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?



Homotopy groups of even-dimensional spheres
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.


