univ. Ser. A 29 (1975), no. 1, 1–42. [Contains $\pi_{n+23}(S^n)$ and $\pi_{n+24}(S^n)$ for all $n$ at the prime $2$]
  • N. Oda, On the 2-components of the unstable homotopy groups of spheres Parts I, II, Proc. Japan Acad. 53, Ser. A(1977), no. 6/7, 202-205/215-218. [Contains $\pi_{n+k}(S^n)$ for $25\leq k\leq 30$ for all $n$ at the prime $2$, and partial results for $31\leq k\leq 33$]

The methods are as in Toda's book. I would be interested to learn of any later references, or errata to the above.

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