I need to understand the structures of $\pi_{4n}(S^{2n})$ and $\pi_{r}(S^{2n};\mathbb{Z}_k) (r\geq 4n-1)$, where the latter group is the homotopy group with coefficient defined as $[P^r(k), S^{2n}]$. Does anyone know their computations and any references are available?
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The $2$-primary parts of the first few groups are as follows: $$\begin{array}{ll} n & \pi_{4n}(S^{2n}) \\ 1 & \mathbb{Z}_2 \\ 2 & \mathbb{Z}_2^2 \\ 3 & \mathbb{Z}_2\\ 4 & \mathbb{Z}_2^4\\ 5 & \mathbb{Z}_2\oplus\mathbb{Z}_4\\ 6 & \mathbb{Z}_2^2\\ 7 & \mathbb{Z}_2\oplus\mathbb{Z}_8\\ 8 & \mathbb{Z}_2^4\\ 9 & \mathbb{Z}_2\oplus\mathbb{Z}_4\oplus\mathbb{Z}_8. \end{array}$$ These are all in Toda's book "Composition methods in the homotopy groups of spheres". As you can see, there is no obvious pattern, and no general description is known. The difficulty of describing $\pi_i(S^j)$ is primarily controlled by $i-j$; the fact that $i=2j$ and $j$ is even is not going to help very much.
answered Aug 7, 2017 at 9:59
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$\begingroup$ Well, this is probably the strong law of the small number, but in the range you gave, we have $Z_2^2$ when $n=4k+2$ and $Z_2^4$ when $n=4k$. $\endgroup$ Commented Aug 8, 2017 at 6:50